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+{"text":"\\section{The Mu2e Experiment}\n\\label{sec:intro}\n\nThe Mu2e experiment \\cite{TDR} at Fermilab will search for the\ncharged-lepton flavor violating neutrino-less coherent conversion\nof a negatively charged muon into an electron in the field of an\naluminum nucleus. The process produces a mono-energetic electron\nwith an energy slightly below the muon rest mass (104.967 MeV). If\nno events are observed, Mu2e will set a\nlimit on the ratio between the conversion rate and the muon capture\nrate of $R_{\\mu e}$~$\\leq 8.4\\ \\times\\ 10^{-17}$ (@ 90$\\%$ C.L.).\nThis will improve the current limit \\cite{Sindrum-II} by four\norders of magnitude.\nOn the other hand, an observation of Charge Lepton Flavour Violation\n(CLFV) events will provide a clear indication of New Physics (NP)\nbeyond the Standard Model up to mass scales of nearly $10^4$ TeV,\nfar beyond the direct search reach at colliders, complementing and\nextending other CLFV searches on a wide range of NP scenarios\n\\cite{CLFV-theory}.\n\nThe Mu2e design is based on the MELC concept \\cite{MELC}. An\nintense pulsed muon beam ($\\sim 10^{10} \\mu\/$sec) is produced by\n8 GeV, 8 kW protons hitting a tungsten target and it is stopped\non an aluminum target after travelling inside a very long, curved\nseries of solenoids (Fig.~\\ref{Fig:Mu2e}).\nThe strong negative gradient of the Production Solenoid, from 4 to\n2.5 T, confines soft pions and increases the yield through magnetic\nreflection. The S-shaped Transport Solenoid efficiently transfers\nlow energy, negatively charged particles while allowing a large\nfraction of pions to decay into muons.\nThe Detector Solenoid has a graded field from 2 to 1 Tesla in\nthe upstream region of the stopping target to increase acceptance\nfor Conversion Electron (CE) events. \n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\includegraphics[width=1.0\\textwidth]{mu2e_layout}\n  \\end{center}\n  \\caption{The Mu2e experiment. Cosmic Ray Veto and Stopping\n    Target Monitor are not shown.}\n  \\label{Fig:Mu2e}\n\\end{figure}\n\nThe Mu2e detector, just downstream of the aluminum target inside\na 1T solenoid, is composed of a tracker and an electromagnetic\ncalorimeter. The Mu2e tracker measures the momentum of the conversion\nelectron and separates it from the background. The crystal\ncalorimeter plays an important role in providing particle\nidentification capabilities and a fast online trigger filter, while\nalso aiding the track reconstruction capabilities. The detector\nsolenoid is in vacuum, at $10^{-4}$ Torr, and in a high radiation\nenvironment. The entire detector region and part of the transport\nsolenoid are surrounded by a Cosmic Ray Veto (CRV) that reduces the\ncosmic ray background.\nA High Purity Germanium Detector and a Lanthanum Bromide crystal\nconstitute the  Stopping Target Monitor, placed $\\sim 35$ m after\nthe stopping target, which provides normalization to CLFV events by\ndetecting $\\gamma$-rays emitted from muon capture in the aluminum\ntarget.\n\nIn order to reach the required sensitivity, control of the background\nto the level of less than 0.5 expected events is required.\nThe background coming from the\nbeam is reduced by means of a pulsed beam structure with a proton\nextinction lower than 10$^{-10}$: a delay in the start of the live\nwindow of $\\sim$ 700 ns after the bunch arrival time removes the\nprompt background from the acquired data.\nThe extinction level is monitored by detecting scattered protons from\nthe production target to evaluate the fraction of out-of-time beam.\n\n\\section{The tracking system}\n\\label{sec:tracker}\n\nThe Mu2e tracker system \\cite{Tracker} is designed to maximize\nacceptance for conversion electrons while minimizing the contamination\nfrom the muon Decay-In-Orbit (DIO) background, where nuclear\nmodifications push the DIO spectrum towards the CE signal\n(Fig.~\\ref{Fig:tracker} left). Energy loss and detector resolution\nproduce an overlap of the two processes. The selected design is based\non nearly 20,000 low mass straw drift tubes of 5 mm in diameter,\nwith a 15 $\\mu$m Mylar wall and 25 $\\mu$m sense wire.\nStraws of lengths ranging from 430 to 1220 mm are oriented transversely\nto the solenoid axis and arranged in 18 stations (Fig.~\\ref{Fig:tracker}\nright), for a total length of 3.2 metres along the solenoid axis. A\ncentral hole, 38 cm in diameter, makes the device blind to low momentum\nbackground particles ($p<55$ MeV\/c) which are constrained to low radius\nby the solenoidal field.\n\n\\begin{figure}[!th]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\textwidth]{dio_nores} &\n    \\includegraphics[width=0.5\\textwidth]{trk-station.png} \\\\\n  \\end{tabular}\n  \\caption{Left: energy spectrum for electrons produced from\n    free muon decays (blue), muon decays in orbit (red) and\n    conversion electrons (purple). Right: Sketch of the Mu2e\n    straw tracker system. The basic element is the panel, where\n    straws are organized in two staggered layers. Six panels\n    arranged as shown in in the middle figure above form a plane;\n    two planes rotated by $30^\\circ$ constitute a station, right.\n    The tracker, containing 18 stations, is 3.2 meters long.}\n  \\label{Fig:tracker}\n\\end{figure}\n\nAn eight channel tracker prototype was built and tested with\ncosmics rays to measure performances and tune detector simulations.\nIn Fig.~\\ref{Fig:trk-proto}, the position resolution and straw\nefficiency are compared with Monte Carlo expectations.\nGood reproducibility of data is observed.\n\n\\begin{figure}[!th]\n  \\vspace{0.3cm}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{prototypedata3.png}\n  \\caption{Longitudinal (left), transverse (center) position\n    resolution and efficiency (right) for an eight channel\n    prototype of the tracker.\n    Data from minimum ionizing particles (blue triangles) are\n    compared with Monte Carlo simulation (red crosses). \n    Resolution is extracted with Gaussian fits to the spectra.}\n  \\label{Fig:trk-proto}\n\\end{figure}\n\nThe tracker performance is studied with Monte Carlo using the\nfull Mu2e simulation. Results are reported in Fig.~\\ref{Fig:trk-reso}.\nThe core momentum resolution of 159 keV\/c is well within physics\nrequirements and stable when increasing accidental hit rate.\nThe total track efficiency of $\\sim 9\\%$ is fully dominated by\ngeometric acceptance.\n\n\\begin{figure}[!th]\n  \\vspace{0.3cm}\n  \\centering\n  \\includegraphics[width=0.5\\textwidth,height=6cm]{trk-reso}\n  \\caption{Momentum resolution evaluated with the fully tuned\n    Mu2e simulation.}\n  \\label{Fig:trk-reso}\n\\end{figure}\n\nAt the moment of writing, twelve pre-production panels are under\nconstruction and testing. In Fig.~\\ref{Fig:trk-plane}, three\npanels are assembled to form a tracking plane.\nA vertical slice test on fully instrumented panels with the entire\nFront-End Electronics chain will be performed.\n\n\\begin{figure}[!th]\n  \\centering\n  \\includegraphics[width=\\textwidth]{3panels.png}\n  \\caption{Tracking plane being assembled with pre-production\n    panels.}\n  \\label{Fig:trk-plane}\n\\end{figure}\n\n\\section{The calorimeter system}\n\\label{sec:calo}\n\nThe Mu2e calorimeter \\cite{TDR-EMC} has to provide confirmation for\nCE signal events, a powerful $e\/\\mu$ separation - with a muon\nrejection factor of $\\sim 200$, a standalone trigger and seeding for\ntrack reconstruction. An energy resolution of $O(10\\%$) and a time\nresolution of $500$ ps for 100 MeV electrons are sufficient to fulfil\nthese requirements.\nThe calorimeter design consists of two disks made from 674 undoped\nCsI scintillating crystals with ($34\\times 34\\times 200$) mm$^3$\ndimension. Each crystal is read-out by two custom array large area\n($2\\times 3$ of $6\\times 6$ mm$^2$ cells) UV-extended Silicon\nPhoto-Multipliers (SiPMs).  Each SiPM is connected to a Front-End\nElectronics (FEE) board providing amplification and shaping of the\nsignal. Groups of 20 signals are sent to a custom digitizer module\n(DIRAC, DIgitizer and ReAdout Controller) where they are sampled at\n200 Mega samples per second and transferred to the Mu2e data acquisition\nsystem. A radioactive source and a laser system allow setting the energy\nscale and monitor the fast changes of response and resolution.\nThe crystals will receive an ionizing dose of 90 krad and a fluence\nof $3\\times10^{12}$ n\/cm$^2$.\nThe photosensors,\nbeing shielded by the crystals, will get a three times smaller dose.\nThe layout of the calorimeter system and pictures of crystals and a\nreadout channel are shown in Fig.~\\ref{Fig:calo}.\n\n\\begin{figure}[!th]\n  \\begin{center}\n      \\includegraphics[width=1.0\\textwidth]{mu2e_calo.png}\n  \\end{center}\n  \\caption{Left: sketch of the calorimeter system. The cooling\n    pipes and the on board racks containing the DIRAC boards\n    are visible. Right: pure CsI calorimeter crystals (top) and\n    a readout channel, composed by two UV-extended SiPMs and the\n    corresponding analog FEE boards (bottom).}\n\\label{Fig:calo} \n\\end{figure}\n\nA long R\\&D phase with small prototypes demonstrates that the calorimeter\ndesign easily satisfies the requirements\n\\cite{NIM-LYSO1,NIM-LYSO2,NIM-BaF2,Proto-EMC}.\nPre-production components have been used to build a large size calorimeter \nprototype, Module-0 (Fig.~\\ref{Fig:emc-mod0}), with 51 crystals and 102\nSiPMs and front end boards \\cite{Module-0}. It represents a portion of the\nfinal disk and has been used to test the integration and assembly procedures\nand to evaluate the operations of running in vacuum and at low temperatures.\nModule-0 performance was tested with an electron beam of 60-120 MeV at the\nINFN Beam Test Facility in Frascati \\cite{BTF}. The energy distribution for\n100 MeV electrons is well reproduced by the calorimeter simulation,\nFig.~\\ref{Fig:calo-testbeam} left. Energy and time resolution are evaluated\nwith particles impinging on the calorimeter surface both at 0 and 50 degrees.\nThe latter is the expected incidence angle for conversion electrons in Mu2e.\nAn energy resolution of 5\\% (7\\%) and a time resolution of 120 ps (150 ps)\nare obtained for 100 MeV particles impinging at $0^{\\circ}$ ($50^{\\circ}$),\nFig.~\\ref{Fig:calo-testbeam} center and right. Results satisfy\nphysics requirements and are well reproduced by simulation.\n\n\\begin{figure}[!th]\n  \\vspace{0.3cm}\n  \\centering\n  \\begin{tabular}{cc}\n  \\includegraphics[width=0.5\\textwidth]{mod0-front} &\n  \\includegraphics[width=0.5\\textwidth]{mod0-testbeam} \\\\\n  \\end{tabular}\n  \\caption{Module-0, a large size prototype of the Mu2e calorimeter.\n    Left: front view before mounting the source panel, where the\n    staggered crystal structure is visible. Right: rear side, with\n    readout channels and cooling circuit.}\n  \\label{Fig:emc-mod0}\n\\end{figure}\n\n\\begin{figure}[!t]\n  \\centering\n  \\begin{tabular}{ccc}\n  \\includegraphics[width=0.33\\textwidth]{calo_ene}  &\n  \\includegraphics[width=0.33\\textwidth]{calo_eres} &\n  \\includegraphics[width=0.33\\textwidth]{calo_tres} \\\\\n  \\end{tabular}\n  \\caption{Calorimeter performance evaluated with a large-scale prototype,\n    Module-0, using 60-120 MeV electron beam. Left: data-MC comparison of\n    the energy distribution for 100 MeV beam. Energy (center) and time\n    (right) resolution for orthogonal and 50$^\\circ$ impinging electrons.\n    The energy resolution is compared with results expected from simulated\n    data.}\n  \\label{Fig:calo-testbeam}\n\\end{figure}\n\nThe complete production components for SiPMs and 85\\% of production\ncrystals have been received and characterized. For all of the 4000\nsensors, the breakdown voltage and the dark current are measured\nat different temperatures. The spread of these quantities over\nthe six cells of each sensor is used as quality control parameter\n(Fig.~\\ref{Fig:calo-qc} bottom). The overall rejection factor is\n1.2\\%, dominated by those sensors whose dark current RMS is too large.\nThe Quality Control of CsI crystals foresees a dimensional control,\nwith 0.1 mm tolerance with respect to nominal values, and a\nmeasurement of the optical properties \\cite{QAcrystals}.\nIn Fig.~\\ref{Fig:calo-qc} (top) the number of photoelectrons and\nthe uniformity response along the crystals are reported for both\nof the CsI producers. About 10\\% of the crystals have been rejected,\nmostly due to problems with mechanical tolerances.\nIrradiation tests have been carried out for small CsI and SiPM\nproduction subsamples. Results show that the calorimeter will be\nable to operate at the end of the Mu2e lifetime at a temperature\nbelow $0^\\circ$ C.\nMean Time To Failure tests on photosensors demonstrate an MTTF\nvalue 10 times larger than the experiment needs.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{calo-qa}\n  \\caption{Summary of Quality Control measurements for production\n    CsI crystals and Silicon Photo-Multipliers: CsI light yield\n    (top left) and longitudinal response uniformity (top right);\n    RMS of the breakdown voltage (bottom left) and of the dark\n    current over the six cells of SiPMs (bottom right).\n    Vertical lines represent the Quality Control acceptance cuts.}\n  \\label{Fig:calo-qc}\n\\end{figure}\n\nThe prototypes of FEE and DIRAC have been exposed to a large ionization\ndose and neutron fluence to qualify rad-hard components. A slice test\nwith the whole calorimeter electronic chain provides results comparable\nto those achieved using a commercial digitizer. A DIRAC prototype is\ncurrently used to read 16 channels of Module-0.\n\n\\section{Cosmic Ray Veto}\n\\label{sec:crv}\n\nIn absence of the vetoing system, cosmic ray muons interacting with\nthe detector materials produce false signal CE candidates at a rate\nof approximately one\/day. In order to maintain the background under\nthe required level, the CRV has to provide a vetoing efficiency of at\nleast 99.99\\% for cosmic ray tracks while withstanding an intense\nradiation environment.\nThe Cosmic Ray Veto system \\cite{CRV} is made by four staggered layers\nof extruded plastic scintillation counters with two embedded 1.4 mm\ndiameter Wavelength Shifting Fibers\/counter, alternated with absorber\nslabs (Fig.~\\ref{Fig:crv-design}).\nEach fiber is readout by means of 2$\\times$2 mm$^2$ SiPMs. To achieve\nthe required coverage, a total of 5,504 counters are needed, organized\nin 86 modules of six different lengths for a total surface coverage of\n327 m$^2$.\n\n\\begin{figure}[!th]\n  \\vspace{0.5cm}\n  \\centering\n  \\begin{tabular}{cc}\n  \\includegraphics[width=0.5\\textwidth]{crvnew.png} &\n  \\includegraphics[width=0.5\\textwidth]{crv_layout} \\\\\n  \\end{tabular}\n  \\caption{Left: the Mu2e Cosmic Ray Veto system, covering\n    the detector solenoid and part of the transport solenoid.\n    Right: layout of a CRV module.}\n  \\label{Fig:crv-design}\n\\end{figure}\n\nMeasurements on a full size prototype with 120 GeV protons in the\nFermilab test beam area was carried out (Fig.~\\ref{Fig:crv-testbeam})\ndemonstrating that the needed light yield can be reached: the number\nof photo-eletrons obtained at 1 meter from the readout end provides\na safety factor of $\\sim 40\\%$ with respect to the requirements\n\\cite{CRVnpe}.\nIn Fig.~\\ref{Fig:crv-pe} test beam results are compared with the\nresults obtained from the CRV counter simulation, which includes\nscintillation and Cerenkov photon production\/transport, SiPM and\nelectronics responses. Good agreement is obtained after tuning\nthe Monte Carlo parameters.\nIrradiation of CRV SiPMs with neutrons was also tested to\nunderstand the maximum level of fluence acceptable for operations\n\\cite{CRVradhard}: neutrons could deteriorate the sensors response\nand increase the detector occupancy and dead-time so that shielding\nis mandatory.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{crvtestbeam.png}\n  \\caption{Set-up of the CRV test beam. Protons are tracked with\n    multi-wire proportional chambers. Front-End Boards (FEB) are\n    visible on the top of the counter.}\n  \\label{Fig:crv-testbeam}\n\\end{figure}\n\n\\begin{figure}[!t]\n  \\vspace{0.5cm}\n  \\centering\n  \\begin{tabular}{cc}\n  \\includegraphics[width=0.5\\textwidth]{crvtestbeampe} &\n  \\includegraphics[width=0.5\\textwidth]{crv_pe_y} \\\\\n  \\end{tabular}\n  \\caption{Comparison of number of photo-electrons between simulated\n    and test beam data for 120 GeV protons normally incident at\n    different locations along (left) and across (right) a CRV\n    counter.}\n  \\label{Fig:crv-pe}\n\\end{figure}\n\nThe assembly of CRV di-counters started in June 2018 and about\nhalf of them have been produced. Production of photosensors and\nelectronics are also underway and 6\\% of the modules have been\nassembled.\nA test stand with cosmic rays is used to control the modules after\nproduction. An example of a cosmic ray event, as recorded by the\ntest stand and by the CRV module under test, is shown in\nFig.~\\ref{Fig:crv-display}.\n\n\n\\begin{figure}[!thb]\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{CRVscreen.png} \n  \\end{center}\n  \\caption{Example of event display at the cosmic ray test stand\n    used to qualify CRV modules.} \n  \\label{Fig:crv-display} \n  \\vspace{1cm}\n\\end{figure}\n\n\\section{Conclusions and perspectives}\n\\label{sec:theend}\n\nThe Mu2e experiment will exploit the world's highest intensity muon\nbeams of the Fermilab Muon Campus to search for CLFV, improving current\nsensitivity by a factor $10^4$ and with a discovery capability over a\nwide range of New Physics models.\nA low mass straw tube tracker, a pure CsI crystal calorimeter with\nSiPM readout and a high efficiency cosmic ray veto have been selected\nto satisfy the demanding requirements.\nTests on prototypes and pre-production modules meet the experimental\nneeds.\nDetector construction is in progress and is expected to be completed\nby the end of 2020.\nInstallation will begin in 2021, followed by commissioning, with data\nbeginning in late 2023.\n\n\n\n\\acknowledgments\n\nWe are grateful for the vital contributions of the Fermilab staff\nand the technical staff of the participating institutions.\nThis work was supported by the US Department of Energy; \nthe Istituto Nazionale di Fisica Nucleare, Italy;\nthe Science and Technology Facilities Council, UK;\nthe Ministry of Education and Science, Russian Federation;\nthe National Science Foundation, USA; \nthe Thousand Talents Plan, China;\nthe Helmholtz Association, Germany;\nand the EU Horizon 2020 Research and Innovation Program under the\nMarie Sklodowska-Curie Grant Agreement No.~690835 and 734303. \nThis document was prepared by members of the Mu2e Collaboration\nusing the resources of the Fermi National Accelerator Laboratory\n(Fermilab), a U.S. Department of Energy, Office of Science, HEP\nUser Facility. Fermilab is managed by Fermi Research Alliance, LLC\n(FRA), acting under Contract No. DE-AC02-07CH11359.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of time dependent classical as well as quantum harmonic oscillators \nhas appealed to theoretical physicists since time immemorial. In the literature \nthe work by Lewis {\\it et al.}~\\cite{Lewis} has lead to an upsurge of analysis \nof the Hamiltonian for the time dependent quantum harmonic oscillator using a \nclass of exact invariants designed for such systems~\\cite{Lewis2,Lewis3}. The problem \nbecomes even more fascinating when one has a system of two such oscillators in \ntwo-dimensional space. Now, in order to address practical situations one needs \nto include damping in the system. Although there are several studies on the \none-dimensional damped quantum harmonic oscillator in the past~~\\cite{Sebawe}-\\cite{Pedrosa3}, it's two-dimensional equivalent \nis a less explored system~\\cite{Gouba}. The work by Lawson {\\it et.al.}~\\cite{Gouba} is \none of the very few which analyses a two-dimensional damped quantum harmonic oscillator \nsystem. The solutions obtained by them for the mentioned system provides a \nplatform to explore the construction of various coherent states with \nintriguing properties. \n\nIn the present work we extend the study by Lawson \n{\\it et.al.}~\\cite{Gouba} and consider the two-dimensional damped quantum harmonic \noscillator in noncommutative (NC) space. It has been argued that study of quantum mechanical systems in NC space is essential to ensure the attainment of \ngravitational stability~\\cite{Doplicher} in the present theories of quantum \ngravity, namely, string theory \\cite{amati, sw} and loop quantum gravity \\cite{rov}. \nThe simplest quantum mechanical setting in two dimensional NC space\nconsists of replacing the standard set of commutation relations between the\ncanonical coordinates by NC commutation relations $[X, Y]=i\\theta$, where $\\theta$ is a positive real constant. Quantum mechanical systems in such spaces have been studied extensively in the literature \\cite{suss}-\\cite{fgs}.\nThe study of a two-dimensional quantum harmonic oscillator in NC space with time dependent NC parameters was done in \\cite{Dey}. However, their system was an undamped oscillator. The parametrized form of solutions obtained there offered an interesting possibility for study of generalized version of Heisenberg's \nuncertainty relations. Quantum damped harmonic oscillator \non noncommuting two-dimensional space was studied in \\cite{anto} where the exact propagator of the system was obtained and the thermodynamic properties of the system was investigated using the standard canonical density matrix.\n\nIn this work, a two-dimensional damped quantum harmonic oscillator in NC space\nis considered once again. However, our focus of study is different than the work carried out in \\cite{anto}. We first construct the Hamiltonian and then express it in terms of standard commutative variables. This is done in Section 2. Then we solve the Hamiltonian using the method of invariants~\\cite{Lewis} and obtain the corresponding eigenfunction in \nSection 3. In doing so, although we start with the Hamiltonian and corresponding \ninvariant in Cartesian coordinates, eventually we transform our operators to \npolar coordinates (following closely the procedure suggested in \\cite{Dey}) for ease of solution. The form of the Lewis invariant in Cartesian coordinates with a Zeeman term in the Hamiltonian is an interesting result in itself and it also makes it easier to make a transition to it's polar form. It is to be noted that the eigenfunction \nof the Hamiltonian is a product of the eigenfunction of the invariant and a \nphase factor. Both the eigenfunction and phase factor are expressed in terms \nof time dependent parameters which obey the non-linear differential equation known as Ermakov-Pinney (EP) equation~\\cite{Ermakov,Pinney}. \nNext, in Section 4 we judiciously choose the parameters of the damped system \nsuch that they satisfy all the equations representing the system as well as \nprovide us with an exact closed form solution of the Hamiltonian. The \nsolutions of the NC parameters obtained in our analysis turns out to be such that the phase factor in an integral form given in \\cite{Dey} is exactly integrable for various kinds of dissipation. Then \nin Section 5 we device a procedure to calculate the matrix element of a finite arbitrary power of the position operator with respect to the exact solutions \nfor Hamiltonian eigenstates. Using these expressions we proceed to calculate \nthe expectation value of energy and study the \nevolution of the energy expectation value of the system with time \nfor various types of damping. In Section 6 we summarize our results.\n\n\n\n\\section{Model of the two-dimensional harmonic oscillator}\nThe system we consider is a combination of two non-interacting damped harmonic oscillators in two dimensional NC space. The oscillators have equal time dependent \nfrequencies, time dependent coefficients of friction and equal mass in \nNC space. Such a model of damped harmonic oscillator was \nconsidered in an earlier communication~\\cite{Gouba} in commutative space. In this work, we extend the model by considering the system in NC space\\footnote{We shall be considering NC phase space in our work. However, we shall generically refer this as NC space.}.\n\n\\noindent The Hamiltonian of the system has the following form,\n\\begin{equation}\nH(t)=\\dfrac{f(t)}{2M}({P_1}^2+{P_2}^2)+\\dfrac{M\\omega^2(t)}{2f(t)}({X_1}^2+{X_2}^2)\\label{1}\n\\end{equation}\nwhere the damping factor $f(t)$ is given by,\n\\begin{equation}\nf(t)=e^{-\\int_{0}^t\\eta(s)ds} \n\\label{1x}\n\\end{equation}\nwith $\\eta(s)$ being the coefficient of friction. Here $\\omega(t)$ is the \ntime dependent angular frequency of the oscillators and $M$ is their mass.\nIt should be noted that in commutative space, the model with $f(t)=e^{-\\Gamma t}$ and $\\omega(t)=\\omega_0$, with $\\Gamma$ and $\\omega_0$\nbeing positive constants, is said to be the two-dimensional Caldirola and Kanai Hamiltonian \\cite{caldi, kanai}.\nThe position and momentum \ncoordinates $(X_i,P_i)$ are noncommuting variables in NC space, that is, \ntheir commutators are $[X_1,X_2]~\\neq~0$ and $[P_1,P_2]~\\neq~0$. The \ncorresponding canonical variables $(x_i,p_i)$ in commutative space are such \nthat the commutator $[x_i,p_j]=i\\hbar\\delta_{i,j}$, $[x_i,x_j]=0=[p_i,p_j]$; ($i,j=1,2$).\n\nIn order to express the NC Hamiltonian in terms of the standard commutative variables explicitly, we apply the standard Bopp-shift relations \\cite{mez} ($\\hbar=1$): \n\\begin{eqnarray}\n& X_1=x_1-\\dfrac{\\theta(t)}{2}p_2\\,\\,\\,;\\,\\,\\,X_2=x_2+\\dfrac{\\theta(t)}{2}p_1\\\\\n& P_1=p_1+\\dfrac{\\Omega(t)}{2}x_2\\,\\,\\,;\\,\\,\\,P_2=p_2-\\dfrac{\\Omega(t)}{2}x_1 \\,\\,.\n\\label{eqn1}\n\\end{eqnarray}\nHere $\\theta(t)$ and $\\Omega(t)$ are the NC parameters for space \nand momentum respectively, such that $[X_1,X_2]~=i\\theta(t)$, \n$[P_1,P_2]~=i\\Omega(t)$ and $[X_1,P_1]=i[1+\\frac{\\theta(t)\\Omega(t)}{4}]=[X_2,P_2]$; ($X_1\\equiv X$, $X_2 \\equiv Y$, $P_1\\equiv P_x$, $P_2 \\equiv P_y$). \n\n\\noindent The Hamiltonian in terms of $(x_i,p_i)$ coordinates is therefore given by the following \nrelation,\n\\begin{equation}\nH=\\dfrac{a(t)}{2}({p_1}^2+{p_2}^2)+\\dfrac{b(t)}{2}({x_1}^2+{x_2}^2)+c(t)({p_1}{x_2}-{p_2}{x_1})\\,\\,\\,.\\label{eqn2}\n\\end{equation}\nThe time dependent coefficients in the above Hamiltonian are given as,\n\\begin{eqnarray}\na(t)&=&\\dfrac{f(t)}{M}+\\dfrac{M{\\omega^2(t)}\\theta^2(t)}{4f(t)}\\label{3} \\\\\n b(t)&=&\\dfrac{f(t){\\Omega^2(t)}}{4M}+\\dfrac{M{\\omega^2(t)}}{f(t)}\\label{4} \\\\\n c(t)&=&\\dfrac{1}{2}\\left[\\dfrac{f(t)\\Omega(t)}{M}+\\dfrac{M\\omega^2(t)\\theta(t)}{f(t)} \\right]. \n\\label{eqn3}\n\\end{eqnarray}\nHere it must be noted that although our Hamiltonian given by Eqn.(\\ref{eqn2}) has the same form as that in \n\\cite{Dey} to study a system of a two dimensional harmonic oscillator in NC space, the time dependent \nHamiltonian coefficients (given by Eqn(s).(\\ref{eqn3})) have very different form. This is because our system is that of \na damped harmonic oscillator in two-dimensional NC space. Thus, the damping factor $f(t)$ modulates and alters \nthe Hamiltonian coefficients from the form considered in earlier study \\cite{Dey}.\n\n\n\n\\section{Solution of the model Hamiltonian}\nIn order to find the solutions of the model Hamiltonian $H(t)$ (Eqn.(\\ref{eqn2}))\nrepresenting the two-dimensional damped harmonic oscillator in \nNC{ space, we follow the route suggested by Lewis {\\it et.al.}~\\cite{Lewis} in their work. First we \nconstruct the time-dependent Hermitian invariant operator $I(t)$ corresponding to our Hamiltonian operator $H(t)$ \n(given by Eqn.(\\ref{eqn2})). This is because if one can solve for the eigenfunctions of $I(t)$, $\\phi(x_1,x_2)$, such \nthat,\n\\begin{equation}\nI(t)\\phi(x_1,x_2)=\\epsilon \\phi(x_1,x_2)\n\\label{eqnegn}\n\\end{equation}\nwhere $\\epsilon$ is an eigenvalue of $I(t)$ corresponding to eigenstate $\\phi(x_1,x_2)$, one can obtain the \neigenstates of $H(t)$, $\\psi(x_1,x_2,t)$, using the relation given by Lewis {\\it et. al.}~\\cite{Lewis} which is as \nfollows, \n\\begin{equation}\n\\psi(x_1,x_2,t)=e^{i\\Theta(t)}\\phi(x_1,x_2)\n\\label{eqnpsi}\n\\end{equation}\nwhere the real function $\\Theta(t)$ which acts as the phase factor will be discussed in details later. \n\n\n\n\\subsection{The Time Dependent Invariant}\nNext, following the approach taken by Lewis {\\it et.al.}~\\cite{Lewis}, we need to construct the operator $I(t)$ which \nis an invariant with respect to time, corresponding to the Hamiltonian $H(t)$, as mentioned earlier, such \nthat $I(t)$ satisfies the condition,\n\\begin{equation}\n\\dfrac{dI}{dt}=\\partial_t{I}+\\dfrac{1}{i}[I,H]=0.\n\\label{eqn4}\n\\end{equation}\nThe procedure is to choose the Hermitian invariant $I(t)$ to be of the same homogeneous quadratic form defined by Lewis \n{\\it et. al.}~\\cite{Lewis} for time-dependent harmonic oscillators. However, since we are dealing with a \ntwo-dimensional system in the present study, $I(t)$ takes on the following form,\n\\begin{equation}\nI(t)=\\alpha(t)({p_1}^2+{p_2}^2)+\\beta(t)({x_1}^2+{x_2}^2)+\\gamma(t)(x_1{p_1}+p_2{x_2}).\n\\label{eqn5}\n\\end{equation}\nHere we will consider $\\hbar=1$ since we choose to work in natural units. Now, using the form of $I(t)$ defined by \nEqn.(\\ref{eqn5}) in  Eqn.(\\ref{eqn4}) and equating the coefficients of the canonical variables, we get the \nfollowing relations,\n\\begin{eqnarray}\n\\dot{\\alpha}(t)&=&-a(t)\\gamma(t)\\label{eqn6}\\\\\n\\dot{\\beta}(t)&=&b(t)\\gamma(t)\\label{eqn7}\\\\\n\\dot{\\gamma}(t)&=&2\\left[\\,b(t)\\alpha(t)-\\beta(t)a(t)\\,\\right]\n\\label{eqn8}\n\\end{eqnarray}\nwhere dot denotes derivative with respect to time $t$.\n\n\\noindent To express the above three time dependent parameters $\\alpha$,$\\beta$ and $\\gamma$ in terms of a single time \ndependent parameter, we parametrize $\\alpha(t)=\\rho^{2}(t)$. Substituting this in Eqn(s).(\\ref{eqn6}, \\ref{eqn8}), we \nget the other two parameters in terms of $\\rho(t)$ as, \n\\begin{eqnarray}\n\\gamma(t)&=&-\\dfrac{2\\rho\\dot{\\rho}}{a(t)}\\label{eqn9}\\\\\n\\beta(t)&=&\\dfrac{1}{a(t)}\\left[\\dfrac{{\\dot{\\rho}^2}}{a(t)}+{{\\rho}^2}b+\\dfrac{\\rho\\ddot{\\rho}}{a(t)}-\\dfrac{\\rho\\dot{\\rho}\\dot{a}}{a^2} \\right].\\label{eqn10}\n\\end{eqnarray}\nNow, substituting the value of $\\beta$ in Eqn.(\\ref{eqn7}), we get a non-linear equation in \n$\\rho(t)$ which has the form of the non-linear Ermakov-Pinney (EP) equation with a dissipative \nterm~\\cite{Dey, Ermakov, Pinney}. The form of the non-linear equation is as follows, \n\\begin{equation}\n\\ddot{\\rho}-\\dfrac{\\dot{a}}{a}\\dot{\\rho}+ab\\rho={\\xi^2}\\dfrac{a^2}{\\rho^3}~.\n\\label{eqn11}\n\\end{equation} \nwhere ${\\xi^2}$ is a constant of integration.  This equation has similar form to the EP equation obtained in \\cite{Dey}, which is expected since \nour $H(t)$ has the same form as theirs. However, once again we should recall the fact that the explicit form of the time-dependent \ncoefficients are different due to the presence of damping. \n\n\n\\noindent Now, using the EP equation we get a simpler form of $\\beta$ as,\n\\begin{eqnarray}\n\\beta(t)&=&\\dfrac{1}{a(t)}\\left[\\dfrac{{\\dot{\\rho}^2}}{a(t)}+\\dfrac{{\\xi^2}{a(t)}}{\\rho^2} \\right].\n\\label{eqnew}\n\\end{eqnarray}  \n\\noindent Next, substituting the expressions of $\\alpha$, $\\beta$ and $\\gamma$ in \nEqn.(\\ref{eqn5}), we get the following \nexpression for $I(t)$,\n\\begin{equation}\nI(t)=\\rho^2({p_1}^2+{p_2}^2)+\\left(\\dfrac{\\dot{\\rho}^2}{a^2}+\\dfrac{{\\xi^2}}{\\rho^2}\\right)({x_1}^2+{x_2}^2)-\\dfrac{2\\rho\\dot{\\rho}}{a}(x_1{p_1}+p_2{x_2}).\n\\label{eqn12}\n\\end{equation}\nThe form of the Lewis invariant in Cartesian coordinates will be used later to go over to it's polar coordinate form.\nThe solution of the EP equation under various physically significant conditions shall be discussed later.\n\n\n\n \n\\subsection{Construction of Ladder operators} \nNow that we have the required Hermitian invariant $I(t)$, we proceed to calculate it's eigenstates using the operator approach. For \nthis purpose we need to first construct some ladder operators. To do this, we first need to transform the form of $I(t)$ \n(given by Eqn.(\\ref{eqn12})) to a more manageable form. For this we invoke a unitary transformation using a suitable unitary \noperator $\\hat{U}$ having the following form, \n\\begin{eqnarray}\n\\hat{U}=exp\\left[-\\dfrac{i\\dot{\\rho}}{2a(t)\\rho}({x_1}^2+{x_2}^2)\\right],\\,\\,\\,\n\\hat{U^{\\dagger}}\\hat{U}=\\hat{U}\\hat{U^{\\dagger}}=\\textbf{I}.\n\\label{eqn13}\n\\end{eqnarray}\nDefining, \n\\begin{eqnarray}\n\\phi^{'}(x_1,x_2)=\\hat{U}\\phi(x_1,x_2) \\,\\,\\,,\\,\\,\\,\nI^{'}(t)&=\\hat{U}I\\hat{U^\\dagger}\\,\\,\\;\n\\label{eqn14}\n\\end{eqnarray}\nwhere $\\phi(x_1,x_2)$ is an eigenfunction of $I(t)$ as introduced in Eqn.(\\ref{eqnegn}), \nthen, using Eqn(s).(\\ref{eqnegn},\\ref{eqn14}), we get, \n\\begin{align}\nI^{'}\\phi^{'}=\\hat{U}I\\hat{U^\\dagger}\\hat{U}\\phi=\\hat{U}I\\phi=\\hat{U}\\epsilon\\phi=\\epsilon\\phi^{'}.\n\\end{align}\nThe transformed expression of the invariant, $I^{'}(t)$, using Eqn.(\\ref{eqn14}), has the following form, \n\\begin{align}\nI^{'}(t)=\\rho^2({p_1}^2+{p_2}^2)+\\dfrac{{\\xi^2}}{\\rho^2}({x_1}^2+{x_2}^2)\\,\\,.\n\\label{eqn15}   \n\\end{align}\nThis transformed form of the invariant, $I^{'}(t)$, has exactly the same form as that of the Hamiltonian for a time dependent \ntwo-dimensional simple harmonic oscillator. So, we can introduce the corresponding ladder operators for $\\hat{I^{'}}(t)$ to be \ngiven by,\n\\begin{eqnarray}\n{\\hat{a}_j}^{'}=\\dfrac{1}{\\sqrt{2\\xi}}\\left(\\dfrac{\\xi}{\\rho}{\\hat{x}}_j+i\\rho{\\hat{p}}_j\\right)\\,\\,\\,,\\,\\,\\,{{\\hat{a}_j}^{'\\dagger}}=\\dfrac{1}{\\sqrt{ 2\\xi}}\\left(\\dfrac{\\xi}{\\rho}{\\hat{x}}_j-i\\rho{\\hat{p}}_j\\right)\n\\label{eqn16}\n\\end{eqnarray}\nwhere $j=1,2$ and the operators satisfy the commutation relation $[{{\\hat{a}_i}^{'}},{{\\hat{a}_j}^{'\\dagger}}]=\\delta_{ij}$.\n\n\\noindent Now we make the reverse transformation to get the expression of the unprimed ladder operators:\n\\begin{eqnarray}\n\\hat{a_j}(t)&=&{\\hat{U}}^\\dagger{\\hat{a}_j}^{'}\\hat{U}=\\dfrac{1}{\\sqrt{2\\xi}}\\left[\\dfrac{\\xi}{\\rho}x_j+i\\rho{p_j}-\\dfrac{i\\dot{\\rho}}{a(t)}x_j\\right]\\\\\n\\label{eqn17}\n\\hat{{a_j}^{\\dagger}}(t)&=&{\\hat{U}}^\\dagger{{\\hat{a}_j}^{'\\dagger}}\\hat{U}=\\dfrac{1}{\\sqrt{2\\xi}}\\left[\\dfrac{\\xi}{\\rho}x_j-i\\rho{p_j}+\\dfrac{i\\dot{\\rho}}{a(t)}x_j\\right].\n\\label{eqn18} \n\\end{eqnarray}\nIt can be easily checked using the algebra of the primed ladder operators that $[{{\\hat{a}_i}},{{\\hat{a}_j}^{\\dagger}}]=\\delta_{ij}$.\n\n\\noindent We now set $\\xi=1$ and consider two linear combinations of the above two operators such that,\n\\begin{eqnarray}\n\\hat{a}(t)=-\\dfrac{i}{\\sqrt{2}}(\\hat{a}_1-i\\hat{a}_2)\n=\\dfrac{1}{2}\\left[\\rho(\\hat{p_1}-i\\hat{p_2})-\\left(\\dfrac{i}{\\rho}+\\dfrac{\\dot{\\rho}}{a(t)} \\right)(\\hat{x_1}-i\\hat{x_2})\\right]\n\\label{eqn19}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\hat{a}}^\\dagger(t)=\\dfrac{i}{\\sqrt{2}}({\\hat{a}_1}^\\dagger+i{\\hat{a}_2}^\\dagger)\n=\\dfrac{1}{2}\\left[\\rho(\\hat{p_1}+i\\hat{p_2})+\\left(\\dfrac{i}{\\rho}-\\dfrac{\\dot{\\rho}}{a(t)} \\right)(\\hat{x_1}+i\\hat{x_2})\\right].\n\\label{eqn20}\n\\end{eqnarray}\nThese also satisfy the commutation relation $[\\hat{a},\\hat{a}^\\dagger]=1$.\n\n\n\n\n\\subsection{Transformation to polar coordinates}\n\nWith the above results in place, we now transform the invariant $I(t)$ and the corresponding ladder operators to polar \ncoordinates for calculational convenience. For this we invoke the transformation of coordinates of the form,\n\\begin{equation}\nx=rcos\\theta\\,\\,\\,;\\,\\,\\,y=rsin\\theta~.\n\\label{eqn21}\n\\end{equation}\nThe canonical coordinates in polar representation takes the following form, \n\\begin{eqnarray}\np_r&=&\\dfrac{1}{2}\\left(\\dfrac{x_1}{r}{p_1}+{p_1}\\dfrac{x_1}{r}+\\dfrac{x_2}{r}{p_2}+p_2\\dfrac{x_2}{r}\\right)\\nonumber\\\\\n&=&\\dfrac{x_1{p_1}+x_2{p_2}}{r}-\\dfrac{i}{2r}\\nonumber\\\\&=&-i\\left({\\partial}_r+\\dfrac{1}{2r} \\right)\\\\\np_{\\theta}&=&(x_1{p_2}-x_2{p_1})=-i{\\partial_{\\theta}}.\n\\label{eqn22}\n\\end{eqnarray} \nThe commutation relations between ($r$, $p_r$) and \n($\\theta$, $p_\\theta$) have the form\n\\begin{equation}\n[r,p_r]=[\\theta,p_{\\theta}]=[x_1,p_1]=[x_2,p_2]=i.\n\\label{eqn23}\n\\end{equation}\nThe corresponding anticommutation relation can be found to be,\n\\begin{equation}\n[r, p_r]_{+}=[x_1, p_1]_{+}+[x_2, p_2]_{+}\n=2(x_1 p_1+p_2 x_2)\n\\label{eqn24}\n\\end{equation}\nwhere $[A, B]_{+}=AB+BA$ represents anticommutator between operators $A$, $B$.\n\n\\noindent In order to transform the invariant $I(t)$ in polar coordinates, we need to have few other relations which are,\n\\begin{eqnarray}\n({p_1}^2+{p_2}^2)&=&\\left({p_r}^2+\\dfrac{{p_{\\theta}}^2}{r^2}-\\dfrac{1}{4r^2}\\right)\\\\\n(p_1+i{p_2})&=&e^{i\\theta}\\left[p_r+\\dfrac{i}{r}p_{\\theta}+\\dfrac{i}{2r} \\right]      \\\\\n(p_1-i{p_2})&=&e^{-i\\theta}\\left[p_r-\\dfrac{i}{r}p_{\\theta}+\\dfrac{i}{2r} \\right].\n\\label{eqn25}\n\\end{eqnarray}\nHence the invariant in polar coordinate system is given by,\n\\begin{eqnarray}\nI(t)=\\dfrac{\\xi^2}{\\rho^2}r^2+\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a}r\\right)^2+\\left({\\dfrac{\\rho{p_\\theta}}{r}}\\right)^2-\\left({\\dfrac{\\rho\\hbar}{2r}}\\right)^2\n\\label{eqn26}\n\\end{eqnarray}\nand the ladder operators in polar coordinate system have the following form,\n\\begin{eqnarray}\n\\hat{a}(t)&=&\\dfrac{1}{2}\\left[\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a(t)}r \\right)-i\\left(\\dfrac{r}{\\rho}+\\dfrac{\\rho{p_\\theta}}{r}+\\dfrac{\\rho}{2r} \\right)    \\right]e^{-i\\theta}\\nonumber\\\\\n{\\hat{a}}^{\\dagger}(t)&=&\\dfrac{1}{2}e^{i\\theta}\\left[\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a(t)}r \\right)+i\\left(\\dfrac{r}{\\rho}+\\dfrac{\\rho{p_\\theta}}{r}+\\dfrac{\\rho}{2r} \\right) \\right].\n\\label{eqn27}\n\\end{eqnarray}\nNow we note from Eqn(s).(\\ref{eqn26}, \\ref{eqn27}) that both the invariant $I(t)$ \nand the ladder operators have the same form as those used in \\cite{Dey} to study the undamped harmonic oscillator in \nNC space. The time-dependent coefficients involved in the \npresent study however differ due to the damping present in our system. Thus, we \ncan just borrow the expression of eigenfunction and the phase factors \nfrom \\cite{Dey} for our present system.\n\n\n\n\\subsection{Eigenfunction and phase factor}\nWe depict the set of eigenstates of the invariant operator $I(t)$ as $\\ket{n,l} $, following the convention in \n\\cite{Dey}. Here, $n$ and $l$ are integers such that $n+l\\geqslant0$. So we have the condition $l\\geqslant-n$. \nThus, if $l=-n+m$, then $m$ is a positive integer; and the corresponding eigenfunction in polar coordinate system has the following form (restoring $\\hbar$), \n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)&=&\\braket{r,\\theta|n,m-n}\\\\\n&=&\\lambda_{n}\\dfrac{{(i\\sqrt{\\hbar}\\rho)}^m}{\\sqrt{m!}}r^{n-m}e^{i\\theta(m-n)-\\dfrac{a(t)-i\\rho\\dot{\\rho}}{2a(t)\n\\hbar{\\rho}^2}r^2}U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\n\\label{eqn28}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^2=\\dfrac{1}{\\pi{n!}{(\\hbar\\rho^2)}^{1+n}}~. \n\\label{eqn28lam}\n\\end{eqnarray}\nHere, $U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)$ is \nTricomi's confluent hypergeometric function \\cite{Arfken, uva} and the eigenfunction $\\phi_{n,m-n}(r,\\theta)$ satisfies the following \northonormality relation,\n\\begin{equation}\n\\int_0^{2\\pi}d\\theta\\int_0^{\\infty}rdr\\phi^{*}_{n,m-n}(r,\\theta)\\phi_{n^{'},m^{'}-n^{'}}(r,\\theta)=\\delta_{nn^{'}}\\delta_{mm^{'}}.\n\\label{eqn29}\n\\end{equation}\nAgain following \\cite{Dey}, the expression of the phase factor $\\Theta(t)$ is given by,  \n\\begin{equation}\n\\Theta_{\\,n\\,,\\,l}(t)\\,=\\,(\\,n\\,+\\,l\\,)\\,\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT~.\n\\label{eqn30}\n\\end{equation}\nFor a given value of $l=-n+m$, it would be given by \\cite{Dey},\n\\begin{equation}\n\\Theta_{\\,n\\,,\\,m\\,-\\,n\\,}(t)=m\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT~.\n\\label{eqn31}\n\\end{equation}\nWe shall use this expression to compute the phase explicitly as a function of time for various physical cases in the subsequent discussion.\n\n\\noindent The eigenfunction of the Hamiltonian therefore reads (using Eqn(s).(\\ref{eqnpsi}, \\ref{eqn28}, \\ref{eqn31}))\n\\begin{eqnarray}\n\\psi_{n,m-n}(r,\\theta,t)&=&e^{i\\Theta_{n, m-n}(t)}\\phi_{n, m-n}(r,\\theta)\\nonumber\\\\\n&=&\\lambda_{n}\\dfrac{{(i\\sqrt{\\hbar}\\rho)}^m}{\\sqrt{m!}}\\exp{\\left[im\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT \\right]}\n\\nonumber\\\\\n&&\\times~r^{n-m}e^{i\\theta(m-n)-\\dfrac{a(t)-i\\rho\\dot{\\rho}}{2a(t)\\hbar{\\rho}^2}r^2}U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right).\n\\label{eqn32}\n\\end{eqnarray}\n\n\n\n\n\n\n\n\\section{Solutions for the noncommutative damped oscillator}\nIn this paper we are primarily interested in damped oscillators in \nNC space. For this purpose we want to find the eigenfunctions of \nthe corresponding Hamiltonian under various types of damping. The various \nkinds of damping are represented by various forms of the time dependent \ncoefficients of the Hamiltonian, namely, $a(t)$, $b(t)$ and $c(t)$. \nHowever, the various forms must be constructed in such a way that they satisfy \nthe non-linear EP equation given by Eqn.(\\ref{eqn11}). \nThe procedure of this construction of exact analytical solutions is based on the Chiellini integrability condition \\cite{chill} and this formalism was followed in \\cite{Dey}. We shall do the same in this paper.\nSo, for various forms of $a(t)$ and $b(t)$, we get the corresponding form of $\\rho(t)$ \nusing the EP equation together with the Chiellini integrability condition. In other words, the set of values of $a(t)$, $b(t)$ and $\\rho(t)$ that we use must be a solution set of the EP equation consistent with the Chiellini integrability condition. In the subsequent discussion we shall proceed to obtain solutions of the EP equation for the damped NC oscillator.\n\n\n\\subsection{Solution Set-I for Ermakov-Pinney equation : Exponentially \\\\ decaying solutions } \n\\subsubsection{The Solution Set}\nThe simplest kind of solution set of EP equation under damping is the \nexponentially decaying set used in \\cite{Dey}. The solution set is given by the following\nrelations, \n\\begin{eqnarray}\na(t)=\\sigma e^{-\\vartheta{t}}\\,\\,\\,,\\,\\,\\,b(t)=\\Delta e^{\\vartheta{t}}\\,\\,\\,,\\,\\,\\rho(t)={\\mu}e^{-\\vartheta{t\/2}}\\,\\,\\,\\,\\,\n\\label{EPsoln1}\n\\end{eqnarray}\nwhere $\\sigma,\\Delta$ and $\\mu$ are constants. Here, $\\vartheta$ is any \npositive real number. Substituting the expression of $a(t),b(t) \\,$and$\\, \\rho(t)$ in the EP equation, we can easily verify the relation between these constants to be as follows, \n\\begin{equation}\n\\mu^4=\\dfrac{\\xi^2{\\sigma^2}}{\\sigma\\Delta-\\dfrac{1}{4}\\vartheta^2}~.\n\\label{EPreln1}\n\\end{equation}\n\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nWe now write down the eigenfunctions of the Hamiltonian for the choosen set of \ntime-dependent coefficients. For this endeavour we need to choose explicit \nforms of the damping factor $f(t)$ and angular frequency of the oscillator \n$\\omega(t)$. The eigenfunction of the invariant $I(t)$ (which is given by \nEqn.(\\ref{eqn28})) takes on the following form for the solution set-I:\n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)=\\lambda_{n}\\dfrac{{(i{\\mu}e^{-\\vartheta{t\/2}})}^m}{\\sqrt{m!}}    r^{n-m}e^{i\\theta(m-n)-\\dfrac{2\\sigma+i\\mu^2\\vartheta}{4\\sigma\\mu^2{e^{-\\vartheta{t}}}}r^2}U\\left(-m,1-m+n,\\dfrac{r^2{e^{\\vartheta{t}}}}{\\mu^2} \\right)\n\\label{eqn33}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^{\\,2}\\,=\\,\\dfrac{1}{\\pi\\,n!\\,[\\mu^2\\,exp\\,(-\\vartheta{t})]^{1+n}}~.\n\\label{eqn33lam}\n\\end{eqnarray}\nIn order to obtain explicit expressions of the phase factors for various cases of the damping factor, we choose both the \nfunctions $\\omega(t)$ and $\\eta(t)$ as follows.\\\\\n\n\n\n\\noindent{\\bf $\\langle A\\rangle$ Solution Set-Ia}\n\n\\noindent Firstly, we choose the damping factor $f(t)=1$. Thus, in this case the damping in the system is due to the exponentially \ndecaying frequency $\\omega(t)$. For this purpose we set, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)={\\omega_0}\\,exp(-\\Gamma{t}\/2)\\,\\,\\,.\n\\label{eqn34}\n\\end{eqnarray}\nSubstituting the expressions for $a(t)$, $b(t)$, $\\omega(t)$ and $f(t)$ in the \nEqn(s).(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as,\n\\begin{eqnarray}\n\\theta(t)=\\dfrac{2}{M\\omega_0}\\,exp\\,[\\Gamma{t}\/2] \\sqrt{M\\sigma\\,exp(-\\vartheta{t})-1}\\label{eqn35}      \\\\\n\\Omega(t)=2\\sqrt{\\,M[\\Delta\\,exp\\,(\\vartheta{t})-M\\omega_0^2\\,exp\\,(-\\Gamma{t})]}. \\label{eqn36}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\, \\sqrt{M\\sigma\\,-e^{\\Gamma t}}\\label{eqn35b}      \\\\\n\\Omega(t)&=&2\\sqrt{\\,M[\\Delta\\,exp\\,(\\Gamma{t})-M\\omega_0^2\\,exp\\,(-\\Gamma{t})]}. \\label{eqn36b}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), \nwe get an expression for the phase in a closed form as, \n\\begin{equation}\nc(t)=\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t})-M{\\omega_0}^2\\,exp\\,(-\\Gamma{t}) }{M}} \n\\,+\\,\\omega_0\\,exp\\,(-\\Gamma{t}\/2)\\sqrt{\\,M\\sigma\\,exp\\,(-\\Gamma{t})-1 \\,}.\\label{eqn38}\n\\end{equation}\nSubstituting the expressions of $a(t)$, $\\rho(t)$ and $c(t)$ in Eqn.(\\ref{eqn31}), we get,\n\\begin{eqnarray}\n\\Theta_{\\,n\\,,\\,l}(t)&=&\\,(\\,n\\,+\\,l\\,)\\,\\dfrac{\\omega_0}{2\\sqrt{M\\sigma}\\Gamma}\\ \\left[log_{e}\\dfrac{e^{{\\Gamma}t}-2M\\sigma-2\\sqrt{M\\sigma(M\\sigma-e^{{\\Gamma}t})}}{1-2M\\sigma-2\\sqrt{M\\sigma(M\\sigma-1)}}\\right.\\nonumber\\\\ \n&&\\left. -{\\Gamma}t-2\\sqrt{M\\sigma(M{\\sigma}e^{-{2\\Gamma}t}-e^{-{\\Gamma}t})}+2\\sqrt{M\\sigma(M\\sigma-1)}\\ \\right]\\nonumber\\\\\n&&+\\dfrac{2(n+l)}{\\Gamma}\\,\\left[\\sqrt{\\frac{\\Delta}{M}e^{{\\Gamma}t}-{\\omega_0^2}e^{-{\\Gamma}t}}-\\sqrt{\\dfrac{\\Delta}{M}-{\\omega_0^2}}\\right.\\nonumber\\\\\n&&\\left.+2i{\\omega_0}\\left\\{e^{-{\\Gamma}t\/2} {_{2}F_{1}}\\left(-\\frac{1}{4},\\frac{1}{2},\\frac{3}{4},\\frac{{\\Delta}e^{{2\\Gamma}t}}{M\\omega_0^2}\\right)- {_{2}F_{1}}\\left(-\\frac{1}{4},\\frac{1}{2},\\frac{3}{4},\\frac{\\Delta}{M\\omega_0^2}\\right)\\right\\}\\right]\n-\\frac{\\sigma}{\\mu^2}(n+l)t  \\nonumber\\\\\n\\label{eqn39}                      \n\\end{eqnarray}\nwhere $_{2}F_{1}(a,b,c;t)$ is said to be the Gauss hypergeometric function. It is interesting to note that the solutions of the time dependent NC parameters enable us to get an exact analytic expression for the phase factor.\nIt is further interesting to observe that the phase has a complex part which indicates that the wave function decays with time.\\\\\n\n\n\n\\noindent {\\bf $\\langle B\\rangle$ Solution Set-Ib}\n\n\\noindent Here the oscillator is damped due to the damping factor $f(t)$ and the frequency $\\omega(t)$ \nis a constant. This situation can be depicted by the following relations, \n\\begin{eqnarray}\nf(t)= exp\\,(-\\Gamma{t})~;~\\omega(t)={\\omega_0}.\n\\label{10x}\n\\end{eqnarray}\nSubstituting these relations in Eqn(s).(\\ref{3}, \\ref{4}), we get the time dependent NC parameters \nas,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{M{\\sigma}\\,exp\\,(-\\vartheta{t})-exp\\,(-\\Gamma{t})}\\,\\,e^{-\\Gamma t\/2}\\label{eqn40}\\\\\n\\Omega(t)&=&2e^{\\Gamma{t}}\\sqrt{M\\,[\\Delta\\,exp\\,(\\vartheta-\\Gamma)t-M{\\omega_0}^2]}.\n\\label{eqn41}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{M{\\sigma}\\,- 1}\\,\\,e^{-\\Gamma t}\\label{eqn40b}\\\\\n\\Omega(t)&=&2e^{\\Gamma{t}}\\sqrt{M\\,[\\Delta\\,-M{\\omega_0}^2]}.\n\\label{eqn41b}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we \nget,\n\\begin{equation}\nc(t)= \\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\, }{M}} \\,+\\,\\omega_0\\sqrt{M\\sigma-1}\\,=\\,constant ~. \\label{eqn43}\n\\end{equation}\nSubstituting the expressions of $a(t)\\,,\\rho(t)$\\, and $c(t)$\\, in Eqn.(\\ref{eqn31})\\,,\\,we get an expression for the phase in a closed form as,\n\\begin{align}\n\\Theta_{\\,n\\,,\\,l}(t)\\,=&(\\,n\\,+\\,l\\,)\\left[-\\frac{\\sigma}{\\mu^2}\\,+\\,\\sqrt{\\frac{ \\Delta\\,-M{\\omega_0}^2\\,}{M} } \\,+\\omega_0\\sqrt{M\\sigma-1}  \\right]\\,t ~.\\label{eqn44}\n\\end{align}\nOnce again we are able to obtain an exact expression for the phase, in this case varying linearly with time.\nIt is important to note that the reality of the phase in this case depends crucially on the parameters $\\Delta$, $M$, $\\sigma$, $\\omega_0$. The phase $\\Theta_{n,l}$ is real if $\\Delta-M\\omega_{0}^2 \\geq 0$ and $M\\sigma \\geq 1$, else it is complex.\\\\\n\n\n\n\\noindent {\\bf $\\langle C\\rangle$ Solution Set-Ic}\n\n\n\\noindent Here the oscillator is damped due to the damping factor $f(t)$ and the time-dependent \nfrequency $\\omega(t)$; both of which are exponentially decaying. Thus, we set, \n\\begin{eqnarray}\nf(t)=exp\\,(-\\Gamma{t})~;~\\omega(t)={\\omega_0}\\,exp\\,(-\\Gamma{t}\/2).\n\\label{eqn45}\n\\end{eqnarray}\nSubstituting these relations in Eqn.(s)(\\ref{3}, \\ref{4}), we get the time dependent NC \nparameters to be, \n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{(M{\\sigma}e^{-(\\vartheta-\\Gamma)t}\\,- 1)}e^{-\\Gamma{t}\/2}\\label{eqn46}\\\\\n\\Omega(t)&=&2\\sqrt{M\\,[\\Delta\\,exp(\\vartheta{t})-M{\\omega_0}^2\\, \\,]}\\,\\,e^{\\Gamma t\/2}.\\label{eqn47}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{(M{\\sigma}\\,- 1)}e^{-\\Gamma{t}\/2}\\,\\,\\label{eqn46}\\\\\n\\Omega(t)&=&2\\sqrt{M\\,[\\Delta\\,exp(\\Gamma{t})-M{\\omega_0}^2\\, \\,]}\\,\\,e^{\\Gamma t\/2}.\\label{eqn47}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we \nget,\n\\begin{align}\nc(t)= \\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\,\\exp\\left[-\\Gamma{t}\\right] }{M}} \n +  {\\omega_0}\\,e^{-\\Gamma t\/2}\\sqrt{M{\\sigma}\\,-1}~.\\label{eqn49}\n\\end{align}\nSubstituting the expressions of $a(t)$, $\\rho(t)$ and $c(t)$\\, in Eqn.(\\ref{eqn31}), we obtain an expression for the phase in a closed form as,\n\\begin{eqnarray}\n\\Theta_{\\,n\\,,\\,l}(t)\\,&=&\\,\\dfrac{(\\,n+l\\,)}{\\Gamma\\,\\sqrt{M}}\\,\\left[\\,\\sqrt{\\Delta}\\,\\Gamma\\,t\\,+\\,2\\sqrt{\\Delta-M\\omega_0^2}\\,-2\\sqrt{\\Delta-M\\omega_0^2{\\exp\\,(-\\Gamma{t})}}\\right.\\nonumber\\\\\n&&\\left.+\\,2\\sqrt{\\Delta}\\,log\\,\\left(\\frac{\\Delta+\\sqrt{\\Delta[\\Delta-M\\omega_0^2\\,\\exp\\,(-\\Gamma{t}) ]}}{\\Delta+\\sqrt{\\Delta[\\Delta-M\\omega_0^2\\,]}}\\right)     \\right]\\nonumber \\\\\n&&-\\,(\\,n+l\\,)\\left[\\dfrac{\\sigma\\,t}{\\mu^2}\\,+\\,\\dfrac{2}{\\Gamma}\\,\\omega_0\\,\\left(e^{-\\Gamma t\/2}-1\\right)\\sqrt{\\,M\\sigma-1} \\right].\\label{eqn50}\n\\end{eqnarray}\n\n\n\n\n\n\n\\subsection{Solution Set-II for Ermakov-Pinney equation: Rationally decaying solutions}\n\\subsubsection{The Solution Set}\nWe now consider rationally decaying solutions of the EP equation similar to that used in~\\cite{Dey} which is of the form,\n\\begin{eqnarray}\n&a(t)=\\dfrac{\\sigma\\,\\left(1+\\dfrac{2}{k}\\right)^{\\,(k+2)\/k}}{(\\Gamma{t}+\\chi)^{\\,(k+2)\/k}}\\nonumber \\\\ \\nonumber\\\\\n&b(t)=\\dfrac{\\Delta\\,\\left(\\dfrac{k}{k+2} \\right)^{(2-k)\/k} }{(\\Gamma{t}+\\chi)^{\\,(k-2)\/k}}     \\,\\,\\,\\Rightarrow\\, \\,\\,\\,\\dfrac{\\Delta\\,\\left(1+\\dfrac{2}{k}\\right)^{\\,(k-2)\/k} }{(\\Gamma{t}+\\chi)^{\\,(k-2)\/k}}\\nonumber  \\\\ \\nonumber\\\\\n&\\rho(t)=\\dfrac{\\mu\\left(1+\\dfrac{2}{k}\\right)^{1\/k} }{(\\Gamma{t}+\\chi)^{1\/k}}\n\\label{EPsoln2}\n\\end{eqnarray}\nwhere $\\sigma$, $\\Delta$, $\\mu$, $\\Gamma$ and $\\chi$ are constants such that $(\\Gamma{t}+\\chi)~\\neq~0$, and $k$ is an integer. Substituting the expressions of $a(t)$, $b(t)$, and $\\rho(t)$ in the EP equation, we can easily verify the relation between these constants to be as follows, \n\\begin{equation}\n\\Gamma^2\\mu=(k+2)^2\\,(\\sigma\\Delta\\mu-\\frac{\\xi^2\\sigma^2}{\\mu^3}).\n\\label{EPreln2}\n\\end{equation}\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nThe eigenfunction of the invariant operator $I(t)$ (given by \nEqn.(\\ref{eqn28})) for this solution Set-II is given by,\n\\begin{eqnarray}\n\\phi_{n\\,,\\,m-n}(r,\\theta)=\\lambda_{n}\\,\\dfrac{{(i\\mu)}^{\\,m}}{\\sqrt{m!}}\\left[\\dfrac{k+2}{k(\\Gamma{t}+\\chi)}\\right]^{m\/k}    r^{n-m}e^{i\\theta(m-n)-\\dfrac{[\\sigma\\,(k+2)\\,+\\,i\\mu^2\\Gamma]\\,\\,(\\Gamma{t}+\\chi)^{2\/k}\\,\\,\\,k^{2\/k} }{2\\sigma\\,(k+2)^{\\,(k+2)\/k}\\mu^2}r^2}\\nonumber \\\\\n\\times\\,\\,\\,U\\left(-m,1-m+n,\\,\\dfrac{r^2[k(\\Gamma{t}+\\chi)]^{2\/k}}{\\mu^2\\left(k+2\\right)^{2\/k}}\\,\\right)\n\\label{eqn51}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by \n\\begin{eqnarray}\n\\lambda_n^{\\,2}=\\dfrac{1}{\\pi\\,n!\\mu^{2n+2}}\\left[\\dfrac{k(\\Gamma{t}+\\chi)}{k+2}\\right]^{2(1+n)\/k}.\n\\label{eqn51lam}\n\\end{eqnarray}\nIn order to get the eigenfunction of the Hamiltonian $H(t)$, we need to calculate the associated phase factor. Once again for this we need to fix up the forms of the damping factor $f(t)$ and angular frequency $\\omega(t)$ of the oscillator. In order to \nexplore the solution of $H(t)$ for rationally decaying coefficients, we choose a rationally decaying form for $\\omega(t)$ \nand set $f(t)=1$. Thus, we have the following relations, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}~.\n\\end{eqnarray}\nSubstituting these relations in Eqns.(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as, \n\\begin{eqnarray}\n\\theta(t)&=& \\dfrac{2\\,(\\Gamma\\,t+\\chi)}{M\\,\\omega_0}\\,\\sqrt{M\\sigma\\,\\left[\\dfrac{(k+2)}{k\\,(\\Gamma{t}\\,+\\,\\chi)}\\right]^{(k+2)\/k}\\,-\\,1}     \\label{eqn52}                   \\\\ \\nonumber \\\\\n\\Omega(t)&=& \\,2\\,\\sqrt{M\\Delta\\,\\left[\\dfrac{k+2}{k(\\Gamma{t}+\\chi)}\\right]^{\\,(k-2)\/k}\\,-\\,\\dfrac{M^{\\,2}\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^2}}~.\\label{eqn53}\n\\end{eqnarray}\nWe now consider $k=2$. This enables us to integrate the expression for the phase factor (given by Eqn.(\\ref{eqn31})).\nThe simplified forms of $a(t)$, $b(t)$ and $\\rho(t)$ for $k=2$ read,\n\\begin{eqnarray}\na(t)=\\dfrac{4\\sigma}{(\\Gamma{t}+\\chi)^{\\,2}}\\,\\,,\\,\\,b(t)\\,=\\,\\Delta\\,\\,,\\,\\,\\rho(t)=\\left[\\dfrac{2\\mu^{\\,2}}{\\Gamma{t}+\\chi}\\right]^{1\/2}.\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}) gives, \n\\begin{equation}\nc(t)\\,=\\,\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\\,\\sqrt{\\dfrac{4\\sigma\\,M}{(\\Gamma{t}+\\chi)^2}\\,-\\,1}\\,+\\,\\sqrt{\\dfrac{\\Delta}{M}\\,-\\,\\dfrac{\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^{\\,2}}}\\,\\,\\,\\,.\n\\end{equation} \nSubstituting these expressions for $a(t)$, $\\rho(t)$ and $c(t)$ \nfor $k=2$ in Eqn.(\\ref{eqn31}), we get the following expression \nfor the phase factor in a closed form as,\n\\begin{eqnarray}\n\\Theta_{\\,n, l\\,}(t)&=&\\dfrac{(n+l)}{\\Gamma}\\,\\left[\\omega_0\\,\\,tan^{\\,-1}\\left(\\dfrac{\\omega_0}{\\sqrt{\\frac{\\Delta}{M}{(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}}\\right)+\\sqrt{\\dfrac{\\Delta\\,{(\\Gamma\\,t+\\chi)^2}}{M}-\\omega_0^2}\\,-\\,\\frac{2\\sigma}{\\mu^2}\\,log_{e}\\,\\frac{(\\chi+\\Gamma\\,t)}{\\chi}\\right.\\nonumber\\\\\n&&\\left.-{\\sqrt{\\frac{\\Delta}{M}{\\chi^2}-\\omega_0^2}}-\\omega_0\\,\\,tan^{\\,-1}\\left(\\dfrac{\\omega_0}{\\sqrt{\\frac{\\Delta}{M}{\\chi^2}-\\omega_0^2}}\\right)\\right] \\nonumber\\\\\n&& +\\dfrac{\\omega_0(n+l)}{\\Gamma}\\left[\\dfrac{\\sqrt{4\\,\\sigma\\,M-\\chi^2}}{\\chi}-\\dfrac{\\sqrt{4\\,\\sigma\\,M-(\\chi+\\Gamma\\,t)^2}}{(\\chi+\\Gamma\\,t)}\n\\right.\\nonumber\\\\\n&&\\left. +ilog_{e}\\dfrac{(\\chi+\\Gamma\\,t)+{\\sqrt{(\\chi+\\Gamma\\,t)^2-4\\,\\sigma\\,M}}}{\\chi+\\sqrt{\\chi^2-4\\,\\sigma\\,M}} \\right].\n\\label{eqn55}\n\\end{eqnarray}\nWe can now get the eigenfunction of this rationally decaying damped system using \nEqn.(\\ref{eqnpsi}).\n\n\n\n\n\n\n\n\\subsection{Solution Set-III for Ermakov-Pinney equation: Elementary Solution}\n\\subsubsection{The Solution Set}\nWe now propose a simple method of obtaining a solution of the EP equation.\nThe method is as follows. Choosing $\\rho(t)$ to be any arbitrary time dependent function and taking it's time derivative as proportional to $a(t)$, that is,\n$a(t)=constant \\times\\dot{\\rho}$ and setting \n$b(t)=constant \\times \\dfrac{a}{\\rho^4}$, we observe that these would always satisfy the EP equation along with a certain constraint relation among the constants.\n\n\\noindent Here we consider a simple solution which is a special case of the above  solution for the EP equation. We call this the elementary solution which reads,\n\\begin{eqnarray}\na(t)={\\sigma}\\,\\,\\,\\,,\\,\\,\\,b(t)=\\dfrac{{\\Delta}}{{(\\Gamma\\,t\\,+\\,\\chi)^4}}\\,\\,\\,,\\,\\,\\rho(t)=\\mu(\\Gamma{t}\\,+\\,\\chi)\n\\label{eqn57}\n\\end{eqnarray}\nwhere $\\Gamma$, $\\chi$, $\\mu$, $\\sigma$ and $\\Delta$ are constants. The above solution set satisfy the EP equation with the following constraint relation,\n\\begin{equation}\n\\Delta\\mu^4=\\xi^2\\sigma\\,\\,.\n\\end{equation}\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nThe eigenfunctions of the invariant operator $I(t)$ for this solution set is given by,\n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)&=&\\lambda_{n}\\dfrac{{[i\\mu(\\Gamma\\,t+\\chi)]}^m}{\\sqrt{m!}}r^{n-m}e^{i\\theta(m-n)-\\dfrac{\\sigma-i\\mu^2\\Gamma(\\Gamma\\,t+\\chi)}{2\\sigma\\mu^2(\\Gamma\\,t+\\chi)^2}r^2}\\nonumber \\\\\n&&\\times\\,\\,U\\left(-m,1-m+n,\\dfrac{r^2}{\\mu^2(\\Gamma\\,t+\\chi)^2} \\right)\n\\label{eqn58}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^2=\\dfrac{1}{\\pi{n!}{\\left[\\mu(\\Gamma\\,t+\\chi) \\right]}^{2+2n}}~. \n\\label{eqn58lam}\n\\end{eqnarray}\nIn order to get an eigenfunction of the Hamiltonian, we calculate the phase factor for a particular case of the damped harmonic oscillator where the angular frequency $\\omega(t)$ is rationally decaying and the damping factor $f(t)$=1. Thus, we set, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\n\\end{eqnarray}\nwhere $\\Gamma$ and $\\chi$ are real constants.\nSubstituting these relations in Eqns.(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as, \n\\begin{eqnarray}\n\\theta(t)=\\dfrac{2\\,(\\Gamma\\,t+\\chi)}{\\omega_0\\,M}\\,\\sqrt{M\\,\\sigma-1}\\label{eqn59}\\\\\n\\Omega(t)=2\\sqrt{\\dfrac{M\\Delta}{(\\Gamma\\,t+\\chi)^4}-\\dfrac{M^2\\,\\omega_0^2}{(\\Gamma\\,t+\\chi)^{2}}}~.\\label{eqn60}\n\\end{eqnarray}\nSubstituting these relations \nin the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we get,  \n\\begin{align}\nc(t)=\\sqrt{\\dfrac{\\Delta}{M(\\Gamma\\,t+\\chi)^4}-\\dfrac{\\omega_0^2}{(\\Gamma\\,t+\\chi)^2}}\\,+\\,\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\\sqrt{M\\,\\sigma-1}.\\label{eqn61}\n\\end{align}\nSubstituting these expressions of $a(t)$, $\\rho(t)$ and $c(t)$ in Eqn.(\\ref{eqn31}), we obtain an expression for the phase factor in a closed form as, \n\\begin{align}\n\\Theta_{\\,n\\,,\\,l}(t)&=\\,(n+l\\,)\\left[\\omega_0\\dfrac{\\sqrt{M\\,\\sigma-1}}{\\Gamma}\\log\\,\\dfrac{(\\Gamma\\,t+\\chi)}{\\chi}-\\frac{{\\sigma}t}{\\mu^2\\chi(\\Gamma\\,t+\\chi)} \\right] +\\,\\dfrac{(\\,n+l\\,)}{\\Gamma}\\left[\\sqrt{\\dfrac{\\Delta}{M\\chi^2}-\\omega_0^2}\\right.\\nonumber\\\\\n&\\left.-\\sqrt{\\dfrac{\\Delta}{M(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}\n+\\omega_0\\left\\{\\tan^{-1}\\left(\\dfrac{\\omega_0\\chi}{\\sqrt{\\dfrac{\\Delta}{M}-\\chi^2\\omega_0^2}}\\right)-\\tan^{-1}\\left(\\dfrac{\\omega_0(\\Gamma\\,t+\\chi)}{\\sqrt{\\dfrac{\\Delta}{M}-\\omega_0^2(\\Gamma\\,t+\\chi)^2}}\\right)\\right\\}\\,\\, \\right].\n\\label{eqn62}\n\\end{align} \nWe can now get the eigenfunction of this system by using Eqn.(\\ref{eqnpsi}).\n\n\n\n\\section{Expectation Values}\nIn this section, we intend to calculate the expectation value of energy. For \nthis we need to calculate the expectation value of the Hamiltonian $H(t)$ in \nit's own eigenstates. \nThe expectation value $\\langle H \\rangle$ is given by (using Eqn.(\\ref{eqn2})), \n\\begin{equation}\n\\langle H\\rangle = \\dfrac{a(t)}{2}(\\langle{p_1}^2\\rangle+\\langle{p_2}^2\\rangle)+\\dfrac{b(t)}{2}(\\langle{x_1}^2\\rangle+\\langle{x_2}^2\\rangle)+c(t)(\\langle{p_1}{x_2}\\rangle-\\langle{p_2}{x_1}\\rangle)\\,\\,\\,.\\label{eqn63}\n\\end{equation}\nTo calculate this we need to get the expectation value of the individual canonical \noperators. To set up our notation we denote the eigenstates of the Hamiltonian $H(t)$  \nby $|n,l\\rangle_H$. \n\n\n\n\n\\subsection{Matrix elements of the coordinate operators raised to arbitrary finite powers}\n\\noindent We start by calculating the matrix element of an arbitrary power of $x$, $_{H}\\langle n,l|x^k|n,l\\rangle_{H}$, which is given by\n\\begin{eqnarray} \n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\int r dr d\\theta\n~_{H}\\langle n,m-n |r,\\theta\\rangle\\langle r,\\theta|r^k cos^k\\theta|n,m'-n\\rangle_{H}\\nonumber\\\\\n&=&\\dfrac{1}{2^k}e^{i(\\Theta_{n,m'-n}-\\Theta_{n,m-n})}\\int r^{k+1} dr d\\theta~(e^{i\\theta}+e^{-i\\theta})^k \\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times\\phi^*_{n,m-n}(r,\\theta)\\phi_{n,m'-n}(r,\\theta)\\nonumber\\\\ \n\\label{eqn64}\n\\end{eqnarray} \nwhere we have used the relations, \n$|n,l\\rangle_{H}~=~e^{i\\Theta_{n,l}}|n,l\\rangle$ where $|n,l\\rangle_H$ and \n$|n,l\\rangle$ are eigenstates of the Hamiltonian $H(t)$ and Lewis invariant $I(t)$ \nrespectively. We have also used the relation $\\langle r,\\theta|n,m'-n\\rangle$~=~$\\phi_{n,m'-n}(r,\\theta)$, with $\\phi$ being the eigenfunction of $I(t)$.\nNow, Eqn.(\\ref{eqn64}) can be rewritten as,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\dfrac{\\pi}{2^{k-1}} \\sum_{r=0}^{k} {^{k}C_r}\\delta_{m',m+2r-k}  A(n,m,m+2r-k)\\nonumber\\\\\n&\\times& \\int_0^\\infty r~dr~r^{2(n-m-r+k)}\ne^{\\dfrac{-r^2}{\\hbar \\rho^2}} \\nonumber\\\\\n&\\times& U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\\nonumber\\\\\n&\\times& U\\left(-m-2r+k,1-m-2r+k+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\n\\label{eqn67}\n\\end{eqnarray}\nwhere $A(n,m,m+2r-k)=e^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}\\lambda_n^2\\dfrac{(-i\\hbar^{1\/2}\\rho)^{m}(i\\hbar^{1\/2}\\rho)^{m+2r-k}}{\\sqrt{m!(m+2r-k)!}}$~. \n\n\\noindent Now defining $w=-\\dfrac{r^2}{\\hbar \\rho^2}$, we have, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w)\n\\label{eqn68}\n\\end{eqnarray}\nwhere we have used the following result on special functions \\cite{Arfken, uva},\n\\begin{eqnarray}\nL_n^{(\\zeta)}(w)=\\frac{(-1)^n}{n!}U(-n,\\zeta+1,w) \\label{eqn69}\n\\label{eqn70}\n\\end{eqnarray}\nwhere $L_n^{(\\zeta)}(w)$ are associated Laguerre polynomials.\n\n\\noindent Now, we get using the relation for phase given in \\cite{Dey},\n\\begin{eqnarray}\n\\Theta_{n,l}=(n+l)\\int^t \\left[c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)}\\right]d\\tau\n\\label{eqn73}\n\\end{eqnarray} \nthe following relation,\n\\begin{eqnarray}\ne^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}&=&e^{i[\\{(n+m-n+2r-k)-(n+m-n)\\}\\int^t (c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)})d\\tau]} \\nonumber\\\\\n&=& e^{i(0+2r-k)\\int^t (c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)})d\\tau}\\nonumber\\\\\n&=&e^{i\\Theta_{0,2r-k}}.\n\\label{eqn74}\n\\end{eqnarray}\nSo, we finally get the following relation for the matrix element of $x^k$,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i\\Theta_{0,2r-k}}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn75}\n\\end{eqnarray}\nThis is a new result in this paper and can be used to obtain the matrix element or expectation value of any power of $x$.\nFor the sake of completeness, we also write down the matrix element of $x^k$ in the eigenstates of the Lewis invariant $I(t)$, which reads\n\\begin{eqnarray}\n\\langle n,m-n|x^k|n,m'-n\\rangle&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn75I}\n\\end{eqnarray}\nNote that the phase factor does not appear in the above result.\n\n\\noindent Now, we proceed to evaluate the matrix element \n$_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}$ using the expression  \nobtained in Eqn.(\\ref{eqn75}). This reads\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}=\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n+ _{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}.\n\\label{eqn76}\n\\end{eqnarray}\nEvaluating the above matrix elements give,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}= -\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}\\sqrt{m}e^{-i\\Theta_0,1}\\delta_{m,m'+1}\n\\label{eqn77}\n\\end{eqnarray}\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}=\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}\\sqrt{m'}e^{i\\Theta_0,1}\\delta_{m',m+1}.\n\\label{eqn78}\n\\end{eqnarray}\nIn order to obtain Eqn(s).(\\ref{eqn77}, \\ref{eqn78}), we  \nused the following relations involving the associated Laguerre polynomials,\n\\begin{eqnarray}\nL_n^{(\\zeta)}(w)=L_n^{(\\zeta+1)}(w)-L_{n-1}^{(\\zeta+1)}(w) \\nonumber\\\\\n\\int_0^\\infty dw~w^{\\zeta} e^{-w}L_n^{(\\zeta)} (w)L_m^\\zeta (w) =\\frac{(n+\\zeta)!}{n!}\\delta_{n,m}~.\n\\label{eqn80}\n\\end{eqnarray}\nCombining Eqn(s).(\\ref{eqn77}, \\ref{eqn78}), we get the following expression,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}~=~\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}[\\sqrt{m'}e^{i\\Theta_{0,1}}\\delta_{m',m+1}\n-\\sqrt{m}e^{-i\\Theta_{0,1}}\\delta_{m,m'+1}].\n\\label{eqn79}\n\\end{eqnarray}\nNext, we evaluate,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^2|n,m'-n\\rangle_{H}= _{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n+_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}+ _{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H}. \n\\label{eqn81}\n\\end{eqnarray}\nEvaluation of the above matrix elements yield, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}&=& \n-\\frac{1}{4}{(\\hbar\\rho^2)}e^{-i\\Theta_{0,2}}\\delta_{m',m-2}\\sqrt{m(m-1)}\\nonumber\\\\\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}&=&~\\frac{1}{2}{(\\hbar\\rho^2)}e^{-i\\Theta_{0,0}}\\delta_{m,m'}(m+n+1)\\nonumber\\\\\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H}&=& \n-\\frac{1}{4}{(\\hbar\\rho^2)}e^{i\\Theta_{0,2}}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}.\n\\label{eqn82}\n\\end{eqnarray}\nIn order to calculate the above expressions, apart from the relations between \nspecial functions given by Eqn.(\\ref{eqn80}), we need the following relation,\n\\begin{eqnarray}\n\\int_{0}^{\\infty} dw~w^{k+p}e^{-w}L_n^k(w)L_n^k(w)~=~\\frac{(n+k)!}{n!}\\times(2n+k+1)^p ~.\n\\label{eqn83}\n\\end{eqnarray}\nSo we have,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^2|n,m'-n\\rangle_{H}&=&\\frac{(\\hbar\\rho^2)}{2}\\delta_{m,m'}(m+n+1)\\nonumber\\\\\n&&-\\frac{(\\hbar\\rho^2)}{4}\\left[e^{-i\\Theta_{0,2}}\\delta_{m',m-2}\\sqrt{m(m-1)}\\right.\\nonumber\\\\\n&& \\left. +e^{i\\Theta_{0,2}}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}\\right].\n\\label{eqn84}\n\\end{eqnarray}\nIt is to be noted that the matrix elements for $x$ and $x^2$ in the \neigenstates of the Hamiltonian [given by Eqn(s).(\\ref{eqn79}, \\ref{eqn84}) \nrespectively], matches exactly with the corresponding expression \ngiven in \\cite{Dey}, although the result \nquoted in \\cite{Dey} is in the eigenstate of the invariant $I(t)$.\n\n\\noindent The matrix element of $y^k$ in the \neigenstates of the Hamiltonian can be obtained similarly, and reads,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|y^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i\\Theta_{0,2r-k}}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn85}\n\\end{eqnarray}\nOnce again we write down the matrix element of $y^k$ in the eigenstates of the Lewis invariant $I(t)$. This reads\n\\begin{eqnarray}\n\\langle n,m-n|y^k|n,m'-n\\rangle&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn85I}\n\\end{eqnarray}\nUsing Eqn.(\\ref{eqn85}), we may evaluate the matrix element of $y$ and $y^2$ \nin the eigenstate of the Hamiltonian. We find, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|y|n,m'-n\\rangle_{H}&=&\n_{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\&&+ _{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}\n\\nonumber\\\\\n&=&-\\frac{1}{2}\n{(\\rho\\hbar^{1\/2})}[\\sqrt{m}e^{-i\\Theta_{0,1}}\\delta_{m',m-1}\n+\\sqrt{m+1}e^{i\\Theta_{0,1}}\\delta_{m',m+1}].\n\\label{eqn86}\n\\end{eqnarray}\n\\begin{eqnarray}\n_{H}\\langle n,m-n|y^2|n,m'-n\\rangle_{H}&=&_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n&&+_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}\\nonumber\\\\\n&&+_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H} \\nonumber\\\\\n&=&\\frac{\\hbar\\rho^2}{4}\\delta_{m',m-2}\\sqrt{m(m-1)}e^{-i\\Theta_{0,2}}\\nonumber\\\\\n&&+\\frac{1}{2}\\delta_{m,m'}{(\\hbar\\rho^2)}(m+n+1)+\\frac{\\hbar\\rho^2}{4}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}e^{i\\Theta_{0,2}}.\\nonumber\\\\\n\\label{eqn87}\n\\end{eqnarray}\nFrom the above analysis, we find that even the expression for the matrix element of the operator $y^k$ in the eigenstate of $H(t)$ \nmatches with that found in \\cite{Dey} for $k=1, 2$, \nthough again they had inappropriately quoted the results in the \neigenstate of the Lewis invariant. \n\n\n\n\\subsection{Analysis of the expectation value of energy}\nAs we have already seen from Eqn.(\\ref{eqn2}), in order to calculate the expectation value of energy one needs the expectation values $\\braket{{p_1}^2}$, $\\braket{{p_2}^2}$, $\\braket{{x_1}^2}$, $\\braket{{x_2}^2}$, $\\braket{{p_1}{x_2}}$ and \n$\\braket{{p_2}{x_1}}$. As we have seen in the previous subsection, our calculated generalized expressions for matrix elements $_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}$ and $_{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}$ matched exactly with the calculations in \\cite{Dey} for $k=1, 2$. Hence, we use the matrix elements quoted in the said work to calculate \nthe following expectation values,\n\\begin{eqnarray}\n\\braket{x_j^2}=\\dfrac{\\rho^2}{2}(n+m+1)\\,\\,;\\,\\,\\braket{p_j^2}=\\dfrac{1}{2}\\left(\\dfrac{1}{\\rho^2}+\\dfrac{\\dot{\\rho}^2}{a^2} \\right)\\,(n+m+1)\\,\\,;\\,\\,\\braket{x_j\\,p_k}=\\dfrac{1}{2}\\,\\epsilon_{jk}(m-n)\\,\\,;\\label{eqn88}\n\\end{eqnarray}\nwhere $j,k=1,2$ and $\\epsilon_{jk}=-\\epsilon_{kj}$ with $\\epsilon_{12}=1$. So, the expectation value of energy $\\braket{E_{n,m-n}(t)}$ with \nrespect to energy eigenstate $\\psi_{n,m-n}(r,\\theta,t)$ can be expressed as,\n\\begin{align}\n&\\braket{E_{n,m-n}(t)}=\\dfrac{1}{2}\\,(n+m+1)\\left[b(t)\\rho^2(t)+\\dfrac{a(t)}{\\rho^2(t)}+\\dfrac{\\dot{\\rho}^2(t)}{a(t)} \\right]+c(t)\\,(n-m)\\,\\,.\\nonumber\\\\\n&=\\dfrac{1}{2}\\left[\\,(n+m+1)\\left(b(t)\\rho^2(t)+\\dfrac{a(t)}{\\rho^2(t)}+\\dfrac{\\dot{\\rho}^2(t)}{a(t)} \\right)+(n-m)\\left(\\dfrac{f(t)\\Omega(t)}{M}+\\dfrac{M\\omega^2(t)\\theta(t)}{f(t)}\\right) \\right].\n\\label{eqn89}\n\\end{align}\n\n\n\n\n\\noindent It is interesting to note that even when the frequency of oscillation $\\omega{\\rightarrow}0$, the \nexpectation value of energy is non-zero. This is because all the three parameters of the Hamiltonian $a(t)$, $b(t)$ and $c(t)$ are finite even as $\\omega{\\rightarrow}0$, as is clear from the Eqn(s).(\\ref{3},\\ref{4},\\ref{eqn3}). Now we will proceed to study the time-dependent behaviour of $\\braket{E_{n,m-n}(t)}$ for various types of damping.\n\n\n\n\n\\subsubsection{Exponentially decaying solution}\nFor the exponentially decaying solution given by Eqn.(\\ref{EPsoln1}), the energy expectation value takes the following form,\n\\begin{equation}\n\\braket{E_{n,m-n}(t)}=(n+m+1)\\mu^2\\Delta+c(t)\\,(n-m)\n\\label{eqn90}\n\\end{equation}\nwhere we have set the constant $\\xi^2$ to unity and used the constraint relation given by Eqn.(\\ref{EPreln1}).\n\n\\vskip 0.1cm\n\n\n\n\\noindent{\\bf $\\langle A\\rangle$ Solution Set-Ia}\n\\vskip 0.15cm\n\n\\noindent For this case we consider $f(t)=1$ and $\\omega(t)=\\omega_0\\,e^{-\\Gamma\\,t\/2}$. The expectation value of \nenergy for the ground state has the following expression,\n\\begin{eqnarray}\n\\braket{E_{n,-n}(t)}&=&(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t})-M{\\omega_0}^2\\,exp\\,(-\\Gamma{t}) }{M}} \n\\,\\right.\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~\\left. +\\,\\omega_0\\,exp\\,(-\\Gamma{t\/2})\\sqrt{\\,M\\sigma\\,exp\\,(-\\Gamma{t})-1 \\,}\\right].\n\\label{eqn91}\n\\end{eqnarray}\nFrom Eqn.(\\ref{eqn91}), we see that the expectation value of the energy becomes complex beyond a certain time limit. The \ncondition for getting the expectation value of energy to be real is as follows,\n\\begin{eqnarray}\nM\\,\\sigma\\,e^{-\\Gamma\\,t}>1\\,\\,\\Rightarrow\\,t\\leq\\,\\dfrac{ln(M\\,\\sigma)}{\\Gamma}~.\\label{eqn92}\n\\end{eqnarray}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{expdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is  calculated for exponentially decaying \nHamiltonian parameters when $\\langle A\\rangle$ Set-IA $f(t)=1$ and \n$\\omega(t)=\\omega_0 e^{-{\\Gamma}t\/2}$; $\\langle B\\rangle$ Set-IB \n$f(t)=e^{-{\\Gamma}t}$ and $\\omega(t)=\\omega_0$ and $\\langle C\\rangle$ \nSet-IC $f(t)=e^{-{\\Gamma}t}$ and $\\omega(t)=\\omega_0 e^{-{\\Gamma}t\/2}$. While \nfor $\\langle A\\rangle$ the energy first decreases, then increases with time, \nfor $\\langle B\\rangle$ the energy remains constant as we vary time. \nFor $\\langle C\\rangle$ the energy decays off with time.}}  \n\\label{fig1}\n\\end{figure}\n\n\n\n\\noindent We see from Fig.(\\ref{fig1}), that the energy initially decays but then increases with time. This is \nbecause for large time at which $exp\\,(-\\Gamma{t\/2})\\approx\\,0 $, the \napproximated expression of energy reads\n\\begin{equation}\nE_{n,-n}(t)\\approx\\,(n+1)\\mu^2\\Delta+\\,n\\,\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t}) }{M}}\\,\\,\n\\label{e7}\n\\end{equation}\nwhich is still increasing with time. The reason for the increase of energy with time is the form of the \ncoefficient $b(t)$ in the Hamiltonian. Although the coefficient $a(t)$ is exponentially decaying with \ntime, the coefficient $b(t)$ exponentially increases with time in order to satisfy EP equation. However, since there is an upper limit of time within which the energy remains real, so the energy remains finite within the allowed time interval.\n\n\\vskip 0.1cm\n\n\n\n\n\\noindent{{\\bf{$\\langle B\\rangle$ Solution Set-Ib}}}\n\n\\noindent Here we set $f(t)=e^{-\\Gamma\\,t}$ and $\\omega(t)=\\omega_0$. With this the energy expression for the ground state takes the form,\n\\begin{equation}\n\\braket{E_{n,-n}(t)}=(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\, }{M}} \\,+\\,\\omega_0\\sqrt{M\\sigma-1}\\right].\\label{e8}\n\\end{equation}\n\n\n\\noindent We note from Fig.(\\ref{fig1}), that the expectation value of the energy remarkably remains constant as we vary time, as is observed from Eqn.(\\ref{e8}). This must be because the effect of the exponentially decaying Hamiltonian \ncoefficient $a(t)$ and damping term $f(t)$ gets balanced out by the exponentially increasing Hamiltonian \ncoefficient $b(t)$.\n\\vskip 0.1cm\n\n\n\n\\noindent{{\\bf{$\\langle C\\rangle$ Solution Set-Ic}}}\n\n\\noindent Here we set $f(t)=e^{-\\Gamma\\,t}$ and $\\omega(t)=\\omega_0\\,e^{-\\Gamma\\,t\/2}$. \nWith this the expectation value of the energy expression takes the form,\n\\begin{equation}\n\\braket{E_{n,-n}(t)}=(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\,exp\\left[-\\Gamma{t}\\right] }{M}} \n +  {\\omega_0}\\,exp\\,(-\\Gamma{t\/2})\\sqrt{M{\\sigma}\\,-1}\\right].\\label{e9}\n\\end{equation}\nThe above expression gives a very nice decaying expression for the expectation value of energy with respect to time, and finally approaching a constant value in the limit \n$t\\rightarrow\\infty$. This behaviour is also exhibited in the nature of the plot of variation of the expectation value of energy with time seen in Fig.(\\ref{fig1}).\n\n\\vskip 0.1cm\n\n\n\n\\subsubsection{Rationally decaying solution}\n\n\\noindent In this case the expectation value of energy for $k=2$ reads\n\\begin{eqnarray}\nE_{n,-n}(t)&=&\\dfrac{(n+1)}{2(\\Gamma\\,t+\\chi)}\\left[2\\left(\\dfrac{\\sigma}{\\mu^2}+\\Delta\\mu^2\\right)+\\dfrac{\\mu^2\\Gamma^2}{8\\sigma} \\right]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~ +n\\left[\\dfrac{\\omega_0}{\\Gamma\\,t+\\chi}\\,\\sqrt{\\dfrac{4\\sigma\\,M}{(\\Gamma{t}+\\chi)^2}\\,-\\,1}\\,+\\,\\sqrt{\\dfrac{\\Delta}{M}-\\dfrac{\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^{\\,2}}} \\right].\\label{e10}\n\\end{eqnarray}\n\\noindent Note that although it has a nice decaying property like the damping case on commutative plane, there is an upper bound of time above which the energy ceases to be real. The upper bound on time reads,\n\\begin{eqnarray}\n4\\sigma\\,M\\,\\geq\\,(\\Gamma\\,t+\\chi)^2\\,\\,\\Rightarrow\\,\\,t\\,\\leq\\,\\dfrac{1}{\\Gamma}(2\\sqrt{M\\,\\sigma}-\\chi).\n\\label{e11}\n\\end{eqnarray}\n\n\n\\vskip 0.1cm\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{ratdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$, $\\chi=1$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is  calculated for rationally decaying \nHamiltonian parameters. We consider $f(t)=1$ and $\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}$.}} \n\\label{fig2}\n\\end{figure}\n\n\\noindent From Fig.(\\ref{fig2}), we see indeed the expectation value of energy $\\langle E \\rangle$ decays with time \nfollowing power law as expected for the rationally decaying solutions.  \n\n\n\\subsubsection{Elementary solution}\nFor the elementary solution set, the expectation value of the energy reads,\n\\begin{eqnarray}\n\\braket{E_{n,-n}(t)}&=&\\dfrac{1}{2}(n+1)\\left[\\left(\\Delta\\mu^2+\\dfrac{\\sigma}{\\mu^2}\\right)\\dfrac{1}{(\\Gamma\\,t+\\chi)^2}+\\dfrac{\\mu^2\\Gamma^2}{\\sigma}\\right]\n\\nonumber\\\\&&+n\\left[\\dfrac{\\omega_0\\sqrt{M\\sigma-1}}{(\\Gamma\\,t+\\chi)}+\\dfrac{1}{(\\Gamma\\,t+\\chi)}\\sqrt{\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}-\\omega_0^2 } \\right].\\label{e12}\n\\end{eqnarray}\nFurther, the constraint relation $\\Delta\\mu^4=\\xi^2\\sigma$ results in the following form for the expectation value of energy (setting $\\xi^2=1$),\n\\begin{align}\n\\braket{E_{n,-n}(t)}\n&=\\dfrac{1}{2}(n+1)\\left[\\dfrac{2\\sigma}{\\mu^2(\\Gamma\\,t+\\chi)^2}+\\dfrac{\\mu^2\\Gamma^2}{\\sigma}\\right]+n\\,\\left[\\dfrac{\\omega_0\\sqrt{M\\sigma-1}}{(\\Gamma\\,t+\\chi)}+\\dfrac{1}{(\\Gamma\\,t+\\chi)}\\sqrt{\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}\\right].\\label{e130}\n\\end{align}\nThis expression also provides an upper bound of the time limit above which the expectation value of energy would become complex. This upper bound reads,\n\\begin{eqnarray}\n\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}\\,\\geq\\,\\omega_0^2\\,\\Rightarrow\\,t\\,\\leq\\,\\dfrac{1}{\\Gamma}\\left[ \\dfrac{1}{\\omega_0}\\sqrt{\\dfrac{\\Delta}{M}}-\\chi \\right].\\label{e14}\n\\end{eqnarray}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{elmdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$, $\\chi=1$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is  calculated for elementarily decaying \nHamiltonian parameters. We consider $f(t)=1$ and $\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}$.}} \n\\label{fig3}\n\\end{figure}\n\n\\noindent In Fig.(\\ref{fig3}), we observe that the expectation value of energy again undergoes a power law decay with time for the elementary solution.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe now summarize our results. In this paper we have considered a two-dimensional damped harmonic oscillator in noncommutative space\nwith time dependent noncommutative parameters.\nWe map this system in terms of commutative variables by using a shift of variables connecting the noncommutative and commutative\nspace, known in the literature as Bopp-shift. We have then obtained the exact solution of this time dependent system by using the well known Lewis invariant which in turn leads to a non-linear differential equation known as the Ermakov-Pinney equation. We first obtain the Lewis invariant in Cartesian coordinates. \nWe then make a transformation to polar coordinates and write down our results in these coordinates. Doing \nso, we use the operator approach to obtain the eigenstates of the invariant. With this background in place, we make various choices of the parameters in the problem which in turn leads to solutions for the time dependent noncommutative parameters. We have considered three different sets of choices for which solutions have been obtained, namely, exponentially decaying solutions, rationally decaying solutions and elementary solutions. Interestingly, the solutions obtained make it possible to integrate the phase factor exactly thereby giving an exact solution for the eigenstates of the Hamiltonian. We have then computed the matrix elements of operators raised to a finite integer power in both the eigenstates of the Hamiltonian as well as the Lewis invariant. From these results, we are able to compute the expectation value of the Hamiltonian. Expectedly, the expectation value of the energy varies with time. For the exponentially decaying solutions, we get three kinds of behaviour corresponding to the choices of the damping factor and the frequency of the oscillator. For the case where the damping factor is set to unity and the frequency of the oscillator decays with time, the expectation value of the energy first decreases with time and then increases.\nThe reason for this behaviour is due to the particular form of the solutions of the Ermakov-Pinney equation which fixes the forms of the noncommutative parameters. It is these time dependent forms of the noncommutative parameters that results in the above mentioned behaviour of the expectation value of the energy with time. \nIn this case, we also observe that there is an upper bound of time above which the energy expectation value ceases to be real. \nFor the case where the damping factor has a decaying part and the frequency of the oscillator is a constant, we observe that the expectation value of the energy remarkably remains constant with time. This must be the case because the effect of the exponentially decaying coefficient in the Hamiltonian \nand the damping term gets balanced out by the exponentially increasing coefficient in the Hamiltonian. For the case where both the damping term as well as the frequency of the oscillator decays with time, we find an exponentially decaying behaviour of the expectation value of the energy. For the rationally decaying and the elementary solution, we observe a power law decay of the energy expectation value with time together with an upper bound of time above which the energy expectation value ceases to be real. Investigating these cases of damped oscillators, we conclude that the behaviour corresponding to the exponentially decaying solution, where both the frequency and damping term are decaying exponentially with time, is similar to a damped oscillator in commutative space. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nMD would like to thank Ms. Riddhi Chatterjee and Ms.Rituparna Mandal for their helpful assistance to operate the software Mathematica.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The Bloch--Kato conjecture, which relates the dimension of the Selmer group of a $p$-adic geometric Galois representation to the order of vanishing of its $L$-function, is one of the most important open problems in number theory. In a recent paper \\cite{LZ20}, we proved this conjecture for the 4-dimensional Galois representations arising from automorphic representations of $\\GSp_4$, under various technical hypotheses, using the ``method of Euler systems''; this relied crucially on the construction of an Euler system for $\\GSp_4$ in our earlier work \\cite{LSZ17} with Skinner, and the construction of a $p$-adic spin $L$-function for $\\GSp_4$ in the paper \\cite{LPSZ1} with Pilloni and Skinner.\n\n In this paper, we prove new cases of the Bloch--Kato conjecture, for the 8-dimensional tensor product Galois representation $V(\\pi) \\otimes V(\\sigma)$ associated to a $\\GSp_4$ automorphic representation $\\pi$ and a $\\GL_2$ automorphic representation $\\sigma$. The principal ingredients are the the Euler system for these Galois representations constructed in \\cite{HJS20}, and the formula proved in \\cite{LZ20b-regulator} relating these Euler system classes to periods of $p$-adic modular forms for $\\GSp_4 \\times \\GL_2$ (obtained by integrating a class in $H^2$ arising from $\\pi$, restricted to $\\GL_2 \\times_{\\GL_1} \\GL_2 \\subset \\GSp_4$, against the product of a cusp form in $\\sigma$ and a non-classical $p$-adic Eisenstein series). The main result of this paper, \\cref{thm:BKconj}, proves the Bloch--Kato conjecture for a certain twist of $V(\\pi)^* \\otimes V(\\sigma)^*$, corresponding to a critical value of the $L$-function $L(\\pi \\times \\sigma, s)$.\n\n The main new technical input needed in order to prove this theorem is to interpolate the $p$-adic automorphic periods arising from \\cite{LZ20b-regulator} in $p$-adic families, with both of the weights $(r_1, r_2)$ of the $\\GSp_4$ automorphic representation allowed to vary. This is not accessible by the methods of our earlier work \\cite{LPSZ1}, since the version of higher Hida theory used in that paper (based on the earlier work \\cite{pilloni20}) is only applicable to 1-parameter families in which $r_1$ varies for a fixed $r_2$. A similar issue arises in our earlier work \\cite{LZ20}, but in that setting, we were able to bypass the problem by applying the functorial lift from $\\GSp_4$ to $\\GL_4$, and applying the results of \\cite{DJR18, barreradimitrovwilliams} on $p$-adic $L$-functions for $\\GL_{2n}$. However, this does not work for $\\GSp_4 \\times \\GL_2$, since there appears to be no known construction of $p$-adic $L$-functions for $\\GL_4 \\times \\GL_2$.\n\n We therefore develop a direct approach to interpolating these $p$-adic periods in 2-parameter families for $\\GSp_4$, with both $r_1$ and $r_2$ varying, using the new ``higher Coleman theory'' introduced in \\cite{boxerpilloni20}. Our main result in this direction is \\cref{thm:bspairing}, whose proof occupies the majority of the present paper. This result shows that there is a well-defined pairing between the higher Coleman theory spaces for $\\GSp_4$ and spaces of overconvergent modular forms for $\\GL_2 \\times \\GL_2$; and in the final sections of the paper, we use this to define $p$-adic $L$-functions in families for $\\GSp_4 \\times \\GL_2$ by pairing a family of $H^2$ eigenclasses for $\\GSp_4$ with the product of a $\\GL_2$ cusp-form family and an auxiliary Eisenstein series. The existence of this $p$-adic $L$-function then allows us to prove a reciprocity law relating the Euler system of \\cite{HJS20} to critical complex $L$-values, and thus to prove the Bloch--Kato conjecture.\n\n These new methods can also be used to strengthen the results of \\cite{LZ20} for the degree 4 motive of $\\GSp_4$; for reasons of space, we shall pursue this in a forthcoming paper. Our methods also give, as a by-product, the construction of a ``$p$-adic Eichler--Shimura isomorphism in families'' for $\\GSp_4$, interpolating the comparison isomorphisms between de Rham and \\'etale cohomology for all (or almost all) specialisations of a $\\GSp_4$ Hida family. Our results give an interpolation of the comparison isomorphism after projecting to a specific filtration step of de Rham cohomology, corresponding to $H^2$ of automorphic vector bundles. Our results are thus complementary to the recent work of Diao et al \\cite{diao-rosso-wu21} which interpolates the filtration step corresponding to $H^0$.\n\n\\begin{remark}\n The switch from ``Hida'' to ``Coleman'' theory allows us to define $p$-adic $L$-functions for finite-slope families, rather than just for ordinary (i.e.~slope 0) families. However, this comes at a price: the use of Coleman theory requires an overconvergence condition on the Eisenstein series, which does not hold for the 2-parameter family of Eisenstein series used in \\cite{LPSZ1}. So the price we pay for including the second weight variable $r_2$ is that we lose sight of the cyclotomic variable -- for each automorphic representation $\\pi \\times \\sigma$ of $\\GSp_4 \\times \\GL_2$, there is an interval of integers $n$ such that $V(\\pi)^* \\otimes V(\\sigma)^*(-n)$ is critical, but in the present paper we can only prove the Bloch--Kato conjecture for a specific $n$, corresponding to the lower endpoint of this interval. Even in the ordinary case, to prove the Bloch--Kato conjecture for all of the critical twists, we would need a version of higher Hida (rather than Coleman) theory for $\\GSp_4$ with both $r_1$ and $r_2$ varying. Such a theory is not available at present, although analogous results for Hilbert modular groups have been announced by Giada Grossi \\cite{grossi21}.\n\\end{remark}\n\n\\emph{Acknowledgements.} We would like to thank George Boxer and Vincent Pilloni for answering our questions about their beautiful theory. We are very grateful for their patience.\n\n\n\n\n\\section{Preliminaries}\n\n Throughout this paper $p$ is a prime.\n\n \\subsection{The group $G$}\n\n  Let $G=\\GSp(4)$, with respect to the anti-diagonal Hermitian form with matrix $J = \\begin{smatrix} & &&1\\\\&&1\\\\&-1\\\\-1\\end{smatrix}$. Write $B_G$ for the Borel subgroup consisting of upper-triangular matrices, and write $P_{\\Kl}$ and $P_{\\Sieg}$ for the Klingen and Siegel parabolic subgroups containing $B_G$. We then have the Levi decompositions\n  \\[ B_G = T N_{B}, \\qquad P_{\\Sieg}= M_{\\Sieg} N_{\\Sieg},\\qquad P_{\\Kl}= M_{\\Kl} N_{\\Kl},\\]\n  where $T$ is the diagonal torus.\n\n  The Siegel parabolic $P_{\\Sieg}$ and its Levi $M_{\\Sieg}$ plays a distinguished role in our constructions, since it is conjugate to the centraliser of the cocharacter defining the Shimura datum; the Klingen parabolic is less important here (in contrast with our previous paper \\cite{LPSZ1}). Hence we shall often write simply $P_G$, $M_G$ for $P_{\\Sieg}, M_{\\Sieg}$. We identify $M_{G}$ with $\\GL_2 \\times \\GL_1$ via $\\stbt{A}{0}{0}{\\star} \\mapsto (A, \\nu)$, where $\\nu$ is the symplectic multiplier.\n\n  Let $W_G=N_G(T)\/T$ denote the Weyl group of $(G,T)$. The group $W_G$ is generated by the $T$-cosets of the elements\n  $s_1=\\begin{smatrix} 1\\\\ & & 1 \\\\ & -1 \\\\ &&&1 \\end{smatrix}$\n  and\n  $s_2=\\begin{smatrix}  & 1\\\\ -1 \\\\ &&& -1 \\\\ &&1\\end{smatrix}$.\n \n\n  Let $W_{M_G} = \\langle s_2 \\rangle $ denote the Weyl group of $(M_G, T)$, and let ${}^{M} W_G=W_{M_G}\\backslash W_G$. This has a distinguished set of coset representatives (the \\emph{Kostant representatives}) given by\n  \\[ {}^MW_G=\\{ \\mathrm{id}, w_1, w_2, w_3 \\}\\]\n  where $w_1 = s_1$, $w_2 = s_1 s_2$, $w_3 = s_1 s_2 s_1$. These have lengths $\\ell(w_i) = i$. We use $w_G^{\\max}$ for the long Weyl element of $G$, and $w_{M_G}^{\\max} = s_2$ the long Weyl element of $M_G$.\n\n  \\begin{remark}\n   Note that:\n   \\begin{enumerate}[(i)]\n    \\item Since $W_G$ permutes the coordinates of the diagonal torus, we can realize it as subgroup of $S_4$. Then ${}^{M}W_G$ identifies with the permutations $w \\in W_G$ such that $w(1)<w(2)$.\n\n    \\item The map $w \\mapsto w_{M}^{\\max} \\cdot w \\cdot w_{G}^{\\max}$ preserves ${}^M W_G$, and interchanges $w_i$ with $w_{3-i}$.\n\n    \\item The group $\\GSp_4$ has the unusual and convenient property that all the elements of ${}^M W_G$ have distinct lengths, so the numbering $w_i$ makes sense.\n\n    \\item There is a slight conflict here with the notations of \\cite{boxerpilloni20} where $w_0$ is the \\emph{long} Weyl element, hence the use of $w_G^{\\max}$ here.\\qedhere\n   \\end{enumerate}\n  \\end{remark}\n\n\n \\subsection{The group $H$}\n\n  Let $H=\\GL_2\\times_{\\GL_1}\\GL_2$, and write $B_H$ for the Borel subgroup of upper-triangular elements of $H$; it has a Levi decomposition $B_H=T_H N_H$. We observe that the Levi subgroup $M_H\\subset B_H$ identifies with $T_H$. Hence $W_{M_H}=\\{1\\}$, so ${}^{M}W_H=W_H \\cong W_{\\GL_2}\\times W_{\\GL_2}$.\n  We consider $H$ as a subgroup of $G$ via\n  \\[ \\iota: \\left[\\begin{pmatrix} a & b\\\\ c & d\\end{pmatrix}, \\begin{pmatrix} a' & b'\\\\ c' & d'\\end{pmatrix}\\right]\\mapsto \\begin{smatrix} a &&& b\\\\ & a' & b' & \\\\ & c' & d' & \\\\ c &&& d\\end{smatrix}.\\]\n  For future use, we write $H_i$ for the $i$th $\\GL_2$-factor of $H$, and we write $T_i$ for the torus of $H_i$ and $\\Delta$ for the joint subgroup identified with the character $\\nu$, so $T=T_1\\times_{\\Delta}T_2$. For $i=1,2$, write $\\varpi_i:T\\rightarrow T_1$ for the natural projection map.\n\n\n \\subsection{Algebraic weights and roots}\n\n  As in \\cite[\\S 5.1.1]{pilloni20} and \\cite[\\S 2.3.1]{LPSZ1}, we identify characters of $T$ with triples of integers $(r_1, r_2; c)$, with $r_1 + r_2 = c \\bmod 2$, corresponding to the character\n  \\[ \\diag(st_1, st_2, s t_2^{-1}, s t_1^{-1}) \\mapsto t_1^{r_1} t_2^{r_2} s^{c}.\\]\n\n  \\begin{remark}\n   Note that the earlier paper \\cite{LSZ17} uses a slightly different notation, but we shall use the above here.\n  \\end{remark}\n\n  In this notation the simple positive roots with respect to $B_G$ are $(1,-1; 0)$ and $(0, 2; 0)$; and the half-sum is $\\rho_G = (2, 1; 0) \\in \\tfrac{1}{2} X^\\bullet(T)$. A weight $(r_1, r_2; c)$ is dominant for $H$ if $r_1, r_2 \\ge 0$, dominant for $M_G$ if $r_1 \\ge r_2$, and dominant for $G$ if both of these conditions hold.\n\n  The Weyl group acts (on the left) on the group of characters $X^*(T)$ via\\footnote{There is also a ``twisted'' action, but we shall write this out explicitly when it arises, rather than defining general notations for it.}\n  \\[ (w\\cdot \\lambda)(t)=\\lambda(w^{-1}tw). \\]\n  Explicitly, the generators of the Weyl group of $G$ act by $s_1 \\cdot (r_1, r_2; c) = (r_1, -r_2; c)$ and $s_2 \\cdot (r_1, r_2; c) = (r_2, r_1; c)$. The long Weyl element acts by $w_G^{\\max} \\cdot (r_1, r_2; c) = (-r_1, -r_2; c)$; thus $-w_G^{\\max} \\lambda$ and $\\lambda$ coincide on the derived subgroup $\\operatorname{Sp}_4$.\n\n  There is an element $\\rho_{G, nc}$, which is half the sum of the ``non-compact'' roots, i.e.~those appearing in the unipotent radical $N_{P_{\\Sieg}}$. This is given by $\\rho_{G, nc} = (\\tfrac{3}{2}, \\tfrac{3}{2}; 0) = \\tfrac{1}{2}(w_M^{\\max} \\rho_G + \\rho_G)$. Note that $\\rho_{G,nc}+ \\rho_{M_G} = \\rho_G$, where $\\rho_{M_G} = \\left(\\tfrac{1}{2}, \\tfrac{-1}{2}; 0\\right)$ is the half-sum for $M_G$.\n\n  For $H$, all roots are non-compact (the maximal compact in $H(\\mathbf{R})$ is abelian), so $\\rho_{M_H} = 0$ and $\\rho_{H, nc} = \\rho_H = (1, 1; 0)$.\n\n\n\n\n \\subsection{Flag varieties and Shimura varieties}\n\n  Write $\\FL_G=P_G \\backslash G$ for the Siegel flag variety of $G$.\n  For a neat open compact subgroup $K\\subset G(\\AA_{\\mathrm{f}})$, write $S_{G,K}$ for the canonical model over $\\Spec(\\mathbf{Q})$ of the Shimura variety $G(\\mathbf{Q})\\backslash \\mathcal{H}_2\\times G(\\AA_{\\mathrm{f}})\/K$; here $\\mathcal{H}_2$ is the Siegel upper half space.\n  Depending on the choice of a projective cone decomposition $\\Sigma$, let $S_{G,K}^{\\mathrm{tor}}\\rightarrow \\Spec(\\mathbf{Q})$ be the toroidal compactification of $S_{G,K}$ corresponding to $\\Sigma$. Denote by $A$ the universal semi-abelian scheme over $S_{G,K}^{\\mathrm{tor}}$.\n\n  We similarly define the flag variety $\\FL_H=B_H\\backslash H$, the canonical rational model of the Shimura variety $S_{H,L}$ for $L=L^pL_p\\subseteq H(\\AA_{\\mathrm{f}})$ an open compact subgroup, and $S^{\\mathrm{tor}}_{H,L}$ for its toroidal compactification. The universal semi-abelian scheme over $S^{\\mathrm{tor}}_{H,L}$ is given by $\\mathcal{E}_1\\boxtimes \\mathcal{E}_2$, where $\\mathcal{E}_i$ is the generalized elliptic curve over the modular curve corresponding to the $i$th $\\GL_2$-factor of $H$.\n\n  \\begin{notation}\n   We shall abuse notation slightly by writing $S_{G,K}^{\\mathrm{tor}}$ where $K$ is a subgroup of $G(\\QQ_p)$. By this, we mean $S_{G, K^p K}^{\\mathrm{tor}}$ where $K^p$ is some fixed choice of open compact away from $p$. When discussing Shimura varieties for $G$ and $H$ together we shall suppose these tame levels $K^{G, p}$ and $K^{H, p}$ are chosen such that $K^{G, p} \\cap H = K^{H, p}$.\n  \\end{notation}\n\n \\subsection{Coefficient sheaves}\n\n  \\subsubsection{Reminders on vector bundles, Weyl chambers, coherent cohomology}\n\n   We briefly recall the conventions for automorphic vector bundles from \\bp{\\S 4.1.1} and attempt to reconcile them with our earlier work elsewhere.\n   \\newcommand{\\operatorname{Rep}}{\\operatorname{Rep}}\n\n   We have a functor $\\operatorname{Rep}(M_G) \\to VB(S^{G,\\mathrm{tor}}_K)$ defined using the torsor $M^G_{\\mathrm{dR}}$, and we let $\\mathcal{V}_{\\kappa}$, for $\\kappa \\in X^\\bullet(T)$ that is $M_G$-dominant, be the image of the representation of highest weight $\\kappa$ (with respect to $B_G \\cap M_G$). With these conventions:\n\n   \\begin{itemize}\n    \\item The weight $-2\\rho_{nc} = (-3, -3; 0)$ maps to $\\Omega^3_{S^{G,\\mathrm{tor}}_K}(\\log D)$ (see \\bp{\\S 4.2.3}.\n\n    \\item The vector bundles having cohomology only in degree 0 are the $\\mathcal{V}_{\\kappa}$ with $\\kappa = (k_1, k_2; m)$, $-3 \\ge k_1 \\ge k_2$ (sic).\n\n    \\item If $\\kappa = (-, -; m)$ for $m \\in \\mathbf{Z}$, and $\\pi$ contributes to the cohomology of $\\mathcal{V}_{\\kappa}$, then the $\\infty$-type of $\\pi$ is $\\|\\cdot\\|^{-m}$ on the centre, so $\\diag(\\varpi_\\ell, \\dots, \\varpi_\\ell) \\in Z_G(\\mathbf{Q}_\\ell)$ acts on $R\\Gamma(S^{G,\\mathrm{tor}}_K, \\mathcal{V}_{\\kappa})$ as $\\ell^m$ times a finite-order character.\n   \\end{itemize}\n\n   We now recall how these are related to automorphic representations.\n\n   \\begin{itemize}\n\n    \\item For each $M_G$-dominant $\\kappa$, there exists a unique $\\nu \\in X^\\bullet(T)$ such that $\\nu + \\rho$ is dominant and\n    \\begin{align*}\n     \\kappa = -w_{M_G}^{\\max} w (\\nu + \\rho) - \\rho &\\text{ for some $w \\in {}^M W_G$},\\\\\n     \\text{or equivalently }\\kappa = -w w_G^{\\max} (\\nu + \\rho) - \\rho &\\text{ for some $w \\in {}^M W_G$}.\n    \\end{align*}\n    Then the automorphic representations contributing to the cohomology of $\\mathcal{V}_{\\kappa}$ have infinitesimal character $\\nu + \\rho$.\n\n    \\item The character $\\nu + \\rho$ is dominant, but possibly not integral; and $\\nu$ is integral, but possibly not dominant. The cases where $\\nu$ is actually dominant integral correspond to cohomological $\\infty$-types: the infinitesimal character $\\nu + \\rho$ is the same as that of the algebraic representation $W_\\nu$.\n\n    \\item Given $\\kappa$, the element $\\nu$ is uniquely determined but the element $w$ is not. We let\n    \\begin{align*}\n     C(\\kappa)^+ &\\coloneqq \\{ w \\in {}^M W_G: \\kappa = -w_{M_G}^{\\max} w (\\nu + \\rho) - \\rho\\},\\\\\n     C(\\kappa)^- &\\coloneqq \\{ w \\in {}^M W_G: \\kappa = -w w_G^{\\max}(\\nu + \\rho) - \\rho\\}.\n    \\end{align*}\n    The involution $w \\mapsto w_{M}^{\\max} w w_{G}^{\\max}$ interchanges $C(\\kappa)^+$ and $C(\\kappa)^-$.\n\n    \\item The definition of $C(\\kappa)^{\\pm}$ can be rewritten in terms of the dual weights $\\nu^\\vee = -w_G^{\\max} \\nu$ and $\\kappa^\\vee = -w_M^{\\max} \\kappa$; then it takes the form\n    \\begin{align*}\n     C(\\kappa)^+ &\\coloneqq \\{ w \\in {}^M W_G: \\kappa^\\vee = w(\\nu + \\rho) - \\rho \\text{ with $\\nu + \\rho$ dominant}\\},\\\\\n     C(\\kappa)^- &\\coloneqq \\{ w \\in {}^M W_G: \\kappa = w(\\nu^\\vee + \\rho) - \\rho \\text{ with $\\nu^\\vee + \\rho$ dominant}\\}.\n    \\end{align*}\n    This shows, in particular, that $C(\\kappa^\\vee)^\\pm = C(\\kappa)^{\\mp}$ (corresponding to Serre duality).\n\n    \\item If $\\pi$ has infinitesimal character $\\nu+\\rho$, then $\\pi$ contributes to cohomology of $\\mathcal{V}_{\\kappa}$ in degree $i$ if $C(\\kappa)^+$ contains an element of length $i$, or equivalently $C(\\kappa)^-$ contains an element of length $3-i$.\n\n   \\end{itemize}\n\n   \\begin{remark}\n    We have overloaded our notations a little for ``dual weights'', since the notation $\\sigma^\\vee$ for a weight $\\sigma$ will mean $-w_{?}^{\\max} \\sigma$ where $?$ variously denotes $G$, $M_G$, or sometimes $H$ or $M_H$. However, it will (we hope) always be clear from the context which is intended.\n   \\end{remark}\n\n\n  \\subsubsection{The BGG complex}\n\n   Given a weight $\\nu = (r_1, r_2; c)$ as above, with $r_1 \\ge r_2 \\ge -1$, we shall define\n   \\[ \\kappa_i(\\nu) = w_i(\\nu + \\rho) - \\rho, \\qquad 0 \\le i \\le 3. \\]\n   These are the weights $\\kappa$ such that representations of infinitesimal character $\\nu^\\vee + \\rho$ contribute to $R\\Gamma(S^{G,\\mathrm{tor}}_K, \\mathcal{V}_{\\kappa})$. They are explicitly given by\n   \\begin{equation}\n    \\label{eq:ourweights}\n    \\begin{aligned}\n     \\kappa_0(\\nu) &= (r_1, r_2; c) &  \\kappa_1(\\nu) &= (r_1, -r_2 - 2; c)\\\\\n     \\kappa_2(\\nu) &= (r_2 - 1, -r_1 -3; c) & \\kappa_3(\\nu) &= (-r_2-3, -r_1-3; c).\n    \\end{aligned}\n   \\end{equation}\n   Often we shall take $c = (r_1 + r_2)$; the sheaves $\\mathcal{V}_{\\kappa_i(\\nu)}$, where $\\nu$ has the form $(r_1, r_2; r_1+r_2)$, will be the main sheaves of interest in this work. Note that $\\mathcal{V}_{\\kappa_i(\\nu)}$ has cuspidal cohomology in degree $3 - i$, and the centre acts on this cohomology via $\\diag(p,p,p,p) \\mapsto p^{r_1 + r_2}$ up to a finite-order character.\n\n   If $r_1 \\ge r_2 \\ge 0$, so $\\nu$ is dominant, then the sheaves $\\mathcal{V}_{\\kappa_i(\\nu)}$ are distinct, and they are the terms in the BGG complex quasi-isomorphic to $\\mathcal{W}_{\\nu} \\otimes \\Omega^\\bullet(\\log D)$.\n\n   \\begin{note}\n    If $\\nu$ is dominant, then $C(\\kappa_i(\\nu))^- = \\{ w_i\\}$, and dually $C(\\kappa_i(\\nu))^+ = \\{w_{3-i}\\}$. If $\\nu = 0$, then $\\mathcal{V}_{\\kappa_i(\\nu)} = \\Omega^i(\\log D)$, so the BGG complex is simply the de Rham complex.\n   \\end{note}\n\n  \\subsubsection{Comparison with other works}\n\n   \\begin{itemize}\n   \\item Comparison to \\cite{LPSZ1}: the (Hecke-equivariant) vector bundle we attached to $(k_1, k_2; m)$ in \\cite{LPSZ1} is now associated to $(-k_2, -k_1; m)$. That is, the conventions differ by $w_{0, M}\\cdot  w_{0, G}$, and correcting for this, the collection of sheaves \\eqref{eq:ourweights} agrees with \\S 5.2 of \\emph{op.cit.}.\n\n   \\item Comparison to \\cite{boxerpilloni20}:\n    Our notations are consistent with the general theory described in \\cite{boxerpilloni20} but \\textbf{not} with the specific choices of conventions made in the application to symplectic groups in \\S 5.14 of \\emph{op.cit.}. More precisely, we take the upper-triangular Borel subgroup throughout; whereas in \\emph{op.cit.} the Borel subgroup consists of matrices of the form\n    \\( \\begin{smatrix}\n     \\star & \\star &  &  \\\\\n      & \\star &  &  \\\\\n     \\star & \\star & \\star & \\star\\\\\n     \\star & \\star &  & \\star\n    \\end{smatrix}\\),\n    so $B_G \\cap M_G$ is the same in both conventions, but the roles of the parabolics $P_\\mu$ and its opposite $P_{\\mu}^{\\operatorname{std}}$ are swapped.\n\n    Another discrepancy is that in \\emph{op.cit.} the weights considered all have the form $(k_1, k_2; -k_1-k_2)$. This is (to some extent) cancelled out by the fact that in \\S 5.14 of \\emph{op.cit.} the Hecke operators considered are $\\diag(1, 1, p^{-1},p^{-1})$ etc.\n   \\end{itemize}\n\n   \\begin{remark}\n    One can choose to have weights with non-negative parameters (up to a constant shift) indexing the weights of automorphic vector bundles with nontrivial $H^0$, or indexing the dominant weights for $G$; but one can't have both. We have chosen the latter, while \\bp{\\S 5.14} chooses the former.\n   \\end{remark}\n\n \\subsection{Hecke operators}\n  \\label{sect:heckeops}\n\n  Let $\\Iw_G(p)$ denote the Iwahori subgroup $\\{ g \\in G(\\ZZ_p): g \\pmod{p} \\in B_G\\}$. We shall consider the following operators in the Hecke algebra of level $\\Iw_G(p)$, acting on the cohomology of any of the sheaves \\eqref{eq:ourweights}:\n  \\begin{align*}\n   \\mathcal{U}_{\\Sieg} &= [\\diag(p, p, 1, 1)], & \\mathcal{U}'_{\\Sieg} &= [\\diag(1, 1, p, p)]\\\\\n   \\mathcal{U}_{\\Kl} &= p^{-r_2} \\cdot [\\diag(p^2, p, p, 1)], & \\mathcal{U}'_{\\Kl} &= p^{-r_2} \\cdot [\\diag(1, p, p, p^2)]\\\\\n   \\mathcal{U}_B &= \\mathcal{U}_{\\Sieg} \\cdot \\mathcal{U}_{\\Kl}, & \\mathcal{U}'_B &= \\mathcal{U}'_{\\Sieg} \\cdot \\mathcal{U}'_{\\Kl},\\\\\n   \\langle p \\rangle &= p^{-(r_1 + r_2)} \\left[\\diag(p, \\dots, p)\\right].\n  \\end{align*}\n  The factor of $p^{-r_2}$ included with $\\mathcal{U}_{\\Kl}$ and $\\mathcal{U}'_{\\Kl}$, and the factor of $p^0$ for the Siegel operators, implies that all four operators are ``optimally integrally normalised'' (i.e.~the Hecke operators preserve a $\\mathbf{Z}_{(p)}$-lattice, and the power of $p$ is minimal with this property).\n\n \\subsection{Automorphic representations}\n\n  Let $\\pi$ be a cuspidal automorphic representation of $G$. We say $\\pi$ has ``weight $(r_1, r_2)$'' , for integers $r_1 \\ge r_2 \\ge -1$, if its infinitesimal character is $\\nu^\\vee + \\rho$, where $\\nu = (r_1, r_2; r_1 + r_2)$.\n  \\[ \\omega_{\\pi} = |\\cdot|^{-(r_1 + r_2)} \\widehat{\\chi}\\]\n  for some Dirichlet character $\\chi$, where $\\widehat{\\chi}$ is the adelic character mapping a uniformizer at $\\ell$ to $\\chi(\\ell)$ for almost all primes $\\ell$, as in \\cite[\\S 2.2]{LPSZ1}. For brevity, we say that ``$\\pi$ has weight $(r_1, r_2)$ and character $\\chi$''.\n\n  We shall also suppose that $\\pi$ is globally generic. Hence it is \\emph{quasi-paramodular} in the sense of \\cite{okazaki}; that is, there exists an explicit subgroup $K(\\pi) = K(N_\\pi,M_\\pi) \\subset G(\\AA_{\\mathrm{f}})$, the quasi-paramodular subgroup (depending on the conductor $N_\\pi$ of $\\pi$, and the conductor $M_\\pi$ of its central character), such that $\\pi_{\\mathrm{f}}$ has one-dimensional invariants under $K(\\pi)$. (See also \\cite{robertsschmidt07} for the case of trivial central characters.)\n\n  \\begin{definition}\n   Let $S$ be a finite set of primes including all primes such that $\\pi_\\ell$ is ramified, and let $\\mathbb{T}^S$ be the Hecke algebra $E[G(\\AA_{\\mathrm{f}}^S) \/ G(\\widehat{\\mathbf{Z}}^S)]$. Then $\\pi_{\\mathrm{f}}$ determines a ring homomorphism $\\lambda_\\pi^S : \\mathbb{T}^S \\to E$; we write $I^S_\\pi$ for its kernel.\n  \\end{definition}\n\n  We shall consider the localisation at $I^S_{\\pi}$ of various finite-dimensional $E$-vector spaces with $\\mathbb{T}^S$-actions; this can be concretely defined as the maximal $E$-subspace on which the operators $t - \\lambda^S_{\\pi}(t)$ are nilpotent for all $t \\in \\mathbb{T}^S$ (a ``generalised eigenspace''). As with all localisations, this is an exact functor, while the usual eigenspace is not.\n\n  \\begin{proposition}\n   Let $n = 1$ or $n = 2$. Then the localised cohomology groups $H^i(S^{\\mathrm{tor}}_{K(\\pi)}, \\mathcal{V}_{\\kappa_n(\\nu)})_{(I_\\pi^S)}$ and $H^i(S^{\\mathrm{tor}}_{K(\\pi), E}, \\mathcal{V}_{\\kappa_n(\\nu)}(-D))_{(I_\\pi^S)}$ are zero for $i \\ne 3-n$; and for $i = 3-n$, both are 1-dimensional and the natural map between them is an isomorphism. In particular, these localisations are independent of $S$.\n  \\end{proposition}\n\n  \\begin{proof}\n   By results of Su \\cite{su-preprint} (building on earlier works of Harris and others), the coherent cohomology of $S^{\\mathrm{tor}}_{K(\\pi)}$ can be expressed as the $(\\mathfrak{p}, K)$-cohomology of the space of automorphic forms of level $K(\\pi)$. Hence it has a filtration, stable under the Hecke action, whose graded pieces are the $K(\\pi)$-invariants in the finite parts of automorphic representations of $G$.\n\n   The only representations which can contribute to the localisation at $I_\\pi^S$ are those which are locally isomorphic to $\\pi$ at all places outside $S$. By Arthur's classification, we can conclude that $\\sigma_v$ is non-generic for some place $v$, but lies in the same $L$-packet as the generic representation $\\pi_v$. The non-generic, holomorphic representation in the $L$-packet of $\\pi_\\infty$ does not contribute to $H^1$ or $H^2$. For a finite place $v$, by \\cite[Theorem 6.10]{okazaki}\\footnote{Or more precisely its proof: we do not know that $\\sigma_v$ is tempered, but the alternate input that it is a non-generic member of a generic $L$-packet implies that it lands in one of the same Sally--Tadic types considered in op.cit.}, $\\sigma_v$ is not quasi-paramodular (of any level) so it cannot contribute to the cohomology of level $K(\\pi)$. So we are left with the contribution from $\\pi$ itself; and since $\\pi$ has multiplicity one in the discrete spectrum, and the $K(\\pi)$-invariants of $\\pi_{\\mathrm{f}}$ are 1-dimensional, we are done.\n  \\end{proof}\n\n  \\begin{definition}\n   Let $\\mathcal{W}(\\pi_{\\mathrm{f}})_E$ be the $E$-rational part of the Whittaker model of $\\pi_{\\mathrm{f}}$, as in \\cite[\\S 10.2]{LPSZ1}, and define\n   \\[ S^2(\\pi, E) = \\Hom_{E[G(\\AA_{\\mathrm{f}})]}\\left( \\mathcal{W}(\\pi)_E, \\varinjlim_K H^2(S^{\\mathrm{tor}}_{K, E}, \\mathcal{V}_{\\kappa_1(\\nu)}(-D))\\right), \\]\n   which is a 1-dimensional $E$-vector space. We define $S^2(\\pi, F)$ for any extension $F\/E$ similarly.\n  \\end{definition}\n\n  Since the space of $K(\\pi)$-invariants in $\\mathcal{W}(\\pi_{\\mathrm{f}})_E$ has a canonical basis vector $W^{\\mathrm{new}}_\\pi$, normalised so that $W^{\\mathrm{new}}_\\pi(1) = 1$, we can identify $S^2(\\pi, E)$ with $H^i(S^{\\mathrm{tor}}_{K(\\pi), E}, \\mathcal{V}_{\\kappa_n(\\nu)}(-D))_{(I_\\pi^S)}$ via evaluation at the new vector. Given $\\eta \\in S^2(\\pi, E)$, we let $\\eta_{\\mathrm{sph}}$ be its image under this map.\n\n  \\begin{definition}\n   Given a non-zero $\\eta \\in S^2(\\pi, L)$, where $L$ is some $p$-adic field with an embedding from $E$, we define periods $\\Omega_p(\\pi,\\eta) \\in L^\\times$ and $\\Omega_\\infty(\\pi, \\eta) \\in \\mathbf{C}^\\times$ as in \\cite[\\S 6.8]{LPSZ1}.\n  \\end{definition}\n\n  These two periods are only unique up to multiplication by $E^\\times$, but the ratio $\\Omega_p \/ \\Omega_\\infty$ is uniquely determined once $\\eta$ is given.\n\n\n \\subsection{P-stabilisation}\\label{sect:pstab}\n\n  \\begin{definition}\n   Suppose $\\pi$ is unramified at $p$. By a \\emph{$p$-stabilisation} of $\\pi$, we mean a choice of one among the $W_G$-orbit of characters of $T(\\QQ_p)$ from which $\\pi_p$ is induced.\n  \\end{definition}\n\n  Extending $E$ if necessary, we suppose that the $p$-stabilisations take values in $E$. They biject with the orderings of the Hecke parameters $\\alpha, \\beta, \\gamma, \\delta$ of $\\pi_p$ respecting the relation $\\alpha\\delta = \\beta\\gamma = p^{r_1 + r_2 + 3} \\chi_\\pi(p)$, and also with the systems of eigenvalues of $\\mathcal{U}'_{\\Sieg}$ and $\\mathcal{U}'_{\\Kl}$ acting on $(\\pi_p)^{\\Iw(p)}$:  we can order the parameters so that $\\mathcal{U}'_{\\Sieg}$ acts as $\\alpha$, and $\\mathcal{U}_{\\Kl}'$ acts as $\\tfrac{\\alpha\\beta}{p^{r_2 + 1}}$.\n\n  \\begin{definition}\n   Let $\\mathbb{T}^{-}_{\\Iw}$ be the product of $\\mathbb{T}^{S \\cup \\{p\\}}$ and the subalgebra of $E[\\Iw(p) \\backslash G(\\QQ_p) \/\\Iw(p)]$ generated by $\\mathcal{U}'_{\\Sieg}$, $\\mathcal{U}'_{\\Kl}$, and $\\langle p \\rangle$, so that a $p$-stabilisation of $\\pi$ determines a character $\\lambda^-_\\pi: \\mathbb{T}^- \\to E$ with some kernel $I_\\pi^-$.\n  \\end{definition}\n\n  We say that the $p$-stabilisation is \\emph{$p$-regular} if its stabiliser in the Weyl group is trivial; if this holds, the generalised eigenspace in $(\\pi_p)^{\\Iw(p)}$ associated to $\\lambda^-_\\pi$ is 1-dimensional. We say a $p$-stablisation is \\emph{ordinary} if it maps $\\mathcal{U}'_{\\Sieg}$ and $\\mathcal{U}'_{\\Kl}$ to $p$-adic units (with respect to some embedding of $E$ into a $p$-adic field $L$). If $\\pi$ has regular weight, an ordinary $p$-stabilisation is unique if it exists, and if so, it is automatically $p$-regular; this is no longer true if $r_2 = -1$.\n\n  Any choice of $p$-regular $p$-stabilisation determines an isomorphism\n  \\[ S^2(\\pi, E) \\cong H^2\\left(S^{\\mathrm{tor}}_{K^p(\\pi) \\Iw(p), E}, \\mathcal{V}_{\\kappa_n(\\nu)}(-D) \\right)_{(I_\\pi^-)},\\qquad \\eta \\mapsto \\eta_{\\Iw}, \\]\n  given by evaluation at the vector $W_{\\alpha, \\beta}^{\\prime, \\Iw} \\in \\mathcal{W}(\\pi_p)$ defined in \\cite{LZ20b-regulator}.\n\n\n \\section{Summary of higher Coleman theory for $G$}\n\n \\subsection{Outline}\n\n  We briefly survey the results we shall need from \\cite{boxerpilloni20}. Here the group $H$ will not appear, so we drop unnecessary subscripts $G$; we also fix a level $K^p$ away from $p$ (and suppress it from the notation).\n\n  \\begin{itemize}\n   \\item We begin with the \\textbf{classical cohomology} $R\\Gamma(\\kappa_n) = R\\Gamma(S_{K^p\\Iw(p)}^{\\mathrm{tor}}, \\mathcal{V}_{\\kappa_n}) \\otimes_{\\mathbf{Q}} \\QQ_p$, a complex of finite-dimensional $\\QQ_p$-vector spaces, and its cuspidal analogue $R\\Gamma(\\kappa_n, \\cusp)$.\n\n\n   \\item For each $w \\in {}^M W$, each algebraic weight $\\kappa$ dominant for $M_G$, and each sign $\\pm$, we have complexes of $\\QQ_p$-Banach spaces $R\\Gamma_w(\\kappa)^{\\pm}$ \\textbf{(``overconvergent cohomology'')}. These are defined as relative cohomology groups for a stratification of $\\mathcal{S}_{K^p\\Iw(p)}^{\\mathrm{tor}}$, with coefficients in $\\mathcal{V}_{\\kappa}$. We have finite-slope decompositions for the action of $(\\mathcal{U}_{\\Sieg}, \\mathcal{U}_{\\Kl})$ on $R\\Gamma_w(\\kappa)^+$, and for $(\\mathcal{U}'_{\\Sieg}, \\mathcal{U}'_{\\Kl})$ on $R\\Gamma_w(\\kappa)^-$.\n\n   \\item There is a spectral sequence (for either sign) relating the $R\\Gamma_w(\\kappa)^{\\pm}$ for varying $w$ to the classical cohomology. If $\\kappa$ is regular, and we localise at an eigenspace of strictly small slope, then only one $w \\in {}^M W$ contributes to this spectral sequence (the unique one such that $w \\in C(\\kappa)^{\\pm}$) so the localised spectral sequence degenerates.\n\n   \\item There are complexes of $\\QQ_p$-Banach spaces $R\\Gamma_{w, \\an}(\\nu)^{\\pm}$ (\\textbf{``locally analytic cohomology''}), which are defined by replacing the coherent sheaves $\\mathcal{V}_{\\kappa}$ with some Banach sheaves of analytic functions (for sign $+$) or distributions (for sign $-$). These are defined for any locally analytic character $\\nu$.\n\n   \\item For $\\kappa$ algebraic and $M$-dominant, there is a second spectral sequence relating $R\\Gamma_w(\\kappa)^{\\pm}$ to the locally-analytic cohomologies $R\\Gamma_{w, \\an}(\\nu)^{\\pm}$ for a collection of $\\nu$ depending on $\\kappa$ and $w$. Again, for regular weights this spectral sequence will degenerate on the small-slope part.\n\n   \\item The locally-analytic cohomology complexes also make sense with coefficients in an affinoid algebra $A$, giving complexes of Banach $A$-modules $R\\Gamma_{w, \\an}(\\nu_A)^{\\pm}$ (``\\textbf{cohomology in families}''), and there is a Tor spectral sequence relating these to $R\\Gamma_{w, \\an}(\\nu)^{\\pm}$ when $\\nu$ is a specialisation of $\\nu_A$ at some point of $\\operatorname{Max}(A)$.\n  \\end{itemize}\n\n  \\begin{remark}[Levels and overconvergence radii]\n   It is important to note that the complexes $R\\Gamma_w(\\kappa)^{\\pm}$ and $R\\Gamma_{w, \\an}(\\nu_A)^{\\pm}$ are not quite canonical, since they depend on various choices (radii of overconvergence, and levels at $p$). However, the maps arising from changing these parameters induce isomorphisms on the finite-slope parts.\n  \\end{remark}\n\n  All of the above constructions also have a ``cuspidal'' flavour (tensoring all the coefficient sheaves with the ideal sheaf of the toroidal boundary).\n\n \\subsection{The spectral sequences} We now spell out the above constructions in slightly more detail.\n\n  \\begin{remark}\n   In \\emph{op.cit.} the tame level $K^p$ is assumed to be \\emph{neat}, but it will be convenient to relax this: if $K^p$ is any open compact in $G(\\AA_{\\mathrm{f}}^p)$, then we choose an auxiliary smaller subgroup $L^p \\trianglelefteqslant K^p$ which is neat (such subgroups always exist) and define complexes $R\\Gamma_w(K^p, \\kappa)^\\pm$ etc as the $(K_p \/ L_p)$-invariants of the corresponding complexes at level $L_p$. This commutes with taking cohomology, since $K_p\/L_p$ is a finite group and all our complexes will be complexes of $\\mathbf{Q}$-vector spaces. (This will allow us to choose $K^p$ to be a paramodular group, in the sense of \\cite{robertsschmidt07}.)\n  \\end{remark}\n\n  \\subsubsection{Overconvergent to classical cohomology}\n\n   This is the spectral sequence of \\bp{Theorem 5.15}. In the ``minus'' case (which interests us most) it will be\n   \\begin{equation}\n    \\label{eq:bruhatSS}\n    E_1^{ij} = H^{i + j}_{w_{3-i}}(\\kappa, \\cusp)^{-, \\fs} \\Rightarrow H^{i+j}(\\kappa, \\cusp)^{-,\\fs},\n   \\end{equation}\n   and similarly for non-cuspidal cohomology. If we apply the strictly-small-slope condition $(-, \\sss^M)$, then only the terms with $w \\in C(\\kappa)^-$ survive; if $\\kappa = \\kappa_i(\\nu)$ with $\\nu$ dominant, then this is only $w = w_i$, so we have (\\bp{Theorem 5.66}):\n   \\[ R\\Gamma(\\kappa_i(\\nu), \\cusp)^{-,\\sss^M} = R\\Gamma_{w_i}(\\kappa_i(\\nu), \\cusp)^{-, \\sss^M}. \\]\n   The same applies without the cuspidal condition, but for cuspidal cohomology we obtain the additional information that the complex is concentrated in degrees $[0, 3-i]$, by \\bp{Theorem 5.15}.\n\n  \\subsubsection{Locally-analytic to overconvergent}\n\n   We only really care about the edge map of the spectral sequence here. In our notation this map is\n   \\[\n    R\\Gamma_{w_1, \\an}(\\nu, \\cusp)^{-, \\fs} \\to R\\Gamma_{w_1}(\\kappa_1(\\nu), \\cusp)^{-,\\fs}.\n   \\]\n   By \\bp{Corollary 6.36}, this is an isomorphism on the eigenspaces satisfying the slope condition $(-,\\sss_{M, w_1}(\\kappa_1))$ in \\emph{op.cit.}. Moreover, $R\\Gamma_{w_1, \\an}(\\nu, \\cusp)^{-, \\fs}$ is concentrated in degrees $[0,1, 2]$, by \\bp{Theorem 6.29}.\n\n  \\subsubsection{The Tor sequence}\n\n   The last spectral sequence we need takes the form\n   \\[ E_2^{pq} = \\Tor_{-p}^A( H^q_{w_1, \\an}(\\nu_A, \\cusp)^{-, \\fs}, \\nu) \\Longrightarrow H^{p+q}_{w_1, \\an}(\\nu, \\cusp)^{-, \\fs}\\]\n   where $A$ is some affinoid algebra with a continuous character $\\nu_A: T(\\ZZ_p) \\to A^\\times$, and the algebraic character $\\nu$ defines a $K$-point of $\\operatorname{Max}(A)$. Here we cannot rely on ``small slope'' arguments in order to make the sequence degenerate (since there is no reason to expect the slopes on the higher Tor terms to be bigger than the slopes on the $\\Tor_0$ term).\n\n \\subsection{Slope conditions}\n\n  We consider the Hecke operators normalised as in \\cref{sect:heckeops}. on the cohomology of the sheaves $\\mathcal{V}_{\\kappa}$, for $\\kappa$ as in \\eqref{eq:ourweights}. Thus each operator is ``minimally integrally normalised'' acting on the classical cohomology (cuspidal or non-cuspidal), i.e.~its slopes are $\\ge 0$.\n\n  \\subsubsection{Expected and provable slope bounds for overconvergent cohomology}\n\n   \\bp{Conjecture 5.29} predicts lower bounds for the slopes of the Hecke operators acting on the overconvergent cohomology complexes $R\\Gamma_w(K^p, \\kappa)^{\\pm}$ and $R\\Gamma_w(K^p, \\kappa, \\cusp)^{\\pm}$; there is a similar conjecture \\bp{Conjecture 6.33} for the locally-analytic cohomology complexes (where there are more possibilities, since these are defined for weights which might not be $M_G$-dominant).\n\n   These are summarized by the following table, in which we compute for various elements $w \\in W_G$ the character $w^{-1}(\\kappa + \\rho) - \\rho$, and how it pairs with the anti-dominant cocharacters $\\diag(1,1,x,x)$ and $\\diag(1,x,x,x^2)$ defining the operators $\\mathcal{U}'_{\\Sieg}$ and $\\mathcal{U}'_{\\Kl}$. We take $\\kappa = \\kappa_1 = (r_1, -r_2-2; r_1 + r_2)$, and subtract $r_2$ from all entries in the bottom row (since this is our normalising constant for $\\mathcal{U}'_{\\Kl}$). This gives the following table:\n\n   \\begin{center}\n   \\begin{tabular}{c|cccc|c}\n    $w=$  & id & $(w_1)$ & $w_2$ & $w_3$ & $w^{\\max}_M w_1$\\\\\n    \\hline\n    $\\mathcal{U}'_{\\Sieg}$ & $r_2+1$ & $(0)$ & $0$         & $r_1+2$     & $r_1+r_2+3$\\\\\n    $\\mathcal{U}'_{\\Kl}$   & $0$     & $(0)$ & $r_1-r_2+1$ & $r_1-r_2+1$ & $r_1+r_2+3$\\\\\n   \\end{tabular}\n  \\end{center}\n\n  We do not know this conjecture in full, but we know a weaker statement, \\bp{Theorem 5.33 and Theorem 6.35}, in which we replace $w^{-1}(\\kappa_1 + \\rho) - \\rho$ with $w^{-1} \\kappa_1$. This gives the following bounds:\n\n  \\begin{center}\\begin{tabular}{c|cccc|c}\n  $w=$ & id & $(w_1)$ & $w_2$ & $w_3$ & $w^{\\max}_M w_1$\\\\\n   \\hline\n   $\\mathcal{U}'_{\\Sieg}$ & $r_2+1$ & $(-1)$ & $-1$        & $r_1-1$ & $r_1 + r_2 + 1$\\\\\n   $\\mathcal{U}'_{\\Kl}$   & $0$     & $(0) $ & $r_1-r_2-2$ & $r_1-r_2-2$& $r_1 + r_2 + 2$\\\\\n  \\end{tabular}\\end{center}\n\n  \\begin{remark}\n   More precisely: for each $w$ we consider the weight $w^{-1}(\\kappa_1 + \\rho) - \\rho$, and how it pairs with the anti-dominant torus elements $(1, 1, p, p)$ and $(1, p, p, p^2)$. The bottom row gets multiplied by $p^{-r_2}$ from our normalisation of Hecke operators.\n\n   In the second table, we do the same computation with $w^{-1}\\kappa_1$ instead of $w^{-1}(\\kappa_1 + \\rho) - \\rho$.\n  \\end{remark}\n\n\n   \\subsubsection{Small slope conditions} We can now compute which eigensystems satisfy the various small-slope conditions of \\bp{\\S 5.11}.\n\n   \\begin{proposition}\n    For the weight $\\kappa_1 = (r_1, -r_2-2; r_1 + r_2)$, with $r_1 \\ge r_2 \\ge 0$, we have the following:\n\n    \\begin{itemize}\n     \\item The ``small slope'' condition $(-,\\ss^M(\\kappa_1))$ is\n     \\[ \\lambda(\\mathcal{U}'_{\\Sieg}) < 1 + r_2, \\qquad \\lambda(\\mathcal{U}'_{\\Kl}) < 1 + r_1 - r_2.\\]\n     \\item The ``strictly small slope'' condition $(-, \\sss^M(\\kappa_1))$ is\n     \\[ \\lambda(\\mathcal{U}'_{\\Sieg}) < 1 + r_2, \\qquad \\lambda(\\mathcal{U}'_{\\Kl}) < -2 + r_1 - r_2.\\]\n \n     \\item The condition $(-, \\sss_{M, w_1}(\\kappa_1))$ is implied by $(-,\\sss^M(\\kappa_1))$, and similarly for the non-strict versions.\n    \\end{itemize}\n   \\end{proposition}\n\n   \\begin{proof}\n    This follows from the tables of the previous section.\n   \\end{proof}\n\n   Thus, in any regular weight, ordinary classes have small slope (as one might reasonably expect). However, they fail the strict-small-slope condition unless $r_1 - r_2 \\ge 3$.\n  \n\n\n \\subsection{Families of eigenclasses}\n  \\label{sect:families}\n\n  We now briefly indicate the consequences of this theory for a cohomological automorphic representation. We let $\\pi$ be as in \\cref{sect:pstab}, and let $K^p$ denote the paramodular subgroup away from $p$ of the appropriate level. We suppose $\\pi$ is ordinary at $p$, and also that $r_2 \\ge 0$ and $r_1 - r_2 \\ge 3$. Let $\\lambda_\\pi^-$ be its ordinary $p$-stabilisation, and $I_\\pi^-$ the kernel of $\\lambda^-_{\\pi}$.\n\n  Then the classicity theorems of higher Coleman theory recalled above give quasi-isomorphisms\n  \\[\n   R\\Gamma_{w_1, \\an}(\\nu, \\cusp)^{-, \\sss} \\cong R\\Gamma_{w_1}(\\kappa_1, \\cusp)^{-,\\sss} \\cong R\\Gamma(\\kappa_1, \\cusp)^{-,\\sss},\n  \\]\n  and since $\\lambda_{\\pi}^-$ is ordinary, the localisation of $R\\Gamma(\\kappa_1, \\cusp)$ at $I_{\\pi}^-$ is contained in the strictly-small-slope part. So, from the results quoted above, the localisation of these three complexes at $I_\\pi^-$ is 1-dimensional in the $H^2$, and vanishes in all other degrees.\n\n  For the ``family'' cohomology, we use the local criterion for flatness: for a finitely-generated module $M$ over a local ring $(A, I)$, if $\\Tor_1^A(M, A\/I) = 0$, then $M$ is free. From this and the above $\\Tor$ spectral sequence, we conclude that the localisation of $H^i_{w_1, \\an}(\\nu_A^\\vee, \\cusp)^{-, \\fs}$ at $I_\\pi^-$ is zero for $i \\ne 2$, and is locally free of rank 1 over $A$ for $i = 2$.\n\n  Thus the $p$-stabilised new-vector $\\eta_{\\Iw} \\in H^2(K^p, \\kappa_1, \\cusp)_{(I_\\pi)}$ associated to $\\pi$ deforms to an analytic family over some open affinoid in the 2-dimensional weight space $\\mathcal{W} \\times \\mathcal{W}$: for some sufficiently small $A \\ni (r_1, r_2)$, we obtain a class $\\underline{\\eta} \\in H^2_{w_1, \\an}(\\nu_A^\\vee, \\cusp)$, and a homomorphism $\\underline{\\lambda}: \\mathbb{T}^- \\to A$ lifting $\\lambda_{\\pi}$, such that $ \\mathbb{T}^-$ acts on $\\underline{\\eta}$ via $\\underline{\\lambda}$, and the specialisation of $\\underline{\\eta}$ at $(r_1, r_2)$ is $\\eta_{\\Iw}$. These results will be used below to construct our $p$-adic $L$-functions.\n\n  Similar arguments apply to the modules $H^{3-i}_{w_i, \\an}(\\nu_A^\\vee, \\cusp)$ for all $i$ (although in the case $i = 0, 3$ the classical eigenspace associated to $\\pi$ may be zero, if $\\pi$ is a Yoshida lift).\n\n\n  \\begin{remark}\n   Unsurprisingly, we could relax the assumption that $\\pi$ be Borel-ordinary at $p$, as long as it admits some refinement satisfying the $\\sss^M$ and $\\sss_{M, w_1}$ conditions.\n  \\end{remark}\n\n\n\\section{Functoriality of higher Coleman theory (overview)}\n\n The next few sections, which are the main technical context of the present paper, are devoted to constructing maps relating higher Coleman theory spaces for $G$ and for $H$. More precisely, we shall define three maps of complexes\n \\begin{subequations}\n  \\begin{align}\n   R\\Gamma^G(\\kappa_1, \\cusp) &\\longrightarrow R\\Gamma^H(\\tau, \\cusp) &&\\text{(classical)} \\\\\n   R\\Gamma^G_{w_1}(\\kappa_1, \\cusp)^{-, \\fs} &\\longrightarrow R\\Gamma^H_{\\mathrm{id}}(\\tau, \\cusp)^{-} &&\\text{(overconvergent)} \\\\\n   R\\Gamma^G_{w_1, \\an}(\\nu_A, \\cusp)^{-, \\fs} &\\longrightarrow R\\Gamma^H_{\\mathrm{id}, \\an}(\\tau_A, \\cusp)^{-} &&\\text{(locally-analytic)}\n  \\end{align}\n \\end{subequations}\n satisfying appropriate compatibilities. In the first two cases, $\\tau$ is a weight for $H$ in an appropriate range depending on $(r_1, r_2)$ (see \\cref{sect:branchcoeffs}; in the third case, $\\nu_A$ and $\\tau_A$ are families of weights for $G$ and $H$ with an appropriate relation between them (see \\cref{def:compat}).\n\n It is important to note that these maps do \\emph{not} land in the finite-slope part on the right-hand side. In particular, the resulting complexes depend on various auxiliary choices of parameters, and changing these induces an isomorphism on the finite-slope part but not on the whole complex; in order to obtain a uniquely-determined map we shall pass to the inverse limit.\n\n\\section{Maps of flag varieties}\n We now delve a little further into the construction of the overconvergent and locally-analytic cohomology complexes, in order to study how they interact with pullback along $H \\hookrightarrow G$.\n\n \\subsection{Bruhat cells and tubes} We consider the following $\\ZZ_p$-subschemes of $\\FL_G$:\n\n  \\begin{definition}\n   For $w \\in {}^M W_G$, we define:\n   \\begin{itemize}\n   \\item $C^G_{w} = P_G \\backslash P_G w B_G$, a locally closed subscheme;\n   \\item $X^G_w = \\bigcup_{w' \\le w} C^G_{w'}$, a closed subscheme (the closure of $C^G_w$);\n   \\item $Y^G_w = \\bigcup_{w' \\ge w} C^G_{w'}$, an open subscheme.\n   \\end{itemize}\n  \\end{definition}\n\n  We shall mostly be interested in their special fibres $C^G_{w, \\mathbf{F}_p}$ etc. Note that the dimension of $C^G_{w}$, or of $X^G_w$, is $\\ell(w)$.\n\n  \\begin{remark}\n   The double coset $P_G g B_G$ for a given $g \\in G$ depends only on the span of the rows of the $2 \\times 2$ lower-left submatrix of $g$. If this span is zero, $g$ is in the small cell; if it is $(0, \\star)$ then we are in the 1-dimensional cell; if it is 1-dimensional but not equal to $(0, \\star)$, then we are in the 2-dimensional cell; if it is the whole of $\\AA^2$, we are in the big cell.\n  \\end{remark}\n\n  Similar definitions apply for $H$ in place of $G$, although we shall only use $C^H_{\\mathrm{id}}$ in the present work.\n\n\n \\subsection{Maps of flag varieties}\n\n  Let $\\gamma=\\begin{smatrix}1 &&&\\\\1 & 1 &&\\\\&&\\phantom{-}1 &\\\\ &&-1& 1 \\end{smatrix}$, and let $\\hat{\\gamma}=\\gamma w_1 = \\begin{smatrix} 1& & & \\\\ 1 && 1& \\\\ & -1&& \\\\ & \\phantom{-}1& & 1\\end{smatrix}$. Then the map $\\hat\\iota: \\FL_H \\to \\FL_G$ sending $B_H h$ to $P_G h \\hat\\gamma$ is a closed immersion of $\\ZZ_p$-schemes (since it is a translate of the obvious closed embedding induced by $H \\hookrightarrow G$). It maps the identity to $w_1$, since $\\gamma \\in P_{\\Sieg}$.\n\n  \\begin{proposition}\n   \\label{prop:intersectcells}\n   The intersections of the Bruhat cells of $\\FL_G$ with $\\hat\\iota(\\FL_H)$ are as follows:\n   \\begin{itemize}\n   \\item $\\hat\\iota^{-1}\\left(C^G_{\\mathrm{id}}\\right)= \\varnothing$;\n   \\item $\\hat\\iota^{-1}\\left(C^G_{w_1}\\right) = C^H_{\\mathrm{id}}$.\n   \\end{itemize}\n  \\end{proposition}\n\n  \\begin{proof}\n   This is an elementary linear-algebra computation. We can identify $\\FL_H$ with $\\mathbf{P}^1 \\times \\mathbf{P}^1$ in such a way that the strata are $\\infty \\times \\infty, \\infty \\times \\AA^1$, $\\AA^1 \\times \\infty$ and $\\AA^2$. The map to $\\FL_G$ sends $((x:y), (X : Y))$ to the plane spanned by $(x,-y,0,-y)$ and $(X,Y,X,0)$. The projection of this plane to the first two coordinate vectors is never 0, so $\\hat\\iota^{-1}\\left(C^G_{\\mathrm{id}}\\right)= \\varnothing$. Its projection to the first coordinate vector is 0 iff $x = X = 0$, i.e.~$\\hat\\iota^{-1}\\left(C^G_{w_1}\\right) = C^H_{\\mathrm{id}}$.\n  \n  \\end{proof}\n\n  \\begin{remark}\n   Conceptually, this is saying that the subspace $\\hat\\iota(\\FL_H) \\subset \\FL_G$, which is 2-dimensional, intersects the Bruhat strata $C^G_{\\mathrm{id}}$ and $C^G_{w_1}$ in the ``expected'' codimensions, and these codimensions are themselves Bruhat strata in $\\FL_H$: it has empty intersection with the codimension 3 stratum, and its intersection with the codimension 2 stratum is the unique 0-dimensional. (Its intersection with the codimension 1 stratum $C^G_{w_2}$ also has the expected dimension, but it is not a Bruhat stratum; however, this plays no role in our theory.)\n  \\end{remark}\n\n\n \\subsection{Explicit coordinates on the flag variety}\n  \\label{sect:explicitparam}\n\n  For $w \\in {}^M W_G$ we let $U^G_w$ be the ``big cell at $w$'', i.e. the translate of the big cell $P_G \\backslash P_G \\bar{N}_G$ by $w$. This is accordingly an open neighbourhood of $w$ naturally parametrised by $\\bar{N}_G \\cong \\AA^3$, which we identify with $2 \\times 2$ off-symmetric matrices $Z = \\stbt x y z x$, via $Z \\mapsto P_G \\backslash P_G \\stbt{1}{}{Z}{1} w$.\n\n  In these coordinates, the action of $g \\in G$ is given as follows: we have\n  \\[ P \\backslash (P \\stbt{1}{}{Z}{1} w) \\cdot g = P \\backslash (P \\stbt{1}{}{Z}{1} w g w^{-1}) w, \\]\n  and if we write $w g w^{-1} = \\stbt{A}{B}{C}{D}$, then\n  \\[ (P \\stbt{1}{}{Z}{1} w g w^{-1}) = P \\stbt{1}{}{Z'}{1},\\qquad Z' = (Z B + D)^{-1} (ZA + C).\\]\n\n  \\begin{remark}\n   This is a mildly twisted version of the familiar formula for the action of the Siegel modular group on the complex-analytic Siegel half-space, which is of course nothing but an open subset of the $\\mathbf{C}$-points of $\\FL_G$.\n  \\end{remark}\n\n  As a special case of this formula, if $\\delta = \\diag(p^3, p^2, p, 1)$ and $w = w_1$, then\n  \\[ \\stbt x y z x \\cdot \\delta = \\stbt{px}{p^{-1}y}{p^3 z}{px}.\\]\n\n  Similarly, the big cell for $H$ (at $w = \\mathrm{id}$) is isomorphic to $\\AA^2$ via $(z_1, z_2)\\mapsto B_H \\backslash B_H\\left( \\stbt{1}{}{z_1}{1},\\stbt{1}{}{z_2}{1}\\right)$. In these coordinates on $U^H_{\\mathrm{id}}$ and $U^G_w$, we compute that the map $\\hat\\iota$ is given by\n  \\begin{equation}\n   \\label{eq:gamma-param}\n  \n   (z_1, z_2) \\mapsto \\stbt{z_2}{z_2}{z_1 + z_2}{z_2}.\n  \\end{equation}\n\n  \\begin{remark}\n   More conceptually, we can identify the big cells $U^H_{\\mathrm{id}}$ and $U^G_{w_1}$ with the Lie algebras $\\bar{\\mathfrak{n}}_H$ and $\\bar{\\mathfrak{n}}_G$, and this formula is just the composite of the natural inclusion $\\bar{\\mathfrak{n}}_H \\subset \\bar{\\mathfrak{n}}_G$ and the adjoint action of $\\gamma \\in M_G$ on $\\bar{\\mathfrak{n}}_G$.\n  \\end{remark}\n\n\n\n\n\n\n\n \\subsection{Analytic geometry: notations}\n\n\n  Let us write $\\scalebox{1.15}{$\\mathtt{FL}$}^H$ and $\\scalebox{1.15}{$\\mathtt{FL}$}^G$ for the analytic adic spaces (over $\\Spa(\\QQ_p, \\ZZ_p)$) associated to $\\FL_G$ and $\\FL_H$. (We use typewriter letters for the flag varieties and open sets in them, since we shall later use calligraphic letters for subsets of the adic Shimura variety.)\n\n\n  We now describe explicit isomorphisms between certain tubes in $\\scalebox{1.15}{$\\mathtt{FL}$}^G$ and $\\scalebox{1.15}{$\\mathtt{FL}$}^H$ and adic polydiscs. We will need to distinguish between \\emph{four} flavours of ``disc'' inside the adic affine line. This is because the ``closed disc'' $\\{|.| : |z| \\le 1\\}$, where $z$ is a coordinate function, is actually open (but not closed) in the adic topology, whereas the ``open disc'' $\\{|.| : |z| < 1\\}$ is closed (but not open)!\n\n  For $m \\in \\mathbf{Q}$, we define\n  \\[ \\mathcal{B}_m = \\{ |.| : |z| \\le |p|^{m} \\},\\qquad \\overline{\\mathcal{B}}_m = \\bigcap_{m' < m} \\mathcal{B}_m, \\]\n  and\n  \\[ \\mathcal{B}^{\\circ}_m = \\bigcup_{m' > m} \\mathcal{B}_{m'}, \\qquad \\overline{\\mathcal{B}}^\\circ_m = \\{ |.| : |z| < |p|^m\\}.\n  \\]\n  Thus $\\mathcal{B}^{\\circ}_m \\subset \\overline{\\mathcal{B}}^\\circ_{m} \\subset \\mathcal{B}_m \\subset\\overline{\\mathcal{B}}_m$, and (as the notation suggests) $\\overline{\\mathcal{B}}_m$ is the closure of $\\mathcal{B}_m$, and similarly $\\overline{\\mathcal{B}}_m^\\circ$ of $\\mathcal{B}^{\\circ}_m$. Moreover, the sets $\\overline{\\mathcal{B}}_m - \\mathcal{B}_m$ and $\\overline{\\mathcal{B}}^\\circ_m -\\mathcal{B}^{\\circ}_m$ consist entirely of rank $> 1$ points.\n\n  \\begin{remark}\n   Compare the four flavours of root subgroups in \\bp{\\S 3.3.2}. The space $\\overline{\\mathcal{B}}_m$ corresponds to the ``dagger affinoid disc'' in Grosse-Kl\\\"onne's theory of dagger spaces.\n  \\end{remark}\n\n  More generally, if $A$ is a subset of $\\overline{\\mathbf{Q}}_p$, we write $A + \\mathcal{B}_m = \\bigcup_{a \\in A}(a + \\mathcal{B}_m)$ etc; we shall only use this if $A$ is compact, in which case the union is finite.\n\n \\subsection{Level groups at $p$}\n\n  \\begin{notation} Let $t \\in \\mathbf{Z}_{\\ge 1}$.\n   \\begin{itemize}\n   \\item Let $K_{\\Iw}^G(p^t) = \\{ g \\in G(\\ZZ_p): g\\bmod p \\in B_G(\\mathbf{Z}\/p^t)\\}$ be the depth $t$ upper-triangular Iwahori of $G$, and similarly for $H$.\n   \\item Let $K^H_{\\diamondsuit}(p^t)$ denote the group $H(\\QQ_p) \\cap \\hat\\gamma K^G_{\\Iw}(p^t) \\hat\\gamma^{-1}$, which is concretely given by\n   \\[ K^H_{\\diamondsuit} =\n    \\left \\{ h \\in H(\\ZZ_p): h = \\left(\\stbt{x}{y}{0}{z},\\stbt{x}{-y}{0}{z}\\right) \\bmod p^t \\text{ \\textup{for some} $x,y,z$}\\right\\}.\\]\n   \\end{itemize}\n  \\end{notation}\n\n  Note that $K^H_{\\diamondsuit}(p^t)$ is, fortuitiously, a subgroup of $K^H_{\\Iw}(p^t)$.\n\n \\subsection{Tubes of ``radius one''}\n\n  We note that if $\\mathcal{X}$ is the analytic adic space associated to a finite-type $\\ZZ_p$-scheme $X$, then there is a specialisation map $\\sp: \\mathcal{X} \\to X_{\\mathbf{F}_p}$ which is a continuous map of topological spaces. If $Z \\subset X_{\\mathbf{F}_p}$ is a locally closed subset, we let $]Z[$ be the \\emph{interior} of $\\sp^{-1}(Z)$; this is the adic space corresponding to the tube in the sense of classical rigid geometry, while $\\sp^{-1}(Z)$ is not a classical rigid space in general. Of course, if $Z$ is open, then $\\sp^{-1}(Z) =\\  ]Z[$; on the other hand, if $Z$ is closed, then $\\sp^{-1}(Z) =\\  \\overline{]Z[}$.\n\n  \\begin{definition}\n   Let $\\scalebox{1.15}{$\\mathtt{U}$}^G_0 =\\ ]Y^G_{w_1, \\mathbf{F}_p}[$, $\\scalebox{1.15}{$\\mathtt{Z}$}^G_0 = \\ \\overline{]X^G_{w_1, \\mathbf{F}_p}[}$, and $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} = \\scalebox{1.15}{$\\mathtt{Z}$}^G_0 \\cap \\scalebox{1.15}{$\\mathtt{U}$}^G_0$.\n  \\end{definition}\n\n  Note that $\\scalebox{1.15}{$\\mathtt{U}$}^G_0$ is open and $\\scalebox{1.15}{$\\mathtt{Z}$}^G_0$ closed, and both are invariant under the Iwahori $K^G_{\\Iw}(p)$ (since the Borel subgroup of $G_{\\mathbf{F}_p}$ fixes the mod $p$ Bruhat cells). Thus $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0, 0}$ is a ``partial closure'' of the Bruhat cell $]C_{w_1, \\mathbf{F}_p}^G[$.\n\n  We also write $\\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\overline{]X^H_{\\mathrm{id}}[}$ (the preimage of the point $\\{\\mathrm{id}_H\\} \\in \\FL^H_{\\mathbf{F}_p}$) which is stable under $K^H_{\\Iw}(p)$, and we formally set $\\scalebox{1.15}{$\\mathtt{U}$}^H_0 = \\scalebox{1.15}{$\\mathtt{FL}$}^H$.\n\n  \\begin{proposition}\n   We have $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} \\subset U^G_{w_1}$, and in the coordinates on $U^G_{w_1}$ described in \\cref{sect:explicitparam}, we have\n   \\[ \\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} = \\{ \\stbt x y z x : x, z \\in \\overline{\\mathcal{B}}_{0}^\\circ, y \\in \\mathcal{B}_0 \\}.\\]\n   Similarly\n   \\[ \\scalebox{1.15}{$\\mathtt{I}$}^H_{0,0} = \\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\{ (z_1, z_2) : z_i \\in \\overline{\\mathcal{B}}_{0}^\\circ \\}.\\]\n  \\end{proposition}\n\n  \\begin{proof}\n   This is an instance of \\bp{Lemma 3.21 (5)}.\n  \\end{proof}\n\n  \\begin{proposition}\n   \\label{prop:Cartadic}\n   We have a Cartesian diagram of adic spaces\n   \\[\n    \\begin{tikzcd}\n     \\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\scalebox{1.15}{$\\mathtt{I}$}^H_{0, 0} \\dar[\"\\hat\\iota\"] \\rar[hook] &\\scalebox{1.15}{$\\mathtt{U}$}^H_0 \\dar[\"\\hat\\iota\"]\\\\\n     \\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} \\rar[hook] & \\scalebox{1.15}{$\\mathtt{U}$}^G_0\n    \\end{tikzcd}\n   \\]\n   in which all the morphisms are closed embeddings.\n  \\end{proposition}\n\n   \\begin{proof}\n    This follows readily from \\cref{prop:intersectcells} and the definition of the $\\scalebox{1.15}{$\\mathtt{U}$}$'s, $\\scalebox{1.15}{$\\mathtt{I}$}$'s and $\\scalebox{1.15}{$\\mathtt{Z}$}$'s.\n   \\end{proof}\n\n \\subsection{Tubes of smaller radius}\n\n  Let $m, n, t$ be integers with\n  \\begin{equation}\\label{eq:mnt}\n   0 \\le n \\le m < t, \\qquad \\text{$m > n$ if $n \\ne 0$}.\n  \\end{equation}\n\n  \\begin{definition}\n   We define subsets $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G \\subset \\scalebox{1.15}{$\\mathtt{U}$}_n^G$ in $\\scalebox{1.15}{$\\mathtt{FL}$}^G$ as follows: we let\n   \\[ \\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\{ \\stbt x y z x : x, z \\in \\overline{\\mathcal{B}}_{m}^\\circ, y \\in \\mathcal{B}_n + \\ZZ_p \\}. \\]\n   (consistently with the $(m, n) = (0, 0)$ case described above). For $n \\ge 1$ we set\n   \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^G =\\{ \\stbt x y z x : x, z \\in \\mathcal{B}_n^\\circ, y \\in \\mathcal{B}_n + \\ZZ_p \\}.\\]\n   and for $n = 0$ we use the definition above.\n  \\end{definition}\n\n  \\begin{proposition} \\\n   \\begin{enumerate}[(i)]\n   \\item The sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ and $\\scalebox{1.15}{$\\mathtt{U}$}_n^G - \\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G$ are open in $\\scalebox{1.15}{$\\mathtt{FL}$}^G$.\n   \\item If $n > 0$ then $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\scalebox{1.15}{$\\mathtt{I}$}_{m, 0}^G \\cap \\scalebox{1.15}{$\\mathtt{U}$}_n^G$.\n   \\item The sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ and $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G$ are stable under $K_{G, \\Iw}(p^t)$, and in the notation of \\bp{\\S 3.3.3}, we have\n   \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^G =\\ ]C_{w_1, \\mathbf{F}_p}[_{(n, n)}K_{G, \\Iw}(p^t),\\qquad \\scalebox{1.15}{$\\mathtt{I}$}_{m, n} =\\ ]C_{w_1, \\mathbf{F}_p}[_{(\\overline{m}, n)}K_{\\Iw}^G(p^t).\\]\n   \\end{enumerate}\n  \\end{proposition}\n\n  \\begin{proof}\n   The first two statements are obvious. For the stability under $K_{G, \\Iw}(p^t)$, we treat $n = 0$ and $n > 0$ separately: in the $n = 0$ case, the stability of $\\scalebox{1.15}{$\\mathtt{U}$}_0^G$ is already established, and the stability of $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}$ follows from the identification $\\scalebox{1.15}{$\\mathtt{I}$}_{m, 0} = \\, ]C_{w_1, \\mathbf{F}_p}\\![\\,_{\\overline{m}, 0} = \\mathcal{P}\\backslash \\mathcal{P} w \\mathcal{G}_{\\overline{m}, 0}$ of \\bp{\\S 3.3.3}.\n\n   For $n > 0$, we argue as in Lemma 3.18 of \\emph{op.cit.} to describe the spaces $]C_{w_1, \\mathbf{F}_p}[_{(n, n)} K_{\\Iw}^G(p^t)$ and $]C_{w_1, \\mathbf{F}_p}[_{(\\overline{m}, n)}K_{\\Iw}^G(p^t)$ as finite unions of translates of $]C_{w_1, \\mathbf{F}_p}[_{(n, n)}$, indexed by coset representatives for $N_{B_G}(\\mathbf{Z}\/p^n)$ modulo its intersection with $w^{-1}P_G w$. We can take these coset representatives to be of the form\n   \\[ \\begin{smatrix} 1 \\\\ &1 & \\alpha \\\\ &&1 \\\\ &&&1\\end{smatrix}, \\qquad \\alpha \\in \\mathbf{Z} \/ p^n, \\]\n   which act as $\\stbt x y z x \\mapsto \\stbt x {y + \\alpha} z x$.\n  \\end{proof}\n\n  \\begin{remark}\n   One can choose a closed subset $\\scalebox{1.15}{$\\mathtt{Z}$}_m^G$ such that $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\scalebox{1.15}{$\\mathtt{Z}$}_m^G \\cap \\scalebox{1.15}{$\\mathtt{U}$}_n^G$; however, it is a little awkward to choose such a subset which is invariant under $K^G_{\\Iw}(p^t)$, and in any case the choice of such a subset will not matter for our theory, so we shall not privilege any particular choice.\n  \\end{remark}\n\n\n  \\begin{remark}\n   Formally setting $n = 0$ in the formula defining $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ for $n > 0$ gives a well-defined and $K^G_{\\Iw}(p)$-invariant set (in fact it is exactly $]C_{w_1}[\\,$), but this set does not contain $\\scalebox{1.15}{$\\mathtt{I}$}_{0, 0}$. Hence we use the formula of the previous section in which $x, z$ are allowed to ``go off to infinity''.\n  \\end{remark}\n\n\n\n \\subsection{Iwahori-level tubes for $H$}\n\n  For $n < t$ we shall define\n  \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^H = \\hat\\iota^{-1}(\\scalebox{1.15}{$\\mathtt{U}$}_n^G) \\subseteq \\scalebox{1.15}{$\\mathtt{FL}$}^H.\\]\n  This is an open $K_{H, \\Iw}(p^t)$-invariant set, containing $\\{\\mathrm{id}\\}$. For $n = 0$ it is the whole of $\\scalebox{1.15}{$\\mathtt{FL}$}^H$.\n\n  \\begin{proposition}\n   The set $\\hat\\iota^{-1}(\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G)$ is closed in $\\scalebox{1.15}{$\\mathtt{FL}$}_H$ and invariant under $K_{H, \\Iw}(p^t)$, and does not depend on $n$; explicitly it is given by\n   \\[ \\scalebox{1.15}{$\\mathtt{Z}$}_m^H = \\{ (z_1, z_2): z_i \\in \\overline{\\mathcal{B}}^\\circ_m\\}.\\]\n  \\end{proposition}\n\n  \\begin{proof}\n   Clear from \\eqref{eq:gamma-param}.\n  \\end{proof}\n\n  We therefore have a Cartesian diagram of closed embeddings generalising \\cref{prop:Cartadic} above, for any $m, n, t$ as in \\eqref{eq:mnt}:\n  \\begin{equation}\\label{eq:Cartadic2}\n   \\begin{tikzcd}\n    \\scalebox{1.15}{$\\mathtt{Z}$}^H_m \\dar[\"\\hat\\iota\"] \\rar[hook] &\\scalebox{1.15}{$\\mathtt{U}$}^H_n \\dar[\"\\hat\\iota\"]\\\\\n    \\scalebox{1.15}{$\\mathtt{I}$}^G_{m,n} \\rar[hook] & \\scalebox{1.15}{$\\mathtt{U}$}^G_n.\n   \\end{tikzcd}\n  \\end{equation}\n\n  It is convenient to extend the definition of $\\scalebox{1.15}{$\\mathtt{Z}$}_m$ to remove the requirement that $m < t$. For any integers $t \\ge 1$ and $m \\ge 0$, let us define\n  \\[ \\scalebox{1.15}{$\\mathtt{Z}$}_m^H \\coloneqq \\{ (z_1, z_2): z_i \\in \\overline{\\mathcal{B}}^\\circ_m + p^t \\ZZ_p\\}.\\]\n  This is invariant under $K_{H, \\Iw}(p^t)$, and hence also under $K_{H, \\diamondsuit}(p^t)$. If $m < t$ then this agrees with the definitions above. We define an open set $\\scalebox{1.15}{$\\mathtt{U}$}_n^H$ similarly. (However, in the $m \\ge t$ case we do not know if these sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^H$ and $\\scalebox{1.15}{$\\mathtt{Z}$}_m^H$ can be fitted into a diagram like \\eqref{eq:Cartadic2}.)\n\n\n\n\n\\section{Pullbacks in overconvergent cohomology}\n \\subsection{Adic spaces and period maps}\n\n  We consider the analytifications $\\mathcal{S}_{G,K}^{\\an}=(S_K\\times \\Spec(\\QQ_p))^{\\an}$,  $\\mathcal{S}_{G,K}^{\\mathrm{tor}}=(S_{G,K}^{\\mathrm{tor}}\\times\\Spec(\\QQ_p))^{\\an}$ and $\\scalebox{1.15}{$\\mathtt{FL}$}_G=(\\FL_{G}\\times\\Spec(\\QQ_p))^{\\an}$, as well as the groups $\\mathcal{G}^{\\an}=(G\\times \\Spec(\\QQ_p))^{\\an}$, $\\mathcal{P}^{\\an}_{G}=(P_{G}\\times \\Spec(\\QQ_p))^{\\an}$ and $\\mathcal{M}_G^{\\an}=(M_G\\times \\Spec(\\QQ_p))^{\\an}$.\n\n  Since we have fixed an integral model of $G$, we have quasi-compact, affinoid groups $\\mathcal{G}\\hookrightarrow \\mathcal{G}^{\\an}$, $\\mathcal{P}_{G,\\Sieg}\\hookrightarrow \\mathcal{P}_{G,\\Sieg}^{\\an}$ and $\\mathcal{M}_G\\hookrightarrow \\mathcal{M}_G^{\\an}$.\n\n  Write $\\mathcal{S}_{G,K^p}^{\\mathrm{tor}}$ for the perfectoid space $\\varprojlim_{K_p} \\mathcal{S}_{G,K^pK_p}^{\\mathrm{tor}}$. We can then consider the Hodge--Tate period map\n  \\[ \\pi_{\\HT,G}^{\\mathrm{tor}}: \\mathcal{S}_{G,K^p}^{\\mathrm{tor}} \\longrightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G\\]\n  which for every open compact $K_p\\subset G(\\QQ_p)$ descends to a map of topological spaces (c.f. \\cite[\\S 4.5]{boxerpilloni20})\n  \\[ \\pi_{\\HT,G,K_p}^{\\mathrm{tor}}: \\mathcal{S}_{G,K^pK_p}^{\\mathrm{tor}}\\longrightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G\/K_p.\\]\n\n There is an analogous Hodge--Tate period map for $H$ also. To lighten the notation, we shall frequently omit many of the subscripts from $\\pi_{\\HT,G,K_p}^{\\mathrm{tor}}$ when they are clear from context (in particular, we shall almost always omit the ``$\\mathrm{tor}$'', since the non-compactified Shimura variety plays no role here).\n\n  We shall attempt to consistently maintain the convention that subsets of flag varieties are denoted by typewriter letters $\\scalebox{1.15}{$\\mathtt{U}$}$ etc, and the preimages of these spaces under the Hodge--Tate period maps are denoted by calligraphic letters $\\mathcal{U}$ etc.\n\n\n \\subsection{Period maps}\n\n  \\begin{theorem}\n   \\label{thm:HTmaps}\n   There is a commutative diagram of Hodge--Tate period maps (where we have omitted some unimportant subscripts and superscripts for clarity)\n   \\[\n    \\begin{tikzcd}\n    \\mathcal{S}^{\\mathrm{tor}}_{H,\\Iw}(p^t) \\rar[\"\\pi^H_{\\Iw}\"]  & \\scalebox{1.15}{$\\mathtt{FL}$}_H \/ K_{H, \\Iw}(p^t)\\\\\n    \\mathcal{S}^{\\mathrm{tor}}_{H,\\diamondsuit}(p^t) \\rar[\"\\pi^H_{\\diamondsuit}\"] \\dar[\"\\hat\\iota\" left] \\uar[\"\\pr_\\diamondsuit\" left] & \\scalebox{1.15}{$\\mathtt{FL}$}_H \/ K_{H, \\diamondsuit}(p^t) \\dar[\"\\hat\\iota\" right] \\uar[\"\\pr_{\\diamondsuit}\" right]\\\\\n    \\mathcal{S}^{\\mathrm{tor}}_{G,\\Iw}(p^t) \\rar[\"\\pi^G_{\\Iw}\"] & \\scalebox{1.15}{$\\mathtt{FL}$}_G \/ K_{G, \\Iw}(p^t)\n    \\end{tikzcd}\n   \\]\n   in which the maps $\\pr_{\\diamondsuit}$ are the natural quotients, and the downward ones are given by the composite of the natural embedding $H \\hookrightarrow G$ and right-translation by $\\hat\\gamma = \\gamma w_1$.\n  \\end{theorem}\n\n  \\begin{proof}\n   It suffices to check that the Hodge--Tate period maps for $H$ and for $G$ at perfectoid infinite level are compatible; but this is a direct consequence of the construction, since the Hodge--Tate period map for Hodge-type Shimura varieties is defined using an embedding into a Siegel Shimura variety. See \\cite[\\S 4.4.7]{boxerpilloni20}).\n  \\end{proof}\n\n  For $(m, n, t)$ as in \\eqref{eq:mnt}, we define subspaces\n  \\[ \\mathcal{I}^G_{m, n} \\subset \\mathcal{U}^G_n \\subset \\mathcal{S}_{G,\\Iw}(p^t), \\qquad \\mathcal{Z}^H_{m} \\subset \\mathcal{U}^H_n \\subset \\mathcal{S}_{H,\\diamondsuit}(p^t)\n  \\]\n  as the preimages of the subsets $\\scalebox{1.15}{$\\mathtt{I}$}^G_{m, n} \\subset \\scalebox{1.15}{$\\mathtt{U}$}^G_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^G$ under $\\pi^G_{\\Iw}$, respectively $\\mathcal{Z}^H_{m} \\subset \\mathcal{U}^H_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^H$ under $\\pi^H_{\\diamondsuit}$. Combining \\cref{thm:HTmaps} and \\cref{eq:Cartadic2}, we obtain a Cartesian diagram\n  \\begin{equation}\\label{eq:adicSh}\n   \\begin{tikzcd}\n    \\mathcal{Z}^{H}_m \\dar \\rar[hook] &\\mathcal{U}^H_n \\dar\\\\\n    \\mathcal{I}^G_{m, n} \\rar[hook] & \\mathcal{U}^G_n\n   \\end{tikzcd}\n  \\end{equation}\n  in which the horizontal arrows are closed embeddings; and $\\mathcal{Z}^{H}_m$ is closed in $\\mathcal{S}_{H,\\diamondsuit}(p^t)$.\n\n\n \\subsection{Overconvergent pullback}\n  We can now define the pullback map on overconvergent cohomology. We give the definitions for the non-cuspidal cohomology, using the coefficient sheaf $\\mathcal{V}= \\mathcal{V}_\\kappa$ for some $M_G$-dominant integral weight $\\kappa$; the definitions are the same for cuspidal cohomology using $\\mathcal{V}= \\mathcal{V}_\\kappa(-D)$ instead.\n\n  Using the diagram \\eqref{eq:adicSh} and the functoriality of cohomology with support, we get a map\n  \\begin{equation}\n   \\label{eq:iota1}\n   \\hat\\iota^* : R\\Gamma_{\\mathcal{I}^G_{mn}}(\\mathcal{U}_n^G, \\mathcal{V}) \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^{H}}\\left(\\mathcal{U}_n^H, \\hat\\iota^* \\mathcal{V}_{\\kappa}\\right) \\cong R\\Gamma_{\\mathcal{Z}_m^{H}}\\left(\\mathcal{S}_{H,\\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right),\n  \\end{equation}\n  where the final isomorphism comes from excision, using the fact that $\\mathcal{Z}_m^{H}$ is closed in $\\mathcal{S}_{H,\\diamondsuit}(p^t)$.\n\n  As in \\bp{\\S 5.4.1}, for any $t \\ge 1$, we can define the finite-slope overconvergent cohomology for $G$ as\n  \\begin{align*}\n    R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} &\\coloneqq R\\Gamma_{\\mathcal{I}_{00}}(\\mathcal{U}^G_0, \\mathcal{V}_\\kappa)^{-, \\fs}.\n  \\end{align*}\n  So \\eqref{eq:iota1} for $(m, n) = (0,0)$ gives our first definition of the pullback map on overconvergent cohomology, taking values in $R\\Gamma_{\\mathcal{Z}_0^{H}}\\left(\\mathcal{S}_{H,\\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right)$.\n\n  \\begin{proposition}[Comparison with classical pullback]\n   \\label{prop:classicalcomp}\n   We have the following commutative diagram:\n   \\[\n    \\begin{tikzcd}\n     R\\Gamma_{\\mathcal{I}_{00}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\rar[\"\\quad \\hat\\iota^*\\quad\" above, \"\\eqref{eq:iota1}\" below] \\dar[\"\\cores\"] & R\\Gamma_{\\mathcal{Z}_0^{H}}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right) \\dar[\"\\cores\"]\\\\\n    \n     R\\Gamma(\\mathcal{U}_0^G,\\mathcal{V}) \\rar{\\quad \\hat\\iota^*\\quad }  & R\\Gamma\\left(\\mathcal{S}^{\\mathrm{tor}}_{H,\\diamondsuit}(p^t),\\hat\\iota^*\\mathcal{V}\\right)\\\\\n    \n        R\\Gamma\\left(\\mathcal{S}^{\\mathrm{tor}}_{G,\\Iw}(p^t),\\mathcal{V}\\right)\\uar[\"\\mathrm{res}\"] \\arrow[\"\\hat\\iota^*\" below]{ru}\n    \\end{tikzcd}\n   \\]\n   in which the bottom horizontal map corresponds to the classical pushforward via the rigid-analytic GAGA theorem. Moreover, the spaces in the left column of the diagram have actions of the prime-to-$p$ Hecke algebra and the operators $\\mathcal{U}'_{\\Sieg}$, $\\mathcal{U}'_{\\Kl}$ at $p$, and the maps $\\mathrm{res}$ and $\\cores$ are compatible with these actions.\n  \\end{proposition}\n\n  \\begin{proof}\n   The only non-obvious step of the diagram is the existence of the middle horizontal map, which follows from \\cref{prop:Cartadic}. The compatibility with Hecke actions is an easy check, cf.~\\bp{Lemma 5.17}.\n  \\end{proof}\n\n  \\begin{proposition}[Change of support condition]\n   \\label{prop:changesupport}\n   The maps \\eqref{eq:iota1} for $(m, n)$, $(m, 0)$, and $(0, 0)$ fit into a diagram\n   \\[\n    \\begin{tikzcd}\n     R\\Gamma_{\\mathcal{I}_{mn}^G}(\\mathcal{U}_n^G,\\mathcal{V}) \\arrow{r}{\\hat\\iota^*} &\n     R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right)\\\\\n     R\\Gamma_{\\mathcal{I}_{m0}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\arrow{u}{\\mathrm{res}} \\arrow{r}{\\hat\\iota^*} \\dar[\"\\cores\"]& R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right) \\uar[equals] \\arrow{d}{\\cores} \\\\\n     R\\Gamma_{\\mathcal{I}_{00}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\arrow{r}{\\hat\\iota^*} &\n     R\\Gamma_{\\mathcal{Z}_0^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right).\n    \\end{tikzcd}\n   \\]\n   The complexes in the left column have compatible actions of the Hecke operators away from $p$, and of $\\mathcal{U}'_{\\Kl}$ and $\\mathcal{U}'_{\\Sieg}$ at $p$, and the maps $\\mathrm{res}$ and $\\cores$ are compatible with these.\n  \\end{proposition}\n\n  \\begin{proof}\n   Immediate from the fact that $\\mathcal{I}^G_{mn} = \\mathcal{I}_{m0}^G \\cap \\mathcal{U}_n^G$ and standard functoriality properties of cohomology with support.\n  \\end{proof}\n\n  \\begin{proposition}\n   The spaces in the left columns of the diagrams in \\cref{prop:classicalcomp,prop:changesupport} all have actions of the prime-to-$p$ Hecke operators, and of the Hecke operators $\\mathcal{U}'_{\\Sieg}$, $\\mathcal{U}'_{\\Kl}$, $\\mathcal{U}'_B$ at $p$. Moreover, the maps in the left column of \\cref{prop:changesupport} become isomorphisms on the finite-slope part for $\\mathcal{U}'_B$.\n  \\end{proposition}\n\n  \\begin{proof}\n   The compatibility with Hecke operators away from $p$ is clear, since the Hodge--Tate period map is invariant under the action of the prime-to-$p$ Hecke algebra. The fact that the maps in the left column of \\cref{prop:classicalcomp} are maps of Hecke modules is an instance of \\bp{Lemma 5.17}.\n\n   The assertions regarding the finite-slope part follow from \\bp{Theorem 5.66}, since one can check that $(\\mathcal{I}_{m, n}, \\mathcal{U}_n)$ defines an ``allowed support condition'' in the sense of \\bp{\\S 5.4.3}.\n  \\end{proof}\n\n  This shows that we have well-defined maps\n  \\begin{equation}\n   \\label{eq:iota2}\n   R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right)\n  \\end{equation}\n  for any $0 \\le m < t$, compatible under corestriction, extending \\eqref{eq:iota1} for in the $m = 0$ case.\n\n\n\n\n \\subsection{Functoriality of coefficients}\n  \\label{sect:branchcoeffs}\n  \\begin{proposition}\n   Let $\\kappa_1 = (r_1, -r_2-2; r_1 + r_2)$ with $r_1 \\ge r_2 \\ge -1$, and let $\\tau = (t_1, t_2; r_1 + r_2)$ where $t_i \\ge -1$ and $t_1 + t_2 = r_1 - r_2 - 2$. Then there is a nonzero homomorphism of $(\\gamma^{-1} M_H \\gamma)$-representations\n   \\[ V^G_{\\kappa_1} |_{\\gamma^{-1} M_H\\gamma } \\to V^H_{\\tau}, \\]\n   uniquely determined up to scaling.\n  \\end{proposition}\n\n  It will be helpful to fix a normalisation for this map, by choosing a vector $f \\in (V^G_{\\kappa_1})^\\vee$ which transforms by $\\tau^{-1}$ under $\\gamma^{-1} M_H \\gamma$. We have an explicit presentation of $(V^G_{\\kappa_1})^\\vee = V^G_{(-w_{0, M} \\kappa_1)}$ as the space of polynomial functions $f \\in \\mathcal{O}(M_G)$ which satisfy $f(mb) = \\kappa_1(b) f(m)$ for all $b \\in B_{M_G}$ and $m \\in M_G$, with $M_G$ acting by left-translation. Since $\\gamma^{-1} M_H \\gamma \\cdot B_{M_G}$ is open in $M_G$, we can choose a unique $f$ which satisfies $f(\\mathrm{id}) = 1$ and transforms via $\\tau^{-1}$ under the action of $M_H$.\n\n  This map gives a homomorphism of sheaves on $\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t)$,\n  \\[ \\hat\\iota^*\\left(\\mathcal{V}_\\kappa^G\\right) \\longrightarrow \\mathcal{V}_\\tau^H, \\]\n  and combining this with \\cref{eq:iota2}, we obtain maps of complexes\n  \\[ R\\Gamma^G_{w_1}(\\kappa_1)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\mathcal{V}^H_\\tau\\right)\n  \\]\n  for any $\\tau$ in the appropriate range, and similarly for cuspidal cohomology.\n\n  \\begin{remark}\n   The map is formally well-defined for a rather wider range of values of the parameters; but we have restricted to the case when $r_1 \\ge r_2 \\ge -1$ and $t_1, t_1 \\ge -1$, in order that there are interesting cuspidal automorphic representations contributing to $H^2$ for both $\\mathcal{V}_{\\kappa_1}^G$ and $\\mathcal{V}^H_{\\tau}$.\n\n   If we set $k_i = r_i + 3$ and $c_i = t_i + 2$, then $(k_1, k_2, c_1, c_2)$ will define a point lying on the top edge of the region labelled $(f)$ in \\cite[Diagram 2]{LZvista}. Unfortunately, it seems to be difficult to extend our present analysis to points in the interior of this region; this would require some sort of ``nearly version'' of higher Coleman theory, analogous to the theory of nearly-overconvergent families in $H^0$ of modular curves recently introduced by Andreatta--Iovita \\cite{andreattaiovita21}. See \\cite[\\S 6]{LPSZ1} for an analogous theory in the ordinary case (with $r_2$ fixed, rather than varying as here).\n  \\end{remark}\n\n\n \\subsection{Change of level}\n  \\label{sect:changelevel}\n  Finally, we note that for any $t \\ge 1$, we have $[K_{H, \\diamondsuit}(p^t): K_{H, \\diamondsuit}(p^{t+1})] = p^4 = [K_{G, \\Iw}(p^t) : K_{G, \\Iw}(p^{t+1})]$, and hence the natural map\n  \\[ \\mathcal{S}_{H, \\diamondsuit}(p^{t+1}) \\longrightarrow \\mathcal{S}_{H, \\diamondsuit}(p^{t}) \\times_{\\mathcal{S}_{G, \\Iw}(p^{t})} \\mathcal{S}_{G, \\Iw}(p^{t+1})\n  \\]\n  is an isomorphism. So the pushforward (trace) maps arising from changing $t$ on the two spaces are compatible with the pullback $\\hat\\iota^*$, and similarly for the cohomology with supports, for any support condition invariant under $K_{G, \\Iw}(p^t)$.\n\n  Hence, if we temporarily write $\\mathcal{I}_{mn}^G(p^t)$ etc to distinguish our various locally closed subspaces of Shimura varieties at the different levels, then we have trace maps\n  \\[ R\\Gamma_{\\mathcal{I}^G_{mn}(p^{t+1})}(\\mathcal{U}^G_n(p^{t+1}), \\mathcal{V}^G_\\kappa) \\longrightarrow R\\Gamma_{\\mathcal{I}^G_{mn}(p^t)}(\\mathcal{U}^G_n(p^t), \\mathcal{V}^G_\\kappa) \\]\n  and\n  \\[  R\\Gamma_{\\mathcal{Z}^H_{m}(p^{t+1})}(\\mathcal{S}_{H, \\diamondsuit}(p^{t+1}), \\mathcal{V}^H_\\tau) \\longrightarrow R\\Gamma_{\\mathcal{Z}^H_{m}(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^{t}), \\mathcal{V}^H_\\tau).\\]\n  and these are compatible with the pullback maps $\\hat\\iota^*$, and the restriction\/corestriction maps for varying $m, n$. Moreover, the trace maps for $G$ are isomorphisms on the finite-slope part by \\bp{Theorem 5.14}.\n\n  We can thus define a map\n  \\[\n   R\\Gamma_{w_1}(\\kappa)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^t), \\mathcal{V}^H_\\tau).\n  \\]\n  for any $t \\ge 1$, $m \\ge 0$ (not necessarily with $t > m$) by composing with the trace map from level $t'$ for some auxiliary $t' > m$; this allows us to define $\\hat\\iota^*$ as a map\n  \\[\n   R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} \\to \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^t),  \\mathcal{V}^H_\\tau).\n  \\]\n  Since the spaces $\\mathcal{Z}_m^H$ are actually invariant under the Iwahori of $H$, we can trace down further to land in the space\n  \\[ \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\Iw}(p^t),  \\mathcal{V}^H_\\tau),\\]\n  where we have abused notation a little by using $\\mathcal{Z}^H_m(p^t)$ for the preimages of $\\scalebox{1.15}{$\\mathtt{Z}$}_m^H$ at either Iwahori or $\\diamondsuit$ level.\n  \\begin{definition}\n   We define\n   \\[ R\\Gamma_{\\mathrm{id}}\\left(\\mathcal{S}_{H, \\Iw}(p^t), \\tau\\right)^{(-, \\dag)} = \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_{m}(p^t)}\\left( \\mathcal{S}_{H, \\Iw}(p^t), \\mathcal{V}^H_\\tau\\right) .\\]\n  \\end{definition}\n\n  This space can be interpreted as the compactly-supported cohomology of the intersection $\\bigcap_m \\mathcal{Z}^H_m(p^t) = \\pi_{H}^{-1}\\left( \\{\\mathrm{id}_H\\} \\right)$; we shall recall this in a little more detail in the next section, where we shall allow more general coefficients.\n\n\n\n\\section{Torsors}\n\n We now begin constructing the ``locally analytic'' version of the pullback map on higher Coleman theory.\n \\subsection{Torsors on flag varieties}\n\n  The map $x \\mapsto x^{-1}: G\\rightarrow \\FL_G$ (recall that $\\FL_G=P_G\\backslash G$) allows us to regard $G$ as a right $P_G$-torsor over $\\FL_G$, and similarly to regard $G\/ N_G\\rightarrow \\FL_G$ as a right $M_G$-torsor. We consider their analytifications\n  \\[ \\scalebox{1.15}{$\\mathtt{P}$}^G: \\mathcal{G}\\rightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G \\qquad \\text{and}  \\qquad \\scalebox{1.15}{$\\mathtt{M}$}^G:\\mathcal{G}\/ \\mathcal{N}_G\\rightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G.\\]\n  which are torsors over $\\scalebox{1.15}{$\\mathtt{FL}$}_G$ under the (affinoid) analytic groups $\\mathcal{P}_G$ and $\\mathcal{M}_G$ respectively. We similarly define torsors over the flag varieties of $H$ and $H_i$ for $i=1,2$.\n\n  \\begin{definition} Define $\\mathcal{P}^G_{\\HT}$ and $\\mathcal{M}^G_{\\HT}$ to be the pullbacks via $\\pi^G_{\\HT}$ of the torsors $\\scalebox{1.15}{$\\mathtt{P}$}^G$ and $\\scalebox{1.15}{$\\mathtt{M}$}^G$; these are (right) torsors over $\\mathcal{S}_{G,\\Iw}(p^t)$ for the groups $\\mathcal{P}_G$ and $\\mathcal{M}_G$. We similarly define $\\mathcal{P}^H_{\\HT}$ and $\\mathcal{M}^H_{\\HT}$, $\\mathcal{P}^{H_i}_{\\HT}$ and $\\mathcal{M}^{H_i}_{\\HT}$ for $i=1,2$.\n  \\end{definition}\n\n  \\begin{note}\n   It is easy to check that $\\mathcal{M}^H_{\\HT}=\\mathcal{M}^{H_1}_{\\HT}\\times_{\\GL_1}\\mathcal{M}^{H_2}_{\\HT}$, where we take the fibre product with respect to the action of $\\nu$ in the parametrisation of $T$.\n  \\end{note}\n\n\n \\subsection{Reduction of structure}\n\n  \\begin{definition}\n   For $n > 0$, let $\\mathcal{M}^1_{G,n} \\triangleleft \\mathcal{M}_G$ be the (affinoid analytic) group of elements which reduce to the identity $\\pmod{p^n}$. Define\n   \\[ \\mathcal{M}^\\square_{G,n} = \\mathcal{M}^1_{G,n} \\cdot B_{M_G}(\\ZZ_p), \\]\n   which is an affinoid analytic subgroup containing $\\Iw_{M_G}(p^n)$. A similar definition applies to $M_H = T$; we write the group as $\\mathcal{T}^{\\square}_{n} = T(\\ZZ_p) \\mathcal{T}^1_{n}$.\n  \\end{definition}\n\n  \\begin{note}\n   We follow \\cite{boxerpilloni20} here in using affinoid subgroups and affinoid subspaces of flag varieties to develop the locally-analytic theory, rather than the ``mixed'' spaces (products of some copies of $\\mathcal{B}_n$ and some of $\\mathcal{B}^\\circ_n$) used in the previous sections.\n  \\end{note}\n\n  \\begin{note}\\label{note:identifyMGsq}\n   Identifying $M_G$ with $\\GL_2 \\times \\GL_1$ as in the introduction, we have\n   \\[\\mathcal{M}^\\square_{G,  n} =\n    \\left\\{\n     (\\stbt x y z w, \\lambda) :\n     \\begin{array}{c}\n       x,w,\\lambda \\in \\ZZ_p^\\times \\cdot (1 + \\mathcal{B}_n),\\\\ z \\in \\mathcal{B}_n, y \\in \\ZZ_p + \\mathcal{B}_n.\n     \\end{array}\n    \\right\\}\n    \\qedhere\n   \\]\n  \n  \\end{note}\n\n  \\begin{notation}\n   Define\n   \\[ \\mathcal{T}^\\diamondsuit_n = \\{\\diag(t_1, t_2, \\nu t_2^{-1}, \\nu t_1^{-1}) \\in \\mathcal{T}^\\square_n: t_1-t_2 \\in \\mathcal{B}_n\\}. \\]\n  \\end{notation}\n\n  Thus $\\mathcal{T}^\\diamondsuit_n$ and $\\mathcal{T}^\\square_n$ are both disjoint unions of copies of $\\mathcal{T}^1_n$, but $\\mathcal{T}^\\diamondsuit_n$ has fewer of these components than $\\mathcal{T}^\\square_n$.\n\n  \\begin{proposition} \\label{prop:redofstr} Let $t > n > 0$.\n   \\begin{enumerate}\n    \\item Over $\\mathcal{U}^G_n$, the torsor $\\mathcal{M}_{\\HT}^G$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^G_{\\HT,n}$ under the group $\\mathcal{M}^\\square_{G,n}$.\n\n    \\item Over $\\mathcal{U}^H_{\\Iw,n}$, the torsor $\\mathcal{M}_{\\HT}^H$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^H_{\\HT,n,\\Iw}$ under the group $\\mathcal{T}^{\\square}_{n}$.\n\n    \\item Over $\\mathcal{U}^H_n$, the torsor $\\mathcal{M}_{\\HT}^H$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^H_{\\HT,n,\\diamondsuit}$ under the group $\\mathcal{T}^\\diamondsuit_{n}$ (and this refines the pullback of $\\mathcal{M}^H_{\\HT,n,\\Iw}$ to level $K^H_{\\diamondsuit}(p^t)$).\n\n   \n  \\end{enumerate}\n  \\end{proposition}\n\n  \\begin{proof}\n   Part (1) is essentially the result of \\bp{\\S 6.2.1}. The proofs of (2) and (3) are similar.\n  \\end{proof}\n\n  \\begin{lemma}\n   We have the following inclusions of subgroups.\n   \\begin{itemize}\n    \\item As subgroups of $\\mathcal{M}_G$, we have\n    \\[ \\mathcal{T}^\\diamondsuit_{n} = \\mathcal{T} \\cap \\gamma \\mathcal{M}^\\square_{G,n}\\gamma^{-1}. \\]\n    \\item As subgroups of $\\mathcal{G}$,\n    \\[ \\hat\\gamma^{-1} \\cdot K^H_{\\diamondsuit}(p^n)\\mathcal{H}^1_{n} \\cdot \\hat\\gamma \\subset K^G_{\\Iw}(p^n)\\mathcal{G}^1_{n}, \\]\n    where $\\hat\\gamma = \\gamma w_1$ as usual.\n   \\end{itemize}\n  \\end{lemma}\n\n  \\begin{proof}\n  \n   If $\\tau = (\\stbt{t_1}{}{}{t_2}, \\nu)$ is an element of $\\mathcal{T}$ then $\\gamma^{-1} \\tau \\gamma = ( \\stbt{t_1}{}{t_2-t_1}{t_2}, \\nu)$. It is now clear that $\\gamma^{-1}T_{\\diamondsuit}(p^t) \\gamma \\subset \\Iw_{M_G}(p^t)$ and $\\gamma^{-1}\\mathcal{T}^1_{n} \\gamma \\subset \\mathcal{M}^1_{G,n}$, so the required inclusion follows.\n\n   The second statement can be verified similarly; the inclusion on $\\ZZ_p$-points is the definition of $K^H_{\\diamondsuit}$, and the inclusion on $\\mathcal{H}^1_{n}$ follows from the fact that $\\mathcal{H}^1_{n} \\subset \\mathcal{G}^1_n$ and $\\mathcal{G}^1_n$ is normal in $\\mathcal{G}$.\n  \\end{proof}\n\n\n  \\begin{proposition}\\label{prop:pullbackcomp}\n   We have an equality of $\\mathcal{M}^\\square_{G, n}$-torsors over $\\mathcal{U}^H_{n,\\diamondsuit}$:\n   \\[ \\hat\\iota^*\\left( \\mathcal{M}^G_{\\HT,n,\\Iw}\\right) = \\mathcal{M}^H_{\\HT,n,\\diamondsuit}\\times^{\\left[\\mathcal{T}^\\diamondsuit_{n}, \\gamma\\right]} \\mathcal{M}^\\square_{G,n}, \\]\n   where we regard $\\mathcal{T}^{\\diamondsuit}_{n}$ as a subgroup of $\\Iw_{M_G}(p^t)\\mathcal{M}^1_{G,n}$ via conjugation by $\\gamma$.\n  \\end{proposition}\n\n  \\begin{proof}\n   We check the analogous statement on the flag varities. We first observe that we have a commutative diagram of adic spaces\n   \\[\n    \\begin{tikzcd}\n     K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n} \\dar\\rar & K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}\\dar\\\\\n     \\mathcal{B}^H\\backslash \\mathcal{B}^H K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}\\rar[hook, \"\\hat\\iota\"]\n     & \\mathcal{P}^G\\backslash \\mathcal{P}^G w_1K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}.\n    \\end{tikzcd}\n   \\]\n   Here, the vertical maps are given by $h\\mapsto \\mathcal{B}^H\\backslash \\mathcal{B}^H h^{-1}$ on the left, and $g\\mapsto \\mathcal{P}^G\\backslash \\mathcal{P}^G w_1g^{-1}$ on the right; the lower horizontal map $\\hat\\iota$ is $\\mathcal{B}^H h\\mapsto \\mathcal{P}^G h \\gamma w_1$, and the map along the top making the diagram commute is $h \\mapsto \\hat\\gamma^{-1} h\\hat\\gamma$, which is well-defined by the preceding lemma. (Note that the commutativity of the diagram relies on the fact that $\\gamma \\in P_G$.)\n\n   The right-translation action of $\\mathcal{B}^H$ on $\\mathcal{G}$ makes the left-hand column into a torsor for the group $\\mathcal{B}^H \\cap K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}$. Similarly, via right-translation conjugated by $w_1$, $K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}$ becomes a torsor for the group\n   \\(  \\mathcal{P}\\cap  w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1}; \\)\n   and these structures are compatible if we consider $\\mathcal{B}^H \\cap K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}$ as a subgroup of $\\mathcal{P} \\cap w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1} $ via conjugation by $\\gamma$.\n\n   Passing to the $\\mathcal{N}_H$-coinvariants on the left, we obtain a torsor for $T_\\diamondsuit(p^t) \\mathcal{T}^1_{n} = \\mathcal{T}^\\diamondsuit_n$; and passing to $\\mathcal{N}_G$-coinvariants on the right, we obtain a torsor for the projection of $\\mathcal{P} \\cap w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1}$ to the Levi $\\mathcal{M}_G$, which is the group $\\mathcal{M}^\\square_{G, n}$. Moreover, these structures are compatible via the $\\gamma$-conjugation inclusion $\\mathcal{T}^\\diamondsuit_n \\hookrightarrow \\mathcal{M}^\\square_{G,n}$ established in the above lemma.\n\n   We now note that $\\scalebox{1.15}{$\\mathtt{U}$}^G_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^G$ is contained in the subset $\\mathcal{P}^G w_1K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}$, since $\\scalebox{1.15}{$\\mathtt{U}$}^G_n$ is the orbit of $w_1$ under $K^G_{\\Iw}(p^t) \\mathcal{G}^1_{n, n}$ in the notation of \\bp{\\S 3.3.3}, and $\\mathcal{G}^1_{n, n} \\subset \\mathcal{G}^1_n$. So pulling back to $\\mathcal{U}^G_n$ via the Hodge--Tate period map gives the result.\n  \\end{proof}\n\n\\section{Spaces of distributions and branching laws}\n\n \\subsection{Analytic characters}\n\n  \\begin{definition}\n   Let $n \\in \\mathbf{Q}_{> 0}$. We say a continuous character $\\kappa: \\ZZ_p^\\times \\to A^\\times$, for $(A, A^+)$ a complete Tate algebra, is \\textbf{$n$-analytic} if it extends to an analytic $A$-valued function on the affinoid adic space\n   \\[ \\ZZ_p^\\times \\cdot \\mathcal{B}_n \\subset \\mathbf{G}_m^{\\mathrm{ad}}.\\]\n   This definition extends naturally to characters $T(\\ZZ_p) \\to A^\\times$: the $n$-analytic characters are exactly those which extend to $\\mathcal{T}^\\square_n$.\n  \\end{definition}\n\n  \\begin{remark}\n   For compatibility with our notations for algebraic weights, we shall denote a $p$-adic character $\\kappa$ of $T(\\ZZ_p)$ by a triple $(\\rho_1, \\rho_2; \\omega)$ of characters of $\\ZZ_p^\\times$, via\n   \\[ \\kappa(\\diag(st_1, st_2, st_2^{-1}, st_1^{-1})) = \\rho_1(t_1) \\rho_2(t_2) \\omega(s), \\]\n   so that formally\n   \\[ \\kappa(\\diag(t_1, t_2, \\nu t_2^{-1}, \\nu t_1^{-1})) = \\rho_1(t_1) \\rho_2(t_2) \\left(\\frac{\\omega}{\\rho_1\\rho_2}\\right)^{\\tfrac{1}{2}}(\\nu).\\]\n   This is of course not well-defined as written, since $p$-adic characters do not have a unique square root, so we should understand the triple $(\\rho_1, \\rho_2; \\omega)$ as coming with an implicit choice of square root of $\\omega \/\\rho_1 \\rho_2$ which is being suppressed from the notation.\n  \\end{remark}\n\n \\subsection{Analytic inductions}\n\n  We recall some definitions from \\bp{\\S 6.1.2}. Let $(A,A^+)$ be a complete Tate algebra over $(\\QQ_p,\\ZZ_p)$. Let $n_0 > 0$, and assume that $\\kappa_A: T(\\ZZ_p)\\rightarrow A^\\times$ is an $n_0$-analytic character.  For $?\\in \\{G,H\\}$ and $n \\geq n_0$, let $\\mathcal{M}^1_{?,n}$ be the affinoid subgroup of $\\mathcal{M}_?$ defined above, and let $B_{M_G}$ be the Borel of $M_?$.\n\n  \\begin{definition}\n   For $n\\geq n_0$, define\n   \\begin{align*}\n    V^{n-\\an}_{G,\\kappa_A}=&\\, \\an\\Ind^{\\left(\\mathcal{M}^\\square_n\\right) }_{\\left(\\mathcal{M}^\\square_n \\cap \\mathcal{B}_{G}\\right)}(w_{0,M_?}\\kappa_A)\\\\\n    =&\\, \\Big\\{ f \\in \\mathcal{O}(\\mathcal{M}^\\square_{G, n}) \\mathop{\\hat\\otimes} A :  f(mb)=(w_{0,M}\\kappa_A)(b^{-1})f(m),\\\\\n     &\\quad \\forall \\, m\\in \\mathcal{M}^\\square_{G, n},\\, \\forall b\\in \\mathcal{M}^\\square_{G, n} \\cap \\mathcal{B}_G \\Big\\}.\n   \\end{align*}\n   We define a left action of $\\mathcal{M}^\\square_{G, n}$ on $V^{n-\\an}_{G,\\kappa_A}$ by $(h \\cdot f)(m) = f(h^{-1} m)$.\n\n   Write $D^{n-\\an}_{G,\\kappa_A}$ for the dual space, and $\\langle -, - \\rangle$ for the pairing between these; we equip $D^{n-\\an}_{G,\\kappa_A}$ with a left action of the same group $\\mathcal{M}^\\square_{G, n}$, in such a way that $\\langle h\\mu , hf \\rangle = \\langle \\mu ,f \\rangle$.\n  \\end{definition}\n\n  Let us describe $V_{G,\\kappa_A}^{n-\\an}$ explicitly. We use the description of $\\mathcal{M}_{G, n}^{\\square}$ given in \\cref{note:identifyMGsq}. Since $\\mathcal{M}_{G, n}^{\\square}$ has an Iwahori decomposition, restriction to elements of the form $(\\stbt{1}{}{\\star}{1}, 1)$ identifies $V_{G,\\kappa_A}^{n-\\mathrm{an}}$ with the space of analytic functions of $z \\in \\mathcal{B}_n$; this space is independent of $\\kappa_A$, but the action of $\\mathcal{M}^{\\square}_{G,n}$ does depend on $\\kappa_A$, as follows.\n\n  \\begin{propqed}\n   Suppose $\\kappa_A$ is the character $(\\rho_1, \\rho_2; \\omega)$. Then the action of $(\\stbt a b c d, \\nu)$ on $f \\in \\mathcal{O}(\\mathcal{B}_n) \\mathop{\\hat\\otimes} A$ is given by\n   \\[ \\left((\\stbt a b c d, \\nu) f\\right)(z) = f\\left( \\frac{az-c}{-bz + d}\\right) (-bz + d)^{\\rho_1 - \\rho_2} (ad-bc)^{\\rho_2} \\nu^{(\\omega-\\rho_1-\\rho_2)\/2}.\\qedhere\\]\n  \\end{propqed}\n\n  \\begin{note}\n   For $H$ in place of $G$, we can make the same definitions; but the resulting spaces are much simpler, since $\\mathcal{M}_H = \\mathcal{T}$ is commutative and contained in $\\mathcal{B}_H$. Hence any function $f \\in V^{n-\\an}_{H,\\kappa_A}$ is uniquely determined by its value at 1. So $V^{n-\\an}_{H,\\kappa_A}$ is canonically $A$, with $\\mathcal{T}^\\square_n$ acting via $\\kappa_A$; and dually $D^{n-\\an}_{H,\\kappa_A}$ is $A$ with $\\mathcal{T}^\\square_n$ acting via $\\kappa_A^{-1}$.\n  \\end{note}\n\n\n  \\begin{note}\n   If $\\kappa_A$ is an algebraic character $(k_1, k_2; c)$, then $V_{G,\\kappa_A}^{n-\\an}$ naturally contains the algebraic $M_H$-representation of highest weight $\\kappa_A$ (identified with polynomials in $z$ of degree $\\le k_1 - k_2$); and dually, $D^{n-\\an}_{?,\\kappa_A}$ surjects onto the algebraic representation of highest weight $\\kappa_A^\\vee$ (the dual of the weight $\\kappa_A$ representation).\n  \\end{note}\n\n\n\n \\subsection{Branching laws in families}\\label{ss:kraken}\n\n  \\begin{definition}\n   Let $A$ be a Tate algebra endowed with an $n_0$-analytic character $\\kappa_A: T(\\ZZ_p) \\to A^\\times$ as above, and additionally with a character $\\lambda: (1 + \\mathcal{B}_n)^\\times \\to A^\\times$. Define the \\emph{kraken} to be the function\n   \\[ \\mathscr{K}^{\\lambda}(z)=\\lambda(1+z),\\]\n   viewed as an element of $V_{G,\\kappa_{A}}^{n-\\an}$.\n  \\end{definition}\n\n  \\begin{lemma}\n   The function $\\mathscr{K}^{\\lambda}$ is an eigenvector for $\\gamma^{-1} \\mathcal{T}^\\diamondsuit_n \\gamma \\subset \\mathcal{M}^\\square_{G, n}$, with eigencharacter $w_{0, M} \\kappa_A + (\\lambda, -\\lambda; 0)$.\n  \\end{lemma}\n\n  \\begin{proof}\n   We have $\\gamma^{-1}\\left( \\stbt x {} {} y, \\nu\\right) \\gamma=(\\stbt x {} {-x+y} y, \\nu)$.\n   If this condition is satisfied, then (writing $\\kappa = (\\rho_1, \\rho_2; \\omega)$ as before) we have\n   \\[ (\\stbt x {} {-x+y} y, \\nu) \\mathscr{K}^{\\lambda}(z)=  x^{\\rho_2}y^{\\rho_1}\\nu^{(\\omega-\\rho_1-\\rho_2)\/2} \\mathscr{K}^{\\lambda}\\left( \\frac{x}{y}(z+1) - 1 \\right)=x^{\\rho_2+\\lambda}y^{\\rho_1-\\lambda} \\nu^{(\\omega-\\rho_1-\\rho_2)\/2}\\mathscr{K}^{\\lambda}(z).\\qedhere \\]\n  \\end{proof}\n\n  As an immediate consquence, we obtain the following result:\n\n  \\begin{propqed}\\label{prop:krakenpower}\n   Pairing with the element $\\mathscr{K}^{\\lambda}$ defines a homomorphism of $\\mathcal{T}^\\diamondsuit_n$-representations\n   \\[ \\hat\\iota^*( D^{n-\\an}_{G,\\kappa_A}) \\longrightarrow D^{n-\\an}_{H,w_{0, M} \\kappa_A + (\\lambda, -\\lambda;0)}.\\qedhere\\]\n  \\end{propqed}\n\n  \\begin{note}\n   Note that $D^{n-\\an}_{H,w_{0, M} \\kappa_A + (\\lambda, -\\lambda;0)}$ is one-dimensional (and independent of $n$).\n  \\end{note}\n\n  We now consider a special case. Let $A = \\QQ_p$ and take $\\kappa_A$ to be the algebraic weight $(r_2 + 2, -r_1; -r_1-r_2)$, for some integers $r_1 \\ge r_2\\ge 0$, so that $\\kappa_A^\\vee$ is the weight $\\kappa_1$ of \\eqref{eq:ourweights}. If we choose $\\lambda$ to be an integer in the range $[0, r_1 + r_2 + 2]$, then $\\mathscr{K}^\\lambda$ lies in the polynomial subspace $V_{G, \\kappa_A} \\subset V_{G, \\kappa_A}^{n-\\an}$. Its value at the identity element of $\\mathcal{M}_{G, n}^{\\square}$ is 1, by definition.\n\n  So, if $t_i \\ge -1$ are integers with $t_1 + t_2 = r_1 - r_2 - 2$, and we we take $\\lambda$ such that $(r_1 - \\lambda, \\lambda - 2 - r_2) = (t_1, t_2)$, then we obtain a commutative diagram of $\\mathcal{T}^\\diamondsuit_n$-representations\n  \\[\\begin{tikzcd}\n   \\iota^*(D^{n-\\an}_{G,\\kappa_A}) \\rar \\dar &D^{n-\\an}_{H,-\\tau_A} \\dar[\"\\cong\"] \\\\\n   \\iota^*\\left(V_{G,\\kappa_A^\\vee}\\right) \\rar &V_{H, \\tau_A}\n  \\end{tikzcd}\\]\n  where $\\tau_A^\\vee = (t_1, t_2; r_1 + r_2)$. Hence the homomorphism of Proposition \\ref{prop:krakenpower} is compatible with the classical branching law described in \\cref{sect:branchcoeffs}\n\n\n\\section{Sheaves of distributions}\n\n We use the above function spaces and morphisms as ``models'' for sheaves on the Shimura variety.\n\n \\subsection{Labelling of weights}\n  \\label{ss:analyticweights}\n\n  We recall some definitions from \\bp{\\S 6.2} (this theory is a bit messy owing to the need to reconcile various different conventions).\n\n  As above, we let $(A,A^+)$ be a Tate algebra over $(\\QQ_p,\\ZZ_p)$. Given a weight $\\nu_A : T(\\ZZ_p) \\to A^\\times$ for some coefficient ring $A$, following \\bp{\\S 6}, we define $\\kappa_A: T(\\ZZ_p) \\to A^\\times$ by\n  \\[ \\kappa_A = -w_{0, M} w_1 (\\nu + \\rho) - \\rho. \\]\n  Explicitly, if $\\nu_A$ is $(\\nu_1, \\nu_2; \\omega)$ for some $\\nu_i, \\omega: \\ZZ_p^\\times \\to A^\\times$, then\n  \\[ \\kappa_A = (\\nu_2 - 1, -3-\\nu_1; -\\omega).\\]\n\n  We are not so much interested in the linear dual $\\kappa_A^\\vee$ as the ``Serre dual'' $\\kappa_A' = (\\kappa_A + 2\\rho_{nc})^\\vee$. Explicitly this is $(\\nu_1, -2-\\nu_2; c) = w_1(\\nu_A+ \\rho) - \\rho$. So when $A = \\QQ_p$ and $\\nu = (r_1, r_2; r_1 + r_2)$ is an integral algebraic weight, we have $\\kappa_A' = \\kappa_1$ in the notation of \\eqref{eq:ourweights}.\n\n\n \\subsection{Sheaves on $G$}\n\n  Let $1 \\le n < t$ be integers.\n\n  \\begin{definition}\n   We now define two sheaves $\\mathcal{V}^{n-\\an}_{G,\\nu_A}$ and $\\mathcal{D}^{n-\\an}_{G,\\nu_A}$ over $\\mathcal{U}^G_n$. The former can be defined as a subsheaf of $\\pi_* (\\mathcal{M}^G_{\\HT, n,\\Iw})$ transforming like functions in $V^{n-\\an}_{\\kappa_A}$; an alternative, possibly cleaner description is as a coproduct\n   \\[ \\mathcal{V}^{n-\\an}_{G, \\nu_A} = \\mathcal{M}^G_{\\HT, n,\\Iw} \\times^{\\mathcal{M}^{\\square}_{G, n}} V^{n-\\an}_{G, \\kappa_A},\\]\n   and similarly\n   \\begin{equation}\\label{eq:Dsheafascofibreprod}\n    \\mathcal{D}^{n-\\an}_{G, \\nu_A} = \\mathcal{M}^G_{\\HT, n,\\Iw} \\times^{\\mathcal{M}^{\\square}_{G, n}} D^{n-\\an}_{G, (\\kappa_A + 2\\rho_{nc})}.\n   \\end{equation}\n  \\end{definition}\n\n  (The shift by $2 \\rho_{nc}$ is present so that the pairing between $\\mathcal{D}^{n-\\an}_{G, \\nu_A}$ and $\\mathcal{V}^{n-\\an}_{G, \\nu_A}$ lands in the dualizing sheaf of $\\mathcal{S}_G$, rather than in the structure sheaf.)\n\n  \\begin{lemma}\\label{lem:sheafspec}\n   The sheaves $\\mathcal{V}^{n-\\an}_{G,\\nu_A}$ and $\\mathcal{D}^{n-\\an}_{G,\\nu_A}$ are sheaves of $A$-modules, whose formation is compatible with base-change in $A$; and if $A = \\QQ_p$ and $\\nu_A = (r_1, r_2; c)$ for integers $r_1 \\ge r_2 \\ge -1$, we have classical comparison maps\n   \\[  \\mathcal{V}_{G, \\kappa_A} \\hookrightarrow \\mathcal{V}^{n-\\an}_{G, \\nu_A},\n   \\qquad\n   \\mathcal{D}^{n-\\an}_{G, \\nu_A} \\twoheadrightarrow \\mathcal{V}_{G,(\\kappa_A + 2\\rho_{nc})^\\vee} = \\mathcal{V}_{G, \\kappa_1}. \\]\n  \\end{lemma}\n\n  \\begin{proof}\n   See \\bp{Prop. 6.18}.\n  \\end{proof}\n\n\n \\subsection{Sheaves on $H$}\n\n  There are analogous constructions for sheaves for $H$. Here we use the element $\\mathrm{id} \\in {}^M W_H$ in place of $w_1$, and $w_{0, M_H}$ is the identity. So given an $n$-analytic character $\\tau_A$, we define $\\kappa_A^H = -\\tau_A-2\\rho_H$; and we set\n  \\[  \\mathcal{V}^{n-\\an}_{H, \\diamondsuit, \\tau_A} = \\mathcal{M}^H_{\\HT, n,\\diamondsuit} \\times^{\\mathcal{T}^{\\diamondsuit}_n} V^{n-\\an}_{H, \\kappa_A^H},\\]\n  and\n  \\[ \\mathcal{D}^{n-\\an}_{H, \\diamondsuit, \\tau_A} = \\mathcal{M}^G_{\\HT, n, \\diamondsuit} \\times^{\\mathcal{T}^{\\diamondsuit}_{n}} D^{n-\\an}_{H,(\\kappa_A^H + 2\\rho_H)}.\\]\n\n  Thus $\\mathcal{D}^{n-\\an}_{H,\\diamondsuit,\\tau_A}$ for a $\\QQ_p$-valued algebraic character $\\tau_A$ is simply (the restriction to $\\mathcal{U}_n^H$ of) the line bundle $\\mathcal{V}^H_{\\tau_A}$. The same definitions make sense at Iwahori level, of course, giving line bundles $\\mathcal{D}^{n-\\an}_{H,\\Iw,\\tau_A}$ and $\\mathcal{V}^{n-\\an}_{H,\\Iw,\\tau_A}$. These sheaves are in fact independent of $n$ (in the sense that $\\mathcal{D}^{(n+1)-\\an}_{H,\\Iw,\\tau_A}$ is isomorphic to the restriction of $\\mathcal{D}^{n-\\an}_{H,\\Iw,\\tau_A}$ to $\\mathcal{U}_{n+1}^H$), so we shall frequently drop the $n$ and write simply $\\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}$ etc.\n\n  \\begin{remark}\\label{remark:bigsheafsmallt}\n   Note that (for simplicity) we have only attempted to define the locally-analytic sheaves for $G$ when the level group at $p$ is $\\Iw(p^t)$ with $t > n$; thus our functions are defined on $\\mathcal{B}_n$ itself, rather than on a union of translates of $\\mathcal{B}_n$. (This restriction on the levels is inherited from \\bp{\\S 6.3}.)\n\n   However, for $H$ the technical difficulties disappear, and we can make sense of $\\mathcal{V}^{\\an}_{H,\\Iw,\\tau_A}$ and $\\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}$ as vector bundles on $\\mathcal{U}_{n, \\Iw}^H(p^t)$ for any $n, t \\ge 1$.\n  \\end{remark}\n\n\n \\subsection{Branching for sheaves}\n\n  \\begin{definition}\n   \\label{def:compat}\n   We say the $A$-valued, $n$-analytic characters $\\nu_A$ and $\\tau_A$ of $T(\\ZZ_p)$ are \\emph{compatible} if $\\nu_A = (\\nu_1, \\nu_2; \\nu_1 + \\nu_2)$, $\\tau_A = (\\tau_1, \\tau_2; \\nu_1 + \\nu_2)$, for some characters $\\nu_i, \\tau_i$ of $\\ZZ_p^\\times$, and we have the relation\n   \\[\n    \\tau_1 + \\tau_2 = \\nu_1 - \\nu_2 - 2.\n   \\]\n  \\end{definition}\n\n\n\n  Recall the kraken $\\mathscr{K}^\\lambda$ defined in \\cref{ss:kraken}. If $\\nu_A, \\tau_A$ are compatible, then taking $\\lambda = \\nu_1 - \\tau_1 = \\nu_2 + \\tau_2 + 2$, we obtain a homomorphism of $\\mathcal{T}^\\diamondsuit_n$-representations\n  \\[ D^{n-\\an}_{G,(\\kappa_A + 2\\rho_{nc})}\n   \\longrightarrow\n  \n   D^{n-\\an}_{H,-\\tau_A}\\]\n  where $\\mathcal{T}^{\\diamondsuit}_n$ acts on $D^{n-\\an}_{G,(\\kappa_A + 2\\rho_{nc})}$ via $\\gamma$-conjugation. So the following result is an immediate consequence of \\cref{prop:krakenpower} and the results of \\cref{ss:analyticweights}:\n\n  \\begin{propqed}\\label{prop:krakenonsheaves}\n   Pairing with $\\mathscr{K}^\\lambda$ induces a morphism of sheaves over $\\mathcal{U}^H_{n}$:\n   \\[ \\hat\\iota^*( \\mathcal{D}^{n-\\an}_{G,\\nu_A})\\longrightarrow \\mathcal{D}^{\\an}_{H, \\diamondsuit,\\tau_A}.\\]\n   This morphism is compatible with specialisation in $A$, and if $A = \\QQ_p$ and $\\nu = (r_1, r_2; r_1+r_2)$, $\\tau = (t_1, t_2; r_1 + r_2)$ are algebraic weights with $r_1 - r_2 \\ge 0$ and $r_i, t_i \\ge -1$, then this morphism is compatible with the map of finite-dimensional sheaves $\\hat\\iota^*\\left(\\mathcal{V}_{\\kappa_1}\\right) \\to \\mathcal{V}^H_{\\tau}$ defined in \\S\\ref{sect:branchcoeffs}.\n  \\end{propqed}\n\n\n\n \\subsection{Locally analytic overconvergent cohomology}\n\n  Let $m,n,t$ be as in \\cref{eq:mnt}, with $n > 0$; and suppose $\\nu_A$ is an $n$-analytic $A$-valued character of $T(\\ZZ_p)$. We define cuspidal, locally analytic, overconvergent cohomology to be\n  \\begin{equation}\\label{eq:cusplocanaoc}\n   R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs} = R\\Gamma_{\\mathcal{I}_{mn}^G}\\left(\\mathcal{U}^G_n, \\mathcal{D}^{n-\\an}_{G,\\nu_A}(-D_G)\\right)^{-, \\fs},\n  \\end{equation}\n  and similarly for the non-cuspidal version. As shown in \\bp{\\S 6}, this complex is independent of $m$, $n$ and $t$, and is concentrated in degrees $[0, 1, 2]$.\n\n\n  \\begin{proposition}\n   \\label{prop:bigsheafpullback}\n   Given $\\nu_A$ and $\\tau_A$ satisfying the compatibility condition of Definition \\ref{def:compat}, we have a morphism of complexes of $A$-modules\n   \\[\n    \\hat\\iota^*: R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs}\n    \\to  R\\Gamma_{\\mathcal{Z}^H_m}\\left(\\mathcal{U}_n^H, \\mathcal{D}^{n-\\an}_{H,\\diamondsuit,\\tau_A}(-D_H)\\right).\n   \\]\n  \\end{proposition}\n\n  \\begin{proof}\n   Immediate from \\cref{prop:krakenonsheaves}.\n  \\end{proof}\n\n  We have only defined this morphism for $m, n$ small relative to $t$. However, using \\cref{remark:bigsheafsmallt}, we can argue as in \\cref{sect:changelevel} and define\n  \\[ R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag} = \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_{m, \\Iw}(p^t)}\\left( \\mathcal{U}^H_{n, \\Iw}, \\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}(-D_H)\\right).\\]\n  Then we obtain a natural map\n  \\[R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs} \\to R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag}.\\]\n\n  \\begin{note}\n   By construction, this morphism is compatible with derived base-change in $A$. If $A = \\QQ_p$, and $\\nu_A$ and $\\tau_A$ are algebraic weights such that $r_1 \\ge r_2 \\ge -1$ and $t_1, t_2 \\ge -1$, then this map fits into a commutative diagram with the pullback map on overconvergent cohomology defined in \\S\\ref{sect:changelevel}.\n  \\end{note}\n\n \\subsection{Pairings and duality}\n\n  Dually to the above, we define\n  \\[\n   R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A)^{+, \\dag} = \\varinjlim_m R\\Gamma\\left(\\mathcal{Z}^H_{m, \\Iw}(p^t), \\mathcal{V}^{\\an}_{H,\\Iw, \\tau_A}\\right).\n  \\]\n\n  Note that if $\\tau_A = (t_1, t_2; c)$ then $\\mathcal{V}^{\\an}_{H,\\Iw, \\tau_A}$ is the sheaf $\\mathcal{V}^H_{-\\tau-2\\rho_H} = \\mathcal{V}_{(-2-t_1, -2-t_2; -c)}$, which is the sheaf of modular forms of weight $(t_1 + 2, t_2 + 2)$ (with the normalisation of the central character depending on $c$).\n\n  \\begin{proposition}\n   The above complex is concentrated in degree 0 and independent of $t$. It can be identified with the space of $p$-adic overconvergent modular forms for $H$ of tame level $K^{H, p}$ and weight $\\tau_A + (2, 2)$.\n  \\end{proposition}\n\n  \\begin{proof}\n   For simplicity we suppose $K^{H, p}$ is the principal congruence subgroup of level $N$ for some $N$ (the general case reduces easily to this). Then the Shimura variety for $H$ is simply the fibre product (over $\\mu_N$) of two copies of the level $N$ modular curve parametrising elliptic curves with full level $N$ structure and a cyclic subgroup of order $p^t$. Then $\\pi_{HT}^{-1}(\\{\\mathrm{id}_H\\})$ is the ``canonical locus'', where the $p$-subgroups are both multiplicative; and the $\\mathcal{Z}_{m, \\Iw}^H$ are a cofinal family of neighbourhoods of this locus. Via the theory of the canonical subgroup, this space is independent of the choice of levels.\n\n   Since the canonical locus is affinoid (and sufficiently small strict neighbourhoods of it also have this property), its cohomology vanishes above degree 0, and the degree 0 cohomology identifies with overconvergent sections of $\\mathcal{V}^{\\an}_{H, \\Iw, \\tau_A}$. If we choose an extension $\\tilde\\tau_A$ of $\\tau_A$ to the maximal torus of $\\GL_2 \\times \\GL_2$, then $\\mathcal{V}^{\\an}_{H, \\Iw, \\tau_A}$ decomposes as the product of two copies of the corresponding sheaves on the individual modular curves. This is precisely the construction of overconvergent modular forms described in \\cite{pilloni13} (see the discussion following Prop 6.2 of \\emph{op.cit.} for a comparison with Coleman's original approach).\n  \\end{proof}\n\n  \\begin{theorem}[c.f. \\bp{Theorem 6.38}]\n   \\label{thm:bspairing}\n   The cup product induces a pairing\n   \\[ H^2_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag} \\times H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A)^{+, \\dag} \\longrightarrow A, \\]\n   whose formation is compatible with base-change in $A$, and which is compatible with the Serre duality pairing on classical cohomology when $A = \\QQ_p$ and $\\nu$, $\\tau$ are classical weights.\n  \\end{theorem}\n\n  \\begin{proof}\n   We define this pairing by combining the pullback map of \\ref{prop:bigsheafpullback} with the pairing between the cohomology groups $H^2_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag}$ and $H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{+, \\dag}$. By construction, this is compatible with Serre duality for each classical weight.\n  \\end{proof}\n\n\n\n\n\n\n\\section{Construction of the $p$-adic $L$-function}\n\n Let $L$ be a finite extension of $\\QQ_p$.\n\n \\subsection{Families of Eisenstein series}\n\n  We refer to \\cite[\\S 7]{LPSZ1} for the construction of $p$-adic families of Eisenstein series $\\mathcal{E}^{\\Phi^{(p)}}(\\kappa_1, \\kappa_2; \\chi^{(p)})$, depending on a prime-to-$p$ Schwartz function $\\Phi^{(p)}$ and prime-to-$p$ Dirichlet character $\\chi^{(p)}$ (both valued in $L$) and a pair of characters $\\kappa_1, \\kappa_2$ of $\\ZZ_p^\\times$ (valued in some $p$-adically complete $L$-algebra $A$).\n\n  \\begin{note}\n   Note that this Eisenstein series is $p$-depleted, i.e.~lies in the kernel of $U_p$; and it is zero on any components of $\\Spec(A)$ which do not satisfy the parity condition $\\kappa_1(-1) \\kappa_2(-1) = -\\chi^{(p)}(-1)$.\n\n   The construction factors through the projection of $\\Phi^{(p)}$ to the eigenspace where $\\stbt{a}{0}{0}{a}$ for $a \\in \\widehat{\\mathbf{Z}}^{(p)}$ acts as $\\widehat{\\chi}^{(p)}(a)^{-1}$, where $\\widehat{\\chi}^{(p)}$ is the adelic character attached to $\\chi^{(p)}$ as in \\cite[\\S 2.2]{LPSZ1}. We shall henceforth assume, without loss of generality, that $\\Phi^{(p)}$ lies in this eigenspace; thus $\\chi^{(p)}$ is uniquely determined by $\\Phi^{(p)}$ and we sometimes drop it from the notation.\n  \\end{note}\n\n  \\begin{proposition}\n   If $A$ is an affinoid algebra, and one of the $\\kappa_i$ is a finite-order character, then $\\mathcal{E}^{\\Phi^{p}}(\\kappa_1, \\kappa_2)$ is an overconvergent $A$-valued cusp form of weight-character $1 + \\kappa_1 + \\kappa_2$.\n  \\end{proposition}\n\n  \\begin{proof}\n   Since twisting by a finite-order character preserves overconvergence, it suffices to assume $\\kappa_1$ or $\\kappa_2$ is 0. Then our $p$-adic Eisenstein series is the $p$-depletion of a family of \\emph{ordinary} Eisenstein series, cf.~\\cite[\\S 2.3]{ohta99}, and it is well-known that these ordinary Eisenstein series are overconvergent (indeed, this is true by definition in Coleman's approach to overconvergent modular forms).\n  \\end{proof}\n\n  As noted in \\emph{op.cit.}, for $k \\ge 1$, the Eisenstein series $F^{k}_{\\Phi^p \\Phi_{\\mathrm{dep}}}$ described in \\cite[\\S 4.3]{LZ20b-regulator} is (the classical form associated to) $\\mathcal{E}^{\\Phi^p}(k-1, 0)$, and $E^{k}_{\\Phi^p \\Phi_{\\mathrm{dep}}}$ is $\\mathcal{E}^{\\Phi^p}(0, k-1)$. It also implies the following relation:\n\n  \\begin{proposition}[cf.~{\\cite[Prop 16.2.1]{LZ20b-regulator}}]\n   Let $t \\in \\mathbf{Z}_{\\ge 0}$. As overconvergent cusp forms of weight $-t$, we have\n   \\[ \\theta^{-(1+t)}\\left(F^{(t+2)}_{\\Phi^p \\Phi_{\\mathrm{dep}}}\\right) = \\mathcal{E}^{\\Phi^p}(0, -1-t; \\Phi^{(p)}), \\]\n   where $\\theta = q \\tfrac{\\mathrm{d}}{\\mathrm{d}q}$ is the Serre differential operator.\n  \\end{proposition}\n\n \\subsection{Tame test data}\n\n  We fix the following data:\n  \\begin{itemize}\n   \\item $M_0, N_0$ are positive integers coprime to $p$ with $M_0^2 \\mid N_0$, and $\\chi_0$ is a Dirichlet character of conductor $M_0$ (valued in $L$).\n   \\item $M_2, N_2$ are positive integers coprime to $p$ with $M_2 \\mid N_2$, and $\\chi_2$ is a Dirichlet character of conductor $M_2$ (valued in $L$).\n  \\end{itemize}\n  We will consider automorphic representations $\\pi$ of $G$ with conductor $N_0$ and character $\\widehat{\\chi}_0$ up to twists by norm, and similarly $\\sigma$ of $\\GL_2$ with conductor $N_2$ and character $\\widehat{\\chi}_2$ up to twists by norm.\\footnote{This numbering of the parameters comes from the fact that the zeta-integral computations of \\cite{LPSZ1} are simpler to write down if the Eisenstein series lives on the first factor of $H$.}\n\n  Let $S$ denote the set of primes dividing $N_0 N_2$. By \\emph{tame test data} we shall mean a pair $\\gamma_S = (\\gamma_{0, S}, \\Phi_S)$, where:\n  \\begin{itemize}\n  \\item $\\gamma_{0, S} \\in G(\\mathbf{Q}_S)$, where $\\mathbf{Q}_S = \\prod_{\\ell \\in S} \\mathbf{Q}_\\ell$;\n  \\item $\\Phi_S \\in C^\\infty_c(\\mathbf{Q}_S^2, L)$, lying in the $\\left(\\widehat{\\chi}_0 \\widehat{\\chi}_2\\right)^{-1}$-eigenspace for $\\mathbf{Z}_S^\\times$.\n  \\end{itemize}\n  We let $K_S$ be the quasi-paramodular subgroup of $G(\\mathbf{Q}_S)$ of level $(N_0, M_0)$; and we let $\\widehat{K}_S$ be some open compact subgroup of $G(\\mathbf{Q}_S)$ such that:\n  \\begin{itemize}\n   \\item $\\widehat{K}_S \\subseteq \\gamma_{0, S} K_S \\gamma_{0, S}^{-1}$,\n   \\item the projection of $\\widehat{K}_S \\cap H$ to the first factor of $H$ acts trivially on $\\Phi_S$,\n   \\item  the projection of $\\widehat{K}_S \\cap H$ to the second factor of $H$ is contained in $\\{ \\stbt \\star\\star0 1 \\bmod N_2\\}$.\n  \\end{itemize}\n  We define $K^p$ and $\\widehat{K}^p$ to be the products of $K_S$ and $\\widehat{K}_S$ with $G(\\AA_{\\mathrm{f}}^{pS})$, and $\\Phi^{(p)} = \\Phi_S \\cdot \\operatorname{ch}\\left((\\widehat{\\mathbf{Z}}^{S \\cup \\{p\\}})^2\\right)$.\n\n \\subsection{The correction term $Z_S$}\n\n  Let $\\pi$ and $\\sigma$ be cohomological cuspidal automorphic representations of $G$ and of $\\GL_2$, both defined over some number field $E$ contained in the $p$-adic field $L$, and both globally generic and unramified outside $S$. We normalise so these are cohomological with weights $(r_1, r_2; r_1 + r_2)$ and $(t_2; t_2)$ respectively, for some integers $r_1, r_2, t_2$; and we let $\\Pi$ and $\\Sigma$ be the unitary twists of $\\pi$ and $\\sigma$ respectively, so that\n  \\[ L(\\Pi \\times \\Sigma, s) = L(\\pi \\times \\sigma, s + \\tfrac{r_1 + r_2 + t_2}{2}).\\]\n\n  \\begin{definition}\n   For $W_0 \\in \\mathcal{W}(\\pi)_E$, $W_2 \\in \\mathcal{W}(\\sigma)_E$, and $\\Phi \\in \\mathcal{S}(\\mathbf{Q}_S^2, E)$, we consider the zeta-integral\n   \\[ Z(W_0, \\Phi, W_2; s) = \\int_{(Z_G N_H \\backslash H)(\\mathbf{Q}_S)} W_0(h) f^{\\Phi}(h_1; \\omega_{\\pi}\\omega_\\sigma, s) W_2(h_2) \\, \\mathrm{d}h. \\]\n  \\end{definition}\n\n  We shall set\n  \\[\n   Z_S(\\pi \\times \\sigma, \\gamma_S; s) =\n    \\frac{Z(\\gamma_{0, S} \\cdot W_0^{\\mathrm{new}}, \\Phi_S, W_2^\\mathrm{new}; s)}{G(\\chi_2^{-1})\\prod_{\\ell \\in S} L(\\pi_\\ell \\times \\sigma_\\ell, s)},\n  \\]\n  and\n  \\[ Z_S(\\pi \\times \\sigma, \\gamma_S) = Z_S(\\pi \\times \\sigma, \\gamma_S; 1 + \\tfrac{t_1}{2})\\]\n  where $t_1 = r_1 -r_2-2-t_2$ as usual. Here $G(\\chi) = \\sum_{a \\bmod N_\\chi} \\chi(a) \\exp(2\\pi i a \/ N_\\chi)$ is the Gauss sum of the character $\\chi$. One can check that this is a product of polynomials in the variables $\\ell^{\\pm s}$, for $\\ell \\in S$, with coefficients in $E$.\n\n  \\begin{proposition}\n   For any given $\\pi, \\sigma$, one can choose $\\gamma_S$ such that $Z_S(\\pi \\times \\sigma, \\gamma_S; s) \\ne 0$.\n  \\end{proposition}\n\n  \\begin{proof}\n   This follows from the definition of the $L$-factor as a GCD of local zeta-integrals.\n  \\end{proof}\n\n \\subsection{P-adic families for $G$}\n\n  Let $U \\subset \\mathcal{W}^2$ be an open affinoid disc; and let $\\mathbf{r}_1$, $\\mathbf{r}_2: \\ZZ_p^\\times \\to \\mathcal{O}(U)^\\times$ be the universal characters associated to the two factors of $\\mathcal{W}^2$. Let $\\nu_U$ be the character $(\\mathbf{r}_1, \\mathbf{r}_2; \\mathbf{r}_1+\\mathbf{r}_2)$ of $T(\\ZZ_p)$.\n\n  The theory of \\cite{boxerpilloni20} shows that there exists a rigid space $\\mathcal{E} \\xrightarrow{\\kappa} \\mathcal{W}^2$, with a map $\\mathbb{T}^- \\to \\mathcal{O}(\\mathcal{E})$ (the eigenvariety for $G$), and graded coherent sheaves $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$ on $\\mathcal{E}$ for $0 \\le j, k \\le 3$, whose pushforward to any affinoid $U\\subset \\mathcal{W}^2$ as above is $H^k_{w_j,\\an}(K^p, \\nu_U, \\cusp)^{(-,\\fs)}$. By construction, the points of $\\mathcal{E}$ biject with systems of $\\mathbb{T}^-$-eigenvalues appearing in one of these modules.\n\n  \\begin{definition}\n  By a \\emph{family of automorphic representations} $\\underline{\\pi}$ over $U$ (of tame level $N_0$ and character $\\chi_0$), we mean the data of a finite flat covering $\\tilde{U} \\to U$, and a homomorphism $\\tilde{U} \\to \\mathcal{E}$ lifting the inclusion $U \\hookrightarrow \\mathcal{W}$, such that the following conditions hold:\n  \\begin{itemize}\n  \\item $\\tilde{U}$ is 2-dimensional and smooth;\n  \\item the restriction of the sheaf $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$ to $\\tilde{U}$ is zero if $k \\ne 3-j$, and the sheaves $S^k(\\underline{\\pi}) = H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_{3-k}})$ are either free over $\\mathcal{O}(\\tilde{U})$ of rank 1 for all $k$ (a general-type family), or free of rank 1 for $k = 1,2$ and zero for $k = 0, 3$ (a Yoshida-type family);\n  \\item the centre of $G(\\AA_{\\mathrm{f}}^p)$ acts on the modules $S^k(\\underline{\\pi})$ by the character $|\\cdot|^{-(\\mathbf{r}_1 + \\mathbf{r}_2)} \\widehat{\\chi}_0$.\n  \\end{itemize}\n  \\end{definition}\n\n  Such a family determines a $\\mathcal{O}(\\tilde{U})$-valued character $\\lambda_{\\underline{\\pi}}^-$ of $\\mathbb{T}^-$, which is the system of eigenvalues by which $\\mathbb{T}$ acts on the modules $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$; conversely, the character $\\lambda_{\\underline{\\pi}}^-$ and ring extension $\\mathcal{O}(\\tilde U)$ of $\\mathcal{O}(U)$ uniquely determine $\\underline{\\pi}$.\n\n  \\begin{definition}\n  We say a point $P \\in \\tilde{U}(L)$ is ``good for $\\underline{\\pi}$'' if the following conditions hold:\n  \\begin{itemize}\n   \\item the weight of $P$ is $(r_1, r_2) \\in U \\cap \\mathbf{Z}^2$ with $r_1 \\ge r_2 \\ge -1$;\n   \\item the specialisation at $P$ of the system of eigenvalues $\\lambda^-_{\\underline{\\pi}}$ is the character of $\\mathbb{T}^-$ assocated to a $p$-stabilised automorphic representation $\\pi_P$, which is cuspidal, globally generic, and has conductor $N_0$ and character $\\chi_0$;\n   \\item the fibre of $S^2(\\underline{\\pi})$ at $P$ maps isomorphically to the $\\pi_P$-eigenspace in the classical $H^2(K^p, \\kappa_1(\\nu), \\cusp)$; in particular, this eigenspace is 1-dimensional.\n  \\end{itemize}\n  \\end{definition}\n\n  \\begin{remark}\n   Note that we do not suppose that the $\\pi_P$ \\emph{generalised} eigenspace be 1-dimensional, and this will not hold when $\\tilde{U} \\to U$ is ramified at $P$.\n  \\end{remark}\n\n  By the classicity theorems for higher Coleman theory recalled above, given a family $\\underline{\\pi}$, all specialisations of integer weight $(r_1, r_2)$ with $r_1 - r_2$ and $r_2$ sufficiently large relative to the slope of $\\underline{\\pi}$ will be good; and if $\\underline{\\pi}$ is ordinary, it suffices to assume that $r_1-r_2 \\ge 3$ and $r_2 \\ge 0$.\n\n  We shall choose a basis $\\underline{\\eta}$ of $S^2(\\underline{\\pi})$. Since the spaces of higher Coleman theory (of varying levels) have an action of $G(\\AA_{\\mathrm{f}}^p)$, we can make sense of $\\gamma_{0, S} \\cdot \\underline{\\eta}$ as a family of classes at tame level $\\widehat{K}^p$, which is still an eigenfamily for the Hecke operators away from $S$.\n\n \\subsection{Families for $\\GL_2$}\n\n  Similarly, we choose a disc $U' \\subset \\mathcal{W}$, a finite flat covering $\\tilde{U}'\\to U'$ with $\\tilde{U}'$ smooth, and a finite-slope overconvergent $p$-adic family of modular eigenforms $\\mathcal{G}$ over $\\tilde{U}'$ (of weight $\\mathbf{t_2} + 2$ where $\\mathbf{t_2}$ is the universal character associated to $U'$). We suppose that this family is new away from $p$ of tame level $N_2$, and nebentype character $\\chi_2$.\n\n  We say a point $Q \\in \\tilde{U}'$ is ``good for $\\mathcal{G}$'' if it lies above an integer $t \\in U' \\cap \\mathbf{Z}_{\\ge -1}$, and the specialisation of $\\mathcal{G}$ at $Q$, which is \\emph{a priori} an overconvergent form of weight $t + 2$, is in fact a classical form. (This is automatic if $t$ is sufficiently large compared to the slope of $\\mathcal{G}$.) We write $\\sigma_t$ for the corresponding automorphic representation (normalised to have central character $|\\cdot|^{-t} \\widehat{\\chi}_2$); and we formally write $\\underline{\\sigma}$ for the collection of the $\\sigma_t$ for varying $t$.\n\n  A mildly irritating detail is that if $\\mathcal{G}$ is normalised to have $a_1(\\mathcal{G}) = 1$, and $t$ is a good specialisation, then $\\mathcal{G}_t$ has $q$-expansion coefficients in some number field $E$; but the modular form $\\mathcal{G}_t$ is not defined over $E$ as a coherent cohomology class, since the cusp $\\infty$ on $X_1(N)$ is not defined over $\\mathbf{Q}$ (with our conventions). However, the class $G(\\chi_2^{-1})\\mathcal{G}_t$ is $E$-rational. We write $S^0(\\sigma_t, E)$ for the $E$-vector space spanned by this form, and similarly $S^0(\\underline{\\sigma})$ for the $\\mathcal{O}(\\tilde{U}')$-module of overconvergent cusp forms generated by $G(\\chi_2^{-1}) \\mathcal{G}$.\n\n  \\begin{remark}\n   Note that by definition $S^0(\\underline{\\sigma})$ is free of rank 1, and its fibre at any good specialisation is in the image of the classical $H^0$ (because of the $q$-expansion principle for $p$-adic modular forms). Hence we do not need any auxiliary hypotheses about local freeness of sheaves.\n  \\end{remark}\n\n \\subsection{Deforming eigenforms}\n\n  Conversely, we say a classical ($p$-stabilised) automorphic representation $\\pi$, of some weight $\\nu$, is \\emph{deformable} if we can find a disc $U$ containing $\\nu$, a family $\\underline{\\pi}$ over some covering $\\tilde{U} \/ U$, and some $Q \\in \\tilde{U}$ above $\\nu$, such that $Q$ is good for $\\underline{\\pi}$ and the specialisation there is $\\pi$. The arguments of \\cref{sect:families} show that any generic $\\pi$ of cohomological weight, with a regular $p$-stabilisation of sufficiently small slope, will be deformable in the above sense (and we may suppose that $\\tilde{U}=U$); again, if $\\pi$ is ordinary, it suffices to suppose that $r_1 - r_2 \\ge 3$ and $r_2 \\ge 0$.\n\n  For $\\GL_2$ we are in much better shape (partly because $\\GL_2$ is better understood than $\\GSp_4$, and partly because our definition of ``family'' is less restrictive): any classical $p$-stabilised newform of integer weight and Iwahori level at $p$ will be deformable, even in the worst-case scenario of non-$p$-regular weight 1 forms, since we may take $\\tilde{U}'$ to be a neighbourhood of $\\sigma$ in the normalisation of the eigencurve. Moreover, if $\\sigma$ is ordinary and has weight $\\ge 2$, we may suppose $\\tilde{U}' = U'$.\n\n  \\begin{remark}\n   We also expect that there exist interesting examples of deformable $\\pi$ for $G$ which do not satisfy these stringent conditions. It seems likely that the extra generality of a finite flat covering of weight space will be genuinely necessary, at least in the non-regular-weight case $r_2 = -1$. However, for simplicity of notation we shall assume $\\tilde{U} = U$ and $\\tilde{U}' = U'$ henceforth; extending these arguments to the general case is straightforward and we leave this to the reader.\n  \\end{remark}\n\n\n  \\subsubsection*{Families over $U \\times U'$} Let $A = \\mathcal{O}(U \\times U')$. We have two canonical $A$-valued characters of $T(\\ZZ_p)$: the canonical character $\\nu_A = (\\mathbf{r}_1, \\mathbf{r}_2; \\mathbf{r}_1 + \\mathbf{r}_2)$, and the character $\\tau_A = (\\mathbf{t}_1, \\mathbf{t}_2; \\mathbf{r}_1 + \\mathbf{r}_2)$ defined as follows: $\\mathbf{t}_2$ is the canonical character of $U'$ as above, and $\\mathbf{t}_1 = \\mathbf{r}_1 - \\mathbf{r}_2 - 2 - \\mathbf{t}_2$ and the action of the centre are determined by the requirement that $\\nu_A$ and $\\tau_A$ be ``compatible'' in the sense of \\cref{def:compat}. Then we can consider\n  \\[ \\mathcal{E}^{\\Phi^{(p)}}(0, \\mathbf{t}_1 + 1) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]} \\in H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^2), \\tau_{A})^{+, \\dag},\\]\n  where the tame level is taken to be $H \\cap \\widehat{K}^p$.\n\n  \\begin{definition}\n   We let $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})$ denote the element of $A$ defined by\n   \\[ \\left\\langle \\hat\\iota^*\\left(\\gamma_{0, S} \\cdot \\underline{\\eta}\\right), \\mathcal{E}^{\\Phi^{(p)}}(0, \\mathbf{t}_1 + 1) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]}\\right\\rangle.\\]\n  \\end{definition}\n\n  (The product denotes the Serre duality pairing at level $\\widehat{K}^p \\cap H$, normalised by a factor $\\operatorname{vol}(\\widehat{K}^p \\cap H)$ in order to make it independent of the choice of $\\widehat{K}_p$.)\n  \\begin{definition} \\\n   \\begin{itemize}\n    \\item We say a point $(P, Q)$ of $U \\times U'$ is \\emph{good} if $P = (r_1, r_2)$ and $Q = (t_2)$ are integer points, with $P$ good for $\\underline{\\pi}$ and $Q$ good for $\\underline{\\sigma}$.\n\n    \\item We say $(P, Q)$ is \\emph{good critical} if we also have $t_2 \\le r_1 -r_2 - 1$ (i.e.~the specialisation $t_1$ of $\\mathbf{t}_1$ at $(P, Q)$ is $\\ge -1$).\n\n    \\item If instead we have $r_1 - r_2 \\le t_2 \\le r_1$, we say $P$ is \\emph{good geometric}.\n   \\end{itemize}\n  \\end{definition}\n\n  One checks easily that any integer point $(r_1, r_2, t_2)$ is the limit of a sequence of good geometric (or good critical) points, so if we exclude the pathological case when $(U \\times U') \\cap \\mathbf{Z}^3$ is empty, then the sets of good critical points and of good geometric points are both Zariski-dense in $U \\times U'$.\n\n \\subsection{Values in the critical range}\n\n  \\begin{definition}\n   For $(P, Q) = (r_1, r_2, t_2) \\in U \\times U'$ a good critical point, we define a degree 8 Euler factor\n   \\[\n    \\mathcal{E}_p(\\pi_P \\times \\sigma_Q) = \\left(1 - \\tfrac{p^{r_1 + 1}}{\\alpha \\mathfrak{a}}\\right)\\dots \\left( 1- \\tfrac{p^{r_1 + 1}}{\\beta \\mathfrak{b}}\\right) \\left(1 - \\tfrac{\\gamma \\mathfrak{a}}{p^{r_1 + 2}}\\right)\\dots\\left(1 - \\tfrac{\\delta \\mathfrak{b}}{p^{r_1 + 2}}\\right).\n   \\]\n   where $\\alpha, \\dots, \\delta$ are the Hecke parameters of $\\pi_P$, and $\\mathfrak{a},\\mathfrak{b}$ the Hecke parameters of $\\sigma_Q$ (so that $\\mathfrak{a}\\mathfrak{b} = p^{t_2 + 1} \\chi_2(p)$).\n  \\end{definition}\n\n  \\begin{proposition}\n   If $\\pi_P$ is ordinary, then $\\mathcal{E}_p(\\pi_P\\times \\sigma_Q) \\ne 0$.\n  \\end{proposition}\n\n  \\begin{proof}\n   This follows by a (somewhat tedious) explicit check from the bounds on the valuations of the Hecke parameters.\n  \\end{proof}\n\n  \\begin{theorem}\n   The $p$-adic $L$-function $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})$ has the following interpolation property: if $(P, Q)$ is good critical, then\n   \\[ \\frac{\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})(P, Q)}{\\Omega_p(\\pi_P, \\eta_P)}\n   =Z_S(\\pi_{P} \\times \\sigma_{Q}, \\gamma_S) \\cdot \\mathcal{E}_p(\\pi_P \\times \\sigma_Q) \\cdot\n   \\frac{G(\\chi_2^{-1})^2 \\Lambda\\left(\\Pi_P \\times \\Sigma_Q, 1 + \\tfrac{t_1}{2}\\right)}{\\Omega_\\infty(\\pi_P, \\eta_P)}, \\]\n   with both sides lying in the field of rationality of $\\pi_P\\times \\sigma_Q$.\n\n   Here $\\Pi_P$ and $\\Sigma_Q$ are the (unitary) automorphic representations generated by the specialisations of $\\underline{\\eta}$ and $\\mathcal{G}$ at $P$; and $\\Lambda(\\Pi_P\\times \\Sigma_Q, s)$ denotes the $L$-function of these automorphic representations, with its archimedean $\\Gamma$-factors included.\n  \\end{theorem}\n\n  \\begin{remark}\n   Note that $s = 1 + \\tfrac{t_1}{2}$ is the upper endpoint of the interval of critical values (in the sense of Deligne) for the degree 8 $L$-function $L\\left(\\Pi_P \\times \\Sigma_Q, s\\right)$. This critical interval is symmetric about $s = \\tfrac{1}{2}$, so unless $t_2 = r_1 - r_2 - 1$ (so that $t_1 = -1$), there are other critical values which we do not see by this method.\n\n   We optimistically hope that there should be a $p$-adic $L$-function on the 4-dimensional space $U \\times U' \\times \\mathcal{W}$ which interpolates the full range of critical values, and that both the above $p$-adic $L$-function on $U \\times U'$, and the 2-variable $p$-adic $L$-function on $U' \\times \\mathcal{W}$ (for fixed $\\pi$) considered in \\cite[\\S 5]{LZ20b-regulator}, should be ``slices'' of this more general construction. However, this seems beyond reach with our present methods.\n  \\end{remark}\n\n  \\begin{proof}\n   By construction, we have\n   \\[ \\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q) = G(\\chi_2^{-1}) \\left\\langle \\hat\\iota^*\\left(\\gamma_{0, S} \\cdot \\eta_P\\right), \\mathcal{E}^{\\Phi^{(p)}}(0, t_1 + 1) \\boxtimes \\mathcal{G}_P^{[p]}\\right\\rangle. \\]\n   This expands as the product of $G(\\chi_2^{-1})\\Lambda\\left(\\Pi_P \\times \\Sigma_Q, 1 + \\tfrac{t_1}{2}\\right)$ and a product of normalised local zeta-integrals, exactly as in \\cite{LPSZ1}. The local integrals away from $pS$ are all 1. The local zeta-integral at $p$ is evaluated in \\cite{LZ21-zeta2}, and gives the Euler factor $\\mathcal{E}_p(-)$. The product of zeta-integrals at the bad primes is by definition $G(\\chi_2^{-1}) Z_S(\\dots)$ and the result follows.\n  \\end{proof}\n\n \\subsection{Values in the geometric range}\n\n  Suppose $(P, Q)\\in U \\times U'$ is a point in the good geometric range; and let us set $t_1' = -2-t_1 = t_2 - r_1 + r_2$. Then the ``geometric'' condition implies that $0 \\le t_1' \\le r_2$, and the quadruple $(r_1, r_2, t_1', t_2)$ satisfies the branching law for algebraic representations defined in \\cite[Proposition 6.4]{LPSZ1}, which is the condition needed to define motivic cohomology classes associated to $\\pi_P \\otimes \\sigma_Q$, using the pushforward of a $\\GL_2$ Eisenstein class of weight $t_1'$ (see \\cite{HJS20}).\n\n  \\begin{remark}\n   Note that this Euler system class lands in the Galois representation $V_p(\\pi \\times \\sigma)^*(-1-r_1)$, and corresponds to the complex $L$-function $L(\\pi \\times \\sigma, s)$ at $s = -\\tfrac{t_1'}{2} = 1 + \\tfrac{t_1}{2}$; but this is no longer a critical value, and the Archimedean $\\Gamma$-factors force the $L$-function to vanish here to degree exactly one (except in some exceptional cases when $t_1' = 0$ and $\\pi$ is a Yoshida lift, when it can happen that the completed $L$-function has a simple pole at $s = 0, 1$).\n\n   The $L$-values having this property are an interval (disjoint from the critical interval, if any) and the value $s = -\\tfrac{t_1'}{2}$ is the \\emph{upper} end of this interval. So our restriction to using only overconvergent, rather than nearly-overconvergent, Eisenstein series pegs us to the the upper endpoint of the critical interval when $P$ is critical, and to the upper endpoint of the geometric interval when $P$ is geometric.\n  \\end{remark}\n\n  In \\cite[\\S 4]{LZ20b-regulator}, we defined an object $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)$ associated to $\\pi_P \\times \\sigma_Q$, the choice of twist $t_1'$, and the basis vector $\\eta_P \\in S^2(\\pi_P, L)$. This was an $H(\\AA_{\\mathrm{f}}^p)$-equivariant map $\\mathcal{T}^p \\to L$, where $\\mathcal{T}^p = \\mathcal{W}(\\pi_{P, \\mathrm{f}}^p) \\otimes C^\\infty_c( (\\AA_{\\mathrm{f}}^p)^2) \\otimes \\mathcal{W}(\\sigma_{Q, \\mathrm{f}}^p)$. Our choice of $\\gamma_S$ defines a choice of vector\n  \\[ (\\gamma_{0, S} W^{\\mathrm{new}}_{\\pi_P}) \\otimes \\Phi_S \\otimes W^{\\mathrm{new}}_{\\sigma_Q} \\in \\mathcal{T}^p \\]\n  and we write $\\operatorname{Per}_{\\eta}(\\pi \\times \\sigma, \\gamma_S) \\in L$ for the value of $\\operatorname{Per}_{\\eta}(\\pi \\times \\sigma)$ on this vector.\n\n  \\begin{remark}\n   If $t_1 \\ne 0$, then one can check that the space of $H(\\AA_{\\mathrm{f}}^p)$-equivariant maps in which $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)$ lies is in fact 1-dimensional and spanned by the product of zeta integrals used to define $Z_S(\\dots)$. It follows that there is a quantity $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)^{\\mathrm{univ}} \\in L$ such that for all $\\gamma_S$ we have\n   \\[  \\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q, \\gamma_S) = Z_S(\\pi_{P} \\times \\sigma_{Q}, \\gamma_S) \\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)^{\\mathrm{univ}}. \\]\n   Similar results also hold for $t_1 = 0$ under some mild additional conditions on $\\pi_P$ and $\\sigma_Q$; compare Theorem 6.6.2 of \\cite{LZ20} in the $\\GSp_4$ case. However, we do not need this for the proof of our main theorem, so we shall not pursue it further here.\n  \\end{remark}\n\n  \\begin{proposition}\n   \\label{prop:coherentperiod}\n   We have\n   \\[\n    \\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q) =  \\operatorname{Per}_{\\eta_P}(\\pi_P \\times \\sigma_Q, \\gamma_S).\n   \\]\n  \\end{proposition}\n\n  \\begin{proof}\n   By construction, we have\n   \\[\n    \\operatorname{Per}_{\\eta_P}(\\pi_P \\times \\sigma_Q, \\gamma_S) = \\left\\langle \\iota^*_{\\Kl}\\left(\\gamma_{0, S}\\cdot \\eta_{P, \\Kl}\\right), \\theta^{-{t_1'}}\\left(F^{(t_1' + 2)}_{\\Phi^p \\Phi_{\\mathrm{dep}}}\\right) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]}\\right\\rangle, \\]\n   where $\\iota_{\\Kl}$ is an embedding of Shimura varieties at Klingen level. The term on the right-hand side is exactly the specialisation at $P$ of our family of $p$-adic modular forms for $H$. From the zeta-integral computations of \\cite{LZ21-zeta2}, we may replace $\\iota^*_{\\Kl}\\left(\\eta_{\\Kl}\\right)$ with $\\hat\\iota^*\\left(\\eta_{\\Iw}\\right)$ without changing the value of the pairing.\n  \\end{proof}\n\n\n\n\\section{Families of cohomology classes}\n\n We persist with the notation and assumptions of the previous section. We also suppose that the family $\\underline{\\pi}$ is not of Yoshida type, so that for each classical specialisation $P$, the $\\lambda_{P}$-eigenspace in \\'etale cohomology of $S_{G, K^p\\Iw(p)}$ is 4-dimensional. We suppose furthermore that $\\underline{\\pi}$ and the $\\GL_2$ family $\\underline{\\sigma}$ are ordinary at $p$.\n\n \\subsection{Galois representations}\n\n  Associated with the family $\\underline{\\pi}$ we have a family of Galois representations $V(\\underline{\\pi})$, which is a rank 4 $\\mathcal{O}(U)$-module with an action of $\\Gal(\\overline{\\mathbf{Q}}\/\\mathbf{Q})$, unramified outside $pN_0$ and satisfying $\\operatorname{tr}(\\operatorname{Frob}_\\ell^{-1} | V(\\underline{\\pi})) = \\lambda(T_{1,\\ell})$ for $\\ell \\nmid pN_0$.\n\n  The existence of this family is a consequence of the results of \\cite{tilouineurban99}, who also give a canonical realisation of the dual representation $V(\\underline{\\pi})^*$ as a localisation of the module\n  \\[ e'_B \\cdot \\varprojlim_t H^3_{\\text{\\textup{\\'et}}, c}\\left(\\mathcal{S}_{G, K^pK_{p, t}, \\overline{\\mathbf{Q}}}, \\ZZ_p\\right) \\otimes_{\\ZZ_p[[\\ZZ_p^{\\times 2}]]} \\mathcal{O}(U), \\]\n  where $K_{p, t}$ is some family of subgroups of $G(\\ZZ_p)$ and $e^-_B$ is the ordinary projector associated to $\\mathcal{U}'_B$. Similarly, there is a 2-dimensional family of Galois representations over $U'$ associated to $\\underline{\\sigma}$.\n\n  \\begin{remark}\n   If the family $\\underline{\\pi}$ has a classical specialisation whose weight is sufficiently regular, but small relative to $p$ (and some additional hypotheses hold regarding the image of the residual Galois representation), then the results of \\cite{mokranetilouine02} and \\cite{rockwood-control} imply that $V(\\underline{\\pi})$ is free of rank 4 over $\\mathcal{O}(U)$.\n\n   Without this condition, we can only deduce that $V(\\underline{\\pi})$ is locally free in a neighbourhood of each good \\emph{cohomological} weight, but not necessarily elsewhere. One can work around this by replacing $V(\\underline{\\pi})^*$ with its double dual (reflexive hull), which does not change its specialisations in cohomological weights.\n  \\end{remark}\n\n\n  \\begin{definition}\n   We set\n   \\[\\mathbb{V}^* = V(\\underline{\\pi})^* \\times V(\\underline{\\sigma})^*(-1-\\mathbf{r}_1),\\]\n   which is an 8-dimensional family of Galois representations over $U \\times U'$.\n  \\end{definition}\n\n \\subsection{Ordinary filtrations at $p$}\n\n  The Galois representation $V(\\underline{\\pi})$ has a decreasing filtration by $\\mathcal{O}(U)$-submodules stable under $\\Gal(\\overline{\\mathbf{Q}}_p\/\\QQ_p)$ (via results of Urban \\cite{urban05}; see \\cite[Theorem 17.3.1]{LZ20} for the formulation we use). We write $\\mathcal{F}^i V(\\underline{\\pi})$ for the codimension $i$ subspace, and similarly for its dual $V(\\underline{\\pi})^*$. Note that $\\Gr^0 V(\\underline{\\pi})^*$ is unramified, with arithmetic Frobenius acting as the $U_{\\Sieg}$-eigenvalue. Abusing notation slightly\\footnote{What we really mean is that $\\Gr^1 V(\\underline{\\pi})^*$ is isomorphic to the tensor product of $\\chi_{\\mathrm{cyc}}^{(1 + \\mathbf{r}_2)}$ and an unramified character.}, we may say that $\\Gr^1 V(\\underline{\\pi})^*$ has ``Hodge--Tate weight $1 + \\mathbf{r}_2$''.\n\n  Similarly, there is a 2-step filtration of $V(\\underline{\\sigma})^*$, with $\\Gr^1V(\\underline{\\sigma})^* = \\mathcal{F}^1 V(\\underline{\\sigma})^*$ having Hodge--Tate weight $1 + \\mathbf{t}_2$.\n\n  \\begin{definition}\n   We set\n   \\[\\mathbb{V}^* = V(\\underline{\\pi})^* \\times V(\\underline{\\sigma})^*(-1-\\mathbf{r}_1);\\]\n   and we let\n   \\[ \\mathcal{F}^{(f)} V(\\underline{\\pi} \\times \\underline{\\sigma})^* = \\mathcal{F}^2 V(\\underline{\\pi}) \\otimes V(\\underline{\\sigma})^*,\\]\n   and\n   \\[ \\mathcal{F}^{(e)} V(\\underline{\\pi} \\times \\underline{\\sigma})^* = \\left(\\mathcal{F}^2 V(\\underline{\\pi})^* \\otimes V(\\underline{\\sigma})^*\\right) + \\left(\\mathcal{F}^1 V(\\underline{\\pi})^* \\otimes \\mathcal{F}^1 V(\\underline{\\sigma})^*\\right). \\]\n   For a good weight $(P, Q)$ we write $\\mathbb{V}_{P, Q}^*$ for the specialisation of $\\mathbb{V}^*$ at $(P, Q)$, so $\\mathbb{V}_{P, Q}^* = V(\\pi_P)^* \\otimes V(\\sigma_Q)^*(-1-r_1)$ if $P = (r_1, r_2)$.\n  \\end{definition}\n\n  (For the significance of the labels (e) and (f), see Figure 2 of \\cite{LZvista}.) Thus $\\mathcal{F}^{(e)}$ has rank 5, $\\mathcal{F}^{(f)}$ has rank 4, and the quotient $\\Gr^{(e\/f)} \\cong \\left(\\Gr^1 V(\\pi)^*\\right)\\otimes\\left(\\mathcal{F}^1 V(\\underline{\\sigma})^*\\right)(-1-\\mathbf{r}_1)$ has Hodge--Tate weight $\\mathbf{t}_1' = -2-\\mathbf{t}_1$.\n\n  \\begin{remark}\n   Note that\n   \\[ \\mathcal{E}_p(\\pi_P\\times \\sigma_Q) = \\det \\left(1 - \\varphi: \\mathbf{D}_{\\mathrm{cris}}(\\mathcal{F}^{(f)} \\mathbb{V}_P^*)\\right) \\cdot \\det\\left( 1 - p^{-1}\\varphi^{-1}: \\mathbf{D}_{\\mathrm{cris}} \\left(\\mathbb{V}_P^* \/ \\mathcal{F}^{(f)}\\right) \\right).\\qedhere\\]\n  \\end{remark}\n\n\n \\subsection{P-adic periods}\n\n  The representations $\\Gr^1V(\\underline{\\pi})^*(-1-\\mathbf{r_2})$ and $\\Gr^1 V(\\underline{\\sigma})^*(-1-\\mathbf{t}_2)$ are unramified, and hence crystalline as $\\mathcal{O}(U)$ (resp.~$\\mathcal{O}(U')$)-linear representations. Since $\\mathbf{D}_{\\mathrm{cris}}(\\QQ_p(1))$ is canonically $\\QQ_p$, we can therefore define $\\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*)$ to be an alias for the rank 1 $\\mathcal{O}(U \\times U')$-module\n  \\[\n   \\mathbf{D}_{\\mathrm{cris}}\\left(\\Gr^1 V(\\pi)^*(-1-\\mathbf{r_2})\\right) \\mathop{\\hat\\otimes} \\mathbf{D}_{\\mathrm{cris}}\\left( \\Gr^1 V(\\underline{\\sigma})^*(-1-\\mathbf{t}_2)\\right).\n  \\]\n  As in \\cite[\\S 8.2]{KLZ17}, we can define a Coleman\/Perrin-Riou big logarithm map for $\\Gr^{(e\/f)} \\mathbb{V}^*$, which is a morphism of $\\mathcal{O}(U \\times U')$-modules\n  \\[ \\mathcal{L}^{\\mathrm{PR}}: H^1(\\QQ_p, \\Gr^{(e\/f)} \\mathbb{V}^*) \\to \\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*). \\]\n  By construction, for good geometric weights $P$, this specialises to the Bloch--Kato logarithm map, up to an Euler factor; and for good critical weights it specialises to the Bloch--Kato dual exponential.\n\n \\subsection{P-adic Eichler--Shimura isomorphisms}\n\n  Let $P$ be a good weight. Then the Faltings--Tsuji comparison isomorphism of $p$-adic Hodge theory gives an identification between $\\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P))$ and the $\\pi_P$-eigenspace in de Rham cohomology (compatibly with the Hodge filtration); and the graded pieces of this filtration are identified with the coherent cohomology groups $S^i(\\pi_P, L)$.\n\n  Since the Hodge and Newton filtrations on $\\mathbf{D}_{\\mathrm{cris}}$ must be complementary to each other (by weak admissibility), we deduce that there is an \\emph{Eichler--Shimura} isomorphism\n  \\[\n   \\ES^2_{\\pi_P}: S^2(\\pi_P, L)\\cong \\Gr^{(r_2 + 1)}_{\\mathrm{Hdg}} \\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P))  \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\pi_P)).\n  \\]\n  Concretely, the isomorphism is given by mapping an element in $\\Gr^{(r_2 + 1)}_{\\mathrm{Hdg}} \\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P))$ to its unique lifting to $\\Fil^{(r_2 + 1)}_{\\mathrm{Hdg}}\\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P)) \\cap \\ker( (\\varphi - \\alpha_P)(\\varphi - \\beta_P))$.\n\n  \\begin{remark}\n   More generally, we have isomorphisms $\\ES^i: S^i(\\pi_P, L)\\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^i V(\\pi_P))$ for each $0 \\le i \\le 3$, where $S^i(\\pi_P, L)$ is the $\\pi_P$-eigenspace in coherent $H^i$.\n\n   We caution the reader that although the source and target of $\\ES^i_{\\pi_P}$ are the specialisations at $P$ of rank-one $\\mathcal{O}(U)$-modules, it is \\textbf{by no means obvious} that the isomorphisms $\\ES^i_{\\pi_P}$ for varying $P$ are the specialisations of a single $\\mathcal{O}(U)$-module isomorphism ``$\\ES^i_{\\underline{\\pi}}$''. We shall establish (a slightly weakened form of) this below, under some additional hypotheses, as a by-product of our main Euler system argument.\n\n   It would be very interesting to have a direct construction of the maps $\\ES^i_{\\underline{\\pi}}$ by methods of arithmetic geometry. For $i = 0$ (corresponding to classical holomorphic Siegel modular forms) this has been achieved in the recent preprint \\cite{diao-rosso-wu21}. One can also obtain $\\ES^3_{\\underline{\\pi}}$ from this via Serre duality; but it seems to be more difficult to construct the ``intermediate'' filtration steps $i = 1, 2$.\n  \\end{remark}\n\n  \\subsubsection{Analogue for $\\GL_2$} Similarly, for $\\GL_2$ we have an isomorphism\n  \\[ \\ES^0_{\\sigma_Q}: S^0(\\sigma_Q, L) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^0 V(\\sigma_Q)). \\]\n  In this setting the existence of comparison isomorphisms in families is known:\n\n  \\begin{theorem}[Ohta, Kings--Loeffler--Zerbes]\n   There exists an isomorphism of $\\mathcal{O}(U')$-modules\n   \\[ \\ES^0_{\\underline{\\sigma}}: S^0(\\underline{\\pi}) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^0 V(\\underline{\\sigma}))\\]\n   interpolating the isomorphisms $\\ES^0_{\\sigma_Q}$ for varying $P$, where $\\mathcal{S}^0(\\underline{\\pi})$ is the $\\mathcal{O}(U')$-module spanned by $\\underline{\\omega} = G(\\chi_2^{-1}) \\cdot \\mathcal{G}$.\n  \\end{theorem}\n  \\begin{proof}\n   This is a restatement of \\cite[Proposition 10.1.1(1)]{KLZ17}, where it is derived from results of Ohta \\cite{ohta00}. For an alternative derivation applying to possibly non-ordinary Coleman families, see \\cite{andreattaiovitastevens,loefflerzerbes16}.\n  \\end{proof}\n\n \\subsection{Euler system classes}\n  Let us suppose that the character $\\chi_0 \\chi_2$ is non-trivial (this allows us to get rid of a ``smoothing factor'' $c$ appearing in the Euler system constructions). Then, associated to the data $\\gamma_S$, we also have a family of cohomology classes\n  \\[ \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\in H^1(\\mathbf{Q}(\\mu_{m}), \\mathbb{V}^*), \\]\n  for all square-free integers coprime to some finite set $T \\supseteq S \\cup \\{p\\}$.\n  By construction, the image of $ \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)$ under localisation at $p$ lands in the image of the (injective) map from the cohomology of $\\mathcal{F}^{(e)} \\mathbb{V}^*$. So we may make sense of\n  \\[  \\mathcal{L}^{\\mathrm{PR}}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right) \\in \\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*). \\]\n  We denote its image under specialisation at $(P,Q)$ by $ \\mathcal{L}^{\\mathrm{PR}}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right)(P,Q)$.\n  Combining \\cref{prop:coherentperiod} with the main result of \\cite{LZ20b-regulator}, which relates the periods $\\operatorname{Per}_\\eta(\\dots)$ to the Euler system classes, we have the following result:\n\n  \\begin{theorem}\\label{prop:geomreg}\n   For each $P$ in the good geometric range, we have\n   \\[ \\left\\langle \\mathcal{L}\\left( \\mathbf{z}_{1}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right)(P,Q), \\mathrm{ES}^2_{\\pi_P}(\\eta_P) \\otimes \\mathrm{ES}^0_{\\sigma_Q}(\\omega_P)\\right\\rangle = \\mathcal{L}_{p,\\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q). \\]\n  \\end{theorem}\n\n \\subsection{Reciprocity laws and meromorphic Eichler--Shimura}\n\n  \\begin{definition}\n   Let $\\mathfrak{S}(\\underline{\\pi}; \\underline{\\sigma})$ denote the set of points $P = (r_1, r_2) \\in U \\cap \\mathbf{Z}^2$ which are good for $\\underline{\\pi}$, and satisfy the following condition: there exists some $t_2 \\in U' \\cap \\mathbf{Z}_{\\ge 0}$, and some local data $\\gamma_S$, such that $(P, Q) = (r_1, r_2, t_2)$ is good geometric and $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma};\\underline{\\eta})$ is non-vanishing at $(P, Q)$.\n  \\end{definition}\n\n  \\begin{lemma}\n   Let $\\underline{\\sigma}, \\underline{\\sigma}'$ be two Hida families satisfying our running hypotheses (possibly of different tame levels and characters). Then the set $\\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma}) \\cap \\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma}')$ is Zariski-dense. In particular, $\\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma})$ is itself Zariski-dense.\n  \\end{lemma}\n\n  \\begin{proof}\n   We first note that there exists $\\gamma_S$ for which $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma};\\underline{\\eta})$ is not identically zero. To see this, we choose some good \\emph{critical} point $(P, Q)$ having $t_1 = r_1 - r_2 - t_2 - 2 \\ge 0$, so that $\\Lambda(\\pi_P \\times \\sigma_Q, 1 + \\tfrac{t_1}{2})$ lies outside the strip $0 < \\Re(s) < 1$ and hence cannot vanish. We can then choose $\\gamma_S$ such that $Z_S(\\pi_P \\times \\sigma_Q, \\gamma_S) \\ne 0$ (which is always possible). Thus $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma})$ is non-vanishing at $(P, Q)$, and hence generically non-vanishing on $U \\times U'$.\n\n   Repeating the construction, we can find local data $\\gamma_{S}'$ for $\\underline{\\pi} \\times \\underline{\\sigma}'$ such that $\\mathcal{L}_{p, \\gamma_S'}(\\underline{\\pi} \\times \\underline{\\sigma}')$ is generically non-vanishing. So there is an open subset $V \\subset U$ such that for all $v \\in V$, neither $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma})$ nor $\\mathcal{L}_{p, \\gamma_S'}(\\underline{\\pi} \\times \\underline{\\sigma}')$ vanishes identically along $\\{ v \\} \\times U'$.\n\n   Since $V$ is open, it must contain some $(r_1, r_2) \\in V \\cap \\mathbf{Z}^2$; and we can therefore find an integer $t$ such that both $p$-adic $L$-functions are non-vanishing at $P = (r_1, r_2, t)$. We consider the sequence of weights $P_k = (r_1 + 3(p-1)p^{k}, r_2 + (p-1)p^{k}, t_2 + 2(p-1)p^k)$ for $k \\to \\infty$. For all but finitely many $k$ the weight $P_k$ will be good geometric, and $P_k$ tends to $P$, so $\\mathcal{L}_{p, \\gamma_S}(P_k) \\ne 0$ for sufficiently large $k$. Thus the projection of $P_k$ to $U$ lies in $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma})$, and also in $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}')$. It follows that $(r_1, r_2)$ is a limit point of $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}) \\cap \\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}')$ in the analytic topology. Thus the Zariski-closure of this intersection contains all points of $U \\cap \\mathbf{Z}^2$ outside a proper closed subset, and hence must be all of $U$.\n  \\end{proof}\n\n  Let us write $\\mathcal{Q}(U)$ for the fraction field of $\\mathcal{O}(U)$ (and similarly for $U \\times U'$ etc).\n\n  \\begin{theorem}\n   There exists an isomorphism of $\\mathcal{Q}(U)$-modules\n   \\[ \\ES^2_{\\underline{\\pi}}: S^2(\\underline{\\pi}) \\otimes_{\\mathcal{O}(U)} \\mathcal{Q}(U) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\underline{\\pi})) \\otimes_{\\mathcal{O}(U)} \\mathcal{Q}(U), \\]\n   depending only on $\\underline{\\pi}$, characterised uniquely by the following property: for all Hida families $\\underline{\\sigma}$ as above, and all $P = (r_1, r_2) \\in \\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$, the morphism $\\ES^2_{\\underline{\\pi}}$ is non-singular at $P$ and its fibre at $P$ coincides with the Eichler--Shimura morphism $\\ES^2_{\\pi_P}$. Moreover, we have the explicit reciprocity law\n   \\[\n    \\left\\langle\n     \\mathcal{L}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)\\right),\n     \\mathrm{ES}^2_{\\underline{\\pi}}(\\underline{\\eta}) \\otimes \\mathrm{ES}^0_{\\underline{\\sigma}}(\\underline{\\omega})\\right\\rangle = \\mathcal{L}_{p,\\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta}).\n   \\]\n  \\end{theorem}\n\n  \\begin{proof}\n   We start by choosing a ``random'' isomorphism $\\jmath$ between $S^2(\\underline{\\pi})$ and $\\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\underline{\\pi}))$, which is possible since both are free rank 1 $\\mathcal{O}(U)$-modules.\n\n   As in the proof of the preceding lemma, we choose local data $\\gamma_S$ such that $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi}, \\underline{\\sigma}; \\underline{\\eta})$ is not identically zero, and consider the ratio\n   \\[ \\mathsf{R} = \\frac{1}{\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})}  \\left\\langle\\mathcal{L}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)\\right),\n        \\jmath(\\underline{\\eta}) \\otimes \\mathrm{ES}^0_{\\underline{\\sigma}}(\\underline{\\omega})\\right\\rangle \\in \\mathcal{Q}(U \\times U').\n   \\]\n\n   If we now take a $(P, Q)$ that is good geometric, and such that $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi}, \\underline{\\sigma}; \\underline{\\eta})$ does not vanish at $(P, Q)$, it follows from the \\cref{prop:geomreg} that $\\mathsf{R}$ is regular at $(P, Q)$ and its value there is equal to the ratio $\\jmath_P \/ \\ES^2_{\\pi_P}$ (independent of $Q$).\n\n   We claim that $\\mathsf{R} \\in \\mathcal{Q}(U)$; that is, as a meromorphic function on $U \\times U'$, it is independent of the $U'$ variable. To justify this, we argue as in Proposition 17.7.3 of \\cite{LZ20}: we consider the meromorphic function $\\mathsf{R}(\\mathbf{r}_1, \\mathbf{r}_2, \\mathbf{t}_2) - \\mathsf{R}(\\mathbf{r}_1, \\mathbf{r}_2, \\hat{\\mathbf{t}}_2)$ on $U \\times U' \\times U'$, where $\\hat{\\mathbf{t}}_2$ is the coordinate on a second copy of $U'$. Because of \\cref{prop:geomreg}, this function has to vanish at all points $(r_1, r_2, t_2, \\hat{t}_2)$ such that $(r_1,r_2, t_2)$ and $(r_1, r_2, \\hat{t}_2)$ are both good geometric and neither is in the vanishing locus of $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})$; this set is easily seen to be Zariski-dense in $U \\times U' \\times U'$. The same argument also shows that $\\mathsf{R}$ doesn't depend on $\\gamma_S$.\n\n   Thus $\\mathsf{R}$ is an element of $\\mathcal{Q}(U)^\\times$, regular at all points $P \\in \\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ and coinciding at each such point with the ratio $j_P \/ \\ES^2_{\\pi_P}$. So if we define $\\ES^2_{\\underline{\\pi}} = \\mathsf{R}^{-1} \\jmath$, then $\\ES^2_{\\underline{\\pi}}$ is regular at all points in $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ and coincides at such points with $\\ES^2_{\\pi_P}$. By the preceding lemma, this interpolating property uniquely determines $\\ES^2_{\\underline{\\pi}}$, and is independent of $\\underline{\\sigma}$; and the reciprocity law holds by construction.\n  \\end{proof}\n\n  \\begin{remark}\n   Note that there could, \\emph{a priori}, be points where $\\ES^2_{\\underline{\\pi}}$ is 0 or $\\infty$; or where it is a well-defined isomorphism but this isomorphism does not coincide with $\\ES_{\\pi_P}^2$.\n  \\end{remark}\n\n \\subsection{Application to the Bloch--Kato conjecture}\n\n  Let us now consider the following situation:\n  \\begin{itemize}\n   \\item $\\pi$ and $\\sigma$ are cohomological cuspidal automorphic representations of $\\GSp_4 \\times \\GL_2$, with $p$-stabilisations which are ordinary and $p$-regular, which are ``deformable'' in the above sense.\n\n   \\item If $t_2 = r_1 - r_2 - 1$ (so that $t_1 = -1$), then we suppose that $L(\\Pi \\times \\Sigma, \\tfrac{1}{2}) \\ne 0$. (In all other cases the non-vanishing of $L(\\Pi \\times \\Sigma, 1+\\tfrac{t_1}{2})$ is automatic.)\n\n   \\item The Galois representation $V = V_p(\\pi)^* \\otimes V_p(\\sigma)^*(-1-r_1)$ satisfies the ``big image'' conditions of \\cite[\\S 3.5]{mazurrubin04}.\n\n   \\item None of the eight characters appearing as graded pieces of $V$ as a $\\Gal(\\overline{\\mathbf{Q}}_p \/ \\QQ_p)$-representation are congruent mod $p$ to the trivial character, or to the $p$-adic cyclotomic character (``$p$-distinction'').\n  \\end{itemize}\n\n  (Note that the ``big image'' hypothesis can only be satisfied if $\\chi_0\\chi_2 \\ne 1 \\bmod p$, but is frequently satisfied when this condition does hold; compare the discussion in \\S 11.1 of \\cite{KLZ17} in the Rankin--Selberg case.)\n\n\n  \\begin{theorem}\n  \\label{thm:BKconj}\n   In the above setting, we have\n   \\[ H^1_{\\mathrm{f}}(\\mathbf{Q}, V(\\pi)^* \\otimes V(\\sigma)^*(-1-r_1)) = 0,\\]\n   as predicted by the Bloch--Kato conjecture.\n  \\end{theorem}\n\n  \\begin{proof}\n   If $(r_1, r_2)$ is in the set $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ defined above (or more generally in $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma}')$ for some possibly different Hida family $\\underline{\\sigma}'$), then the theorem of the previous section implies that we have an Euler system for $V(\\pi)^* \\otimes V(\\sigma)^*(-1-r_1)$ whose bottom class is non-zero. Hence we may apply the machinery of ``Euler systems with local conditions'' developed in \\cite[\\S 12]{KLZ17} to deduce the finiteness of the Selmer group.\n\n   The exceptional case which we need to deal with is when the ``family'' Eichler--Shimura isomorphism degenerates at $(r_1, r_2)$. We expect that this never occurs, but we cannot yet rule it out. In this situation, we use a version of the ``leading term argument'' from \\cite{LZ20, LZ20-yoshida}). The construction of the $p$-adic $L$-function (and the proof of the reciprocity law) extend immediately to equivariant $p$-adic $L$-functions over $\\mathbf{Q}(\\zeta_m)$, for all $m$ coprime to $T$. If the Eichler--Shimura isomorphism degenerates at $(r_1, r_2)$, then not only the class $\\mathbf{z}_1(\\pi \\times \\sigma)$, but all the classes $\\mathbf{z}_m$, must satisfy the stronger local condition defined by $\\mathcal{F}^{(f)}$; and this forces all the classes to be zero, as in \\ and this forces all of the classes to be zero. So we may replace the whole Euler system by its first derivative (in some arbitrarily chosen direction in weight space) and rescale the Eichler--Shimura isomorphism accordingly. Proceeding inductively, we eventually obtain an Euler system with non-trivial bottom class, and the argument proceeds as before.\n\n   (A slight complication here is that in the exceptional case, the Euler system we obtain for $V$ does not necessarily extend to classes over the $p$-cyclotomic tower satisfying the extra-strong local condition $\\mathcal{F}^{(f)}$, since our explicit reciprocity law does not ``see'' the cyclotomic variable. Hence we cannot use the arguments of \\cite[\\S 12]{KLZ17} to prove the crucial lemma that this local condition is preserved by the passage from Euler to Kolyvagin systems, as these arguments rely on the presence of the $p$-cyclotomic tower. This is the reason for imposing the rather stringent $p$-distinction hypothesis, which allows us to use the alternative, slightly more direct approach given in the appendix of \\cite{leiloefflerzerbes14b}, in which the cyclotomic extension is not needed.)\n  \\end{proof}\n\n\n\n\\newlength{\\bibitemsep}\n\\setlength{\\bibitemsep}{0.75ex plus 0.05ex minus 0.05ex}\n\\newlength{\\bibparskip}\n\\setlength{\\bibparskip}{0pt}\n\\let\\oldthebibliography\\thebibliography\n\\renewcommand\\thebibliography[1]{%\n \\oldthebibliography{#1}%\n \\setlength{\\parskip}{\\bibparskip}%\n \\setlength{\\itemsep}{\\bibitemsep}%\n}\n\n\\newcommand{\\noopsort}[1]{}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}[1]{}\n\\renewcommand{\\MR}[1]{%\n    MR \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#1}.\n}\n\\providecommand{\\href}[2]{#2}\n\\newcommand{\\articlehref}[2]{\\href{#1}{#2}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe extensive study of Kummer surfaces is explained by their rich\ngeometry and their multiple roles in the theory of K3 surfaces and\nbeyond \\cite{H1,torelli, B3}.\n\nLet $A$ be an abelian surface and consider the involution which\nmaps $a$ to $-a$ for any $a$ in $A$. This involution has sixteen\nfixed points, namely the sixteen two-torsion points of $A$. The\nquotient surface has sixteen ordinary double points and its\nminimal resolution is a smooth K3 surface called the Kummer\nsurface associated to $A$ and denoted by $\\mathrm{Kum}(A)$. Nikulin\nproved that any K3 surface containing sixteen disjoint smooth\nrational curves is a Kummer surface \\cite{N2}.\n\nGiven a Kummer surface $\\mathrm{Kum}(A)$, there is a natural way of\nconstructing new Kummer surfaces from it. One takes the minimal\nmodel of the double cover of $\\mathrm{Kum}(A)$ branched along eight\ndisjoint smooth rational curves $C_1, \\dots, C_8,$ that are even\n(see section 2) and that are orthogonal in $\\mathrm{Pic(Kum}(A))$ to\neight other smooth rational curves. We obtain in this way a new\nKummer surface $\\mathrm{Kum}(B)$ together with a rational map\n$\\mathrm{Kum}(B) \\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$.\n\nIn the second section of the paper, we explain this construction\nin details and show that the abelian surface associated to the new\nKummer surface $\\mathrm{Kum}(B)$ is isogenous to $A$. In fact we prove\nthat the map $\\tau$ is induced by an isogeny of degree two on the\nassociated abelian surfaces.\n\nIn section 3, we describe the geometry of a generic jacobian\nKummer surface and explain its classical double plane model. We\nalso recall a theorem of Naruki \\cite{Naruki} giving explicit\ngenerators of the N\\'eron-Severi lattice of a generic jacobian\nKummer surface.\n\nIn section 4, we apply the construction of section 2 to the\ngeneric jacobian Kummer surface. We obtain in this way, fifteen\nnon isomorphic Kummer surfaces which are associated to\n$(1,2)$-polarized abelian surfaces.\n\nFinally in section 5, we show  that the Kummer surfaces of section\n4 admit an elliptic fibration with twelve singular fibers of the\ntype $I_2$. We also prove that these Kummer surfaces are double\ncover of a week Del Pezzo surface (i.e. the blowup of $\\mathbb\nP^2$ at seven points) and that for each of our Kummer surfaces\nthere exists a decomposition of a very degenerate sextic $\\mathcal\nS$ (see figure \\ref{plane sextic}) into a quartic $Q$ and a conic\n$C$ for which we have the theorem\n\n\\begin{theorem}\\label{San11}\nThe rational double cover $\\mathrm{Kum}(B) \\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$ decomposes as\n$$\\xymatrix{\\mathrm{Kum}(B) \\ar@{-->}[d]^{\\tau}\\ar[r]^{\\varphi}&\nT \\ar@{->}[d]^{\\zeta}\\\\ \\mathrm{Kum}(A) \\ar@{->}[r]^{\\phi}& \\mathbb P^2}$$ where\n$\\phi$ is the canonical resolution of the double cover of $\\mathbb\nP^2$ branched along $\\mathcal S$. The maps $\\zeta$ and $\\varphi$\nare the canonical resolutions of the double covers branched along $Q$ and $\\zeta^*(C)$ respectively.\n\\end{theorem}\n\n\n\\section{Even Eight and Kummer surface}\n\nWe will now introduce the notion of an even eight and the double\ncover construction associated to it. By applying this construction\nto special even eights of a Kummer surface, we obtain new Kummer\nsurfaces.\n\\begin{definition}\\label{eveneight}\nLet $Y$ be a K3 surface, an \\textit{even eight} on $Y$ is a set of\neight disjoint smooth rational curves $C_1, \\dots, C_8,$ for which\n$C_1 + \\cdots + C_8 \\in 2S_Y.$ Here $S_Y$ denotes the\nN\\'eron-Severi group of $Y.$\n\\end{definition}\n\n\nIf $C_1, \\dots, C_8,$ is an even eight on a K3 surface $Y$, then\nthere is a double cover $Z \\stackrel{p}\\to Y$, branched on $C_1 +\n\\dots +C_8$. If $E_i$ denotes the inverse image of $C_i$, then\n$p^*(C_i)=2E_i$ and $E_i^2=-1$. Hence, we may blowdown the $E_i$'s\nto the surface $X$ and obtain the diagram $$\\xymatrix{Z\n\\ar@{->}[d]_{p }\\ar[r]^{\\epsilon} & X \\ar@{-->}^{2:1}[ld]\n\\\\Y & }$$\nIt turns out that the surface $X$ is again a K3 surface and the\ncovering involution $\\iota: X \\to X$ is symplectic with eight\nfixed points \\cite{N2}.\n\n\nSuppose now that the K3 surface $Y$ is a Kummer surface, we want\nto exhibit natural even eights lying on it. For this purpose, we\nrecall a central lemma of Nikulin.\n\\begin{lemma}\\cite{N2}\\label{nikulin}\nLet $Y$ be a Kummer surface and let $E_1, \\dots, E_{16} \\subset Y$\nbe sixteen smooth disjoint rational curves. Denote by $I=\\{ 1,\n\\dots, 16 \\}$ the set of indices for the curves $E_{i}$'s and by\n$Q = \\{ M \\subset I \\vert \\quad \\frac{1}{2} \\sum_{i\\in M}E_{i}\\in\nS_Y\\}$; then for every $M$ in $Q$, we have $\\# |M|=8 \\textrm{ or }\n16$ and there exists on $I$ a unique 4-dimensional affine geometry\nstructure over $\\mathbb F_{2}$, whose hyperplanes consist of the\nsubsets $M \\in Q$ containing eight elements.\n\\end{lemma}\n\nThe existence of such a 4-dimensional affine geometry implies that\n$I \\in Q $ or equivalently that $\\sum_{i=1}^{16}E_{i}\\in 2 S_Y.$\nWe can proceed exactly as for an even eight and take the double\ncover $V \\stackrel{p}\\to Y$ branched along $E_1+ \\dots + E_{16}$.\nAgain we blowdown the preimage of the $E_i$'s to a surface $A$ and\nobtain the diagram $$\\xymatrix{V \\ar@{->}[d]_{p }\\ar[r]^{\\epsilon}\n& A \\ar@{-->}^{\\pi_A }[ld]\n\\\\Y & }$$The difference with this diagram and the one above is\nthat now the surface $A$ is an abelian surface and that the map\n$\\pi_A$ realizes $Y$ as the Kummer surface associated to $A$. We\npoint out that by uniqueness, the affine geometry on $I$\ncorresponds to the one existing on $A_{2}$, the set of 2-torsion\npoints on $A$. \\cite{N2}.\n\nIt follows also from the lemma that there exist on $Y$ ($\\simeq\n\\mathrm{Kum}(A)$) thirty even eights, denoted by $M_1, \\cdots,\nM_{30}$, i.e. the thirty affine hyperplanes of $I$.\n\nLet $M \\in \\{ M_1, \\dots, M_{30} \\}$ be one of these even eights.\nWe can assume that $M$ consists of the curves $E_{1}, \\dots,\nE_{8}$. The curves $E_{9}, \\dots, E_{16}$ are then orthogonal to\n$M$, i.e. $$E_i \\cdot E_j =0 \\textrm{ if } 1 \\leq i \\leq 8\n\\textrm{ and } 9\\leq j \\leq 16.$$ If $X\n\\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A) $ is the double cover\nassociated to $M$, then the K3 surface $X$ contains again sixteen\ndisjoint smooth rational curves. Indeed since the curves $E_{9},\n\\dots, E_{16}$ do not intersect the branch locus of the double\ncover $p: Z \\to Y$, they split under $p$ and define sixteen\ndisjoint smooth rational curves on $Z$. These sixteen curves are\nthen isomorphically mapped by the blowdown $Z \\stackrel{\\epsilon}\n\\to X$ to sixteen curves on $X$. It follows that $X$ contains\nsixteen disjoint smooth rational curves and hence it is a Kummer\nsurface.\n\n\\begin{prop}\\label{prop}\nLet $M$ be an even eight on a Kummer surface $\\mathrm{Kum}(A)$ such as\nabove, then the K3 surface $X$ associated to $M$ is a Kummer\nsurface. Moreover there is an abelian surface $B$ associated to\n$X$ for which we have the commutative diagram\n$$\\xymatrix{B\\ar@{-->}[d]_{\\pi_B } \\ar[r]^{p} & A\n\\ar@{-->}[d]_{\\pi_A}\n\\\\X=\\mathrm{Kum}(B) \\ar@{-->}[r]^{\\tau}& \\mathrm{Kum}(A)}$$where $B\\stackrel{p}\\to\nA$ is an isogeny of degree two.\n\\end{prop}\n\n\\begin{proof}\n\\noindent Since we have already shown that $X$ is a Kummer\nsurface, we only have to prove that $B$ is degree two isogenous to\n$A$. Write the abelian surface $A$ as the complex torus $\\mathbb\nC^2 \/ \\Lambda$ and let $E_{9}, \\dots, E_{16}\\subset \\mathrm{Kum}(A)$\nbe the eight disjoint smooth rational curves orthogonal to $M$.\nThese curves also form an even eight and hence they correspond to\nan affine hyperplane $H$ in $A_2$. Up to translation we can fix\nthe origin on $A$ in $H$. Let $\\frac{[v]}{2}$ be the generator of\n$A_2\/H$, it defines a sublattice $\\Lambda' \\subset \\Lambda$.\nExplicitly we have that $\\Lambda'= \\mathbb Z h_1 \\oplus \\mathbb Z\nh_2 \\oplus \\mathbb Z h_3 \\oplus \\mathbb Z 2v$, where $H=\\langle\n\\frac{[h_1]}{2}, \\frac{[h_2]}{2}, \\frac{[h_3]}{2} \\rangle \\subset\nA_2$.\nThe canonical inclusion $\\Lambda' \\hookrightarrow \\Lambda$,\ninduces the following commutative diagram : $$\\xymatrix{\\mathbb\nC^2\/\\Lambda'\\ar@{-->}[d]_{\\pi'} \\ar[r]^{p} & \\mathbb C^2\/\\Lambda\n\\ar@{-->}[d]_{\\pi}\\\\\n              \\mathrm {\\kum(\\mathbb C^2\/\\Lambda')} \\ar@{-->}[r]^{q}& \\mathrm {\\kum(\\mathbb C^2\/\\Lambda)}}$$\nwhere $p$ is an isogeny of degree two. The covering involution of\n$p$ is given by the translation by the 2-torsion point $[v]$ in\n$\\mathbb C^2\/\\Lambda'$.\nIt induces the symplectic involution on $\\mathrm {\\kum(\\mathbb\nC^2\/\\Lambda')}$ $$\\sigma: \\mathrm {\\kum(\\mathbb C^2\/\\Lambda')} \\to \\mathrm\n{\\kum(\\mathbb C^2\/\\Lambda')}$$which has exactly eight fixed points\n\\cite{N5}, namely\nthe projection of the sixteen points on $\\mathbb C^2\/\\Lambda'$ satisfying$$[z]+[v]=-[z] \\textrm{, or equivalently } 2[z]=[v].$$ \nThe isogeny $p$ maps the set $\\{ [z] \\in \\mathbb C^2\/\\Lambda'\n\\textrm { }| \\textrm { }2[z]=[v]\\}$ to $\\mathrm {A_2-H}$. In other\nwords, the affine hyperplane $A_2-H$ corresponds  to the even\neight $M$ in $\\mathrm {Kum}(\\mathbb C^2\/\\Lambda)$.  Hence the\nresolution of the rational map $q$ is exactly the double cover of\n$\\mathrm{Kum}(A)$ branched along $M$ and the abelian surface $\\mathbb\nC \/\\Lambda'$ is $B$.\n\n  \n\\end{proof}\n\n\n\\section{Jacobian Kummer surface}\nIn this section we briefly expose the classical geometry of a\njacobian Kummer surface and its beautiful $16_6$-configuration. We\ndescribe its double plane model and give explicit generators for\nits N\\'eron-Severi lattice. This description follows a paper of N\naruki \\cite{Naruki}.\n\nA Kummer surface $\\kum(A)$ is said to be a jacobian Kummer surface\nif the surface $A$ is the jacobian of a curve $C$ of genus two.\nMoreover, it is a generic jacobian Kummer surface if its Picard\nrank is 17.\n\nRecall that the degree two map given by the linear system $|2C|$,\n$A \\stackrel{|2C|} \\to \\mathbb P^3$, factors through the\ninvolution $a \\stackrel{i} \\mapsto -a$, and hence defines an\nembedding $A\/\\{ 1, i\\} \\hookrightarrow \\mathbb P^3$. The image of\nthis map is a quartic $Y_0 \\subset \\mathbb P^3$ with sixteen\nnodes. Denote by $L_0$ the class of a hyperplane section of $Y_0$.\nProjecting $Y_0$ from a node defines a rational map $Y_0\n\\stackrel{2:1}\\dashrightarrow \\mathbb P^2$. We blowup the center\nof projection\n\n$$\\xymatrix{Y_1\\ar@{->}[d] \\ar[rd] & \\\\\n              Y_0 \\ar@{-->}[r]& \\mathbb P^2}$$and we call $E_1\\subset Y_1$  the\n              exceptional divisor and $L_1\\subset\n              Y_1$ the pullback of a line in $\\mathbb P^2$.\n              Finally we resolve the remaining fifteen\n              singularities of $Y_1$ and obtain the Kummer surface\n              $\\mathrm{Kum}(A)$ and a map of degree two\n$\\mathrm{Kum}(A) \\stackrel{\\phi} \\to \\mathbb P^2$.  The map $\\phi$ is\ngiven by the linear system $|L-E_0|$, where $L$ and $E_0$ are the\npullback of $L_1$ and $E_1$ respectively.\n\nThe branch locus of the map $\\phi$ is a reducible\n plane sextic $\\mathcal S$, which is the union of six lines, $l_1, \\cdots, l_6$, all tangent to a conic $W$\n\n\\begin{figure}[h]\n$$\n  \\begin{xy}\n   <0cm,0cm>;<1.5cm,0cm>:\n    (2,-.3)*++!D\\hbox{$l_1$},\n    (1.1,0.25)*++!D\\hbox{$l_2$},\n    (1.1,1.3)*++!D\\hbox{$l_3$},\n    (2,1.8)*++!D\\hbox{$l_4$},\n    (2.8,1.3)*++!D\\hbox{$l_5$},\n    (2.9,0.2)*++!D\\hbox{$l_6$},\n    (1.5,1)*++!D\\hbox{$W$},\n   \n    (.5,0.18);(3.5,0.18)**@{-},\n    (0.5,2);(1.6,0)**@{-},\n    (3.1,2);(2.6,0)**@{-},\n    (0.7,.5);(2,2.5)**@{-},\n    (0.5,1.82);(3.5,1.82)**@{-},\n    (2,2.7);(3.45,0)**@{-},\n   \n   \n    (2,1)*\\xycircle(.8,.8){},\n  \\end{xy}\n  $$\n\\caption{The sextic $\\mathcal S$\\label{plane sextic}}\n\\end{figure}\n\n\\noindent Let $p_{ij}=l_i \\cap l_j \\in \\mathbb P^2$, where $1\\leq\ni < j \\leq 6$. Index the ten $(3,3)$-partitions of the set $\\{ 1,\n2, \\dots ,6 \\}$, by the pair $(i,j)$ with $2 \\leq i < j \\leq 6$.\nEach pair $(i,j)$ defines a plane conic $l_{ij}$ passing through\nthe sixtuplet $p_{1i}, p_{1j}, p_{ij},p_{lm},p_{ln},p_{mn}$, where\n$\\{ l,m,n \\}$ is the complement of $\\{ 1,i,j\\}$ in $\\{ 1, 2, \\dots\n,6 \\}$ and where $l < m < n$. The map $\\phi$ factors as\n$$\\mathrm{Kum}(A) \\stackrel{\\tilde{\\phi}} \\longrightarrow\n\\tilde{\\mathbb P}^2\\stackrel{\\eta}\\longrightarrow \\mathbb\nP^2$$where $\\eta$ is the blowup of $\\mathbb P^2$ at the $p_{ij}$'s\nand where $\\tilde{\\phi}$ is the double cover of $\\tilde{\\mathbb\nP}^2$ branched along the strict transform of the plane sextic\n$\\mathcal S$ in $\\tilde{\\mathbb P}^2$. Denote by $E_{ij} \\subset\n\\mathrm{Kum}(A)$ the preimage of the exceptional curves of\n$\\tilde{\\mathbb P}^2$. The ramification of the map $\\tilde{\\phi}$\nconsists of the union of six disjoint smooth rational curves $C_0+\nC_{12} + C_{13}+ C_{14}+ C_{15}+ C_{16}$. The preimage of the ten\nplane conics $l_{ij}$ defines ten more smooth disjoint rational\ncurves $C_{ij}\\subset \\mathrm{Kum}(A), 2 \\leq i < j \\leq 6$. Finally,\nnote that $\\phi(E_0)=W$. The sixteen curves $E_0, E_{ij} \\quad 2\n\\leq i < j \\leq 6$ are called the \\textit{nodes} of $ \\mathrm{Kum}(A)$\nand the sixteen curves $C_0, C_{ij}$, $2 \\leq i < j \\leq 6$ are\ncalled the \\textit{tropes} of $ \\mathrm{Kum}(A)$. These two sets of\nsmooth rational curves satisfy a beautiful configuration called\nthe $16_6$-configuration, i.e. each node intersects exactly six\ntropes and vice versa.\n\n\\noindent It is now possible to fully describe the N\\'eron-Severi\nlattice $S_{ \\mathrm{Kum}(A)}$ of a general jacobian Kummer surface.\n\n\\begin{theorem}\\cite{Naruki} Let $\\mathrm{Kum}(A)$ be a generic jacobian Kummer surface. Its N\\'eron-Severi lattice $S_{ \\mathrm{Kum}(A)}$\nis generated by the classes of $E_{0}, E_{ij}$, $C_{0}, C_{ij}$\nand $L$, with the relations:\n\n\\begin{enumerate}\n\\item $C_{0}= \\frac{1}{2}(L- E_{0} -\n\\sum_{i=2}^{6} E_{1i}),$\n\\item $C_{1j}= \\frac{1}{2}(L - E_{0} - E_{1j}- \\cdots - E_{j-1j}- E_{jj+1}- \\cdots E_{j6}),$ where $2\\leq j \\leq 6,$\n\\item $C_{jk}= \\frac{1}{2}(L - E_{1j} -E_{1k}- E_{jk}- E_{lm}-E_{ln}-\nE_{mn})$ where $2\\leq i < j \\leq 6,$ and $\\{ l, m, n\\}$ are as\ndescribed above.\n\n\\end{enumerate}\n\n\\noindent The intersection pairing is given by:\n\n\\begin{enumerate}\n\n\\item  the $E_{0}, E_{ij}$ are mutually orthogonal,\n\n\\item $\\langle L, L \\rangle =4, \\langle L, E_{0} \\rangle =\\langle L, E_{ij}\\rangle =0,$\n\n\\item $\\langle E_{0}, E_{0} \\rangle= \\langle E_{ij}, E_{ij} \\rangle=-2$,\n\n\\item the $C_{0}, C_{ij}$ are mutually orthogonal,\n\n\\item  $\\langle L, C_{0} \\rangle=  \\langle L, C_{ij} \\rangle=2$.\n\n\\end{enumerate}\n\n\n\\noindent The action on $S_{ \\mathrm{Kum}(A)}$ of the covering\ninvolution $\\alpha$ of the map $\\phi$ is given by:\n\n\\begin{tabular}{lrrlr}\n\n$\\alpha (C_{0})=C_{0}$ & &&\n\n$\\alpha (C_{1j})=C_{1j}$ & $2 \\leq j \\leq 6$\\\\\n\n$\\alpha (E_{ij})=E_{ij}$ & $1 \\leq i < j \\leq 6$ &&\n\n$\\alpha (L)=3 \\mathrm L - 4 E_{0}$ & \\\\\n\n$\\alpha (E_{0})=2 L - 3E_{0}$& &&\n\n$\\alpha (C_{ij})= C_{ij} + L - 2E_{0}$ & $2 \\leq i < j \\leq 6$.\\\\\n\n\\end{tabular}\n\n\\end{theorem}\n\\begin{remark}\nThe minimal resolution of the double cover of $\\mathbb P^2$\nbranched along the sextic $\\mathcal S$ in figure \\ref{plane\nsextic} is a Kummer surface (see \\cite{H1} for a proof).\\\\\n\\end{remark}\n\n\\section{(1,2)-polarized Kummer surfaces}\n\nIn this section, we apply the construction of section 2 to a\ngeneric jacobian Kummer surface. We identify all the even eights\nmade out of its nodes and study the associated Kummer surfaces.\nFirst we recall some standard facts about the polarization of\nabelian varieties.\n\nA polarization on a complex torus $\\mathbb C^g\/ \\Lambda$ is the\nclass of an ample line bundle $L$ in its the N\\'eron-Severi group.\nAs the latter group is equal, for abelian varieties, to the group\nof hermitian forms $H$ on $\\mathbb C^g$, satisfying\n$E=\\mathrm{Im}H(\\Lambda, \\Lambda) \\subset \\mathbb Z$, the ample line\nbundle $L$ corresponds to a positive definite hermitian forms\n$E_L$. According to the elementary divisor theorem, there exists a\nbasis $\\lambda_1, \\dots, \\lambda_g, \\mu_1, \\dots, \\mu_g$ of\n$\\Lambda$, with respect to which $E_L$ is given by the matrix\n$$\\left(\n\\begin{array}{cc}0 & D \\\\ -D & 0\\end{array}\\right) \\textrm{ with   } D=\\left(\n\\begin{array}{cccc} d_1 & 0 & 0 & \\ldots \\\\ 0& d_2 & 0&  \\ldots \\\\ \\vdots & 0&  \\ddots& 0 \\\\ \\vdots & \\vdots & 0 &d_g \\end{array}\\right)$$\nwhere $d_i \\ge 0$ and $d_i | d_{i+1}$ for $i=1, \\dots, g-1.$ The\nvector $(d_1, d_2, \\dots, d_g)$ is the type of the line bundle\n$L$.\n\n\\begin{example}\\cite{BL}\\label{example}\n\n\\begin{enumerate}\n\\item If $J(C)$ is the Jacobian of a curve $C$ of genus two,\nthen\nthe line bundle associated to the divisor $C$ is a polarization of\ntype $(1,1)$.\n\n\\item If $L$ is a polarization of type $(d_1, \\dots, d_g)$ on a complex torus, then\n$\\chi(L)=d_1 \\cdot \\cdot  \\cdot d_g$.\n\n\\item If $X_1 \\stackrel{p} \\to X_2$ is an isogney of degree 2 of\nabelian surfaces and $L$ is a polarization of type $(1,1)$ on\n$X_2$, then $\\chi(p^*(L))=2 \\chi(L)=2\\cdot 1$. Hence $p^*(L)$ is a\npolarization of type $(1,2)$ on $X_1$.\n\\end{enumerate}\n\\end{example}\n\n\n\n\\begin{prop}\\label{fifteen}\nLet $\\mathrm{Kum}(A)$ be a generic jacobian Kummer surface and let\n$E_0, E_{ij},$ $1\\leq i<j\\leq 6$ be its sixteen nodes. There exist\nfifteen even eights made out of its nodes that do not contain\n$E_0$. These even eights are of the form\n$$\\Delta_{i,j}=E_{1i}+\\cdots+ \\widehat{E}_{ij}+ \\cdots + E_{i6} +\nE_{1j}+ \\cdots + \\widehat{E}_{ij} + \\cdots + E_{j6},$$where$\\quad\n1 \\leq i < j \\leq 6$ and $E_{11}=0.$\n\nThe Kummer surface $\\mathrm{Kum}(B_{ij})$ obtained from the double\ncover branched along $\\Delta_{ij}$ is associated to an abelian\nsurface $B_{ij}$ with a (1,2)-polarization.\n\\end{prop}\n\\begin{proof}\nFor any couple $(i,j)$ with $1 \\leq i < j \\leq 6$, consider the\ndivisor $2C_{1i} + 2C_{1j}$, where we set $C_{11}:= C_0$.\nAccording to the description of the N\\'eron-Severi lattice of a\ngeneral jacobian Kummer surface in the previous section, we have\nthe equality $$2C_{1i} + 2C_{1j}= 2(L-E_0)-(E_{1i}+ \\cdots +\nE_{ij}+ \\cdots + E_{i6} + E_{1j}+ \\cdots + E_{ij} + \\cdots +\nE_{j6}).$$ Therefore $$2C_{1i} + 2C_{1j} - 2(L-E_0)+2 E_{ij}=\nE_{1i}+\\cdots+ \\widehat{E}_{ij}+ \\cdots + E_{i6} + E_{1j}+ \\cdots\n+ \\widehat{E}_{ij} + \\cdots + E_{j6}$$ and consequently\n$$E_{1i}+\\cdots+ \\widehat{E}_{ij}+ \\cdots + E_{i6} + E_{1j}+\n\\cdots + \\widehat{E}_{ij} + \\cdots + E_{j6}$$is an even eight not\ncontaining $E_0$. As there are exactly fifteen choices for $i$ and\n$j$, we obtain this way all of the possible even eight.\n\nLet $\\mathrm{Kum}(B_{ij})$ be the Kummer surface obtained by taking\nthe double cover branched along such an even eight. By the\nproposition \\ref{prop}, the surface $B_{ij}$ is degree two\nisogenous to $A$. Since $A$ has a $(1,1)$-polarization, it follows\nfrom the example \\ref{example} that $B_{ij}$ has a\n$(1,2)$-polarization.\n\\end{proof}\n\nThe reason why we only consider the even eights not containing\n$E_0$ is because we would obtain the exact same Kummer surface\nwhether we take the double cover branched along an even eight or\nits complement (see proof of the proposition \\ref{prop}).\n\nIn the remaining of this section, we will prove that no two of the\nKummer surfaces $\\mathrm {Kum}(B_{ij})$ are isomorphic.\n\n\\begin{definition}\\label{Nikulin}\nThe \\textit{Nikulin lattice} is an even lattice $N$ of rank eight\ngenerated by $\\{ c_i\\}_{i=1}^8$ and\n$d=\\frac{1}{2}\\sum_{i=1}^8c_i$, with the bilinear form $c_i \\cdot\nc_j = - 2\\delta_{ij}.$\n\\end{definition}\n\\begin{remark}\nIf $C_1, \\dots, C_8$ is an even eight on a K3 surface, then the\nprimitive sublattice generated by the class of the $C_i$'s in the\nN\\'eron-Severi group of $X$ is a Nikulin lattice.\n\\end{remark}\n\nThe following proposition gives a condition on two even eights to\ngive rise to non isomorphic K3 surfaces.\n\n\\begin{prop}\nLet $Y$ be a Kummer surface and let $\\Delta_1$ and $\\Delta_2$ be\ntwo even eights on $Y$. Denote by $X_1$ and $X_2$ the respective\ndouble covers of $Y$. If $N_1, N_2 \\subset S_Y $ are the two\nNikulin lattices corresponding to $\\Delta_1$ and $\\Delta_2$, then\n$$X_1 \\simeq X_2 \\iff \\exists f \\in \\mathrm{Aut}(Y) \\textrm{ such that\n} f^*(N_1)=N_2.$$\n\\end{prop}\n\n\\begin{proof}\n\n\\noindent We suppose that $X_1$ is isomorphic to $X_2$ and we\ndenote by $X_2 \\stackrel{g}\\to X_1$ an isomorphism between $X_2$\nand $X_1$. Let $X_1 \\stackrel{i_1} \\to X_1$ and $X_2\n\\stackrel{i_2} \\to X_2$ be the covering involutions with respect\nto the rational double covers $X_1 \\stackrel{\\tau_1}\n\\dashrightarrow Y$ and $X_2 \\stackrel{\\tau_2} \\dashrightarrow Y$.\n\n\\noindent \\textit{Claim:} The following diagram is commutative:\n$$\\xymatrix{H^2(X_1, \\mathbb Z) \\ar@{->}[r]^{g^*}\\ar[d]^{i^*_1}&\nH^2(X_2, \\mathbb Z) \\ar@{->}[d]^{i^*_2}\\\\ H^2(X_1, \\mathbb Z)\n\\ar@{->}[r]^{g^*}& H^2(X_2, \\mathbb Z).}$$\n\n\n\\noindent \\textit{Proof of the claim:} Suppose that the above\ndiagram does not commute. Then the surface $X_1$ would admit two\ndistinct symplectic involutions, namely $i_1$ and $g \\circ\ni_2\\circ g^{-1}$. Moreover the quotient of $X_1$ by both of these\ninvolutions would be birational to the same Kummer surface $Y$. In\n\\cite{AM}, it is shown that the rational double cover of a Kummer\nsurface $\\mathrm{Kum}(A)$ is determined by an embedding $T_X\n\\hookrightarrow T_A$ preserving the Hodge decomposition of $T_X$\nand $T_A$. Since there is an unique embedding of $T_X$ into $T_A$\nwhich preserves the Hodge decomposition, it follows that\n$i_1=g^{-1} \\circ i_2\\circ g.$\n\nHence $i_2 \\circ g = g \\circ i_1$ and the isomorphism $g$ descends\nto an isomorphism on the quotients  $$X_2\/ i_2 \\stackrel{g}\\to\nX_1\/i_1.$$ Since this isomorphism maps the eight singular points\nof $X_2\/ i_2$ to the eight singular points of $X_1\/ i_1$, it\nextends to an automorphism $Y \\stackrel{f}\\to Y$, for which\n$f^*(N_1)=N_2.$\n\nConversely, let $Y \\stackrel{f} \\to  Y$ be an automorphism of $Y$\nfor which $ f^*(N_1)=N_2$. Denote by $Z_i \\stackrel{p_i}\\to Y$ the\ndouble cover of $Y$ branched along the even eight $N_i$ for\n$i=1,2$. Consider the fiber product $$\\xymatrix{Z_1 \\times_Y Y\n\\ar@{->}[r]^{q}\\ar[d]^{p}& Z_1 \\ar@{->}[d]^{p_1}\\\\Y\n\\ar@{->}[r]^{f}&Y.}$$ The map $Z_1 \\times_Y Y \\stackrel{p} \\to Y$\nis a double cover of $Y$ branched along the even set $N_2$ or\nequivalently $Z_1 \\times_Y Y=Z_2$. Similarly, by considering the\nfiber product $$\\xymatrix{Z_2 \\times_Y Y\n\\ar@{->}[r]^{h}\\ar[d]^{r}& Z_2 \\ar@{->}[d]^{p_2}\\\\Y\n\\ar@{->}[r]^{f}&Y,}$$ we see that $Z_2 \\times_Y Y =Z_1$. The maps\n$h$ and $q=h^{-1}$ define an isomorphism between $Z_1$ and $Z_2$\nwhich induces  the required isomorphism between $X_1$ and $X_2$.\n\\end{proof}\n\nUsing the same notation as in the proposition \\ref{fifteen}, we\nprove the following theorem\n\n\\begin{prop}\nLet $\\Delta_{ij}$ and $\\Delta_{i'j'}$ be two even eights defined\nas in propostion \\ref{fifteen}. $$\\mathrm{Kum}(B_{ij}) \\simeq\n\\mathrm{Kum}(B_{i'j'}) \\Leftrightarrow \\{i,j\\}=\\{i',j'\\}.$$\n\\end{prop}\n\\begin{proof}\nIt is clear that if $\\{i,j\\}=\\{i',j'\\}$, then the corresponding\nKummer surfaces are equal. Thus we only have to prove the other\ndirection. Without loss of generality, we may assume that\n$\\Delta_{i'j'}=\\Delta_{12}$ and we suppose that there exists $f$\nan automorphism of $\\mathrm{Kum}(A)$ for which\n$f^*(\\Delta_{12})=\\Delta_{ij}$.\n\n\\noindent \\textit{Claim}: $$\\{\nf^*(E_{13}),f^*(E_{14}),f^*(E_{15}),f^*(E_{16}),f^*(E_{23}),f^*(E_{24}),f^*(E_{25}),f^*(E_{26})\\}=$$\n$$ \\{ E_{1i},\\cdots, \\hat{E_{ij}}, \\cdots ,E_{i6},E_{1j},\\cdots,\n\\hat{E_{ij}},\\cdots, E_{j6}\\}$$\n\n\\noindent\\textit{Proof of the claim}: Let $N$ be a Nikulin lattice\nand let $D \\in N$ be a divisor represented by a smooth rational\ncurve.  Note that since $D$ is an effective reduced divisor and\n$N$ is negative definite, then $D^2=-2$. It is therefore\nsufficient to show that the only $-2$-classes in $N$ are the\n$c_i$'s and the claim will follow. We write $D$ as $D=\n\\sum_{i=1}^8 \\lambda_i c_i+\\epsilon d$ where $\\lambda_j \\in\n\\mathbb Z$ and $\\epsilon=0$ or $1$. If $\\epsilon=1$, then the\nequality $$D^2=-2\\sum_{i=1}^8 \\lambda_i^2-2\\sum_{i=1}^8\n\\lambda_i-4=-2$$implies that $\\sum_{i=1}^8\n\\lambda^2_i+\\lambda_i=-1.$ Since the latter equation has no\ninteger solution, we conclude that $\\epsilon=0$. Hence\n$$D^2=-2\\sum_{i=1}^8 \\lambda^2_i=-2$$or equivalently,\n$\\sum_{i=1}^8 \\lambda_i^2=1.$ Therefore there exists an unique\n$\\lambda_k$ for which $\\lambda_k=1$ and $\\lambda_i=0$ for $i\\ne\nk$.\n\nIn \\cite{Keum2}, it is proven that any automorphism of a jacobian\ngeneric Kummer surface induces $\\pm \\textrm{identity}$ on\n$D_{S_{\\mathrm{Kum}(A)}}$ where $D_{S_{\\mathrm{Kum}(A)}}$ is the\ndiscriminant group $S_{\\mathrm{Kum}(A)}^*\/S_{\\mathrm{Kum}(A)}$. We want to\napply this fact to the automorphism $f$. We consider the action of\n$f^*$ on the following two independent elements of\n$D_{S_{\\mathrm{Kum}(A)}}$ $$\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24})\n\\textrm{ and } \\frac{1}{2}(E_{12}+E_{23}+E_{15}+E_{35}).$$ From\nthe claim, we deduce that $$f^*(E_{13}+E_{14}+E_{23}+E_{24})=\nE_{i_1i}+E_{i_2i}+E_{j_1j}+E_{j_2j}$$for some classes\n$E_{i_1i},E_{i_2i},E_{j_1j},E_{j_2j} \\in \\Delta_{ij}$.\n\n\\noindent From the identity $f^*_{D_{S_{\\mathrm{Kum}(A)}}}=\\pm\n\\textrm{id}_{D_{S_{\\mathrm{Kum}(A)}}},$ we also deduce that\n$$f^*(\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24}))= \\pm\n\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24}).$$ Putting these two\ninformations together we find that $$E_{13}+E_{14}+E_{23}+E_{24}+\nE_{i_1i}+E_{i_2i}+E_{j_1j}+E_{j_2j} \\in 2S_Y.$$Since the only even\neights containing $E_{13}, E_{14},E_{23},E_{24}$ are $\\Delta_{12}$\nand $\\Delta_{34}$, we deduce that $\\Delta_{ij}=\\Delta_{34}$. We\nproceed similarly for $f^*(E_{12}+E_{23}+E_{13}+E_{35})$ and find\nthat $\\Delta_{ij}$ must be equal to $\\Delta_{25}$ which yields to\na contradiction.\n\\end{proof}\n\n\\begin{corollary}\nThe fifteen Kummer surfaces $\\mathrm{Kum}(B_{ij})$ are not isomorphic.\n\\end{corollary}\n\n\\section{Elliptic Fibration and weak del Pezzo surface}\n\nIn this section, we provide an alternate description of the Kummer\nsurfaces $\\mathrm{Kum}(B_{ij})$ as the double covers of a weak del\nPezzo surface. We relate this construction to the projective\ndouble plane model of the generic jacobian Kummer surface of\nsection 3. First we note the existence on $\\mathrm{Kum}(B_{ij})$ of an\nelliptic fibration that will be useful later. For simplicity, we\nwill always argue for the Kummer surface $\\mathrm{Kum}(B_{12})$.\n\n\\begin{prop}\\label{fibration}\nLet $\\mathrm{Kum}(B_{12})$ be the Kummer surface constructed in the\nproposition \\ref{fifteen}. The surface $\\mathrm{Kum}(B_{12})$ admits a\nWeierstrass elliptic fibration with exactly twelve singular fibers\nof the type $I_2$.\n\\end{prop}\n\\begin{proof}\nLet $\\mathrm{Kum}(A) \\stackrel{\\phi} \\to \\mathbb P^2$ be the double\nplane model of the generic jacobian Kummer surface introduced in\nsection 3. Consider the pencil of lines passing through the point\n$p_{12}$ in $\\mathbb P^2$. Its preimage in $\\mathrm{Kum}(A)$ defines\nan elliptic fibration, given by the divisor class\n$F=L-E_0-E_{12}.$ The divisors $$F_1=E_{15}+ E_{16}+2C_{0}+\nE_{13}+ E_{14}, \\quad \\textrm{and} \\quad F_2=E_{25}+\nE_{26}+2C_{12}+ E_{23}+ E_{24}$$ define two fibers of type $I^*_0$\nof this fibration. Moreover, the six divisors\n\\begin{center}\n$F_3= L-E_0-E_{12}-E_{45}+E_{45}$,\n\n$F_4= L-E_0-E_{12}-E_{46}+E_{46}$,\n\n$F_5= L-E_0-E_{12}-E_{35}+E_{35}$,\n\n$F_6= L-E_0-E_{12}-E_{36}+E_{36}$,\n\n$F_7= L-E_0-E_{12}-E_{34}+E_{34}$,\n\n$F_8= L-E_0-E_{12}-E_{56}+E_{56}$\n\\end{center}\n\\noindent define six $I_2$ fibers. Since the Euler characteristics\nof the $F_i$'s add up to 24, which is equal to the Euler\ncharacteristic of a K3 surface, we conclude by Shioda's formula\n\\cite{Shioda} that the $F_i$'s are the only singular fibers of the\nelliptic fibration defined by the linear system $|F|$. Note also\nthat the curves $C_{13}$, $C_{14}$, $C_{15}$ and $C_{16}$ are\nsections of this fibration.\n\nWe now analyze the induced fibration $\\tau^*F$ on\n$\\mathrm{Kum}(B_{12})$, where\n$\\mathrm{Kum}(B_{12})\\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$ is\nthe rational double cover defined by the even eight $\\Delta_{12}.$\nWe remark that the even eight $\\Delta_{12}$ satisfies\n$$\\Delta_{12} = F_1 +F_2 -2(C_0 +C_{12})$$ which mean that the\neight components of $\\Delta_{12}$ are exactly the eight components\nof the fibers $F_1$ and $F_{2}$ that appear with multiplicity one.\nHence $\\tau^*F_1$ and $\\tau^*F_2$ are just smooth elliptic curves.\nHowever the six fibers $F_3, \\dots, F_8$ split under the cover and\ndefine twelve $I_2$ fibers of the elliptic fibration on\n$\\mathrm{Kum}(B_{12})$ defined by $\\tau^*F$. Again a computation of\nEuler characteristics shows that these twelve $I_2$ fibers are the\nonly singular fibers of the linear system $|\\tau^*F|$. Also the\nsections $C_{13}$, $C_{14}$, $C_{15}$ and $C_{16}$ of $|F|$ pull\nback to sections of $\\tau^*F$, which is therefore a Weierstrass\nelliptic fibration.\n\n\\end{proof}\n\nWe now proceed to the realization of the surface\n$\\mathrm{Kum}(B_{12})$ as a double cover of a weak del Pezzo surface.\nWe decompose the sextic $\\mathcal S$ (see figure \\ref{plane\nsextic}) into the quartic $Q=l_3+l_4+l_5+l_6$ and the conic\n$C=l_1+l_2$.\n\\begin{theorem}\\label{San11}\nThe rational double cover associated to $\\Delta_{12}$,\n$\\mathrm{Kum}(B_{12}) \\stackrel{\\tau}\\dashrightarrow Y$ decomposes as\n$$\\xymatrix{\\mathrm{Kum}(B_{12}) \\ar@{-->}[d]^{\\tau}\\ar[r]^{\\varphi}&\nT \\ar@{->}[d]^{\\zeta}\\\\ Y \\ar@{->}[r]^{\\phi}& \\mathbb P^2}$$ where\n$\\phi$ is the canonical resolution of the double cover of $\\mathbb\nP^2$ branched along $\\mathcal S$. The maps $\\zeta$ and $\\varphi$\nare the canonical resolutions of the double covers branched along\n$Q$ and $\\zeta^*(C)$ respectively.\n\\end{theorem}\n\\begin{proof}\n\nLet $T_0 \\to \\mathbb P^2$ be the double cover of $\\mathbb P^2$\nramified over the reducible quartic $Q$. Its canonical resolution\ninduces the diagram $$\\xymatrix{T \\ar@{->}[d]\n\\ar[r]\\ar[rd]^{\\zeta}& T_0 \\ar@{->}[d]\\\\ \\tilde{\\mathbb P}^2\n\\ar@{->}[r]& \\mathbb P^2}$$ where $\\tilde{\\mathbb P}^2 \\to \\mathbb\nP^2$ is the the blowup of $\\mathbb P^2$ at the six singular points\nof $Q$. The surface $T$ is a non-minimal rational surface\ncontaining six disjoint smooth rational curves. Indeed by Hurwitz\nformula, the canonical divisor of $T$ is given by\n\n$$K_{T}= \\zeta^*(K_{\\mathbb\nP^2}+\\frac{1}{2}(l_3+l_4+l_5+l_6))=-\\zeta^*(H)$$where $H$ is a\nhyperplane section. Thus $K_T^2=2, H^2 =2$ and $P_2(T)=0$. Denote\nby $\\tilde{Q}$ the proper transform of $Q$ in $T$. Using the\nadditivity of the topological Euler characteristic and the Noether\nformula, we have that $$e(T)= e(T- \\tilde{Q}) + e(\\tilde{Q})=10\n\\Rightarrow \\mathcal X(\\mathcal O_T)=1 \\Rightarrow q(T)=0$$ By\nCastelnuovo's rationality criterion, $T$ is a rational surface. In\nfact, we show that $T$ is a weak del Pezzo surface of degree two,\ni.e. the blow up of $\\mathbb P^2$ at seven points with nef\ncanonical divisor. Indeed we successively blown down the preimages\nin $T$ of the four lines $l_3, l_4, l_5$ and $l_6$ as well as the\npreimages in $T$ of the three ``diagonals'' of the complete\nquadrangle formed by $l_3, l_4, l_5, l_6$. The surface obtained\nafter these seven blow down is a projective plane.\n\nConsider the following curves of $T$\n$$\\zeta^*(C)=\\zeta^*(l_1+l_2)= E_{1} + E_{2}, \\textrm{ where } E_1\n\\textrm{ and } E_2 \\textrm{ are smooth elliptic curves}$$ and\n$$\\zeta^*(W)=W_1+W_2 \\textrm{ where } W_1 \\textrm{ and } W_2\n\\textrm{ are smooth rational curves}$$ with the following\nintersection properties:\n$$E_i^2=2,  \\quad W_i^2=0, \\quad E_1 \\cdot E_2=2, \\quad W_1 \\cdot\nW_2=4, \\quad W_i \\cdot E_j=2 \\quad \\textrm{ for}\\quad i\\neq j.$$\n(recall that $W$ is the plane conic tangent to the six lines $l_1,\n\\cdots,l_6$). The linear system $|E_1|$ defines an elliptic\nfibration on $T$ with six singular fibers of type $I_2$. Take the\ndouble cover branched along the two fibers $E_1 + E_2 \\in\n2\\mathrm{Pic(T)}$. It induces the canonical resolution commutative\ndiagram: $$\\xymatrix{X \\ar@{->}[d]\\ar[r] \\ar[dr]^{\\varphi}& X_0\n\\ar@{->}[d]\\\\ \\tilde{T} \\ar@{->}[r]& T}$$ where $\\tilde{T} \\to T$\nis the blowup of $T$ at the two singular points of $E_1+E_2$.\n\n\\textit{Claim:} X is a Kummer surface.\n\n\\textit{Proof of the Claim:} Clearly $\\mathcal\nK_{X}=\\varphi^*(\\zeta^*(-H)+ \\frac{1}{2}(E_1+E_2))=\\mathcal O_X$.\n\n\n1) The pullback by $\\varphi$ of the six exceptional curves on $T$ define\ntwelve disjoint smooth rational curves on $X$.\n\n2) The two exceptional curves of $X$ give two more rational curves\ndisjoint from 1).\n\n3) Let $\\varphi^*(W_1)=W'_1+W''_1$ and $\\varphi^*(W_2)=W'_2+W''_2$\nand let $\\sigma$ be the lift on $X$ of the covering involution of\n$\\zeta$, then $\\sigma(W'_1)=W'_2$ or $\\sigma(W'_1)=W''_2$. Without\nloss of generality, we can assume that $\\sigma(W'_1)=W'_2$ and\nhence get the following intersection numbers $$W'^2_i=W''^2_i=-2,\n\\quad W'_i\\cdot W''_i=2 \\quad \\textrm{for } i=1,2 $$ $$ W'_1\\cdot\nW'_2=W''_1\\cdot W''_2=4 \\quad \\textrm{and} \\quad W'_1\\cdot\nW''_2=W''_1\\cdot W'_2=0.$$ One easily checks that $W'_1$ and\n$W''_2$ do not intersect the fourteen curves from 1) and 2).\n\n\\noindent In particular, the $K3$ surface $X$ contains sixteen\ndisjoint smooth rational curves. Consequently $X$ is a Kummer\nsurface.\n Moreover, the surface $X$ contains an elliptic fibration with twelve $I_2$ fibers. It\n also admits two non symplectic involutions $\\theta$ and $\\sigma$ where $\\theta$ is the covering involution of\n the map $\\varphi$ and $\\sigma$ is the lift of the covering involution of $\\zeta$ on $T$ encountered earlier.\n The composition $\\iota = \\varphi \\circ \\sigma$ defines a symplectic involution on $X$ whose quotient is a $K3$\n surface admitting an elliptic fibration with singular fibers identical to the one defined by $F$ on $Y$ in the proposition\n \\ref{fibration}.\n\n In fact, we can now recover sixteen disjoint rational curves on the quotient and conclude that\n it is our original general Kummer surface $Y$ and that $X \\simeq \\mathrm{Kum}(B_{12})$.\n\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLow power consumption is a key requirement for modern computational devices.\nNon-volatility is one of the core concepts to reduce \npower consumption for logics and memories\nin normally-off computing\n\\cite{ando_fed_2001, ando_spin-transfer_2014, nakada_normally-off_2017}. \nMagnetoresistive random access memory (MRAM) is a promising\nnon-volatile memory that stores information \nassociated with the direction of\nmagnetization in magnetic tunnel junctions (MTJs)\n\\cite{yuasa_giant_2004, parkin_giant_2004, djayaprawira_230_2005, yuasa_giant_2007,\nkishi_lower-current_2008, kitagawa_impact_2012,\napalkov_magnetoresistive_2016, sbiaa_recent_2017, cai_high_2017}. \nIn order to reduce\npower consumption of MRAM, \nseveral types of writing schemes have been\ndeveloped. The currently used writing scheme is based on the\nspin-transfer-torque (STT) switching phenomena, which were\nproposed by\nSlonczewski \\cite{slonczewski_conductance_1989,slonczewski_current-driven_1996}\nand independently by Berger \\cite{berger_emission_1996}. The write\nenergy of STT-MRAM is of the order of 100 fJ\/bit\n\\cite{kitagawa_impact_2012, cai_high_2017}, which is\nstill 2 orders of magnitude larger than that of static random-access\nmemory.\n\nDiscovery of the voltage control of magnetic anisotropy (VCMA) effect\n\\cite{weisheit_electric_2007, maruyama_large_2009, duan_surface_2008,\n  nakamura_giant_2009, tsujikawa_finite_2009, nozaki_voltage-induced_2010,\n  endo_electric-field_2010,\n  nozaki_magnetization_2014, skowronski_perpendicular_2015, nozaki_large_2016,\n  li_enhancement_2017}\npaved the way for further reduction of write energy in MRAM.\nThe mechanism of VCMA in an MgO-based MTJ is considered to be the\ncombination of the selective electron or hole doping into the\n$d$-electron orbitals and the induction of a magnetic dipole moment,\nwhich affect the electron spin through spin-orbit interaction\n\\cite{duan_surface_2008,nakamura_giant_2009,tsujikawa_finite_2009,miwa_voltage_2017}.\nThe MRAM which uses the VCMA effect to switch magnetization is called\nthe voltage controlled MRAM (VC-MRAM)\n\\cite{shiota_induction_2012, shiota_pulse_2012,\nkanai_electric_2012,shiota_evaluation_2016, grezes_ultra-low_2016, kanai_electric-field-induced_2016,\nshiota_reduction_2017, \nmatsumoto_voltage-induced_2018, yamamoto_thermally_2018, yamamoto_write-error_2019, \nmatsumoto_voltage-induced_2019, imamura_impact_2019, matsumoto_methods_2019}.\nThe writing procedure of a conventional VC-MRAM is as follows.\nThe perpendicularly magnetized MTJ is subjected to an in-plane\nexternal magnetic field ($H_{\\rm ext}$) as shown in Fig. \\ref{fig:schem}(a).\nThe magnetic anisotropy (MA) constant of the free layer can be controlled by\napplying voltage ($V$) as shown in Fig. \\ref{fig:schem}(b).\nHere, $K_{\\rm eff}$ is the effective perpendicular anisotropy constant\nwhere the demagnetization energy is subtracted from the perpendicular anisotropy constant.\nThroughout the paper, the superscript (0) indicates the quantities at $V = 0$.\nThe voltage pulse with critical amplitude $V_{\\rm c}$\neliminates the MA and induces the precession of the magnetization around\nthe external magnetic field. By turning off the voltage at one half\nperiod of precession, the magnetization switching completes.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics [width=\\columnwidth] {fig1.eps}}\n  \\caption{\n  \\label{fig:schem} \n  (a)  Magnetic tunnel junction with circular cylinder shape,\n  and definitions of Cartesian coordinates $(x, y, z)$, polar angle\n  ($\\theta$), and azimuthal angle ($\\phi$).\n  The $x-$axis is parallel to the direction of \n  the external in-plane magnetic field, ${\\bm H}_{\\rm ext}$.\n  The unit vector ${\\bm m} = (m_{x}$, $m_{y}$, $m_{z})$ \n  represents the direction of the magnetization in the free layer.\n  The magnetization in the reference layer (ref.) is fixed to align in the\n  positive $z-$direction.\n  (b) The voltage ($V$) dependence of the effective perpendicular anisotropy constant ($K_{\\rm eff}$).\n  The effective anisotropy constant at $V=0$ is represented by\n  $K_{\\rm eff}^{(0)}$. It takes the value of\n  $K_{\\rm eff} = K_{\\rm eff}^{\\rm  p}$ at $V=V_{\\rm p}$.\n  }\n\\end{figure}\n\nThe write energy of VC-MRAM is estimated from the Joule heating energy loss\nduring the pulse. Assuming that the voltage\npulse with amplitude $V$ and duration $t_{\\rm p}$ is applied to the MTJ with\nresistance $R$, the write energy is given by \n\\begin{equation}\n  E_{\\rm J} = \\frac{V^{2}}{R}t_{\\rm p},\n\\end{equation}\nTo reduce the write energy, \nthe VC-MRAM should be designed to have large\nresistance and short pulse duration. The pulse duration is given by\na half period of precession as\n\\begin{align}\n  \\label{eq:period_Hext}\n  t_{\\rm p} = \\frac{\\pi (1+\\alpha^{2}) }{\\gamma H_{\\rm ext}},\n\\end{align}\nwhere $\\alpha$ is the Gilbert damping constant and $\\gamma$ is the\ngyromagnetic ratio.  For example, $t_{\\rm p}$ = 0.18 ns for $\\alpha$ =\n0.1 and $\\mu_{0} H_{\\rm ext}$ = 100 mT, where $\\mu_{0}$ is the vacuum\npermeability. Recently, Grezes $et$ $al$. demonstrated \na very small write energy of 6 fJ\/bit for the VC-MRAM with $R=330$ k$\\Omega$ \nat $V$ = 1.96 V and $t_{\\rm p}$ = 0.52 ns\n\\cite{grezes_ultra-low_2016}. The similar results were also obtained\nindependently by Kanai $et$\n$al$. \\cite{kanai_electric-field-induced_2016}.\n\n\nIt is difficult to use MTJ with huge $R$ to further reduce\nwrite energy because the read time of the VC-MRAM increases \nwith increase of $R$. \nAdopting a scheme of decreasing  \npulse duration\nby increasing external magnetic field \nshould also be avoided \nsince the application of a\nstrong in-plane magnetic field $H_{\\rm ext}$ deteriorates the thermal stability\nfactor defined as\n\\begin{align}\n  \\label{eq:delta}\n  \\Delta^{(0)}\n  =\n  \\frac{\n    \\left(2 K_{\\rm eff}^{(0)}\n    - \\mu_{0}M_{\\rm s} H_{\\rm ext}\n    \\right)^{2}\n    V_{\\rm F}\n  }{4K_{\\rm  eff}^{(0)} k_{\\rm B}T},\n\\end{align}\nwhere \n$k_{\\rm B}$ is the Boltzmann constant, $T$ is the temperature,\n$M_{\\rm s}$ is the saturation magnetization, \nand $V_{\\rm F}$ is the volume of the free layer.\n\n\n\n\nIn this paper, we propose another switching scheme which could reduce the pulse\nduration and therefore the write energy of a VC-MRAM. The main\ndifference between the conventional scheme and the proposed switching scheme is\nthe polarity of the voltage pulse. Application of the voltage pulse\nwith the polarity that is opposite to the conventional switching can\nenhance the magnetic anisotropy and induce the precession around the\naxis close to the easy axis. After turning off the voltage \nproximately at a\nhalf of precession period, the magnetization relaxes to the opposite\nequilibrium direction and the switching completes.\nWe perform numerical simulations and demonstrate that the pulse\nduration of the proposed switching scheme is as  short as a few tens\nof pico seconds. We also evaluate the write error rate (WER) and\nshow that the WER is minimized if the pulse duration is about half the\nperiod of precession similar to the conventional switching scheme.\n\n\n\\section{Theoretical model}\nThe system we consider is schematically shown in Fig. \\ref{fig:schem}(a).\nThe macrospin model is employed to describe the magnetization dynamics.\nThe direction of the magnetization in the free layer is\nrepresented by the unit vector ${\\bm m} = (m_{x}$, $m_{y}$, $m_{z})\n= (\\sin \\theta \\cos \\phi$, $\\sin \\theta \\sin \\phi$, $\\cos \\theta$), \nwhere $\\theta$ and $\\phi$ are the polar and azimuthal angles.\nThe $x$ axis is parallel to the direction of external in-plane\nmagnetic field ${\\bm H}_{\\rm ext}$.\n\nThe energy density of the free layer is given by\n\\begin{align}\n  \\label{eq:energy_density}\n  {\\cal E} (m_{x}, m_{y}, m_{z})\n  = \n   -K_{\\rm eff} m_{z}^{2} - \\mu_{0} M_{\\rm s} H_{\\rm ext} m_{x},\n\\end{align}\nThe first term of Eq. \\eqref{eq:energy_density} is the sum of the\nshape, the bulk crystalline and the interfacial\nanisotropies.  Owing to the VCMA effect, \n$K_{\\rm eff}$ can be controlled\nby application of $V$ as shown in Fig. \\ref{fig:schem}(b).\nHere $K_{\\rm eff}^{(0)}$ represents the effective anisotropy constant\nwithout the voltage application. We assume that $K_{\\rm eff}$\ndecreases with increase of $V$ and vanishes at $V=V_{\\rm c}$.\nApplying the voltage $V_{\\rm p} ( < 0)$ increases\n$K_{\\rm eff}$ to $K_{\\rm eff}^{\\rm p}$ and induces\nthe precessional motion of $\\bm{m}$ around the effective magnetic\nfield. The effective field is given by \n$H_{\\rm eff} = (H_{\\rm ext},$ $0,$ $H_{\\rm\n  K} m_{z})$, where ${H}_{\\rm K} =  2 K_{\\rm eff}^{\\rm p} \/ ( \\mu_{0}\nM_{\\rm s} )$ is the anisotropy field.\n\n\nThe magnetization dynamics is simulated by solving \nthe following Landau-Lifshitz-Gilbert equation \\cite{brown_thermal_1963},\n\\begin{equation}\n \\label{eq:LLG}\n \\frac{{\\rm d} {\\bm m}}{{\\rm d}t}\n = -\\gamma_{0} {\\bm m}\\times\n \\left(\\bm{H}_{\\rm eff} + \\bm{h}\\right)\n +\\alpha\n \\bm{m}\\times\n  \\frac{{\\rm d} {\\bm m}}{{\\rm d}t},\n\\end{equation}\nwhere $\\bm{h}$ represents the thermal agitation field satisfying the\nfollowing relations:\n\\begin{align}\n  &\\langle h_{\\iota}(t)\\rangle=0\n  \\\\\n  & \\langle\nh_{\\iota}(t)h_{\\kappa}(t') \n\\rangle\n= \\frac{2\\alpha k_{\\rm B} T }{ \\gamma_{0} \\mu_{0} M_{\\rm s} V_{\\rm F} }\\delta_{\\iota\\kappa}\\delta(t-t').\n\\end{align}\nHere  $\\iota,\\kappa=x,y,z$, \nand $\\langle X \\rangle$ denotes the statistical average of $X$.\n\nThroughout this paper, we assume that the external field is $\\mu_{0}\nH_{\\rm ext}$ = 100 mT and the saturation magnetization of the free\nlayer is $M_{\\rm s}= 1400$ kA\/m.\nAlso the radius of the junction area is assumed as $r=50$ nm and the\nthickness of the free layer, $t_{\\rm F}=1$ nm, and therefore the volume of\nthe free layer as $V_{\\rm F}=\\pi r^{2} t_{\\rm F}=7854$ nm$^{3}$. \nThe initial states are prepared by 10 ns relaxation from the\nequilibrium direction at $T=0$, that is \n$(\\theta^{(0)},$ $\\phi^{(0)})  =  \\left( \\sin^{-1}  \\left[ \\mu_{0} M_{\\rm s} H_{\\rm ext}  \/ (2 K_{\\rm eff}^{(0)} ) \\right] \\right.,$ \n$ 0 \\Bigr)$\n\\cite{matsumoto_voltage-induced_2019}. \nThe write error rates are calculated from\n$10^{6}$ trials with 10 ns relaxation after the pulse.\n\n\n\n\\section{Results and discussions}\nFirst we show the difference between the mechanisms of the conventional\nvoltage controlled switching and the proposed switching that utilizes\nthe enhancement of the magnetic anisotropy.\nThis can be accomplished by analyzing the switching\ntrajectories at $T=0$.\nFigures \\ref{fig:t0}(a) and (b) show the shape of the voltage\npulse and the corresponding time dependence of the effective\nanisotropy constant for the conventional voltage controlled switching.\nThe induced switching dynamics of $\\bm{m}$ at $T=0$ is shown in\nFigs. \\ref{fig:t0}(c) together with the color map of the energy\ndensity of Eq. \\eqref{eq:energy_density} at $V=0$.\nThin black dotted curves represent energy contours. Thick black curves\nrepresent the energy contour crossing\n$\\bm{m}=(1,$ $0,$ $0)$. The initial direction of the magnetization is the\nequilibrium direction with $m_{z}>0$ indicated by the black circle,\nwhich we call as the ``up state''.  \n\nIn Figs. \\ref{fig:t0}(a), (b) and (c),\napplication of the voltage pulse with $V_{\\rm c}$\neliminates the magnetic anisotropy and induce the precession of $\\bm{m}$\naround the external magnetic field as represented by the red\ncurve.  After turning off the voltage at one-half period of precession, \nthe magnetization starts to relax from the point indicated by the orange\ncircle\nto the other equilibrium\ndirection with $m_{z}<0$, i. e. the ``down state'', indicated by the\nblack circle. \nNote that the black circle at $m_{z}<0$ is illustrated \nunder the green curve. \nThe switching is thus completed as represented by the green\ncurve.\n\n\\begin{figure}[H]\n  \\centerline{\n  \\includegraphics [width=0.8\\columnwidth] {fig2.eps}\n  }\n  \\caption{\n    \\label{fig:t0} \n    (a) The shape of the voltage pulse for the {\\em conventional switching\n    scheme}. The amplitude including the polarity of the pulse and duration of the pulse are \n    $V_{\\rm c}$ (positive value) and $t_{\\rm p}$, respectively.\n    (b) The corresponding time dependence of the effective anisotropy\n    constant $K_{\\rm eff}$. At $V=0$, it takes the value $K_{\\rm eff}^{\\rm (0)}$. \n    During the pulse, $K_{\\rm eff} = 0$ because $V=V_{\\rm c}$.\n    (c) The color map of the energy density at $V=0$ on the\n    $\\phi-m_{z}$ plane. Thin black dotted curves represent energy\n    contours. Thick black curves represent the energy contour crossing\n    $\\bm{m}=(1,$ $0,$ $0)$.\n    The trajectory of $\\bm{m}$ during and after the pulse\n    are shown by the red and green curves, respectively. The direction\n    of the trajectory is indicated by the triangle. The orange circle\n    represents the direction of  ${\\bm m}$ at the end of the pulse.\n    We assume that $\\alpha$ = 0.1.\n    (d) The shape of the voltage pulse for the {\\em proposed switching scheme}.\n    The polarity is negative, i. e. $V_{\\rm p}<0$, to enhance $K_{\\rm eff}$.\n    (e) The corresponding time dependence of the effective anisotropy\n    constant.  During the pulse, it is enhanced to $K_{\\rm eff}^{\\rm p}$.\n    (f) The color map of the energy density at $V=0$ on the\n    $\\phi-m_{z}$ plane.  We assume that $K_{\\rm eff}^{\\rm p}$ = 400\n     kJ\/m$^{3}$ and $\\alpha$ = 0.21. The symbols are the same as those\n     in Panel (c).\n     Please note that the left and right boundaries at  $\\phi =\n    \\pm \\pi$ represent the same direction of $\\bm{m}$.\n}\n\\end{figure}\n\nFigures \\ref{fig:t0}(d) and (e) show the shape of the voltage pulse\nand the corresponding time dependence of the effective anisotropy\nconstant for the switching using the enhanced $K_{\\rm eff}$.\nThe induced switching dynamics of $\\bm{m}$ at $T=0$ is shown in\nFig. \\ref{fig:t0}(f) together with the color map of the energy\ndensity at $V$ = 0. \nThe initial state is the up state indicated by the\nblack circle at $m_{z}>0$. Application of the voltage pulse with $V_{\\rm p} (< 0)$ \nenhances the effective anisotropy constant from $K_{\\rm eff}^{(0)}$ to\n$K_{\\rm eff}^{\\rm p}$ and induce the precession of $\\bm{m}$\naround the effective magnetic field as represented by the red curve.\nThe value of $K_{\\rm eff}^{\\rm p}$ is assumed to be 400\nkJ\/m$^{3}$, which gives the anisotropy field of $\\mu_{0} H_{\\rm K}$ = 570\nmT. The effective field is nearly parallel to the easy axis or the $z$\naxis because the directional cosine of the effective field relative to the\neasy axis is 0.98. \nThe voltage is turned off at about a half period\nof the precession, and the magnetization reaches the point, \n$\\phi \\simeq \\pi$ indicated by the orange circle. As will be shown later, \nthe write error rate (WER) is minimized if the pulse duration is set about\nhalf the period of precession. After turning off the pulse, the\nmagnetization relaxes to the down state and completes the switching as\nshown by the green curve. The proposed switching scheme does not\nreduce the thermal stability factor of Eq. \\eqref{eq:delta} \nbecause it\njust enhances $K_{\\rm eff}$ during the voltage pulse.\n\n\n\\begin{figure}[H]\n\\includegraphics [width=\\columnwidth] {fig3.eps}\n\\caption{\\label{fig:mm} \n  (a) The Cartesian components of ${\\bm m} = (m_{x}$, $m_{y}$,\n  $m_{z})$ of a typical switching trajectory \n  are plotted as a function of time\n  during the\n  pulse at $T$ = 300 K.\n  $K_{\\rm eff}^{\\rm p}$ = 400 kJ \/m$^{3}$ and $\\alpha$ = 0.21.\n  The unit of the horizontal axis is ps.  \n  (b) The same as (a) after the pulse. The unit of the horizontal axis is ns.  \n  (c) The pulse duration dependence of the WER at $T$ = 300 K for\n  $K_{\\rm eff}^{\\rm p}$ = 300 kJ \/m$^{3}$ and $\\alpha$ = 0.18.\n  (d) The same as (c) for $K_{\\rm eff}^{\\rm p}$ = 400 kJ \/m$^{3}$ and\n  $\\alpha$ = 0.21.\n  }\n\\end{figure}\n\nNext we discuss the switching properties of the proposed switching\nscheme at $T$ = 300 K by analyzing the numerical simulations results.\nThe time evolution of the Cartesian components of ${\\bm m}$ for a\ntypical switching trajectory during the pulse are\nshown in Fig. \\ref{fig:mm}(a). The value of $K_{\\rm eff}^{(0)} = 100$ kJ\/m$^{3}$\nand $\\alpha$ are \nthe same as in Fig. \\ref{fig:t0}(f),\n$K_{\\rm eff}^{\\rm p}$ = 400 kJ\/m$^{3}$ and $\\alpha$ = 0.21.\nDuring the pulse duration, \n$m_{z}$ increases with the\nincrease of time because the effective anisotropy constant is\nenhanced. The shapes of $m_{x}$ and $m_{y}$ are very similar to\nthe cosine and sine functions, respectively, \nbecause \n$\\bm{m}$ precesses around the effective field which is almost parallel\nto the $z$ axis. \nFigure \\ref{fig:mm}(b) shows the time evolution of $m_{x}$, $m_{y}$\nand $m_{z}$ after the pulse. Please note that the horizontal axis is\nin unit of ns. $m_{z}$ monotonically decreases with the increase of time\nand the switching completes at around 0.4 ns.\n\n\nFigures \\ref{fig:mm}(c) and (d) show the pulse duration, $t_{\\rm p}$,\ndependence of the write error rate (WER) for different\nvalues of $K_{\\rm eff}^{\\rm p}$ and $\\alpha$.\nThe parameters are  $K_{\\rm eff}^{\\rm p}$ = 300 kJ \/m$^{3}$ and\n$\\alpha$ = 0.18 for Fig. \\ref{fig:mm}(c), \nand $K_{\\rm eff}^{\\rm p}$ =\n400 kJ \/m$^{3}$ and $\\alpha$ = 0.21 for Fig. \\ref{fig:mm}(d).\nIn Fig. \\ref{fig:mm}(c), the WER takes a minimum value of $7.6 \\times\n10^{-3}$  at $t_{\\rm p}$ = 46 ps.\nIn Fig. \\ref{fig:mm}(d), the WER takes a minimum value of $3.2 \\times\n10^{-3}$  at $t_{\\rm p}$ = 36 ps. These optimal values of $t_{\\rm p}$ at which\nthe WER is minimized are almost the same as one half period of precession around\n$H_{\\rm eff}$.\n\n\n\n\\section{conclusion}\nIn summary, we propose a low power switching scheme of magnetization using\nenhanced magnetic anisotropy by applying a short voltage pulse.\nThe proposed switching scheme can reduce the pulse duration and\ntherefore the write energy substantially without  deteriorating \nthermal stability.\nWe perform numerical simulations and show that the pulse duration of\nthe proposed switching scheme is as short as a few\ntens of pico seconds.  We also calculated the pulse duration\ndependence of the WER, \nand showed that the optimal values of $t_{\\rm\n  p}$ at which the WER is minimized are nearly half the period of\nprecession around the effective field.\n\n\\acknowledgements\nThis work was partly supported by JSPS KAKENHI Grant No. JP19K05259\nand 19H01108.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nNetwork models are nowadays ubiquitous in the natural, information, social,\nand engineering sciences. The last 15 years or so have seen the emergence of the\nvast, multidisciplinary field of Network Science, with contributions\nfrom a wide array of researchers including physicists, mathematicians, \ncomputer scientists, engineers, biologists, and social scientists \n\\cite{Linked,EstradaBook,NewmanBook}. Applications of Network\nScience range from biology to public health, from social network analysis to   \nhomeland security, from economics to the humanities, from marketing\nto information retrieval. Network analysis is\nalso an essential ingredient in the design of information, communication,\nand transportation networks, as well as in energy-related disciplines\nsuch as power grid maintenance, control, and optimization \\cite{Pinar2010}.\nGraph theory and linear algebra provide abstractions and quantitative \ntools that can be employed in the analysis and design of large and\ncomplex network models.\n\nReal-world networks are characterized by structural properties \nthat make them very different from both regular graphs on one hand, and  \ncompletely random graphs on the other. Real networks frequently exhibit\na highly skewed degree distribution (often following a power law), small\ndiameter, high clustering coefficient (the two last properties together are\noften referred to as the {\\em small world} property), the presence of\nmotifs, communities, and other signatures of complexity.   \n\nSome of the basic questions in network analysis concern node and edge centrality, \ncommunity detection, communicability, and diffusion \n\\cite{Brandes,EstradaBook,NewmanBook}. Related to these are the\nimportant notions of network robustness (or its opposite, vulnerability) and \nconnectivity \\cite{Cohen}. These latter properties refer to the degree of resiliency\ndisplayed by the network in the face of random accidental failures or \ndeliberate, targeted attacks, which can be modeled in terms of edge or node\nremoval. Generally speaking, it is desirable to design networks that are at\nthe same time highly sparse (in order to reduce costs) and highly connected,\nmeaning that disconnecting or disrupting the network would require the removal of a large\nnumber of edges. Such networks should not contain bottlenecks, and they\nshould allow for the rapid exchange of communication between nodes.\nExpander graphs \\cite{E06b,HLW06} are an important class of graphs \nwith such properties. \n\n\nIn this paper we describe some techniques that can be brought to bear on the\nproblems described above and related questions.\nOur approach is based on the notion of {\\em total communicability} of a\nnetwork, introduced in \\cite{Benzi2013} on the basis of earlier work\nby Estrada and Hatano \\cite{EH08,EHB12}. Total communicability, defined\nas the (normalized) sum of the entries in the exponential of the adjacency\nmatrix of the network, provides a global measure of how well the nodes\nin a graph can exchange information. \nCommunicability is based on the number and length\nof graph walks connecting pairs of nodes in the network. Pairs of nodes $(i,j)$\nwith high communicability correspond to large entries $[e^{A}]_{ij}$ in the\nmatrix exponential of $A$, the adjacency matrix of the network.\n\nTotal network communicability can also be used to measure the \nconnectivity of the network as a whole. For instance, given two alternative\nnetwork designs (with a similar ``budget\" in terms of number\nof candidate edges), one can compare the two designs by computing the\nrespective total communicabilities and pick the network with the highest one,\nassuming that a well-connected network with high node communicability is\nthe desired goal. \nIt is important to stress that\nthe total communicability of a network can be efficiently computed or estimated\neven for large networks using Lanczos or Arnoldi based algorithms without\nhaving to compute any individual entry of $e^A$ (only the ability to perform\nmatrix-vector products with $A$ is required).\n\n\nIn this paper we consider three different problems. \nLet $G=(V,E)$ be a connected, undirected and sparse graph. \nThe {\\it downdating problem} consists of selecting an edge $(i,j)$ to \nbe removed from the network so as to minimize the decrease in its \ntotal communicability while preserving its connectedness.\n\nThe goal when tackling the {\\it updating problem}, on the other hand, is to select a pair of \nnodes $i\\neq j$ such that $(i,j)\\not\\in E$ in such a way that the increase in \nthe total communicability of the network is maximized. \n\nFinally, the {\\it rewiring problem} has the same goal as the updating problem, but it requires \nthe selection of two modifications which constitute the downdate-then-update step to be performed.\n\nThe importance of the first two problems for network analysis and\ndesign is obvious.\nWe note that an efficient solution to the second problem would also \nsuggest how to proceed if the goal was to identify existing edges whose\nremoval would {\\em maximize} the decrease in communicability, which could\nbe useful, e.g., in planning anti-terrorism operations or public health policies \n(see, e.g., \\cite{Tong, VM2011}).\nThe third problem is motivated by the observation that\nfor transportation networks (e.g., flight routes) it is sometimes \ndesirable to redirect edges in order to improve the performance \n(i.e., increase the number of travellers) without increasing too much \nthe costs. \nHence, in such cases, one wants to eliminate a route used only by a few \ntravellers and to add a route that may be used by a lot of people.\n\nThe above problems may arise not only in the design of infrastructural networks\n(such as telecommunication or transportation networks), but also in other\ncontexts. For instance, in social networks the addition of a friendship\/collaborative\ntie may change dramatically the structure of the network, leading\nto a more cohesive\ngroup, and hence preventing the splitting of the community into smaller\nsubgroups.\n \n\n\nThe work is organized as follows. \nSection \\ref{sec:background} contains some basic facts from linear \nalgebra and graph theory, and introduces the modifications of the adjacency \nmatrix we will perform. \n In this section we also provide further justification\nfor the use of the total network communicability as the objective\nfunction.\nIn section \\ref{sec:bounds}  we describe bounds for the \ntotal communicability \nvia the Gauss--Radau quadrature rule and we show how these bounds \nchange when a rank-two modification of the adjacency matrix is performed.\nSection \\ref{sec:modification} is devoted to the introduction of the \nmethods to controllably modify the graph in order to adjust the value of its \ntotal communicability.\nNumerical studies to assess the effectiveness and performance of the \ntechniques introduced are provided in section \\ref{sec:test_tc} for \nboth synthetic and \nreal-world networks.\nIn section \\ref{sec:nc+gap} we discuss the evolution of \na popular measure of network connectivity, known as\nthe {\\em free energy} (or {\\em natural connectivity}), \nwhen the same modifications are performed. \nThis section provides further evidence that motivates the use of the total \ncommunicability as a measure of connectivity. \nFinally, in section \\ref{sec:conclusions} we draw conclusions and we describe \nfuture directions.\n\n\n\n\\section{Background and definitions}\n\\label{sec:background}\nIn this section we provide some basic\ndefinitions, notations, and properties associated with graphs.\n\nA {\\itshape graph} or {\\itshape network}\n$G=(V,E)$ is defined by a set of $n$ nodes (vertices) \n$V$ and a set of $m$ edges $E=\\{(i,j)|i,j\\in V\\}$ between the nodes. \nAn edge is said to be {\\itshape incident} to a vertex $i$ if there exists \na node $j\\neq i$ such that either $(i,j)\\in E$ or $(j,i)\\in E$.\nThe {\\itshape degree} of a vertex, denoted by $d_i$, is the number of \nedges incident to $i$ in $G$. \nThe graph is said to be {\\itshape undirected} if the edges are formed by \nunordered pairs of vertices. \nA {\\itshape walk} of length $k$ in $G$ is a set of nodes $i_1, i_2,\\ldots,i_k, \ni_{k+1}$ such that for all $1\\leq l\\leq k$, $(i_l,i_{l+1})\\in E$.\nA {\\itshape closed walk} is a walk for which $i_1=i_{k+1}$. \nA {\\itshape path} is a walk with no repeated nodes. \nA graph is {\\itshape connected} if there is a path connecting every pair of nodes.\nA graph with unweighted edges, no self-loops (edges from a node to itself), \nand no multiple edges is said to be {\\itshape simple}. \nThroughout this work, we will consider undirected, simple, and connected networks.\n\nEvery graph can be represented as a matrix \n$A=\\left(a_{ij}\\right)\\in\\mathbb{R}^{n\\times n}$, called the \n{\\itshape adjacency matrix} of the graph. \nThe entries of the adjacency matrix of an unweighted graph $G=(V,E)$ are \n\\begin{equation*}\na_{ij}=\\left\\{\n\\begin{array}{ll}\n1 & \\mbox{if } (i,j)\\in E\\\\\n0 & \\mbox{otherwise}\n\\end{array}\n\\right.\\qquad \\forall i,j\\in V.\n\\end{equation*}\nIf the network is simple, the diagonal elements of the adjacency matrix \nare all equal to zero.\nIn the special case of an undirected network, the associated adjacency matrix \nis symmetric, and thus its eigenvalues are real.\n\nWe label the eigenvalues in non-increasing order: \n$\\lambda_1\\geq\\lambda_2\\geq\\cdots \\geq \\lambda_n$.\nSince $A$ is a real-valued, symmetric matrix, we can decompose $A$ \ninto $A=Q\\Lambda Q^T$ where $\\Lambda$ is a diagonal matrix containing the \neigenvalues of $A$ and $Q=[\\mathbf{q}_1,\\ldots,\\mathbf{q}_n]$ is orthogonal, \nwhere $\\mathbf{q}_i$ is an eigenvector associated with $\\lambda_i$.\nMoreover, if $G$ is connected, $A$ is irreducible and from the \nPerron--Frobenius Theorem \\cite[Chapter 8]{Meyer00} we deduce that $\\lambda_1>\\lambda_2$ \nand that the leading eigenvector $\\mathbf{q}_1$, \nsometimes referred to as the {\\itshape Perron vector}, \ncan be chosen such that its components $q_1(i)$ are positive \nfor all $i\\in V$.\n\nWe can now introduce the basic operations which will be performed \non the adjacency matrix $A$ associated with the network $G=(V,E)$.\nWe define the {\\itshape downdating} of the edge $(i,j)\\in E$ as the \nremoval of this edge from the network.\nThe resulting graph $\\widehat{G}=(V,\\widehat{E})$, which may be disconnected, \nhas adjacency matrix \n\\begin{equation*}\n\\widehat{A}=A-UW^T, \\qquad U=[\\mathbf{e}_i,\\mathbf{e}_j],\n\\quad W=[\\mathbf{e}_j,\\mathbf{e}_i],\n\\end{equation*}\nwhere here and in the rest of this work  the vectors \n$\\mathbf{e}_i$, $\\mathbf{e}_j$ represent\nthe $i$th and $j$th vectors of the standard basis of $\\mathbb{R}^n$, \nrespectively.\n\nSimilarly, let $(i,j)\\in\\overline{E}$ be an element in the complement of $E$. \nWe will call this element a {\\itshape virtual edge} for the graph $G$. \nWe can construct a new graph $\\tilde{G}=(V,\\tilde{E})$ obtained from $G$ \nby adding the virtual edge $(i,j)$ to the graph.\nThis procedure will be referred to as the {\\itshape updating} of the \nvirtual edge $(i,j)$.\nThe adjacency matrix of the resulting graph is\n\\begin{equation*}\n\\tilde{A}=A+UW^T, \\qquad U=[\\mathbf{e}_i,\\mathbf{e}_j],\n\\quad W=[\\mathbf{e}_j,\\mathbf{e}_i].\n\\end{equation*}\nHence, these two operations can both be described as \nrank-two modifications of the adjacency matrix of the original graph.\n\nThe operation of downdating an edge and successively updating a virtual \nedge will be referred to as {\\itshape rewiring}. \n\\begin{rem}\n{\\rm These operations are all performed in a symmetric fashion, since in this paper we \nconsider exclusively undirected networks.}\n\\end{rem}\n\n\\subsection{Centrality and total communicability}\n\\label{subsec:centrality}\nOne of the main goals when analyzing a network is to identify the\nmost influential nodes in the network. \nOver the years, various measures of the importance, or centrality, \nof nodes have been developed \\cite{Brandes,EstradaBook,NewmanBook}. \nIn particular the {\\itshape (exponential) subgraph centrality} \nof a node $i$ (see \\cite{Estrada2005}) is defined as the $i$th diagonal \nelement of the matrix exponential \\cite{Higham2008}:\n$$e^A=I+A+\\frac{A^2}{2!}+\\ldots=\\sum_{k=0}^\\infty\\frac{A^k}{k!},$$\nwhere $I$ is the $n\\times n$ identity matrix.\nAs it is well known in graph theory, given an adjacency matrix $A$ \nof an unweighted network and $k\\in\\mathbb{N}$, \nthe element $\\left(A^k\\right)_{ij}$ counts the total number of walks \nof length $k$ starting from node $i$ and ending at node $j$.\nTherefore, the subgraph centrality of node $i$ counts the total number \nof closed walks centered at node $i$, weighting walks of length $k$ by a factor \n$\\frac{1}{k!}$, hence giving more importance to shorter walks. \nThe subgraph centrality then accounts for the returnability of the \ninformation to the node which was the source of this same information. \nLikewise, the off-diagonal entries of the adjacency matrix \n$\\left(e^A\\right)_{ij}$ ({\\itshape subgraph communicability} of nodes $i$ and $j$) \naccount for the ability of nodes $i$ and $j$ to exchange information \n\\cite{EH08,EHB12}.\n\nStarting from these observations and with the aim of reducing the cost \nof the computation of the rankings, in \\cite{Benzi2013} it was suggested to use\nas a centrality measure \nthe \\emph{total communicability of a node} $i$, defined as\nthe $i$th entry of the vector\n$e^A\\mathbf{1}$, where $\\mathbf{1}$ denotes the vector of all ones:\n\\begin{equation}\\label{tnc}\nTC(i):= [e^A\\mathbf{1}]_i = \\sum_{j=1}^n \\left[e^A\\right]_{ij}.\n\\end{equation}\nThis measure of centrality is given by a weighted sum of walks from\nevery node in the network (including node $i$ itself), and thus quantifies\nboth the ability of a node to\nspread information across the network and the returnability of the information to \nthe node itself.\n\nThe value resulting from summing these quantities over all the nodes\ncan be interpreted as a global measure of how effectively the \ncommunication takes place across the whole network.\nThis index is called {\\itshape total (network) communicability} \n\\cite{Benzi2013} and can be written as\n\\begin{equation}\\label{tc_spec}\nTC(A):=\\mathbf{1}^Te^A\\mathbf{1}=\\sum_{i=1}^n\\sum_{j=1}^n(e^A)_{ij} = \n\\sum_{k=1}^n e^{\\lambda_k}({\\bf q}_k^T{\\bf 1})^2.\n\\end{equation} \nThis value can be efficiently computed, e.g., by means of a \nKrylov method as implemented in S.~G\\\"uttel's Matlab toolbox \\texttt{funm\\_kryl} \nsee \\cite{Krylov1,Guettel} or by Lanczos-based techniques \nas discussed below. In the toolbox \\cite{Guettel}\nan efficient algorithm for evaluating $f(A)\\mathbf{v}$ is implemented; \nwith this method the vector $e^A\\mathbf{1}$ can be constructed in \nroughly $O(n)$ operations (note that the prefactor can vary\nfor different types of networks) and the total communicability is easily derived.\n\nAs it is clear from its definition, the value of $TC(A)$ may be very large. \nSeveral normalizations have been proposed; the simplest is the normalization \nby the number of nodes $n$ (see \\cite{Benzi2013}), which we \nwill use throughout the paper.\nIt is easy to prove that the normalized \ntotal communicability satisfies\n\\begin{equation}\\label{coarse_bounds}\n\\frac{1}{n}\\sum_{i=1}^n\\left(e^A\\right)_{ii}\\leq \\frac{TC(A)}{n}\\leq e^{\\lambda_1},\n\\end{equation}\nwhere the lower bound is attained by the graph with $n$ nodes and no \nedges and the upper bound is attained by the complete graph with $n$ nodes. \n\n\\begin{rem}\\label{rem:lambda1}\n{\\rm The last equality in equation \\eqref{tc_spec} shows that \nthe main contribution to the value of $TC(A)$ is likely to come from \nthe term $e^{\\lambda_1}\\|{\\bf q}_1\\|_1^2$.} \n\\end{rem}\n\n\n\\subsection{Rationale for targeting the total communicability} \\label{why_TM}\nAs already mentioned, the total communicability provides a good measure\nof how efficiently information (in the broad sense \nof the term) is diffused across the network. Typically, very\nhigh values of $TC(A)$ are observed\nfor highly optimized infrastructure networks \n(such as airline routes or computer networks) and for highly cohesive social and\ninformation networks (like certain type of collaboration networks). \nConversely, the total network communicability is relatively low for spatially\nextended, grid-like networks (such as many road networks) or for \nnetworks that consist of two or more communities with poor communication\nbetween them (such as the Zachary network).\\footnote{ Numerical\nvalues of the normalized total network communicability for a broad\ncollection of networks are reported in\nthe experimental sections of this paper, in the Supplementary Material,\nand in \\cite{Benzi2013}.}\nAs a further example,\nreduced values of the communicability between different brain regions have\nbeen detected in stroke patients compared to healthy individuals \n\\cite{CH09}.\nWe refer to \\cite{EHB12} for an extensive survey on communicability,\nincluding applications for which it has been found to be useful.\n\nAnother reason in support of the use of the total communicability as\nan objective function is that it is closely related to the {\\em natural\nconnectivity} (or {\\em free energy}) of the network, while being\ndramatically easier to compute; see section \\ref{sec:nc+gap} below.  \nSparse networks with high\nvalues of $TC(A)$ are very well connected and thus less likely to \nbe disrupted by either random failures or targeted attacks leading\nto the loss of edges. This justifies trying to design sparse networks with\nhigh values of the total communicability.\n\nAn important observation is that the total network communicability $TC(A)$\ncan be interpreted in at least two different ways. Since it is given by the\nsum of all the pairwise communicabilities $C(i,j)=[e^A]_{ij}$, it is a\nglobal measure of the ability of the network to diffuse information.\nHowever, recalling the definition (\\ref{tnc}) of total node communicability,\nthe normalized total communicability can also be seen as ``the average\ntotal communicability\" of the nodes in the network: \n$$\\frac{TC(A)}{n} = \\frac{1}{n}\\sum_{i=1}^n TC(i).$$\nSince the total node\ncommunicability is a centrality measure \\cite{Benzi2013}, our goal  \ncan then be rephrased as the problem of constructing sparse networks\nhaving high average node centrality, where the node centrality is given\nby total node communicability. Since this is merely one of a large number\nof centrality measures proposed in the literature, it is a legitimate \nquestion to ask why the total node communicability should be used instead of\na different centrality index.  In other words, given any node\ncentrality function $f:V \\longrightarrow \\mathbb R_+$, we could consider \ninstead the problem of, say, adding a prescribed number of edges\nso as to maximize the increase in the global average centrality\n$$\\bar f = \\frac{1}{n}\\sum_{i=1}^n f(i).$$\n\nAs it turns out, most other centrality indices are either computationally\ntoo expensive to work with (at least for large networks), or lead to objective\nfunctions which do not make much sense. The following is a brief discussion\nof some of the most popular centrality indices used in the field of\nnetwork science.\n\n\\vspace{0.1in}\n\n\\begin{enumerate}\n\\item {\\bf Degree:} Consider first the simplest centrality index, the degree. \nObviously, adding $K$ edges according to {\\em any} criteria will\nproduce exacty the same variation in the average degree of a network.\nHence, one may as well add edges at random. Doing so, however,\ncannot be expected to be greatly beneficial if the goal is to \nimprove the robustness or efficiency of the network.\n\\item {\\bf Eigenvector centrality:} Let ${\\bf q}_1$ be the principal\neigenvector of $A$, normalized so that $\\|{\\bf q}_1\\|_2 = 1$. The eigenvector\ncentrality of node $i\\in V$ is the $i$th component of ${\\bf q}_1$, denoted\nby $q_1(i)$. It is straightforward to see that the problem of maximizing\nthe average eigenvector centrality\n$$\\frac{q_1(1) + q_1(2) + \\cdots + q_1(n)}{n}$$\nsubject to the constraint $\\|{\\bf q}_1\\|_2 = 1$ has as its only solution\n$$q_1(1) = q_1(2) = \\cdots = q_1(n) = \\frac{1}{\\sqrt n}.$$\nThis implies that $A$ has constant row sums or, in other words, that the\ngraph is regular --- every node in $G$ has the same degree. Hence, any\nheuristic aimed at maximizing the average eigenvector centrality will\nresult in graphs that are close to being regular. However, regular \ngraph topologies are not, {\\em per se}, endowed with any especially good  \nproperties when it comes to diffusing information or being robust: think\nof a cycle graph, for example. Regular graphs {\\em can} be very well connected\nand robust (this is the case of\nexpander graphs), but there is no reason to think that\nsimply making the degree distribution of a given network more regular will\nimprove its expansion properties.\n\\item {\\bf Subgraph centrality}: the average subgraph centrality of a\nnetwork is known in the literature as the normalized {\\em Estrada index}:\n$$\\frac{1}{n}EE(A) = \\frac{1}{n}{\\text Tr} (e^A) = \n\\frac{1}{n}\\sum_{i=1}^n [e^A]_{ii} = \\frac{1}{n}\\sum_{i=1}^n \ne^{\\lambda_i}.$$\nIt can also be interpreted as the average self-communicability of the\nnodes. As we mentioned, this is a lower bound for the average total\ncommunicability.  Evaluation of this quantity requires knowledge\nof all $n$ diagonal entries of $e^A$, or of all the eigenvalues of $A$ \nand is therefore much more expensive to compute. The heuristics we\nderive in this paper have a similar effect on $TC(A)$ and on the\nEstrada index, as we demonstrate in section \\ref{sec:nc+gap}. So, \nusing subgraph centrality instead of total communicability centrality\nwould lead to exactly the same heuristics and results, with the\ndisadvantage that evaluating the objective function, if necessary, would\nbe much more expensive.\n\\item {\\bf Katz centrality}: the Katz centrality of node $i\\in V$ is\ndefined as the $i$th row sum of the matrix resolvent\n $(I - \\alpha A)^{-1}$, where\nthe parameter $\\alpha$ is chosen in the interval $(0,\\frac{1}{\\lambda_1})$,\nso that the power series expansion\n$$(I - \\alpha A)^{-1} = I + \\alpha A + \\alpha^2 A^2 + \\cdots$$\nis convergent \\cite{Katz}.\nSince this centrality measure can be interpreted in terms of walks,\nusing it instead of the total communicability\nwould lead to the same heuristics and very similar\nresults, especially when\n$\\alpha$ is sufficiently close to $\\frac{1}{\\lambda_1}$ or if the\nspectral gap $\\lambda_1 - \\lambda_2$ is large; see \\cite{Benzi2015}.\nUsing Katz centrality, however, requires the careful selection of the\nparameter $\\alpha$, which leads to some complications. For example,\nafter each update one needs to recompute the dominant eigenvalue\nof the adjacency matrix in order to check whether the value of\n$\\alpha$ used is still within the range of permissible values or if\nit has to be reduced, making\nthis approach computationally very expensive. This\nproblem does not arise if the matrix exponential is used instead\nof the resolvent.  \n\\item {\\bf Other centrality measures}: So far we have only discussed\ncentrality measures that can be expressed in terms of the adjacency matrix $A$.\nThese centrality measures are all connected to the notion of walk in a graph,\nand they can often be understood in terms of spectral graph theory.\nOther popular centrality measures, such as betweenness centrality and\ncloseness centrality (see, e.g., \\cite{NewmanBook}) do not have a simple\nformulation in terms of matrix properties. They are based on the assumption\nthat all communication in a graph tends to take place along shortest paths,\nwhich is not always the case (this was a major motivation for the\nintroduction of walk-based measures, which postulate that communication\nbetween nodes can take place along walks of any length, with a preference\ntowards shorter ones). A further disadvantage is that they are quite\nexpensive to compute, although randomized approximations can bring the\ncost down to acceptable levels \\cite{Brandes}. For these reasons we do not\nconsider them in this paper, where the focus is on linear algebraic\ntechniques. It remains an open question whether heuristics  for\nmanipulating graph edges so as to tune some gloabl average of these\ncentrality measures can lead to networks with desirable connectivity\nand robustness properties.\n\\end{enumerate}\n\n\\vspace{0.1in}\n\nFinally, in view of the bounds (\\ref{coarse_bounds}), the evolution of\nthe total communicability under network modifications is closely tied\nto the evolution of \nthe dominant eigenvalue $\\lambda_1$. This quantity plays a\ncrucial role in network analysis, for example in the definition\nof the {\\em epidemic threshold}; see, for instance, \\cite[p.~664]{NewmanBook} and\n\\cite{VM2011}. In particular, a decrease in the total network \ncommunicability can be expected to lead to an increase in the\nepidemic threshold. \nThus, edge modification techniques developed for tuning $TC(A)$\ncan potentially be used to\nalter epidemics dynamics.\n\n\n\\section{Bounds via quadrature rules}\n\\label{sec:bounds}\nIn the previous section we saw the simple bounds (\\ref{coarse_bounds}) on\nthe normalized total network communicability. \nMore refined bounds for this index can be obtained by means of quadrature \nrules as described in \\cite{Benzi2010,Benzi1999,Golub2010,Fenu2013}.\n\nThe following theorem contains our result on the bounds for the normalized total communicability.\n\\begin{theorem}\\label{thm:bounds}\nLet $A$ be the adjacency matrix of an unweighted and undirected network. Then\n\\begin{equation*}\n\\Phi\\left(\\beta,\\omega_1+\\frac{\\gamma_1^2}{\\omega_1-\\beta}\\right)\\leq \n\\frac{TC(A)}{n}\\leq\\Phi\\left(\\alpha,\\omega_1+\\frac{\\gamma_1^2}{\\omega_1-\\alpha}\\right)\n\\end{equation*}\nwhere $[\\alpha,\\beta]$ is an interval containing the spectrum of $-A$ \n(i.e., $\\alpha \\le -\\lambda_1$ and $\\beta\\ge -\\lambda_n$), \n$\\omega_1=-\\mu=-\\frac{1}{n}\\sum_{i=1}^nd_i$ is the negative mean of the degrees, \n$\\gamma_1=\\sigma=\\sqrt{\\frac{1}{n}\\sum_{k=1}^n(d_k-\\mu)^2}$ is the standard deviation, and\n\\begin{equation}\\label{eq:bound}\n\\Phi(x,y)=\\frac{c \\left(e^{-x}-e^{-y}\\right)+xe^{-y}-ye^{-x}}{x-y}, \\qquad c=\\omega_1. \n\\end{equation}\n\\end{theorem}\n\n\nA proof of this result can be found in the Supplementary Materials \naccompanying the paper.\n\nAnalogous bounds can be found for the \nadjacency matrix of the graph after performing \na downdate or an update. \nThese results are summarized in the following Corollaries.\n\n\\begin{corollary}\\label{Dwdt} [Downdating]\nLet $\\widehat{A}=A-UW^T$, where $U=[\\mathbf{e}_i,\\mathbf{e}_j]$ and\n$W=[\\mathbf{e}_j,\\mathbf{e}_i]$ be the adjacency matrix of an unweighted and \nundirected network obtained after the downdate of the edge $(i,j)$ from the matrix $A$.\nLet $\\omega_1=-\\mu=-\\frac{1}{n}\\sum_{i=1}^nd_i$ and $\\gamma_1=\\sigma=\n\\sqrt{\\frac{1}{n}\\sum_{k=1}^n(d_k-\\mu)^2}$, where $d_i$ is the degree of node $i$ in the \noriginal graph. \nThen\n\\begin{equation*}\n\\Phi\\left(\\beta_{-},\\omega_{-}+\\frac{\\gamma_{-}^2}{\\omega_{-}-\\beta_{-}}\\right)\n\\leq\\frac{TC(\\widehat{A})}{n}\\leq\n\\Phi\\left(\\alpha_-,\\omega_-+\\frac{\\gamma_-^2}{\\omega_--\\alpha_-}\\right)\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\left\\{\n\\begin{array}{l}\n\\omega_{-}=\\omega_1+\\frac{2}{n};\\\\[6pt]\n\\gamma_{-}=\\sqrt{\\gamma_1^2-\\frac{2}{n}\\left(d_i+d_j-1+2\\omega_1+\n\\frac{2}{n}\\right)}\n\\end{array}\n\\right. ,\n\\end{equation*}\n$\\alpha_-$ and $\\beta_-$ are approximation of the smallest and largest \neigenvalues of $-\\widehat{A}$ respectively, \nand $\\Phi$ is defined as in equation \\eqref{eq:bound} with $c=\\omega_-$.\n\\end{corollary}\n\nNote that if bounds $\\alpha$ and $\\beta$ for the extremal eigenvalues of \nthe original matrix are known, we can then use $\\alpha_-=\\alpha$ and $\\beta_-=\\beta+1$.\nIndeed, if we order the eigenvalues of $\\widehat{A}$ in non--increasing order \n$\\widehat{\\lambda}_1>\\widehat{\\lambda}_2\\geq \\cdots \\geq\\widehat{\\lambda}_n$ we\nobtain, as a consequence of Weyl's Theorem\n(see \\cite[Section 4.3]{Horn}), that\n\\begin{equation*}\n\\alpha-1 \\leq -\\lambda_1-1 < -\\widehat{\\lambda}_1 < \n-\\widehat{\\lambda}_2 \\leq \\cdots \\leq -\\widehat{\\lambda}_n < -\\lambda_n+1 \\leq \\beta+1.\n\\end{equation*}\n\nFurthermore, the Perron--Frobenius Theorem ensures that, when performing a \ndowndate, the largest eigenvalue of the adjacency matrix cannot increase; \nhence, we deduce the more stringent bounds $\\alpha\\leq-\\widehat{\\lambda}_1\\leq\n-\\widehat{\\lambda}_2\\leq\\cdots\\leq -\\widehat{\\lambda}_n\\leq \\beta +1.$\n\nSimilarly, we can derive bounds for the normalized total communicability \nof the matrix $\\tilde{A}$ \nobtained from the matrix $A$ after performing the update of the virtual edge $(i,j)$.\n\n\\begin{corollary}\\label{Updt} [Updating]\nLet $\\tilde{A}=A+UW^T$, where $U=[\\mathbf{e}_i,\\mathbf{e}_j]$ and\n$W=[\\mathbf{e}_j,\\mathbf{e}_i]$ be the adjacency matrix of an unweighted and \nundirected network obtained after the update of the virtual edge \n$(i,j)$ in the matrix $A$.\nLet $\\omega_1=-\\mu=-\\frac{1}{n}\\sum_{i=1}^nd_i$ and $\\gamma_1=\\sigma=\n\\sqrt{\\frac{1}{n}\\sum_{k=1}^n(d_k-\\mu)^2}$, where $d_i$ is \nthe degree of node $i$ in the \noriginal graph.\nThen\n\n\\begin{equation*}\n\\Phi\\left(\\beta_{+},\\omega_{+}+\\frac{\\gamma_{+}^2}{\\omega_{+}-\\beta_{+}}\\right)\\leq \n\\frac{TC(\\tilde{A})}{n}\n\\leq\\Phi\\left(\\alpha_+,\\omega_++\\frac{\\gamma_+^2}{\\omega_+-\\alpha_+}\\right)\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\left\\{\n\\begin{array}{l}\n\\omega_{+}=\\omega_1-\\frac{2}{n};\\\\[6pt] \n\\gamma_{+}=\\sqrt{\\gamma_1^2+\\frac{2}{n}\\left(d_i+d_j+1+2\\omega_1-\\frac{2}{n}\\right)}\n\\end{array}\n\\right. ,\n\\end{equation*}\n$\\alpha_+$ and $\\beta_+$ are bounds for the smallest and largest eigenvalues of \n$-\\tilde{A}$ respectively, and $\\Phi$ is defined as in equation \n\\eqref{eq:bound} with $c=\\omega_+$.\n\\end{corollary}\n\nNotice that again, if bounds $\\alpha$ and $\\beta$ for the extremal \neigenvalues of $-A$ are known, we can then take $\\alpha_+=\\alpha-1$ and $\\beta_+=\\beta$.\nIn fact, the spectrum of the rank-two symmetric perturbations $UW^T$ and $-UW^T$ is \n$\\{\\pm 1,0\\}$ and hence we can \nuse Weyl's Theorem as before and then improve the upper bound\nusing the Perron--Frobenius Theorem.\n\n\nIn the next section we will see how the new bounds can be used to \nguide the updating and downdating process.\n\n\n\n\\section{Modifications of the adjacency matrix}\n\\label{sec:modification}\nIn this section we develop techniques that allow us to tackle \nthe following problems.\n\n\\begin{itemize}\n\\item[(P1)] Downdate: select $K$ edges that can be downdated from \nthe network without disconnecting it and that cause the smallest\ndrop in the total \ncommunicability of the graph;\n\\item[(P2)] Update: select $K$ edges to be added to the network \n(without creating self--loops or multiple edges) so as to increase as much as \npossible the total communicability of the graph;\n\\item[(P3)] Rewire: select $K$ edges to be rewired in the network \nso as to increase as much as possible the value of $TC(A)$. The\nrewiring process must not disconnect the network or \ncreate self--loops or multiple edges in the graph.\n\\end{itemize}\n\nAs we will show below, (P3) can be solved using combinations of methods \ndeveloped to solve (P1) and (P2).\nHence, we first focus on the downdate and  the update separately. \nNote that to decrease as little as possible the total communicability \nwhen removing an edge it would suffice to select $(i^*,j^*)\\in E$ so as\nto minimize the quantities\n$$\\mathbf{1}^TA^k\\mathbf{1} -\n\\mathbf{1}^T(A-UW^T)^k\\mathbf{1}\\qquad \\forall k=1,2,\\ldots ,$$\nsince $TC(A)=\\sum_{k=0}^\\infty\\frac{\\mathbf{1}^TA^k\\mathbf{1}}{k!}$.\nSimilarly, to increase as much as possible $TC(A)$ by addition of a virtual edge, \nit would suffice to select $(i^*,j^*)\\in\\overline{E}$ \nthat maximizes the differences \n$$\\mathbf{1}^T(A+UW^T)^k\\mathbf{1}-\\mathbf{1}^TA^k\\mathbf{1}\\qquad \\forall k=1,2,\\ldots $$\nHowever, it is easy to show that in general one can not find a choice for\n $(i^*,j^*)$ that works for all such $k$. \nIndeed, numerical experiments on small synthetic graphs   \n(not shown here) show that in general the optimal edge selection \nfor $k=2$ is different from the one for $k=3$.\nBecause of this, \nit is unlikely that one can find a simple\n``closed form solution\" to the problem, and\nwe need to develop approximation techniques.\n\nThe majority of the heuristics we will develop are based on new edge centrality measures. \nThe idea underlying these is that it seems reasonable to assume that an edge is more likely used as communication channel if its adjacent \nnodes are given a lot of information to spread. \nWe thus introduce three new centrality measures for edges based on this principle: edges connecting important nodes are themselves \nimportant. \n\n\n\n\\begin{definition}\nFor any $i,j\\in V$ we define the {\\rm edge subgraph centrality} of \nan existing\/virtual edge $(i,j)$ as\n\\begin{equation}\n^eSC(i,j)=\\left(e^A\\right)_{ii}\\left(e^A\\right)_{jj}.\n\\label{eq:edge_subgraph}\n\\end{equation}\n\\label{def:subgraph}\n\\end{definition}\n\nThis definition, based on the subgraph centrality of nodes, exploits the fact that the matrix exponential is symmetric positive definite \nand hence $(e^A)_{ii}(e^A)_{jj}>(e^A)_{ij}^2$. \nTherefore, the diagonal elements of $e^A$ somehow control its off-diagonal entries, \nhence they may contain \nenough information to infer the ``payload'' of the edges or of the virtual edges of interest. \n\n\\begin{definition}\nFor any $i,j\\in V$ we define the {\\rm edge total communicability centrality}\nof an existing\/virtual\nedge $(i,j)$ as\n\\begin{equation}\n^eTC(i,j) = [e^A{\\bf 1}]_i [e^A{\\bf 1}]_j.\n\\end{equation}\n\\label{def:tc_edge}\n\\end{definition}\n\nIt is important to observe that when\nthe spectral gap $\\lambda_1-\\lambda_2$ is ``large enough'', \nthen the subgraph centrality $\\left(e^A\\right)_{ii}$ \nand the total communicability centrality $[e^A{\\bf 1}]_i$ are\nessentially \ndetermined by \n$e^{\\lambda_1}q_1(i)^2$ and $e^{\\lambda_1}q_1(i)\\|{\\bf q}_1\\|_1$, respectively (see, e.g., \\cite{Benzi2013,Benzi2015,E06b}); it \nfollows that in this case \nthe two centrality measures introduced and a centrality measure based on the eigenvector centrality for nodes can be expected to provide similar rankings. \nThis is especially true when attention is restricted to the top edges (or nodes).\nThis observation motivates the introduction of the following edge centrality measure.\n\n\\begin{definition}\nFor any $i,j\\in V$ we define the {\\rm edge eigenvector centrality} of an existing\/virtual \nedge $(i,j)$ as\n\\begin{equation}\n^eEC(i,j)=q_1(i)q_1(j).\n\\label{eq:edge_eigenvector}\n\\end{equation}\n\\label{def:eigenvector}\n\\end{definition}\nAs a further justification for this definition, note that \n\n$$\\lambda_1-2 \\left( ^eEC(i,j)\\right)\\leq\\widehat{\\lambda}_1\\leq\\lambda_1,\\qquad\n\\tilde{\\lambda}_1\\geq\\lambda_1+2 \\left( ^eEC(i,j)\\right),$$\n\nwhere $\\widehat{\\lambda}_1$ is the leading eigenvalue of the matrix $\\widehat{A}$ and $\\tilde{\\lambda}_1$ \nis the leading eigenvalue of the matrix $\\tilde{A}$, as defined in \nsection \\ref{sec:background}.\nThese inequalities show that the edge eigenvector centrality \nof an existing\/virtual edge $(i,j)$ is strictly connected \nto the change in the value of the leading eigenvalue of the adjacency matrix, \nwhich influences\nthe evolution of the total communicability when we modify $A$ (see Remark \\ref{rem:lambda1}). \n\n\\begin{rem}\n{\\rm The edge eigenvector centrality has been used in \\cite{Tong, VM2011} to \ndevise edge removal techniques aimed to reduce significantly $\\lambda_1$, so as \nto increase the {\\em epidemic threshold} of networks.}\n\\end{rem}\n\n\nNote that we defined these measures of centrality for \nboth existing and virtual edges (as in \\cite{Berry2013}). \nThe reason for this as well as the justification for these definitions \nwill become clear in the next subsections. \n\n\nWe now discuss\n how to use these definitions to tackle the problems previously described. \nThe computational aspects concerning the implementation of the heuristics we are about to introduce \nand the derivation of their computational costs are described in the \nSupplementary Materials. \n\n\\subsection*{(P1) Downdate}\nThe downdate of any edge in the network will result in a reduction of \nits total communicability. \nNote that since we are focusing on the case of connected networks, we \nwill only perform downdates that keep the resulting graph connected.\nIn practice, it is desirable to further\nrestrict the choice of downdates to a subset of\nall existing edges, on the basis of criteria to be discussed shortly.\n\nAn ``optimal\" approach would select at each step of the downdating \nprocess a candidate edge corresponding to the minimum decrease of \ncommunicability.\\footnote{Strictly speaking, this would correspond to\na greedy algorithm which is only locally optimal. In general, this\nis unlikely to result in ``globally optimal\" network communicability.\nIn this paper, the term ``optimal\" will be understood in this limited\nsense only.} \nNote that for large networks this method is too costly to be practical.\nFor this reason we aim to develop inexpensive techniques that will hopefully\ngive close--to--optimal results.\nNevertheless, for small networks we will use the\n``optimal\" approach (where we systematically try all feasible\nedges and delete the one causing the least drop in total communicability) \nas a baseline method against which we compare the various\nalgorithms discussed below. This method will be henceforth\nreferred to as {\\tt optimal}.\n\nThe next methods we introduce perform the downdate of the lowest ranked existing edge according to \nthe edge centrality measures previously introduced whose removal does not disconnect the network. \nWe will refer to these methods as {\\tt subgraph}, {\\tt nodeTC}, and {\\tt eigenvector}, which are \nbased on definitions \\ref{def:subgraph}, \\ref{def:tc_edge}, and \\ref{def:eigenvector}, respectively. \nFrom the point of view of the communicability,\nthese methods downdate an edge connecting two nodes which are peripheral (i.e., have low centrality)\nand therefore are not expected to \ngive a large contribution to the spread of information along the network.\nHence, the selected edge is connecting two nodes whose ability to\nexchange information is already very low, and\nwe do not expect the total communicability to suffer too much from this edge removal.\nThis observation also suggests that such downdates \ncan be repeatedly applied  without the need to recompute the ranking\nof the edges after each downdate. \nAs long as the number of downdates performed remains small compared to the total number of edges, \nwe expect good results at a greatly reduced total cost. \nNote also that such downdates can be performed simultaneously rather than sequentially.\nWe will refer to these variants as {\\tt subgraph.no}, {\\tt nodeTC.no}, and {\\tt eigenvector.no}.\n\n\n\nFinally, we consider a technique motivated by the bounds \nobtained via quadrature rules derived in section \\ref{sec:bounds}.\nFrom the expression for the function $\\Phi$ in the special case of the downdate \n(cf.~Corollary \\ref{Dwdt}),\nwe infer that a potentially good choice may be to remove the edge\nhaving incident nodes $i,j$ for which the sum $d_i+d_j$ is minimal, if its \nremoval does not disconnect the network.\nIndeed, this choice reduces the upper bound only slightly and the total \ncommunicability may mirror this behavior.\nAnother way to justify this strategy is to observe that it is indeed\nthe optimal strategy if we approximate $e^A$ with its second-order\napproximation $I + A + \\frac{1}{2}A^2$ in the definition of total\ncommunicability.\nThis technique will be henceforth referred to as {\\tt degree}. \nWe note that a related measure, namely, the average \nof the out-degrees $\\frac{d_i+d_j}{2}$, \nwas proposed in \\cite{Berry2013} as a measure for the centrality of \nan edge $(i,j)$ in directed graphs.\n\n\\subsection*{(P2) Update}\nMost real world networks are characterized by low average degree. \nAs a consequence, the adjacency matrices of such networks are sparse ($m=O(n)$). \nFor the purpose of selecting a virtual edge to be updated, this implies \nthat we have approximately $\\frac{1}{2}\\left(n^2-cn\\right)$ possible choices  \nif we want to avoid the formation of multiple edges or self--loops (here $c$ is\na moderate constant).\nEach one of these possible updates will result in an increase of the total \ncommunicability of the network, but not every one of these will result in a\nsignificant increment. \n\nOne natural updating technique is to connect two nodes having high \ncentralities, i.e., add the virtual \nedge having the highest ranking according to the corresponding edge centrality. \nIts incident nodes, being quite central, can be expected to have an \nimportant role in the spreading of information along the network; \non the other hand, the communication between them may be relatively poor \n(think for example of the case where the two nodes sit in two distinct\ncommunities). \nHence, giving them a preferential communication channel, such as an \nedge between them, should result in a better spread of information along the \nwhole network. \nAgain, we will use the labels {\\tt subgraph}, {\\tt nodeTC}, and {\\tt eigenvector} to describe these \nupdating strategies. \nAs before, in order to reduce the computational cost, we also test the effectiveness \nof \nthese techniques\nwithout the recomputation of the \nranking of the virtual edges after each update.\nThese variants (referred to as {\\tt subgraph.no}, {\\tt nodeTC.no}, and {\\tt eigenvector.no}) are \nexpected to return good results as well, since \nthe selected update should not radically change the ranking of the edges. \nIndeed, they make central nodes even more central, and the ranking of the \nedges remains consequently almost unchanged.\nNote again that these updates can be performed simultaneously \nrather than sequentially.\n\n\nAs for the case of downdating, \nthe bounds via quadrature rules derived in section \\ref{sec:bounds} \nsuggest an updating technique, i.e., adding the virtual edge $(i,j)$ for which $d_i+d_j$ is maximal. Indeed,\nsuch a choice would maximize the lower bound on the total communicability, see\nCorollary \\ref{Updt}. Again, this choice can also be justified by noting that it\nis optimal if $e^A$ is replaced by its quadratic Maclaurin approximant.\nWe will again use the label {\\tt degree} to refer to this updating strategy.\n\nAll these techniques will be compared with the {\\tt optimal} one, \nbased on systematically trying all feasible virtual edges and selecting\nat each step the one resulting in the largest increase of the\ntotal communicability. Due to the very high cost of this brute force\napproach, we will use it only on small networks.\n\n\\begin{table}\n\\footnotesize\n\\centering\n\\caption{Brief description of the techniques introduced in the paper.}\n\\begin{tabular}{lcc}\n\\hline\n    Method                 & Downdate: $(i,j)\\in E$ & Update: $(i,j)\\not\\in E$\\\\\n\\hline\n    {\\tt optimal}          & $\\arg\\min\\{TC(A)-TC(\\widehat{A})\\}$ & $\\arg\\max\\{TC(\\tilde{A})-TC(A)\\}$ \\\\\n    {\\tt subgraph(.no)}    & $\\arg\\min\\{^eSC(i,j)\\}$             & $\\arg\\max\\{^eSC(i,j)\\}$\\\\\n    {\\tt eigenvector(.no)} & $\\arg\\min\\{^eEC(i,j)\\}$             & $\\arg\\max\\{^eEC(i,j)\\}$\\\\\n    {\\tt nodeTC(.no)}      & $\\arg\\min\\{^eTC(i,j)\\}$             & $\\arg\\max\\{^eTC(i,j)\\}$\\\\\n    {\\tt degree}           & $\\arg\\min\\{d_i+d_j\\}$               & $\\arg\\max\\{d_i+d_j\\}$   \\\\\n\\hline\n\\end{tabular}\n\\label{tab:description}\n\\end{table}\n\nThe heuristics introduced to tackle (P1) and (P2) are summarized in Table \\ref{tab:description}.\n\n\\subsection*{(P3) Rewire}\n\\label{subsec:rewire}\nAs we have already noted, there are situations in which\nthe rewire of an edge may be preferable \nto the addition of a new one.\nThere are various possible choices for the rewiring strategy to follow. \nThe greatest part of those found in literature are variants of random rewiring \n(see for example \\cite{Beygelzimer2005,Louzada2013}). \nIn this paper, on the other hand, we are interested in devising\nmathematically informed rewiring strategies. \nFor comparison purposes, however, we will compare our rewiring methods\nto the random rewire method, {\\tt random},  \nwhich downdates an edge (chosen uniformly at random\namong all edges whose removal does not disconnect the network) and then updates a \nvirtual edge, also chosen uniformly at random.\n\nCombining the various downdating and updating methods previously introduced  \nwe obtain different rewiring strategies based on the centralities of edges and \non the bounds for the total communicability. \nConcerning the methods based on the edge subgraph, eigenvector,\nand total communicability centrality, \nwe note that since a single downdate does not dramatically change \nthe communication capability of the network, we do not need to recompute \nthe centralities and the ranking of the edges after each downdating step,\nat least as long as the number of rewired edges remains relatively small \n(numerical experiments not shown here support this claim).\nOn the other hand, after each update we may or may not recalculate\nthe edge centralities. As before, we use {\\tt subgraph}\/{\\tt subgraph.no},\n{\\tt eigenvector}\/{\\tt eigenvector.no} and {\\tt nodeTC}\/{\\tt nodeTC.no} to\nrefer to these three variants of rewiring.\nAdditionally,\nwe introduce another rewiring strategy, henceforth referred to\nas {\\tt node}, based on the subgraph centrality of the nodes. \nIn this method we disconnect the most central node from the least central node among \nits immediate neighbors;\nthen we connect it to the most central node among those it is not linked to.\nIt is worth emphasizing that this strategy is philosophically different from the \nprevious ones based on the edge subgraph centrality in the downdating phase \n(the updating step is the same). \nIn fact, in those methods we use information on the nodes in order to \ndeduce some information on the edges connecting them; on the other hand, \nthe {\\tt node} algorithm does not take into account the potentially \nhigh ``payload'' of the edges involved, whose removal may result in a \ndramatic drop in the total communicability.\n\n\n\n\n\n\n\n\n\n\\section{Numerical studies}\n\\label{sec:test_tc}\nIn this section we discuss the results of numerical studies performed in order\nto assess the effectiveness and efficiency of the proposed techniques.\nThe tests have been performed on both synthetic and real-world networks, \nas described below. \nWe refer to the Supplementary Materials for the results of computations\nperformed on four small social networks, aimed at comparing our heuristics with\n{\\tt optimal}. These results show that for these small networks, the resulting\ntotal communicabilities are essentially identical to those obtained with the\n{\\tt optimal} strategy.\n\n\\subsection{Real-world networks}\n\\begin{table}[t]\n\\footnotesize\n\\centering\n\\caption{Description of the Data Set.}\n\\label{tab:Datasets}\n\\begin{tabular}{cccccc\n\\hline\nNAME & $n$ & $m$ &  $\\lambda_1$ & $\\lambda_2$ & $\\lambda_1-\\lambda_2$ \\\\\n\\hline\nMinnesota & 2640 & 3302  & 3.2324 & 3.2319 & 0.0005  \\\\\nUSAir97 & 332 & 2126 & 41.233 & 17.308 & 23.925  \\\\\nas--735 & 6474 & 12572 &  46.893 & 27.823 & 19.070  \\\\\nErd\\\"os02 & 5534 & 8472 & 25.842 & 12.330 & 13.512  \\\\\nca--HepTh & 8638 & 24806& 31.034  & 23.004  & 8.031 \\\\\nas--22july06 & 22963  & 48436   & 71.613  & 53.166  & 18.447 \\\\\nusroad--48     & 126146 & 161950  & 3.911    & 3.840    & 0.071 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nAll the\nnetworks used in the tests can be found in the University of \nFlorida Sparse Matrix Collection \\cite{Davis} under different ``groups''.\nThe USAir97 and Erd\\\"os02 networks are from the Pajek group. \nThe USAir97 network describes the US Air flight routes in 1997, while \nthe Erd\\\"os02 network represents the Erd\\\"os collaboration network, \nErd\\\"os included. \nThe network as--735, from the SNAP group, is the communication network \nof a group of autonomous system (AS) measured over 735 days between \nNovember 8, 1997 and January 2, 2000. \nCommunication occurs when routers from two ASs exchange information.\nThe Minnesota network from the Gleich group represents the Minnesota road network. \nThese latter three networks are not connected, therefore the tests were \nperformed on their largest connected component.\nWe point out that the original largest connected component \nof the network as--735 has 1323 ones on the main diagonal \nwhich were retained in our tests.\nThe network ca--HepTh is from the SNAP group and represents the \ncollaboration network of arXiv High Energy Physics Theory;\nthe network as--22july06 is from the Newman group and represents the \n(symmetrized) structure of Internet routers as of July 22, 2006.\nFinally, the network usroad--48, which is from the Gleich group, \nrepresents the continental US road network. For each network,\nTable \\ref{tab:Datasets} reports the number of nodes ($n$), \nthe number of edges ($m$), the \ntwo largest eigenvalues, and the spectral gap.\nWe use the first four networks to test all methods described\nin the previous section (except for {\\tt optimal}, which is only\napplied to the four smallest networks --- see the Supplementary Materials)\n and the last three to illustrate the\nperformance of the most efficient among the methods tested.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\caption{Evolution of the normalized total communicability vs.~number \nof downdates, updates and rewires\nfor networks Minnesota and as735.}\n\\label{fig:Minnesota_as735}\n\\includegraphics[width=.9\\textwidth]{Minnesota_as735.eps}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\caption{Evolution of the normalized total communicability vs.~number\nof downdates, updates and rewires for networks USAir97 and Erd\\\"os02.}\n\\label{fig:USAir97_Erdos02}\n\\includegraphics[width=.9\\textwidth]{USAir97_Erdos02.eps}\n\\end{figure}\n\n\nWe first consider the networks \nMinnesota, as735, USAir97, and Erd\\\"os02, for which we perform $K=50$ modifications. \nFor these networks the set $\\overline{E}$ \n(the complement of the set $E$ of edges) is large enough \nthat performing an extensive search for the \nedge to be updated is expensive. Hence, we form the set $S$ containing the top\n$10\\%$ of the nodes ordered according to the eigenvector centrality \nand we restrict our search to virtual edges incident to these nodes\nonly. An exception is\nthe network USAir97 where we have used the set $S$ corresponding to\nthe top $20\\%$ of the nodes, since in the case of $10\\%$ this set contained \nonly 52 virtual edges.\nIn Figures \\ref{fig:Minnesota_as735} and \\ref{fig:USAir97_Erdos02}\nwe show results for the methods {\\tt eigenvector}, {\\tt eigenvector.no},\n{\\tt subgraph}, {\\tt subgraph.no} and {\\tt degree}.  \nBefore commenting on these, we want to stress the poor performance of {\\tt node} \nwhen tackling (P3); this shows that the use of edge centrality measures (as opposed to \nnode centralities alone) is indispensable in this framework. \nThe results for these networks clearly show \nthe effectiveness of\nthe {\\tt eigenvector} and {\\tt subgraph} algorithms\nand of their less expensive variants {\\tt eigenvector.no} and {\\tt subgraph.no} in\nnearly all cases; similar results were obtained with\n{\\tt nodeTC} and {\\tt nodeTC.no} (not shown). The only exception is in the downdating\nof the Minnesota network,\nwhere the eigenvector-based techniques\ngive slightly worse results.\nThis fact is easily explained in view of the tiny spectral\ngap characterizing this and similar networks\\footnote{Small spectral gaps are typical\nof large, grid-like\nnetworks such as the road networks or the graphs corresponding to\nuniform triangulations or discretizations of physical domains.}\n(see Table \\ref{tab:Datasets}).\nBecause of this property, eigenvector centrality is a poor approximation\nof subgraph centrality and cannot be expected to give\nresults similar to those obtained with {\\tt subgraph}\nand {\\tt subgraph.no}. \n\nThe results for the downdate show that the inexpensive {\\tt degree}\nmethod does not perform as well on these networks, except\nperhaps on Minnesota. The relatively poor performance of\nthis method is due to the fact that the information used by this\nmethod to select an edge for downdating is too local.\n\nNote, however, the scale on the vertical axis in Figures \n\\ref{fig:Minnesota_as735}--\\ref{fig:USAir97_Erdos02}, \nsuggesting that for these networks (excluding perhaps Minnesota)\nall the edge centrality-based methods perform well with only very small relative\ndifferences between the resulting total communicabilities. \n\n\nOverall, these results indicate that the edge centrality-based\nmethods, especially the inexpensive {\\tt eigenvector.no} and {\\tt nodeTC.no} variants, \nare an excellent choice in almost all cases and to tackle all the problems. \nIn the case of downdating\nnetworks with small spectral gaps, \n{\\tt subgraph.no} may be preferable but at a higher cost.\n\nThe behavior of the {\\tt degree} method depends strongly on the \nnetwork on which it is used. Our tests indicate that it behaves\nwell in some cases (for example, P2 for Erd\\\"os02) but poorly in others (P2 for Minnesota).\nWe speculate that this method may perform adequately when tackling (P2) on scale-free\nnetworks (such as Erd\\\"os02) where a high degree is an indication\nof centrality in spreading information across the network.\n\nSome comments on the difference in the results for updating as compared to those \nfor rewiring (downdating followed by updating) are in order.\nRecall that our downdating strategies aim to reduce as little as \npossible the decrease in the value of the total communicability, whereas the \nupdating techniques aim to increase this index as much as possible.\nWith this in mind, it is not surprising to see that the \ntrends of the evolution of the total communicability after rewiring reflect those \nobtained with the updating strategies.\n The values obtained\nusing the updates are in general higher than those obtained using the rewiring \nstrategies, since updating implies the addition of edges whereas in\nrewiring the number of edges remains the same. \nExperiments not reported here indicate that\nthe methods based on the edge eigenvector \nand total communicability\ncentrality are more stable than the others \nunder rewiring and to dampen the effect of the downdates\n\nIn Figures \\ref{fig:large_down}-\\ref{fig:large_up} we show results for \nthe three largest networks in our data set (ca--HepTh, as-22july06 and\nusroad-48). \nIn the case of the updating, we have selected the virtual edges among those in the subgraph \ncontaining the top $1\\%$ of nodes ranked according to the eigenvector centrality.  \nWe compare the following methods:\n{\\tt eigenvector}, {\\tt eigenvector.no}, {\\tt nodeTC}, {\\tt nodeTC.no},\n{\\tt subgraph.no} and {\\tt degree}; \nrandom downdating was also tested and found to give poor results. \nNote that network\nusroad-48 behaves similarly to Minnesota; this is not surprising in view of\nthe fact that these are both road networks with a tiny spectral gap.\nLooking at the scale on the vertical axis, however, it is clear that\nthe decrease in total communicability is negligible with all the methods\ntested here.\nThe results on these networks confirm the general trend observed so far;\nin particular, we note the excellent behavior of {\\tt nodeTC} and\n{\\tt nodeTC.no}.\n\n\\begin{figure}[t]\n\\centering\n\\caption{Downdates for large networks: normalized total communicability vs.~number of modifications.}\n\\includegraphics[width=.9\\textwidth]{Large_down_new.eps}\n\\label{fig:large_down}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\caption{Updates for large networks: normalized total communicability vs.~number of modifications.}\n\\includegraphics[width=.9\\textwidth]{Large_up_new.eps}\n\\label{fig:large_up}\n\\end{figure}\n\n\n\n\\subsection{Synthetic networks}\nThe synthetic examples used in the tests were produced using \nthe CONTEST toolbox for Matlab (see \\cite{Contest,Taylor2009}).\nWe tested two types of graphs: the preferential attachment \n(Barab\\'asi--Albert) model and the small world (Watts--Strogatz) model. \n\n\\begin{figure}[t]\n\\centering\n\\caption{Evolution of the total communicability when 50 downdates,\nupdates or rewires are performed on two synthetic networks with $n=1000$ nodes.}\n\\includegraphics[width=.9\\textwidth]{synthetic.eps}\n\\label{fig:synthetic}\n\\end{figure}\n\nThe preferential attachment model \n\\cite{prefattach} was designed to produce networks with \nscale--free degree distributions as well as the small world property,\ncharacterized by short average path length and relatively high clustering\ncoefficient.\nIn CONTEST, preferential attachment networks are constructed using \nthe command \\texttt{pref(n,d)} where $n$ is the number of nodes and $d\\geq 1$ is the \nnumber of edges each new node is given when it is first introduced to the network.\nThe network is created by adding nodes one by one (each new node with $d$ edges).\nThe edges of the new node connect to nodes already in the network with \na probability proportional to the degree of the already existing nodes.\nThis results in a scale--free degree distribution.\n\n\nThe second class of synthetic test matrices used in our experiments \ncorresponds to Watts--Strogatz small world networks.\nThe small world model was developed as a way to impose a high \nclustering coefficient onto classical random graphs \\cite{Watts1998}.\nThe function used to build these matrices takes the form \\texttt{smallw(n,k,p)}.\nHere $n$ is the number of nodes in the network, originally arranged in a\nring and connected to their $k$ nearest neighbors. Then each node is\nconsidered independently and, with probability $p$, an edge is added\nbetween the node and one of the other nodes in the graph, chosen uniformly\nat random (self-loops and multiple edges are not allowed). \nIn our tests, we have used matrices with $n=1000$ nodes which were \nbuilt using the default values for the functions previously described. \nWe used $d=2$ in the Barab\\'asi--Albert model  \nand $k=2$, $p=0.1$ in the Watts--Strogatz model. \n\n\nThe results for our tests are presented in Figure \\ref{fig:synthetic}.\nThese results agree with what we have seen previously on real-world networks. \nInterestingly, {\\tt degree} does not perform well for the downdate when \nworking on the preferential attachment model; \nthis behavior reflects what we have seen for the networks USAir97, as--735, \nand Erd\\\"os02,\nwhich are indeed scale--free networks.\n\n\\begin{figure}[t]\n\\centering\n\\caption{Timings in seconds for scale-free graphs of increasing size (500 modifications).} \n\\includegraphics[width=.9\\textwidth]{time_nest_new2.eps}\n\\label{fig:timings_nest}\n\\end{figure}\n\n\n\n\n\\subsection{Timings for synthetic networks}\\label{sec:ltime_synth}\nWe have performed some experiments with synthetic networks of increasing\nsize in order to assess the scalability of the various methods introduced\nin this paper.  A sequence of seven adjacency matrices corresponding to\nBarab\\'asi--Albert scale-free graphs was generated using the CONTEST\ntoolbox. The order of the matrices ranges from 1000 to 7000; the average\ndegree is kept constant at 5. A fixed number of modifications ($K=500$)\nwas carried out on each network.\nAll experiments were performed using Matlab Version 7.12.0.635 (R2011a) \non an IBM ThinkPad running Ubuntu 12.04.5 LTS, a 2.5 GHZ Intel Core i5 processor, and \n3.7 GiB of RAM. \nWe used the built-in Matlab function {\\tt eigs} (with the default settings) to\napproximate the dominant eigenvector of the adjacency matrix $A$, \nthe Matlab toolbox {\\tt mmq} \\cite{mmq} to estimate the diagonal \nentries of $e^A$ (with a fixed number of five nodes in the Gauss--Radau\nquadrature rule, hence five Lanczos steps per estimate),\nand the toolbox {\\tt funm\\_kryl} to compute the vector \n$e^A{\\bf 1}$ of total communicabilities, \nalso with the default parameter settings.\n\nThe results are shown in Figure \\ref{fig:timings_nest}. The approximate\n(asymptotic) linear scaling\nbehavior of the various methods (in particular of {\\tt nodeTC.no} and {\\tt eigenvector.no},\nwhich are by far the fastest, see the insets) is clearly displayed \nin these plots. \n\n\n\n\\subsection{Timings for larger networks}\\label{sec:large}\n\nIn Tables \\ref{tab:timings_down}--\\ref{tab:timings_up} we \nreport the timings for various methods \nwhen $K=2000$ downdates and updates are selected for the three largest networks listed \nin Table \\ref{tab:Datasets}. \n\nThe timings presented refer to the selection of the edges to be downdated or updated, which\ndominates the computational effort. For the method {\\tt subgraph.no} in the case\nof downdates, we restricted the search of\ncandidate edges to a subset of $E$ in order to reduce\ncosts. For the three test networks we used $40\\%$, $45\\%$ and $15\\%$ of the nodes,\nrespectively,\nchosen by taking those with lowest eigenvector centrality, and the corresponding\nedges. We found the results to be very close to those obtained working with the\ncomplete set $E$, but at a significantly lower cost (especially for the largest\nnetwork).\n\nThese results clearly show that algorithms\n{\\tt nodeTC.no} and {\\tt eigenvector.no} are orders of magnitude\nfaster than the other methods; method {\\tt subgraph.no}, while significantly\nmore expensive, is still\nreasonably \nefficient\\footnote{It is worth mentioning that in principle it is possible to \ngreatly reduce the cost of this method using parallel processing, since each \nsubgraph centrality can be computed independently of the others.}\nand can be expected to give better results in \nsome cases (e.g.,\non networks with a very small spectral gap). The {\\tt degree} algorithm, on the\nother hand, cannot be recommended in general since it gives somewhat inferior results.\nThe remaining methods {\\tt eigenvector}, {\\tt nodeTC} and {\\tt subgraph} (not\nshown here) are prohibitively expensive for large networks, at least when the\nnumber $K$ of modifications is high (as it is here).\n\n\\begin{table}[t]\n\\footnotesize\n\\centering\n\\caption{Timings in seconds for $K=2000$ downdates performed on the \nthree largest networks in Table \\ref{tab:Datasets}.}\n\\label{tab:timings_down}\n\\begin{tabular}{cccc}\n\\hline\n                         & ca--HepTh & as--22july06 & usroad--48  \\\\\n\\hline\n{\\tt eigenvector}   & 278.13     & 599.83        & 11207.39  \\\\\n{\\tt eigenvector.no}& 0.07       & 1.79          & 4.08      \\\\\n{\\tt nodeTC}        & 553.04     & 1234.49       & 2634.27   \\\\\n{\\tt nodeTC.no}     & 0.34       &   0.83        &   1.34    \\\\\n{\\tt subgraph.no}   & 107.36     & 383.34        & 1774.07  \\\\\n{\\tt degree}         & 29.67      & 53.42         & 153.52    \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[t]\n\\footnotesize\n\\centering\n\\caption{Timings in seconds for $K=2000$ updates performed on the \nthree largest networks in Table \\ref{tab:Datasets}}\n\\label{tab:timings_up}\n\\begin{tabular}{cccc}\n\\hline\n                         & ca--HepTh & as--22july06 & usroad--48\\\\\n\\hline\n{\\tt eigenvector}   & 192.8     & 436.9        & 1599.5    \\\\\n{\\tt eigenvector.no}& 0.19      & 0.33         & 5.85      \\\\\n{\\tt nodeTC}        & 561.9    & 1218.8      & 2932.   \\\\\n{\\tt nodeTC.no}     & 0.30      &  0.55        &  1.59     \\\\\n{\\tt subgraph.no}   & 3.13      & 7.20         & 121.4     \\\\\n{\\tt degree}         & 11.1      & 12.4         & 175.8     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe also observe that downdating is generally a more\nexpensive process than updating, since in the latter case the edges are to be\nchosen among a fairly small subset of all virtual edges, whereas in the downdating process\nwe work on the whole set $E$ of existing edges (or on a large subset of $E$). \nFor some methods the difference in cost becomes significant when\nthe networks are sufficiently large and the number of modifications to be\nperformed is high.\n\nSummarizing, the method labelled {\\tt nodeTC.no} is the fastest and gives excellent\nresults, quite close to those of the more expensive methods, and therefore we can\nrecommend its use for the type of problems considered here. The methods labelled\n{\\tt eigenvector.no} and {\\tt subgraph.no} are also effective and\nmay prove useful in some settings, especially for updating.\n \n \n\n\n\n\\section{Evolution of other connectivity measures}\n\\label{sec:nc+gap}\nIn this section we want to highlight another facet of the methods we \nhave introduced for (approximately) optimizing the total communicability.\nIn particular, we\nlook at the evolution of other network properties under our updating strategies.\nWhen building or modifying a network, there are various \nfeatures that one may want to achieve.\nTypically, there are two main desirable properties: first, the network should\ndo a good job at spreading information, i.e., have a high total communicability;\nsecond, the network should be robust under targeted attacks or random failure, which \nis equivalent to the requirement that it should be difficult to ``isolate'' \nparts of the network, i.e., the network should be ``well connected''.\nThis latter property can be measured by means of various indices. \nOne such measure is the spectral gap $\\lambda_1 - \\lambda_2$. As a consequence\nof the Perron--Frobenius Theorem, adding an edge to a connected network\ncauses the dominant eigenvalue $\\lambda_1$ of $A$ to increase. Test results\n(not shown here) show that when a network is updated using one of our\ntechniques, the first eigenvalue increases rapidly with the number of updates.\nOn the other hand, the second eigenvalue $\\lambda_2$ tends to change little\nwith each update and it may even decrease (recall that the matrix\n$UW^T = {\\bf e}_i{\\bf e}_j^T + {\\bf e}_j{\\bf e}_i^T$ being added to $A$\nin an update is indefinite). Therefore, the spectral gap $\\lambda_1 - \\lambda_2$\nwidens rapidly with the number of updates.\\footnote{This fact, incidentally,\nmay serve as further justification for the effectiveness\nof algorithms like {\\tt nodeTC.no}\nand {\\tt eigenvector.no}.}\n It has been pointed out\nby various\n authors (see, e.g., \\cite{E06b,Puder2013}) that a large spectral gap is typical\nof complex networks with good expansion properties.  \n\nHere we focus on a related measure, the so-called {\\em free energy}\n(also known in the literature as {\\em natural connectivity}) of the network.\nIn particular, we investigate the effect of our proposed methods of network updating\non the evolution of this index.\n\n\\subsection{Tracking the free energy (or natural connectivity)}\nIn \\cite{Jun2010} the authors discuss a measure of network connectivity which \nis based on an intuitive notion of robustness and\nwhose analytical expression has a clear meaning and can be derived from the \neigenvalues of $A$; they refer to it as the {\\em natural connectivity}\nof the network (see also \\cite{Wu2012}).\nThe idea underlying this index is that a network is more robust if there exists \nmore than one route to get from one node to another; this property \nensures that if a route becomes unusable, there is an alternative way to get \nfrom the source of information to the target.\nTherefore, intuitively a network is more robust if it has a lot of (apparently) \nredundant routes connecting its vertices  \nor, equivalently, if each of its nodes is involved in a lot of closed walks.\nThe natural connectivity aims at quantifying\nthis property by using an existing measure for\nthe total number of closed walks in a graph, namely, the \n{\\em Estrada index} \\cite{Estrada2000}.\nThis index, denoted by $EE(G)$, \nis defined as the trace of the matrix exponential.\nNormalizing this value and taking the natural logarithm, \none obtains the {\\itshape natural connectivity} \nof the graph:\n$$\\overline{\\lambda}(A)=\\ln\\left(\\frac{1}{n}\\sum_{i=1}^ne^{\\lambda_i}\\right)=\\ln(EE(G))-\\ln(n).$$\n\n\n\nIt turns out, however, that essentially the same index was already\npresent in the literature. Indeed, the natural connectivity is only one \nof the possible interpretations\nthat can be given to the logarithm of the (normalized) Estrada index.\nAnother, earlier interpretation was given in \\cite{Estrada2007}, where the authors\nrelated this quantity to the Helmholtz free energy of the network $F=-\\ln\\left(EE(G)\\right)$.\nTherefore, since $\\overline{\\lambda}=-F-\\ln(n)$, the behavior of $F$ completely describes that of\n$\\overline{\\lambda}$ (and conversely) as the graph is modified by adding or removing links.\n\n\nThe natural connectivity has been recently used (see \\cite{Chan2014}) to derive manipulation \nalgorithms that directly optimize this robustness measure. \nIn particular, the updating algorithm introduced in \\cite{Chan2014} appears to be \nsuperior to existing heuristics, such as those proposed in \n\\cite{Beygelzimer2005,Frank1970,Shargel2003}. \nThis algorithm, which costs $O(mt+Kd_{max}^2t+Knt^2)$ where $d_{max}=\\max_{i\\in V}d_i$ \nand $t$ is the (user-defined) number of leading eigenpairs, \nselects $K$ edges to be added to the network\nby maximizing a quantity that involves the elements of the leading $t$ \neigenpairs of $A$.\\footnote{A description of the \nalgorithm can be found in the Supplementary Material.}  \n\n \n\nWe have compared our updating techniques with that described in \\cite{Chan2014}.\nResults for four representative networks are shown in Figure \\ref{fig:TCeNC_big}.\nIn our tests, we use the value $t=50$ (as in \\cite{Chan2014}), and we select $K=500$ edges. \nNote that, when $K$ is large, the authors recommend to recompute the set of $t$ \nleading eigenpairs every $l$ iterations. \nThis operation requires an additional effort that our faster methods do not need.\nSince the authors in \\cite{Chan2014} show numerical experiments in which the methods \nwith and without the recomputation return \nalmost exactly the same results, we did not recompute the eigenpairs after \nany of the updates. \n\nFigure \\ref{fig:TCeNC_big} displays the results for both the evolution of the natural \nconnectivity and of the normalized total communicability, where\nthe latter is plotted in a semi--logarithmic scale.\nA total of 500 updates have been performed.\nThe method labelled {\\tt Chan} selects the edges according to\nthe algorithm described in \\cite{Chan2014} choosing from all the virtual edges of the graph. \nFor our methods we used, as before, the virtual edges in the subgraph obtained \nselecting the top $10\\%$ or $20\\%$ of nodes ranked according \nto the eigenvector centrality.\nAs one can easily see, our methods generally outperform the algorithm proposed in \n\\cite{Chan2014}. In particular, {\\tt nodeTC.no} and {\\tt eigenvector.no} \ngive generally better results than {\\tt Chan} and\nare much faster in practice. For instance, the execution time\nwith {\\tt Chan} on the network ca-HepTh was over 531 seconds, and \nmuch higher for the two larger networks.\nWe recall (see Table \\ref{tab:timings_up})\n that the execution times for {\\tt nodeTC.no} and {\\tt eigenvector.no}\nare about three orders of magnitude smaller.\n\n \n\n\\begin{figure}[t]\n\\caption{Evolution of the natural connectivity and of the normalized total \ncommunicability (in a semi--logarithmic scale plot) when up to 500 updates are \nperformed on four real-world networks.}\n\\centering\n\\includegraphics[width=.9\\textwidth]{TCeNC.eps}\n\\label{fig:TCeNC_big}\n\\end{figure}\n\nIt is striking to see how closely the evolution of the natural connectivity \nmirrors the behavior of the normalized total communicability. This is likely\ndue to the fact that both indices depend on the eigenvalues of $A$\n(with a large contribution coming from the terms containing $\\lambda_1$,\nsee (\\ref{tc_spec}) and the subsequent remark), and all the updating strategies used here\ntend to make $\\lambda_1$ appreciably larger.\n\nReturning to the interpretation in terms of statistical physics,  \nfrom Figure \\ref{fig:TCeNC_big} we deduce that the free energy of the\ngraph decreases as we add edges to the network.\nIn particular this means that the network is evolving toward a more stable\nconfiguration and, in the limit, toward equilibrium, which is the configuration\nwith maximum entropy.\\footnote{The relation between the free energy and the \nGibbs entropy is described in more detail in the Supplementary Material.}\n\n\n\\begin{comment}\nIndeed, the free energy of the system is related to the Gibbs entropy $S$ as\n$TS=H-F$, \nwhere $T$ is the absolute temperature and $H$ is the total energy of the graph.\nTherefore, the Gibbs entropy,\nwhich measures the effective number of states sharing the same energy,\nincreases as $F$ decreases.\n\\end{comment}\n\n\nThese findings indicate\nthat the normalized total communicability is equally effective an index as\nthe natural connectivity\n(equivalently, the free energy) for the purpose of \ncharacterizing network connectivity.\nSince the network total communicability can be computed very fast (in $O(n)$ time),\nwe believe that the normalized total communicability should be used instead\nof the natural connectivity, especially for large networks. \n\nIndeed, computing the natural connectivity requires evaluating the trace of \n$e^A$; even when stochastic trace estimation is used \\cite{AT11}, this is \nseveral times more expensive, for large networks, than the total communicability. \n\n\n\\begin{comment}\nIndeed, computing\nthe natural connectivity requires evaluating all the diagonal entries of $e^A$ and is\ntherefore significantly more expensive, for large networks, than the total\ncommunicability.\n\\end{comment}\n\n\\section{Conclusions and future work}\n\\label{sec:conclusions}\nIn this paper we have introduced several techniques that can be used \nto modify an existing network so as to obtain networks that are highly \nsparse, and yet have \na large total communicability.\n\nThese heuristics make use of various measures of edge \ncentrality, a few of which have been introduced in this work. \nFar from being {\\em ad hoc}, these heuristics are widely\napplicable and mathematically justified.\nAll our techniques can be implemented using well-established tools from numerical\nlinear algebra: algorithms for eigenvector computation, Gauss-based quadrature\nrules for estimating quadratic forms, and Krylov subspace methods for computing\nthe action of a matrix function on a vector. At bottom, the Lanczos algorithm\nis the main player. High quality, public domain software exists to perform these\nmodifications efficiently.\n\nAmong all the methods introduced here, the best results are obtained by \nthe {\\tt nodeTC.no} and {\\tt eigenvector.no} algorithms, \nwhich are based on the edge total communicability\nand eigenvector centrality, respectively. These methods are extremely fast \nand returned excellent results in virtually all \nthe tests we have performed. For updating networks characterized by a small\nspectral gap, a viable alternative is the algorithm {\\tt subgraph.no}.\nWhile more expensive than {\\tt nodeTC.no} and {\\tt eigenvector.no}, this\nmethod scales linearly with the number of nodes and yields consistently\ngood results.\n\nFinally, we have shown that the total communicability can be effectively \nused as a measure of network connectivity, which plays an important role\nin designing robust networks.\n Indeed, the total communicability does a very good job at quantifying  \ntwo related properties of networks: \nthe ease of spreading information, and the extent to which the network is \n``well connected''. Our results show that the total communicability\nbehaves in a manner very similar to the natural connectivity (or free\nenergy) under network\nmodifications, while it can be computed much more quickly.\n\nFuture work should include the extension of these techniques\nto other types of networks,\nincluding directed and weighted ones.\n\n\n\\section*{Acknowledgements}\nWe are grateful to Ernesto Estrada for providing some of the networks \nused in the numerical experiments and for pointing out some useful references.\nThe first author would like to thank Emory University for the hospitality \noffered in 2014, when this work was completed.\nWe also thank two anonymous referees for helpful suggestions.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:intro}}\nWith the advent of modern sky surveys that map out significant contiguous fractions of the observable Universe in ever greater detail~(e.g.,~\\cite{6dFGS,BOSS,DES,DESI,EBOSS,EUCLID,LSST,SDSS,SPHEREX,VIPERS,WFIRST}), it has become possible to investigate its least luminous and most extended constituents: cosmic voids, vast regions of relatively empty space. Voids are not only fascinating objects in their own right, they may also hold the keys to resolving some of today's open problems in cosmology, a fact that has come into focus only recently (see references~\\cite{vdWeygaert2009,vdWeygaert2016,Pisani2019} for an overview). Cosmic voids can be thought of pocket universes in which dark energy became important much earlier than elsewhere in the cosmos~\\cite{Bos2012,Pisani2015a,Verza2019}. Making up the bulk of large-scale structure, they play a major role in the formation of its web-like pattern~\\cite{Gregory1978,Zeldovich1982,deLapparent1986,Sheth2004,vdWeygaert2009}. This pattern contains a wealth of information on the fundamental properties of the Universe and voids have been shown to be sensitive probes thereof, such as its initial conditions~\\cite{vdWeygaert1993,Chan2019}, its matter~\\cite{Peebles2001,Nusser2005,Park2007,Lavaux2010,Sutter2012b,Sutter2014b,Hamaus2016,Mao2017} and energy components~\\cite{Granett2008,Biswas2010,Ilic2013,Cai2014,Planck2014,Kovacs2019}. Moreover not only cosmology, but the very nature of gravity can be investigated with voids~\\cite{Clampitt2013,Zivick2015,Cai2015,Achitouv2016,Falck2018,Paillas2019,Perico2019}, because it is gravity that gives rise to their formation and evolution in the first place. This happens via gravitational collapse of initially over-dense regions in the mass distribution into sheets, filaments, and clusters where galaxies form. The remaining space is occupied by voids that are characterized by the coherent flow of (predominantly dark) matter~\\cite{Shandarin2011,Abel2012,Hahn2015}. Baryonic matter is even more scarce inside voids~\\cite{Paillas2017,Pollina2017}, implying a significant advantage in the attempt to model their evolution when compared to the other structure types. This opens up the opportunity to use voids as laboratories for the physics of dark matter~\\cite{Yang2015,Reed2015,Baldi2018} and other elusive particles, such as neutrinos, that freely permeate their interiors~\\cite{Massara2015,Banerjee2016,Kreisch2019,Schuster2019}.\n\nOn the whole, cosmic voids offer radically novel avenues towards probing the fundamental laws of physics that govern our Universe. General Relativity (GR) relates the distribution of matter and energy to the geometry of spacetime via Einstein's field equations. Consequently, observations of the cosmic expansion history allow constraining the material components of the Universe. In this manner supernova distance measurements have inferred the existence of dark energy (in the form of a cosmological constant $\\Lambda$) that dominates the cosmic energy budget today and is responsible for the observed accelerated expansion~\\cite{Riess1998,Perlmutter1999}. Yet, the fundamental nature of dark energy remains mysterious and further efforts are necessary towards explaining its origin. This has been attempted in studying the expansion history by employing standard rulers, such as the Baryon Acoustic Oscillation~(BAO) feature imprinted in the spatial distribution of galaxies on scales of $\\sim105h^{-1}{\\rm Mpc}$~\\cite{Eisenstein2005}. Because the physics of recombination is well understood, the BAO feature can be modeled from first principles and therefore provides a scale of known extent: a standard ruler. Observations of the BAO in the pairwise distribution of galaxies have been successful in constraining the expansion history and so far consistently confirmed the $\\Lambda$CDM paradigm (e.g.,~\\cite{Alam2017,SanchezA2017,Beutler2017a}).\n\nA similar approach can be adopted for objects of known shape: standard spheres. Both methods are based on the cosmological principle, stating the Universe obeys statistical isotropy and homogeneity. A relatively novel technique is the use of cosmic voids in this context. After averaging over all orientations, their shape obeys spherical symmetry, even though individual voids may not~\\cite{Park2007,Platen2008}. Therefore, stacked voids can be considered as standard spheres~\\cite{Ryden1995,Lavaux2012,Sutter2012b,Sutter2014b,Hamaus2015,Hamaus2016,Mao2017} with sizes typically ranging from $10h^{-1}{\\rm Mpc}$ to $100h^{-1}{\\rm Mpc}$. This means that in a finite survey volume one can find a substantially larger number of such spheres than rulers in the form of BAO, allowing a significant reduction of statistical uncertainties and to probe a wider range of scales. Standard spheres can be used to constrain the expansion history: only if the fiducial cosmological model in converting redshifts to distances is correct, stacked voids appear spherically symmetric, a technique known as the Alcock-Paczynski (AP) test~\\cite{Alcock1979}. In principle this test merely involves a trivial rescaling of coordinates. However, in observational data the spherical symmetry is broken by redshift-space distortions (RSD), which are caused by the peculiar motions of galaxies along the line of sight. Therefore, a successful application of the AP test to constrain cosmological parameters from voids crucially relies on the ability to robustly model their associated RSD~\\cite{Ryden1996}. The latter are notoriously complex and difficult to model in the clustering statistics of galaxies, especially on intermediate and small scales, where non-linear clustering and shell crossing occurs. It has been shown that these limitations can be mitigated in voids, which are dominated by a laminar, single-stream flow of matter that is well described even by linear theory~\\cite{Paz2013,Hamaus2014b,Hamaus2014c,Pisani2015b,Hamaus2015,Hamaus2016}. This, and the additional virtue of enabling constraints on the growth rate of structure, has sparked the recent interest for void RSD in the literature~\\cite{Cai2016,Chuang2017,Achitouv2017a,Hawken2017,Hamaus2017,Correa2019,Achitouv2019,Nadathur2019a,Nadathur2019b,Hawken2020}.\n\nIn this paper we present a first cosmological analysis of voids from the combined galaxy sample of the final BOSS~\\cite{BOSS} data. Our model self-consistently accounts for RSD and the AP effect, without the need for any external inputs from simulations or mock catalogs. The detailed derivation of the underlying theory is outlined in section~\\ref{sec:theory}, along with a definition of all relevant observables. Section~\\ref{sec:analysis} presents the observed and simulated data sets considered and our method for the identification and characterization of voids therein. Our analysis pipeline is then validated based on mock data in the first part of section~\\ref{sec:analysis}, the second part is devoted to process the real data. We demonstrate that the AP test with voids offers cosmological constraints that are competitive with other large-scale structure probes. Section~\\ref{sec:discussion} is used to summarize our constraints and to discuss them in the light of previous works on voids (see figure~\\ref{fig:comparison}), representing the strongest such constraints in the literature. Finally, we draw our conclusions in section~\\ref{sec:conclusion}.\n\n\n\\section{Theory \\label{sec:theory}}\n\n\\subsection{Dynamic distortion \\label{subsec:dynamic}}\nIn cosmology, our observables are the redshifts $z$ and angular sky coordinates $\\boldsymbol{\\theta}=(\\vartheta,\\varphi)$ of an astronomical object. The comoving distance of this object is defined as\n\\begin{equation}\n\\chi_\\parallel(z)=\\int_0^z\\frac{c}{H(z')}\\mathrm{d}z'\\;,\n\\label{chi_par}\n\\end{equation}\nwhere $H(z)$ is the Hubble rate and $c$ the speed of light. The observed redshift $z$ can contain contributions from many different physical effects, but the most important ones are the cosmological Hubble expansion $z_h$ and the Doppler effect $z_d$. The total observed redshift $z$ is then given by~\\cite{Davis2014}\n\\begin{equation}\n1+z = (1+z_h)(1+z_d)\\;. \\label{z_tot}\n\\end{equation}\nThe Doppler effect is caused by peculiar motions along the line of sight, $z_d=v_\\parallel\/c$. Because $z_d$ is typically small compared to $z_h$, we can write\n\\begin{equation}\n\\chi_\\parallel(z)\\simeq\\chi_\\parallel(z_h) + \\frac{c(1+z_h)}{H(z_h)}z_d\\;.\n\\label{chi_rsd}\n\\end{equation}\nThe transverse comoving distance for an observed angle $\\theta\\equiv|\\boldsymbol{\\theta}|$ on the sky is defined as\n\\begin{equation}\n\\chi_\\perp(z) = D_\\mathrm{A}(z)\\,\\theta\\;,\n\\label{chi_per}\n\\end{equation}\nwhere the comoving angular diameter distance is given by\n\\begin{equation} D_\\mathrm{A}(z) = \\frac{c}{H_0\\sqrt{-\\Omega_\\mathrm{k}}}\\sin\\left(\\frac{H_0\\sqrt{-\\Omega_\\mathrm{k}}}{c}\\chi_\\parallel(z)\\right)\\;,\n\\label{D_A}\n\\end{equation}\nwith the Hubble constant $H_0\\equiv H(z=0)$ and present-day curvature parameter $\\Omega_\\mathrm{k}$. In a flat universe with $\\Omega_\\mathrm{k}=0$, equation~(\\ref{D_A}) reduces to $D_\\mathrm{A}(z)=\\chi_\\parallel(z)$. Now, given the observed coordinates $(z,\\vartheta,\\varphi)$, we can transform to the comoving space vector $\\mathbf{x}$ via\n\\begin{equation}\n\\mathbf{x}(z,\\vartheta,\\varphi) = D_\\mathrm{A}(z)\\begin{pmatrix}\\cos\\vartheta\\cos\\varphi\\\\\\sin\\vartheta\\cos\\varphi\\\\\\sin\\varphi\\end{pmatrix}\\;,\n\\label{x_comoving}\n\\end{equation}\nwhere $D_\\mathrm{A}(z)\\simeq D_\\mathrm{A}(z_h)+cz_d(1+z_h)\/H(z_h)$, analogously to equation~(\\ref{chi_rsd}). Hence, using $z_d=v_\\parallel\/c$, we can write\n\\begin{equation}\n\\mathbf{x}(z) \\simeq \\mathbf{x}(z_h) + \\frac{1+z_h}{H(z_h)}\\mathbf{v}_\\parallel\\;,\n\\label{x_rsd}\n\\end{equation}\nwhere $\\mathbf{v}_\\parallel$ is the component of the velocity vector $\\mathbf{v}$ along the line-of-sight direction. We describe the location and motion of tracers by vectors in comoving space, upper-case letters are used for void centers, lower-case letters for galaxies. The observer's location is chosen to be at the origin of our coordinate system, the void center position is denoted by $\\mathbf{X}$ with redshift $Z$ and the galaxy position by $\\mathbf{x}$ with redshift $z$. The redshift $Z$ of the void center is not a direct observable, but it is constructed via the redshifts of all the surrounding galaxies that define it (see section~\\ref{subsec:voids}). Moreover, we pick the direction of the void center as our line of sight, i.e. $\\mathbf{X}\/|\\mathbf{X}|$, and adopt the distant-observer approximation, assuming that $\\mathbf{x}$ and $\\mathbf{X}$ are parallel.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim= 0 0 0 40]{fig\/VoidStretch.pdf}}\n\t\\caption{Separation vector between the comoving void center location $\\mathbf{X}$ and the galaxy location $\\mathbf{x}$ in real space ($\\mathbf{r}$, left) and in redshift space ($\\mathbf{s}$, right). The peculiar line-of-sight velocity $\\mathbf{v}_\\parallel$ of every galaxy that defines the void can be decomposed into the peculiar velocity of the void center $\\mathbf{V}_\\parallel$ and the galaxy's relative velocity $\\mathbf{u}_\\parallel$ with respect to this center. For simplicity, the illustration displays $\\mu$ instead of $\\cos^{-1}(\\mu)$ to indicate line-of-sight angles and shows velocity displacements in units of $(1+z_h)\/H(z_h)$. This yields the relation $\\mathbf{s}=\\mathbf{r}+\\mathbf{u}_\\parallel$ between real- and redshift-space separations.}\n\t\\label{fig:voidstretch}\n\\end{figure}\n\nLet us first consider real space, where the Doppler effect is neglected ($z_d=0$) and hence $\\mathbf{x}(z)=\\mathbf{x}(z_h)$. The vector $\\mathbf{r}\\equiv\\mathbf{x}-\\mathbf{X}$ connects the two positions at a comoving distance of $r=|\\mathbf{r}|$. Similarly, we define the relative velocity $\\mathbf{u}$ between a void center of velocity $\\mathbf{V}$ and a galaxy of velocity $\\mathbf{v}$ as $\\mathbf{u}\\equiv\\mathbf{v}-\\mathbf{V}$. Now, if we consider redshift space with $z_d\\ne0$ and use equation~(\\ref{x_rsd}), the separation vector between void center and galaxy becomes\n\\begin{equation}\n\\mathbf{x}(z)-\\mathbf{X}(Z) \\simeq \\mathbf{x}(z_h)-\\mathbf{X}(Z_h) + \\frac{1+z_h}{H(z_h)}\\left(\\mathbf{v}_\\parallel-\\mathbf{V}_\\parallel\\right) = \\mathbf{r} + \\frac{1+z_h}{H(z_h)}\\mathbf{u}_\\parallel \\equiv \\mathbf{s}\\;.\n\\label{s(r)}\n\\end{equation}\nHere we have used the approximation $z_h\\simeq Z_h$ for the Doppler term, which is accurate to $\\mathcal{O}(10^{-3})$ on scales of $r\\sim\\mathcal{O}(10^{1})h^{-1}{\\rm Mpc}$ and velocities of $u_\\parallel\\sim\\mathcal{O}(10^2)\\frac{\\rm km}{\\rm s\\,Mpc}$. This approximation becomes exact when we consider the average of this vector over many different voids -- a so-called void stack. In that case the cosmological principle ensures statistical homogeneity and isotropy such that $\\langle\\mathbf{V}\\rangle=\\langle\\mathbf{V}_\\parallel\\rangle=0$, since there is no preferred observer (or void, for that matter) in the Universe. In other words: the average void moves with the background Hubble flow. Thus, the vector $\\mathbf{s}$, which connects void centers and galaxies in redshift space, does not depend on the individual motions of galaxies or void centers, but only on their relative velocity $\\mathbf{u}_\\parallel$ along the line of sight. Of course this only applies to those galaxies that are part of the same void, not to the relative velocities of galaxies that are associated with distinct voids at larger separation (see~\\cite{Hamaus2014a,Chan2014,Hamaus2014c,Liang2016,Chuang2017,Voivodic2020} for an account on large-scale void-galaxy cross-correlations and~\\cite{Sutter2014c,Ruiz2015,Lambas2016,Ceccarelli2016,Wojtak2016,Lares2017b} on the motions and pairwise velocity statistics of voids). An illustration of this is shown in figure~\\ref{fig:voidstretch}: voids experience a translation and a deformation between real and redshift space, but the translational component does not enter in the separation vector $\\mathbf{s}$ for galaxies that belong to the void. As long as voids can be considered as coherent extended objects, their centers move along with the galaxies that define them from real to redshift space. This property distinguishes voids from galaxies, which are typically treated as point-like particles in the context of large-scale structure. The same reasoning applies to galaxy clusters, the overdense counterparts of voids: the center of mass is defined by the cluster member galaxies, which are observed in redshift space. Their relative (virial) motion with respect to the center of mass results in an elongated stacked cluster shape, irrespective of the movement of the entire cluster~\\cite{Croft1999,Zu2013,Marulli2017,Farahi2016,Cai2017b}.\n\n\\subsection{Geometric distortion \\label{subsec:geometric}}\nGiven the observed angular sky-coordinates and redshifts of galaxies and void centers, the comoving distance $s = (s_\\parallel^2 + s_\\perp^2)^{1\/2}$ in redshift space between any void-galaxy pair is determined by equations~(\\ref{chi_par}) and~(\\ref{chi_per}),\n\\begin{equation}\ns_\\parallel = \\frac{c}{H(z)}\\delta z\\quad\\mathrm{and}\\quad s_\\perp = D_\\mathrm{A}(z)\\delta\\theta\\;, \\label{comoving}\n\\end{equation}\nwhere $\\delta z$ and $\\delta\\theta$ are the redshift and angular separation of the pair, respectively. This transformation requires the Hubble rate $H(z)$ and the comoving angular diameter distance $D_\\mathrm{A}(z)$ as functions of redshift, so a particular cosmological model has to be adopted. We assume a flat $\\Lambda$CDM cosmology with\n\\begin{equation}\nH(z) = H_0\\sqrt{\\Omega_\\mathrm{m}(1+z)^3+\\Omega_\\Lambda}\\;, \\label{H(z)}\n\\end{equation}\nwhere $\\Omega_\\mathrm{m}$ and $\\Omega_\\Lambda=1-\\Omega_\\mathrm{m}$ are today's density parameters for matter and a cosmological constant, respectively. For the low redshift range considered in this paper, we can neglect the radiation density for all practical purposes. Given a cosmology with parameters $\\boldsymbol{\\Omega}$ we can now convert angles and redshifts into comoving distances according to equation~(\\ref{comoving}). However, the precise value of these parameters is unknown, as is the underlying cosmology, so we can only estimate the true $H(z)$ and $D_\\mathrm{A}(z)$ with a fiducial model and parameter set $\\boldsymbol{\\Omega}'$, resulting in the estimated distances\n\\begin{equation}\ns_\\parallel' = \\frac{H(z)}{H'(z)}s_\\parallel\\equiv q_\\parallel^{-1} s_\\parallel\\;\\mathrm{,}\\qquad s_\\perp' = \\frac{D_\\mathrm{A}'(z)}{D_\\mathrm{A}(z)}s_\\perp\\equiv q_\\perp^{-1} s_\\perp\\;,\n\\end{equation}\nwhere the primed quantities are evaluated in the fiducial cosmology and $(q_\\parallel, q_\\perp)$ are defined as the ratios between true and fiducial values of $H^{-1}(z)$ and $D_\\mathrm{A}(z)$, respectively~\\cite{SanchezA2017}. Therefore, both magnitude and direction of the separation vector $\\mathbf{s}$ may differ from the truth when a fiducial cosmology is assumed. Defining the cosine of the angle between $\\mathbf{s}$ and the line of sight $\\mathbf{X}\/|\\mathbf{X}|$ as\n\\begin{equation}\n\\mu_s\\equiv\\frac{\\mathbf{s}\\cdot\\mathbf{X}}{|\\mathbf{s}||\\mathbf{X}|}=\\frac{s_\\parallel}{s}\\;,\n\\label{mu_s}\n\\end{equation}\none can obtain the true $s$ and $\\mu_s$ from the fiducial $s'$ and $\\mu_s'$ via\n\\begin{gather}\ns = \\sqrt{q_\\parallel^2 s_\\parallel'^2+q_\\perp^2s_\\perp'^2} = s'\\mu_s'q_\\parallel\\sqrt{1+\\varepsilon^2(\\mu_s'^{-2}-1)}\\;,\n\\label{s_fid}\n\\\\\n\\mu_s = \\frac{\\mathrm{sgn}(\\mu_s')}{\\sqrt{1+\\varepsilon^2(\\mu_s'^{-2}-1)}}\\;,\n\\label{mu_s_fid}\n\\end{gather}\nwhere\n\\begin{equation}\n\\varepsilon \\equiv \\frac{q_\\perp}{q_\\parallel} =  \\frac{D_\\mathrm{A}(z)H(z)}{D_\\mathrm{A}'(z)H'(z)}\\;.\n\\label{epsilon}\n\\end{equation}\nIf the fiducial cosmology agrees with the truth, $\\varepsilon=q_\\parallel=q_\\perp=1$ and $s=s'$, $\\mu_s=\\mu_s'$. Conversely, if one of these parameters is measured to be different from unity, one may iteratively vary the fiducial cosmology until the true parameter values $\\boldsymbol{\\Omega}$ are found. As apparent from equations~(\\ref{s_fid}) and~(\\ref{mu_s_fid}), absolute distances $s$ in redshift space depend on both $q_\\parallel$ and $q_\\perp$, whereas angles $\\mu_s$ merely depend on their ratio~$\\varepsilon$. Exploiting the spherical symmetry of stacked voids via the AP effect therefore constrains $\\varepsilon$, but $q_\\parallel$ and $q_\\perp$ remain degenerate without calibration of $s$ with a known scale (such as the BAO scale, for example). However, void-centric distances are typically expressed in units of the effective void radius $R$, which is defined via the cubic root of the void volume in redshift space (see section~\\ref{subsec:voids}). The observed volume is proportional to $s'_\\parallel s'^2_\\perp$, implying $R=q_\\parallel^{1\/3}q_\\perp^{2\/3}R'$ to relate true with fiducial void radii. Then, the ratio $s\/R$ only depends on $\\varepsilon$, as it is the case for $\\mu_s$,\n\\begin{equation}\n\\frac{s}{R}=\\frac{s'}{R'}\\mu_s'\\varepsilon^{-2\/3}\\sqrt{1+\\varepsilon^2(\\mu_s'^{-2}-1)}\\;.\n\\label{s_R_fid}\n\\end{equation} \n\n\n\\subsection{Void-galaxy cross-correlation function \\label{subsec:correlation}}\nThe probability of finding a galaxy at comoving distance $r$ from a void center in real space is given by $1+\\xi(r)$, where $\\xi(r)$ is the void-galaxy cross-correlation function. Due to statistical isotropy, it only depends on the magnitude of the separation vector $\\mathbf{r}$, not its orientation. This is no longer the case in redshift space, where peculiar motions break isotropy via the Doppler effect. However, since this causes RSD exclusively along the line-of-sight direction, we can eliminate their impact by projecting the correlation function onto the plane of the sky. This yields the projected correlation function $\\xi_p$,\n\\begin{equation}\n1+\\xi_p(r_\\perp) = \\frac{\\int\\left[1+\\xi(r)\\right]\\mathrm{d}r_\\parallel}{\\int\\mathrm{d}r_\\parallel} = \\frac{\\int\\left[1+\\xi^s(\\mathbf{s})\\right]\\mathrm{d}s_\\parallel}{\\int\\mathrm{d}s_\\parallel} = 1+\\xi^s_p(s_\\perp)\\;,\n\\label{xi_p}\n\\end{equation}\nwhere $r=(r_\\parallel^2+r_\\perp^2)^{1\/2}$ and $\\xi^s(\\mathbf{s})$ is the redshift-space correlation function, which can now be expressed as\n\\begin{equation}\n1+\\xi^s(\\mathbf{s}) = \\left[1+\\xi(r)\\right]\\frac{\\mathrm{d}r_\\parallel}{\\mathrm{d}s_\\parallel}\\;.\n\\label{xi^s}\n\\end{equation}\nEquation~(\\ref{s(r)}) provides the relation between $\\mathbf{s}$ and $\\mathbf{r}$. In particular, its line-of-sight component can be obtained via taking the dot product with $\\mathbf{X}\/|\\mathbf{X}|$,\n\\begin{equation}\ns_\\parallel = r_\\parallel + \\frac{1+z_h}{H(z_h)}u_\\parallel\\;,\n\\label{s_par(r_par)}\n\\end{equation}\nand hence\n\\begin{equation}\n\\frac{\\mathrm{d}r_\\parallel}{\\mathrm{d}s_\\parallel} = \\left(1 + \\frac{1+z_h}{H(z_h)}\\,\\frac{\\mathrm{d}u_\\parallel}{\\mathrm{d}r_\\parallel}\\right)^{-1}\\;.\n\\label{dr_par\/ds_par}\n\\end{equation}\nThe relative peculiar velocity $\\mathbf{u}$ between void centers and their surrounding galaxies can be derived by imposing local mass conservation. At linear order in the matter-density contrast~$\\delta$ and assuming spherical symmetry in real space, the velocity field is given by~\\cite{Peebles1980}\n\\begin{equation}\n\\mathbf{u}(\\mathbf{r}) = -\\frac{f(z_h)}{3}\\frac{H(z_h)}{1+z_h}\\Delta(r)\\,\\mathbf{r}\\;,\n\\label{u(r)}\n\\end{equation}\nwhere $f(z)\\equiv-\\frac{\\mathrm{d}\\!\\ln D(z)}{\\mathrm{d}\\!\\ln(1+z)}$ is the linear growth rate, defined as the logarithmic derivative of the linear growth factor $D(z)$ with respect to the scale factor. In $\\Lambda$CDM, the linear growth rate is well approximated by\n\\begin{equation}\nf(z)=\\left[\\frac{\\Omega_\\mathrm{m}(1+z)^3}{H^2(z)\/H_0^2}\\right]^\\gamma,\n\\label{growth_rate}\n\\end{equation}\nwith a growth index of $\\gamma\\simeq0.55$~\\cite{Lahav1991,Linder2005}. Furthermore, $\\Delta(r)$ is the average matter-density contrast inside a spherical region of comoving radius~$r$,\n\\begin{equation}\n\\Delta(r) = \\frac{3}{r^3}\\int_0^r\\delta(r')r'^2\\,\\mathrm{d}r'\\;. \\label{Delta(r)}\n\\end{equation}\nAlthough the matter-density contrast in the vicinity of void centers is not necessarily in the linear regime (i.e., $|\\delta|\\ll1$), contrary to over-dense structures (such as galaxy clusters and their dark matter halos) it is bounded from below by the value of $-1$. In simulations it has been shown that equation~(\\ref{u(r)}) provides an extremely accurate description of the local velocity field in and around most voids~\\cite{vdWeygaert1993,Hamaus2014b}. While peculiar velocities at the void boundaries can be due to very non-linear structures, spherical averaging over large sample sizes helps to restore the validity of linear theory to a high degree. Only the smallest and most underdense voids exhibit a non-linear behavior close to their centers and may in fact collapse anisotropically under their external mass distribution~\\cite{vdWeygaert1993,Sheth2004}. We can now evaluate the derivative term in equation~(\\ref{dr_par\/ds_par}) as\n\\begin{equation}\n\\frac{1+z_h}{H(z_h)}\\frac{\\mathrm{d}u_\\parallel}{\\mathrm{d}r_\\parallel} = -\\frac{f(z_h)}{3}\\Delta(r)-f(z_h)\\mu_r^2\\left[\\delta(r)-\\Delta(r)\\right]\\;,\n\\label{du_par\/dr_par}\n\\end{equation}\nwhere $\\mu_r=r_\\parallel\/r$ and the identity $\\frac{\\mathrm{d}\\Delta(r)}{\\mathrm{d}r} = \\frac{3}{r}\\left[\\delta(r)-\\Delta(r)\\right]$ was used. Plugging this back into equation~(\\ref{xi^s}) we obtain\n\\begin{equation}\n1+\\xi^s(\\mathbf{s}) = \\frac{1+\\xi(r)}{1-\\frac{f}{3}\\Delta(r)-f\\mu_r^2\\left[\\delta(r)-\\Delta(r)\\right]}\\;.\n\\label{xi^s_nonlin}\n\\end{equation}\nIn order to evaluate this equation at a given observed separation $\\mathbf{s}$, we make use of equations~(\\ref{s_par(r_par)}) and (\\ref{u(r)}),\n\\begin{equation}\nr_\\parallel = \\frac{s_\\parallel}{1 - \\frac{f}{3}\\Delta(r)}\\;,\n\\label{r_par(s_par)}\n\\end{equation}\nand calculate $r=(r_\\parallel^2+r_\\perp^2)^{1\/2}$ with $r_\\perp=s_\\perp$. However, equation~(\\ref{r_par(s_par)}) already requires knowledge of $r$ in the argument of $\\Delta(r)$, so it can only be evaluated by iteration. We therefore start with using $\\Delta(s)$ as initial step, and iteratively calculate $r_\\parallel$ and $\\Delta(r)$ until convergence is reached. In practice we find $5$ iterations to be fully sufficient for that purpose.\n\nFurthermore, in equation~(\\ref{xi^s_nonlin}) both the void-galaxy cross-correlation function $\\xi(r)$, as well as the void-matter cross-correlation function $\\delta(r)$ are required in real space. The former can be obtained via deprojection of equation~(\\ref{xi_p}),\n\\begin{equation}\n\\xi(r) = -\\frac{1}{\\pi}\\int_r^\\infty\\frac{\\mathrm{d}\\xi^s_p(s_\\perp)}{\\mathrm{d}s_\\perp}\\frac{\\mathrm{d}s_\\perp}{\\sqrt{s_\\perp^2-r^2}}\\;,\n\\label{xi_d}\n\\end{equation}\nmaking use of the inverse Abel transform~\\cite{Pisani2014,Hawken2017}. The latter function $\\delta(r)$, also referred to as the void density profile, is not directly observable. However, it has been investigated in $N$-body simulations and can be inferred via the gravitational lensing effect in imaging surveys~\\cite{Krause2013,Higuchi2013,Melchior2014}. The parametric form suggested in reference~\\cite{Hamaus2014b} (HSW profile) has been shown to accurately describe both simulated~\\cite{Sutter2014a,Hamaus2015,Barreira2015,Falck2018,Pollina2017,Perico2019}, as well as observational data~\\cite{Hamaus2016,SanchezC2017,Pollina2019,Fang2019},\n\\begin{equation}\n\\delta_{\\scriptscriptstyle\\mathrm{HSW}}(r) = \\delta_c\\frac{1-(r\/r_s)^\\alpha}{1+(r\/R)^\\beta}\\;. \\label{HSW}\n\\end{equation}\nHere $R$ is the effective void radius, the scale radius $r_s$ determines where $\\delta_{\\scriptscriptstyle\\mathrm{HSW}}(r_s)=0$, the central underdensity is defined as $\\delta_c\\equiv\\delta_{\\scriptscriptstyle\\mathrm{HSW}}(r=0)$, and the power-law indices $\\alpha$ and $\\beta$ control the inner and outer slopes of the profile. In equation~(\\ref{xi^s_nonlin}) these quantities can then be included as free parameters to be constrained by the observed $\\xi^s(\\mathbf{s})$. This approach has been pursued in the framework of the Gaussian streaming model (GSM)~\\cite{Hamaus2015,Hamaus2016}, which incorporates an additional parameter for the velocity dispersion $\\sigma_v$ of galaxies. In the limit of $\\sigma_v\\rightarrow0$, the GSM recovers the result of equation~(\\ref{xi^s_nonlin}) at linear order in $\\delta$~\\cite{Cai2016}. We note that equation~(\\ref{HSW}) describes the spherically averaged density profile with respect to the void center, but a similar parametrization exists for profiles centered on the void boundary~\\cite{Cautun2016}.\n\nAnother option to constrain the void density profile $\\delta(r)$ is through its relation to $\\xi(r)$, which is equivalent to the void density profile in galaxies. Both simulations~\\cite{Pollina2017,Ronconi2019,Contarini2019} and observational approaches~\\cite{Pollina2019,Fang2019} have established robust evidence for the relationship between $\\delta(r)$ and $\\xi(r)$ to be a linear one, such that\n\\begin{equation}\n\\xi(r) = b\\delta(r)\\;,\n\\label{xi(delta)}\n\\end{equation}\nwith a single proportionality constant $b$. This is similar to the relation between the overdensity of tracers and the underlying matter distribution on large scales, where $|\\delta|\\ll1$~\\cite{Desjacques2018}. That condition is not necessarily satisfied in the interiors and immediate surroundings of voids, and the large-scale linear bias does not coincide with the value of $b$ in general, even for the same tracer population. However, the two bias values approach each other for voids of increasing size, and converge in the limit of large effective void radius $R$~\\cite{Pollina2017,Pollina2019,Contarini2019}. Using equation~(\\ref{xi(delta)}) for $\\delta(r)$, we can simply exchange it with $\\xi(r)$ by making the replacements $f\\rightarrow f\/b$ and $\\Delta(r)\\rightarrow\\overline{\\xi}(r)$, with\n\\begin{equation}\n\\overline{\\xi}(r) = \\frac{3}{r^3}\\int_0^r\\xi(r')r'^2\\mathrm{d}r'\\;.\n\\label{xibar}\n\\end{equation}\nNow equation~(\\ref{xi^s_nonlin}) can be written as\n\\begin{equation}\n1+\\xi^s(\\mathbf{s}) = \\frac{1+\\xi(r)}{1-\\frac{1}{3}\\frac{f}{b}\\overline{\\xi}(r)-\\frac{f}{b}\\mu_r^2\\left[\\xi(r)-\\overline{\\xi}(r)\\right]}\\;.\n\\label{xi^s_nonlin2}\n\\end{equation}\nMoreover, we can expand this to linear order in $\\delta$ (or equivalently, $\\xi$) for consistency with the perturbative level of the mass conservation equation~(\\ref{u(r)})~\\cite{Cai2016,Hamaus2017},\n\\begin{equation}\n\\xi^s(\\mathbf{s}) \\simeq \\xi(r) + \\frac{1}{3}\\frac{f}{b}\\overline{\\xi}(r) + \\frac{f}{b}\\mu_r^2\\left[\\xi(r)-\\overline{\\xi}(r)\\right]\\;.\n\\label{xi^s_lin}\n\\end{equation}\nThe function $\\xi^s(\\mathbf{s})$ can be decomposed into independent multipoles via\n\\begin{equation}\n\\xi^s_\\ell(s) = \\frac{2\\ell+1}{2}\\int\\limits_{-1}^1\\xi^s(s,\\mu_s)\\mathcal{L}_\\ell(\\mu_s)\\mathrm{d}\\mu_s\\;,\n\\label{multipoles}\n\\end{equation}\nwith the Legendre polynomials $\\mathcal{L}_\\ell(\\mu_s)$ of order $\\ell$. For equation~(\\ref{xi^s_lin}) the integral can be performed analytically and the only non-vanishing multipoles at linear order in $\\xi$ and $\\overline{\\xi}$ are the monopole ($\\ell=0$) and quadrupole ($\\ell=2$) with\n\\begin{eqnarray}\n\\xi^s_0(s) &=& \\left(1+\\frac{f\/b}{3}\\right)\\xi(r)\\;,\\label{xi_0} \\\\\n\\xi^s_2(s) &=& \\frac{2f\/b}{3}\\left[\\xi(r)-\\overline{\\xi}(r)\\right]\\;.\\label{xi_2}\n\\end{eqnarray}\nThis can be recast into the following form~\\cite{Cai2016,Hamaus2017},\n\\begin{equation}\n\\xi^s_0(s) - \\overline{\\xi}^s_0(s) = \\xi^s_2(s)\\frac{3+f\/b}{2f\/b}\\;,\n\\label{xi_0_2}\n\\end{equation}\nproviding a direct link between monopole and quadrupole in redshift space without reference to any real-space quantity. However, note that equations~(\\ref{xi_0}),~(\\ref{xi_2}) and (\\ref{xi_0_2}) only hold for the case of $\\varepsilon=1$, and multipoles of higher order can be generated via geometric distortions when assuming a fiducial cosmology that is different from the truth, as discussed in section~\\ref{subsec:geometric}.\n\n\n\\section{Analysis \\label{sec:analysis}}\n\n\\subsection{BOSS galaxies and mocks \\label{subsec:galaxies}}\nWe consider galaxy catalogs from the final data release 12 (DR12) of the SDSS-III~\\cite{Eisenstein2011} Baryon Oscillation Spectroscopic Survey (BOSS)~\\cite{Dawson2013}. In particular, we make use of the combined sample of the individual target selections denoted as LOWZ and CMASS~\\cite{Reid2016}. With a total sky area of about $10\\,000$ square degrees from both the northern and southern Galactic hemispheres the sample contains $1\\,198\\,006$ galaxies in a redshift range of $0.20<z<0.75$ with a linear bias of $b=1.85$~\\cite{Alam2017}, making it the largest spectroscopic galaxy catalog available to date. In order to test and validate our analysis pipeline, we additionally consider the MultiDark PATCHY mock galaxy catalogs~\\cite{Kitaura2016a} that are specifically designed to match the properties of the BOSS DR12 galaxy samples and have been calibrated to $N$-body simulations~\\cite{Klypin2016}. The fiducial cosmology of the mocks assumes a flat $\\Lambda$CDM model with parameter values $\\boldsymbol{\\Omega}'=(\\Omega_\\mathrm{m},\\Omega_\\mathrm{b},\\sigma_8,n_s,h)=(0.307115,0.048206,0.8288,0.9611,0.6777)$ adopted from Planck 2013~\\cite{Planck2013} and a linear bias of $b=2.20$. For both real and mock samples we also make use of the random catalogs provided by the BOSS collaboration, which provide an unclustered realization of the galaxy distribution with the same survey geometry, but $50$ times its density at any given redshift. This facilitates an accurate estimation of correlation functions (see section~\\ref{subsec:estimators}).\n\n\\subsection{VIDE voids \\label{subsec:voids}}\nWe identify voids in the galaxy distribution using \\textsc{vide}\\footnote{\\url{https:\/\/bitbucket.org\/cosmicvoids\/vide_public\/}}, a publicly available software repository that can handle simulation snapshots, as well as masked light cones from observations~\\cite{Sutter2015}. \\textsc{vide} is based on \\textsc{zobov}~\\cite{Neyrinck2008}, which implements a watershed transform~\\cite{Platen2007} to delineate local basins in the three-dimensional density field of tracer particles. Assuming a fiducial cosmology $\\boldsymbol{\\Omega}'$, equation~(\\ref{x_comoving}) is used to transform angles and redshifts to comoving space. Then the density field is estimated via Voronoi tessellation, assigning each tracer particle $i$ a unique cell of volume $\\mathcal{V}_i$. A Voronoi cell is defined as the sub-set of space that is closer to its associated tracer particle than to any other particle in the entire catalog. The tessellation of space into Voronoi cells is volume filling and hence allows the assignment of a tracer density to any location inside that volume. The density can be estimated as $1\/\\mathcal{V}_i$ anywhere inside the cell of tracer particle $i$, providing a piecewise constant density field over the entire domain. Local density minima serve as starting points to search for extended watershed basins whose density monotonically increases among their neighboring cells. The extent of a basin is determined as soon as a saddle point is encountered, indicating the occurrence of an adjacent basin whose cell densities start decreasing again. All the Voronoi cells that together make up such a basin already define a void region. The construction of a nested hierarchy of voids and sub-voids can then be defined by merging adjacent basins~\\cite{Neyrinck2008}. For simplicity, in this paper we only consider the non-overlapping leaf nodes of this hierarchy.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.95, trim=32 0 50 60]{fig\/box.pdf}\n\t\\includegraphics[scale=1.15, trim=90 90 90 50, clip]{fig\/slice.pdf}\n\t\\caption{TOP: \\textsc{vide} void centers ($N_\\mathrm{v}=5952$) in comoving space from the BOSS DR12 combined sample. The observer is located at the origin, redshift is color-coded between $0.20$ (blue) and $0.75$ (yellow). BOTTOM: \\textsc{vide} void centers (circles scaled by the effective void radius) and BOSS galaxies (dots) inside a slice of about one degree width in declination within the northern Galactic hemisphere.}\n\t\\label{fig:box}\n\\end{figure}\n\nWe define the center of each void as the volume-weighted barycenter among all of its constituent Voronoi cells at tracer locations $\\mathbf{x}_i$,\n\\begin{equation}\n\\mathbf{X} = \\frac{\\sum_i\\mathbf{x}_i\\mathcal{V}_i}{\\sum_i\\mathcal{V}_i}\\;.\n\\end{equation}\nThe barycenter can be thought of as the geometric center of the void, as it is mostly constrained by its boundaries where the majority of tracers reside. This implies that it does not generally coincide with the minimum density inside the void, due to its non-spherical geometry. Conversely, the location of the minimum density is largely independent of the void boundaries, making it a poor indicator of the void geometry. Thus, the optimal center definition typically depends on the desired void observables. In this paper we are interested in measurements of the shape of stacked voids, so we want to maximize the sensitivity to their boundaries~\\cite{Cautun2016}. To retain knowledge of this defining property of watershed voids, we therefore choose the volume-weighted barycenter. Moreover, in contrast to center definitions that are based on merely a single or a few tracers, the barycenter is more robust against discreteness noise and peculiar motions of individual tracers~\\cite{Hamaus2014c}. In order to calculate redshifts and sky coordinates from our void centers $\\mathbf{X}$, we simply invert the coordinate transformation from equation~(\\ref{x_comoving}). Although watershed basins may exhibit arbitrary geometries, we assign an effective radius\n\\begin{equation}\nR = \\left(\\frac{3}{4\\pi}\\sum\\nolimits_i\\mathcal{V}_i\\right)^{1\/3}\\;,\n\\end{equation}\nto each void, which corresponds to the radius of a sphere of equal volume. Running \\textsc{vide} on the entire light cone of the BOSS DR12 combined sample we find a total of $N_\\mathrm{v}=5952$ voids. This number already accounts for voids that have been discarded due to intersection with the survey mask, redshift boundaries, and other quality cuts (see reference~\\cite{Sutter2015} for more details). Moreover, we impose a purity selection cut on voids based on their effective radius,\n\\begin{equation}\nR > N_\\mathrm{s}\\left[\\frac{4\\pi}{3}n(z)\\right]^{-1\/3}\\;,\n\\label{ats}\n\\end{equation}\nwhere $n(z)$ is the number density of tracers at redshift $z$ and $N_\\mathrm{s}$ determines the minimum considered void size in units of the average tracer separation. The smaller $N_\\mathrm{s}$, the larger the contamination by spurious voids that may arise from Poisson fluctuations~\\cite{Neyrinck2008,Cousinou2019}. This cut also preferentially removes voids that may have been misidentified due to RSDs~\\cite{Pisani2015b}. As a default we assume a conservative value of $N_\\mathrm{s}=4$, which yields a minimum effective void radius of $R=34.9h^{-1}{\\rm Mpc}$ in our catalog. We note that this criterion depends on the specific tracer sample considered for void identification. It is known that Poisson point processes exhibit highly non-Gaussian Voronoi cell volume distributions~\\cite{vdWeygaert2009}, which can cause spurious void detections. Reference~\\cite{Cousinou2019} finds a very low contamination fraction of spurious voids in the BOSS DR12 CMASS sample based on a multivariate analysis of void properties in training and test samples. This is attributed to the relatively high clustering bias of CMASS galaxies, which is very similar to the combined BOSS sample used here.\n\nThe top of figure~\\ref{fig:box} presents a three-dimensional view of the selected void centers from the northern (right) and southern (left) Galactic hemispheres in comoving space, with the observer located at the origin. Below it, a narrow slice of about one degree in declination within the northern Galactic hemisphere visualizes the distribution of void centers together with their tracer galaxies. Despite the sparsity of the BOSS DR12 combined sample, intricate features of the cosmic web-like structure become apparent. Note that due to the extended three-dimensional geometry of voids, their centers do not necessarily intersect with the slice, leaving some seemingly empty regions without associated void center. The left panel of figure~\\ref{fig:nz} shows the redshift distribution of galaxies, randoms, and voids from the data. For visibility, we have rescaled the total number of randoms to the number of galaxies (by a factor of $50$). Voids are roughly two orders of magnitude scarcer than galaxies, but their redshift distribution follows a similar trend. This is because higher tracer densities allow the identification of smaller voids, as expected from simulation studies~\\cite{Jennings2013,Sutter2014a,Chan2014,Wojtak2016}. The right panel of figure~\\ref{fig:nz} shows the distribution of effective void radii with Poisson error bars, also known as the void-size function. The latter is a quantity of interest for cosmology on its own~\\cite{Pisani2015a,Ronconi2019,Contarini2019,Verza2019}, but in this paper we do not investigate it for that purpose any further, it is shown here only as supplementary information. We repeat the void finding procedure on each of the PATCHY mocks, allowing us to scale down all statistical uncertainties by roughly a factor of $\\sqrt{N_\\mathrm{m}}$, where $N_\\mathrm{m}$ is the number of mock catalogs considered.\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=0 -3 0 0]{fig\/nz.pdf}\n\t\t\\includegraphics{fig\/n_rv.pdf}}\n\t\\caption{LEFT: Number density of galaxies, randoms (scaled to the galaxy density), and \\textsc{vide} voids as a function of redshift in the BOSS DR12 combined sample. RIGHT: Number density of all \\textsc{vide} voids as a function their effective radius (void-size function, only shown for illustration).}\n\t\\label{fig:nz}\n\\end{figure}\n\n\\subsection{Estimators \\label{subsec:estimators}}\nWe need to define an estimator to measure the observed void-galaxy cross-correlation function $\\xi^s(\\mathbf{s})$ in redshift space. In order to take into account the survey geometry, we make use of a random catalog that samples the masked survey volume without intrinsic clustering.  We adopt the expression derived in reference~\\cite{Hamaus2017},\n\\begin{equation}\n\\xi^s(\\mathbf{s}) = \\frac{\\langle\\mathbf{X},\\mathbf{x}\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}\\rangle\\langle\\mathbf{x}\\rangle} - \\frac{\\langle\\mathbf{X},\\mathbf{x}_r\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}\\rangle\\langle\\mathbf{x}_r\\rangle}\\;,\n\\label{estimator}\n\\end{equation}\nwith the void center, galaxy, and random positions $\\mathbf{X}$, $\\mathbf{x}$, and $\\mathbf{x}_r$, respectively. Here, the angled brackets with two vectors represent the average pair count between objects at separation $\\mathbf{s}$, whereas the brackets with only one vector indicate the mean number count of an object in the corresponding redshift slice. This estimator is a limiting case of Landy \\& Szalay~\\cite{Landy1993} and has been validated using mock data on the relevant scales for voids in BOSS data~\\cite{Hamaus2017}. We will consider its two-dimensional variant $\\xi^s(s_\\parallel,s_\\perp)$, with explicit dependence on distances along and perpendicular to the line of sight, as well as its multipoles $\\xi^s_\\ell(s)$ with $\\ell=(0,2,4)$, i.e. monopole, quadrupole and hexadecapole. Calculating the multipoles is particularly simple with this estimator, because it allows the direct application of equation~(\\ref{multipoles}) on the pair counts by using Legendre polynomials as weights in the integral without the need to define bins in $\\mu_s$~\\cite{Hamaus2017}. The mean number counts in the denominator of equation~(\\ref{estimator}) can be pulled outside the integral, as they do not depend on $\\mathbf{s}$. This way the estimation of multipoles becomes more accurate, especially at small separations $s$ and when $\\mu_s$ approaches one, where the binning in both quantities inevitably results in a coarse spatial resolution~\\cite{Cai2016}. We have explicitly compared this estimator with other common choices in the literature~\\cite{Vargas2013} and refer the reader to section~\\ref{subsec:systematics} for more details on this.\n\nWatershed voids exhibit an angle-averaged density profile of universal character, irrespective of their absolute size~\\cite{Hamaus2014b,Sutter2014a,Cautun2016}. In order to capture this unique characteristic in the two-point statistics of a void sample with a broad range of effective radii, it is beneficial to express all separations in units of $R$ (see section~\\ref{subsec:comparison}). When this is done for every void individually in the estimation of $\\xi^s$, it is commonly denoted as a void stack. Like the majority of void-related studies in the literature, we adopt this approach in our analysis and refer to $\\xi^s(\\mathbf{s}\/R)$ as the stacked void-galaxy cross-correlation function. For simplicity, we will omit this explicit notation in the following and bear in mind that all separations $\\mathbf{s}$ are expressed in units of~$R$.\n\nThe degree of uncertainty in a measurement of the void-galaxy cross-correlation function can be quantified by its covariance matrix. It is defined as the covariance of $\\xi^s$ at separations $\\mathbf{s}_i$ and $\\mathbf{s}_j$ from $N$ independent observations,\n\\begin{equation}\n\\mathbf{C}_{ij} = \\bigl<\\bigl(\\xi^s(\\mathbf{s}_i)-\\langle\\xi^s(\\mathbf{s}_i)\\rangle\\bigr)\\bigl(\\xi^s(\\mathbf{s}_j)-\\langle\\xi^s(\\mathbf{s}_j)\\rangle\\bigr)\\bigr>\\;,\n\\label{covariance}\n\\end{equation}\nwhere the angled brackets indicate averages over the sample size. Although we can only observe a single universe, we have a large sample of voids at our disposal that enables an estimate of the covariance matrix as well. Note that we are considering mutually exclusive voids, each of which provide an independent patch of large-scale structure. As we are primarily interested in $\\xi^s$ on scales up to the void extent, as opposed to inter-void scales, we can employ a jackknife resampling strategy to estimate $\\mathbf{C}_{ij}$~\\cite{Hamaus2017}. For this we simply remove one void at a time in the estimator of $\\xi^s$ from equation~(\\ref{estimator}), which provides $N_\\mathrm{v}$ jackknife samples in total. These samples can then be used in equation~(\\ref{covariance}) to calculate $\\mathbf{C}_{ij}$, albeit with an additional factor of $(N_\\mathrm{v}-1)$ to account for the statistical weight of the jackknife sample size. We use the square root of the diagonal elements of the covariance matrix to quote error bars on our measurements of $\\xi^s$. The identical procedure can be applied to the multipoles $\\xi^s_\\ell$, except in that case one can use equation~(\\ref{covariance}) to additionally calculate the covariance between multipoles of different order.\n\nThere are several advantages of this jackknife technique over other common methods for covariance estimation, which typically rely on simulated mock catalogs. Most importantly, it is based on the observed data itself and does not involve prior model assumptions about cosmology, structure formation, or galaxy evolution. In addition, a statistically reliable estimation of $\\mathbf{C}_{ij}$ requires large sample sizes, which are expensive in terms of numerical resources when considering realistic mocks from $N$-body simulations. Our void catalog already provides $\\mathcal{O}(10^3)$ spatially independent samples at no additional cost. It has been shown that the jackknife technique provides covariance estimates that are consistent with those obtained from independent mock catalogs in the limit of large jackknife sample sizes~\\cite{Favole2020}.\n\n\n\\subsection{Likelihood \\label{subsec:likelihood}}\nEquipped with the theory from section~\\ref{subsec:correlation} we can now define the likelihood $L(\\hat{\\xi}^s|\\boldsymbol{\\Omega})$ of the measurement given a model, which we approximate to be of Gaussian form,\n\\begin{equation}\n\\ln L(\\hat{\\xi}^s|\\boldsymbol{\\Omega}) = -\\frac{1}{2N_\\mathrm{m}}\\sum\\limits_{i,j}\\Bigl(\\hat{\\xi}^s(\\mathbf{s}_i)-\\xi^s(\\mathbf{s}_i,\\boldsymbol{\\Omega})\\Bigr)\\,\\hat{\\mathbf{C}}_{ij}^{-1}\\Bigl(\\hat{\\xi}^s(\\mathbf{s}_j)-\\xi^s(\\mathbf{s}_j,\\boldsymbol{\\Omega})\\Bigr)\\;.\n\\label{likelihood}\n\\end{equation}\nThe hat symbols indicate a measured quantity to be distinguished from the model, which explicitly depends on the parameters $\\boldsymbol{\\Omega}$. Here we have dropped the normalization term involving the determinant of $\\hat{\\mathbf{C}}_{ij}$, since it only adds a constant. The form of equation~(\\ref{likelihood}) can be applied to either the two-dimensional void-galaxy cross-correlation function $\\xi^s(s_\\parallel,s_\\perp)$, or its multipoles $\\xi^s_\\ell(s)$. We use $\\xi^s(s_\\parallel,s_\\perp)$, which contains the information from the multipoles of all orders. However, we have verified that only including the multipoles of orders $\\ell=(0,2,4)$ yields consistent results. When analyzing mock catalogs we scale their covariance in equation~(\\ref{likelihood}) by the number of mock samples $N_\\mathrm{m}$ used, allowing us to validate the statistical constraining power of the data. When analyzing the data itself, we set $N_\\mathrm{m}=1$. We vary $\\boldsymbol{\\Omega}$ until a global maximum of the likelihood at the best-fit parameter set is found. The quality of the fit can be assessed by evaluation of the reduced chi-square statistic,\n\\begin{equation}\n\\chi^2_\\mathrm{red} = -\\frac{2N_\\mathrm{m}}{N_\\mathrm{dof}}\\ln L(\\hat{\\xi}^s|\\boldsymbol{\\Omega}) \\;,\n\\label{chi2}\n\\end{equation}\nfor $N_\\mathrm{dof}=N_\\mathrm{bin}-N_\\mathrm{par}$ degrees of freedom, where $N_\\mathrm{bin}$ is the number of bins for the data and $N_\\mathrm{par}$ the number of parameters. Moreover, we explore the likelihood surface in the neighborhood of the global maximum using the Monte Carlo Markov Chain (MCMC) sampler \\textsc{emcee}~\\cite{Foreman-Mackey2019}, which enables us to access the posterior probability distribution of the model parameters.\n\n\\subsection{Parameters \\label{subsec:parameters}}\nInstead of using the fundamental cosmological parameters, we express $\\boldsymbol{\\Omega}$ in terms of derived parameters that directly affect the void-galaxy cross-correlation function, namely the linear growth rate to bias ratio $f\/b$, and the AP parameter $\\varepsilon$. To account for potential systematics in the data that can be caused by discreteness noise or selection effects, we further allow for two additional nuisance parameters $\\mathcal{M}$ and $\\mathcal{Q}$. The parameter $\\mathcal{M}$ may adjust for possible inaccuracies arising in the deprojection technique and a contamination of the void sample by Poisson fluctuations, which can attenuate the amplitude of the monopole~\\cite{Cousinou2019}. On the other hand, the parameter $\\mathcal{Q}$ accounts for potential selection effects when voids are identified in anisotropic redshift space~\\cite{Pisani2015b,Nadathur2019b}. A physical origin of this can be violent shell-crossing and virialization events that change the topology of void boundaries~\\cite{Hahn2015}, causing a so-called Finger-of-God (FoG) effect~\\cite{Jackson1972,Hamilton1998,Scoccimarro2004}. It appears around compact overdensities, such as galaxy clusters, generating elongated features along the line of sight that extend over several $h^{-1}{\\rm Mpc}$~\\cite{Peacock2001,Zehavi2011,Pezzotta2017}. A similar effect can be caused by cluster infall regions, leading to closed caustics in redshift space that may be misinterpreted as voids~\\cite{Kaiser1987,Hamilton1998}. Therefore, this can have a non-trivial impact on the identification of voids with diameters of comparable size~\\cite{Stanonik2010,Kreckel2011a,vdWeygaert2011b}, although the tracer sample we use consists of massive luminous red galaxies that typically reside in the centers of clusters and do not sample the entire cluster profile in its outer parts. $\\mathcal{M}$ (monopole-like) is used as a free amplitude of the deprojected correlation function $\\xi(r)$ in real space, and $\\mathcal{Q}$ (quadrupole-like) is a free amplitude for the quadrupole term proportional to $\\mu_r^2$. Hence, equations~(\\ref{xi^s_nonlin2}) and (\\ref{xi^s_lin}) can be extended by the replacements $\\xi(r)\\rightarrow\\mathcal{M}\\xi(r)$ and $\\mu_r^2\\rightarrow\\mathcal{Q}\\mu_r^2$, which results in the following form for the final parametrization of our model at linear perturbation order\n\\begin{equation}\n\\xi^s(\\mathbf{s}) = \\mathcal{M}\\left\\{\\xi(r) + \\frac{1}{3}\\frac{f}{b}\\overline{\\xi}(r) + \\frac{f}{b}\\mathcal{Q}\\mu_r^2\\left[\\xi(r)-\\overline{\\xi}(r)\\right]\\right\\}\\;,\n\\label{xi^s_lin2}\n\\end{equation}\ntogether with the equivalent replacements in equation~(\\ref{r_par(s_par)}) for the mapping from the observed separation $\\mathbf{s}$ to $\\mathbf{r}$: $r_\\parallel = s_\\parallel\/[1-\\frac{1}{3}\\frac{f}{b}\\mathcal{M}\\overline{\\xi}(r)]$. One can think of equation~(\\ref{xi^s_lin2}) as an adaptive template that attempts to extract those anisotropic distortions that match the radial and angular shape as predicted by linear theory. For the nuisance parameters we assign default values of $\\mathcal{M}=\\mathcal{Q}=1$, as they are not known a priori. Note that this extension does not introduce any fundamental parameter degeneracies, due to the different functional forms of $\\xi(r)$ and $\\overline{\\xi}(r)$. For both our model and nuisance parameters we assume uniform prior ranges of $\\left[-10,+10\\right]$, given that each of their expected values is of order unity. We checked that an extension of these boundaries has no impact on our results.\n\n\\subsection{Model validation \\label{subsec:validation}}\nThe theory model derived in section~\\ref{subsec:correlation} requires knowledge of the void density profile $\\delta(r)$ and the void-galaxy cross-correlation function $\\xi(r)$ in real space for equation~(\\ref{xi^s_nonlin}). Its linear version, equation~(\\ref{xi^s_lin}), only requires $\\xi(r)$. $\\delta(r)$ can be accurately modeled with the HSW profile (cf. equation~(\\ref{HSW})), at the price of including additional parameters to be constrained by the data, an approach that has already been successfully applied to BOSS data~\\cite{Hamaus2016}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics{fig\/Xip_test.pdf}\n\t\t\\includegraphics{fig\/Xip_mock.pdf}}\n\t\\caption{LEFT: Projection (dashed red) and deprojection (dotted green) of a model void-galaxy cross-correlation function (solid orange) based on the HSW profile from equation~(\\ref{HSW}), using the Abel transform. RIGHT: Projected void-galaxy cross-correlation function (red wedges, dashed line) of voids from $30$ PATCHY mock catalogs in redshift space, and its real-space counterpart after deprojection (green triangles, dotted line). The redshift-space monopole in the mocks (blue dots) follows the same functional form, in agreement with the linear model (blue solid line) from equation~(\\ref{xi_0}).}\n\t\\label{fig:xi_p_mock}\n\\end{figure}\nIn this paper we instead choose a more data-driven approach and use the real-space profile $\\xi(r)$ obtained by deprojecting the measured projected void-galaxy cross-correlation function $\\xi^s_p(s_\\perp)$ using the inverse Abel transform, equation~(\\ref{xi_d}). We first test this procedure based on the model template from equation~(\\ref{HSW}) and use equation~(\\ref{xi(delta)}) to define $\\xi_{\\scriptscriptstyle\\mathrm{HSW}}(r)\\equiv b\\delta_{\\scriptscriptstyle\\mathrm{HSW}}(r)$. The left panel of figure~\\ref{fig:xi_p_mock} shows $\\xi_{\\scriptscriptstyle\\mathrm{HSW}}$ with the parameter values $(r_s\/R,\\delta_c,\\alpha,\\beta)\\simeq(0.82,-0.36,1.6,9.1)$ and $b=2.2$. These values have been chosen to match the mock data reasonably well (see below). We then use the forward Abel transform to obtain the projected correlation function,\n\\begin{equation}\n\\xi^s_p(s_\\perp) = 2\\int_{s_\\perp}^\\infty\\xi(r)\\frac{r\\mathrm{d}r}{\\sqrt{r^2-s_\\perp^2}}\\;.\n\\label{xi_p2}\n\\end{equation}\nFinally, we apply the inverse Abel transform of equation~(\\ref{xi_d}) to infer the original void-galaxy cross-correlation function $\\xi_{\\scriptscriptstyle\\mathrm{HSW}}$. As evident from the perfectly overlapping lines in the plot, this procedure works extremely well with noiseless data. In reality, however, we have to measure correlation functions with an estimator, which is unavoidably associated with a finite covariance. In order to estimate $\\xi^s_p$ from the data, we adopt equation~(\\ref{estimator}) for pairs on the plane of the sky, which can be achieved by exchanging the three-dimensional position $\\mathbf{x}$ of an object by $x_\\perp=(|\\mathbf{x}|^2-x_\\parallel^2)^{1\/2}$ and counting pairs at a given projected separation $s_\\perp$ over the redshift range. We restrict the line-of-sight projection range to $s_\\parallel=3R$ at the near and far sides from the void center, where $\\xi^s$ has well converged to zero. The right panel of figure~\\ref{fig:xi_p_mock} shows the result when stacking $N_\\mathrm{v}\\simeq2\\times10^5$ voids from $N_\\mathrm{m}=30$ PATCHY mock catalogs. In this case the situation is very similar to the test case from the left panel. The deprojection via the inverse Abel transform results in a smooth curve, only close to the void center we observe mild fluctuations away from the expected shape, due to larger statistical uncertainties~\\cite{Pisani2014}. We verified that reprojecting our result using equation~(\\ref{xi_p2}) agrees well with the original $\\xi^s_p$. We also plot the measured monopole $\\xi^s_0$ from the same PATCHY mocks, including its best-fit model from equation~(\\ref{xi^s_lin2}) using the deprojected $\\xi^s_p$. It provides an excellent agreement with the mock data, and confirms the predicted proportionality between $\\xi^s_0(s)$ and $\\xi(r)$ of equation~(\\ref{xi_0}).\n\nHaving validated the deprojection procedure to obtain $\\xi(r)$ from the data, we are now ready to test our model for the void-galaxy cross-correlation function in redshift space.\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{0.86\\hsize}{!}{\\includegraphics[trim=-40 0 0 10]{fig\/Xi2d_mock.pdf}}\n\t\\resizebox{0.86\\hsize}{!}{\\includegraphics[trim=0 10 0 10]{fig\/Xi_ell_mock.pdf}}\n\t\\caption{TOP: Estimation of the stacked void-galaxy cross-correlation function $\\xi^s(s_\\parallel\/R,s_\\perp\/R)$ from voids in 30 PATCHY mock catalogs (color scale with black contours) and the best-fit model (white contours) from equation~(\\ref{xi^s_lin2}). BOTTOM: Monopole (blue dots), quadrupole (red triangles) and hexadecapole (green wedges) from the same mock data with corresponding model fits (solid, dashed, dotted lines). The mean redshift and effective radius of the void sample is shown at the top.}\n\t\\label{fig:xi_mock}\n\\end{figure}\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{0.7\\hsize}{!}{\\includegraphics{fig\/triangle_mock.pdf}}\n\t\\caption{Posterior probability distribution of the model parameters that enter in equation~(\\ref{xi^s_lin2}), obtained via MCMC from the PATCHY mock data shown in figure~\\ref{fig:xi_mock}. Dark and light shaded areas show $68\\%$ and $95\\%$ confidence regions with a cross marking the best fit, dashed lines indicate fiducial values of the RSD and AP parameters $(f\/b=0.344,\\,\\varepsilon=1)$, and default values for the nuisance parameters $(\\mathcal{M}=\\mathcal{Q}=1)$. The top of each column states the mean and standard deviation of the 1D marginal distributions.}\n\t\\label{fig:triangle_mock}\n\\end{figure}\nFigure~\\ref{fig:xi_mock} presents the corresponding measurement from voids in 30 PATCHY mock catalogs, both its 2D variant with separations along and perpendicular to the line of sight (top panel), as well as the multipoles of order $\\ell=(0,2,4)$ (bottom panel). For the former we use $18$ bins per direction, resulting in $N_\\mathrm{bin}=18^2=324$, whereas for the multipoles we use $25$ radial bins, which yields $N_\\mathrm{bin}=3\\times25=75$ in total. We apply the linear model from equation~(\\ref{xi^s_lin2}) to fit this data (omitting the innermost radial bin) using the $N_\\mathrm{par}=4$ parameters $\\boldsymbol{\\Omega}=(f\/b,\\varepsilon,\\mathcal{M},\\mathcal{Q})$. As apparent from figure~\\ref{fig:xi_mock}, this yields a very good fit to the mock data, with a reduced chi-square value of $\\chi^2_\\mathrm{red}=1.86$. We note that this value corresponds to the statistical power of $30$ mock observations, so it is entirely satisfactory for the purpose of validating our model for a single BOSS catalog. An increase in the number of mock samples marginally affects our summary statistics, which exhibit a negligible amount of statistical noise compared to the real data. The anisotropy of the void-galaxy cross-correlation function is well captured close to the void center, as well as on the void boundaries at $s\\simeq R$. The flattened contours around the void interior are a result of the quadrupole term in equation~(\\ref{xi^s_nonlin}), respectively its linear version~(\\ref{xi^s_lin})~\\cite{Cai2016}. Outside the void boundaries, where the correlation function declines again, this results in elongated contours, as necessary in order to restore spherical symmetry in the limit of large separations. It is worth noting that the coordinate transformation from equation~(\\ref{r_par(s_par)}) causes a line-of-sight elongation from $r_\\parallel$ to $s_\\parallel$ for negative $\\Delta(r)$, acting in the opposite direction. However, this coordinate effect merely accounts for a small correction to the flattening caused by the quadrupole, which we explicitly checked in our analysis. Finally, we emphasize the strong evidence for a vanishing hexadecapole on all scales, in agreement with the theory prediction from section~\\ref{subsec:correlation}.\n\nWe then run a MCMC to sample the posterior distribution of the model parameters. The result is presented in figure~\\ref{fig:triangle_mock} using the \\textsc{getdist} software package~\\cite{Lewis2019}. We recover the fiducial values of the cosmologically relevant parameters $f\/b$ and $\\varepsilon$ to within the $68\\%$ confidence regions, which validates the theory model we use. Moreover, the parameter contours reveal a nearly Gaussian shape of the posterior, only the nuisance parameter $\\mathcal{Q}$ exhibits a slightly non-Gaussian behavior. While $\\mathcal{Q}$ is marginally consistent with unity to within $1\\sigma$, we find clear evidence for the parameter $\\mathcal{M}$ to exceed unity by roughly $14\\%$. This could be attributed to some degree of overdispersion beyond Poisson noise that has been used in the PATCHY algorithm to calibrate the clustering statistics of galaxies in BOSS~\\cite{Kitaura2016a}, resulting in a higher amplitude of random fluctuations inside voids. We also note a strong anti-correlation between $\\mathcal{M}$ and $f\/b$, which can be understood from equation~(\\ref{xi_2}) for the quadrupole, where both parameters enter via multiplication of $\\xi(r)$ and $\\overline{\\xi}(r)$. However, the monopole in equation~(\\ref{xi_0}) breaks their degeneracy.\n\nAs a next step we investigate the dependence on void redshift $Z$. To this end we split our catalog into subsets that contain $50\\%$ of all voids with redshifts below or above their median value. We will refer to these subsets as ``low-z'' and ``high-z'', respectively.  Figure~\\ref{fig:xi_mock_Z} presents the corresponding correlation function statistics, revealing characteristic redshift trends. Note that the effective radii $R$ in our catalog are somewhat correlated with $Z$ due to the variation of tracer density $n(z)$ with redshift, as shown in figure~\\ref{fig:nz}. When $n(z)$ decreases, fewer small voids can be identified, which means that at the high-redshift end our voids tend to be larger in size. On average, smaller voids are emptier and exhibit higher compensation walls at their edges~\\cite{Sheth2004,vdWeygaert2011a,Hamaus2014b,vdWeygaert2016}, which in turn induces a higher amplitude of both monopole and quadrupole~\\cite{Hamaus2017}. A similar trend is manifest in the evolution from high to low redshift, which reveals the growth of structure on the void boundaries over time while the void core continuously deepens~\\cite{Sheth2004,vdWeygaert2011a,Hamaus2014b,vdWeygaert2016}. Both effects are supported by the mock data shown in figure~\\ref{fig:xi_mock_Z}.\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=0 30 0 5, clip]{fig\/Xi2d_mock_Z1.pdf}\n\t\t\\includegraphics[trim=0 30 0 5, clip]{fig\/Xi_ell_mock_Z1.pdf}}\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=0 10 0 5, clip]{fig\/Xi2d_mock_Z2.pdf}\n\t\t\\includegraphics[trim=0 10 0 5, clip]{fig\/Xi_ell_mock_Z2.pdf}}\n\t\\caption{As figure~\\ref{fig:xi_mock} after splitting the PATCHY void sample at its median redshift of $Z=0.51$ into $50\\%$ lowest-redshift (``low-z'', top row) and $50\\%$ highest-redshift voids (``high-z'', bottom row).}\n\t\\label{fig:xi_mock_Z}\n\\end{figure}\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics{fig\/triangle_mock_Z1.pdf}\n\t\t\\includegraphics{fig\/triangle_mock_Z2.pdf}}\n\t\\caption{As figure~\\ref{fig:triangle_mock} for the ``low-z'' (left) and ``high-z'' (right) PATCHY void sample in figure~\\ref{fig:xi_mock_Z}.}\n\t\\label{fig:triangle_mock_Z}\n\\end{figure}\n\nFinally, we repeat the model fits for the subsets in void redshift and run a full MCMC for each of them. The parameter posteriors are shown in figure~\\ref{fig:triangle_mock_Z}. As for the full void sample from before we retrieve the input cosmology of the PATCHY mocks to within $68\\%$ of the confidence levels for $f\/b$ and $\\varepsilon$. Also the posteriors of the nuisance parameters $\\mathcal{M}$ and $\\mathcal{Q}$ look similar as for the full void sample shown in figure~\\ref{fig:triangle_mock}. However, we notice a very mild increase of $\\mathcal{M}$ and a decrease of $\\mathcal{Q}$ towards higher redshifts. Although the shifts remain well within the $1\\sigma$ confidence intervals for these parameters, they may indicate a slightly lower contamination by Poisson noise, but a slightly stronger anisotropic selection effect for the smaller voids at lower redshift. This would indeed agree with previous simulation results that suggest the impact of RSDs on void identification to be more severe for voids with smaller effective radii~\\cite{Pisani2015b}. Moreover, as large-scale structures develop more non-linear over time, we do expect the FoG effect to have a stronger impact on voids at lower redshift.\n\n\\subsection{Data analysis \\label{subsec:fitting}}\nThe successful model validation from the previous section now enables us to perform model fits on the real BOSS data. To this end we simply repeat the analysis steps that have already been performed on the PATCHY mocks above. We first measure the projected void-galaxy cross-correlation function $\\xi^s_p(s_\\perp)$ and apply the deprojection technique using the inverse Abel transform to obtain $\\xi(r)$. The result is shown in figure~\\ref{fig:xi_p_data}, along with the redshift-space monopole $\\xi^s_0(s)$ and its best-fit model. We observe very similar trends as in the mocks, albeit with larger error bars as expected from the smaller sample size of voids available in the BOSS data.\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{0.86\\hsize}{!}{\\includegraphics[trim=0 10 0 10]{fig\/Xip_data.pdf}}\n\t\\caption{Measurement of the projected void-galaxy cross-correlation function (red wedges, dashed line) from the BOSS DR12 combined sample in redshift space, and its real-space counterpart after deprojection (green triangles, dotted line). The measured redshift-space monopole (blue dots) follows the same functional form, in agreement with the linear model (blue solid line) from equation~(\\ref{xi_0}).}\n\t\\label{fig:xi_p_data}\n\\end{figure}\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{0.86\\hsize}{!}{\\includegraphics[trim=-40 0 0 10]{fig\/Xi2d_data.pdf}}\n\t\\resizebox{0.86\\hsize}{!}{\\includegraphics[trim=0 10 0 10]{fig\/Xi_ell_data.pdf}}\n\t\\caption{TOP: Measurement of the stacked void-galaxy cross-correlation function $\\xi^s(s_\\parallel\/R,s_\\perp\/R)$ from voids in the BOSS DR12 combined sample (color scale with black contours) and the best-fit model (white contours) from equation~(\\ref{xi^s_lin2}). BOTTOM: Monopole (blue dots), quadrupole (red triangles) and hexadecapole (green wedges) from the same data with corresponding model fits (solid, dashed, dotted lines). The mean redshift and effective radius of the void sample is shown at the top.}\n\t\\label{fig:xi_data}\n\\end{figure}\nFigure~\\ref{fig:xi_data} presents the two-point statistics for the void-galaxy cross-correlation function and its multipoles. Apart from the larger impact of statistical noise due to the substantially smaller sample size (by a factor of $N_\\mathrm{m}=30$), the results are in excellent agreement with the mock data. Both amplitude and shape of $\\xi^s(s_\\parallel,s_\\perp)$, as well as $\\xi^s_\\ell(s)$ are very consistent in comparison with figure~\\ref{fig:xi_mock}. A mild but noticeable difference can be seen very close to the void center, which appears more flattened in the data. One can also perceive stronger fluctuations of the quadrupole and hexadecapole in this regime, but those are simply due to the sparser statistics of galaxies near the void center and thus fully consistent with the error bars. This fact is further supported by the accurate model fit to the data, resulting in a reduced chi-square value of $\\chi^2_\\mathrm{red}=1.12$.\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{0.7\\hsize}{!}{\n\t\t\\includegraphics{fig\/triangle_data.pdf}}\n\t\\caption{Posterior probability distribution of the model parameters that enter in equation~(\\ref{xi^s_lin2}), obtained via MCMC from the BOSS DR12 data shown in figure~\\ref{fig:xi_data}. Dark and light shaded areas show $68\\%$ and $95\\%$ confidence regions with a cross marking the best fit, dashed lines indicate fiducial values of the RSD and AP parameters $(f\/b=0.409,\\,\\varepsilon=1)$, and default values for the nuisance parameters $(\\mathcal{M}=\\mathcal{Q}=1)$. The top of each column states the mean and standard deviation of the 1D marginal distributions.}\n\t\\label{fig:triangle_data}\n\\end{figure}\n\nThe full posterior parameter distribution obtained from the BOSS data is shown in figure~\\ref{fig:triangle_data}, which qualitatively resembles the mock results from figure~\\ref{fig:triangle_mock}. However, a few important differences are apparent. Firstly, the value of $f\/b$ from the data is significantly higher than in the mocks, which is partly driven by the lower value of $b=1.85$ in the BOSS data, compared to $b=2.20$ of the mocks. Further, the nuisance parameters $\\mathcal{M}$ and $\\mathcal{Q}$ both take on lower values in the data than in the mocks. In particular, $\\mathcal{M}$ is consistent with unity to within $68\\%$ confidence, which could indicate that voids in the BOSS data are less affected by discreteness noise than what was expected from the PATCHY mocks. At the same time, $\\mathcal{Q}$ is consistent with unity only at the $95\\%$ confidence level from below, suggesting an attenuation of the quadrupole amplitude when compared to the mocks. This could be caused by systematics in the BOSS data that have not been taken into account at the same level of complexity in the mocks. One such example is the foreground contamination by stars~\\cite{Reid2016}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=0 30 0 5, clip]{fig\/Xi2d_data_Z1.pdf}\n\t\t\\includegraphics[trim=0 30 0 5, clip]{fig\/Xi_ell_data_Z1.pdf}}\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=0 10 0 5, clip]{fig\/Xi2d_data_Z2.pdf}\n\t\t\\includegraphics[trim=0 10 0 5, clip]{fig\/Xi_ell_data_Z2.pdf}}\n\t\\caption{As figure~\\ref{fig:xi_data} after splitting the BOSS void sample at its median redshift of $Z=0.52$ into $50\\%$ lowest-redshift (``low-z'', top row) and $50\\%$ highest-redshift voids (``high-z'', bottom row).}\n\t\\label{fig:xi_data_Z}\n\\end{figure}\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics{fig\/triangle_data_Z1.pdf}\n\t\t\\includegraphics{fig\/triangle_data_Z2.pdf}}\n\t\\caption{As figure~\\ref{fig:triangle_data} for the ``low-z'' (left) and ``high-z'' (right) BOSS void sample in figure~\\ref{fig:xi_data_Z}.}\n\t\\label{fig:triangle_data_Z}\n\\end{figure}\n\nAs done for the PATCHY mocks, we further investigate the redshift evolution of our constraints with the BOSS data. To this end we split our catalog into two equally sized low- and high-redshift bins again (``low-z'' and ``high-z'' sample), the resulting clustering statistics are shown in figure~\\ref{fig:xi_data_Z}. We observe the same trends as before, namely a deepening of void interiors and an increase of the quadrupole towards lower redshift. Given the lower statistical power of these bins the data evidently look more noisy, but the linear model still provides a good fit overall. Note that we neglect any uncertainties in our theory model, which relies on a measurement of the projected correlation function $\\xi^s_p(s_\\perp)$. Thus, especially for noisy data, this may result in an underestimation of the full covariance and hence a higher reduced chi-square. Nevertheless, our $\\chi^2_\\mathrm{red}$ values are still reasonably close to unity. Figure~\\ref{fig:triangle_data_Z} presents the parameter posteriors of the model fit. Evidently, even these subsets of voids can still provide interesting constraints with a good accuracy. We find our best-fit values for $f\/b$ and $\\varepsilon$ to be in agreement with the fiducial Planck cosmology to within $68\\%$ of the confidence levels. Moreover, we notice that the low amplitude for the nuisance parameter $\\mathcal{Q}$ is driven by the high-redshift bin, otherwise the best-fit values for both $\\mathcal{M}$ and $\\mathcal{Q}$ are consistent with unity to within the $68\\%$ contours.\n\nSo far our analysis has exclusively been based on the observed data without using any prior information. The model ingredients $\\xi(r)$, $\\mathcal{M}$, and $\\mathcal{Q}$ have been derived from this data self-consistently. One may argue, however, that these quantities are already available from the survey mocks to a much higher accuracy (see section~\\ref{subsec:validation}). Hence, making use of the mocks to calibrate those model ingredients allows us to evade marginalization over nuisance parameters and to use the statistical power of the data solely to constrain cosmology. We implement this calibration approach by simply using the $30$ PATCHY mocks to estimate $\\xi(r)$ via equation~(\\ref{xi_d}) and fixing the nuisance parameters $\\mathcal{M}$ and $\\mathcal{Q}$ to their best-fit values from the corresponding void sample in the mock analysis. This leaves us with only two remaining free parameters $f\/b$ and $\\varepsilon$, for which we repeat the MCMC runs.\n\nThe results are presented in figure~\\ref{fig:triangle_data_cal}, showing their posterior distribution for each of our void samples. Evidently, the mock-calibrated analysis (calib.) significantly improves upon the constraints obtained without calibration (free). While the accuracy on the AP parameter $\\varepsilon$ exhibits mild improvements of about $10\\%$ to $30\\%$, the error on $f\/b$ shrinks by roughly a factor of~$4$. This is mainly due to the considerable anti-correlation between $f\/b$ and $\\mathcal{M}$ apparent in figures~\\ref{fig:triangle_data} and~\\ref{fig:triangle_data_Z}, which is removed when $\\mathcal{M}$ is fixed to a fiducial value. We note, however, that the best-fit values for $\\mathcal{M}$ in our uncalibrated analysis differ significantly between the observed data and the mocks. In particular, we found the values of $\\mathcal{M}$ in the mocks to be higher than in the data by about $10\\%$. This may be partly due to the higher bias parameter and the level of overdispersion in the PATCHY mocks as compared to the data, but more fundamentally the mismatch reveals that not all aspects of the data are understood precisely enough to be fully represented by the mocks.\n\\begin{figure}[b]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics{fig\/triangle_data_cal_Z1.pdf}\n\t\t\\includegraphics{fig\/triangle_data_cal_Z2.pdf}\n\t\t\\includegraphics{fig\/triangle_data_cal.pdf}}\n\t\\caption{As figures~\\ref{fig:triangle_data} and~\\ref{fig:triangle_data_Z}, but focused on the RSD and AP parameters $f\/b$ and $\\varepsilon$. The red contours show constraints when the theory model is calibrated with the PATCHY mocks to determine $\\xi(r)$ and the values of the nuisance parameters $\\mathcal{M}$ and $\\mathcal{Q}$. The blue contours show the original constraints when $\\xi(r)$, $\\mathcal{M}$ and $\\mathcal{Q}$ are left free to be jointly estimated from the BOSS data. The top of each column states the mean and standard deviation of the calibrated constraints, the corresponding void sample is indicated above the figure legend of each panel.}\n\t\\label{fig:triangle_data_cal}\n\\end{figure}\n\nTherefore, we caution the use of mocks for model calibration, as such an approach is prone to cause biased constraints on cosmology. This is evident from the significant shifts of the posteriors in figure~\\ref{fig:triangle_data_cal} after performing the calibration. Another consequence is the underestimation of parameter uncertainties, which is caused by mistaking prior information from the mocks as the truth. The mocks merely represent many realizations of a single cosmological model with one fiducial parameter set and one fixed prescription of how dark matter halos are populated by galaxies (halo occupation distribution). A realistic model must therefore either take into account the dependence on these ingredients including their uncertainty, or constrain them from the data directly. Our approach follows the philosophy to exclusively rely on the observed data to obtain most robust constraints.\n\n\n\\section{Discussion\\label{sec:discussion}}\n\n\\subsection{Parameter constraints \\label{subsec:constraints}}\n\\begin{table}[b]\n\t\\centering\n\t\\caption{Constraints on RSD and AP parameters (mean values with $1\\sigma$ errors) from \\textsc{vide} voids in the final BOSS data (top rows). The middle rows show corresponding constraints after model-calibration on the PATCHY mocks, and the bottom rows provide Planck 2018~\\cite{Planck2018} results as reference values, assuming a flat $\\Lambda$CDM model. All constraints on $\\Omega_\\mathrm{m}$ in the last column assume flat $\\Lambda$CDM as well.}\\vspace{10pt}\n\t\\label{tab:constraints}\n\t\\centerline{\n\t\t\\begin{tabular}{lccccc}\n\t\t\t\\toprule\n\t\t\tSample ($\\bar{Z}$) & $f\/b$           & $f\\sigma_8$     & $\\varepsilon$     & $D_\\mathrm{A} H\/c$       & $\\Omega_\\mathrm{m}$\\\\\n\t\t\t\\midrule\n\t\t\tlow-z ($0.43$)     & $0.493\\pm0.105$ & $0.590\\pm0.125$ & $0.9996\\pm0.0081$ & $0.485\\pm0.004$ & $0.306\\pm0.027$\\\\[3pt]\n\t\t\thigh-z ($0.58$)    & $0.538\\pm0.146$ & $0.594\\pm0.162$ & $1.0100\\pm0.0111$ & $0.702\\pm0.008$ & $0.334\\pm0.030$\\\\[3pt]\n\t\t\tall ($0.51$)       & $0.540\\pm0.091$ & $0.621\\pm0.104$ & $1.0017\\pm0.0068$ & $0.588\\pm0.004$ & $0.312\\pm0.020$\\\\[3pt]\n\t\t\t\\midrule\n\t\t\tlow-z calib.       & $0.390\\pm0.025$ & $0.554\\pm0.036$ & $1.0134\\pm0.0075$ & $0.492\\pm0.004$ & $0.353\\pm0.026$\\\\[3pt]\n\t\t\thigh-z calib.      & $0.288\\pm0.033$ & $0.379\\pm0.043$ & $0.9953\\pm0.0084$ & $0.691\\pm0.006$ & $0.295\\pm0.022$\\\\[3pt]\n\t\t\tall calib.         & $0.347\\pm0.023$ & $0.474\\pm0.031$ & $1.0011\\pm0.0060$ & $0.588\\pm0.003$ & $0.310\\pm0.017$\\\\[3pt]\n\t\t\t\\midrule\n\t\t\tlow-z ref.         & $0.398\\pm0.003$ & $0.476\\pm0.006$ & $1.0025\\pm0.0022$ & $0.487\\pm0.001$ & $0.315\\pm0.007$\\\\[3pt]\n\t\t\thigh-z ref.        & $0.425\\pm0.003$ & $0.470\\pm0.005$ & $1.0031\\pm0.0028$ & $0.697\\pm0.002$ & $0.315\\pm0.007$\\\\[3pt]\n\t\t\tall ref.           & $0.412\\pm0.003$ & $0.474\\pm0.006$ & $1.0028\\pm0.0025$ & $0.589\\pm0.001$ & $0.315\\pm0.007$\\\\[3pt]\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\\end{table}\n\nThe final parameter constraints from our \\textsc{vide} void samples found in the BOSS DR12 data are summarized in table~\\ref{tab:constraints}. We distinguish between uncalibrated and mock-calibrated (calib.) samples, both for the subsets at low and high redshift (low-z, high-z), as well as the full redshift range (all). The table presents measured quantities of mean void redshift $\\bar{Z}$, RSD parameter $f\/b$ and AP parameter $\\varepsilon$. Furthermore, it provides derived constraints on $f\\sigma_8$, $D_\\mathrm{A} H$ and $\\Omega_\\mathrm{m}$. For $f\\sigma_8$ we multiply our constraint on $f\/b$ by $b\\sigma_8$, with $b=1.85$ and $\\sigma_8=0.8111$ from Planck 2018~\\cite{Planck2018}. For the calibrated case we use $b=2.20$ from the mocks.\n\nIn principle the parameter combination $f\\sigma_8$ could be constrained from voids directly, but only if the theory model can explicitly account for the dependence of the void-galaxy cross-correlation function $\\xi(r)$ on $\\sigma_8$. Sometimes the assumption $\\xi(r)\\propto\\sigma_8$ is used without further justification, but evidently this must fail in the non-linear regime~\\cite{Juszkiewicz2010} where $\\xi(r)$ approaches values close to $-1$, due to the restriction $\\xi(r)>-1$. The same argument applies to the density profiles of dark matter halos or galaxy clusters, which do not simply scale linearly with $\\sigma_8$~\\cite{Brown2020}. Moreover, while the value of $\\sigma_8$ controls the amplitude of matter fluctuations and thus the formation of halos, its effect on voids identified in the distribution of galaxies that populate those halos is far from trivial. Another approach is to directly estimate $b\\sigma_8$ via an integral over the projected galaxy auto-correlation function~\\cite{Hawken2017}. However, the result contains non-linear contributions from small scales that must be accounted for, which again involves assumptions about a particular cosmological model. The covariance between measurements of $f\/b$ and $b\\sigma_8$ remains inaccessible to that approach as well.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{0.51\\hsize}{!}{\n\t\t\\includegraphics[trim=10 10 10 10]{fig\/triangle_data_fs8_Om.pdf}}\n\t\\caption{Calibration-independent constraints on the cosmological parameters $f\\sigma_8$ and $\\Omega_\\mathrm{m}$ from our full \\textsc{VIDE} void sample in the final BOSS data. They are converted from the RSD and AP parameter posteriors shown in figure~\\ref{fig:triangle_data}, as described in section~\\ref{subsec:constraints}. A white cross indicates our best fit and dashed lines show mean parameter values from the Planck 2018 baseline results as a reference~\\cite{Planck2018}.}\n\t\\label{fig:triangle_data_fs8_Om}\n\\end{figure}\n\nFinally, measurements involving the parameter $\\sigma_8$ commonly ignore its implicit dependence on the Hubble parameter $h$ via the choice of $8h^{-1}{\\rm Mpc}$ as reference scale, and therefore underestimate the uncertainty. Reference~\\cite{Sanchez2020} argued to instead use $\\sigma_{12}$ with $12$Mpc, which yields about the same value as $\\sigma_8$ for a Planck-constrained value of $h$. For these reasons we decided to follow the simpler procedure described above to derive constraints on $f\\sigma_8$, allowing us to compare existing results across the literature. The constraint on $D_\\mathrm{A} H$ can be obtained via equation~(\\ref{epsilon}) by multiplying $\\varepsilon$ and its error with $D_\\mathrm{A}'H'$ from our fiducial flat $\\Lambda$CDM cosmology from section~\\ref{subsec:galaxies}. In this case the only free cosmological parameter in the product $D_\\mathrm{A} H$ is $\\Omega_\\mathrm{m}$, so we can numerically invert this function to obtain the full posterior on $\\Omega_\\mathrm{m}$. Its mean and standard deviation are shown in the last column of table~\\ref{tab:constraints}. Finally, we present our main result for the converted parameter constraints on $f\\sigma_8$ and $\\Omega_\\mathrm{m}$ in figure~\\ref{fig:triangle_data_fs8_Om}. It originates from the calibration-independent analysis of our full void sample in the final BOSS data at mean redshift $\\bar{Z}=0.51$.\n\n\n\\subsection{Systematics tests\\label{subsec:systematics}}\n\n\\subsubsection{Fiducial cosmology}\nIn order to affirm the robustness of our results, we have performed a number of systematics tests on our analysis pipeline. One potential systematic can be a residual dependence on the fiducial cosmology we assumed in section~\\ref{subsec:galaxies} when converting angular sky coordinates and redshifts into comoving space via equation~(\\ref{x_comoving}). This conversion preserves the topology of large-scale structure, but in the presence of statistical noise due to sparse sampling of tracers it can have an impact on void identification~\\cite{Mao2017}. We investigate how a change of the fiducial cosmology affects our final constraints on cosmological parameters by shifting the fiducial value for $\\Omega_\\mathrm{m}'=0.307$ to $0.247$. This shift amounts to three times the standard deviation we obtain from the posterior on $\\Omega_\\mathrm{m}=0.312\\pm0.020$ in the uncalibrated analysis of all voids (see table~\\ref{tab:constraints}). We then repeat our entire analysis including the void-finding procedure and sample the posterior on $f\/b$ and $\\Omega_\\mathrm{m}$ assuming the new value for $\\Omega_\\mathrm{m}'$. The result is presented in the left panel of figure~\\ref{fig:triangle_sys}, showing a very mild impact of the fiducial cosmology on the posterior parameter distribution. The resulting shifts of the posterior mean values are well within the $68\\%$ credible regions of both cases and their relative accuracies practically remain unchanged, suggesting the impact of our fiducial cosmology to contribute a marginal systematic effect to our final constraints.\n\\begin{figure}[b]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics{fig\/triangle_data_Om.pdf}\n\t\t\\includegraphics{fig\/triangle_mock_bias.pdf}}\n\t\\caption{LEFT: Impact of the fiducial parameter $\\Omega_\\mathrm{m}'$ on the final posterior distribution for $f\/b$ and $\\Omega_\\mathrm{m}$ from all voids in the BOSS DR12 data. The top of each column states the mean and standard deviation obtained for the assumed value of $\\Omega_\\mathrm{m}'=0.247$ and dashed lines indicate fiducial values of the default cosmology with $\\Omega_\\mathrm{m}'=0.307$. RIGHT: Impact of the bias of the galaxy sample used in the cross-correlation with all voids from the PATCHY mocks on the derived posterior for $f\\sigma_8$ and $\\varepsilon$. The top of each column states the mean and standard deviation obtained for the new value $b=1.93$.}\n\t\\label{fig:triangle_sys}\n\\end{figure}\n\n\\subsubsection{Galaxy bias}\nThe bias of the galaxy sample we use to estimate the cross-correlation with voids can contribute another systematic effect on our final parameters. This especially so for the derived combination $f\\sigma_8$, which we obtain via multiplying $f\/b$ by the average bias $b$ of the galaxy sample, and the Planck-constrained $\\sigma_8$ value (see section~\\ref{subsec:constraints}). The BOSS data does not readily allow us to define sub-samples of galaxies with known bias values that differ from the sample average. However, the PATCHY mocks provide a bias parameter for every object in the catalog, so we can investigate its influence on our analysis pipeline. As a simple test, we selected $50\\%$ of all PATCHY galaxies with a bias value below the median, which amounts to an average of $b=1.93$. Because the galaxy bias follows its own redshift evolution, we had to re-sample the random catalog in order for it to follow the same density-redshift trend as the selected galaxy sample. We then cross-correlate it with our original PATCHY void sample used in section~\\ref{subsec:validation} and compare its posterior on $f\\sigma_8$ and $\\varepsilon$ to the original one from figure~\\ref{fig:triangle_mock} in the right panel of figure~\\ref{fig:triangle_sys}. The two constraints are very consistent with each other, suggesting that the final result on $f\\sigma_8$ does not depend on the bias of the galaxy sample used for the cross-correlation.\n\n\\subsubsection{Estimator}\nThe main advantage of our clustering estimator from equation~(\\ref{estimator}) is its simplicity, allowing a fast and precise evaluation of the void-galaxy cross-correlation function and its multipoles without angular binning. In order to assess its accuracy, we have compared it with the more common Landy-Szalay estimator~\\cite{Landy1993}\n\\begin{equation}\n\\xi^s(\\mathbf{s}) = \\left(\\frac{\\langle\\mathbf{X},\\mathbf{x}\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}\\rangle\\langle\\mathbf{x}\\rangle} -\\frac{\\langle\\mathbf{X},\\mathbf{x}_r\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}\\rangle\\langle\\mathbf{x}_r\\rangle} -\\frac{\\langle\\mathbf{X}_r,\\mathbf{x}\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}_r\\rangle\\langle\\mathbf{x}\\rangle} +\\frac{\\langle\\mathbf{X}_r,\\mathbf{x}_r\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}_r\\rangle\\langle\\mathbf{x}_r\\rangle}\\right)\\left\/\n\\left(\\frac{\\langle\\mathbf{X}_r,\\mathbf{x}_r\\rangle(\\mathbf{s})}{\\langle\\mathbf{X}_r\\rangle\\langle\\mathbf{x}_r\\rangle}\\right.\\right)\\;,\n\\label{LS_estimator}\n\\end{equation}\nwhich additionally involves the random void-center positions $\\mathbf{X}_r$. From the PATCHY mocks we generate a sample of such void randoms by assigning the same angular and redshift distribution of its voids to a randomly generated set of points with $50$ times the number of objects (in analogy to the galaxy randoms, see section~\\ref{subsec:galaxies}). We also assign an effective radius to each random void, with the same distribution as the one obtained in the mocks. This guarantees a consistent stacking procedure, as described in section~\\ref{subsec:estimators}. We find that the additional terms $\\langle\\mathbf{X}_r,\\mathbf{x}\\rangle(\\mathbf{s})\/\\langle\\mathbf{X}_r\\rangle\\langle\\mathbf{x}\\rangle$ and $\\langle\\mathbf{X}_r,\\mathbf{x}_r\\rangle(\\mathbf{s})\/\\langle\\mathbf{X}_r\\rangle\\langle\\mathbf{x}_r\\rangle$ in the stacked void-galaxy correlation estimator from equation~(\\ref{LS_estimator}) are independent of the direction and magnitude of~$\\mathbf{s}$, in agreement with the findings of reference~\\cite{Hamaus2017}. However, we notice the amplitude of both terms to exceed unity by roughly $20\\%$, while their ratio remains very close to one with deviations in the order of $10^{-3}$. This results in a different overall normalization between equations~(\\ref{estimator}) and~(\\ref{LS_estimator}), making their amplitudes differ by a constant factor of about $1.2$. When we rescale one of the void-galaxy correlation functions by this number, we find the results from these two estimators to be virtually indistinguishable. Because we use the same estimator for $\\xi^s(s_\\parallel,s_\\perp)$ and the projected correlation function $\\xi^s_p(s_\\perp)$, which is used to infer the real-space $\\xi(r)$ via equation~(\\ref{xi_d}), any such normalization constant gets absorbed on both sides of equation~(\\ref{xi^s_lin}) and therefore has no effect on our model parameters.\n\n\n\\subsubsection{Covariance matrix}\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=5 10 5 5, clip]{fig\/cov_data.pdf}\n\t\t\\includegraphics[trim=5 10 5 5, clip]{fig\/cov_mock.pdf}}\n\t\\caption{Covariance matrix (normalized by its diagonal components) of the stacked void-galaxy cross-correlation function $\\xi^s(s_\\parallel,s_\\perp)$ from the BOSS DR12 data (left) and the PATCHY mocks (right).}\n\t\\label{fig:covariance}\n\\end{figure}\nAs a last consistency test we investigate the impact of the covariance matrix on our results. The left panel of figure~\\ref{fig:covariance} shows the covariance matrix estimated using the jackknife technique on the BOSS data as described in section~\\ref{subsec:estimators}, normalized by its diagonal components (i.e., the correlation matrix $\\mathbf{C}_{ij}\/\\sqrt{\\mathbf{C}_{ii}\\mathbf{C}_{jj}}\\,$). Note that this matrix contains $N_\\mathrm{bin}^2=(18^2)^2$ elements for the covariance of the two-dimensional correlation function $\\xi^s(s_\\parallel,s_\\perp)$. In order to overcome the statistical noise in the data covariance, we can measure the same quantity for all voids in our $N_\\mathrm{m}=30$ independent mock catalogs. The result is shown in the right panel figure~\\ref{fig:covariance}, featuring a very similar structure as for the real data. In our main analysis we have used the data covariance for the sake of maintaining an entirely calibration-independent approach. However, when exchanging it by the mock covariance in our likelihood from equation~(\\ref{likelihood}), we obtain posteriors that are consistent with our previous results, which is why we do not show them again. This suggests that the estimation of the covariance matrix from the data itself provides a sufficiently precise method allowing us to obtain fully calibration-independent constraints on cosmology from the observed sample of voids.\n\n\\subsection{Comparison to previous work \\label{subsec:comparison}}\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[trim=30 50 60 80, clip]{fig\/fs8_DAH.pdf}}\n\t\\caption{Comparison of constraints on $f\\sigma_8$ and $D_\\mathrm{A} H$ (mean values with $68\\%$ confidence regions) obtained from cosmic voids in the literature, references are ordered chronologically in the figure legend. To improve readability, $D_\\mathrm{A} H$ is normalized by its reference value $D_\\mathrm{A}'H'$ in the Planck 2018 flat $\\Lambda$CDM cosmology~\\cite{Planck2018} (gray line with shaded error band). Filled markers indicate growth rate measurements without consideration of the AP effect, while open markers include the AP test. The line style of error bars indicates various degrees of model assumptions made: model-independent (solid), calibrated on simulations (dashed), calibrated on mocks (dotted), calibrated on simulations and mocks (dash-dotted). All the employed simulations and mocks assume a flat $\\Lambda$CDM cosmology. Data points at similar redshifts have been slightly shifted horizontally to avoid overlap.}\n\t\\label{fig:comparison}\n\\end{figure}\nThe observational AP test with cosmic voids has already experienced some history since it was first proposed by Lavaux and Wandelt in 2009~\\cite{Lavaux2010} and its first measurement by Sutter et al. in 2012~\\cite{Sutter2012b}. The early measurements of the AP effect did not yet account for RSD distortions by a physically motivated model, but they calibrated its impact using simulations~\\cite{Sutter2014b,Mao2017}. The first joint RSD and AP analysis from observed cosmic voids has been published in 2016 based on BOSS DR11 data~\\cite{Hamaus2016}. It demonstrated for the first time that a percent-level accuracy on $\\varepsilon$ can be achieved with voids, making them the prime candidate for observational AP tests. Since then, a number of papers appeared that either focused on the RSD analysis of voids exclusively in order to constrain the growth rate~\\cite{Achitouv2017a,Hawken2017,Hamaus2017,Achitouv2019,Hawken2020}, or performed further joint analyses including the AP effect~\\cite{Nadathur2019b}.\n\nWe summarize the constraints on $f\\sigma_8$ and $D_\\mathrm{A} H$ that have been obtained from voids throughout the literature in figure~\\ref{fig:comparison}, including the results from this paper. Evidently, the different analysis techniques have progressed over time and achieved significant improvements of accuracy. Moreover, spectroscopic data from a number of surveys covering different redshift ranges has been exploited to this end, including 6dFGS~\\cite{6dFGS}, BOSS~\\cite{BOSS}, eBOSS~\\cite{EBOSS}, SDSS~\\cite{SDSS}, and VIPERS~\\cite{VIPERS}. All of the published results are consistent with a flat $\\Lambda$CDM cosmology, in agreement with the measurements by Planck~\\cite{Planck2018}. However, some of the analyses have been calibrated using simulations and \/ or mocks to determine unknown model ingredients. If such external information has been used and has not been marginalized over, we indicate the calibrated results in figure~\\ref{fig:comparison} by different line styles of error bars, as described in the caption. A comparison based on equal terms can only be made by taking this essential fact into account. In addition to this, there are a number of analysis choices that differ among the published results. In the following we provide a list of aspects that we have investigated in more detail and encountered to be relevant for our results.\n\n\\subsubsection{Void finding in real vs. redshift space}\nLike most papers on the topic of void RSD in the literature, we define voids in observable redshift space. The recent analysis of reference~\\cite{Nadathur2019b} advocates the use of reconstruction techniques to identify voids in real space instead. Their centers in real space are then correlated with the original galaxy positions in redshift space to estimate a hybrid real\/redshift space void-galaxy cross-correlation function. We have investigated this approach using halo catalogs from $N$-body simulations and calculated the resulting two-point statistics. We confirm that this results in a more elongated shape of $\\xi^s(s_\\parallel,s_\\perp)$ along the line of sight and a change of sign in its quadrupole at small void-centric separations. This can be readily understood from the illustration in figure~\\ref{fig:voidstretch}: the separation vector between a void center in real space and one of the void's galaxies in redshift space is now given by the gray dashed line, which is more elongated along the line of sight because it contains a contribution from the velocity of the void center, $\\tilde{\\mathbf{s}}=\\mathbf{r}+\\mathbf{u}_\\parallel+\\mathbf{V}_\\parallel$ (with velocities in units of $(1+z_h)\/H$). Another consequence of this approach is that for void velocities with $|\\mathbf{V}_\\parallel|\\gtrsim R$, a significant number of galaxies from neighboring voids in redshift space will be closer to the void center in real space than the void's own member galaxies. As a result, the void-galaxy cross-correlation function contains different contributions from galaxies of the same and neighboring voids, depending on the magnitude of $\\mathbf{V}_\\parallel$. As this effect is difficult to model from first principles, reference~\\cite{Nadathur2019b} resorts to the use of mock catalogs to calibrate the form of this hybrid correlation function, and restricts its analysis to the largest $50\\%$ of all voids.\n\nThe motivation for velocity field reconstruction was grounded on the claim that the velocities $\\mathbf{V}$ of void centers cannot be accounted for in all existing models for the void-galaxy cross-correlation function in redshift space~\\cite{Nadathur2019b}. This presumption is unfounded, as these models have actually been derived assuming local mass conservation relative to the motion of the void center~\\cite{Paz2013,Hamaus2015,Cai2016}. While it is true that absolute void velocities $\\mathbf{V}$ are difficult to predict, the same holds for the absolute galaxy velocities $\\mathbf{v}$ in the vicinity of void centers. Both $\\mathbf{V}$ and $\\mathbf{v}$ contain bulk-flow contributions sourced by density fluctuations on scales beyond the void's extent. However, local mass conservation provides a very good prediction for their difference $\\mathbf{u}$, as discussed in section~\\ref{subsec:dynamic}. A consequence of this is a vanishing hexadecapole $\\xi^s_4(s)$, as explained in reference~\\cite{Cai2016} and confirmed by our analysis. We further note that the galaxy velocity field $\\mathbf{v}$ is anisotropic around void centers in redshift space, as expected from figure~\\ref{fig:voidstretch}. The model derived in section~\\ref{subsec:correlation} merely assumes statistical isotropy of the field $\\mathbf{u}$ in real space, which follows from the cosmological principle. Reference~\\cite{Nadathur2019a} speculated about a potential selection effect in favor of voids with higher outflow velocities and hence lower observed central densities in redshift space. As explained in section~\\ref{subsec:voids}, our void finder operates on local minima and their surrounding watershed basins, irrespective of any absolute density threshold. Such a selection effect therefore cannot affect voids identified with~\\textsc{vide} or~\\textsc{zobov}.\n\nAnother argument for the use of reconstruction was motivated by the impact of redshift-space distortions on the void-size function~\\cite{Nadathur2019a}. We note that the effective radii for voids of any size are expected to change between real and redshift space due to dynamic distortions, as evident from figure~\\ref{fig:voidstretch}. However, we only use the observed effective void radii as units to express all separations in either space, which leaves the mapping between $\\mathbf{r}$ and $\\mathbf{s}$ unchanged. A problematic impact of this mapping can be the destruction of voids from catastrophic redshift-space distortions, such as the FoG effect, or from shot noise due to the sparsity of tracers that may change the topology of watershed basins. Because the FoG effect is limited to scales of a few $h^{-1}{\\rm Mpc}$ and smaller voids are defined by fewer tracers, this problem becomes more relevant for voids of relatively small size. We account for this potential systematic via marginalization over the nuisance parameters introduced in section~\\ref{subsec:parameters}.\n\nIn conclusion, velocity field reconstruction is not required to account for the dynamic distortions of voids, as evident from this paper and numerous earlier works~\\cite{Paz2013,Hamaus2016,Cai2016,Achitouv2017a,Hawken2017,Hamaus2017,Correa2019,Achitouv2019,Hawken2020}. The velocity field reconstruction technique merely offers an alternative approach to model RSDs around voids, in addition to the existing models. If reconstruction is used in conjunction with another RSD model, dynamic distortions are unnecessarily taken into account twice. The disadvantages of reconstruction include its dependence on a smoothing scale, assumptions on tracer bias and growth rate relations, as well as its sensitivity to survey edges and shot noise~(e.g., \\cite{Sherwin2019,Philcox2020}). Last but not least, reconstruction makes the data a function of the theory model. Vice-versa, calibration of the theory model on survey mocks that are informed by the data generates an inverse dependence. If information from the mocks is used in the model, theory and data are intertwined to a degree that makes a rigorous likelihood analysis much more involved. Moreover, this practice forfeits the criteria necessary for an independent model validation. The authors of reference~\\cite{Nadathur2019b} claim their analysis to be ``free of systematic errors'', but neglect its systematic dependence on the assumed mock cosmology.\n\n\\subsubsection{Void center definition}\nBecause voids are aspherical by nature, the definition of their centers is not unique. In observations, which typically provide the 3D locations (but not the 3D velocities) of tracers that outline each void, there are in practice two options: the point of minimum tracer density inside the void, or the geometric center defined by the void boundary. Minimum-density centers can be defined as maximally extended empty spheres in a tracer distribution~\\cite{Zhao2016}, without requiring the sophistication of a watershed algorithm. The optimal choice of center definition depends on the specific type of application, so it is not possible to make general statements about this. However, for the sake of measuring geometric distortions via the AP effect it is desirable to enhance the amplitude of tracer fluctuations around their background density to increase the signal-to-noise ratio of anisotropic clustering measurements. As described in section~\\ref{subsec:voids}, the geometric center (barycenter) retains information about the void boundary and thereby generates a pronounced compensation wall in its cross-correlation with galaxies at a separation of one effective void radius $R$. On the other hand, the minimum-density center produces a stronger negative amplitude of the void-galaxy cross-correlation function at small separations. The number of tracers in a shell of width $\\mathrm{d}s$ grows as $s^2$ for a constant tracer density, and even faster for increasing density with $s$, as is the case inside voids. Therefore the coherent compensation walls around void barycenters serve as a lever arm to provide significantly higher signal-to-noise ratios for measurements of anisotropic clustering and hence the AP effect. We have checked this explicitly by repeating our analysis using minimum-density centers, which results in less pronounced compensation walls in $\\xi^s(s_\\parallel,s_\\perp)$, a lower amplitude of its quadrupole, and an uncertainty on the AP parameter $\\varepsilon$ of roughly double the size.\n\n\\subsubsection{Void stacking}\nThe method of void stacking is related to the previous aspect, as it affects the void-galaxy cross-correlation function in a similar way. Because voids are objects of finite extent, the correlation of their centers with tracers inside or outside their boundaries is qualitatively different~\\cite{Hamaus2014a,Chan2014,Cai2016,Voivodic2020}. This is analogous to the halo model, which ascribes two different contributions to the clustering properties of matter particles, those inside the same halo, and those among different halos~\\cite{Seljak2000,Peacock2000}. Therefore, in order to capture the characteristic clustering properties of tracers inside a sample of differently sized voids, one typically rescales the tracer coordinates by the effective radius of their respective host void, a method referred to as void stacking. This guarantees that the void boundaries coherently overlap at a separation of $s=R$, and thus creates a strong compensation wall feature in the stacked void-galaxy cross-correlation function. Without the rescaling procedure, compensation walls of different-size voids do not aggregate, which results in a smeared out correlation function with almost no feature remaining at $s=R$. This smearing in turn is disadvantageous for measurements of AP distortions, following the arguments discussed above. A similar effect can be caused by stacking voids of different evolutionary stages from a wide range of redshifts. For example, the properties of void galaxies are expected to be redshift-dependent~\\cite{Kreckel2011a,Kreckel2011b}. This can be accounted for by splitting the void sample into redshift bins. However, in the BOSS DR12 data and the PATCHY mocks we find a very mild evolution with redshift, allowing us to average over the full void sample.\n\n\\subsubsection{Correlation function estimation}\nIt is common practice to estimate correlation functions via counts in shells, i.e. by counting the number of tracers and randoms inside a spherical shell of width $\\mathrm{d}s$ at separation $s$. In the interiors of voids, however, the density of tracers is low by construction, which can result in shells with insufficiently low tracer counts to reliably estimate correlation functions that are intended to infer properties of the density field (such as its growth rate $f$). The previously discussed method on void stacking helps in this respect, as one may collect the tracers that fall into a given shell from all rescaled voids of the entire sample. The convergence of the estimator can then be assessed by increasing the void sample size, as we have done using mocks in section~\\ref{subsec:validation}. Within our approach we find no dependence of the estimators on sample size, in support of the conclusion that our correlation function statistics have converged. However, if shells with very few or no tracers are encountered in every single void, the counts-in-shell estimator yields biased results, even in the limit of infinite sample size~\\cite{Nadathur2015}.\n\nThis is particularly relevant for shells in the vicinity of the minimum-density center, which exhibits no nearby tracers by construction. As a consequence, the counts-in-shell estimator yields a value of $\\xi=-1$ for all empty shells, regardless of the nature of the tracer distribution. In fact, this is even the case for empty shells in a random distribution of tracers, an example that reveals the limitations of this estimator most clearly. As its name suggests, the counts-in-shell estimator is only defined for non-zero object counts, but returns meaningless results otherwise. Towards larger scales, as soon as the first tracers are encountered in a shell at separation $s_\\mathrm{d}$, the correlation estimate abruptly jumps to a value significantly higher than $\\xi=-1$, and finally converges to a smooth curve at separations with sufficient tracer counts~\\cite{Nadathur2019b}. If voids are not rescaled by their effective radius, this results in a kink in the average correlation function at $s=s_\\mathrm{d}$ even for arbitrarily large sample sizes. A similar behavior can be observed when estimating the radial velocity profile of the tracers~\\cite{Nadathur2019a}. The resulting bias in the counts-in-shell estimator on small scales breaks the validity of equations~(\\ref{u(r)}) and~(\\ref{xi(delta)}), which only apply in the continuous limit of high tracer counts, and can be misinterpreted as evidence for an intrinsic non-linearity or stochasticity of the tracer density field.\n\nThe shell at separation $s_\\mathrm{d}$ indicates the discreteness limit of the tracer distribution. It is determined by the average density of tracers and therefore unrelated to cosmologically induced clustering statistics that can be measured on larger scales. Moreover, discreteness artifacts are notoriously difficult to model from first principles due to their unphysical nature, which leaves no option other than to calibrate them via mock catalogs. In reference~\\cite{Nadathur2019a} it is argued that the void-galaxy cross-correlation function exhibits a ``feature'' both in its monopole and quadrupole at separation $s_\\mathrm{d}$, which is calibrated on mocks to be used for AP distortion measurements in a later publication~\\cite{Nadathur2019b}. It yet remains to be demonstrated whether such features at the discreteness limit of an estimator are of any use for the AP test. They similarly arise in scale-free Poisson distributions, which are insensitive to geometric distortions for the lack of spatial correlations and therefore satisfy the condition $\\varepsilon=1$ in any coordinate system. Consequently, in such a scenario the AP test necessarily returns the fiducial cosmology and therefore becomes dominated by confirmation bias. Changing the fiducial cosmology provides a useful sanity check to exclude the presence of confirmation bias, as we have shown in section~\\ref{subsec:systematics}.\n\n\\subsubsection{RSD model}\nWe have performed extensive tests to compare the existing RSD models for voids that are available in the literature. This essentially concerns the GSM for voids as proposed by Paz et al.~\\cite{Paz2013}, the linear model of Cai et al.~\\cite{Cai2016} used in this paper, and variants thereof. We find consistent results with the GSM, albeit with slightly weaker constraints on $f\/b$ and $\\varepsilon$ due to marginalization over the additional velocity dispersion parameter~$\\sigma_v$. Moreover, as the GSM requires an integration over the pairwise velocity probability distribution function in every bin of $\\xi^s(s_\\parallel,s_\\perp)$, it significantly slows down the model evaluation. We find the impact of velocity dispersion to marginally affect our fits to the data, so we settled on the simpler linear model from equation~(\\ref{xi^s_lin2}).\n\nFurthermore, we explored extensions of the linear model, such as the full non-linear expression~(\\ref{xi^s_nonlin2}). We also tested the model extension proposed in equation (14) of reference~\\cite{Nadathur2019a}, which contains terms of linear and second order in $\\delta$. Note that in this model, every occurrence of the parameter $f$ is multiplied by a factor of $1+\\xi(r)$, unfolding an additional degeneracy between the growth rate and the amplitude of $\\xi(r)$, which depends on $b$ and $\\sigma_8$. Moreover, that model requires the void mass density profile $\\delta(r)$ from simulations in addition to $\\xi(r)$ from the mocks, making it even more dependent on the calibration input. However, none of these extensions improves our fits to either data or mocks. We suspect that a rigorous model at the non-linear level must additionally involve an extension of the linear mass conservation relation that leads to equation~(\\ref{u(r)}), as suggested by reference~\\cite{Achitouv2017b}. Nevertheless, our analysis of the final BOSS data does not indicate any limitations of the simplest linear model from equation~(\\ref{xi^s_lin}), in agreement with previous analyses~\\cite{Hamaus2017,Achitouv2019}.\n\n\n\\section{Conclusion\\label{sec:conclusion}}\nWe have presented a comprehensive cosmological analysis of the geometric and dynamic distortions of cosmic voids in the final BOSS dataset. The extracted information is condensed into constraints on two key quantities, the RSD parameter $f\/b$, and the AP parameter $\\varepsilon$. When calibrated on survey mocks, our analysis provides a relative accuracy of $6.6\\%$ on $f\/b$ and $0.60\\%$ on $\\varepsilon$ (at $68\\%$ confidence level) from the full void sample at a mean redshift of $0.51$. This represents the tightest growth rate constraint obtained from voids in the literature. The AP result even represents the tightest constraint of its kind. However, as these results are calibrated by mock catalogs from a fixed fiducial cosmology, they need to be taken with a grain of salt. Without calibration we are still able to self-consistently model the data, obtaining a relative accuracy of $16.9\\%$ on $f\/b$ and $0.68\\%$ on $\\varepsilon$. While the weaker AP constraint still remains unrivaled, the degradation in the uncertainty on $f\/b$ is mainly due to its strong anti-correlation with the amplitude of the real-space void-galaxy cross-correlation function $\\xi(r)$, which we jointly infer from the data. We emphasize that these uncalibrated constraints are entirely independent from any prior model assumptions on cosmology, or structure formation involving baryonic components, and do not rely on mocks or simulations in any way. They exclusively emerge from the observed data and a linear-theory model that is derived from first principles. With the additional validation of this model on external survey mocks with much higher statistical power, these constraints are robust. The quality of the BOSS data even allows us to analyze sub-samples of voids in two redshift bins. This decreases the mean accuracy per bin by roughly a factor of $\\sqrt{2}$, as expected for statistically independent samples.\n\nWe account for potential systematics in our analysis via a marginalization strategy. To this end we include two nuisance parameters in the model: $\\mathcal{M}$ for modulating the amplitude of the deprojected real-space void-galaxy cross-correlation function $\\xi(r)$, and $\\mathcal{Q}$ for adjusting the quadrupole amplitude. The first parameter accounts for inaccuracies in the deprojection technique and a possible contamination of our void sample by random Poisson fluctuations that can be mistaken as voids with a shallow density profile. The second parameter accounts for anisotropic selection effects due to catastrophic RSDs (such as the FoG effect) or shot noise, that can affect the identification of voids. In the PATCHY mocks we find significant evidence for $\\mathcal{M}>1$, and very marginal evidence for $\\mathcal{Q}>1$, while in the BOSS data both parameters are consistent with unity (to within $68\\%$ confidence) in our low-redshift bin. At higher redshift we observe a mild indication of $\\mathcal{M}>1$ and $\\mathcal{Q}<1$, but only at the significance level of $2\\sigma$ at best. This observation leads us to draw the following conclusion: survey mocks do not necessarily account for all aspects of, and systematics in, the data. They typically represent different realizations drawn from one and the same set of cosmological and astrophysical parameters. Therefore, using the mocks for model calibration may lead to biased constraints on cosmology. This is particularly relevant in situations where model extensions from the fiducial mock cosmology are explored, such as curvature, the properties of dark energy, or massive neutrinos. In addition, this practice underestimates parameter uncertainties, by up to a factor of $4$ for constraints on growth from RSDs. \n\nAs a final remark, we emphasize the important role that cosmic voids will play in the cosmological analysis of future datasets with a much larger scope. Both observatories from the ground~\\cite{DESI,LSST} and from space~\\cite{EUCLID,SPHEREX,WFIRST} will soon provide an unprecedented coverage of large-scale structure in the local Universe, increasing the available sample sizes of voids to at least an order of $10^5$ per survey. This implies that currently achievable error bars on RSD and AP parameters will be further reduced by a factor of a few, potentially allowing us to perform precision cosmology at the level of sub-percent accuracy that could bring about deep ramifications concerning the standard model of cosmology. In this work we have merely investigated flat $\\Lambda$CDM, which exhibits no signs of inconsistency with the final BOSS data.\n\n\n\\begin{acknowledgments}\nWe thank David Spergel for reading over our manuscript and providing pertinent feedback and suggestions. NH would like to thank Kerstin Paech for exchange on coding best practices, Giorgia Pollina for useful suggestions to improve the quality of figures, and Enzo Branchini, Chia-Hsun Chuang, Carlos Correa, Luigi Guzzo, Martha Lippich, Nelson Padilla, Ariel S\\'anchez, Ravi Sheth, Sergey Sibiryakov, Rien van de Weygaert, and Simon White for inspiring discussions about voids at the MIAPP 2019 workshop on ``Dynamics of Large Scale Structure Formation'' in Garching. NH and JW are supported by the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC-2094 -- 390783311. AP is supported by NASA grant 15-WFIRST15-0008 to the Nancy Grace Roman Space Telescope Science Investigation Team ``Cosmology with the High Latitude Survey''. GL and BDW acknowledge financial support from the ANR BIG4 project, under reference ANR-16-CE23-0002. The Center for Computational Astrophysics is supported by the Simons Foundation.\n\nFunding for SDSS-III~\\cite{SDSS} has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe nature of dark matter and dark energy remains one of the greatest\nmysteries of modern cosmology. Dark matter is responsible for the flat rotation\ncurves of the galaxies and dark energy is responsible for the accelerated\nexpansion of the Universe. It is found that dark energy represents about $70\\%$\nof the energy content of the present Universe while the proportions of dark\nmatter and baryonic matter are $25\\%$ and $5\\%$ respectively.\n\n\n\nIn a previous paper \\cite{epjp} (see also \\cite{lettre,jcap}) we have\nintroduced a new cosmological\nmodel that we called\nthe logotropic model. In this model, there is no dark matter and no dark energy.\nThere is just a single dark fluid. What we call ``dark matter'' actually\ncorresponds to its rest-mass energy and what we call ``dark energy'' corresponds\nto its internal energy.\\footnote{Many models try to\nunify dark matter and dark energy. They are called  unified dark energy and dark\nmatter (UDE\/M) models. However, the interpretation of dark matter\nand dark energy that we give in Refs. \\cite{epjp,lettre} is new and\noriginal.}\n\nOur model does not contain any arbitrary parameter so that it is totally\nconstrained. It involves a fundamental constant $\\Lambda$ which is the\ncounterpart of Einstein's cosmological constant \\cite{einsteincosmo} in the\n$\\Lambda$CDM (cold dark\nmatter) model and which turns out to have the same value. Still the\nlogotropic model\nis\nfundamentally different from the $\\Lambda$CDM model. \n\n\nOn the large (cosmological) scales, the   logotropic model is indistinguishable\nfrom the $\\Lambda$CDM model up to the present epoch \\cite{epjp,lettre,jcap}. The\ntwo\nmodels\nwill differ in the far future, in about $25\\, {\\rm Gyrs}$ years, after which the\n logotropic model will become phantom (the\nenergy density will increase as\nthe Universe expands)\nand present a Little Rip (the\nenergy density and the scale factor will become infinite in\ninfinite time) contrary to the $\\Lambda$CDM\nmodel in which the energy density tends towards a  constant \n(de Sitter era).\n\nOn the small (galactic) scales, the logotropic model is able to solve\nsome of the problems encountered by the $\\Lambda$CDM model \\cite{epjp,lettre}.\nIn\nparticular, it is able to account, without free parameter, for the\nconstant surface density of the dark matter halos, for their mass-radius\nrelation, and for the Tully-Fisher relation. \n\nIn this paper, we explore other consequences of this model. By advocating a form\nof ``strong cosmic coincidence'', stating that the present value of\nthe dark energy density $\\rho_{\\rm de, 0}$ is equal to the fundamental constant\n$\\rho_\\Lambda$ appearing in the logotropic model, we predict that\nthe present proportion of dark energy in the Universe is $\\Omega_{\\rm\nde,0}=e\/(1+e)=0.731$ which is close to the observed value\n$0.691$ \\cite{planck2016}. The\nconsequences of this result, which implies that our epoch is very special in the\nhistory of the Universe, are intriguing and related to a form of\n anthropic cosmological\nprinciple \\cite{barrow}.\n\nWe also remark that the universal surface density of dark matter halos (found\nfrom the observations \\cite{donato} and predicted by our model\n\\cite{epjp,lettre}) and the surface density of\nthe Universe are of the same order of magnitude as the surface density of the\nelectron.\nThis makes a curious connection between cosmological and atomic scales.\nExploiting this coincidence, we can relate the Hubble constant, the\nelectron mass and the electron charge to the\ncosmological constant $\\Lambda$.  We also argue that the famous\nnumbers $137$ (fine-structure constant) and $123$ (logotropic\nconstant) may actually represents\nthe same thing. This may be a hint for a theory of\nunification of microphysics and cosmophysics. Speculations are made in the\nAppendices to try to relate these  interconnections to a form of holographic\nprinciple \\cite{bousso} stating that the entropy of the electron, the\nentropy of dark matter\nhalos, and the entropy of the Universe scales like their area as in the case of\nthe entropy of black\nholes \\cite{bekenstein,hawking}.\n\n\n\n\n\\section{The logotropic model}\n\\label{sec_lm}\n\n\\subsection{Unification of dark matter and dark energy}\n\nThe Friedmann equations for a flat universe without cosmological constant\nare \\cite{weinbergbook}:\n\\begin{equation}\n\\frac{d\\epsilon}{dt}+3\\frac{\\dot a}{a}(\\epsilon+P)=0,\\quad H^2=\\left\n(\\frac{\\dot a}{a}\\right )^2=\\frac{8\\pi\nG}{3c^2}\\epsilon,\n\\label{lm1}\n\\end{equation}\nwhere $\\epsilon(t)$ is the energy density of the Universe, $P(t)$ is the\npressure, $a(t)$ is the\nscale factor, and $H=\\dot a\/a$ is the Hubble parameter.\n\n\n\nFor a relativistic fluid experiencing an\nadiabatic evolution such that $Td(s\/\\rho)=0$, the first law of\nthermodynamics\nreduces to \\cite{weinbergbook}:\n\\begin{equation}\nd\\epsilon=\\frac{P+\\epsilon}{\\rho}d\\rho,\n\\label{lm2}\n\\end{equation}\nwhere $\\rho$ is the rest-mass density of the Universe. Combined\nwith the equation of continuity\n(\\ref{lm1}), we get\n\\begin{equation}\n\\frac{d\\rho}{dt}+3\\frac{\\dot a}{a}\\rho=0 \\Rightarrow \\rho=\\frac{\\rho_0}{a^3},\n\\label{lm3}\n\\end{equation}\nwhere  $\\rho_0$ is the present value of the rest-mass density (the\npresent\nvalue of the scale factor is taken to be $a=1$). This equation, which\nexpresses\nthe conservation of the rest-mass, is valid\nfor an arbitrary\nequation\nof state. \n\n\nFor an equation of state specified under the form  $P=P(\\rho)$, Eq.\n(\\ref{lm2}) can be integrated to obtain the relation between the energy density\n$\\epsilon$ and the rest-mass density. We obtain \\cite{epjp}:\n\\begin{equation}\n\\epsilon=\\rho c^2+\\rho\\int^{\\rho}\\frac{P(\\rho')}{{\\rho'}^2}\\, d\\rho'=\\rho\nc^2+u(\\rho).\n\\label{lm4}\n\\end{equation}\nWe note that $u(\\rho)$\ncan be interpreted as an internal energy density \\cite{epjp}. Therefore, the\nenergy density $\\epsilon$ is the sum of the rest-mass energy $\\rho c^2$ and the\ninternal energy $u(\\rho)$.\n\n\n\n\\subsection{The logotropic dark fluid}\n\\label{sec_ldf}\n\nWe assume that the Universe is filled with a single dark fluid described by\nthe\nlogotropic equation of state \\cite{epjp}:\n\\begin{equation}\nP=A\\ln\\left (\\frac{\\rho}{\\rho_P}\\right ),\n\\label{lm5}\n\\end{equation}\nwhere $\\rho_P=c^5\/\\hbar G^2=5.16\\times 10^{99}\\, {\\rm g\\, m^{-3}}$ is the Planck\ndensity and $A$ is a new fundamental constant\nof physics, with the dimension of an energy density, which is the\ncounterpart of the cosmological constant $\\Lambda$ in the $\\Lambda$CDM model\n(see below).\nUsing Eqs. (\\ref{lm4}) and (\\ref{lm5}), the relation\nbetween the energy density and the rest-mass density is\n\\begin{equation}\n\\epsilon=\\rho c^2-A\\ln \\left (\\frac{\\rho}{\\rho_P}\\right )-A=\\rho c^2+u(\\rho).\n\\label{lm6}\n\\end{equation}\nThe energy density is the\nsum of two\nterms: a rest-mass energy term $\\rho c^2=\\rho_0c^2\/ a^{3}$ that mimics the\nenergy density $\\epsilon_{\\rm m}$ of dark matter and\nan internal energy term $u(\\rho)=-A\\ln \\left ({\\rho}\/{\\rho_P}\\right )-A\n=-P(\\rho)-A=3A\\ln a-A\\ln(\\rho_0\/\\rho_P)-A$\nthat mimics the\nenergy density $\\epsilon_{\\rm de}$ of dark energy. This\ndecomposition\nleads to a natural, and physical, unification of dark matter and dark energy and\nelucidates their\nmysterious nature. \n\nSince, in our model, the rest-mass energy of the dark fluid  mimics dark matter,\nwe\nidentify $\\rho_0c^2$ with the present energy density of dark matter. We thus\nset $\\rho_0c^2=\\Omega_{\\rm m,0}\\epsilon_0$,\nwhere $\\epsilon_0\/c^2={3H_0^2}\/{8\\pi G}$ is the\npresent energy density of the Universe  and \n$\\Omega_{\\rm m,0}$ is the present fraction of dark matter (we also include\nbaryonic\nmatter). As a result, the present internal energy of the dark\nfluid,  $u_0=\\epsilon_0-\\rho_0c^2$, is identified with the present\ndark energy density $\\epsilon_{\\rm de,0}=\\Omega_{\\rm de,0}\\epsilon_0$\nwhere\n$\\Omega_{\\rm de,0}=1-\\Omega_{\\rm m,0}$ is the present\nfraction of dark energy. Applying Eq. (\\ref{lm6}) at the present epoch ($a=1$),\nwe\nobtain the identity\n\\begin{equation}\nA=\\frac{\\epsilon_{\\rm de,0}}{\\ln\\left(\\frac{\\rho_Pc^2}{\\epsilon_{\\rm\nde,0}}\\right\n)+\\ln\\left (\\frac{\\Omega_{\\rm de,0}}{1-\\Omega_{\\rm de,0}}\\right )-1}.\n\\label{lm7}\n\\end{equation}\nAt that stage, we can have two points of view. We can consider that this\nequation determines the constant $A$ as a function of\n$\\epsilon_{0}$ and $\\Omega_{\\rm de,0}$ that are both obtained from the\nobservations \\cite{planck2016}. This allows us to determine the value of $A$.\nThis is the\npoint of view that we have adopted in our previous papers \\cite{epjp,lettre} and\nthat we adopt in Sec. \\ref{sec_B} below. However, in the following section,\nwe present another point of view leading to an intriguing result.\n\n\n\n\\subsection{Strong cosmic coincidence and\nprediction of $\\Omega_{\\rm de,0}$}\n\nLet us recall that, in our model, $A$ is considered as a fundamental constant\nwhose value is fixed by Nature. As a result, Eq. (\\ref{lm7}) relates\n$\\Omega_{\\rm de,0}$ to $\\epsilon_{0}$ for a given value of $A$. A priori, we\nhave two unknowns for just one equation. However, we can\nobtain the value of $\\Omega_{\\rm de,0}$ by the following argument. \n\n\nWe can always write the constant  $A$ under the form\n\\begin{equation}\nA=\\frac{\\rho_{\\Lambda}c^2}{\\ln\\left(\\frac{\\rho_P}{\\rho_{\\Lambda}}\\right\n)}.\n\\label{lm8}\n\\end{equation}\nThis is just a change of notation. Eq. (\\ref{lm8}) defines a new\nconstant, the cosmological\ndensity $\\rho_{\\Lambda}$, in place of $A$.  From the cosmological density\n$\\rho_{\\Lambda}$,\nwe can define an effective cosmological constant $\\Lambda$ by\\footnote{We stress\nthat our\nmodel is different from the $\\Lambda$CDM model so that $\\Lambda$ is\nfundamentally different from Einstein's cosmological constant\n\\cite{einsteincosmo}. However, it is always possible to  introduce from the\nconstant $A$ an effective\ncosmological density $\\rho_{\\Lambda}$ and an effective cosmological constant\n$\\Lambda$ by Eqs. (\\ref{lm8}) and (\\ref{lm9}).}\n\\begin{equation}\n\\rho_{\\Lambda}=\\frac{\\Lambda}{8\\pi G}.\n\\label{lm9}\n\\end{equation}\nAgain this is just a change of notation. Therefore, the fundamental constant\nof our model is either $A$, $\\rho_{\\Lambda}$ or $\\Lambda$ (equivalently). We now\nadvocate a form of ``strong cosmic coincidence''. We assume that the present\nvalue of the dark energy density is equal to $\\rho_{\\Lambda}c^2$, i.e.,\n\\begin{equation}\n\\epsilon_{\\rm de,0}=\\rho_{\\Lambda}c^2.\n\\label{lm10}\n\\end{equation}\nSince, in the $\\Lambda$CDM model, $\\epsilon_{\\rm de}$ is a constant usually\nmeasured at the present epoch our\npostulate implies that $\\rho_{\\Lambda}c^2$ coincides with the\ncosmological density in the $\\Lambda$CDM model and that $\\Lambda$, as defined\nby Eq. (\\ref{lm9}), coincides with\nthe ordinary cosmological constant. This is why we have used the same\nnotations. Now, comparing Eqs. (\\ref{lm7}), (\\ref{lm8}) and  (\\ref{lm10})  we\nobtain $\\ln\\left\n\\lbrack \\Omega_{\\rm de,0}\/(1-\\Omega_{\\rm de,0})\\right \\rbrack-1=0$\nwhich determines $\\Omega_{\\rm de,0}$. We find that\n\\begin{equation}\n\\Omega_{\\rm de,0}^{\\rm th}=\\frac{e}{1+e}\\simeq 0.731\n\\label{lm11}\n\\end{equation}\nwhich is close to the observed value $\\Omega_{\\rm de,0}^{\\rm\nobs}=0.691$ \\cite{planck2016}.  This agreement is puzzling. It\nrelies on the ``strong cosmic\ncoincidence'' of Eq. (\\ref{lm10}) implying that our epoch is very special. This\nis a form of  anthropic cosmological principle \\cite{barrow}. This may also\ncorrespond to a fixed point of our model. In order to avoid\nphilosophical issues, in the following, we adopt the more conventional\npoint of view discussed at the end of Sec. \\ref{sec_ldf}. \n\n\n\n\\subsection{The logotropic constant $B$}\n\\label{sec_B}\n\n\n\nWe can rewrite Eq. (\\ref{lm8}) as\n\\begin{equation}\nA=B\\rho_{\\Lambda}c^2\\qquad {\\rm\nwith}\\qquad B=\\frac{1}{\\ln\\left({\\rho_P}\/{\\rho_{\\Lambda}}\n\\right\n)}.\n\\label{lm12}\n\\end{equation}\nAgain, this is just a change of notation defining the dimensionless number\n$B$. We shall call it the logotropic constant since it is equal\nto the inverse of the logarithm of the cosmological density normalized by the\nPlanck density (see Appendix \\ref{sec_const}).\nWe note that $A$ can be expressed in terms of $B$ (see below) so that the\nfundamental constant of our model is either $A$, $\\rho_{\\Lambda}$, $\\Lambda$,\nor $B$. In the following, we shall express all the results in terms of $B$.\nFor example, the relation (\\ref{lm6}) between the energy density and the scale\nfactor can be rewritten as \n\\begin{equation}\n\\frac{\\epsilon}{\\epsilon_0}=\\frac{\\Omega_{\\rm\nm,0}}{a^3}+(1-\\Omega_{\\rm m,0})(1+3B\\ln\na).\n\\label{lm13}\n\\end{equation}\nCombined with the Friedmann equation (\\ref{lm1}) this equation determines the\nevolution of the scale factor $a(t)$ of the Universe in the logotropic model.\nThis evolution has been studied in detail\nin  \\cite{epjp,lettre,jcap}. \n\n\n\n{\\it Remark:} Considering Eq. (\\ref{lm13}), we see that the\n$\\Lambda$CDM model is recovered for $B=0$.\nAccording to Eq. (\\ref{lm12}) this implies that\n$\\rho_P\\rightarrow +\\infty$, i.e., $\\hbar\\rightarrow 0$. Therefore, the \n$\\Lambda$CDM model corresponds to the semiclassical limit of the logotropic\nmodel. The fact that $B$ is intrinsically nonzero implies that\nquantum mechanics ($\\hbar\\neq 0$) plays some role in our model in addition to\ngeneral relativity. This may\nsuggest a link with a theory of quantum gravity. \n\n\n\n\\subsection{The value of $B$ from the\nobservations}\n\\label{sec_Bobs}\n\n\n\nThe fundamental constant ($A$, $\\rho_{\\Lambda}$, $\\Lambda$,\nor $B$) appearing in our model can be  determined from the\nobservations by using Eq. (\\ref{lm7}). We take $\\Omega_{\\rm de,0}=0.6911$ and\n $H_0=2.195\\times 10^{-18}\\, {\\rm\ns^{-1}}$ \\cite{planck2016}. This implies $\\epsilon_0\/c^2=3H_0^2\/8\\pi\nG=8.62\\times 10^{-24}\\, {\\rm g\\, m^{-3}}$ and $\\epsilon_{\\rm\nde,0}\/c^2=\\Omega_{\\rm\nde,0}\\epsilon_0\/c^2=5.96\\times 10^{-24}\\, {\\rm g\\, m^{-3}}$. Since $\\ln\\left\n\\lbrack \\Omega_{\\rm de,0}\/(1-\\Omega_{\\rm de,0})\\right \\rbrack-1=-0.195$\nis small as\ncompared to $\\ln(\\rho_Pc^2\/\\epsilon_{\\rm de,0})=283$, we can write in very\ngood approximation $A$ as in Eq. (\\ref{lm8}) with $\\rho_{\\Lambda}\\simeq\n\\epsilon_{\\rm de,0}\/c^2$ as in Eq. (\\ref{lm10}). Therefore, \n\\begin{equation}\n\\rho_{\\Lambda}=\\frac{3\\Omega_{\\rm\nde,0}H_0^2}{8\\pi G}=5.96\\times 10^{-24}\\, {\\rm g\\, m^{-3}}\n\\label{lm14a}\n\\end{equation}\nand\n\\begin{equation}\n\\Lambda= 3\\Omega_{\\rm\nde,0}H_0^2=1.00\\times 10^{-35}\\, {\\rm s^{-2}}\n\\label{lm14b}\n\\end{equation}\nare approximately equal to\nthe cosmological density and to the cosmological constant in the $\\Lambda$CDM\nmodel. From Eq. (\\ref{lm12}) we get\n\\begin{equation}\nB=\\frac{1}{\\ln(\\rho_P\/\\rho_{\\Lambda})}\\simeq\n\\frac{1}{123\\ln(10)}\\simeq 3.53\\times 10^{-3}.\n\\label{lm15}\n\\end{equation}\nAs discussed in our previous papers \\cite{epjp,lettre,jcap}, $B$ is\nessentially the inverse of\nthe\nfamous number $123$ (see Appendix \\ref{sec_const}). Finally,\n\\begin{equation}\nA=B\\,\\rho_{\\Lambda}c^2=1.89\\times 10^{-9}\n\\, {\\rm g}\\, {\\rm m}^{-1}\\, {\\rm s}^{-2}. \n\\label{lm16}\n\\end{equation}\n\n\n\nFrom now on, we shall view $B$ given by Eq. (\\ref{lm15}) as the fundamental\nconstant of the theory. Therefore, everything should be expressed in terms of\n$B$ and the other fundamental constants of physics defining the Planck scales.\nFirst, we have\n\\begin{equation}\n\\frac{\\rho_{\\Lambda}}{\\rho_{P}}=\\frac{G\\hbar\\Lambda}{8\\pi\nc^5}=e^{-1\/B}=1.16\\times 10^{-123}.\n\\label{lm18}\n\\end{equation}\nThen,\n\\begin{equation}\n\\frac{A}{\\rho_{P}c^2}=Be^{-1\/B}=4.08\\times 10^{-126}.\n\\label{lm17}\n\\end{equation}\nThe logotropic equation of state (\\ref{lm5}) can be written as\n$P\/\\rho_Pc^2=Be^{-1\/B}\\ln(\\rho\/\\rho_P)$. Using Eq. (\\ref{lm10}) and\n$\\epsilon_{\\rm\nde,0}=\\Omega_{\\rm de,0}\\epsilon_0$, we get\n\\begin{equation}\n\\frac{\\epsilon_0}{\\rho_{P}c^2}=\\frac{1}{\\Omega_{\\rm de,0}}e^{-1\/B}=1.67\\times\n10^{-123}.\n\\label{lm19}\n\\end{equation}\nFinally, using Eq. (\\ref{lm1}),\n\\begin{equation}\nt_P H_0=\\left (\\frac{8\\pi}{3\\Omega_{\\rm de,0}}\\right\n)^{1\/2}e^{-1\/2B}=1.18\\times 10^{-61},\n\\label{lm20}\n\\end{equation}\nwhere $t_P=(\\hbar G\/c^5)^{1\/2}=5.391\\times 10^{-44}\\, {\\rm s}$ is the Planck\ntime. In the last two expressions,\nwe can either consider that $\\Omega_{\\rm de,0}$ is ``predicted'' by\nEq. (\\ref{lm11}) or take its measured value. To the order of\naccuracy that we\nconsider, this does not change the numerical values.\n\n\n\n\n\\section{Previous predictions of the logotropic model}\n\nThe interest of the logotropic model becomes apparent when it is\napplied to dark matter halos \\cite{epjp,lettre}. We assume that dark matter\nhalos are\ndescribed by the logotropic equation of state of Eq. (\\ref{lm5}) with\n$A=1.89\\times 10^{-9} \\, {\\rm g}\\, {\\rm m}^{-1}\\, {\\rm\ns}^{-2}$ (or $B=3.53\\times 10^{-3}$). At the\ngalactic scale, we can use Newtonian gravity. \n\n\n\\subsection{Surface density of dark matter halos}\n\n\n\nIt is an empirical evidence that the surface density of galaxies has\nthe same value \n\\begin{equation}\n\\Sigma_0^{\\rm obs}\\equiv \\rho_0 r_h\\simeq 295\\, {\\rm g\\, m^{-2}}\\simeq 141\\,\nM_{\\odot}\/{\\rm pc^2}\n\\label{lm21}\n\\end{equation}\neven\nif their sizes and masses vary by several orders of magnitude (up to $14$\norders of magnitude in luminosity) \\cite{donato}. Here $\\rho_0$ is the central\ndensity and $r_h$ is the halo radius at which the density has decreased by a\nfactor of $4$. The logotropic model predicts that the surface density of the\ndark matter halos is the\nsame for all the halos (because $A$ is a universal constant) and that it  is\ngiven by \\cite{epjp,lettre}:\n\\begin{equation}\n\\Sigma_0^{\\rm th}=\\left (\\frac{A}{4\\pi G}\\right )^{1\/2}\\xi_h= \\left\n(\\frac{B}{32}\\right\n)^{1\/2}\\frac{\\xi_h}{\\pi}\\frac{c\\sqrt{\\Lambda}}{G},\n\\label{lm22}\n\\end{equation}\nwhere $\\xi_h=5.8458...$ is a pure number arising from the Lane-Emden equation\nof index $n=-1$ expressing the condition of hydrostatic equilibrium of\nlogotropic\nspheres.\\footnote{The logotropic spheres \\cite{epjp,lettre}, like the isothermal\nspheres \\cite{chandra}, have an infinite mass. This implies that the logotropic\nequation of state cannot\ndescribe dark matter halos at infinitely large distances. Nevertheless, it may\ndescribe the inner region of dark matter halos and this is sufficient to\ndetermine their surface density. The stability of bounded logotropic spheres has\nbeen studied in \\cite{logo} by analogy with the stability of bounded isothermal\nspheres and similar results have been obtained. In particular, bounded\nlogotropic\nspheres are stable\nprovided that the density contrast is not too large.} Numerically,\n\\begin{equation}\n\\Sigma_0^{\\rm th}= 278\\, {\\rm g\\,\nm^{-2}}\\simeq 133\\,\nM_{\\odot}\/{\\rm pc^2},\n\\label{lm23}\n\\end{equation}\nwhich is very close to the observational value\n(\\ref{lm21}). The fact that the surface density of dark matter halos is\ndetermined by the effective cosmological constant $\\Lambda$ (usually related to\nthe dark energy) tends to confirm that dark matter and dark energy are just two\nmanifestations of the {\\it same} dark fluid, as we have\nassumed in our model.\n\n{\\it Remark:} The dimensional term $c\\sqrt{\\Lambda}\/G$ in Eq. (\\ref{lm22}) can\nbe interpreted as representing the surface density of the Universe (see Appendix\n\\ref{sec_w}).  We note that this term alone, $c\\sqrt{\\Lambda}\/G=14200\\, {\\rm g\\,\nm^{-2}}=6800 M_{\\odot}\/{\\rm pc}^2$, is too large to account precisely for the\nsurface density of dark matter halos so that the prefactor\n$(B\/32)^{1\/2}(\\xi_h\/\\pi)=0.01955$ is\nnecessary to reduce this number.  It is interesting to remark\nthat the term $c\\sqrt{\\Lambda}\/G$ arises from classical general relativity while\nthe prefactor $\\propto B^{1\/2}$ has a quantum origin as discussed at the end of\nSec. \\ref{sec_B}. Actually, we will see that it is related to the fine-structure\nconstant $\\alpha$ [see Eq. (\\ref{lm30}) below].\n\n\n\\subsection{Mass-radius relation}\n\n \n\nThere are interesting consequences of the preceding result. For logotropic\nhalos, the mass of the halos calculated at the halo\nradius $r_h$ is given by  \\cite{epjp,lettre}:\n\\begin{equation}\nM_h=1.49\\Sigma_0 r_h^2.\n\\label{lm24a}\n\\end{equation}\nThis\ndetermines the mass-radius relation of dark matter-halos. On the other hand, the\ncircular\nvelocity at the halo radius is $v_h^2=GM_h\/r_h=1.49\\Sigma_0\nG r_h$. Since the surface density of\nthe dark matter halos is constant, we obtain\n\\begin{equation}\n\\frac{M_h}{M_{\\odot}}=198 \\left (\\frac{r_h}{{\\rm pc}}\\right )^2,\\qquad\n\\left (\\frac{v_h}{{\\rm\nkm}\\, {\\rm s}^{-1}}\\right )^2=0.852\\, \\frac{r_h}{\\rm pc}.\n\\label{lm24}\n\\end{equation}\nThe scalings $M_h\\propto r_h^2$ and $v_h^2\\propto\nr_h$ (and also the prefactors) are consistent with the observations.  \n\n\n\n\\subsection{The Tully-Fisher relation}\n\n\nCombining the previous equations, the logotropic model leads to the Tully-Fisher\n\\cite{tf} relation $v_h^4\\propto M_h$ or, more precisely, \n\\begin{equation}\n\\left (\\frac{M_b}{v_h^4}\\right )^{\\rm th}=\\frac{f_b}{1.49\\Sigma_0^{\\rm th}\nG^2}=46.4 M_{\\odot}{\\rm km}^{-4}{\\rm s}^4,\n\\label{lm25}\n\\end{equation}\nwhere\n$f_b=M_b\/M_h\\sim 0.17$ is the cosmic baryon fraction \\cite{mcgaugh}. The\npredicted value from Eq. (\\ref{lm25}) is\nclose to the observed one $\\left ({M_b}\/{v_h^4}\\right )^{\\rm obs}=47\\pm 6\nM_{\\odot}{\\rm\nkm}^{-4}{\\rm\ns}^4$ \\cite{mcgaugh}.\n\n\n\n{\\it Remark:} The Tully-Fisher relation is sometimes justified by\nthe MOND (Modification of Newtonian dynamics) theory \\cite{mond} which predicts\na relation of the form $v_h^4=Ga_0\nM_b$ between the asymptotic circular velocity and the baryon mass, where $a_0$\nis a critical acceleration. Our results imply\n$a_0^{\\rm th}=1.62\\times\n10^{-10}\\, {\\rm m}\\, {\\rm s}^{-2}$ which is\nclose to the value $a_0^{\\rm obs}=(1.3\\pm 0.3)\\times\n10^{-10}\\, {\\rm m}\\, {\\rm s}^{-2}$ obtained from the observations\n\\cite{mcgaugh}.\nCombining Eqs. (\\ref{lm24a}) and (\\ref{lm25}), we first get $a_0^{\\rm\nth}=(1.49\/f_b)\\Sigma_0^{\\rm th}G=GM_h\/(f_br_h^2)$ which shows that $a_0$ can be\ninterpreted as the surface gravity of the galaxies $G\\Sigma_0$ (which\ncorresponds to Newton's acceleration $GM_h\/r_h^2$) or as the surface density of\nthe Universe (see Appendix \\ref{sec_abh}). Then, using Eqs.\n(\\ref{lm14b}) and (\\ref{lm22}), we obtain $a_0^{\\rm\nth}=({1.49}\/{f_b})({B}\/{32})^{1\/2}({\\xi_h}\/{\\pi})c\\sqrt{\\Lambda}\\simeq H_0\nc\/4$  which\nexplains why $a_0$ is of the order of $H_0 c$. We emphasize,\nhowever, that we do\nnot use the MOND theory in our approach and that the logotropic model assumes\nthe existence of a dark fluid.\n\n\n\n\\subsection{The mass $M_{300}$}\n\n\n\n\n\nThe logotropic equation of state also explains the observation of Strigari {\\it\net al.} \\cite{strigari} that all the dwarf spheroidals (dSphs) of the Milky\nWay have the same total dark matter mass $M_{300}$ contained within a radius\n$r_u=300\\, {\\rm pc}$, namely $M_{300}^{\\rm obs}\\simeq 10^7\\, M_{\\odot}$\nThe logotropic model predicts the\nvalue \\cite{epjp,lettre}:\n\\begin{equation}\nM_{300}^{\\rm th}=\\frac{4\\pi \\Sigma_0^{\\rm th} r_u^2}{\\xi_h\\sqrt{2}}=1.82\\times\n10^{7}\\, M_{\\odot},\n\\label{lm26}\n\\end{equation}\nwhich is in very good agreement with the\nobservational value. \n\n\n\\section{A curious connection between atomic and\ncosmological scales}\n\\label{sec_curious}\n\n\n\n\\subsection{The surface density of the electron}\n\\label{sec_sde}\n\n\n\nThe classical radius of the\nelectron $r_e$ can be obtained qualitatively by writing that the electrostatic\nenergy of the electron, $e^2\/r_e$, is equal to its rest-mass energy $m_e c^2$.\nRecalling the value of the charge of the electron $e=4.80\\times 10^{-13}\\, {\\rm\ng^{1\/2}\\, m^{3\/2}\\, s^{-1}}$ and its mass $m_e=9.11\\times 10^{-28}\\, {\\rm g}$,\nwe obtain $r_e=e^2\/m_ec^2=2.82\\times 10^{-15}\\, {\\rm m}$. As a result, the\nsurface density of the electron is\\footnote{We note that the Thomson\ncross-section $\\sigma=(8\\pi\/3)(e^2\/m_e\nc^2)^2$ can be written as $\\sigma=(8\\pi\/3)r_e^2$ giving a physical meaning to\nthe classical electron radius $r_e$. We also note that $r_e$ can be written as \n$r_e=\\alpha\\hbar\/m_ec$ where $\\lambda_C=\\hbar\/m_e c$ is the Compton\nwavelength of the electron and $\\alpha$ is the fine-structure constant $\\alpha$\n[see Eq. (\\ref{const1})]. Similarly, \n$\\Sigma_e=(1\/\\alpha^2)m_e^3c^2\/\\hbar^2$.}\n\\begin{equation}\n\\Sigma_e=\\frac{m_e}{r_e^2}=\\frac{m_e^3c^4}{e^4}=115\\, {\\rm g\/m^2}= 54.9\\,\nM_{\\odot}\/{\\rm pc^2},\n\\label{lm27}\n\\end{equation}\nwhich is of the same order of magnitude as the surface density of dark matter\nhalos from Eq. (\\ref{lm21}). This\ncoincidence is amazing in view of the different scales (atomic\nversus cosmological) involved.  More precisely, we find\n$\\Sigma_e=\\sigma\\Sigma_0^{\\rm th}$ with $\\sigma\\simeq 0.413$. Of\ncourse, the value of $\\sigma$ depends on the\nprecise manner used to define the surface density of the electron, or its\nradius, but the\nimportant point is that this number is of order unity. \n\n\n\n\\subsection{Relation between $\\alpha$ and $B$}\n\\label{sec_Ba}\n\n\n\nBy matching the two formulae (\\ref{lm22}) and (\\ref{lm27}),\nwriting $\\Sigma_e=\\sigma\\Sigma_0^{\\rm th}$, we get\n\\begin{equation}\n\\Lambda=\\frac{32\\pi^2}{B\\xi_h^2\\sigma^2}\n\\frac{m_e^6c^6G^2}{e^8}=\\frac{32\\pi^2}{B\\xi_h^2\\sigma^2\\alpha^4}\n\\frac{m_e^6c^2G^2}{\\hbar^4},\n\\label{lm28}\n\\end{equation}\nwhere we have introduced the fine-structure constant $\\alpha$ in the\nsecond equality (see Appendix \\ref{sec_const}).\nThis expression provides a curious relation between the cosmological constant,\nthe mass of the electron and its charge. This relation is similar to\nWeinberg's empirical relation (see Appendix\n\\ref{sec_w}) which can be written as [combining Eqs. (\\ref{lm14b}) and\n(\\ref{w4})]\n\\begin{equation}\n\\Lambda=192\\pi^2\\mu^2\\Omega_{\\rm de,0}\\frac{m_e^6c^6G^2}{e^8},\n\\label{w4b}\n\\end{equation}\nwhere $\\mu\\simeq 3.42$. Note that in our formula (\\ref{lm28}),\n$\\Lambda$ appears two times: on the left hand side and in $B$ (which depends\nlogarithmically on $\\Lambda$). This will have important consequences in the\nfollowing.\n\n\nB\\\"ohmer and Harko \\cite{bhLambda}, by a completely different approach, found a\nsimilar relation\\footnote{A closely related formula, involving the Hubble\nconstant instead of the cosmological constant, was first found by Stewart\n\\cite{stewart} in 1931 by trial and error.}\n\\begin{equation}\n\\Lambda=\\nu \\frac{\\hbar^2 G^2 m_e^6 c^8}{e^{12}}=\\frac{\\nu}{\\alpha^6} \\frac{G^2\nm_e^6 c^2}{\\hbar^{4}},\n\\label{lm29}\n\\end{equation}\nwhere $\\nu\\simeq 0.816$ is of order unity. Their result can be\nobtained as follows. They first introduce a minimum\nmass $m_{\\Lambda}\\sim\\hbar\\sqrt{\\Lambda}\/c^2$ interpreted as being the mass of\nthe elementary particle of dark energy, called the cosmon. Then, they define a\nradius $R$ by the relation $m_{\\Lambda}\\sim \\rho_{\\Lambda} R^3$ where\n$\\rho_{\\Lambda}= \\Lambda\/8\\pi G$ is the cosmological density considered as being\nthe lowest density in the Universe. Finally, they remark that $R$ has\ntypically the same value as the classical radius of the electron \n$r_e=e^2\/m_ec^2$. Matching $R$ and $r_e$ leads to the scaling of Eq.\n(\\ref{lm29}). We have then added a prefactor $\\nu$ and adjusted its\nvalue\nin order to exactly obtain the measured value of the  cosmological  constant\n\\cite{planck2016}.\nSince  the approach of B\\\"ohmer and Harko \\cite{bhLambda} is essentially\nqualitative, and depends on the precise manner used to define the radius of\nthe\nelectron, their result can be at best valid up to a constant of order unity.\n\nWe would like now to compare the estimates from Eqs. (\\ref{lm28}) and\n(\\ref{lm29}). At that\nstage, we can have two points of view. If we consider that comparing the\nprefactors is meaningless because our approach can only provide ``rough'' orders\nof\nmagnitude, we  conclude that Eqs. (\\ref{lm28}) and\n(\\ref{lm29}) are\nequivalent, and that they are also equivalent to Weinberg's empirical\nrelation (\\ref{w4}). Alternatively, if we take the prefactors seriously into\naccount (in particular the presence of $B$ which depends on $\\Lambda$) and match\nthe formulae (\\ref{lm28}) and\n(\\ref{lm29}), we find an interesting relation between the\nfine-structure constant\n$\\alpha$ and the logotropic constant $B$:\n\\begin{equation}\n\\alpha=\\left (\\frac{\\nu}{32}\\right\n)^{1\/2}\\frac{\\xi_h\\sigma}{\\pi}\\sqrt{B}\\simeq 0.123 \\sqrt{B}.\n\\label{lm30}\n\\end{equation}\nTherefore, the fine-structure constant (electron charge normalized by the\nPlanck charge) is determined by the logotropic  constant $B$\n(cosmological density normalized by the Planck density) by a relation of the\nform $\\alpha\\propto B^{1\/2}$. This makes a connection between atomic scales and\ncosmological scales. This also suggests that the famous numbers\n$137$ and $123$\n(see Appendix \\ref{sec_const}) are related to each other, or may even represent\nthe same thing. From Eq. (\\ref{lm30}), we have\\footnote{We note that the\nprefactors in\nEqs. (\\ref{lm30}) and (\\ref{lm31}) appear to be close to $123\/1000$ and\n$123\/10$, where the number $123$ appears again (!). We do not know whether this\nis\nfortuitous or if this bears a deeper significance than is apparent at first\nsight.} \n\\begin{equation}\n137\\simeq 12.3 \\sqrt{123}.\n\\label{lm31}\n\\end{equation}\n\n\n{\\it Remark:} the logotropic constant $B$ is related to the effective\ncosmological constant $\\Lambda$ by [see Eq.\n(\\ref{lm18})]\n\\begin{equation}\nB=\\frac{1}{\\ln\\left (\\frac{8\\pi c^5}{G\\hbar\\Lambda}\\right )}.\n\\label{lm31b}\n\\end{equation}\nUsing Eqs. (\\ref{lm30}) and (\\ref{lm31b}), we can express the fine-structure\nconstant $\\alpha$ as a function of the effective cosmological\nconstant $\\Lambda$ or, using Eq. (\\ref{lm20}), as a function of the age of the\nUniverse $t_{\\Lambda}=1\/H_0$ as\n\\begin{equation}\n\\alpha=\\frac{0.123}{\\ln\\left (\\frac{8\\pi c^5}{G\\hbar\\Lambda}\\right\n)^{1\/2}}=\\frac{0.123}{\\sqrt{2}\\ln\\left \\lbrack \\left (\\frac{8\\pi}{3\\Omega_{\\rm\nde,0}}\\right )^{1\/2}\\frac{t_{\\Lambda}}{t_P}\\right\\rbrack^{1\/2}}.\n\\label{lm31c}\n\\end{equation}\nWe emphasize the scaling $1\/\\alpha\\propto (\\ln t_{\\Lambda})^{1\/2}$. It is\ninteresting to note that similar relations have been introduced in the past\nfrom pure numerology (see \\cite{kragh}, P. 428). These relations suggest\nthat the fundamental constants may change with time as argued by Dirac\n\\cite{dirac1,dirac2}. \n\n\n\\subsection{The mass and the charge of the electron\nin terms of $B$}\n\nUsing Eqs. (\\ref{lm9}), (\\ref{lm18}), (\\ref{lm28}) and (\\ref{lm30}), we find\nthat the mass and the charge of the electron\nare determined by the logotropic\nconstant $B$ according to\n\\begin{eqnarray}\n\\frac{m_e}{M_P}=\\left (\\frac{8\\pi}{\\nu}\\right )^{1\/6}\\left\n(\\frac{\\nu}{32}\\right\n)^{1\/2}\\frac{\\xi_h\\sigma}{\\pi}\\sqrt{B}e^{-1\/(6B)}\\nonumber\\\\\n=0.217\\sqrt{B}e^{\n-1\/(6B) } =4.18\\times 10^{-23},\n\\label{lm32}\n\\end{eqnarray}\n\\begin{equation}\n\\frac{e^2}{q_P^2}=\\left (\\frac{\\nu}{32}\\right\n)^{1\/2}\\frac{\\xi_h\\sigma}{\\pi}\\sqrt{B}=0.123\\sqrt{B}=7.29\\times 10^{-3},\n\\label{lm33}\n\\end{equation}\nwhere $M_P=(\\hbar c\/G)^{1\/2}=2.18\\times 10^{-5}\\, {\\rm g}$ is the\nPlanck mass and $q_P=(\\hbar c)^{1\/2}=5.62\\times 10^{-12}\\, {\\rm\ng^{1\/2}\\, m^{3\/2}\\, s^{-1}}$ is the Planck charge.\nThese relations suggest that the  mass and the charge of the electron (atomic\nscales) are determined by the effective cosmological constant\n$\\Lambda$ or $B$ (cosmological scales). We emphasize the presence of\nthe exponential factor $e^{-1\/(6B)}$ in Eq. (\\ref{lm32}) explaining why the\nelectron mass is much smaller than the Planck mass while the electron charge is\ncomparable to the Planck charge.  \n\n\\subsection{A prediction of $B$}\n\n\nIf we match Eqs. \n(\\ref{lm22}) and (\\ref{w3}), or equivalently Eqs. \n(\\ref{lm28}) and (\\ref{w4b}), we obtain\n\\begin{equation}\nB^{\\rm app}=\\frac{1}{6\\lambda^2\\xi_h^2\\Omega_{\\rm\nde,0}}.\n\\label{w5}\n\\end{equation}\nTaking $\\lambda^{\\rm app}=1$ (since we cannot predict its value) and\n$\\Omega_{\\rm de,0}^{\\rm th}=e\/(1+e)$ [see Eq. (\\ref{lm11})], we get $B^{\\rm\napp}=6.67\\times 10^{-3}$ instead of $B=3.53\\times\n10^{-3}$. We recall\nthat the value of $B$ was obtained in Sec. \\ref{sec_Bobs} from the\nobservations.\nOn the other hand, Eq. (\\ref{w5}) gives the correct order of magnitude of $B$\nwithout any reference to observations, up to a dimensionless constant\n$\\lambda\\simeq 1.41$ of order unity.\nConsidering that $B$ is predicted by Eq.\n(\\ref{w5}) implies that we can predict the values of $\\Lambda$, $H_0$, $\\alpha$,\n$m_e$ and $e$ without reference to observations, up to dimensionless constants\n$\\lambda\\simeq 1.41$, $\\nu\\simeq 0.816$ and $\\sigma\\simeq 0.413$\nof order unity.  We note, however, that even if these dimensionless\nconstants ($\\lambda$, $\\nu$, $\\sigma$) are of order unity, their precise values\nare of importance since $B$ usually\nappears in exponentials like in Eqs. (\\ref{lm18}), (\\ref{lm20}) and\n(\\ref{lm32}). \n\n\n\n\\section{Conclusion}\n\nIn this paper, we have developed the logotropic model introduced in\n\\cite{epjp,lettre}. In this model, dark matter corresponds to the rest mass\nenergy of a dark fluid and dark energy  corresponds to its internal energy. The\n$\\Lambda$CDM model may be interpreted as the semiclassical limit\n$\\hbar\\rightarrow 0$ of the logotropic model. We have first recalled that the \nlogotropic model is able to predict (without free parameter) the universal value\nof the surface density of dark matter halos $\\Sigma_0$, their mass-radius\nrelation\n$M_h-r_h$, the Tully-Fisher relation $M_b\\sim v_h^4$ and the value of the mass\n$M_{300}$ of dSphs. Then, we have argued that it also predicts the value of the\npresent fraction of dark energy $\\Omega_{\\rm de,0}$. This arises from a\nsort of ``strong cosmic coincidence'' but this could also correspond to a fixed\npoint of the model. Finally, we have observed that the surface density of the\ndark matter halos  $\\Sigma_0$ is of the same order as the surface density of the\nUniverse\n $\\Sigma_\\Lambda$ and of the same order as the surface density of the electron \n$\\Sigma_e$. This makes an\nempirical connection between atomic physics and cosmology. From this connection,\nwe have obtained a relation between the fine-structure constant $\\alpha\\sim\n1\/137$ and the logotropic constant $B\\sim 1\/123$. We have also expressed the\nmass $m_e$  and the charge $-e$ of the electron as a function of $B$\n(or as a function of the effective\ncosmological constant $\\Lambda$). Finally, we have obtained a prediction of the\norder of magnitude of $B$ independent from the observations. In a sense, our\napproach which expresses the mass and the charge of the electron in terms of\nthe cosmological constant is a continuation of the program initiated by\nEddington \\cite{eddington} in his quest for a `{\\it Fundamental Theory}' of\nthe physical world in which the basic interaction strengths and elementary\nparticle masses would be prediced entirely combinatorically by simple counting\nprocesses \\cite{barrow}. In the Appendices, we try to\nrelate these interconnections to a form of holographic\nprinciple \\cite{bousso} (of course not known at the time of Eddington) stating\nthat the entropy of the electron, of dark matter\nhalos, and of the Universe scales like their area as in the case of black\nholes \\cite{bekenstein,hawking}.\n\n\nThis paper has demonstrated that physics  is full of ``magic'' and mysterious\nrelations that are still not fully understood (one of them being the empirical\nWeinberg relation). Hopefully, a contribution of this\npaper is to reveal these ``mysteries'' and propose some tracks so as to  induce\nfurther research towards\ntheir elucidation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Introduction}%\n\\section{Introduction}\nThe isolation and control of the number of carriers in single and\nfew layer graphene flakes\\cite{Netal04t,Netal05t} has lead to a\nlarge research activity exploring all aspects of these\nmaterials\\cite{NGPNG09}. Among others, the application of graphene\nto spintronic\ndevices\\cite{HGNSB06t,CCF07,NG07,HJPJW07t,TTVJJvW08t,WPLCWSK08t,Petal09}\nand to spin qubits\\cite{TBLB07,FTL09,WYMK09} is being intensively\nstudied. The understanding of these devices requires a knowledge of\nthe electronic spin-orbit interaction. In principle, this\ninteraction turns single layer graphene into a topological\ninsulator\\cite{KM05}, which shows a bulk gap and edge states at all\nboundaries. The magnitude of the spin-orbit coupling in single layer\ngraphene has been studied\\cite{HGB06,HMING06t,YYQZF07,GKEAF09}. The\ncalculated couplings are small, typically below 0.1K. The observed\nspin relaxation\\cite{TTVJJvW08t,Jetal09} suggests the existence of\nstronger mechanisms which lead to the precession of the electron\nspins, like impurities or lattice\ndeformations\\cite{NG09,HGB09,EKGF09}.\n\nBilayer graphene is interesting because, among other properties, a\ngap can be induced by electrostatic means, leading to new ways for\nthe confinement of electrons\\cite{MF06}. The spin-orbit interactions\nwhich exist in single layer graphene modulate the gap of a graphene\nbilayer\\cite{GM09}. The unit cell of bilayer graphene contains four\ncarbon atoms, and there are more possible spin-orbit couplings than\nin single layer graphene.\n\nWe analyze in the following the intrinsic and extrinsic spin-orbit\ncouplings in bilayer graphene, using a tight binding model, and\ndescribing the relativistic effects responsible for the spin-orbit\ninteraction by a $\\vec{{\\bf L}}\\vec{{ \\bf S}}$ intraatomic coupling.\nWe use the similarities between the electronic bands of a graphene\nbilayer and the bands of three dimensional graphite with Bernal\nstacking to generalize the results to the latter.\n\n\\section{The model}.\nWe describe the electronic bands of a graphene bilayer using a tight\nbinding model, with four orbitals, the $2s$ and the three $2p$\norbitals,  per carbon atom. We consider hoppings between nearest\nneighbors in the same plane, and nearest neighbors and next nearest\nneighbors between adjacent layers, see\\cite{CLM09}. The couplings\nbetween each pair of atoms is parametrized by four hoppings, $V_{ss}\n, V_{sp} , V_{pp \\pi}$ and $V_{pp \\sigma}$. The model includes also\ntwo intraatomic levels, $\\epsilon_s$ and $\\epsilon_p$, and the\nintraatomic spin-orbit coupling\n\\begin{align}\n{\\cal H}_{so} &\\equiv \\Delta_{so} \\sum_i \\vec{\\bf L}_i \\vec{\\bf S}_i\n\\end{align}\n The parameters used\nto describe the $\\pi$ bands of graphite\\cite{M57,SW58}, $\\gamma_0 ,\n\\gamma_1 , \\gamma_2 , \\gamma_3 , \\gamma_4 , \\gamma_5$ and $\\Delta$,\ncan be derived from this set of parameters. We neglect the\ndifference between different hoppings between atoms which are next\nnearest neighbors in adjacent layers, which are responsible for the\ndifference between the parameters $\\gamma_3$ and $\\gamma_4$. We also\nset the difference in onsite energies between the two inequivalent\natoms, $\\Delta$ to zero. The parameters $\\gamma_2$ and $\\gamma_5$\nare related to hoppings between next nearest neighbor layers, and\nthey do not play a role in the description of the bilayer. The total\nnumber of parameters is 15, although, without loss of generality, we\nset $\\epsilon_p = 0$. We do not consider hoppings and spin orbit\ninteractions which include $d$ levels, although they can contribute\nto the total magnitude of the spin-orbit\ncouplings\\cite{MY62,GKEAF09}. The effects mediated by $d$ orbitals\ndo not change the order of magnitude of the couplings in single\nlayer graphene, and their contribution to interlayer effects should\nbe small.\n\nThe main contribution to the effective spin-orbit at the Fermi level\ndue to the interlayer coupling is due to the hoppings between $p$\norbitals in next nearest neighbor atoms in different layers. This\ninteraction gives rise to the parameters $\\gamma_3$ and $\\gamma_4$\nin the parametrization of the bands in graphite. For simplicity, we\nwill neglect couplings between $s$ and $p$ orbitals in neighboring\nlayers. The non zero hoppings used in this work are listed in\nTable~\\ref{hoppings}.\n\n\\begin{table}\n\\begin{tabular}{||c|c||} \\hline \\hline\n$\\epsilon_s$ &-7.3  \\\\ \\hline  $t^0_{ss}$ & 2.66 \\\\\n\\hline $t^0_{sp}$ & 4.98  \\\\ \\hline  $t^0_{pp \\sigma}$  &2.66\n\\\\ \\hline $t^0_{pp \\pi}$ &-6.38 \\\\ \\hline  $t^1_{pp \\pi}$ &0.4 \\\\ \\hline\n$t^2_{pp \\sigma}$ &0.4 \\\\ \\hline  $t^2_{pp \\pi}$ &-0.4  \\\\ \\hline\n $\\Delta_{so}$ &0.02 \\\\ \\hline \\hline\n\\end{tabular}\n\\caption{Non zero tight binding parameters, in eV, used in the\nmodel. The hoppings are taken from\\cite{TL87,TS91}, and the\nspin-orbit coupling from\\cite{SCR00}. Superindices 0,1, and 2\ncorrespond to atoms in the same layer, nearest neighbors in\ndifferent layers, and next nearest neighbors in different layers.}\n\\label{hoppings}\n\\end{table}\n\nThe hamiltonian can be written as a $32 \\times 32$ matrix for each\nlattice wavevector. We define an effective hamiltonian acting on the\n$\\pi$, or $p_z$, orbitals, by projecting out the rest of the\norbitals:\n\\begin{align}\n{\\cal H}_{\\pi}^{eff} &\\equiv {\\cal H}_{\\pi} + {\\cal H}_{\\pi \\sigma}\n\\left( \\omega - {\\cal H}_{\\sigma \\sigma} \\right)^{-1} {\\cal\nH}_{\\sigma \\pi} \\label{heff}\n\\end{align}\nWe isolate the effect of the spin-orbit coupling by defining:\n\\begin{align}\n{\\cal H}_{\\pi}^{so} \\left( \\vec{\\bf k} \\right) &\\equiv {\\cal\nH}_{\\pi}^{eff} ( \\Delta_{so} ) - {\\cal H}_{\\pi}^{eff} ( \\Delta_{so}=\n0 )\n\\end{align}\nNote that ${\\cal H}_{\\pi}^{so}$ depends on the energy, $\\omega$.\n\nWe analyze ${\\cal H}_{\\pi}^{so}$ at the $K$ and $K'$ points. The two\nmatrices have a total of 16 entries, which can be labeled by\nspecifying the sublattice, layer, spin, and valley. We define\noperators which modify each of these degrees of freedom using the\nPauli matrices $\\hat{\\sigma} , \\hat{\\mu} ,  \\hat{s}$, and\n$\\hat{\\tau}$. The unit cell is described in\nFig.~\\ref{bilayer_lattice_so}.\n\\begin{figure}\n\\includegraphics[width=5cm]{lattice_bilayer_so.pdf}\n\\caption[fig]{(Color online).  Unit cell of a graphene bilayer.\nLabels A and B define the two sublattices in each layer, while\nsubscripts 1 and 2 define the layers.} \\label{bilayer_lattice_so}\n\\end{figure}\n\nThe hamiltonian has inversion and time reversal symmetry, and it is\nalso invariant under rotations by $120^\\circ$. These symmetries are\ndefined by the operators:\n\\begin{align}\n{\\cal I} &\\equiv \\sigma_x \\mu_x \\tau_x \\nonumber \\\\\n{\\cal T} &\\equiv i s_y \\tau_x {\\cal K} \\nonumber \\\\\n{\\cal C}_{120^\\circ} &\\equiv \\left( - \\frac{1}{2}  + i\n\\frac{\\sqrt{3}}{2} s_z \\right) \\times \\left( - \\frac{1}{2} - i\n\\frac{\\sqrt{3}}{2} \\tau_z \\mu_z \\right) \\times \\nonumber \\\\ &\\times\n\\left( - \\frac{1}{2} + i \\frac{\\sqrt{3}}{2} \\tau_z \\sigma_z \\right)\n\\end{align}\nwhere ${\\cal K}$ is complex conjugation.\n\nThe possible spin dependent terms which respect these symmetries\nwere listed in\\cite{DD65}, in connection with the equivalent problem\nof three dimensional Bernal graphite (see below). In the notation\ndescribed above, they can be written as\n\\begin{align}\n{\\cal H}_{\\pi}^{so} &= \\lambda_1 \\sigma_z \\tau_z s_z + \\lambda_2\n\\mu_z \\tau_z s_z +  \\lambda_3 \\mu_z \\left( \\sigma_y s_x - \\tau_z\n\\sigma_x s_y \\right) + \\nonumber \\\\ &+ \\lambda_4 \\sigma_z \\left(\n\\mu_y s_x + \\tau_z \\mu_x s_y \\right) \\label{hamilso}\n\\end{align}\nThe first term describes the intrinsic spin-orbit coupling in single\nlayer graphene. The other three, which involve the matrices $\\mu_i$,\nare specific to bilayer graphene. The term proportional to\n$\\lambda_3$ can be viewed as a Rashba coupling with opposite signs\nin the two layers.\n\n\\section{Results}.\n\\subsection{Bilayer graphene}.\n\\begin{figure}\n\\includegraphics[width=8cm]{couplings_bilayer_so_E.pdf}\n\\caption[fig]{(Color online). Dependence on energy of the spin-orbit\ncouplings, as defined in eq.~\\ref{hamilso}.}\n\\label{couplings_bilayer_so_E}\n\\end{figure}\n\nThe energy dependence of the four couplings in eq.~\\ref{hamilso} is\nshown in Fig.~\\ref{couplings_bilayer_so_E}. The values of the\ncouplings scale linearly with $\\Delta_{so}$. This dependence can be\nunderstood by treating the next nearest neighbor interlayer coupling\nand the intratomic spin-orbit coupling as a perturbation. The\nspin-orbit coupling splits the spin up and spin down states of the\n$\\sigma$ bands in the two layers. The interlayer couplings couple\nthe $\\pi$ band in one layer to the $\\sigma$ band in the other layer.\nTheir value is of order $\\gamma_3$.  The $\\pi$ states are shifted\nby:\n\\begin{align}\n\\delta \\epsilon_{\\pi \\pm} &\\sim - \\frac{\\gamma_3^2}{\\left|\n\\epsilon_{\\sigma \\pm}\\right|} \\propto \\mp \\Delta_{so} \\left(\n\\frac{\\gamma_3}{ \\epsilon_{\\sigma}^0 } \\right)^2\n\\label{spin_hopping}\n\\end{align}\nwhere $\\epsilon_\\sigma^0$ is an average value of a level in the\n$\\sigma$ band.\n\n The model gives for the only\nintrinsic spin-orbit coupling in single layer graphene the value\n\\begin{align}\n\\left| \\lambda_1^{SLG} \\right| &= 0.0065 {\\rm meV} \\label{lambdaSLG}\n\\end{align}\nThis coupling depends quadratically on $\\Delta_{so}$, $\\delta\n\\epsilon_{\\pi \\pm} \\sim \\pm \\Delta_{so}^2 \/\n\\epsilon_\\sigma^0$\\cite{HGB06}.\n\nThe band dispersion of bilayer graphene at low energies, in the\nabsence of spin-orbit couplings is given by four Dirac cones,\nbecause of trigonal warping effects associated with\n$\\gamma_3$\\cite{MF06}. Hence, we must to consider the couplings for\nwavevectors $\\vec{\\bf k}$ slightly away from the $K$ and $K'$\npoints. We have checked that the dependence of the couplings\n$\\lambda_i$ on momentum, in the range where trigonal warping is\nrelevant, is comparable to the changes with energy shown in\nFig.~\\ref{couplings_bilayer_so_E}.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{couplings_bilayer_so_Egap.pdf}\n\\caption[fig]{(Color online). Dependence on interlayer gap, $E_g$,\nof the spin-orbit couplings, as defined in eq.~\\ref{hamilso}.}\n\\label{couplings_bilayer_so_Egap}\n\\end{figure}\n\nA gap, $E_g$, between the two layers breaks inversion symmetry, and\ncan lead to new couplings. The calculations show no new coupling\ngreater than $10^{-6}$meV for gaps in the range $-0.1 {\\rm eV} \\le\nE_g \\le 0.1 {\\rm eV}$. The dependence of the couplings on the value\nof the gap is shown in Fig.~\\ref{couplings_bilayer_so_Egap}. This\ncalculation considers only the effect in the shift of the\nelectrostatic potential between the two layers. The existence also\nof an electric field will mix the $p_z$ and $s$ orbitals within each\natom, leading to a Rashba term similar to the one induced in single\nlayer graphene\\cite{HGB06,HMING06t}.\n\n\n\nThe effect of $\\lambda_1$ is to open a gap of opposite sign in the\ntwo valleys, for each value of $s_z$. The system will become a\ntopological insulator\\cite{H88,KM05}. The number of edge states is\ntwo, that is, even. The spin Hall conductivity is equal to two\nquantum units of conductance. A perturbation which preserves time\nreversal invariance can hybridize the edge modes and open a gap.\nSuch perturbation should be of the form $\\tau_x s_y$.\n\nThe terms with $\\lambda_3$ and $\\lambda_4$ describe spin flip\nhoppings which involve a site coupled to the other layer by the\nparameter $\\gamma_1$. The amplitude of the wavefunctions at these\nsites is suppressed at low energies\\cite{MF06}. The shifts induced\nby $\\lambda_3$ and $\\lambda_4$ in the low energy electronic levels\nwill be of order $\\lambda_3^2 \/ \\gamma_1 , \\lambda_4^2 \/ \\gamma_1$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{couplings_bilayer_so_kz.pdf}\n\\caption[fig]{(Color online). Dependence on momentum perpendicular\nto the layers in Bernal graphite of the spin-orbit couplings, as\ndefined in eq.~\\ref{hamilso}.} \\label{couplings_bilayer_so_kz}\n\\end{figure}\n\\subsection{Bulk graphite}.\nThe hamiltonian of bulk graphite with Bernal stacking can be reduced\nto a set of bilayer hamiltonians with interlayer hoppings which\ndepend on the momentum along the direction perpendicular to the\nlayers, $k_z$. We neglect in the following the (small) hoppings\nwhich describe hoppings between next nearest neighbor layers,\n$\\gamma_2$ and $\\gamma_5$, and the energy shift $\\Delta$ between\natoms in different sublattices. At the $K$ and $K'$ points of the\nthree dimensional Brillouin Zone ($2 k_z c = 0$, where $c$ is the\ninterlayer distance) the hamiltonian is that of a single bilayer\nwhere the value of all interlayer hoppings is doubled. At the $H$\nand $H'$ points, where $2 k_z c = \\pi$, the hamiltonian reduces to\ntwo decoupled layers, and in the intermediate cases the interlayer\ncouplings are multiplied by $| 2 \\cos (k_z c) |$. Carrying out the\ncalculations described in the previous section, $k_z$ dependent\neffective couplings, $\\lambda_i ( k_z )$, can be defined. These\ncouplings are shown in Fig.~\\ref{couplings_bilayer_so_kz}. The\nresults for bilayer graphene correspond to $k_z c = 2 \\pi \/ 3 , 4\n\\pi \/ 3$. The layers are decoupled for $k_z c = \\pi$. In this case,\nthe only coupling is $\\lambda_1$, which gives the coupling for a\nsingle layer, given in eq.~\\ref{lambdaSLG}.\n\nThe significant dispersion as function of momentum parallel to the\nlayers shown in Fig.~\\ref{couplings_bilayer_so_kz} implies the\nexistence of spin dependent hoppings between layers in different\nunit cells. This is consistent with the analysis which showed that\nthe spin-orbit coupling in a bilayer has a contribution from\ninterlayer hopping, see eq.~\\ref{spin_hopping}.\n\nThe spin-orbit couplings can be larger in bulk graphite than in a\ngraphene bilayer. The bands in Bernal graphite do not have\nelectron-hole symmetry. The shift in the Fermi energy with respect\nto the Dirac energy is about $E_F \\approx 20 {\\rm meV} \\gg \\lambda_1\n, \\lambda_3$\\cite{DM64}. Hence, the spin-orbit coupling is not\nstrong enough to open a gap throughout the entire Fermi surface, and\ngraphite will not become an insulator.\n\n\n\\begin{figure}\n\\includegraphics[width=8cm]{couplings_bilayer_so_ky_ortho.pdf}\n\\caption[fig]{(Color online). Dependence on wavevector, $2 k_y $, of\nthe spin-orbit couplings for orthorhombic graphite, as defined in\neq.~\\ref{couplings_ortho}. The point $k_x = 0 , k_y a \\sqrt{3} = 4\n\\pi \/3$ corresponds to the $K$ point ($a$ is the distance between\ncarbon atoms in the plane).} \\label{couplings_bilayer_so_ky_ortho}\n\\end{figure}\n\nA similar analysis applies to orthorhombic graphite, which is\ncharacterized by the stacking sequence $ABCABC \\cdots$\\cite{M69}.\nThe electronic structure of this allotrope at low energies differs\nmarkedly from Bernal graphite\\cite{GNP06,AG08}, and it can be a\nmodel for stacking defects\\cite{BCP88,GNP06,AG08}. If hoppings\nbeyond nearest neighbor layers are neglected, the hamiltonian can be\nreduced to an effective one layer hamiltonian where all sites are\nequivalent. The effective hamiltonian which describes the $K$ and\n$K'$ valleys contains eight entries, which can be described using\nthe matrices $\\sigma_i , s_i$, and $\\tau_i$. Orthorhombic graphene\nis not invariant under inversion, and a Rashba like spin-orbit\ncoupling is allowed. The spin-orbit coupling takes the form:\n\\begin{align}\n{\\cal H}_{ortho}^{so} &\\equiv \\lambda_1^{ortho} \\sigma_z s_z \\tau_z\n+ \\lambda_2^{ortho} \\left( \\sigma_y s_x - \\tau_z \\sigma_x s_y\n\\right) \\label{ortho}\n\\end{align}\nAs in the case of Bernal stacking, the couplings have a significant\ndependence on the momentum perpendicular to the layers, $k_z$, and\ninterlayer hopping terms are induced. For $\\omega = 0, \\vec{\\bf k} =\n0$ and $k_z = 0$, we find:\n\\begin{align}\n\\lambda_1^{ortho} &= 0.134 {\\rm meV} \\nonumber \\\\\n\\lambda_2^{ortho} &= 0.275 {\\rm meV} \\label{couplings_ortho}\n\\end{align}\nIn orthorhombic graphite the Fermi level is away from the $K$ and\n$K'$ points, in the vicinity of a circle defined by $| \\vec{\\bf k} |\n= \\gamma_1 \/ v_F$\\cite{GNP06,AG08}. The variation of the couplings\nas function of wavevector is shown in\nFig.~\\ref{couplings_bilayer_so_ky_ortho}.\n\n\n\n\\section{Conclusions}\nWe have studied the intrinsic spin-orbit interactions in a graphene\nbilayer and in graphite. We assume that the origin of the couplings\nis the intraatomic $\\vec{\\bf L} \\vec{\\bf S}$ interaction, and we use\na tight binding model which includes the $2s$ and $2p$ atomic\norbitals.\n\nThe intrinsic spin-orbit couplings in a graphene bilayer and in\ngraphite are about one order of magnitude larger than in single\nlayer graphene, due to mixing between the $\\pi$ and $\\sigma$ bands\nby interlayer hoppings. Still, these couplings are typically of\norder $0.01 - 0.1$meV, that is, $0.1 - 1$K.\n\nBilayer graphene becomes an insulator with an even number of edge\nstates. These states can be mixed by perturbations which do not\nbreak time reversal symmetry. These perturbations can only arise\nfrom local impurities with strong spin-orbit coupling, as a spin\nflip process and intervalley scattering are required.\n\nThe interplay of spin-orbit coupling and interlayer hopping leads to\nspin dependent hopping terms. The spin-orbit interactions are\nlargest in orthorhombic graphite, which does not have inversion\nsymmetry.\n\n\\section{Acknowledgements}\nFunding from MICINN (Spain), through grants FIS2008-00124 and\nCONSOLIDER CSD2007-00010 is gratefully acknowledged.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{INTRODUCTION}\n\nAutonomous vehicles are a rapidly expanding market and research field. In recent years, the automotive industry has increased its projects and importance in this area in order to reduce the number of deaths caused by crashes \\cite{1_importance}. Furthermore, according to  \\cite{1_economic}.the autonomous vehicle market is expected to grow 36.9\\% between 2017 and 2027 and reach \\$126.8 billion dollar market value. And the fact that it is such a big economy enables it to renew itself as something. In this direction, studies are carried out for smartening and autonomy in the design of automobiles. According to a World Health Organization research, road traffic collisions are the seventh leading cause of mortality across all age categories, accounting for about 1.35 million deaths in 2016, with cyclists, motorcyclists, and pedestrians contributing for more than half of the deaths \\cite{1_who}. Considering the complex traffic situations and the increase in traffic accidents due to carelessness, misbehavior, fatigue or distraction, the improvements seen in different levels of autonomous vehicles are motivated to reduce injuries and deaths caused by complex traffic scenarios and errors. Injuries and deaths caused by traffic accidents can be reduced with more accurate Advanced Driver Assistance System(ADAS) applications. \n\n\n\nIn light of all of this knowledge, in order to be able to prevent crashes or pass through an obstacle, path planning algorithms are being developed for decades. Path planning algorithms are basically used to enable autonomous vehicles to move in environments with obstacles. Also, as mobile robots have recently started to work in dynamic environments where people are also present, path planning algorithms have gained more importance in order for autonomous vehicles to work safely. To achieve autonomous driving skills, different path planning algorithms use different approaches. While algorithms like RRT(Rapidly-Exploring Random Tree), RRT*(Rapidly-exploring random tree (star)) and PRM(Probabilistic Roadmap) use probabilistic methods for path planning, algorithms like APF use geometric approaches.\n\n\n\n\\clearpage\n\n\nWhen it comes to real-world applications, such as autonomous vehicle deployment, testing the boundaries of safety and performance on real cars is expensive, inaccessible, and dangerous. Commercially available mobile systems such as Jackal UGV(Unmanned Ground Vehicle) \\cite{1_jackal} and TurtleBot2 \\cite{1_turtle} have been developed to resolve these concerns. Several small-scale robotic platforms have been built in recent years to further research, particularly in the area of autonomous driving. Many of these platforms are built on a small-scale racecar with a mechanical framework to support the electronic components, generally 1\/10 size of an actual vehicle. In 2014, Mike Boulet, Owen Guldner, Michael Lin, and Sertac Karaman developed RACECAR (Rapid Autonomous Complex-Environment Competing Ackermann steering Robot) that was the first mobile robot with a strong graphics processing unit(GPU)  \\cite{1_sertac}. By offering realistic car dynamics with Ackermann steering and drive trains, the RACECAR platform provides as a robust robotic platform for research and education. It is built on completely open-source and standardized systems that use ROS and its related libraries.\n\n\n\\section{Purpose of The Thesis}\n\nThis project mainly focuses on implementing and testing different path planning algorithms on MIT RACECAR platform. These algorithms are expected to avoid obstacles while following the road on a high curvature road. It is expected to determine the path planning algorithm that implements this scenario most successfully and to create a starting point for the solution of complex problems such as overtaking and escape maneuver using this algorithm in the period after the thesis. A lane finding algorithm based on image processing had to be developed at the same time in order to implement the scenario. Within this project, necessary tools will be developed and, route planning algorithms will be evaluated by creating a scenario environment. \n\n\n\\clearpage\n\n\n\\section{Literature Review}\n\n\nSince path planning is a subject that has been studied for a long time, many algorithms related to path planning have been developed. DWA is one of the most popular algorithms. Fox proposed DWA algorithm in 1997 but, it is still frequently used. Also, algorithms that relies on geometric relations like APF are very successful with a low computational cost \\cite{1_apf}. Also, algorithms based on optimization of dynamic parameters like TEB are proposed by R\u00f6smann and it is shown that it can navigate complex environments \\cite{3_teb_2}. Several vision-based lane detection algorithms have been presented in the literature to reliably detect lanes. These approaches may be divided into two types: feature-based and model-based. Detecting the evident properties of lanes that distinguishes them from the rest of the objects on the lane image is the basis of the feature-based technique. The task of feature detection would be split into two parts: edge detection and line detection. To identify lane edges and model lane borders, edge-based techniques have been developed. The accuracy and computing needs of more sophisticated implementations of these models rise, while the resilience to noise decreases. For feature extraction, the Canny edge detection method is presented, which offers an exact match to lane lines and is adaptable to a complex road environment \\cite{1_canny}. Improvements to Canny edge detection can successfully deal with numerous noises in the road environment, according to \\cite{1_canny2}. Furthermore, defining the targeted region of the picture where lane borders are located, called to as region of interest (ROI), enhances performance efficiency \\cite{1_roi}. To ease the identification procedure, the region of interest would be split into left and right sub-regions based on the width of the lane.\n\n\n\\chapter{Platform Overview}\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=.6\\linewidth]{2.racecar_platform\/figures\/racecar.png}}\n  \\vspace{5mm}\n  \\caption{MIT Racecar Platform}\n  \\label{fig:racecar}\n\\end{figure}\n\nRACECAR is a full stack robotics system that provides high computing power with reliable mechanical structure. RACECAR platform serves as a strong robotic platform for research and education by providing realistic vehicle dynamics with Ackermann steering, and drive trains. It is designed based on fully open-source and standardized systems that take advantage of ROS and its associated libraries \\cite{2_racecar}. MIT RACECAR has sensors that can be listed as:\n\n\\begin{itemize}\n    \\item StereoLabs ZED Stereo Camera\n    \\item RPLidar A2 Lidar\n    \\item Sparkfun 9DoF Razor IMU\n\\end{itemize}\n\n\n\nBecause of its numerous sensors and well documented software, MIT RACECAR was chosen to be used in this project as the robot that trajectory generation algorithms will be tested on. \n\n\n\\clearpage\n\n\n\\section{Hardware}\n\n\\subsection{Chasis}\n\nChassis of MIT Racecar platform is Slash 4\u00d74 Platinum Truck model from Traxxas. These chassis is giving providing enough performance parameters and has enough space for installation of other equipments. The chassis is able to drive 40 mph with its default Velineon 3500 Brushless Motor, but VESC software the maximum speed is limited by 5 mph for safety concerns. Power required by BLDC is provided by a 2S 4200 mah LiPo battery \\cite{2_traxxas}. \n\n\n\n\\subsection{NVIDIA Jetson TX2}\nNVIDIA Jetson TX2 is high power graphical processing unit which is widely used in robotics applications. It has enough processing power for trajectory planning and other tasks. Jetson TX2 has an ARMCortexA57 CPU(Central Process Unit) and 256 CUDA enabled GPU with an 8 GB RAM(Random Access Memory) \\cite{2_jetson}.\n\n\n\\begin{figure}[h]  \n  \\centerline{\\includegraphics[width=.5\\linewidth]{2.racecar_platform\/figures\/jetson.png}}\n  \\caption{NVIDIA Jetson TX2 \\cite{2_jetson}}\n  \\label{fig:racecar}\n\\end{figure}\n\n\n\\subsection{StereoLabs ZED Stereo Camera}\nStereoLabs ZED Stereo Camera is one of the enviromental awareness sensors MIT Racecar. ZED camera comes with dual 4 MP Camera which that provides 110\u00b0 FOV(Field of View). ZED camera provides real-time pointcloud data in addition to getting 1080p HD video at 30 FPS or WVGA at 100 FPS(Frame Per Second) thanks to its SDK(Software Development Kit) \\cite{2_zed}.\n\n\\clearpage\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[height=3cm]{2.racecar_platform\/figures\/zed.png}}\n  \\caption{StereoLabs ZED Stereo Camera \\cite{2_zed}}\n  \\label{fig:racecar}\n\\end{figure}\n\n\n\n\\subsection{RPLidar A2 Lidar}\n\n\nRPLidar A2 Lidar is a low-cost 2d lidar that developed by SLAMTEC will give us the obstacle distance information which is necessary for trajecrory generation. It can measure distances with a 360 degree field of view with maximum 16 meters range \\cite{2_lidar}.\n\n\n\\begin{figure}[h]  \n  \\centerline{\\includegraphics[width=.7\\linewidth]{2.racecar_platform\/figures\/rplidar.png}}\n  \\vspace{5mm}\n  \\caption{RPLidar A2 Lidar \\cite{2_lidar}}\n  \\label{fig:racecar}\n\\end{figure}\n\n\n\\subsection{Sparkfun 9DoF Razor IMU}\n\nSparkfun 9DoF Razor IMU is reprogrammable IMU(Inertial Measurement Unit) with an MPU-9250 9DoF (9 Degrees of Freedom) sensor. Sparkfun IMU is a easy-to-use IMU with its various connection types like UART, I2C and USB. In MIT RACECAR platform it is fully entegrated by a USB connection but in this project it is not actively used \\cite{2_imu}.\n\n\n\\subsection{VESC Electronic Speed Controller}\n\nBLDC(Brushless DC Motor) and steering servo of the MIT Racecar is controlled by Vedder Electronic Speed Controller (VESC). Unlike conventional ESCs, VESC also provides useful data over USB connection like current and voltage data for each motor phase and wheel odometer data. VESC is capable of controlling speed and steering angle of MIT Racecar succeffully. These low level controllers of VESC allow researchers to be more focused on developing solutions to higher level problems.\n\n\n\n\\section{Software}\n\nThe MIT RACECAR software relies on JetPack and ROS as two main components. JetPack is a special software for NVIDIA Jetson computers that specializes Ubuntu for them. Along with specialized OS for Jetson it also provides libraries, APIs, samples and developer tools for AI applications and GPU computing \\cite{2_jetpack}. \n\n\nAs the other main component of The MIT RACECAR software, RACECAR ROS package provides robotic capabilities on ROS as subpackages and it consists of sub-packages like; \\textit{\\textbf{ackermann\\_cmd\\_mux, joy\\_node rplidar\\_ros, sparkfun-9dof-razor-imu, zed\\_ros\\_wrapper.}} These packages are required for collecting sensor data from vehicle, applying commanded control signals,  making necessary transformations between different frames. As an addition to these packages, for data collection and logging \\textit{\\textbf{rosbag}} package is used. The rosbag package allows us to save the sensor data as time-series and then play it back in real time. \\textit{\\textbf{Rosbag}} package also allows us to measure performance of developed algorithm. Also, for visualization purposes, another ROS package \\textit{\\textbf{rqt}} is used. \n\n\n\\subsection{Robot Operating System (ROS)}\n\nROS is an open-source framework that mainly handles communication between nodes and creates a baseline for researchers. Since it also has useful packages like transformation calculations etc., it is very helpful for beginners in robotics programming and accelerate prototyping processes for researchers.\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=.7\\linewidth]{2.racecar_platform\/figures\/ros-master-node.jpg}}\n  \\vspace{5mm}\n  \\caption{ROS Topics Communication System}\n  \\label{fig:ros_communication}\n\\end{figure}\n\n\\clearpage\n\nCommunication architecture of ROS is communication between nodes with a publish-subscribe messaging model. This communication is only possible with a \\textbf{\\textit{master}} which serves as an XML-RPC based server. \\textbf{\\textit{Master}} makes the communication between nodes with an address which is named \\textbf{\\textit{topic}}. \\textbf{\\textit{Topics}} are special messaging addresses with a specific data type and name. \n\nThe ROS nodes of MIT RACECAR can be seen in Table \\ref{ros_node_t}.\n\n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{@{}\n>{\\columncolor[HTML]{EFEFEF}}c \n>{\\columncolor[HTML]{EFEFEF}}c \n>{\\columncolor[HTML]{EFEFEF}}c @{}}\n\\toprule\n\\textbf{NODE NAME} &\n  \\textbf{PACKAGE} &\n  \\textbf{DESCRIPTION} \\\\ \\midrule\n\\begin{tabular}[c]{@{}c@{}}ackermann\\_cmd\\\\ \\_mux\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}ackermann\\_cmd\\\\ \\_mux\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}Ackermann velocity command inputs are \\\\ multiplexed in this node. It controls \\\\ incoming ackermann command topics but,\\\\  based on priority, allows \\\\ one topic command at  a time.\\end{tabular} \\\\ \\midrule\njoy\\_node &\n  joy &\n  \\begin{tabular}[c]{@{}c@{}}This node is used for generic joystick to ROS. \\\\ It publishes joy message that \\\\ includes joystick all possible states \\\\ of buttons and axes.\\end{tabular} \\\\ \\midrule\nrplidarNode &\n  rplidar\\_ros &\n  \\begin{tabular}[c]{@{}c@{}}Driver for RPLIDAR sensor that converts messages \\\\ that comes from lidar sensor to laser scan messages \\\\ and publishes to a topic.\\end{tabular} \\\\ \\midrule\n\\begin{tabular}[c]{@{}c@{}}zed\\_camera\\\\ \\_node\\end{tabular} &\n  zed-ros-wrapper &\n  \\begin{tabular}[c]{@{}c@{}}Driver for ZED camera that outputs pointcloud, \\\\ camera left and right images and pose \\\\ information of robot.\\end{tabular} \\\\ \\midrule\nrosbag\\_node &\n  rosbag &\n  \\begin{tabular}[c]{@{}c@{}}It is a tool for recording the messages published on \\\\ topics to bag files which are specially formatted files \\\\ that stores timestamped ROS messages.\\\\ It also used for replaying the recorded topics.\\end{tabular} \\\\ \\midrule\nrqt\\_node &\n  rqt &\n  \\begin{tabular}[c]{@{}c@{}}This node is used for visualizatioon purpose. \\\\ It provides tools to easily plot data in real-time,\\\\ displays rostopic messages etc.\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The ROS nodes of MIT RACECAR software package}\n\\label{ros_node_t}\n\\end{table}\n\n\\clearpage\n\n\n\\section{Vehicle Kinematic Model}\n\nAckermann steered vehicles like MIT RACECAR are commonly expressed as bicycle kinematic model. Bicycle kinematic model for Ackermann steered vehicles assumes that front wheels and rear wheels are combined amongst themselves and end up with a two-wheel bicycle \\cite{2_bicycle}. In addition to this, bicycle kinematic model also assumes that the vehicle only moves on XY plane and thanks to these assumptions, bicycle kinematic model is a simple geometric relationship between the steering angle and the curvature that the rear axle of the vehicle will follow.\n\n\n\\begin{figure}[h]  \n  \\centerline{\\includegraphics[height=.6\\linewidth]{2.racecar_platform\/figures\/bicycle.png}}\n      \\caption{Simple Bicycle Model \\cite{2_bicycle}}\n  \\label{fig:bicycle}\n\\end{figure}\n\n\\begin{equation}\n    tan(\\gamma) = \\frac{L}{R}\n\\label{eq_bicycle}\n\\end{equation}\n\nThe geometric relationship can be written as equation \\ref{eq_bicycle} where $\\gamma$ is steering angle, $L$ is the wheelbase and $R$ is the turning radius of the vehicle. It should be noted that this geometric relationship gives acceptable results for only low speed and moderate steering angles.\n\n\\clearpage\n\n\n\n\\chapter{PATH PLANNING}\n\nPath planning algorithms are used to enable mobile robots to perform their most basic tasks, which are their ability to move. These algorithms mainly consists of two parts which is path planning and trajectory planning. While path planning is responsible for the higher level motion of the robot, trajectory planning algorithms calculates the required actuator behavior for following via-points generated by path planning \\cite{3_global_local}. This project focuses on trajectory planning algorithms. In ROS methodology, path planning algorithms corresponds to global planner and trajectory planner corresponds to local planner. \n\n\n\\begin{figure}[h!]  \n  \\centering\n  \\centerline{\\includegraphics[width=\\linewidth, height=14cm]{3.trajectory_planning\/figures\/planners.png}}\n  \\caption{An example view of local and global planners \\cite{3_global_local_fig}.}\n  \\label{fig:costmap}\n\\end{figure}\n\n\\clearpage\n\n\nIn this project, the performance of different trajectory planning algorithms like DWA, TEB and APF, will be compared by their performance. In the realization phase of the project, the algorithms will be tested on the Racecar Platform in a curvy road. The algorithms are implemented in the ROS environment. Thanks to the useful features of ROS, implementation and testing of the algorithms could be done easily. \n\n\n\\section{Main Elements of Path Planning}\n\nPath planning processes may become complex processes. In order to make path planning more tidy and less complex, some auxiliary ROS packages will be used. Before diving into path planning algorithms, these auxiliary packages will be examined. \n\n\n\\subsection{Global Planner}\n\nGlobal planners are responsible for creating a collision free path for the robot. Global planners generate this path around the obstacles from a geometric approach, and they don't take into account  vehicles dynamics. Unlike local planners, generally they plan for a larger map. A*, Dijkstra's Algorithm, RRT, $RRT^*$, and PRM can be counted as examples of global planning algorithms. The output of Global planners is a roadmap or waypoints to the target point. These roadmap provides local planner the points to follow.  \n\n\nIn this project, \\textbf{\\textit{base\\_global\\_planner}} package from ROS navigation stack will be used as global planner. This package provides a good baseline that can be configured to use different algorithms for global planning. In this project, since global planning is not one of the main tasks of this project, the package is configured for the Dijkstra's algorithm for calculating the shortest path that has the lowest cost without including robot kinematic parameters. Dijkstra's algorithm is mainly chosen for its simplicity and being computationally cheap. Global planners are basic algorithms for choosing the path that has minimum costs. Though, their performance heavily depends on configuration of global costmap. Since there is no mapping in this project, we are using our global planner with respect to the base\\_link frame. \n\n\n\\clearpage\n\n\\subsection{Local Planner}\n\nLocal planner is the planner that calculates the short term plan and executes the plan. Unlike global planner, local planner includes robot kinematic parameters and calculates the plan according to these parameters. This extra calculation comes with a computational cost. In order to decrease the computational cost of local planners, boundaries of the local planner generally being selected small and local planner aims the furthest point that is on the global plan that is inside of local planner boundaries, not the real target point. \n\nGlobal planners produce waypoints or roadmaps as output and local planners get the output of global planners as inputs and calculates the required control signal to follow these waypoints. This control signal is generally a ROS message type named \\textbf{\\textit{\"\/geometry\\_msgs\/Twist\"}}. The ROS message contains the commanded linear and angular velocity data, and another node takes this message and controls the vehicle with respect to these commanded velocities.\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=.8\\linewidth]{3.trajectory_planning\/figures\/local_plan.png}}\n  \\vspace{5mm}\n  \\caption{Local Planner trajectory generation \\cite{3_local_planner}.}\n  \\vspace{5mm}\n  \\label{fig:planner}\n\\end{figure}\n\n\n\n\n\n\n    \n    \n    \n    \n\n\n\n\n\\subsection{Global and Local Costmaps}\n\nCosmaps are basically maps that stores the costs of every point on the map that is calculated based on sensor measurements or provided by a known map. 2D and 3D costmaps can be constructed, but for ground vehicles like the MIT RACECAR 2D costmaps are more suitable since the movement in z-axis can be neglected. \n\n\\clearpage\n\nMainly, the costs of cells in costmaps are calculated based on whether there is an obstacle on that cell. In addition to this, another source of cost is the distance of the cell to an obstacle. This \"distance\" is named \\textbf{\\textit{inflation}} in ROS Navigation stack. The inflation is calculated by parameters called \\textbf{\\textit{inflation\\_radius}} and \\textbf{\\textit{cost\\_scaling\\_factor}}. These parameters are required for deciding whether the cell will be determined as near an obstacle or not, and decay rate of inflation cost. The effect of inflation parameters on the cost of a cell can be seen in Figure \\ref{fig:inflation}.\n    \n\n\\begin{figure}[h!]  \n  \\centering\n  \\includegraphics[width=\\linewidth]{3.trajectory_planning\/figures\/inflation.png}\n  \\vspace{5mm}\n  \\caption{The effect of inflation parameters on cell cost \\cite{3_ros_costmap}}\n  \\vspace{5mm}\n  \\label{fig:inflation}\n\\end{figure}\n\n\nAlso, another important parameters of costmaps are \\textbf{\\textit{cost\\_factor}} and \\textbf{\\textit{neutral\\_cost}}. These parameters determine the smoothness and the curvature of the calculated path. The effects of these parameters can be seen in Figure \\ref{fig:cf_effect} and Figure \\ref{fig:nc_effect}.\n\n\n\\clearpage\n        \n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{3.trajectory_planning\/figures\/cf_0.01_0.55_3.55.png}}\n  \\vspace{5mm}\n  \\caption{The effect of cost factor on generated path (cost\\_factor=0.01, cost\\_factor=0.55, cost\\_factor=3.55) \\cite{3_tuning_guide}.}\n  \\vspace{5mm}\n  \\label{fig:cf_effect}\n\\end{figure}\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{3.trajectory_planning\/figures\/nc_1_66_233.png}}\n  \\vspace{5mm}\n  \\caption{The effect of neutral factor on generated path (neutral\\_factor=1, neutral\\_factor=66, neutral\\_factor=233) \\cite{3_tuning_guide}.}\n  \\vspace{5mm}\n  \\label{fig:nc_effect}\n\\end{figure}\n\n\nAnd lastly, it should be noted that, as the names suggests, global costmaps are for global planners and local costmaps are for local planners. Although, there is no difference between global costmaps and local costmaps in fact, their parameter configurations are differs. The map sizes and sensors are main elements that can be difference between them.\n\n\n\n\\clearpage\n\n\n\n\n\\section{Trajectory Planning Methods}\n\nIn this project, 3 different trajectory planning algorithms is considered and applied on MIT RACECAR platform. These algorithms are; Dynamic Window Approach(DWA), Time Elastic Band(TEB) and Artificial Potential Field(APF).\n\n\\subsection{Dynamic Window Approach}\n\nDWA is a proven concept that is used for a long time in robotics. The method was first proposed in 1999 \\cite{3_dwa}. And it is still a popular method for mobile robots. DWA provides a controller between global path and the robot. It calculates the cost function for different control inputs and searches for the maximum scored trajectory to follow. Thanks to its simplicity and easy implementation, DWA is a good choice as the first algorithm for trajectory planning. Basic working principle of DWA can be expressed as below.\n\n\\begin{enumerate}\n    \\item Take dx, dy and dtheta samples from control space \n    \n    \\item Predict next states from current states based on sampled dx, dy and dtheta.\n    \n    \\item Score each trajectory from predictions with distance to obstacles, distance to the goal, distance to the global path, and speed and remove the trajectories that collide with obstacles.\n    \n    \\item Choose the trajectory with the highest score and send the trajectory to the robot.\n    \n    \\item Repeat until goal is reached.\n\\end{enumerate}\n\n\nIn order to be able to use DWA effectively, DWA needs to be configured properly. DWA has parameters that will configure robot configuration, goal tolerance, forward simulation, trajectory scoring and global plan. One of the important parameters of the DWA is sim time parameter. Sim time is essentially the time length that will DWA plan for, and this parameter heavily affects computation time of the trajectory. In tests, it is seen that when sim time is set to low values like 2.0 seconds or less, the performance of DWA is not sufficient for passing complex pass ways because it can not see what will happen after the sim time. This result in a suboptimal trajectory. Also, it should be noted that, since all trajectories generated by DWA is simple arcs, setting the sim time to high value like 5 seconds will result in long curves that are not very flexible. Thus, in order to achieve good performance with DWA, setting sim time to an optimum value is a must.\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{3.trajectory_planning\/figures\/sim_time.png}}\n  \\caption{The Effect of Simulation Time on Generated Trajectory (sim\\_factor=1.5, sim\\_time=4.0 \\cite{3_tuning_guide}}\n  \\label{fig:planner}\n\\end{figure}\n  \n  \nAside from sim time, there are a few other factors to consider. Samples of velocity vx sample and vy sample are two parameters that determine how many translational velocity samples should be taken in the x and y directions, respectively. The number of rotational velocities samples is controlled by vth sample. The number of samples you want to take is determined by your computer's processing power. Because turning is generally more complicated than moving straight ahead, it is preferred to set vth samples to be higher than translational velocity samples in most cases. Since, our vehicle is non-holonomic there is no need for velocity samples in y direction. Simulation granularity is the step size between points on a trajectory that is referred to as sim granularity. It essentially means how often the points on this trajectory should be examined. A lower value indicates a higher frequency, which necessitates more processing power. \n\nLastly, as an addition to all of these parameters, DWA can also plan both for holonomic and non-holonomic robots, but it does not support Ackermann drive robots as MIT RACECAR. Still, DWA was implemented on MIT RACECAR and got acceptable results.  \n\n\n\n\\clearpage\n\n\n\n\n\n\n\n\n\\subsection{Time Elastic Band}\n\nTime Elastic Band(TEB) planner was proposed by R\u00f6smann as an improved version of elastic band algorithm \\cite{3_teb}. The classic elastic band algorithm is based on optimizing a global path for the shortest path length. However, TEB optimizes the path by the time-optimal objective function. Also, while elastic band does take into account of dynamic constraints, TEB considers kino-dynamic constraints in the trajectory planning. Additionally, TEB planner also supports non-holonomic car-like robots.\n\nTEB planner is essentially a solution to a sparse scalarized multi-objective optimization problem. The multi-objective problem includes constraints like maximum velocity, maximum acceleration, minimum turning radius etc. Since this optimization problem does not always have only one solution, TEB can get stuck in locally optimum points. Thus, sometimes the robot can not pass an obstacle even if there is a possible trajectory which the robot can follow. In order to solve this local minimum problem, an improved version of TEB was proposed \\cite{3_teb_2}. With this improved version, TEB optimizes a globally optimal trajectory parallel to the multi-objective optimization problem that it already solves. The algorithm switches to this new globally optimum solution when necessary, and this way the local minimum problem of TEB is solved.\n\nDue to being numerous kino-dynamic constraints for a car-like robot, TEB has numerous weights for each constraint and effects of weights on the generated trajectory depends on other weights. For this reason, optimizing TEB planner parameters requires good understanding of the concept, attention and controlled experiments. In this project, in  order to simplify this process, initial values of the TEB planner was optimized in simulation environment and final configuration fine-tuned in real time tests. The results showed that while MIT RACECAR can avoid obstacles smoothly with TEB planner, the planner still requires more tuning for getting better results while driving narrow areas. \n\n\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{3.trajectory_planning\/figures\/teb_chart.png}}\n  \\vspace{5mm}\n  \\caption{Flowchart of TEB Algorithm \\cite{3_teb}}\n  \\vspace{5mm}\n  \\label{fig:planner}\n\\end{figure}\n\n\n\\clearpage\n\n\n\n\n\\section{Artificial Potential Field Method}\n\nArtificial Potential Field(APF) Method is a basic approach for trajectory planning that is widely used by both industrial and academical applications. APF basically creates artificial attractive vectors that diverts vehicle to the goal position and repulsive vectors from obstacles that diverts vehicle from obstacles. Basically, addition of these attractive and repulsive vectors results in a target direction and speed for the vehicle.   \n\n\n\n\\begin{equation}\n    U(q) = U_{attractive}(q) + U_{repulsive}(q)\n\\end{equation}\n\n\n\nAnd additionally, magnitudes of these vectors are defined by user. By tuning these magnitudes, the user can tune how close the robot will navigate through obstacles. With the help of this simple calculation, APF calculates how much it should divert from current heading.\n\nAs an addition to determining target orientation of the vehicle, speed of the vehicle can be calculated with the help of APF method by the density of obstacles. Simply, the density of obstacles determines the targeted velocity for the vehicle. \n\n\n\n\\begin{equation}\n    V = V_{max} - K_{gain} \\cdot n_{obstacles}\n\\end{equation}\n\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=.4\\linewidth]{3.trajectory_planning\/figures\/apf.jpg}}\n  \\vspace{5mm}\n  \\caption{Basic trajectory generation of APF Method}\n  \\label{fig:planner}\n\\end{figure}\n\n\nAPF method has a very basic approach for trajectory planning. And this approach provides a non-complex solution to the problem with less computational costs. The main reason for choosing this method is this direct approach, but it has some problems that we will discuss in the next chapter. \n\n\n\\subsection{Problems of APF Method}\n\n\nThe main problem with the APF method is the local minimum problem. Similar to the previous version of TEB algorithm, APF can get stuck in local minimum points. As it can be seen in figure below, when repulsive forces from obstacles are equal in both directions, it calculates the resultant vector as the vehicle moves towards to the obstacle. This can result in a collision for the vehicle. Another problem of the APF method is Goal Non-reachable with Obstacles Nearby(GNRON) problem. In such cases that the goal point is near an obstacle, the repulsive force from the obstacle can prevent the robot from reaching the goal point by resulting in a local minimum that is close to the goal point.\n\nThese two problems are primary problems of APF method and various solutions are proposed like Evolutionary APF \\cite{3_apf_evo}, Fuzzy APF \\cite{3_apf_fuzzy} as modified APF algorithms. \n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=.7\\linewidth]{3.trajectory_planning\/figures\/gnron.png}}\n  \\caption{Goal points with GNRON and local minimum problem \\cite{3_apf_fig}}\n  \\label{fig:planner}\n\\end{figure}\n\n\nWhile implementing APF to the MIT RACECAR, these problems were solved by implementing a local minimum detection method. This method, simply checks the calculated repulsive and attractive forces and if it determines that the robot is stuck in a local minimum point, it adds a vector that will divert the vehicle from going towards to the obstacle.\n\n\n\n\n\n\n\n\\chapter{Testing and Results}\n\nReal life experimentation of trajectory planning algorithms on MIT RACECAR requires additional work. Until this part,  MIT RACECAR platform is inspected by its hardware and software. Also, numerous helper package of ROS is expressed and how trajectory planning algorithms can be applied. For the real time testing of different trajectory planning algorithms, there should be a measurement for determining how well the algorithm works. For that purpose, an environment was set up in Artificial Intelligence and Intelligent Systems(AI2S) laboratory. As the scenery, a double lane curvy road with obstacles in different locations was chosen. MIT RACECAR was expected to stay in lane and when it encounters to an obstacle, to pass the obstacle by changing the lane. \n\n\nAs a result of the chosen scenery, it is needed to detect lanes on the road. For achieving this, ZED Camera was used for image processing. Image processing in this project is achieved by OpenCV library. An image processing algorithm that detects lanes and returns goal points was designed with Python. Goal points are generated from detected lanes with respect to the look ahead distance that is determined by velocity of the vehicle dynamically.\n\nAnother aspect of the project is obstacle detection. Obstacle detection is done with the help of RPLidar A2 2D lidar. RPLidar provides users a ROS package that handles communication between computer and lidar and publishes obstacle information as a ROS topic. Lastly, all of these outputs are given to the path planning algorithm as target points and obstacles.\n\n\n\n\n\\section{Test Environment}\n\nIn order to create the required environment, a double lane curvy road was draw on the floor of AI2S Laboratory by electrical tapes. While making the road for MIT RACECAR, constraints like width of the vehicle, minimum turning radius of the vehicle are considered. Also, it is tried to avoid environmental effects like flare on the floor because of the lights. Since, the reference is being obtained from image processing, the flare is changing the quality of the reference signal in a bad manner and causes the vehicle to go out of the lane. \n\n\nAlso, the turning radius is an important constraint in such small environments, especially when obstacle avoidance is required. Thus, the curvature of the road was tried to be kept small enough to turn but also, big enough to see the limits of the algorithm.\n\nDouble lane structure of the road was chosen for lane changing when encountered by an obstacle. This structure also provides a good use case like lane changing and overtaking problems as future works. Besides, the middle lane of the road was chosen red in order to provide easier recognition of the lane. Since red color is easy to recognize with color filters like HSV, this provides a good starting point for the lane detection and leaves more time to focus on trajectory planning algorithms. The most painful part of the generated scenario for this study was that the changeable light conditions in the environment had a serious effect on the lane recognition algorithm's performance.\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/4_environment.png}}\n  \\vspace{5mm}\n  \\caption{Testing environment}\n  \\label{fig:testing_env}\n\\end{figure}\n\n\\clearpage\n\n\n\n\n\n\n\n\n\\section{Lane Detection Algorithm}\n\nLane detection algorithm, which is a crucial part of the scenery that is chosen for the project, will be explained in this section. Lane detection algorithm consists of OpenCV functions that will not be explained since it is out of the scope of the project. The algorithm is designed to have two main parts for its being modular and easy to understand.\n\n\n\\begin{figure}[h!]  \n  \\vspace{5mm}\n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/main_flow.png}}\n  \\vspace{5mm}\n  \\caption{Main structure of the lane detection algorithm}\n\\end{figure}\n\n\nThe first part of the algorithm, the pre-processing part, is responsible for extracting lane information from camera input by eliminating everything except lanes itself from the image. The first idea for the algorithm was based on the idea to extract black lanes and red lane individually. For this purpose, a complex image processing algorithm that can be seen in Figure \\ref{fig:pre_1} was proposed. However, the computational cost of this proposed method was not affordable for Jetson TX2. As a result of this problem, feedback loop of the system is updated only on 5-8 times average in one second and this situation was resulting in a hard control problem and slow response system. \n\n\n\\clearpage\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/pre_1.png}}\n  \\vspace{5mm}\n  \\caption{Flow of the first proposed image pre-processing algorithm}\n  \\label{fig:pre_1}\n\\end{figure}\n\n\nIn order to overcome these problems, a new, more plain algorithm that can be seen in Figure \\ref{fig:pre_2} was proposed. The idea of the new algorithm is that it is not needed to find all the lanes individually, it is only needed to find red lane which is easier to find respectively. As a result, the new proposed method is less accurate but more effective and faster, and this loss of accuracy is a neglectable amount.\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/pre_2.png}}\n  \\vspace{5mm}\n  \\caption{Flow of the final image pre-processing algorithm}\n  \\label{fig:pre_2}\n\\end{figure}\n\n\n\n\\clearpage\n\n\n\nThe pre-processing part of the algorithm uses mainly HSV masking, morphological operations and filtering contours based on sizes and area. The flow of the algorithm with an example input image can be inspected in Figure \\ref{fig:pre_2}. \n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/pre_example.png}}\n  \\vspace{5mm}\n  \\caption{Example output of pre-processing image with stages}\n  \\vspace{5mm}\n  \\label{fig:pre_2}\n\\end{figure}\n\n\nThe second part of the algorithm takes the binary image that is the output of the pre-processing part as input. This part of the algorithm firstly takes the perspective transformation of the lane image to the Bird-Eye view. After taking transformation, the contour information of the image is extracted and applied some filtering again. After all of these processes, a basic second degree polynomial is fit for the lane. This is required because in some cases, only a small part of the lane is visible and the coordinates of the target point at the look ahead distance is required for path planning. Additionally, some coordinate transformations is applied to the target point and the point is passed as output of the lane detection algorithm. The flow of the entire algorithm can be seen in Figure \\ref{fig:entire_flow} and an example output of the lane detection algorithm is also can be seen in Figure \\ref{fig:detect_out}.    \n\n\n\n\\clearpage\n\n\\begin{landscape}\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/entire_flow.png}}\n  \\vspace{5mm}\n  \\caption{The flow of the entire lane detection algorithm}\n  \\label{fig:entire_flow}\n  \\vspace{5mm}\n\\end{figure}\n\n\n\\clearpage\n\n\\end{landscape}\n\n\n\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/detect_out.png}}\n  \\vspace{5mm}\n  \\caption{An example output of the lane detection algorithm}\n  \\vspace{5mm}\n  \\label{fig:detect_out}\n\\end{figure}\n\n\n\\clearpage\n\n\n\\section{Testing and Results}\n\n\nIn this chapter, an overall qualitative assessment of the trajectory planning algorithms is given. This assessment relies on how successfully the vehicle followed the lanes, how many obstacles the vehicle pass through without collision and some special comments on the algorithm behavior. Since there is no global position data source in AI2S laboratory environment, an overall numeric error can not be calculated.  \n\n\nWhile testing the algorithms, it is tried to keep the environment same for all the algorithms. Nevertheless, some environmental circumstances like lighting condition can be changed. During the testing a Rviz is used which can be seen in Figure \\ref{fig:rviz} for visualizing the vehicle condition from the vehicle's perspective, global plan and local plan that is manipulated by trajectory algorithm. \n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth]{4.testing_results\/figures\/global_local.png}}\n  \\vspace{5mm}\n  \\caption{An example Rviz view with global plan (red) and local plan (purple)}\n  \\vspace{5mm}\n  \\label{fig:rviz}\n\\end{figure}\n\n\n\nIn addition, how the trajectory planning manipulates the target points which are determined by lane detection can be seen in Figure \\ref{fig:manipulator}. At the top side of the figure, how the vehicle sees the environment can be seen and below the image from camera and lane detection algorithm can be seen. While the red arrow is the output of the lane detection algorithm, the purple arrows indicates the planned trajectory that will avoid the obstacle without leaving the road.\n\n\n\\clearpage\n\n\n\n\\begin{figure}[h!]  \n  \\centerline{\\includegraphics[width=\\linewidth, height=23cm]{4.testing_results\/figures\/manipulator.png}}\n  \\vspace{5mm}\n  \\caption{An example view that trajectory planning is running (red arrow is the output of the lane detection, the purple arrows are the planned trajectory)}\n  \\label{fig:manipulator}\n\\end{figure}\n\n\n\\clearpage\n\n\nThe final assessment about the trajectory planning algorithms can be inspected in Table \\ref{tab:assessment}. But as a matter of fact, it should be said that this assessment is only according to the chosen scenery, it is not about which trajectory planning algorithm is better than the others. Besides, determining which algorithm is superior to others depends on the application scenery. As a conclusion, artificial potential field algorithm has better results according to the project requests and it is chosen to continue to the project with APF algorithm from now on. Since it provides reasonable performance with low cost, when considered possible future development of the project, the APF method is chosen.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{@{}\n>{\\columncolor[HTML]{EFEFEF}}c \n>{\\columncolor[HTML]{EFEFEF}}c \n>{\\columncolor[HTML]{EFEFEF}}c @{}}\n\\toprule\n\\textbf{\\begin{tabular}[c]{@{}c@{}}Trajectory Planning\\\\ Algorithm\\end{tabular}} &\n  \\textbf{Obstacles Avoided} &\n  \\textbf{Comments} \\\\ \\midrule\n\\begin{tabular}[c]{@{}c@{}}Dynamic Window\\\\ Approach\\end{tabular} &\n  5 out of 7 &\n  \\begin{tabular}[c]{@{}c@{}}DWA planner is easy to implement and tune. \\\\ Its performance was acceptable and stable. \\\\ Also, while avoiding obstacles, \\\\ it could stay in the course, but sometimes, \\\\ its effort was not enough to avoid obstacles. \\\\ Also, it gets stuck in complex situations\\\\  like narrow pass ways\\end{tabular} \\\\ \\midrule\n\\begin{tabular}[c]{@{}c@{}}Time-Elastic Band \\\\ Planner\\end{tabular} &\n  7 out of 7 &\n  \\begin{tabular}[c]{@{}c@{}}TEB planner is very successful to model the \\\\ vehicle, and it is suitable for complex \\\\ environments. But, it is very hard to tune \\\\ effectively and sometimes having so many\\\\  parameters to tune is turning into \\\\ a disadvantage instead of an advantage. \\\\ Also, since it heavily relies on \\\\ an optimization problem, computational \\\\ cost gets very high, especially \\\\ in complex environments.\\end{tabular} \\\\ \\midrule\n\\begin{tabular}[c]{@{}c@{}}Artificial Potential\\\\ Field\\end{tabular} &\n  5 out of 7 &\n  \\begin{tabular}[c]{@{}c@{}}APF is the easiest to implement and tune \\\\ by far. Thanks to its basic logic based \\\\ on simple math, it provides an acceptable \\\\ result with low cost. And also, \\\\ in line with the application scenery that is \\\\ selected for this project, it is not \\\\ too much deviates the vehicle from the road.\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The assessment of the experiments of the trajectory generation algorithms}\n\\label{tab:assessment}\n\\end{table}\n\\chapter{Conclusion and Future Works}\n\n\nPath planning algorithms are gaining more and more importance as autonomous vehicles become more widespread and important nowadays. There are many types of path planning applications used as the navigation unit of autonomous vehicles or for security purposes in semi-autonomous vehicles. In this project, which aims to implement and test a few of the path planning applications in real-time, a testing environment was first established and an image processing-based lane tracking algorithm was developed for this environment as reference signal to the vehicle. After the test environment was created, DWA, TEB and APF methods were implemented and tested separately. Since problems such as overtaking will be studied in the later parts of the project, it is of great importance to choose an algorithm that is predictable and low in cost. For this reason, it was deemed appropriate to continue the next stages of the project with APF. However, this does not mean that APF is better than others. However, all route planning applications have advantages and disadvantages over each other. \n\nIn addition, in the next stages of the project, it is aimed to establish an end-to-end neural network structure and to carry out an end-to-end control with this structure. It is aimed to give sensor data such as camera and lidar as input to neural network and to obtain control signals such as vehicle speed and steering angle. \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nBetavoltaic effect refers to the electric power production by a p-n junction bombarded by beta-particles that ionize the semiconductor material. Among the advantages of beta-batteries are their long service duration, amounting to years or even decades, and the possibility to use in the hard-to-reach areas. Betavoltaics and photovoltaics are related disciplines. In both cases, electric power results from the separation of electron-hole pairs produced by beta-electrons or photons by a p-n junction in the presence of a load in the circuit. In comparison to photovoltaics, publications in the field of the basic principles and applications of betavoltaic elements have been less numerous initially (see, e.g., Refs.~\\cite{Rap54, Pfa54, Rap56, Fli64, Ols73, Ols74, Olstech}), but started to attract the attention of the researchers in the recent years \\cite{And00, Bow02, Ada12, Ols12}. \n\nThe main task in betavoltaic design is the choice of a beta-source\/semiconductor combination, which should meet certain requirements. In particular, the beta-particles produced by the source must be absorbed efficiently by the semiconductor. Within the semiconductor, the diffusion length of the electron-hole pairs generated by the beta-flux should be large enough to allow them to reach the p-n junction with as little losses as possible. Because only the relatively low-energy beta-electrons are utilized effectively (with energies varying between 5 and 70 keV) for the realistic semiconductor thicknesses, three main beta-sources are presently employed in betavoltaic applications: Tritium \\Tr, Nickel $^{63}$Ni, and Promethium \\Pm. The respective mean energies of the electrons produced by these sources are 5.7, 18, and 62 keV.\n\nThe efficiency, $\\eta$, of a betavoltaic converter is proportional to the collection coefficient, $Q$, of the electron-hole pairs generated by the beta-flux.  In Refs.~\\cite{Pfa54, Olstech}, $Q$ was calculated under the assumption that the generation function of electron-hole pairs by a beta-flux $g(x) \\propto \\exp(-\\alpha x)$. In reality, the generation function is close to zero within the so-called ``dead layer'' under the front surface, and exhibits a maximum at some distance $x_m$ from the surface \\cite{Dmi78}. This implies that this exponential approximation is correct starting from some $x$-value greater than $x_m$. The emergence of the maximum in the $g(x)$ curve is due to the fact that, initially, the primary electrons pass through the semiconductor with only weak scattering. The dead layer thickness $x_m$ increases with the energy of the incident beta-electrons. For GaAs, $x_m$ is in the range 0.1 -- 1 $\\mu$m \\cite{Dmi78}.\n\nAlthough the works \\cite{Pfa54, Olstech} do report analytical expressions for $Q$ (obtained under the assumption of the absence of the dead layer), the values of $Q = 1$ and 0.7 were used in the calculations of beta-conversion efficiency \\cite{Olstech, Ols12}. While the value $Q = 1$ corresponds to the limiting conversion efficiency that is maximal in principle, the choice $Q = 0.7$ was not explained in \\cite{Olstech, Ols12}.\n\nIn this work, we derive an expression for $Q$ taking the dead layer into account, and also using the realistic values of the nonradiative Shockley-Reed-Hall (SRH) recombination lifetime, $\\tau_{SR}$, for direct-bandgap semiconductors. In such materials, the values of $\\tau_{SR}$ are usually short, and are in the range of $10^{-9}-10^{-7}$ s. We use the so obtained collection coefficient to derive the expression for the realistically attainable beta-conversion efficiency $\\eta$ of various combinations of beta-sources and direct-bandgap semiconductors. When calculating the efficiency, we focus on GaAs as a typical example. We show that decreasing $\\tau_{SR}$ and increasing the dead layer thickness leads to a strong reduction of $Q$ below 1, and to the corresponding reduction of the beta-conversion efficiency.\n\n\\section{Analysis of the collection coefficient}\nWe assume that the electron-hole pairs are generated only weakly within the dead layer, $x < x_m$, while for $x > x_m$, the generation function has the form $g(x) = I_0\\,\\exp(-\\alpha (x-x_m))$, where $I_0$ is the electron-hole pair generation rate in the $x_m$-plane, and $\\alpha^{-1}$ is the characteristic decay length. Furthermore, we assume that $d_p < x_m$ and $S_d \\ll D\/L$, $d_p$ being the junction depth, $S_d$ the recombination rate on the back surface of the base, and $L$ and $D$ the diffusion length and coefficient of the excess electron-hole pairs generated in the base region. The sketch of our structure is summarized in Fig.~\\ref{fig1}.\n\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.4]{fig1.eps}\n\\caption{Schematic illustration of a p-n junction of thickness $d = d_p + d_b$, where $d_p$ is emitter depth and $d_b$ is base thickness. The dead layer of thickness $x_m$ extends into the base region.}\n\\label{fig1}\n\\end{figure}\n\nApart from the SRH mechanism with the lifiteme $\\tau_{SR}$, the electron-hole pairs in GaAs also recombine radiatively; the characteristic time of this process is $\\tau_r = (AN_d)^{-1}$, where $A$ is the radiative recombination coefficient, and $N_d$ is the base doping concentration. Therefore, the diffusion length can be written as\n\\begin{equation}\nL = (D\\tau_b)^{1\/2}\\ ,\n\\label{1}\n\\end{equation}\nwith $\\tau_b = \\left(\\tau_{SR}^{-1} + \\tau_r^{-1}\\right)^{-1}$ being the effective lifiteme in the neutral base region.\n\nContinuity equation for the excess concentration of the electron-hole pairs, $\\Delta p_1$, within the dead layer (i.e., for $x < x_m$, region 1), where generation is negligible, has the form\n\\begin{equation}\n\\frac{d^2\\Delta p_1}{dx^2} - \\frac{\\Delta p_1}{L^2} = 0\\ ,\n\\label{2}\n\\end{equation}\nIn the rest of the semiconductor ($x > x_m$, region 2), the continuity equation for the excess electron-hole pair density, $\\Delta p_2$, is\n\\begin{equation}\n\\frac{d^2\\Delta p_2}{dx^2} - \\frac{\\Delta p_2}{L^2} = -\\frac{\\alpha I_0\\,e^{-\\alpha (x-x_m)}}{D}\\ .\n\\label{3}\n\\end{equation}\nThe equations (\\ref{2}) and (\\ref{3}) are supplemented by the boundary conditions\n\\begin{eqnarray}\n&&\\Delta p_1(x = d_p) = 0\\ ,\\ \\ \\frac{d\\Delta p_2}{dx}(x = d) = 0\\ , \\nonumber \\\\ \n&&\\Delta p_1(x = x_m) = \\Delta p_2(x = x_m)\\  ,\\nonumber \\\\\n&&\\frac{d\\Delta p_1}{dx}(x = x_m) = \\frac{d\\Delta p_2}{dx}(x = x_m)\\ .\n\\label{4}\n\\end{eqnarray}\nThe first condition reflects the fact that the electron-hole pairs are separated at the junction depth. The second one indicates the absence of surface recombination at the back of the base. The remaining two expressions are the usual continuity conditions for $\\Delta p(x)$ and $d\\Delta p(x)\/dx$ at $x = x_m$. The collection coefficient is then defined as the ratio of the current at the junction depth, $d_p$, to the pair generation rate in the plane of highest generation at $x = x_m$:\n\\begin{equation}\nQ = \\frac{D}{I_0}\\frac{d\\Delta p_1}{dx}(x = d_p)\\ .\n\\label{5}\n\\end{equation}\nThe solution of (\\ref{2}) and (\\ref{3}) that satisfies the first two conditions (\\ref{4}) can be written as\n\\begin{eqnarray}\n&&\\Delta p_1(x) = C\\sinh\\frac{x - d_p}{L}\\ ,\\nonumber \\\\\n&&\\Delta p_2(x) = C'\\cosh\\frac{x - d}{L} \\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + B\\left(e^{-\\alpha(x - x_m)} - \\beta\\,e^{-x\/L}\\right)\\ ,\\nonumber \\\\\n&&B = \\frac{\\alpha\\,I_0\\,L^2}{D\\left(1 - \\alpha^2L^2\\right)}\\ ,\\ \\ \\beta = \\alpha L\\exp{\\left[\\left(\\frac{1}{L} - \\alpha\\right)d\\right]}\n\\end{eqnarray}\nwith constants $C$, $C'$ to be determined from the remaining two conditions (\\ref{4}). This procedure yields:\n\\begin{eqnarray}\n&&Q = \\alpha L\\,\\times \\nonumber \\\\\n&&\\frac{\\alpha L\\left(\\cosh\\frac{d - x_m}{L} - e^{-\\alpha(d - x_m)}\\right) -\\sinh\\frac{d - x_m}{L}}{\\left[(\\alpha L)^2-1\\right]\\cosh\\frac{d - d_p}{L}}\\ .\n\\label{8}\n\\end{eqnarray}\nIf $d - x_m \\gg L$ and $\\alpha(d - x_m) \\gg 1$, this expression simplifies to\n\\begin{equation}\nQ = \\frac{\\alpha L}{1 + \\alpha L}e^{(d_p - x_m)\/L}\\ .\n\\label{8a}\n\\end{equation}\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig2.eps}\n\\caption{(a) Collection coefficient, $Q$, as a function of the diffusion length, $L$,  for different absorption coefficients, $\\alpha$, in the limit $\\alpha (d - x_m) \\gg 1$, $d - x_m \\gg L$, see Eq.~(\\ref{8a}). The values used, $\\alpha = 10^5, 6\\cdot 10^3$, and $6\\cdot 10^2$\\,cm$^{-1}$, approximately correspond to the respective mean beta-energies of $5.7, 20$, and $60$\\,keV for GaAs-based p-n junction \\cite{Tri67}. The dashed curves are calculated for different dead layer thicknesses, $x_m$, and $d_p = 10^{-5}$ cm. The solid curves are from the standard relation $Q = \\alpha L\/(1 + \\alpha L)$, valid in the absence of the dead layer. (b) Collection coefficient (\\ref{8a}) for different junction depth values for $x_m = 10^{-5}$\\,cm and $\\alpha = 10^5$\\,cm$^{-1}$, corresponding to the beta-particle energy of about 5.7\\,eV in the \\Tr\/GaAs combination.}\n\\label{fig2}\n\\end{figure}\n\nFig.~\\ref{fig2} shows the dependence of the collection coefficient $Q$ on the diffusion length from Eq.~(\\ref{8a}). As seen in this figure, the strongest reduction of $Q$ due to the presence of the dead layer is for the case of the \\Tr\\  beta-source. The smallest discrepancy in the $Q$-values obtained with and without taking into account the dead layer  is found for the curves corresponding to $\\alpha = 6\\cdot 10^2$\\,cm$^{-1}$, realized in the case of the \\Pm-source. In this case, to obtain $Q > 1\/2$, one would need the diffusion length $L > 35\\ \\mu$m. The values $Q \\approx 1$ can be achieved only in Si p-n junctions with long minority carrier lifetimes \\cite{Gor00}.\n\nIn Fig.~\\ref{fig2}(b), the junction depth was varied at a fixed electron energy (and thus constant $\\alpha$) and dead layer thickness. As seen in this figure, the collection coefficient increases not only upon increasing $L$, but also upon approaching the junction depth to the $x_m$-value. This effect is especially important for small diffusion length $L$.\n\nA further conclusion from Fig.~\\ref{fig2} is that collection of the electron-hole pairs generated by the electron flux will be quite efficient in the case when the diffusion length exceeds the dead layer thickness, $L > x_m$. An alternative way to increase $Q$ is to use deeper junctions with $d_p \\approx x_m$.\n\nLet us find the relation between the diffusion length and SHR lifetime $\\tau_{SR}$ for the case of GaAs. The radiative recombination coefficient $A$ in GaAs is an effective parameter defined by the relation $A = A_0(1 - \\gamma_r)$ \\cite{Din11}, where $A_0 \\approx 6\\cdot 10^{-10}$ cm$^3$\/s \\cite{Sach14}, and $\\gamma_r$ is the photon re-absorption coefficient. In our calculations, we assumed the value $A = 2\\cdot 10^{-10}$\\,cm$^3$\/s, as can be derived for poorly reflecting GaAs-based plane-parallel p-n structures without multiple reflection using the approach from \\cite{Din11}. In the work \\cite{Sach14}, it was shown that for realistic lifetimes  $\\tau_{SR}$, the open-circuit voltage $V_{OC}$ of GaAs-based p-n junctions increases with the base doping level, $N_d$, and, taking into account the interband Auger recombination, it has a maximum at $N_d \\approx 10^{17}$\\,cm$^{-3}$.\n\nLet us first assume that the GaAs p-n junction base is of p-type, and the diffusion coefficient of electron-hole pairs is 50 cm$^2$\/s. Then, for $A \\approx 2\\cdot 10^{-10}$\\,cm$^3$\/s, $N_d = 10^{17}$ cm$^{-3}$, and lifetimes $\\tau_{SR} = 10^{-9}, 10^{-8}$, and $10^{-7}$ s, diffusion length $L$ has the respective values of 2.2, 6.45, and 12.9 $\\mu$m. \n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig3.eps}\n\\caption{Collection coefficient $Q$ of a \\Tr\/GaAs betavoltaic pair as a function of the junction depth for (a) p-type base and (b) n-type base.}\n\\label{fig3}\n\\end{figure}\n\nFig.~\\ref{fig3}(a) shows the dependence of the collection coefficient, $Q$, of a pair \\Tr\/GaAs as a function of the junction depth, $d_p$, for these three values of $\\tau_{SR}$ at $x_m = 0.15\\,\\mu$m \\cite{Dmi78} and junction thickness $d = 10\\,\\mu$m. As seen in this figure, $Q$ is close to 1 for $\\Delta x = x_m - d_p < 0.1\\,\\mu$m. For $\\Delta x > 0.1\\,\\mu$m, the $Q$-value decreases with $\\Delta x$, but remains rather large.\n\nPresented in Fig.~\\ref{fig3}(b) is the collection coefficient vs. $d_p$ for the case when the base region of the p-n junction is of the n-type. In this case, for $\\tau_{SR} = 10^{-9}, 10^{-8}$, and $10^{-7}$\\,s, and $A \\approx 2\\cdot 10^{-10}$\\,cm$^3$\/s and $N_d = 10^{17}$\\,cm$^{-3}$, and taking into account that $D = 7$\\,cm$^2$\/s, the diffusion length $L = 0.83, 2.41$, and $4.83\\,\\mu$m, respectively. As seen in the figure, in this case $Q$ is also quite large. For $\\tau_{SR} = 10^{-7}$ and $10^{-8}$\\,s, $Q$ is still close to 1, while for $\\tau_{SR} = 10^{-9}$\\,s, $Q$ exceeds 0.75 even for small $\\Delta x$.\n\nIt should be noted that, because of rather strong absorption of the electrons emitted by the \\Tr-source by the auxiliary layers of a betavoltaic element (such as protection coating or contact layers), additional reduction of the beta-generated current can take place, leading to the efficiency reduction.\n\nLet us now analyze the collection coefficient for the \\Pm\/GaAs pair. In this case, according to \\cite{Tri67}, $\\alpha \\approx 600$\\,cm$^{-1}$, i.e., excess electron-hole density decays much more slowly than in the \\Tr\/GaAs case. For this the inequality $\\alpha L \\gg 1$ is alway satisfied even for the shortest lifeteme of $10^{-9}$\\,s. In contrast, for \\Pm\\ source, $\\alpha L = 1.5$ for $L = 25\\,\\mu$m, while $\\alpha L = 0.06$ for $L = 1\\,\\mu$m, so that $Q$ is always notably smaller than 1.\n\nBut this is not the only reason for the reduction of $Q$ in realistic \\Pm\/GaAs structures. When manufacturing solar cells based on the direct-bandgap semiconductors, such as GaAs, full thicknesses of p-n junctions are chosen rather small (of the order of a few $\\mu$m). Such structures were used in \\cite{And00}. In contrast, for the \\Pm\/GaAs pair used in betavoltaics, the situation might be very different, especially for large values of $L$. In this case, the product $\\alpha d$ will be small, so that for full absorption of beta-flux much thicker p-n junctions are required compared to those typically used in photovoltaics.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig4.EPS}\n\\caption{Collection coefficient of a \\Pm\/GaAs betavoltaic element as a function of junction depth for different Shockley-Reed lifetimes and element thicknesses for the case of (a) p-type base and (b) n-type base. }\n\\label{fig4}\n\\end{figure}\n\nShown in Fig.~\\ref{fig4} is the collection coefficient as a function of $d_p$ for a \\Pm\/GaAs pair calculated for different lifetimes $\\tau_{SR}$ and junction thicknesses $d$ of 10 and 100\\,$\\mu$m. In this case, according to \\cite{Dmi78}, $x_m = 3\\cdot 10^{-4}$\\,cm$^3$\/s. Panels (a) and (b) correspond to the cases of p- and n-base conduction types, respectively. As seen in the figure, rather high values of $Q \\ge 0.4$ for the \\Pm\/GaAs pair can be achieved only for the junction thickness $d \\approx 100\\,\\mu$m. Also, collection coefficient decreases dramatically as $\\tau_{SR}$ decreases.\n\nIt should be noted that similar results for the attainable $Q$ are expected for other direct-bandgap  A$_3$B$_5$ semiconductors, in particular, the ones based on the three-component compounds.\n\n\\section{Open-circuit voltage analysis}\nWhen estimating the limiting efficiency value \\cite{Olstech, Ada12}, we used the Shockley-Queisser approach \\cite{Sho61}, in which not only the current density, but also the open-circuit voltage, $V_{OC}$, is assumed to be maximal. Therefore, our next task is to calculate the open-circuit voltage, $V_{OC}$, with realistic values of $\\tau_{SR}$. It is given by the standard expression\n\\begin{equation}\nV_{OC} = \\frac{k_BT}{q}\\ln\\frac{N_d\\Delta p^*}{n_i^2}\\ ,\n\\label{11}\n\\end{equation}\nwhere $\\Delta p^* = \\Delta p(x = d_p + w)$ is the excess minority carrier density in the base at the boundary between the space-charge region and quasilinear region of thickness $w$, $N_d$ is the equilibrium density of the majority carriers in the quasineutral base region, and $n_i$ is the intrinsic charge carrier density. It is related to the effective densities of states in the conduction and valence bands, $N_c$ and $N_v$, as\n\\begin{equation}\nn_i = \\sqrt{N_cN_v}\\exp\\left(-\\frac{E_g}{2k_BT}\\right)\\ .\n\\label{17}\n\\end{equation}\n\nWe assume that both $d_p$ and $w$ are much smaller than the diffusion length $L$. This allows us to approximate \n\\begin{equation}\n\\Delta p(x = 0) \\approx \\Delta p^*\\ .\n\\end{equation} Such an approximation introduces a negligible error into $V_{OC}$ from Eq.~(\\ref{11}) in view of its logarithmic dependence on $\\Delta p^*$.\n\nWe will assume that recombination dominates in the quasineutral base region and in the space-charge region. Then, $V_{OC}$ can be obtained using the approach from \\cite{Sach14}. Taking into account the generation-recombination processes, we first write the continuity equation for the excess carrier density supplemented by the boundary conditions:\n\\begin{eqnarray}\n&&\\frac{d^2\\Delta p}{dx^2} - \\frac{\\Delta p}{L^2} - r(x)\\,\\Delta p(x)  + g(x) = 0\\ , \\nonumber \\\\\n&&\\frac{d\\Delta p}{dx}(x = d) = 0\\ ,\\nonumber\\\\\n&& D\\frac{d\\Delta p}{dx}(x = 0) = S_0\\,\\Delta p^*\\ ,\n\\label{9}\n\\end{eqnarray}\nwhere the third term describes recombination processes in the space-charge region of the abrupt junction, and the last one corresponds to the beta-induced generation. The first boundary condition is consistent with our assumption $S_d \\ll D\/L$ from the beginning of the previous section, and the second one is responsible for recombination effects in the $x = d_p + w$ plane.\n\nIntegration of the continuity equation results in the balance equation for the generation-recombination currents, according to which the current density for electronic excitation is proportional to the integral of the generation term,\n\\begin{equation}\nJ_\\beta = q\\,\\int_0^d dx\\,\\frac{\\Delta p(x)}{\\tau_b} + q\\left(S_0 + R_{SC}\\right)\\,\\Delta p^*\\ ,\n\\label{10}\n\\end{equation}\nwhere $q$ is the elementary charge. The right-hand side in (\\ref{10}) is responsible for the recombination in the bulk and on the front side of the emitter and within the space-charge region. The space-charge region recombination rate is given by \\cite{Sze}\n\\begin{eqnarray}\n&&R_{SC}(\\Delta p^*) = \\frac{L_D}{\\sqrt{2}\\tau_{SR}}\\int_{y_{pn}}^{-1} dy\\,N_d\\,\\left(1 - y + e^y\\right)^{-1\/2}\\times \\nonumber\\\\\n&&\\Big[N_d e^y + n_i e^{E_r\/k_BT} + b\\left(\\frac{n_i^2}{N_d} + \\Delta p^*\\right)e^{-y} \\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ + b n_i e^{-E_r\/k_BT}\\Big]^{-1}\\ ,\n\\nonumber\n\\end{eqnarray} \nwhere $b = \\sigma_p\/\\sigma_n$ is the ratio of the capture cross-sections of holes and electrons by a recombination level, $E_r$ is the recombination level energy measured from the middle of the bandgap,  $y_{pn}$ is the dimensionless potential at the p-n boundary, $L_D$ is the Debye length.\n\nTo evaluate the first integral in (\\ref{10}), we have employed the following approximative procedure. First, we write the solution of the continuity equation (\\ref{9}) as a sum of homogeneous and inhomogeneous parts,\n\\begin{equation}\n\\Delta p(x) = \\frac{e^{-x\/L} + e^{(x-2d)\/L}}{1 + e^{-2d\/L}}\\Delta p^* + \\Delta p_i(x)\\ ,\n\\end{equation}\nwhere the homogeneous term satisfies the first boundary condition in (\\ref{9}) and gives the value $\\Delta p(x = 0) = \\Delta p^*$. The inhomogeneous contribution $\\Delta p_i(x)$, with $\\Delta p_i(x = 0) = 0$, is notably different from zero only within a relatively thin layer below the front surface of the emitter, where the generation-recombination processes take place. Therefore, the contribution to the integral of the second term can be neglected in comparison to the integral of the homogeneous term, allowing us to write\n\\begin{equation}\n\\int_0^d dx\\Delta p(x) \\approx \\Delta p^*L\\tanh(d\/L)\\ .\n\\end{equation}\nThis approximation should produce a negligible error in $V_{OC}$ in view of its logarithmic dependence on $\\Delta p^*$. Substitution of this result into Eq.~(\\ref{11}) taking into account that $L^2 = D\\tau_b$ yields\n\\begin{equation}\nJ_\\beta = q\\Delta p^*\\left[\\frac{D}{L}\\tanh\\left(\\frac{d}{L}\\right) +S_0 + R_{SC}(\\Delta p^*)\\right]\\ .\n\\label{14}\n\\end{equation}\n\nThe current density $J_\\beta$ is inversely proportional to the energy required to create one electron-hole pair, $\\varepsilon$, which is approximately related to the bandgap $E_g$ as \\cite{Klein68}\n\\begin{equation}\n\\varepsilon = 2.8\\,E_g + 0.5\\,\\text{eV}\\ .\n\\label{10a}\n\\end{equation}\nDenoting is the current density in the case of Si ($E_g = 1.12$\\,eV) by $J_0$, the current density in the case of arbitrary bandgap can be approximated as \n\\begin{equation}\nJ_\\beta =J_0\\,Q\\cdot 3.64\\,\\text{eV}\/\\varepsilon\\ .\n\\end{equation}\nWe note that, usually, $J_0$ is in the $1$ -- $10^2\\,\\mu$A\/cm$^2$ range \\cite{Olstech}. The value of $\\Delta p^*$ found from Eq.~(\\ref{14}) should be substituted into Eq.~(\\ref{11}) to obtain the open-circuit voltage $V_{OC}$.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig5.EPS}\n\\caption{Open-circuit voltage as a function of the base doping level for different Shockley-Reed lifetimes in the case of (a) p-type base and (b) n-type base for $E_g$ = 1.43\\,eV, $T = 300$\\,K, and $J_0 = 10$\\,$\\mu$A\/cm$^2$.}\n\\label{fig5}\n\\end{figure}\n\nFig.~\\ref{fig5} shows the dependence of $V_{OC}$ of a GaAs-based p-n junction on the base doping level, $N_d$, neglecting the surface recombination, that is, $S_0 \\approx 0$. As seen in Fig.~\\ref{fig5}, $V_{OC}$ increases with $N_d$. On the one hand, the values of $V_{OC}$ for the pair \\Tr\/GaAs is notably smaller than in the solar cells \\cite{Sach14}, because the beta-produced current densities are at least two order of magnitude smaller than the short-circuit current densities in photovoltaic cells. On the other hand, the open-circuit voltages in Fig.~\\ref{fig5} exceed the values obtained experimentally in \\cite{And00}. The reason is that, in \\cite{And00}, the current density $J_0$ was of the order of $1\\,\\mu$A\/cm$^2$, whereas in our calculations, we have taken $J_0 = 10\\,\\mu$A\/cm$^2$. If the values $J_0 = 1\\,\\mu$A\/cm$^2$, $N_d = 5\\cdot 10^{16}$\\,cm$^{-3}$, and $\\tau_{SR} = 10^{-9}$\\,s are used, we obtain $V_{OC} = 0.44$\\,V, which practically coincides with the value given in \\cite{And00}.\n\n\n\\section{Refined calculation of the limiting betaconversion efficiency}\nAccording to Olsen \\cite{Olstech}, the efficiency of a betavoltaic element, $\\eta$, is\n\\begin{equation}\n\\eta = \\eta_\\beta\\,\\eta_C\\,\\eta_S\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\eta_\\beta = N_\\beta\/N_0\n\\end{equation}\nis the fraction of beta-flux that reaches the semiconductor,\n\\begin{equation}\n\\eta_C = (1 - r)\\,Q\n\\end{equation}\nis the coupling efficiency, given by the product of absorption probability of a beta-particle ($r$ is the electron reflection coefficient from the semiconductor surface) and collection efficiency $Q$ of electron-hole pairs, and, finally, the semiconductor efficiency\n\\begin{equation}\n\\eta_S = q\\,V_{OC}\\,FF\/\\varepsilon\\ ,\n\\end{equation}\nwhere $q$ is the elementary charge, $V_{OC}$ is the open-circuit voltage, $FF$ is the fill factor, $\\varepsilon$ is the energy necessary to generate one electron-hole pair from Eq.~(\\ref{10a}).\n\nLet us obtain $V_{OC}$ within the Shockley-Queisser approximation, where $\\tau_{SR} \\to \\infty$, $S_0 and R_{SC} \\to 0$, and the only recombination mechanism present is radiative recombination, characterised by the coefficient $A$. In this case, $V_{OC}^{lim}$ can be found analytically from (\\ref{10}) and (\\ref{14}):\n\\begin{equation}\nV_{OC}^{lim} = \\frac{k_BT}{q}\\ln\\frac{J_\\beta}{qAdn_i^2}\\ , \\\\\n\\label{16}\n\\end{equation}\n\nThe fill factor can be found using the expression from \\cite{Olstech}\n\\begin{equation}\nFF = \\left[v_{OC} - \\ln(v_{OC} + 0.72)\\right]\/(v_{OC} + 1)\\ ,\n\\label{18}\n\\end{equation}\nwhere $v_{OC} = V_{OC}\/k_BT$.\n\nTo calculate the limiting beta-conversion efficiency, we take $Q = 1$, $r = 0$, $\\eta_\\beta = 1$, corresponding to the bidirecional source in the terminology of \\cite{Olstech}. In this case\n\\begin{equation}\n\\eta_{lim} = \\frac{q\\,V_{OC}^{lim}\\,FF_{lim}}{2.8\\,E_g + 0.5}\\ ,\n\\label{19}\n\\end{equation}\nwhere $V_{OC}^{lim}$ is given by (\\ref{16}). \n\nWhen calculating $\\eta_{lim}$, several issues may arise. First, the parameters $N_c$, $N_v$, and $A$ are material-specific in every semiconductor. Second, when evaluating $V_{OC}^{lim}$ and $FF_{lim}$, Olsen had used, for each source, concrete current density $J_0$ of the order of $10^2\\,\\mu$A\/cm$^2$ for \\Pm\\ and $1\\,\\mu$A\/cm$^2$ for \\Tr. Finally, $V_{OC}^{lim}$ depends on the p-n junction thickness $d$. Therefore, all parameters in (\\ref{19}) must be specified. Since such key parameters as $A$, $N_c$, and $N_v$ are known only for concrete semiconductors and concrete bandgap values $E_g$, in the best-case scenario, the dependence $\\eta_{lim}(E_g)$ can be found as a set of support points for the known semiconductors with different $E_g$. Fitting this with a smooth curve might not be accurate enough.\n\nIn this work, we calculated $\\eta_{lim}$ only for the case of GaAs using Eq.~(\\ref{19}). For $A = 2\\cdot 10^{-10}\\,cm^3$\/s and $d = 10\\,\\mu$m gives for $J_0 = 10^2\\,\\mu$A\/cm$^2$ the value $\\eta_{lim} \\approx 17$\\,\\%, and for $J_0 = 1\\,\\mu$A\/cm$^2$, $\\eta_{lim} \\approx 14$\\,\\%. Note that the values of $\\eta_{lim}$ obtained here notably exceed the ones obtained by Olsen in \\cite{Olstech, Ols12}. In the rest of this work, we will use the values obtained for the \\Pm\/GaAs and \\Tr\/GaAs combinations, respectively.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig6.EPS}\n\\caption{Beta-conversion efficiency of a \\Tr\/GaAs pair as a function of junction depth for different Schokley-Reed lifetimes for the case of (a) p-type base and (b) n-type base.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig7.EPS}\n\\caption{Beta-conversion efficiency of a \\Pm\/GaAs element vs. junction depth for different Schokley-Reed lifetimes and element thcknesses for (a) p-type base and (b) n-type base.}\n\\label{fig7}\n\\end{figure}\n\n\\section{Calculation of the attainable betaconversion efficiency}\nFig.~\\ref{fig6} shows the attainable efficiency as a function of $d_p$ for the \\Tr\/GaAs combination, obtained from\n\\begin{equation}\n\\eta = \\eta_{lim}Q\\frac{V_{OC}}{V^{lim}_{OC}}\\ ,\n\\label{20}\n\\end{equation}\nwhere $\\eta_{lim} \\approx 14$\\,\\%, $Q$ is given by Eq.~(\\ref{8}), $V_{OC}$ is found from Eq.~(\\ref{11}), and $V_{OC}^{lim}$  from Eq.~(\\ref{16}).\n\nWhen plotting Fig.~\\ref{fig6}, we varied the lifetime at a constant $d = 10\\,\\mu$m. Panels (a) and (b) correspond to the base of the p- and n-type, respectively. As seen in Fig.~\\ref{fig6}, the attainable efficiency values are rather high and are in the range of (6.4 - 12.5)\\%.\n\nIt should be noted that our results for \\Tr\/GaAs pair agree well with those given in the review \\cite{Ols12} citing Refs.~\\cite{And00, Bow02, Ada12}, namely, $\\eta =$ (4 - 7) \\%. In these works, a \\Tr-source was used with the $A_3B_5$-based semiconductors. But, as evident from the figures shown, the possibilities of increasing the efficiency of \\Tr\/$A_3B_5$ betaconversion are far from being exhausted.\n\nShown in Fig.~\\ref{fig7} is the attainable beta-efficiency (\\ref{20}) as a function of $d_p$ for \\Pm\/GaAs pair with $\\eta_{lim}$ = 17\\,\\%. The $\\tau_{SR}$ values used were $10^{-9}, 10^{-8}$, and $10^{-7}$\\,s, and GaAs thicknesses were 10 and 100 $\\mu$m. Fig.~\\ref{fig7}(a) and (b) correspond to the p- and n-types of the base conductivity. As seen in this figure, $\\eta$ reduces rather strongly as $\\tau_{SR}$ is decreased. For the highest $\\tau_{SR} = 10^{-7}$\\,s, $\\eta$ decreases with decreasing $d$. The highest efficiency attainable, $\\eta = 7.25$\\,\\%, is achieved for $\\tau_{SR} = 10^{-7}$ s and $d = 100\\,\\mu$m, and the lowest value of $0.51$ \\% is realized for $\\tau_{SR} = 10^{-9}$ s and $d = 10\\,\\mu$m.\n\nThus, we conclude that a \\Pm\/GaAs-based betaconverter is not as efficient as a \\Tr\/GaAs-based one. Perhaps, the very small efficiency of the \\Pm\/GaAs battery obtained in \\cite{Fli64} is due to the small thickness of GaAs and small lifiteme $\\tau_{SR}$. The same applies also to the cases when, instead of GaAs, other direct-bandgap semiconductors are used.\n\n\\section{Conclusions}\nOur analysis, focusing on the attainable collection coefficient $Q$ and open-circuit voltage values $V_{OC}$, has revealed the following features of current collection of the GaAs-based beta-elements.\n\nEfficient collection of the electron-hole pairs generated by a beta-flux can be achieved when the diffusion length exceeds the dead layer thickness, $L > x_m$. An alternative way to increase collection coefficient is to use deep junctions, for which $d_p \\simeq x_m$.\n\nAdditional mechanisms responsible for the reduction of current generated by beta-electrons are possible, leading to smaller betaconversion efficiency. They may be due, for instance, to the strong absorption of the beta-electrons by auxiliary layers of a betavoltaic element.\n\nUsing the Shockley-Queisser approximation, we have derived the limiting betaconversion efficiency, $\\eta_{lim}(E_g)$. Our analysis has shown that, because the main parameters affecting the efficiency are very different for different semiconductors, the $\\eta_{lim}(E_g)$ curve can be build as a set of support points for semiconductors with different bandgaps, and not as a smooth curve.\n\n\\Pm\\ beta-source performs more poorly than \\Tr-source, because the electron-hole pair generation depth in the case of \\Pm-source is large, whereas the diffusion length of GaAs is small. Therefore, the majority of electron-hole pairs generated in the base recombine before reaching the p-n junction.\n\nIn the case of \\Tr-source, the picture is different. The collection coefficient is rather high, because of the small generation depth of electron-hole pairs. Therefore, the realistic betaconversion efficiency for the \\Tr\/GaAs pair will be rather high for relevant parameters (lifitemes and diffusion coefficients) of the semiconductor.\n\nSimilar results are expected also in the case, when other direct-bandgap semiconductors are used instead of GaAs.\n\n\\acknowledgments\nM.E. would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe cost of lattice QCD simulations with dynamical fermions is dominated by the solution of the Dirac equation in both the ensemble generation phase, where configurations of gauge fields are generated, and the measurement phase, where expectation values of physical observables are measured. \nThe Dirac matrix, which is the gauge field dependent discretization of the fermionic part of the continuous QCD action, is a large sparse linear system and inverting the corresponding Dirac equation poses tremendous numerical difficulty. \nFor domain wall fermions(DWF) the conjugate gradient(CG) algorithm proves to be a stable algorithm to solve the Dirac equation but the convergence rate is limited by the condition number of the Dirac matrix, which is typically large in simulations with physical pion mass.  \n\nFor the measurement phase various eigen-space methods, including EigCG\\cite{Stathopoulos2010} and implicitly restarted Lanczos algorithm with Chebyshev polynomial\\cite{YSaad1980}, have been developed successfully to speed up the inversion.\nLow-lying eigenvectors(eigenvectors corresponding to small eigenvalues) of the Dirac matrix are generated and the previously large condition number is effectively reduced to improve the convergence rate of CG.\nIn this phase for one gauge field configuration typically a large number of Dirac equations with the same Dirac matrix but different right hand sides(RHS, or sources) are solved. The large number of sources amortizes the cost of eigenvector generation and the total computation time is reduced.\n\nThis is not the case for the ensemble generation phase. During a typical hybrid Monte Carlo(HMC) evolution of a gauge field as few as one Dirac equation is solved for a single Dirac matrix. This renders it not worthwhile to generate the low-lying eigenvectors for a particular Dirac matrix.\n\nThe development of supercomputers has greatly increased the number of floating point operations per second(flops) that can be performed on each processor(node).\nModern lattice simulations usually divide the gauge field and pseudo-fermion fields into sub-fields that are stored and computed locally on different processors of a large parallel computer.\nThis increases the total theoretical floating point operation capability.\nInter-processor data transfer(communication), however, is needed to perform coherent operations, including the Dirac matrix multiplication. Computations locally performed on one processor require contents of the sub-fields that are stored and updated on other processors.\nFor a specific operation if the rate of communication could not keep up with the local flops then communication becomes the bottleneck and the high flops are not utilized. \n\nFor standard CG solver with DWF one Dirac matrix multiplication is performed for each iteration.  The precise requirement varies with the size of the lattice and processor grid, but roughly this requires one byte of communication for each local floating point operation.\nOn some of the newest machines, for example the SUMMIT machine at Oak Ridge National Laboratory(ORNL), inter-processor communication speed is much less than the  requirement set by their high local floating point operation capability.\n\nIn \\cite{Luscher2004} a domain decomposition algorithm is proposed for Dirac equation with Wilson fermion. Local inversions are performed on two halves of the lattice iteratively. However, attempts to apply the same or similar algorithms to the inversion of the DWF Dirac equation have not been successful. \n\n\nIn this work we report on our investigation into a preconditioned CG solver for solving the DWF Dirac equation for the ensemble generation phase of the simulation. We find a preconditioner that decreases the number of CG iterations needed for a solution, while increasing the local computation required per iteration, thus changing the balance of local computation to off-processor communication.\n\n\n\n\\section{Method}\n\\subsection{Multisplitting Algorithm}\nIn \\cite{OLeary1985} a \\textit{multisplitting} algorithm is proposed for solving generic large linear systems distributed across a parallel computer.\nCompared to the domain decomposition algorithm in \\cite{Luscher2004}, it does not require checkerboarding.\nBefore each iteration the boundary content of the solution field on each of the processors is communicated to its neighbors.\nDuring each iteration, the algorithm uses this communicated neighboring solution field as the Dirichlet boundary condition to perform the inversion of a local matrix on each processor. \nAfter each iteration, the updated boundary content is again communicated to prepare for the next iteration.\n\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{ms_dec.pdf}\n\t\\caption{Decomposition of the matrix $A$, the solution vector $x$ and the right-hand-side(RHS) vector $b$ into local parts on each node.}\\label{fig:ms_dec}\n\\end{figure}\nFollowing \\cite{Jezequel2012}, suppose the equation to be solved is $Ax=b$. For a \\textit{particular} processor the matrix $A$ and vectors $x$ and $b$ are decomposed according to figure \\ref{fig:ms_dec}, where $x_s$ and $b_s$ are the part that is locally stored on this processor. On each processor the original equation turns into\n\\begin{equation}\\label{local}\n\tA_sx_s+A_lx_l+A_rx_r=b_s.\n\\end{equation} \nThe $A_lx_l+A_rx_r$ part involves off-processor content and is calculated before each iteration via communication. $A_s$ is the part of the matrix that requires only the locally stored part of $x$ on a certain processor $s$, i.e. $x_s$. Then for each iteration the algorithm solves the equation \n\\begin{equation}\\label{eq:ms}\n\tA_sx_s=b_s-A_lx_l-A_rx_r\n\\end{equation}\nlocally for $x_s$ on this processor. The updated solution $x_s$ will then be communicated to the neighboring processors. This whole procedure can be done concurrently on all nodes once the communication work to calculate $A_lx_l+A_rx_r$ is done.\n\n\n\n\\subsection{Domain Wall Fermions}\nThe domain wall fermion(DWF)\\cite{Jansen1996} formulation is based on Wilson fermion and a fictitious fifth dimension. Modern numerical implementations of DWF utilize the fact that only the matrix elements that connect the \\textit{even} sites to \\textit{odd} sites and those connecting \\textit{odd} sites to \\textit{even} sites depend the gauge field. The matrix entries that connect \\textit{even} sites to \\textit{even} sites and those connect \\textit{odd} sites to \\textit{odd} sites are constant. Here the even-odd parity is defined by the 4D components of a site:\n\\begin{equation}\n\\mathrm{parity}\\equiv (x+y+z+t)\\mod 2.  \n\\end{equation}\nIn the 4D even-odd preconditioning form the M\\\"obius DWF Dirac equation can be written as, \n\\begin{equation}\n \\begin{pmatrix}\n  M_5 & M^4_{eo} \\\\\n  M^4_{oe} & M_5 \\\\\n \\end{pmatrix}\n  \\begin{pmatrix}\n  \\psi_e \\\\\n  \\psi_o\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n  \\phi_e \\\\\n  \\phi_o\n  \\end{pmatrix},  \n\\end{equation}\nwhere the subscript $e\/o$ refer to even and odd sites. This is equivalent to solving the following even-odd preconditioned equation,\n\\begin{equation}\\label{dirac_equation}\nD_{PC}\\psi_e=\\hat\\phi_e,\\ D_{PC}\\equiv M_5-M^4_{eo}M_5^{-1}M^4_{oe}, \\hat\\phi_e\\equiv\\phi_e-M^4_{eo}M_5^{-1}\\phi_o. \n\\end{equation}\nHere $M^4_{eo\/oe}$ includes the Wilson hopping term $D^w_{x,y}$ that connects 4D space-time sites to their nearest neighbors,\n\\begin{equation}\nM^4_{oe\/eo}=D^w_{x,y}M_\\phi,\\ D^w_{x,y}\\equiv\\sum_\\mu\\left[(1+\\gamma_\\mu)U^\\dagger_{x-\\hat{\\mu},\\mu}\\delta_{x-\\hat\\mu,y}+(1-\\gamma_\\mu)U^\\dagger_{x,\\mu}\\delta_{x+\\hat{\\mu},y}\\right],\n\\end{equation}\nand $M_5$ and $M_\\phi$ are constant matrices that are diagonal in the four Euclidean space-time dimensions. Details of these matrices can be found in \\cite{Brower2014}.\n\nThe CG algorithm requires the matrix to be hermitian and positive definite. A common practice is to multiply both sides of (\\ref{dirac_equation}) with $D^\\dagger_{PC}$ and solve the equation with the normal operator $D^\\dagger_{PC}D_{PC}$ and the new RHS $D_{PC}^\\dagger\\hat\\phi_e$ instead,\n\\begin{equation}\\label{normal_equation}\nD^\\dagger_{PC}D_{PC}\\psi_e=D_{PC}^\\dagger\\hat\\phi_e.  \n\\end{equation}\n\n\\subsection{Dirichlet Boundary Condition on the 4-Hop Normal Operator}\nThere are four Wilson hopping terms, one in each $M_{eo\/oe}^4$, in the normal operator $D^\\dagger_{PC}D_{PC}$,\n\\begin{equation}\\label{eq:DdagD}\n  D^\\dagger_{PC}D_{PC}=\\big[M_5-\\textcolor{red}{M^4_{eo}}M_5^{-1}\\textcolor{red}{M^4_{oe}}\\big]^\\dagger\\big[M_5-\\textcolor{red}{M^4_{eo}}M_5^{-1}\\textcolor{red}{M^4_{oe}}\\big].\n\\end{equation}\n\nTo apply the multisplitting algorithm to equation (\\ref{normal_equation}) Dirichlet boundary conditions are to be enforced on the normal operator $D^\\dagger_{PC}D_{PC}$, i.e. the local part(the $A_s$ in (\\ref{local})) of this normal operator needs to be constructed. As the vector content is distributed across the processors according to its 4D space-time location, this local part for $D^\\dagger_{PC}D_{PC}$ includes \\textit{snake} terms that hop out of the boundary and hop back in as the various components in (\\ref{eq:DdagD}) are evaluated. Figure \\ref{fig:snake} illustrates this and gives some examples of the snake terms. These terms are truncated if Dirichlet boundary conditions are enforced on each of the four $M^4_{eo\/oe}$ hopping terms sequentially. Our simulation results show that the inclusion of these snake terms is crucial to the convergence.\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{snake.pdf}\n\t\\caption{The normal operator $D^\\dagger_{PC}D_{PC}$ has as many as $4$ Wilson hopping terms. Enforcing Dirichlet boundary condition on it requires the inclusion of the \\textit{snake} terms, e.g. the black arrows.}\\label{fig:snake}\n\\end{figure}\n\n\\subsection{Multisplitting Algorithm as a Preconditioner of CG}\nIn \\cite{Luscher2004} to achieve faster convergence the domain decomposition algorithm is eventually used as a preconditioner of GCR. In this work we use the multisplitting algorithm as a preconditioner of CG.\n\nPseudocode for a generic preconditioned CG is shown below, where we are solving $Ax=b$ and $M$ is the preconditioning matrix. The preconditioning step is marked with blue background. The overall convergence rate of preconditioned CG is estimated by the condition number of $AM^{-1}$. If the condition number of $AM^{-1}$ is smaller then that of the original matrix $A$, faster convergence rate is achieved.\n\\begin{algorithm}\n\\setstretch{1.15}\n\\caption{Preconditioned Conjugate Gradient $Ax=b$}\n\\begin{algorithmic}\n\\State ${r}_0 = {b} - {A x}_0$\n\\State ${z}_0 = {M}^{-1} {r}_0$ \n\\State ${p}_0 = {z}_0$ \n\\State $k = 0$  \n\\While {have not converged}\n\\State $\\alpha_k = {\\langle{r}_k,{z}_k\\rangle}\/{\\langle{p}_k,{A p}_k \\rangle}$ \n\\State ${x}_{k+1} = {x}_k + \\alpha _k {p}_k$ \n\\State ${r}_{k+1} = {r}_k - \\alpha _k {A p}_k$ \n\\State \\colorbox{blue!30}{${z}_{k+1} = {M}^{-1} {r}_{k+1}$\n\\State $\\beta _k = {\\langle {z}_{k+1}, {r}_{k+1}\\rangle}\/{\\langle {z}_k,{r}_k \\rangle}$ \n\\State ${p}_{k+1} = {z}_{k+1} + \\beta _k {p}_k$ \n\\State $k = k + 1$ \n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nNow for this preconditioning step we use the multisplitting algorithm to solve for $z_{k+1}$ in\n\\begin{equation}\n  Az_{k+1}=r_{k+1}.\n\\end{equation}\nTo avoid inter-processor communication, a zero initial guess($x_l=x_r=0$) is used in (\\ref{eq:ms}) and only the first iteration is performed. With $r_{k+1}$ as the RHS and $z_{k+1}$ the solution,\n\\begin{equation}\n  A_s x_s = b_s -A_lx_l - A_r x_r \\rightarrow A_s z_{k+1,s} = r_{k+1,s}.\n\\end{equation}\nThis is equivalent to using the local part of the matrix $A$, $A_s$, on each processor as the preconditioner $M$ in the preconditioned CG,\n\\begin{equation}\n  M=\\bigoplus_s A_s,\\ s=\\mathrm{node\\ index}.\n\\end{equation}\nThe local nature of $A_s$ makes it possible to perform the preconditioning step concurrently on all the processors without communication. We refer to this as multisplitting preconditioned CG(MSPCG).\n\n\n\\section{Results}\n\nThe multisplitting preconditioned CG is applied to solve Dirac equations on three 2+1 flavor lattice ensembles generated with M\\\"obius domain wall fermions, all with physical input quark masses. Standard CG is used to perform the inversion in the preconditioning step. Instead of adopting a precision based stopping condition, a fixed number of CG iterations, which will be referred as \\textit{inner iterations}, are performed for these preconditioning solves. The iterations performed in the overall preconditioned CG will be referred as \\textit{outer iterations}. In table \\ref{table:result} the numbers of outer iterations needed for the preconditioned CG to converge are reported on the different lattice ensembles, together with the stopping condition for the outer CG(precision) and the processor grid size used. The numbers of iterations to reach the same precision with standard CG are also included for comparison, where the inner iteration number is marked with \\textit{plain}.\n\nTypically on these ensembles with $6$ inner iterations the preconditioned CG reduces the outer iteration count by a factor of $3$. More inner iterations reduce the outer iteration count more but the reduction saturates as the inner iteration count increases: with large number of inner iterations the inner CG solves the preconditioning inversion completely and no further numerical benefit can be exploited.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.5}\n\\centering\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\nlattice size & $a^{-1}[\\mathrm{GeV}]$ & precision & processor grid size & inner iterations & outer iterations \\\\\n\\hline\n\\hline\n\\multirow{4}{*}{$32^3\\times 64$} & \\multirow{4}{*}{$1.37$} & \\multirow{4}{*}{$10^{-8}$} & $-$ & plain & $13594$ \\\\\n&& & $2^3\\times 4$ & $3$ & $9106$ \\\\\n&& & $2^3\\times 4$ & $4$ & $6020$ \\\\\n&& & $2^3\\times 4$ & $6$ & $5126$ \\\\\n\\hline\n\\hline\n\\multirow{4}{*}{$64^3\\times 128$} & \\multirow{4}{*}{$2.36$} & \\multirow{4}{*}{$10^{-10}$} & $-$ & plain & $18092$ \\\\\n&&& $4^3\\times 8$ & $6$ & $6008$ \\\\\n&&& $4^3\\times 8$ & $12$ & $5083$ \\\\\n&&& $4^3\\times 8$ & $18$ & $4948$ \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{$80^2\\times96\\times 192$} & \\multirow{2}{*}{$3.00$} & \\multirow{2}{*}{$10^{-10}$} & $-$ & plain & $16783$ \\\\\n&&& $4^2\\times 8^2$ & $6$ & $5719$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Number of outer iterations need to converge the multisplitting preconditioned  CG for the lattice ensembles tested in this work. \\textit{Inner iterations} refers to the fixed number of CG iterations performed for the preconditioning inversion.  Rows marked with \\textit{plain} indicate the iteration count for the same standard CG to converge.}\\label{table:result}\n\\end{table}\n\n\\section{Conclusion}\nOur results show the MSPCG reduces the number of outer iterations needed to solve the DWF Dirac equation, reducing the inter-processor communication at the expense of performing more local inner iterations. \nWe observe that executing a fixed number of inner CG iterations for the preconditioning inversion, instead of using a precision based stopping condition, does not jeopardize the convergence of the outer CG. \nThis is true even when as few as $3$ inner iterations are performed.\nAs a consequence the inner iteration count is a parameter that can be tuned to achieve maximum speed up in the trade-off between inter-processor communication and local computation. \n\nWe note that while the multisplitting algorithm can split the general matrix $A$ in a variety of ways, the splitting presented here, used as a preconditioner in CG, makes it equivalent to the additive Schwarz algorithm. (The additive Schwarz algorithm has been used for the Dirac equation inversion for the  fermions\\cite{Osaki2010, Babich2011}.) We use the name MSPCG, as it is through the process of applying the multisplitting algorithm to the DWF Dirac equation that we realize the necessity of including the snake terms in the local matrix.\n\n\n\n\\urlstyle{tt}\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCoagulation processes arise in various areas of physics; one may think of\npolymerisation, growth of ordered domains in non-equilibrium magnetic systems \n\\cite{bray}, dynamics of droplets when water condenses on non-wetting \nsurfaces \\cite{dgy}, etc. \nThe substance, or ``mass'' that aggregates is very frequently not conserved\nduring the process: for example, \nagglomerating insoluble inclusions in molten metal\nmay be lost from the melt by attachment to the wall of the vessel \\cite{wmg}.\nTherefore the theoretical investigation of the kinetics of such non-conserving\ncoagulation processes is of great importance.  \nMoreover, the models developed for the description of such systems may \nshow interesting behaviour: the Smoluchowsky equation with certain coagulation\nkernels exhibits gelation transition and, in general, even the simplest\nmodels with conserved mass may have non-trivial solutions, see\ne.g. Ref. \\cite{wmg} and references therein. \nBeside quite realistic ones, \nthere is a special class of (possibly non-conserving) coagulation models  \nwhere only the actually smallest one among the masses is active \nwhile the other masses are temporarily inert.   \nThis type of {\\it extremal dynamics} can be regarded as a rough \napproximation for models where the reaction rates are \ndecreasing functions of the mass of particles.  \nIn what follows, we shall survey three processes with extremal\ndynamics in detail. \nWe mention, however, that models of this type have also been \nintroduced in the\ncontext of dynamics of growing and coalescing droplets \\cite{dgy} or\nmultispecies pair annihilation reactions \\cite{dht}.  \n\nIn the one-dimensional Glauber-Ising model started from a random initial state\nat zero temperature, the domain walls move as independent random walkers and\nannihilate upon meeting. While the closest pairs of walls come together and\nannihilate, the other domain walls hardly move. A simplified model of\nevolution of distances $X_i$ between adjacent walls can be formulated as\nfollows \\cite{nagai,kon}. The shortest interval $X_m$ is eliminated\ntogether with the two adjacent intervals $X_1$ and $X_2$ and replaced by \n$\\tilde X=X_1+X_2+X_m$. \nAs the density of walls tends to zero,\nthe distributions of intervals at different times become self-similar,\ndepending on a single time-dependent length scale, and the corresponding\nscaling function can be calculated exactly \\cite{nagai,bdg,rutenberg}.  \nAnother quantity of interest is the fraction of space which has never been\ntraversed by a domain wall. The length $Y_i$ of such parts of intervals\ntransforms in the way $\\tilde Y=Y_1+Y_2$ when the shortest interval is\neliminated. \nThe characteristic value of $X$ depends on\nthe fraction $c$ of the initial intervals that have not yet been eliminated \nas $X\\sim c^{-\\alpha}$, obviously, with $\\alpha=1$, while \nit has been found that $Y\\sim c^{-\\beta}$, where the persistence \nexponent $\\beta=0.824924 12\\dots$ is the zero of a \ntranscendental equation \\cite{bdg}.   \nIn addition to this, the autocorrelation exponent has also been exactly\ncalculated in this model \\cite{bd}. To obtain this quantity, the overlap $Z_i$\nof an interval with its initial state that transforms as $\\tilde\nZ=Z_1+Z_2-Z_m$ had to be considered.\nLater, a generalisation of persistence has been studied in the\nsame model, which required to introduce an auxiliary variable transforming \nas $\\tilde Y=Y_1+Y_2+pY_m$ \\cite{mb}. Here, \nthe generalised persistence exponent has\nbeen found to vary monotonically with the partial survival factor $p$\nin the range $-1\\le p\\le 1$.\n\nThe next example is the strong disorder renormalisation group transformation \nof inhomogeneous quantum spin chains \\cite{mdh}. Here, the degrees of freedom\nrelated to the largest coupling (a bond between neighbouring spins or a local\nexternal field) are eliminated one after the other. \nIn terms of logarithmic couplings, $X_i$, the\nrenormalisation rule generally reads as $\\tilde X=X_1+X_2-X_m$, \nwhere $\\tilde X$ is a newly formed effective variable and $X_1$,$X_2$ \nare variables adjacent to the smallest one, $X_m$.\nFor the relation between these variables and the couplings in the particular \n Hamiltonians we refer the reader to Ref. \\cite{fisher}.   \nA variable $Y_i$ that transforms according to  the rule \n$\\tilde Y=Y_1+Y_2$ under such a\nrenormalisation step can be interpreted in the case of a particular model, the\ntransverse field Ising chain, as the magnetic moment of a spin. \nFor this process with i.i.d. random initial variables\n$X_i$, which corresponds to critical spin chains, the distribution of $X$\nflows again to a fixed point where it shows scaling behaviour. \nThe characteristic value of $X$ increases in the course of the process \nas $X\\sim c^{-\\alpha}$ with \n$\\alpha=1\/2$, while the variable $Y$ grows as $Y\\sim c^{-\\beta}$ with\n$\\beta=(1+\\sqrt{5})\/4=0.809016\\dots$ \\cite{fisher}.         \nNote that the coagulation rules in the above two models \ndiffer only in the sign of $X_m$, \nwhich leads to different exponents $\\alpha$ and $\\beta$.\n \nOur third example is a random graph where three edges emanate from each\nnode, and which is built on a regular one-dimensional lattice by adding long\nedges in the following way. To each edge of the one-dimensional\nlattice that we call short edges, a random weight $X_i$ is assigned. \nDefining the length of a path as the sum\nof weights of the edges it contains, the closest pair of nodes of degree 2\nwith respect to this metric is chosen and connected by an edge of unit\nweight. This step is then iterated until all nodes become of degree 3 \n\\cite{juhasz}. \nFor this graph, a renormalisation procedure can be formulated \nwhere loops are eliminated \nstep by step in reversed order compared to the construction procedure. \nFormally, the short edge with the minimal weight $X_m$ is eliminated\ntogether with the nodes it connects, as well as with the \nneighbouring short edges with \nweights $X_1$, $X_2$ and a new effective short edge is formed with a weight\ncalculated asymptotically as $\\tilde X=X_1+X_2$.  \nAccording to numerical results, \nthe characteristic value of effective weights grows as \n$X\\sim c^{-\\alpha}$ with $\\alpha=0.826(1)$ \\cite{juhasz}.\nThis exponent characterises at the same time the \ndiameter of finite graphs with $N$ nodes with respect \nto the above metric via $D(N)\\sim N^{\\alpha}$. \n\nAs can be seen, these seemingly different problems can be treated in a common\nframework and can be interpreted as coagulation\nprocesses with extremal dynamics. \nIn the first example, the total sum of the variables $X_i$ is\nconserved while in the latter two cases it is not. \nWe will study in this work a coagulation model \ncontrolled by a parameter $\\omega$ that interpolates \ncontinuously between the first two models and incorporates the third one as a\nspecial case, as well. \nWe are interested in the exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$\nfor intermediate values of the parameter $\\omega$ and shall\nprovide accurate estimates for $\\alpha_{\\omega}$ that is obtained \nas the root of a \ntranscendental equation while $\\beta_{\\omega}$ is accurately determined \nby the numerical analysis of a system of non-linear differential equations. \nWe shall see that $\\alpha_{\\omega}$ varies monotonically between \nthe corresponding values of\nthe two marginal models, while, \nunlike the generalised persistence exponent of the model with\npartial survival mentioned above \\cite{mb}, the exponent $\\beta_{\\omega}$\nshows a maximum when $\\omega$ is varied. \nAs can be seen, the transformation rule of the variable \n$Y$ does not depend directly on the\nparameter $\\omega$ but it is influenced indirectly via the correlations \nemerging between $X$ and $Y$, the strength of which is controlled\nby $\\omega$. Therefore our results may contribute to the\nunderstanding of the role of correlations in such models.       \nMoreover, these investigations provide an accurate estimate for the diameter\nexponent of the graph quoted above, for which we obtain $\\alpha=0.82617561$ in\nagreement with the previous numerical result.   \n\nThe rest of the paper is organised as follows. In Section \\ref{model},\nthe model and its continuum description is introduced. \nIn Sections \\ref{asec} and \\ref{bsec}, the way of approximative\ndetermination of the exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$\nis presented. Some calculations are given in the Appendix. Finally,\nresults are discussed in Section \\ref{disc}. \n\n\\section{The model and its continuum formulation}\n\\label{model}\n\n\\subsection{Definition of the model}\n\nLet us consider a finite set of positive \nvectors $V_i=(X_i,Y_i)$ indexed by the\nintegers $i=1,2,\\dots,N$. We assume, moreover, that $N$ is odd. \nThe vectors are independent, identically distributed random variables \ndrawn from a continuous distribution $\\rho(X,Y)dXdY$, for which we require \nthat all moments exist.\nThe first components $X_i$ and the second components $Y_i$ are called\nprimary and secondary variables, respectively.  \nAssume, furthermore, that $\\omega\\in [-1,1]$ is a fixed real number. \nNow, the following procedure is considered on this set. \nThe vector $V_m$ with the smallest\nprimary variable is chosen and, at the same time, two further vectors $V_i$ and\n$V_j$ are chosen at random from the set. These three vectors are removed\nand a new vector $\\tilde V$ with components\n\\begin{eqnarray}\n\\tilde X=X_i+X_j+\\omega X_m \\nonumber \\\\\n\\tilde Y=Y_i+Y_j \n\\label{rules}\n\\end{eqnarray}\nis added to the set.\nThereby the number of vectors in the set is reduced by\ntwo. Note that the vectors remain independent after such an operation and\nthat \n\\begin{equation}\n\\tilde X\\ge X_i,X_j,X_m\n\\label{ineq}\n\\end{equation}\neven for $\\omega=-1$.   \nThis step is then iterated until a single vector $V_N=(X_N,Y_N)$ is left in\nthe set.\nIn this general formulation, the cases $\\omega=1,-1,0$ correspond to\nthe three models in the order as they were quoted in the Introduction.  \nBased on the known asymptotical behaviour of $X_N$ and $Y_N$ for \nlarge $N$ in the marginal cases $\\omega=-1,1$, \nwe expect\n\\begin{equation}\nX_N\\sim N^{\\alpha_{\\omega}} \\quad {\\rm and} \\quad Y_N\\sim N^{\\beta_{\\omega}}\n\\label{powerlaw}\n\\end{equation} \nto hold also for intermediate parameter values $-1<\\omega<1$\nwith some exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$ that may \ndepend on $\\omega$. \n\n\\subsection{Continuum formulation}\n\nNow, we consider the continuum limit $N\\to\\infty$ and introduce \nthe probability density $P_{\\Gamma}(X)$ of the primary variable \nthat has the support \n$\\Gamma\\le X<\\infty$ and that depends on the lower boundary \n$\\Gamma$ as a parameter. \nThe function $P_{\\Gamma}(X)$ is normalised as $\\int_{\\Gamma}^{\\infty}P_{\\Gamma}(X)dX=1$ for any $\\Gamma$.\nFollowing Ref. \\cite{bdg}, we consider, furthermore, the expected value \n$\\overline{Y}_{\\Gamma}(X)$ of the secondary\nvariable under the condition that the primary variable is $X$.        \nIn the continuum limit, the system is described by these two\nfunctions of $X$, which depend on the lower boundary of the support \n$\\Gamma$ as a parameter.\nThe inequality (\\ref{ineq}) implies that,\nas the fraction of vectors $c_{\\Gamma}$ that have not yet been eliminated decreases in\nthe course of the coagulation process, the lower edge $\\Gamma$ \nof the distribution continuously increases.          \nAs it is shown in the Appendix, one may write \nthe following differential equation for $P_{\\Gamma}(X)$:\n\\begin{equation}\n\\frac{\\partial P_{\\Gamma}(X)}{\\partial \\Gamma}=P_{\\Gamma}(\\Gamma)\n\\Theta[X-(2+\\omega)\\Gamma]\\int_{\\Gamma}^{X-(1+\\omega)\\Gamma}P_{\\Gamma}(X')P_{\\Gamma}(X-X'-\\omega\\Gamma)dX',\n\\label{Pdiff}\n\\end{equation}\nwhere $\\Theta(X)$ is the Heaviside step function. \nThe fraction $c_{\\Gamma}$ \nis related to $\\Gamma$ as \n$dc_{\\Gamma}\/c_{\\Gamma}=-2P_{\\Gamma}(\\Gamma)d\\Gamma$ or, equivalently,\n\\begin{equation} \n\\frac{dc_{\\Gamma}}{d\\Gamma}=-2P_{\\Gamma}(\\Gamma)c_{\\Gamma}.\n\\label{cdiff}\n\\end{equation}\nThe function $Q_{\\Gamma}(X)$ defined as \n\\begin{equation}\nQ_{\\Gamma}(X)\\equiv P_{\\Gamma}(X)\\overline{Y}_{\\Gamma}(X),\n\\label{Qfunc}\n\\end{equation}\ncan be shown to obey the differential equation  \n\\begin{equation}\n\\frac{\\partial Q_{\\Gamma}(X)}{\\partial \\Gamma}=2P_{\\Gamma}(\\Gamma)\\Theta[X-(2+\\omega)\\Gamma]\\int_{\\Gamma}^{X-(1+\\omega)\\Gamma}Q_{\\Gamma}(X')P_{\\Gamma}(X-X'-\\omega\\Gamma)dX'.\n\\label{Qdiff}\n\\end{equation} \nThe derivation of this equation is given again in the Appendix. \n\n\\subsection{Fixed point solution}\n\nIn the marginal cases $\\omega=-1,1$, it is known that, for any well-behaving\ninitial distributions $\\rho(X,Y)$ with finite moments, the \nsolutions of Eqs. (\\ref{Pdiff}) and (\\ref{Qdiff}) tend to a universal \nfixed point solution $P^*_{\\Gamma}(X)$, $Q^*_{\\Gamma}(Y)$ \nin the limit $\\Gamma\\to\\infty$ that has the scaling property \n\\begin{eqnarray} \nP^*_{\\Gamma}(X)=\\Gamma^{-1}f(X\/\\Gamma)  \\nonumber \\\\\nQ^*_{\\Gamma}(X)=\\Gamma^{\\delta_{\\omega}-1}g(X\/\\Gamma),\n\\label{fp} \n\\end{eqnarray}\nwith some number $\\delta_{\\omega}$ that is related to the growth\nexponents as\\footnote{This can be seen from the equation \n$\\overline{Y}^*_{\\Gamma}(\\Gamma)\\equiv\nQ^*_{\\Gamma}(\\Gamma)\/P^*_{\\Gamma}(\\Gamma)=\\Gamma^{\\delta_{\\omega}}g(1)\/f(1)$\nthat indicates the asymptotical relation $Y\\sim X^{\\delta_{\\omega}}$ between the\ntypical values of primary and secondary variables.} \n\\begin{equation} \n\\delta_{\\omega}=\\beta_{\\omega}\/\\alpha_{\\omega}.\n\\label{delta}\n\\end{equation} \nTherefore we expect this to hold also for intermediate parameter values  \n$-1<\\omega<1$ with some (a priori unknown) exponent $\\delta_{\\omega}$ \nthat may depend on $\\omega$. \nIndeed, the functions in Eq. (\\ref{fp}) solve Eqs. (\\ref{Pdiff}) and\n(\\ref{Qdiff}) provided that the universal \nscaling functions $f(x)$ and $g(x)$\nsatisfy the following differential equations:\n\\begin{eqnarray}\n\\frac{d[xf(x)]}{dx}=-f_1\\Theta(x-2-\\omega)\\int_1^{x-1-\\omega}f(x')f(x-x'-\\omega)dx'\n\\label{fdiff}\n\\\\\n\\frac{d[x^{1-\\delta_{\\omega}}g(x)]}{dx}x^{\\delta_{\\omega}}=-2f_1\\Theta(x-2-\\omega)\\int_1^{x-1-\\omega}g(x')f(x-x'-\\omega)dx',\n\\label{gdiff}\n\\end{eqnarray}\nwhere the notation $f_1\\equiv f(1)$ has been used. \nFor an alternative derivation of these equations in the case $\\omega=1$, \nsee Ref. \\cite{bdg}.\nUsing the fixed point solution, Eq. (\\ref{cdiff}) can be integrated yielding\nthe asymptotic relation in the large $\\Gamma$ limit: \n\\begin{equation} \n\\Gamma\\sim c_{\\Gamma}^{-\\frac{1}{2f_1}}.\n\\end{equation} \nComparing this with Eq. (\\ref{powerlaw}), we obtain the relation:\n\\begin{equation} \n\\alpha_{\\omega}=\\frac{1}{2f_1}.\n\\label{exprel}\n\\end{equation}\n\n\\section{Approximative determination of $\\alpha_{\\omega}$} \n\\label{asec}\n\nAs can be seen, Eq. (\\ref{fdiff}) does not contain $g(x)$ and together\nwith Eq. (\\ref{exprel}) it constitutes an autonomous problem for the calculation of the exponent $\\alpha_{\\omega}$. \nFor the special case $\\omega=-1$, the solution of Eq. (\\ref{fdiff}) is of\nsimple form: $f(x)=e^{-x+1}$; this yields $\\alpha_{-1}=\\frac{1}{2}$. \nIn the other marginal case, $\\omega=1$, \nthe Laplace transform of the solution is\nknown \\cite{nagai,rutenberg} and $\\alpha_{1}=1$. \nIn the case $-1<\\omega<1$, where Eq. (\\ref{fdiff}) is not soluble, we shall\nconstruct an approximative solution that enables us to give an \naccurate estimate\nof $\\alpha_{\\omega}$. An alternative way related to the numerical analysis of\nthe Laplace transforms is presented in the next section. \n\nSome properties of the scaling function $f(x)$ can be easily established\nby investigating Eq. (\\ref{fdiff}) without knowing the exact solution.\nApparently, the r.h.s. of Eq. (\\ref{fdiff}) and, as a consequence, \n$f(x)$ is non-analytical at $x=x_1\\equiv 2+\\omega$. \nBut, as $f(x)$ itself appears on the r.h.s. as a convolution with a shifted\nargument $x-1-\\omega$, the r.h.s. as well as $f(x)$ \nmust be non-analytical also at \n$x=x_2\\equiv x_1+1+\\omega$. Iterating this argument, it turns out that there\nare infinitely many points where $f(x)$ is non-analytical.\nTo be precise, one can show by recursion that the \n$2n$th derivative of $f(x)$ is discontinuous at\\footnote{For a more direct way to this result in the case $\\omega=1$, where the explicit\nform of the scaling function $f(x)$ is available, see Ref. \\cite{rutenberg}.} \n\\begin{equation} \nx_n=1+(1+\\omega)n, \\qquad n=0,1,2,\\dots.\n\\end{equation} \nFurthermore, the function value of $f(x)$ at some $x'$ is determined \nby $f(x)$ in the restricted domain $(1,x'-1-\\omega)$. \nDue to this property, $f(x)$ can be constructed \nin the intervals $[x_n,x_{n+1}]$ step by step \nstarting with $n=0$. However, the solution is more and more complicated for\nincreasing $n$ as it contains multiple integrals that \ncannot be evaluated analytically. \nIn the domains $[x_n,x_{n+1}]$, $n=1,2,\\dots$, the function $f(x)$ can be written in the following form:\n\\begin{equation} \nf(x)=\\frac{1}{x}\\sum_{i=0}^nf_1^{2i+1}C^{(2i+1)}_{\\omega}(x),\n\\quad  x_n\\le x\\le x_{n+1},   \n\\label{series}\n\\end{equation}\nwhereas $f(x)=0$ if $x<x_0$.\nThe functions $C^{(2i+1)}_{\\omega}(x)$ are independent of $f_1$ and the first\nthree of them read as \n\\begin{eqnarray} \nC^{(1)}_{\\omega}(x)=1, \\\\ \\nonumber \nC^{(3)}_{\\omega}(x)=-2\\int_{x_1}^x\\frac{\\ln (x'-\\omega\n  -1)}{x'-\\omega}dx', \\\\ \\nonumber\nC^{(5)}_{\\omega}(x)=-2\\int_{x_2}^x\\int_{x_0}^{x'-x_2+1}\\frac{C^{(3)}_{\\omega}(x'-x''-\\omega)}{x''(x'-x''-\\omega)}dx''dx'.\n\\end{eqnarray}\nSubstituting an exponential trial function in Eq. (\\ref{fdiff}), we obtain that\nthe asymptotical solution $f_{\\infty}(x)$ in the limit $x\\to\\infty$ is \nof the form  \n\\begin{equation} \nf_{\\infty}(x)=\\frac{a}{f_1}e^{-a(x+\\omega)},\n\\label{asymp}\n\\end{equation}\nwith the some number $a$ that is not fixed by this substitution. \nThe graph of $f(x)$ for $\\omega=0$ is shown in Fig.\\ref{fig1}.\n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig1.ps}\n\\caption{\\label{fig1} Graphs of the functions $f(x)$, $g(x)$ and $f_{\\infty}(x)$ for $\\omega=0$. The former two are obtained by\nnumerical integration of Eqs. (\\ref{fdiff}) and (\\ref{gdiff}), whereas the\nlatter is given in Eq. (\\ref{asymp}).}  \n\\end{figure}\nAs can be seen, $f(x)$ tends rapidly to $f_{\\infty}(x)$ for\nincreasing $x$. \nThis suggests an approximation for $f(x)$ in which $f(x)$ is replaced by\n the simple asymptotical function $f_{\\infty}(x)$ for large $x$. \nTo be precise, the $n$th ($n=0,1,2,\\dots$) approximant $f^{(n)}(x)$ is defined as \n\\begin{eqnarray}\nf^{(n)}(x)=f(x), \\qquad {\\rm if} \\quad x\\le x_n \\nonumber \\\\\nf^{(n)}(x)=f_{\\infty}(x), \\qquad {\\rm otherwise}.\n\\end{eqnarray}      \nThe two unknown parameters $f_1$ and $a$ are determined by the requirements\nthat $f^{(n)}(x)$ is continuous at $x=x_n$, i.e. \n\\begin{equation} \nf(x_n)=f_{\\infty}(x_n),\n\\end{equation}\nand that it is normalised as \n\\begin{equation} \n\\int_{x_0}^{x_n}f(x)dx+\\int_{x_n}^{\\infty}f_{\\infty}(x)dx=1.\n\\end{equation}\nUsing the expression in Eq. (\\ref{series}), straightforward calculations\nresult in \nthat the $n$th approximant  $f_1^{(n)}$ ($n>0$) is the root of the\nfollowing transcendental equation: \n\\begin{eqnarray} \n\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+1}C^{(2i+1)}_{\\omega}(x_n)+ \\nonumber \\\\\n+\\frac{x_n}{x_n+\\omega}\n\\left[1-\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+1}N^{(2i+1)}_{\\omega}(x_n)\\right]\n\\ln\\left[f_1^{(n)}-\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+2}N^{(2i+1)}_{\\omega}(x_n)\\right]=0,\n\\nonumber \\\\\n\\label{trans}\n\\end{eqnarray}\nwhere the function $N_{\\omega}^{(2i+1)}(x)$ has been introduced as\n\\begin{equation} \nN_{\\omega}^{(2i+1)}(x)\\equiv\\int_{x_i}^{x}\\frac{C^{(2i+1)}_{\\omega}(x')}{x'}dx'.\\label{nint}\n\\end{equation}\nWe have numerically calculated the root of Eq. (\\ref{trans}) \nand the $n$th approximant $\\alpha_{\\omega}^{(n)}$ of $\\alpha_{\\omega}$ \nby using Eq. (\\ref{exprel}) for $n=1,2,3$ and\nfor several values of $\\omega$. This has necessitated \nthe numerical evaluation of\nthe integrals in Eq. (\\ref{nint}) for $n>1$. \nResults are shown in Fig. \\ref{fig2} and some numerical values are given in\nTable I. As can be seen, the approximants $\\alpha_{\\omega}^{(n)}$ converge\nrapidly with increasing $n$ and they increase monotonically with $\\omega$. \nThe best estimate for the diameter exponent of the graph cited in the\nIntroduction is $\\alpha_0^{(3)}=0.82617561$. \n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig2.ps}\n\\caption{\\label{fig2} The third approximant $\\alpha_{\\omega}^{(3)}$ \nof the exponent $\\alpha_{\\omega}$ plotted against $\\omega$.}\n\\end{figure}\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|r||c|c|c|c|c|}\n\\hline $\\omega$ & $\\alpha_{\\omega}^{(1)}$ &$\\alpha_{\\omega}^{(2)}$\n&$\\alpha_{\\omega}^{(3)}$ & $\\delta_{\\omega}$ \n& $\\beta_{\\omega}=\\delta_{\\omega}\\alpha_{\\omega}^{(3)}$\\\\\n\\hline\n\\hline -0.9 & 0.54752760 & 0.54752815 & 0.54752815 & 1.48973578 & 0.81567227\\\\\n\\hline -0.8 & 0.59036797 & 0.59037862 & 0.59037860 & 1.38841226 & 0.81968889\\\\\n\\hline -0.7 & 0.62906729 & 0.62911723 & 0.62911708 & 1.30687751 & 0.82217896\\\\\n\\hline -0.6 & 0.66421085 & 0.66434418 & 0.66434376 & 1.23995279 & 0.82375490\\\\\n\\hline -0.5 & 0.69632781 & 0.69659263 & 0.69659189 & 1.18399594 & 0.82476197\\\\\n\\hline -0.4 & 0.72586754 & 0.72630756 & 0.72630667 & 1.13643762 & 0.82540221\\\\\n\\hline -0.3 & 0.75320237 & 0.75385266 & 0.75385199 & 1.09543864 & 0.82579860\\\\\n\\hline -0.2 & 0.77863838 & 0.77952410 & 0.77952421 & 1.05965756 & 0.82602872\\\\\n\\hline -0.1 & 0.80242716 & 0.80356405 & 0.80356560 & 1.02809664 & 0.82614309\\\\\n\\hline  0.0 & 0.82477635 & 0.82617193 & 0.82617561 & 0.99999999 & 0.82617561\\\\\n\\hline 0.1 &  0.84585830 & 0.84751339 & 0.84751989 & 0.97478484 & 0.82614953\\\\\n\\hline 0.2 &  0.86581708 & 0.86772721 & 0.86773715 & 0.95199472 & 0.82608118\\\\\n\\hline 0.3 &  0.88477397 & 0.88693068 & 0.88694457 & 0.93126697 & 0.82598219\\\\\n\\hline 0.4 &  0.90283179 & 0.90522369 & 0.90524198 & 0.91230963 & 0.82586098\\\\\n\\hline 0.5 &  0.92007834 & 0.92269199 & 0.92271501 & 0.89488491 & 0.82572374\\\\\n\\hline 0.6 &  0.93658910 & 0.93940970 & 0.93943769 & 0.87879701 & 0.82557503\\\\\n\\hline 0.7 &  0.95242937 & 0.95544130 & 0.95547441 & 0.86388324 & 0.82541834\\\\\n\\hline 0.8 &  0.96765597 & 0.97084323 & 0.97088154 & 0.85000684 & 0.82525595\\\\\n\\hline 0.9 &  0.98231868 & 0.98566518 & 0.98570869 & 0.83705175 & 0.82508918\\\\\n\\hline 1.0 &  0.99646128 & 0.99995110 & 0.99999976 & 0.82492447 & 0.82492427\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table1} Approximants of the exponents $\\alpha_{\\omega}$, \n$\\delta_{\\omega}$ and $\\beta_{\\omega}$ for different values of $\\omega$.} \n\\end{table}\n\n\n\\section{Numerical determination of $\\beta_{\\omega}$} \n\\label{bsec}\n\nNext, we turn to the determination of the exponent $\\delta_{\\omega}$ (and, at\nthe same time, $\\beta_{\\omega}$ through Eq. (\\ref{delta})), which\nrequires the analysis of the full problem, i.e. the system of differential\nequations (\\ref{fdiff}) and (\\ref{gdiff}). \nPrior to this, \na few remarks concerning the scaling function $g(x)$ are in order. \nFirst, as a consequence of the definition in Eq. (\\ref{Qfunc}), $g(x)$ apparently \ninherits the singularity properties of $f(x)$ discussed in the previous\nsection. Furthermore, it can be written in a form analogous to\nEq. (\\ref{series}). In the domain $[x_0,x_1]$, it has a simple form:\n\\begin{equation} \ng(x)=g(1)x^{\\delta_{\\omega}-1}, \\qquad  x_0\\le x\\le x_1.\n\\label{power}\n\\end{equation}\nSecond, the differential equation (\\ref{gdiff}) gives the scaling function\n$g(x)$ only up to a multiplicative constant. \nThis non-universal constant depends\non the initial distribution $\\rho(X,Y)$ and it is fixed in a \nnon-trivial way by the original equations (\\ref{Pdiff}) and \n(\\ref{Qdiff}) that are valid for any $\\Gamma$.  \nThird, the equation (\\ref{gdiff}) contains the a priori unknown \nparameter $\\delta_{\\omega}$ that must be fixed by physical\nconsiderations about the solution that depends on $\\delta_{\\omega}$.\nNamely, the physically acceptable solution must be nonnegative and must have\nthe only reasonable asymptotics allowed by Eq. (\\ref{gdiff}): \n\\begin{equation}\ng_{\\infty}(x)\\simeq const\\cdot xe^{-ax},\n\\end{equation}\nwhere the number $a$ is the same as that appears in Eq. (\\ref{asymp}). \nNumerical analysis of Eq. (\\ref{gdiff}) shows that these requirements are\nfulfilled only for a single value of the parameter $\\delta_{\\omega}$. \n\nFollowing Ref. \\cite{bdg}, it is, however, simpler to analyse \nthe Laplace transform of\nthe equations (\\ref{fdiff}) and (\\ref{gdiff}). \nIntroducing the functions \n\\begin{equation}\n\\phi(p)=\\int_1^{\\infty}e^{-px}f(x)dx, \\qquad \n\\psi(p)=\\int_1^{\\infty}e^{-px}g(x)dx,\n\\label{laplace}\n\\end{equation}\nthe equations (\\ref{fdiff}) and (\\ref{gdiff}) transform to  \n\\begin{eqnarray}\np\\phi'(p)=f_1[e^{-\\omega p}\\phi^2(p)-e^{-p}], \n\\label{phidiff} \\\\\np\\psi'(p)=-\\delta_{\\omega}\\psi(p) -g_1e^{-p} + 2f_1e^{-\\omega p}\\psi(p)\\phi(p),\n\\label{psidiff}\n\\end{eqnarray}\nwhere the prime denotes derivation by $p$ and $g_1\\equiv g(1)$. \nThese equations are not soluble in the parameter range $-1<\\omega<1$ but\nasymptotical expressions of the solution can be established. \nThe functions $\\phi(p)$ and $\\psi(p)$ have the small-$p$ expansions:\n\\begin{equation} \n\\phi(p)=\\sum_{n=0}^{\\infty}a_np^n, \\qquad \n\\psi(p)=g_1\\sum_{n=0}^{\\infty}b_np^n. \n\\label{psiseries}\n\\end{equation}\nSubstituting these into Eqs. (\\ref{phidiff}) and (\\ref{psidiff}), we obtain  \nthat the expansion coefficients for $-1<\\omega<1$ are given by \n$a_0=1$, $b_0=\\frac{1}{2f_1-\\delta_{\\omega}}$ and by the following recursion\nrelations for $n>0$\\footnote{These series expansions are also valid for $\\omega=-1$ with \n$a_2=5\/2$, and for $\\omega=1$ with  $a_1=-2e^{\\gamma}$ \\cite{bdg}, where $\\gamma$ is Euler's constant, given by $\\gamma=-\\int_0^{\\infty}\\ln t e^{-t}dt=0.577215\\dots$.}: \n\\begin{eqnarray}\na_n=\\frac{\\frac{(-1)^n}{n!}(\\omega^n-1) + \n\\sum_{0\\le i,j,k<n;~ i+j+k=n}\\frac{(-\\omega)^i}{i!}a_ja_k}{\\frac{n}{f_1}-2} \\\\\nb_n=\\frac{\\frac{(-1)^n}{n!}\\left(\\frac{\\omega^n}{2f_1-\\delta_{\\omega}}-\\frac{1}{2f_1}\\right)\n  + \\frac{1}{2f_1-\\delta_{\\omega}}a_n +\\sum_{0\\le i,j,k<n;~\n    i+j+k=n}\\frac{(-\\omega)^i}{i!}a_jb_k}{\\frac{n+\\delta_{\\omega}}{2f_1}-1}. \n\\end{eqnarray}\n\n\nNext, we discuss the large-$p$ behaviour of $\\psi(p)$. \nThe differential equation (\\ref{psidiff}) can have two kinds of \nasymptotical solutions depending on the parameter $\\delta_{\\omega}$.\nIf the second term on the r.h.s. dominates, we obtain \n\\begin{equation} \n\\psi'(p)\\simeq -g_1\\frac{e^{-p}}{p},\n\\label{a1}\n\\end{equation}  \nwhile, if the first term dominates, we obtain\n\\begin{equation} \n\\psi(p)\\simeq const\\cdot p^{-\\delta_{\\omega}}.\n\\label{a2}\n\\end{equation}\nOn the other hand, it follows from Eq. (\\ref{laplace}) that \n$\\psi(p)$ must have the large-$p$ asymptotics:\n$\\psi(p)\\simeq g_1e^{-p}\/p[1+O(1\/p)]$. Using that the function $g(x)$ is\nexplicitely known in the domain $1\\le x\\le 2+\\omega$ (see \nEq. (\\ref{power})), we can obtain a more accurate asymptotical form \nfor $1\/p\\ll 1+\\omega$. Replacing $g(x)$ by the function in Eq. (\\ref{power})\nin the entire domain $x\\ge 1$, the integral in Eq. (\\ref{laplace}) can be\nevaluated, yielding\n\\begin{equation} \n\\psi_{\\infty}(p)= g_1e^{-p}\\sum_{k=0}^{\\infty}\n\\left(\\begin{array}{c}\n\\delta_{\\omega}-1 \\\\ \nk\n\\end{array}\\right)\n\\frac{k!}{p^{k+1}}, \\qquad p\\gg \\frac{1}{1+\\omega}.\n\\label{asympfull}\n\\end{equation}\nThis shows that the physically acceptable asymptotics is that given \nin Eq. (\\ref{a1}) whereas that in Eq. (\\ref{a2}) is non-physical. \nNumerical analysis of the differential equations (\\ref{phidiff}) and\n(\\ref{psidiff}) with the correct value of $f_1$ \nshows the following behaviour of the solution when the parameter $\\delta_{\\omega}$ is varied: \nFor small enough $\\delta_{\\omega}$, the function \n$\\psi(p)$ is non-monotonous and tends to zero from below \nfor increasing $p$ as given in\nEq. (\\ref{a2}); for large enough $\\delta_{\\omega}$, the function $\\psi(p)$  \ndecays monotonically to zero again with the asymptotics given in\nEq. (\\ref{a2}). These parameter regimes are separated by a ``critical'' value\nof $\\delta_{\\omega}$. At this value, the solution decays monotonically to zero\nwith the physically acceptable asymptotics given in Eq. (\\ref{a1}).\n  \nThe numerical estimation of $\\delta_{\\omega}$ is based on this scenario: \nThe differential equations (\\ref{phidiff}) and\n(\\ref{psidiff}) are integrated from $p=0$ to some large $p$ \nand the true value of\n$\\delta_{\\omega}$ is selected by the condition that $\\psi(p)$ has the correct\nasymptotics. \nWe have assumed here that $f_1$ is already at our disposal. This can be\nobtained either by the approximative procedure described in the \nprevious section or, analogous to the above method, from the condition that\n$\\phi(p)$ has the correct asymptotics given by Eq. (\\ref{asympfull}) \nwith $\\delta_{\\omega}=0$.           \n\nBefore presenting numerical results on $\\delta_{\\omega}$, we show that,  \nin the case $\\omega=0$, the assumption on the uniqueness of the value\n$\\delta_{\\omega}$ that corresponds to the correct asymptotics \nimplies that $\\delta_0=1$.\nFor $\\omega=0$, the primary and the secondary variables coagulate according to\nthe same rules, see Eq. (\\ref{rules}). If these variables are initially\nperfectly correlated, i.e. $X_i=bY_i$ with some common constant \n$b$ for all $i$,  it is obvious that $\\alpha_0=\\beta_0$. \nNevertheless, this equality holds for general initial distributions, as well. \nIndeed, it is easy to check that for $\\omega=0$, the function\n\\begin{equation} \n\\psi(p)=-\\frac{g_1}{f_1}\\phi'(p)\n\\label{psi0}\n\\end{equation}\nsolves Eq. (\\ref{psidiff}) provided that\n$\\delta_0=1$ and $\\phi(p)$ is the solution of\nEq. (\\ref{phidiff}). \nIn terms of the scaling functions, equation (\\ref{psi0}) reads as \n$g(x)=xf(x)\\frac{g_1}{f_1}$. \nAs the asymptotics of the solution in Eq. (\\ref{psi0}) is \nphysically acceptable, we conclude that \n\\begin{equation} \n\\delta_0=1.\n\\end{equation}  \n\nThe details of the numerical determination of $\\delta_{\\omega}$ are the\nfollowings. The best approximant that we have, $f_1^{(3)}$, has been\nsubstituted in Eqs. (\\ref{phidiff}) and (\\ref{psidiff}) and a trial value \nfor $\\delta_{\\omega}$ has been chosen.  \nAs the derivatives $\\phi'(p)$ and $\\psi'(p)$ calculated from these equations \nare of the form $0\/0$ at $p=0$, the functions $\\phi(p)$ and $\\psi(p)$ \nwere first calculated at $p=0.05$ by using the small-$p$ expansion in\nEq. (\\ref{psiseries}) in order to avoid numerical uncertainties of the\nintegration in the vicinity of $p=0$. \nThen, starting from $p=0.05$, the differential equations were integrated by\nthe Bulirsch-Stoer method \\cite{numrec} to some $p_f$, where $\\psi(p_f)$ is\ncompared to the asymptotical form in Eq. (\\ref{asympfull}). \nIn practice, we have monitored the derivative $\\psi'(p_f)$ rather than\n$\\psi(p_f)$ and $p_f=12$ was sufficiently large so that the asymptotical\nvalue (at the critical $\\delta_{\\omega}$) is reached within the numerical\naccuracy of the integration. \nThe true $\\delta_{\\omega}$ was then\nselected by the condition $\\psi'(p_f)=\\psi_{\\infty}'(p_f)$. \nThe exponent $\\delta_{\\omega}$ determined in this way is plotted against \n$\\omega$ in Fig. \\ref{fig3} whereas \n$\\beta_{\\omega}=\\delta_{\\omega}\\alpha^{(3)}_{\\omega}$ is plotted against\n$\\omega$ in Fig. \\ref{fig4}.\nSome numerical values can be found in Table \\ref{table1}. \nAs can be seen, $\\delta_{\\omega}$ decreases monotonically with $\\omega$ but \n$\\beta_{\\omega}$ has a maximum at $\\omega=0$. The latter observation will\nbe explained in the next section. \n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig3.ps}\n\\caption{\\label{fig3} Numerically calculated exponent $\\delta_{\\omega}$ \nplotted against $\\omega$.}\n\\end{figure}\n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig4.ps}\n\\caption{\\label{fig4} Numerically calculated exponent \n$\\beta_{\\omega}=\\delta_{\\omega}\\alpha^{(3)}_{\\omega}$ \nplotted against $\\omega$. The horizontal lines indicate known values in the\nmarginal cases $\\omega=-1,1$.}\n\\end{figure}\n \n\\section{Discussion}\n\\label{disc}\n\nWe have shown that two problems, the renormalisation group procedure of certain\ndisordered quantum spin chains and a simple model describing coarsening in\nthe Glauber-Ising model can be deformed into each other by varying a single\nparameter. \nThe interpolating model is a special type of non-conserving \ncoagulation process where only the actually smallest mass is active.   \nIn the range $-1\\le\\omega\\le 1$, the exponents $\\alpha_{\\omega}$ and \n$\\beta_{\\omega}$ that characterise the\ngrowth of the primary and the secondary variables, respectively,  \nvary continuously with $\\omega$. \nThe latter exponent exhibits a maximum, which contrasts with the \nmonotonous dependence of the generalised persistence exponent on \nthe partial survival factor that appears directly \nin the transformation rule of the secondary variable \\cite{mb}.  \nAlthough, we have focused on the range $-1\\le\\omega\\le 1$,\nthe equations written down in this work are valid also for $\\omega >1$. \nIn that case, the growth of the primary variable becomes super-linear, meaning\nthat $\\alpha_{\\omega}>1$. \n\nAn intriguing feature of the process studied in this work is the universality\nwith respect to the initial distribution of the variables:\nFor a fixed $\\omega$,\nany sufficiently rapidly decaying initial distribution tends \nat late times to a\nuniversal distribution that displays scaling.\nAlthough, the process is universal in this sense, we have pointed out that\nit is sensitive to the variations of the reaction rules parameterised by \n$\\omega$. \nThe dependence of $\\alpha_{\\omega}$ on $\\omega$ is obvious since the\ntransformation rule of the primary variable contains $\\omega$ explicitely.\nThe growth of the secondary variable is, however, \naffected by $\\omega$ in a more subtle way.    \nFocusing on the secondary variables, the difference to the process\nof primary variables with $\\omega=0$ \nis that, here, not exactly the smallest variable is\nremoved from the set. \nThis is the reason for \nthat $\\beta_{\\omega}$ is unequal to $\\alpha_0$ for $\\omega\\neq 0$. \nNevertheless, for any $\\omega$, the removed secondary variable\nis typically relatively small since $X_i$ and $Y_i$ become\npositively correlated in the course of the process. \nDue to these correlations, the strength of which is controlled by \n$\\omega$, the variation of $\\beta_{\\omega}$ is relatively slight. \nIndeed, it is by an order of magnitude\nsmaller than that of $\\alpha_{\\omega}$.\n\nFor $\\omega=0$, we have shown that $\\alpha_0=\\beta_0$ even if the primary and\nthe secondary variables are initially not perfectly correlated. This can be\nunderstood also on a microscopic level since, in this case,  \nthe vectors in the set are sums of an increasing number of \ninitial vectors. Thus, the ratios $\\tilde X_i\/\\tilde Y_i$ tend\nstochastically to a common constant in the limit $\\Gamma\\to\\infty$ \nfor all $i$. In words, the two types of variables become asymptotically\nperfectly correlated for $\\omega=0$.      \nNow, we are in a position to understand why the exponent $\\beta_{\\omega}$ \nis maximal at $\\omega=0$.\nAt that point, the correlations are (at least asymptotically) perfect and\nalmost always the smallest one among the secondary variables is removed. \nFor $\\omega\\neq 0$, however, the correlations are no longer perfect and, \nas a consequence, not strictly the smallest secondary variables are eliminated.\nTherefore the fastest growth of $Y$ is realized at $\\omega=0$.\n\nIn a general aspect, the benefit of the analysis carried out in this\nwork is that\nthe numerical technique developed here for obtaining accurate estimates of \nthe growth exponents may also apply to other\nnon-soluble coagulation processes with extremal dynamics. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the challenging problems in quantum gravity is to understand the\nmicroscopic properties of black holes, in particular, the statistical\norigin of the Bekenstein-Hawking entropy.\nNew idea for an explanation of the \norigin of the Bekenstein-Hawking entropy has been provided by recent\ndevelopment in our understanding of non-perturbative superstring theory.\nIt is based on the D-brane description of black\nholes\\cite{Strominger-Vafa} and the AdS\/CFT \ncorrespondence\\cite{Maldacena1,Witten,GKP}.\nThese are much related with each other under\nthe Maldacena duality\\cite{Maldacena1}.\n\nThree-dimensional Einstein equation with negative\ncosmological constant has the solutions called \nthe BTZ black holes\\cite{BTZ1,BTZ2}.\nThese black holes have locally ${\\rm AdS}_{3}$ geometry.\nVia the ${\\rm AdS}_{3}\/{\\rm CFT_{2}}$ correspondence, one can also \nhope to be able to analyze the microscopic properties of \nthe BTZ black holes based on a local field theory on\nthe boundary. Infinite dimensional algebra of two-dimensional conformal\nsymmetry, that is, the Virasoro algebra, provides an important clue\nfor our understanding the correspondence and the Maldacena duality.\nThe pioneering work is Strominger's counting of microscopic states\nof the BTZ black holes\\cite{Strominger}.\nBut the qualitative aspects of this counting still remain obscure .\n\nIn this paper, we will discuss the three-dimensional extremal BTZ \nblack holes in the context of the Maldacena duality.\nAlthough this duality has been conjectural yet, various checks have been\ncarried out. (See \\cite{Maldacena2} and references therein.)\nIn this perspective, the extremal BTZ black holes can be identified\nwith the primary states which are 1\/2 BPS states in the N=(4,4)\ntwo-dimensional supersymmetric $\\sigma$-model.\nThis $\\sigma$-model has a quantity called elliptic genus \nconvenient to count the degeneracy of these states.\nWe explicitly count the microscopic states of the extremal BTZ black\nholes with this identification by using the elliptic genus and\nthe unitary representation theory of the N=4 superconformal\nalgebra. The microscopic entropy of these black holes obtained by\nthis counting agrees with the entropy \\'a la Bekenstein-Hawking.\n\nThis paper is organized as follows.\nIn section 2, we will summarize the\nprevious results about the BTZ black holes from the perspective of the\n${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence in a pure quantum gravity and\nin non-perturbative superstring theory, i.e., the Maldacena duality.\nIn section 3, after a brief introduction of N=4 superconformal algebra,\nblack hole states are discussed in the unitary representation\ntheory.\nIn section 4, some facts about the elliptic\ngenus of the N=(4,4) supersymmetric $\\sigma$-model are reviewed.\nIn section 5, we count the number of 1\/8 BPS states in the D1-D5 brane\nsystem in IIB supergravity via the elliptic genus of this\n$\\sigma$-model and then finally count the microscopic states of the\nextremal BTZ black holes.\nIn section 6, some other related topics are discussed.\n  \n\n\\section{BTZ black holes and ${\\bf AdS_{3}\/CFT_{2}}$\ncorrespondence}\n\n\\subsection{BTZ black holes in a three-dimensional pure quantum \ngravity}\n\nThe BTZ black holes\n\\footnote{Exact solutions of the vacuum \nEinstein equation with a negative cosmological \nconstant $\\Lambda=-1\/l^2$.}\nare three-dimensional black holes specified \nby their mass $M$ and angular momenta $J$, where $|J| \\leq Ml$.   \nIn terms of the Schwarzschild coordinates \n$(t,\\phi,r)$, with the ranges \n$-\\infty<t<+\\infty$, \n$0\\leq \\phi <2\\pi$ \nand $0<r<+\\infty$, \nthe black hole metric $ds_{{\\rm BTZ}_{(J,M)}}^2$ \nhas the form \n\\begin{eqnarray}\nds^{2}_{{\\rm BTZ}_{(J,M)}} \n\\equiv \n-N^2(dt)^2+N^{-2}(dr)^2\n+r^2(d\\phi+N^{\\phi}dt)^2, \n\\label{dsX(J,M)}\n\\end{eqnarray}\nwhere $N$ and $N^{\\phi}$ are the functions of the radial coordinate $r$ \n\\begin{eqnarray}\nN^2 =\n\\frac{(r^2-r_+^2)(r^2-r_-^2)}{l^2r^2},~~~~\nN^{\\phi} =  \n\\left\\{ \n\\begin{array}{cc}\n\\frac{r_+r_-}{lr^2} & \\mbox{when}~~J \\geq 0, \n\\\\ \n-\\frac{r_+r_-}{lr^2} & \\mbox{when}~~J < 0.  \n\\end{array}\n\\right. \n\\label{N-Nphi}\n\\end{eqnarray}\nThe outer and inner horizons are located respectively \nat $r=r_+$ and $r=r_-$. Information about the mass \nand angular momentum is encoded in $r_{\\pm}$ by  \n\\begin{eqnarray} \nr_{\\pm}^2 \\equiv \n4GMl^2 \\left( \n1\\pm \\sqrt{1-\\frac{J^2}{M^2l^2}} \n\\right), \n\\label{rpm}\n\\end{eqnarray}\nwhere $G$ is Newton constant. \nIn the case of $Ml=|J|$,  the black hole is called extremal. \nIt holds $r_+=r_-$  \nand then the outer and inner horizons coincide  \nwith each other. \nWhen $Ml>|J|$, it is called non-extremal.\nAnd in the case of $J = 0$ and $Ml = -l\/8 G$,\nthe geometry corresponds to the global ${\\rm AdS}_{3}$.\n\nThe outer horizon of these solutions has finite area.\nThe semiclassical argument leads to\nthe finite Bekenstein-Hawking entropy:\n\\begin{eqnarray}\nS \\equiv \\frac{A}{4 G} = \\frac{2 \\pi r_{+}}{4G}. \\qquad (A : {\\rm area\n\\ of \\ the \\ outer \\ horizon})  \\label{eq:entropy}\n\\end{eqnarray}\n\n\nQuantization of three-dimensional\npure gravity with negative cosmological constant is discussed in \n\\cite{Nakatsu}.\nIt is prescribed, through the detailed analysis of \nBrown-Henneaux's asymptotic Virasoro symmetry\\cite{Brown-Henneaux}, \nas the geometric quantization of the Virasoro coadjoint orbits\nof the Virasoro central charge $c=3 l\/2 G$.\n\nThe BTZ black holes and the ${\\rm AdS}_{3}$ correspond to the primary\nstates (highest weight states) \nof the Virasoro algebra of \nBrown-Henneaux\\footnote{Strictly speaking,\nthese states correspond to the geometry of the exterior of outer\nhorizon of the BTZ black holes and the geometry without the origin of \nthe ${\\rm AdS_{3}}$ respectively}:\n\\begin{eqnarray}\n{\\rm BTZ}_{(J, M)} &\\Longleftrightarrow& |J, M\\rangle \\equiv \n|h\\rangle \\otimes |\\tilde{h}\\rangle,  \\label{eq:pribtz}\\\\  \n{\\rm AdS}_{3} &\\Longleftrightarrow& |vac\\rangle\n\\equiv |0\\rangle\\otimes|0\\rangle, \n\\label{eq:priads}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nh = \\frac{1}{16Gl}(r_{+}+r_{-})^2 + \\frac{c}{24}, \\quad {\\tilde h} = \n\\frac{1}{16Gl}(r_{+}-r_{-})^2 + \\frac{c}{24}.\n\\end{eqnarray}\nThe extremal BTZ black holes correspond to\n\\begin{eqnarray}\n{\\rm BTZ}_{(Ml, M)} \\Longleftrightarrow |Ml, M\\rangle \\equiv \n|h \\rangle \\otimes |\\frac{c}{24} \\rangle. \\label{eq:priext}\n\\end{eqnarray}\n \nThe total Hilbert space of the theory, which includes \nexcited states (secondary states), \nis obtained by the tensor products ${\\cal V}_{h}\\otimes{\\tilde {\\cal\nV}_{{\\tilde h}}}$ of the Verma modules of the Virasoro algebra.\n(${\\cal V}_{h}$ and ${\\tilde {\\cal V}}_{{\\tilde h}}$ are respectively\nthe Verma modules of the left-moving and right-moving sectors.) \nThese Verma modules constitute the unitary irreducible \nrepresentations of the Virasoro algebra. \nWe can identify the states excited by $L_{-n}$ in the Verma module\nwith massive gravitons on the corresponding background geometry.\n\nIn view of the ${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence,\nthis Hilbert space should be realized by the corresponding boundary CFT. \nIn fact it was done\\cite{Nakatsu} based on \nthe Liouville field $X$ with a specific\nbackground charge. The action is given by\n\\begin{eqnarray}\nS[X]=\\frac1{4\\pi i}\\int_{{\\bf P}^1} \n\\bar{\\partial} X \\wedge \\partial X\n     + \\frac{\\alpha_0}{2\\pi}\\int_{{\\bf P}^1} RX, \n\\qquad \\left( \\alpha_0 \\equiv \\sqrt{\\frac{l}{8G}} \\right)\n\\end{eqnarray}\nwhere $R$ is the Riemann tensor of \na fixed K\\\"ahler metric on ${\\bf P}^1$.\nThe stress tensor $T(z)$ has the form\n\\begin{eqnarray}\nT(z)=-\\frac{1}{2} \n\\partial X \\partial X(z)+\\alpha_0\\partial^2 X(z), \n\\end{eqnarray}\nand provides the generators \nof the Virasoro algebra  \nwith the central charge $1+12\\alpha_0^2 = 1+3l\/2G$.\nThis central charge is the same as that of Virasoro algebra of \nBrown-Henneaux in the semiclassical limit, i.e.,  $l\/G \\gg 1$.\nThe Fock space ${\\cal F}_{k}$\\footnote{Similar\narguments hold for the anti-holomorphic (right-moving) part.} is built on \nthe Fock vacuum $|k\\rangle$, which is introduced as \nthe state obtained from the ordinary $SL_{2}({\\bf C})$-invariant vacuum \n$|0\\rangle$ by the relation\n$|k\\rangle = \\lim_{z\\rightarrow 0} e^{i k X(z)}|0\\rangle$.\n\nThe BTZ black hole states (\\ref{eq:pribtz}) can be \nidentified with the following Fock vacuum:\n\\begin{eqnarray}\n{\\rm BTZ}_{(J, M)} \\Longleftrightarrow |J, M\\rangle \\equiv \n  |k_{(J, M)}\\rangle \\otimes |\\tilde{k}_{(J, M)}\\rangle, \n\\label{black hole state in 2d}\n\\end{eqnarray}\nwhere $k_{(J,M)}$ and $\\tilde{k}_{(J,M)}$ are given by \n\\begin{eqnarray}\nk_{(J, M)} &\\equiv& \n-i\\sqrt{\\frac l{8G}}+\\frac{r_+ +r_-}{\\sqrt{8Gl}},\n\\nonumber \\\\ \n\\tilde{k}_{(J, M)} &\\equiv& \n -i\\sqrt{\\frac l{8G}}+\\frac{r_+ -r_-}{\\sqrt{8Gl}}.   \n\\label{k(J,M)}\n\\end{eqnarray}\n${\\rm AdS}_{3}$ state (\\ref{eq:priads}) can be  \nidentified with the $SL_{2}({\\bf C})$-invariant vacuum:\n\\begin{eqnarray}\n{\\rm AdS}_{3} \\Longleftrightarrow |vac\\rangle\n\\equiv |0\\rangle\\otimes|0\\rangle.\n\\end{eqnarray}\n\nThe Fock spaces ${\\cal F}_{k} \\otimes {\\tilde {\\cal\nF}}_{{\\tilde k}}$ built on these primary\nstates give the unitary irreducible representations of the Virasoro\nalgebra with $c=1+3l\/2G$, and coincide with the physical Hilbert\nspace of the previous quantization of three-dimensional pure\ngravity.\n\nTo summarize, in this correspondence of three-dimensional pure gravity\nand the boundary CFT, the BTZ black holes appear as the primary\nstates of the Virasoro algebra with $c = 3l\/2G$ in both descriptions.\nThis result may not be desirable for the counting of microscopic states of\nthe BTZ black holes. We cannot count in principle the\ndegeneracy of these primary states with this boundary theory,\nsince this Liouville field theory has continuum\nspectrum of primary states.\n\n\\subsection{BTZ black holes and Maldacena duality}\n\nNext, we consider the ${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence \nin superstring theory.\nThrough the analysis\nof the near horizon limit of the BPS solitonic solution of \nIIB supergravity, which describes\nthe bound state of $Q_{1}$ D1-branes and $Q_{5}$ D5-branes,\nMaldacena has conjectured in \\cite{Maldacena1},\n\\begin{center}\nIIB superstring theory on (${\\rm AdS}_{3} \\times {\\rm S}^{3})_{Q_{1}Q_{5}} \n\\times M_{4}$ \\quad ($M_{4}$ = $K3$ or $T^4$)\\\\ \n$\\Updownarrow$ dual  \\\\ two-dimensional N=(4,4) supersymmetric\n$\\sigma$-model \\\\ on the Higgs branch of world volume\ntheory of the D1-D5 system.\n\\end{center}\nHere, we indicated the dependence of the radius of \n${\\rm AdS}_{3}$ and ${\\rm S}^{3}$ on $Q_{1}Q_{5}$ \n(see below). We call this duality simply the Maldacena duality.\n\nWe will discuss mainly the case of $M_{4}$ = $K3$ in the following.\nThe N=(4,4) $\\sigma$-model can be regarded as\nthe $\\sigma$-model on the target space of the $k$-th symmetric product of\n$K3$ \\cite{Vafa1,Dijkgraaf}, where\n\\begin{eqnarray}\nk = Q_{1}Q_{5} + 1.\n\\end{eqnarray} \nSince the symmetric product \nis 4$k$-dimensional hyper-K\\\"ahler manifold, it\nhas automatically the N=(4,4) superconformal symmetry.\nThe Virasoro subalgebra and zero mode of the\nSU(2) current algebra may\nbe identified with the Virasoro algebra of \nBrown-Henneaux on the boundary of ${\\rm AdS}_{3}$ and \nthe isometry of ${\\rm S}^{3}$, respectively. We will discuss \nthe N=4 superconformal algebra in more detail in section 3.\n\nThe extremal BTZ black holes can also be obtained\nas the near horizon limit of the similar BPS solitonic solutions of \nIIB supergravity.\nTherefore we can expect that the extremal BTZ black holes \ncan be analyzed by this N=(4,4) $\\sigma$-model.\n\nWe summarize some related facts of IIB supergravity here.\nIIB supergravity on $S^{1} \\times K3$ whose radius and volume are $R$\nand $(2 \\pi)^4 \\alpha^{'2} v$ has the BPS solitonic solution\n(see for example \\cite{Skenderis} and references therein):\n\\begin{eqnarray}\nds_{10}^{2} &=& f_{1}^{-\\frac{1}{2}}f_{5}^{-\\frac{1}{2}}\\{ - dt^2 +\ndx_{5}^{2} + f_{N}(dt+dx_{5})^2\\} \\nonumber \\\\ && + f_{1}^{\\frac{1}{2}}\nf_{5}^{\\frac{1}{2}}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}) + \nf_{1}^{\\frac{1}{2}}f_{5}^{-\\frac{1}{2}} ds_{K3}^{2}, \\label{eq:10dim} \n\\end{eqnarray}\nwith periodic identification $x_{5} \\sim x_{5} + 2 \\pi R$\nin string frame and\n\\begin{eqnarray}\ne^{-2 (\\phi - \\phi_{\\infty})} &=& f_{5}f_{1}^{-1}, \n\\quad C_{05}^{(R)} = \\frac{1}{2}(f_{1}^{-1} - 1), \\nonumber \\\\\nH_{ijk}^{(R)} &=& (*_{6} \\ dC^{(R)})_{ijk} =\n\\frac{1}{2}\\epsilon_{ijkl}\\partial_{l}f_{5} \\qquad (i,j,k,l =\n1,2,3,4),\n\\end{eqnarray}\nwhere $C^{R}$ is Ramond-Ramond 2-form and $*_{6}$ is Hodge dual in\n6-dimension ($t,x_{1},\\cdots,x_{5}$).\nAnd $f_{1}$, $f_{5}$ and $f_{N}$ are following functions with respect\nto radial coordinate, \\\\\n$r = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$:\n\\begin{eqnarray}\nf_{1} &=& 1 + \\frac{{\\tilde Q}_{1}}{r^2}, \\qquad {\\tilde Q}_{1} =\n\\frac{\\alpha^{'} g_{st}}{v} Q_{1}, \\\\\nf_{5} &=& 1 + \\frac{{\\tilde Q}_{5}}{r^2}, \\qquad {\\tilde Q}_{5} = \n\\alpha^{'} g_{st}Q_{5}, \\\\\nf_{N} &=& \\frac{{\\tilde N}}{r^2}, \\qquad \\qquad {\\tilde N} =\n\\frac{\\alpha^{'2} g_{st}^{2}}{R^{2} v} N.\n\\end{eqnarray}\n\nThis solution corresponds to the configuration\nof the bound state of $Q_{5}$ D5-branes wrapping on $K3 \\times S^{1}$\nand $Q_{1}$ D1-branes wrapping on $S^{1}$ with $N$ units of KK momenta\nalong $S^{1}$, i.e., $x_{5}$-direction, and preserves four\nsupercharges. So it is a 1\/8 BPS state\\cite{Strominger-Vafa}.\n\nHere we can get the extremal BTZ black hole as the near horizon limit of\nthe geometry (\\ref{eq:10dim})\\cite{Maldacena-Strominger,Skenderis}.  \nThe near horizon limit is defined as\n\\begin{eqnarray*}\n\\alpha{'} \\rightarrow 0, \\qquad {\\rm with} \\qquad  \nU \\equiv \\frac{r}{\\alpha^{'}}, \\  R  \\  {\\rm and}  \\   v  \\\n  {\\rm fixed}.\n\\end{eqnarray*}\nIn this limit, the metric (\\ref{eq:10dim})\ndescribes $({\\rm BTZ}_{(Ml,M)} \\times S^{3})_{Q_{1}Q_{5}}\n\\times K3$ with $Ml = J = N$.\nThe radius of ${\\rm AdS}_{3}$ and ${\\rm S}^{3}$ coincide and become\n$l = g_{st}^{1\/2} \\alpha^{' 1\/2} (Q_{1}Q_{5}\/v)^{1\/4}$. \\\\\nAnd the three-dimensional effective Newton\nconstant on ${\\rm BTZ}_{(N,N\/l)}$ is given by\n$G_{{\\rm eff}}^{(3)} = l\/(4 Q_{1} Q_{5})$.\nThere exists on\nthis background the asymptotic Brown-Henneaux's Virasoro symmetry\nwith central charge,\n\\begin{eqnarray}\nc = \\frac{3l}{2G_{{\\rm eff}}^{(3)}} = 6 Q_{1}Q_{5}.\n\\end{eqnarray}\nThis is the same as the central charge of the N=(4,4) $\\sigma$-model \nat the semiclassical limit $Q_{1}Q_{5} \\gg 1$. \nThe Bekenstein-Hawking entropy becomes\n\\begin{eqnarray}\nS = \\frac{2 \\pi r_{+}}{4 G_{{\\rm eff}}^{3}} = 2 \\pi\n\\sqrt{Q_{1}Q_{5}N}, \\label{eq:extentropy}\n\\end{eqnarray}\nwhich is valid in the semiclassical region $N \\gg Q_{1}Q_{5} \\gg 1$.\n\nIf one accepts the Maldacena duality, the extremal black hole should be\nidentified with the primary state\n\\begin{eqnarray}\n|N+\\frac{c}{24}\\rangle \\otimes |\\frac{c}{24} \\rangle, \\label{eq:extprimary}\n\\end{eqnarray}\nof the N=(4,4) $\\sigma$-model.\nOne can ask whether the entropy (\\ref{eq:extentropy}) can be regarded as \nthe degeneracy of the primary state (\\ref{eq:extprimary}) of this N=(4,4)\n$\\sigma$-model. In the sequel, we will discuss this question and\nanswer in the affirmative. \n\n\\section{N=4 superconformal symmetry}\nN=(4,4) $\\sigma$-model is known to be finite to all orders of\nperturbation and to be conformally invariant at the quantum\nlevel. Thus the states of the $\\sigma$-model on the $k$-th symmetric\nproduct\\footnote{We will\ndenote $k$-th symmetric product of $K3$ as $S^k K3 \\equiv\nK3^{\\otimes k}\/S_{k} \\ (S_{k}$ is a $k$-dimensional symmetric\ngroup).}, $S^k K3$, constitute the unitary\nirreducible representations of the underlying N=4 superconformal\nalgebra (N=4 SCA).\n\n\\subsection{Basics of N=4 superconformal algebra}\nN=4 SCA is generated by $L_{n},\\ J_{n},\\ G_{r}^{i}$ and ${\\bar\nG}_{r}^{i}$ with \n\\begin{eqnarray*}\n[L_{m},L_{n}] &=& (m-n) L_{m+n}+\\frac{k}{2}m(m^2-1)\\delta_{n+m,0}, \\quad\n\\{G_{r}^{i},G_{s}^{j}\\} = \\{\\bar{G}_{r}^{i},\\bar{G}_{s}^{j}\\} = 0, \\\\\n\\{G_{r}^{i},{\\bar G}_{s}^{j}\\} &=& 2\n\\delta^{ij}L_{r+s}-2(r-s)\\sigma_{ij}^{a}J_{r+s}^{a} + \\frac{k}{2}(4\nr^2-1) \\delta_{r+s,0}, \\\\\n\\left[ J_{m}^{a},J_{n}^{b} \\right] &=& i \\epsilon^{abc} J_{m+n}^{c} + \n\\frac{k}{2}m\\delta_{m+n,0}, \\\\\n\\left[ J_{m}^{a},G_{r}^{i} \\right] &=&\n-\\frac{1}{2}\\sigma_{ij}^{a}G_{m+r}^{j}, \\quad \n\\left[ J_{m}^{a},{\\bar G}_{r}^{i} \\right] =\n\\frac{1}{2}\\left(\\sigma_{ij}^{a}\\right)^{*}{\\bar G}_{m+r}^{j}, \\\\\n\\left[ L_{m},G_{r}^{i} \\right] &=&\n(\\frac{m}{2}-r)G_{m+r}^{i}, \\quad \\left[ L_{m},{\\bar G}_{r}^{i} \\right] =\n(\\frac{m}{2}-r){\\bar G}_{m+r}^{i}, \\\\\n\\left[ L_{m},J_{n}^{a} \\right] &=& -n J_{m+n}^{a}. \n\\end{eqnarray*}\n($\\sigma_{ij}^{a}$ is Pauli matrix and $J_{n}^{(\\pm)} \\equiv \nJ_{n}^{1}\\pm i J_{n}^{2}$.) $L_{n},J_{n}^{a}$ and \n$G_{r}^{i}({\\bar G}_{r}^{i})$ represent the\nFourier components of the energy momentum tensor, SU(2) current and four\nsupercurrents, respectively. \n$G_{r}^{i} ({\\bar G}_{r}^{i})$ transforms as SU(2) doublet (its conjugate) \nunder the global SU(2) symmetry which is generated by $J_{0}^{a}$.\nThe level $k$ of the SU(2) current algebra ($\\widehat{{\\rm SU}(2)}_{k}$)\nmust be positive integer for \nunitary representations. The central charge $c$ of the Virasoro subalgebra is \n$6k$. \n\nTwo different boundary conditions of the supercurrents\nprovide two different sectors of this algebra. \nIf $G_{r}^{i}({\\bar G}_{r}^{i})$ has $r \\in\n{\\bf Z}+ \\frac{1}{2} $, it is called the Neveu-Scwarz (NS) sector \nand if $r \\in {\\bf Z}$, it is called the Ramond (R) sector.\nThese sectors are related by the automorphism of the algebra\ncalled spectral flow.\nWe will discuss the R-sector in the following.\n\nUnitary irreducible representations of N=4 SCA have\ntwo distinct types\\cite{Eguchi1}.\nThey are built on highest weight states $|h,l\\rangle$ called massive\nprimary and massless primary in the R-sector.\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item Massive primary state\n      \\begin{eqnarray}\n      L_{n}|h,l\\rangle &=& G_{n}^{i}|h,l\\rangle = \n      {\\bar G}_{n}^{i}|h,l\\rangle = J_{n}^{a}|h,l\\rangle = 0, \\qquad n\n      \\geq 1 \\nonumber \\\\\n      J_{0}^{(+)}|h,l\\rangle &=& G_{0}^{2} |h,l\\rangle = \n      {\\bar G}_{0}^{1}|h,l\\rangle = 0, \\nonumber \\\\\n      L_{0}|h,l\\rangle &=& h |h,l\\rangle, \\quad J_{0}^{3} |h,l\\rangle =\n      l |h,l\\rangle, \\nonumber \\\\\n      h &>& \\frac{k}{4} = \\frac{c}{24}, \n      \\qquad l = \\frac{1}{2},1,\\cdots,\\frac{k}{2}-\\frac{1}{2},\\frac{k}{2}. \n      \\label{eq:massive}\n      \\end{eqnarray}\n\\item Massless primary state\n      \\begin{eqnarray}\n      L_{n}|h,l\\rangle &=& G_{n}^{i}|h,l\\rangle = \n      {\\bar G}_{n}^{i}|h,l\\rangle = J_{n}^{a}|h,l\\rangle = 0, \\qquad n\n      \\geq 1 \\nonumber \\\\\n      J_{0}^{(+)}|h,l\\rangle &=& G_{0}^{i} |h,l\\rangle = \n      {\\bar G}_{0}^{i}|h,l\\rangle = 0, \\qquad i=1,2 \\nonumber \\\\\n      L_{0}|h,l\\rangle &=& h |h,l\\rangle, \\quad J_{0}^{3} |h,l\\rangle =\n      l |h,l\\rangle, \\nonumber \\\\\n      h&=&\\frac{k}{4}=\\frac{c}{24},\\qquad l =0,\\frac{1}{2},\\cdots,\n      \\frac{k}{2}-\\frac{1}{2},\\frac{k}{2}. \\label{eq:massless} \n      \\end{eqnarray}\n\\end{enumerate}\nThese representations are called massive representation ${\\cal M}^{k}_{(h,l)}$ \nand massless representation ${\\cal M}^{k}_{0 (l)}$ respectively.\n\nThe massive representations have the same number of bosonic and\nfermionic states at each level and the Witten index is equal to \nzero. \nThese representations correspond to\nthe representations which have spontaneously broken supersymmetry.\nThe Witten index of the massless representations is non-zero. \nThese representations are the representations which have\nunbroken supersymmetry. \nThe primary states of massless representations have dimension \n$h=k\/4=c\/24$. These are the ground states of the R-sector.\n\nCharacter of the representation is introduced by\n${\\rm ch}^{(R)}(\\tau,z) = \n{\\rm Tr} (q^{L_{0}-\\frac{c}{24}} y^{2 J_{0}^{3}})$.\n($q=e^{2 \\pi i \\tau}$ and $y=e^{2 \\pi i z}$.)  \nTheir explicit form is given in \\cite{Eguchi2}.     \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item The character of the massive representation ${\\cal M}^{k}_{(h,l)}$: \n      \\begin{eqnarray}\n      {\\rm ch}^{(R) k}(h,l ; \\tau,z) =\n      q^{h-\\frac{k}{4}-\\frac{l^2}{k+1}}\n      \\frac{\\theta_{2}(\\tau,z)^2}{\\eta(\\tau)^3}\n      \\chi_{k-1}^{l-\\frac{1}{2}}(\\tau,z),  \\label{eq:charmass}\n      \\end{eqnarray}\n      where $\\chi_{k}^{l}(\\tau,z)$ is the character of \n      $\\widehat{{\\rm SU}(2)}_{k}$ of isospin $l$,\n      \\begin{eqnarray}\n      \\chi_{k}^{l}(\\tau,z) &=& \\frac{q^{\\frac{(l+\\frac{1}{2})^2}{k+2}\n      -\\frac{1}{8}}}{\\prod_{n=1}^{\\infty}(1-q^n)(1-y^2\n      q^n)(1-y^{-2}q^{n-1})} \\nonumber \\\\\n      &&\\times \\sum_{m=0}^{\\infty}q^{(k+2)m^2 + (2 l + 1)}\\left(y^{2\n      \\{(k+2)m + l\\}}-y^{-2 \\{ (k+2)m+l+1 \\}}\\right) \\nonumber \\\\\n      && = \\frac{\\Theta_{2l+1, k+2}(\\tau, 2z) - \\Theta_{-2l-1, k+2}(\\tau,\n        2z)}{\\Theta_{1,2}(\\tau,2z) - \\Theta_{-1,2}(\\tau,2z)}.\n      \\label{eq:charaffine}\n      \\end{eqnarray}\n      $\\Theta_{l,k}(\\tau,z) = \\sum_{n=-\\infty}^{\\infty} q^{k\n      (n+\\frac{l}{2k})^2} y^{k (n+\\frac{l}{2k})}$ is the theta\n      function associated with $\\widehat{{\\rm SU}(2)}_{k}$ of isospin $l$.\n\\item The character of the massless representation ${\\cal M}^{k}_{0 (l)}$:\n      \\begin{eqnarray}\n      {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z) &=&\n      q^{-\\frac{1}{8}}\\frac{\\theta_{2}(\\tau,z)^2}{\\eta(\\tau)^3} \n      \\frac{1}{\\prod_{n=1}^{\\infty}(1-q^n)(1-y^2\n      q^n)(1-y^{-2}q^{n-1})} \\nonumber \\\\\n      \\times \\sum_{m=-\\infty}^{\\infty}&& \\hskip-1cm q^{(k+1)m^2 + 2 l m} \n      \\left( \\frac{y^{2 \\{(k+2)m+l-\\frac{1}{2}\\}}}{(1+y^{-1}q^{-m})^2} - \n      \\frac{y^{-2 \\{ (k+2)m+l+\\frac{1}{2}\\}}}{(1+y q^{-m})^2} \\right).\n      \\label{eq:charless}\n      \\end{eqnarray}\n\\end{enumerate}\n\nThese characters enjoy the following properties.\nThe Witten index of the representation \ncan be obtained, if one sets $z=1\/2$, i.e., $y=-1$:\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h,l;\\tau,z=\\frac{1}{2}) &=& 0, \\\\\n{\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z=\\frac{1}{2}) &=& \n(-1)^{2 l}(2 l+1).\n\\end{eqnarray}\nThe characters of massive and massless\nrepresentations are related by\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h=\\frac{k}{4},l;\\tau,z) &=&  \n{\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z)  \n+ 2 \\ {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l-\\frac{1}{2};\\tau,z)\n\\nonumber \\\\\n&&+ \\ {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l-1;\\tau,z).\n\\label{eq:relation}\n\\end{eqnarray}\n\n\\subsection{Identification of the black hole states}\n\nAs argued in section 2.2, the extremal BTZ black hole will correspond\nto the primary state (\\ref{eq:extprimary}) of the N=(4,4)\n$\\sigma$-model.\nIt is a Virasoro primary state of the underlying N=4\nSCA. The fact that the extremal BTZ black holes are the 1\/2 \nBPS states with respect to the Poincare supersymmetry in \nthree dimensions\\footnote{These correspond to the 1\/4\nBPS states in Anti-de Sitter supersymmetry in 3-dimension.} implies\nthat this primary state is in the tensor\nproduct of massive and massless representations ${\\cal\nM}^{k}_{(h,l)} \\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$ which\nis built on \n\\begin{eqnarray}\n|h=N+\\frac{k}{4},l\\rangle \\otimes |{\\tilde h}=\\frac{k}{4},{\\tilde\n l}\\rangle. \\label{eq:prisigma}\n\\end{eqnarray}\n\nActually we can proceed further. Since the extremal \nBTZ black holes do not have the conserved \ncharge corresponding to isospin $l$ and ${\\tilde l}$,\nthe primary state (\\ref{eq:extprimary}) may be identified with the\nVirasoro primary state having vanishing isospins $l={\\tilde\nl}=0$ in \n${\\cal M}^{k}_{(h,l)} \\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$.\nThe primary states with $l=0$ \nin ${\\cal M}^{k}_{(h,l)}$\nare as follows:\nwhen $l \\in {\\bf Z}$, they are given by \n\\begin{eqnarray}\n&&J_{0}^{(-) \\ l} |h=N+\\frac{k}{4},l\\rangle, \\nonumber \\\\\n{\\rm and} && J_{0}^{(-) \\ l-1} ({\\bar G}_{0}^{2} G_{0}^{1} - \n\\frac{h-k\/4}{l}) |h=N+\\frac{k}{4},l \\rangle, \\label{eq:integer}\n\\end{eqnarray}\nand when $l \\in {\\bf Z}+\\frac{1}{2}$, they are \n\\begin{eqnarray}\n&&J_{0}^{(-) \\ l-\\frac{1}{2}} G_{0}^{1} |h=N+\\frac{k}{4},l\\rangle,\n\\nonumber \\\\\n\\hskip-1.4cm{\\rm and} \\qquad \n&&J_{0}^{(-) \\ l-\\frac{1}{2}}{\\bar\nG}_{0}^{2}|h=N+\\frac{k}{4},l\\rangle.\n\\label{eq:halfint} \n\\end{eqnarray}\nThe primary states with ${\\tilde l}=0$ in\n${\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$ is identified with\n${\\bar J}_{0}^{(-) \\ {\\tilde l}} |{\\tilde h}=k\/4,{\\tilde\nl}\\rangle.$\nThe primary state (\\ref{eq:extprimary}) can be identified with the\ntensor product of these states. So, the degeneracy of the state is\nalmost same as the degeneracy of the representation ${\\cal M}^{k}_{(h,l)} \n\\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$. \n  \n\\section{Elliptic genus for $\\sigma$-model on symmetric product\nof K3}\nTo count the number of 1\/2 BPS states in the N=(4,4)\n$\\sigma$-model,\nthe so-called ``elliptic genus'' is a convenient tool.\nWe summarize some properties of the elliptic genus\nemphasizing its modular transform and examine it from the perspective\nof N=4 SCA. \n\\subsection{Elliptic genus as a weak Jacobi form}\nThe elliptic genus of target space $M$ \nis defined by the following trace in the R-R sector of the underlying\nN=(2,2) superconformal field theory\\footnote{One can obtain the following\ntopological indices of the target space $M$, if one sets $z$ to be\nspecific value.\n\\begin{eqnarray}\nZ[M](\\tau,0) &:& {\\rm Elliptic \\ extension \\ of \\ Euler \\ number}\n\\nonumber \\\\\nZ[M](\\tau,\\frac{1}{2})&:& {\\rm Elliptic \\ extension \\ of \\\nHirzebruch \\ signature} \\nonumber \\\\\nq^{\\frac{c}{24}}Z[M](\\tau,\\frac{\\tau+1}{2})&:& \n{\\rm Elliptic \\ extension \\ of \\ Dirac \\ genus} \\nonumber\n\\end{eqnarray}}.\n\\begin{eqnarray}\nZ[M](\\tau,z) = {\\rm Tr}_{{\\rm R}\\textrm{-}{\\rm R}}(-1)^{J_{0}-{\\bar\nJ}_{0}} q^{L_{0}-\\frac{c}{24}} {\\bar q}^{{\\bar\nL}_{0}-\\frac{c}{24}} y^{J_{0}}, \\label{eq:genus}\n\\end{eqnarray}\nwhere $J_{0}$ and ${\\bar J}_{0}$ are the integral N=2 U(1) charges\nof the left-moving and the right-moving sectors\\footnote{N=2 \nSCA can be embedded into N=4 SCA by\n$G_{r} = G_{r}^{1} + {\\bar G}_{r}^{2}$, ${\\bar G}_{r} = \nG_{r}^{2} + {\\bar G}_{r}^{1}$ and $J_{n}=2 J_{n}^{3}$.}.   \nThe elliptic genus is\nindependent of ${\\bar \\tau}$ by virtue of supersymmetry of the R-sector.\nThe contribution of the right-moving sector is only from the\nground states.\nBut all states in the left-moving sector contribute to $Z[M](\\tau,z)$.\nSo the elliptic genus $Z[M](\\tau,z)$ is a useful quantity for \ncounting of the 1\/2 BPS states.\n\nThe following theorem is known about this elliptic genus. (See\n \\cite{Kawai2} for detail.)\n\n\\noindent\n{\\bf Theorem 1.} If the target space $M$ of the $\\sigma$-model is an\neven-dimensional Calabi-Yau manifold, then the elliptic genus\n$Z[M](\\tau,z)$ is a weak Jacobi form of weight 0 and index\n$d\/2 (d=\\dim_{{\\bf C}} M)$ without character.\n\nWeak Jacobi form in the above theorem is defined as\nfollows\\cite{Eichler}.\n\n\\noindent \n{\\bf Definition.} A function \n$\\phi(\\tau,z)$ is called a weak Jacobi form of weight $k \\in {\\bf Z}$ \nand index $m \\in {\\bf Z}_{>0}\/2$ without character, if it satisfies\n(i)$\\sim$(iv): \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item $\\phi(\\tau,z)$ is a holomorphic function with respect to $\\tau\n      \\in {\\bf H}^{+} \\ ({\\bf H}^{+}:$ upper half plane) and $z \\in\n      {\\bf C}$.\n\\item $\\phi(\\frac{a \\tau + b}{c \\tau + d},\\frac{z}{c \\tau + d}) = \n      (c \\tau + d)^{k}e^{\\frac{2 \\pi i m c z^{2}}{c \\tau +\n      d}}\\phi(\\tau,z). \\quad (a,b,c,d \\in {\\bf Z} \\ {\\rm and} \\ ad-bc=1)$\n\\item $\\phi(\\tau,z + \\lambda \\tau + \\mu) = e^{- 2\\pi i m (\\lambda^2\n      \\tau + 2 \\lambda z)} \\phi(\\tau,z). \\quad (\\lambda, \\mu \\in {\\bf Z})$\n\\item $\\phi(\\tau,z)$ has the Fourier expansion of the form \\\\\n      \\begin{eqnarray*}\n      \\phi(\\tau,z) = \\sum_{n=0}^{\\infty} \\sum_{r = - \\infty}^{\\infty}\n      c(n,r) q^n y^r \\quad (q= e^{2 \\pi i \\tau},y = e^{2 \\pi i z}).\n      \\end{eqnarray*}\n\\end{enumerate}\n\nWhen $M$ is $K3$, the elliptic genus $Z[K3](\\tau,z)$\nbecomes a weak Jacobi form of weight 0 and index 1 without character.\nAn actual calculation of the N=(4,4) supersymmetric\n$\\sigma$-model on $K3$\\cite{Eguchi4,Kawai1} determines it explicitly as\n\\begin{eqnarray}\nZ[K3](\\tau,z) = 24 \\wp(\\tau,z)K^2(\\tau,z), \\label{eq:pfnK3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\wp(\\tau,z) &:& {\\rm Weierstrass's} \\ \n\\wp {\\rm -function}, \\nonumber \\\\\nK(\\tau,z) &=& i \\frac{\\theta_{1}(\\tau,z)^2}{\\eta(\\tau)^3}.\n\\label{eq:kfn} \n\\end{eqnarray} \n\nWe need the following theorem about weak Jacobi form\\cite{Kawai2,Eichler}. \n\n\\noindent\n{\\bf Theorem 2}. If we assign weights 4, 6 and 2 respectively to\n$E_{4}(\\tau), E_{6}(\\tau)$\\footnote{$E_{4}(\\tau)$ and $E_{6}(\\tau)$\nare the Eisenstein series.} and $\\wp(\\tau,z)$, any weak Jacobi\nform of weight $2l$ ($l \\in {\\bf Z}_{\\geq 0}$) and index $k$ ($k \\in {\\bf\nZ}_{\\geq 0}$) can be expressed as \n\\begin{eqnarray*}\n{\\cal G}_{2 l + 2 k}(E_{4}(\\tau), E_{6}(\\tau), \\wp(\\tau,z))\nK^{2k}(\\tau,z),\n\\end{eqnarray*}\nwhere ${\\cal G}_{2l + 2k}(E_{4},E_{6},\\wp)$ is a homogenious \npolynomial of weight (2$l$ + 2$k$) and its degree as a\npolynomial in $\\wp$ is at most $k$.\n\n$S^k K3$ is 2$k$-dimensional Calabi-Yau manifold. \nThe elliptic genus\n$Z[S^{k}K3](\\tau,z)$ becomes a weak Jacobi form of weight 0 and index $k$.   \nAccording to theorem 2, it has the\nfollowing form:\n\\begin{eqnarray}\nZ[S^k K3](\\tau,z) = {\\cal G}_{2k}(E_{4}(\\tau), E_{6}(\\tau),\n\\wp(\\tau,z))K^{2k}(\\tau,z). \\label{eq:generalgenus}\n\\end{eqnarray}  \n\nThe homogenious polynomial ${\\cal G}_{2k}$ is determined for lower\nvalues of $k$\\cite{Kawai2}\\footnote{It is worth commenting that \nthe coefficient of the first term in the\nbracket is the Euler number of $S^k K3$, $\\chi(S^k K3)$.}.\n\\begin{eqnarray}\nk=1 &:& 24\\wp K^{2} \\nonumber \\\\\nk=2 &:& (324 \\wp^{2} + \\frac{3}{4} E_{4})K^{4} \\nonumber \\\\\nk=3 &:& (3200 \\wp^{3} + \\frac{64}{3}E_{4}\\wp +\n\\frac{10}{27}E_{6})K^{6}\n\\label{eq:pfnprod}\n\\end{eqnarray}\n\n\\subsection{Elliptic genus of $S^k K3$ and characters of N=4\nsuperconformal algebra}\nThe N=(4,4) $\\sigma$-model on the target space of $K3$ has been\nanalyzed in detail by Eguchi et.al.\\cite{Eguchi4} in the context of \na compactification of string theory on $K3$. The elliptic extension\nof the Hirzebruch signature of $K3$ \ncan be represented by the characters of the N=4\nSCA as\n\\begin{eqnarray}\n&&\\hspace{-2cm} Z[K3](\\tau,\\frac{1}{2}) \\nonumber \\\\\n&=&-2 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0) +\n20 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=0;\\tau,0) \\nonumber \\\\\n&&+ F(\\tau) \\ {\\rm ch}^{(R) k=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0)\n\\nonumber \\\\\n&=& 24 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=0;\\tau,0) +\n {\\tilde F}(\\tau) \\ {\\rm ch}^{(R)\nk=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0),\n\\label{eq:genusK3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\tilde F}(\\tau) = -2 + F(\\tau) = \\sum_{n=0}^{\\infty}a_{n}q^{n} \\qquad\n(a_{0} = -2, \\ a_{n} \\in {\\bf Z}_{\\geq 0} (n>0)).\n\\end{eqnarray} \nWe have used eq.(\\ref{eq:relation}) to obtain the last equality in \n(\\ref{eq:genusK3}).\nThe degeneracy of the massive primary states in the left-moving sector\nis encoded in ${\\tilde F}(\\tau)$. The coefficient $a_{n}$ is the\ndegeneracy of the massive primary states of $h=n+1\/4$.\n\nThe function ${\\tilde F}(\\tau)$ can be\ndetermined by combining\ntwo expressions of elliptic genus (\\ref{eq:pfnK3}) and (\\ref{eq:genusK3}),\n\\begin{eqnarray}\nZ[K3](\\tau,\\frac{1}{2}) &=& 24 \\ {\\rm ch}_{0}^{(R) k=1}\n(h=\\frac{1}{4},l=0;\\tau,0) \\nonumber \\\\\n&& + {\\tilde F}(\\tau) \\ {\\rm ch}^{(R)\nk=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0) \\nonumber \\\\\n&=& 24 \\ \\wp(\\tau,\\frac{1}{2})K^{2}(\\tau,\\frac{1}{2}).\n\\end{eqnarray} \nThis gives\n\\begin{eqnarray}\n{\\tilde F}(\\tau) &=& 2 \\ \\frac{\\theta_{2}(\\tau,0)^4 -\n\\theta_{4}(\\tau,0)^4}{\\prod_{n=1}^{\\infty}(1-q^n)^3} - 24 \\  \n{\\tilde h}_{3}(\\tau), \n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \n{\\tilde h}_{3}(\\tau) &=&\n\\frac{1}{\\theta_{3}(\\tau,0)}\\sum_{m=-\\infty}^{\\infty}\n\\frac{q^{\\frac{m^2}{2}}}{1+q^{m-\\frac{1}{2}}}.\n\\end{eqnarray}\n\nNow, we will turn to the case of the symmetric product.\nThe elliptic genus of $S^k K3$ can be \nalso expanded by the characters of the underlying N=4 SCA.\nIn general, it has the following expansion:\n\\begin{eqnarray}\n&&\\hspace{-2cm}Z[S^k K3](\\tau, z+\\frac{1}{2}) \n\\qquad \\left( = {\\rm Tr}(-1)^{- 2 {\\bar\nJ}_{0}^{3}}{\\bar q}^{{\\bar\nL}_{0}-\\frac{c}{24}}q^{L_{0}-\\frac{c}{24}}y^{2 J_{0}^{3}} \\right) \\nonumber \\\\\n&&\\hspace{-1cm}= \\chi(S^k K3) \\ {\\rm ch}_{0}^{(R) k}\n(h=\\frac{k}{4},l=0;\\tau,z) + \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}}F_{l}(\\tau) \\ {\\rm ch}^{(R) k}\n(h=\\frac{k}{4},l;\\tau,z), \\label{eq:charfnprod}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nF_{l}(\\tau) = \\sum_{n=0}^{\\infty}a_{n}^{(l)}q^n, \\qquad \\quad (a_{0}^{(l)}\n\\in {\\bf Z}, \\ a_{n}^{(l)} \\in {\\bf Z}_{\\geq 0} \\ (n>0)). \\label{eq:ffun}\n\\end{eqnarray}        \nAlthough the characters of the massless representations with $l \\neq\n0$ may appear in $Z[S^k K3](\\tau,z+1\/2)$, we can reduce them\nto $l=0$ by applying (\\ref{eq:relation}) recursively. $a_{n}^{(l)}$ in\n(\\ref{eq:ffun}) provides the degeneracy of the representation \n${\\cal M}_{(h,l)}^{k} \\otimes {\\tilde {\\cal M}}_{0 ({\\tilde l})}^{k}$ \nwith $h=n+k\/4$ and isospin $l$. $\\sum_{l}a_{n}^{(l)}$ provides \nthe number of the representations ${\\cal M}_{(h,l)}^{k} \n\\otimes {\\tilde {\\cal M}}_{0 ({\\tilde l})}^{k}$ with $h=n+k\/4$. \n\nBecause of eq.(\\ref{eq:generalgenus}), \nthe spectrum of the massive primary states of this $\\sigma$-model on \n$S^k K3$ also\nbecomes discrete and all states have dimension $h = {\\bf Z}_{\\geq 0}\n+k\/4$ in $Z[S^k K3](\\tau,z+1\/2)$. This corresponds to the fact \nthe extremal BTZ black holes have the discrete mass $N\/l$ in the context of\nthe Maldacena duality. \n\nThe function $F_{l}(\\tau)$ \\ (or $\\sum_{l}F_{l}(\\tau)$) in\neq.(\\ref{eq:charfnprod}) can be determined in principle by an \nanalogous way with the case of $K3$.\nBut it is a hard task to obtain the exact functional form of \n$Z[S^k K3](\\tau,z)$ such as (\\ref{eq:pfnprod}) for the case of \ngeneral $k$. \nHowever, as discussed in section 2.2, what we need for \nthe counting of the microscopic states comparable with \nthe Bekenstein-Hawking entropy is   \nthe asymptotic form of $a_{n}^{(l)}$ (or $\\sum_{l} a_{n}^{(l)}$) \nat $n \\rightarrow \\infty$, since eq.(\\ref{eq:extentropy}) is valid\nfor the region $N \\gg Q_{1}Q_{5} \\gg 1$.\nWe will consider this asymptotic form in the next section.\n\n\\section{State counting via the N=(4,4) $\\sigma$-model}   \nIn this section, we will discuss the degeneracy of  \n1\/8 BPS states of the D1-D5\nsystem in IIB superstring theory and the degeneracy of the primary states \ncorresponding to the extremal BTZ black holes.\nIn the previous section, we obtain two different expressions\nof the elliptic genus of $S^k K3$. \nWe will first use the expression in terms of a weak\nJacobi form and count the number of the 1\/8 BPS states by using a\nTauberian theorem. Then using the expression in terms of the \ncharacters of N=4 SCA, we will discuss the\ndegeneracy of the massive primary states and obtain the\nmicroscopic entropy of the corresponding extremal BTZ black holes.\n\n\\subsection{Counting the 1\/8 BPS states}\nLet us start by studying the asymptotic behavior of the elliptic genus\n$Z[S^k K3](\\tau,1\/2)$ as $\\tau \\downarrow 0$\\footnote{\n$\\tau \\downarrow 0 \\stackrel{{\\rm def}}{\\Longleftrightarrow} \\tau = iT\n\\ (T \\in {\\bf R}_{>0}),\\  {\\rm and} \\ T \\rightarrow 0.$}. \nDue to the structure theorem the elliptic genus\nhas the form (\\ref{eq:generalgenus})\n\\begin{eqnarray*}\nZ[S^k K3](\\tau,\\frac{1}{2}) = \\\n{\\cal G}_{2 k}\\left( E_{4}(\\tau),E_{6}(\\tau),\\wp(\\tau,\\frac{1}{2})\n\\right) K^{2 k}(\\tau,\\frac{1}{2}). \n\\end{eqnarray*}\nThe asymptotics can be obtained \nfrom those of the constituents in (\\ref{eq:generalgenus}).\n$\\wp(\\tau,1\/2)$, $E_{4}(\\tau)$ and\n$E_{6}(\\tau)$ behave as \n$\\wp(\\tau,1\/2) \\rightarrow (-1\/12)(-i \\tau)^{-2}, \\\nE_{4}(\\tau) \\rightarrow 1 (-i \\tau)^{-4}$, and   \n$E_{6}(\\tau) \\rightarrow (-1)(-i \\tau)^{-6}$. \nTherefore the asymptotics of ${\\cal G}_{2k}$ becomes\n\\begin{eqnarray}\n{\\cal G}_{2k}(\\tau,\\frac{1}{2}) \\rightarrow {\\tilde c}(k) (-i \\tau)^{-2 k}\n\\qquad {\\rm as} \\ \\tau \\downarrow 0,\n\\end{eqnarray}\nwhere ${\\tilde c}(k)$ is a constant which depends on $k$ and the\npolynomial form of ${\\cal G}_{2k}$.\nThe asymptotics of $K^{2}(\\tau,1\/2)$ can be read from \n$\\theta_{2}(\\tau,0) \\rightarrow 1 (-i \\tau)^{-\\frac{1}{2}}$ and \n$\\eta(\\tau) \\rightarrow 1 (-i \\tau)^{-\\frac{1}{2}} e^{-\\frac{\\pi i}{12\n\\tau}}$,\n\\begin{eqnarray}\nK^{2}(\\tau,\\frac{1}{2}) \\rightarrow (-1)(-i \\tau)^{2} e^{\\frac{\\pi i}{2\n\\tau}} \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n\\end{eqnarray}\n\nTherefore, gathering these asymptotics, we obtain\n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2}) \\rightarrow c(k)(-i \\tau)^{0}\ne^{\\frac{\\pi i k}{2 \\tau}} \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n \\quad \\left( c(k) = (-1)^k {\\tilde c}(k) \\right) \\label{eq:asymz}\n\\end{eqnarray}\n\nThe elliptic genus has the Fourier expansion of the form\n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2})=\\sum_{n=0}^{\\infty} a_{n}q^n.\n\\label{eq:expgenus}\n\\end{eqnarray}\nAgain, due to the structure theorem, the coefficients satisfy $a_{n}\n\\leq a_{n+1}$. (See Appendix A for the explicit Fourier expansions of \nvarious functions.) Each coefficient $a_{n}$ represent the number of\nthe 1\/2 BPS states of $h=n+k\/4$ in the N=(4,4) $\\sigma$-model.\nThis is the number of\nthe $1\/8$ BPS states of the mass specified by\n$h=n+k\/4$ in the D1-D5 system \\cite{Strominger-Vafa,Maldacena-Strominger}.\nWe can estimate the asymptotic form of $a_{n}$ by using the following\nTauberian theorem\\cite{Kac1,Kac2}.\n\n\\noindent \n{\\bf Theorem 3.} Let $f(\\tau)$ be a function \n\\begin{eqnarray*}\nf(\\tau) = q^{\\lambda} \\sum_{n=0}^{\\infty} a_{n} q^n \\qquad (q=e^{2 \\pi\ni \\tau})\n\\end{eqnarray*}\nwhich satisfies following conditions:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item $f(\\tau)$ is a holomorphic function on ${\\bf H}^{+}$.\n\\item $a_{n} \\in {\\bf R}$ and $a_{n} \\leq a_{n+1}$ for all $n$.\n\\item There exist $c \\in {\\bf C}$, $d \\in {\\bf R}$ and $N \\in \n      {\\bf R}_{>0}$ such that \n      \\begin{eqnarray*}\n      f(\\tau) \\rightarrow c (-i \\tau)^{-d} e^{\\frac{2 \\pi i N}{\\tau}}\n      \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n      \\end{eqnarray*}\n\\end{enumerate}\nThen, the behavior of $a_{n}$ at large $n$ is\n\\begin{eqnarray*}\na_{n} \\sim \\frac{c}{\\sqrt{2}}N^{-\\frac{1}{2}(d-\\frac{1}{2})}\nn^{\\frac{1}{2}(d-\\frac{3}{2})} e^{2 \\pi \\sqrt{4 N n}}  \\qquad {\\rm as} \\ \nn \\rightarrow \\infty, \n\\end{eqnarray*}\nwhere $a_{n} \\sim b_{n}$ as $n \\rightarrow \\infty$ means $\\lim_{n\n\\rightarrow \\infty} b_{n}\/a_{n} = 1$.  \n\nDue to this theorem, the asymptotic form of $a_{n}$ can be read from\nthe estimation (\\ref{eq:asymz})\\footnote{We can\nexpect $c(k)$ is not large number due to the explicit\nexample of lower $k$. (See \\cite{Kawai2}.)}\n\\begin{eqnarray}\na_{n} \\sim \\frac{c(k)}{\\sqrt{2}} \\left( \n\\frac{k}{4} \\right)^{\\frac{1}{4}} n^{-\\frac{3}{4}} e^{2 \\pi \\sqrt{k\nn}}. \\label{eq:bpsnum}\n\\end{eqnarray}  \n\n\\subsection{Counting the massive primary states}\nWe can also expand the elliptic genus by the characters of N=4 SCA\n\\begin{eqnarray}\n&&\\hspace{-2cm}Z[S^k K3](\\tau,\\frac{1}{2}) \\nonumber \\\\ \n=&&\\hspace{-0.5cm}\\chi(S^k K3) \\ {\\rm ch}_{0}^{(R) k} \n(h=\\frac{k}{4},l=0;\\tau,0) + \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}}F_{l}(\\tau) \\ {\\rm ch}^{(R) k}\n(h=\\frac{k}{4},l;\\tau,0).\n\\label{eq:charexp}\n\\end{eqnarray}\nEach coefficient of $F_{l}(\\tau) =\n\\sum_{n=0}^{\\infty}a_{n}^{(l)} q^{n}$ counts the number of the \nmassive representation ${\\cal M}_{(h{\\rm =}n+k\/4,l)}^{k}$ \nin $Z[S^k K3](\\tau,z)$\nof the N=(4,4) $\\sigma$-model. \nIn particular $\\sum_{l}a_{N}^{(l)}$ will be identified\nwith the number of the primary state (\\ref{eq:extprimary}).\nTo obtain the asymptotic form $\\sum_{l}a_{n}^{(l)}$, we may again\nutilize the Tauberian theorem. For this purpose we need to know the\nasymptotic behavior of $\\sum_{l}F_{l}(\\tau)$ as $\\tau \\downarrow 0$.\n\nSince we have obtained the asymptotics of the elliptic genus\n(\\ref{eq:asymz}), the asymptotics of $\\sum_{l}F_{l}(\\tau)$ becomes\ntractable if we can properly estimate the constituents of the\nmassless and massive characters. \nLet us remind that the character of \nmassive representation ${\\cal M}_{(h,l)}^{k}$ is\ngiven by eq.(\\ref{eq:charmass}).\nThe asymptotic behavior of\nthe character of $\\widehat{{\\rm SU}(2)}_{k}$ \nof the isospin $l$, eq.(\\ref{eq:charaffine}), is given by\\cite{Kac1,Kac2}\n\\begin{eqnarray}\n\\chi_{k}^{l}(\\tau, 0) \\rightarrow a(k,l) \\exp \\left( \n\\frac{\\pi i}{12 \\tau} c_{k} \\right)\n\\qquad {\\rm as} \\ \\tau \\downarrow 0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na(k,l) &=& \\sqrt{\\frac{2}{k+2}}\\sin\\left(\\frac{(2 l + 1) \\pi}{k+2}\\right),\n\\nonumber \\\\\nc_{k} &=& \\frac{3 k}{k+2}.\n\\end{eqnarray}\nTherefore, combining those of $\\theta_{2}(\\tau,0)$ and $\\eta(\\tau)$,\nwe obtain\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h=\\frac{k}{4},l;\\tau,0) \\rightarrow \na(k{\\rm -}1,l{\\rm -}\\frac{1}{2}) (-i \\tau)^{\\frac{1}{2}} \\exp \\left(\n\\frac{\\pi i}{12 \\tau} (3 + c_{k-1}) \\right) \\quad \n{\\rm as} \\ \\tau \\downarrow 0. \\label{eq:asymchar}\n\\end{eqnarray}\n \nAs for the character of massless\nrepresentation ${\\cal M}_{0 (l{\\rm =}0)}^{k}$,\nwe can obtain the upper bound of the asymptotic\nbehavior by means of eq.(\\ref{eq:relation}):\n\\begin{eqnarray}\n\\hspace{-1cm}{\\rm ch}_{0}^{(R) k}\n(h=\\frac{k}{4},l=0;\\tau,0)|_{\\tau \\downarrow 0}\n&\\leq& {\\rm ch}^{(R) k}(h=\\frac{k}{4},l=\\frac{1}{2};\\tau,0)|_{\\tau\n\\downarrow 0} \\nonumber \\\\\n\\hspace{-1cm}&=& a(k,l{\\rm =}0)(-i \\tau)^{\\frac{1}{2}}\\exp\n\\left( \\frac{\\pi i}{12 \\tau} (3 + c_{k-1}) \\right),\n\\end{eqnarray}\nwhere $f(\\tau)|_{\\tau \\downarrow 0}$ means the leading asymptotic of\n$f(\\tau)$ as $\\tau \\downarrow 0$.\nFrom this estimation, the dominant contribution of the \nasymptotic behavior of $Z[S^k K3](\\tau,1\/2)$ turns out to come\nfrom the part of the massive representations. \nWe can neglect the contribution of the massless representations \nin the asymptotics.\n\nNow we can obtain the asymptotic behavior of \n$\\sum_{l}F_{l}(\\tau)$.\nTaking the limit $\\tau \\downarrow 0$ in eq.(\\ref{eq:charexp}), \n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2}) \\rightarrow \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau)|_{\\tau\n\\downarrow 0} \\times (-i \\tau)^{\\frac{1}{2}}\\exp \\left( \\frac{\\pi i}{12 \n\\tau} (3+c_{k-1}) \\right) \\qquad {\\rm as} \\ \\tau\n\\downarrow 0, \\label{eq:asymf} \n\\end{eqnarray}\nwhere ${\\tilde F}_{l}(\\tau) = \\sum_{n=0}^{\\infty} {\\tilde\na}_{n}^{(l)}q^{n} = a(k{\\rm -1},l{\\rm -1\/2})F_{l}(\\tau)$.\nThe asymptotic behavior of \n$\\sum_{l}{\\tilde F}_{l}(\\tau)$\nas $\\tau \\downarrow 0$ can be read by comparing (\\ref{eq:asymf}) with\n(\\ref{eq:asymz}) \n\\begin{eqnarray}\n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau) \\rightarrow\n\\ c(k)(-i \\tau)^{- \\frac{1}{2}} \\exp \\left( \\frac{\\pi i (6 k -\n(3+c_{k-1}))}{12 \\tau} \\right) \\quad {\\rm as} \\ \\tau \\downarrow 0.\n\\end{eqnarray}\n\n$\\sum_{l}{\\tilde F}_{l}(\\tau)$ has the\nFourier expansion of the form\\footnote{These coefficients also satisfy \n$b_{n} \\leq b_{n+1}$.} \n\\begin{eqnarray}\n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau) = \n\\sum_{n=0}^{\\infty} b_{n}q^n.\n\\qquad \\left( b_{n} = \\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde\na}_{n}^{(l)} \\right) \\label{eq:expf}\n\\end{eqnarray} \nDue to the Tauberian theorem, the asymptotic form of\n$b_{n}$ becomes:\n\\begin{eqnarray} \nb_{n} = \\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde a}_{n}^{(l)} \\sim\n\\frac{c(k)}{\\sqrt{2}}n^{- \\frac{1}{2}} \\exp \\left( 2 \\pi\n\\sqrt{(k-\\frac{(3+c_{k-1})}{6}) n} \\right) \\qquad {\\rm as} \\ n \\rightarrow\n\\infty. \\label{eq:numpri}\n\\end{eqnarray}\nTherefore, we conclude that the degeneracy of the massive\nprimary state of the dimension $h=n+k\/4$ \nat $n\\rightarrow\\infty$ is given by eq.(\\ref{eq:numpri}).\n\nAccording to the argument in section 3.2, the degeneracy of the \nprimary state (\\ref{eq:prisigma}) corresponds to the degeneracy \nof the microscopic states of the extremal BTZ black hole \n${\\rm BTZ}_{(N,N\/l)}$. At the limit $N\\gg k \\gg 1$\\footnote{The \ndifference between $\\sum_{l}{\\tilde a}_{n}^{(l)}$ and\n$\\sum_{l}a_{n}^{(l)}$ is \nirrelevant in this semiclassical limit.}, that is, the semiclassical limit of\nthree-dimensional gravity, the degeneracy of the state \n(\\ref{eq:extprimary}) becomes   \n\\begin{eqnarray} \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde a}_{N}^{(l)} \\sim\n\\frac{c(k)}{\\sqrt{2}}N^{- \\frac{1}{2}} \\exp (2 \\pi\n\\sqrt{(k-1) N}) \\qquad {\\rm as} \\ N \\rightarrow \\infty, \\label{eq:microent}\n\\end{eqnarray}\nwhere $k=Q_{1}Q_{5}+1$.\nThe logarithm of eq.(\\ref{eq:microent}) can be regarded as\nthe microscopic entropy of the extremal BTZ black hole with $Ml=J=N$.\nIt becomes\n\\begin{eqnarray}\nS_{{\\rm micro}} = 2 \\pi \\sqrt{Q_{1}Q_{5}N} + {\\cal O}(\\log N, \\\n\\log c(k)).\n\\end{eqnarray} \nThis completely agrees with the entropy formula\neq.(\\ref{eq:extentropy}). \nThis provides a justification of the\nidentification of the extremal BTZ black hole states with\nthe primary states (\\ref{eq:extprimary}) of the N=(4,4) $\\sigma$-model.\n\n\\section{Discussion} \n\nUntil now, our study is limited to the case of the extremal BTZ black holes.\nThe non-extremal BTZ black holes can also appear as the near\nhorizon geometry of the non-BPS solitonic solutions in IIB supergravity.\nAccording to the duality, the non-extremal BTZ black\nholes may be also identified with the Virasoro primary states \n\\begin{eqnarray}\n   |h,l=0\\rangle \\otimes |{\\tilde h},{\\tilde l}=0\\rangle  \\ \\quad {\\rm with}\n   \\quad h, {\\tilde h} > \\frac{k}{4}, \\label{eq:nonextpri}\n\\end{eqnarray}\nin the corresponding N=(4,4) $\\sigma$-model.\n\nThese states are in the tensor product of the massive representations both in\nthe left and right moving sectors.\nSo we must consider not the elliptic genus but the full\npartition function of the N=(4,4) $\\sigma$-model for the counting of\nthe degeneracy of the states.\nIt is known that the full partition function, which depends on the moduli\nof $S^k K3$, has the contributions from the massive primary states of\n$h={\\bf Q}_{>0}+k\/4$ (${\\bf Q}$ : rational numbers)\\cite{Eguchi4}. \nTherefore, the patition\nfunction and the counterparts of $\\sum_{l}F_{l}(\\tau)$ can not have the\nforms (\\ref{eq:expgenus}) and (\\ref{eq:expf}). So the Tauberian\ntheorem can not be applied to the counting of the primary state \n(\\ref{eq:nonextpri}). \nWe need the further investigations\nfor the well-defined counting of the microscopic states of \nthe non-extremal BTZ black holes. \n\nThrough this paper, we have discussed only the case of $M_{4}=K3$. \nThe similar arguments in section 2.2 hold for        \nthe case of $M_{4}=T^4$. However the elliptic genus of the\ncorresponding N=(4,4) $\\sigma$-model vanishes identically, since\nthis $\\sigma$-model has the extra $U(1)^{4}$ symmetry other than\nN=4 superconformal symmetry. So one cannot count the degeneracy\nof the state (\\ref{eq:extprimary}) by means of the elliptic genus.\nIn \\cite{Maldacena3}, the counting of the 1\/8 BPS states has been\nargued by using another topological index called new supersymmetric\nindex. And they pointed out that the representations\nof large N=4 superconformal algebra must be considered.\nWe may expect that the similar argument in this paper can be carried out\nvia the relation between this new supersymmetric\nindex and the characters of large N=4 superconformal algebra.\n\n{\\bf Acknowledgements} \\\\\nThe authors thank H. Umetsu and D. Tomino \nfor their collaboration at the early stage of this work and \nalso for their useful discussions and comments. \nThe authors thank also T. Kawai for his useful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSuppose $X$ is a smooth\ncompact manifold and $\\varphi_t:X\\to X$ is an Anosov flow\ngenerated by a smooth vector field $ V $, $ \\varphi_t := \\exp t V $.\nCorrelation functions for a flow are defined as\n\\begin{equation}\n\\label{eq:corr}\n\\rho_{f,g} ( t ) := \\int_X f ( \\varphi_{-t} ( x ) ) g ( x ) dx , \\ \\\nf , g \\in C^\\infty ( X ) , \\ \\ t > 0 ,\n\\end{equation}\nwhere $ dx $ is a Lebesgue density on $ X $.\nThe power spectrum is defined as the (inverse) Fourier-Laplace transform\nof $ \\rho_{f,g} $:\n\\begin{equation}\n\\label{eq:powersp}\n\\widehat \\rho_{f,g} ( \\lambda ) := \\int_0^\\infty \\rho_{f,g} ( t ) e^{ i \\lambda t } dt,\n\\ \\ \\ \\Im \\lambda > 0 .\n\\end{equation}\nFaure--Sj\\\"ostrand \\cite{FaSj} proved that\n\\[   ( P - \\lambda)^{-1}  : C^\\infty ( X )\n\\to {\\mathcal D}' ( X ) , \\ \\ P := \\frac 1 i V , \\ \\ \\Im \\lambda \\gg 1 , \\]\ncontinues to a meromorphic family of operators on all of $ \\mathbb C $. Using the fact\nthat $ f ( \\varphi_{-t} ( x ) ) = [\\exp ( - i t P ) f ] ( x ) $ this easily shows\nthat $ \\widehat \\rho_{f,g} ( \\lambda ) $ has a meromorphic continuation.\nThe poles of this continuation depend only on $ P $ and their study\nwas initiated in the work of Ruelle \\cite{Rue} and Pollicott \\cite{Po}.\nThey are called {\\em Pollicott--Ruelle resonances} and their set is denoted\nby $ \\Res ( P )$. The finer properties of the correlations are then\nrelated to the distribution of these resonances. This is particularly\nclear in the work of Liverani \\cite{Liv} and Tsujii \\cite{Ts} on contact\nAnosov flows, see also Nonnenmacher--Zworski \\cite{NZ} for semiclassical generalizations.\n\nAn equivalent definition of Pollicott--Ruelle resonances was given\nby Dyatlov--Zworski \\cite{DZ2}: they are limits (with multiplicities)\nof the eigenvalues of $ P + i \\epsilon \\Delta_g $, $ \\epsilon \\to 0 + $,\nwhere $ - \\Delta_g \\geq 0 $ is a Laplacian for some { Riemannian metric $g$} on $ X $.\nBecause of a connection to Brownian motion this shows stochastic stability of these resonances.\n\nIn this note we address the basic question about the size of the set of\nresonances: is their number always infinite? Despite the long\ntradition of the subject this appeared to be unknown for arbitrary\nAnosov flows on compact manifolds.\nIn Theorem \\ref{thm2}, we show that\nin sufficiently large strips the counting function of resonances cannot\nbe bounded by $r^\\delta$, $ \\delta < 1 $.\n\n\\begin{figure}\n\\includegraphics[width=6in]{gaps}\n\\caption{The {\\em spectral gap} $ \\nu_0 $ is the supremum of $ \\nu $\nsuch that there are {\\em no} resonances with $ - \\nu < \\Im \\lambda $,\n$ \\lambda \\neq 0 $.\nFor contact Anosov flows it is known that $ \\nu_0 > 0 $ \\cite{Liv},\\cite{NZ},\\cite{Ts}.\nThe {\\em essential spectral gap}, $  \\nu_1 $, is the supremum of $ \\nu $\nsuch that there are only finitely many resonances with $ \\Im \\lambda > - \\nu $.\nOur result states that the essential spectral gap is finite for any\nAnosov flow on a compact manifold.}\n\\end{figure}\n\n{General upper bounds\non the number of resonances in strips were established by\nFaure--Sj\\\"ostrand \\cite{FaSj} (and with a sharper \nexponent in the case of contact flows by Datchev--Dyatlov--Zworski \\cite{DDZ}):\nfor any $A > 0 $ there exists $ C $ such that\n\\begin{equation}\n\\label{eq:counting}\n\\#(\\Res(P)\\cap\\{\\Im\\mu>-A, |\\Re\\mu - r |\\leq \\sqrt r\\})\n\\leq Cr^{n- \\frac12} .\n\\end{equation}}\nOn the other hand, for contact Anosov flows satisfying certain pinching conditions on Lyapunov exponents, Faure--Tsujii \\cite{FaTs} showed that\nthe resonances satisfy a precise counting law in strips, agreeing with\nthe upper bound of \\cite{DDZ}. That is a far reaching generalization of the results known in constant curvature: see Dyatlov--Faure--Guillarmou \\cite{DFG} for recent results in that case and references.\n\nThe new counting result is proved by\nestablishing a local trace formula relating resonances to periods of\nclosed trajectories and to the their Poincar\\'e maps. Hence we denote\nby  $\\mathcal{G}$ periodic orbits $\\gamma$ of the flow, by  $T_\\gamma$ the period of $\\gamma$ and by $T_\\gamma^\\#$ the primitive period. We let $\\mathcal{P}_\\gamma$ be the linearized Poincar\\'{e} map -- see \\S \\ref{pr}.\nWith this notation we can state our {\\em local trace formula}:\n\\begin{thm}\n\\label{thm1}\nFor any $A>0$ there exists\na distribution $F_A\\in\\mathcal {S}'(\\mathbb{R})$ supported in $[0,\\infty)$ such that\n\\begin{equation}\n\\label{localtrace}\n\\sum_{\\mu\\in\\Res(P), \\Im\\mu>-A}e^{-i\\mu t}+F_A(t)\n=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^{\\#}\\delta(t-T_\\gamma)}\n{|\\det(I-\\mathcal{P}_\\gamma)|},\\;\\;\\;\\; t>0\n\\end{equation}\nin the sense of distribution on $(0,\\infty)$.\nMoreover, the Fourier transform of $ F_A $ has an analytic extension\nto $\\Im\\lambda <  A$ which satisfies,\n\\begin{equation}\n\\label{error}\n|\\widehat{F}_A(\\lambda) |=\\mathcal{O}_{A,\\epsilon}( \\langle \\lambda \\rangle^{2n+1}), \\ \\\n\\Im\\lambda < A - \\epsilon, \\text{ for any $\\epsilon>0$.}\n\\end{equation}\n\\end{thm}\n\nThe trace formula \\eqref{localtrace} can be motivated as follows.\nFor the case of geodesic flows of compact Riemann surfaces,\n$X=S^*(\\Gamma\\backslash\\mathbb{H}^2)$, where $ \\Gamma $ is\nco-compact subgroup of $ {\\rm{SL}}_2 ( \\mathbb R) $,\nand $\\varphi_t$ is the geodesic flow, we have a global trace formula:\n\\begin{equation}\n\\label{globaltrace}\n\\sum_{\\mu\\in\\Res(P)}e^{-i\\mu t}=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|},\\;\\;\\; t>0.\n\\end{equation}\nHere the set of resonances is given by\n$\\Res(P)=\\{\\mu_{j,k}=\\lambda_j-i(k+\\frac{1}{2}),j,k\\in\\mathbb{N}\\}$ (up to exceptional values on the imaginary axis),\nwhere $\\lambda_j$'s are the eigenvalues\nof the Laplacian on $ \\Gamma \\backslash \\mathbb H^2 $.  This follows\nfrom the Atiyah--Bott--Guillemin trace formula and\nthe Selberg trace formula -- see \\cite{DFG} and references given there.\n{  The bound $ \\langle \\lambda \\rangle^{ 2 n +1 } $ in \n\\eqref{error} is probably not\noptimal and comes from very general estimates presented in \\S \\ref{estflt}.\nIt is possible that \\eqref{globaltrace} is valid for all Anosov flows.}\n\n\n\nMelrose's Poisson formula for\nresonances valid for Euclidean infinities \\cite{Me,SZ,Z1} and some\nhyperbolic infinities \\cite{GZ} suggests that \\eqref{globaltrace}\ncould be valid for general Anosov flows but that is unknown.\n\nIn general, the validity of \\eqref{globaltrace} follows from, but is not equivalent to, the finite order (as an entire function) of the analytic continuation of\n\\begin{equation}\n\\label{zeta}\n\\zeta_1(\\lambda) { :=} \\exp\\left(-\\sum_\\gamma\\frac{T_\\gamma^\\#e^{i\\lambda T_\\gamma}}{T_\\gamma|\\det(I-\\mathcal{P}_\\gamma)|}\\right).\n\\end{equation}\nThis finite order property is only known under certain analyticity assumptions on $X$ and $\\varphi_t$ -- see Rugh \\cite{Ru} and Fried \\cite{Fr}. {  \nThe notation $ \\zeta_1 $ is motivated by the factorization of the Ruelle\nzeta function -- see \\cite[(2.5)]{DZ}.}\n\nAs a consequence of the local trace formula \\eqref{localtrace},\nwe have the following weak lower bound on the number of resonances in a sufficiently wide strip near the real axis. It is formulated using the Hardy-Littlewood notation: $f=\\Omega(g)$ if it is\nnot true that $|f|=o(|g|)$.\n\n\\begin{thm}\n\\label{thm2}\nFor every $\\delta\\in(0,1)$ there exists a constant $A_\\delta>0$ such that if $A>A_\\delta$, then\n\\begin{equation}\n\\label{lowerbound}\n\\#(\\Res(P)\\cap\\{\\mu \\in \\mathbb C \\, : \\, |\\mu|\\leq r, \\ \\Im\\mu>-A\\})=\\Omega(r^\\delta).\n\\end{equation}\nIn particular, there are infinitely many resonances in any strip $\\Im\\mu>-A$ for $A$ sufficiently large. \n\\end{thm}\n\n\\medskip\n\\noindent{\\bf Remarks.} 1. { An explicit bound for the constant $A_\\delta$ is given by \\eqref{adelta} in the proof. This also gives an explicit bound $A_0=\\inf\\{A_\\delta:0<\\delta<1\\}$ for the essential spectral gap.} In the case of analytic semiflows (see \\cite{Naud1}) Fr\\'ederic Naud \\cite{Naud} pointed out that a better estimate of the essential spectral gap is possible: there are infinitely many resonances in any strip\n$ \\Im \\lambda > - \\frac32 P ( 2 ) - \\epsilon $, where \n{  $ P  (s ) := P ( s \\psi^u ) $ is the\ntopological pressure associated to the unstable Jacobian -- see \\eqref{eq:press} and \\eqref{eq:psiu}}.\nIn Appendix \\ref{weakmix}, Fr\\'ed\\'eric Naud shows how similar methods and Theorem \\ref{thm1} give\na narrower strip with infinitely many resonances for weakly mixing\nAnosov flows.\n\n\\noindent\n2. In the case of flows obtained by suspending Anosov maps the \ngrowth of the number of resonances in strips is linear -- see\nAppendix \\ref{suspe} by Fr\\'ederic Naud \nfor a detailed discussion of analytic perturbations of linear \nmaps. That means that the exponent $ \\delta $ close to one is\nclose to be optimal in general.\n\n\\medskip\n\n\nThe proof of Theorem \\ref{thm1} uses the microlocal approach to\nAnosov dynamics due to Faure--Sj\\\"ostrand \\cite{FaSj} and\nDyatlov--Zworski \\cite{DZ}. In particular we use the fact that\n\\begin{equation*}\n\\frac{d}{d\\lambda} \\log \\zeta_1 ( \\lambda ) = \\tr^\\flat e^{ i \\lambda t_0 } \\varphi_{-t_0}^* ( P - \\lambda)^{-1} ,\n\\end{equation*}\nand that the right hand side continues meromorphically with poles\nwith integral residues. Here the flat trace, $ \\tr^\\flat $,  is defined using a formal integration over the diagonal,  see \\S \\ref{flat},  with the justification provided by the crucial wave front set relation, see \\S \\ref{wavefront}. Some of the techniques are also related to the proof of Sj\\\"{o}strand's local trace formula for scattering resonances in the semiclassical limit \\cite{S}. It is possible that an alternative\nproof of Theorem \\ref{thm1} could be obtained using the methods of Giulietti--Liverani--Pollicott \\cite{GLP} employed in their proof of Smale's conjecture about zeta function (\\cite{DZ} provided a simple microlocal proof of that conjecture).\n\nThe proof of Theorem \\ref{thm2} is based on the proof of a similar\nresult in Guillop\\'{e}--Zworski \\cite{GZ} which in turn was inspired\nby the work of Ikawa \\cite{Ik} on existence of resonances in\nscattering by several convex bodies.\n\n\\def\\smallsection#1{\\smallskip\\noindent\\textbf{#1}.}\n\n\\smallsection{Acknowledgements}\nWe would like to thank Semyon Dyatlov for helpful discussions and in particular for suggesting the decomposition \\eqref{decomposition} which simplified the wave front arguments. We are also grateful to Fr\\'ed\\'eric Naud for sharing his\nunpublished work \\cite{Naud} with us and to the anonymous referee for\nuseful suggestions. This material is based\nupon work supported by\nthe National Science Foundation under the grant and DMS-1201417.\n\n\\smallsection{Notation} We use the following notational\nconventions: $ \\langle x \\rangle := ( 1 + |x|^2 )^{\\frac12} $,\n$ \\langle u , \\varphi \\rangle$, for the the distributional\npairing of $ u \\in \\mathcal D' ( X ) $ (distributions on a compact\nmanifold $ X $), and $ \\langle u , v \\rangle_{ H}$ for\nthe Hilbert space inner product on $ H $.\nWe write $ f =  \\mathcal O_\\ell ( g)_B $ to mean that\n$ \\|f \\|_B  \\leq C_\\ell  g $ where the norm (or any seminorm) is in the\nspace $ B$, and the constant $ C_\\ell  $ depends on $ \\ell $. When either $ \\ell $ or $ B $ are absent then the constant is universal or the estimate is scalar, respectively. When $ G = \\mathcal O_\\ell ( g )_{B_1\\to B_2 } $ then the operator $ B : H_1  \\to H_2 $ has its norm bounded by $ C_\\ell g $. By $ \\neigh_U ( \\rho ) $\nwe mean a (small) neighbourhood of $ \\rho$ in the space $ U$.\nWe refer to \\cite{DZ} and \\cite{Z2} for the notational conventions from microlocal\/semiclassical analysis as they appear in the text.\n\n\\section{Preliminaries}\n\\label{pr}\n\\subsection{Anosov flows}\nLet $X$ be a compact Riemannian manifold, $V\\in C^\\infty(X;TX)$ be\na smooth non vanishing vector field and\nand $\\varphi_t=\\exp tV:X\\to X$ the corresponding flow.\n\nThe flow is called an {\\em Anosov flow}\nif the tangent space to $X$ has a continuous decomposition\n$T_xX=E_0(x)\\oplus E_s(x)\\oplus E_u(x)$ which is invariant under the flow: $d\\varphi_t(x)E_\\bullet(X)=E_\\bullet(\\varphi_t(X))$,\n$\\bullet=s,u$, $E_0(x)=\\mathbb{R}V(x)$,\nand satisfies\n\\begin{equation*}\n\\begin{split}\n|d\\varphi_t(x)v|_{\\varphi_t(x)}\\leq Ce^{-\\theta|t|}|v|_x,&\\;\\;\\; v\\in E_u(x), \\ \\ t<0\\\\\n|d\\varphi_t(x)v|_{\\varphi_t(x)}\\leq Ce^{-\\theta|t|}|v|_x,&\\;\\;\\; v\\in E_s(x),\n\\ \\ t>0,\n\\end{split}\n\\end{equation*}\nfor some fixed $C$ and $\\theta>0$.\n\n\\subsection{Anisotropic Sobolev spaces}\nLet us put $P=-iV:C^\\infty(X)\\to C^\\infty(X)$;\nthen the principal symbol of $ P $, $p\\in S^1(T^*X)$ (see\n\\cite[\\S 18.1]{H3} or \\cite[\\S 14.2]{Z2} for this standard notation; an\noverview of semiclassical and microlocal preliminaries needed in this\npaper can be found in \\cite[\\S 2.3]{DZ}) is given by\n$p(x,\\xi)=\\xi(V(x))$ which is homogeneous of degree 1.\nThe Hamilton flow of $ p $ is\nthe symplectic lift of $ \\varphi_t $ to the cotangent bundle:\n$e^{tH_p}(x,\\xi)=(\\varphi_t(x),({}^Td\\varphi_t(x))^{-1}\\xi)$.\nWe can define the dual decomposition $T^\\ast_xX=E^\\ast_0(x)\\oplus E^\\ast_s(x)\\oplus E^\\ast_u(x)$ where $E^\\ast_0(x),E^\\ast_s(x),E^\\ast_u(x)$ are dual to $E_0(x),E_u(x),E_s(x)$, respectively. Then\n\\begin{equation*}\n\\begin{split}\n\\xi\\not\\in E^\\ast_0(x)\\oplus E^\\ast_s(x)\\Rightarrow d(\\kappa(e^{tH_p}(x,\\xi)),\\kappa(E^\\ast_u))\\to0 \\text{ as } t\\to+\\infty\\\\\n\\xi\\not\\in E^\\ast_0(x)\\oplus E^\\ast_u(x)\\Rightarrow d(\\kappa(e^{tH_p}(x,\\xi)),\\kappa(E^\\ast_s))\\to0 \\text{ as } t\\to-\\infty.\n\\end{split}\n\\end{equation*}\nHere $\\kappa:T^\\ast X\\setminus0\\to S^\\ast X := T^*X \/ \\mathbb R_+ $\nis the natural projection.\n\nA microlocal version of anisotropic Sobolev spaces of\nBlank--Keller--Liverani~\\cite{BKL}, Baladi--Tsujii~\\cite{BT}\nand other authors was provided by Faure-Sj\\\"{o}strand \\cite{FaSj}.\nHere we used a simplified version from Dyatlov-Zworski \\cite{DZ}. For that\nwe construct a function $m_G\\in C^\\infty(T^\\ast X\\setminus0;[-1,1])$ which is homogeneous of degree 0, is supported in a small neighbourhood of $E_s^\\ast\\cup E_u^\\ast$ and satisfies\n\\begin{equation*}\nm_G=1 \\text{ near } E_s^\\ast;\\;\\;\\  m_G=-1 \\text{ near } E_u^\\ast;\\;\\;\\ H_pm_G\\leq0 \\text{ everywhere. }\n\\end{equation*}\nNext, we choose a pseudodifferential operator\n$ G\\in\\Psi^{0+}(X)$, {  $ \\sigma(G)=m_G(x,\\xi)\\log\\langle\\xi\\rangle $}.\nThen for $ s > 0 $, $\\exp(\\pm sG)\\in\\Psi^{s+}(X)$ -- see \\cite[\\S 8.2]{Z2}. The anisotropic Sobolev spaces are defined as\n\\begin{equation*}\nH_{sG}:=\\exp(-sG)L^2(X), \\ \\ \\ \\|u\\|_{H_{sG}}: =\\|\\exp(sG)u\\|_{L^2}.\n\\end{equation*}\nBy the construction of $G$, we have $H^s\\subset H_{sG}\\subset H^{-s}$.\n\n\\subsection{Properties of Resolvent}\nWe quote the following results about the resolvent of $P$, see \\cite[Propositions 3.1, 3.2]{DZ}:\n\\begin{lem}\nFix a constant $C_0>0$. Then for $s>0$ large enough depending on $C_0$, $P-\\lambda:D_{sG}\\to H_{sG}$ is a Fredholm operator of index 0 in the region $\\{\\Im\\lambda>-C_0\\}$. Here the domain $D_{sG}$ of $P$ is the set of $u\\in H_{sG}$ such that $Pu$ (in the distribution sense) is in $H_{sG}$ and it is a Hilbert space with norm $\\|u\\|_{D_{sG}}^2=\\|u\\|_{H_{sG}}^2+\\|Pu\\|_{H_{sG}}^2$.\n\\end{lem}\n\n\\begin{lem}\nLet $s>0$ be fixed as above. Then there exists a constant $C_1$ depending on $s$, such that for $\\Im\\lambda>C_1$, the operator $P-\\lambda:D_{sG}\\to H_{sG}$ is invertible and\n\\begin{equation}\n\\label{resolvent}\n(P-\\lambda)^{-1}=i\\int_0^\\infty e^{i\\lambda t}\\varphi_{-t}^\\ast dt,\n\\end{equation}\nwhere $\\varphi_{-t}^\\ast=e^{-itP}$ is the pull back operator by $\\varphi_t$. The integral converges in operator norm $H^s\\to H^s$ and $H^{-s}\\to H^{-s}$.\n\\end{lem}\n\nThe analytic Fredholm theory now shows that the resolvent $ \\lambda\n\\mapsto R(\\lambda)=(P-\\lambda)^{-1}:H_{sG}\\to H_{sG}$ forms a meromorphic family of operators with poles of finite rank. In the region $\\Im\\lambda>-C_0$, the Ruelle-Pollicott resonances are defined as the poles of $R(\\lambda)$. They can be described as the meromorphic continuation of the Schwartz kernel of the operator on the right-hand side, thus are independent of the choice of $s$ and the weight $G$.\nThe mapping properties of $ ( P - \\lambda)^{-1}$ and formula \\eqref{resolvent}\nshow that the power spectrum \\eqref{eq:powersp} has a meromorphic\ncontinuation with the same poles. We note here that our definition\n\\eqref{eq:powersp} is different from the definition in \\cite{Rue} but\nthe formula there can be expressed in terms of \\eqref{eq:powersp}.\n\n{We recall the following general upper bounds on the number of resonances from \nFaure--Sj\\\"ostrand \\cite{FaSj}:\n\\begin{prop}\nLet $\\Res(P)$ be the set of Ruelle-Pollicott resonances. Then for any $C_0>0$,\n\\begin{equation}\n\\label{hupperbound}\n\\#(h\\Res(P))\\cap D(1,C_0 h^{\\frac12} )=\\mathcal{O}(h^{-n+ \\frac12}),\n\\end{equation}\nwhich is equivalent to \\eqref{eq:counting}. In particular,\n\\begin{equation}\n\\label{upperbound}\n\\#\\Res(P)\\cap\\{\\mu:|\\Re\\mu|\\leq r,\\Im\\mu>-C_0\\}=\\mathcal{O}(r^n). \n\\end{equation}\n\\end{prop}}\n\n\\subsection{Complex absorbing potentials}\nIt is convenient to introduce a semiclassical parameter $h$ and to consider the operator $hP\\in\\Psi_h^1(X)$ (for the definitions of pseudodifferential\noperators and wave front sets we\nrefer to \\cite[\\S 14.2]{Z2} and  \\cite[\\S 2.3, Appendix C]{DZ})\n with semiclassical principal symbol $p=\\sigma_h(hP)(x,\\xi)=\\xi(V_x)$.\nThen we introduce a semiclassical adaption $G(h)\\in\\Psi_h^{0+}(X)$ of the operator $G$ with\n\\begin{equation*}\n\\sigma_h(G(h))=(1-\\chi(x,\\xi))m_G(x,\\xi)\\log|\\xi|,\n\\end{equation*}\nwhere $\\chi\\in C_0^\\infty(T^\\ast X)$ is equal to 1 near the zero section. In this way, $H_{sG(h)}=H_{sG}$ but with a new norm depending on $ h $. We also\ndefine an $h$-dependent norm on the domain of $hP$,  $D_{sG(h)} = D_{s G} $:\n\\begin{equation*}\n\\|u\\|_{D_{sG(h)}} :=\\|u\\|_{H_{sG(h)}}+\\|hPu\\|_{H_{sG(h)}}.\n\\end{equation*}\n\nNow we modify $hP$ by adding a semiclassical pseudodifferential complex absorbing potential $-iQ_\\delta\\in\\Psi_h^0(X)$ which is localized to a neighbourhood of the zero section:\n\\begin{equation}\n\\label{eq:WFQd}\n\\WF_h(Q_\\delta)\\subset\\{|\\xi|<\\delta\\}, \\ \\ \\sigma_h(Q_\\delta)>0 \\text{ on } \\{|\\xi|\\leq\\delta\/2\\}, \\ \\ \\sigma_h(Q_\\delta)\\geqslant0 \\text{ everywhere}.\n\\end{equation}\n{  (For the definition of $ \\WF_h ( A ) \\subset \\overline T^* X $ and\nof the compactified cotangent bundle $ \\overline T^* X $, \nsee \\cite[\\S C.2]{DZ}.)}\nInstead of $P_h(z)=hP-z$, we consider the operator $P^\\delta_h(z)=hP-iQ_\\delta-z$ acting on $H_{sG(h)}$ which is equivalent to the conjugated operator\n\\begin{equation}\n\\label{eq:conj}\nP^{\\delta,s}_h(z)=e^{sG(h)}P_\\delta(z)e^{-sG(h)}=P^\\delta_h (z)+s[G(h),hP]+\\mathcal{O}(h^2)_{\\Psi_h^{-1+}}\n\\end{equation}\nacting on $L^2$. We recall the crucial \\cite[Proposition 3.4]{DZ}:\n\\begin{lem}\n\\label{l:3.4}\nFix a constant $C_0>0$ and $\\delta >0$. Then for $s>0$ large enough depending on $C_0$ and $h$ small enough, the operator\n\\begin{equation*}\nP^\\delta_h(z):D_{sG(h)}\\to H_{sG(h)}, \\ \\ \\ -C_0h\\leq\\Im z\\leq1, \\ \\\n |\\Re z|\\leq 2h^{1\/2},\n\\end{equation*}\nis invertible, and the inverse $R^\\delta_h(z)$, satisfies\n$\\|R^\\delta_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}$.\n\\end{lem}\n\n\\subsection{Finite rank approximation}\n\\label{fra}\nFor our application we need to make $ Q_\\delta $ a finite rank\noperator.\nIt is also convenient to make a further assumption on the\nsymbol of $ Q_\\delta$.\nAs long as \\eqref{eq:WFQd} holds,\nLemma \\ref{l:3.4} still applies.\n\nFrom now on we fix $ \\delta > 0 $ and put\n\\[  Q = Q_\\delta = f ( - h^2 \\Delta_g ), \\ \\ f \\in  {C^\\infty_{\\rm{c}}} ( ( -2 \\delta, 2 \\delta ),\n[ 0 , 1 ]) , \\ \\\nf ( s ) = 1, \\ \\ |s |\\leq \\delta . \\]\nThen (see for instance \\cite[Theorem 14.9]{Z2})\n\\begin{equation}\n\\label{finiterank}\n\\rank Q=\\mathcal O (h^{-n}),  \\ \\ Q \\geq 0 , \\ \\ \\sigma_h ( Q ) = f ( |\\xi|_g^2 ).\n\\end{equation}\nFor technical convenience only (so that we can\ncite easily available results in the proof of Proposition \\ref{flattracees}\nin the appendix)\nwe make an additional assumption on $ f$: for some $ 0 < \\alpha < \\frac12$,\n\\begin{equation}\n\\label{eq:condfk}\n| f^{(k)} ( x ) | \\leq C_k f ( x ) ^{1- \\alpha} .\n\\end{equation}\nThis can be achieved by building $ f$ from functions of the form equal to $\ne^{-1\/x} $ for $ x > 0 $ and $ 0 $ for $ x \\leq 0 $. (In that case\n\\eqref{eq:condfk} holds for all $ \\alpha > 0 $.)\n\n\nLemma \\ref{l:3.4} shows that\nfor  $-C_0h\\leq\\Im z\\leq 1\n$, $|\\Re z|\\leq 2h^{1\/2}$, \n\\begin{equation}\n\\label{eq:wideP} \n\\widetilde{P}_h(z): =hP-iQ-z, \n\\end{equation} \nis also invertible and its inverse $\\widetilde{R}_h(z)$ satisfies\n\\begin{equation}\n\\label{modifiedresolvent}\n\\|\\widetilde{R}_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}.\n\\end{equation}\nIn the upper half plane we have the following estimate on the original resolvent:\n\\begin{equation}\n\\label{eq:orres}\n \\|R_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}, \\ \\ \\\nC_1h\\leq\\Im z\\leq1, \\ \\ |\\Re z|\\leq 2h^{1\/2},\n\\end{equation}\nprovided that $ C_1 $ is large enough. This follows from the\nFredholm property and the estimate $ \\Im \\langle e^{ s G ( h ) } P_h ( z )\ne^{ -s G ( h ) } u , u \\rangle_{L^2 }\n\\geq h \\| u\\|_{L^2} $, $ \\Im z > C_1 h $ -- see \\eqref{eq:conj}.\n\n\\subsection{Wavefront set condition}\n\\label{wavefront}\nWe need to study the wavefront set and semiclassical wavefront set of $R_h(z)$ and $\\widetilde{R}_h(z)$. For the definitions and notations of the wavefront sets and the semiclassical wavefront sets, we refer to \\cite[Chapter VIII]{H}, \\cite[Section 8.4]{Z2} and \\cite[Appendix C]{DZ} and \\cite{A}.\n\nWe recall the following wavefront set condition and semiclassical wavefront set conditions for the resolvent $R(\\lambda)$ and $\\widetilde{R}_h(z)$ from \\cite[Proposition 3.3]{DZ}. {  (For the definition of the standard wave front set $ \\WF $\nsee \\cite[\\S C.1]{DZ} and for the definition of the twisted \nwave front set $ \\WF' $, \\cite[(C.2)]{DZ} -- the reason for the twist\nis to have $ \\WF ( I ) $ equal to the diagonal in $ T^*X \\times T^* X $.)}\n\\begin{prop}\n\\label{p:3.3}\nLet $C_0$ and $s$ be as above and assume $\\lambda$ is not a resonance with $\\Im\\lambda>-C_0$, then\n\\begin{equation}\n\\label{wavefrontset}\n\\WF'(R(\\lambda))\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast),\n\\end{equation}\nwhere $\\Delta(T^\\ast X)$ is the diagonal in $T^\\ast X$ and $\\Omega_+$ is the positive flow-out of $e^{tH_p}$ on $\\{p=0\\}$:\n\\begin{equation}\n\\label{eq:Omegapl}\n\\Omega_+=\\{(e^{tH_p}(x,\\xi),x,\\xi) \\, : \\, t\\geqslant0, \\ p(x,\\xi)=0\\}.\n\\end{equation}\nAlso, if $R_h(z)=h^{-1}R(z\/h)$, then\n\\begin{equation}\n\\label{semiwf1}\n\\WF_h'(R_h(z))\\cap T^\\ast(X\\times X)\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast),\n\\end{equation}\nand\n\\begin{equation}\n\\WF_h'(R_h(z))\\cap S^\\ast(X\\times X)\\subset\\kappa(\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}).\n\\end{equation}\n\\end{prop}\n\nNow we determine the wavefront set and the semiclassical wavefront set of $\\widetilde{R}_h(z)$. First, \n{  by inserting the resolvent formula\n$$\\widetilde{R}_h(z)=R_h(z)+iR_h(z)Q\\tilde{R}_h(z)$$\ninto another resolvent formula\n$$\\widetilde{R}_h(z)=R_h(z)+i\\tilde{R}_h(z)QR_h(z),$$}\nwe write\n$$\\widetilde{R}_h(z) = R_h ( z ) + i R_h ( z ) Q R_h ( z ) -\nR_h ( z ) Q \\widetilde{R}_h ( z ) Q R_h ( z ).$$\nThen since $Q$ is a smoothing operator, $\\WF(Q)=\\emptyset$, we have\n$$\\WF'(R_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast.$$\nSimilarly, since $Q\\widetilde{R}_h(z)Q$ is also a smoothing operator,\n$$\\WF'(R_h(z)Q\\widetilde{R}_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast.$$\nTherefore we get the same wavefront set condition as $R_h(z)$:\n\\begin{equation}\n\\WF'(\\widetilde{R}_h(z))\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast).\n\\end{equation}\n\nFor the semiclassical wavefront set, we already know from \\cite[Proposition 3.4]{DZ} that\n\\begin{equation}\n\\label{eq:WFRh} \\WF_h'(\\widetilde{R}_h(z))\\cap T^\\ast(X\\times X)\\subset\\Delta(T^\\ast X)\\cup\\Omega_+.\n\\end{equation}\nMoreover, since $\\WF_h'(Q)\\cap S^\\ast(X\\times X)=\\emptyset$, we have\n$$\\WF_h'(R_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast,$$\nand similarly, $\\WF_h'(Q\\widetilde{R}_h(z)Q)\\cap S^\\ast(X\\times X)=\\emptyset$. Therefore\n\\begin{equation}\n\\label{eq:WFhR}\n\\WF_h'(\\widetilde{R}_h(z))\\cap S^\\ast(X\\times X)\\subset\n\\kappa(\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}).\n\\end{equation}\n\n\\subsection{Flat trace}\n\\label{flat}\nConsider an operator $B:C^\\infty(X)\\to\\mathcal{D}'(X)$ with\n\\begin{equation}\\label{ftc}\n\\WF'(B)\\cap\\Delta(T^\\ast X)=\\emptyset.\n\\end{equation}\nThen we can define the flat trace of $B$ as\n\\begin{equation}\n\\label{eq:flat}\n\\tr^\\flat B=\\int_X(\\iota^\\ast K_B)(x)dx:=\\langle\\iota^\\ast K_B,1\\rangle\n\\end{equation}\nwhere $\\iota:x\\mapsto (x,x)$ is the diagonal map, $K_B$ is the Schwartz kernel of $X$ with respect to the density $dx$ on $X$. The pull back $\\iota^\\ast K_B\\in\\mathcal{D}'(X)$ is well-defined under the condition \\eqref{ftc} (see \\cite[Section 8.2]{H}).\n\n\\subsection{Dynamical zeta function and Guillemin's trace formula}\nThe zeta function $ \\zeta_1 $ defined in \\eqref{zeta}\nis closely related to the Ruelle zeta function -- see \\cite{GLP},\\cite{DZ}\nand references given there.\nThe right hand side in \\eqref{zeta} converges for $\\Im\\lambda>C_1$ and\nit continues analytically to the entire plane.\n The Pollicott-Ruelle resonances are exactly the zeros of $ \\zeta_1 $.\n  We recall the (Atiyah--Bott--)Guillemin's trace formula \\cite{Gu} (see \\cite[Appendix B]{DZ} for a proof):\n\\begin{equation}\n\\label{Guillemin}\n\\tr^\\flat e^{-itP}=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}, \\ \\ \\   t>0.\n\\end{equation}\nTherefore we have\n\\begin{equation*}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)=\\frac{1}{i}\\sum_\\gamma\\frac{T_\\gamma^\\# e^{i\\lambda T_\\gamma}}{|\\det(I-\\mathcal{P}_\\gamma)|}\n=\\frac{1}{i}\\int_0^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt.\n\\end{equation*}\nFrom \\eqref{Guillemin}, $\\tr^\\flat e^{-itP}=0$ on $(0,t_0)$ if $t_0<\\inf\\{T_\\gamma:\\gamma\\in\\mathcal{G}\\}$. Formally, we can write (see \\cite[\\S 4]{DZ} for the\njustification)\n\\begin{equation*}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)\n=\\frac{1}{i}\\int_{t_0}^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt\n=\\tr^\\flat\\left(\\frac{1}{i}e^{-it_0(P-\\lambda)}\n\\int_0^\\infty e^{it\\lambda}e^{-itP}dt\\right).\n\\end{equation*}\nTherefore by \\eqref{resolvent} we have\n\\begin{equation}\n\\label{zetaresolvent}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)=\\tr^\\flat(e^{-it_0(P-\\lambda)}(P-\\lambda)^{-1}).\n\\end{equation}\nThe wavefront set condition \\eqref{wavefrontset} shows that\n\\begin{equation*}\n\\begin{split}\n& \\WF'(e^{-it_0(P-\\lambda)}(P-\\lambda)^{-1})\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\}\n\\end{split}\n\\end{equation*}\nwhich does not intersect $\\Delta(T^\\ast X)$. This justifies taking the\nflat trace \\eqref{eq:flat}.\n\nTherefore $\\frac{d}{d\\lambda} \\log\\zeta_1$ has a meromorphic continuation to all of $\\mathbb{C}$ with simple poles and\npositive integral residues. That is equivalent to having a holomorphic\ncontinuation of $ \\zeta_1 $. This strategy for proving Smale's conjecture\non the meromorphy of Ruelle zeta functions is the starting point of\nour proof of the local trace formula.\n\n\\section{Estimates on flat traces}\n\\label{estflt}\n\nThe key step in the proof of the trace formula is the following estimate on the flat trace of the propagated resolvent.\n\n\\begin{prop}\n\\label{flattracees}\nLet $\\widetilde{P}_h(z)$ and $\\widetilde{R}_h(z)$ {  be given by \n\\eqref{eq:wideP} and \\eqref{modifiedresolvent},} and let $t_0\\in(0,\\inf T_\\gamma)$. Then\n\\begin{equation}\n\\label{eq:Tofz}  T(z ) := \\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)),\n\\end{equation}\nis well defined and holomorphic in $ z $ when $ - C_0 h \\leq \\Im z \\leq  1, $\n$ |\\Re z | \\leq C_1 h^{\\frac12} $. Moreover, in that range of $ z $,\n\\begin{equation}\n\\label{eq:flattr1}\n T ( z ) = \\mathcal O_{C_0, C_1 } (h^{-2n-1}).\n\\end{equation}\n\\end{prop}\n\nThe proof is based on a quantitative study of the proof of \\cite[Theorem 8.2.4]{H} and on the wave front properties established\nin \\cite[\\S 3.3,3.4]{DZ}. The general idea is the following: the wave\nfront set condition shows that the trace is well defined. The analysis\nbased on the properties of the semiclassical wave front set shows more:\nthe contribution from a microlocal neighbourhood of fiber infinity is $ \\mathcal O ( h^\\infty ) $. The contribution away from fiber infinity can be controlled\nusing the norm estimate on $ \\widetilde R_h ( z ) $. Since the weights\ndefining the $ H_{sG} $ spaces are supported near infinity, the norm\nestimates are effectively $ L^2 $ estimates.\n\nFor the proof of \\eqref{eq:flattr1} we first review\nthe construction of the flat trace under the wave front set\ncondition. Suppose that $ u \\in \\mathcal D' ( X \\times X ) $\nsatisfies the (classical) wave front condition\n\\begin{equation}\n\\label{eq:wfcond}   \\WF ( u ) \\cap N^* \\Delta ( X ) = \\emptyset, \\ \\  \\Delta ( X )\n= \\{ ( x, x ) :  x\\in X \\} \\subset X \\times X.\n\\end{equation}\nIf $ u $ is a Schwartz kernel of an operator $ T $ then\n$ \\tr^\\flat T :=  \\langle \\iota^* u , 1 \\rangle $, where $ \\iota :\n\\Delta ( X ) \\hookrightarrow X \\times X $. We will recall why \\eqref{eq:wfcond}\nallows the definition $ \\iota^* u $.\nFor any $x_0\\in X$, we can choose a neighbourhood $U$ of $x_0$ in $X$ equipped with a local coordinate patch. For simplicity, we abuse the notation and assume $x_0\\in U\\subset \\mathbb{R}^n$. Then $\\iota(x_0)=(x_0,x_0)\\in U\\times U\\subset \\mathbb{R}^n\\times\\mathbb{R}^n$. The conormal bundle to the diagonal is locally given by\n$$N_\\iota=\n\\{(x,x,\\xi,-\\xi)\\in ( U\\times U) \\times\n( \\mathbb{R}^n\\times \\mathbb{R}^n) \\}.$$\nPut $ \\Gamma := \\WF ( u ) $ and\n$ {\\Gamma}_{(x,y)}=\\{(\\xi,\\eta) : (x, y , \\xi, \\eta )\\in\\Gamma\\}$. Then\n\\[ {\\Gamma}_{(x_0,x_0)}\\cap\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}=\\emptyset.\n\\]\nSince ${\\Gamma}_{(x_0,x_0)}$ is closed, we can find a conic neighbourhood, $V$, of $ {\\Gamma}_{(x_0,x_0)}$ in $\\mathbb{R}^n\\times\\mathbb{R}^n\\setminus0$ such that\n$$V\\cap\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}=\\emptyset.$$\nWe can also find a compact neighbourhood $Y_0$ of $(x_0,x_0)$ such that $V$ is a neighbourhood of ${\\Gamma}_{(x,y)}$ for every $(x,y)\\in Y_0$.\nNext we choose a\nneighbourhood $X_0$ of $x_0$ such that $X_0\\times X_0\\Subset Y_0$. Then we have for every $x\\in X_0, (\\xi,\\eta)\\in V$,\n\\begin{equation}\n\\label{eq:nonstat} {}^t\\iota'(x)\\cdot(\\xi,\\eta)=\\xi+\\eta\\neq0.\n\\end{equation}\n\nMoreover, we can choose $ V$ so that its complement, $\\complement V $, is\na small conic neighbourhood of $\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}$. In particular\nthere exists a constant $C>0$ such that in ${\\complement V}$, $C^{-1}|\\eta|\\leq|\\xi|\\leq C|\\eta|$. We can also assume that\n\\begin{equation}\n\\label{eq:CV}\n\\complement V = - \\complement V .\n\\end{equation}\n\nFinally we choose $\\psi(x)\\in C^\\infty(U)$ equal to 1 on $X_0$ such that $\\varphi(x,y)=\\psi(x)\\psi(y)\\in C_0^\\infty(Y_0)$, then for any $\\chi\\in C_0^\\infty(X_0)$, $u\\in C^\\infty(X\\times X)$, we have\n\\begin{equation}\n\\label{eq:extend} \\langle\\iota^\\ast u,\\chi\\rangle =\\langle\\iota^\\ast(\\varphi u),\\chi\\rangle\n=(2\\pi)^{-2n}\\int\\widehat{\\varphi u}(\\xi,\\eta)I_\\chi(\\xi,\\eta)d\\xi d\\eta,\n\\end{equation}\nwhere\n$$I_\\chi(\\xi,\\eta)=\\int \\chi(x)e^{i\\langle\\iota(x),(\\xi,\\eta)\\rangle}dx\n=\\int\\chi(x)e^{ix\\cdot(\\xi+\\eta)}dx.$$\nWe claim that as long as \\eqref{eq:wfcond} holds the right hand\nside of \\eqref{eq:extend} is well defined and hence the pull back\n$ \\iota^* u $ is a well defined distribution.\n\nTo see this, we first notice that if $(\\xi,\\eta)\\in V$, then\n\\eqref{eq:nonstat} shows that the phase is not stationary and hence,\n$|I_\\chi(\\xi,\\eta)|\\leq C_{N,\\chi}(1+|\\xi|+|\\eta|)^{-N} $, for all $N$.\nOn the other hand, we have\n\\begin{equation}\n\\label{eq:widehat} |\\widehat{\\varphi u}(\\xi,\\eta)|=\\left|\\int\\psi(x)\\psi(y)u(x,y)e^{-i(x\\cdot\\xi+y\\cdot\\eta)}dxdy\\right|.\\end{equation}\nThe construction of $ V $ and \\eqref{eq:wfcond} imply  that if $(\\xi,\\eta)\\not\\in V$, then\n$|\\widehat{\\varphi u}(\\xi,\\eta)|\\leq C_N(1+|\\xi|+|\\eta|)^{-N}$, for all $ N$.\nWhen $ ( \\xi, \\eta ) \\in V $ then, there exists $M>0$ such that\n$|\\widehat{\\varphi u}(\\xi,\\eta)|\\leq C_N(1+|\\xi|+|\\eta|)^M$.\nTherefore $\\langle\\iota^\\ast u,\\chi\\rangle$ is well defined. Now to define $\\langle\\iota^\\ast u,1\\rangle$, we first choose a finite partition of unity $1=\\sum\\chi_j$ where $\\chi_j$ is constructed as above for some $x_j\\in X$ (playing the role of $x_0$)\nand then choose the corresponding $\\psi_j$'s (playing the role of $ \\psi $).\nThis concludes our review of the proof\nthat\n$ \\tr^\\flat T = \\langle \\iota^* u  , 1\\rangle  $ is well defined when \\eqref{eq:wfcond}\nholds.\n\nAll of this can be applied to $ u = K$, the\nSchwartz kernel of  $e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)$,\nwith quantitative bounds in terms of $ h $.\nWe first estimate the wave front set of\n$e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)$. For that we\nneed the following\n\\begin{lem}\n\\label{l:prop}\nFor $ t \\geq 0 $,\n\\begin{equation*}\n\\begin{split}\n \\WF_h' ( e^{ - i t h^{-1} \\widetilde P_h ( z ) } )\n\\cap T^\\ast(X\\times X)&\n\\subset \\{ ( e^{ t_0 H_p } ( x, \\xi ) , ( x, \\xi ) ) : ( x ,\\xi ) \\in T^\\ast X \\},\\\\\n\\WF_h' ( e^{ - i t h^{-1} \\widetilde P_h ( z ) } )\n\\cap S^\\ast(X\\times X)&\n\\subset \\kappa(\\{ ( e^{ t_0 H_p } ( x, \\xi ) , ( x, \\xi ) ) : ( x ,\\xi ) \\in T^\\ast X\\setminus\\{0\\} \\}).\n\\end{split}\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nWe first note that the inclusion is obviously true for the\n$ \\WF_h' ( e^{ - i t P } ) $ since the operator is the pull back by\n$ \\varphi_{-t}^* $. Hence the statement above will follow from showing\nthat $ V ( t ) := e^{ i t P} e^{ - i t h^{-1} \\widetilde P_h ( z ) } $ is a pseudodifferential\noperator. If $ B \\in \\Psi^0_h $ satisfies\n\\[ \\WF_h ( B ) \\cap\n\\cup_{ 0 \\leq |t'| \\leq t }\\,  e^{t'H_p}  (\\WF_h ( Q ) ) = \\emptyset , \\]\nthen\n$ B e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) }  =  B + {\\mathcal O} ( h^\\infty )_{\n\\mathcal D' \\to {C^\\infty} } $.\n\nIn fact, we can use Egorov's theorem (a trivial case since $ e^{ it P } =\n\\varphi_t ^* $) to see that\n\\begin{equation}\n\\begin{split}\n h D_t  \\left( B e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) } \\right) & =\ni B e^{ it P } Q  e^{ - i t h^{-1} ( h P - i Q ) } \\\\\n& = i e^{ i t P } B ( t ) Q e^{ - i t h^{-1} ( h P - iQ ) } =\n\\mathcal O ( h^\\infty )_{ \\mathcal D' \\to {C^\\infty} } ,\n\\end{split}\n\\end{equation}\nwhere $ B ( t ) := e^{-  i t P } B e^{ i t P  } $ satisfies\n$ \\WF_h ( B ( t ) ) \\cap \\WF_h ( Q ) = \\emptyset $.\nBy switching the sign of $ P $ and taking adjoints we see\nthat the we also have\n$$  e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) }  B =  B + {\\mathcal O} ( h^\\infty )_{\n\\mathcal D' \\to {C^\\infty} } .$$\nHence it is enough to prove that, for $ \\alpha $ in \\eqref{eq:condfk},\n$ e^{ i tP } e^{ - i h^{-1} t ( P - i Q ) } A \\in \\Psi_\\alpha ( X ) $,\n$ \\alpha < {\\frac12} $, for $ A \\in \\Psi^{\\rm{comp}}( X ) $. But that\nis included in \\cite[Proposition { A}.3]{NZ}.\n\\end{proof}\n\n\\medskip\n\\noindent\n{\\bf Remark.} {  \nThe assumption \\eqref{eq:condfk} in the construction of $ \\widetilde P_h ( z ) $ and used in the proof of Lemma \\ref{l:prop}\nis made for convenience only as we can then cite\n\\cite[Proposition { A}.3]{NZ}. }\n\n\\medskip\n\nInclusions \\eqref{eq:WFRh} and \\eqref{eq:WFhR} and\nLemma \\ref{l:prop} show that\n\\begin{equation*}\n\\begin{split}\n& \\WF_h'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n\\cap T^\\ast(X\\times X)\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\}\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n& \\WF_h'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n\\cap S^\\ast(X\\times X)\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\kappa\\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}\\}.\n\\end{split}\n\\end{equation*}\nIn particular, for all $0 < h < 1 $,\n\\begin{equation*}\n\\begin{split}\n& \\WF'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\},\n\\end{split}\n\\end{equation*}\n{  satisfying \\eqref{ftc},} that is, does not intersect with $\\Delta(T^\\ast X)$ and hence $\\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))$ is well-defined.\n\n\nUsing a microlocal partition of unity,\n$ I=\\sum_{j=1}^J B_j + \\mathcal O ( h^\\infty )_{ \\mathcal D'\n\\to {C^\\infty} } $, $ B_j\\in\\Psi^0_h(X)$ {  (see for instance \\cite[Proposition E.34]{res})}\nwe only need to prove that\n\\[\n\\begin{split}\n& {\\rm(i)}  \\WF_h(B) \\subset \\neigh_{ T^* X } ( x_0, \\xi_0 ) ,  \\\n(x_0,\\xi_0)\\in T^\\ast X  \\Rightarrow\n\\tr^\\flat  e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B\n= \\mathcal O ( h^{-{  2 } n-1} ) , \\\\\n& {\\rm(ii)}  \\WF_h(B) \\subset \\neigh_{ \\overline T^* X } ( x_0, \\xi_0 ) ,  \\\n(x_0,\\xi_0)\\in S^\\ast X  \\Rightarrow\n\\tr^\\flat e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B = \\mathcal O ( h^{\\infty} ) ,\n\\end{split} \\]\nIn case (ii), $ W :=\\neigh_{ \\overline T^* X } ( x_0, \\xi_0 ) $ is the image of\nthe closure of a conic neighbourhood of $(x_0,\\xi_0)$ in $T^\\ast X$, under\nthe map $ T ^* X \\to \\overline T^* X $.\n{  In fact, given (i) and (ii), we can use a microlocal partition of unity to write\n$$ e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)=\\sum_{j=1}^J\n e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B_j+O(h^\\infty)_{\\mathcal{D}'\\to {C^\\infty}}$$\nwhere each $B_j$ satisfies either (i) or (ii) and this proves \n\\eqref{eq:flattr1}.}\n\nFor each case, we repeat the construction with the Fourier transform replaced by the semiclassical Fourier transform. Let\n$u=K_h$ be the Schwartz kernel\nof $e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B$.\nThen, in the notation of \\eqref{eq:extend},\n\\begin{equation}\n\\label{eq:iota}\n\\langle\\iota^\\ast u,\\chi\\rangle =\\langle\\iota^\\ast(\\varphi u),\\chi\\rangle\n=(2\\pi h)^{-2n}\\int\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)I_{\\chi,h}(\\xi,\\eta)d\\xi d\\eta,\n\\end{equation}\nwhere now\n\\begin{equation}\n\\label{eq:Ichih} I_{\\chi,h}(\\xi,\\eta)=\\int\\chi(x)e^{i\\langle\\iota(x),(\\xi,\\eta)\\rangle\/h}dx\n=\\int\\chi(x)e^{ix\\cdot(\\xi+\\eta)\/h}dx. \n\\end{equation}\n\n If $\\WF_h(B)$ is contained in a small compact neighbourhood $W$ of $(x_0,\\xi_0)$, we can assume in the partition of unity $1=\\sum\\chi_j$ (see the\nargument following \\eqref{eq:widehat}), $\\pi(W)\\subset X_0$ for some coordinate patch $X_0$ and $\\pi(W)\\cap\\supp\\chi_j=\\emptyset$ except for the one in this coordinate patch, say $\\psi=\\psi_0$. For $j\\neq0$,\nsince\n\\begin{equation*}\n\\WF_h'(\\varphi_j u)\\subset\\WF_h' ( u ) \\cap[(\\overline{T}^\\ast X)\\times\\WF_h(B)]\n\\cap[(\\overline{T}^\\ast\\supp\\psi_j )\\times(\\overline{T}^\\ast\\supp\\psi_j)]=\\emptyset,\n\\end{equation*}\nwe have\n\\begin{equation*}\n \\mathcal{F}_h(\\varphi_j u)(\\xi,\\eta)=\\mathcal{O}(h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty}),\n\\end{equation*}\nand thus\n$ \\langle\\iota^\\ast u,\\chi_j \\rangle=\\mathcal{O}(h^\\infty)$.\nTherefore we only need to consider the coordinate patch $X_0$ centered at $x_0$ and the corresponding $\\chi,\\psi$ constructed as before.\nWe note that  $ I_{\\chi, h } (\\xi, \\eta ) = \\mathcal O (h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty})$ uniformly for $(\\xi,\\eta)\\in V$ ($ I_{\\chi, h } $ is defined in \\eqref{eq:Ichih}\nand again we use the notation introduced before \\eqref{eq:extend}).\nHence we only need to to estimate\n\\begin{equation*}\n\\left|\\int_{{\\complement V}}\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)I_{\\chi,h}(\\xi,\\eta)d\\xi d\\eta\\right|\\leq\\int_{{\\complement V}}|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|d\\xi d\\eta.\n\\end{equation*}\nHere\n\\begin{equation}\n\\begin{split}\n\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n=&\\int \\psi(x)\\psi(y)u(x,y)e^{-i(x\\cdot\\xi+y\\cdot\\eta)\/h}dxdy\\\\\n=&\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), \\psi(x)e^{- ix\\cdot\\xi\/h}\\rangle,\n\\end{split}\n\\end{equation}\nwhere $ \\langle \\bullet, \\bullet \\rangle $ denotes distributional pairing.\nWe also note that  \n\\[ \\WF_h(\\psi(x)e^{ix\\cdot\\xi\/h})=\\supp\\psi\\times\\{\\xi\\}, \\ \\ \n\\WF_h(\\psi(y)e^{-iy\\cdot\\eta\/h})=\\supp\\psi\\times\\{-\\eta\\}, \\ \\ \n ( \\xi , \\eta ) \\in \\complement V .\\]\n\nIn case (i), we assume \n\\[ \\WF_h(B)\\subset W=W_1 \\times W_2 \\text{ where $W_1=\\pi(W)\\subset X_0$\nand  $W_2 \\subset\\mathbb{R}^n$ are compact.} \\]\n We make the following observation:\nif $ \\widetilde W_2 = \\{ \\xi' : \\exists \\, \\eta' \\in W_2  \\ ( \\xi', \\eta' )\n\\in \\complement V \\} $, then either $ - \\eta \\notin W_2 $ or $ - \\xi \\in \\widetilde W_2 $.\n(Here we used the symmetry \\eqref{eq:CV}.)\nHence if $A\\in\\Psi_h^{{\\rm{comp}}} (X)$, $ \\WF_h ( I - A ) \\cap\n\\widetilde{W}_1\\times\\widetilde{W}_2 = \\emptyset$,\nwhere $\\widetilde{W}_1$ is a small neighbourhood of $\\supp\\psi$, then\n\\begin{equation*}\n\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)=\n\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), A(\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty).\n\\end{equation*}\nTherefore\n\\begin{equation*}\n\\begin{split}\n|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|\n=&\\;|\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), A(\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle|+\\mathcal{O}(h^\\infty)\\\\\n\\leq &\\; C\\|Ae^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B\\|_{L^2\\to L^2}+\\mathcal{O}(h^\\infty)\\leq Ch^{-1}\\\\\n\\end{split}\n\\end{equation*}\nwhere we use the estimate \\eqref{modifiedresolvent} and the fact that microlocally on $\\WF_h(A)\\times\\WF_h(B)$ which is a compact set in $T^\\ast(X\\times X)$, $H_{sG(h)}$ is equivalent to $L^2$ uniformly.\nCombined with \\eqref{eq:iota} this finishes the proof for case (i).\n\nIn case (ii), we again assume that \n$ \\WF_h(B)\\subset W=W_1\\times W_2$ where $W_1=\\pi(W)\\subset X_0$ is a small compact neighbourhood of $x_0$ but now $W_2\\subset\\bar{\\mathbb{R}}^n=\\mathbb{R}^n\\cup\\partial \\, \\bar {\\mathbb{R}}^n$ is a small conic neighbourhood of $\\xi_0\\in\n\\partial \\,\\bar{ \\mathbb{R}}^n$ intersecting with $\\{|\\xi|\\geqslant C\\}$. As in case (i),\nwe put\n$ \\widetilde W =\\widetilde{W}_1\\times\\widetilde{W}_2$ such that $\\widetilde{W}_1$ is a small neighbourhood of $\\supp\\psi$ and $\\widetilde{W}_2$ is a small neighbourhood of ${\\complement V}(W_2)$, which is again a small conic neighbourhood of $\\xi_0$.\n\nWe then  choose $A\\in\\Psi_h^0(X)$ such that $ \\WF_h ( I - A) \\cap \\widetilde{W}\n= \\emptyset $,  and  $\\WF_h(A)$ is contained in a small neighbourhood of $\\widetilde{W}$.\nWe have\n\\[ (\\xi,\\eta)\\in {\\complement V} \\ \\Longrightarrow  \\ \\text{ (a)  $-\\eta\\notin W_2$\nor (b) $- \\xi\\in\\widetilde{W}_2$.}\n\\]\nIn the case (a)  we have\n\\begin{equation*}\n|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|=\\mathcal{O}(h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty}).\n\\end{equation*}\nIn the case\n (b) we  need a uniform estimate for $\\langle\\xi\\rangle^N|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|$ where $N $ is large.\nTo do this, we use the notation from the proof of Lemma \\ref{l:prop} and write\n\\begin{equation*}\n\\begin{split}\n\\langle\\xi\\rangle^N\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n=&\\;\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h}\\rangle\\\\\n=&\\;\\langle \\varphi_{-t_0}^\\ast V (t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty)\\\\\n=&\\;\\langle V(t_0)\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty).\n\\end{split}\n\\end{equation*}\nWe notice that $\\WF_h(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})=\\supp\\psi\\times\\{-\\xi\\}$, and\n\\begin{equation*}\n\\|\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h}\\|_{H_h^{-N}}=\\mathcal{O}(1)\n\\end{equation*}\nuniformly in $\\xi$. Since $ t_0 $ is small we can choose $W$ and $\\widetilde{W}$ small\nenough, so that $e^{-t_0 H_p}\\widetilde{W}\\cap\\widetilde{W}=\\emptyset$.\nThen we choose a microlocal partition of unity,  $A_1^2+A_2^2=I +\n\\mathcal O ( h^\\infty )_{ \\mathcal D' \\to {C^\\infty}} $,\nsuch that $e^{-tH_p}\\widetilde{W}\\subset\\el_h(A_1)$,\n$\\WF_h(A_1)$ is a small neighbourhood of $e^{-tH_p} \\widetilde W $ and\n $\\WF_h(A_2)\\cap e^{-t_0H_p}(\\WF_h(A))=\\emptyset$. We have\n\\begin{equation}\n\\label{eqa}\n\\begin{split}\n& \\langle\\xi\\rangle^N\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n= \\langle A_1 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_1\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle\\\\\n&  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + \\langle A_2 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_2\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle+\n\\mathcal \\mathcal{O}(h^\\infty).\n\\end{split}\n\\end{equation}\n\nWe recall the following propagation estimate \\cite[Propositon 2.5]{DZ} which\nis essentially the clasical result of Duistermaat--H\\\"ormander:\n\\begin{prop}\n\\label{ppg}\nAssume that $P_0\\in\\Psi_h^1(X)$ with semiclassical principal symbol $p-iq\\in S^1_h(X)\/hS_h^0(X)$ where $p\\in S^1(X;\\mathbb{R})$ is independent of $h$ and $q\\geqslant0$ everywhere. Assume also that $p$ is homogeneous of degree 1 in $\\xi$ for $|\\xi|$ large enough. Let $e^{tH_p}$ be the Hamiltonian flow of $p$ on $\\overline{T}^\\ast X$ and $u(h)\\in\\mathcal{D}'(X)$, then if $A_0,B_0,B_1\\in\\Psi_h^0(X)$ and for each $(x,\\xi)\\in\\WF_h(A_0)$, there exists $T\\geqslant0$ with $e^{-TH_p}(x,\\xi)\\in\\el_h(B_0)$ and $e^{tH_p}(x,\\xi)\\in\\el_h(B_1)$ for $t\\in[-T,0]$. Then for each $m$,\n\\begin{equation}\n\\|A_0u\\|_{H^m_h(X)}\\leq C\\|B_0u\\|_{H^m_h(X)}+Ch^{-1}\\|B_1P_0u\\|_{H_h^m(X)}+\\mathcal{O}(h^\\infty).\n\\end{equation}\n\\end{prop}\n\nWe apply the proposition to $u=\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h})$, $P_0=\\widetilde{P}_h(z)$, $A_0=A_1V(t_0)$ with $\\el_h(B_0)$ containing $e^{-TH_p}(\\WF_h(A_0))$ for $T>0$ small enough and $e^{tH_p}(x,\\xi)\\in\\el_h(B_1)$ for $t\\in [-T,0]$. Furthermore, we can choose $B_1$ so that $\\WF_h(B_1)\\cap\\WF_h(B)=\\emptyset$.\n\\[ \\begin{split} \\|A_0u\\|_{H_h^N} & \\leq C\\|B_0u\\|_{H_h^N}+Ch^{-1}\\|B_1B(\\psi(y)e^{-iy\\cdot\\eta})\\|_{H_h^N}+\\mathcal{O}(h^\\infty)\\\\\n& =C\\|B_0u\\|_{H_h^N}+\\mathcal{O}(h^\\infty). \\end{split} \\]\nHowever, the semiclassical wavefront set condition of $\\widetilde{R}_h(z)$ shows that $\\WF_h(B_0)\\cap\\WF_h(u)=\\emptyset$, thus\n$\\|A_0u\\|_{H_h^N}=\\mathcal{O}(h^\\infty)$\nand\nwe know the term corresponding to $A_1$ in the sum of \\eqref{eqa} is $\\mathcal{O}(h^\\infty)$. For the other term involving $ A_2 $, we use\n$$\\|A_2\\varphi_{t_0} ^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{ix\\cdot\\xi\/h})\\|_{H^{P}_h }\n\\leq \\mathcal{O}(h^\\infty)\\|\\varphi_{t_0} ^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{ix\\cdot\\xi\/h})\\|_{H_h^{-N}}=\\mathcal{O}(h^\\infty),$$\nfor any $ P $.\nThis is paired with the term estimated by\n$$\\|A_2 V ( t_0 ) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h})\\|_{H_h^{-P }}\\leq\nC\\|\\psi(y)e^{-iy\\cdot\\eta\/h}\\|_{H_h^P } \\leq C \\langle \\eta \\rangle^P , $$\nfor some $ P$. Hence\n\\[\n\\langle A_2 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_2\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{- ix\\cdot\\xi\/h}) \\rangle = \\mathcal O ( h^\\infty \\langle\n\\eta \\rangle^P ). \\]\nReturning to \\eqref{eqa} we see that\n\\[ \\langle \\xi \\rangle^N \\mathcal F_h ( \\varphi u ) (\\xi, \\eta ) =\n\\mathcal O ( h^\\infty \\langle\n\\eta \\rangle^P ). \\]\nSince $ |\\xi | $ is comparable $ |\\eta | $ in $ \\complement V $, we have\n$$ \\mathcal F_h ( \\varphi u ) (\\xi, \\eta ) = \\mathcal O ( h^\\infty\n\\langle ( \\xi , \\eta ) \\rangle^{ -N + P } ) $$ \nand that concludes the\nproof of \\eqref{eq:flattr1}.\n\n\n\n\n\n\n\\section{Proof of the trace formula}\n\\subsection{Sketch of the proof}\nWe first indicate basic ideas of the proof before we go into the details -- the\nprinciple is quite simple but the implementation involves the use of\nthe results of \\cite{DZ} and of some ideas from \\cite{S}.\n\nIn general, a trace formula such as \\eqref{localtrace} follows from the finite order of the analytic continuation of $\\zeta_1(\\lambda)$ in the strip $\\Im\\lambda\\geqslant-A$, that is,\nfrom having the following estimate valid away from small neighbourhoods of resonances:\n\\begin{equation}\n\\label{finiteorder}\n\\left|\\frac{d}{d\\lambda} \\log\\zeta_1(\\lambda)\\right|=\\mathcal{O}(\\langle\\lambda\\rangle^{2n+1}).\n\\end{equation}\nTo obtain the distributional identity \\eqref{localtrace}\nwe take $\\psi\\in C_0^\\infty(0,\\infty)$ and compute the following integral in two different ways\n\\begin{equation*}\n\\int_{\\mathbb{R}}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}  \\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation*}\nOn one hand, we pass the integral contour to $\\mathbb{R}+iB$, where $B>C_1$ so that \\eqref{zeta} converges. Since there are no resonances in the upper half plane, we have\n\\begin{equation*}\n\\begin{split}\n\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\\left(\\frac{1}{i}\\int_0^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt\\right)d\\lambda\n=&\\frac{1}{i}\\int_0^\\infty\\left(\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)e^{it\\lambda}d\\lambda\\right)\\tr^{\\flat}e^{-itP}dt\\\\\n=&\\int_0^\\infty\\psi(t)\\tr^{\\flat}e^{-itP}dt.\n\\end{split}\n\\end{equation*}\nGuillemin's trace formula \\eqref{Guillemin} gives\n\\begin{equation}\n\\int_{\\mathbb{R}}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda=\\left\\langle \\sum_\\gamma\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}, \\psi \\right\\rangle.\n\\end{equation}\nOn the other hand, we pass the integral contour to $\\mathbb{R}-iA$ and we get the contribution from the poles of $\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)$ which are exactly the Pollicott-Ruelle resonances,\n\\begin{equation}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\psi}(\\mu)=\\left\\langle\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}e^{-i\\mu t}, \\psi \\right\\rangle.\n\\end{equation}\nThe remainder is exactly\n\\begin{equation}\n\\langle F_A, \\psi\\rangle: =\\int_{\\mathbb{R}-iA}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda,\n\\end{equation}\nand we want to show that $ F_A $ can be extended to a tempered distribution\nsupported on $ [ 0, \\infty ) $ and that it satisfies \\eqref{error}. The\nestimate \\eqref{finiteorder} is crucial here.\n\nTo see \\eqref{finiteorder}, we decompose\n\\begin{equation}\n\\label{decomposition}\n\\begin{split}\ne^{-it_0(P-\\lambda)}(P-\\lambda)^{-1} & =\ne^{-it_0(P-i \\widetilde Q-\\lambda)}(P-i \\widetilde Q-\\lambda)^{-1}\n+[(P-\\lambda)^{-1}-(P-i \\widetilde Q-\\lambda)^{-1}]\\\\\n& \\ \\ \\ \\ - i\\int_0^{t_0}[e^{-it(P-\\lambda)}-e^{-it(P-i \\widetilde Q-\\lambda)}]dt,\n\\end{split}\n\\end{equation}\nwhere $ \\widetilde Q = h^{-1} Q $ for a suitably chosen $ h$ depending on the range of $ \\lambda $'s. This is valid from $ \\Im \\lambda \\gg 0 $ and then continues analytically to $ \\mathbb C $ on the level of distributional Schwartz kernels.\n\nThe first term is holomorphic in $\\lambda$ and can be estimated by Proposition \\ref{flattracees} in the semiclassical setting.\n\nThe second term on the right hand side of \\eqref{decomposition} is of trace class if $\\lambda$ is not a resonance. To see this, we use the following formula\n\\begin{equation}\n\\label{res1}\n\\begin{split}\n(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}  & =\n[(P-\\lambda)^{-1}(P-i \\widetilde Q-\\lambda)-I](P-i\\widetilde Q-\\lambda)^{-1}\\\\\n& =-(P-\\lambda)^{-1}i \\widetilde Q(P-i \\widetilde Q-\\lambda)^{-1}\\\\\n\\end{split}\n\\end{equation}\nto get\n\\begin{equation}\n\\label{res2}\n(P-\\lambda)^{-1}=(P-i \\widetilde Q-\\lambda)^{-1}[I+i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}.\n\\end{equation}\nBy using \\eqref{res2} in  \\eqref{res1} we obtain\n\\begin{equation}\n(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}=\n-(P-i\\widetilde Q-\\lambda)^{-1}[I+i \\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}\ni\\widetilde Q(P-i \\widetilde\nQ-\\lambda)^{-1}.\n\\end{equation}\nIf we denote { $F(\\lambda)=I+i\\widetilde{Q}(P-i\\widetilde Q-\\lambda)^{-1}$}, then\n\\begin{equation*}\nF'(\\lambda)=\\frac{d}{d\\lambda}F(\\lambda)=i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-2}.\n\\end{equation*}\nMoreover, $F(\\lambda) - I $ and $ F' ( \\lambda ) $ are operators\nof finite rank. By the cyclicity of the trace, we have\n\\begin{equation*}\n\\begin{split}\n\\tr[(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}]=&\n-\\tr[I+i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-2}\\\\\n=&-\\tr F'(\\lambda)F(\\lambda)^{-1}=-\\frac{d}{d\\lambda}\\log\\det F(\\lambda).\n\\end{split}\n\\end{equation*}\nTherefore it can be controlled by the rank of $\\widetilde Q$ and the norm of $F(\\lambda)$.\n\nThe third term in \\eqref{decomposition} can be handled by Duhamel's principle: if $u(t): =e^{-it(P-i\\widetilde Q-\\lambda)}f$, then\n\\begin{equation*}\n\\partial_t u(t)=-i(P-i\\widetilde Q-\\lambda)u(t), \\ \\ \\ u(0)=f.\n\\end{equation*}\nRewriting the equation as\n$\\partial_t u(t)+i(P-\\lambda)u(t)=-\\widetilde Qu(t)$ ,\nwe get\n\\begin{equation*}\nu(t)=e^{-it(P-\\lambda)}f-\\int_0^t e^{-i(t-s)(P-\\lambda)}\\widetilde Qu(s)ds.\n\\end{equation*}\nTherefore\n\\begin{equation*}\ne^{-it(P-\\lambda)}-e^{-it(P-i\\widetilde Q-\\lambda)}=\\int_0^te^{-i(t-s)(P-\\lambda)}\\widetilde Qe^{-is(P-i\\widetilde Q-\\lambda)}ds.\n\\end{equation*}\nThis shows that the left hand side is also of trace class and its trace class norm is controlled by the trace class norm of $\\widetilde Q$.\n\n{  To carry out the strategy above} we need to choose correct contours and to obtain\na local version of \\eqref{finiteorder} using $ \\det F ( \\lambda ) $.\nFor that we break the infinite contour into a family of finite contours and use the semiclassical reduction to treat the zeta function on each contour separately.\nThat involves choices of $ h $ so that $ z = h \\lambda $ is in an\nappropriate range.\n\n\\subsection{The contours for integration}\nIn this section, we choose contours for integration. First, we decompose the region $\\Omega=\\{\\lambda\\in\\mathbb{C}:-A\\leq \\Im\\lambda\\leq B\\}$ into dyadic pieces: fix $ E > 0 $ and put $\\Omega=\\bigcup_{k\\in\\mathbb{Z}}\\Omega_k$, where $\\Omega_0=\\Omega\\cap\\{-E\\leq\\Re\\lambda\\leq E\\}$ and\n\\begin{gather*}\n\\Omega_k:=\\Omega\\cap\\{2^{k-1}E\\leq\\Re\\lambda\\leq 2^kE\\}, \\ \\ k>0\\\\\n\\Omega_{-k}:=\\Omega\\cap\\{-2^kE\\leq\\Re\\lambda\\leq - 2^{k-1}E\\}, \\ \\ k>0.\n\\end{gather*}\nFor each $k$, we write $\\gamma_k=\\partial\\Omega_k=\\bigcup_{j=1}^4\\gamma^j_k$ with counterclockwise orientation.\n\n\\begin{figure}[ht]\n\\includegraphics[width=6.5in]{contours}\n\\caption{Integration contours}\n\\end{figure}\n\nNext, we shall modify $\\gamma^2_k,\\gamma^3_k$ and $\\gamma^4_k$ to avoid the resonances. For simplicity, we only work for $k>0$ as the case for $k<0$ can be handled by symmetry. We choose $\\widetilde{\\gamma}^2_k,\\widetilde{\\gamma}^3_k$ and $\\widetilde{\\gamma}^4_k$ lying in\n\\begin{equation*}\n([2^{k-1}E-1,2^kE+1]+i[-A-1,B])\\setminus([2^{k-1}E+1,\n2^kE-1]+i[-A,B])\n\\end{equation*}\nso that $\\widetilde{\\gamma}^2_k\\subset[2^{k-1}E-1,2^{k-1}E+1]\n+i[-A,B]$ connects $2^{k-1}E+iB$ with a point $w_k$ which lies\non $[2^{k-1}E-1,2^{k-1}E+1]-iA$, $\\widetilde{\\gamma}^4_k=-\\widetilde{\\gamma}^2_{k+1}$; $\\widetilde{\\gamma}^3_k\\subset[2^{k-1}E-1,2^kE+1]+i[-A-1,-A]$ connects $w_k$ with $w_{k+1}$. The region bounded by  $\\widetilde{\\gamma}_k :=\\bigcup_{j=1}^4\\widetilde{\\gamma}^j_k$\n is denoted as $\\widetilde{\\Omega}_k$, (we write $\\widetilde{\\gamma}^1_k=\\gamma^1_k$). Then we have { \n\\begin{equation*}\n\\Omega\\subset\\widetilde{\\Omega}=\\bigcup_{k\\in\\mathbb{Z}}\\widetilde{\\Omega}_k\n\\subset\\{\\lambda\\in\\mathbb{C},-A-1\\leq\\Im\\lambda\\leq B\\}\n\\end{equation*}}\nand all $\\widetilde{\\Omega}_k$ have disjoint interiors.\n\nFor convenience, we turn into the semiclassical setting.\nLet $W_h=h\\widetilde{\\Omega}_k$ where $h^{-1\/2}=2^kE$, then\n\\begin{equation*}\n\\textstyle [\\frac{1}{2}h^{1\/2}+h,h^{1\/2}-h]+i[-Ah,Bh]\\subset W_h\\subset\n[\\frac{1}{2}h^{1\/2}-h,h^{1\/2}+h]+i[(-A-1)h,Bh].\n\\end{equation*}\nMoreover, $\\rho_h:=\\partial W_h=\\bigcup_{j=1}^4\\rho_h^j$ where $\\rho_h^1$ is the horizontal segment $[\\frac{1}{2}h^{1\/2},h^{1\/2}]+iBh$ with negative orientation; $\\rho_h^2\\subset[\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h]+i[-Ah,Bh]$ connects $\\frac{1}{2}h^{1\/2}+iBh$ with a point $z_h\\in[\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h]-iAh$; $\\rho_h^4\\subset[h^{1\/2}-h,h^{1\/2}+h]+i[-Ah,Bh]$ connects a point $z'_h\\in[h^{1\/2}-h,h^{1\/2}+h]-iAh$ with $h^{1\/2}+iBh$;\nand $\\rho_h^3\\subset[\\frac{1}{2}h^{1\/2}-h,h^{1\/2}+h]$ connects $z_h$ with $z'_h$.\n\nWe have the following contour integration\n\\begin{equation}\n\\label{contour}\n\\oint_{\\rho_h}\\widehat{\\psi}_h(z)\\frac{d}{dz}\\log\\zeta_h(z)dz=\n\\sum_{z_j\\in \\Res_h(P)\\cap W_h}\\psi_h(z_j).\n\\end{equation}\nHere we write $\\widehat{\\psi}_h(z)=\\widehat{\\psi}(z\/h)$, $\\zeta_h(z)=\\zeta_1(z\/h)$, $\\Res_h(P)=h\\Res(P)$.\n\nWe rewrite the decomposition \\eqref{decomposition} in this scaling:\n\\begin{equation}\n\\label{hdecom}\n\\begin{split}\n\\frac{d}{dz}\\log\\zeta_h(z) & = h \\tr^\\flat(e^{-it_0h^{-1}P_h(z)}R_h(z))\\\\\n& = \\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n+ \\tr(R_h(z)-\\widetilde{R}_h(z))\\\\\n& -\\frac{i}{h}\\tr\\int_0^{t_0}[e^{-ith^{-1}P_h(z)}-e^{-ith^{-1}\\widetilde{P}_h(z)}]dt.\n\\end{split}\n\\end{equation}\nThen as in the discussion after \\eqref{decomposition}, in the region $-C_0h\\leq \\Im z\\leq 1, |\\Re z|\\leq 2h^{1\/2}$, we can apply Proposition\n\\ref{flattracees} to obtain\n\\begin{equation}\n\\label{hes1}\n\\left|\\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\\right|=\\mathcal{O}(h^{-2n-1}).\n\\end{equation}\nAlso we have\n\\begin{equation}\n\\label{hes2}\n\\left \\|\\int_0^{t_0}[e^{-ith^{-1}P_h(z)}-e^{-ith^{-1}\\widetilde{P}_h(z)}]dt \\right\\|_{\\rm{tr}} =\\mathcal{O}(h^{-n-1}).\n\\end{equation}\nFor the second term, we have\n\\begin{equation*}\n\\tr(R_h(z)-\\widetilde{R}_h(z))=-\\frac{d}{dz}\\log\\det F(z),\n\\end{equation*}\nwhere\n$F(z)=I+iQ\\widetilde{R}_h(z)$ is a Fredholm operator and the poles for $F(z)^{-1}$ coincides with the resonances. Moreover, by \\eqref{finiterank}, \\eqref{modifiedresolvent} and Weyl's inequality, we have\n\\begin{equation}\n\\label{detes1}\n|\\det F(z)|\\leq (Ch^{-1})^{Ch^{-n}}\\leq Ce^{Ch^{-n-1}}.\n\\end{equation}\n\nMoreover, when $\\Im z\\geqslant C_1h$, we have $F(z)=I+iQ\\widetilde{R}_h(z)=P_h(z)\\widetilde{R}_h(z)$, so $F(z)$ is invertible and $F(z)^{-1}=\\widetilde{P}_h(z)R_h(z)$. Therefore\n\\begin{equation*}\n\\begin{split}\n\\|F(z)^{-1}\\|_{H_{sG(h)}\\to H_{sG(h)}}\n\\leq\\|\\widetilde{P}_h(z)\\|_{D_{sG(h)}\\to H_{sG(h)}}&\\|R_h(z)\\|_{H_{sG(h)}\\to D_{sG(h)}}\\\\\n\\leq\\|\\widetilde{P}_h(z)\\|_{D_{sG(h)}\\to H_{sG(h)}}&(\\|R_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\\\\n&+\\|hPR_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}})\n\\leq Ch^{-1}.\n\\end{split}\n\\end{equation*}\nWe can also write $F(z)^{-1}=I-iQR_h(z)$ which gives the estimate\n\\begin{equation}\n\\label{detes2}\n|\\det F(z)^{-1}|\\leq (Ch^{-1})^{Ch^{-n}}\\leq Ce^{Ch^{-n-1}}.\n\\end{equation}\n\nWe recall a lower modulus theorem due to H. Cartan { (see \\cite[\\S 11.3, Theorem 4]{Le})} : Suppose that $g$ is holomorphic in $D(z_0,2eR)$ and $g(z_0)=1$. Then for any $\\eta>0$,\n\\begin{equation}\n\\label{lowermod}\n\\log|g(z)|\\geqslant-\\log(15e^3\/\\eta)\\log\\max_{|z-z_0|<2eR}|g(z)|,\\;\\;\nz\\in D(z_0,R)\\setminus\\mathcal{D},\n\\end{equation}\nwhere $\\mathcal{D}$ is a union of discs with the sum of radii less than $\\eta R$. With the help of this lower modulus theorem, we can make a suitable choice of integration contour.\n\n\\begin{lem}\nWe can choose $\\widetilde{\\gamma}_k$ suitably such that in addition to the assumptions above, we have\n\\begin{equation}\n\\label{detbound}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation}\nwhen $z\\in\\rho_h$.\n\\end{lem}\n\\begin{proof}\nWe shall apply the lower modulus theorem with $z_0=\\frac{1}{2}h^{1\/2}+iBh\/2$ and $R=C_0'h$ where $C_0'$ is large enough, so that\n\\begin{equation*}\n\\begin{split}\n[\\textstyle{\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h}]+i[(-A-1)h,Bh]   & \\subset D(z_0,R) \\subset D(z_0,2eR) \\\\\n & \\subset[-2h^{1\/2},2h^{1\/2}]+i[-C_0h,1].\n\\end{split}\n\\end{equation*}\nIn addition, we let $\\eta$ be small enough, so that $\\eta R<h$.\nSince $\\widetilde{R}_h(z)$ is holomorphic in $D(z_0,2eR)$, so is $\\det F(z)$. Moreover, from \\eqref{detes1}, we see\n\\begin{equation*}\n\\log\\max_{|z-z_0|<2eR}|\\det F(z)|\\leq Ch^{-n-1}.\n\\end{equation*}\n\nOn the other hand, at $z=z_0$, by \\eqref{detes2}\n\\begin{equation*}\n|\\det F(z_0)^{-1}|\\leq e^{-Ch^{-n-1}}.\n\\end{equation*}\nApplying the lower modulus theorem with $g(z)=\\det F(z)\\det F(z_0)^{-1}$, we can choose $\\widetilde{\\gamma}^2_k$ so that\n\\begin{equation*}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation*}\nwhen $z\\in h\\widetilde{\\gamma}^2_k$. Taking $\\widetilde{\\gamma}^4_k=-\\widetilde{\\gamma}^2_{k+1}$ as required, it is clear that we also have\n\\begin{equation*}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation*}\nwhen $z\\in h\\widetilde{\\gamma}^4_k$.\n\n{  Finally, we can apply the lower modulus theorem to a sequence of balls $D(z_{0j},R)$ which are translations of $D(z_0,R)$ as above by $jh$, $0\\leqslant j\\leqslant \\frac{1}{2}h^{-1\/2}$. In particular, $z_{0j}=z_0+jh$ and we have\n\\begin{equation*}\n\\begin{split}\n[\\textstyle{\\frac{1}{2}h^{1\/2}+jh-\\frac{1}{2}h,\\frac{1}{2}h^{1\/2}+jh+\\frac{1}{2}h}]+i[(-A-1)h,-Ah]   & \\subset D(z_0,R) \\subset D(z_0,2eR) \\\\\n & \\subset[-2h^{1\/2},2h^{1\/2}]+i[-C_0h,1].\n\\end{split}\n\\end{equation*}\nWe can now repeat the argument above and notice that all the estimates hold uniformly in $j$ to conclude that we can choose a curve in $[\\frac{1}{2}h^{1\/2}+jh-\\frac{1}{2}h,\\frac{1}{2}h^{1\/2}+jh+\\frac{1}{2}h]$ for each $j$ such that $|\\log\\det F(z)|=O(h^{-n-1})$ uniformly in $j$ on these curves and these curves form the curve $h\\tilde{\\gamma}_3$ connecting $z_h$ and $z_h'$.}\n\\end{proof}\n\nNow by the upper bound on the number of resonances, we have\n\\begin{equation}\n\\label{resw}\n\\#\\Res_h(P)\\cap W_h=\\mathcal{O}(h^{-\\frac{n}{2}-1}).\n\\end{equation}\nFrom the proof of the lemma above, we can actually construct $\\rho_h$ so that \\eqref{detbound} holds in a neighborhood of $\\rho_h$ of size $\\sim h^{\\frac{n}{2}+2}$. By Cauchy's inequality, we see that for $z\\in\\rho_h$,\n\\begin{equation}\n\\label{hes3}\n\\left|\\frac{d}{dz}\\log\\det F(z)\\right|=\\mathcal{O}(h^{-\\frac{3n}{2}-3}).\n\\end{equation}\nNow combining \\eqref{hes1}, \\eqref{hes2} and \\eqref{hes3}, we have the estimate for $z\\in\\rho_h$,\n\\begin{equation}\n\\label{zetabound}\n\\left|\\frac{d}{dz}\\log\\zeta_h(z)\\right|=\\mathcal{O}(h^{-2n-1}),\n\\end{equation}\nwhich is equivalent to \\eqref{finiteorder}.\n\n\\subsection{End of the proof}\nNow we finish the proof of the local trace formula by summing the contour integrals \\eqref{contour}.\n\nFirst, since $\\widehat{\\psi}(\\lambda)=\\mathcal{O}(\\langle\\lambda\\rangle^{-\\infty})$ as $\\Re\\lambda\\to\\pm\\infty$ as well as all of its derivatives when $\\Im\\lambda$ is bounded, we have $\\widehat{\\psi}_h(z)=\\mathcal{O}(h^\\infty)$ as well as all its derivatives. Therefore by \\eqref{zetabound},\n\\begin{equation*}\n\\int_{\\widetilde{\\gamma}_k^2}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda) d\\lambda =\\int_{h\\widetilde{\\gamma}_k^2}\n\\widehat{\\psi}_h(z)\\frac{d}{dz}\\log\\zeta_h(z)dz=\\mathcal{O}(h^\\infty).\n\\end{equation*}\nMoreover, both of the sums\n\\begin{equation*}\n\\sum_{k\\in\\mathbb{Z}}\\int_{\\widetilde{\\gamma}_k^1}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n=-\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n\\end{equation*}\nand\n\\begin{equation*}\n\\sum_{k\\in\\mathbb{Z}}\\int_{\\widetilde{\\gamma}_k^3}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda=\\int_{\\Gamma}\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n\\end{equation*}\nconverges absolutely. Here $\\Gamma=\\bigcup_{k=-\\infty}^\\infty\\widetilde{\\gamma}_k^3$.\nOn the other hand, by the upper bound of the resonance \\eqref{resw}, the sum\n\\begin{equation*}\n\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega}}\\widehat{\\psi}(\\mu_j)\n=\\sum_{k\\in\\mathbb{Z}}\\sum_{z_j\\in\\Res_h(P)\\cap W_k}\\widehat{\\psi}_h(z_j)\n\\end{equation*}\nalso converges absolutely. Hence we have the following identity\n\\begin{equation*}\n\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n=\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega}}\\widehat{\\psi}(\\mu_j)+\n\\int_{\\Gamma}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation*}\nIn other words,\n\\begin{equation*}\n\\sum_{\\mu\\in\\Res(P), \\Im\\mu>-A}\\widehat{\\psi}(\\mu)\n=\\sum_{\\gamma}\\frac{T_\\gamma^{\\#}\\delta(t-T_\\gamma)}\n{|\\det(I-\\mathcal{P}_\\gamma)|}+\\langle\\psi,F_A\\rangle\n\\end{equation*}\nwhere\n\\begin{equation}\n\\label{errorform}\n\\langle\\psi, F_A\\rangle=-\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega},\\Im\\mu_j\\leq-A}\\widehat{\\psi}(\\mu_j)+\\int_{\\Gamma}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation}\nThis proves \\eqref{localtrace}.\n\nSo far, the distribution $F_A$ is only defined in $\\mathcal{D}'(0,\\infty)$. However, the right-hand side in \\eqref{localtrace} has an obvious extension to $\\mathbb{R}$ by zero on the negative half line as it is supported away from 0. By the polynomial upper bounds \\eqref{upperbound} on the number of resonances in the strip $\\Im\\mu>-A$, the sum\n\\begin{equation*}\nu_A(t)=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}e^{-i\\mu t}\n\\end{equation*}\nalso has an extension to $\\mathbb{R}$ which has support in $[0,\\infty)$. We only need to show that $u_A$ is of finite order: For any $\\varphi\\in C_0^\\infty(0,\\infty)$,  $k\\geqslant0$, we have\n$\\widehat{\\varphi^{(k)}}(\\lambda)=(i\\lambda)^k\\widehat{\\varphi}(\\lambda)$. Therefore we can write\n\\begin{equation*}\n\\langle u_A,\\varphi\\rangle=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}(\\mu)=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}(i\\mu)^{-k}\\widehat{\\varphi^{(k)}}(\\mu)\n\\end{equation*}\nWhen $k$ is large, the sum\n\\begin{equation*}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}|\\mu|^{-k}\n\\end{equation*}\nconverges absolutely. Therefore we have the finite order property of $u_A$. Moreover, any two such extensions of $u_A$ are only differed by a distribution $v$ supported at $\\{0\\}$, that is,  a linear combination of delta function and its derivatives.\n\nNow we can certainly extend $F_A$ to a distribution on $\\mathbb{R}$ with support in $[0,\\infty)$. Since $\\check{v}$ is a polynomial in the whole complex plane. Therefore choice of the extension of $u_A$ does not affect the estimate on $\\widehat{F}_A$.\n\nFinally, we give the {  desired} estimate on $\\widehat{F}_A$. This follows from the fact $e^{\\eta t}F_A\\in\\mathcal{S}'$ for any $\\eta<A$ and \\cite[Theorem 7.4.2]{H}. To see this, we only need to show that we can extend \\eqref{errorform} to $\\psi\\in C^\\infty(\\mathbb{R})$ with support in $(0,\\infty)$ such that $\\varphi=e^{-\\eta t}\\psi\\in\\mathcal{S}$. If $\\psi$ has compact support, then\n\\begin{equation*}\n\\widehat{\\psi}(\\lambda)=\\int\\psi(t)e^{-it\\lambda}dt\n=\\int\\varphi(t)e^{-it(\\lambda+i\\eta)}dt=\\widehat{\\varphi}(\\lambda+i\\eta).\n\\end{equation*}\nTherefore \\eqref{errorform} can be rewritten as\n\\begin{equation}\n\\label{deffa}\n\\langle\\psi, F_A\\rangle=-\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega},\\Im\\mu_j\\leq-A}\\widehat{\\varphi}(\\mu_j+i\\eta)+\\int_{\\Gamma}\\widehat{\\varphi}(\\lambda+i\\eta)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation}\nAgain, by the estimate \\eqref{finiteorder} and the upper bound on the resonances \\eqref{upperbound}, this converges as long as $\\supp\\varphi\\subset(0,\\infty)$ which gives the definition of $e^{\\eta t}F_A$ in $\\mathcal{S}'$. The order of $\\widehat{F}_A(\\lambda)$ comes from the formula \\eqref{deffa} and \\eqref{finiteorder}, \\eqref{upperbound}.\n\n\\section{Proof of the weak lower bound on the number of resonances}\nNow we prove the weak lower bound on the number of resonances. The strategy is similar to the proof in \\cite{GZ}  and we proceed by contradiction. Let\n\\begin{equation*}\nN_A(r)=\\#(\\Res(P)\\cap\\{|\\mu|\\leq r,\\Im\\mu>-A\\})\n\\end{equation*}\nand assume that\n\\begin{equation}\n\\label{contra}\nN_A(r)\\leq P(\\delta,A)r^\\delta.\n\\end{equation}\n\nWe fix a test function $\\varphi\\in C_0^\\infty(\\mathbb{R})$ with the following properties:\n\\begin{equation*}\n\\varphi\\geqslant0, \\ \\ \\varphi(0)>0, \\ \\ \\supp\\varphi\\subset[-1,1].\n\\end{equation*}\nNext we set $\\varphi_{l,d}(t)=\\varphi(l^{-1}(t-d))$ where $d>1$ and $l<1$, so that $\\varphi_{l,d}\\in C_0^\\infty(0,\\infty)$. Therefore we can apply the local trace formula to get\n\\begin{equation}\n\\label{tr1}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}_{l,d}(\\mu)\n+\\langle F_A,\\varphi_{l,d}\\rangle=\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}.\n\\end{equation}\n\nFirst, we note that by Paley-Wiener theorem,\n\\begin{equation}\n\\label{PW}\n|\\widehat{\\varphi}_{l,d}(\\zeta)|=|l\\widehat{\\varphi}(l\\zeta)e^{-id\\zeta}|\\leq\nC_N l e^{(d-l)\\Im\\zeta}(1+|l\\zeta|)^{-N},\n\\end{equation}\nfor $\\Im\\zeta\\leq0$ and any $ N \\geq 0 $.\n\nBy the assumption, we have the following estimate on the sum on the left-hand side of \\eqref{tr1},\n\\begin{equation}\n\\label{es1}\n\\begin{split}\n\\left|\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}_{l,d}(\\mu)\\right|\\leq&\\; Cl\\int_0^\\infty(1+l r)^{-N}dN_A(r)\\\\\n\\leq&\\; Cl\\int_0^\\infty\\frac{d}{dr}[(1+l r)^{-N}]N_A(r)dr\\\\\n\\leq&\\; CP(\\delta,A)l\\int_0^\\infty\\frac{d}{dr}[(1+l r)^{-N}]r^\\delta dr\\leq Cl^{1-\\delta}.\n\\end{split}\n\\end{equation}\n\nThe remainder term $\\langle F_A,\\varphi_{l,d}\\rangle$ on the left-hand side of \\eqref{tr1} can be rewritten as\n\\begin{equation*}\n\\langle\\check{F}_A,\\widehat{\\varphi}_{l,d}\\rangle\n=\\int_{\\mathbb{R}}\\widehat{F}_A(-\\zeta)\\widehat{\\varphi}_{l,d}(\\zeta)d\\zeta.\n\\end{equation*}\nBy \\eqref{error}, we can pass the contour to $\\mathbb{R}+i(\\epsilon-A)$ to get\n\\begin{equation}\n\\label{es2}\n\\begin{split}\n|\\langle F_A,\\varphi_{l,d}\\rangle|\n\\leq &\n\\int_{\\mathbb{R}+i(\\epsilon-A)}|\\widehat{F}_A(-\\zeta)|\n|\\widehat{\\varphi}_{l,d}(\\zeta)|d\\zeta\\\\\n\\leq &\\;\nCl e^{(d-l)(\\epsilon-A)}\n\\int_{\\mathbb{R}+i(\\epsilon-A)}\\langle\\zeta\\rangle^{2n+1}(1+l|\\zeta|)^{-2n-3}d\\zeta\\\\\n\\leq &\\; Cl^{-2n-1}e^{(d-l)(\\epsilon-A)}\n\\end{split}\n\\end{equation}\nwhere we use \\eqref{PW} with $ N =2n+3$.\n\n{ \nOn the other hand, to get a lower bound of the right-hand side of \\eqref{tr1}, we fix one primitive periodic orbit $\\gamma_0$ and let $d=kT_{\\gamma_0}$, $k\\in\\mathbb{N}$. Since every term there is nonnegative, we ignore all but the term corresponding to $\\gamma_d$ which is the $k$-times iterate of $\\gamma_0$ and get\n\\begin{equation*}\n\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}\n\\geq\\frac{T_{\\gamma_d}^\\#\\varphi(0)}{|\\det(I-\\mathcal{P}_{\\gamma_d})|}=\\frac{T_{\\gamma_0}\\varphi(0)}{|\\det(I-\\mathcal{P}_{\\gamma_0}^k)|}.\n\\end{equation*}\nLet $\\lambda_1,\\ldots,\\lambda_{n-1}$ be the eigenvalues of $\\mathcal{P}_{\\gamma_0}$, then for some $\\alpha$ depending only on $\\lambda_j$'s,\n\\begin{equation*}\n|\\det(I-\\mathcal{P}_{\\gamma_0}^k)|=|(1-\\lambda_1^k)\\cdots(1-\\lambda_{n-1}^k)|\\leq Ce^{k\\alpha}=Ce^{\\theta_0d},\n\\end{equation*}\nif $\\theta_0=\\alpha\/T_{\\gamma_0}$.} \nThis gives the lower bound\n\\begin{equation}\n\\label{es3}\n\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}\\geq Ce^{-\\theta_0d}.\n\\end{equation}\n\n\n\nCombining \\eqref{es1},\\eqref{es2},\\eqref{es3}, we have the following inequality\n\\begin{equation*}\nCl^{1-\\delta}+Cl^{-2n-1}e^{(d-l)(\\epsilon-A)}\\geq Ce^{-\\theta d}.\n\\end{equation*}\nWe first choose $l=e^{-\\beta d}$, then we have\n\\begin{equation*}\nCe^{-\\beta d(1-\\delta)}+Ce^{(d-l)(\\epsilon-A)+(2n+1)\\beta d}\\geq Ce^{-\\theta_0d}.\n\\end{equation*}\nNotice that the constants $C$'s may depend on $A$, but not on $d$. If we choose $\\beta$ and $A$ large while $\\epsilon$ small so that $\\beta(1-\\delta)>\\theta_0$ and $A-\\epsilon-(2n+1)\\beta>\\theta_0$, then we get a contradiction as $d\\to\\infty$. This can be achieved when $A>A_\\delta$ where\n\\begin{equation}\n\\label{adelta}\nA_\\delta=\\theta_0(1+(2n+1)(1-\\delta)^{-1}).\n\\end{equation}\nThis finishes the proof of Theorem \\ref{thm2}.\n\n\\medskip\n\n\\noindent\n{\\bf Remark.} { \nFrom the proof, we see that the essential gap is bounded by $A_0=\\theta_0(2n+2)$, where $\\theta_0$ given above only depends on the Poincar\\'{e} map associated to a primitive periodic orbit $\\gamma_0$. More explicitly, \n$$\\theta_0=\\frac{1}{T_{\\gamma_0}}\\sum_{\\lambda\\in\\sigma(\\mathcal{P}_{\\gamma_0}):|\\lambda|>1}\\log|\\lambda|.$$\nA weaker bound not depending on the specific orbit is given by\n$\\theta_0\\leq\\theta d_u$ where $d_u=\\dim E_u$ is the dimension of the unstable fiber and $\\theta$ is the Lyapunov constant of the flow given in \\S 2.1.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nMany complex fluids display shear banding, in which a state of\ninitially homogeneous shear flow gives way to the formation of\ncoexisting bands of differing shear rate, with layer normals in the\nflow-gradient direction. For recent reviews,\nsee~\\cite{Olmsted2008,Manneville2008a,Fielding2014,Divoux2015}.\nFollowing its early observation in wormlike micellar surfactant\nsolutions~\\cite{Britton1997}, over the past two decades shear banding\nhas been seen in virtually all the major classes of complex fluids and\nsoft solids. Examples include microgels~\\cite{Divoux2010},\nclays~\\cite{Martin2012}, emulsions~\\cite{Coussot2002}\nfoams~\\cite{Rodts2005}, lamellar surfactant\nphases~\\cite{Salmonetal2003a}, triblock\ncopolymers~\\cite{Berretetal2001a,Mannevilleetal2007a}, star\npolymers~\\cite{Rogers2008}, and -- subject to ongoing\ncontroversy~\\cite{Wangetal2003a,Wangetal2006c,Wangetal2008a,Li2013,Wang2014,Li2015,Wang2011,Wangetal2008a} -- \nlinear polymers.\n\nPrior to about 2010, the majority of studies of shear banding focused\non conditions of a steadily applied shear flow. The criterion for the\npresence of steady state banding in this case is well known: that the\nunderlying homogeneous constitutive curve of shear stress as a\nfunction of shear rate has a regime of negative slope. (In some cases\nof strong concentration coupling shear banding can arise even for a\nmonotonic constitutive curve~\\cite{Fielding2003a}, but we\ndo not consider that case here.)  Such a regime is predicted by the\noriginal tube theory of Doi and Edwards for non-breakable\npolymers~\\cite{DoiEdwards}, and by the reptation-reaction model of\nwormlike micellar surfactants~\\cite{Cates1990}.  It is\nstraightforward to show that a state of initially homogeneous shear\nflow is linearly unstable, in this regime of negative constitutive\nslope, to the formation of shear bands~\\cite{Yerushalmi1970}.  The\ncomposite steady state flow curve of shear stress as a function of\nshear rate then displays a characteristically flat plateau regime, in\nwhich shear bands are observed.\n\nFrom an experimental viewpoint, the evidence for steady state shear\nbanding under a steadily applied shear flow is now overwhelming in the case\nof wormlike micelles. For reviews,\nsee~\\cite{Cates2007,Berret2005}. For linear unbreakable\npolymers the issue remains controversial, as recently reviewed in\nRef.~\\cite{Snijkers2015}. In particular the original\nDoi-Edwards model did not account for a process known as convective\nconstraint release\n(CCR)~\\cite{Marrucci1996,Ianniruberto2014a,Ianniruberto2014}.\nSince CCR (which we describe below) was proposed, there has been an\nongoing debate about its efficacy in potentially eliminating the\nregime of negative constitutive slope and restoring a monotonic\nconstitutive curve, thereby eliminating steady state banding.\nHowever, a non-monotonic constitutive curve and associated steady\nstate shear banding has been seen in molecular dynamics simulations of\npolymers~\\cite{Likhtmanetal2012}, for long enough chain lengths.  It\nis important to note, though, that the polydispersity that is often\npresent in practice in unbreakable polymers also tends to restore\nmonotonicity.\n\nBesides the conditions of steady state flow just described, many flows\nof practical importance involve a strong time dependence.  In view of\nthis, a natural question to ask is whether shear banding might also\narise in these time-dependent flows and, if so, under what conditions.\nOver the past decade, a body of experimental data has accumulated to\nindicate that it does indeed occur: in shear\nstartup~\\cite{Divoux2010, Divoux2011a, Wangetal2009a,\n  Huetal2007a, Wangetal2008a, Martin2012}, following a step\nstrain (in practice a rapid strain ramp)~\\cite{Wangetal2010a,\n  Boukany2009a, Wangetal2006a, Fang2011, Wangetal2007a,\n  Archer1995, Wangetal2008c}, and following a step\nstress~\\cite{Gibaud2010,Divoux2011,\n  Wangetal2009a,Huetal2007a, Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010}.\n\n\nConsistent with this growing body of experimental evidence,\ntheoretical\nconsiderations~\\cite{Moorcroft2013,Moorcroft2014,Moorcroft2011,Adamsetal2011,Manningetal2009a}\nalso suggest that shear banding might arise rather generically in\nflows with a sufficiently strong time-dependence, even in fluids that\nhave a monotonically increasing constitutive curve and so do not\ndisplay steady state banding under conditions of a continuously\napplied shear.  Indeed, the calculations to date suggest that the set\nof fluids that show banding in steady state is only a subset of those\nthat exhibit banding in time-dependent flows. In view of this,\nalthough the question concerning the existence or otherwise of steady\nstate shear banding in polymers remains an important one, the\nresolution of that controversy is likely to be of less practical\nimportance to the broader issue of whether shear banding arises more\ngenerally in time-dependent flows.\n\nIn the last five years progress has been made in establishing\ntheoretically, separately for each of the time-dependent flow\nprotocols listed above (shear startup, step strain and step stress), a\nfluid-universal criterion~\\cite{Moorcroft2013} for the onset of\nshear banding, based on the shape of the time-dependent rheological\nresponse function for the particular protocol in question. We now\nbriefly review these criteria as backdrop to understanding the results\nthat follow below for shear banding in large amplitude oscillatory\nshear (LAOS).\n\nIn shear startup (the switch-on at some time $t=0$ of a constant shear\nrate $\\dot{\\gamma}$), the onset of banding is closely associated with the\npresence of an\novershoot~\\cite{Moorcroft2013,Moorcroft2014,Moorcroft2011,Adamsetal2011,Manningetal2009a}\nin the startup signal of stress as a function of time (or equivalently\nas a function of strain), as it evolves towards its eventual steady\nstate on the material's flow curve. This concept builds on the early\ninsight of Ref.~\\cite{Marrucci1983}. The resulting bands may, or may\nnot, then persist to steady state, according to whether or not the\nunderlying constitutive curve of stress as a function of strain rate\nis non-monotonic. This tendency of a startup overshoot to trigger\nbanding was predicted on the basis of fluid-universal analytical\ncalculations in Ref.~\\cite{Moorcroft2013}, and has been\nconfirmed in numerical simulations of polymeric fluids (polymer solutions, polymer melts and\nwormlike micelles)~\\cite{Moorcroft2014,Adamsetal2011},\npolymer glasses~\\cite{Fielding2013} and soft glassy materials\n(dense emulsions, microgels, foams, {\\it\n  etc.})~\\cite{Moorcroft2011,Manningetal2009a,Jagla2010}.\nIt is consistent with experimental observations in wormlike micellar\nsurfactants~\\cite{Hu2008, Wangetal2008c},\npolymers~\\cite{Wangetal2008a, Wangetal2008b, Wangetal2009a,\n  Wangetal2009b, Boukany2009a, Wangetal2003a, Huetal2007a,\n  Wangetal2009d}, carbopol\ngels~\\cite{Divoux2011a,Divoux2010} and Laponite\nclay suspensions~\\cite{Martin2012}.\n\nFollowing the imposition of a step stress in a previously undeformed\nsample, the onset of shear banding is closely associated with the\nexistence of a regime of simultaneous upward slope and upward\ncurvature in the time-differentiated creep response curve of shear\nrate as a function of\ntime~\\cite{Moorcroft2013,Fielding2014}.  This criterion\nwas also predicted on the basis of fluid-universal analytical\ncalculations in Ref.~\\cite{Moorcroft2013}, and has been\nconfirmed in numerical simulations of polymeric\nfluids~\\cite{Moorcroft2014} and soft glassy\nmaterials~\\cite{Fielding2014}.  It is consistent with\nexperimental observations in polymers~\\cite{Wangetal2009a,Huetal2007a,Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010, Wangetal2009d}, carbopol\nmicrogels~\\cite{Divoux2011} and carbon black\nsuspensions~\\cite{Gibaud2010}.\n\nIn the shear startup and step stress experiments just described, the\ntime-dependence is inherently transient in nature: after (typically)\nseveral strain units, the system evolves to its eventual steady state\non the material's flow curve.  In any such protocol, for a fluid with\na monotonic constitutive curve that precludes steady state banding,\nany observation of banding is predicted to be limited to this regime\nof time-dependence following the inception of the flow.  That poses an\nobvious technical challenge to experimentalists: of imaging the flow\nwith sufficient time-resolution to detect these transient bands.  This\nis particularly true for a polymeric fluid with a relatively fast\nrelaxation spectrum.  For soft glassy materials, in contrast, the\ndynamics are typically much slower and any bands associated with the\nonset of flow, though technically transient, may persist for a\nsufficiently long time to be mistaken for the material's ultimate\nsteady state response for any practical\npurpose~\\cite{Moorcroft2011,Fielding2014}.\n\nIn the past decade, the rheological community has devoted considerable\nattention to the of study large amplitude oscillatory shear (LAOS).\nFor a recent review, see Ref.~\\cite{Hyun2011}. In this\nprotocol, the applied flow has the form of a sustained oscillation and\nis therefore perpetually time-dependent, in contrast to the transient\ntime-dependence of the shear startup and step stress protocols just\ndescribed.  But by analogy with the predictions of transient shear\nbanding in shear startup and step stress, a sustained oscillatory flow\nmight (in certain regimes that we shall discuss) be expected to\nrepeatedly show banding at certain phases of the cycle, or even to\nshow sustained banding round the whole cycle.  Importantly, again by\nanalogy with our knowledge of shear startup and step stress, this\neffect need not be limited to fluids with a non-monotonic constitutive\ncurve that show steady state banding in a continuously applied shear\nflow, but might instead arise as a natural consequence of the\ntime-dependence inherent to the oscillation.\n\nIndeed, a particularly attractive feature of LAOS is that the severity\nof the flow's time-dependence, relative to the fluid's intrinsic\ncharacteristic relaxation timescale $\\tau$, can be tuned by varying\nthe frequency $\\omega$ of the applied oscillation. A series of LAOS\nexperiments can thereby explore the full range between steady state\nbehaviour in the limit $\\omega\\to 0$, where the oscillation\neffectively corresponds to a repeated series of quasi-static sweeps up\nand down the flow curve, and strongly time-dependent behaviour for\n$\\omega > 1\/\\tau$. A fluid with a non-monotonic underlying\nconstitutive curve that admits steady state banding is then clearly\nexpected to exhibit banding in the limit of $\\omega\\to 0$, as the\nshear rate quasi-statically transits the plateau in the steady state\nflow curve. In contrast, a monotonic constitutive curve precludes\nbanding for $\\omega\\to 0$.  Crucially, though, as noted above, the\nabsence of banding in steady state conditions does not rule out the\npossibility of banding in flows with a strong enough time-dependence,\n$\\omega \\gtrapprox O(1\/\\tau)$.\n\nIndeed, intuitively, a square-wave caricature of a large amplitude\noscillatory shear strain (LAOStrain) experiment points to a perpetual\nswitching between a shear startup like process in the forward\ndirection, followed by `reverse startup' in the opposite direction.\nAny regime in which these startup-like events are associated with an\novershoot in the associated curve of stress as a function of strain\nthen strongly suggests the possibility of shear banding during those\nquasi-startup parts of the cycle, by analogy with the criterion for\nbanding in a true shear startup from rest.  In the same spirit, a\nsquare-wave caricature of a large amplitude oscillatory shear stress\n(LAOStress) experiment indicates a perpetually repeated series of step\nstress events, jumping between positive and negative stress values,\nand so admitting the possibility of shear banding if the criterion for\nbanding following a step stress is met.\n\nIn practice, of course, LAOS is more complicated than the caricatures\njust described and the criteria for banding in shear startup and step\nstress might only be expected to apply in certain limiting regimes.\nNonetheless, in what follows we shall show that many of our results\nfor banding in LAOStrain and LAOStress can, to a large extent, be\nunderstood within the framework of these existing criteria for the\nsimpler time-dependent protocols.\n\nExperimentally, shear banding has indeed been observed in LAOS: in\npolymer solutions~\\cite{Wangetal2006c}, dense\ncolloids~\\cite{Cohen2006}, and also in wormlike micellar\nsurfactants that are known to shear band in steady\nstate~\\cite{Dimitriou2012,KateGurnon2012,Gurnon2014a}.\n\nFrom a theoretical viewpoint, several approaches to the interpretation\nof LAOS data have been put forward in the\nliterature~\\cite{Hyun2011}. These include Fourier transform\nrheology~\\cite{Wilhelm2002}; measures for quantifying\nLissajous-Bowditch curves (defined below) in their elastic\nrepresentation of stress versus strain, or viscous representation of\nstress versus strain rate~\\cite{Tee1975}; a decomposition\ninto characteristic sine, square and triangular wave prototypical\nresponse functions~\\cite{Klein2007,Klein2008};\ndecomposition into elastic and viscous stress contributions using\nsymmetry arguments~\\cite{Cho2005}; Chebyshev series\nexpansions of these elastic and viscous\ncontributions~\\cite{Ewoldt2008a}; and interpretations of the\nLAOS cycle in terms of a sequence of physical\nprocesses~\\cite{Rogers2012,Rogers2011}.\n\nHowever, many of these existing theoretical studies assume either\nexplicitly or implicitly that the flow remains homogeneous, and\nthereby fail to take account of the possibility of shear banding. An\nearly exception can be found in\nRefs.~\\cite{Zhou2010,Zhou2008}, which studied a\nmodel of wormlike micellar surfactants with a non-monotonic\nconstitutive curve in LAOStrain.  Another exception is in the paper of\nAdams and Olmsted~\\cite{Adams2009}, which recognised that\nshear banding can arise even in the absence of any non-monotonicity in\nthe underlying constitutive curve.\n\nThe work that follows here builds on the remarkable insight of these\nearly papers, in carrying out a detailed numerical study of shear\nbanding in LAOStrain and LAOStress within the Rolie-poly\nmodel~\\cite{Grahametal2003a} of polymers and wormlike\nmicellar surfactant solutions.  Consistent with the above discussion,\nin LAOStrain we observe banding at low frequencies $\\omega\\to 0$ and\nsufficiently high strain rate amplitudes $\\dot{\\gamma}\\gtrapprox 1\/\\tau$ in\nfluids for which the underlying constitutive curve of shear stress as\na function of shear rate is non-monotonic. At higher frequencies\n$\\omega=O(1\/\\tau)$ and for sufficiently high strain amplitudes\n$\\gamma\\gtrapprox 1$ we instead see `elastic' shear banding associated\nwith an overshoot in the elastic curve of stress as a function of\nstrain, in close analogy with the elastic banding predicted in a fast\nshear startup\nexperiment~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011,Adams2009}.\nImportantly, we show that this elastic banding arises robustly even in\na wide range of model parameter space for which the underlying\nconstitutive curve is monotonic, precluding steady state banding.\n\nIn LAOStress we observe banding in fluids that shear thin sufficiently\nstrongly to have either a negatively, or weakly positively, sloping\nregion in the underlying constitutive curve. We emphasise again that\nfluids in the latter category do not display steady state banding, and\ntherefore that, for such fluids, the banding predicted in LAOStress is\na direct result of the time-dependence of the applied flow. In this\ncase the banding is triggered in each half cycle as the stress magnitude\ntransits in an upward direction the region of weak slope and the\nstrain rate magnitude increases dramatically such that the material effectively\nyields. This is strongly reminiscent of the transient banding\ndiscussed previously in step\nstress~\\cite{Moorcroft2013,Moorcroft2014}.\n\nWhile it would be interesting to interpret our findings within one (or\nmore) of the various mathematical methodologies for analysing LAOS\ndiscussed above (and in particular to consider the implications of\nbanding for the presence of higher harmonics in the output rheological\ntime series), in the present manuscript we focus instead on the\nphysical understanding that can be gained by considering the shapes of\nthe signals of stress versus strain or strain rate (in LAOStrain) and\nstrain rate versus time (in LAOStress).  In that sense, this work is\nclosest in spirit to the sequence of physical processes (SPP) approach\nof Refs.~\\cite{Rogers2012,Rogers2011} (which did\nnot, however, explicitly consider heterogeneous response).  In\nparticular, we seek to interpret the emergence of shear banding in\nLAOS on the basis of the existing criteria for the onset of banding in\nthe simpler time-dependent protocols of shear startup and step\nstress~\\cite{Moorcroft2013}.\n\nThe paper is structured as follows. In Sec.~\\ref{sec:models} we\nintroduce the model, flow geometry and protocols to be considered.\nSec.~\\ref{sec:methods} outlines the calculational methods that we\nshall use.  Sec.~\\ref{sec:recap} contains a summary of previously\nderived linear instability criteria for shear banding in steady shear,\nfast shear startup and step shear stress protocols, with the aim of providing a\nbackdrop to understanding shear banding in oscillatory protocols.  In\nSecs.~\\ref{sec:LAOStrain} and~\\ref{sec:LAOStress} we present our\nresults for LAOStrain and LAOStress respectively, and discuss their\npotential experimental verification.  Finally\nSec.~\\ref{sec:conclusions} contains our conclusions and an outlook for\nfuture work.\n\n\n\\section{Model, flow geometry and protocols}\n\\label{sec:models}\n\nWe write the stress $\\tens{\\Sigma}(\\tens{r},t)$ at any time $t$ in a\nfluid element at position $\\tens{r}$ as the sum of a viscoelastic\ncontribution $\\tens{\\sigma}(\\tens{r},t)$ from the polymer chains or\nwormlike micelles, a Newtonian contribution characterised by a\nviscosity $\\eta$, and an isotropic contribution with pressure\n$p(\\tens{r},t)$:\n\\begin{equation}\n\\tens{\\Sigma} = \\tens{\\sigma} + 2 \\eta \\tens{D} - p\\tens{I}.\n\\label{eqn: total_stress_tensor}\n\\end{equation}\nThe Newtonian stress $2 \\eta \\tens{D}(\\tens{r},t)$ may arise from the\npresence of a true solvent, and from any polymeric\ndegrees of freedom considered fast enough not to be ascribed their own\nviscoelastic dynamics. The symmetric strain rate tensor $\\tens{D} =\n\\frac{1}{2}(\\tens{K} + \\tens{K}^T)$ where $K_{\\alpha\\beta} =\n\\partial_{\\beta}v_{\\alpha}$ and $\\tens{v}(\\tens{r},t)$ is the fluid\nvelocity field.  \n\nWe consider the zero Reynolds number limit of creeping flow, in which\nthe condition of local force balance requires the stress field\n$\\tens{\\Sigma}(\\tens{r},t)$ to be divergence free:\n\\begin{equation}\n\\vecv{\\nabla}\\cdot\\,\\tens{\\Sigma} = 0.\n\\label{eqn: force_balance}\n\\end{equation}\nThe pressure field $p(\\tens{r},t)$ is determined by enforcing that the\nflow remains incompressible:\n\\begin{equation}\n\\label{eqn: incomp}\n\\vecv{\\nabla}\\cdot\\vecv{v} = 0.\n\\end{equation}\n\nThe viscoelastic stress is then written in terms of a constant elastic\nmodulus $G$ and a tensor $\\visc(\\tens{r},t)$ characterising the\nconformation of the polymer chains or wormlike micelles,\n$\\tens{\\sigma} = G\\, (\\visc - \\tens{I})$.  We take the dynamics of\n$\\visc$ to be governed by the Rolie-poly (RP)\nmodel~\\cite{Grahametal2003a} with\n\\begin{widetext}\n\\begin{eqnarray}\n\\partial_t{\\visc}+\\tens{v}\\cdot\\nabla\\tens{\\visc} &=& \\tens{K} \\cdot \\visc + \\visc \\cdot \\tens{K}^T - \\frac{1}{\\tau_d}\\left(\\visc - \\tens{I}\\right) \n- \\frac{2(1-A)}{\\tau_R}\\left[\\, \\visc + \\beta A^{-2\\delta}\\left(\\visc - \\tens{I}\\right) \\right] + D\\nabla^2\\visc,\n\\label{eqn: rolie-poly_tensor}\n\\end{eqnarray}\n\\end{widetext}\nin which $A = \\sqrt{3\/T\\,}$ with trace $T = \\text{tr}\\,\\tens{\\visc}$.\nThis RP model is a single mode simplification of the GLAMM model\n\\cite{GLAMM}, which provides a microscopically derived stochastic\nequation for the dynamics of a test chain (or micelle) in its mean\nfield tube of entanglements with other chains.  The timescale $\\tau_d$\nsets the characteristic time on which a chain escapes its tube by\nmeans of 1D curvilinear diffusion along the tube's contour, known as\nreptation, allowing the molecular orientation to refresh itself. The\nRouse timescale $\\tau_R$ sets the shorter time on which chain stretch,\nas characterised by $T = \\text{tr}\\,\\tens{\\visc}$, relaxes.  The ratio\n$\\tau_d\/\\tau_R=3Z$, where $Z$ is the number of entanglements per chain.\nThe parameters $\\beta$ and $\\delta$ govern a phenomenon known as\nconvective constraint\nrelease~\\cite{Marrucci1996,Ianniruberto2014a,Ianniruberto2014}\n(CCR), in which the relaxation of the stretch of a test chain has the\neffect of also relaxing entanglement points, thereby facilitating the\nrelaxation of tube orientation.  The diffusive term $D\\nabla^2\\visc$\nadded to the right hand side of Eqn.~\\ref{eqn: rolie-poly_tensor} is\nrequired to account for the slightly diffuse nature of the interface\nbetween shear bands~\\cite{Luetal2000a}: without it the shear\nrate would be discontinuous across the interface, which is unphysical.\n\nUsing this model we will consider shear flow between infinite flat\nparallel plates at $y = \\{0,L\\}$, with the top plate moving in the\n$\\vecv{\\hat{x}}$ direction at speed $\\overline{\\dot{\\gamma}}(t) L$. We assume\ntranslational invariance in the flow direction $\\vecv{\\hat{x}}$ and\nvorticity direction $\\vecv{\\hat{z}}$ such that the fluid velocity can\nbe written as $\\vecv{v} = v(y,t)\\vecv{\\hat{x}}$. The local shear rate\nat any position $y$ is then given by\n\\begin{equation}\n\\dot{\\gamma}(y,t) = \\partial_{y}v(y,t),\n\\end{equation}\nand the spatially averaged shear rate\n\\begin{equation}\n\\overline{\\dot{\\gamma}}(t) = \\frac{1}{L}\\int_{0}^{L} \\dot{\\gamma}(y,t)dy.\n\\end{equation}\nSuch a flow automatically satisfies the constraint of\nincompressibility, Eqn.~\\ref{eqn: incomp}.  The force balance\ncondition, Eqn.~\\ref{eqn: force_balance}, further demands that the\ntotal shear stress is uniform across the cell, in the planar flow\nsituation considered here, giving $\\partial_{y}\\Sigma_{xy} =0$. The\nviscoelastic and Newtonian contributions may, however, each depend on\nspace provided their sum remains uniform:\n\\begin{equation}\n\\Sigma_{xy}(t) = GW_{xy}(y,t) + \\eta \\dot{\\gamma}(y,t).\n\\label{eqn: shear_stress}\n\\end{equation}\n\nFor such a flow, the RP model can be written componentwise as\n\\begin{widetext}\n\\begin{eqnarray}\n\\dot{W}_{xy}  &=& \\dot{\\gamma} W_{yy} - \\frac{W_{xy}}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}(1+ \\beta A)W_{xy} + D\\partial_y^2 W_{xy}, \\nonumber\\\\\n\\dot{W}_{yy}  &=& - \\frac{W_{yy}-1}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}\\left[W_{yy}+ \\beta A(W_{yy}-1)\\right]+ D\\partial_y^2W_{yy},\\nonumber\\\\\n\\dot{T}     &=&  2\\dot{\\gamma}W_{xy}  - \\frac{T-3}{\\tau_d} - \\frac{2(1-A)}{\\tau_R}\\left[T + \\beta A(T - 3)\\right]+ D\\partial_y^2 T.\\quad \\quad\n\\label{eqn: sRP_components}\n\\end{eqnarray}\n(The other components of $\\tens{W}$ decouple to form a separate\nequation set, with trivial dynamics.)  In the limit of fast chain\nstretch relaxation $\\tau_R \\to 0$ we obtain the simpler\n`non-stretching' RP model in which the trace $T=3$ and\n\\begin{eqnarray}\n\\dot{W}_{xy} &=& \\dot{\\gamma} \\left[W_{yy} - \\frac{2}{3} (1+\\beta)W_{xy}^2\\right]\\;\\;\\;\\;\\;\\;\\; -\\frac{1}{\\tau_d}W_{xy},+ D\\partial_y^2 W_{xy}\\nonumber\\\\\n\\dot{W}_{yy} &=& \\frac{2}{3}\\dot{\\gamma}\\left[\\betaW_{xy}-(1+\\beta)W_{xy}W_{yy} \\right] - \\frac{1}{\\tau_d}(W_{yy}-1)+ D\\partial_y^2W_{yy}. \\quad \\quad\n\\label{eqn: nRP_components}\n\\end{eqnarray}\n\\end{widetext}\nFor convenient shorthand we shall refer to this simpler non-stretching\nform as the nRP model. We refer to the full `stretching' model of\nEqns.~\\ref{eqn: sRP_components} as the sRP model.\n\nFor boundary conditions at the walls of the flow cell we assume no\nslip and no permeation for the fluid velocity, and zero-gradient\n$\\partial_y W_{\\alpha\\beta}=0$ for every component $\\alpha\\beta$ of\nthe polymeric conformation tensor.\n\nIn what follows we consider the behaviour of the Rolie-poly model in the following two flow protocols:\n\n\\begin{itemize}\n\n\\item LAOStrain, with an imposed strain \n\\begin{equation}\n\\gamma(t)=\\gamma_0\\sin(\\omega t),\n\\end{equation}\nto which corresponds the strain rate\n\\begin{equation}\n\\dot{\\gamma}(t)=\\gamma_0\\omega\\cos(\\omega t)=\\dot{\\gamma}_0\\cos(\\omega t).\n\\end{equation}\n\n\\item LAOStress, with an imposed stress\n\\begin{equation}\n\\Sigma(t)=\\Sigma_0\\sin(\\omega t).\n\\end{equation}\n\n\\end{itemize}\n\nThe model, flow geometry and protocol just described are characterised\nby the following parameters: the polymer modulus $G$, the reptation\ntimescale $\\tau_d$, the stretch relaxation timescale $\\tau_R$, the CCR\nparameters $\\beta$ and $\\delta$, the stress diffusivity $D$, the\nsolvent viscosity $\\eta$, the gap size $L$, the frequency $\\omega$ and\nthe amplitude $\\gamma_0$ (for LAOStrain) or $\\Sigma_0$ (for\nLAOStress). We are free to choose units of mass, length and time,\nthereby reducing the list by three: we work in units of length in\nwhich the gap size $L = 1$, of time in which the reptation time $\\tau_d\n= 1$ and of mass (or actually stress) in which the polymer modulus\n$G=1$.  We then set the value of the diffusion constant $D$ such that\nthe interface between the bands has a typical width $\\ell =\n\\sqrt{D\\tau_d}=2\\times 10^{-2}L$, much smaller than the gap size. This\nis the physically relevant regime for the macroscopic flow cells of\ninterest here, and we expect the results we report to be robust to\nreducing $l$ further.  Following Ref.~\\cite{Grahametal2003a} we set\n$\\delta = -\\frac{1}{2}$.\n\nAdimensional quantities remaining to be explored are then the model\nparameters $\\eta$, $\\beta$ and (for the sRP model only) $\\tau_R$; and\nthe protocol parameters $\\omega$ and $\\gamma_0$ or $\\Sigma_0$. For\neach set of model parameters we explore the whole plane of feasibly\naccessible values of protocol parameters $\\omega$ and $\\gamma_0$ or\n$\\Sigma_0$.\n\nAmong the model parameters the CCR parameter has the range $0 \\leq\n\\beta \\leq 1$. Within this there is no current consensus as to its\nprecise value, and we shall therefore explore widely the full range\n$0\\to 1$. For the fluids of interest here the Newtonian viscosity is\ntypically much smaller than the zero shear viscosity of the\nviscoelastic component, giving $\\eta\\ll 1$ in our units. Based on a\nsurvey of the experimental data, a range of $10^{-7}$ to $10^{-3}$ was\nsuggested by Graham et al. in Ref.~\\cite{Graham2013}\nConsistentwith comments made in Ref.~\\cite{Agimelen2013} we find values\nless than $10^{-5}$ unfeasible to explore numerically, due to a\nresulting large separation of timescales between $\\tau_d$ and $\\eta\/G$.\nTherefore we adopt typical values $\\eta=10^{-4}$ and $10^{-5}$.  Given\nthat that the susceptibility to shear banding increases with\ndecreasing $\\eta$, we note that the levels of banding reported in what\nfollows are likely, if anything, to be an underestimate of what might\nbe observed experimentally. We return in our concluding remarks to\ndiscuss this issue further.\n\nWe explore a wide range of values of the stretch relaxation time\n$\\tau_R$, or equivalently of the degree of entanglement\n$Z=\\tau_d\/3\\tau_R$: we consider $Z=1$ to $350$ for the sRP model (and\nnote that the nRP model has $Z\\to\\infty$ by definition).\nExperimentally, values of $Z$ in the range of $50$ appear commonplace\nand $100$ towards the upper end of what might currently be used\nexperimentally in nonlinear rheological studies.  One of the\nobjectives of this work is to provide a roadmap of values of $Z$ and\n$\\beta$ in which shear banding is expected to be observed, for typical\nsmall values of $\\eta$, in a sequence of LAOS protocols that scan\namplitude and frequency space.\n\n\\section{Calculation methods} \n\\label{sec:methods}\n\nIn this section we outline the theoretical methods to be used\nthroughout the paper. In order to develop a generalised framework\nencompassing both the nRP and sRP models, we combine all the relevant\ndynamical variables (for any given model) into a state vector\n$\\vecv{s}$, with $\\vecv{s}=(W_{xy},W_{yy})^T$ for the nRP model and\n$\\vecv{s}=(W_{xy},W_{yy},T)^T$ for the sRP model.  Alongside this we\ndefine a projection vector $\\vecv{p}$ of corresponding dimensionality\n$d$, with $\\vecv{p}=(1,0)$ for the nRP model and $\\vecv{p}=(1,0,0)$\nfor sRP.\n\nThe total shear stress $\\Sigma_{xy}=\\Sigma$, from which we drop the\n$xy$ subscript for notational brevity, is then given by\n\\begin{equation}\n\\label{eqn: governing_eqn_force}\n\\Sigma(t) = G\\vecv{p} \\cdot \\vecv{s}(y,t) + \\eta \\dot{\\gamma}(y,t),\n\\end{equation}\nand the viscoelastic constitutive equation has the generalised form\n\\begin{equation}\n\\partial_{t\\,}\\vecv{s}(y,t) = \\vecv{Q}(\\vecv{s},\\dot{\\gamma}) + D\\partial_y^2\\vecv{s}.\n\\label{eqn: governing_eqn_diffusive}\n\\end{equation}\nThe dimensionality and functional form of $\\vecv{Q}$ then specify the\nparticular constitutive model. In this way our generalised notation in\nfact encompasses not only the nRP model (for which $d=2$) and sRP\nmodel (for which $d=3$) but many more besides, including the Johnson\nSegalman, Giesekus and Oldroyd B models~\\cite{Larson1988}.\n\n\\subsection{Homogeneous base state}\n\\label{sec:base}\n\nFor any given applied flow our approach will be first to calculate the\nfluid's response within the simplifying assumption that the\ndeformation must remain homogeneous across the cell. While this is an\nartificial (and indeed incorrect) constraint in any regime where shear\nbanding is expected, it nonetheless forms an important starting point\nfor understanding the mechanism by which shear banding sets in.  (We\nalso note that most papers in the literature make this assumption\nthroughout, thereby disallowing any possibility of shear banding\naltogether.)\n\nWithin this assumption of homogeneous flow, the response of the system\nfollows as the solution to the set of ordinary differential equations\n\\begin{equation}\n\\label{eqn: governing_eqn_force_local}\n\\base{\\Sigma}(t) = G\\vecv{p} \\cdot \\base{\\vecv{s}}(t) + \\eta \\base{\\dot{\\gamma}}(t),\n\\end{equation}\nand\n\\begin{equation} \n\\dot{\\base{\\vecv{s}}}(t) = \\vecv{Q}(\\base{\\vecv{s}},\\base{\\dot{\\gamma}}).\n\\label{eqn: governing_eqn_diffusive_local}\n\\end{equation}\nIn these either $\\base{\\dot{\\gamma}}(t)$ or $\\base{\\Sigma}(t)$ is imposed, in\nLAOStrain and LAOStress respectively, and the other dynamical\nquantities are calculated numerically using an explicit Euler\nalgorithm~\\cite{NumRecipes}. We use the `hat' notation to denote\nthat the state being considered is homogeneous.\n\n\\subsection{Linear stability analysis}\n\\label{sec:lsa}\n\nHaving calculated the behaviour of the fluid within the assumption\nthat the flow remains homogeneous, we now proceed to consider whether\nthis homogeneous `base state' flow will, at any point during an applied\noscillatory protocol, be unstable to the formation of shear bands. To\ndo so we add to the base state, for which we continue to use the hat\nnotation, heterogeneous perturbations of (initially) small amplitude:\n\\begin{eqnarray}\n\\Sigma(t) &=& \\base{\\Sigma}(t),\\nonumber\\\\\n\\dot{\\gamma}(y,t)&=& \\base{\\dot{\\gamma}}(t) + \\sum_{n=1}^\\infty \\delta \\dot{\\gamma}_n(t) \\cos(n\\pi y\/L),\\nonumber\\\\\n\\vecv{s}(y,t) &=& \\vecv{\\base{s}}(t) + \\sum_{n=1}^\\infty \\delta\\vecv{s}_n(t) \\cos(n\\pi y\/L).\n\\label{eqn: LSA}\n\\end{eqnarray}\nNote that the total stress $\\Sigma$ is not subject to heterogeneous\nperturbations because the constraint of force balance decrees that it\nmust remain uniform across the gap, at least in a planar shear cell.\nSubstituting Eqns.~\\ref{eqn: LSA} into Eqns.~\\ref{eqn:\n  governing_eqn_force} and~\\ref{eqn: governing_eqn_diffusive}, and\nexpanding in successive powers of the magnitude of the small\nperturbations $\\delta{\\dot{\\gamma}_n},\\vecv{\\delta s_n}$, we recover at\nzeroth order Eqns.~\\ref{eqn: governing_eqn_force_local} and~\\ref{eqn:\n  governing_eqn_diffusive_local} for the dynamics of the base state. At first order the heterogeneous perturbations obey\n\\begin{eqnarray}\n\\label{eqn: perturbation}\n0&=&G\\tens{p}\\cdot \\delta\\vecv{s}_n(t)+\\eta\\delta\\dot{\\gamma}_n(t),\\nonumber\\\\\n\\dot{\\delta\\vecv{s}}_n &=& \\tens{M}(t) \\cdot \\delta\\vecv{s}_n + \\tens{q}\\delta{\\dot{\\gamma}}_n,\n\\end{eqnarray}\nin which  $\\tens{M} =\n\\partial_{\\vecv{s}\\,}\\vecv{Q}|_{\\vecv{\\base{s}},\\base{\\dot{\\gamma}}}-\\tens{\\delta}D(n\\pi\/L)^2$\nand $\\vecv{q}\n= \\partial_{\\dot{\\gamma}}\\vecv{Q}|_{\\vecv{\\base{s}},\\base{\\dot{\\gamma}}}$. Combining\nthese gives\n\\begin{equation}\n\\label{eqn: one}\n\\dot{\\delta\\vecv{s}}_n = \\tens{P}(t) \\cdot \\delta\\vecv{s}_n,\n\\end{equation}\nwith\n\\begin{equation}\n\\tens{P}(t) = \\tens{M}(t) - \\frac{G}{\\eta}\\vecv{q}(t)\\, \\vecv{p}.\n\\label{eqn: two}\n\\end{equation}\nIn any regime where the heterogeneity remains small, terms of second\norder and above can be neglected.\n\nTo determine whether at any time $t$ during an imposed oscillatory\nflow the heterogeneous perturbations\n$\\delta{\\dot{\\gamma}}_n,\\delta\\vecv{s}_n(t)$ have positive rate of growth,\nindicating linear instability of the underlying homogeneous base state\nto the onset of shear banding, we consider first of all the\ninstantaneous sign of the eigenvalue $\\lambda(t)$ of $\\tens{P}(t)$ that\nhas the largest real part. A positive value of $\\lambda(t)$ is clearly\nsuggestive that heterogeneous perturbations will be instantaneously\ngrowing at that time $t$. We note, however, that the concept of a\ntime-dependent eigenvalue must be treated with caution.  In view of\nthis we cross check predictions made on the basis of the eigenvalue by\nalso directly numerically integrating the linearised Eqns.~\\ref{eqn:\n  one} using an explicit Euler algorithm. This allows us to determine\nunambiguously whether the heterogeneous perturbations will be at any\ninstant growing (taking the system towards a banded state) or decaying\n(restoring a homogeneous state), at the level of this linear\ncalculation.\n\nIn these linear stability calculations we neglect the diffusive term\nin the viscoelastic constitutive equation, setting $D=0$. Reinstating\nit would simply transform any eigenvalue $\\lambda \\to \\lambda_n=\\lambda -\nD n^2\\pi^2\/L^2$ and provide a mechanism whereby any heterogeneity with\na wavelength of order the microscopic lengthscale $l$, or below,\ndiffusively decays. Accordingly the results of this linear calculation\nonly properly capture the dynamics of any heterogeneous perturbations\nthat have macroscopically large wavelengths, which are the ones of\ninterest in determining the initial formation of shear bands starting\nfrom a homogeneous base state.\n\nAs a measure of the degree of flow heterogeneity at any time $t$ in\nthis linear calculation, we shall report in our results sections below\n$\\delta\\dot{\\gamma} (t)$ normalised by the amplitude of the imposed\noscillation $\\dot{\\gamma}_0$ in LAOStrain, or by $1+|\\dot{\\gamma}(t)|$ in LAOStress,\nwhere $\\dot{\\gamma}(t)$ is the instantaneous value of the shear rate. (We\nfind numerically that bands tend to form in LAOStress when\n$|\\dot{\\gamma}(t)|\\gg 1$. The additional 1 in the normalisation is used\nsimply to prevent the divergence of this measure when $\\dot{\\gamma}(t)$ passes\nthrough 0 in each half cycle.) Note that we no longer need to specify\nthe mode number $n$ for $\\delta\\dot{\\gamma}$, because within the assumption\n$D=0$ just described, we are confining our attention to the limit of\nlong wavelength modes only and noting them all to have the same\ndynamics, to within small corrections set by $D$.\n\n\\subsection{Full nonlinear simulation}\n\\label{sec:nonlinear}\n\nWhile the linear analysis just described provides a calculationally\nconvenient method for determining whether shear banding will arise in\nany given oscillatory measurement, enabling us to quickly build up an\noverall roadmap of parameter space, it cannot predict the detailed\ndynamics of the shear bands once the amplitude of heterogeneity has\ngrown sufficiently large that nonlinear effects are no longer\nnegligible. Therefore in what follows we shall also perform full\nnonlinear simulations of the model's spatio-temporal dynamics by\ndirectly integrating the full model Eqns.~\\ref{eqn:\n  governing_eqn_force} and~\\ref{eqn: governing_eqn_diffusive} using a\nCrank-Nicolson algorithm \\cite{NumRecipes}, with the system's\nstate discretised on a grid of $J$ values of the spatial coordinate\n$y$, checked in all cases for convergence with respect to increasing\nthe number of grid points.\n\nAs a measure of the degree of shear banding at any time $t$ in this\nnonlinear calculation we report the difference between the maximum and\nminimum values of the shear rate across the cell:\n\\begin{equation}\n\\Delta_{\\dot{\\gamma}}(t) = \\frac{1}{N}\\Big[|\\dot{\\gamma}_{\\rm max}(t) - \\dot{\\gamma}_{\\rm min}(t)|\\Big],\n\\label{eqn: dob}\n\\end{equation}\nagain normalised depending upon the employed protocol, by $N$, where $N$ is the amplitude of the imposed oscillation $\\dot{\\gamma}_0$ in LAOStrain, and $1+|\\dot{\\gamma}(t)|$ in LAOStress.\n\n\\subsection{Seeding the heterogeneity}\n\\label{sec:seed}\n\nWhen integrating the model equations to determine the time evolution\nof any flow heterogeneity, whether linearised or in their full\nnonlinear form, we must also specify the way in which whatever\nheterogeneous perturbations that are the precursor to the formation of\nshear bands are seeded initially. Candidates include any residual\nheterogeneity left in the fluid by the initial procedure of sample\npreparation; imperfections in the alignment of the rheometer plates;\ntrue thermal noise with an amplitude set by $k_{\\rm B}T$; and\nrheometer curvature in cone-and-plate or cylindrical Couette devices.\nWe consider in particular the last of these because it is likely to be\nthe dominant source of heterogeneity in commonly used flow cells,\nwhich typically have a curvature of about $10\\%$.\n\nWhile modelling the full effects of curvature is a complicated task,\nits dominant consequence can be captured simply by including a slight\nheterogeneity in the total stress field. (The assumption made above of\na uniform stress across the gap only holds in an idealised planar\ndevice.) Accordingly we set $\\Sigma(t)\\to\\Sigma(t)\\left[1+q\n  h(y)\\right]$ where $q$ sets the amplitude of the curvature and\n$h(y)$ is a function with an amplitude of $O(1)$ that prescribes its\nspatial dependence. The detailed form of $h(y)$ will differ from\ndevice to device: for example in a cylindrical Couette it is known to\nhave a $1\/r^2$ dependence, where $r$ is the radial coordinate.\nHowever, the aim here is not to model any particular device geometry\nin detail, but simply to capture the dominant effect of curvature in\nseeding the flow heterogeneity. Accordingly we set\n$h(y)=\\cos(\\pi\/L)$ which is the lowest Fourier mode to fit into the\nsimulation cell while still obeying the boundary conditions at the\nwalls.\n\n\\section{Shear banding in other time dependent protocols}\n\\label{sec:recap}\n\nAs a preamble to presenting our results for shear banding in\noscillatory flow protocols in the next two sections below, we first\nbriefly collect together criteria derived in previous work for linear\ninstability to the formation of shear bands in simpler time-dependent\nprotocols: slow shear rate sweep, fast shear startup, and step stress.\n\n\\subsection{Slow shear rate sweep}\n\\label{sec:recapSweep}\n\nA common experimental protocol consists of slowly sweeping the shear\nrate $\\dot{\\gamma}$ upwards (or downwards) in order to measure a fluid's\n(quasi) steady state flow curve. In this protocol the criterion for\nlinear instability to the onset of shear banding, given a base state\nof initially homogeneous shear flow, has long been known to\nbe~\\cite{Yerushalmi1970}\n\\begin{equation}\n\\label{eqn:criterionSteady}\n\\frac{\\partial\\Sigma}{\\partial\\dot{\\gamma}} < 0.\n\\end{equation}\n\n\\subsection{Fast shear startup}\n\\label{sec:recapStartup}\n\nAnother common experimental protocol consists of taking a sample of\nfluid that is initially at rest and with any residual stresses well\nrelaxed, then suddenly jumping the strain rate from zero to some\nconstant value such that $\\dot{\\gamma}(t)=\\dot{\\gamma}_0\\Theta(t)$, where\n$\\Theta(t)$ is the Heaviside function. Commonly measured in response\nto this applied flow is the time-dependent stress signal $\\Sigma(t)$\nas it evolves towards its eventual steady state value, for that\nparticular applied shear rate, on the fluid's flow curve. This\nevolution typically has the form of an initial elastic regime with\n$\\Sigma\\approx G\\gamma$ while the strain $\\gamma$ remains small,\nfollowed by an overshoot in the stress at a strain of $O(1)$, then a\ndecline to the final steady state stress on the flow curve. In\nRef.~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011}\nwe gave evidence that the presence of an overshoot in this stress\nstartup signal is generically indicative of a strong tendency to form\nshear bands, at least transiently. These bands may, or may not, then\npersist for as long as the shear remains applied, according to whether\nor not the underlying constitutive curve of stress as a function of\nstrain rate is non-monotonic.\n\nSuch behaviour is to be expected intuitively. Consider a shear startup\nrun performed at a high enough strain rate that the material's\nresponse is initially elastic, with the stress startup signal\ndepending only on the accumulated strain $\\gamma=\\dot{\\gamma} t$ and not\nseparately on the strain rate $\\dot{\\gamma}$. The decline in stress following\nan overshoot in the stress startup signal corresponds to a negative\nderivative\n\\begin{equation}\n\\label{eqn:criterionSimpleElastic}\n\\frac{\\partial \\Sigma}{\\partial\\gamma}<0.\n\\end{equation}\nThis clearly has the same form as~(\\ref{eqn:criterionSteady}) above,\nwith the strain rate now replaced by the strain.  As such it is the\ncriterion that we might intuitively expect for the onset of strain\nbands in a nonlinear elastic solid, following the early intuition of\nRef.~\\cite{Marrucci1983}\n\nIn close analogy to this intuitive expectation, for a complex fluid\nsubject to a fast, elastically dominated startup the criterion for the\nonset of banding was shown in Ref.~\\cite{Moorcroft2013} to be\nthat the stress signal $\\Sigma(\\gamma=\\dot{\\gamma} t)$ of the initially\nhomogeneous startup flow obeys\n\\begin{equation} \n\\label{eqn:criterionStartup}\n-\\textrm{tr}\\tens{M} \\frac{\\partial \\Sigma}{\\partial\\gamma} +\n\\dot{\\gamma}\\frac{\\partial^2\\Sigma}{\\partial\\gamma^2} < 0,\n\\end{equation}\nwhere $\\textrm{tr}\\tens{M}<0$ in this startup protocol.  This result\nholds exactly for any model whose equations are of the generalised\nform in Sec.~\\ref{sec:methods} above, and have only two relevant\ndynamical variables, $d=2$. (Recall that for the nRP model these two\nvariables are the shear stress $W_{xy}$ and one component of normal\nstress $W_{yy}$, in units in which the polymer modulus $G=1$.)  The\ncriterion~(\\ref{eqn:criterionStartup}) closely resembles the simpler\nform~(\\ref{eqn:criterionSimpleElastic}) motivated intuitively above,\nwith an additional term informed by the curvature in the signal of\nstress as a function of strain. The effect of this additional term is\nto trigger the onset of banding just {\\em before} overshoot, as the\nstress startup signal starts to curve downwards from its initial\nregime of linear elastic response.\n\nWhat this criterion tells us is that the presence of an overshoot in the\nstress signal of an underlying base state of initially homogeneous shear startup acts as a causative trigger for the formation of shear bands. A common misconception is that instead it is the onset of shear\nbanding that causes the stress drop.  While it is true that the onset\nof banding may reduce the stress further compared to that expected on\nthe basis of a homogeneous calculation, we emphasise that the\ndirection of mechanistic causality here is that the stress drop\nfollowing overshoot causes shear banding and not (primarily) vice\nversa.\n\nWith criterion (\\ref{eqn:criterionStartup}) in mind, theorists should\nbe alert that any model predicting startup stress overshoot in a\ncalculation in which the flow is artificially constrained to remain\nhomogeneous is likely to further predict the formation of shear bands\nin a full heterogeneous calculation that allows bands to form. Likewise\nexperimentalists should be alert that any observations of stress\novershoot in shear startup is strongly suggestive of the presence of\nbanding in the material's flow profile.\n\nIn Ref.~\\cite{Moorcroft2014} the analytically derived\ncriterion~(\\ref{eqn:criterionStartup}) was confirmed numerically for\nfast shear startup in the nRP model, where it should indeed apply\nexactly due to the presence of just $d=2$ relevant dynamical variables\n$W_{xy}$ and $W_{yy}$ in that model. It was also shown to apply to\ngood approximation in the sRP model, for which $d=3$, for strain rates\nlower than the inverse stretch relaxation time (where the dynamics of\nthe sRP model indeed well approximate those of the nRP model).\n\nBanding associated with startup stress overshoot has also been\ndemonstrated in several numerical studies of soft glassy materials\n(SGMs)~\\cite{Moorcroft2011,Fielding2014,Manningetal2009a}.\n(The term SGM is used to describe a broad class of materials including\nfoams, emulsions, colloids, surfactant onion phases and microgels, all\nof which show structural disorder, metastability, a yield stress, and\noften also rheological ageing below the yield stress.)  In these soft\nglasses, however, it should be noted that the decrease in stress\nfollowing the startup overshoot arises from increasing plasticity\nrather than falling elasticity. This makes it more difficult to derive\nan analytical criterion analogous to~(\\ref{eqn:criterionStartup}).\nAccordingly the theoretical evidence for shear banding following\nstartup overshoot in these soft glasses, while very convincing,\nremains primarily numerical to date.\n\nConsistent with these theoretical predictions, experimental\nobservations of banding associated with startup stress overshoot are\nwidespread: in wormlike micellar surfactants~\\cite{Hu2008,\n  Wangetal2008c}, polymers~\\cite{Wangetal2008a, Wangetal2008b,\n  Wangetal2009a, Wangetal2009b, Boukany2009a, Wangetal2003a,\n  Huetal2007a, Wangetal2009d}, carbopol\ngels~\\cite{Divoux2011a,Divoux2010} and Laponite clay\nsuspensions~\\cite{Martin2012}. Nonetheless, we also note other\n  studies of polymer solutions~\\cite{Li2015} where stress overshoot is\n  seen without observable banding. It would be particularly\n  interesting to see further experimental work on polymeric fluids to\n  delineate more fully the regimes, for example of entanglement number\n  and degree of polydispersity, in which banding arises with\n  sufficient amplitude to be observed experimentally.\n\n\n\n\\subsection{Step stress}\n\\label{sec:recapCreep}\n\nBesides the strain-controlled protocols just described, a fluid's\nrheological behaviour can also be probed under conditions of imposed\nstress.  In a step stress experiment, an initially well relaxed fluid\nis suddenly subject to the switch-on of a shear stress $\\Sigma_0$ that\nis held constant thereafter, such that $\\Sigma(t)=\\Theta(t)\\Sigma_0$.\nCommonly measured in response to this applied stress is the material's\ncreep curve, $\\gamma(t)$, or the temporal derivative of this,\n$\\dot{\\gamma}(t)$. In Ref.~\\cite{Moorcroft2013} the criterion for\nlinear instability to the formation of shear bands, starting from a\nstate of initially homogeneous creep shear response, was shown to be\nthat\n\\begin{equation}\n\\frac{\\partial^2\\dot{\\gamma}}{\\partial t^2}\/\\frac{\\partial\\dot{\\gamma}}{\\partial t}>0.\n\\end{equation}\nThis tells us that shear banding should be expected in any step stress\nexperiment in which the differentiated creep response curve\nsimultaneously curves upwards and slopes upwards. (Indeed it should\nalso be expected in any experiment where that response function\nsimultaneously curves downwards and slopes downwards, though we do not\nknow of any instances of such behaviour.) This prediction has been\nconfirmed numerically in the Rolie-poly model of polymers and wormlike\nmicelles~\\cite{Moorcroft2014}, as well as in the soft glassy\nrheology model of foams, dense emulsions, microgels, {\\it\n  etc}~\\cite{Fielding2014}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure1.eps}\n\\caption{LAOStrain: sketch of regions of shear rate amplitude and\n  frequency space in which we expect limiting low frequency `viscous'\n  and high frequency `elastic' behaviours, and regimes of linear and\n  nonlinear response. LAOStrain runs at the locations marked $X_L$ and\n  $X_H$ are explored in Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} for\n  the nRP model with non-monotonic and monotonic underlying\n  constitutive curve respectively.}\n\\label{fig:sketch}\n\\end{figure}\n\nExperimentally, shear banding associated with a simultaneously\nupwardly curving and upwardly sloping differentiated creep response\ncurve has indeed been seen in in entangled\npolymers~\\cite{Wangetal2009a,Huetal2007a,Wangetal2003a, Hu2008,\n  Wangetal2008c, Hu2005, Hu2010, Wangetal2009d}, carbopol\nmicrogels~\\cite{Divoux2011} and carbon black\nsuspensions~\\cite{Gibaud2010}.\n\n\\section{Large amplitude oscillatory strain}\n\\label{sec:LAOStrain}\n\n\nWe now consider shear banding in the time-dependent strain-imposed\noscillatory protocol of LAOStrain. Here a sample of fluid, initially\nwell relaxed at time $t=0$, is subject for times $t>0$ to a strain of\nthe form\n\\begin{equation}\n\\gamma(t)=\\gamma_0\\sin(\\omega t),\n\\end{equation}\nto which corresponds the strain rate\n\\begin{equation}\n\\dot{\\gamma}(t)=\\gamma_0\\omega\\cos(\\omega t)=\\dot{\\gamma}_0\\cos(\\omega t).\n\\end{equation}\nAfter an initial transient, once many cycles have been executed, the\nresponse of the system is expected to attain a state that is\ntime-translationally invariant from cycle to cycle, $t\\to\nt+2\\pi\/\\omega$. All the results presented below are in this long-time\nregime, usually for the $N=20$th cycle after the flow\ncommenced. The dependence of the stress on the cycle number was carefully studied in wormlike micelles in Ref.~\\cite{Fujii2015}.\n\nTo characterise any given applied LAOStrain we must clearly specify\ntwo quantities: the strain amplitude and the frequency\n$(\\gamma_0,\\omega)$, or alternatively the strain rate amplitude and\nthe frequency $(\\dot{\\gamma}_0,\\omega)$, where $\\dot{\\gamma}_0=\\gamma_0\\omega$. In\nwhat follows we usually choose the latter pairing $(\\dot{\\gamma}_0,\\omega)$.\nAny given LAOStrain experiment is then represented by its location in\nthat plane of $\\dot{\\gamma}_0$ and $\\omega$. See Fig.~\\ref{fig:sketch}.\n\nIn any experiment where the applied strain rate remains small,\n$\\dot{\\gamma}_0 \\ll 1$, a regime of linear response is expected.  (Recall\nthat in dimensional form this condition corresponds to $\\dot{\\gamma}_0\\tau_d\n\\ll 1$.)  But even in an experiment where the strain rate does not\nremain small, linear response can nonetheless still be expected if the\noverall applied strain remains small, $\\gamma_0 \\ll 1$. Accordingly,\nlinear response should obtain in the region below the long-dashed line\nmarked in Fig.~\\ref{fig:sketch}. Because shear banding is an\ninherently nonlinear phenomenon, we expect the interesting region of\nthis $(\\dot{\\gamma}_0,\\omega)$ plane from our viewpoint to be in the\nnonlinear regime, above the long-dashed line, and we focus our\nattention mostly on this in what follows.\n\n\n\nBesides considering whether any given applied LAOStrain will result in\nlinear or nonlinear response, also relevant is the characteristic\ntimescale $1\/\\omega$ of the oscillation compared to the fluid's\nintrinsic terminal relaxation timescale $\\tau_d=1$. For low\nfrequencies $\\omega\\ll 1$, to the left of the leftmost dotted line in\nFig.~\\ref{fig:sketch}, we expect the material's reconfiguration\ndynamics to keep pace with the applied deformation. This will lead to\nquasi steady state response in which the stress slowly sweeps up\nand down the steady state flow curve as the shear rate varies through\na cycle. In contrast for high frequencies $\\omega\\gg 1$, to the right\nof the rightmost dotted line in Fig.~\\ref{fig:sketch}, the material's\nrelaxation dynamics cannot keep pace with the applied deformation and\nwe expect elastic-like response.\n\nWe illustrate these two limiting regimes by studying the response of\nthe nRP model to an imposed LAOStrain at each of the two locations\nmarked $X_L$ and $X_H$ in Fig.~\\ref{fig:sketch}.  For simplicity, for\nthe moment, we artificially constrain the flow to remain homogeneous\nand confine ourselves to calculating the uniform `base state' as\noutlined in Sec.~\\ref{sec:base}. The results are shown in\nFig.~\\ref{fig:nonmon} for the nRP model with parameters for which the\nunderlying constitutive curve is non-monotonic, such that (in any\nheterogeneous calculation) the fluid would show shear banding under\nconditions of steady applied shear.  Fig.~\\ref{fig:mon} shows results\nwith model parameters for which the constitutive curve is monotonic,\nsuch that no banding would be expected in steady shear flow.\n\n\nThe left panels of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} contain\nresults for the low frequency oscillation marked $X_L$ in\nFig.~\\ref{fig:sketch}. Here we choose to plot the stress response\n$\\Sigma(t)$ in a Lissajous-Bowditch figure as a parametric function of\nthe time-varying imposed strain rate $\\dot{\\gamma}(t)$, consistent with the\nexpectation of fluid-like response in this low-frequency regime.\n(Throughout the paper we shall describe such a plot of stress versus strain rate as being in the\n`viscous' representation.) As can be seen, in each case the fluid\nindeed tracks up and down its (quasi) steady state homogeneous\nconstitutive curve $\\Sigma(\\dot{\\gamma})$ in the range $-\\dot{\\gamma}_0 < \\dot{\\gamma} <\n\\dot{\\gamma}_0$.  For any set of model parameters, several of these LAOStrain\nresponse curves $\\Sigma(\\dot{\\gamma})$ collected together for different\n$\\dot{\\gamma}_0$ and low frequency $\\omega$ would all collapse onto this\nmaster constitutive curve.\n\nAlso shown by the colour scale in the left panels of\nFigs.~\\ref{fig:nonmon} and~\\ref{fig:mon} is the eigenvalue as\nintroduced in Sec.~\\ref{sec:lsa}. Recall that a positive eigenvalue at\nany point in the cycle strongly suggests that the homogeneous base\nstate is linearly unstable to the development of shear banding at that\npoint in the cycle.  (In any region where this scale shows black the\neigenvalue is either negative, or so weakly positive as to cause only\nnegligible banding growth.) As expected, a regime of instability is\nindeed seen in Fig.~\\ref{fig:nonmon}, in the region\nwhere the constitutive curve has negative slope,\n\\begin{equation}\n\\frac{\\partial\\Sigma}{\\partial\\dot{\\gamma}} <0.\n\\end{equation}\nFor a fluid with a monotonic constitutive curve, no instability is\nobserved at this low frequency (Fig.~\\ref{fig:mon}, left).\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure2.eps}\n\\caption{LAOStrain in the nRP model with a non-monotonic underlying\n  constitutive curve. Model parameters $\\beta=0.4$, $\\eta=10^{-5}$.\n  {\\bf Left:} Viscous Lissajous-Bowditch\n  figure shows stress $\\Sigma$ versus strain rate $\\dot{\\gamma}$ for an\n  imposed frequency and strain rate $(\\omega,\\dot{\\gamma}_0)=(0.001,50.0)$\n  marked as $X_L$ in the low frequency regime of\n  Fig.~\\ref{fig:sketch}.  {\\bf Right:} Elastic Lissajous-Bowditch figure shows\n  stress $\\Sigma$ versus strain $\\gamma$ for an imposed frequency and\n  strain rate $(\\omega,\\dot{\\gamma}_0)=(31.6,200.0)$ marked as $X_H$ in the\n  high frequency regime of Fig.~\\ref{fig:sketch}.  Colourscale shows\n  eigenvalue.}\n\\label{fig:nonmon}\n\\end{figure}\n\n\n\n\nThe corresponding results for the high frequency run marked $X_H$ in\nFig.~\\ref{fig:sketch} are shown in the right panels of\nFigs.~\\ref{fig:nonmon} and~\\ref{fig:mon}. Here we choose to plot the\nstress response $\\Sigma(t)$ in a Lissajous-Bowditch figure as a\nparametric function of the time-varying strain $\\gamma(t)$, in the\nso-called `elastic' representation. Indeed, just as in the low\nfrequency regime the material behaved as a viscous fluid with the\nstress response falling onto the steady state master constitutive\ncurve in the viscous representation $\\Sigma(\\dot{\\gamma})$, for a high\nfrequency cycle we might instead expect a regime of elastic response\nin which only the accumulated strain is important, and not\n(separately) the strain rate, giving a master response curve of stress\nversus strain, $\\Sigma(\\gamma)$.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure3.eps}\n\\caption{As in Fig.~\\ref{fig:nonmon}, but for a value of the CCR\n  parameter $\\beta=1.0$, for which the underlying homogeneous\n  constitutive curve is monotonic.}\n\\label{fig:mon}\n\\end{figure}\n\nWe might further have intuitively expected this curve to be the same\nas that obtained in a fast shear startup from rest, with (in the\npositive strain part of the cycle) elastic response $\\Sigma\\approx G\\gamma$\nat low strain $\\gamma\\ll 1$, followed by stress overshoot at a typical\nstrain $\\gamma=+O(1)$, then decline towards a constant stress at\nlarger strains (with the symmetric curve in the negative-strain part\nof the cycle, such that $\\Sigma\\to-\\Sigma$ for $\\gamma\\to-\\gamma$). In\nother words, in LAOStrain at high frequency we might have expected the\nsystem to continuously explore its elastic shear startup curve\n$\\Sigma(\\gamma)$ between $\\gamma=-\\gamma_0$ and $\\gamma=+\\gamma_0$.\n\nHowever, this intuition is not met in a straightforward way. In the\nright panels of Fig.~\\ref{fig:nonmon} and~\\ref{fig:mon} we observe\ninstead an open cycle that is explored in a clockwise sense as time\nproceeds through an oscillation: the stress transits the upper part of\nthe loop (from bottom left to top right) in the forward part of the\ncycle as the strain increases from left to right, and the\nsymmetry-related lower part of the loop in the backward part, where\nthe strain decreases from right to left.\n\nThis can be understood as follows. For any LAOStrain run at high\nfrequency $\\omega \\gg 1$ but in the linear regime with strain\namplitude $\\gamma_0\\ll 1$, we do indeed find the stress response to\nfall onto a closed master curve $\\Sigma(\\gamma)$, which also\ncorresponds to that obtained in a fast stress startup from rest, with\nlinear elastic response $\\Sigma\\approx G\\gamma$. (Data not shown.) In\ncontrast, for amplitudes $\\gamma_0 > 1$ the system only explores this\nstartup-from-rest curve in the first half of the {\\em first} cycle\nafter the inception of flow. (This has the usual form, with elastic\nresponse for small strains, stress overshoot at a strain\n$\\gamma=O(1)$, then decline to a constant stress.)  In the second half\nof the cycle, when the strain rate reverses and the strain decreases,\nthe stress response departs from the startup-from-rest curve. With\nhindsight this is in fact obvious: as this backward shear part of the\ncycle commences the initial condition is not that of a well-relaxed\nfluid, but one that has just suffered a large forward strain.\n\n\n\n\nThe same is true for the next forward half cycle: its initial\ncondition is that of a fluid that has just suffered a large negative\nstrain, corresponding to the lower left point in the right panel of\nFigs.~\\ref{fig:nonmon} or~\\ref{fig:mon}. Starting from that initial\ncondition the stress evolution nonetheless thereafter resembles that\nof a fast startup, with an initial fast rise followed by an overshoot\nthen decline to constant stress, before doing the same in reverse\n(with a symmetry-related `negative overshoot') during the next half\ncycle, giving the open curves as described. Associated with this\novershoot in each half cycle is a positive eigenvalue indicating\ninstability to the onset of shear banding. Importantly, we note that\nthis arises even in the case of a monotonic underlying constitutive\ncurve (Fig.~\\ref{fig:mon}, right), and therefore even in a fluid that would\nnot display steady state banding under a steadily applied shear flow.\nIt is the counterpart for LAOStrain of the `elastic' banding triggered\nby stress overshoot in a fast shear startup from rest, as explored\npreviously in\nRef.~\\cite{Moorcroft2013,Moorcroft2014,Adamsetal2011}.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure4.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  nRP model with a non-monotonic constitutive curve. Each point in this\n  $\\dot{\\gamma}_0,\\omega$ plane corresponds to a particular LAOStrain run\n  with strain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding that occurs at any point in the\n  cycle, after many cycles.  Model parameters: $\\beta=0.4$,\n  $\\eta=10^{-5}$.  Cell curvature $q=10^{-4}$. Crosses indicate the\n  grid of values of $\\dot{\\gamma}_{0}$ and $\\omega$ in Pipkin diagram of\n  Fig.~\\ref{fig:PipkinNonMon}.}\n\\label{fig:pinPointNonMon}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.0cm]{figure5.eps}\n\\caption{As in Fig.~\\ref{fig:pinPointNonMon}, but with a CCR parameter\n  $\\beta=1.0$, for which the fluid has a monotonic underlying\n  constitutive curve.  Crosses indicate the grid of values of\n  $\\dot{\\gamma}_{0}$ and $\\omega$ used in the Pipkin diagram of\n  Fig.~\\ref{fig:PipkinMon}.}\n\\label{fig:pinPointMon}\n\\end{figure}\n\n\\begin{figure}[tp!]\n\\includegraphics[width=9cm]{figure6.eps}\n\\caption{Lissajous-Bowditch curves in LAOStrain for the nRP model with\n  a non-monotonic constitutive curve. Results are shown in the elastic\n  representation in (a), and the viscous representation in\n  (b). Columns of fixed frequency $\\omega$ and rows of fixed\n  strain-rate amplitude $\\dot{\\gamma}_0$ are labeled at the top and\n  right-hand side. Colourscale shows the time-dependent degree of\n  shear banding.  Model parameters: $\\beta=0.4, \\eta=10^{-5}$. Cell\n  curvature: $q=10^{-4}$. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by\n  the thicker box is shown in Fig.~\\ref{fig:portraitNonMon}.}\n\\label{fig:PipkinNonMon}\n\\end{figure}\n\n\\begin{figure}[tp!]\n\\includegraphics[width=9cm]{figure7.eps}\n\\caption{As in Fig.~\\ref{fig:PipkinNonMon} but for a value of the CCR\n  parameter $\\beta=1.0$, for which the fluid's underlying constitutive\n  curve is monotonic. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by the\n  thicker box is shown in Fig.~\\ref{fig:portraitMon}.}\n\\label{fig:PipkinMon}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8cm]{figure8.eps}\n\\caption{LAOStrain in the nRP model with a non-monotonic constitutive\n  curve. Strain rate amplitude $\\dot{\\gamma}_{0}=10.0$ and frequency\n  $\\omega=3.16$. Model parameters $\\beta=0.4, \\eta=10^{-5}$.  Cell\n  curvature $q=10^{-4}$. Number of numerical grid points $J=1024$.\n  {\\bf Left:} stress response in the elastic representation. Solid\n  black and red-dashed line: calculation in which the flow is\n  constrained to be homogeneous.  Red-dashed region indicates a\n  positive eigenvalue showing instability to the onset of shear\n  banding. Green dot-dashed line: stress response in a full nonlinear\n  simulation that allows banding. {\\bf Right:} Velocity profiles\n  corresponding to stages in the cycle indicated by matching symbols\n  in the left panel. Each profile is normalised by the speed of the\n  moving plate.}\n\\label{fig:portraitNonMon}\n\\end{figure}\n\nIndeed, following the calculation first set out in\nRef.~\\cite{Moorcroft2013}, it is straightforward to show that\nthe condition for a linear instability to banding in this `elastic'\nhigh frequency regime $\\omega\\gg 1$ is the same as in fast shear\nstartup:\n\\begin{equation} \n\\label{eqn:criterionElastic}\n-\\textrm{tr}\\tens{M} \\frac{\\partial \\Sigma}{\\partial\\gamma} +\n\\dot{\\gamma}\\frac{\\partial^2\\Sigma}{\\partial\\gamma^2} < 0.\n\\end{equation}\nAs already discussed, this gives a window of instability setting in\njust before the stress overshoot (or negative overshoot) in each half\ncycle in the right panels of Fig.~\\ref{fig:nonmon} and~\\ref{fig:mon}\ndue to (in the positive $\\dot{\\gamma}$ part of the cycle in which the stress\ntransits from bottom left to top right) the negatively sloping and\ncurving $\\Sigma(\\gamma)$. An analogous statement applies in the other\npart of the cycle, with the appropriate sign reversals. Note that\nthese overshoots are sufficiently weak as to be difficult to resolve\nby eye on the scale of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon}.\n\nInterestingly,~(\\ref{eqn:criterionElastic}) also predicts a region of\n(weaker) instability immediately after the reversal of strain in each\nhalf cycle, as also seen in the right panels of Figs.~\\ref{fig:nonmon}\nand~\\ref{fig:mon}.  Analytical considerations show that this\nadditional regime of instability is not driven by any negative slope or\ncurvature in $\\Sigma(\\gamma)$, but instead arises from a change in\nsign of $\\textrm{\\tens{M}}$.  This instability has no counterpart that\nwe know of in shear startup. Its associated eigenvector is dominated\nby the normal stress component $W_{yy}$ rather than the strain rate or\nshear stress. Heterogeneity in this quantity could feasibly be\naccessed in birefringence experiments.\n\nThe results of Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} can be\nsummarised as follows. At low frequencies the system sweeps slowly up\nand down the underlying constitutive curve $\\Sigma(\\dot{\\gamma})$ as the\nshear rate varies through the cycle. If that curve is non-monotonic,\nthis homogeneous base state is unstable to shear banding in the region\nof negative constitutive slope, $d\\Sigma\/d\\dot{\\gamma}<0$. At high\nfrequencies the system instead executes a process reminiscent of\nelastic shear startup in each half cycle, but with an initial\ncondition corresponding to the state left by the shear of opposite\nsense in the previous half cycle. Associated with this is a stress\novershoot in each half cycle, giving instability to elastic shear\nbanding. Crucially, this elastic instability occurs whether or not the\nunderlying constitutive curve is non-monotonic or monotonic, and\ntherefore whether or not the fluid would shear band in steady shear.\n\nFrom a practical experimental viewpoint it is important to note that,\nwhereas in a single shear startup run these `elastic' strain bands\nwould form transiently then heal back to homogeneous flow (unless the\nsample has a non-monotonic underlying constitutive curve and so also\nbands in steady state), in LAOStrain the banding will recur in each\nhalf cycle and so be potentially easier to access experimentally.\nAny time-averaging measurement should of course only take data in the\nforward part of each cycle, to avoid averaging to zero over the\ncycle.\n\nHaving explored in Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon} the\ntendency to form shear bands in two particular LAOStrain runs (one in\nthe limit of low frequency, $X_L$ in Fig.~\\ref{fig:sketch}, and one in\nthe limit of high frequency, $X_H$), we now explore the full\n$(\\dot{\\gamma}_0,\\omega)$ plane of Fig.~\\ref{fig:sketch} by showing in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon} colour maps of\nthe extent of shear banding across this plane.  Recall that each\npinpoint in this plane corresponds to a single LAOS experiment with\nstrain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  To build up\nthese colour maps, we sweep over a grid of 100x100\nvalues of $\\dot{\\gamma}_0,\\omega$ and execute a LAOStrain run at each point.\nSolving the model's full nonlinear dynamics on such a dense grid would\nbe unfeasibly time-consuming computationally. Therefore at each\n$\\dot{\\gamma}_0,\\omega$ we instead integrate the linearised equations set out\nin Sec.~\\ref{sec:lsa}. In each such run we record the degree of\nbanding $\\delta\\dot{\\gamma}$, maximised over the cycle after many cycles.  It\nis this quantity, normalised by the maximum strain rate amplitude $\\dot{\\gamma}_{0}$, that is represented by the colourscale in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}.\n\nFig.~\\ref{fig:pinPointNonMon} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is\nnon-monotonic. As expected, significant banding (bright\/yellow region)\nis observed even in the limit of low frequency $\\omega\\to 0$ for\nstrain rate amplitudes $\\dot{\\gamma}_0$ exceeding the onset of negative slope\nin the underlying constitutive curve. This region of banding is the\ndirect (and relatively trivial) analogue of banding in a slow strain\nrate sweep along the steady state flow curve.\nFig.~\\ref{fig:pinPointMon} shows results for the nRP model with\nparameters such that the underlying constitutive curve is monotonic.\nHere steady state banding is absent in the limit $\\omega\\to 0$. In\nboth Fig.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}, however,\nsignificant banding is observed at high frequencies for a strain\namplitude $\\gamma_0>1 $: this is the elastic banding associated with\nthe stress overshoot in each half cycle, described in detail above for\nthe $(\\dot{\\gamma}_0,\\omega)$ values denoted by $X_H$ in\nFig.~\\ref{fig:sketch}.\n\nIt is important to emphasise, therefore, that even a fluid with a\npurely monotonic constitutive curve, which does not shear band in\nsteady flow, is still nonetheless capable of showing strong shear\nbanding in a time-dependent protocol of high enough frequency\n(Fig.~\\ref{fig:pinPointMon}). Also important to note is that for a\nfluid with a non-monotonic constitutive curve the region of steady\nstate `viscous' banding at low frequencies crosses over smoothly to\nthat of `elastic' banding as the frequency increases\n(Fig.~\\ref{fig:pinPointNonMon}). \n\nCorresponding to the degree of banding in the shear rate\n$\\delta\\dot{\\gamma}$, as plotted in Figs.~\\ref{fig:pinPointNonMon}\nand~\\ref{fig:pinPointMon}, is an equivalent degree of banding $G\\delta\nW_{xy}=-\\eta\\delta\\dot{\\gamma}$ (to within small corrections of order the\ncell curvature, $q$) in the shear component of the polymeric\nconformation tensor. This follows trivially by imposing force balance\nat zero Reynolds number. Counterpart maps for the degree of banding in\nthe component $\\delta W_{yy}$ of the polymeric conformation tensor can\nalso be built up. These reveal closely similar regions of banding to\nthose shown in Figs.~\\ref{fig:pinPointNonMon}\nand~\\ref{fig:pinPointMon}. (Data not shown.) Experimentally,\nheterogeneities in the polymeric conformation tensor can be accessed\nby flow birefringence.\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8cm]{figure9.eps}\n\\caption{As in Fig.~\\ref{fig:portraitNonMon} but for the nRP model with\n  a CCR parameter $\\beta=1.0$ for which the underlying homogeneous\n  constitutive curve is monotonic, and for a LAOStrain with strain\n  rate amplitude $\\gamma_{0}=56.2$ and frequency\n  $\\omega=10.0$. Number of numerical grid points $J=512$.}\n\\label{fig:portraitMon}\n\\end{figure}\n\n\nAs noted above, to build up such comprehensive roadmaps as in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon} in a\ncomputationally efficient way, we omitted all nonlinear effects and\nintegrated instead the linearised equations of Sec.~\\ref{sec:lsa}.\nThese are only strictly valid in any regime where the amplitude of the\nheterogeneity remains small. In omitting nonlinear effects, they tend\nto overestimate the degree of banding in any regime of sustained\npositive eigenvalue, in predicting the heterogeneity to grow\nexponentially without bound, whereas in practice it would be cutoff by\nnonlinear effects. We now remedy this shortcoming by exploring the\nmodel's full nonlinear spatiotemporal dynamics. To do so within\nfeasible computational run times, we focus on a restricted grid of\nvalues in the $\\dot{\\gamma}_0,\\omega$ plane, marked by crosses in\nFigs.~\\ref{fig:pinPointNonMon} and~\\ref{fig:pinPointMon}.\n\nThe results are shown in Fig.~\\ref{fig:PipkinNonMon} for the nRP model\nwith model parameters for which the underlying constitutive curve is\nnon-monotonic.  At low frequencies the results tend towards the\nlimiting fluid-like behaviour discussed above, in which the the stress\nslowly tracks up and down the steady state flow curve $\\Sigma(\\dot{\\gamma})$.\n(Progression towards this limit can be seen by following the top row\nof panels in Fig.~\\ref{fig:PipkinNonMon}b) to the left.)  Viscous\nbanding is seen for sufficiently high strain rate amplitudes $\\dot{\\gamma}_0$\ndue to the negatively sloping underlying homogeneous constitutive\ncurve.  At high frequencies the response tends instead towards the\nlimiting elastic-like behaviour discussed above. For large enough\nstrain amplitudes the stress then shows an open cycle $\\Sigma(\\gamma)$\nas a function of strain, with an overshoot in each half cycle that\ntriggers the formation of `elastic' banding.  (Progression towards\nthis limit is seen by following the top row of panels in\nFig.~\\ref{fig:PipkinNonMon}a to the right.)\n\nOvershoots in the elastic Lissajous-Bowditch curve of stress as a\nfunction of strain have been identified in earlier\nwork~\\cite{Ewoldt2009loops} as leading to self-intersection of the\ncorresponding viscous Lissajous-Bowditch curve of stress as a function\nof strain-rate. Such an effect is clearly seen here in the Rolie-poly\nmodel: see for example the runs highlighted by the thicker boxes in\nFig.~\\ref{fig:PipkinNonMon} and in Fig.~\\ref{fig:PipkinMon}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure10.eps}\n\\caption{Effect of CCR parameter $\\beta$ and entanglement number $Z$\n  (and so of chain stretch relaxation time $\\tau_R=\\tau_d\/3Z$) on shear\n  banding in LAOStrain.  (Recall that the non-stretching version of\n  the model has $\\tau_R\\to 0$ and so $Z\\to\\infty$.)  Empty circles: no\n  observable banding.  Hatched circles: observable banding, typically\n  $\\Delta_{\\dot{\\gamma}}\/\\dot{\\gamma}_0 \\approx 10\\%-100\\%$. Filled circles: significant\n  banding $\\Delta_{\\dot{\\gamma}}\/\\dot{\\gamma}_0 \\ge 100\\%$.  For hatched and filled\n  symbols we used the criterion that banding of the typical magnitude\n  stated is apparent in a region spanning at least half a decade by\n  half a decade in the plane of $\\dot{\\gamma}_0,\\omega$, by examining maps as\n  in Fig.~\\ref{fig:stretchPinPoint} in by eye. The square shows the\n  parameter values explored in detail in\n  Fig.~\\ref{fig:stretchPinPoint}.}\n\\label{fig:stretchMaster}\n\\end{figure}\n\nFor intermediate frequencies the stress is a complicated function of\nboth strain rate and also, separately, the strain. The three\ndimensional curve $(\\Sigma,\\dot{\\gamma},\\gamma)$ is then best shown as two\nseparate projections: first in the elastic representation of the\n$\\Sigma,\\gamma$ plane (Fig.~\\ref{fig:PipkinNonMon}a), and second, in\nthe viscous representation of the $(\\Sigma,\\dot{\\gamma})$ plane\n(Fig.~\\ref{fig:PipkinNonMon}b).  Collections of these\nLissajous-Bowditch curves on a grid of $(\\dot{\\gamma}_0,\\omega)$ values as in\nFig.~\\ref{fig:PipkinNonMon} are called Pipkin diagrams.\n\n\\begin{figure}[tbp]\n   \\includegraphics[width=10.0cm]{figure11.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  sRP model with a monotonic constitutive curve. Each point in this\n  $\\dot{\\gamma}_0,\\omega$ plane corresponds to a particular LAOStrain run\n  with strain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding at any point in the cycle, after\n  many cycles.  Model parameters: $\\beta=0.7$, $Z=75$ (and so\n  $\\tau_R=0.0044$), $\\eta=10^{-5}$.  Cell curvature $q=2\\times\n  10^{-3}$. Note the different colour scale from Figs.~\\ref{fig:nonmon} and~\\ref{fig:mon}. The model's full nonlinear dynamics for the ($\\dot{\\gamma}_{0}, \\omega$) value marked by the cross are explored in Fig.~\\ref{fig:stretchPortrait}.}\n\\label{fig:stretchPinPoint}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure12.eps}\n\\caption{sRP model with a monotonic constitutive curve in LAOStrain of\n  strain rate amplitude $\\dot{\\gamma}_{0}=20.0$ and frequency $\\omega=8.0$.\n  Model parameters $\\beta=0.7, Z=75, \\eta=10^{-5}$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} stress response in the elastic representation. Solid black\n  and red-dashed line: calculation in which the flow is constrained to\n  be homogeneous.  Red-dashed region indicates a positive eigenvalue\n  showing instability to the onset of shear banding. Green dot-dashed\n  line: stress response in a full nonlinear simulation that allows\n  banding (almost indistinguishable from homogeneous signal in this\n  case.) {\\bf Right:} Velocity profiles corresponding to stages in the\n  cycle indicated by matching symbols in left panel.}\n\\label{fig:stretchPortrait}\n\\end{figure}\n\nFor the particular LAOStrain run highlighted by a thicker box in\nFig.~\\ref{fig:PipkinNonMon}, a detailed portrait of the system's\ndynamics is shown in Fig.~\\ref{fig:portraitNonMon}. Here the stress is\nshown in the elastic representation, as a function of strain $\\gamma$\n(left hand panel). Two curves are shown here. The first shows the\nstress signal in a calculation in which the flow is artificially\nconstrained to remain homogeneous. A linear instability analysis for\nthe dynamics of small heterogeneous perturbations about this\ntime-evolving homogeneous base state then reveals instability towards\nthe formation of shear bands (a positive eigenvalue) in the portion of\nthat curve shown as a red dashed line. A full nonlinear simulation\nthen reveals the formation of shear bands, and leads to a stress\nsignal (green dot-dashed line) that deviates from that of the\nhomogeneously constrained calculation, in particular in having a much\nmore precipitous stress drop due to the formation of bands.\n\nThe associated velocity profiles at four points round the part of the\ncycle with increasing strain are shown in the right hand panel. Before\nthe stress overshoot, no banding is apparent (black circles). The\novershoot then triggers strong shear banding (red squares), with most\nof the shear concentrated in a thin band at the left hand edge of the\ncell. Interestingly, the shear in the right hand part of the cell is\nin the opposite sense to the overall applied shear. This is consistent\nwith the fact that the stress is a decreasing function of strain in\nthis regime: the material is being unloaded, and an elastic-like\nmaterial being unloaded will shear backwards. As the overall applied\nstrain increases towards the end of the window of instability, the\nflow heterogeneity gradually decays away. This process repeats in each\nhalf cycle (with the obvious sign reversals in the part of the cycle\nin which the strain is decreasing).\n\nThe corresponding Pipkin diagram for a fluid with a monotonic\nconstitutive curve (Fig.~\\ref{fig:PipkinMon}) likewise confirms its\ncounterpart linear diagram in Fig.~\\ref{fig:pinPointMon}. Here\n`viscous' banding is absent at low frequencies, because the fluid is\nnot capable of steady state banding. Crucially, however, a strong\neffect of elastic banding is still seen at high frequencies.  A\ndetailed portrait of the system's dynamics in this elastic regime, for\nthe strain rate amplitude and frequency marked by the thicker box in\nFig.~\\ref{fig:PipkinMon}, is shown in Fig.~\\ref{fig:portraitMon}. As\ncan be seen, it shows similar features to those just described in\nFig.~\\ref{fig:portraitNonMon}. We emphasise, then, that even polymeric\nfluids that do not band under conditions of steady shear are still\ncapable of showing strong banding in a time-dependent protocol at high\nenough frequency. This important prediction is consistent with the\nearly insight of Adams and Olmsted in Ref.~\\cite{Adams2009}.\n\nSo far we have presented results for the nRP model, which assumes an\ninfinitely fast rate of chain stretch relaxation compared to the rate\nof reptation, such that the ratio $\\tau_R\/\\tau_d\\to 0$. This corresponds\nto assuming that the polymer chains are very highly entangled, with a\nnumber of entanglements per chain $Z=\\tau_d\/3\\tau_R\\to\\infty$. We now\nconsider the robustness of these results to reduced entanglement\nnumbers, and accordingly increased chain stretch relaxation time\n$\\tau_R$ (in units of $\\tau_d$). \n\nThe results are summarised in Fig.~\\ref{fig:stretchMaster}, which\nshows the regions of the plane of the CCR parameter $\\beta$ and\nentanglement number $Z$ in which significant banding (filled circles),\nobservable banding (hatched circles), and no banding (open circles)\noccur.  (Recall that results presented for the nRP model above pertain\nto the values $\\beta=0.4,1.0$ in the limit $Z\\to\\infty$.) As can be\nseen, by reducing the number of entanglements per chain the effect of\nshear banding is reduced and eventually eliminated.  However it is\nimportant to note that, depending on the value of $\\beta$, significant\nbanding is still observed for experimentally commonly used\nentanglement numbers, typically in the range $20-100$. Furthermore, significant banding is seen in a large region of the ($\\beta, Z$) plane for which the material's underlying constitutive curve is monotonic. As discussed\nabove, there is no current consensus as to the value of the CCR\nparameter $\\beta$ in the range $0<\\beta<1$. Using the routemap\nprovided in Fig.~\\ref{fig:stretchMaster}, a study of shear banding in\nLAOStrain experiments could provide one way to obtain a more accurate\nestimate of the value of this parameter.\n\nFor the pairing of $\\beta$ and $Z$ values marked by the square in\nFig.~\\ref{fig:stretchMaster}, we show in\nFig.~\\ref{fig:stretchPinPoint} a colour map of the degree of banding\nexpected in LAOStrain in the space of strain rate amplitude $\\dot{\\gamma}_0$\nand frequency $\\omega$. This figure, which is for the sRP model,\n  is the counterpart of the earlier Figs.~\\ref{fig:pinPointNonMon}\n  and~\\ref{fig:pinPointMon} discussed above for the nRP model, with\nthe degree of banding calculated for computational\nefficiency within the assumption of linearised dynamics. Recall that\neach pinpoint in this plane corresponds to a single LAOStrain run with\napplied strain rate $\\dot{\\gamma}(t)=\\dot{\\gamma}_0\\cos(\\omega t)$.\n\nConsistent with the underlying constitutive curve being monotonic for\nthese parameters, `viscous' banding is absence in the limit of low\nfrequencies $\\omega\\to 0$.  However significant banding is still\nobserved for runs with strain rate amplitudes $O(10-100)$ and\nfrequencies $O(1-10)$. This is the counterpart of the `elastic'\nbanding reported above in the nRP model, though the effect of finite\nchain stretch in the sRP model is to moderate the degree of banding. A\ndetailed portrait of the model's nonlinear banding dynamics at a\nstrain rate amplitude $\\dot{\\gamma}_0$ and frequency $\\omega$ marked by the\ncross in Fig.~\\ref{fig:stretchPinPoint} is shown in\nFig.~\\ref{fig:stretchPortrait}. Significant shear banding associated with the stress overshoot is apparent in each half cycle.\n\n\\section{Large amplitude oscillatory stress}\n\\label{sec:LAOStress}\n\nWe now consider the time-dependent stress-imposed oscillatory protocol\nof LAOStress. Here the sample is subject for times $t>0$ to a stress\nof the form\n\\begin{equation}\n\\Sigma(t)=\\Sigma_0\\sin(\\omega t),\n\\end{equation}\ncharacterised by the frequency $\\omega$ and stress amplitude\n$\\Sigma_0$. As for the case of LAOStrain above, all the results\npresented below are in the long-time regime, once many ($N=20$) cycles\nhave been executed and the response of the system has settled to be\ntime-translationally invariant from cycle to cycle, $t\\to\nt+2\\pi\/\\omega$.\n\nIn Sec.~\\ref{sec:recapCreep} we reviewed existing work demonstrating\nthe tendency to form shear bands in a {\\rm step} stress experiment.\nHere an initially well relaxed sample is suddenly subject at time\n$t=0$ to the switch-on of a shear stress of amplitude $\\Sigma_0$,\nwhich is held constant for all subsequent times.  The criterion for an\nunderlying base state of initially homogeneous creep response to\nbecome linearly unstable to the formation of shear bands is then that\nthe time-differentiated creep response curve $\\dot{\\gamma}(t)$\nobeys~\\cite{Moorcroft2013}:\n\\begin{equation}\n\\label{eqn:critCreep}\n\\frac{\\partial^2\\dot{\\gamma}}{\\partial t^2}\/\\frac{\\partial\\dot{\\gamma}}{\\partial t}>0.\n\\end{equation}\nTherefore, shear banding is expected in any regime where the\ntime-differentiated creep curve simultaneously slopes up and curves\nupwards; or instead simultaneously slopes down and curves downwards.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure13.eps}\n\\caption{LAOStress in the nRP model with a non-monotonic constitutive\n  curve. Model parameters: $\\beta=0.1$, $\\eta=10^{-4}$. Frequency\n  $\\omega=0.01$ and stress amplitude $\\Sigma_0=0.7$.  {\\bf Left:}\n  stress versus strain rate (shown on a log scale) in the positive\n  stress part of the cycle. Colour scale shows eigenvalue, with\n  negative values also shown as black. Green dashed line: underlying\n  constitutive curve. {\\bf Right:} corresponding stress versus time\n  plot.}\n\\label{fig:eigenNonMon}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.0cm]{figure14.eps}\n\\caption{As in Fig.~\\ref{fig:eigenNonMon} but at a higher imposed\n  frequency $\\omega=1.0$ and for a value of the CCR $\\beta=0.9$, for\n  which the nRP model has a monotonic underlying constitutive curve.\n  {\\bf Right:} corresponding stress versus time plot.}\n\\label{fig:eigenMon}\n\\end{figure}\n\n\n\nHaving been derived within the assumption of an imposed stress that is\nconstant in time, criterion (\\ref{eqn:critCreep}) would not {\\it a\n  priori} be expected to hold for the case of LAOStress. Nonetheless\nit might reasonably be expected to apply, to good approximation, in any\nregime of a LAOStress experiment where a separation of timescales\narises such that the shear rate $\\dot{\\gamma}(t)$ evolves on a much shorter\ntimescale than the stress. In this case, from the viewpoint of the\nstrain rate signal, the stress appears constant in comparison and the\nconstant-stress criterion (\\ref{eqn:critCreep}) is expected to hold.\nIndeed, in what follows we shall show that many of our results for\nLAOStress can be understood on the basis of this simple piece of\nintuition.\n\nWe start in Fig.~\\ref{fig:eigenNonMon} by considering the nRP model in\na parameter regime for which the underlying constitutive curve is\nnon-monotonic (see the dotted line in the left panel), such that shear\nbanding would be expected under conditions of a steadily applied shear\nrate.  With the backdrop of this constitutive curve we consider a\nLAOStress run at low frequency $\\omega\\to 0$. For definiteness we will\nfocus on the part of the cycle where the stress is positive, but\nanalogous remarks will apply to the other half of the cycle, with\nappropriate changes of sign.\n\nConsider first the regime in which the stress is slowly increased from\n$0$ towards its maximum value $\\Sigma_0$. In this part of the cycle,\nat the low frequencies of interest here, we expect the system to\ninitially follow the high viscosity branch of the constitutive curve.\nIn any experiment for which the final stress $\\Sigma_0$ exceeds the\nlocal maximum in the constitutive curve, the system must at some stage\nduring this increasing-stress part of the cycle transit from the high\nto low viscosity branch of the constitutive curve. This transition is\nindeed seen in Fig.~\\ref{fig:eigenNonMon}: it occurs via ``top\njumping'' from the stress maximum across to the low viscosity branch.\nConversely, on the downward part of the sweep as the stress decreases\nfrom its maximum value $\\Sigma_0$, the system initially follows the\nlow viscosity branch until it eventually jumps back to the high\nviscosity branch. (We return in our concluding remarks to discuss the\npossible effect of thermal nucleation events, which are not included\nin these simulations, on these process of jumping between the two\nbranches of the constitutive curve.)\n\n\n\nThe corresponding signal of strain rate versus time during this slow\nup-then-down stress oscillation is shown in the right panel of\nFig.~\\ref{fig:eigenNonMon}. As can be seen, the regimes where the\nshear rate transits between the two different branches of the\nconstitutive curve occur over relatively short time intervals. (The\nduration of this process is informed by the short timescale $\\eta\/G$,\nwhereas the stress evolves on the much longer timescale $2\\pi\/\\omega$.)\nThis separation of timescales renders the stress signal approximately\nconstant in comparison to the fast evolution of the strain rate.\nCriterion (\\ref{eqn:critCreep}) might therefore be expected to apply\nin this regime of transition, at least to good approximation.\n\n\n\n\nFurthermore, during the transition from the high to low viscosity\nbranch we see a regime in which the shear rate signal simultaneously\nslopes up and curves up as a function of time: criterion\n(\\ref{eqn:critCreep}) not only applies but is also met, and we\ntherefore expect an instability to banding.  Plotting, by means of a\ncolourscale, the eigenvalue as defined in Sec.~\\ref{sec:lsa}, we find\nthat it is indeed positive. Likewise, during the rapid transition back\nfrom the low to high viscosity branch, we find a regime in which the\nshear rate signal simultaneously slopes down and curves down. As seen\nfrom the colourscale, the eigenvalue is also positive in this regime\n(although more weakly than during the upward transition). In what\nfollows, we will confirm the expectation of shear band formation\nduring these times of positive eigenvalue by simulating the model's\nfull nonlinear spatiotemporal dynamics.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.5cm]{figure15.eps}\n\\caption{Colour map of the normalised degree of shear banding for the\n  nRP model with a non-monotonic constitutive curve. Each point in\n  this $\\Sigma_0,\\omega$ plane corresponds to a particular LAOStress\n  run with stress amplitude $\\Sigma_0$ and frequency $\\omega$.  For\n  computational efficiency, these calculations are performed by\n  integrating the linearised equations in Sec.~\\ref{sec:lsa}. Reported\n  is the maximum degree of banding that occurs at any point in the\n  cycle, after many cycles.  Model parameters: $\\beta=0.4$,\n  $\\eta=10^{-4}$.  Cell curvature $q=2\\times 10^{-3}$. Crosses\n  indicate the grid of values of $\\Sigma_{0}$ and $\\omega$ in\n  Fig.~\\ref{fig:PipkinNonMon2}.}\n\\label{fig:pinPointNonMon2}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9.5cm]{figure16.eps}\n\\caption{ As in Fig.~\\ref{fig:pinPointNonMon}, but with a CCR\n  parameter $\\beta=0.9$, for which the fluid has a monotonic\n  underlying constitutive curve.  Crosses indicate the grid of values\n  of $\\Sigma_{0}$ and $\\omega$ used in the Pipkin diagram of\n  Fig.~\\ref{fig:PipkinMon2}.}\n\\label{fig:pinPointMon2}\n\\end{figure}\n\nThese processes of rapid transition between different branches of the\nconstitutive curve are of course not expected in a LAOStress\nexperiment at low frequency for a fluid with a monotonic constitutive\ncurve. In this case, for a LAOStress run in the limit $\\omega\\to 0$\nthe system instead sweeps quasi-statically along the monotonic\nconstitutive curve, with no associated banding.  As in the case of\nLAOStrain, however, it is crucial to realise that the absence of\nbanding in an experiment at zero frequency does not preclude the\npossibility of banding in a time-dependent protocol at finite\nfrequency, even in a fluid with a monotonic constitutive curve.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure17.eps}\n\\caption{Lissajous-Bowditch curves in LAOStress for the nRP model with\n  a non-monotonic constitutive curve. Results are shown as shear-rate\n  versus time in (a), and in the viscous representation of stress\n  versus strain rate in (b). Columns of fixed frequency $\\omega$ and\n  rows of fixed strain-rate amplitude $\\dot{\\gamma}_0$ are labeled at the top\n  and right-hand side. Colourscale shows the time-dependent degree of\n  shear banding.  Model parameters: $\\beta=0.4, \\eta=10^{-4}, l=0.02$.\n  Cell curvature: $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. A detailed portrait of the run\n  outlined by the thicker box is shown in\n  Fig.~\\ref{fig:portraitNonMon2}.}\n\\label{fig:PipkinNonMon2}\n\\end{figure}\n\n\n\\begin{figure}[tp!]\n\\includegraphics[width=8.5cm]{figure18.eps}\n\\caption{As in Fig.~\\ref{fig:PipkinNonMon2} but for a value of the CCR\n  parameter $\\beta=0.9$, for which the fluid's underlying constitutive\n  curve is monotonic. Number of numerical grid points $J=512$. A detailed portrait of the run outlined by the thicker box is shown in Fig.~\\ref{fig:portraitMon2}.}\n\\label{fig:PipkinMon2}\n\\end{figure}\n\nWith that in mind, we show in Fig.~\\ref{fig:eigenMon} the counterpart\nof Fig.~\\ref{fig:eigenNonMon}, but now for the nRP model with a\nmonotonic constitutive curve subject to a LAOStress run at a finite\nfrequency $\\omega=1$, of order the fluid's reciprocal stress\nrelaxation timescale.  The key to understanding the emergent dynamics\nin this case is the existence in the underlying zero-frequency\nconstitutive curve (shown by a dotted line in the left panel) of a\nregion in which the stress is a relatively flat (though still\nincreasing) function of the strain rate. As the system transits this\nregion during the increasing stress part of a finite-frequency stress\ncycle, we again observe a regime of quite sudden progression from low\nto high strain rate. This is seen in the left to right transition in\nthe stress versus strain rate representation in the left panel of\nFig.~\\ref{fig:eigenMon}, and (correspondingly) in the rapid increase\nof strain rate versus time in the right panel.\n\n\n\nDuring this regime of rapid transit we again have conditions in which\nthe strain rate evolves rapidly compared to the stress, such that the\nconstant-stress criterion (\\ref{eqn:critCreep}) should apply to good\napproximation. Furthermore, during the first part of the transition,\nthe strain rate signal simultaneously slopes and curves upward as a\nfunction of time.  The eigenvalue is therefore positive, indicating\nlinear instability of an initially homogeneous base state to the\nformation of shear bands. We will again confirm this prediction by\nsimulating the model's full nonlinear spatiotemporal dynamics below.\n\nIn the context of Figs.~\\ref{fig:eigenNonMon} and~\\ref{fig:eigenMon}\nwe have discussed the dynamics of the nRP model with a non-monotonic\nand monotonic constitutive curve respectively, focusing in each case\non one particular value of the imposed frequency $\\omega$ and stress\namplitude $\\Sigma_0$. We now consider the full plane of\n$(\\Sigma_0,\\omega)$ by showing in Figs.~\\ref{fig:pinPointNonMon2}\nand~\\ref{fig:pinPointMon2} colour maps of the extent of banding across\nit.  Recall that each point in this plane corresponds to a single LAOS\nexperiment with stress amplitude $\\Sigma_0$ and frequency $\\omega$.\nTo build up these maps we sweep over a grid of 20x20 values of\n$\\Sigma_0,\\omega$ and execute at each point a LAOStress run,\nintegrating the model's linearised equations set out in\nSec.~\\ref{sec:lsa}. We then represent the degree of banding\n$\\delta\\dot{\\gamma}$, maximised over the cycle after many cycles, by the\ncolourscale.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure19.eps}\n\\caption{ LAOStress in the nRP model with a non-monotonic constitutive\n  curve. Stress amplitude $\\Sigma_{0}=0.8$ and frequency $\\omega=1.0$.\n  Model parameters $\\beta=0.4, \\eta=10^{-4}, l=0.02$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} strain rate response as a function of time, focusing on the\n  region in which the system transits from the high to low viscosity\n  branch of the constitutive curve. Solid black and red-dashed line:\n  calculation in which the flow is constrained to be homogeneous.\n  Red-dashed region indicates a positive eigenvalue showing\n  instability to the onset of shear banding. Green dot-dashed line:\n  stress response in a full nonlinear simulation that allows banding.\n  {\\bf Right:} Velocity profiles corresponding to stages in the cycle\n  indicated by matching symbols in the left panel.}\n\\label{fig:portraitNonMon2}\n\\end{figure}\n\n\n\nFig.~\\ref{fig:pinPointNonMon2} shows results with model parameters for\nwhich the underlying constitutive curve is non-monotonic. As expected,\nfor stress amplitudes $\\Sigma_0$ exceeding the local maximum in the\nunderlying constitutive curve, significant banding is observed even in\nthe limit of low frequency $\\omega\\to 0$. This is associated with the\nprocesses of jumping between the two different branches of the\nconstitutive curve discussed above.  \n\nFig.~\\ref{fig:pinPointMon2} shows results for the nRP model with a\nmonotonic underlying constitutive curve. Here steady state banding is\nabsent in the limit $\\omega\\to 0$, as expected.  However, significant\nbanding is still nonetheless observed at frequencies of order the\nreciprocal reptation time, for imposed stress amplitudes exceeding the\nregion of weak slope in the constitutive curve, consistent with our\ndiscussion of Fig.~\\ref{fig:eigenMon} above.\n\nTo obtain the comprehensive roadmaps of\nFigs.~\\ref{fig:pinPointNonMon2} and~\\ref{fig:pinPointMon2} in a\ncomputationally efficient way, we discarded any nonlinear effects and\nintegrated the linearised model equations set out in\nSec.~\\ref{sec:lsa}. However, these linearised equations tend to\noverestimate the degree of banding in any regime of sustained positive\neigenvalue.  Therefore in Figs.~\\ref{fig:PipkinNonMon2}\nand~\\ref{fig:PipkinMon2} we now simulate the model's full nonlinear\nspatiotemporal dynamics, restricting ourselves for computational\nefficiency to grids of 4x4 values of $\\Sigma_0,\\omega$ as marked by\ncrosses in Figs.~\\ref{fig:pinPointNonMon2} and~\\ref{fig:pinPointMon2}.\n\n\nFig.~\\ref{fig:PipkinNonMon2} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is\nnon-monotonic.  At low frequencies the results tend towards the\nlimiting behaviour discussed above, in which the stress slowly tracks\nup and down the steady state flow curve $\\Sigma(\\dot{\\gamma})$ in between\nregimes of sudden transition between the two branches of the curve,\nduring which shear bands form. This is most pronounced in the case of\nthe jump between the high and low viscosity branches during the upward\nsweep.  Banding on the downward sweep is only apparent in a relatively\nmore limited region of $\\Sigma_0,\\dot{\\gamma}$ space, consistent with the\ntransition of $\\dot{\\gamma}_0$ being more modest in this part of the cycle\nduring which the stress decreases.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm]{figure20.eps}\n\\caption{As in Fig.~\\ref{fig:portraitNonMon2} but for the nRP model\n  with a CCR parameter $\\beta=0.9$ for which the underlying\n  homogeneous constitutive curve is monotonic. Number of numerical\n  grid points $J=512$.}\n\\label{fig:portraitMon2}\n\\end{figure}\n\nFor the particular run highlighted by the thicker box in\nFig.~\\ref{fig:PipkinNonMon2}, a detailed portrait of the system's\ndynamics is shown in Fig.~\\ref{fig:portraitNonMon2}. The left panel\nshows the strain rate signal as a function of time, zoomed on the\nregion in which the strain rate makes its transit from the high to low\nviscosity branch of the constitutive curve. The black and red-dashed\nline show the results of a calculation in which the flow is\nartificially constrained to remain homogeneous. The red-dashed region\nindicates the regime in which the criterion (\\ref{eqn:critCreep}) for\nlinear instability to the formation of shear bands is met, which also\ncorresponds to the regime in which the strain rate signal\nsimultaneously slopes up and curves upwards.\n\n\n\nIn a simulation that properly takes account of flow heterogeneity,\nshear bands indeed develop during this regime where the criterion is\nmet, then decay again once the strain rate signal curves down and\nstability is restored. This sequence can be seen in the velocity\nprofiles in the right hand panel. The stress signal associated with\nthis run that allows bands to form is shown by the green dot-dashed\nline in the left panel, and is only barely distinguishable from that\nof the run in which the flow is constrained to remain homogeneous.\n\nFig.~\\ref{fig:PipkinMon2} pertains to the nRP model with model\nparameters for which the underlying constitutive curve is monotonic,\nwith the grid of $(\\Sigma_0,\\omega)$ values that it explores marked by\ncrosses in Fig.~\\ref{fig:pinPointMon2}. True top-jumping events are\nabsent here, and no shear banding arises in the limit of zero\nfrequency. As discussed above, however, a similar rapid transition\nfrom low to high shear rate is seen in runs at a frequency $O(1)$, as\nthe stress transits the region of weak slope in the constitutive curve\nduring the increasing-stress part of the cycle.  Associated with this\ntransit is a pronounced tendency to form shear bands.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure21.eps}\n\\caption{Effect of CCR parameter $\\beta$ and entanglement number $Z$\n  (and so of chain stretch relaxation time $\\tau_R=\\tau_d\/3Z$) on shear\n  banding in LAOStress.  (Recall that the non-stretching version of\n  the model has $\\tau_R\\to 0$ and so $Z\\to\\infty$.)  Empty circles: no\n  observable banding.  Hatched circles: observable banding,\n  $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) = 10\\%-31.6\\%$. Dot-filled circles:\n  significant banding, $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) = 31.6\\% - 100\\%$.\n  Filled circles: strong banding, $\\Delta_{\\dot{\\gamma}}\/(1+|\\dot{\\gamma}(t)|) > 100\\%$.\n  For the hatched, dot-filled and filled symbols we used the criterion\n  that banding of the typical magnitude stated is apparent for any of\n  $\\omega=0.1,0.316$ or $1.0$, given a stress amplitude $\\Sigma_0$\n  exceeding the region of weak slope in the constitutive curve. The\n  square shows the parameter values explored in detail in\n  Fig.~\\ref{fig:stretchPortrait1}. The solvent viscosity $\\eta$ is $3.16\\times10^{-5}$.}\n\\label{fig:stretchMaster2}\n\\end{figure}\n\n\nThis can be seen for the run highlighted by the thicker box in\nFig.~\\ref{fig:PipkinMon2}, of which a detailed portrait is shown in\nFig.~\\ref{fig:portraitMon2}. This shows very similar features to its\ncounterpart for a non-monotonic underlying constitutive curve. In\nparticular, the regime of simultaneous upward slope and upward\ncurvature in the strain rate signal as the stress transits the region\nof weak positive constitutive slope triggers pronounced shear banding.\n\nThese results illustrate again the crucial point: that shear bands can\nform in a protocol with sufficiently strong time-dependence, even in a\nfluid for which the underlying constitutive curve is monotonic such\nthat banding is forbidden in steady state flows.\n\nSo far, we have restricted our discussion of LAOStress to the nRP\nmodel, for which the stretch relaxation time $\\tau_R$ is set to zero\nupfront so that any chain stretch relaxes to zero instantaneously,\nhowever strong the applied flow. The results of these calculations are\nexpected to apply, to good approximation, to experiments performed in\nflow regimes where chain stretch remains small. This typically imposes\nthe restriction $\\dot{\\gamma}\\tau_R\\ll 1$. We now turn to the sRP model to\nconsider the effects of finite chain stretch in experiments where this\nrestriction is not met.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5cm]{figure22.eps}\n\\caption{sRP model with a monotonic constitutive curve in LAOStress of\n  stress amplitude $\\Sigma_{0}=0.8$ and frequency $\\omega=0.1$.\n  Model parameters $\\beta=0.7, Z=100, \\eta=3.16\\times10^{-5}$.  Cell curvature\n  $q=2\\times 10^{-3}$. Number of numerical grid points $J=512$. {\\bf\n    Left:} strain rate signal versus time. Solid black\n  and red-dashed line: calculation in which the flow is constrained to\n  be homogeneous.  Red-dashed region indicates a positive eigenvalue\n  showing instability to the onset of shear banding. Green dot-dashed\n  line: stress response in a full nonlinear simulation that allows\n  banding (indistinguishable from homogeneous signal in this\n  case.) {\\bf Right:} Velocity profiles corresponding to stages in the\n  cycle indicated by matching symbols in left panel.}\n\\label{fig:stretchPortrait1}\n\\end{figure}\n\nFig.~\\ref{fig:stretchMaster2} shows the regions of the plane of the CCR\nparameter $\\beta$ and entanglement number $Z$ in which banding can be\nexpected even with chain stretch.  As for the case of LAOStrain above\nwe note that, depending on the value of $\\beta$, significant banding\nis still observed for experimentally commonly used entanglement\nnumbers, typically in the range $20-100$. Furthermore, observable\nbanding is clearly evident over a large region of this plane in which\nthe underlying constitutive curve is monotonic, precluding steady\nstate banding. Again, we hope that this figure might act as a roadmap\nto inform the discussion concerning the value of the CCR parameter\n$\\beta$.\n\n\nFor the pairing of $\\beta$ and $Z$ values marked by the square in\nFig.~\\ref{fig:stretchMaster2}, we show in\nFig.~\\ref{fig:stretchPortrait1} a detailed portrait of the model's nonlinear dynamics at a stress amplitude $\\Sigma_0$ and frequency $\\omega$ for which observable banding occurs.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have studied theoretically the formation of shear bands in large\namplitude oscillatory shear (LAOS) in the Rolie-poly model of polymers and wormlike micellar surfactants, with the particular aims of\nidentifying the regimes of parameter space in which shear banding is\nsignificant, and the mechanisms that trigger its onset.\n\n\nAt low frequencies, the protocol of LAOStrain effectively corresponds\nto a repeating series of quasi-static sweeps up and down the steady\nstate flow curve.  Here, as expected, we see shear banding in those\nregimes of parameter space for which the fluid's underlying\nconstitutive curve is non-monotonic, for strain rate amplitudes large\nenough to enter the banding regime in which the stress is a\ncharacteristically flat function of strain rate.\n\nIn LAOStrain at higher frequencies we report banding not only in the\ncase of a non-monotonic constitutive curve, but also over a large\nregion of parameter space for which the constitutive curve is\nmonotonic and so precludes steady state banding. We emphasise that\nthis is an intrinsically time-dependent banding phenomenon that would\nbe absent under steady state conditions, and we interpret it as the\ncounterpart of the `elastic' banding predicted recently in the context\nof shear startup experiments at high strain rates~\\cite{Moorcroft2013}.\n\nIn LAOStress we report shear banding in those regimes of parameter\nspace for which the underlying constitutive curve is either negatively\nor weakly positively sloping. In this case, the bands form during the\nprocess of yielding associated with the dramatic increase in shear\nrate that arises during that part of the cycle in which the stress magnitude\ntransits the regime of weak constitutive slope in an upward direction.\nAlthough the banding that we observe here is dramatically apparent\nduring those yielding events, these events are nonetheless confined to\na relatively small part of the stress cycle as a whole and would\ntherefore need careful focus to be resolved experimentally. (A\npossible related protocol, more focused on the banding regime,\ncould be to perform a shifted stress oscillation\n$\\Sigma(t)=\\Sigma_{\\rm plat} + \\Delta\\Sigma\\sin(\\omega t)$ where\n$\\Sigma_{\\rm plat}$ is a characteristic stress value in the region of\nweak slope in the constitutive curve and $\\Delta\\Sigma$ smaller in\ncomparison.) \n\nThe dramatic increase in strain rate associated with transiting to the\nhigh shear branch in LAOStress is likely to present practical\nexperimental difficulties in open flow cells such as Couette or\ncone-and-plate. To circumvent this, flow in a closed microfluidic channel provides an attractive alternative to those macroscopic\ngeometries in seeking to access this effect experimentally. \n\nIn each case we have demonstrated that the onset of shear banding can,\nfor the most part, be understood on the basis of previously derived\ncriteria for banding in simpler time-dependent\nprotocols~\\cite{Moorcroft2013}. In particular, the trigger for\nbanding in LAOStrain at low frequencies is that of a negatively\nsloping stress versus strain rate, which has long been recognised as\nthe criterion for banding under conditions of a steady applied shear\nflow. The trigger in LAOStrain at high frequencies is instead that of\nan overshoot in the signal of stress as a function of strain, in close\nanalogy to the criterion for banding onset during a fast shear startup\nrun. The trigger for banding in LAOStress is that of a regime of\nsimultaneous upward slope and upward curvature in the\ntime-differentiated creep response curve of strain rate as a function\nof time. This again is a close counterpart to the criterion for\nbanding following the imposition of a step stress.\n\nFor both LAOStrain and LAOStress we have provided a map of shear\nbanding intensity in the space of entanglement number $Z$ and CCR\nparameter $\\beta$. We hope that this will provide a helpful roadmap\nexperimentalists, and might even help to pin down the value of the CCR\nparameter, for which no consensus currently exists. \n\nWe have also commented that the value of the Newtonian viscosity\n$\\eta$ is typically much smaller than the zero shear viscosity $G\\tau$\nof the viscoelastic contribution, giving $\\eta\\ll 1$ in our units.\n\nExperimental literature suggests a range $\\eta=10^{-7}$ to $10^{-3}$.\nWe have typically used $\\eta=10^{-5}$ or\n$\\eta=10^{-4}$ in our numerics, and noted that the degree of banding\ntends to increase with decreasing $\\eta$. We also noted that the\ntimescale to transit from the high to low viscosity branch during\nyielding in each half cycle in LAOStress decreases with decreasing\n$\\eta\/G$. In view of these facts, a study of time-dependent banding in\nfluids with smaller values of $\\eta$ than those used here might\nwarrant the inclusion of inertia, because the small timescale for the\npropagation of momentum might exceed the short timescale $\\eta\/G$ in\nthose cases\n\nIn all our numerical studies the initial seed triggering the formation\nof shear bands was taken to be the weak curvature that is present in\ncommonly used experimental flow cells. In order to demonstrate the\nprinciple that the banding we report requires only a minimal seed,\nrather than being an artefact of strong cell curvature, all our runs\nhave assumed a curvature that is much smaller than that of most flow\ncells in practice. We also neglected stochastic noise altogether in\nall the results presented here. (We have nonetheless also performed\nruns with small stochastic noise instead of cell curvature and find\nqualitatively all the same effects.)\n\nHowever, one obvious shortcoming to this approach of taking only a very\nsmall initial seed is that it tends to suppress the nucleation events\nthat are, in a real experimental situation, likely to trigger banding\neven before the regime of true linear\ninstability~\\cite{Grand1997}, particularly in low frequency\nruns.  It would therefore be interesting in future work to study the\neffect of a finite temperature with particular regard to the\nnucleation kinetics to which it would give rise.\n\nThe calculations performed in this work all assumed from the outset\nthat spatial structure develops only in the flow gradient direction,\nimposing upfront translational invariance in the flow and vorticity\ndirections. We defer to future work a study of whether, besides the\nbasic shear banding instabilities predicted here, secondary\ninstabilities~\\cite{Fardin2014} of the interface between the\nbands~\\cite{Nghe2010,Fielding2005}\nor of the high shear band itself~\\cite{Fielding2010} will have\ntime to form in any given regime of amplitude and frequency space.\n\nWe have ignored throughout the effects of spatial variations in\n  the concentration field. However, it is well known that in a\n  viscoelastic solution heterogeneities in the flow field, and in\n  particular in the normal stresses, can couple to the dynamics of\n  concentration fluctuations via a positive feedback mechanism that\n  enhances the tendency to form shear\n  bands~\\cite{Milner1993,Schmitt1995,Fielding2003a,Fielding2003b,Fielding2003c}.\n  In the calculations performed here in LAOS we have observed\n  significant differences in the viscoelastic normal stresses between\n  the bands (approaching $50-70\\%$ of the cycle-averged value of the\n  same quantity, at least in the calculations without chain stretch).\n  It would therefore clearly be interesting in future work to consider\n  the effects of concentration coupling on the phenomena reported\n  here.\n\nThroughout we have ignored the possibility of edge fracture,\n  because the one-dimensional calculations performed here lack any\n  free surfaces and are unable to address it. It would clearly be\n  interesting in future work to address the effects of edge fracture\n  with regards the phenomena considered here~\\cite{Skorski2011,\n    Li2013, Li2015}.\n\nAll the calculations performed here have adopted what is\n  essentially a single-mode approach, taking account of just one\n  reptation relaxation timescale $\\tau_d$ and one stretch relaxation\n  timescale $\\tau_R$. It would be interesting in future work to\n  consider the effect of multiple relaxation timescales, which is\n  likely to be an important feature of the dynamics of unbreakable\n  polymers. (In wormlike micelles, in contrast, chain breakage and\n  recombination narrows the relaxation spectrum significantly such\n  that the single-mode approach adopted here is already likely to\n  provide a reasonably full picture.)\n\nWe hope that this work will stimulate further experimental studies of\nshear banding in time-dependent flows of complex fluids, with a\nparticular focus on the concept that banding is likely to arise rather\ngenerically during yielding-like events (following a stress overshoot\nin strain controlled protocols, or during a sudden increase in strain\nrate in stress controlled protocols) even in fluids with a monotonic\nconstitutive curve that precludes steady state banding in a\ncontinuously applied shear. In polymers this could form part of the\nlively ongoing debate concerning the presence or otherwise of shear\nbanding in those materials. In wormlike micelles it would be\ninteresting to see a study of LAOS across the full phase diagram (as\nset out, for example, in Ref.~\\cite{Berret1997}), from\nsamples that band in steady state to those above the dynamical\ncritical point, which don't.\n\n{\\it Acknowledgements} The authors thank Alexei Likhtman, Elliot\nMarsden, Peter Olmsted, Rangarajan Radhakrishnan, Daniel Read and\nDimitris Vlassopoulos for interesting discussions. The research\nleading to these results has received funding from the European\nResearch Council under the European Union's Seventh Framework\nProgramme (FP\/2007-2013), ERC grant agreement number 279365.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMyocardial Contrast Echocardiography (MCE) is a cardiac ultrasound imaging technique that utilizes vessel-bound microbubbles as contrast agents. In contrast to conventional B-mode echocardiography which only captures the structure and motion of the heart, MCE also allows for the assessment of myocardial perfusion through the controlled destruction and replenishment of microbubbles \\cite{wei_quantification_1998}. The additional perfusion information gives it great potential for the detection of coronary artery diseases. However, current perfusion analysis of MCE data mainly relies on human visual assessment which is time consuming and not reproducible \\cite{DBLP:conf\/fimh\/MaSRBL09}. There is generally a lack of automatic computerized algorithms and methods to help clinician perform accurate perfusion quantification \\cite{DBLP:conf\/fimh\/MaSRBL09}. One major challenge is the automatic segmentation of the myocardium before subsequent perfusion analysis can be carried out.\n\nIn this paper, we extend the Random Forests (RF) framework \\cite{DBLP:journals\/ml\/Breiman01} to segment the myocardium in our MCE data. RF is a machine learning technique that has gained increasing use in the medical imaging field for tasks such as segmentation \\cite{lempitsky_random_2009} and organ localization \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}. RF has been successful due to its accuracy and computational efficiency. Promising results of myocardial delineation on 3D B-mode echo has also been demonstrated in \\cite{lempitsky_random_2009}. However, classic RF has two limitations. First, our MCE data exhibit large sources of intensity variations \\cite{tang_quantitative_2011} due to factors such as speckle noise, low signal-to-noise ratio, attenuation artefacts, unclear and missing myocardial borders, presence of structures (papillary muscle) with similar appearance to the myocardium. These intensity variations reduce the discriminative power of the classic RF that utilizes only local intensity features. Second, RF segmentation operates on a pixel basis where the RF classifier predicts a class label for each pixel independently. Structural relationships and contextual dependencies between pixel labels are ignored \\cite{DBLP:conf\/iccv\/KontschiederBBP11,DBLP:conf\/ipmi\/MontilloSWIMC11} which results in segmentation with inconsistent pixel labelling leading to unsmooth boundaries, false detections in the background and holes in the region of interest. To overcome the above two problems, we need to incorporate prior knowledge of the shape of the structure and use additional contextual and structural information to guide the RF segmentation.\n\\\\\n\\indent\nThere are several works which have incorporated local contextual information into the RF framework. Lempitsky et al. \\cite{lempitsky_random_2009} use the pixel coordinates as position features for the RF so that the RF learns the myocardial shape implicitly. Tu et Bai \\cite{DBLP:journals\/pami\/TuB10} introduce the concept of auto-context which can be applied to RF by using the probability map predicted by one RF as features for training a new RF. Montillo et al. \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} extend the auto-context RF by introducing entanglement features that use intermediate probabilities derived from higher levels of a tree to train its deeper levels. Kontschieder et al. \\cite{DBLP:conf\/iccv\/KontschiederBBP11} introduce the structured RF that builds in structural information by using RF that predicts structured class labels for a patch rather than the class label of an individual pixel. Lombaert et al. \\cite{DBLP:conf\/miccai\/LombaertCA15} use spectral representations of shapes to classify surface data.\n\\\\\n\\indent\nThe above works use local contextual information that describes the shape of a structure implicitly. The imposed structural constraint are not strong enough to guide the RF segmentation in noisy regions of MCE data. In this paper, we proposed the Shape Model guided Random Forests (SMRF) which provides a new way to incorporate global contextual information into the RF framework by using a statistical shape model that captures the explicit shape of the entire myocardium. This imposes stronger, more meaningful structural constraints that guide the RF segmentation more accurately. The shape model is learned from a set of training shapes using Principal Component Analysis (PCA) and is originally employed in Active Shape Model (ASM) where the model is constrained so that it can only deform to fit the data in ways similar to the training shapes \\cite{DBLP:journals\/cviu\/CootesTCG95}. However, ASM requires a manual initialization and the final result is sensitive to the position of this initialization. Our SMRF is fully automatic and enjoys both the local discriminative power of the RF and the prior knowledge of global structural information contained in the statistical shape model. The SMRF uses the shape model to guide the RF segmentation in two ways. First, it directly incorporates the shape model into the RF framework by introducing a novel Shape Model (SM) feature which has outperformed the other contextual features and produced a more accurate RF probability map. Second, the shape model is fitted to the probability map to generate a smooth and plausible myocardial boundary that can be used directly for subsequent perfusion analysis.\n\n\\section{Method}\nIn this section, we first review some basic background on statistical shape model and RF. We then introduce the two key aspects of our SMRF---the novel SM feature and the fitting of the shape model.\n\\subsubsection{\\textit{Statistical Shape Model:}}\nA statistical shape model of the myocardium is built from 89 manual annotations using PCA \\cite{DBLP:journals\/cviu\/CootesTCG95}. Each annotation has $N=76$ landmarks comprising 4 key landmarks with 18 landmarks spaced equally in between along the boundary of manual tracing (Fig. \\ref{fig:Model} left). The shape model is represented as:\n\\begin{equation}\\label{eq:Model}\n  \\boldsymbol{x}=\\bar { \\boldsymbol{x} } + \\boldsymbol{Pb}\n\\end{equation}\n\\noindent where $\\boldsymbol{x}$ is a 2$N$-D vector containing the 2D coordinates of the $N$ landmark points, $\\bar{\\boldsymbol{x}}$ is the mean coordinates of all training shapes, $\\boldsymbol{b}$ is a set of $K$ shape parameters and $\\boldsymbol{P}$ contains $K$ eigenvectors with their associated eigenvalues ${ \\lambda  }_{ i }$. $K$ is the number of modes and set to 10 to explain 98\\% of total variance so that fine shape variations are modeled while noise is removed. Values of ${b}_{i}$ are bounded between $\\pm s\\sqrt { \\lambda _{ i } } $ so that only plausible shape similar to the training set is generated (Fig. \\ref{fig:Model} right). Refer to \\cite{DBLP:journals\/cviu\/CootesTCG95} for details on statistical shape model.\n\\subsubsection{\\textit{Random Forests:}}\nMyocardial segmentation can be formulated as a problem of binary classification of image pixels. An RF classifier \\cite{DBLP:journals\/ml\/Breiman01} is developed that predicts the class label (myocardium or background) of a pixel using a set of features. The RF is an ensemble of decision trees. During training, each branch node of a tree learns a pair of feature and threshold that results in the best split of the training pixels into its child nodes. The splitting continues recursively until the maximum tree depth is reached or the number of training pixels in the node falls below a minimum. At this time, a leaf node is created and the class distribution of the training pixels reaching the leaf node is used to predict the class label of unseen test pixels. The average of the predictions from all the trees gives a segmentation probability map. Refer to \\cite{DBLP:journals\/ml\/Breiman01}, \\cite{lempitsky_random_2009} for details on RF.\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.5\\textwidth}\n  \\centering\n  \\includegraphics[height=0.16\\textheight]{Fig1}\n  \\caption{}\n  \\label{fig:Model}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.5\\textwidth}\n  \\centering\n  \\includegraphics[height=0.16\\textheight]{Fig2}\n  \\caption{}\n  \\label{fig:Feature}\n\\end{subfigure}\n\\caption{(a) Left: A manual annotation from training set showing key landmarks (\\textit{red}) and other landmarks in between (\\textit{green}). Right: First two modes of variations of the shape model. (b) Left: Landmarks $\\boldsymbol{x}$ (\\textit{blue dots}) generated randomly by the shape model in (\\ref{eq:Model}). Right: $d_1$($d_2$) is the SM feature value measuring the signed shortest distance from pixel $\\boldsymbol{p_1}$($\\boldsymbol{p_2}$) to the myocardial boundary $B(\\boldsymbol{x})$ (\\textit{blue contour}). $d_1$ is positive and $d_2$ is negative.}\n\\label{fig:Result}\n\\end{figure}\n\\subsubsection{\\textit{Shape Model Feature:}}\nThe classic RF uses local appearance features which are based on surrounding image intensities of the reference pixel \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}. We introduced an additional novel SM feature that is derived from the shape model. The SM feature randomly selects some values for the shape model parameters $\\boldsymbol{b}$ and generates a set of landmarks $\\boldsymbol{x}$ using (\\ref{eq:Model}) (Fig. \\ref{fig:Feature} left). The landmarks can be joined to form a myocardial boundary. Let $B(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb})$ be the myocardial boundary formed by joining the landmarks generated using some values of $\\boldsymbol{b}$. The SM feature value is then given by the signed shortest distance $d$ from the reference pixel $\\boldsymbol{p}$ to the boundary $B$ (Fig. \\ref{fig:Feature} right). The distance is positive if $\\boldsymbol{p}$ lies inside the boundary and negative if it lies outside. The SM feature is essentially the signed distance transform of a myocardial boundary generated by the shape model. Each SM feature is defined by the shape parameters $\\boldsymbol{b}$. During training, an SM feature is created by random uniform sampling of each $b_{i}$ in the range of $\\pm s_{feature}\\sqrt { \\lambda _{ i } }$ where $s_{feature}$ is set to 1 in all our experiments. The binary SM feature test, parameterized by $\\boldsymbol{b}$ and a threshold $\\tau$, is written as:\n\\begin{equation}\n{ t }_{ \\textrm{SM} }^{ \\boldsymbol{b},\\tau  }(\\boldsymbol{p})=\\begin{cases} 1,\\qquad \\textrm{if}\\quad D(\\boldsymbol{p},B(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb}))>\\tau  \\\\ 0,\\qquad \\textrm{otherwise}. \\end{cases}\n\\end{equation}\n\\noindent where $D(.)$ is the function that computes $d$. Depending on the binary test outcome, pixel $\\boldsymbol{p}$ will go to the left (1) or right (0) child node of the current split node. During training, the RF learns the values of $\\boldsymbol{b}$ and $\\tau$ that best split the training pixels at a node. The SM features explicitly impose a global shape constraint in the RF framework. The random sampling of $\\boldsymbol{b}$ also allows the RF to learn plausible shape variations of the myocardium.\n\\subsubsection{\\textit{Shape Model Fitting:}}\nThe RF output is a probability map which cannot be used directly in subsequent analysis and application. Simple post-processing on the probability map such as thresholding and edge detection can produce segmentations with inaccurate and incoherent boundaries due to the nature of the pixel-based RF classifier. Our SMRF fits the shape model to the RF probability map to extract a final myocardial boundary that is smooth and which preserves the integrity of the myocardial shape. The segmentation accuracy is also improved as the shape constraint imposed by the shape model can correct some of the misclassifications made by the RF.\n\nLet ${ T }_{ \\boldsymbol{\\theta}}$ be a pose transformation defined by the pose parameter $\\boldsymbol{\\theta}$ which includes translation, rotation and scaling. The shape model fitting is then an optimization problem where we want to find the optimal values of $(\\boldsymbol{b},\\boldsymbol{\\theta})$ such that the model best matches the RF probability map under some shape constraints. We minimize the following objective function:\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\boldsymbol{b},\\boldsymbol{\\theta}}{\\text{min}}\n& & { { \\left\\| { \\boldsymbol{I} }_{ \\textrm{RF} }-{ \\boldsymbol{I} }_{ \\textrm{M} }({ T }_{ \\boldsymbol{\\theta}  }(\\bar { \\boldsymbol{x} } +\\boldsymbol{Pb})) \\right\\|  }^{ 2 }+\\alpha \\frac { 1 }{ K } \\sum _{ i=1 }^{ K }{ \\frac { \\left| { b }_{ i } \\right|  }{ \\sqrt { { \\lambda  }_{ i } }  }  }  } \\\\\n& \\text{subject to}\n& & -{ s }_{ fit }\\sqrt { { \\lambda  }_{ i } } <{ b }_{ i }<{ s }_{ fit }\\sqrt { { \\lambda  }_{ i } }, \\; i = 1, \\ldots, K.\n\\end{aligned}\n\\end{equation}\nThe first term of the objective function compares how well the match is between the model and the RF probability map $\\boldsymbol{I}_{\\textrm{RF}}$. $\\boldsymbol{I}_{\\textrm{M}}(.)$ is a function that converts the landmarks generated by the shape model into a binary mask of the myocardial shape. This allows us to evaluate a dissimilarity measure between the RF segmentation and the model by computing the sum of squared difference between the RF probability map and the model binary mask. The second term of the objective function is a regularizer which imposes a shape constraint. It is related to the probability of a given shape \\cite{DBLP:journals\/pr\/CristinacceC08} and ensures that it does not deviate too much away from its mean shape. $\\alpha$ is the weighting given to the regularization term and its value is determined empirically. Finally, an additional shape constraint is imposed on the objective function by limiting the upper and lower bounds of $b_i$ to allow for only plausible shapes. $s_{fit}$ is set to 2 in all our experiments. The optimization is carried out using direct search which is a derivative-free solver from the MATLAB global optimization toolbox. At the start of the optimization, each $b_i$ is initialized to zero. Pose parameters are initialized such that the model shape is positioned in the image center with no rotation and scaling.\n\\section{Experiments}\n\\subsubsection{\\textit{Datasets:}}\n2D+t MCE sequences were acquired from 15 individuals using a Philips iE33 ultrasound machine and SonoVue as the contrast agent. Each sequence is taken in the apical 4-chamber view under the triggered mode which shows the left ventricle at end-systole. One 2D image was chosen from each sequence and the myocardium manually segmented by two experts to give inter-observer variability. This forms a dataset of 15 2D MCE images for evaluation. Since the appearance features of the RF are not intensity invariant, all the images are pre-processed with histogram equalization to reduce intensity variations between different images. The image size is approximately 351$\\times$303 pixels.\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.55\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{Fig3}\n  \\caption{}\n  \\label{fig:ResultVisual}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{Fig4}\n  \\caption{}\n  \\label{fig:ResultQuant}\n\\end{subfigure}\n\\caption{(a) Visual segmentation results with one MCE example on each row. First three columns: Probability maps from classic RF, position feature RF and SMRF respectively. Last column: Ground truth boundary (\\textit{red}) and the SMRF boundary (\\textit{blue}) obtained from fitting the shape model to the SMRF probability map in the third column. \\textit{Black arrows} indicate papillary muscle. (b) Segmentation accuracy of different RF classifiers at different tree depths.}\n\\label{fig:Result}\n\\end{figure}\n\\subsubsection{\\textit{Validation Methodology:}}\nWe compared our SMRF segmentation results to the classic RF that uses appearance features \\cite{DBLP:journals\/mia\/CriminisiRKSPWS13}, as well as RFs that use other contextual features such as entanglement \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} and position features \\cite{lempitsky_random_2009}. We also compared our results to repeated manual segmentations and the Active Shape Model (ASM) method \\cite{DBLP:journals\/tmi\/GinnekenFSRV02}. Segmentation accuracy is assessed quantitatively using pixel classification accuracy, Dice and Jaccard indices, Mean Absolute Distance (MAD) and Hausdorff Distance (HD). To compute the distance error metrics (MAD and HD), a myocardial boundary is extracted from the RF probability map using the Canny edge detector. This is not required for the SMRF in which the shape model fitting step directly outputs a myocardial boundary.\n\\\\\n\\indent\nWe performed leave-one-out cross-validation on our dataset of 15 images. The RF parameters are determined experimentally and then fixed for all experiments. 20 trees are trained with maximum tree depth of 24. 10\\% of the pixels from the training images are randomly selected for training. The RF and the shape model fitting were implemented in C\\# and MATLAB respectively. Given an unseen test image, RF segmentation took 1.5min with 20 trees and shape model fitting took 8s on a machine with 4 cores and 32GB RAM. RF training took 38mins.\n\\section{Results}\nFig. \\ref{fig:ResultVisual} qualitatively shows that our SMRF probability map (column 3) has smoother boundary and more coherent shape than the classic RF (column 1) and position feature RF (column 2). Fitting the shape model to the SMRF probability map produces the myocardial boundary (\\textit{blue}) in column 4. The fitting guides the RF segmentation especially in areas where the probability map has a low confidence prediction. In the example on the second row, our SMRF predicts a boundary that correctly excludes the papillary muscle (\\textit{black arrows}). This is often incorrectly included by the other RFs due to its similar appearance to the myocardium.\n\\begin{table}[]\n\\centering\n\\caption{Quantitative comparison of segmentation results between the proposed SMRF and other methods. Results presented as (Mean $\\pm$ Standard Deviation).}\n\\label{table:Result}\n\\begin{tabular}{|M{10.5em}|c|c|c|c|r|}\n\\hline\n & \\multicolumn{1}{c|}{Accuracy} & \\multicolumn{1}{c|}{Dice} & \\multicolumn{1}{c|}{Jaccard} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}MAD\\\\   (mm)\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}HD\\\\   (mm)\\end{tabular}} \\\\ \\hline\nIntra-observer & 0.96$\\pm$0.01 & 0.89$\\pm$0.02 & 0.80$\\pm$0.03 & 1.02$\\pm$0.26 & 3.75$\\pm$0.93 \\\\\nInter-observer & 0.94$\\pm$0.02 & 0.84$\\pm$0.05 & 0.72$\\pm$0.07 & 1.59$\\pm$0.57 & 6.90$\\pm$3.24 \\\\\nASM \\cite{DBLP:journals\/tmi\/GinnekenFSRV02} & 0.92$\\pm$0.03 & 0.77$\\pm$0.08 & 0.64$\\pm$0.11 & 2.23$\\pm$0.81 & 11.44$\\pm$5.23 \\\\ \\hline\nClassic RF & 0.91$\\pm$0.04 & 0.74$\\pm$0.12 & 0.60$\\pm$0.14 & 2.46$\\pm$1.36 & 15.69$\\pm$7.34 \\\\\nEntangled RF \\cite{DBLP:conf\/ipmi\/MontilloSWIMC11} & 0.91$\\pm$0.05 & 0.75$\\pm$0.13 & 0.62$\\pm$0.15 & 2.43$\\pm$1.62 & 15.06$\\pm$7.92 \\\\\nPosition Feature RF \\cite{lempitsky_random_2009} & \\textbf{0.93$\\pm$0.03} & \\textbf{0.81$\\pm$0.10} & 0.69$\\pm$0.13 & 1.81$\\pm$0.84 & 9.51$\\pm$3.80 \\\\\nSMRF & \\textbf{0.93$\\pm$0.03} & \\textbf{0.81$\\pm$0.10} & \\textbf{0.70$\\pm$0.12} & \\textbf{1.68$\\pm$0.72} & \\textbf{6.53$\\pm$2.61} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nTable. \\ref{table:Result} compares the quantitative segmentation results of our SMRF to other methods. Both SM features and position features encode useful structural information that produces more accurate RF probability maps than the classic RF and entangled RF. This is reflected by the higher Dice and Jaccard indices. For MAD and HD metrics, SMRF outperforms all other RF methods because the shape model fitting step in SMRF produces more accurate myocardial boundaries than those extracted using the Canny edge detector. In addition, SMRF also outperforms ASM \\cite{DBLP:journals\/tmi\/GinnekenFSRV02} and comes close to the inter-observer variations.\n\nFig. \\ref{fig:ResultQuant} compares the segmentation accuracy of the probability maps of different RF classifiers. Our SMRF obtained higher Jaccard indices than the classic and entangled RFs at all tree depths. At lower tree depths, SMRF shows notable improvement over the position feature RF. The SM features have more discriminative power than the position features as it captures the explicit geometry of the myocardium using the shape model. The SM feature binary test partitions the image space using more complex and meaningful myocardial shapes as opposed to position feature which simply partitions the image space using straight lines. This provides a stronger global shape constraint than the position feature and allows a decision tree to converge faster to the correct segmentation at lower tree depths. This gives the advantage of using trees with smaller depths which speeds up both training and testing.\n\\section{Conclusion}\nWe presented a new method SMRF for myocardial segmentation in MCE images. We showed how our SMRF utilizes a statistical shape model to guide the RF segmentation. This is particular useful for MCE data whose image intensities are affected by many variables and therefore prior knowledge of myocardial shape becomes important in guiding the segmentation. Our SMRF introduces a new SM feature which captures the global myocardial structure. This feature outperforms other contextual features to allow the RF to produce a more accurate probability map. Our SMRF then fits the shape model to the RF probability map to produce a smooth and coherent final myocardial boundary that can be used in subsequent perfusion analysis. In future work, we plan to validate our SMRF on a larger, more challenging dataset which includes different cardiac phases and chamber views.\n\\subsubsection*{Acknowledgments.} The authors would like to thank Prof. Daniel Rueckert, Liang Chen and other members from the BioMedIA group for their help and advice. This work was supported by the Imperial College PhD Scholarship.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\\subsection{Background}\nThe present paper studies phase transitions from an ergodic theory and dynamical system viewpoint. It investigates the relations between renormalization, substitutions and phase transition initiated in \\cite{baraviera-leplaideur-lopes} and continued in \\cite{BL}.\n\nPhase transitions are an important topic in statistical mechanics and also in probability theory (see {\\em e.g.\\ } \\cite{georgii,grimmett}). \nThe viewpoint presented here is different for several reasons. One of them is that, here, the geometry of the lattice is not relevant\\footnote{and we only consider a one-dimensional lattice.}, whereas in statistical mechanics, \nthe geometry of the lattice is the most relevant part. \n\nA {\\em phase transition} is characterized by a lack of analyticity of the \npressure function. \nThis definition of phase transition is inspired by statistical mechanics \nand is now standard for dynamical systems, see \\cite{bowen, ruelle, sinai}. \nGiven a dynamical systems, say $(X,T)$, and a potential $\\varphi:X\\to{\\mathbb R}$, the pressure function is given by \n$$\n\\CP(\\beta):=\\sup\\left\\{h_{\\mu}(T)+\\beta \\int\\varphi\\,d\\mu\\right\\},\n$$\nwhere the supremum is taken over the invariant probability measures $\\mu$, $h_{\\mu}(T)$ is the Kolmogorov entropy and $\\beta$ is a real parameter. \n\nFor a uniformly hyperbolic dynamical system $(X,T)$ and \na H\\\"older continuous potential $\\varphi$, the pressure function $\\beta\\mapsto \\CP(\\beta)$ is analytic (see {\\em e.g.\\ } \\cite{bowen, ruelle, keller}). Even if analyticity is usually considered as a very rigid property and thus quite rare, it turns out that proving non-analyticity for the pressure function is not so easy. \nCurrently, this has become an important challenge in smooth ergodic theory to produce and study phase transitions, see {\\em e.g.\\ } \\cite{Makarov-Smirnov, coronel-Rivera-Letelier, BL, iommi-todd}. \n\nTo observe phase transitions, one has to weaken hyperbolicity of the system\nor of regularity of the potential; it is the latter one that we continue to investigate here. Our dynamical system is the full shift, which is uniformly hyperbolic. \nThe first main question we want to investigate is thus what potentials $\\varphi$ will produce phase transitions. More precisely, we are looking for a machinery to produce potentials with phase transitions. \n\nThe main purpose of \\cite{baraviera-leplaideur-lopes} was to investigate possible relation between {\\em renormalization} and phase transition. In the shift\nspace  $(\\{0,1\\}^{{\\mathbb N}}, \\sigma)$, a renormalization is a function $H$ satisfying an equality of the form \n\\begin{equation}\\label{equ1-renorm}\n\\sigma^{k}\\circ H=H\\circ \\sigma.\n\\end{equation}\n The link with potentials was made in \\cite{baraviera-leplaideur-lopes} by introducing a renormalization operator $\\CR$ acting on potentials and related to a solution $H$ for \\eqref{equ1-renorm}.\n\nIt is easy to check that constant length $k$ substitutions are \nsolutions to the renormalization equation.\nIn \\cite{BL}, we studied  the Thue-Morse case substitution,\nwhich has constant length $2$. \nHere we investigate the Fibonacci substitution, which is not of constant length. Several reasons led us to study the Fibonacci case:\n\n$\\bullet$ Together with the Thue-Morse substitution, the Fibonacci substitutions is the most ``famous'' substitution and it has been well-studied. In particular, the dynamical properties of their respective attracting sets are well-known and this will be\nused extensively in this paper for the Fibonacci shift. As a result, we were \nable to describe the relevant fixed point of renormalization exactly. \nInformation of the left and right-special words in these attractors is \na key ingredient to prove existence of a phase transition; it \nis a crucial issue in the relations between \nsubstitutions and phase transitions.  \n\n$\\bullet$ The type of phase transition we establish is a {\\em freezing phase transition}. This means that beyond the phase transition ({\\em i.e.,\\ } for large $\\beta$), \nthe pressure function is affine and equal to its asymptote, and the equilibrium state ({\\em i.e.,\\ } ground state) is the unique shift-invariant measure\nsupported on an aperiodic subshift space, sometimes called {\\em quasi-crystal}.\nOne open question in statistical mechanics (see \\cite{vanEnter?}) is whether freezing phase transitions can happen and whether {\\em quasi-crystal ground state}\ncan be reach at {\\em positive temperature}. An affirmative answer was given for the Thue-Morse quasi-crystal in \\cite{BL}; we show here this also holds for the Fibonacci quasi-crystal. \n\n$\\bullet$ \nWe think that Fibonacci shift opens the door to study more cases. One natural question is whether any quasi-crystal can be reached as a ground state at positive temperature. In this context we emphasize that the Fibonacci substitution also has a Sturmian shift, that is, it is related to the irrational rotation with angle the golden mean $\\gamma:=\\frac{1+\\sqrt5}2$. \nWe expect that the machinery developed here for the Fibonacci substitution can\nbe extended to the Sturmian shift associated to general irrational rotation\nnumbers \n(although those with bounded entries in the continued fraction expansion\nwill be the easiest), possibly to rotations on higher-dimensional tori, and also to more general substitutions. \n\n\n \n\n\\subsection{Results}\n\nLet $\\S = \\{0,1\\}^{{\\mathbb N}}$ be the full shift space; points in $\\S$ \nare sequences $x:=(x_{n})_{n\\ge 0}$ or equivalently infinite {\\em words} $x_{0}x_{1}\\ldots$. \nThroughout we let $\\overline x_j = 1-x_j$ denote the opposite symbol.\nThe dynamics is the left-shift \n$$\n\\sigma: x=x_{0}x_{1}x_{2}\\ldots\\mapsto x_{1}x_{2}\\ldots.\n$$\nGiven a word $w=w_{0}\\ldots w_{n-1}$ of {\\em length} $|w|=n$, the corresponding\n{\\em cylinder} (or {\\em $n$-cylinder}) is the set of infinite words starting as $w_{0}\\ldots w_{n-1}$. We use the notation\n$C_n(x)=[x_0\\dots x_{n-1}]$\nfor the $n$-cylinder containing $x=x_{0}x_{1}\\ldots$ \nIf $w=w_{0}\\ldots w_{n-1}$ is a word with length $n$ and $w'=w'_{0}\\ldots$ a word of any length, the {\\em concatenation} $ww'$ is the word $w_{0}\\ldots w_{n-1}w'_{0}\\ldots$. \n\nThe Fibonacci substitution on $\\S$ is defined by: \n$$\nH: \\begin{cases}\n0 \\to 01\\\\\n1 \\to 0.\n\\end{cases}\n$$\nand extended for words by the concatenation rule $H(ww')=H(w)H(w')$.\nIt is convenient for us to count the Fibonacci numbers starting with index $-2$:\n\\begin{equation}\\label{eq:Fibo}\nF_{-2} = 1,\\ F_{-1} = 0,\\ F_0=1, \\ F_1 = 1,\\ F_2 = 2,\\ F_{n+2} = F_{n+1} + F_n,\n\\end{equation}\nWe have\n\\begin{equation}\\label{eq:Fiboa}\nF_{n}^{a}:=|H^{n}(a)| = \\begin{cases}\nF_{n+1} & \\text{ if } a = 0,\\\\\nF_{n} & \\text{ if } a = 1.\n\\end{cases}\n\\end{equation}\nThe Fibonacci substitution has a unique fixed point\n$$\n\\rho = 0\\ 1\\ 0\\ 01\\ 010\\ 01001\\ 01001010\\ 0100101001001\\dots\n$$\nWe define the orbit closure ${\\mathbb K} = \\overline{\\cup_n \\sigma^n(\\rho)}$; it forms a subshift\nof $(\\S, \\sigma)$ associated to $\\rho$. More properties on ${\\mathbb K}$ are given in Section~\\ref{sec-H-K-R}.\n\n\\bigskip\nWe define the renormalization operator \nacting on potentials $V:\\S\\to {\\mathbb R}$ by \n$$\n(\\CR V)(x)= \\begin{cases}\n V\\circ \\sigma\\circ H(x)+V\\circ H(x) & \\text{ if }x\\in[0],\\\\\n V\\circ H(x) & \\text{ if }x\\in[1].\n\\end{cases}\n$$\nWe are interested in finding fixed points for $\\CR$ and, where possible, studying their stable leaves,\n{\\em i.e.,\\ } potentials converging to the fixed point under iterations of $\\CR$. \nContrary to the Thue-Morse substitution, \nthe Fibonacci substitution is not of constant length. This is the source of several complications, in particular for the correct expression for $\\CR^{n}$. \n\nFor  $\\alpha>0$, let $\\CX_{\\alpha}$ be the set of functions $V: \\S \\to {\\mathbb R}$ \nsuch that $}%{\\displaystyle V(x) \\sim n^{-\\alpha}$ if $d(x,{\\mathbb K})=2^{-n}$. \nMore precisely, $\\CX_{\\alpha}$ is the set of functions $V$ such that:\n\\begin{enumerate}\n\\item $V$ is continuous and non-negative. \n\\item There exist two continuous functions $g,h:\\S \\to {\\mathbb R}$, satisfying $}%{\\displaystyle h_{|{\\mathbb K}}\\equiv 0$ and $g>0$, such that \n$$\nV(x) = \\frac{g(x)}{n^\\alpha} + \\frac{h(x)}{n^\\alpha} \\quad \\text{ when } \\quad \nd(x,{\\mathbb K}) = 2^{-n}.\n$$\n\\end{enumerate}\nWe call $g$ the \\textit{$\\alpha$-density}, or just the \\textit{density} \nof $V\\in \\CX_{\\alpha}$. Continuity and the assumption $h_{|{\\mathbb K}}\\equiv 0$ imply \nthat $h(x)\/n^{\\alpha} = o(n^{-\\alpha})$.\n\nOur first theorem achieves the existence of a fixed point for $\\CR$ and shows that the germ of $V$ close to ${\\mathbb K}$, {\\em i.e.,\\ } its $\\alpha$-density, allows us to \ndetermine the stable leaf of that fixed point. \n\nGiven a finite word $w$, let\n$\\kappa_a(w)$ denote the number of symbols $a\\in\\{0,1\\}$ in $w$.\nIf $x \\in \\S \\setminus {\\mathbb K}$, we denote by $\\widetilde\\kappa_{a}(x)$ the number of symbols $a$ in the finite word $x_{0}\\ldots x_{n-1}$ where $d(x,{\\mathbb K})=2^{-n}$. \n\n\\begin{theorem} \\label{theo-fixedpoint}\nIf $V \\in \\CX_\\alpha$, with $\\alpha$-density function $g$, then\n$$\n\\lim_{k\\to\\infty} \\CR^{k}V(x) =\n\\begin{cases}\n\\quad \\infty & \\text{ for all  } x\\in \\S\\setminus {\\mathbb K} \\text{ if } \\alpha < 1; \\\\[2mm]\n\\quad 0 & \\text{ for all  } x\\in \\S \\text{ if } \\alpha > 1; \\\\[2mm]\n\\quad }%{\\displaystyle\\int g \\ d\\mu_{\\mathbb K} \\cdot \\widetilde V(x)\n& \\text{ for all  } x\\in \\S \\text{ if } \\alpha = 1,\n\\end{cases}\n$$\nwhere  $\\widetilde V\\in\\CX_{1}$ is a fixed point for $\\CR$, given by \n\\begin{equation}\\label{eq:tildeV}\n\\widetilde V(x) = \n\\begin{cases}\n \\log\\left(}%{\\displaystyle\\frac{\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma}{}%{\\displaystyle\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma-1}\\right) & \\text{ if } x \\in [0];\\\\[4mm]\n \\log\\left(}%{\\displaystyle\\frac{\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}}{}%{\\displaystyle\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}-1}\\right) & \\text{ if } x \\in [1].\n\\end{cases}\n\\end{equation}\n\\end{theorem}\nThis precise expression of $\\widetilde V$ corresponds to a $\\alpha$-density\n$\\tilde g(x) = \\gamma^2\/(2\\gamma-1)$ if $x \\in [0] \\cap K$ and $\\tilde g(x) =\n\\gamma\/(2\\gamma-1)$ if $x \\in [1] \\cap {\\mathbb K}$,\nand $\\int \\widetilde V(x) d\\mu_{\\mathbb K} = 1$.\n\nOur second theorem suggests that renormalization for potentials is a machinery to produce potentials with phase transition. \nWe recall that a {\\em  freezing phase transition} is characterized\nby the fact that the pressure is of the form \n$$\n\\CP(\\beta)=a\\beta+b \\quad \\text{ for } \\beta\\ge\\beta_{c}\n$$\nand that the equilibrium state is fixed for $\\beta\\ge\\beta_c$.\nThe word ``freezing'' comes from the fact that in statistical mechanics \n$\\beta$ is the inverse of the temperature (so the temperature \ngoes to $0$ as $\\beta\\to+\\infty$) and that a {\\em ground-state} \nis reached at positive temperature $1\/\\beta_c$, see \\cite{CLT, Dyson}. \n\n\\begin{theorem}\\label{theo-pt}\nAny potential $\\varphi:=-V$ with $V\\in \\CX_{1}$ admits a freezing phase transition at finite $\\beta$: there exists $\\beta_{c}>0$ such that \n\\begin{enumerate}\n\\item for $0\\le \\beta<\\beta_{c}$ the map $\\CP(\\beta)$ is analytic, there exists a unique equilibrium state for $\\beta \\varphi$ and this measure has full support;\n\\item for $\\beta>\\beta_{c}$, $\\CP(\\beta)=0$ and  $\\mu_{{\\mathbb K}}$ is the unique \nequilibrium state for $\\beta \\varphi$.  \n\\end{enumerate}\n\\end{theorem}\n\nThese two theorems explain a link between substitution, renormalization and phase transition on quasi-crystals: a substitution generates a quasi-crystal but also allows to define a renormalization operator acting on the potentials. This operator has some fixed point, and the stable leaf of that fixed point furnishes a family of potentials with freezing phase transition. \n\n\\subsection{Outline of the paper}\nIn Section~\\ref{sec-H-K-R} we recall and\/or prove various\nproperties of the Fibonacci subshift and its special words.\nWe establish the form of $H^n$ and $\\CR^nV$ for arbitrary $n$ and relate this\nto (special words of) the Fibonacci shift.\nIn Section~\\ref{sec-prooftheofix}, after  clarifying the role of accidents on\nthe computation of $\\CR^nV$, we prove Theorem~\\ref{theo-fixedpoint}.\nSection~\\ref{sec-proffthpt} deals with the thermodynamic formalism.\nFollowing the strategy of \\cite{leplaideur-synth} we specify and estimate the required (quite involved) quantities that are the core of the proof of Theorem~\\ref{theo-pt}.\n\n\\section{Some more properties of $H$, ${\\mathbb K}$ and $\\CR$}\\label{sec-H-K-R}\n\\subsection{The set \\boldmath ${\\mathbb K}$ as Sturmian subshift  \\unboldmath}\nIn addition to being a substitution subshift, $({\\mathbb K},\\sigma)$ is the Sturmian subshift associated to the golden mean rotation, $T_{\\gamma}:x \\mapsto x+\\gamma \\pmod 1$. The golden mean is\n$\\gamma=\\frac{1+\\sqrt5}2$  and it satisfies $\\gamma^{2}=\\gamma+1$. \n\nFixing an orientation on the circle, \nlet $\\arc{ab}$ denote the arc of points between $a$ and $b$ in the circle \nin that orientation. \nIf we consider the itinerary of $2\\gamma$ under the action of $T_{\\gamma}$ with the code $0$ if the point belongs to $\\arc{0\\gamma}$ and $1$ if the point belongs to $\\arc{\\ga0}$ (see Figure~\\ref{fig-codi-fibo}), we get $\\rho$, the fixed point of the substitution.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{fibocode3.pdf}\n\\caption{Coding for Fibonacci Sturmian subshift.}\n\\label{fig-codi-fibo}\n\\end{center}\n\\end{figure}\n\nThere is an almost ({\\em i.e.,\\ } up to a countable set) one-to-one correspondence between points in ${\\mathbb K}$ and codes of orbits of $(\\SS, T_\\gamma)$,\n expressed by the commutative diagram\n$$\n\\begin{array}{ccc}\\SS & \\stackrel{}%{\\displaystyle T_\\gamma}{\\longrightarrow} & \\SS \\\\\\pi\\downarrow & \\circlearrowleft & \\downarrow\\pi \\\\{\\mathbb K} & \\stackrel{}%{\\displaystyle\\sigma}{\\longrightarrow} & {\\mathbb K}\\end{array}\n$$\nand $\\pi$ is a bijection, except at points $T_\\gamma^{-n}(\\gamma) \\in \\SS$, $n \\ge 0$.\nSince Lebesgue measure is the unique $T_\\gamma$-invariant probability measure,\n$\\mu_{\\mathbb K} := \\text{Leb} \\circ \\pi^{-1}$ is the unique\ninvariant probability measure of $({\\mathbb K},\\sigma)$.  \n\nWe will use the same terminology for both ${\\mathbb K}$ and $\\SS$. \nFor instance, a cylinder $C_{n}(x)$ for $x \\in \\SS$ is an interval\\footnote{Some work has to be done to check that it actually is an interval.}, \nwith the convention that $C_{n}(x)=\\pi^{-1}(C_{n}(\\pi(x)))$, and we may confuse \na point $x \\in \\SS$ and its image $\\pi(x) \\in {\\mathbb K}$. \n\n\\begin{definition}\\label{def-words}\nLet $\\CA_{{\\mathbb K}}$ denote the set of finite words that appear in $\\rho$.\n A word $\\omega:=\\omega_{0}\\ldots \\omega_{n-1}\\in \\CA_{{\\mathbb K}}$ is said to be {\\em left-special} if $0w$ and $1w$ both appear in $\\CA_{{\\mathbb K}}$. It is {\\em right-special} if $w0$ and $w1$ both appear in $\\CA_{{\\mathbb K}}$.\nA left and right-special word is called {\\em bi-special}. A {\\em special} word is either left-special or right-special. \n\\end{definition}\n \nSince $\\rho$ has $n+1$ words of length $n$ \n(a characterization of Sturmian words), there is exactly one left-special \nand one right-special word of length $n$.\nThey are of the form $\\rho_{0}\\ldots \\rho_{n-1}$ and $\\rho_{n-1}\\ldots \\rho_{0}$\nrespectively, which can be seen from the forward itinerary\nof $x \\approx \\gamma$ and backward itinerary of $x \\approx 0$ in the circle. \nSometimes the left and right-special word merge into a single bi-special \nword $\\omega$, but only\none of the two words $0\\omega0$, and $1\\omega1$ appears in $\\CA_{{\\mathbb K}}$, see \\cite[Section 1]{arnoux-rauzy}, the construction of $\\Gamma_{n+1}$ from $\\Gamma_{n}$.\n\n\\begin{proposition}\\label{prop-bispecialfibo}\nBi-special words in $\\CA_{{\\mathbb K}}$ are of the form $\\rho_{0}\\ldots \\rho_{F_m-3}$\nand for each $m \\ge 3$, $\\rho_{0}\\ldots \\rho_{F_m-3}$ is bi-special.\n\\end{proposition}\n\nWe prove this proposition at the end of Section~\\ref{subsec:Hn}\n\n\\subsection{Results for \\boldmath $H^{n}$ \\unboldmath}\\label{subsec:Hn}\nWe recall that $\\kappa_{a}(w)$ is the number of symbol $a$ in the finite word $w$. \n\n\\begin{lemma}\\label{lem-lengthHn}\nFor any finite word $w$, the following recursive relations hold:\n\\begin{eqnarray*}\n\\kappa_0(H^n(w)) &=& F_n \\kappa_0(w) + F_{n-1} \\kappa_1(w);\\\\\n\\kappa_1(H^n(w)) &=& F_{n-1} \\kappa_0(w) + F_{n-2} \\kappa_1(w);\\\\\n|H^n(w)| &=& F_{n+1} \\kappa_0(w) + F_n \\kappa_1(w) = |H^{n-1}(w)|+|H^{n-2}(w)|, \n\\end{eqnarray*}\nwhere $|H^{0}(w)|=|w|,\\ |H^{1}(w)|=|H(w)|$.\n\\end{lemma}\n\nSince we have defined $F_{-2} = 1$ and $F_{-1} = 0$, see \\eqref{eq:Fibo},\nthese formulas hold for $n = 0$ and $n=1$ as well.\n\n\\begin{proof}\nSince $H^n(0)$ contains $F_{n+1}$ zeroes and $F_{n-1}$ ones,\nwhile  $H^n(0)$ contains $F_{n-1}$ zeroes and $F_{n-2}$ ones,\nthe first two lines follow from concatenation.\nThe third line is the sum of the first two, and naturally\nthe recursive relation follows from the same recursive relation\nfor Fibonacci numbers.\n\\end{proof}\n\nSince $({\\mathbb K}, \\sigma, \\mu_{\\mathbb K})$ is uniquely ergodic, and isomorphic to \n$(\\SS, T_\\gamma, \\text{Leb})$, we immediately get that\n\\begin{equation}\\label{eq:symfreq}\n\\lim_{n \\to +\\infty} \\frac{\\kappa_a(H^n(w))}{ |H^n(w)| } =\n\\begin{cases} \n|\\arc{0\\gamma}| = \\frac1\\gamma & \\text{ if } a = 0, \\\\\n|\\arc{\\ga0}| = 1-\\frac1\\gamma & \\text{ if } a = 1. \n\\end{cases}\n\\end{equation}\n\n\\iffalse\n\\begin{definition}\\label{def-stopblock}\nWe say that  $x, y \\in \\S$ coincide up to a {\\em stopping block} if there exists a word $w$ such that \n$$x=w01\\text{ and }y=w10.$$\nWe shall also write that $(x,y)$ has a stopping block, and\/or that $x$ and $y$ have a stopping block. \n\\end{definition}\n\nThe Fibonacci substitution is not of constant length, and this \ncomplicates the calculation how long two points $H^{n}(x)$ and $H^{n}(y)$ coincide if we know how long $x$ and $y$ coincide. \nThe main interest of stopping-block is that it propagates itself under the iterations of $H^{n}$. \n\\fi\n\n\\begin{lemma}\\label{lem-prop-stopbloc}\nAssume that $x$ and $y$ have a maximal common prefix $w$.\nThen $H^{n}(x)$ and $H^{n}(y)$ coincide for $T_{n}(w)+F_{n+2}-2$ digits, where $T_{n}(w)$ is defined by \n\\begin{equation}\\label{eq:Tnw}\nT_{0}= |w|,\\ T_{1}=|H(w)|,\\ T_{n+2}(w) = T_{n+1}(w)+T_{n}(w).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor $x = w0\\dots$ and $y = w1\\dots$, we find\n\\begin{align*}\n\\begin{array}{c} w \\\\ w \\\\\\end{array}\n& \\begin{array}{|c|}\\hline 0  \\ {\\hbox{{\\it 1}} \\!\\! I}   \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow}\n\\begin{array}{cc} H(w) & 0\\ \\\\ H(w) & 0\\ \\\\\\end{array}\n\\begin{array}{|c|}\\hline 1 \\\\0 \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow}\\begin{array}{cccc} H^2(w) & 0 & 1 & 0\\ \\\\ H^2(w) & 0 & 1 & 0\\ \\end{array}\n\\begin{array}{|c|}\\hline 0 \\\\ 1 \\\\\\hline \\end{array} \\\\[3mm]\n& \\stackrel{H}{\\longrightarrow}\n\\begin{array}{ccccccc} H^3(w) & 0 & 1 & 0 & 0 & 1 & 0\\ \\\\\nH^3(w) & 0 & 1 & 0 & 0 & 1 & 0\\ \\end{array}\n\\begin{array}{|c|}\\hline 1 \\\\0  \\\\\\hline \\end{array}\n\\ \\stackrel{H}{\\longrightarrow} \\ \\cdots\n\\end{align*}\nwhere we used that $H(a)$ starts with $0$ for both $a=0$ and $a=1$.\nWe set $T_n(w) = |H^n(w)|$, then the recursive formula \\eqref{eq:Tnw}\nfollows as in Lemma~\\ref{lem-lengthHn}.\n\nIterating $H$ on the words $01$ and $10$, we find:\n\\begin{equation}\\label{eq:Hn0110}\n\\begin{array}{|cc|}\\hline 0 & 1 \\ {\\hbox{{\\it 1}} \\!\\! I}  & 0 \\\\\\hline \\end{array}\\stackrel{H}{\\longrightarrow}\\begin{array}{c} 0 \\\\0 \\\\\\end{array}\\begin{array}{|cc|}\\hline 1 & 0 \\\\0 & 1 \\\\\\hline \\end{array}\\stackrel{H}{\\longrightarrow}\\begin{array}{ccc}0 & 1 & 0 \\\\0 & 1 & 0\\end{array}\\begin{array}{|cc|}\\hline 0 & 1 \\ {\\hbox{{\\it 1}} \\!\\! I}  & 0 \\\\\\hline \\end{array}\n\\stackrel{H}{\\longrightarrow}\\begin{array}{cccccc}0 & 1 & 0 & 0 & 1 & 0 \\\\0 & 1 & 0 & 0 & 1 & 0\\end{array}\n\\begin{array}{|cc|}\\hline 1 & 0 \\\\0 & 1 \\\\\\hline \\end{array}\\ .\n\\end{equation}\nThus $|H^n(10)| = |H^n(01| = F_{n+2}$ and the common prefix\nof $H^n(10)$ and $H^n(01)$ has length $F_{n+2}-2$ is precisely the same as\nthe common block of $H^n(w0)$ and $H^n(w0)$ between $H^n(w)$ and\nthe first difference.\n\nTherefore, $H^n(x)$ and $H^n(y)$\ncoincide for $T_n(w) + F_{n+2}-2$ digits.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-coinc-hn-rho}\nFor $x \\in {\\mathbb K}$ and $n\\in{\\mathbb N}$, $H^{n}(x)$ and $\\rho$ coincide for at least $F_{n+3}-2$ digits if $x\\in [0]$ and for at least $F_{n+2}-2$ digits if $x\\in[1]$. \n\\end{corollary}\n\n\\begin{proof}\nIf $x \\in [0]$, then, by Lemma~\\ref{lem-prop-stopbloc}, \n$H^n(x)$ coincides with $H^n(\\rho) = \\rho$\nfor at least $T_n(0) + F_{n+2}-2$ digits.\nBut $T_n(0) = |H^n(0)| = F_{n+1}$, so $T_n(0) + F_{n+2}-2 = F_{n+3}-2$.\n\nIf $x \\in [1]$, then $H(x) \\in [0]$ and the previous argument gives that \n$H^n(x)$ coincides with $H^n(\\rho) = \\rho$\nfor at least $F_{n+2}-2$ digits.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop-bispecialfibo}]\nWe iterate the blocks $0\\cdot01$, $0\\cdot10$ and $1\\cdot01$ under $H$:\\\\\n$$\n\\begin{array}{|cccc|}\\hline 0 & \\cdot & 0 & 1 \\\\ 0 & \\cdot & 1 & 0 \\\\ 1 & \\cdot & 0 & 1 \\\\ \\hline \\end{array}\n\\stackrel{H}{\\longrightarrow}\n\\begin{array}{|cc|}\\hline 0 & 1 \\\\ 0 & 1 \\\\ & 0  \\\\ \\hline \\end{array}\\\n\\begin{array}{c} 0 \\\\0 \\\\ 0 \\end{array}\\\n\\begin{array}{|cc|}\\hline 1 & 0 \\\\ 0 & 1 \\\\  1 & 0 \\\\ \\hline \n\\end{array}\\stackrel{H}{\\longrightarrow}\n\\begin{array}{|ccc|}\\hline \\dots\\!\\!\\! & 1 & 0 \\\\ \\dots\\!\\!\\! & 1 & 0 \\\\ & 0 & 1 \\\\ \\hline \\end{array}\\\n\\begin{array}{ccc} 0 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 0 \\end{array}\\\n\\begin{array}{|cc|}\n\\hline 1 & 0 \\\\ 0 & 1 \\\\  1 & 0 \\\\ \\hline  \n\\end{array}\n\\stackrel{H}{\\longrightarrow} \\cdots\\ , \\\\\n$$\nso the common central block here is bi-special, and it is the same \nas the common block $v$ of $H^n(01)$ and $H^n(10)$ \nof length $F_{n+2}-2$ in the proof of Lemma~\\ref{lem-prop-stopbloc}.\nThus we have found the bi-special word of length $F_{n+2}-2$, and every\nprefix and suffix of $v$ is left and right-special respectively.\nThe fact that these are the only bi-special words can be derived from \nthe Rauzy graph for this Sturmian shift, see\n{\\em e.g.\\ } \\cite[Sec. 1]{arnoux-rauzy}.\nIn their notation, there is a bi-special word of length $k$\nif the two special nodes in the Rauzy graph coincide: $D_k = G_k$.\nThe lengths of the two ``buckles'' of non-special nodes between $D_k = G_k$\nare two consecutive Fibonacci numbers minus one, as follows from the \ncontinued fraction expansion \n$$\n\\gamma=1+\\frac1{1+\\frac1{1+\\ddots}}.\n$$\nTherefore, the complexity satisfies\n$$\nk+1 = p(k) = \\#\\{ \\text{nodes of Rauzy graph of order } k\\}\n= F_n-1 + F_{n-1}-1 + 1,\n$$\nso indeed only the numbers $k = F_{n+1}-2$ can be the lengths of bi-special \nwords.\n\\end{proof}\n\n\n\\subsection{Iterations of the renormalization operator}\n\nThe renormalization operator  for potentials can be rewritten under as \n(recall the definition of $F_{n}^{a}$, $a=0,1$, from \\eqref{eq:Fiboa})\n\\begin{equation}\\label{equ-def-cr}\n\\CR V|_{[a]} = \\sum_{j=0}^{F^{a}_{1}-1} V \\circ \\sigma^j \\circ H|_{[a]}.\n\\end{equation}\nThis general formula may be extended to other substitutions and leads to an expression for $\\CR^{n}V$.  The main result here is Lemma~\\ref{lem:RkV}, where we show that\n\\begin{equation}\\label{eq:RnV}\n(\\CR^n V)(x) = \\sum_{j=0}^{F_{n^{*}}-1} V \\circ \\sigma^j \\circ H^n(x),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:n*}\nn^* = \\begin{cases}\nn+1 & \\text{ if } x \\in [0],\\\\\nn & \\text{ if } x \\in [1].\n\\end{cases}\n\\end{equation}\nThe substitution $H$ solves a renormalization equation of the form \\eqref{equ1-renorm}. If $x=0x_{1}\\ldots$, then $H(x)=01H(x_{1})\\ldots$ and $\\sigma^{2}\\circ H(x)=H\\circ \\sigma(x)$. If $x=1x_{1}\\ldots$ then we simply have \n$\\sigma\\circ H(x)=H\\circ \\sigma(x)$. The renormalization equation is thus more complicated than for the constant length case. We need an expression for iterations of $H$ and $\\sigma$. \n\\begin{lemma}\\label{lem:commute_shift_H}\nGiven $k \\ge 0$ and $a=0,1$, let \n$w = w_1w_2\\dots w_{F^{a}_{k}} = H^k(a)$.\nThen for every $0 \\le i < F^{a}_{k}$ we have\n$$\nH \\circ \\sigma^i \\circ H^k|_{[a]} = \\sigma^{| H(w_1\\dots w_i)|} \\circ H^{k+1}|_{[a]}.\n$$\n\\end{lemma}\n\\begin{proof}\nFor $k = 0$ this is true by default and for $k= 1$, this is precisely\nwhat is done in the paragraph before the lemma.\nLet us continue by induction, assuming that the statement is true for $k$.\nThen $\\sigma^i$ removes the first $i$ symbols of $w = H^k(a)$,\nwhich otherwise, under $H$, would be extended to a word of\nlength $|H(w_1\\dots w_i)|$. We need this number of shifts\nto remove $H(w_1\\dots w_i)$ from $H([w]) = H^{k+1}([a])$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:RkV}\nFor every $k \\ge 0$ and $a=0,1$, we have\n$$\n\\CR^kV|_{[a]} = S_{F^{a}_{k}}V \\circ H^k|_{[a]},\n$$\nwhere $S_nV = \\sum_{i=0}^{n-1} V \\circ \\sigma^i$ denotes the $n$-th ergodic sum.\n\\end{lemma}\n\n\\begin{proof}\nFor $k = 0$ this is true by default. For $k = 1$, this follows by the definition of the renormalization operator $\\CR$.\nLet us continue by induction, assuming that the statement is true for $k$.\nWrite $w = H^k(a)$ and $t_i = |H(w_i)| = F_{w_i}$.\nThen\n\\begin{eqnarray*}\n\\CR^{k+1}V|_{[a]} &=& (\\CR V) \\circ S_{F^a_k}V \\circ H^k|_{[a]} \\hskip 1cm \\text{\\small (Induction assumption)}\\\\\n &=& \\sum_{i=0}^{F^{a}_{k}-1}  \\left( \\sum_{j=0}^{t_i-1} V \\circ \\sigma^j \\circ H \\right)  \\sigma^i\\circ H^k|_{[a]}\\hskip 1cm \\text{\\small (by formula \\eqref{equ-def-cr})}\\\\\n &=& \\sum_{i=0}^{F^{a}_{k}-1} \\left( \\sum_{j=0}^{t_i-1} V \\circ \\sigma^{j+|H(w_1\\dots w_i)|} \\circ H \\right) \\circ H^k|_{[a]}  \\hskip 1cm\\text{\\small (by Lemma~\\ref{lem:commute_shift_H})}\\\\\n &=& \\sum_{l=0}^{F^{a}_{k+1}-1} V \\circ \\sigma^l \\circ H^{k+1}|_{[a]}, \n\\end{eqnarray*}\nas required.\n\\end{proof}\n\n\n\n\n\n\\subsection{Special words are sources of accidents}\nOverlaps of  $\\rho$ with itself are strongly related to bi-special words. They are of prime importance to determine the fixed points of $\\CR$ and their \nstable leaves, see {\\em e.g.\\ } formula \\eqref{equ-crkV} below. Dynamically, they correspond to what we call {\\em accident} in the time-evolution of the distance between the orbit and ${\\mathbb K}$.  \n\n\n\n\nFor most $x$ close to ${\\mathbb K}$, $d(\\sigma(x),{\\mathbb K}) = 2d(x,{\\mathbb K})$, but\nthe variation of $d(\\sigma^{j}(x),{\\mathbb K})$ is not always monotone with respect to $j$. When it decreases, it generates an accident:\n\\begin{definition}\\label{def-accident}\nLet $x\\in\\S$ and $d(x,{\\mathbb K})=2^{-n}$. If $d(\\sigma(x),{\\mathbb K})\\le 2^{-n}$, we say that we have an {\\em accident} at $\\sigma(x)$. \nIf there is an accident at $\\sigma^{j}(x)$, then we shall simply say we have an accident at $j$. \n\\end{definition}\n\nThe next lemma allows us to detect accidents. \n\\begin{lemma}\\label{lem-accident-bispecial}\nLet $x=x_{0}x_{1}\\ldots$ coincide with some $y\\in{\\mathbb K}$ for $d$ digits. Assume that the first accident occurs at $b$. Then $x_{b}\\ldots x_{d-1}$ is a bi-special word in $\\CA_{{\\mathbb K}}$. Moreover, the word $x_{0}\\ldots x_{d-1}$ is not right-special. \n\\end{lemma}\n\\begin{proof}\nBy definition of accident, there exists $y$ and $y'$ in ${\\mathbb K}$ such that $d(x,{\\mathbb K})=d(x,y)$ and $d(\\sigma^{b}(x),{\\mathbb K})=d(\\sigma^{b}(x),y')$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.7]{accident.pdf}\n\\caption{Accident and bi-special words}\n\\label{fig-accident}\n\\end{center}\n\\end{figure}\nFigure~\\ref{fig-accident} shows that the word $x_{b}\\ldots x_{d-1}$ is bi-special because its two extensions $y$ and $y'$ in ${\\mathbb K}$ \nhave different suffix and prefix for this word. \n\nIt remains to prove that $x_{0}\\ldots x_{d-1}$ is not right-special. If it was, then $x_{0}\\ldots x_{d-1}x_{d}=y_{0}\\ldots y_{d-1}\\overline{y_{d}}$ \nwould a ${\\mathbb K}$-admissible word, thus $d(x,{\\mathbb K})\\le 2^{-(d+1)}\\neq 2^{-d}$. \n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{theo-fixedpoint}}\\label{sec-prooftheofix}\n\\subsection{Control of the accidents under iterations of $\\CR$}\nNext we compute $\\CR^{n}V$ and show that accidents do not crucially\nperturb the Birkhoff sum involved. This will follow from Corollaries~\\ref{coro-dHnK} and \\ref{cor-coinc-hn-rho}.\n\nNote that Lemma~\\ref{lem-prop-stopbloc} shows that $H$ is one-to-one. The next proposition explains the relation between the attractor ${\\mathbb K}$ and its image by $H$. \n\n\\begin{proposition}\\label{prop-K-H}\nThe subshift ${\\mathbb K}$ is contained in $H({\\mathbb K}) \\cup \\sigma\\circ H({\\mathbb K})$. More precisely, if $[0]\\cap {\\mathbb K} \\subset H({\\mathbb K})$ and $x\\in[1]\\cap {\\mathbb K} \\subset \\sigma\\circ H({\\mathbb K})$.  \n\\end{proposition}\n\n\\begin{proof}\nFirst note that Lemma~\\ref{lem-prop-stopbloc} shows that $H$ is one-to-one. We also recall that the word $11$ is forbidden in ${\\mathbb K}$. Hence, each digit $1$\nin $x=x_{0}x_{1}x_{2}\\ldots\\in {\\mathbb K}$ is followed and preceded by a digit $0$ (unless the $1$ is in first position). \n\nAssuming $x_{0}=0$, we can unique split $x$ into blocks of the form \n$0$ and $01$.\nIn this splitting, we replace \neach single $0$ by $1$ and each pair $01$ by $0$. \nThis produces a new word, say $y$, and by construction, $H(y)=x$. \nThis operation is denoted by $H^{-1}$. \nIt can be used on finite words too, provided that the last digit is $1$. \nIf $x_{0}=1$, we repeat the above construction with $0x$, and $x=\\sigma\\circ H(y)$. \n\nIt remains to prove that $y\\in{\\mathbb K}$. \nFor every $x\\in{\\mathbb K}$, there is a sequence $k_{n}\\to\\infty$ \nsuch that $\\sigma^{k_{n}}(\\rho)\\to x$. Assume again that $x_0 = 0$.\nThen we can find a sequence $l_n \\sim k_n\/\\gamma$ such that \n$H \\circ \\sigma^{l_n}(\\rho)=\\sigma^{k_{n}}(\\rho)$.\nTherefore $\\lim_n \\sigma^{l_n}(\\rho) \\in {\\mathbb K}$, and this limit is indeed the sequence\n$y$ that satisfies $H(y) = x$.\nFinally, for $x_0 = 1$, we repeat the argument with $0x$.\n\\end{proof}\n\n\n\\begin{corollary}\\label{coro-dHnK}\nIf $d(x,{\\mathbb K})=d(x,y)$ with $y\\in {\\mathbb K}$, then $d(H^{n}(x),{\\mathbb K})=d(H^{n}(x),H^{n}(y))$ for $n\\ge 0$.\n\\end{corollary}\n\\begin{proof}\nWrite $x=wa$ and $y=w\\overline a$ where $a$ is an unknown digit and $\\overline a$ its opposite. Note that $H^{n}(x)$ starts with $0$ for any $n\\ge 1$.\nAssume that there is some $z\\in {\\mathbb K}$ such that $d(H(x),z)<d(H(x),H(y))$.\n\n\\paragraph{{\\it Case 1:}} $x=w0\\ldots$ and $y=w1\\ldots$. Necessarily, $y=w10$. Therefore $H(x)=H(w)01\\ldots$ and $H(y)=H(w)001\\ldots$. By assumption, $z$ coincide with $H(x)$ longer than $H(y)$, which shows that $z$ starts as \n$z=H(w)01\\ldots$\nConsequently $H^{-1}(z)=w0\\dots$ and this contradicts that $d(x,{\\mathbb K})=d(x,y)$. \n\n\\paragraph{{\\it Case 2:}} $x=w1b\\ldots$ and $y=w0\\ldots$. Then $H(x)=H(w)00\\ldots$  (since $H(b)$ starts with $0$ regardless what $b$ is)\nand $H(y)=H(w)01\\ldots$ Again $z$ coincides with $H(x)$ longer than $H(y)$ and thus $z$ starts as $H(w)00$. The $0$ before last position is necessarily a single zero for the $H^{-1}$-procedure and thus $H^{-1}(z)$ coincide with  $x$ for longer than $y$. This is a contradiction.\n\nConsequently for both cases we have shown $d(H(x),{\\mathbb K})=d(H(x),H(y))$. \nThe result follows by induction. \n\\end{proof}\n\nWe recall that by \\eqref{eq:RnV}, $\\CR^{n}V$ is given by a Birkhoff sum, of\n$F_{n^*}$ where $n^* = n+1$ or $n$ as in \\eqref{eq:n*}.\nTo compute $(\\CR^{n}V)(x)$, we need an estimate for $d(\\sigma^{j}(H^{n}(x)),{\\mathbb K})$, for $0 \\le j\\le F_{n+1}-1$ or $0 \\le j\\le F_{n}-1$. The key point is that no accident can occur for these $j$. This follows from the next lemma. \n\n\\begin{lemma}\\label{lem-no-accident}\nThe sequence $H^n(x)$ has no accident in the first $F_{n^*}$ entries.\n\\end{lemma}\n\n\\begin{proof}\nWe give the proof for $x \\in [1]$, so $n^* = n$. The proof for $x \\in [0]$ is analogous.\nBy Corollary~\\ref{cor-coinc-hn-rho}, $H^n(x)$ coincides for at least $F_{n+2}-2$\ndigits with $\\rho$.\nIf an accident happens in the first $F_n$ digits, say at entry $0 \\le j < F_n$,\nthen by Lemma~\\ref{lem-accident-bispecial}, a bi-special word starts\nat $j$, which by Proposition~\\ref{prop-bispecialfibo} is a suffix\nof $\\rho$ of length $F_m-2$ for some $m$.\nSince we have an accident, $j+F_m-2 \\ge F_{n+2}-1$, so $m > n+1$.\n\n\n\n\nHence $\\rho_{0}\\ldots\\rho_{F_{n+2}-1}$ can be written as $BBB'$ where $B$ is the suffix of $\\rho$ of length $j$ and $B'$ is a suffix of $\\rho$ of length $\\ge |B|\/\\gamma$.\nClearly $B$ starts with $0$. We can split it uniquely into blocks $0$ and $01$, and $B$ fits an integer number of such blocks, because if\nthe final block would overlap with the second appearance of $B$, then $B$ \nwould start with $1$, which it does not.\n\nTherefore we can perform an inverse substitution $H^{-1}$, for  each block $B$ and also for $B'$ because we cal globally do $H^{-1}$ for $\\rho_{0}\\ldots\\rho_{F_{n+2}-1}$. We  find $H^{-1}(BBB') = CCC'$ which  has the same characteristics.\nRepeating this inverse iteration, we find that $\\rho$ starts with $0101$, or with $00$, \na contradiction.\n\\end{proof}\n\nLet $N(x,n)$ be the integer such that $2^{-N(x,n)} = d(H^{n}(x),{\\mathbb K})$.\nBy the previous lemma $d(\\sigma^{j}(H^{n}(x)){\\mathbb K})=2^{-(N(x,n)-j)}$\nfor every $j<F_{n^{*}}$. For the largest value $j = F_{N^*}$,\nwe have  $d(\\sigma^{j}(H^{n}(x)){\\mathbb K}) = 2^{-(T_{n}+F_{n+2}-2-F_{n^{*}})}$.\nTherefore, if $g$ is the $\\alpha$-density function for $V$, then we obtain\n\\begin{equation}\\label{equ-crkV}\n(\\CR^{n}V)(x)=\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{(N(x,n)-j)^{\\alpha}} +o\\left(\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{(N(x,n)-j)^{\\alpha}}\\right).\n\\end{equation}\n\n\\subsection{Proof of Theorem~\\ref{theo-fixedpoint}}\n\\subsubsection{$\\widetilde V$ is a fixed point}\nA simple computation shows that $\\CR$ fixes $\\widetilde V$ from\n\\eqref{eq:tildeV}. Assume $x\\notin{\\mathbb K}$ is such that $\\widetilde\\kappa_{0}(x)=n$ and $\\widetilde\\kappa_{1}(x)=m$ (see the definition of $\\widetilde\\kappa_{a}$ above  the statement of Theorem~\\ref{theo-fixedpoint}).\nThen, by Lemmas~\\ref{lem-lengthHn} and \\ref{lem-prop-stopbloc}, \nand the fact that $H(x)$ starts with $0$, we get\n\\begin{eqnarray*}\n\\widetilde\\kappa_{0}(H(x)) = n+m+1\\ & \\qquad &\\widetilde\\kappa_{0}(\\sigma \\circ H(x))  =  n+m,\\\\\n\\widetilde\\kappa_{1}(H(x))  =  n \\qquad \\qquad & \\qquad & \\widetilde\\kappa_{1}(\\sigma \\circ H(x)) =  n.\n\\end{eqnarray*}\n \\begin{itemize}\n\\item If $x$ starts with $0$, then $H(x)$ starts with $01$ and \n\\begin{eqnarray*}\n(\\CR\\widetilde V)(x)&=& \\widetilde V(H(x))+\\widetilde V\\circ \\sigma(H(x))\\\\\n&=& \\log\\left(\\frac{(n+m+1)+\\frac1\\gamma n+\\gamma}{(n+m+1)+\\frac1\\gamma n+\\gamma-1}\\right)+\\log\\left(\\frac{\\gamma(n+m)+ n+\\gamma^{2}}{\\gamma(n+m)+ n+\\gamma^{2}-1}\\right)\\\\\n&=& \\log\\left(\\frac{(n+m+1)+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma}\\right)+\\log\\left(\\frac{n+m+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma-\\frac1\\gamma}\\right)\\\\\n&=& \\log\\left(\\frac{n+m+1+\\frac1\\gamma n+\\gamma}{n+m+\\frac1\\gamma n+\\gamma-\\frac1\\gamma}\\right)\\\\\n&=& \\log\\left(\\frac{n(1+\\frac1\\gamma)+m+\\gamma+1}{n(1+\\frac1\\gamma)+m+\\gamma-\\frac1\\gamma}\\right)\\qquad\\qquad  \\text{\\small since } \\gamma^{2}=\\gamma+1\\\\\n&=& \\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma(\\gamma-1)}\\right)=\\log\\left(\\frac{n+\\frac1\\gamma m+\\gamma}{n+\\frac1\\gamma m+\\gamma-1}\\right)=\\widetilde V(x).\n\\end{eqnarray*}\n \n\\item If $x$ starts with $1$, then $H(x)$ starts with $0$ and \n\\begin{eqnarray*}\n(\\CR\\widetilde V)(x)&=& \\widetilde V(H(x))=\\log\\left(\\frac{(n+m+1)+ \\frac1\\gamma n+\\gamma}{(n+m+1)+\\frac1\\gamma n+\\gamma-1}\\right)\\\\\n&=&\n\\log\\left(\\frac{\\gamma(n+m+1)+ n+\\gamma^{2}}{\\gamma(n+m+1)+ n+\\gamma^{2}-\\gamma}\\right)\\\\\n&=&\n\\log\\left(\\frac{n(\\gamma+1)+\\gamma m+\\gamma+\\gamma^{2}}{n(\\gamma+1)+\\gamma m+\\gamma^{2}}\\right)\\\\\n&=& \\log\\left(\\frac{\\gamma^{2}n+\\gamma m+\\gamma^{3}}{\\gamma^{2}n+\\gamma m+\\gamma^{2}}\\right)\\\\\n&=&\n\\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma}\\right)\n=\\log\\left(\\frac{\\gamma n+m+\\gamma^{2}}{\\gamma n+m+\\gamma^{2}-1}\\right) =\\widetilde V(x).\n\\end{eqnarray*}\n\\end{itemize}\n\n\n\\subsubsection{A Toeplitz summation}\nNext, we consider $V\\in\\CX_{1}$. We shall see in the proof that the convergence for these $V$ automatically implies that for $V\\in\\CX_{\\alpha}$ with $\\alpha\\ne1$, the same computations work but with either too light or too heavy denominators. \n\nIn the same spirit, we shall see that $}%{\\displaystyle \\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{N(x,n)-j}$ actually converges, which immediately yields that  $}%{\\displaystyle o\\left(\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{N(x,n)-j}\\right)$  converges to $0$. We can thus forget any term of that kind. \n\nRecall that by Lemmas~\\ref{lem-lengthHn} and \\ref{lem-prop-stopbloc}\nand Corollary~\\ref{coro-dHnK} we have\n$$\nN(x,n) := \\log_2 d(H^n(x), {\\mathbb K}) = T_n+F_{n+2}-2 \\quad \\text{ for } \\\nT_{n}:=F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x).\n$$\nWe thus have to compute the limit of \n$$\n\\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)+F_{n+2}-(j+2)}\n$$\nas $n^* \\to +\\infty$, where $n^{*} = n+1$ if $x \\in [0]$ and $n^* = n$ if $x\\in[1]$. Moreover $g$ is a non-negative continuous function, hence uniformly continuous. Furthermore, for $y \\in {\\mathbb K}$ closest to $x$,\nthe point $\\sigma^{F_{n^{*}}} \\circ H^{n}(x)$ coincides with $\\sigma^{F_{n^{*}}} \\circ H^{n}(y)$ for at least $F_{n}-2$ digits.\nThere exists a sequence  $\\varepsilon_{n}\\downarrow 0$ such that \n$$\\left|g\\circ \\sigma^{j}(H^{n}(x))-g\\circ \\sigma^{j}(H^{n}(y))\\right|\\le \\varepsilon_{n},$$\nfor every $j\\le F_{n^{*}}-1$.\n\nFinally, Binet's formula $F_{n+1}-\\gamma F_n = \\sqrt{5} \\gamma^{-(n-1)}$ shows that \n\\begin{eqnarray*}\nF_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)\\!&+&\\!F_{n+2}-(j+2)\\\\\n&=& F_{n+1}(\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma-\\frac{j}{F_{n+1}})(1+\\varepsilon'_{n}) \\\\\n&=& F_{n}(\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}-\\frac{j}{F_{n}})(1+\\varepsilon''_{n}),\n\\end{eqnarray*}\nwhere $\\varepsilon'_{n}$ and $\\varepsilon''_{n}$ tend to $0$ as $n \\to +\\infty$. \n\nCombining $\\varepsilon$, $\\varepsilon'$ and $\\varepsilon''$ in a single $o(1)$, \nwe can rewrite the above equalities to \n\\begin{equation}\\label{equ1-toeplitz}\n \\sum_{j=0}^{F_{n^*}-1}\\frac{g \\circ \\sigma^j \\circ H^n(x)}{F_{n+1}\\widetilde\\kappa_{0}(x)+F_{n}\\widetilde\\kappa_{1}(x)+F_{n+2}-(j+2)}=\n\\frac{1+o(1)}{F_{n^{*}}} \\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ\\sigma^{j}\\circ H^{n}(y)}{X_{n}-\\frac{j}{F_{n^{*}}}},\n\\end{equation}\nwhere $X_{n}=\\widetilde\\kappa_{0}(x)+\\frac1\\gamma\\widetilde\\kappa_{1}(x)+\\gamma$ if $x \\in [0]$ \nand $X_{n}=\\gamma\\widetilde\\kappa_{0}(x)+\\widetilde\\kappa_{1}(x)+\\gamma^{2}$ if $x\\in[1]$. \n\n\\subsubsection{Convergence of the weighted sum in \\eqref{equ1-toeplitz}}\nThe reader can verify that we are here considering a Toeplitz summation method, with a regular matrix (see \\cite[ Definition 7.5]{kuipers-niederreiter}  and \\cite{weber-crass}), up to a renormalization factor, which is the limit of \n$$\n\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac1{X_{n}-\\frac{j}{F_{n^{*}}}}.\n$$\nThis expression is a Riemann sum, and converges to $\\widetilde V(x)$\nas $n^* \\to \\infty$. \n\nFrom \\cite{weber-crass} we have that the M\\\"uller (see \\cite{muller}) criterion applies, and we get that for $\\mu_{{\\mathbb K}}$-almost every $z\\in{\\mathbb K}$, \n\\begin{equation}\n\\label{equ-cv-toepli}\n\\lim_{n\\to+\\infty}\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(z)}{X_{n}-\\frac{j}{F_{n^{*}}}}=\\widetilde V(x)\\int g\\,d\\mu_{{\\mathbb K}}.\n\\end{equation}\n\nNevertheless, our expression in \\eqref{equ1-toeplitz} is different, because the point $z$ we are considering is $H^{n}(y)$ and thus depends on $n$.\nA priori, this may generates fluctuations in the convergence, but\nwe prove here that this is not the case.\n\nThe main argument is that $({\\mathbb K},\\sigma)$ is uniquely ergodic. We claim that this implies that the convergence in \\eqref{equ-cv-toepli} is uniform in $z$. \n\nIndeed, if it is not uniform, we can find $\\varepsilon > 0$ and \na sequence of $z_{n}$ such that for every $n$, \n$}%{\\displaystyle |\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(z_{n})}{X_{n}-\\frac{j}{F_{n^{*}}}} - \\widetilde V(x)\\int g\\,d\\mu_{{\\mathbb K}}| > \\varepsilon$ \nfor every $n$. \nThen any accumulation point $\\mu_{\\infty}$ of the family of measures \n$$\n\\mu_{n}:=\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{1}{X_{n}-\\frac{j}{F_{n^{*}}}}\\delta_{\\sigma^{j}(z_{n})}\n$$\nis $\\sigma$-invariant (because $F_{n^{*}}\\to+\\infty$), supported on ${\\mathbb K}$, and \n$\\int g\\,d\\mu_{\\infty}\\neq \\int g\\,d\\mu_{{\\mathbb K}}$. This would contradict the unique ergodicity for $({\\mathbb K}, \\sigma)$. \n\nTherefore, the convergence in \\eqref{equ-cv-toepli} is uniform in $z$ and this shows that \n$$\n\\frac1{F_{n^{*}}}\\sum_{j=0}^{F_{n^{*}}}\\frac{g\\circ \\sigma^{j}(H^{n}(y))}{X_{n}-\\frac{j}{F_{n^{*}}}} \\to \\widetilde V(x) \\cdot \\int g\\,d\\mu_{{\\mathbb K}}.\n$$ \nThis finishes the proof of Theorem~\\ref{theo-fixedpoint}.\n\n\n\\section{Proof of Theorem~\\ref{theo-pt}}\\label{sec-proffthpt}\n\n\\subsection{The case \\boldmath $-\\log \\frac{n+1}{n}$ \\unboldmath}\n\\label{subsec-logcase}\nWe first consider the potential $\\varphi(x) = -\\log \\frac{n+1}{n}$ if $d(x, {\\mathbb K}) = 2^{-n}$, leaving the general potential in $\\CX_1$ for later.\n\n\\subsubsection{Strategy, local equilibria}\nFix some cylinder $J$ such that the associated word, say $\\omega_{J}$, does not appear in $\\rho$ (as {\\em e.g.\\ } 11). We follow the induction method presented in \\cite{leplaideur-synth}. Let $\\tau$ be the first return time into $J$ (possibly $\\tau(x)=+\\infty$), and consider the family of transfer operators \n\\begin{eqnarray*}\n\\CL_{Z,\\beta}:\\psi&\\mapsto& \\CL_{Z,\\beta}(\\psi)\\\\\nx&\\mapsto& \\CL_{Z,\\beta}(\\psi)(x):=\\sum_{n=1}^{+\\infty}\\sum_{\\stackrel{y\\in J\\ \\tau(y)=n}{ \\sigma^{n}(y)=x}}e^{\\beta \\cdot (S_{n}\\varphi)(y)-nZ}\\psi(y),\n\\end{eqnarray*}\nwhich acts on the set of continuous functions $\\psi:J\\to{\\mathbb R}$.  \nFollowing \\cite[Proposition 1]{leplaideur-synth}, for each $\\beta$ there exists $Z_{c}(\\beta)$ such that $\\CL_{Z,\\beta}$ is well defined for every $Z>Z_{c}(\\beta)$. \nBy \\cite[Theorem 1]{leplaideur-synth}, $Z_{c}(\\beta)\\ge 0$ because the pressure of the dotted system \n(which in the terminology of \\cite{leplaideur-synth} is the system restricted to the trajectories that avoid $J$)\nis larger (or equal) than the pressure of ${\\mathbb K}$ which is zero. \n\nWe shall prove\n\\begin{proposition}\n\\label{prop-spectral&zc}\nThere exists $\\beta_{0}$ such that $\\CL_{0,\\beta}({1\\!\\!1}_{J})(x)<1$ for every $\\beta>\\beta_{0}$ and $x\\in J$.\n\\end{proposition}\nWe claim that if Proposition~\\ref{prop-spectral&zc} holds, then\n\\cite[Theorem 4]{leplaideur-synth} proves that $\\CP(\\beta)=0$ for every $\\beta>\\beta_{0}$, and $\\mu_{{\\mathbb K}}$ is the unique equilibrium state for $\\beta \\varphi$. \n\nTo summarize \\cite{leplaideur-synth} (and adapt it to our context), the \npressure function satisfies (see Figure~\\ref{fig-graphes-press}\\footnote{We will see that $\\CL_{0,\\beta}({1\\!\\!1}_{J})(x)$ is a constant function on $J$.}),\n$$\nZ_{c}(\\beta) \\le \\CP(\\beta) \\le \\max(\\log(\\CL_{0,\\beta}({1\\!\\!1}_{J})),0).\n$$\n\nAs long as $\\CP(\\beta)>0$, there is a unique equilibrium state and it has full support. In particular this shows that the construction does not depend on the choice of $J$. If Proposition~\\ref{prop-spectral&zc} holds, then \n$$Z_{c}(\\beta)=\\CP(\\beta)=\\max(\\log(\\CL_{0,\\beta}({1\\!\\!1}_{J})),0)=0,\n\\text{ for } \\beta>\\beta_{c}$$\nand $\\mu_{{\\mathbb K}}$ is the unique equilibrium state because $\\CL_{0,\\beta}({1\\!\\!1}_{J})<1$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=.5]{graphe1na.pdf}\n\\caption{The Pressure between $Z_{c}(\\beta)$ and $\\log\\lambda_{0,\\beta}:=\\CL_{0,\\beta}({1\\!\\!1}_{J})$}\n\\label{fig-graphes-press}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 1}\\label{subsubsec-upper=series}\nWe reduce the problem to the computation of a series depending on $\\beta$. \nNote that $\\varphi(x)$ only depends on the distance from $x$ to ${\\mathbb K}$. This shows that if $x, x' \\in J$ and $y, y' \\in J$ are such that \n$$y=\\omega x\\ , \\quad \u00a0y'=\\omega x',$$\nwith $\\omega\\in \\{0,1\\}^{n}$, $\\tau(y)=\\tau(y')=n$, then \n$$(S_{n}\\varphi)(y)=(S_{n}\\varphi)(y').$$\nIn other words, $\\CL_{Z,\\beta}({1\\!\\!1}_{J})$ is a constant function, and then equal to the spectral radius $\\lambda_{Z,\\beta}$ of $\\CL_{Z,\\beta}$.  \n\nConsequently, to compute $\\lambda_{Z,\\beta}$, it suffices to compute the sum of all \n$}%{\\displaystyle e^{\\beta \\cdot (S_{n}\\varphi)(\\omega)-nZ}$, where $\\omega$ is a word of length $n+|\\omega_{J}|$, starting and finishing as $\\omega_{J}$. Such a word $\\omega$ can also be seen as a path of length $n$ starting from $J$ and returning (for the first time) to $J$ at time $n$. \n\nWe split such a path in several sub-paths. We fix an integer $N$ and say that the path is {\\em free} at time  $k$ if $\\omega_{k}\\ldots \\omega_{n-1}\\omega_{J}$ is at distance  larger than $2^{-N}$ to ${\\mathbb K}$. Otherwise, we say that we have an {\\em excursion}. The path is thus split into intervals of free moments and excursions. \nWe assume that $N$ is chosen so large that $0$ is a free moment. This also shows that for every $k\\le n$, $d(\\sigma^{k}(\\omega\\omega_{J}),{\\mathbb K})$ is determined by $\\omega_{k}\\ldots \\omega_{n-1}$. \n\nIf $k$ is a free time, $}%{\\displaystyle \\varphi(\\sigma^{k}(\\omega\\omega_{J}))\\le A_{N}:=-\\log\\left(1+\\frac 1N\\right)$. Denote by $k_{0}$ the maximal integer such that \n$k$ is a free time for every $k\\le k_{0}$. Then $S_{k_{0}+1}\\varphi\\le (k_{0}+1)A_{N}$ and there are fewer than $2^{k_{0}+1}$ such prefixes of length $k_0+1$. \n\nNow, assume that every $j$  for $k_{0}+1\\le j\\le k_{0}+k_{1}$ is an excursion time, and assume that $k_{1}$ is the maximal integer with this property. To the contribution $(S_{k_{0}+1}\\varphi)(\\omega\\omega_{J})$ we must add the contribution $(S_{k_{1}}\\varphi)(\\sigma^{k_{0}+1}(\\omega\\omega_{J}))$ of the excursion. \nThen we have a new interval of free times, and so on. \nThis means that we can compute $\\CL_{0,\\beta}({1\\!\\!1}_{J})$ by gluing together paths with the same decompositions  of free times and excursion times. If we denote by $C_{E}$ the total contribution of all paths with exactly one excursion (and only starting at the beginning of the excursion), then we have \n\\begin{equation}\n\\label{equ1-upperboundCL0}\n\\lambda_{0,\\beta}=\\CL_{0,\\beta}({1\\!\\!1}_{J}) \\le \\sum_{k=1}^{+\\infty}\\left(\\sum_{k_{0}=0}^{+\\infty}e^{(k_{0}+1)(\\beta A_{N}+\\log2)}\\right)^{k+1}C_{E}^k.\n\\end{equation}\nThe sum in $k$ accounts for $k+1$ intervals of free moment with $k$ intervals of excursions times between them. The sum in $k_{0}$ accounts for the possible length $k_{0}+1$ for an interval of free times. \nThese events are maybe not independent but the sum in \\eqref{equ1-upperboundCL0} includes all paths, possible or not, and therefore yields an upper bound. \n\nThe integer $N$ is fixed, and we can take $\\beta$ so large that $\\beta A_{N}<-\\log2$. This shows that the sum in $k_{0}$ in \\eqref{equ1-upperboundCL0}  converges and is as close to $0$ as we \nwant if $\\beta$ is taken sufficiently large. \n\nTo prove Proposition~\\ref{prop-spectral&zc}, it is thus sufficient to prove that $C_{E}$ can be made as small as we want if $\\beta$ increases. \n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 2}\\label{subsubsec-splitCE}\nWe split excursions according to their number of accidents, see Definition~\\ref{def-accident}. \nLet $x$ be a point at a beginning of an excursion. \n \nLet $B_{0}:=0=b_{0}$, $B_1 := b_1>b_{0},\\ B_2 := b_1+b_2>b_{1},\\ \nB_3 := b_1+b_2+b_3, \\dots ,\nB_M := b_1+b_2+ \\dots + b_M$\nbe the times of accidents in the excursion.\nThere is $y_0 \\in {\\mathbb K}$ such that $x$ shadows $y_0$ at the beginning\nof the excursion, say for $d_0$ iterates.\nLet $y_i \\in {\\mathbb K}$, $i = 1, \\dots, M$, be the points that $x$ starts to shadow \nat the $i$-th accident, for $d_i$ iterates.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{excursion.pdf}\n\\caption{Accidents during an excursion.}\n\\label{fig-excursion}\n\\end{center}\n\\end{figure}\n\nThen by Lemma~\\ref{lem-accident-bispecial}, $x_{b_{i+1}}\\ldots x_{d_{i}}$ is bi-special and by Proposition~\\ref{prop-bispecialfibo}, $d_i-b_{i+1} = F_{n_{i+1}}-2$ for some $n_{i+1}$. \n\n\\begin{remark}\\label{rem-yispe}\nWe emphasize that the first $d_i$ entries of $y_i$\ndo not form a special word. Indeed, it is neither right-special (due to\nLemma~\\ref{lem-accident-bispecial}) nor left-special, because otherwise there would be an accident earlier. \n\\hfill $\\blacksquare$\\end{remark}\n\nIf there are $M+1$ accidents (counting the first as 0), the ergodic sums  for $\\varphi$ are \n\\begin{eqnarray*}\n(S_{b_{i+1}}\\varphi)(\\sigma^{B_{i}} (x) ) &=& \n \\sum_{k=0}^{b_{i+1}-1}\\varphi \\circ \\sigma^{B_{i}+k} (x)\\\\\n &=& \n \\sum_{k=0}^{b_{i+1}-1} -\\log \\frac{d_{i}+1-k}{d_{i}-k} \\\\\n&=& -\\log \\frac{d_{i}+1}{d_{i}+1-b_{i+1}} = -\\log(1+  \\frac{b_{i+1}}{d_i+1-b_{i+1}}),\n\\end{eqnarray*}\nfor $0\\le i\\le M-1$, while the ergodic sum of the tail of the excursion is\n\n\\begin{equation}\n\\label{equ-estiEM+1}\n(S_{d_M}\\varphi)(\\sigma^{B_M} (x)) =  \n\\sum_{k=0}^{d_M-1} \\varphi \\circ \\sigma^{B_M+k}( x) =\n-\\log \\frac{d_M+1}{N+1}.\n\\end{equation}\n\nWe set $}%{\\displaystyle \\mathbf{ e}_{i} := e^{\\beta \\cdot (S_{b_{i}}\\varphi)(\\sigma^{B_{i-1}} (x) )}$ for $i=1\\ldots M$ and $}%{\\displaystyle \\mathbf{ e}_{M+1} := e^{\\beta \\cdot (S_{d_M}\\varphi)(\\sigma^{B_M} (x))}$. \nComputing $C_{E}$, we can order excursions according to their number of accidents ($M+1$) and then according to the contribution of each accident. Let $E_{i}$ stand for the total contribution of all possible $\\mathbf{ e}_{i}$'s \nbetween accidents $i-1$ and $i$. \nThen\n\\begin{equation}\\label{equ-defCE}\nC_{E}=\\sum_{M=0}^{+\\infty}\\prod_{i=1}^{M+1}E_{i}.\n\\end{equation}\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 3}\\label{subsubsec-computCE}\nLet us now find an upper bound for $E_{i}$. \nBy definition, $E_{i}$ is the sum over the possible $d_{i-1}$ and $b_{i}$ of $\\mathbf{ e}_{i}$. \n\nRecall $d_{i-1}-b_{i}=F_{n_{i}}-2$, so $b_{i}$ and $F_{n_{i}}$\ndetermine $d_{i-1}$. \nThe key idea is that $F_{n_{i}}$ and $F_{n_{i+1}}$ determine the possible values of $b_{i}$. \nThis implies that $E_{i}$ can be written as an expression over the $F_{n_{i}}$ and $F_{n_{i+1}}$. \n\n\\medskip\n$\\bullet$ For $2\\le i\\le M$ each $\\mathbf{ e}_{i}$ depends on $F_{n_{i}}$ and $b_{i}$. \nLet us show that  for $2\\le i\\le M$, $b_{i}$ depends on $n_{i}$ and $n_{i-1}$. \nIndeed, the sequence $y_{i} \\in {\\mathbb K}$ coincides for\n$F_{n_{i}}-2$ initial symbols with $\\rho$, and from entry $b_{i+1}$ has another\n$d_{i} - b_{i+1} = F_{n_{i+1}}-2$ symbols in common with the head of $\\rho$,\nbut differs from $x_{B_{i}+ d_i}$ at entry $d_i$, see Figure~\\ref{fig-excursion}.\nThus we need to find all the values of $d_i > F_{n_i}-2$ such that\n$\\rho_0 \\dots \\rho_{d_i-1}$ ends the bi-special word $\\rho_0 \\dots \\rho_{F_{n_{i+1}-3}}$ but is itself not bi-special.\nThe possible starting positions of this appearance\nof $\\rho_0 \\dots \\rho_{F_{n_{i+1}-3}}$ are the required numbers $b_{i+1}$.\n\n\\begin{lemma}\\label{lem:bij}\nLet us denote by $b_{i+1}(j)$, $j \\ge 1$, the $j$-th value that\n $b_{i+1}$ can assume. Then \n\\begin{equation}\\label{equ-estibj}\nb_{i+1}(j)\\ge\\max(F_{n_{i}}-F_{n_{i+1}},F_{n_{i}-1})+j F_{n_{i+1}-2}.\n\\end{equation}\n\\end{lemma}\n\nThis will allow us to find an upper bound for $E_{i}$ for $1\\le i\\le M-1$\nlater in this section. \n\n\\begin{proof}\nWe abbreviate the bi-special words $L_k = \\rho_0 \\dots \\rho_{F_k-3}$\nfor $k \\ge 4$. \nFor the smallest value $d_i \\ge F_{n_i}-2$ so that\n$\\rho_0 \\dots \\rho_{d_i-1}$ ends in (but is not identical to) a block $L_{n_{i+1}}$,\nthis block starts at entry:\n$$\nb_{i+1}(0) =\n\\begin{cases}\nF_{n_i}-F_{n_{i+1}} &\\text{ if } n_{i+1} < n_i \\text{ and } n_i-n_{i+1} \\text{ is even,}\\\\\nF_{n_i}-F_{n_{i+1}-1} &\\text{ if } n_{i+1} < n_i  \\text{ and } n_i-n_{i+1} \\text{ is odd,}\\\\\nF_{n_{i+1}-1} &\\text{ if } n_{i+1} \\ge n_i.\n\\end{cases}\n$$\nHowever, if $n_{i+1} < n_i$ then $d_i = F_{n_i}-2$ and if \n $n_{i+1} \\ge n_i$ then $d_i = F_{n_{i+1}+1}-2$ in this case, and thus\n$\\rho_0 \\dots \\rho_{d_i-1}$ is right-special, contradicting\nLemma~\\ref{lem-accident-bispecial}.\nTherefore we need to wait for the next appearance of $L_{n_{i+1}}$.\nFor the Rauzy graph of the Fibonacci shift, the bi-special word $L_k$\nis the single node connecting loops of length $F_{k-1}$ and $F_{k-2}$,\nsee \\cite[Section 1]{arnoux-rauzy}. Therefore the gap between two \nappearances of $L_k$ is always $F_{k-2}$ or $F_{k-1}$.\nThis gives $b_{i+1}(j+1) \\ge b_{i+1}(j) + F_{n_{i+1}-2}$ for all $j \\ge 0$\nand \\eqref{equ-estibj} follows.\n\\end{proof}\n\n$\\bullet$ For $i=1$, formula \\eqref{equ-estibj} can be applied, if we introduce the quantity $n_{0}$, coinciding with the overlap of the end of the previous \n``fictitious'' word, say $y_{-1}$. The point is that $y_{0}$ is the ``beginning'' of the excursion, thus the first accident. Then, $F_{n_{0}}\\le N$ and $F_{n_{1}}>N$, which yields $n_{0}<n_{1}$. Therefore $b_{1}=F_{n_{0}}-2+}%{\\displaystyle \\frac{j}\\gamma(F_{n_{1}}-2)$ with $j\\ge 0$. \n\n$\\bullet$ The estimation \n$$\nE_{M+1} = \\sum_{d \\ge 1} e^{-\\beta \\log(\\frac{F_{N_m}+d}{N+1})}\n= \\sum_{d \\ge 1} \\left( \\frac{F_{N_m}+d}{N+1} \\right)^{-\\beta}\n\\le \\frac{N+1}{\\beta-1} \\left( \\frac{F_{N_m}}{N+1} \\right)^{1-\\beta}\n$$ \nfollows from by \\eqref{equ-estiEM+1}, with $d_M=F_{n_M}+d$ and $d\\ge 1$. \n\n\\medskip\nRecall that within excursions, all $F_{n_{j}} \\ge N+1$\nfor all $j$, where $N$ can be chosen as large as we want.\nBy Binet's formula $F_n = \\frac{1}{\\sqrt{5}}(\\gamma^{n+1}-(-1\/\\gamma)^{n+1})$, \nwe can replace $F_{n}$ by $}%{\\displaystyle\\frac{\\gamma^{n+1}}{\\sqrt 5}$, and  \nwe can also treat the quantities $-1$ as negligible compared to $\\gamma^{n_i}$.  \nTherefore, assuming that $\\beta > 1$, we obtain\n\\begin{eqnarray*}\nE_{i} &=& \\sum_{j \\ge 1} \ne^{}%{\\displaystyle-\\beta  \\log\\left(1 +  \\frac{ \\max(F_{n_{i}}-F_{n_{i+1}},F_{n_{i}-1}) + j F_{n_{i+1}-2}}{F_{n_{i+1}}-1}\\right)} \\\\\n&\\simeq& \\sum_{j \\ge 1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )+ j\/\\gamma^2\\right)^{-\\beta}\\\\\n&\\le& \\frac{\\gamma^2}{\\beta-1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}.\n\\end{eqnarray*} \nfor $2\\le i\\le M$\n\nLet $P\\approx }%{\\displaystyle \\frac{\\log\\frac{N}{\\sqrt5}}{\\log\\gamma}$ be the largest integer $n$ such that $F_{n}\\le N$. Then \\eqref{equ-defCE} yields \n\\begin{align}\\label{eq:est0}\n\\nonumber C_{E}\\le & \\sum_{M=0}^{+\\infty}\\left(\\frac{\\gamma^2}{\\beta-1}\\right)^{M}\\frac{(N+1)}{\\beta-1} \\ \\cdot \\\\\n& \\sum_{\\stackrel{n_{1},\\ldots n_{M}> P}{n_{0}\\le P}} E_{1}\\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\gamma^{(P-n_{M})(\\beta-1)}.\n\\end{align}\n\n\n\\subsubsection{Proof of Proposition~\\ref{prop-spectral&zc}-Step 4}\\label{subsubsec-CEsmall}\nWe show that $C_{E}\\to0$ as $\\beta\\to+\\infty$. \n\n\n\\begin{proposition}\n\\label{prop-Cezero}\nThere exists $A=A(\\beta)\\in (0,1)$ with $\\lim_{\\beta\\to+\\infty} A = 0$ such that \n$$\nC_{E}\\le 2P\\, \\frac{N+1}{\\beta-1}\\sum_{n=1}^{+\\infty}\\gamma^{-n(\\beta-1)}\\sum_{M=0}^{+\\infty}A^{M}\\sum_{i=0}^{M}\\frac{n^{i}}{i!}.\n$$\n\\end{proposition}\n\nBefore proving this proposition, we show that is finishes the proof of Proposition~\\ref{prop-spectral&zc}. \nThe series has only positive terms. Clearly, $}%{\\displaystyle \\sum_{M=0}^{+\\infty}A^{M}\\sum_{i=0}^{M}\\frac{n^{i}}{i!}\\le \\frac1{1-A}e^{n}$, so the main sum converges if\n$\\gamma^{\\beta-1}>e$. Thus Proposition~\\ref{prop-Cezero} implies that $C_{E}\\to 0$ as $\\beta\\to+\\infty$. \n\nTherefore, inequality \\eqref{equ1-upperboundCL0} shows that if $\\beta\\to+\\infty$, \nthen $\\lambda_{0,\\beta}\\to0$ too, and hence Proposition~\\ref{prop-spectral&zc} is proved. \n\nThe rest of this subsection is then devoted to the proof of Proposition~\\ref{prop-Cezero}. \n\\begin{lemma}\n\\label{lem-gamma-n}\nLet $\\eta$ and $y$ be positive real numbers. Then for every $n$, \n$$\\int_{y}^{\\infty}x^{n}e^{-\\eta (x-y)}dx=\\sum_{j=0}^{n}\\frac{n!}{j!}\\frac{y^{j}}{\\eta^{n+1-j}}.$$\n\\end{lemma}\n\\begin{proof}\nSet $u_{n}:=}%{\\displaystyle \\int_{y}^{\\infty}x^{n}e^{-\\eta (x-y)}dx$. \nThen\n\\begin{eqnarray*}\nu_{n}&=& \\int_{0}^{\\infty}(x+y)^{n}e^{-\\eta x}dx\\\\\n&=&\\left[\\frac{-1}{\\eta}(x+y)^{n}e^{-\\eta x}\\right]_{0}^{\\infty}+\\frac{n}\\eta\\int_{0}^{\\infty}(x+y)^{n-1}e^{-\\eta x}\\\\\n&=& \\frac{y^{n}}\\eta+\\frac{n}\\eta u_{n-1}.\n\\end{eqnarray*}\nThe formula follows by induction.\n\\end{proof}\n\n\nLet $n$ be some positive integer and $\\xi$ and $\\zeta$ two positive real numbers. We consider a matrix $D_{n} = (d_{n,i,j})_{i=1, j=1}^{n+1, n}$ with $n+1$ rows and $n$ columns defined by \n$$\nd_{n,i,j}:= \\begin{cases}\n \\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1} & \\text{ if } i \\le j,\\\\[1mm]\n \\frac\\xi{j} & \\text{ if }i=j+1,\\\\[1mm]\n 0 &\\text{ if }i>j+1.\n\\end{cases}\n$$\nor in other words:\n$$\nD_n=\\left(\\begin{array}{cc cc ccc}\n0!\\zeta & 1!\\zeta^2 & 2!\\zeta^3 & \\ldots & (j-1)!\\zeta^j & \\ldots &   (n-1)!\\zeta^{n} \\\\\n\\xi & \\zeta & \\ldots &  &  &  & (n-2)!\\zeta^{n-1} \\\\\n0 & \\frac\\xi2 & \\zeta &  &  &  & \\vdots \\\\\n0 & 0 & \\frac\\xi3 & \\ddots & \\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1} & &  \\vdots\\\\\n\\vdots &  & 0 & \\ddots & \\ddots &  & \\vdots\\\\\n\\vdots &  & & 0 & \\frac{\\xi}{j} & \\zeta & \\zeta^2 \\\\\n0 &  &  &  & 0 & \\frac{\\xi}{n-1} & \\zeta \\\\\n0 & 0 & \\ldots & \\ldots & 0 & 0 & \\frac{\\xi}{n} \n\\end{array}\\right).\n$$\nWe call $\\mathbf{ w}$ non-negative (and write $\\mathbf{ w}\\succeq 0$) if all a entries of $\\mathbf{ w}$ are non-negative. This defines a partial ordering on vectors by \n$$\\mathbf{ w}' \\succeq \\mathbf{ w} \\iff \\mathbf{ w}'-\\mathbf{ w}\\succeq 0.$$  \n\n\\begin{lemma}\n\\label{lem-matrix}\nAssume $0<\\zeta<1$ and \nset $K:=\\frac1{1-\\zeta}$. Then, for every $n$,\n$$\nD_n \\cdot \\left(\\begin{array}{c}}%{\\displaystyle\\frac{K^{n-1}}{0!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{1!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{2!} \\\\[1mm]\n\\vdots \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n-1}}{(n-1)!}\\end{array}\\right)\n\\preceq\n\\left(\\begin{array}{c}}%{\\displaystyle\\frac{K^{n}}{0!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{1!} \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{2!} \\\\[1mm]\n\\vdots \\\\[1mm]\n}%{\\displaystyle\\frac{K^{n}}{n!}\\end{array}\\right).$$\n\\end{lemma}\n\n\\begin{proof}\nThis is just a computation. For the first row we get \n$$\n\\sum_{j=1}^{n}(j-1)!\\zeta^{j}.\\frac{K^{n-1}}{(j-1)!}\\le K^{n-1}.\\frac\\zeta{1-\\zeta}\\le K^{n}.$$\nFor row $i>1$ we get \n$$\\frac1{(i-1)}\\frac{K^{n-1}}{(i-2)!}+\\sum_{j=i}^{n}\\frac{(j-1)!}{(i-1)!}\\zeta^{j-i+1}\\frac{K^{n-1}}{(j-1)!}=\\frac{K^{n-1}}{(i-1)!}\\left(1+\\zeta+\\zeta^{2}\\ldots\\right)\\le \\frac{K^{n}}{(i-1)!}.$$\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{prop-calcul-matrix-majo}\nSet $\\zeta:=}%{\\displaystyle\\frac1{(\\beta-1)\\log\\gamma}$ and $K=\\frac{1}{1-\\zeta}$. \nConsider $M$ integers $n_{1},\\ldots n_{M}$, with $n_{M}> P$. Then, \nfor every $M\\ge 2$, \n$$\n\\sum_{n_1, \\dots, n_{M-1} > P} \\prod_{i=1}^{M-1} \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} ) \\right)^{1-\\beta}\\le K^{M-1}\\sum_{i=0}^{M-1}\\frac{(n_{M}-P)^{i}}{i!}$$\n\\end{proposition}\n\\begin{proof}\nNote that \n\\begin{align*}\n\\sum_{n_1, \\dots, n_{M-1} > P} & \\prod_{i=1}^M \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\\\\n= & \\sum_{n_{M-1}=1}^{\\infty}\\left(\\ldots\\left(\\sum_{n_{2}=1}^{\\infty}\\left(\\sum_{n_{1}=1}^{\\infty}\\right.\\right.\\right.\n\\left(1+\\max(\\gamma^{n_{1} - n_{2}}-1,\\gamma^{n_{1}-n_{2}-1} ))^{1-\\beta}\\right)\\cdot\\\\\n& \\left.\\left((1+\\max(\\gamma^{n_{2} - n_{3}}-1,\\gamma^{n_{2}-n_{3}-1} ))^{1-\\beta}\\right)\\ldots\\right) \\\\\n& \\left(1+\\max(\\gamma^{n_{M-1} - n_{M}}-1,\\gamma^{n_{M-1}-n_{M}-1} \\right)^{1-\\beta}.\n\\end{align*}\nThis means that we can proceed by induction. Now \n\\begin{align*}\n\\sum_{n_{1}=P+1}^{\\infty} & (1+\\max(\\gamma^{n_{1} - n_{2}}-1,\\gamma^{n_{1}-n_{2}-1} ))^{1-\\beta}\\\\\n&\\le\\int_{P}^{n_{2}} (1+\\gamma^{x-n_{2}-1} ))^{1-\\beta}dx\n+\\int_{n_{2}}^{\\infty} (\\gamma^{x - n_{2}})^{1-\\beta}dx\\\\\n&\\le n_{2}-P+\\int_{n_{2}}^{\\infty}e^{-(\\beta-1)\\,(x-n_{2})\\,\\log\\gamma}\\,dx\\\\\n&= n_{2}-P+\\int_{n_{2}}^{\\infty}e^{-\\frac{x-n_{2}}{\\zeta}}\\,dx,\n\\end{align*}\nbecause $\\zeta=\\frac1{(\\beta-1)\\log\\gamma}$. This shows that the result holds for $M=2$.\n\nAssuming that the sum for $M=p$ is of the form \n$\\sum_{j=0}^{p-1}a_{j}(n_{p}-P)^{j}$, we compute the sum for $M=p+1$.\n\\begin{align*}\n\\sum_{n_{p}=P+1}^{\\infty} & \\sum_{j=0}^{p-1}  a_{j}\\frac{(n_{p}-P)^{j}}{(1+\\max(\\gamma^{n_{p}-n_{p+1}}-1,\\gamma^{n_{p}-n_{p+1}-1} ))^{\\beta-1}} \\\\ \n&\\le\\ \\sum_{j}a_{j}\\int_{P}^{n_{p+1}}\\frac{(x-P)^{j}}{(1+\\gamma^{x-n_{p+1}-1})^{\\beta-1}}\\,dx+\n\\sum_{j}a_{j}\\int_{n_{p+1}}^{\\infty}\\frac{(x-P)^{j}}{(\\gamma^{x-n_{p+1}})^{\\beta-1}}\\,dx\\\\\n&\\le \\sum_{j}\\frac{a_{j}(n_{p+1}-P)^{j+1}}{(j+1)}+\\int_{n_{p+1}}^{\\infty}(x-P)^{j}e^{-\\frac{x-n_{p+1}}{\\zeta}}\\,dx.\n\\end{align*}\nSet $}%{\\displaystyle \\mathbf{ w}\\cdot\\mathbf{ w}'=\\sum w_{i}w'_{i},$\nfor vectors $\\mathbf{ w}=(w_{1},\\ldots, w_{p+1})$ and $\\mathbf{ w}'=(w'_{1},\\ldots, w'_{p+1})$. \nLemma~\\ref{lem-gamma-n} yields \n \\begin{align*}\n\\sum_{n_{p}=P+1}^{\\infty} & \\sum_{j=0}^{p-1} a_{j}\\frac{(n_{p}-P)^{j}}{(1+\\max(\\gamma^{n_{p}-n_{p+1}}-1,\\gamma^{n_{p}-n_{p+1}-1}))^{\\beta-1}}\\\\\n&\\le \\sum_{j}\\frac{a_{j}}{(j+1)}(n_{p+1}-P)^{j+1} + \n\\sum_{i=0}^{j}\\frac{j!}{i!}\\zeta^{j-i+1}(n_{p+1}-P)^{i}\\\\\n&\\le D_{p}\\left(\\begin{array}{c}a_0 \\\\a_1 \\\\ \\vdots \\\\ a_{p-1}\n\\end{array}\\right) \\cdot \n\\left(\\begin{array}{c}1 \\\\n_{p+1} \\\\\\vdots \\\\n_{p+1}^{p}\\end{array}\\right).\n\\end{align*}\nLemma~\\ref{lem-matrix} concludes the proof of the induction. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop-Cezero}]\nWe have just proven that \n\\begin{align*}\n\\sum_{n_{1},\\ldots n_{M}>P} & \\left(1+\\max(\\gamma^{n_{i} - n_{i+1}}-1,\\gamma^{n_{i}-n_{i+1}-1} )\\right)^{1-\\beta}\\gamma^{(P-n_{M})(\\beta-1)} \\\\\n& \\le K^{M-1}\\sum_{n_{M}=P+1}^{+\\infty}\\sum_{j=0}^{M-1}\\frac{(n_{M}-P)^{j}}{j!}\\gamma^{(n_{M}-P)(\\beta-1)}.\n\\end{align*}\n\nIt remains to sum over $n_{0}$. Note that in that case, there are only $P$ terms of the form \n$}%{\\displaystyle \\sum_{j=0}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}+\\frac{j}{\\gamma}\\right)^{\\beta}}$ because $n_{0}\\le P<n_{1}$ for each possible $n_{0}$, \n\\begin{eqnarray*}\n}%{\\displaystyle \\sum_{j=0}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-1}+\\frac{j}{\\gamma^2}\\right)^{\\beta}}&=&\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}\\right)^{\\beta}}+}%{\\displaystyle \\sum_{j=1}^{+\\infty}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}+\\frac{j}{\\gamma^2}\\right)^{\\beta}}\\\\\n&\\le & 1+\\frac{\\gamma^2}{\\beta-1}\\frac1{\\left(1+\\gamma^{n_{0}-n_{1}-2}\\right)^{\\beta-1}}\\\\\n&\\le & 1+\\frac{\\beta-1}2,\n\\end{eqnarray*}\nfor $\\beta \\ge \\sqrt{2} \\gamma$.\n Finally, inequality \\eqref{eq:est0} yields \n$$C_{E}\\leq 2P\\, \\frac{N+1}{\\beta-1}\\sum_{M=0}^{+\\infty}A^{M}\\sum_{n=1}^{+\\infty}\\gamma^{n(1-\\beta)}\\sum_{j=0}^{M-1}\\frac{n^{j}}{j!},$$\nwith $}%{\\displaystyle A:=\\frac{\\gamma}{\\beta-1}K=\\frac{\\gamma}{\\beta-1-\\frac1{\\log\\gamma}}$. This tends to $0$ as $\\beta \\to +\\infty$. \n\\end{proof}\n\n\n\\subsection{End of the proof of Theorem~\\ref{theo-pt}}\n\n\\subsubsection{End of the case $-\\log\\frac{n+1}n$}\nProposition~\\ref{prop-spectral&zc} shows that there exists some minimal $\\beta_{0}$ such that $\\lambda_{0,\\beta}<1$ for every $\\beta>\\beta_{0}$. \nThis also shows that $\\CP(\\beta)=0$ for $\\beta>\\beta_{0}$. \nSince $\\CP(\\beta)$ is a continuous and convex function, it is constant for \n$\\beta>\\beta_{0}$. As $\\CP(0)=\\log2$, there exists \na minimal $\\beta_{c} > 0$ such that $\\CP(\\beta)>0$ for every $0\\le \\beta<\\beta_{c}$. \nClearly, $\\beta_{c}\\le \\beta_{0}$. \n\nWe claim that for $\\beta<\\beta_{c}$, there exists a unique equilibrium state and that it has full support. Indeed, there exists at least one equilibrium state, say $\\mu_{\\beta}$, and at least one cylinder, say $J$, has positive $\\mu_{\\beta}$-measure. \nTherefore, we can induce on this cylinder, and the form of potential (see \\cite[Theorem 4]{leplaideur-synth}) shows that there exists a unique local equilibrium state. It is a local Gibbs measure and therefore $\\mu_{\\beta}$ is uniquely determined on each cylinder, and unique  and with full support (due to the mixing property). \n\nWe claim that the pressure function $\\CP(\\beta)$ is analytic on $[0,\\beta_{c}]$. \nIndeed, each cylinder $J$ has positive $\\mu_{\\beta}$-measure and the associated $Z_{c}(\\beta)$ is the pressure of the dotted system (that is: restricted to the trajectories that avoid $J$). This set of trajectories has a pressure strictly smaller than $\\CP(\\beta)$ because otherwise, several equilibrium states would coexist. \nTherefore $\\CP(\\beta)$ is determined by the implicit equation $\\lambda_{\\CP(\\beta),\\beta}=1$ and $\\CP(\\beta)>Z_{c}(\\beta)$ for $\\beta \\in [0,\\beta_{c}]$. \nThe Implicit Function Theorem shows that $\\CP(\\beta)$ is analytic. \n \nFor $\\beta\\ge \\beta_{c}$, the pressure $\\CP(\\beta)=0$ and for cylinders $J$ as above, we have $Z_{c}(\\beta)\\ge 0$. This shows that $Z_{c}(\\beta)=0$ for every $\\beta\\ge \\beta_{c}$. Due to the form of the potential, $\\lambda_{0,\\beta}$ is continuous and decreasing  in $\\beta$. \n\nWe claim that $\\beta_{c}=\\beta_{0}$. \nIndeed, assume by contradiction $\\beta<\\beta_{c}$. Then $\\lambda_{0,\\beta_{c}}>1$, since  otherwise (because $\\lambda_{0,\\beta}$ being strictly decreasing in $\\beta$), \n$\\lambda_{0,\\beta_{c}}\\le 1$ would yield that $\\lambda_{0,\\beta}<1$ for every $\\beta>\\beta_{c}$.\nThis would imply $\\beta_{c} \\ge \\beta_{0}$ (recall that $\\beta_{0}$ is minimal with this property). \nNow, for fixed $\\beta$, $Z\\mapsto \\lambda_{Z,\\beta}$ is continuous and strictly decreasing and goes to $0$ at $Z\\to+\\infty$. Therefore, if $\\lambda_{0,\\beta_{c}}>1$ then there exists $Z>0$ such that $\\lambda_{Z,\\beta_{c}}=1$. The local equilibrium state for this $Z$ generates\n some $\\sigma$-invariant probability measure\\footnote{Since $Z_{c}(\\beta_{c})=\\CP(\\beta_{c})=0<Z$, the expectation of the return time is comparable to $\\left|\\frac{\\partial\\CL_{Z,\\beta_{c}}({1\\!\\!1}_{J})}{\\partial Z}\\right|$, which converges.} with pressure for $\\beta \\varphi$ equal to $Z$, thus positive, and this is in contradiction with $\\CP(\\beta_{c})=0$. \nThis proves that $\\beta_{c}=\\beta_{0}$ and finishes the proof of Theorem~\\ref{theo-pt} in the case that $V(x) = -\\log \\frac{n+1}{n}$. \n\n\\subsection{The general case \\boldmath $V\\in \\CX_{1}$ \\unboldmath}\nFor $V \\in \\CX_1$, there exists $\\kappa>0$ such that \n$$-V \\le \\kappa\\varphi.$$\n This shows that the pressure function is constant equal to zero for $\\beta\\ge \\beta_{0}\/\\kappa$. Again, the pressure is convex, thus non-increasing and continuous. We can define $\\beta'_{c}$ such that $\\CP(\\beta)>0$ for$0\\le\\beta\\le \\beta'_{c}$ and $\\CP(\\beta)=0$ for $\\beta\\ge \\beta'_{c}$. \n \n The rest of the argument is relatively similar to the previous discussion. We deduce that for $\\beta<\\beta'_{c}$, there exists a unique equilibrium state, it has full support and $\\CP(\\beta)$ is analytic on this interval. For $\\beta\\ge \\beta'_{c}$, it is not clear that $\\lambda_{0,\\beta}$ decreases in $\\beta$. However,\nwe do not really need this argument, because if $\\lambda_{0,\\beta}>1$, then the decrease of $Z\\mapsto \\lambda_{Z,\\beta}$ (which follows from convexity argument and $}%{\\displaystyle\\lim_{Z\\to+\\infty}\\lambda_{Z,\\beta}=0)$, is sufficient to produce a contradiction. \n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\n\nThe main paradigm for adapting pretrained models for classification~\\cite{GPT, UniLM, BERT} is fine-tuning via an explicit classifier head. However, an alternative approach has arisen: adapting the pretrained language model directly as a predictor through autoregressive text generation~\\cite{GPT2} or completion of a cloze task~\\cite{ASimpleMethod}. This method is notably used in T5 fine-tuning~\\cite{T5} leading to state-of-the-art results on the SuperGLUE benchmark~\\cite{SuperGLUE}.\n\n\\blfootnote{Code available at \\url{https:\/\/github.com\/TevenLeScao\/pet}}\n\n\nOne argument made for classification by direct language generation is that it allows us to pick custom \\textit{prompts} for each task~\\cite{Decathlon}. \nWhile this approach can be used for zero-shot classification~\\cite{Zeroshot} or priming~\\cite{GPT3}, it can also be used in fine-tuning to provide extra task information to the classifier, especially in the low-data regime~\\cite{PET, PET2}.\n\nIf this argument is indeed true, it is natural to ask how it impacts the sample efficiency of the model, or more directly, \\textit{how many data points is a prompt worth?} As with many low-data and pretraining-based problems, this \nquestion is complicated by the fine-tuning setup, training procedure, and prompts themselves. We attempt to isolate these variables through diverse prompts, multiple runs, and best practices in low-training data fine-tuning. We introduce a metric, the \\textit{average data advantage}, for quantifying the impact of a prompt in practice.\n\n\nOur experiments find that the impact of task-targeted prompting can nicely be quantified in terms of direct training data, and that it varies over the nature of different tasks. On MNLI~\\cite{MNLI}, we find that using a prompt contributes approximately 3500 data points. On SuperGLUE, it adds approximately 280 data points on RTE~\\cite{RTE} and up to 750 on BoolQ~\\cite{BoolQ}. In low- to medium-data settings, this advantage can be a real contribution to training a model. \n\n\n\\section{Related Work}\n\n\nPrompting has been used both for zero-shot and fine-tuning based methods. Zero-shot  approaches attempt to answer a task with a prompt without fine-tuning through generation~\\cite{GPT2}. GPT3~\\cite{GPT3} extends this approach to a supervised priming method by taking in training data as priming at inference time, so it can attend to them while answering. T5~\\cite{T5} and other sequence-to-sequence pretrained models use standard word-based fine-tuning with a marker prompt to answer classification tasks with strong empirical success. Our setting differs in that we are interested in using task-based prompts and fine-tuning, in-between the T5 and GPT2 setting. \n\nOur setting most closely resembles PET~\\cite{PET, PET2}, which claims that task-specific prompting helps transfer learning, especially in the low-data regime. However, in order to reach the best possible results on SuperGLUE, PET introduces several other extensions: semi-supervision via additional pseudo-labeled data, ensembling models trained with several different prompts, and finally distilling the ensemble into a linear classifier rather than a language model. Our aim is to isolate the specific contributions of prompting within supervised fine-tuning. \n\nFinally, recent papers have experimented with discovering prompts through automated processes tailored to the language model~\\cite{HowCanWeKnow, AutomaticVerbalizer}. We limit ourselves to human-written prompts, as we are interested into \nwhether prompting itself specifically adds information to the supervised task. It is an interesting question as to whether automatic prompts can have this same impact (relative to the training data they require). \n\n\\section{Comparison: Heads vs Prompts}\n\n\n\nConsider two transfer learning settings for text classification: \\textit{head-based}, where a generic head \nlayer takes in pretrained representations to predict an output class; \\textit{prompt-based}, where a \ntask-specific pattern string is designed to coax the model into producing a textual output corresponding to \na given class. Both can be utilized for fine-tuning with supervised training data, but prompts further allow\nthe user to customize patterns to help the model. \n\nFor the \\textit{prompt} model we follow the notation from PET~\\cite{PET} and decompose a prompt into a \\textit{pattern} and a \\textit{verbalizer}. The \\textit{pattern} turns the input text into a cloze task, i.e. a sequence with a masked token or tokens that need to be filled. Let us use as example an excerpt from SuperGLUE task BoolQ~\\cite{BoolQ}, in which the model must answer yes-or-no binary questions. In order to let a language model answer the question in \\textit{italics}, our pattern is in \\textbf{bold}~\\cite{PET2}:\n\n\\begin{quote}\n\\small\n \"Posthumous marriage -- Posthumous marriage (or necrogamy) is a marriage in which one of the participating members is deceased. It is legal in France and similar forms are practiced in Sudan and China. Since World War I, France has had hundreds of requests each year, of which many have been accepted.\n\\textbf{Based on the previous passage, \\textit{can u marry a dead person in france ?} <MASK>}\"\n\\end{quote}\n\n\n\nThe masked word prediction is mapped to a \\textit{verbalizer} which produces a class. (here \"Yes\": True. \"No\": False\\footnote{The correct answer here is, of course, \\textit{yes}. Originated in 1803 as Napoleon rose to power, this practice was mainly to the benefit of war widows.}). \nSeveral \\textit{pattern-verbalizer pairs} (\\textit{PVPs}) could be used for a single task, differing either through the pattern, the verbalizer, or both. Fine-tuning is done by training the model to produce the correct verbalization. The loss is the cross-entropy loss between the correct answer and the distribution of probabilities amongst the tokens in the verbalizer. We re-use pattern choices from~\\citet{PET2}; examples are available in Appendix~\\ref{prompts}.\n\n\n\n\n\n\n\n\\section{Experimental Setting}\n\n\nWe run all experiments with the same pretrained checkpoint, \\textit{roberta-large} (355M parameters) from RoBERTa~\\cite{Roberta}, which we load from the \\textit{transformers}~\\cite{Transformers} library.\\footnote{After experimenting with RoBERTa, AlBERT~\\cite{Albert} and BERT~\\cite{BERT}, we found \\textit{roberta-large} to have the most consistent performance.}\nIn line with previous observations~\\cite{Feather,Finetuning,Mixout}, head-based fine-tuning performance varies considerably. We follow recommendations of~\\citet{Stability} and~\\citet{Revisiting} to train at a low learning rate ($10^{-5}$) for a large number of steps (always at least $250$, possibly for over 100 epochs).\n\n\n\n\n\n\n\\begin{figure*}[h!]\n\\centering\n\\hspace*{-1cm}\\includegraphics[width=1.1\\textwidth]{Graphs\/Master_figure.png}\n\\caption{Prompting vs head (classifier) performance across data scales, up to the full dataset, for six SuperGLUE tasks. Compares the best prompt and head performance at each level of training data across 4 runs. Highlighted region shows the accuracy difference of the models. Cross-hatch region highlights the lowest- and highest- accuracy matched region in the curves. The highlighted area in this region is used to estimate the data advantage. }\n\\label{main_figure}\n\\end{figure*}\n\nWe perform our evaluation on SuperGLUE and MNLI~\\cite{MNLI}. These datasets comprise a variety of tasks, all in English, including entailment (MNLI, RTE~\\cite{RTE}, CB~\\cite{CB}), multiple choice question answering (BoolQ~\\cite{BoolQ}, MultiRC~\\cite{MultiRC}), and common-sense reasoning (WSC~\\cite{WSC}, COPA~\\cite{COPA}, WiC~\\cite{WiC}). We do not include ReCoRD~\\cite{ReCORD} in our comparisons as there is no head model to compare with, since it is already a cloze task. Data sizes range from $250$ data points for CB to $392,702$ for MNLI. As test data is not publicly available for SuperGLUE tasks, we set aside part of training (from $50$ for CB, COPA and MultiRC to $500$ for BoolQ) to use for development, and evaluate on their original validation sets. For MNLI, we use the available matched validation and test sets. \n\nWe compare models across a scale of available data, starting with $10$ data points and increasing exponentially (as high-data performance tends to saturate) to the full dataset. For example, for MultiRC, which has 969 data points initially, we start by reserving 50 data points for development. This leaves us with 919 training points, and we train models with 10, 15, 20, 32, 50, 70, 100, 150, 200, 320, 500, 750, and 919 training points. We run every experiment 4 times in order to reduce variance, for a total of 1892 training runs across all tasks. At every point, we report the best performance that has been achieved at that amount of data or lower. Full graphs are presented in Appendix~\\ref{reduction}.\n\n\n\n\\section{Results}\n\n\n\n\n\nFigure~\\ref{main_figure} shows the main results comparing head- and prompt-based fine-tuning with the best-performing pattern on that task. \nPrompting enjoys a substantial advantage on every task, except for WiC as is reported in previous results~\\cite{PET2}.\nBoth approaches improve with more training data, but prompting remains better by a varying amount. Many tasks in SuperGLUE have relatively few data points, but we also see an advantage in large datasets like BoolQ and MNLI.\n\nTo quantify how many data points the prompt is worth, we first isolate the $y$-axis band of the lowest- and highest- accuracy where the two curves match in accuracy.\\footnote{We assume asymptotically the two curves would match, but are limited by data.} The horizontal line at these points represents the advantage of prompting. We then take the integral in this region, i.e. area between the linearly-interpolated curves\\footnote{In areas where the head model is better, if any, get subtracted from the total.}, divided by the height of the band. The area has the dimension of a quantity of data points times the metric unit, so dividing by the performance range yields a \\# of data points advantage. \nAs low data training is sensitive to noise, in addition to following best training practices we \nrun several different experiments for each $x$-point. We use a bootstrapping approach to estimate confidence over these runs. Specifically, we hold out one of the 4 head runs and 4 prompt runs (16 combinations total), and compute the standard deviation of those outcomes.\n\n\nWe report these quantities for every task in Table~\\ref{main_table} as \\textit{Average advantage}. For almost all the tasks, we see that prompting gives a substantial advantage in terms of data efficiency, adding the equivalent of hundreds of data points on average.\n\n\n\n\n\n\n\n\\begin{table*\n\n\\hspace*{-0.7cm}\\begin{tabular}{@{}l rrrrrrrr@{}}\n\\toprule \n& \\multicolumn{8}{c}{Average Advantage (\\# Training Points)} \\\\\n& MNLI & BoolQ & CB & COPA & MultiRC* & RTE & WiC & WSC\\\\\\midrule \\multicolumn{1}{l}{\\textit{P vs H}} & $3506\\pm536$ & $752\\pm46$ & $90\\pm2$ & $288\\pm242$ & $384\\pm378$  & $282\\pm34$ & $-424\\pm74$ & $281\\pm137$ \\\\\n\\hline\n\\multicolumn{1}{l}{\\textit{P vs N}} & $150\\pm252$ & $299\\pm81$ & $78\\pm2$ & -& $74\\pm56\\phantom{0}$ & $404\\pm68$ & $-354\\pm166$ & -\\\\\n\\multicolumn{1}{l}{\\textit{N vs H}} & $3355\\pm612$ & $453\\pm90$ & $12\\pm1$ & -& $309\\pm320$ & $-122\\pm62$ & $-70\\pm160$ & -\\\\\n\\bottomrule\n\\end{tabular}\n\n\\caption{Average prompting advantage in number of data points for MNLI \\& SuperGLUE tasks. \\textit{P} denotes the prompt model, \\textit{H} the head model. On average across performance levels, an MNLI prompt model yields the results of an MNLI head model trained with 3500 additional data points. Confidence levels are based on a multiple random runs (see text). \\textit{N} indicates a null-verbalizer prompting task that replaces the verbalizer with a non-sensical mapping.  *The comparison band of MultiRC is too small as the head baseline fails to learn beyond majority class; we use the full region for a lower-bound result.}\n\\label{main_table}\n\\end{table*}\n\n\n\n\n\n\\section{Analysis}\n\n\\paragraph{Impact of Pattern vs Verbalizer}\n\nThe intuition of prompts is that they introduce a task description in natural language,\neven with few training points. \nTo better understand the zero-shot versus adaptive nature of prompts,\nwe consider a \\textit{null verbalizer}, a control with a verbalizer that cannot yield semantic information without training. For every task that requires filling in one word (which excludes the more free-form COPA and WSC), we replace the verbalizers, for example, \"yes\", \"no\", \"maybe\", \"right\" or \"wrong\",  with random first names.\n\nTable~\\ref{main_table} shows the advantage of the standard prompt over the null verbalizer to estimate this control.\nWe see that for small data tasks such as CB, the null verbalizer removes much of the benefits of prompting. However, with more training data, the model seems to adapt the verbalizer while still gaining the inductive bias benefits of the pattern. Figure~\\ref{neutral_run_figure} showcases this dynamic on MNLI. This result further shows that prompting yields data efficiency even if it is not directly analogous to the generation process of training. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Graphs\/MNLI_partial_neutral_run.png}\n\\caption{Comparison of full prompt and null verbalizer advantage on MNLI at lower data scales.}\n\\label{neutral_run_figure}\n\\end{figure}\n\n\\paragraph{Impact of Different Prompts}\n\nIf the prompt acts as a description of the task, one would expect different valid descriptions to vary in their benefits. In order to compare the different prompts we used on each task, we chart the median performance for each of them under different runs. In nearly every experiment, we find that the confidence intervals of those curves largely overlap, implying that prompt choice is not a dominant hyperparameter, i.e. the variance across random seeds usually outweighs the possible benefits of prompt choice. One exception is the low-data regime of BoolQ, where one of the prompts enjoys a significant few-shot advantage over the others. We plot this curve for MultiRC in Figure~\\ref{median_comparison} and the rest in Appendix~\\ref{all_results}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Graphs\/median_f1_multirc.png}\n\\caption{Median performance on MultiRC across runs for three prompts. Differences are inconsistent and eclipsed by the variance within one prompt's runs.}\n\\label{median_comparison}\n\\end{figure}\n\n\\paragraph{Metric sensitivity}\n\n\nWe treat each metric linearly in calculating advantage; alternatively, we could re-parameterize the $y$ axis for each task. This choice does not have a consistent effect for or against prompting. For example, emphasizing gains close to convergence increases prompting advantage on CB and MNLI but decreases it on COPA or BoolQ. \n\n\n\n\n\n\\section{Conclusion}\n\nWe investigate prompting through a systematic study of its data advantage. Across tasks, prompting consistently yields a varying improvement throughout the training process. Analysis shows that prompting is mostly robust to pattern choice, and can even learn without an informative verbalizer. On large datasets, prompting is similarly helpful in terms of data points, although they are less beneficial in performance. In future work, we hope to study the mechanism and training dynamics of the prompting benefits.\n\n\\section{Impact statement}\n\nSignificant compute resources were used to run this paper's experiments. A single experiment (defined as one model run, at one data level, on one task) was quite light-weight, taking usually a little under an hour on a single Nvidia V100. However, as we computed a little under two thousand runs, this adds up to about 1800 GPU hours, to which one must add around 400 GPU hours of prototyping and hyper-parameter searching. Those 2200 GPU hours would usually have necessitated the release of about 400kg of CO2, about 40\\% of a transatlantic flight for a single passenger, in the country where we ran the experiments, although we used a carbon-neutral cloud compute provider.\n\nThe main benefit of prompting, rather than compute efficiency, is data efficiency. Although we ran all of our experiments on English, we hope that this property will be especially helpful in low-resource language applications. In a sense, a practitioner could then remedy the lack of task-specific data in their language by introducing information through a prompt. However, this comes with the inherent risk of introducing human biases into the model. Prompt completion also suffers from biases already present within the language model~\\cite{Babysitter}. This could cause a prompted model to repeat those biases in classification, especially in the few-shot setting where prompting mostly relies on the pretrained model.\n\n\\section{Acknowledgments}\n\nWe thank Steven Cao and Joe Davison for the discussions about prompting that initially spurred this paper. We further thank Timo Schick for making the code for PET available and for discussions about performance replication. We lastly thank Canwen Xu, Yacine Jernite, Victor Sanh, Dimitri Lozeve and Antoine Ogier for their help with the figures and writing of this draft.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe propagation and acceleration\nof cosmic rays (CRs) are governed by their interactions\nwith magnetic fields. Astrophysical magnetic fields are turbulent and, \ntherefore, the resonant and non-resonant (e.g. transient time damping, or TTD)\ninteraction of cosmic rays with MHD turbulence is the accepted\n principal mechanism to scatter and isotropize\ncosmic rays \\citep[see][]{Schlickeiser02}. In addition, efficient scattering is essential for the acceleration of cosmic rays. \nFor instance, scattering of cosmic rays back into the shock is a\nvital component of the first order Fermi acceleration \\citep[see][]{Longairbook}. At the same time, stochastic acceleration by turbulence is \nentirely based on scattering. The dynamics of cosmic rays in MHD turbulence holds the key to all high energy astrophysics and related problems. \n\nWe live in an exciting era when we are starting to test fundamental processes taking place at the beginning of the Universe, at the event horizon of black holes, when the nature of dark matter and dark energy is being probed etc. Using computers many researchers make sophisticated complex models to confront the observations in unprecedented details. In the mean time, with the launching of the new facilities, we have much more observational data available than ever before. For instance, CHANDRA observations of supernova\nremnants provide a strong constraint to diffusion coefficients and\/or magnetic fields near the shock \\citep[see, e.g.][]{Bamba05, PYL05}; \nthe diffuse\ngamma-ray measurements from Fermi from the Galactic disc have been successfully used to\nphenomenologically constrain numerical modeling of cosmic rays, e.g., with GALPROP \\citep{Ackermann12}; observations of solar\nenergetic particles (SEP) have been also fruitful over the past decades and lead to better understanding of transport in the solar\nwind \\citep[see a review by][and references therein]{Horbury05_SEP}. These developments make it urgent that we understand the key physical processes underlying astrophysical phenomena, can parameterize them and, if necessary, use as a subgrid input in our computer models.\n\n\nAt present, the propagation of the CRs is an advanced theory, which makes\nuse both of analytical studies and numerical simulations. However,\nthese advances have been done within the turbulence paradigm which\nis being changed by the current research in the field.\nInstead of the empirical 2D+slab model of turbulence, numerical\nsimulations suggest anisotropic Alfv\\'enic modes following \\cite[GS95]{GS95} scalings (an analog of 2D, but not an\nexact one, as the anisotropy changes with the scale involved) + fast modes \\citep{CL02_PRL}. These progresses resulted in important revisions on the theory of cosmic ray transport (see review by \\citealt{LBYO} and references therein). The GS95 turbulence injected on large scales and its extensions to compressible medium is less efficient in scattering of CRs compared to the estimates made assuming that magnetic turbulence consists of plane waves moving parallel to magnetic field \\citep{Chandran00, YL02}. Fast compressible modes, on the other hand, are demonstrated as the dominant scattering agent in spite of various damping processes they are subjected to \\citep{YL02, YL04, YL08}\n\nAt the same time, one should not disregard the possibilities of generation of additional perturbations on small scales by CR themselves. For instance,  the slab Alfv\\'enic perturbation can be created, e.g., via streaming instability \\citep[see][]{Wentzel74, Cesarsky80}. Instabilities induced by anisotropic distribution of CRs were also suggested as a possibility to scatter CRs \\citep[]{Lerche, Melrose74}. Particularly at shock front, studies of instabilities have been one of the major efforts since the acceleration efficiency is essentially determined by the confinement at the shock front and magnetic field amplifications. Examples of the new developments in the field include, current driven instability \\citep{Bell2004}, vorticity generation at curved shock \\citep{Giac_Jok2007}, through Baroclinic effect \\citep{Inoue09}, through precursor \\citep{BJL09}, etc. This field is rich in its own and we shall not dwell upon it in this chapter.\n\nIn fact, the small scale instabilities and large scale turbulence are not independent of each other. {\\em First} of all, the instability generated waves can be damped through nonlinear interaction with the large scale turbulence \\citep[henceforth YL02, YL04]{YL02, YL04}. In the case of anisotropic GS95 turbulence, the efficiency is reduced \\citep{FG04}. Nonetheless, owing to the non-linear damping, the instabilities can only grow in a limited range, e.g., $\\sim< 100$GeV in interstellar medium for the streaming instability \\citep{FG04, YL04}. \n{\\em Secondly}, the large scale compressible turbulence also generate small scale waves through firehose, gyroresonance instability, etc \\citep{Schek06, LB06, YL11, Santos-Lima}. \n\nPropagation of CRs perpendicular to mean magnetic field\nis another important problem for which one needs to take into account both large and small scale interactions in tested models of turbulence. Indeed, if one takes only the diffusion along the magnetic field line and field line random walk \\citep[FLRW][]{Jokipii1966, Jokipii_Parker1969, Forman1974}, compound (or subdiffusion) would arise. Whether the subdiffusion is realistic in fact depends on the models of turbulence chosen \\citep{YL08, Yan:2011valencia}. In this chapter we review current understandings to this question within the domain of numerically tested models of MHD turbulence.\n\nIn what follows, we introduce the basic mechanisms for the interactions between particles and turbulence in \\S2. We discuss the cosmic ray transport in large scale turbulence, including both analytical and numerical studies in \\S3. Applications to cosmic ray propagation is presented in \\S4. In \\S5, we consider the perpendicular transport of cosmic rays on both large and small scales. We shall also discuss the issue of super-diffusion and the applicability of sub-diffusion. In \\S6, we concentrate on the issue of self-confinement in the presence of preexisting turbulence and dwell on, in particular, the streaming instability at supernova remnant shocks and its implication for CR acceleration. \\S7, we address the issue of gyroresonance instability of CRs and its feedback on large scale compressible turbulence. Summary is provided in \\S8.\n\n\\section{Interactions between turbulence and particles}\n\\label{basics}\nBasically there are\ntwo types of resonant interactions: gyroresonance acceleration\nand transit acceleration (henceforth TTD). The resonant condition is $\\omega-k_{\\parallel}v\\mu=n\\Omega$ ($n=0, \\pm1,2...$),\nwhere $\\omega$ is the wave frequency, $\\Omega=\\Omega_{0}\/\\gamma$\nis the gyrofrequency of relativistic particle, $\\mu=\\cos\\xi$,\nwhere $\\xi$ is the pitch angle of particles. TTD formally corresponds to $n=0$ and it requires compressible perturbations.  \n\nThe Fokker-Planck equation is generally used to describe\nthe evolvement of the gyrophase-averaged distribution function $f$,\n\n\\[\n\\frac{\\partial f}{\\partial t}=\\frac{\\partial}{\\partial\\mu}\\left(D_{\\mu\\mu}\\frac{\\partial f}{\\partial\\mu}+D_{\\mu p}\\frac{\\partial f}{\\partial p}\\right)+\\frac{1}{p^{2}}\\frac{\\partial}{\\partial p}\\left[p^{2}\\left(D_{\\mu p}\\frac{\\partial f}{\\partial\\mu}+D_{pp}\\frac{\\partial f}{\\partial p}\\right)\\right],\\]\n where $p$ is the particle momentum. The Fokker-Planck coefficients\n$D_{\\mu\\mu},D_{\\mu p},D_{pp}$ are the fundamental physical parameters\nfor measuring the stochastic interactions, which are determined by\nthe electromagnetic fluctuations \\citep[see][]{SchlickeiserMiller}:\n\nGyroresonance happens when the Doppler shifted wave frequency matches the Larmor frequency of a particle. In quasi-linear theory (QLT), the Fokker-Planck\ncoefficients are given by  \\citep[see][]{SchlickeiserMiller, YL04}\n\n\\begin{eqnarray}\n\\left(\\begin{array}{c}\nD_{\\mu\\mu}\\\\\nD_{pp}\\end{array}\\right)  =  {\\frac{\\pi\\Omega^{2}(1-\\mu^{2})}{2}}\\int_{\\bf k_{min}}^{\\bf k_c}dk^3\\delta(k_{\\parallel}v_{\\parallel}-\\omega \\pm \\Omega)\n\\left[\\begin{array}{c}\n\\left(1+\\frac{\\mu V_{ph}}{v\\zeta}\\right)^{2}\\\\\nm^{2}V_{A}^{2}\\end{array}\\right]\\times\\nonumber\\\\\n\\times\\left\\{ \\left[J_{2}^{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)+J_{0}^{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)\\right]\n\\left[\\begin{array}{c}\nM_{{\\mathcal{RR}}}({\\mathbf{k}})+M_{{\\mathcal{LL}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{RR}}}({\\mathbf{k}})+K_{{\\mathcal{LL}}}({\\mathbf{k}})\\end{array}\\right]\\right.\\nonumber\\\\\n\\left.-2J_{2}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)J_{0}\\left({\\frac{k_{\\perp}v_{\\perp}}{\\Omega}}\\right)\n\\left[e^{i2\\phi}\\left[\\begin{array}{c}\nM_{{\\mathcal{RL}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{RL}}}({\\mathbf{k}})\\end{array}\\right]+e^{-i2\\phi}\\left[\\begin{array}{c}\nM_{{\\mathcal{LR}}}({\\mathbf{k}})\\\\\nK_{{\\mathcal{LR}}}({\\mathbf{k}})\\end{array}\\right]\\right]\\right\\} ,\\label{gyro}\n\\end{eqnarray}\nwhere $\\zeta=1$ for Alfv\\'{e}n modes and $\\zeta=k_{\\parallel}\/k$\nfor fast modes, $k_{min}=L^{-1}$, $k_c=\\Omega_{0}\/v_{th}$\ncorresponds to the dissipation scale, $m=\\gamma m_{H}$ is the relativistic\nmass of the proton, $v_{\\perp}$ is the particle's velocity component\nperpendicular to $\\mathbf{B}_{0}$, $\\phi=\\arctan(k_{y}\/k_{x}),$\n${\\mathcal{L}},{\\mathcal{R}}=(x\\pm iy)\/\\sqrt{2}$ represent left and\nright hand polarization. $M_{ij}$ and $K_{ij}$ are the correlation tensors of magnetic and velocity fluctuations.  \n\nFrom the resonance condition, we know that the most important interaction\noccurs at $k_{\\parallel}=k_{\\parallel,res}=\\Omega\/v_{\\parallel}$.\nThis is generally true except for small $\\mu$ (or scattering near\n$90^{\\circ}$). \n\nTTD happens due to the resonant interaction with parallel magnetic mirror force. Particles can be accelerated by when they are in phase with the waves either by interacting with oscillating parallel electric field (Landau damping), or by moving magnetic mirrors (TTD). When particles are trapped by moving in the same speed with waves, an appreciable amount of interactions can occur between waves and particles. Since head-on collisions are more frequent than that trailing collisions, particles gain energies. Different from gyroresonance, the resonance function of TTD is broadened even for CRs with small pitch angles. The formal resonance peak $k_{\\parallel}\/k=V_{ph}\/v_{\\parallel}$ favors quasi-perpendicular modes. However, these quasi-perpendicular modes cannot form an effective mirror to confine CRs because the gradient of magnetic perturbations along the mean field direction $\\nabla_{\\parallel}\\mathbf{B}$ is small. As we will show later in \\S\\ref{NLT_sec}, the resonance is broadened in nonlinear theory \\citep[see][]{YL08}.    \n\n\n\n\\section{Scattering of cosmic rays}\n\\label{scattering}\n\n\\subsection{Scattering by Alfv\\'{e}nic turbulence}\n\\label{Alf_scatter}\nAs we discussed in $\\S$2, Alfv\\'{e}n modes are anisotropic, eddies\nare elongated along the magnetic field, i.e., $k_{\\perp}>k_{\\parallel}$.\nThe scattering of CRs by Alfv\\'{e}n modes is suppressed first because\nmost turbulent energy goes to $k_{\\perp}$ due to the anisotropy of\nthe Alfv\\'{e}nic turbulence so that there is much less energy left\nin the resonance point $k_{\\parallel,res}=\\Omega\/v_{\\parallel}\\sim r_{L}^{-1}$.\nFurthermore, $k_{\\perp}\\gg k_{\\parallel}$ means $k_{\\perp}\\gg r_{L}^{-1}$\nso that cosmic ray particles have to be interacting with lots of eddies\nin one gyro period. This random walk substantially decreases the scattering\nefficiency. The scattering by Alfv\\'en modes was studied in YL02. In case that the pitch angle $\\xi$ not close to 0, the analytical result is  \\begin{equation}\n\\left[\\begin{array}{c}\nD_{\\mu\\mu}\\\\\nD_{pp}\\end{array}\\right]=\\frac{v^{2.5}\\mu^{5.5}}{\\Omega^{1.5}L^{2.5}(1-\\mu^2)^0.5}\\Gamma[6.5,k_c^{-\\frac{2}{3}}k_{\\parallel,res}L^{\\frac{1}{3}}]\\left[\\begin{array}{c}\n1\\\\\nm^{2}V_{A}^{2}\\end{array}\\right],\\label{ana}\\end{equation}\nwhere $\\Gamma[a,z]$ is the incomplete gamma function. The presence\nof this gamma function in our solution makes our results orders of\nmagnitude larger than those%\n\\footnote{The comparison was done with the resonant term in Chandran (2000) as the nonresonant term is spurious %\n} in \\cite{Chandran00}, who employed\nGS95 ideas of anisotropy, but lacked the quantitative\ndescription of the eddies. However,\nthe scattering frequency,\n\n\\begin{equation}\n\\nu=2D_{\\mu\\mu}\/(1-\\mu^{2}),\\label{nu}\n\\label{nu}\n\\end{equation}\nare nearly $10^{10}$ times lower than the estimates for isotropic and slab model (see Fig.~\\ref{impl} {\\em left}). {\\em It is clear that for most interstellar circumstances, the scattering by Alfv\\'enic turbulence is suppressed.} As the anisotropy of the Alfv\\'{e}n modes is increasing with the\ndecrease of scales, the interaction with Alfv\\'{e}n modes becomes\nmore efficient for higher energy cosmic rays. When the Larmor radius\nof the particle becomes comparable to the injection scale, which is\nlikely to be true in the shock region as well as for very high energy cosmic\nrays in diffuse ISM, Alfv\\'{e}n modes get important.\n\n\\subsection{Cosmic ray scattering by compressible MHD turbulence}\n\nAs we mentioned earlier, numerical simulations of MHD turbulence supported the GS95 model of turbulence,\nwhich does not have the \"slab\" Alfv\\'enic modes that produced most of the scattering in the earlier models\nof CR propagation. Can the turbulence that does not appeal to CRs back-reaction (see \\S 4) produce \nefficient scattering? \n\nIn the models of ISM turbulence \\citep[]{Armstrong95, Mckee_Ostriker2007}, where the injection happens at large scale, \nfast modes were identified as a scattering agent for cosmic rays in interstellar medium \\cite[]{YL02,YL04}.\nThese works made use of the quantitative description of turbulence\nobtained in \\cite{CL02_PRL}  to calculate\nthe scattering rate of cosmic rays. \n\nDifferent from Alfv\\'en and slow modes, fast modes are isotropic \\citep{CL02_PRL}. Indeed they are subject to both collisional and collisionless damping. The studies in \\cite{YL02, YL04} demonstrated, nevertheless, that the scattering by fast modes dominates in most cases in spite of the damping\\footnote{On the basis of weak turbulence theory, \\cite{Chandran2005} has argued that high-frequency \nfast waves, which move mostly parallel to magnetic field, generate Alfv\\'en waves also moving mostly parallel to magnetic field. We expect\nthat the scattering by thus generated Alfv\\'en modes to be similar to the scattering by the fast modes created by them. Therefore\nwe expect that the simplified approach adopted in \\cite{YL04} and the papers that followed to hold.} (see Fig.\\ref{impl} {\\em right}).\n\\begin{figure*} [h!t] \n{\\includegraphics[width=0.45\\textwidth]{YL_fig1a.eps} \n\\includegraphics[width=0.45\\textwidth]{comp.eps}\n} \n\\caption{\\small {\\em Left:} rate of CR scattering by\nAlfv\\'en waves versus CR energy.  The lines at the top of the figure are\nthe accepted estimates obtained for Kolmogorov turbulence. The dotted\ncurve is from \\cite{Chandran00}. The analytical calculations are given\nby the solid line with our numerical calculations given by\ncrosses;  {\\em Right:} the scattering by fast modes,  dashed line represents the case without damping for fast modes included, the solid and dash-dot line are the results taking into account collisionless damping.}\n\\label{impl}\n\\end{figure*}\nMore recent studies of cosmic ray propagation and acceleration that explicitly appeal to the effect of\nthe fast modes include \\citet{Cassano_Brunetti, Brunetti_Laz, YL08, YLP08}.\nIncidentally, fast modes have been also identified as primary agents for the acceleration of charged dust particles \\cite{YL03,YLD04}.\n\n\n\\subsection{Nonlinear theory of diffusion}\n\\label{NLT_sec}\n\nWhile QLT allows easily to treat the CR dynamics in a local magnetic\nfield system of reference, a key assumption in QLT, that the particle's orbit is unperturbed, makes one wonder about the limitations of the approximation. Indeed, while QLT provides simple physical insights into scattering, it is known to have problems. For instance, it fails in treating $90^\\circ$ scattering  \\citep[see][]{Volk:1973, Volk:1975, Jones:1973, Jones:1978, Owens:1974, Goldstein:1976, Felice90degree} and perpendicular transport \\citep[see][]{Kota_Jok2000, Matthaeus:2003}. \n\nIndeed, many attempts have been made to improve the QLT and various non-linear\n theories have been attempted (see \\citealt{Dupree:1966}, V\\\"olk 1973, 1975, \nJones, Kaiser \\& Birmingham 1973, Goldstein 1976). Currently we observe a surge\nof interest in finding way to go beyond QLT. Examples include the nonlinear guiding center theory \\citep[see][]{Matthaeus:2003}, second-order \nquasilinear theory \\citep{Shalchi_SQT, Qin_NLT, LeRoux:2007}, etc. Most of the analysis were limited to traditional 2D+slab models of MHD turbulence. An important step was taken in Yan \\& Lazarian (2008), where non-linear effect was accounted for in treating CR scattering in the type of MHD turbulence that are supported by numerical simulations. The results have been applied to both solar flares (Yan, Lazarian \\& Petrosian 2008) and grain acceleration \\citep{HLS12}. Below, we introduce the nonlinear theory and their applications to both particle transport and acceleration in incompressible and compressible turbulence based on the results from Yan \\& Lazarian (2008).\n\nThe basic assumption of the quasi-linear theory is that particles follow unperturbed orbits. In reality, particle's pitch angle varies gradually with the variation of the magnetic field due to conservation of adiabatic invariant $v_\\bot^2\/B$, where $B$ is the total strength of the magnetic field \\citep[see][]{Landau:1975}. Since B is varying in turbulent field, so are the projections of the particle speed $v_\\bot$ and $v_\\|$.\n This results in broadening of the resonance. The variation of the velocity is mainly caused by the magnetic perturbation $\\delta B_\\|$ in the parallel direction. This is true even for the incompressible turbulence we discussion in this section. For the incompressible turbulence, the parallel perturbation arises from the pseudo-Alfv\\'en modes. The perpendicular perturbation $\\delta B_\\bot$ is higher order effect, which we shall neglect here.\n\nThe propagation of a CR can be described as a combination of a motion of its guiding center and CR's motion about its guiding center. \nBecause of the dispersion of the pitch angle $\\Delta\\mu$ and therefore of the parallel speed $\\Delta v_\\|$, the guiding center is perturbed about the mean position $<z>=v\\mu t$ as they move along the field lines. As a result, the perturbation $\\delta B({\\bf x},t)$  that the CRs view when moving along the field gets a different time dependence. The characteristic phase function $e^{ik_\\|z(t)}$ of the perturbation $\\delta B({\\bf x},t)$ deviates from that for plane waves. Assuming the guiding center has a Gaussian distribution along the field line, \\begin{equation}\nf(z)=\\frac{1}{\\sqrt{2\\pi}\\sigma_z}e^{-\\frac{(z-<z>)^2}{2\\sigma_z^2}},\n\\label{gauss}\n\\end{equation}\none gets by integrating over z, \\begin{equation}\n\\int_{-\\infty}^{\\infty} dze^{ik_\\| z}f(z)= e^{ik_\\|<z>}e^{-k_\\|^2\\sigma_z^2\/2}. \n\\label{phase}\n\\end{equation}\nThe first adiabatic invariant gives us \n\\begin{equation}\n\\sigma_z^2=<\\Delta v_\\|^2>t^2=\\frac{v^4}{v_\\|^2}\\left(\\frac{<\\delta B_\\parallel^2>}{B_0^2}\\right)t^2.\n\\end{equation}\n\n\nInsert the Eq.(\\ref{phase}) into the expression of $D_{\\mu\\mu}$ (see V\\\"olk 1975, \\citealt{YL04}), we obtain\n\n\\begin{eqnarray}\nD_{\\mu\\mu}&=&\\frac{\\Omega^2(1-\\mu^2)}{B_0^2}\\int d^3k\\sum_{n=0}^{\\infty}R_n(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)\\nonumber\\\\\n&&\\left[I^A({\\bf k})\\frac{n^2J_n^2(w)}{w^2}+\\frac{k_\\|^2}{k^2}J^{'2}_n(w)I^M({\\bf k})\\right],\n\\label{general}\n\\end{eqnarray} \nFollowing are the definitions of the parameters in the above equation. $\\Omega, \\mu$ are the Larmor frequency and pitch angle cosine of the CRs. $J_n$ represents Bessel function, and $w=k_\\bot v_\\bot\/\\Omega=k_\\bot LR\\sqrt{1-\\mu^2}$, where $R=v\/(\\Omega l)$ is the dimensionless rigidity of the CRs, $L$ is the injection scale of the turbulence. $k_\\bot, k_\\|$ are the components of the wave vector ${\\bf k}$ perpendicular and parallel to the mean magnetic field, $\\omega$ is the wave frequency. $I^A({\\bf k})$ is the energy spectrum of the Alfv\\'en modes and $I^M({\\bf k})$ represents the energy spectrum of magnetosonic modes. In QLT, the resonance function $R_n=\\pi\\delta(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)$. Now due to the perturbation of the orbit, it should be   \n\\begin{eqnarray}\n&&R_n(k_{\\parallel}v_{\\parallel}-\\omega\\pm n\\Omega)\\nonumber\\\\\n&=&\\Re\\int_0^\\infty dt e^{i(k_\\|v_\\|+n\\Omega-\\omega) t-\\frac{1}{2}k_\\|^2<\\Delta v_\\|^2>t^2}\\nonumber\\\\\n&=&\\frac{\\sqrt{\\pi}}{|k_\\|\\Delta v_\\||}\\exp\\left[-\\frac{(k_\\|v \\mu-\\omega+n\\Omega)^2}{k_\\|^2\\Delta v_\\|^2}\\right]\\nonumber\\\\\n&\\simeq&\\frac{\\sqrt{\\pi}}{|k_\\||v_\\bot \\sqrt{M_A}}\\exp\\left[-\\frac{(k_\\|v \\mu-\\omega+n\\Omega)^2}{k_\\|^2v_\\bot^2M_A}\\right]\n\\label{resfunc}\n\\end{eqnarray}\nwhere $M_A\\equiv \\delta V\/v_A=\\delta B\/B_0$ is the Alfv\\'enic Mach number and $v_A$ is the Alfv\\'en speed. We stress that Eqs.~(\\ref{general},\\ref{resfunc}) are generic, and applicable to both incompressible and compressible medium. \n\nFor gyroresonance ($n=\\pm 1,2,...$), the result is similar to that from QLT for $\\mu\\gg \\Delta \\mu=\\Delta v_\\|\/v$. In this limit, Eq.(\\ref{general}) represents a sharp resonance and becomes equivalent to a $\\delta$-function when put into Eq.(\\ref{general}).  \nIn general, the result is different from that of QLT, especially at $\\alpha\\rightarrow 90^\\circ$, the resonance peak happens at $k_{\\|,res}\\sim \\Omega\/\\Delta v$ in contrast to the QLT result \n$k_{\\|,res}\\sim\\Omega\/v_\\|\\rightarrow \\infty$. We shall\nshow below, that due to the anisotropy, the scattering coefficient $D_{\\mu\\mu}$ is still very small if the Alfv\\'en and the pseudo-Alfv\\'en modes are concerned. \n\nOn the other hand, the dispersion of the $v_\\parallel$ means that CRs with a much wider range of pitch angle can be scattered by the compressible modes through TTD \n($n=0$), which is marginally affected by the anisotropy and much more efficient than the gyroresonance. In QLT, the projected particle speed should be comparable to phase speed of the magnetic field compression according to the $\\delta$ function for the TTD resonance.. This means that only particles with a specific pitch angle \ncan be scattered. For the rest of the pitch angles, the interaction is still dominated by gyroresonance, which efficiency is negligibly small for the Alfv\\'enic anisotropic turbulence (see \\S\\ref{Alf_scatter}). With the resonance broadening, however, wider range of pitch angle can be scattered through TTD, including $90^\\circ$. \n\n\n\\subsection{Results from test particle simulations}\n\nWe live in an era when we can test various processes in astrophysics and numerical studies have become an important part of theoretical efforts. Test particle simulation has been used to study CR scattering and\ntransport \\cite{Giacalone_Jok1999, Mace2000}. The aforementioned studies, however, used synthetic\ndata for turbulent fields, which have several disadvantages.\nCreating synthetic turbulence data which has scale-dependent\nanisotropy with respect to the local magnetic field (as observed\nin \\citealt{CV00} and \\citealt{MG01}) is difficult\nand has not been realized yet.  Also,\nsynthetic data normally uses Gaussian statistics and\ndelta-correlated fields, which is hardly appropriate for\ndescription of strong turbulence. \n\nUsing the results of direct numerical MHD simulations as the input data, \\cite{BYL2011} and \\cite{Xu_Yan} performed test particle simulations. Their results show good correspondence with the analytical predictions. We briefly summarize the results here. As shown in Fig.\\ref{xx_yy}, particles' motion is diffusive both along the magnetic field (x direction) and across the field (y direction). Moreover, the scattering coefficient shows the same pitch angle dependence as that predicted in \\cite{YL08}, namely the scattering is most efficient for large pitch angles due to the TTD mirror interaction (see Fig. \\ref{xx_yy} {\\em left}).  \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{duu.eps}\n\\includegraphics[width=0.45\\textwidth]{ratio.jpg}\n\\caption{{\\em Left}: dimensionless CR scattering coefficient $D_{\\mu\n    \\mu}\/\\Omega$ vs the pitch angle $\\mu$. It is dominated by TTD resonant mirror interaction with compressible modes; {\\em right:} Diffusive behavior of the particles displayed in the tracing\n  simulations.  Both the parallel and perpendicular transport are normal diffusion, and the ratio of their diffusion coefficients is $\\sim M_A^4$, consistent with the analytical prediction in \\cite{YL08} \\citep*[from][]{Xu_Yan}.}\n\\label{xx_yy}\n\\end{figure}\n\n\n\\section{Cosmic ray propagation in Galaxy}\n\\label{results}\nThe scattering by fast modes is influenced by the medium properties as the fast modes are subject to linear damping, e.g., Landau damping.\n Using the approach above we revisit the problem of the CR propagation in the selected phases of the ISM (see Table~\\ref{ch1t1} for a list of fiducial parameters appropriate for the idealized phases\\footnote{The parameters of idealized interstellar phases are a subject of debate. Recently, even the entire concept of the phase being stable\n entities has been challenged \\citep[see][and ref. therein]{Gazol:2007}. Indeed different parts\n of interstellar medium can exhibit variations of these parameters \\citep[see][and ref. therein]{Wolfire:2003}}) assuming that turbulence is injected on large scales.\n\\begin{table*}\n{\\footnotesize \\begin{tabular}{ccccccc}\n\\hline\n\\hline \n ISM&\nhalo&\n HIM&\n WIM&\n WNM&\n CNM&\n DC\\tabularnewline\n\\hline\nT(K)&\n $2\\times 10^6$&\n $1\\times10^{6}$&\n 8000&\n 6000&\n 100&\n 15\\tabularnewline\n$c_S$(km\/s)&\n130&\n91&\n8.1&\n7&\n0.91&\n0.35\\tabularnewline\nn(cm$^{-3}$)&\n $10^{-3}$&\n $4\\times10^{-3}$&\n 0.1&\n 0.4&\n 30&\n 200\\tabularnewline\n$l_{mfp}$(cm)&\n$4\\times 10^{19}$&\n$2\\times10^{18}$&\n$6\\times10^{12}$&\n$8\\times10^{11}$&\n$3\\times10^{6}$&\n$10^{4}$\\tabularnewline\nL(pc)&\n 100&\n 100&\n 50&\n 50&\n 50&\n 50\\tabularnewline\nB($\\mu$G)&\n5&\n2&\n5&\n5&\n5&\n15\\tabularnewline\n$\\beta$&\n0.28&\n3.5&\n0.11&\n0.33&\n0.42&\n0.046\\tabularnewline\ndamping&\n collisionless&\n collisional&\n collisional&\n neutral-ion&\n neutral-ion&\n neutral-ion\\tabularnewline\n\\hline\n\\hline\n\\end{tabular}\n\\caption{The parameters of idealized ISM phases and relevant damping. The\ndominant damping mechanism for turbulence is given in the last line. HIM=hot ionized medium, CNM=cold neutral medium, WNM=warm neutral\nmedium, WIM=warm ionized medium, DC=dark cloud.}}\n\\label{ch1t1}\n\\end{table*}\n\n\\subsection{Halo}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{f5.eps}\n\\includegraphics[width=0.45\\textwidth]{f9.eps}\n\\caption{\\small {\\em Left}: The turbulence truncation scales in Galactic halo and warm ionized medium (WIM). The damping curves flattens around $90^\\circ$ due to field line wandering (dotted lines, see \\citealt*{YL04, LVC04}); For WIM, both viscous and collisionless damping are applicable; {\\em right}: The mean free paths in two different phases of ISM: halo (solid line) and WIM (dashed line). At lower energies ($\\sim<100$GeV), the different dependence in WIM is owing to the viscous damping \\citep[from][]{YL08}.}\n\\label{mfp}\n\\end{figure}\n\nIn Galactic halo (see Table~\\ref{ch1t1}), the Coulomb collisional mean free path is $\\sim 10$ pc, the plasma is thus in a collisionless regime. The cascading rate \nof the fast modes is \\citep{CL02_PRL}\n\\begin{equation}\n\\tau_k^{-1}=(k\/L)^{1\/2}\\delta V^2\/V_{ph}.\n\\label{tcasfast}\n\\end{equation}\n\nBy equating it with the collisionless damping rate \n\\begin{equation}\n\\Gamma_{c} = \\frac{\\sqrt{\\pi\\beta}\\sin^{2}\\theta}{2\\cos\\theta}kv_A\\times \\left[\\sqrt{\\frac{m_e}{m_i}}\\exp\\left(-\\frac{m_e}{\\beta m_i\\cos^2\\theta}\\right)+5\\exp\\left(-\\frac{1}{\\beta\\cos^{2}\\theta}\\right)\\right],\n\\label{Ginz}\n\\end{equation}\nwe obtain the turbulence truncation scale $k_c$:\n\\begin{equation}\nk_c L\\simeq \\frac{4M_A^4m_i\\cos^2\\theta}{\\pi m_e\\beta\\sin^4\\theta}\\exp\\left(\\frac{2m_e}{\\beta m_i\\cos^2\\theta}\\right).\n\\label{landauk}\n\\end{equation}\nwhere $\\beta=P_{gas}\/P_{mag}$.\n\nThe scale $k_c$ depends on the {\\it wave pitch angle} $\\theta$, which makes\nthe damping anisotropic. As the turbulence undergoes cascade and the waves propagate in a turbulent medium, the angle $\\theta$ is changing.\nAs discussed in YL04 the field wandering defines the spread of angles. During one cascading time, the fast modes propagate a distance \n$v\\tau_{cas} $ and see an angular deviation $\\tan \\delta \\theta \\simeq \\sqrt{\\tan^2\\delta \\theta_\\parallel+\\tan^2 \\delta\\theta_\\perp}$, which is\n\\begin{equation}\n\\tan \\delta\\theta \\simeq  \\sqrt{\\frac{M_A^2\\cos\\theta}{27(kL)^{1\/2}}+\\left(\\frac{M_A^2\\sin^2\\theta}{kL}\\right)^{1\/3}}\n\\label{dthetaB}\n\\end{equation}\nAs evident, the damping scale given by Eq.(\\ref{landauk}) varies considerably especially when $\\theta\\rightarrow 0$ and $\\theta\\rightarrow 90^\\circ$. For the quasi-parallel modes, the randomization ($\\propto (kL)^{-1\/4}$) is negligible since the turbulence cascade continues to very small scales. On small scales, most energy of the fast modes is contained in these quasi-parallel modes \\citep*{YL04, Petrosian:2006}.\n\nFor the quasi-perpendicular modes, the damping rate (Eq.\\ref{Ginz}) should be averaged over the range $90^\\circ-\\delta\\theta$ to $90^\\circ$. Equating Eq.(\\ref{tcasfast}) and Eq.(\\ref{Ginz}) averaged over $\\delta\\theta$, we get the averaged damping wave number (see Fig.\\ref{mfp} {\\em left}). The field line wandering has a marginal effect on the gyroresonance, whose interaction with the quasi-perpendicular modes is negligible (YL04). However, TTD scattering rates of moderate energy CRs ($<10$TeV) will be decreased owing to the increase of the damping around the $90^\\circ$ (see Fig.\\ref{mfp} {\\em left}). For higher energy CRs, the influence of damping is marginal and so is that of field line wandering. \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{f7.eps}\n\\includegraphics[width=0.45\\textwidth]{f8.eps}\n\\caption{Pitch angle diffusion coefficients in halo and WIM. Upper lines in the plots represent the contribution from TTD and lower lines are for gyroresonance \\citep[from][]{YL08}.}\n\\label{fastcompr}\n\\end{figure}\n\nThe QLT result on gyroresonance in the range $\\mu>\\Delta \\mu$ provides a good approximation to the non-linear results \\citep{YL08}. For CRs with sufficiently small rigidities, the resonant fast modes ($k_{res}\\approx 1\/(R\\mu)$) are on small scales with a quasi-slab structure (see Fig.\\ref{mfp} {\\em left}). For the scattering by these quasi-parallel modes, the analytical result that follows from QLT approximation \\citep[see][]{YL04} for the gyroresonance is\\footnote{It can be shown that the QLT result follows from our more general results (see Eqs.\\ref{general}, \\ref{resfunc}) if we put $\\Delta \\mu \\rightarrow 0$. This justifies our use of the analytical approximation.} \n\n\\begin{equation}\n\\left[\\begin{array}{c}\nD^{G}_{\\mu\\mu}\\\\\nD^{G}_{pp}\\end{array}\\right]=\\frac{\\pi v \\mu^{0.5}(1-\\mu^{2})}{4LR^{0.5}}\\left[\\begin{array}{c}\n\\frac{1}{7}[1+(R\\mu)^2]^{-\\frac{7}{4}}-(\\tan^{2}\\theta_c+1)^{-\\frac{7}{4}}\\\\\n\\frac{m^{2}V_{A}^{2}}{3}\\left\\{[1+(R\\mu)^2]^{-\\frac{3}{4}}-(\\tan^{2}\\theta_c+1)^{-\\frac{3}{4}}\\right\\}\\end{array}\\right]\n\\label{lbgyro}\\end{equation}\nwhere $\\tan\\theta_c={k_{\\perp,c}}\/{k_{\\parallel,res}}$.\n\nOnce we know the functional form of the $D_{\\mu\\mu}$, we can obtain the corresponding mean free path \\citep{Earl:1974}:\n\\begin{equation}\n\\lambda_\\|\/L=\\frac{3}{4}\\int^1_0 d\\mu \\frac{v(1-\\mu^2)^2}{(D^T_{\\mu\\mu}+D^G_{\\mu\\mu})L},\n\\end{equation}\nwhere $D^T_{\\mu\\mu}$ is the contribution from TTD interaction and can be obtained using the nonlinear theory (see \\citealt{YL08}, and also \\S\\ref{NLT_sec}) with the inertial range of fast modes determined for the local medium (see, e.g. \\ref{landauk} in the case of collisionless damping). \n\nThe mean free path is sensitive to the scattering by gyroresonance at small pitch angles, due to the influence of damping on the fast modes on small scales. Fig.\\ref{fastcompr} shows the pitch angle diffusion of CRs with different energies due to the TTD and gyroresonance.\n \nThe weak dependence of the mean free path (see Fig.\\ref{mfp} {\\em right}) of the moderate energy (e.g$<1$TeV) CRs in halo results from the fact that gyroresonance changes marginally with the CR energy (see Fig.\\ref{fastcompr}). This is associated with the damping in collisionless medium. We expect that similar flat dependence can happen in any collisionless medium. This can be a natural explanation of the puzzling ``Palmer Concensus\" \\citep{Palmer:1982}, the same trend observed in solar wind.\n\n\\subsection{Warm Ionized Medium}\n\nIn warm ionized medium, the Coulomb collisional mean free path is $l_{mfp}=6\\times 10^{12}$ cm and the plasma $\\beta\\simeq0.11$. Suppose that the turbulence energy is injected from large scale, then the compressible turbulence is subjected to the viscous damping besides the collisionless damping. \nBy equating the viscous damping rate with the cascading rate (Eq.\\ref{tcasfast}), we obtain the following truncation scale, \n\\begin{eqnarray}\nk_{c}L=x_c\\left\\{\\begin{array}{rl}(1-\\xi^2)^{-\\frac{2}{3}} & \\beta\\ll 1\\\\\n(1-3\\xi^2)^{-\\frac{4}{3}} & \\beta\\gg 1\\end{array}\\right.\n\\end{eqnarray}\nwhere $x_c=\\left[\\frac{6\\rho\\delta V^2L}{\\eta_0V_A}\\right]^{\\frac{2}{3}}$,\n $\\eta_0$ is the longitudinal viscosity. In the low $\\beta$ regime, the motions are primarily perpendicular to the magnetic field so that $\\partial v_{x}\/\\partial x=\\dot{n}\/n\\sim\\dot{B}\/B$. The longitudinal viscosity enters here as the result of distortion of the Maxiwellian distribution \\citep[see][]{Braginskii:1965}. The transverse energy of the ions increases during compression because of the conservation of adiabatic invariant $v_{\\perp}^{2}\/B$. If the rate of compression is faster than that of collisions, the ion distribution in the momentum space is bound to be distorted from the Maxiwellian isotropic sphere to an oblate spheroid with the long axis perpendicular to the magnetic field. As a result, the transverse pressure gets greater than the longitudinal pressure, resulting in a stress $\\sim\\eta_{0}\\partial v_{x}\/\\partial x$.\nThe restoration of the equilibrium increases the entropy and causes the dissipation of energy.\n\n\nThe viscous damping scale is compared to collisionless cutoff scale (Eq.\\ref{landauk}) in Fig.\\ref{mfp} {\\it left}. \nAs shown there, both viscous damping and collisionless damping are important in WIM. Viscous damping is dominant for small $\\theta$ and\n collisionless damping takes over for large $\\theta$ except for $\\theta=90^\\circ$.\nThis is because collisionless damping increases with $\\theta$ much faster than the viscous damping. For sufficiently small wave pitch angles, the viscous damping is too small to prevent the fast modes to cascade down to scales smaller than the mean free path $l_{mfp}$. Because of the similar quasi-slab structure on small scales, \nEq.(\\ref{lbgyro}) can be also applied in WIM. The results are illustrated in Fig.\\ref{fastcompr}. Compared to the case in halo, we see that the qualitative difference stands in the gyroresonance. This is because gyroresonance is sensitive to the quasi-slab modes whose damping differs in halo and WIM.  \n\n\\subsection{Other phases}\n\nIn hot ionized medium (HIM), the plasma is also in collisionless regime, but the density is higher and the plasma beta is larger than 1. The damping by protons thus becomes substantial especially at small pitch angles. The damping truncates the turbulence at much larger scales than the gyroscales of the CRs of the energy range we consider. No gyroresonance can happen and some other mechanisms are necessary to prevent CRs streaming freely along the field. The turbulence injected from small scales might play an important role (see \\S6). \n\n\nIn partially ionized gas one should take into account an additional damping that arises from ion-neutral collisions \\citep[see][]{Kulsrud_Pearce, LG01, LVC04}. In the latter work a viscosity-damped regime of turbulence was predicted at scales less the scale $k_{c, amb}^{-1}$ at which the ordinary magnetic turbulence is damped by ionic viscosity. The corresponding numerical work, e.g., \\cite{CLV_newregime} testifies that for the viscosity-damped regime the parallel scale stays equal to the scale of the ambipolar damping, \ni.e., $k_{\\|}=k_{c, amb}$, while $k_{\\bot}$ increases. In that respect, the scattering by such magnetic fluctuations is analogous to the scattering induced by the weak turbulence (see \\S 2.3, \\citealt{YL08}). The difference stems from the spectrum\nof $k_{\\bot}$ is shallower than the spectrum of the weak turbulence. The predicted values of the spectrum for the viscosity-damped turbulence $E(k_\\bot)\\sim\nk_\\bot^{-1}$ \\citep{LVC04} are in rough agreement with simulations. More detailed studies of scattering in partially ionized gas will be necessary.    \n\n\\section{Perpendicular transport}\n\nIn this section we deal with the diffusion perpendicular\nto {\\it mean} magnetic field. \n\nPropagation of CRs perpendicular to the mean magnetic field is another important problem in which QLT encounters serious difficulties.\nCompound diffusion, resulting from the convolution of diffusion along the magnetic field line and diffusion of field line perpendicular to mean field direction, has been invoked to discuss transport of cosmic rays in the Milky Way \\citep*{Getmantsev, Lingenfelter:1971,Allan:1972}. The role of compound diffusion in the acceleration\n of CRs at quasi-perpendicular shocks were investigated by \\cite{Duffy:1995} and \\cite{Kirk:1996}.  \n\nIndeed, the idea of CR transport in the direction perpendicular to the mean magnetic field being dominated by the field line random walk \n(FLRW, \\citealt{Jokipii1966, Jokipii_Parker1969, Forman1974}) can be easily justified\nonly in a restricted situation where the turbulence perturbations are small and CRs do not scatter backwards to retrace their trajectories. If the latter is not true, the particle motions are subdiffusive, \ni.e., the squared distance diffused growing as not as $t$ but as $t^{\\alpha}$, $\\alpha<1$, e.g., $\\alpha=1\/2$ \\citep{Kota_Jok2000, Mace2000, Qin2002}.\nIf true, this could indicate a substantial shift in the paradigm of CR transport, a shift that surely dwarfs a modification of magnetic turbulence model from the 2D+slab to a more simulation-motivated model that we deal here.\n\n \nIt was also proposed that with substantial transverse structure, {\\it i.e.}, transverse displacement of field lines, perpendicular diffusion is recovered \\citep{Qin2002}. Is it the case of the MHD turbulence models we deal with? \n\n\nHow realistic is the subdiffusion in the presence of turbulence? The answer for this question apparently depends on the models of turbulence chosen. \n\nCompound diffusion happens when particles are restricted to the magnetic field lines and perpendicular transport is solely due to the random walk of field line wandering \\citep[see][]{Kota_Jok2000}. \nIn the three-dimensional turbulence, field lines are diverging away due to shearing by the Alfv\\'en modes \\citep[see][]{LV99, Narayan_Medv, Lazarian06}.\n Since the Larmor radii of CRs are much larger than the minimum scale of eddies $l_{\\bot, min}$, field lines within the CR Larmor orbit are effectively diverging away owing to shear by the Alfv\\'enic turbulence.\nThe cross-field transport thus results from the deviations of field lines at small scales, as well as field line random walk at large scale ($>{\\rm min}[L\/M^3_A,L]$).\n\nBoth observation of Galactic CRs and solar wind indicate that the diffusion of CRs perpendicular to magnetic field is normal diffusion \\citep[]{Giacalone_Jok1999, Maclennan2001}. Why is that?\n\nMost recently the diffusion in magnetic fields was considered for thermal particles in Lazarian (2006), for cosmic rays in \\cite{YL08}. In what follows we present the results based on the studies in  \\cite{YL08}.\n\n\\subsection{Perpendicular diffusion on large scale}\n\nFor perpendicular diffusion, the important issue is the reference frame. We emphasize that we consider the diffusion perpendicular to the {\\emph mean} field direction in the global reference of frame.  \n\n{\\it High $M_A$ turbulence}: High $M_A$ turbulence corresponds to the field that is easily bended by\nhydrodynamic motions at the injection scale as the hydro energy at the\ninjection scale is much larger than the magnetic energy, i.e.\n$\\rho V_L^2\\gg B^2$. In this case\nmagnetic field becomes dynamically important on a much smaller scale, i.e. the \nscale $l_A=L\/M_A^3$ \\citep[see][]{Lazarian06}. If the parallel mean free path of CRs $\\lambda_\\|\\ll l_A$, the stiffness of B field is negligible so that the perpendicular diffusion coefficient is the same as the parallel one, i.e., $D_\\bot=D_\\|\\sim 1\/3 \\lambda_{\\|} v$. If $\\lambda_\\|\\gg l_A$, the\n diffusion is controlled by the straightness of the field lines, and $\nD_\\bot=D_{\\|}\\approx 1\/3l_Av.\n\\label{dbb}\n$ The diffusion is isotropic if scales larger than $l_A$ are\nconcerned. \n\n{\\it Low $M_A$ turbulence}: In the magnetically dominated case, i.e. the field that cannot be easily bended at\nthe turbulence injection scale, individual magnetic field lines are aligned\nwith the mean magnetic field. The diffusion in this case is anisotropic.\nIf turbulence is injected at scale $L$ it stays \nweak for the scales larger than $LM_A^2$ and it is \nstrong at smaller scales. Consider first the case of $\\lambda_\\|>L$.\nThe time of the individual step is $L\/v_\\|$, then $D_\\perp\\approx 1\/3Lv M_A^4.$\nThis is similar to the case discussed in the FLRW model (Jokipii 1966). However, we obtain the dependence of $M_A^4$ instead of their $M_A^2$ scaling. In the opposite case of $\\lambda_\\|<L$, the perpendicular diffusion coefficient is $\nD_{\\bot}\\approx D_{\\|}M_A^4,\n\\label{diffx}$ which coincides with the result\nobtained for the diffusion of thermal electrons in magnetized plasma \\citep{Lazarian06}. This is due to the anisotropy of the Alfv\\'enic turbulence.\n\n\\subsection{Superdiffusion on small scales}\n\nThe diffusion of CR on the scales $\\ll L$ is different and it is determined by how fast field lines are diverging away from each other. The mean deviation of a field in a distance $\\delta x$ is proportional to $[\\delta z]^{3\/2}$  \\citep{LV99, Lazarian06}, same as Richardson diffusion in the case of hydrodynamic turbulence \\cite[see][]{Eyink2011}. Following the argument, we showed in \\cite{YL08} that the cosmic ray perpendicular transport is superdiffusive. The reason is that there is no random walk on small scales up to the injection scale of strong MHD turbulence ($LM_A^2$ for $M_A < 1$\n and $l_A$ for $M_A>1$). This can well explain the recently observed super-diffusion in solar wind \\citep{Perri2009}. Superdiffusion can have important implications for shock acceleration as discussed in details in \\cite{LY13}.\n\n\n\\subsection{Is there subdiffusion?}\nThe diffusion coefficient $D_{\\|}M_A^4$ we obtained in the case of $M_A<1$, means that the transport\nperpendicular to the dynamically strong magnetic field is a normal diffusion, rather\nthan the subdiffusion as discussed in a number of recent papers. This is also supported by test particle simulations (\\citealt*{BYL2011, Xu_Yan}, see Fig.\\ref{xx_yy} {\\it right}). Let us\nclarify this point by obtaining the necessary conditions for the subdiffusion\nto take place.\n\nThe major implicit assumption in subdiffusion (or compound diffusion) is that the particles trace back \ntheir \ntrajectories in x direction on the scale $\\delta z$. When is it possible to talk about retracing of particles? In the case of random motions at a single scale {\\it only}, the distance over \nwhich the particle\ntrajectories get uncorrelated is given by the \\cite{RR1978}\nmodel. Assuming that the damping scale of the turbulence  is larger\nthat the CR Larmor radius, the \\cite{RR1978}\nmodel, when generalized to anisotropic turbulence provides \\citep{Narayan_Medv, Lazarian06} $L_{RR}=l_{\\|, min}\\ln(l_{\\bot, min}\/r_{Lar})$\nwhere $l_{\\|, min}$ is the parallel scale of the cut-off of turbulent motions, \n$l_{\\bot, min}$ is the corresponding perpendicular scale, $r_{Lar}$ is the\nCR Larmor radius. The assumption of $r_{Lar}<l_{\\bot, min}$ can be\n valid, for instance, for Alfv\\'enic motions in partially ionized gas.\nHowever, it is easy to see that, even in this case,\n the corresponding scale is rather\nsmall and therefore subdiffusion is not applicable for the transport\nof particles in Alfv\\'enic turbulence over scales $\\gg l_{\\|,min}$.\n\nIf  $r_{Lar}>l_{\\bot, min}$, as it is a usual case for Alfv\\'en motions in the\nphase of ISM with the ionization larger than $\\approx 93\\%$, where the\nAlfv\\'enic motions go to the thermal particle gyroradius \n\\citep[see estimates in][]{LG01, LVC04}, \nthe subdiffusion of CR is not an applicable concept for Alfv\\'enic turbulence.  \nThis does\nnot preclude subdiffusion from taking place\nin particular models of magnetic perturbations,\ne.g. in the slab model considered in \\cite{Kota_Jok2000}, but we believe in the omnipresence of Alfv\\'enic turbulence in interstellar gas \\citep[see][]{Armstrong95}.\n\n\\section{Streaming Instability in the Presence of Turbulence}\n\n\\begin{table}\n\\caption{The notation we used in this section}\n\\label{notations}\n\\begin{tabular}{|c|r|}\n\\hline\nA  & normalized wave amplitude $\\delta B\/B_0$\\\\\na& hardening of the CR spectrum at the shock front\\\\\n$B_0$ & mean magnetic field at the shock in the later Sedov phase\\\\\n$B_{cav}$ &inercloud magnetic field strength\\\\\n$\\delta B$& wave amplitude\\\\\nc & light speed\\\\\nd& distance of the molecular cloud from observer\\\\\nD& diffusion coefficient of CRs\\\\\nE& CR energy\\\\\n$E_{SN}$& supernova explosion\\\\\nf& distribution function of CRs\\\\\n$f_\\pi$& fraction of energy transferred from parent protons to pions\\\\\nk& wave number\\\\\nK(t) & Normalization factor of CR distribution function\\\\\nL& the injection scale of background turbulence\\\\\nm& proton rest mass\\\\\n$M_c$ & cloud mass\\\\ \nn& intercloud number density\\\\ \n$N_\\gamma$ & $\\gamma$ ray flux\\\\\np& CR's momentum\\\\\n$p_{max}$& the maximum momentum accelerated at the shock front\\\\\n$P_{CR}$ & CR pressure\\\\\nq & charge of the particle\\\\\nr& distance from SNR centre\\\\\n$R_c$& the distance of the molecular cloud from the SNR centre\\\\\n$r_g$ & Larmor radius of CRs\\\\\n$R_d$ & diffusion distance of CRs\\\\\n$R_{sh}$ & shock radius\\\\\n$R_{esp},\\,t_{esp}$& the escaping distance\/time of CRs\\\\ \ns& 1D spectrum index of CR distribution\\\\\nt& time since supernova explosion\\\\\n$t_{age}$ & the age of SNR\\\\\n$t_{sed}$ & the time at which SNR enters the Sedov phase\\\\\nU& shock  speed\\\\\n$U_i$ &initial shock velocity\\\\\nv& particle speed\\\\\n$v_s$ & streaming speed of CRs\\\\\nW& wave energy\\\\\n$\\alpha$& power index of D with respect to particle momentum p\\\\\n$\\chi$&reduction factor of D with respect to $D_{ISM}$\\\\\n$\\delta$& power index of $p_{max}$ with respect to t\\\\\n $\\eta$&fraction of SN energy converted into CRs\\\\\n $\\eta_A$& a numerical factor in Eq.\\ref{pp}\\\\\n$\\Gamma_{cr}$& the growth rate of streaming instability\\\\\n$\\Gamma_d$& wave damping rate\\\\\n $ \\kappa$&ratio of diffusion length to shock radius\\\\\n $\\Omega_0$ & the Larmor frequency of non-relativistic protons\\\\\n $\\sigma_{pp}$ &cross section for pp collision\\\\\n $\\xi$& the ratio of CR pressure to fluid ram pressure\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n                \nWhen cosmic rays stream at a velocity much larger than Alfv\\'{e}n\nvelocity, they can excite by gyroresonance MHD modes which in turn scatter\ncosmic rays back, thus increasing the amplitude of the resonant mode. This \nrunaway process is known as streaming instability. It was claimed\nthat the instability could provide confinement for cosmic rays with\nenergy less than $\\sim 10^2$GeV \\citep{Cesarsky80}. However, this was calculated\nin an ideal regime, namely, there was no background MHD turbulence.\nIn other words, it was thought that the\nself-excited modes would not be appreciably damped in fully ionized gas\\footnote{We neglect the nonlinear Landau damping, which is suppressed in turbulence due to decrease of mean free path.}. \n\nThis is not true for turbulent medium, however. \\citet{YL02}\npointed out that the streaming instability is partially suppressed\nin the presence of background turbulence \\citep[see more in][]{LCY02}. More recently, detailed calculations of the \nstreaming instability in the presence of background Alfv\\'enic turbulence \nwere presented in \\cite{FG04}. The growth rate of the modes of wave number $k$ is \\citep{Longairbook}.\n\\begin{equation}\n\\Gamma_{cr}(k)=\\Omega_0\\frac{N(\\geq E)}{n_{p}}(-1+\\frac{v_{stream}}{V_{A}}),\n\\label{instability}\n\\end{equation}\n where $N(\\geq E)$ is the number density of cosmic rays with energy\n$\\geq E$ which resonate with the wave,\n$n_{p}$ is the number density of charged particles in the medium.\nThe number density of cosmic rays near the sun is $N(\\geq E)\\simeq 2\\times10^{-10}(E\/$GeV$)^{-1.6}$\ncm$^{-3}$sr$^{-1}$ \\citep{Wentzel74}.\n\nInteraction with fast modes was considered by \\citet{YL04}. \nSuch an interaction happens at the rate \n$\\tau_k\\sim (k\/L)^{-1\/2}V_{ph}\/V^2$. \nBy equating it with the growth rate Eq.(\\ref{instability}), we can find that the streaming instability\nis only applicable for particles with energy less than\n\\begin{equation}\n\\gamma_{max}\\simeq1.5\\times 10^{-9}[n_{p}^{-1}(V_{ph}\/V)(Lv\\Omega_0\/V^2)^{0.5}]^{1\/1.1},\n\\end{equation}\nwhich for HIM, provides $\\sim 20$GeV if taking the injection speed to be $V\\simeq 25$km\/s. Similar result was obtained with Alfv\\'en modes by \\citet{FG04}. \n\n\nMagnetic field itself is likely to be amplified through an inverse \ncascade of magnetic energy at which perturbations created at a particular\n$k$ diffuse in $k$ space to smaller $k$ thus inducing inverse cascade. \nAs the result, the magnetic perturbations at smaller $k$ get larger than the \nregular field. Consequently, even if the instability is suppressed\nfor the growth rate given by Eq. (\\ref{instability}) it gets efficient\ndue to the increase of perturbations of magnetic field stemming from the\ninverse cascade. The precise picture of the process depends on yet not completely\nclear details of the inverse cascade of magnetic field. \n\nBelow, we present the application of the current understanding of the interaction between the streaming instability and the background turbulence to the modeling of the gamma ray emission from molecular clouds near SNRs \\citep[see more details in][]{YLS12}. We shall treat the problem in a self-consistent way by comparing the streaming level that is allowed by the preexisting turbulence and the required diffusion for the CRs. \n\n\\subsection{Application to CR acceleration at the shocks }\n\\label{pmax}\nDiffusive shock acceleration of energetic CR particles relies on the crucial process of amplification of MHD turbulence so that particles can be trapped at the shock front long enough to be accelerated to the high energy observed. One of the most popular scenarios that has been adopted in the literature is the streaming instability generated by the accelerated particles. However, in the highly nonlinear regime the fluctuations of magnetic field arising from the streaming\ninstability get large and the classical treatment of the streaming instability is not applicable. We circumvent the\nproblem by proposing that the field amplification we consider does not arise from the streaming\ninstability, but is achieved earlier through other processes, e.g. the interaction of the shock precursor with density perturbations preexisting in the interstellar medium \\citep*{BJL09}. Due to the\nresonant nature of the streaming instability, the perturbations $\\delta B$ arising from it are more efficient\nin scattering CRs compared to the large scale fluctuations produced by non-resonant mechanisms, e.g.\nthe one in \\citet{BJL09}. Therefore in this chapter, we limit our discussions to the regime of $\\delta B \\sim< B_0$, where $B_0$ is the magnetic field that has already been amplified in the precursor region\\footnote{The effective $B_0$ is therefore renormalized and can be much larger than the typical field in ISM (see, e.g., \\citealt{Diamond_Makov}).}.  \n\nWhen particles reach the maximum energy at a certain time, they escape and the growth of the streaming instability stops. Therefore we can obtain the maximum energy by considering the stationary state of the evolution. The steady state energy density of the turbulence $W(k)$ at the shock is determined by\n\n\\begin{equation}\n(U\\pm v_A)\\nabla W(k) = 2 (\\Gamma_{cr}-\\Gamma_d)W(k),\n\\label{wave}\n\\end{equation}\nwhere $U$ is the shock speed, and the term on the l.h.s. represents the advection of turbulence by the shock flow. $v_A\\equiv B_0\/\\sqrt{4\\pi nm}$ and $n$ are the Alfv\\'en speed and the ionized gas number density of the precursor region, respectively. The plus sign represents the forward propagating Alfv\\'en waves and the minus sign refers to the backward propagating Alfv\\'en waves. The terms on the r.h.s. describes the wave amplification by the streaming instability and damping with $\\Gamma_d$ as the corresponding damping rate of the wave.  The distribution of accelerated particles at strong shocks is $f(p)\\propto p^{-4}$. If taking into account the modification of the shock structure by the accelerated particles, the CR spectrum becomes harder. Assume the distribution of CRs at the shock is $f_0(p)\\propto p^{-4+a}$. The nonlinear growth was studied by \\citet{Ptuskin:2005}. \n\nThe generalized growth rate of streaming instability is\n\n\\begin{eqnarray}\n\\Gamma_{cr}&=&\\frac{12\\pi^2 q^2v_A\\sqrt{1+A^2}}{c^2k}\\nonumber\\\\\n&\\times& \\int^\\infty_{p_{res}} dp p\\left[1-\\left(\\frac{p_{res}}{p}\\right)^2\\right]D\\left|\\frac{\\partial f}{\\partial x}\\right|, \n\\label{general_growth}\n\\end{eqnarray}\nwhere $q$ is the charge of the particle, c is the light speed, $p_{res}=ZeB_0\\sqrt{1+A^2}\/c\/k_{res}$ is the momentum of particles that resonate with the waves.  $A=\\delta B\/B_0$ is wave amplitude normalized by the mean magnetic field strength $B_0$.\n\\begin{equation}\nD=\\sqrt{1+A^2}v r_g\/3\/A^2(>k_{res})\n\\label{crdiff}\n\\end{equation}\nis the diffusion coefficient of CRs, $v$ and $r_g$ are the velocity and Larmor radius of the CRs.  \nIn the planar shock approximation, one gets the following growth rate of the upstream forward moving wave at x=0,\n\\begin{equation}\n\\Gamma_{cr}(k)=\\frac{C_{cr}\\xi U^2(U+v_A)k^{1-a}}{(1+A^2)^{(1-a)\/2}cv_A\\phi(p_{max})r_0^a} \n\\label{growth}\n\\end{equation}\nwhere $C_{cr}=4.5\/(4-a)\/(2-a)$, $r_0=m c^2\/q\/B_0$, where $\\xi$ measures the ratio of CR pressure at the shock and the upstream  momentum flux entering the shock front, $m$ is the proton rest mass, and $p_{max}$ is the maximum momentum accelerated at the shock front. $H(p)$ is the Heaviside step function.\n\nThe linear damping is negligible since the medium should be highly ionized. In fully ionized gas, there is nonlinear Landau damping, which, however, is suppressed due to the reduction of particles' mean free path in the turbulent medium \\citep[see][]{YL11}. We therefore neglect this process here. Background turbulence itself can cause nonlinear damping to the waves \\citep{YL02}. Unlike hydrodynamical turbulence, MHD turbulence is anisotropic with eddies elongated along the magnetic field. The anisotropy increases with the decrease of the scale \\citep{GS95}. Because of the scale disparity, $k_\\| > k_\\bot \\gg k^t_\\|$, the nonlinear damping rate in MHD turbulence is less than the wave frequency $k_\\| v_A$, and it is given by \\citep{FG04, YL04}\n\n\\begin{equation}\n\\Gamma_d \\sim \\sqrt{k\/L} v_A,\n\\label{damping}\n\\end{equation}\nwhere L is the injection scale of background turbulence, and the $k$ is set by the resonance condition $k \\sim k_\\| \\sim 1\/r_L$.\n\nThere are various models for the diffusive shock acceleration. We consider here the escape-limited acceleration. In this model, particles are confined in the region near the shock where turbulence is generated. Once they propagate far upstream at a distance $l$ from the shock front, where the self-generated turbulence by CRs fades away, the particles escape and the acceleration ceases.  The characteristic length that particles penetrate into the upstream is $D(p)\/U$. The maximum  momentum is reached when $D(p)\/U\\simeq l\/4$\\footnote{The factor 1\/4 arises from the following reason. As pointed out by \\cite{Ostrowski:1996}, the spectrum is steepened for small l, i.e., $l U\/D(p) \\sim< 4$}. Assuming $l=\\kappa R_{sh}$, where $\\kappa<1$ is a numerical factor, one can get\n\\begin{equation}\n\\frac{p_{max}}{mc} = \\frac{3\\kappa A^2 U R_{sh}}{\\sqrt{1+A^2}v r_0}.\n\\label{gmax}\n\\end{equation}\n\nIn particular, for $A<1$\n\\begin{eqnarray}\\frac{p_{max}}{mc}&=&\\left[\\left(-v_A\\sqrt{\\frac{1}{r_0 L}}+\\sqrt{\\frac{v_A^2}{r_0 L}+\\frac{2C_{cr}a\\xi U^3(U+v_A)}{\\kappa r_0R_{sh}cv_A}}\\right) \\left(\\frac{\\kappa R_{sh}}{U}\\right)\\right]^2,\\nonumber\\\\\nA&=&\\frac{p_{max}r_0}{\\sqrt{18}\\kappa mU R_{sh}}\\sqrt{1+\\sqrt{1+36 \\left(\\frac{\\kappa mU R_{sh}}{p_{max}r_0}\\right)^2}},\n\\label{gmax_general}\n\\end{eqnarray}\n\nIn the limit of low shock velocity, \n\\begin{eqnarray\nv_A\\ll U\\ll  c\\left[\\left(\\frac{v_A}{c}\\right)^3\\frac{\\kappa R_{sh}}{2 L C_{cr}a\\xi}\\right]^{1\/4},\n\\label{lowshockU}\n\\end{eqnarray}\nwe get \n\\begin{eqnarray\n\\frac{p_{max}}{mc}&=&(C_{cr}\\xi U^3)^2\\frac{a^2 L}{r_0c^2v_A^4}\n\\label{gmax_solution}\n\\end{eqnarray}\nfor the Sedov phase ($t>t_{sed}\\equiv 250(E_{51}\/(n_0U_9^5))^{1\/3}$yr), where $E_{51}=E_{SN}\/10^{51}$erg and $U_9=U_i\/10^9$cm\/s are the total energy of explosion and the initial shock velocity. In Fig.\\ref{Emax}, we plot the evolution of $p_{max}\/(mc)$ during the Sedov phase. The solid line represents the results from Eqs.(\\ref{gmax_general}). As we see, at earlier epoch when advection and streaming instability are both important, the evolution of $p_{max}$ does not follow a power law. For comparison, we also put a power law evolution in the same figure as depicted by Eq.(\\ref{gmax_solution})  (dashed line). \nOur result is also larger than that obtained by \\cite{Ptuskin:2005} since the wave dissipation rate is overestimated in their treatment.\n\n\\begin{figure}\n\\includegraphics[width=0.47\\textwidth]{Emax.eps}\n\\includegraphics[width=0.47\\textwidth]{spectrum0.eps}\n \\caption{\\small {\\em Left}: The energy of CRs that are released at the shock at time t in the Sedov phase. Our result shows that the often assumed power law solution \\citep[see][]{Gabici:2009, Ohira:2010} is only realized in asymptotic regime as described in Eqs.(\\ref{lowshockU},\\ref{gmax_solution}). It is also larger than the earlier result (dotted line) in Ptuskin \\& Zirakashvili (2005) where the damping of the waves by background turbulence is overestimated. {\\em Right}: The spectrum of CRs at a distance $r=12$ pc after 1800 (solid line), 6000 (dotted line), 12000 (dashdot line), 50000 years (cross line). The Galactic mean is plotted as a reference (dashed line). From \\cite{YLS12}.}\n\\label{Emax} \n\\end{figure}\n\n\\subsection{Enhanced scattering and streaming instability near SNRs}\n\\label{nearby}\n\nThe result from \\cite{YLS12} show that the local scattering of CRs has to be enhanced by an order of magnitude $\\chi =0.05$ in order to produce the amount of $\\gamma$ ray emission observed. A natural way to increase the scattering rate is through the streaming instability. The enhanced flux of the CRs are demonstrated to generate strong enough instability to overcome nonlinear damping by the background turbulence \\citep{YLS12}. The growth rate in the linear regime is\n\n\\begin{equation}\n\\Gamma_{gr}=\\Omega_0\\frac{N(\\geq E)}{n}\\left(\\frac{v_s}{v_A}-1\\right),\n\\end{equation}\nwhere $v_s$ is the streaming speed of CRs. The growth rate should overcome the damping rate (eq.\\ref{damping}) for the instability to operate. The condition $\\Gamma_{gr}>\\Gamma_d$ leads to\n\\begin{equation}\nv_s > v_A \\left(1+\\frac{n v_A}{N \\Omega_0\\sqrt{r_gL}}\\right)\n\\end{equation}\n\nThe spatial diffusion coefficient adopted here, $D \\approx v_s L = \\chi D_{ISM}$, satisfies  this requirement. The growth and damping rates are compared in Fig.\\ref{rates} {\\em right}. We see that the streaming instability works in the energy range needed to produce the observed $\\gamma$ ray emission, proving that our results are self-consistent.\n\nNote that the case we consider here is different from the general interstellar medium discussed in \\cite{YL04} and \\cite{FG04}, namely, the local cosmic ray flux near SNRs is much enhanced (see Fig.\\ref{Emax} {\\em right}). Consequently, the growth rate of the streaming instability becomes high enough to overcome the damping rate by the preexisting turbulence in the considered \nenergy range.\n\n\\begin{figure}\n\\includegraphics[width=0.47\\textwidth]{gmray.eps}\n\\includegraphics[width=0.47\\textwidth]{stream.eps}\n \\caption{\\small {\\em Left}: The spectrum of Gamma ray emission from W28. The Fermi data are shown as dotted points \\citep{Abdo:2010}, and the H.E.S.S. data are plotted as 'x' points \\citep{Aharonian:2008} with error bars. Solid line is our result. {\\em Right}: The growth and nonlinear damping rates of streaming instability. With the locally enhanced flux, the growth rate of streaming instability becomes much larger than the mean Galactic value so that it can overcome the nonlinear damping by turbulence for a wide energy range. This is consistent with our earlier treatment in which streaming instability plays an essential role in the cosmic ray diffusion near SNRs. From \\cite{YLS12}.}\n \\label{rates}\n\\end{figure}\n\n\\begin{table}\n\\caption{Model parameters adopted}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\na&$\\chi$&$\\eta$&$ \\kappa$&$\\xi$&$\\alpha$\\\\\n\\hline\n0.1$\\sim 0.3$&$\\sim$0.05&$\\sim 0.3$&$0.04\\sim 0.1$& $0.2\\sim$0.4& 0.5\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Gyroresonance Instability of CRs in Compressible Turbulence}\n\n\\begin{figure}\n\\includegraphics[width=0.55\\columnwidth,\n  height=0.25\\textheight]{feedback.eps}\n\\includegraphics[width=0.4\\columnwidth,\n  height=0.25\\textheight]{YL_fig1c.eps}\n\\caption{{\\em Left}: The spectral energy density of slab waves that is transferred from the large scale compressible turbulence via the gyroresonance instability of CRs. In the case that the instability grows up to the maximum energy rate allowed by the turbulence cascade, large scale turbulence is truncated at $\\lambda_{fb}$ and the wave amplitude $E(k)dk\\sim \\epsilon_N^u$ is given by Eq.(\\ref{energy}). Note that the picture is different from LB06, namely, the feedback on the large scale turbulence occurs only in some cases when the scattering is not sufficient to prevent the waves from growing to the maximum values \\citep{YL11}; {\\em right}: CR scattering is dominated by compressible modes. For high energy CRs ($>\\sim$10GeV), the scattering is due to direct interaction with fast modes; For low energy CRs, the interaction is mainly due to the gyroresonance instability induced by compression of magnetic fields.}\n\\label{feedback_fig}\n\\end{figure}\n\nUntil recently, test particle approximation was assumed in most of earlier studies in which turbulence cascade is established from large scales and no feedback of CRs is included. This may not reflect the reality as we know the energy of CRs is comparable to that in turbulence and magnetic field \\citep[see][]{Kulsrudbook}. It was suggested by \\cite{LB06} that the gyroresonance instability of CRs can drain energy from the large scale turbulence and cause instability on small scales by the turbulence compression induced anisotropy on CRs (see Fig.\\ref{feedback_fig} {\\em left}). And the wave generated on the scales, in turn, provides additional scattering to CRs. In \\cite{YL11}, quantitative studies was provided based on the nonlinear theory of the growth of the instability and the feedback of the growing waves on the distributions of CRs.\n\nIn the presence of background compressible turbulence, the CR distribution is bound to be anisotropic because of the conservation of the first adiabatic invariant $\\mu\\equiv v_\\bot^2\/B$. Such anisotropic distribution is subjected to various instabilities. Waves are generated through the instabilities, enhancing the scattering rates of the particles, their distribution will be relaxed to the state of marginal state of instability even in the collisionless environment.  While the hydrodynamic instability requires certain threshold, the kinetic instability can grow very fast with small deviations from isotropy. Here, we focus on the gyroresonance instability. Both the qualitative and quantitative studies in \\cite{YL11} show that the isotropization rate is roughly $\n\\tau^{-1}_{scatt} \\sim \\frac{\\Gamma_{gr}\\epsilon_N}{\\beta_{CR} A}\\label{nu_est}$, where $\\Gamma_{cr}, \\epsilon_N$ are the instability growth rate and the wave energy normalized by magnetic energy, respectively. $\\beta_{CR}$ is the ratio of CR pressure to magnetic pressure, $A$ is the degree of anisotropy of the CR momentum distribution.\n\nBy balancing the rate of decrease in anisotropy due to scattering and the growth due to compression, one can get\n\\begin{equation}\n\\epsilon_N\\sim \\frac{ \\beta_{CR}\\omega\\delta v}{\\Gamma_{gr} v_A },~~~\\lambda_{CR}=r_p\/\\epsilon_N.\n\\label{epsilon_est},\n\\end{equation}\nwhere $v_A$ is the Alfv\\'en speed, $\\omega, \\delta v$ are the wave frequency and amplitude at the scale that effectively compresses the magnetic field and create anisotropy in CRs' distribution \\citep{YL11}. Since the growth rate decreases with energy, the instability only operates for low energy CRs ($\\sim<$ 100GeV, see Fig.\\ref{feedback_fig} {\\em right}) due to the damping by the preexisting turbulence \\citep{YL11} .\n  \n\\subsection{Bottle-neck for the growth of the instability and feedback on turbulence}\n\\label{feedback}\nThe creation of the slab waves through the CR resonant instability is another channel to drain the energy of large scale turbulence. This process, on one hand, can damp the turbulence. On the other hand, it means that the growth rate is limited by the turbulence cascade. The energy growth rate  cannot be larger than the turbulence energy cascading rate, which is $1\/2 \\rho V_L^4\/v_A\/L$ for fast modes in low $\\beta$ medium and $\\rho v_A^3\/l_A$ for slow modes in high $\\beta$ medium. This places a constraint on the growth, thus the upper limit of wave energy is given by\n\\begin{eqnarray}\n\\epsilon^u_N=\\cases{ M_A^2 L_i\/(L A)\\gamma^{\\alpha-1},& $\\beta<1$ \\cr\n  L_i\/(l_A A)\\gamma^{\\alpha-1}, & $\\beta>1$, \\cr}\n\\label{energy}\n\\end{eqnarray}\nwhere $\\gamma$ is the Lorentz factor and $L_i\\simeq 6.4\\times 10^{-7}(B\/5{\\rm \\mu G})(10^{-10}{\\rm cm}^3\/n_{cr})$ pc. The growth is induced by the compression at scales $\\sim< \\lambda_{CR}$. Therefore, in the case that $\\Gamma_{gr} \\epsilon$ reaches the energy cascading rate, fast modes are damped at the corresponding maximum turbulence pumping scale $\\lambda_{fb}=r_p\/\\epsilon_N$ (see Fig.\\ref{feedback_fig} {\\em left}).  If $\\lambda_{fb}$ is larger than the original damping scale $l_c$, then there is a feedback on the large scale compressible turbulence. This shows that test particle approach is not adequate and feedback should be included in future simulations.\n\n\n\\section{Summary}\n\nIn this chapter, we reviewed recent development on cosmic ray transport theories based on modern understanding of MHD turbulence. The main conclusions from both analytical study and test particle simulations in MHD turbulence are:\n\\begin{itemize}\n\\item Compressible fast modes are most important for CR scattering. CR transport therefore varies from place to place.\n\\item Nonlinear mirror interaction is essential for pitch angle scattering (including 90 degree).\n\\item Cross field transport is diffusive on large scales and super-diffusive on small scales.\n\\item Subdiffusion does not happen in 3D turbulence.\n\\item Self - generated waves are subject to damping by preexisting turbulence \n\\item Small scale waves can be generated in compressible turbulence by gyroresonance instability. Feedback of CRs on turbulence need to be included in future simulations. \n\\end{itemize}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe site where r-process nuclei above A=90 have been synthesized\nremains a major unsolved problem in nucleosynthesis theory\n\\citep[e.g.,][]{Arnould07}.  Historically, many possibilities have\nbeen proposed \\citep[see][]{Meyer94}, but today, there are two\nprincipal contenders - neutron star mergers\n\\citep{Lattimer77,Freiburghaus99} and the NDW\n\\citep{Woosley94,Qian96,Hoffman97,Otsuki00,Thompson01}.  Observations\nof ultra-metal-poor stars suggest that many r-process isotopes were\nalready quite abundant at early times in the galaxy\n\\citep{Cowan95,Sneden96,Frebel07}, suggesting both a primary origin\nfor the r-process and an association with massive stars. NDWs would\nhave accompanied the first supernovae that made neutron stars and,\ndepending upon what is assumed about their birth rate and orbital\nparameters, the first merging neutron stars could also have occurred\nquite early.\n\nBoth the merging neutron star model and the NDW have problems\nthough. In the simplest version of galactic chemical evolution,\nmerging neutron stars might be capable of providing the necessary\nintegrated yield of the r-process in the sun, but they make it too\nrarely in large doses and possibly too late to be consistent with\nobservations \\citep{Argast04}.  On the other hand, making the\nr-process in NDWs requires higher entropies, shorter time-scales, or\nlower electron mole numbers, $Y_e$, than have been demonstrated in any\nrealistic, modern model for a supernova explosion \\citep[though\n  see][]{Burrows06}.\n\nPrevious papers and models for nucleosynthesis in the NDW have focused\non the production of nuclei heavier than iron using either greatly\nsimplified dynamics \\citep{Beun08,Farouqi09} or nuclear\nphysics \\citep{Qian96,Otsuki00,Arcones07,Fischer09,Huedepohl09}.\nPost processing nuclear network calculations have been performed using \nthermal histories from accurate models of the dynamics \n\\citep{Hoffman97,Thompson01,Wanajo06}, but the calculations sampled only a \nlimited set of trajectories in the ejecta. No one has yet combined the \ncomplete synthesis of a realistic NDW with that of the rest of the \nsupernova.\n\nTo address this situation, and to develop a framework for testing the\nnucleosynthesis of future explosion models, we have calculated\nnucleosynthesis using neutrino luminosity histories taken from two PNS\ncalculations found in the literature \\citep{Woosley94,Huedepohl09}.\nThis was done using a modified version of the implicit one-dimensional\nhydrodynamics code Kepler, which includes an adaptive nuclear network\nof arbitrary size.  This network allows for the production of both\nr-process nuclei during neutron-rich phases of the wind and production\nof light p-elements during proton-rich phases.  Since the results of wind\nnucleosynthesis depend sensitively on the neutrino luminosities and\ninteraction rates \\citep{Qian96,Horowitz02}, we have included accurate\nneutrino interaction rates that contain both general relativistic and\nweak magnetism corrections.  \n\nThe synthesis of all nuclei from carbon through lead is integrated\nover the history of the NDW and combined with the yield from the rest\nof the supernova, and the result is compared with a solar distribution.\nIf a nucleus produced in the NDW is greatly overproduced relative to\nthe yields of abundant elements in the rest of the supernova, there is\na problem. If it is greatly underproduced, its synthesis in the NDW is\nunimportant.  If it is co-produced, the NDW may be responsible for the\ngalactic inventory of this element.  An important outcome of this\nstudy are the yields expected from a ``plain vanilla'' model for the\nNDW. Are there any elements that are robustly produced and thus might\nbe used as diagnostics of the wind in an early generation of stars?\n\nIn \\S\\ref{wind_physics}, we discuss the general physics of neutrino\ndriven winds and analytically delineate the regions in neutrino\ntemperature space were different modes of nucleosynthesis occur.  We\nthen discuss our numerical model in \\S\\ref{computational_method}.  In\n\\S\\ref{results}, the results of the time dependent models are\npresented.  We conclude with a discussion of how the NDW might affect\ngalactic chemical evolution and consider if this allows the strontium\nabundance in low metallicity halo stars to be used as a tracer of\nsupernova fallback at low metallicity.  Additionally, we investigate\nif observed abundances in SN 1987A can put constraints on late time\nneutrino luminosities from PNSs.  Finally, we discuss some possible\nmodifications of the basic model that might improve the r-process\nproduction. These ideas will be explored more thoroughly in \na subsequent paper.\n\n\\section{General Concepts and Relevant Physics} \\label{wind_physics}\n\nAfter collapse and bounce in a core collapse supernova, a condition of\nnear hydrostatic equilibrium exists in the vicinity of the\nneutrinospheres. The temperature of the outer layers is changing on a\ntime scale determined by the Kelvin-Helmholtz time of the PNS,\n$\\tau_{KH} \\approx 10 s$ \\citep{Burrows86,Pons99}.  This is much\nlonger than the dynamical time scale of the PNS envelope, so the\nhydrostatic part of the envelope is in an approximate steady state.\nThe neutrino heating rate, which is determined by the neutrino\nluminosities from the neutrino sphere, must then balance the local\nneutrino cooling rate.  Heating and cooling are dominated by the\ncharged current processes $(\\nu_e + n) \\rightleftharpoons (e^- + p)$\nand $(\\bar \\nu_e + p) \\rightleftharpoons (e^+ + n)$ \\citep{Qian96}.\nEquating these rates, while neglecting the neutron-proton mass\ndifference and weak magnetism corrections and assuming the geometry\ncan be approximated as close to plane-parallel gives the temperature\nstructure of the neutron star atmosphere as a function of radius,\n$T_{atm} \\approx 1.01 \\, \\textrm{MeV} \\, R_{\\nu,6}^{-1\/3}\nL_{\\nu,51}^{1\/6}\\epsilon_{\\nu,MeV}^{1\/3} \\left(y_\\nu\/y\\right)^{1\/3} $,\nwhere $L_{\\nu,51}$ and $\\epsilon_{\\nu,MeV}$ are the electron neutrino\nluminosity and average neutrino energy at the neutrino sphere in units\nof $10^{51} \\, \\textrm{ergs s}^{-1}$ and MeV, respectively.  The\ngravitational redshift factor is $y = \\sqrt{1-2GM_{NS}\/r c^2}$ which, \nwhen evaluated at the neutrino sphere, $R_\\nu$, is $y_\\nu$.  Notice that\nthe only dependence on radius is carried in the redshift factor, so that\nthe atmosphere is close to isothermal.\n \nAt the radius, $r_c$, where the pressure in the envelope becomes\nradiation dominated, the material becomes unstable to outflow\n\\citep{Salpeter81}.  Since the neutrino luminosity is significantly\nlower than the neutrino Eddington luminosity, a thermally driven wind\nresults \\citep{Duncan86}.  The density at which this wind begins can\nbe found approximately by equating the radiation pressure to the\nbaryonic pressure.  This results in a critical density, $ \\rho_c\n\\approx \\sci{8.3}{7} \\, \\textrm{g} \\, \\textrm{cm}^{-3} \\,\nR_{\\nu,6}^{-1} L_{\\nu,51}^{1\/2}\\epsilon_{\\nu,MeV}\n\\left(y_\\nu\/y\\right), $ at which significant outflow begins and the\nkinetic equilibrium of weak interactions ceases to hold.  Under these\nconditions, nuclear statistical equilibrium is maintained on a time\nscale much shorter than the dynamical time scale and, for these\ntemperatures and densities, there will be no bound nuclei present.\nSince the electron fraction is set by kinetic equilibrium, the\ncomposition of the wind does not depend on any previous nuclear\nprocessing, so any nucleosynthesis from the wind will be primary.\n\nAssuming that most neutrino heating occurs near $r_c$, the entropy is\nconstant once the temperature cools to the nucleon recombination\ntemperature, $kT \\approx 0.5$ MeV.  Therefore, the final nuclear\nabundances in the wind depend mainly on the wind entropy, electron\nfraction, and the dynamical timescale at the radius where alpha \ncombination occurs \\citep{Qian96,Hoffman97}.  To\ndetermine the contribution of the wind to the nucleosynthesis of the\nentire supernova, the mass loss rate must also be known.  Estimates\nfor these quantities are given in the Appendix along with a discussion\nof the effect of general relativistic corrections.\n\nIntegrating the mass loss rate (equation \\ref{eq:mdot}) for a typical\nneutrino luminosity history implies that the wind will eject\napproximately $10^{-3} \\, M_\\odot$ of material.  This in turn means\nthat for the wind to contribute to the integrated yields of the\nsupernova for a particular isotope, that isotope needs to overproduced\nrelative to its solar mass fraction by a factor of at least $10^5$ in\nthe wind, assuming the rest of the supernova ejects $\\sim10 \\, M_\\odot$ and has\nover production factors of its most abundant metals of order 10.\n\nAt early times in the wind, the PNS is losing lepton number\n\\citep[e.g.][]{Burrows86}.  Neutrino interactions in the wind then tend to\nincrease the lepton number of the wind, and, to maintain charge\nneutrality, cause the wind to become proton rich.  Under these\nconditions, the wind may synthesize some of the light p-process\nelements via the so called $\\nu$p-process \\citep{Frohlich06,Pruet06}.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.7]{fig1.eps}\n\\end{center}\n\\caption{Neutrino two-color plot produced using the analytic relations\n  in the Appendix.  A neutron star with gravitational mass 1.4\n  $M_\\odot$ \\ has been assumed with a neutrinosphere radius of 10 km.\n  The total neutrino luminosity is assumed to scale as $L_{\\nu_e,tot}\n  = 10^{51} (\\langle T_{\\nu} \\rangle\/ 3.5 \\, \\textrm{MeV})^4 \\,\n  \\textrm{erg} \\textrm{s}^{-1}$.  This luminosity is split between\n  neutrinos and anti-neutrinos so as to ensure that the net\n  deleptonization rate of the PNS is zero.  The thick black line\n  corresponds to an electron fraction of $Y_e = 0.5$.  Above this\n  line, neutron-rich conditions obtain and below it the matter is\n  proton-rich.  The white region is where there no free neutrons\n  remain after charged particle reactions cease.  The N = 50 (tan)\n  region corresponds to final neutron-to-seed ratios between 0.01 and\n  15.  The ``first peak'' (yellow) region corresponds to a neutron-to-seed\n  ratio between 15 and 70, and the ``second peak'' (orange) region is where\n  the neutron-to-seed ratio is greater than 70.  The dashed lines\n  correspond to the base ten logarithm of the mass loss rate in solar\n  masses per second. }\n\\label{fig:tcp_base}\n\\end{figure*}\n\nAfter the initial deleptonization burst though, the net lepton number\ncarried by the wind will be small. Since the anti-electron\nneutrinosphere sits deeper in the PNS than the electron\nneutrinosphere, the electron neutrinos will be cooler than the\nelectron anti-neutrinos \\citep{Woosley94}.  If this asymmetry is large\nenough, the wind can become neutron rich at later times.  Combined\nwith an $\\alpha$-rich freeze out, this can give conditions favorable\nfor the r-process \\citep{Woosley92}.\n\nIn both cases, the resulting nucleosynthesis is characterized by the\nintegrated neutron to seed ratio after charged particle reactions\nfreeze out.  For proton-rich winds, alpha-particles recombine into\n$^{12}$C by the standard triple alpha reaction and then alpha-capture\nand proton-capture reactions carry the nuclear flow up to\napproximately mass 60 \\citep{Woosley92}.  The slowest reaction in this\nsequence is $^4$He($2\\alpha$,$\\gamma$)$^{12}$C, so the total number of\nseed nuclei produced is equal to the number of $^{12}$C nuclei\nproduced.  When only free protons are present, the integrated neutron density is\ndetermined by the rate of anti-neutrino capture on free protons.  An\nestimate for the neutron to seed ratio for proton-rich winds is given\nby equation \\ref{eq:nsp}.  A neutron to seed ratio of only a few is\nrequired to bypass the few long-lived waiting points which hinder\nproduction of the light p-isotopes \\citep{Pruet06}.  Still, it is\nchallenging to produce even a small neutron to seed ratio, since the\ndynamical time scale of the wind is short compared to the anti-neutrino\ncapture time scale.\n\nIn the neutron-rich case, seed nuclei are produced by a different\nreaction sequence $^4$He($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C\n\\citep{Woosley92}.  For the conditions encountered in the wind, the\nneutron catalyzed triple-alpha reaction proceeds about ten times as\nquickly as $^4$He($2\\alpha$,$\\gamma$)$^{12}$C.  This increases the\nseed number compared with that obtained in a proton-rich wind with\nsimilar dynamical properties.  Also, since there are free neutrons,\ncapture can proceed up to the N=50 closed shell isotopes $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr.  Here, the neutron density is just determined\nby the number of free neutrons left after charged particle reactions\nfreeze out.  The neutron to seed ratio in neutron-rich winds can be\napproximated using equation \\ref{eq:nsn}.  Charged particle reactions\ncontinue up to the N=50 closed shell, at which point it becomes\nunfavorable to capture alpha particles due to small separation\nenergies and large coulomb barriers \\citep{Hoffman96}.  If neutrons are exhausted during\nthis process, the wind will mainly produce the isotopes $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr \\citep{Hoffman97}.  This happens when the\ncondition\n\\begin{equation} \n\\label{eq:n50_prdod} \n\\frac{\\bar Z}{\\bar A} \\approx 0.42-0.49 = \\frac{Y_e\n  f_\\alpha}{2Y_e(f_\\alpha-1) + 1}\n\\end{equation} \nis met.  Here, $f_\\alpha \\approx 14 Y_s\/Y_{\\alpha,i}$ is the fraction\nof the initial helium abundance that gets processed into heavy nuclei.\nNeutron to seed ratios of approximately 30 and 110 are required to\nproduce first and second peak r-process nucleosynthesis, respectively.\n\nUsing the analytic results for the wind dynamics and nucleosynthesis\ngiven in the Appendix (equations \\ref{eq:rcrit}, \\ref{eq:ent},\n\\ref{eq:mdot}, \\ref{eq:tau},\\ref{eq:ye}, \\ref{eq:nsp}, \\ref{eq:nsn},\nand using the neutrino interaction rates given in\n\\S\\ref{sec:neutrino_rates} to fix the thermodynamic state at $r_c$),\none can easily explore the neutrino temperature parameter space to\ndetermine the neutrino temperatures and fluxes that are most conducive\nto the r-process or the production of the light p-process.  Figure\n\\ref{fig:tcp_base} is a neutrino two-color plot where it is assumed\nthat the deleptonization rate is zero and that the neutrino luminosity\nscales with the temperature to the fourth power ($L_{\\nu_e,tot} =\n10^{51} (\\langle T_{\\nu} \\rangle\/ 3.5 \\, \\textrm{MeV})^4 \\,\n\\textrm{erg} \\textrm{s}^{-1}$).  The different nucleosynthetic regions\nare delineated by the final calculated neutron to seed ratio.  To give\na feeling for how a particular point in parameter space might\ncontribute to the integrated nucleosynthesis of the wind, the mass\nloss rate is also shown.\n\nFor a significant amount of material to move past the N = 50 closed\nshell during neutron-rich conditions, the anti-neutrino temperature\nmust be approximately $60$\\% higher than the neutrino temperature.\nFor second peak r-process nucleosynthesis to occur, the asymmetry must\nbe greater than $100$\\%.  Modern PNS cooling calculations do not give\nsuch large asymmetries \\citep{Pons99,Huedepohl09}.\n\nUnder proton-rich conditions, only a small region of the\nparameter space at high neutrino and low anti-neutrino temperature is\nfavorable for the $\\nu$p-process.  There will be a small amount of\nneutron production in the white region, but it is unlikely that\nsignificant production of the light p-process elements $^{74}$Se,\n$^{78}$Kr, $^{84}$Sr, and $^{92}$Mo will occur.  The region in\nneutrino temperature space where there is significant neutron\nproduction is unlikely to be reached.  This region is small due to the\nshort dynamical time scale of the wind, which reduces the time over\nwhich anti-neutrinos can capture on free neutrons.  One should note\nthat, very soon after shock formation in the supernova, a wind\nsolution may not be appropriate and material will be entrained closer\nto the PNS for a longer period of time.  This scenario would be similar\nto the the conditions used in \\cite{Pruet06}.\n  \nTherefore, based upon simple principles, it seems unlikely that the\nstandard wind scenario will produce r-process or light p-process\nisotopes in solar ratios, as is required by observations of metal poor\nhalo stars \\citep{Sneden96}.  This same conclusion has been reached by\nother authors \\citep{Hoffman97,Thompson01}, but is repeated\nhere in simple terms.  We will find that our numerical calculations\ngive similar results and that there is no significant r-process\nnucleosynthesis associated with the wind.  Still, the wind can produce \nsome isotopes that may have an observable signature.  For standard PNS \nluminosities, the wind will spend a significant amount of time in the \nregion of parameter space were N = 50 closed shell nucleosynthesis occurs.  \n\n\n\n\n\\section{Computational Method}\n\\label{computational_method}\n\nTo more accurately investigate the integrated nucleosynthesis of the\nNDW, we have updated the implicit Lagrangian hydrodynamics code Kepler\n\\citep{Weaver78,Woosley02} to carry out time-dependent simulations of\nthe wind dynamics and nucleosynthesis.  Kepler has been used\npreviously to study time-independent winds\\citep{Qian96}, but the weak\nand nuclear physics employed there was rudimentary and nucleosynthesis\nwas not tracked.  Trajectories from Kepler were used for post-processing\ncalculations of nucleosynthesis in \\cite{Hoffman97}.\n\nKepler solves the non-relativistic hydrodynamic equations in\nLagrangian coordinates assuming spherical symmetry.  First order\ngeneral relativistic corrections are included in the gravitational\nforce law (cf. \\cite{Shapiro83}).  All order $v\/c$ effects are\nneglected.  This is justified since the maximum wind speeds\nencountered are, at most, a few percent of the speed of light.  The\nmomentum equation is then \n\\begin{equation} \n\\td{v_r}{t} = -4 \\pi r^2 \\pd{P}{m} - \\frac{Gm}{r^2}\\left( 1 +\n\\frac{P}{\\rho c^2} + \\frac{4 \\pi P r^3}{m c^2} \\right) \\left(1 -\n\\frac{2 G m}{r c^2} \\right)^{-1}\n\\end{equation}\nwhere the symbols have their standard meanings.  As has been shown by\nprevious studies \\citep{Qian96,Cardall97,Otsuki00,Thompson01}, general\nrelativistic corrections to the gravitational force can have an\nappreciable effect on the entropy and dynamical time scale of the\nwind.  The equation of state includes a Boltzmann gas of nucleons and\nnuclei, an arbitrarily relativistic and degenerate ideal electron gas,\nand photons.\n\n\\subsection{Weak Interaction Physics}\n\\label{sec:neutrino_rates}\n\nEnergy deposition from electron neutrino capture on nucleons, neutrino\nannihilation of all neutrino flavors, and neutrino scattering of \nall flavors on electrons is included in the\ntotal neutrino heating rate.  Neutrino ``transport'' is calculated in the \nlight-bulb approximation.  The energy deposition rate is dominated\nby neutrino captures on nucleons.  The neutrino annihilation rates\ngiven in \\cite{Janka91} are employed.  For the scattering rates, the\nrates given in \\cite{Qian96} are used, but we include general\nrelativistic corrections.  Standard neutrino capture rates are\nemployed in the limit of infinitely heavy nucleons with first order\ncorrections.  In this limit, the cross section is \n(Y.Z. Qian, private communication)\n\\begin{equation}\n  \\sigma_{ \\nu n \\atop \\bar \\nu p} =\\frac{G_F^2 cos^2(\\theta_C)}{ \\pi\n    (\\hbar c)^4}\\left[g_V^2+3g_A^2 \\right] \\left(\\epsilon_{\\nu} \\pm\n  \\Delta \\right)^2\\left(1 \\pm W_{M, {\\nu \\atop {\\bar \\nu}}}\n  \\epsilon_{\\nu}\\right) \n\\end{equation} \nHere, $G_F$ is the Fermi coupling constant, $\\theta_C$ is the Cabibo angle,\n$g_V$ and $g_A$ are the dimensionless vector and axial-vector coupling \nconstants for nucleons, $\\Delta$ is the proton neutron mass difference, \n$\\epsilon_\\nu$ is the neutrino energy, and $W_{M, {\\nu \\atop {\\bar \\nu}}}$ \naccounts for the weak magnetism\nand recoil corrections to the neutrino-nucleon cross section when the\nbase cross section is derived in the limit of infinitely heavy\nnucleons \\citep{Horowitz02}.  This correction reduces the\nanti-neutrino cross section and increases the neutrino cross section\n(by about a total of 10\\% at the energies encountered in NDWs), which,\nfor a given incident neutrino spectrum, significantly increases the\nasymptotic electron fraction.  Assuming a thermal distribution, these\ncross sections result in the neutrino energy deposition rate for\nanti-electron neutrino capture \n\\begin{equation} \n\\begin{array}{rl}\n    \\dot q_{\\bar \\nu p}=&\\sci{4.2}{18} \\textrm{ergs\n      s}^{-1}\\textrm{g}^{-1}\\frac{Y_p L_{\\bar \\nu,51}}{ \\langle\n      \\mu\\rangle r_6^2} \\\\ \\times & \\biggl[-W_M^{\\bar\\nu\n        p}\\frac{\\langle \\epsilon_{\\bar \\nu}^4 \\rangle }{\\langle\n        \\epsilon_{\\bar \\nu} \\rangle } + (1 + 2W_M^{\\bar\\nu p} \\Delta)\n      \\frac{\\langle \\epsilon_{\\bar \\nu}^3 \\rangle }{\\langle \\epsilon_{\\bar\\nu}\n        \\rangle } \\\\ & - (2 \\Delta + W_M^{\\bar\\nu p} \\Delta^2\n      )\\frac{\\langle \\epsilon_{\\bar \\nu}^2 \\rangle }{\\langle \\epsilon_{\\bar \\nu}\n        \\rangle } + \\Delta^2 \\biggr]\n\\end{array}\n\\end{equation} \nand a similar expression for electron neutrino capture.  The neutrino\nenergy distributions are parameterized by assuming a Fermi-Dirac\nspectrum.  The neutrino energy averages, $\\avg{\\epsilon_\\nu^n}$, are\nevaluated using this distribution.  The neutrino energy moments and\nluminosity are evaluated in the rest frame of the fluid.  With general\nrelativistic corrections for the bending of null geodesics, the\naverage neutrino angle is given by\n\\begin{equation} \n\\langle \\mu \\rangle = \\frac{1}{2} +\n\\frac{1}{2}\\sqrt{1-\\left( \\frac{R_\\nu y_\\nu}{r y}\\right)^2}.\n\\end{equation} \nSpecial relativistic corrections are negligible in the regions where\nneutrino interactions are important.\n\nThe lepton capture rates used are calculated in the limit of\ninfinitely heavy nucleons.  This results in a positron capture energy\nloss rate\n\\begin{equation}\n\\begin{array}{rl}\n\\dot q_{e^+ n} =&\\sci{6.9}{15}\\, \\textrm{ergs g}^{-1} \\, \\textrm{s}^{-1} \\, Y_n T_{10}^6 \\\\\n\\times & \\int_0^\\infty du f_{e}(u,-\\eta)  \\left( u^5 + 3 \\delta u^4 + 3 \\delta^2 u^3 + \\delta^3 u^2  \\right)\n \\end{array}\n\\end{equation}\nhere $f_e(u,\\eta) = (\\exp(u-\\eta)+1)^{-1}$, $\\eta$ is the electron degeneracy parameter, \n$\\delta$ is the proton neutron mass difference divided by $k_bT$, and $Y_n$ is the neutron \nfraction.  A similar rate is employed for electron capture.\n\nFor the neutrino losses, we include electron and positron capture on\nnucleons and include thermal losses as tabulated in \\cite{Itoh96}.\nThe energy loss rate in the wind is dominated by the electron\ncaptures.\n\n\\subsection{Nuclear Physics}\n\nDuring a hydrodynamic time step in Kepler, the nuclear energy generation\nrate and the changing nuclear composition are calculated\nusing a modified version of the 19-isotope network described in\n\\cite{Weaver78}.  Neutrino and electron capture rates on nucleons are\ncoupled to the network, which are calculated under the same\nassumptions as the charged current energy deposition\/loss rates\ndescribed above.  Therefore, non-equilibrium evolution of the electron\nfraction is accurately tracked.\n\nAlthough this network is appropriate for calculating energy generation\nthroughout the entire wind, it is not large enough to accurately track\nthe nucleosynthesis once alpha recombination begins at $ T \\approx\n0.5$ MeV.  Therefore, for temperatures below $20 \\textrm{GK}$ an\nadaptive network is run alongside the hydrodynamics calculation.  The\ndetails of this network can be found in \\cite{Woosley04} and \n\\cite{Rauscher02}.  As a fluid\nelement passes the temperature threshold, the composition from the\n19-isotope network is mapped into the adaptive network.  Typically,\nthe network contains approximately 2000 isotopes.  Where available,\nexperimental nuclear reaction rates are employed, but the vast\nmajority of the rates employed in the network come from the\nstatistical model calculations of \n\\cite{Rauscher00}.  In general, the nuclear physics employed in these\ncalculations is the same as that used in \\cite{Rauscher02}.  The\nnucleon weak interaction rates employed in the 19-isotope network are\nalso used in the adaptive network.\n\n\n\\subsection{Problem Setup and Boundary Conditions}\n\nTo start the neutrino driven wind problem, an atmosphere of mass 0.01\n$M_\\odot$ is allowed to relax to hydrostatic equilibrium on top of a\nfixed inner boundary at the neutron stars radius.  The mass enclosed\nby the inner boundary is the neutron star's mass.  The photon\nluminosity from the neutron star is assumed to be nearly Eddington,\nbut we have found that the properties of the wind are insensitive to\nthe the luminosity boundary condition.  Once hydrostatic equilibrium\nis achieved, the neutrino flux is turned on and a thermal wind forms.\nThis wind is allowed to relax to a quasi-steady state, and then the 19\nisotope network is turned on and the wind is, once again, allowed to\nreach a quasi-steady state.  After this point, the neutrino flux is\nallowed to vary with time, and the adaptive network is turned on.  \n\nAs the calculation proceeds, the mass of the envelope being followed\ndecreases and could eventually all be blown away.  To prevent this,\nmass is added back to the innermost mass elements at a rate equal to\nthe mass loss rate in the wind.  The mass added to a fluid element at\neach time step is a small fraction of its total mass.  We find that\nmass recycling has no effect on the properties of the wind. It is\nsimply a way of treating a problem that is essentially Eulerian in a\nLagrangian code.\n\nFor most runs, a zero outer boundary pressure and temperature are \nassumed.  To investigate the effect of a wind termination shock, a\ntime dependent outer boundary condition is included in some of \nthe simulations detailed below.  The pressure \nof the radiation dominated region behind the supernova \nshock is approximately given by \\citep{Woosley02}\n\\begin{equation}\n\\label{eq:Pbound}\nP_{ps} \\approx \\frac{E_{sn}}{4 \\pi (v_{sn}  t)^3}\n\\end{equation}   \nwhere $E_{sn}$ is the explosion energy of the supernova, $v_{sn}$ is\nthe supernova shock velocity, and $t$ is the time elapsed since the \nshock was launched.  As was discussed in \\cite{Arcones07}, this \nresults in a wind termination shock at a radius where the condition\n$\\rho_w v_w^2 + P_w \\approx P_{ps}$\nobtains, where $v_w$ is the wind velocity and $\\rho_w$ is the \nwind density.  To avoid an accumulation of too many zones, \nmass elements are removed from the calculation once they \nexceed a radius of $10,000 \\, \\textrm{km}$.  This is well \noutside the sonic point and nuclear burning has \nceased by this radius in all calculations .\n \n\\section{Numerical Results}\n\\label{results}\n\nTo survey both low and intermediate mass core collapse supernovae, \nneutrino emission\nhistories were taken from two core collapse calculations, one from a\n$20 M_\\odot$ \\citep{Woosley94} supernova calculation and the other \nfrom a $8.8 M_\\odot$ \\citep{Huedepohl09}\nsupernova calculation.  Since the PNSs studied have significantly\ndifferent masses and neutrino emission characteristics, one is able to\nget a rough picture of how integrated nucleosynthesis in the NDW\nvaries with progenitor mass.\n\n\\subsection{Neutrino Driven Wind from a $20 M_\\odot$ Supernova}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig2.eps}\n\\end{center}\n \\caption{Neutrino luminosities and temperatures taken from the model\n  of \\cite{Woosley94}.  The top panel is the neutrino luminosities.  The \n  bottom panel is the average neutrino energies.  The solid line corresponds \n  to $\\nu_e$, the dashed line corresponds to $\\bar \\nu_e$, the dot-dashed \n  line corresponds to $\\nu_{\\mu,\\tau}$.}\n \\label{fig:woos94_neut}\n\\end{figure}\n\nThe first set of neutrino luminosities and temperatures are taken from\n\\cite{Woosley94}.  This calculation began with a $20 M_\\odot$ progenitor\nmeant to model the progenitor of 1987A \\citep{Woosley88}.  The\nresulting neutron star had a gravitational mass of $1.4 M_\\odot$ and\nthe neutrino sphere was taken to be at 10 km.  The neutrino\nluminosities and average energies as a function of time from this\nmodel are shown in figure \\ref{fig:woos94_neut}.  After about 4\nseconds, the neutrino energies become constant and the large\ndifference between the electron neutrino and anti-neutrino energies\nimplies that the wind will be neutron rich.  This supernova model had\nsome numerical deficiencies (Sam Dalhed, Private Communication). The\nentropy calculated for the wind in \\cite{Woosley94} ($S\/N_Ak \\approx 400$)\nwere unrealistically large due to some problems with the equation of\nstate. Here, that is not so important because the NDW is being\ncalculated separately, but this study did rely on older neutrino\ninteraction rates and did not include weak magnetism corrections (see \\S\n\\ref{sec:neutrino_rates}).  Therefore, the results obtained using\nthese neutrino histories are only suggestive of what might happen in a\nmore massive star.  If weak magnetism were taken into\naccount, the calculated electron and anti-electron neutrino\ntemperatures would probably be somewhat further apart.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.5]  {fig3.eps}\n\\end{center}\n\\caption{Wind structure after two seconds in the model using \nthe neutrino luminosities from \\cite{Woosley94}.  The top panel \nshows the density in units of $10^8 \\, \\textrm{g} \\, \\textrm{cm}^{-3}$ \n(solid line) and the radial velocity in units of $10^3 \\, \\textrm{km} \\, \n\\textrm{s}^{-1}$ (dot-dashed line).  The middle panel shows the net \nenergy deposition rate from weak and strong interactions in units of \n$10^{20} \\, \\textrm{erg} \\, \\textrm{g}^{-1} \\, \\textrm{s}^{-1}$ (dot-dashed line) \nand the entropy (solid line).  The bottom panel shows the temperature \nin units of $\\sci{2}{9} \\, \\textrm{K}$ (solid line), the electron \nfraction (dot-dahsed line), and the fraction of material contained in \nnuclei (dotted line). }\n \\label{fig:wnd_struct}\n\\end{figure}\n\nThe calculation was run for a total of 18 seconds.  During this time,\nthe mass loss rate decreased by almost three orders of magnitude while\na total mass of $\\sci{2}{-3} M_\\odot$ was lost in the wind.  A\nsnapshot of the wind structure two seconds after bounce is shown in\nfigure \\ref{fig:wnd_struct}.  Note that the wind velocity stays very\nsub-luminal throughout the calculation.  Therefore, the neglect of\nspecial relativistic effects is reasonable.  The secondary bump in the\nenergy deposition rate occurs at the same radius where nucleons and\nalpha-particles assemble into heavy nuclei.  This increases the\nentropy by about 10 units.  Clearly, the electron fraction is set\ninterior to were nuclei form.  The radius where nuclei form is at a\nlarge enough value that the alpha effect \\citep{Fuller95} is not\nsignificant at early times in the wind.  However, as the neutrino luminosity\ndecreases with time, nucleon recombination occurs at a smaller radius,\nand the alpha effect becomes increasingly important.\n\nThe time evolution of the wind as calculated by Kepler is shown in\nfigure \\ref{fig:woos94_wndprop}.  The increase in asymptotic entropy is mainly\ndriven by the decrease in neutrino luminosity, since the average\nneutrino energies do not vary greatly.  The analytic approximation\n(calculated using equation \\ref{eq:ent} and the neutrino interaction\nrates given in \\S\\ref{sec:neutrino_rates}) to the entropy tracks the\nentropy calculated in Kepler fairly well.  This implies that the\nvariation in the neutrino luminosity with time does not significantly alter the\ndynamics from a steady state wind. In contrast to the high entropies\nreported in \\citet{Woosley94}, the entropy here never exceeds 130.\nFor the time scales and electron fractions also obtained, such a low\nvalue of entropy is not sufficient to give a strong r-process (see\nbelow).\n\nThe electron neutrino and anti-neutrino energies do move further apart\nas a function of time though, which causes the wind to evolve from\nproton-rich conditions at early times to neutron-rich conditions\nlater.  A transition occurs from the synthesis of proton-rich isotopes\nvia the $\\nu p$-process at early times to the $\\alpha-$process mediated\nby the reaction sequence\n$\\alpha$($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C later.  The\nslight difference between the analytic approximation and the Kepler\ncalculation of $Y_e$ is due to the alpha effect \\citep{Fuller95}.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.43]  {fig4.eps}\n\\end{center}\n\\caption{Properties of the neutrino driven wind from the \n\\cite{Woosley94} supernova model as a function of time. The thick \nlines correspond to the numerical results from Kepler and the thin \nlines correspond to the predictions of the analytic estimates described \nin the appendix.  The solid line is the dimensionless entropy per baryon, \nthe dashed line is the electron fraction, the dash dotted line is the \ndynamical timescale, and the dotted line is the mass loss rate.  All \nof the quantities are taken extracted from where the wind temperature \nreaches 2 GK.}\n \\label{fig:woos94_wndprop}\n\\end{figure}\n\nIntegrated production factors for the wind are shown in figure\n\\ref{fig:WWsolopf}.  The production factor for the species $i$ is\ndefined as\n\\begin{equation} \nP_i = \\frac{X_{i,w}M_{w}}{X_{i,\\odot}(M_{w}+M_{sn})},\n\\end{equation} \nwhere $X_{i,w}$ is the mass fraction of species $i$  in the wind\nafter all material has decayed to stable isotopes, $M_w$\nis the mass ejected in the wind, and $M_{sn}$ is the amount of mass\nejected by the entire supernova.  $X_{i,\\odot}$ is the mass fraction\nof isotope $i$ in the sun for which the values of \\citet{Lodders03}\nwere used. The only isotopes that are co-produced in the wind\nalone are $^{87}$Rb, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr, with\nproduction factor of $^{88}$Sr about a factor of 3 higher than the\nother two N = 50 closed shell isotopes.  Before eight seconds, the\nproduction factors had been much closer.  After eight seconds though,\nthe wind is dominated by $^{88}$Sr because $Y_e \\sim 0.45$ and only\n53\\% of alpha particles are free after freeze out which puts\n$\\frac{\\bar Z}{\\bar A}\\approx 0.41 $ of the heavy nuclei just\nbelow the range given in equation \\ref{eq:n50_prdod}.  There are not\nenough free neutrons to make any significant amount of heavier nuclei,\nand this results in significant production of the stable N = 50 closed\nshell isotope with the lowest $\\frac{\\bar Z}{\\bar A}$.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig5.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model when  \nthe neutrino luminosities from \\cite{Woosley94} are used.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.  The top dashed line\ncorresponds to the greatest production factor in the wind, the solid line is a\nfactor of two below that, and the bottom dashed line is a factor of two below the\nsolid line.  These lines specify an approximate coproduction band for the wind alone.}\n \\label{fig:WWsolopf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.42]  {fig6.eps}\n\\end{center}\n \\caption{Neutrino two-color plot when the anti-neutrino luminosity \n is 1.2 times neutrino luminosity, and the total luminosity scales \n with average temperature to the fourth.  Similar to figure \n \\ref{fig:tcp_base}.  The red line is the neutrino temperatures \n from \\cite{Woosley94}.}\n \\label{fig:tcp_woos}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig7.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model \nemploying the neutrino luminosities from \\cite{Woosley94} with \nthe anti-electron neutrino temperature reduced by $15\\%$.  The \nproduction factors are calculated assuming that $18.4 \\, M_\\odot$ \nof material was ejected in the supernova in addition to the wind.\nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWredpf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig8.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model \nemploying the neutrino luminosities from \\cite{Woosley94} \nwith weak magnetism corrections turned off.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.\nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWnowmpf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig9.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model when  \nthe neutrino luminosities from \\cite{Woosley94} are used and an external boundary \npressure is specified as described in the text, which results in a wind termination\nshock.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.  The horizontal lines are \nsimilar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWpbd}\n\\end{figure}\n\n\nDuring the first four seconds, the wind is proton rich and the isotopes\n$^{69}$Ga, $^{70,72}$Ge, $^{74,76}$Se, and $^{78,80,82}$Kr \nare produced by proton captures on seed nuclei produced by the triple-alpha \nreaction and subsequent ($\\alpha$,p) reactions.  Although the mass \nloss rate is much higher when the wind \nis proton rich, the alpha-fraction freezes out at 98\\% of its initial value, \nwhich results in significantly decreased production of heavy nuclei.  \nThe difference in final alpha fraction between the neutron- and proton-rich \nphases of the wind is due mainly to the difference in speed of the reaction \nchains $\\alpha$($2\\alpha$,$\\gamma$)$^{12}$C and\n$\\alpha$($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C, but also \nto the decreased entropy at early times.\n\nWe can compare this with the analytic predictions for nucleosynthesis\nby plotting the neutrino temperature evolution from this model on a\nneutrino ``two-color plot'' (figure \\ref{fig:tcp_woos}).  Here we have\nset $L_{\\bar \\nu_e} = 1.2 L_\\nu$ which is approximately correct at\nlate times in the calculation of \\cite{Woosley94}.  The wind never\nreaches a region in which r-process nucleosynthesis is expected, but\nspends a significant amount of time making nuclei in the N = 50 closed\nshell isotones.\n\n\\subsubsection{Variations in Neutrino Properties}\nSince the neutrino temperatures from the original model were\nuncertain, several other models were calculated. One had a reduced (by $15\\%$)\nelectron antineutrino temperature; another had the weak\nmagnetism corrections to the neutrino interaction rates turned off.  A\nsmaller antineutrino temperature is more in line with recent\ncalculations of PNS cooling \\citep{Pons99,Keil03}.  Because the\nmodel of \\cite{Woosley94} did not include weak magnetism corrections,\nour model with weak magnetism corrections turned off is more\nconsistent with the original supernova model.\n\nThe production factors for the model with a reduced electron\nantineutrino temperature are shown in figure \\ref{fig:WWredpf}.  The\nyield of $^{88}$Sr is reduced by almost a factor of ten from the base\ncase, while the production factors of $^{89}$Y and $^{90}$Zr are\nreduced by a factor of three. In this case, the wind also produces the\nproton-rich isotopes $^{74}$Se, $^{78}$Kr, and $^{84}$Sr.  The\ncoproduction line for lighter elements like oxygen in a $20 M_\\odot$\nsupernova at solar metallicity is around 18, so the wind could contribute\nto the total nucleosynthesis if the antineutrino temperature was\nreduced, but its contribution would be small.  \n\nThe yields when weak magnetism corrections are ignored are shown in\nfigure \\ref{fig:WWnowmpf}.  Without weak magnetism, the electron\nfraction drops below 0.4 at late times when the entropy is fairly\nhigh.  Equation \\ref{eq:n50_prdod} is no longer satisfied and material\nmoves past the N = 50 closed shell towards A $\\approx$ 110.  Some\nr-process isotopes are produced, such as $^{96}$Zr and $^{100}$Mo, but\nnot anywhere near solar ratios, and no material reaches the first\nr-process peak.\n\n\\begin{figure*}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.75]  {fig10.eps}\n\\end{center}\n\\caption{Combined isotopic production factors of the neutrino \ndriven wind with unaltered neutrino temperatures and including weak magnetism\ncorrections added to those of a $20 M_\\odot$ stellar model from \\cite{Woosley95}.  \nThe solid black line is the coproduction line with $^{16}$O.  \nThe dashed lines are a factor of two above and below the coproduction line.  The \nneutrino driven wind is responsible for the production of $^{88}$Sr, \n$^{89}$Y, and $^{90}$Zr.}\n \\label{fig:comb_pf}\n\\end{figure*}\n\n\\subsubsection{Effect of a Wind Termination Shock}\nTo investigate the possible effect of a wind termination shock on nucleosynthesis,\nanother model was run with a boundary pressure and temperature \ndetermined by equation \\ref{eq:Pbound}.  An explosion energy of \n$10^{51} \\, \\textrm{erg}$ was assumed and the shock velocity was taken as\n$\\sci{2}{9} \\, \\textrm{cm s}^{-1}$.  This resulted in a wind termination\nshock that was always at a radius greater than $10^3 \\, \\textrm{km}$.  \nThe production factors from this model are shown in figure \n\\ref{fig:WWpbd}.  Similar to the simulation without a wind termination\nshock, the N=50 closed shell elements dominate the wind's nucleosynthesis.\n\n\n\nThe main difference between the case with and without a wind termination \nshock is a shift in the mass of isotopes produced during the proton-rich\nphase.  This can be seen in the increased production of Mo. \nDuring this phase, the post shock temperature varied from \n2.5 GK down to 0.8 GK and the density varied from\n$\\sci{5}{4} \\, \\textrm{g cm}^{-3}$ to $\\sci{5}{2} \\, \\textrm{g cm}^{-3}$.  \nThese conditions are very favorable for \ncontinued proton capture once the long lived waiting point isotopes \n$^{56}$Ni and $^{64}$Ge are bypassed by (n,p) reactions.  Because \nthese conditions persist for at least a second after a fluid element passes\nthrough the wind termination shock, significantly more proton captures \ncan occur on seed nuclei that have moved past mass $\\sim 64$ relative\nto the case with no termination shock.  Still, not many more neutrons \nare produced per seed nucleus relative to the base run.  Therefore, the net \nnumber of seeds that get past the long lived waiting points remains small\nand the proton-rich wind does not contribute to the integrated nucleosynthesis.     \nIt should also be noted that a different treatment of the wind's \ninteraction with the supernova shock might result in a breeze solution \nwhich may supply more favorable conditions for $\\nu$p-process \nnucleosynthesis \\citep{Wanajo06}.    \n\n\\subsubsection{Total Supernova Yields}\n\n\nIn figure \\ref{fig:comb_pf}, the production factors from a $20\nM_\\odot$ supernova model from \\cite{Woosley95} have been combined with\nthe production factors we calculated in the NDW with the unaltered\nneutrino histories of \\citep{Woosley94} with weak magnetism corrections\nincluded.  The wind could be\nresponsible for synthesizing the isotopes $^{87}$Rb, $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr.  $^{88}$Sr production is above the\nco-production band, but the rest are in agreement with the stellar\nyields.  This overproduction of $^{88}$Sr is similar to the result of\n\\cite{Hoffman97}.\n\nFor the model with a reduced anti-electron neutrino temperature\ncombined with the yields from the $20 M_\\odot$ supernova model, the\nwind contributes 28\\%, 42\\%, 35\\%, 75\\%, 75\\%, and 80\\% of the total\n$^{74}$Se, $^{78}$Kr, $^{84}$Sr, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr\nabundances in the supernova, respectively.  This wind model does not result in any\nisotopes being overproduced relative to the rest of the yields of the\nsupernova.  For the case with weak magnetism turned off, the nuclei\nproduced by the wind are overproduced relative to those made in the rest of \nthe star by factor of nearly 100, hence this\nwould need to be a very rare event if this model were realistic.\n\nClearly, weak magnetism corrections and variations in the neutrino\ntemperatures have a very significant effect on nucleosynthesis in the\nwind.  Aside from the effects of an extra source of energy\n(\\ref{modifications}), the neutrino spectra are the largest current\ntheoretical uncertainty in models of the NDW.\n\n\\subsection{Neutrino Driven Wind from a $8.8 M_\\odot$ Supernova}\n\\label{eight_results}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig11.eps}\n\\end{center}\n \\caption{Neutrino luminosities and temperatures taken from the model \n of \\cite{Huedepohl09}.  The line styles are the same as in figure \n \\ref{fig:woos94_neut}.  }\n \\label{fig:jnk_neut}\n\\end{figure}\n\nThe second PNS model is a more modern one-dimensional calculation of an\nelectron-capture supernova \\citep{Huedepohl09} that started from an\n$8.8 M_\\odot$ progenitor model \\citep{Nomoto84}. This resulted in a\nPNS with a gravitational mass of $1.27 M_\\odot$ and a radius of 15 km.\nTogether the lower mass and increased radius imply a lower\ngravitational potential at the neutrinosphere.  This work employed\nneutrino interaction rates which took weak magnetism and ``in-medium''\neffects into account.  The neutrino luminosities and average energies\nas a function of time are shown in figure \\ref{fig:jnk_neut}. The\nmaximum difference between the electron and anti-electron neutrino\naverage energies is significantly less than in the model of\n\\cite{Woosley94}.  This is likely due in part to both the decreased\ngravitational potential of the PNS and the more accurate neutrino\ninteraction rates in the newer model.\n\nThe calculation was run for a total of nine seconds, at which point\nthe mass loss rate had dropped by two orders of magnitude.  The total\namount of mass ejected in the wind was $\\sci{3.8}{-4} \\, M_\\odot$.  In\nfigure \\ref{fig:jnk_wndprop}, the properties of the NDW calculated\nusing Kepler are plotted as a function of time.  Notice that the\nentropy never reaches above 100 in this model, which diminishes the\nlikelihood of significant nucleosynthesis.  For comparison, we also\ninclude the analytic estimates detailed above.  There is reasonable\nagreement between the analytic and the numerical calculations, but not\nnearly as good as in the $20 M_\\odot$ model.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig12.eps}\n\\end{center}\n\\caption{Properties of the neutrino driven wind from the \n\\cite{Huedepohl09} supernova model as a function of time.  \nThe lines have the same meaning as in figure \\ref{fig:woos94_wndprop}.}\n \\label{fig:jnk_wndprop}\n\\end{figure}\n\n\nIn contrast to the simulation run with the neutrino luminosities of\n\\citet{Woosley94}, the electron fraction continues to increase with\ntime.  The difference between the average electron neutrino energy and\nelectron anti-neutrino energy is, at most, about 3 MeV, compared to a\nmaximum of 8 MeV in the \\citet{Woosley94} calculations.  Also, the\ndifference between the average neutrino energies decreases as a\nfunction of time, compared to an increase with time in\n\\citet{Woosley94}.  Finally, the energies of all kinds of neutrinos\nare lower in the \\citet{Huedepohl09} calculation, so that the\nproton-neutron rest mass difference significantly suppresses the\nanti-neutrino capture rate relative to the neutrino capture rate.\nThese differences are presumably due to both the different neutron\nstar masses and neutrino interaction rates employed.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig13.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model employing \nthe neutrino luminosities from \\cite{Huedepohl09}.  The production \nfactors are calculated assuming that $7.4 \\, M_\\odot$ of material was \nejected in the supernova in addition to the wind.  \nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.\nNotice that none of the production factors are significantly greater \nthan one.}\n \\label{fig:jnkpf}\n\\end{figure}\n\nThe conditions in this model thus preclude {\\sl any} r-process\nnucleosynthesis, but they are potentially favorable for production of\nsome low mass p-process isotopes by the $\\nu$p-process.  The\nintegrated isotopic production factors are shown in figure\n\\ref{fig:jnkpf}.  The total ejected mass was take as 7.4 $M_\\odot$, as\n1.4 $M_\\odot$ neutron star is left behind in the calculation of\n\\cite{Huedepohl09}.  During the calculation a maximum network size of\n988 isotopes is reached.  The p-process elements $^{74}$Se and\n$^{78}$Kr are co-produced with $^{63}$Cu, $^{67}$Zn, and $^{69}$Ga,\nbut the maximum production factor for any isotope is 1 when weighted\nwith the total mass ejected in the supernova.  Therefore, in this\nsimple model, the proton-rich wind from low mass neutron stars \nwill not contribute significantly to galactic chemical evolution.\n\nThe entropies encountered when the mass loss rate is high are low\n($\\sim 50$), so that there is more production of $^{56}$Ni by\ntriple-alpha and a subsequent $\\alpha$p-process.  As the neutron\nabundance available for the $\\nu$p-process is given by\n\\begin{equation} \nY_n \\approx \\frac{\\lambda_\\nu Y_p}{\\rho N_A \\sum_i\nY_i \\avg{\\sigma v}_{i(n,p)j}}, \n\\end{equation} \nincreased seed production reduces the available neutron abundance and\ntherefore hinders production of the p-process elements $^{74}$Se,\n$^{78}$Kr, $^{84}$Sr, and $^{92}$Mo.  Additionally, at early times,\nthe dynamical time scale is short which implies a smaller integrated neutron \nto seed ratio, $\\Delta_n$ (see the appendix).        \n\nThe yields of from this model cannot be combined with the yields from \nthe rest of the supernova because they are not published.  As \\cite{Nomoto84}\nhas discussed, the mass inside the helium burning shell was close to the \nmass of the neutron star that was left after the explosion.  Therefore \nthe ejecta of the supernova is expected to have small production factors.\nThis implies that, even when the yields of the NDW are combined with the \nrest of the supernova, it is unlikely that these low mass core collapse\nsupernovae will contribute significantly to galactic chemical evolution.\n\n\\subsubsection{Effect of a Wind Termination Shock}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig14.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model employing \nthe neutrino luminosities from \\cite{Huedepohl09}, but including a time\ndependent external boundary pressure which results in a wind \ntermination shock.  The production \nfactors are calculated assuming that $7.4 \\, M_\\odot$ of material was \nejected in the supernova in addition to the wind.  \nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.\nNotice that the production factors are almost unchanged when an \nexternal boundary pressure is added.}\n \\label{fig:jnkpfpbd}\n\\end{figure}\n\n\nAs was mentioned above, it is very possible that a transonic wind solution \nmay not be appropriate this early in the supernovas evolution.  \\cite{Fischer09} \nhave found that a wind termination shock is not present in a one-dimensional\nsupernova model using the progenitor from \\cite{Nomoto84}.  Still, it is \ninteresting to consider the effect of a reverse shock on the wind nucleosynthesis.\n\nA second simulation was run with a time dependent boundary pressure given\nby equation \\ref{eq:Pbound}, with $E_{sn} = 10^{50} \\, \\textrm{erg}$ and \n$v_{sn} = \\sci{2}{9} \\, \\textrm{cm s}^{-1}$.  This results in a wind termination\nshock at a radius of approximately $\\sci{3}{8} \\, \\textrm{cm}$ throughout the \nsimulation.  Inside the wind termination shock the wind dynamics are very similar\nto those in the run with no boundary pressure.  The production factors from \nthis model are shown in figure \\ref{fig:jnkpfpbd}.  Clearly, there is almost no\ndifference in the nucleosynthesis in the runs with and without a wind termination\nshock.\n \nAfter $0.75 \\, \\textrm{s}$, the post shock temperature drops below 1 GK and \nthe wind termination shock has little effect on subsequent nucleosynthesis.\nBecause the post shock temperature is high for less than one second, the \nwind termination shock has very little effect on the integrated nucleosynthesis.  \nA larger explosion energy would likely result in a larger effect on the nucleosynthesis, \nbut there are still very few neutrons available to bypass the long lived waiting points\nand it seems unlikely that the production factors would be increased by \nmore than a factor of a few.\n\n\\section{Discussion}\n\\label{discussion}\n\n\\subsection{Comparison with SN 1987A}\n\nSince the progenitor model used in \\cite{Woosley94} was a model for SN\n1987A, it is interesting to compare our predicted abundances with\nthose observed in the ejecta of that event.  $^{88}$Sr, produced by\nthe NDW, dominates the elemental strontium yield when the results of\nthe 20 $M_\\odot$ wind model are combined with those predicted by\n\\cite{Woosley88}, who ignored the wind.  For the base NDW model, [Sr\/Fe]$=\n0.8$, if weak magnetism corrections are neglected, [Sr\/Fe]$= 1.6$; and\nif the anti-neutrino temperature is reduced in the base model by 15\\%,\n[Sr\/Fe]$=0.2$.\n\n\\cite{Mazzali92} found [Sr\/Fe]$\\approx 0.3$ in the ejecta of SN 1987A\n2-3 weeks after the explosion. This observation has a substantial\nerror bar due to the uncertainty of modelling the spectrum of the\nexpanding ejecta. Of even greater concern is comparing of our models\nfor bulk yields with supernova photospheric abundances observed on a\ngiven day when the observations are probably not even probing the\ninnermost ejecta.  Still, if one assumes that the [Sr\/Fe] ratio found\nby \\cite{Mazzali92} represents the value for all of the ejecta (for\ninstance by assuming that ``mixing'' was extremely efficient), a weak\nconstraint can be put on the neutrino fluxes and energies predicted by\nthe model of \\cite{Woosley94}. The NDW model with a reduced\nanti-neutrino temperature is much closer to the observed value of\n[Sr\/Fe] than the other two models.  This suggests the anti-neutrino\ntemperature may have been overestimated in the original model, a\nchange that would be more consistent with more modern calculations of\nneutrino spectra formation in PNS atmospheres \\citep{Keil03}. But\nobviously, a meaningful constraint will require a more complete modeling\nof the multi-dimensional explosion and time-dependent spectrum of SN\n1987A.\n\n\n\\subsection{Strontium and Yttrium in Halo Stars} \n\nSince strontium and yttrium are abundantly produced in our models, it\nmay be that the NDW has contributed to their production throughout\ncosmic history. An interesting possibility is that the abundances of\nthese elements might trace the birth rate of neutron stars at an early\ntime.  Taking a standard r-process abundance pattern from metal poor\nstars with strong r-process enhancments, \\cite{Travaglio04} find that\n8\\% and 18\\% of solar strontium and yttrium, respectively, are not\nproduced by either the ``standard'' r-process or any component of the\ns-process.  It therefore seems plausible that charged particle\nreactions in the NDW could make up this ``missing'' component.\n\nAny nucleosynthesis that happens in the NDW will be primary,\ni.e. provided that the mass function of neutron stars at birth does\nnot itself scale with metallicity, similar nucleosynthesis will occur\nfor stars of any population.  Below [Fe\/H]$\\sim -1.5$, no component of\nthe s-process contributes to the abundances of N = 50 closed shell\nisotopes \\citep{Serminato09}.  If the NDW escapes the potential well\nof the PNS, and contributes to the galactic budget of N = 50 closed\nshell isotopes, it should provide a floor to [Sr\/Fe] and [Y\/Fe].  Based\nupon the arguments of \\cite{Travaglio04}, this floor would be at\n[Sr\/Fe]$\\approx-0.18$ and [Y\/Fe]$\\approx -0.16$.  These numbers assume\nthat when the main r-process source contributes in addition to the NDW,\n[Sr\/Fe] and [Y\/Fe] approach their solar values even though the\ns-process has yet to contribute.  This is consistent with\nobservations.\n\nIn defining this floor, one must assume that the abundances in a\nparticular star sample a large number of individual supernovae.\nThis is because the production of N=50 closed shell elements likely\ndepends on the PNS mass and therefore the progenitor mass.  As we have\nfound, [Sr\/Fe]$=0.8$ in the $20 M_\\odot$ model with reduced\nanti-neutrino temperatures, but the $8.8 M_\\odot$ model produces no\nstrontium.  Observations show that below [Fe\/H]$\\sim -3$, the spreads\nin [Sr\/Fe] and [Y\/Fe] increase significantly and the mean values\nfalloff some \\citep{Francois07,Cohen08,Lai08}.  Single stars have\nvalues of [Sr\/Fe] below the predicted floor.  This could be because, \nat this metallicity, the metals in a particular\nstar come from only a handful of supernovae.\n\nAnother possible explanation of this variation is that supernova fall\nback varies with metallicity.  Since the NDW is the\ninnermost portion of the supernova ejecta, it will be the most\nsusceptible to fallback.  It has been found that the amount of\nsupernova fallback depends strongly on the metallicity of the\nprogenitor, especially going between zero and low metallicity\n\\citep{Zhang08}.  Additionally, mixing is also greatly reduced in zero\nmetallicity stars compared to solar metallicity stars due to the formers \ncompact structure \\citep{Joggerst09}. \n\nThe current understanding of supernova fallback suggests that the\nnucleosynthetic contribution of the NDW will be suppressed at very low\nmetallicity.  Of course, the ejection of iron by the supernova is also\nvery susceptible to fallback, so the effect of fallback on the\nevolution of [Sr,Y\/Fe] is complicated and may require fine tuning to\ngive the observed decrease.\nA somewhat different explanation was offered by \\citet{Qian08} who\nattributed the fall off of [Sr\/Fe] at low metallicity to the evolution\nof the ``hypernova'' rate with metallicity. For their purposes,\nhypernovae were stellar explosions that contributed iron without\nmaking much strontium. \n\nGiven the sensitivity of strontium and yttrium yields to uncertain NDW\ncharacteristics, especially neutrino fluxes and temperatures, it may\nbe some time before the complex history of these elements is even\nqualitatively understood. It is likely though that their abundances in\nhalo stars will ultimately be powerful constraints upon the evolution\nof supernova physics as a function of metallicity.\n\n\\subsection{Possible Modifications of the Basic Model}\n\\label{modifications}\n\nAs is clear from figures \\ref{fig:WWsolopf} and \\ref{fig:jnkpf}, the\nsimplest case of a non-magnetic non-rotating NDW from a neutron star\nwithout additional energy deposition does not produce r-process nuclei in\nsignificant abundances.  Are there extensions to this simple scenario\nthat {\\sl could} make the wind a site of the r-process?\n\nAs was pointed out by \\cite{Metzger07}, the combination of rotation\nand magnetic fields can decrease the dynamical time scale by magnetic\n``flinging''.  This is not particularly effective.  Adding a\nnon-thermal source of kinetic energy means that less thermal energy\nmust be put into the wind for it to escape the potential well.\nTherefore, lower entropies are achieved.  It seems unlikely that this\nmechanism, by itself, will salvage the NDW as a site for the full\nr-process.  If there were a way to make the rotation rate of the PNS\nhigh enough, it might be possible that there would be a centrifugally\ndriven outflow.  Then the electron fraction would be determined by kinetic\nequilibrium much deeper in the PNS envelope, and the material in the\noutflow would have an electron fraction much lower than that seen in\nthe wind.\n\nTo test this possibility, we ran calculations with a centrifugal force\nterm added and corotation with the PNS enforced out to $10^3$ km.\nUnfortunately, for reasonable PNS spin rates (20 ms period), \nwe found this had little effect on the nucleosynthesis.  These\ncalculations were in a regime were the electron fraction was still set\nby neutrino interactions.\n\nMany authors have discussed the possible effects of both\nmatter-enhanced\\citep{Qian95,Sigl95} and collective neutrino\n\\citep{Pastor02,Duan06} oscillations on NDW nucleosynthesis.  If\nelectron antineutrinos could undergo a collective oscillation near\nthe launch radius while the electron neutrinos did not, this would\nincrease the average energy of the antineutrinos if the $\\mu$\nand $\\tau$ neutrinos have a significantly higher temperature,\nfacilitating a reduction in the electron fraction.  For a normal mass hierarchy however,\nmatter enhanced neutrino oscillations would probably cause electron\nneutrino flavor conversion, which would {\\sl increase} the electron fraction\nand decrease the probability of significant r-process nucleosynthesis\n\\citep{Qian95}.\n\nCollective neutrino oscillations can cause antineutrino oscillations\nin the region were the electron fraction is set, and thereby decrease\nthe electron fraction where pure MSW oscillations would have predicted\nan increased electron fraction\\citep{Duan06}.  Clearly, the main\neffect of oscillations would be on the composition of the wind, not\nthe dynamics.  As can be seen in the neutrino two color plots,\noscillations would have to change the effective temperature of the\nanti-neutrinos by a very large amount to move from a region where N=50\nclose shell nucleosynthesis occurs to a region where the second\nr-process peak can be produced.\n\nThese effects are based upon the assumption that $\\mu$- and\n$\\tau$-neutrinos are significantly more energetic than the electron \nneutrinos.  In the calculation\nof \\cite{Woosley94}, this is the case, as can be seen in figure\n\\ref{fig:woos94_neut}.  Interestingly, the $\\mu$ and $\\tau$\ntemperatures are almost the same as the electron anti-neutrino\ntemperature in the \\cite{Huedepohl09} calculation, which can be seen\nin figure \\ref{fig:jnk_neut}.  It is not clear wether this difference\nobtains because of the difference in the PNS masses or the\nsignificantly different neutrino physics employed in the calculations.\nA detailed study of neutrino transport in static backgrounds showed\nthat the inclusion of all relevant neutrino interactions brings the\naverage energies of the $\\mu$- and $\\tau$- neutrinos closer to the\ntemperature of the anti-electron neutrinos \\citep{Keil03}.  Therefore\nit is uncertain wether or not neutrino oscillations could effect\nnucleosynthesis significantly.  Clearly, the uncertainties here are\nnot in the wind itself but in the formation of the spectra in the PNS\nand the details of neutrino transport with neutrino oscillations.\n\nFinally, it has been suggested \\citep{Qian96,Nagataki05} that adding a\nsecondary source of volumetric energy deposition can significantly\nincrease the entropy of the wind, which results in a more alpha-rich\nfreeze out and conditions that would be more favorable for r-process\nnucleosynthesis.  The addition of energy to the wind also decreases\nthe dynamical timescale.  Since the important quantity to consider for\nthe r-process is $s^3\/\\tau_d$ \\citep{Hoffman97}, \nboth effects increase the chance of having a significant neutron to\nseed ratio after freeze out.  If the NDW model is to be salvaged, this\nseems to us the minimal necessary extension.  Of course, the physical\nprocess contributing this extra energy is very uncertain.  One\npossibility is that oscillations of the PNS power sound waves\nwhich produce shocks and deposit energy in the wind, similar to the supernova\nmechanism of \\cite{Burrows06}, but smaller in magnitude.  We will\nexplore this possibility in some detain in a subsequent paper.\n\nFrom a chemical evolution standpoint, it is important to consider what\neffect neutron star mergers will have on the evolution of the\nr-process abundances.  It seems unavoidable that r-process nuclei will\nbe produced in the tidal tails ejected during these mergers\n\\citep{Freiburghaus99} and the amount of material ejected in these\nevents is approximately enough to account for the galactic inventory\nof r-process elements given the expected merger rate\n\\citep{Lattimer76,Rosswog99}.\n\nNeutron star mergers have been largely discounted because inferred\nmerger rates are small at low metallicity due to the long in spiral time\nand therefore they cannot account for the r-process enrichment seen in\nlow metallicity halo stars \\citep{Argast04}.  Of course, the inferred\nmerger rates are very uncertain as are models for the early evolution\nof the Milky Way, so that both the conclusion that neutron star\nmergers can account for the r-process inventory of the galaxy and that\nthey are not consistent with producing the r-process at low\nmetallicity are very uncertain.  Clearly, there is significant room\nfor more work in this area.\n\nTherefore, it seems reasonable that the galactic r-process abundances\ncould be accounted for by a combination of mergers and winds with an\nextra source of energy, either from an acoustically and\/or magnetically\nactive PNS.  Since not every supernova will have the requisite\nconditions for an r-process, there will be significant variation in\nthe yields from single supernovae.  This, along with the contribution\nfrom neutron star mergers, will give significant variation in the\n[r-process\/$\\alpha$-element] values found in single stars at low\nmetallicity but averaged over many stars these should track one\nanother, which is consistent with observations \\citep{Sneden08}.\n\n\\section{Conclusions}\n\nWe have performed calculations of the dynamics and nucleosynthesis in\ntime dependent neutrino driven winds.  This was done for two sets of\nneutrino spectra calculated in one-dimensional supernova models taken\nfrom the literature.  The nucleosynthesis in these models was compared\nwith supernova yields to determine if these models were consistent\nwith observations.  Additionally, we compared the results of these\nnumerical models to analytic models of the neutrino driven wind and\nfound good agreement.\n\nSimilar to most of the work on the NDW after \\cite{Woosley94}, we find\nthat it is unlikely that the r-process occurs in the neutrino driven\nwind unless there is something that causes significant deviation from\na purely neutrino driven wind.  Additionally, in the simplest case,\nthere is little production of p-process elements at early times in the\nwind.  In our calculation that used spectra from a more massive\nneutron star, the wind only produces the N=50 closed shell elements\n$^{87}$Rb, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr.\n\nThis result is sensitive to small changes in the neutrino interaction rates \n(i.e. the inclusion of weak magnetism) and changes to the neutrino temperature \nof order 10\\%.  Comparing our models with the over abundance of strontium \nseen in SN 1987A suggests that the difference between the electron and \nanti-electron neutrino temperatures in the model of \\cite{Woosley94} \nmay have been to large.  We also find that the effect of a wind termination\nshock on the wind nucleosynthesis is small.\n\nUsing neutrino spectra from an $8.8 M_\\odot$ supernova that drives a\nwind which is proton rich throughout its duration \\citep{Huedepohl09}, we\nfind that no significant $\\nu$p-process occurs and the wind does not\ncontribute to the yields of the supernova. The neutrino spectra from\nthis model are probably more accurate than the spectra from the model\nof \\cite{Woosley94}.  We also investigated\nthe effect of an outer boundary pressure which resulted in a wind \ntermination shock.  This had a negligible effect on the nucleosynthesis.\n\nHowever, one also expects that the nucleosynthesis in the NDW will\nvary considerably from event to event, especially with the mass and\npossibly the rotation rate of the PNS. The winds from more massive PNS\nhave greater entropy and might, in general, be expected to produce\nheavier elements and more of them. The neutrino spectral histories of\nPNS as a function of mass have yet to be determined over a wide \nrange of parameter space.  Currently, the neutrino luminosities and \ntemperatures are the largest uncertainties in models of the NDW.\n\n\n\\begin{acknowledgements}\n\nWe would like to thank Alex Heger, David Lai, Enrico Ramirez-Ruiz,\nSanjay Reddy, and Yong-Zhong Qian for useful discussions about issues\nrelating to this work.  L. R. was supported by an NNSA\/DOE Stewardship\nScience Graduate Fellowship (DE-FC52-08NA28752) and the University of\nCalifornia Office of the President (09-IR-07-117968-WOOS).  S. W.  was\nsupported by the US NSF (AST-0909129), the University of California\nOffice of the President (09-IR-07-117968-WOOS), and the DOE SciDAC\nProgram (DEFC-02-06ER41438).  R. H. was supported by the DOE SciDAC\nProgram (DEFC-02-06ER41438) and under the auspices of the Department of \nEnergy at Lawrence Livermore National Laboratory under contract \nDE-AC52-07NA27344.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Results and Discussion}\n\nWe first illustrate the method in some detail for the case $\\epsilon_{BB}=1.0$, and then present the general results. We also proceed to rigorously characterize the motifs and identify them in the lattice structures.\n\nIn order to name the different phases we searched the Material Project Database\\cite{Jain2013} to find a prototype isostructural phase and name the GA calculated lattice accordingly. If no match is found, then we name the phase according to the following convention:\n\\begin{equation}\\label{Eq:naming_convention}\n{\\mbox{A}_m \\mbox{B}_n}^{\\mbox{space group}}_{\\mbox{identifier}} .\n\\end{equation}\nHere $m$ and $n$ are the number of A and B particles within the unit cell. The space group is determined using the FINDSYM package\\cite{Stokes2005}, with the tolerance for lattice and atomic positions set to $0.05$. The identifier is necessary as multiple phases with the same stoichiometry and space group, differing only in Wyckoff number and positions, are found.\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{fig1.png}\n\\caption{Structures searched by GA in $\\epsilon_{BB}=1.0$ and $\\gamma = 0.8$. (a) Formation energies ($E_{form}$) of structures searched by GA as a function of stoichiometry ($x$). Each point corresponds to a structure. The color of points are assigned by the type of motifs in the corresponding structure. The black solid line is the convex hull of the system, while the black dash line is the threshold for metastable structures. (b) Structure of the FCC motif (c) Structure of the MgZn$_2$ motif, which is Frank-Kasper Z$_{16}$.}\\label{Fig:epsilon_1}\n\\end{figure}\n\n\n\\textbf{The case $\\boldsymbol{\\epsilon_{BB}=1.0}$.} Here we consider $\\epsilon_{BB} = \\epsilon_{AA} = 1.0$, while $0.3 \\le \\gamma \\le 0.9$. We first compute the energy of the ground state for the pure A and B states, which previous calculations\\cite{Stillinger2001,Travesset2014} have shown to be the hcp phase. Here, however, because of the finite cut-off of LJ potentials, the fcc phase has lower energy. The identification of equilibrium phases proceeds by comparing their energy against phase separation into pure $A$ and $B$. Then, out this list of putative binary phases that are stable against phase separation, the energies are compared to establish the resulting true phase diagram equilibrium. This is how the phase diagram Fig.~\\ref{Fig:epsilon_1} is built, where there is only one stable BNSL, the MgZn$_2$ Frank-Kasper phase at $\\gamma=0.8$. We should note that maximum of the packing fraction for this phase occurs for $\\gamma_c=\\sqrt{2\/3}=0.8165$\\cite{Travesset2017a}, which is very close.\n\nSince it is common that structures that are metastable at 0 K can be observed in experiments at finite temperatures, we also considered metastable phases defined to be those within $0.1\\epsilon$\/particle in energy above the convex hull. As shown in Fig.~\\ref{Fig:epsilon_1}, there are a number of metastable phases at $x=0.333$, which are minor variations of MgZn$_2$ as we analyze further below in the context of motifs.\n\n\\textbf{General $\\epsilon_{BB}$.} On physical grounds, it is expected that the smaller the particle the weaker the interaction, hence we consider $\\epsilon_{BB} \\le 1$. In Fig.~\\ref{Fig:epsilon_gen_energy}, we provide a typical calculation for fixed $\\gamma=0.6$ as a function of both $\\epsilon_{BB}$ and $x$. As expected, see  Fig.~\\ref{Fig:epsilon_1}, the phase diagram is trivial for $\\epsilon_{BB}=1$. However, three phases TiCu$_3$, AlB$_2$ and CrB at $x=0.25, 0.333, 0.5$ are found for $\\epsilon_{BB}= 0.8$.\n\nBy repeating the calculations shown in Fig.~\\ref{Fig:epsilon_gen_energy} for the other values of $\\epsilon_{BB}$ at a fixed $\\gamma=0.6$ (see Table~\\ref{tab:my_label}), we constructed the phase diagram shown in Fig.~\\ref{Fig:epsilon_gen}. In Fig.~\\ref{Fig:epsilon_gen} we note the appearance of seven additional phases for $\\epsilon_{BB}< 0.6$ that could not be matched to any prototype: Detailed description for these and all other equilibrium phases are collected in Supporting Information Table S1. A database for all the structures is included in Supporting Information. \n\n\n\n\nSimilarly, the phase diagrams for all other values of $\\gamma$ are also presented in Supporting Information Fig. S2. Common to all these phase diagrams is the appearance of many diffusionless (martensitic), usually incongruent transformations, as a function of the  energy parameter $\\epsilon_{BB}\/\\epsilon_{AA}$. In Supporting Information Fig. S3, we have also included phase diagrams for all values of $\\epsilon_{BB}\/\\epsilon_{AA}$ in $x$ and $\\gamma$.\n\n\\textbf{Motifs.} We define motifs as the polyhedron consisted of a center particle and its first-shell neighbors. The motifs are generated according to the analysis of bond length table from neighboring particles to the center (see details in Supporting Information Fig. S6). In this study, we only include motifs with the larger A-particles as the center. We will name motifs according to\n\\begin{equation}\\label{Eq:motif_name}\n    \\mbox{Motif}-\\mbox{CN}-\\mbox{Identifier} \\ ,\n\\end{equation}\nwhere CN is the Coordination (the number of particles) and identifier discriminates among motifs with the same coordination number.\n\n\\onecolumngrid\n\n\n\n\\begin{figure}[b]\n    \\includegraphics[width=1.0\\textwidth]{fig2.pdf}\n    \\caption{Two examples of GA results for $\\gamma = 0.6$. In each figure, the solid line is the convex hull, while the dashed line is the threshold for metastable structures, see the discussion above. (a) Structures searched by GA as a function of x when $\\epsilon_{BB} = 1.0$: There are no stable binary structures between x=0 and x=1. (b) Structures searched by GA as a function of x for $\\epsilon_{BB}=0.8$. There are three stable structures which appear at $x = 0.25$ (TiCu$_3$),  $x = 0.333 $ (AlB$_2$) and $x = 0.5$ (CrB).}\\label{Fig:epsilon_gen_energy}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[]\n\\includegraphics[width=1.0\\textwidth]{fig3.pdf}\n\\caption{Phase diagram in $x$ and $\\epsilon_{BB}\/\\epsilon_{AA}$ for $\\gamma = 0.6$.}\\label{Fig:epsilon_gen}\n\\end{figure}\n\n\\twocolumngrid\n\n\nWe identified 187 equilibrium and 102,822 metastable structures. Out of the 187 equilibrium structures, we removed redundancies by a cluster alignment algorithm\\cite{Fang2010,Sun2016a}leading to only 53 equilibrium structures. Out of these 53 structures we identified 42 motifs, which are listed in the order of increasing CN in Supporting Information Fig. S4. 416,391 motifs can be found in the 102,822 metastable structures. Among them, a vast majority (312,891) of the motifs of the metastable structures also exist in the equilibrium phases. In Tab~\\ref{tab:motif}, we list the name, CN and the percentage fraction of the ten most frequent motifs present in meta-stable structures. Note that these ten already account for more than 95\\% of the 312,891 motifs. The details about how to identify the motif from a crystal and how to identify if a crystal has the motif inside have been included in the Supporting Information.\n\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Ten most frequent Motifs in metastable structures}\n    \\begin{tabular}{c|c|c}\n    \\hline\n    \\hline\n        Motif & CN & Frequency \\\\ \\hline\n        FCC & 12 & 31.4\\% \\\\ \\hline\n        HCP & 12 & 18.9\\% \\\\ \\hline\n        Octahedron (Motif-6-4) & 6 & 9.8\\% \\\\ \\hline\n        Half Hexagonal Prism 1 (Motif-6-2) & 6 & 9.1\\% \\\\ \\hline\n        Triangular Prism (Motif-6-3) & 6 & 6.4\\% \\\\ \\hline\n        Half Hexagonal Prism 2 (Motif-6-1) & 6 & 5.3\\% \\\\ \\hline\n        BCC & 8 & 5.0\\% \\\\ \\hline\n        Hexagonal Prism (Motif-12-3) & 12 & 4.8\\% \\\\ \\hline\n        Half Truncated Cube (Motif-12-1) & 12 & 2.3\\% \\\\ \\hline\n        MoB (Motif-13-1) & 13 & 2.2\\% \\\\ \\hline\n        Total & & 95\\% \\\\ \\hline\n        \\hline\n    \\end{tabular}\n    \\label{tab:motif}\n\\end{table}\n\nAs an illustrative example, we consider the case of $\\epsilon_{BB}=1$ and $\\gamma=0.8$, where in Fig.~\\ref{Fig:epsilon_1} we have shown the two relevant motifs are the FCC and the Frank-Kasper Z$_{16}$.\nBy coloring each structure according to the motif, we can confirm that the metastable phases (all in red) have motifs which are small variations of the Frank-Kasper Z$_{16}$ and that the vast majority of the structures found in other searches have motifs which are variations of either FCC or Frank-Kasper Z$_{16}$.\n\n\\onecolumngrid\n\n\\begin{figure}[]\n\\includegraphics[width=0.7\\textwidth]{fig4.png}\n\\caption{(a) Map of MgZn$_2$ in $\\gamma$ and $\\epsilon_{BB}$. The red regime indicates that the structure of MgZn$_2$ is thermodynamically stable while in blue regime it is metastable. (b) Map for the Z$_{16}$ motif with red stable, blue metastable. The red regime is where the stable structure has Z$_{16}$ motif inside. Note that the motif has a wider range of both stability and metastability, as it also appears in other Laves phase, such as the MgCu$_2$ and MgNi$_2$.}\\label{Fig:mgzn2_motif_map}\n\\end{figure}\n\n\\twocolumngrid\n\n\\onecolumngrid\n\n\\begin{figure}[]\n    \\includegraphics[width=0.8\\textwidth]{fig5.png}\n    \\caption{Map for first four frequent motifs in $\\gamma$ and $\\epsilon_{BB}$ excludes the general motif FCC and HCP. (a) Octahedron (b) Half hexagonal prism 1. (c) Triangular Prism (d) Half hexagonal prism 2. Red indicates stable structures and blue indicates metastable.\n    }\\label{Fig:req_motif}\n\\end{figure}\n\n\\twocolumngrid\n\nIn Fig.~\\ref{Fig:mgzn2_motif_map} we show the domain of stability and metastability for the MgZn$_2$ phase and the Z$_{16}$ motif. The GA searches were performed on a mesh of $\\gamma$ and $\\epsilon_{BB}$ with an increment of 0.1. Here, to improve the resolution of the stability range, we examined the stability of all GA-found structures and motifs on a finer mesh in the $\\gamma$-$\\epsilon_{BB}$ plane with an increment of 0.02. Rather interestingly, the stability range of the Z$_{16}$ motif is larger than that of the MgZn$_2$ phase, indicating this motif is not unique to MgZn$_2$, but shared by other Laves and Frank-Kasper phases. Similar plots for the four more frequent motifs are shown in Fig.~\\ref{Fig:req_motif}.\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Comparison between packing phases\\cite{Hopkins2012} and our study for $ 0.3 \\le \\gamma  \\le 1$. (SG=Space group), The $\\ast$ indicates there are small distortions in the LJ phase, compared with the packing phase. Motifs in the LJ column indicates that they are not stable in the GA result, but they have the motif inside in the corresponding $\\gamma$ regime.}\n    \\begin{tabular}{c c c c | c c}\n    \\hline\n    \\hline\n     Phase   & $\\gamma$-range        &  Ref  &  SG &  LJ & Distortion \\\\ \\hline\n     A$_3$B  & $[0.618, 0.660]$ & \\cite{OToole2011} &  59  & TiCu$_3$  &  \\\\\n     AlB$_2$  & $[0.528, 0.620]$  &       & 191 & AlB$_2$ & \\\\\n     AuTe$_2$ & $[0.488, 0.528]$ & \\cite{Filion2009} & 12 & Motif-6-2 &   \\\\\n     (2-2)$^{\\ast}$ & $[0.480, 0.497]$ &\\cite{Marshall2010} & 11 & Motif-6-1 & \\\\\n     (4-2) & $[0.488, 0.483]$ &\\cite{Hopkins2012} & 191 & Motif-12-3 & \\\\\n     (5-2) & $[0.480, 0.483]$ &\\cite{Hopkins2012} & 44 & & \\\\\n     (7-3) & $[0.468, 0.480]$ &\\cite{Hopkins2012} & 71 & Motif-12-3 & \\\\\n     HgBr$_2$ & $[0.443, 0.468]$ &\\cite{Filion2009} & 36 & Motif-6-4 & \\\\\n     (6-6) & $[0.414, 0.457]$ &\\cite{Hopkins2012} & 11 & Motif-6-4 &  \\\\\n     XY & $[0.275, 0.414]$ &\\cite{Hopkins2012} &  & & \\\\\n     (6,1)$_4$ & $[0.352, 0.321]$ &\\cite{Hopkins2012} & 69 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     (6,1)$_6$ & $[0.321, 0.304]$ &\\cite{Hopkins2012} & 139 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     (6,1)$_8$ & $[0.302, 0.292]$ &\\cite{Hopkins2012} & 139 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     \\hline\\hline\n     \\\\\n    \\end{tabular}\n\n    \\label{tab:pg_lj_comp}\n\\end{table}\n\nQuite generally, the motifs are far more sensitive to $\\gamma$ than they are to $\\epsilon_{BB}\/\\epsilon_{AA}$, confirming that the particle size is more important than the actual intensity of the interactions. It is consistent with all calculations that stable structures with the same values of $\\gamma$ tend to share motifs. As found for MgZn$_2$ and Z$_{16}$, the regions for stability and metastability is wider than the corresponding structures, thus indicating that motifs define very general families of structures, like Laves phases. A classification of motifs by Renormalized Angle Sequences (RAS)\\cite{Lv2017,Lv2018} has been included in Supporting Information. \n\nThis study has identified 53 equilibrium lattices and 42 motifs (with the larger particle A as reference). We now discuss the relevance of these results for packing models\\cite{Filion2009,Hopkins2012}, their connection to the motifs reported in Quasi Frank-Kasper phases\\cite{Travesset2017} and their implications for binary superlattices.\n\n\n\\textbf{Packing Phase Diagram.} We consider the study of Hopkins \\textit{et al.}\\cite{Hopkins2012} as the reference phase diagram for packing problems, although it only includes unit cells containing up to 12 particles. Consistently with this study we concentrate on the range $0.3\\le \\gamma \\le 1$, also because for smaller $\\gamma$ there are many phases with narrow stability ranges that are less relevant in actual experimental systems.\n\n\n\nFrom Table~\\ref{tab:pg_lj_comp}, the packing of binary phase diagram contains 13 phases for the $0.3 \\le \\gamma \\le 1$ range. For large $\\gamma > 0.528$ only two phases exist; AlB$_2$ and A$_3$B, which are both found in binary LJ systems  (if allowing for small differences in A$_3$B). For $0.488 < \\gamma < 0.528$, however, the AuTe$_2$ phase is reported; We did not find such phase, but we do report the Motif-6-2 as stable for the same range of $\\gamma$, see Supporting Information, which is present in the equilibrium phases at $\\gamma=0.5$ ${{\\mbox A}_4{\\mbox B}_6}^{(166)}_{(9)}$, BaCu and TePt. Some other phases, which are reported as packing phases \\cite{Hopkins2012} but not stable in the GA search, are also identified to have the motif in the corresponding $\\gamma$ regime. This indicates that these packing phases may be meta-stable in our calculation. For smaller $\\gamma$, there is also overlap if allowing for small distortions. \n\nOther phases that have large packing fractions, such as CrB and S74e\/h(KHg$_2$ in our notation)\\cite{Filion2009}, that are metastable in the packing phase diagram become equilibrium, thus showing that the LJ system augments the number of stable phases as compared with packing models.\n\n\\textbf{Motifs and Quasi Frank Kasper Phases.} In Ref.~\\cite{Travesset2017} it was shown that all experimental BNSLs could be described as disclinations of the $\\{3,3,5\\}$ polytope, thus generalizing well known four Frank-Kasper motifs Z$_{12}$,Z$_{14}$,Z$_{15}$, Z$_{16}$\\cite{Frank1958,Frank1959} to include other motifs. \n\n\\onecolumngrid\n\n\\begin{table}[]\n    \\centering\n    \\caption{Motifs in Quasi Frank Kasper phases\\cite{Travesset2017} compared to the ones described in this work.} \n    \\begin{tabular}{c | c c c c c c}\n    \\hline\\hline\n    QFK\\cite{Travesset2017}  & $\\mbox{Z}_6$ & $\\mbox{Z}_{12}^{\\prime\\prime}$ & $\\mbox{Z}_{14}^{\\prime}$  & $\\mbox{Z}_{16}$ & $\\mbox{Z}_{18}^{\\prime\\prime}$ & $\\mbox{Z}_{24}$ \\\\\n    \\hline\n    This work & Motif-6-4 & Motif-12-2 & Motif-14-1 & Motif-16-2 & Motif-18-3 & Motif-24-1 or  \\\\\n              &        &         &         &         & & Motif-24-3 \\\\\n    \\hline\n    \\hline\n\n    \\end{tabular}\n    \\label{tab:qfk_lj_comp}\n\\end{table}\n\n\\twocolumngrid\n\nIn Table~\\ref{tab:qfk_lj_comp} we show the equivalence between Quasi Frank Kasper motifs and the ones obtained in this work, which only include those with the A-particle as reference. It should be pointed that the motifs are not completely the same, as in Ref.~\\cite{Travesset2017} the motifs were defined by the Voronoi cell and its corresponding neighbors, which is a slightly different definition than the one used in this paper. \n\n\\textbf{Experimental Results.} The list of experimentally reported BNSLs is taken from Ref.~\\cite{Travesset2017a}, where we have excluded two dimensional superlattices and those where nanocrystals cannot be approximated as spherical, see Ref.~\\cite{Boles2016}. The comparison between the results obtained in this paper and experimental BNSLs is provided in Table~\\ref{tab:exp_lj_comp}.\n\n\\begin{table}[]\n    \\centering\n    \\caption{Experimentally determined structures. NA: Phase not available in this study. NF: Phase not found in this study. The DDQC\/AT is a quasicrystal phase. The bccAB$_6$ phase is also known as C$_{60}$K$_6$ and is denoted as ${\\mbox{A}\\mbox{B}_6}^{(229)}_{(1)}$ in this paper.} \n    \\begin{tabular}{c c | c c c}\n    \\hline \\hline\n    \\multicolumn{2}{c|}{Experiment} &  \\multicolumn{3}{c}{Binary LJ}\\\\ \\hline\n    BNSL & $\\gamma$-range &     & $\\gamma$-range & $\\varepsilon_{BB}$-range  \\\\\n    NaCl &  $[0.41,0.60]$  &    &  $[0.2,0.5]$    & $[0.1,0.8]$     \\\\\n    CsCl &  $[0.71,0.90]$  &    &   NF   &      \\\\\n    AuCu &  $[0.58,0.71]$  &    &    NF  &      \\\\\n    DDQC\/AT&  $[0.41,0.43]$  &    &  NA   &      \\\\\n    AlB$_2$ &  $[0.45,0.70]$  &    & $[0.4,0.7]$    & $[0.1,0.9]$     \\\\\n    MgZn$_2$ & $[0.60,0.81]$  &    & $[0.7,1.0]$     & $[0.1,1.0]$  \\\\\n    AuCu$_3$ & $[0.40,0.60]$  &   &   NF   &      \\\\\n    Li$_3$Bi & $[0.53,0.56]$  &   &   NF   &      \\\\\n    Fe$_4$C  & $[0.55,0.65]$  &    &   NF   &     \\\\\n    CaCu$_5$ & $[0.60,0.80]$  &    &  $[0.6,0.8]$    & $[0.1,0.9]$    \\\\\n    CaB$_6$  & $[0.43,0.47]$  &    & $[0.3,0.5]$    & $[0.1,0.8]$    \\\\\n    bccAB$_6$  & $[0.45,0.50]$  &    & $[0.4,0.6]$      & $[0.1,0.5]$     \\\\\n    cubAB$_{13}$  & $[0.55,0.60]$  &   &  NF    &     \\\\\n    NaZn$_{13}$  & $[0.47,0.70]$  &    & $[0.6]$  & $[0.1,0.6]$     \\\\\n    \\hline \\hline\n    \\end{tabular}\n    \\label{tab:exp_lj_comp}\n\\end{table}\n\nSeven of the experimentally reported BNSLs, namely NaCl, AlB$_2$, MgZn$_2$, CaCu$_5$, CaB$_6$, bccAB$_6$ and NaZn$_{13}$ are found as equilibrium phases in the LJ system essentially for the same range of $\\gamma$. The fact that in our results the stability is roughly independent of $\\varepsilon_{BB}$ in certain regions provides some support for the idea that microscopic details of the potential are unimportant in this region (``universality''). Further making this point is that the same phases are stable for soft repulsive potentials in the same $\\gamma$-range \\cite{Travessetpnas2015,HorstTravesset2016,LaCour2019}. \n\nWe now analyze the phases reported in experiments that are not equilibrium in our study. One of them is beyond the scope of our calculation; DDQC\/AT, which is a quasicrystal. The Li$_3$Bi and also the AuCu$_3$ are stabilized by large deformations of the ligands, i.e. vortices\\cite{Travesset2017}, and therefore are not possible to obtain from a quasi HS approximation. The Fe$_4$C phase was observed in 2006\\cite{Shevchenko2006}, and since then, it has not been reported in any further study, which may suggest is metastable, and furthermore, it can only be stabilized by vortices\\cite{Travesset2017a}. The CsCl phase has a very narrow range of stability around $\\gamma_c = \\sqrt{3}-1=0.732$\\cite{Travesset2017a}, which is likely missed by the discretization of $\\gamma$ values in our study.\nFinally, AuCu occurs when there is ligand loss\\cite{Travesset2017,Boles2019} and is stabilized through a different mechanism involving the non-spherical shape of the nanocrystal. We therefore conclude that the binary LJ model successfully predicts those experimentally reported phases that can be described as quasi-hard spheres. This is in contrast to packing models, where MgZn$_{2}$ or CaCu$_5$ phases, widely reported in experiments are not equilibrium phases (maximum of the packing fractions). See Fig.~\\ref{Fig:summary} for a visual summary of this discussion.\n\n\\begin{figure}[]\n    \\includegraphics[width=0.48\\textwidth]{fig6.pdf}\n    \\caption{Summary of the main results of the paper: The experimental phases are classified according to: Hard sphere, OTM\/hard sphere (exist when NCs are modeled as hard spheres but are stabilized by vortices)\\cite{Travesset2017}, pure OTM(only stable with vortices), and other (observed in special cases, such as ligand detachment\\cite{Boles2019}). See also Table~\\ref{tab:exp_lj_comp}. Consistent with the LJ assumptions, only the hard sphere phases are found in our work. The Experiment Pred includes those strong candidates to be found experimentally, as discussed below.}\\label{Fig:summary}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nBy the use of Genetic Algorithm (GA), we have been able to predict stable structures under different sizes of particles and strengths of interaction ($\\gamma \\in$  [0.3 to 0.9], $\\epsilon_{BB} \\in$ [0.1 to 1.0]). We report 53 stable phases, which cover a significant part of currently reported structures.  Besides that, we also predict 35 stable structures which are not in Material Project database. We find that the type of stable structures strongly depends on $\\gamma$, but weakly on $\\epsilon_{BB} < 1$, providing evidence that the stability of the lattices has a weak dependence on the potential details (universality). By comparing our results with other theoretical and experimental works, it is shown that regardless of potential details, the same $\\gamma$ regime has the same stable structure, which reinforce that the stable structure has a weak dependence on the potential details.\n\n\nThere are two aspects about the limitations of the hard sphere description: The first is that it does \\textit{not} provide a free energy: the observed phases are not the ones with maximum packing fraction\\cite{Hopkins2012}, but rather, ones where the packing fractions is maximum for the particular structure. This is where the Binary LJ becomes important: the stable phases are the ones that minimize the free energy (modeled as the LJ potential). The second limitation is that it does not model large deformations of the ligand shell: these cases go beyond the LJ model and is evident from Fig.~\\ref{Fig:summary}, showing that these phases are absent.\n\nThe crystalline motifs are employed to describe the large amount of metastable structures. We find that metastable structures mostly can be described from the motifs present in equilibrium structures, thus suggesting the possibility of building superlattices by patching all motifs that can tile the 3D space, as similarly done in the more restricted case of Frank-Kasper phases\\cite{DutourSikiric2010}. It also raises the possibility of motifs being present within the liquid\\cite{Damasceno2012} as a way to anticipate the emergent crystalline structure.\n\nComparing with available experimental results, see Table~\\ref{tab:exp_lj_comp} and Fig.~\\ref{Fig:summary}, the binary LJ model captures all the equilibrium phases where nanocrystals can be faithfully described as quasi hard spheres: NaCl, AlB$_2$, MgZn$_2$, CaCu$_5$, CaB$_6$, bccAB$_6$ and NaZn$_{13}$. The other phases reported in experiments either require the presence of vortices, as predicted by the OTM\\cite{Travesset2017,Travesset2017a}, or are stable over a very narrow range of $\\gamma$ values, likely missed by the necessary discrete number considered in our study.\n\nPacking phase diagram models reported 14 equilibrium phases in the interval $\\gamma \\in [0.3,1)$, see Table~\\ref{tab:pg_lj_comp}, while our study reports 53, thus showing that binary LJ have a more complex phase diagram. Rather interestingly, phases such as MgZn$_2$ or CaCu$_5$, which are very common in experiments, are absent in the packing phase diagram; Although very useful in identifying at which $\\gamma$ values a phase is likely to appear, packing models give very poor predictions on which, among all possible phases, will actually be observed.\n\nThe two guiding principles for stability of BNSLs in experiments are high packing fraction (or low Lennard-Jones Energy) and tendency towards icosahedral order, as reflected in the motifs\\cite{Travesset2017a,Coropceanu2019}. Therefore, we expect that those equilibrium Lennard-Jones phases with Quasi Frank-Kasper motifs, \nfor example, the BNSLs ${\\mbox{A}_{2}\\mbox{B}_{4}}^{(227)}_{(1)}$ and\n${\\mbox{A}_{2}\\mbox{B}_{12}}^{(139)}_{(1)}$\n(Motif-16-2), or $\\mbox{Zr}_{2}\\mbox{Cu}^{(139)}_{(1)}$ (Motif-14-1), will be excellent candidates to search for BNSLs,  see Fig.~\\ref{Fig:summary}. Definitely, these ideas will be developed further in the near future, where the 53 stable lattices will be studied with more realistic nanocrystal models described at the atomic level. \n\nIn this work we focused on spherically symmetric potentials with additive interactions, as described by relations like\n\\begin{equation}\n\\epsilon_{AB}=\\frac{1}{2}(\\epsilon_{AA}+\\epsilon_{BB}) \\ .\n\\end{equation}\nIt is of interest to consider more general models, where these restrictions are lifted. This, however, will be the subject of another study.\n\n\\section{Methods}\nThe crystal structure searches with GA were only constrained by stoichiometry, without any assumption on the Bravais lattice type, symmetry, atom basis or unit cell dimensions (up to a maximum of particles per unit cell). During the GA search, energy was used as the only criteria for optimizing the candidate pool. At each GA generation, 64 structures are generated from the parent structure pool \\textit{via} the mating procedure described in Ref.~\\cite{Deaven1995,Oganov2006,Ji2010}. The mating process was based on real-space \"cut-and-paste\" operations that was first introduced to optimize cluster structures~\\cite{Deaven1995}. This process was extended to predict low-energy crystal structures by Oganov~\\cite{Oganov2006} and reviewed in Ref.~\\cite{Ji2010}. Here, we follow the same procedure that was described in detail in Ref.~\\cite{Ji2010} and was implemented in the Adaptive Genetic Algorithm (AGA) software.\n\nWith a given set of LJ parameters, we performed three GA searches independently, with each GA search running for 1000 generations. The maximum number of particles per unit cell used in each search was 20, and thus, phases with large unit cells, the most relevant being NaZn$_{13}$, could not be included. Therefore, we include NaZn$_{13}$ into our calculation manually. All energy calculations and structure minimizations were performed by the LAMMPS code \\cite{Plimpton1995} with some cross checks using HOOMD-Blue\\cite{AndersonMe2008a} with FIRE minimization\\cite{Bitzek2006}. The database of binary lattices in HOODLT\\cite{Travesset2014} was also used.\n\n\n\\section{Supporting Information}\nSupporting information contains: \nList and maps of structures searched by genetic algorithm;\nphase diagrams of equilibrium structures; equilibrium motif database; maps of motifs; algorithms for motif identification and renormalized angle sequence\n\n\\section{acknowledgement}\n\nA.T acknowledges discussions with I. Coropceanu and D. Talapin. We also thank Prof. Torquato for facilitating the data of his group packing studies. Work at Ames Laboratory was supported by the US Department of Energy, Basic Energy Sciences, Materials Science and Engineering Division, under Contract No. DE-AC02-07CH11358, including a grant of computer time at the National Energy Research Supercomputing Center (NERSC) in Berkeley, CA. The Laboratory Directed Research and Development (LDRD) program of Ames Laboratory supported the use of GPU-accelerated computing. Y. S. was partially supported by National Science Foundation award EAR-1918134 and EAR-1918126.\n\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\setcounter{page}{2}\n\n\nAn interesting application of gauge\/gravity duality to condensed matter physics arises in the study of \nmomentum relaxation. This is so mainly because the resulting zero frequency conductivities are finite, \nallowing us to study transport in a more realistic way.\nTo this end, one must find gravitational solutions which break translational \ninvariance along the boundary directions due to the presence of one or more spatially dependent sources. \nGenerically this involves numerically solving the Einstein equations which give an elliptic PDE problem in this context.\nSome studies focus on configurations which describe a lattice in the dual field theory, for instance when the chemical potential is a periodic function \nof a spatial direction on the boundary.\nGravitational solutions of this kind have been successfully constructed \n\\cite{Horowitz:2012ky, Horowitz:2012gs, Horowitz:2013jaa, Donos:2014yya, Rangamani:2015hka}\nand they indeed reproduce the expected\nlow frequency dynamics, i.e.\\ the zero frequency delta functions in the conductivities are resolved into finite width\nDrude peaks.   \n\nA considerable technical simplification arises if a certain global symmetry is present, say, in the matter sector \nof the bulk theory. Then, translational invariance can be broken along that symmetry direction while \npreserving homogeneity of the geometry, which in turn \nimplies that the construction of such solutions only requires solving ODEs. Examples of this kind include\n\\cite{Donos:2012js, Donos:2013eha}, which have been shown to yield finite DC conductivities as well as to possess a rich \nstructure which displays transitions between different metallic and \ninsulating regimes.\\footnote{A simplification of this type arises in holography with massive gravity in the bulk \\cite{Vegh:2013sk}.}\n\n\nWe can  simplify the problem even further by arranging the bulk matter content in such a way that the \nblack branes of interest are not only homogeneous but also isotropic. This fact was exploited in \\cite{Andrade:2013gsa} \n(see also \\cite{Taylor:2014tka}), which considered a particular configuration of a set of massless scalars, termed linear axions since they \nare shift-symmetric, that allows for an analytical black brane \nsolution at non-zero chemical potential in an arbitrary number of dimensions. As expected, the DC \nelectric and thermal conductivities are finite \\cite{Andrade:2013gsa, Gouteraux:2014hca,Davison:2014lua,Donos:2014cya}, and they can be evaluated \nanalytically. However, the conductivities at non-zero frequency need to be computed numerically since this \ninvolves solving coupled fluctuation equations around the background solution. Interestingly, it has been noted that \ndeviations from Drude physics are present when the strength of momentum relaxation is large \n\\cite{Kim:2014bza, Davison:2014lua, Davison:2015bea}. \n\n\nRecently, it has been noted that the equations of General Relativity simplify considerably in the limit in \nwhich the number of spacetime dimensions, $D$, is taken to be large, which provides an efficient tool to \napproximate finite $D$ results, as a perturbative calculation in $1\/D$  \\cite{Emparan:2013moa}. \nThe main ingredient of the construction is the fact that when the number of spacetime dimensions is large, \nthe gravitational potential becomes very steep near the horizon which yields to a natural separation of the \ndynamics that localise near the horizon and the ones that probe regions far away from it. \nIn particular, this implies that the quasi-normal mode (QNM) spectrum splits into near-horizon {\\it decoupled} modes,\nand {\\it coupled} modes that are delocalised \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}. \nAs explained in these references, while the coupled modes are quite generic, i.e.\\ \nshared by many black holes, the decoupled modes are sensitive to the particularities of different \nsolutions. \n\nIn this paper we initiate the study of holographic inhomogeneities and the resulting momentum relaxation using large $D$ techniques.\nOur first goal is to study the decoupled quasi-normal modes which control the characteristic decay rate of the  \nelectric and thermal conductivities in the linear axion model at non-zero chemical potential. \nWe find that momentum conservation is restored unless we scale the strength of the axions with $D$, i.e.\\ the decoupled QNM frequency vanishes to leading order. \nTherefore we scale the axion strength appropriately and obtain a QNM capturing the momentum decay rate at leading order in large $D$. We calculate corrections to this in $1\/D$.\nWe see that in certain regimes the QNM frequencies are well-described using the large $D$ approximation. \nSecond, we will compute the AC thermal conductivity for zero chemical potential, which at leading order is exactly of Drude form. These results are consistent with the corresponding QNM calculation. To our knowledge, this is the first \nanalytical realisation of Drude behaviour outside of the hydrodynamic regime in the context of holography.\nWith these results at hand, we will comment on the signature of the transition from coherent to incoherent regimes, \ni.e.\\ the breakdown of Drude physics, in the large $D$ approximation.   \n\nThis paper is organised as follows. In section \\ref{sec:model} we review the axion model, its transport properties and the appropriate master fields for the conductivity calculation. In section \\ref{sec:QNM} we compute the large $D$ decoupled QNMs which control momentum relaxation, giving analytical expressions for their frequencies. In section \\ref{sec:conductivity} we compute the AC thermal conductivity as an expansion in $1\/D$. We conclude in section \\ref{sec:conclusions}. \n\n\n\\section{Momentum relaxation in arbitrary D}\n\\label{sec:model}\n\n\nIn this section we review the holographic model of momentum relaxation proposed in~\\cite{Andrade:2013gsa}. \nWe will discuss the main properties of the background solution and describe the computation of the two-point \nfunctions using a gauge invariant master field formalism. \n\n\n\\subsection{Linear axion background}\n\\label{sec:axion}\n\n\nThe holographic model of momentum relaxation in $D = n+3$ bulk dimensions which we consider throughout this \npaper is given by the action \\cite{Andrade:2013gsa}\n\\begin{equation}\\label{S0}\n\tS_0 = \\int d^{n+3} x \\sqrt{- g} \\left( R + (n+1)(n+2) \\ell^{-2} - \\frac{1}{4} F^2 - \\frac{1}{2} \\sum_{I=1}^{n+1} (\\partial \\psi_I)^2  \\right)\n\\end{equation}\n\\noindent where $F = dA$ is the field strength of a $U(1)$ gauge field, $\\psi_I$ are $(n+1)$ massless scalar fields \nand $\\ell$ is the AdS radius which we set to one henceforth. \n\nThis model admits the following analytical black brane solution\\footnote{This solution was previously derived in \\cite{Bardoux:2012aw} \nin a different context.}\n\\begin{equation}\\label{NL ansatz}\n\tds^2 = - f(r) dt^2 + \\frac{dr^2}{f(r)} + r^2 \\delta_{a b} dx^a dx^b, \\qquad A = A_t(r) dt, \\qquad \\psi_I = \\delta_{I a} x^a\n\\end{equation}\n\\noindent where $a$ labels the $(n+1)$ boundary spatial directions $x^a$ and\n\\begin{align}\n\tf(r) &= r^2 - \\frac{\\alpha^2}{2 n} - \\frac{m_0}{r^n} + \\frac{n \\mu^2}{2(n+1)} \\frac{r_0^{2n}}{r^{2n}}\\label{axion soln} \\\\\n\tA_t(r) &= \\mu \\left(1  - \\frac{r_0^n}{r^n} \\right)\\label{axion gauge field}\n\\end{align}\nHere $r_0$ is the horizon of the brane, $\\mu$ is the chemical potential in the dual theory and $m_0$ is related to the total energy\nof the solution. \nThe Hawking temperature is given by \n\\begin{equation}\n\tT = \\frac{f'(r_0)}{4 \\pi} = \\frac{1}{4 \\pi} \\left( (n+2) r_0 - \\frac{\\alpha^2}{2 r_0} - \\frac{n^2 \\mu^2}{2 (n+1) r_0} \\right).\n\\end{equation}\nNote that, despite the fact that the geometry is isotropic and homogeneous, the solution manifestly breaks translational\ninvariance due to the explicit dependence of $\\psi_I$ on $x^a$. This feature is reflected in the its thermoelectric DC conductivities, \nwith the $\\delta$-function at zero frequency present in the Reissner-Nordstr\\\"om solution removed due to the breaking of translational invariance. \n\n\\subsection{Transport}\n\\label{sec:Trans}\nConductivities can be computed in terms of two-point functions, which are given in AdS\/CFT by studying linear fluctuations\naround the black holes under consideration.  Here we are interested in the electric and thermal conductivities at zero spatial \nmomentum, which can be obtained in terms of the retarded two-point functions\n\\begin{equation}\\label{2pt fns}\n\tG_{JJ} (\\omega) = \\langle J^1 J^1 \\rangle (\\omega), \\quad G_{QJ} (\\omega) = \\langle Q^1J^1 \\rangle (\\omega) \\quad G_{QQ} (\\omega) = \\langle Q^1Q^1 \\rangle (\\omega)\n\\end{equation}\n\\noindent where $Q^i = T^{ti} - \\mu J^i$ and $T^{ij}$ and $J^i$ are the the stress tensor and $U(1)$ current of the field theory, respectively. \nHere we have chosen to compute the conductivities along the axis $x^1$. Because the black holes of interest are isotropic, this does not \nresult in loss of generality. \nWe can then express the electric conductivity $\\sigma(\\omega)$, the thermo-electric conductivity $\\beta(\\omega)$ and the thermal conductivity $\\kappa(\\omega)$ \nin terms of the two-point functions \\eqref{2pt fns} by means of the Kubo formulae:\n\\begin{align}\n\\nonumber\n\t\\sigma(\\omega) &= \\frac{i}{\\omega} (G_{JJ} (\\omega) - G_{JJ} (0))  , \\\\\n\\nonumber\n\t\\beta(\\omega)   &= \\frac{i}{\\omega T} (G_{QJ} (\\omega) - G_{QJ} (0))  , \\\\ \n\\label{conductivityDefs}\n\t\\kappa(\\omega) &= \\frac{i}{\\omega T} ( G_{QQ} (\\omega) - G_{QQ} (0) )  \n\\end{align}\n\nAnalytical traction may be gained in the DC limit, where these conductivities can be computed. \nAs shown in \\cite{Andrade:2013gsa} the DC electrical conductivity is given by\n\\begin{equation}\n\t\\sigma(0) = r_0^{n-1} \\left(1 + n^2\\frac{\\mu^2}{\\alpha^2} \\right)\n\\end{equation}\nwhilst the thermal and thermo-electric conductivities for general $n$ are given in \\cite{Donos:2014cya}\n\\begin{equation}\n\t\\kappa(0) =  r_0^{n+1}\\frac{(4\\pi)^2 T}{\\alpha^2}, \\qquad \\beta(0) =  r_0^{n}\\frac{4\\pi \\mu}{\\alpha^2}. \\label{kappaDC} \n\\end{equation}\n\nSeparately, the conductivities may be approximated analytically for small $\\alpha$ by the Drude formula. \nFor instance, at $n=1$ the thermal conductivity is given by \n\\begin{equation}\\label{drude}\n\t\\kappa(\\omega) = \\frac{\\kappa(0)}{1 - i \\omega \\tau}, \\qquad \\omega \\ll T\n\\end{equation}\nwhere $\\tau$ is the characteristic time of momentum relaxation, set by $\\alpha$. Since there is only \none characteristic time scale, we say that transport is {\\it coherent} in this regime.\\footnote{See \\cite{Hartnoll:2014lpa} for a discussion on this terminology.}\nIncreasing $\\alpha$, the deviations from \\eqref{drude} become large, driving the system into an {\\it incoherent} \nphase. This transition was first observed in this holographic system by a numerical analysis in $n=1$ \\cite{Kim:2014bza}, \nand later on also noticed in the presence of a charged scalar condensate in \\cite{Andrade:2014xca}. A closely related coherent\/incoherent\ntransition has been reported for the thermal conductivity at zero chemical potential for $n=1$ in \\cite{Davison:2014lua}, which \nfocussed on an explanation in terms of QNM: the system behaves coherently when there is an isolated, long-lived, \npurely dissipative excitation in the spectrum. Moreover, this analysis was extended in perturbation theory to include \nchemical potential \\cite{Davison:2015bea}, with qualitatively similar results. \n \n\n\\subsection{Master fields}\n\\label{sec:MF}\nA general approach to computing the conductivities in the background \\eqref{NL ansatz}-\\eqref{axion gauge field} utilises a minimal, consistent set of perturbations,\n\\begin{equation}\\label{linear perts}\n  \\delta A = e^{- i \\omega t} a(r) dx^1, \\qquad \t\\delta (ds^2) = 2 e^{- i \\omega t} r^2 h(r) dt dx^1 , \n  \\qquad \\delta \\psi_1 = e^{- i \\omega t} \\alpha^{-1} \\chi(r).\n\\end{equation}\nThe linearised equations of motion which govern the perturbations \\eqref{linear perts} can be written as\n\\begin{align}\n\ta'' + \\left[ \\frac{f'}{f} + \\frac{(n-1)}{r} \\right] a' + \\frac{\\omega^2}{f^2} a + \\frac{\\mu n}{f} \\frac{r_0^n}{r^{n-1}} h' &= 0 \\\\\n\t\\chi'' + \\left[ \\frac{f'}{f} + \\frac{(n+1)}{r} \\right] \\chi' + \\frac{\\omega^2}{f^2} \\chi - \\frac{i \\omega \\alpha^2}{f^2} h &=0 \\\\\n\t\\frac{i \\omega r^2}{f} h' + \\frac{i \\omega n \\mu}{f} \\frac{r_0^n}{r^{n+1}} a - \\chi' &=0 \n\\end{align}\nwhere primes denote derivatives with respect to $r$. For odd $n$, the near boundary expansions for the physical fields are given by \n\\begin{align}\n\\label{UV phys1}\n\th &= h^{(0)} + \\ldots + \\frac{h^{(n+2)}}{r^{n+2}}  \t  + \\ldots\\\\\n\ta &= a^{(0)} + \\ldots + \\frac{a^{(n)}}{r^n}   + \\ldots \\\\\n\\label{UV phys3}\n\t\\chi &= \\chi^{(0)} + \\frac{\\chi^{(1)}}{r} + \\frac{\\chi^{(2)}}{r^2}  + \\ldots\n\\end{align}\nFor even $n$, the expansions \\eqref{UV phys1}-\\eqref{UV phys3} contain logarithms, as a result of the Weyl anomaly \npresent in even boundary dimensions \\cite{Henningson:1998gx}. These terms will play no role in the following, so we \nshall omit them. \nThe terms $\\chi^{(1)} $, $\\chi^{(2)} $ are fixed by the equations of motion as\n\\begin{equation}\\label{chi 1 2}\n\t\\chi^{(1)} = 0 , \\qquad \\chi^{(2)}  = \\frac{\\omega( \\omega \\chi^{(0)} - i \\alpha^2 h^{(0)})}{2 n}\n\\end{equation}\nThe gauge invariant sources for the electric and thermal conductivity are $a^{(0)}$ and \n$ s^{(0)} =\\omega \\chi^{(0)} - i \\alpha^2 h^{(0)} $, respectively (see e.g.\\ \\cite{Donos:2013eha}).\n\nAs shown in \\cite{Andrade:2013gsa}, the perturbation equations can be decoupled in terms of two gauge invariant master \nfields $\\Phi_\\pm$, given by \n\\begin{equation}\\label{MF def}\n   f  r \\chi'  = \\frac{\\omega}{\\mu} ( \\tilde c_+ \\Phi_+ + \\tilde c_- \\Phi_-  ),  \\qquad\n\ta = - i (\\Phi_+ + \\Phi_-)\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{cpm}\n\t\\tilde c_\\pm = \\frac{1}{2 r_0^n} \\left\\{ (n+2) m_0 \\pm [ (n+2)^2 m_0^2 + 4 r_0^{2n} \\mu^2 \\alpha^2  ]^{1\/2} \\right \\}.\n\\end{equation}\n\n\\noindent The master fields are governed by the equations  \n\\begin{equation}\\label{MF eqs pm}\n\tr^{3-n} ( f r^{n-1} \\Phi_\\pm' )' + \\left(  \\frac{r^2 \\omega^2}{f} - \\frac{n^2 \\mu^2 r_0^{2n}}{r^{2n}} + \n\tn \\tilde c_\\pm \\frac{r_0^n}{r^n} \\right) \\Phi_\\pm  = 0.\n\\end{equation}\n\\noindent As shown in \\cite{Son:2002sd}, in order to obtain the retarded correlators the fluctuations must satisfy ingoing boundary conditions \nat the black hole horizon. These can be implemented by simply imposing the ingoing condition on the master field \n\\cite{Berti:2009kk}, which amounts to\n\\begin{equation}\\label{ingoing bc}\n\t\\Phi_\\pm(r) = (r - r_0)^{- i \\omega\/(4 \\pi T)} ( \\Phi_\\pm^H + \\ldots  ), \\qquad {\\rm near } \\; r = r_0\n\\end{equation}\n\\noindent where $\\Phi_\\pm^H$ are arbitrary constants and the ellipsis denotes regular subleading terms. \n\nThe near boundary asymptotics of the master fields are given by \n\\begin{equation}\\label{UV MF}\n\t\\Phi_\\pm = \\Phi^{(0)}_\\pm + \\ldots + \\frac{1}{r^{n}} \\Phi^{(n)}_\\pm + \\ldots\n\\end{equation}\nFrom \\eqref{UV phys1}-\\eqref{UV phys3} and \\eqref{MF def}, we learn that the asymptotic data in \\eqref{UV MF} \nis related to the physical asymptotic data as\n\\begin{align}\n\t\\Phi^{(0)}_\\pm &= \\pm \\frac{1}{\\omega(\\tilde c_- - \\tilde  c_+)} ( 2 \\mu \\chi^{(2)} + i \\omega \\tilde c_- a^{(0)}  ) \\\\\n\t\\Phi^{(n)}_\\pm &= \\pm \\frac{i}{\\omega^2(\\tilde c_- - \\tilde  c_+)} ( - \\alpha^2 (n+2) \\mu h^{(n+2)} + \\omega^2 \\tilde c_- a^{(n)}  ) \n\\end{align}\n\\noindent where $\\chi^{(2)}$ is related to the gauge invariant source for the stress tensor by \\eqref{chi 1 2}.\nIn order to compute the two-point functions at $\\mu\\neq 0$, a detailed computation of the on-shell action is needed due to the \nnon-trivial interplay between the physical sources and vevs in $\\Phi_\\pm$\\footnote{This computation was carried out\nfor $n=1$ in \\cite{Davison:2015bea}.}. However,  it is easy to see that in order to obtain the poles in such correlators \nit suffices to solve for the spectra of $\\Phi_\\pm$ with Dirichlet boundary conditions $\\Phi^{(0)}_\\pm =0$. \n\n\n\n\n\\subsubsection{The neutral case}\n\\label{neutral master}\n\nFor $\\mu = 0$, all the gauge invariant information is contained in the thermal conductivity. To compute it, the \nrelevant fluctuations are \\eqref{linear perts} with $a(r) = 0$. The physical boundary data satisfies \\eqref{chi 1 2} \nand the gauge invariant source for the stress tensor is again $ s^{(0)}$. \nVia simple manipulations of the equations of motion, we can derive the master field equation\n\\begin{equation}\\label{MF eq neutral}\n\tr^{3-n} ( f r^{n-1} \\Phi' )' + \\left(  \\frac{r^2 \\omega^2}{f}  + n (n+2) \\frac{m_0}{r^n} \\right) \\Phi = 0 \n\\end{equation}\n\\noindent where the master field $\\Phi$ is given by\n\\begin{equation}\\label{Phi def}\n\t\\Phi =   \\frac{f r \\chi'}{i\\omega}\n\\end{equation}\nNote that \\eqref{MF eq neutral} is the $\\mu \\to 0$ limit of the equation for $\\Phi_+$ \\eqref{MF eqs pm} with $m_0 \\geq 0$.\nEquation \\eqref{MF eq neutral} has been previously derived\nfor $n=1$ in \\cite{Davison:2014lua}.\nAs in the $\\mu \\neq 0$ case, the UV asymptotics for $\\Phi$ can be written as\n\\begin{equation}\\label{UV Psi}\n\t\\Phi = \\Phi^{(0)} + \\ldots + \\frac{\\Phi^{(n)}}{r^{n}}  + \\ldots\n\\end{equation}\n\\noindent where once again we are not writing down the terms involving $ \\log r $ which are present for \neven $n$. The independent coefficients in \\eqref{UV Psi} are related to the boundary data \\eqref{UV phys1} and \\eqref{UV phys3}\nby\n\\begin{equation}\\label{Phi UV data}\n\t\\Phi^{(0)} = \\frac{ i\\omega \\chi^{(0)} + \\alpha^2 h^{(0)}}{n}, \\qquad \n\t\\Phi^{(n)} = \\frac{ (n+2) \\alpha^2}{\\omega^2}   h^{(n+2)}\n\\end{equation}\n\nUp to an overall $\\omega$-independent factor, $\\xi$, which we will fix later using the DC results, the two-point function $G_{QQ}$ can be written as \n\\begin{equation}\\label{G2 neutral}\n\tG_{QQ} = \\xi\\frac{\\Phi^{(n)}}{\\Phi^{(0)}} \n\\end{equation}\nHere we have chosen a renormalization scheme in which all local contributions to \\eqref{G2 neutral} are removed by \ncounterterms \\cite{deHaro:2000vlm}. \n\n\n\\section{QNM frequencies}\n\\label{sec:QNM}\n\n\nFinding analytical solutions to the master field equations \\eqref{MF eqs pm} and \\eqref{MF eq neutral} for general \nvalues of the parameters seems out of reach. Closely following \\cite{Emparan:2013moa, Emparan:2014cia, Emparan:2014aba, \nEmparan:2015rva},  we obtain perturbative \nsolutions using $1\/n$ as the expansion parameter. \nIn this section we will find expressions for the decoupled QNM for $\\mu \\neq 0$ \nto order $n^{-1}$ and for $\\mu = 0 $ to order $n^{-3}$, finding good agreement with numerical calculations at finite $n$\nin a certain region of parameter space.  \nIn  section~\\ref{sec:conductivity}  we will carry out the computation of the AC thermal conductivity to  order $n^{-2}$, obtaining a result consistent with our \nQNM calculation. \n\n\nAs explained in \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}, the spectrum of QNM in the large $n$ limit splits into decoupled \nmodes, which are normalisable in the near horizon geometry, and non-decoupled modes,  which are not. The latter \nare shared by many black holes so we do not expect to obtain information about the conductivities in this set of \nmodes, since, in particular, they are part of the spectra of black holes which are translationally invariant along \nthe boundary directions.\nWe focus on the decoupled modes and find that they indeed correspond to `Drude poles', i.e.\\ they are the purely \nimaginary modes which control the relaxation time of the system. As stated in \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}, \na necessary condition for the existence of decoupled QNMs is the presence of negative minima in the effective potential $V_{\\pm}$ defined by recasting the master field \nequation as\n\\begin{equation}\n\t\\left( \\frac{d^2}{d r_*^2} + \\omega^2 - V_{\\pm} \\right) \\Psi_\\pm = 0\n\\end{equation}\n\\noindent where $d r_* = dr \/f(r)$ is the tortoise coordinate. This form can be achieved by letting $ \\Phi_\\pm(r) = r^{(1-n)\/2} \\Psi_\\pm(r)$ \nin the master field equations \\eqref{MF eqs pm} and \\eqref{MF eq neutral}. By examining $V_{-}$, we conclude that there are \nno decoupled QNMs for $\\Phi_-$.  \n\nWhen taking the $n \\to \\infty$ limit, it is important to assign the scaling with $n$ of different physical \nquantities. Our goal is to capture the effects of momentum relaxation, so we will rescale quantities as appropriate so that $\\alpha$ appears at infinite $n$. More concretely, we will \ntake the $n \\to \\infty$ limit holding $r_0$, $\\mu$ and $\\hat \\alpha$ fixed, where\n\\begin{equation}\n  \t\\hat \\alpha = \\frac{\\alpha}{\\sqrt{n}}.\n\\end{equation}  \nThis scaling mirrors the scaling of momenta required in \\cite{Emparan:2015rva}.\nIt is convenient to define the radial variable $\\rho$ by\n\\begin{equation}\n \t\\rho = \\left( \\frac{r}{r_0} \\right)^n.\n\\end{equation} \nHere we work with $r_0=1$. We will keep $\\rho$ finite as $n \\to \\infty$, performing expansions of\n\\begin{equation}\nr = \\rho^{1\/n} = 1 + \\frac{1}{n}\\log{\\rho} + \\ldots\n\\end{equation} \nwhich takes us into the horizon region.\nIn order to obtain the perturbative solution we are after, we postulate the following expansions for the fields and the \nQNM frequency $\\omega = \\omega_\\pm$,\n\\begin{equation}\\label{n expansions}\n\t\\Phi_\\pm(\\rho) = \\sum_{i = 0} \\frac{\\Phi_{\\pm,i}(\\rho)}{n^i}, \\qquad \\omega_\\pm = \\sum_{i = 0} \\frac{\\omega_{\\pm,i}}{n^i}\n\\end{equation}\nOur boundary conditions are normalisability in the near horizon, i.e.\\ \n\\begin{equation}\n\t\\Phi_{\\pm,i}(\\rho) \\to 0 , \\qquad {\\rm at} \\, \\, \\rho \\to \\infty \n\\end{equation}\n\\noindent and ingoing boundary conditions at the horizon. These can be written as boundary conditions for the $\\Phi_{\\pm,i}$\nin \\eqref{n expansions} by expanding \\eqref{ingoing bc} in powers of $1\/n$. \nThe remainder of the computation of the decoupled QNM proceeds in close parallel to the one described in \n\\cite{Emparan:2015rva}, and we shall simply quote our results.\n\nFor $\\Phi_+$, we find decoupled QNMs with frequencies\n\\begin{align}\n\\omega_{+} &= -i \\hat\\alpha ^2\\left\\{\\frac{ \\left(2-\\hat\\alpha ^2\\right)}{2-\\hat\\alpha ^2+\\mu ^2}  - \\frac{1}{n}\\left[\\frac{2 \\hat\\alpha^2}{\\left(2-\\hat\\alpha ^2+\\mu ^2\\right) } \\,  \\log \\left(\\frac{2-\\hat\\alpha ^2}{2-\\hat\\alpha ^2-\\mu ^2}\\right) \\right.\\right. \\nonumber\\\\\n &\\phantom{=\\ }\\left.\\left. +\\frac{2 \\left(2 - \\hat\\alpha^2\\right)^3 + \\left(12 - \n    8 \\hat\\alpha^2 + \\hat\\alpha^4\\right) \\mu^2 + \\left(2 - 3 \\hat\\alpha^2\\right) \\mu^4 }{\\left(2-\\hat\\alpha ^2+\\mu ^2\\right)^3 }  \\right]+O\\left(n^{-2}\\right)\\right\\}. \\label{eq:QNMcharged}\n\\end{align}\nWe find that it is impossible to satisfy the  boundary conditions for $\\Phi_-$, so we conclude that there are no decoupled QNMs for this field, as argued above. \nIn the $\\mu=0$ case we are able to obtain two higher orders in the expansion for the $\\Phi$ frequency:\\footnote{Interestingly, these QNM frequencies can be obtained directly from the QNMs of black branes in AdS without momentum relaxation \\cite{Emparan:2015rva} by mapping the momenta $\\hat{q}^2 = \\hat\\alpha^2\/2$ and the spatial metric curvature parameter $K=-\\hat\\alpha^2\/2$.}\n\\begin{align}\n\\omega &= -i \\hat{\\alpha}^2  \\left\\{1 - \\frac{2}{n} - \\frac{2\\left(12+(\\pi^2-6)\\hat{\\alpha}^2\\right)}{3\\left(\\hat{\\alpha}^2-2\\right)n^2} \\right. \\nonumber\\\\\n &\\phantom{=\\ }\\left.+\\frac{8\\left[-12+\\hat{\\alpha}^2\\left((\\hat{\\alpha}^2 -4)(\\pi^2-3)-3(\\hat{\\alpha}^2 +2)\\zeta(3)\\right)\\right]}{3\\left(\\hat{\\alpha}^2-2\\right)^2 n^3}+O\\left(n^{-4}\\right)\\right\\} \\label{eq:QNMneutral}.\n\\end{align} \n\nA notable feature of the frequencies \\eqref{eq:QNMcharged} and \\eqref{eq:QNMneutral} is a breakdown of the expansion when $\\hat\\alpha^2+\\mu^2 =2$. In fact this behaviour could have been predicted by examining a large~$n$ expansion of the DC thermal conductivity, \\eqref{kappaDC},\n\\begin{equation}\n\\kappa(0) = \\kappa(0)|_{n\\to\\infty} \\left(1 + \\frac{4 + \\mu^2}{(2-\\hat\\alpha^2 - \\mu^2)n}+ O\\left(n^{-2}\\right)\\right).\n\\end{equation}\nThis breakdown can be traced back to a change in the way that the temperature scales with $n$ at large $n$:\n\\begin{equation}\nT = \\frac{(2 - \\hat\\alpha^2 -\\mu^2)n}{8 \\pi} + O\\left(n^{0}\\right).\n\\end{equation}\nConsequently, in order to examine the point $\\hat\\alpha^2+\\mu^2 =2$ we must repeat our large $n$ analysis there. For $\\mu=0$ and $\\hat\\alpha^2 =2$ the master field equation \\eqref{MF eq neutral} can be solved exactly for any $n$. This generalises the analysis performed at $n=1$ in \\cite{Davison:2014lua}. The additional divergence in the $\\mu\\neq 0$ case \\eqref{eq:QNMcharged} at $2+\\mu^2 = \\hat\\alpha^2$ coincides with the change of $n$ scaling of the mass parameter of the background solution, and occurs at a higher value of $\\hat\\alpha^2$ than the divergence discussed above. \n\nMoving on, we would like to compare these large $n$ analytical expressions \\eqref{eq:QNMcharged} and \\eqref{eq:QNMneutral} with finite $n$ numerics. For clarity we focus on $\\mu=0$, for which the comparison is presented in figure~\\ref{QNMplot} for values $n=1,11$ and $101$.  The $n=1$ case was previously analysed numerically in \\cite{Davison:2014lua} wherein it was noted that a pole collision occurred as $\\alpha$ was dialled. In this figure~\\ref{QNMplot}, we demonstrate that there are numerous such pole collisions at $n=1$, indicated by each extremum of the curve. For $n>1$ we find only one pole collision. Interestingly, the oscillations are centred on the critical value $\\hat\\alpha = \\sqrt{2}$ discussed above, and the locations where the crossings occur coincide with the existence of analytic regular, normalisable modes whose frequencies have integer imaginary part as discussed in appendix \\ref{appendixCritical}. We discuss these collisions in the context of a transition from coherent to incoherent behaviour in the conclusions, section \\ref{sec:conclusions}.\n\nFinally, we note that there is excellent agreement at sufficiently large finite $n$ between the numerical results and the large $n$ expansion, which interestingly includes the breakdown near $\\hat\\alpha = \\sqrt{2}$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{qnmTripA}\\\\\n\\includegraphics[width=0.45\\textwidth]{qnmTripB}\\\\\n\\begin{picture}(0.1,0.1)(0,0)\n\\put(-160,240){\\makebox(0,0){$n=1$}}\n\\put(45,240){\\makebox(0,0){$n=11$}}\n\\put(-55,120){\\makebox(0,0){$n=101$}}\n\\put(-215,225){\\makebox(0,0){$-\\Im(\\omega)$}}\n\\put(-115,100){\\makebox(0,0){$-\\Im(\\omega)$}}\n\\put(0,10){\\makebox(0,0){$\\hat\\alpha$}}\n\\put(-105,135){\\makebox(0,0){$\\hat\\alpha$}}\n\\put(105,135){\\makebox(0,0){$\\hat\\alpha$}}\n\\end{picture}\n\\caption{Purely imaginary quasi-normal mode frequencies of the linear axion black brane at $\\mu=0$ as a function of $\\hat\\alpha = \\alpha\/\\sqrt{n}$ computed numerically for various $n$ as labelled (solid curves). The dashed lines show the large $n$ analytical counterpart \\eqref{eq:QNMneutral} computed to order $n^{-2}$. The vertical dotted line is the `critical' value $\\hat\\alpha = \\sqrt{2}$, where the master field equation can be solved analytically for any $n$, giving the integer crossing frequencies. The $n=1$ case shows several pole collisions,  indicated by each turn over of the curve, whilst for $n>1$ we see only one collision for this range. Units are given by $r_0=1$.\\label{QNMplot}}\n\\end{center}\n\\end{figure}\n\n\n\\section{AC conductivity}\n\\label{sec:conductivity}\n\nIn this section we compute the AC thermal conductivity for the neutral theory to order $n^{-2}$, and compare the resulting expressions with numerics. Specifically we look at the frequency range which captures the decoupled mode describing the essential momentum relaxation physics, i.e.\\ we take $\\omega = O\\left(n^0\\right)$.\n\nThe computation begins with the approach outlined in \\cite{Emparan:2013moa}. The basic structure of the calculation is a matched asymptotic expansion made possible by the new small scale $r_0\/n$, corresponding to the localisation of gradients near the horizon. This new scale allows us to separate the bulk geometry into a near and far zone, defined as follows:\n\\begin{eqnarray}\n\\text{near zone:}\\quad & r-r_0\\ll r_0, & \\quad \\log \\rho \\ll n\\\\\n\\text{far zone:}\\quad & r-r_0\\gg \\frac{r_0}{n}, &\\quad \\log \\rho \\gg 1.\n\\end{eqnarray}\nAs we have previously, we shall take $r_0=1$. Note that these zones overlap: in particular, the overlap zone is described by $\\log \\rho \\gg 1$ in the near zone, and $\\log \\rho \\ll n$ in the far zone. Thus the calculation proceeds by solving in both zones and matching at the overlap; the near zone will allow us to imprint the ingoing horizon boundary conditions on the solution, whilst the far zone will enable us to read off the normalisable and non-normalisable data and allow the computation of the two-point function. \n\n\\subsection{Near zone}\nThe near zone is reached by taking the large $n$ limit whilst working at fixed $\\rho$. The calculation proceeds similarly to the QNM calculation and so we will be brief. A key difference is that we do not wish to impose normalisablity, and so the frequencies are not quantised. As before, we expand $\\Phi(\\rho) = \\sum_{i = 0} \\frac{\\Phi_i(\\rho)}{n^i}$, but we do not expand $\\omega$. After imposing the ingoing boundary conditions we obtain, \n\\begin{eqnarray}\n\\Phi_{0} &=& \\frac{a_0}{\\rho}\\\\\n\\Phi_{1} &=& \\frac{2 i a_0 \\omega}{(\\hat\\alpha^2-2)}\\frac{\\log{(\\rho-1)}}{\\rho} - \\frac{2 a_0(\\hat\\alpha^2 - i \\omega)}{\\hat\\alpha^2 -2}\\frac{\\rho-1}{\\rho}\n\\end{eqnarray}\ntogether with explicit expressions for $\\Phi_{2}$ and $\\Phi_{3}$ which we have omitted here. $a_0$ is an unconstrained  integration constant. Finally, in the overlap region, we have\n\\begin{eqnarray}\n\\Phi_{0} &=& \\frac{a_0}{\\rho}\\\\\n\\Phi_{1} &=& -2a_0\\frac{(\\hat\\alpha^2-i\\omega)}{\\hat\\alpha^2 -2} \\left(1-\\frac{1}{\\rho}\\right) + 2 a_0 \\frac{i \\omega}{\\hat\\alpha^2 -2}\\frac{\\log\\rho}{\\rho} - a_0 \\frac{2 i \\omega}{\\hat\\alpha^2-2}\\frac{1}{\\rho^2}.\n\\end{eqnarray}\nwhere again we have evaluated the overlap expressions for $\\Phi_{2}$ and $\\Phi_{3}$ but we omit them here in the interest of keeping the presentation concise.\n\n\\subsection{Far zone and matching}\nAt leading order the far zone equations can be obtained by removing any terms which decay exponentially fast with $n$ \\cite{Emparan:2013moa}. More generally, an expansion can be formed by counting powers of $r^{-n}$ in the equations of motion after inserting\n\\begin{equation}\n\\Phi = \\phi + r^{-n} \\psi + \\ldots.\n\\end{equation}\nLet us introduce a counting parameter $\\lambda$ for this purpose, i.e. we count $\\phi$ as order $\\lambda^0$. In order to obtain the conductivity we need the coefficient of $r^{-n}$, and so we need to go to order $\\lambda^1$.\nAt order $\\lambda^0$ the master field equation becomes\n\\begin{equation}\n{\\cal D}\\phi = 0,\\qquad {\\cal D} \\equiv \\partial_r^2 - \\frac{(n-1)\\hat{\\alpha}^2 - 2(n+1) r^2}{2r^3 - \\hat{\\alpha}^2 r} \\partial_r + \\frac{4 \\omega^2}{(\\alpha^2-2r^2)^2}.  \\label{lambda0}\n\\end{equation}\nThis equation can be solved explicitly in terms of Gauss hypergeometric functions,\n\\begin{eqnarray}\n\\phi &=&  \\left(1-\\frac{\\hat{\\alpha}^2}{2 r^2}\\right)^{-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}} \\Bigg(A\\; _2F_1\\left(-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1- \\frac{n}{2} - \\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1-\\frac{n}{2}, \\frac{\\hat{\\alpha}^2}{2r^2}\\right)\\nonumber\\\\\n&& + r^{-n} B\\; _2F_1\\left(1-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, \\frac{n}{2} - \\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1+\\frac{n}{2}, \\frac{\\hat{\\alpha}^2}{2r^2}\\right) \\Bigg)\\label{phifar}\n\\end{eqnarray}\nwhere $A$ and $B$ are integration constants; $A$ will contribute to the non-normalisable part of $\\Phi$ at infinity, $\\Phi^{(0)}$, whilst $B$ will contribute to the normalisable part, $\\Phi^{(n)}$. To find this in the overlap region we need to use expressions for the Gauss hypergeometric functions for large parameters. Expanding $\\phi$ in powers of $n$ and similarly for the integration constants $A$ and $B$, we find, \n\\begin{eqnarray}\n\\phi_0 &=& A_0 - \\frac{2B_0}{(\\hat\\alpha^2 - 2)\\rho}\\label{far0overlap}\\\\\n\\phi_1 &=& A_1 - \\frac{A_0\\omega^2}{\\hat\\alpha^2-2} + \\frac{1}{\\rho} \\left(-\\frac{2B_1}{\\hat\\alpha^2-2} -2 B_0\\frac{2\\hat\\alpha^2 + \\omega^2}{(\\hat\\alpha^2-2)^2} -4 B_0\\frac{\\hat\\alpha^2}{(\\hat\\alpha^2-2)^2} \\log\\rho\\right)\\label{far0overlapB}\n\\end{eqnarray}\ntogether with similar expressions for $\\phi_2$ and $\\phi_3$. Subscripts denote the power of $1\/n$ for which it is a coefficient.\n\nAt next order in $\\lambda$, the equation for $\\psi$  is sourced by $\\phi$:\n\\begin{eqnarray}\nr^n{\\cal D} \\left(\\frac{\\psi}{r^n} \\right) &=& {\\cal S} \\label{DS}\n\\end{eqnarray}\nwhere the operator ${\\cal D}$ is defined in \\eqref{lambda0} and where\n\\begin{align}\n{\\cal S} &= r^{-1} \\frac{(2-\\hat\\alpha^2) \\left(n \\hat\\alpha^2  - 2(2+n) r^2\\right)}{(\\hat\\alpha^2-2r^2)^2} \\phi' \\nonumber\\\\\n&\\phantom{=\\ }+ r^{-2} \\frac{2-\\hat\\alpha^2}{2r^2-\\hat\\alpha^2}\\left(-n(2+n) - \\frac{8 \\omega^2 r^2}{(2r^2-\\hat\\alpha^2)^2}\\right)\\phi.\n\\end{align}\nUnlike for $\\phi$ we have not directly integrated this equation. However, we can do so order-by-order in a large $n$ expansion at fixed $r$ provided we include the correct non-perturbative contributions. To the order of $n$ considered these turn out to be,\n\\begin{eqnarray}\n\\psi &=& \\left(\\psi_{B,0}(r) + \\frac{\\psi_{B,1}(r)}{n} + \\frac{\\psi_{B,2}(r)}{n^2} + \\frac{\\psi_{B,3}(r)}{n^3}  +O\\left(n^{-4}\\right) \\right)\\nonumber\\\\\n&&+ r^{-n}\\left(\\psi_{C,0}(r) + \\frac{\\psi_{C,1}(r)}{n} + \\frac{\\psi_{C,2}(r)}{n^2} + \\frac{\\psi_{C,3}(r)}{n^3} + O\\left(n^{-4}\\right) \\right). \\label{nonpert}\n\\end{eqnarray}\nWe can solve for each $\\psi_{B,i}$ and $\\psi_{C,i}$ provided $A_0=0$, which as we shall see shortly is consistent with the required value from the matching calculation.  Each term is required for the matching calculation to work and is straightforward to obtain. This method is more efficient than solving \\eqref{DS} at arbitrary $n$ and then expanding, as in \\cite{Emparan:2013moa}. Applied to $\\phi$ above, this method gives the same result as the expansion of \\eqref{phifar}.\n\nExpressing \\eqref{nonpert} in the overlap region and combining with \\eqref{far0overlap} and \\eqref{far0overlapB} gives us $\\Phi$ in the overlap zone, which can be matched with the expression coming from the near zone calculation. This fixes the coefficients appearing in \\eqref{far0overlap},\\eqref{far0overlapB} together with additional integration constants which arise in each of the $\\psi_{B,i}$. For example, \n\\begin{align}\nA_0 &= 0,\\\\\nA_1 &= \\frac{2 a_0(\\hat\\alpha^2-i\\omega)}{2-\\hat\\alpha^2}\\\\\nA_2 &= -\\frac{2 a_0(-i \\omega^3 + \\hat\\alpha^2(4-2i\\omega+\\omega^2))}{(2-\\hat\\alpha^2)^2}\\\\\n\\nonumber\nA_3 &= \\frac{a_0}{3\\left(\\hat{\\alpha }^2-2\\right)^3} \n\\bigg\\{ \\hat{\\alpha }^4 \\left(-6 \\omega ^2-4 i \\pi ^2 \\omega \\right) \\\\\n   &-\\hat{\\alpha }^2 \\left[ 3 \\left(\\omega ^4-6 i \\omega\n   ^3+8 \\omega ^2+32\\right)+4 \\pi ^2 \\omega  (\\omega +2 i)\\right] + i \\omega ^2 \\left(3 \\omega ^3-4 \\pi ^2 \\omega +48 i\\right) \\bigg\\}\n\\end{align}\nThe $B_i$ coefficients are given in relation to the coefficients appearing in $\\psi_{B,i}$. These coefficients determine $\\Phi^{(0)}$ and $\\Phi^{(n)}$ to order $n^{-3}$. Note that since $\\psi$ does not contribute to $\\Phi^{(0)}$, and $A_0=0$, the non-normalisable data $\\Phi^{(0)}$ vanishes to leading order in $n$ and so the Green's function will grow with $n$.\n\n\\subsection{Results}\nCombining the asymptotic results for $\\phi$ and $\\psi$ discussed above brings us to the main result of this section --- the thermal conductivity \\eqref{conductivityDefs} to order $n^{-2}$: \n\\begin{eqnarray}\n\\nonumber\n&& \\kappa(\\omega) =  2\\pi \\frac{2-\\hat\\alpha^2}{\\hat\\alpha^2-i \\omega} \\\\\n\\nonumber\n&&+\\frac{4 \\pi  \\left(\\hat\\alpha^4 (2+\\omega  (\\omega -i))-i \\hat\\alpha^2 \\omega ^3+2 i \\omega  \\left(\\hat\\alpha^2-i\n   \\omega \\right)^2 \\log \\left(2-\\hat\\alpha^2\\right)-2 i \\omega  \\log (2) \\left(\\hat\\alpha^2-i \\omega \\right)^2\\right)}{n\n   \\left(\\hat\\alpha^3-i \\hat\\alpha \\omega \\right)^2}\\nonumber\\\\\n\\nonumber\n&&-\\frac{4 \\pi \\omega }{3 n^2 \\hat{\\alpha }^4 \\left(\\hat{\\alpha }^2-2\\right)^2 \\left(\\omega +i \\hat{\\alpha }^2\\right)^3}\\bigg\\{\\\\\n\\nonumber\n&& +\\hat{\\alpha }^6 \\bigg[ \\hat{\\alpha }^6 (-(6 \\log (2-\\hat{\\alpha }^2 )+\\pi ^2+6-6 \\log (2) )) \\\\\n\\nonumber\n&& +4 \\hat{\\alpha }^2 (6 \\log (2-\\hat{\\alpha }^2 )+\\pi ^2+12-6 \\log (2) )-48 \\bigg]\\\\\n\\nonumber\n&& +i \\hat{\\alpha }^6 \\bigg[  (\\hat{\\alpha }^2-2 ) (3 (\\hat{\\alpha }^2+\\pi ^2+4 ) \\hat{\\alpha }^2+2 (\\pi ^2-6)) \\\\\n\\nonumber\n&&+18 (\\hat{\\alpha }^4-4 ) \\log (2-\\hat{\\alpha }^2 ) -18 (\\hat{\\alpha }^4-4 ) \\log (2)  \\bigg] \\omega\\\\\n\\nonumber\n&& +\\hat{\\alpha}^4 \\bigg[ -12 \\hat{\\alpha }^6 (1+\\log (2))+2 \\hat{\\alpha }^4 (\\pi ^2+15 (1+\\log (2))) \\\\\n\\nonumber\n&& -4 \\hat{\\alpha }^2 (3+\\pi ^2+\\log (4096))\n +6 (\\hat{\\alpha }^2-2) (2 \\hat{\\alpha }^4-\\hat{\\alpha }^2+6) \\log (2-\\hat{\\alpha }^2 )+72 \\log (2) \\bigg] \\omega ^2\\\\\n\\nonumber\n&& -3 i \\hat{\\alpha }^2 \\bigg[ \\hat{\\alpha }^8-3 \\hat{\\alpha }^6 (5+\\log (16))+\\hat{\\alpha }^4 (28+46 \\log (2))-4 \\hat{\\alpha }^2\n (1+\\log (4096)) \\\\\n \\nonumber\n&& +2 (\\hat{\\alpha }^2-2) (6 \\hat{\\alpha }^4-11 \\hat{\\alpha }^2+2) \\log (2-\\hat{\\alpha }^2)+\\log (256) \\bigg] \\omega ^3\\\\\n\\nonumber\n&& +6 \\hat{\\alpha }^2 \\left(\\hat{\\alpha }^2-2\\right) \\bigg[-\\hat{\\alpha }^4+6 \\hat{\\alpha }^2 (1+\\log (2))-6 \\left(\\hat{\\alpha }^2-2\\right) \\log \\left(2-\\hat{\\alpha }^2\\right)-12 \\log (2)\\bigg] \\omega ^4\\\\\n\\nonumber\n&& +3 i \\bigg[ \\hat{\\alpha }^6-2 \\hat{\\alpha }^4 (3+\\log (4))+8 \\hat{\\alpha }^2 (1+\\log (4)) \\\\\n&& +4(\\hat{\\alpha }^2-2)^2  \\log (2-\\hat{\\alpha }^2 )-16 \\log (2) \\bigg ] \\omega ^5 \\bigg\\} +O\\left(n^{-3}\\right). \\label{conductivityfinal}\n\\end{eqnarray}\nWe have fixed the overall normalisation given by $\\xi$ in \\eqref{G2 neutral} by comparing with the DC value \\eqref{kappaDC}:\n\\begin{equation}\n\\kappa(0) = 2\\pi\\left(\\frac{2}{\\hat\\alpha^2}-1\\right) \\left(1+ \\frac{4}{(2-\\hat\\alpha^2)n}\\right). \\label{kappaDCNetural} \n\\end{equation}\nWe find  $\\xi = -\\hat\\alpha^2+ O(n)^{-3}$ . Interestingly, at $\\mu=0$, the expansion for $\\kappa(0)$ truncates at order $1\/n$.\n\n \nWe note that at leading order the result takes Drude form.  A comparison with numerical integration for finite $n=1,3, 11,101$ is given in figure \\ref{kappaplot}, truncating at orders $n^0$, $n^{-1}$ and $n^{-2}$.\n\nLet us start with the $n^0$ approximation (black dash in figure \\ref{kappaplot}).  Even at this leading order the broad features of the conductivity are well captured by the large $n$ expansion. We note that the agreement at larger frequencies is excellent. The width of the peak, which is related to the relaxation timescale, is also very good, in agreement with the approximation of the QNMs by the large $n$ expansion. The only notable discrepancy is the height of the peak. This can be easily understood: $\\kappa(0)$ receives a $1\/n$ correction \\eqref{kappaDCNetural} and no further corrections, thus the DC limit is not expected to agree at this order. By $n=101$ the agreement is good everywhere.\n\nAt order $n^{-1}$ (blue dots in figure \\ref{kappaplot}) the DC limit now agrees, as anticipated. Overall the approximation is good even at $n=1$, but now we see even for modest values of $n$ the analytical result is in excellent agreement with the numerical result, e.g.\\ at $n=11$. \n\nAt order $n^{-2}$ (red dash in figure \\ref{kappaplot}) the $n=11$ and $n=101$ results are not visibly affected. However for the lower values of $n=1,3$ the agreement becomes worse than at order $n^{-1}$. This is similar to the large $D$ approximation applied to the Gregory-Laflamme instability \\cite{Emparan:2015rva}, where it was argued that the series is asymptotic due to the existence of non-perturbative contributions.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{kappaTripA}\\\\\n\\includegraphics[width=0.9\\textwidth]{kappaTripB}\\\\\n\\begin{picture}(0.1,0.1)(0,0)\n\\put(-40,250){\\makebox(0,0){$n=1$}}\n\\put(155,250){\\makebox(0,0){$n=3$}}\n\\put(-40,120){\\makebox(0,0){$n=11$}}\n\\put(155,120){\\makebox(0,0){$n=101$}}\n\\put(-210,230){\\makebox(0,0){$\\Re(\\kappa)$}}\n\\put(-210,100){\\makebox(0,0){$\\Re(\\kappa)$}}\n\\put(-105,10){\\makebox(0,0){$\\omega$}}\n\\put(105,10){\\makebox(0,0){$\\omega$}}\n\\end{picture}\n\\caption{The AC thermal conductivity at $\\mu=0$ computed analytically to orders $n^0$ (black dashed), $n^{-1}$ (blue dotted) and $n^{-2}$ (red dashed) as given in \\eqref{conductivityfinal}, compared to the finite $n$ numerical result (solid) for $\\hat\\alpha = 1\/2$.  Units are given by $r_0=1$.\\label{kappaplot}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nWe have studied the linear axion model defined via \\eqref{S0} when the number of spacetime dimensions\nis large, focusing on the quasi-normal modes governing momentum relaxation and the AC thermal conductivity. We have kept the horizon radius $r_0$ and the chemical potential $\\mu$ fixed in this limit. We \nfound that the influence of the momentum relaxation parameter, $\\alpha$, vanished at large $n$, but could be restored by scaling $\\alpha$ \nsuch that $\\hat\\alpha \\equiv \\alpha\/ \\sqrt{n}$ is held fixed. This places the physics of momentum relaxation at leading order in the large $D$ expansion.\n\nAn important technical point which simplifies our analysis is the existence of \nmaster field equations in arbitrary dimensions \\eqref{MF eqs pm}, \\eqref{MF eq neutral}. These decoupled wave equations contain \nall the gauge invariant information required to compute the two-point functions of interest, reducing our problem \nto calculations closely related to those already carried out in the context of General Relativity in large $D$. \n\nWe obtain analytical expressions for the QNM which control the electric and heat transport as a power series in $1\/D$. \nIn the language of \\cite{Emparan:2014aba}, these are of the decoupled kind, meaning that they are normalisable \nin the near horizon geometry. \nFor $\\mu \\neq 0$, we have computed the QNM which controls the electric conductivity for   \nup to order $n^{-1}$, while for $\\mu = 0$ we obtain the thermal QNM to order $n^{-3}$. Furthermore, \nwe calculate the AC thermal conductivity for $\\mu = 0$ to order $n^{-2}$. At leading order it takes Drude form, \nand at order $n^{-1}$ it provides a good approximation even for small values of $n$, illustrating the practicality of this technique for such systems.\n\nInterestingly, our perturbative series for the QNM breaks down due to the growth of the \ncoefficients as $\\hat\\alpha \\to  \\hat \\alpha_{c} \\equiv  \\sqrt{2 r_0^2-\\mu^2}$. \nNumerically, we observe that the structure of lowest lying QNM \nchanges significantly as we approach $\\hat \\alpha_{c}$. For very small values of $\\hat \\alpha$ there exists an isolated, \npurely dissipative excitation which governs transport, i.e.\\ the system is in a coherent regime. Increasing $\\alpha$ towards \n$\\hat \\alpha_{c}$, the characteristic time scale of this mode decreases and it mixes with the rest of the QNM in the spectrum, so \nthat we enter  an incoherent phase.\\footnote{In this regime there can be numerous pole collisions at finite $n$, including a collision between the Drude mode and a higher lying excitation.}\nWe thus interpret the breakdown of the perturbative expansion as a large $D$ signature \nof the coherent\/incoherent transition. It would be interesting to revisit our analysis with transverse wavevector $k \\neq 0$ and investigate the interplay of these QNMs with diffusion.\n\nMore generally, it is interesting to observe that in the case where we do not scale the sources of the axions with $D$, momentum conservation is restored at infinite $D$. This suggests that the large $D$ expansion may be used to improve analytical control over more generic setups incorporating inhomogeneity. We leave this possibility for future work.\n\n\\acknowledgments\n\nWe are pleased to thank Marco Caldarelli, Richard Davison, Roberto Emparan, Blaise Gout\\'eraux and Kostas Skenderis for valuable comments. \nT.A.\\ is supported by the European Research Council under the European Union's Seventh Framework Programme\n(ERC Grant agreement 307955). He also thanks the Institute of Physics at University of Amsterdam for their hospitality \nduring the completion of this work. \nS.A.G.\\ is supported by National Science Foundation grant PHY-13-13986.\nB.W.\\ is supported by European Research Council grant ERC-2014-StG639022-NewNGR.  He also thanks the Department of Physics at the University of Oxford for their hospitality during the completion of this work.\nWe are grateful to Centro de Ciencias Pedro Pascual, Benasque where this work was initiated.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOne important topic in many branches of biology is to understand evolutionary events and forces leading to current biological systems, such as a group of species or strains of a virus. To this end, evolutionary relationships among the biological system under investigation are typically represented by \na phylogenetic tree, that is, a binary tree whose leaves are labelled by the taxon units in the system.  As these events and forces, such as rates of speciation and expansion, are often not directly observable~\\cite{mooers2007some,heath2008taxon},  one popular approach is to compare empirical shape indices computed from trees inferred from real datasets with those predicted by a null tree growth model\n~\\cite{blum2006random,hagen2015age}. Furthermore, topological tree shapes are also closely related to several fundamental statistics in population genetics~\\cite{ferretti2017decomposing,arbisser2018joint} and certain important parameters in \nthe dynamics of virus evolution and propagation~\\cite{colijn2014phylogenetic}. \n\n\n\nOne important family of tree shapes are balance indices, such as Colless' index, Sackin's index and the number of subtrees (see, e.g.~\\cite{fischer2021tree} and the references therein).  Various properties concerning  these  statistics have been established in the past decades on the following two fundamental random phylogenetic tree models: \nthe Yule model (aka the Yule-Harding-Kingman (YHK) model)~\\cite{rosenberg06a,disanto2013exact,Janson2014} and the uniform model (aka the proportional to distinguishable arrangements (PDA) model)~\\cite{McKenzie2000,chang2010limit, WuChoi16,CTW19}.\nHowever, for phylogenetic trees inferred from real datasets, the Yule or uniform model may not always be a good fit~\\cite{blum2006random}, and several general classes of random trees have been proposed  for modelling and analysing the observed data,\ntwo popular ones being Ford's alpha model~\\cite{Ford2006} and Aldous' beta model~\\cite{aldous96a}.  \n\n\nIn this paper, we confine ourselves to  Ford's alpha model, a one-parameter family  of random tree growth models introduced by Daniel J. Ford in his PhD thesis \\cite{Ford2006}. \n More precisely, under the Ford model with a fixed parameter $0\\le \\alpha \\le 1$, a random tree of a given number of leaves is generated such that at any step in which a tree $T_n$ with $n$ leaves has been constructed from previous steps, a new leaf attaches to  an internal edge of $T_n$ with probability $\\frac{\\alpha}{n-\\alpha}$ and  to  a leaf edge in $T_n$ with probability $\\frac{1-\\alpha}{n-\\alpha}$. The resulting random tree model will be referred to as the Ford model (with parameter $\\alpha$) in this paper, which is also known as the alpha tree model~(see, e.g.~\\cite{coronado2019balance}). Note that the Ford model is a family of random tree models which includes the Yule model with $\\alpha=0$,   the uniform model with $\\alpha=1\/2$, and the Comb model with $\\alpha=1$.  \n\nThe tree shape indices studied in this paper are the number of cherries and that of pitchforks. Here a cherry is a subtree with precisely two leaves and a pitchfork a subtree with three leaves. \nThe asymptotic properties of the number of cherries was first studied by McKenzie and Steel  \\cite{McKenzie2000}, who showed that the number of cherries is asymptotically normal for the Yule and the uniform models as the number of leaves tends to infinity. Later, similar properties of the number of cherries are  extended to the Ford model~\\cite[Theorem 57]{Ford2006} and to the Crump-Mode-Jagers branching process~\\cite{plazzotta2016asymptotic}. \nFor the number of pitchforks, Rosenberg~\\cite{rosenberg06a} obtained its  mean and variance and Chang and Fuchs~\\cite{chang2010limit} proved that  the number of pitchforks is also  asymptotically normal for the Yule and the uniform models. For the joint distributions, Holmgren and Janson showed that~\\cite{Janson2014} the joint distribution is asymptotically normal for the Yule model. This was recently extended by us to the uniform model based on a uniform version of the extended urn models in which negative entries are permitted for their replacement matrices~\\cite{Paper1}. \n\n\nIn this paper,  we establish the strong law of large numbers and the central limit theorem \nfor the joint distribution of cherries and pitchforks under the Ford model (Theorem~\\ref{Thm:Convg-ChPh}) \nby considering an associated nonuniform urn model (Theorem~\\ref{thm:urn:edge}). These results are presented in Section~\\ref{sec:limiting}, following Section~\\ref{sec:preliminary} in which we  collect  background concerning the Ford model and  limiting theorems on uniform urn models.\nFurthermore, we derive a recurrence  formula for computing the exact joint distribution under the Ford model~(Theorem~\\ref{jointpmf}) in Section~\\ref{sec:exact}, generalizing the results in~\\cite{WuChoi16,CTW19} for the Yule and the uniform model. This recurrence formula enables us to obtain exact expressions for the mean and variance of the number of cherries and that of pitchforks and their covariance under the Ford model. This, in-particular, generalises the exact expressions of mean and variance for the number of cherries and that of pitchforks for the Yule and the uniform models~\\cite{McKenzie2000,rosenberg06a,chang2010limit}  and the number of cherries for the Ford model~\\cite[Theorem 60]{Ford2006}. \nAs an application, in Section~\\ref{sec:expansion} we obtain higher order expansions of the first and second moments of the joint distributions. \n\n\n\n\n\\section{Ford Model and Urn Model}\n\\label{sec:preliminary}\nIn this section, we first introduce  the Ford model, which is a one-parameter family of random phylogenetic tree models. Next we present a nonuniform version of the extended urn models  associated with the Ford tree model. Finally, we recall certain conditions on the related uniform version of the extended urn model under which the strong law of large numbers and the central limit theorem are obtained.\n\n\\subsection{Ford model}\n\nA rooted binary tree is a finite connected simple graph without cycles that contains a unique vertex of degree 1 designated as the root and all the remaining vertices are of degree 3 (interior vertices) or 1 (leaves). A phylogenetic tree with $n$ leaves is a rooted binary tree whose leaves are bijectively labelled by the elements in $\\{1,\\dots,n\\}$. Edges incident with leaves are referred to as pendant edges. \n\nUnder the Ford model with parameter $0\\le \\alpha \\le 1$,  a random phylogenetic tree $T_n$ with $n$ leaves is constructed recursively by adding one leaf at a  time as follows. Fix a random permutation $(x_1,\\dots,x_n)$ of $\\{1,\\dots,n\\}$.  The initial tree $T_2$  contains precisely two leaves (e.g. one cherry) which are labelled as $x_1$ and $x_2$. For the recursive step, given a tree $T_m$ with $m$ leaves constructed so far, choose a random edge in $T_m$ according to the distribution that assigns weight $1-\\alpha$ to each pendant edge (i.e., those incident with a leaf)  and weight $\\alpha$ to each of the other edges. The new leaf labelled $x_{m+1}$ bifurcates the selected edge and joins in the middle. \nEvery single addition of a leaf in the tree results into a replacement of the selected edge with two new edges.\nFinally, we let $A_n$ and $C_n$ denote the numbers of pitchforks and cherries in tree $T_n$, respectively.\n\n \\begin{figure}[ht]\n\t\\begin{center}\n\t\t{\\includegraphics[width=0.9\\textwidth]{1_Fig_example.pdf}}\n\t\\end{center}\n\t\\caption{A sample path of the Ford model and the associated trajectory under the urn model. (i) A sample path of the Ford model evolving from $T_2$ with two leaves to $T_6$ with six leaves. The labels of the leaves are omitted for simplicity. The type of an edge is indicated by the circled number next to it. For $2\\le i \\le 5$, the edge selected in $T_i$ to generate $T_{i+1}$ is highlighted in bold and the associated edge type is indicated in the circled number above the arrow. (ii) The associated urn model with six colours, derived from the types of pendants edges in the trees. \n\t\tIn vector form, $U_0=(0,2,0,0,1,0), U_1=(2,0,1,0,0,2), U_2=(0,4,,0,0,2,1), U_3=(2,2,1,0,1,3)$, and $U_4=(2,2,1,1,1,4)$. \t\n\t}\n\t\\label{fig:example}\n\\end{figure}\n\n\n\n\\subsection{An urn model associated with trees }\n\\label{subsect:urn:tree}\n\nConsider an urn containing balls of $d$ different colours where colours are denoted by integers $\\{1,2,\\dots, d\\}$. Let $U_n=(U_{n,1},\\dots, U_{n,d})$ be the configuration vector of length $d$ such that the $i$-th element of $U_n$ is the number of balls of colour $i$ at time $n$.  Let $U_0$ be the initial vector of colour configuration, then at every time $n\\geq 1$, a ball is selected uniformly at random from the urn and if the colour of the selected ball is $i$ then the ball is replaced along with $R_{i,j}$ many balls of colour $j$,  for every $1 \\leq j \\leq d$. The dynamics of the urn configuration depends on its initial configuration $U_0$ and the $d \\times d$ replacement matrix $R = (R_{i,j})_{1\\leq i,j\\leq d}$.\n\n\nWe  study the limiting properties of the numbers of cherries and pitchforks via an equivalent urn process. Towards this, we use six different colours and assign one colour to each type of edges of the tree in the following scheme introduced in~\\cite{Paper1}:  colour $1$ for all pendant edges of a cherry in a pitchfork; colour $2$ for pendant edges of a cherry not contained in a pitchfork; colour $3$ for pendant edges in a pitchfork but not in any cherry; colour $4$ for pendant edges in neither a cherry nor a pitchfork; colour $5$ for internal edges adjacent to a cherry but not in a pitchfork (i.e., those adjacent to colour $2$ edges), and colour $6$ for all other (necessarily internal) edges (including the one incident with the root). See Fig.~\\ref{fig:example} for an illustration of the scheme. \n\n\nConsider an urn with colour configuration at time $n$ as\n$U_n = (U_{n,1},\\dots, U_{n,6})$, where $U_{n,i}$ denotes the number of edges of colour $i$ in the tree at time $n$, which has precisely $n+2$ leaves. Then $U_0 = (0,2,0,0,1,0)$, since at the initial time step $(n=0)$ there is one internal edge and one essential cherry in a rooted tree; see $T_2$ in Fig.~\\ref{fig:example}. Based on the colouring scheme of the edges, at any time $n\\geq 0$, we have\n\\begin{equation}\\label{urn-pf-ch}\n(A_{n+2}, C_{n+2}) = \\frac{1}{2} \\left(U_{n,1}, U_{n,1}+U_{n,2}\\right),\n\\end{equation}\nwhere $A_{n+2}$ and $C_{n+2}$ are the numbers of pitchforks and cherries in $T_{n+2}$, respectively.\nUnder the alpha tree model, the dynamics of the corresponding urn process evolves according to the following replacement matrix\n\\[ R = \\begin{bmatrix}\n0&0&0&1&0&1\\\\\n2&-2&1&0&-1&2\\\\\n-2&4&-1&0&2&-1\\\\\n0&2&0&-1&1&0\\\\\n2&-2&1&0&-1&2\\\\\n0&0&0&1&0&1\n\\end{bmatrix}. \\]\nLet $e_i$, $1\\le i \\le 6$, denote a $6$-vector in which the $i$-th component is $1$ and $0$ elsewhere; and   $\\chi_n$   the random vector taking value $e_i$ if, at time $n$, speciation happens at an edge with type  $i$.\nThus,  we have the following recursion\n\\[U_n= U_{n-1} +\\chi_n R, \\qquad n\\geq 1, \\]\nwhere\n\\begin{equation}\\label{SelecProb}\nP(\\chi_n =e_i|\\mbox{${\\mathcal F}$}_{n-1}) \\propto \\begin{cases}\n(1-\\alpha) U_{n-1,i}, &\\text{ for } i \\in\\{1,2,3,4\\}, \\\\[1ex]\n\\alpha \\,U_{n-1,i}, & \\text{ for } i\\in \\{5,6\\}.\n\\end{cases}\n\\end{equation}\nObserve that the process $(U_n)_{n \\ge 0}$, which  describes the dynamics of the numbers of cherries and pitchfork,\nis a {\\em nonuniform urn model} since the balls are not selected uniformly at random from the urn, which is different from the classical {\\em uniform} urn models in which  the balls are selected uniformly at random from the urn (see, e.g.~\\cite[Chapter 7]{hofri2019algorithmics}).\n\n\\subsection{Limiting theorems on uniform urn models}\n\\label{limit:uniform:urn}\nIn this subsection, we recall the strong laws of large numbers and the central limit theorems on a version of uniform urn models developed in~\\cite{Paper1}, which will be related to the nonuniform urn process in Subsection~\\ref{subsect:urn:tree} later using the urn coupling idea  in \\cite{Kaur2018}. \n\n\nFor the classical uniform urn models, it has been shown (see \\cite{BaiHu2005}) that the random process $U_n\/n$ converges almost surely to the left eigenvector of $R$ corresponding to the maximal eigenvalue and the asymptotic normality holds with a known limiting variance matrix under certain assumptions on $R$.  Standard assumptions made in the urn model theory are  that the replacement matrix is  irreducible with a constant row sum and all the off-diagonal elements are non-negative~(see, e.g.~\\cite{Hosam2009}).  In  \\cite{Paper1}, we extend this to the case when off-diagonal elements of a replacement matrix can be negative satisfying  the following set of assumptions {\\bf (A1)--(A4)}, which was slightly rephrased from~\\cite{Paper1}. Let $\\text{diag}(a_1, \\dots , a_d)$ denote the diagonal matrix whose diagonal elements are $a_1,\\dots , a_d$. \n\n\\noindent\n{\\bf (A1):} {\\em Tenable:} It is always possible to draw balls and follow the replacement rule.\n\\\\\n{\\bf (A2):} {\\em Small:} \nAll eigenvalues of $R$ are real; the maximal eigenvalue $\\lambda_1$, called the {\\em principal eigenvalue} is positive with $\\lambda_1>2\\lambda$ holds for all other eigenvalues $\\lambda$ of $R$. \\\\\n{\\bf (A3):} {\\em Strictly balanced:}  \tThe column vector $\\mathbf{u}_1=(1,1,\\dots,1)^\\top$,  is a right eigenvector \nof $R$ corresponding to $\\lambda_1$;   and it has a principal left eigenvector $\\bf{v}_1$ (i.e., the \nleft eigenvectors corresponding to $\\lambda_1$) that is also a  probability  vector.\n\\\\\n{\\bf (A4):} {\\em Diagonalisable:} There exists an invertible matrix $V$ with real entries whose first row is $\\bf{v}_1$ such that the first column of $V^{-1}$ is $\\mathbf{u}_1$ and \n\\begin{equation}\n\\label{eq:R:diagonal}\nVRV^{-1} = \\text{diag}(\\lambda_1, \\lambda_2,\\dots, \\lambda_d) =: \\Lambda,\n\\end{equation}\nwhere $\\lambda_1> \\lambda_2 \\ge \\dots \\ge \\lambda_d$ are  eigenvalues of $R$. \n\n\n\\medskip\n\nLet $\\mathcal{N}(\\mathbf{0}, \\Sigma)$ be the multivariate normal distribution with mean vector $\\mathbf{0}=(0,\\dots,0)$ and covariance matrix $\\Sigma$. Then we have the following result from~\\cite[Theorems 1 \\& 2]{Paper1}, which can also be alternatively derived from~\\cite[Theorems 3.21 \\& 3.22 and Remark 4.2]{Janson2004}. \n\n\\begin{theorem}\n\t\\label{thm1&2:paper1}\n\tUnder assumptions {\\em \\bf{(A1)--(A4)}}, we have\n\t\\begin{equation}\n\t\\label{eq:asconv:urn}\n\t(n\\pev)^{-1} U_n \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\mathbf{v}_1\n\t~\\quad~\\mbox{and}~\\quad~ \n\tn^{-1\/2} (U_n - n\\pev \\mathbf{v}_1) \\xrightarrow{~d~}  \\mbox{${\\mathcal N}$}(\\mathbf{0}, \\Sigma),\n\t\\end{equation}\n\twhere $\\pev$ is the principal eigenvalue and $\\mathbf{v}_1$ is the principal left eigenvector of $R$, and \n\t\\begin{equation}\n\t\\Sigma = \\sum_{i,j=2}^d \\frac{\\pev \\lambda_i \\lambda _j {\\mathbf u}_i^\\top \\mbox{\\em diag}(\\mathbf{v}_1) {\\mathbf u}_j }{\\pev-\\lambda_i -\\lambda_j} \\mathbf{v}_i^\\top \\mathbf{v}_j,\n\t\\end{equation}\n\twhere  ${\\mathbf v}_j$ is the $j$-th row of $V$  and ${\\mathbf u}_j$ the $j$-th column of $V^{-1}$ for $2\\le j \\le d$.\n\\end{theorem}\n\n\n\n\\section{Limit Theorems for the Joint Distribution}\n\\label{sec:limiting}\n\nIn this section, we present the strong laws of large numbers and the central limit theorems on the joint distribution of the number of cherries and that of pitchforks under the Ford model.\n\n\n\\subsection{Main convergence results}\n\n\nFor later use, we consider the following polynomials in $\\alpha$: \n\\begin{equation}\n\\label{eq:main:cov:poly}\n\\begin{matrix*}[l]\n\\gp_1 = 8\\alpha^3-32\\alpha^2+45\\alpha-23, & \\quad\\quad\\quad \\gp_4 = 8\\alpha^3-40\\alpha^2+37\\alpha+13, \\\\\n\\gp_2 = 40\\alpha^3-164\\alpha^2+221\\alpha-97, &  \\quad\\quad\\quad  \\gp_5 = 40\\alpha^3-112\\alpha^2-31\\alpha+181, \\\\\n\\gp_3 = 56\\alpha^3-248\\alpha^2+367\\alpha-181, &  \\quad\\quad\\quad  \\gp_6 = 8\\alpha^3+4\\alpha^2-71\\alpha+71; \n\\end{matrix*}\n\\end{equation}\nand for simplicity of notation, we do not indicate the $\\phi_i$'s as  functions of $\\alpha$. Moreover, it can be verified directly that $\\phi_1, \\phi_2, \\phi_3 <0$ and $\\phi_4, \\phi_5, \\phi_6 > 0$ for $\\alpha \\in (0, 1).$\nThen, we have the following result on the joint asymptotic properties of the urn model process associated with the $\\alpha$-tree model.\n\n\n\\begin{theorem}\n\t\\label{thm:urn:edge}\n\tSuppose $(U_n)_{n\\geq 0}$  is the urn process associated with the Ford model with parameter $\\alpha\\in(0,1)$. \n\tThen, \n\t\\begin{equation}\n\t\\label{eq:conv:urn}\n\t\\frac{U_n}{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \n\t\\mathbf{v}\t\n\t~\\quad~\\mbox{and}~\\quad~\n\t\\frac{ U_n- n\\mathbf{v} }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N}\\left (\\mathbf{0},\\Sigma \\right),\n\t\\end{equation}\n\tas $n \\to \\infty$, where \n\t\\begin{equation}\\label{main:Leftev1}\n\t{\\mathbf v}= \\frac{1}{2(3-2\\alpha)} \\left(2(1-\\alpha), \\,2(1-\\alpha),\\, (1-\\alpha),\\, 1+\\alpha,\\, 1-\\alpha, \\,5-3\\alpha \\right)\n\t\\end{equation}\nand with the polynomials $\\gp_1,\\dots,\\gp_6$ defined in~\\eqref{eq:main:cov:poly},\n\\begin{equation}\n\\Sigma=\\frac{1-\\alpha}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\\begin{bmatrix*}[r]\n-12\\gp_1 & 4\\gp_2 & -6\\gp_1  & -2\\gp_4 & 2 \\gp_2 & -2 \\gp_2 \\\\\n4\\gp_2 & -4\\gp_3 & 2\\gp_2  & -2\\gp_6 & -2 \\gp_3 & 2 \\gp_3   \\\\\n-6\\gp_1 & 2\\gp_2  & -3\\gp_1  & -\\gp_4 &  \\gp_2 & - \\gp_2  \\\\\n-2\\gp_4  &  -2\\gp_6  &  -\\gp_4 & \\gp_5 & - \\gp_6 &  \\gp_6  \\\\\n2 \\gp_2   &  -2 \\gp_3  &  \\gp_2& - \\gp_6 & - \\gp_3 &  \\gp_3  \\\\\n-2 \\gp_2    &  2 \\gp_3   &  - \\gp_2 & \\gp_6  &   \\gp_3 & -\\gp_3  \n\\end{bmatrix*}.\n\\end{equation}\n\\end{theorem}\nThe proof of Theorem~\\ref{thm:urn:edge} is given at  the end of this section.\n\n\n\\begin{remark} \n\\label{rem3.1} For later use, here we present the limiting results on the urn model using a scaling factor relating to the time $n$ (which is motivated by noting that the number of leaves in the tree  at time $n$ is $n+2$). However, the results can be readily rephrased using the proportion of color balls in the urn process.\n\n\\end{remark}\n\n\\begin{remark} \n\tUsing the approach outlined in~\\cite{Paper1},  Theorem \\ref{thm:urn:edge} \tcontinues to hold for  the unrooted $\\alpha$-tree models. \n\\end{remark}\n\nWith Theorem~\\ref{thm:urn:edge}, we are ready to present one of our main results in this paper \nconcerning limit theorems on the joint distribution of the number of cherries $C_n$ and the number of pitchforks $A_n$\nunder the Ford model. \n\n\n\\begin{theorem} \\label{Thm:Convg-ChPh}\nUnder the Ford model with parameter $\\alpha \\in [0,1]$, we have\n\t\\[\\frac{1}{n} (A_n,C_n) \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  (\\nu, \\mu) := \\frac{1-\\alpha}{2(3-2\\alpha)} (1,2),\\]\n\tand\n\t\\[ \\frac{(A_n, C_n) -n (\\nu, \\mu) }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N}\\big((0,0), S \\big), \\]\n\twhere \\begin{equation}\n\tS= \\begin{bmatrix} \\tau^2 & \\rho \\\\ \\rho & \\si^2 \\\\ \\end{bmatrix} =\n\t\\frac{1-\\alpha}{(3- 2\\alpha)^2 (5-4\\alpha) } \\begin{bmatrix}\n\t\\frac{-24 \\alpha^3 +96\\alpha^2 -135\\alpha +69 }{4(7-4\\alpha)} & \\frac{-(2-\\alpha) (1-2\\alpha)}{2} \\\\[1ex]\n\t\\frac{-(2-\\alpha) (1-2\\alpha)}{2} & 2-\\alpha\n\t\\end{bmatrix}.\n\t\\end{equation}\n\\end{theorem}\n\\medskip\n \\begin{remark} \nWe consider special cases of $\\alpha$-tree model, which are commonly studied in phylogenetics. The first two have been established in ~\\cite{Paper1}.\n\n\\begin{enumerate}\n\t\\item The uniform model corresponds to  $\\alpha=1\/2$, where all edges, internal or leaf, are selected with equal weight and the limit results hold with\n\t\\[(\\nu, \\mu) = \\frac{1}{8}(1, 2)\n\t\\quad \\text{and } \\quad   \\begin{bmatrix} \\tau^2 & \\rho \\\\ \\rho & \\si^2 \\\\ \\end{bmatrix}   =\n\t\\frac{1}{64}\\begin{bmatrix} 3&0\\\\0&4\\end{bmatrix}. \\]\n\t\\item The Yule model corresponds to  $\\alpha =0$, where only leaf edges are selected with equal weight and the limit results hold with\n\t\\[ (\\nu, \\mu) = \\frac{1}{6} (1,2)\n\t\\quad \\text{and } \\quad  \\begin{bmatrix}\n\t\\tau^2 & \\rho \\\\\n\t\\rho & \\si^2 \\\\\n\t\\end{bmatrix}  =  \\frac{1}{45}  \\begin{bmatrix} 69\/28  &-1  \\\\  -1 & 2 \\end{bmatrix}.  \\]\n\t\n\\item The  Comb model corresponds to $\\alpha =1$,  a degenerate case. It is easy to see that  $(\\nu, \\mu) = (0,0) $ and $\\tau^2= \\rho = \\si^2  = 0$.\n\t\n\\end{enumerate}\n \\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:Convg-ChPh}]\n\tFirst note that the case $\\alpha=1$ reduces to a degenerate case of Comb model and therefore we only consider $\\alpha\\in[0,1)$. The limiting results for the case $\\alpha=0$ has been obtained in \\cite{Paper1}, which agree with the above results when $\\alpha=0$. Thus, it is enough to prove the result for $\\alpha\\in (0,1)$.\n\t\n\t By~\\eqref{urn-pf-ch}, we have \t \n\t$ (A_n, C_n) = U_n Q $\n\twith\n\t\\begin{equation}\\label{Def-T}\n\tQ^\\top = \\frac{1}{2}\\begin{bmatrix} 1 &0&0&0&0&0\\\\\n\t1&1&0&0&0&0\n\t\\end{bmatrix}.\n\t\\end{equation}\t\t\nSince\n\t\\begin{equation}\n\t\\frac{U_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\   {\\mathbf v}  = \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha), 2(1-\\alpha), 1-\\alpha, 1+\\alpha, 1-\\alpha, 5-3\\alpha \\big),\n\t\\end{equation}\n\tusing the relation from equation \\eqref{urn-pf-ch} we get\n\t\\[\\frac{1}{n} (A_n,C_n) =\\left(\\frac{U_n}{n} \\right) Q \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  {\\mathbf v}\\,Q = \\frac{1-\\alpha}{2(3-2\\alpha)} (1,2). \\]\n\tThis concludes the proof of the almost sure convergence. We now prove the central limit theorem and obtain the expression for the limiting variance matrix.\n\t\n\t\n\tDenoting covariance matrix $\\Sigma$ by $(\\sigma_{i,j})$ for $1\\le i,j\\le 6$, we   consider the matrix \n\t\\begin{align*}\n\tS&= Q^\\top  \\Sigma Q\n\t=\\frac{1}{4} \\begin{bmatrix}\n\t\\sigma_{1,1} & \\sigma_{1,1}+\\sigma_{1,2}\\\\\n\t\\sigma_{1,1}+\\sigma_{2,1}& \\sigma_{1,1}+\\sigma_{2,1}+\\sigma_{1,2}+\\sigma_{2,2}\n\t\\end{bmatrix} \\\\[1ex]\n\t&=\\frac{1-\\alpha}{16(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\t\\begin{bmatrix}\n\t-12\\gp_1 & -12\\gp_1+4\\gp_2\\\\\n\t-12\\gp_1+4\\gp_2 & -12\\gp_1+8\\gp_2-4\\gp_3\n\t\\end{bmatrix} \\\\[1ex]\n\t\t&=\\frac{1-\\alpha}{(3- 2\\alpha)^2 (5-4\\alpha) } \\begin{bmatrix}\n\t\\frac{-24 \\alpha^3 +96\\alpha^2 -135\\alpha +69 }{4(7-4\\alpha)} & \\frac{-(2-\\alpha) (1-2\\alpha)}{2} \\\\[2ex]\n\t\\frac{-(2-\\alpha) (1-2\\alpha)}{2} & 2-\\alpha\n\t\\end{bmatrix}.\n\t\\end{align*}\nSince  $(A_n, C_n) = U_n Q$, where $Q$ is as defined in \\eqref{Def-T}, we get\n\t\\[ \\frac{(A_n, C_n) -n (\\nu, \\mu) }{\\sqrt{n}} = \\frac{1}{\\sqrt{n}} \\left(U_n -n {\\mathbf v} \\right)Q \\xrightarrow{~d~}   \\mathcal{N}\\left(\\mathbf{0}, Q^\\top  \\Sigma Q \\right)\n\t=\\mathcal{N}\\left(\\mathbf{0}, S \\right)\n\t.\\]\n This completes the proof.\n\\end{proof}\n\n\nWe end this subsection with the following results on the behaviour of the first and second moments of the limiting joint distribution of cherries and pitchforks in the parameter region, as indicated by their plots in Figure~\\ref{fig:limit:cov}.\n\n\\begin{corollary}\n\t\\begin{enumerate}[\\normalfont(i)]\n\t\t\\item For $0< \\alpha <1$,  $A_n\/C_n  \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  1\/2$ as $n\\to \\infty$. That is, the number of pitchforks is asymptotically equal to the number of essential cherries.\n\t\t\n\t\t\\item $ A_n\/n \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\   \\frac{1-\\alpha}{2(3-2\\alpha)}, $ which decreases strictly from $1\/6$ to $0$, as $\\alpha$ increases from $0$ to $1$.\n\t\t\n\t\t\n\t\t\\item  The limiting variance of $A_n\/\\sqrt{n}$,  $\\tau^2$, decreases strictly from $23\/420$ to $0$, as $\\alpha$ increases from $0$ to $1$.\n\t\t\\item The limiting variance of $C_n\/\\sqrt{n} $, $\\si^2$, increases strictly from $2\/45$\n\t\tto $0.0695$  over $(0, a_0)$ and decreases from $0.0695$ to $0$ over $(a_0, 1)$, where $a_0 =0.7339$,\n\t\t the unique root of $19-48\\alpha+36\\alpha^2-8\\alpha^3 =0$ in $(0,1)$.\n\t\t\n\t\t\\item  The limiting covariance of $A_n\/\\sqrt{n}$ and $C_n\/\\sqrt{n}$ changes sign from negative to positive at $\\alpha =1\/2$. Specifically, it increases from $-1\/45$ \n\t\tto $0.0225$ \n\t\tover $(0, a_1)$ and decreases from $0.0225$\n\t\tover $(a_1, 1)$, where $a_1=0.8688$,\n\t\tthe unique root of $-24\\alpha^4+160\\alpha^3 -370\\alpha^2 +358\\alpha -123=0$ in $(0,1)$.\n\t\\end{enumerate}\n\\end{corollary}\n\n\\begin{figure}[H] \n\t\\label{fig:limit:cov}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth,height=0.38\\textheight]{Rplot_New.png}\n\t\\caption{Plot of the limiting  covariances of the joint distribution of cherries and pitchforks with respect to the parameter $\\alpha$ under the Ford model.}\n\\end{figure}\n\n\\subsection{A uniform urn model derived from $U_n$}\n\nFor $\\alpha\\in (0,1)$, consider the diagonal $6\\times 6$ matrix\n$\nT_\\alpha=\\text{diag}(1-\\alpha,1-\\alpha,1-\\alpha,1-\\alpha,\\alpha,\\alpha)\n$ \nand \\[ \\widetilde{U}_n := U_nT_\\alpha = \\left((1-\\alpha) U_{n,1}, \\dots, (1-\\alpha) U_{n,4}, \\alpha U_{n,5}, \\alpha U_{n,6}\\right). \\]\nClearly, there is a one to one correspondence between $U_n$ and $\\widetilde{U}_n= U_nT_\\alpha$ for $\\alpha\\in (0,1)$ and therefore it is sufficient to obtain the limiting results for the urn process $\\widetilde{U}_n$. Note that the off-diagonal elements of the replacement matrix $R_\\alpha$ are not all  non-negative, therefore we will use the limit results from \\cite{Paper1} to  obtain the convergence results for  the urn process $\\widetilde{U}_n$.\n\n\n\\begin{theorem}\\label{Thm:Urn}\nSuppose $\\alpha\\in(0,1)$. Then \t$(\\widetilde{U}_n)_{n\\geq 0}$ is an uniform urn process with replacement matrix $ R_\\alpha = RT_\\alpha$ and\n\\begin{equation}\n\\label{eq:cas:nonuniform}\n\\frac{\\widetilde{U}_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\widetilde{{\\mathbf v}}_1,\n\\end{equation}\nwhere \\begin{equation}\\label{Leftev1}\n\\widetilde{{\\mathbf v}}_1= \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha)^2, 2(1-\\alpha)^2, (1-\\alpha)^2, 1-\\alpha^2, \\alpha(1-\\alpha), \\alpha(5-3\\alpha) \\big)\n\\end{equation}\nis the normalized left eigenvector of $R_\\alpha$ corresponding to the largest eigenvalue $\\lambda_1=1$. Furthermore,\n\\begin{equation}\n\\label{eq:cweak:nonuniform}\n\\frac{\\widetilde{U}_n -n \\widetilde{{\\mathbf v}}_1}{\\sqrt{n}} \\xrightarrow{~d~}   \\mathcal{N}(\\mathbf{0}, \\widetilde{\\Sigma}),\n\\end{equation}\nwith the polynomials $\\gp_1,\\dots,\\gp_6$ defined in~\\eqref{eq:main:cov:poly} and $\\beta=1-\\alpha$,\n\\begin{equation}\n\\label{eq:covariance:nonuniform}\n\\widetilde{\\Sigma}=\\frac{\\beta}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\\begin{bmatrix*}[r]\n-12\\beta^2\\gp_1 & 4\\beta^2\\gp_2 & -6\\beta^2\\gp_1  & -2\\beta^2\\gp_4 & 2\\alpha\\beta \\gp_2 & -2\\alpha\\beta \\gp_2 \\\\\n4\\beta^2\\gp_2 & -4\\beta^2\\gp_3 & 2\\beta^2\\gp_2  & -2\\beta^2\\gp_6 & -2\\alpha\\beta \\gp_3 & 2\\alpha\\beta \\gp_3   \\\\\n-6\\beta^2\\gp_1 & 2\\beta^2\\gp_2  & -3\\beta^2\\gp_1  & -\\beta^2\\gp_4 & \\alpha\\beta \\gp_2 & -\\alpha\\beta \\gp_2  \\\\\n-2\\beta^2\\gp_4  &  -2\\beta^2\\gp_6  &  -\\beta^2\\gp_4 & \\beta^2\\gp_5 & -\\alpha\\beta \\gp_6 & \\alpha\\beta \\gp_6  \\\\\n2\\alpha\\beta \\gp_2   &  -2\\alpha\\beta \\gp_3  &  \\alpha\\beta \\gp_2& -\\alpha\\beta \\gp_6 & -\\alpha^2 \\gp_3 & \\alpha^2 \\gp_3  \\\\\n-2\\alpha\\beta \\gp_2    &  2\\alpha\\beta \\gp_3   &  -\\alpha\\beta \\gp_2 & \\alpha\\beta \\gp_6  &  \\alpha^2 \\gp_3 & -\\alpha^2 \\gp_3  \n\\end{bmatrix*}.\n\\end{equation}\n\n\\end{theorem}\n\\begin{proof}[Proof of Theorem \\ref{Thm:Urn}]\n\tFirst, observe that at any time $n$, there are $n+2$ pendant edges and $n+1$ internal edges in a rooted tree. That is, \n\\[ U_{n,1}+ U_{n,2}+ U_{n,3}+ U_{n,4}= n+2 \\quad \\text{and } \\quad U_{n,5}+ U_{n,6} = n+1.\\]\nThis gives\n\\begin{align*}\n\\|\\widetilde{U}_n\\|_1\n= (1-\\alpha) \\sum_{j=1}^4 U_{n,j}+ \\alpha  \\sum_{j=5}^6 U_{n,j}  \n= (1-\\alpha) (n+2) + \\alpha (n+1) = n+2 -\\alpha.\n\\end{align*}\nTherefore, from \\eqref{SelecProb} we get,\n\\[ \\mbox{${\\mathbb E}$}[\\chi_{n}| \\mbox{${\\mathcal F}$}_{n-1}] = \\dfrac{U_{n-1} T_{\\alpha}}{\\|U_{n-1}T_{\\alpha}\\|_1}= \\dfrac{U_{n-1} T_{\\alpha}}{ n+1 -\\alpha},\n\\]\nand\n\\begin{align*}\n\\mbox{${\\mathbb E}$}[U_{n}|\\mbox{${\\mathcal F}$}_{n-1}]\n= U_{n-1} + \\mbox{${\\mathbb E}$}[\\chi_{n}|\\mbox{${\\mathcal F}$}_{n-1}] R\n= U_{n-1} + \\dfrac{1}{n+1 -\\alpha} U_{n-1} T_\\alpha R.\n\\end{align*}\nMultiplying both sides by $T_\\alpha$,  we get\n\n\\[\\mbox{${\\mathbb E}$}[\\widetilde{U}_{n}|\\mbox{${\\mathcal F}$}_{n-1}] = \\widetilde{U}_{n-1} + \\left(\\dfrac{1}{\\|\\widetilde{U}_{n-1} \\|_1} \\widetilde{U}_{n-1} \\right) R T_\\alpha. \\]\nHence, $(\\widetilde{U}_n)_{n\\geq 0}$ is a classical uniform urn model with replacement matrix $ R_\\alpha = RT_\\alpha$.\n\n\nNote that  {\\bf (A1)} holds because the general Ford's dynamics on a rooted tree is well defined at every time $n$, thus the corresponding urn model satisfies the assumption of tenability. That is, it is always possible to draw balls  without getting stuck  with the replacement rule.\nNote that $R_\\alpha$ is diagonalisable as \n\\[V R_\\alpha V^{-1}=\\Lambda \\]\nholds with  $ \\Lambda = \\text{diag} \\big(1,0,0,0, -2(1-\\alpha),-(3-2\\alpha)\\big)$,\n\\begin{equation} \\label{Reigen:AalphaR}\nV^{-1}= \\begin{bmatrix}\n1& \\frac{1}{\\beta}&0&0&1& 1-\\alpha\\\\[1ex]\n1& 0&\\frac{1}{\\beta}&0&1& 3-\\alpha \\\\[1ex]\n1& \\frac{-2}{\\beta}&0& \\frac{3}{\\beta}&\\frac{-(2-\\alpha)}{\\beta}&-5+\\alpha \\\\[1ex]\n1 &0&0&\\frac{1}{\\beta}&\\frac{-(2-\\alpha)}{\\beta}&-3+\\alpha \\\\[1ex]\n1 & 0&\\frac{-2}{\\alpha}&\\frac{1}{\\alpha}&1&3-\\alpha \\\\[1ex]\n1 & 0&0&\\frac{-1}{\\alpha} & 1& 1-\\alpha\n\\end{bmatrix}\n\\end{equation}\n\nand \n\t\\setlength{\\arraycolsep}{2.5pt}\n\\medmuskip = 1mu\n\\begin{align}\nV =\\frac{1}{2(3-2\\alpha) }\n\\begin{bmatrix}\\label{Leigen:AalphaR:1}\n2\\beta^2 & 2\\beta^2 & \\beta^2 & (1+\\alpha)\\beta & \\alpha\\beta& \\alpha(5-3\\alpha)\\\\[1ex]\n2\\beta(1+\\alpha-\\alpha^2) & 2\\beta^3 &-(2-\\alpha)\\beta^2 & (2-\\alpha)\\beta^2 & -\\alpha\\beta^2& -\\alpha\\beta(5-3\\alpha) \\\\[1ex]\n2\\alpha\\beta^2 & 2\\alpha(2-\\alpha)\\beta & \\alpha\\beta^2 &-\\alpha\\beta^2 & -\\alpha(3- \\alpha)\\beta & -3\\alpha\\beta^2 \\\\[1ex]\n2\\alpha(2-\\alpha)\\beta & 2\\alpha\\beta^2 & \\alpha(2-\\alpha)\\beta &-\\alpha(2-\\alpha)\\beta & \\alpha^2 \\beta & -3\\alpha(2-\\alpha)\\beta \\\\[1ex]\n2(2-\\alpha)\\beta & -2\\beta^2 & (2-\\alpha)\\beta & -(4-\\alpha)\\beta & -\\alpha\\beta &\\alpha\\beta\\\\[1ex]\n-2\\beta & 2\\beta & -\\beta & \\beta & \\alpha & -\\alpha\n\\end{bmatrix}.\n\\end{align}\nTherefore, $R$ satisfies condition {\\bf (A4)}. Next, {\\bf (A2)} holds because $R_\\alpha$ has eigenvalues \n$$1,\\quad 0,\\quad 0, \\quad 0, \\quad -2(1-\\alpha),\\quad -(3-2\\alpha)$$ which are all real. The maximal eigenvalue $\\lambda_1=1$ is positive with $\\lambda_1>2\\lambda$ holds for all other eigenvalues $\\lambda$ of $R_\\alpha$.\nFurthermore, put $\\mathbf{u}_i={V^{-1}}\\mathbf{e}^\\top_i$ and $\\mathbf{v}_i=\\mathbf{e}_iV$ for $1\\le i \\le 4$. Then {\\bf (A3)} follows by noting that $\\mathbf{u}_1=(1,1,1,1,1,1)^\\top$ is the principal right eigenvector, and\n\\[\\widetilde{{\\mathbf v}}_1= \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha)^2, 2(1-\\alpha)^2, (1-\\alpha)^2, 1-\\alpha^2, \\alpha(1-\\alpha), \\alpha(5-3\\alpha) \\big)\\]  is the principal left eigenvector. \n\n\nSince all the assumptions {\\bf (A1)--(A4)} are satisfied  by the replacement matrix $R_\\alpha$, by Theorem~\\ref{thm1&2:paper1},\n\\eqref{eq:cas:nonuniform} holds. Furthermore, since\n\\begin{equation} \n\\widetilde{\\Sigma} = \\sum_{i,j=2}^6 \\frac{ \\lambda_i \\lambda _j {\\mathbf u}_i^\\top \\mbox{diag}(\\mathbf{v}_1) {\\mathbf u}_j }{1-\\lambda_i -\\lambda_j} \\mathbf{v}_i^\\top \\mathbf{v}_j,\n\\end{equation}\nby~\\eqref{eq:cas:nonuniform} it follows that~\\eqref{eq:cweak:nonuniform} holds.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:urn:edge}}\n\n\n\n\n\\begin{proof}\n\t\\noindent\n\tObserve that  $\\sum_{i=1}^6U_{n,i} = 3+2n$ (since $2$ balls are added into the urn at every time point), thus the vector of color proportions is $U_n \/(3+2n)$.\n\n\tSince $\\alpha \\in (0,1)$, it follows that  $T_\\alpha$ is invertible and its inverse is \n\t$$\n\tT_\\alpha^{-1}=\\frac{1}{\\alpha(1-\\alpha)}\\mbox{diag}(\\alpha,\\alpha,\\alpha,\\alpha,1-\\alpha,1-\\alpha),\n\t$$\n\twhich is also a diagonal matrix, and so $(T_\\alpha^{-1})^\\top=T_\\alpha^{-1}$.\n\tNote that  we have $U_n = \\widetilde{U}_n T_\\alpha^{-1}$ and consider \n\t$$\n\t{{\\mathbf v}} = \\widetilde{{\\mathbf v}}_1 (T_\\alpha)^{-1} = \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha), 2(1-\\alpha), 1-\\alpha, 1+\\alpha, 1-\\alpha, 5-3\\alpha \\big).\n\t$$\n\tSince $ \\dfrac{\\widetilde{U}_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\widetilde{{\\mathbf v}}_1$ holds in view of~\\eqref{eq:cas:nonuniform} in Theorem \\ref{Thm:Urn}, \n\n\t\\begin{equation}\n\t\\frac{U_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  {{\\mathbf v}},\n\t\\end{equation}\n\twhich concludes the proof of the almost sure convergence in~\\eqref{eq:conv:urn}. \n\t\n\tConsider the covariance matrix $\\widetilde{\\Sigma}$ for $ \\widetilde{U}_n$ as stated in~\\eqref{eq:covariance:nonuniform}, then by straightforward calculation we have \n\t$$\n\t\\Sigma=(T_\\alpha^{-1})^\\top \\widetilde{\\Sigma} T_\\alpha^{-1}=T_\\alpha^{-1}  \\widetilde{\\Sigma} T_\\alpha^{-1}. \n\t$$\n\tTherefore, since\n\t\\[\\frac{\\widetilde{U}_n -n \\widetilde{{\\mathbf v}}_1}{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N} (\\mathbf{0}, \\widetilde{\\Sigma} ) \\]\n\tin view of Theorem \\ref{Thm:Urn}, we get\n\t\\[\\frac{U_n -n {\\mathbf v} }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N} \\big(\\mathbf{0}, (T_\\alpha^{-1})^\\top \\widetilde{\\Sigma}\\, T_\\alpha^{-1}\\big)=\\mathcal{N} (\\mathbf{0},  \\Sigma). \\]\n\tThis completes the proof.\n\\end{proof}\n\n\n\n\\section{Exact Distributions }\n\\label{sec:exact}\n\nIn this section, we present recursion formulas for exact computation of  the joint distributions of cherries and pitchforks, their means, variances and covariance for fixed $n$ under the Ford model.\n\n\n\nWe begin with the following notation. Given a phylogenetic tree $T$, let $E_{1}(T)$ be the set of pendant edges that are contained in a pitchfork but not a cherry; $E_{2}(T)$ the set of edges in $T$\nthat are contained in a cherry but not in a pitchfork (note that in our notation a cherry contains three leaves);\n$E_{3}(T)$ the set of pendant edges that are contained in neither a cherry nor a pitchfork; and $E_{4}(T)=E(T)\\setminus (E_{1}(T)\\cup E_{2}(T) \\cup E_{3}(T))$.\nIn addition, $E(T)$ can be decomposed into the disjoint union of these four sets of edges.\ni.e., $E(T)=E_1(T)\\sqcup E_2(T) \\sqcup E_3(T) \\sqcup E_4(T)$, where $\\sqcup$ denotes disjoint union.\nLet $C(T), A(T)$ be the number of cherries and pitchforks in a tree $T$. The following result presented in \\cite{WuChoi16}  will be useful later. \n\n\\begin{lemma}\t\\label{lem:edge-set}\n\tSuppose that $T$ is a phylogenetic tree with $n$ leaves. Then we have\n\t\\begin{equation}\n\t\\label{eq:edge:dec}\n\tE(T)=E_{1}(T)\\,\\sqcup\\,E_{2}(T)\\,\\sqcup\\,E_{3}(T)\\,\\sqcup\\,E_{4}(T).\n\t\\end{equation}\n\tIn addition, we have\n\n\t$|E_{1}(T)|=A(T)$,\n\t$|E_{2}(T)|=3(C(T)-A(T))$,\n\t$|E_{3}(T)|=n-A(T)-2C(T)$,\n\tand\n\t$|E_{4}(T)|=n-1+3A(T)-C(T)$.\nFurthermore, suppose that $e$ is an edge in $T$ and $T'=T[e]$. Then\n\t\twe have\n\t\t\\small{\n\t\t\t\\begin{equation*}\n\t\t\tA(T') = \\begin{cases}\n\t\t\tA(T) & \\text{if } e\\in E_3(T)\\cup E_4(T), \\\\\n\t\t\tA(T)-1 & \\text{if } e\\in E_1(T),\\\\\n\t\t\tA(T)+1 & \\text{if } e\\in E_2(T);\n\t\t\t\\end{cases}\n\t\t\t\\mbox{and} \\quad\n\t\t\tC(T') = \\begin{cases}\n\t\t\tC(T) & \\text{if } e \\in E_2(T)\\cup E_4(T), \\\\\n\t\t\t{} & \\\\\n\t\t\tC(T)+1 & \\text{if } e \\in E_1(T)\\cup E_3(T).\n\t\t\t\\end{cases}\n\t\t\t\\end{equation*}\n\t\t}\n\n\\end{lemma}\n\nWe start with the following result on the exact computation of the  joint probability mass function (pmf)  of $A_n$ and $C_n$, \nwhich can be regarded as a generalization of the previous results on the Yule model (e.g. when $\\alpha=0$~\\cite[Theorem 1]{WuChoi16}) and  the uniform model (e.g. $\\alpha=1\/2$~\\cite[Theorem 4]{WuChoi16}).\nA related result for unrooted  trees is presented in~\\cite{CTW19}.\n\n\\begin{theorem} \\label{jointpmf}\nFor $n \\ge 3$, $0 \\le  a\\le n\/3$ and $1\\le b\\le n\/2$, under the Ford model with parameter $\\alpha\\in [0,1]$  we have\n\\begin{eqnarray*}\n&& \\mathbb{P}(A_{n+1}=a, C_{n+1}=b) \\\\&=&  \\frac{2a+ \\alpha(n-a-b-1)}{n-\\alpha} \\mathbb{P}(A_n=a, C_n=b)\n+ \\frac{(1-\\alpha)(a+1)}{n-\\alpha} \\mathbb{P}(A_n=a+1, C_n=b-1) \\\\\n&& \n\\quad\n+ \\frac{(2-\\alpha)(b-a+1)}{n-\\alpha} \\mathbb{P}(A_n=a-1, C_n=b) + \\frac{(1-\\alpha)(n-a-2b+2)}{n-\\alpha} \\mathbb{P}(A_n=a, C_n=b-1).\n\\end{eqnarray*}\n\\end{theorem}\n\\begin{proof}[Proof of Theorem \\ref {jointpmf}]\nFix $n> 3$, and let $T_2,\\dots,T_n,T_{n+1}$ be a sequence of random trees generated by the Ford process, that is, $T_2$ contains two leaves and $T_{i+1}=T_i[e_i]$ for a random edge $e_i$ in $T_i$  chosen according to the Ford model for $2\\leq i \\leq n$. \n\tThen we have\n\t\\begin{align}\n\t\\label{eq:total:yule}\n\t\\mathbb{P}(A_{n+1}=a, C_{n+1}=b) &=\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b) \\notag \\\\\t\n\t&\\hspace{-2cm}\n\t=\\sum_{p,q} \\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b\\,|\\,A(T_n)=p,C(T_n)=q) \\mathbb{P}(A(T_n)=p,C(T_n)=q) \\notag \\\\\n\t\\hspace{-1cm}\n\t&\\hspace{-2cm}\n\t=\\sum_{p,q} \\mathbb{P} (A(T_{n+1})=a, C(T_{n+1})=b\\,|\\,A(T_n)=p,C(T_n)=q) \\mathbb{P}(A_{n}=p, C_{n}=q),\n\t\\end{align}\n\twhere the first and second equalities follow from the law of total probability, and the definition of random variables $A_n$ and $C_n$.\n\t\nLet $e_n$ be the edge in $T_n$ chosen in the above Ford process for generating $T_{n+1}$, that is, $T_{n+1}=T_n[e_n]$. Since Lemma~\\ref{lem:edge-set} implies that\n\t\\begin{equation}\n\t\\label{eq:yule:5}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=p,C(T_n)=q)=0 \n\t\\end{equation}\n\tfor $(p,q) \\not \\in\\{(a,b),(a+1,b-1),(a-1,b),(a,b-1)\\}$,\n\tit suffices to consider the following four cases in the summation in (\\ref{eq:total:yule}): case (i): $p=a, q=b$; case (ii): $p=a+1, q=b-1$; case (iii): $p=a-1, q=b$; and case (iv): $p=a, q=b-1$.\n\t\n\t\n\tFirstly, Lemma~\\ref{lem:edge-set} implies that case (i) occurs if and only if $e_n\\in E_4(T_n)$. Using Lemma~\\ref{lem:edge-set} again, it follows that $E_4(T_n)$ contains precisely $2A(T_n)$ pendent edges and $(n-1)+A(T_n)-C(T_n)$ interior edges. Therefore we have\n\t\\begin{align}\n\t&\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a,C(T_n)=b)  \\nonumber \\\\\n\t&\\quad \\quad =\\frac{2A(T_n)(1-\\alpha)+\\alpha(n-1+A(T_n)-C(T_n))}{n-\\alpha}=\n\t\\frac{2a+ \\alpha(n-a-b-1)}{n-\\alpha}.\\label{eq:yule:1}\n\t\\end{align}\n\n\n\t\n\tSimilarly, Lemma~\\ref{lem:edge-set} implies that case (ii)\n\toccurs if and only if $e_n\\in E_1(T_n)$.\n\t Using Lemma~\\ref{lem:edge-set} again, it follows that $E_1(T_n)$ contains precisely $A(T_n)$ pendent edges and no interior edges. Therefore we have\n\t\\begin{align}\n\t\\label{eq:yule:2}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a+1,C(T_n)=b-1)\n\t=\\frac{(a+1)(1-\\alpha)}{n-\\alpha}.\n\t\\end{align}\n\t\n\tNext, Lemma~\\ref{lem:edge-set} implies case (iii) occurs if and only if $e_n\\in E_2(T_n)$. \t Using Lemma~\\ref{lem:edge-set} again, it follows that $E_2(T_n)$ contains precisely $2(A(T_n)-C(T_n))$ pendent edges and $A(T_n-C(T_n)$ interior edges. Thus we have\n\t\\begin{align}\n\t&\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a-1,C(T_n)=b) \\nonumber \\\\\n\t&\t\\quad \\quad\n\t=\\frac{2(a-1-b)(1-\\alpha)+\\alpha(a-1-b)}{n-\\alpha}=\\frac{(2-\\alpha)(b-a+1)}{n-\\alpha}.\\label{eq:yule:3}\n\t\\end{align}\n\n\t\n\tFinally, Lemma~\\ref{lem:edge-set} implies case (iv) occurs\n\tif and only if $e_n$ is contained in $E_3(T_n)$. Using Lemma~\\ref{lem:edge-set} again, it follows that $E_3(T_n)$ contains precisely $n-A(T_n)-2C(T_n)$ pendent edges and no interior edges. Hence, it follows that\n\t\\begin{equation}\n\t\\label{eq:yule:4}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1}=b)~|~A(T_n)=a,C(T_n)=b-1)\n=\\frac{(1-\\alpha)(n-a-2b+2)}{n-\\alpha}. \n\t\\end{equation}\n\t\n\tSubstituting Eq.~\\eqref{eq:yule:1}--\\eqref{eq:yule:4} into Eq.~\\eqref{eq:total:yule} completes the proof of the theorem.\n\t\\end{proof}\n\n\nTo study the moments of $A_n$ and $C_n$, we present below a functional recursion form of Theorem~\\ref{jointpmf}, whose proof is straightforward and hence omitted here. \n\n\n\\begin{theorem}\n\\label{jointrr}\nLet $\\varphi: \\mathbb{N}\\times \\mathbb{N} \\to \\mathbb{R}$ be an arbitrary function. For $n \\ge 3$,\nunder the Ford model with parameter $\\alpha\\in [0,1]$ we have\n\\begin{eqnarray*}\n(n-\\alpha) \\mbox{${\\mathbb E}$} \\varphi(A_{n+1}, C_{n+1}) &=& \\mbox{${\\mathbb E}$}\\bigg[ \\big\\{\\alpha(n-A_n-C_n-1) +2A_n \\big\\} \\varphi(A_n, C_n) \\\\\n&& +(1-\\alpha)A_n \\varphi(A_n-1, C_n+1) +(2-\\alpha) (C_n-A_n) \\varphi(A_n+1, C_n) \\\\\n&& + (1-\\alpha)(n-A_n-2C_n) \\varphi(A_n, C_n+1) \\bigg ].\n\\end{eqnarray*}\n\\end{theorem}\n\nFor a fix integer $k$, consider the indicating function $I_k(x,y)$ that equals to 1 if $y=k$, and $0$ otherwise. Then by Theorem~\\ref{jointrr} the following result on the distribution of cherries follows.\n\n\\begin{corollary}\n\\label{cherrypmf}\nFor integers $n \\ge 3$ and $0\\le k \\le n\/2$, under the Ford model with parameter $\\alpha\\in [0,1]$ we have\n$$\n(n-\\alpha) \\mathbb{P}(C_{n+1}=k) = [(n-1) \\alpha +2(1-\\alpha) k ] \\mathbb{P}(C_{n}=k)\n+(1-\\alpha)(n-2k +2) \\mathbb{P}(C_{n+1}=k-1).\n$$\n\\end{corollary}\n\nFor the purpose of next section, we end this section by writing the recurrence relation in the following form in the next Corollary.\n\\begin{corollary}\n\\label{recurrence}\nFor $n \\ge 3$, under the Ford model with parameter $\\alpha\\in [0,1]$ we have\n\\begin{eqnarray}\n(n-\\alpha) \\mbox{${\\mathbb E}$} [C_{n+1}] - (n -2+\\alpha) \\mbox{${\\mathbb E}$} [C_n] &=& n(1-\\alpha), \\label{cherrymean}\\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$}[A_{n+1}] - (n-3+\\alpha) \\mbox{${\\mathbb E}$}[A_n] &=& (2-\\alpha) \\mbox{${\\mathbb E}$} [C_n], \\label{forkmean} \\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$} [C_{n+1}^2] - (n-4+3\\alpha) \\mbox{${\\mathbb E}$} [C_n^2] &=& 2(n-1)(1-\\alpha) \\mbox{${\\mathbb E}$} [C_n] + n(1-\\alpha), \\label{cherry2nd}\\\\\n(n-\\alpha)\\mbox{${\\mathbb E}$}[A_{n+1}C_{n+1}] - (n-5+3\\alpha) \\mbox{${\\mathbb E}$} [A_nC_n] &=& (n-1)(1-\\alpha)\\mbox{${\\mathbb E}$}[A_n] + (2-\\alpha) \\mbox{${\\mathbb E}$}[ C_n^2], \\label{cov}\\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$}[A_{n+1}^2]- (n-6+3\\alpha)\\mbox{${\\mathbb E}$}[A_n^2] &=& 2(2-\\alpha)\\mbox{${\\mathbb E}$} [A_nC_n] + (2-\\alpha)\\mbox{${\\mathbb E}$} [C_n] - \\mbox{${\\mathbb E}$} [A_n] \\label{fork2nd}\n\\end{eqnarray}\nwith initial conditions $\\mbox{${\\mathbb E}$}[A_3]= \\mbox{${\\mathbb E}$}[C_3]=\\mbox{${\\mathbb E}$}[A_3^2]=\\mbox{${\\mathbb E}$}[C_3^2]=\\mbox{${\\mathbb E}$}[A_3C_3]=1. $\n\\end{corollary}\n\n\\begin{remark}\nLet $\\mu_n = \\mbox{${\\mathbb E}$} [C_n]$ and \t$\\si_{n}^2=\\mathrm{var}(C_n)$. \nSubstituting $\\mbox{${\\mathbb E}$}[C_{n}^2]=\\si_{n}^2 + \\mu_{n}^2$ into (\\ref{cherry2nd}) and applying (\\ref{cherrymean}), we obtain below a recurrence relation of the $\\si_n^2$, which was also obtained in Ford's thesis (Theorem 60, \\cite{Ford2006}):\n\\begin{eqnarray*}\n(n-\\alpha) \\si_{n+1}^2 - (n-4+3\\alpha) \\si_n^2 \n&=& -\\frac{4(1-\\alpha)^2}{n-\\alpha} \\mu_n^2 + \\frac{2(1-\\alpha)[(1-2\\alpha)n +\\alpha]}{n-\\alpha} \\mu_n\n+ \\frac{\\alpha (1-\\alpha)n(n-1)}{n-\\alpha}. \\label{cherryvar}\n\\end{eqnarray*}\n\\end{remark}\n\n\\section{Higher Order Asymptotic Expansion of the Joint Moments}\n\\label{sec:expansion}\n\nAlthough the leading terms of the first and second moments of the distributions of cherries and pitchforks, $\\mbox{${\\mathbb E}$} [A_n], \\mbox{${\\mathbb E}$} [C_n], \\mathrm{var}(A_n),\\mathrm{var}(C_n)$ and $cov(A_n, C_n)$, can be identified from  Theorem \\ref{Thm:Convg-ChPh}, for better understanding of their asymptotic behaviour \n we derive their  higher order expansions in this Section. \n\n We start with the following result on the first moments. Note that Proposition~\\ref{Prop:firstm} (i) has been obtained in  \\cite{Ford2006}.\n\\begin{proposition\n\\label{Prop:firstm} Under the Ford model with parameter $\\alpha\\in [0,1]$, the following exact expansions hold for $\\mbox{${\\mathbb E}$} [C_n] $ and $\\mbox{${\\mathbb E}$} [A_n]$.\n\\begin{enumerate}[\\normalfont(i)]\n\\item $ \\mbox{${\\mathbb E}$} [C_n]  = \\dfrac{1-\\alpha}{3-2\\alpha} \\ n + \\dfrac{\\alpha}{2(3-2\\alpha)} +x_n, $\nwhere\n\\[x_2= \\frac{(2- \\alpha)}{2(3-2\\alpha)}, \\quad x_3 = \\frac{\\alpha}{2(3-2\\alpha)}, \\quad x_n = \\frac{\\alpha }{2(3-2\\alpha)} \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha},  \\quad n \\ge 4.  \\]\nFurther, as $n \\to \\infty$,\n\\begin{equation}\\label{Order:xn}\nx_n = \\frac{\\alpha\\Ga(3-\\alpha)}{2(3-2\\alpha)\\Ga(1+\\alpha)}n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\n\n\\item $\\mbox{${\\mathbb E}$} [A_n]  = \\dfrac{1-\\alpha}{2(3-2\\alpha)} \\ n + \\dfrac{\\alpha}{2(3-2\\alpha)} + y_n,$\nwhere\n\\[y_2 = \\frac{\\alpha-2}{2(3-2\\alpha)}, ~ y_3=\\frac{1}{2}, ~  y_n =\\frac{1}{2} \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha }{2(3-2\\alpha)} \\frac{n-3}{n-3+\\alpha} \\prod_{i=3}^{n-1}\\frac{i-2+\\alpha}{i-\\alpha }, ~~ n \\geq 4.\\]\nFurther, as $n \\to \\infty$,\n\\begin{equation}\\label{Order:yn}\ny_n  = \\frac{(2- \\alpha) \\Gamma(3-\\alpha) }{2(3-2\\alpha)\\Gamma(\\alpha)} n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition\n\\label{Prop:secondm}\nUnder the Ford model with parameter $\\alpha\\in [0,1]$, the following asymptotic expansions hold for $\\mathrm{var} (C_n)$,  $cov(A_n, C_n)$ and $\\mathrm{var} (A_n)$:\n\\begin{enumerate}[\\normalfont(i)]\n\\item $$\\mathrm{var} (C_n) = \\frac{(1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} \\ n -  \\frac{\\alpha (1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\item\n$$cov(A_n, C_n)  = \\frac{-(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(3-2\\alpha)^2(5-4\\alpha) } \\ n -\\frac{ \\alpha (1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)}  +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\item\n$$ \\mathrm{var}(A_n) =  \\frac{(1-\\alpha)(69-135\\alpha+96\\alpha^2-24\\alpha^3)}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \\ n   + \\frac{3\\alpha(1-\\alpha)(1-2\\alpha)(5-3\\alpha)}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)}  +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\end{enumerate}\n\\end{proposition}\n\n\n\\begin{remark}\nWhen $n$ is large,  $Cov(A_n, C_n)$ changes sign. Specifically, for $ \\alpha \\in (0, 1\/2)$, $A_n$ and $C_n$ are negatively correlated, which is expected; and  for $ \\alpha \\in (1\/2, 1)$, $A_n$ and $C_n$ are positively correlated, which is unexpected.\n\\end{remark}\n\n\n\n\n\\subsection{Proofs of Propositions~\\ref{Prop:firstm} and~\\ref{Prop:secondm}} \nWe need  the  lemmas below to prove the two propositions.\n\n\\begin{lemma}\\label{Lemma1}\nSuppose a real sequence $\\{X_n, n \\ge n_0\\}$ satisfies the recursion\n\\[ X_{n+1}  = f_n X_n +g_n, \\qquad n \\geq n_0,\\]\nwhere  $\\{f_n, n \\ge n_0\\}$ and $\\{g_n, n \\ge n_0\\}$ are sequences such that for every $ \\ell \\geq n_0$,  $ \\left|\\prod_{i=\\ell}^n f_i \\right| \\leq C (n\/\\ell)^{-a} $ and  $|g_\\ell| \\leq C \\ell^{-b}$,  for some finite $a, b$ and $C>0$. Then, there exists a finite positive constant $C'$ (which depends on $|X_{n_0}|$ and $C$) such that  $|X_n | \\leq C' n^{-q_{a,b}}$ where $q_{a,b} \\coloneqq  \\min\\{a, b-1\\}$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma1}]\nIt is easy to verify that the  solution to the given recursion is given by\n\\[X_n =X_{n_0} \\prod_{i= n_0}^{n-1} f_i + \\sum_{i=n_0}^{n-1} g_i  \\prod_{j=i+1}^{n-1}f_j, \\quad n \\ge n_0.\\]\nTherefore,\n\\[ |X_n| \\leq  |X_{n_0}| \\left| \\prod_{i= n_0}^{n-1} f_i \\right| + \\sum_{i=n_0}^{n-1} |g_i| \\left| \\prod_{j=i+1}^{n-1} f_j  \\right|.\\]\nUnder the assumptions of the Lemma,\n\\[ |X_{n_0}| \\left| \\prod_{i= k}^{n-1}f_i \\right|  \\leq  C |X_{n_0}| n^{-a}\n\\leq C' n^{-a}; \\]\nand\n\\begin{align*}\n \\sum_{i=n_0}^{n-1} |g_i| \\left| \\prod_{j=i+1}^{n-1} f_j \\right|\n &\\leq C  \\sum_{i=n_0}^{n-1} |g_i|   (n\/i)^{-a}  \\leq   C^2  \\, n^{-a} \\sum_{i=n_0}^{n-1} i^{-b}  i^{a}  \\leq  C'  \\, n^{-a}   \\, n^{-b+a +1} = C' n^{-b+1}.\n\\end{align*}\nThus\n\\[ |X_n| \\leq  C' \\max( n^{-a} , n^{-b+1})  = C' n^{-q_{a,b}},\\]\nwhere $q_{a,b} = \\min\\{a, b-1\\}$. This completes the proof.\n\\end{proof}\n\n\\begin{lemma} \\label{Lemma2}\nFor finite non-negative integers $l,k$ such that $l\\geq k$, $m\\geq 1$ and  $\\alpha\\in [0,1]$, there exists a positive constant $K=K(\\alpha,l)$  such that\n\\begin{equation} \\label{eq:bound1}\n\t\\left|\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}  \\right|   \\leq\n\tK \\left(n\/l\\right)^{-k+(m+1)\\alpha}\n\t~~\\mbox{for all $1\\le l\\le n-1$.}\n\t\\end{equation}\nand as $n\\to \\infty$\n\\begin{equation}\\label{Order:prod}\n\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}   =   \\frac{\\Gamma(l-\\alpha) }{\\Gamma(l-k+m\\alpha)} n^{-k+(m+1)\\alpha} \\left(1+o(1) \\right).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma2}]\nThe bound in \\eqref{eq:bound1} follows from Lemma 2 of \\cite{Paper1}. We now prove \\eqref{Order:prod}. Note that, we can write\n\\[ \\frac{i-k+m\\alpha}{i-\\alpha}  = \\frac{\\Gamma(i+1-k+m\\alpha)\\Gamma(i-\\alpha) }{\\Gamma(i-k+m\\alpha)\\Gamma(i+1-\\alpha)}.\\]\nThus\n\\begin{align}\n\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}\n& =  \\prod_{i=l }^{n-1} \\frac{\\Gamma(i+1-k+m\\alpha)\\Gamma(i-\\alpha) }{\\Gamma(i-k+m\\alpha)\\Gamma(i+1-\\alpha) } \\nonumber\\\\\n& =   \\frac{\\Gamma(n-k+m\\alpha) }{\\Gamma(l-k+m\\alpha)}   \\frac{\\Gamma(l-\\alpha) }{\\Gamma(n-\\alpha)} \\label{Prod:Gamma}\\\\\n& =   \\frac{\\Gamma(l-\\alpha)}{\\Gamma(l-k+m\\alpha)}  \\frac{\\Gamma(n+m\\alpha)  }{\\Gamma(n-\\alpha)}   \\prod_{j=1}^k \\frac{1}{n-j+m\\alpha}.\\nonumber\n\\end{align}\n\n\\begin{equation}\\label{Eq:prodk}\n\\prod_{j=1}^k \\frac{1}{n-j+m\\alpha} =n^{-k}\\left(1+o(1) \\right).\n\\end{equation}\nBy  Stirling's approximation formula, $\\Ga(x) = \\sqrt{2 \\pi} \\ x^{x-1\/2} e^{-x} \\left(1+o(1)\\right)$, we have\n\\begin{eqnarray}\n\\frac{\\Ga(n+m\\alpha)}{\\Ga(n-\\alpha)}\n&=& \\frac{\\sqrt{2\\pi} (n+m\\alpha)^{n+m\\alpha-1\/2} \\, e^{-(n+m\\alpha)}}{\\sqrt{2\\pi} (n-\\alpha)^{n-\\alpha-1\/2} \\, e^{-(n-\\alpha)}}\\left(1+o(1) \\right) \\nonumber \\\\\n&=& n^{(m+1)\\alpha} \\frac{(1+ m\\alpha\/n)^{n+m\\alpha-1\/2}}{(1-\\alpha\/n)^{n-\\alpha-1\/2}} e^{-(m+1)\\alpha} \\left(1+o(1) \\right) \\nonumber \\\\\n&=& n^{(m+1)\\alpha} \\left(1+o(1) \\right). \\label{Eq:Str}\n\\end{eqnarray}\nCombining \\eqref{Eq:prodk} and \\eqref{Eq:Str}, we get \\eqref{Order:prod}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{Prop:firstm}]\nRecall  $\\mu_n= \\mbox{${\\mathbb E}$}[C_n]$. By Theorem \\ref{Thm:Convg-ChPh}, $\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} \\ n + \\mbox{${\\mathcal O}$}(1)$. Thus, we write $\\mu_n$  as\n\\begin{equation}\\label{Exp:mu}\n\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} \\ n + \\frac{\\alpha}{2(3-2\\alpha)} +x_n.\n\\end{equation}\nFor simplicity, the dependence of $\\mu_n$ and $x_n$ on $\\alpha$ are  suppressed.\n\nSince $\\mu_2= \\mu_3=1$, we get $x_2 = 1-\\frac{4-3\\alpha}{2(3-2\\alpha)}=\\frac{2-\\alpha}{2(3-2\\alpha)}$\nand\n$x_3 = 1- \\frac{6-5\\alpha}{2(3-2\\alpha)} = \\frac{\\alpha}{2(3-2\\alpha)}.$\nSubstituting (\\ref{Exp:mu}) into (\\ref{cherrymean}) leads to\n$$(n-\\alpha) x_{n+1}- (n-2+\\alpha)x_n=0, \\quad n \\ge 2,  $$\nand hence,\n\\[ x_n=\\begin{cases}\n\\frac{\\alpha }{2(3-2\\alpha)} \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha} & \\quad n \\ge 4, \\\\\n\\frac{\\alpha}{2(3-2\\alpha)} & \\quad n=3, \\\\\n\\frac{(2- \\alpha)}{2(3-2\\alpha)} & \\quad n=2.\n\\end{cases} \\]\nTo prove \\eqref{Order:xn}, we rewrite $x_n$ as follows\n\\[x_n = x_3  \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha} = x_3 \\frac{ \\Gamma(3-\\alpha)}{ \\Ga(1+\\alpha)} \\frac{\\Ga(n-2+\\alpha)}{\\Ga(n-\\alpha)}, \\quad n \\ge 4. \\]\nApply  Lemma \\ref{Lemma2}, \\eqref{Order:xn} holds.\nConsequently,\n\\begin{equation}\\label{Exp:mu_n}\n\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} n + \\frac{\\alpha}{2(3-2\\alpha)} + \\frac{ \\alpha\\Ga(3-\\alpha)}{2(3-2\\alpha) \\Ga(1+\\alpha)} n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\nThis completes the proof of part (i).\n\nThe same method of proof can be used to prove part (ii).\nRecall $\\nu_n =  \\mbox{${\\mathbb E}$}[A_n]$. By Theorem  \\ref{Thm:Convg-ChPh}, $\\nu_n = \\frac{1-\\alpha}{2(3-2\\alpha)} \\ n +\\mbox{${\\mathcal O}$}(1)$, and we write it as\n\n\n\\begin{equation}\\label{Exp:nu}\n\\nu_n = \\frac{1-\\alpha}{2(3-2\\alpha)} \\ n + \\frac{\\alpha}{2(3-2\\alpha)} + y_n,\n\\end{equation}\nwhere, again,  the dependence of $\\nu_n$ and $y_n$ on $\\alpha$ are suppressed.  Substituting (\\ref{Exp:nu}) into (\\ref{forkmean}) leads to\n\\[ y_{n+1}= \\frac{n-3+\\alpha}{n-\\alpha } y_n + \\frac{2- \\alpha}{n-\\alpha } x_n, \\quad n \\ge 4.\\]\nThe solution to this  recurrence relation is given by \n\\[ y_n =y_3 \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha }   + \\sum_{i=3}^{n-1} \\frac{2- \\alpha}{i-\\alpha } x_i \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\\]\nSince  $y_3=1\/2$ and the expression for $x_i$ from part (i), we get\n\\begin{align*}\ny_n\n& =y_3 \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\sum_{i=3}^{n-1} \\frac{2- \\alpha}{i-\\alpha } \\times \\frac{\\alpha }{2(3-2\\alpha)} \\prod_{j=3}^{i-1} \\frac{j-2+\\alpha}{j-\\alpha} \\times \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }  \\\\\n& =\\frac{1}{2}  \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } +  \\frac{(2- \\alpha) \\alpha }{2(3-2\\alpha)} \\sum_{i=3}^{n-1} \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha } \\times \\frac{1}{i-\\alpha}\\times \\prod_{j=3}^{i-1} \\frac{j-2+\\alpha}{j-\\alpha} \\\\\n& =\\frac{1}{2} \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\sum_{i=3}^{n-1}  \\frac{1}{3-\\alpha}\\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha } \\\\\n& =\\frac{1}{2}  \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\frac{(n-3)}{(3-\\alpha)} \\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\n\\end{align*}\nThus, for $n \\geq 5$,\n\\begin{equation}\\label{sol:yn}\ny_n = \\left( \\frac{1}{2}  + \\frac{(2- \\alpha)\\alpha }{2(3-2\\alpha)} \\frac{(n-3)}{(3-\\alpha)}  \\right) \\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\n\\end{equation}\nBy  Lemma \\ref{Lemma2},\n\\begin{align*}\ny_n  &=  \\left( \\frac{1}{2}  + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\, \\frac{(n-3)}{(3-\\alpha)} \\right) \\frac{\\Gamma(4-\\alpha) }{\\Gamma(1+\\alpha)} n^{-3+2\\alpha} \\left(1+o(1) \\right)\\\\\n&=  \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\frac{\\Gamma(3-\\alpha) }{\\Gamma(1+\\alpha)} n^{-2+2\\alpha} \\left(1+o(1) \\right) \\\\\n&=\\frac{(2- \\alpha)\\Gamma(3-\\alpha) }{2(3-2\\alpha)\\Gamma(\\alpha)}  n^{-2(1-\\alpha)} \\left(1+o(1) \\right)\n\\end{align*}\nas $n \\to \\infty$.\nThis completes the proof of part (ii) and hence the Proposition.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{Prop:secondm}]\n\nThe method of proof is similar to that of Proposition \\ref{Prop:firstm}. \n\nRecall $\\si_n ^2= \\mathrm{var}(C_n)$. From Theorem \\ref{Thm:Convg-ChPh},  we have  $ \\si_n^2 = \\frac{(1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} n+ \\mbox{${\\mathcal O}$}(1)$. \nWe first consider $\\mbox{${\\mathbb E}$}[C_n^2]$. As \n$$\\mbox{${\\mathbb E}$}[C_n^2] = \\mu_n^2 + \\si_n^2 = \\frac{(1-\\alpha)^2}{(3-2\\alpha)^2} n^2 +\\mbox{${\\mathcal O}$}(n),$$ we rewrite it as\n\\begin{equation}\n\\label{CM2}\n\\mbox{${\\mathbb E}$}[C_n^2]  = \\frac{(1-\\alpha)^2}{(3-2\\alpha)^2} \\ n^2 + \\frac{2(1-\\alpha)(1+ 2\\alpha -2\\alpha^2)}{(5-4\\alpha)(3-2\\alpha)^2} \\ n - \\frac{\\alpha (8-17\\alpha+8\\alpha^2)}{4(5-4\\alpha)(3-2\\alpha)^2} +z_n,\n\\end{equation} \nand derive below a recursion on $z_n$. Substituting (\\ref{CM2}) into (\\ref{cherry2nd}) and after   straightforward algebraic simplification, we have  \n$$ (n-\\alpha) z_{n+1} - (n-4+3\\alpha) z_n = 2(1-\\alpha)(n-1) x_n, \\quad n \\ge 2.  $$\nSince $C_2=C_3=1$, we get $ z_2 = \\frac{3(2-\\alpha) (8\\alpha^2 -21\\alpha+14)}{4(3-2\\alpha)^2 (5-4\\alpha)}$  and\n$ z_3 = \\frac{88 \\alpha^3 - 213 \\alpha^2 + 152 \\alpha - 24}{4(3-2\\alpha)^2 (5-4\\alpha)}.$ Consequently, \n\\[ \\si_n^2 = \\frac{(1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} \\ n -  \\frac{\\alpha (1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} + v_n -x_n^2,\\]\nwhere\n \\[v_n=z_n  - \\frac{2(1-\\alpha)}{3-2\\alpha} n x_n - \\frac{\\alpha}{3-2\\alpha} x_n = z_n - \\frac{[2(1-\\alpha) n +\\alpha]}{3-2\\alpha}x_n.\\]\nThen,  for $n\\geq 6$,\n\\begin{align*}\n(n-\\alpha) v_{n+1} \n&= (n-\\alpha)z_{n+1}  - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-\\alpha)x_{n+1}\\\\\n&=  (n-4+3\\alpha) z_n + 2(1-\\alpha)(n-1) x_n   - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-2+\\alpha)x_{n}\\\\\n& = (n-4+3\\alpha) v_n  + (n-4+3\\alpha) \\frac{[2(1-\\alpha) n +\\alpha]}{3-2\\alpha}x_n  \\\\\n& \\quad + 2(1-\\alpha)(n-1) x_n   - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-2+\\alpha)x_{n}\\\\\n&= (n-4+3\\alpha)v_n  -\\frac{2(1-\\alpha)}{3-2\\alpha} x_n.\n\\end{align*}\nEquivalently, \n\\[  v_{n+1} =\\frac{n-4+3\\alpha}{n-\\alpha} v_n  -\\frac{2(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)}.\\]\nApplying Lemma \\ref{Lemma1}, with $f_n = \\frac{(n-4+3\\alpha)}{(n-\\alpha)}$,  $g_n  = -\\frac{2(1-\\alpha)x_n}{(3-2\\alpha)(n-\\alpha)}$,  $a = 4-3\\alpha$ and $b= 3-2\\alpha$,  we get  $v_n = \\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$.\nThis proves part (i) of the proposition.\n\nPart (ii) is proved in a similar fashion.   By Theorem \\ref{Thm:Convg-ChPh}, $ Cov(A_n, C_n) = -\\frac{(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(3-2\\alpha)^2(5-4\\alpha)} n+ \\mbox{${\\mathcal O}$}(1).$ \nSince $\\mbox{${\\mathbb E}$}[A_nC_n]= Cov(A_n, C_n)+\\mu_n \\nu_n $, with $\\mu_n$ and $ \\nu_n $ found in Proposition \\ref{Prop:firstm}, we write   \n\\begin{eqnarray}\n\\mbox{${\\mathbb E}$}[A_nC_n] &=& \\frac{(1-\\alpha)^2}{2(3-2\\alpha)^2} \\ n^2 -\\frac{(1-\\alpha)(4-25\\alpha+16\\alpha^2)}{4(5-4\\alpha)(3-2\\alpha)^2} n  -\\frac{\\alpha (8- 17\\alpha +8\\alpha^2 )}{4(5-4\\alpha)(3-2\\alpha)^2}+ t_n. \\label{Eq:E[AnCn]}\n\\end{eqnarray}\nCombining   (\\ref{cov}) and \\eqref{Eq:E[AnCn]}, $t_n$ satisfies the recursion, \n\\begin{equation}\n(n-\\alpha)t_{n+1}-(n-5+3\\alpha)t_n = (2-\\alpha) z_n + (1-\\alpha)(n-1)y_n, \\quad n\\geq 6.\n\\end{equation}\nBy \\eqref{Exp:mu}, \\eqref{Exp:nu} and \\eqref{Eq:E[AnCn]},\n\\begin{eqnarray*}\nCov(A_n, C_n) &=& \\frac{-(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(5-4\\alpha)(3-2\\alpha)^2} \\ n -\\frac{ \\alpha (1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2}  + w_n -x_n y_n,\n\\end{eqnarray*}\nwhere $w_n = t_n  -\\dfrac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}$. Consider\n\\begin{align*}\n&(n-\\alpha) w_{n+1} \\\\\n&=(n-\\alpha) t_{n+1}  -(n-\\alpha)\\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}\\\\\n&= (n-5+3\\alpha) t_n +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n  \\\\\n& \\quad - (n-\\alpha) \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)} \\\\\n&= (n-5+3\\alpha) \\left( t_n -\\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\right) \\\\\n& \\quad +(n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\\\\n&\\quad +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n -(n-\\alpha)  \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}\\\\\n&= (n-5+3\\alpha)w_n +(n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\\\\n&\\quad +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n -(n-\\alpha)  \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}.\n\\end{align*}\nThus $w$ satisfies the following recursion for $n\\geq 6$\n\\begin{align*}\n(n-\\alpha) w_{n+1}  -  (n-5+3\\alpha)w_n &= {\\rm RHS},\n\\end{align*}\nwhere  RHS  is given by \n\\begin{align*}\n& (n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}  +(1-\\alpha) (n-1)y_n \\\\\n&\\quad -  (n-\\alpha)\\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)} +(2-\\alpha)z_n \\\\\n& = (2-\\alpha)v_n - \\frac{1-\\alpha}{3-2\\alpha} x_n.\n\\end{align*}\nWe omit the straightforward but tedious algebraic simplification steps. Hence,\n\\begin{equation}\\label{Rec:w}\n w_{n+1}  =\\frac{ n-5+3\\alpha}{n-\\alpha} w_n + (2-\\alpha) \\frac{v_n}{n-\\alpha} - \\frac{(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)}.\n\\end{equation}\nApplying Lemma \\ref{Lemma1} with $f_n = \\frac{n-5+3\\alpha}{n-\\alpha} $,  $g_n  = \\ (2-\\alpha) \\frac{v_n}{n-\\alpha} - \\frac{(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)} $,  $a = 5-4\\alpha$ and $b= 3-2\\alpha$,  we get  $v_n = \\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$. This proves part (ii).\n\nTo prove (iii), we let $\\tau_n^2 =\\mathrm{var}(A_n) = \\frac{(1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} n +\\mbox{${\\mathcal O}$}(1)$ by  Theorem \\ref{Thm:Convg-ChPh}. So, \n$ \\mbox{${\\mathbb E}$}[A_n^2] = \\nu_n^2 + \\tau_n^2= \\frac{(1-\\alpha)^2}{4(3-2\\alpha)^2} n^2 +\\mbox{${\\mathcal O}$}(n)$. We write \n\\[ \\mbox{${\\mathbb E}$}[A_n^2]  = \\frac{(1-\\alpha)^2}{4(3-2\\alpha)^2} n^2 + \\frac{2(1-\\alpha)(1+ 2\\alpha -2\\alpha^2)}{(5-4\\alpha)(3-2\\alpha)^2} n +   \\frac{\\alpha( 5-3\\alpha+\\alpha^2)}{4(3-2\\alpha) (5-4\\alpha)(7-4 \\alpha)} + s_n. \\]\n\nLet $u_n = s_n -  \\dfrac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha}$. Then,\n\\begin{align*}\n(n-\\alpha) u_{n+1} & = (n-\\alpha)  s_{n+1} -  \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)s_n + (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)[ s_n-\\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha}]  +  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} \\\\\n&\\quad +  (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)u_n +  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} \\\\\n&\\quad +  (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha}.\n\\end{align*}\nThus,\n\\begin{align*}\n&(n-\\alpha) u_{n+1} - (n-6+3\\alpha)u_n \\\\\n& =  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} -y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} +(2-\\alpha)x_n +  2(2-\\alpha)t_n\\\\\n& =  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha]y_n }{3-2\\alpha} -y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] [ (n-3+\\alpha) y_n + (2-\\alpha) x_n] }{3-2\\alpha}\\\\\n& \\quad +  2(2-\\alpha)w_n +(2-\\alpha)x_n + (2-\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{3-2\\alpha}\\\\\n& = y_n \\left\\{  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] }{3-2\\alpha} -1 +(2-\\alpha) \\frac{[2(1-\\alpha)n +\\alpha]}{3-2\\alpha}   - (n-3+\\alpha) \\frac{[(1-\\alpha)(n+1) + \\alpha]   }{3-2\\alpha} \\right\\} \\\\\n&  \\quad +2(2-\\alpha)w_n +(2-\\alpha)  x_n \\left\\{  \\frac{[(1-\\alpha)n +\\alpha] }{3-2\\alpha}  - \\frac{[(1-\\alpha)(n+1) + \\alpha]}{3-2\\alpha}+1\\right\\} \\\\\n& =2(2-\\alpha)w_n - y_n \\frac{\\alpha(2-\\alpha)}{3-2\\alpha} +\\frac{(2-\\alpha)^2}{3-2\\alpha}  x_n.\n\\end{align*}\nEquivalently, \n\\[ u_{n+1} =  \\frac{n-6+3\\alpha}{n-\\alpha} u_n  +  \\frac{\\alpha(2-\\alpha)}{(3-2\\alpha)} \\frac{y_n}{(n-\\alpha)} +\\frac{(2-\\alpha)^2}{(3-2\\alpha)}  \\frac{x_n}{(n-\\alpha)}.\\]\nApply Lemma \\ref{Lemma1} as in the proofs of parts (i) and (ii),  \nwe can conclude that $u_n =\\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$. This completes the proof of (iii) and hence the Proposition.\n\\end{proof}\n\n\\section*{Acknowledgements} \nThe work of Gursharn Kaur was supported by NUS Research Grant R-155-000-198-114, and that of Kwok Pui Choi by Singapore Ministry of Education Academic Research Fund\nR-155-000-188-114. We thank Chris Greenman and Ariadne Thompson for stimulating discussions on the Ford model. \n\n\n\n\\bibliographystyle{abbrv}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Dirac semi-metals, whose low energy physics can be described by three dimensional (3D) pseudorelativistic Dirac equation with the linear dispersion around the Fermi level \\cite{Burkov2011}, have attracted lots of attention in recent days, owing to their exotic physical properties \\cite{WangZJ-2012-Na3Bi,WangZJ-2013,li2010dynamical,potter2014quantum,Parameswaran2014} and large application potentials in the future \\cite{Abrikosov1998,liang2014,He2014}. Current studies mainly focus on two types of Dirac semi-metals with both inversion symmetry and time-reversal (TR) symmetry. One is achieved at the critical point of a topological phase transition. This type of Dirac semi-metal is not protected by any topology and can be gapped easily via small perturbations \\cite{sato2011-TlBiSSe,wu2013sudden,LiuJP2013}. In contrast, the other type is protected by the uniaxial rotation symmetry \\cite{ChenF2012}, so is quite stable. And according to even or odd parity of the states at the axis of $C_n$ rotation, the symmetry protected Dirac semi-metals can be further classified as two subclasses \\cite{YangBJ2014}. The first subclass has a single Dirac point (DP) at a time-reversal invariant momentum (TRIM) point on the rotation axis protected by the lattice symmetry \\cite{YoungSM2012,Steinberg2014}, while the second one possesses non-trivial band inversion and has a pair of DPs on the rotation axis away from the TRIM points. For the materials of the second subclass (such as Na$_3$Bi \\cite{WangZJ-2012-Na3Bi,liu2014discovery}, Cd$_3$As$_2$ \\cite{WangZJ-2013,liu2014stable,Borisenko2014,yi2014evidence,liang2014,JeonSJ2014,He2014,Narayanan2015}, and some charge balanced compounds \\cite{gibson20143d,du2014dirac}) the non-zero $\\mathbb{Z}_{2}$ number can be well defined at the corresponding two dimensional (2D) plane of the Brillouin zone (BZ) \\cite{Morimoto2014,Gorbar2015}. And due to the non-trivial topology, these stable Dirac semi-metals are regarded as a copy of Weyl semi-metals \\cite{YangBJ2014}. Thus, its Fermi arcs are observed on the specific surfaces \\cite{xu2015}, and a quantum oscillation of the topological property is expected to be achieved in the thin film with the change of thicknesses \\cite{WangZJ-2013}.\n\nIn spite of these successful progresses, the 3D Dirac semi-metal materials either take uncommon lattice structures or contain heavy atoms, which are not compatible with current semiconductor industry. On the other hand, the group \\uppercase\\expandafter{\\romannumeral4} elements, including C, Si, Ge, Sn and Pb, have been widely used in electronics and microelectronics. Generally, for some of the group \\uppercase\\expandafter{\\romannumeral4} elements, the diamond structure is one of the most stable 3D forms at ambient conditions. However, under specific experimental growth conditions, various allotropes with exotic phyiscal and chemical properties are discovered experimentally. For example, the new orthorhombic allotrope of silicon, Si$_{24}$, is found to be a semiconductor with a direct gap of 1.3 eV at the $\\Gamma$ point \\cite{kim2015Si24}; and the 2D forms of silicene \\cite{Seymur2009,Seymur2013-Sil,Seymure-sil-2014}, germanene \\cite{Daviala2014,Chensi-2014} and stanene \\cite{TangPz2014-stanene,Yong2013,zhu2015epitaxial} have been theoretically predicted to exist or experimentally grown on different substrates, which can be 2D topological insulators (TIs) and used as 2D field-effect transistors \\cite{tao2015silicene}.\n\nIn this article, by using \\emph{ab initio} density functional theory (DFT) with hybrid functional \\cite{heyd2003hybrid}, we predict new 3D metastable allotropes for Ge and Sn with staggered layered dumbbell (SLD) structure, named as germancite and stancite; and discover that they are stable Dirac semi-metals with a pair of gapless DPs on the rotation axis of $C_3$ protected by the lattice symmetry. Similar to the conventional Dirac semi-metals, such as Na$_3$Bi and Cd$_3$As$_2$, the topologically non-trivial Fermi arcs can be observed on the surfaces parallel to the rotation axis in the germancite and stancite. And via tuning the Fermi level, we can observe a Lifshitz transition in the momentum space. More importantly for future applications, the thin film of the germancite is found to be an intrinsic 2D TI, and the ultrahigh mobility and giant magnetoresistance can be expected in these compounds due to the 3D linear dispersion.\n\n\\section{Methods}\nThe calculations were carried out by using DFT with the projector augmented wave method \\cite{PhysRevB.50.17953,PhysRevB.59.1758}, as implemented in the Vienna \\textit{ab initio} simulation package \\cite{PhysRevB.54.11169}. Plane wave basis set with a kinetic energy cutoff of $\\mathrm{250~eV}$ and $\\mathrm{150~eV}$ was used for germancite and stancite respectively. The structure is allowed to fully relax until the residual forces are less than $1\\times 10^{-3}~\\mathrm{eV\/\\AA}$. The Monkhorst-Pack $k$ points are $9\\times 9\\times 9$. With the relaxed structure, the electronic calculation of germancite and stancite using hybrid functional HSE06 \\cite{heyd2003hybrid} has been done with and without SOC. The maximally localized Wannier functions \\cite{Mostofi2008685} are constructed to obtain the tight-binding Hamiltonian for the Green's function method \\cite{0305-4608-15-4-009}, which is used to calculate the surface electronic spectrum and surface states.\n\n\\section{Results}\nAs shown in Fig. \\ref{fig:1}, the germancite and stancite share the same rhombohedral crystal structure with the space group of $D_{3d}^6$ ($R\\bar{3}c$) \\cite{PhysRevB.90.085426}, which contains the spacial inversion symmetry and $C_3$ rotation symmetry along the trigonal axis (defined as $z$ axis). In one unit-cell, fourteen atoms bond with each others to form six atomic layers; and in each layer, one dumbbell site can be observed. To clearly visualize the SLD structure in the germancite and stancite, we plot the side view of the hexagonal lattice shown in Fig. \\ref{fig:1}(b) and the top view from (111) direction in Fig. \\ref{fig:1}(c). As the grey shadow shown, the layers containing dumbbell sites stack along (111) direction in the order of $\\cdots B\\bar{A}C\\bar{B}A\\bar{C}\\cdots$. The interlayer interaction is the covalent bonding between adjacent layers, whose bond lengths are almost equal to those of intralayer bonding (the difference is about $0.03$\\AA). Meanwhile, different from the diamond structure, the tetrahedral symmetry is absent in the SLD structure and the coupling here is not typical $sp^3$ hybridization. Furthermore, in order to test the structural stability, we calculate the phonon dispersion for the germancite and stancite shown in Fig. \\ref{fig:1}(e). It can be seen that the frequencies of all modes are positive over the whole Brillouin zone, which indicates that the SLD structures are thermodynamically stable. Furthermore, compared with the other experimentally discovered metastable allotropes of Ge and Sn \\cite{guloy2006guest,kiefer2010synthesis,PhysRevLett.110.085502,ja304380p,Ceylan2015407,PhysRevB.34.362}, the germancite and stancite share the same order of magnetite of the mass density and cohesive energies (see Supplemental Information for details), so we expect the germancite and stancite could be composed in the future experiments.\n\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure1.eps}}\n   \\caption{(Color online) (a) The unit cell of the SLD structure with three private lattice vectors set as \\textbf{a$_{1,2,3}$}. The balls in different colors stand for the same kind of atoms in different layers. (b) The side view and (c) top view of the SLD structure. The layers containing dumbbell (DB) structures are labelled. The letters ($A,B,C$) denote the positions of DB sites and the sign of bar is applied to distinguish between two trigonal lattices transformed to each other by inversion. As an example, the top view of two adjacent layers (marked by dashed blue lines) is shown. The DB structures are labeled by the grey shadow shown in the top view of a single layer, and the atoms in one DB structure are represented by grey balls. (d) The 3D Brillouin zone (BZ) of germancite and stancite.  The four inequivalent TRIM points are $\\Gamma$ (0,0,0), $L$ (0,$\\pi$,0), $F$ ($\\pi$,$\\pi$,0) and T ($\\pi$,$\\pi$,$\\pi$). The hexagon and square, connected to $\\Gamma$ by blue lines, show the 2D BZs projected to (111) and (2$\\bar{1}\\bar{1}$) surfaces respectively, and the high-symmetry $k$ points are labelled. (e) The phonon dispersion of germancite and stancite along high symmetry lines of 3D BZ.}\n\\label{fig:1}\n\\end{figure}\n\nThe calculated electronic structures of the germancite and stancite around the Fermi level are shown in Fig. \\ref{fig:2}(a), in which the solid lines and the yellow shadow stand for the bulk bands with and without spin-orbit coupling (SOC) respectively. It could be observed that: when the SOC effect is not included, the germancite is a conventional semi-metal whose bottom of the conduction bands and top of valence bands touch at the $\\Gamma$ point with the parabolic dispersions; while for stancite, it is a metal whose band touching at the $\\Gamma$ point is higher than the Fermi level. When the SOC effect is fully considered, our calculations indicate both germancite and stancite to be 3D Dirac semi-metals with a pair of DPs in the trigonal rotation axis (DP at (0,0,$\\pm k_{z0}$)). Therefore, the low energy physics of this kind of materials can be described by the 3D Dirac-type Hamiltonian. And the schematic band structure based on the effective $k\\cdot p$ model (see Supplemental Information for details) for germancite and stancite is shown in Fig. \\ref{fig:2}(c), in which the pair of 3D DPs is clear.\n\nTo understand the physical origin of the 3D gapless Dirac Fermions in the SLD structure, we plot the schematic diagram of the band evolution for the germancte and stancite in Fig. \\ref{fig:2}(b). In contrast to isotropic coupling in the diamond structure, the hybridizations in the layered SLD structure are anisotropic, in which the inter-layer couplings are relatively weaker than intra-layer couplings and the $p_z$ and $p_{x\\pm iy}$ states are splited.  Furthermore, based on our calculations, the kind of anisotropic coupling will further shift down the anti-bonding state of $s$ orbital which is even lower than the bonding states of the $p_{x\\pm iy}$ orbitals at the $\\Gamma$ point. So the band inversion occurs at the $\\Gamma$ point even without SOC effect, and the SOC herein just removes the degeneracy of $p_{x\\pm iy}$ orbitals around the Fermi level. In the 2D BZ which contains the $\\Gamma$ point and is perpendicular to the $\\Gamma$-$\\text{T}$ direction, the non zero $\\mathbb{Z}_{2}$ topological number can be well defined. On the other hand, the $C_{3v}$ symmetry along the $\\Gamma$-$\\text{T}$ line contains one 2D ($\\Lambda_{4}$) and two degenerate 1D ($\\Lambda_{5}$, $\\Lambda_{6}$) irreducible representations for its double space group \\cite{koster1963properties}. As shown in the Fig. \\ref{fig:2}(b), the two crossing bands at the Fermi level belong to $\\Lambda_{5}+\\Lambda_{6}$ and $\\Lambda_{4}$ respectively.  So there is no coupling and a TR pair of 3D DPs can be observed at the Fermi level along the $\\Gamma$-$\\text{T}$ direction.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure2.eps}}\n         \\caption{(Color online) (a) The band structures of germancite (left) and stancite (right) along high symmetry lines with the corresponding DOS around the Fermi level (dashed horizontal line). In the k-path $\\text{T}$-$\\Gamma$, the size of the red dots represents the contribution from the atomic $s$ and $p_z$ orbitals. The cyan dots are the Dirac points at (0,0,$k_{z0}$), where $k_{z0}\\approx 0.08 $ \\AA$^{-1}$ and $\\approx 0.18 $ \\AA$^{-1}$ respectively. Shaded regions denote the calculated energy spectrum without SOC. (b) Schematic diagrams of the lowest conduction bands and highest valence bands from the $\\text{T}$ point to the $\\Gamma$ point for germancite and stancite. The black lines present the SOC effect at the $\\text{T}$ and $\\Gamma$ point. Between them, the red and blue lines denote doubly degenerate bands belonging to different irreducible representations, where the solid\/dashed red line is for germancite\/stancite. And the crossing points (solid cyan dots) correspond to those gapless Dirac points in (a) respectively. (c) Schematic band dispersion based on the effective $k\\cdot p$ model for germancite and stancite. The $k_{\\perp}$ direction refers to any axis perpendicular to the $k_{z}$ direction in the momentum space and the color becomes warmer, as the energy increases.}\n\\label{fig:2}\n\\end{figure}\n\nDue to the non-trivial topology of 3D Dirac semi-metals, the projected 2D DPs and Fermi arcs are expected to be observed on some specific surfaces for the germancite and stancite. As shown in the Fig. \\ref{fig:3}, by using the surface Green's function method \\cite{0305-4608-15-4-009}, we study the electronic spectrum on the (111) and (2$\\bar{1}\\bar{1}$) surface whose BZs are perpendicular and parallel to the $\\Gamma$-$\\text{T}$ direction respectively. For the BZ of (111) surface, the pair of 3D DPs project to the $\\widetilde{\\Gamma}$ point as 2D Dirac cones (see Fig. \\ref{fig:3}(a) and (d)); when the coupling between two projected 2D DPs is considered, a finite band gap could be easily obtained. Furthermore, besides the projected Dirac cones, we also observe the trivial surface states in the germancite and stancite ($\\alpha_{1,2}$ states in the Fig. \\ref{fig:3}(a) and (d)) which mainly originate from the dangling bonds on the (111) surface.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure3.eps}}\n         \\caption{(Color online) The electronic spectrum on the $(111)$ surface and its corresponding Fermi surface for (a) germanctie and (d) stancite respectively. Two bulk DPs are projected to the $\\widetilde{\\Gamma}$ point. The electronic spectrum on the $(2\\bar{1}\\bar{1})$ surface and its corresponding Fermi surface for (b) germanctie and (e) stancite respectively. The cyan dots label the projected DPs and the yellow dot represents the band crossing at the $\\bar{\\Gamma}$ point. On the Fermi surface, the Fermi arcs connect two projected DPs (cyan dots). For stancite $(2\\bar{1}\\bar{1})$ surface, the constant-energy contour is at $\\epsilon_f-5.2$ meV, slightly away from the Fermi level, to distinguish the Fermi arcs. Stacking plots of constant-energy contours at different energies on its $(2\\bar{1}\\bar{1})$ surface of (c) germanctie and (f) stancite respectively. The Fermi level is set to be zero.}\n\\label{fig:3}\n\\end{figure}\n\nFor the (2$\\bar{1}\\bar{1}$) surface of the germancite and stancite, the electronic structures are quite different. Because the BZ of (2$\\bar{1}\\bar{1}$) surface is parallel to the $\\Gamma$-$\\text{T}$ direction, the pair of 3D DPs are projected to different points (0,0,$\\bar{\\pm k_{z0}}$) which are marked by the cyan dots in the Fig. \\ref{fig:3}(b) and (e). Between the projected DPs, a pair of the Fermi arcs could be observed clearly, which share the helical spin-texture and are not continuous at the projected points. This Fermi arcs originate from the non-trivial $\\mathbb{Z}_{2}$ topology in the Dirac semi-metals. On any 2D plane in the bulk whose BZ is perpendicular to the $\\Gamma$-$\\text{T}$ direction with $-k_{z0}<k_z<k_{z0}$, the $\\mathbb{Z}_{2}$ number is +1. Thus, in real space, the corresponding ``edge state\" exist on the boundary. In the moment space, the BZ of the ``edge state'' corresponds to the line parallel to $\\bar{Y}$-$\\bar{\\Gamma}$-$\\bar{Y}$ with $-\\bar{k_{z0}}$$<$$\\bar{k_z}$$<$$\\bar{k_{z0}}$, and its Fermi surface should be two points. After concluding all the contributions of planes with  $\\mathbb{Z}_{2}$=1, the Fermi surface becomes a pair of the Fermi arcs on the BZ of (2$\\bar{1}\\bar{1}$) surface which connect the projected DPs. At the same time, on the (2$\\bar{1}\\bar{1}$) surface, the other surface states contributed by the dangling bond also exist. Via tuning the Fermi level, we could observe the hybridization between the non-trivial surface states and Fermi arcs (see Fig. \\ref{fig:3}(c) and (f)), so a Lifshitz transition is found on the Fermi surface. Additionally, because the Fermi surface contours on the (2$\\bar{1}\\bar{1}$) surface contain roughly the same wave vector (see the yellow arrow in Fig. \\ref{fig:3} (e)), the charge density wave or surface reconstruction is possible to be observed here. However, the surface coupling will not break the TR symmetry or change the bulk topology, the pair of Fermi arcs always exist.\n\n\n\\section{Discussion and Conclusion}\nBecause of the compatibility with the traditional semiconductor devices and dissipationless edge transport, the realization of the quantum spin Hall (QSH) effect in the thin film of Ge attracts lots of attention recently. In a recent proposal \\cite{Zhang2013-Ge-TI}, the non-trivial topology of the 2D thin film is induced by the large build-in electric field in the semiconductor interface, which may be difficult to control in real experiments. However, owing to the non-trivial topology of the Dirac semi-metal, the germancite (111) film may provide an opportunity for obtaining the QSH insulator. As discussed above, the topologically non-trivial band inversion occurs around the $\\Gamma$ point in the germancite. So if we build 2D film with the proper thickness along the (111) direction, the band inversion may be restored at the $\\widetilde{\\Gamma}$ point and this thin film would become a QSH insulator. Figure \\ref{fig:4} shows the electronic structure for germancite (111) film with the thickness of 14.5 nm (i.e., 16 layer). A small band gap (5.6 meV) opens at the $\\widetilde{\\Gamma}$ point due to the quantum confinement. To confirm our prediction, we calculate its $\\mathbb{Z}_{2}$ number from the evolution of the Wannier charge centers (see Supplemental Information for details). It is found that the (111) thin film of the germancite is a 2D TI without applying the external electric field.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure4.eps}}\n         \\caption{(Color online) (a) Band structure of 16 layer germancite (111) film. The topologically nontrivial gap at the $\\widetilde{\\Gamma}$ can be seen in the inset. (b) Schematic device consisting of three germancite thin films with different thickness. The middle one is a QSH insulator, whereas the other two are topologically trivial. In the lower panel, the purple and green vectors stand for the spin-polarized current at the interfaces.}\n\\label{fig:4}\n\\end{figure}\nIn conclusion, from DFT calculations with the hybrid functional, we predict the germancite and stancite with SLD structure are stable topological Dirac semi-metals protected by the rotation symmetry. And it is found that the Fermi arcs coexist with the trivial surface states on the surface plane parallel to the rotation axis of $C_3$, and a Lifshitz transition is observed when the Fermi level is tuned. Furthermore, we discover the (111) thin film of the germancite is a 2D TI without applying the external electric field which is important for future applications. Experimentally, the metastable allotropes of germanium has been synthesized through the oxidation of Ge$_9^{4-}$ Zintl anions in ionic liquids under ambient conditions \\cite{guloy2006guest}. And owing to similar density and cohesive energy, we expect the germancite and stancite could be synthesized via the similar methods in the future.\n\n\n\\begin{acknowledgments}\nWe would like to thank S. Cahangirov and L. Xian for useful discussions. W.C. and W.D. acknowledge support from the Ministry of Science and Technology of China (Grant Nos. 2011CB606405 and 2011CB921901) and the National Natural Science Foundation of China (Grant No. 11334006). A.R. acknowledges financial support from the European Research Council Grant DYNamo (ERC-2010-AdG No. 267374) Spanish Grants (FIS2010-21282-C02-01), Grupos Consolidados UPV\/EHU del Gobierno Vasco (IT578-13) and EC project CRONOS (280879-2 CRONOS CP-FP7). P.T. and S.-C.Z. acknowledge NSF under grant numbers DMR-1305677 and FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA. The calculations were done on the ``Explorer 100'' cluster system of Tsinghua University.\n\\end{acknowledgments}\n\nW.C., P.T., W.D., S.-C.Z, and A.R. conceived and designed the project. W.C. and P.T. performed the $ab~initio$ calculations and theoretical analysis. P.T. and W.C. wrote the manuscript with the help from other authors. All authors contributed to the discussions. W.C. and P.T. contribute equally to this work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe idea to use noncommutative coordinates is quite old - it goes\nback to Heisenberg and was firstly formalized by Snyder in\n\\cite{snyder}. Recently, however,  there were  found new formal\narguments based mainly on Quantum Gravity \\cite{2}, \\cite{2a} and\nString Theory models \\cite{recent}, \\cite{string1}, indicating that\nspace-time at Planck scale  should be noncommutative, i.e. it should\nhave a quantum nature. On the other side, the main reason for such\nconsiderations follows from many phenomenological considerations,\nwhich state that relativistic space-time symmetries should be\nmodified (deformed) at Planck scale, while  the classical Poincare\ninvariance still remains valid at larger distances\n\\cite{1a}, \\cite{1d}.\n\n\nIt is well-known that  a proper modification of the Poincare and\nGalilei Hopf algebras can be realized in the framework of Quantum\nGroups \\cite{qg1}, \\cite{qg3}. Hence, in accordance with the\nHopf-algebraic classification  of all deformations of relativistic\nand nonrelativistic symmetries (see \\cite{class1}, \\cite{class2}),\none can distinguish three\ntypes of quantum spaces \\cite{class1}, \\cite{class2} (for details see also \\cite{nnh}):\\\\\n\\\\\n{ \\bf 1)} Canonical ($\\theta^{\\mu\\nu}$-deformed) type of quantum space \\cite{oeckl}-\\cite{dasz1}\n\\begin{equation}\n[\\;{ x}_{\\mu},{ x}_{\\nu}\\;] = i\\theta_{\\mu\\nu}\\;, \\label{noncomm}\n\\end{equation}\n\\\\\n{ \\bf 2)} Lie-algebraic modification of classical space-time \\cite{chiqft}-\\cite{lie1}\n\\begin{equation}\n[\\;{ x}_{\\mu},{ x}_{\\nu}\\;] = i\\theta_{\\mu\\nu}^{\\rho}{ x}_{\\rho}\\;,\n\\label{noncomm1}\n\\end{equation}\nand\\\\\n\\\\\n{ \\bf 3)} Quadratic deformation of Minkowski and Galilei  spaces \\cite{chiqft}, \\cite{chiqft1}, \\cite{lie1}-\\cite{paolo}\n\\begin{equation}\n[\\;{ x}_{\\mu},{ x}_{\\nu}\\;] = i\\theta_{\\mu\\nu}^{\\rho\\tau}{\nx}_{\\rho}{ x}_{\\tau}\\;, \\label{noncomm2}\n\\end{equation}\nwith coefficients $\\theta_{\\mu\\nu}$, $\\theta_{\\mu\\nu}^{\\rho}$ and  $\\theta_{\\mu\\nu}^{\\rho\\tau}$ being constants.\\\\\n\\\\\nBesides, it has been demonstrated in \\cite{nnh}, that in the case of\nso-called N-enlarged Newton-Hooke Hopf algebras\n$\\,{\\mathcal U}^{(N)}_0({ NH}_{\\pm})$ the twist deformation\nprovides the new  space-time noncommutativity of the\nform\\footnote{$x_0 = ct$.},\\footnote{ The discussed space-times have been  defined as the quantum\nrepresentation spaces, so-called Hopf modules (see e.g. \\cite{oeckl}, \\cite{chi}), for quantum N-enlarged\nNewton-Hooke Hopf algebras.}\n\\begin{equation}\n{ \\bf 4)}\\;\\;\\;\\;\\;\\;\\;\\;\\;[\\;t,{ x}_{i}\\;] = 0\\;\\;\\;,\\;\\;\\; [\\;{ x}_{i},{ x}_{j}\\;] =\nif_{\\pm}\\left(\\frac{t}{\\tau}\\right)\\theta_{ij}(x\n\\;, \\label{nhspace}\n\\end{equation}\nwith time-dependent  functions\n$$f_+\\left(\\frac{t}{\\tau}\\right) =\nf\\left(\\sinh\\left(\\frac{t}{\\tau}\\right),\\cosh\\left(\\frac{t}{\\tau}\\right)\\right)\\;\\;\\;,\\;\\;\\;\nf_-\\left(\\frac{t}{\\tau}\\right) =\nf\\left(\\sin\\left(\\frac{t}{\\tau}\\right),\\cos\\left(\\frac{t}{\\tau}\\right)\\right)\\;,$$\n$\\theta_{ij}(x) \\sim \\theta_{ij} = {\\rm const}$ or\n$\\theta_{ij}(x) \\sim \\theta_{ij}^{k}x_k$ and  $\\tau$ denoting the time scale parameter\n -  the cosmological constant. It should be also noted that different relations  between all mentioned above quantum spaces ({\\bf 1)}, { \\bf 2)}, { \\bf 3)}\nand { \\bf 4)}) have been summarized in paper \\cite{nnh}.\n\n\nRecently, there appeared a lot of papers dealing with classical\n(\\cite{deri}-\\cite{daszwal}) and quantum (\\cite{qm1}-\\cite{oscy})\nmechanics, Doubly Special Relativity frameworks (\\cite{dsr1a},\n\\cite{dsr1b}), statistical physics (\\cite{maggiore}, \\cite{rama})\nand field theoretical models (see e.g. \\cite{przeglad}), defined on\nquantum\n space-times (\\ref{noncomm}),\n (\\ref{noncomm1})\\footnote{For earlier studies see \\cite{lukiluk1} and\n\\cite{lukiluk2}.}. Particulary, there was\n investigated the impact of the mentioned\nabove   deformations  on dynamics of basic classical and quantum\nsystems. Consequently, in  papers \\cite{romero}, \\cite{romero1}, the\nauthors considered  classical particle moving in central\ngravitational field defined on canonically deformed space-time\n(\\ref{noncomm}). They  demonstrated, that in such a case there\nis generated Coriolis force acting additionally on the moving\nparticle. Besides, in articles \\cite{lodzianieosc}, \\cite{romero}\nand \\cite{oscy} there was analyzed classical and quantum oscillator\nmodel formulated on canonically and Lie-algebraically deformed\nspace-time respectively. Particulary, there has been found its\ndeformed energy spectrum as well as the corresponding equation of\nmotion. Interesting results have been also obtained in two papers\n\\cite{qm1} and \\cite{qm2} concerning the  hydrogen atom model defined\non quantum space (\\ref{noncomm}). Besides,\nit should be noted that  there appeared article \\cite{toporzelek},\nwhich provides the link between Pioneer anomaly phenomena\n\\cite{piophen} and classical mechanics defined on $\\kappa$-Galilei\nquantum space. Precisely,  there has been demonstrated that\nadditional force term acting on moving satellite can be identified\nwith the force generated  by space-time noncommutativity. The value\nof deformation parameter $\\kappa$ has been fixed  by comparison of\nobtained theoretical results with observational data.\n\nUnfortunately, in all mentioned above articles there were analyzed\nonly the one-particle classical and quantum dynamics in\nthe field of forces. Here, we extend  such a kind of investigations\nto the quantum mechanics of many particles, which move in the\nmodified twist-deformed N-enlarged Newton-Hooke space-time\n\\begin{equation}\n[\\,t,x_{iA}\\,] = 0\\;\\;\\;,\\;\\;\\;[\\,x_{iA},x_{jB}\\,] =\nif(t) = if_{\\pm}\\left(\\frac{t}{\\tau}\\right)\\theta_{ij}\\;\\;\\;;\\;\\;\\;i, j = 1,2, 3\\;, \\label{canamm}\n\\end{equation}\nwith  indices $A, B = 1,2, \\ldots ,M$ labeling the particle.\nFurther, we indicate that as in the case of one-particle quantum system\nthere appeared additional dynamical terms generated by space-time\nnoncommutativity. Of course,  in the case of Coulomb potential for $M=1$ and $f(t) = \\theta_{ij}$ our results become the same\nas the ones obtained in \\cite{qm1} and \\cite{qm2}  respectively.\n\nThe motivations for present studies are manyfold. First of all we\nextend in natural way the  results for quantum one-particle model to the much more complicated\nmany-particle system. Secondly, such investigations permit to\nanalyze the deformations of wide class of physical models such as, for\nexample, the noncommutative many-electron atoms or noncommutative many-atomic molecules \\cite{24}, \\cite{24a}. Finally, it gives a starting\npoint for the construction of Dirac quantum mechanics for\nmany particles defined on the relativistic counterpart of modified space-time (\\ref{canamm}).\n\nThe paper is organized as follows. In Sect. 2 we recall basic facts\nconcerning the twisted N-enlarged Newton-Hooke   space-times\nprovided in article \\cite{nnh}. The third section is devoted to the short review of quantum mechanics of many particles moving in\ncommutative (classical) space. In Sect. 4 we construct the quantum\nmany-particle model defined on modified N-enlarged Newton-Hooke space-time\n(\\ref{canamm}).  The\nfinal remarks are presented in the last section.\n\n\n\\section{Twisted N-enlarged Newton-Hooke space-times}\n\nIn this section we turn to the twisted N-enlarged Newton-Hooke space-times  equipped with two spatial directions\ncommuting to classical time, i.e. we consider  spaces of the form \\cite{nnh}\n\\begin{equation}\n[\\;t,\\hat{x}_{i}\\;] =[\\;\\hat{x}_{1},\\hat{x}_{3}\\;] = [\\;\\hat{x}_{2},\\hat{x}_{3}\\;] =\n0\\;\\;\\;,\\;\\;\\; [\\;\\hat{x}_{1},\\hat{x}_{2}\\;] =\nif({t})\\;\\;;\\;\\;i=1,2,3\n\\;. \\label{spaces}\n\\end{equation}\nAs it was already mentioned in Introduction  such a kind of quantum spaces provides the most general deformation of nonrelativistic systems. It should be noted, however,\nthat this type of noncommutativity  has  been  constructed explicitly  only in the case of 6-enlarged Newton-Hooke Hopf algebra, with\n\\begin{eqnarray}\nf({t})&=&f_{\\kappa_1}({t}) =\nf_{\\pm,\\kappa_1}\\left(\\frac{t}{\\tau}\\right) = \\kappa_1\\,C_{\\pm}^2\n\\left(\\frac{t}{\\tau}\\right)\\;, \\nonumber\\\\\nf({t})&=&f_{\\kappa_2}({t}) =\nf_{\\pm,\\kappa_2}\\left(\\frac{t}{\\tau}\\right) =\\kappa_2\\tau\\, C_{\\pm}\n\\left(\\frac{t}{\\tau}\\right)S_{\\pm} \\left(\\frac{t}{\\tau}\\right) \\;,\n\\nonumber\\\\\n&~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\nonumber\\\\\n&~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\cdot \\nonumber\\\\\n&~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\cdot \\label{w2}\\\\\n&~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\cdot \\nonumber\\\\\n&~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\nonumber\\\\\nf({t})&=&\nf_{\\kappa_{35}}\\left(\\frac{t}{\\tau}\\right) = 86400\\kappa_{35}\\,\\tau^{11}\n\\left(\\pm C_{\\pm} \\left(\\frac{t}{\\tau}\\right)  \\mp \\frac{1}{24}\\left(\\frac{t}{\\tau}\\right)^4 - \\frac{1}{2}\n\\left(\\frac{t}{\\tau}\\right)^2 \\mp 1\\right) \\,\\times \\nonumber\\\\\n&~~&~~~~~~~~~~~~~~~~\\times~\\;\\left(S_{\\pm} \\left(\\frac{t}{\\tau}\\right)  \\mp \\frac{1}{6}\\left(\\frac{t}{\\tau}\\right)^3 - \\frac{t}{\\tau}\\right)\\;,\n\\nonumber\\\\\nf({t})&=&\nf_{\\kappa_{36}}\\left(\\frac{t}{\\tau}\\right) =\n518400\\kappa_{36}\\,\\tau^{12}\\left(\\pm C_{\\pm} \\left(\\frac{t}{\\tau}\\right)  \\mp \\frac{1}{24}\\left(\\frac{t}{\\tau}\\right)^4 - \\frac{1}{2}\n\\left(\\frac{t}{\\tau}\\right)^2 \\mp 1\\right)^2\\;, \\nonumber\n\\end{eqnarray}\nand\n$$C_{+\/-} \\left(\\frac{t}{\\tau}\\right) = \\cosh\/\\cos \\left(\\frac{t}{\\tau}\\right)\\;\\;\\;{\\rm and}\\;\\;\\;\nS_{+\/-} \\left(\\frac{t}{\\tau}\\right) = \\sinh\/\\sin\n\\left(\\frac{t}{\\tau}\\right) \\;.$$\nBesides, one can easily check that in $\\tau$ approaching infinity limit the above quantum spaces reproduce the canonical (\\ref{noncomm}),\nLie-algebraic (\\ref{noncomm1}) and quadratic (\\ref{noncomm2})  type of\nspace-time noncommutativity, i.e. for $\\tau \\to \\infty$ we get\n\\begin{eqnarray}\nf_{\\kappa_1}({t}) &=& \\kappa_1\\;,\\nonumber\\\\\nf_{\\kappa_2}({t}) &=& \\kappa_2\\,t\\;,\\nonumber\\\\\n&\\cdot& \\nonumber\\\\\n&\\cdot& \\label{qqw2}\\\\\n&\\cdot& \\nonumber\\\\\nf_{\\kappa_{35}}({t}) &=& \\kappa_{35}\\,t^{11}\\;, \\nonumber\\\\\nf_{\\kappa_{36}}({t}) &=& \\kappa_{36}\\,t^{12}\\;. \\nonumber\n\\end{eqnarray}\nOf course, for all  parameters $\\kappa_a$ $(a=1,...,36)$ running to zero the above deformations disappear.\n\nFinally, let us notice that the spaces (\\ref{spaces}) can be extended to the case of multiparticle systems as follows\n\\begin{equation}\n[\\;t,\\hat{x}_{iA}\\;] =[\\;\\hat{x}_{1A},\\hat{x}_{3B}\\;] = [\\;\\hat{x}_{2A},\\hat{x}_{3B}\\;] =\n0\\;\\;\\;,\\;\\;\\; [\\;\\hat{x}_{1A},\\hat{x}_{2B}\\;] =\nif({t})\\delta_{AB}\\;\\;;\\;\\;i=1,2,3\n\\;, \\label{spaces300}\n\\end{equation}\nwith $A, B = 1,2, \\ldots ,M$. It should be also observed that  such an extension (blind in $A$, $B$ indecies) is compatible with canonical deformation (\\ref{noncomm}).\nPrecisely, in $\\tau$ approaching infinity limit the space (\\ref{spaces300}) with function $f(t) = f_{\\pm,\\kappa_1}\\left(\\frac{t}{\\tau}\\right) = \\kappa_1\\,C_{\\pm}^2\n\\left(\\frac{t}{\\tau}\\right)$ passes into the well-known multiparticle canonical space-time proposed in \\cite{fiore}\\footnote{It should be noted that  modification of the relation (\\ref{spacesfiore}) (blind in $A$, $B$ indieces as well) is in accordance with the formal arguments proposed in \\cite{fiore}. Precisely, the relations (\\ref{spacesfiore}) are constructed with adopt so-called braided tensor algebra procedure, dictated by structure of quantum $R$-matrix for canonical deformation \\cite{qg1}, \\cite{qg3}. We would like to mention, however, that in \\cite{fiore} an\nerroneous conclusion has been stated that based on such a twisted symmetry the noncommutative\nquantum field theory (QFT) on the quantum space satisfying the relation in (\\ref{noncomm}), and the usual commutative QFT are\nidentical. This conclusion has been reached by a misuse of the proper transformation properties of the\nfields in the corresponding noncommutative space time \\cite{chaia}.} (see also \\cite{odynswiatowid})\n\\begin{equation}\n[\\;t,\\hat{x}_{iA}\\;] =[\\;\\hat{x}_{1A},\\hat{x}_{3B}\\;] = [\\;\\hat{x}_{2A},\\hat{x}_{3B}\\;] =\n0\\;\\;\\;,\\;\\;\\; [\\;\\hat{x}_{1A},\\hat{x}_{2B}\\;] =\ni\\kappa_1\\delta_{AB}\n\\;. \\label{spacesfiore}\n\\end{equation}\n\n\n\n\\section{Quantum mechanics of many particles moving in commutative space-time  - short review}\n\nIn this section we recall basic facts concerning the  many-particle quantum mechanics defined on commutative space. First of all, we start with the following hamiltonian function\nfor M interacting particles\n\\begin{equation}\nH(\\bar{{p}}_1, \\ldots ,\\bar{{p}}_M;\\bar{{r}}_1, \\ldots ,\\bar{{r}}_M) =\n\\sum_{A=1}^{M}\\left(\\frac{\\bar{p}_A^2}{2m_A}  +V_A(\\bar{r}_A)\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}(\\bar{r}_A,\\bar{r}_B)\\;, \\label{grom0}\n\\end{equation}\nwhere $\\bar{{p}}_{A}=[\\;{{x}_{1A},{p}_{2A},{p}_{3A}}\\;]$ and $\\bar{{r}}_{A}=[\\;{{x}_{1A},{x}_{2A},{x}_{3A}}\\;]$ denote the positions and momenta operators such that\n\\begin{equation}\n[\\;x_{iA},x_{jB}\\;] = 0 =[\\;p_{iA},p_{jB}\\;]\\;\\;\\;,\\;\\;\\; [\\;x_{iA},p_{jB}\\;]\n={i\\hbar}\\delta_{ij}\\delta_{AB}\\;. \\label{classpoisson}\n\\end{equation}\nBesides, present in the above formula symbol $V_A(\\bar{r}_A)$ denotes the single-particle stationary potential while $V_{AB}(\\bar{r}_A,\\bar{r}_B)$ describes the correlations of particles. Hence, the corresponding Schroedinger equation in so-called  position representation  looks as follows\\footnote{$p_{iA} = -i\\hbar \\frac{\\partial}{\\partial x_{iA}}$.}\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) &=& \\left[\\;\\sum_{A=1}^{M}\\left(\\frac{1}{2m_A} \\Delta_A +V_A(\\bar{r}_A)\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}(\\bar{r}_A,\\bar{r}_B)\n\\;\\right] \\times \\nonumber \\\\\n&\\times& \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;,\\label{grom1}\n\\end{eqnarray}\n and, if one neglects\nthe potential functions  $V_{AB}(\\bar{r}_A,\\bar{r}_B)$ then, it takes the form\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\left[\\;\\sum_{A=1}^{M}\\left(\\frac{1}{2m_A} \\Delta_A +V_A(\\bar{r}_A)\\right)\n\\;\\right]  \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;.\\label{grom2}\n\\end{eqnarray}\nMoreover, it is easy to see that the solution of equation (\\ref{grom2}) is given by\n\\begin{eqnarray}\n \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\psi_1(\\bar{r}_1,t) \\cdots \\psi_M(\\bar{r}_M,t)\\;,\\label{grom3}\n\\end{eqnarray}\nwith wave functions $\\psi_A(\\bar{r},t)$ satisfying  the standard (one-particle) differential equation\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t} \\psi_A(\\bar{r},t) = \\left(\\frac{1}{2m_A} \\Delta +V_A(\\bar{r})\\right)\\psi_A(\\bar{r},t)\\;.\\label{grom4}\n\\end{eqnarray}\n\n Usually, the potentials $V_A(\\bar{r}_A)$ and $V_{AB}(\\bar{r}_A,\\bar{r}_B)$ remain spherically symmetric, i.e. they depend on the length of vector $\\bar{r}$ and the relative positions of particles respectivel\n\\begin{eqnarray}\nV_A(\\bar{r}_A) = V_A(|\\bar{r}_A|)\\;\\;\\;,\\;\\;\\;V_{AB}(\\bar{r}_A,\\bar{r}_B) = V_{AB}(|\\bar{r}_A - \\bar{r}_B|)\\;.\\label{grom5}\n\\end{eqnarray}\nSuch a situation appears (for example) in the case of M electrons moving in the Coulomb field   of single nucleon with charge $Ze$ and interacting each other also by means\nCoulomb potential; then, we have\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) &=& \\left[\\;\\sum_{A=1}^{M}\\left(\\frac{1}{2m_A} \\Delta_A -\\frac{Ze^2}{{|\\bar{r}_A|}}\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}\\frac{e^2}{{|\\bar{r}_A - \\bar{r}_B|}}\n\\;\\right] \\times \\nonumber \\\\\n&\\times& \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;.\\label{grom6}\n\\end{eqnarray}\n~~~~~Finally, it should be noted that the function\n\\begin{eqnarray}\n \\rho(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = |\\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)|\\;,\\label{grom7}\n\\end{eqnarray}\ncan be interpreted as the density of  probability of finding first particle at point $\\bar{r}_1$, second - at point $\\bar{r}_2$, etc. in time-moment $t$. Besides, the average value of quantum mechanical observable ${A}$ is defined as follows\n\\begin{eqnarray}\n<{A}> =  \\int d^3{r}_1 \\ldots d^3{r}_M\\; \\psi^*(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;{A}\\;\\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;.\\label{grom8}\n\\end{eqnarray}\n\n\\section{Many-body quantum mechanics for twisted N-enla-rged Newton-Hooke space-times}\n~~\\\\\n~~\\\\\n~~\\\\\nLet us now turn to the main aim of our investigations - to the quantum mechanical  model of many particles defined on quantum space-times (\\ref{spaces300}).\nIn first step of our construction we extend the described in second section spaces to the whole algebra of momentum and position operators as follows\n\\begin{eqnarray}\n&&[\\;\\hat{ x}_{1A},\\hat{ x}_{2B}\\;] = if_{\\kappa_a}({t})\\delta_{AB}\\;\\;\\;,\\;\\;\\;[\\;\\hat{ x}_{1A},\\hat{ x}_{3B}\\;] =\n [\\;\\hat{ x}_{2A},\\hat{ x}_{3B}\\;] =\n [\\;\\hat{ p}_{iA},\\hat{ p}_{jB}\\;] =0\\;,\\label{phasespaces1}\\\\\n&&~~~~~~~~~~~~~~~~~[\\;\\hat{ x}_{iA},\\hat{ p}_{jB}\\;] = {i\\hbar}\\delta_{ij}\\delta_{AB}\\;\\;;\\;\\;i,j=1,2,3\\;. \\label{phasespaces2}\n\\end{eqnarray}\nOne can check that relations (\\ref{phasespaces1}), (\\ref{phasespaces2}) satisfy the Jacobi identity and for deformation parameters\n$\\kappa_a$ approaching zero become classical. \\\\\nNext, by analogy to the commutative case (see formula (\\ref{grom0})) we define the following multi-particle hamiltonian operator\n\\begin{eqnarray}\nH(\\bar{\\hat{p}}_1, \\ldots ,\\bar{\\hat{p}}_M;\\bar{\\hat{r}}_1, \\ldots ,\\bar{\\hat{r}}_M) =\n\\sum_{A=1}^{M}\\left(\\frac{\\bar{\\hat{p}}_A^2}{2m_A}  +V_A(\\bar{\\hat{r}}_A)\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)\\;,\n\\label{gromek1}\n\\end{eqnarray}\nwith $\\bar{\\hat{p}}_{A}=[\\;{\\hat{x}_{1A},\\hat{p}_{2A},\\hat{p}_{3A}}\\;]$ and $\\bar{\\hat{r}}_{A}=[\\;{\\hat{x}_{1A},\\hat{x}_{2A},\\hat{x}_{3A}}\\;]$. \\\\\nIn order to analyze the above system we represent the\nnoncommutative operators $({\\hat x}_{iA}, {\\hat p}_{iA})$ by classical\nones $({ x}_{iA}, { p}_{iA})$ as  (see e.g.\n\\cite{lodzianieosc}, \\cite{lukiluk2})\n\\begin{eqnarray}\n&~~&{\\hat x}_{1A} = { x}_{1A} - \\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{2A}\\;\\;\\;,\\;\\;\\;{\\hat x}_{2A} = { x}_{2A} +\\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{1A}\\;,\\\\\n&~~&~~~~~~~~~~~~~~~~ {\\hat x}_{3A}= x_{3A} \\;\\;\\;,\\;\\;\\; {\\hat p}_{iA}=\np_{iA}\\;. \\label{rep}\n\\end{eqnarray}\nThen, the  hamiltonian (\\ref{gromek1}) takes the form\n\\begin{eqnarray}\n&~~&H(\\bar{p}_1, \\ldots ,\\bar{p}_M;\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\nonumber \\\\\n&=&\\sum_{A=1}^{M}\\left[\\;\\frac{\\bar{{p}}_A^2}{2m_A}  +\nV_A\\left(\\bar{\\hat{r}}_A = \\left({ x}_{1A} - \\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{2A},{ x}_{2A} +\\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{1A},x_{3A}\\right)\\right)\\right. + \\nonumber \\\\\n&~~&~~~~+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}\\left(\\bar{\\hat{r}}_A = \\left({ x}_{1A} - \\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{2A},{ x}_{2A} +\\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{1A},x_{3A}\\right), \\right.\\label{gromek2}\\\\\n&,&\\left.\\left.\\bar{\\hat{r}}_B = \\left({ x}_{1B} - \\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{2B},{ x}_{2B} +\\frac{1}{2\\hbar}f_{\\kappa_a}(t)\np_{1B},x_{3B}\\right)\\right)\\;\\right]\n\\;,\\nonumber\n\\end{eqnarray}\nand, consequently, the corresponding Schroedinger equation in the position representation looks as follows\n\\begin{eqnarray}\n&~~&i\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\nonumber \\\\\n&=&\\left\\{\\;\\sum_{A=1}^{M}\\left[\\;\\frac{1}{2m_A}\\Delta_A  +\nV_A\\left(\\bar{\\hat{r}}_A = \\left({ x}_{1A} + \\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{2A}},{ x}_{2A} -\\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{1A}},x_{3A}\\right)\\right)\\right. \\right.+ \\nonumber \\\\\n&~~&~~~~~~~+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}\\left(\\bar{\\hat{r}}_A = \\left({ x}_{1A} + \\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{2A}},{ x}_{2A} -\\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{1A}},x_{3A}\\right), \\right.\\label{gromek2}\\\\\n&,&\\left.\\left. \\left.\\bar{\\hat{r}}_B = \\left({ x}_{1B} + \\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{2B}},{ x}_{2B} -\\frac{i}{2}f_{\\kappa_a}(t)\n{\\partial_{1B}},x_{3B}\\right)\\right)\\;\\right] \\;\\right\\}\\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;.\\nonumber\n\\end{eqnarray}\nFurther, we  expand the hamiltonian function (\\ref{gromek2}) in Taylor series up to the terms linear in deformation parameter $\\kappa_a$, i.e. to the terms linear in\nfunction $f_{\\kappa_a}(t)$; then, we have\\footnote{We denote by ${\\cal O}(\\kappa_a)$ the higher order terms in deformation parameter $\\kappa_a$.}\n\\begin{eqnarray}\n&~~&H(\\bar{p}_1, \\ldots ,\\bar{p}_M;\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\nonumber \\\\\n&=&\\sum_{A=1}^{M}\\left(\\frac{\\bar{{p}}_A^2}{2m_A}  +V_A(\\bar{{r}}_A)\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}(\\bar{{r}}_A,\\bar{{r}}_B) +\\nonumber \\\\\n&~~&~~~~+\\left[\\;\\sum_{A=1}^{M}\\left(\\left.-\\frac{\\partial V_A(\\bar{\\hat{r}}_A)}{\\partial \\hat{x}_{1A}}\\cdot \\frac{1}{2 \\hbar}p_{2A}\\right.\\right.\n+\\left.\\left.\\frac{\\partial V_A(\\bar{\\hat{r}}_A)}{\\partial \\hat{x}_{2A}}\\cdot \\frac{1}{2 \\hbar}p_{1A}\\right.\\right)\\right.+\\nonumber\\\\\n&+&\\frac{1}{2}\\sum_{A\\ne B}\\left.\\left.\\left(-\\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{1A}}\\cdot \\frac{1}{2 \\hbar}p_{2A}\\right.\n+ \\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{2A}}\\cdot\\frac{1}{2 \\hbar}p_{1A}\\right.\\right. + \\label{gromek4}\\\\\n&-&\\left.\\left.\\left.\\left.\\left.\\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{1B}}\\cdot \\frac{1}{2 \\hbar}p_{2B}\\right.\n+ \\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{2B}}\\cdot \\frac{1}{2 \\hbar}p_{1B}\\right.\\right)\\;\\right]\\right|_{f_{\\kappa_a}(t)=0}\\cdot f_{\\kappa_a}(t) +\\nonumber \\\\\n&+& {\\cal O}(\\kappa_a) \\;, \\nonumber\n\\end{eqnarray}\nwith the corresponding  wave equation  given by\n\\begin{eqnarray}\n&~~&i\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = \\nonumber \\\\\n&=&\\left\\{\\;\\sum_{A=1}^{M}\\left(\\frac{1}{2m_A}\\Delta_A  +V_A(\\bar{{r}}_A)\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}V_{AB}(\\bar{{r}}_A,\\bar{{r}}_B)\\right. +\\nonumber \\\\\n&~~&~~~~+\\left[\\;\\sum_{A=1}^{M}\\left(\\left.\\frac{\\partial V_A(\\bar{\\hat{r}}_A)}{\\partial \\hat{x}_{1A}}\\cdot \\frac{i}{2 }\\partial_{2A}\\right.\\right.\n\\left.\\left.-\\frac{\\partial V_A(\\bar{\\hat{r}}_A)}{\\partial \\hat{x}_{2A}}\\cdot \\frac{i}{2 }\\partial_{1A}\\right.\\right)\\right.+\\label{gromek5}\\\\\n&+&\\frac{1}{2}\\sum_{A\\ne B}\\left.\\left.\\left(\\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{1A}}\\cdot \\frac{i}{2 }\\partial_{2A}\\right.\n- \\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{2A}}\\cdot \\frac{i}{2}\\partial_{1A}\\right.\\right. + \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n&+&\\left.\\left.\\left.\\left.\\left.\\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{1B}}\\cdot \\frac{i}{2}\\partial_{2B}\\right.\n -\\frac{\\partial V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)}{\\partial \\hat{x}_{2B}}\\cdot \\frac{i}{2}\\partial_{1B}\\right.\\right)\\;\\right]\\right|_{f_{\\kappa_a}(t)=0}\\cdot f_{\\kappa_a}(t) +\\nonumber \\\\\n&+& \\left.{\\cal O}(\\kappa_a) \\;\\right\\}\\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;. \\nonumber\n\\end{eqnarray}\nConsequently, we see that space-time noncommutativity (\\ref{spaces}) generates in the hamiltonian (\\ref{gromek1}) two types of additional dynamical terms. First of them arises from\nthe single-particle potential $V_A(\\bar{\\hat{r}}_A)$ while the second one corresponds to the correlations $V_{AB}(\\bar{\\hat{r}}_A,\\bar{\\hat{r}}_B)$. Of course, for deformation parameters $\\kappa_a$ approaching zero all additional \"potential\" terms  disappear.\n\nLet us now turn to the mentioned in pervious section the system of M particles moving \"in\" and interacting \"by\" the Coulomb potential. Then, in accordance with formulas (\\ref{gromek4}) and (\\ref{gromek5}) the corresponding hamiltonian function as well as the corresponding Schroedinger equation take the form\n\\begin{eqnarray}\nH(\\bar{p}_1, \\ldots ,\\bar{p}_M;\\bar{r}_1, \\ldots ,\\bar{r}_M,t) &=& \\sum_{A=1}^{M}\\left(\\frac{\\bar{{p}}_A^2}{2m_A}  -\\frac{Ze^2}{|{\\bar{{r}}_A}|}\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}\\frac{e^2}{{|\\bar{{r}}_A - \\bar{{r}}_B|}} + \\nonumber\\\\\n&~~&~~~~- \\sum_{A=1}^{M} \\frac{Ze^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A|^3}}\\cdot L_{3A} +\n\\label{gromek6}\\\\\n&+&\\frac{1}{2}\\sum_{A\\ne B}  \\frac{e^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A-\\bar{r}_B|^3}} \\cdot \\left(L_{3B}+L_{3A}\\right) + \\nonumber\\\\\n&-&\\frac{1}{2}\\sum_{A\\ne B}  \\frac{e^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A-\\bar{r}_B|^3}} \\cdot \\left(G_{AB}+G_{BA}\\right) + {\\cal O}(\\kappa_a)\\;,\\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t} \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) &=& \\left[\\;\\sum_{A=1}^{M}\\left(\\frac{1}{2m_A}\\Delta_A  -\\frac{Ze^2}{|{\\bar{{r}}_A}|}\\right)\n+\\frac{1}{2}\\sum_{A\\ne B}\\frac{e^2}{{|\\bar{{r}}_A - \\bar{{r}}_B|}} + \\right.\\nonumber\\\\\n&~~&~~~~ -\\sum_{A=1}^{M} \\frac{Ze^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A|^3}}\\cdot L_{3A} +\n\\label{gromek7}\\\\\n&+&\\left.\\frac{1}{2}\\sum_{A\\ne B} \\frac{e^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A-\\bar{r}_B|^3}}\\cdot \\left(L_{3B}+L_{3A}\\right) +\\;\\right.  \\nonumber \\\\\n&-&\\left.\\frac{1}{2}\\sum_{A\\ne B} \\frac{e^2{f_{\\kappa_a}(t)}}{2\\hbar{|\\bar{r}_A-\\bar{r}_B|^3}}\\cdot \\left(G_{AB}+G_{BA}\\right) + {\\cal O}(\\kappa_a)\\;\\right]\\times \\nonumber \\\\\n&\\times& \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t)\\;.\\nonumber\n\\end{eqnarray}\nrespectively, with $L_{3A} =x_{1A}p_{2A} - x_{2A}p_{1A}$ and $G_{AB} = x_{1B}p_{2A} - x_{2B}p_{1A}$. Particulary, in the case of single particle, for canonical deformation $f_{\\kappa_a}(t) = \\kappa_a$  we reproduce\nthe noncommutative model of hydrogen atom proposed in \\cite{qm1} and \\cite{qm2}\n\\begin{eqnarray}\nH(\\bar{p},\\bar{x}) &=& \\frac{\\bar{p}^2}{2m} -\\frac{Ze^2}{{|\\bar{r}|}} - \\frac{Ze^2{\\kappa_a}}{2\\hbar{|\\bar{r}|^3}}\\cdot L_3 + \\mathcal{O}\n(\\kappa_a) \\;,\\label{gromek8}\n\\end{eqnarray}\nwhile for more complicated (time-dependent) functions $f_{\\kappa_a}(t)$, we get the one-particle system described by\n\\begin{eqnarray}\nH(\\bar{p},\\bar{x},t) &=& \\frac{\\bar{p}^2}{2m} -\\frac{Ze^2}{{|\\bar{r}|}} - \\frac{Ze^2f_{\\kappa_a}(t)}{2\\hbar{|\\bar{r}|^3}}\\cdot L_3 + \\mathcal{O}\n(\\kappa_a) \\;. \\label{gromek9}\n\\end{eqnarray}\nIt is well-known, that the solution of the  corresponding (associated with (\\ref{gromek9})) Schroedinger equation  can be found  with use of time-dependent perturbation theory \\cite{24}. It looks as follows\n\\begin{eqnarray}\n {\\psi}(\\bar{r},t) = \\sum_{n=0}^{\\infty}\\sum_{l=0}^{n-1}\\sum_{m=-l}^{l} c_{nlm}(t){\\rm e}^{iE_n(t-t_0)}\\psi_{nlm}(\\bar{x})\\;, \\label{gromek10}\n\\end{eqnarray}\nwhere symbols $E_n$ and $\\psi_{nlm}$ denote eigenvalues and eigenfunctions for hydrogen atom, while coefficients $c_{nlm}(t)$ are defined as the solutions of the following differential equations\n\\begin{eqnarray}\n\\frac{dc_{nlm}(t)}{dt} &=& -\\frac{1}{i\\hbar} \\sum_{n'=0}^{\\infty}\\sum_{l'=0}^{n-1}\\sum_{m'=-l}^{l} \\left(\\psi_{nlm}(\\bar{r}),\\frac{Ze^2f_{\\kappa_a}(t)}{2\\hbar{|\\bar{r}|^3}}\\cdot L_3\\psi_{n'l'm'}(\\bar{r})\\right)c_{n'l'm'}(t_0)\\;\\cdot \\cr\n&\\cdot&{\\rm e}^{i\\omega_{nn'}(t-t_0)}\\;\\;\\;;\\;\\;\\;\\omega_{nn'} \\;=\\; \\frac{1}{\\hbar}(E_n-E_{n'})\\;.\\label{gromek11}\n\\end{eqnarray}\nHence, in accordance with prescription (\\ref{grom3}),  the solution of  multiparticle wave equation (\\ref{gromek7}) with neglected correlation potential $V_{AB}(|\\bar{r}_A-\\bar{r}_B|)$ and vanishing $\\mathcal{O}\n(\\kappa_a)$-terms takes the form\n\\begin{eqnarray}\n \\psi(\\bar{r}_1, \\ldots ,\\bar{r}_M,t) = {\\psi}_1(\\bar{r}_1,t) \\cdots {\\psi}_M(\\bar{r}_M,t)\\;,\\label{gromek3}\n\\end{eqnarray}\nwith one-particle functions ${\\psi}_A(\\bar{r}_A,t)$ given by (\\ref{gromek10}).\n\nFinally, it should be noted that the average values of energy operators (\\ref{gromek2}), (\\ref{gromek4}) and (\\ref{gromek6}) can be found with use of the formula (\\ref{grom8}).\n\n\\section{Final remarks}\n\nIn this article we construct the quantum model of M\nnonrelativistic particles moving in noncommutative space-time\n(\\ref{spaces300}). The corresponding Schroedinger equation for arbitrary stationary\npotential is provided and, in\nparticular, there is analyzed the distinguished example of such\nsystem - the set of M particles moving \"in\" and interacting \"by\" the Coulomb potential. It\nshould be noted, however, that by analogy to the investigations performed in article \\cite{qm1}, one can\nask about more physical features (such as for example the energy spectrum or the Lamb shift) of the model defined by Hamiltonian (\\ref{gromek6}).\nBesides,  it should be added, that the presented  considerations give a starting\npoint for the construction of Dirac quantum mechanics for\nmany particles defined on the relativistic counterpart of modified space-time (\\ref{canamm}).\n The studies in these directions  already started and\nare in progress.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nThe author would like to thank J. Lukierski\nfor valuable discussions. This paper has been financially  supported  by Polish\nNCN grant No 2011\/01\/B\/ST2\/03354.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\columnwidth, angle=0]{figures\/fig_Intro_DCA.pdf}\n\\caption{\\textbf{An Illustration of the Dynamic Context Augmentation process.} A traditional global EL model jointly optimizes the linking configuration after iterative calculations over all mentions, which is computationally expensive. In contrast, the DCA process only requires one pass of the document to accumulate knowledge from previously linked mentions to enhance fast future inference.\n}\n\n\\label{fig:DCA}\n\\end{figure}\n\nLinking mentions of entities in text to knowledge base entries (i.e., entity linking, or EL) is critical to understanding and structuring text corpora. In general, EL is approached by first obtaining candidate entities for each mention, and then identifying the true referent among the candidate entities. \nPrior distribution and local contexts, either in the form of hand-crafted features~\\cite{ratinov2011local,shen2015entity} or dense embeddings~\\cite{he2013learning,nguyen2016joint,francis2016capturing}, play key roles in distinguishing different candidates. However, in many cases, local features can be too sparse to provide sufficient information for disambiguation.\n\nTo alleviate this problem, various collective EL models have been proposed to globally optimize the inter-entity coherence between mentions in the same document\n~\\cite{hoffart2011robust,cheng2013relational,nguyen2014aida,alhelbawy2014graph,pershina2015personalized}.\nDespite of their success, existing global EL models try to optimize the entire linking configuration of all mentions, with extra assumptions of either all-mention coherence or pairwise coherence \\cite{phan2018pair}.\nSuch assumptions are against human intuitions, as they imply that no inference can be made until all mentions in a document have been observed.\nAlso, there usually exists a trade-off between accuracy and efficiency: state-of-the-art collective\/global models suffer from high time complexity.\nFrom the perspective of computational efficiency, optimal global configuration inference is NP-hard.\nApproximation methods, such as loopy belief propagation \\cite{ganea2017deep} or iterative substitutions \\cite{shen2015entity}, are still computationally expensive due to the huge hypothesis space, and thus can hardly be scaled to handle large corpus.\nMany previous works have discussed the urgent needs of more efficient linking system for production, both in time complexity~\\cite{hughes2014trading} and memory consumption~\\cite{blanco2015fast}. \n\n\nIn this paper, we propose a simple yet effective \\textbf{D}ynamic \\textbf{C}ontext \\textbf{A}ugmentation (DCA) process to incorporate global signal for EL. \nAs Figure \\ref{fig:DCA} shows, in contrast to traditional global models, DCA only requires one pass through all mentions to achieve comparable linking accuracy.\nThe basic idea is to accumulate knowledge from previously linked entities as dynamic context to enhance later decisions.\nSuch knowledge come from not only the inherent properties (e.g., description, attributes) of previously linked entities, but also from their closely related entities, which empower the model with important associative abilities. \nIn real scenarios, some previously linked entities may be irrelevant to the current mention. Some falsely linked entities may even introduce noise.\nTo alleviate error propagation, we further explore two strategies: (1) soft\/hard attention mechanisms that favour the most relevant entities; (2) a reinforcement learning-based ranking model, which proves to be effective as reported in other information extraction tasks.\n\n\\smallskip\n\\noindent\n\\textbf{Contributions.} \nThe DCA model forms a new linking strategy from the perspective of data augmentation and thus can serve as a plug-and-enhance module of existing linking models. The major contributions of this work are as follows:\n(1) DCA can introduce topical coherence into local linking models without reshaping their original designs or structures; (2) Comparing to global EL models, DCA only requires one pass through all mentions, yielding better efficiency in both training and inference; (3) Extensive experiments show the effectiveness of our model under different learning settings, base models, decision orders and attention mechanisms.\n\\section{Background}\n\n\\subsection{Problem Definition}\nGiven a set of entity mentions $\\mathcal{M} = \\{m_1, ..., m_T\\}$ in corpus $\\mathcal{D}$, Entity Linking aims to link each mention $m_t$ to its corresponding gold entity $e_t^*$. Such a process is usually divided into two steps:\n\\emph{Candidate generation} first collects a set of possible (candidate) entities $\\mathcal{E}_t = \\{e_t^1, ..., e_t^{|\\mathcal{E}_t|}\\}$ for $m_t$;\n\\emph{Candidate ranking} is then applied to rank all candidates by likelihood. The linking system selects the top ranked candidate as the predicted entity $\\hat{e}_t$.\nThe key challenge is to capture high-quality features of each entity mention for accurate entity prediction, especially when local contexts are too sparse to disambiguate all candidates.\n\nWe build our DCA model based on two existing local EL models. In this section, we first introduce the architecture of the base models, then present the proposed DCA model under the standard supervised learning framework. Since the DCA process can be naturally formed as a sequential decision problem, we also explore its effectiveness under the Reinforcement Learning framework. Detailed performance comparison and ablation studies are reported in Section \\ref{sec:result}.\n\\subsection{Local Base Models for Entity Linking}\n\\label{sec:base}\nWe apply the DCA process in two popular local models with different styles: the first is a neural attention model named ETHZ-Attn~\\cite{ganea2017deep}, the other is the Berkeley-CNN~\\cite{francis2016capturing} model which is made up of multiple convolutional neural networks (CNN).\n\n\\medskip\n\\noindent\\textbf{ETHZ-Attn.}\nFor each mention $m_t$ and a candidate $e^j_t \\in \\mathcal{E}_t$, three local features are considered: \n(1) \\emph{Mention-entity Prior} $\\hat{P}(e_t^j | m_t)$ is the empirical distribution estimated from massive corpus (e.g. Wikipedia); \n(2) \\emph{Context Similarity} $\\Psi_C(m_t, e^j_t)$ measures the textual similarity between $e^j_t$ and the local context of $m_t$;\n(3) \\emph{Type Similarity} $\\Psi_T(m_t, e^j_t)$ considers the similarity between the type of $e_t^j$ and contexts around $m_t$.\n$\\hat{P}(e_t^j | m_t)$ and $\\Psi_C(m_t, e^j_t)$ are calculated in the same way as \\cite{ganea2017deep}.\nFor $\\Psi_T(m_t, e^j_t)$, we first train a typing system proposed by \\cite{xu2018neural} on AIDA-train dataset, yielding 95\\% accuracy on AIDA-A dataset. In the testing phase, the typing system predicts the probability distribution over all types (PER, GPE, ORG and UNK) for $m_t$, and outputs $\\Psi_T(m_t, e^j_t)$ for each candidate accordingly. All local features are integrated by a two-layer feedforward neural network with 100 hidden units, as described in \\cite{ganea2017deep}.\n\n\\medskip\n\\noindent\\textbf{Berkeley-CNN.} The only difference between \\textbf{ETHZ-Attn} and \\textbf{Berkeley-CNN} is that, this model utilizes CNNs at different granularities to capture \\emph{context similarity} $\\Psi_C(m_t, e^j_t)$ between a mention's context and its target candidate entities.\n\\section{Dynamic Context Augmentation}\n\\label{sec:DCA}\n\\label{sec:dca}\nAs Figure \\ref{fig:DCA} demonstrates, the basic idea of DCA is to accumulate knowledge from previously linked entities as dynamic context to enhance later decisions.\nFormally, denote the list of previously linked entities as $S_t = \\{\\hat{e}_1, ..., \\hat{e}_t\\}$, where each $\\hat{e}_i$ is represented as an embedding vector.\nThe augmented context can be represented by accumulated features of all previous entities and their neighbors (e.g. by averaging their embeddings, in the simplest way).\nIn actual scenarios, some entities in $S_t$ are irrelevant, if not harmful, to the linking result of $m_{t+1}$. \nTo highlight the importance of relevant entities while filtering noises, we also try to apply a neural attention mechanism on dynamic contexts (Figure \\ref{fig:attention}).\nFor mention $m_{t+1}$, candidates that are more coherent with $S_t$ are preferred.\nMore specifically, we calculate the relevance score for each $\\hat{e}_i \\in S_t$ as\n\\begin{equation}\n    u(\\hat{e}_i) = \\max_{e^j_{t+1} \\in \\mathcal{E}_{t+1}} {e_{t+1}^j}^\\top \\cdot A \\cdot \\hat{e}_i,\n\\end{equation}\nwhere $A$ is a parameterized diagonal matrix. Top $K$ entities in $S_t$ are left to form dynamic context while the others are pruned. The relevance scores are transformed to attention weights with\n\\begin{equation}\n    a(\\hat{e}_i) = \\frac{\\exp[u(\\hat{e}_i)]}{\\sum_{\\hat{e}_j \\in S_t} \\exp[u(\\hat{e}_j)]}~.\n\\end{equation}\nThus, we can define a weighted coherence score between $e_{t+1}^j \\in \\mathcal{E}_{t+1}$ and $S_t$ as\n\\begin{equation}\n    \\Phi(e_{t+1}^j, S_t) = \\sum_{\\hat{e}_i \\in S_t} a(\\hat{e}_i) \\cdot {e^j_{t+1}}^\\top \\cdot R \\cdot \\hat{e}_i,\n\\end{equation}\nwhere $R$ is a learnable diagonal matrix. Such a coherence score will be later incorporated in the final representation of $e_{t+1}^j$.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.45\\textwidth, angle=0]{figures\/fig_method_attention.pdf}\n\\caption{Neural attention mechanism on the dynamic context. \nThe soft attention module assigns higher weights to entities that are more relevant to the target mention.\nThe hard attention module only considers top $K$ entities as dynamic contexts.\n}\n\n\\label{fig:attention}\n\\end{figure}\n\nTo empower the linking model with associative ability, aside from previously linked entities, we also incorporate entities that are closely associated with entities in $S_t$. Specifically, for each $\\hat{e}_i \\in S_t$, we collect its neighborhood $\\mathcal{N}(\\hat{e}_i)$ consisting of Wikipedia entities that have inlinks pointing to $\\hat{e}_i$. Denoting $S'_t$ as the union of $\\{\\mathcal{N}(\\hat{e}_i) | \\hat{e}_i \\in S_t\\}$, we define a similar weighted coherence score between $e_{t+1}^j \\in \\mathcal{E}_{t+1}$ and $S'_t$ as\n\\begin{equation}\n    \\Phi'(e_{t+1}^j, S'_t) = \\sum_{\\hat{e}_i \\in S'_t} a'(\\hat{e}_i) \\cdot {e^j_{t+1}}^\\top \\cdot R' \\cdot \\hat{e}_i,\n\\end{equation}\nwhere $a'$ is defined similarly to $a$, and $R'$ is a learnable diagonal matrix. \nThe final representation $\\vec{h_0}(m_{t+1}, e^j_{t+1})$ is the concatenation of $\\Phi(e_{t+1}^j, S_t)$, $\\Phi'(e_{t+1}^j, S'_t)$, $\\Psi_T(m_t, e_{t+1}^j)$, $\\Psi_C(m_t, e^j_{t+1})$ and $\\log\\hat{P}(e^j_{t+1} | m_{t+1})$. \n\\section{Model Learning for DCA}\nIn this section, we explore different learning strategies for the linking model. Specifically, we present a Supervised Learning model, where the model is given all gold entities for training, and a Reinforcement Learning model, where the model explores possible linking results by itself in a long-term planning task.\n\n\\subsection{Supervised Ranking Method}\nGiven a mention-candidate pair $(m_t, e_t^j)$, the ranking model parameterized by $\\theta$ accepts the feature vector $\\vec{h_0}(m_{t}, e^j_{t})$ as input, and outputs the probability $P_\\theta(e^j_t | m_t)$.\nIn this work, we use a two-layer feedforward neural network as the ranking model. We apply the max-margin loss as\n\\begin{equation*}\n\\begin{split}\n    \\mathcal{L_\\theta} &= \\sum_{D \\in \\mathcal{D}} \\sum_{m_t \\in D} \\sum_{e_t^j \\in \\mathcal{E}_t} g_\\theta(e_t^j, m_t),\\\\\n    g_\\theta(e_t^j, m_t) &= \\max(0, \\gamma - P_\\theta(e^*_t | m_t) + P_\\theta(e^j_t | m_t)).\n\\end{split} \n\\end{equation*}\nThe learning process is to estimate the optimal parameter such that $\\theta^* = \\arg\\min_\\theta \\mathcal{L}_\\theta$.\n\nNote that, in the Supervised Ranking model, dynamic contexts are provided by previous gold entities: $S_t = \\{e_1^*, ..., e_t^*\\}$, $S'_t = \\bigcup\\limits_{i=1}^{t} \\mathcal{N}(e_i^*)$. In the testing phase, however, we do not have access to gold entities. Wrongly linked entities can introduce noisy contexts to future linking steps. To consider such long-term influences, we introduce an alternative Reinforcement Learning model in the next section.\n\\subsection{Reinforcement Learning Method}\nNaturally, the usage of dynamic context augmentation forms a sequential decision problem, as each linking step depends on previous linking decisions.\nCorrect linking results provide valuable information for future decisions, while previous mistakes can lead to error accumulation. \nReinforcement Learning (RL) algorithms have proven to be able to alleviate such accumulated noises in the decision sequence in many recent works \\cite{narasimhan2016improving, feng2018relation}.\nIn this work, we propose an RL ranking model for DCA-enhanced entity linking.\n\n\\vspace{0.15cm}\n\\noindent\\textbf{Agent}: The Agent is a candidate ranking model that has a similar architecture to~\\cite{clark2016deep}, aiming to output the action preference $H_\\theta(S_{t-1}, S'_{t-1}, A_t^j)$ of each linking action $A_t^j = (m_t \\rightarrow e_t^j)$. It is a 2-layer feedforward neural network with following components:\n\n\\vspace{0.15cm}\n\\noindent{\\bf Input Layer}: For each $( m_t, e_t^j )$ pair, DCA-RL extracts context-dependent features from $S_{t-1}, S'_{t-1}$, and concatenates them with other context-independent features to produce an $I$-dimensional input vector $\\vec{h_0}(m_{t}, e^j_{t})$.\n\n\\vspace{0.15cm}\n\\noindent{\\bf Hidden Layers}: Let $Drop(\\vec x)$ be the dropout operation~\\cite{srivastava2014dropout} and $ReLU(\\vec x)$ be the rectifier nonlinearity \\cite{nair2010rectified}. So the output $\\vec {h}_1$ of the hidden layer is defined as:\n\\begin{equation}\n\\vec {h}_1 = Drop(ReLU(\\vec{W}_1 \\cdot \\vec{h}_0 + \\vec{b}_1)),\n\\end{equation}\nwhere $\\vec{W}_1$ is a $H_1 \\times I$ weight matrix.\n\n\\vspace{0.15cm}\n\\noindent{\\bf Output Layers}: This scoring layer is also fully connected layer of size 1.\n\\begin{equation}\n\\vec {h}_2 = \\vec{W}_2 \\cdot \\vec{h}_1 + \\vec{b}_2,\n\\end{equation}\nwhere $\\vec{W}_2$ is a $1 \\times H_1$ weight matrix. In the end, all action preference would be normalized together using an exponential softmax distribution, getting their action probabilities $\\pi_\\theta(A_t^j|S_{t-1},S'_{t-1})$:\n\n\nAccording to policy approximating methods, the best approximate policy may be stochastic. So we randomly sample the actions based on the softmax distribution during the training time, whereas deliberately select the actions with the highest ranking score at the test time.\n\n\\vspace{0.15cm}\n\\noindent\n\\textbf{Reward.}\nThe reward signals are quite sparse in our framework. For each trajectory, the Agent can only receive a reward signal after it finishes all the linking actions in a given document. Therefore the immediate reward of action $t$, $R_t = 0$, where $0 \\leq t <T$, and $R_T = - (|\\mathbb{M}_{e}|\/T)$, where $T$ is total number of mentions in the source document, and $|\\mathbb{M}_{e}|$ is the number of incorrectly linked mentions. Then the value $G_t$ (expected reward) of each previous state $S_t$ can be retraced back with a discount factor $\\rho$ according to $R_T$:\n\\begin{equation}\nG_t=\\rho^{T-t} R_T\n\\end{equation}\n\n\nTo maximize the expected reward of all trajectories, the Agent utilizes the REINFORCE algorithm~\\cite{sutton1998reinforcement} to compute Monte Carlo policy gradient over all trajectories, and perform gradient ascent on its parameters:\n\\begin{equation}\n\\theta \\leftarrow \\theta + \\alpha \\sum_t G_t \\nabla_\\theta \\ln \\pi_\\theta(A_t^j|S_{t-1},S'_{t-1})\n\\end{equation}\n\n\nIn following sections, to fully investigate the effectiveness of the proposed method, we report and compare the performances of both the Supervised-learning model and the Reinforcement-learning model.\n\\section{Analysis of Computational Complexity}\n\\label{sec:complexity}\nFor each document $D$, the train and inference of the global EL models are heavily relied on the inter-entity coherence graph~$\\Phi_g$. Many studies~\\cite{ratinov2011local,globerson2016collective,yamada2016joint,ganea2017deep,le2018improving} obtain $\\Phi_g$ by calculating all pairwise scores between two arbitrary elements ${e_i}^x$ and ${e_j}^y$ sampled independently from candidate sets $\\mathcal{E}_i$ and $\\mathcal{E}_j$ in the given document. It is obvious that $\\Phi$ is intractable, and the computational complexity of $\\Phi_g$ is\n\\begin{equation}\n\\mathcal{O}(\\Phi_g)=\\mathcal{O}(\\sum_{i=1}^{T}\\sum_{j=1,j\\neq i}^{T}\\sum_{{e_i}^x\\in\\mathcal{E}_i}^{|\\mathcal{E}_i|}\\sum_{{e_j}^y\\in\\mathcal{E}_j}^{|\\mathcal{E}_j|}\\Phi({e_i}^x,{e_j}^y))\n\\end{equation}, where $\\Phi({e_i}^x,{e_j}^y)$ is a learnable score function. Thus, $\\mathcal{O}(\\Phi_g)$ is approximate to $\\mathcal{O}(T^2 \\times |\\mathcal{E}|^2 \\times \\mathcal{I})$, where $|\\mathcal{E}|$ is the average number of candidates per mention and $\\mathcal{I}$ is the unit cost of pairwise function $\\Phi$. In order to reduce $\\mathcal{O}(\\Phi_g)$, most previous models~\\cite{hoffart2011robust,ganea2017deep,le2018improving,fang2019joint} have to hard prune their candidates into an extremely small size (e.g. $|\\mathcal{E}|$=5). This will reduce the gold recall of candidate sets and also unsuitable for large scale production (e.g. entity disambiguation for dynamic web data like Twitter).\n\nIn contrast, the computational complexity of our model is $\\mathcal{O}(T \\times |\\mathcal{E}| \\times \\mathcal{I} \\times K )$, where $K$ is the key hyper-parameter described in Section \\ref{sec:DCA} and is usually set to a small number. This indicates the response time of our method grow linearly as a function of $T \\times |\\mathcal{E}| $.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\scalebox{1}{\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{lcccc}\n    \\toprule[2pt]\n    \\textbf{Dataset} & \\textbf{\\begin{tabular}[x]{@{}c@{}} \\#{} \\\\ mention\\end{tabular}} &  \\textbf{\\begin{tabular}[x]{@{}c@{}} \\#{} \\\\ doc\\end{tabular}} & \\textbf{\\begin{tabular}[x]{@{}c@{}} Mentions\\\\per doc\\end{tabular}} & \\textbf{\\begin{tabular}[x]{@{}c@{}} Gold\\\\recall\\end{tabular}} \\\\\n   \n    \\toprule[2pt]\n   \n   \n    AIDA-train & 18448 & 946 & 19.5 & -\\\\\n    AIDA-A & 4791 & 216 & 22.1 & 97.3 \\\\\n    AIDA-B & 4485 & 231 & 19.4 & 98.3 \\\\\n   \n    \\midrule\n    MSNBC & 656 & 20 & 32.8 & 98.5\\\\\n    AQUAINT & 727 & 50 & 14.5 & 94.2 \\\\\n    ACE2004 & 257 & 36 & 7.1 & 90.6 \\\\\n    CWEB & 11154 & 320 & 34.8 & 91.1 \\\\\n    WIKI & 6821 & 320 & 21.3 & 92.4\\\\\n   \n    \\bottomrule[2pt]\n\\end{tabular}%\n}\n}\n\\caption{\\textbf{Dataset Statistics.} \\emph{Gold recall} is the percentage of mentions for which the candidate entities contain the ground truth entity. \n}\n\\label{tab:dataset}\n\\end{table}\n\n\n\n\\section{Experiment}\n\\label{sec:result}\n\n\\subsection{Experiment Setup}\n\\label{sec:setup}\n\\textbf{Datasets.} Following our predecessors, we train and test all models on the public and widely used AIDA CoNLL-YAGO dataset \\cite{hoffart2011robust}. The target knowledge base is Wikipedia. The corpus consists of 946 documents for training, 216 documents for development and 231 documents for testing (AIDA-train\/A\/B respectively). To evaluate the generalization ability of each model, we apply cross-domain experiments following the same setting in~\\cite{ganea2017deep, le2018improving, yang2018collective}. Models are trained on AIDA-train, and evaluated on five popular public datasets: AQUAINT~\\cite{milne2008learning}, MSNBC~\\cite{cucerzan2007large}, ACE2004~\\cite{ratinov2011local}, CWEB~\\cite{guo2016robust} and WIKI \\cite{guo2016robust}. The statistics of these datasets are available in Table~\\ref{tab:dataset}. In the candidate generation step, we directly use the candidates provided by the Ment-Norm system \\cite{le2018improving}\\footnote{https:\/\/github.com\/lephong\/mulrel-nel}, and their quality is also listed in Table \\ref{tab:dataset}.\\\\\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{lc}\n  \\toprule[2pt]\n  \\textbf{System}  & \\textbf{\\begin{tabular}[x]{@{}c@{}} In-KB acc. (\\%)\\end{tabular}} \\\\\n  \\toprule[2pt]\n  AIDA-light~\\cite{nguyen2014aida} & 84.8 \\\\\n  WNED~\\cite{guo2016robust} & 89.0 \\\\\n  Global-RNN~\\cite{nguyen2016joint} & 90.7  \\\\\n  MulFocal-Att~\\cite{globerson2016collective} & 91.0 \\\\\n  Deep-ED~\\cite{ganea2017deep} & 92.22\\\\\n  Ment-Norm~\\cite{le2018improving} & 93.07 \\\\\n  \\midrule\n  Prior ($p(e|m)$)~\\cite{ganea2017deep} &  71.51 \\\\\n  \\midrule\n  \\emph{Berkeley-CNN (Sec.~\\ref{sec:base})}  & 84.21 \\\\\n  \\emph{Berkeley-CNN~+~DCA-SL} & 92.72 $\\pm$ 0.3\\\\\n  \\emph{Berkeley-CNN~+~DCA-RL} & 92.37 $\\pm$ 0.1 \\\\\n  \\emph{ETHZ-Attn (Sec.~\\ref{sec:base})} & 90.88 \\\\\n  \\emph{ETHZ-Attn~+~DCA-SL} &  \\textbf{{94.64}} $\\pm$ 0.2 \\\\\n  \\emph{ETHZ-Attn~+~DCA-RL} & {93.73} $\\pm$ 0.2 \\\\\n  \\bottomrule[2pt]\n\\end{tabular}%\n}\n\\caption{\\textbf{In-domain Performance Comparison on the AIDA-B Dataset.} For our method we show 95\\% confidence intervals obtained over 5 runs. DCA-based models achieve the best reported scores on this benchmark.\n}\n\\label{tab:conll}\n\\end{table}\n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\scalebox{1}{\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lccccc}\n  \\toprule[2pt]\n  \\textbf{System} & \\textbf{MSBNC} & \\textbf{AQUAINT} & \\textbf{ACE2004} & \\textbf{CWEB} &\\textbf{WIKI} \\\\\n  \\toprule[2pt]\n  AIDA~\\cite{hoffart2011robust} & 79 & 56 & 80 & 58.6 & 63 \\\\\n  GLOW~\\cite{ratinov2011local} & 75 & 83 & 82 & 56.2 & 67.2 \\\\\n  RI~\\cite{cheng2013relational} & 90 & \\textbf{90} & 86 & 67.5 & 73.4 \\\\ \n  WNED~\\cite{guo2016robust} & 92 & 87 & 88 & 77 & \\textbf{84.5} \\\\\n  Deep-ED~\\cite{ganea2017deep} &  93.7 & 88.5 & 88.5 & \\textbf{77.9} & 77.5 \\\\\n  Ment-Norm~\\cite{le2018improving} & 93.9 & 88.3 & 89.9 & 77.5 & 78.0 \\\\\n  \\midrule\n  Prior ($p(e|m)$)~\\cite{ganea2017deep} & 89.3 & 83.2 & 84.4 & 69.8 & 64.2 \\\\\n  \\midrule\n  \\emph{Berkeley-CNN (Section \\ref{sec:base})} & 89.05 & 80.55 & 87.32 & 67.97 & 60.27 \\\\\n  \\emph{Berkeley-CNN~+~DCA-SL} & 93.38 $\\pm$ 0.2 & 85.63 $\\pm$ 0.3 & 88.73 $\\pm$ 0.3 & 71.01 $\\pm$ 0.1 & 72.55  $\\pm$ 0.2 \\\\\n  \\emph{Berkeley-CNN~+~DCA-RL} & 93.65 $\\pm$ 0.2 & 88.53 $\\pm$ 0.3 & 89.73 $\\pm$ 0.4 & 72.66 $\\pm$ 0.4 & 73.98 $\\pm$ 0.2 \\\\\n  \\emph{ETHZ-Attn (Section  \\ref{sec:base})} & 91.97 & 84.06 & 86.92 & 70.07 & 74.37  \\\\\n  \\emph{ETHZ-Attn~+~DCA-SL} &  \\textbf{94.57} $\\pm$ 0.2 & 87.38 $\\pm$ 0.5 & 89.44 $\\pm$ 0.4 & 73.47 $\\pm$ 0.1 & 78.16 $\\pm$ 0.1 \\\\\n  \\emph{ETHZ-Attn~+~DCA-RL} & 93.80 $\\pm$ 0.0 & 88.25 $\\pm$ 0.4 & \\textbf{90.14} $\\pm$ 0.0 & 75.59 $\\pm$ 0.3 & 78.84 $\\pm$ 0.2 \\\\\n  \\bottomrule[2pt]\n\\end{tabular}%\n}\n}\n\\caption{\\textbf{Performance Comparison on Cross-domain Datasets using F1 score (\\%)}. The best results are in bold. Note that our own results all retain two decimal places. Other results with uncertain amount of decimal places are directly retrieved from their original paper.}\n\\label{tab:soa-cross}\n\\end{table*}\n\n\n\n\\medskip\n\\noindent\n\\textbf{Compared Methods.}\nWe compare our methods with following existing systems that report state-of-the-art results on the test datasets: \\textbf{AIDA-light}~\\cite{nguyen2014aida} uses a kind of two-stage collective mapping algorithm and designs several domain or category related coherence features.\n\\textbf{WNED}~\\cite{guo2016robust} applies random walks on carefully built disambiguation graphs and uses a greedy, iterative and global disambiguation algorithm based on Information Theory.\n\\textbf{Global-RNN}~\\cite{nguyen2016joint} develops a framework based on convolutional neural networks and recurrent neural networks to simultaneously model the local and global features.\n\\textbf{MulFocal-Att}~\\cite{globerson2016collective} adopts a coherence model with a multi-focal attention mechanism.\n\\textbf{Deep-ED}~\\cite{ganea2017deep} leverages learned neural representations, and uses a deep learning model combined with a neural attention mechanism and graphical models.\n\\textbf{Ment-Norm}~\\cite{le2018improving} improving the \\textbf{Deep-ED} model by modeling latent relations between mentions.\n\n\nFor a fair comparison with prior work, we use the same input as the \\textbf{WNED}, \\textbf{Deep-ED} and \\textbf{Ment-Norm} (models proposed after 2016), and report the performance of our model with both Supervised Learning (\\textbf{DCA-SL}) and Reinforcement Learning (\\textbf{DCA-RL}). We won't compare our models with the \\textbf{RLEL}~\\cite{fang2019joint} which is a deep reinforcement learning based LSTM model. There are two reasons: 1) \\textbf{RLEL} uses optimized candidate sets with smaller candidate size and higher gold recall than ours and the listed baselines. 2) \\textbf{RLEL} uses additional training set from Wikipedia data. \\cite{fang2019joint} doesn't release either their candidate sets or updated training corpus, so the comparison with their work would be unfair for us.\n\n\\medskip\n\\noindent\\textbf{Hyper-parameter Setting.} We coarsely tune the hyper-parameters according to model performance on AIDA-A. We set the dimensions of word embedding and entity embedding to 300, where the word embedding and entity embedding are publicly released by \\cite{pennington2014glove} and  \\cite{ganea2017deep} respectively. Hyper-parameters of the best validated model are: $K=7$, $I=5$, $H_1=100$, and the probability of dropout is set to 0.2. Besides, the rank margin $\\gamma=0.01$ and the discount factor $\\rho=0.9$. We also regularize the \\emph{Agent} model as adopted in \\cite{ganea2017deep} by constraining the sum of squares of all weights in the linear layer with $MaxNorm=4$. When training the model,  we use Adam~\\cite{kingma2014adam} with learning rate of 2e-4 until validation accuracy exceeds 92.8\\%, afterwards setting it to 5e-5. \\\\\n\n\\begin{table}[]\n\\centering\n\\scalebox{0.88}{\n\\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{c|cc}\n   \n    \\toprule[2pt]\n    \\multirow{2}{*}{\\textbf{System}} & \\multicolumn{2}{c}{\\textbf{In-KB acc. (\\%)}} \\\\ \\cline{2-3} \n                                     & \\textbf{SL}           & \\textbf{RL}           \\\\ \\hline\n    ETHZ-Attn (Section \\ref{sec:base})                   & 90.88                 & -                     \\\\\n    ETHZ-Attn + 1-hop DCA           & 93.69                 & 93.20                 \\\\\n    ETHZ-Attn + 2-hop DCA           & 94.47                 & 93.76                 \\\\ \n    \\bottomrule[2pt] \n   \n    \\end{tabular}%\n}\n}\n\\caption{\\textbf{Ablation Study on Neighbor Entities.} We compare the performance of DCA with or without neighbor entities (i.e., 2-hop vs. 1-hop).}\n\\label{tab:abl_hop}\n\\end{table}\n\n\\subsection{Overall Performance Comparison}\nStarting with an overview of the end-task performance, we compare DCA (using SL or RL) with several state-of-the-art systems on in-domain and cross-domain datasets. We follow prior work and report in-KB accuracy for AIDA-B and micro F1 scores for the other test sets.\n\nTable \\ref{tab:conll} summarizes results on the AIDA-B dataset, and shows that DCA-based models achieve the highest in-KB accuracy and outperforms the previous state-of-the-art neural system by near 1.6\\% absolute accuracy. Moreover, compared with the base models, dynamic context augmentation significantly improve absolute in-KB accuracy in models Berkeley-CNN (more than 8\\%) and ETHZ-Attn (3.3\\% on average). Note that, our DCA model outperforms existing global models with the same local model (\\textbf{Global-RNN} uses Berkeley-CNN as base model, \\textbf{Deep-ED} and \\textbf{Ment-Norm} use ETHZ-Attn as the local model).\n\nTable \\ref{tab:soa-cross} shows the results on the five cross-domain datasets. As shown, none of existing methods can consistently win on all datasets. DCA-based models achieve state-of-the-art performance on the MSBNC and the ACE2004 dataset. On remaining datasets, DCA-RL achieves comparable performance with other complex global models. In addition, RL-based models show on average 1.1\\% improvement on F1 score over the SL-based models across all the cross-domain datasets. At the same time, DCA-based methods are much more efficient, both in time complexity and in resource requirement. Detailed efficiency analysis will be presented in following sections.\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=1.0\\textwidth, angle=0]{figures\/fig_ablation.pdf}\n\\caption{(a) \\textbf{In-KB accuracy} as a function of decision length on AIDA-B dataset; (b) \\textbf{Memory usage} as a function of the number of candidates on AIDA-B dataset; (c) \\textbf{Runtime cost} (at inference time) as a function of the number of mentions. (each dot represents a document of AIDA-B with $|\\mathcal{E}|=35$).}\n\\label{fig:abl_total}\n\\end{figure*}\n\n\\begin{table}[]\n\\centering\n\\scalebox{0.9}{\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{c|cc}\n\\toprule[2pt]\n\\multirow{2}{*}{\\textbf{System}} & \\multicolumn{2}{c}{\\textbf{In-KB acc. (\\%)}} \\\\ \\cline{2-3} \n                                 & \\textbf{SL}           & \\textbf{RL}           \\\\ \\hline\nDCA + Average Sum                & 94.05                 & 93.56                 \\\\\nDCA + Soft Attention             & 94.23                 & 93.69                 \\\\\nDCA + Soft\\&Hard Attention     & 94.47                 & 93.76                 \\\\ \\bottomrule[2pt]\n\\end{tabular}%\n}\n}\n\\caption{\\textbf{Study on Different Attention Mechanisms on the Dynamic Context.} Results (using ETHZ-Attn as local model) show that different attention mechanisms have similar impact on the performance.}\n\\label{tab:abl_attention}\n\\end{table}\n\n\n\\subsection{Performance Analysis}\n\\noindent\n\\textbf{1. Impact of decision order.}\nAs the DCA model consecutively links and adds all the mentions in a document, the linking order may play a key role in the final performance. In this work, we try three different linking orders: \\emph{Offset} links all mentions by their natural orders in the original document; \\emph{Size} first links mentions with smaller candidate sizes, as they tend to be easier to link; The baseline method is to link all mentions in a \\emph{Random} order.\nFigure \\ref{fig:abl_order} shows the performance comparison on the AIDA-B and the CWEB dataset. As shown, in general, \\emph{Size} usually leads to better performance than \\emph{Offset} and \\emph{Random}. However, the DCA-SL model shows poor performance on the CWEB dataset with \\emph{Size} order. This is mainly because the CWEB dataset is automatically generated rather than curated by human, and thus contains many noisy mentions. Some mentions in CWEB with less than three candidates are actually bad cases , where none of the candidates is the actual gold entity. Thus, such mentions will always introduce wrong information to the model, which leads to a worse performance. In contrast, the AIDA-B dataset does not have such situations. The DCA-RL model, however, still has strong performance on the CWEB dataset, which highlights its robustness to potential noises.\\\\\n\n\\noindent\n\\textbf{2. Effect of neighbor entities.}\nIn contrast to traditional global models, we include both previously linked entities and their close neighbors for global signal. Table \\ref{tab:abl_hop} shows the effectiveness of this strategy. We observe that incorporating these neighbor entities (2-hop) significantly improve the performance (compared to 1-hop) by introducing more related information. And our analysis shows that on average 0.72\\% and 3.56\\% relative improvement of 2-hop DCA-(SL\/RL) over 1-hop DCA-(SL\/RL) or baseline-SL (without DCA) is statistically significant (with P-value $<$ 0.005). This is consistent with our design of DCA.\\\\\n\n\\noindent\n\\textbf{3. Study of different attention mechanisms.}\nTable \\ref{tab:abl_attention} shows the performance comparison by replacing the attention module described in Section \\ref{sec:DCA} with different variants. \\emph{Average Sum} treats all previously linked entities equally with a uniform distribution. \\emph{Soft Attention} skips the pruning step for entities with low weight scores. \\emph{Soft\\&Hard Attention} stands for the strategy used in our model. It is obvious that the attention mechanism does show positive influence on the linking performance compared with \\emph{Average Sum}. Hard pruning brings slight further improvement.\\\\\n\n\\noindent\n\\textbf{4. Impact of decision length.}\nAs wrongly linked entities can introduce noise to the model, there exists a trade-off in DCA: involving more previous entities (longer historical trajectory) provides more information, and also more noise. Figure (\\ref{fig:abl_total}.a) shows how the performance of DCA changes with the number of previous entities involved. We observe that longer historical trajectories usually have a positive influence on the performance of DCA. The reason is that our attention mechanism could effectively assess and select relevant contexts for each entity mention on the fly, thus reducing potential noise.\n\n\\smallskip\n\\noindent\n\\subsection{Analysis on Time Complexity}\nAs discussed in Sec.~\\ref{sec:complexity}, the running time of a DCA enhanced model may rise linearly when the average number of candidates per mention (i.e., $|\\mathcal{E}|$) increases, while the global EL model increases exponentially. To validate the theory we empirically investigate the scalability of DCA, and carefully select two global EL models Ment-Norm~\\cite{le2018improving} and Deep-ED~\\cite{ganea2017deep} as our baselines. The reason for this choice is that our final model shares the same local model as their models, which excludes other confounding factors like implementation details. As Figure (\\ref{fig:abl_total}.c) shows, when  $|\\mathcal{E}|$  increases, the running time of these two global EL models increases shapely, while our DCA model grows linearly. On the other hand, we also observed that the resources required by the DCA model are insensitive to $|\\mathcal{E}|$. For example, as shown in Figure (\\ref{fig:abl_total}.b), the memory usage of Ment-Norm and Deep-ED significantly rises as more candidates are considered, while the DCA model remains a relatively low memory usage all the time. We also measure the power consumption of Ment-Norm and DCA models, and we find that the DCA model saves up to 80\\% of the energy consumption over the Ment-Norm, which is another advantage for large scale production. \n\n\n\\section{Related Work}\n\\label{sec:related_work}\nLocal EL methods disambiguate each mention independently according to their local contexts \\cite{yamada2016joint, Chen2017Bilinear, globerson2016collective, raiman2018deeptype}. \nThe performance is limited when sparse local contexts fail to provide sufficient disambiguation evidence.\n\nTo alleviate this problem, global EL models jointly optimize the entire linking configuration.\nThe key idea is to maximize a global coherence\/similarity score between all linked entities \\cite{hoffart2011robust,ratinov2011local,cheng2013relational,nguyen2014aida,alhelbawy2014graph,pershina2015personalized,guo2016robust,globerson2016collective,ganea2017deep,le2018improving,yang2018collective,fang2019joint,xue2019neural}.\nDespite of its significant improvement in accuracy, such global methods suffer from high complexity.\nTo this end, some works try to relax the assumption of all-mention coherence, e.g. with pairwise coherence, to improve efficiency \\cite{phan2018pair}, but exact inference remains an NP-hard problem.\nApproximation methods are hence proposed to achieve reasonably good results with less cost.\n\\cite{shen2012linden} propose the iterative substitution method to greedily substitute linking assignment of one mention at a time that can improve the global objective.\nAnother common practice is to use Loopy Belief Propagation for inference \\cite{ganea2017deep,le2018improving}.\nBoth approximation methods iteratively improve the global assignment, but are still computationally expensive with unbounded number of iterations.\nIn contrast, the proposed DCA method only requires one pass through the document. Global signals are accumulated as dynamic contexts for local decisions, which significantly reduces computational complexity and memory consumption.\n\\section*{Acknowledgments}\n\nThis work has been supported in part by National Key Research and Development Program of China (SQ2018AAA010010), NSFC (No.61751209, U1611461), Zhejiang University-iFLYTEK Joint Research Center, Zhejiang University-Tongdun Technology Joint Laboratory of Artificial Intelligence, Hikvision-Zhejiang University Joint Research Center, Chinese Knowledge Center of Engineering Science and Technology (CKCEST), Engineering Research Center of Digital Library, Ministry of Education. Xiang Ren's research has been supported in part by National Science Foundation SMA 18-29268, DARPA MCS and GAILA, IARPA BETTER, Schmidt Family Foundation, Amazon Faculty Award, Google Research Award, Snapchat Gift and JP Morgan AI Research Award. Finally, we would like to thank Qi Zhang from Sogou, Inc. and all the collaborators in INK research lab for their constructive feedback on this work.\n\\section{Conclusions}\nIn this paper we propose Dynamic Context Augmentation as a plug-and-enhance module for local Entity Linking models. In contrast to existing global EL models, DCA only requires one pass through the document. \nTo incorporate global disambiguation signals, DCA accumulates knowledge from previously linked entities for fast inference.\nExtensive experiments on several public benchmarks with different learning settings, base models, decision orders and attention mechanisms demonstrate both the effectiveness and efficiency of DCA-based models. The scalability of DCA-based models make it possible to handle large-scale data with long documents.\nRelated code and data has been published and may hopefully benefit the community.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nModels are used in all stages of the security process: from security requirements elicitation and organisational risk assessment to run-time verification and business process compliance audit. Often these models are inter-connected. For example, if a security requirements model for a software system was elicited, on the later stage it may be re-used to design the security testing process for this system. At the same time, as manual production of security models is very tedious and error-prone, many researchers and practitioners look into automating the model creation and transformation processes.\n\nRecently security researchers have looked at systematic design \\cite{Paul-GraMSec-2014} and automated generation of attack models \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Ou-CCS-2006}, \\cite{Vigo-CSF-2014}, such as attack graphs and attack trees, from system models. This model transformation allows to switch the view from the system description perspective to a compact representation of possible attacker actions. At the same time, given the generated attack model, the system defender is interested to find the weakest links: the spots in the model where additional security controls can be introduced to improve protection and eliminate potential attacks. Therefore, automated generation of defences is an obvious next step in the process. \n\nIn this paper we look at socio-technical models as succinct abstractions of large organisations. Such models capture simultaneously locations, actors and objects in the system. They often take into account both physical and digital domains and offer to a human analyst the means to represent ``the world as it is''. That means that the designer of socio-technical systems does not need to be a security or risk analysis expert. She only needs to know the intricacies of her own company (department) to be able to model it. With the system model at hand, at the next step the attack generation tools aim at automatic creation of attack scenarios that can be further discussed by security professionals. The overall idea of this process is to automate threat scenarios identification (an important aspect of risk analysis) as much as possible. \n\nIn this paper we would like to push the envelope even further. Our main question is: \\emph{given a socio-technical system model, how to find and capture, possibly automatically, the security controls that will counteract the discovered threats?} Indeed, the main goal of risk analysis is to improve the existing system by introducing new security controls, so that the most dangerous or easily executed attacks are thwarted. Therefore, automated creation of attack scenarios only is not yet a full solution. \n\nWe want to look at perspectives and limitations of automated defence generation from socio-technical models. It seems that the main obstacle to rich defensive strategies generation directly from the model is the fact that socio-technical models do not capture many security controls.  \n\nTo find an answer to the main question, we start from investigating the security controls (defences) already present in an advanced socio-technical model and propose a scheme to extract these controls, together with the attack steps, in the compact format of attack-defence bundles. We then evaluate the limitations of the extracted defences inherent from the socio-technical model and discuss how to overcome these limitations. We argue that an attack-defence model needs to be maintained (in parallel with the socio-technical model) that can capture not only the attacker's view but also the defender's view of the system. In this paper we have chosen attack-defence trees \\cite{Kordy-JLC-2014} as the basis for the attack-defence model. As an alternative to this model, one can choose, for example, attack-countermeasure trees \\cite{Roy-SCN-2011}.\n\n\n\nThe goal of this paper is to propose an attack-defence view for socio-technical models that can capture simultaneously attacker's options and available\/proposed countermeasures in the system. The main idea is that given that view it can be easily synchronised with the model (but it contains richer defence information than the model), and it can be used to synthesise attack-defence trees and evaluate different interesting attributes. \n\n\\section{Socio-Technical Models versus Attack-Defence Models}\\label{sec:motivation}\n\nAs socio-technical models are abstractions, they do not capture all defensive mechanisms that can be available in an organisation, but only a subset of them. Indeed, it is impossible to model all security-relevant devices, protocols and behaviours in a single model. Typically, socio-technical models look at capturing organisational infrastructure (e.g., \\cite{D1.3.1}, \\cite{ANKH}, \\cite{Exasym-2008}, \\cite{Lenzini-2015}), but sometimes they can focus only on some aspects of human-computer interactions (e.g., \\cite{Radomirovic-2015}, \\cite{STEAL-2014}).\n\n\n\nSince all security aspects cannot be captured by a socio-technical model without overcomplicating it, we argue that there is a need to maintain a separate view of attack and defence capabilities of the system together with the socio-technical model. Preferably, we should be able to trace the objects in the socio-technical model into the attack-defence model and back. \n\n\\textbf{Requirements for the attack-defence model}\n\nThe first requirement for the chosen attack-defence model is that \\emph{the defences that are already captured by the model need to be represented explicitly in the attack-defence model}. Indeed, we would like to faithfully represent the system security state. So, if some security control is captured by the system model, it should be translated into the generated attack-defence model. \n\nSecondly, we want to propose a way to \\emph{update the generated defender's view (the security controls obtained directly from the system model) with more security controls and countermeasures} of the organisation. This \\emph{update needs to be consistent}: once a security control is captured in the attack-defence model, it should be traced to an object in the system model. For example, if our approach identifies that a security camera is to be placed in a certain location in the system, all attack scenarios that involve that location should be updated to take the camera into account. In this way later on one can investigate automated defence generation process that will maintain consistency of the socio-technical system. \n\n\n\\textbf{Background}\n\nIn this paper we use the TREsPASS socio-technical model \\cite{D1.3.1} that is graph-based. We can briefly summarise this model as follows. Locations in the system represent physical and network locations; actors model humans and processes; and items can be physical or digital objects. Edges among locations represent connectedness (e.g., adjacent rooms), and all actors and items are located somewhere in the system. Actors can possess items, and items can be embedded into other items. Some locations have access control policies attached to them. These policies specify a set of credentials (items in the system) an actor needs to possess to enter the location or access the object. These policies can also be formalized by more complex predicates capturing, e.g., role-based access control or trust relationships among actors. \n\nAs the starting point for the attack-defence model, we consider the process of attack trees generation by policy invalidation that relies on structural information about the system \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Kammuller-2013}. This process was initially designed for the TREsPASS socio-technical model \\cite{D1.3.1}, but it can be applied to other socio-technical models capturing systems as graphs, e.g., \\cite{Exasym-2008}, \\cite{ANKH}, \\cite{Portunes-2010}, \\cite{Lenzini-2015}, because it is reachability-based.\n\n\n\nIn short, this process is started by choosing an asset among the entities in the system. The attacker is also selected among actors in the system (the main goal of the attacker is to invalidate the security policy, e.g., confidentiality or integrity policy, associated with this asset). Then, based on the reachability reasoning, the process systematically searches for the ways for the attacker to access the asset. For example, consider the asset to be a sensitive document located in a locker in the manager's office, and the attacker to be an insider (an employee) working on the same floor. To access the document, the attacker can try to access the locker and open it (an AND-decomposition \\cite{Kordy-JLC-2014}). This might require possession of the key to the locker that needs to be obtained elsewhere in the system.  Alternatively (an OR-decomposition with the previous attack), also the manager has access to the locker and the document. Thus, the attacker can get access the document by influencing the manager. This can be implemented through, e.g, social engineering (for instance, befriending the manager, or hiring an external actor to pretend to be a higher executive who needs the document), bribing, or coercing the manager.\n\nIn this small motivating example we see that two general attack strategies come into play: the attacker can actively pursue moving across the system and collecting items that will open him the way to the desired asset, or the attacker can attempt to orchestrate actions of other actors in the model so that they will do the necessary actions for him. Irrespectively of the chosen strategy, the process of attack trees generation by policy invalidation will systematically identify available (reachable) steps, add them to the tree, and refine those steps further, producing a complete attack tree in the end \\cite{Ivanova-Generation-2014}. Notice, that this summary is a simplification of the overall process, and we encourage the reader to refer to the original articles about the approach for more details \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Kammuller-2013}.\n\n\\section{Attack-Defence Model}\\label{sec:extraction}\n\n\\textbf{Extraction of defences from the model}\\\\\nThe only security controls the TREsPASS socio-technical model captures are access control policies that restrict access to certain locations. These policies can correspond to physical (locks) or digital (password check) means (policy enforcement mechanisms) implemented in the system to restrict access to assets. Therefore, we propose to make explicit in the attack generation process the fact that the attacker needs to overcome the restrictions imposed by security policies. To achieve that we will use attack-defence bundles that are based on the attack-defence tree formalism \\cite{Kordy-JLC-2014}. \n\nIntuitively, the attacker can chose from two approaches to deal with security policies in the system. He can attempt to satisfy the access control policy (for example, by collecting the necessary credentials or coercing someone with the right credentials) or he can try to circumvent the policy (e.g., by forcing the lock). The first approach is in line with the attack tree generation by policy invalidation process, because it can be automatically designed based on reachability. If we want to refine the second approach, we need to understand how exactly different policies (more precisely -- enforcement mechanisms for these policies) can be circumvented. There is a need to represent the human expert knowledge in circumventing different security mechanism in such a way that it is useful for automated generation process. To achieve that, one can use, for example, the hierarchical approach to attack representation suggested in \\cite{Pinchinat-WFMDS-2014}.\n\n  \nIndeed, the enforcement mechanisms for access control policies defined in the socio-technical model can be automatically introduced into attack-defence trees. If the knowledge about breaking certain kinds of enforcement mechanisms is available in a suitable format (e.g., the hierarchical representation), then the attack-defence trees can be further refined based on that information. Further analysis based on the attack-defence trees produced at this stage (e.g., computation of the most probable or the most cheap attack for the attacker \\cite{Bagnato-IJSSE-2012}) can identify the missing enforcement mechanisms. For example, if in the sensitive document scenario the attacker can directly access the document because the locker does not require any key (no access control is enforced for the document), it might be the first recommendation for improving security of the organisation: to introduce some appropriate access control mechanism (e.g., an actual lock with the key) to protect access to the document.\n\n\n\\subsection{Simplified Socio-Technical Model}\nWe introduce a simplified TREsPASS socio-technical model to exemplify the attack-defence model creation. The simplified model allows to reason only about potential reachability. However, this is already very useful for risk analysis, as quantitative evaluation of the possibility that an attacker accesses some system elements can simplify risk analysis for human analysts \\cite{Othmane-CS-2015}.\n\nThe simplified model captures simultaneously organisation's infrastructure topology for both physical and digital locations, as well as actors moving around this infrastructure (these can be persons or processes).  In the model these entities are represented as a set of model elements $N$ that is a union of a set of infrastructure locations $N_i$, actors $N_a$, and objects $N_o$. We consider two domains: $Ph$ is the physical space (model elements in this domain are physical entities, including, e.g.\\ rooms, persons, and items), while $Dg$ is the digital space (network locations and processes are in this domain), such that $N = Ph \\cup Dg$, and $Ph \\cap Dg = \\emptyset$. \n\n\nSome model elements are connected. We denote as $E \\subseteq N \\times N$ the set of directed connections. All edges $e$ in $E$ are of the following types:\n\\begin{itemize}\n\\item $e \\in E_{ii} \\subseteq N_i \\times N_i$: connections between infrastructure locations (rooms, corridors, etc.). These connections are assumed bi-directional. More precisely, if $(i_1,i_2) \\in E_{ii}$ then $(i_2,i_1) \\in E_{ii}$.\n\\item $e \\in E_{ai} \\subseteq N_a \\times N_i$: placement of actors in the infrastructure; \t\n\\item $e \\in E_{oi} \\subseteq N_o \\times N_i$: placement of objects in the infrastructure;\n\\item $e \\in E_{oa} \\subseteq N_o \\times N_a$: placement of objects that are carried around by actors; \n\\item $e \\in E_{oo} \\subseteq N_o \\times N_o$: placement of objects that are inside other objects; here $e = (o_1, o_2)$ denotes an object $o_1$ located within an object $o_2$.\n\\end{itemize}\nMutual intersections of $E_{ii}, E_{ai}, E_{oi}, E_{oa}, E_{oo}$ are empty sets. Elements of the same domain can be connected liberally. However, some self-evident restrictions apply when connections between elements of the physical and digital domains are considered. For example, a data file cannot be located in an office or inside a cupboard. We allow multiple locations for the same actor and object. This corresponds to the possibility of actors to move in the model, and represents that some items can appear in several locations. \n\nWe define a location function \\loc{}: $N$ $\\times$ $N$ as follows: \n$\\forall n \\in N $ \\loc{n}: = $\\{ l \\in N | (n, l) \\in E \\}$.\n\nNotice that for infrastructure locations or actors the function \\loc{} returns infrastructure locations where these model items are accessible from. However, as objects can be accessible from actors or other objects, \\loc{} may return any type of items in the model. \n\n\\textbf{Policies}\n\nLet $P$ be a set of policies defined in the model. We consider access control policies represented as tuples restricting access to element $n$. The \\emph{local policy} $\\delta_n$ is a set of individual access control configurations. Each access control configuration $p \\in \\delta_n$ is a tuple $\\langle Cred, atLocation, EM\\rangle $, where $Cred \\subseteq N_o$ is a set of credentials required to get access, $atLocation \\in N$ s.t. $(n, atLocation) \\in E$ is a model element from which access to $n$ is granted (e.g. access from the office to the locker is granted with the key in the example in Sec. \\ref{sec:motivation}), and $EM \\in N$ is a reference to the mechanism enabled in the model to enforce the policy. $EM$ can be the same as $atLocation$, meaning that the enforcement mechanism is implemented right at the spot (e.g., a lock), it can be an actor (e.g., a security guard checking identity documents or a process implementing access control), or an object. Notice that we assume that $c \\in Cred \\subseteq N_o$ is an asset present in the model, which can be either an item or data. \n\nIn theory, different access control configurations of the same local policy $\\delta_n$ can be enforced by different enforcement mechanisms. For example, to access a building employees might use a badge applying it to an RFID-reader, or they might show their IDs to a security guard. \n\n\n\\subsection{AD-Bundles Generation}\\label{subsec:generation}\n\n\n\n\n\nWe will now show how to generate attack-defence bundles (AD-bundles) that can be used to capture the attack-defence state of the system.\nAD-bundles are generated for individual assets. They consist of attack nodes that correspond to gaining access to items in the model and attacking these items, and defence nodes that represent protections offered by the local policies in place. Notice that the bundles are attacker-agnostic, and they refer only to the system configuration regarding some particular item. Our notation abuses the standard notation for attack-defence trees, as we use AD-terms to represent both the tree structure and to refer to concrete attacker goals. We also define different types of AD-terms. This is syntactic sugar to ease the type representation, as types are used to put bundles together and synthesise AD-trees.\n\n\n\\paragraph{Attack node types.}\nWe consider attack nodes can be of the following types. \n\\begin{itemize}\n\\item \\taccess$_n$ is an attack node that represents that the attacker gains access to item $n$. \n\\item \\taccfrom$_{n,l}$ represents the goal of the attacker to access item $n$ from specific model element $l$. This node type explicitly states the way $n$ is accessed in the model, thus allowing us to understand immediately what access control policy is applicable (by looking at the $atLocation$ attribute). \n\\item \\tbreak$_n$ represents the goal of the attacker to somehow disable an access control mechanism implemented in $n$ (this enforcement mechanism can protect assets not located in $n$).\n\\item \\tbreakpol$_p$ represents the goal of the attacker to overcome protection of an individual access control configuration $p$.\n\\item \\tsatpol$_p$ represents attacker's goal to satisfy access control configuration $p$ (by collecting all necessary credentials).\n\n\\end{itemize}\n\n\n\n\\paragraph{Defence node types.}\nThe defence nodes can be of the following types:\n\\begin{itemize}\n\\item \\tdefence$_{n,l}$ represents the defence of enforcement mechanisms enforcing policies at $l$ to control access to $n$ (notice that the enforcement mechanism itself can be located elsewhere).\n\\item  \\tdefpolicy$_p$ represents protection offered by an individual access control configuration for some $p \\in \\delta_{n}$. \n\\end{itemize}\nNotice that term types \\tbreakpol\\ and \\tdefpolicy\\ are required to satisfy the requirement of AD trees for the unique child of the opposite type \\cite{Kordy-JLC-2014}. \n\n\\paragraph{Bundle construction.}\nLet $n \\in N$ be an item in the model. An AD-bundle \\baccess$_n$ that characterises accessing $n$ is constructed as follows.\n\nWe start by setting the root of the bundle to \\taccess$_n$, as this is the desired attacker's goal. \n\nNext, \\taccess$_n$ is refined:\\\\\n\\taccess$_n$ := \\OR$^p$ $\\Big( \\taccfrom_{n,l} | l \\in \\loc{n}  \\Big)$ \\ \\ \/\/ \\emph{$n$ can be accessed only from an adjacent element in the model. Any of these elements is suitable for the attacker}\\\\\n\nIf $\\nexists p = \\langle Cred, l, EM \\rangle \\in \\delta_n$ then\\\\\n \\taccfrom$_{n,l}$ := $ \\taccess_{l}$ \\ \\ \/\/ \\emph{access to $n$ from $l$ can be implemented by simply accessing $l$. No access control policy is set up to guard this connection.}\\\\\n\n\nIf $\\exists p = \\langle Cred, l, EM \\rangle \\in \\delta_n$ then\\\\\n\\taccfrom$_{n,l}$ := $c^p$ $\\Big( \\taccess_{l}, \\tdefence_{n,l} \\Big)$ \\ \\ \/\/ \\emph{access to $n$ from $l$ can be implemented by accessing $l$. However, as there is an enforcement mechanism that controls access, the defence node is also added.}\\\\\n\n\\tdefence$_{n,l}$ := $\\AND^o \\Big( \\tdefpolicy_p | \\forall p \\in \\delta_n$ s.t. $ p = \\langle Cred, l, s \\rangle \\Big)$ \\ \\ \/\/ \\emph{Protection of access from $l$ to $n$ is implemented via individual policy configurations.}\n\n\n\\tdefpolicy$_p$ := $c^p \\Big( \\tbreakpol_p \\Big)$ \\ \\ \/\/ \\emph{syntactic sugar to switch back to attacker's view} \n\n\\tbreakpol$_p$ := $\\OR^p \\Big( \\tsatpol_p, \\tbreak_{s} \\Big)$, where $p = \\langle Cred, l, s \\rangle$ \\ \\ \/\/ \\emph{Attacker can either satisfy the individual policy configuration $p$, or he can break the enforcement mechanism $s$ that enforces this configuration $p$. }\n\n\\tsatpol$_p$ := $\\AND^p \\Big( \\taccess_{cred} | \\forall cred \\in Cred \\Big)$, where $p = \\langle Cred, l, s \\rangle$ \\ \\ \/\/ \\emph{To satisfy the configuration the attacker needs to access all credentials in the set $Cred$ identified in this configuration}.\n\nWe provide an example of an AD bundle in Fig. \\ref{fig:bundle}.\nBy construction, for each bundle $\\baccess_n$ its leaf nodes are either terms of the same type (\\taccess$_l$ for some $l$), or terms $\\tbreak_s$. We do not refine terms of the type \\tbreak$_{s}$ because the model itself lacks the knowledge how enforcement mechanisms can be broken. If an additional knowledge on breaking enforcement mechanisms will be available (e.g., as a hierarchy of attacks \\cite{Pinchinat-WFMDS-2014}), this term can further expanded. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{figure}\n\\caption{An AD bundle.}\n\\label{fig:bundle}\n\\end{figure}\n\n\\subsection{Approach to Synthesise AD-Trees}\n\nAD-bundles represent attacks on individual assets in the model. They can be ``glued'' together to form AD-trees, in the spirit of attack generation by policy invalidation. In this subsection we outline an approach to synthesis of attack-defence trees.\n\nThe main requirement for AD-trees synthesis is that it should terminate. Indeed, it is easy to see that any simple loop in the infrastructure will create infinite trees if bundles are composed naively. Moreover, some bundles may appear more than once in the generated tree, creating duplicate subtrees. To avoid this, we introduce a system state that will keep track of already achieved progress and will allow to terminate the synthesis process when the attacker has achieved the goal. \n\n\n\n\\paragraph{State.}\nWe define now two functions that identify the state of the system. These functions will be updated as the attack tree is generated in order to keep track with the attack development.\n\n\\iffalse{\nWe define a boolean function \\Posesses{}{} $\\subseteq$ $N_a$ $\\times$ $N_o$:\n\\begin{itemize}\n\\item If $(o,a) \\in E_{oa}$, and  \\Posesses{a}{o} := \\texttt{True}.\n\\item If for some $o_2 \\in N_o$ $(o_1,o_2) \\in E_{oo}$, and \\Posesses{a}{o_2} = \\texttt{True}, then \\Posesses{a}{o_1} := \\texttt{True}.\n\\item Else \\Posesses{a}{n} = \\texttt{False}.\t\n\\end{itemize}\nIntuitively, the function \\Posesses{}{} refers to objects that an actor has.\n}\\fi\n\n\\begin{definition}[\\Reachable{}{}]\t\nLet \\model\\ = $(N,E)$ be a model. We define a boolean function \\Reachable{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item If $(a,n) \\in E$, \\Reachable{a}{n} := \\texttt{True}.\n\\item If for some $l \\in N_i$ $(a,l) \\in E_{ai}$ and $(o,l) \\in E_{oi}$, then \\Reachable{a}{o} := \\texttt{True}. \n\\item If for some $l \\in N_i$ $(a,l) \\in E_{ai}$ and $(a_1,l) \\in E_{ai}$, then \\Reachable{a}{a_1} := \\texttt{True} and \\Reachable{a_1}{a} := \\texttt{True}. \n\n\\item Else \\Reachable{a}{n} := \\texttt{False}.\t\n\\end{itemize}\n\\end{definition}\n\nThis function initially captures for a given actor all items immediately reachable in the model. These items can be objects or actors located in the same location as the actor. Let \\Reach{a} := \\{$\\forall n \\in N$ s.t. \\Reachable{a}{n} = \\texttt{True}\\}.\n\n\n\\begin{definition}[\\Granted{}{}]\nWe define a boolean function \\Granted{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item If for an item $n$ $\\delta_n = \\emptyset $ then \\Granted{a}{n} := \\texttt{True}.\n\\item If for an item $n$ there is a tuple $p = \\langle Cred, atLocation\\rangle$ $\\in \\delta_n =$ s.t. $Cred \\subseteq \\Reach{a} \\cap N_o$ then \\Granted{a}{n} := \\texttt{True}.\n\\item Else \\Granted{a}{n} := \\texttt{False}.\n\\end{itemize}\n\\end{definition}\n\nIntuitively, this function refers to some policy configuration that grants access to $n$. If \\Granted{a}{n} = \\texttt{True}, then there is a way for this actor to satisfy the access control policy for $n$ (possibly under condition that he arrives at the right location). \n\nLet us define a model state.\n\\begin{definition}[State]\nA generated state for a model \\model\\ is a tuple $\\langle \\Reachable{}{}, \\Granted{}{} \\rangle$.\t \n\\end{definition}\n\n\\begin{definition}[\\Accessible{}{}]\nWe define a boolean function \\Accessible{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item \\Accessible{a}{n} := \\Reachable{a}{n} \\AND\\ \\Granted{a}{n}\n\\end{itemize}\n\n\t\n\\end{definition}\n\n\n\n\\paragraph{Bootstrapping}\nGiven a model \\model= $\\langle N,E \\rangle$ produced by a modeller, the functions \\Reachable{}{}, \\Granted{}{} and \\Accessible{}{} are initially computed from \\model. First we compute a transitive closure of reachable locations:\n\\begin{itemize}\n\\item \\Reachable{a}{n} \t:= \\Reachable{a}{n} \\OR\\ ($\\exists l \\in N:$ \\Accessible{a}{l} \\AND ($(l,n) \\in E$ \\OR $(n,l) \\in E$) )\n\\end{itemize}\nNotice that here we do not re-compute the function \\Granted{}{}, and thus, eventually, the reachable objects set for each actor will increase only with locations that are not guarded by access control policy. Once \\Reachable{}{} is recomputed, it can be used to quickly evaluate whether an actor can reach certain locations in the original model (where may he end up). \n\n\n\n\\subsubsection{Synthesis of AD-trees from Bundles}\n\nWe now discuss composition of generated attack-defence trees. An attack-defence tree $\\tree(\\eta,\\alpha)$ is synthesised for a chosen attacker $\\eta \\in N_a$ and a target asset $\\alpha \\in N_o$. The root node is the bundle \\taccess$_\\alpha$. For each leaf node of the type \\taccess$_b$ we can compute its value by referring to the corresponding AD bundle $\\baccess_b$. \n\n\\paragraph{Bundle Value.}\nIn the simplest case we use propositional semantics for evaluating AD-bundles and, eventually, AD-trees \\cite{Kordy-JLC-2014}. For leaf nodes of the type \\taccess$_n$, \\taccess$_n$ $\\equiv$ \\Accessible{\\eta}{n}. For leaf nodes of the type \\tbreak$_s$, \\tbreak$_s$ $\\equiv$ \\texttt{False} in the current synthesis approach. Thus, given a bundle for asset $n$, we can evaluate its value based on the values of the leaf nodes available. By updating the model state as attack progresses (more items become reachable to the attacker) we can eventually evaluate the target bundle, once all its descendants become evaluated. As state changes monotonically, the process will eventually terminate.\n\n\n\n\n\n\n\\section{Introducing New Defences}\\label{sec:2ndstep}\nThe enforcement mechanisms for access control policies are not the only type of security controls that organisations use. Moreover, access control is not the only remedy that can be advised to improve security. Indeed, the existing risk analysis standards and security catalogues that guide practitioners in risk analysis identify many types of security controls and countermeasures. Many of those (for example, security cameras) cannot be captured by socio-technical models directly, because it will introduce unnecessary complications to the model. Some countermeasures can be introduced as properties of system elements (e.g., after a security training the employees might become less susceptible to social-engineering), but not as independent elements of the system. \n\nWe want to be able to update the attack-defence model of our system, captured by the suite of attack-defence bundles, after the first stage of automated generation. At this second stage we would like to obtain more complete attack-defence bundles with new defence nodes added that can capture additional security countermeasures (either existing in the model already or newly proposed once). We have two main questions associated with the newly introduced defences: \\emph{how to generate\/propose new defences} and \\emph{where to place them in the attack-defence model} to keep the consistency across many attack scenarios and system updates. We start by addressing the second question first.\n\n\n\\textbf{Where to put new countermeasures}\\\\\nGiven an AD bundle representing the goal of an attacker to access asset $n$, two types of attack nodes are the candidates to be protected from by some countermeasures: the root node \\taccess$_n$ and its children \\taccfrom$_{n,l}$. Indeed, for the connectors to other bundles (the leaf nodes \\taccess$_b$) it will make sense to introduce defences at the corresponding bundle to ensure the consistency requirement. For the nodes \\tsatpol$_p$, the attacker's goal is to satisfy the policy by finding the right credentials. It is not obvious what can be done as a protective measure besides protecting the credentials themselves. As for the nodes representing circumventing the enforcement mechanism, \\tbreak$_{s}$, we do not have enough details for the moment how the attacker is going to break it. If this node is to be refined using some attack pattern library, it is better to create a separate AD bundle for treating the scenarios and assign defences there.\n\nNow we have candidate attack nodes to be assigned countermeasures. To select the countermeasures that could be assigned, we first review existing types of security controls. It is well-established in the security industry to classify controls as preventive, detective and corrective \\cite{NIST-800-30}:\n\\begin{itemize}\n\\item \\emph{Preventive}\tcontrols focus on preventing security incidents from occurring. \n\\item \\emph{Detective} controls focus on detecting occurrences of security incidents.\n\\item \\emph{Corrective} controls focus on aiding the organisation to recover from a security incident. \n\\end{itemize}\n\nFrom the implementation perspective, it is traditional to divide controls into the following categories \\cite{NIST-800-30}:\n\\begin{itemize}\n\\item \\emph{Technical} controls that are implemented typically as software controls.\n\\item \\emph{Management}, or administrative, controls that are implemented as procedures and guidelines.\n\\item \\emph{Operational} controls that focus on ensuring security and dependability of operations. These controls include physical security controls (physical access control, fire and water damage protection, etc.) and some controls that are difficult to classify as fully technical or physical (e.g., protection of personal computers).   \t\n\\end{itemize}\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{figure1}\n\\caption{An updated AD bundle with defence nodes in the designated positions (children of EM$_(n,l)$ not shown).}\n\\label{fig:updatedbundle}\n\\end{figure}\n\n\nFrom this classification, we propose a way to update AD bundles with security controls in a consistent manner. The preventive controls can be added as children to the attack nodes \\taccfrom$_{n,l}$, because they correspond to preventive measures for certain directed actions of the attacker. Access control policies present in the model are already embedded in the bundles at this position. To satisfy the attack-defence trees requirement of only one child of the opposite type, we will modify the bundle as in Fig.~\\ref{fig:updatedbundle} (now the node \\tpreventive$_{n,l}$ is a parent of the node \\tdefence$_{n,l}$). \n\nThe detective and corrective measures can be added as children to the root node \\taccess$_n$ (see Fig.~\\ref{fig:updatedbundle}). In this position the defence nodes are directly linked to the system object in question, be it a location, a person, or an object. The semantics of the controls placed in this position are clear: assuming the attacker has already gained access to his target, is this detectable or what can be the remedy for this? Notice that some controls in practice can be both detective and preventive (e.g., security guards). In this case, it is safe to classify them as preventive controls.\n\n\\textbf{What defences to choose}\\\\\nThe choice of security controls is a tough question in practice. Not only it requires the human analyst to know possible attacks and countermeasures, but also the analyst needs to solve a complex multi-parameter optimisation problem. Indeed, the controls addressing the same threat can have different cost, efficiency and effectiveness. They can be more or less compliant with the industry standards and best practices. They can be more or less easy to implement and easy to verify. Finally, they can be more or less desired by the organisation because of personal views of the top-management. Thus, if we just consider the baseline controls listed by NIST Special Publication 800-53 \\cite{NIST-800-53}, and try to evaluate all the above-mentioned parameters in order to fully automate the defence selection (now that we know where to place them), we already will face a very complex problem. Moreover, a single mistake in evaluation of some of the values will likely make the full analysis invalid. Therefore, it is likely that human-assisted control selection cannot be fully replaced by automated defence generation, at least for some time.\n\nYet, we can try to facilitate the defence selection problem by further categorising the security controls based on applicability to the scenarios in question and usefulness in attribute-based computations. \n\nIn the socio-technical model we have clear categories of objects: locations, actors and items that can belong to either physical or digital space. Thus controls can be chosen based already on simple considerations such as ``access to digital objects by processes can be protected by using technical preventive controls'', or ``access of humans to humans can be protected by administrative and physical preventive controls''. Table~\\ref{tab:preventive} summarises these choices of controls. In this table, it is expected that respective controls will be introduced in the dedicated AD bundles (following the template in Fig.~\\ref{fig:updatedbundle}).\n\n\n\\begin{table}[t!]\n\\centering\n\\caption{Controls selection based on system elements. }\n\\label{tab:preventive}\n\\scriptsize\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Entity}  &  \\textbf{Physical space}  & \\textbf{Digital space}   \\\\\n\\hline\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Preventive}}\\\\\n\\hline\nLocation & Physical access control  & Technical access control, firewall  \\\\\n\\hline\nActor & Physical access control, Security trainings, Email filter  & Technical access control and authentication\\\\\n\\hline\nObject & Physical access control  & Technical access control \\\\\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Detective}}\\\\\n\\hline\nLocation & \\multirow{3}{*}{ Security cameras, visitor logs}  & \\multirow{3}{*}{System logs, IDS} \\\\\nActor & &\\\\\nObject & & \\\\\n\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Corrective}}\\\\\n\\hline\nLocation & \nInsurance, liability limitation,   & Insurance, liability limitation, \\\\\nActor &  business continuity plan & secure state restoring mechanisms, \\\\\nObject &   & business continuity plan \\\\\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nFurthermore, following the investigation of attribute decoration on attack-defence trees by Bagnato et al. \\cite{Bagnato-IJSSE-2012}, we can look at what controls contribute to computations of certain attributes. For instance, if the analyst is interested in the probability of an attack to succeed, the minimal cost of attack for an attacker, or time of executing an attack, then (under the assumption that detection cannot stop the attack) she would like to look at her preventive measures. If she is interested in the impact the attack has on her organisation (how business continuity is affected after the attack was executed), she would like to consider the preventive and corrective controls, especially the latter ones, because these ensure business continuity. Thus, if she is interested only in some attributes, the computation on AD bundles does not need to take into account all controls at once. Table~\\ref{tab:attributes} summarises the control types that are the most relevant for some selected attributes. \n\n\\begin{table}[t!]\n\\centering\n\\caption{Relevant controls for example attributes}\n\\label{tab:attributes}\n\\scriptsize\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Attribute}  &  \\textbf{Preventive}  & \\textbf{Detective}  & \\textbf{Corrective}  \\\\\n\\hline\n\\hline\nRisk of detection &  & \\checkmark & \\\\\nCost of attack (for attacker) & \\checkmark &  & \\\\\nProbability of attack success & \\checkmark &  & \\\\\nTime of attack & \\checkmark &  & \\\\\nImpact of attack & \\checkmark & \\checkmark & \\checkmark\\\\\n\n\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nNotice that the controls added at this stage will probably not follow the AD bundle notation, but will be expressed in the natural language (e.g, \\emph{security training} or \\emph{ID check}). This is understandable, because, as we have mentioned, models are not rich enough by nature. Yet, this is acceptable for the format, because the attack-defence model does not need to be fully formal. On the contrary, it is used to assist the human analyst to create and maintain the attack-defence view on the system. The only requirement that we have for it is the consistency, which is ensured by adding each control only to the bundle representing the attack-defence view of a particular (unique) entity in the model. Some controls can require a notion of \\emph{perimeter} to be defined in the model, so that they can be uniquely assigned to the bundle corresponding to the perimeter, and not to each entity belonging to that perimeter. This is easily implementable in any socio-technical model.\n\n\n\n\n\\section{Related Work}\\label{sec:discussion}\nThe question of attack trees generation from system models has been tackled in \\cite{Ivanova-Generation-2014}. Similarly, \\cite{Vigo-CSF-2014} and \\cite{Ou-CCS-2006} worked on generating attack models from a system model. While we follow the same approach for  attacker's view, our main focus is on keeping both attacker's and defender's views consistent with the main socio-technical model.  \n\nAttack-countermeasure trees (ACTs) is an alternative model to attack-defence trees in keeping both views simultaneously \\cite{Roy-SCN-2011}. In \\cite{Roy-DSN-2012} the authors have investigated optimal countermeasure selection for ACTs when a set of possible countermeasures to be implemented is already predefined. It will be interesting to investigate ACTs suitability for the attack-defence model, because they support explicit detection and mitigation countermeasure nodes (but not corrective). \n\nIn \\cite{Lenzini-2015} the authors work on directly applying model checking to a socio-technical model in order to evaluate some reachability-based security properties. \n\nFerreira et al. \\cite{STEAL-2014} have discussed defences suggestion in the context of the socio-technical model STEAL. They propose to apply defences at the technical and social levels of the system, what is in line with our proposal for applying security control categories in selecting defences.\n\n\\section{Conclusion and the Next Steps}\n\nIn this work we have approached the question of creating and maintaining the security controls representation in parallel to the socio-technical model. Our solution creates a set of attack-defence bundles (small attack-defence trees) that can be maintained with a socio-technical model as its separate view. The bundles are generated from the model in the beginning, but afterwards they are enriched consistently alongside the new security controls identified by a human analyst. We have also discussed how new controls can be selected based on the model entities and the attributes of interest to the analyst. This work attempts to bridge the gap between the approach of automated attack generation from system model and the manual security control selection in the traditional risk analysis. The next step is to look into the compositional attack-defence tree synthesis for more complex attribute domains.  After that, it will be possible to investigate optimal countermeasure selection, based, e.g., on the approaches suggested in \\cite{Aslanyan-POST-2015} and \\cite{Roy-DSN-2012}. Another further research direction is practical validation of the proposed approach on realistic case studies and evaluation of its usefulness and scalability.\n\n\\subsubsection*{Acknowledgements}\nThis work was partially supported by the European Commission through the FP7 project TREsPASS (grant agreement n. 318003) and by Fonds National de la Recherche Luxembourg through the ADT2P project (grant n. C13\/IS\/5809105).\n\n\n\\bibliographystyle{splncs03}\n\n\\section{Introduction}\\label{sec:introduction}\n\nModels are used in all stages of the security process: from security requirements elicitation and organisational risk assessment to run-time verification and business process compliance audit. Often these models are inter-connected. For example, if a security requirements model for a software system was elicited, on the later stage it may be re-used to design the security testing process for this system. At the same time, as manual production of security models is very tedious and error-prone, many researchers and practitioners look into automating the model creation and transformation processes.\n\nRecently security researchers have looked at systematic design \\cite{Paul-GraMSec-2014} and automated generation of attack models \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Ou-CCS-2006}, \\cite{Vigo-CSF-2014}, such as attack graphs and attack trees, from system models. This model transformation allows to switch the view from the system description perspective to a compact representation of possible attacker actions. At the same time, given the generated attack model, the system defender is interested to find the weakest links: the spots in the model where additional security controls can be introduced to improve protection and eliminate potential attacks. Therefore, automated generation of defences is an obvious next step in the process. \n\nIn this paper we look at socio-technical models as succinct abstractions of large organisations. Such models capture simultaneously locations, actors and objects in the system. They often take into account both physical and digital domains and offer to a human analyst the means to represent ``the world as it is''. That means that the designer of socio-technical systems does not need to be a security or risk analysis expert. She only needs to know the intricacies of her own company (department) to be able to model it. With the system model at hand, at the next step the attack generation tools aim at automatic creation of attack scenarios that can be further discussed by security professionals. The overall idea of this process is to automate threat scenarios identification (an important aspect of risk analysis) as much as possible. \n\nIn this paper we would like to push the envelope even further. Our main question is: \\emph{given a socio-technical system model, how to find and capture, possibly automatically, the security controls that will counteract the discovered threats?} Indeed, the main goal of risk analysis is to improve the existing system by introducing new security controls, so that the most dangerous or easily executed attacks are thwarted. Therefore, automated creation of attack scenarios only is not yet a full solution. \n\nWe want to look at perspectives and limitations of automated defence generation from socio-technical models. It seems that the main obstacle to rich defensive strategies generation directly from the model is the fact that socio-technical models do not capture many security controls.  \n\nTo find an answer to the main question, we start from investigating the security controls (defences) already present in an advanced socio-technical model and propose a scheme to extract these controls, together with the attack steps, in the compact format of attack-defence bundles. We then evaluate the limitations of the extracted defences inherent from the socio-technical model and discuss how to overcome these limitations. We argue that an attack-defence model needs to be maintained (in parallel with the socio-technical model) that can capture not only the attacker's view but also the defender's view of the system. In this paper we have chosen attack-defence trees \\cite{Kordy-JLC-2014} as the basis for the attack-defence model. As an alternative to this model, one can choose, for example, attack-countermeasure trees \\cite{Roy-SCN-2011}.\n\n\n\nThe goal of this paper is to propose an attack-defence view for socio-technical models that can capture simultaneously attacker's options and available\/proposed countermeasures in the system. The main idea is that given that view it can be easily synchronised with the model (but it contains richer defence information than the model), and it can be used to synthesise attack-defence trees and evaluate different interesting attributes. \n\n\\section{Socio-Technical Models versus Attack-Defence Models}\\label{sec:motivation}\n\nAs socio-technical models are abstractions, they do not capture all defensive mechanisms that can be available in an organisation, but only a subset of them. Indeed, it is impossible to model all security-relevant devices, protocols and behaviours in a single model. Typically, socio-technical models look at capturing organisational infrastructure (e.g., \\cite{D1.3.1}, \\cite{ANKH}, \\cite{Exasym-2008}, \\cite{Lenzini-2015}), but sometimes they can focus only on some aspects of human-computer interactions (e.g., \\cite{Radomirovic-2015}, \\cite{STEAL-2014}).\n\n\n\nSince all security aspects cannot be captured by a socio-technical model without overcomplicating it, we argue that there is a need to maintain a separate view of attack and defence capabilities of the system together with the socio-technical model. Preferably, we should be able to trace the objects in the socio-technical model into the attack-defence model and back. \n\n\\textbf{Requirements for the attack-defence model}\n\nThe first requirement for the chosen attack-defence model is that \\emph{the defences that are already captured by the model need to be represented explicitly in the attack-defence model}. Indeed, we would like to faithfully represent the system security state. So, if some security control is captured by the system model, it should be translated into the generated attack-defence model. \n\nSecondly, we want to propose a way to \\emph{update the generated defender's view (the security controls obtained directly from the system model) with more security controls and countermeasures} of the organisation. This \\emph{update needs to be consistent}: once a security control is captured in the attack-defence model, it should be traced to an object in the system model. For example, if our approach identifies that a security camera is to be placed in a certain location in the system, all attack scenarios that involve that location should be updated to take the camera into account. In this way later on one can investigate automated defence generation process that will maintain consistency of the socio-technical system. \n\n\n\\textbf{Background}\n\nIn this paper we use the TREsPASS socio-technical model \\cite{D1.3.1} that is graph-based. We can briefly summarise this model as follows. Locations in the system represent physical and network locations; actors model humans and processes; and items can be physical or digital objects. Edges among locations represent connectedness (e.g., adjacent rooms), and all actors and items are located somewhere in the system. Actors can possess items, and items can be embedded into other items. Some locations have access control policies attached to them. These policies specify a set of credentials (items in the system) an actor needs to possess to enter the location or access the object. These policies can also be formalized by more complex predicates capturing, e.g., role-based access control or trust relationships among actors. \n\nAs the starting point for the attack-defence model, we consider the process of attack trees generation by policy invalidation that relies on structural information about the system \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Kammuller-2013}. This process was initially designed for the TREsPASS socio-technical model \\cite{D1.3.1}, but it can be applied to other socio-technical models capturing systems as graphs, e.g., \\cite{Exasym-2008}, \\cite{ANKH}, \\cite{Portunes-2010}, \\cite{Lenzini-2015}, because it is reachability-based.\n\n\n\nIn short, this process is started by choosing an asset among the entities in the system. The attacker is also selected among actors in the system (the main goal of the attacker is to invalidate the security policy, e.g., confidentiality or integrity policy, associated with this asset). Then, based on the reachability reasoning, the process systematically searches for the ways for the attacker to access the asset. For example, consider the asset to be a sensitive document located in a locker in the manager's office, and the attacker to be an insider (an employee) working on the same floor. To access the document, the attacker can try to access the locker and open it (an AND-decomposition \\cite{Kordy-JLC-2014}). This might require possession of the key to the locker that needs to be obtained elsewhere in the system.  Alternatively (an OR-decomposition with the previous attack), also the manager has access to the locker and the document. Thus, the attacker can get access the document by influencing the manager. This can be implemented through, e.g, social engineering (for instance, befriending the manager, or hiring an external actor to pretend to be a higher executive who needs the document), bribing, or coercing the manager.\n\nIn this small motivating example we see that two general attack strategies come into play: the attacker can actively pursue moving across the system and collecting items that will open him the way to the desired asset, or the attacker can attempt to orchestrate actions of other actors in the model so that they will do the necessary actions for him. Irrespectively of the chosen strategy, the process of attack trees generation by policy invalidation will systematically identify available (reachable) steps, add them to the tree, and refine those steps further, producing a complete attack tree in the end \\cite{Ivanova-Generation-2014}. Notice, that this summary is a simplification of the overall process, and we encourage the reader to refer to the original articles about the approach for more details \\cite{Ivanova-Generation-2014}, \\cite{D3.4.1-2014}, \\cite{Kammuller-2013}.\n\n\\section{Attack-Defence Model}\\label{sec:extraction}\n\n\\textbf{Extraction of defences from the model}\\\\\nThe only security controls the TREsPASS socio-technical model captures are access control policies that restrict access to certain locations. These policies can correspond to physical (locks) or digital (password check) means (policy enforcement mechanisms) implemented in the system to restrict access to assets. Therefore, we propose to make explicit in the attack generation process the fact that the attacker needs to overcome the restrictions imposed by security policies. To achieve that we will use attack-defence bundles that are based on the attack-defence tree formalism \\cite{Kordy-JLC-2014}. \n\nIntuitively, the attacker can chose from two approaches to deal with security policies in the system. He can attempt to satisfy the access control policy (for example, by collecting the necessary credentials or coercing someone with the right credentials) or he can try to circumvent the policy (e.g., by forcing the lock). The first approach is in line with the attack tree generation by policy invalidation process, because it can be automatically designed based on reachability. If we want to refine the second approach, we need to understand how exactly different policies (more precisely -- enforcement mechanisms for these policies) can be circumvented. There is a need to represent the human expert knowledge in circumventing different security mechanism in such a way that it is useful for automated generation process. To achieve that, one can use, for example, the hierarchical approach to attack representation suggested in \\cite{Pinchinat-WFMDS-2014}.\n\n  \nIndeed, the enforcement mechanisms for access control policies defined in the socio-technical model can be automatically introduced into attack-defence trees. If the knowledge about breaking certain kinds of enforcement mechanisms is available in a suitable format (e.g., the hierarchical representation), then the attack-defence trees can be further refined based on that information. Further analysis based on the attack-defence trees produced at this stage (e.g., computation of the most probable or the most cheap attack for the attacker \\cite{Bagnato-IJSSE-2012}) can identify the missing enforcement mechanisms. For example, if in the sensitive document scenario the attacker can directly access the document because the locker does not require any key (no access control is enforced for the document), it might be the first recommendation for improving security of the organisation: to introduce some appropriate access control mechanism (e.g., an actual lock with the key) to protect access to the document.\n\n\n\\subsection{Simplified Socio-Technical Model}\nWe introduce a simplified TREsPASS socio-technical model to exemplify the attack-defence model creation. The simplified model allows to reason only about potential reachability. However, this is already very useful for risk analysis, as quantitative evaluation of the possibility that an attacker accesses some system elements can simplify risk analysis for human analysts \\cite{Othmane-CS-2015}.\n\nThe simplified model captures simultaneously organisation's infrastructure topology for both physical and digital locations, as well as actors moving around this infrastructure (these can be persons or processes).  In the model these entities are represented as a set of model elements $N$ that is a union of a set of infrastructure locations $N_i$, actors $N_a$, and objects $N_o$. We consider two domains: $Ph$ is the physical space (model elements in this domain are physical entities, including, e.g.\\ rooms, persons, and items), while $Dg$ is the digital space (network locations and processes are in this domain), such that $N = Ph \\cup Dg$, and $Ph \\cap Dg = \\emptyset$. \n\n\nSome model elements are connected. We denote as $E \\subseteq N \\times N$ the set of directed connections. All edges $e$ in $E$ are of the following types:\n\\begin{itemize}\n\\item $e \\in E_{ii} \\subseteq N_i \\times N_i$: connections between infrastructure locations (rooms, corridors, etc.). These connections are assumed bi-directional. More precisely, if $(i_1,i_2) \\in E_{ii}$ then $(i_2,i_1) \\in E_{ii}$.\n\\item $e \\in E_{ai} \\subseteq N_a \\times N_i$: placement of actors in the infrastructure; \t\n\\item $e \\in E_{oi} \\subseteq N_o \\times N_i$: placement of objects in the infrastructure;\n\\item $e \\in E_{oa} \\subseteq N_o \\times N_a$: placement of objects that are carried around by actors; \n\\item $e \\in E_{oo} \\subseteq N_o \\times N_o$: placement of objects that are inside other objects; here $e = (o_1, o_2)$ denotes an object $o_1$ located within an object $o_2$.\n\\end{itemize}\nMutual intersections of $E_{ii}, E_{ai}, E_{oi}, E_{oa}, E_{oo}$ are empty sets. Elements of the same domain can be connected liberally. However, some self-evident restrictions apply when connections between elements of the physical and digital domains are considered. For example, a data file cannot be located in an office or inside a cupboard. We allow multiple locations for the same actor and object. This corresponds to the possibility of actors to move in the model, and represents that some items can appear in several locations. \n\nWe define a location function \\loc{}: $N$ $\\times$ $N$ as follows: \n$\\forall n \\in N $ \\loc{n}: = $\\{ l \\in N | (n, l) \\in E \\}$.\n\nNotice that for infrastructure locations or actors the function \\loc{} returns infrastructure locations where these model items are accessible from. However, as objects can be accessible from actors or other objects, \\loc{} may return any type of items in the model. \n\n\\textbf{Policies}\n\nLet $P$ be a set of policies defined in the model. We consider access control policies represented as tuples restricting access to element $n$. The \\emph{local policy} $\\delta_n$ is a set of individual access control configurations. Each access control configuration $p \\in \\delta_n$ is a tuple $\\langle Cred, atLocation, EM\\rangle $, where $Cred \\subseteq N_o$ is a set of credentials required to get access, $atLocation \\in N$ s.t. $(n, atLocation) \\in E$ is a model element from which access to $n$ is granted (e.g. access from the office to the locker is granted with the key in the example in Sec. \\ref{sec:motivation}), and $EM \\in N$ is a reference to the mechanism enabled in the model to enforce the policy. $EM$ can be the same as $atLocation$, meaning that the enforcement mechanism is implemented right at the spot (e.g., a lock), it can be an actor (e.g., a security guard checking identity documents or a process implementing access control), or an object. Notice that we assume that $c \\in Cred \\subseteq N_o$ is an asset present in the model, which can be either an item or data. \n\nIn theory, different access control configurations of the same local policy $\\delta_n$ can be enforced by different enforcement mechanisms. For example, to access a building employees might use a badge applying it to an RFID-reader, or they might show their IDs to a security guard. \n\n\n\\subsection{AD-Bundles Generation}\\label{subsec:generation}\n\n\n\n\n\nWe will now show how to generate attack-defence bundles (AD-bundles) that can be used to capture the attack-defence state of the system.\nAD-bundles are generated for individual assets. They consist of attack nodes that correspond to gaining access to items in the model and attacking these items, and defence nodes that represent protections offered by the local policies in place. Notice that the bundles are attacker-agnostic, and they refer only to the system configuration regarding some particular item. Our notation abuses the standard notation for attack-defence trees, as we use AD-terms to represent both the tree structure and to refer to concrete attacker goals. We also define different types of AD-terms. This is syntactic sugar to ease the type representation, as types are used to put bundles together and synthesise AD-trees.\n\n\n\\paragraph{Attack node types.}\nWe consider attack nodes can be of the following types. \n\\begin{itemize}\n\\item \\taccess$_n$ is an attack node that represents that the attacker gains access to item $n$. \n\\item \\taccfrom$_{n,l}$ represents the goal of the attacker to access item $n$ from specific model element $l$. This node type explicitly states the way $n$ is accessed in the model, thus allowing us to understand immediately what access control policy is applicable (by looking at the $atLocation$ attribute). \n\\item \\tbreak$_n$ represents the goal of the attacker to somehow disable an access control mechanism implemented in $n$ (this enforcement mechanism can protect assets not located in $n$).\n\\item \\tbreakpol$_p$ represents the goal of the attacker to overcome protection of an individual access control configuration $p$.\n\\item \\tsatpol$_p$ represents attacker's goal to satisfy access control configuration $p$ (by collecting all necessary credentials).\n\n\\end{itemize}\n\n\n\n\\paragraph{Defence node types.}\nThe defence nodes can be of the following types:\n\\begin{itemize}\n\\item \\tdefence$_{n,l}$ represents the defence of enforcement mechanisms enforcing policies at $l$ to control access to $n$ (notice that the enforcement mechanism itself can be located elsewhere).\n\\item  \\tdefpolicy$_p$ represents protection offered by an individual access control configuration for some $p \\in \\delta_{n}$. \n\\end{itemize}\nNotice that term types \\tbreakpol\\ and \\tdefpolicy\\ are required to satisfy the requirement of AD trees for the unique child of the opposite type \\cite{Kordy-JLC-2014}. \n\n\\paragraph{Bundle construction.}\nLet $n \\in N$ be an item in the model. An AD-bundle \\baccess$_n$ that characterises accessing $n$ is constructed as follows.\n\nWe start by setting the root of the bundle to \\taccess$_n$, as this is the desired attacker's goal. \n\nNext, \\taccess$_n$ is refined:\\\\\n\\taccess$_n$ := \\OR$^p$ $\\Big( \\taccfrom_{n,l} | l \\in \\loc{n}  \\Big)$ \\ \\ \/\/ \\emph{$n$ can be accessed only from an adjacent element in the model. Any of these elements is suitable for the attacker}\\\\\n\nIf $\\nexists p = \\langle Cred, l, EM \\rangle \\in \\delta_n$ then\\\\\n \\taccfrom$_{n,l}$ := $ \\taccess_{l}$ \\ \\ \/\/ \\emph{access to $n$ from $l$ can be implemented by simply accessing $l$. No access control policy is set up to guard this connection.}\\\\\n\n\nIf $\\exists p = \\langle Cred, l, EM \\rangle \\in \\delta_n$ then\\\\\n\\taccfrom$_{n,l}$ := $c^p$ $\\Big( \\taccess_{l}, \\tdefence_{n,l} \\Big)$ \\ \\ \/\/ \\emph{access to $n$ from $l$ can be implemented by accessing $l$. However, as there is an enforcement mechanism that controls access, the defence node is also added.}\\\\\n\n\\tdefence$_{n,l}$ := $\\AND^o \\Big( \\tdefpolicy_p | \\forall p \\in \\delta_n$ s.t. $ p = \\langle Cred, l, s \\rangle \\Big)$ \\ \\ \/\/ \\emph{Protection of access from $l$ to $n$ is implemented via individual policy configurations.}\n\n\n\\tdefpolicy$_p$ := $c^p \\Big( \\tbreakpol_p \\Big)$ \\ \\ \/\/ \\emph{syntactic sugar to switch back to attacker's view} \n\n\\tbreakpol$_p$ := $\\OR^p \\Big( \\tsatpol_p, \\tbreak_{s} \\Big)$, where $p = \\langle Cred, l, s \\rangle$ \\ \\ \/\/ \\emph{Attacker can either satisfy the individual policy configuration $p$, or he can break the enforcement mechanism $s$ that enforces this configuration $p$. }\n\n\\tsatpol$_p$ := $\\AND^p \\Big( \\taccess_{cred} | \\forall cred \\in Cred \\Big)$, where $p = \\langle Cred, l, s \\rangle$ \\ \\ \/\/ \\emph{To satisfy the configuration the attacker needs to access all credentials in the set $Cred$ identified in this configuration}.\n\nWe provide an example of an AD bundle in Fig. \\ref{fig:bundle}.\nBy construction, for each bundle $\\baccess_n$ its leaf nodes are either terms of the same type (\\taccess$_l$ for some $l$), or terms $\\tbreak_s$. We do not refine terms of the type \\tbreak$_{s}$ because the model itself lacks the knowledge how enforcement mechanisms can be broken. If an additional knowledge on breaking enforcement mechanisms will be available (e.g., as a hierarchy of attacks \\cite{Pinchinat-WFMDS-2014}), this term can further expanded. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{figure}\n\\caption{An AD bundle.}\n\\label{fig:bundle}\n\\end{figure}\n\n\\subsection{Approach to Synthesise AD-Trees}\n\nAD-bundles represent attacks on individual assets in the model. They can be ``glued'' together to form AD-trees, in the spirit of attack generation by policy invalidation. In this subsection we outline an approach to synthesis of attack-defence trees.\n\nThe main requirement for AD-trees synthesis is that it should terminate. Indeed, it is easy to see that any simple loop in the infrastructure will create infinite trees if bundles are composed naively. Moreover, some bundles may appear more than once in the generated tree, creating duplicate subtrees. To avoid this, we introduce a system state that will keep track of already achieved progress and will allow to terminate the synthesis process when the attacker has achieved the goal. \n\n\n\n\\paragraph{State.}\nWe define now two functions that identify the state of the system. These functions will be updated as the attack tree is generated in order to keep track with the attack development.\n\n\\iffalse{\nWe define a boolean function \\Posesses{}{} $\\subseteq$ $N_a$ $\\times$ $N_o$:\n\\begin{itemize}\n\\item If $(o,a) \\in E_{oa}$, and  \\Posesses{a}{o} := \\texttt{True}.\n\\item If for some $o_2 \\in N_o$ $(o_1,o_2) \\in E_{oo}$, and \\Posesses{a}{o_2} = \\texttt{True}, then \\Posesses{a}{o_1} := \\texttt{True}.\n\\item Else \\Posesses{a}{n} = \\texttt{False}.\t\n\\end{itemize}\nIntuitively, the function \\Posesses{}{} refers to objects that an actor has.\n}\\fi\n\n\\begin{definition}[\\Reachable{}{}]\t\nLet \\model\\ = $(N,E)$ be a model. We define a boolean function \\Reachable{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item If $(a,n) \\in E$, \\Reachable{a}{n} := \\texttt{True}.\n\\item If for some $l \\in N_i$ $(a,l) \\in E_{ai}$ and $(o,l) \\in E_{oi}$, then \\Reachable{a}{o} := \\texttt{True}. \n\\item If for some $l \\in N_i$ $(a,l) \\in E_{ai}$ and $(a_1,l) \\in E_{ai}$, then \\Reachable{a}{a_1} := \\texttt{True} and \\Reachable{a_1}{a} := \\texttt{True}. \n\n\\item Else \\Reachable{a}{n} := \\texttt{False}.\t\n\\end{itemize}\n\\end{definition}\n\nThis function initially captures for a given actor all items immediately reachable in the model. These items can be objects or actors located in the same location as the actor. Let \\Reach{a} := \\{$\\forall n \\in N$ s.t. \\Reachable{a}{n} = \\texttt{True}\\}.\n\n\n\\begin{definition}[\\Granted{}{}]\nWe define a boolean function \\Granted{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item If for an item $n$ $\\delta_n = \\emptyset $ then \\Granted{a}{n} := \\texttt{True}.\n\\item If for an item $n$ there is a tuple $p = \\langle Cred, atLocation\\rangle$ $\\in \\delta_n =$ s.t. $Cred \\subseteq \\Reach{a} \\cap N_o$ then \\Granted{a}{n} := \\texttt{True}.\n\\item Else \\Granted{a}{n} := \\texttt{False}.\n\\end{itemize}\n\\end{definition}\n\nIntuitively, this function refers to some policy configuration that grants access to $n$. If \\Granted{a}{n} = \\texttt{True}, then there is a way for this actor to satisfy the access control policy for $n$ (possibly under condition that he arrives at the right location). \n\nLet us define a model state.\n\\begin{definition}[State]\nA generated state for a model \\model\\ is a tuple $\\langle \\Reachable{}{}, \\Granted{}{} \\rangle$.\t \n\\end{definition}\n\n\\begin{definition}[\\Accessible{}{}]\nWe define a boolean function \\Accessible{}{} $\\subseteq$ $N_a$ $\\times$ $N$:\n\\begin{itemize}\n\\item \\Accessible{a}{n} := \\Reachable{a}{n} \\AND\\ \\Granted{a}{n}\n\\end{itemize}\n\n\t\n\\end{definition}\n\n\n\n\\paragraph{Bootstrapping}\nGiven a model \\model= $\\langle N,E \\rangle$ produced by a modeller, the functions \\Reachable{}{}, \\Granted{}{} and \\Accessible{}{} are initially computed from \\model. First we compute a transitive closure of reachable locations:\n\\begin{itemize}\n\\item \\Reachable{a}{n} \t:= \\Reachable{a}{n} \\OR\\ ($\\exists l \\in N:$ \\Accessible{a}{l} \\AND ($(l,n) \\in E$ \\OR $(n,l) \\in E$) )\n\\end{itemize}\nNotice that here we do not re-compute the function \\Granted{}{}, and thus, eventually, the reachable objects set for each actor will increase only with locations that are not guarded by access control policy. Once \\Reachable{}{} is recomputed, it can be used to quickly evaluate whether an actor can reach certain locations in the original model (where may he end up). \n\n\n\n\\subsubsection{Synthesis of AD-trees from Bundles}\n\nWe now discuss composition of generated attack-defence trees. An attack-defence tree $\\tree(\\eta,\\alpha)$ is synthesised for a chosen attacker $\\eta \\in N_a$ and a target asset $\\alpha \\in N_o$. The root node is the bundle \\taccess$_\\alpha$. For each leaf node of the type \\taccess$_b$ we can compute its value by referring to the corresponding AD bundle $\\baccess_b$. \n\n\\paragraph{Bundle Value.}\nIn the simplest case we use propositional semantics for evaluating AD-bundles and, eventually, AD-trees \\cite{Kordy-JLC-2014}. For leaf nodes of the type \\taccess$_n$, \\taccess$_n$ $\\equiv$ \\Accessible{\\eta}{n}. For leaf nodes of the type \\tbreak$_s$, \\tbreak$_s$ $\\equiv$ \\texttt{False} in the current synthesis approach. Thus, given a bundle for asset $n$, we can evaluate its value based on the values of the leaf nodes available. By updating the model state as attack progresses (more items become reachable to the attacker) we can eventually evaluate the target bundle, once all its descendants become evaluated. As state changes monotonically, the process will eventually terminate.\n\n\n\n\n\n\n\\section{Introducing New Defences}\\label{sec:2ndstep}\nThe enforcement mechanisms for access control policies are not the only type of security controls that organisations use. Moreover, access control is not the only remedy that can be advised to improve security. Indeed, the existing risk analysis standards and security catalogues that guide practitioners in risk analysis identify many types of security controls and countermeasures. Many of those (for example, security cameras) cannot be captured by socio-technical models directly, because it will introduce unnecessary complications to the model. Some countermeasures can be introduced as properties of system elements (e.g., after a security training the employees might become less susceptible to social-engineering), but not as independent elements of the system. \n\nWe want to be able to update the attack-defence model of our system, captured by the suite of attack-defence bundles, after the first stage of automated generation. At this second stage we would like to obtain more complete attack-defence bundles with new defence nodes added that can capture additional security countermeasures (either existing in the model already or newly proposed once). We have two main questions associated with the newly introduced defences: \\emph{how to generate\/propose new defences} and \\emph{where to place them in the attack-defence model} to keep the consistency across many attack scenarios and system updates. We start by addressing the second question first.\n\n\n\\textbf{Where to put new countermeasures}\\\\\nGiven an AD bundle representing the goal of an attacker to access asset $n$, two types of attack nodes are the candidates to be protected from by some countermeasures: the root node \\taccess$_n$ and its children \\taccfrom$_{n,l}$. Indeed, for the connectors to other bundles (the leaf nodes \\taccess$_b$) it will make sense to introduce defences at the corresponding bundle to ensure the consistency requirement. For the nodes \\tsatpol$_p$, the attacker's goal is to satisfy the policy by finding the right credentials. It is not obvious what can be done as a protective measure besides protecting the credentials themselves. As for the nodes representing circumventing the enforcement mechanism, \\tbreak$_{s}$, we do not have enough details for the moment how the attacker is going to break it. If this node is to be refined using some attack pattern library, it is better to create a separate AD bundle for treating the scenarios and assign defences there.\n\nNow we have candidate attack nodes to be assigned countermeasures. To select the countermeasures that could be assigned, we first review existing types of security controls. It is well-established in the security industry to classify controls as preventive, detective and corrective \\cite{NIST-800-30}:\n\\begin{itemize}\n\\item \\emph{Preventive}\tcontrols focus on preventing security incidents from occurring. \n\\item \\emph{Detective} controls focus on detecting occurrences of security incidents.\n\\item \\emph{Corrective} controls focus on aiding the organisation to recover from a security incident. \n\\end{itemize}\n\nFrom the implementation perspective, it is traditional to divide controls into the following categories \\cite{NIST-800-30}:\n\\begin{itemize}\n\\item \\emph{Technical} controls that are implemented typically as software controls.\n\\item \\emph{Management}, or administrative, controls that are implemented as procedures and guidelines.\n\\item \\emph{Operational} controls that focus on ensuring security and dependability of operations. These controls include physical security controls (physical access control, fire and water damage protection, etc.) and some controls that are difficult to classify as fully technical or physical (e.g., protection of personal computers).   \t\n\\end{itemize}\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{figure1}\n\\caption{An updated AD bundle with defence nodes in the designated positions (children of EM$_(n,l)$ not shown).}\n\\label{fig:updatedbundle}\n\\end{figure}\n\n\nFrom this classification, we propose a way to update AD bundles with security controls in a consistent manner. The preventive controls can be added as children to the attack nodes \\taccfrom$_{n,l}$, because they correspond to preventive measures for certain directed actions of the attacker. Access control policies present in the model are already embedded in the bundles at this position. To satisfy the attack-defence trees requirement of only one child of the opposite type, we will modify the bundle as in Fig.~\\ref{fig:updatedbundle} (now the node \\tpreventive$_{n,l}$ is a parent of the node \\tdefence$_{n,l}$). \n\nThe detective and corrective measures can be added as children to the root node \\taccess$_n$ (see Fig.~\\ref{fig:updatedbundle}). In this position the defence nodes are directly linked to the system object in question, be it a location, a person, or an object. The semantics of the controls placed in this position are clear: assuming the attacker has already gained access to his target, is this detectable or what can be the remedy for this? Notice that some controls in practice can be both detective and preventive (e.g., security guards). In this case, it is safe to classify them as preventive controls.\n\n\\textbf{What defences to choose}\\\\\nThe choice of security controls is a tough question in practice. Not only it requires the human analyst to know possible attacks and countermeasures, but also the analyst needs to solve a complex multi-parameter optimisation problem. Indeed, the controls addressing the same threat can have different cost, efficiency and effectiveness. They can be more or less compliant with the industry standards and best practices. They can be more or less easy to implement and easy to verify. Finally, they can be more or less desired by the organisation because of personal views of the top-management. Thus, if we just consider the baseline controls listed by NIST Special Publication 800-53 \\cite{NIST-800-53}, and try to evaluate all the above-mentioned parameters in order to fully automate the defence selection (now that we know where to place them), we already will face a very complex problem. Moreover, a single mistake in evaluation of some of the values will likely make the full analysis invalid. Therefore, it is likely that human-assisted control selection cannot be fully replaced by automated defence generation, at least for some time.\n\nYet, we can try to facilitate the defence selection problem by further categorising the security controls based on applicability to the scenarios in question and usefulness in attribute-based computations. \n\nIn the socio-technical model we have clear categories of objects: locations, actors and items that can belong to either physical or digital space. Thus controls can be chosen based already on simple considerations such as ``access to digital objects by processes can be protected by using technical preventive controls'', or ``access of humans to humans can be protected by administrative and physical preventive controls''. Table~\\ref{tab:preventive} summarises these choices of controls. In this table, it is expected that respective controls will be introduced in the dedicated AD bundles (following the template in Fig.~\\ref{fig:updatedbundle}).\n\n\n\\begin{table}[t!]\n\\centering\n\\caption{Controls selection based on system elements. }\n\\label{tab:preventive}\n\\scriptsize\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Entity}  &  \\textbf{Physical space}  & \\textbf{Digital space}   \\\\\n\\hline\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Preventive}}\\\\\n\\hline\nLocation & Physical access control  & Technical access control, firewall  \\\\\n\\hline\nActor & Physical access control, Security trainings, Email filter  & Technical access control and authentication\\\\\n\\hline\nObject & Physical access control  & Technical access control \\\\\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Detective}}\\\\\n\\hline\nLocation & \\multirow{3}{*}{ Security cameras, visitor logs}  & \\multirow{3}{*}{System logs, IDS} \\\\\nActor & &\\\\\nObject & & \\\\\n\n\\hline\n\\multicolumn{3}{|c|}{\\textbf{Corrective}}\\\\\n\\hline\nLocation & \nInsurance, liability limitation,   & Insurance, liability limitation, \\\\\nActor &  business continuity plan & secure state restoring mechanisms, \\\\\nObject &   & business continuity plan \\\\\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nFurthermore, following the investigation of attribute decoration on attack-defence trees by Bagnato et al. \\cite{Bagnato-IJSSE-2012}, we can look at what controls contribute to computations of certain attributes. For instance, if the analyst is interested in the probability of an attack to succeed, the minimal cost of attack for an attacker, or time of executing an attack, then (under the assumption that detection cannot stop the attack) she would like to look at her preventive measures. If she is interested in the impact the attack has on her organisation (how business continuity is affected after the attack was executed), she would like to consider the preventive and corrective controls, especially the latter ones, because these ensure business continuity. Thus, if she is interested only in some attributes, the computation on AD bundles does not need to take into account all controls at once. Table~\\ref{tab:attributes} summarises the control types that are the most relevant for some selected attributes. \n\n\\begin{table}[t!]\n\\centering\n\\caption{Relevant controls for example attributes}\n\\label{tab:attributes}\n\\scriptsize\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Attribute}  &  \\textbf{Preventive}  & \\textbf{Detective}  & \\textbf{Corrective}  \\\\\n\\hline\n\\hline\nRisk of detection &  & \\checkmark & \\\\\nCost of attack (for attacker) & \\checkmark &  & \\\\\nProbability of attack success & \\checkmark &  & \\\\\nTime of attack & \\checkmark &  & \\\\\nImpact of attack & \\checkmark & \\checkmark & \\checkmark\\\\\n\n\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nNotice that the controls added at this stage will probably not follow the AD bundle notation, but will be expressed in the natural language (e.g, \\emph{security training} or \\emph{ID check}). This is understandable, because, as we have mentioned, models are not rich enough by nature. Yet, this is acceptable for the format, because the attack-defence model does not need to be fully formal. On the contrary, it is used to assist the human analyst to create and maintain the attack-defence view on the system. The only requirement that we have for it is the consistency, which is ensured by adding each control only to the bundle representing the attack-defence view of a particular (unique) entity in the model. Some controls can require a notion of \\emph{perimeter} to be defined in the model, so that they can be uniquely assigned to the bundle corresponding to the perimeter, and not to each entity belonging to that perimeter. This is easily implementable in any socio-technical model.\n\n\n\n\n\\section{Related Work}\\label{sec:discussion}\nThe question of attack trees generation from system models has been tackled in \\cite{Ivanova-Generation-2014}. Similarly, \\cite{Vigo-CSF-2014} and \\cite{Ou-CCS-2006} worked on generating attack models from a system model. While we follow the same approach for  attacker's view, our main focus is on keeping both attacker's and defender's views consistent with the main socio-technical model.  \n\nAttack-countermeasure trees (ACTs) is an alternative model to attack-defence trees in keeping both views simultaneously \\cite{Roy-SCN-2011}. In \\cite{Roy-DSN-2012} the authors have investigated optimal countermeasure selection for ACTs when a set of possible countermeasures to be implemented is already predefined. It will be interesting to investigate ACTs suitability for the attack-defence model, because they support explicit detection and mitigation countermeasure nodes (but not corrective). \n\nIn \\cite{Lenzini-2015} the authors work on directly applying model checking to a socio-technical model in order to evaluate some reachability-based security properties. \n\nFerreira et al. \\cite{STEAL-2014} have discussed defences suggestion in the context of the socio-technical model STEAL. They propose to apply defences at the technical and social levels of the system, what is in line with our proposal for applying security control categories in selecting defences.\n\n\\section{Conclusion and the Next Steps}\n\nIn this work we have approached the question of creating and maintaining the security controls representation in parallel to the socio-technical model. Our solution creates a set of attack-defence bundles (small attack-defence trees) that can be maintained with a socio-technical model as its separate view. The bundles are generated from the model in the beginning, but afterwards they are enriched consistently alongside the new security controls identified by a human analyst. We have also discussed how new controls can be selected based on the model entities and the attributes of interest to the analyst. This work attempts to bridge the gap between the approach of automated attack generation from system model and the manual security control selection in the traditional risk analysis. The next step is to look into the compositional attack-defence tree synthesis for more complex attribute domains.  After that, it will be possible to investigate optimal countermeasure selection, based, e.g., on the approaches suggested in \\cite{Aslanyan-POST-2015} and \\cite{Roy-DSN-2012}. Another further research direction is practical validation of the proposed approach on realistic case studies and evaluation of its usefulness and scalability.\n\n\\subsubsection*{Acknowledgements}\nThis work was partially supported by the European Commission through the FP7 project TREsPASS (grant agreement n. 318003) and by Fonds National de la Recherche Luxembourg through the ADT2P project (grant n. C13\/IS\/5809105).\n\n\n\\bibliographystyle{splncs03}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec_intro}\nIn the past decade, Demand-Side Management (DSM) with price-based demand response has been widely considered to be an efficient method of incentivising users to collectively achieve better grid welfare \\cite{vardakas15}. In a typical DSM scenario, an independent utility operator plans the preferred demand volume for a certain metering time duration within the considered micro (e.g., district) grid. In order to achieve this, the operator adopts a time-varying pricing scheme and announces the prices to the users on a rolling basis. In general, the price may be determined by factors such as the demand for the utility, the supply of the utility, and the history of their volatility. The adjusted price is released in advance at some time window to the consumers. In response to the updated electricity prices, the users voluntarily schedule or shift their appliance loads with the aim of minimising their individual costs, hence making it possible to reduce the energy waste or load fluctuation of the considered microgrid (see the example in~\\cite{deng15}).\n\nIn the framework of dynamic pricing (a.k.a. real-time pricing), dynamic demand response on the user side (e.g., at residential households) relies on the premise of the availability of smart meters and automatic energy-consumption-manager modules installed in each household. This is mainly due to two characteristics of residential utility demand. Firstly, as noted in~\\cite{vardakas15}, residential consumers are usually risk-averse and shorted-sighted; it would be very difficult for consumers to schedule for long-term cost optimisation based solely on a  dynamic price signal. Secondly, compared with the commercial or industry consumers, individual residential consumers often appear to exhibit near-random consumption patterns ~\\cite{barker12}. Real-time sensing and communication of data between each home's energy-consumption manager and the utility operator thus becomes essential.\n\nUnder the framework of automatic price-demand scheduling, a variety of dynamic-response problem formulations have been proposed in the literature. One major category of the existing schemes focuses on direct load control to optimise a certain objective with (possibly) the constraints on grid capacity and appliance requirement. Typical objectives in this scenario can be to minimise energy consumption\/costs~\\cite{ruelens17,chen12,wen15,kim16,mocanu19,mohsenian10}, maximise the utility of either middlemen or end-users~\\cite{cao18,wang19,reddy11,lu18,yang18}, or minimise the peak load demand of the overall system~\\cite{deng15,mohsenian10,barker12}. With such a setup, DSM is usually assumed to be performed in a slotted manner (e.g., with a time scale measured in hours~\\cite{deng15}) within finite stages. The DSM problem can be formulated under a centralised framework as a scheduling problem~\\cite{barker12}, or an optimisation problem such as a linear program~\\cite{mohsenian10} or a stochastic programs~\\cite{cao18}. Alternatively, under a decentralised framework, the DSM problem can also be developed in the form of non-cooperative games with specifically designed game structure to guarantee desired social welfare indices~\\cite{deng15}. Apart from the programming\/planning oriented schemes, another major category of design emphasises the uncertainties in the DSM process, e.g., due to the random arrival of appliance tasks or the unpredictability of the future energy prices. In such a scenario, DSM planning based on the premise of known system model is typically replaced by DSM strategy-learning schemes, which usually address the uncertainty in the system by formulating the considered dynamics to be Markov Decision Processes (MDPs) with partially observability~\\cite{hansen18}. By assuming infinite plays in a stationary environment, optimal DSM strategies are typically learned in various approaches of reinforcement learning without knowing the exact system models~\\cite{ruelens17,wen15}.\n\nIn this paper, we propose a DSM problem formulation with dynamic prices in a microgrid of residential district, where a number of household users are powered by a single energy aggregator. By equipping the households with energy manager-enabled smart meters, we aim at distributedly minimising the cost of energy in the grid, under real-time pricing, over the long term with deep reinforcement learning. The main advantage of decentrally controlled energy-managing smart meters is that, after the training phase, each household agent is able to act independently in the execution phase, without requiring specific information about other households --- each household agent will not be able to observe the number or nature of devices operated by other households, the amount of energy they individually consume, and when they consume this energy. Our aforementioned aim addresses at least two key challenges. Firstly, the model of the microgrid system and its pricing methodology is not available to each household. Secondly, the limitation in the observation available to each household and the lack of visibility regarding the strategy of other households creates the local impression of a non-stationary environment. We introduce details of the system model in Section~\\ref{sec_system_model}.\n\nTo overcome the first mentioned key challenge, we propose a model-free policy-based reinforcement learning method in Section~\\ref{sec_rl_method} for each household to learn their optimal DSM strategies. To address the second key challenge, we propose a centralised critic network which allows the aggregator to provide an allowable level of messaging to guide the learning processes of the households. Our experiments, presented in Section~\\ref{sec_experiments}, show that the proposed model is able to not only address the scenarios of cooperative learning with shared utility, but also able to adapt themselves with non-cooperative goals of load demands. By comparing to an offline method that plans consumption based on a predicted price schedule \\cite{mohsenian10}, we show that our method is better able to handle the uncertainties associated with real-time pricing.\n\n\\section{Related Work and Preliminaries}\nA number of previous works have applied reinforcement learning to train decision-making entities in smart grid settings \\cite{wang19,zhao20,duan19,lu18, kara12, kim16, reddy11}. Of these, \\cite{lu18, kara12, kim16, reddy11} have applied Q-learning, a canonical value-based reinforcement learning algorithm, to optimise energy costs. While they vary in the exact method of defining the state and action spaces in the MDPs, a common theme was the presence of electricity producers, consumers, and service providers that brokered trading between them. The reward function in these studies were designed to minimise the cost to the service provider and\/or consumer, minimise the discrepancy between the supply and demand of energy or, in the case of \\cite{yang18}, maximise the profit achieved by the service provider. A common limitation between \\cite{lu18, kara12, kim16, reddy11} was the use of tabular Q-learning limited the algorithm to processing only discretised state and action spaces.\n\nRecently, there have also been studies that have introduced methods utilising deep neural networks to multi-agent reinforcement learning, albeit to different classes of problems \\cite{RN242,RN243,RN244}. These newly introduced methods are extensions of the now-canonical deep Q-network (DQN) algorithm for the multi-agent environment, and a common technique is to have a central value-function approximator to share information. However, as explained in section \\ref{sec_system_model}, a value-based RL algorithm would not suit the problem introduced in this paper because it requires the Markov assumption that the current state and action are sufficient to describe both the state transition model and the reward function.\n\nIn this paper, we propose a policy-based algorithm based on the Proximal Policy Optimisation (PPO) algorithm \\cite{RN136} for a multi-agent framework. We consider the partially observable Markov game framework \\cite{RN260}, which is an extension of the Markov Decision Process (MDP) for multi-agent systems. A Markov game for $N$ agents can be mathematically described by a tuple $\\langle\\mathcal{N}, \\mathcal{S}, \\{\\mathcal{A}_n\\}_{n\\in\\mathcal{N}}, \\{\\mathcal{O}_n\\}_{n\\in\\mathcal{N}}, \\mathcal{T}, \\{r_n\\}_{n\\in\\mathcal{N}} \\rangle$, where $\\mathcal{N}$ is the set of individual agents and $s \\epsilon \\mathcal{S}$ describes the overall state of all agents in the environment.\n\nAt each discrete time step $k$, each agent $n$ is able to make an observation $O_n$ of its own state. Based on this observation, each agent takes an action that is chosen using a stochastic policy $\\pi_{\\theta_n}: \\mathcal{O}_n \\mapsto \\mathcal{A}_n$, where $\\mathcal{A}_n$ is the set of all possible actions for agent $n$.  The environment then determines each agent's reward as a function of the state and each agent's action, such that $r_n(k): \\mathcal{S \\times A_1 \\times \\cdots \\times A}_n \\mapsto \\Re$. It moves on to the next state according to its state transition model $\\mathcal{T: S \\times A_1 \\times \\cdots \\times A}_n \\to \\mathcal{S}'$.\n\n\\section{System Model}\n\\label{sec_system_model}\nWe consider a DSM problem with a set of $N$ households that each have $M$ household appliances. Each household is equipped with a smart meter that is capable of communicating and controlling all household appliances. The problem that we present is a more advanced form of the demand-side scheduling problem than that presented in~\\cite{mohsenian10} in that each household has no advanced knowledge of upcoming tasks. We primarily consider interactive background appliances. These appliances are actively turned on by the household, but the actual time of operation poses a less immediate impact to the user. The appliances that we consider in each household can be, for example, a washing machine, a clothes dryer, a storage water heater, a dish washer and a refrigerator.\n\nEach household's local operational state can be expressed as $\\mathbf{s}_n=[\\mathbf{x}_n^{\\top},\n\\mathbf{t}_n^{\\top}, \\mathbf{l}_n^{\\top}, \\mathbf{q}_n^{\\top}]^{\\top}$, where $\\mathbf{t}_n=[t^1_n,\\ldots,t^M_n]^{\\top}$ denotes the vector of the length of time before the next task for appliance $m$ can commence, $\\mathbf{l}_n=[l^1_n,\\ldots,l^M_n]^{\\top}$ denotes the vector of task periods, and $\\mathbf{q}_n=[q^1_n,\\ldots,q^M_n]^{\\top}$ denotes the number of tasks queued for each appliance. The private observation of each household consists of its local state augmented with the price of electricity at the previous time step $p(k-1)$, such that $\\mathbf{o}_n=[\\mathbf{x}_n^\\top, \\mathbf{t}_n^\\top, \\mathbf{l}_n^\\top, \\mathbf{q}_n^\\top, p(k-1)]^\\top$. The overall state is simply the joint operational state of all the users:\n\\begin{equation}\n  \\label{eq_joint_state}\n  \\mathbf{s} = [\\mathbf{s}_1,\\ldots, \\mathbf{s}_N].\n\\end{equation}\n\nThe task arrival for each appliance corresponds to the household's demand to turn it on, and is based on a four-week period of actual household data from the Smart* dataset \\cite{barker12data}. We first discretise each day into a set of $H$ time intervals $\\mathcal{H}=\\{0,1,2,...,H-1\\}$ of length $T$, where each interval corresponds to the wall-clock time in the simulated system. The wall-clock time interval for each time-step in the Markov game is determined with the following relation:\n\\begin{equation}\n  \\label{eq_time_map}\n  h(k) = \\mod (k, H).\n\\end{equation}\n\nNext, we count the number of events in each interval where the above-mentioned appliances of interest were turned on. This is used to compute an estimate for the probability each appliance receives a new task during each time interval, $p_n^m(h(k))$. The task arrival for each appliance ${q'}_n^m$ at each time interval $k$ is thus modelled with a Bernoulli distribution. Each task arrival ${q'}_n^m$ for a particular appliance $m$ in household $n$ is added to a queue $q^m_n$.\n\nThe duration of the next task to be executed is denoted by $l^m_n$. Its value is updated by randomly sampling from an exponential distribution whenever a new task is generated (i.e. ${q'}_n^m = 1$). The rate parameter $\\lambda_n^m$ for the distribution for each appliance is assigned to be the reciprocal of the approximate average duration of tasks in the above-mentioned four-week period selected from the Smart* dataset \\cite{barker12data}. We choose to keep the task duration constrained to $l \\in [T, \\infty]$.\n\nThe use of the described probabilistic models for generation of task arrivals and task lengths enable us to capture the variation in household energy demand throughout the course of an average day while preserving stochasticity in energy demand at each household.\n\nWe consider that each household's scheduler is synchronised to operate with the same discrete time steps of length $T$. At the start of a time step, should there be a task that has newly arrived into an empty queue ($q^m_n = 1$), the naive strategy of inaction would have the task start immediately such that $t_n^m = 0$. This corresponds with the conventional day-to-day scenario where a household appliance turns on immediately when the user demands so. However, in this paper we consider a setting where each household has an intelligent scheduler that is capable of acting by delaying each appliance's starting time by a continuous time period $a_n^m \\in [0,T]$. The joint action of all users is thus:\n\\begin{equation}\n  \\label{eq_joint_action}\n  \\mathbf{a} = [\\mathbf{a}_1,\\ldots, \\mathbf{a}_N]^{\\top}.\n\\end{equation}\n\nOnce an appliance has started executing a task from its queue, $q_n^m$ is shortened accordingly. We consider that the appliance is uninterruptible and operates with a constant power consumption $P_n^m$. If an appliance is in operation for any length of time during a given time step, we consider its operational state to be $x^m_n=1$. Otherwise, $x_n^m=0$.\n\nLet $d^m_n(k)$ denote the length of time that an appliance $m$ is in operation during time step $k$ for household $n$. The energy consumption of a household $E_n(k)$ during time step $k$ is thus:\n\\begin{equation}\n  \\label{eq_power_load}\n  E_n(k) = \\sum_{m=1}^{M}d^m_n(k) P_n^m.\n\\end{equation}\n\nOn the grid side, we consider a dynamic energy price $p(k)$ that is a linear function of the Peak-Average-Ratio \\cite{mohsenian10,5540263} of the energy consumption in the previous $\\kappa$ time steps:\n\\begin{equation}\n  \\label{eq_price}\n  p(k)=\\frac{\\kappa T\\max\\limits_{k-\\kappa +1\\le k}\\sum\\limits_{n=1}^{N}E_n(k)}{\\sum\\limits_{n=1}^{N}E_n(k)}.\n\\end{equation}\n\nThe reward signal then sent to each household is a weighted sum of a cost objective and a soft constraint. \n\\begin{equation}\n    \\label{eq_reward}\n    r_{n}(k) = -r_{c,n}(k) + w r_{e,n}(k)\n\\end{equation}\n\nThe first component of the reward function $r_{c,n}$ is the monetary cost incurred by each household at the end of each time step. The negative sign of the term (\\ref{eq_cost}) in  (\\ref{eq_reward}) encourages the policy to delay energy consumption until the price is sufficiently low.\n\\begin{equation}\n    \\label{eq_cost}\n    r_{c,n}(k) = p(k) \\times E_n(k)\n\\end{equation}\n\nThe second component of the reward is a soft constraint $r_{e,n}$ tunable by weight $w$. The soft constraint encourages the policy to schedule household appliances at a rate that matches the average rate of task arrival into the queue.\n\\begin{equation}\n    \\label{eq_constraint}\n    r_{e,n}(k) = E_n(k)\n\\end{equation}\n\nWe adopt a practical assumption that each household can only observe its own internal state along with the published price at the previous time step, and receive its own cost incurred as the reward signal. For each household $n$, the system is thus partially observed, where $\\mathbf{o}_n(k) = [\\mathbf{s}_n(k)^\\top, p(k-1)]^\\top$.\n\nWe note that the state transition for all households can be considered to be Markovian because the transition to the next overall system state is dependent on the current state and the actions of all household agents: $\\mathcal{T}: \\mathcal{S} \\times \\mathcal{A}_1 \\times \\cdots \\times \\mathcal{A}_n \\to \\mathcal{S}'$. In contrast, we recall from (\\ref{eq_price}) and (\\ref{eq_reward}) that the price of energy, and consequently each agent's reward, is a function of not only the current state and actions, but also of the history of previous states. Our pricing-based energy demand management process is, therefore, a generalisation of the Markov game formulation \\cite{LITTMAN1994157}.\n\n\n\\section{Semi-Distributed Deep Reinforcement Learning for Task Scheduling}\n\\label{sec_rl_method}\nThe DSM microgrid problem that we introduce in the previous section presents various constraints and difficulties in solving it: (i) each household can choose its action based only on its local observation and the last published price, (ii) the reward function is dependent on the states and actions beyond the last timestep, (iii) the state and observation spaces are continuous, (iv) the household agents have no communication channels between them, and (v) a model of the environment is not available. Constraint (iv) is related to the general problem of environmental non-stationarity: concurrent training of all agents would cause the environmental dynamics to constantly change from the perspective of a single agent, thus violating Markov assumptions.\n\nRequirements (i) and (ii) and environmental non-stationarity mean that a value-based algorithm such as Q-learning is not suitable, because these algorithms depend on the Markov assumption that the state transition and reward functions are dependent only on the state and action of a single agent at the last time step. Furthermore, requirement (v) rules out model-based algorithms. Thus, in this section, we propose an extension of the actor-critic algorithm Proximal Policy Optimisation (PPO) \\cite{RN136} for the multi-agent setting which we will refer to as Multi-Agent PPO (MAPPO).\n\nLet $\\mathbf{\\pi}=\\{\\pi_1, \\pi_2, ..., \\pi_N\\}$ be the set of policies, parameterised by $\\mathbf{\\theta}=\\{\\theta_1, \\theta_2, ..., \\theta_N\\}$, for each of the $N$ agents in the microgrid. The objective function that we seek to maximise is the expected sum of discounted reward  $J(\\theta) = E_{s\\sim p^\\pi,a\\sim \\pi_\\theta}[\\sum^K_{k=i}\\gamma^{k-i}r_k]$, where $r_k$ is obtained based on (\\ref{eq_reward}). In canonical policy gradient algorithms such as A2C, gradient ascent is performed to improve the policy:\n\\begin{equation}\n    \\label{eq_policy_grad}\n    \\nabla_{\\theta_n} J_n(\\theta_n)=\\\\\n    E_{{s}\\sim p^{\\mathbf{\\pi}}, {a}_n\\sim \\pi_n}\n    \\left[\\nabla_{\\theta_n} \\log\\pi_n({s},{a}_n) Q^{\\pi_n}_n({s}, {a}_n) \\right].\n\\end{equation}\n\nThis policy gradient estimate is known to have high variance. It has been shown that this problem is exacerbated in multi-agent settings \\cite{RN242}. For this reason, we propose the implementation of PPO as it limits the size of each policy update through the use of a surrogate objective function. Additionally, we augment the individually trained actor networks with a central critic network $\\hat{V}^\\pi$ that approximates the value function for each household $V^\\pi_n$ based on the overall system state, thus receiving information for training of cooperative actions by individual households. The objective function we propose is the following:\n\\begin{equation}\n    \\label{eq_obj_}\n    \\begin{split}\n    L(\\theta_n)=E_k\n    \\Big[ & \\min \\Big( \\rho_k(\\theta_n)\\hat{A}^\\pi_n(k),\\max \\big( 1-\\epsilon, \\\\ & \\min(\\rho_k(\\theta_n),1+\\epsilon) \\big) \\hat{A}^\\pi_n(k)  \\Big) +\\lambda H({\\pi_n}) \\Big],\n    \\end{split}\n\\end{equation}\nwhere $\\rho_k(\\theta_n) = \\frac{\\pi_n( \\mathbf{a}_k|\\mathbf{o}_k)}{\\pi_{n,old(\\mathbf{a}_k|\\mathbf{o}_k)}}$, and both $\\epsilon$ and $\\lambda$ are hyperparameters. $\\hat{A}^\\pi_n(k)$ is the estimated advantage value, which is computed using the critic network:\n\\begin{equation}\n    \\label{eq_adv}\n    \\hat{A}^\\pi_n(\\mathbf{s}, \\mathbf{a}_n) \\approx r(\\mathbf{s}, \\mathbf{a}_n) + \\gamma \\hat{V}^\\pi(\\mathbf{s}(k+1))_n - \\hat{V}^\\pi(\\mathbf{s}(k))_n.\n\\end{equation}\n\nWe fit the critic network, parameterised by $\\phi$, with the following loss function using gradient descent:\n\\begin{equation}\n    \\label{eq_critic_obj}\n    L(\\phi) = {\\sum_{n} \\sum_{k} \\left( \\hat{V}^\\pi( \\mathbf{s}(k))_n -\\mathbf{y}_n(k) \\right)^2}.\n\\end{equation}\nwhere the target value $\\mathbf{y}(k)=r_n(k) + \\gamma \\hat{V}^\\pi (\\mathbf{s}(k+1))_n$. As the notation suggests, this update is based on the assumption that the policies of the other agents, and hence their values, remain the same. In theory, the target value must be re-calculated every time the critic network is updated. However, in practice, we take a few gradient steps at each iteration of the algorithm.\n\nThe schematic of the proposed actor-critic framework is illustrated in Figure \\ref{fig_ac}. The centralised critic-network itself utilises a branched neural network architecture shown in Figure \\ref{centralised_critic}. To compute the estimated state value of a particular household $n$, $V_n$, we provide the following input vector to the centralised critic-network:\n$[\\begin{smallmatrix}\n\\mathbf{s}_n^\\top & \\mathbf{s}_1^\\top & \\mathbf{s}_2^\\top & \\dots & \\mathbf{s}_1^\\top & \\mathbf{s}_{n-1}^\\top & \\mathbf{s}_{n+1}^\\top & \\dots \\mathbf{s}_N^\\top\n\\end{smallmatrix}]^\\top$. The critic network would thus provide an estimate of the quantity $V^\\pi(\\mathbf{s})_n = \\sum_{k} E_{\\pi_\\theta} [r(\\mathbf{s}_n, \\mathbf{a}_n) | \\mathbf{s}_1, \\mathbf{s}_2, \\ldots, \\mathbf{s}_{n-1}, \\mathbf{s}_{n+1}, \\ldots, \\mathbf{s}_N]$.\n\nWe compare the MAPPO algorithm to a similar algorithm that uses a decentralised critic-network instead. The architecture we utilise for the decentralised critic-network is shown in Figure \\ref{decentralised_critic}. The main consideration for such a design is to maintain a similar number of learnable parameters across all the $N$ decentralised critic networks to the centralised critic, on the assumption that this would result in a similar computation resource requirements.\n\n\\begin{figure}[t]\n\\centering    \n\\includegraphics[width=.35\\textwidth]{NN_model.png}\n\\caption{The high-level architecture of the reinforcement learning setup with a centralised critic. Each actor and the critic is implemented as a feedforward multi-layer perceptron.}\n\\label{fig_ac}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[height=2in]{CentralisedCritic3.png}\n        \\caption{}\n        \\label{centralised_critic}\n    \\end{subfigure}%\n    ~ \n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[height=2in]{DecentralisedCritic3.png}\n        \\caption{}\n        \\label{decentralised_critic}\n    \\end{subfigure}\n    \\caption{The architectures of a (\\subref{centralised_critic}) centralised critic network and a (\\subref{decentralised_critic}) decentralised critic network respectively.}\n    \\label{fig_NNarchitecture}\n\\end{figure*}\n\n\n\\section{Experiments}\n\\label{sec_experiments}\nIn this section, we present the settings and environmental parameters that we use for experimentation, evaluate our chosen reinforcement learning method against non-learning agents and the offline Energy Consumption Scheduler (ECS) method by \\cite{mohsenian10}, examine the effect of augmenting local observations in proposed system model, and consider the effects of household agents learning shared policies.\n\n\\subsection{System Setup}\nWe present experiments on a simulated system of identical households, each with 5 household appliances. As mentioned, their parameters relating to power consumption $P^m_n$, task arrival rates $p^m_n(k)$ and lengths of operation per use $\\lambda^m_n$ are chosen to be a loose match to data sampled from the Smart* 2017 dataset \\cite{barker12data}, and thus resemble typical household usage patterns.\n\nThe period for each time step $T$ is chosen to be 0.5 hours to achieve sufficient resolution, while limiting the amount of computation required for each step of `real time' in the simulation. To reasonably limit the variance of trajectories generated, the maximum length of each simulation episode is limited to 240 time steps or 5 days. For each iteration of the training algorithm, 5040 time steps of simulation experience is accumulated for each batch. Each experiment is run with a number of random seeds; the extent of their effect is shown in the range of the lightly shaded regions on either side of the lines in the graphs shown.\n\nFor this chosen experimental setup, we run two baseline performance benchmarks that were non-learning: (i) Firstly, an agent that assigned delay actions of zero for all time steps. This represents the default condition in current households where the user dictates the exact time that all appliances turn on. (ii) A second baseline agent randomly samples delay actions from a uniform distribution with range $[0, T]$. We compare these non-learning agents to agents trained using our proposed Multi-Agent PPO (MAPPO) algorithm as described in Section \\ref{sec_rl_method}.\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{AverageRewardperDay32Comparison2.png}\n        \\caption{Average reward obtained per day}\n        \\label{av_reward}\n    \\end{subfigure}%\n    ~ \n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{AverageCostperDay32Comparison2.png}\n        \\caption{Average monetary cost incurred per day}\n        \\label{av_cost}\n    \\end{subfigure}\n    \\caption{Results for a 32-household microgrid system. (\\subref{av_reward}) and (\\subref{av_cost}) show the MAPPO agent's progress during training, compared against an agent that samples from a uniform random action distribution, and another that chooses no delays.}\n    \\label{fig_32households}\n\\end{figure*}\n\nThe results of the above-mentioned agents are compared in Figures~\\ref{av_reward} and~\\ref{av_cost}. The y-axis for both graphs shows the average reward gained and average cost incurred by all households for each day within each batch of samples. The x-axis shows the iteration index in the actor-critic training process. (Note that each batch consists of 105 days of simulation.) All trained policies demonstrate a sharp improvement in reward achieved from the onset of training. The monetary cost decreases with increasing reward, showing that the reward function is suitably chosen for the objective of minimising cost.\n\n\\begin{figure}[t]\n\\centering    \n\\includegraphics[width=.45\\textwidth]{EnergyTimeGraph32houses.png}\n\\caption{Average aggregate energy consumed by 32-household microgrid for each timestep of the day.}\n\\label{av_energy_con}\n\\end{figure}\n\nIn Figure~\\ref{av_energy_con}, we plot the average total energy consumed by the entire microgrid system for each half-hour time step. Note that the x-axis shows the wall-clock in the simulated environment using the 24 hour format. The blue line shows the energy-time consumption history if all home appliance were to be run immediately on demand; it reveals the peaks in demand in the morning and evening, presumably before and after the home occupants are out to work, and lows in demand in the night time. The uniform-random-delay agent dampens these peaks in demand by shifting a number of tasks forward into the following time steps (note the lag in energy consumption compared to the zero-delay agent). In contrast, the agent trained using reinforcement learning is able to not just delay the morning peak demand, but spread it out across the afternoon. The result is a lower PAR, and hence a lower monetary cost per unit energy.\n\nAlthough the morning peak in energy demand is smoothed out by the MAPPO trained households, the same effect is less pronounced for the evening peak. We hypothesise that this was due to the training procedure: the batch of experience gathered in each iteration of the algorithm began at 00:00 hrs, and terminated at the end of a 24 hour cycle, providing fewer opportunities for the agents to learn that the evening demand can be delayed to the early hours of the subsequent day. Moreover, delaying tasks on the last cycle of each batch would have led to a lower value of soft-constraint reward $r_{e,n}$, but an un-materialised cost improvement (higher $r_{c,n}$) for the subsequent morning (because of the midnight cut-off).\n\nThe results discussed above are significant for our particular application because it shows that while each household is controlled by different policies, all households are able to learn a cooperative strategy that flattens energy demand and consequently decrease the average cost for all users. The ability of the agents to learn in a decentralised manner is compatible with a microgrid system where there may be limited bandwidth to both receive and transmit information centrally. The individual user, or a small cluster of users, could improve their policy using local computing resources, while receiving only the output signal from the centralised critic network. Conversely, if computing resources for neural network training are only available centrally, update gradients can be transmitted to individual actor networks and result in similar performance.\n\n\\subsection{Comparison with ECS}\n\\begin{figure}[t]\n\\centering    \n\\includegraphics[width=.45\\textwidth]{EnergyTimeGraph10houses.png}\n\\caption{Average aggregate energy consumed by 10-household microgrid for each timestep of the day - a comparison between MAPPO and ECS}\n\\label{av_energy_consumed_10houses}\n\\end{figure}\n\nWe compare our proposed MAPPO algorithm-based method to the Energy Consumption Scheduler (ECS) method proposed by \\cite{mohsenian10}, which is also a DSM-based solution. Instead of making decisions online as with MAPPO, ECS requires the following information in advance to schedule all energy consumption for the next day: (i) the total planned energy consumption for each household appliance for the entire day, and (ii) the mathematical function used to determine the price of electricity for a given time period. For a better comparison, we adapted the experimental setup to match the method used for ECS in its original paper: we considered a smart grid system with $N=10$ households, split each day into $H=24$ equal time intervals, fixed the length of operation of each task such that $l^m_n = 1 \/ \\lambda_n^m$, and used a quadratic price function as described in (\\ref{eq_price_quad}). We also consider different values of the multiplier $w$ for the cost constraint in the reward function (\\ref{eq_reward}).\n\\begin{equation}\n  \\label{eq_price_quad}\n  p_n(k)=b(h) \\times E_n(k)^2.\n\\end{equation}\n\nInput (i) required by the ECS method is thus computed as follows:\n\\begin{equation}\n  \\label{eq_day_energy}\n  \\sum_{h=1}^{H}E_n^m = \\sum_{h=1}^{H} \\left( p^m_n(h) \\times l^m_n \\times P^m_n \\right).\n\\end{equation}\n\nIn Figure \\ref{av_energy_consumed_10houses}, we plot the average aggregate energy consumed by the microgrid for each hour-long time step. Once again, the agents trained using reinforcement learning are able to delay the peaks in demand and spread the energy demand across subsequent time steps. The MAPPO agents trained with a coefficient of $w=2.2$ consume almost the same energy over the entire day as the naive agents with zero delay; this constitutes a shift in energy demand that results in a decrease in the cost of energy.\n\n\\begin{figure}[t]\n\\centering    \n\\includegraphics[width=.45\\textwidth]{AverageRewardperDay10households2.png}\n\\caption{Average reward obtained per day in a 10-household microgrid - a comparison with ECS}\n\\label{av_reward_10households}\n\\end{figure}\n\nIn comparison to the MAPPO agents, ECS produces a flatter energy demand profile. Because the ECS algorithm receives both the energy consumption requirements and the price function before the day commences, it is able to schedule appliances to operate in the early hours of the day to exploit lower prices. However, since the energy consumption requirements received by ECS are average values, each household experiences a mismatch between its actual energy demand for the day and the schedule produced by ECS. Consequently, we compute a reward according to (\\ref{eq_reward}), where the value $r_{c,n}$ is taken to be the actual energy demand that is fulfilled by the ECS-generated schedule. The results plotted in Figure \\ref{av_reward_10households} show that the MAPPO agents with $w=2.2$ achieve an average daily reward that is more than $10\\%$ higher than that achieved by ECS.\n\nThe results reveal the MAPPO method's key advantage, in that it can be utilised in a fully online scenario without requiring the price function or the total daily consumption in advance. It is able to rely on instantaneous observations of the local state and the price signal to determine how tasks should be scheduled by delaying them. This online approach and reliance on mainly locally observable information is more consistent with the nature of domestic electricity use.\n\nAnother advantage of MAPPO is that, in addition to shifting the demand of energy, we are able to implement energy reduction by tuning the coefficient $w$. Figure \\ref{av_energy_consumed_10houses} shows that the MAPPO agents trained with a smaller coefficient of $w=0.4$ exhibit lower energy demand than both the MAPPO agents with $w=2.2$ and the ECS method.\n\n\\subsection{Comparison with Methods using Decentralised Critics}\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{AverageRewardperDay8households3-2.png}\n        \\caption{Average reward obtained per day}\n        \\label{av_reward_8households}\n    \\end{subfigure}%\n    ~ \n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{AverageCostperDay8households4.png}\n        \\caption{Average monetary cost incurred per day}\n        \\label{av_cost_8households}\n    \\end{subfigure}\n    \\caption{Results for simulation in an 8-household microgrid system. (\\subref{av_reward}) and (\\subref{av_cost}) compare the best results obtained in our experiments on the effectiveness of augmented observations and policy sharing.}\n    \\label{fig_8households}\n\\end{figure*}\nFigures~\\ref{av_reward} and~\\ref{av_cost} also compare our proposed MAPPO algorithm to a similar PPO-based algorithm with decentralised critics, and the more traditional Advantage Actor-Critic (A2C) \\cite{pmlr-v48-mniha16} algorithm. The results show that all the agents trained with deep reinforcement learning perform better than the non-learning agents.\n\nOur proposed MAPPO algorithm achieves the most sample-efficient learning speed and best performance, while the performance of the decentralised PPO algorithm is close. This demonstrates that good performance can also be achieved by learning in a completely decentralised manner. This would be useful in scenarios where connectivity of the smart meters outside of the household is limited, but local computing power can be utilised.\n\nThe canonical A2C algorithm is also able to reach the same level of performance, although it learns less efficiently with the same number of samples. Experiments with higher learning rates showed instabilities that manifested in exploding gradients -- this was in spite of policy-based algorithms such as A2C being generally known to be more stable than value-based algorithms. The reason is that, in our proposed microgrid environment, the updates to the policies for other households under A2C have a large effect on the overall system dynamics, possibly worsening the non-stationarity of the environment as perceived by each household. In contrast, the nature of the actor network update in the PPO algorithm limits the change in the actions taken by the updated policy, thus leading to stable performance increments.\n\n\\subsection{Effect of augmenting local observations}\nWe perform experiments to determine the necessity of including the simulation clock time and the previous price as part of the observations $o_n$ perceived by each agent. In an 8-household microgrid system, we train MAPPO agents that use observations without the time in one experiment, and train agents with the time but without the previous price in another experiment. The results are shown in Figure~\\ref{fig_8households}.\n\nThe agents trained without knowledge of the simulation clock time achieve a performance that significantly overlaps with the agents trained with the full observation vector. On the other hand, depriving the agents of knowledge of the price of energy on the previous time step shows both a marked decrease in average reward accrued and an increase in the average cost of electricity. The reason is that knowledge of the price at the previous time step is useful because it indirectly reveals information about the strategy used by the other household agents. In making this information observable, the price-augmented observation ameliorates the problem of perceived environmental non-stationarity. In contrast, knowledge of the simulation clock time may be less useful because features within the local state, such as the appliance status $x_n$ and queue lengths $q_n$, already provide time-related queues on how high the current demand for energy is.\n\n\\subsection{Effect of Policy Sharing}\n\\begin{figure}[t]\n\\centering    \n\\includegraphics[width=.45\\textwidth]{PercentDone1-critic21-8houses-xpolicies2.png}\n\\caption{Percentage of incoming tasks to the queue that were completed - household 1.}\n\\label{percent_done1}\n\\end{figure}\n\nWe consider a scenario where groups of household agents may share a common policy. This may be useful when computing resources are unavailable in individual households, but accessible from a more centralised location. Instead of having each household agent controlled by a single policy, we conduct experiments where each policy is shared by two specific household agents. In these setups, each household always follows the same policy network.\n\nThe results are shown in Figure~\\ref{fig_8households}. While the reward achieved is similar, the systems where agents shared policies show a slightly lower resulting cost in energy. This is consistent with the findings of~\\cite{RN242}, which reported that sharing \"policy ensembles\" resulted in better scores. However, it should be noted that in the mentioned study, the policies are assigned to randomly chosen agents at each episode.\n\nWe investigated further by counting both the number of tasks generated in the household appliance queues $q^m_n$ and the number of tasks done in each time step to compute the percentage of tasks done by each agent for each episode. Figure~\\ref{percent_done1} shows the percentage of tasks that were completed by household 1 throughout the training process. It reveals that the shared policies are able to achieve a lower energy cost by strategically delaying tasks in the queue (which decreases energy consumption). On the other hand, single policies are unable to employ the same task-delaying strategy without overly sacrificing the second component of the reward that encourages task completion.\n\n\\section{Conclusion}\n\\label{sec_conclusion}\nIn this paper, we proposed and implemented a smart grid environment, as well as a multi-agent extension of a deep actor-critic algorithm for training decentralised household agents to schedule household appliances with the aim of minimising the cost of energy under a real-time pricing scheme. Our approach allows for an online approach to household appliance scheduling, requiring only the local observation and a price signal from the previous time step to act. A joint critic is learned to coordinate training centrally.\n\nOur results show that our proposed algorithm is able to train agents that achieve a lower cost and flatter energy-time profile than non-learning agents. Our algorithm also achieves quicker learning than independent agents trained using the canonical Advantage Actor-Critic (A2C) algorithm. The successful results show that the choice of the observed variables and reward function design in our proposed environment contained sufficient information for learning.\n\nFuture work can involve extension of the smart grid environment to include more features and real-world complexities; we identify two such possibilities. Firstly, the architecture of the microgrid can be expanded to include more energy sources, which may be operated by producers or households. Secondly, we may expand the overall grid to include more agents and more types of agents, such as energy producers and brokers.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\renewcommand*{\\thefootnote}{$^\\ast$}\n\t\\footnotetext[1]{Last three authors are listed in alphabetical order.}\n    \\renewcommand*{\\thefootnote}{\\arabic{footnote}}\n    \\setcounter{footnote}{0}\n\n\tCan we devise a learning algorithm for general deep neural networks (DNNs) featuring inertia and Newtonian directional intelligence only by means of backpropagation? In an optimization jargon: can we use second-order ideas in time and space for {\\em nonsmooth nonconvex} optimization by only using a subgradient oracle?\n    Before providing some answers to this  question, let us have a glimpse at some of the fundamental optimization algorithms for training DNNs.\n\n\tThe backpropagation algorithm  is, to this day, the fundamental block to compute gradients in deep learning (DL). It is used in most instances of the Stochastic Gradient Descent (SGD) algorithm  \\citep{robbins1951stochastic}. The latter is powerful, flexible, capable of handling large-size problems, noise, and further comes with theoretical guarantees of many kinds. We refer to \\cite{bottou2008tradeoffs,moulines2011non} in a convex machine learning context and \\cite{bottou2018optimization} for a recent account highlighting the importance of DL applications and their challenges. In the nonconvex setting, recent works of \\cite{adil,davis2018stochastic} follow the \\textit{Ordinary Differential Equations (ODE) approach} introduced in \\cite{ljung1977analysis}, and further developed in \\cite{benaim1999dynamics,kushner2003stochastic,benaim2005stochastic,borkar2009stochastic}. \n\tTwo research directions have been explored in order to improve SGD's training efficiency:\n    \\begin{itemize}\n    \\renewcommand\\labelitemi{--}\n        \\item using local geometry of empirical loss functions to obtain {steeper} descent directions,\n        \\item using past steps history to design more {larger step sizes} in the present.\n    \\end{itemize}\n\tThe first approach is akin to quasi-Newton methods while the second revolves around Polyak's inertial method \\citep{polyak1964some}.\n\tThe latter is inspired by the following appealing mechanical thought-experiment. Consider a heavy ball evolving on the graph of the loss (the loss \\textit{landscape}), subject to gravity and stabilized by some friction effects. Friction generates energy dissipation, so that the particle will eventually reach a steady state which one hopes to be a local minimum. These two approaches are already present in the DL literature: among the most popular algorithms for training DNNs, ADAGRAD \\citep{duchi2011adaptive} features  local geometrical aspects while ADAM \\citep{kingma2014adam} combines inertial ideas with step sizes similar to the ones of ADAGRAD. Stochastic Newton and quasi-Newton algorithms have been considered by \\cite{martens2010deep,byrd2011use,byrd2016stochastic} and recently reported to perform efficiently on several problems \\citep{berahas2017investigation,xu2017second}. The work of \\cite{wilson2017marginal} demonstrates that carefully tuned SGD and heavy-ball algorithms are competitive with concurrent methods.\n\t\n\n\tHowever, deviating from the simplicity of SGD also comes with major challenges because of {the high dimensionality of DL problems} and the severe absence of regularity in DL (differential regularity is generally absent, but even weaker regularity such as semi-convexity or Clarke regularity are not {always} available). All sorts of practical and theoretical hardships are met: {computing and even defining the Hessians is delicate}, inverting them is unthinkable to this day, first and second-order Taylor approximations are unavailable due to nonsmoothness, and finally one has to deal with ``shocks'' which are inherent to inertial approaches in a nonsmooth context (``corners'' and ``walls'' {in the landscape} of the loss generate velocity discontinuity). This makes {in particular the study} of the popular algorithms ADAGRAD and ADAM in full generality quite difficult, with recent progresses reported in \\cite{barakat2018convergence}.\n\n Our approach is inspired by the following {continuous-time} dynamical system introduced in \\cite{alvarez2002second} and referred to as DIN (standing for ``dynamical inertial Newton''):\n \\begin{align}\n \\label{eq:physicalIntuitionSmooth}\n \\underbrace{\\ddot\\theta(t)}_{\\text{\\rm Inertial term}}+\\underbrace{\\alpha\\,\\dot\\theta(t)}_{\\text{\\rm Friction term}}+\\underbrace{\\beta\\,\\nabla^2\\mathcal{J}(\\theta(t))\\dot\\theta(t)}_{\\text{\\rm Newtonian effects}}+\\underbrace{\\nabla \\mathcal{J}(\\theta(t))}_{\\text{\\rm Gravity effect}}= \\ 0, \\quad \\text{for $t\\in [0,+\\infty)$},\n \\end{align}\n where $t$ is the time parameter which acts as a continuous epoch counter, $\\mathcal{J}$ is a given loss function (e.g., the empirical loss in DL applications), for now assumed $C^2$ (twice-differentiable), with gradient $\\nabla \\mathcal{J}$ and Hessian $\\nabla^2\\mathcal{J}$. \\new{It can be shown that solutions to \\eqref{eq:physicalIntuitionSmooth} converge to critical points of $\\cal{J}$ \\citep{alvarez2002second}. As such the discretization of \\eqref{eq:physicalIntuitionSmooth} can support the design of algorithms that optimize $\\cal{J}$ and that leverage inertial properties with Newton's method. To adapt this dynamics to DL we must first overcome the computational or conceptual difficulties raised by the second-order objects $\\ddot \\theta$ and $\\nabla^2\\mathcal{J}(\\theta)$ appearing in \\eqref{eq:physicalIntuitionSmooth}. To do this, we propose in this paper to combine a phase-space lifting method introduced in \\cite{alvarez2002second} with the use of Clarke subdifferential $\\partial  \\mathcal{J}$. Clarke subdifferential defines a notion of differentiability for nonconvex and nonsmooth functions. This approach  results in the study of a first-order differential inclusion in place of \\eqref{eq:physicalIntuitionSmooth}, namely,\n\\begin{equation}\n\t\t\\begin{cases}\n\t\t\\dot{\\theta}(t) + \\beta  \\partial \\mathcal{J}(\\theta(t)) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0\\\\\n\t\t\\dot{\\psi}(t) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0  \\end{cases} \\mbox{,\\quad for a.e. $t\\in (0,+\\infty)$}.\n\t\t\\end{equation}\nThis differential inclusion can then be discretized to obtain the practical algorithm INDIAN that is introduced in Section~\\ref{sec:algo}, together with a rigorous presentation of the concepts mentioned above.}\n\n\\new{Computation of the (sub)gradients and convergence proofs (in batch or mini-batch settings) typically rely on the sum-rule in smooth or convex settings, i.e., $\\partial ({\\cal J}_1 + {\\cal J}_2) = \\partial{\\cal J}_1 + \\partial{\\cal J}_2$. Unfortunately this sum-rule does not hold in general in the nonconvex setting using the standard Clarke subdifferential. Yet, many DL studies ignore the failure of the sum rule: they use it in practice, but circumvent the theoretical problem by modeling their method through simple dynamics (e.g., smooth or convex). We tackle this difficulty as is, and show that such practice  can create additional spurious stationary points that are not Clarke-critical. To address this question, we introduce the notion of $D$-criticality. It is less stringent than Clarke-criticality and   it describes more accurately real-world implementation. We  then show convergence of INDIAN to such $D$-critical points. Our theoretical results are general, simple and allow for aggressive step-sizes in $o(1\/\\log k)$.} {We first provide adequate calculus rules and tame nonsmooth Sard's-like results for the new steady states we introduced. We then combine these results with a Lyapunov analysis from \\cite{alvarez2002second} and} the differential inclusion approximation method \\citep{benaim2005stochastic} to characterize the asymptotics of our algorithm similarly to \\cite{davis2018stochastic,adil}. This provides a strong theoretical ground to our study since we can prove that our method converges to a connected component of the set of steady states {even for networks with ReLU or other nonsmooth activation functions}. For the smooth deterministic dynamics, we also show that convergence in values is of the form $O(1\/t)$ where $t$ is the running time. \nFor doing so we provide a general result  for the solutions of a family of differential inclusions having a certain type of favorable Lyapunov functions.\n\nOur algorithm INDIAN shows great efficiency in practice. It has similar computational complexity to state-of-the-art methods SGD, ADAGRAD and ADAM, often achieves better training accuracy and shows good robustness to hyper-parameters selection. INDIAN can avoid parasitic oscillations and produce acceleration; a first illustration of the behavior of the induced dynamics is given in Figure \\ref{fig::rosenbrockexp} for a simple nonsmooth and nonconvex function in $\\mathbb{R}^2$.\n\nThe rest of the paper is organized as follows. INDIAN is introduced in details in Section \\ref{sec:algo} and its convergence is established in Section \\ref{sec:proof}. Convergence rates of the underlying continuous time differential inclusion are obtained in Section~\\ref{sec::rateofCV}. Section \\ref{sec:numerics} describes experimental DL results on synthetic and real data sets (MNIST, CIFAR10, CIFAR100).\n\n\n\n\n\t\\begin{figure}[t]\n    \\begin{center}%\n    \\begin{tabular}{cc}\n       \\includegraphics[width=0.47\\textwidth]{Figures\/EXP_2D\/nonsmooth_rosenalpha03_beta_001_g2.pdf} & \\includegraphics[width=0.47\\textwidth]{Figures\/EXP_2D\/nonsmooth_rosenalpha03_beta_01_g2.pdf}\\\\\n       (a) $\\alpha=0.5,\\ \\beta=0.01$  & (b) $\\alpha=0.5,\\ \\beta=0.1$ \\medskip \\\\\n       \\includegraphics[width=0.47 \\textwidth]{Figures\/EXP_2D\/nonsmooth_rosenalpha1_3_beta_01_g2.pdf} & \\includegraphics[width=0.47\\textwidth]{Figures\/EXP_2D\/2D_exp_loss.pdf}\\\\\n       (c) $\\alpha=1.3,\\ \\beta=0.1$ & (d) Objective function $\\mathcal{J}$ (in log-scale)\n    \\end{tabular}\n    \\caption{Illustration of INDIAN applied to the nonsmooth function $\\mathcal{J}(\\theta_1,\\theta_2)= 100(\\theta_2-\\vert \\theta_1 \\vert)^2 +\\vert 1-\\theta_1\\vert $. Subplots (a-c) represent the trajectories of the parameters $\\theta_1$ and $\\theta_2$ in $\\mathbb{R}^2$ for three choices of hyper-parameters $\\alpha$ and $\\beta$, see \\eqref{eq:physicalIntuitionSmooth} for an intuitive explanation. Subplot (d) displays the values of the objective function $\\mathcal{J}(\\theta_1,\\theta_2)$ for the three settings considered.}\\label{fig::rosenbrockexp\n    \\end{center}%\n    \\end{figure}\n\n\n\n\n\n\n\n   \n\\section{INDIAN: an Inertial Newton Algorithm for Deep Neural Networks}\n\t\\label{sec:algo}\n\tWe first introduce our functional framework and then we describe step by step the process of building INDIAN from \\eqref{eq:physicalIntuitionSmooth}.\n\t\\subsection{Neural Networks with Lipschitz Continuous Prediction Function and loss functions}\\label{sec::structofDL}\n    \n    We consider DNNs of a general type represented by a function $f:(x,\\theta)\\in \\mathbb{R}^M\\times \\mathbb{R}^P\\mapsto y\\in \\mathbb{R}^D$ that is a {\\em locally Lipschitz continuous} in $\\theta$. This includes for instance feed-forward, convolutional or residual networks used with ReLU, sigmoid, or tanh activation functions. Recall that a function $F:\\mathbb{R}^P\\to \\mathbb{R}$ is {locally Lipschitz continuous}, if for any $\\theta \\in\\mathbb{R}^P$, there exists a neighborhood $V$ of $\\theta$ and a constant $C>0$ such that for any $\\theta_1,\\theta_2\\in V$,\n        \\begin{equation*}\n            \\vert F(\\theta_1) - F(\\theta_2)\\vert \\leq C \\left\\|\\theta_1 - \\theta_2\\right\\|,\n        \\end{equation*}\n        where $\\left\\| \\cdot \\right\\|$ is any norm on $\\mathbb{R}^P$. A function $F:\\mathbb{R}^P\\to \\mathbb{R}^D$ is locally Lipschitz continuous if each of its coordinates is locally Lipschitz continuous. The variable $\\theta \\in \\mathbb{R}^P$ is the parameter of the model ($ P $ can be very large), while $ x\\in \\mathbb{R}^M $ and $y\\in \\mathbb{R}^D$ represent input and output data. For instance, the vector $ x $ may embody an image while $y$ is a label explaining its content. Consider further a data set of $ N$ samples $ (x_n,y_n)_{n=1,\\ldots,N} $. Training the network amounts to finding a value of the parameter $ \\theta $ such that, for each input data $ x_n $ of the data set, the output $ f(x_n,\\theta)$ of the model predicts the real value $ y_n $ with good accuracy. To do so, we follow the traditional approach of minimizing an empirical risk loss function, \\begin{equation}\\label{eq::loss}\n\t\\mathbb{R}^P\\ni\\theta \\mapsto \\mathcal{J}(\\theta)=\\sum_{n=1}^N l(f(x_n,\\theta),y_n),\n\t\\end{equation} where $l:\\mathbb{R}^D\\times \\mathbb{R}^D\\to \\mathbb{R}$ is a locally Lipschitz continuous dissimilarity measure. \\new{In the sequel, for $n\\geq1$, we will sometimes denote by $\\mathcal{J}_n$ the $n$-th term of the sum: $\\mathcal{J}_n(\\theta) \\triangleq l(f(x_n,\\theta),y_n)$, so that $\\mathcal{J}=\\sum_{n=1}^N \\mathcal{J}_n$. Despite the nonsmoothness and the nonconvexity of typical DL loss functions, they generally possess a very strong  property sometimes called tameness. We now introduce this notion which is essential to obtain the convergence results of Section~\\ref{sec:proof}.}\n\n\t\\subsection{Neural Networks and Tameness in a Nutshell}\\label{sec::favstruct}\n\nTameness  refers to an ubiquitous geometrical property of loss functions and constraints encompassing most finite dimensional optimization problems met in practice. Prominent classes of tame objects are piecewise-linear or piecewise-polynomial objects (with finitely many pieces), and more generally, semi-algebraic objects. However, the notion is much more general, as we intend to convey below. The formal definition is given at the end of this subsection (Definition~\\ref{DEF}).\n\nInformally, sets or functions are called  tame  when they can be described by a finite number of basic formulas, inequalities, or Boolean operations involving standard functions such as polynomial, exponential, or max functions. We refer to \\cite{attouch} for illustrations, recipes and examples within a general optimization setting or  \\cite{davis2018stochastic} for illustrations in the context of neural networks. The reader is referred to \\cite{van1998tame,coste2000introduction,shiota} for foundational material. To apprehend the strength behind tameness it is convenient to remember that it models nonsmoothness by confining the study to sets and functions which are union of smooth pieces. This is the so-called {\\em stratification} property of tame sets and functions. It was this property which motivated the term of {\\em tame topology},\\footnote{``{\\em La topologie mod\\'er\\'ee}'' wished for by Grothendieck.} see \\cite{van1998tame}. In a nonconvex optimization settings, the stratification property is crucial to generalize qualitative algorithmic results to nonsmooth objects.\n\n{\\em All finite dimensional DL optimization models we are aware of yield tame loss functions $\\mathcal{J}$}. To understand this assertion and illustrate the wide scope of tameness assumptions, let us provide concrete examples (see also \\cite{davis2018stochastic}).\nAssume that the DNNs under consideration are built from the following traditional components:\n\\begin{itemize}\n    \\renewcommand\\labelitemi{--}\n    \\item the network architecture describing $ f $ is fixed with an arbitrary number of layers of arbitrary dimensions and arbitrary Directed Acyclic Graph (DAG) representing computation,\n\\item  the activation functions are among classical ones: ReLU, sigmoid, SQNL, RReLU, tanh, APL, soft plus, soft clipping, and many others including multivariate activations (norm, sorting), or activations defined piecewise with polynomials, exponential and logarithm,\n\\item  the dissimilarity function $l$ is a standard loss such as $\\ell_p$ norms, logistic loss or cross-entropy or more generally a function defined piecewise using polynomials, exponentials and logarithms,\n\\end{itemize}\nthen one can easily show, by elementary quantifier elimination arguments (property (iii) below), that the corresponding loss, $\\mathcal{J}$,  is tame.\n\\medskip\n\n\nFor the sake of completeness, we provide below the formal definition of tameness and o-minimality. \n\\begin{definition}\\label{DEF}\n{\\rm {\\bf [o-minimal structure] }}{\\rm\n\\cite[Definition\\,1.5]{coste2000introduction}} \\label{Domin}\\rm{An {\\it\no-minimal } structure on $(\\mathbb{R},+,.)$ is a countable collection of sets ${\\cal O}=\\{\\mathcal{O}_{q}\\}_{q\\geq 1}$ where each $\\mathcal{O}_{q}$ is itself a collection of subsets of $\\mathbb{R}^q$, called {\\em definable} subsets. They must have the following properties,  for each $q\\geq 1$:}\n\\begin{enumerate}\\itemsep=1mm\n\\item[(i)] (Boolean properties) $\\mathcal{O}_{q}$ contains the empty set, is stable by finite union, finite intersection and complementation;\n\\item[(ii)] (Lifting property) \n{\\rm if $A$ belongs to $\\mathcal{O}_{q}$, then $A\\times\\mathbb{R}$ and\n$\\mathbb{R}\\times A$ belong to $\\mathcal{O}_{q+1}$.}\n\\item[(iii)] (Projection or quantifier elimination property) \n{\\rm if $\\Pi:\\mathbb{R}^{q+1}\\rightarrow\\mathbb{R}^q$ is the canonical\nprojection onto $\\mathbb{R}^q$ then for any $A$ in $\\mathcal{O} _{q+1}$,\nthe set $\\Pi(A)$ belongs to $\\mathcal{O}_{q}$.}\n\\item[(iv)] (Semi-algebraicity)\n{\\rm $\\mathcal{O}_{q}$ contains the family of algebraic\nsubsets of $\\mathbb{R}^q$, that is, every set of the form\n\\[\n\\{\\theta\\in\\mathbb{R}^q\\mid\\zeta(\\theta)=0\\},\n\\]\nwhere $\\zeta:\\mathbb{R}^q\\rightarrow\\mathbb{R}$ is a polynomial function.}\n\\item[(v)] (Minimality property), \n{\\rm  the elements of $\\mathcal{O}_{1}$ are exactly the finite\nunions of intervals and points. }\n\\end{enumerate}\n\\end{definition}\n\n\n\\noindent A mapping $F:S\\subset \\mathbb{R}^m\\rightarrow \\mathbb{R}^q$ is said to be\n{\\em definable in ${\\cal O}$} if its graph is  definable in $\\cal O$ as a subset of~$\\mathbb{R}^m\\times\\mathbb{R}^q$. For illustration of o-minimality in the context of optimization one is referred to \\cite{attouch,davis2018stochastic}.\n\n\n\n\\begin{center}\n\\fbox{\\begin{minipage}{14cm}  From now on we fix an o-minimal structure $\\mathcal{O}$ and a set or a mapping definable in $\\cal O$ will be called {\\em tame}.\n\\end{minipage}}\n\\end{center}\n\n\\subsection{From DIN to INDIAN}\\label{sec::optpart}\n        \n        \n        We describe in this section the construction of our proposed algorithm INDIAN from the discretization of the second-order ODE~\\eqref{eq:physicalIntuitionSmooth}.\n        \n        \n        \n        \n\t\t\\subsubsection{Handling Nonsmoothness and Nonconvexity}\\label{sec::nonsmoothnonconv}\n\n\t\t We first show how the formalism offered by Clarke's subdifferential can be applied to generalize \\eqref{eq:physicalIntuitionSmooth} to the nonsmooth nonconvex setting. Recall that the dynamical system \\eqref{eq:physicalIntuitionSmooth} is described by,\n\t\t %\n\t\t\\begin{equation}\\label{eq::secondorderDIN}\n\t\t\\ddot{\\theta}(t)+\\alpha\\dot{\\theta}(t) + \\beta \\nabla^2 \\mathcal{J}(\\theta(t))\\dot{\\theta}(t) +\\nabla \\mathcal{J}(\\theta(t))=0,\n\t\t\\end{equation}\n\t\t%\n\t\twhere $\\mathcal{J}$ is a twice-differentiable potential,  $\\alpha>0$, $\\beta>0$ are two hyper-parameters and $\\theta: \\mathbb{R}_+ \\rightarrow \\mathbb{R}^P$. We cannot exploit \\eqref{eq::secondorderDIN} directly since in most DL applications $\\mathcal{J}$ is not twice differentiable (and even not differentiable at all). We first overcome the explicit use of the Hessian matrix $\\nabla^2 \\mathcal{J}$ \\new{by introducing an auxiliary variable $\\psi: \\mathbb{R}_+ \\rightarrow \\mathbb{R}^P$ like in \\cite{alvarez2002second}. Consider the following dynamical system (defined for $\\mathcal{J}$ merely differentiable),}\n\t\t\\begin{equation}\\label{eq::contdinSmooth}\n\t\t\\begin{cases}\n\t\t\\dot{\\theta}(t) + \\beta  \\nabla \\mathcal{J}(\\theta(t)) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) = 0\\\\\n\t\t\\dot{\\psi}(t) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) = 0  \\end{cases} \\mbox{,\\quad for a.e. $t\\in (0,+\\infty)$}.\n\t\t\\end{equation}\n\t\t\\new{As explained in \\cite{alvarez2002second}, \\eqref{eq::secondorderDIN} is equivalent to \\eqref{eq::contdinSmooth} when $\\mathcal{J}$ is twice differentiable. Indeed, one can rewrite \\eqref{eq::secondorderDIN} into \\eqref{eq::contdinSmooth} by introducing $\\psi = -\\beta\\dot{\\theta}-\\beta^2\\nabla\\mathcal{J}(\\theta)-(\\alpha\\beta-1)\\theta$. Conversely, one can substitute the first line of \\eqref{eq::contdinSmooth} into the second one to retrieve \\eqref{eq::secondorderDIN}.} Note however that \\eqref{eq::contdinSmooth} does not require the existence of second-order derivatives.\n\t\t\n\t\tLet us now introduce a new {\\em nonconvex nondifferentiable} version of \\eqref{eq::contdinSmooth}.\n\t\tBy Rademacher's theorem, locally Lipschitz continuous functions $\\mathcal{J}:\\mathbb{R}^P\\to \\mathbb{R}$ are differentiable almost everywhere. Denote by $\\mathsf{R}$ the set of points where $\\mathcal{J}$ is  differentiable. Then, $\\mathbb{R}^P\\setminus \\mathsf{R}$ has zero Lebesgue measure. It follows that for any $\\theta^\\star\\in\\mathbb{R}^P\\setminus \\mathsf{R}$, there exists a sequence of points in $\\mathsf{R}$ whose limit is this $\\theta^\\star$. This motivates the introduction of the subdifferential due to \\cite{clarke1990optimization}, defined next.\n\t\t\\begin{definition}[Clarke subdifferential of Lipschitz functions] \\label{def:clarke}\n\t\t\tFor any locally Lipschitz continuous function $F: \\mathbb{R}^P\\to \\mathbb{R}$, the Clarke subdifferential of $F$ at $\\theta\\in\\mathbb{R}^P$, denoted $\\partial F(\\theta)$, is the set defined by,\n\t\t\t\\begin{equation}\n\t\t\t\t\\partial F(\\theta) = \\mathrm{conv}\\left\\{ v\\in\\mathbb{R}^P \\mid \\exists (\\theta_k)_{k\\in\\mathbb{N}}\\in \\mathsf{R}^\\mathbb{N},\\text{ such that } \\theta_k \\xrightarrow[k\\to\\infty]{}\\theta \\text{ and }  \\nabla F(\\theta_k) \\xrightarrow[k\\to\\infty]{} v \\right\\},\n\t\t\t\\end{equation}\n\t\t\twhere $\\mathrm{conv}$ denotes the convex hull operator. \\new{The elements of the Clarke subdifferential are called Clarke subgradients.}\n\t\t\\end{definition}\n\t\tThe Clarke subdifferential is a nonempty compact convex set. \\new{It coincides with the gradient for smooth functions and with the traditional subdifferential for nonsmooth convex functions. As already mentioned, and contrarily to the (sub)differential operator, it does not enjoy a sum rule.} \n\t\t\n\t\tThanks to Definition~\\ref{def:clarke}, we can extend \\eqref{eq::contdinSmooth} to nondifferentiable functions. Since $\\partial \\mathcal{J}(\\theta)$ is a set, we no longer study a differential equation but rather a {\\em differential inclusion}, given by,\n\t\t \\begin{equation}\\label{eq::contdinClarke}\n\t\t \\begin{cases}\n\t\t \\dot{\\theta}(t) + \\beta  \\partial \\mathcal{J}(\\theta(t)) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0\\\\\n\t\t \\dot{\\psi}(t) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0  \\end{cases} \\mbox{,\\quad for a.e. $t\\in (0,+\\infty)$}.\n\t\t \\end{equation}\n\t\t For a given initial condition $(\\theta_0,\\psi_0)\\in \\mathbb{R}^P\\times\\mathbb{R}^P$, we call {\\em solution} (or {\\em trajectory}) of this system any absolutely continuous curve $(\\theta,\\psi)$ from $\\mathbb{R}_+$ to $\\mathbb{R}^P\\times\\mathbb{R}^P$ for which $(\\theta(0),\\psi(0)) = (\\theta_0,\\psi_0)$ and \\eqref{eq::contdinClarke} holds. We recall that absolute continuity amounts to the fact that $\\theta$ is differentiable almost everywhere with integrable derivative and,\n\t\t$$\\theta(t) -\\theta(0)=\\int_0^t\\dot\\theta(s)\\diff s, \\mbox{ for $t\\in [0,+\\infty)$.}$$\n\t\tDue to the properties of the Clarke subdifferential, existence of a solution to differential inclusions such as \\eqref{eq::contdinClarke} is ensured, see \\cite{aubin}; note however that uniqueness of the solution does not hold in general. We will now use the structure of \\eqref{eq::contdinClarke} to build a new algorithm to train DNNs.\n\n\t\t \\subsubsection{Discretization of the Differential Inclusion}\n\n        To obtain the basic form of our algorithm, we discretize \\eqref{eq::contdinClarke} according to the classical explicit Euler method. Given $(\\theta,\\psi)$ a solution of \\eqref{eq::contdinClarke} and any time $t_k$, set $\\theta_k = \\theta(t_k)$ and $\\psi_k = \\psi(t_k)$. Then, at time $t_{k+1}=t_{k}+\\gamma_k$ with $\\gamma_k$ positive small, one can approximate $\\dot{\\theta}(t_{k+1})$ and $\\dot{\\psi}(t_{k+1})$ by\n        \\[ \\dot{\\theta}(t_{k+1})\\simeq \\frac{\\theta_{k+1}-\\theta_k}{\\gamma_k},\\quad \\quad \\dot{\\psi}(t_{k+1})\\simeq \\frac{\\psi_{k+1}-\\psi_k}{\\gamma_k}. \\]\n        This discretization yields the following algorithm,\n        \\begin{equation}\\,\\,\\,\\label{eq::notindian}\n        \t\\begin{cases}\n        \t    v_k &\\in \\partial\\mathcal{J}(\\theta_k)\\\\\n            \t\\theta_{k+1}&=  \\theta_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k -\\frac{1}{\\beta}\\psi_k - \\beta v_k \\right)\\\\\n            \t\\psi_{k+1}& = \\psi_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k - \\frac{1}{\\beta} \\psi_k \\right)\n        \t\\end{cases}\n        \t\\end{equation}\n\n        Although the algorithm above is well defined for our problem, \\new{it is not suited to  DL. First, the computation of $\\partial\\mathcal{J}(\\theta_k)$ is generally not possible since there is no general operational calculus for Clarke's subdifferential; secondly, a mini-batch strategy must be designed to cope with the large dimensionality of DL problems which makes the absence of sum rule even more critical.\n        \n        The next section is meant to address these issues and design  a practical algorithm.}\n        \n        \n\n\n        \\subsubsection{INDIAN Algorithm with a New Notion of Steady States}\n\n\n        In order to compute or approximate the subdifferential of $\\mathcal{J}$ at each iteration and to cope with large data sets, $\\mathcal{J}$ can be approximated by mini-batches, reducing the memory footprint and computational cost of evaluation.  For any $\\mathsf{B} \\subset \\{1,\\ldots, N\\}$, let us define\n\t    \\begin{align}\\label{eq::minibatch}\n\t        \\mathcal{J}_{\\mathsf{B}} \\colon \\theta \\mapsto \\sum_{n \\in \\mathsf{B}} l(f(x_n,\\theta),y_n).\n\t    \\end{align}\n        Unlike in the differentiable case, subgradients do not in general sum up to a subgradient of the sum, that is $\\partial \\mathcal{J}_\\mathsf{B}(\\theta) \\neq \\sum_{n \\in \\mathsf{B}} \\partial l(f(x_n,\\theta),y_n) $ in general. To see this, take for example $ 0 = |\\cdot| - |\\cdot|$, the Clarke subgradient of this function at $0$ is $\\{0\\}$, whereas $\\partial(|0|) + \\partial(-|0|) = [-1,1] + [-1,1] = [-2,2]$. \\new{Standard DL solvers use backpropagation algorithms which implement smooth calculus on nonsmooth and nonconvex objects. Due to the absence of qualification conditions, {\\em the resulting objects are not Clarke subgradients in general}.  In order to match the real-world practice of DL,} we introduce a notion of steady states that corresponds to the stationary points generated by a generic mini-batch approach. As we shall see, this  allows both for practical applications and convergence analysis (despite the sum rule failure for Clarke subdifferential). \\new{ We emphasize once more that our goal is to {\\em capture the  stationary points that are actually met in practice}.}\n        \n        For any $\\mathsf{B} \\subset \\{1,\\ldots, N\\}$, we introduce the following objects,\n\t    \\begin{equation}\n\t\t\tD\\mathcal{J}_{\\mathsf{B}}=\\sum_{n\\in \\mathsf{B}} \\partial\\left[  l(f(x_n,\\cdot),y_n)\\right], \\quad D\\mathcal{J}=\\sum_{n=1}^N \\partial\\left[ l(f(x_n,\\cdot),y_n)\\right].\n\t\t\\end{equation}\n\t    Observe that, for each $\\mathsf{B}$, we have $ D\\mathcal{J}_{\\mathsf{B}}\\supset \\partial \\mathcal{J}_{\\mathsf{B}} $ and that $\\mathcal{J}_\\mathsf{B}$ is differentiable almost everywhere with $D\\mathcal{J}_{\\mathsf{B}}= \\partial \\mathcal{J}_{\\mathsf{B}}=\\{\\nabla \\mathcal{J}_{\\mathsf{B}}\\}$, see \\cite{clarke1990optimization}. In particular $D\\mathcal{J} = \\partial \\mathcal{J}$ almost everywhere so that the potential differences with the Clarke subgradient occur on a negligible set. When $\\mathcal{J}$ is tame the equalities even hold on the complement of a finite union of manifolds of dimension strictly lower than $P$---use the classical stratification results for o-minimal structures, \\cite{coste2000introduction}. A point satisfying $D\\mathcal{J}(\\theta)\\ni 0$ will be called {\\em $D$-critical}. \\new{Note that Clarke-critical points ($0 \\in \\partial\\mathcal{J}$) are  $D$-critical points but that the converse is not true.} This terminology is motivated by favorable properties: sum and chain rules along curves (see Lemmas \\ref{lem::chainrule} and \\ref{lem:chainRuleSum} below) and  the existence of a tame Sard's theorem (see Lemma \\ref{lem:sard}). To our knowledge, this notion of a steady state has not previously been used in the literature.{\\footnote{In a follow-up of this paper, \\cite{BP} have further developed the present ideas and in particular the connection to the backpropagation algorithm.} To the best of our knowledge a direct approach modelling the mini-batch practice has never been considered before.} While this notion is needed for the theoretical analysis, one should keep in mind that $D\\mathcal{J}$ is actually what is computed numerically provided that the automatic differentiation library returns a Clarke subgradient. This computation is usually done with a backpropagation algorithm, similarly to the seminal method of \\cite{rumelhart1986learning}. \n\n\n\t    Ultimately, one can rewrite \\eqref{eq::contdinClarke} by replacing $\\partial\\mathcal{J}$ by $D\\mathcal{J}$, which yields a differential inclusion adapted to study mini-batch approximations of nonsmooth loss functions $\\mathcal{J}$. This reads, \n\t    \\begin{equation}\\label{eq::contdin}\n\t    \\begin{cases}\n\t    \\dot{\\theta}(t) + \\beta  D \\mathcal{J}(\\theta(t)) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0\\\\\n\t    \\dot{\\psi}(t) &+(\\alpha -\\frac{1}{\\beta})\\theta(t) + \\frac{1}{\\beta} \\psi(t) \\ni 0  \\end{cases} \\mbox{,\\quad for a.e. $t\\in (0,+\\infty)$}.\n\t    \\end{equation}\n\t    Discretizing this system gives a workable version of INDIAN.\n\t    Let us consider a sequence $(\\mathsf{B}_k)_{k \\in \\mathbb{N}}$  of nonempty  subsets of $\\{1,\\ldots,N\\}$, chosen independently and uniformly at random with replacement, and a sequence of positive step sizes $(\\gamma_k)_{k \\in \\mathbb{N}}$.\n\t    For a given initialization $(\\theta_0,\\psi_0)\\in\\mathbb{R}^P\\times\\mathbb{R}^P$, at iteration $k\\geq 1$, our algorithm reads,\n\t    \t\\begin{equation}\\hspace{-2cm}\\textrm{(INDIAN)}\\,\\,\\,\\label{eq::discdin}\n        \t\\begin{cases}\n        \t    v_k &\\in D \\mathcal{J}_{\\mathsf{B}_k}(\\theta_k)\\\\\n            \t\\theta_{k+1}&=  \\theta_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k -\\frac{1}{\\beta}\\psi_k - \\beta v_k \\right)\\\\\n            \t\\psi_{k+1}& = \\psi_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k - \\frac{1}{\\beta} \\psi_k \\right)\n        \t\\end{cases}\n        \t\\end{equation}\n        Here again $\\alpha > 0$ and $\\beta > 0$ are hyper-parameters of the algorithm. {The mini-batch procedure forms} a stochastic approximation of the deterministic dynamics obtained by choosing $\\mathsf{B}_k=\\{1,\\ldots,N\\}$, i.e., when $\\mathcal{J}_{\\mathsf{B}_k}=\\mathcal{J}$ (batch version). This can be seen by observing that the vectors $v_k$ above may be written as $v_k = \\tilde{v}_k+\\eta_k$, where $\\tilde{v}_k \\in D\\mathcal{J}(\\theta_k)$ and $\\eta_k$ compensates for the missing subgradients and can be seen as a zero-mean noise. \t\n    %\n\tHence, INDIAN admits the following general abstract stochastic formulation,\n\t\\begin{equation}\\,\\,\n        \\begin{cases}\n        \t  w_k &\\in D \\mathcal{J}(\\theta_k)\\\\\n            \\theta_{k+1}&=  \\theta_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k -\\frac{1}{\\beta}\\psi_k - \\beta w_k +\\xi_k\\right)\\\\\n            \\psi_{k+1}& = \\psi_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k - \\frac{1}{\\beta} \\psi_k \\right)\n        \t\\end{cases}\\label{eq::indiang}\n    \\end{equation}\n where $(\\xi_k)_{k \\in \\mathbb{N}}$ is a martingale difference noise sequence adapted to the filtration induced by (random) iterates up to $k$. While~\\eqref{eq::discdin} is the version implemented in practice, its equivalent form~\\eqref{eq::indiang} is more convenient for the convergence analysis of the next section. We stress that the equivalence between \\eqref{eq::discdin} and \\eqref{eq::indiang} relies on the use of $D\\mathcal{J}$ and would not hold with with the use of $\\partial \\mathcal{J}$ as in~\\eqref{eq::notindian}.\n\nINDIAN in its general and practical form is summarized in Table~\\ref{tab:indian}.\n\n\n\n \\begin{table}[ht]\n    \\centering\n\\begin{mdframed}[style=MyFrame]\n \\begin{center}\n {\\bf Inertial Newton Algorithm for Deep Learning (INDIAN)}\n \\end{center}\n\\bigskip\n\n\\noindent\n\\textbf{Objective function:} $\\mathcal{J} = \\sum_{n = 1}^N \\mathcal{J}_n$, with $\\mathcal{J}_n\\colon \\mathbb{R}^P \\mapsto \\mathbb{R}$ locally Lipschitz.\\\\\n\\textbf{Hyper-parameters:} $(\\alpha,\\beta)$ positive.\\\\\n\\textbf{Mini-batches:} $(\\mathsf{B}_k)_{k \\in \\mathbb{N}}$, nonempty  subsets of $\\{1,\\ldots,N\\}$.\\\\\n\\textbf{Step sizes:} $(\\gamma_k)_{k \\in \\mathbb{N}}$ positive.\\\\\n\\textbf{Initialization:}  $(\\theta_0,\\psi_0)\\in\\mathbb{R}^P\\times\\mathbb{R}^P$.\\\\\n\n\n\n\n\n\\textbf{For $k\\in \\mathbb{N}$:}\n\\begin{equation*}\n        \t\\begin{cases}\n        \t    v_k &\\in\\quad \\sum_{n\\in \\mathsf{B}_k} \\partial\\left[  \\mathcal{J}_n(\\theta_k)\\right]\\\\\n            \t\\theta_{k+1}&=\\quad  \\theta_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k -\\frac{1}{\\beta}\\psi_k - \\beta v_k \\right)\\\\\n            \t\\psi_{k+1}& =\\quad \\psi_k + \\gamma_k \\left( (\\frac{1}{\\beta}-\\alpha)\\theta_k - \\frac{1}{\\beta} \\psi_k \\right)\n        \t\\end{cases}\n\\end{equation*}\n\\end{mdframed}\n\\caption{INDIAN in a nutshell. \\label{tab:indian}}\n\\end{table}\n\n\n\n\n\n    \\section{Convergence Results for INDIAN}\n\\label{sec:proof}\n\n\n\t\\subsection{Main result: Accumulation Points of INDIAN are Critical}\n\n\n\nWe now study the convergence of INDIAN. The main idea here is to prove that the discrete algorithm~\\eqref{eq::discdin} asymptotically behaves like the solutions of the continuous differential inclusion~\\eqref{eq::contdin}. Besides tameness, our main assumption is the following:\n\t\\begin{assumption}[Stochastic approximation]\\label{ass:itbound}\n\t    The sets $(\\mathsf{B}_k)_{k \\in \\mathbb{N}}$ are taken independently uniformly at random with replacement.\n\t\tThe step-size sequence $\\gamma_k$ is positive with $\\sum_k\\gamma_k=+\\infty$ and satisfies $\\gamma_k = o \\left( \\frac{1}{\\log k}\\right)$, that is  $\\displaystyle\\limsup_{k\\to+\\infty} |\\gamma_k \\log k|=0$.\t       \\label{ass:mainAssumption}\n\t\\end{assumption}\nTypical admissible choices are $\\gamma_k=C(k+1)^{-a}$ with $a\\in (0,1]$, $C > 0$. \n\tThe main theoretical result of the paper follows. \n\n\t\\begin{theorem}[\\textrm{INDIAN } converges to the set of $D$-critical points of $\\mathcal{J}$]\\label{th::thmDIN}\n\t\tAssume that for \\new{$n\\in\\{1,\\ldots,N\\}$, each $\\mathcal{J}_n$ is locally Lipschitz continuous}, tame and that the step sizes satisfy Assumption~\\ref{ass:mainAssumption}. Set an initial condition $(\\theta_0,\\psi_0)$ and assume that there exists $M>0$ such that $\\sup_{k\\geq 0}\\|(\\theta_k,\\psi_k)\\|\\leq M$ almost surely, where $(\\theta_k,\\psi_k)_{k\\in\\mathbb{N}}$ are generated by INDIAN.\n\t    Then, almost surely, any accumulation point $\\bar\\theta$ of the sequence $(\\theta_k)_{k\\in \\mathbb{N}}$ satisfies $D\\mathcal{J}(\\bar\\theta)\\ni 0 $. In addition $ (\\mathcal{J}(\\theta_{k}))_{k\\in\\mathbb{N}}$ converges.\t    \n\t\\end{theorem}\n\n\t\\new{Before proving Theorem~\\ref{th::thmDIN}, we will first make some comments and illustrate this result.}\n    \\subsection{Comments on the Results of Theorem~\\ref{th::thmDIN}}\\label{sec::comRes}\n    \n    \\new{\n    \n    \\paragraph{On the step sizes.}\n    First, Assumption~\\ref{ass:mainAssumption} offers much more flexibility than the usual $O(1\/\\sqrt{k})$ assumption commonly used for SGD. We leverage the boundedness assumption on the norms of $(\\theta_k,\\psi_k)$, the local Lipschitz continuity and finite-sum structure of $\\mathcal{J}$, so that the noise is actually uniformly bounded and hence sub-Gaussian, allowing for much larger step sizes than in the more common bounded second moment setting. See \\citet[Remark 1.5]{benaim2005stochastic} and \\cite{benaim1999dynamics} for more details. The interest of this aggressive strategy is highlighted in Figure~\\ref{fig::decay} of the experimental section. \n    \n    \\paragraph{On the scope of the theorem.}\n    Our result actually holds for general locally Lipschitz continuous tame functions with finite-sum structure and for the general stochastic process under uniformly bounded martingale increment noise. We do not use any other specific structure of DL loss functions. Other variants could be considered depending on the assumptions on the noise, see \\cite{benaim2005stochastic}.}\n    \n    \\new{\n    \\paragraph{On $D$-criticality.}\n    The result of Theorem~\\ref{th::thmDIN} states that the bounded discrete trajectories of INDIAN are attracted by the $D$-critical points.  Recall that $D$-critical points include local minimizers and thus our theoretical finding agrees with our empirical observations that most initializations lead to ``valuable weights'' $\\theta$ and to efficient training. In particular for smooth networks where $\\mathcal{J}$ is differentiable, limit points of INDIAN are simply critical points of $\\cal J$.\n    The reader should however remember that when the algorithm is initialized on the $D$-critical set, the algorithm is stationary as well, {\\em even when the initialization is non-Clarke critical}.  This last point shows that $D$-points are not introduced to simplify the analysis but to {\\em sharply model the use of mini-batch methods on nonconvex and nonsmooth problems}.  Hopefully, in practice one can expect to avoid such points with overwhelming probability. Indeed, following our initial introduction of the $D$-points, \\citet{bolte2020mathematical} proved  for SGD, under very mild conditions on the initialization, that $D$-critical points that are not Clarke-critical are not reached with probability one (see also \\cite{bianchi2020convergence}).\n    \n   \n \n  \n  \n  \\paragraph{On the boundedness assumption.} The bounded assumption on the iterates is a classical assumption for first or second-order algorithms, see for instance \\cite{davis2018stochastic,duchi2018stochastic}. When using deterministic algorithms (i.e., without mini-batch approximations), properties such as the coercivity of $\\mathcal{J}$ can be sufficient to remove the boundedness assumption for descent algorithms. This does not remain true when dealing with mini-batch approximations, yet, in the case of INDIAN, the coercivity of $\\mathcal{J}$ would guarantee at least that the solutions of the continuous underlying differential inclusion \\eqref{eq::contdin} remain bounded. Indeed, we will prove in Section~\\ref{sec::proofofConv} that for any solution $(\\theta,\\psi)$ of \\eqref{eq::contdin}, the function $E(\\theta(t),\\psi(t)) \\triangleq 2(1+\\alpha\\beta)\\mathcal{J}(\\theta(t)) + \\left\\Vert (\\alpha-\\ovb)\\theta(t) +\\ovb \\psi(t) \\right\\Vert^2 $ is decreasing in time (see Lemma~\\ref{lem::Edec} hereafter). As a consequence, we cannot have $\\mathcal{J}(\\theta(t))\\xrightarrow[t\\to\\infty]{}\\infty$ so $\\Vert\\theta(t)\\Vert\\not\\to\\infty$ due to the coercivity of $\\mathcal{J}$. In addition this guarantees $\\Vert\\psi(t)\\Vert\\not\\to\\infty$ as well. However DL loss functions are not coercive in general and studying the boundedness issue in DL or even for nonconvex semi-algebraic problems is far beyond the scope of this paper. Let us however mention that it is not uncommon to project the iterates on a given large ball to ensure boundedness;  this is a matter for future research.}\n  \n  \n \\subsection{Preliminary Variational Results}\n\t\\label{sec:ResultsOnD}\n    Prior to proving Theorem 3, we extend some results known for the Clarke subdifferential of tame functions to the operator $D$ that we previously introduced. First, we recall a useful result of \\cite{davis2018stochastic} which follows from the projection formula in \\cite{bolte2007clarke}.\n\t\\begin{lemma}[Chain rule for the Clarke subdifferential]\\label{lem::chainrule}\n\t    Let $\\mathcal{J}:\\mathbb{R}^P\\to \\mathbb{R}$ be a locally Lipschitz continuous tame function, then $\\mathcal{J}$ admits a chain rule, meaning that for all absolutely continuous curves $\\theta:\\mathbb{R}_+\\to\\mathbb{R}^P$, \\new{$\\mathcal{J}\\circ\\theta$ is differentiable a.e.} and for a.e. $t\\geq 0$,\n\t    \\begin{equation}\\label{eq::chainruleJ}\n\t        \\frac{\\diff \\mathcal{J}}{\\diff t} (\\theta(t)) = \\langle \\dot{\\theta}(t),\\partial \\mathcal{J}(\\theta(t)) \\rangle =  \\langle \\dot{\\theta}(t) ,v \\rangle,\\ \\ \\forall v\\in \\partial \\mathcal{J}(\\theta(t)).\n\t    \\end{equation}\n\t\\end{lemma}\n\\medskip\n\\new{Note that, even though $\\mathcal{J}$ is nondifferentiable on $\\mathbb{R}^P$, the function $t\\mapsto \\mathcal{J}(\\theta(t))$ is differentiable for a.e. $t>0$. Indeed, as introduced in Section~\\ref{sec::nonsmoothnonconv}, an absolutely continuous curve from $t\\geq 0$ to $\\mathbb{R}^P$ is differentiable for a.e. $t>0$. This, combined with the chain-rule of Lemma~\\ref{lem::chainrule} allows to differentiate $\\mathcal{J}\\circ\\theta$ for a.e. $t>0$ whenever $\\mathcal{J}$ is tame and locally Lipschitz continuous. \\new{Besides, notice that the value of $\\frac{\\diff \\mathcal{J}}{\\diff t} (\\theta(t))$ in \\eqref{eq::chainruleJ} does not depend on the element $v$ taken in $\\partial\\mathcal{J}(\\theta(t))$ which justifies the notation $\\langle \\dot{\\theta}(t),\\partial \\mathcal{J}(\\theta(t)) \\rangle$}.}\n\nConsider now a function with an additive finite-sum structure (such as in DL):\n\\begin{align}\\label{eq::fsumofTame}\n\t\t\\mathcal{J} \\colon\\mathbb{R}^P \\ni \\theta \\mapsto \\sum_{n=1}^N \\mathcal{J}_n(\\theta),\n\\end{align}\nwhere each $\\mathcal{J}_n \\colon \\mathbb{R}^P \\mapsto \\mathbb{R}$ is locally Lipschitz continuous and tame. We define for any $\\theta \\in \\mathbb{R}^P$\n\\begin{align*}\n\t\t\t\tD\\mathcal{J}(\\theta) = \\sum_{n=1}^N \\partial \\mathcal{J}_n(\\theta).\n\\end{align*}\nThe following lemma is a direct generalization of the above chain rule.\n\\begin{lemma}[Chain rule for $D\\mathcal{J}$]\\label{lem::chainruleD}\n\t\t\t\tLet $\\mathcal{J}$ be a sum of tame functions like in \\eqref{eq::fsumofTame}. Let $c \\colon [0,1] \\mapsto \\mathbb{R}^P$ be an absolutely continuous curve so that $t \\mapsto \\mathcal{J}(c(t))$ is differentiable almost everywhere. For a.e. $t \\in [0,1]$, and for all $v \\in D\\mathcal{J}(c(t))$,\n\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\t\t\\frac{d}{dt} \\mathcal{J}(c(t)) = \\left\\langle v, \\dot{c}(t) \\right\\rangle.\n\t\t\t\t\\end{align*}\n\t\t\t\t\\label{lem:chainRuleSum}\n\n\\end{lemma}\n    \\begin{proof}\n\t\t\t\tBy local Lipschitz continuity and absolute continuity, each $\\mathcal{J}_n$ is differentiable almost everywhere and Lemma \\ref{lem::chainrule} can be applied:\n\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\t\t\\frac{d}{dt} \\mathcal{J}_n(c(t)) = \\left\\langle v_n, \\dot{c}(t) \\right\\rangle, \\mbox{for all $v_n \\in \\partial \\mathcal{J}_n(c(t))$ and for a.e. $t\\geq 0$.}\n\t\t\t\t\\end{align*}\n\t\t  Thus\n\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\t\t\\frac{d}{dt} \\mathcal{J}(c(t)) = \\sum_{n=1}^N \\frac{d}{dt} \\mathcal{J}_n(c(t)) = \\sum_{n=1}^N\\left\\langle v_n, \\dot{c}(t) \\right\\rangle,\n\t\t\t\t\\end{align*}\n\t\t\tfor any $v_n \\in \\partial \\mathcal{J}_n(c(t))$, for all $n=\\{1,\\ldots, N\\}$, and for a.e. $t\\geq 0$. This proves the desired result.\n    \\end{proof}\n    We finish this section with a Sard lemma for $D$-critical values, in the spirit of \\cite{bolte2007clarke}.\n    \\begin{lemma}[A Sard's theorem for tame D-critical values]\n\t\t\t\tLet, $$ \\mathsf{S}= D\\mbox{\\rm -crit\\,}\\triangleq\\left\\{ \\theta \\in \\mathbb{R}^P\\mid D\\mathcal{J}(\\theta) \\ni 0 \\right\\},$$ then $\\mathcal{J}(\\mathsf{S})$ is finite. \n\t\t\t\t\\label{lem:sard}\n    \\end{lemma}\n    \\begin{proof}\n\t\t\t\tThe set $\\mathsf{S}$ is tame and hence it has a finite number of connected components. It is sufficient to prove that $\\mathcal{J}$ is constant on each connected component of $\\mathsf{S}$. Without loss of generality, assume that $\\mathsf{S}$ is connected and consider $\\theta_0,\\theta_1 \\in \\mathsf{S}$. By Whitney regularity \\cite[4.15]{van1998tame}, there exists a tame continuous path $\\Gamma$ joining $\\theta_0$ to $\\theta_1$. Because of the tame nature of the result, we should here conclude with only tame arguments and use the projection formula in \\cite{bolte2007clarke}, but for convenience of readers who are not familiar with this result we use  Lemma \\ref{lem::chainrule}. Since $\\Gamma$ is tame, the monotonicity lemma (see for example \\citet[Lemma 2]{kurdyka1998gradients}) gives the existence of a finite collection of real numbers $0 = a_0 < a_1 < \\ldots < a_q = 1$, such that $\\Gamma$ is $C^1$ on each segment $(a_{j-1}, a_{j})$, $j = 1,\\ldots,q$. Applying Lemma \\ref{lem::chainrule} to each $\\Gamma_{|(a_i,a_{i+1})}$, we see that $\\mathcal{J}$ is constant except perhaps on a finite number of points, thus $\\mathcal{J}$ is constant by continuity.\n    \\end{proof}\n\n\n\t\\subsection{Proof of Convergence for INDIAN}\n\t\\label{sec::proofofConv}\n  Our approach follows the stochastic method for differential inclusions developed in \\cite{benaim2005stochastic} for which the differential system \\eqref{eq::contdin}  and its Lyapunov properties play fundamental roles. \n  \n    \n\tThe steady states of \\eqref{eq::contdin}\\, are given by,\n\t\\begin{equation}\\label{S} \\mathsf{S} = \\left\\{ (\\theta,\\psi)\\in \\mathbb{R}^P\\times \\mathbb{R}^P \\mid 0\\in D \\mathcal{J}(\\theta), \\psi=(1-\\alpha\\beta) \\theta \\right\\}.\\end{equation}\n\tThese points are initialization values for which the system does not evolve and remains constant.\n    Observe that the first coordinates of these points are $D$-critical for $\\mathcal{J}$ and that conversely any $D$-critical point of $\\mathcal{J}$ corresponds to a unique rest point in $\\mathsf{S}$. \n    \n\n\t\\begin{definition}[Lyapunov function]\\label{def::lyap}\n\t\tLet $ \\mathsf{A} $ be a subset of $\\mathbb{R}^P \\times \\mathbb{R}^P $, we say that $ E : \\mathbb{R}^P\\times \\mathbb{R}^P\\to \\mathbb{R} $ is a Lyapunov function for the set $ \\mathsf{A} $ and the dynamics \\eqref{eq::contdin} if\n\t\t\\renewcommand{\\theenumi}{(\\roman{enumi})}%\n\n\t\t\\begin{enumerate}\n\t\t    \\item for any  solution $ (\\theta,\\psi)$ of  \\eqref{eq::contdin} with initial condition $(\\theta_0,\\psi_0) $, we have:\\\\\n\t\t    $E(\\theta(t),\\psi(t) ) \\leq E(\\theta_0,\\psi_0)$ a.e. on $\\mathbb{R}$.\n\t\t     \\item for any  solution $ (\\theta,\\psi)$ of  \\eqref{eq::contdin} with initial condition $(\\theta_0,\\psi_0) \\notin \\mathsf{A}$, we have:\\\\\n\t\t     $E(\\theta(t),\\psi(t) ) < E(\\theta_0,\\psi_0)$ a.e. on $\\mathbb{R}$.\n\t\t\\end{enumerate}\n\t\\end{definition}\n\n\tIn practice, to establish that a functional is Lyapunov, one can simply use differentiation through chain rule results, with in particular Lemma \\ref{lem::chainrule}. In the context of INDIAN, we will use Lemma \\ref{lem:chainRuleSum}. To build a Lyapunov function for the dynamics \\eqref{eq::contdin} and the set $ \\mathsf{S} $, consider the two following energy-like functions,\n\t\\begin{equation}\n\t\\begin{cases}\n\tE_{\\min}(\\theta(t),\\psi(t)) &= (1-\\sqrt{\\alpha\\beta})^2 \\mathcal{J}(\\theta(t)) + \\frac 1 2 \\left\\Vert (\\alpha-\\ovb)\\theta(t) +\\ovb \\psi(t) \\right\\Vert^2 \\\\\n\tE_{\\max}(\\theta(t),\\psi(t)) &= (1+\\sqrt{\\alpha\\beta})^2 \\mathcal{J}(\\theta(t)) + \\frac 1 2 \\left\\Vert (\\alpha-\\ovb)\\theta(t) +\\ovb \\psi(t) \\right\\Vert^2.\n\t\\end{cases}\n\t\\end{equation}\n\nThen the following lemma applies.\n\t\\begin{lemma}[Differentiation along DIN trajectories]\\label{lem::dEdt}\n\t\tLet $(\\theta,\\psi)$ be a solution of \\eqref{eq::contdin} with initial condition $(\\theta_0,\\psi_0)$. For a.e. $t>0$, $\\theta$ and $\\psi$ are differentiable at $t$, \\eqref{eq::contdin} holds, $\\frac{\\dt(t)-\\ddp(t)}{\\beta} \\in D \\mathcal{J}(\\theta(t))$ and%\n\t\t\\begin{align*}\n\t\t%\n\t\t\\frac{\\diff E_{\\min}}{\\diff t}\\left(\\theta(t),\\psi(t)\\right) &= -\\left\\Vert \t\\sqrt{\\alpha}\\dot{\\theta}(t) -\\frac{1}{\\sqrt{\\beta}}\\left(\\dot{\\psi}(t)-\\dot{\\theta}(t)\\right)\\right\\Vert^2 \\\\\n\t\t%\n\t\\frac{\\diff E_{\\max}}{\\diff t}\\left(\\theta(t),\\psi(t)\\right) &= -\\left\\Vert \t\\sqrt{\\alpha}\\dot{\\theta}(t) +\\frac{1}{\\sqrt{\\beta}}\\left(\\dot{\\psi}(t)-\\dot{\\theta}(t)\\right) \\right\\Vert^2\n\\end{align*}\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tDefine $\\displaystyle E_\\lambda(\\theta,\\psi) = \\lambda {\\mathcal J}(\\theta) + \\frac 1 2 \\left\\Vert (\\alpha-\\ovb)\\theta +\\ovb \\psi \\right\\Vert^2 $. We aim at choosing $ \\lambda $ so that $ E_\\lambda$ is a Lyapunov function. Because $\\mathcal{J}$ is tame and locally Lipschitz continuous, using Lemma~\\ref{lem:chainRuleSum} we know that for any absolutely continuous trajectory $ \\theta:\\mathbb{R}_+\\to \\mathbb{R}^P $ and for a.e. $ t>0 $,\n\t\t\\begin{equation}\n\t\t\\frac{\\diff \\mathcal{J}}{\\diff t} (\\theta(t)) = \\langle \\dot{\\theta}(t), D \\mathcal{J}(\\theta(t)) \\rangle = \\langle \\dot{\\theta}(t) ,v(t) \\rangle,\\ \\ \\forall v(t)\\in D \\mathcal{J}(\\theta(t)).\n\t\t\\end{equation}\n\t\tLet $\\theta$ and $\\psi$ be solutions of (DIN). For a.e. $t\\geq 0$, we can differentiate $ E_\\lambda(\\theta,\\psi) $ to obtain\n\\begin{equation}\\begin{split}\n\\frac{\\diff E_\\lambda}{\\diff t}(\\theta(t),\\psi(t)) = & \\lambda \\langle \\dot{\\theta}(t) ,v(t) \\rangle + (\\alpha-\\ovb) \\langle \\dot{\\theta}(t) ,(\\alpha-\\ovb)\\theta(t)+\\ovb\\psi(t) \\rangle \\\\\n\t& + \\ovb \\langle \\dot{\\psi(t)} ,(\\alpha-\\ovb)\\theta(t)+\\ovb\\psi(t)\\rangle\n\t\\end{split}\n\t\\end{equation}\n\tfor all $v(t)\\in D \\mathcal{J}(\\theta(t))$. Using \\eqref{eq::contdin}, we get $\\frac{1}{\\beta}(\\dot\\theta(t)-\\dot \\psi(t))\\in D \\mathcal{J}(\\theta(t))$ and $-\\dot{\\psi}(t)=(\\alpha-\\ovb)\\theta(t)+\\ovb \\psi(t)$ a.e. Choosing $v(t)=\\frac{1}{\\beta}(\\dot\\theta(t)-\\dot \\psi(t))$ yields:\\begin{align*}\n\t\t  \\frac{\\diff E_\\lambda}{\\diff t}(\\theta(t),\\psi(t)) &= \\lambda \\left\\langle \\dot{\\theta}(t) ,\\frac{\\dt(t)-\\ddp(t)}{\\beta} \\right\\rangle - (\\alpha-\\ovb) \\left\\langle \\dot{\\theta}(t) ,\\dot{\\psi}(t) \\right\\rangle - \\ovb \\left\\langle \\dot{\\psi}(t) ,\\dot{\\psi}(t) \\right\\rangle.\n\t\t\\end{align*}\n\t\tThen, expressing everything as a function of $\\dot{\\theta}$ and $\\frac{1}{\\beta}(\\psi-\\theta)$, one can show that a.e. on $\\mathbb{R}_+$:\n\t\t\\begin{align*}\n\t\t\\frac{\\diff E_\\lambda}{\\diff t}&(\\theta,\\psi)(t) =  - \\alpha \\Vert \\dot{\\theta}(t)\\Vert^2 -\\beta \\left\\Vert  \\frac{\\dt(t)-\\ddp(t)}{\\beta} \\right\\Vert^2 + \\left(\\lambda - \\alpha\\beta -1 \\right)\\langle \\dot{\\theta}(t) ,\\frac{\\dt(t)-\\ddp(t)}{\\beta} \\rangle\\\\\n\t\t&= -\\left\\Vert \\sqrt{\\alpha}\\dot{\\theta}(t) + \\frac{\\alpha\\beta +1 - \\lambda }{2\\sqrt{\\alpha}}\\frac{\\dt(t)-\\ddp(t)}{\\beta}\\right\\Vert^2 - \\left (  \\beta -\\frac{(\\alpha\\beta +1 - \\lambda)^2}{4\\alpha}  \\right)\\left\\Vert \\frac{\\dt(t)-\\ddp(t)}{\\beta} \\right\\Vert^2.\n\t\t\\end{align*}\n\t\tWe aim at choosing $ \\lambda $ so that $ E_\\lambda $ is decreasing that is $\\left (  \\beta -\\frac{(\\alpha\\beta +1 - \\lambda)^2}{4\\alpha}  \\right) > 0$. This holds whenever $ \\lambda \\in\\, \\left[(1-\\sqrt{\\alpha\\beta})^2,(1+\\sqrt{\\alpha\\beta})^2\\right] $. We choose $ \\lambda_{\\min} =(1-\\sqrt{\\alpha\\beta})^2 $, and $ \\lambda_{\\max} = (1+\\sqrt{\\alpha\\beta})^2 $, for these two values we obtain for a.e. $ t>0 $ ,\n\t\t\\begin{equation}\\begin{cases}\n\t\t\\dot E_{\\lambda_{\\min}}(\\theta(t),\\psi(t)) &= -\\left\\Vert \\sqrt{\\alpha}\\dot{\\theta}(t) + \\frac{1}{\\sqrt{\\beta}}\\left(\\dot{\\theta}(t)-\\dot{\\psi}(t)\\right) \\right\\Vert^2 \\vspace{0.2cm} \\\\\n\t\t\\dot E_{\\lambda_{\\max}} (\\theta(t),\\psi(t))&= -\\left\\Vert \\sqrt{\\alpha}\\dot{\\theta}(t) -  \\frac{1}{\\sqrt{\\beta}}\\left(\\dot{\\theta}(t)-\\dot{\\psi}(t)\\right)  \\right\\Vert^2\n\t\t\\end{cases}\n\t\t\\end{equation}\n\t\tRemark finally that by definition $E_{\\min}=E_{\\lambda_{\\min}}$ and $E_{\\max}=E_{\\lambda_{\\max}}$.\n\t\\end{proof}\n\tDefine $ E = E_{\\min} + E_{\\max} $ and recall that $ \\displaystyle \\mathsf{S}  = \\left\\{ (\\theta,\\psi)\\in \\mathbb{R}^P\\times \\mathbb{R}^P \\mid 0\\in D J(\\theta), \\psi=(1-\\alpha\\beta) \\theta) \\right\\}$. By a direct integration argument, we obtain the following lemma.\n\t\\begin{lemma}[$E$ is Lyapunov function for INDIAN with respect to $\\mathsf{S}$]\\label{lem::Edec}\n\t\tFor any $(\\theta_0,\\psi_0)\\notin \\mathsf{S}$ and any solution $(\\theta,\\psi)$ with initial condition $(\\theta_0,\\psi_0)$,\n\t\t\\begin{equation}\n\t\tE(\\theta(t),\\psi(t))<E(\\theta_0,\\psi_0), \\text{ for a.e. }t>0.\n\t\t\\end{equation}\n\t\\end{lemma}\n\n\\bigskip\n\n\\noindent\nWe are now in position to provide the desired proof.\n\n\\smallskip\n\n\\noindent\n\t{\\bf Proof of Theorem \\ref{th::thmDIN}} Lemmas~\\ref{lem::dEdt} and~\\ref{lem::Edec} state that\n\t\t$ E $ is a Lyapunov function for the set $ \\mathsf{S} $ and the dynamics \\eqref{eq::contdin}. Let $\\mathsf{C}=\\{\\theta\\in \\mathbb{R}^P\\mid(\\theta,\\psi)\\in \\mathsf{S}\\}$ which is actually the set of $D$-critical points of $\\mathcal{J}$. Using Lemma \\ref{lem:sard} of Section \\ref{sec:ResultsOnD}, $\\mathcal{J}(\\mathsf{C})$ is finite. Moreover, since $E(\\theta,\\psi)=2(1+\\alpha\\beta)\\mathcal{J}(\\theta) $ for all $(\\theta,\\psi)\\in\\mathsf{S}$, $ E $ takes a finite number of values on $ \\mathsf{S} $, and in particular, $E(\\mathsf{S})$ has empty interior.\n\n\n\t\t  Denote by $\\mathsf{L}$ the set of accumulation points of a the sequences $((\\theta_k, \\psi_k))_{k \\in \\mathbb{N}}$ produced by \\eqref{eq::discdin} starting at $(\\theta_0,\\psi_0)$ and $\\mathsf{L}_1$ its projection on $\\mathbb{R}^P\\times\\{0\\}$. We have the 3 following properties:\\smallskip\\\\\n    \\noindent\n\t$\\quad -$ By assumption, we have $\\|(\\theta_k, \\psi_k)\\| \\leq M$ almost surely, for all $k \\in \\mathbb{N}$.\\\\\n\t$\\quad -$ By local Lipschitz continuity $\\partial \\mathcal{J}_\\mathsf{B}(\\theta)$ is uniformly bounded for $\\|\\theta\\| \\leq M$ and any $\\mathsf{B} \\subset \\{1,\\ldots,N\\}$, hence the centered noise $(\\xi_k)_{k \\in \\mathbb{N}}$ is a uniformly bounded martingale difference sequence.\\\\\n    $\\quad -$ By Assumption \\ref{ass:mainAssumption}, the sequence $ (\\gamma_k)_{k \\in \\mathbb{N}} $ are chosen such that $ \\gamma_{k} = o(\\frac{1}{\\log k}) $ (see Section~\\ref{sec::comRes}).\n\n        \\new{Then the sufficient conditions of Remark 1.5 of \\cite{benaim2005stochastic} state that the discrete process $(\\theta_k,\\psi_k)_{k\\in\\mathbb{N}}$ asymptotically behaves like the solutions of \\eqref{eq::contdin}. We can then combine Proposition 3.27 and Theorem 3.6 of \\cite{benaim2005stochastic}, \n\t\tto obtain that the limit set $\\mathsf{L}$ of the discrete process is contained in the set $\\mathsf{S}$ where the Lyapunov has vanishing derivatives. Thus the set $\\mathsf{L}_1$ (the set of the first coordinates of all accumulation points) contains only $D$-critical points of $\\mathcal{J}$. In addition, $E(\\mathsf{L})$ is a singleton, and for all  $(\\theta,\\psi)\\in\\mathsf{S}$, we have $E(\\theta,\\psi) = \\mathcal{J}(\\theta)$, so $\\mathcal{J}(\\mathsf{L}_1) $ is also a singleton and the theorem follows.}\n\n\n\n    \\section{Towards Convergence Rates for INDIAN}\\label{sec::rateofCV}\n\n\n    In the previous section, connecting INDIAN to \\eqref{eq::contdin} was one of the keys to prove the convergence of the discrete dynamics. Let us now focus on the continuous dynamical system \\eqref{eq::contdinClarke} in the deterministic case where $\\mathcal{J}$ and $\\partial\\mathcal{J}$ are not approximated anymore -- we thus no longer use $D\\mathcal{J}$ although this would be possible but would require more technical proofs. In this section and in this section only, we pertain to loss functions $\\mathcal{J}$ that are real semi-algebraic (a particular case of tame functions).\\footnote{We could extend the results of this section to more general objects including analytic functions on bounded sets. The semi-algebraicity assumption is made here for the sake of clarity.} Recall that a set is called semi-algebraic if it is a finite union of sets of the form,\n    $$\\{\\theta\\in \\mathbb{R}^P\\mid \\zeta(\\theta)=0,\\zeta_i(\\theta)<0\\}$$\n    where  $\\zeta,\\zeta_i$ are real polynomial functions. A function is called semi-algebraic if its graph is semi-algebraic. \n    \n    We will prove that the continuous time system \\eqref{eq::contdinClarke} is actually a quasi-gradient dynamic, and that we can characterize the convergence rate to critical points. Let us first introduce an essential mechanism to obtain such convergence rates: the Kurdyka-{\\L}ojasiewicz (KL) property.\n\n\n\\subsection{The Nonsmooth Kurdyka-{\\L}ojasiewicz  Property for the Clarke Subdifferential}\n\nThe nonsmooth Kurdyka-{\\L}ojasiewicz (KL) property, as introduced in \\citep{bolte2010}, is a measure of ``amenability to sharpness'' \\new{(as illustrated at the end of Section~\\ref{sec::rateIND})}. Here we provide a uniform version for the Clarke subdifferential of semi-algebraic functions as in \\cite{bolte2007clarke} and \\cite{bolte2014proximal}. In the sequel we denote by ``$\\dist$'' any given distance on $\\mathbb{R}^P$.\n\\begin{lemma}[Uniform Nonsmooth KL Property for the Clarke Subdifferential]\\label{lem::UKL}\n     Let $\\mathsf{K}$ be a nonempty compact set and let $L:\\mathbb{R}^P\\to \\mathbb{R}$ be a semi-algebraic locally Lipschitz continuous function. Assume that $L$ is constant on $\\mathsf{K}$, with value $L^\\star$. Then there exist $\\varepsilon>0$,  $\\delta>0$, $a\\in(0,1)$ and $\\rho>0$ such that, for all\n     \\begin{equation*}\n         v\\in \\left\\{ v\\in\\mathbb{R}^P \\mid \\dist(v,\\mathsf{K}) < \\varepsilon \\right\\} \\cap \\left\\{  v\\in\\mathbb{R}^P \\mid L^\\star < L(v) < L^\\star+\\delta  \\right\\},\n     \\end{equation*}\n     it holds that,\n     \\begin{equation}\\label{eq::SemiAlgKL}\n         \\rho(1-a)\\left(L(v) - L(\\bar{v})\\right)^{-a}\\dist\\left( 0, \\partial L(v)\\right)  > 1.\n     \\end{equation}\n \\end{lemma}\n\n    \\new{The proof directly follows from the general inequality provided in \\cite{bolte2007clarke} or the local result of \\cite{bolte2007clarke} with the compactness arguments of \\citet[Lemma 6]{bolte2014proximal}.} In the sequel, we make an abuse of notation by writing $\\Vert \\partial\\mathcal{J}(\\cdot)\\Vert \\triangleq \\dist(0,\\partial \\mathcal{J}(\\cdot))$.\n    To obtain a convergence rate we will use inequality \\eqref{eq::SemiAlgKL} on the Lyapunov function $E$. But first we state a general result of convergence that is built around the KL property.\n\n\\subsection{A General Asymptotic Rate Result}\n    We state a general theorem that leads to the existence of a convergence rate. This theorem will hold in particular for \\eqref{eq::contdinClarke}. We start by stating the result.\n\n   \n     \\begin{theorem}\\label{th::genlemma}\n        Let $X:[0,+\\infty)\\to \\mathbb{R}^P$ be a bounded absolutely continuous trajectory and let $L:\\mathbb{R}^P\\to \\mathbb{R}$ be a semi-algebraic locally Lipschitz continuous function. If there exists $c_1>0$ such that for a.e. $t>0$,\n        \\begin{equation}\\label{it::2}\\tag{i}\n             \\frac{\\diff L}{\\diff t} (X(t)) \\leq -c_1 \\Vert(\\partial L)(X(t))\\Vert^2,\n        \\end{equation}\n        then $L(X(t))$ converges to a limit value $L^\\star$ and,\n        \\[ \\vert L(X(t)) - L^\\star \\vert = O\\left(\\frac{1}{t}\\right). \\]\n        If in addition there exists $c_2>0$ such that for a.e. $t>0$,\n        \\begin{equation}\\label{it::3}\\tag{ii}\n            c_2 \\Vert \\dot{X}(t)\\Vert  \\leq \\Vert (\\partial L)(X(t))\\Vert,\n        \\end{equation}\n    \tthen, $X$ converges to a critical point of $L$ with a rate of the form $O(1\/t^b)$ with $b>0$.\\footnote{In some cases we even have linear rates or finite convergence as detailed in the proof.}\n    \\end{theorem}\n    \n    \\begin{proof}\n    We first prove the convergence of $L(X(\\cdot))$. Suppose that \\eqref{it::2} holds. Since $X$ is bounded and $L$ is continuous, $L(X(\\cdot))$ is bounded. Moreover from \\eqref{it::2}, $L(X(\\cdot))$ is decreasing, so it converges to some value $L^\\star$. To simplify suppose $L\\geq 0$ and $L^\\star=0$. Define,\n    \\begin{equation*}\n\t\t\t\\mathsf{I} = \\left\\{ x\\in\\mathbb{R}^P \\ \\mid \\ L(x)=0 \\right\\}.\n\t\\end{equation*}\n\tSuppose first that there exists $s\\geq 0$, such that $X(s)\\in \\mathsf{I}$. Since $L(X(\\cdot))$ is decreasing with limit $0$, then for all $t\\geq s$, $L(X(t)) = 0$ and the convergence rate holds true.\n\t\n\tLet us thus assume that for all $t\\geq 0$, $L(X(t))>0$. The trajectory $X$ is bounded in $\\mathbb{R}^P$, hence there exists a compact set $\\mathsf{C}\\subset\\mathbb{R}^P$ such that $X(t)\\in\\mathsf{C}$ for all $t\\geq 0$. Define $\\mathsf{K}=\\mathsf{I}\\cap \\mathsf{C}$. It is a compact set since $\\mathsf{I}$ is closed (by continuity of $L$) and $\\mathsf{C}$ is compact. Moreover, $L$ is constant on $\\mathsf{K}$. As such by Lemma~\\ref{lem::UKL}, there exist $\\varepsilon>0,\\ \\delta>0$, $a\\in(0,1)$ and a constant $\\rho>0$ such that for all\n\t\t\\[ v\\in \\left\\{ v\\in\\mathbb{R}^P, \\dist(v,\\mathsf{K}) < \\varepsilon \\right\\} \\cap \\left\\{ 0 < L(v) < \\delta  \\right\\},  \\]\n\t\tit holds that\n\t\t\\[\\rho(1-a)\\left(L(v)\\right)^{-a}\\dist\\left( 0, \\partial L(v)\\right)  > 1.\\]\n    We have $L(X(t))\\to 0$ so there exists $t_0\\geq 0$ such that for all $t\\geq t_0$, $0<L(X(t))<\\delta$. Without loss of generality, we assume $t_0=0$. Similarly, we have $\\dist(X(t),K) \\to 0$, so we may assume that for all $t\\geq 0$, $\\dist\\left(X(t),\\mathsf{K}\\right)<\\varepsilon$. Thus, for all $t\\geq 0$,\n\t\t\\begin{equation*}\n\t\t\\rho(1-a) L(X(t))^{-a}\\Vert \\partial L(X(t)) \\Vert > 1.\n\t\t\\end{equation*}\n\tGoing back to assumption~\\eqref{it::2}, for a.e. $t>0$, one has\n\t\t\\[ \\frac{\\diff L}{\\diff t} (X(t)) \\leq -c_1 \\Vert(\\partial L)(X(t))\\Vert^2, \\]\n\t\tbut the KL property implies that for a.e. $t>0$,\n\t\t$$- \\Vert \\partial L (X(t)) \\Vert^2 < -\\frac{1}{\\rho^2(1-a)^2} L(X(t))^{2a}.$$ Therefore,\n\t\t\\[\\frac{\\diff L}{\\diff t} (X(t)) < -\\frac{c_1}{\\rho^2(1-a)^2} L(X(t))^{2a}. \\]\n\t\t%\n\t\tWe consider two cases depending on the value of $a$. If $0<a\\leq 1\/2$, then for $t$ large, $L(X(t))<1$ so $-L(X(t))^{2a} < -L(X(t))$ and hence, $$\\frac{\\diff L}{\\diff t} (X(t)) < -\\frac{c_1}{\\rho^2(1-a)^2} L(X(t)),$$ so we obtain a linear rate.\n\tWhen $1\/2<a<1$, we have for a.e. $t>0$,\n\t\t\\begin{equation}\n\t\tL(X(t))^{-2a} \\frac{\\diff }{\\diff t}L(X(t)) =\\frac{1}{1-2a}\\frac{\\diff}{\\diff t} L(X(t))^{1-2a} < - \\frac{c_1}{ (\\rho^2 (1 - a)^2)},\n\t\t\\end{equation}\n\t\twith $1-2a<0$. We can integrate from $0$ to $t>0$:\n\t\t\\begin{equation*}\n\t\t \tL(X(t))^{1-2a}> \\frac{(2a -1)c_1}{\\rho^2(1-a)^2}t + L(X(0))^{1-2a}>\\frac{(2a -1)c_1}{\\rho^2(1-a)^2}t.\n\t\t\\end{equation*}\n\t\tSince $\\frac{1}{1-2a}<-1$, one obtains a convergence rate of the form $O\\left(t^\\frac{1}{1-2a}\\right)$. In both cases the rate is at least $O\\left(\\frac{1}{t}\\right)$.\n\t\n\t\n\t\t\n\tWe assume now that both \\eqref{it::2} and \\eqref{it::3} holds and prove convergence of the trajectory with a convergence rate. \n    Let $t>s>0$, by the fundamental theorem of calculus and the triangular inequality,\n    \t\\begin{equation}\\label{eqpr::dotX}\n    \t\\Vert X(t) - X(s)\\Vert \\leq \\left\\|  \\int_s^{t} \\dot{X}(\\tau) \\diff \\tau \\right\\| \\leq  \\int_s^{t} \\Vert\\dot{X}(\\tau)\\Vert \\diff \\tau.\n    \t\\end{equation}\n    \tWe wish to bound $\\Vert \\dot{X}\\Vert$ using $L$. Using the chain rule (Lemma~\\ref{lem::chainrule} of Section~\\ref{sec:ResultsOnD}), for a.e. $\\tau>0$,\n    \t\\begin{align}\n    \t\\begin{split}\n    \t\\frac{\\diff }{\\diff \\tau}L(X(\\tau))^{1-a} = (1-a)L(X(\\tau))^{-a} \\langle \\dot{X}(\\tau),(\\partial L)(X(\\tau))\\rangle.\n    \t\\end{split}\n    \t\\end{align}\n    \tThen, from \\eqref{it::2}, we deduce that for a.e. $\\tau>0$,\n    \t\\begin{align}\n    \t\\begin{split}\n    \t\\langle \\dot{X}(\\tau),(\\partial L)(X(\\tau))\\rangle = \\frac{\\diff L}{\\diff \\tau}(X(\\tau)) \\leq -c_1 \\Vert (\\partial L)(X(\\tau)) \\Vert^2,\n    \t\\end{split}\n    \t\\end{align}\n    \tso\n    \t\\begin{align}\\label{eq::dtL}\n    \t\\begin{split}\n    \t\\frac{\\diff }{\\diff \\tau}L(X(\\tau))^{1-a} \\leq -c_1(1-a)L(X(\\tau))^{-a} \\Vert (\\partial L)(X(\\tau)) \\Vert^2.\n    \t\\end{split}\n    \t\\end{align}\n    \tThe KL property \\eqref{eq::SemiAlgKL} implies that for a.e. $\\tau>0$,\n    \t\\begin{equation}\\label{eq::KLofL}\n    \t-(1-a)L(X(\\tau))^{-a}\\Vert (\\partial L)(X(\\tau))\\Vert< -\\frac{1}{\\rho}.\n    \t\\end{equation}\n    \tPutting this in \\eqref{eq::dtL} and using assumption \\eqref{it::3} we finally obtain\n    \t\\begin{align}\\label{eq::dtL2}\n    \t\\begin{split}\n    \t\\frac{\\diff }{\\diff t}L(X(\\tau))^{1-a} < -\\frac{c_1}{\\rho} \\Vert (\\partial L)(X(\\tau)) \\Vert\\leq -\\frac{c_1c_2}{\\rho} \\Vert \\dot{X}(\\tau)\\Vert.\n    \t\\end{split}\n    \t\\end{align}\n    \tWe can use that in \\eqref{eqpr::dotX},\n    \t\\begin{equation}\n    \t\\begin{split}\\label{eq::boundX}\n    \t\\Vert X(t) - X(s)\\Vert &\\leq  - \\frac{\\rho}{c_1 c_2}\\int_s^{t} \\frac{\\diff }{\\diff t}L(X(\\tau))^{1-a} \\diff \\tau\\\\\n    \t&=  \\frac{\\rho}{c_1 c_2 } (L(X(s))^{1-a}-L(X(t))^{1-a}).\n    \t\\end{split}\n    \t\\end{equation}\n    \tThen, using the convergence rate we already proved for $L$, we deduce that the Cauchy criterion holds for $X$ inside the compact (hence complete) subset $\\mathsf{C}\\subset\\mathbb{R}^P$ containing the trajectory. Thus, $X$ converges, and from (i) we have that $\\lim\\inf_{t \\to +\\infty} \\|\\partial L(X(t))\\| = 0$ because $\\partial L$ has closed graph. This shows that the limit is a critical points of $L$. Finally, taking the limit in \\eqref{eq::boundX} and using the convergence rate of $L$ we obtain a rate for $X$ as well.\n    \\end{proof}\n\n  \\begin{remark}\\label{rem::shift}\n     {\\rm   Theorem \\ref{th::genlemma} takes the form of a general recipe to obtain a convergence rate since it may be applied in many cases, to curves or flows, provided that a convenient Lyapunov function is given. \n     Note also that it is sufficient for assumptions \\eqref{it::2} and \\eqref{it::3} to hold only after some time $t_0>0$ as in such case, one could simply do a time shift to use the theorem.} \n    \\end{remark}\n\n\n    \\subsection{Application to INDIAN}\\label{sec::rateIND}%\nWe now  apply Theorem~\\ref{th::genlemma} to the deterministic continuous dynamical model of INDIAN \\eqref{eq::contdinClarke}.\n\\begin{theorem}[Convergence rates]\\label{cor::IndianRate}\n        Suppose that $\\mathcal{J}$ is semi-algebraic locally Lipschitz continuous and lower bounded. Then, any bounded trajectory $(\\theta,\\psi)$ that solves \\eqref{eq::contdinClarke} converges to a point $(\\bar{\\theta},\\bar{\\psi}) \\in \\mathsf{S}$,  with a convergence rate of the form $O\\left(t^{-b}\\right)$ with $b>0$. Moreover $\\mathcal{J}(\\theta(t))$ converges to its limit $\\bar \\mathcal{J}$ with rate $\\left\\vert\\mathcal{J}(\\theta(t))-\\bar \\mathcal{J}\\right\\vert=O\\left(\\frac{1}{t}\\right)$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $(\\theta,\\psi)$ be a bounded solution of \\eqref{eq::contdinClarke}. We would like to use Theorem~\\ref{th::genlemma} with $X =(\\theta,\\psi)$, and a well chosen function. In the proof of Theorem~\\ref{th::thmDIN} we proved a descent property along the trajectory for the function $E(\\theta,\\psi) = 2(1+\\alpha\\beta)\\mathcal{J}(\\theta) + \\left\\Vert (\\alpha-\\ovb)\\theta +\\ovb \\psi \\right\\Vert^2 $. This function is semi-algebraic, locally Lipschitz continuous, so it remains to prove that \\eqref{it::2} and \\eqref{it::3} hold for $E$ along $(\\theta,\\psi)$.\n\n For $t\\geq 0$, denote $w(t) = (\\alpha-\\frac{1}{\\beta})\\theta(t) +\\frac{1}{\\beta}\\psi(t)$, then according to Lemma~\\ref{lem::dEdt} for a.e. $t>0$,\n    \\begin{align}\\label{eq::valofDT}\n        \\begin{split}\n        \\frac{\\diff E}{\\diff t}(\\theta(t),\\psi(t)) &=- \\Vert \\sqrt{\\alpha} \\dot{\\theta}(t) - \\frac{1}{\\sqrt{\\beta}} \\left( \\dot{\\psi}(t) - \\dot{\\theta}(t) \\right)\\Vert^2 - \\Vert \\sqrt{\\alpha} \\dot{\\theta}(t) + \\frac{1}{\\sqrt{\\beta}} \\left( \\dot{\\psi}(t) - \\dot{\\theta}(t) \\right)\\Vert^2\\\\\n        &=  -2\\alpha \\Vert \\dot{\\theta}(t)\\Vert^2 - \\frac{2}{\\beta} \\Vert \\dot{\\psi}(t) -\\dot{\\theta}(t) \\Vert^2\n        = -2\\alpha \\Vert \\dot{\\theta}(t) \\Vert^2 - \\frac{2}{\\beta} \\Vert \\beta \\partial \\mathcal{J}(\\theta(t)) \\Vert^2\\\\\n        &= -2\\alpha \\Vert -\\beta\\partial \\mathcal{J}(\\theta(t)) -w(t) \\Vert^2 - 2\\beta \\Vert \\partial \\mathcal{J}(\\theta(t)) \\Vert^2.\n        \\end{split}\n    \\end{align}\n    On the other hand, by standard results on the sum rule, we have for all $(\\theta,\\psi)\\in\\mathbb{R}^P\\times\\mathbb{R}^P$,\n     \\begin{equation}\n         \\partial E(\\theta,\\psi) = 2\\begin{pmatrix}(1+\\alpha\\beta)\\partial \\mathcal{J}(\\theta) + (\\alpha-\\frac{1}{\\beta})\\left( (\\alpha-\\frac{1}{\\beta})\\theta + \\frac{1}{\\beta}\\psi\\right)\n         \\\\  \\frac{1}{\\beta} \\left( (\\alpha-\\frac{1}{\\beta})\\theta + \\frac{1}{\\beta}\\psi\\right)\n         \\end{pmatrix},\n     \\end{equation}\n    so for a.e. $t>0$,\n    \\begin{align}\\label{eq::NormOfGrad}\n        \\frac{\\left\\| \\partial E(\\theta(t),\\psi(t))\\right\\|^2}{4} = &\\left\\| (1+\\alpha\\beta)\\partial \\mathcal{J}(\\theta(t)) + (\\alpha-\\frac{1}{\\beta})w(t) \\right\\|^2 + \\left\\| \\frac{1}{\\beta} w(t) \\right\\|^2.\n    \\end{align}\n    We wish to find $c_1>0$, such that $\\frac12\\frac{\\diff E}{\\diff t} + \\frac{c_1}{4}\\Vert \\partial E\\Vert^2 < 0$. This follows from the following claim.\n\n        \\textit{Claim}: let $r_1>0$, $r_2\\in\\mathbb{R}$, $r_3>0$, then there exist $C_1$ and $C_2$ two positive constants such that for any $a,b\\in\\mathbb{R}$,\n        \\begin{equation}\\label{eq::analysisinequality}\n                C_1(a^2+b^2) \\leq (r_1 a +r_2 b)^2 + r_3 b^2 \\leq C_2(a^2+b^2).\n        \\end{equation}\n        Indeed, the function $Q:(a,b) \\mapsto (r_1 a +r_2 b)^2 + r_3 b^2$ is a positive definite quadratic form, $C_1$ and $C_2$ can be taken to be two eigenvalues of the positive definite matrix which represents $Q$. Hence  \\eqref{eq::analysisinequality} holds for all $a$ and $b$.\n\n\n    Applying the previous claim to \\eqref{eq::NormOfGrad} and \\eqref{eq::valofDT} leads to the existence of $c_1>0$ such that for a.e. $t>0$,\\begin{equation*}\n        \\frac{\\diff E}{\\diff t}(\\theta(t),\\psi(t))\\leq - c_1 \\Vert \\partial E (\\theta(t),\\psi(t))\\Vert^2,\n    \\end{equation*} so assumption \\eqref{it::2} holds for INDIAN.\n\n    It now remains to show that \\eqref{it::3} of Theorem~\\ref{th::genlemma} holds i.e. that there exists $c_2>0$ such that for $(\\theta,\\psi)$ solution of \\eqref{eq::contdinClarke} and for a.e. $t>0$, $\\Vert \\partial E(\\theta(t),\\psi(t))\\Vert^2\\geq c_2 \\left( \\Vert \\dot{\\theta}(t)\\Vert^2 + \\Vert \\dot{\\psi}(t)\\Vert^2 \\right)$.\n    Using \\eqref{eq::contdinClarke} and \\eqref{eq::NormOfGrad} we obtain:\n    \\begin{align}\\label{eq::assum3}\n        \\begin{split}\n            \\frac{\\Vert \\partial E(\\theta(t),\\psi(t))\\Vert^2}{4} =\\left\\| \\frac{1}{\\beta}(1+\\alpha\\beta)\\dot{\\theta}(t) + \\left[(\\alpha-\\frac{1}{\\beta}) -\\frac{1}{\\beta}(1+\\alpha\\beta)\\right]\\dot{\\psi}(t)\\right\\|^2 + \\frac{1}{\\beta^2}\\Vert\\dot{\\psi}(t)\\Vert^2 \n        \\end{split},\n    \\end{align}\n    and applying the claim \\eqref{eq::analysisinequality} again to \\eqref{eq::assum3} one can show that there exist $c_2>0$, such that for a.e. $t>0$, \\[\\Vert \\partial E(\\theta(t),\\psi(t))\\Vert^2\\geq c_2 \\left( \\Vert \\dot{\\theta}(t)\\Vert^2 + \\Vert \\dot{\\psi}(t)\\Vert^2 \\right).\\] So assumption \\eqref{it::3} holds for \\eqref{eq::contdinClarke}. To conclude, we can apply Theorem~\\ref{th::genlemma} to \\eqref{eq::contdinClarke} and the proof is complete.\n    \\end{proof}\n\n\n    \\begin{remark}\\label{rem::rateofIND}\n       \\rm{(a)  \\new{Since the discrete algorithm INDIAN asymptotically resembles to its continuous time version (see the proof of Theorem~\\ref{th::thmDIN}), the results above suggest that similar behaviors and rates could be hoped for INDIAN itself. Yet, these results remain difficult to obtain in the case of DL, in particular in the mini-batch setting because of the noise $(\\xi_k)_{k\\in\\mathbb{N}}$.}\\\\\n       (b) The proof above is significantly simpler when $\\alpha\\beta>1$ since \\cite{alvarez2002second} proved that in this case, \\eqref{eq::contdinClarke} is equivalent to a gradient system, thus assumptions \\eqref{it::2} and \\eqref{it::3} of Theorem~\\ref{th::genlemma} instantly holds.}\\\\\n       \\rm{ (c)  Theorems \\ref{th::genlemma} and \\ref{cor::IndianRate} can be adapted to the case when the Clarke subdifferential is replaced by $D\\mathcal{J}$, but we do not state it here for the sake of simplicity.}\\\\\n       \\rm{ (d) \\new{Theorems~\\ref{th::genlemma} and~\\ref{cor::IndianRate} are actually valid by assuming that $\\mathcal{J}$ belongs to a polynomially bounded o-minimal structure. One of the most common instance of such structures is the one given by globally subanalytic sets (as illustrated in a example below). We refer to \\cite{bolte2007lojasiewicz} for a definition and further references.}}\n    \\end{remark}\n    \n    \\new{Let us now comment the results of Theorem~\\ref{cor::IndianRate}. First, we restrained here to semi-algebraic loss functions $\\mathcal{J}$, which are a subset of tame loss functions. Most networks, activations and dissimilarity measures mentioned in Section~\\ref{sec::favstruct} fall into this category. Nonetheless, the loss functions of the DL experiments of Section~\\ref{sec::trainDL} are not semi-algebraic.  Indeed, the dissimilarity measure $l$ used is the cross-entropy: $l(f(x_n,\\theta),y_n)= - \\sum_{d=1}^D \\mathbf{1}_{[y_{n}]_d=1}\\log( [f(x_n,\\theta)]_d)$. Such a function cannot be described by polynomials and presents a singularity whenever $[f(x_n,\\theta)]_d=0$. Fortunately, due to the numerical precision but also to the ``soft-max'' functions often used in classification experiments, the outputs of the network $f$, for inputs restricted to a compact set, have values in $[\\varepsilon,1]$ for some small $\\varepsilon>0$. Therefore, the singularity at $0$ is harmless and the cross-entropy acts as a globally subanalytic function. As a consequence the nonsmooth {\\L}ojasiewicz inequality holds, and the theorems  apply (see also numerical experiments).\n        \n         The rate of convergence of the trajectory in Theorem~\\ref{cor::IndianRate} is non-explicit in the sense that the exponent $b>0$  is unknown in general. In the light of the proof of Theorem~\\ref{th::genlemma}, this exponent depends on the KL exponent $a$ of the Lyapunov function, which is itself hard to determine in practice. However the intuition is that small exponents $a$ may yield faster convergence rates (indeed, when $a\\in(0,1\/2)$ we actually have a linear rate). As an example, for the function: $t\\in\\mathbb{R}\\mapsto \\vert t \\vert^c$ with $c>1$, the exponent at $0$ is $a=1-\\frac{1}{c}$ and thus, the closer \n         $c$ is to $1$, the smaller $a$ is, and the faster the convergence becomes.\n    }\n    \n\n    \\section{Experiments}\n\t\\label{sec:numerics}\n\nIn this section we first discuss the role and influence of the hyper-parameters of INDIAN using the 2D example given in Figure~\\ref{fig::rosenbrockexp}. We then compare INDIAN with SGD, ADAGRAD and ADAM on deep learning problems for image recognition.\n\n\t\\subsection{Understanding the Role of the Hyper-parameters of INDIAN}\\label{sec::HPmeaning}\n Both  hyper-parameters $\\alpha$ and $\\beta$ can be seen as damping coefficients from the viewpoint of mechanics as discussed by \\cite{alvarez2002second} and sketched in the introduction. Recall the second-order time continuous dynamics which served as a model to the design INDIAN:\n\t   \\begin{equation*}\n    \t  \\ddot{\\theta}(t)+\\alpha\\,\\dot{\\theta}(t) + \\beta\\, \\nabla^2 \\mathcal{J}(\\theta(t))\\dot{\\theta}(t) + \\nabla \\mathcal{J}(\\theta(t))=0.\n\t   \\end{equation*}\n\tThis differential equation was inspired by Newton's Second Law of dynamics asserting that the acceleration of a material point   coincides with the sum of forces applied to the particle. As recalled in the introduction three forces are at stake: the gravity and two friction terms. The parameter $\\alpha$ calibrates the {\\em viscous damping} intensity as in the Heavy Ball friction method of \\cite{polyak1964some}. It acts as a dissipation term but it can also be seen as a proximity parameter of the system with the usual gradient descent: the higher $\\alpha$ is, the more DIN behaves like a pure gradient descent.\\footnote{This is easier to see when one rescales $\\mathcal{J}$ by $\\alpha$.} On the other hand the parameter $\\beta$ can be seen as a \\textit{Newton damping} which takes into account the geometry of the landscape to brake or accelerate the dynamics in an adaptive anisotropic fashion, see \\cite{felipe,alvarez2002second} for further insights.\n\n\n\tWe now turn our attention to INDIAN, and illustrate the versatility of the hyper-parameters $\\alpha$ and   $\\beta$ in this case. We proceed on a 2D visual nonsmooth ill-conditioned example \\`a la Rosenbrock, see Figure~\\ref{fig::rosenbrockexp}. For this example, we aim at finding the minimum of the function $\\mathcal{J}(\\theta_1,\\theta_2)= 100(\\theta_2-\\vert \\theta_1 \\vert)^2 +\\vert 1-\\theta_1\\vert $. This function has a {\\em V}-shaped valley, and a unique critical point at $(1,1)$ which is also the global minimum. Starting from the point $(-1,1.5)$ (the black cross), we apply INDIAN with constant steps $\\gamma_k=10^{-4}$. Figure \\ref{fig::rosenbrockexp} shows that when $\\beta$ is too small, the trajectory presents many transverse oscillations as well as longitudinal ones close to the critical point (subplot (a)). Then, increasing $\\beta$ significantly reduces  transverse oscillations (subplot (b)). Finally, the longitudinal oscillations are reduced by choosing a higher $\\alpha$ (subplot (c)). In addition, these behaviors are also reflected in the values of the objective function (subplot (d)).\n\tThe orange curve (first setting) presents large oscillations. Moreover, looking at the red curve, corresponding to plot (c), there is a short period between $20,000$ and $60,000$ iterations when the decrease is slower than for the other values of $\\alpha$ and $\\beta$, but still it presents fewer oscillations. In the longer term, the third choice ($\\alpha=1.3$, $\\beta=0.1$) provides remarkably good performance\n    \n    \\new{The choice of these hyper-parameters may come with rates of convergence for convex and strongly convex smooth functions \\citep{attouch2019first}. Following this work, one may also consider to make $\\alpha$ and $\\beta$ vary in time (for example like the famous Nesterov damping $\\frac{\\alpha}{t}$). In our DL experiments we will however keep these parameters constant so that our theorems still hold. Yet, different behaviors depending on $(\\alpha,\\beta)$ can also be observed for DL problems as illustrated on Figure~\\ref{fig::Multiparam} and described next.}\n    \n\n\t\\subsection{Training a DNN with INDIAN}\\label{sec::trainDL}\n\t\n\n\tBefore comparing INDIAN to state-of-the-art algorithms in DL, we first describe the methodology we followed.\n\t\n\t\n\t\\subsubsection{Methodology}\n\t\\begin{itemize}\n\t    \\renewcommand\\labelitemi{--}\n\n\t    \\item We train a DNN for classification using the three most common image data sets (MNIST, CIFAR10, CIFAR100) \\citep{lecun1998gradient,krizhevsky2009learning}. These data sets are composed of $60,000$ small images associated with a label (numbers, objects, animals, etc.). We split the data sets into $50,000$ images for training and $10,000$ for testing.\n\t    \\item \tRegarding the network, we use a slightly modified version of the popular Network in Network (NiN) \\citep{lin2013network}. It is a reasonably large convolutional network with $P\\sim10^6$ parameters to optimize. We use ReLU activation functions.\n\t    \\item The dissimilarity measure $l$ that is used in the empirical loss $\\mathcal{J}$ given by \\eqref{eq::loss} is set to the cross-entropy.\n\t    The loss $\\mathcal{J}$ is optimised with respect to $\\theta$ (the weights of the DNN) on the training data. The classification accuracy of the trained DNN is measured using the test data of $10,000$ images. Measuring the accuracy boils down to counting how many of the $10,000$ were correctly classified (in percentage).\n\t    \\item \tBased on the results of Section \\ref{sec::HPmeaning}, we run INDIAN for four different values of $(\\alpha,\\beta)$: \\[(\\alpha,\\beta)\\in \\left\\{(0.1,0.1),(0.5,0.1),(0.5,0.5),(0.5,1)\\right\\}.\\]\n    \tGiven an initialisation of the weights $\\theta_0$, we \n    \tinitialize $\\psi_0$ such that the initial\n    \tvelocity is in the direction of $-\\nabla \n    \t\\mathcal{J}(\\theta_0)$. More precisely, we use $ \\psi_0 = \n    \t(1-\\alpha\\beta)\\theta_0 \n    \t-(\\beta^2-\\beta)\\nabla \\mathcal{J}(\\theta_0)$.\n    \t\\item We compare our algorithm INDIAN with the classical SGD algorithm and the popular ADAGRAD \\citep{duchi2011adaptive} and ADAM\\citep{kingma2014adam} algorithms. At each iteration $k$, we compute the approximation of $\\partial \\mathcal{J}(\\theta)$ on a subset $\\mathsf{B}_k\\subset\\left\\{1,\\ldots,50,000\\right\\}$ of size $32$. The algorithms are initialized with the same random weights (drawn from a normal distribution). Five random initializations are considered for each experiment.\n    \t%\n    \t\\item Regarding the selection of step sizes, ADAGRAD and ADAM both use an adaptive procedure based on past gradients, see \\cite{duchi2011adaptive,kingma2014adam}. For the other two algorithms (INDIAN and SGD), we use the classical step size schedule $\\gamma_k=\\frac{\\gamma_0}{\\sqrt{k+1}}$, which meets Assumption~\\ref{ass:mainAssumption}.  For all four algorithms, choosing the right initial step length $\\gamma_0$ is often critical in terms of efficiency. \\new{We choose this $\\gamma_0$ using a grid-search: for each algorithm we select the initial step size that most decreases the training error $\\mathcal{J}$ after fifteen epochs (one epoch consisting in a complete pass over the data). The test data is not used to chose the initial step size nor other hyper-parameters.} Note that we could use more flexible step size schedules but chose a standard schedule for simplicity. Other decay schemes are considered in Figure~\\ref{fig::decay}.\n    \t\n    \t\n\t\\end{itemize}\n     For these experiments, we used \\texttt{Keras 2.2.4} \\citep{chollet2015} with \\texttt{Tensorflow 1.13.1} \\citep{abadi2016tensorflow} as backend. The INDIAN algorithm is available in Pytorch, Keras and Tensorflow: \\url{https:\/\/github.com\/camcastera\/Indian-for-DeepLearning\/} \\citep{castera2019github}.\n\n\t\\subsubsection{Results}\\label{sec:numreslenet}\n    \n    Figure~\\ref{fig::Multiparam} displays the training loss $\\mathcal{J}$ and test accuracy with respect to epochs for INDIAN in its four considered hyper-parameter configurations and for the three data sets considered. Figure~\\ref{fig::vsothalgo} displays the performance of INDIAN \\new{with the hyper-parameter configuration that led to smallest average training error in Figure~\\ref{fig::Multiparam}}, with comparison to SGD, ADAGRAD and ADAM. In these two figures (and also in subsequent Figure~\\ref{fig::decay}), solid lines represent mean values and pale surfaces represent the best and worst runs in terms of training loss and validation accuracy over five random initializations.\n    \n     \n    \n    \\begin{figure}[t]\n     \t\\begin{center}\n                \\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (a) CIFAR-10\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (b) CIFAR-100\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (c) MNIST\n \t\t\t\t\\end{minipage}%\n\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n                \\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.45cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_CIFAR100.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n\t\t\t\t\t\t\\centering\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_MNIST.pdf}\n\t\t\t\t\\end{minipage}%\n\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.2cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_VAL_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.5cm 0.5cm 0.3cm 0.5cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_VAL_CIFAR100.pdf}\n\t\t\t\t\\end{minipage}%\n\t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.5cm 0.5cm 0.3cm 0.5cm, width=0.95\\linewidth]{Figures\/EXP_DL_2\/NiN_otheralgo_VAL_MNIST.pdf}\n \t\t\t\t\\end{minipage}%\n\n \t\t\t \t\\begin{minipage}[c]{0.8\\linewidth}\n \t\t\t \t\\centering\\includegraphics[width=0.45\\linewidth]{Figures\/EXP_DL_2\/legend_indian_hor.pdf}\n \t\t\t \t\\end{minipage}%\n \t\t\t \t\\caption{Analysis of the sensibility of INDIAN to the choice of $\\alpha$ and $\\beta$ using NiN for three different image classification problems. Top:  logarithm of the loss function $\\mathcal{J}(\\theta)$ during the training. Bottom: classification accuracy on the test set.}\\label{fig::Multiparam}\n \t\t\t \t\\end{center}\n \t\t\\end{figure}\n\n \t\tFigure~\\ref{fig::Multiparam} suggests that the tuning of the hyper-parameters $\\alpha$ and $\\beta$ is not crucial to obtain satisfactory results both for training and testing. It mostly affects the training speed. Thus, INDIAN looks quite stable with respect to these hyper-parameters. Setting $(\\alpha,\\beta)=(0.5,0.1)$ appears to be a good default choice when necessary. Nevertheless, tuning these hyper-parameters is of course advised to get the most from INDIAN.\n \t\t\n \t\t\\begin{figure}[t]\n     \t\\begin{center}\n                \\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (a) CIFAR-10\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (b) CIFAR-100\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (c) MNIST\n \t\t\t\t\\end{minipage}%\n\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n                \\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.45cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_CIFAR100.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n\t\t\t\t\t\t\\centering\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_MNIST.pdf}\n\t\t\t\t\\end{minipage}%\n\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.2cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_VAL_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.5cm 0.5cm 0.3cm 0.5cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_VAL_CIFAR100.pdf}\n\t\t\t\t\\end{minipage}%\n\t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\\centering\\includegraphics[clip, trim=0.5cm 0.5cm 0.3cm 0.5cm, width=0.95\\linewidth]{Figures\/EXP_DL_1\/NiN_otheralgo_VAL_MNIST.pdf}\n \t\t\t\t\\end{minipage}%\n\n \t\t\t \t\\begin{minipage}[c]{0.8\\linewidth}\n \t\t\t \t\\centering\\includegraphics[width=0.45\\linewidth]{Figures\/EXP_DL_1\/legend_other_hor.pdf}\n \t\t\t \t\\end{minipage}%\n \t\t\t \t\\caption{Comparison of INDIAN with state-of-the-art algorithms SGD, ADAM and ADAGRAD. Top: logarithm of the loss function $\\mathcal{J}(\\theta)$ during the training. Bottom: classification accuracy on the test set.\n \t\t\t \t}\n \t\t\t \t\\label{fig::vsothalgo}\n \t\t\t \t\\end{center}\n \t\t\\end{figure}\n \t\nFigure~\\ref{fig::vsothalgo} shows that best performing methods achieve state-of-the art accuracy using NiN and represent what can be achieved with a moderately large network and coarse grid-search tuning of the initial step size. In our comparison, INDIAN and ADAM outperform SGD and ADAGRAD for training. While ADAM seems to be faster in the early training phase, INDIAN achieves the best accuracy almost every time especially on CIFAR-100 (Figure~\\ref{fig::vsothalgo}(b)). Thus INDIAN appears to be competitive in comparison to the other algorithms with the advantage of having solid theoretical foundations and a simple step-size rule as compared to ADAM and ADAGRAD.\n\n \t\t\n \t\t\\begin{figure}[t]\n     \t\\begin{center}\n                \\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (a) CIFAR-10\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (b) CIFAR-100\n \t\t\t\t\\end{minipage}%\n                \\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering \\footnotesize (c) MNIST\n \t\t\t\t\\end{minipage}%\n\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n                \\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.45cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_3\/NiN_decay_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_3\/NiN_decay_CIFAR100.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_3\/NiN_decay_MNIST.pdf}\n \t\t\t\t\\end{minipage}%\n\n \t\t\t \t\\begin{minipage}[c]{0.8\\linewidth}\n \t\t\t \t\\centering\\includegraphics[width=0.4\\linewidth]{Figures\/EXP_DL_3\/legend_decres_hor.pdf}\n \t\t\t \t\\end{minipage}%\n \t\t\t \t\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n                \\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.45cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_4\/indian_vs_adam_CIFAR10.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_4\/indian_vs_adam_CIFAR100.pdf}\n \t\t\t\t\\end{minipage}%\n \t\t\t\t\\begin{minipage}[c]{0.33\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\t\t\\includegraphics[clip, trim=0.40cm 0.5cm 0.35cm 0.3cm, width=0.95\\linewidth]{Figures\/EXP_DL_4\/indian_vs_adam_MNIST.pdf}\n \t\t\t\t\\end{minipage}%\n\n \t\t\t \t\\begin{minipage}[c]{0.8\\linewidth}\n \t\t\t \t\\centering\\includegraphics[width=0.4\\linewidth]{Figures\/EXP_DL_4\/legend_ind_vs_adam_hor.pdf}\n \t\t\t \t\\end{minipage}%\n \t\t\t \t\\caption{On top: Training loss of INDIAN on three image classification problems with various step size decays. In the legend, $k^{-q}$ means a step size decay at iteration $k$ of the form $\\gamma_k = \\gamma_0 k^{-q}$. The bottom row show the comparison between INDIAN with a well-chosen step size decay and ADAM.}\\label{fig::decay}\n \t\t\t \t\\end{center}\n \t\t\\end{figure}\n\n        Finally let us point out that although ADAM was faster in the experiments of Figure~\\ref{fig::vsothalgo}, INDIAN can outperform ADAM using the slow step size decay discussed in Section~\\ref{sec::comRes}. Indeed, in the previous experiments we used a standard decreasing step size of the form $\\gamma_0 \/ \\sqrt{k+1}$ for simplicity, but Assumption~\\ref{ass:mainAssumption} allows for step sizes decreasing much slower. As such, we also considered decays of the form $\\gamma_0 (k+1)^{-q}$ with $q\\leq 1\/2$. The results are displayed on top of Figure~\\ref{fig::decay}. \n        Except when $q$ is too small (too slow decay, e.g., $q=1\/16$), these results show that some decays slower than $q=1\/2$ make INDIAN a little bit faster than any of the other algorithms we tried. In particular, with a step size decay proportional to $k^{-1\/4}$, INDIAN outperforms ADAM (bottom of Figure~\\ref{fig::decay}). This suggests that tuning $q$ can also significantly accelerate the training process.\n        \n        \n\n\t\\section{Conclusion}\n\tWe introduced a novel stochastic optimization algorithm featuring inertial and Newtonian behavior motivated by applications to deep learning. We provided a powerful algorithmic convergence analysis under weak hypotheses applicable to most DL problems. We also provided new general results to study differential inclusions on Clarke subdifferential and obtain convergence rates for the continuous time counterpart of our algorithm. We would like to point out that, apart from SGD \\citep{davis2018stochastic}, the convergence of concurrent methods in such a general setting is still an open question. Our result seems moreover to be the first one to be able to rigorously handle the analysis of mini-batch subsampling for ReLU DNNs via the introduction of the $D$-critical points. Our experiments show that INDIAN is very competitive with state-of-the-art algorithms for DL but also very malleable. We stress that these numerical manipulations were performed on substantial DL benchmarks with only limited algorithm tuning. \n\t\n\tThis facilitates reproducibility and allows to stay as close as possible to the reality of DL applications in machine learning.\n\t\n \t\\acks{\n     The authors acknowledge the support of the European Research Council (ERC FACTORY-CoG-6681839), the Agence Nationale de la Recherche (ANR 3IA-ANITI, ANR-17-EURE-0010 CHESS, ANR-19-CE23-0017 MASDOL) and the Air Force Office of Scientific Research (FA9550-18-1-0226).\n\n\n     Part of the numerical experiments were done using the OSIRIM platform of IRIT, supported by the CNRS, the FEDER, R\u00e9gion Occitanie and the French government (\\url{http:\/\/osirim.irit.fr\/site\/en}). We thank the development team of the following libraries that were used in the experiments: Python \\citep{rossum1995python}, Numpy \\citep{walt2011numpy}, Matplotlib \\citep{hunter2007matplotlib}, Pytorch \\citep{paszke2019pytorch}, Tensorflow and Keras \\citep{abadi2016tensorflow,chollet2015}.\n         \n    \n \t We thank Hedy Attouch and Sixin Zhang for useful discussions.\n    }%\n\n    \n\n\n\n\n\n    \\DeclareRobustCommand{\\VAN}[3]{#3}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nFree probability theory was introduced by Voiculescu around 1983 motivated by \nthe isomorphism problem of von Neumann algebras of free groups. He developed a noncommutative probability theory, on a noncommutative\nprobability space, in which a new notion of freeness plays the\nrole of independence in classical probability. Around 1991, Voiculescu \\cite{V}  threw a bridge connecting random matrix theory with free probability \nsince he realized that the freeness property is also present for many classes of random matrices, in the\nasymptotic regime when the size of the matrices tends to infinity. Since then, random matrices have played a key role in operator  algebra  whereas tools developed in operator algebras and free\nprobability theory could now be applied to random matrix problems.\\\\\nFor the reader's convenience, we recall the following basic definitions from free probability theory. For a thorough introduction to free probability theory, we refer to \\cite{VDN}.\n\\begin{itemize}\n\\item A ${\\cal C}^*$-probability space is a pair $\\left({\\cal A}, \\tau\\right)$ consisting of a unital $ {\\cal C}^*$-algebra ${\\cal A}$ and \na linear map $\\tau: {\\cal A}\\rightarrow \\mathbb{C}$ such that $\\tau(1_{\\cal A})=1$ and $\\tau(aa^*)\\geq 0$ for all $a \\in {\\cal A}$. $\\tau$ is a trace if it satisfies $\\tau(ab)=\\tau(ba)$ for every $(a,b)\\in {\\cal A}^2$. A trace is said to be faithful if $\\tau(aa^*)>0$ whenever $a\\neq 0$. \nAn element of ${\\cal A}$ is called a noncommutative random variable. \n\\item The noncommutative distribution of a family $a=(a_1,\\ldots,a_k)$ of noncommutative random variables in a ${\\cal C}^*$-probability space $\\left({\\cal A}, \\tau\\right)$ is defined as the linear functional $\\mu_a:P\\mapsto \\tau(P(a,a^*))$ defined on the set of polynomials in $2k$ noncommutative indeterminates, where $(a,a^*)$ denotes the $2k$-tuple $(a_1,\\ldots,a_k,a_1^*,\\ldots,a_k^*)$.\nFor any self-adjoint element $a_1$ in  ${\\cal A}$,  there exists a probability  measure $\\nu_{a_1}$ on $\\mathbb{R}$ such that,   for every polynomial P, we have\n$$\\mu_{a_1}(P)=\\int P(t) \\mathrm{d}\\nu_{a_1}(t).$$\nThen,  we identify $\\mu_{a_1}$ and $\\nu_{a_1}$. If $\\tau$ is faithful then the  support of $\\nu_{a_1}$ is the spectrum of $a_1$  and thus  $\\|a_1\\| = \\sup\\{|z|, z\\in \\rm{support} (\\nu_{a_1})\\}$. \n\\item A family of noncommutative random variables $(a_i)_{i\\in I}$ in a ${\\cal C}^*$-probability space  $\\left({\\cal A}, \\tau\\right)$ is free if for all $k\\in \\mathbb{N}$ and all polynomials $p_1,\\ldots,p_k$ in two noncommutative indeterminates, one has \n\\begin{equation}\\label{freeness}\n\\tau(p_1(a_{i_1},a_{i_1}^*)\\cdots p_k (a_{i_k},a_{i_k}^*))=0\n\\end{equation}\nwhenever $i_1\\neq i_2, i_2\\neq i_3, \\ldots, i_{k-1}\\neq i_k$ and $\\tau(p_l(a_{i_l},a_{i_l}^*))=0$ for $l=1,\\ldots,k$.\n\\item A family $(x_i)_{i\\in I}$ of noncommutative random variables in a ${\\cal C}^*$-probability space  $\\left({\\cal A}, \\tau\\right)$ is a semicircular system if\n $x_i=x_i^*$ for all $i\\in I$, $(x_i)_{i\\in I}$\nis a free family and for any $k\\in \\mathbb{N}$, $$\\tau(x_i^k)= \\int t^k d\\mu_{sc}(t)$$\nwhere $d\\mu_{sc}(t)=\n\\frac{1}{2\\pi} \\sqrt{4-t^2}{\\1}_{[-2;2]}(t) dt$ is the semicircular standard distribution.\n\\item Let $k$ be a nonnull integer number. Denote by ${\\cal P}$ the set of polynomials in $2k $ noncommutative indeterminates.\nA sequence of families of variables $ (a_n)_{n\\geq 1}  =\n(a_1(n),\\ldots, a_k(n))_{n\\geq 1}$ in ${\\cal C}^* $-probability spaces \n$\\left({\\cal A}_n, \\tau_n\\right)$ converge, when n goes to infinity, respectively  in distribution if the map \n$P\\in {\\cal P} \\mapsto\n\\tau_n(\nP(a_n,a_n^*))$ converges pointwise\nand strongly in distribution if moreover the map \n$P\\in {\\cal P} \\mapsto  \\Vert P(a_n,a_n^*) \\Vert$  converges pointwise.\n\\end{itemize}\n\n\nVoiculescu  considered random matrices in this noncommutative probability context. Let ${\\cal A}_n$ be \nthe algebra of $n\\times n$ matrices\nwhose entries are random variables with finite moments and endow this algebra with\nthe expectation of the normalized trace defined for any $M\\in {\\cal A}_n$ by \n$\\tau_n(M) = \\mathbb{E}[\\frac{1}{n}\\Tr(M)]$. Let us \nconsider r  independent $n\\times n$ so-called G.U.E matrices, that is to say  random  Hermitian matrices $X_n^{(v)} =  [X^{(v)}_{jk}]_{j,k=1}^n$, $v=1,\\ldots,r$,   for which the  random variables $(X^{(v)}_{ii})$,\n$(\\sqrt{2} Re(X^{(v)}_{ij}))_{i<j}$, $(\\sqrt{2} Im(X^{(v)}_{ij}))_{i<j}$ are independent centred gaussian distribution with variance $1$.\nVoiculescu  discovered that these independent G.U.E  matrices  are asymptotically free and provide a model for a free semi-circular system since he established that\nfor any polynomial P in r noncommutative indeterminates,\n\\begin{equation}\\label{sansD}\\tau_n\\left\\{  P\\left(\\frac{ X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{ X_n^{(r)}}{\\sqrt{n}}\\right)\\right\\} \\rightarrow_{n\\rightarrow +\\infty} \\tau \\left( P(x_1,\\ldots,x_r)\\right)\\end{equation}\nwhere $(x_1,\\ldots,x_r)$ is a semicircular system in some ${\\cal C}^*$-probability space $({\\cal A}, \\tau)$. \nIt turns out that this result still holds on the one hand  almost surely dealing with the normalized trace $\\tau_n(M) = \\frac{1}{n}\\Tr(M)$ instead of the mean normalized trace \\cite{T}  and on the other hand\nfor non-gaussian Wigner matrices \\cite{D}.\\\\\n\nIn \\cite{HT}, Haagerup and Thorbj{\\o}rsen proved the strong asymptotic freeness of independent G.U.E matrices, namely: almost surely,\nfor any polynomial P in r noncommutative indeterminates,\n\\begin{equation}\\label{strong}\\left\\| P\\left(\\frac{ X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{ X_n^{(r)}}{\\sqrt{n}}\\right)\\right\\| \\rightarrow_{n\\rightarrow +\\infty} \\Vert P(x_1,\\ldots,x_r) \\Vert.\\end{equation}\nNote that this result led to the proof of the very important result in the theory of operator algebras that $Ext({\\cal C}^*_{red}(F_2))$ is not  a group.\n\\eqref{strong} was proved for Gaussian random matrices with real or symplectic entries \n by Schultz \\cite{Schultz05}, for Wigner matrices with symmetric distribution of the entries satisfying a Poincar\\'e inequality  by Capitaine and Donati-Martin \\cite{CD07} and for Wigner matrices under i.i.d assumptions and  fourth moment hypotheses by Anderson \\cite{A}.\\\\\n\nThe  result \\eqref{sansD} of Voiculescu is actually more general since Voiculescu \\cite{V,Voiculescu97} proved the asymptotic freeness of independent \nG.U.E matrices with an extra family of deterministic  matrices  with limiting distribution. \nMale \\cite{CamilleM} established the strong  asymptotic freeness of independent \nG.U.E matrices with an extra independent family of  matrices  with strong limiting distribution. Note that in \\cite{MCo}, Collins and Male established the  strong asymptotic freeness of Haar-distributed random unitary matrices \nand deterministic matrices with strong limiting distribution. \\\\\nIn this paper, on the one hand,  we \nprove the strong asymptotic freeness of independent non-Gaussian Wigner matrices and an extra family of deterministic matrices with strong limiting distribution (see Theorem \\ref{thprincipal}). \\\\\n\n\n\n\nOn the other hand we prove that a sequence of deterministic measures plays a central role in the study of the\nspectrum of a Hermitian  polynomial in independent Wigner matrices and deterministic matrices. Roughly speaking, these measures\nare in\nsome sense obtained by  taking partially the limit when n goes to infinity, only for the Wigner matrices. \nWe establish  that almost surely, for large n, each interval lying at  some distance from  the supports of these  deterministic measures contains no eigenvalue of the Hermitian  polynomial (see Theorem \\ref{noeigenvalue}). This type of result was first established\nby Bai and Silverstein in  \\cite{BaiSil98} for sample covariance matrices and in \\cite{BaiSil12} for Information-plus-noise type models, and by Capitaine,  Donati-Martin, F\\'eral and F\\'evrier  in \\cite{CDMFF} for additive deformations of a Wigner matrix.\n\\\\\n\n Here are the matricial models we deal with. Let $t$ and $r$ be fixed nonull integer numbers independent from $n$.\\\\\n\n\n\\noindent\n\n \n\\noindent $\\bullet$  $(A_n^{(1)},\\ldots,A_n^{(t)})$ is a $t-$tuple of  $n\\times n$   deterministic matrices   such that\n  for any $u=1,\\ldots,t$, \\begin{equation}\\label{normeAn} \\sup_n \\Vert A_n^{(u)} \\Vert< \\infty,\\end{equation} where $\\Vert \\cdot \\Vert$ denotes the spectral norm. \\\\\n  \n\n\\noindent $\\bullet$  We consider r  independent $n\\times n$ random  Hermitian Wigner matrices $X_n^{(v)} =  [X^{(v)                }_{ij}]_{i,j=1}^n$, $v=1,\\ldots,r$,  where, for each $v$,\n$ [X^{(v)  }_{ij}]_{i\\geq1,j\\geq 1}$ is an infinite array of  random variables such that $X^{(v)}_{ii}$,\n$\\sqrt{2}\\Re(X^{(v)}_{ij}), i<j$, $\\sqrt{2} \\Im(X^{(v)}_{ij}), i<j$, are independent,  centred with variance 1 and satisfy:\n\\begin{enumerate} \\item There exists a $K_v$ and a random variable $Z^{(v)}$ with finite fourth moment for which there exists $x_0>0$ and  an  integer number $n_0>0$ such that, for any $x >x_0$ and any integer number $n >n_0$, we have\n\\begin{equation}\\label{condition}\\frac{1}{n^2} \\sum_{1\\leq i,j\\leq n}\\mathbb{P}\\left( \\vert X^{(v)}_{ij}\\vert >x\\right) \\leq K_v\\mathbb{P}\\left(\\vert Z^{(v)} \\vert>x\\right).\\end{equation}\n\\item $$\\sup_{1\\leq i<j}\\mathbb{E}(\\vert X^{(v)}_{ij}\\vert^3)<+\\infty. $$\n\\end{enumerate}\n\\begin{remarque}\nThe independence of the real and imaginary parts of the entries of the Wigner matrices is necessary in our approach since, after conditioning, we use Lemma \\ref{lem1} for the real and imaginary parts of each entry.\n\\end{remarque}\n\\begin{remarque}\nNote that assumption such as \\eqref{condition} appears in \\cite{CSBD}. It obviously holds if the $X^{(v)}_{ii}$,\n$ \\sqrt{2}\\Re(X^{(v)}_{ij}),{i<j}$, $\\sqrt{2} \\Im(X^{(v)}_{ij}),{i<j}$, are identically distributed with finite fourth moment.\n\\end{remarque}\n\\noindent Here are our main results.\n\\begin{theoreme}\\label{noeigenvalue}\nLet $({\\cal A},  \\tau)$ be a ${\\cal C}^*$-probability space\n equipped with a faithful tracial state and \n $x=(x_1,\\ldots,x_r)$ be  a semi-circular system in  $({\\cal A},  \\tau)$.\nLet $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative   random variables which is free from  $x$ in $({\\cal A},\\tau)$ and such that the distribution of  $a_n$  in $({\\cal A},\\tau)$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}), \\frac{1}{n}\\Tr)$. Let $P$ be a self-adjoint  polynomial  in $2t+r$ noncommutative indeterminates $X_1, \\ldots, X_r$, where for any $i=1,\\ldots,r$, $X_i=X_i^*$, and $X_{r+1},\\ldots, X_{r+t}, X_{r+1}^*,\\ldots, X_{r+t}^*$. Let $[b,c]$ be a real interval  such  that there exists $\\delta>0$ such that, for any large  $n$,   $[b-\\delta,c+\\delta]$ lies outside the support of the distribution of  the noncommutative random variable  $ P\\left(x_1,\\ldots,x_r, a_n^{(1)},\\ldots,a_n^{(t)},(a_n^{(1)})^*,\\ldots,(a_n^{(t)})^*\\right)$ in $({\\cal A},\\tau)$.\n  Then, \nalmost surely,  for all large $n$, there is no eigenvalue of  the $n\\times n$  matrix $ P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)$ in $[b,c].$\n\\end{theoreme}\n\\begin{remarque}\nWhen $r=t=1$,  $A_n^{(1)}=(A_n^{(1)})^*$ and $P(X_1,X_2,X_2^*)=X_1+\\frac{X_2+X_2^*}{2}$, the distribution of $P(x_1,a_n^{(1)},(a_n^{(1)})^*)$ is the so-called free convolution $\\mu_{sc}\\boxplus \\mu_{A_n^{(1)}}$ where  $\\mu_{A_n^{(1)}}=\\frac{1}{n} \\sum_{i=1}^n \\lambda_i(A_n^{(1)})$, denoting by \n$ \\lambda_i(A_n^{(1)})$, $i=1,\\ldots,n$, the eigenvalues of  $A_n^{(1)}$.\n\\end{remarque}\n\\begin{theoreme}\\label{thprincipal} Let $({\\cal A},  \\tau)$ be  a ${\\cal C}^*$-probability space\n equipped with a faithful tracial state.\nLet  $x=(x_1,\\ldots,x_r)$ be  a semi-circular system and  $a=(a_1,\\ldots,a_t)$ be  a t-tuple of noncommutative   random variables which is free from $x$ in $({\\cal A},\\tau$).\\\\\nAssume moreover that  $(A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*)$  converges strongly towards $a=(a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*)$ in\n $({\\cal A},  \\tau)$, that is\nfor any polynomial P in 2t noncommutative indeterminates,\\\\\n\n$\\frac{1}{n} \\Tr P\\left( A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)$ $$ \\rightarrow_{n\\rightarrow +\\infty} \\tau \\left( P(a_1,\\ldots,a_t,a_1^*,\\ldots,a_t^*\\right)$$ and \\\\\n\n$\\left\\|P\\left( A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)\\right\\| $ $$\\rightarrow_{n\\rightarrow +\\infty} \\left\\| P(a_1,\\ldots,a_t,a_1^*,\\ldots,a_t^*)\\right\\|_{\\cal A}.$$\n Then, almost surely, for any  polynomial $P$ \\hspace*{-0.1cm}in $r+2t $ \\hspace*{-0.16cm} noncommutative variables, \n$$ \\lim_{n\\rightarrow +\\infty} \\frac{1}{n}\\Tr P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right) $$\\begin{equation}\\label{af} =\\tau \\left( P\\left(x_1,\\ldots,x_r, a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*\\right) \\right)\\end{equation} and \n$$\\lim_{n\\rightarrow +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)},(A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right) \\right\\|$$\\begin{equation} \\label{saf} =\\left\\| P\\left(x_1,\\ldots,x_r, a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*\\right) \\right\\|_{\\cal A}.\\end{equation}\n\\end{theoreme}\n\\begin{remarque}{ Note that it is sufficient to prove Theorem \\ref{noeigenvalue} and  Theorem \\ref{thprincipal} for Hermitian matrices $A_n^{(1)},\\ldots,A_n^{(t)}$ by considering their Hermitian and anti-Hermitian parts, so that throughout the paper we assume that the $A_n^{(i)}$'s are Hermitian.}\\end{remarque}\n\n\n\\noindent We adopt the   strategy  from  \\cite{HT} and \\cite{Schultz05} based on a linearization trick and  sharp estimates on matricial Stieltjes transforms. More precisely,  both proofs of Theorem \\ref{thprincipal} and Theorem \\ref{noeigenvalue} are based on  the following key Lemma \\ref{inclu2}.  First, note that the algebra $M_m(\\C)\\otimes {\\cal A}$ formed by the $m\\times m $ matrices with coefficients in ${\\cal A}$, inherits the structure of ${\\cal C}^*$-probability space with trace $\\frac{1}{m} \\Tr_m \\otimes \\tau$ and norm \n$$\\Vert b \\Vert = \\lim_{k\\rightarrow +\\infty} \\left( \\frac{1}{m} \\Tr_m \\otimes \\tau \\left[(b^*b)^k\\right]\\right)^{\\frac{1}{2k}}, \\; \\forall b \\in M_m(\\C)\\otimes {\\cal A} .$$\n\n\n\\begin{lemme}  \\label{inclu2} Let $({\\cal A},  \\tau)$ be a ${\\cal C}^*$-probability space\n equipped with a faithful tracial state and \n $x=(x_1,\\ldots,x_r)$ be  a semi-circular system in  $({\\cal A},  \\tau)$. Let $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint  random variables which is free from $x$ in $({\\cal A},\\tau$), such that the distribution of  $a_n$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}), \\frac{1}{n}\\Tr_n)$. Then, for all $m \\in \\N$, all self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$ of size $m\\times m$ and\nall $\\epsilon >0$, almost surely, for all large $n$, we have\\\\\n\n\\noindent $\nsp(\\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}+  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})$ \\begin{equation} \\label{spectre3} \\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\n Here, $sp(T)$ denotes the spectrum of the operator $T$, $I_n$ the identity matrix and $1_{\\cal A}$ denotes the unit of ${\\cal A}$.\n\\end{lemme}\n\n\\begin{remarque}\\label{remarqueinversible}\nBy a density argument, it is sufficient to prove Lemma \\ref{inclu2} for invertible  self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$. This invertibility will be used in the proof of Lemma \\ref{inversion}.\n\\end{remarque}\n The proof of \\eqref{spectre3} requires sharp estimates of  $g_n(z)-\\tilde g_n(z)$\nwhere for $z\\in \\mathbb{C}\\setminus \\mathbb{R}$, $$g_n(z) =\\mathbb{E} \\frac{1}{m} \\Tr_m \\otimes \\frac{1}{n} \\Tr_n [ (zI_m \\otimes I_n - \\gamma \\otimes I_n - \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}(\\omega)-  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})^{-1}]$$ and $$\\tilde g_n(z) = \\frac{1}{m} \\Tr_m \\otimes \\tau [ (zI_m \\otimes 1_{\\cal A} - \\gamma \\otimes 1_{\\cal A} - \\sum_{v=1}^r \\alpha_v \\otimes x_v - \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)})^{-1}].$$\n More precisely we are going to establish that \n there exists a polynomial $Q$ with nonnegative coefficients such that, for $z \\in \\C \\setminus \\mathbb{R}$, \n\\begin{equation} \\label{estimdiffeqno}\n\\left|g_n(z)-\\tilde g_n(z)+{\\tilde{E}_n(z)}\\right|\\leq  \\frac{Q(\\vert \\Im  z\\vert^{-1})}{n\\sqrt{n}},\n\\end{equation}\nwhere $\\tilde{E}_n$ is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$ and such that $\\nabla_n(1)=0$.\\\\\n But this required sharp estimate  makes necessary a technical piece of work and  a fit  use of free operator-valued subordination maps (see Section \\ref{free}). In particular, we need an explicit development of the Stieltjes transform up to the order $\\frac{1}{n\\sqrt{n}}$ but the stability under perturbation argument used in \\cite{CamilleM} does not provide  this development from the approximate matricial subordination equation. Therefore we use a strategy based on an invertibility property of   matricial subordination maps related to semi-circular system (see Lemma \\ref{inversion}).\\\\\n \n Theorem \\ref{thprincipal} can be deduced from Lemma \\ref{inclu2} by  following the proofs in  \\cite{HT} and \\cite{Schultz05}. \nGiven a noncommutative polynomial $P$, choosing in Lemma \\ref{inclu2} the $\\gamma$, $(\\alpha_v)_{v=1,\\ldots,r}$, $(\\beta_u)_{u=1,\\ldots,t}$ corresponding to a self-adjoint linearization of $P$ as defined in \\cite{A} allows to deduce  Theorem \\ref{noeigenvalue}. \\\\\n\n\nIn  Section \\ref{troncation}, we explain why, using a truncation and Gaussian convolution procedure, it is sufficient  to prove  Theorem \\ref{thprincipal}  and Theorem \\ref{noeigenvalue}\n when we assume that the $X_{ij}^{(v)}$'s satisfy:\n\\begin{itemize} \n\\item[(H)] the  variables $\\sqrt{2}\\Re  X_{ij}^{(v)}$, $\\sqrt{2}\\Im   X_{ij}^{(v)}$, $1\\leq i<j$, $X_{ii}^{(v)}$, $i\\geq1$, $v=1,\\ldots,r$, are independent, centred with variance 1 and satisfy a Poincar\\'e inequality with common constant  $C_{PI}$.\n\\end{itemize}\n\n\n\\noindent Note that, according to Corollary 3.2 in \\cite{L},  (H) implies that  for any $p\\in \\mathbb{N}$,  \\begin{equation}\\label{moments}\\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j\\geq 1} \\mathbb{E}\\left(\\vert  X_{ij}^{(v)}\\vert^p\\right) <+\\infty.\\end{equation}\n\n\\noindent We also explain how the proof of Theorem \\ref{thprincipal} \n may be  reduced to prove that\nfor any  polynomial $P$ in $r+t $ noncommutative variables, almost surely, \n\\begin{equation} \\label{safbis}\\lim_{n\\rightarrow +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}\\right) \\right\\| =\\left\\| P\\left(x_1,\\ldots,x_r, a_1,\\ldots,a_t\\right) \\right\\|_{\\cal A}.\\end{equation}\nSuch a truncation and Gaussian convolution procedure also allows us  to relax the assumption of uniform boundness of moments of entries of Wigner matrices in the almost sure asymptotic freeness result of  Theorem 5.4.5 in \\cite{AGZ} (see Proposition \\ref{sansmoment} below). \nNote that Proposition \\ref{sansmoment} does not need  the independence  of  real and imaginary parts of the entries of the Wigner matrices or  the strong convergence of $\\{A_n^{(1)},\\ldots, A_n^{(t)}\\}$.\n\nIn Section \\ref{Notations}, we introduce numerous notations and objects  that will be used in the proofs. Section \\ref{free} provides required preliminaries on free  operator-valued subordination  properties.  The main section is Section \\ref{lemmefonda} where \nwe establish Lemma \\ref{inclu2}.\n     In Sections \\ref{sectionnoeigenvalue} and \\ref{strategie}, we show how to deduce Theorem \\ref{noeigenvalue} and Theorem \\ref{thprincipal} respectively from Lemma \\ref{inclu2}.\n We end with an Appendix which gathers several useful basic algebraic  lemmas and variance estimates.\n\\section{Truncation and Gaussian convolution}\\label{troncation}\nWe start with a slightly modified version of Bai-Yin's theorem  (see \\cite{BaiYin} for the symmetric case and Theorem 5.1 in \\cite{BaiSil06} for the Hermitian case).\n\\begin{lemme}\\label{baiyin}\n\nLet $Z_n =  [Z_{ij}]_{i,j=1}^n$ be a Hermitian $n\\times n$ matrix such that \n$( Z_{ij})_{i\\leq  j}$,  are independent, centred random variables  such that \n$$\\sup_{1\\leq i<j}\\mathbb{E}(\\vert Z_{ij}\\vert^3)<+\\infty $$  \nand there exists a real number $K>0$ and a nonnegative random variable $Y$ with finite fourth moment for which there exists $x_0>0$ and an  integer number $n_0>0$ such that, for any $x >x_0$ and any integer number $n >n_0$, we have \\begin{equation}\\label{majquatreZ}\\frac{1}{n^2} \\sum_{1\\leq i\\leq n,1\\leq  j\\leq n}\\mathbb{P}\\left( \\vert Z_{ij}\\vert >x\\right) \\leq K \\mathbb{P}\\left( Y >x\\right).\\end{equation}\nThen, setting $\\sigma^*=\\{\\sup_{1\\leq i<j}\\mathbb{E}(\\vert Z_{ij}\\vert^2)\\}^{1\/2},$\nwe have that almost surely $$\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{Z_n}{\\sqrt{n}}\\right\\| \\leq 2\\sigma^*.$$\n\\end{lemme}\n\\begin{proof} \nNoting that, for any nonnegative increasing sequence $ (u_l)_{l\\geq 1}$, we have \n $$\\mathbb{E}\\left( \\left(\\frac{Y}{\\sigma^*}\\right)^4\\right)=4 \\int_{0}^{+\\infty} t^3 \\mathbb{P} \\left( \\frac{Y}{\\sigma^*}>t\\right) dt \\geq \\sum_{l=1}^{+\\infty} u_l^3 (u_{l+1}-u_l) \\mathbb{P} \\left( \\frac{Y}{\\sigma^*}>u_{l+1}\\right), $$\n it readily follows  that for any $\\delta>0$, choosing $u_l= \\delta 2^{(l-1)\/2}$, we have\n $$\\sum_{l=1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\delta 2^{l\/2}\\right) <\\infty.$$\n In particular, there exists an increasing sequence $(N_k)_{k\\geq 1}$ of integer numbers such that \n for any $k \\geq 1$, $$\\sum_{l=N_k+1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\frac{1}{2^{\\frac{k}{8}}} 2^{l\/2}\\right) \\leq \\frac{1}{2^k}.$$\n Set for any $l \\in ]0,N_{1}]$, $\\epsilon_l={1}$ and for any $l \\in   ]N_k,N_{k+1}]$, $\\epsilon_l=\\frac{1}{2^{\\frac{k}{8}}}$. Then, \n $$\\sum_{l=N_1 +1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\epsilon_l 2^{l\/2}\\right)= \\sum_{k=1}^{+\\infty}\\sum_{l=N_k+1}^{N_{k+1}} 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\frac{1}{2^{\\frac{k}{8}} }2^{l\/2}\\right)\\leq \\sum_{k=1}^{+\\infty}\\frac{1}{2^k} <\\infty.$$\n Define $\\delta_n= \\sqrt{2}\\epsilon_l$ for $2^{l-1} < n \\leq 2^{l}$. \n Thus, we exhibited a sequence of nonnegative numbers such that $\\delta_n \\downarrow 0$,\n \n   $\\delta_n^2 \\sqrt{n} \\rightarrow +\\infty$ and  $$\\sum_{n=1}^\\infty 2^{2n} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\delta_{2^n} 2^{(n-1)\/2}\\right) <\\infty.$$\nLet us consider $X=Z_n\/\\sigma^*$.\nDefine for any $i\\geq 1, j\\geq 1$,  $\\check X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} $.\nWe have for any $k$ large enough, \n \\begin{eqnarray*}\n\\mathbb{P}\\left(X\\neq \\check X \\; i.o \\right) &\\leq & \\sum_{l=k}^\\infty \\mathbb{P} \\left( \\bigcup_{2^{l-1}< n \\leq  2^{l}} \\bigcup_{1\\leq i,j\\leq n} \\left\\{\\vert X_{ij} \\vert > \\sqrt{n}\\delta_n\\right\\} \\right)\\\\\n &\\leq & \\sum_{l=k}^\\infty \\mathbb{P} \\left( \\bigcup_{2^{l-1}< n \\leq  2^{l}} \\bigcup_{1\\leq i,j\\leq n} \\left\\{\\vert X_{ij} \\vert > 2^{\\frac{l-1}{2}}\\delta_{2^l}\\right\\} \\right)\\\\\n &\\leq & \\sum_{l=k}^\\infty \\sum_{1\\leq i,j\\leq 2^{l}}  \\mathbb{P} \\left( \\vert X_{ij} \\vert > 2^{\\frac{l-1}{2}}\\delta_{2^l} \\right)\\\\\n  &\\leq & K\\sum_{l=k}^\\infty 2^{2l} \\mathbb{P} \\left( Y > 2^{\\frac{l-1}{2}} \\delta_{2^l } \\sigma^*\\right) \\rightarrow_{k \\rightarrow +\\infty} 0.\n\\end{eqnarray*}\n\n\n\n\\noindent Define for $i\\neq j$, $1\\leq i,j\\leq n,$ $ \\hat X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} $ and for any $i$, $\\hat X_{ii}=0$. Then, we have $\\left\\| \\frac{\\check X}{\\sqrt{n}}- \\frac{\\hat X}{\\sqrt{n}}\\right\\| \\leq \\delta_n\\rightarrow_{n\\rightarrow +\\infty} 0.$\n\\noindent Finally define for $i\\neq j$, $1\\leq i,j\\leq n$, $\\tilde X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} -\\mathbb{E}\\left( X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n}\\right)$ and for any $i$, $\\tilde X_{ii}=0$.\nWe have for large $n$ (denoting by $\\Vert\\cdot\\Vert_2$ the Hilbert-Schmidt norm)\\begin{eqnarray*} \n\\left\\| \\frac{\\hat X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|&\\leq  &\\left\\| \\frac{\\hat X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|_{2}\n\\\\&=&\\left( \\frac{1}{n}\\sum_{i,j=1;i\\neq j }^n \\left|\\mathbb{E}\\left(  X_{ij}{\\1}_{\\vert X_{ij} \\vert > \\sqrt{n} \\delta_n} \\right)\\right|^2\\right)^{1\/2}\\\\\n&\\leq&\\left( \\frac{1}{n^3 \\delta_n^4}\\sum_{i,j=1}^n \\mathbb{E}\\left( \\left| X_{ij}^2\\right| \\right)\\mathbb{E}\\left( \\left| X_{ij}\\right|^4{\\1}_{\\vert X_{ij} \\vert > \\sqrt{n} \\delta_n} \\right)\\right)^{1\/2}\\\\\n&\\leq &\\left( \\frac{K}{n \\delta_n^4}\\mathbb{E}\\left( \\left| \\frac{Y}{\\sigma^*} \\right|^4{\\1}_{\\vert \\frac{Y}{\\sigma^*} \\vert > \\sqrt{n} \\delta_n} \\right)\\right)^{1\/2} \\rightarrow_{n\\rightarrow +\\infty} 0.\n\\end{eqnarray*}\nThus, we have that almost surely \\begin{equation}\\label{normetilde}\\left\\| \\frac{ X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|\\rightarrow_{n\\rightarrow +\\infty} 0.\\end{equation}\nNote that the entries of $\\tilde X$ satisfy \n\\begin{itemize}\n\\item $\\tilde X_{ii}=0$;\n\\item $\\tilde X_{ij}$, $i<j$, are independent;\n\\item $\\mathbb{E}\\left( \\tilde X_{ij}\\right) =0, \\mathbb{E}\\left( \\vert \\tilde X_{ij}\\vert^2\\right) \\leq 1, \\mbox{\\; for \\;} i\\neq j$;\n\\item $\\vert \\tilde X_{ij}\\vert\\leq 2\\delta_n \\sqrt{n}, \\mbox{\\; for \\;} i\\neq j$;\n\\item $ \\mathbb{E}\\left( \\vert \\tilde X_{ij}\\vert^l\\right) \\leq b \\left(2\\delta_n \\sqrt{n}\\right)^{l-3}, \\mbox{for some constant } b>0 \\mbox{\\;and all \\;} i\\neq j, \\; l\\geq~3$.\n\\end{itemize}\nThen, sticking to the end of  the proof of Theorem 5.1  pages 87-93    in \\cite{BaiSil06}, one can prove that for any even integer $k$, one has \n$$\\mathbb{E}\\left(\\Tr \\left(\\frac{\\tilde X}{\\sqrt{n}}\\right)^k \\right)\\leq n^2\\left[2+ (10(2\\delta_n)^{1\/3} k\/\\log n)^3\\right]^k.$$\nChebychev's inequality yields that for any $\\eta >2$, \n\\begin{equation}\\label{chebychev}\\mathbb{P}\\left(\\left\\|\\frac{\\tilde X}{\\sqrt{n}}\\right\\| >\\eta\\right)\\leq \\frac{1}{\\eta^k}\\mathbb{E}\\left(\\Tr \\left(\\frac{\\tilde X}{\\sqrt{n}}\\right)^k \\right)\n\\leq \nn^2\\left[\\frac{2}{\\eta}+ \\frac{(10 (2\\delta_n)^{1\/3} k\/\\log n)^3}{\\eta}\\right]^k.\\end{equation}\nSelecting the sequence of even integers $k_n=2 \\left\\lfloor \\frac{\\log n }{\\delta_n^{1\/6}}\\right\\rfloor$ with the properties $k_n\/\\log n \\rightarrow +\\infty$ and $k_n \\delta_n^{1\/3}\/\\log n \\rightarrow 0$, we obtain that the right hand side of \\eqref{chebychev} is summable.\nUsing Borel-Cantelli's Lemma, we easily deduce  that almost surely $$\\limsup_{n\\rightarrow +\\infty} \\left\\|\\frac{\\tilde X}{\\sqrt{n}}\\right\\| \\leq 2$$ and then, using \\eqref{normetilde}, that almost surely $$\\limsup_{n\\rightarrow +\\infty} \\left\\|\\frac{ X}{\\sqrt{n}}\\right\\| \\leq 2.$$\n\\end{proof}\nAccording to Lemma \\ref{baiyin}, for any $v=1,\\ldots,r$, almost surely, \n$\\sup_n \\left\\| \\frac{X_n^{(v)}}{\\sqrt{n}} \\right\\| < +\\infty.$ Therefore by a simple approximation argument using polynomials with coefficients in $\\mathbb{Q} +i \\mathbb{Q}$,  to establish Theorem \\ref{thprincipal}, it is sufficient to prove that for any polynomial, almost surely \\eqref{af} and \\eqref{saf} hold. Now, we are going to show that the proof of Theorem \\ref{thprincipal}  can be reduced to the proof of \\eqref{saf} in the  case where the $X_{ij}^{(v)}$'s satisfy $(H)$.\\\\\nLet $X =  [X_{jk}]_{j,k=1}^n$ be a Hermitian $n\\times n$ matrix such  that the  random variables $X_{ii}$,\n$\\sqrt{2} \\Re (X_{ij})$, $\\sqrt{2} \\Im (X_{ij}), {i<j},$ are independent, centred  with variance 1 such that \n$$\\theta^*=\\sup_{1\\leq i<j}\\mathbb{E}(\\vert X_{ij}\\vert^3)<+\\infty, $$\nand there exists a $K$ and a random variable $Z$ with finite fourth moment for which there exists $x_0>0$ and  an  integer number $n_0>0$ such that, for\nany $x>x_0$ and  any integer number $n>n_0$, we have \\begin{equation}\\label{majquatre}\\frac{1}{n^2} \\sum_{i\\leq n, j\\leq n}\\mathbb{P}\\left( \\vert  X_{ij}\\vert >x\\right) \\leq K\\mathbb{P}\\left(\\vert Z \\vert>x\\right).\\end{equation}\n\n\n\\noindent Define for any  $C>0$, for any $1\\leq i , j \\leq n$,\n\\begin{eqnarray}Y_{ij}^C &=&\\Re  X_{ij}\\1_{\\vert  \\Re X_{ij} \\vert \\leq C} - \\mathbb{E}\\left( \\Re  X_{ij}\\1_{\\vert \\Re  X_{ij} \\vert \\leq C} \\right) \\nonumber \\\\&&+ \\sqrt{-1} \\left\\{\n\\Im  X_{ij}\\1_{\\vert  \\Im  X_{ij} \\vert \\leq C} - \\mathbb{E}\\left( \\Im  X_{ij}\\1_{\\vert \\Im  X_{ij} \\vert \\leq C} \\right) \\right\\}.\\label{ycdef}\\end{eqnarray}\nWe have \n\\begin{eqnarray*}\\mathbb{E} \\left( \\vert   X_{ij}-Y_{ij}^C\\vert^2 \\right)&=&\\mathbb{E} \\left( \\vert \\Re   X_{ij}\\vert^2 \\1_{\\vert \\Re  X_{ij} \\vert > C} \\right)\n+ \\mathbb{E} \\left( \\vert \\Im  X_{ij}\\vert^2 \\1_{\\vert \\Im  X_{ij} \\vert > C} \\right) \\\\&&\n-\\left\\{ \\mathbb{E} \\left( \\Re  X_{ij} \\1_{\\vert \\Re  X_{ij} \\vert > C} \\right)\\right\\}^2-\\left\\{ \\mathbb{E} \\left( \\Im   X_{ij} \\1_{\\vert \\Im  X_{ij} \\vert > C} \\right)\\right\\}^2\\\\& \\leq & \\frac{\\mathbb{E} \\left(\\vert \\Re  X_{ij} \\vert^3\\right)+\\mathbb{E} \\left(\\vert \\Im  X_{ij} \\vert^3\\right)}{C}\n\\end{eqnarray*}\nso that $$\\sup_{i\\geq 1,j\\geq 1}\\mathbb{E} \\left( \\vert  X_{ij}-Y_{ij}^C\\vert^2 \\right) \\leq \\frac{2\\theta^*}{C}.$$\nAccording to Lemma \\ref{baiyin}, we have almost surely \\begin{equation}\\label{centragebis}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X}{\\sqrt{n}}-\\frac{Y^C}{\\sqrt{n}}\\right\\| \\leq 2\\frac{\\sqrt{2\\theta^*}}{\\sqrt{C}}.\\end{equation}\nNote that \n\\begin{eqnarray*}1 - 2 \\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right) &=& 1-2 \\mathbb{E}   \\left\\{\\left(\\Re   X_{ij} \\1_{\\vert \\Re X_{ij} \\vert \\leq  C} -  \\mathbb{E} \\left(  \\Re    X_{ij} \\1_{\\vert \\Re X_{ij} \\vert \\leq  C}\\right)\\right)^2\\right\\}\n\\\\&=& 2\\left[ \\frac{1}{2} -\\mathbb{E} \\left( \\vert  \\Re  X_{ij}\\vert^2 \\1_{\\vert \\Re X_{ij} \\vert \\leq  C} \\right)\\right] \n+2\\left\\{ \\mathbb{E} \\left( \\Re X_{ij} \\1_{\\vert \\Re  X_{ij} \\vert \\leq  C} \\right)\\right\\}^2\\\\&=& \n2 \\mathbb{E} \\left( \\vert \\Re   X_{ij}\\vert^2 \\1_{\\vert\\Re  X_{ij} \\vert > C} \\right)+ 2\\left\\{\\mathbb{E} \\left(  \\Re  X_{ij} \\1_{\\vert \\Re  X_{ij} \\vert > C}\\right) \\right\\}^2. \\end{eqnarray*}\nso that $$\\sup_{i\\geq 1, j\\geq 1}\\vert 1 - 2\\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right) \\vert \\leq \\frac{4\\theta^*}{C}.$$\nSimilarly $$\\sup_{i\\geq 1,j\\geq 1}\\vert 1 - 2\\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right) \\vert \\leq \\frac{4\\theta^*}{C}.$$\nLet us assume that $C>8\\theta^*.$ Then, we have \n$$\\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right)> \\frac{1}{4} \\; \\mbox{and}\\; \\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right)> \\frac{1}{4}.$$\nNow define \\begin{equation}\\label{defxc}{X}_{ij}^C =\\frac{\\Re  Y_{ij}^C}{\\sqrt{2\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right)}} +\\sqrt{-1}  \\frac{\\Im  Y_{ij}^C}{\\sqrt{2\\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right)}}.\\end{equation}\nNote that \\begin{eqnarray*} {X}_{ij}^C-Y_{ij}^C&=& \\Re  X_{ij}^C \\left( 1-\\sqrt{2}  \\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right)^{1\/2}\\right)\n+\\sqrt{-1} \\Im  X_{ij}^C \\left( 1-\\sqrt{2}  \\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right)^{1\/2}\\right)\n\\\\&=&\\Re  X_{ij}^C\\frac{1 - 2\\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right) }{1 + \\sqrt{2}\\mathbb{E} \\left( \\vert \\Re  Y_{ij}^C\\vert^2 \\right)^{1\/2}}+\n\\sqrt{-1} \\Im  X_{ij}^C\\frac{1 - 2\\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right) }{1 + \\sqrt{2}\\mathbb{E} \\left( \\vert \\Im  Y_{ij}^C\\vert^2 \\right)^{1\/2}}.\\end{eqnarray*} \n\\noindent Thus, according to Lemma \\ref{baiyin}, we have almost surely \\begin{equation}\\label{normal}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X^C}{\\sqrt{n}}-\\frac{Y^C}{\\sqrt{n}}\\right\\| \\leq \\frac{{8\\theta^*}}{{C}}.\\end{equation}\nThus, by \\eqref{centragebis} and \\eqref{normal}, we obtain that almost surely \\begin{equation}\\label{xc}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X}{\\sqrt{n}}-\\frac{X^C}{\\sqrt{n}}\\right\\| \\leq 2\\frac{\\sqrt{2\\theta^*}}{\\sqrt{C}}+ \\frac{{8\\theta^*}}{{C}}.\\end{equation}\nLet $[{\\cal G}_{ij}]_{i\\geq 1, j\\geq 1}$ be an infinite array which is  independent of the $X_{ij}'s$ and such that $\\sqrt{2} \\Re {\\cal G}_{ij}$, $ \\sqrt{2} \\Im {\\cal G}_{ij}$, $i<j$, \nand  ${\\cal G}_{ii}$ are independent centred standard real Gaussian variables and for any $i\\geq 1, j\\geq 1$,  ${\\cal G}_{ji}= \\overline{{\\cal G}_{ij}}$ .\nThus,  ${\\cal G} = [{\\cal G}_{ij}]_{ i,j =1}^n $ is  a random G.U.E matrix being independent of $X$. Define \nfor any $\\delta>0$,\n\\begin{equation}\\label{xcdelta}X^{C,\\delta}= \\frac{ X^C +\\delta {\\cal G}}{\\sqrt{1+\\delta^2}}.\\end{equation}\nNote that  the random variables $\\sqrt{2}\\Re  (X^{(v)})^{C,\\delta}_{ij}$, $\\sqrt{2}\\Im   (X^{(v)})^{C,\\delta}_{ij}$, $ i<j$, $(X^{(v)})_{ii}^{C,\\delta}$ are independent, centred with variance 1.\\\\\nFrom Bai-Yin's theorem  (Theorem 5.8 in \\cite{BaiSil06}),\nwe have that almost surely, \\begin{equation}\\label{by}\\left\\| \\frac{{\\cal G}}{\\sqrt{n}} \\right\\|=2+o(1)\\end{equation} and by  Lemma  \\ref{baiyin},\n for any $C > 8\\theta^*$, we have almost surely\n\\begin{equation}\\label{bybis} \\limsup_{n\\rightarrow+\\infty}\\left\\| \\frac{{X^C}}{\\sqrt{n}} \\right\\|\\leq 2.\\end{equation}\nNow, \\eqref{xc}, \\eqref{by}  and \\eqref{bybis} yield that for any $C > 8 \\theta^*$, any $\\delta >0$, almost surely   $ \\limsup_{n\\rightarrow +\\infty}\\left\\|\\frac{X -{X^{C,\\delta}}}{\\sqrt{n}} \\right\\| \\leq u_C +v_\\delta$ where $u_C$ and $v_\\delta$ are deterministic positive functions tending to zero when respectively $C$ goes to infinity and $\\delta$ goes to zero. \nHence, it is easy to see that for any $0<\\epsilon<1$, there exists $C_\\epsilon$ and $\\delta_\\epsilon$ such that almost surely for all large $n$, \n$$\\left\\|\\frac{X -{X^{C_\\epsilon,\\delta_\\epsilon}} }{\\sqrt{n}}\\right\\|\\leq \\epsilon.$$\nWe can deduce that for any  polynomial $P$ in $r+t$ noncommutative variables, there exists some constant $L>0$ such that the following holds: for any $0<\\epsilon<1$, there exists $C_\\epsilon$ and $\\delta_\\epsilon$ such that almost surely for all large $n$, \\\\\n\n\\noindent $\\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}\\right)\\right. $ \\begin{equation}\\label{approxnorme} \\left.- P\\left(\\frac{(X_n^{(1)})^{C_\\epsilon,\\delta_\\epsilon}}{\\sqrt{n}},\\ldots,\\frac{(X_n^{(r)})^{C_\\epsilon,\\delta_\\epsilon}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}\\right)\\right\\|\\leq L \\epsilon.\\end{equation} \nThen, it is clear that it is sufficient to establish Theorem \\ref{thprincipal}  and Theorem \\ref{noeigenvalue} for the $(X_n^{(v)})^{C_\\epsilon,\\delta_\\epsilon}$'s. \\\\Moreover, we obviously have that for any $\\epsilon$ and  for any $p\\in \\mathbb{N}$,  \\begin{equation}\\label{moments}\\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j\\geq 1} \\mathbb{E}\\left(\\vert  (X_{ij}^{(v)})^{C_\\epsilon,\\delta_\\epsilon}\\vert^p\\right) <+\\infty.\\end{equation}\n Then, \\eqref{af}  is a consequence of Theorem 5.4.5 in \\cite{AGZ}.\\\\\n\n\n Moreover, note that, by definition,   the distributions of the random variables\n$\\sqrt{2}\\Re  (X^{(v)})_{ij}^{C_\\epsilon,\\delta_\\epsilon}, \\sqrt{2}\\Im  (X^{(v)})_{ij}^{C_\\epsilon,\\delta_\\epsilon},  i<j,  ( X^{(v)})_{ii}^{C_\\epsilon,\\delta_\\epsilon}$,  $v=1\\ldots,r$, are all    a convolution of a centred   Gaussian distribution with variance $\\delta_\\epsilon^2\/(1+\\delta_\\epsilon^2)$,  with  some  law with bounded support\nin the ball of radius $2C_\\epsilon$; thus, according to Lemma \\ref{zitt},    \nthey satisfy a  Poincar\\'e inequality with some common  constant $ C_{PI}(C_\\epsilon,\\delta_\\epsilon)$.\\\\\n\n\n{\\it Hence in the following we will focus on establishing Theorem \\ref{noeigenvalue} and \\eqref{safbis} when  the $X_{ij}^{(v)}$'s satisfy  (H).}\n\\\\\n\nNote that a similar truncation and Gaussian convolution procedure also allows to establish the following asymptotic freeness result.\n\\begin{proposition}\\label{sansmoment}\nAssume that $(A_n^{(1)},\\ldots,A_n^{(t)})$ is a $t-$tuple of  $n\\times n$   Hermitian deterministic matrices   such that\n $\\max_{u\\in\\{1,\\ldots,t\\}} \\sup_n \\Vert A_n^{(u)} \\Vert< \\infty$ and  \n $(A_n^{(1)},\\ldots,A_n^{(t)})$ converges  in distribution in $(M_n(\\C), \\frac{1}{n} \\Tr_n )$ towards some t-tuple of noncommutative random variables $(a_1,\\ldots a_t)$ in some noncommutative probability space $({\\cal A},\\tau)$.\n  Let  $X_n^{(v)} =  [X^{(v) }_{ij}]_{i,j=1}^n$, $v=1,\\ldots,r$, be  r  independent $n\\times n$ random  Hermitian matrices such that, for each $v$,\n$ [X^{(v)  }_{ij}]_{i\\geq1,j\\geq 1}$ is an infinite array of  random variables such that $X^{(v)}_{ii}$,\n$X^{(v)}_{ij}, i<j$,  are independent, centred and $\\mathbb{E}(\\vert X_{ij}\\vert^2)=1$. Let  $(x_1,\\ldots,x_r)$ be  a semi-circular system in  $({\\cal A},  \\tau)$ which is free from $(a_1,\\ldots a_t)$.\nThen, $\\left( X^{(1)},\\ldots, X^{(r)}, A_n^{(1)},\\ldots,A_n^{(t)}\\right)$ converges in distribution in $(M_n(\\C), \\frac{1}{n} \\Tr_n )$ towards\n$(x_1,\\ldots,x_r, a_1,\\ldots a_t)$ \\end{proposition}\n\\begin{proof}\nIn the arguments above the present proposition, one  can consider $$\\tilde Y_{ij}^C =X_{ij}\\1_{\\vert X_{ij} \\vert \\leq C} - \\mathbb{E}\\left( X_{ij}\\1_{\\vert X_{ij} \\vert \\leq C} \\right),\\mbox{~~and~~}\\tilde{X}_{ij}^C =\\frac{\\tilde Y_{ij}^C}{\\sqrt{\\mathbb{E} \\left( \\vert \\tilde Y_{ij}^C\\vert^2 \\right)}},$$\ninstead of $Y^C$ in \\eqref{ycdef} and $X^C$ in \\eqref{defxc} and similarly  obtain that almost surely \\begin{equation}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X}{\\sqrt{n}}-\\frac{\\tilde X^C}{\\sqrt{n}}\\right\\|\\leq 2\\frac{\\sqrt{\\theta^*}}{\\sqrt{C}}+ \\frac{{4\\theta^*}}{{C}}.\\end{equation}\nThen,  one can similarly prove that \\eqref{approxnorme} holds with \n$$\\tilde X^{C,\\delta}=\\frac{ \\tilde X^C +\\delta {\\cal G}}{\\sqrt{1+\\delta^2}}$$ instead of $X^{C,\\delta}$ in \\eqref{xcdelta}, so that the result  follows from Theorem 5.4.5 in \\cite{AGZ}.\n\\end{proof}\n\\section{Notations }\\label{Notations} \nThe main part of the following  consists in proving Lemma \\ref{inclu2}. For that purpose, we fix some integer number $m$ and Hermitian $ m\\times m$ matrices $\\{\\alpha_v\\}_{v=1,\\ldots,r}, \\{\\beta_u\\}_{u=1,\\ldots,t}, \\gamma$.\nTo begin with, we introduce\nsome notations on the set of matrices.\n\\begin{itemize}\n\\item $M_p(\\C)$ is the set of $p\\times p$ matrices  with complex entries,  $M_p(\\C)_{sa}$ the subset of self-adjoint elements of $M_p(\\C)$ and $I_p$ the identity matrix. In the following, we shall consider two sets of matrices with $p=m$ ($m$ fixed) and $p=n$ with $n\\vers \\infty$.\n\\item $\\Tr_p$ denotes the trace and $\\tr_p = \\frac{1}{p} \\Tr_p$ the normalized trace on $M_p(\\C)$.\n\\item $|| . ||$ denotes the spectral norm on $M_p(\\C)$ and $||M||_2 = (\\Tr_p (M^*M))^{1\/2}$ the Hilbert-Schmidt norm.\n\\item ${\\rm id}_p$ denotes the identity operator from $M_p(\\C)$ to $M_p(\\C)$.\n \\item  Let $(E_{ij})_{i,j =1}^n$ be the canonical basis of $M_n(\\C)$ and define a basis of the real vector space of the self-adjoint matrices $M_n(\\C)_{sa}$ by:\n \\begin{eqnarray*}\n e_{jj} &= &E_{jj}, 1\\leq j \\leq n \\\\\n e_{jk}& = &\\frac{1}{\\sqrt{2}}(E_{jk} + E_{kj}), 1 \\leq j < k \\leq n \\\\\n f_{jk} &=& \\frac{\\sqrt{-1}}{\\sqrt{2}}(E_{jk} - E_{kj}) , 1 \\leq j < k \\leq n\n \\end{eqnarray*}\n \\item[-]  $(\\hat{E}_{ij})_{i,j =1}^m$ is the canonical basis of $M_m(\\C)$.\n \\item[-] For a matrix $M$ in $M_m(\\C) \\otimes M_n(\\C)$, we set\n $$M_{ij} : = ({\\rm id}_m \\otimes \\Tr_n)(M (1_m \\otimes E_{ji})) \\in M_m(\\C), \\ 1\\leq i,j \\leq n.$$\n\\item[-] For any matrix $M$ (or, more generally, any operator $M$),\nwe define its real part to be $\\Re  M=(M+M^*)\/2$ and its imaginary part to be $\\Im  M=(M-M^*)\/2i$.\nWe write $M>0$ if the matrix $M$ is positive definite and $M\\ge0$ if it is nonnegative definite.\nIn general $M>P$ means that $M-P$ is positive definite.\n  \\end{itemize}\n\\noindent We now define   the random variables of interest. \nLet $({\\cal A},  \\tau)$ be a ${\\cal C}^*$-probability space with unit $1_{\\cal A}$,\n equipped with a faithful tracial state and \n $x=(x_1,\\ldots,x_r)$ be  a semi-circular system in  $({\\cal A},  \\tau)$.\n Let $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint  random variables which is free from $x$  in $({\\cal A},\\tau$) and  such that the distribution of  $a_n$ in $({\\cal A},\\tau)$  coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\n\\begin{itemize}\n\n \\item[-] \n We define the random variable $S_n$ with values in $M_m(\\C) \\otimes M_n(\\C)$ by:\n \\begin{equation} \\label{defSn}\n S_n = \\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v  \\otimes \\frac{X^{(v)}_n}{\\sqrt{n}}+ \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\n \\end{equation}\nand $s_n\\in M_m(\\C) \\otimes {\\cal A}$ by \n$$s_n=\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r\\alpha_v \\otimes x_v+ \\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}.$$\n \\item[-] For any matrix $\\lambda$ in $ M_m(\\C) $ such that $ \\Im(\\lambda)$ is positive definite,\n  we define the $M_m(\\C)\\otimes M_n(\\C)$-valued respectively $M_m(\\C) \\otimes {\\cal A}$-valued  random variables:\n \\begin{equation}\\label{rn}\n\tR_n(\\lambda)=(\\lambda \\otimes I_n - S_n)^{-1},\n\t\\end{equation}\n\t$$r_{n}(\\lambda)= \\left(\\lambda \\otimes 1_{\\cal A} -s_n \\right)^{-1},$$\n\tand \n\tthe $M_m(\\C)$-valued random variables:\n \\begin{equation} \\label{defSt}\n H_n(\\lambda) = ({\\rm id}_m \\otimes \\tr_n) [ (\\lambda \\otimes I_n - S_n)^{-1}],\n \\end{equation}\n  \\begin{equation} \\label{espSt}\nG_n(\\lambda) =\\mathbb{ E}[ H_n(\\lambda) ]\n\\end{equation}\nand\n\\begin{equation}\\label{defGntilde}\\tilde G_{n}(\\lambda)={\\rm id}_m \\otimes \\tau \\left(r_n (\\lambda ) \\right).\\end{equation}\nSince $\\sum_{v=1}^r\\alpha_v \\otimes x_v$ is an $ M_m(\\mathbb C)$-valued semicircular of variance $\\eta\\colon b\\mapsto\\sum_{v=1}^r\\alpha_vb\\alpha_v$ which is free over $ M_m(\\mathbb C)$ from $\\gamma\\otimes 1_{\\cal A}+\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$, we know from \\cite{ABFN} (see proof of Theorem 8.3) that $\\tilde G_{n}$ satisfies (see Section \\ref{free} Theorem \\ref{resusub} for $p=1$) \\begin{equation}\\label{subor}\\tilde G_{n}(\\lambda)= G_{\\sum_{u=1}^t\\beta_s \\otimes a_n^{(u)}}(\\omega_n(\\lambda))\n\\end{equation}\nwhere \\begin{equation}\\label{omegan}\n\\omega_n(\\lambda)=\\lambda-\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde  G_n(\\lambda)\\alpha_v\n\\end{equation} and $$G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}(\\lambda)={\\rm id}_m \\otimes \\tau \\left(\\lambda\\otimes 1_{\\cal A} -\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)} \\right)^{-1}.$$\n\\noindent For $z\\in \\C \\setminus  \\R$, we also define\n \\begin{equation}\\label{defpetitg}g_n(z) = \\tr_m(G_n(zI_m))\\end{equation}and \\begin{equation}\\label{defpetitgtilde}\\tilde g_n(z) = \\tr_m(\\tilde G_n(z I_m)).\\end{equation}\n\n\n\n\n\n\n\n\n\n\n \n\n \n  \\end{itemize}\n\n In the sequel, we will say that a random  term in some $M_p(\\C)$, depending on $n$, $\\lambda \\in M_m(\\mathbb{C})  $ such that $\\Im  \\lambda$ is positive definite,  the $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$ and  $\\gamma$, is  $O\\left(\\frac{1}{n^k}\\right)$ if its  operator  norm  is smaller than\n $\\frac{ Q\\left(\\Vert (\\Im \\lambda)^{-1} \\Vert\\right)}{n^k}$ for some deterministic  polynomial $Q$\n whose coefficients are nonnegative real numbers  and can depend on $m$,  $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$, $\\gamma$.\\\\\n For a family of random terms $I_{pq}$, $(p,q) \\in \\{1,\\ldots,n\\}^2$, we will set $I_{pq}=O_{p,q}^{(u)} \\left(\\frac{1}{n^k}\\right)$ if for each $(p,q)$, $I_{pq}=O \\left(\\frac{1}{n^k}\\right)$ and moreover one can find a bound of \n\nthe norm of each $I_{p,q}$  as above involving a common polynomial $Q$.\\\\\n\nThroughout the paper, $K$, $C$ denote some positive constants and  $Q$  denotes some deterministic  polynomial in one variable\n whose coefficients are nonnegative real numbers; they can depend on $m$,   $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$ and $\\gamma$ and they  may  vary from line to line.\n\n\n\n\n\n\n\n\n\n\\section{Operator-valued subordination}\\label{free}\n\n\nIn this section we introduce one of the main tools used in describing joint distributions of\nrandom variables that do not necessarily commute. It was a crucial insight of Voiculescu\n\\cite{V2000, FreeMarkov, V1} that J. L. Taylor's theory of free noncommutative functions \\cite{taylor} \nprovides the appropriate analogue of the classical Stieltjes transform for encoding\noperator-valued distributions \\cite{V1995}, and hence joint distributions of $q$-tuples of\nnoncommuting random variables. We shall only present below the case of relevance to our\npresent work, and refer the reader to \\cite{V1995,V2000,V1} for the general case and for the\nproofs of the main results.\n\nGiven a tracial $\\mathcal C^*$-probability space $(\\mathcal A,\\tau)$ and a trace and order preserving\nunital $\\mathcal C^*$ inclusion $M_m(\\mathbb C)\\subseteq\\mathcal A$ (i.e. such that $I_m\\in \nM_m(\\mathbb C)$ identifies with the unit of $\\mathcal A$ and ${\\rm tr}_m(b)=\\tau(b)$ for all $b\\in \nM_m(\\mathbb C)$), there exists a conditional expectation $E\\colon\\mathcal A\\to M_m(\\mathbb C)$, i.e. \na linear map sending the unit to itself and such that $E(b_1yb_2)=b_1E(y)b_2$ for all $b_1,b_2\\in M_m(\\mathbb C),y\n\\in\\mathcal A$ - see \\cite[Section II.6.10.13]{Bruce}. We will only be concerned with the trivial case\nof the canonical inclusion $M_m(\\mathbb C)\\subseteq M_m(\\mathbb C)\\otimes\\mathcal A$ \ngiven by $b\\mapsto b\\otimes 1_{\\cal A}$, when the conditional expectation is the partial trace:\n$E(b\\otimes y)=({\\rm id}_m\\otimes\\tau)(b\\otimes y)=\\tau(y)b$.\nThe $M_m(\\mathbb C)$-valued distribution of an element $y\\in\\mathcal A$ with respect to $E$ is by \ndefinition the family of multilinear maps \n$\\mu_y=\n\\{\\Psi_q\\colon\\underbrace{M_m(\\mathbb C)\\times\\cdots\\times M_m(\\mathbb C)}_{q-1\\text{ times}}\n\\to M_m(\\mathbb C)\\colon \\Psi_q(b_1,\\dots,b_{q-1})=E[yb_1y\\cdots b_{q-1}y],q\\in\\mathbb N\\}$. \nBy convention, $\\Psi_0=1\\in M_m(\\mathbb C)$, $\\Psi_1=E[y]$.\n\nFor a given $y=y^*\\in\\mathcal A$ with distribution $\\mu_y$, define its noncommutative \nStieltjes transform to be the {\\em countable family} of maps\n$G_{\\mu_y,p}(b)=(E\\otimes{\\rm id}_{p})\\left[(b-y\\otimes I_p)^{-1}\\right], p\\in \\mathbb{N}\\setminus\\{0\\}.$\nThus, $G_{\\mu_y,1}(b)=E\\left[(b-y)^{-1}\\right],b\\in M_m(\\mathbb C)$,\n$G_{\\mu_y,2}(b)=(E\\otimes{\\rm id}_{2})\\left[\\left(\\begin{bmatrix}\nb_{11} & b_{12}\\\\\nb_{21} & b_{22}\n\\end{bmatrix}-\\begin{bmatrix}\ny & 0\\\\\n0 & y\n\\end{bmatrix}\\right)^{-1}\\right],$ $b_{11}, b_{12},\nb_{21}, b_{22}\\in M_m(\\mathbb C)$ etc. These maps are clearly analytic on the open sets $\\{b\\in \nM_m(\\mathbb C)\\otimes M_p(\\mathbb C)\\colon b-y\\otimes I_p\\text{ invertible in }\\mathcal A\\otimes M_p(\\mathbb C)\\}$. Two \nsuch sets will be important in this paper: the noncommutative upper half-plane $H^+_p(M_m(\\mathbb \nC))=\\{b\\in M_m(\\mathbb C)\\otimes M_p(\\mathbb C)\\colon \\Im  b:=(b-b^*)\/2i>0\\}$, $p\\in \\mathbb{N}\\setminus\\{0\\}$, \nand the ``ball around infinity'' $\\{b\\in M_m(\\mathbb C)\\otimes M_p(\\mathbb C) \\colon b\\text{ invertible, \n}\\|b^{-1}\\|<\\|y\\|^{-1}\\}$ (actually only for $p=1$). The maps $b\\mapsto G_{\\mu_y,p}(b^{-1})$ have thus an analytic extension \naround zero, which maps zero to zero and has the identity as first (Frechet) derivative. While \n$G_{\\mu_y,p}$ does not map these ``balls around infinity'' into themselves, it does map \n$H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$, and moreover $G_{\\mu_y,p}(b)^{-1}$ maps \n$H^+_p(M_m(\\mathbb C))$ into itself (see \\cite[Section 3.6]{V2000}). (In addition, the family of maps \n$\\{G_{\\mu_y,p}\\}_{p\\in \\mathbb{N}\\setminus\\{0\\}}$ satisfy certain compatibility conditions that make them into free \nnoncommutative maps - see \\cite{KVV}. It is known \\cite{W} that there is a bijection between such families \nof maps $G$ that send for any $p \\in \\mathbb{N}\\setminus\\{0\\}$, $H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$ and have the \nabove-described behavior on ``balls around infinity'' and $M_m(\\mathbb C)$-valued distributions of \nself-adjoint elements; however, since we will not make use of this correspondence, we chose to only \nmention it in order to illustrate the parallel to the case of classical Stieltjes transforms, and direct the \ninterested reader to \\cite{W} for details.)\n\n\n\nAs in scalar-valued free probability, one defines \\cite{V1995} {\\em freeness with amalgamation}\nover $M_m(\\mathbb C)$ via an algebraic relation similar to \\eqref{freeness}, but involving $E$\ninstead of $\\tau$ and noncommutative polynomials with coefficients in $M_m(\\mathbb C)$.\nSince it is not important for us here, we refer the interested reader to \\cite{V1995} for \nmore details. The essential result of Voiculescu that we will need in this paper is the following\nanalytic subordination result:\n\\begin{theoreme}\\label{resusub}\nWith the above notations, assume that $y_1=y_1^*,y_2=y_2^*\\in\\mathcal A$ are free with amalgamation\nover $M_m(\\mathbb C)$. For any $p\\in\\mathbb N$ there exist analytic maps $\\omega_{1,p},\n\\omega_{2,p}\\colon H^+_p(M_m(\\mathbb C))\\to H^+_p(M_m(\\mathbb C))$ such that:\n\\begin{enumerate}\n\\item For all $b\\in H^+_p(M_m(\\mathbb C)),$ $\\Im \\omega_{j,n}(b)\\ge\\Im  b$, $j=1,2$;\n\\item For all $b\\in H^+_p(M_m(\\mathbb C))$\n$$G_{\\mu_{y_1+y_2},p}(b)=G_{\\mu_{y_1},p}(\\omega_{1,p}(b))=G_{\\mu_{y_2},p}(\\omega_{2,p}(b))\n=\\left[\\omega_{1,p}(b)+\\omega_{2,p}(b)-b\\right]^{-1}$$ \n\\item $\\omega_{1,p},\n\\omega_{2,p}$ are noncommutative maps in the sense of $\\cite{taylor}$ $($see $\\cite{KVV})$.\n\\end{enumerate}\n\\end{theoreme}\nThe result as phrased here is a combination of parts of \\cite[Theorem 3.8]{V2000} and\n\\cite[Theorem 2.7]{BMS}. We shall use this theorem in the particular case when $y_2=s$ is a centred \n{\\em operator-valued semicircular} random variable (and actually only for $p=1$). As for the scalar-valued semicircular\ncentred random variables, it is  uniquely determined by its variance $\\eta\\colon b\\mapsto E(sbs)$,\nwhich is a completely positive self-map of $M_m(\\mathbb C)$. A characterization in terms of moments\nand cumulants via $\\eta$ is provided by Speicher in \\cite{SMem}. Given the context of our paper,\nwe find it more useful to provide a characterization in terms of the noncommutative \nStieltjes transform \\cite{HRS}: the functions $G_{\\mu_s,p}$ are  the unique solutions mapping\n$H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$ of the functional equations\n$$\nG_p(b)^{-1}=b-(\\eta\\otimes{\\rm id}_p)(G_p(b)),\\quad b\\in H^+_p(M_m(\\mathbb C)),\np\\in\\mathbb N.\n$$\nStarting from this equation, it can be shown (see \\cite{ABFN}) that the subordination function associated \nto a semicircular operator-valued random variable is particularly nice: if $y_1$ and $s$ are free with\namalgamation over $M_m(\\mathbb C)$, then \n\\begin{equation}\n\\omega_{1,p}(b)=b-(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1},p}(\\omega_{1,p}(b)))=b-\n(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1+s},p}(b)),\n\\end{equation}\nfor $b\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N,$ or, equivalently,\n\\begin{equation}\nG_{\\mu_{y_1},p}(b-(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1+s},p}(\\omega_{1,p}(b)))=G_{\\mu_{y_1+s},p}(b),\n\\quad b\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n\\end{equation}\nThis indicates that the $\\omega_{1,p}$'s are injective maps on $H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n$ Their left inverses are defined as\n\\begin{equation}\\label{tauto}\n{\\Lambda}_{1,p}(w)=w+(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1},p}(w)),\n\\quad w\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n\\end{equation}\n\nLet us  explain how all this relates to the joint distributions of free\nrandom variables. It turns out (see, for example, \\cite{NSS}, but it can be easily verified directly)\nthat if $\\{x_1=x_1^*,\\dots,x_r=x_r^*\\}, \\{y_1=y_1^*,\\dots,y_t=y_t^*\\}\\subset\\mathcal A$ are free over \n$\\mathbb C$ and for $v=1,\\ldots,r$, $\\alpha_v=\\alpha_v^*$, for $u=1,\\ldots,t$, $\\beta_u=\\beta_u^*\\in M_m(\\mathbb C)$,\nthen $\\{\\alpha_1\\otimes x_1,\\dots,\\alpha_r\\otimes x_r\\}$ and $\\{\\beta_1\\otimes y_1,\\dots,\\beta_t\\otimes y_t\\}$ are free \nwith amalgamation over $M_m(\\mathbb C)$. Thus, they can be treated with the tools described above.\nMoreover, if $x_1,\\dots,x_r$ are free $\\mathbb C$-valued semicircular centred random variables \nof variance one and $\\alpha_1,\\dots,\\alpha_r$ are self-adjoint $m\\times m$ complex matrices, then\n$\\alpha_1\\otimes x_1+\\cdots+\\alpha_r\\otimes x_r$ is a centred $M_m(\\mathbb C)$-valued semicircular of \nvariance $b\\mapsto\\sum_{j=1}^r \\alpha_jb\\alpha_j$. These simple facts together with a linearization trick\n(see Section \\ref{linearisation} and  Step 1 of Section \\ref{strategie}) will allow us in principle to treat, from the point of view of\nthe Stieltjes transform, an $r$-tuple of Wigner matrices and deterministic  matrices as we would treat a \nsingle Wigner matrix together with a single deterministic  matrix. \\\\\n~~\n\n\\noindent Let us conclude this section with the following  invertibility property of   matricial subordination maps related to semi-circular system that will fundamental in our approach.\n\n\\begin{lemme}\\label{inversion}\nUsing the notations of Section \\ref{Notations}, define  for any $\\rho$ in $M_m(\\mathbb{C})$ such that $\\Im  \\rho>0$,\n\\begin{equation}\\Lambda_n(\\rho)= \\gamma +\\rho + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\rho) \\alpha_v.\\end{equation}\nWith $\\omega_n$  defined in \\eqref{omegan},\nwhen $\\Im \\rho>0$ and $\\Im \\Lambda_n(\\rho)>0$ we have \n$$\\omega_n(\\Lambda_n(\\rho))=\\rho.$$\n\\end{lemme}\n\\begin{proof}\nThe equality $\\Lambda_n(\\omega_n(\\rho))=\\rho$ holds tautologically for all $\\rho$ with $\\Im  \\rho>0$ (see \\eqref{tauto}).\nLet us first show that the equality $\\omega_n(\\Lambda_n(\\rho))=\\rho$ holds when $\\rho$ has \na small enough inverse. \nThe map $\\Lambda_n$ has a power series expansion\n$$\n\\Lambda_n(\\rho)=\\rho+\\gamma+\\sum_{v=1}^r\\alpha_v\\left(\n\\sum_{k=0}^\\infty({\\rm id}_m\\otimes\\tau)\n\\left(\\rho^{-1}\\left[\\sum_{u=1}^t(\\beta_u\\otimes a_n^{(u)})\\rho^{-1}\\right]^k\\right)\\right)\\alpha_v,\n$$\nconvergent when $\\|\\rho^{-1}\\|<\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$. For simplicity\nwe let $h(\\lambda)=\\sum_{v=1}^r\\alpha_v\\left(\n\\sum_{k=0}^\\infty({\\rm id}_m\\otimes\\tau)\n\\left(\\lambda\\left[\\sum_{u=1}^t(\\beta_u\\otimes a_n^{(u)})\\lambda\\right]^k\\right)\\right)\\alpha_v,$ norm\nconvergent on a ball of radius $\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$ and fixing zero. \nPerforming the change of variable $\\lambda=\\rho^{-1}$, we obtain $\\Lambda_n(\\rho)=\n\\Lambda_n(\\lambda^{-1})=\\lambda^{-1}+\\gamma+h(\\lambda)$. Then\n$(\\Lambda_n(\\lambda^{-1}))^{-1}=(\\lambda^{-1}+\\gamma+h(\\lambda))^{-1}=\n\\lambda(1+(\\gamma+h(\\lambda))\\lambda)^{-1}$, which is analytic on the set of all $\\lambda\\in\n M_m(\\mathbb C)$ such that $\\|\\lambda\\|<\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$\nand $\\|\\gamma+h(\\lambda)\\|<\\|\\lambda\\|^{-1}.$ \n\nDefine ${\\check\\Lambda_n}(\\rho)=(\\Lambda_n(\\rho^{-1}))^{-1}$ and ${\\check\\omega_n}\n(\\rho)=(\\omega_n(\\rho^{-1}))^{-1}$. We have established above that ${\\check\\Lambda_n}$ is analytic\non a neighbourhood of zero, and a direct computation shows that ${\\check\\Lambda_n}(0)=0,\n{\\check\\Lambda_n}'(0)={\\rm id}$. The inverse function theorem for analytic maps allows us to\nconclude that there exists a neighbourhood of zero on which ${\\check\\Lambda_n}$ has a unique inverse\nwhich fixes zero and whose derivative at zero is equal to the identity. \nThe map ${\\check\\omega_n}$ is shown precisely the same way to satisfy the same properties as\n${\\check\\Lambda_n}$. In particular, for $\\|\\rho\\|$ small enough, \n${\\check\\Lambda_n}({\\check\\omega_n}(\\rho))=(\\Lambda_n({\\check\\omega_n}(\\rho)^{-1}))^{-1}=\n(\\Lambda_n(((\\omega_n(\\rho^{-1}))^{-1})^{-1}))^{-1}=(\\Lambda_n(\\omega_n(\\rho^{-1}))^{-1}=\n(\\rho^{-1})^{-1}=\\rho$ for any $\\rho$ with strictly positive imaginary part. Since zero is in the\nclosure of $\\{\\rho\\in M_m(\\mathbb C)\\colon\\Im  \\rho>0\\}$, it follows that ${\\check\\omega_n}$\nand ${\\check\\Lambda_n}$ are compositional inverses to each other on a small enough neighbourhood of\nzero. We conclude that for all $\\rho$ such that the lower bound of the spectrum of $\\Im  \\rho$ is \nsufficiently large, $\\omega_n(\\Lambda_n(\\rho))=\\rho$.\n\n\nLet now $\\rho$ be fixed in $ M_m(\\mathbb{C})$ such that $\\Im \\rho>0$ and $\\Im \\Lambda_n(\\rho)>0$.\nLet $\\phi$ be  a positive linear functional on $ M_m(\\mathbb{C})$ such that\n$\\phi(1)=1$ (i.e. a state). Define $\\varphi_\\rho(\\cdot )=\\phi(\\cdot )\/\\phi(\\Im \\rho)$. It is linear and \npositive (well defined because $\\Im  \\rho\\ge\\frac{1}{\\|(\\Im  \\rho)^{-1}\\|}1$, so that $\\phi(\\Im  \\rho)\\ge\n\\frac{1}{\\| (\\Im  \\rho)^{-1}\\|}>0$). Define \n$$\nf_\\rho(z)=\\varphi_\\rho\\left(\\Lambda_n(\\Re \\rho+z\\Im \\rho)\\right),\\quad z\\in\\mathbb C^+.\n$$\nNote that $$f_\\rho(z)=z+ \\varphi_\\rho(\\gamma+\\Re \\rho))+F(z)$$\nwhere $$F(z)=\\frac{\\phi\\left[ \\sum_{v=1}^r \\alpha_v {\\rm id}_m\\otimes \\tau \\left\\{\\left( (\\Re  \\rho +z \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\alpha_v\\right] }{\\phi(\\Im \\rho)}.$$\n$F(z)$ is analytic on $\\mathbb{C}\\setminus \\mathbb{R}$ and satisfies $\\overline{F(z)}=F(\\bar{z}).$\nLet $z\\in \\mathbb{C}^+$.\nWe have \\\\\n\n\n\\noindent $\\Im  F(z)$ $$= \\frac{\\phi\\left[ \\sum_{v=1}^r \\alpha_v{\\rm id}_m\\otimes \\tau \\left\\{ \\Im  \\left\\{\\left( (\\Re  \\rho +z \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\} \\right\\}\\alpha_v\\right] }{\\phi(\\Im \\rho)}$$\nwhere \\\\\n\n\\noindent $\\Im  \\left\\{\\left( (\\Re  \\rho +z \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}$ \\begin{eqnarray*}&=&\n-\\Im  z \\left(( \\Re  \\rho +z \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1} (\\Im  \\rho \\otimes 1_{\\cal A})\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\left( (\\Re  \\rho +\\bar{z} \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}<0.\n\\end{eqnarray*}\nIt follows by the complete positivity of the trace $\\tau$ that $${\\rm id}_m\\otimes \\tau\\left\\{ \\Im  \\left\\{\\left(( \\Re  \\rho +z \\Im  \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\right\\}<0.$$\nNow, according to Remark \\ref{remarqueinversible}, we can assume the $\\alpha_v$'s invertible so that \n$\\sum_{v=1}^r \\alpha_v{\\rm id}_m\\otimes \\tau \\left\\{\\Im  \\left\\{\\left( (\\Re  \\rho +z \\Im  \\rho) \\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\right\\}\\alpha_v <0$ and then $\\Im  F(z) <0$.\nThus for any $z\\in \\mathbb{C}\\setminus \\mathbb{R}$, we have $\\Im z \\Im  F(z)<0.$\nFinally \n$$\\lim_{y \\rightarrow +\\infty} iy F(iy)= \\varphi_\\rho \\left( \\sum_{v=1}^r\\alpha_v (\\Im  \\rho )^{-1} \\alpha_v \\right):=c_\\rho >0. $$\nThus, by Akhiezer-Krein's Theorem (\\cite{AK} page 93), there exists a probability measure $\\mu$ on $\\mathbb {R}$ such that\n$$\nF(z)= c_\\rho \\int_\\mathbb R\\frac{d\\mu(s)}{z-s}\n,\\quad z\\in\\mathbb C^+.\n$$\nThen\n$$\nf_{\\rho}(z)=z+\\varphi_\\rho(\\gamma+\\Re \\rho)+c_\\rho \\int_\\mathbb R\\frac{d\\mu(s)}{z-s}\n,\\quad z\\in\\mathbb C^+.\n$$\nThus $\\Im f_{\\rho}(u+iv)=v\\left(1-c_\\rho\\int_\\mathbb R\\frac{d\\mu(s)}{(u-s)^2+v^2}\\right)$. We observe \nthat what's under parenthesis is strictly increasing in $v$. Since by hypothesis, we have $\\Im  \\Lambda_n(\\rho)>0$ and thus  $\\Im f_{\\rho}(i)=\\Im \\varphi_\\rho\\left(\\Lambda_n(\\Re \\rho+i\\Im \\rho)\\right)\n>0$, we obtain \nimmediately that $\\Im f_{\\rho}(iv)>0$ for all $v\\ge1$. This means that $\\Im \n\\phi(\\Lambda_n(\\Re \\rho+iv\\Im \\rho))>0$ for all $v\\in[1,+\\infty)$ and all states $\\phi$, so that \\begin{equation}\\label{ray}\\Im \\Lambda_n (\\Re \\rho+iv\\Im \\rho)\n>0, {\\rm~ for~ all~} v\\ge1.\\end{equation}\nNow it is clear that $$\\Omega=\\{z \\in \\C^+, \\Im \\Lambda_n(\\Re  \\rho + z\\Im  \\rho) >0\\}$$\nis an open set which contains $d=\\{iv, v\\geq 1\\}$.\nLet $\\Omega_d$ be the connected component of $\\Omega$ which contains $d$. Note that $\\Omega_d $ is an open set.\n\nAs we have shown at the beginning of our proof, for given $\\rho,\\Im   \\rho>0$, there exists an $M>0$ (possibly depending on $\\rho$) such that $\\omega_n(\\Lambda_n(\\Re  \\rho + iv\\Im  \\rho))=\\Re  \\rho + iv\\Im  \\rho$ for all $v>M.$\nBy the identity principle for analytic functions, we immediately obtain that \n$ \\omega_n(\\Lambda_n(\\Re \\rho +z \\Im \\rho))=\\Re  \\rho +z \\Im  \\rho$ for all $z\\in \\Omega_d$ and in particular for $z=i$. The \nproof of Lemma \\ref{inversion} is complete.\n\\end{proof}\n\n\n\\section{Proof of Lemma \\ref{inclu2}}\\label{lemmefonda} \\subsection{ Sharp estimates of Stieltjes transforms} The proof of \\eqref{spectre3} requires the  sharp estimate \\eqref{estimdiffeqno} we are going to prove here.\n\n\n\\noindent According to Section \\ref{troncation}, from now on, we assume that the $X_{ij}^{(v)}$'s satisfy (H).\nNote that this assumption  implies that for any  $v\\in \\{1,\\ldots,r\\}$, \n$$\\forall i\\geq 1, \\forall j \\geq  1, \\;\\kappa_1^{i,j,v}=0, \\;\\kappa_2^{i,j,v}=1,$$\n $$\\forall i\\geq 1, \\forall j \\geq  1,\\;,  i\\neq j, \\; \\tilde \\kappa_1^{i,j,v}=0,\\; \\tilde \\kappa_2^{i,j,v}=1$$ and  \n for any $p\\in \\mathbb{N}\\setminus\\{0\\}$, \\begin{equation}\\label{cumulants}\\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j \\geq 1} \\vert \\kappa_p^{i,j,v}\\vert<+\\infty, \\; \\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j\\geq 1} \\vert \\tilde  \\kappa_p^{i,j,v}\\vert<+\\infty, \\end{equation}\nwhere for $i\\neq j$,  $(\\kappa_p^{i,j,v})_{p\\geq1}$ and $(\\tilde \\kappa_p^{i,j,v})_{p\\geq 1}$ denote  the classical cumulants of $\\sqrt{2}\\Re  X_{ij}^{(v)}$ and $\\sqrt{2}\\Im   X_{ij}^{(v)}$ respectively and $(\\kappa_p^{i,i,v})_{p\\geq 1}$ denotes  the classical cumulants of $ X_{ii}^{(v)}$ (we set  $(\\tilde \\kappa_p^{i,i,v})_{p\\geq 1}\\equiv 0$).\\\\\n\n\n\n\\noindent Now, we present our  main technical tool (see \\cite{KKP}):\n \\begin{lemme} \\label{lem1}\nLet $\\xi$ be a real-valued random variable such that  $\\mathbb{E}(\\vert \\xi\n\\vert^{p+2})<\\infty$. Let  $\\phi$ be a function from  $\\R$ to $\\C$\nsuch that the first $p+1$ derivatives are  continuous and bounded. Then,\n\\begin{equation}\\label{IPP}\\mathbb E(\\xi \\phi(\\xi)) = \\sum_{a=0}^p\n\\frac{\\kappa_{a+1}}{a!}\\mathbb{E}(\\phi^{(a)}(\\xi)) + \\epsilon\\end{equation}\nwhere  $\\kappa_{a}$ are the classical  cumulants of  $\\xi$, $\\epsilon \\leq C\n\\sup_t \\vert \\phi^{(p+1)}(t)\\vert \\mathbb{E}(\\vert \\xi \\vert^{p+2})$, $C$\ndepends  on $p$ only.\n\\end{lemme}\nIn the following, we shall apply this identity with a function\n$\\phi(\\xi)$ given by the entries  of the resolvent of $S_n$. It\nfollows from  Lemma \\ref{lem2} and (\\ref{resolvente}) below\nthat the conditions of Lemma \\ref{lem1} (bounded derivatives) are\nfulfilled.\nWe first need the following preliminary lemma.\n \n\\begin{lemme}\\label{inverseY}\nFor any $\\lambda \\in M_m(\\mathbb{C})  $ such that $\\Im  \\lambda$ is positive definite, $(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}$ and $(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda)\\alpha_v)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}$ are invertible.\nSet  \\begin{equation}\\label{y}Y_n(\\lambda)=\\left((\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\end{equation}\nand  \\begin{equation}\\label{ytilde}\\tilde Y_n(\\lambda)=\\left((\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda)\\alpha_v)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}.\\end{equation}\nWe have \\begin{equation} \\label{Y}\\left\\| Y_n(\\lambda) \\right\\| \\leq \\Vert ( \\Im  \\lambda)^{-1} \\Vert\\end{equation}\nand \\begin{equation} \\label{Y2}\\left\\| \\tilde Y_n(\\lambda) \\right\\| \\leq \\Vert ( \\Im  \\lambda)^{-1} \\Vert.\\end{equation}\n\n\\end{lemme}\n\\begin{proof}\nWe only present the proof for $Y_n(\\lambda)$ since the proof is similar for $\\tilde Y_n(\\lambda)$.\nNote that \n\\begin{eqnarray*}\n\\Im  \\left[ \\left( \\lambda\\otimes I_n -S_n\\right)^{-1}\\right]\n&=& \\frac{1}{2i}\\left[\\left( \\lambda\\otimes I_n -S_n\\right)^{-1}-\\left( \\lambda^*\\otimes I_n -S_n\\right)^{-1}\\right]\\\\\n&=& -\\left( \\lambda\\otimes I_n -S_n\\right)^{-1}\\left( \\Im  \\lambda \\otimes I_n\\right) \\left( \\lambda^*\\otimes I_n -S_n\\right)^{-1}.\n\\end{eqnarray*}\nThis yields that  $-\\Im  R_n(\\lambda) $ is positive definite. Since the map ${\\rm id }_m\\otimes \\tr_n$ is positive we can deduce that $-\\Im  H_n(\\lambda)$ is positive and then that $-\\Im  G_n(\\lambda)$ is positive. It readily follows that\n\\begin{equation}\\label{image} \\Im  \\left[\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v \\right] \\geq \\Im  \\lambda\\end{equation} and then\n$$\\Im  \\left[\\left(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v\\right)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\\right] \\geq \\Im  \\lambda\\otimes I_n.$$ Hence Lemma \\ref{inverseY} follows  by lemma 3.1 in \\cite{HT}.\n\\end{proof}\n\\begin{theoreme}\\label{resolvante}\nFor any $\\lambda \\in M_m(\\mathbb{C})  $ such that $\\Im \\lambda$ is positive definite, we have \n\\begin{equation}\\label{mast}\\mathbb{E} \\left(R_n(\\lambda)\\right)=Y_n(\\lambda)+Y_n(\\lambda)\\Xi(\\lambda)\\end{equation}\nwhere $Y_n(\\lambda)$ is defined in Lemma \\ref{inverseY}\n\\noindent and  $\\Xi(\\lambda)=\\sum_{l,j}\\Xi_{lj}(\\lambda)\\otimes E_{lj}$ satisfies that   for all $l,j\\in \\{1, \\ldots,n\\}$, $$ \\Xi_{lj}(\\lambda)=\\Psi_{lj}(\\lambda) +O_{lj}^{(u)}( \\frac{1}{n^2})$$\nwhere\n \\begin{eqnarray}\\Psi_{lj}(\\lambda)& =&\\sum_{v=1}^r\\bigg\\{\\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v    [R_n(\\lambda ) ]_{lj} \\right]\\nonumber\n\\\\&&+ \\frac{1}{2\\sqrt{2}n\\sqrt{n}}  \\sum_{i=1}^n  \\left\\{1-\\delta_{il}\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\right\\}M^{(3)}(v,i,l,j)\\nonumber\\\\&&+  \\frac{1}{4n^2} \\sum_{i=1}^n \\left(1-\\frac{1}{2}\\delta_{il}\\right) M^{(4)}(v,i,l,j)\\nonumber\\\\&&\\left.\n+  \\frac{1}{4\\sqrt{2}n^2\\sqrt{n}}\\sum_{i=1}^n  \\left[1-\\delta_{il}\\left(1-\\frac{1}{2\\sqrt{2}}\\right)\\right]M^{(5)}(v,i,l,j)\\right\\},\\label{psi}\\end{eqnarray}\nwith \\\\\n\n\n\\noindent $M^{(3)}(v,i,l,j)$\n\\begin{eqnarray}=& \\mathbb{E} \\{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{lj}\\nonumber \\\\\n &+(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ij}\\label{except}\n  \\\\\n&+(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{lj} \\nonumber \\\\\n&+(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ij}\\}, \\nonumber \\end{eqnarray}\n\n\n\n\\noindent $M^{(4)}(v,i,l,j)$\n\\begin{eqnarray}=&(\\kappa_4^{i,l,v}+\\tilde \\kappa_4^{i,l,v}) \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n(R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij}\\label{premierk4}\n\\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{lj}\\label{deuxcas4} \\\\&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v(R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{lj}\\label{troiscas4}\\\\& + \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v(R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{ij}\\} \\label{quatrecas4}\\\\\n&\\hspace*{-0.4cm}+(\\kappa_4^{i,l,v}~\\hspace*{-0.4cm}-\\tilde \\kappa_4^{i,l,v}) \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n(R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{lj} \\label{tilde1}\n\\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{ij} \\label{tilde2} \\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v(R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{lj} \\label{tilde3}\\\\& + \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v(R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ij}\\}, \\label{tilde4}\n\\end{eqnarray}\n\n\\noindent $M^{(5)}(v,i,l,j)$\n\\begin{eqnarray*}=& \\mathbb{E} \\{(\\kappa_5^{i,l,v}+\\tilde\\kappa_5^{i,l,v}\\sqrt{-1}) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\& \\times \\left[\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ij}\\right.\\\\\n &+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v  (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{lj} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{lj} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij} \\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{li}  \\alpha_v  (R_n(\\lambda))_{li}\\alpha_v  (R_n(\\lambda))_{lj} \\\\\n&+  \\alpha_v  (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ij} \\\\\n&+ \\alpha_v  (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ij} \\\\\n&+\\left. \\alpha_v  (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{lj} \\right]\\\\\n&+(\\kappa_5^{i,l,v}-\\tilde\\kappa_5^{i,l,v}\\sqrt{-1}) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\&\\left[\n\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v  (R_n(\\lambda))_{lj}\\right.\\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v  (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ij} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda))_{ij} \\\\\n&+\\left. \\alpha_v  (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{lj} \\right]\\\\\n&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij} \\\\\n&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{ll}  \\alpha_v  (R_n(\\lambda))_{ii}\\alpha_v  (R_n(\\lambda))_{lj} \\\\\n&+  \\alpha_v  (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v  (R_n(\\lambda))_{li} \\alpha_v  (R_n(\\lambda))_{lj} \\\\\n&+\\left. \\alpha_v  (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{il} \\alpha_v  (R_n(\\lambda))_{ij}\\right]\\}.\n\\end{eqnarray*}\n\\end{theoreme}\n \n\n \\begin{proof}\n We shall apply formula \\eqref{IPP} to the ${ M}_m(\\C)$-valued function $\\phi(\\xi) = (R_n(\\lambda))_{ij}$ for $1 \\leq i,j \\leq n$ and $\\xi$ is one of the variable $\\frac{X^{(v)}_{kk}}{\\sqrt{n}}$,\n$\\sqrt{2} \\frac{Re(X^{(v)}_{kl})}{\\sqrt{n}}$, $\\sqrt{2} \\frac{Im(X^{(v)}_{kl})}{\\sqrt{n}}$ for $1 \\leq k<l \\leq n$ and $1\\leq v \\leq r$. \\\\\nWe notice that\n$$\\frac{\\partial \\phi}{\\partial Re(\\frac{X^{(v)}_{kk}}{\\sqrt{n}})}  = \\phi'_{\\frac{X_n^{(v)}}{\\sqrt{n}}} . e_{kk}, \\;\\frac{X^{(v)}_{kk}}{\\sqrt{n}} = \\Tr(\\frac{X_n^{(v)}}{\\sqrt{n}} e_{kk}), \\ 1 \\leq k \\leq n,$$\n$$\\frac{\\partial \\phi}{\\partial \\sqrt{2} Re(\\frac{X^{(v)}_{kl}}{\\sqrt{n}})}  = \\phi'_{\\frac{X_n^{(v)}}{\\sqrt{n}}} . e_{kl}, \\; \\sqrt{2} Re(\\frac{X^{(v)}_{kl}}{\\sqrt{n}}) = \\Tr(\\frac{X_n^{(v)}}{\\sqrt{n}} e_{kl}), \\ 1 \\leq k <l \\leq n, $$\n$$\\frac{\\partial \\phi}{\\partial \\sqrt{2} Im(\\frac{X^{(v)}_{kl}}{\\sqrt{n}})}  = \\phi'_{\\frac{X_n^{(v)}}{\\sqrt{n}}} . f_{kl} , \\;\\sqrt{2} Im(\\frac{X^{(v)}_{kl}}{\\sqrt{n}}) = \\Tr(\\frac{X_n^{(v)}}{\\sqrt{n}} f_{kl}), \\ 1 \\leq k <l \\leq n. $$\n\n\n\\noindent Let  $1\\leq k < l \\leq n$  be fixed.\nFor simplicity, we write $\\phi'$, $\\phi''$, $\\phi'''$ and  $\\phi''''$ for the\nfirst derivatives of $\\phi$ with respect to $\\sqrt{2}\nRe(\\frac{X^{(v)}_{kl}}{\\sqrt{n}})$. Then, according to (\\ref{resolvente}),\n\\begin{eqnarray*}\n\\phi' &=& \\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\right]_{ij} \\\\\n\\phi''&=&2 \\left[R_n (\\lambda) \\alpha_v \\otimes e_{kl}R_n (\\lambda)  \\alpha_v \\otimes e_{kl}R_n (\\lambda)\\right]_{ij} \\\\\n\\phi'''&=&6 \\left[ R_n (\\lambda) \\alpha_v \\otimes e_{kl}R_n (\\lambda)  \\alpha_v \\otimes e_{kl} R_n (\\lambda)  \\alpha_v \\otimes e_{kl} R_n(\\lambda)\\right]_{ij}\\\\\n\\phi''''&=&24 \\left[R_n (\\lambda) \\alpha_v \\otimes e_{kl}R_n (\\lambda)  \\alpha_v \\otimes e_{kl} R_n (\\lambda)  \\alpha_v \\otimes e_{kl} R_n(\\lambda)\\alpha_v \\otimes e_{kl} R_n (\\lambda)\\right]_{ij}\n\\end{eqnarray*}\nWriting \\eqref{IPP} in this setting gives\n$$\\mathbb{E}[ \\Tr(\\frac{X_n^{(v)}}{\\sqrt{n}}e_{kl}) (R_n (\\lambda))_{ij} ] = \\frac{1}{n} \\mathbb{E}[ \\phi'] +\n\\frac{\\kappa_3^{k,l,v}}{2n\\sqrt{n}} \\mathbb{E}[\\phi'']  +\n\\frac{\\kappa_4^{k,l,v}}{6n^2} \\mathbb{E}[\\phi''']+ \\frac{\\kappa_5^{k,l,v}}{24 n^2 \\sqrt{n}}\\mathbb{E}[\\phi'''']$$\n\\begin{equation} \\label{1}+O_{k,l,i,j,v}\\left(\\frac{1}{n^3}\\right)\n\\end{equation}\nwhere there exists $C>0$ such that for every $k,l,i,j \\in\\{1,\\ldots,n\\}$ and every $v\\in \\{1,\\ldots,r\\}$, \\begin{equation}\\label{majuniv}\\left\\|  O_{k,l,i,j}\\left(\\frac{1}{n^3}\\right)\\right\\|\\ \\leq \\frac{C\\Vert \\alpha_v \\Vert^5 \\Vert ( \\Im \\lambda)^{-1} \\Vert^6}{n^3},\\end{equation}\nwith the analogous equations with $f_{kl}$  and $e_{kk}$ replacing  the $\\kappa_i^{k,l,v}$'s by the  $\\tilde \\kappa_i^{k,l,v}$'s and $\\kappa_i^{k,k,v}$'s respectively. \\\\\n\n\\noindent\nNoticing that for $ k<l$, $$E_{kl} =\\frac{e_{kl} -\\sqrt{-1}f_{kl}}{\\sqrt{2}}\\;\n\\mbox{and }\\; E_{lk} =\\frac{e_{kl} +\\sqrt{-1}f_{kl}}{\\sqrt{2}},$$\n  we deduce that :\\\\\n\n$\\mathbb{E}[ \\Tr(\\frac{X_n^{(v)}}{\\sqrt{n}} E_{kl}) (R_n(\\lambda))_{ij} ] $\n\\begin{eqnarray*} &=& \\frac{1}{n}\\mathbb{E}\\left[ R_n(\\lambda) \\alpha_v \\otimes E_{kl} R_n(\\lambda)\\right]_{ij}\\\\\n&&+ \\frac{\\kappa_3^{k,l,v}}{  \\sqrt{2}n\\sqrt{n}}\\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha_v \\otimes e_{kl}  R_n(\\lambda) \\right]_{ij}\\right\\}\\\\ && -\\sqrt{-1} \\frac{\\tilde \\kappa_3^{k,l,v}}{  \\sqrt{2}n\\sqrt{n}}\\mathbb{E} \\left\\{ \\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) )\\right]_{ij}\\right\\}\\\\\n&&+ \\frac{\\kappa_4^{k,l,v}}{  \\sqrt{2}n^2}\\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha_v \\otimes e_{kl}  R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\ && -\\sqrt{-1}  \\frac{\\tilde \\kappa_4^{k,l,v}}{  \\sqrt{2}n^2}\\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&+ \\frac{\\kappa_5^{k,l,v}}{  \\sqrt{2}n^2\\sqrt{n}} \\\\&&~~~~\\times \\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha_v \\otimes e_{kl}  R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\ && -\\sqrt{-1} \\frac{\\tilde \\kappa_5^{k,l,v}}{  \\sqrt{2}n^2\\sqrt{n}}  \\\\&&~~~ \\times \\mathbb{E} \\left\\{ \\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&+ O_{l,k,i,j,v}\\left(\\frac{1}{n^3}\\right),\n \\end{eqnarray*}\n\n$\\mathbb{E}[ \\Tr(\\frac{X^{(v)}_n }{\\sqrt{n}}E_{lk}) (R_n(\\lambda))_{ij} ]$\n\\begin{eqnarray*}  &=& \\frac{1}{n}\\mathbb{E}\\left[ R_n(\\lambda) \\alpha_v \\otimes E_{lk} R_n(\\lambda)\\right]_{ij} \\\\\n&&+ \\frac{\\kappa_3^{k,l,v}}{  \\sqrt{2}n\\sqrt{n}}\\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha_v \\otimes e_{kl}  R_n(\\lambda) \\right]_{ij}\\right\\}\\\\ &&+\\sqrt{-1}\\frac{\\tilde \\kappa_3^{k,l,v}}{  \\sqrt{2}n\\sqrt{n}}\\mathbb{E} \\left\\{ \\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) )\\right]_{ij}\\right\\}\\\\\n&& + \\frac{\\kappa_4^{k,l,v}}{  \\sqrt{2}n^2}\\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha \\otimes e_{kl}  R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\ && +\\sqrt{-1}  \\frac{\\tilde\\kappa_4^{k,l,v}}{  \\sqrt{2}n^2}\\mathbb{E} \\left\\{ \\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&+ \\frac{\\kappa_5^{k,l,v}}{  \\sqrt{2}n^2\\sqrt{n}}\\\\ &&~~\\times \\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kl} R_n(\\lambda) \\alpha_v \\otimes e_{kl}  R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda) \\alpha_v \\otimes e_{kl}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\ &&+\\sqrt{-1}\n\\frac{\\tilde \\kappa_5^{k,l,v}}{  \\sqrt{2}n^2\\sqrt{n}} \\\\ &&~~ \\times \\mathbb{E} \\left\\{ \\left[ R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda) \\alpha_v \\otimes f_{kl} R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&+ O_{k,l,i,j,v}\\left(\\frac{1}{n^3}\\right)\n \\end{eqnarray*}\nand \\\\\n\n\\noindent $\\mathbb{E}[ \\Tr(\\frac{X^{(v)}_n}{\\sqrt{n}} E_{kk}) (R_n(\\lambda))_{ij} ]$\n\\begin{eqnarray*}  &=& \\frac{1}{n}\\mathbb{E}\\left[ R_n(\\lambda) \\alpha_v \\otimes E_{kk} R_n(\\lambda)\\right]_{ij}\\\\&&  + \\frac{\\kappa_3^{k,k,v}}{  n\\sqrt{n}} \\mathbb{E} \\left\\{\\left[  R_n(\\lambda) \\alpha_v \\otimes e_{kk}  R_n(\\lambda) \\alpha_v \\otimes e_{kk}R_n(\\lambda)\\right]_{ij}\\right\\}\n\\\\&&  + \\frac{\\kappa_4^{k,k,v}}{  n^2} \\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kk} R_n(\\lambda) \\alpha_v \\otimes e_{kk}  R_n(\\lambda) \\alpha_v \\otimes e_{kk}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&  + \\frac{\\kappa_5^{k,k,v}}{  n^2 \\sqrt{n}} \\mathbb{E} \\left\\{\\left[ R_n(\\lambda) \\alpha_v \\otimes e_{kk} R_n(\\lambda) \\alpha_v \\otimes e_{kk}  R_n(\\lambda) \\alpha_v \\otimes e_{kk}R_n(\\lambda)e_{kk}R_n(\\lambda)\\right]_{ij}\\right\\}\\\\\n&&+  O_{k,k,i,j,v}\\left(\\frac{1}{n^3}\\right)\n\\end{eqnarray*}\nwhere the $O_{k,l,i,j}\\left(\\frac{1}{n^3}\\right)$'s satisfy \\eqref{majuniv}.\nWe then obtain:\n  \\begin{eqnarray}\\mathbb{E}[(\\alpha_v \\otimes \\frac{X^{(v)}_n}{\\sqrt{n}}) R_n(\\lambda ) ]_{lj}& =& \\mathbb{E}\\left[ \\alpha_v H_n(\\lambda) \\alpha_v    [R_n(\\lambda ) ]_{lj}  \\right]\\nonumber\n\\\\&&+ \\frac{1}{2\\sqrt{2}n\\sqrt{n}}  \\sum_{i=1}^n  \\left\\{1-\\delta_{il}\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\right\\}M^{(3)}(v,i,l,j)\\nonumber \\\\&&+  \\frac{1}{4n^2} \\sum_{i=1}^n \\left(1-\\frac{1}{2}\\delta_{il}\\right) M^{(4)}(v,i,l,j)\\nonumber\\\\&&\n+  \\frac{1}{4\\sqrt{2}n^2\\sqrt{n}}\\sum_{i=1}^n  \\left\\{1-\\delta_{il}\\left(1-\\frac{1}{2\\sqrt{2}}\\right)\\right\\}M^{(5)}(v,i,l,j)\\nonumber\\\\&&+O_{l,j,v}(1\/n^2),\\label{4}\\end{eqnarray}\n\n\\noindent  where the $M^{(q)}(v,i,l,j)$ for $ q=3,4,5$ are defined in Theorem \\ref{resolvante} and \n there exists $C>0$ such that for every $l,j \\in\\{1,\\ldots,n\\}$ and any $v \\in \\{1,\\ldots,r\\}$, \\begin{equation}\\label{majuniv2}\\left\\|  O_{l,j,v}\\left(\\frac{1}{n^2}\\right)\\right\\|\\ \\leq \\frac{C\\Vert \\alpha_v \\Vert^5 \\Vert ( \\Im \\lambda)^{-1} \\Vert^6}{n^2}.\\end{equation}\n\n\\noindent Now,\n\\begin{eqnarray*}\n\\sum_{v=1}^r(\\alpha_v   \\otimes X_n^{(v)})R_n (\\lambda)\n &=&  (S_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}-\\gamma \\otimes I_n) (\\lambda \\otimes I_n - S_n)^{-1}  \\\\\n &=& -I_m\\otimes I_n  +\\left[ (\\lambda - \\gamma) \\otimes I_n -\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right] R_n(\\lambda)\n \\end{eqnarray*}\n implying\n\\begin{eqnarray}\\sum_{v=1}^r \\mathbb{E}[(\\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}) R_n(\\lambda ) ]_{lj} &=& -\\delta_{jl} I_m +(\\lambda-\\gamma) \\mathbb{E} (R_n(\\lambda))_{lj}\\nonumber\n\\\\&&- \\left[\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}  \\mathbb{E}( R_n(\\lambda))\\right]_{lj}. \\label{identiteresol}\\end{eqnarray}\nOn the other hand, we have \n\\begin{eqnarray}\\mathbb{E}\\left[ \\alpha_v H_n(\\lambda) \\alpha_v    [R_n(\\lambda ) ]_{lj}  \\right]&= &  \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v    [R_n(\\lambda ) ]_{lj} \\right]\\nonumber\\\\ &&\n+\n\\alpha_v G_n(\\lambda) \\alpha_v \\mathbb{E}\\left[ [R_n(\\lambda ) ]_{lj} \\right]\n.\\label{centrage}\\end{eqnarray}\nHence \\eqref{4}, \\eqref{identiteresol} and \\eqref{centrage} yield \\\\\n\n $ -\\delta_{jl}I_m +(\\lambda-\\gamma) \\mathbb{E} (R_n(\\lambda))_{lj}\n- \\left[\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}  \\mathbb{E}( R_n(\\lambda))\\right]_{lj}$ \\begin{equation}\\label{me} = \\sum_{v=1}^r\\alpha_v G_n(\\lambda) \\alpha_v \\mathbb{E}\\left[ [R_n(\\lambda ) ]_{lj} \\right] +\n\\Xi_{lj}(\\lambda) \\end{equation}\nwhere $$\\Xi_{lj}(\\lambda)=\\Psi_{lj}(\\lambda)\n+O^{(u)}_{l,\nj}(1\/n^2)$$ and  $\\Psi_{lj}$ is defined in Theorem \\ref{resolvante}.\nThus, we have \n$$ \\left[\\left(\\lambda-\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda) \\alpha_v\\right)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right]  \\mathbb{E} (R_n(\\lambda))\n =  I_n\\otimes I_m +\n\\Xi(\\lambda).$$\n\\eqref{mast} readily follows. \\end{proof}\n\n\n\\begin{proposition}\\label{55}\nFor any $p,q\\in \\{1,\\ldots,n\\}^2$, for any $mn\\times mn $ deterministic matrix $F_n(\\lambda)$ such that  \n$  F_n(\\lambda)=O(1)$, setting $\\Psi(\\lambda)=\\sum_{l,j}\\Psi_{lj}(\\lambda)\\otimes E_{lj}$ where $\\Psi_{lj}$ is defined by \\eqref{psi}, we have\\\\\n\n$\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$\n\\begin{eqnarray} &=&\\frac{1}{2\\sqrt{2}n\\sqrt{n}}  \\sum_{v=1}^r\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n)_{pl}\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq}\\} \\nonumber\\\\&&+O^{(u)}_{p,q}(\\frac{1}{{n}}). \\label{YPSIF}\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nLet us fix $v \\in \\{1,\\ldots,r\\}$. Using \\eqref{cumulants}, \\eqref{Y} and  \\eqref{norme}, one can easily deduce from \\eqref{Oderacine} (respectively from \\eqref{Oden}) that  all  the terms  in $\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$ corresponding to the $M^{(3)}(v,i,l,j)$'s  in \\eqref{psi} excluding $$(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v  (R_n(\\lambda))_{ij} $$  (respectively all  the terms corresponding to the $M^{(4)}(v,i,l,j)$'s)  are equal to $O(1\/n)$.\n\n\n\n\n\n\n\n\nLet  $C$ be   some constant such that $\\sup_{i,l,v} \\{\\vert \\kappa_5^{i,l,v}\\vert +\\vert \\tilde \\kappa_5^{i,l,v} \\vert\\}\\leq C$.\nFor the terms in $\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$ corresponding to $M^{(5)}(v,i,l,j)$'s, note that using \\eqref{norme} they can be all obviously bounded by \n\\\\\n\n\\noindent $\n\\frac{C\\Vert F_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^5  \\Vert (Im(\\lambda))^{-1}\\Vert^5}{n\\sqrt{n}}\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl}\\right\\|$\n\\begin{eqnarray*}&\\leq &\n\\frac{C\\Vert F_n (\\lambda)\\Vert \\Vert \\alpha_v\\Vert^5  \\Vert (Im(\\lambda))^{-1}\\Vert^5}{n}\\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl}\\right\\|^2\\right\\}^{\\frac{1}{2}}\\\\&\\leq & \\frac{\\sqrt{m}C\\Vert F_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^5  \\Vert (Im(\\lambda))^{-1}\\Vert^6}{n}\\\\&=&O(1\/n)\\end{eqnarray*}\nwhere we used \\eqref{l} and \\eqref{Y}.\\\\\n\n\\noindent Finally define\n$$\\hat {\\cal R}= \\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v\\right] \\otimes I_n $$\nso that there exists some constant $C>0$ such that \n\\\\\n\n\\noindent $\\sum_{j,l=1}^n (Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v    [R_n(\\lambda ) ]_{lj}(F_n(\\lambda)_{jq} \\right]\n$\\begin{eqnarray*}&=& \\mathbb{E}\\left[  [Y_n (\\lambda)\\hat {\\cal R} R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]\\\\\n&\\leq&\\Vert F_n(\\lambda) \\Vert  \\Vert (Im(\\lambda))^{-1}\\Vert^2 \\mathbb{E}\\left[  \\Vert \\hat  {\\cal R} \\Vert \\right]\\\\\n&\n\\leq&\\frac{C \\sqrt{m} \\Vert F_n (\\lambda)\\Vert \\Vert \\alpha_v\\Vert^2}{n}\n \\Vert (Im(\\lambda))^{-1}\\Vert^4\\\\&=&O(1\/n).\\end{eqnarray*}\n where we used \\eqref{Y}, \\eqref{norme} and \\eqref{varhn} in the last lines.\\\\\n\\noindent It is moreover clear that one can find a common polynomial to bound  the involved $nO_{p,q}(1\/n)$.  \\eqref{YPSIF} follows.\n\\end{proof}\n\n\\begin{corollaire} \\label{estimenunsurn}\nFor any  $mn\\times mn$ deterministic  matrix  $F_n(\\lambda)$ such that $F_n(\\lambda)=O(1)$, we have  \\\\\n\n\\noindent \n$\\mathbb{E}\\left[  [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray*}&=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \n\\\\&&+\\ \\sum_{v=1}^r \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl}  \\alpha_v(Y_n(\\lambda))_{ii}\\alpha_v (Y_n(\\lambda))_{ll}\\alpha_v\\\\~&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E}\\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]\\\\&& +O^{(u)}_{p,q}(1\/n).\\end{eqnarray*}\n\\end{corollaire}\n\\begin{proof} \nNoticing that\n\n\\noindent $\\left\\|\n\\sum_{l,j=1}^n \\left(Y_n(\\lambda)\\right)_{pl} O^{(u)}_{l,j}\\left(1\/n^2\\right) \\left(F_n(\\lambda)\\right)_{jq} \\right\\|$\n \\begin{eqnarray*}&\\leq & \n\n O\\left(\\frac{1}{n}\\right)\\left\\{ \\sum_{l=1}^n \\Vert \\left(Y_n(\\lambda)\\right)_{pl} \\Vert^2 \\right\\}^{1\/2} \\left\\{ \\sum_{j=1}^n \\Vert \\left(F_n(\\lambda)\\right)_{jq} \\Vert^2 \\right\\}^{1\/2}\\\\\n\n &=&O_{p,q}^{(u)}(1\/n) ,\n\\end{eqnarray*}\n(using Lemma \\ref{majcarre} and \\eqref{Y} in the last line)\nit readily follows from Theorem \\ref{resolvante} and Proposition \\ref{55} that \\\\\n\n\n\n\\noindent $\\mathbb{E}\\left[  [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray*}&\\hspace*{-0.5cm}=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \n\\\\&&+ \\sum_{v=1}^r  \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl}    \\alpha_v\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]\\\\&& +O^{(u)}_{p,q}(1\/n).\\end{eqnarray*}\n\nTo simplify the writing let us set $U_i=\\alpha_v (R_n(\\lambda))_{ii}$, $V_l=\\alpha_v(R_n(\\lambda))_{ll}$ and $W_i= \\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq}$.\nWe have \\\\\n\n$\\mathbb{E}\\left[U_i V_l W_i\\right]$\n\\begin{eqnarray}&=&\\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i)) (V_l-\\mathbb{E}(V_l))W_i\\right]  \\nonumber\\\\&&+ \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i))\\mathbb{E}(V_l) (W_i-\\mathbb{E}(W_i))\\right] \\nonumber\n\\\\&&+ \\mathbb{E}(U_i) \\mathbb{E}\\left[ (V_l-\\mathbb{E}(V_l))(W_i-\\mathbb{E}(W_i))\\right]+\\mathbb{E}\\left[U_i\\right]\n\\mathbb{E}\\left[V_l\\right]\\mathbb{E}\\left[W_i\\right]. \\label{decomposition}\n\\end{eqnarray}\n\n\\noindent Now,\\\\\n\n\\noindent $\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl}    \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i)) (V_l-\\mathbb{E}(V_l))W_i\\right]\\right\\|$ \n\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert \\Vert F_n(\\lambda)\\Vert\n\\\\&&~~~~~~\\times  \\sum_{i,l=1}^n \\left\\| \n(Y_n(\\lambda))_{pl} \\right\\|\\left\\{\n\\mathbb{E} \\left(  \\left\\| U_{i} -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left(  \\left\\| V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \\Vert F_n(\\lambda)\\Vert \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&&~~~~~~~~~~~~~~~~~\\times  \n\\left\\{\\sum_{l=1}^n \\mathbb{E} \\left(  \\left\\|V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\left\\{\\sum_{i=1}^n \\mathbb{E} \\left(  \\left\\|U_i -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\n\n\\noindent Moreover,\n$$\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl}    \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i))\\mathbb{E}(V_l) (W_i-\\mathbb{E}(W_i))\\right]\\right\\|$$\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert  \\sum_{i,l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|\\\\&&~~~~~~~~~~~~~~~~~\\times  \\left\\{\n\\mathbb{E} \\left(  \\left\\| U_{i} -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left(  \\left\\| W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \n \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&& \\times \\left\\{\\sum_{l=1}^n \\mathbb{E} \\left(  \\left\\|W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2} \\left\\{\\sum_{i=1}^n \\mathbb{E} \\left(  \\left\\|U_i -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\nFinally\n$$\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl}    \\mathbb{E}\\left[ U_i\\right]\\mathbb{E}\\left[ (V_l-\\mathbb{E}(V_l)) (W_i-\\mathbb{E}(W_i))\\right]\\right\\|$$\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert  \\sum_{i,l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|\n\\\\&& \\times \\left\\{\n\\mathbb{E} \\left(  \\left\\| V_{l} -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left(  \\left\\| W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \n \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&& \\times \\left\\{\\sum_{l=1}^n \\mathbb{E} \\left(  \\left\\|V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\left\\{\\sum_{i=1}^n \\mathbb{E} \\left(  \\left\\|W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\n\n\n\n\n \\noindent Using Lemma \\ref{var},   (\\ref{l}) and \\eqref{Y}, we readily deduce that \\\\\n\n\n\\noindent $\\mathbb{E}\\left[  [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray}&=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \\nonumber\n\\\\&&+\\frac{1}{2\\sqrt{2}n\\sqrt{n}}\\sum_{v=1}^r  \\sum_{i,l=1}^n  (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda)_{pl}  \\alpha_v\\\\&&~~~~~~~~~~~~~~~~~~~~~~~\\times\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]\n\\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v \\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\nonumber \\\\&& +O^{(u)}_{p,q}(1\/{n}). \\label{eq}\\end{eqnarray}\n\\noindent Now, define \n$${\\cal R}= \\sum_{v=1}^r\\sum_{i,l=1}^n  (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})    \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]\\alpha_v\\mathbb{E}\n\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v\\otimes E_{li},$$\nIt is easy to see that  $\\Vert {\\cal R}\\Vert\\leq C \\Vert (\\Im \\lambda)^{-1}\\Vert^2 n.$\nWe have \\\\\n\n\\noindent $ \\sum_{v=1}^r \\sum_{i,l=1}^n(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})  (Y_n(\\lambda))_{pl}   \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]  $ $$ ~~~~~~~~~~\\times \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]= \\left[ Y_n(\\lambda) {\\cal R}  \\mathbb{E} (R_n(\\lambda) F_n(\\lambda))\\right]_{pq}.$$\nSo that if we define $$T=\\sum_{p,q=1}^n T_{pq}\\otimes E_{pq}$$ where \n$$T_{pq}\n= \\frac{1}{2\\sqrt{2}n\\sqrt{n}}\\sum_{v=1}^r \\sum_{l,i=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})    (Y_n(\\lambda))_{pl}    \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times\\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\n\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right],$$\nwe have \\begin{equation}\\label{Q}\\Vert T\\Vert=\\left\\| \\frac{1}{2\\sqrt{2}n\\sqrt{n}} Y_n(\\lambda) {\\cal R}  \\mathbb{E} (R_n(\\lambda) F_n(\\lambda)) \\right\\|=O(1\/\\sqrt{n}).\\end{equation}\nHence, in particular we have \\begin{equation}\\label{estimenracinen} \\mathbb{E}\\left[  [ R_n(\\lambda ) F_n]_{pq} \\right]= \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} +O^{(u)}_{p,q}(1\/\\sqrt{n}). \\end{equation}\n\n \n\\noindent Now, \nwe have \n$$\\left\\| \\sum_{i,l=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}}  (Y_n(\\lambda))_{pl}  \\alpha_vO^{(u)}_{i,i}(\\frac{1}{\\sqrt{n}})\\alpha_v(Y_n(\\lambda))_{ll}\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$\n\\begin{eqnarray}& \\leq& \\frac{C \\Vert(\\Im \\lambda)^{-1} \\Vert }{n} \\left(  \\sum_{l=1}^n \\Vert (Y_n(\\lambda))_{pl} \\Vert^2 \\right)^{1\/2} \\left\\{\\sum_{i=1}^n \n\\left\\| \\mathbb{E} \\left[(R_n(\\lambda)F_n(\\lambda))_{iq}\\right] \\right\\|^2 \\right\\}^{1\/2}\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~\\times  \\left\\{\\sum_{i=1}^n \\Vert O^{(u)}_{i,i}(1\/{\\sqrt{n}})\\Vert^2\\right\\}^{1\/2} \\nonumber \\\\ & =& O^{(u)}_{p,q}(1\/n) \\label{eq2} \n\\end{eqnarray}\nwhere we used (\\ref{l}) twice and \\eqref{Y} and \\eqref{norme} in the last line.\nSimilarly $$\n\\left\\|  \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl}  \\alpha_v(Y_n(\\lambda))_{ii}\\alpha_vO_{l,l}(\\frac{1}{\\sqrt{n}})\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$ \\begin{equation}\\label{eq3}=O^{(u)}_{p,q}(\\frac{1}{{n} }), \\end{equation} \n$$\n\\left\\|  \\sum_{i,l=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}}(Y_n(\\lambda))_{pl}  \\alpha_vO_{i,i}(\\frac{1}{\\sqrt{n}})\\alpha_vO_{l,l}(1\/\\sqrt{n})\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$ \\begin{equation}\\label{eq4}=O^{(u)}_{p,q}(\\frac{1}{{n \\sqrt{n}} }) \\end{equation} \n\n\n(\\ref{eq}), (\\ref{estimenracinen}) and (\\ref{eq2}), (\\ref{eq3}), \\eqref{eq4} readily yields Corollary \\ref{estimenunsurn}.\n\\end{proof}\n\\begin{corollaire} \\label{ME} With the notations of Section \\ref{Notations},\n \n \\begin{eqnarray*}G_n(\\lambda) &=&\\mathbb{E} \\left({\\rm id}_m\\otimes tr_n  R_n(\\lambda)\\right)\\\\\n&=&G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right)\n\\\\&&+L_n(\\lambda)\n+\\epsilon_n(\\lambda)\n \\end{eqnarray*}\n where \n$$L_n(\\lambda)=\\frac{1}{n}\\sum_{p=1}^n  \\left[Y_n(\\lambda)\n\\Psi(\\lambda) \\right]_{pp},$$\n(with $\\Psi(\\lambda)$ defined in Theorem \\ref{resolvante} and $Y_n(\\lambda)$ defined in Lemma \\ref{inverseY})\n and $$ \\epsilon_n(\\lambda) =O \\left( \\frac{1}{n\\sqrt{n}}\\right).$$\n Moreover \\begin{equation} \\label{L}   L_n(\\lambda) =O \\left( \\frac{1}{\\sqrt{n}}\\right).\\end{equation}\n\n\n \\end{corollaire}\n\\begin{proof}\nFirst note that, since  the distribution of  $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$  in $({\\cal A},\\tau)$  coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$, we have \\begin{eqnarray*}{\\rm id}_m\\otimes tr_n Y_n(\\lambda)&= &{\\rm id}_m\\otimes \\tau \\left((\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v)\\otimes 1_{\\cal A}-\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)} \\right)^{-1}\n\\\\&=& G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right).\\end{eqnarray*}\nThen, the corollary readily follows from Theorem \\ref{resolvante} and Proposition \\ref{55} by noting that \n\\begin{itemize}\n\\item \n\n\\noindent  \n$\\frac{1}{n}\\Vert \\sum_{p,l=1}^n (Y_n(\\lambda))_{pl}O_{lp}^{(u)}(1\/n^2)\\Vert$  \\begin{eqnarray*} & \\leq &\\frac{1}{n}\\left(\\sum_{p,l=1}^n \\Vert (Y_n(\\lambda))_{pl}\\Vert ^2 \\right)^{1\/2}\\left( \\sum_{p,l=1}^n \\Vert O^{(u)}_{lp}(1\/n^2)\\Vert ^2 \\right)^{1\/2}\\\\&=&O(\\frac{1}{n\\sqrt{n}})\\end{eqnarray*}\nwhere we used Lemma \\ref{majcarre} and \\eqref{Y} in the last line.\n\\item $$ \\Vert \\sum_{i,l,p=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{n^2\\sqrt{n}}(Y_n(\\lambda))_{pl}    \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda))_{ip} \\Vert $$\n$$\\leq   \\frac{C\\Vert \\alpha_v \\Vert^3 \\Vert (\\Im  \\lambda) ^{-1}\\Vert^2}{n^2} \\sum_{l,p=1}^n \\Vert (Y_n(\\lambda))_{pl}   \\Vert \\left(\\sum_{i=1}^n  \\Vert(R_n(\\lambda))_{ip} \\Vert^2\\right)^{1\/2}$$\n$$ \\leq   \\frac{ C\\sqrt{m}\\Vert \\alpha_v \\Vert^3 \\Vert (\\Im \\lambda )^{-1}\\Vert^3}{n}  \\left( \\sum_{p,l=1}^n \\Vert (Y_n(\\lambda))_{pl}\\Vert ^2 \\right)^{1\/2} =O(1\/\\sqrt{n})$$\nwhere we used Lemma \\ref{majcarre}, \\eqref{norme} and \\eqref{Y} in the two last lines.\n\\end{itemize}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\begin{theoreme}\\label{difftilde}\nLet $\\lambda$ be in $M_m(\\mathbb{C})$ such that \n $\\Im  \\lambda>0$, and $\\tilde G_n(\\lambda)$ as defined in \\eqref{defGntilde}. We have \n\\begin{equation} \\label{prediff}\nG_n(\\lambda)-\\tilde G_n(\\lambda)+{E_n(\\lambda)}= O(\\frac{1}{n\\sqrt{n}}),\n\\end{equation}\nwhere $E_n(\\lambda)$ is given by\\\\\n\n\\noindent $E_n(\\lambda) =$\n\\begin{equation}\n \\sum_{v=1}^r \\tilde G_n'(\\lambda) \\cdot \\alpha_v L_n(\\lambda) \\alpha_v -\\frac{1}{2} \\tilde G_n''(\\lambda) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v,  \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right) -L_n(\\lambda) \n\\end{equation}\nwith $L_n(\\lambda)$ defined in Corollary \\ref{ME}.\n\\end{theoreme}\n\\begin{proof}\nLet $\\lambda$ be in $M_m(\\mathbb{C})$ such that \n $\\Im  \\lambda>0$. Note that according to \\eqref{image}, we have $\\Im \\left( \\lambda-\\gamma - \\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v\\right)>0.$\nDefine  (using the notations of Section \\ref{Notations})\n\\begin{equation}\\label{lambdan}\\Lambda_n(\\lambda)= \\gamma +\\lambda + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda) \\alpha_v\\end{equation}\nand  $\\lambda'=\\Lambda_n(\\lambda-\\gamma - \\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v).$\nUsing Corollary \\ref{ME}, we have \n\\begin{eqnarray}\\lambda'-\\lambda &= & - \\sum_{v=1}^r \\alpha_v G_n(\\lambda) \\alpha_v + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda ) \\alpha_v) \\alpha_v \\nonumber \\\\&=& -\\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v +O(\\frac{1}{n\\sqrt{n}})\\label{lambdaprimemoinslambdaavant}\\\\&=& O(1\/\\sqrt{n}). \\label{lambdaprimemoinslambda}\n\\end{eqnarray}\n\n\n\n\\noindent Thus there exists  a polynomial $Q$ with nonnegative coefficients \nsuch that $$\\left\\|\\lambda'-\\lambda\\right\\|\\leq \\frac{Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}.$$\n-On the one hand, if $$\\frac{Q(\\Vert (\\Im  \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\geq \\frac{1}{2\\Vert (\\Im \\lambda )^{-1}\\Vert},$$ \nor equivalently \n\\begin{equation} \\label{1=O(1\/n)}\n1\\leq \\frac{2\\Vert (\\Im \\lambda )^{-1}\\Vert Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}},\n\\end{equation}\nto prove \\eqref{prediff}\nit is enough to prove that \n\\begin{equation} \\label{O(1)}\nG_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) = O(1).\n\\end{equation}\nIndeed, if we assume that \\eqref{1=O(1\/n)} and \\eqref{O(1)} hold, \nthen there exists a polynomial $\\tilde Q$ with nonnegative coefficients \nsuch that \n\\begin{eqnarray*}\n\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) \\right\\|&\\leq &\\tilde Q(\\Vert (\\Im  \\lambda )^{-1}\\Vert)\\\\\n&\\leq &\\tilde Q(\\Vert ( \\Im \\lambda )^{-1}\\Vert)\\frac{2\\Vert ( \\Im \\lambda )^{-1}\\Vert Q(\\Vert( \\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\\\\n&\\leq &\\tilde Q(\\Vert (\\Im  \\lambda )^{-1}\\Vert)(\\frac{2\\Vert (\\Im \\lambda) ^{-1}\\Vert Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}})^4.\n\\end{eqnarray*}\nHence, $$G_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) = O(\\frac{1}{n^2})$$\nso that \\eqref{prediff} holds.\nTo prove \\eqref{O(1)}, one can notice that, using \\eqref{norme} and \\eqref{normeG},\n both $G_n(\\lambda)$ and $\\tilde G_n(\\lambda)$ \nare bounded by $\\Vert (\\Im  \\lambda) ^{-1}\\Vert$, and that \\\\\n\n\\noindent \n$\\left\\|E_n(\\lambda)\\right\\|$ $$\\leq \\left\\{r \\max_{v=1}^r\\Vert \\alpha_v\\Vert^2 \\Vert (\\Im  \\lambda )^{-1}\\Vert^2 +1\\right\\} \\left\\|L_n(\\lambda)\\right\\|+r^2 \\max_{v=1}^r\\Vert \\alpha_v\\Vert^4 \\Vert (\\Im \\lambda )^{-1}\\Vert^3 \\left\\|L_n(\\lambda)\\right\\|^2 ,$$\nwhere $L_n(\\lambda)=O(1\/\\sqrt{n})$ according to \\eqref{L}.\\\\\n~~\n\n\\noindent -On the other hand, if $$\\frac{Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\leq \\frac{1}{2\\Vert (\\Im \\lambda )^{-1}\\Vert},$$ \none has : \n\\begin{equation}\\label{lambdaprime}\\left\\|\\Im \\lambda'-\\Im \\lambda\\right\\|\\leq \\left\\|\\lambda'-\\lambda \\right\\|\\leq \\frac{1}{2\\Vert ( \\Im \\lambda) ^{-1}\\Vert}\\end{equation}\nDenoting for any Hermitian matrix $H$ by $l_1(H)$ the smallest eigenvalue of $H$, we readily deduce from \\eqref{lambdaprime} that \n $l_1(\\Im \\lambda')\\geq \\frac{l_1(\\Im \\lambda)}{2}$ and \ntherefore \\begin{equation}\\label{lambdaprimepositif}\\Im \\lambda' >0. \\end{equation}\nThen, it makes sense to consider $\\tilde G_{n}(\\lambda')$ which satisfies according to \\eqref{subor}\n\\begin{eqnarray}\\tilde G_{n}(\\lambda')&= &G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda'-\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde  G_n(\\lambda')\\alpha_v) \\nonumber\\\\&=&\nG_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\omega_n(\\lambda')\\right) \\nonumber\\\\& =&G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\omega_n(\\Lambda_n(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v ))\\right). \\label{subordin}\\end{eqnarray}\n\n\n\n\n\n\n\n\n\nApplying Lemma \\ref{inversion} to $\\rho=\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v $ ( using \\eqref{image} and \\eqref{lambdaprimepositif}) we obtain that \n since $\\Im \\lambda'=\\Im  \\Lambda_n(\\rho)>0$ we have $\\omega_n(\\Lambda_n(\\rho))=\\rho$ and according to \\eqref{subordin}\n$$ \\tilde G_{n}\\left(\\lambda'\\right)\n=G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right). $$\nHence, by Corollary \\ref{ME}, we have\n\\begin{equation} \\label{termone}\nG_n(\\lambda)-\\tilde G_n(\\lambda')-{L_n(\\lambda)}=O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\nNow, we have \\\\\n\n\\noindent $\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)$\n\\begin{eqnarray*}\n&=& {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda')\\left[ (\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\}\\\\\n&=& {\\rm id}_m\\otimes \\tau \\left\\{ r_n(\\lambda)\\left[(\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\} \\\\&&+ \n{\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda) \\left[ (\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda) \\right\\}\n\\\\&&+ \n{\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda')\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda)\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda)  \\left[(\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda) \\right\\}\\\\\n&=&\\sum_{v=1}^r {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[\\alpha_v L_n(\\lambda)\\alpha_v \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\} \\\\&&+\\sum_{v,v'=1}^r {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[\\alpha_v L_n(\\lambda)\\alpha_v \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\left[\\alpha_{v'} L_n(\\lambda)\\alpha_{v'} \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\}\n\\\\&& \n+O(\\frac{1}{n\\sqrt{n}})\n\\end{eqnarray*}\nwhere we used \\eqref{lambdaprimemoinslambdaavant}, \\eqref{lambdaprimemoinslambda} and \\eqref{normeG} in the last line.\nHence we have \n$$\n\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)+\\sum_{v=1}^r\\tilde G_n'(\\lambda) \\cdot \\alpha_v L_n(\\lambda) \\alpha_v\n-\\frac{1}{2}\\tilde G_n''(\\lambda) \\cdot\\left(\\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v, \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right)$$\n\\begin{equation} \\label{termtwo}~~~~~~~~~~~~~~~~~~~~~~~~~~=O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\n(\\ref{prediff}) follows from \\eqref{termone} and \\eqref{termtwo} since \\\\\n\n\\noindent \n$\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda)+{E_n(\\lambda)}\\right\\|\\leq \n\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda')-{L_n(\\lambda)}\\right\\|$\n$$+\\left\\|\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)+\\sum_{p=1}^r\\tilde G_n'(\\lambda)\\cdot \\alpha_v{L_n(z)}\\alpha_v -\\frac{1}{2}\\tilde G_n''(\\lambda) \\cdot\\left(\\sum_{p=1}^r \\alpha_v L_n(\\lambda) \\alpha_v, \\sum_{p=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right)\\right\\|.\n$$\n\n\\end{proof}\n\\begin{remarque}\n\\eqref{L} and \\eqref{normeG} readily yield that $E_n(\\lambda)=O(\\frac{1}{\\sqrt{n}})$. Thus,  we can deduce from (\\ref{prediff}) that \\begin{equation}\\label{difgngntilde}G_n( \\lambda)-\\tilde G_n(\\lambda) =O(\\frac{1}{\\sqrt{n}}).\\end{equation}\n\\end{remarque}\n\n\n\n\\begin{proposition} \\label{estimdiff}\n For $z \\in \\C  \\setminus \\R$, let $g_n(z)$ and $\\tilde g_n(z)$ as defined in \\eqref{defpetitg} and \\eqref{defpetitgtilde} respectively. We have \n\\begin{equation} \\label{estimdiffeqn}\ng_n(z)-\\tilde g_n(z)+{\\tilde{E}_n(z)} = O(\\frac{1}{n\\sqrt{n}}),\n\\end{equation}\nwhere $\\tilde{E}_n(z)$ is given by\n\\begin{equation}\\label{defentilde}\n\\tilde{E}_n(z) = \\sum_{v=1}^r tr_m\\left( \\tilde G_n'(zI_m)\\cdot \\alpha_v \\tilde{L}_n(z)\\alpha_v\\right)  - \\tr_m \\tilde L_n(z)\n\\end{equation}\n{with }\\\\\n\n$\\displaystyle{\\tilde{L}_n(z)=\\sum_{v=1}^r}$ \\begin{eqnarray*}  && \\bigg\\{\n\\sum_{i,l,p=1}^n  \\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})}{4n^3}  \\tilde Y_n(zI_m)_{pl} \\alpha_v (\\tilde Y_n(zI_m))_{ii}   \\alpha_v (\\tilde Y_n(zI_m))_{ll} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v \\left(\\tilde  Y_n(zI_m)\\right)_{lp}\\\\\n&&+ \\sum_{i,l,p=1}^n \\frac{(\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\tilde Y_n(zI_m)_{pl} \\alpha_v  (\\tilde Y_n(zI_m))_{ii} \\alpha_v  (\\tilde Y_n(zI_m))_{li} \\alpha_v  \\left( \\tilde Y_n(zI_m)\\right)_{lp} \\\\\n&&+\\sum_{i,l,p=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} }  \\tilde Y_n(zI_m)_{pl} \\alpha_v  (\\tilde Y_n(zI_m))_{ii} \\alpha_v  (\\tilde Y_n(zI_m))_{ll} \\alpha_v  \\left(\\tilde Y_n(zI_m)\\right)_{ip}\\\\\n&&+\\sum_{i,l,p=1}^n  \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }   \\tilde Y_n(zI_m)_{pl} \\alpha_v  (\\tilde Y_n(zI_m))_{il}  \\alpha_v  (\\tilde Y_n(zI_m))_{ii}  \\alpha_v  \\left (\\tilde Y_n(zI_m)\\right)_{lp}\\bigg\\}\n\\end{eqnarray*}\nwhere  $\\tilde Y_n$ and $\\tilde G_n$ were defined in \\eqref{ytilde} and \\eqref{defGntilde} respectively  so that \n \\begin{eqnarray*}\\tilde Y_n(zI_m)&= &\\left((zI_m -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde  G_n(zI_m)\\alpha_v)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\\\&=&\\left(\\omega_n(zI_m)\\otimes I_n -  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\end{eqnarray*}\nand  \\begin{eqnarray*}\\tilde G_n(zI_m)&=&{\\rm id}_m\\otimes \\tau\\left( (zI_m-\\gamma) \\otimes 1_{\\cal A} - \\sum_{v=1}^r \\alpha_v \\otimes x_v-\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}\\right)^{-1}\\\\&=&{\\rm id}_m \\otimes \\tau\\left( zI_m\\otimes  1_{\\cal A} - s_n\\right)^{-1} .\\end{eqnarray*}\n\\end{proposition}\n\\begin{proof}\nLet $z\\in \\mathbb{C}\\setminus \\mathbb{R}$ such that $\\Im  z >0$.\nTheorem \\ref{difftilde} yields $$ g_n(z)- \\tilde g_n(z)+\\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v -tr_mL_n(zI_m)\n$$ \\begin{equation}\\label{doubleetoile}-\\frac{1}{2} \\tr_m \\tilde G_n''(zI_m) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v,  \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v\\right)=O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\nFirst note that, by Riesz-Fr\\'echet's Theorem (and using \\eqref{normeG} and \\eqref{L}), there exists $B_n^{(1)}(z)$ and $B_n^{(2)}(z)$ in $M_m(\\C)$ such that  $$\\Vert B_n^{(1)}(z) \\Vert_2 \\leq \\vert \\Im  z \\vert^{-2} \\left( \\sum_{v=1}^r \\Vert \\alpha_v\\Vert^2 \\right)=O(1), $$  \\begin{equation}\\label{B2}\\Vert B_n^{(2)}(z) \\Vert_2=O(1\/\\sqrt{n}), \\end{equation} and \n$$\\sum_{v=1}^r \\tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v = \\Tr_m \\left[ B_n^{(1)}(z) L_n(zI_m) \\right],$$\n\\begin{equation}\\label{etoilehat}\\tr_m \\tilde G_n''(zI_m) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v,  \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v\\right)= \\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right].\\end{equation}\n\n\\noindent Recall that for $\\lambda \\in M_m(\\C)$ such that $\\Im  \\lambda>0$,  $$L_n(\\lambda)=\\frac{1}{n}\\sum_{p=1}^n  \\left[Y_n(\\lambda)\n\\Psi(\\lambda) \\right]_{pp},$$ where $\\Psi$ is defined in \\eqref{psi}.\nFirst, note that according to \\eqref{YPSIF}, we have (setting  $c_{l,i,v}= \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}} $)\\\\\n~~\n\n\\noindent $\\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right]$\n\\begin{eqnarray*} \n\\hspace*{-1.6cm}=&\\hspace*{-0.3cm}\n\\displaystyle{ \\sum_{v=1}^r \\sum_{i,l=1}^n} \\frac{c_{l,i,v}}{n^2\\sqrt{n}}  \\mathbb{E} \\Tr_m \\left\\{\\alpha_v(R_n(zI_m))_{ii}\\alpha_v(R_n(zI_m))_{ll}\\alpha_v \n \\left(R_n(zI_m) (B_n^{(2)}(z)\\otimes I_n)Y_n(zI_m)\\right)_{il}\\right\\}\n\\\\&+O(\\frac{1}{n\\sqrt{n}}).\n\\end{eqnarray*}\nMoreover, we have \n $$  \\sum_{v=1}^r\\sum_{i,l=1}^n \\frac{ c_{l,i,v}}{n^2\\sqrt{n}} \\mathbb{E} \\Tr_m \\left\\{\\alpha_v(R_n(zI_m))_{ii}\\alpha_v(R_n(zI_m))_{ll}\\alpha_v \n \\left(R_n(zI_m) (B_n^{(2)}(z)\\otimes I_n)Y_n(zI_m)\\right)_{il}\\right\\}$$ \n$$= O(\\frac{1}{n\\sqrt{n}}),$$\nwhere we used  \\eqref{Odenracinepas}, \\eqref{norme}, \\eqref{Y} and  \\eqref{B2}. \nHence \\begin{equation}\\label{etoiletilde}\\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right]= O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\n\\eqref{doubleetoile}, \\eqref{etoilehat} and \\eqref{etoiletilde} yield  \\begin{equation}\\label{doubleetoilehat}g_n(z)- \\tilde g_n(z)+\\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v -tr_mL_n(zI_m)=O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\n Thus, in the following we will consider $\\frac{1}{n}\\sum_{p=1}^n \\tr_m B_n(\\lambda) \\left[Y_n(\\lambda)\nT(\\lambda) \\right]_{pp}$  for any $\\lambda\\in M_m\\left( \\mathbb{C}\\right)$ such that $\\Im \\lambda >0$,  for each term $T(\\lambda)$ involving in \\eqref{psi} and any $m\\times m$ matrix $B_n(\\lambda)=O(1)$ (in the interests of simplifying notations, we deal with any $\\lambda\\in M_m\\left( \\mathbb{C}\\right)$  such that $\\Im \\lambda >0$ instead of $zI_m$).\nWe set   ${\\cal B}(\\lambda)= B_n(\\lambda) \\otimes I_{n}$. \n\n\n\\noindent First, for any fixed $v \\in \\{1,\\ldots,r\\}$,\\\\\n \n$\\left|\\frac{1}{n}\\sum_{p,l=1}^n \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v    [R_n(\\lambda ) ]_{lp} \\right]\\right|$\n\\begin{eqnarray*}\n&=& \\left|\\tr_m \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v id\\otimes  tr_n  [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)] \\right]\\right|\\\\\n&=& \\left|\\tr_m \\mathbb{E}\\left\\{ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v\\right. \\right.\\\\&&\\left.\\left.~~~~~~\\times \\left[id\\otimes  tr_n  [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)]\n-\\mathbb{E}(id\\otimes  tr_n  [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)]) \\right]\\right\\}\\right|\\\\&\n\\leq&\\frac{\\Vert B_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^2 C m}{n^2}\n \\Vert (Im(\\lambda))^{-1}\\Vert^5\\\\&=& O(1\/n^2).\\end{eqnarray*}\nwhere we used Cauchy Schwarz's inequality, \\eqref{norme}, \\eqref{Y} and Lemma \\ref{var} in the last line.\\\\\nWe also have\n$$\\left|\\sum_{i,p,l=1}^n  \\frac{ (\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{n^2\\sqrt{n}} \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl}    \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n (R_n(\\lambda))_{ip}\\}\\right|$$\n\\begin{eqnarray*}&=&\\frac{1}{n^2\\sqrt{n}} \\left| \\sum_{i,l=1}^n  (\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})  \\mathbb{E}\\tr_m \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n (R_n(\\lambda){\\cal B}(\\lambda) Y_n(\\lambda))_{il}\\}\\right|\n\\\\&=&O(\\frac{1}{n\\sqrt{n}})\n\\end{eqnarray*}\nwhere we used \\eqref{Odenpas}, \\eqref{norme} and  \\eqref{Y}.\n\n\n\\noindent Now, let us investigate  the terms corresponding to the the  $M^{(4)}(v,i,l,j)$'s in \\eqref{psi}. We have \n$$  \\frac{1}{4n^3} \\sum_{i,p,l=1}^n  (\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})\\mathbb{E} \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl}  \\alpha_v (R_n(\\lambda))_{il}  \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il}\\alpha_v  (R_n(\\lambda))_{ip}$$\n\\begin{eqnarray*}\n&=& \\frac{1}{4n^3} \\sum_{i,l=1}^n  (\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) \\mathbb{E}\\tr_m    \\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il} \\alpha_v  \\left(R_n(\\lambda){\\cal B}(\\lambda)Y_n(\\lambda)\\right)_{il}\n\\\\&=&O(1\/n^2) \\end{eqnarray*}\nwhere we used \\eqref{Odenpas}, \\eqref{norme} and \\eqref{Y}.\nSimilarly the terms corresponding to  \\eqref{deuxcas4}, \\eqref{quatrecas4}, \\eqref{tilde1},   \\eqref{tilde2}, \\eqref{tilde3} and  \\eqref{tilde4}\nare $O(1\/n^2).$\\\\\n\n\n\n\\noindent Since moreover each term in \\eqref{psi} corresponding to the $M^{(3)}(v,i,i,j)$, $M^{(4)}(v,i,i,j)$ and $M^{(5)}(v,i,l,j)$ leads obviously to a term which is a $O(\\frac{1}{n\\sqrt{n}})$, it readily follows from \\eqref{doubleetoilehat} that $$ g_n(z)- \\tilde g_n(z)+ \\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v \\hat L_n(zI_m) \\alpha_v -tr_m \\hat L_n(zI_m)=O(\\frac{1}{n\\sqrt{n}}),$$\nwhere \\\\\n\n\\noindent $\\hat L_n (zI_m)=\\sum_{v=1}^r \\sum_{i,p,l=1}^n $ $$\\bigg\\{  \\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})}{4n^3} ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{(R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v(R_n(zI_m))_{ii}\\alpha_v (R_n(zI_m))_{lp}\\right\\}$$\n$$+\\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } ( Y_n(zI_m)) _{pl} \\alpha_v  \\mathbb{E} \\left\\{(R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{li} \\alpha_v (R_n(zI_m))_{lp}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }  ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{ (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v (R_n(zI_m))_{ip}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }  ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{(R_n(zI_m))_{il} \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{lp}\\right\\}\\bigg\\}$$\nFor any $m\\times m$ deterministic matrix $B_n(z)$, \\\\\n~~\n\n\\noindent $\\tr_m B_n (z)\\hat L_n (zI_m)=\\sum_{v=1}^r\\sum_{i,l=1}^n$ $$\\bigg\\{\\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) }{4n^3}  \\mathbb{E} \\left\\{\\tr_m  \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll}\\alpha_v (R_n(zI_m))_{ii}\\alpha_v (R_n(zI_m){\\cal B}(z)Y_n(zI_m))_{ll}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m \\left\\{ \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{li} \\alpha_v \\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right\\}$$\n$$+ \\frac{ (\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m\\left\\{ \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v \\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{il}\\right\\}$$\n$$+\\frac{ (\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m\\left\\{ \\alpha_v (R_n(zI_m))_{il} \\alpha_v (R_n(zI_m))_{ii} \\alpha_v\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right\\}\\bigg\\}$$\n\n\\noindent Hence (using a decomposition similar to \\eqref{decomposition}) Lemma \\ref{var} readily yields that \\\\\n\n\n\\hspace*{-0.8cm} $\\tr_m B_n \\hat L_n (zI_m) + O(\\frac{1}{n\\sqrt{n}})= \\sum_{v=1}^r \\sum_{i,l=1}^n \\tr_m$ $$\\hspace*{-0.8cm}\\bigg\\{\\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) }{4n^3} \\alpha_v \\mathbb{E}\\left[ (R_n(zI_m))_{ii} \\right]  \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ll}\\right] \\alpha_v\\mathbb{E}\\left[ (R_n(zI_m))_{ii} \\right]\\alpha_v [ \\mathbb{E}\\left[\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right]$$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }   \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right]  \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{li}\\right] \\alpha_v   \\mathbb{E}\\left[\\left(R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right] $$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }     \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right]  \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{ll}\\right]  \\alpha_v  \\mathbb{E}\\left[ \\left(R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{il}\\right]$$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} }  \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{il} \\right] \\alpha_v  \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right]  \\alpha_v  \\mathbb{E}\\left[\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right]\\bigg\\}.$$\nNote that, with $Y_n$ and $\\tilde Y_n$  defined in \\eqref{y} and \\eqref{ytilde}, we have  $$Y_n(zI_m)-\\tilde Y_n(zI_m) =\\sum_{v=1}^r Y_n(zI_m) \\left[\\alpha_v (G_n(zI_m)-\\tilde G_n(zI_m)) \\alpha_v \\otimes I_n\\right] \\tilde Y_n(zI_m)$$ so that, using (\\ref{difgngntilde}), \\eqref{Y} and \\eqref{Y2}, we can deduce that \\begin{equation}\\label{YmoinsYtilde} \\Vert Y_n(zI_m) -\\tilde Y_n(zI_m)\\Vert =O(1\/\\sqrt{n}). \\end{equation}\nNow, (\\ref{estimenracinen}) and (\\ref{YmoinsYtilde}) obviously yield that, up to a $O(\\frac{1}{n\\sqrt{n}})$ correction term, one can replace any $R_n(zI_m)$  and  $Y_n(zI_m)$ by $\\tilde Y_n(zI_m)$ in any term in the sum corresponding to the fourth cumulants. Now, using (\\ref{lp}), (\\ref{estimenracinen}) and (\\ref{YmoinsYtilde}) also  yield that, up to a $O(\\frac{1}{n\\sqrt{n}})$ correction term, for any $p=1,\\ldots,n$, one can replace  $(R_n(zI_m))$ and  $(Y_n(zI_m))$ by $\\tilde Y_n(zI_m)$ in any diagonal term $(R_n(zI_m))_{pp}$ or $(R_n(zI_m){\\cal B}(z)Y_n(zI_m))_{pp} $ in the sums corresponding to the third cumulants.\\\\\nFinally, \nassume that   for $i=1,2,$ $Q^{(i)}= \\tilde Y_n(zI_m) $ or $\\tilde Y_n(zI_m){\\cal B}(z)\\tilde Y_n(zI_m)$. Let us consider any  $ Q^{(3)}=  \\sum_{i,l=1}^nQ^{(3)}_{il}\\otimes E_{il}$.\nIt is clear that if there exists some polynomial $Q$ with nonnegative coefficients such that for any $i,l \\in \\{1,\\ldots,n\\}^2$, $\\Vert Q^{(3)}_{il}\\Vert \\leq \\frac{Q(\\vert \\Im  z \\vert^{-1})}{n}$, then \n$$\\frac{1}{n^2 \\sqrt{n}} \\sum_{i,l=1}^n\\Vert Q^{(1)}_{ii}\\Vert \\Vert Q^{(2)}_{ll}\\Vert \\Vert Q^{(3)}_{il}\\Vert =O(1\/n\\sqrt{n}).$$\nNow, if $\\Vert Q^{(3)}\\Vert =O(1\/\\sqrt{n})$,\nwe have $$\\frac{1}{n^2 \\sqrt{n}} \\sum_{i,l=1}^n\\Vert Q^{(1)}_{ii}\\Vert \\Vert Q^{(2)}_{ll}\\Vert \\Vert Q^{(3)}_{il}\\Vert $$ $$\\leq \\frac{1}{n \\sqrt{n}} \\vert (\\Im  z)^{-1}\\vert^q\\left(\\sum_{i,l=1}^n\\Vert Q^{(3)}_{il}\\Vert^2 \\right)^{1\/2} \\leq  \\frac{\\sqrt{m}}{n } \\vert (\\Im  z)^{-1}\\vert^q\\Vert Q^{(3)}\\Vert =O(1\/n\\sqrt{n}) $$ for some $q\\in \\mathbb{N}\\setminus{\\{0\\}},$ where we used \\eqref{lp}.\n\\\\\nIt is then clear that using Corollary \\ref{estimenunsurn}, \\eqref{Q} and (\\ref{YmoinsYtilde}), \nup to a $0(\\frac{1}{n\\sqrt{n}})$ correction term, for any $(i,l)\\in \\{1,\\ldots,n\\}^2$, one can replace   $R_n(zI_m)$ and  $Y_n(zI_m)$ by $\\tilde Y_n(zI_m)$ in any non-diagonal term $(R_n(zI_m))_{il}$, $(R_n(zI_m))_{li}$ or  $(R_n(zI_m){\\cal B}Y_n)_{il}$ in the sums corresponding to the third cumulants.\nHence \\eqref{estimdiffeqn} is proved for any $z\\in \\mathbb{C}$ such that $\\Im  z >0$. \\\\Set $\\alpha=(\\alpha_1,\\ldots \\alpha_r)$, $\\beta=(\\beta_1,\\ldots, \\beta_t)$. Let us denote for a while $g_n=g_n^{\\alpha,\\beta,\\gamma}$, $\\tilde g_n=\\tilde g_n^{\\alpha,\\beta,\\gamma}$ and $\\tilde E_n=\\tilde E_n^{\\alpha,\\beta,\\gamma}$. Note that we have similarly for  any $z\\in \\mathbb{C}$ such that $\\Im  z >0$, \n\\begin{equation} \\label{estimdiffeqnmoins}\ng_n^{-\\alpha,-\\beta,-\\gamma}(z)-\\tilde g^{-\\alpha,-\\beta,-\\gamma}_n(z)+{\\tilde{E}^{-\\alpha,-\\beta,-\\gamma}_n(z)} = O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation} Thus, since $g_n^{-\\alpha,-\\beta,-\\gamma}(z)=-g_n^{\\alpha,\\beta,\\gamma}(-z)$, $\\tilde g_n^{-\\alpha,-\\beta,-\\gamma}(z)=-\\tilde g_n^{\\alpha,\\beta,\\gamma}(-z)$ and  $\\tilde E_n^{-\\alpha,-\\beta,-\\gamma}(z)=-\\tilde E_n^{\\alpha,\\beta,\\gamma}(-z)$, it readily follows that  \n\\eqref{estimdiffeqn} is also valid for any $z \\in \\mathbb{C}$ such that $\\Im  z <0$.\n\\end{proof}\n\n\n\n\\subsection{ From Stieltjes transform estimates to spectra.}\nWe start with the following key lemma.\n\\begin{lemme}\\label{LSt}\nFor any fixed large n, $ \\tilde E_n$ defined in Proposition \\ref{estimdiff} is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$ and such that $\\nabla_n(1)=0$ .\n\\end{lemme}\n\n\nThe proof relies on the following characterization already used in \\cite{Schultz05}. \n\n\\begin{theoreme}\\label{TS}\\cite{Tillmann53}\n\\begin{itemize}\n\\item Let $\\Lambda $ be a distribution on $\\R$ with compact support. \nDefine the Stieltjes transform of $\\Lambda $, \n$ l:\\C\\setminus \\R \\rightarrow \\C$ by \n$$l(z)=\\Lambda \\left( \\frac{1}{z-x}\\right) .$$\n\\noindent Then $l$ is analytic on $\\C\\setminus \\R$\nand has an analytic continuation to $\\C\\setminus {\\rm supp}(\\Lambda )$. \nMoreover\n\\begin{itemize}\n\\item[($c_1$)] $l(z)\\rightarrow 0$ as $|z|\\rightarrow \\infty ,$\n\\item[($c_2$)] there exists a constant $C > 0$, \nan integer $q\\in \\N$ and a compact set $K\\subset \\R$ containing ${\\rm supp}(\\Lambda )$, \nsuch that for any $z\\in \\C\\setminus \\R$, \n$$|l(z)|\\leq C\\max \\{ {\\rm dist}(z,K)^{-q}, 1\\} ,$$\n\\item[($c_3$)] for any $\\phi \\in \\cal C^\\infty (\\R, \\R)$ with compact support\n$$\\Lambda (\\phi )=\\frac{i}{2\\pi }\\lim _{y\\rightarrow 0^+} \\int _\\R\\phi (x)[l(x+iy)-l(x-iy)]dx.$$\n\\end{itemize}\n\\item Conversely, if $K$ is a compact subset of $\\R$ \nand if $l:\\C \\setminus K\\rightarrow \\C$ is an analytic function\nsatisfying ($c_1$) and ($c_2$) above, \nthen $l$ is the Stieltjes transform of a compactly supported distribution $\\Lambda $ on $\\R$. \nMoreover, ${\\rm supp}(\\Lambda )$ is exactly the set of singular points of $l$ in $K$. \n\\end{itemize}\n\\end{theoreme}\n\\begin{lemme} The singular points of $\\tilde{E}_N$ defined in \\eqref{defentilde} are included in the \nspectrum of  $s_n=\\gamma \\otimes 1_{\\cal A}+\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$. \n\\end{lemme}\n\\begin{proof}\nLet us start by noting that, as it follows from the definition \\eqref{defentilde} of $\\tilde{E}_N$,\nit is enough to show that $\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\n\\in GL_{nm}(\\mathbb C)$ (the group of invertible $nm\\times nm$ complex matrices) for any $z$\nin the domain of definition of $\\mathbb C\\ni z\\mapsto\\omega_n(zI_m)\\in M_{m}(\\mathbb C).$\n\nAssume towards contradiction that $x_0\\in\\mathbb C$ is in the domain of $\\omega_n(\\cdot I_m)$, and \nyet $\\omega_n(x_0I_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}$ is not invertible. First,\nobserve that a point $x_0$ with this property must be isolated and real. Indeed, otherwise the zeros of\nthe analytic map $\\mathbb C\\ni z\\mapsto\\det\\left(\\omega_n(zI_m)\\otimes I_n-\n\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)\\in\\mathbb C$ would have $x_0$ as a cluster point\nin the interior of its domain (which coincides with the domain of $\\omega_n(\\cdot I_m)$), and thus\nit would be identically equal to zero. However,  \n$\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}$ is invertible when $\\Im  z\\neq0$,\nproviding us with a contradiction. Consider now such an isolated $x_0$. Recall \n\\begin{eqnarray*}\n\\tilde{g}_n(z) & = & (\\tr_m\\otimes\\tau)\\left(\\left(\n\\omega_n(zI_m)\\otimes1_\\mathcal A-\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}\\right)^{-1}\\right)\\\\\n& = & (\\tr_m\\otimes\\tr_n)\\left(\\left(\n\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)^{-1}\\right),\n\\end{eqnarray*}\nfrom before (the second equality is justified by the hypothesis that the distribution of $(A_n^{(1)},\n\\dots,A_n^{(t)})$ with respect to $\\tr_n$ coincides with the distribution of $(a_n^{(1)},\\dots,a_n^{(t)})$\nwith respect to $\\tau$). We have seen that $\\tilde{g}_n$ is defined exactly on the complement of the \nspectrum of $s_n$. Thus, it is enough to show that, given an analytic function \n$f\\colon\\mathbb C^+\\to H^{-}(M_p(\\mathbb C))=\\{b\\in M_p(\\mathbb{C}), \\Im  b <0\\}$, and $x_0\\in\\mathbb R$ with the property that there\nexists some $\\epsilon>0$ such that $f$ extends analytically through $(x_0-\\epsilon,x_0)\\cup(x_0,\nx_0+\\epsilon)$ with self-adjoint values, then either both or none of $f$ and $\\tr_p\\circ f$ extend \nanalytically to $x_0$. We shall then apply this to $p=mn$ and $f(z)=\\left(\\omega_n(zI_m)\\otimes I_n-\n\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)^{-1}$ to conclude.\n \nIt is clear that if $f$ extends analytically through $x_0$, then so does $\\tr_p\\circ f$. Assume \ntowards contradiction that $\\tr_p\\circ f$ extends analytically through $x_0$, but $f$ does not.\nConsider an arbitrary system $\\{e_1,\\dots,e_p\\}$ of minimal mutually orthogonal projections\nin $M_p(\\mathbb C)$. It is clear that $z\\mapsto e_jf(z)e_j\\in e_jM_p(\\mathbb C)e_j\\simeq\\mathbb C$ is \nanalytic wherever $f$ is. Moreover, $\\Im  e_jf(z)e_j\\leq0$ whenever $z\\in\\mathbb C^+$, and \n$e_jf(z)e_j\\in\\mathbb R$ whenever $f(z)$ is self-adjoint. If $z\\mapsto e_jf(z)e_j$ extends analytically\nthrough $x_0$ for any $e_j\\in\\{e_1,\\dots,e_p\\}$ and all systems $\\{e_1,\\dots,e_p\\}$, then \n$z\\mapsto\\varphi(f(z))$ extends analytically through $x_0$ for all linear functionals $\\varphi\\colon\nM_p(\\mathbb C)\\to\\mathbb C$. This implies that $f$ itself is analytic around $x_0$. On the other hand,\nif there exists a system $\\{e_1,\\dots,e_p\\}$ of minimal mutually orthogonal projections\nin $M_p(\\mathbb C)$ which contains an $e_j$ such that $z\\mapsto e_jf(z)e_j$ does not \nextend analytically through $x_0$, then, as $z\\mapsto e_jf(z)e_j$ does extend analytically \nwith real values through $(x_0-\\epsilon,x_0)\\cup(x_0,x_0+\\epsilon)$ and maps $\\mathbb C^+$\ninto $\\mathbb C^-\\cup\\mathbb R$, it follows that $x_0$ is a simple pole of \n$z\\mapsto e_jf(z)e_j$ and $\\lim_{y\\to0}iye_jf(x_0+iy)e_j\\in(0,+\\infty)$ by the Julia-Carath\\'eodory\nTheorem applied to $1\/e_jf(z)e_j$ (see (2) Theorem 2.1 in \\cite{Serban}). But then\n$\\tr_p(f(z))=\\sum_{k=1}^pe_kf(z)e_k$, so that \n$$\n\\lim_{y\\to0}iy\\tr_p(f(x_0+iy))=\\sum_{k=1}^p\\lim_{y\\to0}iye_kf(x_0+iy)e_k>0,\n$$\nwhich contradicts the assumption that $\\tr_p\\circ f$ extends analytically through $x_0$.\n\n\\end{proof}\n\n\nNow, we are going to show that for any fixed large $n$, \n$\\tilde{E}_n$ satisfies ($c_1$) and ($c_2$) of Theorem \\ref{TS}. \n\n\n\n\\noindent First note that there exists a polynomial Q in two variables  with positive coefficients such that \n\\begin{equation}\\label{normedetildeEn} \\vert \\tilde {E}_n(z)\\vert  \\leq \\Vert \\tilde Y(zI_m))\\Vert^4 Q(\\Vert \\tilde Y(zI_m))\\Vert, \\Vert r_n(zI_m)\\Vert ).\n\\end{equation}\n\n\\noindent Let $C > 0$ be such that, for all  $n$, $\\mbox{sp}(\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)})\\subset [-C;C]$ and\n$\\mbox{sp}(\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)})\\subset [-C;C]$.\n\n\\noindent Let $d > C + \\sqrt{r}\\max_{v=1}^r\\Vert \\alpha_v \\Vert $. \nFor any $z\\in \\C$ such that $|z| > \\Vert \\gamma \\Vert +d $, \n\\begin{eqnarray*}\\Vert \\gamma +\\sum_{v=1}^r \\alpha_v \\tilde G_n(zI_m) \\alpha_v \\Vert &\\leq& \\Vert \\gamma \\Vert + \\frac{r \\max_{v=1}^r\\Vert \\alpha_v \\Vert ^2}{\\vert z\\vert -C}\\\\&\\leq &\\Vert \\gamma \\Vert + \\frac{r \\max_{v=1}^r\\Vert \\alpha_v \\Vert ^2}{d-C} \\\\&\n<& \\Vert \\gamma \\Vert + \\frac{(d -C)^2}{d -C} \\\\&=& \\Vert \\gamma \\Vert + d-C\\end{eqnarray*}\n Thus, $$\\Vert  \\gamma+\\sum_{v=1}^r\\alpha_v\\tilde G_n(zI_m) \\alpha_v +\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\\Vert \\leq\\Vert \\gamma  \\Vert +d $$\nso that we get that for any $z\\in \\C$ such that $|z| > \\Vert \\gamma\\Vert +d $, \n\\begin{eqnarray*}\\Vert \\tilde Y_n(zI_m)\\Vert &=&\n\\Vert ((zI_m- \\gamma -\\sum_{v=1}^r \\alpha_v \\tilde G_n(zI_m) \\alpha_v)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)})^{-1}\\Vert\\\\\n& \\leq &\\frac{1}{\\vert z\\vert -\\Vert \\gamma \\Vert -d }.\n\\end{eqnarray*}\nWe get readily from \\eqref{normedetildeEn} that, for $|z| > \\Vert \\gamma\\Vert +d $, \n\\begin{equation}\\label{bound} \\vert \\tilde{E}_n(z)\\vert \\leq \\frac{1} {(\\vert z\\vert -\\Vert \\gamma \\Vert -d )^4}Q\\left(\\frac{1} {(\\vert z\\vert -\\Vert \\gamma \\Vert -d )},\\frac{1}{(|z|-C)} \\right) .\\end{equation}\nThen, it is clear than $\\vert\\tilde{E}_n(z)\\vert \\rightarrow 0$ \nwhen $|z|\\rightarrow +\\infty $ and ($c_1$) is satisfied.\\\\\n\\noindent \nNow we are going to prove ($c_2$) using  the approach of \\cite{Schultz05}(Lemma 5.5). \nDenote by $\\mathcal {E}_n$ the convex envelope of the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$\nand define $$K_n:=\\left\\{ x\\in \\R; {\\rm dist}(x, \\mathcal {E}_n)\\leq 1\\right\\} $$\n\\noindent \nand $$D_n=\\left\\{ z\\in \\C; 0 < {\\rm dist}(z, K_n)\\leq 1\\right\\} .$$\n\\begin{itemize}\n\\item Let $z\\in D_n\\cap (\\C\\setminus \\R)$ with $\\Re  (z)\\in K_n$. \nWe have ${\\rm dist}(z, K_n)=|\\Im  z|\\leq 1$. \nWe have from \\eqref{normedetildeEn},  (\\ref{normeG}) and  \\eqref{Y2} that \n$$\\vert \\tilde{E}_n(z)\\vert \\leq {\\vert \\Im  z \\vert^{-4}} Q\\left(\\vert \\Im  z \\vert^{-1},\\vert \\Im  z \\vert^{-1}\\right).$$\nNoticing that $1\\leq {\\vert \\Im  z\\vert^{-1}}$, \nwe easily deduce that there exists some constant $C_0$ and some  number $q_0 \\in \\mathbb{N}\\setminus \\{0\\}$ such that \nfor any $z\\in D_n\\cap \\C\\setminus \\R$ with $\\Re  (z)\\in K_n$, \n\\begin{eqnarray*}\n\\vert \\tilde{E}_n(z)\\vert &\\leq &C_0|\\Im  z|^{-q_0}\\\\\n&\\leq &C_0{\\rm dist}(z, K_n)^{-q_{0}}\\\\\n&\\leq &C_0\\max ({\\rm dist}(z, K_n)^{-q_{0}}; 1)\n\\end{eqnarray*}\n\\item Let $z\\in D_n\\cap (\\C\\setminus \\R)$ with $\\Re (z)\\notin K_n$. \nThen ${\\rm dist}(z,{\\rm sp}(s_n))\\geq 1$. \nSince $\\tilde{E}_n$ is bounded on compact subsets of $\\C\\setminus {\\rm sp}(\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)})$,\nwe easily deduce that there exists some constant $C_1(n)$ \nsuch that for any $z\\in D_n$ with $\\Re  (z)\\notin K_n$, \n$$\\vert \\tilde{E}_n(z)\\vert \\leq C_1(n)\\leq C_1(n)\\max ({\\rm dist}(z, K_n)^{-q_0}; 1).$$\n\\item Since $\\vert \\tilde{E}_n(z)\\vert \\rightarrow 0$ when $|z|\\rightarrow +\\infty $, \n$\\tilde{E}_n$ is bounded on $\\C\\setminus \\overline{D_n}$. \nThus, there exists some constant $C_2(n)$ such that for any $z\\in \\C\\setminus \\overline{D_n}$, \n$$\\vert \\tilde{E}_n(z)\\vert \\leq C_2(n)=C_2(n)\\max ({\\rm dist}(z, K_n)^{-q_0}; 1).$$\n\\end{itemize}\nHence ($c_2$) is satisfied with $C(n)=\\max (C_0, C_1(n), C_2(n))$ and $l=q_0$.  Thus, Theorem \\ref{TS} implies that for any fixed large n, $ \\tilde E_n$ defined in Proposition \\ref{estimdiff} is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$.  Following the proof of Lemma 5.6 in \\cite{Schultz05} and using (\\ref{bound}), \none can show that $\\nabla _n(1)=0$. \nThe proof of Lemma \\ref{LSt} is complete. $\\Box$\\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent Now,  \\eqref{spectre3} can be deduced from \\eqref{estimdiffeqno} by an approach inspired by \\cite{HT} and \\cite{Schultz05} as follows.  \\\\\nUsing the inverse Stieltjes tranform, we get respectively that, \nfor any $\\varphi _n $ in ${\\cal C}^\\infty (\\R, \\R)$ with compact support, \n$$\\mathbb{E} [\\tr_m \\otimes \\tr _n(\\varphi _n(S_n))]-\\tr_m\\otimes \\tau(\\varphi_n(s_n))\n+{\\nabla _{n}(\\varphi _n)}$$\n$$=\\frac{1}{\\pi }\\lim _{y\\rightarrow 0^+}\\Im  \\int _\\R\\varphi _n(x)\\epsilon_n(x+iy)dx,$$\nwhere $\\epsilon_n(z)=\\tilde g_n(z)-g_n(z)-\\tilde{E}_n(z)$ satisfies, \naccording to Proposition \\ref{estimdiff}, for any $z\\in \\C\\setminus \\R$, \n\\begin{equation*}\\label{estimgdif}\n\\vert \\epsilon_n(z)\\vert \\leq \\frac{1}{n\\sqrt{n}}P(\\vert \\Im z \\vert ^{-1}) .\n\\end{equation*} \nWe refer the reader to the Appendix of \\cite{CD07} \nwhere it is proved using the ideas of \\cite{HT} that if  $h$ is an analytic function on $\\C\\setminus \\R$ which satisfies\n\\begin{equation*}\\label{nestimgdif}\n\\vert h(z)\\vert \\leq P(\\vert \\Im  z\\vert ^{-1})\n\\end{equation*} \n\\noindent for some polynomial $P$ with nonnegative coefficients and degree $k$, then\nthere exists a polynomial $Q$ such that\n$$\\limsup _{y\\rightarrow 0^+}\\vert \\int _\\R\\varphi _n(x)h(x+iy)dx\\vert $$\n$$\\leq \\int _\\R\\int _0^{+\\infty }\\vert (1+D)^{k+1}\\varphi _n(x)\\vert Q(t)\\exp(-t)dtdx$$\nwhere $D$ stands for the derivative operator.\nHence, if there exists $K > 0$ such that, for all large $n$, \nthe support of $\\varphi _n$ is included in $[-K, K]$ and \n$\\sup _n\\sup _{x \\in [-K, K]}\\vert D^p\\varphi _n(x)\\vert =C_p < \\infty$ for any $p\\leq k+1$, \ndealing with $h(z) =n\\sqrt{n}\\epsilon_n(z)$, we deduce that there exists $C>0$ such that for all large $n$,\n\\begin{equation*} \\label{majlimsup1} \n\\limsup _{y\\rightarrow 0^+}\\vert \\int _\\R \\varphi _n(x)\\epsilon_n(x+iy)dx\\vert \\leq \\frac{C}{n\\sqrt{n}}\n\\end{equation*} \nand then \n\\begin{equation}\\label{StS} \n\\mathbb{E} [\\tr_m \\otimes \\tr _n(\\varphi _n(S_n))]-\\tr_m\\otimes \\tau(\\varphi_n(s_n))\n+{\\nabla _{n}(\\varphi _n)}=O(\\frac{1}{n\\sqrt{n}}).  \n\\end{equation}\nLet $\\rho \\geq 0$ be in ${\\cal C}^\\infty (\\R, \\R)$ \nsuch that its support is included in $[-1;1]$ and $\\int \\rho (x)dx=1$. \nLet $0 < \\epsilon < 1$. \nDefine for any $x\\in \\mathbb{R}$, $$\\rho _{\\frac{\\epsilon }{2}}(x)=\\frac{2}{\\epsilon }\\rho(\\frac{2x}{\\epsilon }).$$\nSet $$K_n(\\epsilon )=\\{ x, {\\rm dist}(x, {\\rm sp}(s_n))\\leq \\epsilon \\}$$ \nand define for any $x\\in \\mathbb{R}$, $$f_n(\\epsilon )(x)=\\int _\\mathbb{R} \\1 _{K_n(\\epsilon )}(y)\\rho _{\\frac{\\epsilon }{2}}(x-y)dy.$$\nThe function $f_{n}(\\epsilon )$ is in ${\\cal C}^\\infty (\\mathbb{R}, \\mathbb{R})$, \n$f_{n}(\\epsilon )\\equiv 1$ on $K_n(\\frac{\\epsilon }{2})$; \nits support is included in $K_n(2\\epsilon )$. \nSince there exists $K$ such that, for all large $n$, the spectrum of $s_n$ \nis included in $[-K;K]$, for all large $n$ the support of $f_n(\\epsilon )$ is included in $[-K-2;K+2]$ \nand for any $p > 0$, \n$$\\sup _{x\\in [-K-2;K+2]}\\vert D^pf_n(\\epsilon )(x)\\vert \\leq \n\\sup _{x\\in [-K-2;K+2]}\\int _{-K-1}^{K+1} \\vert D^p \\rho _{\\frac{\\epsilon }{2}}(x-y)\\vert dy \\leq C_p(\\epsilon ).$$\nThus, according to \\eqref{StS}, \n\\begin{equation} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(f_n(\\epsilon )(S_n))]-\\tr_m \\otimes \\tau f_n(\\epsilon )(s_n)\n+{\\nabla _n(f_n(\\epsilon ))}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}})\n\\end{equation}\nand \n\\begin{equation}\\label{prime} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n((f_n'(\\epsilon ))^2(S_n))]-\\tr_m \\otimes \\tau (f_n'(\\epsilon )(s_n))^2\n+{\\nabla _n((f_n'(\\epsilon ))^2)}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\nAccording to Lemma \\ref{LSt}, we have  $\\nabla _n(1)=0$. \nThen, the function $\\psi _n(\\epsilon )\\equiv 1-f_n(\\epsilon )$ also satisfies\n\\begin{equation} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))]-\\tr_m \\otimes \\tau \\left(\\psi _n(\\epsilon )(s_n)\\right)\n+\\nabla _n(\\psi _n(\\epsilon ))=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}). \n\\end{equation}\nMoreover, since $\\psi _n'(\\epsilon )=-f_n'(\\epsilon )$, \nit comes readily from \\eqref{prime} that \n$$\\mathbb{E} [\\tr _n((\\psi _n'(\\epsilon ))^2(S_n))]-\\tr_m \\otimes \\tau (\\psi _n'(\\epsilon )(s_n))^2\n+{\\nabla_n((\\psi _n'(\\epsilon ))^2)}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}).$$\nNow, since $\\psi _n(\\epsilon )\\equiv 0$ on the spectrum of $s_n$, \nwe deduce that \n\\begin{equation}\\label{psi2} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))]=O_{\\epsilon }\\left(\\frac{1}{n\\sqrt{n}}\\right)\n\\end{equation}\nand \n\\begin{equation} \\label{psiprime} \\mathbb{E} [\\tr_m\\otimes \\tr _n((\\psi _n'(\\epsilon ))^2(S_n))]=O_{\\epsilon }\\left(\\frac{1}{n\\sqrt{n}}\\right).\n\\end{equation}\nBy Lemma \\ref{variance} (sticking to the proof of Proposition 4.7 in \\cite{HT} with $\\varphi=f_n(\\epsilon)$), \nwe have\n$$\\mathbf{V}{\\left[ \\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))\\right] }\n\\leq \\frac{C }{n^2}\\mathbb{E} \\left[\\tr_m \\otimes  \\tr _n\\{ (\\psi _n'(\\epsilon )(S_n))^2\\} \\right] .$$\nHence, using \\eqref{psiprime}, one can deduce that \n\\begin{equation} \\label{variancepsi}\n\\mathbf{V}{\\left[ \\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))\\right] }=O_{\\epsilon }\\left(\\frac{1}{n^3\\sqrt{n}}\\right).\n\\end{equation}\nFix $0<\\delta<\\frac{1}{4}$.\nSet $$Z_{n, \\epsilon }:=\\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))$$ \nand $$\\Omega _{n, \\epsilon }=\\{ \\left| Z_{n, \\epsilon }-\\mathbb{E}\\left(Z_{n, \\epsilon }\\right)\\right| > n^{-(1+\\delta)}\\}.$$\nHence, using  \\eqref{variancepsi}, we have $$\\mathbb{P}(\\Omega _{n, \\epsilon })\\leq n^{2+2\\delta}\\mathbf{V}\\{ Z_{n, \\epsilon }\\}\n=O_{\\epsilon }\\left(\\frac{1}{n^{1+\\frac{1}{2}-2\\delta}}\\right).$$\nBy Borel-Cantelli lemma, we deduce that, almost surely for all large $n$, \\begin{equation}\\label{concentrezn} \\left|Z_{n, \\epsilon }-\n\\mathbb{E}\\left(Z_{n, \\epsilon }\\right)\\right| \\leq n^{-(1+\\delta)}.\\end{equation}\nFrom \\eqref{psi2} and \\eqref{concentrezn}, we deduce that there exists some constant $C_\\epsilon$ such that, almost surely for all large $n$,\n$$\\left| Z_{n, \\epsilon }\\right|\\leq n^{-1}\\left(n^{-\\delta}+C_\\epsilon n^{-1\/2}\\right).$$\nSince $\\psi_{n}( \\epsilon )\\geq \\1 _{\\mathbb{R}\\setminus K_n({2\\epsilon })}$, \nit readily follows that, almost surely for all large $n$, \nthe number of eigenvalues of $S_n$ which are in $\\R\\setminus K_n({2\\epsilon })$ \nis lower than $m\\left(n^{-\\delta}+C_\\epsilon n^{-1\/2}\\right)$ and thus obviously,  almost surely for all large $n$, the number of eigenvalues of $S_n$ which are in $\\R\\setminus K_n({2\\epsilon })$ \n has \nto be equal to zero. Thus we have the following\n\\begin{theoreme} Let $\\epsilon>0$.\nAlmost surely for all large $n$, the spectrum of $S_n$ is included in $K_n(\\epsilon )=\\{ x, {\\rm dist}(x, {\\rm sp}(s_n))\\leq \\epsilon \\}$. \n\\end{theoreme}\nSince the  above theorem holds for any  $m\\times m$ Hermitian matrices $\\gamma$, $\\{\\alpha_v\\}_{v=1,\\ldots,r}$,  $\\{\\beta_u\\}_{u=1,\\ldots,t}$, the proof of Lemma \\ref{inclu2} is complete.\n\n\\section{Proof of Theorem \\ref{noeigenvalue}}\\label{sectionnoeigenvalue}\n\\subsection{Linearization}\\label{linearisation}\n\nLinearization procedures are by no means unique, and no agreed upon definition of what a linearization\nis exists in the literature. We use the procedure introduced in \\cite[Proposition 3]{A}, which has several \nadvantages, to be described below.\n\n\n\n\nIt is shown in \\cite{A} that, given a polynomial $P\\in\\mathbb C\\langle X_1,\\dots,X_k\\rangle$,\nthere exist $m\\in\\mathbb N$ and matrices $\\zeta_1,\\dots,\\zeta_k,\\gamma\\in M_m(\\mathbb C)$\nsuch that $(z-P(X_1,\\dots,X_k))^{-1}=\\left[\\left(z\\hat{E}_{11}\\otimes1-\\gamma\\otimes 1-\\sum_{j=1}^k\n\\zeta_j\\otimes X_j\\right)^{-1}\\right]_{11}$. Moreover, if $P=P^*$, then $\\gamma$ and $\\zeta_1,\\dots,\n\\zeta_k$ can be chosen to be self-adjoint. We denote $L_P=\\gamma\\otimes1+\\sum_{j=1}^k\\zeta_j\n\\otimes X_j\\in M_m(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$ and call it a {\\em linearization} of $P$. \nThe size $m$ and the matrix coefficients $\\gamma,\\zeta_1,\\dots,\\zeta_k$ aren't unique. Following \\cite{BMS} (see also \\cite{Mai}), we provide \na very brief outline of a recursive construction for a linearization  $L_P$ such that $$L_P := \\begin{pmatrix} 0 & u\\\\v & Q \\end{pmatrix} \\in M_m(\\mathbb{C}) \\otimes \\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$$\nwhere\n\\begin{enumerate}\n\\item $ m \\in \\mathbb{N}$,\n\\item $ Q \\in M_{m-1}(\\mathbb{C})\\otimes \\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$ is invertible,\n\\item \n  u is a row vector and v is a column vector, both of size $m-1$ with\nentries in $\\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$,\n\\item  the polynomial entries in $Q, u$ and $v$ all have degree $\\leq 1$,\\\\\n\\item\n$${P=-uQ^{-1}v} ,$$\n\\item and moreover, if $P$ is self-adjoint, $L_P$ is self-adjoint.\n\\end{enumerate}\nThus, to linearize a monomial $P=X_{i_1}X_{i_2}X_{i_3}\\cdots X_{i_{k-1}}X_{i_l}$, write\n$$\nL_P=-\\begin{bmatrix}\n0 & 0 & \\cdots & 0 & 0 & X_{i_1}\\\\\n0 & 0 & \\cdots & 0 & X_{i_2} & -1\\\\\n0 & 0 & \\cdots & X_{i_3} & -1 & 0\\\\\n\\vdots&\\vdots& \\cdots&\\vdots&\\vdots&\\vdots\\\\\n0 & X_{i_{l-1}}&\\cdots&0&0&0\\\\\nX_{i_l}&-1&\\cdots&0&0&0\n\\end{bmatrix},\n$$\nwith the obvious adaptations if $l=1,2$. The $(l-1)\\times(l-1)$ lower right corner of the \nabove matrix is invertible in the algebra $M_{l-1}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$ and \nits inverse has as entries polynomials of degree up to $l-1$ (again with the obvious modifications when \n$l\\leq 2$). The constant term in the inverse's formula is simply the matrix having $-1$ on its second \ndiagonal, and its spectrum included in $\\{-1,1\\}$.\n If the matrices $\\begin{bmatrix}\n0 & u_j\\\\\nv_j & Q_j\\end{bmatrix},$ \nwith $u_j\\in M_{1\\times n_j}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),v_j\\in M_{n_j\\times1}\n(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),Q_j\\in M_{n_j}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$\nare linearizations for $P_j,$ $j=1,2$, then \n$$\nL_{P_1+P_2}=\\begin{bmatrix}\n0 & u_1 & u_2\\\\\nv_1 & Q_1 & 0\\\\\nv_2 & 0 &Q_2\n\\end{bmatrix}\\in M_{n_1+n_2+1}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle).\n$$\nIn particular, we have again that $\\begin{bmatrix} Q_1 & 0\\\\ 0 & Q_2\\end{bmatrix}^{-1}\\in\nM_{n_1+n_2}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),$ with entries of degrees no more $\\max\\{n_1,n_2\\}$. \nAgain, its constant term has all its eigenvalues of absolute value equal to one.\nThe above construction does not necessarily provide a self-adjoint $L_P$, even if\n$P=P^*$. However, any self-adjoint polynomial $P$ is written as a sum $P=P_0+P_0^*$ for some \nother polynomial $P_0$ (self-adjoint or not) of the same degree. Let $L_{P_0}=\\begin{bmatrix}\n0 & u_0\\\\\nv_0 & Q_0\n\\end{bmatrix}.$  To insure that the linearization we obtain is self-adjoint, we write\n$$\nL_P=L_{P_0+P_0^*}=\\begin{bmatrix}\n0 & u_0 & v_0^*\\\\\nu_0^* & 0 & Q_0^* \\\\\nv_0 & Q_0 & 0\n\\end{bmatrix},\n$$\nwhich satisfies $L_P=L_P^*$ and linearizes $P$. Moreover, since $\\begin{bmatrix} 0 & Q_0^* \\\\\nQ_0 & 0\n\\end{bmatrix}^{-1}=\\begin{bmatrix} 0 & Q_0^{-1} \\\\\n(Q_0^*)^{-1} & 0\n\\end{bmatrix}$, the matrix $\\begin{bmatrix} 0 & Q_0^* \\\\\nQ_0 & 0\n\\end{bmatrix}^{-1}$ has entries which are polynomials in $X_1,\\dots,X_k$, all of them of degree \nmajorized by the degree of $P$, and its constant term is a complex matrix having \nspectrum included in the unit circle of the complex plane. These remarks, which the reader can find in\n\\cite{Mai}, will be most useful in our analysis below.\n\n\n\nIt follows from the above that if $X_1,\\dots,X_k$ are elements in some complex algebra $\\mathcal R$ \nwith unit $1$, then $z1-P(X_1,\\dots,X_k)$ is invertible in $\\mathcal R$ if and only if $z\\hat{E}_{11}-L_{P}\n(X_1,\\dots,X_k)$ is invertible in $M_m(\\mathcal R)$ ($m$ being the size of the matrix $L_P$). Moreover, \nthe construction above guarantees that the matrix $Q$ in the linearization $L_{P}(X_1,\\dots,X_k)\n=\\begin{bmatrix}\n0 & u\\\\\nv & Q\n\\end{bmatrix}$ is invertible independently of the elements $X_1,\\dots,X_k\\in\\mathcal R$.\nWe apply this to the case when $\\mathcal R\\subseteq\\mathcal B(\\mathcal H)$ is \na unital ${\\cal C}$${}^*$-algebra of bounded linear operators on the Hilbert space  $\\mathcal H$, \nfor various separable Hilbert spaces $\\mathcal H$. We formalize this result in the following\n\n\n\\begin{lemme}\\label{inversible}\nLet  $P=P^*\\in\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle$ and let\n$L_P \\in M_m(\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle)$\nbe a linearization of P with the properties outlined above. Let $y = (y_1,\\ldots, y_k)$  be a k-tuple of self-adjoint operators in a ${\\cal C}^*$-algebra ${\\cal A}$. Then, for any $z\\in \\mathbb{C}$,  $z\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y)$\nis invertible if and only if $z 1_{\\cal A}-P(y)$ is invertible.\n\\end{lemme}\n\nBeyond the property described above, we want also to compare the norms of the inverses of \n$z\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y)$ and $z 1_{\\cal A}-P(y)$ when one (and hence the other) exists.\n\n\\begin{lemme}\\label{distanceauspectre} Let  $P=P^*\\in\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle$ and let\n$L_P \\in M_m(\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle)$\nbe a linearization of P constructed as above. Let $y_n = (y_1^{(n)},\\ldots, y_k^{(n)})$  be a k-tuple of self-adjoint operators in a ${\\cal C}^*$-algebra ${\\cal A}$ such that $\\sup_n \\max_{i=1}^k \\Vert y_n^{(i)}\\Vert=C<+\\infty$. Let $z_0 \\in \\mathbb{C}$ be such that, for all large $n$,  the distance from $z_0$\nto $sp(P(y_n))$ is greater than $\\delta$. Then, there exists a constant $\\epsilon > 0$, depending only on $ \\delta$, $L_P$ and $C$ such that the distance from $0$ to $sp(z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y_n))$\nis at least  $\\epsilon$.\n\\end{lemme}\n\\begin{proof}\nIn this proof we only consider $P,u$ and $Q$ evaluated in $y_n$, so we will suppress $y_n$ from the \nnotation without any risk of confusion. Let $L_P=\\begin{bmatrix} 0 & u^*\\\\ u & Q\\end{bmatrix}$,\nso that $z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P=\\begin{bmatrix} z_0 &- u^*\\\\ -u & -Q\\end{bmatrix}$, as \nabove. We seek an $\\epsilon>0$ such that \n$z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P-z(I_m\\otimes 1_{\\cal A})=\\begin{bmatrix} z_0-z &- u^*\\\\ -u & -Q-\nz\\end{bmatrix}$ is invertible for all $|z|<\\epsilon$. We naturally require first that $-Q-z$ remains\ninvertible. As $Q=Q^*$, it follows by functional calculus that the spectrum of $-Q$ is at \ndistance equal to $\\|Q^{-1}\\|^{-1}$ from zero. As noted before, $Q^{-1}\\in M_{m-1}(\\mathbb C\n\\langle y_1^{(n)},\\dots,y_k^{(n)}\\rangle)$, with entries depending only on $P$, so that we can \nmajorize $\\|Q^{-1}\\|$ by a constant $\\kappa>0$ depending only on $C$ and $L_P$ (and independent of\nthe particular $y_n$). Thus, our first condition on $\\epsilon$ is $\\epsilon\\leq\\kappa^{-1}\/2$. \nNote that it follows that $\\|(Q+z)^{-1}\\|< 2 \\kappa.$\nNext,\nwe require that in addition $(z_0-z-u^*(-Q-z)^{-1}u)$ is invertible for all $|z|<\\epsilon$. By the openness\nof the resolvent set, we do know that such an $\\epsilon>0$ exists. More precisely, by a geometric series \nargument, if $a$ is invertible, then $b=((b-a)a^{-1}+1)a$ is invertible whenever $\\|b-a\\|<\\|a^{-1}\\|^{-1}\n$. We apply this to $a=z_0-P=z_0+u^*Q^{-1}u$ (so that $\\|a^{-1}\\|^{-1}>\\delta$) and \n$b=z_0-z-u^*(-Q-z)^{-1}u$.\nWe have \\\\\n\n\\noindent $\\|z_0+u^*Q^{-1}u-z_0+z+u^*(-Q-z)^{-1}u\\|$ \n\\begin{eqnarray*}\n& \\leq  &|z|+ \\|u^*[(-Q-z)^{-1}+Q^{-1}]u\\|\\\\\n& \\leq &|z|+ |z|\\|u\\|^2\\|Q^{-1}\\|\\|(Q+z)^{-1}\\|\\\\\n&\\leq & |z|(1+ 2\\kappa^2 \\|u\\|^2).\n\\end{eqnarray*}\nSince the norm of  $u$ is majorized in terms of $C$ and $L_P$ only by \nsome  constant $\\ell>0$, we deduce that \n $\\|z_0+u^*Q^{-1}u-z_0+z+u^*(-Q-z)^{-1}u\\|\\leq |z|(1+ 2\\kappa^2 \\ell^2).$\nThus, we require $|z|(1+ 2\\kappa^2 \\ell^2)<\\delta$.  \n This yields that , if $|z|<\\min\\{\\kappa^{-1}\/2,\\frac{\\delta}{(1+2\n\\kappa^2\\ell^2)}\\}$, then $z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P-z(I_m\\otimes 1_{\\cal A})$ is invertible. This\nconcludes the proof of our lemma.\n\n\\end{proof}\n\n\\subsection{From Lemma \\ref{inclu2} to Theorem  \\ref{noeigenvalue}}\n\nLet $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint  random variables which \nis free with the semicircular system $x=(x_1,\\ldots,x_r)$ in $({\\cal A},\\tau)$, such that the distribution of  \n$a_n$  in $({\\cal A},\\tau)$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\nLet $P$ be a Hermitian  polynomial  in t+r noncommutative indeterminates.\nLet\n$L_P\\in M_m\\left(\\mathbb{C}\\langle X_1,\\ldots,X_{t+r}\\rangle\\right)$ be a linearization of $P$ as constructed in Section \\ref{linearisation}.  Fix $\\delta>0$ and let $z\\in \\mathbb{R}$ be such that  for all large $n$, the distance from $z$ to the spectrum of  $\\left(P\\left(x_1,\\ldots,x_r, a_n^{(1)},\n\\ldots,a_n^{(t)}\\right)\\right)$ is greater than $\\delta$. According to Lemma  \\ref{distanceauspectre} (and \n\\eqref{normeAn}),  there exists a constant $\\epsilon > 0$, depending only on $\\delta,L_P$ and $\\sup_{n}\n\\max_{1\\le u\\le t}\\|A_n^{(u)}\\|$ such that the distance from 0 to $sp(z\\hat E_{11}\\otimes 1_{\\cal A}-\nL_P(x_1,\\ldots,x_r, a_n^{(1)},\\ldots,a_n^{(t)}))$ is as least $\\epsilon$.  Now, according to Lemma \n\\ref{inclu2}, almost surely, for all large $n$, the distance from 0 to  the spectrum of $(z\\hat E_{11}\\otimes I_n-L_P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}))$ is as least \n$\\epsilon\/2$. Hence, \nfor any $z'\\in ]z-\\epsilon\/4;z+\\epsilon\/4[$, 0 is not in the spectrum of $(z' \\hat E_{11}\\otimes I_n-\nL_P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}))$.\n Finally, according to Lemma \\ref{inversible}, almost surely, for large $n$, $]z-\\epsilon\/4;z+\\epsilon\/4[$ lies outside the spectrum of $P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)})$.\nA compacity argument readily yields Theorem \\ref{noeigenvalue}.\n \n \n \n \n \n \n \n \n\\section{Proof of \\eqref{safbis}}\\label{strategie}\n\n Our approach is then very similar to that of \\cite{HT} and \\cite{Schultz05}. Therefore, we will recall the main steps.\\\\ First,  the almost sure  minoration\n$$ \\liminf_{n \\vers +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}},A_n^{(1)},\\ldots,A_n^{(t)}\\right) \\right\\| \\geq  \\left\\| P(x_1, \\ldots x_r,a_1,\\ldots,a_t)\\right\\| \\  $$\ncomes rather easily from \\eqref{af}; this can be proved by closely following the proof of  Lemma 7.2 in \\cite{HT}. So, the main\ndifficulty is the proof of the almost sure reverse inequality:\n\\begin{equation}\\label{limsup}\n\\limsup_{n \\vers +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}},,A_n^{(1)},\\ldots,A_n^{(t)}\\right) \\right\\|\\leq  \\left\\| P(x_1, \\ldots x_r,,a_1,\\ldots,a_t)\\right\\|_{\\cal A}. \\ \n\\end{equation}\nThe proof of\n(\\ref{limsup}) consists in two steps.\\\\\n\n\n  \\noindent\n{\\bf Step 1: A linearization trick} (see Section 2 and the proof of Proposition 7.3 in  \\cite{HT}) \\\\\nIn order to prove (\\ref{limsup}), it is sufficient to prove:\n\\begin{lemme} \\label{inclu} For all $m \\in \\N$, all self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$ of size $m\\times m$ and\nall $\\epsilon >0$, almost surely for all large $n$,\n$$ \nsp(\\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}+  \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})$$ \\begin{equation} \\label{spectre}\\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_u) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\n\n\\end{lemme}\n\\noindent {\\bf Step 2: An intermediate inclusion}\nLet $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint  random variables which is free with the semicircular system $x=(x_1,\\ldots,x_r)$ in $({\\cal A},\\tau$), such that the distribution of  $a_n$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\nMale \\cite{CamilleM} proved that if $(A_n^{(1)},\\ldots, A_n^{(t)})$ converges strongly to $(a_1,\\ldots, a_t)$, then, for any $\\epsilon >0$, for all large $n$,\\\\\n\n\\noindent \n$\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)})$ \\begin{equation} \\label{spectre2} \\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_i + \\sum_{u=1}^t \\beta_u \\otimes a_u) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\nTherefore,  Lemma \\ref{inclu} can be deduced from Lemma \\ref{inclu2}.\n\n \n\n\\section{Appendix}\n\\subsection{Basic identities and inequalities}\n\\begin{lemme}\\label{majcarre}\nFor any matrix $M \\in M_m(\\mathbb{C})\\otimes M_n(\\mathbb{C})$ \nand for any fixed $k$, we have \\begin{equation}\\label{O}  \\sum_{l =1}^n ||M_{lk} ||^2 \\leq m ||M ||^2\\end{equation}\n(or equivalently \\begin{equation}\\label{l} \\sum_{l =1}^n ||M_{kl} ||^2 \\leq m ||M ||^2.)\\end{equation}\nTherefore, we have \n \\begin{equation}\\label{lp}\n \\frac{1}{n} \\sum_{k,l =1}^n ||M_{kl} ||^2 \\leq m ||M ||^2.\n \\end{equation}\n\\end{lemme}\n\\begin{proof}\nNote that \\begin{eqnarray*} \\sum_{l =1}^n ||M_{lk} ||^2 &\\leq& \\sum_{l =1}^n ||M_{lk} ||_2^2\\\\&=& \\Tr M ( I_m\\otimes E_{kk}) M^*\\\\&=&\\Tr  ( I_m\\otimes E_{kk}) M^*M ( I_m\\otimes E_{kk})\\\\&\\leq& \n \\Vert M \\Vert^2 \\Tr ( I_m\\otimes E_{kk})=m \\Vert M \\Vert^2  .\\end{eqnarray*}\nNow, since  $$ \\sum_{l =1}^n ||M_{kl} ||^2=\\sum_{l =1}^n ||M_{kl}^* ||^2 = \\sum_{l =1}^n ||(M^*)_{lk} ||^2  \\; \\mbox{and} \\; \\Vert M^* \\Vert= \\Vert M \\Vert,$$\n\\eqref{O} and \\eqref{l} can be deduced from each other thanks to conjugate transposition.\nFinally \\eqref{l} readily yields \\eqref{lp}.\n\\end{proof}\n\\begin{lemme}\nLet $k\\geq 1$. Let $M^{(0)}, M^{(1)},\\ldots, M^{(k)}, M^{(k+1)}$, be $nm\\times nm$ matrices  depending on $\\lambda \\in \\{\\rho\\in M_m(\\mathbb{C}), \\Im \\rho>0\\}$ such that \n$\\forall w=0, \\ldots, k+1, \\; \\left\\| M^{(w)} \\right\\| =O(1).$ Assume that for any $ (i,l)\\in \\{1,\\ldots,n\\}^2$,\n$z_{i,l}$ are complex numbers such that  $\\sup_{i,l} \\vert z_{i,l} \\vert \\leq C$ for some constant $C$ \nand $\\{ i_w(i,l) \\}_{w=1,\\ldots,k+1}$ and \n $\\{j_w(i,l)\\}_{w=0,\\ldots,k}$ are equal to either $i$ or $l$.\\\\\n Then, \n \\begin{itemize} \\item for any $(p,q)\\in \\{1,\\ldots,n\\}^2,$ \\begin{equation}\\label{Oden}\\sum_{i,l=1}^n z_{i,l} M^{(0)}_{pj_0} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} M^{(k+1)}_{i_{k+1}q}=O_{p,q}^{(u)}(n),\\end{equation}\n\\item  if there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$, then for any $(p,q)\\in \\{1,\\ldots,n\\}^2,$ \\begin{equation}\\label{Oderacine}\\sum_{i,l=1}^n z_{i,l} M^{(0)}_{pj_0} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} M^{(k+1)}_{i_{k+1}q} =O_{p,q}^{(u)}(\\sqrt{n}),\\end{equation}\n\\item  if there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$, then\n \\begin{equation}\\label{Odenracinepas}\\sum_{i,l=1}^n z_{i,l}  M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k}  =O(n\\sqrt{n}),\\end{equation}\n \\item  if there exists $(w_0, w_1)\\in \\{1,\\ldots,k\\}^2$, $w_0\\neq w_1$,  such that \n $\\{(i_{w_{0}}, j_{w_0}), (i_{w_{1}}, j_{w_1})\\}$ is a subset of $\\{(i,l),(l,i)\\}^2$\n then \n  \\begin{equation}\\label{Odenpas}\\sum_{i,l=1}^n z_{i,l}  M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k}  =O(n).\\end{equation}\n\\end{itemize}\n\\end{lemme}\n\\begin{proof}\nIf $(j_0, i_{k+1}) \\in \\{(i,l),(l,i)\\}$, noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*}\\sum_{i,l=1}^n  \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| M^{(k+1)}_{lq}\\right\\|\n&\\leq& \\left( \\sum_{i=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \\sqrt{n}  \\left( \\sum_{l=1}^n  \\left\\|M^{(k+1)}_{lq} \\right\\|^2 \\right)^{1\/2} \\sqrt{n} \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oden} follows. \\\\\nNow, if $(j_0, i_{k+1}) \\in \\{(i,i),(l,l)\\}$, noticing  (using Lemma \\ref{majcarre}) that \\begin{eqnarray*}\\sum_{i,l=1}^n  \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| M^{(k+1)}_{iq}\\right\\|\n&\\leq& n \\left( \\sum_{i=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2}  \\left( \\sum_{i=1}^n  \\left\\|M^{(k+1)}_{iq} \\right\\|^2 \\right)^{1\/2} \n \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oden} follows. The proof of \\eqref{Oden} is complete.\\\\\n\n\\noindent Now, assume that  there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$. Let $\\tilde  M^{(w_0)}$ be either $ M^{(w_0)}$ or $ (M^{(w_0)})^*$.\\\\\nIf $(j_0, i_{k+1}) \\in \\{(i,l),(l,i)\\}$, noticing (using Lemma \\ref{majcarre}) that\n\\\\\n\n$\\sum_{i,l=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|\\left\\| \\tilde M^{(w_0)}_{il}\\right\\| \\left\\| M^{(k+1)}_{lq}\\right\\|$  \\begin{eqnarray*}\n&\\leq&  \\sum_{l=1}^n  \\left\\|M^{(k+1)}_{lq} \\right\\| \\left( \\sum_{i=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2}   \\left( \\sum_{i=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2} \\\\&\\leq & \\left( \\sum_{i=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \n\\left( \\sum_{i=1}^n  \\left\\|M^{(k+1)}_{lq} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2}\n \\\\&=&O_{p,q}^{(u)}(\\sqrt{n}),\n\\end{eqnarray*}\n\\eqref{Oderacine} follows. \\\\\nNow, if $(j_0, i_{k+1}) \\in \\{(i,i),(l,l)\\}$, noticing  (using Lemma \\ref{majcarre}) that \\\\\n\n\\noindent $\\sum_{i,l=1}^n  \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| \\tilde M^{(w_0)}_{il}\\right\\| \\left\\| M^{(k+1)}_{iq}\\right\\|$ \\begin{eqnarray*}\n&\\leq& n  \\left\\| \\tilde M^{(w_0)}\\right\\| \\left( \\sum_{i=1}^n  \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2}  \\left( \\sum_{i=1}^n   \\left\\|M^{(k+1)}_{iq} \\right\\|^2 \\right)^{1\/2} \n \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oderacine} follows. The proof of \\eqref{Oderacine} is complete.\\\\\n\n\\noindent  Assume that  there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$; then  noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*} \n\\sum_{i,l=1}^n \\left\\| \\tilde M^{(w_0)}_{il} \\right\\|& \\leq& n  \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2}\\\\&=& O(n\\sqrt{n}),\n\\end{eqnarray*}\n\\eqref{Odenracinepas} follows.\\\\\n\n\\noindent Now, assume that there exists $(w_0, w_1)\\in \\{1,\\ldots,k\\}^2$, $w_0\\neq w_1$,  such that \n $\\{(i_{w_{0}}, j_{w_0}), (i_{w_{1}}, j_{w_1})\\}$ is a subset of $\\{(i,l),(l,i)\\}^2$.  Let for $h=0,1$, $\\tilde  M^{(w_h)}$ be either $ M^{(w_h)}$ or $ (M^{(w_h)})^*$; then  noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*} \n\\sum_{i,l=1}^n \\left\\|  \\tilde M^{(w_0)}_{il} \\right\\| \\left\\|  \\tilde M^{(w_1)}_{il} \\right\\|& \\leq&   \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i,l=1}^n \\left\\| \\tilde  M^{(w_1)}_{il} \\right\\|^2 \\right)^{1\/2}\\\\&=& O(n),\n\\end{eqnarray*}\n\\eqref{Odenpas} follows.\\\\\n\\end{proof}\n We end  by recalling some properties of  resolvents.\n First, one can easily see that for any $\\lambda$ and $\\lambda^{'}$ in $M_m(C)$ such that $\\Im(\\lambda)$ and $\\Im(\\lambda^{'})$ are positive\ndefinite,\n \\begin{equation}\\label{difStieljes}(\\lambda \\otimes 1_{{\\cal A}} - s)^{-1}-(\\lambda^{'} \\otimes 1_{{\\cal A}} -\n s)^{-1}=(\\lambda \\otimes 1_{{\\cal A}} - s)^{-1}(\\lambda^{'} -\n \\lambda)(\\lambda^{'} \\otimes 1_{{\\cal A}} - s)^{-1}.\n \\end{equation}\nFor a Hermitian matrix $M$, the derivative w.r.t $M$ of the resolvent $R(z) = (z-M)^{-1}$ satisfies:\n \\begin{equation} \\label{resolvente}\n R'_M(z) . H = R(z) H R(z)   \\mbox{ for all  Hermitian matrix $H$}. \\end{equation}\n\\begin{lemme} \\label{G}\nLet $\\lambda$ in $M_m(C)$ such that $\\Im(\\lambda)$ is positive\ndefinite and $h$ be a self-adjoint element in $M_m(\\C)\\otimes {\\cal A}$ where ${\\cal A}$ is a   ${\\cal C}^*$-algebra endowed with some state $\\tau$. Then\n \\begin{equation}\\label{normeG}\n \\Vert (\\lambda \\otimes 1_{{\\cal A}} - h)^{-1}\\Vert \\leq ||\n \\Im(\\lambda)^{-1}\\Vert \\mbox{~~and~~~}\n  || G(\\lambda ) || \\leq || \\Im(\\lambda)^{-1}\n  ||,\\end{equation}\nwhere  $G(\\lambda) = ({\\rm id}_m \\otimes \\tau) [ (\\lambda \\otimes 1_{{\\cal A}} - h)^{-1}].$\n\\end{lemme}\n\n\n\n \\begin{lemme} \\label{lem2}\n Let $\\lambda$ in $M_m(C)$ such that $\\Im(\\lambda)$ is positive definite, then for any $mn\\times mn$ Hermitian matrix $H$\n \\begin{equation}\\label{norme}\n  || (\\lambda \\otimes I_n - H)^{-1} || \\leq || \\Im(\\lambda)^{-1}\n  ||,\\end{equation}\n $$  \\forall 1 \\leq k,l \\leq n, || (\\lambda \\otimes I_n - H)^{-1}_{kl} || \\leq || \\Im(\\lambda)^{-1} ||,$$\n  and for $p \\geq 2$,\n \\begin{equation}\\label{pplusgrand}\n \\frac{1}{n} \\sum_{k,l =1}^n ||(\\lambda \\otimes I_n - H)^{-1}_{kl} ||^p \\leq m ||\\Im(\\lambda)^{-1} ||^p.\n \\end{equation}\n\n\n\n \\end{lemme}\n \\subsection{Variance estimates}\nWe refer the reader to the book \\cite{Tou}. \nA probability measure $\\mu$ on $\\mathbb{R}$ is said to satisfy the Poincar\\' e inequality with constant $C_{PI}$ if\n for any\n${\\cal C}^1$ function $f: \\R\\rightarrow \\C$  such that $f$ and\n$f' $ are in $L^2(\\mu)$,\n$$\\mathbf{V}(f)\\leq C_{PI}\\int  \\vert f' \\vert^2 d\\mu ,$$\n\\noindent with $\\mathbf{V}(f) = \\int \\vert\nf-\\int f d\\mu \\vert^2 d\\mu$. \\\\\n\n\n\\begin{remarque}\\label{multiple} If the law of a random variable $X$ satisfies the Poincar\\'e inequality with constant $C_{PI}$ then, for any fixed $\\alpha \\neq 0$, the law of $\\alpha X$ satisfies the Poincar\\'e inequality with constant $\\alpha^2 C_{PI}$.\\\\\nAssume that  probability measures $\\mu_1,\\ldots,\\mu_M$ on $\\mathbb{R}$ satisfy the Poincar\\'e inequality with constant $C_{PI}(1),\\ldots,C_{PI}(M)$ respectively. Then the product measure $\\mu_1\\otimes \\cdots \\otimes \\mu_M$ on $\\mathbb{R}^M$ satisfies the Poincar\\'e inequality with constant $\\displaystyle{C_{PI}^*=\\max_{i\\in\\{1,\\ldots,M\\}}C_{PI}(i)}$ in the sense that for any differentiable function $f$ such that $f$ and its gradient ${\\rm grad} f$ are in $L^2(\\mu_1\\otimes \\cdots \\otimes \\mu_M)$,\n$$\\mathbf{V}(f)\\leq C_{PI}^* \\int \\Vert {\\rm grad} f \\Vert_2 ^2 d\\mu_1\\otimes \\cdots \\otimes \\mu_M$$\n\\noindent with $\\mathbf{V}(f) = \\int \\vert\nf-\\int f d\\mu_1\\otimes \\cdots \\otimes \\mu_M \\vert^2 d\\mu_1\\otimes \\cdots \\otimes \\mu_M$ (see Theorem 2.5 in  \\cite{GuZe03}) .\n\\end{remarque}\n\n\n\n\n\n\n\n\n\\begin{lemme}\\label{zitt}[Theorem 1.2 in \\cite{BGMZ}]\nAssume that the distribution of  a random variable $X$ is  supported in  $[-C;C]$ for some constant $C>1$. Let $g$ be an independent standard real  Gaussian random variable. Then  $X+\\delta g$ satisfies a Poincar\\'e inequality with constant \n$C_{PI}\\leq \\delta^2 \\exp \\left( 4C^2\/\\delta^2\\right)$.\n\\end{lemme}\n\nConsider the linear isomorphism $\\Psi_0$ between $M_n(\\C)_{sa}$ and $ \\mathbb{R}^{n^2}$ defined for any  $[b_{kl}]_{k,l=1}^{n} \\in M_n(\\C)_{sa}$  by $$\\Psi_0([b_{kl}]_{k,l=1}^{n}) = ((b_{kk})_{1\\leq k\\leq n}, (\\sqrt{2} \\Re  (b_{kl}))_{1\\leq k<l \\leq n},  (\\sqrt{2} \\Im  (b_{kl}))_{1\\leq k<l \\leq n}).$$ Denote by $\\Psi$ the natural extension of $\\Psi_0$ to a linear isomorphism between $(M_n(\\C)_{sa})^r$ and $ \\mathbb{R}^{rn^2}$ defined for any $(B_1, \\ldots, B_r)$ in $(M_n(\\C)_{sa})^r$ by \n$$\\Psi (B_1,\\ldots, B_r)=(\\Psi_0(B_1),\\ldots, \\Psi_0(B_r)).$$\n \\begin{lemme}\\label{variance}\nConsider   r  independent $n\\times n$ random  Hermitian matrices $X_n^{(v)} =    [X^{(v)}_{jk}]_{j,k=1}^n$, $v=1,\\ldots,r$,   such that  the  random variables $ X^{(v)}_{ii}$,\n$ \\sqrt{2}Re(X^{(v)}_{ij})$, $\\sqrt{2} Im(X^{(v)}_{ij}),{i<j}$, are independent and satisfies\na  Poincar\\'e\ninequality with common  constant $C_{PI}$. Let $\\tilde f: \\mathbb{R}^{rn^2}\\rightarrow \\mathbb{C}$ be \na  ${\\cal C}^1$ function such\nthat $\\tilde f$ and ${\\rm grad} \\tilde f $ are  bounded. Consider the ${\\cal C}^1$ function $f: (M_n(\\C)_{sa})^r \\rightarrow \\mathbb{C}$ given by $f=\\tilde f\\circ \\Psi$. Then \n\\begin{equation}\\label{Poincare}\\mathbf{V}{f(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots,\n\\frac{X_n^{(r)}}{\\sqrt{n}})} \\leq \\frac{C_{PI}}{n}\\mathbb{E}\\{\\Vert {\\rm grad}  f(\\frac{X_n^{(1)}}{\\sqrt{n}},\n\\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}) \\Vert_e^2 \\}\\end{equation}\nwhere for any Hermitian matrices $ Z_1,\\ldots,Z_r, W_1,\\ldots, W_r$, $$\\langle (Z_1,\\ldots,Z_r), (W_1,\\ldots,W_r)\\rangle_e = \\sum_{v=1}^r \\Tr (Z_vW_v).$$\n\\end{lemme}\n\\begin{proof}\n(\\ref{Poincare}) follows from  Remark \\ref{multiple}, using that $\\Psi$ \nis a linear isometry from  $\\left((M_n(\\C)_{sa})^r, \\langle,\\rangle_e\\right)$ to \n$ \\mathbb{R}^{rn^2}$ with usual Euclidean inner product.\n\\end{proof}\n\n\\noindent Thanks to (\\ref{Poincare}),  sticking to\n  the proof (from ``(4.6)'' to ``(4.13)'') of Theorem 4.5 in \\cite{HT}, we similarly\nget the following  variance estimates.\n\\begin{lemme}\\label{var}\nLet $X_n^{(v)}, v=1,\\ldots,r$ be as defined in Lemma \\ref{variance} and $A_n^{(u)}$, $u=1,\\ldots,t$, be $n\\times n$ deterministic Hermitian matrices such that for any $u=1,\\ldots,t$, $\\sup_n \\left\\| A_n^{(u)} \\right\\| <\\infty$. Let $R_N$ and $H_N$ be as defined in \\eqref{rn} and \\eqref{defSt}.\nThere exists some constant $C>0$ such that, for any $n\\geq 1$ and  for any  $\\lambda$  in $M_m(\\mathbb{C})$ such that $\\Im  \\lambda$ is positive definite,\nwe have, \n   for any deterministic $nm\\times nm$ matrices $F_n^{(1)}$ and $F_n^{(2)}$ such that $\\Vert F_n^{(1)}\\Vert \\leq K$ and $ \\Vert F_n^{(2)}\\Vert \\leq K$, for any $(p,q)\\in \\{1,\\ldots,n\\}^2$,\\\\\n \n \\noindent \n$ \\mathbb{E}\\{\\Vert{\\rm id}_m\\otimes tr_n( F_n^{(1)}R_n(\\lambda)F_n^{(2)})-\\mathbb{E}({\\rm id}_m\\otimes tr_n(F_n^{(1)}R_n(\\lambda)F_n^{(2)}))\\Vert^2\\}$ \\begin{equation}\\label{varhn}\\leq\\frac{K^4C m^3}{n^2}\n \\Vert (\\Im (\\lambda))^{-1}\\Vert^4,\\end{equation}\n $ \\mathbb{E}\\{\\Vert (F_n^{(1)}R_n(\\lambda)F_n^{(2)})_{pq}-\\mathbb{E}((F_n^{(1)}R_n(\\lambda)F_n^{(2)})_{pq})\\Vert^2\\}$ $$~~~~~~~~~~\\leq\\frac{K^4C m^3}{n}\n \\Vert (\\Im(\\lambda))^{-1}\\Vert^4.$$\n\n \\end{lemme}\n\n{\\bf Acknowledgements.} The authors  wish to thank an anonymous referee for pertinent  comments\nwhich led to an improvement of this paper.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere are several reasons to increase the still rather\nmeagre data on very high-$z$, powerful radio galaxies (e.g.\nMcCarthy \\cite{mcc}, Pariskij et al. \\cite{pari:kop}). \nHigh-$z$ radio galaxies are unique laboratories for investigating the early\nstages of galaxy and AGN evolution at look-back times corresponding to \nmore than 90\\% of the age of the universe derived from\nFriedmann models. \nThey may be used as tracers of the first generation of galaxy clusters\n(Peacock \\& Nicholson \\cite{pea:nic}; Peacock \\cite{peac}) and of the\nphysical state of the intergalactic space\n(Parijskij at al. \\cite{pari:goss}). By high-resolution \none may study important morphological\nfeatures related to e.g. \nmerging activity and ``star burst'' regions. \n\nAt redshift $z>2$, 120 radio galaxies are known at present\n(de Brueck et al. \\cite{debr:br}),\nin comparison with about 250 radio loud quasars\n($S_{5GHz}>$0.03 Jy in Veron-Cetty \\& Veron \\cite{ver:ver}),\n though the former are intrinsically more abundant. \nAccording to the popular unified scheme, both classes\nare the same thing. One can study the host galaxies and \nclose environments of radio galaxies, but this is difficult\nfor QSOs at a similar redshift.\n\n\n\n\nOne aspect, where even a single galaxy may be decisive, is the question of\nhow close in time to the cosmological singularity it is possible\nto find galaxies, with normal stellar population\nand supermassive compact objects in their nuclei.\nThough the use of high-$z$ objects\nin classical cosmological tests is hampered by\nsevere problems, development of such tests is still\none aim of observational cosmology.\nTo identify  selection effects and\nevolution, large samples are required.\nOne must increase identifications of very remote galaxies, also\nin view of the new generation ground and space telescopes,\nwhich will allow their  study at high resolution.\n\n\n\n\\subsection{\nExtension of identified USS sources to fainter fluxes}\n\nIt has been known since the late 70's \n(Tielens et al., \\cite{tie:mi}; Blumenthal \\& Miley \\cite{blu:mi}) \nthat radio sources with steep \nspectra are optically fainter (and hence probably more distant) \nthan sources with flatter spectra. \nLater it was established \nthat observing faint radio sources\nwith ultra-steep spectra (USS) is an efficient\nway  to detect radio galaxies at high redshifts\n(see e.g. McCarthy \\cite{mcc}).\nAs the USS Fanaroff-Riley type II \n(FRII-type; Fanaroff \\& Riley \\cite{fan:ri}) \nradio galaxies are not\ngood ``standard radio candles''  and as \nthe reason  for the success of\nthe spectrum criterium is not known \n(see e.g. R\\\"{o}ttgering et al. \\cite{rott:lacy}),\nit is not clear what the outcome will be when USS samples are extended to \na progressively fainter flux limit.\nFainter flux may\nimply 1) larger redshifts, 2) similar redshifts, though weaker\nradio luminosity, or 3) smaller redshifts and still weaker luminosities.\nThe first alternative is most interesting, though cases 2 and 3 \nare also important: extension of the luminosity range will help\none to uncover the influence of radio luminosity on the classical\ncosmological tests (angular size--redshift; \nNilsson et al. \\cite{nils:val}\nand Hubble diagram; Eales et al. \\cite{eal:ra}) and\nto decide whether alignment effect depends primarily\non redshift or luminosity.\n\n\nThe flux range where differential normalized source counts \nshow steepening is generally regarded as the most promising \nhunting place for high redshift objects.\nParijskij et al. (\\cite{pari:bur})  pointed out that the bulk of the\nRATAN-600 sample (see below) has fluxes in the range \nof 10-50 mJy at 3.9 GHz where the\nnormalized counts show a maximum steepening, usually interpreted as\na cosmological effect.\nA similar steepening in the counts is seen  separately for steep\nspectrum sources (Fig. 6 in Kellermann \\& Wall 1987).\nIt has been suggested (e.g. R\\\"{o}ttgering et al. \\cite{rott:lacy}) \n that the most effective way to \nfind distant galaxies would be a USS sample with \n$S_{408}\\sim 0.2-1$ Jy.\nIndeed, this has proven to be so\nsince about 50\\% of the R\\\"{o}ttgering et al. (\\cite{rott:lacy})\nUSS objects have $z>2$ (van Ojik et al. \\cite{vanojik:rott}).\nThe bright end of the USS sources is well studied\n(e.g. $4C\/USS, B2\/1Jy, MRC\/1Jy$ McCarthy \\cite{mcc}\nand references therein) and\nrecently fainter flux limits have been  reached \n(e.g. $B3\/VLA$ $S_{408}>0.8$ Jy Thompson et al. \\cite{thompson};\nESO\/Key-Project $S_{365}>0.3$ Jy\nR\\\"{o}ttgering at el. \\cite{rott:lacy}). \nHowever, in the  R\\\"{o}ttgering et al. (\\cite{rott:lacy})\nsample 365 MHz flux density distribution  peaks at about 1 Jy. \n\n\n\n\\subsection{RATAN-600 (RC) and UTRAO catalogues}\n\nThis paper is part of a programme initiated\nat the Special Astrophysical Observatory (Russia) with\nthe aim of searching  distant radio galaxies and\ninvestigating the early evolutionary stages of\nthe universe (Goss et al. \\cite{goss:par}).\nWe wish to extend the steep-spectrum criteria to fainter fluxes\nthan previously.\nThis is accomplished by RC and UTRAO catalogues (see Fig. ~\\ref{fig1})\n\n\n\n\\begin{figure}\n\\begin{center}\n\\hspace*{0.5cm}\n\\epsfig{file=astrds7556f1.ps, height=5.0cm}\n\\end{center}\n\\caption[]{\nFrequency - flux limit diagram with the positions of the RC sample\nand some other major radio catalogues.  \nUTRAO (-36$\\degr < \\delta <$ 72$\\degr$) is the optimum \nlow frequency catalogue presently available,  \nwhich can be used for calculating\nthe spectral index for a large part of the RC sample \n($\\delta \\sim 5\\degr$).\nNote that the 6C sample has $\\delta >$ 20$\\degr$.\nThe lines correspond to a source with $\\alpha$=1.\n}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\hspace*{0.5cm}\n\\epsfig{file=astrds7556f2.ps, height=5.0cm}\n\\caption[]{\nHubble diagram in R-band for various radio galaxies \nfrom the literature.\nThe triangles are from the  complete Molonglo sample\n(McCarthy et al. \\cite{mcc:kap2}),\nthe open boxes are from  \nAllington-Smith et al.(\\cite{all:spi}),\nMaxfield et al. (\\cite{max:tho}),\nMcCarthy et al. (\\cite{mcc:spi}),\nMcCarthy et al. (\\cite{mcc:kap1}),\nMcCarthy et al. (\\cite{mcc:van}),\nThompson et al. (\\cite{thompson}),\nWindhorst et al. (\\cite{wind}).\nFilled dots are from\nCarilli et al. (\\cite{car:ro}),\nChambers et al. (\\cite{cham:mi}),\nDjorgovski et al. (\\cite{djor:spin}),\nDunlop\\&Peacock (\\cite{dun}),\nEales et al. (\\cite{eal:raw}), \nHammer\\&LeFevre (\\cite{ham:le}), \nKristian et al. (\\cite{kri:san}),\nLacy et al. (\\cite{lac:mi}), \nLeFevre et al. (\\cite{lefev:ham1}),\nLeFevre\\&Hammer (\\cite{lefev:ham2}),\nLilly (\\cite{lil1}),\nLilly (\\cite{lil2}),\nOwen\\&Keel (\\cite{owen:keel}),\nMiley et al. (\\cite{mi:ch}), \nSpinrad et al. (\\cite{spin}) \nand filled stars are from the  ESO\/Key-Project\n(R\\\"{o}ttgering et al. \\cite{rott:miley}\nR\\\"{o}ttgering et al. \\cite{rott:west}\nvan Ojik et al. \\cite{van:rott}).\nOpen symbols are  $r$-magnitudes, which are transformed\nas $R=r-0.4$.\nThe histogram of RC\/USS sources\n$R$-magnitudes is shown above.\nThe magnitudes are from K95b.\nPresent NOT-observations are concerned with $R\\la24$. \n}\n\\label{fig2}\n\\end{figure}\n\n\nOur high frequency\ncatalogue is based on  a sample of faint radio sources\noriginally discovered using the RATAN-600 radio telescope\nin the \"Kholod\" (\"Cold\") experiment in 1980-81\n(Parijskij et al. \\cite{pari:bur}; Parijskij et al. \\cite{pari:bur2}\nParijskij \\& Korolkov \\cite{par:kor}).\nIn the experiment, performed at 7.6 cm (3.9 GHz),\nthe strip around the sky at $\\delta$=5$\\degr \\pm$ 20$\\arcmin$\nwas surveyed with a limiting flux of about 4 mJy.\nThe RC catalogue resulted in  containing 1145 objects.\nWithin the inner strip of $\\pm$ 5$\\arcmin$\nthe completeness of the catalogue reaches 80\\% at\nthe flux limit S$_{3.9} =7.5$ mJy and is almost\n100\\% at 15mJy (Parijskij et al. \\cite{pari:bur}).\nSuch flux limits are really quite faint and allow one\nto identify a large number of steep spectrum sources,\nif a low frequency catalogue with\nsufficiently faint flux limit is available.\nThe  UTRAO (Douglas et al. \\cite{douglas})\nis such a catalogue with a\nflux limit of $\\sim$ 100mJy at 365 MHz  (see Fig. ~\\ref{fig1}).\nThe RATAN-600 catalogue (RC) provided the first sample \nwhich allowed one\nto calculate the spectral index for practically all\nUTRAO sources within the region covered by the \"Kholod\"\nexperiment (Soboleva et al. \\cite{sobo:pari}).\nOf the original sample of 840 sources (Parijskij et al. \\cite{pari:bur}), \n491 sources matched those of the UTRAO catalogue.  \nSoboleva et al. (\\cite{sobo:pari}) could identify optically from\nPOSS (Palomar Optical Sky Survey) 240 sources at galactic \nlatitude $>$ 20$\\degr$.\n\n\n\n\n\\begin{table*}\n\\caption[]{RC\/USS source parameters. \nThe IAU name is in the first column followed by the\nequatorial, then galactic coordinates and galactic extinction in R-band. \nThe radio spectral index is in the seventh column, followed by\n3.9 GHz flux density  and the LAS of the radio source.\nThe results of optical identification are in the last column.\nThe data have been taken from \nKopylov et al. \\cite{kop:goss1}, \\cite{kop:goss2} \nand Parijskij et al. \\cite{pari:goss}. }\n\\begin{flushleft}\n\\begin{center}\n\\begin{tabular}{lllrrrlrrr}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nName& R.A. &Decl &$l$&$b$&A$_{R}$& $\\alpha^{365}_{3900}$ & S$_{3900}$ & LAS & m$_{R}$\\\\\n    & B1950 &B1950& \\degr  &\\degr   & &         &mJy  & $[\\arcsec]$    &   \\\\  \n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\object{J0406+0453}&04 03 48.22&4 39 49.7&186 &-33&0.41 &1.02 & 79 & 21.8  & 24.9\\\\\n\\object{J0444+0501}&04 41 38.68&4 55 55.79&192&-25&0.41 &1.09 & 69  & 10.8  &23.0 \\\\\n\\object{J0457+0452}&04 55 15.09&4 49 13.77&194&-22&0.23 &1.12 & 56  & 34    &19.4 \\\\\n\\object{J0459+0456}&04 56 25.51&4 51 30.45&194&-22&0.23 &0.95 & 76  & 63.8  &22.1 \\\\\n\\object{J0506+0508}&05 03 45.56&5 04 21.1 &195&-20&0.27 &0.88 & 70  & 0.8   &21.6 \\\\\n\\object{J0552+0451}&05 50 16.92&4 46 49.9 &201&-10&1.51 &1.18 & 65 & 1.6 &$>$25.5 \\\\\n\\object{J0743+0455}&07 40 36.54&5 03 02.88&214& 13&0.12 &1.07 & 37  & 20.5  &23.5 \\\\\n\\object{J0756+0450}&07 53 31.2 &4 47 17.1 &216& 16&0.07 &1.16 & 14 & $<$1&$>$25.0 \\\\\n\\object{J0837+0446}&08 34 51.28&4 54 51.82&221& 25&0.07 &1.0  & 54  & 3.9   &22.4 \\\\\n\\object{J0845+0444}&08 42 53.28&4 53 52.9 &222& 27&0.09 &1.14 & 135  & 4.6  &21.4 \\\\\n\\object{J0909+0445}&09 07 13.51&4 56 37.0 &225& 32&0.08 &1.0  & 64  & 1     &20.6 \\\\\n\\object{J0934+0505}&09 31 48.21&5 17 10.76&229& 38&0.06 &1.07 & 36  & 5     &24.4 \\\\\n\\object{J1031+0443}&10 28 43.01&4 58 33.53&240& 49&0.06 &1.2  & 191 & 33    &22.5 \\\\\n\\object{J1043+0443}&10 41 10.27&4 56 12.62&243& 52&0.06 &1.14 & 37  & 48    &23.0 \\\\\n\\object{J1113+0436}&11 11 24.05&4 54 20.12&252& 57&0.10 &0.98 & 52  & 29    &22.4 \\\\\n\\object{J1152+0449}&11 49 49.71&5 04 56.75&268& 63&0.07 &1.0  & 29  & 7  &$>$24.0 \\\\\n\\object{J1155+0444}&11 52 45.43&5 00 13.86&269& 63&0.07 &1.0  & 54  & 13    &18.6 \\\\\n\\object{J1219+0446}&12 17 06.94&5 04 02.84&282& 66&0.06 &1.23 & 23  & 118   &22.0 \\\\\n\\object{J1235+0435}&12 33 16.52&4 49 26.7 &292& 67&0.06 &0.98 & 45  & 7     &21.5 \\\\\n\\object{J1322+0449}&13 19 31.84&5 04 28.13&322& 66&0.06 &0.96 & 47  & 7     &20.4 \\\\\n\\object{J1333+0451}&13 30 32.35&5 07 08.5 &328& 65&0.05 &1.3  & 11  & 1     &23.4 \\\\\n\\object{J1333+0452}&13 30 54.66&5 07 21.17&328& 65&0.05 &1.4 & 16  & 54    &23.3 \\\\\n\\object{J1339+0445}&13 37 06.5 &5 10 15.85&332& 64&0.06 &1.07 & 41  & 34    &22.7 \\\\\n\\object{J1347+0441}&13 44 37.58&4 57 16.48&336& 63&0.06 &0.98 & 43  & 1.4   &23.5 \\\\\n\\object{J1429+0501}&14 26 45.73&5 14 43.41&353& 57&0.06 &0.92 & 82 &11.1 &$>$24.0\\\\\n\\object{J1436+0501}&14 34 04.66&5 15 10.8 &356& 56&0.06 &1.25 & 48  & 15    &22.9 \\\\\n\\object{J1439+0455}&14 37 15.64&5 08 38.68&357& 55&0.06 &1.15 & 40 & 17.9&$>$24.0 \\\\\n\\object{J1510+0438}&15 07 43.00&4 50 51.72&  4& 50&0.06 &0.9  & 67  & 3.4   &22.1 \\\\\n\\object{J1609+0456}&16 06 54.69&5 07 50.48& 16& 38&0.14 &1.15 & 30 & 6.3 &$>$24.5 \\\\\n\\object{J1626+0448}&16 24 21.72&4 55 33.4 & 19& 34&0.18 &1.26 & 39  & 2.4   &22.9 \\\\\n\\object{J1646+0501}&16 44 24.94&5 06 28.92& 22& 29&0.27 &0.92 & 54  & 15.7  &21.2 \\\\\n\\object{J1658+0454}&16 55 43.34&4 58 04.9 & 23& 27&0.27 &1.25 & 31&$<$0.3&$>$24.5\\\\\n\\object{J1703+0502}&17 01 01.3 &5 06 20.0 & 24& 26&0.27 &1.18 & 175 & 1.8   &23.6 \\\\\n\\object{J1720+0455}&17 17 36.0 &4 56 48.0 & 26& 22&0.27 &1.22 & 19  & $<$0.5&20.6 \\\\\n\\object{J1725+0457}&17 23 04.58&5 00 05.0 & 27& 21&0.27 &1.26 & 27  & 1  &$>$24.0 \\\\\n\\object{J1735+0454}&17 33 13.52&4 57 07.37& 28& 19&0.39 &1.0  & 30  & 4     &23.5 \\\\\n\\object{J1740+0502}&17 38 06.03&5 04 11.1 & 29& 18&0.41 &1.2  & 32  & 4     &22.5 \\\\\n\\object{J2013+0508}&20 10 54.69&5 01 24.78& 47&-15&0.46 &0.96 & 51  & 10    &21.1 \\\\\n\\object{J2036+0451}&20 34 27.46&4 39 22.7 & 50&-20&0.27 &1.02 & 75  & 56    &19.0 \\\\\n\\object{J2144+0513}&21 41 56.65&4 57 26.1 & 61&-34&0.18 &1.06 & 72  & $<$5.5&18.8 \\\\\n\\noalign{\\smallskip}\t\t\t\t\t\t       \n\\hline\t\t\t\t\t\t\t\t       \n\\end{tabular}\t\t\t\t\t\t\t       \n\\end{center}\n\\end{flushleft}\n\\end{table*}\t\t\t\t\t\t\t       \n\t\t\t\t\t\t\t\t       \n\t\t\t\t\t\t\t\t       \n\n\n\\subsection{\nConstruction and properties of the RC\/USS sample}\n\n\nThe present study is concerned with\nsources in the range 4$^{h}<RA<22^{h}$ \n(Parijskij et al. \\cite{pari:bur}).\nThe first RC\/USS sample consists of 40 \nsteep spectrum ($\\alpha >$ 0.9, \\, $f_{\\nu}\\propto\\nu^{-\\alpha}$),\ndouble or triple FRII sources,\nand optically fainter than the POSS limit. The radio morphology\ncomes from observations with the VLA \n(Kopylov et al. \\cite{kop:goss1}).\nThe largest angular size of the radio source (LAS) \nwas not used as a criterion, because\nonly eight sources had LAS larger than 30\".\nThe median LAS of the sample is 7$\\arcsec$. \nThe median 365 MHz flux density is 0.5 Jy (average 0.7 Jy) \nranging from 0.2 Jy to 3 Jy.\n\n\nOptical identifications were made from deep observations at\nthe 6 m telescope, down to about $m_{R}$=24.  These results\nand the optical fields around the sources have been reported\nby Kopylov et al. (\\cite{kop:goss2}, here after K95b).  \nTable 1 contains information on the basic RC\/USS sample:\nsource name, equatorial and galactic coordinates, spectral index,\nflux, LAS and $m_{R}$. \nFrom this list we selected\nobjects which are not unreasonably faint\n($m_{R} <$ 24 mag) for a medium sized telescope.\n\n\n\nFig.~\\ref{fig2} gives a representative\n m$_{R}$-$z$ Hubble diagram for radio galaxies\ncollected from the literature, together with the magnitude distribution\nof the RC\/USS objects.  \nThe Hubble diagram allows one to estimate\na lower limit to redshift, because of the rather sharp lower envelope,\nespecially above $m_{R}$=21. Where the bulk of the RC\/USS galaxies \nare situated, redshift is expected to be $\\ga$0.7 as shown\nin Fig. 2.   \nSoboleva et al. (\\cite{sobo:pari}) estimated the maximum \nphotometric redshifts for the RC\/USS objects from \nthe requirement that radio\nluminosity is not higher than optical luminosity: when radio flux\nis known, the minimum optical magnitude may be calculated,\nhence the rough maximum $z_{ph}$, which is usually large, $>$1.\n\n\nIt should be mentioned that one optically\nbright ($m_{R}$=19) object RC2036+0451  was measured at the\n6 m telescope  to have $z$=2.95 (Pariskij et al. \\cite{pari:sobo}).\nThough for a quasar, this large $z$ also supports the view that\npresent  selection criteria lead to high average redshift.\n\nThe aim of the NOT imaging\nwas to study the morphology\nof the RC\/USS sources with high resolution\nand confirm the optical identifications.\nThis paper is organised as follows. \nIn Sect. 2  we describe our observations\nand data reduction. Morphology of individual galaxies \nis discussed\nin Sect. 3. The results are  summarised  in Sect. 4.\n\n\n\n\n\\section{Observations and reductions}\n\n\\subsection{Observations}\n\n\nOptical images were obtained with the 2.56 m\nNordic Optical Telescope (NOT)\nat La Palma during three observing runs in March,\nMay and December 1994.\nTable 2 summarises the instrumentation used. \nIn addition, we have\nsome supplementary observations from other observing runs.\nI-band observations of RC1510+0438 were made with \n``Stockholm'' CCD in July 1994 and\nRC2013+0508 was observed with Brocam1 in September 1994.\nThe complete log of observations is given in Table 3.\nFor each observed object it contains the filter used, \nnumber of separate images,\ntotal integration time, seeing, and date.  \nCalibration stars from Landolt (\\cite{landolt}) were\nobserved several times each night at a range of air masses.\n\n\n\n\\begin{table*}\n\\caption[]{Instruments}\n\\begin{flushleft}\n\\begin{tabular}{lllll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nCCD & Date & Field size (pixels) & Field size ($\\arcmin$)&\nPixel size ($\\arcsec$) \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nAstromed &&&&\\\\\nEEV P88200  & March 1994 & 1152x770 & 3.1x2.1& 0.163\\\\\nIAC CCD  &&&&\\\\\nTHX31156 & May 1994 & 1024x1024 & 2.4x2.4 &\n0.14\\\\\nBrocam 1 &&&&\\\\\nTK1024A& December 1994 & 1024x1024& 3x3 & 0.176\\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{flushleft}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\caption[]{Journal of observations.    }\n\\begin{flushleft}\n\\begin{tabular}{cccrll|cccrll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nObject & Band & No. of & T$_{int}$ & Seeing & Date & \nObject & Band & No. of & T$_{int}$ & Seeing & Date     \\\\\n     &      & images     & $[sec]$  & $[\\arcsec]$ & 1994  & \n&     & images   & $[sec]$  & $[\\arcsec]$&1994 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nRC0444+0501 & R& 5 & 3300 & 2.0 &3.12. &RC1219+0446 & R &5 & 3000 & 0.5&18.5 \\\\\n            & I &3 & 2700 & 2.0 & 4.12.&RC1235+0435 & V &5 & 3000 & 3 & 14.3 \\\\\nRC0457+0452 & B &3 & 3200 & 3.2-3.9& 3.\\&5.12.&    & V &3 & 2100 & 0.6 &17.5 \\\\\n            & V &2 & 1800 & 0.7 & 2.12.& \t & R &3 & 1800 & 3 & 14.3\\\\ \n            & R &4 & 3600 & 0.8 & 2.12.&\t& R &3 & 1800 & 0.6&17.5\\\\ \n  & I &4 & 2100 & 2.0-2.9 & 4.\\& 5.12. & RC1322+0449 &V&6 &3600 & 1.8 & 14.3\\\\\nRC0459+0456 & V &2 & 1800 & 0.7& 2.12.&\t       & R &3 & 1800 & 1.8 & 14.3\\\\\n            & R &2 & 1800 & 0.6 & 2.12.& RC1333+0451 & V &1 & 600 &0.6 & 18.5\\\\\n            & I &3 & 2700 & 2.0 & 3.12.& \t    & R &3 & 2100 & 0.5&18.5 \\\\\nRC0506+0558 & V &2 & 1200 & 0.8& 2.12.&\tRC1339+0445 &V &5 & 3000 & 1.4 & 15.3\\\\\n            & R &6 & 1800 & 0.8 & 2.12.&\t & R &3 & 1800 & 1.4 & 15.3\\\\ \n            & I &2 & 1200 & 1.5 & 4.12.& RC1347+0441  & V &5 &3300 & 1.1 &16.\\\\\nRC0743+0455 & V &3 & 1800 & 1.3 & 15.3&\t           & R &3 & 2400 & 0.7 &16.5 \\\\\n            & R &3 & 1800 & 1.1 & 15.3&\t RC1510+0438& V &3 & 2100 & 0.6 &17.5\\\\\n            & R &1 &  600 & 0.8 & 2.12.&\t& R &7 & 4100 & 0.5&17\\&18.5 \\\\\nRC0837+0446 & V &3 & 1800 & 0.9 & 14.3&\tRC1609+0456 & V &1& 600 & 0.6  &18.5 \\\\\n            & R &4 & 2400 & 1.0 & 14.3&\t        & R &2 &  600 & 0.5&18.5   \\\\\n            & I &3 & 2400 & 1.6 & 4.12.&RC1626+0448& V &2 & 1200 & 1.7 & 15.3\\\\\nRC0845+0444 & V &1 &  600 & 1.2 & 15.3&\t       & R &2 & 1200 & 1.7 & 15.3\\\\\n            & R &1 &  600 & 1.1 & 15.3&\tRC1646+0501&V&3&1800 & 1.5-2.0 & 13.3\\\\\n            & I &2 & 2800 & 1.2 & 4.12.&     & R &3 & 1800 & 1.5-2.0 & 13.3\\\\  \nRC0909+0445 & B &3 & 2700 & 1.8& 3.12.&\tRC1703+0502 &R &4 & 2400 & 0.6 & 17.5\\\\\n       & B &4 & 4500 & 3.0 & 5.12.&RC1720+0455& V &2 & 1200 & 1.5-2.0 & 13.3\\\\ \n     & V &2 & 1200 & 1.7 & 14.3        &  & V &1 &  600 &0.7   &16.5   \\\\      \n            & V &2 & 1800 & 1.6 & 3.12.&  & R &1 &  600 & 1.5-2.0 & 13.3\\\\ \n            & R &2 & 1200 & 1.7 & 14.3&\t   & R &2 & 1200 & 0.7 &16.5     \\\\ \n            & R &4 & 3600 & 2.0 & 3.12.& RC1735+0454&R&12& 3080& 0.6&17\\&18.5\\\\\n           & I &1 &  900 & 1.5 & 4.12. & RC1740+0502&V &3 & 1800 & 0.6 &16.5\\\\\n            & I &3 & 2700 & 3.0 & 5.12.&           & R &2& 1200 & 0.6 &16.5 \\\\\nRC1031+0443 & V &6 & 3600 &1.5-2.0&13.3&RC2013+0508 & V &2&1800&1.5&3.\\&4.12.\\\\\n      & V &3 & 1800 & 1.5 & 15.3 &         & R &1 & 900  & 1.2 & 4.12.\\\\\n            & R &3 & 1800 & 1.6 & 13.3 &   & R &1 &  600 & 0.6 & 4.9. \\\\\n            & R &3 & 1800 & 0.9 & 15.3 &RC2036+0451&B&2 &1800 & 1.4 & 2.12.\\\\\n            & I &2 & 1800 & 1.4 & 4.12.& & V &2 & 1800 & 1.5 & 3.\\& 4.12.\\\\ \nRC1043+0443 & V &5 & 3000 & 1.7 & 14.3 & & R &1 &  900 & 1.1 & 4.12.\\\\ \n            & R &4 & 2400 & 1.7 & 14.3 & & I &1 &  900 & 3.0 & 5.12.\\\\ \nRC1113+0436 & V &5 & 3000 & 1.6 & 15.3 &RC2144+0513 & B &2 & 1800&1.8 & 3.12.\\\\\n        & R &3 & 1800 & 1.6 & 15.3 &     & V &2 & 1800 & 1.0 & 2.12.\\\\ \nRC1152+0449 & V &5 & 3000 & 1.0 &16.5&   & R &3 & 1920 & 1.0 & 2.12.\\\\  \n            & R &3 & 1800 & 0.8 &16.5&   & I &2 & 1800 & 1.2 & 4.12.\\\\   \nRC1155+0444 & V &1 &  600 & 0.6 &17.5  \\\\\n            & R &1 &  600 & 0.6 &17.5    \\\\  \n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{flushleft}\n\\end{table*}\n\n\n\nIn this paper,\nwe shall restrict the discussion  to observations\nmade under excellent or good seeing (FWHM $\\la 1\\farcs$1) \nconditions,  totaling 22 objects.\nWe present only R-band images except for  a few cases where\nthe morphology has a strong wavelength dependence and the\nS\/N-ratio in other passbands is high enough.\nAll observations presented were made under\nphotometric conditions. \n\n``Blooming'' of the CCD was a serious problem with the IAC CCD.\nSome objects were close to bright stars which limited\nthe longest possible exposure time, or a bright star had to be\nmoved outside the CCD.\nRC1735+0454 lies close to the galactic plane, hence\nthe field is  crowded with bright stars. \nWe could not obtain long exposures of this faint object \nand had to move it close to the edge of the CCD.\nNote that the exposure time of the greyscale image is \n900 seconds and for the contour image 3080 seconds.\nThe bright star northeast from the centre of gravity of RC1219+0446\nhampered the observations and the northern part of the radio\nsource was not observed.\n\n\\subsection{Reductions}\n\nThe reductions were carried out using standard IRAF routines\n(bias subtraction, trimming, flat fielding).\nThe average bias frame was constructed for each night.\nThe flatfielding was made by twilight flats\nobtained each evening and morning.\nAll the scientific frames were flattened at better than a 1\\%-level.\nThe exposures of each object were registered in position\nusing several stars in the field and then averaged. The number of\nreference stars varied from three up to a  dozen. \n\n\nThe astrometric calibration was carried out using the\nAPM Catalogue (Irwin et al. \\cite{irwin}) whenever  possible.\nFor the objects near the galactic plane\nthe Guide Star Catalog (GSC) (Lasker et al. \\cite{lasker})\nwas used. \nDue to the small field of view of the CCDs there\nwere typically only a few reference stars in the frame.\nThe number of stars and hence the accuracy\nof the astrometry  strongly depends on\ngalactic latitude. We estimate the accuracy\nof the astrometric calibration to be typically better than\n1 second of arc. This is enough for the current study,\nbecause the typical resolution of the radio map is\nabout 1$\\farcs$5\nand most of the radio sources are so compact that the optical\nidentification is straightforward. \n\n\nAs a check of our photometry in the March and May 1994 run\nwe measured comparison stars\nof OJ287 (Fiorucci \\& Tosti \\cite{fiorucci}). \nThe derived brightnesses\nwere consistent with each other within 0.1 magnitudes.\n\n\n\n\n\n\\begin{table*}\n\\caption[]{NOT imaging data. The diameter of the aperture is \nindicated in arcseconds. The magnitudes are without \ncorrection of galactic extinction.  Ellipticity\nand position angle of resolved sources is measured\nwith the same aperture as the magnitudes.\nThe radio position angles are measured from Kopylov et al.\n(\\cite{kop:goss1}).\n}\n\\begin{flushleft}\n\\begin{center}\n\\begin{tabular}{llllllcrr}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nName & Aperture & $m_{R}$ & merr&$m_{V-R}$&merr&$e$&PA$_{opt}$ & PA$_{radio}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip }\nRC0457+0452 &7   &19.72 &0.01 &0.93  &0.05   &0.13    &13    &  58 \\\\\nRC0506+0558 &3.5 &21.54 &0.03 &1.29  &0.13   &unresolved & .. & -75 \\\\\nRC0837+0446 &6.5 &22.19 &0.08 &0.74  &0.13   &0.23    & 76   & -67 \\\\\nRC0845+0444 &8.2 &20.75 &0.08 &1.15  &0.17   &0.25    & -78  &9  \\\\\nRC1031+0443 &6.5 &22.32 &0.12 &1.12  &0.23   &0.57    & 88   &-36  \\\\\nRC1152+0449 &3   &22.23 &0.07 & 0.84 &0.15   &0.11    & -33  &-15  \\\\\nRC1155+0444 &8.4 &18.81 &0.02 & 1.09 &0.05   &0.28    & -54  &-76 \\\\\nRC1235+0453 &4.2 &21.70 &0.06 & 0.96 &0.17   &0.26    & 43   &-50 \\\\\nRC1347+0441 &2   &23.99 &0.19 & 0.75 &0.43   &0.32    & -29  &-49 \\\\\nRC1510+0438 &3   &22.20 &0.04 & 1.57 &0.25   &0.08    & -79  &62 \\\\\nRC1703+0502 &3   &23.69 &0.18 & ..   & ..    &0.49    & -86  &-82 \\\\\nRC1720+0455 &4.2 &20.34 &0.02 & 1.15 &0.07   &unresolved & ..&point \\\\\nRC1740+0502 &3   &22.19 &0.08 & 0.74 &0.12   &0.05  & 66  &64 \\\\\nRC2013+0508 &5.3 &20.70 &0.04 & 0.22 &0.06   &unresolved& ..  &-55  \\\\\nRC2036+0451 &4.2 &19.06 &0.02 & 0.35 &0.04   &unresolved & ..  &-2  \\\\\nRC2144+0513 &3.5 &18.89 &0.02 &0.24  &0.03   &unresolved& .. &point \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{flushleft}\n\\end{table*}\n\n\\section{Reduced images: Overview of the morphology}\n\n\n\nIn this section  \nwe give a greyscale image and a contour map\nfor each identified  object (Fig.~\\ref{fig4}) \nThe close environment and faint features can be \nstudied from the greyscale image and\nthe confidence level of the features and light distribution\nfrom the contour map.  \nThe field of view of the greyscale image is indicated in the\nupper left hand corner. \nThe images are slightly smoothed with a Gaussian function\n($FWHM=\\frac{1}{2}seeing$) in order to enhance the low \nsurface brightness features and maintain the resolution\nof the original images.\nThe centre of gravity of the radio source and\nthe positions of the radio lobes are indicated with a cross\nand circles, respectively. The images are presented in\nlinear scale from 0$\\sigma$ to 10$\\sigma$ above\nthe background of the image.\nIn contour maps the object is in the origin and \nthe numbers on both the vertical and horizontal\naxes refer to distance in  arcsecond.\nThe contour interval is 0.5 mag arcsec$^{-2}$ and\nthe surface brightness of the lowest\ncontour is indicated in the upper right hand corner. \nThe limiting surface brightness at which objects can be detected\nis typically between 25 and 26 mag arcsec$^{-2}$.\nFor RC1510+0438 we also give V and I band images.\nUncertain identifications are presented in Fig. \\ref{fig5}\nand ``faint objects'' in Fig. ~\\ref{fig6}.\n\n\nThe magnitudes are based on aperture photometry using\nDAOPHOT. The size of the aperture was selected\nin such a way that 1) as much light as possible was included\nin the aperture while keeping the errors reasonable but \n2) the companions were excluded.\nThe image shapes are determined by the moments of the \nbrightness distribution\nusing IMEXAMINE (eq. 4 in Valdes et al. \\cite{valdes})\nwith the size of the aperture the same as in photometry.  \nThe estimation is vague for small \nellipticity (e.g. RC1152+0449) or when the inner regions  \nhave a different position angle than the outer regions\n(e.g. RC1031+0443). The results of  photometry and\n image shape analysis are given in Table 4.\nThe magnitudes between this work and K95b\nare generally in agreement.\nThe differences are primarily caused by\nthe better seeing conditions at NOT, as compared with the 6 m-telescope\nand the different size of the aperture.\n\n\n\\subsection{Identified objects}\n\nIn this section we give notes on individual objects (Fig. ~\\ref{fig4}).\nFor the strongest sources alternative names are given in brackets.\n\n\\noindent\n{\\bf \\object{RC0457+0452}}\\\\\nThis is one of the brightest objects in this  sample.\nThe new VLA map shows that the radio  source has FRI \nstructure (an unpublished radio map).\nThis agrees with a high optical to\nradio luminosity ratio (Parijskij et al. \\cite{pari:goss}).\nThere is a near companion $\\sim$2.5$\\arcsec$ northwest from the nucleus.\nIn addition there are four faint companions in the southeast (5$\\arcsec$).\nThe position angle of the galaxy is not uniform,\nwhich is possibly due the close companions.\nThe inner isophotes are roughly perpendicular to the radio axis\nand the outermost fuzz is roughly aligned with the radio axis. \nThe same trend can be seen in both R and V-band images.\nThis object is possibly situated in a cluster of galaxies and\nthere are a few companions with similar brightness. The \ncompanion 20$\\arcsec$ to the east has a double nucleus\n($m_{R}=19.26, m_{V-R}=0.69$), hence it is \napparently a merger, and the companion 22$\\arcsec$ to the \nsouth has distorted morphology ($m_{R}=19.94, m_{V-R}=0.93$) .\n\n\n\\noindent\n{\\bf \\object{RC0506+0508}}\\\\\nThis is a faint point-like source, even under excellent seeing\nconditions.  \nIt might be a high redshift ($z>1, M_{R}< -23$) quasar\nbecause\nin the quasar catalogue by Veron-Cetty \\& Veron (\\cite{ver:ver})\nthere are only a few  quasars fainter than    \n21 magnitudes with $z<1$.\nThe companions 14$\\arcsec$ to the northwest have a multicomponent\nstructure with an extended  diffuse emission. \n\n\\noindent\n{\\bf \\object{RC0837+0446}}\\\\\nThe galaxy is marginally resolved and lies\nin or behind a galaxy cluster.\nIn the lower left hand corner of the grey scale image is a\ntrail of a solar system object.\n\n\\noindent\n{\\bf \\object{RC0845+0444}}\\\\\nThe optical counterpart of this radio source\ncoincides with the western radio component.\nThis object is optically extended \nand there is a \nfaint extension towards the  southwest.   \n\n\\noindent\n{\\bf \\object{RC1031+0443}}\\\\\n(\\object{4C +05.43} \\& \\object{PKSB1028+049})\nThe galaxy is near the centre of  gravity of the radio source. \nThe object is extended and has a multicomponent structure. \nThe strongest optical emission is aligned with the radio source,\nbut the position angle of the outermost isophotes is\nalmost perpendicular to the radio axis.\n\n\\noindent\n{\\bf \\object{RC1152+0449}}\\\\\nThis galaxy \nhas two or possibly three components.\nThe outer isophotes of the galaxy are box-like,\nbut the separate components are roughly aligned with the\nradio axis. \nThe faint companion 4$\\arcsec$ to  the west is near to the \nwestern radio lobe. This blue companion may be related \nto the radio source ($m_{R}=23.0, m_{V-R}\\sim$0.3).\n\nThis is the only source where the astrometry of the present work\nis not consistent with the result of K95b, but coincides with \ncurrent identification in Parijskij et al. (\\cite{pari:goss}).\n\n\n\\noindent\n{\\bf \\object{RC1155+0444}}\\\\\nThis is the brightest galaxy in our sample.\nThe galaxy is elliptical and it is clearly aligned \nwith the radio source.\nThe two neighbouring galaxies are possibly interacting\nforeground galaxies.\n\n\n\\noindent\n{\\bf \\object{RC1235+0453}}\\\\\nThis faint galaxy is spatially extended in our R-band images. \nThe fuzz $\\sim 1 \\arcsec$ northwest from the nucleus \nhas a clumpy structure.\nOur deep images show faint low surface brightness  \ncompanions near the object $\\sim 10 \\arcsec$ to the east, north and west.\nIn the V-band images only the \ncore of the galaxy is detected.\n\n \n\\noindent\n{\\bf \\object{RC1347+0441}}\\\\\nThis galaxy is the faintest of our sample. \nThe object is elongated and  \nthe size of the optical object is roughly the same as the\nradio source. \n\n\\noindent\n{\\bf \\object{RC1510+0438}}\\\\\nThis is the most spectacular object in our sample,\nlying in a group of galaxies and having apparently\nwavelength dependent properties.\nIn the R- and I-band image the object is almost round\n(Table 4) but in the V-band the object is weakly elongated with\nthe same position angle as the radio source.\nThe redshift estimation from BVRI colours suggest\n$z\\sim0.6$ (Pariskij et al. \\cite{pari:sobo}).\nIf this is the case, the strong emission lines $[$\\ion{O}{ii}$]$\n3727 and $[$\\ion{O}{iii}$]$ 5007 would be shifted into R and I band,\nrespectively, possibly causing the wavelngth dependence of morphology.\nHowever, new colours from the 6 m-telescope\ndo not agree with strong line contribution.  \nAnother consequence of such redshift would be that\nthis  would be one of the faintest\n(intrinsically) radio galaxies in the Hubble diagram (Fig. ~\\ref{fig2}).\nThere are three relatively bright galaxies and one faint companion galaxy\nwithin 5$\\arcsec$ of the object. \nAll the companions are bluer than the object ($m_{R-I}=1.52$).\nThe objects towards the east, \nC1 ($m_{R}=22.54, m_{V-R}=0.51, m_{R-I}=0.27$), \nnorth, C2 ($m_{R}=22.34, m_{V-R}=0.81, m_{R-I}=0.87$) \nand west,C3 ($m_{R}=23.24, m_{R-I}=0.83$) could be foreground galaxies. \nIn addition there is a faint companion 1$\\farcs$5 north from the \nobject.\n\n\\noindent\n{\\bf \\object{RC1703+0502}}\\\\\n(\\object{PKS B1701+051})\nThis is one of the strongest and most compact radio sources\nof the present sample.\nThe optical and radio axes are clearly aligned and the sizes are\nalmost the same.\nThere are a few faint galaxies in the field, but no\nclose companions.  \nThis object is possibly located behind \na foreground galaxy cluster although some of the\nfield objects might be faint galactic stars.\n\n\n\n\n\\noindent\n{\\bf \\object{RC1720+0455}}\\\\\nThis object is compact in radio and optical.\nThis suggests that it is a QSO and\nthe same conclusion may be drawn from radio-optical luminosity\nconsideration (Parijskij et al. 1996a).\nThe extension towards the southeast, seen in K95b, was an artifact. \nThere are a few companions close to the object.\nThe southern companion either has a double nucleus or a dust lane. \nThe wide field image shows several faint companions,\nhence this galaxy is either in a cluster of galaxies or behind one.\n\n\n\\noindent\n{\\bf \\object{RC1740+0502}}\\\\\nThis source was identified by K95b \nand it is marginally resolved in the R-band image.  \nIn the V-band image the object has an extension to the\nsouth west in contrast to almost round morphology in R-band\n(see Table 4).\n\n\n\\noindent\n{\\bf \\object{RC2013+0508}}\\\\\nThis unresolved object could possibly be a quasar.\nBecause of its low galactic latitude ($b\\sim$-15),\nthe field is crowded with stars.  \n\n\n\\noindent\n{\\bf \\object{RC2036+0451}}\\\\\n(\\object{MRC 2034+046}).\nThis is the second of the two triple radio sources in this sample. \nA point source coincides with the central component fairly well.   \nThis is the only object with known redshift ($z$=2.95\nPariskij et al. \\cite{pari:goss}).\nThis indicates that the\nabsolute magnitude of this quasar is M$_{R}\\approx$-29.\n\n\n\\noindent\n{\\bf \\object{RC2144+0513}}\\\\\nThis object is unresolved. \nThe companion 3$\\arcsec$ to the southwest is most likely a\ngalactic star ($m_{R}=20.90, m_{V-R}=1.1$). \nThe profile of this object matches \nperfectly with the average stellar profile from the same field.\n\n\n\n\\subsection{Uncertain identifications}\n\n\\noindent\n{\\bf \\object{RC0459+0456}}\\\\\n(\\object{MRC 0456+048})\nThis source has two candidates for optical identification in\nK95b. Id1 ($m_{R}=22.08$) is an elongated galaxy with  roughly the same \nposition angle as the radio source. This object has a \ncompanion to the west (Id2). This is a  marginally resolved \npoint-like source, hence it might be\na quasar with a host galaxy ($m_{R}=21.12$).\nFWHM of the id2 0$\\farcs$66 compares with FWHM of a field star\n0$\\farcs$62.\n\n\n\\noindent\n{\\bf \\object{RC1219+0446}}\\\\\nThis is the largest radio source in our  sample\n(118$\\arcsec$). The nature of the source remains unclear\nand it is possible that there are indeed two independent radio sources.\nIf this is one source, then a possible identification would be a\nfaint, rather round galaxy (Id1) 5$\\arcsec$ from the centre of \nradio source ($m_{R}=21.9$).\nOn the other hand if the southern radio lobe is an independent\nobject, the identification could be an unresolved \nobject (Id2) 2$\\arcsec$ southeast from the radio source ($m_{R}=17.88$).   \n\n\n\\noindent\n{\\bf \\object{RC1735+0454}}\\\\\nThe possible optical counterpart is $\\sim 3 \\arcsec$ to the east of \nthe radio source. The galaxy has several components and it is elongated\nin a north south direction. \nThe identification should be confirmed by future observations.\n\n\n\\subsection{Faint objects}\n\n\n\\noindent\n{\\bf \\object{RC0743+0455}}\\\\\nThis object is very faint and hardly visible in the 30 min. exposure. \n\n\\noindent\n{\\bf \\object{RC1333+0451}}\\\\\nThe radio source is compact. There is a  faint \nextended emission exactly at the position of the radio source. \n\n\n\\noindent\n{\\bf \\object{RC1609+0456}}\\\\\nNew 6-m telescope measurements find an object with $m_{R}\\sim25.5$ \nexactly at the position of the radio source.\nOur 600 second exposures are not deep enough to detect \nthis object.\nThe bright nearby object is unresolved and BVRI photometry by K95b\nsuggests it to be a star.\n\n\n\\section{Concluding remarks}\n\nExcepting the quasar RC2036+0451,\nthe present galaxies do not have measured redshifts as yet. Hence,\nit is interesting to ask whether optical\nmorphology\nprovides information on the redshift\ndistribution of the sample.\nOther properties, for example\nthe Hubble diagram of the RC\/USS sample, suggest (Sec. 1.3) \nthat it contains   galaxies with $z\\ga$0.7.\n\n\nOf the 22 observed objects, three are extremely faint and  \nthree others are either faint or have several possible \nidentifications. Of the remaining 16 objects, 5 are unresolved.\nThis is roughly the same fraction of point sources \nwhich R\\\"{o}ttgering et al. (\\cite{rott:miley})\nfound, but slightly more than \nLu et al. (\\cite{lu:hof}) found from their \n``distant'' sample ($S_{1.4GHz}>35$mJy), which does not have \nradio spectral index selection criteria.\nTypically RC\/USS objects have a multicomponent\nstructure with extended emission.\nThis compares with\n$HST$ images which have shown that about 30\\%  of intermediate\nredshift 3CR galaxies  have distorted morphology\n(De Koff et al. \\cite{dekoff}).\nMore distant ($z\\sim1$) 3CR galaxies have typically \nmulticomponent structure with diffuse extended emission \n(Best et al.  \\cite{best:long}).\nThe ellipticity ($e$) of the current sample (11 objects) ranges from 0.05\nto 0.57, with a mean of 0.25.\nTaking into account the measurement errors\nthese values agree with studies\nby Rigler et al. (\\cite{rigler}) for 3C galaxies (e=0.19)\n and by R\\\"{o}ttgering et al. (\\cite{rott:miley}) USS sample (e=0.33).\n\n\n\nVisual inspection of the images in Fig. ~\\ref{fig4} suggests \nthat about half of the objects have a companion \nwith comparable brightness within 10$\\arcsec$.\nWe examined  the excess of companion galaxies \nalong the radio axis suggested by R\\\"{o}ttgering et al.\n(\\cite{rott:west}). Our sample  has 7 resolved objects with   \n3$\\arcsec<LAS<20\\arcsec$. \nOnly two of these have companion galaxies (RC1152+0449 and RC1510+0438). \nRC1152+0449 has one companion almost exactly on the extension of \nthe radio axis.  \nRC1510+0438 has two companions, the brighter one being \nalong the radio axis and the fainter one being perpendicular.\nIt is also interesting to note that both of these ``aligned'' \ncompanions are about 1 magnitude fainter than the object.\nFour out of 16 objects have an apparent excess of companions\nand hence may be situated in a cluster of galaxies.\n\n\nWe can conclude that the morphology\nof the present  weak radio flux USS population \nis close to that observed by e.g HST in high-$z$ radio \ngalaxies.\nThe results of this study make it imperative to measure\nspectroscopic redshifts for the RC\/USS galaxies.  \nEspecially, the fainter flux limit makes it very interesting to see\nwhere in the Hubble diagram (both R and K, the latter magnitudes\nstill lacking) the RC-sources are found (c.f. Eales et al.\n\\cite{eal:ra}).\nNaturally, the number of well observed RC\/USS galaxies\nis still small. A more conclusive discussion of morphological\nfeatures must wait until we complete the observations \nin the range $22^{h}<RA<4^{h}$ for which 6 m telescope identifications\nare now available. Also,  we intend to reobserve the cases\nof poor seeing in the present sample.\n\n\n\\begin{acknowledgements}\nThe authors thank the anonymous referee for useful comments. \nThis work has been\npartially supported by grants from the  Russian Foundation\nof Basic Research  95-02-03783 and 96-02-16597\nand the ``Astronomy'' programme project 2-296 and 1.2.1.2.,1.2.2.4.\nYu.V.B. acknowledges the support of the Russian \n``Integration'' project N. 578.\nTP acknowledges financial support from the Wihuri foundation.\nThis work has been supported by the Academy of Finland (project\nCosmology in the local galaxy Universe).\nThis research has made use of the NASA\/IPAC Extragalactic \nDatabase (NED) which is operated by the \nJet Propulsion Laboratory, California\nInstitute of Technology, under contract with the \nNational Aeronautics and Space Administration.\nThe National Radio Astronomy Observatory is a facility of the\nNational Science Foundation operated under cooperative agreement by\nAssociated Universities,Inc.  \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe cosmological inflation is an accelerated expansion \nof the universe at the early universe which can solve the \nhorizon problem and the flatness problem at the same time. \nSuch expanding universe is realized by the vacuum energy \ndensity of the scalar field, so called inflaton, whose \nquantum fluctuations produce the origin of the density \nperturbation of the universe. \n\nThe current cosmological observations, especially, \nthe Planck satellite report that the inflation scenario is \nwell consistent with its data and the primordial density \nfluctuations are almost Gaussian and the spectrum index of the \nscalar density perturbation is nearly scale-invariant~\\cite{Ade:2013uln}. \nRecently, the BICEP2 collaboration reported that the \nprimordial tensor modes can be measured as B-mode \npolarization of the Cosmic Microwave Background (CMB), \nwhich leads to the tensor-to-scalar ratio, $r = {\\cal O}(0.1)$, \nalthough there is a tension between the Planck and BICEP2 \ncollaborations.\n To achieve the tensor-to-scalar-ratio, $r = {\\cal O}(0.1)$, \nwe require that inflaton will roll slowly down to the minimum \nof its scalar potential from its Planckian field value which is \nproblematic from the theoretical point of view, especially in \nthe string theory which will be expected as the unified theory \nof gauge and gravitational interactions. \n\nIn the higher dimensional theories as well as the string theories, \nthere are a lot of axions associated with the internal cycles of \nthe internal manifolds such as the Calabi-Yau (CY) manifold which \nkeeps the only ${\\cal N}=1$ supersymmetry (SUSY) in the \nfour-dimensional ($4$D) spacetime. \nWhen we consider such axions as the candidate of inflaton, \nthe natural inflation~\\cite{Freese:1990rb} is an attractive \nscenario as one of the large-field inflation, which is \noriginally proposed by identifying the inflaton as the pseudo-Nambu \nGoldstone boson \\footnote{The axion monodromy inflation is another interesting possibility.\n See e.g. \\cite{Silverstein:2008sg,Kaloper:2008fb,Kobayashi:2014ooa}.}. \nHowever, the natural inflation compatible with the observed \nPlanck~\\cite{Ade:2013uln} and\/or BICEP2~\\cite{Ade:2014xna} \ndata requires the trans-Planckian axion decay constant, \nsee Ref.~\\cite{Freese:2014nla} and references therein. \nSo far, there are several approaches to realize the natural \ninflation with trans-Planckian axion decay constant \nin the framework of supergravity models or \nType IIB superstring theory~\\cite{Kallosh:2007ig,Czerny:2014xja,Choi:2014rja,Tye:2014tja,Kappl:2014lra,Ben-Dayan:2014zsa,Long:2014dta,Abe:2014vca,Li:2014lpa}, \nalthough, in the string theory, the fundamental axion decay \nconstants are typically in the range \n$10^{16}- 10^{17} {\\rm GeV}$~\\cite{Choi:1985je}.  \n\nIn this paper, we propose the single-field natural inflation in the framework \nof heterotic string theory on ``Swiss-Cheese\" CY \nmanifold with multiple $U(1)$ fluxes induced from the anomalous \n$U(1)$ symmetries. \nBy employing such multiple $U(1)$ fluxes, the linear combination of \nthe universal axion and K\\\"ahler axions except for the inflaton are \nabsorbed by the multiple $U(1)$ gauge bosons and they get the mass terms \nfrom these $U(1)$ fluxes. From the phenomenological point of view, \n$U(1)$ fluxes may be important tools to realize the $4$D standard \nmodel gauge groups from the heterotic string \ntheory~\\cite{Witten:1984dg,Blumenhagen:2006ux} \nas well as the Type IIB superstring theory~\\cite{Cremades:2004wa}. \nDuring and after the inflation, the dilaton and real parts of K\\\"ahler \nmoduli have to be stabilized and decoupled from the inflaton \ndynamics, otherwise \nthe oscillations of these moduli would lead to the sizable isocurvature \nperturbations. In our model, the stabilization of the dilaton and real \nparts of K\\\"ahler moduli are realized by the non-perturbative corrections to \nthe K\\\"ahler potential, superpotential and a nature of the structure of \n ``Swiss-Cheese\" CY manifold whose manifolds \nare also well studied as several topics about the particle phenomenology \nand cosmology based on the Type IIB string theory~\\cite{Balasubramanian:2005zx} \nor F-theory~\\cite{Blumenhagen:2009gk} and moduli stabilization based \non the heterotic string theory~\\cite{Cicoli:2013rwa}. \n\nThis paper is organized as follows. In Sec.~\\ref{sec:hetero}, \nwe review the heterotic string theory on CY manifold with \nmultiple $U(1)$ magnetic fluxes. \nWe propose two inflation models in Secs.~\\ref{subsec:naturalinf1} \nand \\ref{subsec:naturalinf2}. \nBoth are consistent with the WMAP, Planck and\/or BICEP2 \ndata. Sec.~\\ref{sec:con} is devoted to the conclusion. \nIn Appendix~\\ref{app:mass}, we show the mass matrices of fields \nfor inflation model $1$ in Sec.~\\ref{subsec:naturalinf1}.\n\n\\section{Heterotic string on CY manifolds with multiple \n$U(1)$ magnetic fluxes}\n\\label{sec:hetero}\nWe consider the $E_8\\times E_8$ or $SO(32)$ heterotic string theory \non Calabi-Yau manifold with multiple $U(1)$ magnetic fluxes (in other \nwords, multiple line bundles). The low-energy effective theory of the \nheterotic string is given by the following Lagrangian at the string \nframe,\n\\begin{align}\nS_{\\rm bos}&=\\frac{1}{2\\kappa_{10}^2}\\int_{M^{(10)}} \ne^{-2\\phi_{10}} \\left[ R+4d\\phi_{10} \\wedge  \\ast \nd\\phi_{10} -\\frac{1}{2}H\\wedge \\ast H \\right] \n\\nonumber\\\\\n&-\\frac{1}{2g_{10}^2} \\int_{M^{(10)}} e^{-2\\phi_{10}} \n{\\rm tr}(F\\wedge \\ast F),\n\\label{eq:heterob}\n\\end{align}\nwhich is the bosonic part of the Lagrangian in the notation \nof~\\cite{Polchinsky} and $\\phi_{10}$ is the dilaton and $F$ \nis the field strength of $E_8\\times E_8$ or $SO(32)$ \ngauge groups and then ``tr\" denotes in the vector \nrepresentation of these gauge groups. As will be mentioned later, \nthe $U(1)$ magnetic fluxes are inserted in these \ngauge groups. $H$ is the three-form field strength defined \nby \n\\begin{align}\nH&=dB^{(2)} -\\frac{\\alpha^{'}}{4}(w_{\\rm YM} -w_{L}),\n\\end{align}\nwhere $w_{\\rm YM}$ and $w_{L}$ are the gauge and \ngravitational Chern-Simons three-form, respectively.\nThe gravitational and Yang-Mills couplings are \nset by $2\\kappa_{10}^2 =(2\\pi)^7 (\\alpha^{'})^4$ and \n$g_{10}^2 =2(2\\pi)^7 (\\alpha^{'})^3$. \n\nThroughout this work, we focus on the \n$E_8^{\\rm vis}\\times E_8^{\\rm hid}$ heterotic string with \nnon-standard embedding, that is, the visible $E_8^{\\rm vis}$ \ngauge group decomposes into the product group of \n$G_{\\rm vis}$ and multiple $U(1)$s where $G_{\\rm vis}$ \nis the Grand Unified Group (GUT) or just the stand model (SM)\ngauge groups and we do not consider the charged scalar \nfields under the multiple $U(1)$s \n\\footnote{An extension to the cases for $SO(32)$ \nheterotic string theory is straightforward.}. \nIn addition to it, we assume that the hidden gauge groups \nare just non-abelian gauge groups, for simplicity.\n\nAfter the dimensional reduction on the CY manifold with \nmultiple $U(1)$ magnetic fluxes, \nwe get the following $4$D $U(1)$ \ninvariant effective tree-level K\\\"ahler potential,\n\\begin{align}\n{\\cal K}=&-M_{\\rm Pl}^2\n\\Biggl[\n\\ln \\left( S+\\bar{S} -\\sum_{m} \\frac{{\\cal Q}_S^m}{16\\pi^2}V_m \\right) \n\\nonumber\\\\\n&+\\ln \\Biggl\\{\\frac{d_{ijk}}{48} \n\\left( T_i +\\bar{T}_i -\\sum_{m} \\frac{{\\cal Q}_{T_i}^m}{2\\pi} V_m \\right)\n\\left( T_j +\\bar{T}_j -\\sum_{m} \\frac{{\\cal Q}_{T_j}^m}{2\\pi} V_m \\right)  \n\\left( T_k +\\bar{T}_k -\\sum_{m} \\frac{{\\cal Q}_{T_k}^m}{2\\pi} V_m \\right) \n\\Biggl\\}\\Biggl],\n\\label{eq:Kahler}\n\\end{align} \nwhere $M_{\\rm Pl}^2=\\frac{e^{-2\\phi_{10}}{\\cal V}}{\\kappa_{10}^2}$, \n$m$ labels the number of anomalous $U(1)$ vector multiplets \n$V_m$, $d_{ijk}$ are the intersection \nnumbers of the Calabi-Yau manifold.\n$S$ and $T_i$ for $i=1,2,\\cdots h^{1,1}$ are the superfield \ndescriptions of the dilaton and the K\\\"ahler moduli, respectively,\n\\begin{align}\nS&=\\frac{1}{4\\pi}\\left[ \\frac{e^{-2\\phi_{10}}{\\cal V}}{l_s^6} \n+i b_S^{(0)}\\right],\\nonumber\\\\\\\nT_i&=t_i+i b_{T_i}^{(0)},\n\\end{align} \nwhere ${\\cal V}=\\frac{1}{6}\\int_{\\rm CY}J\\wedge J\\wedge J$ with \n$J=l_s^2\\sum_{i}t_iw_i$ is the volume of the \nCY manifold, $J$ is the K\\\"ahler form expanded by the base of \ntwo-form $w_i$, $i=1,\\cdots, h^{1,1}$ and $l_s=2\\pi\\sqrt{\\alpha^\\prime}$ is the string length. \nThe imaginary parts of $S$ and $T_i$, $b_S^{(0)}$ and \n$b_{T_i}^{(0)}$ are the universal and K\\\"ahler axions given by the \ndimensional reduction of the Kalb-Ramond two-form $B^{(2)}$ \nand six-form $B^{(6)}$ as \n\\begin{align}\n&B^{(2)} =b_S^{(2)} +l_s^2\\sum_{k=1}^{h_{11}} b_{T_k}^{(0)} w_k,\n\\nonumber\\\\\n&B^{(6)} =l_s^6b_S^{(0)}{\\rm vol}_6 +l_s^4\\sum_{k=1}^{h_{11}} \nb_{T_k}^{(2)} \\hat{w}_k, \n\\end{align} \nwhere $b_S^{(2)}$ and $b_{T_i}^{(2)}$ are the $4$D tensor fields, \n${\\rm vol}_6$ is the normalized volume form, \n$\\int_{\\rm CY}{\\rm vol}_6=1$, and \n$\\hat{w}_i$ are the Hodge dual four-form of the two-form $w_i$. \nThe two-form $B^{(2)}$ and six-form $B^{(6)}$ are related \nby the Hodge duality, $\\ast_{10} dB^{(2)}=e^{2\\phi_{10}}dB^{(6)}$ \nand $\\ast_{4}db_I^{(2)}=e^{2\\phi_{10}}db_I^{(0)}$, \n$I=S,T_1,\\cdots T_{h^{1,1}}$. \n\nThe $U(1)$ charges of the dilaton and K\\\"ahler moduli for \nthe $U(1)^m$ symmetries, ${\\cal Q}_I^m$, $I=S,T_1,\\cdots T_{h^{1,1}}$ \nare defined via \nthe following couplings of the $U(1)$ gauge bosons $A_m$,\n\\begin{align}\nS \\supset \n\\sum_{m}\\frac{{\\cal Q}_S^m}{4l_s^2}\n\\int_{R^{1,3}}b_S^{(2)} \\wedge F_m +\n\\sum_{i,m}\\frac{{\\cal Q}_{T_i}^m}{2l_s^2}\n\\int_{R^{1,3}}b_{T_i}^{(2)} \\wedge F_m,\n\\label{eq:masscp}\n\\end{align} \nwhere \n\\begin{align}\n{\\cal Q}_S^m\\equiv {\\rm tr}(T^mT^m)\\int_{\\rm CY}\\frac{{\\rm tr}\\bar{F}_m}{2\\pi} \n\\wedge \\frac{1}{16\\pi^2} \n\\left( {\\rm tr}\\bar{F}^2 -\\frac{1}{2}{\\rm tr}\\bar{R}^2\\right),\n\\,\\,\\,\n{\\cal Q}_{T_i}^m\\equiv {\\rm tr}(T^mT^m)\\int_{T_i}\\frac{{\\rm tr}\\bar{F}_m}{2\\pi}, \n\\label{eq:charge}\n\\end{align} \n$T^m$ are the $U(1)^m$ generators embedded in the visible $E_8$ \ngauge group, ${\\bar F}_m$ and ${\\bar F}$ are the internal field \nstrengths of the $U(1)^m$ symmetry and $E_8^{\\rm vis}$ symmetry. \nSuch couplings are obtained from the dimensional reduction \nof the $10$D kinetic terms of $H$ given by Eq.~(\\ref{eq:heterob}) \nand the one-loop Green-Schwarz (GS) \ncounter term~\\cite{Ibanez:1986xy} \nwhich is determined by the S-dual of the \ntype I theory as shown in Appendix of Ref.~\\cite{Blumenhagen:2006ux}, \n\\begin{align}\nS_{\\rm GS}=\\frac{1}{24(2\\pi)^5\\alpha'}\\int B^{(2)}\\wedge X_8,\n\\label{eq:GS}\n\\end{align}\nwhere the eight-form $X_8$ reads,\n\\begin{align}\nX_8=\\frac{1}{24}{\\rm Tr}F^4-\\frac{1}{7200}({\\rm Tr}F^2)^2-\n\\frac{1}{240}({\\rm Tr}F^2)({\\rm tr}R^2) +\\frac{1}{8}{\\rm tr}R^4 \n+\\frac{1}{32}({\\rm tr}R^2)^2.\n\\end{align}\n\nThe mass terms of the $U(1)$ gauge \nbosons are derived by expanding the K\\\"ahler \npotential to second order on the vector multiplets,\n\\begin{align}\nS_{\\rm mass}&=-\\sum_{m,n} \\frac{M_{\\rm Pl}^2}{4}\n\\left( \\frac{K_{S\\bar{S}}{\\cal Q}_S^m {\\cal Q}_S^n}{(16\\pi^2)^2}\n+\\sum_{i,j} \\frac{K_{T_i\\bar{T_j}}{\\cal Q}_{T_i}^m {\\cal Q}_{T_j}^n}{(2\\pi)^2}\\right)\n\\int_{R^{1,3}} A_{m} \\wedge \\ast_4 A_n,\n\\label{eq:mass}\n\\end{align}\nwhich is typically of order the \nstring scale $M_s^2=1\/l_s^2$ with $l_s=2\\pi\\sqrt{\\alpha^{\\prime}}$, \nsee Refs.~\\cite{Blumenhagen:2005ga} for $E_8 \\times E_8$ and~\\cite{Blumenhagen:2005pm} for $SO(32)$ \nheterotic string theories and references therein.\n\nFrom the $U(1)$ invariant K\\\"ahler potential given by \nEq.~(\\ref{eq:Kahler}), $U(1)$ magnetic fluxes generate \nthe moduli-dependent Fayet-Iliopoulos terms~\\cite{Atick:1987gy}, \n\\begin{align}\n\\xi_m &=\n\\frac{\\partial{\\cal K}}{\\partial V_m}\\Biggl|_{V_m=0} \n=-\\frac{{\\cal Q}_S^m}{16\\pi^2} K_S -\\sum_{i=1}^{h^{1,1}} \n\\frac{{\\cal Q}_{T_i}^m}{2\\pi} K_{T_i},\n\\end{align} \nwhere $K_I=\\partial K\/\\partial Z^I$ for $Z^I=S,T_1,\\cdots, T_{h^{1,1}}$. \nFinally, we comment on the gauge kinetic function of the \nnon-abelian gauge groups obtained from the decomposition \nof the $E_8^{({\\rm vis})}\\times E_8^{({\\rm hid})}$ heterotic \nstring theory. They receive the one-loop \ncorrections originating from the one-loop GS \nterm as shown in Eq.~(\\ref{eq:GS}),\n\\begin{align}\n&f_{\\rm vis} =S+\\beta_i T_i, \\nonumber\\\\\n&f_{\\rm hid} =S-\\beta_i T_i,\n\\label{eq:gaugekin}\n\\end{align} \nwhere \n\\begin{align}\n\\beta_i \\equiv \\frac{1}{8\\pi}\\int_{\\rm CY}\n\\frac{1}{16\\pi^2} \n\\left( {\\rm tr}\\bar{F}^2 -\\frac{1}{2}{\\rm tr}\\bar{R}^2\\right)\n\\wedge \\hat{w}_i.\n\\label{eq:oneloop}\n\\end{align} \nBoth gauge kinetic functions in the visible and \nhidden sector are correlated by the tadpole \ncancellation condition of the \n$E_8^{({\\rm vis})}\\times E_8^{({\\rm hid})}$ heterotic string theory. \nFor the $SO(32)$ heterotic string theory, the \nnon-abelian gauge groups included in $SO(32)$ have \nthe nonuniversal gauge kinetic functions depend \non the decomposition of $SO(32)$.\n\n\n\n\\section{Natural inflation from heterotic string theory}\n\\label{sec:naturalinf}\nIn this section, we propose two natural inflation scenarios \nin the framework of the weakly coupled heterotic string \ntheory on ``Swiss-Cheese\" Calabi-Yau manifold with multiple $U(1)$ \nfluxes induced from the anomalous $U(1)$ symmetries. \nAs pointed out in the introduction, \nboth natural inflation scenarios are the single-filed \ninflation models whose inflaton are identified as the single \nK\\\"ahler axion with trans-Planckian axion decay constant. \nThe trans-Planckian axion decay constant is originating \nfrom the one-loop corrections to the gauge kinetic function \nof the hidden gauge groups to achieve the successful natural \ninflation which is different from the natural \ninflation scenarios by employing two axions with sub-Planckian \naxion decay constants~\\cite{Kim:2004rp}. \n\nOn the other hand, the other K\\\"ahler \naxions are absorbed by the multiple $U(1)$ gauge bosons \nand become massive. The real \npart of dilaton is stabilized at the finite value \nby the contributions from the non-perturbative effect \nto the dilaton K\\\"ahler potential and gaugino condensation \nterm as shown in Secs.~\\ref{subsec:naturalinf1} \nand \\ref{subsec:naturalinf2}, respectively. \nOne of the real parts of K\\\"ahler moduli is stabilized by \nthe world sheet instanton effect which leads to the \nstabilization of other real parts of K\\\"ahler moduli. \n \n\\subsection{Model $1$ (Single gaugino condensation)}\n\\label{subsec:naturalinf1}\nIn this section, we will show the inflaton potential along \nthe following three steps. First, the universal axion \nand K\\\"ahler axions except for the inflaton \nare absorbed by the multiple $U(1)$ gauge bosons \nat the string scale as shown in Eq.~(\\ref{eq:mass}). \nNext, the dilaton and all real parts of K\\\"ahler moduli \nare stabilized at the SUSY breaking minimum \nby the inclusion of non-perturbative corrections \nto the dilaton K\\\"ahler potential and superpotential \nwhich is the world sheet instanton effect. \nFinally, below the SUSY breaking scale, we get \nthe effective scalar potential for the light K\\\"ahler \naxion which is identified as the inflaton later. \n\n\\subsubsection{Setup}\n\\label{subsubsec:naturalinf1}\nWe consider the following K\\\"ahler potential with five \nK\\\"ahler moduli and four \nanomalous $U(1)$ symmetries, \n\\begin{align}\n{\\cal K}=&K\n\\left( S+\\bar{S}, V^1, V^2, V^3\\right)\n\\nonumber\\\\\n-&\\ln \\left\\{ k_1(T_1+\\bar{T}_1)^3 \n-k_2\\left( T_2 +\\bar{T}_2 -\\sum_{n=1}^3q_{T_2}^n V^n\\right)^3 \n-k_3\\left( T_3 +\\bar{T}_3 -\\sum_{n=1}^3q_{T_3}^n V^n\\right)^3 \\right.\n\\nonumber\\\\\n&\\left. -k_4\\left(T_4 +\\bar{T}_4 -q_{T_4}^4 V^4 \\right)^3\n-k_5\\left(T_5 +\\bar{T}_5 -q_{T_5}^4 V^4 \\right)^3 \n\\right\\},\n\\label{eq:m1K}\n\\end{align} \nin the unit $M_{\\rm Pl}=1$ \nwith $M_{\\rm Pl}=2.4\\times 10^{18} {\\rm GeV}$, \nwhere we choose $h^{1,1}=5$, $q_{T_i}^m={\\cal Q}_{T_i}^m\/2\\pi$, \n$i=1,2,3,4,5$, $m=1,2,3,4$ and $k_i$, $i=1,2,3,4,5$ \nare the positive constants determined by the intersection \nnumbers $d_{t_1t_1t_1}$, $d_{t_2t_2t_2}$, $d_{t_3t_3t_3}$, \n$d_{t_4t_4t_4}$,$d_{t_5t_5t_5}$. (The reason why we \nchoose five K\\\"ahler moduli and four anomalous $U(1)$s are \nshown later.)\n\nThe K\\\"ahler potential of dilaton consists of the tree-level \nand the non-perturbative part,\n\\begin{align}\nK^0&=-\n\\ln \\left( S+\\bar{S}-\\sum_{n=1}^3q_S^n V^n\\right),\n\\nonumber\\\\\nK^{\\rm np}&=\nd\\,g^{-p}\ne^{-b\/g},\n\\label{eq:dilaton}\n\\end{align} \nwhere $b$, $p$, and $d$ are the real constants, \n$q_s^n={\\cal Q}_S^n\/16\\pi^2$, $n=1,2,3$ and \n$g=({\\rm Re}\\,S-\\sum_{i\\neq 1} \\beta_i{\\rm Re}\\,T_i)^{-1\/2}$ \nis the gauge coupling in the hidden sector as \nshown in the superpotential (\\ref{eq:gaugekin}). \n$K^{\\rm np}$ denotes the non-perturbative correction to the \nK\\\"ahler potential~\\cite{Shenker:1990uf,Burgess:1995aa,Casas:1996zi}. \nThere are known ansatz to write the dilaton K\\\"ahler \npotential as \n\\begin{align}\nK&=K^0+K^{\\rm np}\\,\\,\\,{\\rm or}\\,\\,\\,\\,\\,\nK=\\ln \\left( e^{K^0}+e^{K^{\\rm np}}\\right),\n\\label{eq:dilatonK}\n\\end{align} \netc.\\footnote{In~\\cite{Burgess:1995aa}, the non-perturbative \nK\\\"ahler potential of dilaton is discussed in the effective field \ntheory approaches.}. Anyway, we assume that the dilaton is stabilized at \nthe finite value due to such corrections to the K\\\"ahler \npotential as discussed in~\\cite{Higaki:2003jt}.\nNote that our following moduli stabilization as well as the \ninflation mechanism do not depend on the detailed structure \nof the non-perturbative K\\\"ahler potential, $K^{\\rm np}$. \n\nIn addition to the K\\\"ahler potential, we consider the \nfollowing $U(1)^m$, $m=1,2,3,4$, invariant superpotential,\n\\begin{align}\nW=&W_0+A\\,e^{-\\frac{8\\pi^2}{a}(S-\\beta_2 T_2-\\beta_3 T_3\n-\\beta_4 T_4-\\beta_5 T_5)} \n+B\\,e^{-\\mu_1 T_1}, \n\\label{eq:m1W}\n\\end{align} \nwhere $W_0$ is the Neveu-Schwarz (NS) three-form flux induced constant \nterm which stabilizes \nthe $h^{1,2}$ complex structure moduli of the CY manifold, \nthe second term of the right handed side (r.h.s.) shows the \nhidden sector gaugino condensation which \nreceive the one-loop corrections originating from the one-loop \nGreen-Schwarz counter term given by Eq.~(\\ref{eq:oneloop}). \nThe third term of the (r.h.s.) shows the world-sheet instanton \neffects on the two cycle $T_1$.  \n\nAs the first step to obtain the inflaton potential, \nwe comment on the anomalous $U(1)^m$ \nvector multiplets $V^m$, $m=1,2,3,4$. \nSuch $U(1)^m$ vector multiplets are massive due to the \n$U(1)^m$ magnetic fluxes as shown in Eq.~(\\ref{eq:mass}) and then \n$U(1)^m$ gauge bosons absorb the linear \ncombination of imaginary component of the dilaton and the K\\\"ahler \nmoduli. \n$U(1)^1$, $U(1)^2$ and $U(1)^3$ gauge bosons absorb \nthe following linear combination of the moduli, \n\\begin{align}\nX^1&=\\frac{1}{N^1}\\left( \\frac{{\\rm Im}\\,S}{q_S^1\\sqrt{K_{S\\bar{S}}}} \n+\\frac{{\\rm Im}\\,T_2}{q_{T_2}^1\\sqrt{K_{T_2\\bar{T}_2}}}\n+\\frac{{\\rm Im}\\,T_3}{q_{T_3}^1\\sqrt{K_{T_3\\bar{T}_3}}}\\right), \n\\nonumber\\\\\nX^2&=\\frac{1}{N^2}\\left( \\frac{{\\rm Im}\\,S}{q_S^2\\sqrt{K_{S\\bar{S}}}} \n+\\frac{{\\rm Im}\\,T_2}{q_{T_2}^2\\sqrt{K_{T_2\\bar{T}_2}}}\n+\\frac{{\\rm Im}\\,T_3}{q_{T_3}^2\\sqrt{K_{T_3\\bar{T}_3}}}\\right), \n\\nonumber\\\\\nX^3&=\\frac{1}{N^3}\\left( \\frac{{\\rm Im}\\,S}{q_S^3\\sqrt{K_{S\\bar{S}}}} \n+\\frac{{\\rm Im}\\,T_2}{q_{T_2}^3\\sqrt{K_{T_2\\bar{T}_2}}}\n+\\frac{{\\rm Im}\\,T_3}{q_{T_3}^3\\sqrt{K_{T_3\\bar{T}_3}}}\\right), \n\\end{align} \nwhere $N^i=\\sqrt{(1\/q_S^n\\sqrt{K_{S\\bar{S}}})^2+(1\/q_{T_2}^n\\sqrt{K_{T_2\\bar{T}_2}})^2+(1\/q_{T_3}^n\\sqrt{K_{T_3\\bar{T}_3}})^2}$, $n=1,2,3$,\nrespectively. Here the dilaton and K\\\"ahler moduli \nare canonically normalized and their K\\\"ahler metric \nare summarized in Appendix~\\ref{app:mass}.\n\nThus, the imaginary component of $S$, $T_2$ and $T_3$ \nare absorbed by the $U(1)^1$, $U(1)^2$, $U(1)^3$ gauge bosons \nand their mass-squared matrices are given by\n\\begin{align}\nM_{m,n}^2\\simeq \\frac{M_{\\rm Pl}^2}{4\\sqrt{\\langle{\\rm Re}\\,f_{m,m}\\rangle}\n\\sqrt{\\langle{\\rm Re}\\,f_{n,n}\\rangle}}\n\\left( K_{S\\bar{S}}q_S^m q_S^n\n+\\sum_{i,j} K_{T_i\\bar{T_j}}q_{T_i}^m q_{T_j}^n\\right)\n\\label{eq:m1mass}\n\\end{align} \nfor $m,n=1,2,3$, where the $U(1)$ gauge bosons are canonically \nnormalized. The gauge kinetic functions \nof $U(1)$s, $f_{m,n}$ are given by \n$f_{m,n}={\\rm tr}(T^mT^n)S\\delta_{m,n} +{\\cal O}(\\beta T)$. \nThese $U(1)^{1}$, $U(1)^{2}$ charges of the moduli \n$S$, $T_2$ and $T_3$ are related by the $U(1)^{1}$, $U(1)^{2}$ \nand $U(1)^{3}$ gauge invariance of the superpotential~(\\ref{eq:m1W}), \n\\begin{align}\nq_S^1=q_{T_2}^1\\,\\beta_2 +q_{T_3}^1\\,\\beta_3,\\,\\,\\,\nq_S^2=q_{T_2}^2\\,\\beta_2 +q_{T_3}^2\\,\\beta_3,\\,\\,\\,\nq_S^3=q_{T_2}^3\\,\\beta_2 +q_{T_3}^3\\,\\beta_3,\n\\label{eq:cdcharge}\n\\end{align} \nUnder the $U(1)$ gauge invariance condition~(\\ref{eq:cdcharge}), \nthe full-rank mass matrices~(\\ref{eq:m1mass}) \nare realized if the number of $U(1)$s are bigger than three. \nThus we can stabilize the K\\\"ahler axion and \nuniversal axion. The universal axion cannot be identified as\nthe candidate of inflaton, because its decay constant is \nmuch less than the Planck scale as shown in the \nsuperpotential~(\\ref{eq:m1W}).  \n\nThe other $U(1)^4$ gauge boson absorbs the following \ncombination of the moduli, \n\\begin{align}\nX^4=\\frac{1}{N^4}\\left( \\frac{{\\rm Im}\\,T_4}{q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}}} \n+\\frac{{\\rm Im}\\,T_5}{q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}}}\\right), \n\\end{align} \nwhere $N^4=\\sqrt{(1\/q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}})^2+(1\/q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}})^2}$, and \nthe orthogonal direction of $X_4$ (which is \nidentified as the inflaton later), \n\\begin{align}\nY^4=\\frac{1}{N^4}\\left(-\\frac{{\\rm Im}\\,T_4}{q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}}} \n+\\frac{{\\rm Im}\\,T_5}{q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}}} \\right), \n\\label{eq:infdef}\n\\end{align} \ncannot be absorbed by the anomalous $U(1)$ gauge \nbosons and the mass of $Y^4$ is obtained from the \ngaugino condensation term in Eq.~(\\ref{eq:m1W}). \nIn summary, \nthe four imaginary parts of the moduli $X^m$, $m=1,2,3,4$ are \nabsorbed by four anomalous $U(1)$ vector multiplets \nafter following the above procedure. \nAs the second step to obtain the inflaton potential, \nlet us discuss the F-term potential derived from the \nK\\\"ahler potential~(\\ref{eq:m1K}) \nand the superpotential~(\\ref{eq:m1W}). \n\nFirst, we redefine the linear combination of \nthe dilaton and the K\\\"ahler moduli as,\n\\begin{align}\n\\Phi =S-\\beta_2 T_2-\\beta_3 T_3-\\beta_4 T_4-\\beta_5 T_5,\n\\end{align} \nand then the K\\\"ahler potential and superpotential given by \nEqs.~(\\ref{eq:m1K}) and (\\ref{eq:m1W}) are rewritten by \n\\begin{align}\n{\\cal K}=&K\\left( \\Phi +\\bar{\\Phi}, T_2+\\bar{T}_2, T_3 +\\bar{T}_3, \nT_4+\\bar{T}_4, T_5 +\\bar{T}_5, V^1, V^2, V^3\\right) \\nonumber\\\\ \n-&\\ln \\left\\{ k_1(T_1+\\bar{T}_1)^3 \n-k_2\\left( T_2 +\\bar{T}_2 -\\sum_{n=1}^3q_{T_2}^n V^n\\right)^3 \n-k_3\\left( T_3 +\\bar{T}_3 -\\sum_{n=1}^3q_{T_3}^n V^n\\right)^3 \\right.\n\\nonumber\\\\\n&\\left. -k_4\\left(T_4 +\\bar{T}_4 -q_{T_4}^4 V^4 \\right)^3\n-k_5\\left(T_5 +\\bar{T}_5 -q_{T_5}^4 V^4 \\right)^3 \n\\right\\},\n\\nonumber\\\\\nW=&\\,W_0+A\\,e^{-\\frac{8\\pi^2}{a}\\,\\Phi} \n+B\\,e^{-\\mu_1 T_1}, \n\\label{eq:m1KW}\n\\end{align} \nand we assume that the gaugino condensation term \nin Eq.~(\\ref{eq:m1KW}) is much smaller than the other \nterms in Eq.~(\\ref{eq:m1KW}) at least at the minimum, that is, \n$W_0$, $B\\,e^{-\\mu_1 T_1}$ \n$\\gg Ae^{-\\frac{8\\pi^2}{a}\\,\\Phi}$. \nTherefore, at the moment, we ignore the contribution \nof the gaugino condensation term as we will mention later. \n\nSecond, we stabilize the moduli $T_1$, ${\\rm Re}\\,T_2$, \n${\\rm Re}\\,T_3$, ${\\rm Re}\\,T_4$, ${\\rm Re}\\,T_5$ and ${\\rm Re}\\,\\Phi$\nby imposing the supersymmetric conditions,\n\\begin{align}\n&D_{T_1}W=0,\\nonumber\\\\\n&D_{T_2}W=K_{T_2}W=0,\\,\\,\nD_{T_3}W=K_{T_3}W=0, \\,\\,\nD_{T_4}W=K_{T_4}W=0,\\,\\,\nD_{T_5}W=K_{T_5}W=0 \\nonumber\\\\\n&D_{\\Phi}W=K_{\\Phi}W=0,\n\\label{eq:m1cd1}\n\\end{align} \nwhere $D_I W=W_I +K_I W$ and $W_I=\\partial W\/\\partial Z_I$ \nwith $Z_I=T_1,T_2,T_3,T_4,T_5,\\Phi$. \nThen the D-term potential induced from the \nK\\\"ahler potential~(\\ref{eq:m1K}) \nare automatically vanished under \nthe above supersymmetric conditions, \n$K_{T_2}=K_{T_3}=K_{T_4}=K_{T_5}=K_{\\Phi}=0$. \n${\\rm Re}\\,\\Phi$ is stabilized by the contribution from the \nnon-perturbative correction to the dilaton,\n\\begin{align}\nK_{\\Phi}=0.\n\\end{align}  \nIn the same way for ${\\rm Re}\\,\\Phi$, the real parts of moduli \n$T_j$, $j=2,3,4,5$, are stabilized by the following conditions,\n\\begin{align}\nK_{T_j}\\simeq \\frac{3k_j(T_j+\\bar{T}_j)^2}{k_1(T_1+\\bar{T}_1)^3}\n+\\frac{\\partial K^0}{\\partial T_j} \n\\simeq \\frac{3k_j(T_j+\\bar{T}_j)^2}{k_1(T_1+\\bar{T}_1)^3}\n-\\frac{\\beta_j}{\\Phi +{\\bar \\Phi}}\n+{\\cal O}\\left(\\beta_j\\frac{\\sum_{k=2}^5 \\beta_{k}{\\rm Re}\\,T_k}{{\\rm Re}\\,\\Phi}\\right) \n=0,\n\\end{align}  \nwhere the dilaton K\\\"ahler potential is approximated as \nits tree-level part $K^0$ in Eq.~(\\ref{eq:dilatonK}). \nIn the limit of ${\\rm Re}\\,T_1 >{\\rm Re}\\,T_j$, \n$j=2,3,4,5$, the above equations are rewritten by \n\\begin{align}\n{\\rm Re}\\,S\\simeq {\\rm Re}\\,\\Phi \\simeq \n\\frac{k_1({\\rm Re}\\,T_1)^3}{3k_j{\\rm Re}\\,T_j^2}\\beta_j \\gg \\beta_j {\\rm Re}\\,T_j,\n\\end{align}  \nfor $j=2,3,4,5$. Thus the tree-level part of \nthe gauge kinetic function is always bigger than the one-loop \ncorrections of one under the condition that \n${\\rm Re}\\,T_1 >{\\rm Re}\\,T_j$ ($j\\neq 1$) \nas shown in Eq.~(\\ref{eq:gaugekin}), that is, the \nperturbative expansion is valid. \nThis property is an important feature of the ``Swiss-Cheese\" \nCalabi-Yau manifold. \n\nFrom the scalar potential given by using the formula of $4$D $N=1$ \nsupergravity, \n\\begin{align}\nV=e^K\\left( K^{I\\bar{J}} D_I WD_{\\bar{J}} \\bar{W} -3|W|^2\\right),\n\\label{eq:scalarpo}\n\\end{align} \nwe get the supersymmetric AdS minimum at the minimum \ngiven by Eq.~(\\ref{eq:m1cd1}),\n\\begin{align}\n\\langle V\\rangle =-3e^K|W|^2.\n\\end{align} \nThere are several approaches to uplift such AdS vacuum by the \nF-terms with dynamical SUSY breaking \nsector~\\cite{Dudas:2006gr,Abe:2006xp,Kallosh:2006dv,Abe:2007yb} or D-terms with anti-heterotic five branes~\\cite{Buchbinder:2004im}, \netc.. \nHere we assume that the SUSY \nis broken by the dynamical SUSY breaking sector \nwhose K\\\"ahler potential and superpotential are given by\n\\begin{align}\n&\\Delta K=|X|^2 -\\frac{|X|^4}{\\Lambda^2},\\nonumber\\\\\n&\\Delta W=\\mu X,\n\\label{eq:m1KW1loop}\n\\end{align} \nwhere X is gauge singlet chiral superfield under the non-abelian \ngroups in the visible sector $G_{\\rm vis}$ and anomalous $U(1)^{m}$ symmetries, $m=1,2,3,4$, \n$\\Lambda$ is the dynamical SUSY breaking scale and we omit the \nmoduli dependence of $X$, because they do not affect the following \nmoduli stabilization. Then the Minkowski minimum is realized by choosing the parameter $\\mu$ as \n\\begin{align}\n\\langle V\\rangle +\\Delta V\\simeq e^{\\langle K\\rangle}\\left( \n-3|\\langle W\\rangle|^2 +K^{X\\bar{X}}|\\mu|^2\n\\right)=0 \\Leftrightarrow |\\mu|^2 =3|\\langle W\\rangle|^2.\n\\end{align} \n\nFinally, we consider the contribution of the omitted term \n$Ae^{-\\frac{8\\pi^2}{a}\\,\\Phi}$ in the superpotential~(\\ref{eq:m1KW}) \nwhich is ignored on the previous analysis. \nSince we assume that such omitted term is much smaller \nthan the other terms in the superpotential~(\\ref{eq:m1KW}) at the minimum, \nthe moduli ${\\rm Re}\\,\\Phi$, $T_1$, ${\\rm Re}\\,T_2$, ${\\rm Re}\\,T_3$, ${\\rm Re}\\,T_4$ and \n${\\rm Re}\\,T_5$ \nare stabilized at the minimum close to the values given by \nEq.~(\\ref{eq:m1cd1}) and are decoupled from the \ninflaton dynamics if their masses are heavier \nthan the inflation scale. \nThe mass scales of these moduli are determined by \nthe constant term, the world-sheet instanton \neffect of the superpotential~(\\ref{eq:m1KW}) \nand the D-term contribution~(\\ref{eq:m1K}) \nwhich is heavier than the inflation scale as shown later. \n(The mass matrices of them \nare summarized in the Appendix~\\ref{app:mass}.). \nAs mentioned before, the imaginary parts of the moduli \nexcept for the inflaton $Y^4$ \nare absorbed by the four $U(1)$ gauge bosons \nwhose mass scale is of order the string scale $M_s$. \n\n\\subsubsection{Inflaton potential and its dynamics}\nNow we are ready to write down the inflation potential. \nAs discussed in the previous section~\\ref{subsubsec:naturalinf1}, \nafter integrating out these heavy \nmoduli and substituting the field values given by Eq.~(\\ref{eq:m1cd1}), \nwe get the effective scalar potential \nfor the light moduli $Y^4$ which is the linear combination of \n${\\rm Im}\\,T_4$ and ${\\rm Im}\\,T_5$ given by Eq.~(\\ref{eq:infdef}), \n\\begin{align}\nV_{\\rm eff} \n\\simeq \\Lambda^4 (1-{\\rm cos}\\,(\\beta\\,\\hat{Y}^4)),\n\\label{eq:ninf_po}\n\\end{align} \nin the limit of $Ae^{-\\frac{8\\pi^2}{a}\\,\\langle {\\rm Re}\\Phi\\rangle}\n\\ll W_0, Be^{-\\mu_1 \\langle T_1\\rangle}$, \nwhere the energy scale of the scalar potential $\\Lambda^4$ \nand the axion decay constant $\\beta$ are defined as \n\\begin{align}\n\\Lambda^4 &\\equiv 6\\,e^{K}e^{-\\frac{8\\pi^2}{a}\\,{\\rm Re}\\Phi}\nA(W_0+Be^{-\\mu_1 T_1}),\n\\nonumber\\\\\n\\beta &\\equiv \\frac{8\\pi^2}{a\\,N^4{\\hat N}^4}\\,\n\\left(\\frac{\\beta_5}{q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}}}\n-\\frac{\\beta_4}{q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}}}\\right),\n\\label{eq:decct}\n\\end{align} \nand \n\\begin{align}\n{\\hat Y^4}\\simeq \n\\frac{1}{N^4}\\sqrt{2\\left(\\frac{K_{T_4\\bar{T}_4}}{(q_{T_5}^4)^2K_{T_5\\bar{T}_5}}\n+\\frac{K_{T_5\\bar{T}_5}}{(q_{T_4}^4)^2K_{T_4\\bar{T}_4}}\\right)}\\,Y^4 \n\\equiv \\hat{N}^4\\,Y^4\n\\end{align} \nis the canonically normalized axion field. \nHere we employed the following redefinitions of the \nmoduli,\n\\begin{align}\n&{\\rm Im}\\,T_4=\n\\frac{1}{N^4}\\left( \n\\frac{X^4}{q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}}}\n-\\frac{Y^4}{q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}}}\\right),\n\\nonumber\\\\\n&{\\rm Im}\\,T_5=\n\\frac{1}{N^4}\\left( \n\\frac{X^4}{q_{T_5}^4\\sqrt{K_{T_5\\bar{T}_5}}}\n-\\frac{Y^4}{q_{T_4}^4\\sqrt{K_{T_4\\bar{T}_4}}}\\right),\n\\end{align} \nand $U(1)^4$ gauge invariance of the superpotential~(\\ref{eq:m1KW}), \n\\begin{align}\nq_{T_4}^4\\,\\beta_4+ q_{T_5}^4\\,\\beta_5 =0.\n\\end{align} \n\nWhen we identify the axion ${\\hat Y}^4$ as the inflaton, \nthe effective scalar potential~(\\ref{eq:ninf_po}) is considered as the inflation potential for the single-field ${\\hat Y}^4$, \nsince the mass of the other moduli \nare much heavier than the inflaton. Thus we can realize the \nscalar potential of the type of natural inflation. \nThe power spectrum of the scalar density perturbation is \nexplained by choosing the parameter,\n$\\Lambda^4 \\sim {\\cal O}(10^{-9})$ in the $M_{\\rm Pl}$ unit, and \nthe spectral index of the scalar density perturbation and \nthe tensor-to-scalar ratio is also consistent with the \ncosmological observations reported by WMAP, Planck and\/or \nBICEP2 collaborations. This is because we can realize the \ntrans-Planckian axion decay constant $\\beta$ originating \nfrom the one-loop corrections to the gauge kinetic function \nas shown in Eq.~(\\ref{eq:decct}). \n\nNext, we estimate the cosmological observables constrained \nby the observations. We choose the dilaton K\\\"ahler potential \nas the type of $K^0+K^{\\rm np}$~\\footnote{The stabilization of \nmoduli are discussed in Appendix~\\ref{app:mass}.} and the following input parameters \nin the K\\\"ahler potential given by Eqs.~(\\ref{eq:m1K}) \nand~(\\ref{eq:dilaton}) as \n\\begin{align} \n&k_1=k_2=k_3=k_4=k_5=\\frac{1}{8},\\nonumber\\\\\n&d=7,\\,b=1,\\,p=2,\\nonumber\\\\\n&\\beta_2\\simeq\\beta_3\\simeq\\beta_4\\simeq\\beta_5\\simeq 0.01,\n\\label{eq:inputK}\n\\end{align}  \nand in the superpotential given by Eqs.~(\\ref{eq:m1W}) \nand~(\\ref{eq:m1KW1loop}) as,\n\\begin{align} \nA\\,=\\frac{1}{300},\\,\\,\\,a=30,\\,\\,\\,B=-\\frac{1}{2},\n\\,\\,\\,\\mu_1= 2\\pi,\\,\\,\\, W_0=6\\times \n10^{-4},\\,\\,\\,\\mu\\simeq 1\\times 10^{-3},\n\\label{eq:inputW}\n\\end{align}  \nin the unit $M_{\\rm Pl}=1$ and \nthe $U(1)$ charges of the moduli are of ${\\cal O}(1)$. \nFrom these input parameters, \nwe get the field values of the moduli at the minimum,\n\\begin{align} \nT_1\\,\\simeq 1.3,\\,\\,\\,T_2\\simeq T_3\\simeq T_4\\simeq T_5\\simeq 0.06,\n\\,\\,\\,\nS\\simeq \\Phi\\simeq 2,\n\\end{align}  \nwhich yield the gauge coupling unification of the \ngrand unified theory (GUT) at the Kaluza-Klein (KK) scale, \n\\begin{align} \nM_{KK}\\simeq \\frac{M_s}{{\\cal V}^{1\/6}} \\simeq 1.2\\times 10^{17}\n\\,{\\rm GeV},\n\\end{align}  \nwith \n\\begin{align} \nM_s=\\frac{M_{\\rm Pl}}{\\sqrt{4\\pi \\alpha^{-1}}} \\simeq 1.4\\times 10^{17}\n\\,{\\rm GeV},\n\\end{align}  \nwhere $\\alpha^{-1}\\simeq 24$ is the gauge coupling \nof visible gauge group $G_{\\rm vis}$ at the string \nscale. \n\nBy employing the input parameters given by \nEqs.~(\\ref{eq:inputK}) and~(\\ref{eq:inputW}), \nthe energy scale of the scalar potential,\n\\begin{align} \n\\Lambda^4 \\simeq 3.22 \\times 10^{-9}, \n\\end{align}  \nand the axion decay constant,\n\\begin{align} \n\\beta^{-1} \\simeq 7.8,\n\\end{align}  \nin the unit $M_{\\rm Pl}=1$, are obtained which leads \nto the desired trans-Planckian axion decay constant. \n\nTo estimate the cosmological observables, we define \nthe slow-roll parameters,\n\\begin{align} \n\\epsilon &\\equiv \\frac{M_{\\rm Pl}^2}{2}\n\\left(\\frac{\\partial_{\\hat{Y}^4}V_{\\rm eff}}{V_{\\rm eff}}\\right)^2,\n\\nonumber\\\\\n\\eta &\\equiv M_{\\rm Pl}^2 \n\\frac{\\partial_{\\hat{Y}^4}^2 V_{\\rm eff}}{V_{\\rm eff}},\n\\nonumber\\\\\n\\xi^2 &\\equiv M_{\\rm Pl}^4 \n\\frac{\\partial_{\\hat{Y}^4} V_{\\rm eff} \n\\partial_{\\hat{Y}^4}^3 V_{\\rm eff}}{V_{\\rm eff}^2},\n\\end{align}  \nand then the e-folding number from the time $t_\\ast$ \nto the inflation end $t_{\\rm end}$ is estimated as \n\\begin{align} \nN_e=-\\int_{t_{\\rm end}}^{t_\\ast} dt H(t) \\simeq \n\\frac{1}{M_{\\rm Pl}}\\int_{\\hat{Y}^4_\\ast}^{\\hat{Y}^4_{\\rm end}}\n\\frac{d\\hat{Y}^4}{\\sqrt{2\\epsilon}},  \n\\end{align}  \nwhere the Hubble parameter $H(t)$ is defined as \n$H(t)=\\frac{\\dot{a}(t)}{a(t)}$, $a(t)$ is the \nscale factor of the $4$D spacetime. \n$\\hat{Y}^4_\\ast$ and $\\hat{Y}^4_{\\rm end}$ are the \nfield values of the inflaton $\\hat{Y}^4$ at the time \n$t_\\ast$ and $t_{\\rm end}$, respectively.\n\\footnote{The end of inflation is estimated when the \nslow-roll condition is violated as \n${\\rm max}\\{ |\\epsilon|, |\\eta|\\}=1$.}\nThe observables such as the power spectrum \nof the scalar density perturbation~$P_{\\zeta}$, \nthe spectral index of it~$n_s$, the tensor-to-scalar ratio~$r$ are written in terms of the \nslow-roll parameters as  \n\\begin{align} \nP_\\zeta &=\\frac{1}{24\\pi^2}\n\\frac{V_{\\rm eff}}{\\epsilon M_{\\rm Pl}^4},\n\\nonumber\\\\\nn_s&= 1-6\\epsilon +2\\eta,\n\\nonumber\\\\\nr&=16\\epsilon.\\,\\,\\,\n\\end{align}  \nAt the field value $\\hat{Y}^4_\\ast \\simeq 13M_{\\rm Pl}$, \nwe find the numerical values of observables and the \ne-folding number as \n\\begin{align} \nP_\\zeta \\simeq 2.2\\times 10^{-9},\\,\\,\\,\nn_s\\simeq 0.956,\\,\\,\\,\nr\\simeq 0.11,\\,\\,\\,\nN_e\\simeq 48,\n\\end{align}  \nwhich are consistent with the WMAP, Planck \ndata~\\cite{Ade:2013uln},\n\\begin{align} \nP_\\zeta = 2.196^{+0.051}_{-0.060}\\times 10^{-9},\\,\\,\\,\nn_s= 0.9583\\pm 0.0080,\n\\end{align}  \nat the pivot scale $k_{\\ast}=0.05 {\\rm Mpc}^{-1}$ \nand BICEP2 data~\\cite{Ade:2014xna},\n\\begin{align} \nr=0.16^{+0.06}_{-0.05},\n\\end{align}  \nafter considering the foreground dust. \nNow we choose the hidden gauge group \nas $E_8$ which leads to the dual Coxeter number $a=30$ and \n$\\beta_3\\simeq \\beta_4\\simeq \\beta_5\\simeq 0.01$. \n\nNote that we can realize smaller tensor-to-scalar ratio \nwhich is more consistent with the WMAP and Planck data, since \nthe size of the axion decay constant \n$\\beta$ depends on the dual Coxeter number of the hidden \ngauge group~$a$ in Eq.~(\\ref{eq:decct}) and the size of one-loop \ncorrection to the gauge kinetic function of the hidden gauge \ngroup in Eq.~(\\ref{eq:oneloop}). \n\n\n\\subsection{Model $2$ (Double gaugino condensations)}\n\\label{subsec:naturalinf2}\nIn this section, we propose the natural inflation based on \nthe other type of the K\\\"ahelr potential and superpotential. \nThe main difference between the model $1$ in the previous \nSec.~\\ref{subsec:naturalinf1} and the model $2$ in this section \nis the stabilization mechanism of dilaton. In the model $1$, the \ndilaton is stabilized at the finite value by the non-perturbative \ncorrections to its K\\\"ahler potential given by \nEq.~(\\ref{eq:dilatonK}). \nHowever, in the model $2$, the dilaton is stabilized \nby using one of the gaugino condensation terms which will be \nmentioned later. \nIn the same way as the model $1$, the trans-Planckian \naxion decay constant is realized from the one-loop correction \nto the gauge kinetic function of the hidden gauge group.\n\n\\subsubsection{Setup}\nWe consider the CY manifold expressed by the following \nK\\\"ahler potential with one $U(1)$ anomalous symmetry,  \n\\begin{align}\n{\\cal K}=-&\n\\ln \\left( S+\\bar{S} \\right)\n-\\ln \n\\left( k_b(T_b+\\bar{T}_b)^3 -k_s\\left( T_s +\\bar{T}_s -\\frac{{\\cal Q}_s}{2\\pi} V_s \\right)^3  \n-k_s^\\prime\\left(T_s^\\prime +\\bar{T}_s^\\prime -\\frac{{\\cal Q}_s^\\prime}{2\\pi} V_s \\right)^3 \n\\right) ,\n\\label{eq:m2K}\n\\end{align} \nin the unit $M_{\\rm Pl}=1$, where $h^{1,1}=3$, \n$k_b$, $k_s$, $k_s^\\prime$ are positive constants \ndetermined by the triple intersection number \n$d_{t_bt_bt_b}$, $d_{t_st_st_s}$, \n$d_{t_s^\\prime t_s^\\prime t_s^\\prime}$ \nand $V_s$ is an anomalous $U(1)_s$ vector multiplet \nunder which only two moduli \n$T_s$ and $T_s^\\prime$ have $U(1)_s$ charge. \n$U(1)_s$ vector multiplet absorbs the linear \ncombination of K\\\"ahler axions, while the other massless axion \nis identified as the inflaton. We further assume \nthat the dilaton K\\\"ahler potential is approximated by its tree-level \nK\\\"ahler potential. \n\nNext, we consider the following $U(1)_s$ invariant superpotential,\n\\begin{align}\nW=&w_0+A_2\\,e^{-\\frac{8\\pi^2}{a_2}(S-\\beta_s^{(1)} T_s-\\beta_s^{\\prime (1)} T_s^\\prime)} \n+B_2\\,e^{-\\frac{8\\pi^2}{b_2}(S-\\beta_s^{(2)} T_s-\\beta_s^{\\prime (2)} T_s^\\prime)} \n+C_2\\,e^{-\\mu_b T_b}, \n\\label{eq:m2W}\n\\end{align} \nwhere $w_0$ is the NS flux induced constant term which stabilizes \nthe $h^{1,2}$ complex structure moduli of the CY manifold, \nthe second and third term of the right handed side (r.h.s.) show the gaugino condensations on two hidden sectors, the fourth term of the (r.h.s.) shows the world-sheet instanton effect on the two-cycle $T_b$.  \nLet us discuss the moduli stabilization and the inflaton potential. \n\nFirst, $U(1)_s$ vector multiplet becomes massive whose mass scale is \nof order the string scale due to the $U(1)_s$ magnetic fluxes as shown \nin Eq.~(\\ref{eq:mass}), and then $U(1)_s$ gauge boson absorbs the linear \ncombination of ${\\rm Im}~T_s$ and ${\\rm Im}~T_s^\\prime$ as,\n\\begin{align}\nX_s=\\frac{1}{N_s}\\left( \\frac{{\\rm Im}\\,T_s}{q_s\\sqrt{K_{T_s\\bar{T}_s}}} \n+\\frac{{\\rm Im}\\,T_s^\\prime}{q_s^\\prime\\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}}}\\right), \n\\end{align} \nwhere $N_s=\\sqrt{(1\/q_s\\sqrt{K_{T_s\\bar{T}_s}})^2\n+(1\/q_s^\\prime\\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}})^2}$ with \n$q_s={\\cal Q}_s \/2\\pi$ and $q_s^\\prime={\\cal Q}_s^\\prime \/2\\pi$ \nand two K\\\"ahler moduli are canonically normalized under \nthe condition that their K\\\"ahler mixing are neglected, because \ntheir stabilization is also the same as the previous model $1$ in \nSec.~\\ref{subsec:naturalinf1}. \nIts orthogonal direction (which is identified as the inflaton later), \n\\begin{align}\nY_s=\\frac{1}{N_s}\n\\left( -\\frac{{\\rm Im}\\,T_s}{q_s^\\prime\\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}}}\n+\\frac{{\\rm Im}\\,T_s^\\prime}{q_s\\sqrt{K_{T_s\\bar{T}_s}}}\\right), \n\\end{align} \nremains massless. The $U(1)_s$ charges of the moduli are related as \n\\begin{align}\n&q_s\\,\\beta_s^{(1)}+q_s^\\prime\\,\\beta_s^{^\\prime (1)} =0,\n\\nonumber\\\\\n&q_s\\,\\beta_s^{(2)}+q_s^\\prime\\,\\beta_s^{^\\prime (2)} =0,\n\\end{align} \ndue to the $U(1)_s$ gauge invariance of the \nsuperpotential~(\\ref{eq:m2W}). \n\nSecond, we redefine a linear combination of dilaton and K\\\"ahler \nmoduli as,\n\\begin{align}\n\\Phi =S-\\beta_s^{(1)} T_s-\\beta_s^{\\prime (1)} T_s^\\prime,\n\\end{align} \nand then the K\\\"ahler potential and superpotential are rewritten by \n\\begin{align}\n{\\cal K}=-&\n\\ln \\left( \\Phi +\\bar{\\Phi} +\\beta_s^{(1)} (T_s+\\bar{T}_s) \n+\\beta_s^{\\prime (1)} (T_s^\\prime +\\bar{T}_s^\\prime) \\right) \\nonumber\\\\ \n-&\\ln \\left( k_b(T_b+\\bar{T}_b)^3 -k_s\\left( T_s +\\bar{T}_s -\\frac{{\\cal Q}_s}{2\\pi} V_s \\right)^3  \n-k_s^\\prime\\left(T_s^\\prime +\\bar{T}_s^\\prime -\\frac{{\\cal Q}_s^\\prime}{2\\pi} V_s \\right)^3 \n\\right),\\nonumber\\\\\nW=&w_0+A_2\\,e^{-\\frac{8\\pi^2}{a_2}\\,\\Phi} \n+B_2\\,e^{-\\frac{8\\pi^2}{b_2}(\\Phi +(\\beta_s^{(1)} -\\beta_s^{(2)}) T_s \n+(\\beta_s^{\\prime (1)} -\\beta_s^{\\prime (2)}) T_s^\\prime)} +C_2\\,e^{-\\mu_b T_b}, \n\\label{eq:KW2}\n\\end{align} \nand we assume that first and second and fourth terms of the (r.h.s.) \nin Eq.~(\\ref{eq:KW2}) are much larger than the third term of the (r.h.s.) \nin Eq.~(\\ref{eq:KW2}) at least at the minimum, that is, \n$w_0$, $A_2\\,e^{-\\frac{8\\pi^2}{a_2}\\,\\Phi}$, $C_2\\,e^{-\\mu_b T_b}$ \n$\\gg B_2\\,e^{-\\frac{8\\pi^2}{b_2}(\\Phi +(\\beta_s^{(1)} -\\beta_s^{(2)}) T_s \n+(\\beta_s^{\\prime (1)} -\\beta_s^{\\prime (2)}) T_s^\\prime)}$. \nSuch hierarchies between two gaugino condensation terms are \nrealized by the differences between the rank of the two hidden \ngauge groups whose gauginos condensate. \nTherefore, at the moment, we ignore the term \n$B_2\\,e^{-\\frac{8\\pi^2}{b_2}(\\Phi +(\\beta_s^{(1)} -\\beta_s^{(2)}) T_s \n+(\\beta_s^{\\prime (1)} -\\beta_s^{\\prime (2)}) T_s^\\prime)}$ \nin the superpotential~(\\ref{eq:KW2}). \n\nThird, we stabilize the moduli $\\Phi$, $T_b$, ${\\rm Re}\\,T_s$ \nand ${\\rm Re}\\,T_s^\\prime$ by imposing the supersymmetric conditions,\n\\begin{align}\n&D_{\\Phi}W=0,\\nonumber\\\\\n&D_{T_b}W=0,\\nonumber\\\\\n&K_{T_s}=K_{T_s^\\prime}=0,\n\\label{eq:m2cd1}\n\\end{align} \nwhich leads to the vanishing $D$-terms induced from the K\\\"ahler \npotential~(\\ref{eq:KW2}). \nFrom the scalar potential in the framework of $4$D $N=1$ \nsupergravity given by Eq.~(\\ref{eq:scalarpo}), \nwe get the supersymmetric AdS minimum at the minimum \ngiven by Eq.~(\\ref{eq:m2cd1}), \n\\begin{align}\n\\langle V\\rangle =-3e^K|W|^2.\n\\end{align} \nIn the same way as the previous model $1$ in the Sec.~\\ref{subsec:naturalinf1}, \nhere we assume that the dynamical \nSUSY breaking sector uplift this AdS minimum. \nTheir K\\\"ahler potential and superpotential are given by\n\\begin{align}\n&\\Delta K=|X|^2 -\\frac{|X|^4}{\\Lambda^2},\\nonumber\\\\\n&\\Delta W=\\mu X,\n\\label{eq:m2KW1loop}\n\\end{align} \nwhere X is the gauge singlet chiral superfield under the non-abelian \ngauge groups $G_{\\rm vis}$ and anomalous $U(1)_s$ symmetry, \n$\\Lambda$ is \nthe dynamical SUSY breaking scale and we omit the moduli \ndependence of $X$, because they do not affect the following \nmoduli stabilization. \nThe Minkowski minimum is realized by choosing the parameter \n$\\mu$ as \n\\begin{align}\n\\langle V\\rangle +\\Delta V\\simeq e^K\\left( -3|W|^2 +K^{X\\bar{X}}|\\mu|^2\n\\right)=0,\\,\\,\\,\n\\Leftrightarrow |\\mu|^2= 3|\\langle W\\rangle|^2.\n\\end{align} \n\nFinally, we consider the term \n$B_2\\,e^{-\\frac{8\\pi^2}{b_2}(\\Phi +(\\beta_s^{(1)} -\\beta_s^{(2)}) T_s \n+(\\beta_s^{\\prime (1)} -\\beta_s^{\\prime (2)}) T_s^\\prime)}$ in the \nsuperpotential~(\\ref{eq:KW2}). Since we assume that \nsuch term is much smaller than the other terms \nin the superpotential~(\\ref{eq:KW2}), the moduli \n$\\Phi$, $T_b$, ${\\rm Re}T_s$, ${\\rm Re}T_s^\\prime$ \nare stabilized at the values close to the minimum given by \nEq.~(\\ref{eq:m2cd1}) and they become massive \ndue to the constant term of the superpotential \nfor $\\Phi$, ${\\rm Re}T_s$ and ${\\rm Re}T_s^\\prime$ \nand the world-sheet instanton effect for $T_b$. \nAs ${\\rm Re}T_s$ and ${\\rm Re}T_s^\\prime$, they also obtain \nthe D-term contributions from the K\\\"ahler \npotential~(\\ref{eq:KW2}). \n\n\\subsubsection{Inflaton potential}\nLet us discuss the inflaton potential. \nAfter integrating out these heavy moduli and \nsubstituting the field values \ngiven by Eq.~(\\ref{eq:m2cd1}), we get the effective scalar \npotential for the light modulus $Y_s$ which is the linear \ncombination of ${\\rm Im}\\,T_s$ and ${\\rm Im}\\,T_s^\\prime$,\n\\begin{align}\nV_{\\rm eff} \n\\simeq \\Lambda_s^4 (1-{\\rm cos}\\,(\\beta_s\\,\\hat{Y}_s)),\n\\end{align} \nin the limit of $B_2e^{-\\frac{8\\pi^2}{b_2}\\,\\langle{\\rm Re}\\Phi\\rangle} \n\\ll w_0, A_2e^{-\\frac{8\\pi^2}{a_2}\\,\\langle{\\rm Re}\\Phi\\rangle}, \nC_2e^{-\\mu_b \\langle T_b\\rangle}$, \nwhere \n\\begin{align}\n\\Lambda_s^4\\equiv 6e^{K}e^{-\\frac{8\\pi^2}{b_2}\\,{\\rm Re}\\Phi}B_2\n(w_0+A_2\\,e^{-\\frac{8\\pi^2}{a_2}\\,\\Phi}+C_2\\,e^{-\\mu_b T_b}),\n\\end{align}\nand the axion decay constant $\\beta_s$ is defined \nby \n\\begin{align}\n\\beta_s \\equiv \\frac{8\\pi^2}{b_2N_s{\\hat N}_s}\\,\n\\left( -\\frac{\\beta_s^{(1)}-\\beta_s^{(2)}}\n{q_{s}^\\prime \\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}}}+\n\\frac{\\beta_s^{\\prime (1)}-\\beta_s^{\\prime (2)}}\n{q_s\\sqrt{K_{T_s\\bar{T}_s}}}\\right).\n\\end{align} \n${\\hat Y}_s$ is the canonically normalized axion field, \n\\begin{align}\n{\\hat Y}_s\\simeq \n\\frac{1}{N_s}\\sqrt{2\\left (\\frac{K_{T_s\\bar{T}_s}}{(q_{s}^\\prime)^2 \nK_{T_s^\\prime\\bar{T}_s^\\prime}} \n+\\frac{K_{T_s^\\prime\\bar{T}_s^\\prime}}{(q_{s})^2 \nK_{T_s\\bar{T}_s}}\\right)}\\,Y_s \n\\equiv \\hat{N}_s\\,Y_s.\n\\end{align} \nHere we employed the following redefinitions of the \nmoduli,\n\\begin{align}\n&{\\rm Im}\\,T_s=\n\\frac{1}{N_s}\\left( \\frac{X_s}{q_{s}\\sqrt{K_{T_s\\bar{T}_s}}}\n-\\frac{Y_s}{q_{s}^\\prime\\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}}}\n\\right),\n\\nonumber\\\\\n&{\\rm Im}\\,T_s^\\prime=\n\\frac{1}{N_s}\\left( \n\\frac{X_s}{q_{s}^\\prime\\sqrt{K_{T_s^\\prime\\bar{T}_s^\\prime}}} \n+\\frac{Y_s}{q_{s}\\sqrt{K_{T_s\\bar{T}_s}}}\\right),\n\\end{align} \nand $U(1)_s$ gauge invariance of the superpotential~(\\ref{eq:m2W}), \n\\begin{align}\n&q_s\\,\\beta_s^{(1)}+q_{s}^\\prime\\,\\beta_s^{^\\prime (1)} =0,\n\\nonumber\\\\\n&q_s\\,\\beta_s^{(2)}+q_{s}^\\prime\\,\\beta_s^{^\\prime (2)} =0.\n\\end{align} \n\nThus when we consider the axion ${\\hat Y}_s$ as the inflaton, \nthe effective scalar potential is the type of natural inflation. \nThe power spectrum of the scalar density perturbation is \nexplained by choosing the parameter,\n\\begin{align} \n\\Lambda_s^4 \\sim {\\cal O}(10^{-9})\n\\end{align}  \nin the $M_{\\rm Pl}$ unit, and \nthe spectral index of the scalar density perturbation and \nthe tensor-to-scalar ratio are also consistent with the \ncosmological observations reported by WMAP, Planck and \nBICEP2 collaborations. This is because we can realize the \ntrans-Planckian axion decay constant $\\beta_s$ originating \nfrom the one-loop corrections to the gauge kinetic function. \n\nHowever, $E_8\\times E_8$ or $SO(32)$ heterotic string \ntheories have the rank $16$ gauge groups which have to \nincorporate the rank $4$ SM gauge groups. Then the energy \nscales which two gaugino condense are constrained since \nthe total rank of their gauge groups are taken up to $12$ included \nin $E_8\\times E_8$ or $SO(32)$. Thus we would need tuning \nsome  parameters to realize the correct inflation scale. \n\n\n\\section{Conclusion}\n\\label{sec:con}\nIn this paper, we proposed two natural inflation scenarios \nbased on the weakly coupled $E_8\\times E_8$ or $SO(32)$ \nheterotic string theory on the ``Swiss-cheese\" Calabi-Yau \nmanifold with multiple $U(1)$ magnetic fluxes. \nThe natural inflation is consistent with the WMAP, Planck \nand\/or BICEP2 data, only if the size of axion decay constant \nbecomes the trans-Planckian. \nHowever, such trans-Planckian axion decay \nconstant is problematic from the theoretical point of view, \nespecially on the supergravity models or the string theory. \nSo far, there are known scenarios to get the \ntrans-Planckian axion decay constant from the \nsub-Planckian axion decay constants~\\cite{Kim:2004rp}. \n\nWe identified the inflaton as one of the linear combination of \nK\\\"ahler axions associated with the two-cycles of the \nCalabi-Yau manifold. When the gauginos of the hidden \ngauge group are condensed, the gaugino condensation terms \nare generated on the superpotential in the framework \nof $4$D ${\\cal N}=1$ supergravity. \nIn this case, we can realize the trans-Planckian axion \ndecay constant originating from the one-loop \ncorrections to the gauge kinetic function of the hidden \ngauge group derived from one-loop Green-Schwarz \nterm~\\cite{Ibanez:1986xy} which is the feature of \nthe weakly coupled heterotic string theory. \nOn the other hand, in type II superstring theory \nsuch as the intersecting D-models or magnetized \nD-branes, the gauge kinetic function has the \n${\\cal O}(1)$ moduli mixing induced from the winding \nnumber of D-brane, magnetic fluxes or \ninstanton effects.\n\nTo realize the single-field inflaton potential, we have to \nstabilize the dilaton and the other K\\\"ahler moduli. \nAt the same time, their masses should be heavier than \nthe inflation scale, \notherwise these moduli would be oscillated during and \nafter the inflation which may lead to the sizable \nisocurvature perturbations and cosmological moduli \nproblem. Therefore, we \nconsidered two stabilization scenarios \ncategorized as the model $1$ and $2$ based on the \n$E_8\\times E_8$ or $SO(32)$ \nheterotic string theory with multiple \n$U(1)$ magnetic fluxes.\n\nIn the case of model $1$ discussed in the \nSec.~\\ref{subsec:naturalinf1}, \nthe dilaton is stabilized at the finite value by the \ncontributions from its non-perturbative \ncorrections to the K\\\"ahler potential. The volume \nmoduli is also stabilized by the world-sheet instanton \neffect which leads to the stabilization of the other \nreal parts of K\\\"ahler moduli by using the nature of \n``Swiss-cheese\" Calabi-Yau manifold. \nBy employing the multiple $U(1)$ magnetic fluxes, \nthe imaginary parts of the moduli except for the inflaton \nare absorbed by the corresponding anomalous $U(1)$ \ngauge bosons and then they become massive which \nis of order the string scale. \nThus we can realize the single-field axion potential \nwith trans-Planckian axion decay constant \ndetermined by the one-loop corrections to the \ngauge kinetic function of the hidden gauge group. \n\nThe essential difference between the model $1$ and $2$ \nis the stabilization mechanism of dilaton. In model $2$ \ndiscussed in the Sec.~\\ref{subsec:naturalinf2}, \nthe dilaton is stabilized by one of the gaugino condensation \nterms and we get the effective scalar potential for a linear \ncombination of the K\\\"ahler axions. \nThese two proposed inflation scenarios are consistent \nwith the WMAP, Planck and\/or BICEP2 data, although \nwe need to tune the parameters in the model $2$. \n\nWe can also realize smaller tensor-to-scalar ratio \nwhich is more consistent with the WMAP and Planck data, since \nthe size of axion decay constant depends on the dual \nCoxeter number and the size of one-loop \ncorrection to the gauge kinetic function of the hidden gauge \ngroup in the heterotic string theory. \n\n\n\\subsection*{Acknowledgement}\nH.A. was supported in\npart by the Grant-in-Aid for Scientific Research No. 25800158 from the\nMinistry of Education,\nCulture, Sports, Science and Technology (MEXT) in Japan. T.K. was\nsupported in part by\nthe Grant-in-Aid for Scientific Research No. 25400252 from the MEXT in\nJapan.\nH.~O. was supported in part by a Grant-in-Aid for JSPS Fellows \nNo. 26-7296 and a Grant for \nExcellent Graduate Schools from the MEXT in Japan. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nVideo action recognition requires strong temporal reasoning.\nTemporal reasoning in deep networks can be implemented by 3D (space+time) convolutions, temporal (average) pooling, or recurrent layers that aggregate frame-level or spatio-temporal representations of an input sequence. The effectiveness of temporal reasoning hereby depends on the representation upon which the aggregation is performed.\n\nWhile spatio-temporal features are more effective than frame-level features in video recognition tasks, they are also more costly to compute and harder to train, including a large number of parameters. Furthermore, even with a deep backbone the receptive field of spatio-temporal features are confined to the small fixed-size frame window from which they were computed. Since more abstract representations benefit from larger temporal context, a gradual increase of the temporal extend of receptive fields in the higher layers of the feature extraction backbone would be desired instead.\n\nA feasible solution to increase the temporal size of receptive fields is to let features from adjacent branches interact. \\ac{tsn}~\\cite{tsn} showed that it is useful to apply the difference of frames as input to a CNN for video action recognition. \\ac{tdn}~\\cite{tdn} extended this idea to compute the difference of frame level features obtained from each layer in a CNN. Such hardcoded approach considers that there is a significant variation of information in adjacent frames. This is however not always guaranteed in practice. It is therefore desirable to let a network decide when to subtract the features from adjacent frames.\n\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.35\\textwidth}\n\t\t\\includegraphics[scale=0.3]{.\/figures\/hfnet}\n\t\t\\caption{HF-Net}\n\t\t\\label{fig:hfnet} \n\t\\end{subfigure} \\hspace{3cm}\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[scale=0.42]{.\/figures\/hf}\n\t\t\\caption{HF module}\n\t\t\\label{fig:hfmod}\n\t\\end{subfigure}\n\t\\caption{HF-Net: We introduce hierarchical feature aggregation in video deep architectures with CNN backbone. At each layer the feature from a CNN block is gated and its residual is transferred to the adjacent branch. Gating is learnt jointly with the network during training, and decides locally whether, and to what extend, a feature differencing or feature averaging is performed. The temporal extend of receptive fields increases with network depth, which helps developing spatio-temporal representations without\n\texpensive 3D convolutions\n}\n\t\\label{fig:basic_block}\n\\end{figure}\n\nIn this paper we present a hierarchical feature aggregation scheme that is lightweight and can be plugged into any deep architecture with CNN backbone. In particular, when plugged into \\ac{tsn}, a performance gain of 24.2\\% is obtained on Something-v1~\\cite{goyal2017something} dataset with an addition of only 1.14\\% parameters and 1.04\\% Floating Point Operations (FLOPs). The main idea, as illustrated in Fig.~\\ref{fig:hfnet}, is to transfer features between adjacent branches at each layer in a way that adjacent features interact as they develop into the higher level representation. The amount of feature transfer is controlled through a convolutional gate that decides to what amount a feature is subtracted or added into the adjacent branch, as detailed in Fig.~\\ref{fig:hfmod}. Through end-to-end training, the network learns to route features through the hierarchy, hereby causing the temporal extend of receptive fields to grow with depth and adapt to the input. We evaluate our scheme on a number of existing models, TSN, TRN and ECO, and show its flexibility and effectiveness in improving action recognition performance.\n\nThe rest of the paper is organized as follows. Sec.~\\ref{sec:related_works} reviews work on action recognition most related to our proposal. Sec.~\\ref{sec:hfnet_main} presents the hierarchical feature aggregation network. Experimental results are reported in Sec.~\\ref{sec:results} and Sec.~\\ref{sec:conclusions} concludes the paper.\n\n\n\n\n\n\n\n\n\\section{Related work}\n\n\\label{sec:related_works}\n\nInspired by the performance improvements obtained by CNNs on image recognition~\\cite{resnet, szegedy2016rethinking}, many deep learning based techniques have been developed for video action recognition. The most effective and simple extension was developed by Simonyan and Zisserman~\\cite{twoStream}. Their method consists of two different CNNs trained on a single RGB image frame and a stack of optical flow images followed by a late fusion of the prediction scores. The image stream encodes the appearance information while the optical flow stream encodes the motion information. Several works followed this approach to find a suitable fusion of the two streams~\\cite{feichtenhofer2016convolutional} and exploring residual connections between them~\\cite{feichtenhofer2016spatiotemporal}. The downside of this approach is its reliance on externally computed optical flow which is computationally intensive\n\nIn order to address the aforementioned problem, researchers have explored techniques for extracting spatio-temporal features from the RGB frames itself. Karpathy~\\emph{et al}.~\\cite{karpathy2014large} examined several fusion approaches at various levels of the CNN hierarchy. They found that a slow fusion approach where the features from adjacent frames are fused at multiple hierarchical levels of the CNN results in the best performance. Their fusion approach stacks the convolutional features from adjacent frames and perform temporal convolutions for extracting the temporal features. Later, Tran~\\emph{et al}.~\\cite{tran2015learning} developed 3DCNN with 3D convolutional layers so that spatio-temporal features can be extracted from a set of multiple RGB frames. Their approach showed that 3D convolutions are capable of extracting spatio-temporal features from small video segments consisting of a small number of frames. Several approaches have been later proposed for exploiting the parameters learned by a 2DCNN for image recognition~\\cite{carreira2017quo, qiu2017learning, tran2018closer}. Carreira and Zisserman~\\cite{carreira2017quo} developed a 3DCNN by inflating the 2D filters to 3D. Qiu~\\emph{et al}.~\\cite{qiu2017learning} and Tran~\\emph{et al}.~\\cite{tran2018closer} proposed to factorize the 3D convolutions to a 2D convolution for spatial encoding followed by a 1D convolution for temporal encoding. Wang~\\emph{et al}. developed an architecture based on 3DCNNs which decouples the spatial and temporal encoding. Their approach performs a linear encoding on individual frame features and a multiplicative encoding between a set of features from multiple frames. Such approaches using 3DCNNs have two major drawbacks, firstly, the massive increase in the number of trainable parameters and secondly, they extract spatio-temporal features from just a small sample of adjacent frames. The first problem makes such approaches difficult to be trained on smaller datasets and the second problem makes them incapable of extracting long range spatio-temporal features from videos.\n\nIn order to extract long-range spatio-temporal features, several techniques that perform sparse sampling of the video frames, as opposed to the dense sampling used in 3DCNNs, have been proposed. Such approaches use a 2DCNN for extracting the frame level features followed by a late fusion using \\acp{rnn}~\\cite{yue2015beyond, lrcn}, pooling techniques such as average pooling or max pooling~\\cite{tsn}, \\ac{fc} layers~\\cite{trn}, or 3D convolutions~\\cite{eco}, among others. Such late fusion based approaches ignore the information extracted at the different layers of a CNN. Several approaches have been proposed to address this drawback by fusing the features from consecutive frames at different layers of hierarchy of a CNN~\\cite{tdn, off, lee2018motion}. Ng and Davis~\\cite{tdn} propose to use an extra CNN that accepts the difference of feature maps from adjacent frames at different layers of a separate CNN. The final prediction is done by average pooling of the scores obtained from the two networks. Sun~\\emph{et al}. improves this approach by applying a Sobel filter on the features in addition to the temporal differencing operation. Lee~\\emph{et al}.~\\cite{lee2018motion} improves this method by forwarding the spatial and temporal features across a single network. This is achieved by applying a set of fixed filters followed by differencing of the features from adjacent frames.\nLin~\\emph{et al}.~\\cite{tsm} proposes a plug and play module that shifts the channels in the feature tensor across the temporal dimension for transferring information across frames.\n\nOur approach is related to~\\cite{tdn}, \\cite{off}, \\cite{lee2018motion} and \\cite{tsm}. These approaches perform a fixed interaction between feature maps from adjacent frames whereas our approach learns a set of filters that can perform point wise difference or average operations across the feature maps.\n\n\n\\section{HF-Net: Hierarchical Feature Aggregation Networks}\n\\label{sec:hfnet_main}\nIn this section we describe our hierarchical feature aggregation for video representation learning architectures that utilize a CNN backbone for frame-level or spatio-temporal feature extraction. We then present details of our action recognition model used in the experiments.\n\n\\subsection{Hierarchical Feature Aggregation}\n\\label{sec:hfmodule}\n\nDeep architectures with CNN backbone by design do not account for correlations that\nmay exist between frame-level or spatio-temporal features at the earlier layers of the CNN.\nIndeed, in a CNN backbone a feature $F_{t,l}$ for input $t$ at layer $l$ is computed from $F_{t,l-1}$ independently of the features from inputs other than $t$, that is, $F_{t,l} = \\verb+block+(F_{t,l-1})$, where \\verb+block+ represents a set of convolutional layers with non-linearities. This may limit performance by design when the input sequence exhibits strong temporal dependence such as video frames in action recognition. In such a setting, a feature aggregation in the early layers may better capture the temporal dependencies in the data sequence.\n\nOur hierarchical feature aggregation is based on the assumption that adjacent features will benefit from interacting in order to produce a more abstract representation at each layer of the architecture. We want this interaction to be pairwise and feed-forward computable, that is, our backbone hierarchical aggregation scheme is realized with a re-designed $F_{t,l} = \\verb+block2+(F_{t,l-1},F_{t-1,l-1})$.\n\nIn order to define \\verb+block2+ we consider two elements that are pervasive in the state-of-the-art architectures. First, most architectures perform feature pooling at the higher level, most often, average pooling. Second, many networks follow a two stream structure where optical flow is late fused from a separate branch. Some works~\\cite{tsn, tdn} already reported performance improvements even with frame differencing as a simplified version of optical flow. In our design of \\verb+block2+ both feature differencing and feature averaging are considered. We want the network to select or interpolate between these two modes, and maximise flexibility by learning to decide so locally in the feature tensor. This way, we provide the network with the capability to selectively route features through the hierarchy and build up discriminative representations integrating temporal context. That is, growing temporal receptive fields as we go up in the hierarchy. \n\nWe define \\verb+block2+ as:\n\n\\begin{equation}\nF_{t,l} = F'_{t,l} + G_{t,l} \\odot F'_{t+1, l} - G_{t-1,l} \\odot F'_{t, l}, \\label{eq:new_feat_t+1}\n\\end{equation}\nwhere $F'_{t,l} = \\verb+block+(F_{t,l-1})$, `$*$' and `$\\odot$' represent convolution operation and Hadamard product, respectively, and $G_t$ acts as a gating tensor that chooses between averaging and differencing operations. Fig.~\\ref{fig:hfmod} illustrates \\verb+block2+. Note that we subtract a gated $G_{t-1,l} \\odot F'_{t,l}$ from the backbone feature $F'_{t,l}$ but we add it when computing $F_{t-1,l}$. Thus, the total feature flow on the sequence is preserved. Gating $G_t$ is generated by performing a 3D convolution on the stacked features with a $2\\times3\\times3$ kernel followed by $\\tanh$ non-linearity:\n\n\\begin{equation}\nG_{t,l} = \\tanh(W_l * [F'_{t,l} , F'_{t+1, l}] + B_{l}). \\label{eq:gate}\n\\end{equation}\nSince the range of $\\tanh$ non-linearity is [-1, 1], the network is capable of selecting from both averaging and differencing operations.\n$W_l$ and $B_l$ represent the weights and the bias of the 3D convolution. We use a $2\\times3\\times3$ kernel with $1$ output channel in the 3D convolution. \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Layer Implementation}\n\nThe proposed hierarchical aggregation across each layer can be realized in parallel, thereby allowing for a fast training and inference. Assuming that sequences of frames are applied as a batch to the network, the parallel implementation of hierarchical aggregation is realized as:\n\n\\begin{eqnarray}\nF^{\\verb:+:} &=& \\verb+block+(F_{:,l-1}),\\\\\nF^{\\verb:-:} &=& \\verb+shift_left+(F^{\\verb:+:}), \\\\\nG^{\\verb:-:} &=& \\tanh(\\verb+conv3+([F^{\\verb:+:},F^{\\verb:-:}])), \\\\\nG^{\\verb:+:} &=& \\verb+shift_right+(G^{\\verb:-:}), \\\\\nF_{:,l} &=& G^{\\verb:-:} \\odot F^{\\verb:-:} + (1- G^{\\verb:+:}) \\odot F^{\\verb:+:}.\n\\end{eqnarray}\n\n\n\nAt layer $l$ of the backbone, we first feed the input sequence $F_{:,l-1}$ through the corresponding convolutional \\verb+block+ of the backbone to obtain a new sequence $F^{\\verb:+:}$. We then left-shift and zero-pad to obtain $F^{\\verb:-:}$ and compute the gating tensor $G^{\\verb:-:}$ through a 3D convolution on the stacked $[F^{\\verb:+:},F^{\\verb:-:}]$. This is followed by $\\tanh$ to map the resulting tensor to values in the range [-1, 1]. In order to obtain the sequence $F_{:,l}$ of output features, the Hadamard product of the gating tensor $G^{\\verb:-:}$ and the left shifted feature tensor $F^{\\verb:-:}$ is added to the input feature tensor $F^{\\verb:+:}$ while the Hadamard product of the right shifted gating tensor $G^{\\verb:+:}$ and the input feature tensor $F^{\\verb:+:}$ is subtracted.\n\n\n\\subsection{Hierarchical Feature Aggregation Networks}\n\\label{sec:hfanet}\nHierarchical Feature Aggregation Network (HF-Net) is obtained by adding the Hierarchical Feature Aggregation Module, presented in Sec.~\\ref{sec:hfmodule}, at various levels of the backbone CNN used in the network. \nOur approach is generalizable to any CNN architecture (e.g, HF-TSN). In the experiments, we use Inception with \\ac{bn}~\\cite{inception_bn} as our backbone CNN. We plug in our Hierarchical Feature Aggregation module after each Inception block in the CNN. At each iteration, a set of sparsely sampled frames $K$ from a video is passed to the network. In addition to the common feed forward flow of information in conventional CNNs, the hierarchical feature aggregation modules enable a horizontal flow of information across the features at different levels of the CNN hierarchy. The output features corresponding to all the frames from the final layer of the CNN can then be pooled together for encoding the long range temporal features. Our architecture is highly flexible and can be combined with a number of temporal pooling techniques such as \\ac{tsn}~\\cite{tsn}, \\ac{trn}~\\cite{trn}, \\ac{gru}~\\cite{dwibedi2018temporal}, or 3DCNN~\\cite{eco}. In Sec.~\\ref{sec:ablation}, we evaluate the performance of our proposal on a number of existing approaches and show its flexibility and effectiveness in improving action recognition performance. \n\n\n\\section{Experiments and Results}\n\\label{sec:results}\nWe evaluate the proposed hierarchical feature aggregation strategy on a number of existing models and standard action recognition datasets. We first briefly describe the considered datasets, evaluation metrics, and implementation details. Then, we perform an ablation analysis to quantitatively evaluate the impact of different HF properties and components. Finally, we compare the performance of various HF-Net architectures against state-of-the-art results on different public action recognition datasets. We show HF enhances recognition performance of all models where it is applied, with negligible additional complexity.\n\n\\subsection{Datasets and evaluation protocol}\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/something_lifting_something}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/something_turn_something_upside_down}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/something_throwing}\n\t\t\\caption{Something-v1}\n\t\t\\label{fig:something}\n\t\t\\end{subfigure}\\hskip3mm\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n    \t\\includegraphics[width=35px,height=25px]{.\/figures\/epic_stir_vegetable}\n    \t\\includegraphics[width=35px,height=25px]{.\/figures\/epic_putdown_knife}\n    \t\\includegraphics[width=35px,height=25px]{.\/figures\/epic_close_tap}\n    \t\\caption{EPIC-KITCHENS}\n    \t\\label{fig:epic}\n    \t\\end{subfigure}\\hskip3mm\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/hmdb_handstand}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/hmdb_golf}\n\t\t\\includegraphics[width=35px,height=25px]{.\/figures\/hmdb_shootbow}\n\t\t\\caption{HMDB51}\n\t\t\\label{fig:hmdb}\n\t\t\\end{subfigure}\n\t\t\\caption{Samples from the datasets used for evaluating the performance of HF-Nets.\n\t\\label{fig:dataset_samples}\n\\end{figure}\n\nWe evaluate our proposed hierarchical feature aggregation technique on three standard action recognition benchmarks, Something-v1~\\cite{goyal2017something}, EPIC-KITCHENS~\\cite{damen2018scaling} and HMDB51~\\cite{kuehne2013hmdb51}.  Something-v1 consists of 86K and 11K videos in the training and validation sets from 174 action classes. We report the performance on the validation set.\nEPIC-KITCHENS dataset comprises of egocentric videos with fine-grained activity labels. The labels are provided as verb and nouns and the dataset consists of 24K videos in the training set and 10K videos in the test set. We report the performance obtained on the test set.\nHMDB51 dataset consists of videos collected from Youtube and contains around 6000 video clips from 51 action categories. The dataset is provided with three standard train\/test splits and the final recognition accuracy is reported as the average of the accuracy obtained on the three splits.\nBoth Something-v1 and EPIC-KITCHENS datasets consists of crowd collected videos with actions involving objects. Something-v1 gives importance to the actions rather than the objects involved in the action while EPIC-KITCHENS gives relevance to both actions and objects. HMDB51 is a smaller dataset with less complex action categories of shorter temporal span, which can be identified with simple appearance cues. Some sample frames from the datasets are shown in Fig.~\\ref{fig:dataset_samples}. In addition to the recognition accuracy, we also compare the complexity of the models in terms of number of parameters and Floating Point Operations (FLOPs).\n\n\n\n\n\\subsection{Implementation Details}\n\nAs explained in Sec.~\\ref{sec:hfanet}, we choose BNInception as the backbone. The proposed HF module is added after each Inception block of the CNN. The entire network, including the \\ac{bn} layers, is trained for 60 epochs using \\ac{sgd} optimization algorithm with a batch size of 32. The learning rate is fixed as 0.001 and is reduced by a factor of 0.1 after 25 and 40 epochs. Dropout at a rate of 0.5 is applied before the final classification layer to avoid overfitting. Random scaling and cropping, as proposed in~\\cite{tsn} is used as data augmentation. During inference, only the center crop of the frames are used. In all experiments, we use 16 frames as the input to the model.\n\n\n\\begin{table}[t]\n\t\\centering \n\t\\begin{tabular}{|l|c|c|c|}\n\t\t\\hline\n\t\t\\textbf{Model} & \\textbf{Accuracy (\\%)} & \\textbf{Params} & \\textbf{FLOPs}\\\\   \\hline\n\t\tBaseline & 17.77 & 10.47M & 32.73G\\\\ \\hline\n\t\tBaseline+1HF & 19.37 & 10.47M & 32.81G \\\\ \\hline\n\t\tBaseline+5HF & 34.15 & 10.53M & 32.90G \\\\ \\hline\n\t\tBaseline+10HF & 41.97 & 10.59M & 33.07G \\\\ \\hline\n\t\tBaseline+10HF (non-Conservative) & 38.81 & 10.69M & 33.47G\\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Ablation analysis conducted on the validation set of Something-v1 dataset.\n    }\n\t\\label{tab:ablation}\n\\end{table}\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\begin{subfigure}[b]{1\\textwidth}\n\t\t\t\\hspace{2cm}\n\t\t\\includegraphics[angle=90, scale=0.35]{.\/figures\/graph_TSN_HF_TSN.pdf}\n\t\t\\caption{Improved classes over \\ac{tsn}}\n\t\t\\label{fig:classes_imp}\n\t\\end{subfigure} \\\\ \\vspace*{.5cm}\n\t\\begin{subfigure}[b]{0.35\\textwidth}\n\t\t\\includegraphics[scale=0.4]{.\/figures\/tsn_feats2_feats.png}\n\t\t\\caption{\\ac{tsn} t-SNE plot}\n\t\t\\label{fig:tsn_tsne}\n\t\\end{subfigure} \\hspace{3cm}\n\t\\begin{subfigure}[b]{0.35\\textwidth}\n\t\t\\includegraphics[scale=0.4]{.\/figures\/hftsn_feats2_feats.png}\n\t\t\\caption{HF-\\ac{tsn} t-SNE plot}\n\t\t\\label{fig:hftsn_tsne}\n\t\\end{subfigure}\n\t\t\\caption{(a) Action classes in Something-v1 dataset with the highest improvement over \\ac{tsn} by adding HF on the backbone. Y-axis labels are in the format true label (HF-TSN)\/predicted label (TSN). X-axis shows the number of corrected samples for each class. (b) and (c) show the t-SNE plots of the output layer features preceding the \\ac{fc} layer. The 10 improved classes shown in (a) are visualized in the t-SNE plots.\n\t\\label{fig:tsne_plots}\n\n\\end{figure}\n\n\n\\subsection{Ablation Analysis}\n\\label{sec:ablation}\n\n\nIn this section, we report the ablation analysis performed on the validation set of Something-v1 dataset. We compare the performance improvement by adding the proposed HF module on the CNN backbone of a standard action recognition technique. We choose \\ac{tsn}~\\cite{tsn} as the baseline since it is one of the standard techniques for video action recognition. \\ac{tsn} divides each video into a pre-defined number of segments and applies one frame from each of the segments as the input to the network. The average of the output from each of the frames is used for computing the prediction scores. Thus, \\ac{tsn} fails to encode the temporal relations between video frames and hence acts as a suitable baseline for showing the capability of the proposed hierarchical feature aggregation approach in extracting spatio-temporal features.\n\n\nTab.~\\ref{tab:ablation} shows the result of the ablation study conducted. We first evaluated the performance of the model when a single module is added after the final inception block of the backbone. We obtained an improvement of $1.6\\%$. A further gain of $14.78\\%$ is obtained by adding 5 modules after each of the final 5 inception blocks. Finally, we add 10 modules after each of the inception blocks which resulted in an improvement of $24.2\\%$ over the \\ac{tsn} baseline. We also evaluated the performance of the model when two independent 3D convolutional layers are used to compute the gating, thereby breaking the conservative flow of features. This increases the number of parameters and complexity slightly, albeit the performance of the network is reduced thereby proving that feature flow conservation is useful for spatio-temporal feature extraction in HF-Nets.\n\n\nIn Fig.~\\ref{fig:classes_imp}, we show the top 10 action classes that improved the most by adding HF module to the backbone CNN of \\ac{tsn}. From the figure, it can be seen that the network has enhanced its ability to distinguish between action classes that are similar in appearance, such as \\verb+Unfolding something+ and \\verb+Folding something+, \\verb+Dropping something next+ \\verb+to something+ and \\verb+Showing something next+ \\verb+to something+, etc.\nThe t-SNE plots of the features from the final layer of the CNN corresponding to these 10 action classes are shown in Fig.~\\ref{fig:tsn_tsne} and~\\ref{fig:hftsn_tsne}. It can be seen that the features from HF-\\ac{tsn} show a lower intra-class and higher inter-class variability compared to those from \\ac{tsn}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}[h]\n\t\\centering\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\t\\hline\n\t\\multirow{2}{*}{\\textbf{Model}} & \\multirow{2}{*}{\\textbf{Backbone}} & \\multirow{2}{*}{\\textbf{Pre-training}} & \\multicolumn{2}{c|}{\\textbf{Accuracy (\\%)}} \\\\ \\cline{4-5}\n\t& & & Something-v1 & HMDB51\\\\ \\hline\n\t    TDN~\\cite{tdn} & ResNet-50 & ImageNet & - & 55.5 \\\\ \\hline\n\t\tECO~\\cite{eco} & BNInception+3D ResNet-18 & ImageNet+Kinetics & 41.4 & 68.2\\\\ \\hline\n\t\tMFNet-C~\\cite{lee2018motion} & ResNet-101 & Scratch & 43.92 & -\\\\ \\hline\n\t\tARTNet~\\cite{wang2018appearance} & 3D ResNet-18 & Kinetics & - & 70.9 \\\\ \\hline \\hline\n\t\tI3D~\\cite{carreira2017quo} & 3D ResNet-50 & ImageNet+Kinetics & 41.6 & 74.8\\\\ \\hline\n\t\tC3D~\\cite{tran2015learning} & 3D ResNet-18 & Sports-1M & - & 62.1 \\\\ \\hline\n\t\t3D-ResNeXt-101~\\cite{hara2018can} & ResNeXt-101 & Kinetics & - & 70.2 \\\\ \\hline\n\t\tR(2+1)D~\\cite{tran2018closer} & 3D ResNet-34 & ImageNet+Kinetics & - & 74.5 \\\\ \\hline\n\t\tNon-local I3D~\\cite{wang2018videos} & 3D ResNet-50 & ImageNet+Kinetics & 44.4 & -\\\\ \\hline\n\t\tNon-local I3D+GCN~\\cite{wang2018videos} & 3D ResNet-50+GCN & ImageNet+Kinetics & 46.1 & -\\\\ \\hline\n\t\tTSM~\\cite{tsm} & ResNet-50 & ImageNet+Kinetics & 44.8 & 73.2\\\\ \\hline\n\t\tS3D-G~\\cite{xie2018rethinking} & Inception-V1 & ImageNet & 48.2 & -\\\\ \\hline \\hline\n\t\tTSN~\\cite{tsn} & BNInception & ImageNet & 17.77 & 53.7 \\\\ \\hline\n\t\tTRN~\\cite{trn} & BNInception & ImageNet & 31.32 & 54.93 \\\\ \\hline \n\t\tECOLite~\\cite{eco} & BNInception+3D ResNet-18 & ImageNet+Kinetics & 42.2 & 68.5\\\\ \\hline \\hline\n\t\tHF-TSN & BNInception & ImageNet & 41.97 & 55.92 \\\\ \\hline\n\t\tHF-TRN & BNInception & ImageNet & 39.17 & 57.61 \\\\ \\hline\n\t\tHF-ECOLite & BNInception+3D ResNet-18 & ImageNet+Kinetics & - & 71.13\\\\ \\hline\n\t\\end{tabular\n\t\\caption{Comparison of proposed method with state-of-the-art techniques on the validation set of something-v1 and HMDB51 datasets.}\n\t\\label{tab:something_sota}\n\\end{table}\n\n\n\\begin{figure}[t\n    \\centering\n    \\begin{subfigure}[b]{.45\\textwidth}\\hspace{-6.5mm}\n    \\includegraphics[scale=0.08]{.\/figures\/complexity_something.png}\n    \\caption{Something-v1 dataset}\n    \\label{fig:complexity_something}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{.45\\textwidth}\\hspace{1.5mm}\n    \\includegraphics[scale=0.08]{.\/figures\/complexity_hmdb.png}\n    \\caption{HMDB51 dataset}\n    \\label{fig:complexity_hmdb}\n    \\end{subfigure}\n    \\caption{Recognition accuracy (\\%) of various state-of-the-art techniques on (a) Something-v1 and (b) HMDB51, plotted against the computational complexity in terms of GFLOPs. Color of the data point indicates the number of parameters (M, in millions).}\n    \\label{fig:complexity}\n\\end{figure}\n\n\n\n\\subsection{State-of-the-Art Comparison}\n\nIn order to compare methods at the same conditions, we compare only with those models using raw RGB frames as input. However, note that the proposed approach is extendable to optical flow images as well.\n\n{\\noindent \\textbf{Something-v1 and HMDB51:}} Tab.\\ref{tab:something_sota} compares the proposed approach with state-of-the-art techniques on Something-v1 and HMDB51 datasets. For Something-v1 dataset, in addition to \\ac{tsn}, we also evaluate the performance of the proposed approach on \\ac{trn}. In both methods, by adding the proposed HF module to the backbone, a large improvement in the performance is observed. From the table, one can see that the proposed approach results in comparable performance to other approaches that use a bigger backbone CNN such as ResNet-50 or I3D. It should also be noted that other methods which achieve superior performance use strong pre-training using Kinetics dataset. Importantly, the HF augmented models achieve this improved performance at a fraction of the number of parameters and FLOPs compared to other methods, allowing for faster inference and smaller memory footprint, e.g. rendering them to be deployed in mobile devices. Fig.~\\ref{fig:complexity_something} illustrates the accuracy vs complexity of state-of-the-art techniques. From the figure, it can be seen that HF-Net results in a large boost in the recognition accuracy over the baseline models.\n\nFor HMDB51, we augment three baselines, \\ac{tsn}~\\cite{tsn}, \\ac{trn}~\\cite{trn} and ECOLite~\\cite{eco}, with HF module and observe a gain of more than $2\\%$ on recognition accuracy over all the three baselines. As mentioned previously, HMDB51 consists of actions with shorter temporal duration. As a result, 3D CNNs that perform dense sampling of frames resulted in the best performance on this dataset~\\cite{tran2018closer, carreira2017quo} due to the ability of 3D convolution layers in extracting spatio-temporal features from a short temporal window. ECOLite consists of a smaller number of 3D convolution layers on top of a 2D CNN backbone and thus is a middle ground between \\ac{tsn} and 3D CNNs.\nEven though ECOLite uses a more powerful (3D convolution) consensus layer than TSN (average pooling) and TRN (fully connected) the spatio-temporal features provided by HF aggregation are found to be beneficial.\nFrom the plot showing accuracy against complexity comparison of state-of-the-art approaches shown in Fig.~\\ref{fig:complexity_hmdb}, one can see that HF-ECOLite performs on par with the bigger 3D CNN models. \n\n\n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\begin{tabular}{|c|l|c c c|c c c| c c c|c c c|}\n\t\t\\hline\n\t\t& \\textbf{Method} & \\multicolumn{3}{c|}{\\textbf{Top-1 Accuracy (\\%)}} & \\multicolumn{3}{c|}{\\textbf{Top-5 Accuracy (\\%)}} & \\multicolumn{3}{c|}{\\textbf{Precision (\\%)}} & \\multicolumn{3}{c|}{\\textbf{Recall (\\%)}} \\\\\n\t\t\\cline{3-14}\n\t\t& & \\parbox{0.7cm}{Verb} & \\parbox{0.7cm}{Noun} & \\parbox{0.8cm}{Action} & \\parbox{0.7cm}{Verb} & \\parbox{0.7cm}{Noun} & \\parbox{0.8cm}{Action} & \\parbox{0.7cm}{Verb} & \\parbox{0.7cm}{Noun} & \\parbox{0.8cm}{Action} & \\parbox{0.7cm}{Verb} & \\parbox{0.7cm}{Noun} & \\parbox{0.8cm}{Action}\\\\\n\t\t\\hline \n\t\t\\multirow{5}{*}{\\rotatebox[origin=t]{90}{S1}} & 2SCNN (FUSION)~\\cite{twoStream} & 42.16 & 29.14 & 13.23 & 80.58 & 53.70 & 30.36 & 29.39 & 30.73 & 5.35 & 14.83 & 21.10 & 4.46\\\\ \\cline{2-14}\n\t\t& TSN (RGB)~\\cite{tsn} & 45.68 & 36.80 & 19.86 & 85.56 & 64.19 & 41.89 & \\textbf{61.64} & 34.32 & 9.96 & 23.81 & 31.62 & 8.81\\\\ \\cline{2-14}\n\t\t& TSN (FLOW)~\\cite{tsn} & 42.75 & 17.40 & 9.02 & 79.52 & 39.43 & 21.92 & 21.42 & 13.75 & 2.33 & 15.58 & 9.51 & 2.06\\\\ \\cline{2-14}\n\t\n\t\t& TSN (FUSION)~\\cite{tsn} & 48.23 & 36.71 & 20.54 & 84.09 & 62.32 & 39.79 & 47.26 & 35.42 & 10.46 & 22.33 & 30.53 & 8.83\\\\ \\cline{2-14}\n\t\t& HF-TSN (RGB) & \\textbf{57.57} & \\textbf{39.9} & \\textbf{28.09} & \\textbf{87.83} & \\textbf{65.37} & \\textbf{48.63} & 49.12 & \\textbf{35.83} & \\textbf{11.38} & \\textbf{39.37} & \\textbf{37.04} & \\textbf{13.84}\\\\\n\t\t\\hline \\hline\n\t\t\\multirow{5}{*}{\\rotatebox[origin=t]{90}{S2}} & 2SCNN (FUSION)~\\cite{twoStream} & 36.16 & 18.03 & 7.31 & 71.97 & 38.41 & 19.49 & 18.11 & 15.31 & 2.86 & 10.52 & 12.55 & 2.69\\\\ \\cline{2-14}\n\t\t& TSN (RGB)~\\cite{tsn} & 34.89 & 21.82 & 10.11 & 74.56 & 45.34 & 25.33 & 19.48 & 14.67 & 4.77 & 11.22 & 17.24 & 5.67\\\\ \\cline{2-14}\n\t\t& TSN (FLOW)~\\cite{tsn} & 40.08 & 14.51 & 6.73 & 73.40 & 33.77 & 18.64 & 19.98 & 9.48 & 2.08 & 13.81 & 8.58 & 2.27 \\\\ \\cline{2-14}\n\t\t& TSN (FUSION)~\\cite{tsn}& 39.4 & 22.70 & 10.89 & 74.29 & 45.72 & 25.26 & 22.54 & 15.33 & 5.60 & 13.06 & 17.52 & 5.81\\\\ \\cline{2-14}\n\t\t& HF-TSN (RGB) & \\textbf{42.40} & \\textbf{25.23} & \\textbf{16.93} & \\textbf{75.76} & \\textbf{48.96} & \\textbf{33.32} & \\textbf{24.25} & \\textbf{20.48} & \\textbf{6.29} & \\textbf{15.77} & \\textbf{21.96} & \\textbf{10.05}\\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Comparison of recognition accuracies with state-of-the-art on EPIC-KITCHENS dataset. HF-TSN is trained for verb, noun and action classification.}\n\t\\label{tab:epic_kitchens}\n\n\\end{table}\n\n{\\noindent\\textbf{EPIC-KITCHENS:}} HF-TSN model is compared against the baseline methods on Tab.~\\ref{tab:epic_kitchens}. Results on all other models are provided by the dataset developers~\\cite{damen2018scaling}. The model is trained for verb, noun and action classification. We also apply the action prediction score as a bias to the verb and noun classifiers. We report the scores on the test set obtained from the evaluation server. It can be seen that HF-TSN obtained a gain of $12\\%$ and $8\\%$ for verb classification on S1 and S2 settings over \\ac{tsn} (RGB) baseline. In fact, HF-TSN from RGB surpasses the performance of the baselines that use both RGB and optical flow, showing its capacity for extracting highly discriminative short and long term spatio-temporal features.\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe presented a hierarchical aggregation scheme for video understanding architectures with CNN backbone that is lightweight and effective. In HF-Nets, adjacent feature branches interact between feature differencing and averaging as they compile higher level representations, thereby providing cheap spatio-temporal features for competitive performance.\nWe plugged HF on top of different baseline models (TSN, TRN, ECO) and evaluated on three public action recognition datasets, obtaining consistent performance improvements. We improve action recognition accuracy of TSN on video clips of complex human-object relationships~\\cite{goyal2017something} by more than 24\\% while introducing only about 1\\% additional trainable parameters and 1\\% FLOPs of computation overhead in inference. \nSince HF-Net scheme can be plugged into any deep video architecture with CNN backbone, our future work includes the evaluation of additional baselines and two-stream solutions.\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCoronal holes (CHs) are regions of predominantly unipolar coronal magnetic \nfields with a significant component of the magnetic field open into the heliosphere. \nThey are visible in spectral lines emitting at coronal temperatures as dark areas when \ncompared to the quiet Sun, while in the chromospheric He~{\\sc i}~10830~\\AA\\ line they \nappear bright. For detailed introduction on coronal holes see \\citet{2009arXiv0906.2556M} (hereafter paper I). \nCHs are identified as the source of the fast solar wind \nwith velocities of up to $\\approx$~800~\\rm km~\\;s$^{-1}$\\  \\citep{1973SoPh...29..505K}.  In \ncontrast, the slow wind has velocities around 400~\\rm km~\\;s$^{-1}$\\ and  is more dense and \nvariable in nature when compared to the fast solar wind. \\citet{1996ASPC..109..491V} \nfound from Ulysses satellite data  that the elemental composition of the fast \nwind is similar to the elemental composition of the photosphere. The slow solar  \nwind is enriched with low first ionization potential (FIP) elements by a factor \nof 3--5 greater than in the photosphere (with respect to hydrogen) while higher \nFIP elements were found at solar surface abundances. The FIP effect describes \nthe element abundance anomalies  (the enhancement of elements with low FIP such \nas Fe, Mg and Si over those with high FIP like Ne and Ar) in the upper solar \natmosphere and solar wind, and can give a clue on the origin of both the fast \nand the slow solar winds. \\citet{1996ASPC..109..491V} concluded that the fast and \nslow solar winds not only differ in their kinetics but also in their composition \nof elements. \n\n\\citet{2004ApJ...612.1171W} suggested that the release of trapped plasma in closed \nloop structures by magnetic reconnection could play a significant role in the solar wind flow. Such reconnection between the open and  closed magnetic field \nlines presumably happens continuously at coronal hole boundaries. \\citet{1998ApJ...498L.165W} investigated the ejection of plasma blobs from the streamer belt linked to \nthe slow wind and concluded that magnetic reconnection between the distended \nstreamer loops and the open magnetic field lines might be behind the plasma \nejection. They also suggested that this ejection cannot account for all the \nslow solar wind and a major component should, therefore, originate outside the helmet \nstreamers, i.e. from inside the coronal holes. \\citet{2004ApJ...603L..57M} \nfound non-Gaussian profiles  along the boundaries in an equatorial extension \nof a polar  CH in  the mid- and high-transition region lines N~{\\sc iv}~765~\\AA\\ \nand Ne~{\\sc viii}~770~\\AA, respectively, recorded with the Solar Measurement of Emitted \nRadiation (SUMER) spectrometer on-board the Solar and Heliospheric Observatory (SoHO). \nThe authors suggested that these profiles are the signature of magnetic \nreconnection occurring between the closed magnetic field lines of the quiet Sun \nand the open of the coronal hole. Similar activity was reported by \n\\citet{2006A&A...446..327D} along the boundary of a polar CH.\n\n\\begin{figure*}[!ht]\n    \\centering\n    \\vspace{16cm}\n      \\special{psfile=fig1_11.eps  hoffset=-20 voffset=240 hscale=65 vscale=65 angle=0}\n      \\special{psfile=fig1_2.eps  hoffset=220 voffset=240 hscale=65 vscale=65 angle=0}  \n        \\special{psfile=fig1_31.eps  hoffset=-20 voffset=20 hscale=65 vscale=65 angle=0}\n          \\special{psfile=fig1_4.eps hoffset=220 voffset=20 hscale=65 vscale=65 angle=0}\n\\vspace*{-0.7cm}\n      \\caption{ Equatorial coronal hole (top left),  polar CH (top right), \n      quiet Sun (bottom left) and quiet Sun with TCHs (bottom right)  \n      with the positions of all the corresponding identified brightening \n      pixels over-plotted. The CH boundaries are outlined with a black line. The \n      over-drawn rectangles correspond to the field-of-views shown in \n      Figs.~\\ref{fig5}, \\ref{fig6} and \\ref{fig7}. }\n      \\label{fig1}\n\\end{figure*}\n\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Description of the  XRT  data used in the present study (ECH -- equatorial \ncoronal hole, PCH -- polar coronal hole, TCH -- transient coronal hole). }\n\\label{table1}\n\\begin{tabular} {ccccc}\n\\hline\nDate & Remark & Observing period & Field-of-view   & Exposure time \t\\\\\n     & \t      &\tof time (UT)\t            & (arcsec) &  (sec)\t\t\\\\\n\\hline\n09\/11\/07 & ECH & 06:35-14:59  & $366\\times366$\t & 16\t\\\\\n12\/11\/07 & ECH & 01:17-10:59  & $374\\times374$\t & 16\t\\\\\n14\/11\/07 & ECH & 00:12-11:11  & $374\\times374$\t & 16\t\\\\\t\n16\/11\/07 & ECH & 18:07-23:58  & $370\\times370$\t & 16\t\\\\\n16\/12\/08 & PCH\t & 10:02-17:52  & $362\\times337$\t & 23\t\\\\\n20\/09\/07 & PCH \t & 12:22-18:05  & $1018\\times291$ \t & 16\t\\\\\n10\/01\/09 & QS\t & 11:30-17:27  & $370\\times370$\t & 23\t\\\\\n13\/01\/09 & QS& 11:22-17:41  & $370\\times370$\t & 23\t\\\\\n29\/11\/07 & QS with TCH& 18:17-23:59  & $506\\times506$\t & 23\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\nIn paper I we demonstrated that although isolated  equatorial  CH and \nequatorial extension of polar CH maintain their general shape during several \nsolar rotations, a closer look at their day-by-day and even hour-by-hour \nevolution demonstrates significant dynamics. We showed that small-scale \nloops which are abundant along coronal hole boundaries contribute to the \nsmall-scale evolution of coronal holes. We suggested that these dynamics \nare triggered by continuous magnetic reconnection already proposed by\n\\citet{2004ApJ...603L..57M}. The next step of our research was to analyse  \nimages taken with the X-ray Telescope (XRT) on-board Hinode.\n \nSeen in XRT images, CHs are highly structured and dynamic at small scales.  \nHigh cadence  XRT data reveal in great detail the fine structure of coronal \nbright points (BPs) and  X-ray jets associated with them.  X-ray jets \nare collimated transient ejection of coronal plasma, first reported with the \nSolar X-ray telescope (SXT) onboard Yohkoh \\citep{1992PASJ...44L.173S}. They \nare believed to result from magnetic reconnection \\citep{1994xspy.conf...29S} and represent \nplasma outflows from the reconnection site. Recently,  \\citet{2008ApJ...673L.211M}  presented  \nthree--dimensional simulations of flux emergence in CH combined with spectroscopic  and imager \nobservations from XRT and EIS\/Hinode of an X-ray jet.  The authors  report that a jet resulting from \nmagnetic reconnection  is  expelled upward along the open\nreconnected field lines with values of temperature, density, and velocity in agreement with the XRT and EIS\nobservations. \\citet{1996PASJ...48..123S} reported 100 jets over 6 months in SXT \nimages from the Yohkoh while \\citet{2007PASJ...59S.771S} upgraded \nthis number to an average of 60 jet events per day in polar coronal holes. The \nauthors concluded that jets preferably occur inside polar coronal holes (PCH).  \n\nWe should note that although TRACE images  which have higher spatial \nresolution were used in paper I, the detailed structure of the dynamic \nchanges along CH boundaries was hard to distinguish. A reason for that is \nthe effect of  stray light in the TRACE  extreme-ultraviolet (EUV) telescope \nreported recently by \\citet{2009ApJ...690.1264D}. The authors found that 43\\% of the light \nwhich enters TRACE through the Fe~{\\sc ix\/x}  171~\\AA\\ filter is scattered\neither through diffraction off the entrance filter grid or through other \nnon-specific effects. This creates a haze effect and  especially effects  \nthe visibility of small-scale bright structures. \n\n\\begin{figure}[!h]\n\\vspace{6cm}\n\\special{psfile= fig_app_5.eps hoffset=0  voffset=40 hscale=55 vscale=55 angle=0}\n\\vspace{-2.9cm}\n\\caption{Polar coronal hole observed by XRT on 2007  September 20 with  the positions of all the identified brightening pixels over-plotted.}\n\n\\label{fig2}\n\\end{figure}\n\n\n  \\begin{figure}[!h]\n  \\vspace{7cm}\n\\special{psfile= fig_app_4.eps hoffset=-10  voffset=-10 hscale=65 vscale=65 angle=0}\n \n \n      \\caption{Quiet Sun observed by XRT on 2009  January  13 with  the positions \n      of all the identified brightening \n      pixels over-plotted.}\n      \\label{fig3}\n \n\\end{figure}\n\nOther transient structures seen in coronal holes  are the so-called plumes \nobserved off-limb above the North and South polar coronal holes.  They were \nfirst observed in white light as ray like structures \n\\citep{1965PASJ...17....1S}. They are also observed at EUV \nand soft X-ray temperature \\citep{1978SoPh...58..323A} as coronal outflow \nstructures similar to coronal jets, but hazy in nature with no sharp boundaries unlike jets.\nThey represent denser and cooler outflows with respect to the surrounding media and are \nobserved to extend from coronal BPs. They can extend up to 30 R$_{\\odot}$  from the solar disk center in a plane \nimage \\citep{2001ApJ...546..569D} and are observed to be in a steady state \nfor at least 24 hours \\citep{1997SoPh..175..393D}. The X-ray jets have been \nidentified as precursors for the plume formation  \\citep{2008ApJ...682L.137R}. \nRecently, \\citet{2008SoPh..249...17W} identified coronal plumes inside equatorial \ncoronal holes. They found that the plumes are analogous to polar coronal \nplumes. On the disk they are seen as a diffuse structure with a bright core \nand associated with EUV BPs. \n\nThe present study is a continuation of paper I and presents results from \nthe analysis of high-cadence\/high-resolution images of coronal holes (equatorial, \npolar and transient) and quiet Sun from XRT\/Hinode. We aim to establish which type of \nevent generates the non-Gaussian profiles registered at CH boundaries by  \n\\citet{2004ApJ...603L..57M} and how they are related to the small-scale BPs \nevolution along coronal hole boundaries as reported in paper I. In \nSect.~\\ref{data} we describe the data used for our study. Sect.~\\ref{ip} \noutlines an automatic brightening identification  procedure. In \nSect.~\\ref{results}, we give the obtained results and draw some \nconclusions on the outcome of our study. Finally, in Sect.~\\ref{conclusions} \nwe discuss the implication of our result to the understanding of the nature \nof coronal hole boundaries evolution at small scale and the possible \ncontribution of these events to the formation of the slow solar wind.  \n\n\\begin{figure*}[!ht]\n    \\centering\n    \\vspace{7cm}\n      \\special{psfile=fig_app_1.eps  hoffset=-40 voffset=25 hscale=50 vscale=50 angle=0}\n      \\special{psfile=fig_app_2.eps  hoffset=130 voffset=25 hscale=50 vscale=50 angle=0}  \n        \\special{psfile=fig_app_3.eps  hoffset=300 voffset=20 hscale=50 vscale=50 angle=0}\n\\vspace*{-0.75cm}\n     \\caption{ Equatorial coronal hole observed by XRT on 2007  November  9, 14 and 16 with the positions of all the identified brightening pixels over-plotted. The CH boundaries  are over-plotted with a black solid line.}\n      \\label{fig4}\n\\end{figure*}\n\\section{Data, reduction and preparation}\n\\label{data}\n\nWe used images from the X-ray Telescope  \\citep{2007SoPh..243...63G} on-board \nHinode taken during a dedicated observing run of an isolated equatorial \ncoronal hole (ECH), a Southern polar coronal hole and quiet Sun regions. \nThe ECH was tracked from the West to the East limb from 8 to 10 hours per day \nfor 4 days. The Southern polar CH was \nobserved for one day while the quiet Sun regions over 2 days. All data were \ntaken with an Al Poly filter which has a well pronounced temperature response \nat logT$_{max}~\\approx$~6.9~K. XRT images have an angular pixel size of\n1$^{\\prime\\prime}$~$\\times$~1$^{\\prime\\prime}$\\ at full resolution. They were recorded with 16~{\\rm s} and \n23~{\\rm s} exposure time and  a cadence of about 40~{\\rm s}. We also used \nrandomly selected  quiet Sun data with transient coronal holes (TCHs) and a Northern \npolar coronal hole observation. Further details on the data can be found \nin Table~\\ref{table1}.\n\nThe data were reduced using the standard procedures, which include flat-field \nsubtraction, dark current removal, despiking, normalisation to  data number \nper second to account for the variations in exposure time, satellite jitter \nand orbital variation corrections. The images were then de-rotated to a \nreference time to compensate for the solar rotation. A common field-of-view \n(FOV) was selected from all the images for each day. We then prepared an \narray with dimensions (nx, ny, nf), where nx is the number of Solar\\_X pixels, \nny is the number of Solar\\_Y pixels and nf  is the number of images. Each image was \nbinned to   4~$\\times$~4 pixels$^2$ in order to improve the signal-to-noise ratio and \n reduce the data points (and subsequently \nthe computational time).  The binned images were used to produce light \ncurves of nf points for each pixel. These light curves were the input for \nan identification procedure which will be discussed  in the next section.\n\n\n\\section{Brightening identification procedure}\n\\label{ip}\n\nWe developed an automatic identification procedure to distinguish small-scale \nintensity enhancements in XRT images. While the visual identification of large \nevents such as jets from bright points give good results, it is difficult to \nidentify and track small-scale events, especially on the quiet Sun high \nbackground emission or over pre-existing bright coronal loop structures \n(e.g. bright points, active regions etc). We eliminated all light curves which show no activity or minimum activity comparable to the noise level.  \n\n\\begin{table*}\n\\begin{center}\n\\caption{Number of events  over  24 hours per 100 $\\times$ 100 arcsec$^2$ identified inside the  coronal holes (CH), in coronal hole boundary regions (CHBR)  and  in the quiet Sun (QS).}\n\\label{table2}\n\\begin{tabular} {lcccccc}\n\\hline \\\\\n & \\multicolumn{3}{c}{No of events identified} & \\multicolumn{3}{c}{No of unresolved events } \\\\\nDate  & \\multicolumn{3}{c}{with plasma outflows} & \\multicolumn{3}{c}{identified with no plasma outflows} \\\\\n\\hline\n & CH & CHBR & QS & CH & CHBR & QS \\\\ \n\\hline \n09\/11\/07 & 29  & 40   &    6    &   6  & 26  &  8      \\\\\n12\/11\/07 & 16  & 57   &    6    &   2  & 17  &  7      \\\\\n14\/11\/07 & 10  & 51   &    1    &  16  & 41  &  7      \\\\\n16\/11\/07 & 56  & 32   &    -    &  44  & 33  & 12      \\\\\n16\/12\/08 & 99  & 86   &    6    &  31  & 14  & 33      \\\\\n20\/09\/07 & 57  & 72   &    4    &   5  &  7  &  9      \\\\\n10\/01\/09 &  -  &  -   &    6    &   -  &  -  & 17      \\\\\n13\/01\/09 &  -  &  -   &    9    &   -  &  -  & 14      \\\\\n29\/11\/07 &  -  & 64   &    7    &   -  & 62  & 14      \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n  \nThe first step of the identification procedure was to define the background \nemission for each light curve. The light curves were smoothed over a  window \nof width 5 to remove the spikiness in the background. Due to a difference in the \nbackground emission between the quiet Sun and the CHs, it was necessary to set two \ndifferent thresholds for further analysis. The threshold we used was \n1.8 times the mean emission value  for the CHs and 1.3 times the  \nmean emissivity for the QS. The comparatively higher threshold set for the \nCH light curves helped to eliminate the high fluctuations of the low emission \nbackground. Light curves with a maximum value less than these thresholds were \nneglected. Any point in the light curve was considered as a peak if its \nvalue was greater than the threshold and also greater than the average of the\ntwo preceding points, and the average of two  successive points. All the \nvalues below the threshold were considered as local minima. Each identified \npeak was traced back on either side to identify the minimum from the local \nminima. The value of all the points between the two identified minima for each \npeak were set to zero in the light curve and thereby from the average over \nthe rest of the light curve (I$_{av}$) we computed the standard deviation (SD). \nThe new background (BG) was obtained as BG = I$_{av}$ +1.1~$\\times$~SD.\n\nThe next step was the actual  identification of intensity enhancements. A  \nnew threshold of 2~$\\times$~BG for CHs and 1.3~ $\\times$~BG for QS was set \nusing the above  calculated background. Intensity increases above these \nthresholds with  corresponding minima less than  BG and duration less than 45 \nminutes were identified. A pixel brightening  was considered only if all the \nabove mentioned conditions were satisfied. The threshold was calculated with \na trial and error method.\n\nThe peaks having a duration of more than 45 minutes were examined separately. \nThe closest local minimum on either side of the peak were traced back. If \nthe difference between the peak and the minimum were greater than the BG for \nthe CHs and 0.3$\\times$BG for the QS with the duration less than 45 minutes, then \nthey were considered. Also the peaks which have one minimum that was either \nin the beginning or at the end of the light-curve were evaluated with the same \ncriteria, in order not to miss any real event. Any intensity enhancements \nin the coronal holes, the quiet Sun or over pre-existing bright loop structures \n which satisfy the above criteria could be \nidentified by our procedure. \n\n\\section{Results and discussion}\n\\label{results}\n\nAs it has been described in Sect.~2, we made a selection of data which comprised \nobservations of different features on the  Sun: equatorial  and polar coronal holes \nas well as  quiet Sun regions  with and without transient coronal holes.  \nIn Fig.~\\ref{fig1}--\\ref{fig4} we display examples of an X-ray image from each different region.  \nOur intention was to find out whether the changes we have seen so far along CHBs \n(\\citet{2004ApJ...603L..57M} and paper I) are unique for CH regions, i.e. regions \nof open magnetic field lines.  These data also permit to resolve the fine \nstructure of individual features and follow their dynamics at high cadence. \n To each dataset we applied the identification procedure described above. This \nprocedure provided us with the following information:  (i) light curves which \ncontain one or more  radiance enhancement identified as brightenings following \nthe criteria given in Sect.~ 3; (ii)  the start  and end time of each radiance \nenhancement; (iii)  the brightening positions in pixel numbers. As we produced \nlight curves by binning over 4~$\\times$~4 pixels$^2$, imprints of brightening \nevents with spatial scales larger than 4~$\\times$~4 pixels$^2$ were observed \nin more than one light curve. This made visual grouping of identified bright \npixels essential to distinguish each event.  Grouping of the features into \nindividual events was done by playing the image sequence of each dataset with \nthe identified brightenings over-plotted at corresponding times (see the online \nmovies). Clusters of bright pixels identified next to each other with similar \nlightcurves were grouped into events. The events showing plasma outflows (i.e. \nplasma moving along quasi-straight trajectories) were classified as jets, while \nevents exhibiting plasma blobs moving along curved trajectories  or just \nbrightening increase in a group of pixels were classified as unresolved \nbrightenings events. The so-called space-time plot was also used to \ninvestigate the plasma motion in the form of a jet and to determine their proper \nmotion. A space-time plot was produced by averaging  over a slice of 3 pixels wide and 100 \npixels long from each image, cut along the jet, i.e. in the direction of plasma propagation and then plotting that in time \n\\citep[][and Subramianian, PhD thesis \n2010]{2007PASJ...59S.745S} .  We were able to group more \nthan $95\\%$ of the identified bright pixels. The ungrouped pixels ($\\leq5\\%$) \ncomprise  bright pixels identified at the edges of images and above bad \npixels. The pixels identified in the beginning of each dataset which \ncould not be classified due to the lack of coverage of the whole event and \nthe pixels identified with a time lapse over their lifetimes were also rejected \nfrom counting. \n\nThe visual grouping of identified bright pixels into \nevents can be found in Table \\ref{table2}. We defined a coronal hole boundary \nregion (CHBR) as the region  $\\pm$15$^{\\prime\\prime}$\\ on both sides of the contour line \ndefining the CH boundary.  Additionally, animated image sequences with over-plotted  \nidentified brightenings at  corresponding times can be seen online as \nmovie\\_qs.mp4 (quiet Sun region on 2009 January 10), movie\\_ech.mp4 (ECH on \n2007 November 12,), movie\\_pch.mp4 (PCH on 2008 December 16) and movie\\_tch.mp4  \n(TCH on 2007 November 29). \n\nThe first and the most important result of this study is easily noticeable \nfrom Figs.~\\ref{fig1}--\\ref{fig4} the boundaries of coronal holes are abundant with \nbrightening events which appear much larger than the same phenomena in the \nquiet-Sun region. We separated the events visually into two groups, events \nwith plasma outflows or jet-like events  and events without outflows or \nsimple brightenings. The equatorial coronal hole data, observed near the disk \ncenter, shows  twice  as many jet-like and simple brightening events in \nthe CHBRs (as defined above) as compared to the CH regions. In contrast, polar \ncoronal hole data and ECH close to the West limb (2007 November 16) show a higher \nnumber of events inside the CH as well as in the CHBRs suggesting that this  \ncan be due to the line-of-sight effect. However, further investigation is \nneeded on a larger number of datasets.  \n \nIf we assume that the magnetic reconnection responsible for the occurrence \nof jet-like events takes place predominantly between closed and open \nmagnetic field lines, then the number of reconnection events producing \noutflows will be always higher in the CH boundary region since open and \nclosed magnetic field lines are continuously pushed together by different \nprocesses such as convection, differential rotation, meridian motions \netc. Inside coronal holes, where the number of bipolar systems and the \ncorresponding closed loop structures are limited, the number of jet-like \nevents will therefore be lower. For the transient coronal hole regions, the \nseparation of the coronal hole boundaries region  (30$^{\\prime\\prime}$\\ wide) from the coronal holes, \nfor estimating the number of events in each region, is more difficult due to the very small size of these coronal holes. Therefore,  here\n we consider that these CHs represent entirely a boundary region. The number of events found in these \n TCHs is several times larger than in the quiet Sun (both with and without outflows).  \n\nThe plasma ejected during the outflow events always originates from \npre-existing or newly emerging (at X-ray temperatures) bright points\nboth inside CHB regions and CHs. They typically start with a brightening \nin just a few XRT  pixels (4-6) somewhere in a pre-existing BP which \nwe believe to be the reconnection site.  \\cite{1996PASJ...48..123S} \nreported from the statistical study of 100 jets in SXT\/Yohkoh observations \nthat most of them were associated with micro-flares in the foot-points of \nthe jets.  \\citet{2007PASJ...59S.745S} resolved the fine structure of a \nquiet Sun X-ray jet to be the expansion and eruption of loop structures colliding \nwith the ambient magnetic field, similar to the CH jet.  Madjarska (2009) studied in great detail \none of the  jet-like phenomena identified in the datasets analysed here. The author estimated that the \nreconnection site reaches temperatures of up to 12 MK from observations \nof Fe~{\\sc xxiii}~263~\\AA\\ from EIS\/Hinode, which confirms that some of these \nevents are very similar to large flares but on a much smaller spatial scale. \nSeconds after the reconnection takes place, a cloud of plasma is blown \nout from the BP. Madjarska (2009) reported that \nalthough  the event appears more like a jet (although some expanding loops \ncan also be distinguished) in X-ray images as observed in projection on \nthe solar disk, the two additional view points from STEREO\/SECCHI reveal \nthat the phenomenon evolves as an expulsion of BP loops followed \nby a collimated flow along the quasi-open field lines of the expanded \nloops. The escaping plasma reaches temperatures of around 2~MK \n\\citep[][Madjarska 2009]{2007PASJ...59S.751C}. \n\n\\citet{2009SoPh..tmp..120N} found 5 out of 79 jets analysed from STEREO\/SECCHI \nobservations exhibiting a three part structure typically of coronal mass \nejections (CMEs) - bright leading edge, a dark void and \nbright trailing edge (e.g. the prominence material). The authors named \nthem micro-CMEs. The rest were called Eiffel tower-type jets, where the \nreconnection appears to happen on top of the loops and lambda-type jets \nwith reconnection occurring in the jet foot-points. As most of the events \nstudied here were seen in projection on the solar disk, this visual \ndivision into different groups is not possible. However, the visual examination \nof the phenomena analysed  here confirms that expulsion of BP loops \n describes these features best, hence, we will  further refer to them as EBPLs. \n\\begin{figure*}[!ht]\n\\vspace{4.6cm}\n\\special{psfile= fig2.eps hoffset=0  voffset=0 hscale=100 vscale=100 angle=0}\n\\caption{An example of a typical jet-like happening on 2007 November 12 at the \ncoronal hole boundaries. A refers to the jet while B refers to the BP.}\n\\label{fig5}\n\\end{figure*}\n\n\n\\begin{figure*}[!ht]\n\\vspace{3.2cm}\n\\special{psfile= fig4.eps hoffset=0  voffset=0 hscale=100 vscale=100 angle=0}\n\\caption{A bright point which produced several jet-like events at the boundaries of a transient coronal hole on 2007 November 29.}\n\\label{fig6}\n\\end{figure*}\n\n\n\\begin{figure*}[!ht]\n\\vspace{3.2cm}\n\\special{psfile= fig3.eps hoffset=0  voffset=0 hscale=100 vscale=100 angle=0}\n\\caption{An example of a brightening event identified in quiet Sun data  obtained on 2009 January 10.}\n\\label{fig7}\n\\end{figure*}\nFig.~\\ref{fig5} presents an example of a typical EBPL event  \nfrom a BP happening along an ECH boundary. The EBPL ends with the BP  \nvanishing at X-ray temperatures triggering the coronal hole to expand.  \nThe plasma ejected from the BP seems to collide with structures on its way  \n(propagating towards the QS region seen in projection on the solar disk) \nsetting off a brightening in a pre-existing BP (denoted with B in \nFig.~\\ref{fig5}) with no obvious plasma outflows. \n\nWe also found that the presence of a transient CH in the quiet Sun triggers \nthe occurrence of EBPL-like phenomena, similar to the equatorial and \npolar CH ones. Yet again, the evolution of the BPs along the boundaries \nchanged  the CHBs  (Fig.~\\ref{fig6}).  Due to the smaller size of the TCH, \nthese changes lead to a large expansion or contraction of the TCH, in some \ncases even a disappearance. In Fig.~\\ref{fig7} we give a series of images \nwhich exhibit one of the quiet Sun events. In comparison to CHs, the \nevents identified in  the quiet Sun images rarely show outflows \n(Table~\\ref{table2}). Neither expanding loops nor collimated flows were \ndistinguishable in the XRT images (i.e. no EBPL). \n\nThis brings us to the numerical comparison of CHBRs and CHs with QS areas. We \nfind an average of 75 brightening events per 100 $\\times$ 100 arcsec$^2$ per day \n for the coronal holes and boundaries and 20 events in the quiet Sun. Approximately \n70$\\%$ of these brightenings in CHs and CHBRs showed plasma outflows while only \n30$\\%$ of the brightenings seen  in the quiet Sun exhibit jet-like structures. \nPrevious works indicated fewer events per day either because of the poorer spatial resolution of the \ninstrument used or because of the visual identification methods. \n\n The identified brightening events with no plasma outflows could either \nbe driven by two sided loop reconnection \\citep{1994xspy.conf...29S} between \nemerging fluxes and overlying coronal fields, in which the ejected plasma flow \nalong the closed loop structures, or reconnection driven brightenings with \nplasma outflows at much lower temperatures. They can also represent flows in \nloop structures, perhaps triggered by reconnection shocks from the \nneighbourhood as seen in the example brightening event B in Fig.~\\ref{fig5}. \n\\citet{2007PASJ...59S.745S}  showed that a QS jet appears to be guided by the \nclosed magnetic field lines (loops), unlike the jets in the CHs and CHBRs which \nare guided by the open magnetic field lines. This result immediately \n raises the question whether the presence of open magnetic field lines is \ncrucial for the generation of outflow phenomena. \n\nComparison of the physical properties such as duration and size of the \njet-like events in ECHs and polar CHs show no difference. A good \ncorrespondence was found between the duration and the size of the events, \nirrespective of their position. The larger events have longer duration \nof around 40 min and are mostly associated with pre-existing coronal \nbright points or at least with features becoming visible in X-rays just \nbefore the eruption. The smaller events have a shorter duration of around \n20 min (mostly with no pre-existing features at coronal temperatures).\nHowever, for most events the actual duration from the moment the \nreconnection occurs (i.e. when the plasma is ejected) until the plasma \noutflow is no longer evident is usually between 10 and 15~{\\rm min}.  \nThe amount of plasma ejected entirely depends on the magnetic \nenergy available before the reconnection and it will, therefore, differ  \nfrom event to event. Repetitive occurrence of jets in the same bright \npoints are more common in CHs than in the QS. No periodicity was found, \nalthough a large number of BPs produced several jet-like events (from 2 \nto 5 times) during the course of the observations until all the stored magnetic \nenergy was exhausted and the bright point fully disappeared.  \n\n The proper motions of the outflows obtained from space-time plots are in\n  the range of 100$-$500~\\rm km~\\;s$^{-1}$ for most of the events. Because of the projection \n  effect of the jets with respect to the solar disk, the velocity we obtain gives the lower \n  boundary of the real velocity of the ejected plasma. Based \non their plasma velocity we divide X-ray jets into two groups: (i) jets \nwith pre-existing coronal structures (X-ray BPs) which have velocities \n$\\approx$350~\\rm km~\\;s$^{-1}$\\ or greater (i.e. in the range of the Alfv\\'en velocity \nin the lower corona) and (ii) jets with no pre-existing structures at X-ray \ntemperatures, showing velocities of around 150~\\rm km~\\;s$^{-1}$\\ \n(i.e. close to the Alfv\\'en velocity in the transition region). Stereo EUV images \ntaken with the 171~\\AA\\ filter confirm the presence of a corresponding \nreconnection jet-like structures at transition region temperatures.  \n\n\\section{Conclusions}\n\\label{conclusions}\n\nThe present article confirms our findings from  paper I that the  evolution \nof loop structures known as coronal bright points is associated with the \nsmall-scale changes of CHBs.  We were able to identify the true nature of these \nchanges which represent plasma outflows \nassociated with the expansion of the bright point loop structures. The plasma \ntrapped in the loop structures is consequently released along the ``quasi'' open \nmagnetic field lines. These \nejections appear to be triggered by magnetic reconnection, most probably  \nthe so-called interchange reconnection \\citep{2004ApJ...612.1196W} between \nthe closed magnetic field lines (BPs) and the open magnetic fields of coronal \nholes. The ejected plasma is guided and accelerated further away from the \nSun by the open magnetic field lines with some jets reaching \nseveral solar radii \\citep{2009SoPh..tmp..120N}. Contrary, in the quiet Sun \nthe plasma ejected as a result of two or one sided loop reconnection, is \ncontained in the corona by the closed magnetic field lines. \n\n Tall and extended coronal loops are very rare in coronal holes \n\\citep{2004SoPh..225..227W}, while closed (loop) magnetic structures of \nvarying physical properties are ubiquitous in the quiet Sun corona as seen \nin EUV and X-ray observations. Coronal holes with predominant open magnetic \nfields and minority closed loop structures (BPs) are encompassed by these loop \nstructures seen at transition region and coronal temperatures. \n\\citet{2004ApJ...612.1171W}, \\citet{2001ApJ...555..426W} and \n\\citet{2005JGRA..11007109F} showed that the elemental abundance of  \ntrapped plasma is proportional to the confinement time of the plasma in \nloop structures. Newly emerged active region loop structures were found \nto have initial photospheric abundances (FIP bias $\\approx$1--2)\n which increased with time, reaching $\\approx$~5 in 1--3 days \n\\citep{2001ApJ...555..426W}. \\citet{1998Natur.394..152S} concluded that \n$\\approx$1.5 days  is the reconfiguration time-scale for the super-granulation \nnetwork magnetic fields  where coronal BPs have their foot-points rooted.  \nThis time period  is in the range of the confinement time-scale needed for \nthe enhancement of the FIP bias. \n\nCoronal BPs electron densities were derived from CDS\/SoHO in the temperature \nrange 1.3--2$\\times 10^6$~K by \\citet{2005A&A...435.1169U}. The authors \nconcluded that the bright points plasma have properties which are more \nsimilar to active region plasma rather than quiet Sun plasma, although BPs do \nnot show the increase of electron density at temperature over \nLog$T_e$~$\\sim$~6.2~K, observed in the core of active regions \n\\citep{2005A&A...435.1169U}. These results were later confirmed from data taken \nwith the Extreme-ultraviolet Imaging Spectrometer (EIS) for Log$T_e \\sim$6.1 \nand 6.2~K \\citep {2008A&A...492..575P}. The BP lifetime in EUV were found to be \non average 20~hrs in the EUV \\citep{2001SoPh..198..347Z} and on average 8~hrs \nin X-rays with some BPs lasting up to 40~hrs \\citep{1974ApJ...189L..93G}. The \nresults of individually studied BPs by \\citet{2004A&A...418..313U} give for two \nBPs a lifetime of 38 and 51~hrs detected in  Fe~{\\sc xii}~195~\\AA\\  images from \nExtreme-ultraviolet Imaging Telescope (EIT) on-board SoHO and by P\\'erez-Suarez \n(PhD thesis 2009) in five BPs: BP1 -- 48~hrs, BP2 more than 54~hrs, BP3 -- \n37~hrs, BP4 -- 45.2~hrs and BP5 -- 35 h (on the limb). The study by  \n\\citet{1974ApJ...189L..93G} on BPs lifetime in X-rays has not been updated so \nfar using Hinode X-ray observations. The BPs properties given above strongly \nsuggest that their plasma can become enriched on low FIP elements. \n\n \\citet{2008ApJ...682L.137R} concluded that X-ray jets are the precursors \nof polar plumes, and jets happening in pre-existing polar plumes enhance the \nbrightness of the plume haze. Polar plumes are observed even at several \nsolar radii \\citep{1997SoPh..175..393D} and were found to contribute to the \nsolar wind stream. They have been reported to occur even at low latitudes \n\\citep{1995ApJ...446L..51W}.  Jets, associated with BPs, were also recently \nregistered with the Large Angle and Spectrometric Coronagraph on-board SoHO \n\\citep{2008SoPh..249...17W} and SECCHI\/STEREO  \\citep{2009SoPh..tmp..120N}. \nHence, the  BP plasma cloud, which is ejected as a result of magnetic \nreconnection, will therefore, escape from the Sun having the plasma \ncharacteristics of the slow solar wind. We asked ourselves whether the plasma \nejections we observe can possibly be a source of the fast solar wind? This \npossibility cannot be fully rejected, although it is a  fact that these jets \nhappen sporadically rather than continuously, which is in contradiction with \nthe nature of the fast solar wind.\n\nOur specially designed observing programs provided us with spectroscopic \nco-observations from SUMER, CDS and EIS along with the XRT and SOT. In a  \nfollow up paper we will derive the physical properties such as velocity, \ndensity, temperature and others of a large number of events happening in the \nFOV of the spectrometers. \n\n\\begin{acknowledgements} The authors thank ISSI, Bern for the support of \nthe team ``Small-scale transient phenomena and their contribution to coronal \nheating''. Research at Armagh Observatory is grant-aided by the N.~Ireland \nDepartment of Culture, Arts and Leisure. We also thank STFC for support via \ngrants ST\/F001843\/1 and PP\/E002242\/1. Hinode is a Japanese mission developed \nand launched by ISAS\/JAXA, with NAOJ as domestic partner and NASA and STFC \n(UK) as international partners. It is operated by these agencies in \nco-operation with ESA and NSC (Norway). The STEREO\/ SECCHI data used here are \nproduced by an international consortium of the Naval Research Laboratory (USA), \nLockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight \nCenter (USA), Rutherford Appleton Laboratory (UK), University of Birmingham \n(UK), Max-Planck-Institut f\\\"{u}r Sonnensystemforschung (Germany), Centre Spatiale\nde Li\\`{e}ge (Belgium), Institut d'Optique Th\\'{e}orique et Appliqu\\'{e}e (France), and \nInstitute Astrophysique Spatiale (France).\n\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n\\input CHB_madjarska.bbl\n\n\\end{document}   \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{s:intro}\n\nHydrogen peroxide (\\ho) is a well-studied system because of its\nunusual properties, particularly the almost freely rotating\nOH moieties.\nThe role of \\ho\\ in the chemistry of the Earth's atmosphere \\cite{74Davis.H2O2,03ViMaMi.H2O2, 13AlAbBeBo.H2O2} as well as of the Martian atmosphere \\cite{04EnBeGr.HOOH,12EnGrLe.HOOH}  has been widely acknowledged. It has also\nrecently been detected in the interstellar medium \\cite{11BePaLi.H2O2}.\nThe vibration-rotation spectra of hydrogen peroxide has attracted\nsignificant attention, both experimental\nand theoretical. For example it has been used\n                                                    as a benchmark system with large amplitude motion for\ntesting different variational nuclear motion codes \\cite{93.BrCarrin.HOOH,00CaHaxx.HOOH,00Luckhaus.HOOOH,01ChMaGu.HOOH,02YuMuck.HOOH,02Mladenovic.HOOH,03LiGuxx.HOOH,09CaHaBo.h2o2,11CaShBo.H2O2}.\n\n\nVery recently Ma{\\l}yszek and Koput \\cite{Koput3} presented a highly accurate \\ai\\\npotential energy surface (PES) of HOOH which was shown to reproduce the known vibrational band origins\nwith the average accuracy of 1 \\cm. Other \\ai\\ PESs of HOOH were reported by Harding \\cite{91Harding.HOOH},\nKuhn \\etal\\cite{99Kuhn.HOOH}, Senent \\etal \\cite{00Senent.HOOH}, and Koput \\etal\\cite{Koput1}.\n\n\nHydrogen peroxide has three important properties from the viewpoint of\nvariational calculations. Firstly, the large amplitude motion\nof the OH internal rotors that has already been mentioned. Secondly, its\nrelatively low dissociation energy of about 17,000 \\cm\\\nhas made \\ho\\ a benchmark tetratomic molecule for experimental study\nof the dissociation process \\cite{Rizo}. Thirdly, \\ho\\  is a tetratomic\nsystem where variational calculations can really aid the analysis\nof spectra.\n\nFor triatomic molecules, accurate\ncalculation of the rotation-vibration levels to high\naccuracy using variational nuclear motion\nmethods has become routine \\cite{jt309,jt512,12HuScTaLe.CO2}. For tetratomic\nmolecules this process is just beginning; it is natural for initial\nhigh accuracy studies to focus on molecules with large\namplitude motion such as  ammonia \\cite{Huang:Schwenke:Lee:2010:I,Huang:Schwenke:Lee:2010:II} and hydrogen\nperoxide.\n\nThe advantages of using variational calculations to assign\nvibration-rotation spectra of triatomic molecules has been\ndemonstrated for several molecules. Initial studies focused\non \\hp\\ \\cite{jt80,jt102,jt193} and water\n\\cite{jt200,jt218}, systems for which the use of variational\ncalculations to analyse spectra is now the accepted procedure.\nIn particular, spectra involving hot molecules, and hence high rotational\nstates, and large amplitude motion,\nsuch as \\hp\\ on Jupiter \\cite{jt80}  and water on the Sun\n\\cite{jt200}, assignments using the traditional, effective\nHamiltonian approach are almost impossible.\n\nA significant advantage of variational calculations over effective\nHamiltonian techniques is the automatic allowance for accidental\nresonances between vibrations. Whereas for most triatomic molecules such\nresonances become significant at fairly high vibrational\nenergies, for tetratomic molecules accidental resonances\ncan even make the analysis of low-lying vibrational states intractable using effective Hamiltonians. \\ho\\ is a good example of this situation.  Although \\ho\\ spectral lines\nare strong and were first observed more than seventy years ago\nwith spectrometers  much less sophisticated then those available\nnowadays \\cite{41ZuGi.H2O2,50Giguer.H2O2}, the analysis of experimental spectra involving high $J$ transitions for \\ho\\ is only complete up to\n2000 \\cm\\ \\cite{perin1,perin2,perin3}, significantly lower\nin frequency than transitions to the OH stretching fundamentals.\nOne reason for this is the\ncomplication of the analysis by accidental resonances.\nAccurate variational calculations on \\ho\\ offer a way out of this impasse.\n\n\n\nRecent advances in variational calculations suggest that they can be\nused for  systems larger than triatomic.  High accuracy variational\ncalculations of the spectra and line lists for tetratomic molecules\nsuch as ammonia \\cite{jt466,jt500,Huang:Schwenke:Lee:2010:II} have been\nperformed.\n         These NH$_3$ line lists have been used both for the assignment\nof transitions involving higher vibrational states \\cite{12SuBrHu.NH3},  hot rovibrational spectra involving high $J$ levels \\cite{jt508}\nas well as  for correcting and improving the analysis\nof more standard transitions \\cite{Huang:Schwenke:Lee:2010:II,jt543}.  Numerical\ncalculations of wavefunctions for high $J$ states of tetratomic molecules\nare possible not only because modern computers have the ability to diagonalise\nlarger matrices but also because, as illustrated below, the accuracy\nof calculations employing\napproximate kinetic energy operators \\cite{multimode,trove-paper}\nbecomes comparable with those using an exact kinetic energy approach \\cite{jt312}.  While high $J$ calculations within the exact kinetic\nenergy approach  are still\ncomputationally challenging for tetratomic molecules,\ncalculations with $J \\sim 50$ are feasible with approaches such as\nTROVE \\cite{trove-paper}.  Furthermore, the possibility of calculating \\ai\\\ndipole moment surfaces of extremely high accuracy \\cite{jt509}\nenhances the value of using variational calculations since they can also be\nused to\ncreate line lists.\nThese factors raise the possibility of creating accurate line lists for\n\\ho.  However, the presence of the large amplitude, torsional motion\nof the two OH fragments in \\ho\\ complicates the problem. This requires an appropriate\nnuclear motion programme for calculation of the\nrovibrational energy levels by solving the corresponding Schr\\\"odinger\nequation; this programme should be able to compute high $J$ levels\nwithin the limitations of the modern computers.\n\nIn this paper we compute high accuracy rovibrational energy\nlevels going to high $J$ for\n\\ho\\  using the                          \\ai\\       PES due to\nMa{\\l}yszek and Koput \\cite{Koput3}.\n To do this we\ntest two nuclear motion programmes:\nthe exact kinetic energy (EKE)\nprogramme WAVR4 \\cite{jt339}  and approximate kinetic energy programme TROVE\n\\cite{trove-paper}. It is  shown that use of\nTROVE allows us to calculate very high $J$ energy levels which are\nin excellent agreement with observation.\n The paper is organised as follows.\nSection~\\ref{s:method}  describes the modifications of the TROVE programme\nnecessary to make it suitable for the calculation of spectra\nof such a nonrigid molecule.\nSection~\\ref{s:results} describes the details of computations performed.\nSection~\\ref{s:concl} presents our results which is followed by the concluding section\nwhich discusses prospects for further work on this system.\n\n\n\n\\section{Methods of calculation }\n\\label{s:method}\n\n\nThe accuracy of a calculation of  rovibrational energy levels depends\nfirst of all on the accuracy of the potential energy surface (PES) used\nas input to the nuclear motion Schr\\\"odinger equation. Until recently the most\naccurate PES for \\ho\\ was  the one due to Koput \\etal\\cite{Koput1} which gave a\ntypical discrepancy between theory and experiment for vibrational band\norigins of about 10 \\cm\\ \\cite{Koput2}.  However,\ntwo of us\\cite{Koput3} recently determined a very accurate PES computed using the explicitly correlated coupled-cluster method  [CCSD(T)-F12] method,\n\\cite{CC,CC2} in the F12b form \\cite{F12a} as implemented in\nthe  MOLPRO package \\cite{molpro}.\nVarious correlation-consistent basis sets were used for various parts of the PES, the largest being aug-cc-pV7Z. The  CCSD(T)-F12 results were augmented with the Born-Oppenheimer diagonal, higher-order valence-electron correlation, relativistic, and core-electron correlation corrections. The 1762 \\ai\\ points obtained were fitted to the functional form\n\\begin{equation}\n\\label{e:koput}\nV(q_1,q_2,q_3,q_4,q_5,q_6) = \\sum_{ijklmn} c_{ijklmn} q_1^{i} q_2^{j}q_3^{k} q_4^{l} q_5^{m} \\cos{n q_6}\n\\end{equation}\nwhere $q_{i}$  ($i = 1,2,3$)  are the Simons-Parr-Finlan stretching OO and OH coordinates  \\cite{73SimParrFin.SPF} $q_1= (R-R_{\\rm e})\/R $ and\n$q_i = (r_i - r_{\\rm e})\/r $ ($i=1,2$), $q_4 = \\theta_1 - \\theta_{\\rm e} $ and $q_5 = \\theta_2 - \\theta_{\\rm e} $ are the two OOH bending coordinates,  $q_6 = \\tau$ is the torsional angle $\\angle$HOOH (see Fig.~\\ref{f:tau}),\nand $R_{\\rm e}$, $r_{\\rm e}$, and $\\theta_{\\rm e}$ are the\ncorresponding equilibrium values. The\nexpansion coefficients  $c_{ijklmn}$  used in this work are\ngiven in the supplementary material \\cite{supl} to this\narticle (see also Ref. \\citenum{Koput3}).\n\nRecent calculations\\cite{Koput3} using this PES gave, for the 30 observed vibrational band origins of \\ho, a standard deviation for the observed minus calculated (obs $-$ calc) wavenumbers of about 1 \\cm, an order of \nmagnitude improvement over the previous results.\\cite{Koput2} An \\ai\\ line list with this accuracy  could be useful for a number of applications. \nHowever, for most applications it is also necessary to accurately compute  highly excited rotational levels.  This is done in this work. Before looking at high $J$ rotational levels, we reconsidered the $J=0$ results of  Ma\\l yszek and  Koput \\cite{Koput3} using both EKE programme WAVR4 \\cite{jt339} and approximate kinetic energy programme TROVE \\cite{trove-paper}.\n\n\nDiatom-diatom HO--OH coordinates were employed in the programme WAVR4; these coordinates were one of those used to consider acetylene -- vinylidene isomerisation \\cite{jt346}.  The calculations used a discrete variable\nrepresentations (DVR) based on a grid of 10 radial functions for each OH coordinate\nand 18 radial functions for the OO coordinate. The parameters used for\nOH stretch Morse-oscillator like functions were $r_{\\rm e} = 0.91$ \\AA,\n$\\omega_{\\rm e} = 2500$ \\cm\\ and $D_{\\rm e} = 35000$ \\cm, and $r_{\\rm e} =1.53$ \\AA,\n$\\omega_{\\rm e} = 1500$ \\cm\\ and $D_{\\rm e} = 45 000$ \\cm\\ for the OO stretch.\nThe bending basis set consists of coupled angular functions\n\\cite{jt339} defined by $j^{\\rm max}=l^{\\rm max} =22$ and $k^{\\rm max}$ =\n12. The resulting energy levels with $J$ = 0 were within\n0.1 \\cm\\ of the previous calculations \\cite{Koput3}, see Table~\\ref{tab:J1}.\nHowever for WAVR4 calculations of the same accuracy for levels with  $J = 1$\nrequire about 10 times more computer time. This is a consequence\nof the $J$ -- $K$ coupling used in the EKE procedure. This coupling\nis essential for the linear HCCH system \\cite{jt346} and very floppy\nmolecules \\cite{jt312}, but not for \\ho.\nThe use of WAVR4 to calculate energies of high $J$ levels is\ncomputationally unrealistic at present and we note that indeed    corresponding\nstudies on acetylene have thus far been confined to low $J$ values\n\\cite{jt479}.\n\nTROVE is a computer suite for rovibrational calculations of energies and intensities for molecules of (at least in principle) arbitrary structures. TROVE uses a multilevel contraction scheme for constructing the rovibrational basis set. The primitive basis functions are given by products of six 1-dimensional (1D) functions $\\phi_i(\\xi_i)$, where $\\xi_i$ represents one of the six internal coordinates. For HOOH we choose $\\xi_1$, $\\xi_2$,  and $\\xi_3$ to be the linearized versions of the three stretching  internal displacements\n$R-R_{\\rm e}$, $r_1-r_{\\rm e}$, and $r_2-r_{\\rm e}$, respectively,\n$\\xi_{4}$ and $\\xi_{5}$\nare the linearized versions of the two bending\ndisplacements $\\theta_1-\\theta_{\\rm e}$\nand $\\theta_2-\\theta_{\\rm e}$; $\\xi_6$ is\nthe torsional coordinate $\\tau$, see Fig.~\\ref{f:tau}.\n\nThe kinetic energy operator in TROVE is given by an\nexpansion in terms of the five coordinates $\\xi_{i}$,\nrepresenting the rigid modes $i=1\\ldots 5$. The\npotential energy function is also expanded but using three Morse-type\nexpansion variables $1-\\exp(-a_i\\xi_i)$ ($i=1,2,3$) and two bending\ncoordinates $\\xi_4$ and $\\xi_5$.\nHere $a_1 = 2.2$ \\AA$^{-1}$, $a_2=a_3=2.3$ \\AA$^{-1}$\nwere selected to match closely the shape of the \\ai\\\nPES along the stretching modes. In the present work we\nemploy  6th and 8th order expansions to represent,\nrespectively, the  kinetic energy operator and potential\nenergy function.\n\nThe rovibrational motion of the non-rigid molecule HOOH is\nbest represented by the extended $C_{\\rm 2h}^{+}$(M)\nmolecular symmetry group \\cite{Hoxxxx.HOOH}, which is\nisomorphic to $D_{\\rm 2h}$(M) as well as to the extended\ngroup $G(4)$(EM).\\cite{BJ} As explained in detail by Bunker\nand Jensen\\cite{BJ}, the extended group is needed to describe\nthe torsional splitting due both the \\textit{cis}-\nand \\textit{trans}-tunnelings. In the present work we use\nthe $D_{\\rm 2h}$(M) group to classify the symmetry of the HOOH states.\nThis group is given by the eight irreducible\nrepresentations $A_{\\rm g}, A_{\\rm u}, B_{\\rm 1g},\nB_{\\rm 1u}, B_{\\rm 2g}, B_{\\rm 2u}, B_{\\rm 3g}, B_{\\rm 3u}$.\nIn order to account for the extended symmetry properties of the\nfloppy HOOH molecules an extended range for torsion motion,\nfrom 0$^{\\circ}$ to 720$^\\circ$\nwas introduced into TROVE. In this representation\n$\\tau = 0$ and 720$^\\circ$ correspond to the \\textit{cis} barrier, while at $\\tau=360^{\\circ}$ the molecule has the \\textit{trans} configuration.\n\n\n\nTROVE's primitive basis functions, $\\phi_{v_i}^{(i)}$ ($i=1\\ldots 6$),\nare generated numerically by solving six 1D vibrational Schr\\\"{o}dinger\nequations for each vibrational mode $i$ employing the\nNumerov-Cooley method~\\cite{Numerov,Cooley}. The corresponding\nreduced 1D Hamiltonian operators $H^{\\rm 1D}_i$\n($i=1\\ldots 6$) are obtained by freezing the five remaining modes at\nthe corresponding equilibriums. The integration ranges are\nselected to be large enough to accommodate all basis functions\nrequired (see below for the discussion of the basis set sizes).\nThe torsional functions are obtained initially on the\nrange $\\tau=0\\ldots 360^{\\circ}$ and transform according\nwith the $C_{\\rm 2h}$(M) group. A very fine grid of 30,000 points\nand the quadruple numerical precision [real(16)] was used for\ngenerating the eigenfunctions of the corresponding Schr\\\"{o}dinger\nequation in order to resolve the $trans$ splittings up to $v_6=42$.\nThe wavefunctions are then extended to $\\tau=360\\ldots 720^\\circ$\nthrough the $+\/-$ reflection of the $C_{\\rm 2h}$(M) values and\nclassified according to $D_{\\rm 2h}$(M). The extended primitive\ntorsion functions are able to account for the torsional splitting\ndue to both the \\textit{cis} and \\textit{trans} tunneling.\n\nThe 6D primitive basis functions are then formed from different\nproducts of the 1D functions $\\phi_{v_i}^{(i)}$. The size of the basis set\nis controlled in TROVE by the so-called polyad number $P$, which in the present case is given by\n\\begin{equation}\\label{e:polyad}\n  P = 4 v_1 + 8 (v_2+v_3) + 8 (v_4 + v_5) + v_6 \\le P_{\\rm max},\n\\end{equation}\nwhere $v_i$, $i=1\\ldots 6$ are the local mode quantum numbers\ncorresponding to the primitive functions $\\phi_{v_i}^{(i)}(\\xi_{i})$.\nThe primitive basis set is then processed through a number of\ncontractions, as described in detail previously \\cite{jt466},\nto give a  final basis set in the $J=0$ representation.\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{HOOH.eps}\n\\vspace*{0.5cm}\n\\caption{\\label{f:opt:geom} The internal coordinates for the HOOH molecule. }\n\\label{f:tau}\n\\end{center}\n\\end{figure}\n\n\n\n\nCalculations using programme TROVE started with a search for a basis\nset and operator expansions which would give results close to\nthe EKE ones.  The final values of the basis set parameters were\nchosen to try to meet two conflicting requirements: the best possible\nconvergence and a compact enough calculation  to\nallow high $J$ energy levels to be computed. The\nfinal basis set parameters were the following: the maximum polyad\nnumber $P_{\\rm max}=42$, which also corresponds to the highest excitation of the torsional mode $v_6$.\nFor O-O stretch the maximal number was 8, for OH stretches - 8 and for\nthe bending modes - 10. These parameters control the size of the basis\nset  used in the TROVE calculations according with Eq.~(\\ref{e:polyad}). For the $J=0$ levels\nthey give good agreement with the previous studies, see Table~\\ref{tab:J0}.\n\n\\section{Results}\n\\label{s:results}\n\nThe updated version of TROVE was used to calculate excited\nrotational levels for $J$ up to 35, the highest assigned thus far experimentally.\nInitial calculations were performed with the equilibrium distances and\nangles obtained \\ai\\ in Ref. \\citenum{Koput3}. In this case\nthe discrepancies between theory and experiment increased\nquadratically with increasing $J$: the $J = 1$ levels\nwere calculated with an accuracy around 0.001 \\cm, but those for $J = 35$\ndiffer from experiment by about 1 \\cm.\n\nWe therefore chose to adjust the equilibrium parameters $R_{\\rm e}$\nand $r_{\\rm e}$ to better reproduce the experimental values.  Only\nvery small changes were needed to make the $J = 35$ levels accurate to\nabout 0.03 \\cm\\ for the ground vibrational state.  In particular, the\noriginal value of $R_{\\rm e}$ of 1.45539378 \\AA\\ was shifted to\n1.45577728 \\AA\\ and $r_{\\rm e} = 0.96252476$ \\AA\\ moved to 0.96253006 \\AA.\nThe rotational structure\nwithin the excited vibrational states is of similar accuracy, meaning\nthat these levels are essentially shifted just by the discrepancy in\nthe vibrational band origin.  In practice this geometry shift not only\nmeant that low $J$ energy levels were reproduced with an accuracy of 1\n\\cm, reproducing the accuracy of the vibrational band origin, but\nalso resulted in pseudo-resonance artifacts. To illustrate\nthis consider the interaction between the ground vibrational state and\nthe low-lying $v_4 = 1$ torsional vibrational state, which lies about\n2 \\cm\\ too low in the calculations.  This results is an artificial\ncloseness, and interaction, between levels with the same $J$  and\n$K_a= 8$ for $v=0$ and $K_a = 6$ for $v_4 = 1$.  The resulting shift\nin the energy levels is significant; it grows with $J$ and reaches\nabout 1 \\cm\\ at $J$ = 30. We call this a  pseudo-resonance artifact\nsince no such interaction is seen in the experimentally-determined energy levels.\n\nThere are several ways to avoid this artificial pseudo-resonances.  One\nwould be to fit the PES to experimental data, which would remove this\nartificial near-degeneracy. This is likely to be a topic of future\nwork. An alternative possibility, which is already available within\nTROVE \\cite{jt466},  is to simply adjust the calculated values of the\nvibrational band origins given by the $J = 0$ calculation to the\nobserved ones prior to their use in calculations of the $J>0$ levels.\nThis option, which is not available in EKE codes which couple the\nbending basis with the rotational functions \\cite{jt14}, not only\nshifts the energies, it also rearranges the matrix elements so that the\nartificial resonances disappear.  With this adjusted calculation the\nenergy levels vary smoothly with the increasing $J$ and $K_a$ quantum\nnumbers, see Tables \\ref{tab:J1}, \\ref{tab:J30} and \\ref{tab:J35}, as one would expect\n\\cite{jt205} from purely \\ai\\ levels.\n\nOne other problem remained when comparing our rotationally excited\nenergy levels with the observed ones.  This concerned rotational\nlevels with the quantum number $K_a = 1$ which did not behave as\nlevels associated with other values of $K_a$. The error for the\ntwo levels with $K_a = 1$ increases disproportionately to that of\nother levels as $J$ increases. This error was about 0.1 \\cm\\ for $J$ =\n30. A series of test calculations revealed the reason for such\ndiscrepant behaviour of the levels with $K_a = 1$. It transpires that a\nsmall change in the height of the torsional barrier, which is strongly\ninfluenced by the linear expansion coefficient $c_{000001}$, does not affect other $K_a$ levels, but significantly influences only those with $K_a$ = 1.\nVarying this expansion coefficient can both increase and decrease the splitting of the $K_a$ = 1 doublet.  As this splitting is overestimated in calculations\nusing the \\ai\\ value of $c_{000001}$, its reduction by about 1~\\%\\ from\nthe \\ai\\ value of 0.00487 to 0.00483 $E_h$\nresults in roughly a fourfold   improvement of the obs -- calc value for the $K_a=1$ levels. This change affects the value of\nthe  ground-state torsional splitting of 11 \\cm\\ and also the values of the other torsional energy levels, all of which move significantly closer to the observed values, than the purely \\ai\\ levels given in the Table~\\ref{tab:J0}.  In\nparticular, this small adjustment improves the calculated  ground-state splitting to 11.4 \\cm\\ and the first torsional level to 255.2 \\cm.\nThus adjusting $c_{000001}$\nnot only improves  significantly the values of  levels with $K_a=1$ levels,\n      it\nimproves the overall agreement with experiment for the band origins.               The\nunderlying reason for this is that the expansion coefficient  $c_{000001}$\ncontrols the height of the torsional barrier.\n\nThese minor adjustments result in very accurate values for rotational\nenergy levels a sample of which are presented in Tables~\\ref{tab:J1},\n\\ref{tab:J30} and \\ref{tab:J35}. A more comprehensive set of energy\nlevels is given in the supplementary material \\cite{supl}. From these\ntables one can see that the discrepancy between observed and\ncalculated energy values increases both gently and smoothly with\nrotational quantum number $J$. Such calculations therefore provide an\nexcellent starting point for assigning high $J$ transitions both\nwithin the ground state and to excited vibrational states, as the\ndensity of observed transitions is orders of magnitude smaller  than the accuracy of calculations.\n\n\\section{Conclusions}\n\\label{s:concl}\n\nWe present results of \\ai\\ and slightly adjusted \\ai\\ calculations for\nthe vibrational and rovibrational energy levels of the \\ho\\ molecule.\nUse of the accurate \\ai\\ PES calculated by Ma\\l yszek and Koput\n\\cite{Koput3} reproduces the known vibrational band origin with a\nstandard deviation of about 1 \\cm.  The use of programme TROVE\n\\cite{Trove} for the nuclear motion calculations allowed us compute\nhigh rotational levels up to $J=35$. Indeed,\nenergy levels with $J=50$\ncould be calculated on a high-end workstation, and the accuracy of prediction will be very\nhigh - better than 0.5 \\cm. However, we have not yet performed such calculations,\nas no comparison with  experimental values is currently possible.\nExperimentally derived energy levels up to $J=35$ are compared with our\ncalculations. These are reproduced with an unprecedented accuracy of\n0.001 \\cm\\ for the levels up to $J=10$ and 0.02 \\cm\\ for all the known\nlevels above this.  Variational calculations using this\nslightly adjusted \\ai\\ PES\nresults in very smooth variation in the discrepancies between the\nobserved and calculated levels as a function of the rotational quantum\nnumbers $J$ and $K_a$. This smoothness and accuracy is the key to the\nsuccessful analysis of previously unassignable spectra\n\\cite{jt200,jt205} as, in particular, the accidental resonances, which\nseriously complicate any analysis based on the use an effective\nHamiltonian, are automatically allowed for in such calculations.\n\nThere is one other important aspect of the \\ho\\ rotation-vibration\nproblem which we should mention. The detection of extrasolar planets and,\nin particular, our ability to probe  the molecular composition of these\nbodies using spectroscopy \\cite{jt400}, has led to demand for\naccurate, comprehensive line lists over an extended range of both\ntemperature and wavelength for all species of possible importance in\nexoplanet atmospheres \\cite{jt528}. The accuracy\nof the calculations  presented here and, especially their\nability to reliably predict highly excited rotational levels which\nare of increasing importance at higher temperatures, suggests\nthat the present work will provide an excellent starting point\nfor the calculation of a comprehensive line list for \\ho. In this\nwe will be following the recent work of Bowman and co-workers who have\ncomputed similar line lists for somewhat more rigid hydrocarbon\nsystems \\cite{09WaScSh.CH4,12CaShBo.C2H4}.\n\n\\section*{Acknowledgment}\nThis work was performed as part of ERC Advanced Investigator Project 267219.\nWe  also thank the Russian Fund for Fundamental Studies\nfor their support for aspects of this project.\n\n\\newpage\n\n\n\\begin{table}\n\\caption{ Calculated and observed energy levels, in \\cm, for $J=0$  using WAVR4,  TROVE and\npublished by Ma{\\l}yszek and Koput (MK) \\cite{Koput3}; ``tr-shift'' results are computed with an adjusted height\nfor the torsional barrier. The observed values are taken from Ref.~\\protect\\onlinecite{perin2,perin3,perin4,95PeVaFl.H2O2}.}\n\\label{tab:J0}\n\\begin{tabular}{cccccccrrrrr}\n\\hline\n\\hline\n$v_1$  & $v_2$ & $v_3$ &    $v_4$   & $v_5$    & $v_6$  & Sym&   Obs   &   WAVR4  &  TROVE &tr-shift & MK\\\\\n\\hline\n\n0 & 0   &  0  &  1       &  0     &  0   & $A_{\\rm g}$           &  254.550 &  256.406 & 256.419 & 255.490 & 255.43\\\\\n 0 & 0   &  0  &  2       &  0     &  0  &  $A_{\\rm g}$         &  569.743 &  570.334 & 570.251 & 570.690 & 570.45 \\\\\n 0 & 0   &  1  &  0       &  0     &  0  &    $A_{\\rm g}$      &  865.939 &  865.547 & 865.652 & 865.468  & 866.02\\\\\n 0 & 0   &  0  &  3       &  0     &  0  &    $A_{\\rm g}$      & 1000.882 & 1001.227 &1001.073 & 1002.493 & 1001.92 \\\\\n 0 & 0   &  0  &  0       &  0     &  1  &  $B_{\\rm u}$         & 1264.583 & 1264.819 &1265.121 & 1264.868  & 1264.54\\\\\n 0 & 0   &  0  &  1       &  0     &  1  &   $B_{\\rm u}$        & 1504.872 & 1505.977 &1506.283 & 1505.634 & \\\\\n 0 & 0   &  0  &  2       &  0     &  1  &  $B_{\\rm u}$                  & 1853.634 & 1853.949 &1854.424 & 1855.305 & \\\\\n 0 & 0   &  0  &  0       &  0     &  0  &   $A_{\\rm u}$       &   11.437 &   11.014 &  10.997 & 11.289 & 11.28 \\\\\n 0 & 0   &  0  &  1       &  0     &  0  &     $A_{\\rm u}$      &  370.893 &  371.247 & 371.203 & 371.478& 371.32 \\\\\n 0 & 0   &  0  &  2       &  0     &  0  &    $A_{\\rm u}$       &  776.122 &  776.465 & 776.320 & 777.336 & 776.93 \\\\\n 0 & 0   &  1  &  0       &  0     &  0  &   $A_{\\rm u}$        &  877.934 &  877.094 & 877.200 &  877.303 &  \\\\\n 0 & 0   &  0  &  0       &  0     &  1  &  $B_{\\rm g}$           & 1285.121 & 1284.889 &1285.249 &  1285.457&  \\\\\n 0 & 0   &  0  &  1       &  0     &  1  &   $B_{\\rm g}$          & 1648.367 & 1648.553 &1649.012 &  1649.485&  \\\\\n 0 & 0   &  0  &  2       &  0     &  1  &   $B_{\\rm g}$           & 2072.404 & 2072.384 &2072.949 &  2074.231&   \\\\\n\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe vibrational ground state (left hand column) and the (000 100 $A_{\\rm g}$) state - (right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from  Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J1}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n1 & 0  & 1   &   1.71154 &    1.71152 &  0.00002 &   256.255 &  256.255 &  0.000 \\\\\n1 & 1  & 1   &  10.90677 &   10.9068  &  0.0000  &   265.427 &  265.427 &  0.000 \\\\\n1 & 1  & 0   &  10.9426  &   10.9426  &  0.0000  &   265.474 &  265.475 &  0.001 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n3 & 0  & 3   &  10.2683  &   10.2682  &  0.0001  &   264.777 &  264.777 &  0.000 \\\\\n3 & 1  & 3   &  19.374   &   19.374   &  0.000   &   273.830 &  273.831 & -0.001 \\\\\n3 & 1  & 2   &  19.589   &   19.589   &  0.000   &   274.119 &  274.117 &  0.002 \\\\\n3 & 2  & 2   &  47.115   &   47.115   &  0.000   &   301.555 &  301.556 &  0.001 \\\\\n3 & 2  & 1   &  47.115   &   47.115   &  0.000   &   301.556 &  301.557 &  0.001 \\\\\n3 & 3  & 1   &  93.155   &   93.155   &  0.000   &   347.509 &  347.512 &  0.003 \\\\\n3 & 3  & 0   &  93.155   &   93.155   &  0.000   &   347.509 &  347.512 &  0.003 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n5 & 0  & 5   &  25.6667  &   25.6664  &  0.0003  &   280.113 &  280.112 &  0.001 \\\\\n5 & 1  & 5   &  34.613   &   34.613   &  0.000   &   288.954 &  288.954 &  0.000 \\\\\n5 & 1  & 4   &  35.151   &   35.150   &  0.001   &   289.671 &  289.671 &  0.000 \\\\\n5 & 2  & 4   &  62.513   &   62.513   &  0.000   &   316.893 &  316.893 &  0.000 \\\\\n5 & 2  & 3   &  62.517   &   62.517   &  0.000   &   316.899 &  316.899 &  0.000 \\\\\n5 & 3  & 3   & 108.551   &  108.551   &  0.000   &   362.845 &  362.847 &  0.002 \\\\\n5 & 3  & 2   & 108.551   &  108.551   &  0.000   &   362.845 &  362.847 &  0.002 \\\\\n5 & 4  & 2   & 172.968   &  172.968   &  0.000   &   427.143 &  427.146 &  0.003 \\\\\n5 & 4  & 1   & 172.968   &  172.968   &  0.000   &   427.143 &  427.146 &  0.003 \\\\\n5 & 5  & 1   & 255.733   &  255.733   &  0.000   &   509.757 &  509.764 &  0.007 \\\\\n5 & 5  & 0   & 255.733   &  255.733   &  0.000   &   509.757 &  509.764 &  0.007 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (000 200    $A_{\\rm g}$) (left hand column) and the (000 300   $A_{\\rm g}$) state - right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n1 & 0  & 1   & 571.448   &  571.449   & -0.001   &  1002.584 & 1002.584 &  0.000 \\\\\n1 & 1  & 1   & 580.550   &  580.549   &  0.001   &  1011.664 & 1011.614 &  0.050 \\\\\n1 & 1  & 0   & 580.577   &  580.576   &  0.001   &  1011.678 & 1011.628 &  0.050 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n3 & 0  & 3   & 579.976   &  579.975   &  0.001   &  1011.098 & 1011.097 &  0.001 \\\\\n3 & 1  & 3   & 589.009   &  589.007   &  0.002   &  1020.143 & 1020.093 &  0.050 \\\\\n3 & 1  & 2   & 589.174   &  589.172   &  0.002   &  1020.225 & 1020.176 &  0.049 \\\\\n3 & 2  & 2   & 616.427   &  616.427   &  0.000   &  1047.246 & 1047.244 &  0.002 \\\\\n3 & 2  & 1   & 616.427   &  616.427   &  0.000   &  1047.246 & 1047.244 &  0.002 \\\\\n3 & 3  & 1   & 661.974   &  661.975   &  0.001   &  1092.462 & 1092.411 &  0.051 \\\\\n3 & 3  & 0   & 661.974   &  661.975   &  0.001   &  1092.462 & 1092.411 &  0.051 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n5 & 0  & 5   & 595.324   &  595.324   &  0.000   &  1026.421 & 1026.021 &  0.008 \\\\\n5 & 1  & 5   & 604.235   &  604.232   &  0.003   &  1035.404 & 1035.354 &  0.050 \\\\\n5 & 1  & 4   & 604.645   &  604.644   &  0.001   &  1035.610 & 1035.561 &  0.049 \\\\\n5 & 2  & 4   & 631.773   &  631.773   &  0.000   &  1062.566 & 1062.565 &  0.001 \\\\\n5 & 2  & 3   & 631.775   &  631.775   &  0.000   &  1062.566 & 1062.565 &  0.001 \\\\\n5 & 3  & 3   & 677.317   &  677.315   &  0.002   &  1107.779 & 1107.729 &  0.050 \\\\\n5 & 3  & 2   & 677.317   &  677.315   &  0.002   &  1107.779 & 1107.729 &  0.050 \\\\\n5 & 4  & 2   & 741.046   &  741.044   &  0.002   &  1170.935 & 1170.930 &  0.005 \\\\\n5 & 4  & 1   & 741.046   &  741.044   &  0.002   &  1170.935 & 1170.930 &  0.005 \\\\\n5 & 5  & 1   & 822.934   &  822.930   &  0.004   &  1252.191 & 1252.140 &  0.051 \\\\\n5 & 5  & 0   & 822.934   &  822.930   &  0.004   &  1252.191 & 1252.140 &  0.051 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (001 000    $A_{\\rm g}$) (left hand column) and the (000 000 $A_{\\rm u}$)\n state - right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from  Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n1 & 0  &  1  &  867.628  &  867.628   &  0.000   &    13.150 &   13.149 &  0.001 \\\\\n1 & 1  &  0  &  876.815  &  876.816   &  0.001   &    22.337 &   22.337 &  0.000 \\\\\n1 & 1  &  0  &  876.851  &  876.850   &  0.001   &    22.369 &   21.368 &  0.001 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n3 & 0  &  3  &  876.074  &  875.073   &  0.001   &    21.712 &   21.711 &  0.001 \\\\\n3 & 1  &  3  &  885.171  &  884.171   &  0.000   &    30.821 &   30.820 &  0.001 \\\\\n3 & 1  &  2  &  885.386  &  885.387   & -0.001   &    31.009 &   31.009 &  0.000 \\\\\n3 & 2  &  2  &  912.886  &  912.885   &  0.001   &    58.518 &   58.518 &  0.000 \\\\\n3 & 2  &  1  &  912.887  &  912.887   &  0.000   &    58.518 &   58.518 &  0.000 \\\\\n3 & 3  &  1  &  958.884  &  958.887   & -0.003   &   104.508 &  104.508 &  0.000 \\\\\n3 & 3  &  0  &  958.884  &  958.887   & -0.003   &   104.508 &  104.508 &  0.000 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n5 & 0  &  5  &  891.273  &  891.2712  &  0.002   &    37.121 &   37.120 &  0.001 \\\\\n5 & 1  &  5  &  900.210  &  900.209   &  0.001   &    46.089 &   46.088 &  0.001 \\\\\n5 & 1  &  4  &  900.749  &  900.748   &  0.001   &    46.561 &   46.560 &  0.001 \\\\\n5 & 2  &  4  &  928.085  &  928.085   &  0.000   &    73.926 &   73.926 &  0.000 \\\\\n5 & 2  &  3  &  928.089  &  928.088   &  0.001   &    73.929 &   73.928 &  0.001 \\\\\n5 & 3  &  3  &  974.080  &  974.082   & -0.002   &   119.913 &  119.913 &  0.000 \\\\\n5 & 3  &  2  &  974.080  &  974.082   & -0.002   &   119.913 &  119.913 &  0.000 \\\\\n5 & 4  &  2  & 1038.438  & 1038.440   & -0.002   &   184.260 &  184.261 & -0.001 \\\\\n5 & 5  &  1  & 1121.127  & 1121.133   & -0.006   &   266.936 &  266.938 & -0.002 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (000 100    $A_{\\rm u}$) (left hand column) and the (000 200    $A_{\\rm u}$)\nstate - (right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n1 & 0  &  0  &  372.601  &  372.602   &  0.001   &   777.826 &  777.827 & -0.001 \\\\\n1 & 1  &  1  &  381.733  &  381.733   &  0.000   &   786.881 &  786.891 & -0.010 \\\\\n1 & 1  &  1  &  381.764  &  381.764   &  0.000   &   786.901 &  786.911 & -0.010 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n3 & 0  &  3  &  381.140  &  381.139   &  0.001   &   786.350 &  786.350 &  0.000 \\\\\n3 & 1  &  3  &  390.195  &  390.195   &  0.000   &   795.355 &  795.365 & -0.010 \\\\\n3 & 1  &  2  &  390.378  &  390.376   &  0.002   &   795.472 &  795.482 & -0.010 \\\\\n3 & 2  &  2  &  417.723  &  417.722   &  0.001   &   822.639 &  822.640 & -0.001 \\\\\n3 & 2  &  1  &  417.723  &  417.722   &  0.001   &   822.639 &  822.640 & -0.001 \\\\\n3 & 3  &  1  &  463.434  &  463.434   &  0.000   &   867.976 &  867.986 & -0.010 \\\\\n3 & 3  &  0  &  463.434  &  463.434   &  0.000   &   867.976 &  867.986 & -0.010 \\\\\n  &    &     &           &            &          &           &          &        \\\\\n5 & 0  &  5  &  396.505  &  396.504   &  0.001   &   801.689 &  801.689 &  0.000 \\\\\n5 & 1  &  5  &  405.426  &  405.423   &  0.003   &   810.607 &  810.617 & -0.010 \\\\\n5 & 1  &  4  &  405.881  &  405.879   &  0.002   &   810.900 &  810.909 & -0.009 \\\\\n5 & 2  &  4  &  433.088  &  433.088   &  0.000   &   837.977 &  837.977 &  0.000 \\\\\n5 & 2  &  3  &  433.090  &  433.089   &  0.001   &   837.977 &  837.977 &  0.000 \\\\\n5 & 3  &  3  &  478.796  &  478.794   &  0.002   &   883.310 &  883.320 & -0.010 \\\\\n5 & 3  &  2  &  478.796  &  478.794   &  0.002   &   883.310 &  883.320 & -0.010 \\\\\n5 & 4  &  2  &  542.756  &  542.756   &  0.000   &   946.769 &  946.770 &  0.001 \\\\\n5 & 5  &  1  &  624.938  &  624.937   &  0.001   &  1028.289 & 1028.299 & -0.010 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for $J=30$. Results are for\nthe ground vibrational state (left hand column) and the (000 100    $A_{\\rm g}$)\nstate - (right hand column). Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J30}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n\\hline\n30 & 0   & 30  &  789.577 &  789.581 & -0.004 & 1038.983 & 1038.925 &  0.058 \\\\\n30 & 1   & 30  &  793.053 &  793.065 & -0.012 & 1041.540 & 1041.481 &  0.039 \\\\\n30 & 1   & 29  &  809.594 &  809.565 &  0.029 & 1063.053 & 1063.007 &  0.046 \\\\\n30 & 2   & 29  &  829.292 &  829.282 &  0.010 & 1080.396 & 1080.345 &  0.051 \\\\\n30 & 2   & 28  &  832.547 &  832.521 &  0.026 & 1086.079 & 1086.036 &  0.043 \\\\\n30 & 3   & 28  &  876.030 &  876.017 &  0.013 & 1127.862 & 1127.815 &  0.047 \\\\\n30 & 3   & 27  &  876.191 &  876.174 &  0.017 & 1128.304 & 1128.262 &  0.042 \\\\\n30 & 4   & 27  &  940.027 &  940.015 &  0.012 & 1191.838 & 1191.794 &  0.044 \\\\\n30 & 4   & 26  &  940.029 &  940.013 &  0.016 & 1191.850 & 1191.807 &  0.043 \\\\\n30 & 5   & 26  & 1022.323 & 1022.311 &  0.012 & 1274.332 & 1274.293 &  0.039 \\\\\n30 & 5   & 25  & 1022.324 & 1022.312 &  0.012 & 1274.332 & 1274.293 &  0.039 \\\\\n30 & 6   & 25  & 1122.848 & 1122.837 &  0.011 & 1371.047 & 1371.024 &  0.023 \\\\\n30 & 6   & 24  & 1122.849 & 1122.838 &  0.011 & 1371.047 & 1371.024 &  0.023 \\\\\n30 & 7   & 24  & 1241.304 & 1241.294 &  0.010 & 1491.957 & 1491.933 &  0.024 \\\\\n30 & 7   & 23  & 1241.304 & 1241.294 &  0.010 & 1491.957 & 1491.933 &  0.024 \\\\\n30 & 8   & 23  & 1381.938 & 1381.916 &  0.022 & 1628.749 & 1628.731 &  0.018 \\\\\n30 & 9   & 21  & 1534.588 & 1534.578 &  0.010 & 1782.735 & 1782.739 & -0.004 \\\\\n30 & 10  & 21  & 1707.358 & 1707.349 &  0.009 & 1958.577 & 1958.558 &  0.019 \\\\\n30 & 11  & 19  & 1898.169 & 1898.159 &  0.010 & 2148.182 & 2148.186 & -0.004 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for $J=30$ levels of the (000 000    $A_{\\rm u}$)\nvibrational state. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     \\\\\n\\hline\n30 & 0   & 30  &  802.349 &  802.340 &  0.009         \\\\\n30 & 1   & 30  &  806.280 &  806.276 &  0.004         \\\\\n30 & 1   & 29  &  820.768 &  820.746 &  0.022         \\\\\n30 & 2   & 29  &  841.344 &  841.330 &  0.013         \\\\\n30 & 2   & 28  &  843.911 &  843.892 &  0.019         \\\\\n30 & 3   & 28  &  887.854 &  887.839 &  0.015         \\\\\n30 & 3   & 27  &  887.964 &  887.949 &  0.015         \\\\\n30 & 4   & 27  &  951.826 &  951.815 &  0.011         \\\\\n30 & 4   & 26  &  951.828 &  951.817 &  0.011         \\\\\n30 & 5   & 26  & 1034.096 & 1034.085 &  0.011         \\\\\n30 & 5   & 25  & 1034.096 & 1034.085 &  0.011         \\\\\n30 & 5   & 25  & 1134.609 & 1134.599 &  0.010         \\\\\n30 & 5   & 24  & 1253.291 & 1253.284 &  0.007         \\\\\n30 & 5   & 24  & 1390.060 & 1390.055 &  0.005         \\\\\n30 & 5   & 23  & 1544.803 & 1544.803 &  0.000         \\\\\n30 & 5   & 23  & 1717.164 & 1717.167 & -0.003         \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{   Variationally calculated and  observed or predicted\nusing effective Hamiltonian)  energy levels, in \\cm, for $J=35$. Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J35}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n\\hline\n35 & 0   & 35  & 1067.027 & 1067.037 & -0.010   & 1314.205 & 1314.124 &  0.081  \\\\\n35 & 1   & 35  & 1069.466 & 1069.484 & -0.018   & 1315.869 & 1315.787 &  0.082  \\\\\n35 & 1   & 34  & 1091.715 & 1091.680 &  0.035   & 1344.395 & 1344.338 &  0.057  \\\\\n35 & 2   & 34  & 1108.893 & 1108.882 &  0.011   & 1358.709 & 1358.637 &  0.052  \\\\\n35 & 2   & 33  & 1114.564 & 1114.523 &  0.041   & 1368.256 & 1368.203 &  0.053  \\\\\n35 & 3   & 33  & 1156.257 & 1156.241 &  0.016   & 1407.374 & 1407.309 &  0.065  \\\\\n35 & 3   & 32  & 1156.655 & 1156.628 &  0.027   & 1408.434 & 1408.382 &  0.052  \\\\\n35 & 4   & 32  & 1220.062 & 1220.044 &  0.018   & 1471.314 & 1471.252 &  0.062  \\\\\n35 & 4   & 31  & 1220.070 & 1220.051 &  0.019   & 1471.355 & 1471.296 &  0.059  \\\\\n35 & 5   & 31  & 1302.102 & 1302.088 &  0.014   & 1553.875 & 1553.821 &  0.054  \\\\\n35 & 5   & 30  & 1302.114 & 1302.098 &  0.016   & 1553.876 & 1555.822 &  0.054  \\\\\n35 & 6   & 30  & 1402.366 & 1402.353 &  0.013   & 1648.318 & 1648.288 &  0.030  \\\\\n35 & 6   & 29  & 1402.369 & 1402.350 &  0.019   & 1648.318 & 1648.288 &  0.030  \\\\\n35 & 7   & 29  & 1520.294 & 1520.282 &  0.012   & 1769.879 & 1769.844 &  0.035  \\\\\n35 & 8   & 28  & 1662.739 & 1662.707 &  0.032   & 1906.483 & 1906.459 &  0.024  \\\\\n35 & 9   & 27  & 1814.222 & 1814.207 &  0.015   &          &          &         \\\\\n35 & 10  & 26  & 1986.565 & 1986.552 &  0.013   &          &          &         \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{   Variationally calculated and  observed or predicted using effective Hamiltonian)  energy levels, in \\cm, for $J=35$}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$  & $K_a$ & $K_c$ &    Obs   &    Calc    &  o-c     &   Obs     &   Calc   &  o-c   \\\\\n\\hline\n35 & 0   & 35  & 1933.163 & 1932.118 &  0.045   & 1080.587 & 1080.574 &  0.013  \\\\\n35 & 1   & 35  & 1936.042 & 1935.996 &  0.046   & 1083.472 & 1083.465 &  0.007  \\\\\n35 & 1   & 34  & 1955.496 & 1955.415 &  0.081 u & 1102.961 & 1102.929 &  0.032  \\\\\n35 & 2   & 34  & 1973.810 & 1973.743 &  0.067 u & 1121.270 & 1121.247 &  0.023  \\\\\n35 & 2   & 33  & 1978.296 & 1978.228 &  0.068 u & 1125.774 & 1125.743 &  0.031  \\\\\n35 & 3   & 33  & 2020.807 & 2020.749 &  0.058 u & 1168.268 & 1168.245 &  0.023  \\\\\n35 & 3   & 32  & 2021.078 & 2021.020 &  0.058 u & 1168.540 & 1168.517 &  0.023  \\\\\n35 & 4   & 32  & 2084.733 & 2084.691 &  0.042 u & 1232.076 & 1232.060 &  0.016  \\\\\n35 & 4   & 31  & 2084.738 & 2084.768 &  0.042 u & 1232.081 & 1232.055 &  0.026  \\\\\n35 & 5   & 31  & 2165.503 & 2165.420 &  0.083 u & 1314.156 & 1314.138 &  0.018  \\\\\n35 & 5   & 30  & 2165.503 & 2165.420 &  0.083 u & 1314.156 & 1314.138 &  0.018  \\\\\n   &     &     &          &          &          & 1414.475 & 1414.459 &  0.016  \\\\\n   &     &     &          &          &          & 1532.940 & 1532.928 &  0.012  \\\\\n   &     &     &          &          &          & 1669.447 & 1669.438 &  0.009  \\\\\n   &     &     &          &          &          & 1823.841 & 1823.839 &  0.002  \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\bibliographystyle{rsc}\n\n\\providecommand*{\\mcitethebibliography}{\\thebibliography}\n\\csname @ifundefined\\endcsname{endmcitethebibliography}\n{\\let\\endmcitethebibliography\\endthebibliography}{}\n\\begin{mcitethebibliography}{69}\n\\providecommand*{\\natexlab}[1]{#1}\n\\providecommand*{\\mciteSetBstSublistMode}[1]{}\n\\providecommand*{\\mciteSetBstMaxWidthForm}[2]{}\n\\providecommand*{\\mciteBstWouldAddEndPuncttrue}\n  {\\def\\EndOfBibitem{\\unskip.}}\n\\providecommand*{\\mciteBstWouldAddEndPunctfalse}\n  {\\let\\EndOfBibitem\\relax}\n\\providecommand*{\\mciteSetBstMidEndSepPunct}[3]{}\n\\providecommand*{\\mciteSetBstSublistLabelBeginEnd}[3]{}\n\\providecommand*{\\EndOfBibitem}{}\n\\mciteSetBstSublistMode{f}\n\\mciteSetBstMaxWidthForm{subitem}\n{(\\emph{\\alph{mcitesubitemcount}})}\n\\mciteSetBstSublistLabelBeginEnd{\\mcitemaxwidthsubitemform\\space}\n{\\relax}{\\relax}\n\n\\bibitem[Davis(1974)]{74Davis.H2O2}\nD.~Davis, \\emph{Can. 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+{"text":"\\section{Introduction and Summary}\nWitten diagrams are the AdS analogue of position space Feynman diagrams in flat space, and appear as the building blocks in the holographic computation of conformal correlators in the supergravity limit. These diagrams were extensively studied  in the early days of AdS\/CFT in position space, but their simplicity was only later appreciated in Mellin space \\cite{Mack:2009mi,Penedones:2010ue,Paulos:2011ie,Fitzpatrick:2011ia}. Recently there has been another resurgence of interest in the community to revisit the Witten diagrams.\n The renewed interest is partly due to the curious appearance of these diagrams in the Mellin bootstrap method \\cite{Gopakumar:2016wkt,Gopakumar:2016cpb,Gopakumar:2018xqi}, where the tree-level exchange Witten diagrams (modulo certain ambiguities in adding contact terms) are used as an expansion basis for conformal correlators. Relatedly, the usual conformal blocks admit a natural AdS bulk description in terms of a variation of the exchange Witten diagrams. They are the so-called geodesic Witten diagrams where the integration regions of the cubic vertex points are restricted to the AdS geodesics connecting the boundary operators \\cite{Hijano:2015zsa}.\\footnote{See also \\cite{Rastelli:2017ecj,Goncalves:2018fwx} for generalization of geodesic Witten diagrams to CFTs with conformal boundaries and defects.}  Moreover, one is also motivated to revisit these diagrams because of the remarkable simplicity recently discovered in the  holographic one-half BPS four-point functions \\cite{Rastelli:2016nze,Alday:2017xua,Aprile:2017bgs,Aprile:2017xsp,Rastelli:2017udc,Alday:2017vkk,Aprile:2017qoy,Rastelli:2017ymc,Zhou:2017zaw,Zhou:2018ofp,Aprile:2018efk,Caron-Huot:2018kta,Alday:2018pdi}.\\footnote{Modern methods have been recently invented to efficiently compute  holographic correlators at the tree level. See \\cite{Rastelli:2016nze,Rastelli:2017udc,Rastelli:2017ymc,Zhou:2017zaw,Zhou:2018ofp} for bootstrap-like methods of computing holographic correlators which does not require the detailed knowledge of the effective Lagrangian. See also \\cite{Arutyunov:2017dti,Arutyunov:2018neq,Arutyunov:2018tvn} for an improved version of the original algorithm and \\cite{Caron-Huot:2018kta} for a method based on the inversion formula \\cite{Caron-Huot:2017vep} and crossing symmetry.} In particular an interesting $SO(10,2)$ symmetry \\cite{Caron-Huot:2018kta}  was shown to exist in the general formula for one-half BPS four-point functions from tree-level IIB supergravity in $AdS_5\\times S^5$ \\cite{Rastelli:2016nze,Rastelli:2017udc}, unifiying the correlators of all Kaluza-Klein modes into a single ten-dimensional object. The simplicity in these results requires remarkable conspiracy of individual Witten diagrams, and some of its aspects still remain to be better understood.  Finally, one is led to study Witten diagrams from the reconstruction of AdS physics using CFT principles \\cite{Heemskerk:2009pn}. Witten diagrams emerge as solutions to the crossing equation, at tree level \\cite{Heemskerk:2009pn,Alday:2017gde,Li:2017lmh} as well as at  loop level \\cite{Aharony:2016dwx}. \n \n However, some basic properties of Witten diagrams still remain to be better understood, even at tree level. In particular, the following seemingly simple problem still appears to lack a satisfactory solution: how do we perform the conformal block decomposition of a tree-level exchange Witten diagram in the crossed channel? In the crossed channel an exchange Witten diagram is known to decompose into two towers of double-trace conformal blocks.\\footnote{Here we assume that the external conformal dimensions are generic such that the spectra of the two towers of double-trace operators do not overlap. In the degenerate case, both conformal blocks and their derivative with respect to the conformal dimension will appear in the expansion. We will have more discussions on this point later in the paper.} However obtaining the crossed channel OPE coefficients turns out to be very non-trivial, and no method is currently available to extract efficiently {\\it all} the coefficients. This should be contrasted with the decomposition in the direct channel, where it contains a single-trace conformal block and infinitely many double-trace blocks with bounded spins. The problem in the direct channel is much easier and can be solved using a variety of methods. For example one can obtain all the decomposition coefficients in closed forms using the split representation of propagators \\cite{Costa:2014kfa}, or the geodesic Witten diagram techniques \\cite{Hijano:2015zsa}.\\footnote{In Section \\ref{Secdirectchan} we will offer another approach to obtain these coefficients from studying the contact diagrams related to the exchange diagram.} \n \n \n\n \n A number of recent papers have appeared that revisit this problem, and the methods have both advantages and disadvantages. In  \\cite{Liu:2018jhs,Cardona:2018dov} methods based on the Lorentizian inversion formula \\cite{Caron-Huot:2017vep} (see also \\cite{Simmons-Duffin:2017nub,Kravchuk:2018htv}) are introduced, which allow one to extract the crossed channel CFT data for operators with spins greater than the spin of the exchanged single-trace operator. However the inversion formula is not valid for lower spins due to the Regge behavior of the Witten diagrams, and one therefore cannot use this method to probe the rest of the operators. This difficulty is absent in Mellin space \\cite{Mack:2009mi,Penedones:2010ue}. The OPE coefficients of the double-trace operators with the leading conformal twist can be obtained by taking the residue of the Mellin amplitude at the leading double-trace pole, and then projecting the residue into  continuous Hahn polynomials of different spins \\cite{Costa:2014kfa,Gopakumar:2016cpb,Sleight:2018epi,Sleight:2018ryu,Gopakumar:2018xqi} -- analogues to projecting flat space amplitudes into partial waves using Gegenbaur polynomials. But in this approach one encounters a different difficulty in obtaining the OPE coefficients of double-trace operators with sub-leading twists. The residue of the Mellin amplitude at a sub-leading double-trace pole receives contributions from both the double-trace primary operators, as well as the conformal descendants of the double-trace operators whose twists are smaller. Therefore there is a mixing of contributions between the two. The degeneracy problem must be first solved in order to extract the OPE coefficients. One method of disentangling the contributions was suggested in \\cite{Sleight:2018epi,Sleight:2018ryu}.   \nIn this method one first acts on the correlator with a quartic differential operator and then repeats the Mellin space analysis of performing the projections on the residue. The differential operator can be chosen such that its kernel contains the conformal blocks of the leading double-trace twist  \\cite{Alday:2016njk}, the sub-leading double-trace operators therefore become leading in the new correlator. Unfortunately, the complexity of this algorithm quickly grows upon increasing the twists of the double-trace operators. Applying this method to low orders also reveals few general patterns.\n\nIn this paper, we will attack the problem from a different angle. The main result of our analysis is  a set of simple constraining linear relations satisfied by the crossed channel decomposition coefficients. These linear relations will lead us to a recursive algorithm for solving the coefficients. In our analysis we emphasize the pivotal role played by the contact Witten diagrams, and highlight the importance of an ``equation of motion'' operator which relates an exchange Witten diagram to a sum of contact Witten diagrams. More precisely, the equation of motion operator is given by the quadratic conformal Casimir operator in the exchange channel with a constant shift, and is closely related to the equation of motion for the bulk-to-bulk propagator. The direct consequence of this relation between these two types of diagrams is that the direct channel decomposition coefficients of an exchange Witten diagram are completely fixed in terms of the coefficients of the related contact Witten diagrams. This relation has important implications in the crossed channel too. Using properties of conformal blocks \\cite{Dolan:2011dv}\\footnote{See also \\cite{Karateev:2017jgd}.}, we find that the equation of motion operator admits simple actions on conformal blocks. In one dimension, the action of this operator on a conformal block with dimension $\\Delta$ produces three conformal blocks with new dimensions $\\Delta-1$, $\\Delta$ and $\\Delta+1$. In higher dimensions, the three-term relation becomes a five-term one. The action of the  operator  on a conformal block with dimension $\\Delta$ and spin $\\ell$  contains the original conformal block, as well as four other conformal blocks with shifted quantum numbers $(\\Delta\\mp1,\\ell\\pm1)$, $(\\Delta\\pm1,\\ell\\mp1)$. Moreover, the coefficients of the three-term and five-term relations vanish for conformal blocks of unphysical double-trace operators\\footnote{These are the operators with conformal dimensions $\\Delta=\\Delta_1+\\Delta_2+\\ell-1, \\Delta_3+\\Delta_4+\\ell-1$ or negative spins $\\ell=-1$ for $d>1$, and $\\Delta=\\Delta_1+\\Delta_2-1,\\Delta_3+\\Delta_4-1$ for $d=1$.}. These zeros of the coefficients guarantee that  when we restrict $(\\Delta,\\ell)$ to be those of the double-trace operators, the spectrum is preserved after applying these relations. The simple action of the equation of motion operator on double-trace conformal blocks makes it possible to formulate a recursive algorithm for solving the crossed channel decomposition coefficients. Thanks to the equation of motion identity, these crossed channel coefficients satisfy simple linear equations with the decomposition coefficients of the contact Witten diagrams as inhomogeneous terms. We can solve the linear equations recursively, in terms of certain seed decomposition coefficients. In one dimension, the seed coefficients are just the OPE coefficients of the double-trace operators with the lowest conformal dimension. In higher dimensions, the seed coefficients are the OPE coefficients of the leading twist double-trace operators. Therefore the recursion relations give us a very efficient way to obtain OPE coefficients of sub-leading double-trace operators. \n\nThe decomposition of exchange Witten diagrams in the crossed channel is also closely related to the $6j$ symbol (or the crossing kernel) of the conformal group. In the crossed channel, a conformal partial wave is decomposed into infinitely many double-trace conformal blocks. The various decomposition coefficients of the double-trace operators can be viewed as the residues of the $6j$ symbol  \\cite{Liu:2018jhs}. Because a conformal partial wave can be identified with the difference of two exchange Witten diagrams with opposite quantizations \\cite{Penedones:2007ns,Costa:2014kfa,Giombi:2018vtc}, our analysis of the Witten diagrams extends easily to conformal partial waves. \n\n\n\n\n\nThe paper is organized as follows. We start by defining the Witten diagrams in Section \\ref{SecWD} and introducing the equation of motion operator in \\ref{SecExtoCon}. In Section \\ref{CPW} we review some basic facts of conformal partial waves. In Section \\ref{SecCBdecompContact} we discuss the conformal block decomposition of contact Witten diagrams. We show that the decomposition of a generic contact diagram can always be recursively reduced to the simplest contact diagram with zero derivatives in the quartic vertex. In Section \\ref{Secdirectchan} we show how the direct channel decomposition of exchange Witten diagrams can be fixed by the decomposition of the relevant contact Witten diagrams. In Section \\ref{SecRecurcross}, we discuss the crossed channel decomposition of exchange Witten diagrams and conformal partial waves. We first outline the strategy in Section \\ref{SecStrat}. We discuss the simpler problem in $\\mathrm{CFT}_1$ in Section \\ref{1d}, and then extend the story to $\\mathrm{CFT}_d$ in Section \\ref{higherd}. We end with a brief discussion in Section \\ref{SecDiscuss}. Further technical details are relegated to the three appendices. In Appendix \\ref{appcontact} we make further comments on the contact Witten diagrams. In Appendix \\ref{appeqweight} we discuss the special case where the external conformal dimensions are degenerate. In Appendix \\ref{appseed} we discuss how to compute the seed coefficients for $AdS_2$ exchange Witten diagrams.   \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Witten Diagrams and Conformal Partial Waves}\n\\subsection{Witten Diagrams}\\label{SecWD}\nIn this paper we study tree-level four-point Witten diagrams, {\\it i.e.}, contact Witten diagrams and exchange Witten diagrams. We focus on scalar Witten diagrams where the external operators have zero spins and conformal dimensions $\\Delta_i$, $i=1,2,3,4$. This restricts the operator in the internal line of the exchange Witten diagram to be in the rank-$\\ell_E$ symmetric-traceless representation under the Lorentz group. We will denote the conformal dimension of the exchange operator by $\\Delta_E$. After making a choice for the cubic and quartic vertices, the Witten diagrams are built from the scalar bulk-to-boundary propagators $G_{B\\partial}^{\\Delta_i}(z,\\vec{x}_i)$, and the spin-$\\ell_E$ bulk-to-bulk propagator $\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)$. The scalar bulk-to-boundary propagator is explicitly given by \\begin{equation}\nG_{B\\partial}^{\\Delta_i}(z,\\vec{x}_i)=\\left(\\frac{z_0}{z_0^2+(\\vec{z}-\\vec{x}_i)^2}\\right)^{\\Delta_i}\n\\end{equation}\nwhere the dimension $\\Delta_i$ is associated to the scalar field mass in $AdS_{d+1}$ via $M_i^2=\\Delta_i(\\Delta_i-d)$. The spin-$\\ell_E$ bulk-to-bulk propagators are defined to satisfy the equation of motion and have vanishing divergence\n\\begin{equation}\n(\\bigtriangledown_1^2-M_E^2)\\,\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)=-g^{\\mu_1\\{\\nu_1}\\ldots g^{|\\mu_{\\ell_E}|\\nu_{\\ell_E}\\}}\\delta(z_1,z_2)+\\ldots\\;,\\label{EOMofBtoBprop}\n\\end{equation}\n\\begin{equation}\n\\bigtriangledown_1^{\\mu_1}\\,\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)=0+\\ldots\\;,\\label{divergenceless}\n\\end{equation}\nup to local source terms denoted by $\\ldots$. These terms introduce ambiguities to the exchange Witten diagrams\\footnote{These terms will only change the exchange Witten diagram by contact Witten diagrams \\cite{Costa:2014kfa}.}, but such ambiguities are not important for the propagating degrees of freedom. The squared mass of the bulk field is given by\n\\begin{equation}\nM_E^2=\\Delta_E(\\Delta_E-d)-\\ell_E\\;.\n\\end{equation}\nAn explicit example of the bulk-to-bulk propagator is given by that of  a scalar field\n\\begin{equation}\\small\n\\Pi^{\\Delta_E}(z_1,z_2)=\\frac{\\Gamma(\\Delta_E)}{2\\pi^{\\frac{d}{2}}\\Gamma(\\Delta_E-\\frac{d}{2}+1)}u^{-\\Delta}{}_2F_1\\left(\\Delta_E,\\frac{2\\Delta_E-d+1}{2},2\\Delta_E-d+1,-\\frac{4}{u}\\right)\n\\end{equation}\nwhere\n\\begin{equation}\nu=\\frac{(z_1-z_2)^2}{z_{10}z_{20}}\\;.\n\\end{equation}\n\nA generic contact Witten diagram is built in terms of the bulk-to-boundary propagators only\n\\begin{equation}\\label{Wcontact}\nW^{contact}=\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^4 (\\bigtriangledown^{\\mu})^{j_i}G^{\\Delta_i}_{B\\partial}(z,x_i)\\;.\n\\end{equation}\nIn this formula, $(\\bigtriangledown^{\\mu})^{j_i}$ is schematic for the product of covariant derivatives $\\bigtriangledown^{\\mu_1}\\ldots \\bigtriangledown^{\\mu_{j_i}}$, and the indices are appropriately contracted on the RHS of (\\ref{Wcontact}). Such a contact diagram arises from a quartic AdS contact vertex\n\\begin{equation}\n(\\bigtriangledown^{\\mu})^{j_1}\\phi_1(\\bigtriangledown^{\\mu})^{j_2}\\phi_2(\\bigtriangledown^{\\mu})^{j_3}\\phi_3(\\bigtriangledown^{\\mu})^{j_4}\\phi_4\n\\end{equation}\nwhich contains a total number of $j_1+j_2+j_3+j_4$ derivatives. When $j_1=j_2=j_3=j_4=0$, we have the simplest contact diagram which is denoted in the literature as a $D$-function\n\\begin{equation}\\label{Dfunction}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^4 G^{\\Delta_i}_{B\\partial}(z,x_i)\\;.\n\\end{equation} \n\nWe also define a spin-$\\ell_E$ exchange Witten diagram (in the s-channel) as\n\\begin{equation}\\label{sWitten}\n\\begin{split}\nW^{s,\\, exchange}_{\\Delta_E,\\ell_E}={}&\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}\\frac{d^{d+1}z_2}{z_{20}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)((\\bigtriangledown^{\\mu})^{\\ell_E}G^{\\Delta_2}_{B\\partial}(z_1,x_2))\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)\\\\\n{}&\\times G^{\\Delta_3}_{B\\partial}(z_2,x_3) ((\\bigtriangledown^{\\nu})^{\\ell_E}G^{\\Delta_4}_{B\\partial}(z_2,x_4))\n\\end{split}\n\\end{equation}\nwhere we assumed the cubic couplings are \n\\begin{equation}\n\\phi_1\\bigtriangledown^{\\mu_1}\\ldots\\bigtriangledown^{\\mu_{\\ell_E}}\\phi_2 h_{\\mu_1\\ldots\\mu_{\\ell_E}}\\;,\\quad\\text{and}\\quad \\phi_1\\bigtriangledown^{\\mu_1}\\ldots\\bigtriangledown^{\\mu_{\\ell_E}}\\phi_2 h_{\\mu_1\\ldots\\mu_{\\ell_E}}\\;.\n\\end{equation}\nOther distributions of the derivatives in the cubic vertices can be obtained from the above choice by using integration by parts. Because of (\\ref{EOMofBtoBprop}) and (\\ref{divergenceless}), the different choices of the cubic vertices will only affect $W^{s,\\, exchange}$ by a finite number of contact diagrams.\n\nThe main goal of this paper is to study the conformal block decomposition of these diagrams. From the Mellin representation of Witten diagrams \\cite{Penedones:2010ue,Paulos:2011ie,Fitzpatrick:2011ia,Costa:2012cb}, it is clear that a contact Witten diagram (\\ref{Wcontact}) decomposes only into double-trace operators. For example, in the s-channel the decomposition reads\\footnote{In this paper we will abuse the terminology by calling the coefficients in front of the conformal blocks, such as $a^{12}_{n,J}$, $a^{34}_{n,J}$, the ``OPE coefficients''. }\n\\begin{equation}\\label{Wcontactins}\nW^{contact}(x_i)=\\sum_{J=0}^{J_{\\rm max}}\\sum_{n=0}^\\infty a^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{J_{\\rm max}}\\sum_{n=0}^\\infty a^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nNotice that the spin of the conformal blocks has a finite support $0\\leq J\\leq J_{\\rm max}\\leq j_1+j_2+j_3+j_4$. \n\nDecomposing an exchange Witten diagram into the direct channel, one finds a single-trace conformal block and infinitely many double-trace blocks\n\\begin{equation}\\label{Wexchangeins}\nW^{s,\\,exchange}_{\\Delta_E,\\ell_E}=A\\, g^{(s)}_{\\Delta_E,\\ell_E}(x_i) +\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty A^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty A^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nAgain, the support of spins is finite. In contrast, when we decompose an exchange Witten diagram into the crossed channel ({\\it i.e.}, the t-channel and u-channel), we find only double-trace operators of which the spins are unbounded\n\\begin{equation}\\label{Wexchangeint}\nW^{s,\\,exchange}_{\\Delta_E,\\ell_E}=\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty B^{14}_{n,J} g^{(t)}_{\\Delta_1+\\Delta_4+2n+J,J}(x_i)+\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty B^{23}_{n,J} g^{(t)}_{\\Delta_2+\\Delta_3+2n+J,J}(x_i)\\;.\n\\end{equation}\nThe above discussion is for {\\it generic} external conformal dimensions. When $\\Delta_1+\\Delta_2-\\Delta_3-\\Delta_4\\in 2\\mathbb{Z}$, we will also encounter derivative conformal blocks $\\partial_\\Delta g^{(s)}_\\Delta(x_i)$ in the s-channel.\\footnote{\\label{FNderivativeblock}We can understand this fact from just large $N$ counting. For concreteness we use the counting of 4d $\\mathcal{N}=4$ SYM, then the tree level Witten diagrams are all of order $\\mathcal{O}(1\/N^2)$. If $\\Delta_1+\\Delta_2- \\Delta_3-\\Delta_4\\neq 2\\mathbb{Z}$, there is no order $\\mathcal{O}(1)$ overlap between the double-trace spectra of operators $:O_1\\square^{n_{12}}\\partial^{\\ell_{12}}O_2:$ and operators $:O_3\\square^{n_{34}}\\partial^{\\ell_{34}}O_4:$. The double trace operators $:O_1\\square^{n_{12}}\\partial^{\\ell_{12}}O_2:$ therefore can only appear in the $O_3\\times O_4$ OPE with a suppression power of $1\/N^2$ (the same for $:O_3\\square^{n_{34}}\\partial^{\\ell_{34}}O_4:$ to appear in the $O_1\\times O_2$ OPE). The conformal dimensions of the double-trace operators are corrected at order $\\mathcal{O}(1\/N^2)$. But their effect is invisible in the tree diagrams because the correction is of order $\\mathcal{O}(1\/N^4)$ due to the $\\mathcal{O}(1\/N^2)$ suppression in the OPE coefficients. On the other hand, when $\\Delta_1+\\Delta_2- \\Delta_3-\\Delta_4$ is an even integer (which we can further assume to be non negative), the double-trace operators with twists $\\tau\\geq\\Delta_1+\\Delta_2$ appear in both OPEs with $\\mathcal{O}(1)$ coefficients. The correction due to the anomalous dimensions of these operators are therefore now visible. We can view the appearance of derivative double-trace blocks $\\partial g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}$ as expanding the anomalous dimension in $g^{(s)}_{\\Delta_1+\\Delta_2+2n+J+\\frac{1}{N^2}\\gamma_{n,J},J}$ to $\\mathcal{O}(1\/N^2)$.\n } Similarly,  when $\\Delta_1+\\Delta_4-\\Delta_2-\\Delta_3$ or $\\Delta_1+\\Delta_3-\\Delta_2-\\Delta_4$ is an even integer, there will be derivative conformal blocks in the t or u-channel. \n\nFor $d=1$, the decomposition of Witten diagrams has the same qualitative features, except that there is no spin. In the s-channel decomposition, we have\n\\begin{equation}\\label{Wcontactins1d}\nW^{contact}(x_i)=\\sum_{n=0}^\\infty a^{12}_{n} g^{(s)}_{\\Delta_1+\\Delta_2+2n}(x_i)+\\sum_{n=0}^\\infty a^{34}_{n} g^{(s)}_{\\Delta_3+\\Delta_4+2n}(x_i)\\;,\n\\end{equation}\n\\begin{equation}\\label{Wexchangeins1d}\nW^{s,exchange}(x_i)=Ag^{(s)}_\\Delta(x_i)+\\sum_{n=0}^\\infty A^{12}_{n} g^{(s)}_{\\Delta_1+\\Delta_2+2n}(x_i)+\\sum_{n=0}^\\infty A^{34}_{n} g^{(s)}_{\\Delta_3+\\Delta_4+2n}(x_i)\\;,\n\\end{equation}\nand only double-trace operators with one ``parity'' will show up, {\\it i.e.}, $\\Delta_1+\\Delta_2+n$ and $\\Delta_3+\\Delta_4+n$ with $n$ even. In the t-channel decomposition of the exchange Witten diagram,\n\\begin{equation}\\label{Wexchangeint1d}\nW^{s,\\,exchange}_{\\Delta_E}=\\sum_{n=0}^\\infty B^{14}_{n} g^{(t)}_{\\Delta_1+\\Delta_4+n}(x_i)+\\sum_{n=0}^\\infty B^{23}_{n} g^{(t)}_{\\Delta_2+\\Delta_3+n}(x_i)\\;,\n\\end{equation}\ndouble-trace operators of both parities will appear, {\\it i.e.}, $n\\in \\mathbb{Z}$.\n\n\\subsection{Relating Exchange Diagrams to Contact Diagrams}\\label{SecExtoCon}\nThe exchange Witten diagrams are related  to the contact Witten diagrams by the a second order differential operator, as a result of the fact that the bulk-to-bulk propagators are Green's functions in AdS. To see this explicitly, let us first focus on the $z_1$ integral inside\n the s-channel exchange Witten diagram (\\ref{sWitten})\\begin{equation}\nI^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}(x_1,x_2;z_2)=\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)((\\bigtriangledown^{\\mu})^{\\ell_E}G^{\\Delta_2}_{B\\partial}(z_1,x_2))\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)\\;.\n\\end{equation}\nThis integral is manifestly invariant under $SO(d,2)$, and therefore satisfies the identity \n\\begin{equation}\n(\\mathbf{L}_1+\\mathbf{L}_2+\\mathfrak{L}_{z_2})_{AB}\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}(x_1,x_2;z_2)=0\\;.\n\\end{equation}\nHere $\\mathbf{L}_1$ and $\\mathbf{L}_2$ are the conformal generators which act on $x_1$ and $x_2$, and $\\mathfrak{L}_{z_2}$ is the $AdS_{d+1}$ isometry generator which acts on a spin-$\\ell_E$ field at $z_2$. Using this identity twice, we obtain the following action of the conformal Casimir operator with respect to $x_1$ and $x_2$\n\\begin{equation}\n-\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}=-\\frac{1}{2}\\mathfrak{L}_{z_2}^2\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}=(\\bigtriangledown^2_{z_2}+\\ell_E(\\ell_E+d-1))I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}\\;.\n\\end{equation}\nIn the second equality, we have used that the conformal Casimir is equal to the Laplacian up to a constant shift $\\ell_E(\\ell_E+d-1)$ \\cite{Pilch:1984xx}. We now apply the equation of motion (\\ref{EOMofBtoBprop}) to get rid of $\\bigtriangledown^2_{z_2}$ that acts on the bulk-to-bulk propagator, and perform the $z_2$ integral. We find the expression gives the following relation between an exchange diagram and a sum of contact Witten diagrams\n\\begin{equation}\\label{EOMWexWcon}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right]W^{s,\\, exchange}_{\\Delta_E,\\ell_E}=\\sum_I c_IW^{contact}_I\\;.\n\\end{equation}\nHere $\\sum_I c_IW^{contact}_I$ is a collection of contact Witten diagrams, obtained by replacing the bulk-to-bulk propagator with the RHS of (\\ref{EOMofBtoBprop}).\\footnote{One might worry about the contact term ambiguities in defining an exchange Witten diagram, introduced by the ``\\ldots'' in (\\ref{EOMofBtoBprop}) and (\\ref{divergenceless}). What is the action of the operator $\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}$ on a contact Witten diagram? It turns out that the action of the equation of motion operator on a contact Witten can again be expressed as the linear combination of finitely many contact Witten diagrams, as we will show at the end of Appendix \\ref{appcontact}. Therefore the relation (\\ref{EOMWexWcon}) holds for exchange Witten diagrams independent of the choice of contact terms. \\label{footnoteEOMonContact}} $C^{(2)}_{\\Delta_E,\\ell_E}$ is the eigenvalue of the quadratic conformal Casimir for an operator with dimension $\\Delta_E$ and spin $\\ell_E$\n\\begin{equation}\nC^{(2)}_{\\Delta_E,\\ell_E}=\\Delta_E(\\Delta_E-d)+\\ell_E(\\ell_E+d-2)\\;.\n\\end{equation}\n\nLet us work out how the operator on the LHS acts as a differential operator in terms of the cross ratios. It is standard to extract a kinematic factor from the four-point function\n\\begin{equation}\\label{GUV}\n\\langle O_1(x_1)\\ldots O_4(x_4) \\rangle\\equiv G(x_i)=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^{a}\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^{b}\\mathcal{G}(U,V)\n\\end{equation}\nsuch that it becomes a function of the conformal cross ratios  $U$ and $V$\n\\begin{equation}\nU=\\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\\;,\\quad V=\\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}\\;.\n\\end{equation}\nWe have also defined\n\\begin{equation}\\small\na=\\frac{\\Delta_2-\\Delta_1}{2}\\;,\\quad b=\\frac{\\Delta_3-\\Delta_4}{2}\\;.\n\\end{equation}\nThe action of $\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}$ on a function of $x_i$ defines an operator acting on $\\mathcal{G}(U,V)$. Let us denote this operator as $\\mathbf{EOM}^{(s)}$, then (\\ref{EOMWexWcon}) becomes\n\\begin{equation}\\label{EOMWexWconcrossratio}\n\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{\\Delta_E,\\ell_E}(U,V)]=\\sum_I c_I\\mathcal{W}^{contact}_I(U,V)\\;.\n\\end{equation}\n To give an explicit expression for this operator, it is convenient to make a change of variables for the conformal cross ratios\n\\begin{equation}\nU=z\\bar{z}\\;,\\quad V=(1-z)(1-\\bar{z})\\;,\n\\end{equation}\nand define a second order differential operator\n\\begin{equation}\n\\mathbf{D}_{z}(a,b)=(1-z)z^2\\frac{d^2}{dz^2}-(1+a+b)z^2\\frac{d}{dz}-abz\\;.\n\\end{equation}\nThe s-channel equation of motion operator can be written as\n\\begin{equation}\\label{EOMszzb}\n\\mathbf{EOM}^{(s)}[\\mathcal{G}(z,\\bar{z})]=-2\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]+C^{(2)}_{\\Delta_E,\\ell_E}\\, \\mathcal{G}(z,\\bar{z})\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{\\Delta}_\\epsilon(a,b)=\\mathbf{D}_z(a,b)+\\mathbf{D}_{\\bar{z}}(a,b)+2\\epsilon \\frac{z\\bar{z}}{z-\\bar{z}}\\bigg((1-z)\\frac{d}{dz}-(1-\\bar{z})\\frac{d}{d\\bar{z}}\\bigg)\\;,\n\\end{equation}\nand we have defined\n\\begin{equation}\n\\epsilon=\\frac{d}{2}-1\\;.\n\\end{equation}\nNote that the s-channel conformal blocks are eigenfunctions of this differential operator \\cite{Dolan:2003hv} \\footnote{In this paper we slightly abuse the notation to let $g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})$ also denote the conformal block as a function of the cross ratios where a kinematic factor of $x_{ij}^2$ is extracted from $g^{(s)}_{\\Delta,\\ell}(x_i)$ according to (\\ref{GUV}). The meaning should be clear from their different arguments and the context.}\n\\begin{equation}\n\\mathbf{EOM}^{(s)}[g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})]=(C^{(2)}_{\\Delta,\\ell}-C^{(2)}_{\\Delta_E,\\ell_E})g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})\\;.\n\\end{equation}\nIn particular, the operator annihilates the single-trace block $g^{(s)}_{\\Delta_E,\\ell_E}(z,\\bar{z})$.\n\n\nFor $d=1$, the bulk space is $AdS_2$ and only scalar fields propagate in the internal line of the exchange Witten diagrams. Acting with the Casimir operator on the scalar exchange Witten diagram, we get the $D$-function (\\ref{Dfunction})\n\\begin{equation}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+\\Delta_E(\\Delta_E-1)\\right]W^{s,\\,exchange}_{\\Delta_E}=D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}\\;.\n\\end{equation}\nMoreover, because there is only one cross ratio in one dimension\n\\begin{equation}\nz=\\frac{x_{12}x_{34}}{x_{13}x_{24}}\\;,\n\\end{equation}\nthe operator $\\mathbf{EOM}^{(s)}$ is a differential operator of $z$, and is given by\n\\begin{equation}\n\\mathbf{EOM}^{(s)}=-\\mathbf{D}_{z}(2a,2b)+\\Delta_E(\\Delta_E-1)\\;.\n\\end{equation}\n\n\\subsection{Conformal Partial Waves}\\label{CPW}\nIt is sometimes useful to think of the conformal block decomposition of conformal correlators as arising from a more primitive formula, in terms of the so-called conformal partial waves $\\Psi_{\\Delta,J}^{(s)}(x_i)$.  We review in this section some basic properties of conformal partial waves for reader's convenience.\n\n\nThe conformal partial wave takes the form as the sum of the conformal blocks and its shadow block with $\\Delta\\to\\widetilde{\\Delta}=d-\\Delta$\n\\begin{equation}\\label{CPWdef}\n\\Psi_{\\Delta,J}^{(s)}(x_i)=K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)+K^{\\Delta_1,\\Delta_2}_{\\Delta,J}g^{(s)}_{\\widetilde{\\Delta},J}(x_i)\n\\end{equation}\nwhere the coefficients are given by\n\\begin{equation}\nK^{\\Delta_1,\\Delta_2}_{\\Delta,J}=\\left(-\\frac{1}{2}\\right)^J\\frac{\\pi^{\\frac{d}{2}}\\Gamma(\\Delta-\\frac{d}{2})\\Gamma(\\Delta+J-1)\\Gamma(\\frac{\\widetilde{\\Delta}+\\Delta_1-\\Delta_2+J}{2})\\Gamma(\\frac{\\widetilde{\\Delta}+\\Delta_2-\\Delta_1+J}{2})}{\\Gamma(\\Delta-1)\\Gamma(d-\\Delta+J)\\Gamma(\\frac{\\Delta+\\Delta_1-\\Delta_2+J}{2})\\Gamma(\\frac{\\Delta+\\Delta_2-\\Delta_1+J}{2})}\\;.\n\\end{equation}\nConformal partial waves with all integer spins $J$ and unphysical complex dimensions $\\Delta=\\frac{d}{2}+i \\nu$, $\\nu\\geq0$, form a complete set of functions \\cite{Dobrev:1977qv}, and are usually referred to as the principal series representation. In terms of conformal partial waves, a four-point correlation function can be written as a contour integral \\cite{Dobrev:1975ru}\\footnote{More precisely, for $d=1$ the principal series is also supplemented by a discrete series with $\\Delta=2n$, $n=1,2,\\ldots$. Here we assume that the spectral function $\\rho(\\Delta)$ does not contain physical poles at $\\Delta=2n$. Then after closing the contour and taking the residues, poles from the conformal partial wave will precisely cancel the contribution from the discrete series.}\n\\begin{equation}\n\\begin{split}\nG(x_i)={}&\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\,\\Psi_{\\Delta,J}^{(s)}(x_i)\\\\\n={}&\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\,(K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)+K^{\\Delta_1,\\Delta_2}_{\\Delta,J}g^{(s)}_{\\widetilde{\\Delta},J}(x_i))\n\\end{split}\n\\end{equation}\nUsing the symmetry that \n\\begin{equation}\n\\rho(\\Delta,J)\\,K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}=\\rho(\\widetilde{\\Delta},J)K^{\\Delta_1,\\Delta_2}_{\\widetilde{\\Delta},J}\\;,\n\\end{equation}\nwe can eliminate the shadow blocks from the above integral and rewrite it as\n\\begin{equation}\nG(x_i)=\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}-i\\infty}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\, K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)\\;.\n\\end{equation}\nAfter closing the contour to the right and picking up the poles, we get the usual conformal block decomposition.\n\nThe combination (\\ref{CPWdef}) of the conformal block with its shadow is special because it makes the conformal partial wave a single-valued function in Euclidean space ({\\it i.e.}, when $\\bar{z}=z^*$). By contrast, each individual conformal block is not\\footnote{For example this can be explicitly seen from the 2d conformal block\n\\begin{equation}\ng^{(s)}_{\\Delta,J}(z,\\bar{z})=\\frac{k_{\\Delta-J}(z)k_{\\Delta+J}(\\bar{z})+k_{\\Delta+J}(z)k_{\\Delta-J}(\\bar{z})}{1+\\delta_{J,0}}\\;,\\quad k_{\\beta}(z)=z^{\\beta\/2}{}_2F_1(\\beta\/2+a,\\beta\/2+b,\\beta;z)\\;.\n\\end{equation}\nAround $z=1$ and $\\bar{z}=1$, we can use the property of ${}_2F_1$ to write the conformal block as\n\\begin{equation}\ng^{(s)}_{\\Delta,J}(z,\\bar{z})=f_{1}(z,\\bar{z})(z-1)^{-a-b}(\\bar{z}-1)^{-a-b}+f_{2}(z,\\bar{z})+f_{3}(z,\\bar{z})(z-1)^{-a-b}+f_4(z,\\bar{z})(\\bar{z}-1)^{-a-b}\n\\end{equation}\nwhere $f_i$ are regular at $z=\\bar{z}=1$. In the Euclidean regime $\\bar{z}=z^*$, the first two terms are single-valued while the latter two terms fail to be.\n\n\n}. We can most easily see this single-valuedness of the conformal partial wave from its integral representation \\cite{Ferrara:1972xe,Ferrara:1973vz,Ferrara:1972uq,Ferrara:1972ay,SimmonsDuffin:2012uy}\n\\begin{equation}\\label{CPWintegralrep}\n\\Psi_{\\Delta,J}^{(s)}(x_i)=\\int d^dx^5\\langle O_1(x_1)O_2(x_2)O_5^{\\mu_1\\ldots\\mu_J}(x_5)\\rangle \\langle \\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}(x_5)O_3(x_3)O_4(x_4)\\rangle\\;,\n\\end{equation}\nwhere $\\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}$ is the shadow operator of $O_5$. This integral is manifestly single-valued in Euclidean space. \n\nThe above integral representation can also be lifted into the AdS space \\cite{Costa:2014kfa}\n\\begin{equation}\\label{AdSlift}\n\\begin{split}\n\\Psi_{\\Delta,J}^{(s)}(x_i)\\propto{}& \\int d^d x_5 \\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}} G^{\\Delta_1}_{B\\partial}(z_1,x_1)(\\bigtriangledown^\\mu)^JG^{\\Delta_2}_{B\\partial}(z_1,x_2)\\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z_1,x_5)\\\\\n{}& \\times \\int \\frac{d^{d+1}z_2}{z_{20}^{d+1}} \\Pi^{d-\\Delta}_{\\nu_1\\ldots\\nu_J,\\rho_1\\ldots\\rho_J}(z_2,x_5) G^{\\Delta_3}_{B\\partial}(z_2,x_3)(\\bigtriangledown^\\nu)^J  G^{\\Delta_4}_{B\\partial}(z_2,x_4)\n\\end{split}\n\\end{equation}\nwhere $\\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z,x_5)$ is the spin-$J$ bulk-to-boundary propagator. It is clear that the first AdS integral over $z_1$ gives the three-point function $\\langle O_1(x_1)O_2(x_2)O_5^{\\mu_1\\ldots\\mu_J}(x_5)\\rangle$ and the second AdS integral over $z_2$ gives $\\langle \\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}(x_5)O_3(x_3)O_4(x_4)\\rangle$. Further performing the $x_5$ integral therefore reproduces the conformal partial wave $\\Psi_{\\Delta,J}^{(s)}(x_i)$ in the integral representation (\\ref{CPWintegralrep}).\n\n\nThe above lift of the conformal block into AdS is closely related to the split representation of AdS propagators \\cite{Costa:2014kfa}. It is convenient to first define the AdS harmonic function\n\\begin{equation}\\label{Omegasplit}\n\\Omega^{\\Delta,J}_{\\mu_1\\ldots\\mu_J,\\nu_1\\ldots\\nu_J}(z_1,z_2)=-\\frac{(\\Delta-\\frac{d}{2})^2}{\\pi J!(\\frac{d}{2}-1)_J}\\int d^dx_5 \\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z_1,x_5)\\Pi^{d-\\Delta}_{\\nu_1\\ldots\\nu_J,\\rho_1\\ldots\\rho_J}(z_2,x_5)\n\\end{equation}\nwhich splits a function of two bulk points into a product of two bulk-to-boundary propagators with a common  integrated boundary point. \nThe bulk-to-bulk propagator can then be expanded in terms of these AdS harmonic functions\n\\begin{equation}\\label{PiinOmega}\n\\Pi^{\\Delta,J}_{\\mu_1\\ldots\\mu_J,\\nu_1\\ldots\\nu_J}(z_1,z_2)=\\sum_{l=0}^J\\int d\\Delta' \\,a_\\ell(\\Delta') (\\bigtriangledown_1^\\mu)^{J-\\ell}(\\bigtriangledown_2^\\nu)^{J-\\ell}\\Omega^{\\Delta',\\ell}_{\\mu_1\\ldots\\mu_l,\\nu_1\\ldots\\nu_l}(z_1,z_2)\\;\n\\end{equation}\nwhere explicit coefficients $a_\\ell(\\Delta')$ can be found in \\cite{Costa:2014kfa}. The above split representation of the AdS harmonic function (\\ref{Omegasplit}) gives the split representation of the bulk-to-bulk propagator.\n\nRelatedly, a conformal partial wave can also be represented in AdS as the difference of two exchange Witten diagrams with opposite quantizations\n\\begin{equation}\\label{Psiasdifference}\n\\Phi^{(s)}_{\\Delta,J}(x_i)\\propto W^{s,exchange}_{\\Delta,J}(x_i)-W^{s,exchange}_{d-\\Delta,J}(x_i)\\;.\n\\end{equation}\nThis fact was pointed out, {\\it e.g.}, in \\cite{Penedones:2007ns,Costa:2014kfa,Giombi:2018vtc}, and can be understood in two steps as follows. Firstly, all the double-trace conformal blocks in $W^{s,exchange}_{\\Delta,J}$ and $W^{s,exchange}_{d-\\Delta,J}$ cancel out in (\\ref{Psiasdifference}). We can see this by noting that both $W^{s,exchange}_{\\Delta,J}$ and $W^{s,exchange}_{d-\\Delta,J}$ satisfy the same equation of motion identity\n\\begin{equation}\n\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{\\Delta_E,\\ell_E}(U,V)]=\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{d-\\Delta_E,\\ell_E}(U,V)],\n\\end{equation}\nbecause the contact diagrams are determined by vertices which are independent of the quantization. Moreover, the double-trace conformal blocks are diagonal under the equation of motion operator, with non vanishing and quantization-independent eigenvalues. This guarantees the cancellation of double-trace blocks in the difference and implies that (\\ref{Psiasdifference}) is just the linear combination of the single-trace conformal block and its shadow. The second step is therefore to show that the linear combination is proportional to (\\ref{CPWdef}). However, this is guaranteed by single-valuedness. Because the exchange Witten diagrams are single-valued in the Euclidean regime, there is only one way, {\\it i.e.}, in the fashion of (\\ref{CPWdef}), to combine the non-single-valued conformal block and its shadow in order to achieve single-valuedness .\n\n\n\n\n\n\n\n\n\n\nAs a final comment, let us mention that a single conformal block in $d\\geq2$ {\\it cannot} be decomposed into the crossed channel in terms of conformal blocks. This was noticed in, {\\it e.g.}, \\cite{ElShowk:2011ag,Liu:2018jhs}, and follows from the non single-valuedness of conformal block. In the Euclidean regime, an s-channel conformal block is not single-valued around $z=\\bar{z}=1$ while the t-channel conformal blocks are. It is not possible to sum over infinitely many single-valued functions to obtain a non-single-valued function. On the other hand, a conformal partial wave can always be decomposed into the crossed channel in terms of double-trace conformal blocks. This follows from (\\ref{Psiasdifference}) where the conformal partial wave is written as the difference of two s-channel exchange Witten diagrams, and each exchange Witten diagram admits decomposition into double-trace conformal blocks in the crossed channel. In 1d, the above comments do not apply. There is only one cross ratio and one can explicitly show that the conformal block can be decomposed into the crossed channel as infinitely many double-trace conformal blocks.\n\n\\section{Recursion Relations in Contact Witten Diagrams}\n\\subsection{Conformal Block Decomposition of Contact Witten Diagrams}\\label{SecCBdecompContact}\nIn this subsection we focus on the conformal block decomposition of contact Witten diagrams (\\ref{Wcontact}). We organize the contact Witten diagrams in terms of the total number of covariant derivatives in the quartic vertex. We start with the simplest contact Witten diagram where there is no derivative, {\\it i.e.}, $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ defined in (\\ref{Dfunction}).  The conformal block decomposition of higher-derivative contact Witten diagrams, as we will see, can be recursively related to the decomposition of the zero-derivative contact Witten diagrams. In this section, we will assume the external conformal dimensions are generic such that we will not encounter derivative conformal blocks. The conformal block decomposition for the special cases satisfying $\\Delta_1+\\Delta_2=\\Delta_3+\\Delta_4+2m$, {\\it etc}, can be obtained from the generic case by taking the limit. We give more details of taking the limit in Appendix \\ref{appcontact}.\n\n\\subsubsection*{The decomposition the zero-derivative contact diagram}\nTo obtain the conformal block decomposition of $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$, we use a special case of the split representation, namely the split representation of the delta-function \\cite{Penedones:2010ue,Costa:2014kfa,Bekaert:2015tva}\n\\begin{equation}\n\\delta(z_1,z_2)=\\int d^dx_5\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i} \\,\\rho_\\delta(c)\\,G_{B\\partial}^{\\frac{d}{2}+c}(z_1,x_5)\\,G_{B\\partial}^{\\frac{d}{2}-c}(z_2,x_5)\\;\n\\end{equation}\nwhere \n\\begin{equation}\n\\rho_\\delta(c)=\\frac{\\Gamma(\\frac{d}{2}+c)\\Gamma(\\frac{d}{2}-c)}{2\\pi^d\\Gamma(-c)\\Gamma(c)}\\;.\n\\end{equation}\nInserting this identity into (\\ref{Dfunction}), we have \n\\begin{equation}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int d^dx_5\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,\\rho_\\delta(c)\\, D_{\\Delta_1\\,\\Delta_2\\, \\frac{d}{2}+c}(x_1,x_2,x_5)D_{ \\frac{d}{2}-c\\, \\Delta_3\\,\\Delta_4}(x_5,x_3,x_4)\n\\end{equation}\nwhere $D_{\\Delta_1\\Delta_2\\Delta_3}(x_1,x_2,x_3)$ is a three-point function\n\\begin{equation}\\label{D3function}\n\\begin{split}\nD_{\\Delta_1\\Delta_2\\Delta_3}={}&\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^3G^{\\Delta_i}_{B\\partial}(z,x_i)=\\frac{a_{\\Delta_1\\Delta_2\\Delta_3}}{x_{12}^{\\Delta_1+\\Delta_2-\\Delta_3}x_{13}^{\\Delta_1+\\Delta_3-\\Delta_2}x_{23}^{\\Delta_2+\\Delta_3-\\Delta_1}}\\;,\\\\\na_{\\Delta_1\\Delta_2\\Delta_3}={}& \\frac{\\pi^{\\frac{d}{2}}\\Gamma(\\frac{\\Delta_1+\\Delta_2-\\Delta_3}{2})\\Gamma(\\frac{\\Delta_1+\\Delta_3-\\Delta_2}{2})\\Gamma(\\frac{\\Delta_2+\\Delta_3-\\Delta_1}{2})}{2\\Gamma(\\Delta_1)\\Gamma(\\Delta_2)\\Gamma(\\Delta_3)}\\Gamma(\\frac{\\Delta_1+\\Delta_2+\\Delta_3-d}{2})\\;.\n\\end{split}\n\\end{equation}\nWe can integrate out $x_5$ using (\\ref{CPWintegralrep}), and obtain the conformal partial wave decomposition of the $D$-function\n\\begin{equation}\\label{DfunctioninCPW}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,\\rho_D(c)\\,\\Psi^{(s)}_{\\frac{d}{2}+c,0}(x_i)\\;.\n\\end{equation}\nThe spectral density is\n\\begin{equation}\n\\rho_D(c)=\\rho_\\delta(c)\\, a_{\\Delta_1\\,\\Delta_2\\,\\frac{d}{2}+c}\\,a_{\\Delta_3\\,\\Delta_4\\,\\frac{d}{2}-c}\\;.\n\\end{equation}\nWe can use the shadow symmetry in (\\ref{DfunctioninCPW})to write it as a spectral representation with respect to the conformal blocks \n\\begin{equation}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,2\\,\\rho_D(c)\\, K^{\\Delta_3,\\Delta_4}_{\\frac{d}{2}-c,0}g^{(s)}_{\\frac{d}{2}+c,0}(x_i)\\;.\n\\end{equation}\nBy closing the contour to the right and taking the residues, we arrive at the conformal block decomposition of $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$\n\\begin{equation}\\label{Dfunctioningxi}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\sum_{n=0}^\\infty a^{12}_{n,0} g^{(s)}_{\\Delta_1+\\Delta_2+2n,0}(x_i)+\\sum_{n=0}^\\infty a^{34}_{n,0} g^{(s)}_{\\Delta_3+\\Delta_4+2n,0}(x_i)\\;\n\\end{equation}\nwhere \n\\begin{equation}\\label{Dfunctiondecomcoe}\\small\n\\begin{split}\na^{12}_{n,0}={}&\\frac{\\pi ^{d\/2} (-1)^{-n} \\Gamma (n+\\Delta_1) \\Gamma (n+\\Delta_2) \\Gamma \\left(-\\frac{d}{2}+n+\\Delta_1+\\Delta_2\\right) \\Gamma \\left(\\frac{-d+2 n+\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4}{2}\\right)}{2 n! \\Gamma (\\Delta_1) \\Gamma (\\Delta_2) \\Gamma (\\Delta_3) \\Gamma (\\Delta_4) \\Gamma (2 n+\\Delta_1+\\Delta_2)}\\\\\n{}&\\times\\frac{\\Gamma \\left(\\frac{2 n+\\Delta_1+\\Delta_2+\\Delta_3-\\Delta_4}{2}\\right) \\Gamma \\left(\\frac{2 n+\\Delta_1+\\Delta_2-\\Delta_3+\\Delta_4}{2}\\right) \\Gamma \\left(\\frac{-2 n-\\Delta_1-\\Delta_2+\\Delta_3+\\Delta_4}{2}\\right)}{ \\Gamma \\left(-\\frac{d}{2}+2 n+\\Delta_1+\\Delta_2\\right)}\\;, \n\\end{split}\n\\end{equation}\nand $a^{34}_{n,0}$ can be obtained from $a^{12}_{n,0}$ by replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$.\n\n\\subsubsection*{The decomposition of higher-derivative contact diagrams}\nNow let us consider a general contact Witten diagram (\\ref{Wcontact}) with derivatives in the quartic vertex.  We first notice that a generic contact Witten diagram $W^{contact}$ can always be written as a linear combination of finitely many building blocks \\cite{Penedones:2010ue}\n\\begin{equation}\\label{Dnijdef}\nD^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)\\equiv\\prod_{i<j}(x_{ij}^2)^{n_{ij}}D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}(x_i)\n\\end{equation}\nwith {\\it position-independent} coefficients. Here $\\Delta^{n_{ij}}_k$ are the shifted dimensions\n\\begin{equation}\n\\Delta^{n_{ij}}_k=\\Delta_k+\\sum_{l\\neq k}n_{kl}\\;,\n\\end{equation}\nand $n_{ij}=n_{ji}$ are {\\it non negative} integers.  Note that under a conformal transformation, the function $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)$ transforms with conformal dimensions $\\Delta^{n_{ij}}_k$. The conformal block decomposition of $W^{contact}$ is thus the {\\it sum} of the decompositions of the building blocks $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$. \n\n\n\nThe second step is then to obtain the decomposition of $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$. Let us rewrite it in a different form\n\\begin{equation}\\label{buildingblockandDshifted}\nD^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\frac{P(x_i,\\Delta_1,\\Delta_2,\\Delta_3,\\Delta_4)}{P(x_i,\\Delta_1^{n_{ij}},\\Delta_2^{n_{ij}},\\Delta_3^{n_{ij}},\\Delta_4^{n_{ij}})}(U^{-\\frac{1}{2}}V)^{n_{23}}(U^{-\\frac{1}{2}})^{n_{13}+n_{14}+n_{24}}D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}\n\\end{equation}\nwhere\n\\begin{equation}\nP(x_i,\\Delta_1,\\Delta_2,\\Delta_3,\\Delta_4)=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^{\\frac{\\Delta_2-\\Delta_1}{2}}\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^{\\frac{\\Delta_3-\\Delta_4}{2}}\n\\end{equation}\nis the prefactor of $x_{ij}$ we extracted in (\\ref{GUV}). From the above discussion we have already known how to decompose the $D$-function  $D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}\n$ into conformal blocks, when we view it as a four-point correlation function with external conformal dimensions $\\Delta^{n_{ij}}_1$, $\\Delta^{n_{ij}}_2$, $\\Delta^{n_{ij}}_3$, $\\Delta^{n_{ij}}_4$. The decomposition is given by (\\ref{Dfunctioningxi}), but it will be helpful in the following to rewrite the decomposition in such a way that only the cross ratios are involved. Explicitly, we have\n\\begin{equation}\\label{Dshifteddecomp}\n\\begin{split}\n\\frac{D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}}{P(x_i,\\Delta_1^{n_{ij}},\\Delta_2^{n_{ij}},\\Delta_3^{n_{ij}},\\Delta_4^{n_{ij}})}={}&\\sum_{n=0}^{\\infty} a^{12,n_{ij}}_{n,0} g^{(s)}_{\\Delta_1^{n_{ij}}+\\Delta_2^{n_{ij}}+2n,0}(\\epsilon,a^{n_{ij}},b^{n_{ij}};z,\\bar{z})\\\\\n{}&+\\sum_{n=0}^{\\infty} a^{34,n_{ij}}_{n,0} g^{(s)}_{\\Delta_3^{n_{ij}}+\\Delta_4^{n_{ij}}+2n,0}(\\epsilon,a^{n_{ij}},b^{n_{ij}};z,\\bar{z})\\;\n\\end{split}\n\\end{equation}\nwhere $a^{n_{ij}}=(\\Delta_2^{n_{ij}}-\\Delta_1^{n_{ij}})\/2$ and $b^{n_{ij}}=(\\Delta_3^{n_{ij}}-\\Delta_4^{n_{ij}})\/2$. In the above decomposition, we have restored all the parameters in a conformal block $g^{(s)}_{\\Delta,\\ell}(\\epsilon,a,b;z,\\bar{z})$ to emphasize its parameter dependence.  For a four-point correlator with external dimensions $\\Delta_i$, the conformal block $g^{(s)}_{\\Delta,\\ell}(\\epsilon,a,b;z,\\bar{z})$ is a function of $\\Delta$, $\\ell$, the spacetime dimension, $z$, $\\bar{z}$, and depends on the external conformal dimensions only via the combinations $a=(\\Delta_2-\\Delta_1)\/2$, $b=(\\Delta_3-\\Delta_4)\/2$. The latter fact has immediate interesting consequences. Consider for a moment the special example when $n_{13}=n_{14}=n_{24}=n_{23}=0$, it follows from (\\ref{buildingblockandDshifted}) that we have the following identity\n\\begin{equation}\n\\frac{D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}}{P(x_i,\\Delta_1,\\Delta_2,\\Delta_3,\\Delta_4)}\\bigg|_{n_{13}=n_{14}=n_{24}=n_{23}=0}=\\frac{D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}}{P(x_i,\\Delta_1^{n_{ij}},\\Delta_2^{n_{ij}},\\Delta_3^{n_{ij}},\\Delta_4^{n_{ij}})}\\;.\n\\end{equation}\nThe conformal block decomposition (\\ref{Dshifteddecomp}) of the RHS expresses the LHS as a sum of double-trace conformal blocks. To confirm that it really gives the conformal block decomposition for the LHS, we still need verify that the conformal blocks have the correct parameters, {\\it i.e.}, $a^{n_{ij}}=a$, $b^{n_{ij}}=b$. We can easily check that this is indeed true. Therefore we conclude that when $n_{13}=n_{14}=n_{24}=n_{23}=0$, $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ as a correlator with external dimensions $\\Delta_k$, has the {\\it same} decomposition as $D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}$ as a correlator with dimensions $\\Delta^{n_{ij}}_k$.\n\n More generally, we can relate the decomposition coefficients of a higher-derivative contact diagram to the those of a zero-derivative contact diagram using recursion relations of the conformal blocks. We can show that  (\\ref{Dnijdef}) as a basis for decomposing $W^{bulk}$ is in fact {\\it over complete}, and we can always restrict to $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ with $n_{13}=n_{24}=n_{23}=n_{34}=0$ (we prove this in Appendix \\ref{appcontact}). By inserting  (\\ref{Dshifteddecomp}) into  (\\ref{buildingblockandDshifted}) and setting $n_{13}=n_{24}=n_{23}=n_{34}=0$, we get \n\\begin{equation}\\label{Dnijdecomp}\n\\begin{split}\n\\frac{D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}}{P(x_i,\\Delta_1,\\Delta_2,\\Delta_3,\\Delta_4)}={}&(U^{-\\frac{1}{2}})^{n_{14}} \\sum_{n=0}^{\\infty} a^{12,n_{ij}}_{n,0} g^{(s)}_{\\Delta_1^{n_{ij}}+\\Delta_2^{n_{ij}}+2n,0}(\\epsilon,a^{n_{ij}},b^{n_{ij}};z,\\bar{z})\\\\\n+{}&(U^{-\\frac{1}{2}})^{n_{14}}  \\sum_{n=0}^{\\infty} a^{34,n_{ij}}_{n,0} g^{(s)}_{\\Delta_3^{n_{ij}}+\\Delta_4^{n_{ij}}+2n,0}(\\epsilon,a^{n_{ij}},b^{n_{ij}};z,\\bar{z})\\;.\n\\end{split}\n\\end{equation}\nThis is not a conformal block decomposition yet, due to the multiplicative factor $(U^{-\\frac{1}{2}})^{n_{14}}$. To expand the RHS in conformal blocks, we use the following  recursion relation for conformal blocks\\footnote{In this paper, we normalize our conformal blocks such that $g^{(s)}_{\\Delta,\\ell}\\sim z^{\\frac{\\Delta-\\ell}{2}}\\bar{z}^{\\frac{\\Delta+\\ell}{2}}$ when $z\\ll\\bar{z}\\ll1$. This differs from the normalization used in \\cite{Dolan:2011dv} by a factor: $\\frac{(\\epsilon)_\\ell}{(2\\epsilon)_\\ell}g^{(s),{\\bf Here}}_{\\Delta,\\ell}=g^{(s),{\\bf DO}}_{\\Delta,\\ell}$, and $g^{(s),{\\bf DO}}_{\\Delta,\\ell}$ was denoted as $F_{\\lambda_1\\lambda_2}$, with $\\lambda_1=\\frac{\\Delta+\\ell}{2}$, $\\lambda_2=\\frac{\\Delta-\\ell}{2}$.} \\cite{Dolan:2011dv} (we suppress $\\epsilon$ and $z$, $\\bar{z}$ which are not changed)\n\\begin{equation}\\label{Uhalfrecur}\n\\begin{split}\nU^{-\\frac{1}{2}}g^{(s)}_{\\Delta,\\ell}(a,b)={}&g^{(s)}_{\\Delta-1,\\ell}(a+\\frac{1}{2},b+\\frac{1}{2})+\\Lambda_1\\, g^{(s)}_{\\Delta,\\ell+1}(a+\\frac{1}{2},b+\\frac{1}{2})\\\\\n{}&+\\Lambda_2\\, g^{(s)}_{\\Delta,\\ell-1}(a+\\frac{1}{2},b+\\frac{1}{2})+\\Lambda_3\\, g^{(s)}_{\\Delta+1,\\ell}(a+\\frac{1}{2},b+\\frac{1}{2})\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n\\Lambda_1={}&-\\frac{(2 a+\\Delta +\\ell ) (2 b+\\Delta +\\ell )}{4 (\\Delta +\\ell -1) (\\Delta +\\ell )}\\;,\\\\\n\\Lambda_2={}&\\frac{\\ell  (\\ell +2 \\epsilon -1) (2 a+\\Delta -\\ell -2 \\epsilon ) (-2 b-\\Delta +\\ell +2 \\epsilon )}{4 (\\ell +\\epsilon -1) (\\ell +\\epsilon ) (-\\Delta +\\ell +2 \\epsilon ) (-\\Delta +\\ell +2 \\epsilon +1)}\\;,\\\\\n\\Lambda_3={}&\\frac{(\\Delta -1) (\\Delta -2 \\epsilon ) (2 a+\\Delta +\\ell ) (2 b+\\Delta +\\ell ) (2 a+\\Delta -\\ell -2 \\epsilon ) (2 b+\\Delta -\\ell -2 \\epsilon )}{16 (\\Delta -\\epsilon -1) (\\Delta -\\epsilon ) (\\Delta +\\ell -1) (\\Delta +\\ell ) (-\\Delta +\\ell +2 \\epsilon ) (-\\Delta +\\ell +2 \\epsilon +1)}\\;.\n\\end{split}\n\\end{equation}\nAfter using the recursion relation for $n_{14}$ times, we can get rid of all powers of $U^{-\\frac{1}{2}}$ in (\\ref{Dnijdecomp}). Notice that the recursion coefficient $\\Lambda_2$ contains a factor of $\\ell$. This guarantees that when we apply this recursion relation to a spin-0 conformal block, we will not generate unphysical blocks with negative spins. Let us also note that each application of the recursion relation changes the values of $a$ and $b$ by $\\frac{1}{2}$. Starting with $a^{n_{ij}}$, $b^{n_{ij}}$ and applying the recursion relation $n_{14}$ times, we can check that their values are shifted to\n\\begin{equation}\na^{n_{ij}}\\to a^{n_{ij}}+\\frac{n_{14}}{2}=\\frac{\\Delta_2-\\Delta_1}{2}\\;,\\quad b^{n_{ij}}\\to a^{n_{ij}}+\\frac{n_{14}}{2}=\\frac{\\Delta_3-\\Delta_4}{2}\\;,\n\\end{equation}\nmatching precisely the $a$ and $b$ values that are assigned to a correlator with external dimensions $\\Delta_1$, $\\Delta_2$, $\\Delta_3$, $\\Delta_4$. \n\nTo summarize,  we have the following algorithm for performing  conformal block decomposition of a generic contact Witten diagram $W^{contact}$\n\\begin{enumerate}\n\\item We write $W^{contact}$ as a linear combination of $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)$ with $n_{13}=n_{24}=n_{23}=n_{34}=0$.\\footnote{It is easiest to find this linear combination in Mellin space (see Appendix \\ref{appcontact}).}\n\\item Each $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)$ is associated with a $D$-function $D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}$. We use (\\ref{Dfunctioningxi}) to get the conformal block decomposition of $D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}$ for all the $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ that appear in the first step.\n\\item The conformal block decomposition of $D_{\\Delta^{n_{ij}}_1\\Delta^{n_{ij}}_2\\Delta^{n_{ij}}_3\\Delta^{n_{ij}}_4}$ in step 2 gives the RHS of (\\ref{Dnijdecomp}). Using the relation (\\ref{Uhalfrecur}) $n_{14}$ times, we obtain the conformal block decomposition of $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$.\n\\item We obtain the conformal block decomposition $W^{contact}$ by summing up the decompositions of $D^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ with coefficients obtained from step 1.\n\\end{enumerate}\n\n\n\n\n\n\n\\subsection{Direct Channel Decomposition of Exchange Diagrams from Contact Diagrams}\\label{Secdirectchan}\nThe direct channel decomposition of an exchange Witten diagram can be obtained by using the split representation (\\ref{PiinOmega}) (\\ref{Omegasplit}) for the bulk-to-bulk propagator, and this method has already been streamlined in \\cite{Costa:2014kfa}. Schematically, because of (\\ref{AdSlift}) and (\\ref{Omegasplit}), each $\\Omega^{\\Delta',\\ell}$ becomes proportional to a spin-$\\ell$ conformal partial wave after integrations in position space. One obtains from this procedure the spectral representation of the exchange Witten diagram\n\\begin{equation}\nW^{s,exchange}=\\sum_{\\ell=0}^{\\ell_E}\\int_{-i\\infty}^{i\\infty}\\frac{dc}{2\\pi i}\\rho_\\ell^{s,exchange}(c)\\Psi^{(s)}_{\\frac{d}{2}+c,\\ell}(x_i)\\;,\n\\end{equation}\nwhich can further be written as \n\\begin{equation}\nW^{s,exchange}=\\sum_{\\ell=0}^{\\ell_E}\\int_{-i\\infty}^{i\\infty}\\frac{dc}{2\\pi i}\\,2\\rho_\\ell^{s,exchange}(c)K^{\\Delta_3,\\Delta_3}_{\\frac{d}{2}-c,\\ell}g^{(s)}_{\\frac{d}{2}+c,\\ell}(x_i)\\;,\n\\end{equation}\nThe OPE coefficients of the single-trace operator and double-trace operators can be obtained by closing the contour to the right and taking the residues.\\footnote{For $\\ell<J$, $\\rho_\\ell^{s,exchange}(c)$ also contains spurious poles which do not correspond to physical operators, see \\cite{Costa:2012cb,Costa:2014kfa}.} We refer the reader to \\cite{Costa:2014kfa} for details of implementing this algorithm.\n\nThe purpose of this subsection is to point out an alternative way of obtaining the direct channel decomposition, namely from the contact diagrams. We recall that the exchange Witten diagram satisfies  the following equation of motion identity (\\ref{EOMWexWcon})\n\\begin{equation}\\label{EOMWexWconb}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right]W^{s,\\, exchange}_{\\Delta_E,\\ell_E}=\\sum_I c_IW^{contact}_I\\;.\n\\end{equation}\nIn the previous subsection, we discussed how to decompose a generic contact Witten diagram into conformal blocks. Using the algorithm in Section \\ref{SecCBdecompContact}, we can write the RHS of (\\ref{EOMWexWconb}) as a sum of double-trace conformal blocks\n\\begin{equation}\n\\sum_I c_IW^{contact}_I=\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nBecause the double-trace conformal blocks are diagonal under the equation of motion operator with non vanishing eigenvalues, the double-trace coefficients of the exchange diagram are simply proportional to those of the contact Witten diagrams. This gives us right away all the double-trace coefficients $A^{12}_{n,J}$, $A^{34}_{n,J}$  in (\\ref{Wexchangeins})\n\\begin{equation}\\label{A12andA34}\nA^{12}_{n,J}=\\frac{\\tilde{a}^{12}_{n,J}}{C^{(2)}_{\\Delta_E,\\ell_E}-C^{(2)}_{\\Delta_1+\\Delta_2+2n+J,J}}\\;,\\quad A^{34}_{n,J}=\\frac{\\tilde{a}^{34}_{n,J}}{C^{(2)}_{\\Delta_E,\\ell_E}-C^{(2)}_{\\Delta_3+\\Delta_4+2n+J,J}}\\;.\n\\end{equation}\n We are therefore only one coefficient short for obtaining the full direct channel decomposition (\\ref{Wexchangeins}). Naively, the information of the remaining single-trace block is lost because the single-trace block is annihilated by the equation of motion operator. However, knowing all these double-trace coefficients is in fact {\\it enough} to uniquely fix the single-trace coefficient $A$. We can intuitively understand this statement from single-valuedness. As we alluded to in Section \\ref{CPW}, a single conformal block is not single-valued in the Euclidean regime ($\\bar{z}=z^*$), in contrast to an exchange Witten diagram. This fact implies that given we have obtained the correct double-trace coefficients $A^{12}_{n,J}$, $A^{34}_{n,J}$, there must be a unique choice for the single-trace coefficient $A$ such that the sum (\\ref{Wexchangeins}) is single-valued.\\footnote{One may also ask the following opposite question: with a fixed coefficient for the single-trace block, what should the double-trace coefficients be in order for the whole diagram to be single-valued? The answer to this question is not unique as one can always add contact diagrams which are single-valued.} More conceretely, we can relate the single-trace coefficient to the double-trace coefficients as follows. We start from the equation of motion identity (\\ref{EOMWexWconb}). The algorithm in Section \\ref{SecCBdecompContact} gives us the decomposition the contact diagrams on the RHS into double-trace operators, which can also be rewritten in terms of a spectral function\n\\begin{equation}\n\\sum_I c_IW^{contact}_I=\\sum_{\\ell=0}^{\\ell_E}\\int_{-i\\infty}^{i\\infty}\\frac{dc}{2\\pi i}\\rho_\\ell^{contact}(c)\\Psi^{(s)}_{\\frac{d}{2}+c,\\ell}(x_i)\\;.\n\\end{equation}\nNotice that the conformal partial waves are diagonal under the equation of motion operator\n\\begin{equation}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right]\\Psi^{(s)}_{\\frac{d}{2}+c,\\ell}(x_i)=\\left(C^{(2)}_{\\Delta_E,\\ell_E}-C^{(2)}_{\\frac{d}{2}+c,\\ell}\\right)\\Psi^{(s)}_{\\frac{d}{2}+c,\\ell}(x_i)\\;.\n\\end{equation} \nThis implies that the spectral function $\\rho_\\ell^{s,exchange}(c)$ of the exchange Witten diagram is related to $\\rho_\\ell^{contact}(c)$ by\n\\begin{equation}\n\\rho_\\ell^{s,exchange}(c)=\\frac{\\rho_\\ell^{contact}(c)}{C^{(2)}_{\\Delta_E,\\ell_E}-C^{(2)}_{\\frac{d}{2}+c,\\ell}}\\;.\n\\end{equation}\nIn particular, the case with $\\ell=\\ell_E$\n\\begin{equation}\n\\rho_{\\ell_E}^{s,exchange}(c)=\\frac{\\rho_{\\ell_E}^{contact}(c)}{(\\frac{d}{2}-\\Delta_E)^2-c^2}\\;,\n\\end{equation}\ngives the coefficient for the spin-$\\ell_E$ single-trace operator, upon taking the residue of the simple pole at $c=-\\frac{d}{2}+\\Delta_E$. \n\n\\section{Recursion Relations in the Crossed Channel}\\label{SecRecurcross}\n\\subsection{The Strategy for Crossed Channel Decomposition}\\label{SecStrat}\nIn this subsection we outline a recursive method to compute the decomposition coefficients of exchange Witten diagrams and conformal partial waves in the crossed channel.\\footnote{More precisely, we will consider the conformal block decomposition of a t-channel exchange Witten diagram in the s-channel. The decomposition of a u-channel exchange diagram in the s-channel is similar.} The technical details of this algorithm and further comments are left to the ensuing subsections. \n\nOur strategy exploits three useful facts.\n Firstly, it is well known that the exchange Witten diagrams can be decomposed into the crossed channel in terms of only double-trace conformal blocks. Secondly, as we  discussed in Section \\ref{SecExtoCon}, acting on an exchange Witten diagram with the equation of motion operator, we can collapse it into contact Witten diagrams. The conformal block decomposition of the contact Witten diagrams is easy, and we have streamlined the method in Section \\ref{SecCBdecompContact}. These two facts so far are not sufficient for making progress. We need to further combine them with the following third observation: the action of the t-channel equation of motion operator admits a very simple action on an s-channel conformal block. More precisely, the action produces a linear combination of finitely many conformal blocks with shifted quantum numbers. In particular, the double-trace spectra are preserved by the action of the equation of motion operator. Taken together, these three facts imply that the crossed channel decomposition coefficients of an exchange Witten diagram are not independent. Rather, the equation of motion operator reshuffles the  crossed channel double-trace operators of the exchange Witten diagram, and equates them to the decomposition of the contact Witten diagrams. The decomposition coefficients thus satisfy linear equations among themselves with the decomposition coefficients of contact diagrams as inhomogeneous terms. As we will soon see in detail, these linear equations and sufficiently simple thanks to the simple action of the equation of motion operator. They allow us to recursively solve all the crossed channel decomposition coefficients, when appropriate seed coefficients are inputted.  In one dimension, the seed coefficients are simply those of the lowest dimension operator in each double-trace tower, {\\it i.e.}, the double-trace operators with dimension $\\Delta_1+\\Delta_2$\n and $\\Delta_3+\\Delta_4$.  In higher dimensions, the seed coefficients are those of the double-trace operators with minimal conformal twists, {\\it i.e.}, $\\tau=\\Delta_1+\\Delta_2$ and $\\tau=\\Delta_3+\\Delta_4$. The same algorithm also applies to the crossed channel decomposition of conformal partial waves, since they also decompose into double-trace operators in the crossed channel and are annihilated by the equation of motion operator (therefore there are no inhomogeneous terms in the linear recursion relations).\n \nIn Section \\ref{1d}, we flesh out the above schematic comments in the simplest case of $\\mathrm{CFT}_1$. We derive the recursion relation for the crossed channel equation of motion operator acting on conformal blocks, and use these recursion relations to formulate a recursive algorithm. We discuss how to solve the crossed channel decomposition coefficients from the recursion relations. In particular we solve in a closed form the crossed channel decomposition of conformal partial waves, in terms of the Wilson polynomials. In Section \\ref{higherd}, we detail the parallel story for $\\mathrm{CFT}_d$ with $d>1$. \n \n \n\\subsection{One Dimension}\\label{1d}\nLet us consider an $AdS_2$ scalar exchange Witten diagram in the t-channel \n\\begin{equation}\nW^{t,exchange}_{\\Delta_E}=\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}\\frac{d^{d+1}z_2}{z_{20}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)G^{\\Delta_4}_{B\\partial}(z_1,x_4)\\Pi^{\\Delta_E}(z_1,z_2)G^{\\Delta_2}_{B\\partial}(z_2,x_2)G^{\\Delta_3}_{B\\partial}(z_2,x_3)\\;.\n\\end{equation}\nAs we reviewed in Section \\ref{SecExtoCon}, the diagram $W^{t,exchange}_{\\Delta_E}$ satisfies the following t-channel equation of motion identity \n\\begin{equation}\\label{tchannelCasimir}\n\\left[\\frac{1}{2}(\\mathbf{L}_2+\\mathbf{L}_3)^{AB}(\\mathbf{L}_2+\\mathbf{L}_3)_{AB}+M_E^2\\right]W^{t,exchange}_{\\Delta_E}=D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}\n\\end{equation}\nwhere $M_E^2=\\Delta_E(\\Delta_E-1)$. It is convenient to extract a kinematic factor in the convention of (\\ref{GUV}), so that we work with functions of the cross ratio $z$\n\\begin{equation}\n G(x_i)=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^{a}\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^{b}\\mathcal{G}(z)\\;.\n\\end{equation}\nThe equation of motion identity (\\ref{tchannelCasimir}) then becomes\n\\begin{equation}\\label{EOMtWexeqWcon}\n\\mathbf{EOM}^{(t)}[\\mathcal{W}^{t,exchange}](z)=\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)\n\\end{equation}\nwhere $\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)$ is $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)$ after stripping off the kinematic factor. It is straightforward to work out the action of the differential operator $\\mathbf{EOM}^{(t)}$ on a generic function $\\mathcal{G}(z)$. The action takes the following form\n\\begin{equation}\n\\begin{split}\n\\mathbf{EOM}^{(t)}[\\mathcal{G}]{}&=\\frac{1}{2}\\mathbf{D}_z(2a,2b)[\\mathcal{G}(z)]-\\mathbf{f}_0(2a,2b)[\\mathbf{D}_z(2a,2b)[\\mathcal{G}(z)]]+\\Delta_E(\\Delta_E-1)\\mathcal{G}(z)\\\\\n{}&-\\left(1-\\sum_{i=1}^4\\Delta_i\\right)\\mathbf{f}_1(2a,2b)[\\mathcal{G}(z)]-(\\Delta_1+\\Delta_2)(\\Delta_3+\\Delta_4)\\mathbf{f}_0(2a,2b)[\\mathcal{G}(z)]\\\\\n{}&- \\left(2ab+\\frac{1}{2}\\sum_{i=1}^4\\Delta_i(\\Delta_i-1)\\right)\\mathcal{G}(z)\\;.\n\\end{split}\n\\end{equation}\nIn the above expression, we have additionally defined two operators\n\\begin{equation}\n\\mathbf{f}_0(a,b)=\\frac{1}{z}-\\frac{1}{2}\\;, \\quad \\mathbf{f}_1(a,b)=(1-z)\\frac{d}{dz}-\\frac{1}{2}(a+b)\\;.\n\\end{equation}\n\nLet us now consider the action of $\\mathbf{EOM}^{(t)}$ when $\\mathcal{G}(z)$ is an s-channel conformal block. To proceed, the following properties of $\\mathbf{f}_0$ and $\\mathbf{f}_1$ \\cite{Dolan:2011dv}  will be useful to us\\footnote{Let us manifest the normalization by writing down the explicit expression for the 1d conformal block\n\\begin{equation}\ng^{(s)}_\\Delta(z)=z^\\Delta{}_2F_1(\\Delta+\\Delta_2-\\Delta_1,\\Delta+\\Delta_3-\\Delta_4;2\\Delta;z)\\;.\n\\end{equation}}\n\\begin{equation}\n\\begin{split}\n\\mathbf{f}_0(a,b)[g_\\Delta^{(s)}(z)]={}&g^{(s)}_{\\Delta-1}(z)+\\alpha_\\Delta(a,b) g^{(s)}_{\\Delta}(z)+\\beta_\\Delta(a,b) g^{(s)}_{\\Delta+1}(z)\\;,\\\\\n\\mathbf{f}_1(a,b)[g^{(s)}_\\Delta(z)]={}&g^{(s)}_{\\Delta-1}(z)+\\alpha_\\Delta(a,b) g^{(s)}_{\\Delta}(z)-(\\Delta-1)\\beta_\\Delta(a,b) g^{(s)}_{\\Delta+1}(z)\\;\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{split}\n\\alpha_\\Delta(a,b)={}&-\\frac{ab}{2\\Delta(\\Delta-1)}\\;,\\\\\n\\beta_\\Delta(a,b)={}&\\frac{(\\Delta+a)(\\Delta+b)(\\Delta-a)(\\Delta-b)}{4\\Delta^2(2\\Delta-1)(2\\Delta+1)}\\;.\n\\end{split}\n\\end{equation}\nUsing these two relations and the Casimir equation\n\\begin{equation}\n\\mathbf{D}_z(2a,2b)[g^{(s)}_{\\Delta}(z)]=\\Delta(\\Delta-1)g^{(s)}_{\\Delta}(z)\\;,\n\\end{equation}\nwe find the following three-term recursion relation for $\\mathbf{EOM}^{(t)}$ acting on an s-channel conformal block\n\\begin{equation}\\label{crosseomong}\n\\mathbf{EOM}^{(t)}[g^{(s)}_{\\Delta}(z)]=\\mu\\,g^{(s)}_{\\Delta-1}(z)+\\nu\\,g^{(s)}_{\\Delta}(z)+\\rho\\,g^{(s)}_{\\Delta+1}(z)\\;.\n\\end{equation}\nThe recursion coefficients are given by\n\\begin{equation}\n\\begin{split}\n\\mu={}&-(\\Delta -\\Delta_1-\\Delta_2) (\\Delta -\\Delta_3-\\Delta_4)\\;,\\\\\n\\nu={}&\\frac{2 a b (\\Delta_1+\\Delta_2-1) (\\Delta_3+\\Delta_4-1)}{(\\Delta -1) \\Delta }+(\\Delta_E -1) \\Delta_E+\\frac{1}{2} (\\Delta -1) \\Delta\\\\\n{}&-\\frac{1}{2} \\left((\\Delta_1-1)\\Delta_1+(\\Delta_2-1)\\Delta_2+(\\Delta_3-1)\\Delta_3+(\\Delta_4-1)\\Delta_4\\right)\\;,\\\\\n\\rho={}&-\\frac{(\\Delta -2 a) (2 a+\\Delta ) (\\Delta -2 b) (2 b+\\Delta ) (\\Delta +\\Delta_1+\\Delta_2-1) (\\Delta +\\Delta_3+\\Delta_4-1)}{4 \\Delta ^2 (2 \\Delta -1) (2 \\Delta +1)}\\;.\n\\end{split}\n\\end{equation}\n\nLet us take the conformal dimension $\\Delta$ to be the dimensions of the double-trace operators $\\Delta_1+\\Delta_2+n$. The recursion relation now reads\n\\begin{equation}\\label{recurblock1d}\n\\mathbf{EOM}^{(t)}[g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)]=\\mu^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n-1}(z)+\\nu^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\rho^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n+1}(z)\n\\end{equation}\nwhere $\\mu^{12}_n$, $\\nu^{12}_n$, $\\rho^{12}_n$ are $\\mu$, $\\nu$, $\\rho$ with $\\Delta=\\Delta_1+\\Delta_2+n$. From the explicit expression of $\\mu^{12}_n$ we can see that when $n=0$,\n\\begin{equation}\n\\mu^{12}_0=0\\;,\n\\end{equation}\nleaving only two double-trace blocks on the RHS of the recursion relation. The conformal block with dimension $\\Delta_1+\\Delta_2-1$ (which is not part of the double-trace spectrum) will not be generated. Therefore the action of $\\mathbf{EOM}^{(t)}$ preserves the double-trace spectrum $\\Delta_1+\\Delta_2+n$ with $n\\geq0$ and $n\\in \\mathbb{Z}$. A similar recursion relation also exists for double-trace operators with dimensions $\\Delta_3+\\Delta_4+n$, and can be obtained from the above relation by replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$.\n\nThese recursion relations give us an efficient way to compute the crossed channel decomposition. For simplicity, we will assume that the external conformal dimensions are generic such that no derivative conformal blocks appear. Special cases, such as $\\Delta_i=\\Delta_\\phi$, can be obtained from the generic case by taking a limit, and will be discussed in Appendix \\ref{appeqweight}.  Let us insert in (\\ref{EOMtWexeqWcon}) the s-channel decomposition of the t-channel exchange diagram $\\mathcal{W}^{t,exchange}$ \n\\begin{equation}\\label{texchange1dB12B34}\n\\mathcal{W}^{t,exchange}(z)=\\sum_{n=0}^\\infty B^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^\\infty B^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;.\n\\end{equation}\nThe relation (\\ref{recurblock1d}) for $g^{(s)}_{\\Delta_1+\\Delta_2+n}$ and its counterpart for $g^{(s)}_{\\Delta_3+\\Delta_4+n}$ allows us to express the action of $\\mathbf{EOM}^{(t)}$ on $\\mathcal{W}^{t,exchange}(z)$ again in terms of the sum of double-trace conformal blocks. By the identity (\\ref{EOMtWexeqWcon}), this decomposition should be equal to the decomposition (\\ref{Dfunctioningxi}) for the contact diagram $\\mathcal{D} _{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$\\footnote{The  decomposition result (\\ref{Dfunctioningxi}) also applies to the $AdS_2$ contact diagram. We just need to set $d=1$, and identify $g^{(s)}_{\\Delta_1+\\Delta_2+2n,0}$, $g^{(s)}_{\\Delta_3+\\Delta_4+2n,0}$ with the 1d conformal blocks $g^{(s)}_{\\Delta_1+\\Delta_2+2n}$, $g^{(s)}_{\\Delta_3+\\Delta_4+2n}$.} which takes the following form\n\\begin{equation}\n\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)=\\sum_{n=0}^\\infty a^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+2n}(z)+\\sum_{n=0}^\\infty a^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+2n}(z)\\;.\n\\end{equation}\nWe therefore arrive at the following linear recursion relations among the crossed channel decomposition coefficients\n\\begin{equation}\\label{recureq12inhomo}\n\\rho^{12}_{n-1}B^{12}_{n-1}+\\nu^{12}_{n}B^{12}_{n}+\\mu^{12}_{n+1}B^{12}_{n+1}=\n\\begin{cases}\na^{12}_{\\frac{n}{2}}\\;,\\quad n\\text{ even}\\;,\\\\\n0\\;,\\quad n\\text{ odd}\\;,\n\\end{cases}\\quad \\quad B^{12}_{-1}\\equiv 0\\;,\n\\end{equation}\n\\begin{equation}\\label{recureq34inhomo}\n\\rho^{34}_{n-1}B^{34}_{n-1}+\\nu^{34}_{n}B^{34}_{n}+\\mu^{34}_{n+1}B^{34}_{n+1}=\n\\begin{cases}\na^{34}_{\\frac{n}{2}}\\;,\\quad n\\text{ even}\\;,\\\\\n0\\;,\\quad n\\text{ odd}\\;,\n\\end{cases}\\quad \\quad B^{34}_{-1}\\equiv 0\\;.\n\\end{equation}\nIt is straightforward to solve this relation. As we have commented already, when $n=0$ the three-term relations reduce to two-term relations with only $B^{12}_0$, $B^{12}_1$ and $B^{34}_0$, $B^{34}_1$. Once the seed coefficients $B^{12}_0$, $B^{34}_0$ are determined, the sub-leading coefficients $B^{12}_n$, $B^{34}_n$ with $n\\geq 1$ can be recursively computed from the above three-term relations. In Appendix \\ref{appseed}, we will show how to compute the seed coefficients.  \n\n\n\\subsubsection*{Homogeneous Solution: the Wilson Polynomial}\nThe linear recursion equations (\\ref{recureq12inhomo}) and (\\ref{recureq34inhomo}) are inhomogeneous, and their solution gives the s-channel decomposition of a t-channel exchange Witten diagram. On the other hand, it is also interesting to consider the homogenous recursion equations for which the RHS' are  zero\n\\begin{equation}\\label{recureq12homo}\n\\rho^{12}_{n-1}\\widetilde{B}^{12}_{n-1}+\\nu^{12}_{n}\\widetilde{B}^{12}_{n}+\\mu^{12}_{n+1}\\widetilde{B}^{12}_{n+1}=0\\;,\n\\end{equation}\n\\begin{equation}\\label{recureq34homo}\n\\rho^{34}_{n-1}\\widetilde{B}^{34}_{n-1}+\\nu^{34}_{n}\\widetilde{B}^{34}_{n}+\\mu^{34}_{n+1}\\widetilde{B}^{34}_{n+1}=0\\;.\n\\end{equation}\n Such homogenous equations arise from a function $\\mathcal{G}(z)$ that satisfies\n\\begin{equation}\n\\mathbf{EOM}^{(t)}[\\mathcal{G}](z)=0\\;,\n\\end{equation}\nand admits an s-channel decomposition in terms of double-trace conformal blocks\n\\begin{equation}\n\\mathcal{G}(z)=\\sum_{n=0}^\\infty \\widetilde{B}^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^\\infty \\widetilde{B}^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;.\n\\end{equation}\nIn one dimension, such $\\mathcal{G}(z)$ can be either the t-channel conformal block $g^{(t)}_{\\Delta_E}(z)$, or the conformal partial wave $\\Psi^{(t)}_{\\Delta_E}(z)$. The solution to the homogenous recursion equations (\\ref{recureq12homo}), (\\ref{recureq34homo}) gives the crossed channel decomposition of the conformal block or partial wave (as we will see, their decomposition coefficients are only different by an overall factor). We will also see in the next subsection an extension of the story to $d>1$. There the solution to the homogenous recursion equations has to be interpreted as the crossed channel decomposition of a conformal partial wave. This is because Euclidean single-valuedness dictates that a single conformal block cannot be expressed in the crossed channel as infinitely many double-trace blocks, as we have already commented on at the end of Section \\ref{CPW}.\n \nLet us now solve the homogeneous equation (\\ref{recureq12homo}). The solution to (\\ref{recureq34homo}) can be obtained by simply replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$. Notice that a three-term recursion relation like (\\ref{recureq12homo}) is characteristic of systems of orthogonal polynomials. Indeed, after making the change of variables \n\\begin{equation}\n\\mathfrak{a}=\\frac{1}{2}+\\Delta_1-\\Delta_4\\;,\\quad \\mathfrak{b}=\\frac{1}{2}+\\Delta_2-\\Delta_3\\;,\\quad \\mathfrak{c}=-\\frac{1}{2}+\\Delta_1+\\Delta_4\\;,\\quad \\mathfrak{d}=-\\frac{1}{2}+\\Delta_2+\\Delta_3\\;, \n\\end{equation}\nand writing $\\widetilde{B}^{12}_n$ as\n\\begin{equation}\n\\widetilde{B}^{12}_n=-\\frac{\\Gamma (\\mathfrak{a}+\\mathfrak{c}+n) \\Gamma (\\mathfrak{a}+\\mathfrak{d}+n) \\Gamma (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)}{\\Gamma (n+1) \\Gamma (\\mathfrak{a}+\\mathfrak{c}) \\Gamma (\\mathfrak{a}+\\mathfrak{d}) \\Gamma (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1)}p_n\\;,\n\\end{equation}\nwe find that the recursion relation (\\ref{recureq12homo}) can be cast into the following form\n\\begin{equation}\\label{Wilsonrecur}\n-\\left(\\mathfrak{a}^2+x^2\\right)p_n(x)=\\mathcal{A}_n(p_{n+1}(x)-p_n(x))+\\mathcal{B}_n(p_{n-1}(x)-p_n(x))\n\\end{equation}\nwhere\n\\begin{equation}\nx=i\\left(\\Delta_E-\\frac{1}{2}\\right)\\;,\n\\end{equation}\n\\begin{equation}\\label{Wilsonrecurcoe}\n\\begin{split}\n\\mathcal{A}_n={}&\\frac{(\\mathfrak{a}+\\mathfrak{b}+n) (\\mathfrak{a}+\\mathfrak{c}+n) (\\mathfrak{a}+\\mathfrak{d}+n) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)}{(\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n)}\\;,\\\\\n\\mathcal{B}_n={}&\\frac{n (\\mathfrak{b}+\\mathfrak{c}+n-1) (\\mathfrak{b}+\\mathfrak{d}+n-1) (\\mathfrak{c}+\\mathfrak{d}+n-1)}{(\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-2) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1)}\\;.\n\\end{split}\n\\end{equation}\nThis is precisely the recursion relation that defines the Wilson polynomial \\cite{Wilson1977,Wilson:1980aa}. Since the solution to the recursion relation is unique up to an overall rescaling, we can set $p_0(x)=1$ for convenience. Then $p_n(x)$ can be expressed compactly as a ${}_4F_3$ function \\cite{Wilson1977,Wilson:1980aa}\n\\begin{equation}\\label{Wilsonpolyn}\np_n(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})={}_4F_3\\left(\\left.\\begin{array}{c}-n,n+\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}-1, \\mathfrak{a}+ix,\\mathfrak{a}-ix\\\\\\mathfrak{a}+\\mathfrak{b},\\mathfrak{a}+\\mathfrak{c},\\mathfrak{a}+\\mathfrak{d}\\end{array}\\right.;1\\right)\\;.\n\\end{equation}\nThis function $p_n(x)$ is a polynomial in $x^2$ of degree $n$, as is already clear from the recursion relation (\\ref{Wilsonrecur}). \n\nNow let us use this solution in the s-channel decomposition of a 1d t-channel conformal block. The decomposition takes the following form\n\\begin{equation}\\label{tconfblockdecomp}\n\\begin{split}\n\\frac{z^{\\Delta_1+\\Delta_2}}{(1-z)^{\\Delta_2+\\Delta_3}}g^{(t)}_{\\Delta_E}(z)={}&\\frac{z^{\\Delta_1+\\Delta_2}}{(1-z)^{\\Delta_2+\\Delta_3}}{}_2F_1(\\Delta_E-\\Delta_3+\\Delta_2,\\Delta_E+\\Delta_1-\\Delta_4,2\\Delta_E,1-z)\\\\\n={}&\\sum_{n=0}^{\\infty}b^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^{\\infty}b^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;,\n\\end{split}\n\\end{equation}\nand one can check that the LHS of (\\ref{tconfblockdecomp}) is annihilated by the equation of motion operator $\\mathbf{EOM}^{(t)}$. The action on the RHS thus gives rise to the recursion equations (\\ref{recureq12homo}) and (\\ref{recureq34homo}). The above solution to the homogenous equations (\\ref{recureq12homo}) then determines the ratios $b^{12}_n\/b^{12}_0$ of the decomposition coefficients to be\n\\begin{equation}\\label{b12ratio}\n\\frac{b^{12}_n}{b^{12}_0}=\\frac{\\Gamma (n+1) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)_n}{(\\mathfrak{a}+\\mathfrak{c})_n (\\mathfrak{a}+\\mathfrak{d})_n}p_n(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})\\;.\n\\end{equation}\nBy further comparing the two sides of (\\ref{tconfblockdecomp}) expanded at $z=1^-$, we can also determine the leading coefficient\n\\begin{equation}\nb^{12}_0=\\frac{\\Gamma (2 \\Delta_E) \\Gamma (-\\Delta_1-\\Delta_2+\\Delta_3+\\Delta_4)}{\\Gamma (-\\Delta_1+\\Delta_4+\\Delta_E) \\Gamma (-\\Delta_2+\\Delta_3+\\Delta_E)}\\;,\n\\end{equation}\nand thereby obtaining the crossed channel decomposition. Similarly, $b^{34}_n$ can also be obtained by solving recursion relations. Its solution is simply the solution for $b^{12}_n$ with $\\Delta_1$, $\\Delta_2$ replaced by $\\Delta_3$, $\\Delta_4$. The above crossed channel decomposition coefficients can also be obtained from the alpha space techniques \\cite{Hogervorst:2017sfd}.\n\nNote that the coefficient ratio (\\ref{b12ratio}) depends on $\\Delta_E$ only via the shadow symmetric combination $x^2$. This implies that the decomposition coefficients for the shadow conformal block have the same ratio (\\ref{b12ratio}). Furthermore, since the conformal partial wave (\\ref{CPWdef}) is just the linear combination of the conformal block and its shadow, its crossed channel decomposition coefficients will also have the same ratio.\n\n\n\\subsection{$d>1$ Dimensions}\\label{higherd}\nThe story for $d>1$ is similar to what we have seen for $d=1$. We will organize this subsection as the follows.  We start by obtaining the action of the t-channel equation of motion operator on a generic conformal block. We will find the result can be expressed as the linear combination of five conformal blocks with shifted dimensions and spins. Then we restrict the quantum numbers of the conformal blocks to those of the double-trace operators which appear in the crossed channel decomposition. The spectra of these operators are preserved by the recursion relation. This gives us an efficient algorithm for recursively solving the crossed channel decomposition coefficients. We can consider two different problems depending on whether the recursion equations are homogenous or inhomogeneous. The latter corresponds to the s-channel decomposition of a t-channel exchange Witten diagram, while the former corresponds to the decomposition of an t-channel conformal partial wave. We also consider the special case of equal external weights, where our results imply recursion relations for the anomalous dimensions.\n\n\n\\subsubsection*{Action of the t-channel equation of motion operator}\nConsider a t-channel exchange Witten diagram with dimension $\\Delta_E$ and spin $\\ell_E$. The t-channel equation of motion give the identity \n\\begin{equation}\\label{higherdtchEOM}\n\\left(\\frac{1}{2}(\\mathbf{L}_2+\\mathbf{L}_3)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right)W^{t,exchange}_{\\Delta_E,\\ell_E}=\\sum_I c_I W^{contact}_I\\;.\n\\end{equation}\nThe LHS of the identity defines a second order differential operator $\\mathbf{EOM}^{(t)}$ which acts on $\\mathcal{G}(z,\\bar{z})$\n\\begin{equation}\\small\n\\begin{split}\n{}&\\mathbf{EOM}^{(t)}[\\mathcal{G}(z,\\bar{z})]=\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]-\\mathbf{F}_0(a,b)[\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]]+M_E^2\\, \\mathcal{G}(z,\\bar{z})\\\\\n{}&+\\mathbf{F}_2(a,b)[\\mathcal{G}(z,\\bar{z})]+\\left(\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4-2+2\\epsilon\\right)\\mathbf{F}_1(a,b)[\\mathcal{G}(z,\\bar{z})]\\\\\n{}&-\\frac{1}{2}(\\Delta_1+\\Delta_2)(\\Delta_3+\\Delta_4)\\mathbf{F}_0(a,b)[\\mathcal{G}(z,\\bar{z})]\\\\\n{}&+\\left(2+\\epsilon(\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4)-\\frac{(\\Delta_1-1)^2+(\\Delta_2-1)^2+(\\Delta_3-1)^2+(\\Delta_4-1)^2}{2}-2ab\\right)\\mathcal{G}(z,\\bar{z})\n\\end{split}\n\\end{equation}\nwhere $\\mathbf{F}_i$ is a differential operator of order $i$\n\\begin{equation}\n\\begin{split}\n\\mathbf{F}_0(a,b)={}&\\frac{1}{z}+\\frac{1}{\\bar{z}}-1\\;,\\\\\n\\mathbf{F}_1(a,b)={}&(1-z)\\frac{\\partial}{\\partial z}+(1-\\bar{z})\\frac{\\partial}{\\partial \\bar{z}}\\;,\\\\\n\\mathbf{F}_2(a,b)={}&\\frac{z-\\bar{z}}{z\\bar{z}}(\\mathbf{D}_z(a,b)-\\mathbf{D}_{\\bar{z}}(a,b))\\;.\n\\end{split}\n\\end{equation}\nHere it will turn out to be more convenient to use the normalization of \\cite{Dolan:2011dv} for the conformal blocks, which are denoted by $F_{\\lambda_1,\\lambda_2}(z,\\bar{z})$ \n\\begin{equation}\n\\begin{split}\nF_{\\lambda_1,\\lambda_2}(z,\\bar{z})={}&\\mathcal{N}_{\\epsilon,\\ell}g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})\\;,\\\\\n\\mathcal{N}_{\\epsilon,\\ell}={}&\\frac{(\\epsilon)_\\ell}{(2\\epsilon)_\\ell}\\;,\\\\\n\\lambda_1=\\frac{\\Delta+\\ell}{2}\\;,{}&\\quad \\lambda_2=\\frac{\\Delta-\\ell}{2}\\;.\n\\end{split}\n\\end{equation}\nThe $\\mathbf{F}_i$ operators have the following recursion relations on conformal blocks \\cite{Dolan:2011dv}\n\\begin{equation}\\label{Ffivetermrecur}\n\\mathbf{F}_i(a,b)[\\mathcal{F}_{\\lambda_1,\\lambda_2}]=r_i F_{\\lambda_1,\\lambda_2-1}+s_iF_{\\lambda_1-1,\\lambda_2}+t_iF_{\\lambda_1+1,\\lambda_2}+u_iF_{\\lambda_1,\\lambda_2+1}+w_iF_{\\lambda_1,\\lambda_2}\\;,\n\\end{equation}\nwith various coefficients defined as follows. The coefficients with label $i=0$ are\n\\begin{equation}\n\\begin{split}\n{}&r_0=\\frac{\\lambda_1-\\lambda_2+2 \\epsilon }{\\lambda_1-\\lambda_2+\\epsilon }\\;,\\quad s_0=\\frac{\\lambda_1-\\lambda_2}{\\lambda_1-\\lambda_2+\\epsilon }\\;,\\\\\n{}&t_0=\\frac{(\\lambda_1+\\lambda_2-1) (\\lambda_1+\\lambda_2-2 \\epsilon ) (\\lambda_1-\\lambda_2+2 \\epsilon )}{(\\lambda_1+\\lambda_2-\\epsilon -1) (\\lambda_1+\\lambda_2-\\epsilon ) (\\lambda_1-\\lambda_2+\\epsilon )}\\beta_{\\lambda_1}(a,b)\\;,\\\\\n{}&u_0=\\frac{(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1) (\\lambda_1+\\lambda_2-2 \\epsilon )}{(\\lambda_1+\\lambda_2-\\epsilon -1) (\\lambda_1+\\lambda_2-\\epsilon ) (\\lambda_1-\\lambda_2+\\epsilon )}\\beta_{\\lambda_2-\\epsilon}(a,b)\\;,\\\\\n{}&w_0=-\\frac{(c_{\\lambda_1\\lambda_2}+2\\epsilon)}{2\\lambda_1(\\lambda_1-1)(\\lambda_2-\\epsilon)(\\lambda_2-1-\\epsilon)} ab\\;.\n\\end{split}\n\\end{equation}\nIn terms of the $i=0$ coefficients, the higher $i$ coefficients are given by\\footnote{There are two typos in (4.29) of \\cite{Dolan:2011dv}. The first typo is in the expression for $s_2$ and is corrected below. The other one is in $s_3$, and the correct expression should be\n\\begin{equation}\ns_3=(\\lambda_1+\\epsilon)(\\lambda_1-\\lambda_2+2\\epsilon)(\\lambda_1+\\lambda_2-1)s_0\\;.\n\\end{equation}}\n\\begin{equation}\n\\begin{split}\n{}&r_1=\\lambda_2 r_0\\;,\\quad s_1=(\\lambda_1+\\epsilon)s_0\\;,\\quad t_1=-(\\lambda_1-1-\\epsilon)t_0\\;,\\quad u_1=-(\\lambda_2-1-2\\epsilon)u_0\\;,\\\\\n{}& w_1=(1+\\epsilon)w_0\\;,\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n{}& r_2=(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1)r_0\\;,\\quad s_2= -(\\lambda_1+\\lambda_2-1) (\\lambda_1-\\lambda_2+2\\epsilon )s_0\\;,\\\\\n{}& t_2=-(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-2 \\epsilon -1)t_0\\;,\\quad u_2=(\\lambda_1-\\lambda_2+2 \\epsilon ) (\\lambda_1+\\lambda_2-2 \\epsilon -1)u_0\\;,\\\\\n{}& w_2=-\\frac{(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1) (\\lambda_1-\\lambda_2+2 \\epsilon ) (\\lambda_1+\\lambda_2-2 \\epsilon -1)}{2\\lambda_1 (\\lambda_1-1) (\\lambda_2-\\epsilon ) (\\lambda_2-\\epsilon -1)}ab\\;.\n\\end{split}\n\\end{equation}\nUsing the recursion relations (\\ref{Ffivetermrecur}) and the Casimir equation\n\\begin{equation}\n2\\mathbf{\\Delta}_\\epsilon(a,b)[F_{\\lambda_1,\\lambda_2}]=C^{(2)}_{\\Delta,\\ell}F_{\\lambda_1,\\lambda_2}\\;,\n\\end{equation}\n the action of $\\mathbf{EOM}^{(t)}$ on a conformal block $F_{\\lambda_1,\\lambda_2}$ can be expressed as the linear combination of five conformal blocks with shifted dimensions and spins\n\\begin{equation}\\label{EOMrecuranyd}\n\\mathbf{EOM}^{(t)}[\\mathcal{F}_{\\lambda_1,\\lambda_2}]=\\mathfrak{R}\\, F_{\\lambda_1,\\lambda_2-1}+\\mathfrak{S}\\,F_{\\lambda_1-1,\\lambda_2}+\\mathfrak{T}\\,F_{\\lambda_1+1,\\lambda_2}+\\mathfrak{U}\\,F_{\\lambda_1,\\lambda_2+1}+\\mathfrak{W}\\,F_{\\lambda_1,\\lambda_2}\\;.\n\\end{equation}\nThe coefficients are determined to be\\footnote{Here we can appreciate the advantage of using the normalization of \\cite{Dolan:2011dv}: the coefficients $\\mathfrak{S}$ and $\\mathfrak{U}$ are simply related to $\\mathfrak{R}$ and $\\mathfrak{T}$ with $\\ell\\to-\\ell-2\\epsilon$. Otherwise there will be an additional factor which depends on $\\epsilon$ and $\\ell$. }\n\\begin{equation}\\label{EOMrecuranydcoe}\n\\begin{split}\n\\mathfrak{R}={}&-\\frac{(\\ell +2 \\epsilon ) (-\\Delta +\\Delta_1+\\Delta_2+\\ell ) (-\\Delta +\\Delta_3+\\Delta_4+\\ell )}{2 (\\ell +\\epsilon )}\\,\\\\\n\\mathfrak{T}={}&-\\frac{(2 a+\\Delta +\\ell ) (-2 a+\\Delta +\\ell ) (2 b+\\Delta +\\ell ) (-2 b+\\Delta +\\ell )}{32 (\\Delta -\\epsilon )  (\\Delta +\\ell )^2 (\\Delta -\\epsilon -1) }\\\\\n{}&\\times \\frac{(\\Delta +\\Delta_1+\\Delta_2+\\ell -2 \\epsilon -2) (\\Delta +\\Delta_3+\\Delta_4+\\ell -2 \\epsilon -2)(\\Delta -1) (\\Delta -2 \\epsilon )(\\ell +2 \\epsilon)}{(\\Delta +\\ell -1)(\\Delta +\\ell +1) (\\ell +\\epsilon )}\\,\\\\\n\\mathfrak{S}={}&\\mathfrak{R}\\big|_{\\ell\\to-\\ell-2\\epsilon}\\;,\\\\\n\\mathfrak{U}={}&\\mathfrak{T}\\big|_{\\ell\\to-\\ell-2\\epsilon}\\;.\\\\\n\\mathfrak{W}={}&C^{(2)}_{\\Delta_E,\\ell_E}+\\frac{C^{(2)}_{\\Delta,\\ell}-\\sum_{i=1}^4C^{(2)}_{\\Delta_i,\\ell_i=0}}{2}+\\frac{2ab(\\Delta_1+\\Delta_2-2\\epsilon-2)(\\Delta_3+\\Delta_4-2\\epsilon-2)(C^{(2)}_{\\Delta,\\ell}+4\\epsilon)}{(\\Delta +\\ell -2) (\\Delta +\\ell ) (-\\Delta +\\ell +2 \\epsilon ) (-\\Delta +\\ell +2 \\epsilon +2)}\n\\end{split}\n\\end{equation}\nNote that the above recursion relation has the following desirable features similar to those of (\\ref{crosseomong}). We first look at the factor $\\mathfrak{R}$ multiplying the conformal block $F_{\\lambda_1,\\lambda_2-1}$ which has conformal dimension $\\Delta-1$ and twist $\\Delta-\\ell$. This factor vanishes $\\mathfrak{R}$ when the conformal block $F_{\\lambda_1,\\lambda_2}$ has the minimal twist for a double-trace operator formed with $\\mathcal{O}_1$ and $\\mathcal{O}_2$ or with $\\mathcal{O}_3$ and $\\mathcal{O}_4$, {\\it i.e.}, \n\\begin{equation}\n\\mathfrak{R}=0\\;,\\quad\\text{when}\\quad\\Delta=\\ell+\\Delta_1+\\Delta_2\\;,\\quad \\text{or}\\quad \\Delta=\\ell+\\Delta_3+\\Delta_4\\;.\n\\end{equation}\nLet us also notice that the coefficients $\\mathfrak{S}$ and $\\mathfrak{U}$, which multiply conformal blocks with shifted spin $\\ell-1$, both  contain a factor $\\ell$. It implies that when the spin of $F_{\\lambda_1,\\lambda_2}$ is zero, both $\\mathfrak{S}$ and $\\mathfrak{U}$ vanish and no blocks with negative spins will be generated on the RHS. Moreover, the twists of conformal blocks are always shifted by an even integer. It is therefore not hard to see that these properties of the coefficients guarantee that the double-trace spectra, labelled by $\\{\\Delta,\\ell\\}$ with\n\\begin{equation}\n\\Delta=\\Delta_1+\\Delta_2+2n+\\ell\\quad \\text{and}\\quad \\Delta=\\Delta_3+\\Delta_4+2n+\\ell\\;,\\quad n,\\ell=0,1,2,\\ldots,\n\\end{equation}\nare preserved by the recursion relation (\\ref{EOMrecuranyd}). \n\n\\subsubsection*{Reduction to 1d}\nWe can show that the recursion relation in $d>1$ reduces to the 1d recursion relation (\\ref{crosseomong}) in an appropriate limit. Setting $\\epsilon=-\\frac{1}{2}$, $\\ell_E=0$, $\\ell=0$ and restricting to $z=\\bar{z}$ we find from  (\\ref{EOMrecuranydcoe}) that\n\\begin{equation}\n\\mathfrak{S}=\\mathfrak{U}=0\\;,\\quad \\mathfrak{R}=\\mu\\;,\\quad \\mathfrak{W}=\\nu\\;,\\quad \\mathfrak{T}=\\rho\\;.\n\\end{equation}\nWe then use the following identity for conformal blocks\\footnote{In the second identity, $\\ell=1$ for the conformal block $F_{\\lambda_1\\lambda_2}$. Note that $\\ell$ appears in the Casimir eigenvalue as $\\ell(\\ell+2\\epsilon)=\\ell(\\ell-1)$, both $\\ell=0$ and $\\ell=1$ give the same 1d conformal block.} \\cite{Dolan:2011dv}\n\\begin{equation}\nF^{(-\\frac{1}{2})}_{\\frac{\\Delta}{2}\\frac{\\Delta}{2}}(z,z)=g_\\Delta(z)\\;,\\quad F^{(-\\frac{1}{2})}_{\\frac{\\Delta+1}{2}\\frac{\\Delta-1}{2}}(z,z)=g_\\Delta(z)\\;,\n\\end{equation}\nto reduce the higher dimensional conformal blocks to one dimensional blocks $g_\\Delta(z)$. We find that the 1d recursion relation (\\ref{crosseomong}) is precisely reproduced.\n\n\n\\subsubsection*{The recursive algorithm for the crossed channel decomposition}\nLet us now use the recursion relation (\\ref{EOMrecuranyd}) to formulate an algorithm for the crossed channel decomposition of an exchange Witten diagram.  We start with the s-channel decomposition for $W^{t,\\,exchange}_{\\Delta_E,\\ell_E}$, with the assumption that the external dimensions $\\Delta_i$ are generic\n\\begin{equation}\nW^{t,\\,exchange}_{\\Delta_E,\\ell_E}=\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\underbrace{\\mathcal{B}^{12}_{n,J} \\mathcal{N}_{\\epsilon,J}}_{B^{12}_{n,J}} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\underbrace{\\mathcal{B}^{34}_{n,J}\\mathcal{N}_{\\epsilon,J}}_{B^{34}_{n,J}} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nHere $\\mathcal{N}_{\\epsilon,J}=\\frac{(\\epsilon)_J}{(2\\epsilon)_J}$ is a normalization factor such that $\\mathcal{N}_{\\epsilon,J} g^{(s)}_{\\Delta,J}=F_{\\frac{\\Delta+J}{2},\\frac{\\Delta-J}{2}}$. We insert this decomposition in (\\ref{higherdtchEOM}). Thanks to the recursion relation (\\ref{EOMrecuranyd}), the action of $\\mathbf{EOM}^{(t)}$ can be rewritten as the linear combination of double-trace blocks. The resulting expansion should be equal to the conformal block decomposition of the contact terms on the RHS\n\\begin{equation}\n\\sum_I c_I W_I^{contact}=\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nFor convenience, let us define $\\mathfrak{R}^{12}_{n,\\ell}$, $\\mathfrak{S}^{12}_{n,\\ell}$, $\\mathfrak{T}^{12}_{n,\\ell}$, $\\mathfrak{U}^{12}_{n,\\ell}$, $\\mathfrak{W}^{12}_{n,\\ell}$ to be $\\mathfrak{R}$, $\\mathfrak{S}$, $\\mathfrak{T}$, $\\mathfrak{U}$, $\\mathfrak{W}$ with $\\Delta=\\Delta_1+\\Delta_2+2n+\\ell$, and  $\\mathfrak{R}^{34}_{n,\\ell}$, $\\mathfrak{S}^{34}_{n,\\ell}$, $\\mathfrak{T}^{34}_{n,\\ell}$, $\\mathfrak{U}^{34}_{n,\\ell}$, $\\mathfrak{W}^{34}_{n,\\ell}$ to be $\\mathfrak{R}$, $\\mathfrak{S}$, $\\mathfrak{T}$, $\\mathfrak{U}$, $\\mathfrak{W}$ with $\\Delta=\\Delta_3+\\Delta_4+2n+\\ell$.\nThen we arrive at the following recursion relations for the crossed channel decomposition coefficients. The recursion relation for $\\mathcal{B}^{12}_{n,\\ell}$ is\n\\begin{equation}\\label{Recur12nleqatilde12nl}\n\\begin{split}\nRecur^{12}_{n,\\ell}\\equiv\\mathfrak{R}^{12}_{n+1,\\ell-1}\\mathcal{B}^{12}_{n+1,\\ell-1}+\\mathfrak{S}^{12}_{n,\\ell+1}\\mathcal{B}^{12}_{n,\\ell+1}+\\mathfrak{T}^{12}_{n,\\ell-1}\\mathcal{B}^{12}_{n,\\ell-1}{}&\\\\\n+\\mathfrak{U}^{12}_{n-1,\\ell+1}\\mathcal{B}^{12}_{n-1,\\ell+1}+\\mathfrak{W}^{12}_{n,\\ell}\\mathcal{B}^{12}_{n,\\ell}{}&=\\tilde{a}^{12}_{n,\\ell}\\;,\n\\end{split}\n\\end{equation}\nwith $n\\geq0$, $\\ell\\geq0$, and \n\\begin{equation}\n\\mathcal{B}^{12}_{n,-1}=\\mathcal{B}^{12}_{-1,\\ell}=0\\;.\n\\end{equation}\nThe recursion relation for $\\mathcal{B}^{34}_{n,\\ell}$ takes the same form, and can be obtained by replacing 1, 2 with 3, 4.\n\nLet us discuss how we can solve these recursion relations. Without the loss of generality, we focus on the coefficients $\\mathcal{B}^{12}_{n,\\ell}$ and their recursion relations. Just as in the 1d case where we need to input the seed coefficient for the double-trace operator with minimal conformal dimension, here we need to input the OPE coefficients of the double-trace operators with minimal {\\it twist}, {\\it i.e.}, $\\mathcal{B}^{12}_{0,\\ell}$. These seed coefficients can be obtained from applying the inversion formula \\cite{Caron-Huot:2017vep} for $\\ell>\\ell_E$ \\cite{Liu:2018jhs,Cardona:2018dov}, or from Mellin space \\cite{Costa:2014kfa,Sleight:2018epi,Sleight:2018ryu}.  After inputting the seed coefficients, the coefficients $\\mathcal{B}^{12}_{n,\\ell}$ of operators with higher twists are uniquely determined. More precisely,  the equation \n\\begin{equation}\\label{eqn12nllgeq1}\nRecur^{12}_{n-1,\\ell+1}=\\tilde{a}^{12}_{n-1,\\ell+1}\\;,\\quad     n\\geq1\\;, \\ell\\geq0\\;,\n\\end{equation}\ndetermines the coefficient $\\mathcal{B}^{12}_{n,\\ell}$ in terms of $\\mathcal{B}^{12}_{n',\\ell}$ with $n'<n$ \n\\begin{equation}\\label{solB12nl}\n\\mathcal{B}^{12}_{n,\\ell}=\\frac{\\tilde{a}^{12}_{n-1,\\ell+1}-\\mathfrak{S}^{12}_{n-1,\\ell+2}\\mathcal{B}^{12}_{n-1,\\ell+2}-\\mathfrak{T}^{12}_{n-1,\\ell}\\mathcal{B}^{12}_{n-1,\\ell}-\\mathfrak{U}^{12}_{n-2,\\ell+2}\\mathcal{B}^{12}_{n-2,\\ell+2}-\\mathfrak{W}^{12}_{n-1,\\ell+1}\\mathcal{B}^{12}_{n-1,\\ell+1}}{\\mathfrak{R}^{12}_{n,\\ell}}\\;.\n\\end{equation}\nWe can start with $Recur^{12}_{0,\\ell+1}=\\tilde{a}^{12}_{0,\\ell+1}$ to solve $\\mathcal{B}^{12}_{1,\\ell}$ in terms of $\\mathcal{B}^{12}_{0,\\ell}$ (note $\\mathcal{B}^{12}_{-1,\\ell}=0$), and work our way up to any $\\mathcal{B}^{12}_{n,\\ell}$ by considering equations $Recur^{12}_{i,\\ell+1}=\\tilde{a}^{12}_{i,\\ell+1}$ with increasing $i$. This solving strategy is pictorially illustrated by Figure \\ref{recuralgo}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.68]{recuralgo}\n\\caption{Illustration of the recursive algorithm for solving the OPE coefficient $\\mathcal{B}^{12}_{n,\\ell}$. The equation $Recur^{12}_{n-1,\\ell+1}=\\tilde{a}^{12}_{n-1,\\ell+1}$ solves $\\mathcal{B}^{12}_{n,\\ell}$ in terms of $\\mathcal{B}^{12}_{n-1,\\ell}$, $\\mathcal{B}^{12}_{n-1,\\ell+1}$, $\\mathcal{B}^{12}_{n-1,\\ell+2}$ and $\\mathcal{B}^{12}_{n-2,\\ell+2}$ which have smaller values of $n$. This procedure is iterated until one reaches the $n=0$ data which are the seed coefficients one inputs.}\n\\label{recuralgo}\n\\end{center}\n\\end{figure}\n\nAlthough the recursion relations (\\ref{Recur12nleqatilde12nl}) are not enough to fix the leading twist OPE coefficients $\\mathcal{B}^{12}_{0,\\ell}$, we should note that they still impose nontrivial constraints on the possible values that $\\mathcal{B}^{12}_{0,\\ell}$ can take as a function of $\\ell$. We note that the equations (\\ref{eqn12nllgeq1}) have not exhausted all the cases of (\\ref{eqn12nllgeq1}). The remaining set of equations \n\\begin{equation}\nRecur^{12}_{n,0}=\\tilde{a}^{12}_{n,0}\\;,\\quad n\\geq0\n\\end{equation}\ngive the following identities\n\\begin{equation}\\label{relation12leq0}\n\\mathfrak{S}^{12}_{n,1}\\mathcal{B}^{12}_{n,1}+\\mathfrak{U}^{12}_{n-1,1}\\mathcal{B}^{12}_{n-1,1}+\\mathfrak{W}^{12}_{n,0}\\mathcal{B}^{12}_{n,0}=\\tilde{a}^{12}_{n,0}\\;.\n\\end{equation}\nBecause $\\mathcal{B}^{12}_{n,\\ell}$ have been solved by (\\ref{eqn12nllgeq1}) in terms $\\mathcal{B}^{12}_{0,\\ell}$, the relations (\\ref{relation12leq0}) impose infinitely many constraints on $\\mathcal{B}^{12}_{0,\\ell}$. More precisely, it follows from (\\ref{solB12nl}) that $\\mathcal{B}^{12}_{n,\\ell}$ depends on $\\mathcal{B}^{12}_{0,\\ell'}$ with $\\ell'=\\ell,\\ell+1,\\ldots,\\ell+2n$.The $n$-th relation (\\ref{relation12leq0}) therefore gives a constraint for the first $2n+2$ coefficients (see Figure \\ref{recurconstrs})\n\\begin{equation}\n\\mathcal{B}^{12}_{0,\\ell}\\;,\\quad \\ell=0,1,\\ldots 2n+1\\;.\n\\end{equation}\nThe number of constraints is ``half as many'' as the number of $\\mathcal{B}^{12}_{0,\\ell}$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.68]{recurconstrs}\n\\caption{Illustration of the constraints on $\\mathcal{B}^{12}_{0,0}$, $\\mathcal{B}^{12}_{0,1}$, \\ldots, $\\mathcal{B}^{12}_{0,2n+1}$ imposed by the equation $Recur^{12}_{n,0}=\\tilde{a}^{12}_{n,0}$.}\n\\label{recurconstrs}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection*{The homogenous equations}\nBy setting the RHS' of (\\ref{Recur12nleqatilde12nl}) to zero, we obtain the homogenous equations \n\\begin{equation}\\label{Recur12nleq0nl}\n\\begin{split}\n\\widetilde{Recur}^{12}_{n,\\ell}\\equiv\\mathfrak{R}^{12}_{n+1,\\ell-1}\\tilde{\\mathcal{B}}^{12}_{n+1,\\ell-1}+\\mathfrak{S}^{12}_{n,\\ell+1}\\tilde{\\mathcal{B}}^{12}_{n,\\ell+1}+\\mathfrak{T}^{12}_{n,\\ell-1}\\tilde{\\mathcal{B}}^{12}_{n,\\ell-1}{}&\\\\\n+\\mathfrak{U}^{12}_{n-1,\\ell+1}\\tilde{\\mathcal{B}}^{12}_{n-1,\\ell+1}+\\mathfrak{W}^{12}_{n,\\ell}\\tilde{\\mathcal{B}}^{12}_{n,\\ell}{}&=0\\;,\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{Recur34nleq0nl}\n\\begin{split}\n\\widetilde{Recur}^{34}_{n,\\ell}\\equiv\\mathfrak{R}^{34}_{n+1,\\ell-1}\\tilde{\\mathcal{B}}^{34}_{n+1,\\ell-1}+\\mathfrak{S}^{34}_{n,\\ell+1}\\tilde{\\mathcal{B}}^{34}_{n,\\ell+1}+\\mathfrak{T}^{34}_{n,\\ell-1}\\tilde{\\mathcal{B}}^{34}_{n,\\ell-1}{}&\\\\\n+\\mathfrak{U}^{34}_{n-1,\\ell+1}\\tilde{\\mathcal{B}}^{34}_{n-1,\\ell+1}+\\mathfrak{W}^{34}_{n,\\ell}\\tilde{\\mathcal{B}}^{34}_{n,\\ell}{}&=0\\;.\n\\end{split}\n\\end{equation}\nThese equations constrain the s-channel decomposition coefficients of a t-channel conformal partial wave, which takes the following form\n\\begin{equation}\n\\begin{split}\n\\frac{(z\\bar{z})^{\\frac{\\Delta_1+\\Delta_2}{2}}}{((1-z)(1-\\bar{z}))^{\\frac{\\Delta_2+\\Delta_3}{2}}}\\Psi^{(t)}_{\\Delta_E,\\ell_E}(z,\\bar{z})={}&\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\tilde{\\mathcal{B}}^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(z,\\bar{z})\\\\\n{}&+\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\tilde{\\mathcal{B}}^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(z,\\bar{z})\\;.\n\\end{split}\n\\end{equation}\nAs in the inhomogeneous case,  the recursion equations \n\\begin{equation}\n\\widetilde{Recur}^{12}_{n,\\ell}=\\widetilde{Recur}^{34}_{n,\\ell}=0\\;, \\quad \\ell\\geq1\n\\end{equation}\nallow us to solve $\\tilde{\\mathcal{B}}^{12}_{n,\\ell}$ in terms of $\\tilde{\\mathcal{B}}^{12}_{0,\\ell}$. On the other hand, the remaining equations with $\\ell=0$\n\\begin{equation}\n\\widetilde{Recur}^{12}_{n,0}=\\widetilde{Recur}^{34}_{n,0}=0\n\\end{equation}\nput constraints on the values of $\\tilde{\\mathcal{B}}^{12}_{0,\\ell}$. For example, by looking at the equations with $n=0$ we find that \nsimple rational ratios \n\\begin{equation}\\label{simpleratio12}\n\\frac{\\tilde{\\mathcal{B}}^{12}_{0,1}}{\\tilde{\\mathcal{B}}^{12}_{0,0}}=\\frac{1}{(d+\\Delta_1+\\Delta_2-\\Delta_3-\\Delta_4)}\\left(C^{(2)}_{\\Delta_E,\\ell_E}+\\Delta_1\\Delta_2+\\frac{\\Delta_1\\Delta_3(d-\\Delta_3)+\\Delta_2\\Delta_4(d-\\Delta_4)}{(\\Delta_1+\\Delta_2)}\\right)\\;,\n\\end{equation} \n\\begin{equation}\\label{simpleratio34}\n\\frac{\\tilde{\\mathcal{B}}^{34}_{0,1}}{\\tilde{\\mathcal{B}}^{34}_{0,0}}=\\frac{1}{(d+\\Delta_3+\\Delta_4-\\Delta_1-\\Delta_2)}\\left(C^{(2)}_{\\Delta_E,\\ell_E}+\\Delta_3\\Delta_4+\\frac{\\Delta_1\\Delta_3(d-\\Delta_1)+\\Delta_2\\Delta_4(d-\\Delta_2)}{(\\Delta_3+\\Delta_4)}\\right)\n\\end{equation} \nfor the leading twist coefficients for any spacetime dimension $d>1$. \n\n\nBefore we end this subsection let us consider a special case where the t-channel conformal partial wave has $\\ell_E=0$, and the external weights are degenerate $\\Delta_i=\\Delta_\\phi$. In this case the solution to (\\ref{Recur12nleq0nl}) and (\\ref{Recur34nleq0nl}) has the interpretation of anomalous dimensions (see Appendix \\ref{appeqweight}). The seed coefficients for the scalar exchange was obtained first in \\cite{Giombi:2018vtc}\\footnote{A straightforward exercise is to check that the expressions satisfy the relations (\\ref{simpleratio12}), (\\ref{simpleratio34}).}\n\\begin{equation}\n\\begin{split}\n\\tilde{\\mathcal{B}}^{12}_{0,\\ell}={}&\\frac{4 \\Gamma (\\Delta_E) \\Gamma \\left(\\frac{d-\\Delta_E}{2}\\right)^2 \\Gamma (d+\\ell -2) \\Gamma (\\ell +\\Delta_\\phi )^2 \\Gamma (\\ell +2 \\Delta_\\phi -1)}{\\Gamma (d-1) \\Gamma \\left(\\frac{\\Delta_E}{2}\\right)^2 \\Gamma (\\Delta_\\phi )^2 \\Gamma (\\ell +1) \\Gamma \\left(\\frac{d}{2}-\\Delta_E\\right) \\Gamma \\left(\\frac{d}{2}+\\ell -1\\right) \\Gamma (2 \\ell +2 \\Delta_\\phi -1)}\\\\\n{}&\\times p_\\ell\\left[i\\left(\\frac{\\Delta_E}{2}-\\frac{d}{4}\\right);\\frac{d}{4},\\Delta_\\phi-\\frac{d}{4},\\Delta_\\phi-\\frac{d}{4},\\frac{d}{4}\\right]\n\\end{split}\n\\end{equation}\nwhere $p_\\ell(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})$ is the Wilson polynomial defined in (\\ref{Wilsonpolyn}),\\footnote{These Wilson polynomials also appear in the crossed channel decomposition of conformal partial waves of spinning operators \\cite{Sleight:2018ryu}.} and we will suppress its argument in the following to write it as $p_\\ell$. With this input, our method allows us to efficiently compute all the subleading coefficients. From $\\widetilde{Recur}^{12}_{0,\\ell}=0$, we find \n\\begin{equation}\\small\n\\begin{split}\n{}&\\tilde{\\mathcal{B}}^{12}_{1,\\ell}=-\\frac{2^{\\Delta_E-2 \\Delta_\\phi -2 \\ell -2}\\Gamma \\left(\\frac{\\Delta_E+1}{2}\\right) \\Gamma \\left(\\frac{d-\\Delta_E}{2}\\right)^2  \\Gamma (\\ell +\\Delta_\\phi +1) \\Gamma (\\ell +2 \\Delta_\\phi )}{\\Gamma \\left(\\frac{d}{2}\\right) \\Gamma \\left(\\frac{\\Delta_E}{2}\\right) \\Gamma (\\Delta_\\phi )^2 \\Gamma (\\ell +1) \\Gamma \\left(\\frac{d}{2}-\\Delta_E\\right) \\Gamma \\left(\\ell +\\Delta_\\phi +\\frac{3}{2}\\right)}\\\\{}&\\times\\bigg[\\frac{(\\Delta_\\phi +\\ell ) (-d+2 \\Delta_\\phi +\\ell +2) (-d+4 \\Delta_\\phi +2 \\ell )}{-d+4 \\Delta_\\phi +2 \\ell +2}p_{\\ell}+\\frac{(d+\\ell -1) (d+2 \\ell +2) (\\Delta_\\phi +\\ell +1) (2 \\Delta_\\phi +\\ell )}{(\\ell +1) (d+2 \\ell )}p_{\\ell+2}\\\\\n{}&-\\frac{(2 \\Delta_\\phi +2 \\ell +1) (C^{(2)}_{\\Delta_E,0}+\\Delta_\\phi  (d+2 \\ell +2)+\\ell ^2+\\ell)}{\\ell +1}p_{\\ell+1}\\bigg]\\;.\n\\end{split}\n\\end{equation}\nSubstituting the solution into $\\widetilde{Recur}^{12}_{1,\\ell}=0$, we get an expression for $\\tilde{\\mathcal{B}}^{12}_{2,\\ell}$ which has the following form \n\\begin{equation}\n\\tilde{\\mathcal{B}}^{12}_{2,\\ell}=(\\ldots)p_{\\ell}+(\\ldots)p_{\\ell+1}+(\\ldots)p_{\\ell+2}+(\\ldots)p_{\\ell+3}+(\\ldots)p_{\\ell+4}\\;.\n\\end{equation}\nThe coefficients in this expression are more complicated and we will refrain from writing down their explicit expressions. The above two sets of subleading coefficients were computed in \\cite{Sleight:2018ryu} using a different method. We have checked that these expressions are equivalent to their results.\\footnote{We thank Massimo Taronna for providing the Mathematica notebook that contains their relevant results.} It is totally straightforward to iterate and get $\\tilde{\\mathcal{B}}^{12}_{n,\\ell}$ with higher $n$. The result is expressed as a linear combination of $p_{\\ell}$, $p_{\\ell+1}$, \\ldots $p_{\\ell+2n+2}$, and the coefficients can be efficiently computed using the recursion relations. However, in contrast to the 1d case, we have not found an obvious way to write the coefficients $\\tilde{\\mathcal{B}}^{12}_{n,\\ell}$ for any $n$ in a closed form.\n\n\n\n\\section{Discussion and Outlook}\\label{SecDiscuss}\nIn this paper we performed a systematic position space analysis of the conformal block decomposition of Witten diagrams and conformal partial waves. In our analysis we emphasized the use of the equation of motion operator and the contact Witten diagrams. These objects played important roles in the decomposition of exchange Witten diagrams, both in the direct channel and in the crossed channel. Our main finding is a recursive algorithm for computing the crossed channel decomposition coefficients of exchange Witten diagrams and conformal partial waves. This algorithm allows us to efficiently obtain the OPE coefficient of any double-trace operator with sub-leading conformal twist, in terms of the coefficients of the double-trace operators with the minimal twist. At face value, our results provide a useful tool to extract CFT data from tree-level holographic correlators for all internal spins, especially when the exchange Witten diagrams do not admit a truncated representation in terms of finitely many $D$-functions.\n\n\n\nLet us also mention other problems where our results might be useful.\n\\subsubsection*{Mellin bootstrap}\nOne use of our results is to simplify the Mellin bootstrap program. The Mellin bootstrap approach uses  the crossing symmetrized exchange Witten diagrams as an expansion basis. The use of the exchange Witten diagrams introduces spurious double-trace operators in the conformal block decomposition of the correlator ansatz. The method hence derives the bootstrap conditions by requiring the vanishing of all the double-trace coefficients when summing over the physical spectrum. In the Mellin bootstrap method one obtains the double-trace coefficients of a single Witten diagram by taking residues of the Mellin amplitude at the double-trace poles, and then projecting them to different spins using continuous Hahn polynomials. As we already mentioned in the introduction, the projection at a certain pole gives only the mixed OPE coefficients between the primary double-trace operators and the descendant double-trace operators for which the primaries have smaller twists. For the Mellin bootstrap method, it is not necessary to solve the mixing problem. This is because the descendant contributions always vanish in the bootstrap equations if the conditions on the primary operators are already satisfied. Nevertheless, it might be useful to clarify the structures of the Mellin bootstrap equations by eliminating the redundant descendant contributions. Using our method, such contributions are absent automatically because we work with the primary operators only. Furthermore, because the decomposition coefficients satisfy recursion relations, it should be possible to recursively derive sub-leading bootstrap conditions from the leading ones. These sub-leading conditions are crucial for probing operators with sub-leading twists, for example in the $4-\\epsilon$ expansion \\cite{Gopakumar:2018xqi}. One should however notice that there still is a subtlety in the Mellin bootstrap method which is to fix the contact term ambiguity in the basis (see \\cite{Dey:2017fab,Gopakumar:2018xqi} for recent progress in general $d$, and \\cite{Mazac:2018ycv} for $d=1$). This issue must be  addressed separately.\n\n\\subsubsection*{Analytic functionals in $\\mathrm{CFT}_1$}\nIt was pointed out in \\cite{Mazac:2018ycv} that the decomposition coefficients of the following crossing symmetric combination of $AdS_2$ Witten diagrams\\footnote{In \\cite{Mazac:2018ycv} the external operators are restricted to be identical and all have the same conformal dimension $\\Delta_\\phi$. The coefficient $\\lambda$ can be fixed such that the coefficient of $\\partial g^{(s)}_{2\\Delta_\\phi}$ is zero in the s-channel decomposition. This is one of the infinitely many equivalent choices of $\\lambda$, see \\cite{Mazac:2018ycv} for more details.}\n\\begin{equation}\nW^{s,exchange}_\\Delta+W^{t,exchange}_\\Delta+W^{u,exchange}_\\Delta+\\lambda W^{contact}\n\\end{equation}\nencode the information of the complete set of functionals for $\\mathrm{CFT}_1$ (see \\cite{Mazac:2016qev,Mazac:2018mdx} for earlier related work, also \\cite{Mazac1dinversion}). A basis of analytic functionals is given by $\\alpha_n$ and $\\beta_m$, labelled by integers $n=0,1,2,\\ldots$, $m=1,2,3,\\ldots$, and the application of the functionals to the crossing equation gives the complete set of constraints. It is sufficient to know the action of the functionals on the bootstrap vector function $F_\\Delta(z)$ defined by\n\\begin{equation}\nF_\\Delta(z)=\\frac{g^{(s)}_\\Delta(z)}{z^{2\\Delta_\\phi}}-\\frac{g^{(s)}_\\Delta(1-z)}{(1-z)^{2\\Delta_\\phi}}\\;.\n\\end{equation}\nThe action of the $n$-th functional on the function $F_\\Delta$ can be expressed as the ratio of decomposition coefficients of the Witten diagrams\n\\begin{equation}\\label{functionalaction}\n\\begin{split}\n\\alpha_n[F_\\Delta]={}&-\\frac{A_n+2B_{2n}+\\lambda a_n}{A}\\;,\\\\\n\\beta_n[F_\\Delta]={}&-\\frac{C^{(s)}_n+2C^{(t)}_{2n}+\\lambda c_n}{A}\\;.\n\\end{split}\n\\end{equation}\nWe have computed all these decomposition coefficients in this paper, and we briefly remind the reader of our notations. $A$, $A_n$, $C^{s}_n$ are respectively the coefficient of $g^{(s)}_\\Delta$, $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of  the exchange diagram $W^{s,exchange}_\\Delta$ in the s-channel\\footnote{The explicit expressions are given in (\\ref{DirectcoeA}), (\\ref{DirectcoeAn}), (\\ref{DirectcoeCns}).}; $B_{2n}$, $C^{(t)}_{2n}$ are the coefficient of $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of the exchange diagrams $W^{t,exchange}_\\Delta$  or $W^{u,exchange}_\\Delta$ in the t and u-channel; and finally $a_n$ and $c_n$ are the coefficient of $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of the contact diagram $W^{contact}=D_{\\Delta_\\phi\\Delta_\\phi\\Delta_\\phi\\Delta_\\phi}$. In \\cite{Mazac:2018ycv}, the actions of the functionals are constructed as contour integrals against certain weight functions. For general $\\Delta_\\phi$ and $n$, evaluating these integrals to yield explicit expressions is still technically challenging. On the other hand, our methods for computing the decomposition coefficients of Witten diagrams have no restrictions on quantum numbers.\n By exploiting the relation (\\ref{functionalaction}) between the two, our techniques therefore provide a complementary way to compute the analytic functional actions. Our results for the decomposition coefficients can be easily assembled to give the general analytic functionals for arbitrary external dimension $\\Delta_\\phi$, and recursively for all $n$, which should be particularly useful for the numeric bootstrap application using the analytic functionals \\cite{Paulos1dnumeric}.\n\n\\subsubsection*{$6j$ symbols}\nMoreover, the recursive algorithm we formulated here may provide some help to the computation of the $6j$ symbol in general dimensions. In \\cite{Liu:2018jhs}, the $6j$ symbols in $d=1,2,4$ were computed using the Lorentzian inversion formula \\cite{Caron-Huot:2017vep}, and expressed in terms of hypergeometric functions ${}_4F_3$. However evaluating $6j$ symbols in other dimensions still remains a challenging open problem.\\footnote{The $6j$ symbols  also admit an Mellin Barnes integral representation \\cite{Sleight:2018ryu}. The integral representation is in general quite complicated but simplifies in certain cases.} This is due to the fact that the explicit form of the conformal partial waves is not known in odd spacetime dimensions, while in even dimensions $d>4$ the Lorentzian inversion integral does not factorize. The $6j$ symbols are intimately related to the crossed channel decomposition of conformal partial waves. More precisely the $6j$ symbol can be viewed as the spectral density function for decomposing an t-channel conformal partial wave into the s-channel conformal partial waves. By eliminating the s-channel shadow conformal block and closing the contour, the encircled poles of the $6j$ symbol correspond to the double-trace operators and their residues give the crossed channel decomposition coefficients. Since we can in principle obtain all the double-trace OPE coefficients using our recursive algorithm, once we input the coefficients of the leading twist double-trace operators ({\\it e.g.} from Mellin space), it is natural to ask the following question: knowing the poles and residues, is there a convenient way to reverse engineer the $6j$ symbol? We will not further explore this problem in this paper, but the apparent advantage of such a method is that it is insensitive to the spacetime dimensions. \n\n\\subsubsection*{Boundary conformal field theories}\nThe techniques we discussed in this paper also admit a straightforward extension to boundary conformal field theories. The simplest holographic setup for BCFT is obtained by taking a half of the $AdS_{d+1}$ space which ends on a $AdS_d$ boundary. We further require fields in $AdS_{d+1}$ to satisfy Neumann boundary condition on $AdS_d$. Two-point functions on the conformal boundary $AdS_{d+1}$ now become the simplest non-trivial objects to study. There are two types of exchange Witten diagrams for two-point functions, namely, the bulk channel exchange Witten diagram and the boundary channel exchange Witten diagrams. These exchange Witten diagrams have similar decomposition properties to those of the four-point functions, {\\it i.e.}, only double-trace operators appear in the crossed channel and both the single-trace operator and double-trace operators appear in the direct channel \\cite{Rastelli:2017ecj}. The equation of motion operators and properties of conformal blocks allow us to similarly formulate recursive algorithms for solving the crossed channel decomposition coefficients of exchange Witten diagrams. We discuss the BCFT extension in a separate publication \\cite{bcftpolyakov}, where we also use the decomposition coefficients to perform Polyakov style bootstrap \\cite{Polyakov:1974gs}.\n\n\n\nThere are also other extensions worth exploring.  One is to repeat the analysis for four-point Witten diagrams with external spinning operators. Another question is whether one can also find similar recursive techniques for decomposing AdS loop diagrams. Finally, it would also be interesting to incorporate supersymmetry which presumably will further facilitate the extraction of CFT data from holographic correlators in supersymmetric backgrounds.    \n\n\\acknowledgments\nI thank Dalimil Maz\\'a\\v{c} and for discussions and collaboration on a related project \\cite{bcftpolyakov}. I am grateful to Dalimil Maz\\'a\\v{c}, Wolfger Peelaers, Leonardo Rastelli and especially Massimo Taronna for carefully reading the draft and helpful comments.  I also thank  Rajesh Gopakumar, Eric Perlmutter, Jo\\~ao Penedones, David Simmons-Duffin, Massimo Taronna, Balt van Rees and other participants of the Bootstrap 2018 for useful conversations and comments on the work. I wish to thank the California Institute of Technology for the great hospitality during the Bootstrap 2018 workshop where part of this work was finished. This work is supported in part by NSF Grant PHY-1620628.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA \\emph{fiber graph} is a graph on the finitely many lattice points\n$\\cF\\subset\\ZZ^d$ of a polytope where two lattice points are connected\nby an edge if their difference lies in a finite set of allowed moves\n$\\cM\\subset\\ZZ^d$. The implicit\nstructure of these graphs makes them a useful tool to explore the set of lattice\npoints randomly: At the current lattice point $u\\in\\cF$, an\nelement $m\\in\\pm\\cM$ is sampled and the random walk moves along $m$ if\n$u+m\\in\\cF$ and stays at $u$ otherwise. The corresponding Markov chain\nis irreducible if the underlying fiber graph is connected and the set\n$\\cM$ is called a \\emph{Markov basis} for $\\cF$ in this case. \nThis paper investigates the \\emph{heat-bath} version of this random\nwalk: At the current lattice point $u\\in\\cF$, we sample $m\\in\\cM$ and\nmove to a random element in the integer ray $(u+\\ZZ\\cdot m)\\cap\\cF$.\nThe authors of~\\cite{Diaconis1998a} discovered that this random\nwalk can be seen as a discrete version of the \\emph{hit-and-run}\nalgorithm~\\cite{lovasz1999,Vempala2005,Lovasz2006} that has been used frequently\nto sample from all the points of a polytope -- not only from its\nlattice points. The popularity of the continuous version of\nthe hit-and-run algorithm has not spread to its discrete analog, and\nnot much is known about its mixing behaviour. One reason is that it is\nalready challenging to guarantee that all points\nin the underlying set $\\cF$ can be reached by a random walk that uses\nmoves from $\\cM$, whereas for the continuous version, a random sampling\nfrom the unit sphere suffices. However, in many situations where a\nMarkov basis is known, the heat-bath random walk is evidently fast.\nFor instance, it was shown in~\\cite{Cryan2002} that the heat-bath\nrandom walk on contingency tables mixes rapidly when the number of\ncolumns is fixed.\nTo work around the connectedness issue, a \\emph{discrete\nhit-and-run} algorithm was introduced in~\\cite{Baumert2009} for\narbitrary finite sets $\\cF\\subset\\ZZ^d$. \nAt each step in this random walk, a subordinate and unrestricted\nrandom walk starts at the current lattice point $u \\in \\cF$ and uses\nthe unit vectors to collect a set of proposals $S \\subset \\ZZ^d$. The\nrandom walk then moves from $u$ to a random point in $S \\cap \\cF$.\n\nRandom walks of the heat-bath type, such as the one presented above, have been\nstudied recently in~\\cite{Dyer2014} in a more general context. In this\npaper, we explore the mixing behaviour of heat-bath random walks on\nthe lattice points of polytopes with Markov bases. Throughout, we\nassume that a Markov basis has been found already and refer to the\nrelevant literature for their\ncomputation~\\cite{Sturmfels1996,sullivant2003,hemmecke2005,malkin2007,hara2010,Rauh2014}.\nWe call the underlying graph of the heat-bath\nrandom walk a \\emph{compressed fiber graph}\n(Definition~\\ref{d:CompressedFibergraph}) and determine in\nSection~\\ref{s:Diameter} bounds on its graph-diameter. We prove that \nfor any $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$, the\ndiameter of compressed fiber graphs on $\\{u\\in\\NN^d:\nAu=b\\}$ that use a fixed Markov bases\n$\\cM\\subset\\ker_\\ZZ(A)$ is bounded from above by a constant as $b$ varies\n(Theorem~\\ref{t:DiameterCompressedFiberGraphs}). In contrast, we \nshow that the diameter of conventional fiber graphs grow linearly\nunder a dilation of the underlying polytope\n(Remark~\\ref{r:RaysOfMatrices}). This gives\nrise to slow mixing results for conventional fiber walks as observed\nin~\\cite{windisch2015-mixing}.\nIn Section~\\ref{s:HeatBath}, we study in more detail the combinatorial\nand analytical structure of the transition matrices of heat-bath random\nwalks on lattice points and prove upper and lower bounds on their\nsecond largest eigenvalues. We also discuss how the distribution on\nthe moves $\\cM$ affects the speed of convergence\n(Example~\\ref{ex:SolvingSLEMOpti}).\nTheorem~\\ref{t:MixingofAugmentingMarkovBases} establishes with\nthe \\emph{canonical path approach} from~\\cite{Sinclair1992} an upper\nbound on the second largest eigenvalue when the Markov basis is\n\\emph{augmenting} (Definition~\\ref{d:Augmentation}) and the stationary\ndistribution is uniform. From that, we conclude fast mixing\nresults for random walks on lattice points in fixed dimension.\n\n\\subsection*{Acknowledgements}\nCS was partially supported by the US National Science Foundation (DMS\n0954865).  TW gratefully acknowledges the support received from the\nGerman National Academic Foundation. \n\n\\subsection*{Conventions and Notation}\nThe natural numbers are $\\NN:=\\{0,1,2,\\ldots\\}$ and for any $N\\in\\NN$,\n$\\NN_{> N}:=\\{n\\in\\NN: n> N\\}$ and $\\NN_{\\ge N}:=\\{N\\}\\cup \\NN_{> N}$.\nFor $n\\in\\NN_{>0}$, let $[n]:=\\{1,\\ldots,n\\}$. Let $\\cM\\subset\\QQ^d$\nbe a finite set, then $\\ZZ\\cdot\\cM:=\\{\\lambda m: m\\in\\cM,\n\\lambda\\in\\ZZ\\}$ and $\\NN\\cM$ is the affine semigroup in $\\ZZ^d$\ngenerated by $\\cM$. For an integer matrix $A\\in\\ZZ^{m\\times d}$ with\ncolumns $a_1,\\dots,a_d\\in\\ZZ^m$, we write $\\NN\nA:=\\NN\\{a_1,\\ldots,a_d\\}$. A\ngraph is always undirected and can have multiple loops.\nThe distance of two nodes $u,v$ which are contained in the same\nconnected component of a graph $G$, i.e. the number of\nedges in a shortest path between $u$ and $v$ in $G$, is denoted by\n$\\dist{G}{u}{v}$. We set $\\dist{G}{u}{v}:=\\infty$ if $u$\nand $v$ are disconnected. A mass\nfunction on a finite set $\\Omega$ is a map $f:\\Omega\\to[0,1]$ such\nthat $\\sum_{\\omega\\in\\Omega}f(\\omega)=1$. A mass function $f$ on\n$\\Omega$ is \\emph{positive} if $f(\\omega)>0$ for all\n$\\omega\\in\\Omega$.  A set $\\cF\\subset\\ZZ^d$ is \\emph{normal} if it there\nexists a polytope $\\cP\\subset\\QQ^d$ such that $\\cP\\cap\\ZZ^d=\\cF$. \n\n\\section{Graphs and statistics}\n\nWe first introduce the statistical framework in which this paper\nlives and recall important aspects of the interplay between graphs\nand statistics. A \\emph{random walk} on a graph $G=(V,E)$ is a map\n$\\cH:V\\times V\\to[0,1]$ such that for all $v\\in V$, $\\sum_{u\\in\nV}\\cH(v,u)=1$ and such that $\\cH(v,u)=0$ if $\\{v,u\\}\\not\\in E$. \nWhen there is no ambiguity, we represent a random walk as an\n$|V|\\times|V|$-matrix, for example when it is clear how the elements\nof $V$ are ordered. Fix a random walk $\\cH$ on $G$.  Then $\\cH$ is\n\\emph{irreducible} if for all $v,u\\in V$ there exists $t\\in\\NN$ such\nthat $\\cH^t(v,u)>0$. The random\nwalk $\\cH$ is \\emph{reversible} if there exists a mass function\n$\\mu:V\\to[0,1]$ such that $\\mu(u)\\cdot \\cH(u,v)=\\mu(v)\\cdot \\cH(v,u)$ for\nall $u,v\\in V$ and \\emph{symmetric} if $\\cH$ is a symmetric map. A\nmass function $\\pi:V\\to[0,1]$ is a \\emph{stationary distribution}\nof $\\cH$ if $\\pi\\circ \\cH =\\pi$. For symmetric random walks, the uniform\ndistribution on $V$ is always a stationary distribution. \nIf $|V|=n$, then we denote the eigenvalues of $\\cH$ by\n$1=\\lambda_1(\\cH)\\ge\\lambda_2(\\cH)\\ge\\dots\\ge\\lambda_n(\\cH)\\ge -1$ and we\nwrite $\\lambda(\\cH):=\\max\\{\\lambda_2(\\cH),-\\lambda_n(\\cH)\\}$ for the\n\\emph{second largest eigenvalue modulus} of $\\cH$.  Any irreducible\nrandom walk has a unique stationary\ndistribution~\\cite[Corollary~1.17]{levin2008} and $\\lambda(\\cH)\\in[0,1]$\nmeasures the convergence rate: the smaller $\\lambda(\\cH)$,\nthe faster the convergence.\n\nThe aim of this paper is to study random walks on lattice points that\nuse a set of moves. Typically, this is achieved by constructing a\ngraph on the set of lattice points as follows (compare\nto~\\cite[Section~1.3]{drton2008} and~\\cite[Chapter~5]{Sturmfels1996}).\n\n\\begin{defn}\\label{d:FiberGraphs}\nLet $\\cF\\subset\\ZZ^d$ be a finite set and $\\cM\\subset\\ZZ^d$. The graph\n$\\cF(\\cM)$ is the graph on $\\cF$ where two nodes $u,v\\in\\cF$ are adjacent\nif $u-v\\in\\cM$ or $v-u\\in\\cM$. \n\\end{defn}\n\nA normal set $\\cF\\subset\\ZZ^d$ is finite and satisfies\n$\\cF=\\conv{\\cF}\\cap\\ZZ^d$. A canonical class of normal sets that arise\nin many applications, is given by the fibers of an integer matrix:\n\n\\begin{defn}\\label{d:Fibers}\nLet $A\\in\\ZZ^{m\\times d}$ and $b\\in\\cone{A}$. The set\n$\\fiber{A}{b}:=\\{u\\in\\NN^d: Au=b\\}$ is the \\emph{$b$-fiber} of $A$.\nThe collection of all fibers of $A$ is $\\cP_A:=\\{\\fiber{A}{b}:\nb\\in\\NN A\\}$. For $\\cM\\subset\\ker_\\ZZ(A)$, the graph\n$\\fibergraph{A}{b}{\\cM}$ is a \\emph{fiber graph}.\n\\end{defn}\n\nLet $\\cF,\\cM\\subset\\ZZ^d$ be finite. If the membership in $\\cF$ can be\nverified efficiently  -- for instance when $\\cF$ is given implicitly by\nlinear equations and inequalities -- then it is possible to explore\n$\\cF$ randomly using $\\cM$ as follows: At a given node $v\\in\\cF$, a\nuniform element $m\\in\\cM$ is selected. If $v+m\\in\\cM$, then the random\nwalk moves along $m$ to $v+m$ and if $v+m\\not\\in\\cM$, the we stay at\n$v$. Formally, we obtain the following random walk.\n\n\\begin{defn}\\label{d:SimpleFiberWalk}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be two finite sets. The\n\\emph{simple walk} is the random walk on $\\cF(\\cM)$ where the\nprobability to traverse between to adjacent nodes $u$ and $v$ is\n$|\\pm\\cM|^{-1}$ and the probability to stay at a node $u$ is \n$|\\{m\\in\\pm\\cM: u+m\\not\\in\\cF\\}|\\cdot|\\pm\\cM|^{-1}$.\n\\end{defn}\n\nThe simple walk is symmetric and hence the uniform distribution is a\nstationary distribution (see also~\\cite[Section~2]{windisch2015-mixing}). To\nensure convergence, the random walk has to be irreducible, that is,\nthe underlying graph has to be connected. The following definition is\na slight adaption of the generalized Markov basis as defined\nin~\\cite[Definition~1]{Rauh2014}.\n\n\\begin{defn}\\label{d:Markovbasis}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$. A finite set\n$\\cM\\subset\\ZZ^d$ is a \\emph{Markov basis} of $\\cP$, if for all\n$\\cF\\in\\cP$, $\\cF(\\cM)$ is a connected graph. \n\\end{defn}\n\nWe refer to~\\cite[Theorem~3.1]{Diaconis1998a} for a proof that for collections $\\cP_A$,\na finite Markov basis always exists and can be computed with tools\nfrom commutative algebra (see also~\\cite{hemmecke2005} for more on the\ncomputation of Markov bases).  We now introduce a construction of\ngraphs on lattice points that also give rise to implementable random\nwalks, but whose edges have far more reach.\n\n\\begin{defn}\\label{d:CompressedFibergraph}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets. The\n\\emph{compression} of the graph $\\cF(\\cM)$ is the graph\n$\\cF^c(\\cM):=\\cF(\\ZZ\\cdot\\cM)$.\n\\end{defn}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[b]{0.45\\textwidth} \n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n         \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n      \\foreach \\X in {0,1,2,3,4,5}{\n         \\draw[dotted](\\X,0) -- (\\X,5);\n      }\n\n      \\foreach \\Y in {0,1,2,3,4,5}{\n         \\draw[dotted](0,\\Y) -- (5,\\Y);\n      }\n\n        \n        \n        \n\n        \n        \n\n\n      \\draw[thick](1,3) --(1,4);\n      \\draw[thick](2,2) --(2,3);\n      \\draw[thick](3,1) --(3,2);\n\n\n       \\draw[thick](4,1) -- (1,4);\n       \\draw[thick](3,1) -- (1,3);\n\n\n         \\draw [fill=gray, opacity=0.25] (1,3) --(1,4) -- (4,1) --(3,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n         \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n      \\foreach \\X in {0,1,2,3,4,5}{\n         \\draw[dotted](\\X,0) -- (\\X,5);\n      }\n\n      \\foreach \\Y in {0,1,2,3,4,5}{\n         \\draw[dotted](0,\\Y) -- (5,\\Y);\n      }\n\n        \n        \n        \n\n        \n        \n\n\n      \\draw[thick](1,3) --(1,4);\n      \\draw[thick](2,2) --(2,3);\n      \\draw[thick](3,1) --(3,2);\n\n\n\n       \\draw[thick,bend angle=40, bend right](4,1) to (1,4);\n       \\draw[thick,bend angle=40, bend right](4,1) to (2,3);\n       \\draw[thick,bend angle=40, bend right](3,2) to (1,4);\n       \\draw[thick,bend angle=40, bend left](3,1) to (1,3);\n\n       \\draw[thick] (4,1) -- (1,4);\n       \\draw[thick] (3,1) -- (1,3);\n\n\n\n\n\n         \\draw [fill=gray, opacity=0.25] (1,3) --(1,4) -- (4,1) --(3,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n         \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n      \\foreach \\X in {0,1,2,3,4,5}{\n         \\draw[dotted](\\X,0) -- (\\X,5);\n      }\n\n      \\foreach \\Y in {0,1,2,3,4,5}{\n         \\draw[dotted](0,\\Y) -- (5,\\Y);\n      }\n\n        \n        \n        \n\n        \n        \n\n\n          \\draw[thick](1,1) to (1,4);\n          \\draw[thick](2,1) to (2,3);\n          \\draw[thick](3,1) to (3,2);\n\n          \\draw[thick](1,1) to (4,1);\n          \\draw[thick](1,2) to (3,2);\n          \\draw[thick](1,3) to (2,3);\n\n\n\n          \\draw [fill=gray, opacity=0.25] (1,1) --(1,4) -- (4,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n         \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n      \\foreach \\X in {0,1,2,3,4,5}{\n         \\draw[dotted](\\X,0) -- (\\X,5);\n      }\n\n      \\foreach \\Y in {0,1,2,3,4,5}{\n         \\draw[dotted](0,\\Y) -- (5,\\Y);\n      }\n\n        \n        \n        \n\n        \n        \n\n\n          \\draw[thick](1,1) to (1,4);\n          \\draw[thick](2,1) to (2,3);\n          \\draw[thick](3,1) to (3,2);\n\n          \\draw[thick](1,1) to (4,1);\n          \\draw[thick](1,2) to (3,2);\n          \\draw[thick](1,3) to (2,3);\n\n          \\draw[thick,bend angle=40, bend left](1,1) to (1,4);\n          \\draw[thick,bend angle=25, bend left](1,1) to (1,3);\n          \\draw[thick,bend angle=25, bend left](1,2) to (1,4);\n\n          \\draw[thick,bend angle=40, bend right](1,1) to (4,1);\n          \\draw[thick,bend angle=25, bend right](1,1) to (3,1);\n          \\draw[thick,bend angle=25, bend right](2,1) to (4,1);\n\n          \\draw[thick,bend angle=25, bend right](1,2) to (3,2);\n          \\draw[thick,bend angle=25, bend left](2,1) to (2,3);\n\n\n\n          \\draw [fill=gray, opacity=0.25] (1,1) --(1,4) -- (4,1) --cycle; \n\n\t\\end{tikzpicture}\n   \\end{minipage}\n   \\end{minipage}\n\t\\caption{\\label{f:CompressedFibergraph}Compressing graphs.}\n\\end{figure}\n\nCompressing a graph $\\cF(\\cM)$ preserves its connectedness: $\\cF(\\cM)$\nis connected if and only if $\\cF^c(\\cM)$ is connected.\n\n\\section{Bounds on the diameter}\\label{s:Diameter}\n\nIn general knowledge of the diameter of the graph underlying a Markov\nchain can provide information about the mixing time. For random walks\non fiber graphs, the chains which we consider, the underlying graph\ncoincides with the fiber graph. In this section, we determine lower\nand upper bounds on the diameter of fiber graphs and their compressed\ncounterparts.  For a finite set\n$\\cM\\subset\\ZZ^d$ and any norm $\\|\\cdot\\|$ on $\\RR^d$, let $\\|\\cM\\|:=\\max_{m\\in\\cM}\\|m\\|$.\n\n\\begin{lemma}\\label{l:LowerBound}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets, then\n\\begin{equation*}\n\\diam{\\cF(\\cM)}\\ge\\frac{1}{\\|\\cM\\|}\\cdot\\max\\{\\|u-v\\|: u,v\\in\\cF\\}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIf $\\cF(\\cM)$ is not connected, then the statement holds\ntrivially, so assume that $\\cM$ is a Markov basis for $\\cF$.  Let $u',v'\\in\n\\cF$ such that $\\|u'-v'\\|=\\max\\{\\|u-v\\|: u,v\\in\\cF\\}$ and let\n$m_1,\\dots,m_r\\in\\cM$ so that $u'=v'+\\sum_{i=1}^rm_i$ is a path of\nminimal length, then $\\|u'-v'\\|\\le r\\cdot\\|\\cM\\|$ and the claim follows\nfrom $\\diam{\\cF(\\cM)}\\ge\\dist{\\cF(\\cM)}{u'}{v'}=r$.\n\\end{proof}\n\n\\begin{remark}\\label{r:LatticeWidth}\nLet $\\cF\\subset\\ZZ^d$ be a normal set. For all\n$l\\in\\{-1,0,1\\}^d$ and $u,v\\in\\cF$ we have $(u-v)^Tl\\le\\|u-v\\|_1$ and\nthus $\\mathrm{width}_l(\\cF):=\\max\\{(u-v)^Tl:\nu,v\\in\\cF\\}\\le\\max\\{\\|u-v\\|_1: u,v\\in\\cF\\}$. \nSuppose that $u',v'\\in\\cF$ are such that $\\|u'-v'\\|_1=\\max\\{\\|u-v\\|_1:\nu,v\\in\\cF\\}$ and let\n$l'_i:=\\mathrm{sign}(u'_i-v'_i)$ for $i\\in[d]$, then\n\\begin{equation*}\n\\|u'-v'\\|_1= (u'-v')^T\\cdot l'\\le\\mathrm{width}_{l'}(\\cF)\\le\\max\\{\\|u-v\\|_1:\nu,v\\in\\cF\\}=\\|u'-v'\\|_1.\n\\end{equation*}\nThe \\emph{lattice width} of $\\cF$ is\n$\\mathrm{width}(\\cF):=\\min_{l\\in\\ZZ^d}\\mathrm{width}_l(\\cF)$ and thus\nLemma~\\ref{l:LowerBound} gives\n\\begin{equation*}\n\\|\\cM\\|_1\\cdot\\diam{\\cF(\\cM)}\\ge\\mathrm{width}(\\cF).\n\\end{equation*}\n\\end{remark}\n\n\\begin{defn}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$. A\nfinite set $\\cM\\subset\\ZZ^d$ is\n\\emph{norm-like} for $\\cP$ if there exists a constant $C\\in\\NN$ such\nthat for all $\\cF\\in\\cP$ and all $u,v\\in\\cF$, $\\dist{\\cF(\\cM)}{u}{v}\\le\nC\\cdot\\|u-v\\|$. The set $\\cM$ is $\\|\\cdot\\|$-\\emph{norm-reducing} for $\\cP$ if for\nall $\\cF\\in\\cP$ and all $u,v\\in\\cF$ there exists $m\\in\\cM$ such that\n$u+m\\in\\cF$ and $\\|u+m-v\\|<\\|u-v\\|$.\n\\end{defn}\n\nThe property of being norm-like does not depend on the norm, whereas\nbeing norm-reducing does. Norm-reducing sets are always norm-like, and\nnorm-like sets are in turn always Markov bases, but the reverse of both\nstatements is false in general (Example~\\ref{ex:NonNormLikeMB}\nand Example~\\ref{ex:NonNormReducingNormLike}). For collections $\\cP_A$\nhowever, every Markov basis is norm-like\n(Proposition~\\ref{p:EveryMBIsNormLike}).\n\n\\begin{example}\\label{ex:NonNormLikeMB}\nFor any $n\\in\\NN$, consider the normal set\n$\\cF_n:=([2]\\times[n]\\times\\{0\\})\\cup\\{(2,n,1)\\}$\nwith the Markov basis $\\{(0,1,0),(0,0,1),(-1,0,-1)\\}$.  The distance\nbetween $(1,1,0)$ and $(2,1,0)$ in $\\cF_n(\\cM)$ is $2n$ and thus $\\cM$ is\nnot norm-like for $\\{\\cF_n:n\\in\\NN\\}$ (see also\nFigure~\\ref{f:NonNormLikeMB}).\n\\end{example}\n\n\\begin{example}\\label{ex:NonNormReducingNormLike}\nLet $d\\in\\NN$ and consider $A:=(1,\\dots,1)\\in\\ZZ^{1\\times d}$, then the\nset $\\cM:=\\{e_1-e_i: 2\\le i\\le d\\}$ is a Markov basis for\nthe collection $\\cP_{A}$. However, $\\cM$ is not\n$\\|\\cdot\\|_p$-norm-reducing for any $d\\ge\n3$ and any $p\\in[1,\\infty]$. For\ninstance, consider $e_2$ and $e_3$ in\n$\\fibergraph{A}{1}{\\cM}$. The only move from $\\cM$ that can be applied\non $e_2$ is $e_1-e_2$, but $\\|(e_2+e_1-e_2)-e_3)\\|_p=\\|e_2-e_3\\|_p$.\nOn the other hand,\nin the case we cannot find a move that decreases the\n$1$-norm of two nodes $u,v\\in\\fiber{A}{b}$ by $1$, we can\nfind instead two moves $m_1,m_2\\in\\cM$ such that\n$u+m_1,u+m_1+m_2\\in\\fiber{A}{b}$ and $\\|u+m_1+m_2-v\\|=\\|u-v\\|-2$.\nThus, the graph-distance of any two elements $u$ and $v$ in\n$\\fibergraph{A}{b}{\\cM}$ is at most $\\|u-v\\|_1$ and hence $\\cM$ is\nnorm-like for $\\cP_A$.\n\\end{example}\n\n\\begin{figure}[htbp]\n\\tdplotsetmaincoords{70}{110}\n\\begin{tikzpicture}[scale=0.6,tdplot_main_coords]\n\n\\def3{3}\n\\def8{8}\n\\def1{1}\n\n\\draw[thick,->] (0,0,0) -- (3+0.25,0,0) node[anchor=north east]{};\n\\draw[thick,->] (0,0,0) -- (0,8+0.25,0) node[anchor=north west]{};\n\\draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{};\n\n\\foreach \\X in {0,...,3\n   {\\draw[dotted,color=black] (\\X,0,0) --(\\X,8,0) node[anchor=north]{};}\n\\foreach \\Y in {0,...,8} \n   {\\draw[dotted,color=black] (0,\\Y,0) --(3,\\Y,0) node[anchor=north]{};}\n\n\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,1,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,2,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,3,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,4,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,5,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,6,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,7,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,1,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,2,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,3,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,4,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,5,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,6,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,7,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,7,1) {};\n\n\n\\draw[thick] (1,1,0) -- (1,7,0) ;\n\\draw[thick] (1,7,0) -- (2,7,1) ;\n\\draw[thick] (2,7,1) -- (2,7,0) ;\n\\draw[thick] (2,7,0) -- (2,1,0) ;\n\n\\end{tikzpicture}\n\\caption{The graph from\nExample~\\ref{ex:NonNormLikeMB}}\\label{f:NonNormLikeMB}\n\\end{figure}\n\n\\begin{remark}\\label{r:NormLikeUpperDiamBound}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$ and\n$\\cM\\subset\\ZZ^d$ be norm-like for $\\cP$. It follows\nfrom the definition that there exists a constant $C\\in\\QQ_{\\ge 0}$ such that \nfor all $\\cF\\in\\cP$\n$$\\diam{\\cF(\\cM)}\\le C\\cdot\\max\\{\\|u-v\\|: u,v\\in\\cF\\}.$$\n\\end{remark}\n\nThe proof of our next results uses the \\emph{Graver basis}\n$\\graver{A}\\subset\\ZZ^d$ for an integer matrix $A\\in\\ZZ^{m\\times d}$\nwith $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$. We refer to\n\\cite[Chapter~3]{Loera2013} for a precise definition.\n\n\\begin{prop}\\label{p:EveryMBIsNormLike}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and\n$\\cM\\subset\\ker_\\ZZ(A)$ be a Markov basis of $\\cP_A$. Then $\\cM$ is\nnorm-like for $\\cP_A$.\n\\end{prop}\n\\begin{proof}\nLet $\\cM$ be a Markov basis for $\\cP_A$.\nThe Graver basis $\\graver{A}$ for $A$ is a finite set which is\n$\\|\\cdot\\|_1$-norm-reducing for $\\cP_A$. \nThus, define\n$C:=\\max_{g\\in\\graver{A}}\\diam{\\fibergraph{A}{Ag^+}{\\cM}}$.\nNow, pick $u,v\\in\\fiber{A}{b}$ arbitrarily and let\n$u=v+\\sum_{i=1}^r g_i$ be a walk from $u$ to $v$ in\n$\\fibergraph{A}{b}{\\graver{A}}$ of minimal length. Since the Graver\nbasis is norm-reducing for $\\fiber{A}{b}$, there always exists a path\nof length at most $\\|u-v\\|_1$ and hence $r\\le\\|u-v\\|_1$. Every $g_i$\ncan be replaced by a path in $\\fibergraph{A}{Ag_i^+}{\\cM}$ of length\nat most $C$ and these paths stay in $\\fiber{A}{b}$. This gives a path\nof length $C\\cdot r$, hence $\\dist{\\fibergraph{A}{b}{\\cM}}{u}{v}\\le\nC\\|u-v\\|_1$.\n\\end{proof}\n\n\n\n\\begin{prop}\\label{p:DiamBounds}\nLet $\\cP\\subset\\ZZ^d$ be a polytope with $\\dim(\\cP\\cap\\ZZ^d)>0$ and let $\\cM$ be a Markov\nbasis for $\\cF_i:=(i\\cdot\\cP)\\cap\\ZZ^d$ for all $i\\in\\NN$.\nThere exists a constant $C'\\in\\QQ_{>0}$ such that for all $i\\in\\NN$,\n$C'\\cdot i\\le\\diam{\\cF_i(\\cM)}$. If $\\cM$ is norm-like for $\\{\\cF_i:\ni\\in\\NN\\}$, then there exists a constant $C\\in\\QQ_{>0}$ such that \n$\\diam{\\cF_i(\\cM)}\\le C\\cdot i$ for all $i\\in\\NN$.\n\\end{prop}\n\\begin{proof}\nFor the lower bound on the diameter, it suffices to show the existence\nof $C'$ such that $C'\\cdot i\\le\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}$ for all\n$i\\in\\NN$ due to Lemma~\\ref{l:LowerBound}. Since\n$\\dim(\\cP\\cap\\ZZ^d)>0$, we can pick distinct $w,w'\\in\n\\cP\\cap\\ZZ^d$. For all $i\\in\\NN$, $i\\cdot w,i\\cdot w'\\in\\cF_i$\nand hence $i\\cdot\\|w-w'\\|\\le\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}$. \n\nTo show the upper bound, assume that $\\cM$ is norm-like. It suffices\nto show that there exists $C\\in\\QQ_{\\ge 0}$ such that $\\max\\{\\|u-v\\|:\nu,v\\in \\cF_i\\}\\le i\\cdot C$ by Remark~\\ref{r:NormLikeUpperDiamBound}.\nNow, let $v_1,\\ldots,v_r\\in\\QQ^d$ such that\n$\\cP=\\conv{v_1,\\ldots,v_r}$ and define $C:=\\max\\{\\|v_s-v_t\\|: s\\neq\nt\\}$. \nSince $\\cF_i=(i\\cdot\n\\cP)\\cap\\ZZ^d\\subset\\conv{iv_1,\\ldots,iv_r}$ for all\n$i\\in\\NN$, we have\n$\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}\\le\\max\\{\\|iv_s-iv_t\\|: s\\neq\nt\\}\\le C\\cdot i$. \n\\end{proof}\n\n\\begin{remark}\\label{r:RaysOfMatrices}\nLet $A\\in\\ZZ^{m\\times n}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let $\\cM$ be a Markov\nbasis for $\\cP_A$. Then $\\cM$ is norm-like due to\nProposition~\\ref{p:EveryMBIsNormLike} and thus for all $b\\in\\cone{A}$\nthere exists $C,C'\\in\\QQ_{\\ge 0}$ such that \n$$i\\cdot C'\\le\\diam{\\fibergraph{A}{ib}{\\cM}}\\le i\\cdot C$$ \nfor all $i\\in\\NN$. This generalizes for instance\n\\cite[Proposition~2.10]{potka2013}\nand~\\cite[Example~4.7]{windisch2015-mixing}, where linear diameters\non a ray in $\\cone{A}$ have been observed.\nThis also implies that the construction of\nexpanders from~\\cite[Section~4]{windisch2015-mixing} works for every\nright-hand side $b\\in\\cone{A}$.\n\\end{remark}\n\n\\begin{remark}\\label{r:Mixing}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$,\n$b\\in\\cone{A}$, and let $\\cM$ be a Markov basis\nfor $\\cP_A$. Proposition~\\ref{p:DiamBounds} provides a new proof that\nthe simple walk on $(\\fibergraph{A}{ib}{\\cM})_{i\\in\\NN}$ cannot mix\nrapidly. The lower bound on the diameter from Proposition~\\ref{p:DiamBounds}\nimplies, in general, the following upper bound on the edge-expansion\n(see for example~\\cite[Proposition~1.30]{gardam2012}):\n\\begin{equation*}\nh(\\fibergraph{A}{i\\cdot b}{\\cM})\\le\n|\\cM|\\left(\\exp\\left(\\frac{\\log|\\fiber{A}{i\\cdot b}|}{D\\cdot\ni}\\right)-1\\right).\n\\end{equation*}\nIn particular, the edge-expansion cannot be bounded from below by\n$\\Omega(\\frac{1}{p(i)})_{i\\in\\NN}$ for a polynomial $p\\in\\QQ[t]$ and\nsince  $(|\\fiber{A}{i\\cdot\nb}|)_{i\\in\\NN}\\in\\mathcal{O}(i^r)_{i\\in\\NN}$, the simple walk cannot\nmix rapidly. In~\\cite{windisch2015-mixing}, it was shown that the\nedge-expansion can be bounded from above by\n$\\mathcal{O}(\\frac{1}{i})_{i\\in\\NN}$, which cannot be concluded from\nthe upper expression. \n\\end{remark}\n\nWe now turn our attention to the diameter of compressed fiber graphs.\nIn particular, we want to know for which collections of normal sets is\ntheir diameter bounded. In general, compressing a fiber\ngraph does not necessarily have an effect on the diameter\n(Example~\\ref{ex:StairCase}). \n\nAlthough a low diameter is a necessary\ncondition for good mixing, it is not sufficient.  For instance, let\n$G_n$ be the disjoint union of two complete graphs $K_n$ connected by\na single edge. Then $\\diam{G_n}=3$, but $h(G_n)\\le\\frac{1}{n}$ implies\nthat the simple walk does not mix rapidly. \n\n\\begin{example}\\label{ex:StairCase}\nFor any $n\\in\\NN$, let\n$\\cF_n:=\\{(0,0),(0,1),(1,1),(1,2),\\dots,(n,n)\\}\\subset\\ZZ^2$. The\nunit vectors $\\cM=\\{e_1,e_2\\}$ are a Markov basis for\n$\\{\\cF_n:n\\in\\NN\\}$. However,\n$\\cF_n^c(\\cM)=\\cF_n(\\cM)$ and thus $\\diam{\\cF^c_n(\\cM)}=\\diam{\\cF_n(\\cM)}=2n$\nis unbounded.\n\\end{example}\n\n\\begin{lemma}\\label{l:SignCompatibility}\nLet $A\\in\\ZZ^{m\\times d}$ and $z\\in\\ker_\\ZZ(A)$. There exists\n$r\\in[2d-2]$, distinct elements $g_1,\\dots,g_r\\in\\cG_A$, and\n$\\lambda_1,\\dots,\\lambda_r\\in\\NN_{>0}$ such that\n$z=\\sum_{i=1}^r\\lambda_ig_i$ and $g_i\\sqsubseteq z$ for all $i\\in[r]$\n\\end{lemma}\n\\begin{proof}\nThis is \\cite[Lemma~3.2.3]{Loera2013}, although it only becomes clear\nfrom the original proof of~\\cite[Theorem~2.1]{Sebo1990} that the\nappearing elements are all distinct.\n\\end{proof}\n\n\\begin{prop}\\label{p:DiamCompressedGraverFiberGraphs}\nLet $A\\in\\ZZ^{m\\times d}$ and\n$\\cP:=\\left\\{\\{x\\in\\ZZ^d: Ax=b, l\\le x\\le u\\}:\nl,u\\in\\ZZ^d,b\\in\\ZZ^m\\right\\}$.\nThen for all $\\cF\\in\\cP$, \n$\\diam{\\cF^c(\\graver{A})}\\le 2d-2$.\n\\end{prop}\n\\begin{proof} Let $s,t\\in\\{x\\in\\ZZ^d: Ax=b, l\\le x\\le u\\}$, then\n$s-t\\in\\ker_\\ZZ(A)$ and thus $s=t+\\sum_{i=1}^r\\lambda_ig_i$ with $r\\le\n2d-2$, $\\lambda_1,\\dots,\\lambda_r\\in\\NN_{>0}$, and distinct\n$g_1,\\dots,g_r\\in\\graver{A}$ such that $g_i\\sqsubseteq s-t$ according\nto Lemma~\\ref{l:SignCompatibility}. It's now a\nconsequence from~\\cite[Lemma~3.2.4]{Loera2013} that all intermediate\npoints $t+\\sum_{i=1}^k\\lambda_ig_i$ for $k\\le r$ are in $\\{x\\in\\ZZ^d:\nAx=b, l\\le x\\le u\\}$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:DistOfScaledMoves}\nLet $\\cF\\subset\\ZZ^d$ be finite and let \n$\\cF_i:=(i\\cdot\\conv{\\cF})\\cap\\ZZ^d$ for $i\\in\\NN$. For all $u,v\\in\\cF$,\n$\\dist{\\cF^c_i(\\cM)}{iu}{iv}\\le\\dist{\\cF(\\cM)}{u}{v}$ for all\n$i\\in\\NN$.\n\\end{lemma}\n\\begin{proof}\nThe statement is trivially true if $u$ and $v$ are disconnected in\n$\\cF(\\cM)$. Thus, assume the contrary and let $u=v+\\sum_{j=1}^k m_j$\nwith $m_j\\in\\cM$ be a path in $\\cF(\\cM)$ of length\n$k=\\dist{\\cF(\\cM)}{u}{v}$ and let $i\\in\\NN$. Clearly, $i\\cdot u=i\\cdot\nv+i\\cdot\\sum_{j=1}^km_j=i\\cdot v+ \\sum_{l=1}^ki\\cdot m_j$, so it is\nleft to prove that the elements traversed by this paths are in $\\cF_i$.\nLet $l\\in[k]$, since $v+\\sum_{j=1}^lm_j\\in\\cF$, we have $i\\cdot v+\\sum_{j=1}^l\ni\\cdot m_j\\in i\\cdot \\cF\\subseteq\\cF_i$. Hence, this is a path in\n$\\cF^c_i(\\cM)$ of length $k=\\dist{\\cF(\\cM)}{u}{v}$.\n\\end{proof}\n\nWe are ready to prove that the diameter of compressed fiber graphs\ncoming from an integer matrix can be bounded for all right-hand sides\nsimultaneously.\n\n\\begin{thm}\\label{t:DiameterCompressedFiberGraphs}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let\n$\\cM$ be a Markov basis for $\\cP_A$.  There exists a constant\n$C\\in\\NN$ such that $\\diam{\\cF^c(\\cM)}\\le C$ for all $\\cF\\in\\cP_A$.\n\\end{thm}\n\\begin{proof}\nOur proof relies on basic properties of the Graver basis $\\cG_A$ of\n$A$. For any $g\\in\\graver{A}$, let $\\cF_g:=\\fiber{A}{Ag^+}$\nand let\n$K:=\\max\\{\\dist{\\cF_g(\\cM)}{g^+}{g^-}: g\\in\\graver{A}\\}$. We\nshow that the diameter of any compressed fiber graph of $A$ is\nbounded from above by $(2d-2)\\cdot K$. \nLet $b\\in\\cone{A}$ arbitrary and choose elements $u,v\\in\\fiber{A}{b}$. \nAccording to Proposition~\\ref{p:DiamCompressedGraverFiberGraphs},\nthere exists $r\\in[2d-2]$, \n$g_1,\\ldots,g_{r}\\in\\graver{A}$ and\n$\\lambda_1,\\ldots,\\lambda_{r}\\in\\ZZ$ such that\n$u=v+\\sum_{i=1}^{r}\\lambda_i g_i$, and\n$v+\\sum_{i=1}^l\\lambda_ig_i\\in\\NN^d$ for all $l\\in[r]$. According to\nLemma~\\ref{l:DistOfScaledMoves}, for any $i\\in[r]$ there are\n$m_1^i,\\ldots,m_{k_i}^i\\in\\cM$ and\n$\\alpha_1,\\ldots,\\alpha_{k_i}\\in\\ZZ$ such that\n$\\lambda_ig_i^+=\\lambda_ig_i^-+\\sum_{j=1}^{k_i}\\alpha_j m_j^i$ is a\npath in the compression of $\\fiber{A}{A\\lambda_ig_i^+}(\\cM)$ of length $k_i\\le K$. Lifting these\npaths for every $i\\in[r]$ yields a path\n$u=v+\\sum_{i=1}^{r}\\sum_{j=1}^{k_i}\\alpha_jm_j^i$\nin $\\scfibergraph{A}{b}{\\cM}$ of length $r\\cdot K\\le(2d-2)\\cdot K$.\n\\end{proof}\n\n\\section{Heat-bath random walks}\\label{s:HeatBath}\n\nIn this section, we establish the heat-bath random walk on compressed\nfiber graphs. We refer to~\\cite{Dyer2014} for a more general\nintroduction on random walks of heat-bath type. Let\n$\\cF\\subset\\ZZ^d$ be finite set. For any $u\\in\\cF$ and\n$m\\in\\ZZ^d$, the\nray in $\\cF$ through $u$ along $m$ is denoted by\n$\\polyray{u}{m}{\\cF}:=(u+m\\cdot\\ZZ)\\cap\\cF$. \nAdditionally, given a mass function\n$\\pi:\\cF\\to[0,1]$, we define\n\\begin{equation*}\n\\heatbathmove{\\pi}{\\cF}{m}(x,y):=\\begin{cases}\n\\frac{\\pi(y)}{\\pi(\\polyray{x}{m}{\\cF})}&,\\text{ if\n}y\\in\\polyray{x}{m}{\\cF}\\\\\n0&,\\text{ otherwise}\n\\end{cases}\n\\end{equation*}\nfor $x,y\\in\\cF$. For $\\cM\\subset\\ZZ^d$ and a mass function $f:\\cM\\to[0,1]$, the\n\\emph{heat-bath random walk} is \n\\begin{equation}\\label{equ:HeatBath}\n\\heatbath{\\pi}{f}{\\cF}{\\cM}=\\sum_{m\\in\\cM}f(m)\\cdot\\heatbathmove{\\pi}{\\cF}{m}.\n\\end{equation}\nThe underlying graph of the heat-bath random walk is the compression\n$\\cF^c(\\cM)$ and in this section, we assume throughout that for all\n$m\\in\\cM$ and $\\lambda\\in\\ZZ\\setminus\\{-1,1\\}$, $\\lambda\\cdot\nm\\not\\in\\cM$. Let us first recall the basic properties of this random\nwalk (compare also to~\\cite[Lemma~2.2]{Diaconis1998a}).\n\n\\begin{algorithm}[h]\n\\caption{Heat-bath random walk on compressed fiber\ngraphs}\\label{a:HeatBath}\n\\begin{algorithmic}[1]\n\\Require{$\\cF\\subset\\ZZ^d$, $\\cM\\subset\\ZZ^d$,\n$v\\in\\cF$, mass functions\n$f:\\cM\\to[0,1]$ and $\\pi:\\cF\\to[0,1]$, $r\\in\\NN$}\n\n\\Procedure{HeatBath:}{}\n\\State $v_0:=v$  \n\\State \\texttt{FOR} $s=0$; $s=s+1$, $s<r$  \n\\State\\hspace{\\algorithmicindent}Sample $m\\in\\cM$ according to $f$\n\\State\\hspace{\\algorithmicindent}Sample\n$v_{s+1}\\in\\polyray{v_{s}}{m}{\\cF}$ according to $\\polyray{v_{s}}{m}{\\cF}\\to[0,1]$,\n$y\\mapsto\\frac{\\pi(y)}{\\pi(\\polyray{v_{s}}{m}{\\cF})}$\n\\State \\texttt{RETURN} $v_1,\\dots,v_r$\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{prop}\\label{p:HeatBath}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets. Let\n$f:\\cM\\to[0,1]$ and $\\pi:\\cF\\to(0,1)$ be mass functions. Then\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}$ is aperiodic, has stationary\ndistribution $\\pi$, is reversible with respect to $\\pi$, and all of\nits eigenvalues are non-negative. The random walk is irreducible if\nand only if $\\{m\\in\\cM: f(m)>0\\}$ is a Markov basis for $\\cF$.\n\\end{prop}\n\\begin{proof}\nSince for any $u\\in\\cF$ and any $m\\in\\cM$,\n$\\heatbathmove{\\pi}{\\cF}{m}(u,u)>0$, there are halting states and thus\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}$ is aperiodic.  By definition,\n$\\pi(x)\\heatbathmove{\\pi}{\\cF}{m}(x,y)=\\pi(y)\\heatbathmove{\\pi}{\\cF}{m}(y,x)$\nand thus $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ is reversible with\nrespect to $\\pi$ and $\\pi$ is a stationary distribution.  The\nstatement on the eigenvalues is exactly \\cite[Lemma~1.2]{Dyer2014}.\nLet $\\cM'=\\{m\\in\\cM: f(m)>0\\}$ and $f'=f|_{\\cM'}$, then\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}=\\heatbath{\\pi}{f'}{\\cF}{\\cM'}$ and thus the\nheat-bath random walk is irreducible if and only if $\\cM'$ is a Markov\nbasis for $\\cF$.\n\\end{proof}\n\n\\begin{remark}\\label{r:ExecutionHeatBath}\nAnalyzing the speed of convergence of random walks with second largest\neigenvalues does not take the computation time of a single transition\ninto account. From a computational point of view, the\ndifference of the simple walk and the heat-bath random walk is \nStep~4 of Algorithm~\\ref{a:HeatBath}. However, we argue that Step~4\ncan be done efficiently in many cases. For instance, a hard\nnormalizing constant of $\\pi$ cancels out. If $\\pi$ is the uniform\ndistribution, then one needs to sample uniformly from\n$\\polyray{v}{m}{\\cF}$ in Step~4, which can be done efficiently. If the\ninput of Algorithm~\\ref{a:HeatBath} is a normal set $\\cF=\\{u\\in\\ZZ^d:\nAu\\le b\\}$ that is given in $\\mathcal{H}$-representation, then the\nlength of the ray $\\polyray{v}{m}{\\cF}$ can be computed with a number\nof rounding, division, and comparing operations that is linear in the\nnumber of rows of $A$.  \n\\end{remark}\n\nThere are situations in which the heat-bath random walk provides no\nspeed-up compared with the simple walk (Example~\\ref{ex:NoSpeedup}).\nIntuitively, adding more moves to the set of allowed moves should\nimprove the mixing time of the random walk. In general, however, this\nis not true for the heat-bath walk\n(Example~\\ref{ex:AddingMovesSlowingChain}).\n\n\\begin{example}\\label{ex:NoSpeedup}\nFor $n\\in\\NN$, consider the normal set\n\\begin{equation*}\n\\cF_n:=\n\\left\\{\n\\begin{bmatrix}\n0 & 1 & 1 &\\cdots& 1\\\\\n1 & 0 & 0 &\\cdots& 0\\\\\n\\end{bmatrix},\n\\begin{bmatrix}\n1 & 0 & 1 &\\cdots& 1 \\\\\n0 & 1 & 0 &\\cdots& 0\\\\\n\\end{bmatrix},\n\\ldots,\n\\begin{bmatrix}\n1 & 1 &\\cdots & 1 & 0 \\\\\n0 & 0 &\\cdots & 0 & 1 \\\\\n\\end{bmatrix}\\right\\}\\subset\\QQ^{2\\times n}.\n\\end{equation*}\nIn the language of~\\cite[Section~1.1]{drton2008}, $\\cF_n$ is precisely\nthe fiber of the $2 \\times n$ independence model where row sums are\n$(n-1,1)$ and column sums are $(1,1,\\ldots,1)$. The minimal Markov\nbasis of the independence model, often referred to as the \\emph{basic\nmoves}, is precisely the set $\\cM_n:=\\{v-u: u,v\\in\n\\cF_n\\}\\setminus\\{0\\}$.  In particular, the fiber graph $\\cF_n(\\cM_n)$ is\nthe complete graph on $n$ nodes. All rays along basic moves have\nlength $2$ and thus the transition matrices of the simple random walk\nand the heat-bath random walk coincide. There are $n\\cdot(n-1)$ many\nbasic moves and the transition matrix of both random walks is\n\\begin{equation*}\n\\frac{1}{n(n-1)}\n\\begin{bmatrix}\n1 & \\dots & 1 \\\\ \n\\vdots & & \\vdots \\\\ \n1 & \\dots & 1 \\\\ \n\\end{bmatrix}\n+\n\\frac{(n(n-1)-n)}{n(n-1)}\\cdot I_n.\n\\end{equation*}\nThe second largest eigenvalue is $1-\\frac{1}{n-1}$ which implies that\nfor $n\\to\\infty$, neither the simple walk nor the heat-bath random walk\nare rapidly mixing.\n\\end{example}\n\n\n\n\\begin{example}\\label{ex:AddingMovesSlowingChain}\nLet $\\cF=[2]\\times[5]\\subset\\ZZ^2$, $\\cM=\\{e_1,e_2,2e_1+e_2\\}$, and let\n$\\pi$ be the uniform distribution on $\\cF$.  Since $\\{e_2,2e_1+e_2\\}$ is\nnot a Markov basis for $\\cF$, any mass function $f:\\cM\\to[0,1]$ must\nhave $f(e_1)>0$ in order to make the corresponding heat-bath random\nwalk irreducible. Comparing the second largest eigenvalue modulus of\nheat-bath random walks that sample from $\\{e_1,e_2\\}$ and $\\cM$\nuniformly, we obtain\n\\begin{equation*}\n\\lambda\\left(\\frac{1}{2}\\heatbathmove{\\pi}{\\cF}{e_1}+\\frac{1}{2}\\heatbathmove{\\pi}{\\cF}{e_2}\\right)=\\frac{1}{2}<\n\\frac{2}{3}=\\lambda\\left(\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{e_1}+\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{e_2}+\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{2e_1+e_2}\\right).\n\\end{equation*}\nSo, adding $2e_1+e_2$ to the set of allowed moves slows the\nwalk down. This phenomenon does not appear for the simple walk on\n$\\cF$, where the second largest eigenvalue modulus improves from\n$\\approx 0.905$ to $\\approx 0.888$ when adding the move $2e_1+e_2$ to\nthe Markov basis. \n\\end{example}\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\n\t\t\\node [fill, circle, inner sep=2pt](D00) at (-6,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D10) at (-5,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D20) at (-4,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D30) at (-3,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D40) at (-2,0) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](D01) at (-6,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D11) at (-5,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D21) at (-4,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D31) at (-3,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D41) at (-2,1) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](D02) at (-6,2) {};\n\t\t\\node [fill, 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{};\n\t\t\\node [fill, circle, inner sep=2pt](C10) at (13,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](C20) at (14,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](C30) at (15,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](C40) at (16,0) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](C01) at (12,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](C11) at (13,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](C21) at (14,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](C31) at (15,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](C41) at (16,1) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](C02) at (12,2) {};\n\t\t\\node [fill, circle, inner sep=2pt](C12) at (13,2) {};\n\t\t\\node [fill, circle, inner sep=2pt](C22) at (14,2) {};\n\t\t\\node [fill, circle, inner sep=2pt](C32) at (15,2) {};\n\t\t\\node [fill, circle, inner sep=2pt](C42) at (16,2) {};\n\n\\draw[thick] (C00) -- (C21);\n\\draw[thick] (C10) -- (C31);\n\\draw[thick] (C20) -- (C41);\n\\draw[thick] (C01) -- (C22);\n\\draw[thick] (C11) -- (C32);\n\\draw[thick] (C21) -- (C42);\n\n\t\\end{tikzpicture}\n   \\caption{Decomposition of the graph in\n   Example~\\ref{ex:AddingMovesSlowingChain}}\\label{f:AddingMovesSlowingChain}\n\\end{figure}\n\n\n\\begin{remark}\\label{r:GlauberDynamics}\nLet $\\cF\\subset\\ZZ^d$ be finite and $\\cM=\\{m_1,\\dots,m_d\\}\\subset\\ZZ^d$ be a\nlinearly independent Markov basis of $\\cF$. If the moves are selected\nuniformly, then the heat-bath random walk on $\\cF$ coincides with the\n\\emph{Glauber dynamics} on $\\cF$. To see it, choose $u\\in\\cF$ and let\n\\begin{equation*}\n\\cF':=\\{\\lambda\\in\\ZZ^d: u+\\lambda_1m_1+\\dots+\\lambda_dm_d\\in\\cF\\}.\n\\end{equation*}\nIt is easy to check that $\\cF'$ is unique up to translation and\ndepends only on $\\cF$, $u$, and $\\cM$. Since the vectors in $\\cM$ are linearly\nindependent, every element of $\\cF$ can be represented by a\nunique choice of coefficients in $\\cF'$. Thus, the heat-bath random walk\non $\\cF$ using $\\cM$ is equivalent to the heat-bath random walk on\non $\\cF'$ using the unit vectors as moves. For any unit vector\n$e_i\\in\\ZZ^d$, the ray through an element $v\\in\\cF'$ is $\\{w\\in\\cF:\nw_j=v_j \\forall j\\neq i\\}$ and this is precisely the form desired in\nthe Glauber dynamics~\\cite[Section~3.3.2]{levin2008}.\n\\end{remark}\n\nFor the remainder of this section, we primarily focus on heat-bath\nrandom walks $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ that converge to the\nuniform distribution $\\pi$ on a finite, but not necessarily normal, set $\\cF$. We particularly\naim for bounds on its second largest eigenvalue by\nmaking use of the decomposition from equation~\\ref{equ:HeatBath}. Our\nfirst observations consider its summands $\\heatbathmove{\\pi}{\\cF}{m}$\nthat can be well understood analytically\n(Proposition~\\ref{p:MoveMatrix}) and combinatorially\n(Proposition~\\ref{p:IntersectionTheorem}). \n\n\\begin{prop}\\label{p:MoveMatrix}\nLet $\\cF\\subset\\ZZ^d$ be a finite set, $m\\in\\ZZ^d$, and $\\pi:\\cF\\to[0,1]$\nbe the uniform distribution. Let $\\cR_1,\\dots,\\cR_k$ be the disjoint\nrays through $\\cF$ along $m$. Then\n\\begin{enumerate}\n\\item $\\heatbathmove{\\pi}{\\cF}{m}$ is symmetric and idempotent.\n\\item\n$\\img(\\heatbathmove{\\pi}{\\cF}{m})=\\spann{\\sum_{x\\in\\cR_1}e_x,\\sum_{x\\in\\cR_2}e_x,\\dots,\\sum_{x\\in\\cR_k}e_x}$.\n\\item $\\ker(\\heatbathmove{\\pi}{\\cF}{m})=\\bigoplus_{i=1}^k\\spann{e_x-e_y:\nx,y\\in\\cR_i, x\\neq y}$.\n\\item $\\rank(\\heatbathmove{\\pi}{\\cF}{m})=k$ and\n$\\dim\\ker(\\heatbathmove{\\pi}{\\cF}{m})=|\\cF|-k$.\n\\item The spectrum of $\\heatbathmove{\\pi}{\\cF}{m}$ is $\\{0,1\\}$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nSymmetry of $\\heatbathmove{\\pi}{\\cF}{m}$ follows from the\ndefinition. By assumption, $\\cF$ is the disjoint union of\n$\\cR_1,\\dots,\\cR_k$ and hence there exists a permutation matrix\n$S$ such that $S\\heatbathmove{\\pi}{\\cF}{m}S^T$ is a block matrix whose\nbuilding blocks are the matrices \n\\begin{equation*}\n\\frac{1}{|\\cR_i|}\\begin{bmatrix}\n1 & \\dots & 1 \\\\ \n\\vdots & & \\vdots \\\\ \n1 & \\dots & 1 \\\\ \n\\end{bmatrix}\\in\\QQ^{|\\cR_i|\\times|\\cR_i|}.\n\\end{equation*}\nThus, $\\heatbathmove{\\pi}{\\cF}{m}$ is\nidempotent and the rank of $\\heatbathmove{\\pi}{\\cF}{m}$ is $k$. A basis\nof its image and its kernel can be read off directly and\nidempotent matrices can only have the eigenvalues $0$ and~$1$. \n\\end{proof}\n\n\\begin{prop}\\label{p:IntersectionTheorem}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets,\n$\\pi:\\cF\\to[0,1]$ be the uniform distribution, and let\n$V_1,\\dots,V_c\\subseteq\\cF$ be the nodes of the connected components of\n$\\cF(\\cM)$, then\n$$\\bigcap_{m\\in\\cM}\\img(\\heatbathmove{\\pi}{\\cF}m)=\\spann{\\sum_{x\\in\nV_1}e_x,\\dots,\\sum_{x\\in V_c}e_x}.$$\n\\end{prop}\n\\begin{proof}\nIt is clear by Proposition~\\ref{p:MoveMatrix} that the set\non the right-hand side is contained in any\n$\\img(\\heatbathmove{\\pi}{\\cF}m)$ since any $V_i$ decomposes disjointly into rays\nalong $m\\in\\cM$.\nTo show the other inclusion, write $\\cM=\\{m_1,\\dots,m_k\\}$ and let for any $i\\in[k]$,\n$\\cR^i_1,\\dots,\\cR^i_{n_i}$ be the disjoint rays through $\\cF$ parallel\nto $m_i$. In particular, $\\{\\cR^i_1,\\dots,\\cR^i_{n_i}\\}$ is a\npartition of $\\cF$ for any $i\\in[k]$. Let $v\\in\\bigcap_{m\\in\\cM}\\img(\\heatbathmove{\\pi}{\\cF}m)$.\nAgain by Proposition~\\ref{p:MoveMatrix}, there exists for any\n$i\\in[k]$, $\\lambda^i_1,\\dots,\\lambda^i_{n_i}\\in\\QQ$ such that\n\\begin{equation*}\nv=\\sum_{j=1}^{n_i}\\sum_{x\\in\\cR^i_j}\\lambda_j^ie_x.\n\\end{equation*}\nNotice that if two distinct Markov moves $m_i$ and $m_{i'}$\nand two indices $j\\in[n_i]$ and $j'\\in[n_{i'}]$\nsatisfy $\\cR_{j}^i\\cap\\cR^{i'}_{j'}\\neq\\emptyset$,\nthen $\\lambda^i_j=\\lambda^{i'}_{j'}$. We show that for\nany $i\\in[k]$ and any $a\\in[c]$, $\\lambda_{j}^i=\\lambda_{j'}^i$ when\n$\\cR^i_j$ and $\\cR^i_{j'}$ are a subset of $V_a$. This implies the\nproposition. So take distinct $x,x'\\in V_a$\nand assume that $x$ and $x'$ lie on different rays of\n$m_i$ and let that be $x\\in\\cR^i_{j}$ and $x'\\in\\cR^i_{j'}$ with\n$j\\neq j'$. Since $x$ and $x'$ are in the same connected component\n$V_a$ of $\\cF(\\cM)$, let $y_{i_0},\\dots,y_{i_r}\\in\\cF$ be the nodes on a\nminimal path in $\\cF^c(\\cM)$ with $y_{i_0}=x$ and $y_{i_r}=x'$. For any $s\\in[r]$,\n$y_{i_s}$ and $y_{i_{s-1}}$ are contained in the same ray\n$\\cR^{k_s}_{t_s}$ coming from a\nMarkov move $m_{k_s}$. In particular,\n$\\cR_{k_{s-1}}^{t_{s-1}}\\cap\\cR^{k_s}_{t_s}\\neq\\emptyset$ and due to\nour observation made above\n$\\lambda_{j}^i=\\lambda_{t_1}^{k_1}=\\lambda_{t_2}^{k_2}=\\dots=\\lambda_{t_r}^{k_r}=\\lambda_{j'}^i$\nwhich finishes the proof.\n\\end{proof}\n\n\\begin{defn}\\label{d:RayMatrix}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and\n$\\cM'\\subseteq\\cM$. Let $\\mathcal{V}$ be the set of\nconnected components of $\\cF(\\cM\\setminus\\cM')$ and\n$\\cR$ be the set of all rays through $\\cF$ along all elements\nof $\\cM'$. The \\emph{ray matrix} of $\\cF(\\cM)$ along $\\cM'$\nis\n$\\rayMat{\\cF}{\\cM}{\\cM'}:=(|R\\cap\nV|)_{R\\in\\cR,V\\in\\cV}\\in\\NN^{\\cR\\times\\cV}$.\n\\end{defn}\n\n\\begin{example}\nLet $\\cF =[3]\\times [3]$, $\\cM=\\{e_1, e_2, e_1+e_2 \\}$, and $\\cM'= \\{\ne_1, e_2\\}$. Then $\\cF(\\cM\\setminus \\cM')$ has five connected\ncomponents and the ray matrix of $\\cF(\\cM)$ along\n$\\cM'$ is \n\\[ \\rayMat{\\cF}{\\cM}{\\cM'}=\n\\begin{bmatrix}\n1 & 1 & 1 & 0 & 0 \\\\ \n0 & 1 & 1 & 1 & 0 \\\\ \n0 & 0 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 1 & 1\\\\\n0 & 1 & 1 & 1 & 0\\\\\n1 & 1 & 1 & 0 & 0\n\\end{bmatrix}.\n\\]\n\\end{example}\n\n\\begin{remark}\\label{r:RayMatUnitVecTranspose}\nLet $\\cF\\subset\\ZZ^2$, then the rays through $\\cF$ along $e_1$ are the\nconnected components of $\\cF(\\{e_1,e_2\\}\\setminus\\{e_2\\})$ and the rays through $\\cF$\nalong $e_2$ are the connected components of\n$\\cF(\\{e_1,e_2\\}\\setminus\\{e_1\\})$, thus\n$\\rayMat{\\cF}{\\cM}{e_1}=\\rayMat{\\cF}{\\cM}{e_2}^T$.\n\\end{remark}\n\n\\begin{prop}\\label{p:RaysAndCC}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets,\n$\\pi:\\cF\\to[0,1]$ be the uniform distribution, and $\\cM'\\subseteq\\cM$.\nThen\n\\begin{equation*}\n\\ker(\\rayMat{\\cF}{\\cM}{\\cM'})\\cong\n\\bigcap_{m\\in\\cM\\setminus\\cM'}\\img(\\heatbathmove{\\pi}{\\cF}{m})\n\\cap\\bigcap_{m\\in\\cM'}\\ker(\\heatbathmove{\\pi}{\\cF}{m}).\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nLet $V_1,\\dots,V_c$ be the connected components of $\\cF(\\cM\\setminus\\cM')$ and\n$\\cR_1,\\dots,\\cR_r$ be the rays along elements in $\\cM'$. \nLet $I:=\\bigcap_{m\\in\\cM\\setminus\\cM'}\\img(\\heatbathmove{\\pi}{\\cF}{m})$ and\n$K:=\\bigcap_{m\\in\\cM'}\\ker(\\heatbathmove{\\pi}{\\cF}{m})$. By\nProposition~\\ref{p:IntersectionTheorem}, any element of $I$ has the\nform $v=\\sum_{i=1}^c(\\lambda_i\\sum_{x\\in V_i}e_x)$ for\n$\\lambda_1,\\dots,\\lambda_c\\in\\QQ$. Assume additionally that\n$v\\in\\ker(\\heatbathmove{\\pi}{\\cF}{m})$ for $m\\in\\cM'$ and let\n$\\cR_{i_1},\\dots\\cR_{i_j}$ be the rays which belong to $m$,\nthen for any $k\\in[j]$,\n$0=\\sum_{x\\in\\cR_{i_k}}v_x=\\sum_{j=1}^c\\lambda_j|\\cR_{i_k}\\cap V_j|$. Put \ndifferently, a vector $\\lambda\\in\\RR^c$ is in the kernel\nof $(|\\cR_i\\cap V_j|)_{i\\in[r],j\\in[c]}$ if and only if\n$\\sum_{i=1}^c(\\lambda_i\\sum_{x\\in V_i}e_x)\\in I\\cap K$.  \n\\end{proof}\n\nConditions on the kernel of the ray matrix allow us to give a lower bound on the \nsecond largest eigenvalue of the heat-bath random walk.\n\n\\begin{prop}\\label{p:KernelRayMat}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and\n$\\pi$ be the uniform distribution.  Let\n$\\cM'\\subseteq\\cM$ such that $\\ker(\\rayMat{\\cF}{\\cM}{\\cM'})\\neq\\{0\\}$, then $\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\ge\n1-\\sum_{m\\in\\cM'}f(m)$ for any mass function $f:\\cM\\to[0,1]$.\n\\end{prop}\n\\begin{proof}\nUsing the isomorphism from Proposition~\\ref{p:RaysAndCC}, we can\nchoose a non-zero $v\\in\\QQ^P$ such that\n$\\heatbathmove{\\pi}{\\cF}{m}v=v$ for all $m\\in\\cM\\setminus\\cM'$ and\n$\\heatbathmove{\\pi}{\\cF}{m}v=0$ for all $m\\in\\cM'$. In particular\n\\begin{equation*}\n\\heatbath{\\pi}{f}{\\cF}{\\cM}v=\\sum_{m\\in\\cM}f(m)\\heatbathmove{\\pi}{\\cF}{m}v=\n\\sum_{m\\in\\cM\\setminus\\cM'}f(m)\\heatbathmove{\\pi}{\\cF}{m}v=\\sum_{m\\in\\cM\\setminus\\cM}f(m)v.\n\\end{equation*}\nSince $f$ is a mass function, $1-\\sum_{m\\in\\cM'}f(m)$ is an\neigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$.\n\\end{proof}\n\n\\begin{defn}\\label{d:IntersectingRayProperty}\nLet $\\cF\\subset\\ZZ^d$ and $m,m'\\in\\ZZ^d$ not collinear. The pair\n$(m,m')$ has the \\emph{intersecting\nray property} in $\\cF$ if the following holds:\nFor any pair of rays $\\cR_1,\\cR_2$ parallel to $m$ and any pair of rays\n$\\cR_1',\\cR_2'$ parallel to $m'$ where both $\\cR_1\\cap\\cR_1'$ and\n$\\cR_2\\cap\\cR_2'$ are not empty, then\n$\\cR_1\\cap\\cR_2'\\neq\\emptyset$ implies $\\cR_1'\\cap\\cR_2\\neq\\emptyset$\nand $|\\cR_1|\\cdot|\\cR_1'|^{-1}=|\\cR_2|\\cdot|\\cR_2'|^{-1}$. For a\nfinite set $\\cM\\subset\\ZZ^d$, the graph $\\cF^c(\\cM)$ has the\n\\emph{intersecting ray property} if all $(m,m')$ have the intersecting\nray property in $\\cF$.\n\\end{defn}\n\n\\begin{example}\nThe compressed fiber graph on\n$[n_1]\\times\\cdots\\times[n_d]\\subset\\ZZ^d$ that uses the unit vectors\n$\\{e_1,\\dots,e_d\\}$ as moves has the intersecting ray property. On the\nother hand, consider $\\cF=\\{u\\in\\NN^2: u_1+u_2\\le 1\\}$ and take the\nrays $\\cR_1:=\\{(0,0),(0,1)\\}$ and $\\cR_2:=\\{(1,0)\\}$ that\nare parallel to $e_2$ and the rays $\\cR_1':=\\{(0,1)\\}$ and\n$\\cR_2':=\\{(0,0),(1,0)\\}$ that are parallel to $e_1$. Then\n$\\cR_1\\cap\\cR_1'=\\{(1,0)\\}$ and $\\cR_2\\cap\\cR_2'=\\{(0,1)\\}$, but\n$\\cR_1\\cap\\cR_2'=\\{(0,0)\\}\\neq\\emptyset$ and\n$\\cR_1'\\cap\\cR_2=\\emptyset$.\n\\end{example}\n\n\\begin{prop}\\label{p:CommutingMoveMatrices}\nLet $m,m'\\in\\ZZ^d$ not collinear and $\\cF\\subset\\ZZ^d$ be a finite set. The\nmatrices $\\heatbathmove{\\pi}{\\cF}{m}$ and $\\heatbathmove{\\pi}{\\cF}{m'}$\ncommute if and only if $(m,m')$ have the intersecting ray property in\n$\\cF$.\n\\end{prop}\n\\begin{proof}\nLet $u_1,u_2\\in\\cF$. Then \n\\begin{equation*}\n(\\heatbathmove{\\pi}{\\cF}{m}\\cdot\\heatbathmove{\\pi}{\\cF}{m'})_{u_1,u_2}=\n\\begin{cases}\n|\\polyray{u_1}{m}{\\cF}|^{-1}\\cdot|\\polyray{u_2}{m'}{\\cF}|^{-1},&\\text{ if\n}\\polyray{u_1}{m}{\\cF}\\cap\\polyray{u_2}{m'}{\\cF}\\neq\\emptyset\\\\\n0,&\\text{ otherwise}\n\\end{cases}.\n\\end{equation*}\nLet $\\cR_1:=\\polyray{u_1}{m}{\\cF}$, $\\cR_1':=\\polyray{u_1}{m'}{\\cF}$,\n$\\cR_2:=\\polyray{u_2}{m}{\\cF}$, and $\\cR_2':=\\polyray{u_2}{m'}{\\cF}$\nThus,\n$(\\heatbathmove{\\pi}{\\cF}{m}\\cdot\\heatbathmove{\\pi}{\\cF}{m'})_{u_1,u_2}=(\\heatbathmove{\\pi}{\\cF}{m'}\\cdot\\heatbathmove{\\pi}{\\cF}{m})_{u_1,u_2}$.\nIt is easy to see that the matrices commute if and only if $(m,m')$\nhave the intersecting ray property.\n\\end{proof}\n\n\\begin{lemma}\\label{l:DiagonizableCommutingMatrices}\nLet $H_1,\\dots,H_n\\in\\RR^{n\\times n}$ be pairwise\ncommuting matrices. Then any eigenvalue of $\\sum_{i=1}^nH_i$ has the\nform $\\lambda_1+\\dots+\\lambda_n$ where $\\lambda_i$ is an eigenvalue\nof~$H_i$.\n\\end{lemma}\n\\begin{proof}\nThis is a straightforward extension of the case $n=2$\nin~\\cite[Theorem~2.4.8.1]{Horn2013} and relies on the fact that\ncommuting matrices are simultaneously triangularizable.\n\\end{proof}\n\n\\begin{prop}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and suppose\nthere exists $m\\in\\cM$ such that $(m,m')$ has the intersecting ray\nproperty in $\\cF$ for all $m'\\in\\cM':=\\cM\\setminus\\{m\\}$. Let\n$\\cV_1,\\dots,\\cV_c$ be the connected components of\n$\\cF(\\cM')$, $\\pi_i:\\cV_i\\to[0,1]$ the uniform distribution, and\n$f'=(1-f(m))^{-1}\\cdot f|_{\\cM'}$, then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le\nf(m)+(1-f(m))\\cdot\\max\\{\\lambda(\\heatbath{\\pi_i}{f'}{\\cV_i}{\\cM'}):i\\in\n[c]\\}.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nLet $\\cH:=\\heatbath{\\pi}{f'}{\\cF}{\\cM'}$\nbe the heat-bath random walk on $\\cF(\\cM)$ that samples moves from\n$\\cM'$ according to $f'$, then\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}=f(m)\\cdot\\heatbathmove{\\pi}{\\cF}{m}+(1-f(m))\\cdot\n\\cH$.\nBy assumption, all pairs $(m,m')$ with $m'\\in\\cM'$ have the intersecting ray\nproperty and thus the matrices $\\heatbathmove{\\pi}{\\cF}{m}$ and\n$\\cH$ commute according to\nProposition~\\ref{p:CommutingMoveMatrices}. The eigenvalues of all\ninvolved matrices are non-negative and thus\nLemma~\\ref{l:DiagonizableCommutingMatrices} implies that the second largest\neigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ has the form\n$\\lambda+\\lambda'$ where $\\lambda\\in\\{0,f(m)\\}$ by\nProposition~\\ref{p:MoveMatrix} and where $\\lambda'$\nis an eigenvalue of $(1-f(m))\\cdot\\cH$.\nThe matrix $\\cH$ is a block matrix\nwhose building blocks are the matrices\n$\\heatbath{\\pi}{f'}{\\cV_i}{\\cM'}=\\heatbath{\\pi_i}{f'}{\\cV_i}{\\cM'}$\nand thus the statement follows.\n\\end{proof}\n\n\\begin{prop}\\label{p:UpperSLEMBound}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^k$ be finite sets. If\n$\\cF(\\cM)$ has the intersecting ray\nproperty, then $\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le 1-\\min(f)$.\n\\end{prop}\n\\begin{proof}\nLet $\\cM=\\{m_1,\\dots,m_k\\}$. The intersecting ray property and\nProposition~\\ref{p:CommutingMoveMatrices} give that\nthe matrices\n$f(m_1)\\cdot\\heatbathmove{\\pi}{\\cF}{m_i},\\dots,f(m_k)\\cdot\\heatbathmove{\\pi}{\\cF}{m_k}$\ncommute pairwise. According to Proposition~\\ref{p:MoveMatrix}, the\neigenvalues of\n$f(m_i)\\cdot\\heatbathmove{\\pi}{\\cF}{m_i}$ are $\\{0,f(m_i)\\}$.\nLemma~\\ref{l:DiagonizableCommutingMatrices} gives that the second\nlargest eigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$, which equals the\nsecond largest eigenvalue modulus since all of its eigenvalues are\nnon-negative, fulfills\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})=\\sum_{i\\in\nI}f(m_i)$ for a subset $I\\subseteq[k]$. Since\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})<1$ and $\\sum_{i=1}^kf(m_i)=1$, we\nhave $I\\neq[k]$ and the claim follows.\n\\end{proof}\n\n\\begin{prop}\\label{p:HeatBathOnHyperrectangle}\nLet $n_1,\\dots,n_d\\in\\NN_{>1}$,\n$\\cF=[n_1]\\times\\cdots\\times[n_d]$, and\n$\\cM=\\{e_1,\\dots,e_d\\}$. Then for any positive\nmass function $f:\\cM\\to[0,1]$,\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})=1-\\min(f)$. \n\\end{prop}\n\\begin{proof}\nIt is easy to verify that $\\cF^c(\\cM)$ has the intersecting ray property\nand thus Proposition~\\ref{p:UpperSLEMBound} shows\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le 1-\\min(f)$. Assume that\n$\\min(f)=f(e_i)$. The connected\ncomponents of $\\cF^c(\\{e_1,\\dots,e_d\\}\\setminus\\{e_i\\})$ are the layers $V_j:=\\{u\\in\n\\cF: u_i=j\\}$ for any $j\\in[n_i]$ and the rays through $\\cF$ parallel are\n$\\cR_{k}:=\\{(0,k)+s\\cdot e_i: s\\in[n_i]\\}$ for\n$k=(k_1,\\dots,k_{i-1},k_{i+1},\\dots,k_d)\\in[n_1]\\times\\cdots\\times[n_{i-1}]\\times[n_{i+1}]\\times\\cdots\\times[n_d]$. In particular, any\nray intersects any connected component exactly once. Thus, the matrix\n$(|R_k\\cap V_j|)_{k,j}$ is the all-ones matrix, which has a\nnon-trivial kernel. Proposition~\\ref{p:KernelRayMat} implies \n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\ge 1-f(e_i)$.\n\\end{proof}\n\n\\begin{remark}\\label{r:RooksWalk}\nIn the special case $n:=n_1=\\dots=n_d$ and\n$f:\\{e_1,\\dots,e_d\\}\\to[0,1]$ the uniform distribution in\nProposition~\\ref{p:HeatBathOnHyperrectangle}, the heat-bath\nrandom walk on $[n]^d$ is known as \\emph{Rook's walk} in the\nliterature. In this case, Proposition~\\ref{p:HeatBathOnHyperrectangle}\nis exactly~\\cite[Proposition~2.3]{Kim2012}.\nIn~\\cite{Mcleman2015}, upper bounds on the mixing time of the Rook's\nwalk were obtained with \\emph{path-coupling}.\n\\end{remark}\n\nThe stationary distribution of the heat-bath random walk is\nindependent of the actual mass function on the Markov moves. \nThe problem of finding the mass function which leads to the fastest\nmixing behaviour can be formulated as the following optimization\nproblem:\n\\begin{equation}\\label{equ:SLEMOpti}\n\\arg\\min\\left\\{\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM}): f:\\cM\\to(0,1),\n\\sum_{m\\in\\cM}f(m)=1\\right\\}.\n\\end{equation}\nIt follows from\nProposition~\\ref{p:HeatBathOnHyperrectangle} that the optimal value of\n\\eqref{equ:SLEMOpti} for $\\cF=[n_1]\\times\\cdots\\times[n_d]$,\n$\\cM=\\{e_1,\\dots,e_d\\}$, and the uniform distribution $\\pi$ on $\\cF$\nis the uniform distribution on $\\cM$. Another example where the\nuniform distribution is the optimal solution to~\\eqref{equ:SLEMOpti},\nbut where the verification is more involved, is presented in\nExample~\\ref{ex:SolvingSLEMOpti}.\n\n\\begin{example}\\label{ex:SolvingSLEMOpti}\nLet $\\cF=[2]\\times[5]$ as in Example~\\ref{ex:AddingMovesSlowingChain}\nand consider $\\cM=\\{e_1,2e_1+e_2\\}$.\nWe investigate for which $\\mu\\in(0,1)$, the transition matrix\n$\\mu\\heatbathmove{\\pi}{\\cF}{e_1}+(1-\\mu)\\heatbathmove{\\pi}{\\cF}{2e_1+e_2}$ has the\nsmallest second largest eigenvalue modulus. Its characteristic polynomial in $\\QQ[\\mu,x]$ is\n$$-\\frac{1}{25}x^4(x-1)(\\mu+x-1)^6(-5x^2+5x+2\\mu^2-2\\mu)(-5x^2+5x+4\\mu^2-4\\mu)$$\nand hence its eigenvalues are \n\\begin{equation*}\n\\begin{split}\nx_1(\\mu):=1,& \\quad x_2(\\mu):=1-\\mu,\\\\\nx_3(\\mu):=\\frac{1}{2}\\left[1+\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right],\n&\\quad\nx_4(\\mu):=\\frac{1}{2}\\left[1-\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right],\\\\\nx_5(\\mu):=\\frac{1}{2}\\left[1+\\sqrt{1+4(\\mu^2-\\mu)}\\right],&\\quad\nx_6(\\mu):=\\frac{1}{2}\\left[1-\\sqrt{1+4(\\mu^2-\\mu)}\\right].\\\\\n\\end{split}\n\\end{equation*}\nIt is straightforward to check that $x_5(\\mu)>\\frac{1}{2}> x_6(\\mu)$, $x_3(\\mu)>\\frac{1}{2}>\nx_4(\\mu)$. Since $\\mu^2-\\mu<0$ for $u\\in(0,1)$ and \n$x_3(\\mu)\\ge x_6(\\mu)$. We can show that\n$x_4(\\mu)\\ge x_2(\\mu)$ and thus\n\\begin{equation*}\n\\lambda(\\mu\\heatbathmove{\\pi}{\\cF}{e_1}+(1-\\mu)\\heatbathmove{\\pi}{\\cF}{2e_1+e_2})=\\frac{1}{2}\\left[1+\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right].\n\\end{equation*}\nThe fastest heat-bath random walk on $\\cF(\\cM)$ which converges to\nuniform is thus obtained for\n$\\mu=\\frac{1}{2}$, i.e. when the moves are selected uniformly. The\nsecond largest eigenvalue in this case is\n$\\frac{1}{10}(5+\\sqrt{15})\\approx 0.887$, which is larger than the\nsecond largest eigenvalue of the heat-bath walk that selects uniformly\nfrom $\\{e_1,e_2\\}$ (see Proposition~\\ref{p:HeatBathOnHyperrectangle}).\n\\end{example}\n\n\\section{Augmenting Markov bases}\n\nIt follows from our investigation in Section~\\ref{s:Diameter} that the\ndiameter of all compressed fiber graphs coming from a fixed integer matrix\n$A\\in\\ZZ^{m\\times d}$ can be bounded from above by a constant.\nHowever, Markov moves can be used twice in a minimal path which can\nmake the diameter of the compressed fiber graph larger than the size\nof the Markov basis. The next definition puts more constraints on the\nMarkov basis and postulates the existence of a path that uses every\nmove from the Markov basis at most once.\n\n\\begin{defn}\\label{d:Augmentation}\nLet $\\cF\\subset\\ZZ^d$ be a finite set and\n$\\cM=\\{m_1,\\dots,m_k\\}\\subset\\ZZ^d$. An \\emph{augmenting path} between\ndistinct $u,v\\in\\cF$ of length $r\\in\\NN$ is a path in $\\cF^c(\\cM)$ of\nthe form\n\\begin{equation*}\nu\\to u+\\lambda_{i_1}m_{i_1}\\to u+\\lambda_{i_1}m_{i_1}+\\lambda_{i_2}m_{i_2}\n\\to\\cdots \\to u+\\sum_{k=1}^r\\lambda_{i_k}m_{i_k}=v\n\\end{equation*}\nwith distinct indices $i_1,\\dots,i_r\\in[k]$. An augmenting path is\n\\emph{minimal} for $u,v\\in\\cF$ if there exists no shorter augmenting\npath between $u$ and $v$ in $\\cF^c(\\cM)$. \nA Markov basis $\\cM$ for $\\cF$ is \\emph{augmenting} if there is an\naugmenting path between any distinct nodes in $\\cF$. The\n\\emph{augmentation length} $\\auglen{\\cM}{\\cF}$ of an augmenting Markov\nbasis $\\cM$ is the maximum length of all minimal augmenting paths in\n$\\cF^c(\\cM)$.\n\\end{defn}\n\nNot every Markov basis is augmenting (see Example~\\ref{ex:StairCase}),\nbut the diameter of compressed fiber graphs that use an augmenting\nMarkov basis is at most the number of the moves. For fiber graphs\ncoming from an integer matrix, an augmenting Markov basis for all of its\nfibers can be computed (Remark~\\ref{r:GraverAugmentation}).\n\n\n\\begin{remark}\\label{r:GraverAugmentation}\nLet $A\\in\\ZZ^{m\\times d}$  with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let\n$b\\in\\cone{A}$. The\nGraver basis is an augmenting Markov basis for\n$\\fiber{A}{b}$ for any $b\\in\\cone{A}$. We claim that when $A$ is\ntotally unimodular, then $\\auglen{\\graver{A}}{\\fiber{A}{b}}\\le d^2(\\rank(A)+1)$. In\nparticular, the augmentation length is independent of the right-hand\nside $b$. Let $u,v\\in\\fiber{A}{b}$ be arbitrary and for $i\\in\\NN$, let\n$l_i:=\\min\\{u_i,v_i\\}$, $w_i:=\\max\\{u_i,v_i\\}$, and\n$c_i:=\\mathrm{sign}(u_i-v_i)\\in\\{-1,0,1\\}$.\nThen $v$ is the unique optimal value of the linear integer optimization\nproblem\n\\begin{equation*}\n\\min\\{c^Tx: Ax=b, l\\le x\\le w,x\\in\\ZZ^d\\}.\n\\end{equation*}\nA \\emph{discrete steepest decent} as defined in\n\\cite[Definition~3]{loera-augmentation} using Graver moves needs at most\n$\\|c\\|_1\\cdot d\\cdot(\\rank(A)+1)\\le d^2\\cdot(\\rank(A)+1)$ many\naugmentations from $u$ to reach the optimal value $v$. We refer to \n\\cite[Corollary~8]{loera-augmentation} which ensures that every Graver move is used\nat most once. Note that in~\\cite{loera-augmentation}, $x$ is constrained to\n$x\\ge 0$ instead to $x\\ge l$, but their argument works for any lower\nbound.\n\\end{remark}\n\n\\begin{example}\\label{ex:AugmentingMarkovBasis}\nFix $d\\in\\NN$ and consider $A$ and $\\cM$ from\nExample~\\ref{ex:NonNormReducingNormLike}. We show that $\\cM$ is an\naugmenting Markov basis for $\\fiber{A}{b}$ for any $b\\in\\NN$.\nLet $u,v\\in\\fiber{A}{b}$ be distinct, then there exists $i\\in[d]$ such\nthat $u_i>v_i$ or $u_i<v_i$, thus, we can walk from $u$ to\n$u':=u+(u_i-v_i)(e_1-e_i)$ or from $v$ to $v':=v+(v_i-u_i)(e_1-e_i)$.\nIn any case, after that augmentation, the pairs $(u',v)$ and $(v',u)$\ncoincide in the $i$th coordinate and thus we find an augmenting path\nby induction on the dimension $d$. We have used at most $d-1$ many\nedges in these paths and hence $\\auglen{\\cM}{\\fiber{A}{b}}\\le d-1$ for\nall $b\\in\\NN$.\n\\end{example}\n\nWe now show that the augmentation length is essentially bounded from\nbelow by the dimension of the node set and hence the bound observed in\nExample~\\ref{ex:AugmentingMarkovBasis} cannot be improved. We first need the\nfollowing lemma.\n\n\\begin{lemma}\\label{l:VectorSpace}\nLet $v_1,\\dots,v_k\\in\\QQ^d$ such that any\n$v\\in\\spannQQ{v_1,\\dots,v_k}$ can be represented by a linear\ncombination of $r$ vectors. Then\n$\\dim(\\spannQQ{v_1,\\dots,v_k})\\le r$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathfrak{B}\\subset\\mathfrak{P}(v_1,\\dots,v_k)$ the set of all\nsubsets of cardinality $r$. By our assumption,\n$\\cup_{B\\in\\mathfrak{B}}\\spannQQ{B}=\\spannQQ{v_1,\\dots,v_k}$.\nSince $\\dim(\\spannQQ{B})\\le r$ for all $B\\in\\mathfrak{B}$ and\nsince $\\mathfrak{B}$ is finite, the claim follows.\n\\end{proof}\n\n\\begin{prop}\\label{p:AugmentingDilatation}\nLet $\\cP\\subset\\QQ^d$ be polytope and let $\\cM\\subset\\ZZ^d$ be an\naugmenting Markov basis for\n$\\cF_i:=(i\\cdot\\cP)\\cap\\ZZ^d$ for all $i\\in\\NN$. Then\n$\\dim(\\cP)\\le\\max_{i\\in\\NN}\\auglen{\\cM}{\\cF_i}$.\n\\end{prop}\n\\begin{proof}\nWithout restricting generality, we can assume that\n$0\\in\\cP$. Let $V:=\\spannQQ{\\cP}$ be the $\\QQ$-span of\n$\\cP$, then $\\dim(\\cP)=\\dim(V)$. \nWe must have $\\dim(\\spannQQ{\\cM})=\\dim(V)$ since \n$\\dim(\\cP)=\\dim(\\conv{\\cF_i})$ for $i$ sufficiently large and since $\\cM$ is a Markov basis\nfor $\\cF_i$. Define $r:=\\max_{i\\in\\NN}\\auglen{\\cM}{\\cF_i}$\nand choose any\nnon-zero $v\\in V$ and $u\\in\\mathrm{relint}(\\cP)\\subset\\QQ^d$. Then there exists\n$\\delta\\in\\QQ_{>0}$ such that $u+\\delta v\\in\\cP$. Thus,\n$\\frac{1}{\\delta}u+v\\in\\frac{1}{\\delta}\\cP$. Let $c\\in\\NN_{\\ge 1}$\nsuch that $i:=\\frac{c}{\\delta}\\in\\NN$ and \n$w:=\\frac{c}{\\delta}u\\in\\ZZ^d$. Then\n$w+c\nv=c(\\frac{1}{\\delta}u+v)\\in(i\\cdot\\cP)\\cap\\ZZ^d=\\cF_{i}$. By\nassumption, there exists an augmenting path from $w$ to $w+c v$ using\nonly $r$ elements from $\\cM$. Put differently, the element $c v$ from\n$V$ can be represented by a linear combination of $r$ vectors from $\\cM$.\nSince $v$ was chosen arbitrarily, Lemma~\\ref{l:VectorSpace} implies\n$\\dim(\\cP)=\\dim(V)\\le r$.\n\\end{proof}\n\n\\begin{remark}\nIt is a consequence from Proposition~\\ref{p:AugmentingDilatation}\nthat for any matrix $A\\in\\ZZ^{m\\times d}$ with\n$\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and an augmenting Markov\nbasis $\\cM$, there exists $\\cF\\in\\cP_A$ such that\n$\\auglen{\\cM}{\\cF}\\ge\\dim(\\ker_\\ZZ(A))$.\n\\end{remark}\n\nLet us now shortly recall the framework from~\\cite{Sinclair1992} which\nis necessary to prove our main theorem. Let $G=(V,E)$ be a graph. For\nany ordered pair of distinct nodes $(x,y)\\in V\\times V$, let\n$p_{x,y}\\subseteq E$ be a path from $x$ to $y$ in $G$ and let\n$\\Gamma:=\\{p_{x,y}: (x,y)\\in V\\times V, x\\neq y\\}$ be the collection\nof these paths, then $\\Gamma$ is \\emph{a set of canonical paths}.\nLet for any edge $e\\in E$, $\\Gamma_e:=\\{p\\in\\Gamma: e\\in p\\}$ be the\nset of paths from $\\Gamma$ that use $e$. Now, let $\\cH:V\\times V\\to[0,1]$ be a symmetric\nrandom walk on $G$ and define\n\\begin{equation*}\n\\rho(\\Gamma,\\cH):=\\frac{\\max\\{|p|:\np\\in\\Gamma\\}}{|V|}\\cdot\\max_{\\{u,v\\}\\in\nE}\\frac{|\\Gamma_{\\{u,v\\}}|}{\\cH(u,v)}.\n\\end{equation*}\nObserve that symmetry of $\\cH$ is needed to make $\\rho(\\Gamma,\\cH)$\nwell-defined. This can be used to prove the following upper bound on\nthe second largest eigenvalue.\n\n\\begin{lemma}\\label{l:CanonicalPaths}\nLet $G$ be a graph, $\\cH$ be a symmetric random walk on $G$, and\n$\\Gamma$ be a set of canonical paths in $G$. Then\n$\\lambda_2(\\cH)\\le 1-\\frac{1}{\\rho(\\Gamma,\\cH)}$.\n\\end{lemma}\n\\begin{proof}\nThe stationary distribution of $\\cH$ is the uniform distribution and\nthus the statement is a direct consequence of~\\cite[Theorem~5]{Sinclair1992},\nsince $\\rho(\\Gamma,\\cH)$ is an upper bound on the constant defined\nin~\\cite[equation~4]{Sinclair1992}.\n\\end{proof}\n\n\\begin{thm}\\label{t:MixingofAugmentingMarkovBases}\nLet $\\cF\\subset\\ZZ^d$ be finite and let\n$\\cM:=\\{m_1,\\dots,m_k\\}\\subset\\ZZ^d$ be an augmenting Markov\nbasis. Let $\\pi$ be the uniform and $f$ be a positive\ndistribution on $\\cF$ and $\\cM$ respectively. For $i\\in[k]$, let\n$r_i:=\\max\\{|\\polyray{u}{m_i}{\\cF}|:u\\in\\cF\\}$ \nand suppose that $r_1\\ge r_2\\ge\\dots\\ge r_k$. Then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cM}{\\cF})\\le1-\\frac{|\\cF|\\cdot\\min(f)}{\\auglen{\\cM}{\\cF}\\cdot\\auglen{\\cM}{\\cF}!\\cdot\n3^{\\auglen{\\cM}{\\cF}-1}\\cdot2^{|\\cM|}\\cdot r_1r_2\\cdots\nr_{\\auglen{\\cM}{\\cF}}}.\n\\end{equation*}\n\\end{thm}\n\\begin{proof}\nChoose for any distinct\n$u,v\\in\\cF$ an augmenting path $p_{u,v}$ of minimal length in\n$\\cF^c(\\cM)$ and let $\\Gamma$ be the collection of all\nthese paths. Let $u+\\mu m_k=v$ be an edge in\n$\\cF^c(\\cM)$, then our goal is to bound $|\\Gamma_{\\{u,v\\}}|$ from\nabove. Let $\\mathcal{S}:=\\{S\\subseteq[r]: |S|\\le\\auglen{\\cM}{\\cF},\nk\\in S\\}$ and take any path $p_{x,y}\\in\\Gamma_{\\{u,v\\}}$. Then there exists\n$S:=\\{i_1,\\dots,i_s\\}$ with $s:=|S|\\le\\auglen{\\cM}{\\cF}$ such that\n$x+\\sum_{k=1}^s\\lambda_{i_k}m_{i_k}=y$. Since $p_{x,y}$ uses the edge\n$\\{u,v\\}$, there is $j\\in[s]$ such that $i_j=k$ and\n$\\lambda_{i_j}=\\mu$. Since $|\\lambda_{i_k}|\\le r_{i_k}$, there are at\nmost  \n$$s!\\cdot (2r_{i_1}+1)\\cdots (2r_{i_{j-1}}+1)\\cdot(2r_{i_{j+1}}+1)\\cdots\n(2r_{i_s}+1)\\le\ns!\\cdot 3^{s-1}\\prod_{t\\in S\\setminus\\{k\\}}r_{t}$$\npaths in\n$\\Gamma_{\\{u,v\\}}$ that uses the edge $\\{u,v\\}$ and the moves\n$m_{i_1},\\dots,m_{i_{j-1}},m_{i_{j+1}}\\dots,m_{i_s}$. Since all the\npaths are minimal, they have length at most $\\auglen{\\cM}{\\cF}$ so\nindeed every path in $\\Gamma$ has that form.\n\\begin{equation*}\n\\frac{|\\Gamma_{u,v}|}{\\heatbath{\\pi}{f}{\\cF}{\\cM}(u,v)}\\le\n3^{\\auglen{\\cM}{\\cF}-1}\\frac{\\sum_{S\\in\\mathcal{S}}\\left(|S|!\\prod_{t\\in\nS\\setminus\\{k\\}}r_{t}\\right)}{f(m_{i_j})\\cdot\\frac{1}{|\\ray{u}{m_{i_j}}|}}\\le\\frac{3^{\\auglen{\\cM}{\\cF}-1}\\cdot\n\\auglen{\\cM}{\\cF}!\\cdot|\\mathcal{S}|\\cdot\nr_1r_2\\dots r_{\\auglen{\\cM}{\\cF}}}{f(m_{i_j})},\n\\end{equation*}\nwhere we have used the assumption $r_1\\ge r_2\\ge \\dots\\ge r_k$.\nBounding $|\\mathcal{S}|$ rigorously from above by $2^{|\\cM|}$, the claim follows\nfrom Lemma~\\ref{l:CanonicalPaths}.\n\\end{proof}\n\n\\begin{defn}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets. The longest\nray through $\\cF$ along vectors of $\\cM$ is\n$\\longestRay{\\cF}{\\cM}:=\\arg\\max\\{|\\polyray{u}{m}{\\cF}|: m\\in\\cM, u\\in\n\\cF\\}$.\n\\end{defn}\n\n\\begin{cor}\\label{c:AugmentingExpander}\nLet $(\\cF_i)_{i\\in\\NN}$ be a sequence of finite sets in $\\ZZ^d$ and let\n$\\pi_i$ be the uniform distribution on $\\cF_i$. Let $\\cM\\subset\\ZZ^d$ be\nan augmenting Markov basis for $\\cF_i$ with $\\auglen{\\cM}{\\cF_i}\\le\\dim(\\cF_i)$\nand suppose that\n$(|\\longestRay{\\cF_i}{\\cM}|)^{\\dim(\\cF_i)})_{i\\in\\NN}\\in\\mathcal{O}(|\\cF_i|)_{i\\in\\NN}$.\nThen for any positive mass function $f:\\cM\\to[0,1]$, there exists $\\epsilon>0$ such that\n$\\lambda(\\heatbath{\\pi_i}{f}{\\cF_i}{\\cM})\\le 1-\\epsilon$ for all\n$i\\in\\NN$.\n\\end{cor}\n\\begin{proof}\nThis is a straightforward application of\nTheorem~\\ref{t:MixingofAugmentingMarkovBases}. \n\\end{proof}\n\n\\begin{cor}\\label{c:ExpanderDilatation}\nLet $\\cP\\subset\\ZZ^d$ be a polytope, $\\cF_i:=(i\\cdot\\cP)\\cap\\ZZ^d$\nfor $i\\in\\NN$, and let $\\pi_i$ be the uniform distribution on $\\cF_i$.\nSuppose that $\\cM\\subset\\ZZ^d$ is an augmenting Markov basis\n$\\{\\cF_i:i\\in\\NN\\}$ such that $\\auglen{\\cM}{\\cF_i}\\le\\dim(\\cP)$ for\nall $i\\in\\NN$. \nThen for any positive mass function $f:\\cM\\to[0,1]$, there exists\n$\\epsilon>0$ such that $\\lambda(\\heatbath{\\pi_i}{f}{\\cF_i}{\\cM})\\le\n1-\\epsilon$ for all $i\\in\\NN$.\n\\end{cor}\n\\begin{proof}\nLet $r:=\\dim(\\cP)$. We first show that\n$(|\\longestRay{\\cF_i}{\\cM}|)_{i\\in\\NN}\\in\\mathcal{O}(i)_{i\\in\\NN}$.\nWrite $\\cM=\\{m_1,\\dots,m_k\\}$ and denote by\n$l_i:=\\max\\{|(u+m_i\\cdot\\ZZ)\\cap\\cP|: u\\in\\cP\\}$ be the length of the\nlongest ray through the polytope $\\cP$ along $m_i$.  It suffices to\nprove that $i\\cdot(l_k+1)$ is an upper bound on the length of any ray\nalong $m_k$ through $\\cF_i$. For that, let $u\\in\\cF_i$ such that\n$u+\\lambda m_k\\in\\cF_i$ for some $\\lambda\\in\\NN$, then\n$\\frac{1}{i}u+\\frac{\\lambda}{i} m_k\\in\\cP$ and thus\n$\\lfloor\\frac{\\lambda}{i}\\rfloor\\le l_k$, which gives $\\lambda\\le\ni\\cdot(l_k+1)$. With $C:=\\max\\{l_1,\\dots,l_k\\}+1$ we have\n$|\\longestRay{\\cF_i}{\\cM}|\\le C\\cdot i$.\nEhrhart's theorem~\\cite[Theorem~3.23]{Beck2007} gives\n$(|\\cF_i|)_{i\\in\\NN}\\in\\Omega(i^r)_{i\\in\\NN}$ and\nsince $|\\longestRay{\\cF_i}{\\cM}|\\le C\\cdot i$, we have\n$(|\\longestRay{\\cF_i}{\\cM}|^r)_{i\\in\\NN}\\in\\mathcal{O}(|\\cF_i|)_{i\\in\\NN}$.\nAn application of Corollary~\\ref{c:AugmentingExpander} proves the claim.\n\\end{proof}\n\n\\begin{example}\\label{e:CrossPoly}\nFix $d,r\\in\\NN$ and let $\\cC_{d,r}:=\\{u\\in\\ZZ^d: \\|u\\|_1\\le r\\}$ be\nthe set of integers of the $d$-dimensional cross-polytope with radius\n$r$. The set $\\cM_d=\\{e_1,\\dots,e_d\\}$\nis a Markov basis for\n$\\cC_{d,r}$ for any $r\\in\\NN$. We show that $\\cM_d$ is an augmenting\nMarkov basis whose augmentation length is at most $d$. For that, let \n$u,v\\in\\cC_{d,r}$ distinct elements. We claim that there exists\n$i\\in[d]$ such that $x_i\\neq v_i$ and $u_i+(v_i-u_i)\\in\\cC_{d,r}$.\nLet $S\\subseteq[d]$ be the set of indices where $u$ and $v$ differ and let\n$s= r- ||u||_1$. If $|S|=1$, then the result is clear so\nsuppose $|S| \\geq 2$. If the result doesn't hold then for all\n$i\\in S$, $|v_i|-|u_i| > s$. It follows that \n\\begin{equation*} \n\\|v\\|_1 = \\sum_{i \\notin S} |u_i| + \\sum_{i \\in S} |v_i|\n > \\sum_{i \\notin S} |u_i| + \\sum_{i \\in S} s + |u_i| = |S_{uv}|\\cdot s + \\|u\\|_1\n = (|S|-1) \\cdot s + r.\n\\end{equation*}\nBut we assumed that $v \\in\\cC_{d,r}$. It follows that for any pair of\npoints $u,v$ in $\\cC_{d,r}$, there is a walk, using the unit vectors\nas moves, that uses each move at most once.\nCorollary~\\ref{c:AugmentingExpander} yield that for any $d\\in\\NN$, the\nsecond largest eigenvalue modulus of the heat-bath random walk on\n$\\cC_{d,r}$ with uniform as stationary distribution can be strictly\nbounded away from $1$ for $r\\to\\infty$.\n\\end{example}\n\nThe bound on the second largest eigenvalue in\nTheorem~\\ref{t:MixingofAugmentingMarkovBases} is quite general and can\nbe improved vastly, provided one has better control over the paths.\nFor example, this can be achieved for hyperrectangles intersected with\na halfspace.\n\n\\begin{prop}\\label{p:Hyperrectangles}\nLet $a\\in\\NN^d_{>0}$, $b\\in\\NN$,\n$\\cF=\\{u\\in\\NN^d: a^T\\cdot u\\le b\\}$, and\n$\\cM:=\\{e_1,\\dots,e_d\\}$. If $\\pi$ and $f$ are the uniform\ndistributions on $\\cF$ and $\\cM$ respectively, then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le\n1-\\frac{|\\cF|}{d^2}\\prod_{i=1}^d\\frac{a_i}{b}.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nObserve that $\\cM$ is a Markov basis for $\\cF$ since all nodes are connected\nwith $0\\in\\cF$. Let $u,v\\in\\cF$ be distinct. We first show\nthat there exists $k\\in[d]$ such that $u_k\\neq v_k$ and\n$u+(v_k-u_k)e_k\\in\\cF$. If $u\\le v$, the statement trivially holds.\nOtherwise, there exists $k\\in [d]$ such that $u_k>v_k$ and the vector\nobtained by replacing the $k$th coordinate of $u$ by $v_k$ remains in $\\cF$.\nNow, consider for the following path between $u$ and $v$: Choose the smallest index\n$k\\in[d]$ such that $u_k\\neq v_k$ and such that $u+(v_k-u_k)\\cdot\ne_k\\in\\cF$ and proceed recursively with $u+(v_k-u_k)$ and $v$. This\ngives a path $p_{u,v}$ between $u$ and $v$ of length at most $d$. Let\n$\\Gamma$ be the collection of all these paths.\nWe want to apply Lemma~\\ref{l:CanonicalPaths}. Thus, let $x\\in\\cF$ and\nconsider the edge $x\\rightarrow x+c\\cdot e_s$. Let us count the paths\n$p_{u,v}$ that use that edge. Let $u,v\\in\\cF$ and let $k_1,\\dots,k_r\\in\n[d]$ be distinct indices such that\n\\begin{equation*}\nu\\rightarrow u+(v_{k_1}-u_{k_1})e_{k_1}\\rightarrow\nu+(v_{k_1}-u_{k_1})e_{k_1}+(v_{k_2}-u_{k_2})e_{k_2}\\rightarrow\\cdots\\rightarrow\nv\n\\end{equation*}\nrepresents the path $p_{u,v}$ constructed by the upper rule. Assume\nthat $p_{u,v}$ uses the edge $\\{x,x+ce_s\\}$ and let $k_l=s$ and\n$(v_{k_l}-u_{k_l})=c$. In particular,\n\\begin{equation*}\n\\begin{split}\nu&+(v_{k_1}-u_{k_1})e_{k_1}+\\cdots +(v_{k_{l-1}}-u_{k_{l-1}})e_{k_{l-1}}=x\\\\\nx&+(v_{k_l}-u_{k_l})e_{k_l}+\\cdots +(v_{k_r}-u_{k_r})e_{k_r}=v.\n\\end{split}\n\\end{equation*}\nWe see that $v_{k_t}=x_{k_t}$ for all $t< l$ and that\n$u_{k_t}=x_{k_t}$ \nfor all $t\\ge l$. In particular, $v_{k_l}=u_{k_l}+c=x_{k_l}+c$ is also\nfixed. The coordinates $u_{k_t}$ and $v_{k_t}$ are bounded\nfrom above by $\\frac{b}{a_{k_t}}$ for\nall $t\\in[r]$, and hence there can be at most \n\\begin{equation*}\n\\left(\\prod_{t=1}^{l-1}\\frac{b}{a_{k_t}}\\right)\\cdot\n\\left(\\prod_{t=l+1}^{r}\\frac{b}{a_{k_t}}\\right).\n\\end{equation*}\nSince $k_1,\\dots,k_t$ are distinct coordinate indices, we have\n\\begin{equation*}\n\\frac{|\\Gamma_{x,x+c\\cdot e_s}|}{\\heatbath{\\pi}{f}{\\cF}{\\cM}(x,x+c\\cdot\ne_s)}\\le d\\cdot\\prod_{i=1}^d\\frac{b}{a_i}.\n\\end{equation*}\nLemma~\\ref{l:CanonicalPaths} finishes the proof.\n\\end{proof}\n\nIn fixed dimension, Proposition~\\ref{p:Hyperrectangles} leads to rapid\nmixing, but for $d\\to\\infty$, no statement can be made.\nIn~\\cite{Morris2004}, it was shown that the simple walk with an\nadditional halting probability on $\\{u\\in\\NN^d: a^tu\\le\nb\\}\\cap\\{0,1\\}^d$ has mixing time in $\\mathcal{O}(d^{4.5+\\epsilon})$.\nFor zero-one polytopes, simple and heat-bath walk coincide and we are\nconfident that a similar statement holds without the restriction on\nzero-one polytopes.\n\nThe heat-bath random walk mixes rapidly when an augmenting Markov\nbasis with a small augmentation length is used. We think that it is\ninteresting to question how might an augmenting Markov bases be obtained and\nhow their augmentation length can be improved.\n\n\\begin{question}\nLet $\\cM$ be an augmenting Markov basis of $A$.  Can we find finitely\nmany moves $m_1,\\dots,m_k$ such that the augmentation length of\n$\\cM\\cup\\{m_1,\\dots,m_k\\}$ on $\\fiber{A}{b}$ is at most\n$\\dim(\\ker_\\ZZ(A))$ for all $b\\in\\cone{A}$?\n\\end{question}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    Since the lack of observational evidence of WIMP (Weakly interacting massive particle), primordial black hole (PBH) gradually becomes one of the promising dark matter (DM) candidate~\\cite{Hawking:1971ei,Chapline:1975ojl,Khlopov:2008qy,Carr:2016drx,Carr:2020gox,Carr:2020xqk,Green:2020jor} and provides convincing explanation~\\cite{Clesse:2016vqa,Bird:2016dcv,Sasaki:2016jop} for LIGO\/Virgo~\\cite{LIGOScientific:2016aoc,LIGOScientific:2016sjg,LIGOScientific:2017bnn} gravitational wave (GW) signals.\nSeveral production mechanisms have been proposed.\nAfter inflation, PBHs can be generated through the collapse\nof the overdensity regions,\nwhich are developed from the primordial fluctuation re-entering the horizon~\\cite{Carr:1974nx,Sasaki:2018dmp}.\nOther papers discussed PBHs directly form following the bubble collisions of first-order phase transition (FOPT)~\\cite{Hawking:1982ga,Moss:1994iq,Konoplich:1999qq,Kodama:1982sf,Gross:2021qgx,Baker:2021nyl}.\n\nIn this work, we consider an alternative PBHs production mechanism from FOPT in the early Universe.\nDuring FOPT,\nan intermediate state, dubbed as Fermi balls (FBs), were formed\nvia the expanding bubbles.\nThen, the attractive Yukawa interaction triggers a FB collapsing into a PBH~\\cite{Kawana:2021tde}.\nThe success of FB formation requires the fermionic particles $\\chi$'s\nbeing trapped and aggregated\ninside a false vacuum surrounding by the expanding bubbles of true vacuum.\n~\\cite{Hong:2020est,Marfatia:2021twj,Witten:1984rs,Bai:2018dxf}.\nSince the Standard Model (SM)\nelectroweak and QCD (quantum chromodynamics) phase transitions are smooth crossovers,\nwe therefore invoke the FOPT being generated in the dark sector.\nSpecifically, a commonly discussed\nquartic effective thermal potential of a scalar field $\\phi$ is adopted.\nIn addition, the scalar field couples to the $\\chi$ through Yukawa interaction,\nwhich serves two purposes.\nFirst, the non-zero VEV (vacuum expectation value) of the scalar increases the mass $\\chi$ in the true vacuum, due to the energy-momentum conservation at bubble wall, which keeps the $\\chi$'s in the false vacuum\nif the mass difference is larger than the critical temperature of the FOPT.\nThe other purpose, the Yukawa interaction induces attractive force between $\\chi$'s\nand triggers the collapse of a FB into a PBH.\nThis instability of FB happened when the range of interaction becomes comparable with the mean separation distance of $\\chi$'s\nin a FB~\\cite{Kawana:2021tde}.\n\n\n\nOn the astrophysical observations perspective,\nan excess amount of photons with energy 511 keV has been confirmed\nfrom the central region of Milky Way by SPI\/INTEGRAL~\\cite{Bouchet:2010dj}.\nIf someone identified this as a result of electron-positron annihilation,\nit requires the injection rate of $2\\times 10^{43}$ non-relativistic positrons per second\n~\\citep{Weidenspointner:2004my,Churazov:2004as,Weidenspointner:2007rs,Jean:2005af,Prantzos:2005pz}.\nVarious potential astrophysical sources have been proposed for the excess,\nbut encountered difficulties explaining the characteristic of the signal~\\cite{Prantzos:2010wi}. In addition, the MeV scale DM annihilation is hardly consistent with cosmic microwave background observation~\\cite{Wilkinson:2016gsy}.\nRecently, it has been suggested that the positrons responsible for the 511 keV line excess\nmight be produced through the Hawking evaporation\n~\\cite{Hawking:1975vcx,Gibbons:1977mu} of PBHs\n~\\cite{Keith:2021guq,Frampton:2005fk,Bambi:2008kx,Cai:2020fnq,Laha:2019ssq,DeRocco:2019fjq}.\nAssuming the PBHs distribution follows the NFW halo profile\nand concentrates in the inner Galaxy.\nThe PBH mass of $M_{\\rm PBH}\\sim \\mathcal{O}(5\\times 10^{16})~{\\rm g}$ and the $\\mathcal{O}(10^{-3})$ fraction of total DM abundance, has been shown,\nnot only produces the right amount of positron flux\nbut is also consistent with existing limits of\nCOMPTEL\/INTEGRAL gamma-ray observations~\\cite{Keith:2021guq}.\n\nIn this work, we aim to find the corresponding parameters space under the FOPT scenario, which generates the PBH mass and relic abundance to explain the 511 keV excess.\nAs a result, the correlated gamma-ray spectra form PBH evaporation and GW signals from FOPT are predicted.\nThis paper is organized as following: In section~\\ref{sec:PBH_511keV}, base on NFW and isothermal distributions,\nwe scrutinize the preferred PBH mass and abundance fraction for 511 keV excess.\nThe realization of PBH formation and correlated signals production\nare discussed in section~\\ref{sec:PBH_FOPT} and \\ref{sec:BP}, respectively.\nFinally, we summarize the results in section~\\ref{sec:conclusion}.\n\n\n\n\\bigskip\n\n\n\\section{PBH evaporation}\n\\label{sec:PBH_511keV}\n    Hawking emission describes a PBH thermally produce primary particles with masses lighter than the PBH temperature $T_{\\rm PBH}=M^2_{\\rm Pl}\/M_{\\rm PBH}$, which numerically formulated as\n    \\begin{eqnarray}\n    T_{\\rm PBH}\\simeq 5.3~{\\rm MeV}\\times \\left( \\frac{10^{-18}M_\\odot}{M_{\\rm PBH}} \\right)\\,.\n    \\end{eqnarray}\n    The secondary particles productions come from the decays or fragmentation\n    of primary particles (e.g $e^\\pm,\\mu^\\pm,\\pi^\\pm,\\pi^0$).\n    A PBH dominantly emits photons and neutrinos,\n    but for $M_{\\rm PBH}\/M_\\odot \\lesssim 10^{-16}$,\n    PBH can significantly produces $e^\\pm$.\n    %\n    The emission rate of primary particle $i$ is given by~\\cite{Hawking:1974rv,Hawking:1975vcx}\n    \\begin{eqnarray}\n    \\frac{dN_i}{dE dt}=\n    \\frac{n^{\\rm d.o.f}_i  \\Gamma_i(E,M_{\\rm PBH})}{2\\pi (e^{E\/T_{\\rm PBH}}\\pm 1)}\\,,\n    \\end{eqnarray}\n    where $n^{\\rm d.o.f}_i$ indicates the degrees of freedom of $i$ particle,\n    and the fermions (bosons) are distinguished by the $+(-)$ in the denominator.\n    The graybody factor $\\Gamma_i(E,M)$ varies from particle to particle\n    and is derived from considering a wave packet scattering\n    in the PBH spacetime geometry from the PBH horizon to an observer\n    at infinity.\n\n    %\n    Technically, we use the software package BlackHawk v2.1~\\cite{Arbey:2019mbc,Arbey:2021mbl}\n    to compute the $\\gamma$ and $e^\\pm$ production rates\n    including both primary and secondary components of PBH evaporation.\n    The positron emission rate per single PBH can be numerically obtained after integrating out the evaporation spectrum\n    \\begin{eqnarray}\n    L_{e^+}= \\int dE\\, \\frac{dN_{e^+}}{dE dt}\\,.\n    \\end{eqnarray}\n\n\\bigskip\n\n\\subsection{Galaxy center 511 keV line}\n\n\n    The galaxy center 511 keV line observed by SPI\/INTEGRAL~\\cite{Bouchet:2010dj}\n     exceeds the astrophysical contributions.\n    In this paper, we focus on the explanation of PBHs in mass range from $10^{15}\\,$g\n    to $2\\times 10^{17}\\,$g, which significantly produce particles by Hawking evaporation.\nPositrons are abundantly emitted from a PBH and more than 95\\% of them become non-relativistic via ionization. Among the non-relativistic positrons, fraction of $(1-f)$ directly annihilate with an electron to produce two 511 keV photons, while the others fraction of $f\\approx0.967$ form a positronium bound state with an electron~\\cite{Keith:2021guq}. Among the later case, 25\\% of positroniums annihilate to form pair of 511 keV photons, and the rest 75\\% yield three photons with energy less than 511 keV. As a Result, the number of 511 keV photons produced per positron is\n    \\begin{eqnarray}\n    2(1-f)+\\frac{2f}{4}\\approx0.55\\,.\n    \\end{eqnarray}\n    The flux of 511 keV photons from PBHs near the galactic center,\n    covers solid angle $\\Delta\\Omega$, is formulated as\n    \\begin{eqnarray}\n    \\Phi_{\\mathrm{PBH}}(\\Delta\\Omega)=\\frac{0.55L_{e^+}(M_{\\textrm{PBH}})f_{\\textrm{PBH}}}{4\\pi M_{\\textrm{PBH}}}\\int_{\\Delta\\Omega}{\\int_{\\textrm{l.o.s}}{\\rho(\\ell,\\Omega)d\\ell}d\\Omega} \\,,\n    \\end{eqnarray}\n    where $L_{e^+}(M_{\\textrm{PBH}})$ is the positron production rate from a PBH, $f_{\\textrm{PBH}}$ be the fraction of PBHs abundance to the DM relic abundance. The integrals are performed over solid angle and along the line-of-sight.\n    We assume the PBHs distribution follows the Navarro-Frenk-White (NFW) DM halo profile:\n    \\begin{eqnarray}\n    \\rho(r)=\\frac{\\rho_0}{(r\/R_s)^\\gamma[1+(r\/R_s)]^{3-\\gamma}}\\,,\n    \\end{eqnarray}\n    where $r$ is the distance from the galactic center,\n    and the normalization parameter $\\rho_0$ adjusts the DM density near our solar system,\n    $\\rho=0.4$ $\\textrm{GeV}\/\\textrm{cm}^3$ at $r=8.25$ kpc.\n    Here, we fix $\\gamma=1.6$ and scale radius $R_s=20$ kpc in this analysis.\n    Combining with the galactic disk contribution,\n    the angular distribution of 511 keV line predicted from PBHs emission\n    is shown in Fig.~\\ref{fig:1}.\n    It distributes as a function of galactic latitude $b$ after averaging over the longitude profile $-8^\\circ<\\ell<+8^\\circ$ and is compared with the INTEGRAL data of photon energy from 508.25 keV to 513.75 keV~\\cite{Bouchet:2010dj}.\n\n    \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=13cm]{FIG\/1.jpg}\n      \\caption{The orange curve is combined 511 keV gamma-ray flux\n      from PBH evaporation with $M_{\\rm PBH}=10^{16}$ g, $f_{\\textrm{PBH}}=10^{-4}$ and astrophysical source~\\cite{Robin:2003uus}(blue curve).\n      Comparing to the INTEGRAL data from 508.25 keV to 513.75 keV~\\cite{Bouchet:2010dj}}\\label{fig:1}\n    \\end{figure}\n\n    The best-fit normalization for the 511 keV INTEGRAL data is obtained by the $\\chi^2$ test, which is defined by\n    \\begin{eqnarray}\n    \\chi^2(f_\\mathrm{PBH})=\\sum_{i}{\\left[\\frac{\\Phi_{s}(b_i,f_\\mathrm{PBH})+\\Phi_{\\mathrm{disk}}(b_i)-\\Phi_{\\textrm{data}}(b_i)}{\\sigma_{\\Phi}(b_i)}\\right]^2}\\,,\n    \\end{eqnarray}\n    and $i$ runs over the INTEGRAL data points in Fig.\\ref{fig:1}.\n    $\\Phi_{\\mathrm{disk}}$ stems from astrophysical positron emission in the disk. $\\Phi_{s}$ represents the PBHs contribution after taking into account the angular resolution of INTEGRAL, which is obtained by Gaussian smearing as\n    \\begin{eqnarray}\n    \\Phi_s(b_i,f_\\mathrm{PBH})=\\int^{b_i+3\\sigma_{b_i}}_{b_i-3\\sigma_{b_i}}{ F(b,b_i,\\sigma_{b_i})\\Phi_{\\mathrm{PBH}}(b,f_\\mathrm{PBH})db} \\,,\n    \\end{eqnarray}\n    where $b_i$ and $\\sigma_i$ respectively correspond to the central value and the horizontal error of each INTEGRAL data point, and then the smearing function is\n    \\begin{eqnarray}\n    F(b,b_i,\\sigma_{b_i})=\\frac{1}{\\sqrt{2\\pi \\sigma^2_{b_i}}}\\exp{\\left[-\\frac{(b_i-b)^2}{\\sigma^2_{b_i}}\\right]}\\,.\n    \\end{eqnarray}\n    The yellow region of Fig. \\ref{fig:2} shows\n    the best-fit parameter region of $(M_{\\rm PBH},f_{\\rm PBH})$,\n    which requires $\\chi^2-\\chi_{\\mathrm{min}}^2\\leq4$.\n    For $M_{\\rm PBH}\\simeq 2\\times 10^{17}~{\\rm g}$,\n    PBHs can serve as 100\\% DM abundance\n    and explain the 511 keV line excess simultaneously.\n    For $M_{\\rm PBH}\\lesssim 3\\times 10^{16}~{\\rm g}$,\n    PBHs overproduce the extragalactic gamma-ray,\n    and thus is excluded by current upper bound of COMPTEL.\n\n\n    \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=13cm]{FIG\/2.jpg}\n      \\caption{The yellow and cyan bands,\n      which assume the NFW and Isothermal profiles respectively, can produce the 511 keV excess observed by INTEGRAL via the PBH evaporation.\n      The constraints of COMPTEL+EGRET and sensitivities of eASTROGAM and AMEGO are shown by the solid curves, requiring the extragalactic photon from PBHs contribution does not exceed the observation data.\n      The benchmark points ({\\bf BP}s) label the PBHs produced from the FOPT scenario, and their parameters are listed in Table~\\ref{table:1}.\n      }\\label{fig:2}\n    \\end{figure}\n\n\n    \\bigskip\n\n\\subsection{Extragalactic gamma-ray}\n\n\n    The PBH evaporation also produces extragalactic gamma-ray background, thus the current gamma-ray observational data can be converted into the upper limit of the PBHs abundance.\n    To find the experiment limits in the parameter space, we calculate the PBH extragalactic photon contributions and compare with the current experimental data of COMPTEL\/EGRET\/FermiLAT and future sensitivities of AMEGO\/e-ASTROGAM as well.\n    The extragalactic photon contributions due to PBH is given by~\\cite{Marfatia:2021hcp}\n    \\begin{eqnarray}\n    \\frac{d\\Phi}{dE}=\\int^{\\mathrm{min}(t_{\\mathrm{eva}},t_0)}_{t_\\mathrm{CMB}}{c[1+z(t)]\\frac{f_\\mathrm{PBH}\\rho_{\\mathrm{DM}}}{M_\\mathrm{PBH}}\\frac{d^2N_\\gamma}{d\\tilde{E}dt}\\bigg|_{\\tilde{E}=[1+z(t)]E}dt}\\,,\n    \\end{eqnarray}\nwhere $\\rho_\\mathrm{DM}=1.27\\mkern5mu \\mathrm{GeV }\\mathrm{m}^{-3}$ is the average dark matter density in the extragalactic medium. From this expression,\nwe can see the extragalactic photon flux is independent from the uncertainties of DM density profile near the galactic center,\nthereby it provides more robust prediction than the galactic one.\nThe lower limit of temporal integral starts at the time of last scattering, $t_\\mathrm{CMB}=3.8\\times10^5\\mkern5mu\\mathrm{yr}$, after that photons decoupled from thermal plasma and can propagate freely. Meanwhile, the upper limit of integral is set by the shorter period of the lifetime of PBH ($t_\\mathrm{eva}$) or the age of Universe ($t_0=13.77\\times10^9\\mkern5mu\\mathrm{yr}$). Applying matter-dominated approximation to the evolution of Universe, the time-redshift relation is given by\n$1+z(t)=\\left(t_0\/t\\right)^{\\frac{2}{3}}$.\n    For different masses of PBH,\n    we show the upper limits of $f_\\mathrm{PBH}$ in Fig.~\\ref{fig:2},\n    which are compatible with the present COMPTEL\/EGRET\/FermiLAT upper bounds (blue curve)\n    and future AMGEGO\/e-ASTROGAM gamma-ray sensitivities (red\/black curves).\n\n\n\\bigskip\n\n\\section{First-order phase transition}\n\\label{sec:PBH_FOPT}\nThe PBH can be produced through a two-step processes during FOPT\nwhich is induced by the effective thermal potential of a scalar $\\phi$ in the hidden sector.\nThe FB stems from the aggregation of dark fermions $\\chi$'s and plays the role as intermediate state before PBH formation.\nBecause the $\\chi$'s inside a FB couple through $\\phi$, which provides an attractive Yukawa potential.\nThe length of Yukawa interaction increases as FB temperature decreases,\nand thus Yukawa energy eventually dominates the total FB energy\nwhich causing the FB collapses into a PBH~\\cite{Kawana:2021tde}.\n\n\n\n\n\n    Specifically, the formation of PBHs can be realized by the Lagrangian~\\cite{Marfatia:2021hcp}\n    \\begin{eqnarray}\n    \\mathcal{L}\\supset  \\bar{\\chi}(i\\slashed{\\partial}-M_i)\\chi  -g_{\\chi}\\phi \\bar{\\chi}\\chi-V_{\\mathrm{eff}}(\\phi,T)\\,,\n    \\end{eqnarray}\n    where $M_i$ represents the bare mass of $\\chi$ in the false vacuum,\n    $V_{\\mathrm{eff}}$ is the finite-temperature quartic effective potential\n    commonly appears from theoretical models\n    ~\\cite{Dine:1992wr,Adams:1993zs}\n    \\begin{eqnarray}\n    V_{\\mathrm{eff}}(\\phi,T)=D(T^2-T^2_0)\\phi^2-(AT+C)\\phi^3+\\frac{\\lambda}{4}\\phi^4\\,,\n    \\end{eqnarray}\n    which induces the cosmological FOPT.\n    The FOPT happened when the temperature becomes lower than critical temperature $T_c$,\n    which is determined by the condition $V_{\\rm eff}(0,T_c)=V_{\\rm eff}(v_\\phi(T_c),T_c)$,\n    at this moment, the false vacuum ($\\langle \\phi \\rangle=0$)\n    was tunnelling to the true vacuum ($\\langle \\phi \\rangle = v_\\phi$).\n    We define the parameter $B$ to be the zero-temperature potential energy density difference\n    between the false and true vacuum~\\cite{Marfatia:2021twj}.\n    Finally, the quartic potential can be described by these input parameters~\\cite{Marfatia:2021twj}\n    \\[ \\lambda,  A,  B,  C,  D\\,. \\]\n\n    We use the analytical expression for the Euclidean action $S_3(T)\/T$\n    of quartic potentials~\\cite{Adams:1993zs}\n    to compute the bubble nucleation rate per unit volume $\\Gamma(T)$\n    and then obtain the fraction of space in the false vacuum $F(t)$~\\cite{Marfatia:2021twj}.\n    The phase transition temperature $T_\\star$ identified as percolation\n    temperature can be obtained when a fraction $1\/e$ of the space\n    remains in the false vacuum. Such that the corresponding time of phase transition $t_\\star$\n    is given by $F(t_\\star)=1\/e\\simeq 0.37$.\n\n   \nIn general, GW signals accompany the FOPT from bubble collisions, sound waves, and Magnetohydrodynamic turbulence.~\\cite{Huber:2008hg,Nakai:2020oit,Espinosa:2010hh,Caprini:2015zlo,Kehayias:2009tn}. In our case, since the bubble wall reaches a relativistic terminal velocity thus corresponds to a non-runaway bubble, the sound waves dominates the GW production~\\cite{Caprini:2015zlo}. We follow the semi-analytical approaches to evaluate the GW spectra~\\cite{Marfatia:2020bcs,Caprini:2015zlo}.\n    GW characteristic frequency associates with\n    the\n   \n    inverse time duration of the FOPT\n    $\\beta\/H_\\star\\simeq T_\\star\\,\n    \\left[d(S_3\/T)\/dT\\right]$~\\cite{Nakai:2020oit},\n    which is normalized to Hubble time scale. In addition to this parameter,\n    the GW spectrum also depends on $\\alpha$, the strength of FOPT,\n    which is defined as the ratio between\n    false vacuum energy to the total radiation energy.\n\n\n    To find $n_{\\rm FB}$ the number density of FB, we need to know\n    the critical volume $V_\\star$ which is defined from a volume in false vacuum satisfying the\n    condition $\\Gamma(T_\\star)V_\\star R_\\star\\simeq v_w$~\\cite{Hong:2020est},\n    and thus this volume would not further separate into smaller one during its shrinking.\n    Here $R_\\star$ and $v_w$\n    are the radius of critical volume and the bubble wall velocity, respectively.\n    Under the assumption that each critical volume corresponds to one FB,\n    the volume fraction of false vacuum implies the relation\n    $n_{\\rm FB}|_{T_\\star}= F(t_\\star)\/V_\\star$ at $t_\\star$.\n\n    Inside a FB, to avoid complete annihilation $\\bar{\\chi} \\chi \\to \\phi \\phi$,\n    there must be a nonzero number density asymmetry\n    $\\eta_{\\rm DM} \\equiv (n_\\chi-n_{\\bar{\\chi}})\/s$ (normalized to the entropy density).\n    Then we define $Q_{\\rm FB}$ the total number of $\\chi$'s comprising a FB:\n    $Q_{\\rm FB}\\equiv \\eta_{\\rm DM}(s\/n_{\\rm FB})|_{T_\\star}$~\\cite{Hong:2020est}.\n\n    In general, the dark and SM sectors need not to be in thermal equilibrium,\n    otherwise the dark radiation ($\\phi$ and $\\chi$) would contributes\n    the effective number of extra neutrino species\n    $\\Delta N_{\\rm eff}$\n    and exceed the current observational upper bound.\n    In fact, lowering the dark sector temperature than SM sector\n    alleviates the tension between $\\Delta N_{\\rm eff}$ and observations~\\cite{Marfatia:2021twj}.\n    Thus, we include the temperature ratio $r_T\\equiv T_\\star\/T_{\\rm SM_{\\star}}$\n    at $t_\\star$ as one of the input parameter for FOPT.\n\n    The dark fermion particles $\\chi$'s aggregated to form\n    macroscopic FBs due to the FOPT.\n    Subsequently, the attractive Yukawa force between $\\chi$'s mediated by\n    $\\phi$ destabilizes the FBs to form PBHs.\n    In the following section, we quantitatively discuss these processes.\n   \n\n  \n\n \n\n\n\n\n  \\bigskip\n\n \\subsection{FB properties and PBH formations}\n    The derivation for the FB mass and radius can be found in Ref.~\\cite{Marfatia:2021hcp}. Including the Fermi gas kinetic energy, Yukawa potential energy, and the temperature-dependent potential energy difference between the false and true vacua; the total energy of a FB is approximately written as\n\n    \\begin{eqnarray}\n        \\label{eq:FB_energy}\n        E_{\\rm FB}&=&\n        \\frac{3\\pi}{4} \\left( \\frac{3}{2\\pi} \\right)^{2\/3} \\frac{Q^{4\/3}_{\\rm FB}}{R}\n        \\left[1+\\frac{4 \\pi}{9}\\left(\\frac{2\\pi}{3} \\right)^{1\/3} \\frac{R^2 T^2}{Q^{2\/3}_{\\rm FB}}\n        \\left(1+\\frac{3}{2\\pi^2}\\frac{M^2_i}{T^2} \\right)\n         \\right] \\nonumber \\\\\n        &&\n        -\\frac{3 g^2_\\chi}{8 \\pi} \\frac{Q^2_{\\rm FB} L^2_\\phi}{R^3}\n        + \\frac{4 \\pi}{3} V_0 R^3 \\left(1+\\frac{T^2 M^2_i}{12 V_0} \\right) \\,,\n    \\end{eqnarray}\n    where $V_0(T)\\equiv V_{\\rm eff}(0,T) - V_{\\rm eff}(v_\\phi(T),T)$, which at zero temperature returns into $B$. Since the FB is a macroscopic object, we ignore the contribution from the surface tension.\n    Inside a FB, each $\\chi$ has mass $M_i$ and couples to others via the attractive Yukawa interaction $g_{\\chi}\\phi \\bar{\\chi}\\chi$. The length of Yukawa interaction is given as\n    \\begin{eqnarray}\n     L_\\phi(T)\\equiv \\left(\\frac{d^2 V_\\mathrm{eff}}{d\\phi^2}\\bigg|_{\\phi=0}\\right)^{-1\/2}=(2D(T^2-T^2_0))^{-1\/2} \\,.\n    \\end{eqnarray}\n\n    To find $R_{\\rm FB}$, we require $dE_{\\rm FB}\/dR=0$,\n    which yields a cubic polynomial equation with three solutions.\n    The largest among the three roots\n    gives the FB radius $R_{\\rm FB}$.\n    Evaluating $E_{\\rm FB}$ at $R_{\\rm FB}$ gives rise the mass of the FB,\n    we have\n    \\begin{eqnarray}\n        R_{\\rm FB} &=& \\left[ \\frac{1}{4}\\left( \\frac{3}{2\\pi} \\right)^{2\/3} \\frac{Q^{4\/3}_{\\rm FB}}{V_0} \\right]^{1\/4} X^{1\/2} \\\\\n        M_{\\rm FB} &=& \\frac{3}{4} Q_{\\rm FB} \\left( 9 \\pi^2 V_0 \\right)^{1\/4} X^{-3\/2} \\nonumber \\\\\n        &\\times& \\left\\lbrace X+ \\frac{4}{9}\\left(1+\\frac{T^2 M^2_i}{12V_0} \\right)X^3\n        + \\left(  \\frac{2\\pi T^2}{9 V^{1\/2}_0}+ \\frac{M^2_i}{3\\pi V^{1\/2}_0} \\right) X^2\n        - \\frac{2g^2_\\chi L^2_\\phi V^{1\/2}_0}{3\\pi}\\right\\rbrace\\,,\n        \\label{eq:FB_mass_radius}\n    \\end{eqnarray}\n    where denote\n    \\begin{eqnarray}\n        X&\\equiv& \\left(1+\\frac{T^2 M^2_i}{12V_0} \\right)^{-1}  \\nonumber\\\\\n        & \\times &\\Bigg[\\left( 1+ \\frac{13}{108}\\frac{T^2M^2_i}{V_0}+ \\frac{\\pi^2}{81} \\frac{T^4}{V_0} + \\frac{1}{36\\pi^2}\\frac{M^4_i}{V_0} \\right)^{1\/2} \\cos\\theta-\\frac{\\pi}{18} \\frac{T^2}{V^{1\/2}_0}\\left(1+\\frac{3}{2\\pi^2}\\frac{M^2_i}{T^2} \\right) \\Bigg]\\,. \\nonumber\n       \\end{eqnarray}\nTo make a FB stable in the false vacuum, it further requires the conditions~\\cite{Kawana:2021tde}\n\\begin{eqnarray}\n\\label{eq:FB_stable}\n\\frac{dM_{\\rm FB}}{dQ_{\\rm FB}} \\leq g_\\chi v_\\phi+M_i\\,,~~~~\\frac{d^2M_{\\rm FB}}{dQ^2_{\\rm FB}}\\leq 0\\,,\n\\end{eqnarray}\nwhere the second one is automatically satisfied by the surface tension.\nHowever, in our region of interest, non-zero $M_i$ is always necessary to full fill the first condition with perturbative Yukawa coupling\n$|g_\\chi|\\leq \\sqrt{4 \\pi}$.\n\n\n    If we neglect the Yukawa energy, $X$ reduces into\n    \\begin{eqnarray}\n        X\n        &\\simeq& \\frac{\\sqrt{3}}{2}\\left(1-\\frac{1}{4\\sqrt{3}\\pi}\\frac{M^2_i}{V^{1\/2}_0}- \\frac{\\pi}{6\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{5}{216} \\frac{M^2_iT^2}{V_0}  \\right)\\,.\n    \\end{eqnarray}\n    Under the assumption $V_0\\gg T$ and $M_i$, Eq.~(\\ref{eq:FB_mass_radius}) can be simplified into\n    \\begin{eqnarray}\n        \\label{eq:FB_mass_radius_simple}\n        R_{\\rm FB}&=& \\left[ \\frac{3}{16} \\left( \\frac{3}{2\\pi} \\right)^{2\/3}\n        \\frac{Q^{4\/3}_{\\rm FB}}{V_0} \\right]^{1\/4}\n        \\left(1-\\frac{1}{8\\sqrt{3}\\pi}\\frac{M^2_i}{V^{1\/2}_0}\n        - \\frac{\\pi}{12\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{5}{432} \\frac{M^2_iT^2}{V_0} \\right)\\,,  \\\\\n        M_{\\rm FB} &=& Q_{\\rm FB}\\left( 12 \\pi^2 V_0 \\right)^{1\/4}\n        \\left(1+ \\frac{\\sqrt{3}}{8\\pi} \\frac{M^2_i}{V^{1\/2}_0}\n        +\\frac{\\pi}{4\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{1}{16} \\frac{M^2_i T^2}{V_0} \\right)\\,.\n    \\end{eqnarray}\n\n\n\nDue to the fact that the magnitude of Yukawa energy\n    \\begin{eqnarray}\n        |E_Y|\\simeq \\frac{3g^2_\\chi}{8 \\pi} \\frac{Q^2_{\\rm FB}}{R}\\,  \\left(\\frac{L_\\phi}{R} \\right)^2\n    \\end{eqnarray}\nincreases as the temperature decreases, at temperature $T_\\phi$ when $|E_Y|$ equals to the Fermi-gas kinetic energy, the FB becomes unstable and starts collapsing to a PBH. This roughly coincides when $L_\\phi$ equals to the mean separation distance of $\\chi$'s, i.e., $L_\\phi\\simeq R_{\\rm FB}\/Q^{1\/3}_{\\rm FB}$.\nTherefore, there are two scenarios of PBH formation:\ni) If $T_\\phi \\leq T_\\star$, the FB forms as an intermediate state before collapsing into PBH.\nii) If $T_\\phi > T_\\star$, the $\\chi$'s enclosed in a critical volume directly collapse into PBH without FB formation.\nIn the first scenario, we can adopt the above formulas of FB to compute the mass of PBH, i.e. $M_{\\rm PBH}(T_\\phi)=M_{\\rm FB}(T_\\phi)$, and the\nnumber density follows the adiabatic evolution of Universe $n_{\\rm PBH}|_{T_\\phi}=n_{\\rm FB}|_{T_\\star} s(T_\\phi)\/s(T_\\star)$~\\cite{Kawana:2021tde}.\nConsequently, the PBH relic abundance and fraction at present Universe is given by\n\\begin{eqnarray}\n\\Omega_{\\rm PBH}h^2 = \\frac{M_{\\rm PBH}|_0 n_{\\rm PBH}|_0}{3 M^2_{\\rm Pl}(H_0\/h)^2}\\,,~ f_{\\rm PBH}\\equiv \\frac{\\Omega_{\\rm PBH}h^2}{\\Omega_{\\rm DM}h^2}\\,,\n\\end{eqnarray}\nwhere the Hubble constant $H_0=2.13h\\times 10^{-42}~{\\rm GeV}$.\nIn the following section,\nwe will select the benchmark points belong to the first scenario.\n\n\n\n\n\n\\bigskip\n\n\\section{Correlated signals}\n\\label{sec:BP}\n\n    \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=13cm]{FIG\/3.jpg}\n      \\caption{The extragalactic photon contributions of {\\bf BP}s in Table~\\ref{table:1}. }\\label{fig:3}\n    \\end{figure}\n\n     \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=13cm]{FIG\/4.jpg}\n      \\caption{\n      Gravitational wave power spectra for {\\bf BP}s in Table~\\ref{table:1}.\n      }\\label{fig:4}\n    \\end{figure}\n\n    Since the FOPT behaves as common origin for both PBHs and GW,\n    it predicts the correlated signals among the 511 keV line,\n    extragalactic photon spectrum,\n    and GW spectrum.\n    More specifically, the quartic potential dictates the FOPT\n    and then the Yukawa interaction determines the period of PBHs formation.\n    %\n    Therefore, we select and scan the input parameters: including the coefficients of the effective potential,\n    the asymmetry parameter $\\eta_\\mathrm{DM}$,\n    the temperature ratio of the dark and SM sectors $r_T$,\n    the Yukawa coupling $g_\\chi$, and the bare DM mass $M_i$,\n    in the ranges\n\\begin{eqnarray}\n&& 0.1\\leq B^{1\/4}\/{\\rm MeV}\\leq 10^4, \\mkern15mu 0.1\\leq D\\leq 10, \\mkern15mu 0.05\\leq \\lambda \\leq 0.2 \\nonumber \\\\\n&& 0.01\\leq C\/{\\rm MeV} \\leq 10^4 , \\mkern15mu 0.3\\leq r_T\\leq 1, \\mkern15mu 0.01\\leq g_\\chi \\leq \\sqrt{4\\pi} \\nonumber \\\\\n&& 10^{-3} \\leq M_i\/B^{1\/4} \\leq 10\\,,\n\\end{eqnarray}\n    where $A$ is fixed to be $0.1$.\n    We pick six benchmark points (BPs) satisfying the 511 keV excess with the corresponding PBH mass and $f_\\mathrm{PBH}$ are listed in Table~\\ref{table:1} and Fig.~\\ref{fig:2}.\n    \\begin{table}[h]\n   \n        \\centering\n        \\resizebox{\\textwidth}{!}{\n        \\begin{tabular}{c|c c| c c| c c }\n            \\hline\n            \\hline\n            & \\bf{BP-1} & \\bf{BP-2} & \\bf{BP-3} & \\bf{BP-4}& \\bf{BP-5} & \\bf{BP-6} \\\\\n           \\hline\n           \\hline\n            $B\/\\rm MeV^4$ & $5.94\\times10^{4}$ & $4.13\\times10^{5}$ & $5.69\\times10^{2}$ & $4.80\\times10^{5}$ & $3.99\\times10^{7}$ & $5.19\\times10^{6}$ \\\\\n            $\\lambda$ & $0.1693$ & $0.0888$ & $0.0999$ & $0.0795$ & $0.0685$ & $0.1903$\\\\\n            $D$ & $1.9103$ & $0.5360$ & $3.5249$ & $1.6074$ & $1.3413$ & $0.8706$ \\\\\n            $\\eta_{\\mathrm{DM}}$ & $4.70\\times10^{-14}$ & $3.60\\times10^{-13}$ & $6.70\\times10^{-11}$ & $7.60\\times10^{-11}$ & $4.35\\times10^{-11}$ & $1.08\\times10^{-9}$ \\\\\n            $r_T$ & $0.4377$ & $0.3512$ & $0.3149$ & $0.3845$ & $0.3891$  & $0.4707$\\\\\n            $C\/\\rm MeV$ & $1.5257$ & $1.3976$ & $0.0920$ & $0.5830$ & $2.8082$ & $3.6441$\\\\\n            $g_{\\chi}$ & $1.2728$ & $1.1556$ & $1.0275$ & $1.1958$ & $0.9141$ & $1.3278$\\\\\n            $M_i\/B^{1\/4}$ & $0.5570$ & $0.0516$ & $0.4973$ & $0.0047$ & $0.3048$ & $1.4341$ \\\\\n            \\hline\n            $M_\\mathrm{PBH}\/M_{\\bigodot}$ & $9.32\\times 10^{-19}$ & $4.39\\times 10^{-18}$ & $1.52\\times 10^{-17}$ & $3.72\\times 10^{-17}$ & $4.90\\times 10^{-17}$ & $7.04\\times 10^{-17}$\\\\\n            $f_\\mathrm{PBH}$ & $4.74\\times 10^{-6}$ & $8.20\\times 10^{-5}$ & $0.0013$ & $0.0100$ & $0.0235$ & $0.4350$\\\\\n          \\hline\n        \\end{tabular}\n        }\n        \\caption{\n        The benchmark points form PBHs after FOPT with $A=0.1$ fixed.\n        }\n        \\label{table:1}\n    \\end{table}\n\n\n    According to the BPs in Table~\\ref{table:1}, in order to produce the desired PBH mass,\n    one requires the energy scale $\\mathcal{O}(1)\\lesssim B^{1\/4}\/{\\rm MeV} \\lesssim \\mathcal{O}(100)$ of FOPT. By incorporating $\\eta_{\\rm DM}$, we can obtain the correct values of $f_{\\rm PBH}$ to generate 511 keV line.\n    Under our scenario,\n    FB was formed as an intermediate state and treated as a progenitor of PBH.\n    In order to satisfy the stability condition of FB, i.e Eq.(\\ref{eq:FB_stable}), all BPs tend to have large values of $g_\\chi$, close to the perturbative limit,\n    and nonzero values of $M_i$ are necessary.\n    In addition, the observational upper bound of $\\Delta N_{\\rm eff}\\leq 0.5$\n    was imposed to all the BPs,\n    and thereby restricts the temperature ratio to be $0.3\\leq r_T\\leq 0.5$.\n    Here, we assumed the dark sector and SM sector are thermally decoupled,\n    such that it suppresses the dark sector contribution to the light degree of freedom.\n\n\n    The extragalactic photon contributions of these BPs are shown\n    in Fig.~\\ref{fig:3}. {\\bf{BP-4}}, {\\bf{BP-5}} and {\\bf{BP-6}} haven't\n    been rule out by the present observations COMPTEL\/EGRET\/FermiLAT.\n    This is because the associated PBH masses from {\\bf{BP-4}}, {\\bf{BP-5}} and {\\bf{BP-6}} are heavier\n    comparing with {\\bf{BP-1}}, {\\bf{BP-2}}, and {\\bf{BP-3}}.\n    Base on the property of Hawking radiation,\n    the gamma-ray spectra of them peak\n    at lower energy window between\n    0.1 MeV to 1 MeV, where the present observations do not have sufficient sensitivities.\n    However, this window will be explored\n    by future AMEGO and e-ASTROGAM observations.\n    %\n    On the other hand, the GW spectra of BPs, mainly arising from sound waves in the plasma after bubbles collision during the FOPT~\\cite{Caprini:2015zlo},\n    are shown in Fig.~\\ref{fig:4}.\n    They cover the frequency region from $10^{-5}$ Hz to $10^{-3}$ Hz,\n    and their amplitudes can be substantially detected by future $\\mu$Ares telescope.\n\n\n\\bigskip\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nPBH with mass $3\\times 10^{-17} \\lesssim M_{\\rm PBH}\/m_\\odot \\lesssim 7\\times 10^{-17}$ and fractional abundance\n$0.01 \\lesssim f_{\\rm PBH} \\lesssim 0.5$ is favored to explain\nthe galaxy center 511 keV line excess.\nPBH evaporation emits not only positrons but also photons\ncontributing to the extragalactic gamma-ray flux.\nWe investigated the novel PBH production mechanism from the cosmological FOPT\naggregating $\\chi$'s into FB as intermediate state.\nThrough the attractive Yukawa interaction, FBs become unstable as temperature decreasing and eventually collapse to form PBHs.\nTo realise this scenario,\nwe selected the BPs satisfying\nseveral conditions: $\\Delta N_{\\rm eff}\\leq 0.5$,\nstability of FB in the false vacuum,\nand upper limits of gamma-ray flux from EGERT\/COMPTEL\/FermiLAT.\nIncorporate the $\\chi$ asymmetry $\\eta_{\\rm DM}\\simeq 10^{-10}$,\nthe phase transition with vacuum energy\n$\\mathcal{O}(1)\\lesssim B^{1\/4}\/{\\rm MeV} \\lesssim \\mathcal{O}(100)$\nproduces the desired PBH mass and abundance for 511 keV excess.\nConsequently, correlated signals of extragalactic $\\mathcal{O}({\\rm MeV})$ gamma-ray and  GW spectrum with peak frequency\nfrom $10^{-5}$ Hz to $10^{-3}$ Hz are predicted.\nIn the future, these signals can be either confirmed or ruled out by AMEGO\/e-ASTROGAM\nand $\\mu$Ares.\n\n\n\n\\bigskip\n\n\\section*{Acknowledgment}\nP.Y.Tseng is supported in part by the Ministry of Sciences and Technology with\nGrant No. MoST-111-2112-M-007-012-MY3.\nY.M.Yeh is supported in part by Ministry of Education with Grant No. 111J0382I4.\n\n\n\n\n\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\n\nIn \\cite{Ni:Siena} considering a second order operator in ${\\mathbb R}^{1+2}$\n \\begin{equation}\n \\label{eq:model}\n P_{mod}=-D_0^2+2x_1D_0D_2+D_1^2+x_1^3D_2^2,\\quad x=(x_0, x')=(x_0, x_1, x_2)\n \\end{equation}\n %\n we have proved\n\\begin{them}\n\\label{thm:main}{\\rm(\\cite[Theorem 1.1]{Ni:Siena})}\nThe Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable at the origin in the Gevrey class of order $s$ for any $b_0, b_1, b_2\\in {\\mathbb C}$ if $s>6$. In particular the Cauchy problem for $P_{mod}$ is $C^{\\infty}$ ill-posed near the origin for any $b_0, b_1, b_2\\in {\\mathbb C}$.\n\\end{them}\nRecall that the Gevrey class of order $s$, denoted by $\\gamma^{(s)}({\\mathbb R}^n)$, is the set of all $f(x)\\in C^{\\infty}({\\mathbb R}^n)$ such that for any compact set $K\\subset {\\mathbb R}^n$, there exist $C>0$, $h>0$ such that\n\\begin{equation}\n\\label{eq:gevrey}\n|\\partial_x^{\\alpha}f(x)|\\leq Ch^{|\\alpha|}|\\alpha|!^s,\\quad x\\in K,\\;\\;\\alpha\\in {\\mathbb N}^{n}.\n\\end{equation}\nWe say that the Cauchy problem for $P=P_{mod}+\\sum_{j=0}^2b_jD_j$ is locally solvable in $\\gamma^{(s)}$ at the origin if for any $\\Phi=(u_0, u_1)\\in (\\gamma^{(s)}({\\mathbb R}^2))^2$ there exists  a neighborhood $U_{\\Phi}$, may depend on $\\Phi$,  of the origin such that the Cauchy problem \n\\begin{equation}\n\\label{eq:CP}\n\\begin{cases}\nPu=0\\quad \\text{in}\\;\\;U_{\\Phi},\\\\\nD_0^ju(0, x')=u_j(x'),\\quad x'\\in U_{\\Phi}\\cap\\{x_0=0\\},\\;\\;j=0, 1\n\\end{cases}\n\\end{equation}\nhas a solution $u(x)\\in C^2(U_{\\Phi})$.\nIn this note we remark that one can improve Theorem \\ref{thm:main} so that\n\\begin{them}\n\\label{thm:main:bis}\nThe Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable in $\\gamma^{(s)}$ at the origin  for any $b_0, b_1, b_2\\in {\\mathbb C}$ if $s>5$. \n\\end{them}\n$P_{mod}$ plays a special role in the well-posedness of the Cauchy problem for differential operators with non-effectively hyperbolic characteristics. Before explaining it we give a short introduction to the general context.  Let $P$ be a differential operator of order $m$ with principal symbol $p(x, \\xi)$.  At a singular point $\\rho$ of $p=0$, the Hamilton map $F_p(\\rho)$ is defined as the linearization at $\\rho$ of the Hamilton vector field $H_p$. The first fundamental result is\n\\begin{them}\n\\label{thm:IPH}{\\rm(\\cite{IP}, \\cite{Ho1})} Let $\\rho$ be a singular point of $p=0$  and assume that $F_p(\\rho)$ has no non-zero real  eigenvalues. If the Cauchy problem for $P$ is $C^{\\infty}$ well posed it is necessary that\n\\begin{equation}\n\\label{eq:IPH}\n{\\mathsf{Im}}P_{sub}(\\rho)=0,\\quad \\big|P_{sub}(\\rho)\\big|\\leq {\\rm Tr^+}F_p(\\rho)\/2\n\\end{equation}\nwhere ${\\rm Tr^+}F_p(\\rho)=\\sum|\\mu_j|$ and $\\mu_j$ are the eigenvalues of $F_p(\\rho)$, counted with multiplicity, and $P_{sub}$ is the subprincipal symbol of $P$. \n\\end{them}\nWe call \\eqref{eq:IPH} the Ivrii-Petkov-H\\\"ormander condition (IPH condition, for short). If the strict inequality holds in \\eqref{eq:IPH} we call it the strict Ivrii-Petkov-H\\\"ormander condition (strict IPH condition, for short). For the sufficiency of the IPH condition, we assume that the set of singular points $\\Sigma$ of $p=0$ is a $C^{\\infty}$ manifold and the following conditions are satisfied:\n\\begin{equation}\n\\label{eq:jyoken}\n\\begin{cases}\n\\text{$F_p$ has no non-zero real eigenvalues at each point of $\\Sigma$},\\\\\n\\text{Near  each point of $\\Sigma$, $p$ vanishes exactly of order $2$} \\\\\n\\text{and the rank of  $\\sum_{j=0}\n^nd\\xi_j\\wedge dx_j$ is constant. }\n\\end{cases}\n\\end{equation}\nAccording to the spectral type of $F_p(\\rho)$, two different possible cases may arise\n\\begin{align}\n\\label{eq:type:1}\n{\\rm Ker}\\,F_p^2(\\rho)\\cap {\\rm Im}\\,F_p^2(\\rho)= \\{0\\}, \\\\\n\\label{eq:type:2}\n{\\rm Ker}\\,F_p^2(\\rho)\\cap {\\rm Im}\\,F_p^2(\\rho)\\neq \\{0\\}.\n\\end{align}\nWe say that $p$ is of spectral type $1$ (resp. type $2$) near $\\rho\\in\\Sigma$ if there is a conic neighborhood $V$ of $\\rho$ such that \\eqref{eq:type:1} (resp. \\eqref{eq:type:2}) holds in $V\\cap \\Sigma$. We say that there is no transition (of spectral type) if for any $\\rho\\in \\Sigma$ one can find a conic neighborhood $V$ of $\\rho$ such that either \\eqref{eq:type:1} or \\eqref{eq:type:2} holds in $V\\cap\\Sigma$. \n\\begin{them}\n\\label{thm:jyubun} {\\rm(\\cite[Theorem 5.1]{Ni:book})} Assume \\eqref{eq:jyoken} and that there is no transition and no bicharacteristic tangent to $\\Sigma$. Then the Cauchy problem for $P$ is $C^{\\infty}$ well posed  under the strict IPH condition.\n\\end{them}\nThe principal symbol  $p=-\\xi_0^2+2x_1\\xi_0\\xi_2+\\xi_1^2+x_1^3\\xi_2^2$ of $P_{mod}$  is a typical example of spectral type $2$ with tangent bicharacteristics (note that there is no tangent bicharacteristic if $p$ is of spectral type $1$, see \\cite{Iv2}). The set of singular point of $p=0$ with $\\xi\\neq 0$ is $\\Sigma=\\{x_1=0, \\xi_0=\\xi_1=0\\}$ which is a $C^{\\infty}$ manifold on which $p$ vanishes exactly of order $2$ and a tangent bicharacteristic is given explicitly  by\n\\begin{equation}\n\\label{eq:bitokusei}\n(x_1,  x_2)=(-x_0^2\/4,  x_0^5\/80),\\quad \n(\\xi_0, \\xi_1, \\xi_2)=(0, x_0^3\/8, 1)\n\\end{equation}\nwhich is  parametrized by $x_0$. The operator $P_{mod}$ shows how the situation becomes to be complicated when a tangent bicharacteristic exists. We now give some  such results: The Cauchy problem for $P_{mod}$  is not locally solvable in $\\gamma^{(s)}$ for $s>5$, in particular ill-posed in $C^{\\infty}$ while the Cauchy problem for general second order operator $P$ of spectral type $2$ satisfying $P_{sub}=0$ on $\\Sigma$ (note that the subprincipal symbol of $P_{mod}$ is identically zero) is well posed in the Gevrey class of order $1<s\\leq 5$ (\\cite{BN:jdam} see also \\cite{Ni:book}). The Cauchy problem for $P_{mod}+SD_2$ with $0\\neq S\\in {\\mathbb C}$ is not locally solvable in $\\gamma^{(s)}$ for $s>3$ (Proposition \\ref{pro:main} below) while the Cauchy problem for general second order operator $P$ of spectral type $2$ with ${\\rm codim}\\,\\Sigma=3$ is well posed in the Gevrey class of order $1<s<3$  for any lower order term even if a tangent bicharacteristic exists (\\cite{BN:JHDE}).\n\n\n\\section{A family of exact solutions}\n\n\n\n\nFirst recall\n\\begin{lemma}\n\\label{lem:b2:zero}{\\rm(\\cite[Proposition 8.1]{Ni:book})}The Cauchy problem for $P_{mod}+\\sum_{j=0}^1b_jD_j$ is not locally solvable in $\\gamma^{(s)}$ at the origin  for any $b_0, b_1\\in {\\mathbb C}$ if $s>5$.\n\\end{lemma}\nThus in order to prove Theorem \\ref{thm:main:bis} it suffices to prove the following result which also improves \\cite[Theorem 1.3]{BN:JHDE}.\n\\begin{pro}\n\\label{pro:main}The Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable in $\\gamma^{(s)}$ at the origin  for any $b_0, b_1\\in {\\mathbb C}$ and $0\\neq b_2\\in{\\mathbb C}$ if $s>3$.\n\\end{pro}\nTo prove Proposition \\ref{pro:main} we repeat the proof of \\cite[Theorem 1.3]{BN:JHDE} with obvious minor changes. Look for a family of solutions to  $(P_{mod}+\\sum_{j=0}^2b_jD_j)U_{\\lambda}=0$ in the form\n\\begin{equation}\n\\label{eq:teigi:U}\nU_{\\lambda}(x)=e^{ i\\xi_0\\lambda x_0}V_{\\lambda}(x'),\\quad V_{\\lambda}(x')=e^{\\pm \\lambda^5x_2-i(b_1\/2)x_1}u(\\lambda^2x_1),\\;\\;\\xi_0=\\xi_0(\\lambda)\n\\end{equation}\nthat is, we look for $u(x)$ satisfying\n\\begin{equation}\n\\label{eq:a8}\nu''(x)=\\big(x^3+ 2\\xi_0 x+ b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4\\big)u(x).\n\\end{equation}\nTo study solutions to \\eqref{eq:a8} we consider \n\\begin{equation}\n\\label{eq:aa1}\nu''(x)=(x^3+a_2x+a_3)u(x),\\quad x\\in{\\mathbb C},\\;\\;a_j\\in{\\mathbb C},\\;\\;j=2, 3.\n\\end{equation}\nLet ${\\mathcal Y}_0(x;a)$, $a=(a_2,a_3)$ be the  solution given in \\cite[Chapter 2]{Si} to (\\ref{eq:aa1}) which has asymptotic representation\n\\begin{equation}\n\\label{eq:zenkin:a}\n{\\mathcal Y}_0(x; a)\\simeq x^{-3\/4}\\big[1+\\sum_{N=1}^{\\infty}B_Nx^{-N\/2}\\big]e^{-E(x,a)}\n\\end{equation}\nas $x$ tends to infinity in any closed subsector of the open sector $|\\arg x|<3\\pi\/5$ where\n\\[\nE(x,a)=\\frac{2}{5}x^{5\/2}+a_2x^{1\/2}\n\\]\nand $A_N$, $B_N$ are polynomials in $(a_2,a_3)$. Let $\\omega=e^{2\\pi i\/5}$ and set\n\\begin{equation}\n\\label{eq:setuzoku}\n{\\mathcal Y}_k(x;a)={\\mathcal Y}_0(\\omega^{-k}x; \\omega^{-2k}a_2,\\omega^{-3k}a_3),\\quad k=0, 1, 2, 3, 4\n\\end{equation}\nwhich are also solutions to (\\ref{eq:aa1}).  Recall that \\cite[Chpater 17]{Si}\n\\[\n{\\mathcal Y}_k(x;a)=C_k(a){\\mathcal Y}_{k+1}(x;a)-\\omega {\\mathcal Y}_{k+2}(x;a)\n\\]\nwhere $C_k(a)$ are entire analytic in $a=(a_2,a_3)$ and $C_k(a_2,a_3)=C_0(\\omega^{-2k}a_2,\\omega^{-3k}a_3)$. Choose \n\\begin{equation}\n\\label{eq:aa2}\n\\begin{split}\n&u(x)={\\mathcal Y}_0(x; a)=C_0(a){\\mathcal Y_1}(x; a)-\\omega{\\mathcal Y_2}(x; a),\\\\\n &a_2=2\\xi_0,\\quad a_3= b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4\n\\end{split}\n\\end{equation}\nwhich solves \\eqref{eq:a8}. We require that  $V_{\\lambda}(x')$  is bounded  as  $\\lambda\\to +\\infty$ when $|x'|$ remains in a bounded set. Note  \\eqref{eq:setuzoku} and\n\\begin{equation}\n\\label{eq:heiho}\n\\begin{split}\n(\\omega^{-1}x)^{5\/2}=-i|x|^{5\/2},\\quad 2\\omega^{-2} \\xi_0(\\omega^{-1}x)^{1\/2}=-2i \\xi_0|x|^{1\/2},\\\\\n(\\omega^{-2}x)^{5\/2}=i|x|^{5\/2},\\quad 2\\omega \\xi_0(\\omega^{-2}x)^{1\/2}=2i \\xi_0|x|^{1\/2}\n\\end{split}\n\\end{equation}\nfor $x<0$. Then $|{\\mathcal Y}_1(\\lambda^2x; a_2, a_3)|$ and $|{\\mathcal Y}_2(\\lambda^2x;a_2, a_3)|$ behaves like $e^{2{\\mathsf{Re}}(i\\xi_0)\\lambda |x|^{1\/2}}$ and $e^{-2{\\mathsf{Re}}(i\\xi_0)\\lambda |x|^{1\/2}}$  respectively as $x\\to -\\infty$.  Since $\\omega\\neq 0$, taking \\eqref{eq:aa2} into account, the requirement for boundedness implies that\n\\begin{align}\n\\label{eq:Stokes}\nC_0(2\\xi_0, a_3)=0,\\\\\n\\label{eq:expo}\n-{\\mathsf{Re}}(i\\xi_0)={\\mathsf{Im}}\\,\\xi_0<0.\n\\end{align}\n\nInstead of solving directly the ``Stokes equation\" (\\ref{eq:Stokes}) we go rather indirectly. Let us consider\n\\[\nH(\\beta)=p^2+x^2+i\\beta x^3\n\\]\nas an operator in $L^2({\\mathbb R})$ with the domain $D(H(\\beta))=D(p^2)\\cap D(x^3)$. Here $p^2$ denotes the self-adjoint realization of $-d^2\/dx^2$ defined in $H^2({\\mathbb R})$ and by $D(x^3)$ we mean the domain of the maximal multiplication operator by the function $x^3$.\n\n\\begin{pro}\n\\label{pro:Caliceti_1}{\\rm \\cite[Corollary 2.16, Lemma 3.1]{CGM})}\n Let $k\\in {\\mathbb N}_0$ and $\\epsilon>0$ be given. Then there is a $B>0$ such that for $|\\beta|<B$, ${\\mathsf {Re}}\\,\\beta>0$, $H(\\beta)$ has exactly one eigenvalue $E_k(\\beta)$ near $2k+1$. Such eigenvalues are analytic functions of $\\beta$ for $|\\beta|<B$, ${\\mathsf{Re}}\\,\\beta>0$, and admit an analytic continuation across the imaginary axis to the whole sector $|\\beta|<B$, $|\\arg{\\beta}|<\\frac{5\\pi}{8}-\\epsilon$ and \n uniformly asymptotic to the following  formal Taylor expansion in powers of $\\beta^2$\n\\begin{equation}\n\\label{eq:Eigenvalue}\n\\sum_{j=0}^{\\infty}a_{2j}\\beta^{2j},\\quad a_0=2k+1\n\\end{equation}\n near $\\beta=0$ in any closed subsector in $|\\arg{\\beta}|<\\frac{5\\pi}{8}-\\epsilon$.\n\\end{pro}\nFor the proof we refer to \\cite{CGM}, \\cite{Simon}. Now we have\n\\begin{lemma}\n\\label{lem:a1}{\\rm(\\cite[Lemma 6.1]{BN:JHDE})}\nAssume that ${\\mathsf{Re}}\\,\\beta>0$ and $E(\\beta)$ is an eigenvalue of the problem\n\\begin{equation}\n\\label{eq:a1}\n-u''(x)+( x^2+i\\beta x^3)u(x)=E(\\beta)u(x)\n\\end{equation}\nthat is $(\\ref{eq:a1})$ has a solution $0\\neq u\\in D(H(\\beta))$. Then we have\n\\begin{equation}\n\\label{eq:a2}\nC_0\\Big(-\\frac{\\omega^2}{3}\\beta^{-\\frac{8}{5}},\\, \\omega^3\\big\\{\\frac{2}{27}\\beta^{-\\frac{12}{5}}+\\beta^{-\\frac{2}{5}}E(\\beta)\\bigl\\}\\Big)=0\n\\end{equation}\nwhere $\\beta^{-j\/5}=(\\beta^{-1\/5})^{j}$ and the branch $\\beta^{\\pm1\/5}$ is chosen such that $|\\arg \\beta^{\\pm1\/5}|<\\pi\/10$.\n\\end{lemma}\nLet $E(\\beta)$, $|\\beta|<B$, ${\\mathsf Re}\\,\\beta>0$ be an eigenvalue which is analytically continued to the sector $|\\beta|<B$, $|{\\rm arg}\\,\\beta|<5\\pi\/8$ by Proposition \\ref{pro:Caliceti_1} (though when $|{\\rm arg}\\,\\beta|=\\pi\/2$, $H(\\beta)$ admits  infinitely many distinct self-adjoint extensions, see \\cite{CGM}). Since $C_0(a_2, a_3)$ is entire analytic in $(a_2, a_3)$ then \\eqref{eq:a2} holds in this sector. Thanks to Lemma \\ref{lem:a1}, if $\\xi_0$ satisfies\n\\begin{equation}\n\\label{eq:a9}\n\\left\\{\\begin{array}{l}\n\\displaystyle{2\\xi_0=-\\frac{\\omega^2}{3}\\beta^{-\\frac{8}{5}},}\\\\[8pt]\n\\displaystyle{a_3=\\omega^3\\big\\{\\frac{2}{27}\\beta^{-\\frac{12}{5}}+\\beta^{-\\frac{2}{5}}E(\\beta)\\big\\}}\n\\end{array}\\right.\n\\end{equation}\nthen \\eqref{eq:Stokes} is satisfied hence we have\n\\[\nu(\\lambda^2 x)={\\mathcal Y}_0(\\lambda^2x;2\\xi_0,a_3)\n=-\\omega{\\mathcal Y}_2(\\lambda^2x;2\\xi_0, a_3).\n\\]\nTherefore  we look for $\\xi_0$ satisfying \\eqref{eq:a9} and   \\eqref{eq:expo}. Plugging $\\xi_0=-\\omega^2\\beta^{-8\/5}\/6$ into the second equation, (\\ref{eq:a9}) is reduced to\n\\begin{equation}\n\\label{eq:10}\n\\begin{split}\n\\frac{2}{27}\\omega^3\\beta^{-\\frac{12}{5}}+E(\\beta)\n\\omega^3\\beta^{-\\frac{2}{5}}+\\frac{1}{36}\n\\omega^4\\beta^{-\\frac{16}{5}}\\lambda^{-2}\\\\\n+ \\frac{b_0}{6}\\omega^2\\beta^{-8\/5}\\lambda^{-3}= b_2\\lambda-\\frac{b_1^2}{4}\\lambda^{-4}.\n\\end{split}\n\\end{equation}\n\n\n\\subsection{Solving the equation (\\ref{eq:10}) }\n\\label{sec:solv}\n\nSolve (\\ref{eq:10}) with respect to $\\beta$ where $E(\\beta)$ is one of $E_k(\\beta)$ which is analytic in $|\\arg{\\beta}|<5\\pi\/8-\\epsilon$ and admits a uniform asymptotic expansion (\\ref{eq:Eigenvalue}) there by Proposition \\ref{pro:Caliceti_1}. Put $\\zeta=\\beta^{-2\/5}=(\\beta^{-1\/5})^{2}$ so that \nthe equation leads to\n\\begin{equation}\n\\label{eq:reduced}\n\\omega^3\\zeta^6+\\frac{27}{2}E(\\zeta^{-5\/2})\\omega^3\\zeta\n+\\frac{3}{8}\\omega^4\\zeta^8\\lambda^{-2}+ \\frac{9}{4}b_0\\omega^2\\zeta^4\\lambda^{-3}= \\frac{27}{2}b_2\\lambda-\\frac{27}{8}b_1^2\\lambda^{-4}\n\\end{equation}\nand we look for a solution $\\zeta(\\lambda)$ to \\eqref{eq:reduced} verifying\n\\begin{equation}\n\\label{eq:cond}\n|\\arg \\zeta(\\lambda)|<\\pi\/4-\\epsilon,\\quad {\\mathsf{Im}}\\;\\omega^2\\zeta(\\lambda)^4>0\n\\end{equation}\nwhere the second requirement comes from \\eqref{eq:expo}. Assume that \n\\begin{equation}\n\\label{eq:argA}\n0<\\arg b_2<\\pi \\quad \\text{or}\\quad -\\pi \\leq \\arg b_2<-\\pi\/2\n\\end{equation}\nand denote $27b_2 \/2$ by  $A$  for notational simplicity and look for $\\zeta(\\lambda)$ in the form\n\\[\n\\left\\{\\begin{array}{l}\n\\zeta(\\lambda)=e^{-\\pi i\/5}A^{1\/6}\\big(1+\\lambda^{-5\/6}z\\big)\\lambda^{1\/6}\n\\quad (0<\\arg A<\\pi)\n,\\\\[5pt]\n\\zeta(\\lambda)=e^{2\\pi i\/15}A^{1\/6}\\big(1+\\lambda^{-5\/6}z\\big)\\lambda^{1\/6}\\quad \n(-\\pi\\leq \\arg A<-\\pi\/2).\\end{array}\\right.\n\\]\n It is clear that $\\zeta(\\lambda)$ verifies (\\ref{eq:cond}) provided that $z$ is bounded and $\\lambda$ is large. Note that   $E(\\zeta^{-5\/2})$ is analytic in $|\\arg \\zeta|<\\pi\/4-\\epsilon$ for large $|\\zeta|$ and verifies\n\\[\n\\big|E(\\zeta^{\n-5\/2})-a_0\\big|\\leq C|\\zeta|^{-5}\n\\]\nwith some $a_0=2k+1$, $k\\in{\\mathbb N}$ uniformly in $|\\arg \\zeta|<\\pi\/4-\\epsilon$ when $|\\zeta|\\to \\infty$. We insert $\\zeta$ into (\\ref{eq:reduced}) to get\n\\begin{equation}\n\\label{eq:A:la}\n\\begin{split}\nA\\lambda(1+\\lambda^{-5\/6}z)^6+\\lambda^{1\/6}H(z)+d_1\\lambda^{-2\/3}(1+\\lambda^{-5\/6}z)^8\\\\\n+d_2\\lambda^{-7\/3}(1+\\lambda^{-5\/6}z)^4=A\\lambda+d_3\\lambda^{-4},\\quad d_i\\in{\\mathbb C}\n\\end{split}\n\\end{equation}\nwhere $H(z)$ is analytic  in $|z|<B$ for $\\lambda\\geq R$ and \n\\[\n|H(z)-a_0|\\leq C\\lambda^{-5\/6},\\quad \\lambda\\geq R.\n\\]\nNote that $B$ can be chosen to be arbitrarily large taking $R$ large. Thus one can write \\eqref{eq:A:la} as\n\\begin{equation}\n\\label{eq:zed}\n6Az+a_0+\\lambda^{-5\/6} F(z,\\lambda)=0\n\\end{equation}\nwhere $F(z,\\lambda)$ is analytic in $|z|<B$ and bounded uniformly in  $|\\lambda|\\geq R$. We may assume that $|a_0\/6A|<B\/2$ taking $R$ large as noted above. By Rouch\\'e's thoerem we conclude that the equation (\\ref{eq:zed}) has a solution $z(\\lambda)$ with $|z|<B$ for any $|\\lambda|\\geq R_1$. Returning to $\\beta$ ($\\zeta=\\beta^{-2\/5}$) we conclude that (\\ref{eq:10}) has \n a solution of the form\n\\[\n\\left\\{\\begin{array}{l}\n\\beta(\\lambda)=iA^{-5\/12}\\lambda^{-5\/12}\\big(1+\\lambda^{-5\/6}z(\\lambda)\\big)\n\\quad (0<\\arg A<\\pi),\\\\[5pt]\n\\beta(\\lambda)=e^{-\\pi i\/3}A^{-5\/12}\\lambda^{-5\/12}\\big(1+\\lambda^{-5\/6}z(\\lambda)\\big) \\quad \n(-\\pi\\leq \\arg A<-\\pi\/2)\\end{array}\\right.\n\\]\nwhere $|z(\\lambda)|<B$ for $\\lambda\\geq R_1$. \nPlugging this $\\beta(\\lambda)$ into \\eqref{eq:a9} we get\n\\begin{pro}\n\\label{pro:sol} Assume \\eqref{eq:argA}. Then there exists $\\xi_0=\\xi_0(\\lambda)$ such that\n\\[\n\\left\\{\\begin{array}{l}\n2\\xi_0=c\\,\\lambda^{2\/3}(1+\\lambda^{-5\/6}z(\\lambda)),\\quad {\\mathsf{Im}}\\;c<0,\\\\[4pt]\nC_0(2\\xi_0,\\,  b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4)=0\n\\end{array}\\right.\n\\]\nwhere $|z(\\lambda)|<B$ for $\\lambda>R$.\n\\end{pro}\nIf we write \n\\begin{equation}\n\\label{eq:a:b:teigi}\n2\\xi_0=\\lambda^{2\/3}a(\\lambda),\\quad b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4=\\lambda b(\\lambda)\n\\end{equation}\n it is clear that $|a(\\lambda)|$,  $|b(\\lambda)|<M$ with some $M>0$ for $\\lambda\\geq R_1$.\n\n\n\n\\subsection{Asymptotics of ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda )$ for large $|x|$ and $\\lambda$ }\n\nRecalling \\eqref{eq:a:b:teigi} we study  how ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda )$ behaves for large $|x|$ and large $\\lambda$ where $|a|, |b| \\leq M$ is assumed. \nIn what follows $f=o_{a}(1)$ means that there are positive constants $C_{a}>0$ and $\\delta_a>0$ such that\n\\[\n|f|\\leq C_a\\lambda^{-\\delta_a}, \\quad \\lambda\\to+\\infty.\n\\]\nWe make the asymptotic representation \\eqref{eq:zenkin:a} slightly precise. \n\\begin{pro}\n\\label{pro:Esti:Y}\nLet $\\rho>1\/3$ be given. Then one can write\n\\begin{gather*}\n{\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)\n\\simeq (1+p_{\\rho}(x,\\lambda))e^{R_{\\rho}(x,\\lambda)}x^{-3\/4}\\exp{\\{-E_{\\rho}(x; a, b, \\lambda)\\}},\\\\\n{\\mathcal Y}_0'(x;a\\lambda^{2\/3},b\\lambda)\n\\simeq (-1+p_{\\rho}(x,\\lambda))e^{R_{\\rho}(x,\\lambda)}x^{3\/4}\\exp{\\{-E_{\\rho}(x; a, b,  \\lambda)\\}}\n\\end{gather*}\nas $x$ tends to infinity in any closed subsector of\n\\[\nS_{\\lambda}=\\{x;|\\arg x|<3\\pi\/5, |x|> \\lambda^{\\rho}\\}\n\\]\nwhere  \n\\[\nE_{\\rho}(x; a, b, \\lambda)=\\frac{2}{5}x^{5\/2}+a\\lambda^{2\/3} x^{1\/2}-b\\lambda x^{-1\/2}+r_{\\rho}(x,\\lambda)\n\\]\nand $r_{\\rho}$ is a polynomial in $x^{-1\/2}$ such that\n\\[\n|r_{\\rho}(x,\\lambda)|=C\\lambda^{4\/3-3\\rho\/2}(1+o_{\\rho}(1)),\\quad  x\\in S_{\\lambda}\n\\]\nand $p_{\\rho}(x,\\lambda)$, $R_{\\rho}(x,\\lambda)$ are holomorphic in $S_{\\lambda}$ and in any closed subsector of $S_{\\lambda}$\n\\begin{gather*}\n|p_{\\rho}(x,\\lambda)|\\leq C_{\\rho}\\lambda^{-2(\\rho-1\/3)},\\quad |R_{\\rho}(x,\\lambda)|\\leq C_{\\rho}\\lambda^{-1},\\quad x\\in S_{\\lambda}\n\\end{gather*}\nand $R_{\\rho}(x,\\lambda)\\to 0$, $p_{\\rho}(x,\\lambda)\\to 0$  as $|x|\\to\\infty$, $x\\in S_{\\lambda}$.\n\\end{pro}\n\\begin{proof} \nWe follow Sibuya \\cite[Chapter 2]{Si} and \\cite[Proposition 2.3]{BN:JHDE} with needed modifications. \n\\end{proof}\n\\if0\n\\begin{proof} \nWe follow Sibuya \\cite{Si} and \\cite{BN2} with needed modifications. \nLet  $y(x,\\lambda)$ satisfy\n\\begin{equation}\n\\label{eqn:1}\ny''(x,\\lambda)=(x^3+a\\lambda^{2\/3} x+b\\lambda)y(x,\\lambda)\n\\end{equation}\nwhere $|a|,|b|\\leq M$. \nSet\n\\[\nx=\\xi^2,\\quad \\left[\\begin{array}{l}\ny\\\\\ny'\\end{array}\\right]=\\left[\\begin{array}{cc}\n1&0\\\\\n0&\\xi^3\\end{array}\\right]v\n\\]\nthen $v$ satisfies \n\\[\n\\frac{dv}{d\\xi}=\\xi^4\\left[\\begin{array}{cc}\n0&2\\\\\n2+2a\\lambda^{2\/3}\\xi^{-4}+2b\\lambda\\xi^{-6}&-3\\xi^{-5}\\end{array}\\right]v.\n\\]\nLet us set again\n\\[\nv=\\left[\\begin{array}{cc}\n1&1\\\\\n-1&1\\end{array}\\right]w\n\\]\nso that $w$ verifies\n\\[\n\\frac{dw}{d\\xi}=\\xi^4\\left[\\begin{array}{cc}\n\\alpha_1(\\xi)&\\beta_1(\\xi)\\\\\n\\beta_2(\\xi)&\\alpha_2(\\xi)\\end{array}\\right]w\n\\]\nwhere\n\\[\n\\left\\{\n\\begin{array}{ll}\n\\alpha_1(\\xi)=-2-a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}-b\\lambda\\xi^{-6},&\\beta_1(\\xi)=-a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}-b\\lambda\\xi^{-6}\n,\\\\\n\\alpha_2(\\xi)=2+a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6},&\n\\beta_2(\\xi)=a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6}.\n\\end{array}\\right.\n\\]\nLet us set \n\\[\nw=\\left[\\begin{array}{ll}\n1\\\\\np\\end{array}\\right]e^{\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta}\n\\]\nwhere now $p$ and $\\gamma$ are unknowns. It is clear that\n\\[\n\\left\\{\\begin{array}{ll}\n\\gamma(\\xi)=\\alpha_1(\\xi)+\\beta_1(\\xi)p,\\\\\n\\displaystyle{\\frac{dp}{d\\xi}=\\xi^4(\\beta_2(\\xi)+(\\alpha_2(\\xi)-\\gamma(\\xi))p)}\\end{array}\\right.\n\\]\nwhich gives\n\\[\n\\frac{dp}{d\\xi}=\\xi^4(\\beta_2+(\\alpha_2-\\alpha_1)p-\\beta_1p^2).\n\\]\nDenoting\n\\[\n\\left\\{\n\\begin{array}{lll}\nf(\\xi)=\\beta_2(\\xi)=a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6},\\\\[3pt]\nh(\\xi)=\\alpha_2(\\xi)-\\alpha_1(\\xi)=4+2a\\lambda^{2\/3}\\xi^{-4}+2b\\lambda \\xi^{-6},\\\\[3pt]\ng(\\xi)=-\\beta_1(\\xi)=a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6}\n\\end{array}\\right.\n\\]\none can write\n\\begin{equation}\n\\label{eqn:2}\n\\frac{dp}{d\\xi}=\\xi^4\\bigl[f(\\xi)+h(\\xi)p+g(\\xi)p^2\\bigr].\n\\end{equation}\nPutting \n\\[\n\\theta=\\xi^{-5},\\quad \\phi=a\\lambda^{2\/3}\\xi^{-4},\\quad \\psi=b\\lambda\\xi^{-6}\n\\]\nsuch that $f(\\xi)=\\phi+\\frac{3}{2}\\theta+\\psi$, $h(\\xi)=4+2\\phi+2\\psi$, $g(\\xi)=\\phi-\\frac{3}{2}\\theta+\\psi$ we look for a formal solution to (\\ref{eqn:2}) of the form\n\\[\n{\\hat p}=\\sum_{k+\\ell+m\\geq 1}C_{k,\\ell,m}\\theta^k\\phi^{\\ell}\\psi^m.\n\\]\nSince \n\\[\n\\xi^{-4}\\frac{d}{d\\xi}(\\theta^k\\phi^{\\ell}\\psi^m)=-(5k+4\\ell+6m)\\theta^{k+1}\\phi^{\\ell}\\psi^m\n\\]\nit is easy to see that $C_{k,\\ell,m}$ are uniquely determined by \n\\begin{eqnarray*}\nC_{100}=-3\/8,\\;\\;C_{010}=-1\/4,\\;\\;C_{001}=-1\/4,\\\\\n4C_{k,\\ell,m}=F_j(C_{k,\\ell,m}; k+\\ell+m \\leq j-1),\\;\n k+\\ell+m=j,\\;\\; j\\geq 2.\n\\end{eqnarray*}\nLet us take\n\\[\n{\\hat p}=\\sum_{1\\leq k+\\ell+m<N}C_{k,\\ell,m}\\theta^k\\phi^{\\ell}\\psi^m\n\\]\nso that \n\\[\nf(\\xi)+h(\\xi){\\hat p}+g(\\xi){\\hat p}^2-\\xi^{-4}\\frac{d}{d\\xi}{\\hat p}=O(|\\theta|+|\\phi|+|\\psi|)^N.\n\\]\nLook for a solution to (\\ref{eqn:2}) in the form $p=q+{\\hat p}$ then $q$ verifies \n\\[\n\\frac{dq}{d\\xi}=\\xi^4\\bigl[\\bigl(f+h{\\hat p}+g{\\hat p}^2-\\xi^{-4}\\frac{d{\\hat p}}{d\\xi}\\bigr)+(h+2g{\\hat p})q+gq^2\\bigr].\n\\]\nLet us set\n\\begin{eqnarray*}\n\\mu(\\xi)=f(\\xi)+h(\\xi){\\hat p}(\\xi)+g(\\xi){\\hat p}^2(\\xi)-\\xi^{-4}\\frac{d{\\hat p}}{d\\xi},\\\\\n\\lambda(\\xi)=h(\\xi)+2g(\\xi){\\hat p}(\\xi),\\quad\n\\nu(\\xi)=g(\\xi)\n\\end{eqnarray*}\nso that we have\n\\begin{equation}\n\\label{eqn:3}\n\\frac{dq}{d\\xi}=\\xi^4\\bigl[\\mu(\\xi)+\\lambda(\\xi)q+\\nu(\\xi)q^2\\bigr].\n\\end{equation}\nNote that\n\\begin{equation}\n\\label{eq:teigi:mu}\n\\left\\{\n\\begin{array}{lll}\n\\mu(\\xi)=O(|\\theta|+|\\phi|+|\\psi|)^N,\\\\[3pt]\n\\lambda(\\xi)-4=2\\phi+2\\psi+2\\nu(\\xi){\\hat p}(\\xi),\\\\[3pt]\n\\nu(\\xi)=\\phi-\\frac{3}{2}\\theta+\\psi.\n\\end{array}\\right.\n\\end{equation}\nOne can rewrite (\\ref{eqn:3}) as an integral equation;\n\\begin{equation}\n\\label{eqn:4}\nq(\\xi)=\\int_{\\infty}^{\\xi}\\eta^4\\bigl[\\mu(\\eta)+{\\tilde \\lambda}(\\eta)q(\\eta)+\\nu(\\eta)q(\\eta)^2\\bigr]\\exp{\\frac{4(\\xi^5-\\eta^5)}{5}}d\\eta\n\\end{equation}\nwhere ${\\tilde \\lambda}(\\xi)=\\lambda(\\xi)-4$. With $s=\\frac{4}{5}\\xi^5$, $\\sigma=\\frac{4}{5}\\eta^5$ one has\n\\[\nq(\\xi)=\\frac{1}{4}\\int_{\\infty}^s\\bigl[\\mu(\\eta)+{\\tilde \\lambda}(\\eta)q(\\eta)+\\nu(\\eta)q(\\eta)^2\\bigr]e^{s-\\sigma}d\\sigma.\n\\]\nLet $\\rho>1\/6$ and note that \n\\[\n|\\psi|=|b\\lambda\\xi^{-6}|\\leq M|\\xi|^{-(6-1\/\\rho)},\\quad |\\phi|=|a\\lambda^{2\/3}\\xi^{-4}|\\leq M|\\xi|^{-2(6-1\/\\rho)\/3}\n\\]\nprovided $\n|\\xi|\\geq \\lambda^{\\rho}$\nand, taking $N$ large so that $2(N-1)(6-1\/\\rho)\/3\\geq 1$  one has\n\\begin{gather*}\n|\\mu(\\xi)|\\leq C|\\xi|^{-1-2(6-1\/\\rho)\/3},\\; |{\\tilde \\lambda}(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3},\\\\\n |\\nu(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3},\\;|{\\hat p}(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3}.\n\\end{gather*}\nWe now repeat the same argument as in Section 10 of \\cite{Si} \nin $ {\\mathcal S}_{\\delta,M}$ (for the notation ${\\mathcal S}_{\\delta,M}$, \nsee \\cite{Si} and here $M$ depends on $\\lambda$) then we get a solution $q(\\xi)$ to (\\ref{eqn:4}) \nsuch that $|q(\\xi)|\\leq C|\\xi|^{-1}$ for $|\\xi|\\geq \\lambda^{\\rho}$. Since we can choose arbitrarily large $N=N(\\rho)$ in \\eqref{eq:teigi:mu} we conclude that\n\\begin{equation}\n\\label{eqn:6}\n|q(\\xi)|\\leq C|\\xi|^{-N},\\quad |\\xi|\\geq \\lambda^{\\rho}.\n\\end{equation}\n Thus we obatain a solution $p(\\xi)=q(\\xi)+{\\hat p}(\\xi)$. We turn back to $v$;\n\\[\nv=\\left[\\begin{array}{cc}\n1&1\\\\\n-1&1\\end{array}\\right]\\left[\\begin{array}{c}\n1\\\\\np(\\xi)\\end{array}\\right]e^{\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta}.\n\\]\nConsider\n\\[\n\\eta^4\\gamma(\\eta)=-\\{2\\eta^4+a\\lambda^{2\/3}+\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2}+(a\\lambda^{2\/3}-\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2})p(\\eta)\\}\n\\]\nand hence\n\\begin{eqnarray*}\n\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta=-\\{\\frac{2}{5}\\xi^5+a\\lambda^{2\/3}\\xi+\\frac{3}{2}\\log\\xi-b\\lambda\\xi^{-1}\\}\\\\\n-\\int^{\\xi}(a\\lambda^{2\/3}-\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2})p(\\eta)d\\eta.\n\\end{eqnarray*}\nDenoting\n\\[\n{\\hat p}=\\sum_{1\\leq k+\\ell+m<N_1-1}C_{k,\\ell,m}\\theta^k\\phi^{\\ell}\\psi^m+\\sum_{N_1\\leq k+\\ell+m<N}C_{k,\\ell,m}\\theta^k\\phi^{\\ell}\\psi^m={\\hat p}_1+{\\hat p}_2\n\\]\nwe define\n\\begin{gather*}\n\\int^{\\xi}(a\\lambda^{2\/3}-3\\eta^{-1}\/2+b\\lambda\\eta^{-2}){\\hat p}_j(\\eta)d\\eta=r_j(\\xi, \\lambda)\n\\end{gather*}\nwhich are polynomials in $\\xi^{-1}$. Since \n\\[\n\\Big|\\int^{\\xi}\\lambda^i\\eta^{-j}\\theta^k\\phi^{\\ell}\\psi^md\\eta\\Big|\\leq C\\lambda^{i+\\rho(1-j)-(4\\rho-2\/3)(k+\\ell+m)},\\quad |\\xi|\\geq \\lambda^{\\rho}\n\\]\n it is also easy to see that\n\\begin{equation}\n\\label{eq:hyoka:r:bis}\n\\begin{split}\n&|r_1(\\xi, \\lambda)|\\leq C\\lambda^{4\/3-3\\rho}(1+o_{\\rho}(\\lambda)),\\\\\n&|r_2(\\xi, \\lambda)|\\leq C_{N_1,N}\\lambda^{2\/3+\\rho-(4\\rho-2\/3)N_1},\\quad |\\xi|\\geq \\lambda^{\\rho}.\n\\end{split}\n\\end{equation}\nOn the other hand, taking $N=N(\\epsilon)$ large, we see that\n\\[\nr_3(\\xi,\\lambda)=-\\int^{\\xi}(a\\lambda^{2\/3}-\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2})q(\\eta)d\\eta\n\\]\nis holomorphic in $|{\\rm arg} \\xi|<3\\pi\/10$ and $|r_3(\\xi,\\lambda)|\\leq C_N |\\xi|^{-N}$ in any closed subsector of $| \\arg \\xi |<3\\pi \/10$ if $|\\xi|\\geq \\lambda^{\\rho}$. \nIt is also clear that for fixed $\\lambda$ \n\\[\nE(\\xi, \\lambda)=r_2(\\xi, \\lambda)+r_3(\\xi,\\lambda)\\to 0,\\quad \\xi\\to\\infty,\\quad |\\arg \\xi|<3\\pi\/10.\n\\]\nNote that\n\\[\n|p(\\xi, \\lambda)|\\leq C_{\\rho}\\lambda^{-4(\\rho-1\/6)},\\quad |\\xi|\\geq \\lambda^{\\rho}.\n\\]\nThen one can write, recalling $x=\\xi^2$\n\\begin{eqnarray*}\ny(x,\\lambda)=(1+p(\\sqrt{x},\\lambda))e^{E(\\sqrt{x},\\lambda)}x^{-3\/4}\\\\\n\\times\\exp{\\big\\{-\\bigl(\\frac{2}{5}x^{5\/2}+a\\lambda^{2\/3} x^{1\/2}-b\\lambda x^{-1\/2}+r_1(\\sqrt{x},\\lambda)\\bigr)\\big\\}},\\\\\ny'(x,\\lambda)=(-1+p(\\sqrt{x},\\lambda))e^{E(\\sqrt{x},\\lambda)}x^{3\/4}\\\\\n\\times\\exp{\\big\\{-\\bigl(\\frac{2}{5}x^{5\/2}+a\\lambda^{2\/3} x^{1\/2}-b\\lambda x^{-1\/2}+r_1(\\sqrt{x},\\lambda)\\bigr)\\big\\}}\n\\end{eqnarray*}\nfor $\n|\\arg x|<\\frac{3\\pi}{5}$, $|x|\\geq |\\lambda|^{2\\rho}$ \nwhere $p(\\sqrt{x},\\lambda)\\to 0$ as $x \\to \\infty$ in any closed subsector of $|\\arg x |<3\\pi\/5$ and $|p(\\sqrt{x},\\lambda)|\\leq C|\\lambda|^{-4(\\rho-1\/6)}$ for $|x|\\geq \\lambda^{2\\rho}$. \nSince $y(x,\\lambda)$ is subdominant in the sector $|\\arg x|<\\pi\/5$ one can write ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)=c(a,b,\\lambda)y(x,\\lambda)$. Fixing $a$, $b$, $\\lambda$ and letting $x\\to\\infty$ in the equality\n\\[\n\\Bigl[{\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)-c(a,b,\\lambda)y(x,\\lambda)\\Bigr]x^{3\/4}\\exp{\\big\\{\\bigl(\\frac{2}{5}x^{5\/2}+a\\lambda^{2\/3} x^{1\/2}\\bigr)\\big\\}}=0\n\\]\n we see that $c(a,b,\\lambda)=1$. Replacing $2\\rho$ by $\\rho$ we conclude the assertion. \n \\end{proof}\n %\n %\n %\n \\fi\n\n\\begin{lemma}\n\\label{lem:1:ten} Assume that ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)$ verifies\n\\begin{equation}\n\\label{eq:rela}\n{\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)=-\\omega {\\mathcal Y}_2(x;a\\lambda^{2\/3},b\\lambda)\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:Im:a}\n {\\mathsf{Im}}\\,a= -\\delta (1+o_a(1))\n \\end{equation}\n %\nwith some $\\delta>0$. Let $X>0$. There exist $\\ell$, $c>0$, $C>0$ such that for any $0\\leq \\mu<5\/6$ we have\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C(1+o_{\\mu}(1))\\lambda^{\\ell}\\\\\n\\times\\exp{\\big\\{-\\delta \\lambda^{5\/3}|x|^{1\/2}(1+o_{a}(1))+C_{\\mu}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\\big\\}}\n\\end{gather*}\nfor $k=0, 1$ and $|x|\\geq \\lambda^{-2\\mu}X$.\n\\end{lemma}\n\\begin{proof} In Proposition \\ref{pro:Esti:Y} choose $\\rho=2(1-\\mu)$ and estimate ${\\mathcal Y_0}$ in $x\\leq -\\lambda^{-2\\mu}X$ first. Recall that for $x<0$ we have\n\\begin{gather*}\n{\\mathcal Y}_0(x; a\\lambda^{2\/3},b\\lambda)=-\\omega{\\mathcal Y_2}(x; a\\lambda^{2\/3}, b\\lambda)\n=-\\omega {\\mathcal Y}_0(e^{\\pi i\/5}|x|;\\omega a\\lambda^{2\/3},\\omega^{-1}b\\lambda).\n\\end{gather*}\nDenote\n\\[\n\\phi^{-}(x,\\lambda)=E_{\\rho}(e^{\\pi i\/5}\\lambda^2|x|; \\omega a, \\omega^{-1}b, \\lambda),\\quad\\phi^{+}(x,\\lambda)=E_{\\rho}(\\lambda^2 x; \\omega a, \\omega^{-1}b, \\lambda)\n\\]\nthen we have\n\\begin{equation}\n\\label{eq:phi:x:fu}\n\\begin{split}\n\\phi^{-}(x,\\lambda)=\\frac{2}{5}i\\lambda^5 |x|^{5\/2}+i a\\lambda^{5\/3}|x|^{1\/2}\n+ib|x|^{-1\/2}+r_{\\rho}(e^{\\pi i\/5}\\lambda^{2}|x|,\\lambda).\n\\end{split}\n\\end{equation}\nSince $\\big|r_{\\rho}(e^{\\pi i\/5}\\lambda^{2}|x|,\\lambda)\\big|\\leq C_{\\rho}\\lambda^{-5\/3+3\\mu}$ for $|x|\\geq \\lambda^{-2\\mu}X$ and then\n\\begin{equation}\n\\label{eq:x:fu}\n-{\\mathsf{Re}}\\,\\phi^{-}(x,\\lambda)\n\\leq -\\delta\\lambda^{5\/3}|x|^{1\/2}(1+o_{a}(1))+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu}).\n\\end{equation}\n  For $x\\geq \\lambda^{-2\\mu}X>0$ note that\n\\begin{equation}\n\\label{eq:x:sei}\n\\begin{split}\n-{\\mathsf{Re}}\\;\\phi^{+}(x,\\lambda)\\leq -c\\lambda^5 x^{5\/2}+C\\lambda^{5\/3}x^{1\/2}\n+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\\\\\n=-c\\lambda^{5-4\\mu} x^{1\/2}(1+o_{a}(1))+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\n\\end{split}\n\\end{equation}\nfor $\\mu<5\/6$. Then the assertion follows from \\eqref{eq:x:fu} and \\eqref{eq:x:sei}. \n\\end{proof}\n\\begin{lemma}\n\\label{lem:X:1:2}Assume that \\eqref{eq:rela} and \\eqref{eq:Im:a} hold with some $\\delta>0$ and that $\\mu< 5\/6$. Let $0<X_1<X_2<1$. Then there exist $C>0$, $\\ell$ and $c>0$ such that \n\\[\n\\big|{\\mathcal Y}_0(\\lambda^{2} x; a\\lambda^{2\/3}, b\\lambda)\\big|\\geq C\\lambda^{\\ell} e^{-c\\lambda^{5\/3-\\mu}},\\quad \\lambda^{-2\\mu}X_1\\leq -x\\leq \\lambda^{-2\\mu}X_2.\n\\]\n\\end{lemma}\n\\begin{proof}It is clear from \\eqref{eq:phi:x:fu} that there exists $C_1>0$ such that\n\\[\n-\\lambda^{5\/3-\\mu}\/C_1\\leq -{\\mathsf{Re}}\\,\\phi^{-}(x, \\lambda)\n\\leq -C_1\\lambda^{5\/3-\\mu}\n\\]\nwhen $\\lambda^{-2\\mu}X_1\\leq -x\\leq \\lambda^{-2\\mu}X_2$. This proves the assertion.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:outside:many} Under the same assumptions as in Lemma \\ref{lem:1:ten} there exist $c>0$, $A>0$ such that \n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C_{\\mu}A^{k+1}(1+k^{3}+\\lambda^{5\/2})^{k}e^{c\\lambda^{5\/6}},\\;\\;k\\in {\\mathbb N}\n\\end{gather*}\nfor  $|x|\\geq \\lambda^{-2\\mu}X$. \n\\end{lemma}\n\\begin{proof} We first estimate $\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b)\\big|$ in $x\\leq -\\lambda^{-2\\mu}X$. \nFrom Proposition \\ref{pro:Esti:Y} with $\\rho=2(1-\\mu)$ we have\n\\[\n{\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b\\lambda)=C(1+p_{\\mu}(x))\\lambda^{-3\/2}x^{-3\/4}e^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\n\\]\nwhere $p_{\\mu}(x)$ and $R_{\\mu}(x)$ are holomorphic and bounded in $|x|>\\lambda^{-2\\mu}X$, $|{\\rm arg}\\, x|<3\\pi\/5$. Since $|x|^{-1}\\leq \\sqrt{X}\\lambda^{2\\mu}\\leq \\sqrt{X}\\lambda^{5\/3}$ we have\n\\begin{gather*}\n|d^k(-\\phi^{-}(x, \\lambda)+R_{\\mu}(x))\/dx^k|\\\\\n\\leq C_{\\mu}A^kk!(\\lambda^5 |x|^{3\/2}+\\lambda^{5\/3}|x|^{-1\/2}+C_{\\mu}|x|^{-3\/2})|x|^{1-k}\\\\\n\\leq C_{\\mu}A^kk!(1+\\lambda^5 |x|^{3\/2}+\\lambda^{5\/2})\\lambda^{5(k-1)\/3}\\\\\n\\leq C_{\\mu}A^kk!\\lambda^{5k\/3}(\\lambda^{10\/3}|x|^{3\/2}+\\lambda^{5\/6}),\\quad |x|\\geq \\lambda^{-2\\mu}X,\\;\\;k\\geq 1.\n\\end{gather*}\nTherefore it follows that for $x\\leq -\\lambda^{-2\\mu}X$ \n\\[\n\\big|d^ke^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\/dx^k|\\leq CA^k\\lambda^{5k\/3}(\\lambda^{10\/3}|x|^{3\/2}+\\lambda^{5\/6}+k)^ke^{-{\\mathsf{Re}}\\,(-\\phi(x)+R_{\\mu}(x))}.\n\\]\nSince $-{\\mathsf{Re}}\\,(-\\phi^{-}(x, \\lambda)+R_{\\mu}(x))\\leq -c\\lambda^{5\/3}|x|^{1\/2}+C_{\\mu}\\lambda^{5\/6}$ for $ x\\leq -\\lambda^{-2\\mu }X$  \nwith $c>0$ independent of $\\mu$ and \n\\[\n|x|^{3k\/2}e^{-c\\lambda^{5\/3}|x|^{1\/2}}\\leq C^{k+1}\\lambda^{-5k}k^{3k}\n\\]\nwe conclude that\n\\[\n|d^ke^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\/dx^k|\\leq CA^k(\\lambda^{5\/2}+k^3)^ke^{c\\lambda^{5\/6}}\n\\]\nwhich proves the assertion for $x\\leq -\\lambda^{-2\\mu}X$. For $x\\geq \\lambda^{-2\\mu}X$ it is enough to repeat the same arguments noting that \\eqref{eq:x:sei} and $5-4\\mu>5\/3$.\n\\end{proof}\n\\begin{cor}\n\\label{cor:outside} Assume that ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)$ verifies \\eqref{eq:rela} and \\eqref{eq:Im:a}.  Then there exist $c>0$, $A>0$, $C>0$ such that for any $\\epsilon>0$ there is $\\lambda_{\\epsilon}$ such that\n\\[\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C_{\\epsilon}A^{k+1}(1+k^{3}+\\lambda^{5\/2})^{k}e^{c\\lambda^{5\/6}},\\;\\;k\\in {\\mathbb N}\n\\]\nfor $\\lambda^{-5\/3+\\epsilon}\\leq |x|$, $\\lambda\\geq \\lambda_{\\epsilon}$. \n\\end{cor}\n\\begin{proof} Choose $\\mu=5\/6-\\epsilon\/2$ in Lemma \\ref{lem:outside:many}.\n\\end{proof}\n\n\n\n\nNext we estimate ${\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b\\lambda)$ for $|x|\\leq \\lambda^{-5\/3+\\epsilon}$. \n\n\n\\begin{lemma}\n\\label{lem:time:T}{\\rm (\\cite[Lemma 6.5, Lemma 6.7]{BN:JHDE})}  Assume that $y(x,\\lambda)$ satisfies \n\\begin{equation}\n\\label{eqn:1}\ny''(x,\\lambda)=(x^3+a\\lambda^{2\/3} x+b\\lambda)y(x,\\lambda),\\quad |a|,\\;|b|\\leq M.\n\\end{equation}\nThen there are $c>0$, $C>0$ and $\\ell_i>0$ such that for any $T>0$ we have\n\\begin{eqnarray*}\n\\big|(d\/dx)^ky(x,\\lambda)\\big| \\leq C^{k+1}(k+\\lambda^{1\/3}+|x|)^{3k\/2}\\lambda^{\\ell_1}(1+T)^{\\ell_2\n}e^{c\\lambda^{5\/6}(1+\\lambda^{-1\/3}T)^{5\/2}}\\\\\n\\times \\big\\{|y(T,\\lambda)|+|y'(T,\\lambda)|\\big\\},\\quad |x|\\leq T,\\;\\;k\\in{\\mathbb N},\\;\\;\\lambda\\geq 1.\n\\end{eqnarray*}\n\\end{lemma}\n\n\n\\begin{pro}\n\\label{pro:inside:many} Assume that ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)$ verifies \\eqref{eq:rela}  and \\eqref{eq:Im:a}.\nThen there are $\\ell$, $c>0$, $A>0$, $C>0$ such that for any $\\epsilon>0$ there is $\\lambda_{\\epsilon}$ such that\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x; a\\lambda^{2\/3},b\\lambda)\\big|\\leq CA^{k+1}\n \\lambda^{\\ell}\n(k^{3}+\\lambda^{4})^{k}e^{c\\lambda^{5\/6+\\epsilon}},\\;\\;k\\in{\\mathbb N}\n\\end{gather*}\nfor $\\lambda\\geq \\lambda_{\\epsilon}$.\n\\end{pro}\n %\n \\begin{proof}\n Applying Lemma \\ref{lem:1:ten} with $\\mu=5\/6-\\epsilon\/2$.  we have\n %\n \\[\n  \\big|(d\/dx)^k{\\mathcal Y}_0(\\pm\\lambda^{1\/3+\\epsilon})\\big|\\leq C\\lambda^{\\ell}e^{c_1\\lambda^{5\/6}},\\;\\;\\lambda\\geq \\lambda_{\\epsilon},\\;\\; k=0, 1.\n  \\]\n  %\nSince ${\\mathcal Y}_0(x)={\\mathcal Y}_0(x, a\\lambda^{2\/3}, b\\lambda)$  satisfies \\eqref{eqn:1}, choosing $T=\\lambda^{1\/3+\\epsilon}$ in Lemma \\ref{lem:time:T} we get\n\\[\n\\big|(d\/dx)^k{\\mathcal Y}_0(x)\\big|\\leq C^{k+1}\\lambda^{\\ell}(k+\\lambda^{1\/3+\\epsilon})^{3k\/2}e^{c\\lambda^{5\/6+3\\epsilon}},\\;\\;|x|\\leq \\lambda^{1\/3+\\epsilon},\\;\\;  k\\in {\\mathbb N}\n\\]\nfor $\\lambda\\geq \\lambda_{\\epsilon}$. This proves that\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2}x)\\big|\\leq C^{k+1}\\lambda^{\\ell}(\\lambda^{2}k^{3\/2}+\\lambda^{5\/2+2\\epsilon})^{k}e^{c\\lambda^{5\/6+3\\epsilon}}\n\\end{gather*}\nfor $|x|\\leq \\lambda^{-5\/3+\\epsilon}$. Since $\\lambda^{k(5\/2+2\\epsilon)}e^{-\\lambda^{5\/6+3\\epsilon}}\\leq k^{3k}$ and $\n\\lambda^{2}k^{3\/2}\\leq C(\\lambda^{4}+k^3)$ \ncombining Corollary \\ref{cor:outside} with the above-obtained estimates we conclude the assertion.\n\\end{proof}\nRecalling\n\\[\nV_{\\lambda}(x')=e^{i\\lambda^5x_2-i(b_1\/2)x_1}{\\mathcal Y_0}(\\lambda^2 x_1; \\lambda^{2\/3}a(\\lambda), \\lambda b(\\lambda))\n\\]\nwith $\\lambda^{2\/3}a(\\lambda)=2\\xi_0$ and $b(\\lambda)=b_2+b_0\\xi_0\\lambda^{-4}-\\xi_0^2\\lambda^{-3}-b_1^2\\lambda^{-5}\/4$ one has\n\\begin{lemma}\n\\label{lem:V:bibun} There exist $c>0$, $A>0$, $C>0$  and $\\lambda_0>0$  such that \n\\begin{equation}\n\\label{eq:V:bibun}\n\\big|\\partial_{x_1}^{k}V_{\\lambda}(x')\\big|\\leq CA^{k}(k!)^3e^{c\\lambda^{4\/3}},\\quad k\\in{\\mathbb N},\\;\\lambda\\geq \\lambda_0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nNoting $\\lambda^{4k}\\leq C_1^{k}(k!)^3e^{3\\lambda^{4\/3}}$ the assertion follows from Proposition \\ref{pro:inside:many}.\n\\end{proof}\n \n\\section{Proof of Theorem \\ref{thm:main:bis}}\n\n\nAssume that $b_2\\neq 0$ satisfies \\eqref{eq:argA}. Following Section \\ref{sec:solv} we have a family of exact solutions $\\{U_{\\lambda}\\}$ satisfying $(P_{mod}+\\sum_{j=0}^2b_jD_j)U_{\\lambda}=0$. We show that $\\{U_{\\lambda}\\}$   \ndoes not satisfy apriori estimates derived from $\\gamma^{(s)}$ local solvability of the Cauchy problem if $s>3$. Let $h>0$ and a compact set $K$ be fixed and denote by $\\gamma_0^{(s),h}(K)$ the set of all $f(x')\\in \\gamma^{(s)}({\\mathbb R}^2)$ such that ${\\rm supp}\\,f\\subset K$ and \\eqref{eq:gevrey} holds with some $C>0$ for all $\\alpha\\in {\\mathbb N}^2$. Note that $\\gamma_0^{(s), h}(K)$ is a Banach space with the norm\n\\[\n\\sup_{\\alpha, x}\\frac{|\\partial_x^{\\alpha}f(x')|}{h^{|\\alpha|}|\\alpha|!^s}.\n\\]\n\\begin{pro}[Holmgren]\n\\label{pro:Holmgren}Denote $\nD_{\\epsilon}=\\{x\\in {\\mathbb R}^3\\mid |x'|^2+|x_0|<\\epsilon\\}$. \nThere exists $\\epsilon_0>0$ such that for $\\epsilon$ satisfying  $0<\\epsilon<\\epsilon_0$  if $u(x)\\in C^2(D_{\\epsilon})$ satisfies\n\\[\n\\begin{cases}\n\\big(P_{mod}+\\sum_{j=0}^2b_jD_j\\big)u=0\\;\\;\\text{in}\\;\\; D_{\\epsilon},\\\\\\;D_0^ju(0, x')=0\\;\\;(j=0,1),\\;\\;x\\in D_{\\epsilon}\\cap\\{x_0=0\\}\n\\end{cases}\n\\]\nthen $u(x)\\equiv 0$ in $D_{\\epsilon}$.\n\\end{pro}\n\\begin{lemma}\n\\label{lem:futosiki}{\\rm(e.g.\\cite[Proposition 4.1, Theorem 4.2]{Miz}, \\cite{Lax})}\nAssume that the Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is locally solvable in $\\gamma^{(s)}$ at the origin. Then there exists $\\delta>0$ such that for any  $0<\\epsilon_1<\\delta$ and any $\\Phi=(u_j(x'))\\in \\gamma_0^{(s),h}(\\{|x'|\\leq \\epsilon_1\\})$ there is a unique solution $u(x)\\in C^2(D_{\\delta})$ to the  Cauchy problem \\eqref{eq:CP} with $U_{\\Phi}=D_{\\delta}$ and for any compact set $L\\subset D_{\\delta}$ there exists $C>0$ such that \n\\begin{equation}\n\\label{eq:G:hyoka}\n|u(x)|_{C^2(L)}\\leq C\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}u_j(x')|}{h^{|\\alpha|}|\\alpha|!^s}\n\\end{equation}\nholds.\n\\end{lemma}\nSince $\\lambda^{5k}\\leq k!^se^{s\\lambda^{5\/s}}$ and $U_{\\lambda}(0, x')=V_{\\lambda}(x')$, $D_0 U_{\\lambda}(0, x')= \\xi_0(\\lambda)\\lambda V_{\\lambda}(x')$ it is clear from Lemma \\ref{lem:V:bibun} that one can find $c_1>0$, $C>0$ such that\n\\begin{equation}\n\\label{eq:U:futosiki}\n\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}D_0^jU_{\\lambda}(0, x')|}{h^{|\\alpha|}|\\alpha|^{s|\\alpha|}}\\leq Ce^{c_1\\lambda^{\\max{\\{5\/s, 4\/3\\}}}},\\quad s\\geq 3.\n\\end{equation}\nOn the other hand thanks to Lemma \\ref{lem:X:1:2} \nand Proposition \\ref{pro:sol} there is $c_0>0$ such that\n\\begin{equation}\n\\label{eq:U:x:0}\n\\big|U_{\\lambda}( x_0, -\\lambda^{-2\\mu}, 0)\\big|\\geq C\\lambda^{\\ell}e^{c_0\\lambda^{5\/3} x_0-c\\lambda^{5\/3-2\\mu}},\\quad x_0>0\n\\end{equation}\nwhere $\\mu$ is chosen such that $0<\\mu<5\/6$. Let $\\chi(x')\\in \\gamma^{(s)}({\\mathbb R}^2)$ be such that $\\chi(x')=0$  for $|x'|\\geq \\sqrt{\\epsilon_1}$ and $\\chi(x')=1$ for $|x'|\\leq \\sqrt{\\epsilon_2}<\\sqrt{\\epsilon_1}$. Since  $\\Phi_{\\lambda}=\\chi(x')(U_{\\lambda}(0, x'), D_0U_{\\lambda}(0, x'))\\in \\gamma_0^{(s), h}(\\{|x'|\\leq \\epsilon_1\\})$, thanks to Lemma \\ref{lem:futosiki},  there is a unique solution $u_{\\lambda}(x)\\in C^{2}(D_{\\delta})$ to the Cauchy problem  with Cauchy data  $\\Phi_{\\lambda}(x')$ which satisfies \\eqref{eq:U:futosiki}. \nThanks to Proposition \\ref{pro:Holmgren} we see that $u_{\\lambda}=U_{\\lambda}$ in $D_{\\epsilon_2}$. Take a compact set $L\\subset D_{\\epsilon_2}$ that contains $( x_0, -\\lambda^{-2\\mu}, 0)$ with small $x_0>0$ and large $\\lambda$.  If $s>3$ hence  $\\max{\\{5\/s, 4\/3\\}}<5\/3$ the inequalities \\eqref{eq:U:futosiki} and \\eqref{eq:U:x:0} are not compatible  which proves Theorem \\ref{thm:main:bis}.\n\nWhen $b_2\\neq 0$ does not satisfy \\eqref{eq:argA} we make a change of local coordinates $(x_0, x_1, x_2)\\mapsto (-x_0, x_1, -x_2)$ such that $P_{mod}+\\sum_{j=0}^2b_jD_j$ will be\n\\begin{equation}\n\\label{eq:sin:eq}\nP_{mod}-b_0D_0+b_1D_1-b_2D_2.\n\\end{equation}\nin the new local coordinates. Since the local solvability in $\\gamma^{(s)}$ at the origin is invariant under (analytic) change of local coordinates and $-b_2$ obviously satisfies \\eqref{eq:argA}, we conclude the same assertion also in this case.\n\n\n\n\\begin{comment}\n\n\\subsection{Remaining case}\n\nIn this section we treat the case that $0\\neq b_2$ does not satisfy \\eqref{eq:iti}, that is\n\\[\n-4\\pi\/5\\leq \\arg b_2\\leq -\\pi\/5.\n\\]\n  Look for a family of exact solutions $U_{\\lambda}$ to $P_{mod}+\\sum_{j=0}^2b_jD_j$ in the form\n\\begin{equation}\n\\label{eq:teigi:U:bis}\nU_{\\lambda}(x)=e^{i\\xi_0\\lambda(\\tau-x_0)-i\\lambda^5x_2}W_{\\lambda}(x_1),\\quad W_{\\lambda}(x_1)=e^{-i(b_1\/2)x_1}u(\\lambda^2x_1)\n\\end{equation}\nwhere $\\tau>0$ is a constant, that is we look for $u(x)$ satisfying\n\\begin{equation}\n\\label{eq:a8:bis}\nu''(x)=\\big(x^3+ 2\\xi_0 x-b_2\\lambda-b_0\\xi_0\\lambda^{-3}-\\xi_0^2\\lambda^{-2}-b_1^2\\lambda^{-4}\/4\\big)u(x).\n\\end{equation}\nSince  $-b_2$ clearly satisfies \\eqref{eq:argA} one can apply Proposition \\ref{pro:sol} to find $\\xi_0(\\lambda)$ satisfying\n\\begin{equation}\n\\label{eq:case:b}\n\\begin{split}\n2\\xi_0=c\\,\\lambda^{2\/3}(1+\\lambda^{-5\/6}z(\\lambda))=\\lambda^{2\/3}a(\\lambda),\\quad {\\mathsf{Im}}\\quad c<0,\\\\\nC_0(2\\xi_0,\\, -b_2\\lambda-\\xi_0^2\\lambda^{-2}-b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4)=0.\n\\end{split}\n\\end{equation}\nChoose $u(x)={\\mathcal Y_0}(x; 2\\xi_0, a_2)$ with $a_2=-b_2\\lambda-\\xi_0^2\\lambda^{-2}-b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4$. Suppose that the Cauchy problem \\eqref{eq:CP} is locally solvable in $\\gamma^{(s)}$ at the origin with some $s>3$. \nTake $\\chi(t)\\in \\gamma^{(s_1)}({\\mathbb R})$ with $0\\leq \\chi(t)\\leq 1$ such that $\\chi=1$ on  $|t|\\leq1\/4$ and $\\chi=0$ in $|t|\\geq 1\/2$ and $\\int \\chi(t)ds=1$ and denote\n\\[\n\\chi_{\\lambda}(x_1)=\\chi(\\lambda^{2\\mu}(x_1+\\lambda^{-2\\mu})),\\quad \\varphi_{\\lambda}(x')=\\psi(x_2)\\chi_{\\lambda}(x_1)\n\\]\nwhere $\\psi\\in \\gamma^{(s)}({\\mathbb R})$ is supported in a small neighborhood of $x_2=0$.  Choosing $s_1>1$ so that $s-s_1>5\/4$ and hence  $k!^{s_1}\\lambda^{2\\mu k}\\leq k!^{s}e^{c\\lambda^{4\/3}}$ for $\\mu<5\/6$ it is clear that\n\\[\n|\\partial_{x'}^{\\alpha}\\varphi_{\\lambda}(x')|\\leq CA^{|\\alpha|}|\\alpha|!^{s}e^{c_1\\lambda^{4\/3}},\\quad \\alpha\\in {\\mathbb N}^2.\n\\]\nWe define $v_{\\lambda}(x')$ by\n\\[\nv_{\\lambda}(x')=\\varphi_{\\lambda}(x')W_{\\lambda}(x_1)\n\\]\nthen  it follows easily from Lemma \\ref{lem:V:bibun} that\n\\begin{equation}\n\\label{eq:shokiti}\n\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}v_{\\lambda}(x')|}{h^{|\\alpha|}|\\alpha|!^s}\\leq Ce^{c_2\\lambda^{4\/3}},\\quad s\\geq 3.\n\\end{equation}\nLet $u_{\\lambda}(x)$ be a solution to the Cauchy problem \\eqref{eq:CP} with the Cauchy data $(u_0, u_1)=(0, v_{\\lambda}(x'))$. \nThanks to Lemma \\ref{lem:futosiki}  and \\eqref{eq:shokiti} we have\n\\begin{equation}\n\\label{eq:u:lam}\n\\big|u_{\\lambda}(x)\\big|_{C^1(L)}\\leq Ce^{c_2\\lambda^{4\/3}}.\n\\end{equation}\nFrom the Holmgren's uniqueness theorem (e.g. \\cite[Theorem 4.2]{Miz}), taking $\\tau>0$ and the support of $\\psi$ enough small, one can assume that the support  of $u_{\\lambda}(x_0, x')$ with respect to $x'$ is small uniformly in large $\\lambda$ and  $0\\leq t\\leq \\tau$. Noting that $P=P_{mod}+\\sum_{j=0}^2b_jD_j$ verifies $P^*=P$ we have\n\\begin{gather*}\n\\int_0^{\\tau}(U_{\\lambda}, Pu_{\\lambda})dx_0\n=\\int_0^{\\tau}(PU_{\\lambda}, u_{\\lambda})dx_0-i(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))\\\\\n-i(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))-2i\\lambda^5(x_1 U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))+i(U_{\\lambda}(0), D_0u_{\\lambda}(0)).\n\\end{gather*}\nFrom this, it follows that\n\\begin{gather*}\n(U_{\\lambda}(0), D_0u_{\\lambda}(0))=e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\\\\n=(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))\n+(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))-2\\lambda^5(x_1U_{\\lambda}(\\tau), u_{\\lambda}(\\tau)).\n\\end{gather*}\nFrom Lemma \\ref{lem:V:bibun} and \\eqref{eq:u:lam} it is clear that \n\\begin{gather*}\n|(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))|\n+|(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))|\n+|2\\lambda^5(x_1U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))|\n\\end{gather*}\nis bounded by $Ce^{c_3\\lambda^{4\/3}}$ with some $c_3>0$, $C>0$ which proves\n\\begin{equation}\n\\label{eq:u:0:U}\n\\Big|e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\Big|\\leq Ce^{c_3\\lambda^{4\/3}}.\n\\end{equation}\nHere note that\n\\begin{gather*}\n\\int e^{-\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'={\\hat \\psi}(\\lambda^5)\\int\\chi_{\\lambda}(x_1)|W_{\\lambda}(x_1)|^2dx_1\\\\\n={\\hat \\psi}(\\lambda^5)\\int\\chi_{\\lambda}(x_1)e^{({\\mathsf{Im}}\\,b_2)x_1}\\big|{\\mathcal Y_0}(\\lambda^2x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|^2dx_1\n\\end{gather*}\nwhere ${\\hat \\psi}$ is the Fourier transform of $\\psi$. Thanks to Lemma \\ref{lem:X:1:2} it follows that $\\big|{\\mathcal Y}_0(\\lambda^{2} x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|\\geq C\\lambda^{\\ell}\\exp{(-c\\lambda^{5\/3-2\\mu})}$ on the support of $\\chi_{\\lambda}$. Therefore we have\n\\[\n\\int\\chi_{\\lambda}(x_1)e^{({\\mathsf{Im}}\\,b_2)x_1}\\big|{\\mathcal Y_0}(\\lambda^2x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|^2dx_1\\geq C\\lambda^{2\\ell-2\\mu}e^{-2c\\lambda^{5\/3-2\\mu}-3|{\\mathsf{Im}}\\,b_2|\\lambda^{-2\\mu}}.\n\\]\nSince $\\xi_0(\\lambda)$ verifies \\eqref{eq:case:b} we have from \\eqref{eq:u:0:U} that\n\\begin{equation}\n\\label{eq:psi:gevrey}\n\\begin{split}\nCe^{c_3\\lambda^{4\/3}}\\geq \\Big|e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\Big|\\\\\n\\geq C\\lambda^{2\\ell'}e^{-{\\mathsf{Im}}\\xi_0(\\lambda)\\lambda\\tau-c'\\lambda^{5\/3-2\\mu}}|{\\hat \\psi}(\\lambda^5)|\\geq C\\lambda^{2\\ell'}e^{{\\hat c}\\,\\lambda^{5\/3}\\tau}|{\\hat \\psi}(\\lambda^5)|\n\\end{split}\n\\end{equation}\nwith some ${\\hat c}>0$. If we choose $\\psi$ to be real and even we conclude from \\eqref{eq:psi:gevrey} that \n\\[\n|{\\hat \\psi}(\\xi)|\\leq Ce^{-c|\\xi|^{1\/3}}\n\\]\nwith some $c>0$ which implies $\\psi\\in \\gamma^{(3)}$. This gives a contradiction since $s>3$. This completes the proof.\n\n\\end{comment}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\nDelfs \\cite{Delfs85} considered a sheaf cohomology theory for\n(abstract) semialgebraic sets over arbitrary closed fields and proved\na semialgebraic version of homotopy invariance. In \\cite{EdmundoJP05}\nthis was generalized to definable sets and maps over an o-minimal\nexpansion of a group. If one further assumes that the o-minimal\nstructure expands a field, then one can use the triangulation theorem\nto show that the cohomology groups of a definable set are finitely\ngenerated and invariant under both elementary extensions and\nexpansions of the language. We prove that this continues to hold for\narbitrary o-minimal expansions of groups, provided we restrict\nourselves to definably compact sets. \n\nWorking without the field assumption entails various difficulties. \nTo begin with one cannot make use of the apparatus of singular cohomology.\nSo we work, following the above authors, with sheaf cohomology. More\nprecisely, given a definable set $X\\subset M^n$, the set of types $\\widetilde\nX$ of $X$ with the ``spectral topology'' is quasi-compactification of\n$X$, and we define $\\Hom^i(X;\\ca F ) := \\Hom^i(\\widetilde X ; \\ca F)$, where $\\ca\nF$ is a sheaf of Abelian groups on $\\widetilde X$.\n\nNow consider the case when $G$ is an Abelian group and $\\ca F$ is the\nconstant sheaf $G$ (i.e.\\xspace\\ the sheaf generated by the presheaf with\nconstant value $G$). Assuming that $X$ is {\\bf definably compact}\n(i.e.\\xspace\\ closed and bounded) we prove that $\\Hom^i(X;G)$ is finitely\ngenerated and invariant under both elementary extensions $N\\succ M$\n(i.e.\\xspace\\ $\\Hom^i(X;G) = \\Hom^i(X(N);G)$) and expansions of the language of $M$\n(note that expanding the language leaves $X$ invariant but alters $\\widetilde\nX$).\n\nThis would be easy to prove if $M$ expands a field. In fact in this\ncase by the triangulation theorem $X$ is definably homeomorphic to the\ngeometrical realization $|K|$ (in $M$) of a finite simplicial complex\n$K$, and a routine Mayer-Vietoris argument (together with the\nacyclicity of simplexes) shows that $\\Hom^i(X;G) \\cong \\Hom^i(K;G)$, where\nthe latter is the $i$-th simplicial cohomology group of $K$.\n\nIf $M$ does not expand a field we do not have the triangulation\ntheorem but we still have the cell decomposition theorem (see\n\\cite{Dries98}).  One could then be tempted to invoke the uniqueness\ntheorem for cohomology functors satisfying the (appropriate form) of\nthe Eilenberg-Steenrod axioms.  However, despite the cell\ndecomposition theorem, we have no guarantee that a definable set in an\no-minimal expansion of a group is a sort of definable CW-complex, so\nthe uniqueness theorem does not apply.  The problem is that in the\ndefinition of a CW-complex one requires that the cells come equipped\nwith an attaching map that extends continuously to the boundary, while\nfor the o-minimal cells we do not have any such control of the\nboundary.  \n\nAs standard references on sheaf cohomology and \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology we\nuse \\cite{Godement73} and \\cite{Bredon97}. For the reader's\nconvenience we give in the appendixes the relevant definitions and\nresults.  Sheaf cohomology is defined for arbitrary topological\nspaces, but many results are proved in the quoted texts under the\nadditional assumption that the space (or the family of supports) is\nHausdorff and paracompact. This is potentially a source of problems\nsince, given a definable set $X$, the spectral space $\\widetilde X$ associated\nto it is in general not Hausdorff. In some cases we can reduce to the\ncompact Hausdorff situation using the fact that $\\widetilde X$ has a continuous\nretraction onto a compact Hausdorff subspace ${\\widetilde X}^{\\max}$ with the\nsame cohomology groups (see \\cite{CarralC83}), but in some other cases\nit is more convenient to show that the proofs of the relevant results\nin \\cite{Godement73} or \\cite{Bredon97} work with the Hausdorff\nhypothesis being replaced by normality (according to our convention\nnormality does not imply Hausdorff).  The latter approach has the\nadvantage that one can do without the spectrality hypothesis. In\nparticular it can be shown that for normal paracompact spaces (not\nassumed to be Hausdorff or spectral) sheaf cohomology coincides with\n\\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology with coefficients in the given sheaf. This is\nreported in appendix~\\ref{AP:D},\nbut we shall not really need it.\nInstead we do need some results connecting the cohomology of\na subspace with the cohomology of its neighbourhoods.  Such results\nhave been proved in \\cite{Delfs85,Jones06,EdmundoJP05} in the spectral\nsituation (see Corollary \\ref{taut}), but they can also been\nestablished under more general hypothesis (Theorem\n\\ref{LEM:TAUT-PCN}). As a corollary we obtain that\n$\\Hom^*(\\bigcap_{t>0} Y_t) \\cong \\varinjlim_t \\Hom^*(Y_t)$, whenever\n$(Y_t \\mid t>0)$ is a definable decreasing\nfamily  of definably compact sets~$Y_t$\n(Corollary~\\ref{LEM:TAUT-DEFINABLE}).  We will also show\n(Theorem~\\ref{THM:DEFINABLE}) that if the o-minimal structure $M$\nexpands a field, then $\\varinjlim_t \\Hom^*(Y_t) \\cong \\Hom^*(Y_{t_0})$ for\nall sufficiently small $t_0$, but we are not able to prove this fact\nwithout the field assumption.\n\n\\section{Topological preliminaries}\\label{preliminaries} \n\nLet $X$ be a topological space. $X$ is {\\bf normal} if every pair of\ndisjoint closed subsets of $X$ can be separated by open\nneighbourhoods. $X$ is {\\bf paracompact} if every open covering of $X$\nhas a locally finite refinement. Unlike other authors, in both\ndefinitions we {\\bf do not require that $X$ be Hausdorff}.  We shall\ncall a space {\\bf PcN\\xspace} if it is paracompact and normal. Note that a\nparacompact Hausdorff space is PcN\\xspace.\n\nLet us recall that a {\\bf quasi-compact} space is a topological space $X$ in\nwhich every open covering has a finite refinement, or equivalently every\nfamily of closed sets with the finite intersection property has a non-empty\nintersection. So a compact space is an Hausdorff quasi-compact space.\n\nNote that a quasi-compact space is \\emph{a fortiori} paracompact.\nMoreover, it is a well-known fact that a in a PcN\\xspace space $X$\nevery open covering $\\Cov{U}$ admits a shrinking $\\Cov{V}$:\ni.e.\\xspace, $\\Cov{V}$ is an open covering of~$X$,\nand for every $V \\in \\Cov{V}$ there exists $U \\in \\Cov{U}$ containing~$\\overline V$, the closure of~$V$.\n\nA {\\bf spectral space} is a\nquasi-compact space having a basis of quasi-compact open sets stable\nunder finite intersections and such that every irreducible closed set\nis the closure of a unique point. The prime spectrum of a commutative\nring with its Zariski topology is an example of a spectral\nspace. Another example is the set of prime filters of a lattice (see\n\\cite{CarralC83}). The set of $n$-types (ultrafilters of definable\nsets) of a first order topological structure $M$ in the sense of\n\\cite{Pillay87} can also be endowed with a spectral topology (see\n\\cite{Pillay88}). In particular if $M$ is a real closed field one\nobtains in this way the real spectrum of the polynomial ring $M[x_1,\n\\ldots, x_n]$ (see \\cite{CosteR82}).\n\n\\section{Compactifications}\n\nIn this section we discuss a variant of the Wallman (or\nStone-\\texorpdfstring{\\v{C}ech}{Cech}\\xspace) compactification of a normal topological space. The\nvariant depends on the particular choice of a basis of open sets. \n\n\\begin{definition} \\label{comp} \nGiven a topological space $X$ with a fixed basis of open sets $\\basis$\n(that we always assume closed under finite intersections) we can\ndefine a spectral space $\\widetilde X = \\widetilde {(X,\\basis)}$ (depending on $\\basis$) as follows.  A\n{\\bf constructible set} is a boolean combination of basic open sets\n$U\\in \\basis$.  Let $\\widetilde X$ be the set of {\\bf\nultrafilters} of constructible sets (i.e.\\xspace\\ maximal families of\nconstructible sets closed under finite intersections and not\ncontaining the empty set).  For $b\\subset X$ constructible, let\n$\\tilde b = \\{p\\in \\widetilde X \\mid b\\in p\\}$.  So $p\\in \\tilde b \\Longleftrightarrow b\\in\np$. The {\\bf spectral topology} on $\\widetilde X$ is defined as follows. As a\nbasis of open sets of $\\widetilde X$ we take the sets of the form $\\tilde b$\nwith $b$ an open constructible subset of $X$.\n\\end{definition}\n\n\\begin{lemma} $\\widetilde X$ is a spectral space. \\end{lemma} \n\n\\begin{proof} To prove that $\\widetilde X$ is quasi-compact consider a family $\\{C_i\n\\mid i\\in I\\}$ of closed sets $C_i \\subset \\widetilde X$ with the finite\nintersection property. We must prove that $\\bigcap_{i\\in I}C_i$ is\nnon-empty. Without loss of generality we can assume that $\\{C_i\\mid\ni\\in I\\}$ is closed under finite intersections. Let $x\\in \\widetilde X$ be an\nultrafilter containing all the closed constructible sets $b$ with $\\widetilde\nb \\supset C_i$ for some $i$. Then $x\\in \\bigcap_i C_i$, so $\\widetilde X$ is\nquasi-compact. The same argument shows that the sets $\\widetilde b$ with $b$\nconstructible are quasi-compact.  So the sets $\\widetilde b$, with $b$ open\nand constructible, form a basis of quasi-compact open sets of $\\widetilde X$\nstable under finite intersections. To finish the proof we must show\nthat given an irreducible closed set $C$ of $\\widetilde X$, there is a unique\npoint $x\\in C$ with $C = Cl(x)$.  To this aim, let $x$ be an\nultrafilter of constructible sets containing all the closed\nconstructible sets $b$ with $\\widetilde b \\supset C$ and the complements of\nthe closed constructible sets $c$ such that $\\widetilde c \\cap C$ is a proper\nsubset of $C$ (this family has the finite intersection property by the\nirreducibility of $C$).  Then clearly $Cl(x) = C$.  To prove that $x$\nis unique, suppose $Cl(x) = Cl(y)$. Then $x$ and $y$ contain the same\nclosed constructible sets. But the closed constructible sets generate\nthe boolean algebra of all the constructible sets. So $x$ and $y$ must\ncontain the same constructible sets, and are therefore equal. \\end{proof}\n\n\\begin{remark}\nNote that a point $x \\in \\widetilde X$ is closed if and only if $x$\ncontains a maximal family of closed constructible sets with the finite intersection property. \nSo by Zorn's lemma every closed subset $C$ of $\\widetilde X$ contains a closed point. \n\\end{remark}\n\n\\noindent For $x\\in X$ let \\[\\type x := \\{b \\mid x \\in b\\} \\in \\widetilde X.\\]\nSince a constructible set $b$ is empty if and only if $\\widetilde b$ is empty,\nthe map \\[\\type {~} \\colon X\\to \\widetilde X\\] has dense image.  Moreover this\nmap is injective whenever $X$ is a $T_1$-space. So in this case,\nidentifying $x$ with $\\type x$, we have $X\\subset \\widetilde X$, and it is\neasy to see that the original topology on $X$ coincides with the\ntopology induced by $\\widetilde X$ (use the fact that for $A$ constructible,\n$\\widetilde A \\cap X = A$). Therefore:\n\n\\begin{lemma} \\label{T1} If $X$ is $T_1$, then for every open basis $\\basis$ of~$X$,\n$\\widetilde{(X, \\basis)}$ is a quasi-compactification of~$X$. \\end{lemma}\n\nWe say that $(X,\\basis)$ is {\\bf constructibly\nnormal} if any pair of disjoint constructible open sets can be\nseparated by closed constructible sets. \n\n\\begin{lemma} If $(X,\\basis)$ is constructibly normal, then $\\widetilde X$ is\nnormal (not necessarily Hausdorff).  \\end{lemma}\n\n\\begin{proof} Indeed given two disjoint closed subsets $A$ and $B$ of $\\widetilde X$, by\nquasi-compactness there are disjoint closed constructible sets $A',B'$\nin $X$ with $\\widetilde {A'} \\supset A$ and $\\widetilde {B'} \\supset B$. By the\nassumption $A',B'$ can be separated by disjoint open constructible\nsets $U\\supset A'$ and $V \\supset B'$. So $\\widetilde U$ and $\\widetilde V$ are open\nsets separating $A,B$. \n\\end{proof} \n\nIn a normal spectral space $Y$, the subset $Y^{\\max}$ of the closed\npoints of $Y$ is compact Hausdorff (see \\cite{CarralC83}). Also note\nthat, if $x$ is a closed point of $X$ and the singleton $\\{x\\}$ is\nconstructible, then $\\type x$ is a closed point of $\\widetilde X$. So we have:\n\n\\begin{lemma} If $(X,\\basis)$ is constructibly normal and the points of $X$ are\nclosed and constructible, then ${\\widetilde X}^{\\max}$ is a compactification\nof $X$, namely it is a compact Hausdorff space containing $X$ as a\ndense subspace.\\end{lemma}\n\nIf $X$ is a normal Hausdorff space, and we take as a basis of $X$ the family\n$\\basis$ of {\\em all} its open subsets, then ${\\widetilde X}^{\\max}$ is the\n{\\bf Wallman compactification} of $X$ \\cite{Wallman38}. For\nnormal Hausdorff spaces it coincides with the {\\bf Stone-\\texorpdfstring{\\v{C}ech}{Cech}\\xspace\ncompactification} (see \\cite[Thm. 3.6.22]{Engelking89}).\n\n\\begin{example} \nConsider the space $\\Q$ with a basis $\\basis$ of open sets given by the open intervals \n$(a,b)\\subset \\Q$, where we allow $a=-\\infty$ or $b = +\\infty$. Then \n\\[\\widetilde {(\\Q,\\basis)} = \\{a^-, a^+\\}_{a\\in \\Q} \\cup \\R \\cup \\{\\pm \\infty\\}\\]\nwhere: $a^+$ is the unique ultrafilter containing all sets of the form\n$(a,b)$ with $b>a$; similarly $a^-$ contains all sets $(b,a)$ with\n$b<a$; $+\\infty$ is the ultrafilter containing $(a,\\infty)$ for all $a\\in \\Q$; finally\n$-\\infty$ contains $(-\\infty,a)$ with $a\\in \\Q$. We have $Cl(a^-) =\n\\{a^-,a\\}$ and $Cl(a^+) = \\{a^+,a\\}$. The set of closed points is\n\\[\\widetilde {(\\Q,\\basis)}^{\\max} = \\R \\cup \\{\\pm \\infty\\},\\]\na two point-compactification of $\\R$. \n\\end{example}\n\n\\begin{example} \nConsider the space $\\Q$ with a basis $\\basis$ of open sets given by the bounded open intervals \n$(a,b)\\subset \\Q$. Then \n\\[\\widetilde {(\\Q,\\basis)}^{\\max} = \\R \\cup \\{\\infty\\},\\]\nthe one-point compactification of $\\R$. \n\\end{example}\n\n\\section{Cohomology of definable sets} \nLet $M = (M,<,\\ldots)$ be an o-minimal structure (cf.\\xspace~\\cite{Dries98}) \nexpanding a dense linear order $(M,<)$.\nFor instance $M$ may be a divisible ordered\ngroup or a real closed field. We put on $M$ the topology generated by\nthe open intervals and on $M^n$ the product topology. If $X$ is a\nsubset of $M^n$ we put on $X$ the induced topology from $M^n$ unless\notherwise stated. By a {\\bf definable} set we mean a first-order\ndefinable (with parameters) subset of $M^n$ for some $n$. For instance\nif $M$ is a real closed field, the definable sets are the\nsemialgebraic sets. Let $X\\subset M^n$ be a definable set. Let\n$\\basis$ be the basis of $X$ consisting of the open definable subsets\nof $X$. Define $\\widetilde X = \\widetilde {(X,\\basis)}$ as in Definition \\ref{comp}.\nNote that, by the o-minimality assumption, the definable sets\ncoincide with the constructible ones, namely every definable set is a\nboolean combination of open definable sets. It follows that $\\widetilde X$ is\nthe set of types of $X$ (a type $p\\in \\widetilde X$ can be identified\nwith an ultrafilter of definable sets such that $X\\in p$) endowed with\nthe spectral topology: as a basis of open sets we take the sets of\nthe form $\\widetilde U$ with $U$ a definable open subset of $X$.\nSince $X$ is Hausdorff, by Lemma \\ref{T1} $\\widetilde X$ is a\nquasi-compactification of $X$, in general not Hausdorff.\n\nLet us now make the further assumption that $M$ is an o-minimal\nexpansion of an ordered group.  In this case one can use the $M$-valued\nmetric  $\\abs{x-y}$ to show that\nevery definable set $X$\nis {\\bf definably normal} (any pair of disjoint definable closed sets\ncan be separated by definable open sets). Therefore in this case\n$\\widetilde X$ is {\\bf normal} (not necessarily Hausdorff), and the\nsubspace ${\\widetilde X}^{\\max}$ of its closed points is compact Hausdorff\n(and contains $X$ as a topological subspace).\n\nGiven a definable set $X\\subset M^n$ and a sheaf of Abelian groups\n$\\cal F$ on $\\widetilde X$, we define, following\n\\cite{Delfs85,Jones06,EdmundoJP05}:\n\\begin{equation} \\Hom^i(X;{\\cal F} ):= \\Hom^i(\\widetilde X ; {\\cal F}),\n\\end{equation}\nsee Definition~\\ref{DEF:COM}.\nEquivalently one can work with sheaves directly on $X$ rather than\n$\\widetilde X$ by considering $X$ not as a topological space but as a\nsite in the sense of Groethendieck (see \\cite[\\S 1.3]{CarralC83}). \nIf $A$ is a definable subset of~$X$, we write $\\Hom^*(A; \\cal F)$  for\n$\\Hom^*(\\widetilde A; \\cal F \\rest {\\widetilde A})$.\n\n\nIf $f\\colon X\\to Y$ is a definable function, then $f$ induces a\nfunction $\\widetilde f \\colon \\widetilde X \\to \\widetilde Y$ by $f(p)\n= \\{Z \\mid f^{-1}(Z) \\in p\\}$ where $Z$ ranges over the definable\nsubsets of $Y$ and we identify $p$ with an ultrafilter of definable\nsets.  We have $\\widetilde{f(X)} = \\widetilde f (\\widetilde X)$ and\n${\\widetilde f}^{-1}(\\widetilde Z ) = \\widetilde {f^{-1}(Z)}$. It\nfollows that if $f$ is continuous, $\\widetilde f$ is continuous. So if\n$f$ is an homeomorphism, then $\\widetilde f$ is an homeomorphism.\n\nIf $G$ is an Abelian group and $X$ is a topological space, the\nconstant sheaf on $X$ with stalk $G$ will also be denoted $G$.  \n\nWe recall that a {\\bf definable homotopy} between two definable functions\n$f\\colon X \\to Y$ and $g\\colon X\\to Y$ is a definable continuous function\n$F\\colon I\\times X\\to Y$, where $I = [a,b]$ is some closed bounded interval in\n$M$, such that $F(a,x) = f(x)$ and $F(b,x) = g(x)$ for every $x\\in X$. \nNote that $F$ induces a map $\\widetilde F \\colon \\widetilde {I \\times X} \\to \\widetilde Y$,\nbut in general $\\widetilde {I \\times X} \\neq \\widetilde I \\times \\widetilde X$, so we cannot consider $\\widetilde F$\nas a sort of ``homotopy'' parametrised by $\\widetilde I$. \nNevertheless we have the following definable version of the homotopy axiom:\n\n\\begin{fact}\\label{homotopy}\n{\\rm(\\cite{Delfs85,Jones06,EdmundoJP05})}  If $X,Y$ are definable\nsets and $f,g\\colon X \\to Y$ are definably homotopic definable maps,\nthen $\\widetilde f, \\widetilde g \\colon \\widetilde X \\to \\widetilde Y$\n(although not necessarily homotopic) induce the same homomorphism in\ncohomology, namely for every Abelian group $G$ we have\n\\[f^* = g^* \\colon \\Hom^i(Y; G) \\to \\Hom^i(X;G).\\] Similarly \nfor maps $f,g\\colon (X,A) \\to (Y,B)$ of pairs.  \\end{fact}\n\nThis has been proved by \\cite{Delfs85} in the semialgebraic case (i.e.\\xspace\\ when $M$ is a real\nclosed field and $X,Y,f,g$ are semialgebraic). Jones \\cite{Jones06}\nextended it to the case of definable sets and maps in an o-minimal\nexpansion $M$ of an ordered field. In \\cite{EdmundoJP05} it is shown\nthat it suffices that $M$ is an o-minimal expansion of an ordered\ngroup. All proofs make use of the following lemma:\n\n\\begin{lemma} {\\rm(\\cite[Prop.~4.7]{Delfs85})} Let $I\\subset M$ be a closed and bounded interval $[a,b]$.\nThen $\\Hom^p(I; G) = 0$ for all $p>0$\nand every Abelian group $G$. \\end{lemma}\n\nIndeed it is easy to see the lemma holds for an arbitrary interval $I$, not necessarily closed and bounded\n(note that in any case $\\widetilde I$ is quasi-compact). \n\n\\section{Contractibility of cells}\n\nLet $M$ be an o-minimal expansion of a group. \n\n\\begin{lemma} \\label{contraction1} Let $I$ be a bounded interval in $M$ \\rom(closed,\nhalf-closed, or open\\rom). Then $I$ is definably contractible to a\npoint. \\end{lemma}\n\n\\begin{proof} Let us consider the case $I = (a,b)$. Given $0< t \\leq \\frac{b-a}\n2$, there is a (unique) definable continuous function $f_t \\colon\n(a,b) \\to (a,b)$ such that $f_t$ is the identity on $[a+t,b-t]$ and it is\nconstant on both $[a,a+t]$ and $[b-t,b]$, with values $a+t$ and $b-t$\nrespectively (so the image of $f_t$ is $[a+t,b-t]$). Define $f_0 =\nf$. Then $(f_t)_{0\\leq t \\leq \\frac{b-a} 2}$ is a deformation retract\nof $(a,b)$ to the point $\\frac {a+b} 2$. The other cases are\nsimilar. \\end{proof}\n\nWe will employ the following notation: given $B \\subseteq M^{n-1}$ and \\mbox{$f,g : B \\to M$},\n\\[\\begin{aligned}{}\n(f,g)_B &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) < y < g(x)},\\\\\n[f,g]_B &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) \\leq y \\leq g(x)},\\\\\n\\Gamma(f) &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) = y},\n\\text{ the graph of } f.\n\\end{aligned}\\]\n\n\\begin{lemma} \\label{contraction2} If $C$ is a bounded cell of dimension $m>0$\nin $M^n$ then there is a deformation retract of $C$ onto a cell of\nstrictly lower dimension.  So by induction every bounded cell is\ndefinably contractible to a point. \\end{lemma}\n\n\\begin{proof} If $C$ is the graph of a function we can reason by induction on\nthe dimension of the ambient space. So the only interesting case is\nwhen $m>1$ and\n$C = (f,g)_B$.\nLet $h = \\frac {f+g} 2$.\nWe will define\na deformation retract from $C$ to $\\Gamma(h)$.  We can assume that\n$h$ is a constant function, since we can reduce to this case by a\ndefinable homeomorphism which fixes all but the last coordinate (just\ntake any constant function $h_1\\colon B \\to M$, and define $f_1,g_1$\nso that they differ from $h_1$ by the same amount in which $f,g$\ndiffer from $h$). Since $C$ is bounded, there are constants $a,b\\in M$\nsuch that $h$ is the constant function $\\frac {a+b} 2$ and $(f,g)_B\n\\subset B\\times (a,b)$.  By (the proof of) Lemma \\ref{contraction1}\nthere is deformation retract of $(a,b)$ onto $\\{\\frac {a+b} 2\\}$,\nwhich induces a deformation retract of $B\\times (a,b)$ onto $B \\times\n\\{\\frac {a+b} 2\\}$, namely onto the graph of $h$. \\end{proof}\n\nBy Fact \\ref{homotopy} we obtain:\n\n\\begin{corollary} \\label{acyclic-cell} If $C$ is a bounded cell of dimension $m$ in\n$M^n$ then $\\Hom^p(C;G) = 0$ for all $p>0$ and every Abelian group~$G$.\n\\end{corollary}\n\nIf we generalize slightly the definition of definable homotopy and allow the parameter of a homotopy to vary in the interval $[-\\infty, + \\infty]$, we get that Fact~\\ref{homotopy} is still true, and therefore in Lemmata~\\ref{contraction1} and~\\ref{contraction2} we can drop the ``boundedness'' hypothesis.\nThus, Corollary~\\ref{acyclic-cell} is true also for unbounded cells.\n\n\\section{Cells with non-acyclic closure} \n\nLet $M$ be an o-minimal expansion of a group and let $X\\subset M^n$ be\na definable set. We will prove (Theorem \\ref{fin-gen}) that the\ncohomology groups $\\Hom^p(X;G)$ of $X$ are finitely generated.  An\nimportant special case is when $X$ is the closure $\\overline C$ of a bounded\ncell $C$. One may be tempted to conjecture that $\\Hom^i(\\overline C ; G)=0$ in\ndimension $i>0$, but Example \\ref{strange-cell} shows that in general\nthis is false.  Indeed similar examples show that $\\Hom^1(\\overline C ; G)$ can\nhave arbitrarily large finite rank.\n\n\\begin{example} \\label{strange-cell} Let $M=(\\R,<,+,\\cdot)$. There is a bounded\ncell $C$ of dimension $2$ in $\\R^4$ whose closure $\\overline C$ has an ``hole'',\nnamely it is definably homotopic to a circle. \\end{example}\n\n\n\\begin{floatingfigure}[r]{15ex}\n\\input{annulus.latex} \n\\caption{\n\\label{annulus}\n\\end{floatingfigure}\nBefore giving the example, let us observe that\nin Figure~\\ref{annulus} (an annulus with a ray removed) we have\na space homeomorphic to a disk whose closure is homotopic to a circle.\nHowever the space of Figure~\\ref{annulus} is not a cell in\nthe sense of o-minimal cell decompositions. So we have to proceed\ndifferently.  \n\nAs a preliminary step we show that there is a\ntwo-dimensional cell $D$ in $\\R^3$ which is homeomorphic to an open\ndisk minus a ray via an homeomorphism which extends to the\nclosures. An example is the cell $D$ depicted on the top-right part of\nFigure~\\ref{cell}.\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{cell5.latex} \n\\caption{A two-dimensional cell in $\\R^3$}\n\\label{cell}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{cell} must be interpreted as follows.  The projection of\n$D$ on the first two coordinates is the open square on the\nbottom-right part of Figure~\\ref{cell}, with the five triangles\npartitioning the square corresponding to the five regions in $D$. So $D$\nis the graph of a function $f$ from the square to $\\R$.  $D$ is\nhomeomorphic to an open disk with a ray removed via an homeomorphism\nthat extends to the closures and sends the ray of the disk to the\nvertical dashed line $I \\subset \\partial D$ depicted in the Figure.\nThe three triangular regions of the cell $D$ are level sets of $f$,\nwith the central triangle being in a lower level, and the two top\ntriangles being on a common higher level.\n\nWe are now ready to describe the cell $C$ of Example\n\\ref{strange-cell}. The idea is the following.  Given a disk without a\nray as in Figure~\\ref{cell}, we can ``bend'' it going in a higher\ndimension, so as to create a ``hole'' (as in Figure~\\ref{annulus}) by\nseparating the two sides of the middle part of the ray. More\nprecisely, consider a definable continuous function $g\\colon D \\to \\R$\nwith the following properties: 1) $g$ takes non-zero values only for\npoints $x\\in D$ sufficiently close to the middle point $x_0$ of the\ndashed segment $I \\subset \\partial D$; 2) if $g(x) > 0$, then $x$ is\non the ``left hand side'' of $I$ and $g(x) \\to 1$ for $x\\to x_0$ from\nthe left; 3) if $g(x) < 0$, then $x$ is on the ``right-hand-side'' of\n$I$ and $g(x) \\to -1$ for $x\\to x_0$ from the right. Let $C$ be the\ngraph of $g$. Then $H^1(\\overline C; G) \\neq 0$.\n\n\\begin{remark} It is possible to show that $\\Hom^1(X;G)$ can have arbitrarily high\nrank even for a definable set $X$ that can be decomposed as a disjoint\nunion of only two bounded cells. \nTherefore, unlike the case of triangulations,\nthe abstract structure of a cell decomposition\n(namely the dimensions and the adjacency relation between cells) by no\nmeans determines the cohomology of a definable set. It is not however\nexcluded that by a refined version of the cell decomposition theorem one \ncould obtain decompositions that do determine the cohomology groups.  \\end{remark}\n\n\\section{Acyclic coverings}\nIn this section we give a sufficient condition, not based on\ndeformation retracts, to prove that an inclusion $X\\subset Y$ of\ntopological spaces induces an isomorphism in cohomology (see Lemma\n\\ref{iso}).\n\nGiven an open cover $\\ca U = \\{U_i \\mid i \\in I\\}$ of a topological\nspace $X$, and a subset $J\\subset I$, we write $U_J$ for the\nintersection $\\bigcap_{i\\in J} U_i$. The following theorem of Leray\nsays that, given an ``acyclic covering'', the cohomology of a sheaf\ncan be computed as the \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of the covering\n(cf.\\xspace~Appendix~\\ref{AP:D}). \n\n\\begin{fact}\\label{Leray}\n{\\rm(\\cite[Thm. 4.13, p. 193]{Bredon97}, \\cite[\\S 5.4, p. 213]{Godement73})}\nLet $\\ca F$ be\na sheaf on a topological space $X$ and let $\\ca U = \\{U_i \\mid i \\in\nI\\}$ be an open covering of $X$ having the property that $\\Hom^p(U_J; \\ca\nF) = 0$ for every $p>0$ and every finite $J\\subset I$. \nThen the canonical homomorphism $\\check{\\Hom}\\vphantom{\\Hom}^*(\\ca U ; \\ca F)\\to \\Hom^*(X;\\ca F)$\nis an isomorphism.\n\\end{fact}\n\nIf moreover we assume that $\\ca F$ is the constant sheaf $G$ and $\\Hom^0(U_J; \\ca F ) = G$ for every $J$ with $U_J$ non-empty, then from the\ndefinition of \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology it follows that $\\check{\\Hom}\\vphantom{\\Hom}^*(\\ca U;G)$\ncoincides with the $i$-th simplicial cohomology group with\ncoefficients in $G$ of the nerve $N({\\ca U})$ of $\\ca U$. So we have:\n\n\\begin{corollary} \\label{good-cover} \nIf there exists a finite open cover $\\ca U= \\{U_i \\mid i \\in I\\}$ of $X$ with \n $\\Hom^p(U_J; G ) = 0$ for every $p > 0$ and $\\Hom^0(U_J; G) = G$ for every finite\n$J\\subset I$ with $U_J$ non-empty, then $\\Hom^i(X;G)$ is isomorphic to the $i$-th \nsimplicial cohomology group of the nerve $N({\\ca U})$ of the covering, so in particular it is \nfinitely generated. \n\\end{corollary} \n\nUnfortunately we do not know the answer to the following:\n\n\\begin{question} Let $X$ be a definably compact set in an o-minimal expansion $M$ of a\ngroup. Does $\\widetilde X$ have a cover as in Corollary \\ref{good-cover}? \\end{question} \n\nThe answer is positive if $M$ expands a field, as a simple application\nof the triangulation theorem shows.\n\n\\begin{remark} \\label{inducedh}\nLet $f\\colon X\\to Y$ be a continuous function, let ${\\cal V} =\n\\{V_i \\mid i\\in I\\}$ be an (indexed) open cover of $Y$ and consider\nthe (indexed) open cover $f^{-1}({\\cal V}):= \\{ f^{-1}(V_i)\\mid i \\in I\\}$ of\n$X$. It follows easily from the definition of the induced\nhomomorphism $\\check{f}$ in \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology (see \\cite[IX, \\S 4]{EilenbergS52} or \\cite[III.4.1.4]{Bredon97}) that there is a commutative\ndiagram:\n\n\\DIAGV{90} {\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V};G)} \\n{\\Ear{a}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(f^{-1}({\\cal V});G)} \\nn \n           {\\Sar{b}}        \\n{}       \\n{\\Sar{c}} \\nn \n           {\\check{\\Hom}\\vphantom{\\Hom}^p(Y; G)}    \\n{\\Ear{\\check{f}}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(X; G)}        \n\\diag \n\n\n\\noindent where $b,c$ are the natural morphisms (see \\cite{Godement73})\nand $a$ is induced by the simplicial map on the nerves of\nthe indexed coverings sending $f^{-1}(V_i)$ to $V_i$.  \\end{remark}\n\n\n\\begin{lemma} \\label{iso} Let $X\\subset Y$ be definable sets. Suppose that there\nare coverings ${\\cal U} = \\{ U_i \\mid i\\in I\\}$ of $X$ and ${\\cal V} =\n\\{ V_i \\mid i \\in I\\}$ of $Y$ indexed by the same finite set $I$ such\nthat:\n\\begin{enumerate}\n\\item  $U_i \\subset V_i$ for all $i,j\\in I$.\n\\item For all finite $F\\subset I$, $U_F$ is non-empty iff $V_F$ is\nnon-empty (i.e.\\xspace\\ the natural map among the nerves of the coverings is\nan isomorphism).\n\\item For each finite $F \\subset I$ the sets $U_F$ and $V_F$ are either empty or connected, \nand for all $q>0$, $\\Hom^q(U_F; G) = \\Hom^q(V_F; G) = 0$.  \n\\end{enumerate}\nThen the inclusion map $X\\subset Y$ induces an isomorphism $\\Hom^*(Y;\nG) \\to \\Hom^*(X; G)$ for any Abelian group $G$. \\end{lemma}\n\nNote that we do not require that $V_i \\cap X = U_i$. \n\n\\begin{proof} We are going to apply Remark \\ref{inducedh} to the case when\n$X\\subset Y$ and $f$ is the inclusion map. In this case $f^{-1}({\\cal\nV}) = {\\ca V} \\cap X$ (by definition).  Consider the following\ncommutative diagram, where $a,b,c$ are as in Remark \\ref{inducedh},\n$d$ is induced by the natural simplicial isomorphism on the nerves of\nthe coverings, $e$ is the natural morphism form the\n\\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of a covering to the \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of the\nspace, and $p,q$ are the natural maps from \\texorpdfstring{\\v{C}ech}{Cech}\\xspace to sheaf\ncohomology.\n\n\\DIAGV{90} {\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V};G)} \\n{\\Ear{a}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V}\\cap X;G)} \\n{\\Ear{d}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal U}; G)}\\nn           \n           {\\Sar{b}} \\n{} \\n{\\Sar{c}} \\n{\\raisebox{1em}{\\SWAR{\\raisebox{1em}{e}}}} \\n{} \\n{} \\nn\n           {\\check{\\Hom}\\vphantom{\\Hom}^p(Y; G)} \\n{\\Ear{\\check{f}}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(X; G)} \\n{} \\n{} \\nn\n           {\\Sar{p}} \\n{} \\n{\\Sar{q}} \\n{} \\n{} \\nn\n           {\\Hom^p(Y; G)} \\n{\\Ear{f^* }} \\n{\\Hom^p(X; G)} \\n{} \\n{} \n\\diag\n\n\\noindent By our assumptions on the coverings, $a$ and $d$ are isomorphisms. \nBy Fact \\ref{Leray} $p\\circ b$ and $q\\circ e$ are isomorphisms. So \n $f^*: \\Hom^*(Y;G) \\to \\Hom^*(X; G)$  is\nan isomorphism.  \\end{proof}\n\n\\section{Finiteness results for cohomology} \n\nAs usual let $M$ be an o-minimal expansion of group. Given a definably\ncompact set $X$ we want to prove that $\\Hom^i(X;G)$ is finitely generated\nfor every $i$ and every Abelian group $G$. An important special case\nis when $X$ is the closure $\\overline C$ of a bounded cell $C$. We will show\n(as a consequence of Corollary \\ref{X-C}) that there is a point $a\\in\nC$ such that $\\Hom^p(\\overline C \\setminus \\{a\\};G) \\cong \\Hom^p(\\partial C;G)$,\nwhere $\\partial C := \\overline C \\setminus C$ is the boundary of\n$C$. Granted this, since $\\partial C$ has smaller dimension than $\\overline\nC$, by induction on the dimension we can assume that the cohomology\ngroups of $\\partial C$ are finitely generated and carry on with the\ninductive proof. At first sight the fact that $\\Hom^p(\\overline C \\setminus\n\\{a\\};G) \\cong \\Hom^p(\\partial C;G)$ looks rather intuitive: one may even\nbe tempted to conjecture that $\\partial C$ is a definable deformation\nretract of $\\overline C \\setminus \\{a\\}$ (as it would be the case for the\ncells of a CW-complex), or at least that these two spaces are\ndefinably homotopy equivalent. However we are not able to prove this.\nWe proceed instead in a different manner, with the role of definable\ndeformation retracts being taken by Lemma \\ref{iso}.  There is however\na further complication. We are not able to apply Lemma~\\ref{iso}\ndirectly to the pair of sets $(X,Y) = (\\overline C \\setminus \\{a\\}, \\partial\nC)$, but only to pairs of sets of the form $(C \\setminus \\{a\\},\nC\\setminus C_t)$, where $(C_t)_{t>0}$ is a suitable definable\ncollection of sets with $\\bigcup_{t>0} C_t = C$, and consequently $\\bigcap_{t>0}(\\overline C \\setminus C_t) = \\partial C$ (the singleton $\\{a\\}$ is one of the\n$C_t$). So we first prove $\\Hom^p(C \\setminus \\{a\\}) \\cong \\Hom^p(C\\setminus\nC_t)$. Then we deduce $\\Hom^p(\\overline C \\setminus \\{a\\}) \\cong \\Hom^p(\\overline\nC\\setminus C_t)$ by the excision theorem. Finally, with the help of\nCorollary \\ref{LEM:TAUT-DEFINABLE}, we let $t\\to 0$ to obtain $\\Hom^p(\\overline\nC \\setminus \\{a\\}) \\cong \\Hom^p(\\partial C)$. This is the idea. Let us\nnow come to the details.\n\n\\begin{lemma} \\label{cell-covers}\nLet $C\\subset M^n$ be a bounded cell of dimension $m$.\nThere is a definable family $\\{ C_t \\mid t>0 \\}$ of definably compact sets\n$C_t\\subset C$ such that:\n\\begin{enumerate}\n\\item $C = \\bigcup_{t>0}C_t$.\n\\item If $0<t'<t$, then $C_t\\subset C_{t'}$ and the inclusion\n$C\\setminus C_{t'} \\subset C\\setminus C_{t}$ induces an isomorphism\n$\\Hom^p(C \\setminus C_t; G) \\to \\Hom^p(C \\setminus C_{t'}; G)$.\n\\item For every Abelian group~$G$,\n$C\\setminus C_t$ has the same cohomology groups of an $m-1$\ndimensional sphere, namely $\\Hom^p(C\\setminus C_t; G) = 0$ for $p \\not\\in\n\\{0,m-1\\}$, and $\\Hom^p(C\\setminus C_t; G) = G$ for $p \\in \\{0,m-1\\}$.\n\\end{enumerate}\n\\end{lemma}\n\nNote that the result would be obvious if $M$ expands a field, since in\nthat case $C$ is definably homeomorphic to an open ball, and each open\nball is the increasing union of its concentric closed sub-balls.\n\n\\noindent\\textit{Proof}.\nWe define $C_t$ as follows. \n\n\\begin{enumerate}\n\\item If $n=1$ and $C = (a,b)$, then $C_t = [a+ \\gamma_t, b-\n\\gamma_t]$ where $\\gamma_t = \\min{\\{\\frac {a+b} 2, t \\}}$ (so $C_t$ is\nnon-empty).\n\\item If $n=1$ and $C$ is a singleton in $M$, $C_t = C$.\n\\item Let $n>1$ and $C = \\Gamma(f)$, where $f\\colon B\\to M$. By\ninduction $B_t$ is defined and we set $C_t = \\Gamma(f \\rest{B_t})$. \n\\item \nLet $n>1$ and $C = (f,g)_B$. By induction $B_t$ is defined.\nWe put\n$C_t = [f+\\gamma_t, g-\\gamma_t]_{B_t}$,\nwhere $\\gamma_t := \\min(\\frac{f - g}{2}, t)$.\n\\end{enumerate} \n\nWith this definition we have:\n\n\\begin{claim} \\label{cell-cover} For each $t>0$ \nthere is a covering $\\ca U = \\{U_i\\mid i\\in I\\}$ of\n$C\\setminus C_t$ such that:\n\\begin{enumerate}\n\\item \nThe index set $I$ is the family of the closed faces of an\n$m$-dimensional cube, where $m= \\dim(C)$. \\rom(So $|I|=2m$\\rom).\n\\item If $F\\subset I$, then $U_F := \\bigcap_{i\\in F}U_i$ is either\nempty or a cell. \\rom(So in particular $\\Hom^p(U_F; G) = 0$ for all $p>0$\nand, if $U_F \\neq \\emptyset$, $\\Hom^0(U_F;G) = G$.\\rom)\n\\item For $F\\subset I$, $U_F \\neq \\emptyset$ iff the faces of the\ncubes belonging to $F$ have a non-empty intersections. \\rom(So the nerve of\n$\\ca U$ is isomorphic to the nerve of a covering of an $m$-cube by its\nclosed faces.\\rom)\n\\end{enumerate} \n\\end{claim} \n\\begin{minipage}{\\textwidth}\n\\begin{floatingfigure}[r]{25ex}\n\\input{cover2.latex} \n\\caption{}\n\\label{cover}\n\\end{floatingfigure}\nFor example for $m=2$ we have four open sets $U_i$ which intersect each other\nas in Figure~\\ref{cover}.\nNote that the claim implies, by Corollary~\\ref{good-cover}, that $C\n\\setminus C_t$ has the same cohomology groups of an $m-1$ dimensional\nsphere.  To prove the claim we define $\\ca U$ by induction on the\ndimension $n$ of the ambient space. We distinguish four cases\naccording to the definition of~$C_t$.\n\\end{minipage}\n\\begin{enumerate}\n\\item{If $n=1$ and $C$ is a singleton, then $\\ca U$ is the covering\nconsisting of one open set (given by the whole space $C$).}\n\\item{If $n=1$ and $C = (a,b)$, then $C\\setminus C_t$ is the union of\nthe two open subsets $(a, a + \\gamma_t)$ and $(b- \\gamma_t, b)$,\nand we define $\\ca U$ as the covering consisting of these two sets.}\n\\item Let $n>1$ and $C = \\Gamma(f)$, where $f\\colon B\\to M$. By\ninduction we have a covering $\\ca V$ of $B\\setminus B_t$ with the\nstated properties, and we define $\\ca U$ to be the covering of\n$C\\setminus C_t$ induced by the natural homeomorphism between the\ngraph of $f$ and its domain.\n\\item\nLet $n > 1$ and $C = (f,g)_B$. By definition $C_t = (f + \\gamma_t, g -\n\\gamma_t)_{B_t}$.  By induction $B\\setminus B_t$ has a covering $\\ca V\n= \\{V_j \\mid j\\in J\\}$ with the stated properties, where $J$ is the\nset of closed faces of the cube $[0,1]^{m-1}$.  Define a covering $\\ca\nU = \\{U_i \\mid i\\in I\\}$ of $C\\setminus C_t$ as follows. As index set\n$I$ we take the closed faces of the cube $[0,1]^m$. Thus $|I| = |J| +\n2$, the two extra faces corresponding to the ``top'' and ``bottom''\nface of $[0,1]^m$. \nWe associate to the top face \nthe open set $(g- \\gamma_t,g)_B \\subset C\\setminus C_t$ and to the\nbottom face the open set $(f, f + \\gamma_t)_B \\subset C \\setminus\nC_t$. The other open sets of the covering are the preimages of the sets $V_j$ \nunder the projection $M^n\\to M^{n-1}$.  This\ndefines a covering of $C\\setminus C_t$ with the stated properties.\n\\end{enumerate}\n\nIt remains to show that the inclusion map $C\\setminus C_{t'} \\subset\nC\\setminus C_{t}$ induces an isomorphism $\\Hom^p(C \\setminus C_t; G) \\to\n\\Hom^p(C \\setminus C_{t'}; G)$.  To this aim it suffices to observe that\nby (the proof of) Claim 1 there are coverings $\\ca U$ of $C \\setminus\nC_{t'}$ and $\\ca V$ of $C \\setminus C_t$ satisfying the assumptions of\nLemma \\ref{iso}. \n\\hfill\\textsquare\n\n\\begin{lemma} \\label{x-c} Let $X$ be a definably compact set, $C$~be a cell of\nmaximal dimension in~$X$, and $G$ be an Abelian group.\nThen for each $0<t'<t$ the inclusion map\n$X\\setminus C_{t'} \\subset X\\setminus C_t$\ninduces an isomorphism\n\\[\\Hom^*(X\\setminus C_{t}; G) \\cong \\Hom^*(X\\setminus C_{t'}; G).\\] \\end{lemma}\n\n\\begin{proof} \nBy the excision theorem  (\\cite[Thm. 12.9]{Bredon97})\nthe inclusion of pairs $(C\\setminus C_{t'}, C\\setminus C_t) \\to (X\n\\setminus C_{t'}, X \\setminus C_{t})$ induces an isomorphism \\[\\Hom^*(X\n\\setminus C_{t'}, X \\setminus C_{t}; G) \\cong \\Hom^*(C\\setminus C_{t'},\nC\\setminus C_t; G).\\] The right-hand side is zero by\nLemma \\ref{cell-covers} and the long cohomology sequence of the pair (see\n\\cite[eqn. 24, p. 88]{Bredon97}). So the left-hand side is also zero and therefore\n$\\Hom^*(X\\setminus C_{t}; G) \\to \\Hom^*(X\\setminus C_{t'}; G)$ is an isomorphism. \\end{proof}\n\nIn the above theorem it is not necessary that $X$ is definably\ncompact: it suffices that $C$ is bounded.\n\n\\begin{remark} \\label{lim} Let $R$ be the limit of a direct system\n$(R_i \\mid f_{i,j})_{i,j\\in I}$ of Abelian groups and morphisms\n$f_{i,j}\\colon R_i \\to R_j$.\nSuppose that each $f_{i,j}$ is an isomorphism. Then for each\n$i$ the natural morphism $R_i \\to R$ is an isomorphism. \\end{remark}\n\n\\begin{corollary} \\label{X-C} Suppose that $X$ is a definably compact set such that\n$C$ is a cell of maximal dimension of $X$ \\rom(for instance $X = Cl(C)$\\rom).\nThen for every $t>0$ the inclusion map\n$X\\setminus C \\subset X\\setminus C_t$ induces an isomorphism \\[\\Hom^p(X\n\\setminus C_t; G) \\cong \\Hom^p(X \\setminus C; G).\\]  \\end{corollary}\n\n\\begin{proof}  For $t>0$ we have\n\\[ \\begin{array}{ccc}\n\\Hom^*(X\\setminus C_t;G) & \\cong & \\varinjlim_s \\Hom^*(X \\setminus C_s;G) \\\\\n & \\cong & \\Hom^*(X \\setminus C;G)\n\\end{array} \\]\nwhere the first isomorphism follows from Lemma~\\ref{x-c} and\nRemark~\\ref{lim}, and the second one follows from the equality $X\n\\setminus C = \\bigcap_s (X \\setminus C_s)$ and\nCorollary~\\ref{LEM:TAUT-DEFINABLE}. \\end{proof}\n\nNote that for $t$ big enough, $C_t$ is a singleton. So in particular,\napplying the theorem to $X = \\overline{C}$, we have proved that there is a\npoint $a\\in C$ such that:\n \\begin{equation} \\label{partial} \\Hom^p(\\overline{C}\n\\setminus \\{a\\}; G) \\cong \\Hom^p(\\partial C; G). \\end{equation} We do not\nknow however whether $\\partial C$ is a definable deformation retract of $\\overline C\n\\setminus \\{a\\}$.\n\n\\begin{theorem} \\label{fin-gen} Let $X \\subset M^n$ be a definably compact\nset and $G$ be an Abelian group.\nThen, for each $p$, $\\Hom^p(X;G)$ is finitely generated. Moreover\n$\\Hom^p(X;G)=0$ for $p>\\dim(X)$.  \\end{theorem}\n\n\\begin{proof}\nBy a Mayer-Vietoris argument. Decompose $X$ into cells. Let $C$ be\na cell of $X$ of maximal dimension. Let $t>0$ be such that $C_t \\neq\n\\emptyset$ and write $X$ as the union of $X\\setminus C_t$ and\n$C$. Consider the Mayer-Vietoris sequence (see \\cite[eqn. 32,\np. 98]{Bredon97}) associated to this union:\n\\begin{equation} \\label{EQ:MAYER}\n\\ldots \\to \\Hom^{p-1}(C \\setminus C_t) \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C_t)\n\\oplus \\Hom^p(C) \\to \\Hom^p(C\\setminus C_t) \\to \\ldots\n\\end{equation}\nwhere we have omitted the coefficients $G$ in the notation. By Corollary \\ref{X-C} the\ninclusion $X\\setminus C_t \\subset X\\setminus C$ induces an isomorphism\nin cohomology, so composing with this isomorphism we obtain \n\\begin{equation} \\label{mv} \n\\ldots \\to \\Hom^{p-1}(C \\setminus C_t) \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to \\Hom^p(C\\setminus C_t) \\to \\ldots\n\\end{equation}\nNow $C$ has the cohomology of a point and $C\\setminus C_t$ has the\ncohomology of an $(m-1)$-dimensional sphere. So the displayed part of\nthe sequence above has the form:\n\\begin{align}\\label{mv1} \n& G \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to 0,\\\\\n\\text{ or}\\quad &\n\\label{mv2} \n0 \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to G,\\\\\n\\text{ or} \\quad &\n\\label{mv3} \n0 \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to 0;\n\\end{align}\nwhere (\\ref{mv3}) applies for $p \\in \\{m, m-1\\}$. \n By induction on the number of cells \n$\\Hom^p(X\\setminus C)$ is finitely generated, and vanishes for $p\\geq m$.\n From the above sequences it then follows that the same holds for $\\Hom^p(X)$.\n\\end{proof}\n \n\n\\section{Elementary extensions and change of language}\n\nLet $M$ be an o-minimal expansion of a group and let $X\\subset M^n$ be\na definable set.  We have seen that we can associate to $X$ the\nspectral space $\\widetilde X$ of all its types over $M$ (such a type can\nbe identified with an ultrafilter of $M$-definable sets that contains \n$X$). The cohomology of $X$ has been defined as the cohomology of $\\widetilde X$. \nIf $N\\succ M$ is an elementary extension we may also associate to $X$ the spectral\nspace $\\widetilde{X(N)}$ (ultrafilters of $N$-definable sets that contains $X(N)$). \n\n\\begin{theorem} \\label{theta} Let $\\theta \\colon \\widetilde{X(N)} \\to \\widetilde X$ be\nthe map that sends a type over $N$ to its restriction over $M$. If $X$\nis definably compact then $\\theta$ induces an isomorphism\n$\\Hom^*(\\widetilde X; G) \\to \\Hom^*(\\widetilde{X(N)}; G)$ for any Abelian group $G$. \\end{theorem}\n\n\\begin{proof}\nFor both $X$ and $X(N)$ we have an exact sequence as in\nequation~\\eqref{EQ:MAYER}\nabove. The terms of the two exact sequences so obtained are connected by the\nhomomorphisms induced by $\\theta$ in such a way that the resulting \ndiagram commutes (it is important that in equation~\\eqref{EQ:MAYER}\nwe take the parameter $t$ in the small model~$M$).\nThe desired result then follows\narguing as in Theorem \\ref{fin-gen} by induction on the number of cells. \\end{proof} \n\n  \nWe now extend the above result to certain type-definable sets. As above let $N\\succ M$. \n\n\\begin{theorem} Let $X\\subset M^n$ be a definably compact set. Let $A\\subset\n\\widetilde X$ be a type-definable closed subset and let $A(N) := \\theta^{-1}(A)\n\\subset \\widetilde{X(N)}$. Then $\\theta$ induces an isomorphism\n$\\Hom^*(A; G) \\to \\Hom^*(A(N) ; G)$. In particular if\n$p$ is a closed type in $\\widetilde X$, then the set $\\theta^{-1}(p)$ of all\ntypes of $\\widetilde{X(N)}$ which restrict to $p$ has the same\ncohomology of a point \\rom(so in particular it is connected\\rom). \\end{theorem}\n\\begin{proof} Each closed type-definable set $A\\subset \\widetilde X$ can be written as an\nintersection $\\bigcap_{i\\in I}X_i$ of definably compact sets $X_i\n\\subset M^n$. Now observe that $A(N) =\n\\bigcap_{i\\in I}\\widetilde{X_i(N)}$.  By Corollary \\ref{LEM:TAUT-DEFINABLE} and Theorem\n\\ref{theta} $\\Hom^*(A(N);G) = \\varinjlim_{i\\in I}\\Hom^*(\\widetilde{X_i(N)};G) =\n\\varinjlim_{i\\in I}\\Hom^*(X_i;G) = \\Hom^*(A;G)$.\n\\end{proof}\nIn similar way we can prove: \n\n\\begin{theorem} If $M_1$ is an o-minimal expansion of $M$ to a bigger language and\n$X\\subset {M}^n$ is a definably compact set in $M$, then the map\n$\\widetilde{X(M_1)} \\to \\widetilde X$ sending each type in the language\n$L_1$ to its restriction to $L$ induces an isomorphism $\\Hom^*(\\widetilde\nX; G) \\to \\Hom^*(\\widetilde{X(M_1)}; G)$ for any Abelian group $G$.\\end{theorem}\n\n\\section{Definable families in expansions of fields}\n\\begin{definition}\nGiven definable maps $f : X \\to Y$ and $f' : X' \\to Y'$, we say that $f$ and $f'$ have the same definable topological type iff there exist definable homeomorphisms $\\lambda: X \\to X'$ and $\\mu: Y \\to Y'$ making the following diagram commute:\n\\DIAGV{50}\n{}\\n{X} \\n{\\Ear{f}} \\n{X'} \\nn\n{}\\n{\\Sar{\\lambda}} \\n{} \\n{\\saR{\\mu}} \\nn\n{}\\n{Y} \\n{\\Ear{g}} \\n{Y'}\n\\diag\n\\end{definition}\n\n\\begin{definition}\nLet $(Y_t)_{t > 0}$ be a family of subsets of~$M^n$.\nWe will say that $(Y_t)_{t>0}$ is decreasing if $Y_t \\subseteq Y_{t'}$ for every $t \\leq t'$.\n\\end{definition}\n\nIf $M$ is an o-minimal expansion of a\nfield, Corollary \\ref{LEM:TAUT-DEFINABLE} can be strengthened as follows. \n\n\\begin{theorem} \\label{THM:DEFINABLE}\nAssume that $M$ is an o-minimal expansion of a field.\nLet $\\Pa{Y_t}_{t > 0}$ be a definable decreasing\nfamily of definably compact subsets of some definable set~$Y$.\nLet $A := \\bigcap_{t > 0} Y_t$. \nThen for every sufficiently small $t$ we have a natural isomorphism\ninduced by the inclusion\n\\[\n\\Hom^*\\Pa{ A; {\\ca F}} \\cong \\Hom^*\\Pa{{Y_t}; {\\ca F}},\n\\]\nfor every sheaf ${\\ca F}$ on~$\\widetilde Y$.\n\\end{theorem}\nIn the above theorem we cannot weaken the hypothesis to $Y_t$ closed\n(instead of definably compact).  For instance, let $Y_t :=\n[\\frac{1}{t}, \\infty) \\subseteq M$. \n\nNote that Corollary \\ref{X-C} is a special case of Theorem \\ref{THM:DEFINABLE}.\nTo prove the theorem, we need the following lemmata.\nNote that for the lemmata we do not need that $M$ expands a field,\nbut only that it expands a group.\n\\begin{lemma} \\label{REM:MINIMA} Let $M$ be an o-minimal expansion of a group. \nLet $Y \\subseteq M^n$ be definable, and $f: Y \\to M$ be a definable function \\rom(not necessarily continuous\\rom).\nLet $A \\subseteq Y$ be the set of local minima of~$f$.\nThen, $f(A)$ is finite.\n\\end{lemma} \n\\begin{proof}\nIf not, let $a < b \\in M$ such that, for every $t \\in (a,b)$ there\nexists $\\gamma(t) \\in Y$ such that $f(\\gamma(t)) = t$ and $\\gamma(t)$\nis a local minimum for~$f$.  By definable choice, we can assume that\n$\\gamma$ is a definable continuous function. It follows that in\n any neighbourhood of $\\gamma(t)$ there are points of the\nform $\\gamma(t')$ with $t'<t$. But $f(\\gamma(t')) < f(\\gamma(t))$, contradicting the fact that $\\gamma(t)$ is a local minimum.\n\\end{proof}\n\n\\begin{lemma}\\label{LEM:FAMILY}\nLet $M$ be an o-minimal expansion of a group. \nLet $(Y_t)_{t > 0}$ be a decreasing definable family of closed subsets of~$M^n$.\nThen, there exists $t_0 >0$ such that, for every $t \\in M$ with $0 < t < t_0$ we have\n\\[\nY_t = \\bigcap_{t < u < t_0} Y_u\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $Y := \\bigcup_{t > 0} Y_t$.\nDefine\n\\begin{align*}\nf &: Y \\to M^{\\geq 0}\\\\\nf(x) &:= \\inf \\set{t: x \\in Y_t}.\n\\end{align*}\nDefine also $Z_t := f^{-1}([0,t])$.\nNote that $Z_t = \\bigcap_{t < u < t_0} Y_u$, and\n$Y_t \\subseteq Z_t$.\nTherefore,\nthe conclusion is equivalent to saying that, for every $0 < t < t_0$, we have that $Y_t = Z_t$.\n\n\\begin{claim}\nFor every $t \\geq 0$, \nif $x \\in Z_t \\setminus Y_t$, then $x$ is a local minimum for~$f$.\n\\end{claim}\nIf not, let  $x \\in Z_t \\setminus Y_t$ such that $x$ is not a local minimum for~$f$. Note that $f(x) = t$.\nLet $\\gamma: (0, \\varepsilon) \\to Y$ be a definable function such that $\\lim_{s \\to 0}\\gamma_s = x$ and $f(\\gamma_s) < f(x)$.\nThen, $\\gamma_s \\in Y_t$.\nSince $Y_t$ is closed, we have that $x \\in Y_t$, absurd.\n\nBy Lemma~\\ref{REM:MINIMA}, there exists $t_0 >0$ such that, for every \n$x \\in Y$, if $0 < f(x) < t_0$, then $x$ is not a local minimum for~$f$.\nThe claim implies the conclusion.\n\\end{proof}\n\n\\begin{remark}\nA few remarks about the above Lemma and its proof.\n\\begin{enumerate}\n\\item The function $f$ is lower semi-continuous, because $Z_t$ is closed for every~$t$.\n\\item The hypothesis that the $Y_t$ are closed is necessary.\n\\item Even if the function $f$ in the proof is continuous, we cannot conclude that $Y_t = Z_t$ always.\n\\item It is not true that there exists $t_0 > 0$ such that $f$ is continuous on~$Y_{t_0}$.\n\\end{enumerate}\n\\end{remark} \n\n\\begin{proof}[Proof of Theorem~\\ref{THM:DEFINABLE}]\nBy Corollary~\\ref{LEM:TAUT-DEFINABLE},\n\\[\n\\Hom^*(\\widetilde A; {\\ca F}) = \\varinjlim_{t \\to 0} \\Hom^*(\\widetilde{Y_t}; {\\ca F}).\n\\]\nSince $\\Pa{Y_t}_{t > 0}$ is a definable family, and $M$ expands a\nfield, the (definable) topological type of the $Y_t$ is eventually\nconstant (this is a consequence of the trivialization theorem holding\nin o-minimal expansions of fields, see \\cite{Dries98}). Hence we can\nassume w.l.o.g.\\xspace\\ that the $Y_t$ are all definably homeomorphic, and\ntherefore that their cohomology groups over $G$ are isomorphic to each\nother; let us denote by $P$ such group.  Thus, for every $t \\leq t'$,\nthe map induced by the inclusion\n\\[\n\\phi^{t'}_{t}: \\Hom^*(\\widetilde{Y}_{t'}; {\\ca F}) \\to \\Hom^*(\\widetilde{Y}_{t}; {\\ca F})\n\\]\nis an endomorphism of~$P$.\nIt suffices to prove that $\\phi^{t'}_{t}$ is an isomorphism for all sufficiently small $t'$ to prove the theorem.\nBy Lemma~\\ref{LEM:FAMILY}, w.l.o.g.\\xspace we can assume that, for every $t >0$,\n\\[\nY_t = \\bigcap_{t' >t} Y_{t'}.\n\\]\n\nFor convenience, define $\\phi^{t'}_t := \\phi^{t}_{t'}$  if $t' \\leq t$, and $I := M^{>0}$.\nLet $E$ be the equivalence relation on $I$ given by\n\\[\nt E t' \\Leftrightarrow \\phi^{t'}_{t} \\text{is an isomorphism}.\n\\]\n\\begin{claim}\n$E$~is a definable subset of $I^2$.\n\\end{claim}\nIn fact, by the trivialization theorem, there exists a finite\npartition $\\{ C_1, \\dotsc, C_n \\}$ of $\\{(t,t') \\in M^2: 0 < t \\leq\nt' \\}$ into definable sets, such that the definable topological type of\nthe inclusion map $Y_t \\to Y_{t'}$ is constant on each~$C_k$.\nHence, the isomorphism type of the map $\\phi^{t'}_t$ is constant on each\n$C_k$, and thus $E$ is a finite union of some of the~$C_k$.\n\n\\begin{claim}\\label{CL:OPEN}\nFor every $t \\in I$ there exists $t_0 >t$ such that, for every $t' \\in I$,\n\\[\nt \\leq t' < t_0 \\Rightarrow t E t'.\n\\]\n\\end{claim}\\noindent\nSince $Y_t = \\bigcap_{t' >t} Y_{t'}$, by Corollary\n\\ref{LEM:TAUT-DEFINABLE} we have $\\Hom^*(\\widetilde{Y_t}; {\\ca F}) =$%\n\\linebreak[2]%\n\\raisebox{0pt}[2ex][0pt]{$\\displaystyle{\\varinjlim_{t' \\to t^+}} \\Hom^*(\\widetilde{Y}_{t'}; {\\ca F})$};\nnamely,\n\\[\nP = \\varinjlim_{t' \\to t^+} \\set{P, \\phi^{t'}_{t}}.\n\\]\nBy definition of inductive limit, for every $p \\in P$ there exists $t'>t$ and $p' \\in P$ such that $p = \\phi^{t'}_{t}(p')$.\nHowever, $P$ is finitely generated, and therefore there exists $t_0 >t$ such that $\\phi^{t'}_t$ is surjective for every $t_0 > t' >t$.\nSince $P$ is finitely generated, using Nakayama's lemma, we can conclude that $\\phi^{t'}_t$ is an isomorphism.\n\n\\begin{claim}\\label{CL:E-FINITE}\n$I\/E$ is finite\n\\end{claim}\nClaim~\\ref{CL:OPEN} implies that each equivalence class of $E$ has\nnon-empty interior. Since $E$ is definable, $I\/E$ must be finite.\n\nIt follows from Claim~\\ref{CL:E-FINITE} that  there exists a left\nneighbourhood $J$ of $0$ such that, for every $t,t' \\in J$, $t E t'$.\nTherefore, for every $t \\leq t'$ in that neighbourhood $\\phi^{t'}_t$\nis an isomorphism, and we conclude by Remark~\\ref{lim}.\n\\end{proof}\n\n\\begin{remark}\nIn the situation of Theorem~\\ref{THM:DEFINABLE}, let\n\\begin{align*}\nf &: Y \\to M^{\\geq 0}\\\\\nf(x) &:= \\inf \\set{t: x \\in Y_t}.\n\\end{align*}\nWe have remarked that $f$ might not be continuous.\nAssume that there exists $t_0 >0$ such that $f$ is continuous on $Y_{t_0}$.\nThen, the proof of Theorem~\\ref{THM:DEFINABLE} can be simplified.\nIn fact, by the trivialization theorem, we can assume that there exists a continuous definable map $\\lambda: Y_{t_0} \\setminus A \\to F$ (where $F := f^{-1}(t_0)$), such that the map\n\\begin{align*}\n\\mu:= (f,\\lambda) &: Y_{t_0}\\setminus A \\to (0,t_0) \\times F\n\\end{align*}\nis a homeomorphism.\nLet $0 < t \\leq t' \\leq t_0$.\nLet $\\theta: [0,1] \\times [0,t'] \\to [0,t']$ be a definable \nstrong deformation retraction between $[0,t']$ and $[0,t]$.\nDefine\n\\[\\begin{array}{r@{\\,}ll}\n\\Lambda: [0,1] \\times Y_{t'} &\\to Y_{t'}\\\\\n\\Lambda\\Pa{s, \\mu^{-1}(u,x)} &= \\mu^{-1}\\Pa{\\theta(s,u),x} &\\text{if }  0 <u <t',\\\\\n\\Lambda(s,a) &= a &\\text{if } a \\in A.\n\\end{array}\\]\nNote that $\\Lambda$ gives a definable strong deformation retraction between\n$Y_t'$ and~$Y_t$.\nTherefore the inclusion $Y_t \\subseteq Y_{t'}$ induces an isomorphism in cohomology.\n\\end{remark}\nThe fact that in the situation of the above remark $Y_t$ is a deformation retract of $Y_{t'}$ follows from a stronger results in \\cite[Thm.~A.5]{PeterzilS07}, where however it is assumed that $0$ is a regular value for $f$.\n\n\\begin{question} Under the hypothesis of Theorem~\\ref{THM:DEFINABLE}, is there some\n$t_0 >0$ such that $A$ is a deformation retract of $Y_t$ for every $0\n< t < t_0$?  \\end{question} \n\n\nNote that in the proof of Theorem~\\ref{THM:DEFINABLE}\nwe have used heavily the trivialization theorem, which holds only for\nexpansions of fields.  \n\n\\begin{question} Can we weaken the hypothesis of\nTheorem~\\ref{THM:DEFINABLE} dropping the condition that $M$ expands a\nfield \\rom(but saying instead that it expands a group\\rom)?  \\end{question}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n   Resolved stellar populations and star formation histories (SFHs) provide a powerful tool to understand galaxy formation and evolution in all scales (e.g., Cignoni \\& Tosi \\cite{sl_cignoni10}). Detailed SFHs of Local Group (LG) dwarf galaxies have shown that these are complex systems (e.g., Mateo \\cite{sl_mateo98}; Tolstoy, Hill \\& Tosi \\cite{sl_tolstoy09}). The complexity of SFHs observed in LG dwarfs has been explored by means of numerical simulations and the results show that their diversity can be understood by invoking the action of ram pressure stripping, tidal stirring, and\/or cosmic ultraviolet (UV) background radiation upon gas-rich dwarf galaxies (e.g., Mayer \\cite{sl_mayer10}; Kazantzidis et al.~\\cite{sl_kazantzidis11}; Nichols \\& Bland-Hawthorn \\cite{sl_nichols11}; and references therein). From an observational point of view, the comparison of SFHs between dwarf galaxies in the LG and outside the LG shows that the former are representative of their kind (Weisz et al.~\\cite{sl_weisz11a}), so that a deeper study of individual SFHs is valuable to uncover and understand the intrinsic and environmental differences affecting their evolution (e.g., Weisz et al.~\\cite{sl_weisz11b}). \n\n   One such interesting galaxy to focus on is the Sculptor Dwarf Irregular Galaxy (SDIG). First discovered by Laustsen et al.~(\\cite{sl_laustsen77}), SDIG is a low-luminosity (Heisler et al.~\\cite{sl_heisler97}), gas-rich dwarf irregular (dIrr) galaxy (Cesarsky, Falgarone \\& Lequeux \\cite{sl_cesarsky77}; Lequeux \\& West \\cite{sl_lequeux81};  C\\^ot\\'e et al.~\\cite{sl_cote97}). Atomic hydrogen observations show that SDIG has a smooth distribution without the presence of structures such as shells, holes or bubbles (C\\^ot\\'e et al.~\\cite{sl_cote00}). SDIG contains no $\\ion {H}{ii}$ regions and this is suggestive of a quiescent phase with very low current star formation rate (SFR; Miller \\cite{sl_miller96}; Heisler et al.~\\cite{sl_heisler97}; Skillman, C\\^ot\\'e \\& Miller \\cite{sl_skillman03}). Bouchard et al.~(\\cite{sl_bouchard09}) detect two H$\\alpha$ point sources and estimate a very low current SFR (3.8$\\times$10$^{-5}$~M$_{\\sun}$ yr$^{-1}$); however, Lee et al.~(\\cite{sl_lee11}) detect far-ultraviolet (FUV) and near-ultraviolet (NUV) emission that appears to have a clumpy morphology and suggest main-sequence B stars as the source of this emission. \n\n  With a distance of 3.2~Mpc and based on its radial velocity, SDIG is a member of a loose galaxy triplet (Karachentsev et al.~\\cite{sl_karachentsev06}), of which the other two galaxy members are NGC\\,7793 at a distance of 3.61$\\pm$0.53~Mpc (Vlajic, Bland-Hawthorn \\& Freeman \\cite{sl_vlajic11}) and UGCA442 at a distance of 3.8$^{+0.8} _{-0.3}$~Mpc (Mould \\cite{sl_mould05}). This triplet, the NGC\\,7793 subgroup (Karachentsev et al.~\\cite{sl_karachentsev03}) lies at the far side of the Sculptor group. SDIG therefore lies near the limit of the ACS Nearby Galaxy Survey Treasury galaxy sample (ANGST; Dalcanton et al.~\\cite{sl_dalcanton09}), so a study of the SFH of this subgroup represents an opportunity to learn more about the Sculptor group, a low-density and extended filamentary structure with several embedded subgroups (e.g., Jerjen, Freeman \\& Binggeli \\cite{sl_jerjen98}). The filamentary structure of the Sculptor group extends from 1.5~Mpc to 4~Mpc, and points to an unvirialised state, with galaxies still falling in (Jerjen, Freeman \\& Binggeli \\cite{sl_jerjen98,sl_jerjen00}; Karachentsev \\cite{sl_karachentsev05}). The early evolutionary state and the low density of the Sculptor group provide an ideal environment in which its galaxy members may be considered as still evolving in isolation.\n\n   In the present work, we examine the resolved stellar populations and model the color-magnitude diagram (CMD) of SDIG using archival Hubble Space Telescope (HST) observations. The structure of this work is as follows. In \\S2 we present the observations and photometry, including photometric error analysis. In \\S3 we show the CMD and discuss the stellar content of SDIG, including an examination of the spatial distribution of stellar populations selected in several stellar evolutionary phases. In \\S4 we briefly review the CMD modelling technique. In \\S5 we show the results for the SFH of SDIG. We summarise and discuss our findings in \\S6.\n\n\\section{Observations and photometry}\n\n   We use archival observations taken with the Advanced Camera for Surveys (SNAP\\,9771; Karachentsev et al.~\\cite{sl_karachentsev06}) with a total exposure time of 900~sec in the F814W-band, and 1200~sec in the F606W-band. We retrieve the pre-reduced images through the ST-ECF Hubble Science Archive and perform stellar point source photometry using the ACS module of DOLPHOT, a modified version of the photometry package HSTphot (Dolphin \\cite{sl_dolphin00}). We apply point source photometry simultaneously on all the individual, calibrated and flat-fielded \"FLT\" images, and we use as a reference image the drizzled image in the F814W filter. We proceed with the photometry reduction steps as described in the ACS module of DOLPHOT. DOLPHOT provides magnitudes in both the ACS\\,\/\\,WFC and the Landolt UBVRI photometric systems and we choose to use the ACS\\,\/\\,WFC filter system for our study. The final photometric catalogue includes 11363~stars selected to have ${\\rm S\/N} >$~5, sharpness $|$sharp$_{\\rm F606W}+$sharp$_{\\rm F814W}|<$1, and crowding $($crowd$_{\\rm F606W}+$crowd$_{\\rm F814W})<$1 (e.g., Williams et al.~\\cite{sl_williams09}). \n\n   We conduct artificial star tests in order to quantify the photometric errors and completeness of our data, using the utilities provided within DOLPHOT. To that end, we run artificial star tests with 10$^5$~stars per ACS\\,\/\\,WFC chip, with range in F606W from 21~mag to 30~mag, and in (F606W-F814W) from $-$1.5~mag to 3.5~mag. In order to better quantify the photometric errors in the faintest magnitudes, which we need for the CMD modelling, we run complementary artificial star tests with an additional number of 2$\\times$10$^5$ stars per ACS\\,\/\\,WFC chip and covering magnitudes fainter than 25~mag. The high number of stars inserted during the artificial star tests is not an issue for self-induced crowding, since each star is inserted and photometered one at a time (Dolphin \\cite{sl_dolphin00}; see also Perina et al.~\\cite{sl_perina09}). \n\n \\begin{figure}\n  \\centering\n      \\includegraphics[width=8cm,clip]{20193fg1a.eps}\n      \\includegraphics[width=8cm,clip]{20193fg1b.eps}\n  \\caption{Photometric errors derived from artificial star tests and estimated as the difference between output and input magnitudes, as a function of the output magnitude in the F814W filter (upper panel) and in the F606W filter (lower panel).}\n  \\label{sl_figure1}%\n \\end{figure}\n  The photometric errors derived using the artificial star tests are shown in Fig.~\\ref{sl_figure1}. The mean photometric error and standard deviation is 0.03$\\pm$0.1~mag in F814W, and 0.02$\\pm$0.09~mag in F606W. \n \\begin{figure}\n  \\centering\n      \\includegraphics[width=8cm,clip]{20193fg2a.eps}\n      \\includegraphics[width=8cm,clip]{20193fg2b.eps}\n  \\caption{Completeness factors as a function of the F606W-band magnitude, upper panel, and F814W-band magnitude, lower panel. The red arrow in each panel indicates the 50\\% completeness factor.}\n  \\label{sl_figure2}%\n \\end{figure}\nIn Fig.~\\ref{sl_figure2} we show the completeness factors as a function of magnitude. The 50\\% completeness factor occurs at the magnitude F606W~$=\\sim$26.9~mag, and F814W~$=\\sim$25.9~mag. \n\n\\begin{table}[t]\n      \\caption[]{Properties of SDIG.}\n      \\label{table1} \n      \\begin{tabular}{lc}\n\\hline\\hline\n  Quantity                            & Value                   \\\\\n\\hline\n  Type                                & dIrr                     \\\\\n  RA~(J2000.0)                        & $00^h08^m13.36^s$         \\\\\n  Dec~(J2000.0)                       & $-34^\\circ 34' 42.0''$    \\\\\n  $\\rm E(B-V)$~(mag)                  & 0.012                   \\\\\n  $A_{\\rm F606W}, A_{\\rm F814W}$~(mag)   & 0.034, 0.022            \\\\\n  $F814W_{\\rm TRGB}$~(mag)             & 23.44$\\pm$0.05          \\\\\n  $(m-M)_0$~(mag)                     & 27.51$\\pm$0.06          \\\\\n  Distance~(Mpc)                      &3.2$\\pm$0.1              \\\\\n  M$_{\\rm V}$~(mag)                   &$-11.87$                 \\\\\n  M$_{\\rm {\\ion {H}{i}}}$~(M$_{\\sun}$)   &2.5$\\times$10$^7$        \\\\\n  x$_{\\rm C}$~(arcsec)                & 105.1$\\pm$1.7           \\\\\n  y$_{\\rm C}$~(arcsec)                & 114.5$\\pm$1.3           \\\\\n  $e$                                & $0.05$                  \\\\\n  PA~(degrees)                       & 0                       \\\\\n  $r_{\\rm eff}$~(arcsec)              & $29.4\\pm1.6$            \\\\\n\\hline\n\\end{tabular} \n\\end{table}\n Table~\\ref{table1} lists the basic properties for SDIG. Each row shows: (1) the galaxy type; (2) and (3) equatorial coordinates of the field center (J2000.0); (4) the reddening adopted from Schlegel et al.~(\\cite{sl_schlegel98}); (5) the galactic foreground extinction in the F606W and F814W filters, computed using the extinction ratios for a G2 star listed in Sirianni et al.~(\\cite{sl_sirianni05}; their Table~14); (6) the magnitude of the tip of the red giant branch (TRGB) we derive from the data; (7) the distance modulus we derive from the data; (8) the derived distance in Mpc; (9) the absolute V-band magnitude adopted from Georgiev et al.~(\\cite{sl_georgiev09}), who use the distance of Karachentsev et al.~\\cite{sl_karachentsev06}; (10) the {\\ion {H}{i}} mass adopted from Koribalski et al.~(\\cite{sl_koribalski04}); (11) and (12) the magnitude-weighted coordinates of the center, x$_{\\rm C}$ and y$_{\\rm C}$, respectively, which we compute from the data (see $\\S$3); (13) and (14) the ellipticity and position angle adopted from Kirby et al.~(\\cite{sl_kirby08}); (15) the effective radius adopted from Kirby et al.~(\\cite{sl_kirby08}).\n\n\\section{Color-magnitude diagram and stellar content}\n\n \\begin{figure}\n  \\centering\n      \\includegraphics[width=8cm,clip]{20193fg3.eps}\n  \\caption{Color-magnitude diagram of SDIG. The gray dashed line shows the 50\\% completeness factors. The error-bars correspond to the photometric errors derived using artificial star tests.}\n  \\label{sl_figure3}%\n \\end{figure}\n   The CMD of SDIG shown in Fig.~\\ref{sl_figure3} reveals the presence of several stellar evolutionary features. The most prominent feature is the red giant branch (RGB). The RGB indicates that stars with ages from $\\sim$1.5~Gyr and older are present (e.g, Salaris, Cassisi \\& Weiss \\cite{sl_salaris02}). Intermediate-age stars, with ages between 1~Gyr and 10~Gyr, are revealed in the presence of luminous  asymptotic giant branch (AGB) stars, while the young main-sequence (MS) and blue-loop (BL) stars indicate the presence of stars with ages younger than 1~Gyr. Therefore, SDIG contains stars spanning a wide range in age, from $\\sim$Myr to older ages. \n\n   In the following analyses and unless otherwise noted, we use the TRGB magnitude and distance modulus we measure in the ACS\\,\/\\,WFC filter system. We derive the distance modulus for SDIG using the absolute F814W-band magnitude of the TRGB computed through the calibration of Rizzi et al.~(\\cite{sl_rizzi07}): $M_{\\rm F814W} = -4.06 + 0.20\\times [({\\rm F606W-F814W}) - 1.23]$, and the F814W-band magnitude of the TRGB computed through the Sobel-filtering technique on the F814W-band luminosity function (e.g., Lee, Freedman \\& Madore \\cite{sl_lee93}; Sakai et al.~\\cite{sl_sakai96}). We find a TRGB F814W-band magnitude of 23.44$\\pm$0.05~mag, while measuring a dereddened color at the TRGB magnitude level of 1.05$\\pm$0.03~mag, we derive ${\\rm M}_{\\rm F814W}=-$4.09$\\pm$0.02~mag. The quoted uncertainties take into account the photometric errors from the artificial star tests and the width of the Sobel filter response. These values along with the extinction listed in Table~\\ref{table1} give a distance modulus of 27.51$\\pm$0.06~mag, which translates to a distance of 3.2$\\pm$0.1~Mpc. The quoted uncertainties take into account the width of the Sobel filter response for the TRGB magnitude detection, uncertainties related to the TRGB calibration, and the photometric errors. Our derived distance modulus and distance are consistent with the values of 27.53~mag and 3.21~Mpc, respectively, found in Karachentsev et al.~(\\cite{sl_karachentsev06}), who use the same method on the transformed V and I filters and assign an uncertainty to their distance of 8\\%. We list the derived F814W-band magnitude of the TRGB, distance modulus, and distance in Table~\\ref{table1}.\n\n \\begin{figure*}\n  \\centering\n     \\includegraphics[width=6cm,clip]{20193fg4a.eps}\n     \\includegraphics[width=6cm,clip]{20193fg4b.eps}\n     \\includegraphics[width=6cm,clip]{20193fg4c.eps}\n  \\caption{Color-magnitude diagrams within three elliptical radius annuli. The first annulus corresponds to an elliptical radius equal to 1~$r_{\\rm eff}$, the second between 1 and 2~$r_{\\rm eff}$, and the third beyond 2~$r_{\\rm eff}$, shown from left to right, respectively. In the left panel, we overplot Padova isochrones with ages of 20~Myr, 40~Myr, 60~Myr, 80~Myr, and 150~Myr, and for all isochrones a Z metallicity of 0.001. Also shown in the left panel with the error bars the photometric errors. In the right panel, we show with open circles the galactic foreground contamination.}\n  \\label{sl_figure4}%\n \\end{figure*}\n  Fig.~\\ref{sl_figure4} shows the CMD of stars distributed within three elliptical radius annuli: within 1 effective radius, $r_{\\rm eff}$; between 1~$r_{\\rm eff}$ and 2~$r_{\\rm eff}$; and beyond 2~$r_{\\rm eff}$. The elliptical radius is defined as: $r = [{\\rm x}^{2}+{\\rm y}^{2}\/(1 - e)^{2}]^{1\/2}$, where x and y are the distances along the major and minor axis, respectively, and $e$ is the ellipticity. We adopt the ellipticity, position angle, and effective radius from Kirby et al.~(\\cite{sl_kirby08}), listed in Table~\\ref{table1}. Also listed in Table~\\ref{table1} are the x$_{\\rm C}$ and y$_{\\rm C}$ coordinates of the magnitude-weighted center of SDIG, which we compute from our data using the equation: ${\\rm x_{C}}=\\Sigma({\\rm mag}_{i}\\times {\\rm x}_{i})\/\\Sigma {\\rm mag}_{i}$, where ${\\rm mag}_{i}$ corresponds to the F814W-band magnitude of each star, and ${\\rm x}_{i}$ to their x (or y for the  y$_{\\rm C}$) coordinates, in pixels. We use stars with magnitudes brighter than the magnitude with 50\\% completeness factors. In the right panel of Fig.~\\ref{sl_figure4} we show with the open circles the Galactic foreground contamination, which was estimated using the TRILEGAL code (Girardi et al.~\\cite{sl_girardi05}; Vanhollebeke, Groenewegen \\& Girardi \\cite{sl_vanhollebeke09}). We expect a total of 58 foreground stars in the same color and magnitude location of the SDIG stars, which translates to less than 0.5\\% of the total number of SDIG stars. In the left panel of Fig.~\\ref{sl_figure4} we overplot Padova isochrones (Marigo et al.~\\cite{sl_marigo08}; Girardi et al.~\\cite{sl_girardi10}), with ages between 20~Myr and 150~Myr. Fig.~\\ref{sl_figure4} indicates that the bulk of the young MS stars are confined within the central parts of SDIG. \n\n  We examine the cumulative distribution functions of the MS, BL, luminous AGB, and RGB stars, versus the elliptical radius, r, defined in the preceding paragraph. We select those MS stars with their color ranging from $-$0.5~mag to $-$0.1~mag and their F814W-band magnitude ranging from 23.8~mag to 25.25~mag; the BL stars are selected with their color ranging from $-$0.05~mag to 0.35~mag and with their F814W-band magnitude ranging from 21~mag to 25~mag. The RGB stars are selected between the magnitude of the TRGB and 1~mag below, while the color selection ranges such as to follow the slope of the RGB. The luminous AGB stars are selected to be brighter by 0.15~mag than the TRGB (Armandroff et al.~\\cite{sl_armandroff93}) and to lie within 1~mag above (${\\rm I_{TRGB}}-0.15$)~mag, and within the color range of 1~$<({\\rm V-I})_0<$~3.5~(mag). We count a number of 80 luminous AGB stars, which translates into a fraction, i.e. number of luminous AGB stars versus the number of RGB stars within 1~mag below the TRGB magnitude,  N(AGB)~\/~N(RGB)=13\\%, comparable to that of Sculptor group dwarf galaxies (Lianou et al.~\\cite{sl_lianou12}).\n \\begin{figure}\n  \\centering\n       \\includegraphics[width=6cm,clip]{20193fg5.eps}\n       \\caption{Cumulative distribution functions for the selected stellar populations. The vertical line shows the effective radius, r$_{eff}$.} \n  \\label{sl_figure5}%\n \\end{figure}\nThe cumulative distribution functions in Fig.~\\ref{sl_figure5} show that the MS and BL stars are confined to the central parts of SDIG, while the luminous AGB and the RGB stars are more spatially extended. Performing a two-sided Kolmogorov-Smirnov (K-S) test between each of the cumulative distributions for the MS, BL, and luminous AGB stars with the cumulative distribution of the RGB stars, we derive results consistent with spatially separated populations at the 99\\% confidence level in all cases.\n\n \\begin{figure}\n  \\centering\n       \\includegraphics[width=4.4cm,clip]{20193fg6a.eps}\n       \\includegraphics[width=4.4cm,clip]{20193fg6b.eps}\n       \\includegraphics[width=4.4cm,clip]{20193fg6c.eps}\n       \\includegraphics[width=4.4cm,clip]{20193fg6d.eps}\n  \\caption{Gaussian-smoothed stellar density maps for stars selected in the MS, BL, luminous AGB, and RGB phases, expressed in terms of angular offsets from the dwarf's center. The density maps are color coded such that black corresponds to a density of 0.16 stars per sq.~arcsec and the density declines toward lighter grey. The ellipse has a major axis equal to 1~$r_{\\rm eff}$, while the crosses emphasise the location of the center of the dwarf. The red arrows in the panel of the RGB population indicates the directions of North and East, and is the same for all panels.}\n  \\label{sl_figure6}%\n \\end{figure}\n  Comparing the spatial distribution of the luminous AGB stars with that of the RGB stars, in the stellar density maps shown in Fig.~\\ref{sl_figure6}, the former are more confined in the inner parts of SDIG. The selected BL stars are confined to the central parts of the dwarf when we compare their spatial distribution with the one of the luminous AGB and RGB stars. The stellar density map of the young MS stars shows a population concentrated towards the central parts of SDIG, while in the same time its spatial location is not coincident with the very center of SDIG, having an offset of 8.2~arcsecs, or 130~pc at a distance of 3.2~Mpc. We note that a similar characteristic of off-center local sites of young star formation characterise the majority of the transition-type dwarfs in the Sculptor group (Jerjen \\& Rejkuba \\cite{sl_jerjen01}; Lianou et al.~\\cite{sl_lianou12}), while the same is also observed for the young stellar populations of other LG dwarf galaxies (e.g, Dohm-Palmer et al.~\\cite{sl_dohm-palmer97}; Gallagher et al.~\\cite{sl_gallagher98}; Mateo \\cite{sl_mateo98}; Cole et al.~\\cite{sl_cole99}; McConnachie et al.~\\cite{sl_mcconnachie06}).\n\n\\section{Color-magnitude diagram modelling}\n\n  The basic idea using the modelling of the CMD to uncover the evolutionary history of a galaxy is that its composite stellar populations can be regarded as a linear combination of simple stellar populations of different ages and metallicities. Thus, the SFH can be obtained by comparing the observed features in the CMD with those in a model CMD. There are several features in a CMD that can be used as age and metallicity indicators (e.g., Gallart, Zoccali \\& Aparicio \\cite{sl_gallart05}). Synthetic populations are constructed using a set of stellar evolution models, adopting an initial mass function (IMF) and applying a star formation and a chemical enrichment law (e.g., Gallart, Zoccali \\& Aparicio \\cite{sl_gallart05}; Cignoni \\& Tosi \\cite{sl_cignoni10}). In practice, the density of stars within a defined color and magnitude grid along the observed CMD is compared with the density of the synthetic stars drawn from the model CMD (for a review see Tolstoy, Hill \\& Tosi \\cite{sl_tolstoy09}; Aparicio \\cite{sl_aparicio02}), and the best model is adopted as the SFH.  \n\n  The approach taken here was developed by Cole to study the stellar populations of dIrr galaxies in the LG (see, e.g., Skillman et al.~\\cite{sl_skillman03b}; Cole et al.~\\cite{sl_cole07}; Cignoni et al.~\\cite{sl_cignoni12}). As with any such approach, the power of the method is greatly reduced by lack of photometric depth (e.g., Weisz et al.~\\cite{sl_weisz11b}). Detailed inferences about the SFH will be restricted to age ranges for which the age-sensitive MS, BL, and to a lesser extent AGB stars are visible. It is still possible to place constraints on long-term average SFRs and mean metallicities based on inferences from the RGB and the regions of the CMD that are badly affected by incompleteness. However, these inferences are by necessity more model-dependent and of lower precision, and the resulting uncertainties are dominated by systematic effects.\n\n  The SFH-fitting starts with theoretical isochrones interpolated to a fine grid of age and metallicity in order to create a synthetic CMD with no gaps. The most recent models from the Padova models are used. The synthetic CMDs are binned in age and metallicity (Z) to avoid ``overfitting'' noise in the CMDs. The bins are initially spaced by a default value of $\\Delta$log(age) = 0.10, but adjacent bins are merged when lack of information in the CMDs leads to instability in the solutions. In this case, ages greater than 1~Gyr are only represented on the RGB and AGB, and the age resolution is decreased to $\\Delta$log(age) = 0.30 (factor of two). There are not many stars younger than 30~Myr in the CMD, so the youngest age bin spans the entire range from 4--30~Myr as a way to populate the CMD bins sufficiently to draw SFH inferences.  \n\n  The CMD is divided into a regular grid of color-magnitude cells, and the expectation value of the number of stars in each cell for a SFR of 1 M$_{\\odot}$ yr$^{-1}$ is calculated from the isochrones.  Several parameters are assumed to be fixed during the solution: the distance modulus and reddening, IMF, fraction of binaries, and the binary mass ratio distribution function. The adopted IMF is from Chabrier (\\cite{sl_chabrier03}), and the binary fraction and mass ratios are parameterized based on Duquennoy \\& Mayor (\\cite{sl_duquennoy91}) and Mazeh et al.~(\\cite{sl_mazeh92}). We take 35\\% of stars to be single and the rest to be binary. The binaries are divided into ``wide'' and ``close'' binaries in a 3:1 ratio; the secondaries in the wide systems are drawn from the same IMF as the primaries, but in the close systems the secondary masses are drawn from a flat IMF. The distance and reddening are initially constrained to the values given in Table~1, but are varied if the resulting synthetic CMDs are mismatched to the data.\n\n  No age-metallicity relation is explicitly assumed, but a range of metallicities at each age is allowed, constrained by the color range of the data. Because of the shallow photometric depth of the data, there are few strong constraints on the metallicity, and those that exist are model-dependent. The RGB is affected by age-metallicity degeneracy, but indicates a range of metallicities between 0.0004 $\\leq$ Z $\\leq$ 0.002. These numbers are consistent with the Padova AGB star tracks, and the color separation between the BL stars and MS. There is no metallicity constraint from $\\ion {H}{ii}$ regions, but the inferred values are consistent with the metallicity-luminosity relationship for dIrr galaxies in Lee, Zucker \\& Grebel (\\cite{sl_lee07}) (see discussion below).\n\n  The IMF-weighted, color-magnitude binned isochrones are convolved with color and magnitude error distributions derived from the artificial star tests in order to create synthetic CMDs with the same properties as the data. Linear combinations of the synthetic CMDs are created by the fitting routine, which uses a maximum likelihood test based on the Poisson distribution of counts in each bin (Cash \\cite{sl_cash79}) to find the combination that is most likely to produce the observed CMD. We use a simulated annealing technique to find the best fit while avoiding false, local maxima in likelihood space. The SFH errorbars are calculated by successive perturbation of each age-metallicity bin; because the total number of stars is fixed by the observations, a decrease in SFR in one bin results in an increase in adjacent bins; this gives anticorrelations between adjacent bins. In general, the code is driven by the most populous cells of the CMD to its preferred solution; because most stars are observed on the MS and low-mass stars vastly outnumber those of higher mass, the most significant cells in determining the SFH are frequently below the 50\\% completeness threshold, and the solution depends sensitively on the artificial star test results.\n\n  Our data are not deep enough to reach the oldest MS turn-offs, and due to the age-metallicity degeneracy on the RGB, the solution of the CMD modelling for ages higher than $\\sim$1.5~Gyr bears larger uncertainties. \n \\begin{figure*}\n  \\centering\n      \\includegraphics[width=17cm, clip]{20193fg7.eps}\n  \\caption{Left panel: logarithmically-scaled, binned CMD used in the maximum-likelihood fit for SFH; right panel: best-fit model. Without differential reddening, there is clear separation between the MS and BL stars, which are mingled together in the data. The AGB stands out as being poorly-fit; the mismatch occurs because of the large number of stars fainter than m814 $\\approx$26, which are part of the same stellar population as the AGB stars.}\n  \\label{sl_figure7}%\n \\end{figure*}\nThe observed and modelled CMDs for SDIG are shown in Fig.~\\ref{sl_figure7}. From the CMD modelling, we derive a distance modulus of 27.54~mag, while the reddening required is E(B-V)=0.04. The higher reddening, as compared to the value from Schlegel et al.~(\\cite{sl_schlegel98}), is consistent with the one that Heisler et al.~(\\cite{sl_heisler97}) derive for the very central parts of SDIG.\n\n  The best-fit solution under the restrictions of the fit procedure matches the mean stellar density and colors of the major stellar sequences well, but there are some notable differences between the simulated CMD and the data. The MS and BL in the data are blurred, with no clear color gap between them; however, in the models these features are well-separated. Presumably this is due to differential reddening, with a tendency for young stars to be associated with dust clouds and circumstellar material. \n\n \\section{Star formation history}\n\n \\begin{figure}\n  \\centering\n      \\includegraphics[width=8cm,clip]{20193fg8a.eps}\n      \\includegraphics[width=8cm,clip]{20193fg8b.eps}\n  \\caption{Star formation rate as a function of age for SDIG based on the maximum-likelihood fit, shown in the upper panel with a linear scale on age, and in the lower panel with a logarithmic scale with age. Note in the upper panel the break in age scale at 750~Myr that reflects the decreasing time resolution of the CMD with increasing age.}\n  \\label{sl_figure8}%\n \\end{figure}\n  We show the best-fit SFH for SDIG in Fig.~\\ref{sl_figure8}. Overall, the best-fit SFH is characterised by the following features:\n\n\\begin{itemize}\n\n\\item An average age for formation of stars of $\\langle \\tau \\rangle$ = 6.4$^{+1.6}_{-1.4}$~Gyr and a total astrated mass (time-integrated SFR) of 17.7$\\times$10$^6$M$_{\\sun}$.\n\n\\item The average metallicity is inferred to be Z $\\approx$0.0006 ([M\/H] $\\approx$ $-$1.5~dex), with some tendency to increase with time, as the BL stars are more consistent with a metallicity closer to Z = 0.001.\n\n\\item A long-term SFH that is consistent with a constant SFR over several Gyr. The average SFR is 1.3$^{+0.4}_{-0.3}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$. Any bursts or fallow periods are of short enough duration and\/or amplitude that they do not strongly perturb the mean SFR averaged over factor of two periods in age. This is broadly consistent with the typical dIrr galaxy found in Weisz et al.~(\\cite{sl_weisz11b}); however the HST SDIG observations analysed here were not deep enough to reveal factor of 2--3 variation in SFR over sub-Gyr intervals at intermediate or old ages, and these are not excluded by the CMD analysis.\n\n\\item The age resolution improves dramatically for ages less than $\\approx$800--1000~Myr, but strong variations over intervals of a few hundred Myr are not observed, with two exceptions:\n\n\\begin{enumerate}\n\\item Over the past 100~Myr the mean SFR increases by a factor of $\\approx$2 over its long-term average value, but then declines slightly. Variations of this duration and amplitude would be completely hidden by photometric incompleteness and age-metallicity degeneracy for ages older than $\\approx$1~Gyr; this behaviour may be typical of the history of low-luminosity dIrrs (e.g., Cole \\cite{sl_cole10}). Because the fraction of evolved stars is low, the derived SFR is susceptible to uncertainties in the modelling of incompleteness and possible variations in the IMF. Stars in this age range would contribute to the SFR as measured from FUV data, but not to the H$\\alpha$ luminosity owing to the low absolute SFR with a value of $\\sim$2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ from 0--10~Myr.\n\n\\item A similar-amplitude SFR enhancement is inferred to have taken place from 600--1100~Myr ago. The younger stars in this interval are directly observable in the CMD as BL stars at F606W-band magnitude $\\approx$ 26.5--27~mag on the blue side of the RGB. The older stars are confused with the RGB and highly subject to incompleteness, resulting in the large uncertainties in the SFR at this age range. Because of the high incompleteness, the best-fit SFR has high uncertainty (reflected in the model overproduction of AGB stars associated with this age range). Based on this data alone the balance of evidence favours the existence of a $\\lesssim$500-Myr long episode of increased SFR $\\approx$1~Gyr ago, but is consistent with a flat SFH given the small number of stars and the statistical nature of the problem. As above, this type of event would be undetectable at older ages based on this data set alone. \n\\end{enumerate}\n\\end{itemize}\n\n  Integrating the SFR over the entire history gives a total astrated stellar mass M$_{\\star}$=~1.77$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$. For ages less than 1~Gyr, we estimate a stellar mass M$_{\\star, <1 {\\rm Gyr}} =$~1.68$^{+0.69}_{-0.75}$~$\\times$10$^{6}$~M$_{\\sun}$, which consists a fraction of $\\sim$10\\% of the total stellar mass. Because the maximum mass of surviving stars decreases with age, only $\\approx$85\\% of the total astrated mass is still present in the form of stars, i.e., the current stellar mass M$_{\\star, {\\rm current}} =$~1.50$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$. Kirby et al.~(\\cite{sl_kirby08}) derive a total stellar mass equal to  1.6$\\times$10$^{7}$~M$_{\\sun}$, using deep near-infrared observations that trace the old stellar populations and assuming an H-band stellar mass-to-light ratio of 1. The agreement between our current stellar mass and the total stellar mass of Kirby et al.~(\\cite{sl_kirby08}) is very good. \n\n  The current stellar mass is comparable to the {\\ion {H}{i}} mass in SDIG (see Table~\\ref{table1}; Koribalski et al.~\\cite{sl_koribalski04}) with a ratio of M$_{\\rm {\\ion {H}{i}}}$~\/~M$_{\\star} =$~0.6. We compute the baryonic gas fraction, M$_{\\rm gas}$~\/~M$_{\\rm baryonic}$, and compare it with the baryonic gas fraction of the ANGST sample galaxies (Weisz et al.~\\cite{sl_weisz11b}; their Fig.~9). The baryonic gas fraction of SDIG is estimated equal to M$_{\\rm gas}$~\/~M$_{\\rm baryonic} =$0.7, assuming a gas mass M$_{\\rm gas}=$1.4$\\times$M$_{\\rm {\\ion {H}{i}}}$ and a baryonic mass M$_{\\rm baryonic}=$M$_{\\rm gas} +$M$_{\\star}$. These values are listed in Table~\\ref{table2}. Placing SDIG in Fig.~9 of Weisz et al.~(\\cite{sl_weisz11b}) locates it in the upper left region occupied by dIrrs, with SDIG slightly above ESO321-G014. \n\n  Based on our current stellar mass estimate and the V-band absolute magnitude listed in Table~\\ref{table1}, we compute a V-band stellar mass-to-light ratio equal to M$_{\\star}$~\/~L$_{\\rm V} =$ 3.2~M$_{\\sun}$~\/~L$_{\\sun}$. This is higher than transition-type LG galaxies such as LGS3 and DDO\\,216 (Hunter, Elmegreen \\& Ludka \\cite{sl_hunter10}). Using the B-band magnitude listed in Karachentsev et al.~(\\cite{sl_karachentsev04}; column (6) in their Table~1), we compute a B-band stellar mass-to-light ratio equal to M$_{\\star}$~\/~L$_{\\rm B} =$1.4~M$_{\\sun}$~\/~L$_{\\sun}$. The B-band stellar mass-to-light ratio of SDIG is lower than the average B-band stellar mass-to-light ratio (equal to 2.4~M$_{\\sun}$~\/~L$_{\\sun}$) of the sample of star-burst dwarf galaxies listed in McQuinn et al.~(\\cite{sl_mcquinn10}), but is higher than the average value of $\\sim$1.1~M$_{\\sun}$~\/~L$_{\\sun}$ for dIrrs with $-$11~mag $\\leq$ M$_B$ $\\leq$ $-$13~mag (Weisz et al.~\\cite{sl_weisz11b}). \n\n  SDIG is a late-type dwarf galaxy and the dominant population has intermediate ages, with an average age of 6.4$^{+1.6}_{-1.4}$~Gyr. Thus, the age-metallicity degeneracy on the RGB makes it difficult to derive individual stellar metallicities. On the other hand, the {\\em mean} stellar metallicity of mixed stellar populations remains rather robust against the age-metallicity degeneracy on the RGB (Lianou et al.~\\cite{sl_lianou11}). Therefore, we use the average age of the stellar populations in SDIG \n \\begin{figure}\n  \\centering\n     \\includegraphics[width=7.5cm,clip]{20193fg9.eps}\n  \\caption{CMD focused on the RGB stars. The grey circles indicate the RGB stars used to compute the fiducial line, with the latter shown with the open black circles. The solid red line shows the Dartmouth 6.4~Gyr, solar-scaled isochrone with metallicity $-$1.6~dex in [M\/H] that best matches the RGB fiducial, while the black dotted line shows the Padova 6.4~Gyr isochrone with metallicity $-$1.5~dex in [M\/H]. Towards fainter magnitudes, the two isochrones overlap. }\n  \\label{sl_figure9}%\n \\end{figure}\nin order to constrain its {\\em mean} stellar metallicity. To that end, we use the fiducial of RGB stars, with magnitudes ranging from the TRGB to 1~mag fainter and shown in Fig.~\\ref{sl_figure9} with the open circles, to which we overplot the best-matched isochrone from the Dartmouth (Dotter et al.~\\cite{sl_dotter07,sl_dotter08}) and Padova Stellar Evolution Databases, with an age equal to the average age of the stellar population in SDIG. Here, we use the colour excess and distance modulus we derive from the SFH modelling (see $\\S$5). The best-matched isochrones are shown in Fig.~\\ref{sl_figure9}, with the red solid line and black dotted line, respectively, for the Dartmouth and Padova isochrones. The metallicity of the best-matched Dartmouth isochrone is $-$1.6~dex, while the metallicity of the best-matched Padova isochrone is $-$1.5~dex. The metallicity of the Padova isochrone that best matches the RGB fiducial is consistent with the average metallicity we derive from the SFH-modelling. The inferred stellar metallicity translates to an [O\/H] value of $-$1.13~dex (Mateo \\cite{sl_mateo98}), or to a 12+${\\rm log}$[O\/H] value of 7.78, therefore is consistent with the metallicity-luminosity relation in Lee et al.~(\\cite{sl_lee07}). Moreover, the stellar metallicity of SDIG is consistent with the stellar metallicity-luminosity relation in Cloet-Osselaer et al.~(\\cite{sl_cloet-osselaer12}). \n\n\\begin{table}[t]\n      \\caption[]{Derived properties of SDIG.}\n      \\label{table2} \n      \\begin{tabular}{lc}\n\\hline\\hline\n Quantity                           & Value                                                       \\\\\n\\hline\n$\\langle {\\rm SFR} \\rangle$         & 1.3$^{+0.4}_{-0.3}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$     \\\\\nSFR$_{\\rm current}$                   & 2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$     \\\\\nSFR$_{\\rm recent}$                    & 2.7$\\pm$0.5$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$             \\\\\n$\\langle {\\rm \\tau} \\rangle$        & 6.4$^{+1.6}_{-1.4}$~Gyr                                        \\\\ \nM$_{\\star}$                          & 1.77$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$              \\\\\nM$_{\\star, <1 {\\rm Gyr}}$              & 1.68$^{+0.69}_{-0.75}$~$\\times$10$^{6}$~M$_{\\sun}$              \\\\\nM$_{\\star, {\\rm current}}$             & 1.50$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$              \\\\\nM$_{\\rm gas}$                         & 3.5$\\times$10$^{7}$~M$_{\\sun}$                                \\\\\nM$_{\\rm baryonic}$                     & 5$\\times$10$^{7}$~M$_{\\sun}$                                  \\\\\nM$_{\\rm gas}$~\/~M$_{\\rm baryonic}$      & 0.7                                                          \\\\\nM$_{\\star}$~\/~L$_{\\rm V}$              & 3.2~M$_{\\sun}$~\/~L$_{\\sun}$                                    \\\\\nM$_{\\star}$~\/~L$_{\\rm B}$              & 1.4~M$_{\\sun}$~\/~L$_{\\sun}$                                    \\\\\n${\\rm [M\/H]}_{\\rm SFH}$               & $-$1.5~dex                                                    \\\\\n${\\rm [M\/H]}_{\\rm Padova}$             & $-$1.5~dex                                                    \\\\\n${\\rm [M\/H]}_{\\rm Dartmouth}$          & $-$1.6~dex                                                    \\\\\n\\hline\n\\end{tabular} \n\\end{table}\nThe derived properties of SDIG are listed in Table~\\ref{table2}. Each row lists the following information: (1) the average SFR; (2) and (3) the current and recent SFRs, respectively (see below); (4) the average age of the stellar populations; (5) the astrated stellar mass; (6) the stellar mass for stars younger than 1~Gyr; (7) the current stellar mass, M$_{\\star, {\\rm current}}$; (8) the gas mass, computed as 1.4$\\times$M$_{\\rm {\\ion {H}{i}}}$; (9) the baryonic mass, computed as M$_{\\rm gas}+$M$_{\\star, {\\rm current}}$; (10) the baryonic gas fraction, computed as M$_{\\rm gas}\/$M$_{\\rm baryonic}$; (11) the current V-band stellar mass-to-light ratio; (12) the current B-band stellar mass-to-light ratio; (13) the average metallicity based on the SFH results; (14) the stellar metallicity based on the RGB fiducial and Padova isochrones; (15) the stellar metallicity based on the RGB fiducial and Dartmouth isochrones. \n\nWe can compare the SFR in our youngest age bin to current SFR indicators like H$\\alpha$ and FUV emission.\nThe current SFR is estimated from our data assuming it is equal to the average SFR in our first age bin (4--30~Myr), 2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$. H$\\alpha$ emission is sensitive to SFR over the past 10~Myr but can be biased to lower values than the true current SFR when the SFR is less than $\\approx$3$\\times$10$^{-3}$~M$_{\\odot}$~yr$^{-1}$ (e.g., Lee et al.~\\cite{sl_lee09}) because it relies on ionising photons from O stars, which may not be fully sampled by the IMF in cases of low SFR.  We detect of order a dozen stars bright enough and blue enough to be candidate OV stars, so stochastic effects are expected to be important. Despite this caveat, the current SFR we derive from the CMD modelling is consistent with the upper limit of 2$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ that Heisler et al.~(\\cite{sl_heisler97}) derive based on their 3~$\\sigma$ detection of the H$\\alpha$ surface brightness across the continuum-subtracted image of SDIG.  However, Bouchard et al.~(\\cite{sl_bouchard09}) derive a lower current SFR of 3.8$\\times$10$^{-5}$~M$_{\\sun}$~yr$^{-1}$ based on the H$\\alpha$ flux in two detected knots. Their SFR is just 3\\% of the value we find. This could be taken as evidence for a quiescent period of low SFR, or for random variance due to the low numbers of massive stars formed. \n\nWe can estimate the number of O-type stars expected to be produced by the recent SFR using the stellar population synthesis code Starburst99 (Leitherer et al.~\\cite{sl_leitherer99}). As a rough estimate, we assume continuous star formation at a constant rate equal to the recent SFR (see Table~\\ref{table2}), a Salpeter IMF, and solar metallicity. The results provide us with the total number of ionising photons emitted during the most recent time step of 2~Myr, and over a longer period of 100~Myr. Using the ionising flux Q$_0$ for an O5V star (Q$_0$ = 10$^{49.22}$ s$^{-1}$, M $\\approx$37~M$_{\\sun}$; Martins, Schaerer \\& Hillier~\\cite{sl_martins05}), we estimate that of order 18 such stars would have been produced over 100~Myr given our CMD-derived SFR.  Of these, only a small fraction would be expected to be observed today owing to their short lifetimes. As an additional way to estimate the number of massive stars present, we employ a Monte Carlo approach to estimate the variance in massive star numbers given an overall low SFR. To that end, we assume a Salpeter IMF (Salpeter~\\cite{sl_salpeter}) with lower and upper mass limits of 0.5 to 150~M$_{\\sun}$, and draw from this distribution until 2.7$\\times$10$^{5}$~M$_{\\sun}$ total mass of stars are formed; this number corresponds to our recent SFR integrated over 10$^8$ yr. The process is repeated 100 times, allowing us to calculate that on average 1216$\\pm$40 stars with masses appropriate to O stars (here assumed to be 18--100~M$_{\\sun}$) are formed over this time period. At any given moment we would expect to see $\\approx$120 O stars if the IMF was being fully sampled, assuming an average lifetime of 10~Myr.  The fact that we observe about an order of magnitude less confirms that SDIG is in a ``down'' state of star formation lasting at least the duration of a typical O star lifetime.  However, given the SFR determined over our youngest age bin (4--30~Myr) is higher than the lifetime average for the galaxy and the high gas content of SDIG, it seems more likely that this represents a combination of short-term SFR fluctuation and stochastic effects due to IMF sampling than to any physically meaningful suppression of star formation on longer timescales.\n\nBecause H$\\alpha$ is known to be problematic in cases of very low SFR, the FUV emission provides a better estimator of the recent SFR, sampling stars down to a few solar masses, and ages of order 100~Myr (Lee et al.~\\cite{sl_lee11}). We use the FUV asymptotic magnitude from Table~2 of Lee et al.~\\cite{sl_lee11} in conjunction with the FUV-SFR relation from Lee et al.~(\\cite{sl_lee09}; their $\\S$3) to estimate a recent SFR of 9.6$\\times$10$^{-4}$~M$_{\\sun}$~yr$^{-1}$. For comparison, the SFR we derive from the CMD is 2.7$\\pm$0.5~$\\times$10$^{-3}$~M$_{\\sun}$~yr$^{-1}$. We therefore find improved agreement between the recent SFR indicator and the CMD modelling, but a factor of 2--3 discrepancy remains, presumably due to the fact that a galaxy that is deficient in O stars will also be deficient in FUV flux.\n\n\\section{Summary and discussion}\n\n  SDIG is a gas-rich, low-luminosity dIrr (C\\^ot\\'e et al.~\\cite{sl_cote97}; Heisler et al.~{\\cite{sl_heisler97}}). It is dominated by intermediate-age stars with an average age ${\\rm \\tau}=$6.4$^{+1.6}_{-1.4}$~Gyr, similar to the mean mass-weighted age of LG dIrrs (e.g., Orban et al.~\\cite{sl_orban08}). The average SFR is similar to other Sculptor group dwarfs, listed in Weisz et al.~(\\cite{sl_weisz11b}; their Table~2), with ESO540-G030 and ESO540-G032 having the lowest and highest average SFRs, respectively. The non-detection of $\\ion {H}{ii}$ regions and low H$\\alpha$-based SFR have been regarded as indicative of SDIG experiencing a currently quiescent phase (Miller \\cite{sl_miller96}; Heisler et al.~\\cite{sl_heisler97}; Skillman et al.~\\cite{sl_skillman03}; Bouchard et al.~\\cite{sl_bouchard09}). However, our SFH modelling shows elevated star formation for ages younger than 100~Myr, consistent with the star formation levels detected in the FUV emission. Our current data do not allow us to resolve star formation events lasting $\\sim$10$^8$~Myr for ages older than $\\sim$1~Gyr.  There are intriguing hints of a strong previous enhancement of SFR at ages of 600--1100~Myr, although the amplitude and timing of this event are model-dependent. Similar patterns of globally decreasing SFR during the past 1--2~Gyr have been observed in a number of LG galaxies with similar luminosity to SDIG, e.g. Leo~A (Cole et al.~\\cite{sl_cole07}).\n\n  SDIG is a member of the NGC\\,7793 subgroup located at the far-side of the Sculptor group (Jerjen et al.~\\cite{sl_jerjen98}; Karachentsev et al.~\\cite{sl_karachentsev06}), and NGC\\,7793 is identified as its main disturber, i.e. the neighbouring galaxy producing the maximum tidal action (Karachentsev et al.~\\cite{sl_karachentsev04}). NGC\\,7793 is a spiral galaxy and stellar population analyses of several halo and disk fields detect a metallicity gradient (Vlajic et al.~\\cite{sl_vlajic11}), as well as indications for substantial stellar radial migration (Radburn-Smith et al.~\\cite{sl_radburn12}). The SFH of a halo field in NGC\\,7793 indicates that a burst of star formation has occurred within the last 20~Myr and with an amplitude three times that of the average SFR, most likely associated with the spiral arm feature contaminating the halo field (Dalcanton et al.~\\cite{sl_dalcanton12}; Figs.~1, 2, and 9 for NGC\\,7793). The deprojected distance between SDIG and NGC\\,7793 is 450$^{+340}_{-270}$~kpc, estimated using the method in Karachentsev et al.~(\\cite{sl_karachentsev04}), i.e. ${\\rm R}^{2}={\\rm D}^{2} + {\\rm D}^{2}_{{\\rm MD}} - 2 {\\rm D} {\\rm D}_{{\\rm MD}}cos\\Theta$, where ${\\rm D}_{{\\rm MD}}$ is the distance in Mpc of NGC\\,7793 adopted from Vlajic et al.~(\\cite{sl_vlajic11}), ${\\rm D}$ is the distance in Mpc of SDIG, and $\\Theta$ is their angular separation, calculated using NASA\/IPAC Extragalactic Database (NED). The uncertainty in the deprojected distance takes into account the uncertainties in the distance of NGC\\,7793 and SDIG, as well as their angular projected separation as the minimum physical distance between them. We note that many gas-rich irregular and transition dwarfs are similarly distant from the Milky Way, for example NGC\\,6822 or Leo\\,T (McConnachie 2012).  Given the mass of NGC\\,7793 from Carignan \\& Puche (\\cite{sl_car90}) the orbital timescale for SDIG around NGC\\,7793 would appear to be too long to be responsible for star formation activity on timescales shorter than a Hubble time, but other signs of disturbance are present.\n\n  C\\^ot\\'e et al.~(\\cite{sl_cote00}) discuss how the low value of SDIG's maximum rotational velocity implies that random motions may significantly contribute to the dynamical support and may thus lead to the puffing-up of an initially thin disk. To that end, the twisted {\\ion {H}{i}} disk of SDIG (C\\^ot\\'e et al.~\\cite{sl_cote00}) may indicate an interaction having occurred in this dwarf's past. NGC\\,7793 has been mapped with Chandra (Pannuti et al.~\\cite{sl_pannuti11}), and the results indicate an asymmetry in the detected X-ray sources. Pannuti et al.~(\\cite{sl_pannuti11}) discuss an interaction between SDIG and NGC\\,7793 as a plausible way to explain this asymmetry in the X-ray sources, although they do not regard it as strong a possibility. Radburn-Smith et al.~(\\cite{sl_radburn12}) discuss that the detected stellar radial migration in NGC\\,7793 may also be due to a past interaction event having occurred in NGC\\,7793. While the deprojected distance of SDIG shows its current location and its actual orbit remains unknown (e.g. Bellazzini et al.~\\cite{sl_bellazzini96}), the timescale of the star formation enhancements and the crossing timescale of SDIG around NGC\\,7793 suggest that the elevated SFRs in SDIG are not due to gravitational interactions between them. The SF enhancements and disturbed {\\ion {H}{i}} kinematics seen in SDIG may be due to the effect of gravitational interactions with a lower luminosity, gas-poor dwarf galaxy companion that has thus far remained elusive, as such tidal interactions between dwarf galaxies now start to become revealed (e.g., Martinez-Delgado et al.~\\cite{sl_delgado12}; Rich et al.~\\cite{sl_rich12}; Hunter et al.~\\cite{sl_hunter98}). Tidal interactions between dwarf galaxies may also provide an important link to understanding the evolution and characteristics of dwarf galaxies in the isolated far-outskirts of the LG, as the case of the transition-type dwarf galaxy VV124 shows (e.g., Kirby et al. 2012).\n\n  It is interesting in this respect that based on photometric arguments alone, Heisler et al.~(\\cite{sl_heisler97}) conclude that it is not possible to rule out that SDIG-like systems may evolve to dSphs. It is thus intriguing that the off-center local sites of star formation observed in the young MS stars of SDIG, as well as other LG dwarfs (Dohm-Palmer et al.~\\cite{sl_dohm-palmer97}; Gallagher et al.~\\cite{sl_gallagher98}; Mateo \\cite{sl_mateo98}; Cole et al.~\\cite{sl_cole99}; McConnachie et al.~\\cite{sl_mcconnachie06}), are observed in the majority of the transition-type dwarfs in the Sculptor group (Lianou et al.~\\cite{sl_lianou12}; Jerjen \\& Rejkuba \\cite{sl_jerjen01}). Environmental effects on gas-rich dwarfs provide a possible avenue to their evolution into gas-poor systems (e.g., Mayer \\cite{sl_mayer10}; and references therein), and such environmental effects seem to affect their recent star formation activity (e.g., Bouchard et al.~\\cite{sl_bouchard09}; Thomas et al.~\\cite{sl_thomas10}; Weisz et al.~\\cite{sl_weisz11b}). However, the detection of a recent modest upturn in SFR, and hints of a similar episode $\\approx$0.6-1~Gyr ago is further indication that small galaxies are able to sustain complex star formation histories without strong tidal interactions from larger neighbours.\n\n\\begin{acknowledgements} The authors would like to thank an anonymous referee for the thoughtful comments. We are grateful to Margrethe Wold and Claus Leitherer for useful discussions on Starburst 99. \\\\\n  This research has made use of the following facilities: NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration; NASA's Astrophysics Data System Bibliographic Services; SAOImage DS9 developed by Smithsonian Astrophysical Observatory; Aladin.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThin metal films of nanometric size can have physical properties different from their bulk counterparts due to quantum size effects (QSE) caused by \nthe valence electron confinement in the direction normal to the film surfaces \\cite{chiang,tringides,jia}. Since thin films are among the basic \ncomponents of modern nanodevices an increasing attention has been addressed to the quantum vacuum fluctuations forces (van der Waals and Casimir \nforces) between them. Theoretical determinations of these forces have been based mainly on a continuum description of the material dielectric \nproperties, neglecting modifications of the electronic structures due to the boundaries, i.e. under the assumption that the film dielectric \nproperties be the same of the bulk material \\cite{esquivel,benassi,lambrecht2,pirozhenko2,lenac}. This assumption is not valid for small systems with low electron \ndensity and\/or large surface to volume ratio. In such cases a significant size dependence of the basic properties, like the Fermi energy and the \ndensity of electron states, arises as a consequence of the electron confinement \\cite{rogers1}.\\\\\nIn a previous paper \\cite{benassi3} we have shown that the inclusion of QSE in the plasma model of a free electron metal leads to modifications of the \nforce intensity between nanometric size films with respect to the bulk plasma model which range from several to few percent depending upon the film \nelectron density and the separation distance. The calculated forces show quantum size oscillations and are less intense compared \nto those determined with the bulk plasma model. The calculations were performed using the \\emph{particle in a box} model \\cite{wood} (hereafter indicated as PBM) in which independent electrons \nare confined along the direction normal to the film surfaces by hard walls and behave as a two dimensional gas parallel to the \nsurfaces. Such model represents a simplified picture of the one electron potential in the film.\\\\\nIn this paper we improve our description of the film dielectric properties along two lines of development: \nfirst we introduce a finite well model for the one-electron potential along the surface normal, second we include intraband absorption by introducing the relaxation time in a manner that allows to keep number conservation.\nAs to the first issue, we notice that the use of a soft confining potential like a finite well, besides being closer to the real shape of the one-electron potential as determined by first \nprinciples calculations \\cite{schulter,feibelman,ciraci,ogando}, allows for a better treatment of QSE for films of small thickness, since it does not \nintroduce a priori a distance \nwithin which electrons are to be confined and leaves the Fermi energy free to oscillate with the film size \\cite{rogers1,rogers2}, a feature that is \nnot present in the PBM model, where the Fermi energy is kept equal to the bulk value. \nThe inclusion of Drude absorption in vacuum force calculations has been a subject of considerable debate, mainly in relation with finite temperature \ncorrections \\cite{boestrom3,bezerra,brevik,bezerra2,hoye2,brevik2}. Very accurate measurements have been reported that shows that the simple plasma model of bulk dielectric can give better agreement with the experiments than the model with finite relaxation times \\cite{decca2}.\nOur results refer to the $T=0^{\\circ}$K case where difficulties do not seem to be present \\cite{lamoreaux}. Our purpose is to understand how QSE may affect the force between thin films when intraband absorption is included.\nIn this paper we introduce such corrections and we show their importance in the \ncalculation of the intensity of the force between metallic films.\\\\\n\\section{Theoretical framework}\nWe consider two identical metal films of thickness $D$ with plane boundaries separated by a distance $\\ell$ (see figure \\ref{fig1} (a)). $D$ represents \nthe extension, along the $z$-direction normal to the surface, of the positive ion distribution, which is supposed to be uniform with the same density of the bulk system. By extending \nprevious results for isotropic slabs to the case of films with anisotropic dielectric tensor we can write the expression of the force per unit area \nas \\cite{lifshitz,dzyaloshinskii,zhou,bordag}:\n\\begin{equation}\nF=-\\frac{\\hbar }{2 \\pi^2}\\int_{0}^{\\infty} k\\: dk\\int_{0}^{\\infty}d\\omega\\: \\gamma(\\omega)\n\\bigg[\\frac{Q_{TM}(i\\omega)^2}{1-Q_{TM}(i\\omega)^2}+\n+\\frac{Q_{TE}(i\\omega)^2}{1-Q_{TE}(i\\omega)^2}\\bigg] \n\\end{equation}\n\\begin{equation}\nQ_{TM}=\\frac{\\rho_{TM}(1-e^{-2 \\gamma_{TM} D})}{1-\\rho_{TM}^2e^{-2 \\gamma_{TM} D}}e^{-\\gamma \\ell}\\qquad\nQ_{TE}=\\frac{\\rho_{TE}(1-e^{-2 \\gamma_{TE} D})}{1-\\rho_{TE}^2e^{-2 \\gamma_{TE} D}}e^{-\\gamma \\ell}\n\\end{equation}\n\\begin{equation}\n\\rho_{TM}=\\frac{\\gamma_{TM}(\\omega)-\\gamma\\epsilon_{xx}(\\omega)}{\\gamma_{TM}(\\omega)+\\gamma\\epsilon_{xx}(\\omega)}    \n\\qquad\\rho_{TE}=\\frac{\\gamma_{TE}(\\omega)-\\gamma(\\omega)}{\\gamma_{TE}(\\omega)+\\gamma(\\omega)}\n\\end{equation}\n\\begin{equation}\n\\gamma_{TE}(\\omega)=\\sqrt{k^2-\\frac{\\omega^2}{c^2}\\epsilon_{xx}(\\omega)}\\qquad\n\\gamma_{TM}(\\omega)=\\sqrt{\\bigg(\\frac{k^2}{\\epsilon_{zz}(\\omega)}-\\frac{\\omega^2}{c^2}\\bigg)\\epsilon_{xx}(\\omega)}\n\\end{equation}\n\\begin{equation}\n\\gamma(\\omega)=\\sqrt{k^2-\\frac{\\omega^2}{c^2}}\n\\end{equation}\nHere $\\epsilon_{xx}$ and $\\epsilon_{zz}$ are the diagonal components of the dielectric tensor along the planar directions and along the surface normal respectively. We assume $\\epsilon_{xx}=\\epsilon_{yy}$ and the off diagonal components to be\nzero. \n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm,angle=0]{fig1.eps}\n\\caption{\\label{fig1} (a) Sketch of the two identical interacting films. (b) Infinitely deep quantum well with artificial spill-out. (c) Finite quantum well with natural spill-out.}\n\\end{figure}\nThis assumption is consistent with the two dimensional gas behavior of the electrons parallel to the surface. The anisotropy of the dielectric tensor is a consequence of the finite extension of the film along the $z$-direction and it is the main feature introduced by the size quantization. To calculate the force one needs the expression of the dielectric tensor, which has to be derived from the film electronic structure. \nWe assume an independent particle model with the one electron potential $V(z)$. The electron energies are given by\n\\begin{equation}\nE_{\\textbf{k}_{\\parallel},n}=\\frac{\\hbar^2}{2 m}\\textbf{k}_{\\parallel}^2+E_{n}\n\\end{equation} \ni.e. they are described by the continuous quantum number $\\textbf{k}_{\\parallel}$ and by the discrete sub-band index $n$ coming from the quantization of the perpendicular wavevector, $m$ being the electron mass. The corresponding wavefunctions are given by\n\\begin{equation}\n\\psi_{\\textbf{k}_{\\parallel},n}(\\textbf{r}_{\\parallel},z)=\\frac{1}{\\sqrt{A}}e^{i\\textbf{k}_{\\parallel}\\cdot\\textbf{r}_{\\parallel}}\\phi_{n}(z)\n\\label{wfc}\n\\end{equation} \nhere $A$ is the surface area, $\\textbf{k}_{\\parallel}$ and $\\textbf{r}_{\\parallel}$ are two dimensional wavevectors parallel to the surface, and $\\phi_{n}(z)$ is supposed to be independently normalized. The functions \n$\\phi_{n}(z)$ are the solutions of the equation\n\\begin{equation}\n\\bigg\\{\\frac{\\hbar^2}{2 m}\\frac{\\partial^2}{\\partial z^2}+V(z)\\bigg\\}\\phi_{n}(z)=E_n\\phi_{n}(z)\n\\label{schrod}\n\\end{equation}\nwith the proper boundary conditions. The Fermi energy $E_{F}$ is obtained through the \\emph{aufbau} procedure i.e. by arranging the eigenvalues in \nascending numerical order and counting until the number of states needed to accommodate all the electrons in the film is reached. This procedure \nleads to a Fermi energy that depends upon the film size and is generally different from the bulk value \\cite{rogers2}. This can be understood \nby noting that, to ensure charge neutrality, the number of electrons and the number of ions per unit area have to be equal. The electron density \n$n(z)$ can be simply obtained from the wavefunctions:\n\\begin{equation}\nn(z)=\\frac{1}{2 \\pi}\\sum_{m=1}^{m_0}\\bigg(\\frac{2 m E_F}{\\hbar^2}-E_n\\bigg)\\vert\\phi_{m}(z)\\vert^2\n\\end{equation} \nwhere $m_0$ is the label of the last occupied state, while the ion density is simply given by $n_0=k_{F_{B}}^3\/3 \\pi^2$, where $k_{F_{B}}$                                               is the bulk Fermi wavevector. By integrating the densities along the $z$ axis and imposing that both give the same number of charges per unit area, one gets the relation\n\\begin{equation}\n\\frac{1}{2 \\pi}\\sum_{m=1}^{m_0}\\bigg(\\frac{2 m E_F}{\\hbar^2}-E_n\\bigg)=n_0 D\n\\label{neucond}\n\\end{equation}\nwhich, for finite $D$ values, is generally not satisfied if one replaces $E_F$ with its bulk counterpart $E_{F_{B}}=\\hbar^2 k_{F_{B}}^2\/2 m$. \nIn the case of the PBM model this equation is not satisfied, since one assumes that the Fermi level is the same as in the bulk. To obtain charge neutrality one has to \nimpose the additional condition that the electron density be confined on a length $d$ larger than $D$. This artificially introduces the electronic charges spill-out but has the consequence that the average \nelectronic density is lower than the ion density \\cite{rogers1,benassi3,czoschke,czoschke2}.\\\\  \nFor the purpose of the present study we assume the potential to be a finite well $V(z) = -V_0$ inside the film and zero outside. For such finite well\nmodel (FWM) the energies of the bound states can be written as \n\\begin{equation}\nE_{\\textbf{k}_{\\parallel},n}=\\frac{\\hbar^2}{2 m}\\textbf{k}_{\\parallel}^2+\\frac{\\hbar^2}{2 m}k_{zn}^2-V_0\n\\end{equation}\nwhere $k_{zn}$ are the quantized transverse wavevectors. They are obtained from the equation giving the condition for the existence of bound states \nin a quantum well \\cite{davydov}:\n\\begin{equation}\nk_{zn}=\\frac{n \\pi}{D}-\\frac{2}{D}sin^{-1}\\bigg(\\frac{k_{zn}}{k_0}\\bigg)\n\\label{russi}\n\\end{equation}\nwith $k_0=\\sqrt{2 m V_0}\/\\hbar$.  Notice that the first term at the second member is the value of the transverse wavevector for an infinite well \nmodel of size $D$ and the second term goes to zero as $V_0$ goes to infinity. This implies that, for given film size and number of electrons, \nthe Fermi energy referred to the well bottom is higher in the infinite well model.\nNotice that when $V_0$ goes to infinity one does not recover the PBM, since the Fermi energy is varied with respect to the bulk value in order to \nsatisfy the charge neutrality condition (\\ref{neucond}). Figure \\ref{fig1} (b) and (c) illustrate the difference: in the PBM the electronic charge density is \nconfined within a distance $d=D+\\Delta$, larger than the size of the ionic charge distribution, to allow for the electronic charge spill-out and to \nensure global neutrality for $E_F=E_{F_{B}}$. In the FWM the charge spill-out results naturally from the behaviour of the single particle \nstates while the charge neutrality is achieved by varying the Fermi energy with respect to the bulk value. Obviously in the limit of infinitely deep well \n(hereafter indicated as IWM) the electronic charge turns out to be entirely localized within the length $D$ and the Fermi energy is strongly \nincreased compared to its bulk value.\\\\\nOnce the electron energies and wavefunctions have been obtained, one can calculate the dielectric tensor from the expression \\cite{wood}:\n\\begin{eqnarray}\n\\nonumber\n\\epsilon_{\\alpha\\alpha}(\\omega)&=1-\\frac{\\omega_{P}^{2}}{\\omega^{2}}-\\frac{8\\pi e^{2}}{A d m^{2}\n\\omega^{2}}\\sum_{{\\bf k}_{\\parallel},n}\\sum_{{\\bf k}_{\\parallel}',n'}\nf(E_{\\bf{k}_{\\parallel},n})(E_{\\bf{k}_{\\parallel},n}-E_{\\bf{k}_{\\parallel}',n'})\\times\\\\\n&\\times\\frac{\\vert\\langle\\psi_{{\\bf k}_{\\parallel},n}\\vert\\hat{p_{\\alpha}}\\vert\\psi_{{\\bf k}_{\\parallel}',n'}\n\\rangle\\vert^{2}}{(E_{\\bf{k}_{\\parallel},n}-E_{\\bf{k}_{\\parallel}',n'})^2-\\hbar^2\\omega^2}\n\\label{dieltens}\n\\end{eqnarray}\nhere $\\alpha=x,y,z$ labels the cartesian component of the tensor, $\\hat{p}_{\\alpha}$ indicates the component of the electron linear momentum,\n$\\omega_{P}=\\Omega_{P} n(D)\/n_0$ is the plasma frequency of the quantized electron gas ($\\Omega_{P}=\\sqrt{4\\pi e^2 N_0\/m}$ is the free electron plasma frequency and $n(D)$ is the average electron density of the film) and $f(E_{\\bf{k}_{\\parallel},n})$ is the occupation factor of the $(\\bf{k}_{\\parallel},n)$ state. \nIn the IWM and the FWM $\\omega_P = \\Omega_P$, while in PBM $n(D)$ is smaller than $n_0$, since the electronic charge is distributed over a larger distance than the ionic charge (see Fig.\\ref{fig1} (b)).\nThe off-diagonal component are equal to zero. This expression differs from the \nplasma model dielectric function adopted in previous studies in that: \n(i) it has a tensor character with $\\epsilon_{xx}=\\epsilon_{yy}\\neq\\epsilon_{zz}$, (ii) through the double sum in the second member it accounts for transitions between lateral sub-bands, whose probability amplitude is expressed by the momentum matrix element between the one electron wavefunctions (\\ref{wfc}). It can be easily shown that these \ntransitions do not affect the lateral components of the dielectric tensor, which are given by the simple expression of the plasma dielectric function\n\\begin{equation}\n\\epsilon_{xx}(\\omega)=\\epsilon_{yy}(\\omega)=1-\\frac{\\omega_{P}^{2}}{\\omega^2}\n\\label{drudeded}\n\\end{equation}\nbecause the momentum matrix element for $x$ and $y$ component vanishes.\n\\section{Results for finite well potential}\nIn this section we present the results of calculations for finite well potentials. We take $Al$, $Ag$ and $Cs$ corresponding to a radius per electron \nin Bohr units $r_s\/a_0$ equal to $2.07$, $3.02$ and $5.62$ respectively, in order to illustrate QSE at different densities and potential depths. The \nvalue of the potential depth is obtained by summing the metal work function $W$ with the calculated Fermi energy. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig2.eps}\n\\caption{\\label{fig2} Fermi energy normalized to its bulk value for $Al$, $Ag$ and $Cs$ using the IWM (dashed line) and the FWM (continuous line). The bulk Fermi energies and the work functions have been taken from ref. \\cite{ashcroft}.}\n\\end{figure}\nFigure \\ref{fig2} reports the calculated Fermi energies as a function of the product between the Fermi wavevector and the film thickness. This allows \nto better point out the oscillations and the cusps arising from the crossing  of the Fermi energy by new subbands upon varying the film size. \nThe figure shows the $V_0$ value appropriate to the bulk and to the large $D$ limit. \nWe give in the same figure the results obtained by assuming an infinitely deep potential (IWM). The comparison allows to illustrate the effects of the potential softening. In agreement with previously published results \\cite{schulter,rogers2,sernelius3} we find that: \n\\begin{itemize}\n\\item  $E_F$ is systematically larger than $E_{F_{B}}$ and goes to the bulk value as $D$ goes to infinity. The difference is more pronounced and the bulk \nlimit is achieved at larger size in the low density systems, as it is clearly shown by the comparison between $Cs$ and $Al$ curves;\n\\item  As expected from the discussion of the previous section, the softening of the potential leads to less pronounced deviations from the bulk \nvalues. Because of the stronger electron confinement, the IWM has a larger Fermi energy that the FWM;\n\\item  The cusps correspond to integer values of half of the Fermi wavelength in the IWM case. This feature is only approximately verified for the FWM.\n\\end{itemize}    \nAs a consequence of size quantization, the $z$-component of the dielectric tensor is expected to go to a finite value $\\epsilon_{zz}(0)$ as the \nfrequency goes to zero. This value increases proportionally to $D^2$ in the large size limit \\cite{wood}. In Figure \\ref{fig3} we plot the quantity  \n$\\epsilon_{zz}(0)\/D^2$ for the three cases under study. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig3.eps}\n\\caption{\\label{fig3} Static value of the $zz$ component of the dielectric tensor for $Al$, $Ag$ and $Cs$ using the IWM (dashed lines), the PBM (dotted lines) and the FWM (continuous lines).}\n\\end{figure}\nAgain the results show the cusps due to the filling of new subbands as $D$ increases \n\\cite{benassi3}. The asymptotic limit is obtained for $k_{F}D\/\\pi$ of the order of $5\\div 6$ in the three cases. Significant differences appear in the \nlarge $D$ behaviour when the FWM results are compared with those from the IWM model: $\\epsilon_{zz}(0)$ is larger for finite wells of small \nsize, while it is smaller at high $D$ values. The convergence to the asymptotic limit is considerably slower for the infinite well, specially in the \nlow density metals. This behaviour reflects the differences in the distribution of the eigenvalues of equation (\\ref{schrod}). For the infinite well \nthere are infinite bound states whose energy scales like $n^2$, see equation (\\ref{russi}), and the separation between two successive levels \nincreases linearly with $n$. Such behaviour is not present in the FWM, for which equation (\\ref{schrod}) has a finite number of eigenvalues \ncorresponding to bound states and a continuum spectrum at positive energies. The PBM curve takes values closer to the FWM than to the IWM. This is primarily a consequence of the fact that PBM allows for electron charge spill-out, while in IWM the electron distribution is confined within $D$ i.e. it has the same size of the positive charge.\\\\\nIn the following we show the changes in the force caused by the size quantization with respect to the results obtained by using the isotropic continuum \nplasma model, where the $z$-component of the film dielectric tensor is equal to the planar components i.e. is given by equation (\\ref{drudeded}). We \nuse the symbol $F_Q$ to indicate the force per unit area calculated for the quantized film, while $F_P$ is the force per unit area calculated in the \nisotropic plasma model. To better illustrate the results, in figures \\ref{fig4} and \\ref{fig5} we plot the quantity\n\\begin{equation}\n\\delta_{P}=\\frac{F_P-F_Q}{F_P}\n\\end{equation}\nas a function of the separation distance $\\ell$ for films of $1$ and $5$ nm thickness respectively. In each figure we display the results for the \nthree cases under study and we compare the finite well with the IWM at the same density. This allows us to point out the modifications caused by the \npotential softening.  We also show the curves appropriate to the PBM.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig4.eps}\n\\caption{\\label{fig4} Relative percentual difference for the force between two identical films of thickness $D=1$ nm as a function of the films separation $\\ell$, for $Al$, $Ag$ and $Cs$. Dashed lines have been obtained using the PBM, dotted lines have been obtained using the IWM and the continuous lines represents the finite well results.}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig5.eps}\n\\caption{\\label{fig5} Relative percentual difference for the force between two identical films of thickness $D=5$ nm as a function of the films separation $\\ell$, for $Al$, $Ag$ and $Cs$. Dashed lines have been obtained using the PBM, dotted lines have been obtained using the IWM and the continuous lines represents the finite well results.}\n\\end{figure}\nIn agreement with our previous findings we observe that \n\\begin{itemize}\n\\item  QSE tend to reduce the intensity of the force;\n\\item  the reduction is more significant at low density ($Cs$) than at high density ($Al$); \n\\item  it may be considerably higher than $10\\%$ for $1$ nm thickness and reduce to few per cent at $5$ nm;\n\\item  it can be appreciable over a distance interval up to $10\\div 50$ nm.\n\\end{itemize}\nThe most important conclusion that can be drawn from the figures is that the potential shape is important and can lead to a substantial  \nmodifications of the quantum size effects both at small and at large distances. The models which confine the electronic charge tend to overestimate \nthe force reduction induced by size quantization. The curves for the PBM show large force reduction (greater $\\delta_P$ values) over a wide interval of distances. \nOn passing to the IWM case one notice that the \nremoval of the constriction that the Fermi energy be equal to the bulk value, still keeping an infinitely deep potential, leads to smaller $\\delta_P$ \nvalues and to a more rapid decay of the curves at large distances. Reducing the well depth to finite values has a similar effect: it causes a general \ndecrease of $\\delta_P$ and a narrowing of the distance interval over which QSE are appreciable. \nThis also implies that any increase of the confining potential depth at fixed ion density leads to higher $\\delta_P$ values and to more significant \nQSE.\nThe large values taken by $\\delta_{P}$ in the PBM case do not arise from the charge confinement only, since, as pointed out before, the constraint on \nthe charge distribution is weaker that in the IWM. To a large extent they are a consequence of the plasma frequency normalization caused by the \ndecrease in the average electron charge density that it is necessary in order to achieve global charge neutrality \\cite{benassi3,czoschke,czoschke2}. \nIn the isotropic plasma model one takes $\\omega_{P}=\\Omega_{P}$. In the PBM this value is obtained only at large film thickness. Neglecting this \nnormalization i.e. taking the free electron plasma frequency in the parallel components of the dielectric tensor (but not in $\\epsilon_{zz}$) would \nlead to $\\delta_{P}$ values  closer to the well potential models.\n\\section{Intraband absorption effects}\nTo introduce intraband absorption we have to modify the dielectric tensor in a way that allows to include relaxation time effects in the parallel \ncomponents and to recover  the Drude behaviour in the large $D$ limit. This cannot be done by simply introducing an imaginary part of  the frequency \n$\\omega$, since this violates the continuity equation locally \\cite{mermin,garik}. The appropriate recipe is to replace into equation \n(\\ref{dieltens}) $\\omega^2$  with $\\omega(\\omega+i 2 \\pi\/\\tau)$, where $\\tau$ is the relaxation time.\nFor the parallel components this leads to the Drude dielectric function\n\\begin{equation}\n\\epsilon(\\omega)=1+\\frac{\\omega_{P}^2}{\\omega(\\omega+i \\gamma)}\n\\end{equation}\nwhere $\\gamma=2 \\pi \/\\tau$\nThe results of the calculation of the force per unit area with the modified dielectric tensor are displayed in Figures \\ref{fig6} and \\ref{fig7} for \n$1$ nm and $5$ nm films. We report the quantity\n\\begin{equation}\n\\delta_{D}(\\gamma)=\\frac{F_D(\\gamma)-F_{QD}(\\gamma)}{F_D(\\gamma)}\n\\end{equation}\nwhere $F_D$ is the force calculated using the bulk Drude model with a given relaxation time and $F_{QD}$ in the one obtained by the calculation with \nthe same relaxation time and with size quantization included. For $\\tau\\rightarrow\\infty$ we recover the plasma model so that \n$\\delta_P=\\delta_D(0)$.\nThe calculations have been performed assuming a finite potential well and for two values of  the relaxation frequency $\\gamma$. We have taken values \nthat approximately correspond to those reported for the metals under consideration \\cite{ashcroft}. The figures show a comparison with the curves \nobtained by the plasma model with QSE. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig6.eps}\n\\caption{\\label{fig6} Relative percentual difference for the force between two identical films of thickness $D=1$ nm as a function of the films \nseparation $\\ell$, for $Al$, $Ag$ and $Cs$. Continuous lines represent the results for $\\gamma=0$. Dotted lines have been obtained using \n$\\gamma=5\\times 10^{13}$ rad\/s for $Ag$ and $Cs$ and $\\gamma=10^{14}$ rad\/s for $Al$. Dashed lines have been obtained using $\\gamma=10^{14}$ rad\/s \nfor $Ag$ and $Cs$ and $\\gamma=10^{15}$ rad\/s for $Al$.}\n\\end{figure}\nIt is clear from these results that the main effect of the inclusion of intraband absorption is to increase $\\delta$ i.e. to increase the difference \nwith respect to the calculations with the bulk dielectric function. The smaller is the relaxation time the larger is the reduction of the force. The \neffect is qualitatively the same in the three metals under study, but it depends upon the well depth and the film size.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig7.eps}\n\\caption{\\label{fig7} Relative percentual difference for the force between two identical films of thickness $D=5$ nm as a function of the films \nseparation $\\ell$, for $Al$, $Ag$ and $Cs$. Continuous lines represent the results for $\\gamma=0$. Dotted lines have been obtained using \n$\\gamma=5\\times 10^{13}$ rad\/s for $Ag$ and $Cs$ and $\\gamma=10^{14}$ rad\/s for $Al$. Dashed lines have been obtained using $\\gamma=10^{14}$ rad\/s \nfor $Ag$ and $Cs$ and $\\gamma=10^{15}$ rad\/s for $Al$.}\n\\end{figure}\nThe influence of the shape of the potential is illustrated in Fig.\\ref{fig8} showing the curves of $Ag$ films at a given relaxation frequency for the \nvarious models. Again it should be noticed that the FWM gives the lower $\\delta_D$ values. The PBM results show large $\\delta_D$ values over a very \nwide interval of distances. To a large extent this behaviour has to be imputed to the renormalization of the plasma frequency.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig8.eps}\n\\caption{\\label{fig8} Relative percentual difference for the force between two identical $Ag$ films of thickness $D=5$ nm as a function of the films \nseparation $\\ell$, $\\gamma=10^{14}$ rad\/s. Continuous lines represent the FWM, dotted line the IWM and dashed line the PBM.}\n\\end{figure}\n \\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig9.eps}\n\\caption{\\label{fig9} Relative percentual difference for the force between two identical $Ag$ films separated by a distance $\\ell=5$ nm as a function of the films thickness $D$. Continuous line represents the result for $\\gamma=0$ whereas the dashed line has been obtained with $\\gamma=10^{14}$ rad\/s. }\n\\end{figure}\nFigure \\ref{fig9} shows typical curves of $\\delta_D$ as a function of the film thickness for different values of the relaxation frequencies at a \ngiven separation distance of $5$ nm. As expected $\\delta_D$ decreases with $D$, but the slope at large thicknesses ($D$ of the order of $10\\div50$ \nnm) depends significantly upon the relaxation frequency. \n\\section{Conclusions and lines of development}\nWe have presented a rather complete set of theoretical results illustrating the possible role of size quantization effects in the electromagnetic \nvacuum force between very thin films and showing how the determination of these effects depends upon the description of the film electronic structure \nand upon the inclusion of intraband effects. However we want to point out that our analysis is still a mere indication of the corrections to the \nsimple picture that assumes the same dielectric function for films and bulk solids. Although the basic features of size quantization (confinement of \nthe electronic charge, anisotropy of the dielectric tensor, presence of inter-subbands transitions in the dielectric function) are already present in \nthe models we have studied, there is room for substantial improvements before an accurate comparison with experimental data, like those obtained by \nLisanti et al. for $Pd$ films \\cite{lisanti}, can be done. A more detailed treatment should include (i) band structure effects, (ii) non-locality of the dielectric response and (iii) non-local treatment of the reflectivity.\nA self consistent first principles calculation of the inverse dielectric matrix for a slab of the appropriate size and symmetry, from which a \nmacroscopic dielectric function can be derived with band structure and non-local effects included, could provide the appropriate treatment of the \nfirst two issues \\cite{li,sernelius2}. \nCorrections to the Fresnel optics, along the lines indicated by several authors \\cite{appel,feibelman2,kempa}, can lead to important modifications \nof the reflectivity even in the case of a free electron gas film. We are currently investigating these matters and the results will be presented \nelsewhere.\nStill when comparing theory with experiments for thin films one should consider the fact that measured relaxation frequency turn out to depend upon \nthe film size and morphology \\cite{jalochowski2,fahsold,sotelo,pribil,hoffmann}. This is largely due to the so called classical size effects arising from the scattering of the electrons at the \nfilm boundaries. In view of the sensitivity of QSE to the value of the relaxation time, comparison with experimental data may be possible only if \na realistic estimate of the modifications in the relaxation times caused by surface scattering is available.\n\\ack AB thanks \\emph{CINECA Consorzio Interuniversitario} ({\\tt\nwww.cineca.it}) for funding his Ph.D. fellowship.\n\\section*{Reference}\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}{-3.5ex plus-1ex minus\n    -.2ex}{2.3ex plus.2ex}{\\reset@font\\small\\bf\\uppercase}}\n\\makeatother\n\n\\title{\\large \\bf Electroweak symmetry breaking in Higgs mechanism with\ncomposite operators and solution of naturalness }\n\\author{V.V. Kiselev,\\\\\n{\\small\\it State Research Center of Russia \"Institute for High\nEnergy Physics\"},\\\\ {\\small\\sl Protvino, Moscow region, 142281 Russia}\\\\\n{\\small\\sl E-mail: kiselev@th1.ihep.su, Fax: +7-(0965)-744739 }}\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n{Introducing a source for a bi-local composite operator motivated by the\nperturbative expansion in gauge couplings, we calculate its effective potential\nin the renormalization group of Standard Model with no involvement of\ntechnicolor. The potential indicates the breaking of electroweak symmetry below\na scale $M$ due to a nonzero vacuum expectation value of neutral component for\nthe ${\\sf SU(2)}$-doublet operator. At virtualities below a cut off $\\Lambda$\nwe introduce the local higgs approximation for the effective fields of sources\ncoupled to the composite operators. The value of $\\Lambda\\approx 600$ GeV is\nfixed by the measured masses of gauge vector bosons. The exploration of\nequations for infrared fixed points of calculated Yukawa constants allows us to\nevaluate the masses of heaviest fermion generation with a good accuracy, so\nthat $m_t(m_t) = 165\\pm 4$ GeV, $m_b(m_b) = 4.18\\pm 0.38$ GeV and\n$m_\\tau(m_\\tau) = 1.78\\pm 0.27$ GeV. After a finite renormalization of\neffective fields for the sources of composite operators, the parameters of\neffective Higgs field potential are calculated at the scale of matching with\nthe local theory $\\Lambda$. The fixed point for the Yukawa constant of $t$\nquark and the matching condition for the null effective potential at $M$ drive\nthe $M$ value to the GUT scale. The equation for the infrared fixed point of\nquartic self-action allows us to get estimates for two almost degenerate scalar\nparticles with $m_H= 306\\pm 5$ GeV,  while third scalar coupled with the $\\tau$\nlepton is more heavy: $m_{H_\\tau} = 552\\pm 9$ GeV. Some phenomenological\nimplications of the offered approach describing the effective scalar field, and\na problem on three fermion generations are discussed.\n}\n\\end{abstract}\n\n\\vspace*{1cm}\nPACS numbers:   12.60.Rc, 11.15.Ex, 12.60.Fr,  14.80.Cp\n\n\\section{Introduction}\nAt present, the Standard Model exhibits almost a total success in experimental\nmeasurements \\cite{SM}. The only question being a white spot on its body, is\nthe empirical verification of mechanism for the spontaneous breaking of\nelectroweak symmetry. In this respect,  the minimal model involving a single\nlocal Higgs field brings a disadvantage: the stability of potential under the\nquantum loop corrections requires a restriction of quadratic divergency in the\nself-action by the introduction of ``low'' energy cut-off $\\Lambda\\sim 10^3$\nGeV, which is not a natural physical scale standing far away from what can be\ndesirable \\cite{tHoft}: the GUT scale, $M_{\\rm GUT}\\sim 10^{16}$ GeV\n\\cite{GUT}, or even the Planck mass, $M_{\\rm Pl}\\sim 10^{19}$ GeV. The reason\nfor putting the $\\Lambda$ so small, has to originate beyond the Standard\nModel. Two highways to a ``new physics'' merit the most popularity. The first\none is a technicolor \\cite{tc} postulating an extra-strong interaction for new\ntechnifermions, which form some ``QCD-like'' condensates, breaking down the\nelectroweak symmetry and giving the masses to the ordinary gauge bosons.\nDespite some problems with the generation of realistic mass values for the\nquarks and leptons and suppression of flavor changing neutral currents, the\nextended technicolor \\cite{etc} provides quite a clear picture for what happens\nin the region deeper than $10^3$ GeV. However, the most strict objection\nagainst such the way is the comparison with the current measurements, which\ndisfavor the technicolor models possessing the calculability \\cite{PEP}.\nA general consideration of models with the condensation of heavy fermions is\nreviewed in ref.\\cite{cvet}, while a brilliant presentation of both the\nideas on the electroweak symmetry breaking with composite operators and\ntechniques as well as results is given in a comprehensive survey by C.T.Hill\nand E.H.Simmons \\cite{HiSi}. However, the condensation, in general, does not\nprovide us with the solution of naturalness. In fact, this approach\nreformulates the problem as a fine-tuning phenomenon, since the separation of\ndynamics responsible for the composite operators at a high scale from the\nlow-energy electroweak physics takes place at effective couplings tuned to some\ncritical values. Therefore, we need an additional argumentation in order to\naddress the naturalness in the framework of condensation mechanism with\ncomposite operators. We present an idea toward this direction below.\n\nThe second way is a supersymmetry \\cite{SUSY} reforming the\nquadratic divergency in the self-action of Higgs field into the\nlogarithmic one, so that it prescribes the scale $\\Lambda$ to be a\nsplitting between the particles of Standard Model and their\nsuper-partners. Therefore, the supersymmetry has to be broken in a\nmanner conserving the logarithmic behavior of renormalization,\nwhich is an additional challenge to study and a degree of\nambiguity. However, the advantage is the stability of Higgs\npotential, so that $\\Lambda$ certainly is a reasonable scale\nreflecting the physics in the supersymmetric theory. What remains\nis the question: why the basic SUSY scale is so ``low'' in\ncomparison with the GUT scale? Hence, the naturalness is again the\nproblem standing in the higher-quality context.\n\nIf the ultraviolet cut off energy in the loop calculations is placed close to\nthe Planck scale (see Fig.\\ref{SM}\\footnote{The figure originally appeared in\nref. \\cite{HR}, and it is taken from ref. \\cite{Bj}, while the two-loop\nconsideration recently was done in ref. \\cite{PZ}.}), the\nStandard Model suffers from the inherent inconsistency except a narrow window\nin the range of possible values of Higgs particle mass: $m_H = 160\\pm 20$ GeV,\nwhich does not contradict the value following from the precise measurements of\nelectroweak parameters in the electron-positron annihilation at the $Z$ boson\npeak.\n\n\\begin{figure}[th]\n\\setlength{\\unitlength}{1mm}\n\\begin{center}\n\\begin{picture}(80,80)\n\\put(0,0){\\epsfxsize=9cm \\epsfbox{7.eps}}\n\\end{picture}\n\\end{center}\n\\caption{The region of higgs mass constrained by requirements of the SM\nconsistency.}\n\\label{SM}\n\\end{figure}\n\n\nThe reasons for such the inconsistency are the following: At lower masses the\nvacuum stability is broken, i.e., the quartic coupling constant of scalar field\nchanges its sign \\cite{AI}. At higher masses the theory enters the strong\nself-interaction regime, which indicates that the quartic coupling constant\nbecomes infinite (alike the Landau pole) at a scale less than the offered cut\noff \\cite{Elias}. If the scale of ultraviolet cut off in the SM is much lower\nthan the Planck scale, then the region of higgs masses providing the SM\nconsistency, is more wide. However, such the scales are not natural. A low cut\noff scale should indicate a new dynamics. While the vacuum instability is an\nunavoidable physical constraint, the phase of strong higgs self-interaction\ncould be treated in the framework of the following representation: The scalar\nhiggs can be described in terms of the local field below an ultraviolet cut off\n$\\Lambda$ placed close to the region of strong regime. At virtualities higher\nthan $\\Lambda$, the strongly self-coupled higgs is not fundamental and local\nquantum. The dynamics should be described by means of weakly interacting\nparticles, so that a composite operator with appropriate quantum numbers has to\ncorrespond to the higgs in the `dual' limit implying that the effective\npotential of the composite operator yields a development of vacuum expectation\nvalue for the global (independent of space-time point) source of operator. This\nstrong self-interaction regime could be realized with no involvement of an\nextended underlying theory alike the technicolor dynamics, since some composite\noperators can develop an appropriate effective potential in the framework of\nstandard electroweak symmetry.\n\nOur assumptions are the followings:\n\n1. We choose a form of composite operators describing the nonlocal phase of\nhiggs in the strong self-interaction regime (SSIR) and suppose the connection\nof such the operators to the higgses. The suggestion on the nonlocality of\nhiggses allows us to replace the strong self-interaction regime in the theory\nwith the local Higgs fields by the weak self-interaction regime (WSIR) of\nsources for the composite operators.\n\n2. The interactions of fermions and gauge bosons in the SSIR are given by the\ndynamics of SM with no local scalar higgses as well as no extensions like a\ntechnicolor or so.\n\n3. Concerning the position of scale $\\Lambda$ denoting the infrared cut off in\nthe calculations with the composite operators as well as the ultraviolet cut\noff in the local theory with the scalar higgs, we put it into the (infrared)\nfixed point for the Yukawa coupling constants of heaviest fermion in the local\ntheory. the numerical value of $\\Lambda$ is given by the masses of\nweak-interaction gauge bosons.\n\n4. We consider Yukawa couplings of the only heaviest fermion generation in the\nSM.\n\n5. In the SSIR we introduce the ultraviolet cut off $M\\gg \\Lambda$. At $M$ the\nelectroweak symmetry is exactly restored.\n\n6. We match the effective potential of sources for the composite operators with\nthe potential of corresponding local scalar fields at the scale $\\Lambda$.\n\nThe corresponding divisions of virtualities are presented in Fig.\n\\ref{division}.\n\n\\begin{figure}[th]\n\\setlength{\\unitlength}{1mm}\n\\begin{center}\n\\begin{picture}(100,45)\n\\put(0,0){\\epsfxsize=11cm \\epsfbox{8.eps}}\n\\put(15,-4){$\\Lambda$}\n\\put(86,-4){$M$}\n\\put(100,0){log$_{\\rm 10}\\mu [{\\rm GeV}]$}\n\\put(-10,30){Local higgs}\n\\put(-4,25){phase}\n\\put(27,27){`Strong self-interaction regime'}\n\\put(34,33){Composite operators:}\n\\put(92,33){GUT or}\n\\put(92,27){Quantum Gravity}\n\\end{picture}\n\\end{center}\n\\caption{The division of virtualities as accepted in the calculations\nthroughout of this paper.}\n\\label{division}\n\\end{figure}\n\nIt is important to stress that the global sources of composite operators\ndevelop the Higgs-like potential in the region of $[\\Lambda;M]$, so that the\ncorresponding couplings of self-interaction as well as Yukawa constants fall\noff to zero under the increase of virtuality from $\\Lambda$ to $M$. Therefore,\nthe dynamics of local interactions is perturbative in the region of $[\\Lambda;\nM]$, while the notion on the ``strong self-interaction regime'', strictly\nspeaking, concerns for a theory with the local Higgs field, i.e. we replace\n$\\left.{\\rm SSIR}\\right|_{\\rm local}\\to \\left.{\\rm WSIR}\\right|_{\\rm\ncomposite}$.\n\nPostponing a supersymmetric extension in a time, in this paper we develop a\nnew insight into the breaking of electroweak symmetry by means of exploring the\ndynamics of SM to calculate an effective potential for a source of bi-local\noperator with no technicolor interactions. The physical reasoning for the\nchoice of operator under study was hinted in ref.\\cite{nz}. So, in the second\norder of perturbation theory we write down the following contribution to the\naction:\n\\begin{equation}\n i S_{2m} = - \\int dx dy\\;\\;  {\\bf T}[\\bar L_L(x) \\slashchar{B}(x) L_L(x) \\cdot\n\\bar L_R(y) \\slashchar{B}(y) L_R(y)]\\cdot 4\\pi \\alpha_Y \\cdot\n\\frac{Y_L}{2}\\cdot  \\frac{Y_R}{2},\n  \\label{eq:2}\n\\end{equation}\nwhere we have introduced the notations $L_L$ for the left-handed doublets and\n$L_R$ for the right-handed singlets, $B$ is the gauge field of weak hypercharge\n$Y$, $\\alpha_Y$ is its coupling constant. Note, that the gauge field of local\n${\\sf U(1)}$-group is the only one interacting with both the left-handed and\nright-handed fermions. If we suggest a nontrivial vacuum correlators with the\ncharacteristic distance $r\\sim 1\/v$\n\\begin{eqnarray}\n  \\langle 0|{\\bf T}[ \\slashchar{B}(x) L_L(x)\\cdot\n\\bar L_R(y) \\slashchar{B}(y)] |0\\rangle  &\\Rightarrow&\n\\frac{\\delta(x-y)}{v^4} \\langle 0| {\\bf T}[\\slashchar{B}(x) L_L(x)\\cdot\n\\bar L_R(x) \\slashchar{B}(x)] |0\\rangle \\nonumber \\\\ & \\sim &\n\\delta(x-y)\\; v ,\n  \\label{eq:3}\n\\end{eqnarray}\nsupposing that the scales of expectations for $BB$ and $L_L \\bar L_R$ are\ndriven by $v^2$ and $v^3$, respectively, then the Dirac masses of fermions are\ndetermined by the action\\footnote{By the way, Eq.(\\ref{eq:3a}) implies that\nin the SM the neutrino is massless since its right-handed component is\ndecoupled, $Y_R=0$.}\n\\begin{equation}\n  \\label{eq:3a}\n  S_{fm} \\sim \\int dx\\; \\bar L_L(x) L_R(x)\\cdot v\\cdot\n4\\pi \\alpha_Y \\cdot\n\\frac{Y_L}{2}\\cdot  \\frac{Y_R}{2} +{\\rm h.c.}\n\\end{equation}\nIn this way we extend the SM action by the initial bi-local bare $J$-term\n$$\nS_{ib}=\\int dx dy\\; N_J\\cdot J(x,y)\\; [\\bar L_R(x)\\;\n\\underbrace{\\slashchar{B}^{\\perp}(x) \\slashchar{B}^{\\perp}(y)} \\;\nL_L(y)] - \\int dx \\phi(x) J(x,x)+{\\rm h.c.},\n$$\nwhere $N_J=\\pi\\alpha_Y\\cdot Y_l\\cdot Y_R$, and\n$\\underbrace{\\cdot\\;\\cdot} $ denotes the propagation of\ntransversal U(1)-gauge field $B_{\\mu}^{\\perp}\n=(g_{\\mu\\nu}-\\partial_\\mu\\partial_\\nu\/\\partial^2)B^{\\nu}$, which\nis independent of the longitudinal mode, so that\n$$\n\\underbrace{B_\\mu^{\\perp}(x) B_\\nu^{\\perp}(0)} = -i g_{\\mu\\nu}\\int\n\\frac{d^4 p}{(2\\pi)^4} e^{ipx} \\frac{1}{p^2}\n$$\nto the leading order of perturbative theory. To the bare order the\nequation of motion for the bi-local field results in the\nstraightforward substitution of local field $\\phi$, as it stands\nin the above consideration for the correlators, developing the\nvacuum expectation values. After the analysis of divergences in\nthe $J$-dependent Green functions, the corresponding contra-terms\nmust be added to the action. Then the $J$-source can be integrated\nout or renormalized, that results in a Higgs-like action,\ncontaining some couplings to fermions as well as a suitable\npotential to develop the spontaneous breaking of electroweak\nsymmetry.\n\nWe stress that there are no other suitable composite operators appearing in the\nsecond order of SM gauge symmetry with the quantum numbers relevant to the\nHiggs interactions providing the generation of fermion masses through the\nYukawa-like couplings except the operators described above.\n\nIn this paper we calculate the effective potential up to the quartic term for\nthe sources corresponding to the bi-local composite operators of quarks and\nleptons to the one-loop accuracy of renormalization in the SM. The\nnormalization condition of potential parameters: $\\mu^2$ and $\\lambda$ standing\nin\n$$\nV(J^\\dagger,J) = -\\mu^2\\cdot J^\\dagger J+\\lambda \\cdot (J^\\dagger J)^2,\n$$\nis strictly defined in the SM, since we do not involve some additional\ninteractions. Therefore, both $\\mu^2$ and $\\lambda$ for a nonfundamental source\nmust be equal to zero, exactly, i.e. $V=0$, which, however, can be satisfied at\na single scale $M$ because of logarithmic renormalization for couplings, so\nthat\n\\begin{equation}\n\\mu^2(M) =0,\\;\\;\\; \\lambda(M)=0.\n\\label{matchnull}\n\\end{equation}\nIt is essential that the choice of composite operators is conformed to the\neffective action of SM in the second order over the gauge couplings. Otherwise,\nthe introduction of arbitrary composite operators with the given properties\nwith respect to the gauge symmetry generally does not imply the imposition of\nmatching condition in (\\ref{matchnull}), which is extremely important, since it\nremoves an uncertainty of the potential due to a finite renormalization of\nparameters.\n\nBelow $M$, the mass parameter $\\mu^2(\\Lambda)$ depending on the ``infrared\ncut-off $\\Lambda$, is positive, and the electroweak symmetry is broken down.\nSo, we suppose that the bi-local representation is valid in the range of\nvirtualities: $[\\Lambda;M]$, and below $\\Lambda$ we can explore the local\nHiggs fields.\n\nAs was shown in ref. \\cite{HaHa}, a variety of composite operators appropriate\nfor the Higgs quantum numbers can be rearranged so that practically arbitrary\nvalues of higgs mass or $t$ quark mass could be derived. In other words, in\norder to get a definite description of higgs sector, one should to suppress\ncontributions by a lot of composite operators except the special ones. this\ndominance of several composite operators is usually motivated by an extended\ndynamics beyond the SM, for instance, by the thechnicolor providing the\ndominance of some bound channels.\n\nIn the present paper, the choice of dominant composite operators is dictated by\nthe SM gauge symmetry, since we isolate the only composite structure in the\nsecond order over the gauge couplings, while the appearance of other operators\ntakes place at higher orders, and, hence, their contributions must be\nsuppressed\\footnote{This suppression becomes even better at higher virtualities\nbecause of the asymptotic freedom of nonabelian interactions, while the abelian\ncharge remains small up to the GUT scale.}. Moreover, this motivation on the\nform of composite operators makes us to add the matching condition of\n(\\ref{matchnull}), which is a new idea for the composite models, and it is\ncertainly due to the electroweak nature of composite operators.\n\nThus, for a walker travelling from low scales to  higher ones, the\nwhole picture of electroweak symmetry breaking looks as the\nfollowing:\n\\begin{itemize}\n\\item[1.] The SM extension with several Higgs fields is the local theory with\nthe ultraviolet cut-off $\\Lambda$.\n\\item[2.] The parameters of Higgs potential at the scale $\\Lambda$ is matched\nwith the effective potential of bi-local source, calculated in the range\n$[\\Lambda; M]$, so that $M$ denotes the scale, where the potential is exactly\nzero.\n\\end{itemize}\nThe value of $\\Lambda$, hence, can be related with the masses of\ngauge bosons, or the vacuum expectation value ({\\sc vev}) $v_{\\rm\nSM}$ for the Higgs field in the SM. The value of $M$ with respect\nto $\\Lambda$ is fixed by two simple requirements: at the matching\npoint $\\Lambda$ the Yukawa constant of $t$-quark calculated with\nthe composite operators is determined by the condition of infrared\nfixed point in the local theory \\cite{Hill}, while the Yukawa\nconstant is expressed in terms of abelian gauge coupling at the\nscales of $\\Lambda$ and $M$ due to the consistent matching\ncondition of potential (\\ref{matchnull}). At fixed $\\Lambda$ this\nsupposition makes $M$ to grow to the GUT scale, that implies the\nsolution of naturalness. So, we can read off the third point:\n\\begin{itemize}\n\\item[3.] The Yukawa constants of heaviest fermions in the local theory have\nthe matching conditions at $\\Lambda$ to the couplings given by the bi-local\nrepresentation, so that the infrared fixed point for the $t$-quarks is exactly\nreached.\n\\end{itemize}\nThe masses of $b$-quark and $\\tau$-lepton can be also calculated after the use\nof both the definite matching at $\\Lambda$ and infrared fixed points in the RG\nequations below $\\Lambda$.\n\nFinally, the potential of Higgs fields at $\\Lambda$ can serve to estimate the\nmasses of neutral scalar particles by means of RG evolution and the infrared\nfixed point for the quartic vertex $\\lambda$.\nThe important property of fixed points under consideration is that the Yukawa\nconstants and quartic coupling are given by appropriate combinations of gauge\ncoupling constants.\n\nThus, the local theory with the local Higgs fields and the electroweak symmetry\nbreaking can be certainly matched to the effective potential of sources for the\nbi-local composite operators of quarks and leptons at the scale $\\Lambda$ and\nto the corresponding Yukawa constants, which are calculable in the region of\nvirtualities $[\\Lambda; M]$, so that the fixed point matching of $t$-quark\ncoupling and the symmetry matching-condition of null effective potential\n(\\ref{matchnull}) result in $M$ living in the GUT area.\n\nThen we find the following general results:\n\n1. Three bi-local composite operators formed by the fermions of heaviest\ngeneration, develop the effective potential of their sources, so that nonzero\n{\\sc vev}'s break the electroweak symmetry. We treat these dynamics above the\nscale $\\Lambda$ as the strong self-interaction regime for three independent\nscalar higgses as equivalent to the weak self-interaction regime for three\nindependent sources of composite operators.\n\n2. The position of matching point $\\Lambda=633$ GeV is fixed by the measured\nmasses of gauge bosons, after the higgs sector is given by three independent\nscalar fields.\n\n3. At $\\Lambda$ the infrared fixed point condition is satisfied for the\nYukawa coupling of $t$ quark, only, while the couplings of $b$ quark and $\\tau$\nlepton evolve to the fixed points at lower scales in agreement with the current\ndata available. The masses of higgses evolve too.\n\n4. Under the item 3, the position of ultraviolet cut off $M\\sim\n10^{12}-10^{19}$ GeV with respect to $\\Lambda$ is given by the condition of\nzero effective potential for the sources of composite operators, that is\ngoverned by the renormalization group for the ${\\sf U}(1)$ hypercharge, so that\nwe find the natural hierarchy for $M$ and $\\Lambda$.\n\nThe paper is organized in the following way: Section II is devoted to the\ndefinition of sources for the bi-local composite operators and calculation of\neffective potential to the one-loop accuracy. The masses of gauge bosons and\nYukawa constants of fermions are evaluated in Section III at the scale\n$\\Lambda$. The exploration of infrared fixed point conditions for the Yukawa\nconstants and quartic Higgs coupling is considered in Section IV. Numerical\nestimates of masses for the heaviest fermions as well as the Higgs fields are\ngiven in Section V. In Section VI we shortly discuss the problem of generations\nand the vacuum structure. The obtained results and the points of discussion are\nsummarized in Conclusion.\n\n\\section{Sources of composite operators and effective potential}\n\nLet us define the following bare actions for the sources of bi-local operators\n\\begin{eqnarray}\nS_\\tau &=& \\int dx dy\\; N_\\tau\\cdot J_{\\tau}^\\dagger (x,y)\\; [\\bar\n\\tau_R(x)\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)}\n\\; \\tau_L(y)] +{\\rm h.c.},\\nonumber \\\\\nS_t &=& \\int dx dy\\; N_t\\cdot J_t^\\dagger (x,y)\\; [\\bar\nt_R(x)\\cdot n\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)}\n\\; \\bar n\\cdot t_L(y)] +{\\rm h.c.}, \\label{act}\\\\\nS_b &=& \\int dx dy\\; N_b\\cdot J_b^\\dagger (x,y)\\; [\\bar\nb_R(x)\\cdot n\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)} \\; \\bar n\\cdot b_L(y)] +{\\rm\nh.c.},\\nonumber\n\\end{eqnarray}\nwhere we have introduced the ${\\sf SU(3)}$-triplet unit-vector $n_i$, so that\n$\\bar n\\cdot n=1$, and the $n$-dependent terms in the effective action after\nthe account for the loop-corrections have to be averaged over $n_i$ to restore\nthe explicit invariance under the transformations of ${\\sf SU(3)}$. For\ninstance, since we generally have $n_i\\bar n_j= \\frac{1}{3} \\delta_{ij}+\n\\frac{1}{\\sqrt{3}}\\lambda^a_{ij} F_a$, we can straightforwardly check that\n$$\n\\langle n_i \\bar n_j\\rangle = \\frac{1}{3}\\delta_{ij},\\;\\;\\;\n\\langle F_a\\rangle=0,\\;\\;\\; \\langle F_a F_b\\rangle = \\frac{1}{8}\\delta_{ab},\n$$\nand so on.\n\nFor nonzero $Y_L$ and $Y_R$, which are under consideration, we can redefine\nthe factors $N_J$ to include the hypercharges into the definition of sources,\nso that $\\tilde N_p = \\alpha_Y$ and $\\tilde J_p =\\pi\\cdot Y_L \\cdot Y_R J_p$\nfor $p=t,\\; b,\\; \\tau$, which will not change the final results concerning for\nthe physical quantities: masses and couplings. In what follows we will omit the\ntildes for the sake of briefness.\n\n\\setlength{\\unitlength}{1mm}\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(70,40)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{1.ps}}\n\\put(33.5,30){$J^\\dagger$}\n\\put(10,10){$L$}\n\\put(60,10){$R$}\n\\end{picture}\n\\end{center}\n\\caption{The vertex of global source $J^\\dagger$ for the bi-local operator of\nleft-handed and right-handed fermions, where the huge dot denotes the\npropagation of hypercharge gauge boson.}\n\\label{vert}\n\\end{figure}\n\nIn the calculations of effective potential we consider the global values of\nsources independent of local coordinates: $\\partial_{x,y} J(x,y) \\equiv 0$. The\ncorresponding vertex derived from actions (\\ref{act}) is shown in Fig.\n\\ref{vert}.\nFor the $t$-quark it has the form\n\\begin{equation}\n\\Gamma_t = i\\alpha_Y\\; J^\\dagger\\; \\bar t_R(p)\\cdot n\\; \\frac{-4i}{p^2}\\;\n\\bar n\\cdot t_L(p)+{\\rm h.c.}\n\\end{equation}\n\nThe diagrams for the calculation of quadratic and quartic terms of effective\npotential are shown in Figs. \\ref{mu} and \\ref{lam}, respectively.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(70,40)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{2.ps}}\n\\put(33.5,35){$L$}\n\\put(33.5,0){$R$}\n\\put(33.5,26){$p$}\n\\put(33.5,9){$p$}\n\\put(8,17){$J$}\n\\put(59,17){$J^\\dagger$}\n\\end{picture}\n\\end{center}\n\\caption{The $J^\\dagger J$-term  in the effective potential.}\n\\label{mu}\n\\end{figure}\nThe parameters of potential\n\\begin{equation}\nV(J^\\dagger,J) = -\\mu^2\\cdot J^\\dagger J+\\lambda \\cdot (J^\\dagger J)^2,\n\\end{equation}\ncan be written down in the euclidean space as\n\\begin{eqnarray}\ni\\mu^2_B & = & -i N_J^2 \\int_{\\Lambda^2}^{M^2} \\frac{d^4 p}{(2\\pi)^4}\\;\n\\frac{4^2{\\rm tr}[P_L \\slashchar{p} \\slashchar{p}]}{(p^2)^4},\\\\\n-i4\\lambda_B & = & -2 i N_J^4 \\int_{\\Lambda^2}^{M^2} \\frac{d^4 p}{(2\\pi)^4}\\;\n\\frac{4^4{\\rm tr}[P_L \\slashchar{p} \\slashchar{p} \\slashchar{p}\n\\slashchar{p}]}{(p^2)^8},\n\\end{eqnarray}\n\\noindent\nwhich are independent of the fermion flavor. Here $P_L=\n\\frac{1}{2}(1-\\gamma_5)$ is the projector on the left-handed fermions.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(110,40)\n\\put(7,0){\\epsfxsize=4cm \\epsfbox{3.ps}}\n\\put(43.5,36){$J^\\dagger$}\n\\put(43.5,0){$J$}\n\\put(2,0){$J^\\dagger$}\n\\put(2,36){$J$}\n\\put(2,17.5){$R$}\n\\put(43.5,17.5){$R$}\n\\put(67,0){\\epsfxsize=4cm \\epsfbox{4.ps}}\n\\put(103.5,36){$J^\\dagger$}\n\\put(103.5,0){$J$}\n\\put(62,0){$J^\\dagger$}\n\\put(62,36){$J$}\n\\put(62,17.5){$L$}\n\\put(103.5,17.5){$L$}\n\\end{picture}\n\\end{center}\n\\caption{The $(J^\\dagger J)^2$-term in the effective potential.}\n\\label{lam}\n\\end{figure}\n\nSupposing $M^2\\gg \\Lambda^2$, we find\n\\begin{eqnarray}\n-\\mu^2_B & = &  N_J^2\\; \\frac{2}{\\pi^2}\\; \\frac{1}{\\Lambda^2},\\\\\n\\lambda_B & = & N_J^4\\; \\frac{4}{\\pi^2}\\; \\frac{1}{\\Lambda^8}.\n\\end{eqnarray}\n\nAs we have already mentioned in the Introduction, the effective potential has\nto be subtracted, so that at the scale $M$ it equals zero, exactly, since we\ndeal with the source of composite operators not involving some interactions\nbeyond the gauge interactions of SM. Then, we get\n\\begin{eqnarray}\n\\mu^2_R(\\Lambda) & = &  \\frac{2}{\\pi^2}\\; \\frac{1}{\\Lambda^2}\\; \\alpha_Y^2(M)\n(1-\\varkappa^2(\\Lambda)),\\\\\n\\lambda_R(\\Lambda) & = & \\frac{4}{\\pi^2}\\; \\frac{1}{\\Lambda^8}\\; \\alpha_Y^4(M)\n(1-\\varkappa^2(\\Lambda))^2,\n\\end{eqnarray}\nwhere we have introduced the notation for\n$$\n\\varkappa(\\Lambda) = \\frac{\\alpha_Y(\\Lambda)}{\\alpha_Y(M)},\n$$\nwith the normalization $\\varkappa(M)=1$. The scale-independent factors\n$\\alpha_Y^{2,4}(M)$ can be removed by the redefinition of sources: $J^\\prime =\n\\alpha_Y J$, which we imply below. In addition we introduce $J(\\Lambda) =\n\\frac{1}{\\Lambda^2} J^\\prime$ to obtain more usual notations. Then,\n\\begin{eqnarray}\n\\mu^2(\\Lambda) & = &  \\frac{2}{\\pi^2}\\;\n(1-\\varkappa^2(\\Lambda))\\;{\\Lambda^2},\\\\\n\\lambda(\\Lambda) & = & \\frac{4}{\\pi^2}\\; (1-\\varkappa^2(\\Lambda))^2.\n\\end{eqnarray}\n\nThe vacuum expectation value, {\\sc vev}, is given by $\\langle J^\\dagger\nJ\\rangle = \\frac{\\mu^2}{2\\lambda}$, so that\n\\begin{equation}\n\\langle J^\\dagger(\\Lambda) J(\\Lambda)\\rangle  = \\frac{1}{4}\\;\n\\frac{1}{1-\\varkappa^2(\\Lambda)}\\; \\Lambda^2.\n\\end{equation}\nRemember, that the potential parameters are the same for all charged heavy\nfermions: $t$-quark, $b$-quark and $\\tau$-lepton. The density of vacuum energy\nis independent of flavor, too,\n$$\nV({\\rm vac}) = -\\frac{\\mu^4}{4\\lambda} = -\\frac{1}{4 \\pi^2}\\;\\Lambda^4.\n$$\nThen the action represented as the sum of terms over the space-time intervals\nwith $d^4 x \\sim 1\/\\Lambda^4$, has the form\n$$\nS({\\rm vac}) = -\\int d^4 x\\; V({\\rm vac})\\sim \\sum \\frac{1}{4\\pi^2},\n$$\nand it is independent of $\\Lambda$.\n\n\\section{Masses of gauge bosons and Yukawa constants}\n\nThe diagrams, which result in the masses of gauge bosons, are shown in Fig.\n\\ref{mw}, where the permutations over the gauge bosons are implied.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(140,60)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{5.ps}}\n\\put(38,39){$A_j$}\n\\put(28,39){$A_i$}\n\\put(9,17){$J$}\n\\put(56,17){$J^\\dagger$}\n\\put(67,0){\\epsfxsize=5.5cm \\epsfbox{6.ps}}\n\\put(98,-5){$A_j$}\n\\put(88,39){$A_i$}\n\\put(69,17){$J$}\n\\put(116,17){$J^\\dagger$}\n\\end{picture}\n\\end{center}\n\\caption{The $(A_i A_j)$-terms in the effective potential of gauge bosons.}\n\\label{mw}\n\\end{figure}\n\n\nWe straightforwardly find that the couplings of gauge bosons are proportional\nto the differences of their charges, so that\n$$\nm_{12}^2 A_1^\\mu A_2^\\nu g_{\\mu\\nu} \\sim (Q_1^L-Q_1^R) (Q_2^L-Q_2^R) A_1^\\mu\nA_2^\\nu g_{\\mu\\nu},\n$$\nwhere $Q^{L,R}$ denote the charges of left-handed and right-handed fermions.\nThis implies that the vector-like gauge bosons, i.e. when $Q^L=Q^R$, remain\nmassless.\n\nFor the $W$- and $Z$-bosons after the subtraction procedure of $\\varkappa^2 \\to\n(1-\\varkappa^2)$, we find\n\\begin{eqnarray}\nm_W^2 & = & \\frac{4\\pi \\alpha_2}{2} \\sum_p \\frac{\\Lambda_p^2}{4\\pi^2},\\\\\nm_Z^2 & = & m_W^2 \\; \\frac{1}{\\cos^2 \\theta_W},\n\\end{eqnarray}\nwhere $\\theta_W$ is the Weinberg angle \\cite{Wein}, as usual, and the sum is\ntaken over the heavy flavors $p=t,\\; b,\\; \\tau$. As we have seen in the\nprevious section $\\Lambda_p=\\Lambda$ is independent of flavor, and, hence, we\ncan introduce the Higgs field $h_p$ with the {\\sc vev}, $\\langle h_p\\rangle=v$,\nso that\n$$\nv=\\frac{\\Lambda}{2\\pi},\\;\\;\\;\\; h_p(v) = \\frac{1}{\\pi}\n\\sqrt{1-\\varkappa^2(v)}\\;\nJ_p(v).\n$$\nThus, we get\n\\begin{equation}\nm_W^2 = \\frac{4\\pi \\alpha_2}{2}\\; 3 v^2,\n\\end{equation}\nso that $v_{\\rm SM}^2 = 3 v^2 \\approx (174\\; {\\rm GeV})^2$, when the potential\nat the scale $\\Lambda$ has the form\\footnote{There is a possibility to change\nthe convention on the prescription of scale by replacing $\\Lambda\\to v$.}\n\\begin{equation}\nV(h_p,h_p^\\dagger) = -2 \\Lambda^2\\; h_p^\\dagger\\cdot h_p + (2\\pi)^2\\;\n(h_p^\\dagger\\cdot h_p)^2.\n\\end{equation}\nWe check that the quadratic term $-2\\Lambda^2$ is exactly given by the one-loop\ncalculation in the local $\\phi^4$-theory with $\\lambda=(2\\pi)^2$ and cut-off\n$\\Lambda$.\n\nThe masses of fermions at the same scale can be derived from the diagram shown\nin Fig. \\ref{vert} by putting the fermion momenta to the given virtuality,\n$p^2=\\Lambda^2$. Then, after the appropriate subtraction [$\\varkappa\\to\n(1-\\varkappa)$] we\nget\n$$\nm_p = \\lambda_p \\cdot v,\n$$\nwith\n\\begin{eqnarray}\n\\lambda_t(v) = \\lambda_b(v) &=& \\frac{4\\pi}{3\\sqrt{2}}\\;\n\\sqrt{\\frac{1-\\varkappa(v)}{1+\\varkappa(v)}},\\\\\n\\lambda_\\tau(v) &=& 3 \\lambda_t(v).\n\\end{eqnarray}\nReplacing $v$ by $v_{\\rm SM}$, we find $\\lambda_p^{\\rm SM} =\n\\lambda_p\/\\sqrt{3}$.\n\nThus, we have calculated the masses of gauge bosons, Yukawa constants of\nheaviest fermions and the parameters of Higgs potential at the scale $\\Lambda$,\nwhich have to be matched with the quantities of local theory valid below\n$\\Lambda$.\n\n\\section{Infrared fixed points}\nTo the moment we have the local theory with three neutral Higgs fields, which\nare coupled with the appropriate heavy fermions in each sector, with the\ncut-off $\\Lambda$, where the Yukawa couplings have to be matched with the\nvalues calculated in the effective potential of sources for the composite\noperators.\n\nThe one-loop RG equations for the couplings\\footnote{The corresponding two-loop\nRG equations are given in ref. \\cite{Machacek:1983fi}. We shortly comment the\ninfluence of two-loop corrections below.} have the form \\cite{Pok}\n\\begin{eqnarray}\n\\frac{d\\ln \\lambda_t}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[ \\frac{9}{2}\n\\lambda_t^2 -8 g_3^2 -\\frac{17}{12} g_Y^2 -\\frac{9}{4} g_2^2\\right],\\nonumber\\\\\n\\frac{d\\ln \\lambda_b}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[ \\frac{9}{2}\n\\lambda_b^2 -8 g_3^2 -\\frac{5}{12} g_Y^2 -\\frac{9}{4} g_2^2\\right],\n\\label{RG1}\\\\\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[\n\\frac{5}{2} \\lambda_\\tau^2 -\\frac{15}{4} g_Y^2 -\\frac{9}{4}\ng_2^2\\right],\\nonumber\n\\end{eqnarray}\nwhere $g_3^2 = 4\\pi \\alpha_s$ is the QCD coupling, $g_Y^2 = 4\\pi \\alpha_Y$ is\nthe hypercharge coupling, and $g_2^2 = 4\\pi \\alpha_2$ is the $\\sf SU(2)$-group\ncoupling. At ``low'' virtualities about $v\\sim 100$ GeV, the dominant\ncontribution to the $\\beta$-functions of quark couplings is given by QCD. We\nsuppose that the value of matching point $\\Lambda$ is dictated by the fixed\npoint condition for the $t$-quark: $\\frac{d\\ln \\lambda_t}{d\\ln \\mu}=0$\n\\cite{Hill}, i.e.\n\\begin{equation}\n\\lambda_t^2(v) = \\frac{64\\pi}{9}\\; \\alpha_s(v)+\\frac{34\\pi}{27}\\;\n\\alpha_Y(v)+2\\pi \\alpha_2(v)\\approx \\frac{64\\pi}{9}\\; \\alpha_s(v),\n\\label{app1}\n\\end{equation}\nwhen the matching gives\n\\begin{equation}\n\\lambda_t^2(v) = \\frac{8\\pi^2}{9}\\; \\frac{1-\\varkappa(v)}{1+\\varkappa(v)}.\n\\end{equation}\nTherefore, we find\n\\begin{equation}\n\\varkappa(v) = \\frac{1-\\frac{8\\alpha_s(v)}{\\pi}}{1+\\frac{8\\alpha_s(v)}{\\pi}}.\n\\end{equation}\nDue to the contribution by the hypercharge the difference between the RG\nequations for $\\lambda_b$ and $\\lambda_t$ causes the reach of infrared fixed\npoint for the $b$-quark at a lower scale than for the $t$-quark. Indeed, the\nfixed point condition for the $b$-quark reads off\n\\begin{equation}\n\\frac{9}{2}(\\lambda_b^2(\\mu)-\\lambda_t^2(\\mu)) = -g_Y^2(\\mu).\n\\label{fb}\n\\end{equation}\nMaking use of matching condition $\\lambda_b(v)=\\lambda_t(v)$, we can write down\n$$\n\\frac{d\\ln \\lambda_t\/\\lambda_b}{d\\ln \\mu}= - \\frac{1}{(4\\pi)^2}\\; g_Y^2,\n$$\nfor small changes, so that\n\\begin{equation}\n\\lambda_b(\\mu)-\\lambda_t(\\mu)\\approx -\\frac{\\lambda_t(\\mu)}{(4\\pi)^2}\\; g_Y^2\\;\n\\ln\\frac{\\Lambda}{\\mu}.\n\\label{rb}\n\\end{equation}\nThen, we can derive from (\\ref{fb}) and (\\ref{rb}) the following estimate of\ncurrent mass for the $b$-quark\n\\begin{equation}\n\\ln\\frac{m_t}{m_b(\\hat v_b)} = \\frac{\\pi}{4\\alpha_s(m_b(\\hat v_b))},\n\\end{equation}\nwhere the current mass of $t$-quark is given by\n$$\nm_t(m_t) = \\frac{8}{3} \\sqrt{\\pi \\alpha_s(v)}\\cdot v,\n$$\nsince the evolution of $t$-quark mass above the scale $v$ is determined by the\nrunning of effective constant, which is negligibly small in the interval\n$[v,m_t]$, and, hence, $m_t(m_t)\\approx m_t(v)$ with quite a high accuracy.\nThe scale of $b$-quark normalization is given by the following\n$$\nm_b(\\hat v_b) = \\frac{8}{3}\\sqrt{\\pi\\alpha_s(\\hat v_b)}\\cdot \\hat v_b,\n$$\nand we use the QCD evolution to extract the current mass of $b$-quark at the\nscale of its value\n$$\nm_b(m_b) = m_b(\\hat v_b)\\left(\\frac{\\alpha_s(m_b(m_b))}{\\alpha_s(\\hat\nv_b)}\\right)^{12\/25}.\n$$\n\nNext, we can evaluate the mass of $\\tau$-lepton in the same manner. At low\nenergies we modify the RG equation for the $\\tau$-coupling, neglecting the\nfour-fermion weak interactions and taking into account the photon contribution.\nSo, we have\n\\begin{equation}\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu}  =  \\frac{1}{(4\\pi)^2}\\; \\left[\n\\frac{5}{2} \\lambda_\\tau^2 -24\\pi \\alpha_{em}\\right],\n\\end{equation}\nand the infrared fixed point condition reads off\n\\begin{equation}\n\\lambda_\\tau^2 = \\frac{48\\pi}{5}\\; \\alpha_{em}.\n\\label{ftau}\n\\end{equation}\nThe change of $\\lambda_\\tau$ from the matching value\n$\\lambda_\\tau^2=9\\lambda_t^2 = 64\\pi\\alpha_s$ can be found in the solution of\n\\begin{equation}\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu}  \\approx  \\frac{1}{(4\\pi)^2}\\;\n\\frac{5}{2} \\cdot 9 \\lambda_t^2 \\approx 40\\; \\frac{\\alpha_s}{4\\pi},\n\\label{tauev}\n\\end{equation}\nso that\n\\begin{equation}\n\\lambda_\\tau(\\mu) =\n\\lambda_\\tau(v)\\cdot\\left(\\frac{\\alpha_s(\\mu)}\n{\\alpha_s(v)}\\right)^{-\\frac{40}{2 b_3}},\n\\label{rtau}\n\\end{equation}\nwhere $b_3 = 11-\\frac{2}{3}n_f =9$ at $n_f=3$. From (\\ref{ftau}), (\\ref{rtau})\nand $\\lambda_t^2= 64\\pi \\alpha_s\/9$ we deduce the relation\n\\begin{equation}\n\\alpha_s(m_\\tau) =\n\\alpha_s(v)\\cdot\\left(\\frac{3}{20}\\;\\frac{\\alpha_{em}(m_\\tau)}{\\alpha_s(v)}\n\\right)^{-\\frac{9}{40}}.\n\\end{equation}\nNote, that the one-loop evolution to such the large change of scales is quite a\nrough approximation. To improve the estimate of $\\tau$-lepton mass we integrate\n(\\ref{tauev}) numerically with the same boundary conditions and extract the\nvalue under consideration.\n\nLet us consider the way to estimate the masses of neutral Higgs bosons. The RG\nequations for the quartic couplings of scalar particles with the heaviest\nfermions are represented by the following:\n\\begin{equation}\n\\frac{d \\lambda}{d\\ln\\mu} = \\frac{3}{2\\pi^2}\\left[\\lambda^2-\\frac{a_p}{4}\\;\n\\lambda_p^4\\right],\n\\end{equation}\nwhere $a_t=a_b=1$, $a_\\tau=\\frac{1}{3}$, and we neglect the contribution given\nby the electroweak gauge couplings. This approximation is quite reasonable,\nsince at $\\Lambda(v)$ the quartic couplings $\\lambda(v)=(2\\pi)^2$ dominate. For\nthe Higgs fields coupled to the $t$- and $b$-quarks, the infrared fixed points\ncoincide with each other to the order under consideration, so that\n$$\n\\lambda(\\mu_H) \\approx \\frac{1}{2}\\; \\lambda^2_{t,b}(\\mu_H) = \\frac{32\\pi}{9}\\;\n\\alpha_s(\\mu_H),\n$$\nwhich implies that the corresponding masses of scalars are degenerated with a\nhigh accuracy. Let us evaluate the scale of reaching the infrared fixed point.\nThe evolution can be approximated at large $\\lambda$ by the equation\n$$\n\\frac{1}{\\lambda(\\mu_H)} =\n\\frac{1}{\\lambda(v)}+\\frac{3}{2\\pi^2}\\;\\ln\\frac{v}{\\mu_H},\n$$\nso that we derive\n\\begin{equation}\n\\ln \\frac{v}{\\mu_H} \\approx\n\\frac{3\\pi}{16\\alpha_s(\\mu_H)}-\\frac{1}{6},\n\\label{str-h}\n\\end{equation}\nIf we use the RG evolution for the QCD\ncoupling \\cite{PEP}\n$$\n\\frac{1}{\\alpha_s(\\mu_H)} = \\frac{1}{\\alpha_s(v)}-\n\\frac{b_3}{2\\pi}\\; \\ln\\frac{v}{\\mu_H},\n$$\nat $n_f=5$, we arrive to\n\\begin{equation}\n\\ln \\frac{v}{\\mu_H} \\approx \\frac{6}{55}\\;\\frac{\\pi}{\\alpha_s(v)},\n\\end{equation}\nalthough the straightforward equation for the scale in (\\ref{str-h}) can be\nmore accurate numerically.\n\nThus, following the general relation for the mass of Higgs field,\n$$\nm_H(\\mu) = 2 \\sqrt{\\lambda(\\mu)}\\cdot v,\n$$\nwe have the estimates\n\\begin{eqnarray}\nm_H(v)     &=& 4\\pi\\cdot v,\\\\\nm_H(\\mu_H) &=& \\frac{8}{3}\\sqrt{2\\pi\\alpha_s(\\mu_H)} \\cdot v.\n\\end{eqnarray}\nAs for the Higgs field coupled to the $\\tau$-lepton, it is quite easily\nrecognize that the corresponding scale $\\mu$ is much greater than for the\nscalars coupled with the heaviest quarks, and, hence, its mass is greater than\nwe have considered above. Indeed, we can use the evolution of $\\lambda_\\tau$ at\nlarge scales, where it is driven as $\\lambda_\\tau = 3 \\lambda_t$, so that we\nderive the relation analogous to (\\ref{str-h})\n\\begin{equation}\n\\ln \\frac{v}{\\mu_{H_\\tau}} \\approx\n\\frac{\\pi}{16\\sqrt{3}\\alpha_s(\\mu_{H_\\tau})}-\\frac{1}{6},\n\\label{str-htau}\n\\end{equation}\nand\n$$\nm_{H_\\tau}(\\mu_{H_\\tau}) =\n{8}\\sqrt{\\frac{2}{\\sqrt{3}}\\pi\\alpha_s(\\mu_{H_\\tau})} \\cdot v.\n$$\nTo the moment we are ready to get numerical estimates.\n\n\\section{Numerical evaluation and the naturalness}\n\nFirst of all, the {\\sc vev}'s of Higgs fields are directly given by the masses\nof gauge bosons, so that\n$$\nv = 100.8\\pm 0.1\\; {\\rm GeV},\n$$\nand the cut-off\n$$\n\\Lambda = 2\\pi v= 633.0\\pm 0.6\\; {\\rm GeV,}\n$$\nwhere we use the experimental data shown in Table \\ref{tab}.\n\n\\begin{table}[th]\n\\begin{center}\n\\begin{tabular}{||p{2cm}|p{2.5cm}||}\n\\hline\n$m_W$, GeV      & $80.41\\pm 0.09 $ \\\\\n\\hline\n$\\alpha_2^{-1}$ & $29.60\\pm 0.04 $ \\\\\n\\hline\n$m_t$, GeV      & $174\\pm 5      $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The experimental data on the electroweak parameters\n[5,6].}\n\\label{tab}\n\\end{table}\n\nThe estimates for the masses of fermions depend on the values of QCD coupling\nconstant. We put the value\\footnote{The central value is slightly displaced\nfrom the ``world average'' $\\alpha_s(m_Z)=0.119\\pm 0.002$ \\cite{PDG}, though it\nis within the current uncertainty. However, this parameter corresponds to the\nLEP fit \\cite{SM} as well as to the recent global fit of structure functions\n\\cite{AK}.}\n$$\n\\alpha_s(m_Z)=0.122\\pm 0.003,\n$$\nwhich corresponds to the $\\Lambda^{(5)}_{\\overline{\\rm MS}}=255\\pm 45$ MeV in\nthe three-loop approximation for the $\\beta$-function. We suppose that the\nthreshold values for the changing the number of active quark flavors are equal\nto $\\hat m_b=4.3$ GeV and $\\hat m_c=1.3$ GeV. The variation of threshold values\nis not so important in the estimates in contrast to the uncertainty in\n$\\alpha_s$, which dominates in the error-bars.\n\nThen we can numerically solve the equations in the previous section to find the\ncurrent masses\n\\begin{eqnarray}\nm_t(m_t) & = & 165\\pm 1\\; {\\rm GeV,} \\nonumber\\\\\nm_b(m_b) & = & 4.18\\pm 0.38\\; {\\rm GeV,} \\\\\nm_\\tau(m_\\tau) & = & 1.78\\pm 0.27\\; {\\rm GeV.}\\nonumber\n\\end{eqnarray}\nThe one-loop relation of perturbative QCD for the pole mass of quark \\cite{PEP}\nis given by\n$$\nm^{(p)} = m(m) \\left(1+\\frac{4}{3\\pi}\\; \\alpha_s(m)\\right).\n$$\nThen we estimate\n\\begin{eqnarray}\nm_t^{(p)} &=& 173\\pm 2\\; {\\rm GeV},\\nonumber\\\\\nm_b^{(p)} &=& 4.62\\pm 0.40\\; {\\rm GeV}.\\nonumber\n\\end{eqnarray}\nThe QED correction to the $\\tau$-lepton mass is negligibly small.\n\nWe see that the $t$-quark mass is in a good agreement with the direct\nmeasurements. The $b$-quark mass is in the desirable region. It is close to\nthat of estimated in the QCD sum rules \\cite{SVZ}, where $m_b(m_b)=4.25\\pm\n0.15$ GeV \\cite{BM}, and in the potential approach \\cite{KKO}, where\n$m_b(m_b)=4.20\\pm 0.06$ GeV. It is worth to note that the pole mass is not the\nvalue, which has a good convergency in the OPE approach (see references in\n\\cite{BM,KKO}), so we present it to the first order for the sake of reference.\nHowever, we stress also that the deviations from the central values are caused\nby the uncertainties in the $\\alpha_s$ running.\n\nThe infrared fixed masses of neutral scalars, coupled with the $t$- and\n$b$-quarks and the $\\tau$-lepton, equal\n\\begin{equation}\nm_H = 306\\pm 5\\; {\\rm GeV,}\\;\\;\\;\\; m_{H_\\tau} = 552\\pm 9\\; {\\rm GeV,}\n\\label{hg}\n\\end{equation}\nwhich can be compared with the global fit of SM at LEP yielding $m_H=\n76^{+85}_{-47}$ GeV \\cite{SM}. The central value of this fit was recently\nexcluded by the direct searches at modern LEP energies, where the constraint\nwas obtained $m_H > 95$ GeV \\cite{SM,PEP}. We expect, however, that\nmany-doublet models of Higgs sector have a different connection to the LEP\ndata. Indeed, the fit of SM with the single Higgs particle yields the value for\nthe logarithm $l_H=\\log_{10} m_H^{\\rm SM}[{\\rm GeV}] = 1.88^{+0.33}_{-0.41}$,\nwhereas this correction basically contributes into the observed quantities due\nto the coupling to the massive gauge bosons. Then, we can write down the\nfollowing approximation for this value in the model under consideration:\n$$\nl_H = \\frac{1}{3}\\sum_p \\kappa_p \\log_{10}\nm_{H_p}[{\\rm GeV}],\n$$\nwhere the factor $\\frac{1}{3}$ represents the fraction of scalar coupling in\nthe squares of gauge boson masses, respectively for $p=t,\\; b,\\; \\tau$, and\n$\\kappa_p$ stands for the possible formfactors at high virtualities of the\norder of masses of Higgs fields. To test, we put the simple approximation\n$$\n\\kappa_p \\approx \\frac{1}{1+\\frac{m^2_{H_p}}{\\Lambda^2}},\n$$\nwhich results in $\\kappa_t=\\kappa_b$ close to unit, and $\\kappa_\\tau\\sim\n\\frac{1}{4}$, so that the value under consideration is equal to\n$$\nl_H \\approx 1.86,\n$$\nthat is optimistically close to what was observed at LEP. So, the values in\n(\\ref{hg}) are not in contradiction with the current data.\n\nNext, since we deal with the strongly coupled version of Higgs sector\n(remember, that $m_H(v)\\approx 1267$ GeV), we need more careful consideration\nof effective potential to take into account the higher dimensional operators,\nrepresenting the multi-higgs couplings. So, we keep (\\ref{hg}) as soft\nestimates of masses for the Higgs fields, which implies that the decays into\nthe massive gauge bosons are the dominant modes for these scalar particles .\n\nFinally, we evaluate the scale $M$, where the electroweak symmetry has to be\nexactly restored. The value of $\\varkappa(v)$ is equal to\n$$\n\\varkappa(v) = \\frac{\\alpha_Y(v)}{\\alpha_Y(M)} = 0.532\\pm 0.005,\n$$\nwhich implies $\\alpha_1^{-1}(M) \\approx 32$. The implication of $\\varkappa$ for\n$M$ depends on the running of $\\alpha_Y=\\frac{3}{5}\\alpha_1$ \\cite{PEP}:\n$$\n\\frac{1}{\\alpha_1(M)} = \\frac{1}{\\alpha_1(v)}+\n\\frac{b_1}{2\\pi}\\; \\ln\\frac{M}{v},\n$$\nwhere $b_1$ is model-dependent. So, in the SM $b_1=-\\frac{4}{3}\\;\nn_g-\\frac{1}{10}\\; n_h$ with $n_g=3$ being the number of fermion generations,\n$n_h$ is the number of Higgs doublets, we obtain\\footnote{Numerically, we put\n$\\alpha_1^{-1} = 58.6$ for the order-of-magnitude estimate.}\n$$\nM_{\\rm SM} \\approx 2.5\\cdot 10^{19}\\; {\\rm GeV,}\n$$\nwhen in the SUSY extension $b_1=-2\\; n_g-\\frac{3}{10}\\; n_h$, so that\n$$\nM_{\\rm SUSY} \\approx 7\\cdot 10^{12}\\; {\\rm GeV.}\n$$\nHence, we obtain the broad constraints\n$$\nM = 7\\cdot 10^{12}-2.5\\cdot 10^{19}\\; {\\rm GeV,}\n$$\nand the value strongly depends on the set of fields in the region above the\ncut-off $\\Lambda$. At present, we cannot strictly draw a conclusion on a\npreferable point. However, we can state that the offered mechanism for the\nbreakdown of the electroweak symmetry solves the problem of naturalness, since\nthe observed ``low'' scale of gauge boson masses is reasonably related to the\n``high'' scale of GUT or even Planck mass.\n\nFinally, we comment on possible uncertainties of numerical\nestimates and a role of two-loop corrections. First, we analyze\nthe subleading terms in eq.(\\ref{app1}). The gauge charges\nneglected in the fixed point condition of (\\ref{app1}) result in\nthe displacement of $t$ quark mass by a value about 4 GeV, if we\ndo not change the normalization of QCD coupling constant. In this\nway, we note that under the account of gauge charge corrections in\n(\\ref{app1}) the same central value of $t$ quark mass, i.e. 165\nGeV, is reproduced at $\\alpha_s(m_Z^2) =0.118$, which coincides\nwith the Particle Data Group ``world-average''. Next, the two-loop\ncorrections in the RG equations for the Yukawa couplings\n\\cite{Machacek:1983fi} as applied to the $t$ quark lead to an\nadditional displacement of fixed point value. However, in this\ncase we have to take into account the one-loop correction to the\nrelation between the current mass and the pole mass of $t$ quark\ndue to the Higgs sector, that results in the following additive\nrenormalization of $m_t(m_t)$ \\cite{HK}\n$$\n\\frac{\\delta m_t(m_t)}{m_t(m_t)} = - \\frac{1}{16\\pi^2}\\,\\frac{9}{2}\\,\n\\frac{m_t^2}{v_{\\rm SM}^2},\n$$\nat the higgs mass $m_H\\approx 2 m_t$. The above correction compensates the\ndisplacement due to the two-loop modification of infrared fixed-point condition\nfor the $t$ quark.\n\nSecond, we study the two-loop corrections to the fixed point condition for the\n$b$ quark. The corresponding modification of (\\ref{fb}) reads off\n\\begin{equation}\n\\frac{9}{2}(\\lambda_b^2(\\mu)-\\lambda_t^2(\\mu)) \\approx -g_Y^2(\\mu)\\left(1 -\n\\frac{1}{16\\pi^2}\\,\\frac{4}{3}\\,\\lambda_t^2(\\mu)\\right)+O(g^4),\n\\label{fb2}\n\\end{equation}\nthat results in the appropriate correction in (\\ref{rb}), {\\it\nviz.}, in the small change of slope in front of log about 2\\%. The\nsolution of equations for the running $b$ quark mass under the\nvariations caused by the introduction of two-loop corrections and\nthe uncertainty in the coupling constant of QCD is shown in Fig.\n\\ref{uncert}. We can straightforwardly see that the variation of\nslope in the RG equation for the Yukawa constant of $b$ quark due\nto the two-loop corrections results in uncertainties, which are\nmuch less than the variation of $b$ quark mass caused by the\nuncertainties in the running coupling constant of QCD at moderate\nvirtualities about the $b$ quark mass. Therefore, the dominant\norigin of uncertainty for the $b$ quark mass is the normalization\nof $\\alpha_s$. The same conclusion can be drawn for the mass of\n$\\tau$ lepton. Therefore, the uncertainty in the estimates of\nmasses for the $b$ quark and the $\\tau$ lepton is not essentially\nchanged by the introduction of two-loop corrections, while the\nvalue of $t$ quark mass depends on the normalization of $\\alpha_s$\nas well as the two-loop corrections combined, so that the\nuncertainty in the current mass can reach 4 GeV in $m_t$.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(150,50)\n\\put(0,0){\\epsfxsize=7cm \\epsfbox{9.eps}}\n\\put(80,0){\\epsfxsize=7cm \\epsfbox{10.eps}}\n\\put(0,45){$\\lambda_b\/\\lambda_t$}\n\\put(80,45){$\\lambda_b\/\\lambda_t$}\n\\put(63,-2){$\\ln m_t\/\\mu$}\n\\put(140,-2){$\\ln m_t\/\\mu$}\n\\end{picture}\n\\end{center}\n\\caption{The variation of RG solution for the $b$ quark Yukawa constant under\nthe introduction of two-loop corrections with respect to the one-loop result\n(solid lines) and the uncertainty caused by the change in the normalization of\n$\\alpha_s$ with $\\Lambda_{\\overline{\\rm MS}}^{(5)} = 200$ MeV and\n$\\Lambda_{\\overline{\\rm MS}}^{(5)} = 280$ MeV (the band). At the right figure\nwe scale the square region marked in the left picture.}\n\\label{uncert}\n\\end{figure}\n\nAs for the estimates of masses for the scalar fields, we emphasize that they\ngive preliminary results, and further investigations are in progress, since we\nshould, first, sum up subleading terms with higher powers of the higgs field\nsquared in the effective potential and, second, consider complete RG equations\nfor the quartic self-coupling, including suppressed terms. Nevertheless, our\npreliminary estimates show that the scalar fields should be significantly\nheavy.\n\n\\section{Generations, the number of Higgs fields and vacuum}\n\nIn the previous sections we have introduced three independent global sources\nfor the bi-local operators composed by the fermions of the heaviest generation,\ni.e., $t$ quark, $b$ quark and $\\tau$ lepton. In the SSIR, these sources\nacquire the effective potentials providing the spontaneous breaking of\nelectroweak symmetry. Below the scale $\\Lambda$ we assume the connection of\nsuch the potentials with the potentials of local Higgs fields. Thus, we suppose\nthe introduction of three independent local Higgs doublets at the low energies.\nTherefore, we suggest the condensation of sources related with the heaviest\ngeneration only.\n\nWe have found the `democratic' form for the potentials of independent sources\nin the SSIR. All three potentials have the same values of quadratic and quartic\ncouplings, while we suggest evidently broken `democracy' for the fermion\ngenerations, since we do not introduce the condensation of sources for the\ncomposite operators built of junior fermions.\n\nIn this section we describe a possible development on the problem of fermion\ngenerations and the structure of vacuum in the Higgs sector.\n\nSo, let us introduce the notation of normalized {\\sc vev}'s for the global\nsources connected with the Higgs fields as follows:\n$$\n\\chi_p = h_p\/v, \\;\\;\\; p=\\tau,\\, t,\\, b,\n$$\nand the corresponding vacua $| 0_p\\rangle$, so that\n\\begin{equation}\n\\langle 0_p| \\chi_{p'} |0_{p''}\\rangle = \\delta_{pp'}\\delta_{p'p''}.\n\\label{norm}\n\\end{equation}\nThen we easily find that the mass terms\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot \\chi_\\tau + {\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot \\chi_t\n+ {\\rm h.c.}, \\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot \\chi_b\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\ncould be represented by means of fields\n\\begin{equation}\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\\;\n\\left(\\begin{array}{ccc} 1 & 1 & 1 \\\\ 1 & \\omega & \\omega^2 \\\\ 1 & \\omega^2 &\n\\omega \\end{array}\\right)\n\\cdot\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n = U \\cdot\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right) ,\n\\label{trans}\n\\end{equation}\nas follows\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot\n(\\varphi_1+\\varphi_2+\\varphi_3) +\n{\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot (\\varphi_1+\\omega^2 \\varphi_2+\\omega\n\\varphi_3)\n+ {\\rm h.c.}, \\label{generation}\\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot (\\varphi_1+\\omega \\varphi_2+\\omega^2\n\\varphi_3)\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\nwhere we omit the Yukawa couplings and use the matrix $U$ defined in terms of\n$\\omega = \\exp(i\\frac{2\\pi}{3})$.\n\nSuch the transformation in (\\ref{trans}) relates the `heavy' basis of $\\chi_p$\nwith the `democratic' basis of $\\varphi_i$. So, the definition (\\ref{trans})\ncan be equivalently changed by permutations of $\\chi_\\tau \\leftrightarrow\n\\chi_t \\leftrightarrow \\chi_b$ or permutations of columns in the matrix $U$.\nSuch the permutations correspond to the finite cyclic group ${\\mathbb Z}_3$\nwith the\nbasis $\\omega$, so that complex phases of $\\varphi_i$ are given by\n$e^{iq\\frac{2\\pi}{3}}$ with the charges $q=(0,-1,1)$ of\n$(\\varphi_1,\\varphi_2,\\varphi_3)$.\n\nFurther, we can note that the vacuum fields have simple connections as follows:\n\\begin{eqnarray}\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1^{~} \\\\ 1\\\\ 1\\end{array}\\right) & \\Rightarrow &\n\\langle 0_\\tau| \\boldsymbol \\chi_1 |0_\\tau \\rangle =\n\\langle 0_\\tau|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_\\tau \\rangle =\n\\left(\\begin{array}{c} 1 \\\\ 0\\\\ 0\\end{array}\\right), \\nonumber\\\\[4mm]\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1 \\\\ \\omega\\\\ \\omega^2\\end{array}\\right) & \\Rightarrow &\n\\langle 0_t| \\boldsymbol \\chi_2 |0_t \\rangle =\n\\langle 0_t|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_t \\rangle=\n\\left(\\begin{array}{c} 0 \\\\ 1\\\\ 0\\end{array}\\right), \\\\[4mm]\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1 \\\\ \\omega^2\\\\ \\omega\\end{array}\\right) & \\Rightarrow &\n\\langle 0_b| \\boldsymbol \\chi_3 |0_b \\rangle =\n\\langle 0_b|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_b \\rangle =\n\\left(\\begin{array}{c} 0 \\\\ 0\\\\ 1\\end{array}\\right), \\nonumber\n\\end{eqnarray}\nwhere the conditions of normalization (\\ref{norm}) are reproduced.\n\nLet us {\\it postulate the extended definition of the vacuum}\n\\begin{equation}\n|vac \\rangle = |0_\\tau \\rangle \\otimes |0_t \\rangle \\otimes |0_b \\rangle ,\n\\label{vacuum}\n\\end{equation}\nwhich implies the ${\\mathbb Z}_3$ symmetry of the vacuum. Then, the couplings\nintroduced\nin (\\ref{generation}) are extended to three generations of fermions, whereas\nthe only generation of $\\tau$, $t$ and $b$ is heavy, while two junior\ngenerations are massless.\n\nThe vacuum definition (\\ref{vacuum}) could be treated as the following\nassumption:\n\n\\noindent\n{\\sf The number of generations equals the number of charged flavors in the\ngeneration as well as the number of Higgs fields in the local phase.}\n\nThus, we postulate the ${\\mathbb Z}_3$ symmetry of the vacuum as the\nfundamental\ndynamical principal of the theory. Moreover, we suggest that this symmetry of\nthe vacuum is exact, so that it is conserved under radiative corrections to the\nYukawa constants of fermions.\n\nA realistic description of generations, i.e., a model with nonzero masses of\njunior fermions is not the problem under the current consideration, and it is\nbeyond the scope of this work. Nevertheless, we add two notes.\n\nFirst, a general structure of Yukawa interactions with the ${\\mathbb Z}_3$\nsymmetry of\nthe vacuum has the form\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot (g_{1}^\\tau\n\\varphi_1+g_{2}^\\tau\n\\varphi_2+ g_{3}^\\tau\n\\varphi_3) + {\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot ( g_{1}^t \\varphi_1+ g_{2}^t\n\\omega^2 \\varphi_2+\ng_{3}^t \\omega \\varphi_3)\n+ {\\rm h.c.}, \\label{real}\\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot ( g_{1}^b \\varphi_1+ g_{2}^b \\omega\n\\varphi_2+\ng_{3}^b \\omega^2 \\varphi_3)\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\nwhere the constants $g_i$ can be restricted by the following conditions:\n$g_{2,3}$ are real, while $g_1$ can be complex. So, the symmetric point $$\ng_1=g_2=g_3=1,\n$$\nrestores the hierarchy of single heavy generation and two massless generations.\nThe development of {\\it ansatz} (\\ref{real}) with the realistic values of\nparameters consistent with the current data on the quark masses and mixing CKM\nmatrix of charged quark currents was given in Ref. \\cite{generations}.\n\nSecond, the same form of mass matrix following from (\\ref{real}) could result\nin the leading order symmetry in the sector of neutrinos. Indeed, if we put\n$$\ng_1=e^{i\\frac{2\\pi}{3}},\\;\\;\\;  |g_1|=g_2=g_3=1,\n$$\nthen we get the completely degenerate neutrinos, while small deviations in the\n$g_1$ phase and absolute values of $g_i$ will result in small differences of\nneutrino masses squared as observed in the neutrino oscillations\n\\cite{neutrinos}.\n\n\\section{Conclusion}\n\nIn this work we have argued that there are three starting points for the\npresented consideration of Higgs sector of electroweak theory. These\nmotivations are the following:\n\nFirst, as well known, in the Standard Model the Yukawa coupling constant of $t$\nquark obtained from the measurement of $t$ quark mass, is close to its value in\nthe infrared fixed point derived from the renormalization group equation. If\nthe Higgs sector is extended to three scalar fields separately coupled to\nthree heaviest charged fermions, i.e., the $t$, $b$ quarks and $\\tau$ lepton,\nwith the same vacuum expectation values of Higgs fields, then the Yukawa\ncoupling constant of $t$ quark is {\\bf exactly} posed to the infrared fixed\npoint. The challenge is whether this coincidence is accidental or not. We treat\nthis fact as the fundamental feature of dynamics determining the development of\nmasses. Moreover, the problem acquires an additional insight because the only\nfermion generation is heavy, while two junior generations are approximately\nmassless. These features can be attributed by introducing the fundamental\n${\\mathbb Z}_3$\nsymmetry of the vacuum, so that this symmetry is conserved under the radiative\nloop corrections responsible for the development of nonzero masses of junior\ngenerations.\n\nSecond, the strong self-interaction regime in the Higgs sector of Standard\nModel at large virtualities can be treated as the indication of nonlocality,\ni.e., the compositeness of operators relevant to the electroweak symmetry\nbreaking. We have introduced a separation of virtuality regions: the local\nHiggs phase in the range of $[0;\\, \\Lambda]$, the nonlocal strong\nself-interaction regime in the range $[\\Lambda;\\, M]$, and the symmetric phase\nabove $M$. We have determined a form of composite operators and their\nconnection to the local phase by considering the second order of effective\naction in the SM.\n\nThird, the development of effective potential for the global sources of\ncomposite operators from the point of symmetric phase $M\\sim 10^{12}-10^{19}$\nGeV is stopped in the infrared fixed point $\\Lambda \\approx 633$ GeV for the\nYukawa coupling constant of $t$ quark. If the dynamics of evolution is given by\nthe same electroweak group, then the large logarithm of $\\ln M\/\\Lambda$ is\nclose to the value appearing in the calculation of GUT scale. So, since the\nbreaking of electroweak symmetry and the fermion mass generation involve the\ncomposite operators with both left and right handed fermions, the gauge\ninteraction of ${\\sf U}(1)$ group determines the evolution of parameters of the\neffective potential in the strong self-interaction regime. Then the logarithm\nof $\\ln M\/\\Lambda$ in the coupling $g_1$ has the value depending on the set of\nfundamental fields above the scale $\\Lambda$. Anyway, $M$ should be close to\nthe\nGUT scale, and this fact implies the solution of problem on the naturalness.\n\nThen, following the above motivations, we have evaluated the basic parameters\nof the model. We have calculated the effective potential for the sources of\ncomposite operators, responsible for the breaking down the electroweak symmetry\nand generation of masses for the gauge bosons and heaviest fermions. The\ncorresponding couplings serve as the matching values for the quadratic and\nquartic constants in the potential of local Higgs fields as well as the Yukawa\ninteractions at the scale $\\Lambda$, which is the ultraviolet cut-off for the\nlocal theory and the low boundary of $[\\Lambda;M]$-range for the effective\npotential of sources coupled with the bi-local composite operators of quarks\nand leptons. At $M$ the local gauge symmetry is restored, so that the effective\npotential is exactly equal to zero.\n\nPosing the matching of Yukawa constant for the $t$-quark to the infrared fixed\npoint at the scale $\\Lambda$, related to the gauge boson masses, we have found\nthe null-potential value $M$ in the range of GUT park, which indicates the\nsolution of naturalness. The exploration of fixed points has resulted in the\nfollowing current masses of heaviest fermions: $m_t(m_t) = 165\\pm 4$ GeV,\n$m_b(m_b) = 4.18\\pm 0.38$ GeV and $m_\\tau(m_\\tau) = 1.78\\pm 0.27$ GeV. Two\ndegenerated neutral Higgs fields have the infrared fixed mass $m_H= 306\\pm 5$\nGeV, and the third scalar has the mass $m_{H_\\tau}= 552\\pm 9$ GeV. So, the\nestimates do not contradict with the current constraints, coming from the\nexperimental data.\n\nSome questions need for an additional consideration. To the moment, discussing\nno possible ways to study, we focus on the directions requiring a progress.\n\n\\begin{itemize}\n\\item[1.]\nWhat is a picture for the generation of Yukawa constants, responsible for the\nmasses of ``junior'' fermions?\n\\end{itemize}\nAs we have supposed in the paper, three sectors of Higgs fields are coupled to\nthe appropriate heavy fermions, so that we need speculations based on a\nsymmetry causing the junior generations to be massless to the leading order.\n\\begin{itemize}\n\\item[2.]\nWhat are the constraints on the model parameters as follows from the current\ndata on the flavor changing neutral currents and precision measurements at LEP?\n\\end{itemize}\nSo, we expect that this point is not able to bring serious objections against\nthe model, since we do not involve any interactions distinct from the gauge\nones, composing the SM.\n\\begin{itemize}\n\\item[3.]\nThe most constructive question is a supersymmetric extension of mechanism under\nconsideration. Can SUSY provide new features or yield masses of super-partners?\n\\end{itemize}\nTo our opinion, the SUSY extension is more complicated, since there are many\ndifferent relations between the mixtures of various sparticles, which all are\nexpected to be essentially massive ($\\tilde m\\sim \\Lambda$) in contrast to the\nSM, wherein the junior generations are decoupled from the Higgs fields to the\nleading order.\n\\begin{itemize}\n\\item[4.]\nA simple application, we think, is an insertion of the model into the TeV-scale\nKaluza-Klein ideology, being under intensive progress now \\cite{KK}.\n\\end{itemize}\nSo, $\\varkappa(v)$ transforms its logarithmic behavior to the\npower dependence on the scale. Then, the $M$ returns to a value\nnot far away from the matching point $\\Lambda\\sim 1$ TeV, as it\nshould be in the KK approach.\n\nThus, we have offered the model of electroweak symmetry breaking, which\nprovides a positive connection to the naturalness as well as needs some deeper\nstudies under progress.\n\nThis work is in part supported by the Russian Foundation for Basic Research,\ngrants 99-02-16558, 01-02-99315, 01-02-16585 and 00-15-96645, the Russian\nMinistry on the education, grant E00-3.3-62.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\n\nDynamics on networks is a topic on which extensive works have been\ndone in recent years. Phenomena which have been studied include evolution of spin systems, \nopinion dynamics, disease spreading dynamics, etc. \\cite{Barrat,psen-book}. \nThe dynamical picture is  quite different from that on regular lattices due to \nthe topological features of network. For example, one can define the dynamics in different ways for the voter model on a network while these \nrules become identical on lattices \\cite{caste}. \n\n\nThe study of Ising model\non networks has revealed a number of interesting features  when \nstatic properties are considered.  \nEven in one dimension, when randomly new links are added (or existing links rewired) as in a Watts Strogatz (WS) network \\cite{WS},\n one gets a phase transition \\cite{Weigt,Gitterman,Niko} which occurs\nwith mean field criticality \\cite{Kim,Herrero,Hong}. On Euclidean networks, indications of both mean field type and finite dimensional-like phase transitions have been \nshown to exist by varying the relevant parameter \\cite{achat-psen}. \nOn scale free networks \\cite{Leone,Golt,Igloi}, the transition temperature shows a logarithmic\nincrease with the system size which is perhaps the most surprising result \\cite{Alek,Bian,Herrero1,Viana}.\n\nWhile considering ordering dynamics on regular lattices at zero temperature for the Ising model using Glauber dynamics, \nit is known that for any dimension greater than one, freezing occurs with a probability dependent on the dimension \\cite{redner}. This happens \nwhen one considers a completely random initial condition. On random graphs and \nnetworks, one encounters similar freezing phenomena  which depend on the \ndensity of added links \\cite{svenson,hagg,boyer,pratap}. \n\n\nOn random networks or graphs,  the\n evolution of the Ising model from a completely random state\nshows that the system does not order.\nA freezing effect was observed   and  although there could\nbe an emergent majority of nodes with either spin up or spin down state, domains of nodes with opposing spins\nsurvive \\cite{svenson,hagg}. \nCareful observations show  that  the disordered state is not an absorbing state \\cite{castellano}. \nIt is instead a stationary active state,\nwith some spins flipping, while keeping the energy constant. The number of\ndomains remaining in the system is just two. The qualitative picture is then the same as on\nregular lattices for $d > 2$ \\cite{redner},  the system wanders forever in an iso-energy set of states.\nThe distribution of  the steady state residual energy (which is identical to the number of bonds between oppositely oriented spins apart from a constant) \nfor the Ising model on a random networks\nwas investigated \\cite{baek}.\nIt was found that the distribution typically shows two peaks, one very close to the actual ground state where the residual\nenergy is zero and one far away from it.\n\n\nDynamics of the Ising model on the WS network  with restricted rewiring has also been considered.\n Here  initially\na spin is connected to its four nearest neighbours\n and then only the second nearest neighbour links are rewired  with probability $p$.\nThe system therefore always remains connected.\nUnder the  zero temperature Glauber dynamics,\nfreezing effect was observed\nfor any $p\\neq0$ \\cite{biswas}\n\n\nIn this paper, we have considered in detail the variation of the relevant thermodynamic quantities as \nfunctions of time for the zero temperature dynamics of Ising model\non various networks. Our main emphasis is on the Ising model on  scale free networks at temperature $T=0$ as such studies \nhave not been made  so  far to the best of our knowledge. Apart from the question whether\nthe equilibrium state is reached or not, we have also explored the nature of the state in case it does not.\nWe are  interested to see whether \nany active-absorbing phase transition occurs as the system parameters are\nvaried. \nBoth the random scale free network (RSF) and the Barab\\'{a}si-Albert (BA) network have been considered for the study.\nAlthough many results are known for the WS network, we have explored  specifically the \npossibility of  an active-absorbing phase transition in this network.\nThe BA model is studied to make direct comparison with the random scale free network, where results can be  quite different \\cite{albert1}.\nAlso, comparison with respect to\nissues like freezing and absorbing phase transition \nmay be made for the RSF and the WS networks.\n\nIn section II, we describe the network models and the dynamical evolution.\nThe quantities which have been calculated are defined in Section III. The results are presented in the next section and in the last section we summarise and \ndiscuss the studies made. \n\n\\section{THE NETWORK MODELS and DYNAMICS}\nWe have considered \n three different types of network: (a) Random scale free, (b) Barab\\'{a}si-Albert and (3)\n Watts Strogatz (addition type) \\cite{WS} network. We describe in brief how these networks are generated\nand the dynamical evolution process. In this section we also include a brief discussion of how the numerical data had been analysed in previous studies.\n\n\\subsection{Random scale free (RSF) network} In the random scale free network the degree distribution follows a power law\nbut otherwise the network is random.\nTo generate random scale free network \\cite{albert1,doro,boguna,newman,albert,psen}, we assigned the degree of  each node using\nthe power law:\n\\begin{equation}\n{ \\mathcal P}(k) \\sim k^{-\\gamma}, \n\\end{equation}\nwhere $k$ is the degree of node and $\\gamma$ is the characteristic degree exponent. The minimum value of   $k$ is $1$ and \nthe maximum cut-off value is $\\surd N$, where $N$ number of nodes.\nThis cut-off value ensures that there is no correlation \\cite{cutoff}. \nStarting from the node with the maximum degree, links have been established with   randomly selected  distinct nodes.\n\n\\subsection{ Barab\\'{a}si-Albert (BA) network} \nBarab\\'{a}si-Albert network is a growing network where new nodes are joined to existing nodes\nwith preferential attachment. We start with the three fully connected nodes.\nSubsequently,   a single node is  added at  a time to the network which is linked to \none existing node. The probability that  the new  node is  connected\nto the existing $i$th node with degree $k_i$ is given by \\cite{albert,albert1},  \n\\begin{equation}\n \\Pi (k_i) = k_i\/ {\\sum_{j} {k_j}}.\n\\end{equation}\nDegree distribution of the network is a power law with exponent $\\gamma=3$;\n$ P(k) \\sim k^{-3}$ \\cite{albert1}.\n\n\n\\subsection{Watts Strogatz (addition type) model} Addition type WS network is a one  dimensional regular\nchain with two nearest neighbour links as well as with some extra randomly  connected long range links.\nHere the long range links have been added with probability $q\/N$ (total long range links $\\sim O(N)$), where $N$ is the number of  nodes and $q$ is a parameter, \nwhich denotes the number of extra long range links per node on an average. So the average degree per node of this network is $2+q$,\nwhich is a finite quantity as $q\/N \\to 0$ in the thermodynamic limit. It is known \\cite{Gitterman} that an order-disorder transition occurs at $q=1$.\n\n\\subsection{Dynamics of Ising model on networks}\n\\label{ising-net}\nThe Hamiltonian of the  Ising system in these networks can be expressed as\n\\begin{equation}\nH= -{\\sum_{i<j} {J_{ij}S_iS_j}}\\\\,\n\\end{equation}\nwhere  $S_i = \\pm1$ and $J_{ij} = 1$ when sites $i$ and $j$ are connected and zero otherwise.\nStarting with the random configuration, single spin flip energy minimizing Glauber dynamics has been used\nto update the spin. In this dynamics a randomly selected spin is flipped if the energy of the updated configuration is lowered.\nIt is  flipped with probability $1\/2$ if the energy remains unchanged on flipping. \nFifty different network configurations have been considered and for each network \nhundred different initial configurations have been taken. The results are averaged over these configurations. We have considered system sizes $N$ upto $1600$.\nPeriodic boundary \ncondition has been used for the WS model which is embedded  in real space. \n\nPrevious studies  have shown that for the thermally driven phase transition, not only mean field critical behaviour exists for the Ising model\n on small world networks but finite size scaling is also valid there \\cite{Kim,Herrero,Hong,achat-psen}.\nThe data collapse was obtained by rescaling the data and the scaling argument occurring in the scaling function was found\nto be of the familiar form $\\epsilon N^{1\/ \\tilde{\\nu}}$ (with $\\tilde{\\nu}=2$) where $\\epsilon$ denotes the deviation from the critical point.\n$\\tilde{\\nu}$ is argued to be equal to $\\nu d$ where $\\nu$ is the correlation length exponent and $d$  the effective\ndimension of the system. With mean field critical exponent $\\nu=0.5$,\n $d$ turns out to be equal to the upper critical dimension; $d=4$.\n\n\n\n\n\\section{Quantities calculated}\nWe have estimated the following quantities in the present work.\n\n\n1. Magnetisation: $m(t)$ has been calculated by taking the average of the absolute values of the magnetisation, $m={\\langle| \\sum_{i}S_i|\\rangle}\/{N}$, as a function of time.\nSince evolution to both up spin dominated and down spin dominated configurations are possible, the absolute\nvalue is taken to compute the configuration average. Distribution of saturation value of magnetisation   has also been estimated for random scale free (RSF) network.\n\n2. Residual energy: $E_r (t) = E(t)-E_g$ where $E_g$ is the equilibrium energy of the ground state per spin and $E(t)$ the energy per spin at time $t$.\nResidual energy measurement indicates the closeness to the equilibrium ground state.\nSince we are employing a zero temperature quench, the equilibrium ground state configuration corresponds to either all spins up or down.\n\n\n3. $P_{flip}(t) = N_{flip}(t)\/N$, where $N_{flip}$ is the number of spin flips at time $t$, has been studied as a function of time.\n We count all the\nspin flips, i.e., if the same spin flips more than one time, all these occurrences are taken into account.\n\n4. Freezing probability $F$ is defined as the probability that the system does not reach the true ground state.\nIt has  been calculated as a function of the relevant parameters.\nFor $T=0$, the known ground state is the ferromagnetic state with $m(t\\rightarrow \\infty)=1$.\n A frozen configuration will have an absolute value of magnetisation less than $1$. \n\n\n5. Whether or not the dynamics continue, the system always reaches an iso-energy state. Time  $\\tau$ to reach the iso-energy \n state  has been calculated as a function of the relevant parameter for random scale free (RSF) network.\n\nThe saturation  values have been denoted using appropriate subscript, e.g. $m_{sat}$ for magnetisation.\n\n\n\n\n\n\n\\section{Results}\n\\subsection{Random scale free network}\n\\begin{figure}[!h]\n\\includegraphics[width=5cm,angle=270]{rsf_freeze_scalingn.eps}\n\\caption{(Color online) RSF: Variation of freezing probability $F(\\gamma)$\n as a function of $\\gamma$ for different system size $N$.\nInset shows the data collapse where $F(\\gamma)$ has been plotted against $X=(\\gamma-\\gamma_c^f)N^{1\/{\\tilde\\nu}}$.\n \\label{sat_frz_rsf}\n}\n\\end{figure}\n\\begin{figure}[!h]\n\\resizebox{88mm}{!}{\\includegraphics {rsf_distribution2.eps}}\n\\resizebox{88mm}{!}{\\includegraphics {rsf_fluc_time.eps}}\n\\caption{(Color online) RSF: ((a), (b)) The distribution of saturation value of magnetisation for $N=576$ for different values  \nof $\\gamma$. (c) The variation of fluctuation of saturation value of magnetisation and (d) time to reach the energetically stable state as a\nfunction of $\\gamma$ for different system size $N$.\n \\label{dist_rsf}\n}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\resizebox{52mm}{!}{\\includegraphics {Mag_rsf1_1.5.eps}}\n\\resizebox{52mm}{!}{\\includegraphics {Mag_rsf1_3.2.eps}}\n\\caption{(Color online) RSF: Variation of magnetisation $m(t)$ with time for different system size $N$.\n(a) $\\gamma=1.5$ and (b) $\\gamma=3.2$.\n \\label{dyn_mag_rsf}\n}\n\\end{figure}\nWe first discuss the behaviour of the freezing probability $F$ on the RSF. The variation of $F$\nwith the network parameter $\\gamma$ for different system sizes  \nsuggests that a freezing transition takes place here as the freezing probability is unity\nabove  a value of $\\gamma$ which we denote by $\\gamma_c^f$ (Fig. \\ref{sat_frz_rsf}). Of course, if the initial state is fully\nordered it is not counted as a frozen state. However, barring these two states (all up\/ all down), none of the other initial states reaches the actual equilibrium configuration.\nHence the freezing probability is actually $1-2\/2^N$ which becomes unity in the thermodynamic limit. Below $\\gamma_c^f$, \nthe freezing probability decreases with system size\nand  shows a system size independent behaviour for $\\gamma \\lesssim 1.0$. The freezing probability is non zero for all values of $\\gamma$.\n\n\nAs the freezing probability shows finite size dependence close to $\\gamma_c^f$ and \nsince it is dimensionless, we argue that it should  show a finite size scaling behavior in the following manner\n\\begin{equation}\nF(\\gamma,N)  =   g_1[(\\gamma-\\gamma_c^f)N^{1\/{\\tilde\\nu}}]\n\\end{equation}\nwhere $g_1$ is a scaling function. \n We  estimate  $\\gamma_c^f \\approx 2.0$ and the exponent $\\tilde \\nu \\approx 2.20$ (inset of Fig.\\ref{sat_frz_rsf}). \n The estimates are obtained from the manifestly best collapse of the data after rescaling is done properly.\nAs already discussed in section  \\ref{ising-net}, \nfinite size scaling is valid for the Ising model on networks with $\\tilde \\nu = \\nu d$ where $\\nu$ is the mean field value of the \ncorrelation length exponent and $d$ is equal to $4$. \nAssuming the same to hold good here, we get\n $\\nu \\approx 0.55$ from the fact that $\\tilde \\nu \\approx 2.2$ which is fairly \n close to the mean field value $0.5$ for Ising model \\cite{Stanley}.\n\nWe next show that an order-disorder phase transition also occurs in this network.\nThe distribution of magnetisation $\\Omega(m_{sat})$  has been studied (Fig. \\ref{dist_rsf}a, b). It is unimodal in nature for the values \nof $\\gamma$ far from the  $\\gamma_c^f$. The peaks occur  at $m_{sat}\\sim 1$ for $\\gamma \\ll \\gamma_c^f$ and $m_{sat}\\sim 0$ for $\\gamma \\gg \\gamma_c^f$.\nThe distribution is bimodal close to  $\\gamma_c^f$ with the peak values at $m_{sat}\\sim 0$ and $m_{sat} \\sim 1$.\n\nFluctuations of the saturation value of the magnetisation $\\langle {{\\Delta {m_{sat}}^2}}\\rangle$ shows \na peak which shifts toward $\\gamma_c^f$ and the  the peak value increases \nas the system size is increased (Fig. \\ref{dist_rsf}c). We have also estimated the time $\\tau$ to reach the energetically stable  state.\nIt shows a peak close to $\\gamma_c^f$. \nClearly $\\tau$    diverges near $\\gamma_c^f$,\nas the  peak value increases with system size (Fig. \\ref{dist_rsf}d).\nThese behaviour suggest  that there is an order-disorder transition also taking place as $\\gamma$ is varied.\n In principle the order-disorder transition may take place at a value $\\gamma_c^m \\neq \\gamma_c^f$\nand we make further analysis to estimate $\\gamma_c^m$ more accurately. \nWe also check whether a mean field-like behaviour is   present for the order-disorder transition. \n\n\n\n\n \n\n\n\n\nThe variation with time of the magnetisation $m(t)$ has been shown for different system \nsizes  for two different values of the degree exponent $\\gamma$ (Fig. \\ref{dyn_mag_rsf}a, b).\nThe behaviour of the saturation values of the magnetisation $m_{sat}$ for finite sizes shows the typical characteristics\nof a continuous phase transition with $\\gamma$ acting as the driving parameter (Fig. \\ref{sat_mag_rsf}). \n\n\n\nUsing  finite size scaling method we indeed obtain a collapse of the data points  for $m_{sat}$ for different system sizes. \nThe following scaling form for $m_{sat}$ has been used:\n\\begin{equation}\n m_{sat} = N^{-\\beta\/{\\tilde\\nu}} g_2[(\\gamma-\\gamma_c^m)N^{1\/{\\tilde\\nu}}]\\\\.\n\\end{equation}\nWe obtain $\\gamma_c^m \\approx 2.20$,   \n $\\beta \\simeq 0.47$ and  $\\tilde\\nu \\simeq 2.20$ (inset of Fig. \\ref{sat_mag_rsf}).\nThe values of the exponents $\\beta$ and $\\nu$ are fairly \nclose to the mean field values once again. \n\n\n\n\\begin{figure}[!h]\n\\includegraphics[width=5cm,angle=270]{rsf_mag_scalingn.eps}\n\\caption{(Color online) RSF: Variation of saturation value of magnetisation \n as a function of $\\gamma$ for different system size $N$.\nInset shows the data collapse where $Y=m_{sat}N^{\\beta\/{\\tilde\\nu}}$ has been plotted against $X=(\\gamma-\\gamma_c^m)N^{1\/{\\tilde\\nu}}$.\n \\label{sat_mag_rsf}\n}\n\\end{figure}\n\nNext we discuss the behaviour of the residual energy and the spin flip probability which \nalso attain a saturation value in time ($P_{sat}$ and $E_{sat}$).\nBoth the saturation values $P_{sat}$ and $E_{sat}$ show nonmonotonic\nvariation with  $\\gamma$ (Figs. \\ref{sat_flip_rsf}, \\ref{sat_enr_rsf}), with  a peak which shifts  as the system size is increased.\n$P_{sat}$ in particular shows a very interesting behaviour with finite size. For $\\gamma \\lesssim 2.0$, it \ndecreases with system size indicating an absorbing phase which is actually the ordered phase as\nindicated by the behaviour of the magnetisation discussed above.\nHowever, there is a region between  $\\gamma \\approx 2.0$ and $\\gamma \\approx  3.0$, where $P_{sat}$ increases with system size \nwhich indicates an active state. For $\\gamma \\gtrsim  3.0$, $P_{sat}$\ndecreases with system size indicating an absorbing state once again. Hence we have two absorbing phases\nseparated by an active phase and  two transitions  as the parameter $\\gamma$ is varied.\nThis is reminiscent of two distinct transitions observed in opinion dynamics models \\cite{droz,khaleque}.\n\n\nWe found from the above studies that $\\gamma_c^f \\approx 2.0$\nand $\\gamma_c^m\\approx 2.2$ are close but not exactly the same. However, this could be due to finite size\neffects and  these two might turn out to be\nidentical, i.e., $\\gamma_c^f=\\gamma_c^m=2.0$ in the thermodynamic limit. This possibility  is  supported by the behaviour\nof both $\\tau$ and $E_{sat}$. While $\\tau$ diverges at $\\gamma \\simeq 2.0$, $E_{sat}$  \ndecreases with system size below $\\gamma\\simeq 2.0$ (indicating an order phase) and increases or remains\nconstant above this value (Figs. \\ref{sat_enr_rsf}).\n\n\\begin{figure}[!h]\n\\includegraphics[width=7cm,angle=0]{flip_rsf_satn.eps}\n\\caption{(Color online) RSF: Variation of saturation value of the fraction of spin flips \n as a function of $\\gamma$ for different system size $N$. The vertical arrows separate different phases.\n \\label{sat_flip_rsf}\n}\n\\end{figure}\n\\begin{figure}[!h]\n\\includegraphics[width=7cm,angle=0]{enr_rsf_satn.eps}\n\\caption{(Color online) RSF: Variation of saturation value of the residual energy\n as a function of $\\gamma$ for different system size $N$. \n \\label{sat_enr_rsf}\n}\n\\end{figure}\n\n\n\n\n\n\\subsection{Barab\\'{a}si-Albert network}\nThe BA network has no intrinsic parameter. We calculate the relevant dynamic quantities for different system sizes.\nThe magnetisation $m(t)$ slowly increases for the first few time steps and gets saturated at long times for all the system sizes. \nBoth $E_r(t)$ and $P_{flip}(t)$ decrease with time and then reach a saturation value. Saturation values of all these quantities\nas a function of system size have been plotted in Fig. \\ref{sat_ba}a.\nThe saturation value of magnetisation $m_{sat}(N)$ decreases with system size and the variation shows a power law behaviour.\nWe fitted the variation with the form $m_{sat}(N)=a_mN^{-b_m}$ and the estimated exponents are $a_m \\approx 1.467$ and $ b_m \\approx 0.150$.\n\nWe have also studied the variation of saturation value of the fraction of spin flips $P_{sat}(N)$ and saturation value of\nresidual energy $E_{sat}(N)$. Both of them show a slowly increasing behaviour with system size and these increasing behaviour  also\nfollow power law (Fig. \\ref{sat_ba}a). We fitted the variation of $P_{sat}(N)$ with the form $P_{sat}(N)=a_fN^{b_f}$\nand the exponents are $a_f \\approx 0.001$ and $b_f \\approx 0.346$. For $E_{sat}(N)$, the fitted form is $E_{sat}(N)=a_EN^{b_E}$ and\nthe exponents are $a_E \\approx 0.083$ and $b_E \\approx 0.293$.\nThe important point to note here is this seems to be an active phase as the \nsaturation value of the fraction of spin flips shows increase with system size.\nIn contrast, the same quantities plotted for the RSF with $\\gamma = 3.0$ (Fig. \\ref{sat_rsf}) \nshows that it is an absorbing phase (as already noted in the previous subsection).\nWe will discuss more about this observation in Section \\ref{discussion}. \nThe variation of $E_{sat}(N)$ with $N$ in RSF is however, similar, with $E_{sat}(N)=0.013N^{0.279}$. $m_{sat}(N)$ for RSF decays as $1.17N^{-0.408}$\nwhich signifies a faster decay compared to the BA network.\n.\n\n\nIn the BA  model, the freezing probability $F(N)$ increases with system size (Fig. \\ref{sat_ba}b). \nFreezing probability $\\rightarrow 1$ for $N \\rightarrow \\infty$.\nThus the BA model is in an active disordered state where none of the \nconfiguration reaches the equilibrium ground state.  The variation of\n$F(N)$ with system size fits well with the form $F(N)=1-e^{-a_dN^{b_d}}$ and the \ncalculated exponents are $a_d \\approx 0.113$ and $b_d \\approx 0.548$.\nThe behaviour of the freezing probability as a function of $N$ is also quite \ndifferent for the BA and the RSF networks (with $\\gamma = 3$), in the latter we found a system size independent behaviour. \n\n\\begin{figure}[!h]\n\\resizebox{42mm}{!}{\\includegraphics {Mag_sat_ba.eps}}\n\\resizebox{43mm}{!}{\\includegraphics {freeze_ba.eps}}\n\\caption{(Color online) BA: (a) Variation of saturation value of magnetisation $m_{sat}(N)$, fraction of spin flips $P_{sat}(N)$ \nand residual energy $E_{sat}(N)$ as a function system size. (b) Variation of freezing probability $F(N)$\n as a function of system size $N$.\n \\label{sat_ba}\n}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[width=7cm,angle=0]{Mag_flip_enr_sat_rsf.eps}\n\\caption{(Color online) RSF($\\gamma=3.0)$: Variation of saturation value of magnetisation $m_{sat}(N)$, fraction of spin flips $P_{sat}(N)$ \nand residual energy $E_{sat}(N)$ as a function system size\n \\label{sat_rsf}\n}\n\\end{figure}\n\\subsection{WS network}\n\n\nOn the WS network, all the relevant quantities show a saturation behaviour as already noted previously on\na slightly different version of the WS network \\cite{biswas}.\n\n\n Here in  addition,  we have  studied the spin flip probabilities. \nThe saturation value of  the\nprobability of spin flips $P_{sat}$ has been plotted against the parameter $q$ for different system sizes and this quantity reveals interesting behaviour (Fig. \\ref{sat_ws}a).\nIt is clearly seen that above $q \\simeq 1$,\nthe probability decreases as a function of system size $N$ while below $q\\simeq 1$ it is\nalmost size independent. This  indicates that there is an active absorbing phase\ntransition taking place at this point.\n\nThe value of the freezing probability $F(q)$  either increases with $N$ or remains constant which \nclearly shows that the entire phase is frozen for any $q> 0$. We find that   for $q < 1$ the freezing probability reaches unity in the thermodynamic limit \nwhile it remains fairly constant beyond this value (Fig. \\ref{sat_ws}b). This is consistent with the \nactive-absorbing phase transition stipulated to take place at $q=1$; in the active state, one can never  reach the equilibrium ground state configuration. \n\n\n\n\\begin{figure}[!h]\n\\resizebox{43mm}{!}{\\includegraphics {flip_add_sat.eps}}\n\\resizebox{42mm}{!}{\\includegraphics {freeze_inset.eps}}\n\\caption{(Color online) WS: Variation of (a) saturation value of fraction of spin flips $P_{sat}$\nand (b) freezing probability $F(q)$ as a function of $q$ for different system sizes $N$ (right panel). Inset of (b) shows the variation of peak value of $F(q)$\nwith the system size.\n \\label{sat_ws}\n}\n\\end{figure}\n\n\n\n$F(q)$ shows a peak at $q=0.2$  and the position of the peak is independent of system size which\nshows that the system is maximally disordered here.\nThe peak values of freezing probability as a function of system size $N$ is fitted with\nthe form $F_{peak}(N)=1-e^{-aN^b}$ and the estimated values of the exponents\nare $a\\approx 0.134$ and $b\\approx 0.540$ (inset of Fig. \\ref{sat_ws}b). It may be noted that the same form is obeyed in the BA model.\n\n\n\n\n\\section{Discussions and conclusions}\nWe have studied zero temperature Glauber dynamics of the Ising model on three types of networks and compared the results.\nFrozen state is observed in all the three types of network models. For random scale free network, freezing probability is\n unity for $\\gamma \\gtrsim 2$, i.e., the system never reaches  the global equilibrium\n but for lower value of the parameter $\\gamma$, it decreases with system size and shows a system size independent \nbehaviour for $\\gamma \\lesssim 1$. This behaviour of freezing probability suggests a freezing transition point at $\\gamma_c^f \\simeq 2.0$.\nWe also find an order disorder transition point taking place very close to this point; in fact we believe that they occur at the same point\nand the difference is only a finite size effect. Also close to this point, the first active-absorbing (A-A) phase transition takes place; \nthe disordered phase for $2\\lesssim \\gamma \\lesssim 3$ is active while for $\\gamma \\lesssim 2$, \none gets an absorbing phase. A second A-A transition takes place close to $\\gamma \\approx 3.0$\nand the system evolves to an absorbing disordered state beyond this value. In all probability these two A-A transitions take place at $\\gamma=2.0$ and\n$3.0$ in absence of finite size effects; these two points are significant as  for $\\gamma\\leq 2.0$, the average degree diverges while\nfor  $\\gamma\\leq 3.0$, the degree variance diverges in the thermodynamic limit.\n\nOne can compare the results of RSF network and BA network for the same characteristic degree exponent $\\gamma =3.0$.\nThe residual energy shows an increase with system size $N$ in both cases in a power law manner with an exponent which is fairly\n close. Also, the saturation value of magnetisation decreases with $N$ in both networks in a power law manner, corresponding exponents are\n however quite different. The saturation energy and magnetisation behaviour are consistent with the fact that the state is disordered for both RSF and BA networks.\n  However freezing probability for BA model and RSF network\nshow different behaviour with system size.  For RSF, the  freezing probability  shows a system size independent behaviour while for  BA model, it has  a nonlinear dependence.\nAs far as the spin flip probability is concerned RSF (at $\\gamma = 3.0$) again differs from the BA network.\nThe RSF network and BA model are intrinsically different,  BA is a growing network generated with a particular strategy.\nThere is no loop in this network. In the case of RSF, the structure is completely different, there may be loops. \nAlthough in a numerical study, the value of $\\gamma$ may not be exactly 3 in either the RSF or BA networks\ndue to finite size effects, we believe that this  cannot be the reason for the results being  \nqualitatively different. In fact \nprevious studies on RSF and BA have shown that characteristic features may be quite different for the two networks \\cite{albert1,psen}.\n\nThe results obtained for the WS model can also be compared to those found for the RSF network and BA network.\nIn the WS network an A-A transition is observed at $q \\simeq 1.0$ which is also the order-disorder transition point. Hence this \nis similar to the occurrence of an A-A and an order-disorder transition occurring simultaneously in the RSF. However in the WS network,\nthe entire disordered phase is active. For small average degree the system is in an active and for large degree in an absorbing phase.\nFor the RSF network,  two A-A transitions exist where absorbing phase is observed for {\\it both} large degree and small degree and an active \nstate exists in between these two absorbing phases. In both RSF and WS another interesting feature is present, the ordered \nstate shows a finite freezing probability with negligible system size dependence.\nIn the WS network maximum value of the freezing probability $F(q)$ occurs at $q\\approx 0.2$ and \nshows a behaviour similar to the freezing probability in BA network\nas a function of system size.\n\nTo summarise, a systematic study of ordering dynamics of the Ising model on scale free networks has been made for the first time to the best of our knowledge.\nIt is observed that the system freezes to a non equilibrium steady state for all values of the relevant parameters in the random scale free\nnetworks (RSF) and the Barab\\'{a}si-Albert model (BA).\nThe presence of two active-absorbing phase transitions in the RSF makes it different from the WS network where only one such transition \nis observed. It is also concluded that in RSF one of the active-absorbing phase transition takes place at the order-disorder transition\npoint which is similar to what is observed in the WS network.\n\\label{discussion}\n\nAcknowledgements: The authors thank Soham Biswas for his suggestions and encouragement.\nAK acknowledges financial support from UGC sanction no. F.7-48\/2007(BSR). PS acknowledges financial support from CSIR \nproject.  Computations made on HP cluster  financed by DST (FIST scheme), India.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nStemming from a combination of ideas from nonlinear time series analysis~\\cite{kantz2004nonlinear, bradley2015nonlinear} and information theory~\\cite{shannon1948mathematical}, permutation entropy was first introduced in 2002 by Bandt and Pompe~\\cite{bandt2002permutation} as a simple, robust, and computationally efficient complexity measure for time series. This complexity measure is defined as the Shannon entropy of a probability distribution associated with ordinal patterns evaluated from partitions of a time series -- a procedure known as the Bandt-Pompe symbolization approach. Permutation entropy and its underlying symbolization approach have become increasingly popular among researchers working with time series analysis, leading to successful applications in fields as diverse as biomedical sciences~\\cite{nicolaou2012detection}, econophysics~\\cite{zunino2009forbidden}, physical sciences~\\cite{garland2018anomaly}, and engineering~\\cite{yan2012permutation}. The uses of permutation entropy also span a large variety of goals such as monitoring the dynamical regime of a system~\\cite{yan2012permutation}, detecting anomalies in time series~\\cite{garland2018anomaly}, characterizing time series data~\\cite{nicolaou2012detection}, testing for serial independence~\\cite{garcia2008nonparametric}, and are further documented in review articles by Zanin~\\textit{et al.}~\\cite{zanin2012permutation}, Riedl~\\textit{et al.}~\\cite{riedl2013practical}, Amig{\\'o}~\\textit{et al.}~\\cite{amigo2015ordinal}, and Keller \\textit{et al.}~\\cite{keller2017permutation}.\n\nPermutation entropy's success is not limited to its practical usage as this approach has inspired numerous time series analysis tools. Some of these related methods consider different quantifiers for the ordinal probability distribution~\\cite{rosso2007distinguishing, zunino2008fractional, carpi2010missing, parlitz2012classifying, unakafov2014conditional, liang2015eeg, ruan2019ordinal, zunino2015permutation, bandt2017new, ribeiro2017characterizing, jauregui2018characterization}, generalize the Bandt-Pompe symbolization algorithm to evaluate ordinal structures on multiple temporal scales~\\cite{aziz2005multiscale, zunino2010permutation, morabito2012multivariate, zunino2012distinguishing}, include signal amplitude information~\\cite{fadlallah2013weighted, xia2016permutation, azami2016amplitude, chen2018weighted}, and account for equal values in time series~\\cite{bian2012modified, cuesta2018patterns}. Other works have generalized permutation entropy and its ordinal approach to two-dimensional data such as images~\\cite{ribeiro2012complexity, zunino2016discriminating}. The ordinal patterns underlying permutation entropy have also been used for mapping time series and images into networks known as ordinal networks~\\cite{small2013complex, mccullough2015time, small2018ordinal, pessa2019characterizing, pessa2020mapping, borges2019learning, chagas2020characterization}.\n\nThe original version of permutation entropy and its various generalizations represent an essential and appealing framework for data analysis, especially when considering the increasing availability of large data sets~\\cite{economist2010deluge} and the steady demand for reliable and computationally efficient methods for extracting meaningful information from these data sets~\\cite{mattmann2013vision, blei2017science}. However, most methods emerging from Bandt and Pompe's seminal work lack freely available computational implementation, and the exceptions are limited to a single or very few approaches. Here we help to fill this gap by presenting \\texttt{ordpy}{} -- a simple and open-source Python module implementing several of the principal methods related to Bandt and Pompe's framework. This module has been designed to be easily set up and installed as its only dependency is \\texttt{numpy}{}~\\cite{harris2020array}, a fundamental Python library implementing array objects and fast math functions that operate on these objects. Beyond our preferences, the Python programming language has been chosen for its widespread use in scientific computing~\\cite{harris2020array} and extensive community support~\\cite{perkel2015pickup}.\n\nWe present \\texttt{ordpy}{}'s functions and illustrate their usage along with a review of the pertinent theoretical developments of permutation entropy and its related ordinal methods. Our work alternates between the mathematical description of the different techniques and the presentation of functions and code snippets that implement these data analysis tools. We further use \\texttt{ordpy}{} to replicate several literature results. The source-code of \\texttt{ordpy}{} is freely available on its git repository (\\url{github.com\/arthurpessa\/ordpy}) together with the documentation of all \\texttt{ordpy}{}'s functions (\\url{arthurpessa.github.io\/ordpy}). We can install \\texttt{ordpy}{} using the Python Package Index (PyPI) via:\n\\begin{minted}{shell}\n $ pip install ordpy\n\\end{minted}\nWe further provide the code and data for replicating all analyses presented in this article as a Jupyter notebook~\\cite{shen2014interactive, kluyver2016jupyter} on \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s website.\n\n\\section{An overview of ordinal distributions, permutation entropy, and complexity-entropy plane} \\label{sec:permutation_methods}\n\nWe start by presenting a short review of Bandt and Pompe's seminal permutation entropy~\\cite{bandt2002permutation}. As we have already mentioned, permutation entropy is the Shannon entropy of a probability distribution related to ordinal (or permutation) patterns evaluated using sliding partitions over a time series. This probability distribution is the so-called ordinal distribution or distribution of ordinal patterns, and the symbolization process used to estimate this distribution is the Bandt-Pompe approach. To describe this process, let us consider an arbitrary time series $\\{x_t\\}_{t=1,\\dots,N_x}$. First, we divide this time series into $n_x = N_x - (d_x-1)\\tau_x$ overlapping partitions comprised of $d_x>1$ observations separated by $\\tau_x\\geq1$ time units. For given values of $d_x$ and $\\tau_x$, each data partition can be represented by \n\\begin{equation}\\label{eq:1dpartition}\n    w_p =  (x_{p}, x_{p + \\tau_x}, x_{p + 2\\tau_x}, \\dots, x_{p + (d_x - 2)\\tau_x}, x_{p + (d_x - 1)\\tau_x})\\,,\n\\end{equation}\nwhere $p = 1, \\dots, n_x$ is the partition index. The parameters $d_x$ and $\\tau_x$ are the only two parameters of the Bandt-Pompe method: $d_x$ is the embedding dimension~\\cite{bandt2002permutation} and $\\tau_x$ is the embedding delay~\\cite{zunino2010permutation}. It is worth remarking that Bandt and Pompe's original proposal was restricted to $\\tau_x=1$ (that is, data partitions comprised of consecutive time series elements), and the embedding delay was further introduced by Cao \\textit{et al.}~\\cite{cao2004detecting} and Zunino \\textit{et al.}~\\cite{zunino2010permutation}. As we shall see, the choices of $d_x$ and $\\tau_x$ are important, and there is research exclusively devoted to determining optimal values for these parameters~\\cite{riedl2013practical, cuestafrau2019embedded, myers2020automatic}.\n\nNext, for each partition $w_p$, we evaluate the permutation $\\pi_p = (r_0, r_1, \\dots, r_{d_x-1})$ of the index numbers $(0, 1, \\dots, d_x - 1)$ that sorts the elements of $w_p$ in ascending order, that is, the permutation of the index numbers defined by the inequality $x_{p + r_0} \\leq x_{p + r_1} \\leq \\dots \\leq x_{p + r_{d_x-1}}$. In case of equal values, we maintain the occurrence order of the partition elements, that is, if $x_{p+r_{k-1}} = x_{p+r_k}$ then $r_{k-1} < r_{k}$ for $k=1,\\dots,d_x-1$~\\cite{cao2004detecting}. As an illustration, suppose we have $x_t = (5,3,2,2,7,9)$ and set $d_x = 4$ and $\\tau_x = 1$. The first partition is $w_1 = (5,3,2,2)$, and sorting its elements we find $2 \\leq 2 < 3 < 5$ or $x_{1+2} \\leq x_{1+3} < x_{1+1} <x_{1+0}$. Thus, the permutation symbol (ordinal pattern) associated with $w_1$ is $\\pi_1 = (2,3,1,0)$. \nAnother possibility for dealing with ties among partition elements consists in adding a small noise perturbation without modifying all other ordering relations. This latter scheme was initially proposed in Bandt and Pompe's seminal work, but it is much less used in the literature. In most cases, time series data have enough resolution for making these equalities negligible; however, this issue can become critical for low-resolution signals~\\cite{zunino2017permutation}.\n\nAfter evaluating the permutation symbols associated with all data partitions, we obtain a symbolic sequence $\\{\\pi_p\\}_{p = 1, \\dots, n_x}$. The \\texttt{ordpy}'s function \\texttt{ordinal\\_sequence}{} returns this sequence as illustrated in the following code:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_sequence\n>>> x = [5, 3, 2, 2, 7, 9]\n>>> ordinal_sequence(x, dx=4, taux=1)\narray([[2, 3, 1, 0],\n       [1, 2, 0, 3],\n       [0, 1, 2, 3]])\n>>> ordinal_sequence([1.55, 1.54, 1.53], dx=2)\narray([[1, 0], [1, 0]])\n>>> ordinal_sequence([1.55, 1.54, 1.53], dx=2, \n... tie_precision=1)\narray([[1, 0], [0, 1]])\n\\end{minted}\nThe last two examples illustrate the use of parameter \\texttt{tie\\_precision}{} that defines the number of decimals considered for establishing the ordinal relations. This parameter is available in most \\texttt{ordpy}{}'s functions and is particularly relevant when working with time series presenting equal values that could be mistaken by floating-point number representation. Figure~\\ref{fig:1}(a) illustrates the application of the Bandt-Pompe approach for a simple time series and different values of $d_x$ and $\\tau_x$.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{fig1.pdf}\n\\caption{The symbolization process of Bandt and Pompe. (a) Illustration of Bandt-Pompe method applied to a time series and the resulting ordinal sequences for embedding parameters $d_x=3$ and $\\tau_x=1$ (left) and $d_x=2$ and $\\tau_x=2$ (right). (b) Application of the two-dimensional version of the Bandt-Pompe method to a data array and the resulting ordinal sequences for embedding parameters $d_x=d_y=2$ and $\\tau_x=\\tau_1=1$ (left) and $d_x=d_y=2$ and $\\tau_x=\\tau_1=2$ (right). In both panels, the colored boxes indicate the data partitioning scheme for each set of embedding parameters, and the numbers above them represent the index numbers used for determining the ordinal patterns or permutation symbols.}\n\\label{fig:1}\n\\end{figure*}\n\nThe ordinal probability distribution $P = \\{\\rho_i(\\Pi_i)\\}_{i = 1, \\dots, n_\\pi}$ is simply the relative frequency of all possible permutations within the symbolic sequence, that is, \n\\begin{equation}\\label{eq:permutation_probability}\n    \\rho_i(\\Pi_i) = \\frac{\\text{number of partitions of type} \\ \\Pi_i \\ \\text{in} \\ \n    \\{\\pi_p\\}}{n_x}\\,,\n\\end{equation}\nwhere $\\Pi_i$ represents each one of the $n_\\pi=d_x!$ different ordinal patterns. The following code shows how to obtain the ordinal distribution with \\texttt{ordpy}{}'s function \\texttt{ordinal\\_distribution}{}:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_distribution\n>>> x = [5, 3, 2, 2, 7, 9]\n>>> pis, rho = ordinal_distribution(x, dx=3)\n>>> pis\narray([[0, 1, 2],\n       [1, 2, 0],\n       [2, 1, 0]])\n>>> rho\narray([0.5 , 0.25, 0.25])\n\\end{minted}\nThe two arrays returned by \\texttt{ordinal\\_distribution}{} are the ordinal patterns and their corresponding relative frequencies, respectively. By default, \\texttt{ordinal\\_distribution}{} does not return non-occurring permutations (that is, those with $\\rho_i(\\Pi_i)=0$); however, the parameter \\texttt{return\\_missing} modifies this behavior as in:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_distribution\n>>> x = [5, 3, 2, 2, 7, 9]\n>>> pis, rho = ordinal_distribution(x, dx=3, \n... return_missing=True)\n>>> pis\narray([[0, 1, 2],\n       [1, 2, 0],\n       [2, 1, 0],\n       [0, 2, 1],\n       [1, 0, 2],\n       [2, 0, 1]])\n>>> rho\narray([0.5 , 0.25, 0.25, 0.  , 0.  , 0.  ])\n\\end{minted}\nMissing permutation symbols are always the latest elements of the returned array.\n\nHaving the ordinal probability distribution $P$, we can calculate its Shannon entropy~\\cite{shannon1948mathematical} and define the permutation entropy as\n\\begin{equation}\\label{eq:permutation_entropy}\n    S(P) = -\\sum_{i = 1}^{n_\\pi} \\rho_i(\\Pi_{i})\\log \\rho_i(\\Pi_{i})\\,,\n\\end{equation}\nwhere $\\log(\\dots)$ stands for the base-$2$ logarithm. Permutation entropy quantifies the randomness in the ordering dynamics of a time series such that $S\\approx\\log n_\\pi$ indicates a random behavior, while $S\\approx0$ implies a more regular dynamics. Because the maximum value of $S$ is $S_{\\rm max} = \\log{n_\\pi}$, we can further define the normalized permutation entropy as\n\\begin{equation}\\label{eq:normalized_pe}\n    H(P) = \\frac{S(P)}{\\log{n_\\pi}}\\,,\n\\end{equation}\nwhere the values of $H$ are restricted to the interval $[0,1]$. The \\texttt{ordpy}'s function \\texttt{permutation\\_entropy}{} calculates the values of $S$ and $H$ directly from a time series as illustrated in:\n\\begin{minted}{python}\n>>> from ordpy import permutation_entropy\n>>> x = [5, 3, 2, 2, 7, 9]\n>>> permutation_entropy(x)\n0.5802792108518123\n>>> permutation_entropy(x, normalized=False, \n... base='e')\n1.0397207708399179\n\\end{minted}\nThe \\texttt{permutation\\_entropy}{} function uses the base-2 logarithm function by default; however, the parameter \\texttt{base} can modify this behavior.\n\nThe embedding dimension $d_x$ defines the number of possible permutations $(n_\\pi = d_x!)$, and following Bandt and Pompe's recommendation~\\cite{bandt2002permutation}, it is common to choose the values of $d_x \\in \\{3, 4, 5, 6, 7\\}$ to satisfy the condition $d_x! \\ll N_x$ to obtain a reliable estimate of the ordinal probability distribution. Another less common choice is to use a value of $d_x$ such that $5 d_x! \\leq N_x$~\\cite{amigo2008combinatorial}. More recently, however, Cuesta-Frau \\textit{et al.}~\\cite{cuestafrau2019embedded} have shown that these requirements on $d_x$ can be considerably loosened in several situations related to classification tasks. The embedding delay $\\tau_x$ defines a time scale for the system under analysis and is often set as $1$; however, different values of $\\tau_x$ may inform about delayed feedback mechanisms and time-correlation structures. We present a more detailed discussion about the choices of $d_x$ and $\\tau_x$ in Appendix~\\ref{appendix:parameter}.\n\nThe permutation entropy framework was extended to two-dimensional data by Ribeiro \\textit{et al.}~\\cite{ribeiro2012complexity} and Zunino and Ribeiro~\\cite{zunino2016discriminating}. To present this generalization, let us consider an arbitrary two-dimensional data array $\\{y_t^u\\}_{t = 1,\\dots,N_x}^{u = 1,\\dots,N_y}$ whose elements may represent pixels of an image. We further define the embedding dimensions $d_x$ and $d_y$ along the horizontal and vertical directions (respectively), and the corresponding embedding delays $\\tau_x$ and $\\tau_y$. Similarly to the one-dimensional case, we slice the data array in partitions of size $d_x \\times d_y$ defined by \n\\begin{equation}\\label{eq:matrix_partition}\n    w_{p}^{q} = \n\\begin{pmatrix}\n    y_{p}^{q} & y_{p+\\tau_x}^{q} & \\dots & y_{p+(d_x-1)\\tau_x}^{q} \\\\[.5em]\n    y_{p}^{q+\\tau_y} & y_{p+\\tau_x}^{q+\\tau_y}   & \\dots & y_{p+(d_x-1)\\tau_x}^{q+\\tau_y} \\\\\n    \\vdots & \\vdots & \\ddots & \\vdots\\\\\n    y_{p}^{q+(d_x-1)\\tau_x} & y_{p+\\tau_x}^{q+(d_x-1)\\tau_x}  & \\dots & y_{p+(d_x-1)\\tau_x}^{q+(d_y-1)\\tau_y} \\\\\n\\end{pmatrix},\n\\end{equation}\nwhere the indexes $p = 1,\\dots, n_x$ and $q = 1,\\dots, n_y$, with $n_x = N_x-(d_x-1)\\tau_x$ and $n_y = N_y-(d_y-1)\\tau_y$, cover all $n_x n_y$ data partitions. To associate a permutation symbol with each two-dimensional partition, we flatten the partitions $w_p^q$ line by line, that is,\n\\begin{equation}\\label{eq:partition_index}\n\\begin{split}\n    w_{p}^{q} =  & \\left( y_{p}^{q}, y_{p+\\tau_x}^{q}, \\dots,\n                    y_{p+(d_x-1)\\tau_x}^{q}, \\right.\\\\\n                 & ~~y_{p}^{q+\\tau_y}, y_{p+\\tau_x}^{q+\\tau_y}, \\dots,         \n                     y_{p+(d_x-1)\\tau_x}^{q+\\tau_y},\\dots,\\\\\n                 & \\left. ~y_{p}^{q+(d_x-1)\\tau_x}, y_{p+\\tau_x}^{q+(d_x-1)\\tau_x}, \\dots, y_{p+(d_x-1)\\tau_x}^{q+(d_y-1)\\tau_y}\\right)\\,.\n\\end{split}\n\\end{equation}\nAs this procedure does not depend on a particular partition, we can simplify the notation by representing $w_{p}^{q}$ as\n\\begin{equation}\\label{eq:partition}\n    w_p^q = \\left(\\tilde{y}_0, \\tilde{y}_1, \\dots, \\tilde{y}_{d_x d_y-2}, \\tilde{y}_{d_x d_y-1}\\right)\\,,\n\\end{equation}\nwhere $\\tilde{y}_0 = y_{p}^{q},~\\tilde{y}_1 = y_{p+\\tau_x}^{q}$, and so on. Then, we evaluate the permutation symbol associated with each data partition as in the one-dimensional case to define the symbolic array $\\{\\pi_p^q\\}_{p=1,\\dots,n_x}^{q=1,\\dots,n_y}$ related to the data set (Fig.~\\ref{fig:1}{b} illustrates the Bandt-Pompe approach for two-dimensional data). From this array, we calculate the relative frequency for all $n_\\pi = (d_x d_y)!$ possible permutations $\\Pi_i$ via \n\\begin{equation}\\label{eq:permutation_probability_2d}\n    \\rho_i(\\Pi_i) = \\frac{\\text{number of partitions of type $\\Pi_i$ in } \\{\\pi_p^q\\}}{n_x n_y}\\,,\n\\end{equation}\nwhere $i=1,\\dots,n_\\pi$, and so the ordinal probability distribution is $P = \\{\\rho_i(\\Pi_i)\\}_{i = 1, \\dots, n_\\pi}$. It is worth noticing that the ordering procedure defining the permutation symbols is no longer unique as in the one-dimensional case. For instance, we would find a different symbolic array by flattening the partitions $w_p^q$ column by column. However, different ordering procedures do not modify the set of elements comprising the ordinal probability distribution (only their order is changed)~\\cite{ribeiro2012complexity}.\n\nAs in the one-dimensional case, the two-dimensional permutation entropy is simply the Shannon entropy of the ordinal distribution $P = \\{\\rho_i(\\Pi_i)\\}_{i = 1, \\dots, n_\\pi}$, so we can calculate the two-dimensional permutation entropy and its normalized version using Eqs.~\\ref{eq:permutation_entropy} and \\ref{eq:normalized_pe}, respectively. Only the total number of possible ordinal patterns ($n_\\pi = (d_x d_y)!$ in the two-dimensional case) is modified.\n\nSimilarly to the one-dimensional case, the values of $d_x$ and $d_y$ are usually constrained by the condition $(d_x d_y)! \\ll N_x N_y$ in order to obtain a reliable estimate of the ordinal distribution $P$~\\cite{ribeiro2012complexity, zunino2016discriminating}. Naturally, this two-dimensional formulation recovers the one-dimensional case ($N_y=1$ for time series data) by setting $d_y = \\tau_y = 1$. In \\texttt{ordpy}{}, the functions \\texttt{ordinal\\_sequence}{}, \\texttt{ordinal\\_distribution}{} and \\texttt{permutation\\_entropy}{} automatically implement this two-dimensional generalization when the input data is a two-dimensional array as in:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_sequence, \n... ordinal_distribution, permutation_entropy\n>>> y = [[5, 3, 2], [2, 7, 9]]\n>>> ordinal_sequence(y, dx=2, dy=2)\narray([[[2, 1, 0, 3],\n        [1, 0, 2, 3]]])\n>>> ordinal_distribution(y, dx=2, dy=2)\n(array([[1, 0, 2, 3],\n        [2, 1, 0, 3]]), array([0.5, 0.5]))\n>>> permutation_entropy(y, dx=2, dy=2)\n0.21810429198553155\n\\end{minted}\n\nIn addition to permutation entropy, the complexity-entropy plane proposed by Rosso \\textit{et al.}~\\cite{rosso2007distinguishing} is another popular time series analysis tool directly related to Bandt and Pompe's symbolization approach. This method was initially introduced for distinguishing between chaotic and stochastic time series but has been successfully used as an effective discriminating tool in several other contexts~\\cite{rosso2009detecting, zunino2010complexity, zunino2012efficiency, ribeiro2012songs, sigaki2019estimating}. The complexity-entropy plane combines the normalized permutation entropy $H$ (Eq.~\\ref{eq:normalized_pe}) with an intensive statistical complexity measure $C$ (also calculated using the ordinal distribution) to build a two-dimensional representation space with the values of $C$ versus $H$. The statistical complexity $C$ used by Rosso \\textit{et al.} is inspired by the work of Lopez-Ruiz \\textit{et al.}~\\cite{lopezruiz1995statistical} and is defined by the product of the normalized permutation and a normalized version of the Jensen-Shannon divergence~\\cite{lin1991divergence} between the ordinal distribution $P = \\{\\rho_i(\\Pi_i)\\}_{i = 1, \\dots, n_\\pi}$ and the uniform distribution $U = \\{ 1\/n_\\pi \\}_{i = 1,\\dots,n_\\pi}$ (it is worth remembering that $n_\\pi$ is the number of possible ordinal patterns). Mathematically, we can write this measure as\n\\begin{equation}~\\label{eq:statistical_complexity}\n    C(P) = \\frac{D(P,U)H(P)}{D^{\\rm max}}\\,,\n\\end{equation}\nwhere \n\\begin{equation}\n    D(P,U) = S[(P + U)\/2] - \\dfrac{1}{2}S(P) - \\dfrac{1}{2}S(U) \n\\end{equation}\nis the Jensen-Shannon divergence and \n\\begin{equation*} \n    D^{\\rm max} = -\\dfrac{1}{2}\\left(\\frac{n_\\pi!+1}{n_\\pi!}\\log(n_\\pi!+1)-2\\log(2n_\\pi!)+\\log{n_\\pi!}\\right)\n\\end{equation*}\nis a normalization constant. This latter constant expresses the maximum possible value of $D(P,U)$ occurring for $P = \\{ \\delta_{1,i} \\}_{i = 1,\\dots,n_\\pi}$~\\cite{lamberti2004intensive, martin2006generalized}, where\n$\\delta_{ij} =\n\\begin{cases}\n1 & \\text{if } i = j\\\\\n0 & \\text{if } i \\neq j\n\\end{cases}$ \nis the Kronecker delta function.\n\nDifferently from permutation entropy, the statistical complexity $C$ is zero in both extremes of order (when only one permutation symbol occurs) and disorder (when all permutations are equally likely to happen). The value of  $C$ quantifies structural complexity and provides additional information that is not carried by the\nvalue of $H$. Furthermore, $C$ is a nontrivial function of $H$ in the sense that for a given value of $H$, there exists a range of possible values for $C$~\\cite{lamberti2004intensive, martin2006generalized, rosso2007distinguishing}. This happens because $H$ and $D$ are expressed by different sums of $\\rho_i(\\Pi_i)$ and there is thus no reason for assuming a univocal relationship between $H$ and $C$. \n\nTo better illustrate this feature, let us assume (for simplicity) we replace the Jensen-Shannon divergence by the Euclidean distance between $P$ and $U$ (as in the seminal work of Lopez-Ruiz \\textit{et al.}~\\cite{lopezruiz1995statistical}), that is, $D(P,U) = \\sum_{i=1}^{n_\\pi} (\\rho_i(\\Pi_i)-1\/n_\\pi)^2$. In this case, the statistical complexity is\n\\begin{equation*}\n    C(P) \\propto -\\left(\\sum_{i=1}^{n_\\pi} \\rho_i(\\Pi_i) \\log \\rho_i(\\Pi_i)\\right)\\left(\\sum_{i=1}^{n_\\pi} (\\rho_i(\\Pi_i)-1\/n_\\pi)^2\\right)\\,,\n\\end{equation*}\nand we can readily observe that different ordinal distributions $P=\\{\\rho_i(\\Pi_i)\\}_{i=1,\\ldots,n_\\pi}$ may lead to the same value of $H$ but different values of $C$ (or vice-versa). Let us further consider a particular ordinal distribution with three possible permutation symbols (this would be equivalent to having $d_x!=3$ or $(d_x d_y)!=3$, if possible), that is, $P=\\{a,b,1-(a+b)\\}$, where $a>0$ and $b>0$ are real numbers such that $(a+b)\\leq1$ (to ensure the normalization of $P$). For this case, we have $S = - a \\log a - b \\log b - [1-(a+b)]\\log[1-(a+b)]$ and $D = (a-1\/3)^2 + (b-1\/3)^2 + ([1-(a+b)]-1\/3)^2$. Thus, for instance, if $a=0.79$ and $b=0.18$ or $a=0.80$ and $b=0.16$ we find the same value of $H = S\/\\log{3}\\approx0.55$, but different values for $D$ ($0.32$ in the first case and $0.33$ in the second) and, consequently, for $C$. \n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig2.pdf}\n\\caption{Probability distributions of ordinal patterns for stochastic and deterministic series. (a) Comparison between the empirical probability distribution of ordinal patterns obtained from a simulated Gaussian random walk with $10^6$ steps and the exact distribution $P_{\\rm walk}$ (dashed horizontal lines) for $d_x = 3$ and $\\tau_x = 1$. (b) Comparison between the empirical probability distribution of ordinal patterns obtained from $10^6$ iterations of the logistic map at fully developed chaos and the exact distribution $P_{\\rm logistic}$ (dashed horizontal lines) for $d_x = 3$ and $\\tau_x = 1$. All results in this figure can be replicated by running a Jupyter notebook available at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:2}\n\\end{figure*}\n\nIn \\texttt{ordpy}{}, the \\texttt{complexity\\_entropy}{} function simultaneously returns the values of $H$ and $C$ from time series as illustrated in:\n\\begin{minted}{python}\n>>> from ordpy import complexity_entropy \n>>> complexity_entropy([4,7,9,10,6,11,3],\n... dx=2)\n(0.9182958340544894, 0.06112816548804511)\n\\end{minted}\nFurthermore, the complexity-entropy plane was generalized for two-dimensional data~\\cite{ribeiro2012complexity, zunino2016discriminating} (notice that the only changes are related to the process of estimating the ordinal distribution) and the \\texttt{complexity\\_entropy}{} function also accepts two-dimensional arrays as input as shown in:\n\\begin{minted}{python}\n>>> from ordpy import complexity_entropy\n>>> complexity_entropy([[1,2,1],[8,3,4],\n... [6,7,5]], dx=2, dy=2)\n(0.3271564379782973, 0.2701200547320647)\n\\end{minted}\n\n\\section{Applications of Bandt and Pompe's framework with \\texttt{ordpy}{}}\n\nThis section presents more engaging applications of \\texttt{ordpy}{}'s functions by replicating literature results. We start by determining the ordinal probability distributions of two different time series of stochastic and chaotic nature, namely, a random walk with Gaussian steps and the logistic map at fully developed chaos (see Appendix~\\ref{appendix:chaos} for definitions). We choose these two examples because their ordinal distributions are exactly known for some combinations of the embedding parameters~\\cite{amigo2006order, bandt2007order}. More specifically, for $d_x = 3$ and $\\tau_x = 1$, the probability distributions associated with the permutation symbols $\\{(0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), (2,1,0)\\}$ are $P_{\\rm walk} = \\{{1}\/{4}, {1}\/{8}, {1}\/{8}, {1}\/{8}, {1}\/{8}, {1}\/{4}\\}$ and $P_{\\rm logistic} = \\{{1}\/{3}, {1}\/{15}, {2}\/{15}, {3}\/{15}, {4}\/{15}, 0\\}$ for the random walk~\\cite{bandt2007order} and the logistic map~\\cite{amigo2006order}, respectively.\n\nTo numerically estimate these two ordinal distributions, we generate a time series from a Gaussian random walk process and another time series from iterations of the fully chaotic logistic map. In both cases, we have simulated one realization of each process with $10^6$ observations and used the \\texttt{ordinal\\_distribution}{} function. Figure~\\ref{fig:2} shows that the exact ordinal distributions are in excellent agreement with simulated results obtained with \\texttt{ordpy}{}. It is intriguing to observe that the ordinal pattern $(2,1,0)$ (``descending permutation'') does not occur in the logistic series (it has probability zero). This fact is best understood as a feature directly associated with the intrinsic determinism of the logistic map dynamics~\\cite{amigo2006order, amigo2007true}. As we shall discuss in the next section, investigations about such ``missing ordinal patterns'' are also useful for characterizing time series dynamics.\n\nTo better illustrate the use of the \\texttt{permutation\\_entropy}{} function, we partially reproduce Bandt and Pompe's analysis of the logistic map (Fig.~2 of Ref.~\\onlinecite{bandt2002permutation}). We generate time series consisting of $10^6$ iterations of the logistic map for each value of parameter $r \\in \\{3.5, 3.5001, 3.5002, \\dots, 4.0\\}$ (see Appendix~\\ref{appendix:chaos} for definitions). Next, we calculate the permutation entropy $S$ for each of these 5001 time series using \\texttt{permutation\\_entropy}{} with embedding parameters $d_x = 6$ and $\\tau_x = 1$. We further divide the permutation entropy by $5$ to obtain the permutation entropy per symbol of order 6, that is, $h_6 = S\/5$ as defined in Bandt and Pompe's work~\\cite{bandt2002permutation}. Figure~\\ref{fig:3}{a} depicts the well-known bifurcation diagram for the logistic map, while Fig.~\\ref{fig:3}{b} shows the values of $h_6$ as a function of the parameter $r$. We note that the permutation entropy per symbol has an overall increasing trend with the parameter $r$, marked by abrupt drops in intervals of $r$ related to periodic behaviors. As noticed by Bandt and Pompe, the behavior of the permutation entropy is similar to the one observed for the Lyapunov exponent~\\cite{bandt2002permutation}.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=1\\linewidth]{fig3.pdf}\n\\caption{Permutation entropy of one- and two-dimensional data. (a) Bifurcation diagram of the logistic map for $r$ between $3.5$ and $4$ in steps of size $10^{-4}$. (b) Permutation entropy per symbol of order six $(h_6)$ calculated from logistic time series with $10^6$ observations (random initial conditions) and $r \\in \\{3.5,3.5001,3.5002,\\dots,4.0\\}$. The embedding parameters are $d_x = 6$ and $\\tau_x = 1$. (c) Time series of the transient logistic map obtained from the initial condition $x_0 = 0.65$, and by incrementing the logistic parameter $r$ at each iteration from $2.8$ to $4$ in steps of size $10^{-5}$. Despite appearing very similar to a bifurcation diagram, this result refers to a time series where each observation $x[r(t)]$ corresponds to a value $r(t)$. (d) Dependence of the normalized permutation entropy evaluated within a sliding window with 1024 observations of the original time series. Here $r(t)$ represents the logistic parameter at the end of each sliding window. The different curves show the results for $d_x = 5$ and $\\tau_x = 1$ (red), and $d_x = 5$ and $\\tau_x = 2$ (blue). The vertical line at $r = 3.56$ indicates the period-8 to period-16 bifurcation. (e) Ising surfaces obtained after $10^6$ Monte Carlo steps with reduced temperatures $T_r \\in \\{0.8, 0.9, 1.0, 1.1\\}$. In these surfaces, dark gray shades indicate high lattice sites while light gray regions indicate the opposite. (f) Normalized permutation entropy as a function of the reduced temperature $T_r \\in \\{0.5,0.6,\\dots,3.0\\}$ for Ising surfaces of size $250 \\times 250$ obtained after $10^6$ Monte Carlo steps. The different curves show the results for embedding parameters $d_x = 3$ and $d_y =2$ (red) and $d_x = 2$ and $d_y =3$ (blue), both with $\\tau_x = \\tau_y = 1$. All results in this figure can be reproduced by running a Jupyter notebook available at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:3}\n\\end{figure*}\n\nIn another example with \\texttt{permutation\\_entropy}{}, we replicate a numerical experiment of Cao \\textit{et al.}~\\cite{cao2004detecting} (see their Fig.~1) that searches for dynamical changes in the transient logistic map time series (see Appendix~\\ref{appendix:chaos} for definitions). This problem illustrates the role of the embedding delay $\\tau_x$. As in the original article, we iterate the transient logistic map starting with the initial condition $x_0 = 0.65$ and incrementing the logistic parameter $r$ from $2.8$ to $4$ in steps of size $10^{-5}$. This process generates a time series with $120001$ observations as shown in Fig.~\\ref{fig:3}{c}. Using this time series, we calculate the normalized permutation entropy within a sliding window with 1024 observations for the embedding dimension $d_x = 5$ and two values for the embedding delay ($\\tau_x = 1$ and $\\tau_x = 2$). \n\nAs Cao \\textit{et al.}~\\cite{cao2004detecting}, we denote the permutation entropy values by $H[r(t)]$, where $r(t)$ represents the logistic parameter at the end of the sliding window. Figure~\\ref{fig:3}{d} shows the values of $H[r(t)]$, where abrupt changes are clearly associated with dynamical changes observed in the time series (Fig.~\\ref{fig:3}{c}). Despite the overall similarities, we note that the embedding delay $\\tau_x = 2$ identifies these dynamical changes better than the case with $\\tau_x = 1$; for instance, the transition from period-8 to period-16 (at $r \\approx 3.56$) is missed when $\\tau_x = 1$ but captured when $\\tau_x = 2$~\\cite{cao2004detecting}.\n\nAs we have mentioned, a generalization of permutation entropy to two-dimensional data was first proposed by Ribeiro \\textit{et al.}~\\cite{ribeiro2012complexity}. To illustrate the use of the \\texttt{permutation\\_entropy}{} function with two-dimensional data, we replicate a numerical experiment related to Ising surfaces (see Appendix~\\ref{appendix:stochastic} for definitions) present in that work (Fig.~8 of Ref.~\\onlinecite{ribeiro2012complexity}). These surfaces represent the accumulated sum of spin variables of the canonical two-dimensional Ising model in a Monte Carlo simulation. Figure~\\ref{fig:3}{e} shows four examples of these surfaces (square lattices of size $250 \\times 250$) obtained after $10^6$ Monte Carlo steps for different reduced temperatures $T_r$. We notice non-trivial patterns emerging when the reduced temperature is equal to the critical temperature $(T_r = 1)$ of phase transition for the Ising model~\\cite{landau2015guide}. Following the original article, we generate Ising surfaces (size $250 \\times 250$) for reduced temperatures $T_r \\in \\{0.5,0.6,\\dots,3.0\\}$ and calculate their normalized permutation entropy with $d_x = 3$ and $d_y = 2$, and $d_x = 2$ and $d_y = 3$, both for $\\tau_x = \\tau_ y = 1$. In agreement with Ribeiro \\textit{et al.}~\\cite{ribeiro2012complexity}, Fig.~\\ref{fig:3}{f} shows that the permutation entropy precisely identifies the phase transition of the Ising model (the sudden decrease around the critical temperature) and that these Ising surfaces are symmetric under reversal of the embedding dimensions.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig4.pdf}\n\\caption{Complexity-entropy plane for one- and two-dimensional data. (a) Average values of the statistical complexity $C$ versus the normalized permutation entropy $H$ (over ten realizations) evaluated from time series of chaotic maps and stochastic processes. The embedding parameters are $d_x = 6$ and $\\tau_x = 1$. The solid  lines represent the maximum and minimal possible values of complexity for a given entropy (for $d_x = 6$ and $\\tau_x = 1$). (b) Localization of three art paintings in the complexity-entropy plane with embedding parameters $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$. All data and code necessary to reproduce this figure are available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:4}\n\\end{figure*}\n\nThe \\texttt{complexity\\_entropy}{} function simultaneously calculates the permutation entropy and the statistical complexity from time series and image data. To illustrate its usage, we partially reproduce the results of Rosso \\textit{et al.}~\\cite{rosso2007distinguishing} (Fig.~1 in that work) on distinguishing chaotic from stochastic time series. By following their article, we iterate four discrete maps to generate chaotic series. Specifically, we obtain chaotic time series from skew tent map (parameter $w = 0.1847$), H\\'enon map ($x$-component, parameters $a = 1.4$ and $b = 0.3$), logistic map $(r = 4)$, and Schuster map (parameter $z \\in \\{3\/2, 2, 5\/2\\}$) -- see Appendix~\\ref{appendix:chaos} for definitions. We further generate stochastic series from three stochastic processes: noises with $1\/f^{-k}$ power spectrum (for $k \\in \\{0.00, 0.25, \\dots, 3.00\\}$), fractional Brownian motion (Hurst exponent $h \\in \\{0.1,0.2,\\dots,0.9\\}$), and fractional Gaussian noise (also $h \\in \\{0.1,0.2,\\dots,0.9\\}$) -- see Appendix~\\ref{appendix:stochastic} for definitions. For each of these maps and stochastic processes, we generate ten time series with $2^{15}$ observations and random initial conditions. Next, we use \\texttt{complexity\\_entropy}{} with embedding parameters $d_x = 6$ and $\\tau_x = 1$ to calculate their statistical complexity and permutation entropy (average values over 10 time series realizations).\n\nAs in the original work of Rosso \\textit{et al.}~\\cite{rosso2007distinguishing}, Fig.~\\ref{fig:4}{a} shows that chaotic series usually have high complexity and low entropy values. Stochastic time series, in turn, display high entropy and intermediary complexity values. It is also interesting to note that stochastic time series approach the lower-right corner of the complexity-entropy plane ($H\\to1$ and $C\\to0$) as the serial auto-correlation decreases~\\cite{rosso2007distinguishing}. These results also illustrate that some stochastic and chaotic series have very similar entropy values but different statistical complexity (for instance, fractional Brownian motion with $h=0.9$ and Schuster map with $z=3\/2$), confirming that the statistical complexity extracts additional information from the ordinal distribution. In this figure, we have also included two solid lines delimiting the accessible region of the complexity-entropy plane~\\cite{martin2006generalized}. In \\texttt{ordpy}{}, the functions \\texttt{maximum\\_complexity\\_entropy}{} and \\texttt{minimum\\_complexity\\_entropy}{} generate these curves, as shown in the following code snippet:\n\\begin{minted}{python}\n>>> from ordpy import \n... maximum_complexity_entropy, \n... minimum_complexity_entropy\n>>> maximum_complexity_entropy(dx=4)\narray([[-0.        , -0.        ],\n       [ 0.21810429,  0.19670592],\n       [ 0.34568712,  0.28362016],\n       ...\n       [ 0.98660828,  0.02388382]])\n>>> minimum_complexity_entropy(dx=4)\narray([[-0.00000000e+00, -0.00000000e+00],\n       [ 2.67076969e-02,  2.55212327e-02],\n       ...\n       [ 1.00000000e+00, -3.66606083e-16]])\n\\end{minted}\n\nThe \\texttt{complexity\\_entropy}{} function also works with two-dimensional data, and to illustrate its usage, we follow Sigaki \\textit{et al.}~\\cite{sigaki2018history} and use the complexity-entropy plane to investigate patterns in art paintings. Due to the large-scale of the data analyzed by Sigaki \\textit{et al.} and to keep our examples self-contained, we do not reproduce their original results but simply use their ideas to illustrate how complexity and entropy extract useful information from images. To do so, we handpick three paintings from \\href{https:\/\/www.wikiart.org\/}{wikiart.org} (in the original article, the authors studied 137,364 images obtained from the same webpage). These are a Color Field Painting artwork (Blue, 1953 by Ad Reinhardt, image size $768 \\times 435$~\\cite{reinhardt1953abstract}), a Brazilian Modernist artwork (Abaporu, 1928 by Tarsila do Amaral, image size $1200 \\times 1026$~\\cite{tarsila1928abaporu}), and an American Abstract Expressionist painting (Number 1, 1950 (Lavender Mist), 1950 by Jackson Pollock, image size $749 \\times 1024$~\\cite{pollock1950number}). The three images are in JPEG format with 24 bits per pixel (8 bits for red, green, and blue colors in the RGB color space). We have averaged the pixels over the three RGB layers to represent each image by a usual two-dimensional array. Having these arrays, we calculate the statistical complexity and permutation entropy for the three paintings with embedding parameters $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$.\n\nFigure~\\ref{fig:4}{b} shows the complexity-entropy plane for these images (insets depict the artworks). In agreement with the global trend observed by Sigaki \\textit{et al.}~\\cite{sigaki2018history}, these results show that paintings portraying objects with clearly defined borders (such as the squares in Reinhardt's artwork) tend to present large values of statistical complexity and low values of entropy. On the other extreme, paintings with smudged and diffuse contours (such as Pollock's drip paintings) have high entropy and low complexity values. Between these somewhat opposite behaviors, we have a whole continuum of images, as exemplified here by the work of the Brazilian painter Tarsila do Amaral. As argued by Sigaki \\textit{et al.}~\\cite{sigaki2018history}, the complexity-entropy plane maps the local degree of order of artworks into a scale of order-disorder and simplicity-complexity that is similar to qualitative descriptions of artworks proposed by art historians such as W\\\"olfflin (the linear versus painterly dichotomy) and Riegl (the haptic versus optic dichotomy).\n\n\\section{Missing ordinal patterns}\n\nAs we have commented, the logistic map at fully developed chaos does not exhibit the ``descending permutation'' $(2,1,0)$ for $d=3$ (see Fig.~\\ref{fig:2}{b}). This feature is not a particularity of the logistic map. Indeed, these missing ordinal patterns (also called forbidden patterns) occur in different systems, and simple statistics associated with them have proven to be useful and reliable indicators of a system's dynamics~\\cite{zanin2008forbidden, zunino2009forbidden, sakellariou2016counting2, mcculough2016counting1, kulp2016using}. The works of Amig\\'o \\textit{et al.}~\\cite{amigo2006order, amigo2007true} are seminal in this regard, and by following their classification, we can divide these forbidden ordinal patterns into two categories: true or false~\\cite{amigo2007true}. True forbidden patterns (such as the  $(2,1,0)$ in the logistic map) are a fingerprint of determinism in a time series dynamics and represent an intrinsic feature of the underlying dynamical process~\\cite{amigo2006order}; that is, these patterns are not an artifact related to the finite length of empirical observations. In turn, false forbidden patterns are related to the finite length of time series~\\cite{amigo2007true} and can emerge even from completely random processes.\n\nThis distinction is not straightforward when dealing with empirical data, but a typical analysis in this context consists in investigating the number of missing patterns ($\\eta$) as a function of the time series length ($N_x$). The behavior of this curve is useful for discriminating time series. In \\texttt{ordpy}{}, the \\texttt{missing\\_patterns}{} function identifies missing ordinal patterns and estimates their relative frequency as in:\n\\begin{minted}{python}\n>>> from ordpy import missing_patterns\n>>> missing_patterns([4,7,9,10,6,11,3,5,\n... 6,2,3,1], dx=3)\n(array([[0, 2, 1],\n        [2, 1, 0]]),\n 0.3333333333333333)\n\\end{minted}\n\nTo better illustrate the use of this function, we investigate missing ordinal patterns in time series obtained from the logistic map at fully developed chaos ($r = 4$) and Gaussian random walks. In both cases, we use the embedding dimensions $d_x = 5$ and $d_x = 6$ (with $\\tau_x = 1$) and series lengths $N_x \\in \\{60,150,240,\\dots,6000\\}$. Figure~\\ref{fig:5}{a} shows the results. We observe that the number of missing ordinal patterns approaches zero as the time series length of random walks increases. Conversely, the number of missing permutations related to the logistic map displays an initial decay with the time series length but it rapidly saturates in considerably large numbers, indicating that these missing patterns are intrinsically associated with the underlying determinism of the process~\\cite{amigo2007true}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig5.pdf}\n\\caption{Missing ordinal patterns in time series. (a) Number of missing ordinal patterns ($\\eta$) in random walk (blue) and logistic map (red) time series as a function of sequence length ($N_x$) for embedding parameters $d_x = 5$ and $d_x = 6$, both with $\\tau_x = 1$. Results represent the average number of missing permutations over ten time series replicas for each $N_x\\in \\{60,150,240,\\dots,6000\\}$. (b) Dependence of the number of missing ordinal patterns on the noise intensity ($\\xi$) for noisy logistic time series with 6000 observations. The noise added to the logistic series is uniformly distributed in the interval $[-\\xi, \\xi]$ with $\\xi \\in \\{0,0.001,0.002,\\dots,0.5\\}$. Results represent average values over ten time series replicas for each noise level. The embedding dimensions are indicated within the plot and the embedding delay is $\\tau_x = 1$. We use random initial conditions and set the parameter $r = 4$ in all experiments with the logistic map. The necessary code to reproduce these results is available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:5}\n\\end{figure*}\n\nIn another application with the \\texttt{missing\\_patterns}{} function, we replicate a result of Amig\\'o \\textit{et al.}~\\cite{amigo2007true} (Fig.~4 in their work) to further show that the number of missing patterns is a good indicator of determinism in time series~\\cite{amigo2007true, amigo2008combinatorial}. By following the original work, we generate time series from the logistic map at fully developed chaos (6000 iterations) and add to them uniformly distributed noise in the interval $[-\\xi, \\xi]$, where $\\xi$ is the noise amplitude. Next, we estimate the average number of missing patterns (over ten time series replicas) for each noise level $\\xi \\in \\{0,0.001,0.002,\\dots,0.5\\}$, and embedding dimensions $d_x=5$ and $d_x=6$ (with $\\tau_x=1$). Figure~\\ref{fig:5}{b} shows the number of missing ordinal patterns as a function of noise amplitude $\\xi$ for both embedding dimensions. We observe that the number of missing patterns related to these deterministic series contaminated with noise approaches zero as noise amplitude grows. However, significantly higher noise levels are necessary to remove all signs of determinism expressed by the lack of permutation patterns when $d_x=6$~\\cite{amigo2007true}.\n\n\\section{Tsallis and R\\'enyi entropy-based quantifiers of the ordinal distribution}\n\nIn addition to Shannon's entropy and the statistical complexity, researchers have proposed to use other quantifiers of the ordinal probability distribution~\\cite{bandt2017new, small2018ordinal, zunino2008fractional, liang2015eeg}. As we have explicitly verified for the statistical complexity, these different quantifiers are supposed to extract additional information from a time series' dynamics that is not captured by permutation entropy and statistical complexity. In this context, a productive approach is to consider parametric generalizations of Shannon's entropy, such as those proposed by Tsallis~\\cite{tsallis1988generalization} and R\\'enyi~\\cite{renyi1961measures}. The work of Zunino \\textit{et al.}~\\cite{zunino2008fractional} was the first to consider the Tsallis entropy in place of Shannon's entropy to define the Tsallis permutation entropy as\n\\begin{equation}\n    S_\\beta(P) = \\frac{1}{\\beta-1}\\sum_{i = 1}^{n_\\pi} (\\rho_i(\\Pi_i) - \\rho_i(\\Pi_i)^\\beta)\\,,\n\\end{equation}\nwhere $\\beta$ is a real parameter ($\\beta \\to 1$ recovers the usual Shannon entropy and so the permutation entropy). Tsallis's entropy is also maximized by the uniform distribution, such that $S_\\beta^{\\rm max} = \\frac{1 - (n_\\pi)^{1-\\beta}}{\\beta-1}$. Thus, the normalized Tsallis permutation entropy is\n\\begin{equation}\n    H_\\beta(P) = (\\beta-1)\\frac{S_\\beta(P)}{1 - (n_\\pi)^{1-\\beta}}.\n\\end{equation}\n\nSimilarly, Liang \\textit{et al.}~\\cite{liang2015eeg} have proposed the R\\'enyi permutation entropy\n\\begin{equation}\n    S_\\alpha(P) = \\frac{1}{1-\\alpha}\\ln\\left(\\sum_{i = 1}^{n_\\pi} \\rho_i(\\Pi_i)^\\alpha\\right)\\,,\n\\end{equation}\nwhere $\\alpha > 0$ is a real parameter. R\\'enyi's entropy converges to Shannon's entropy when $\\alpha \\to 1$ and is maximized by the uniform distribution ($S_\\alpha^{\\rm max} = \\ln n_\\pi$, as the usual Shannon entropy). Thus, the normalized R\\'enyi permutation entropy is\n\\begin{equation}\n    H_\\alpha(P) = \\frac{S_\\alpha(P)}{\\ln n_\\pi}.\n\\end{equation}\n\nIn both cases, the generalized entropic form is mono-parametric and has a term where the ordinal probabilities appear raised to the power of the entropic parameter (that is, $\\rho_i(\\Pi_i)^\\beta$ and $\\rho_i(\\Pi_i)^\\alpha$). These parameters assign different weights to the underlying ordinal probabilities, allowing us to access different dynamical scales and produce a family of quantifiers for the ordinal distribution. In \\texttt{ordpy}{}, the \\texttt{tsallis\\_entropy}{} and \\texttt{renyi\\_entropy}{} functions implement these two quantifiers as in:\n\\begin{minted}{python}\n>>> from ordpy import tsallis_entropy, \n... renyi_entropy \n>>> tsallis_entropy([4,7,9,10,6,11,3], \n... q=[1,2], dx=2) #Here q plays the\n... role of beta.\narray([0.91829583, 0.88888889])\n>>> renyi_entropy([4,7,9,10,6,11,3],\n... alpha=[1,2], dx=2)\narray([0.91829583, 0.84799691])\n\\end{minted}\n\nIn a similar direction, there are also the developments of complexity-entropy curves proposed by Ribeiro \\textit{et al.}~\\cite{ribeiro2017characterizing} and Jauregui \\textit{et al.}~\\cite{jauregui2018characterization}. These works have further extended the complexity-entropy plane concept by considering the Tsallis and R\\'enyi entropies combined with proper generalizations of statistical complexity~\\cite{martin2006generalized}. Thus, instead of having a single point in the complexity-entropy plane for a given time series, Ribeiro \\textit{et al.}~\\cite{ribeiro2017characterizing} and Jauregui \\textit{et al.}~\\cite{jauregui2018characterization} have created parametric curves by varying the entropic parameter ($\\beta$ or $\\alpha$) and simultaneously calculating the generalized entropy and the generalized statistical complexity. \n\nTo define the Tsallis complexity-entropy curves~\\cite{ribeiro2017characterizing}, we first extend the statistical complexity (Eq.~\\ref{eq:statistical_complexity}) using the Tsallis entropy, that is,\n\\begin{equation}\\label{eq:statistical_complexity_tsallis}\n    C_\\beta(P) = \\frac{D_\\beta(P,U) H_\\beta(P)}{D^{\\rm max}_\\beta}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n    D_\\beta(P,U) = \\frac{1}{2}K_\\beta\\left( P\\bigg|\\frac{P+U}{2}\\right) + \\frac{1}{2}K_\\beta\\left(U\\bigg|\\frac{P+U}{2}\\right)\n\\end{equation}\nis the Jensen-Tsallis divergence~\\cite{martin2006generalized} written in terms of the corresponding Kullback-Leibler divergence~\\cite{martin2006generalized, tsallis2009introduction}\n\\begin{equation}\n    K_\\beta(V|R) = \\frac{1}{\\beta-1}\\sum_i^{n_\\pi} v_i^\\beta[r_i^{1-\\beta}-v_i^{1-\\beta}]\\,,\n\\end{equation}\nwhere $V=\\{v_i\\}_{i = 1, \\dots, n_\\pi}$ and $R=\\{r_i\\}_{i = 1, \\dots, n_\\pi}$ are two arbitrary distributions. In Eq.~\\ref{eq:statistical_complexity_tsallis}, \\begin{equation*}\n    D^{\\rm max}_\\beta = \\frac{2^{2-\\beta}n_\\pi - (1 + n_\\pi)^{1-\\beta}-n_\\pi(1+1\/n_\\pi)^{1-\\beta}-n_\\pi+1}{2^{2-\\beta}n_\\pi(1-\\beta)}\n\\end{equation*}\nis a normalization constant representing the maximum possible value of $D_\\beta(P,U)$ that occurs for $P = \\{ \\delta_{1,i} \\}_{i = 1,\\dots,n_\\pi}$ (as in the usual Jensen-Shannon divergence). By following Ribeiro \\textit{et al.}~\\cite{ribeiro2017characterizing}, we construct a parametric representation of the ordered pairs $(H_\\beta(P), C_\\beta(P))$ for $\\beta > 0$, obtaining the Tsallis complexity-entropy curves. \n\nSimilarly, to define the R\\'enyi complexity-entropy curves~\\cite{jauregui2018characterization}, we generalize the statistical complexity in R\\'enyi's formalism as \n\\begin{equation}\n    C_\\alpha(P,U) = \\frac{D_\\alpha(P,U) H_\\alpha(P)}{D_\\alpha^{\\rm max}}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n    D_\\alpha(P,U) = \\frac{1}{2}K_\\alpha\\left( P\\bigg|\\frac{P+U}{2}\\right) + \\frac{1}{2}K_\\alpha\\left(U\\bigg|\\frac{P+U}{2}\\right)\n\\end{equation}\nis the Jensen-R\\'enyi divergence~\\cite{martin2006generalized} written in terms of \n\\begin{equation}\n    K_\\alpha(V|R) = \\frac{1}{\\alpha-1}\\ln \\left( \\sum_{i = 1}^{d!} v_i^\\alpha r_i^{1-\\alpha}\\right)\\,,    \n\\end{equation}\nthe corresponding Kullback-Leibler divergence for R\\'enyi's entropy~\\cite{martin2006generalized, vanerven2014renyi}. The normalization constant\n\\begin{equation*}\n    D_\\alpha^{\\rm max} = \\frac{1}{2(\\alpha-1)}\\ln{\\left[\\frac{(n_\\pi+1)^{1-\\alpha} + n_\\pi - 1}{n_\\pi}\\left(\\frac{n_\\pi+1}{4 n_\\pi}\\right)^{1-\\alpha}\\right]}\n\\end{equation*}\ncorresponds to the maximum possible value of $D_\\alpha(P,U)$ occurring for $P = \\{ \\delta_{1,i} \\}_{i = 1,\\dots,n_\\pi}$ (as in the usual Jensen-Shannon divergence). Again, we can construct a parametric representation of the ordered pairs $(H_\\alpha(P), C_\\alpha(P))$ for $\\alpha > 0$, obtaining the R\\'enyi complexity-entropy curves proposed by Jauregui \\textit{et al.}~\\cite{jauregui2018characterization}.\n\nIn \\texttt{ordpy}{}, the functions \\texttt{tsallis\\_complexity\\_entropy}{} and \\texttt{renyi\\_complexity\\_entropy}{} implement the Tsallis and R\\'enyi complexity-entropy curves as shown in the following code snippet:\n\\begin{minted}{python}\n>>> from ordpy import \n... tsallis_complexity_entropy,\n... renyi_complexity_entropy\n>>> tsallis_complexity_entropy(\n... [4,7,9,10,6,11,3], \n... dx=2, q=[1,2]) #Here q plays the\n... role of beta.\narray([[0.91829583, 0.06112817],\n       [0.88888889, 0.07619048]])\n>>> renyi_complexity_entropy(\n... [4,7,9,10,6,11,3],\n... dx=2, alpha=[1, 2])\narray([[0.91829583, 0.06112817],\n       [0.84799691, 0.08303895]])\n\\end{minted}\n\nTo better illustrate the use of these \\texttt{ordpy}{}'s functions, we replicate some numerical experiments involving the logistic map and random walks presented in the original works of Ribeiro \\textit{et al.}~\\cite{ribeiro2017characterizing} (Figs.~1 and 6 in that work) and Jauregui \\textit{et al.}~\\cite{jauregui2018characterization} (Figs.~1 and 3 in that work). We start by generating time series from the logistic map at fully developed chaos ($r=4$, random initial condition) and a Gaussian random walk. For the logistic map series, we discard the first $10^4$ iterations to avoid transient effects and iterate other $10^6$ steps. The random walk series also has $10^6$ observations. By using these time series, we generate their corresponding Tsallis complexity-entropy curves for  $d_x = 3$ and $\\tau_x = 1$ by sampling $10^3$ log-spaced values of the entropic parameter $\\beta$ between $0.01$ and $100$ for the logistic map, and between $0.001$ and $100$ for the random walk. \n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig6.pdf}\n\\caption{Tsallis and R\\'enyi complexity-entropy curves. Tsallis complexity-entropy curves for time series obtained from (a) the logistic map at fully developed chaos and (b) a Gaussian random walk, both with embedding parameters $d_x = 3$ and $\\tau_x = 1$. The solid lines represent the empirical results and the dashed lines indicate the exact form of these complexity-entropy curves. Panels (c) and (d) show the R\\'enyi complexity-entropy curves obtained from the same two time series with embedding parameters $d_x = 4$ and $\\tau_x = 1$. In all panels, star markers indicate the beginning of the curves ($\\beta\\approx0$ or $\\alpha\\approx0$), while circle markers indicate the end of the curves (largest values of $\\beta$ and $\\alpha$). Data and code necessary to reproduce these results are available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:6}\n\\end{figure*}\n\nFigures~\\ref{fig:6}{a} and \\ref{fig:6}{b} show the empirical complexity-entropy curves in comparison with their exact shape (dashed lines). These theoretical curves can be determined for these time series because the ordinal distributions of the logistic map ($P_{\\rm logistic} = \\{{1}\/{3}, {1}\/{15}, {2}\/{15}, {3}\/{15}, {4}\/{15}, 0\\}$) and random walks ($P_{\\rm walk} = \\{{1}\/{4}, {1}\/{8}, {1}\/{8}, {1}\/{8}, {1}\/{8}, {1}\/{4}\\}$) are exactly known for $d_x = 3$~\\cite{amigo2006order, bandt2007order}. We observe that theoretical and empirical results are in excellent agreement. As discussed by Ribeiro \\textit{et al.}~\\cite{ribeiro2017characterizing}, random series tend to form closed complexity-entropy curves (Fig.~\\ref{fig:6}{b}), while chaotic time series are usually represented by open complexity-entropy curves (Fig.~\\ref{fig:6}{a}). These features emerge as a direct consequence of the existence or not of missing ordinal patterns captured by the limiting behavior of $H_\\beta$ as $\\beta \\to 0$ and $\\beta \\to \\infty$~\\cite{ribeiro2017characterizing}.\n\nBy following a similar approach, we also estimate the R\\'enyi complexity-entropy curves for the two previous time series for $d_x = 4$ and $\\tau_x = 1$. Figures~\\ref{fig:6}{c} and \\ref{fig:6}{d} show these R\\'enyi complexity-entropy curves. Differently from the Tsallis case, R\\'enyi complexity-entropy curves are always open~\\cite{jauregui2018characterization}, and the usage of these curves for distinguishing chaotic from stochastic series relies on a more subtle characteristic. Indeed, Jauregui \\textit{et al.}~\\cite{jauregui2018characterization} have found that the initial curvature of R\\'enyi complexity-entropy curves ($dC_\\alpha\/dH_\\alpha$ for small $\\alpha$) can be used as an indicative of determinism in time series. Specifically, they found that positive curvatures are associated with time series of stochastic nature, while negative ones are related to chaotic phenomena. This pattern also occurs in the results of Figs.~\\ref{fig:6}{c} and \\ref{fig:6}{d}.\n\n\\section{Ordinal networks}\n\nAmong the more recent developments related to the Bandt-Pompe framework, we have the so-called ordinal networks. First proposed by Small~\\cite{small2013complex} for investigating nonlinear dynamical systems, and later generalized with his collaboration in a series of works~\\cite{mccullough2015time, mccullough2017multiscale, sun2014characterizing, sakellariou2019markov}, ordinal networks belong to a more general class of methods designed to map time series into networks, collectively known as time series networks~\\cite{zou2019complex}. Beyond counting ordinal patterns, this approach considers first-order transitions among ordinal symbols within a symbolic sequence. In this network representation, the different ordinal patterns occurring in a data set are mapped into nodes of a complex network. The edges between nodes indicate that the associated permutation symbols are adjacent to each other in a symbolic sequence. Furthermore, edges can be directed according to the temporal succession of ordinal symbols and weighted by the relative frequencies in which the corresponding successions occur in a symbolic sequence~\\cite{mccullough2015time}.\n\nAfter applying the Bandt-Pompe method with embedding parameters $d_x$ and $\\tau_x$ to a time series $\\{x_t\\}_{t=1,\\dots,N_x}$ and obtaining the symbolic sequence $\\{\\pi_p\\}_{p = 1, \\dots, n_x}$, we can define the elements of the weighted adjacency matrix of the corresponding ordinal network as~\\cite{mccullough2015time, pessa2019characterizing}\n\\begin{equation}\\label{eq:adj_matrix}\n    \\rho_{i, j} = \\frac{\\text{total of transitions} \\ \\Pi_i \\to \\Pi_j\\ \\text{in}\\ \\{\\pi_p\\}_{p = 1, \\dots, n_x}}{n_x - 1}\\,,\n\\end{equation} \nwhere $i,j=1,2,\\dots,n_\\pi$ (with $n_\\pi=d_x!$), $\\Pi_i$ and $\\Pi_j$ represent all possible ordinal patterns, and the denominator $n_x - 1$ is the total number of ordinal transitions. In \\texttt{ordpy}{}, the \\texttt{ordinal\\_network}{} function returns the nodes, edges, and edge weights of an ordinal network mapped from a time series as in:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_network\n>>> ordinal_network([4,7,9,10,6,11,8,3,7],\n... dx=2, normalized=False)\n(array(['0|1', '1|0'], dtype='<U3'),\n array([['0|1', '0|1'],\n        ['0|1', '1|0'],\n        ['1|0', '0|1'],\n        ['1|0', '1|0']], dtype='<U3'),\n array([2, 2, 2, 1]))\n\\end{minted}\n\nIt is worth noting that the original algorithm of Small~\\cite{small2013complex} for mapping time series into ordinal networks uses a different approach for creating the symbolic sequence. Instead of defining overlapping partitions (Eq.~\\ref{eq:1dpartition}), Small~\\cite{small2013complex} evaluates the ordinal patterns in non-overlapping partitions of size $d_x$ (the embedding delay is also not present in his original formulation). Furthermore, edges are undirected and unweighted in this initial formulation. This implementation is not as popular as the one directly following the Bandt-Pompe symbolization method~\\cite{mccullough2015time, mccullough2017multiscale, sun2014characterizing, sakellariou2019markov}, but is also available in \\texttt{ordpy}{} through the \\texttt{overlapping} parameter in the \\texttt{ordinal\\_network}{} function, as shown in:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_network\n>>> ordinal_network([4,7,9,10,6,11,8,3,7],\n... dx=2, normalized=False,\n... overlapping=False)\n(array(['0|1', '1|0'], dtype='<U3'), \narray([['0|1', '0|1'], ['0|1', '1|0']], \ndtype='<U3'), array([2, 1]))\n\\end{minted}\n\nOrdinal networks have also been recently generalized by Pessa and Ribeiro~\\cite{pessa2020mapping} to account for two-dimensional data sets such as images. In this case, we apply the two-dimensional version of Bandt and Pompe's symbolization approach~\\cite{ribeiro2012complexity} (see Eqs.~\\ref{eq:matrix_partition}, \\ref{eq:partition_index}, and \\ref{eq:partition}) to a data array $\\{y_t^u\\}_{t = 1,\\dots,N_x}^{u = 1,\\dots,N_y}$ for given embedding dimensions ($d_x$ and $d_y$) and embedding delays ($\\tau_x$ and $\\tau_y$), obtaining the corresponding two-dimensional ordinal sequence $\\{\\pi_p^q\\}_{p = 1,\\dots,n_x}^{q = 1,\\dots,n_y}$. Similarly to the one-dimensional case, each permutation symbol $\\Pi_i$ ($i = 1,\\dots,n_\\pi$, with $n_\\pi=(d_x d_y)!$) is associated with a node in the ordinal network, and directed edges connect permutation symbols that are vertically ($\\pi_p^q \\to \\pi_p^{q+1}$ for $q = 1,\\dots,n_y-1$) or horizontally ($\\pi_p^q \\to \\pi_{p+1}^{q}$ for $p = 1,\\dots,n_x-1$) adjacents in the symbolic sequence. The directed link between a pair of permutation symbols ($\\Pi_i$ and $\\Pi_j$) is weighted by the total number of occurrences of this particular transition in the symbolic sequence. Thus, the weighted adjacency matrix representing the ordinal network mapped from two-dimensional data is~\\cite{pessa2020mapping}\n\\begin{equation}\\label{eq:edge_weights}\n    \\rho_{i,j}\\! =\\! \\frac{\\text{total of transitions $\\Pi_i \\to \\Pi_j$ in $\\{\\pi_p^q\\}_{p=1,\\dots,n_x}^{q=1,\\dots,n_y}$}}{2n_x n_y-n_x-n_y}\\,,\n\\end{equation}\nwhere $i, j = 1,\\dots,n_\\pi$ (with $n_\\pi = (d_x d_y)!$) and the denominator represents the total number of horizontal and vertical transitions. The \\texttt{ordinal\\_network}{} function also handles two-dimensional data as in:\n\\begin{minted}{python}\n>>> from ordpy import ordinal_network\n>>> ordinal_network([[1,2,1],[8,3,4],[6,7,5]], \n... dx=2, dy=2, normalized=False)\n(array(['0|1|3|2', '1|0|2|3', '1|2|3|0'], \ndtype='<U7'),\n array([['0|1|3|2', '1|0|2|3'],\n        ['0|1|3|2', '1|2|3|0'],\n        ['1|0|2|3', '0|1|3|2'],\n        ['1|2|3|0', '0|1|3|2']], dtype='<U7'),\n array([1, 1, 1, 1]))\n\\end{minted}\n\nPessa and Ribeiro~\\cite{pessa2020mapping} have also proposed to create ordinal networks by considering only horizontal (horizontal ordinal networks) or only vertical (vertical ordinal networks) transitions among the permutations symbols. They have shown that comparing properties of these two networks is useful for exploring visual symmetries in images. In \\texttt{ordpy}{}, this possibility is available through the \\texttt{connections} parameter in the \\texttt{ordinal\\_network}{} function as in: \n\\begin{minted}{python}\n>>> from ordpy import ordinal_network\n>>> ordinal_network([[1,2,1],[8,3,4],[6,7,5]],\n... dx=2, dy=2, normalized=False, \n... connections='horizontal')\n(array(['0|1|3|2', '1|0|2|3', '1|2|3|0'], \ndtype='<U7'), \narray([['0|1|3|2', '1|0|2|3'],\n       ['1|2|3|0', '0|1|3|2']], dtype='<U7'), \narray([1, 1]))\n\\end{minted}\n\nAn intriguing feature of ordinal networks is the existence of intrinsic connectivity constraints~\\cite{pessa2019characterizing, pessa2020mapping} inherited from Bandt and Pompe's symbolization method. These constraints are directly related to the fact that adjacent partitions share elements, such that ordering relations in one partition are partially carried out to neighboring partitions. For one-dimensional data, these restrictions imply that all nodes in an ordinal network have in-degree and out-degree limited to numbers between $0$ and $d_x$; consequently, the maximum number of edges is $d_x \\times (d_x!)$~\\cite{pessa2019characterizing}. Ordinal networks mapped from one-dimensional data can only have self-loops in nodes associated with solely ascending or solely descending ordinal patterns~\\cite{pessa2019characterizing}. \n\nThe horizontal and vertical transitions related to networks mapped from two-dimensional data impose similar but trickier connectivity constraints~\\cite{pessa2020mapping}. In this case, the maximum number of outgoing connections emerging from horizontal and vertical transitions are $C(d_x d_y, d_y) \\times d_y!$ and $C(d_x d_y, d_x) \\times d_x!$, respectively~\\cite{pessa2020mapping}. However, the sets of horizontal and vertical transitions are not disjoint, and their union defines all possible outgoing edges. Finding a general expression for the latter set operation is cumbersome because it depends on the ordinal pattern associated with the node under analysis. Thus, while limited, the maximum number of edges varies among the ordinal patterns and needs to be numerically obtained~\\cite{pessa2020mapping}. Furthermore, differently from the one-dimensional case, ordinal networks mapped from two-dimensional data can display self-loops in several nodes~\\cite{pessa2020mapping}.\n\nA direct consequence of these intrinsic connectivity constraints is that ordinal networks mapped from completely random arrays (in one or two dimensions) are not random graphs~\\cite{pessa2019characterizing, pessa2020mapping}. Even more counter-intuitive is the existence of different edge weights in random ordinal networks, albeit all permutations are equiprobable in random arrays~\\cite{pessa2019characterizing, pessa2020mapping}. This non-trivial property results from the fact that, among all possible amplitude relations involved in an ordinal transition between a fixed permutation and all its possible neighboring permutations, some permutations appear more than once. For one-dimensional data, random ordinal networks only have two different edge weights: $1\/(d_x+1)!$ and $2\/(d_x+1)!$ (the denominator represents the sum of weights)~\\cite{pessa2019characterizing}. A rule of thumb for determining the edges with double weight is to pick all transitions in which the index number equal to ``$d_x - 1$'' in the next permutation fits the position of the index number ``0'' in the first permutation~\\cite{pessa2019characterizing}. For instance, the edge weight between permutations $(3,2,1,0)$ and $(2,1,0,3)$ has double weight. Ordinal networks mapped from two-dimensional random data have more than two different edge weights, and there is no simple rule (at least up to now) for obtaining these weights~\\cite{pessa2020mapping}. However, these values can be numerically calculated by explicitly considering each possible ordinal pattern~\\cite{pessa2020mapping}.\n\nIn \\texttt{ordpy}{}, the \\texttt{random\\_ordinal\\_network}{} function generates the exact form of ordinal networks expected from the mapping of one- and two-dimensional random data with arbitrary embedding dimensions ($d_x$ and $d_y$). The following code illustrates the usage of \\texttt{random\\_ordinal\\_network}{}:\n\\begin{minted}{python}\n>>> from ordpy import random_ordinal_network\n>>> random_ordinal_network(dx=2)\n(array(['0|1', '1|0'], dtype='<U3'),\n array([['0|1', '0|1'],\n        ['0|1', '1|0'],\n        ['1|0', '0|1'],\n        ['1|0', '1|0']], dtype='<U3'),\n array([0.16666667, 0.33333333, \n        0.33333333, 0.16666667]))\n\\end{minted}\nThe three returned arrays represent nodes, edges, and edge weights of the random ordinal network, respectively. It is worth noticing that these connectivity constraints disappear when considering non-overlapping data partitions as in the initial proposal of Small~\\cite{small2013complex}. In this case, ordinal networks mapped from large enough random data sets are represented by complete graphs with self-loops and all-equal edge weights. The \\texttt{random\\_ordinal\\_network}{} function returns these graphs by changing its \\texttt{overlapping} argument as in: \n\\begin{minted}{python}\n>>> from ordpy import random_ordinal_network\n>>> random_ordinal_network(dx=2,\n... overlapping=False)\n(array(['0|1', '1|0'], dtype='<U3'),\n array([['0|1', '0|1'],\n        ['0|1', '1|0'],\n        ['1|0', '0|1'],\n        ['1|0', '1|0']], dtype='<U3'),\n array([0.25, 0.25, 0.25, 0.25]))\n\\end{minted}\nSimilarly, embedding delays larger than one modify how elements are shared among partitions and impose connectivity constraints to high-order transitions. The \\texttt{random\\_ordinal\\_network}{} function is thus restricted to the case $\\tau_x = \\tau_y =1$ when considering overlapping partitions.\n\nThe primary purpose of mapping time series or images into ordinal networks is to use network measures to characterize data sets. In addition to the many network statistics derived from network science~\\cite{newman2010networks}, the inherent probabilistic nature of nodes and edges in ordinal networks has motivated two entropy-related measures~\\cite{mccullough2017multiscale, small2018ordinal, pessa2019characterizing}. The first one is a local measure defined at the node level known as the local node entropy~\\cite{mccullough2017multiscale, small2018ordinal, pessa2019characterizing, pessa2020mapping}\n\\begin{equation}\\label{eq:local_node_entropy}\n    s_i = -\\sum_{j\\in\\mathcal{O}_i}\\rho'_{i, j}\\log \\rho'_{i,j},\\,\n\\end{equation}\nwhere the index $i$ refers to a node related to a given permutation $\\Pi_i$, $\\rho'_{i,j} = \\rho_{i,j}\/\\sum_{k\\in\\mathcal{O}_i}\\rho_{i,k}$ represents the renormalized probability of transitioning from node $i$ to node $j$ (permutations $\\Pi_i$ and $\\Pi_j$), and $\\mathcal{O}_i$ is the outgoing neighborhood of node $i$ (set of all edges leaving node $i$). This quantity measures the determinism of ordinal transitions at the node level such that $s_i$ is maximum when all edges leaving $i$ have the same weight, while $s_i=0$ if there is only one edge leaving node $i$. Using the local node entropy, we can further define the global node entropy~\\cite{mccullough2017multiscale, small2018ordinal, pessa2019characterizing, pessa2020mapping}\n\\begin{equation}\\label{eq:global_node_entropy}\n    S_{\\rm GN} = \\sum_{i=1}^{n_\\pi} \\rho_i s_i\\,,\n\\end{equation}\nwhere $\\rho_i$ is the probability of finding the permutation $\\Pi_i$ (Eqs.~\\ref{eq:permutation_probability} and \\ref{eq:permutation_probability_2d}). Thus, the value $S_{\\rm GN}$ represents a weighted average of the local determinism over all nodes of an ordinal network (see also Unakafov and Keller~\\cite{unakafov2014conditional} for the definition of conditional entropy of ordinal patterns). In image classification tasks~\\cite{pessa2020mapping}, global node entropy has proven to outperform different image quantifiers derived from gray-level co-occurrence matrices (GLCMs)~\\cite{haralick1973textural, haralick1979statistical}, a traditional technique for texture analysis.\n\nContrarily to permutation entropy~\\cite{bandt2002permutation} and because of the intrinsic connectivity constraints of ordinal networks, the global node entropy is not maximized by random data~\\cite{pessa2019characterizing, pessa2020mapping}. For one-dimensional data, the global node entropy calculated from a random ordinal network is~\\cite{pessa2019characterizing}\n\\begin{equation}\n    S_{\\rm GN}^{\\rm random} = \\log(d_x + 1) - (\\log 4)\/(d_x + 1)\\,.\n\\end{equation}\nWhile there is no equivalent expression for two-dimensional data, it is possible to numerically calculate $S_{\\rm GN}^{\\rm random}$ using random ordinal networks numerically generated~\\cite{pessa2020mapping}. In both cases, the global node entropy can be normalized by the value of $S_{\\rm GN}^{\\rm random}$, that is, $H_{\\rm GN} = {S_{\\rm GN}}\/{S_{\\rm GN}^{\\rm random}}$.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=1\\linewidth]{fig7.pdf}\n\\caption{Ordinal networks mapped from one- and two-dimensional data. (a) Time series obtained from the $x$-coordinate of the R\\\"ossler system (with parameters $a=0.3$, $b = 2$ and $c = 4$). Here we show only the latest 500 of all $10^5$ observations. This time series exhibits a periodic behavior after a short initial transient. (b) Visualization of the ordinal network mapped from the $x$-coordinate of the R\\\"ossler system with embedding parameters $d_x = 16$ and $\\tau_x = 1$. This ordinal network uses Small's original algorithm~\\cite{small2013complex} with non-overlapping data partitions and undirected and unweighted edges. (c) Time series obtained from a realization of the fractional Brownian motion with $h = 0.8$ (only the latest 500 of all $2^{16}$ observations are shown). (d) Ordinal network representation of the fractional Brownian motion time series with $d_x = 3$ and $\\tau_x = 1$. Here we have used overlapping data partitions and made edge thickness proportional to edge weight. (e) Example of a periodic ornament with size $250 \\times 250$. (f) Ordinal network representation of the previous image with embedding parameters $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$. In this visualization, edge thickness is made proportional to edge weight. Data and code necessary to reproduce these results are available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:7}\n\\end{figure*}\n\nIn \\texttt{ordpy}{}, the \\texttt{global\\_node\\_entropy}{} function evaluates $S_{\\rm GN}$ directly from data arrays or using an ordinal network as returned by \\texttt{ordinal\\_network}{}. The following code shows simple usages of \\texttt{global\\_node\\_entropy}{}:\n\\begin{minted}{python}\n>>> from ordpy import global_node_entropy\n>>> global_node_entropy(\n... [1,2,3,4,5,6,7,8,9], dx=2)\n0.0\n>>> global_node_entropy(\n... ordinal_network([1,2,3,4,5,6,7,8,9],\n... dx=2))\n0.0\n>>> global_node_entropy(\n... np.random.uniform(size=100000), dx=3)\n1.4988332319747597\n>>> global_node_entropy(\n... random_ordinal_network(dx=3))\n1.5\n\\end{minted}\n\n\\section{Applications of ordinal networks with \\texttt{ordpy}{}}\n\nTo better illustrate the use of \\texttt{ordpy}{} in the context of ordinal networks, we review and replicate some literature results. Before starting, we remark that \\texttt{ordpy}{} does not have functions for network analysis or graph visualization. The \\texttt{ordinal\\_network}{} function generates output data (nodes, edges and weight lists) that can feed graph libraries such as \\texttt{graph\\_tool}{}~\\cite{peixoto2014graphtool}, \\texttt{networkx}{}~\\cite{hagberg2008networkx}, and \\texttt{igraph}{}~\\cite{csardi2006igraph}. Here, we have used \\texttt{networkx}{} and \\texttt{igraph}{}. \n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig8.pdf}\n\\caption{Global node entropy of one- and two-dimensional data. (a) The initial data points of a sawtooth-like time series defined as $x_t = \\{0,{1}\/{3},{1}\/{6},1,\\dots\\}$. (b) Normalized permutation entropy ($H$) and normalized global node entropy ($H_{\\rm GN}$) as a function of the amplitude ($\\xi$) of the uniform white noise added to the periodic sawtooth-like signals. The different curves represent average values of $H$ and $H_{\\rm GN}$ over ten realizations for $\\xi = \\{0, 0.05,0.1,\\dots,2\\}$. (c) Eight examples (out of 112) of the normalized Brodatz textures. These are grayscale images (256 gray levels) with size $640\\times640$~\\cite{safia2020multiband}. (d) Differences between the global node entropy evaluated from the horizontal and vertical ordinal networks ($S_{\\rm GN}^{\\rm Horizontal} - S_{\\rm GN}^{\\rm Vertical}$) mapped from each Brodatz texture. We highlight eight textures (the same as shown in panel c) with the largest differences. Data and code necessary to reproduce these results are available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:8}\n\\end{figure*}\n\nWe start by partially reproducing Small's~\\cite{small2013complex} pioneering work in which ``ordinal partition networks'' first appeared (see Fig.~3 in that work). By following Small~\\cite{small2013complex}, we numerically solve the differential equations of the R\\\"ossler system (with parameters $a=0.3$, $b = 2$ and $c = 4$, see Appendix~\\ref{appendix:chaos} for definitions) and sample the $x$-coordinate to obtain a time series with $10^5$ observations. Figure~\\ref{fig:7}{a} illustrates the periodic behavior of this time series. We then create the ordinal network from this data set with embedding parameters $d_x = 16$ and $\\tau_x=1$. It is worth remembering that Small's original algorithm uses non-overlapping partitions and the edges of the resulting ordinal network are undirected and unweighted. The parameter \\texttt{overlapping} in \\texttt{ordinal\\_network}{} should be equal to \\texttt{False} to properly use Small's original algorithm. Figure~\\ref{fig:7}{b} shows a visualization of this ordinal network, where the circular structure alludes to the periodicity of the original time series.\n\nIn another simple example with ordinal networks, we partially replicate Pessa and Ribeiro's~\\cite{pessa2019characterizing} results on fractional Brownian motion (see Fig.~6 in their work). To do so, we generate a time series from this stochastic process with Hurst exponent $h = 0.8$ (see Appendix~\\ref{appendix:stochastic} for definitions) and $2^{16}$ observations, as illustrated in Fig.~\\ref{fig:7}{c}. Next, we map this time series into an ordinal network with embedding parameters $d_x = 3$ and $\\tau_x = 1$ (this time using overlapping partitions as in the usual Bandt-Pompe approach). Figure~\\ref{fig:7}{d} shows a visualization of the resulting ordinal network, where the persistent behavior imposed by the Hurst exponent $h = 0.8$ is captured by the quite intense autoloops associated with the ordinal patterns $(0,1,2)$ and $(2,1,0)$ (that is, the upward and downward trends of this time series). Pessa and Ribeiro~\\cite{pessa2019characterizing} have also shown that local properties of ordinal networks (for instance, average weighted shortest path) are quite effective for estimating the Hurst exponent of time series, having performance superior to widely used approaches such as detrended fluctuation analysis (DFA)~\\cite{peng1994mosaic}.\n\nWe also consider ordinal networks mapped from two-dimensional data. We map a periodic ornament previously explored in Ref.~\\onlinecite{pessa2020mapping} (see Fig.~2 in that reference). Figure~\\ref{fig:7}{e} shows the ornament of size $250 \\times 250$ (see Appendix~\\ref{appendix:stochastic} for more details), while Fig.~\\ref{fig:7}{f} presents a visualization of the corresponding ordinal network with embedding parameters $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$. \nWe have made edge thickness proportional to edge weight (Eq. 22) to highlight that a few edges concentrate most of the transition probability of the network. Furthermore, we observe that this network has $12$ nodes and $72$ edges, that is, only a small fraction of all possible nodes ($24$) and edges ($416$) of a ordinal networks with $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$.\n\nIn addition to the previous more qualitative examples, we have also replicated some results related to the global node entropy of ordinal networks. For time series, we follow Pessa and Ribeiro~\\cite{pessa2019characterizing} (see Fig.~5 in their work) and generate a periodic sawtooth-like signal (Fig.~\\ref{fig:8}{a}) with $10^5$ observations and add to it uniform white noise in the interval $[-\\xi, \\xi]$, where $\\xi$ represents the noise amplitude. We generate these noisy sawtooth-like time series for each $\\xi \\in \\{0,0.05,0.1,\\dots,2\\}$ and determine the average values of the normalized permutation entropy ($H$) and the normalized global node entropy ($H_{\\rm GN}$) over ten time series replicas with $d_x = 4$ and $\\tau_x = 1$. \n\nFigure~\\ref{fig:8}{b} shows the average values of $H$ and $H_{\\rm GN}$ as a function of the noise amplitude $\\xi$. We note that both measures approach one with the increase of the noise amplitude. However, permutation entropy saturates for $\\xi\\approx1$, while global node entropy requires significantly higher values of $\\xi$. This result indicates that global node entropy is more robust to noise addition and has a higher discrimination power than permutation entropy~\\cite{pessa2019characterizing}.\n\nTo demonstrate the use of \\texttt{global\\_node\\_entropy}{} with two-dimensional data, we calculate the global node entropy for a set of 112 8-bit images of natural textures known as the normalized Brodatz textures~\\cite{safia2013new, safia2020multiband}. Figure~\\ref{fig:8}{c} shows examples of these images. By following Pessa and Ribeiro~\\cite{pessa2020mapping} (see Fig.~5 in their work), we calculate the global node entropy from the horizontal ($S_{\\rm GN}^{\\rm Horizontal}$) and vertical ($S_{\\rm GN}^{\\rm Vertical}$) ordinal networks mapped from the Brodatz textures with $d_x = d_y = 2$ and $\\tau_x = \\tau_y = 1$. \n\nFigure~\\ref{fig:8}{d} depicts the difference between these two entropy values (that is, $S_{\\rm GN}^{\\rm Horizontal} - S_{\\rm GN}^{\\rm Vertical}$) for each Brodatz texture. We have also highlighted eight textures with extreme values for this difference. Most of these images are characterized by stripes or line segments predominantly oriented in the vertical or horizontal directions which, in turn, suggests that properties of vertical and horizontal ordinal networks can detect simple image symmetries.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig9.pdf}\n\\caption{True and false missing links in ordinal networks. (a) Dependence of the fraction of missing links ($f$) estimated from Gaussian white noise time series as a function of the time series length ($N_x$). (b) Dependence of the fraction of missing links ($f$) estimated from fully chaotic logistic time series ($r = 4$) as a function of the time series length ($N_x$). In both panels, the different curves represent average values over ten realizations (for each series length) and embedding dimension $d_x \\in\\{3,4,5,6\\}$ with $\\tau_x = 1$. We also use 194 values for $N_x$ logarithmically spaced in the interval $[10,10^5]$ in both panels. Data and code necessary to reproduce these results are available in a Jupyter notebook at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s webpage.}\n\\label{fig:9}\n\\end{figure*}\n\nIn a final application with ordinal networks, we explore the concept of missing links or missing transitions among ordinal patterns~\\cite{pessa2019characterizing}. Similarly to the missing ordinal patterns described by Amig\\'o \\textit{et al.}~\\cite{amigo2006order, amigo2007true}, ordinal networks can display true and false forbidden transitions among ordinal patterns. In this case, true missing links are related to the intrinsic dynamics of the process under analysis, while false missing links are associated with the finite size of empirical data sets. Because we know the exact form of random ordinal networks~\\cite{pessa2019characterizing, pessa2020mapping} (here these networks represent all possible connections) we can readily find all missing links of an empirical ordinal network. In \\texttt{ordpy}{}, the \\texttt{missing\\_links}{} function evaluates all missing ordinal transitions directly from a data set or the returned arrays of \\texttt{ordinal\\_network}{} as in: \n\\begin{minted}{python}\n>>> from ordpy import missing_links\n>>> missing_links([4,7,9,10,6,11,3], dx=2, \n... return_fraction=False)\n(array([['1|0', '1|0']], dtype='<U3'), 1)\n>>> missing_links(ordinal_network(\n... [4,7,9,10,6,11,3],  dx=2),\n... dx=2, return_fraction=True)\n(array([['1|0', '1|0']], dtype='<U3'), 0.25)\n\\end{minted}\n\nTo demonstrate the use of \\texttt{missing\\_links}{} in a more engaging example, we replicate the results of Pessa and Ribeiro~\\cite{pessa2019characterizing} about missing links in ordinal networks mapped from Gaussian white noise time series ( Fig.~4 in their work). We generate these time series with length $N_x$ varying logarithmically between $10$ and $10^5$, and for each one, we estimate the average fraction of missing links over ten replicas for embedding dimensions $d_x\\in\\{3,4,5,6\\}$ and $\\tau_x = 1$. Figure~\\ref{fig:9}{a} shows these fractions of missing links as a function of the time series length. We observe that this quantity approaches zero as $N_x$ becomes sufficiently large. Furthermore, the smaller the embedding dimension, the faster the missing links vanish. This pattern is a fingerprint of false missing links. We have also carried out the same analysis with time series generated from logistic map iterations at fully developed chaos. Figure~\\ref{fig:9}{b} shows the corresponding results. Unlike white noise, the logistic map produces ordinal networks with missing links that persist even in considerably long time series. This behavior is typical of true missing links.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe have introduced \\texttt{ordpy}{} -- an open-source Python module for data analysis that implements several ordinal methods associated with the Bandt-Pompe framework. Specifically, \\texttt{ordpy}{} has functions implementing the following methods: permutation entropy, complexity-entropy plane, missing ordinal patterns, Tsallis and R\\'enyi permutation entropies, complexity-entropy curves, ordinal networks, and missing ordinal transitions. All \\texttt{ordpy}{}'s functions automatically deal with one-dimensional (time series) and two-dimensional (images) data. Furthermore, most of these functions are also ready for multiscale analysis via the embedding delay parameters. Along with the description of \\texttt{ordpy}{} functionalities, we have also presented a literature review of several of the principal methods related to Bandt and Pompe's framework. This review further includes a reproduction of several literature results with \\texttt{ordpy}{}'s functions. Beyond the summarized description of \\texttt{ordpy}{}'s functions presented here, we notice that a complete documentation is available at \\url{arthurpessa.github.io\/ordpy}. All data and code used in this work are also freely available at \\href{http:\/\/github.com\/arthurpessa\/ordpy}{\\texttt{ordpy}{}}'s website.\n\nWe believe \\texttt{ordpy}{} will help to popularize ordinal methods even further, particularly in research fields with more limited tradition in scientific computing. In addition to a myriad of possible empirical applications, we also believe \\texttt{ordpy}{} can further promote the development of new methods related to Bandt and Pompe's framework. We remark that some techniques available in \\texttt{ordpy}{} have received little attention or have not even been formally proposed. These possible developments already implemented in \\texttt{ordpy}{} include the use of complexity-entropy curves for two-dimensional data, multiscale complexity-entropy curves, ordinal networks with different embedding delays (particularly for two-dimensional data), analysis of missing patterns in two-dimensional data, and missing ordinal transitions. We also plan to implement more techniques based on the Bandt and Pompe's framework and include them in future versions of \\texttt{ordpy}{}.\n\nFinally, we hope our module helps making research methods more accessible and reproducible~\\cite{baker2016interactive, fanelli2018opinion} as well as other open-source software efforts such as the \\texttt{tisean}{}~\\cite{hegger1999tisean} (nonlinear time series analysis), \\texttt{pyunicorn}{}~\\cite{donges2015unified} (time series networks and recurrence analysis), and \\texttt{powerlaw}{}~\\cite{alstott2014powerlaw} (analysis of heavy-tailed distributions) packages.\n\n\\begin{acknowledgments}\nThis research was supported by Coordena\\c{c}\\~ao de Aperfeicoamento de Pessoal de N\\'ivel Superior (CAPES) and Conselho Nacional de Desenvolvimento Cient\\'ifico e Tecnol\\'ogico (CNPq -- Grants 407690\/2018-2 and 303121\/2018-1).\n\\end{acknowledgments}\n\n\\section*{Data Availability}\nAll data and code necessary to reproduce the results and figures of this work are available at \\url{http:\/\/github.com\/arthurpessa\/ordpy}, Ref.~\\onlinecite{pessa2021ordpygithub}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years, statistical tests based on the reproducing kernel Hilbert space (RKHS) embedding of distributions have attracted much attention because of their flexibility and broad applicability. Like other kernel methods, RKHS embedding based tests present a general and unifying framework for testing problems in arbitrary domains by using appropriate kernels defined on those domains. See, \\textit{e.g.}, \\cite{muandet2017kernel}, for a recent review and detailed discussion about the applications of kernel embeddings. The idea of using kernel embedding for comparing probability distributions was initially introduced by \\cite{Smola-07,gretton2007kernel, gretton2012kernel}. Related extensions were also proposed by \\cite{harchaoui2009kernel, zaremba2013b}. Furthermore,~\\cite{sejdinovic2012equivalence} established a close relationship between kernel-based hypothesis tests and energy distanced based test introduced by \\cite{szekely2007measuring}. See also \\cite{lyons2013distance}. More recently, motivated by several applications based on quantifying the convergence of Monte Carlo simulations,~\\cite{liu2016kernelized}, \\cite{chwialkowski2016kernel} and \\cite{gorham2017measuring} proposed goodness-of-fit tests which were based on combing the kernel based approach with Stein's identity. A linear-time method for goodness-of-fit was also proposed by \\cite{jitkrittum2017linear} recently. Finally, the idea of kernel-embedding has also been used for constructing implicit generative models~\\citep[\\textit{e.g.},][]{dziugaite2015training, li2015generative}.\n\nDespite their popularity, fairly little is known about the statistical performance of these kernel-embedding based tests. Our goal is to fill in this void. In particular, we focus on kernel-embedding based goodness-of-fit tests and investigate their power under a general composite alternative. Our results not only provide new insights on the operating characteristics of these kernel-embedding based tests but also suggest improved testing procedures that are minimax optimal and adaptive over a large collection of alternatives.\n\nThe problem of testing for goodness-of-fit has a long and illustrious history in statistics and is often associated with household names such as \\emph{Kolmogrov-Smirnov tests, Pearson's Chi-square test} or \\emph{Neyman's smooth test}. A plethora of other techniques have also been proposed over the years in both parametric and non-parametric settings \\citep[\\textit{e.g.},][]{ingster2003nptest, lehmann2008testing}. Most of the existing techniques are developed with the domain $\\mathcal X={\\mathbb R}$ or $[0,1]$ in mind and work the best in these cases. Modern applications, however, oftentimes involve domains different from these traditional ones. For example, when dealing with directional data, which arise naturally in applications such as diffusion tensor imaging, it is natural to consider $\\mathcal X$ as the unit sphere in ${\\mathbb R}^3$ \\citep[\\textit{e.g.},][]{jupp2005sobolev}.  Another example occurs in the context of ranking or preference data \\citep[\\textit{e.g.},][]{Ailon2008ranking}. In these cases, $\\mathcal X$ can be taken as the group of permutations. Furthermore, motivated by several applications, combinatorial testing problems have been investigated recently~\\citep[\\textit{e.g.},][]{addario2010combinatorial}, where the spaces under consideration are specific combinatorially structured spaces.\n\nA particularly attractive approach to goodness-of-fit testing problems in general domains is through RKHS embedding of distributions. Specifically, let $K: \\mathcal X\\times \\mathcal X\\to {\\mathbb R}$ be a Mercer kernel that is symmetric, positive (semi-)definite and square integrable. The RKHS embedding of a probability measure $P$ on $(\\mathcal X,\\mathcal B)$, with respect to $K$, is given by\n$$\n\\mu_P(\\cdot) :=\\int_\\mathcal X K(x,\\cdot)P(dx).\n$$\nThe Moore-Aronszajn Theorem indicates that there is an RKHS, denoted by $(\\mathcal H(K), \\langle\\cdot,\\cdot\\rangle_K)$, uniquely identified with the kernel $K$ \\citep[\\textit{e.g.},][]{aronszajn1950theory}. It is clear that $\\mu_P\\in \\mathcal H(K)$, and hence the notion of RKHS embedding. The RKHS embedding of probability measures is closely related to a certain integral probability metric. The so-called \\emph{maximum mean discrepancy} (MMD) between two probability measures $P$ and $Q$ is defined as\n\\begin{equation*}\n\\gamma_K(P,Q):=\\sup_{f\\in \\mathcal H(K): \\|f\\|_K\\le 1} \\int_\\mathcal X fd\\left(P-Q\\right),\n\\end{equation*}\nwhere $\\|\\cdot\\|_K$ is the norm associated with $(\\mathcal H(K), \\langle\\cdot,\\cdot\\rangle_K)$. It is not hard to see \\citep[\\textit{e.g.},][]{gretton2012kernel} that\n$$\n\\gamma_K(P,Q)=\\|\\mu_P-\\mu_Q\\|_K.\n$$ \nThe goodness-of-fit test can be carried out conveniently through RKHS embeddings of $P$ and $P_0$ by first constructing an estimate of $\\gamma_K(P,P_0)$:\n\\begin{equation*}\n\\gamma_K(\\widehat{P}_n,P_0):=\\sup_{f\\in \\mathcal H(K): \\|f\\|_K\\le 1} \\int_\\mathcal X fd\\left(\\widehat{P}_n-P_0\\right),\n\\end{equation*}\nwhere $\\widehat{P}_n$ is the empirical distribution of $X_1,\\cdots,X_n$, and then rejecting $H_0$ if the estimate exceeds a threshold calibrated to ensure a certain significance level, say $\\alpha$ ($0<\\alpha<1$).\n\nIn this paper, we investigate the power of the above discussed testing strategy under a general composite alternative. Following the spirit of \\cite{ingster2003nptest}, we consider in particular a set of alternatives that are increasingly close to the null hypothesis. To fix ideas, we assume hereafter that $P$ is dominated by $P_0$ under the alternative so that the Radon-Nikodym derivative $dP\/dP_0$ is well defined. Recall that the $\\chi^2$ divergence between $P$ and $P_0$ is defined as\n$$\n\\chi^2(P,P_0):=\\int_\\mathcal X\\left({dP\\over dP_0}\\right)^2dP_0-1.\n$$\nWe are particularly interested in the detection boundary, namely how close $P$ and $P_0$ can be in terms of $\\chi^2$ distance, under the alternative, so that a test based on a sample of $n$ observations can still consistently distinguish between the null hypothesis and the alternative. For example, in the parametric setting where $P$ is known up to a finite dimensional parameters under the alternative, the detection boundary of the likelihood ratio test is $n^{-1}$ under mild regularity conditions \\citep[\\textit{e.g.},][]{lehmann2008testing}. We are concerned here with alternatives that are nonparametric in nature. Our first result suggests that the detection boundary for aforementioned $\\gamma_K(\\widehat{P}_n,P_0)$ based test is of the order $n^{-1\/2}$. However, our main results indicate, perhaps surprising at first, that this rate is far from optimal and the gap between it and the usual parametric rate can be largely bridged.\n\nIn particular, we argue that the distinguishability between $P$ and $P_0$ depends on how close $u:=dP\/dP_0-1$ is to the RKHS $\\mathcal H(K)$. The closeness of $u$ to $\\mathcal H(K)$ can be measured by the the distance from $u$ to an arbitrary ball in $\\mathcal H(K)$. In particular, we shall consider the case where $\\mathcal H(K)$ is dense in $L_2(P_0)$, and focus on functions that are polynomially approximable by $\\mathcal H(K)$ for concreteness. More precisely, for some constants $M,\\theta>0$, denote by $\\mathcal F(\\theta; M)$ the collection of functions $f\\in L_2(P_0)$ such that for any $R>0$, there exists an $f_R\\in \\mathcal H(K)$ such that\n$$\n\\|f_R\\|_K\\le R,\\qquad {\\rm and}\\qquad \\|f-f_R\\|_{L_2(P_0)}\\le M R^{-1\/\\theta}.\n$$\nSee, \\textit{e.g.}, \\cite{cucker2007learning} for further discussion on these so-called interpolation spaces and their use in statistical learning. We shall also adopt the convention that\n$$\\mathcal F(0;M)=\\{f\\in \\mathcal H(K): \\|f\\|_K\\le M\\}.$$\nWe investigate the optimal rate of detection for testing $H_0$ against\n\\begin{equation}\n\\label{alter}\nH_1(\\Delta_n,\\theta,M): P\\in \\mathcal P(\\Delta_n,\\theta,M),\n\\end{equation}\nwhere $\\mathcal P(\\Delta_n,\\theta,M)$ is the collection of distributions $P$ on $(\\mathcal X,\\mathcal B)$ satisfying:\n$$\ndP\/dP_0-1\\in \\mathcal F(\\theta; M),\\qquad {\\rm and}\\qquad \\chi^2(P,P_0)\\ge \\Delta_n.\n$$\nWe call $r_n$ the optimal rate of detection if for any $c>0$, there exists no consistent test whenever $\\Delta_n\\le cr_n$; and on the other hand, a consistent test exists as long as $\\Delta_n\\gg r_n$.\n\nAlthough one could consider a more general setup, for concreteness, we assume that the eigenvalues of $K$ with respect to $L_2(P_0)$ decays polynomially in that $\\lambda_k\\asymp k^{-2s}$. We show that the optimal rate of detection for testing $H_0$ against $H_1(\\Delta_n,\\theta,M)$ for any $\\theta\\ge 0$ is $n^{-{4 s\\over 4 s+ \\theta+1}}$. The rate of detection, although not achievable with a $\\gamma_K(\\widehat{P}_n,P_0)$ based test, can be attained via a moderated version of the MMD based approach. A practical challenge to the approach, however, is its reliance on the knowledge of $\\theta$. Unlike $s$ which is determined by $K$ and $P_0$ and therefore known apriori, $\\theta$ depends on $u$ and is not known in advance. This naturally brings about the issue of adaptation -- is there an agnostic approach that can adaptively attain the optimal detection boundary without the knowledge of $\\theta$. We show that the answer is affirmative although a small price in the form of $\\log \\log n$ is required to achieve such adaptation.\n\nThe rest of the paper is organized as follows. We first analyze the power of MMD based tests in Section \\ref{sec:mmd}. This analysis reveals a significant gap between the detection boundary achieved by the MMD based test and the usual parametric $1\/n$ rate. In turn, this prompts us to introduce, in Section \\ref{sec:m3d}, a new class of tests based on a modified MMD. We show that the new tests are rate optimal. To address the practical challenge of choosing an appropriate tuning parameter for these tests, we investigate the issue of optimal adaptation in Section \\ref{sec:adapt}, where we establish the optimal rates of detection for adaptively testing $H_0$ agains a broader set of alternatives and propose a test based on the modified MMD that can attain these rates. Numerical experiments are presented in Section \\ref{sec:num}.\nAll proofs are relegated to Section \\ref{sec:proof}.\n\n\\section{Operating characteristics of MMD based test}\n\\label{sec:mmd}\n\n\\subsection{Background and notation}\nIn this section, we investigate the performance of the MMD based test. As shown in \\citet{gretton2012kernel}, the squared MMD between two probability distributions $P$ and $P_0$ can be expressed as\n\\begin{align}\n\\gamma_K^2(P,P_0)=\\int K(x,x'){d}(P-P_0)(x){d}(P-P_0)(x').\\label{deg}\n\\end{align}\nWrite\n\\begin{align*}\n\\bar{K}(x,x')=K(x,x')-{\\mathbb E} _{P_0}K(x,X)-{\\mathbb E} _{P_0} K(X,x')+{\\mathbb E} _{P_0}K(X,X'),\n\\end{align*}\nwhere the subscript $P_0$ signifies the fact that the expectation is taken over $X, X'\\sim P_0$ independently. By (\\ref{deg}), $\\gamma_K^2(P,P_0)=\\gamma_{\\bar{K}}^2(P,P_0)$. Therefore, without loss of generality, we shall assume in what follows that $K$ is degenerate under $P_0$, \\textit{i.e.}, \n\\begin{align}\n{\\mathbb E} _{P_0}K(X,\\cdot)=0.\\label{degenerate}\n\\end{align}\nFor brevity, we shall omit the subscript $K$ in $\\gamma$ in the rest of the paper, unless it is necessary to emphasize the dependence of MMD on the reproducing kernel.\n\nAssuming that $K$ is square integrable, by Mercer's theorem, it can be decomposed as\n\\begin{align}\nK(x,x')=\\sum\\limits_{k\\geq 1}\\lambda_k\\varphi_k(x)\\varphi_k(x'),\\label{decomp}\n\\end{align}\nwhere the limit is in the sense of $L_2(P_0)$, $\\lambda_1>\\lambda_2>\\cdots>0$ are the positive eigenvalues of the integral operator induced by $K$, and $\\{\\varphi_k: k\\ge 1\\}$ are the corresponding orthonormal eigenfunctions, \\textit{i.e.}, $\\langle \\varphi_k,\\varphi_{k'} \\rangle_{L_2(P_0)}=\\delta_{k,k'}$ and $\\langle \\varphi_k,\\varphi_{k'} \\rangle_K=\\lambda_k^{-1}\\delta_{k,k'}$, with $\\delta$ representing the Kronecker delta. For the sake of concreteness, we shall assume $K$ is universal in that $\\{\\varphi_k: k\\ge 1\\}$ forms an orthonormal basis of $L_2(P_0)$, and has infinitely many positive eigenvalues decaying polynomially, that is,\n\\begin{align}\n0<\\varliminf_{k\\rightarrow \\infty}k^{2s}\\lambda_k\\leq\\varlimsup_{k\\rightarrow \\infty}k^{2s}\\lambda_k<\\infty\\label{summable}\n\\end{align}\nfor some $s>1\/2$. Moreover, we assume the eigenfunctions are uniformly bounded, \\textit{i.e.},\n\\begin{align}\n\\sup_{k\\geq 1}\\|\\varphi_k\\|_{\\infty}<\\infty.\\label{unif}\n\\end{align}\nAssumptions (\\ref{summable}) and (\\ref{unif}) ensure that the spectral decomposition (\\ref{decomp}) holds both pointwisely and uniformly.\n\nNote that (\\ref{degenerate}) implies ${\\mathbb E} _{P_0}\\varphi_k(X)=0$, $\\forall\\ k\\ge 1$, and (\\ref{decomp}) gives\n\\begin{align*}\n\\gamma^2(P,P_0)=\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2\n\\end{align*}\nfor any $P$. Accordingly, when $P$ is replaced by the empirical distribution $\\widehat{P}_n$, the empirical squared MMD can be expressed as\n\\begin{align*}\n\\gamma^2(\\widehat{P}_n,P_0)=\\sum\\limits_{k\\geq 1}\\lambda_k\\left[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)\\right]^2.\n\\end{align*}\nClassic results on the asymptotics of V-statistic \\citep{serfling2009approximation} imply that\n\\begin{align*}\nn\\gamma^2(\\widehat{P}_n,P_0)\\stackrel{d}{\\rightarrow}\\sum\\limits_{k\\geq 1}\\lambda_kZ_k^2:= W\n\\end{align*}\nunder $H_0$, where $Z_k\\stackrel{i.i.d.}{\\sim} N(0,1)$. Let $T_{\\mathrm{MMD}}$ be an MMD based test, which rejects $H_0$ if and only if $n\\gamma^2(\\widehat{P}_n,P_0)$ exceeds the upper $\\alpha$ quantile $q_{w,1-\\alpha}$ of $W$, \\textit{i.e.},\n$$T_{\\mathrm{MMD}}=\\mathds{1}_{\\{n\\gamma^2(\\widehat{P}_n,P_0)>q_{w,1-\\alpha}\\}}.$$\nThe above limiting distribution of $n\\gamma^2(\\widehat{P}_n,P_0)$ immediately suggests that $T_{\\mathrm{MMD}}$ is an asymptotic $\\alpha$-level test.\n\n\\subsection{Power analysis for MMD based tests}\nWe now investigate the power of $T_{\\mathrm{MMD}}$ in testing $H_0$ against $H_1(\\Delta_n,\\theta,M)$ given by (\\ref{alter}). Recall that the type \\uppercase\\expandafter{\\romannumeral2}\\  error of a test $T:\\mathcal{X}^{n}\\rightarrow [0,1]$ for testing $H_0$ against a composite alternative $H_1:P\\in\\mathcal{P}$ is given by\n$$\\beta(T;\\mathcal P)=\\sup\\limits_{P\\in \\mathcal{P}}{\\mathbb E}_{P}[1-T(X_1,\\ldots,X_n)],$$\nwhere ${\\mathbb E}_P$ means taking expectation over $X_1,\\ldots, X_n\\stackrel{i.i.d.}{\\sim}P$.\nFor brevity, we shall write $\\beta(T;\\Delta_n,\\theta,M)$ instead of $\\beta(T;\\mathcal{P}(\\Delta_n,\\theta,M))$ in what follows. The performance of a test $T$ can then be evaluated by its detection boundary, that is, the smallest $\\Delta_n$ under which the type \\uppercase\\expandafter{\\romannumeral2}\\ error converges to $0$ as $n\\rightarrow \\infty$. Our first result establishes the convergence rate of the detection boundary for $T_{\\mathrm{MMD}}$\\ in the case when $\\theta=0$. Hereafter, we abbreviate $M$ in $\\mathcal P(\\Delta_n,\\theta,M)$, $H_1(\\Delta_n,\\theta,M)$ and $\\beta(T;\\Delta_n,\\theta,M)$, unless it is necessary to emphasize the dependence.\n\n\n\\begin{theorem}\\label{mmdthm}\nConsider testing $H_0: P=P_0$ against $H_1(\\Delta_n,0)$ by $T_{\\mathrm{MMD}}$.\n\\renewcommand{\\labelenumi}{(\\roman{enumi})} \n\\begin{enumerate}\n\t\\item If $\\sqrt{n}\\Delta_n\\rightarrow \\infty$, then\n\t$$\\beta(T_{\\mathrm{MMD}};\\Delta_n,0)\\rightarrow 0\\qquad  {\\rm as} \\quad n\\rightarrow \\infty;$$\n\t\\item conversely, there exists a constant $c_0>0$ such that\n\t$$\\varliminf\\limits_{n\\rightarrow \\infty}\\beta(T_{\\mathrm{MMD}};c_0n^{-1\/2},0)>0.$$\n\\end{enumerate}\n\\end{theorem}\n\n\nTheorem \\ref{mmdthm} shows that when the alternative $H_1(\\Delta_n,0)$ is considered, the detection boundary of $T_{\\mathrm{MMD}}$\\ is of the order $n^{-1\/2}$.\nIt is of interest to compare the detection rate achieved by $T_{\\text{MMD}}$ with that in a parametric setting where  consistent tests are available if $n\\Delta_n\\rightarrow \\infty$ \\citep{lehmann2008testing}. It is natural to raise the question to what extent such a gap can be entirely attributed to the fundamental difference between parametric and nonparametric testing problems. We shall now argue that this gap actually is largely due to the sup-optimality of $T_{\\mathrm{MMD}}$, and the detection boundary of  $T_{\\mathrm{MMD}}$\\ could be significantly improved through a slight modification of the MMD.\n\n\\section{Optimal tests based on moderated MMD}\n\\label{sec:m3d}\n\n\\subsection{Moderated MMD test statistic}\nThe basic idea behind MMD is to project two probability measures onto a unit ball in $\\mathcal H(K)$ and use the distance between the two projections to measure the distance between the original probability measures. If the two probability measures are far away from $\\mathcal H(K)$, the distance between the two projections may not honestly reflect the distance between them. More specifically, $\\gamma^2(P,P_0)=\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2$, while the $\\chi^2$ distance between $P$ and $P_0$ is $\\chi^2(P,P_0)=\\sum\\limits_{k\\geq 1}[{\\mathbb E} _{P}\\varphi_k(X)]^2$. Considering that $\\lambda_k$ decreases with $k$, $\\gamma^2(P,P_0)$ can be much smaller than $\\chi^2(P,P_0)$. To overcome this problem, we consider a moderated version of the MMD which allows us to project the probability measures onto a larger ball in $\\mathcal H(K)$. The new class of integral probability metric between two distributions $P$ and $Q$ is given as\n\\begin{equation}\n\\label{def2}\n\\eta_{K,\\varrho}(P,Q;P_0)=\\sup_{f\\in \\mathcal H(K): \\|f\\|_{L_2(P_0)}^2+\\varrho^2\\|f\\|_K^2\\le 1} \\int_\\mathcal X fd(P-Q)\n\\end{equation}\nfor a given distribution $P_0$ and a constant $\\varrho>0$. It should be noted that a related test statistics was proposed previously by \\cite{harchaoui2009kernel} from a completely different viewpoint\n\nIt is worth noting that $\\eta_{K,\\varrho}(P,Q;P_0)$ can also be identified with a particular type of RKHS embedding. Specifically, $\\eta_{K,\\varrho}(P,Q;P_0)=\\gamma_{\\tilde{K}_{\\varrho}}(P,Q)$, where\n\\begin{align*}\n\\tilde{K}_{\\varrho}(x,x'):=\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho^2}\\varphi_k(x)\\varphi_k(x')\n\\end{align*}\nWe shall abbreviate the dependence of $\\eta$ on $K$ and $P_0$ unless necessary. The unit ball in (\\ref{def2}) is defined in terms of both RKHS norm and $L^2$ norm. Recall that $u=dP\/dP_0-1$ so that\n\\begin{align*}\n\\sup\\limits_{\\|f\\|_{L_2(P_0)}\\leq 1}\\int\\limits_{\\mathcal{X}}f{d}(P-P_0)=\\sup\\limits_{\\|f\\|_{L_2(P_0)}\\leq 1}\\int\\limits_{\\mathcal{X}}fu{d}P_0=\\|u\\|_{L_2(P_0)}=\\chi(P,P_0).\n\\end{align*}\nWe can therefore expect that a smaller $\\varrho$ will make $\\eta^2_{\\varrho}(P,P_0)$ closer to $\\chi^2(P,P_0)$, since the unit ball to be considered will become more similar to the unit ball in $L_2(P_0)$. This can also be verified by noticing that $\\eta^2_{\\varrho}(P,P_0)$ could be expressed as\n\\begin{align*}\n\\eta^2_{\\varrho}(P,P_0)=\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho^2}[{\\mathbb E}_P\\varphi_k(X)]^2\n\\end{align*}\nTherefore, we choose $\\varrho$ converging to $0$ when constructing our test statistic.\n\nHereafter we shall attach the subscript $n$ to $\\varrho$ to signify its dependence on $n$. We now argue that letting $\\varrho_n$ converge to $0$ at an appropriate rate indeed results in a test more powerful than $T_{\\mathrm{MMD}}$. The test statistic we propose is the empirical version of $\\eta^2_{\\varrho_n}(P,P_0)$,\n\\begin{align}\n\\eta^2_{\\varrho_n}(\\widehat{P}_n,P_0)=\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\left[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)\\right]^2.\\label{eq:etatest}\n\\end{align}\n\n\\subsection{Operating characteristics of $\\eta^2_{\\varrho_n}(\\widehat{P}_n,P_0)$ based tests}\nAlthough the expression for $\\eta^2_{\\varrho_n}(\\widehat{P}_n,P_0)$ given by \\eqref{eq:etatest} looks similar to that of $\\gamma^2(\\widehat{P}_n,P_0)$, their asymptotic behaviors are quite different. At a technical level, this is due to the fact that the eigenvalues of the underlying kernel\n$$\\tilde{\\lambda}_{nk}:=\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}$$ \ndepend on $n$ and may not be uniformly summable over $n$. As presented in the following theorem, a certain type of asymptotic normality, instead of a sum of chi-squares as in the case of $\\gamma^2(\\widehat{P}_n, P_0)$, holds for $\\eta^2_{\\varrho_n}(\\widehat{P}_n,P_0)$ under $P_0$, which helps determine the rejection region of the $\\eta^2_{\\varrho_n}$ based test.\n\n\\begin{theorem}\\label{asympm3d}\nAssume that $\\varrho_n\\to 0$ as $n\\to \\infty$ in such a fashion that $n\\varrho_n^{{1}\/(2s)}\\rightarrow\\infty$. Then under $H_0$ where $X_1,\\ldots, X_n\\stackrel{i.i.d.}{\\sim}P_0$, \n\\begin{align*}\nv_n^{-1\/2}[n\\eta_{\\varrho_n}^2(\\widehat{P}_n,P_0)-A_n]\\stackrel{d}{\\rightarrow}N(0,2),\n\\end{align*}\nwhere\n$$v_n=\\sum\\limits_{k\\geq 1}\\left(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\right)^2,\\qquad {\\rm and} \\qquad\nA_n=\\frac{1}{n}\\sum\\limits_{i=1}^n\\tilde{K}_{\\varrho_n}(X_i,X_i).$$\n\\end{theorem}\n\nIn the light of Theorem \\ref{asympm3d}, a test that rejects $H_0$ if and only if\n$$2^{-1\/2}v_n^{-1\/2}[n{\\eta}_{\\varrho_n}^2(\\widehat{P}_n,P_0)-A_n]$$\nexceeds $z_{1-\\alpha}$ is an asymptotic $\\alpha$-level test, where $z_{1-\\alpha}$ stands for the $1-\\alpha$ quantile of a standard normal distribution. We refer to this test as $T_{\\mathrm{M}^3\\mathrm{d}}$. The performance of $T_{\\mathrm{M}^3\\mathrm{d}}$\\ under the alternative hypothesis is characterized by the following theorem, showing that its detection boundary is much improved when compared with that of $T_{\\mathrm{MMD}}$.\n\n\\begin{theorem}\\label{crm3d}\nConsider testing $H_0$ against $H_1(\\Delta_n,\\theta)$ by $T_{\\mathrm{M}^3\\mathrm{d}}$\\ with $\\varrho_n=cn^{-{2s(\\theta+1)\\over 4 s+\\theta+1}}$ for an arbitrary constant $c>0$. If $n^{\\frac{4s}{4s+\\theta+1}}\\Delta_n\\rightarrow\\infty$, the\n$$\n\\beta(T_{\\mathrm{M}^3\\mathrm{d}};\\Delta_n, \\theta)\\to 0, \\qquad {\\rm as\\ }n\\to \\infty.\n$$\n\\end{theorem}\n\nTheorem \\ref{crm3d} indicates that the detection boundary for $T_{\\mathrm{M}^3\\mathrm{d}}$\\ is $n^{-{4s}\/({4s+\\theta+1})}$. In particular, when testing $H_0$ against $H_1(\\Delta_n, 0)$, \\textit{i.e.}, $\\theta=0$, it becomes $n^{-4s\/(4s+1)}$. This is to be contrasted with the detection boundary for $T_{\\mathrm{MMD}}$, which, as suggested by Theorem \\ref{mmdthm}, is of the order $n^{-1\/2}$. It is also worth noting that the detection boundary for $T_{\\mathrm{M}^3\\mathrm{d}}$\\ deteriorates as $\\theta$ increases, implying that it is harder to test against a larger interpolation space.\n\n\\subsection{Minimax optimality}\n\nIt is of interest to investigate if the detection boundary of $T_{\\mathrm{M}^3\\mathrm{d}}$\\ can be further improved. We now show that the answer is negative in a certain sense. More specifically, we shall follow the minimax framework for nonparametric hypothesis testing pioneered by Ingster \\citep[see, \\textit{e.g.},][]{ingster1993asymptotically, ingster1995minimax} and show that $T_{\\mathrm{M}^3\\mathrm{d}}$\\ attains the optimal rate of detection for testing $H_0$ against $H_1(\\Delta_n,\\theta)$ in that no consistent test exists if there exists $c>0$ such that $\\Delta_n\\le cn^{-\\frac{4s}{4s+\\theta+1}}$.\n\n\\begin{theorem}\\label{cr}\nConsider testing $H_0: P=P_0$ against $H_1(\\Delta_n,\\theta)$, for some $\\theta<2s-1$. If $\\varlimsup\\limits_{n\\rightarrow \\infty}\\Delta_nn^{4s\\over 4s+\\theta+1}<\\infty$, then\n$$\n\\varliminf\\limits_{n\\rightarrow \\infty}\\inf_{{T}\\in\\mathcal{T}_n} \\left[{\\mathbb E}_{P_0} {T} +\\beta({T};\\Delta_n,\\theta)\\right]>0,\n$$\nwhere $\\mathcal{T}_n$ denotes the collection of all test functions based on $X_1,\\ldots,X_n$.\n\\end{theorem}\n\nRecall that for a test ${T}$, ${\\mathbb E}_{P_0} {T}$ is its Type I error. Theorem \\ref{cr} shows that, if $\\Delta_n=O\\left(n^{-4s\/(4s+\\theta+1)}\\right)$, then the sum of Type I and Type II errors of any test does not vanish as $n$ increases. In other words, there is no consistent test if $\\Delta_n=O\\left(n^{-4s\/(4s+\\theta+1)}\\right)$. Together with Theorem \\ref{crm3d}, this suggests that $T_{\\mathrm{M}^3\\mathrm{d}}$\\ is rate optimal in the minimax sense.\n\n\\section{Adaptation}\n\\label{sec:adapt}\nDespite the minimax optimality of $T_{\\mathrm{M}^3\\mathrm{d}}$, a practical challenge in using it is the choice of an appropriate tuning parameter $\\varrho_n$. In particular, Theorem \\ref{crm3d} suggests that $\\varrho_n$ needs to be taken at the order of $n^{-2s(\\theta+1)\/(4s+\\theta +1)}$ which depends on the value of $s$ and $\\theta$. On the one hand, since $P_0$ and $K$ are known apriori, so is $s$. On the other hand, $\\theta$ reflects the property of $dP\/dP_0$ which is typically not known in advance. This naturally brings about the issue of adaptation \\citep[see, \\textit{e.g.},][]{spokoiny1996adaptive,ingster2000adaptive}. In other words, we are interested in a single testing procedure that can achieve the detection boundary for testing $H_0$ against $H_1(\\Delta_n(\\theta), \\theta)$ simultaneously over all $\\theta\\ge 0$. We emphasize the dependence of $\\Delta_n$ on $\\theta$ since the detection boundary may depend on $\\theta$, as suggested by the results from the previous section. In fact, we should build upon the test statistic introduced before.\n\nMore specifically, write\n$$\n\\rho_\\ast=\\left(\\frac{\\sqrt{\\log \\log n}}{n}\\right)^{2s},\n$$\nand\n$$\nm_\\ast=\\left\\lceil\\log_2 \\left[\\rho_\\ast^{-1}\\left(\\frac{\\sqrt{\\log\\log n}}{n}\\right)^{\\frac{2s}{4s+1}}\\right]\\right\\rceil.\n$$\nThen our test statistics is taken to be the maximum of $T_{n,\\varrho_n}$ for $\\rho_n=\\rho_\\ast, 2\\rho_\\ast, 2^2\\rho_\\ast,\\ldots, 2^{m_\\ast}\\rho_\\ast$:\n\\begin{align*}\n\\tilde{T}_n:= \\sup\\limits_{0\\le k\\le m_\\ast}T_{n,2^k\\varrho_\\ast},\n\\end{align*}\nwhere, with slight abuse of notation,\n$$T_{n,\\varrho_n}=(2v_n)^{-1\/2}[n\\eta_{\\varrho_n}^2(\\widehat{P}_n,P_0)-A_n].$$ \nIt turns out if an appropriate rejection threshold is chosen, $\\tilde{T}_n$ can achieve a detection boundary very similar to the one we have before, but now simultaneously over all $\\theta>0$.\n\n\\begin{theorem}\\label{crtm3d}\n\t\\rm(\\romannumeral1) \\it Under $H_0$,\n\t\\begin{align*}\n\t\\lim\\limits_{n\\rightarrow \\infty}P\\left(\\tilde{T}_n\\geq \\sqrt{3\\log\\log n}\\right)=0;\n\t\\end{align*}\n\t\\rm(\\romannumeral2) \\it on the other hand, there exists a constant $c_1>0$ such that\n\t\\begin{align*}\n\t\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\cup_{\\theta\\ge 0}\\mathcal P(\\Delta_{n}(\\theta))}P\\left(\\tilde{T}_n\\geq \\sqrt{3\\log\\log n}\\right)=1,\n\t\\end{align*}\n\tprovided that $\\Delta_n(\\theta)\\geq c_1(n^{-1}{\\sqrt{\\log\\log n}})^{\\frac{4s}{4s+\\theta+1}}$.\n\\end{theorem}\n\nTheorem \\ref{crtm3d} immediately suggests that a test rejects $H_0$ if and only if $\\tilde{T}_n\\geq \\sqrt{3\\log\\log n}$ is consistent for testing it against $H_1(\\Delta_n(\\theta),\\theta)$ for all $\\theta\\ge 0$ provided that $\\Delta_n(\\theta)\\geq c_1(n^{-1}{\\sqrt{\\log\\log n}})^{\\frac{4s}{4s+\\theta+1}}$. We can further calibrate the rejection region to yield a test at a given significance level. More precisely, let $\\tilde{q}_\\alpha$ be the upper $\\alpha$ quantile of $\\tilde{T}_n$, we can proceed to reject $H_0$ whenever the observed test statistic exceeds $\\tilde{q}_\\alpha$. Denote such a test by $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$. By definition, $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ is an $\\alpha$-level test. Theorem \\ref{crtm3d} implies that the type II error of $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ vanishes as $n\\to\\infty$ uniformly over all $\\theta\\ge 0$. In practice, the quantile $\\tilde{q}_\\alpha$ can be evaluated by Monte Carlo methods as we shall discuss in further details in the next section. We note that the detection boundary given in Theorem \\ref{crtm3d} is similar, but inferior by a factor of $(\\log\\log n)^{\\frac{2s}{4s+\\theta+1}}$, to that from Theorem \\ref{cr}. This turns out be the price one needs to pay for adaptation.\n\n\n\n\n\n\n\n \n\\begin{theorem}\\label{cra}\nLet $0<\\theta_1<\\theta_2<2s-1$. Then there exists a positive constant $c_2$ such that\n\\begin{align*}\n\\varlimsup\\limits_{n\\rightarrow \\infty}\\sup\\limits_{\\theta\\in[\\theta_1,\\theta_2]}\\left\\{\\Delta_n(\\theta)\\left(\\frac{n}{\\sqrt{\\log\\log n}}\\right)^{\\frac{4s}{4s+\\theta+1}}\\right\\}\\leq c_2\n\\end{align*}\nimplies that\n\\begin{align*}\n\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{{T}\\in \\mathcal{T}_n}\\left[{\\mathbb E}_{P_0} {T}+\\sup_{\\theta\\in [\\theta_1,\\theta_2]}\\beta({T};\\Delta_n(\\theta), \\theta)\\right]=1.\n\\end{align*}\n\\end{theorem}\n\nSimilar to Theorem \\ref{cr}, Theorem \\ref{cra} shows that there is no consistent test for $H_0$ against $H_1(\\Delta_n,\\theta)$ simultaneously over all $\\theta\\in [\\theta_1,\\theta_2]$, if $\\Delta_n(\\theta)\\leq c_2\\left(n^{-1}\\sqrt{\\log\\log n}\\right)^{\\frac{4s}{4s+\\theta+1}}$ $\\forall\\ \\theta\\in[\\theta_1,\\theta_2]$ for a sufficiently small $c_2$. Together with Theorem \\ref{crtm3d}, this suggests that the test $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ is indeed rate optimal.\n\n\n\n\\section{Numerical Experiments}\n\\label{sec:num}\n\nTo complement the earlier theoretical development, we also performed several sets of simulation experiments to demonstrate the merits of the proposed adaptive test based on $\\tilde{T}_n$. To do so, we need to first address a practical issue of computing the test statistic $\\tilde{T}_n$:  how to compute ${\\eta}_{\\varrho_n}^2(\\widehat{P}_n, P_0)$ for a given $\\varrho_n$.\n\n\\subsection{Computing $\\tilde{T}_n$}\\label{sec:compute}\nThough the form of ${\\eta}_{\\varrho_n}^2(\\widehat{P}_n, P_0)$ looks similar to that of ${\\gamma}^2(\\widehat{P}_n, P_0)$, from the point of view of computing it numerically, there is a subtle issue. The kernel $\\tilde{K}_{\\varrho_n}(x,x')$ is defined only in its Mercer decomposed form, which is based on the Mercer decomposition of $K(x,x')$. Hence, in order to compute the kernel $\\tilde{K}_{\\varrho_n}(x,x')$, we need to first choose a kernel $K(x,x')$ and compute its Mercer decomposition numerically. Specifically, we use \\emph{chebfun} framework in Matlab (with slight modifications) to compute Mercer decompositions associated with kernels based on their integral operator representations ~\\cite{Driscoll2014, Battles2004}. Once we compute $\\lambda_k$ and the associated $\\varphi_k(\\cdot)$, we approximately compute $\\tilde{K}_{\\varrho_n}(x,x')$ based on the top $K$ eigvevalues and eigenfunctions. This provides a numerical framework for computing $\\tilde{K}_{\\varrho_n}(x,x')$ once we fix a kernel $K(x,x')$. In the cases when the eigenvalues and eigenfunction are known, for example when using polynomial kernels, from our experiments we found that using the top few numerical eigenvalues gives a good approximation to the actual value of the kernel. Given a way to compute kernel evaluations, computing  ${\\eta}_{\\varrho_n}^2(\\widehat{P}_n, P_0)$ follows similarly.\n\n\n\\subsection{ Power comparison}\n\nOnce we are able to compute $\\tilde{T}_n$, we can assess its null distribution by simulating it under the null hypothesis $H_0$. In particular, we repeated for each case 200 runs and estimated the $95\\%$ quantile of $\\tilde{T}_n$ under $H_0$ by the corresponding sample quantile. We then proceeded to reject $H_0$ when an observed test statistic exceeds the estimate $95\\%$ quantile. By construction, the procedure gives a $5\\%$-level test, up to Monte Carlo error.\n\n\\paragraph{Euclidean data:} We first consider using both the test $T_{\\mathrm{MMD}}$\\ and $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ to test the hypothesis $P_0$ is uniform on $[0,1]^d$ given a sample of observations $\\{ X_1, \\ldots, X_n\\}$. The dimensionality used are $100$ and $200$. We followed the examples for densities put forward in~\\cite{marron1992exact} in the context of nonparametric density estimation, for the alternatives.  Specifically we set the alternative hypothesis to be (1) mixture of five Gaussians, (2) skewed unimodal, (3) asymmetric claw density and (4) smooth comb density. The value of $\\alpha$ is set to $0.05$. The sample size $n$ is varied from $200$ to $1000$ (in steps of 200) and for each value of sample size 100 simulations are conducted to estimate the probability of rejecting a false null hypothesis. \n\nWe use a Gaussian kernel $K$ to compute the ${\\gamma}^2(\\widehat{P}_n, P_0)$ and use the procedure outlined in section~\\ref{sec:compute} to compute ${\\eta}_{\\varrho_n}^2(\\widehat{P}_n, P_0)$. The issue of choosing the kernel is subtle when using $T_{\\mathrm{MMD}}$. For simplicity, we fixed the value of bandwidth of Gaussian kernel (which corresponds to choosing the kernel in this case) to a fixed value, that corresponds to the best performance of $T_{\\mathrm{MMD}}$. With the fixed value of the bandwidth, to fix $\\varrho$, we tried values over a grid and set it to the value that performed best. Figure~\\ref{fig:simT2} illustrates a plot of the estimated probability of  accepting the null hypothesis when it is false for different values of sample size $n$ for the proposed test $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ along with $T_{\\mathrm{MMD}}$\\ and the more classical Kolmogorov-Smirnov (K-S, for short) goodness-of-fit test. We note from Figure~\\ref{fig:simT2} that the estimated error probability converges to zero at a faster rate for the adaptive M$^3$D test compared to the MMD test and the Kolmogorov-Smirnov test on all the different simulation settings that are considered. Note that it has been previously observed that MMD test performs better than K-S test in various setting in \\cite{gretton2012kernel}, which we observe in our setting as well.\n\n\\begin{figure*}[!htbp]\n\\centering\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.40 )+- (0.0, 0.03)\n( 600, 0.31 )+- (0.0, 0.00)\n( 800, 0.15 )+- (0.0, 0.00)\n( 1000, 0.05 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.45 )+- (0.0, 0.03)\n( 600, 0.38 )+- (0.0, 0.03)\n( 800, 0.22 )+- (0.0, 0.03)\n( 1000, 0.08 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.98 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.12 )+- (0.0, 0.03)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.40 )+- (0.0, 0.03)\n( 600, 0.30 )+- (0.0, 0.00)\n( 800, 0.20 )+- (0.0, 0.00)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.44 )+- (0.0, 0.03)\n( 600, 0.35 )+- (0.0, 0.03)\n( 800, 0.23 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.53 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.13 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\hfill \\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.04)\n( 400, 0.30 )+- (0.0, 0.03)\n( 600, 0.14 )+- (0.0, 0.00)\n( 800, 0.105 )+- (0.0, 0.00)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.46 )+- (0.0, 0.03)\n( 600, 0.35 )+- (0.0, 0.03)\n( 800, 0.19 )+- (0.0, 0.00)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.59 )+- (0.0, 0.03)\n( 600, 0.48 )+- (0.0, 0.03)\n( 800, 0.23 )+- (0.0, 0.00)\n( 1000, 0.11 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.04)\n( 400, 0.32 )+- (0.0, 0.03)\n( 600, 0.19 )+- (0.0, 0.00)\n( 800, 0.11 )+- (0.0, 0.00)\n( 1000, 0.05 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.46)+- (0.0, 0.03)\n( 600, 0.26 )+- (0.0, 0.03)\n( 800, 0.20 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.59 )+- (0.0, 0.03)\n( 600, 0.41 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.11 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.04)\n( 400, 0.35 )+- (0.0, 0.03)\n( 600, 0.20 )+- (0.0, 0.00)\n( 800, 0.15 )+- (0.0, 0.00)\n( 1000, 0.05 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.41 )+- (0.0, 0.03)\n( 800, 0.25 )+- (0.0, 0.03)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.63 )+- (0.0, 0.03)\n( 600, 0.51 )+- (0.0, 0.03)\n( 800, 0.35 )+- (0.0, 0.00)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.04)\n( 400, 0.48 )+- (0.0, 0.03)\n( 600, 0.30 )+- (0.0, 0.00)\n( 800, 0.15 )+- (0.0, 0.00)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.66 )+- (0.0, 0.03)\n( 600, 0.45 )+- (0.0, 0.03)\n( 800, 0.26 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.69 )+- (0.0, 0.03)\n( 600, 0.51 )+- (0.0, 0.03)\n( 800, 0.31 )+- (0.0, 0.00)\n( 1000, 0.104 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.04)\n( 400, 0.41 )+- (0.0, 0.03)\n( 600, 0.25 )+- (0.0, 0.00)\n( 800, 0.16 )+- (0.0, 0.00)\n( 1000, 0.04 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.54 )+- (0.0, 0.03)\n( 600, 0.40 )+- (0.0, 0.03)\n( 800, 0.22 )+- (0.0, 0.03)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.60 )+- (0.0, 0.03)\n( 600, 0.45 )+- (0.0, 0.04)\n( 800, 0.33 )+- (0.0, 0.00)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.04)\n( 400, 0.41 )+- (0.0, 0.03)\n( 600, 0.25 )+- (0.0, 0.00)\n( 800, 0.16 )+- (0.0, 0.00)\n( 1000, 0.04 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.54 )+- (0.0, 0.03)\n( 600, 0.40 )+- (0.0, 0.03)\n( 800, 0.22 )+- (0.0, 0.03)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.60 )+- (0.0, 0.03)\n( 600, 0.45 )+- (0.0, 0.03)\n( 800, 0.33 )+- (0.0, 0.00)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$KS$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\hfill\\vspace{-0.1in}\n\\caption{Error versus Sample Size: mixture of Gaussian (row 1), skewed unimodal (row 2), asymmetric claw (row 3) and smooth comb (row 4) with dimensionality 100 (left) and 200 (right).}\n\\label{fig:simT2}\n\\end{figure*}\n\n\n\\paragraph{Directional data:} One of the advantages of the proposed RKHS embedding based approach is that it could be used on domains other than the $d$-dimensional Euclidean space. For example when $\\mathcal{X} = \\mathbb{S}^{d-1}$ where  $\\mathbb{S}^{d-1}$ corresponds to the $d$-dimensional unit sphere, one can perform hypothesis testing using the above framework, as long as we can compute the Mercer decomposition of a kernel $K$ defined on the domain. In several applications, like protein folding, often times data are modeled as coming from the unit-sphere and testing goodness-of-fit for such data needs specialized methods different from the standard nonparametric testing methods~\\cite{mardia2009directional,jupp2005sobolev}. \n\nIn order to highlight the advantage of the proposed approach, we assume $P_0$ is uniform distribution on the unit sphere of dimension $100$ and test it against the alternative that data are from: \n\\begin{enumerate}\n\\item[(1)] multivariate von Mises-Fisher distribution (which is the Gaussian analogue on the unit-sphere) given by $f_{vM\\mhyphen F}(x, \\mu, \\kappa) = C_{vM\\mhyphen F}(\\kappa) \\exp(\\kappa \\mu^\\top x)$ for data $x \\in \\mathbb{S}^{d-1}$, where $\\kappa \\geq 0$ is concentration parameter and $\\mu$ is the mean parameter. The term $C_{vM\\mhyphen F}$ is the normalization constant given by $\\frac{\\kappa^{d\/2-1}}{2\\pi^{d\/2}I_{d\/2-1}(\\kappa)}$ where $I$ is modified Bessel function;\n\n\\item[(2)] multivariate Watson distribution (used to model axially symmetric data on sphere) given by  $f_{W}(x, \\mu, \\kappa) = C_W(\\kappa) \\exp(\\kappa (\\mu^\\top x)^2)$ for data $x \\in \\mathbb{S}^{d-1}$, where $\\kappa \\geq 0$ is concentration parameter and $\\mu$ is the mean parameter as before. The term $C_W(\\kappa)$ is the normalization constant given by $\\frac{\\Gamma(d\/2)}{2\\pi^{p\/2} M(1\/2,d\/2,\\kappa)}$ where $M$ is Kummer's confluent hypergeometric function;\n\n\\item[(3)] mixture of five von Mises-Fisher distribution which are used in modeling and clustering spherical data~\\cite{banerjee2005clustering};\n\\item[(4)] mixture of five Watson distribution which are used in modeling and clustering spherical data~\\cite{sra2013multivariate}.\n\\end{enumerate}\n\nNote that in this setup one can analytically compute the Mercer decomposition of the Gaussian kernel on the unit sphere with respect to the uniform distribution. Specifically, the eigenvalues are given by Theorem 2 in~\\cite{minh2006mercer} and the eigenfunctions are the standard spherical harmonics of order $k$ (see section 2.1 in ~\\cite{minh2006mercer} for details). Rest of the simulation setup is similar to the previous setting (of Euclidean data) and we compared $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ against  $T_{\\mathrm{MMD}}$\\ and the Sobolev test approach (denoted as ST hereafter) proposed in~\\cite{jupp2005sobolev}. Figure~\\ref{fig:simT3} illustrates a plot of estimated probability of accepting null hypothesis when it is false for different values of sample size, from which we see the adaptive M$^3$D test performs better.  \n\n\\begin{figure*}[!htbp]\n\\centering\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.04)\n( 400, 0.46 )+- (0.0, 0.03)\n( 600, 0.28 )+- (0.0, 0.00)\n( 800, 0.18 )+- (0.0, 0.00)\n( 1000, 0.08 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.93 )+- (0.0, 0.03)\n( 400, 0.59 )+- (0.0, 0.03)\n( 600, 0.38 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.10 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.98 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.12 )+- (0.0, 0.03)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.92 )+- (0.0, 0.04)\n( 400, 0.50 )+- (0.0, 0.03)\n( 600, 0.34 )+- (0.0, 0.00)\n( 800, 0.24 )+- (0.0, 0.00)\n( 1000, 0.07 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.61 )+- (0.0, 0.03)\n( 600, 0.40 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.07 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.53 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.13 )+- (0.0, 0.00)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\hfill \\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.04)\n( 400, 0.48 )+- (0.0, 0.03)\n( 600, 0.32 )+- (0.0, 0.00)\n( 800, 0.19 )+- (0.0, 0.00)\n( 1000, 0.08 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.97 )+- (0.0, 0.03)\n( 400, 0.60 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.32 )+- (0.0, 0.03)\n( 1000, 0.11 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.98 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.12 )+- (0.0, 0.03)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.45 )+- (0.0, 0.03)\n( 600, 0.30 )+- (0.0, 0.00)\n( 800, 0.18 )+- (0.0, 0.00)\n( 1000, 0.07 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.54 )+- (0.0, 0.03)\n( 600, 0.41 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.11 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.53 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.13 )+- (0.0, 0.00)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.40 )+- (0.0, 0.03)\n( 600, 0.20 )+- (0.0, 0.00)\n( 800, 0.11 )+- (0.0, 0.00)\n( 1000, 0.05 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.56 )+- (0.0, 0.03)\n( 600, 0.34 )+- (0.0, 0.03)\n( 800, 0.23 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.98 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.12 )+- (0.0, 0.03)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.43 )+- (0.0, 0.03)\n( 600, 0.29 )+- (0.0, 0.00)\n( 800, 0.12 )+- (0.0, 0.00)\n( 1000, 0.07 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.63 )+- (0.0, 0.03)\n( 600, 0.38 )+- (0.0, 0.03)\n( 800, 0.25 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.53 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.13 )+- (0.0, 0.00)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\vspace{-0.1in}\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.04)\n( 400, 0.48 )+- (0.0, 0.03)\n( 600, 0.26 )+- (0.0, 0.00)\n( 800, 0.15 )+- (0.0, 0.00)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.93 )+- (0.0, 0.03)\n( 400, 0.59 )+- (0.0, 0.03)\n( 600, 0.42 )+- (0.0, 0.03)\n( 800, 0.31 )+- (0.0, 0.03)\n( 1000, 0.09 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.98 )+- (0.0, 0.03)\n( 400, 0.52 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.28 )+- (0.0, 0.03)\n( 1000, 0.12 )+- (0.0, 0.03)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{tikzpicture}[scale=0.70]\n  \\begin{axis}[\n    xlabel = $\\text{Sample size (n)}$,\nylabel=$P(~\\text{accepting}~H_0~\\text{when false}~)$,\nxmax = 1000,\nxmin = 200,\nymax = 1,\nymin = 0\n]\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.94 )+- (0.0, 0.04)\n( 400, 0.42 )+- (0.0, 0.03)\n( 600, 0.21 )+- (0.0, 0.00)\n( 800, 0.12 )+- (0.0, 0.00)\n( 1000, 0.06 )+- (0.0, 0.00)\n};  \\addlegendentry{$M^3D$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.95 )+- (0.0, 0.03)\n( 400, 0.56 )+- (0.0, 0.03)\n( 600, 0.35 )+- (0.0, 0.03)\n( 800, 0.18 )+- (0.0, 0.03)\n( 1000, 0.08 )+- (0.0, 0.00)\n};  \\addlegendentry{$MMD$} ;\n\\addplot+[error bars\/.cd,\ny dir=both,y explicit]\n coordinates {\n( 200, 0.96 )+- (0.0, 0.03)\n( 400, 0.53 )+- (0.0, 0.03)\n( 600, 0.44 )+- (0.0, 0.03)\n( 800, 0.30 )+- (0.0, 0.00)\n( 1000, 0.13 )+- (0.0, 0.00)\n};  \\addlegendentry{$ST$} ;\n\\end{axis}\n\\end{tikzpicture}\n\\end{minipage\n\\hfill\\vspace{-0.1in}\n\\caption{Error versus Sample Size: von Mises-Fisher distribution (row 1), Watson distribution (row 2), Mixture of von Mises-Fisher distribution (row 3) and mixture of Watson distribution (row 4) on sphere for 100 dimensions (left) and 150 dimensions (right).}\n\\label{fig:simT3}\n\\end{figure*}\n\n\n\n\\subsection{Real data experiments}\nIn addition to the simulation examples, we also performed experiments on several real-world data examples. Similar to before, in the case of Euclidean data, we compared $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ against the standard MMD and Kolmogorov-Smirnov test, and in the case of spherical data, we used the Sobolev test~\\cite{jupp2005sobolev}, instead of the Kolmogorov-Smirnov test.\n\nFor the case of Euclidean data, we used the MINST digits data set from the following webpage: \\hyperref[http:\/\/yann.lecun.com\/exdb\/mnist\/]{http:\/\/yann.lecun.com\/exdb\/mnist\/}. Model-based clustering~ \\cite{fraley2002model} is a widely-used and practical successful clustering technique in the literature.  Furthermore, the MNIST data set is a standard data set for testing clustering algorithms and consists of image of digits. Several works have implicitly assumed that the data come from a mixture of Gaussian distributions, because of the observed superior empirical performance under such an assumption. But the validity of such a mixture model assumption is invariably not tested statistically. In this experiment we selected three digits (which correspond to a cluster) randomly and conditioned on the selected digit (cluster), we test the hypothesis that the data come from a Gaussian distribution (that is, $P_0$ is Gaussian). For our experiments, we down sampled the images and use pixels as feature vectors with dimensionality 64 as is commonly done in the literature. Table~\\ref{tab:realdata1} reports the probability with which the null hypothesis is accepted. The observed result reiterates in a statistically significant way that it is reasonable to make a mixture of Gaussian assumption in this case.\n\nFor the spherical case, we use the Human Fibroblasts dataset from~\\cite{iyer1999transcriptional,dhillon2003diametrical}, Yeast Cell Cycle dataset from~\\cite{spellman1998comprehensive} and the Rosetta yeast gene expression dataset~\\cite{hughes2000functional}.  The Fibroblast data set contains 12 expression data corresponding to 517 samples (genes) report in the response of human fibroblasts following addition of serum to the growth media. We refer to~\\cite{iyer1999transcriptional} for more details about the scientific procedure with which these data were obtained. The Yeast Cell Cycle dataset consists of 82-dimensional data corresponding to 696 subjects. The Rosetta yeast dataset contains 300-dimensional element vector for around 6000 yeast genes. Previous data analysis studies~\\cite{sra2013multivariate, dhillon2003diametrical} have used mixtures of spherical distributions for clustering the above data set. Specifically, it has been observed in~\\cite{sra2013multivariate} that clustering using a mixture of Watson distribution has superior performance. While that has proved to be useful scientifically, it was not statistically tested if such an assumption is valid. Here, we test for goodness of fit of Watson distribution (that is, $P_0$ is a Watson distribution) for the largest cluster from the above data sets.  Table~\\ref{tab:realdata2} shows the estimated probability of acceptance of the null hypothesis when it is assumed to be true. The values reported are averages from 50 random trails of the same dataset. The observed results provide a statistical justification for the use of Watson distribution in modeling the above data sets. \n\nWe note that for both situations, the tests considered tend to agree that the true hypothesis is true when there are more samples as indicated by Table~\\ref{tab:realdata1} and~\\ref{tab:realdata2}. But, the probability of acceptance is higher for low sample sizes for the adaptive M$^3$D test $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\, in all cases, showing that the method works better with finite sample sizes. This highlights the advantage of the $\\tilde{T}_{\\mathrm{M}^3\\mathrm{d}}$\\ in a finite sample setting confirming the better rates of convergence obtained in theory.\n\\begin{table}[t]\n\\centerin\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n Sample size = &  $300$ & $400$& $500$\\\\ \\hline \\hline \nK-S &  0.86&0.91 &0.94\\\\ \\hline\n$MMD$ & 0.90 & 0.93 &0.95 \\\\ \\hline\n$M^3D$& 0.94  &0.96   & 0.98\\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n $300$ & $400$& $500$\\\\ \\hline \\hline \n  0.83& 0.88 & 0.93\\\\ \\hline\n 0.89  &0.92  & 0.95\\\\ \\hline\n    0.93& 0.95  & 0.98\\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n $300$ & $400$& $500$\\\\ \\hline \\hline \n  0.84& 0.88 & 0.92\\\\ \\hline\n 0.88  &0.92  & 0.94\\\\ \\hline\n    0.93& 0.95  & 0.98\\\\ \\hline\n\\end{tabular}\n\\caption{ The values reported are the estimated probability with which the corresponding hypothesis test accepts the null hypothesis when it is true. The level of the test $\\alpha=0.05$.  Digit 4 on left, Digit 6 on the middle and Digit 7 on right, for various values of sample size.}\n\\label{tab:realdata1}\n\\end{table}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n Sample size = &  $75$ & $150$& $200$\\\\ \\hline \\hline \nST &  0.87&0.93 &0.98\\\\ \\hline\n$MMD$ & 0.90 & 0.94 &0.98 \\\\ \\hline\n$M^3D$& 0.92  &0.96   & 0.99\\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n  $150$ & $200$& $250$\\\\ \\hline \\hline \n  0.82& 0.87 & 0.91\\\\ \\hline\n 0.85  &0.92  & 0.94\\\\ \\hline\n    0.88& 0.93  & 0.96\\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n $400$ & $500$& $600$\\\\ \\hline \\hline \n  0.76& 0.84 & 0.91\\\\ \\hline\n 0.79  &0.87  & 0.92\\\\ \\hline\n    0.81& 0.90  & 0.95\\\\ \\hline\n\\end{tabular}\n\\caption{ The values reported are the estimated probability with which the corresponding hypothesis test accepts the null hypothesis when it is true. The level of the test $\\alpha=0.05$.  Human Fibroblasts dataset  on left, Yeast Cell Cycle dataset on the middle and Rosetta Yeast dataset on the right, for various values of sample size.}\n\\label{tab:realdata2}\n\\end{table}\n\n\n\\section{Proofs}\n\\label{sec:proof}\n\n\\begin{proof}[Proof of Theorem \\ref{mmdthm}]\n\n\\noindent{\\bf Part (i).} The proof of the first part consists of two key steps. First, we show that the population counterpart $n\\gamma^2(P,P_0)$ of the test statistic converges to $\\infty$ uniformly, \\textit{i.e.},\n$$n\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\gamma^2(P,P_0)\\rightarrow \\infty.$$\nThen, we argue that the deviation from $\\gamma^2(P,P_0)$ to $\\gamma^2(\\widehat{P}_n,P_0)$ is uniformly negligible compared with $\\gamma^2(P,P_0)$ itself.\n\nIt is not hard to see that\n\t\\begin{align*}\n\t\\gamma(\\widehat{P}_n,P_0)=&\\sqrt{\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)\\Big]^2}\\\\\n\t\\geq&\\sqrt{\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2}-\\sqrt{\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2}.\n\t\\end{align*}\nThus,\n\t\\begin{align*}\n\t&P\\left\\{n\\gamma^2(\\widehat{P}_n,P_0)<q_{w,1-\\alpha}\\right\\}\\\\\\leq &P\\left\\{ \\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2}-\\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2}<\\sqrt{q_{w,1-\\alpha}}\\right\\} \\\\\n\t=&P\\left\\{ \\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2}>\\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2}-\\sqrt{q_{w,1-\\alpha}}\\right\\}.\n\t\\end{align*}\n\t\n\tSuppose that \n\t\\begin{align*}\n\tn\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2>q_{w,1-\\alpha}.\n\t\\end{align*}\nThen\n\t\\begin{align*}\n\tP\\left\\{n\\gamma^2(\\widehat{P}_n,P_0)<q_{w,1-\\alpha}\\right\\}\\leq \\frac{{\\mathbb E} _P\\Big\\{n\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2\\Big\\}}{\\Bigg\\{\\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2}-\\sqrt{q_{w,1-\\alpha}}\\Bigg\\}^2}.\n\t\\end{align*}\nObserve that for any $P\\in\\mathcal{P}(\\Delta_n,0)$,\n\t\\begin{align*}\n\t{\\mathbb E} _P\\Big\\{n\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2\\Big\\}=&\\sum\\limits_{k\\geq 1}\\lambda_k\\mathrm{Var}[\\varphi_k(X)]\\\\\n\t\\leq& \\sum\\limits_{k\\geq 1}\\lambda_k{\\mathbb E} _P\\varphi_k^2(X)\\\\\n\t\\leq& \\Big(\\sup\\limits_{k\\geq 1}\\|\\varphi_k\\|_{\\infty}\\Big)^2\\sum\\limits_{k\\geq 1}\\lambda_k<\\infty.\n\t\\end{align*}\nThis implies that\n\t\\begin{align*}\n\t\\lim\\limits_{n\\rightarrow \\infty}\\beta(T_{\\text{MMD}};\\Delta_n,0)=&\\lim\\limits_{n\\rightarrow \\infty}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}P\\left\\{n\\gamma^2(\\widehat{P}_n,P_0)<q_{w,1-\\alpha}\\right\\}\\\\\n\t\\leq&\\lim_{n\\rightarrow \\infty}\\frac{\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}{\\mathbb E} _P\\Big\\{n\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)\\Big]^2\\Big\\}}{\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\Bigg\\{\\sqrt{n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2}-\\sqrt{q_{w,1-\\alpha}}\\Bigg\\}^2}\\\\\n\t=&0,\n\t\\end{align*}\nprovided that\n\\begin{align}\n\t\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2\\rightarrow \\infty, \\qquad {\\rm as\\ }n\\to\\infty.\\label{conv1.1}\n\t\\end{align}\n\tIt now suffices to show that (\\ref{conv1.1}) holds if $n\\Delta_n^2\\rightarrow \\infty$ as $n\\rightarrow \\infty$.\n\t\n\tTo this end, let $u={{d}P}\/{{d}P_0}-1$ and\n\t$$a_k=\\langle u,\\varphi_k\\rangle_{L_2(P_0)}={\\mathbb E} _P\\varphi_k(X)-{\\mathbb E} _{P_0}\\varphi_k(X)={\\mathbb E} _P(\\varphi_k(X)).$$\n\tIt is clear the that\n\t$$\\sum\\limits_{k\\geq 1}\\lambda_k^{-1}a_k^2=\\|u\\|_K^2,\\qquad {\\rm and}\\qquad \\sum\\limits_{k\\geq 1}a_k^2=\\|u\\|_{L_2(P_0)}^2=\\chi^2(P,P_0).$$ \n\tBy the definition of $\\mathcal{P}(\\Delta_n,0)$,\n\t\\begin{align*}\n\t\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\sum\\limits_{k\\geq 1}\\lambda_k^{-1}a_k^2\\leq M^2, \\qquad {\\rm and} \\qquad \\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\sum\\limits_{k\\geq 1}a_k^2\\geq \\Delta_n.\n\t\\end{align*}\n\tSince $n\\Delta_n^2\\rightarrow \\infty$ as $n\\rightarrow \\infty$, we get\n\t\\begin{align*}\n\t\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}n\\sum\\limits_{k\\geq 1}\\lambda_k[{\\mathbb E} _{P}\\varphi_k(X)]^2=&\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}n\\sum\\limits_{k\\geq 1}\\lambda_ka_k^2\\\\\n\t\\geq& \\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}n\\frac{\\Big(\\sum\\limits_{k\\geq 1}a_k^2\\Big)^2}{\\sum\\limits_{k\\geq 1}\\lambda_k^{-1}a_k^2}\\\\\n\t\\geq& \\frac{n\\Delta_n^2}{M^2}\\rightarrow \\infty\n\t\\end{align*}\n\tas $n\\rightarrow \\infty$.\n\t\n\\paragraph{Part (ii).} In proving the second part, we will make use of the following lemma that can be obtained by adapting the argument in \\citet{gregory1977large}. It gives the limit distribution of V-statistic under $P_n$ such that $P_n$ converges to $P_0$ in the order $n^{-1\/2}$.\n\n\t\\begin{lemma}\\label{lemG}\n\t\tConsider a sequence of probability measures $\\{P_n: n\\geq 1\\}$ contiguous to $P_0$ satisfying $u_n={{d}P_n}\/{{d}P_0}-1\\rightarrow 0$ in $L^2(P_0)$. Suppose that for any fixed $k$,\n\t\t$$\\lim\\limits_{n\\rightarrow \\infty}\\sqrt{n}\\langle u_n,\\varphi_k\\rangle_{L^2(P_0)}=\\tilde{a}_k, \\qquad {\\rm and}\\qquad\\lim\\limits_{n\\rightarrow \\infty}\\sum\\limits_{k\\geq 1}\\lambda_k(\\sqrt{n}\\langle u_n,\\varphi_k\\rangle_{L^2(P_0)})^2=\\sum\\limits_{k\\geq 1}\\lambda_k\\tilde{a}_k^2+\\tilde{a}_0<\\infty,$$\n\t\tfor some sequence $\\{\\tilde{a}_k: k\\ge 0\\}$, then\n\t\t\\begin{align*}\n\t\t\\frac{1}{n}\\sum\\limits_{k\\geq 1}\\lambda_k\\Big[\\sum\\limits_{i=1}^n\\varphi_k(X_i)\\Big]^2\\stackrel{d}{\\rightarrow}\\sum_{k\\geq 1}\\lambda_k(Z_k+\\tilde{a}_k)^2+\\tilde{a}_0,\n\t\t\\end{align*}\nwhere $X_1,\\ldots, X_n\\stackrel{i.i.d}{\\sim}P_n$, and $Z_k$s are independent standard normal random variables.\n\t\\end{lemma}\n\n\t\n\tWrite $L(k)=\\lambda_k k^{2s}$. By assumption $(\\ref{summable})$,\n\t$$0<\\underline{L}:=\\inf\\limits_{k\\geq 1}L(k)\\leq\\sup\\limits_{k\\geq 1}L(k):=\\overline{L}<\\infty.$$ \n\tConsider a sequence of $\\{P_n: n\\geq 1\\}$ such that\n\t$${{d}P_n}\/{{d}P_0}-1={C_1\\sqrt{\\lambda_{k_n}}}[L(k_n)]^{-1}\\varphi_{k_n},$$ where $C_1$ is a positive constant and $k_n=\\lfloor{C_2n^\\frac{1}{4s}}\\rfloor$ for some positive constant $C_2$. Both $C_1$ and $C_2$ will be determined later. Since $\\sup\\limits_{k\\geq 1}\\|\\varphi_k\\|_{\\infty}<\\infty$ and $\\lim\\limits_{k\\rightarrow \\infty}\\lambda_k=0$, there exists $N_0>0$ such that $P_n$'s are well-defined probability measures for any $n\\geq N_0$.\n\t\n\tNote that \\begin{align*}\n\t\\|u_n\\|_K^2=\\frac{C_1^2}{L^2(k_n)}\\leq \\underline{L}^{-2}C_1^2\n\t\\end{align*} and \n\t\\begin{align*}\n\t\\|u_n\\|_{L^2(P_0)}^2=\\frac{C_1^2\\lambda_{k_n}}{L^2(k_n)}=\\frac{C_1^2}{L(k_n)}k_n^{-2s}\\geq \\overline{L}^{-1}{C_1^2}k_n^{-2s}\\sim \\overline{L}^{-1}{C_1^2}C_2^{-2s}n^{-1\/2},\n\t\\end{align*}\n\twhere $A_n\\sim B_n$ means that $\\lim\\limits_{n\\rightarrow \\infty}{A_n}\/{B_n}=1$.\n\tThus, by choosing $C_1$ sufficiently small and $c_0=\\frac{1}{2}\\overline{L}^{-1}{C_1^2}C_2^{-2s}$, we ensure that $P_n\\in \\mathcal{P}(c_0n^{-1\/2},0)$ for sufficiently large $n$.\n\t\n\t\n\t\n\tTo apply Lemma $\\ref{lemG}$, we note that\n\t$$\\lim\\limits_{n\\rightarrow \\infty}\\|u_n\\|_{L^2(P_0)}^2=\\lim\\limits_{n\\rightarrow \\infty}\\frac{C_1^2\\lambda_{k_n}}{L^2(k_n)}=0.$$ \n\tIn addition, for any fixed $k$,\n\t$$\\tilde{a}_{n,k}=\\sqrt{n}\\langle u_n,\\varphi_k\\rangle_{L^2(P_0)}=0$$\n\t for sufficiently large $n$, and \n\t $$\\sum\\limits_{k\\geq 1}\\lambda_k\\tilde{a}_{n,k}^2=\\frac{nC_1^2\\lambda_{k_n}^2}{L^2(k_n)}=nC_1^2k_n^{-4s}\\rightarrow C_1^2C_2^{-4s}$$ \n\t as $n\\rightarrow \\infty$. Thus, Lemma \\ref{lemG} implies that\n\t\\begin{align*}\n\tn\\gamma(\\widehat{P}_n,P_0)\\stackrel{d}{\\rightarrow}\\sum_{k\\geq 1}\\lambda_kZ_k^2+C_1^2C_2^{-4s}.\n\t\\end{align*}\n\t\n\tNow take $C_2=\\left({2C_1^2}\/{q_{w,1-\\alpha}}\\right) ^{{1}\/{4s}}$ so that $C_1^2C_2^{-4s}=\\frac{1}{2}q_{w,1-\\alpha}$. Then\n\t\\begin{align*}\n\t\\varliminf\\limits_{n\\rightarrow \\infty}\\beta(T_{\\text{MMD}};c_0n^{-1\/2},0)\\geq& \\lim_{n\\rightarrow \\infty}P_n(n\\gamma(\\widehat{P}_n,P_0)< q_{w,1-\\alpha})\\\\\n\t=&P\\Big(\\sum_{k\\geq 1}\\lambda_kZ_k^2< \\frac{1}{2}q_{w,1-\\alpha}\\Big)>0,\n\t\\end{align*}\n\twhich concludes the proof.\n\\end{proof}\n\\vskip 25pt\n\n\\begin{proof}[Proof of Theorem \\ref{asympm3d}]\n\tLet $\\tilde{K}_n(\\cdot,\\cdot):=\\tilde{K}_{\\varrho_n}(\\cdot,\\cdot)$. Note that\n\t\\begin{align*}\n\tn\\widehat{\\eta}_{\\varrho_n}^2(P,P_0)=&\\frac{1}{n}\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big[\\sum\\limits_{i=1}^n\\varphi_k(X_i)\\Big]^2\\\\=&\\frac{1}{n}\\sum\\limits_{i,j=1}^n\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\varphi_k(X_i)\\varphi_k(X_j)\\\\=&\\frac{1}{n}\\sum\\limits_{i,j=1}^n\\tilde{K}_n(X_i,X_j).\n\t\\end{align*}\n\tThus\n\t\\begin{align*}\n\tv_n^{-1\/2}[n\\widehat{\\eta}_{\\varrho_n}^2(P,P_0)-A_n]=2(n^2v_n)^{-1\/2}\\sum\\limits_{j=2}^n\\sum\\limits_{i=1}^{j-1}\\tilde{K}_n(X_i,X_j).\n\t\\end{align*}\n\tLet $\\zeta_{nj}=\\sum\\limits_{i=1}^{j-1}\\tilde{K}_n(X_i,X_j)$. Consider a filtration $\\{\\mathcal{F}_j: j\\geq 1\\}$ where $\\mathcal{F}_j=\\sigma\\{X_j:1\\leq i\\leq j\\}$. Due to the assumption that $K$ is degenerate, we have ${\\mathbb E} \\varphi_k(X)=0$ for any $k\\geq 1$, which implies that\n\t\\begin{align*}\n\t{\\mathbb E} (\\zeta_{nj}|\\mathcal{F}_{j-1})=\\sum\\limits_{i=1}^{j-1}{\\mathbb E} [\\tilde{K}_n(X_i,X_j)|\\mathcal{F}_{j-1}]=\\sum\\limits_{i=1}^{j-1}{\\mathbb E} [\\tilde{K}_n(X_i,X_j)|X_i]=0,\n\t\\end{align*}\n\tfor any $j\\geq 2$. \n\t\n\tWrite \n\t\\begin{align*}\n\tU_{nm}=\\begin{cases}\n\t0&m=1 \\\\ \n\t\\sum\\limits_{j=2}^m\\zeta_{nj}& m\\geq 2\n\t\\end{cases}.\n\t\\end{align*}\n\tThen for any fixed $n$, $\\{U_{nm}\\}_{m\\geq 1}$ is a martingale with respect to $\\{\\mathcal{F}_m: m\\geq 1\\}$ and \n\t\\begin{align*}\n\tv_n^{-1\/2}[n\\widehat{\\eta}_{\\varrho_n}^2(P,P_0)-A_n]=2(n^2v_n)^{-1\/2}U_{nn}.\n\t\\end{align*}\n\t\n\tWe now apply martingale central limit theorem to $U_{nn}$. Following the argument from \\citet{hall1984central}, it can be shown that\n\t\\begin{align}\n\t\\Big[\\frac{1}{2}n^2{\\mathbb E} \\tilde{K}_n^2(X,X')\\Big]^{-1\/2}U_{nn}\\stackrel{d}{\\rightarrow}N(0,1),\\label{con2}\n\t\\end{align}\n\tprovided that\n\t\\begin{align}\n\t&[{\\mathbb E} G_n^2(X,X')+n^{-1}{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})+n^{-2}{\\mathbb E} \\tilde{K}_n^4(X,X')]\/[{\\mathbb E} {\\tilde{K}_n^2(X,X')}]^2\\rightarrow 0,\\label{con3}\n\t\\end{align}\n\tas $n\\rightarrow \\infty$, where $G_n(x,x')={\\mathbb E} \\tilde{K}_n(X,x)\\tilde{K}_n(X,x')$.\n\tSince\n\t$${\\mathbb E} \\tilde{K}_n^2(X,X')=\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^2=v_n,$$\n\t(\\ref{con2}) implies that\n\t\\begin{align*}\n\tv_n^{-1\/2}[n\\widehat{\\eta}_{\\varrho_n}^2(P,P_0)-A_n]=\\sqrt{2}\\cdot\\Big(\\frac{1}{2}n^2{\\mathbb E} \\tilde{K}_n^2(X,X')\\Big)^{-1\/2}U_{nn}\\stackrel{d}{\\rightarrow}N(0,2).\n\t\\end{align*}\n\tIt therefore suffices to verify (\\ref{con3}).\n\t\t\n\tNote that\n\t\\begin{align*}\n\t{\\mathbb E} \\tilde{K}_n^2(X,X')=\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^2\n\t\\geq &\\sum\\limits_{\\lambda_k\\geq \\varrho_n^2}\\frac{1}{4}+\\frac{1}{4\\varrho_n^4}\\sum\\limits_{\\lambda_k<\\varrho_n^2}\\lambda_k^2\\\\\n\t=&\\frac{1}{4}|\\{k:\\lambda_k\\geq \\varrho_n^2\\}|+\\frac{1}{4\\varrho_n^4}\\sum\\limits_{\\lambda_k<\\varrho_n^2}\\lambda_k^2\\asymp \\varrho_n^{-1\/s},\n\t\\end{align*}\n\twhere the last step holds by considering that $\\lambda_k\\asymp k^{-2s}$. \tHereafter, we shall write $a_n\\asymp b_n$ if $0<\\varliminf\\limits_{n\\rightarrow \\infty}a_n\/b_n\\leq \\varlimsup\\limits_{n\\rightarrow \\infty}a_n\/b_n <\\infty$, for two positive sequences $\\{a_n\\}$ and $\\{b_n\\}$.\nSimilarly, \n\t\\begin{align*}\n\t{\\mathbb E} G_n^2(X,X')=\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^4\\leq |\\{k:\\lambda_k\\geq\\varrho_n^2\\}|+\\varrho_n^{-8}\\sum\\limits_{\\lambda_k<\\varrho_n^2}\\lambda_k^4\\asymp\\varrho_n^{-1\/s},\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\t{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X'')=&{\\mathbb E} \\Big\\{\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^2\\varphi_k^2(X)\\Big\\}^2\\\\\\leq &\\left(\\sup_{k\\geq 1}\\|\\varphi_k\\|_{\\infty}\\right)^4\\Big\\{\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^2\\Big\\}^2\\asymp \\varrho_n^{-2\/s}.\n\t\\end{align*}\n\tThus there exists a positive constant $C_3$ such that\n\t\\begin{align}\n\t{\\mathbb E} G_n^2(X,X')\/[{\\mathbb E} \\tilde{K}_n^2(X,X')]^2\\leq C_3\\varrho_n^{1\/s}\\rightarrow 0,\\label{conv1}\n\t\\end{align}\n\tand\n\t\\begin{align}\n\tn^{-1}{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X'')\/[{\\mathbb E} {\\tilde{K}_n^2(X,X')}]^2\\leq C_3n^{-1}\\rightarrow \\infty,\\label{conv2}\n\t\\end{align}\n\tas $n\\rightarrow \\infty$. On the other hand,\n\t\\begin{align*}\n\t{\\mathbb E} \\tilde{K}_n^4(X,X')\\leq \\|\\tilde{K}_n\\|^2_{\\infty}{\\mathbb E} \\tilde{K}_n^2(X,X'),\n\t\\end{align*}\n\twhere\n\t\\begin{align*}\n\t\\|\\tilde{K}_n\\|_{\\infty}=\\sup\\limits_{x}\\Bigg\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\varphi_k^2(x)\\Bigg\\}\\leq\\Big(\\sup\\limits_{k\\geq 1}\\|\\varphi_k\\|_{\\infty}\\Big)^2\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\asymp \\varrho_n^{-1\/s}.\n\t\\end{align*}\n\tThis implies that for some positive constant $C_4$,\n\t\\begin{align}\n\tn^{-2}{\\mathbb E} \\tilde{K}_n^4(X,X')\\}\/[{\\mathbb E} {\\tilde{K}_n^2(X,X')}]^2\\leq n^{-2}\\|\\tilde{K}_n\\|^2_{\\infty}\/{\\mathbb E} {\\tilde{K}_n^2(X,X')}\\leq C_4(n^2\\varrho_n^{1\/s})^{-1}\\rightarrow 0.\\label{conv3}\n\t\\end{align}\n\tas $n\\rightarrow \\infty$. Together, (\\ref{conv1}), (\\ref{conv2}) and (\\ref{conv3}) ensure that condition (\\ref{con3}) holds.\n\\end{proof}\n\\vskip 25pt\n\n\\begin{proof}[Proof of Theorem \\ref{crm3d}]\n\tNote that\n\t\\begin{align*}\n\t&n\\eta_{\\varrho_n}^2(\\widehat{P}_n,P_0)-\\frac{1}{n}\\sum_{i=1}^n\\tilde{K}_n(X_i,X_j)\\\\=&\\frac{1}{n}\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\sum_{\\substack{1\\leq i,j\\leq n\\\\ i\\neq j}}\\varphi_k(X_i)\\varphi_k(X_j)\\\\\n\t=&\\frac{1}{n}\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\sum_{\\substack{1\\leq i,j\\leq n\\\\ i\\neq j}}[\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X_j)-{\\mathbb E} _P\\varphi_k(X)]\\\\\n\t&+\\frac{2(n-1)}{n}\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)]\\sum_{1\\leq i\\leq n}[\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)]\\\\\n\t&+\\frac{n(n-1)}{n}\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)]^2\\\\\n\t:=& V_1+V_2+V_3.\n\t\\end{align*}\nObviously, ${\\mathbb E} _PV_1V_2=0$. We first argue that the following three statements together implies the desired result:\n\\begin{eqnarray}\n\\label{as2}\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}v_n^{-1\/2}V_3&=&\\infty,\\\\\n\\label{as1a}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}({\\mathbb E} _PV_1^2\/V_3^2)&=&o(1),\\\\\n\\label{as1b}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}({\\mathbb E} _PV_2^2\/V_3^2)&=&o(1).\n\\end{eqnarray}\n\nTo see this, note that \\eqref{as2} implies that\n\t\\begin{align*}\n\t&\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P(v_n^{-1\/2}[n\\widehat{\\eta}_{\\varrho_n}^2(P,P_0)-A_n]\\geq \\sqrt{2}z_{1-\\alpha})\\\\\n\t\\geq&\\lim_{n\\rightarrow \\infty}\\inf_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P\\Big(v_n^{-1\/2}V_3\\geq 2\\sqrt{2}z_{1-\\alpha}, V_1+V_2+V_3\\geq\\frac{1}{2}V_3\\Big)\\\\\n\t=&\\lim_{n\\rightarrow \\infty}\\inf_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P\\Big(V_1+V_2+V_3\\geq\\frac{1}{2}V_3\\Big).\n\t\\end{align*}\nOn the other hand, \\eqref{as1a} and \\eqref{as1b} imply that\n\t\\begin{align*}\n\\lim_{n\\rightarrow \\infty}\\inf_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P\\Big(V_1+V_2+V_3\\geq\\frac{1}{2}V_3\\Big)\n\t=&1-\\lim_{n\\rightarrow \\infty}\\sup_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P\\Big(V_1+V_2+V_3<\\frac{1}{2}V_3\\Big)\\\\\n\t\\geq&1-\\lim_{n\\rightarrow \\infty}\\sup_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\frac{{\\mathbb E} _P(V_1+V_2)^2}{(V_3\/2)^2}=1.\n\t\\end{align*}\nThis immediately suggests that $T_{\\text{M}^3\\text{d}}$ is consistent. We now show that \\eqref{as2}-\\eqref{as1b} indeed hold.\n\n\\paragraph{Verifying \\eqref{as2}.}\nWe begin with \\eqref{as2}. Since $v_n\\asymp \\varrho_n^{-1\/s}$ and $V_3=(n-1)\\eta_{\\varrho_n}^2(P,P_0)$, (\\ref{as2}) is equivalent to\n\t\\begin{align*}\n\t\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}n\\varrho_n^{\\frac{1}{2s}}\\eta_{\\varrho_n}^2(P,P_0)=\\infty.\n\t\\end{align*}\n\t\n\tFor any $P\\in\\mathcal{P}(\\Delta_n,\\theta)$, let $u={{d}P}\/{{d}P_0}-1$ and $a_k=\\langle u,\\varphi_k\\rangle_{L_2(P_0)}={\\mathbb E} _P\\varphi_k(X)$. Based on the assumption that $K$ is universal, $u=\\sum\\limits_{k\\geq 1}a_k\\varphi_k$. We consider the case $\\theta=0$ and $\\theta>0$ separately\n\t\n\\begin{enumerate}\t\n\\item[(1)] First consider $\\theta=0$. It is clear that\n\t\\begin{align*}\n\t\\eta_{\\varrho_n}^2(P,P_0)=&\\sum_{k\\geq 1}a_k^2-\\sum_{k\\geq 1}\\frac{\\varrho_n^2}{\\lambda_k+\\varrho_n^2}a_k^2\\\\\n\t\\geq&\\|u\\|_{L_2(P_0)}^2-\\varrho_n^2\\sum_{k\\geq 1}\\frac{1}{\\lambda_k}a_k^2\\\\\n\t\\geq&\\|u\\|_{L_2(P_0)}^2-\\varrho_n^2M^2.\n\t\\end{align*}\n\tTake $\\varrho_n\\le \\sqrt{{\\Delta_n}\/(2M^2)}$ so that $\\rho_n^2M^2\\le\\frac{1}{2}\\Delta_n$. Then we have\n\t\\begin{align*}\n\t\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\eta_{\\varrho_n}^2(P,P_0)\\geq\\frac{1}{2}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,0)}\\|u\\|_{L_2(P_0)}^2=\\frac{1}{2}\\Delta_n\n\t\\end{align*}\n\n\t\n\\item[(2)] Now consider the case when $\\theta>0$. For $P\\in\\mathcal{P}(\\Delta_n,\\theta)$, $\\forall$ $R>0$, $\\exists$ $f_R\\in\\mathcal{H}(K)$ such that $\\|u-f_R\\|_{L_2(P_0)}\\leq MR^{-1\/\\theta}$ and $\\|f_R\\|_{K}\\leq R$. Let $b_k=\\langle f_R,\\varphi_k\\rangle _{L_2(P_0)}$.\n\\begin{align*}\n\\eta_{\\varrho_n}^2(P,P_0)=&\\sum_{k\\geq 1}a_k^2-\\sum_{k\\geq 1}\\frac{\\varrho_n^2}{\\lambda_k+\\varrho_n^2}a_k^2\\\\\n\\geq&\\|u\\|_{L_2(P_0)}^2-2\\sum_{k\\geq 1}\\frac{\\varrho_n^2}{\\lambda_k+\\varrho_n^2}(a_k-b_k)^2-2\\sum_{k\\geq 1}\\frac{\\varrho_n^2}{\\lambda_k+\\varrho_n^2}b_k^2\\\\\n\\geq&\\|u\\|_{L_2(P_0)}^2-2\\sum_{k\\geq 1}(a_k-b_k)^2-2\\varrho_n^2\\sum_{k\\geq 1}\\frac{1}{\\lambda_k}b_k^2\\\\\n=&\\|u\\|_{L_2(P_0)}^2-2\\|u-f_R\\|_{L_2(P_0)}^2-2\\varrho_n^2\\|f_R\\|_{K}^2.\n\\end{align*}\n\tTaking $R=({2M}\/{\\|u\\|_{L_2(P_0)}})^{\\theta}$ yields that\n\t\\begin{align*}\n\t\\eta_{\\varrho_n}^2(P,P_0)\\geq \\|u\\|_{L_2(P_0)}^2-2M^2R^{-2\/\\theta}-2\\varrho_n^2R^2=\\frac{1}{2}\\|u\\|_{L_2(P_0)}^2-2\\varrho_n^2R^2.\n\t\\end{align*}\n\tNow by choosing\n\t$$\n\t\\varrho_n\t\\le\\frac{1}{2\\sqrt{2}}(2M)^{-\\theta}\\Delta_n^{\\frac{1+\\theta}{2}},\n\t$$\n\twe can ensure that\n\t$$2\\varrho_n^2R^2\\leq \\frac{1}{4}\\|u\\|_{L_2(P_0)}^2.$$ So that\n\t\\begin{align*}\n\t\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\eta_{\\varrho_n}^2(P,P_0)\\geq\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\frac{1}{4}\\|u\\|_{L_2(P_0)}^2\\geq\\frac{1}{4}\\Delta_n.\n\t\\end{align*} \n\t\n\n\\end{enumerate}\n\n\tIn both cases, with $\\varrho_n\\leq C\\Delta_n^{\\frac{\\theta+1}{2}}$ for a sufficiently small $C=C(M)>0$,  $\\lim\\limits_{n\\rightarrow\\infty}\\varrho_n^{\\frac{1}{2s}}n\\Delta_n=\\infty$ suffices to ensure (\\ref{as2}) holds. Under the condition that $\\lim\\limits_{n\\rightarrow \\infty}\\Delta_nn^{\\frac{4s}{4s+\\theta+1}}=\\infty$, \n$$\n\\varrho_n=cn^{-\\frac{2s(\\theta+1)}{4s+\\theta+1}}\\leq C\\Delta_n^{\\frac{\\theta+1}{2}}\n$$\nfor sufficiently large $n$ and $\\lim\\limits_{n\\rightarrow\\infty}\\varrho_n^{\\frac{1}{2s}}n\\Delta_n=\\infty$ holds as well.\n\n\t\n\\paragraph{Verifying \\eqref{as1a}.}\nRewrite $V_1$ as\n\t\\begin{align*}\n\tV_1=&\\frac{1}{n}\\sum_{\\substack{1\\leq i,j\\leq n\\\\ i\\neq j}}\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[\\varphi_k(X_i)-{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X_j)-{\\mathbb E} _P\\varphi_k(X)]\\\\\n\t:=& \\frac{1}{n}\\sum_{\\substack{1\\leq i,j\\leq n\\\\ i\\neq j}}F_n(X_i,X_j).\n\t\\end{align*}\n\tThen\n\t\\begin{align*}\n\t{\\mathbb E} _PV_1^2=&\\frac{1}{n^2}\\sum_{\\substack{i\\neq j\\\\ i'\\neq j'}}{\\mathbb E} _PF_n(X_i,X_j)F_n(X_{i'},X_{j'})\\\\\n\t=&\\frac{2n(n-1)}{n^2}{\\mathbb E} _PF_n^2(X,X')\\\\\n\t\\leq& 2{\\mathbb E} _PF_n^2(X,X').\n\t\\end{align*}\n\t\n\tRecall that, for any two random variables $Y_1$, $Y_2$ such that ${\\mathbb E} Y_1^2<\\infty$,\n\t\\begin{align*}\n\t{\\mathbb E} [Y_1-{\\mathbb E} (Y_1|Y_2)]^2={\\mathbb E} Y_1^2-{\\mathbb E} [{\\mathbb E} (Y_1|Y_2)^2]\\leq {\\mathbb E} Y_1^2.\n\t\\end{align*}\n\tTherefore,\n\t\\begin{align*}\n\t{\\mathbb E} _PF_n^2(X,X')\\leq {\\mathbb E} _P\\{\\tilde{K}_n(X,X')-{\\mathbb E} _P[\\tilde{K}_n(X,X')|X]\\}^2\\leq {\\mathbb E} _{P}\\tilde{K}_n^2(X,X').\n\t\\end{align*}\n\tThus, to prove \\eqref{as1a}, it suffices to show that\n\t\\begin{align*}\n\t\\lim\\limits_{n\\rightarrow \\infty}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}{\\mathbb E} _{P}\\tilde{K}_n^2(X,X')\/V_3^2=0.\n\t\\end{align*}\n\t\n\t\n\tFor any $g\\in L_2(P_0)$ and positive definite kernel $G(\\cdot,\\cdot)$ such that ${\\mathbb E} _{P_0}G^2(X,X')<\\infty$, let\n\t\\begin{align*}\n\t\\|g\\|_G:= \\sqrt{{\\mathbb E} _{P_0}[g(X)g(X')G(X,X')]}.\n\t\\end{align*}\n\tBy the positive definiteness of $G(\\cdot,\\cdot)$, triangular inequality holds for $\\|\\cdot\\|_G$, \\textit{i.e.}, for any $g_1$, $g_2\\in L_2(P_0)$, \n\t\\begin{align*}\n\t|\\|g_1\\|_G-\\|g_2\\|_G|\\leq \\|g_1-g_2\\|_G,\n\t\\end{align*}\n\twhich implies that\n\t\\begin{align}\n\t\\Bigg|\\sqrt{{\\mathbb E} _P\\tilde{K}_n^2(X,X')}-\\sqrt{{\\mathbb E} _{P_0}\\tilde{K}_n^2(X,X')}\\Bigg|\\leq \\sqrt{{\\mathbb E} _{P_0}[u(X)u(X')\\tilde{K}_n^2(X,X')]}.\\label{tri}\n\t\\end{align}\n\tWe now appeal to the following lemma to bound the right hand side of \\eqref{tri}:\n\t\\begin{lemma}\\label{schur2}\n\tLet $G$ be a Mercer kernel defined over $\\mathcal X\\times \\mathcal X$ with eigenvalue-eigenfunction pairs $\\{(\\mu_k, \\varphi_k): k\\ge 1\\}$ with respect to $L_2(P)$ such that $\\mu_1\\ge \\mu_2\\ge\\cdots$. If $G$ is a trace kernel in that ${\\mathbb E} G(X,X)<\\infty$, then for any $g\\in L_2(P)$\n\t$$\n\t{\\mathbb E} _{P}[g(X)g(X')G^2(X,X')]\\le \\mu_1\\left(\\sum_{k\\ge 1}\\mu_k\\right)\\left(\\sup_{k\\ge 1}\\|\\varphi_k\\|_\\infty\\right)^2\\|g\\|_{L_2(P)}^2.\n\t$$\n\t\\end{lemma}\n\tBy Lemma \\ref{schur2}, we get\n\t\\begin{align*}\n\t{\\mathbb E} _{P_0}[u(X)u(X')\\tilde{K}_n^2(X,X')]\\leq &C_5\\left(\\sum\\limits_k\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\right)\\|u\\|_{L_2(P_0)}^2\\asymp \\varrho_n^{-1\/s}\\|u\\|_{L_2(P_0)}^2.\n\t\\end{align*}\n\tRecall that\n\t$${\\mathbb E} _{P_0}\\tilde{K}^2_n(X,X')=\\sum\\limits_k\\left(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\right)^2\\asymp \\varrho_n^{-1\/s}.$$ In the light of (\\ref{tri}), they imply that\n\t\\begin{align*}\n\t{\\mathbb E} _{P}\\tilde{K}^2_n(X,X')\\leq 2\\{{\\mathbb E} _{P_0}\\tilde{K}^2_n(X,X')+{\\mathbb E} _{P_0}[u(X)u(X')\\tilde{K}_n^2(X,X')]\\}\\leq C_6\\varrho_n^{-1\/s}[1+\\|u\\|_{L_2(P_0)}^2].\n\t\\end{align*}\nOn the other hand, it is not hard to verify that with our choice of $\\varrho_n$,\n\t\\begin{align*}\n\t\\frac{1}{4}\\|u\\|_{L_2(P_0)}^2\\leq \\eta_{\\varrho_n}^2(P,P_0)\\leq \\|u\\|_{L_2(P_0)}^2,\n\t\\end{align*}\n\tfor any $P\\in\\mathcal{P}(\\Delta_n,\\theta)$. Thus\n\t\\begin{align*}\n\t&\\lim\\limits_{n\\rightarrow \\infty}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}{\\mathbb E} _P\\tilde{K}_n^2(X,X')\/V_3^2\\\\\\leq &16C_6\\Big\\{\\Big(\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\varrho_n^{1\/s}n^2\\|u\\|_{L_2(P_0)}^4\\Big)^{-1}+\\Big(\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\varrho_n^{1\/s}n^2\\|u\\|_{L_2(P_0)}^2\\Big)^{-1}\\Big\\}=0\n\t\\end{align*}\n\tprovided that $\\lim\\limits_{n\\rightarrow \\infty}n^{\\frac{4s}{4s+\\theta+1}}\\Delta_n=\\infty$. This immediately implies \\eqref{as1a}.\n\t\n\\paragraph{Verifying \\eqref{as1b}.} Observe that\n\t\\begin{align*}\n\t{\\mathbb E} _PV_2^2\\leq &4n{\\mathbb E} _P\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)-{\\mathbb E} _P\\varphi_k(X)]\\Big\\}^2\\\\\n\t\\leq &4n{\\mathbb E} _P\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2\\\\\n          =& 4n{\\mathbb E} _{P_0}\\left([1+u(X)]\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2\\right).\n\t\\end{align*}\n\t\nIt is clear that\n\t\\begin{align*}\n\t&{\\mathbb E} _{P_0}\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2\\\\=&\\sum\\limits_{k,k'\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\frac{\\lambda_{k'}}{\\lambda_{k'}+\\varrho_n^2}{\\mathbb E} _P\\varphi_k(X){\\mathbb E} _P\\varphi_{k'}(X){\\mathbb E} _{P_0}[\\varphi_k(X)\\varphi_{k'}(X)]\\\\=&\\sum\\limits_{k\\geq 1}\\Big(\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}\\Big)^2[{\\mathbb E} _P\\varphi_k(X)]^2\\leq \\eta_{\\varrho_n}^2(P,P_0).\n\t\\end{align*}\nOn the other hand,\n\\begin{align*}\n\t&{\\mathbb E} _{P_0}\\left(u(X)\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2\\right)\\\\\n\t\\leq&\\sqrt{{\\mathbb E} _{P_0}\\left(u^2(X)\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2\\right)}\\times\\\\\n\t&\\times\\sqrt{{\\mathbb E} _{P_0}\\Big\\{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(X)]\\Big\\}^2}\\\\\n\t\\leq&\\|u\\|_{L_2(P_0)}\\sup\\limits_{x}\\Big|\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)][\\varphi_k(x)]\\Big|\\cdot\\eta_{\\varrho_n}(P,P_0)\\\\\n\t\\leq&\\left(\\sup\\limits_{k}\\|\\varphi_k\\|_{\\infty}\\right)\\|u\\|_{L_2(P_0)}\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}|{\\mathbb E} _P\\varphi_k(X)|\\cdot\\eta_{\\varrho_n}(P,P_0)\\\\\n\t\\leq&\\left(\\sup\\limits_{k}\\|\\varphi_k\\|_{\\infty}\\right)\\|u\\|_{L_2(P_0)}\\sqrt{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}}\\sqrt{\\sum\\limits_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)]^2}\\cdot\\eta_{\\varrho_n}(P,P_0)\\\\\\leq &C_7\\|u\\|_{L_2(P_0)}\\varrho_n^{-\\frac{1}{2s}}\\eta_{\\varrho_n}^2(P,P_0).\n\t\\end{align*}\nTogether, they imply that\n\\begin{align*}\n\t&\\lim\\limits_{n\\rightarrow \\infty}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}{\\mathbb E} _PV_1^2\/V_3^2\\\\\\leq &4\\max\\{1,C_7\\}\\Bigg\\{\\Big(\\lim\\limits_{n\\rightarrow \\infty}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}n\\eta_{\\varrho_n}^2(P,P_0)\\Big)^{-1}+\\lim\\limits_{n\\rightarrow \\infty}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}\\Bigg(\\frac{\\|u\\|_{L_2(P_0)}}{\\varrho_n^{\\frac{1}{2s}}n\\eta_{\\varrho_n}^2(P,P_0)}\\Bigg)\\Bigg\\}=0,\n\t\\end{align*}\nunder the assumption that $\\lim\\limits_{n\\rightarrow \\infty}n^{\\frac{4s}{4s+\\theta+1}}\\Delta_n=\\infty$.\n\\end{proof}\n\\vskip 25pt\n\n\\begin{proof}[Proof of Theorem \\ref{cr}]\nWithout loss of generality, assume $M=1$ and $\\Delta_n=cn^{-\\frac{4s}{4s+\\theta+1}}$ for some $c>0$. The main idea behind our proof is to carefully construct a finite subset of $\\mathcal{P}(\\Delta_n,\\theta)\\setminus \\{P_0\\}$, and show that one can not reliably distinguish $P_0$ from an unknown instance from this subset based on a sample of $n$ observations. We shall consider the cases of $\\theta=0$ and $\\theta>0$ separately.\n\n\\paragraph{The case of $\\theta=0$.} \nWe first treat the case when $\\theta=0$. Let $K_n=\\lfloor{C_8\\Delta_n^{-\\frac{1}{2s}}}\\rfloor$ for a sufficiently small constant $C_8>0$ and $a_n=\\sqrt{{\\Delta_n}\/{K_n}}$. For any $\\xi_n:=(\\xi_{n1},\\xi_{n2},\\cdots,\\xi_{nK_n})^\\top\\in \\{\\pm 1\\}^{K_n}$, write\n$$\nu_{\\xi_n} = a_n\\sum_{k=1}^{K_n}\\xi_{nk}\\varphi_k.\n$$\nIt is clear that\n$$\\|u_{n,\\xi_n}\\|_{L_2(P_0)}^2=K_na_n^2=\\Delta_n$$\nand\t\n$$\n\\|u_{n,\\xi_n}\\|_\\infty\\le a_nK_n\\left(\\sup\\limits_{k}\\|\\varphi_k\\|_{\\infty}\\right)\\asymp \\Delta_n^{2s-1\\over 4s}\\rightarrow 0.\n$$\nBy taking $C_8$ small enough, we can also ensure\n$$\\|u_{\\xi_n}\\|_K^2=a_n^2\\sum\\limits_{k=1}^{K_n}\\lambda_k^{-1}\\leq 1,$$\n\nTherefore, there exists a probability measure $P_{\\xi_n}\\in\\mathcal{P}(\\Delta_n, 0)$ such that $dP\/dP_0=1+u_{n,\\xi_n}$. Following a standard argument for minimax lower bound, it suffices to show that\n\\begin{equation}\n\\label{eq:min}\n\\varlimsup\\limits_{n\\rightarrow \\infty}{\\mathbb E}_{P_0} \\left({1\\over 2^{K_n}}\\sum_{\\xi_n\\in \\{\\pm 1\\}^{K_n}}\\left\\{\\prod\\limits_{i=1}^{n}[1+u_{\\xi_n}(X_i)]\\right\\}\\right)^2<\\infty.\n\\end{equation}\nSee, \\textit{e.g.}, \\cite{ingster2003nptest,tsybakov2008introduction}.\n\nNote that\n\t\\begin{align*}\n&{\\mathbb E}_{P_0} \\left({1\\over 2^{K_n}}\\sum_{\\xi_n\\in \\{\\pm 1\\}^{K_n}}\\left\\{\\prod\\limits_{i=1}^{n}[1+u_{\\xi_n}(X_i)]\\right\\}\\right)^2\\\\\n=&{\\mathbb E}_{P_0} \\left({1\\over 2^{2K_n}}\\sum_{\\xi_n,\\xi_n'\\in \\{\\pm 1\\}^{K_n}}\\left\\{\\prod\\limits_{i=1}^{n}[1+u_{\\xi_n}(X_i)]\\right\\}\\left\\{\\prod\\limits_{i=1}^{n}[1+u_{\\xi_n'}(X_i)]\\right\\}\\right)\\\\\n\t=&{1\\over 2^{2K_n}}\\sum_{\\xi_n,\\xi_n'\\in \\{\\pm 1\\}^{K_n}}\\prod\\limits_{i=1}^n{\\mathbb E}_{P_0} \\Bigg\n\t\\{[1+u_{\\xi_n}(X_i)][1+u_{\\xi_n'}(X_i)]\\Bigg\\}\\\\\n\t=&{1\\over 2^{2K_n}}\\sum_{\\xi_n,\\xi_n'\\in \\{\\pm 1\\}^{K_n}}\\Big(1+a_{n}^2\\sum\\limits_{k=1}^{K_n}\\xi_{nk}\\xi_{nk}'\\Big)^n\\\\\n\t\\leq&{1\\over 2^{2K_n}}\\sum_{\\xi_n,\\xi_n'\\in \\{\\pm 1\\}^{K_n}}\\exp\\Big(na_n^2\\sum\\limits_{k=1}^{K_n}\\xi_{n,k}\\xi_{n,k}'\\Big)\\\\\n\t=&\\left\\{\\frac{\\exp(na_{n}^2)+\\exp(-na_{n}^2)}{2}\\right\\}^{K_n}\\\\\n\t\\leq&\\exp\\Big\n\t\\{K_n\\Big(\\frac{\\exp(na_{n}^2)+\\exp(-na_{n}^2)}{2}-1\\Big)\\Big\\}.\n\t\\end{align*}\nAn application of Taylor expansion shows that there exist $t_0>0$ and $C_9>0$ such that\n\t\\begin{align*}\n\t{t^2}-C_9|t|^3\\leq \\exp(t)+\\exp(-t)\\leq{t^2}+C_9|t|^3\n\t\\end{align*}\n\tfor any $|t|\\leq t_0$. With the particular choice of $K_n$, $a_n$, and the conditions on $\\Delta_n$, this immediately implies \\eqref{eq:min}.\n\t\n\\paragraph{The case of $\\theta>0$.} The main idea is similar to before. To find a set of probability measures in $\\mathcal{P}(\\Delta_n, \\theta)$, we appeal to the following lemma.\n\n\\begin{lemma}\\label{char0}\nLet $u=\\sum\\limits_{k}a_k\\varphi_k$. If\n\\begin{align*}\n\\left(\\sum\\limits_{k=1}^K\\frac{a_k^2}{\\lambda_k}\\right)^{2\/\\theta}\\left(\\sum\\limits_{k\\geq K}a_k^2\\right)\\leq M^2\n\\end{align*}\nthen $u\\in \\mathcal{F}(\\theta, M)$.\n\\end{lemma}\n\nSimilar to before, we shall now take $K_n=\\lfloor{C_{10}\\Delta_n^{-\\frac{\\theta+1}{2s}}}\\rfloor$ and $a_{n}=\\sqrt{{\\Delta_n}\/{K_n}}$. By Lemma \\ref{char0}, we can find $P_{\\xi_n}\\in\\mathcal{P}(\\Delta_n, \\theta)$ such that $dP\/dP_0=1+u_{n,\\xi_n}$, for appropriately chosen $C_{10}$. Following the same argument as in the previous case, we can again verify \\eqref{eq:min}.\n\\end{proof}\n\\vskip 25pt\n\n\\begin{proof}[Proof of Theorem \\ref{crtm3d}]\n\tWithout loss of generality, assume that $\\Delta_n(\\theta)= c_1({n}^{-1}{\\sqrt{\\log\\log n}})^{\\frac{4s}{4s+\\theta+1}}$ for some constant $c_1>0$ to be determined later. \n\n\\noindent{\\bf Type I Error.} We first prove the first statement which shows that the Type I error converges to $0$. Following the same notations as defined in Theorem \\ref{asympm3d}, let\n\t\\begin{align*}\n\tN_{n,2}={\\mathbb E} \\Big\\{\\sum\\limits_{j=2}^n{\\mathbb E} \\Big(\\tilde{\\zeta}_{nj}^2|\\mathcal{F}_{j-1}\\Big)-1\\Big\\}^2,\\quad L_{n,2}=\\sum\\limits_{j=2}^n{\\mathbb E} \\tilde{\\zeta}_{nj}^4\n\t\\end{align*}\nwhere $\\tilde{\\zeta}_{nj}={\\sqrt{2}\\zeta_{nj}}\/({n\\sqrt{v_n}})$. As shown by \\citet{haeusler1988rate},\n\t\\begin{align*}\n\t\\sup\\limits_{t}|P(T_{n,\\varrho_n}>t)-\\bar{\\Phi}(t)|\\leq C_{11}(L_{n,2}+N_{n,2})^{1\/5},\n\t\\end{align*}\n\twhere $\\bar{\\Phi}(t)$ is the survival function of the standard normal, \\textit{i.e.}, $\\bar{\\Phi}(t)=P(Z>t)$ where $Z\\sim N(0,1)$. Again by the argument from \\citet{hall1984central},\n\t\\begin{align*}\n\t{\\mathbb E} \\Big\\{\\sum\\limits_{j=2}^n{\\mathbb E} (\\zeta_{nj}^2|\\mathcal{F}_{j-1})-\\frac{1}{2}n(n-1)v_n\\Big\\}^2\\leq C_{12}[n^4{\\mathbb E} G_n^2(X,X')+n^3{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})],\n\t\\end{align*}\nwhere $G_n(\\cdot,\\cdot)$ is defined in the proof of Theorem \\ref{asympm3d}, and \n\t\\begin{align*}\n\t\\sum\\limits_{j=2}^n{\\mathbb E} \\zeta_{nj}^4\\leq C_{13}[n^2{\\mathbb E} \\tilde{K}_n^4(X,X')+n^3{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})],\n\t\\end{align*}\n\twhich ensure\n\t\\begin{align*}\n\tN_{n,2}=&\\frac{4{\\mathbb E} \\Big\\{\\sum\\limits_{j=2}^n{\\mathbb E} (\\zeta_{nj}^2|\\mathcal{F}_{j-1})-\\frac{1}{2}n(n-1)v_n-\\frac{1}{2}nv_n\\Big\\}^2}{n^4v_n^2}\\\\\\leq &8\\max\\left\\{C_{12},\\frac{1}{4}\\right\\}\\Bigg\\{\\frac{{\\mathbb E} G_n^2(X,X')}{v_n^2}+\\frac{{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})}{nv_n^2}+\\frac{1}{n^2}\\Bigg\\},\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\tL_{n,2}=\\frac{4\\sum\\limits_{j=2}^n{\\mathbb E} \\tilde{\\zeta}_{nj}^4}{n^4v_n^2}\\leq 4C_{13}\\Bigg\\{\\frac{{\\mathbb E} \\tilde{K}_n^4(X,X')}{n^2v_n^2}+\\frac{{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})}{nv_n^2}\\Bigg\\}.\n\t\\end{align*}\n\t\n\tAs shown in the proof of Theorem \\ref{asympm3d},\n\t\\begin{align*}\n\t\\frac{{\\mathbb E} G_n^2(X,X')}{v_n^2}\\leq C_3\\varrho_n^{1\/s},\\quad \\frac{{\\mathbb E} \\tilde{K}_n^4(X,X')}{n^2v_n^2}\\leq C_4n^{-2}\\varrho_n^{-1\/s},\\quad {\\rm and}\\quad \\frac{{\\mathbb E} \\tilde{K}_n^2(X,X')\\tilde{K}_n^2(X,X^{''})}{nv_n^2}\\leq C_3n^{-1}.\n\t\\end{align*}\n\tTherefore,\n\t\\begin{align*}\n\t\\sup\\limits_{t}|P(T_{n,\\varrho_n}>t)-\\bar{\\Phi}(t)|\\leq C_{14}(\\varrho_n^{\\frac{1}{5s}}+n^{-\\frac{1}{5}}+n^{-\\frac{2}{5}}\\varrho_n^{-\\frac{1}{5s}}),\n\t\\end{align*}\nwhich implies that\n\t\\begin{align*}\n\tP\\left(\\sup\\limits_{\\varrho_n\\in\\Lambda_n}T_{n,\\varrho_n}>t\\right)\\leq m_*\\bar{\\Phi}(t)+C_{15}(2^{m_*\\over5s}\\varrho_*^{1\\over 5s}+m_*n^{-\\frac{1}{5}}+n^{-\\frac{2}{5}}\\varrho_*^{-\\frac{1}{5s}}),\\qquad \\forall t.\n\t\\end{align*}\n\tIt is not hard to see, by the definitions of $m_*$,\n$$2^{m_*}\\varrho_*\\leq 2\\left(\\frac{\\sqrt{\\log\\log n}}{n}\\right)^{\\frac{2s}{4s+1}}$$\nand\n\t\\begin{align*}\n\tm_*=&(\\log 2)^{-1}\\{2s\\log n-\\frac{2s}{4s+1}\\log n+o(\\log n)\\}\\\\=&(\\log 2)^{-1}\\frac{8s^2}{4s+1}\\log n+o(\\log n)\\asymp \\log n.\n\t\\end{align*}\nTogether with the fact that $\\bar{\\Phi}(t)\\leq {1\\over 2} e^{-{t^2}\/{2}}$ for $t\\geq 0$, we get\n\t\\begin{align*}\n\t&P\\left(\\sup\\limits_{0\\leq k\\leq m_*}T_{n,2^k\\varrho_*}>\\sqrt{3\\log\\log n}\\right)\\\\\\leq &C_{16}\\left[e^{-\\frac{3}{2}\\log\\log n}\\log n+\\left(\\frac{\\sqrt{\\log n}}{n}\\right)^{\\frac{2}{5(4s+1)}}+n^{-\\frac{1}{5}}\\log\\log n+n^{-\\frac{2}{5}}\\left(\\frac{\\sqrt{\\log \\log n}}{n}\\right)^{-\\frac{2}{5}}\\right]\\rightarrow 0,\n\t\\end{align*}\n\tas $n\\rightarrow \\infty$.\n\t\n\\paragraph{Type II Error.} Next consider Type II error. To this end, write $\\varrho_n(\\theta)=({\\sqrt{\\log\\log n}\\over n})^{2s(\\theta+1)\\over 4s+\\theta+1}$. Let\n$$\\tilde{\\varrho}_ n(\\theta)=\\sup\\limits_{0\\leq k\\leq m_*}\\{2^{k}\\varrho_\\ast: \\varrho_n\\leq\\varrho_n(\\theta)\\}.$$\nIt is clear that $\\tilde{T}_n\\ge T_{n,\\tilde{\\varrho}_n(\\theta)}$ for any $\\theta\\ge 0$. It therefore suffices to show that for any $\\theta\\ge 0$,\n$$\n\\lim_{n\\to\\infty}\\inf\\limits_{\\theta\\geq 0}\\inf_{P\\in\\mathcal{P}(\\Delta_n,\\theta)}P\\left\\{T_{n,\\tilde{\\varrho}_n(\\theta)}\\geq \\sqrt{3\\log\\log n}\\right\\}=1.\n$$\nBy Markov inequality, this can accomplished by verifying\n\t\\begin{align}\n\t\\inf\\limits_{\\theta\\in[0,\\infty)}\\inf\\limits_{P\\in\\mathcal{P}(\\Delta_n(\\theta),\\theta)}{\\mathbb E} _PT_{n,\\tilde{\\varrho}_n(\\theta)}\\geq \\tilde{M}\\sqrt{\\log\\log n}\\label{mean}\n\t\\end{align}\n\tfor some $\\tilde{M}>\\sqrt{3}$; and\n\t\\begin{align}\n\t\\lim\\limits_{n\\rightarrow\\infty}\\sup\\limits_{\\theta\\ge 0}\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n(\\theta),\\theta)}\\frac{\\mathrm{Var}\\left(T_{n,\\tilde{\\varrho}_n(\\theta)}\\right)}{\\left({\\mathbb E} _PT_{n,\\tilde{\\varrho}_n(\\theta)}\\right)^2}=0\\label{mv}.\n\t\\end{align}\n\nWe now show that both (\\ref{mean}) and (\\ref{mv}) hold with $$\\Delta_n(\\theta)=c_1\\left({\\sqrt{\\log\\log n}\\over n}\\right)^{\\frac{4s}{4s+\\theta+1}}$$ for a sufficiently large $c_1=c_1(M,\\tilde{M})$.\n\nNote that $\\forall$ $\\theta\\in[0,\\infty)$, \n\t\\begin{align}\n\t\\frac{1}{2}\\varrho_n(\\theta)\\leq\\tilde{\\varrho_n}(\\theta)\\leq \\varrho_n(\\theta),\\label{lub}\n\t\\end{align},\n\twhich immediately suggests \n\t\\begin{align}\n\t\\eta_{\\tilde{\\varrho}_n(\\theta)}^2(P,P_0)\\geq\\eta_{\\varrho_n(\\theta)}^2(P,P_0).\\label{ineq}\n\t\\end{align}\n\t\nFollowing the arguments in the proof of Theorem \\ref{crm3d},\n$${\\mathbb E} _PT_{n,\\tilde{\\varrho}_n(\\theta)}\\geq C_{17}n[\\tilde{\\varrho}_n(\\theta)]^{1\/(2s)}\\eta_{\\tilde{\\varrho}_n(\\theta)}^2(P,P_0)\\geq 2^{-1\/(2s)}C_{17}n[\\varrho_n(\\theta)]^{1\/2s}\\eta_{\\varrho_n(\\theta)}^2(P,P_0),$$ and $\\forall\\ P\\in\\mathcal P(\\Delta_n(\\theta),\\theta)$, \n\\begin{align}\n\\eta_{\\varrho_n(\\theta)}^2(P,P_0)\\geq {1\\over 4}\\|u\\|_{L_2(P_0)}^2\\label{ineq2}\n\\end{align}\nprovided that $\\Delta_n(\\theta)\\geq C'(M)\\left({\\sqrt{\\log\\log n}\\over n}\\right)^{\\frac{4s}{4s+\\theta+1}}$.\n\nTherefore,\n$$\\inf\\limits_{P\\in\\mathcal P(\\Delta_n(\\theta),\\theta)}{\\mathbb E}_PT_{n,\\tilde{\\varrho}_n(\\theta)}\\geq C_{18}n[\\varrho_n(\\theta)]^{1\/(2s)}\\Delta_n(\\theta)\\geq C_{18}c_1\\sqrt{\\log\\log n}\\geq \\tilde{M}\\sqrt{\\log\\log n}$$\nif $c_1\\geq C_{18}^{-1}\\tilde{M}$. Hence to ensure (\\ref{mean}) holds, it suffices to take\n$$c_1=\\max\\{C'(M),C_{18}^{-1}\\tilde{M}\\}.$$\n\n\\begin{comment}\n\tWe shall write \n\t$$\\tilde{\\chi}_{\\varrho_n}^2(P,P_0):=\\tilde{\\chi}_2^2(P,P_0)=\\sum_{k\\geq 1}\\frac{\\lambda_k}{\\lambda_k+\\varrho_n^2}[{\\mathbb E} _P\\varphi_k(X)]^2$$\n\tto signify its dependence on $\\varrho_n$. It is clear that\n\t\\begin{align*}\n\t\\tilde{\\chi}_{\\tilde{\\varrho}_n(\\theta)}^2(P,P_0)\\geq\\tilde{\\chi}_{\\varrho_n(\\theta)}^2(P,P_0)\n\t\\end{align*}\n\t\n\tAs shown in Theorem \\ref{crm3d}, for any $\\varrho_n$,\n\t\\begin{align*}\n\t{\\mathbb E} _PT_{n,\\varrho_n}\\geq Cn\\varrho_n^{\\frac{1}{2s}}\\tilde{\\chi}_{\\varrho_n}^2(P,P_0)\n\t\\end{align*}\n\tand $\\forall$ $P\\in\\mathcal{P}(\\Delta_n(\\theta),\\theta,M)$,\n\t\\begin{align*}\n\t\\tilde{\\chi}_{\\varrho_n(\\theta)}^2(P,P_0)\\geq C\\|u\\|_{L_2(P_0)}^2\n\t\\end{align*}\n\twhere $u={{d}P}\/{{d}P_0}-1$. Therefore, by (\\ref{lub}),\n\t\\begin{align*}\n\t{\\mathbb E} _PT_{n,\\tilde{\\varrho}_n(\\theta)}\\geq &Cn[\\tilde{\\varrho}_n(\\theta)]^{\\frac{1}{2s}}\\tilde{\\chi}_{\\tilde{\\varrho}_n(\\theta)}^2(P,P_0)\\\\\\geq &Cn[\\varrho_n(\\theta)]^{\\frac{1}{2s}}\\tilde{\\chi}_{\\varrho_n(\\theta)}^2(P,P_0)\\\\\n\t\\geq& Cn[\\varrho_n(\\theta)]^{\\frac{1}{2s}}\\|u\\|_{L_2(P_0)}^2\\\\\n\t\\geq& Cn[\\varrho_n(\\theta)]^{\\frac{1}{2s}}\\Delta_n(\\theta),\n\t\\end{align*}\n\twhich immediately implies that (\\ref{mean}) holds for sufficiently large $n$.\n\\end{comment}\t\n\tWith (\\ref{lub}), (\\ref{ineq}) and (\\ref{ineq2}), the results in Theorem 3 imply that for sufficiently large $n$\n\t\\begin{align*}\n\t\\sup\\limits_{P\\in\\mathcal{P}(\\Delta_n^*(\\theta),\\theta,M)}\\frac{\\mathrm{Var}\\left(T_{n,\\tilde{\\varrho}_n(\\theta)}\\right)}{\\left({\\mathbb E} _PT_{n,\\tilde{\\varrho}_n(\\theta)}\\right)^2}\\leq& C_{19}\\Big\\{\\left([\\varrho_n(\\theta)]^{\\frac{1}{2s}}n\\Delta_n^*(\\theta)\\right)^{-2}+\\left([\\varrho_n(\\theta)]^{\\frac{1}{s}}n^2\\Delta_n^*(\\theta)\\right)^{-1}\\\\&+(n\\Delta_n^*(\\theta))^{-1}+\\left([\\varrho_n(\\theta)]^{\\frac{1}{2s}}n\\sqrt{\\Delta_n^*(\\theta)}\\right)^{-1}\\Big\\}\\\\\n\t\\leq&2C_{19}\\left([\\varrho_n(\\theta)]^{\\frac{1}{2s}}n\\Delta_n^*(\\theta)\\right)^{-1}= 2C_{19}(c_1\\log\\log n)^{-\\frac{1}{2}}\\rightarrow \\infty,\n\t\\end{align*}\n\twhich shows (\\ref{mv}).\n\\end{proof}\n\\vskip 25pt\n\n\\begin{proof}[Proof of Theorem \\ref{cra}]\nThe main idea of the proof is similar to that for Theorem \\ref{cr}. To this end, assume, without loss of generality, that\n$$\\Delta_n(\\theta)=c_2\\left(\\frac{n}{\\sqrt{\\log\\log n}}\\right)^{-\\frac{4s}{4s+\\theta+1}},\\qquad \\forall\\theta\\in[\\theta_1,\\theta_2],$$ \nwhere $c_2>0$ is a sufficiently small constant to be determined later.\n\n\t\n\tLet $r_n=\\lfloor C_{20}\\log n\\rfloor$ and $K_{n,1}=\\lfloor{C_{21}\\Delta_n^{-\\frac{\\theta_1+1}{2s}}(\\theta_1)}\\rfloor$for sufficiently small $C_{20},C_{21}>0$. Set $\\theta_{n,1}=\\theta_1$.  For $2\\leq r\\leq r_n$, let\n\t$$\n\tK_{n,r}=2^{r-2}K_{n,1}\n\t$$\nand $\\theta_{n,r}$ is selected such that the following equation holds.\n\t\\begin{align*}\n\tK_{n,r}=\\left\\lfloor C_{21}[\\Delta_n(\\theta_{n,r})]^{-\\frac{\\theta_{n,r}+1}{2s}}\\right\\rfloor.\n\t\\end{align*}\n\tNote that by choosing $C_{20}$ sufficiently small,\n\t\\begin{align*}\n\tK_{n,r_n}=2^{r_n-2}K_{n,1}\\leq &\\left\\lfloor c_2^{\\frac{2(\\theta_1+1)}{4s+\\theta+1}}\\left(\\frac{n}{\\sqrt{\\log\\log n}}\\right)^{\\frac{2(\\theta_1+1)}{4s+\\theta+1}}\\cdot 2^{r_n-2}\\right\\rfloor\\\\=&\\left\\lfloor c_2^{\\frac{2(\\theta_1+1)}{4s+\\theta+1}}\\exp\\left(\\log\\left(\\frac{n}{\\sqrt{\\log\\log n}}\\right)\\cdot\\frac{2(\\theta_1+1)}{4s+\\theta_1+1}+(r_n-2)\\log 2\\right)\\right\\rfloor\\\\\\leq &\\left\\lfloor C_{21}\\exp\\left(\\log\\left(\\frac{n}{\\sqrt{\\log\\log n}}\\right)\\cdot\\frac{2(\\theta_2+1)}{4s+\\theta_2+1}\\right)\\right\\rfloor=\\lfloor C_{21}[\\Delta_n(\\theta_{2})]^{-\\frac{\\theta_{2}+1}{2s}}\\rfloor\n\t\\end{align*}\n\tfor sufficiently large $n$. Thus, we can guarante that $\\forall$ $1\\leq r\\leq r_n$, $\\theta_{n,r_n}\\in[\\theta_1,\\theta_2]$.\n\t\nWe now construct a finite subset of $\\cup_{\\theta\\in [\\theta_1,\\theta_2]}\\mathcal{P}(\\Delta_n(\\theta),\\theta)$ as follows. For each $\\xi_{n,r}=(\\xi_{n,r,1},\\cdots,\\xi_{n,r,K_{n,r}})\\in\\{\\pm 1\\}^{K_{n,r}}$, let\n\\begin{align*}\nf_{n,r,\\xi_{n,r}}=1+\\sum\\limits_{k=K^*_{n,r-1}+1}^{K^*_{n,r}}a_{n,r}\\xi_{n,r,k}\\varphi_k,\n\\end{align*}\nwhere $K^*_{n,r}=K_{n,1}+\\cdots+K_{n,r}$, and $a_{n,r}=\\sqrt{\\Delta_n(\\theta_{n,r})\/K_{n,r}}$. Following the same argument as that in the proof of Theorem \\ref{cr}, we can verify that with a sufficiently small $C_{21}$, each $P_{n,r,\\xi_{n,r}}\\in \\mathcal{P}(\\Delta_n(\\theta_{n,r}),\\theta_{n,r})$, where $f_{n,r,\\xi_{n,r}}$ is the Radon-Nikodym derivative $dP_{n,r,\\xi_{n,r}}\/dP_0$. With slight abuse of notation, write\n\\begin{align*}\nf_n(X_1,X_2,\\cdots,X_n)={1\\over r_n}{\\sum\\limits_{r=1}^{r_n}f_{n,r}(X_1,X_2,\\cdots,X_n)},\n\\end{align*}\nwhere\n\\begin{align*}\nf_{n,r}(X_1,X_2,\\cdots,X_n)={1\\over 2^{K_{n,r}}}\\sum_{\\xi_{n,r}\\in \\{\\pm 1\\}^{K_{n,r}}}\\prod\\limits_{i=1}^n f_{n,r,\\xi_{n,r}}(X_i).\n\\end{align*}\nIt now suffices to show that\n$$\n\\|f_n\\|_{L_2(P_0)}:={\\mathbb E}_{P_0}f_n^2(X_1,X_2,\\cdots,X_n)\\to 1,\\qquad {\\rm as\\ }n\\to\\infty.\n$$\n\t\n\tNote that\n\t\\begin{align*}\n\\|f_n\\|^2_{L_2(P_0)}=&\\frac{1}{r_n^2}\\sum\\limits_{1\\leq r,r'\\leq r_n}\\langle f_{n,r},f_{n,r'}\\rangle _{L_2(P_0)}\\\\\n\t=&\\frac{1}{r_n^2}\\sum\\limits_{1\\leq r\\leq r_n}\\|f_{n,r}\\|^2_{L_2(P_0)}+\\frac{1}{r_n^2}\\sum\\limits_{\\substack{1\\leq r,r'\\leq r_n\\\\r\\neq r'}}\\langle f_{n,r},f_{n,r'}\\rangle_{L_2(P_0)}.\n\t\\end{align*}\nIt is easy to verify that, for any $r\\neq r'$,\n\t\\begin{align*}\n\t\\langle f_{n,r},f_{n,r'}\\rangle_{L_2(P_0)}=1.\n\t\\end{align*}\nIt therefore suffices to show that\n\\begin{align*}\n\\sum\\limits_{1\\leq r\\leq r_n}\\|f_{n,r}\\|^2_{L_2(P_0)}=o(r_n).\n\t\\end{align*} \n\t\nFollowing the same derivation as that in the proof of Theorem \\ref{cr}, we can show that\n\t\\begin{align*}\n\t\\|f_{n,r}\\|_{L_2(P_0)}^2\\leq&\\left(\\frac{\\exp(na_{n,r}^2)+\\exp(-na_{n,r}^2)}{2}\\right)^{K_{n,r}}\\leq\\exp(K_{n,r}n^2a_{n,r}^4)\n\t\\end{align*}\n\tfor sufficiently large $n$. By setting $c_2$ in the expression of $\\Delta_n(\\theta)$ sufficiently small, we have\n\t\\begin{align*}\n\tK_{n,r}n^2a_{n,r}^4\\leq \\frac{1}{2}\\log r_n\n\t\\end{align*} \n    which ensures that\n\t\\begin{align*}\n\t\\|f_{n,r}\\|_{L_2(P_0)}^2\\leq r_n^{1\/2}=o(r_n).\n\t\\end{align*}\n\\end{proof}\n\n\\bibliographystyle{myplainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDuring the past 20 years a number of methods has been devised for\nstate selective preparation and manipulation of discrete-level\nquantum systems\n\\cite{paramonov1983,chelkowski1990,kaluza1993,bergmann1998,rabitz2003}.\nHowever, simple population oscillations, induced by a resonant\ndriving pulse have received negligible attention as a prospective\npopulation manipulation method. This might be attributed to two\nreasons. The first is that Rabi theory is based on the rotating\nwave approximation (RWA), and all attempts to generalize it\nwithout RWA (e.g. \\cite{shariar2002.1,barata2000,fujii2003}) are\nmathematically very involved. The second is that no attempt has\nbeen made to analytically generalize the original Rabi theory\nbeyond the two-level systems.\n\nIn this paper an analytic extension of Rabi theory to transitions\nin many-level systems is presented. The aim is to 'design' a\ndriving pulse of the form:\n\\begin{equation}\n  F(t)= F_{\\rm 0} \\; m(t) \\; \\cos{(\\omega(t) \\; t)}\n\\label{pulse}\n\\end{equation}\nby establishing analytical optimization relations between its\nparameters: maximum pulse amplitude $F_{0}$, pulse envelope shape\n$m(t)$, and time dependent carrier frequency $\\omega(t)$. The goal\nof this enterprize is twofold: the first is to achieve as complete\nas possible transfer of population between two selected states of\nthe system; the second is to make this transfer as rapid as\npossible. These two requirements, however, are conflicting:\npopulation transfer can be accelerated by using a more intense\ndrive, but at the same time a stronger drive increases involvement\nof remaining system levels in population dynamics hence\ndeteriorating population transfer between a selected pair of\nlevels.\n\nIn the previous paper on this topic \\cite{bonacci2003.2} it was\nshown how, for a pulse of arbitrary shape and duration, the drive\nfrequency can be analytically optimized to maximize the amplitude\nof the population oscillations between the selected two levels in\na general many level quantum system. It was shown how the standard\nRabi theory can be extended beyond the simple two-level systems.\nNow, in order to achieve the quickest and as complete as possible\npopulation transfer between two pre-selected levels, driving pulse\nshould be tailored so that it produces only a single\nhalf-oscillation of the population. In this paper, this second\n(and final) step towards the controlled population transfer using\nmodified (i.e. many level system) Rabi oscillations is discussed.\nThe results presented herein can be regarded as an extension of\nthe standard $\\pi$-pulse theory (see e.g. \\cite{holthaus1994}) -\nalso strictly valid only in the two level systems - to the\ncoherently driven population oscillations in general many level\nsystems.\n\n\\section{Theoretical analysis}\n\nAll the calculations in this section are done in a system of units\nin which $\\hbar=1$.\n\n\\subsection{Calculation setup}\nA quantum system with N discrete stationary levels with energies\n$E_i \\ (i=1,...,N)$ is considered. The system is driven by a time\ndependent perturbation given in Eq. (\\ref{pulse}). In the\ninteraction picture, the dynamics of the system obeys the\nSchroedinger equation:\n\\begin{equation}\n  \\frac{d}{dt}\\mathbf{a}(t)=-i \\oper{V}(t) \\mathbf{a}(t) ,\n\\label{schrodinger}\n\\end{equation}\nwhere $\\mathbf{a}(t)$ is a vector of time-dependent expansion\ncoefficients $a_1(t),..., a_N(t)$. The N$\\times$N Matrix\n$\\oper{V}(t)$ describes interaction between the system and\nperturbation. Explicitly, its elements are given by:\n\\begin{equation}\n  V_{ij}(t) \\equiv \\frac{F_0 \\mu_{ij}}{2} m(t)(e^{i s_{ij}\n  (\\omega(t)-\\omega_{ij}) t}+e^{-i s_{ij} (\\omega(t)+\\omega_{ij}) \\; t}).\n\\end{equation}\n$\\mu_{ij}$ is transition moment between the i-th and the j-th\nlevels induced by the perturbation. $s_{ij} \\equiv sign(E_i-E_j)$\nand $\\omega_{ij} \\equiv |E_i-E_j|$ are respectively the sign and\nthe magnitude of the resonant frequency for the transition between\nthe i-th and the j-th level.\n\nThe aim is to induce population transfer between two arbitrarily\nselected levels, designated by $\\alpha$ and $\\beta$, directly\ncoupled by the perturbation (i.e. such that $\\mu_{\\alpha \\beta}\\ne\n0$). To simplify equations, the time variable t is re-scaled to\n$\\tau$, with transformation between the two given by:\n\\begin{equation}\n  d\\tau \\equiv \\frac{F_0 \\mu_{\\alpha \\beta}}{2} m(t) dt .\n\\label{definition tau}\n\\end{equation}\nThen with following substitutions:\n\\begin{eqnarray}\n  f_{ij}(\\tau)&\\equiv&s_{ij} \\frac{2}{F_0 \\mu_{\\alpha \\beta}} (\\omega(t)-\\omega_{ij}) \\label{definition f} \\\\\n  g_{ij}(\\tau)&\\equiv&s_{ij} \\frac{2}{F_0 \\mu_{\\alpha \\beta}} (\\omega(t)+\\omega_{ij}) \\label{definition g} \\\\\n  x(\\tau)&\\equiv& \\frac{F_0 \\mu_{\\alpha \\beta}}{2} t(\\tau) \\label{definition x} \\\\\n  R_{ij} &\\equiv& \\frac{\\mu_{ij}} {\\mu_{\\alpha \\beta}} \\label{definition R}\n\\end{eqnarray}\nEq. (\\ref{schrodinger}) transforms into:\n\\begin{equation}\n  \\frac{d}{d \\tau}\\mathbf{a}(\\tau)=-i \\oper{W}(\\tau)\\mathbf{a}(\\tau),\n\\label{n level}\n\\end{equation}\nwhere:\n\\begin{equation}\n  W_{ij}(\\tau) \\equiv R_{ij}(e^{i f_{ij}(\\tau) x(\\tau)}+e^{-i g_{ij}(\\tau)x(\\tau)}).\n\\label{wovi}\n\\end{equation}\n\nInitial conditions for the problem of selective population\ntransfer comprise complete population initially (at $t=\\tau=0$)\ncontained in only one of the selected levels, either $\\alpha$ or\n$\\beta$. The other selected level, as well as all the remaining\nN-2 'perturbing' levels of the system are unpopulated at this\ntime.\n\nPopulation evolution $\\Pi_i(t)$ of the i-th level is determined\nfrom $\\Pi_i(t)=|a_i(t)|^2$.\n\\subsection{Rabi-like population transfer in a three level system}\n\nIt was demonstrated in the previous paper on this topic\n\\cite{bonacci2003.2} that the analytical extension of the Rabi\noscillations theory beyond two-level systems is anchored in the\nanalysis of the simplest of the 'many-level' systems - a three\nlevel one. Hence, in this section the impact of the single\nadditional level on the population transfer period is discussed:\nbeyond the 'selected' levels $\\alpha$ and $\\beta$, the system now\ncontains one additional 'perturbing' level, designated with index\n\\textit{p}. The only requirements on the system internal structure\nare that $\\mu_{\\alpha \\beta}, \\mu_{\\beta p} \\ne 0$ and\n$\\mu_{\\alpha p}=0$. While the first two requirements are\nnecessary, the last one does not reduce the generality of the\nfinal results to any significant extent and is introduced for\ncalculational convenience exclusively.\n\nFor the observed three-level system, the dynamical equation\n(\\ref{n level}) reduces to:\n\n\\begin{eqnarray}\n\\label{3 level}\n  \\frac{d}{d \\tau}\n  \\begin{bmatrix}\n  a_\\alpha(\\tau) \\\\\n  a_\\beta(\\tau) \\\\\n  a_p(\\tau) \\\\\n  \\end{bmatrix}\n  &=& \\\\\n  -&i&\n  \\begin{bmatrix}\n  0 & e^{i f_{\\alpha \\beta}(\\tau) x(\\tau)} + e^{-i g_{\\alpha \\beta}(\\tau) x(\\tau)} & 0 \\\\\n  e^{-i f_{\\alpha \\beta}(\\tau) x(\\tau)} + e^{i g_{\\alpha \\beta}(\\tau) x(\\tau)} &  0\n  & R_{\\beta p}(e^{-i f_{\\beta p}(\\tau) x(\\tau)} + e^{i g_{\\beta p}(\\tau) x(\\tau)}) \\\\\n  0 & R_{\\beta p}(e^{i f_{\\beta p}(\\tau) x(\\tau)} + e^{-i g_{\\beta p}(\\tau) x(\\tau)}) & 0 \\\\\n  \\end{bmatrix}\n  \\begin{bmatrix}\n  a_\\alpha(\\tau) \\\\\n  a_\\beta (\\tau) \\\\\n  a_p(\\tau) \\\\\n  \\end{bmatrix} \\nonumber\n\\end{eqnarray}\n\n\\subsubsection{Recapitulation: minimizing the impact of the perturbing level}\n\nAs it was shown in \\cite{bonacci2003.2}, Eq. (\\ref{3 level}), when\nthe driving frequency is near the resonant value for the\ntransition $\\alpha \\leftrightarrow \\beta$, the following\nexpression can be obtained for the population dynamics of level\n$\\textit{p}$:\n\n\\begin{equation}\n\\label{apert}\n  a_p(\\tau) \\approx -  a_{\\beta} (\\tau)\\Big( \\sigma_{\\beta p}\n  \\frac{ m(t(\\tau)) }{1-\\Delta_{\\beta p}(\\tau)} \\Big) \\; e^{ i f_{\\beta\n  p}(\\tau) x(\\tau)}\n\\end{equation}\n\nwhere\n\n\\begin{eqnarray}\n\\label{delta i Delta}\n \\sigma_{\\beta p} \\equiv \\frac{ F_0 \\mu_{\\beta p}}{2(\\omega_{\\alpha \\beta} - \\omega_{\\beta p})} \\, \\nonumber \\\\\n \\Delta_{\\beta p}(\\tau) \\equiv \\frac {\\omega(\\tau)- \\omega_{\\alpha \\beta}} {\\omega_{\\beta p} - \\omega_{\\alpha \\beta}} \\ .\n\\end{eqnarray}\n\nPut in words, with the conditions mentioned, the dynamics of the\nlevel \\textit{p} parametrically depends on the dynamics of the\nlevel $\\beta$ to which it is coupled. The relation between the\namplitudes of the population oscillations for levels \\textit{p}\nand $\\beta$ follows directly from the above expression:\n\n\\begin{eqnarray}\n\\label{beta and p}\n  \\Pi_{p} &\\approx& \\epsilon_{\\beta p}(\\tau) \\Pi_{\\beta}(\\tau)\n\\end{eqnarray}\n\nwhere:\n\n\\begin{equation}\n\\epsilon_{\\beta p}(\\tau) \\equiv \\Big(\\sigma_{\\beta p}\n  \\frac{m(t(\\tau))}{1-\\Delta_{\\beta p}(\\tau)}\\Big) ^2 \\ .\n\\end{equation}\n\n\nNote that, as $\\Delta_{\\beta p}(\\tau)$ is generally very small and\n$|m(\\tau)|\\leq1$, that parameter $\\sigma_{\\beta p}$ actually\ndetermines the effective strength of applied perturbation: if\n$\\sigma_{\\beta p}^2<<1$, then dynamical impact of level p is\nnegligible and perturbation may be considered weak; if\n$\\sigma_{\\beta p}^2\\sim 1$, perturbation is very strong.\n\nFurther, the requirement of the minimization of the dynamical\nimpact of the perturbing level $\\textit{p}$ on the transition\n$\\alpha \\leftrightarrow \\beta$ leads to the following equation for\nthe optimized dynamics of the ($\\alpha$,$\\beta$) subsystem:\n\n\\begin{equation}\n  \\label{beta dynamics}\n  \\frac {d }{d \\tau}\n  \\begin{bmatrix}\n  b_\\alpha(\\tau) \\cr\n  b_\\beta(\\tau)\n  \\end{bmatrix}\n  = - i\n  \\begin{bmatrix}\n  0 & 1 \\cr\n  1 & 0 \\cr\n  \\end{bmatrix}\n  \\begin{bmatrix}\n  b_\\alpha(\\tau) \\cr\n  b_\\beta(\\tau)\n  \\end{bmatrix}\n\\end{equation}\n\nwhere the two-level state vector $\\big(\nb_\\alpha(\\tau),b_\\beta(\\tau)\\big)$ is merely the unitary\ntransformed vector of the subsystem ($\\alpha$,$\\beta$):\n\n\\begin{equation}\n\\label{transformation}\n  \\begin{bmatrix}\n  b_\\alpha(\\tau) \\\\\n  b_\\beta(\\tau) \\\\\n  \\end{bmatrix}\n  = e^{-i \\oper{\\Lambda} (\\tau)}\n  \\begin{bmatrix}\n  a_\\alpha(\\tau) \\\\\n  a_\\beta(\\tau) \\\\\n  \\end{bmatrix}\n\\end{equation}\n\nNote that the precise form of the real transformation matrix\n$\\hat{\\mathbf{\\Lambda}}(\\tau)$ is irrelevant here as it has no\nimpact on the population dynamics.\n\nThe optimization procedure produces the analytic expression for\nthe chirp of the driving frequency, which in the lowest order of\napproximation (suitable for all but the most intensive\nperturbations) amounts:\n\n\\begin{equation}\n\\label{chirp}\n  \\omega(t)=\n  \\omega_{\\alpha \\beta} +\n  (s_{\\beta \\alpha} s_{\\beta p})( \\omega_{\\beta p}- \\omega_{\\alpha\n  \\beta})\\frac{2 \\omega_{\\beta p} }{ \\omega_{\\alpha \\beta} +\\omega_{\\beta p}}\n  \\sigma_{\\beta p}^2 \\frac{1}{t} \\int_0^t \\big( m(t') \\big) ^2 dt'\n\\end{equation}\n\nand from the Eq (\\ref{beta dynamics}) it is found that the time\n$\\Theta$ required for a single population transfer between levels\n$\\alpha$ and $\\beta$, determined from the fundamental relation of\nthe $\\pi$-pulse theory:\n\n\\begin{equation}\n\\label{pi pulse theory}\n  \\int_0^\\Theta d\\tau = \\frac{\\pi}{2}\n\\end{equation}\n\nequals (in units of $\\tau$):\n\\begin{equation}\n\\label{period pi}\n  \\Theta=\\frac{\\pi}{2}\n\\end{equation}\n\nAs was shown in \\cite{bonacci2003.2}, the 'exact' numerical\nsolution to the Eq. (\\ref{3 level}) indeed maximizes the\npopulation oscillations in the $\\alpha-\\beta$ subsystem. However,\nas will be discussed below, the predicted value for the period of\nthe population oscillations (Eq. (\\ref{period pi})) is somewhat\nsmaller than the correct one, with the discrepancy increasing with\nthe increasing population leak to the level \\textit{p}. In the\nfollowing section this issue is resolved and the corrected\nanalytical expression for determination of the population transfer\n(or oscillation) period is obtained.\n\n\\subsubsection{Patching the population conservation of the total system}\n\nTo start the following discussion, notice that the optimized\nsolution for the population transfer between levels $\\alpha$ and\n$\\beta$ (described by the Eq. (\\ref{beta dynamics})) is\nunfortunately too good to be true. Namely, its serious drawback\nlays in the fact that the leak of the population from the\n($\\alpha$,$\\beta$) subsystem into the perturbing level \\textit{p}\ngoes by completely unnoticed!\n\nFormally, the root of the problem hides in the fact that the\nmathematical trick which enabled decoupling of the level\n\\textit{p} dynamics from the rest of the system (i.e. the step\nbetween Eq.(18) and Eq.(19) in \\cite{bonacci2003.2}) destroys the\nunitarity of the full dynamical equation for the three level\nsystem, Eq.(\\ref{3 level}). The consequence is that the dynamical\nequation for the $\\alpha-\\beta$ subsystem, Eq.(\\ref{beta\ndynamics}) itself claims to be unitary, keeping the population of\nthat subsystem fully conserved. This is clearly impossible, as\nlevel \\textit{p} does indeed capture some population - the exact\namount given by Eq.(\\ref{beta and p}).\n\nThis malfunction caused by the decoupling procedure unfortunately\ncannot be remedied within the decoupling procedure itself - the\npatch has to be provided by an independent approach. To do this,\nthe following argument is used: since it is the equation for the\ndynamics of level $\\beta$ that changes due to the decoupling\nprocedure and consequently causes the breakdown of the population\nconservation, it is only the dynamical equation for the level\n$\\beta$ that has to be modified; then, as the dynamics of the\nlevel $\\beta$ is governed by the elements in the lower row of the\ndynamical matrix in Eq.(\\ref{beta dynamics}), a particular ansatz\nintervention precisely into these elements might help rectify the\noverall population dynamics of the whole three level system.\nHence, the correction is sought in the following form:\n\n\\begin{equation}\n  \\label{two level corrected}\n  \\frac{d}{d\\tau}\n  \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix}\n  = - i\n  \\begin{bmatrix}\n  0 & 1 \\\\\n  \\zeta(\\tau) & i \\xi(\\tau) \\\\\n  \\end{bmatrix}\n    \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix}\n\\end{equation}\n\nwhere $\\zeta[\\tau]$ and $\\xi[\\tau]$ are real non-negative\nfunctions. Such an ansatz does not interfere with the\nphase-fitting effect of the driving frequency optimization and Eq.\n(\\ref{chirp}) - forged by the decoupling procedure - which\nmaximizes the population oscillations in the $\\alpha-\\beta$\nsubsystem, is left unharmed. Instead, it merely enables\nphase-independent modification of the \\textbf{amplitudes} of\n$\\alpha$ and $\\beta$ populations.\n\nExpressing requirement of population conservation in the total\nthree-level system as:\n\n\\begin{equation}\nd|b_{\\alpha}(\\tau)|^2+d|b_{\\beta}(\\tau)|^2+d|b_{p}(\\tau)|^2=0 \\ ,\n\\end{equation}\n\nsplitting the phase and amplitude contributions of the three wave\nfunction projections on the three stationary states:\n\n\\begin{eqnarray}\n b_{\\alpha}(\\tau) &\\equiv& B_{\\alpha}(\\tau) e^{i \\phi_{\\alpha}(\\tau)}\\  , \\nonumber \\\\\n b_{\\beta}(\\tau) &\\equiv& B_{\\beta}(\\tau) e^{i \\phi_{\\beta}(\\tau)}\\  ,  \\\\\n b_{p}(\\tau) &\\equiv& B_{p}(\\tau) e^{i \\phi_{p}(\\tau)} \\  \\nonumber ,\n\\end{eqnarray}\n\nand using the known relation between the populations of levels\n$\\beta$ and \\textit{p}, Eq.(\\ref{beta and p}), the following\nresult is obtained:\n\n\\begin{equation}\n\\label{cons condition}\n B_{\\alpha}(\\tau)\\; dB_{\\alpha} + (1+\\epsilon_{\\beta p}(\\tau))\n B_{\\beta}(\\tau)\\; dB_{\\beta} + \\frac{1}{2} B_{\\beta}(\\tau)^2 \\; d\\epsilon_{\\beta p} =\n 0\\ .\n\\end{equation}\n\nNow the corrected equation for the $\\alpha-\\beta$ subsystem,\nEq.(\\ref{two level corrected}), can be used to eliminate\n$dB_{\\alpha}$ and $dB_{\\beta}$ and introduce $\\zeta[\\tau]$ and\n$\\xi[\\tau]$ in their stead:\n\n\\begin{eqnarray}\n\\label{eqn condition}\n  dB_{\\alpha}&=& -i B_{\\beta} \\; Im \\big(\n  e^{i(\\phi_{\\beta}(\\tau)-\\phi_{\\alpha}(\\tau))}\\big)\n  \\; d{\\tau} \\ , \\nonumber \\\\\n  dB_{\\beta}&=& -i \\Big(\\zeta(\\tau) \\;B_{\\alpha}(\\tau) \\; Im \\big( e^{-i(\\phi_{\\beta}(\\tau)-\\phi_{\\alpha}(\\tau))}\\big) + \\xi(\\tau) \\; B_{\\beta}(\\tau) Im \\big(e^{-i\\phi_{\\beta}(\\tau)}\\big) \\Big) \\;\n  d{\\tau} \\ .\n\\end{eqnarray}\n\nFinally, taking together Eq.(\\ref{cons condition}) and\nEq.(\\ref{eqn condition}) it is found that:\n\n\\begin{equation}\n\\label{final condition}\n B_{\\alpha}(\\tau) \\Big(\\zeta(\\tau)-\\frac{1}{1+\\epsilon_{\\beta p}(\\tau)}\\Big)\n \\ sin \\big( \\phi_{\\beta}(\\tau)-\\phi_{\\alpha}(\\tau) \\big)\n+ B_{\\beta}(\\tau) \\Big( \\xi(\\tau) + \\frac{d }{d\\tau}\\ln\n\\big(1+\\epsilon_{\\beta p}(\\tau)\\big)^{\\frac{1}{2}} \\Big) sin \\big(\n\\phi_{\\beta}(\\tau) \\big)=0 .\n\\end{equation}\n\nSince for population oscillations $B_{\\beta}$ and $B_{\\alpha}$ are\n$180^o$ out of phase, this condition can be satisfied only if:\n\n\\begin{eqnarray}\n\\label{zeta xi}\n \\zeta(\\tau) &=& \\frac{1}{1+\\epsilon_{\\beta p}(\\tau)} \\nonumber \\ , \\\\\n \\xi(\\tau) &=& - \\frac{d }{d\\tau}\\ln\n\\big(1+\\epsilon_{\\beta p}(\\tau)\\big)^{\\frac{1}{2}} \\ .\n\\end{eqnarray}\n\n\\subsubsection{Population oscillation period modified}\n\nHence, the correct dynamical equation for the $\\alpha-\\beta$\nsubsystem which both maximizes the population oscillation\namplitudes of these two levels as well as properly conserves the\noverall population of the three-level ($\\alpha-\\beta-p$) system\nis:\n\n\\begin{equation}\n  \\label{two level final}\n  \\frac{d}{d\\tau}\n  \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix}\n  = - i\n  \\begin{bmatrix}\n  0 & 1 \\\\\n  \\frac{1}{1+\\epsilon_{\\beta p}(\\tau)} & - i \\frac{d }{d\\tau}\\ln\n\\big(1+\\epsilon_{\\beta p}(\\tau)\\big)^{\\frac{1}{2}} \\\\\n  \\end{bmatrix}\n    \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix} \\ .\n\\end{equation}\n\nA simple extension of this result to the general many-level system\n(in which level $\\alpha$ also has some perturbing levels - jointly\ndesignated by q - attached to it) yields the total corrected\ndynamical equation for such a system:\n\n\\begin{equation}\n  \\label{two level final}\n  \\frac{d}{d\\tau}\n  \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix}\n  = - i\n  \\begin{bmatrix}\n  - i \\frac{d }{d\\tau}\\ln \\big(1+\\epsilon_{\\alpha q}(\\tau)\\big)^{\\frac{1}{2}} & \\frac{1}{1+ \\epsilon_{\\alpha q}(\\tau)} \\\\\n  \\frac{1}{1+\\epsilon_{\\beta p}(\\tau)} & - i \\frac{d }{d\\tau}\\ln\n\\big(1+\\epsilon_{\\beta p}(\\tau)\\big)^{\\frac{1}{2}} \\\\\n  \\end{bmatrix}\n    \\begin{bmatrix}\n  b_{\\alpha}(\\tau) \\\\\n  b_{\\beta}(\\tau) \\\\\n  \\end{bmatrix} \\ .\n\\end{equation}\n\nNow to finalize the calculation of the corrected population\ntransfer period the following procedure is administered. First,\nthe time variable is transformed $\\tau \\rightarrow \\varphi$\naccording to:\n\n\\begin{equation}\n \\label{var transform}\n d\\tau = \\kappa(\\varphi) d\\varphi\n\\end{equation}\n\nThe goal of this variable transformation is to produce, in the new\ntime variable $\\varphi$, the closed dynamical equations for\n$b_\\alpha(\\varphi)$ and $b_\\beta(\\varphi)$ describing the dynamics\nwhich is as close as possible to the simple harmonic oscillation.\nSecond, and to that end, the transformation Eq.(\\ref{var\ntransform}) is introduced into Eq.(\\ref{two level final}), the\nresulting relation is differentiated with respect to $\\varphi$ and\nall but the lowest order terms in the small parameters\n$\\epsilon_{\\alpha q}(\\tau)$ and $\\epsilon_{\\beta p}(\\tau)$ are\nkept. Hence the following result is established:\n\n\\begin{eqnarray}\n  \\label{many level time correction}\n   \\frac{d^2 b_{\\alpha}(\\varphi)}{d\\varphi^2} +\n  \\frac{1}{2} \\frac{d}{d\\varphi}\\Big(\\ln \\frac{\\big(1+\\epsilon_{\\alpha q}(\\varphi)\\big)^3\n  \\big(1+\\epsilon_{\\beta p}(\\varphi)\\big)}{\\kappa^2} \\Big)\\;\\frac{d b_{\\alpha}(\\varphi)}{d\\varphi}\n  +\\frac{\\kappa^2}{\\big(1+\\epsilon_{\\alpha q}(\\varphi)\\big)\n  \\big(1+\\epsilon_{\\beta p}(\\varphi)\\big)}b_{\\alpha}(\\varphi)= 0\n  \\nonumber\n  \\\\\n  \\frac{d^2 b_{\\beta}(\\varphi)}{d\\varphi^2} +\n  \\frac{1}{2} \\frac{d}{d\\varphi}\\Big(\\ln \\frac{\\big(1+\\epsilon_{\\alpha q}(\\varphi)\\big)\n  \\big(1+\\epsilon_{\\beta p}(\\varphi)\\big)^3}{\\kappa^2} \\Big)\\;\\frac{d b_{\\beta}(\\varphi)}{d\\varphi}\n  +\\frac{\\kappa^2}{  \\big(1+\\epsilon_{\\alpha q}(\\varphi)\\big)\n  \\big(1+\\epsilon_{\\beta p}(\\varphi)\\big)}b_{\\beta}(\\varphi)= 0\n\\end{eqnarray}\n\nIn the third and final step, the appropriate value of the free\nparameter $\\kappa(\\varphi)$ is selected:\n\n\\begin{equation}\n  \\kappa(\\varphi)^2 \\equiv \\big(1+\\epsilon_{\\alpha\n  q}(\\varphi)\\big)\n  \\big(1+\\epsilon_{\\beta p}(\\varphi)\\big)\n\\end{equation}\n\nwhich transforms Eq.(\\ref{many level time correction}) into:\n\n\\begin{eqnarray}\n\\label{approx equations}\n  \\frac{d^2 b_{\\alpha}(\\varphi)}{d\\varphi^2}+ \\frac{d}{d\\varphi}(\\ln(1+\\epsilon_{\\alpha q}(\\varphi)))\\frac{db_{\\alpha}(\\varphi)}{d\\varphi}+\n  b_{\\alpha}(\\varphi)=0 \\nonumber \\\\\n  \\frac{d^2 b_{\\beta}(\\varphi)}{d\\varphi^2}+ \\frac{d}{d\\varphi}(\\ln(1+\\epsilon_{\\beta p}(\\varphi)))\\frac{db_{\\beta}(\\varphi)}{d\\varphi}+\n  b_{\\beta}(\\varphi)=0\n\\end{eqnarray}\n\nBoth these equations are similar to the damped oscillator\nequation. For negligible damping ($\\epsilon_{\\alpha q}(\\varphi),\n\\epsilon_{\\beta p}(\\varphi)<<1$), they reduce to the harmonic\noscillator equations, in which case the population transfer time\n of $\\pi\/2$ is obtained in the variable $\\varphi$. In the the\noriginal time coordinate, $\\tau$, the corrected time $\\Theta$\nrequired for a single population transfer between the levels\n$\\alpha$ and $\\beta$ is then obtained from:\n\n\\begin{equation}\n\\label{transfer time corrected}\n \\int_0^\\Theta \\frac{d\\tau}{\\sqrt{(1+\\epsilon_{\\alpha q}(\\tau))(1+\\epsilon_{\\beta p}(\\tau))}} =\n \\frac{\\pi}{2}\n\\end{equation}\n\nNote the difference between this result, and the result in Eq.\n(\\ref{pi pulse theory}) obtained from the standard $\\pi$-pulse\ntheory: in the lowest order approximation, the corrected\npopulation transfer time is shorter then the one obtained from Eq.\n(\\ref{pi pulse theory}) by an order of\n$\\frac{1}{2}(\\epsilon_{\\alpha q}(\\tau)+\\epsilon_{\\beta p}(\\tau))$.\n\nOn the other hand, taking into consideration the damping factor in\nthe Eq.(\\ref{approx equations}), approximating:\n\n\\begin{equation}\n\\label{corrected period}\n  \\frac{d}{d\\varphi}\\ln(1+\\epsilon_{\\alpha q, \\beta\n  p}(\\varphi))\\approx \\frac{d}{d\\varphi}\\epsilon_{\\alpha q, \\beta\n  p}(\\varphi)\n\\end{equation}\n\nand using the damped oscillator theory \\cite{Kent1996}(p.246), an\nincrease in the damping (expressed as an increase in\n$\\epsilon_{\\alpha q}(\\tau)$ and $\\epsilon_{\\beta p}(\\tau)$) leads\nto the an increase in the population time transfer by an order of\n$\\frac{1}{2}\\frac{d}{d\\tau}\\epsilon_{\\alpha q, \\beta p}^2(\\tau)$.\nAs this correction is an order of magnitude smaller than the\ncorrection obtained from Eq.(\\ref{transfer time corrected}), it\ncan be neglected for all practical purposes. Furthermore, the\nadditional corrections to the transfer period due to the neglected\nhigher order elements in the Eq.(\\ref{many level time correction})\nare of the same order of magnitude as this correction due to the\nfirst derivative component, and as they are impossible to obtain\nanalytically, the value of this whole second order correction for\nthe case of strong perturbations is somewhat shaky. However, this\nis not an issue, as the whole optimization theory developed in\n\\cite{bonacci2003.2} - and on whose applicability the results of\nthe above analysis hinge - assumes rather modest perturbations,\nand is not even expected to work properly for the extreme values.\n\n\\section{Numerical simulations}\n\n\\begin{figure}\n  \\includegraphics[width=8cm]{BonacciFig1.eps}\\\\\n\\caption{In all of the numerical examples the same pulse form was\nused - $m(t)= \\sin(\\Omega t)^2$, but with different values of\nmaximum intensity parameter ($F_0$) and different total pulse\nduration, $T$. The respective values are quoted in each particular\nexample.}\n  \\label{fig1}\n\\end{figure}\n\nIn this section, numerical simulations of system dynamics for\nunoptimized and fully optimized driving pulse of the form of Eq.\n(\\ref{pulse}) are presented and compared. Here, unoptimized\ndriving pulse is the one with driving frequency equal to the pure\nresonant frequency between the two levels selected for population\ntransfer ($\\omega(t)=\\omega_{\\alpha \\beta}$) and with pulse\nduration $T$ determined according to the standard $\\pi$-pulse\ntheory relation, Eq.(\\ref{pi pulse theory}). On the other hand,\nthe parameters of the fully optimized pulse are determined from\nEq.(\\ref{chirp}) and Eq.(\\ref{transfer time corrected}).\n\nIn all cases, a three-level system is considered, with the\nfollowing system parameters ($a.u.\\equiv atomic \\;units$):\n$\\omega_{\\beta \\alpha}=0.017671 \\;a.u.$, $s_{\\beta \\alpha}=1$,\n$\\mu_{\\beta \\alpha}=0.073 \\;a.u.$; $\\omega_{\\beta p}=0.017611\n\\;a.u.$, $s_{\\beta p}=-1$, $\\mu_{\\beta p}=0.098\\; a.u.$. These\nsystem parameters correspond to the three ro-vibrational levels of\nthe HF molecule in the ground electronic state: $\\alpha \\equiv\n(v=0,j=2,m=0)$, $\\beta \\equiv (v=1,j=1,m=0)$, $p \\equiv\n(v=2,j=2,m=0)$. The pulse shape in all of the examples is $m(t)=\n\\sin(\\Omega t)^2$ as shown in Fig 1.\n\n\\subsection{Population oscillations}\n\n\\begin{figure}\n  \\includegraphics[width=12cm]{BonacciFig2.eps}\\\\\n\\caption{Significance of the total analytical correction (in\ndriving frequency and total pulse duration) for the dynamics of a\nmildly disturbed system. In all plots, major oscillations are the\npopulations of the two targeted levels ($\\alpha$ and $\\beta$)\nwhereas the minor oscillations are the population of the\nperturbing level ($p$). For the value of perturbation strength\nparameter $\\sigma_{\\beta p}^2=0.05$, the loss of the final\nunoptimized population transfer amplitude amounts about 2\\%.\nOptimization reduces this loss to below 0.05\\%. The difference\nbetween the pulse duration obtained from standard $\\pi$-pulse\ntheory and the optimized value is 1.6\\%. Right hand-side plots\npresent the details from the left hand-side plots.}\n  \\label{fig2}\n\\end{figure}\n\nAs was demonstrated in \\cite{bonacci2003.2}, frequency\noptimization minimizes the impact of the perturbing levels on the\n\\textit{amplitude} of the population oscillations. In this\nsubsection, the necessity of the inclusion of additional\ncorrection Eq.(\\ref{transfer time corrected}) for the\n\\textit{population transfer time} - alongside the correction for\nthe driving frequency - will be demonstrated. Also, the validity\nand the limitations of the analytically obtained expression for\nthis correction will be discussed.\n\n\\subsubsection{Legitimate perturbation}\nFig.2 presents the dynamics of the system subjected to the\nexternal drive of limiting intensity, $\\sigma_{\\beta p}^2=0.05$,\ncorresponding to the $F_0=2.80534 \\ast 10^{-4} a.u.$. It is just\nstrong enough to noticeably (albeit not significantly) distort the\npure resonant oscillations, but at the same time weak enough so\nthat the theory developed in \\cite{bonacci2003.2} and further in\nthis paper provides the full and precise quantitative corrections.\n\nPulse duration determined according to the standard $\\pi$-pulse\ntheory expression, Eq.(\\ref{pi pulse theory}) is $T_{\\pi}=3077832\na.u.$, whereas the optimized one, obtained from Eq.(\\ref{transfer\ntime corrected}) is $T_{opt}=3126029 a.u.$. The pulse is aimed at\nproducing five complete population oscillations.\n\nThree cases of dynamics are presented: Fig. 2.a shows the\nunoptimized dynamics; Fig 2.b shows the 'semi-optimized' dynamics,\nwith optimized driving frequency, but unoptimized population\ntransfer period; finally, Fig. 2.c shows the fully optimized\ndynamics. Observe that in the unoptimized case, the population\noscillations end somewhat short of the complete cycle, and the\ninitially populated level never achieves complete depopulation.\nOptimizing only the driving frequency does indeed maximize the\npopulation oscillations by inducing the complete depopulation of\nthe initially populated level during oscillations, but at the same\ntime the final population oscillation stops even further from the\nfull cycle than in the unoptimized case. Finally, introducing the\npopulation transfer period correction alongside the driving\nfrequency correction yields the required result: complete cycle of\nmaximized population oscillations.\n\n\\subsubsection{Strong perturbation}\nIncreasing the driving perturbation intensity to somewhat greater\nvalue, $\\sigma_{\\beta p}^2=0.25$ ($F_0=6.11409 \\ast 10^{-4} \\\na.u.$.), the limitations of the analytical theory clearly emerge.\nThis is shown in Fig. 3: Fig. 3.a - Fig. 3.c respectively show the\nunoptimized, analytically optimized (according to Eq.(\\ref{chirp})\nand Eq.(\\ref{transfer time corrected})) and 'manually optimized'\ndynamics. The corresponding pulse duration times, aimed at\nproducing three complete population oscillations, are\n$T_{\\pi}=847324 \\ a.u.$, $T_{opt}=901075 \\ a.u.$ and\n$T_{man}=884300 \\ a.u.$.\n\nNotice that in the analytically optimized case, Fig. 3.b, the\ninitially populated level still fully depopulates, which indicates\nthat even for this rather strong perturbation, the frequency\ncorrection Eq.(\\ref{chirp}) still stands strong. However, the\ncorrected period, although closer to the correct value than in the\nunoptimized case, is still somewhat removed from the correct\nvalue. Unfortunately, this 'optimization error' cannot be remedied\nanalytically. Remember that the analytical result\nEq.(\\ref{transfer time corrected}) is obtained using only the\nfirst order approximation (Eq.(\\ref{many level time correction}))\nto the full dynamical equations Eq.(\\ref{two level final}). With\nperturbation as strong as in this case, the dynamical impact of\nthe neglected elements of that equation begin to show. However, as\nshown in Fig. 3.c, the full cycle of oscillations can still be\nproduced, but this additional correction to the pulse duration had\nto be found by hand, using the trial and error method.\n\n\\begin{figure}\n  \\includegraphics[width=12cm]{BonacciFig3.eps}\\\\\n\\caption{Limitations of the analytical optimization theory. Again,\nmajor oscillations are the population of the two targeted levels\nwhereas the minor oscillations are the population of the\nperturbing level. For the value of perturbation strength parameter\nis now $\\sigma_{\\beta p}^2=0.25$, which is just beyond the\nlimiting value for the full applicability of the presented\noptimization procedure. Although the analytical correction\nimproves the final population transfer from 92\\% to 97\\% (with\npulse duration correction of 6\\%), the theory presented in this\npaper cannot account for an additional 1.6\\% correction in the\nduration of the pulse that further increases the final population\ntransfer to over 99.99\\%.}\n  \\label{fig3}\n\\end{figure}\n\n\\subsection{Population transfer}\nThe two final examples demonstrate the application of the\ndeveloped optimization theory to the most interesting dynamical\ncase regarding the coherent control: that of the single population\ntransfer between the two targeted levels $\\alpha$ and $\\beta$. As\nthe validity and the limitations of the whole theory were already\nexplored in the previous two examples, the following examples will\nonly demonstrate the improvements to the population transfer that\ncan be produced using the above results.\n\n\\subsubsection{Legitimate perturbation}\nAgain as in the previous section, the first example (Fig. 4)\npresents the dynamics of the system subjected to the external\ndrive of limiting intensity. In this case, the perturbation\nstrength parameter amounts $\\sigma_{\\beta p}^2=0.1$, corresponding\nto the $F_0=4.07606 \\ast 10^{-4} \\ a.u.$. Calculated population\ntransfer times are $T_{\\pi}=211831 \\ a.u.$ and $T_{opt}=218483 \\\na.u.$.\n\nThe unoptimized (dotted line) and the optimized dynamics (solid\nline) are plotted on the same graph to facilitate the comparison\nbetween the two. Only the dynamics of the two target levels is\nshown - the plot of the perturbing level's ($p$) dynamics is\nomitted for the sake of clarity of the overall graph. Although the\nloss of the population transfer in the unoptimized case is not\ngreat, it nevertheless is noticeable. On the other hand,\nintroducing the corrections for pulse frequency and pulse duration\nclearly improves the population transfer bringing it very close to\n100\\%.\n\n\\begin{figure}\n  \\includegraphics[width=12cm]{BonacciFig4.eps}\\\\\n\\caption{Applicability of the analytical optimization procedure to\nthe maximization of the population transfer. Yet again, major\noscillations are the population of the two targeted levels whereas\nthe minor oscillations are the population of the perturbing level.\nFor this limiting value of perturbation strength parameter of\n$\\sigma_{\\beta p}^2=0.10$, the optimization almost completely\neradicates the loss of the population transfer due to the\ndynamical impact of the perturbing level, increasing the\npopulation transfer from 96\\% to 99.7\\%. Optimized pulse lasts 3\\%\nlonger than the one obtained from the standard $\\pi$-pulse\ntheory.}\n  \\label{fig4}\n\\end{figure}\n\n\\subsubsection{Extreme perturbation}\nThe final example - presented in Fig. 5. - is qualitative, rather\nthan quantitative, but even as such it is quite indicative of the\noverall usefulness of the whole optimization theory. The\nperturbation is now extreme, with strength parameter\n$\\sigma_{\\beta p}^2=1$ corresponding to the $F_0=1.22282 \\ast\n10^{-3}\\ a.u.$. Calculated population transfer times are\n$T_{\\pi}=70610 \\ a.u.$ and $T_{opt}=82816 \\ a.u.$.\n\nAgain, the unoptimized and the optimized dynamics are plotted on\nthe same graph. Although optimization now clearly does not lead to\nthe complete population transfer, the improvement from the\nunoptimized dynamics is significant demonstrating that even for\nthis perturbation intensity the developed optimization theory\nqualitatively works quite nicely.\n\n\\begin{figure}\n  \\includegraphics[width=12cm]{BonacciFig5.eps}\\\\\n\\caption{Breakdown of the quantitative optimization, but\nqualitatively the theory is still applicable. Half way through the\npulse, in the unoptimized case now the population of the\nperturbing level surpasses the population of the initially\nunpopulated level. Introduction of the optimized driving frequency\nand pulse duration significantly - although not fully - rectifies\nthe population transfer from below 40\\% to almost 90\\%. Optimized\npulse is now almost 20\\% longer then the one obtained from the\nordinary $\\pi$-pulse theory.} \\label{fig5}\n\\end{figure}\n\n\\section{Conclusion}\n\nThe aim of research that led to this paper was to explore the\npossibility of using 'old fashioned' and rather simple phenomenon\nof Rabi oscillations for the controlled manipulation of the\npopulation in general many level system. This paper rounds up the\ntopic of analytical optimization of pulse parameters (frequency\nchirp and pulse duration), opened in the author's previous work\n(\\cite{bonacci2003.2}) that would lead to maximizing the\npopulation transfer between two targeted levels of the system. The\ntheory developed provides the exact quantitative predictions of to\nwhat extent the dynamical impact of the remainder of the many\nlevel system (beyond the two levels selected for the population\ntransfer) begins to interfere with the targeted population\ntransfer. It also provides the closed (albeit recursive)\nanalytical expressions for the optimization of pulse parameters.\n\nAlthough the major correction to the population transfer is\nachieved by optimizing the driving pulse's frequency chirp (given\nin \\cite{bonacci2003.2}), this paper provides the additional fine\ntuning by establishing similarly simple analytical expression for\nthe determination of the optimal pulse duration. It demonstrates\nthat the standard formula of the $\\pi$-pulse theory, Eq. (\\ref{pi\npulse theory}) begins to fail as the perturbation increases to and\nbeyond the well defined limiting value. It also provides some\nremedy to this failure.\n\nThe whole theory presented in \\cite{bonacci2003.2} and this paper\ndeals with only single laser pulse driving one particular\ntransition in the many level system. The further research\ncurrently under way considers the possibility of applying a number\nof distinct but simultaneous optimized pulses to drive the\npopulation through the chain of transitions through the system,\nhence producing as clean as possible transfer between the two\nlevels not coupled by the single photon transition. The\npreliminary results indicate that an analytical optimization\nformula can be developed even for such a case.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro} \n\nThe uniform, random sampling of arbitrarily shaped surfaces  is of\nimportance in several scientific and technological applications.\nFor example, generic surface sampling can be used to create and\ntest more realistic computer graphics models~\\cite{Tur92}.\nIn medical imaging, such sampling can be used to generate a uniform\ndistribution of target points over the surface of tumors~\\cite{Wil87}.\nSurface sampling has also been used to study oxygen production in\nforests~\\cite{Mel04}.\nIn low-background radiation detection, the application for which\nthe algorithm presented here was developed, the simulation of\nradioactive contaminants on various detector surfaces is important\nfor quantifying backgrounds and their impact on detector sensitivity.\nThis algorithm was successfully\nimplemented into the Geant4-based~\\cite{Geant4} simulation toolkit, MaGe~\\cite{MaGe},\nbeing jointly developed by the GERDA~\\cite{GERDA} and {\\sc Majorana}~\\cite{MJ} collaborations \nto simulate germanium detector arrays.\n\nSeveral algorithms exist to perform such generic surface sampling\n(see, for example, Refs.~\\cite{Tur92} - \\cite{Mel04}).\nUnfortunately, some of these methods (such as the retiling of\npolygonal surfaces) are algorithmically complex and computationally\nintensive. Other\nalgorithms require the surfaces to be represented as differentiable\nfunctions. Deriving such a function for each surface-of-interest\ncan be a computationally intensive task, particularly for\ncomplex geometries. Finally, to the authors' knowledge, little is available in the form of\nfree, open-source code for plug-and-play usage.\n\nWe have developed a Monte Carlo algorithm that only requires the\ngeometry modeling software to be able to find\nthe intersection points between an arbitrary line and the surfaces\nof the volumes to be sampled. The algorithm generates a random set of rays that impinge on\nthe surfaces of interest that are isotropic in direction and uniform\nin space.  The intersection points, provided by the geometry modeling software,\nare sampled again to provide the final set of random\nand uniform surface points.\n\nWe demonstrate this generic surface sampling routine using the\nC\\nolinebreak\\hspace{-.05em}\\raisebox{.4ex}{\\tiny\\bf +}\\nolinebreak\\hspace{-.10em}\\raisebox{.4ex}{\\tiny\\bf +}-based Geant4 Monte Carlo simulation toolkit~\\cite{Geant4}.\nGeant 4 is used extensively in high-energy, nuclear and medical\nphysics to simulate the interactions of radiation with matter. In\nGeant4, arbitrary geometries can be constructed by arranging\ncollections of nested solid volumes and boolean combinations\n(intersections, additions, or subtractions) of those volumes in\nspecified positions and orientations relative to each other. The\navailable basic solids include fundamental solids such as spheres,\ncylinders, and polyhedra, as well as more generic and complex\nboundary-representation volumes.  Our sampler relies on the fact\nthat each Geant4 volume class provides a function that finds the\nintersection points between the volume's surface and an arbitrary\nline, if such an intersection exists. Each volume class also defines\na function that returns a bounding radius for the volume in question,\nwhich is used to constrain the parameter space of lines sampled.\n\n\n\\section{Sampling Algorithm}\n\\label{sec:algorithm}\n\nThe principle of the sampling algorithm is based upon uniformly \nsampling the volume within a sphere.  \nWhen the user selects a volume or set of volumes whose surfaces are\nto be sampled, the radius $R$ of a bounding sphere which wholly contains the \nvolume(s) must be determined.  In the\ncase of multiple disjoint volumes, a ``mother'' volume that encompasses\nall the volumes to be sampled must be used. In practice, the radius of this bounding\nsphere is determined by querying the geometry modeling software.   \nTo generate a uniform,\nisotropic flux of rays within this bounding sphere, first a random\nisotropic point ${\\bf r}$ on the sphere is generated, where ${\\bf r} = R {\\bf \\hat{\\Omega}}$ and \n${\\bf \\hat{\\Omega}}$ is the randomly generated direction.  A disk, also of radius\n$R$, is defined tangential to the bounding sphere, with its center\nat position ${\\bf r}$. Figure~\\ref{fig:rayGeneration} shows the position of\nthis disk and the bounding sphere for a sampling trial of an arbitrary example volume.\nThe starting point for another ray $\\boldsymbol{\\rho}$ is\ngenerated on the interior of the disk at point ${\\bf r} + {\\bf b}$,\nwhere ${\\bf b}$ has polar coordinates $(b, \\alpha)$ in the coordinate\nsystem of the disk. The ``impact parameter'' $b$ is generated with\na uniform distribution in $b^2$ between 0 and $R$, and the angle $\\alpha$ is generated\nuniformly between 0 and $2\\pi$.  The direction of $\\boldsymbol{\\rho}$ is taken\nto be $-{\\bf \\hat{\\Omega}}$, normal to the circle and hence pointing\ninto the bounding sphere. The uniformity and isotropy of the\nrays produced in this manner will be discussed in the next section.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.95\\columnwidth]{SamplerGrayScale1}\n\\caption{An schematic of the bounding sphere \n(shown with a section missing for illustrative purposes),\ntangent disk, and ray $\\boldsymbol{\\rho}$ for a sampling trial of\nan arbitrary example volume.  $\\boldsymbol{\\rho}$ originates\nat ${\\bf r} + {\\bf b}$ and continues in the $-{\\bf \\hat{\\Omega}}$\ndirection, in this case intersecting the enclosed volume twice.\nThe determination of the various rays and angles is described in the text.}\n\\label{fig:rayGeneration}\n\\end{figure} \n\nOnce $\\boldsymbol{\\rho}$  has been generated, the geometry modeling software is\nqueried to find the intersection points of the ray with all\nsurfaces among the volumes-of-interest. If no such intersections\nexist, another ray is generated with a new direction and starting point. \nIf $N$ intersections are found, a\nrandom integer $n$ is generated between one and the maximum number\nof intersections possible for the given geometry ($N_{\\textrm{max}}$),\nwhich is input by the user~\\footnote{A guess will suffice\nfor the value of $N_{\\textrm{max}}$, it merely needs to match or exceed\nthe greatest number of intersections encountered in the ouput set of \nsampled points. If the algorithm encounters more \nsurfaces than $N_{\\textrm{max}}$, a warning can be generated and the\nuser can rerun with a larger value of $N_{\\textrm{max}}$.}.  \nIf $n > N$ the ray is discarded, and the algorithm starts over. \nOtherwise, one of the $N$\nintersections is chosen at random.  The set of intersection points\nchosen in this way is the output of the algorithm.  \n\n\\section{Algorithm Properties}\n\\label{sec:properties}\n\nIn the following, it is assumed that we have a random\nnumber generator that can generate a sequence of real numbers\nuniformly distributed between $0$ and $1$, with the standard\nrequirement of randomness~\\cite{Knu81}. Additionally, all vectors,\nvolumes and surfaces are assumed to lie in 3-dimensional Euclidean\nspace.\n\nWe first show that the flux of rays generated as described above\nis uniform and isotropic within the bounding sphere of the\nsurfaces-of-interest. For every point ${\\bf x}$ in the interior of\na sphere of radius $R$, and for every direction ${\\bf \\hat{\\Omega}}$\nfrom ${\\bf x}$, there exists one and only one line passing through\n${\\bf x}$ that is normal to the plane tangent to the sphere at ${\\bf\nr} = R {\\bf \\hat{\\Omega}}$.  The set of all intersections with this\ntangent plane of rays in direction ${\\bf \\hat{\\Omega}}$ originating\nfrom all points ${\\bf x}$ interior to the sphere fill a disk of\nradius R centered at ${\\bf r}$. Since the direction ${\\bf \\hat{\\Omega}}$\nis chosen isotropically, and since the starting point on the disk\n${\\bf b}$ is chosen uniformly across the surface of the disk, then\nthe probability for a ray to pass within a small area ${\\bf \\Delta\nA}$ centered at ${\\bf x}$ with surface normal pointing in direction\n${\\bf \\hat{\\Omega}}$ is independent of ${\\bf x}$ (uniform), and is\nindependent of direction ${\\bf \\hat{\\Omega}}$ (isotropic). Symbolically,\nwe write the normalized vector flux of rays as $\\boldsymbol{\\phi}({\\bf x}, {\\bf \\hat{\\Omega}})\n= {\\bf \\hat{\\Omega}} \/ 4 \\pi^2 R^2$, which is independent\nof ${\\bf x}$.  The randomness of this flux\nis guaranteed as long as a new direction ${\\bf \\hat{\\Omega}}$ and\ndisk position ${\\bf b}$ are chosen for each ray.\n\nThe uniformity and randomness of the set of intersection points\ngenerated by the uniform isotropic flux of rays can be demonstrated\nas follows.  First, divide the surfaces-of-interest into an large number\nof surface elements ${\\bf \\Delta A}({\\bf x})$, where the direction\npoints normal to the surface at point ${\\bf x}$, and the magnitude\n$\\Delta A$ is independent of ${\\bf x}$ (${\\bf \\Delta A}({\\bf x}) =\n\\Delta A {\\bf \\hat{n}}({\\bf x})$). $\\Delta A$ is taken to be small enough\nthat each surface element may be approximated to be flat \\footnote{This\nis equivalent to requiring that the sampled surfaces be differentiable.\nWithin the Geant4 framework, this implies a requirement that the\nradius-of-curvature for any surface be much greater than the tolerance\nparameter, which sets the distance within which a point is considered\nto be ``on'' a volume's surface. This parameter has a default value\nof 1 pm, but can be tuned by the user to be as low as $\\sim$1 fm\nfor typical meter-sized or smaller geometries, at which point one\nis limited by numerical round-off of the 64-bit double-precision\nfloating point data type used to define volume dimensions. The\nassumption of flatness of the surface elements also neglects\ninfinitely sharp corners, which are unphysical.}.  The\nprobability for a surface element to be hit by a ray from our\ngenerated vector flux $\\boldsymbol{\\phi}({\\bf x}, {\\bf \\hat{\\Omega}})$ is\n\\begin{eqnarray*}\n\\int_0^{4\\pi} \\left|\\boldsymbol{\\phi}({\\bf x}, \\Omega) \\cdot {\\bf \\Delta A}({\\bf x})\\right|~d\\Omega & = & \\int_0^{4\\pi} \\left|\\frac{\\Delta A}{4 \\pi^2 R^2} ~{\\bf \\hat{\\Omega}} \\cdot {\\bf \\hat{n}}({\\bf x})\\right|~d\\Omega \\\\ \n& = & \\frac{\\Delta A}{4 \\pi^2 R^2} \\int_0^{4\\pi} \\left|\\cos\\theta\\right|~d\\Omega\\\\ \n& = & \\frac{\\Delta A}{2 \\pi R^2} \\\\\n\\end{eqnarray*}\nwhich is independent of ${\\bf x}$. This implies that all surface\nelements are hit with constant probability. Thus the set of\nintersections of all rays with the surfaces-of-interest gives a\nuniform sampling of those surfaces.\n\nThe randomness of initial flux of rays implies that the set of\nintersection points generated by one ray is statistically independent\nfrom those of other rays. However, intersection points of a single\nray are not statistically independent from each other, as they all\nlie along a single line. For a truly random sampling, at most one\nintersection point can be chosen from each ray.  Note that if a\nsingle point were chosen at random and kept for each ray with\nintersections, those points which lie along rays with fewer\nintersections would be sampled more often than those points lying\nalong lines with more intersections, ruining the uniformity of the\ndistribution.  In essence, rays with $N$ intersections would effectively\nbe given a $1\/N$ weighting.\nFor this reason, the point selection is weighted by\n$N\/N_{\\textrm{max}}$, and uniformity is retained. \n\nThe efficiency of the above method, in terms of the number of surface\npoints generated per geometrical calculation, can be poor when the\nvolumes-of-interest sparsely fill the bounding sphere. If the volumes\nare disjoint, efficiency can be recovered by considering distant\nvolumes independently. Poor efficiency for volumes having needle-like\nor planar geometries, with one dimension much larger or smaller\nthan the other dimensions, can be remedied by considering bounding\nsurface other than a sphere, for example a wide plane or a narrow\ncylinder.  In such cases care must be taken to ensure the generated\nflux of rays is (at least approximately) uniform and isoptropic.\nWe did not consider such cases in this paper.\n\nThe step in which rays with fewer intersections are preferentially\ndiscarded also imposes an efficiency reduction by a factor of roughly\n$\\bar{N}\/N_{\\textrm{max}}$, where $\\bar{N}$ is the average number\nof intersections per ray. This reduction can be significant for\ngeometries with many aligned, repeated volumes, as well as for\ngeometries with regions containing many small components. In such\ncases it may be prudent to simply keep all intersection points of\nall rays. The resulting set of points, taken as a whole, will still\ndistribute with uniform surface density, and with much higher\nefficiency, albeit at the cost of introducing correlations among\nsome consecutive points. For many applications, though, such correlations\nare irrelevant.\n\n\n\\section{Geant4 Implementation}\n\nWe implemented this algorithm within the Geant4 framework by deriving\nclasses from the ``user action'' base classes \\emph{G4VUserPrimaryGeneratorAction} and\n\\emph{G4UserSteppingAction}.  At runtime the user\ninputs a list of volume names whose surfaces are to be sampled,\nwhich are sent to the generator action class.  After geometry\ninitialization, the class queries the \\emph{G4PhysicalVolumeStore}\nto find the smallest volume which contains all volumes-of-interest\nas daughter volumes (this volume may itself be a volume-of-interest).\nThe \\emph{G4VSolid} corresponding to that mother volume is extracted\nfrom its \\emph{G4LogicalVolume}.  A bounding radius for the\nsurfaces-of-interest is then obtained by calling\n\\begin{verbatim}\nG4VSolid::GetExtent().GetExtentRadius();\n\\end{verbatim}\nThe class then sets the primary particle to be a ``geantino'', an\nimaginary neutral, massless utility ``particle\" within the Geant4\nframework which undergoes no interactions, and only travels in\nstraight lines. Geantinos are commonly used for debugging purposes\nand to map out geometries.  The geantino's position and direction\nare selected by our algorithm to give a uniform, isotropic flux of\ngeantinos throughout the interior of the bounding sphere. The energy\nof the geantino can be any value greater than 0.  The choice of\ngeantinos delegates all geometrical calculations to Geant4.\n\nThe stepping action class checks at each step whether the geantino\nis entering or exiting a volume of interest. Each such entrance or\nexit point is added to a list of surface intersections. At the end\nof the event, one of these surface intersections can be chosen at\nrandom, or all surface\nintersections can be kept if efficiency requirements outweigh\nthe necessity for truly uncorrelated sampling. The set of surface\nintersections generated in this way uniformly sample the surfaces\nof interest, and may be saved to disk or used for further processing\nin the program (e.g.~as the vertex for the next event).\n\n\n\\section{Example Application and Verification}\n\nSuch an implementation of our generic surface sampling algorithm\nwas added to MaGe~\\cite{MaGe}, a Geant4-based Monte Carlo simulation\ntoolkit optimized for low-background germanium detector simulations.\nThe output vertices are written to a ROOT~\\cite{ROOT} file, which can then be\nused in simulations involving surface physics, for example\n$\\alpha$-particle backgrounds from natural U and Th decay chain isotopes in settled dust,\nor from Rn decay chain daughters plated out on detector surfaces.\n\nFigure~\\ref{fig:demonstration} demonstrates the usage of the surface sampler \non the 57-detector array design for the {\\sc Majorana} experiment~\\cite{MJ}.\nFigure~\\ref{fig:StringRender} shows a rendering of a 3-Ge-crystal string assembly,\ncomplete with detector supports and electronic connections and components. 19\nsuch strings are arranged in a hexagonal close-pack pattern, suspended from a \nCu cold plate, and housed in a cylindrical low-background cryostat made of \nelectroformed Cu. An imaging of the \nfull simulated geometry (minus the surrounding cryostat) \nby our surface sampling algorithm is shown in \nFigure~\\ref{fig:57BangerSamp}, as viewed from one side. \nWe also show more detailed samplings of two\nspecific detector components in Figures~\\ref{fig:SingXtalCoax} and \\ref{fig:TraysSamp}.\nFigure~\\ref{fig:SingXtalCoax} plots the output of the algorithm for\none close-ended coaxial high-purity germanium detector crystal. The detector is represented\nby a boolean combination of basic volumes. The body is modeled as\ntwo cylinders OR'd with a torus to form the rounded top face. A\nthird, smaller-radius cylinder OR'd with a sphere at one end is\nsubtracted from the body to form the coaxial well along the detector's\nvertical axis.  Figure~\\ref{fig:TraysSamp} shows a surface sampling of one\nof the plastic trays on which the Ge crystals rest in the string assembly. A\nrendering of the simulated tray design is shown to the upper right of the surface \nsampling for comparison.\n\n\\begin{figure*}\n\\centering\n  \\subfigure[~Rendering of a 3-Ge-crystal ``string'' assembly. The entire assembly is about 30~cm in length.]{\\label{fig:StringRender} \\includegraphics[width=.8\\columnwidth]{3CrystalAsm.pdf}}\n  \\subfigure[~Horizontal view of 19 strings hanging from a Cu coldplate, imaged with our surface sampling algorithm.]{\\label{fig:57BangerSamp} \\includegraphics[width=.8\\columnwidth]{mod.png}}\n  \\subfigure[~Surface sampling of a close-ended coaxial high-purity germanium detector crystal.]{\\label{fig:SingXtalCoax} \\includegraphics[width=.8\\columnwidth]{xtal.png}}\n  \\subfigure[~Surface sampling of a crystal support tray. A rendering of the simulated geometry is shown in the upper right corner.]{\\label{fig:TraysSamp} \\includegraphics[width=.8\\columnwidth]{tray.pdf}} \n  \\caption{Demonstration of the uniform surface sampling on various\n  volumes in the {\\sc Majorana} 57-crystal array design.  The\n  2-dimensional projection of 3-dimensional points leads to regions\n  with apparent higher or lower sampling densities, for example at\n  the edges of the displayed geometries.  See Table~\\ref{tab:Verification}\n  for an analytic verification of the sampler. }\n  \\label{fig:demonstration}\n\\end{figure*} \n\nWe ran a high statistics simulation to test the behavior of the surface sampler\nand verify that the surface density of sampled points is\nindependent of surface shape and orientation.  To this end, we sampled a portion of the\n{\\sc Majorana} 57-detector array design.  We chose to \nsample the inner surface of the enclosing\ncylindrical cryostat, the cold plate from which all the crystals hang, two\ncrystal detectors, and a single ``contact ring\" (a thin plastic ring \nthat clamps leads against the crystal surface to make electrical connections\nto the detector) surrounding one of the crystals.\nThe inner cryostat surface and the cold plate are both simple cylinders.\nThe contact ring is an annulus, and the detectors are as described\nabove. All surfaces were sampled simultaneously, so the surface density\nof sampled points should be the same for all five components.\nThe ratio of points on a volume's surface to total number of sampled\npoints in the run were tabulated from the output ROOT file.  These\nratios were then compared with analytical calculations of the surface\narea ratio for each volume to the total surface area of all sampled\nvolumes.  The results are shown in\nTable~\\ref{tab:Verification}. In all cases, the ratios agree within the\nsampled statistics.\n\n\\begin{table}\n\\caption{A comparison of analytically calculated surface area ratios\nto the fractions of sampled points landing on each surface of a\nnumber of volumes sampled simulataneously using our generic surface\nsampling algorithm. In all cases, the ratios agree within the\nstatistics of the simulation.}\n\\label{tab:Verification}\n\\begin{center}\n\\begin{tabular}{|l|r@{.}l|r@{.}l|}\n\\hline\n &  \\multicolumn{2}{|c|}{Analytic [\\%]} & \\multicolumn{2}{|c|}{Sampled [\\%]} \\\\ \\hline\nCryostat        & ~~~~69  & 544  & ~69 & 577 $\\pm$ 0.042~ \\\\\nCold Plate      & 25  & 906  & 25 & 881 $\\pm$ 0.026 \\\\\nDetector 1      & 2   & 173  & 2  & 171 $\\pm$ 0.007 \\\\\nDetector 2      & 2   & 173  & 2  & 167 $\\pm$ 0.007 \\\\\nContact Ring    & 0   & 202  & 0  & 203 $\\pm$ 0.002 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Concluding Remarks}\n\nWe have developed a generic surface sampling algorithm that distributes\nvertices uniformly and randomly over sets of arbitrary surfaces.\nSuch an algorithm has potential application in many scientific and\ntechnical fields. Our implementation within the Geant4 Monte Carlo\nsimulation toolkit and the MaGe simulation framework for germanium\ndetector-based systems is of particular use to nuclear and particle\nphysicists. It may be used, for example, to study surface $\\alpha$\nbackgrounds, a key background in many low-background calorimetry-based\nexperiments in these fields.\n\n\n\\section{Acknowledgments}\n\nThis work was sponsored in part by the US Department of\nEnergy under Grant nos. DE-FG02-97ER41020 and\nDE-AC02-05CH11231. This research used the Parallel\nDistributed Systems Facility at the National Energy\nResearch Scientific Computing Center, which is supported\nby the Office of Science of the U.S. Department of Energy\nunder Contract no. DE-AC02-05CH11231. The authors would \nlike to thank D.\\,Y.\\,Sebe for assistance with some of the \nfigures.  In addition, the authors acknowledge the MaGe group of the\nGERDA and {\\sc Majorana} collaborations for important comments\nand insight.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThis work was supported by the National Science Foundation.\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{Signal Asymmetry}\n\\label{sec:signal_asymmetry}\nAs described in section~\\ref{sec:Measurement_scheme}, the accumulated phase $\\Phi$ was read out by resonantly addressing the $H \\rightarrow C$ transition with linearly polarised light and monitoring the resulting fluorescence. The state readout laser was switched between orthogonal polarisations, $\\hat{X}$ and $\\hat{Y}$, at $100$~kHz (with $1.2~\\upmu$s of dead time between polarisations) in order to normalize against molecular flux variations. By switching at a rate fast enough that each molecule experienced both polarisations, we achieved nearly photon-shot-noise-limited phase measurements \\cite{Kirilov2013}. With a sufficiently wide laser beam, all molecules were completely optically pumped by both laser polarisations during their $\\sim$20~$\\upmu$s fly-through time. We induced approximately one fluorescence photon from each molecule by projecting the molecule state onto the two orthogonal spin states excited by laser beams with orthogonal polarisations.\n\nThe rapid switching of the laser polarisation resulted in a modulated PMT signal, $S(t)$, as shown in figure~\\ref{fig:modulation}. For the following discussion we consider the polarisation state to switch at a time $t=0$. Immediately after, there is a rapid increase in fluorescence as the molecules in the laser beam are quickly excited; while $\\Omega_rt\\ll1$, where $\\Omega_r\\sim2\\pi\\times1$~MHz is the Rabi frequency on the $H$ to $C$ transition, the fluorescence increases as $S(t)\\propto\\Omega_r^2\\times t^2$. At later times, when $\\Omega_rt\\gtrsim1$, population is about evenly mixed between the $H$ and $C$ states (since $\\Omega_r\\gtrsim\\gamma_C$); hence, $S(t)$ decays exponentially with a time constant of roughly $1\/(2\\gamma_C)\\approx1~\\upmu$s. Molecules that were not present at $t=0$ continue to enter the laser beam, causing $S(t)$ to approach a steady state. The laser is then turned off and the signal decays exponentially with time constant $1\/\\gamma_C\\approx0.5~\\upmu$s. The next laser pulse, with orthogonal polarisation, is turned on 1.2~$\\upmu$s $\\approx2.5\/\\gamma_C$ after the end of the previous one to prevent significant overlap of contributions to $S(t)$ induced by different polarisations. A low-pass filter in the PMT voltage amplifier with a cut-off frequency of $2\\pi\\times2$~MHz removed any short timescale dynamics from $S(t)$, and prevented aliasing of high frequency components in the signal given our fixed digitization rate of 5 MSa\/s.\n\nTo determine the fluorescence $F(t)$ produced by each polarisation state, we subtracted a time-dependent background, $B(t^{\\prime})$, taken from data with no molecule fluorescence present, i.e. $F(t)=S(t)-B(t^{\\prime})$. Examples of the extracted $F(t)$ and $B(t^{\\prime})$ time series are shown in figure~\\ref{fig:modulation}A and B, respectively. $B(t^{\\prime})$ was modulated in time due to scattered light from the state readout laser beam and has a DC electronic offset intrinsic to the PMTs. The first millisecond of data, which contains no fluorescence, was used to determine $B(t^{\\prime})$. We assumed that $B(t^{\\prime})$ was periodic with the switching of the laser polarisation but did not depend on the polarisation; we inferred its value by averaging together the recorded PMT signal across all polarisation bins for ${\\approx}1$~ms of data taken before the arrival of the molecule pulse. Since molecule beam velocity variations caused jitter in the temporal position of the molecule pulse within the trace, 9~ms of data were collected per pulse, despite the fact that only the ${\\approx}$2~ms of strong signal with $F(t)\\gg B(t^{\\prime})$ and ${\\approx}1$~ms of background contained useful information for the spin precession measurement. \n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=12.7cm]{fluorescence_background_asymmetry_v2.pdf}\n\\caption{(A) Molecule fluorescence signal $F(t)$ in photoelectrons\/s induced by $\\hat{X}$ (blue) and $\\hat{Y}$ (red) readout laser polarisations. Lines show the raw data for a single trace consisting of an average of 25 molecule pulses. Shaded regions show the waveform averaged over 16 traces. (B) Background signal $B(t^{\\prime})$ in photoelectrons\/s obtained before the arrival of molecules in the state readout region. (C) Integrated fluorescence signals $F_X$ and $F_Y$ throughout the molecule pulse. Dashed lines denote the region with $F=(F_X+F_Y)\/2>3\\times10^5~\\rm{s}^{-1}$, used as a typical cut for inclusion in eEDM data. Points are spaced by 5~$\\upmu$s. (D) Computed asymmetry throughout the molecule pulse. In this example, 18 of the ungrouped asymmetry points are grouped together to compute the mean and uncertainty shown as the grouped asymmetry.}\\label{fig:modulation}\n\\end{figure}\n\nIntegrating $F(t)$ over times associated with pairs of orthogonally polarised laser pulses resulted in signals $F_X, F_Y$. The integration was performed over a specified time window that we denoted as a `polarisation bin'. Figure~\\ref{fig:cuts}B shows two typical choices of polarisation bin and illustrates that the extracted eEDM is not significantly affected by this choice. Figure \\ref{fig:pixel_plot} shows that most of the extracted quantities did not vary linearly within the polarisation bin (Pol.\\ Cycle Time Dependence column).\n\nAfter polarisation binning, the data displayed a fluorescence signal modulated by the envelope of the molecule pulse, as in figure~\\ref{fig:modulation}C. Figure~\\ref{fig:modulation}D shows the asymmetry, $\\mathcal{A}$, computed from these data. \nThe asymmetry is computed for each 10~$\\upmu$s polarisation cycle, so that for the $i^{\\rm{th}}$ cycle we have\n\\begin{equation}\n\\A_i=\\frac{F_{X,i}-F_{Y,i}}{F_{X,i}+F_{Y,i}}.\n\\label{eq:asym_bins}\n\\end{equation}\nThe molecule phase, and hence asymmetry (see equation~\\ref{eq:asymmetry}), had a linear dependence on the time after ablation because the molecules precessed in a magnetic field over a fixed distance; the slower molecules, which arrive later, precessed more than the faster molecules, which arrived earlier. We applied a fluorescence signal threshold cut of around $F=(F_X+F_Y)\/2\\ge3\\times10^5~{\\rm s}^{-1}$, indicated by dashed lines in figure~\\ref{fig:modulation}C,D. Section \\ref{sec:data_cuts} describes the threshold choice in detail.\n\nTo determine the statistical uncertainty in $\\A$, $n\\approx$ 20--30 adjacent asymmetry points were grouped together. For each group, $j$, centred around a time after ablation $t_j$, we calculated the mean, $\\bar{\\mathcal{A}}_j$, and the uncertainty in the mean, $\\delta\\bar{\\mathcal{A}}_j$, depicted as red points and error bars in figure~\\ref{fig:modulation}D. For smaller $n$, the variance in the sample variance in the mean grows, in which case, error propagation that utilises a weighted mean of data ultimately leads to an understimate of the final statistical uncertainty \\cite{Kenny1951}. For larger $n$, the mean significantly varies within the group due to velocity dispersion, and the variance in the mean grows in a manner not determined by random statistical fluctuations. For the range $n=$ 20--30 we observed no significant change in any quantities which were deduced from the measured asymmetry. \n\nAs described earlier in this section, the background, $B(t^{\\prime})$, which we subtracted from the PMT signal, $S(t)$, was observed to be correlated with the fast switching of the readout laser beam polarisation. This can arise, for example, if the two polarisations have different laser beam intensities or pointings. \nWe chose to use a polarisation independent $B(t^{\\prime})$ by averaging over the two polarisation states. This produced an asymmetry offset as per equation~\\ref{eq:asym_bins} and hence a significant $\\Phi^{\\rm nr}$ associated with the polarisation-dependent background. However, we did not consider $\\Phi^{\\rm nr}$ to be a crucial physical or diagnostic quantity.\nWe found that this methodology produced accurate estimates of the uncertainties of quantities computed from the measured asymmetry, as verified by $\\chi^2$ analysis of measurements of $\\Phi^{\\N\\E}$. We also found that none of the phase channels of interest changed significantly dependent on whether a polarisation-dependent $B(t^{\\prime})$ was used.\n\n\\subsection{Computing Contrast and Phase}\nTo compute the measured phase $\\Phi$ we must also measure the fringe contrast $\\mathcal{C}$ and relative laser polarisation angle $\\theta=\\theta_{\\rm read}-\\theta_{\\rm prep}$, as described in section~\\ref{sec:Measurement_scheme}. The $\\hat{X}$ and $\\hat{Y}$ laser polarisations were set by a $\\lambda\/2$ waveplate and were determined absolutely by auxiliary polarimetry measurements \\cite{Hess2014}. The contrast, defined as either $2\\mathcal{C}=-\\partial\\A\/\\partial\\theta$ or $2\\mathcal{C}=\\partial\\A\/\\partial\\phi$\\footnote{Recall that in practice we consider $\\mathcal{C}$ as an unsigned quantity for the purposes of data analysis.}, can be determined by dithering either the accumulated phase $\\phi$ (by varying $\\mathcal{B}_z$) or the relative laser polarisation angle $\\theta$. We chose the latter as it could be changed quickly ($<1$~s) by rotating a half-wave plate with a stepper-motor-driven rotation stage. Figure~\\ref{fig:fringe} shows the asymmetry as a function of $\\theta$, for a range of values of applied magnetic field. We ran the experiment at the steepest part of the asymmetry fringe (where $\\theta=\\theta^{\\rm nr}$) and measured the contrast, $\\mathcal{C}_j$, for each asymmetry group, $\\bar{\\A}_j$, by switching $\\theta$ between two angles, $\\theta=\\theta^{\\rm nr}+\\Delta\\theta\\tilde{\\theta}$, for $\\tilde{\\theta}=\\pm1$ and $\\Delta\\theta=0.05$~rad:\n\\begin{equation}\n\\label{eq:contrast_1}\n\\mathcal{C}_j=-\\frac{\\bar{\\A}_{j}(\\tilde{\\theta}=+1)-\\bar{\\A}_{j}(\\tilde{\\theta}=-1)}{4\\Delta\\theta}.\n\\end{equation}\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=12cm]{pol_fringe.pdf}\n\\caption{Asymmetry vs.\\ relative laser polarisation angle $\\theta=\\theta_{\\rm read}-\\theta_{\\rm prep}$ for several magnetic field values. The value of $\\theta$ was dithered about the value $\\theta^{\\rm nr}$ by $\\pm\\Delta\\theta=\\pm0.05$~rad to measure fringe contrast, $\\mathcal{C}$. To stay on the steepest part of the fringe, we chose $\\theta^{\\rm nr}=0$ rad for $\\B=\\pm20$~mG and $\\theta^{\\rm nr}=\\pi\/4$ rad for $\\B=\\pm1,\\pm40$~mG. For these data $|\\mathcal{C}|<90\\%$ due to low preparation laser power; typically, however, $|\\mathcal{C}|\\approx 95\\%$. Solid lines represent the expected behaviour for a given magnetic field and contrast.}\n\\label{fig:fringe}\n\\end{figure}\n\nBecause the fringe contrast was fairly constant over the duration of the molecule pulse (figure~\\ref{fig:contrast}A), we used a weighted average\\footnote{Each $\\mathcal{C}_j$ measurement is weighted by its computed uncertainty.} of all $\\mathcal{C}_j$ measurements within the cut region for that trace to extract the accumulated phase. We also performed the analysis by fitting $\\mathcal{C}_j$ to a 2nd-order polynomial as a function of time during the ablation pulse; this led a better fit to the data, but had no significant effect on the results. We typically found $|\\mathcal{C}|\\approx95$\\%. We believe that this was limited by a number of effects including: imperfect state preparation\/readout, decay from the $C$ state back to the $H$ state and dispersion in the spin precession. We also observed that this value was constant over a $\\pm2\\pi\\times1$~MHz detuning range of the state preparation laser (figure~\\ref{fig:contrast}B), indicating complete optical pumping over this frequency range. Recall that, as defined, $\\C$ can be positive or negative, depending on the sign of the asymmetry fringe slope (see figure~\\ref{fig:fringe}, or equation~\\ref{eq:contrast_1}). Given that we worked near zero asymmetry where the fringe slope was steepest, and that $\\theta^{\\rm nr}$ was always chosen to be 0 or $\\pi\/4$, we computed the total accumulated phase as\n\\begin{equation}\n\\label{eq:phase_1}\n\\Phi_j=\\frac{\\bar{\\A}_j(\\tilde{\\theta}=+1)+\\bar{\\A}_j(\\tilde{\\theta}=-1)}{4\\C}+q\\frac{\\pi}{4}.\n\\end{equation}\nHere, $q=0,\\pm1$ or $\\pm2$, corresponds to applied magnetic fields of $\\pm1$, $\\pm20$, and $\\pm40$~mG, respectively. We chose to apply a small magnetic field, $\\B=1$ mG when operating at $q=0$ rather than turning off the magnetic field completely so that we would not need to change the experimental switch sequence or data analysis routine for data taken under this condition. Figure~\\ref{fig:fringe} illustrates the correspondence between $\\theta^{\\rm nr}$ and applied magnetic field needed to remain on the steepest part of the asymmetry fringe.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[trim=7mm 0mm 0mm 0mm,scale=0.6]{contrast_combined.pdf}\n\\caption{(A) Contrast vs time after ablation, averaged over 64~traces. The signal threshold window is indicated by dashed lines (cf.\\ figure~\\ref{fig:modulation}). (B) Contrast vs preparation laser detuning. Error bars were computed as the standard error associated with 64 averaged traces. The solid line is a fit of the form $\\mathcal{C}=a\\times{\\rm tanh}(b\\gamma_C^2\/(4\\Delta_{\\rm prep}^2+\\gamma_C^2))$, motivated by solution of a classical rate equation.}\n\\label{fig:contrast}\n\\end{figure}\n\n\\subsubsection{Accounting for Correlated Contrast}\n\\label{sec:corr_contrast}\n\\hspace*{\\fill} \\\\\nIt was possible for the magnitude of the contrast $|\\C|$ to vary between different experimental states. For example, if the state preparation laser detuning or fluorescence signal background were correlated with any of the block switches $\\Nsw$, $\\Esw$, or $\\Bsw$, then contrast would also be correlated with those switches. As described in section~\\ref{sec:systematics}, we observed both $\\Nsw$- and $\\Nsw\\Esw$-correlated contrast. The latter was particularly troubling since it could lead to a systematic offset in the measured eEDM if not properly accounted for: since $\\A=-\\C\\cos[2(\\phi-\\theta)]$, a nonzero $\\A^{\\mathcal{NE}}$ could occur due to either $\\C^{\\N\\E}$ or $\\phi^{\\N\\E}$. We accounted for contrast correlations by calculating $\\C$ separately for each combination of $\\Nsw$, $\\Esw$, and $\\Bsw$ experimental states (`state-averaged' contrast\\footnote{Since there were $2^3=8$ different $\\Nsw$, $\\Esw$, and $\\Bsw$ states in each 64-trace block, 64\/8 = 8 traces were averaged together to determine the contrast for each experimental state.}):\n\\begin{equation}\n\\label{eq:contrast_2}\n\\C_j(\\Nsw,\\Esw,\\Bsw)=-\\frac{\\bar{\\A}_j(\\tilde{\\theta}=+1,\\Nsw,\\Esw,\\Bsw)-\\bar{\\A}_j(-\\tilde{\\theta}=-1,\\Nsw,\\Esw,\\Bsw)}{4\\Delta \\theta}.\n\\end{equation}\nAs previously discussed, we averaged or applied a quadratic fit to all $\\C_j(\\Nsw,\\Esw,\\Bsw)$ within a molecule pulse to compute $\\bar{\\C}_j(\\Nsw,\\Esw,\\Bsw)$. The precession phase was calculated from each state-specific asymmetry and contrast measurement (cf.\\ equation~\\ref{eq:contrast_1}):\n\\begin{equation}\n\\label{eq:phase_2}\n\\Phi_j(\\Nsw,\\Esw,\\Bsw)=\\frac{\\bar{\\A}_j(\\Nsw,\\Esw,\\Bsw)}{2\\bar{\\C}_j(\\Nsw,\\Esw,\\Bsw)}+q \\frac{\\pi}{4},\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:asymmetry_avg}\n\\bar{\\A}_j(\\Nsw,\\Esw,\\Bsw)=\\frac{\\bar{\\A}_j(\\tilde{\\theta}=+1,\\Nsw,\\Esw,\\Bsw)+\\bar{\\A}_j(\\tilde{\\theta}=-1,\\Nsw,\\Esw,\\Bsw)}{2}\n\\end{equation}\nis the average asymmetry over the two $\\tilde{\\theta}$ states in a data block that share identical values of $\\Nsw$, $\\Esw$ and $\\Bsw$. By construction, phases computed from state-averaged contrast are immune to contrast correlations. We also computed phases by ignoring contrast correlations (i.e. treating contrast as independent of $\\Nsw,\\, \\Esw,\\, \\Bsw$) and the result did not change significantly.\n\n\n\\subsubsection{Computing Phase and Frequency Correlations}\n\\label{sec:compute_phase}\n\\hspace*{\\fill} \\\\\nAfter extracting the measured phase $\\Phi_j(\\Nsw,\\Esw,\\Bsw)$, we performed the basis change described in equation \\ref{eq:general_parity}, from this experiment switch \\emph{state} basis to the experiment switch \\emph{parity} basis, denoted by $\\Phi^p_j$, where $p$ is a placeholder for a given experiment switch parity. \n\nWe observed that the molecule beam forward velocity, and hence the spin precession time $\\tau_j$, fluctuated by up to 10\\% over a 10 minute time period. Since $\\B_z$ and $g_1$ are known from auxiliary measurements to a precision of around 1\\%, we were able to extract $\\tau_j$ from each block from the Zeeman precession phase measurement, $\\Phi^\\B_j=-\\mu_{\\rm B}g_1\\B_z\\tau_j$ (see section~\\ref{sec:Measurement_scheme_more_detail}). Velocity dispersion caused $\\tau_j$ to vary across the molecule pulse with a nominally linear dependence on time after ablation, $t$, however we observed significant deviations from linearity. Thus, we fit $\\tau_j$ to a 3rd order polynomial in $t$ in order to evaluate $\\bar{\\tau}_j$. Then, we evaluated the measured spin precession frequencies defined as\n\\begin{equation}\n\\omega^p_j=\\Phi^p_j\/\\bar{\\tau}_j,\n\\label{eq:omega_def}\n\\end{equation}\nfor all phase channels $p$ (see equation \\ref{eq:phase_parity} for definition). We extracted the eEDM from $\\omega_j^{\\N\\E}$, which in the absence of systematic errors would be given by $\\omega^{\\N\\E}=-d_e\\Eeff$ independent of $j$.\n\nFrom here on we will drop the $j$ subscript that denotes a grouping of $n$ adjacent asymmetry points about a particular time after ablation $t_j$; it is implicit that independent phase measurements were computed from many separate groups of data, each with different values of $t_j$ across the duration of the molecule pulse. At the end of the analysis, and whenever it was convenient to do so, we implicitly performed weighted averaging across the $j$ subscript.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.9]{omega_nb_vs_eb.pdf}\n\\caption{The difference between magnetic moments of the two $\\Omega$-doublet levels as measured by $\\omega^{\\N\\B}$. As expected, this phase component scales linearly with $\\E$ and $\\B_z$. The constant of proportionality is $\\eta \\mu_{\\rm B}$. Reproduced with permission from \\cite{Petrov2014}}\n\\label{fig:delta_g}\n\\end{figure}\n\nOther phase channels couuld be used to search for and monitor systematic errors, discussed in detail in section~\\ref{sec:systematics}, or to measure properties of ThO, as is the case with $\\omega^{\\N\\B}$. We discuss the latter case here. This channel provided a measure of $\\Delta g$, the magnetic moment difference between upper and lower $\\Nsw$-levels, arising from perturbations due to other electronic and rotational states \\cite{Bickman2009,Petrov2014}. Because this difference limits the extent to which the $\\Nsw$ reversal can suppress certain systematic errors \\cite{Vutha2010}, it is an important quantity both in our experiment and in other experiments measuring eEDMs in molecules with $\\Omega$-doublet structure \\cite{JILAEDM}. Figure~\\ref{fig:delta_g} illustrates an observed linear dependence $\\Delta g\/2=\\eta \\E$, as predicted \\cite{Bickman2009,Petrov2014}. Since $\\E$ and $\\B_z$ are precisely known from auxiliary measurements, the constant $\\eta$ can be directly calculated from our angular frequency measurements:\n\\begin{equation}\n\\label{eq:eta_2}\n\\eta=-\\frac{\\omega^{\\N\\B}}{\\mu_{\\rm B}\\E\\B_z}.\n\\end{equation}\nOur measured value of $\\eta=-0.79\\pm0.01~\\rm{nm}\/\\rm{V}$ was approximately half of what one would compute using the methods developed to understand the effect in the PbO molecule \\cite{Hamilton2010,Bickman2009}. This discrepancy was subsequently understood as being primarily due to coupling to other fine-structure components in the $^3\\Delta$ manifold \\cite{Petrov2014,HutzlerThesis}. The $\\omega^{\\N\\B}$ channel illustrates the importance of understanding phase channels besides that corresponding to the eEDM.\n\n\\subsection{Data Cuts}\n\\label{sec:data_cuts}\nThree data cuts were applied as part of the analysis: fluorescence rate threshold (see section~\\ref{sec:signal_asymmetry}), polarisation bin (see below), and contrast threshold (see below). These cuts made sure that we only used data taken under appropriate experimental conditions (e.g. only when lasers remained locked etc.) and thus ensured a high signal to noise ratio for the data used to extract the eEDM value. We thoroughly investigated how each of these cuts affected the calculated eEDM mean and uncertainty.\n\nAs previously mentioned, a fluorescence threshold cut of about $F_{\\rm{cut}}=3\\times10^5$~s$^{-1}$, was applied to each trace (average of 25 molecule pulses) to ensure that the fluorescence rate would always be larger than the background rate. This threshold was chosen to include the maximum number of asymmetry points in our measurement while also excluding low signal-to-noise asymmetry measurements that would increase the overall eEDM uncertainty, as described below. We also removed entire blocks (complete sets of $\\Nsw$,$\\Esw$,$\\Bsw$,$\\tilde{\\theta}$) of data from the analysis if \nany of the block's experiment states had $\\lesssim0.5$~ms of fluorescence data above $F_{\\rm{cut}}$. \n\nThe count rates of uncorrelated fluorescence photoelectrons exhibit Poissonian statistics. In each block we averaged together four traces with the same experimental configuration. After such averaging, the number of detected photoelectrons within a pair of laser polarisation bins was ${\\gtrsim}50$, which was large enough that the photoelectron number distribution closely resembled a normal distribution. Because the asymmetry was defined as a ratio of two approximately normally distributed random variables ($F_X-F_Y$ and $F_X+F_Y$), its distribution was not necessarily normal. Rather, it approached a normal distribution in the limit of large $F_X+F_Y$ \n\\cite{HutzlerThesis}. The same followed for all quantities computed from the asymmetry, including the eEDM. The fluorescence threshold cut therefore ensured that the distribution of eEDM measurements was very nearly normally distributed. Including low-signal data would have caused the distribution to deviate from normal and increase the overall uncertainty. To check that this signal size cut did not lead to a systematic error in our determination of $d_e$, the eEDM mean and uncertainty were calculated for multiple $F_{\\rm{cut}}$ values, as shown in figure~\\ref{fig:cuts}. If the cut was increased above $6\\times10^5$~s$^{-1}$ the mean value was seen to move slightly (but within the computed uncertainties), and the uncertainty to increase. However, for all plausible values of the cuts the resulting value of $d_e$ was consistent, within uncertainties, with our final stated value.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=14cm]{edm_vs_cut_new.pdf}\n\\caption{Measured eEDM mean and uncertainty as a function of (A) fluorescence signal threshold and (B) polarisation bin size and position. For the former, a value of $3\\times10^5$~s$^{-1}$ was used for the final result. For the latter, the two leftmost data points correspond to the polarisation bins used.}\n\\label{fig:cuts}\n\\end{figure}\n\nAs described in section~\\ref{sec:signal_asymmetry}, data points within a polarisation bin were averaged together when calculating the asymmetry (cf. figure~\\ref{fig:modulation}A). These data points were separated by 200~ns. Numbering these points from when the readout laser beam polarisation is switched, we binned points 5--20 or points 0--25, depending on the analysis routine (see section~\\ref{ssec:differences_between_data_analysis_routines} below) when reporting our final result. The former choice was made to cut out background signal and overlapping fluorescence between polarisation states while retaining as much of the fluorescence signal as possible whereas the latter was chosen to minimize the statistical uncertainty given the lack of evidence for systematic errors that depended on time within the polarisation switching cycle. As shown in figure~\\ref{fig:cuts}, we checked for systematic errors associated with this choice by also using several different polarisation bins to compute the eEDM. The eEDM uncertainty increased, as expected, for polarisation bins that cut out data with significant fluorescence levels, but the mean values were all consistent with each other within their respective\nuncertainties.\n\nIn order for a block of data to be included in our final measurement, we also required that \neach of the 8 $(\\Nsw,\\Esw,\\Bsw)$ experiment states\nhad a measured fringe contrast above 80\\%. The primary cause of blocks failing to meet this requirement was the state preparation laser becoming unlocked. This cut resulted in less than 1\\% of blocks being discarded. If the contrast cut was lowered, or not applied at all, the eEDM mean and uncertainty change by less than 3\\% of our statistical uncertainty. As with the signal threshold, if this cut threshold was increased to 90\\%, close to the average value of contrast, $\\mathcal{C}$, then a larger fraction of data was neglected and the eEDM uncertainty was seen to increase.\n\nFor all the cuts discussed, we significantly varied the associated cut and in some cases removed it entirely. The eEDM mean and uncertainty were very robust against significant variation of each of these cuts, and the cuts were chosen before the blind offset applied to the eEDM channel was removed.\n\n\\subsection{Differences Between Data Analysis Routines}\n\\label{ssec:differences_between_data_analysis_routines}\nAs a systematic error check, we performed three independent analyses of the data. Each routine followed the general analysis method described above, but varied in many small details such as background subtraction method, cut thresholds, numbers of points grouped together to compute asymmetry, polarisation bin choice, etc. The analyses differed in the polynomial order of the fits applied to both the contrast $\\mathcal{C}$ and the precession time $\\tau$ vs.\\ time after ablation $t$. The analyses also differed in the inclusion of a subset of the eEDM data that featured a particularly large unexplained signal in the $\\omega^\\N$ channel.\n\nEach of the three analyses independently computed the eEDM channel and the systematic error in the eEDM channel. The uncertainties for all three routines were nearly identical, and the means agreed to within $\\Delta\\omega^{\\N\\E}<3~\\rm{mrad\/s}$, which is within the statistical uncertainty of the measurement $\\delta\\omega^{\\mathcal{NE}}=\\pm4.8$~mrad\/s. The eEDM mean and uncertainty were averaged over the three analyses to produce the final result.\n\n\\subsection{EDM Mean and Statistical Uncertainty}\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=15.5cm]{EDM_statistics.pdf}\n\\caption{The data set associated with our reported eEDM limit. \\textbf{(A)} Variations in the extracted eEDM as a function of position within the molecular pulse. \\textbf{(B)} Over 10,000 blocks of data were taken over a combined period of about two weeks. \\textbf{(C)}-\\textbf{(D)} The distribution of $\\sim$200,000 separate eEDM measurements (black) matches very well with a Gaussian fit (red). The same data is plotted with both a linear and a log scale. In these histograms the mean of each individual measurement was normalized to its corresponding error bar.}\n\\label{fig:statistics}\n\\end{figure}\nThe final data set used to report our result is shown in figure~\\ref{fig:statistics}. It consisted of ${\\sim}10^4$ blocks of data taken over the course of $\\sim$2~weeks (figure~\\ref{fig:statistics}B); each block contains ${\\approx}20$ separate eEDM measurements distributed over the duration of the molecule pulse (Figure~\\ref{fig:statistics}A). All ${\\approx}2\\times10^5$~measurements were combined with standard Gaussian error propagation to obtain the reported mean and uncertainty. Figure~\\ref{fig:statistics}C,D shows histograms of all measurements on a linear (C) and log (D) scale, showing the distribution agrees extremely well with a Gaussian fit. The resulting uncertainty was about 1.2 times that expected from the photoelectron shot-noise limit, taking into account the photoelectron rate from molecule fluorescence, background light, and PMT dark current. When the eEDM measurements were fit to a constant value, the reduced $\\chi^2$ was $0.996\\pm0.006$ where this uncertainty represents the $1\\sigma$ width of the $\\chi^2$ distribution for the appropriate number of degrees of freedom.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=10cm]{edm_stats_v3.pdf}\n\\caption{Measured $\\omega^{\\mathcal{NE}}$ values grouped by the states of $\\left|\\mathcal{B}_{z}\\right|,$~$\\left|\\mathcal{E}_{z}\\right|$, $\\hat{k}\\cdot\\hat{z}$, and each superblock switch, before systematic error corrections. Reproduced with permission from \\cite{Baron2014}.}\n\\label{fig:edm_vs_superblock}\n\\end{figure}\n\nWhen computing the eEDM result, data from superblocks were averaged together. The mean could be either weighted or unweighted by the statistical uncertainty in each superblock state. Weighted averaging minimized the resulting statistical uncertainty, but unweighted averaging could  suppress systematic errors that have well-defined superblock parity from entering into the extracted value for $\\omega^{\\N\\E}$.\n\nDue to molecule number fluctuations, each block of data had a different associated uncertainty. However, roughly equal amounts of data were gathered for the $2^4$ superblock states defined by the state readout parity $\\Psw$, field plate lead configuration $\\Lsw$, state readout laser polarisation $\\Rsw$, and global laser polarisation $\\Gsw$. For the reported eEDM value, unweighted averaging (or to be precise, performing the basis change prescribed by equation \\ref{eq:general_parity}) was used to combine data from the different $\\Psw$, $\\Rsw$, $\\Lsw$, $\\Gsw$ experiment states, since there were known systematic errors with well-defined superblock parity that were suppressed by these switches (see, for example, sections \\ref{sssec:stark_interference_between_E1_and_M1_transition_amplitudes} and \\ref{ssec:asymmetry_effects}). Note, however, that figure~\\ref{fig:edm_vs_superblock} shows that these systematic errors produced no significant eEDM shift, and that the overall uncertainty was comparable (within 10\\%) when the data was combined with weighted or unweighted averaging.\n\nUnequal amounts of data were collected for the $\\B_z$, $\\E$, and $\\hat{k}\\cdot\\hat{z}$ experimental states. For example, 40\\% (60\\%) of data were gathered with the state preparation and readout laser beams pointing east (west), $\\hat{k}\\cdot\\hat{z}=-1 (+1)$. To account for this, we performed state-by-state analysis of the systematic errors: the primary systematic errors (described in section \\ref{sssec:correlated_laser_parameters}) were allowed to depend on the magnitude of the magnetic field (though $\\B_z=1,40$ mG were grouped together), and the pointing direction, and separate systematic error subtractions were performed for each ($\\B_z$, $\\hat{k}\\cdot\\hat{z}$) state. After this subtraction, the systematic uncertainties were added in quadrature with the statistical uncertainties for each state, and the data from each state was averaged together weighted by the resulting combined statistical and systematic uncertainties.\n\nThe reported statistical uncertainty was obtained via the method above assuming no systematic uncertainty. The reported systematic uncertainty was defined such that the quadrature sum of the reported statistical and systematic uncertainties gives the same value as when incorporating the state-by-state analysis. A description of the methods used to evaluate the systematic error and the systematic uncertainty in the measurement is provided in section \\ref{ssec:total_systematic_error_budget}.\nTo prevent experimental bias we performed a blind analysis by adding an unknown offset to the mean of the eEDM channel, $\\omega^{\\mathcal{NE}}$. The offset was randomly generated in software from a Gaussian distribution with standard deviation $\\sigma=150$~mrad\/s and mean zero. The mean, statistical error, procedure for calculating the systematic error, and procedure for computing the reported confidence interval were all determined before revealing and subtracting the blind offset.\n\n\\subsection{Confidence Intervals}\n\\label{ssec:confidence_intervals}\nA classical (i.e.\\ frequentist) confidence interval \\cite{Riley2006} is a natural choice for reporting the result of an eEDM measurement.\nFor repeated and possibly different experiments measuring the eEDM, the frequency with which the confidence intervals include or exclude the value $\\de=0$ suggests whether the results are consistent or inconsistent, respectively, with the Standard Model.\nFurthermore, the confidence level (C.L.) represents an objective measure of the \\emph{a priori} probability that the confidence interval assigned to any one of these measurements, selected at random, includes the unknown true value of the eEDM $d_{e,{\\rm true}}$. \nSince no statistically significant eEDM has yet been observed, the recent custom has been for electron eEDM experiments to report an upper limit at the 90\\% C.L. \\cite{Regan2002,Hudson2011}. \nThe proper interpretation of such limits is that if the experiment were performed a large number of times, and the confidence interval were \\emph{computed in the same way} for each experimental trial, $d_{e,\\rm true}$ would fall within the interval 90\\% of the time.\n\nFeldman and Cousins pointed out that in order for this interpretation to be valid, the confidence interval construction must be independent of the result of the measurement \\cite{Feldman1998}. If the procedure for constructing 90\\% confidence intervals is chosen contingent upon the measurement outcome, the resulting intervals may `undercover', i.e. fail to include the true value more than 10\\% of the time. This happens, for example, if an upper bound is reported whenever the measured result falls within a few standard deviations of zero, and a two-sided confidence interval is reported whenever the measured result is significant at more than a few-sigma level. Feldman and Cousins termed this inconsistent approach `flip-flopping'.\n\n\nIn order to avoid flip-flopping, we chose a confidence interval construction, the Feldman-Cousins method described in reference~\\cite{Feldman1998}, that consistently unifies these two limits. We applied this method to a model with Gaussian statistics, in which the measured magnitude of the eEDM channel, $x=|\\omega^{\\N\\E}_{T,{\\rm meas}}|$, is sampled from a folded Gaussian distribution\n\\begin{equation}\\label{eq:foldednormal}\nP(x|\\mu)=\\frac{1}{\\sigma\\sqrt{2\\pi}}\\left(\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]+\\exp\\left[-\\frac{(x+\\mu)^2}{2\\sigma^2}\\right]\\right),\n\\end{equation}\nwhere the location parameter is the unknown true magnitude of the eEDM channel, $\\mu=|\\omega^{\\N\\E}_{T,{\\rm true}}|$, and the scale parameter $\\sigma$ is equal to the quadrature sum of the statistical and systematic uncertainties given in equation~\\ref{eq:wNEt_num_err_comb} and at the bottom of table~\\ref{tbl:syst_error}. \n\nThe central idea of the Feldman-Cousins approach is to use an ordering principle which, for each possible value of the parameter of interest $\\mu$, ranks each possible measurement outcome $x$ by the `strength' of the evidence it provides that $\\mu$ is the true value. The values of $x$ that provide the strongest evidence for each value of $\\mu$ are included in the confidence band for that value. In the Feldman-Cousins method, the metric for the strength of evidence is the likelihood of $\\mu$ given that $x$ is measured [i.e. $\\mathcal{L}(\\mu|x) = P(x|\\mu)$], divided by the largest probability $x$ can possibly achieve for any value of $\\mu$. The denominator in this prescription takes into account the fact that an experimental result that is somewhat improbable under a particular hypothesis can still provide good evidence for that hypothesis if the result is similarly improbable under even the most favorable hypothesis. This approach has its theoretical roots in likelihood ratio testing \\cite{Stuart1999}.\n\nOur specific  procedure for computing confidence intervals was a numerical calculation performed using the following recipe (cf.\\ figure~\\ref{fig:fc_conf_int}):\n\\begin{enumerate}\n\\item{Construct the confidence bands on a Cartesian plane, of which the horizontal axis represents the possible values of $x$ and the vertical axis the possible values of $\\mu$. Divide the plane into a fine grid with $x$-intervals of width $\\Delta_x$ and $\\mu$-intervals of height $\\Delta_{\\mu}$. We will consider only the discrete possible values $x_i = i \\Delta_x$ and $\\mu_j = j \\Delta_{\\mu}$, where the index $i$($j$) runs from $0$ to $n_x$($n_{\\mu}$).}\\label{it:setup}\n\\item{For all values of $i$, maximize $P(x_i|\\mu_j)$ with respect to $\\mu_j$. Label the maximum points $\\mu^{\\mathrm{max},i}$.}\\label{it:ymax}\n\\item{For some value of $j$, say $j=0$, compute the likelihood ratio $R(x_i) = P(x_i|\\mu_j)\/P(x_i|\\mu^{\\mathrm{max},i})$ for every value of $i$.}\\label{it:R}\n\\item{Construct the `horizontal acceptance band' at $\\mu_j$ by including values of $x_i$ in descending order of $R(x_i)$. Stop adding values when the cumulative probability reaches the desired C.L. of 90\\%, i.e., $\\displaystyle \\sum_{x_i}P(x_i|\\mu_j)\\Delta_x = 0.9$.}\\label{it:horiz}\n\\item{Repeat steps (\\ref{it:R})--(\\ref{it:horiz}) for all values of $j$.}\n\\item{To determine the reported confidence interval, draw a vertical line on the plot at $x = |\\omega^{\\N\\E}_{T,{\\rm meas}}|$. The 90\\% confidence interval is the region where the line intersects the constructed confidence band.}\n\\end{enumerate}\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{FC_conf_int.pdf}\n\\includegraphics[width=0.49\\textwidth]{Conf_int_compare.pdf}\n\\caption{Left: Feldman-Cousins confidence bands for a folded Gaussian distribution, constructed as described in the text, for a variety of confidence levels. Each pair of lines indicates the upper and lower bounds of the confidence band associated with each C.L. To the left of the $x$-intercepts, the lower bounds are zero. Confidence bands are plotted as a function of the possible measured central values $x$ scaled by the standard deviation $\\sigma$, and our result is plotted as a vertical dot-dashed line. The $\\mu$-value of the point at which our result line intersects with each of the colored lines gives the upper limit of our measurement at different C.L.'s. Right: Comparison between 90\\% confidence intervals computed using three different methods, described in the text. Confidence bands are plotted as a function of the possible measured central values of a quantity $x$ scaled by the standard deviation $\\sigma$. Our result, $|\\omega^{\\N\\E}_{T,{\\rm meas}}|\/\\sigma=0.46$, is plotted as a vertical dot-dashed line. The $\\mu$-values of the points at which our result line intersects the upper and lower line for each method give the upper and lower bounds of three possible 90\\% confidence intervals for our measurement. To avoid invalidating the confidence interval by flip-flopping, our result should be interpreted using the Feldman-Cousins method, which we chose before unblinding.}\n\\label{fig:fc_conf_int}\n\\end{figure}\n\nThe left-hand plot in figure~\\ref{fig:fc_conf_int} was generated using the prescription above at several different C.L.'s. \nNote that the 90\\% confidence intervals switch from upper bounds to two-sided confidence intervals when the value of $|\\omega^{\\N\\E}_{T,{\\rm meas}}|$ becomes larger than $1.64 \\sigma$. This is the level of statistical significance required to exclude the value $\\de = 0$ from a 90\\% C.L.\\ central Gaussian confidence band.\n\nFrom equation~(\\ref{eq:wNEt_num_err_comb}), we find $|\\omega^{\\N\\E}_{T,{\\rm meas}}|=0.46\\sigma$ with $\\sigma=5.79~\\rm{mrad}\/\\rm{s}$. In our confidence interval construction, this corresponds to an upper bound of $|\\wNEt|<1.9\\sigma=11~\\rm{mrad}\/\\rm{s}$ (90\\% C.L.). \nA comparison between three different 90\\% confidence interval constructions for small values of $\\mu$ is shown in the right-hand plot of figure~\\ref{fig:fc_conf_int}. The black dashed lines represent the central confidence band for the signed values (rather than the magnitude) of $\\mu$ and $x$, where $\\mu$ is the mean of a Gaussian probability distribution in $x$. The blue lines give an upper bound constructed by computing the the value of $\\mu$ such that the cumulative distribution function for the folded Gaussian in equation~\\ref{eq:foldednormal} is equal to $0.9$ for each value of $x$. It should be noted that this upper bound is more conservative than a true classical 90\\% confidence band, as it overcovers for small values of $\\mu$ (e.g., if the true value were $\\mu_{\\rm true} < 1.64 \\sigma$, the confidence intervals of 100\\% of experimental results would include $\\mu_{\\rm true}$). We nevertheless include this construction for comparison because we believe that previous experiments have reported EDM upper bounds using this method \\cite{Hudson2011,Griffith2009,Regan2002}. These intervals have a valid interpretation as Bayesian `credible intervals' conditioned on a uniform prior for $\\mu$ \\cite{Feldman1998}. Finally, the red lines represent the Feldman-Cousins approach described here, which unifies upper limits and two-sided intervals. For our measurement outcome, indicated by the vertical dot-dashed line, the Feldman-Cousins intervals yield a $7\\%$ larger eEDM limit than the folded Gaussian upper bound would have.\n\n\\subsection{Physical Quantities}\n\\label{ssec:physical_quantities}\n\n\nUnder the most general interpretation, our experiment is sensitive to any $P$- and $T$-violating interaction that produces an energy shift $\\wNEt$. The eEDM is not the only such predicted interaction for diatomic molecules \\cite{Kozlov1995}, and in particular a $P$- and $T$-odd nucleon-electron scalar-pseudoscalar interaction would also manifest as a $\\Nsw\\Esw$-odd phase in our experiment. Thus, we write\n\\begin{equation}\n\\wNEt=-d_e\\Eeff + W_{\\rm S}C_{\\rm S},\\footnote{Note that the sign of the $C_{\\rm S}$ term is opposite to that used, incorrectly, in our original paper \\cite{Baron2014}. In addition, here $W_{\\rm S}$ differs in magnitude from the related quantity $W_{\\rm T,P}$ given explictly in \\cite{Denis2016,Skripnikov2016}. A detailed discussion of the sign and notational conventions for this Hamiltonian is provided in \\ref{sec:sign_conventions}.}\n\\end{equation}\nwhere $W_{\\rm S}$ is a (calculated) energy scale specific to the species of study \\cite{Skripnikov2013,Dzuba2011a,DzubaErratum2012,Denis2016,Skripnikov2016} and $C_{\\rm S}$ is a dimensionless constant characterizing the strength of the $T$-violating nucleon-electron scalar-pseudoscalar coupling relative to the ordinary weak interaction. \n\nWe can use our measurement to set an upper limit on $\\de$ by assuming that $C_{\\rm S}=0$ and that $\\wNEt$ is therefore entirely attributable to the eEDM. Taking the effective electric field to be the unweighted mean of the two most recent calculations of this quantity \\cite{Denis2016,Skripnikov2016}, $\\Eeff=78~{\\rm GV\/cm}$, we can interpret our result in equation~(\\ref{eq:wNEt_num_err_comb}) as:\n\\begin{align}\n\\de&=(-2.2\\pm4.8)\\times10^{-29}~e\\cdot{\\rm cm}\\\\\n\\Rightarrow|\\de|&<9.3\\times10^{-29}~e\\cdot{\\rm cm}~(90\\%\\,{\\rm C.L.}),\n\\end{align}\nwhere the second line is obtained by appropriately scaling the upper bound on $\\wNEt$ derived in section \\ref{ssec:confidence_intervals}.\n\nIf, instead, we assume that $d_e=0$, our measurement of $\\wNEt$ in ThO can be restated as a measurement of $C_{\\rm S}$. Using an unweighted mean of the most recent calculations of the interaction coefficient, $W_{\\rm S} = -2\\pi\\times 282~{\\rm kHz}$ \\cite{Denis2016,Skripnikov2016}, we obtain:\n\\begin{align}\nC_{\\rm S}&=(-1.5\\pm3.2)\\times10^{-9}\\\\\n\\Rightarrow|C_{\\rm S}|&<6.2\\times10^{-9} \\:(90\\%\\,{\\rm C.L.}),\n\\end{align}\nwhich, at the time, was an order of magnitude smaller than the existing best limit set by the ${}^{199}$Hg EDM experiment \\cite{Swallows2013}, and is still a factor of 2 smaller than the recently improved limit from the same group \\cite{Heckel2016}.\n\n\\subsection{Determining Systematic Errors and Uncertainties}\n\\label{ssec:determining_systematic_uncertainty}\n\n\\begin{table}[tbp]\n\\caption{\\label{tbl:syst_check}Parameters varied during our systematic error search. Left: Category I Parameters --- These were ideally zero under normal experimental running conditions and we were able to vary them significantly from zero. For each of these parameters direct measurements or limits were placed on possible systematic errors. Right: Category II Parameters --- These had no single ideal value. Although direct limits on these systematic errors could not be derived, they served as checks for the presence of unanticipated systematic errors. See the main text for more details on all the systematic errors referenced.}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\begin{tabular}[t]{l}\n\\br\nCategory I Parameters\\\\\n\\mr\n\\textbf{Magnetic Fields}\\\\\n- Non-reversing $\\B$-field: $\\B_{z}^{\\rm{nr}}$\\\\\n- Transverse $\\B$-fields: $\\B_{x},\\B_{y}$\\\\\n(both even and odd under $\\Bsw$)\\\\\n- $\\B$-field gradients: \\\\\n$\\frac{\\partial\\B_{x}}{\\partial x},\\frac{\\partial\\B_{y}}{\\partial x},\\frac{\\partial\\B_{y}}{\\partial y},\\frac{\\partial\\B_{y}}{\\partial z},\\frac{\\partial\\B_{z}}{\\partial x},\\frac{\\partial\\B_{z}}{\\partial z}$\\\\\n(both even and odd under $\\Bsw$)\\\\\n- $\\Esw$ correlated $\\B$-field: $\\B^{\\E}$\n(to simulate\\\\\n$\\vec{v}\\times\\vecE$\/geometric phase\/leakage current)\\\\\n\\textbf{Electric Fields}\\\\\n- Non-reversing $\\E$-field: $\\E^{\\rm{nr}}$\\\\\n- $\\E$-field ground offset\\\\\n\\textbf{Laser Detunings}\\\\\n- State preparation\/readout lasers: $\\Delta_{{\\rm prep}}^{\\rm nr}$, $\\Delta_{\\rm{read}}^{\\rm nr}$\\\\\n- $\\Psw$ correlated detuning, $\\Delta^{\\mathcal{P}}$\\\\\n- $\\Nsw$ correlated detunings: $\\Delta^{\\N}$\\\\\n\\textbf{Laser Pointings}\\\\\n- Change in pointing of prep.\/read lasers\\\\\n- State readout laser $\\hat{X}\/\\hat{Y}$ dependent pointing\\\\\n- $\\Nsw$ correlated laser pointing\\\\\n- $\\Nsw$ and $\\hat{X}\/\\hat{Y}$ dependent laser pointing\\\\\n\\textbf{Laser Powers}\\\\\n- $\\Nsw\\Esw$ correlated power $\\Omega_{\\rm r}^{\\N\\E}$\\\\\n- $\\Nsw$ correlated power $\\Omega_{\\rm r}^{\\N}$\\\\\n- $\\hat{X}\/\\hat{Y}$ dependent state readout laser power, $\\Omega_{\\rm r}^{XY}$\\\\\n\\textbf{Laser Polarisation}\\\\\n- Preparation laser ellipticity, $S_{{\\rm prep}}$\\\\\n\\textbf{Molecular Beam Clipping}\\\\\n- Molecule beam clipping along $\\hat{y}$ and $\\hat{z}$\\\\\n(changes $\\left\\langle v_{y}\\right\\rangle $,$\\left\\langle v_{z}\\right\\rangle $,$\\left\\langle y\\right\\rangle $,$\\left\\langle z\\right\\rangle $\nof molecule beam)\\\\\n\\br\n\\end{tabular}\n\\end{minipage}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\begin{tabular}[t]{l}\n\\br\nCategory II Parameters \\\\\n\\mr\n\\textbf{Laser Powers}\\\\\n- Power of prep.\/read lasers \\\\\n\\textbf{Experiment Timing}\\\\\n- $\\hat{X}$\/$\\hat{Y}$ polarisation switching rate\\\\\n- Number of molecule pulses averaged \\\\\nper experiment trace\\\\\n\\textbf{Analysis}\\\\\n- Signal size cuts, asymmetry size cuts,\\\\\ncontrast cuts\\\\\n- Difference between two PMT detectors\\\\\n- Variation with time within molecule pulse\\\\\n(serves to check $v_{x}$ dependence)\\\\\n- Variation with time within polarisation \\\\\nswitching cycle\\\\\n- Variation with time throughout the \\\\\nfull data set (autocorrelation)\\\\\n- Search for correlations between all channels \\\\\nof phase, contrast and fluorescence signal \\\\\n- Correlations with auxiliary measurements\\\\\nof $\\B$-fields, laser powers, vacuum pressure\\\\\nand temperature\\\\\n- 3 independent data analysis routines\\\\\n\\br\n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\nIn total, we varied more than 40 separate parameters during our search for systematic errors (see Table \\ref{tbl:syst_check}). These fall into two categories.\nCategory I contains parameters $P$ which are optimally zero; $P\\neq0$ represents an experimental imperfection. We were able to use experimental data to put a direct limit on the size of possible systematic errors proportional to these parameters. Category II contains parameters that have no optimum value and which we could vary significantly without affecting the nature of the spin precession measurement. The variation of these parameters could reveal systematic errors and serve as a check that we understood the response of our system to those parameters, but no quantitative bounds on the associated systematic errors were derived. \n\nFor each Category I parameter $P$, we exaggerated the size of the imperfection by a factor greater than $10$, if possible, relative to the maximum size of the imperfection under normal operating conditions, $\\bar{P}$, which was obtained from auxiliary measurements. Following previous work \\cite{Regan2002,Griffith2009,Hudson2011}, we assumed a linear relationship between $\\omega^{\\mathcal{NE}}$ and $P$, and extracted the sensitivity of the $\\omega^{\\mathcal{NE}}$ to parameter $P$, $\\partial\\omega^{\\mathcal{NE}}\/\\partial P$. The systematic error under normal operating conditions was computed as $\\omega_P^{\\N\\E}=(\\partial\\omega^{\\N\\E}\/\\partial P)\\bar{P}$. The statistical uncertainty in the systematic error (henceforth referred to as the systematic uncertainty) $\\delta\\omega_P^{\\N\\E}$ was obtained from linear error propagation of uncorrelated random variables,\n\\begin{equation}\n\\delta\\omega_P^{\\N\\E}= \\sqrt{\\left(\\frac{\\partial\\omega^{\\N\\E}}{\\partial P}^{\\:}\\delta\\bar{P}\\right)^{2}+\\left(\\bar{P}^{\\:}\\delta\\frac{\\partial\\omega^{\\N\\E}}{\\partial P}\\right)^{2}},\n\\end{equation}\nwhere $\\delta\\bar{P}$ is the uncertainty in $\\bar{P}$ and $\\delta\\partial\\omega^{\\N\\E}\/\\partial P$ is the uncertainty in $\\partial\\omega^{\\N\\E}\/\\partial P$. \n\nFor parameters that had been observed to produce statistically significant shifts in $\\omega^{\\mathcal{NE}}$, such as the non-reversing electric field, $\\E^{\\rm{nr}}$, we monitored the size of the systematic error throughout the reported data set during \\textit{Intentional Parameter Variations} (described in section \\ref{sec:Measurement_scheme_more_detail}) and deducted this quantity from $\\omega^{\\mathcal{NE}}$ to give a value of the spin precession frequency due to T-odd interactions in the H state of ThO, $\\wNEt=\\omega^{\\N\\E}-\\sum_{P}\\omega_P^{\\N\\E}$. Most Category I parameters did not cause a statistically significant $\\omega^{\\N\\E}_P$ and were not monitored. For these parameters, we did not subtract $\\omega_P^{\\N\\E}$ from $\\omega^{\\mathcal{NE}}$, but rather included an upper limit of $\\left[(\\omega_P^{\\N\\E})^{2}+(\\delta\\omega_P^{\\N\\E})^{2}\\right]^{1\/2}$ in the systematic uncertainty on $\\wNEt$, or chose to omit this parameter from the systematic error budget altogether based on the criteria described in section \\ref{ssec:total_systematic_error_budget}.\n\nWhere applicable, we also fit higher-order polynomial functions to $\\omega^{\\mathcal{NE}}$ with respect to $P$ during the systematic error searches. No significant increase in the systematic uncertainty was observed using such fits and hence the contributions to the systematic error budget in Table \\ref{tbl:syst_error} were all estimated from linear fits. We note, however, that certain non-linear dependences of $\\omega^{\\mathcal{NE}}$ on $P$ could lead to underestimates of the systematic uncertainty, for example if $\\omega^{\\mathcal{NE}}$ has a small (large) nonzero value for large (small) values of $P$. In efforts to avoid this, data were taken over as wide a range as possible, it is, however, always possible that such non-linear dependence is present between the parameter values for which we took data. We had no models by which non-linear dependence could manifest by variation of the parameters investigated, so we believe the procedure outlined above produced accurate estimates of the systematic errors.\n\n\n\\subsection{Systematic Errors Due to Imperfect Laser Polarisations}\n\\label{ssec:systematic_errors_due_to_imperfect_laser_polarizations}\n\nThe dominant systematic errors in our experiment were due to imperfections in the laser beams used to prepare the molecular and read out the molecular state. Non-ideal laser polarisations combined with laser parameters correlated with the expected eEDM signal resulted in three distinct systematic errors which we refer to as the $\\E^{\\rm{nr}}$, $\\Omega_{\\rm r}^{\\N\\E}$, and Stark Interference (S.I.) systematic errors. In this section, we model the effects of several types of polarisation imperfections on the measured phase $\\Phi$ (sections~\\ref{sssec:stark_interference_between_E1_and_M1_transition_amplitudes} and \\ref{sssec:AC_stark_shift_phases}) and discuss the correlated laser parameters that couple to these polarisation imperfections to result in systematic errors (section~\\ref{sssec:correlated_laser_parameters}). We then discuss how we were able to suppress and quantify the residual systematic errors in the eEDM experiment (sections~\\ref{sssec:suppression_of_the_AC_stark_shift_phases} and \\ref{sssec:correlated_laser_parameters}).\n\n\\subsubsection{Idealized Measurement Scheme with Polarisation Offsets}\n\\label{sssec:idealized_measurement_scheme_with_polarization_offsets}\n\nAs described in section \\ref{sec:Measurement_scheme}, the molecules initially enter the state preparation laser beam in an incoherent mixture of the two states $|\\pm,\\Nsw\\rangle$. The bright state $|B(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw_{{\\rm prep}})\\rangle$ is then optically pumped away through $|C,\\Psw_{{\\rm prep}}\\rangle$ leaving behind the dark state $|D(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw_{{\\rm prep}})\\rangle$ as the initial state for the spin precession. The molecules then undergo spin precession by angle $\\phi$ evolving to a final state $|\\psi_{f}\\rangle=U(\\phi)|D(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw_{{\\rm prep}})\\rangle$ where $U(\\phi)=\\sum_{\\pm}e^{\\mp i\\phi}|\\pm,\\Nsw\\rangle\\langle\\pm,\\Nsw|$ is the spin precession operator. The molecules then enter the state readout laser that optically pumps the molecules with alternating polarisations $\\hat{\\epsilon}_{X}$ and $\\hat{\\epsilon}_{Y}$\n(which are nominally linearly polarised and orthogonal) between $|\\pm,\\Nsw\\rangle$ and $|C,\\Psw_{\\rm{read}}\\rangle$. For each polarisation, the optical pumping results in a fluorescence count rate proportional to the projection of the state onto the bright state, $F_{X,Y}=fN_{0}|\\langle B(\\hat{\\epsilon}_{X,Y},\\Nsw,\\Psw_{\\rm{read}})|\\psi_{f}\\rangle|^{2}$ where $f$ is the photon detection efficiency, and $N_{0}$ is the number of molecules in the addressed $\\Nsw$ level. We then compute the asymmetry, $\\mathcal{A}=(F_{X}-F_{Y})\/(F_{X}+F_{Y})$, dither the linear polarisation angles in the state readout laser beams to evaluate the fringe contrast, $\\mathcal{C}=(\\partial\\mathcal{A}\/\\partial\\phi)\/2\\approx-(\\partial\\mathcal{A}\/\\partial\\theta_{\\rm{read}})\/2$, and extract the measured phase, $\\Phi=\\mathcal{A}\/(2\\mathcal{C})+q\\pi\/4$.\\footnote{Recall $q$ is chosen to be an integer which depends on the size of the applied magnetic field.} We then report the result of the measurement in terms of an equivalent phase precession frequency $\\omega=\\Phi\/\\tau$ where $\\tau\\approx1^{\\:}\\mathrm{ms}$ is the spin precession time, which was measured for each block as described in section~\\ref{sec:compute_phase}.\n\nLet us first consider the idealized case in which all laser polarisations are exactly linear, $\\Theta_{i}=\\pi\/4$ for each laser $i\\in\\left\\{{\\rm prep},X,Y \\right\\}$, the angle between the state preparation laser polarisation (${\\rm prep}$) and state readout basis ($X,Y$) is $\\pi\/4$, $\\theta_{\\rm{read}}-\\theta_{{\\rm prep}}=-\\pi\/4$, and the accumulated phase is small, $\\left|\\phi\\right|\\ll1$ (i.e. no magnetic field is applied). Under these conditions, the measured phase $\\Phi$ is equal to the accumulated phase $\\phi$. Now consider the effect of adding polarisation offsets $d\\vec{\\epsilon}_{i}$ to each of the three laser beams such that $\\hat{\\epsilon}_{i}\\rightarrow\\hat{\\epsilon}_{i}+\\kappa d\\vec{\\epsilon}_{i}$, where $\\kappa = 1$ is a perturbation parameter. It is useful to cast the polarisation imperfections in terms of linear angle imperfections, $\\theta_{i}\\rightarrow\\theta_{i}+\\kappa d\\theta_{i}$ and ellipticity imperfections, $\\Theta_{i}\\rightarrow\\Theta_{i}+\\kappa d\\Theta_{i}$ where $S_{i}=-2d\\Theta_{i}$ is the laser ellipticity Stokes parameter; these are related by\n\\begin{equation}\n\\frac{\\hat{z}\\cdot(\\hat{\\epsilon}_{i}\\times d\\vec{\\epsilon}_{i})}{\\hat{\\epsilon}_{i}\\cdot\\hat{\\epsilon}_{i}}=d\\theta_{i}-id\\Theta_{i}.\\label{eq:extracting_polarization_imperfection_components}\n\\end{equation}\nNote that laser polarisations can have a nonzero projection in the $\\hat{z}$ direction, but we assume in the discussion above that $\\hat{\\epsilon}_{i}$ represents a normalized projection of the laser polarisation onto the $xy$ plane.\\footnote{The $z$-component of the polarisation can only drive $\\Delta M=0$ transitions, which are far off resonance from the state preparation\/readout lasers.} With these polarisation imperfections in place, the measured phase $\\Phi$ gains additional terms:\n\\begin{equation}\n\\Phi=\\phi+\\kappa(d\\theta_{{\\rm prep}}-\\frac{1}{2}(d\\theta_{X}+d\\theta_{Y}))-\\kappa^{2}\\Psw_{{\\rm prep}}\\Psw_{\\rm{read}}d\\Theta_{{\\rm prep}}(d\\Theta_{X}-d\\Theta_{Y})+O\\left(\\kappa^{3}\\right),\\label{eq:Measured_Phase_with_Polarization_Imperfections}\n\\end{equation}\nup to second order in $\\kappa$. In the eEDM measurement, we switch between two values of $\\Psw\\equiv\\Psw_{\\rm{read}}$, the parity of the excited state addressed during state readout, and we set $\\Psw_{{\\rm prep}}=+1$, the parity of the excited state addressed during state preparation. It is worth dwelling on equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections} for a moment. A rotation of all polarisations by the same angle leaves the measured phase unchanged: $d\\theta_{i}\\rightarrow d\\theta_{i}+d\\theta\\implies\\Phi\\rightarrow\\Phi$, as expected. A deviation in the relative angle between the state preparation and readout beams, $d\\theta_{{\\rm prep}}\\rightarrow d\\theta_{{\\rm prep}}+d\\theta$ and $d\\theta_{X,Y}\\rightarrow d\\theta_{X,Y}-d\\theta$, enters into the phase measurement as $\\Phi\\rightarrow\\Phi+2d\\theta$, but is benign so long as $d\\theta$ is uncorrelated with the expected eEDM signal. The laser ellipticities affect the\nphase measurement only when the state readout beams differ in ellipticity, and this contribution to the phase can be distinguished from the others by switching the excited state parity, $\\Psw$. This last term is particularly interesting because it allows for multiplicative couplings between polarisation imperfections in the state preparation and state readout beams to contribute to the measured phase.\n\nAlthough the polarisation imperfection terms in equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections} are uncorrelated with the $\\Nsw\\Esw$ and hence do not contribute to the systematic error, we will see in later sections that additional imperfections can lead to changes in the molecule state that is prepared or read-out that are equivalent to correlations $d\\theta_{i}^{\\N\\E}$ and $d\\Theta_{i}^{\\N\\E}$. The framework of equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections} is useful for understanding how these correlations result in systematic errors in the eEDM measurement extracted from $\\Phi^{\\N\\E}$.\n\n\n\\subsubsection{Stark interference between E1 and M1 transition amplitudes}\n\\label{sssec:stark_interference_between_E1_and_M1_transition_amplitudes}\n\nIn this section we describe in detail how interference between multipole transition amplitudes can lead to a measured phase that mimicks an eEDM spin precession phase. We develop a general framework illustrating how such phases depend on laser polarisation and pointing.\n\nIn an applied electric field, opposite parity levels are mixed, allowing both odd parity (E1, M2,...) and even parity (M1, E2,...) electromagnetic multipole amplitudes to contribute when driving an optical transition. These amplitudes depend on the orientation of the electric field relative to the light polarisation $\\hat{\\epsilon}$ and the laser pointing direction $\\hat{k}$. This Stark interference (S.I.) effect forms the basis of precise measurements of weak interactions through parity non-conserving amplitudes in atoms and molecules \\cite{Bouchiat1974,Demille2008,Wood1997}. However, it can also generate a systematic error in searches for permanent electric dipole moments which look for spin precession correlated with the orientation of an applied electric field. These Stark interference amplitudes have been calculated and measured for optical transitions in Rb \\cite{Chen1994,Hodgdon1991} and Hg \\cite{Lamoreaux1992,Loftus2011}, and have been included in the systematic error analysis in the Hg EDM experiment \\cite{Griffith2009,Swallows2013}.\n\nIn this section, we consider Stark interference as a source of systematic errors in the ACME experiment. There are two important differences between molecular and atomic systems. First, molecular states such as the $H^{3}\\Delta_{1}$ state in ThO can be highly polarisable and opposite parity states can be completely mixed by the application of a modest laboratory electric field. Second, molecular selection rules can be much weaker than atomic selection rules: the $H^{3}\\Delta_{1}\\rightarrow C^{3}\\Pi_{1}$ transition that we drive is nominally an E1 forbidden spin-flip transition ($\\Delta\\Sigma=1$, where $\\Sigma$ is the projection of the total electron spin $S=1$ onto the intermolecular axis), \nbut these states have significant subdominant contributions from other spin-orbit terms \\cite{Paulovic2003}, between some of which the E1 transition is allowed. Both of these effects significantly amplify the effect of Stark interference in molecules relative to atoms. In this section we will derive the effect of Stark interference on the measured phase $\\Phi$.\n\nConsider a plane wave vector potential $\\vec{A}$ with real amplitude $A_{0}$, oscillating at frequency $\\omega$, that is resonant with a molecular optical transition $\\left|g\\right\\rangle \\rightarrow\\left|e\\right\\rangle$,with wave vector $\\vec{k}=\\left(\\omega\/c\\right)\\hat{k}$, and complex polarisation $\\hat{\\epsilon}$:\n\\begin{align}\n\\vec{A}\\left(\\vec{r},t\\right)= & A_{0}\\hat{\\epsilon}e^{i\\vec{k}\\cdot\\vec{r}-i\\omega t}+\\mathrm{c.c}.\n\\end{align}\nThe interaction Hamiltonian $H_{\\mathrm{int}}$ between this classical light field and the molecular system is given by:\n\\begin{align}\nH_{\\mathrm{int}}\\left(t\\right)= & -\\sum_{a}\\frac{e^{a}}{m^{a}}\\vec{A}\\left(\\vec{r}^{\\,a},t\\right)\\cdot\\vec{p}^{a}\n\\end{align}\nwhere $a$ indexes a sum over all of the particles in the system with charge $e^{a}$, mass $m^{a}$, position $\\vec{r}^{\\,a}$ and momentum $\\vec{p}^{a}$. Typically we apply the multipole expansion on the transition matrix element between states $\\left|g\\right\\rangle $ and $\\left|e\\right\\rangle$; the matrix element can then be written as \n\\begin{equation}\n\\mathcal{M}\\equiv\\langle e|H_{\\mathrm{int}}|g\\rangle=iA_{0}\\omega_{eg}\\sum_{\\lambda=1}^{\\infty}\\langle e|\\hat{\\epsilon}\\cdot\\vec{E}_{\\lambda}+(\\hat{k}\\times\\hat{\\epsilon})\\cdot\\vec{M}_{\\lambda}|g\\rangle,\n\\end{equation}\nwhere $\\vec{E}_{\\lambda}$ describes the electric interaction of order $O((\\vec{k}\\cdot\\vec{r})^{\\lambda-1})$ and $\\vec{M}_{\\lambda}$ describes the magnetic interaction of order $O(\\alpha(\\vec{k}\\cdot\\vec{r})^{\\lambda-1})$\n(where $\\alpha$ is the fine structure constant) such that\n\\begin{align}\n\\vec{E}_{\\lambda}= & \\frac{\\left(i\\right)^{\\lambda-1}}{\\lambda!}\\sum_{a}e^{a}\\vec{r}^{\\,a}\\left(\\vec{k}\\cdot\\vec{r}^{\\,a}\\right)^{\\lambda-1},\\\\\n\\vec{M}_{\\lambda}= & \\frac{\\left(i\\right)^{\\lambda-1}}{\\left(\\lambda-1\\right)!}\\sum_{a}\\left(\\frac{e^{a}}{2m^{a}}\\right)\\left[\\left(\\vec{k}\\cdot\\vec{r}^{\\,a}\\right)^{\\lambda-1}\\left(\\frac{1}{\\lambda+1}\\vec{L}^{a}+\\frac{1}{2}g^{a}\\vec{S}^{a}\\right)+\\left(\\frac{1}{\\lambda+1}\\vec{L}^{a}+\\frac{1}{2}g^{a}\\vec{S}^{a}\\right)\\left(\\vec{k}\\cdot\\vec{r}^{\\,a}\\right)^{\\lambda-1}\\right],\\nonumber \n\\end{align}\nwhere $L^{a}$ is the orbital angular momentum, $S^{a}$ is the spin angular momentum, and $g^{a}$ is the spin g-factor for particle of index $a$ (see e.g.\\ \\cite{Sachs1987}). For typical atomic or molecular optical transitions, if all moments are allowed, we expect the dominant\ncorrections to the leading order E1 transition moment to be on the order of M1\/E1 $\\sim\\alpha\\sim10^{-2}$--$10^{-3}$ and E2\/E1$\\sim ka_{0}\\sim10^{-3}$--$10^{-4}$, where $a_0$ is the Bohr radius. In this work we neglect the higher order contributions beyond E2, though the effects may by evaluated by using the expansion\nabove.\n\n\n\\begin{center}\n\\begin{table}\n\\centering\n\\begin{tabular}{cccc}\n\\hline \n\\multicolumn{4}{c}{}\\tabularnewline\n\\multicolumn{4}{c}{$\\left\\langle e\\left|H_{\\mathrm{int}}\\left(O^{\\lambda}\\right)\\right|g\\right\\rangle =iA_{0}\\omega_{eg}\\left[\\hat{\\epsilon}_{+1}^{*}\\left\\langle e\\left|T_{+1}^{\\lambda}\\left(O^{\\lambda}\\right)\\right|g\\right\\rangle +\\hat{\\epsilon}_{-1}^{*}\\left\\langle e\\left|T_{-1}^{\\lambda}\\left(O^{\\lambda}\\right)\\right|g\\right\\rangle \\right]\\cdot\\vec{V}\\left(O^{\\lambda}\\right)+\\dots$}\\tabularnewline\n\\tabularnewline\n\\hline \n\\multirow{2}{*}{Term} & Tensor & \\multirow{2}{*}{Molecular Operator, $O^{\\lambda}$} & \\multirow{2}{*}{Light Vector, $\\vec{V}\\left(O^{\\lambda}\\right)$}\\tabularnewline\n & rank, $\\lambda$ &  & \\tabularnewline\n\\hline \nE1 & 1 & $\\Sigma_{a}e^{a}r_{i}^{a}$ & $\\hat{\\epsilon}$\\tabularnewline\nM1 & 1 & $\\Sigma_{a}\\frac{e^{a}}{2m^{a}}\\left(L_{i}^{a}+g^{a}S_{i}^{a}\\right)$ & $\\hat{k}\\times\\hat{\\epsilon}$\\tabularnewline\nE2 & 2 & $\\frac{\\omega}{2c}\\sum_{a}e^{a}r_{i}^{a}r_{j}^{a}$ & $\\frac{i}{\\sqrt{2}}\\left[\\hat{\\epsilon}(\\hat{k}\\cdot\\hat{z})+\\hat{k}(\\hat{\\epsilon}\\cdot\\hat{z})\\right]$\\tabularnewline\nM2 & 2 & $\\frac{\\omega}{c}\\Sigma_{a}\\frac{e^{a}}{2m^{a}}\\left\\{ r_{i}^{a},\\frac{1}{3}L_{j}^{a}+\\frac{1}{2}g^{a}S_{j}\\right\\} $ & $\\frac{i}{\\sqrt{2}}\\left[\\hat{k}((\\hat{k}\\times\\hat{\\epsilon})\\cdot\\hat{z})+(\\hat{k}\\times\\hat{\\epsilon})(\\hat{k}\\cdot\\hat{z})\\right]$\\tabularnewline\n\\hline \n\\end{tabular}\n\\par\n\\protect\\caption{Only spherical tensor operators $T_{q}^{\\lambda}$ with projection $q=\\pm1$ contribute to the $\\left|H\\right\\rangle \\rightarrow\\left|C\\right\\rangle$ transition amplitude. With this simplifying assumption, we can write the matrix element for each multipole operator in the form shown at the top of this table, which factors the molecule properties and the light properties (where $\\hat{\\epsilon}_{\\pm}=\\mp\\left(\\hat{x}\\pm i\\hat{y}\\right)\/\\sqrt{2}$ are the spherical basis vectors, and $\\hat{z}$ is the direction of the electric field). Here, the molecular operators $O^{\\lambda}$ and the corresponding light vectors $\\vec{V}\\left(O^{\\lambda}\\right)$ are listed for the E1, M1, E2, and M2 operators.}\n\\label{tab:spher_tens}\n\\end{table}\n\n\\par\\end{center}\n\nDuring the state preparation and readout of the molecule state, transitions are driven between the state $|g\\rangle=\\sum_{\\pm}d_{\\pm}|\\pm,\\Nsw\\rangle$ and $|e\\rangle=|C,\\Psw\\rangle$, where $d_{\\pm}$ are state amplitudes that denote the particular superposition in $\\left|H\\right\\rangle $ that is being interrogated. The particular $d_{\\pm}$ combination that results in $\\mathcal{M}=0$ describes the state that is dark, and the orthogonal state is bright and is optically pumped away.\n\nIt is convenient to expand the Hamiltonian $H_{\\mathrm{int}}$ in terms of spherical tensor operators. Furthermore, the laser is only resonant with $\\Delta M=\\pm1$ transitions, so the spherical tensor\noperators with angular momentum projections other than $\\pm1$ can be reasonably omitted. In table \\ref{tab:spher_tens}, we factor the first 4 multipole operators into products of molecule and light field operators and express the molecular operators in terms of spherical tensors $T_{\\pm1}^{\\lambda}$ of rank $\\lambda=1,2$. The E1 and M1 terms consist of vector operators with $\\lambda=1$. The E2 and M2 operators are rank 2 cartesian operators which can have spherical tensor operator contributions for $\\lambda=0,1,2$. The rank $\\lambda=0$ components of the E2 and M2 operators, and the $\\lambda=1$ component of the E2 operator, vanish. The rank $\\lambda=1$ component of the M2 operator does not vanish, but the light field angular dependence of this operator is equivalent to E1, so we may treat it as such.\n\nUsing well-known properties of angular momentum matrix elements \\cite{Brown2003}, we may write the transition matrix element in the following form,\n\\begin{align}\n\\mathcal{M}= & iA_{0}\\omega_{eg}c_{\\rm E1}\\frac{1}{\\sqrt{2}}\\left[\\left(-1\\right)^{J+1}\\Psw\\right]^{(1-\\mathcal{\\tilde{N}}\\Esw)\/2}\\left(\\hat{\\epsilon}_{-1}^{*}d_{+}+\\Psw\\left(-1\\right)^{J'}\\hat{\\epsilon}_{+1}^{*}d_{-}\\right)\\cdot\\vec{\\varepsilon}_{\\mathrm{eff}},\n\\end{align}\nsuch that $\\vec{\\varepsilon}_{\\mathrm{eff}}$ is the `effective E1 polarisation' (i.e. including the effects of interference between multipole transition matrix elements is equivalent to an E1 transition with this polarisation) with the form\n\\begin{align}\n\\vec{\\varepsilon}_{\\mathrm{eff}}=& \\hat{\\epsilon}-a_{\\rm M1}i\\hat{n}\\times(\\hat{k}\\times\\hat{\\epsilon})+a_{\\rm E2}(\\Psw)i(\\hat{k}(\\hat{\\epsilon}\\cdot\\hat{n})+\\hat{\\epsilon}(\\hat{k}\\cdot\\hat{n}))+\\dots\\label{eq:effective E1 polarization}\n\\end{align}\nwhere $\\hat{n}=\\Nsw\\Esw\\hat{z}$ is the orientation of the internuclear axis in the laboratory frame,  $a_{\\rm E2}(\\Psw)=c_{\\rm E2}(\\Psw)\/(\\sqrt{2}c_{\\rm E1})$ and $a_{\\rm M1}=c_{\\rm M1}\/c_{\\rm E1}$ are real dimensionless ratios describing the strength of the M1 and E2 matrix elements relative to E1, and the $c$ coefficients are matrix elements,\n\\begin{align}\nc_{\\rm E1}= & \\left\\langle C,J,0,1\\left|\\mathrm{E1}\\right|H,J',1,1\\right\\rangle \\\\\nc_{\\rm M1}= & \\left\\langle C,J,0,1\\left|\\mathrm{M1}\\right|H,J',1,1\\right\\rangle \\\\\nc_{\\rm E2}(\\Psw)= & \\left\\langle C,J,0,1\\left|\\mathrm{E2}\\right|H,J',1,1\\right\\rangle +\\nonumber \\\\\n & \\Psw\\left(-1\\right)^{J}\\left\\langle C,J,0,1\\left|\\mathrm{\\rm E2}\\right|H,J',1,-1\\right\\rangle ,\n\\end{align}\nwhich are defined using the state notation $\\left|A,J,M,\\Omega\\right\\rangle $ for electronic state $A$, and `E1, M1, E2' refer to the corresponding molecular operators in table \\ref{tab:spher_tens}. It is useful to define the Rabi frequency $\\Omega_{\\rm r}=|\\mathcal{M}|$ as the magnitude of the amplitude connecting to the bright state, and the unit vector $\\hat{\\varepsilon}_{\\mathrm{eff}}$ corresponding to the projection of $\\vec{\\varepsilon}_{\\mathrm{eff}}$ onto the $xy$ plane,\n\\begin{align}\n\\hat{\\varepsilon}_{\\mathrm{eff}}= & \\frac{\\vec{\\varepsilon}_{\\mathrm{eff}}-(\\vec{\\varepsilon}_{\\mathrm{eff}}\\cdot\\hat{z})\\hat{z}}{\\sqrt{|\\vec{\\varepsilon}_{\\mathrm{eff}}|^{2}-|\\vec{\\varepsilon}_{\\mathrm{eff}}\\cdot\\hat{z}|^{2}}}.\\label{eq:unit vector}\n\\end{align}\nThis completely determines the bright and dark states, which have been previously defined in equations \\ref{eq:bright_state} and \\ref{eq:dark_state} for solely E1 transition matrix elements.\n\nThe odd parity E1 and even parity M1 and E2 contributions to the effective polarisation differ by a factor of $\\Nsw\\Esw$, which is correlated with the expected eEDM signal. Expanding the effective E1 polarisation in terms of switch parity components, $\\hat{\\varepsilon}_{\\mathrm{eff}}=\\hat{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr}}+\\Nsw\\Esw d\\vec{\\varepsilon}_{\\mathrm{eff}}^{\\N\\E}$, and evaluating the effective $\\Nsw\\Esw$ correlated polarisation imperfections using equation \\ref{eq:extracting_polarization_imperfection_components}, we find that the bright and dark states have effective polarisation correlations given by:\n\\begin{align}\n\\frac{\\hat{z}\\cdot(\\hat{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr}}\\times d\\vec{\\varepsilon}_{\\mathrm{eff}}^{\\N\\E})}{\\hat{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr}}\\cdot\\hat{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr}}}\\approx & \\, d\\theta_{\\mathrm{eff}}^{\\N\\E}-id\\Theta_{\\mathrm{eff}}^{\\N\\E}\\\\\n\\approx & -i(a_{M1}-a_{E2}(\\Psw))(\\hat{\\epsilon}\\cdot\\hat{z})((\\hat{k}\\times\\hat{\\epsilon})\\cdot\\hat{z}).\n\\label{eq:state_correlations}\n\\end{align}\nIt is useful to use a particular parameterization of the laser pointing $\\hat{k}$ and polarisation $\\hat{\\epsilon}$ to expand the expression in equation \\ref{eq:state_correlations} in terms of pointing and polarisation imperfections. The state preparation laser $\\hat{k}$-vector is aligned along (or against) the $\\hat{z}$ direction in the laboratory, so it is convenient to parameterize the pointing deviation from normal by spherical angle $\\vartheta_{k}$, and the direction of this pointing imperfection by polar angle $\\varphi_{k}$ in the $xy$ plane, such that:\n\\begin{align}\n\\hat{k}= & \\cos\\varphi_{k}\\sin\\vartheta_{k}\\hat{x}+\\sin\\varphi_{k}\\sin\\vartheta_{k}\\hat{y}+\\cos\\vartheta_{k}\\hat{z}.\n\\label{eq:pointing_imperfection}\n\\end{align}\nWe may use a parameterization for the polarisation $\\hat{\\epsilon}$ that is similar to that in equation \\ref{eq:polarization_parametrization}, but a slight modification is required to ensure that $\\hat{k}\\cdot\\hat{\\epsilon}=0$:\n\\begin{align}\n\\hat{\\epsilon}= & N_{\\epsilon}\\left(-e^{-i\\theta}\\cos\\Theta\\hat{\\epsilon}_{+1}+e^{i\\theta}\\sin\\Theta\\hat{\\epsilon}_{-1}+\\epsilon_{z}\\hat{z}\\right)\\\\\n\\epsilon_{z}= & -\\frac{1}{\\sqrt{2}}\\tan\\theta_{k}\\left(e^{-i\\left(\\theta-\\varphi_{k}\\right)}\\cos\\Theta+e^{i\\left(\\theta-\\varphi_{k}\\right)}\\sin\\Theta\\right)\n\\end{align}\nwhere $N_{\\epsilon}$ is a normalization constant that ensures that $\\hat{\\epsilon}^{*}\\cdot\\hat{\\epsilon}=1$. With these parameterizations in place, and expanding about small ellipticities $d\\Theta$ such that $\\Theta=\\pi\/4+d\\Theta$, and small laser pointing deviation, $\\vartheta_{k}\\ll1$, we find that the $\\Nsw\\Esw$-correlated effective laser polarisation imperfections are given by:\n\\begin{align}\nd\\theta_{\\mathrm{eff}}^{\\N\\E}\\approx & -\\frac{1}{2}(a_{M1}-a_{E2}(\\Psw))\\vartheta_{k}^{2\\:}S^{\\:}\\cos(2(\\theta-\\varphi_{k}))\\\\\nd\\Theta_{\\mathrm{eff}}^{\\N\\E}\\approx & -\\frac{1}{2}(a_{M1}-a_{E2}(\\Psw))\\vartheta_{k}^{2\\:}\\sin(2(\\theta-\\varphi_{k}))\n\\end{align}\nwhere $S_{i}=-2d\\Theta_{i}$ describe the laser ellipticities. Hence, following equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections}, there is a systematic error in $\\omega^{\\mathcal{NE}}$:\n\\begin{align}\n\\omega_{\\mathrm{S.I.}}^{\\N\\E}=&\\frac{1}{\\tau}\\frac{1}{4}\\left(a_{M1}-a_{E2}\\left(\\Psw\\right)\\right)\\times\\\\&\\left[\\vartheta_{k,{\\rm prep}}^{2}\\left(-2S_{{\\rm prep}}c_{{\\rm prep}}+\\Psw s_{{\\rm prep}}\\left(S_{X}-S_{Y}\\right)\\right)+\\right.\\\\&\\left.\\vartheta_{k,X}^{2}\\left(S_{X}c_{X}+\\Psw S_{{\\rm prep}}s_{X}\\right)+\\vartheta_{k,Y}^{2}\\left(S_{Y}c_{Y}-\\Psw S_{{\\rm prep}}s_{Y}\\right)\\right]\n\\end{align}\nwhere $c_{i}\\equiv\\cos\\left(2(\\theta_{i}-\\varphi_{i,k})\\right)$ and $s_{i}\\equiv\\sin\\left(2(\\theta_{i}-\\varphi_{i,k})\\right)$ describe the dependence of the systematic error on the difference between the linear polarisation angle $\\theta_{i}$ and the pointing angle $\\varphi_{i,k}$ in the $xy$ plane. \n\nThere is another contribution to this systematic error that arises when the coupling to the off-resonant\nopposite parity excited state $|C,-\\Psw\\rangle$ is also taken into account. This additional contribution becomes significant when the ellipticities are comparable to or smaller than $\\gamma_{C}\/\\Delta_{\\Omega,C,J=1}\\approx0.5\\%$.\n\nThe eEDM channel, $\\omega^{\\mathcal{NE}}$, was defined to be even under the superblock switches (including $\\Psw$), hence those terms proportional to $\\Psw$ in the equation above do not contribute to our reported result. Additionally, the $\\Gsw$ and $\\Rsw$ switches rotate the polarisation angles for each laser by roughly $\\theta_{i}\\rightarrow\\theta_{i}+\\pi\/2$ periodically and the resulting $\\omega^{\\mathcal{NE}}$ signal is averaged over these states. Provided that the pointing drift is much slower than the timescale of these switches, and to the extent that the laser polarisations constituting the $\\Rsw$ and $\\Gsw$ states are orthogonal, then these systematic errors should dominantly contribute to the $\\omega^{\\mathcal{NEG}}$ and $\\omega^{\\mathcal{NER}}$ channels which were found to be consistent with zero (see Figure~\\ref{fig:pixel_plot}). \n\nAn indirect limit on the size of the systematic error due to Stark interference, $\\omega^{\\mathcal{NE}}_{\\mathrm{S.I.}}$, may be estimated by assuming a reasonable suppression factor by which the effects in $\\omega^{\\N\\E\\R}$ and $\\omega^{\\N\\E\\G}$ may `leak' into $\\omega^{\\mathcal{NE}}$. We monitored the pointing drift on a beam profiler and observed pointing drifts up to $d\\vartheta_k\\sim 50~\\upmu\\rm{rad}$  throughout a full set of superblock states. The absolute pointing misalignment angle was not well known but was estimated to be larger than $\\vartheta_k\\gtrsim0.5~\\rm{mrad}$. Hence we may estimate a conservative suppression factor $d\\vartheta_k\/\\vartheta_k\\lesssim1\/10$ by which pointing drift may contaminate $\\omega^{\\mathcal{NE}}$ from $\\omega^{\\N\\E\\R}$ and $\\omega^{\\N\\E\\G}$. The two $\\mathcal{\\tilde{R}}$ states are very nearly orthogonal, but the $\\Gsw$ states deviate sufficiently from orthogonal (see section~\\ref{sssec:suppression_of_the_AC_stark_shift_phases}) such that the leakage from $\\omega^{\\N\\E\\G}\\rightarrow\\omega^{\\N\\E}$ will dominate the systematic error; we estimate a suppression factor of about $c_p^{\\rm{nr}}\/c_p^{\\mathcal{G}}\\sim s_p^{\\rm{nr}}\/s_p^{\\mathcal{G}}\\sim1\/5$. Based on the upper limits on the measured values for $\\omega^{\\N\\E\\R}$ and $\\omega^{\\N\\E\\G}$ combined with leakage from $\\omega^{\\N\\E\\R}$ and $\\omega^{\\N\\E\\G}$ into $\\omega^{\\N\\E}$ due to pointing drift, and leakage from $\\omega^{\\N\\E\\G}$ into $\\omega^{\\N\\E}$ due to non-orthogonality of the two $\\Gsw$ states, we estimate the possible size of the systematic error to be $\\omega_{\\mathrm{S.I.}}^{\\N\\E}\\lesssim 1^{\\:}\\mathrm{mrad}\/\\mathrm{s}$.\n\nNote that the mechanism for this systematic error was not discovered until after the publication of our result \\cite{Baron2014} and hence was not included in our systematic error analysis there. Furthermore, since we did not observe this effect, this systematic error does not match any of the inclusion criteria outlined in section \\ref{ssec:total_systematic_error_budget} and hence is not included in the systematic error budget in this paper. Since we did not understand the mechanism for this systematic error while running the apparatus, we were not able to place direct limits on the size of this systematic error. We estimate that the absolute pointing deviation from ideal was at most $5^{\\:}\\mathrm{mrad}$ and the ellipticity of each laser was no more than $S_{i}\\approx5\\%$. The E1\/M1 interference coefficient is $a_{M1}\\approx0.1$ for the $H\\rightarrow C$ transition. This gives an estimate of $\\omega_{\\mathrm{S.I.}}^{\\N\\E}\\sim0.1^{\\:}\\mathrm{mrad}\/\\mathrm{s}$ before suppression due to the $\\Rsw$ and $\\Gsw$ switches. Hence, we do not believe that this systematic error significantly shifted the result of our measurement.\n\n\\subsubsection{AC Stark shift phases}\n\\label{sssec:AC_stark_shift_phases}\nIn this section we describe contributions to the measured phase $\\Phi$ that depend on the AC Stark shifts induced by the state preparation and readout lasers. We describe mechanisms by which such phase contributions may arise, and we describe mechanisms by which $\\Nsw\\Esw$ correlated experimental imperfections may couple to these phases to result in eEDM-mimicking phases. Concise descriptions of some of the effects described here can be found in \\cite{Hess2014,SpaunThesis,HutzlerThesis}.\n\n\nDuring our search for systematic errors as described in section \\ref{ssec:determining_systematic_uncertainty}, we empirically found that there was a contribution to the measured phase $d\\Phi(\\Delta,\\Omega_{\\rm r})$ that had an unexpected linear dependence on the laser detuning, $\\Delta$, a quadratic dependence on laser detuning $\\Delta$ in the presence of a nonzero magnetic field, and a linear dependence on small changes to the magnitude of the Rabi frequency, $d\\Omega_{\\rm r}\/\\Omega_{\\rm r}$, in the presence of a nonzero magnetic field,\n\\begin{align}\nd\\Phi\\left(\\Delta,\\Omega_{\\rm r}\\right)=&\\sum_{i}\\left[\\alpha_{\\Delta,i}\\Delta_{i}+\\alpha_{\\Delta^{2},i}\\Delta_{i}^{2}+\\beta_{d\\Omega_{{\\rm r}, i}}(d\\Omega_{{\\rm r}, i}\/\\Omega_{{\\rm r}, i})+\\dots\\right].\n\\label{eq:Empirical_AC_Stark_Shift_Phase_Result}\n\\end{align}\nwhere $i\\in\\left\\{ {\\rm prep},X,Y \\right\\}$ indexes the state preparation and readout lasers. The coefficients we measured were $\\alpha_{\\Delta}\\sim1^{\\:}\\mathrm{mrad}\/(2\\pi\\times\\mathrm{MHz})$, $\\alpha_{\\Delta^{2}}\\sim1^{\\:}\\mathrm{mrad}\/\\left(2\\pi\\times\\mathrm{MHz}\\right)^{2}$ and $\\beta_{d\\Omega_{\\rm r}}\\sim10^{-3}$. We performed these measurements by independently varying the laser detunings $\\Delta_i$ across resonance using AOMs or modulating the laser power using AOMs with the set-up depicted in figure \\ref{fig:HC_transitions_setup} and extracting the measured phase $\\Phi$. Examples of such measurements are given in figure~\\ref{fig:phase_vs_detuning}.\n\nWe determined that this behaviour can be caused by mixing between bright and dark states, due to a small non-adiabatic laser polarisation rotation or Zeeman interaction present during the optical pumping used to prepare and read out the spin state. The mixed bright and dark states differ in energy by the AC Stark shift, which leads to a relative phase accumulation between the bright and dark state components that depends on the laser parameters $\\Delta$ and $\\Omega_{\\rm r}$. We shall now derive the AC Stark shift phase that results in equation \\ref{eq:Empirical_AC_Stark_Shift_Phase_Result}, under simplifying assumptions amenable to analytic calculations.\n\nConsider a three level system consisting of the bright $|B(\\hat{\\varepsilon},\\Nsw,\\Psw)\\rangle$ and dark $|D(\\hat{\\varepsilon},\\Nsw,\\Psw)\\rangle$ states and the lossy excited state $|C,\\Psw\\rangle$ with decay rate $\\gamma_{C}$. For simplicity, assume that there is no applied\nmagnetic field for the time being. In this system, the instantaneous eigenvectors (depicted in figure \\ref{fig:bases}C) are\n\\begin{align}\n|B_{\\pm}\\rangle\\equiv & \\pm\\kappa_{\\pm}|C,\\Psw\\rangle+\\kappa_{\\mp}|B(\\hat{\\varepsilon},\\Nsw,\\Psw)\\rangle,^{\\:\\:}|D\\rangle\\equiv|D(\\hat{\\varepsilon},\\Nsw,\\Psw)\\rangle,\n\\label{eq:inst_eigv}\n\\end{align}\nand the instantaneous eigenvalues are\n\\begin{align}\nE_{B\\pm}= & \\frac{1}{2}\\left(\\Delta\\pm\\sqrt{\\Delta^{2}+\\Omega_{\\rm r}^{2}}\\right),^{\\:}E_{D}=0,\n\\label{eq:inst_eig}\n\\end{align}\nsuch that the mixing amplitudes $\\kappa_{\\pm}$ are given by \n\\begin{align}\n\\kappa_{\\pm}= &\\frac{1}{\\sqrt{2}} \\sqrt{1\\pm\\frac{\\Delta}{\\sqrt{\\Delta^{2}+\\Omega_{\\rm r}^{2}}}}.\n\\end{align}\n\nThe effect of the decay of the excited state (which occurs almost entirely to states outside of the three level system) may be taken into account by adding an anti-Hermitian operator term in the Schrodinger equation, $|\\dot{\\psi}\\rangle=-i(H-i\\frac{1}{2}\\Gamma)|\\psi\\rangle$, where $\\Gamma=\\gamma_{C}|C,\\Psw\\rangle\\langle C,\\Psw|$ is the decay operator. This formulation is equivalent to the Lindblad master equation,\n\\begin{align}\n\\dot{\\rho}= & -i\\left[H,\\rho\\right]-\\frac{1}{2}\\left\\{ \\Gamma,\\rho\\right\\} ,\n\\end{align}\nthat governs the time evolution of the density matrix $\\rho=|\\psi\\rangle\\langle\\psi|$. In practice, we implement this decay term by calculating the time evolution of the system according to $H$, and then making the substitution $\\Delta\\rightarrow\\Delta-i\\gamma_C\/2$ before calculating squares of amplitudes. \n\nIt is useful to work in the dressed state basis, $|D\\rangle$, $|B_\\pm\\rangle$, (basis C in figure \\ref{fig:bases}) because these are nearly stationary states and have simple time evolution in the case that laser polarisation and Rabi frequency are stationary. If we allow the laser polarisation to vary in time, then the dressed state basis varies in time, and the system evolves according to the Hamiltonian,\n\\begin{align}\n\\tilde{H}= & UHU^{\\dagger}-iU\\dot{U}^{\\dagger},\n\\end{align}\nwhere $U$ is the transformation from time independent basis A to time dependent basis C (from figure \\ref{fig:bases}), $UHU^{\\dagger}$ is diagonal, and $-iU\\dot{U}^{\\dagger}$ is a fictitious force term that arises because we are working in a non-inertial frame when the laser polarisation is time dependent \\cite{Budker2008}.\n\nAssuming that the polarisation is nearly linear, $\\Theta\\approx\\pi\/4$, but allowing the polarisation to rotate slightly, and allowing for a nonzero two photon detuning due to the Zeeman shift $\\delta=-g_1\\mu_{\\rm B}\\B_z\\Bsw$, the Hamiltonian in the dressed state picture is:\n\\begin{align}\n\\tilde{H}= & \\left(\\begin{array}{ccc}\n0 & -i\\dot{\\chi}^{*}\\kappa_{+} & -i\\dot{\\chi}^{*}\\kappa_{-}\\\\\ni\\dot{\\chi}\\kappa_{+} & E_{B-} & -\\frac{i}{2}\\frac{\\dot{\\Omega_{\\rm r}}\\Delta-\\Omega_{\\rm r}\\dot{\\Delta}}{\\Delta^{2}+\\Omega^{2}}\\\\\ni\\dot{\\chi}\\kappa_{-} & \\frac{i}{2}\\frac{\\dot{\\Omega_{\\rm r}}\\Delta-\\Omega_{\\rm r}\\dot{\\Delta}}{\\Delta^{2}+\\Omega_{\\rm r}^{2}} & E_{B+}\n\\end{array}\\right)\\begin{array}{c}\n\\ket{D} \\\\\n\\ket{B_{-}} \\\\\n\\ket{B_{+}} \n\\end{array}\n\\end{align}\nwhere $\\dot{\\chi}=\\dot{\\Theta}-i(\\dot{\\theta}+\\delta)$ can be considered to be a complex polarisation rotation rate, $\\dot{\\Omega}_{\\rm r}$ is the rate of change of the Rabi frequency, and $\\dot{\\Delta}$ is\nthe rate of change of the detuning. Note that this Hamiltonian implies that the effect of a two photon detuning arising from the Zeeman shift is equivalent to that of a linear polarisation rotating at a constant rate.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=15cm]{various_bases_used.pdf}\n\\caption{Energy level diagrams depicting the Hamiltonian when the three-level $H\\rightarrow C$ transition is addressed by the state preparation or readout lasers in three different bases. Solid double-sided blue arrows denote strong laser couplings between $H$ and $C$. Wiggly red arrows denote spontaneous emission from $C$ to states outside of the three level system. Dashed orange lines denote weak couplings induced by laser polarisation rotation. Basis A is useful for describing the spin precession phase induced by the Zeeman and eEDM Hamiltonians. Basis B is useful for describing the states that are prepared and read-out in the spin precession measurement. Basis C is useful for evaluating the AC Stark Shift phases induced by laser polarisation rotations.}\n\\label{fig:bases}\n\\end{figure}\n\nWe may then apply first order time-dependent perturbation theory in this picture to determine the extent of bright\/dark state mixing due to $\\dot{\\chi}$ in the time evolution of the system. If we parameterize the time-dependent state as\n\\begin{align}\n\\left|\\psi\\left(t\\right)\\right\\rangle = & c_{D}\\left(t\\right)\\left|D\\right\\rangle +c_{B+}\\left(t\\right)\\left|B_{+}\\right\\rangle +c_{B-}\\left(t\\right)\\left|B_{-}\\right\\rangle,\n\\end{align}\nthen in the case of a uniform laser field $\\dot{\\Omega}_{\\rm r}=0$, of duration $t$ and with a constant detuning $\\dot{\\Delta}=0$, the time evolution of the coefficients is given at first order by:\n\\begin{align}\nc_{D}\\left(t\\right)= & c_{D}\\left(0\\right)-\\sum_{\\pm}\\int_{0}^{t}\\dot{\\chi}^{*}\\left(t'\\right)\\kappa_{\\mp}\\left(t'\\right)e^{-iE_{B\\pm}t'}c_{B\\pm}\\left(0\\right)\\enspace dt'\\label{eq:first order perturbation theory dark state}\\\\\nc_{B\\pm}\\left(t\\right)= & e^{-iE_{B\\pm}t}c_{B\\pm}\\left(0\\right)+e^{-iE_{B\\pm}t}\\int_{0}^{t}\\dot{\\chi}\\left(t'\\right)\\kappa_{\\mp}\\left(t'\\right)e^{iE_{B\\pm}t'}c_{D}\\left(0\\right)\\enspace dt'.\\label{eq:first order perturbation theory bright state}\n\\end{align}\n\n\nIn the state preparation region, the molecules begin in an incoherent mixture of the states $|B(\\hat{\\varepsilon}_{{\\rm prep}},\\Nsw,\\Psw)\\rangle$ and $|D(\\hat{\\varepsilon}_{{\\rm prep}},\\Nsw,\\Psw)\\rangle$ and then enter the state preparation laser beam. In the ideal case of uniform laser polarisation, molecules that were in the bright state are optically pumped out of the three level\nsystem, and molecules that are in the dark state remain there; this results in a pure state, $|D(\\hat{\\varepsilon}_{{\\rm prep}},\\Nsw,\\Psw)\\rangle$. However, if there is a small polarisation rotation by amount $d\\chi\\equiv\\int_{0}^{t}\\dot{\\chi}\\left(t'\\right)dt'\\equiv d\\Theta-i(d\\theta-g_1\\mu_{\\rm B}\\B_z\\Bsw t)$, such that $\\left|d\\chi\\right|\\ll1$, then the dark state obtains a bright state admixture that may not be completely optically pumped away before leaving the laser beam.\\footnote{This is most liable to occur just before a molecule leaves the laser beam, such that complete optical pumping does not occur.} In this case, the final state can be written as\n\\begin{align}\n|D(\\hat{\\varepsilon}'_{{\\rm prep}},\\Nsw,\\Psw)\\rangle= & |D(\\hat{\\varepsilon}_{{\\rm prep}},\\Nsw,\\Psw)\\rangle+d\\chi\\Pi|B(\\hat{\\varepsilon}_{{\\rm prep}},\\Nsw,\\Psw)\\rangle\n\\end{align}\nwhere $\\hat{\\varepsilon}'_{{\\rm prep}}$ is the effective polarisation that parameterizes the initial state in the spin precession region \n\\begin{align}\n\\hat{\\varepsilon}'_{{\\rm prep}}= & \\hat{\\varepsilon}_{{\\rm prep}}+d\\chi\\Pi i\\hat{z}\\times\\hat{\\varepsilon}_{{\\rm prep}}^{*},\n\\end{align}\nand $\\Pi$ is an amplitude that accounts for the AC Stark shift phase and the time dependent dynamics of the non-adiabatic mixing due to the polarisation rotation,\n\\begin{equation}\n\\Pi=\\sum_{\\pm}(\\kappa_{\\mp})^{2}e^{-iE_{B\\pm}t}\\int_{0}^{t}dt'^{\\:}\\frac{\\dot{\\chi}\\left(t'\\right)}{d\\chi}e^{iE_{B\\pm}t'}.\\label{eq:Pi_def}\n\\end{equation}\nThe deviations between the effective polarisation and the actual laser polarisation can be viewed as effective polarisation imperfections,\n\\begin{align}\nd\\theta_{{\\rm prep},\\mathrm{eff}}= & -d\\Theta_{{\\rm prep}}\\text{Im}\\Pi+(d\\theta_{{\\rm prep}}-g_1\\mu_{\\rm B}\\B_z\\Bsw t)\\text{Re}\\Pi,\\\\\nd\\Theta_{{\\rm prep},^{\\:}\\mathrm{eff}}= & -d\\Theta_{{\\rm prep}}\\text{Re}\\Pi-(d\\theta_{{\\rm prep}}-g_1\\mu_{\\rm B}\\B_z\\Bsw t)\\text{Im}\\Pi,\n\\end{align}\nthat lead to shifts in the measured phase $\\Phi$ as described in equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections}.\nFor definiteness, consider the case in which the polarisation rotation rate $\\dot{\\chi}(t')=d\\chi\/t$ is a constant for the duration of the optical pumping pulse $t$. In this case,\n\\begin{align}\n\\Pi= & \\sum_{\\pm}(\\kappa_{\\mp})^{2}e^{-iE_{B\\pm}t\/2}\\mathrm{sinc}(E_{B\\pm}t\/2).\n\\end{align}\n\n\nThis function has the property that $\\text{Im}\\Pi$ is an odd function in $\\Delta$ that can take on values up to order unity across resonance (a frequency range on the order of $\\gamma_{C}$) and is exactly zero on resonance. $\\text{Re}\\Pi$ is an even function quadratic in $\\Delta$ about resonance, and depends on Rabi frequency on resonance. If the laser beam intensity reduces quickly as the molecule leaves it then most of the AC Stark shift phase arises from the last Rabi flopping period before the molecule exits the laser beam (provided $\\dot{\\chi}$ is nonzero during that time). If the intensity reduces slowly, the AC Stark shift phase can be exacerbated since the bright state amplitude is not as effectively optically pumped away while $\\Omega_{\\rm r}<\\gamma_C$. Nevertheless, beamshaping tests shown in figure \\ref{fig:phase_vs_detuning} and numerical simulations indicate that $\\Pi$ is not very sensitive to the shape of the spatial intensity profile of the laser beam or the shape of the spatial variation of the polarisation.\n\nIf we consider only the first order contribution to the shift in the measured phase in equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections}, $d\\theta_{{\\rm prep},\\mathrm{eff}}$, and neglect the second order shift that arises due to $d\\Theta_{{\\rm prep},\\mathrm{eff}}$, then we can relate the parameters in equation \\ref{eq:Empirical_AC_Stark_Shift_Phase_Result} to the amplitude $\\Pi$ accounting for the AC Stark shift phase and the complex polarisation rotation $d\\chi$, by\n\\begin{align}\n\\alpha_{\\Delta,{\\rm prep}}\\approx&-\\frac{\\partial\\text{Im}\\Pi}{\\partial\\Delta_{{\\rm prep}}}d\\Theta_{{\\rm prep}}\\\\\n\\alpha_{\\Delta^{2},{\\rm prep}}\\approx&\\frac{\\partial^{2}\\text{Re}\\Pi}{\\partial\\Delta_{{\\rm prep}}^{2}}\\left(d\\theta_{{\\rm prep}}-g\\mu_{\\rm B}\\B_{z}\\tilde{\\B}t\\right)\\\\\n\\beta_{d\\Omega_{\\rm r},{\\rm prep}}\\approx&\\Omega_{\\rm r}\\frac{\\partial\\text{Re}\\Pi}{\\partial\\Omega_{\\rm r}}\\left(d\\theta_{{\\rm prep}}-g\\mu_{\\rm B}\\B_{z}\\tilde{\\B}t\\right).\\label{eq:bdOmega}\n\\end{align}\nWe can interpret these results as follows. The linear dependence of the measured phase on detuning, $\\alpha_{\\Delta,{\\rm prep}}$, comes from a spatially varying ellipticity in the $x$ direction coupling to the AC Stark shift phase. Similarly, the quadratic dependence of $\\Phi$ on $\\Delta$, $\\alpha_{\\Delta^{2},{\\rm prep}}$, and the dependence of $\\Phi$ on a relative change in $\\Omega_{\\rm r}$, $\\beta_{d\\Omega_{\\rm r}, {\\rm prep}}$, come from either a spatially varying linear polarisation in the $x$ direction or a Zeeman shift, each coupling to the AC Stark shift phase. Here, we only analyzed the phase shift that results from AC Stark shift effects in the state preparation laser beam, but there is an analogous phase shift in the state readout beam.\n\nThere are several other subdominant effects that also contribute to the AC Stark shift phase behavior described in equation \\ref{eq:Measured_Phase_with_Polarization_Imperfections} in the presence of polarisation imperfections. The opposite parity excited state $|C,-\\Psw\\rangle$ couples strongly to the dark state, but the mixing between these two states is weak because the transition frequency is off-resonant by a detuning $\\Delta_{\\Omega,C,J=1}\\approx2\\pi\\times51~\\mathrm{MHz}\\gg\\gamma_{C}$. In the case that an optical pumping laser has nonzero ellipticity, the bright state gains a weak coupling to the opposite-parity excited state proportional to this ellipticity. Then, two-photon bright-dark state mixing ensues in such a way that the mixing amplitude, and hence the measured phase, depends on the laser detuning.\n\nThe rapid polarisation switching of the state readout beam can also introduce AC Stark shift-induced phases in the absence of a polarisation gradient, if the average ellipticity between the two polarisations is nonzero. Suppose a particular molecule is first excited by the $\\hat{\\epsilon}_{X}$ polarised beam. The two bright eigenstates $\\ket{B_{\\pm}}$ are mostly optically pumped away, resulting in a fluorescence signal $F_{X}$. The population remaining in the bright eigenstates acquires a phase relative to the dark state, due to the AC Stark shift. Then the molecules are optically pumped by the $\\hat{\\epsilon}_{Y}$ polarised beam. If there is a nonzero average ellipticity, $\\hat{\\epsilon}_{Y}$ is not quite orthogonal to $\\hat{\\epsilon}_{X}$ and the new bright eigenstates that give rise to the fluorescence signal $F_Y$ are superpositions of the former bright and dark states that acquired a relative AC Stark shift phase. This results in a fluorescence signal, and hence measured phase component, that depends linearly on laser detuning $\\Delta$. \n\n\\subsubsection{Polarisation Gradients from Thermal Stress-Induced Birefringence}\n\\label{sssec:polarization_gradients_from_thermal_stress_induced_birefringence}\n\n\\hspace*{\\fill} \\\\\nThe AC Stark shift phases described in the previous section can be induced by polarisation gradients in $\\hat{x}$ across the state preparation and readout laser beams. In this section we describe a known mechanism by which these arose. Recall that these laser beams passed through transparent, ITO-coated electric field plates. For an absorbance $\\alpha$ and laser intensity $I$, the rate of heat deposition into the plates is $\\dot{Q}\\left(x,y\\right)=\\alpha\\, I\\left(x,y\\right)$. The laser beam profile is stretched in the $y$ direction to ensure that all molecules are addressed. For simplicity we assume that the heating distribution, $\\dot{Q}\\left(x,y\\right)=\\dot{Q}\\left(x\\right)$, is completely uniform in the $y$ direction. We also assume that there are no shear stresses, i.e.\\ local expansion of the glass is isotropic. Under these assumptions, the relationship between the heating rate, $\\dot{Q}$, and the internal stress tensor $\\sigma_{ij}$ (where $i,j$ are Cartesian indices) is\n\\begin{equation}\n\\frac{\\partial^{2}\\sigma_{yy}}{\\partial x^{2}}= \\frac{E\\alpha_{V}}{\\kappa}\\dot{Q}\\left(x\\right),\n\\end{equation}\nwhere $E$, $\\alpha_{V}$ and $\\kappa$ are the Young's modulus, coefficient of thermal expansion, and thermal conductivity, respectively \\cite{Barber2010}. Unit vectors $\\hat{x}$ and $\\hat{y}$ correspond to the principal axes of the stress tensor due to the symmetry of the heating function, hence the off-diagonal (shear) elements are zero, $\\sigma_{xy}=0$. The other diagonal component, $\\sigma_{xx}$, is uniform across the plates, and equal to $\\sigma_{yy}$ far away from the laser. The stress-optical law states that the birefringence and stress are linearly proportional along the principal axes of the stress tensor \\cite{Dally1991}. The difference between the indices of refraction in the $x$ and $y$ directions is then $\\Delta n=K\\left(\\sigma_{xx}-\\sigma_{yy}\\right)$, where $K\\approx4\\times10^{-6}\\,\\mbox{MPa}^{-1}$ is the stress-optical coefficient for Borofloat glass \\cite{Schott2013b}. The retardance of an incident laser beam of index $i$ is $\\Gamma_i=2\\pi\\Delta n\\left(t\/\\lambda\\right)$, where $t$ is the thickness of the field plates (in the $z$ direction), and $\\lambda$ is the wavelength of light. Hence, in this limit, the retardance due to thermal stress-induced birefringence is related to the laser intensity by:\n\\begin{equation}\n\\frac{\\partial^{2}\\Gamma}{\\partial x^{2}}=\\eta\\frac{t}{\\lambda}I\\left(x\\right),\n\\label{eq:retarddiff}\n\\end{equation}\nwhere $\\eta=2\\pi KE\\alpha_{V}\\alpha\/\\kappa\\approx26\\times10^{-6}$~W$^{-1}$ is a material constant of Borofloat glass \\cite{Schott2013b}.\nThe ellipticity imprinted on the nominally linearly polarised laser beam is given by\n\\begin{equation}\nS_i=\\Gamma_i(x)\\sin\\left(2(\\theta_i-\\phi_{\\Gamma,i})\\right),\n\\label{eq:s3}\n\\end{equation}\nwhere $\\theta_i$ is the linear polarisation angle and $\\phi_{\\Gamma,i}$ is the orientation of the fast axis of the birefringent material (nominally $\\hat{x}$ in our case).\n\nAssuming the laser has total power $P$, a Gaussian profile in $x$ with standard deviation $w_{x}$, and a top-hat profile in $y$ with half width $w_{y}$, the intensity is given by\n\\begin{equation}\nI\\left(x\\right)= \\frac{P}{\\sqrt{8\\pi}w_{x}w_{y}}e^{-\\frac{x^{2}}{2w_{x}^{2}}\n\\end{equation}\nwhere $2w_y \\gg w_x$. There is then an analytic solution to equation~\\ref{eq:retarddiff} from which we extracted a retardance gradient in the laser tail, $x=w_{x}$, of\n\\begin{equation}\n\\frac{\\partial\\Gamma}{\\partial x}\\approx\\frac{{\\rm erf}(1\/\\sqrt{2})P\\kappa t}{4w_y\\lambda}\\approx0.03~\\mathrm{rad}\/{\\rm mm}\n\\label{eq:retardance_gradient}\n\\end{equation}\nfor a nominal laser power of ${\\approx}2$~W. Similar results were obtained from numerical finite element analysis. Thermal stress-induced birefringence has been observed in similar systems such as in UHV vacuum windows \\cite{Solmeyer2011}, laser output windows \\cite{Eisenbach1992}, and Nd:YAG rods \\cite{Koechner1970}.\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=10cm]{Ellipticity_and_Thermo_elastic_model.pdf}\n\\par\\end{centering}\n\\caption{Measurement of the ellipticity, $S$, as a function of position along $x$ within the state readout laser beam. A fit to the thermo-elastic model, which assumes a Gaussian laser profile and has the amplitude and offset in $S$ as free parameters, is overlaid.}\n\\label{ite:birefringence}\n\\end{figure}\n\nThe estimates of the ellipticity gradient agree well with measurements of the polarisation of the beam, as shown in figure~\\ref{ite:birefringence}. These polarimetry measurements were adapted from the procedure described in \\cite{Berry1977}; a polarimeter was constructed consisting of a rotating quarter-wave plate, fixed polariser, and fast photodiode.\nThe use of a fast photodetector allows for polarimetry of the probe beam during the 100~kHz polarisation switching. The resolution of the system was such that we could quickly measure the normalized circular Stokes parameter, $S$, to a few percent, which is sufficient to measure typical birefringence gradients of ${\\sim}10\\%$ across the beam.\n\n\n\\subsubsection{Suppression of AC Stark Shift Phases}\n\\label{sssec:suppression_of_the_AC_stark_shift_phases}\n\n\\hspace*{\\fill} \\\\\nWe were able to suppress the magnitude of the AC Stark shift phases in several different ways that are illustrated in figure~\\ref{fig:phase_vs_detuning}. The ellipticity gradient across the state preparation laser beam was suppressed by tuning the linear polarisation angle: as per equation~\\ref{eq:s3}, the ellipticity gradient is proportional to $\\sin(2\\theta_{{\\rm prep}}-2\\phi_{\\Gamma,{\\rm prep}})$, which vanishes when the polarisation is aligned along a birefringence axis, i.e. $\\theta_{{\\rm prep}}=\\phi_{\\Gamma,{\\rm prep}},\\phi_{\\Gamma,{\\rm prep}}+\\pi\/2$. To determine $\\phi_{\\Gamma,{\\rm prep}}$ we measured the total accumulated phase as a function of laser detuning for various $\\theta_{{\\rm prep}}$ and then extracted the slope $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}=\\partial\\Phi^{\\rm{nr}}\/\\partial\\Delta_{{\\rm prep}}$ for small detuning values. Note that when fitting the phase vs.\\ detuning data we found that cubic functions provided significantly better fits over the detuning ranges used (see Figure~\\ref{fig:phase_vs_detuning}(B)). We then selected $\\theta_{{\\rm prep}}$ to minimize $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}$. This suppressed $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}$ by about a factor of 50 relative to its original value, to $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}\\lesssim0.1$~mrad\/(2$\\pi~\\times$~MHz). \n\nAnother method implemented to suppress AC Stark shift phases was to reduce the time-averaged power of the state preparation laser incident on the field plates. We used a chopper wheel to modulate the laser at 50~Hz, synchronous with the molecular beam pulses, with a 50\\% duty cycle. We estimated the time scale for thermal changes to be on the order of $Q\/\\dot{Q}\\sim2\\rho Cw_{x}^{2}\/\\kappa\\sim10^{\\:}\\mathrm{s}$, where $\\rho$ and $C$ are the density and heat capacity of Borofloat respectively, so did not anticipate any significant transient effects\nto be introduced. This modification reduced the retardance gradient, and hence the value of $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}$, by about a factor of two, as shown in Figure~\\ref{fig:phase_vs_detuning}(C).\n\nFinally, $\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}$ was suppressed by shaping of the laser beam intensity profile. AC Stark shift phases were most significant at the downstream edge of the state preparation laser beam. Here, the intensity is such that bright-dark state mixing is still occurring but the bright state is not efficiently optically pumped away. By making the spatial intensity profile drop off more rapidly, we reduced the time that molecules spent in this intermediate intensity regime. This was achieved by taking advantage of the aspherical distortion introduced by misaligning a telescope immediately before the laser beam entered the spin-precession region. This suppressed $\\alpha_{\\Delta,{\\rm prep}}$ and $\\beta_{\\Delta^{2},{\\rm prep}}$ by ${\\approx}2$, as shown in Figure~\\ref{fig:phase_vs_detuning}(C). In addition to a phase suppression, we noticed that the optimal laser polarisation angle changed after implementing the steps described, as can be seen in Figure~\\ref{fig:phase_vs_detuning}(C). The reason for this change is not definitively known, but we suspect that as we suppressed the birefringent contribution to the AC Stark shift phase, the non-birefringent contributions (i.e. the phase due to nonzero ellipticity causing bright-dark state mixing via the off-resonant opposite parity excited state) became fractionally larger, and we needed to tune the polarisation angle to obtain cancellation between these two classes of effects.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=15cm]{phase_vs_detuning.pdf}\n\\caption{(A) Measured molecule phase as a function of preparation laser detuning. The slope agrees with originally observed $\\Phi^{\\N\\E}$ dependence on $\\Delta^{\\N\\E}$. (B) Phase dependence on detuning for multiple preparation laser polarisation angles. (C) $\\partial \\Phi\/\\partial \\Delta^{\\rm{nr}}$ shows clear sinusoidal dependence on preparation laser polarisation. The magnitude of $\\partial \\Phi\/\\partial \\Delta^{\\rm{nr}}$ decreases for all polarisation angles when the Gaussian beam tails are clipped (blue) and when the average laser power is reduced with a chopper wheel (red).}\n\\label{fig:phase_vs_detuning}\n\\end{figure}\n\nWe observed much smaller AC Stark shift phases in the state readout laser beam than in the state preparation laser beam. This is not surprising since the effect is largely birefringent; the contributions to the effective polarisation imperfections for the $\\hat{\\epsilon}_{X}$ and $\\hat{\\epsilon}_{Y}$ polarised lasers should be opposite in sign, $d\\theta_{X}\\propto\\sin(2(\\theta_{\\rm{read}}-\\phi_{\\Gamma,\\rm{read}}))$, $d\\theta_{Y}\\propto\\sin(2(\\theta_{\\rm{read}}-\\phi_{\\Gamma,\\rm{read}}+\\pi\/2))$, such that they cancel each other in the measured phase (cf.\\ equation~\\ref{eq:Measured_Phase_with_Polarization_Imperfections}). The residual AC Stark shift phases measured in the state readout beam gave $\\alpha_{\\Delta,\\rm{read}}^{\\rm{nr}}\\approx0.5^{\\:}\\mathrm{mrad}\/(2\\pi\\times\\mathrm{MHz})$. This was sufficiently small that the methods of suppression described above were only implemented in the state preparation region.\n\n\\subsubsection{Systematic Errors due to Correlated Laser Parameters}\n\\label{sssec:correlated_laser_parameters}\n\nIn the discussion above, we described how polarisation imperfections can lead to contributions to the measured phase that depend on the AC Stark shifts and hence on the laser detunings $\\Delta_{i}$ and Rabi frequencies $\\Omega_{{\\rm r}, i}$. However, these phases only produce a systematic error in $\\omega^{\\mathcal{NE}}$ if there is a nonzero correlation $\\Delta_{i}^{\\N\\E}$ or $\\Omega_{{\\rm r}, i}^{\\N\\E}$ of the laser detuning or Rabi frequency. We observed such correlations and discuss them in this section. We will also describe how we evaluated the associated systematic errors.\n\nIn section~\\ref{sec:state_prep_read} (see figure~\\ref{fig:Enr_wNE}) we discussed how a non-reversing component of the applied electric field, $\\E^{\\rm{nr}}$, could produce a $\\Delta^{\\N\\E}$. In an entirely analogous manner, the Rabi frequency magnitude $\\Omega_{\\rm r}$ of the $H\\rightarrow C$ transition can exhibit the following correlations:\n\\begin{align}\n\\Omega_{{\\rm r}, i}=&\\Omega_{{\\rm r}, i}^{\\rm{nr}}+\\Nsw\\Omega_{{\\rm r}, i}^{\\N}+\\Nsw\\Psw\\Omega_{{\\rm r}, i}^{\\N\\P}+\\Nsw\\Esw\\Omega_{{\\rm r}, i}^{\\N\\E}+\\dots\n\\end{align}\nHere, $\\Omega_{{\\rm r}, i}^{\\rm{nr}}$ is the dominant component of the Rabi frequency for laser $i\\in\\left\\{ {\\rm prep},X,Y\\right\\} $, which could fluctuate in time on the order of 5\\% due to laser power instability. $\\Omega_{{\\rm r}, i}^{\\N}$ is generated by a laser power difference between the $\\tilde{\\N}$ states. This arose because we routed the laser light along different paths through a series of AOMs for each state. We measured this effect with photodiodes and found that the largest fractional power correlation was $\\Omega_{\\rm r}^{\\N}\/\\Omega_{\\rm r}^{\\rm{nr}}\\approx2.5\\times10^{-3}$. An additional contribution to $\\Omega_{{\\rm r}, i}^{\\N}$ and a contribribution to $\\Omega_{{\\rm r}, i}^{\\N\\P}$ on the same order arises due to Stark mixing between rotational levels in $H$ and $C$, leading to $\\Nsw$- and $\\Nsw\\Psw$-correlated transition amplitudes on the $H\\rightarrow C$ transition.\n\nAlthough we did not observe a laser power correlation with $\\Nsw\\Esw$ we did observe signals consistent with a Rabi frequency correlation, $\\Omega_{\\rm r}^{\\N\\E}$. A nonzero $\\Nsw\\Esw$-correlated fluorescence signal (as defined in section \\ref{sec:signal_asymmetry}) that also reversed with the laser propagation direction $\\hat{k}\\cdot\\hat{z}$, $F^{\\N\\E}\/F^{\\rm{nr}}\\approx-(2.4\\times10^{-3})(\\hat{k}\\cdot\\hat{z})$, together with a nonzero $\\omega^{\\N\\E\\B}\\approx(2.5{}^{\\:}\\mathrm{mrad}\/\\mbox{s})(\\B_{z}\/\\mathrm{mG})(\\hat{k}\\cdot\\hat{z})$, provided the first evidence that a nonzero $\\Omega_{\\rm r}^{\\N\\E}$ existed in our system. We believe that this fluorescence correlation arises from a linear dependence of the fluorescence signal size on Rabi frequency, $F^{\\N\\E}=(\\partial F\/\\partial\\Omega_{\\rm r}^{\\rm{nr}})\\Omega_{\\rm r}^{\\N\\E}$, which is nonzero since the state readout transitions were not fully saturated. We believe that the signal in $\\omega^{\\N\\E\\B}$ was caused by a coupling between the Rabi-frequency correlation and the $\\B$-odd AC Stark shift phase, $\\omega^{\\N\\E\\B}=\\frac{1}{\\tau}\\beta_{d\\Omega_{\\rm r}}^{\\B}\\B_{z}(\\Omega_{\\rm r}^{\\N\\E}\/\\Omega_{\\rm r}^{\\rm{nr}})$. We were able to verify a linear dependence of both of these channels on $\\Omega_{\\rm r}^{\\N\\E}$ by intentionally correlating the laser intensity with $\\Nsw\\Esw$ using AOMs; this is shown for the $\\Phi^{\\N\\E\\B}$ channel in Figure~\\ref{fig:Omega_NE}. Varying the size of this artificial $\\Omega_{\\rm r}^{\\N\\E}$ allowed us to measure the value present in the experiment under normal operating conditions, $\\Omega_{\\rm r}^{\\N\\E}\/\\Omega_{\\rm r}^{\\rm{nr}}=(-8.0\\pm0.8)\\times10^{-3}(\\hat{k}\\cdot\\hat{z})$. $\\Omega_{\\rm r}^{\\N\\E}$ can couple to $\\beta_{d\\Omega_{\\rm r},i}^{\\rm{nr}}$ as per equations~\\ref{eq:Empirical_AC_Stark_Shift_Phase_Result} and \\ref{eq:bdOmega} to result in a systematic error in $\\omega^{\\mathcal{NE}}$. A nonzero $\\beta_{d\\Omega_{\\rm r},i}^{\\rm{nr}}$ can be produced by a linear polarisation angle gradient (not observed in the experiment) or by a non-reversing Zeeman shift component $g_1\\mu_{\\rm B}\\B_{z}^{\\rm{nr}}$.\n\nWhile searching for a model to explain the intrinsic $\\Omega_{\\rm r}^{\\N\\E}$, we developed the Stark interference model presented in section \\ref{sssec:stark_interference_between_E1_and_M1_transition_amplitudes}. For unnormalized effective polarisation $\\vec{\\varepsilon}_{\\mathrm{eff}}=\\vec{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr}}+\\Nsw\\Esw d\\vec{\\varepsilon}_{\\mathrm{eff}}^{\\N\\E}$,\nthis model predicts $\\Omega_{\\rm r}^{\\N\\E}\/\\Omega_{\\rm r}^{\\rm{nr}}\\approx\\text{Re}(\\vec{\\varepsilon}_{\\mathrm{eff}}^{\\rm{nr} *}\\cdot d\\vec{\\varepsilon}_{\\mathrm{eff}})\\approx-\\text{Im}\\left[(a_{M1}+a_{E2})\\right](\\hat{k}\\cdot\\hat{z})$, which correctly predicts the dependence of $\\Omega_{\\rm r}^{\\N\\E}$ on the laser propagation direction $\\hat{k}\\cdot\\hat{z}$. However, the factors $a_{\\rm M1}$ and $a_{\\rm E2}$, which correspond to the ratio of M1 and E2 amplitudes to the E1 amplitude, must be real for a plane wave, so $\\text{Im}\\left[(a_{M1}+a_{E2})\\right]=0$. Hence this model fails to explain this Rabi frequency correlation unless there is some additional effect that introduces a phase shift between the E1 and M1 amplitudes. For example, interference between the E1 amplitude due to the incident laser beam, and a phase shifted M1 amplitude due to a (low intensity) reflected beam can lead to a nonzero $\\Omega_{\\rm r}^{\\N\\E}$ by this model. However, this phase factor oscillates spatially on the scale of the light wavelength, which is very small compared to the size of the molecule cloud and hence should average out over the entire molecular beam cloud. The origin of the intrinsic $\\Omega_{\\rm r}^{\\N\\E}$ is still not fully understood, and we are continuing to explore models to understand this effect.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=12cm]{P_NE.pdf}\n\\caption{$\\Phi^{\\N\\E\\B}$ as a function of applied $\\Nsw\\Esw$-correlated laser power, $P^{\\N\\E}$, for both directions of laser pointing, $\\hat{k}\\cdot\\hat{z}$. The artificial $\\Omega_{\\rm r}^{\\N\\E}$ resulting from correlated power $P^{\\N\\E}$ systematically shifts $\\omega^{\\N\\E\\B}$ in accordance with equation~\\ref{eq:Empirical_AC_Stark_Shift_Phase_Result}. $\\Phi^{\\N\\E\\B}$ is zero when the applied $P^{\\N\\E}$ is such that there is no net $\\Nsw\\Esw$-correlated Rabi frequency. The intrinsic $\\Omega_{\\rm r}^{\\N\\E}$ (i.e. that inferred when $P^{\\N\\E}=0$) changed sign with $\\hat{k}\\cdot\\hat{z}$ within the resolution of the measurement. The slopes between the two measurements differ due to differences in the AC Stark shift phase, believed to be due to differences in the spatial intensity profile and polarisation structure between the two measurements.}\n\\label{fig:Omega_NE}\n\\end{figure}\n\nGiven the empirical AC Stark shift phase model in equation~\\ref{eq:Empirical_AC_Stark_Shift_Phase_Result}, the resulting systematic errors in the frequency measurement are given by\n\\begin{align}\n\\omega^{\\N\\E}_{\\E^{\\rm{nr}}} & =\\frac{1}{\\tau}\\sum_{i\\in\\left\\{ {\\rm prep},X,Y\\right\\} }\\alpha_{\\Delta,i}^{\\rm{nr}}D_1\\E^{\\rm{nr}}(x_{i})\\\\\n\\omega^{\\N\\E}_{\\Omega_{\\rm r}^{\\N\\E}} & =\\frac{1}{\\tau}\\sum_{i\\in\\left\\{ {\\rm prep},X,Y\\right\\} }\\beta_{d\\Omega_{\\rm r},i}^{\\rm{nr}}(\\Omega_{\\rm r}^{\\N\\E}\/\\Omega_{\\rm r}^{\\rm{nr}}).\n\\end{align}\nEarly in the experiment, we observed a nonzero systematic shift $\\omega^{\\N\\E}_{\\E^{\\rm{nr}}}$ and took the steps outlined in section~\\ref{sssec:suppression_of_the_AC_stark_shift_phases} to suppress it. To verify that the steps taken were effective, we examined $\\omega^{\\mathcal{NE}}$ as a function of an intentionally applied non-reversing electric field. The resulting data are shown in figure~\\ref{fig:Enr_slope}. The original slope, $\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}}=(6.7\\pm0.4)(\\mathrm{rad\/s})\/(\\mbox{V\/cm})$,\ncorresponded to a systematic shift of $\\omega^{\\N\\E}_{\\E^{\\rm{nr}}}\\approx-34~\\mathrm{mrad}\/\\mathrm{s}$ when combined with the measured $\\E^{\\rm{nr}}\\approx-5^{\\:}\\mathrm{mV\/\\mathrm{cm}}$. Following the modifications described above, the $\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}}$ slope was greatly suppressed, reducing the systematic error to $\\omega^{\\N\\E}_{\\E^{\\rm{nr}}}<1~\\mathrm{mrad}\/\\mathrm{s}$, well below the statistical uncertainty in the measurement of $\\omega^{\\mathcal{NE}}$.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=12cm]{Enr_slope.pdf}\n\\caption{Linear dependence of the $\\omega^{\\mathcal{NE}}$ channel on an applied non-reversing electric field observed before (red) and after (black) we suppressed the known AC Stark shift phase by optimizing the preparation laser beam shape, time-averaged power and polarisation.}\n\\label{fig:Enr_slope}\n\\end{figure}\n\nBecause we observed that the parameters $\\E^{\\rm{nr}}$ and $\\Omega_{\\rm r}^{\\N\\E}$ caused systematic errors in $\\omega^{\\mathcal{NE}}$, we intermittently measured the size of the associated systematic errors throughout the datasets that were used for our reported result. We measured $\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}}$ by applying a range of large non-reversing electric fields, up to around 70 times that present under normal running conditions. The value of $\\partial\\omega^{\\N\\E}\/\\partial\\Omega_{\\rm r}^{\\N\\E}$ was measured by applying a correlated laser power $P^{\\N\\E}$ in the state preparation and state readout beams with a magnitude corresponding to an applied $\\Omega_{\\rm r}^{\\N\\E}$ that was up to 20 times that measured under normal operating conditions. These parameters were measured for multiple values of the magnetic field magnitude, $\\B_{z}$, for which different state readout laser beam polarisations were required. Due to known birefringent behavior of the AC stark shift phases, we allowed for this possibility for all AC stark shift phase systematic errors. We measured  $\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}}$ for both $\\hat{k}\\cdot\\hat{z}=\\pm1$, but the $\\Omega_{\\rm r}^{\\N\\E}$ systematic error was only discovered after the $\\hat{k}\\cdot\\hat{z}=+1$ dataset and hence $\\partial\\omega^{\\N\\E}\/\\partial\\Omega_{\\rm r}^{\\N\\E}$\nwas only monitored during the $\\hat{k}\\cdot\\hat{z}=-1$ dataset. The $\\Omega_{\\rm r}^{\\N\\E}$ systematic error during the $\\hat{k}\\cdot\\hat{z}=+1$ dataset was determined from auxiliary measurements of the AC Stark\nshift phase. As described in section \\ref{ssec:efields}, $\\E^{\\rm{nr}}(x)$ exhibits significant spatial variation along the beam-line axis, $x$. However the $\\E^{\\rm{nr}}$ that was intentionally applied to determine $\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}}$ was spatially uniform, and hence these measurements were insensitive to the difference $(\\E^{\\rm{nr}}(x_{{\\rm prep}})-\\E^{\\rm{nr}}(x_{\\rm{read}}))$ between the state preparation laser beam at $x_{{\\rm prep}}$ and the state readout beam at $x_{\\rm{read}}$. For this reason, we deduced the systematic error proportional to the difference $(\\E^{\\rm{nr}}(x_{{\\rm prep}})-\\E^{\\rm{nr}}(x_{\\rm{read}}))$ from auxiliary measurements of the AC Stark shift phase parameters, $\\alpha_{\\Delta,i}^{\\rm{nr}}$.\n\nIn summary, the systematic errors proportional to $\\E^{\\rm{nr}}$ and $\\Omega_{\\rm r}^{\\N\\E}$ that were evaluated and subtracted from $\\omega^{\\N\\E}$ to report a measured value of $\\wNEt$ can be expressed as\n\n\\begin{align}\n\\omega_{\\E^{\\rm{nr}}}^{\\N\\E}= & \\left(\\frac{\\partial\\omega^{\\N\\E}}{\\partial\\E^{\\rm{nr}}}\\right)\\frac{1}{2}(\\E^{\\rm{nr}}(x_{{\\rm prep}})+\\E^{\\rm{nr}}(x_{\\rm{read}}))\\nonumber\\\\\n&+\\frac{1}{\\tau}(\\alpha_{\\Delta,{\\rm prep}}^{\\rm{nr}}-\\alpha_{\\Delta,X}^{\\rm{nr}}-\\alpha_{\\Delta,Y}^{\\rm{nr}})\\frac{1}{2}(\\E^{\\rm{nr}}(x_{{\\rm prep}})-\\E^{\\rm{nr}}(x_{\\rm{read}}))\\label{eq:Enr_systematic_error_value}\\\\\n\\omega_{\\Omega_{\\rm r}^{\\N\\E}}^{\\N\\E}= & \\begin{cases}\n\\frac{1}{\\tau}\\sum_{i\\in\\left\\{ {\\rm prep},X,Y\\right\\} }\\beta_{d\\Omega_{\\rm r},i}^{\\rm{nr}}\\left(\\frac{\\Omega_{\\rm r}^{\\N\\E}}{\\Omega_{\\rm r}^{\\rm{nr}}}\\right) & (\\hat{k}\\cdot\\hat{z})=+1\\\\\n\\left(\\frac{\\partial\\omega^{\\N\\E}}{\\partial\\Omega_{\\rm r}^{\\N\\E}}\\right)\\Omega_{\\rm r}^{\\N\\E} & (\\hat{k}\\cdot\\hat{z})=-1\n\\end{cases}\n\\end{align}\nwhere $(\\partial\\omega^{\\N\\E}\/\\partial\\E^{\\rm{nr}})$ and $(\\partial\\omega^{\\N\\E}\/\\partial\\Omega_{\\rm r}^{\\N\\E})$ were monitored by \\emph{Intentional Parameter Variations} (see section~\\ref{sec:Measurement_scheme_more_detail}) throughout the dataset used for our reported result, and $\\E^{\\rm{nr}}(x_{{\\rm prep}})$, $\\E^{\\rm{nr}}(x_{\\rm{read}})$,\n$\\Omega_{\\rm r}^{\\N\\E}$, $\\alpha_{\\Delta,i}^{\\rm{nr}}$, and $\\beta_{d\\Omega_{\\rm r},i}^{\\rm{nr}}$ were obtained from auxiliary measurements. These two systematic errors account for almost all of the systematic offset that was subtracted from $\\omega^{\\mathcal{NE}}$ to obtain $\\wNEt$ as described in section~\\ref{ssec:total_systematic_error_budget}.\n\n\n\n\\subsection{\\texorpdfstring{$\\A^{\\N\\E}$}{ANE} asymmetry effects}\n\\label{ssec:asymmetry_effects}\nIn addition to the dependence of the measured phase on laser detuning and Rabi frequency, we observed dependence of the asymmetry $\\A$ (as defined in section~\\ref{sec:signal_asymmetry}) on the laser parameters $\\Delta_{\\rm{read}}$ and $\\Omega_{{\\rm r},\\rm{read}}$, due to differences between the properties of the $X$ and $Y$ readout laser beams. The laser-induced fluorescence signal $F(\\Delta,\\Omega_{\\rm r})$ varies quadratically with detuning (for small detuning) and linearly with Rabi frequency. Under normal conditions, the signal sizes from $X$ and $Y$ are comparable, $F_X \\approx F_Y \\approx F$. If the $X$ and $Y$ beams have different wavevectors, $\\vec{k}_{X,Y} = \\vec{k}^{\\mathrm{nr}} \\pm \\vec{k}^{XY}$, and $\\vec{k}^{XY}$ has some component along $\\hat{x}$, then the two beams will acquire different Doppler shifts. This leads to a linear dependence of the asymmetry on detuning, which in turn can couple to $\\Delta^{\\N\\E}$ to result in a contribution to $\\A^{\\N\\E}$,\n\\begin{equation}\n\\A^{\\N\\E} \\approx\\frac{1}{F} \\frac{\\partial^2 F}{\\partial \\Delta_{\\rm{read}}^2} (\\vec{k}^{XY}\\cdot\\langle\\vec{v}\\rangle)\\Delta^{\\N\\E}.\n\\end{equation}\nSimilarly, if the two readout beams differ in Rabi frequency, $\\Omega_{r, X\/Y} \\approx \\Omega_{\\rm r}^{\\mathrm{nr}} \\pm \\Omega_{\\rm r}^{XY}$, the asymmetry becomes linearly dependent on Rabi frequency, which in turn can couple to $\\Omega_{\\rm r}^{\\N\\E}$ to result in a contribution to $\\A^{\\N\\E}$,\n\\begin{equation}\n\\A^{\\N\\E} \\approx -\\left(\\frac{1}{F} \\frac{\\partial F}{\\partial \\Omega_{\\rm r}}\\right)^2 \\Omega_{\\rm r}^{XY} \\Omega_{\\rm r}^{\\N\\E}.\n\\end{equation}\n\nHowever, these asymmetry effects are very distinguishable from spin precession phases and polarisation misalignments. Since the $\\Psw$ and $\\Rsw$ switches effectively swap the role of the $X$ and $Y$ readout beams, the $\\A^{\\N\\E}$ effects described above do not contribute to $\\omega^{\\mathcal{NE}}$ when summed over these switches. Additionally, asymmetry effects, once converted to an equivalent frequency or phase, depend on the sign of the contrast, $\\C$, unlike true phases. In the $\\B_z\\approx 20$~mG configuration, ${\\rm sgn}(\\C)={\\rm sgn}(\\B_z)$, but ${\\rm sgn}(\\C)$ has no dependence on ${\\rm sgn}(\\B_z)$ for $\\B_z\\approx 1,~40$~mG. Hence asymmetry correlations $\\A^{\\N\\E}$ are mapped onto frequency correlations $\\omega^{\\N\\E\\P\\R}$ or $\\omega^{\\N\\E\\B\\P\\R}$ depending on the magnetic field magnitude. \n\nIf the pointing or Rabi frequency differences between the $X$ and $Y$ beams drift on timescales comparable to or shorter than the $\\Psw$ or $\\Rsw$ switches, these effects can occasionally `leak' into the `adjacent' channels $\\omega^{\\N\\E\\P}$, $\\omega^{\\N\\E\\R}$, $\\omega^{\\N\\E\\B\\P}$, $\\omega^{\\N\\E\\B\\R}$; however, we have not seen any evidence of these effects contributing to the $\\omega^{\\N\\E}$ channel itself, and hence did not include systematic error contributions due to these effects in our systematic error budget.\n\n\n\\subsection{\\texorpdfstring{$\\Esw$}{E}-Correlated Phase}\n\\label{ssec:E_correlated_phase}\n\nPrevious eEDM measurements have often been limited by a variety of systematic errors that would have produced an $\\Esw$-correlated phase precession frequency in our experiment, $\\omega^{\\E}$ \\cite{Khriplovich1997,Murthy1989,Regan2002,Eckel2013}, such as $\\Esw$-correlated leakage currents, geometric phases, and motional magnetic fields. Our ability to spectroscopically reverse the molecular orientation through a choice of $\\Nsw$ distinguished these effects from an eEDM-generated phase. In addition, the aforementioned effects scale with the magnitude of the applied electric field, which was orders of magnitude smaller in our experiment than previous similar eEDM experiments due to the high polarisability of ThO \\cite{Regan2002}. Also, because the molecular polarisation was saturated, the eEDM phase should have been independent of the magnitude of the applied field. We also note that any shifts from leakage currents and motional magnetic fields coupled through the magnetic dipole moment, which is near-zero in the $H$-state of ThO. Thus we expected $\\omega^{\\E}$ to be substantially suppressed, and that it should not enter $\\omega^{\\N\\E}$ at any significant level.\n\nThe reversal of $\\Nsw$ did not, however, entirely eliminate an eEDM-like phase due to $\\omega^{\\E}$. As discussed in section~\\ref{sec:compute_phase}, there was a small and $\\E$-field dependent difference between the $g$-factors of the two $\\Nsw$ levels \\cite{Bickman2009,Petrov2014}, which meant that a systematic error in the $\\omega^\\E$ channel showing up in $\\omega^{\\N\\E}$ at a level given by $\\omega^{\\N\\E}_{\\omega^\\E}=(\\eta\\E\/g_1)\\omega^{\\E}$. \nWe verified this relation by intentionally correlating a 1.4~mG component of our applied magnetic field with $\\Esw$. This deliberate $\\B^{\\E}$ resulted in a large shift in the value of $\\omega^\\E$ and a $\\sim$1000-times smaller offset of $\\omega^{\\mathcal{NE}}$, as illustrated in figure~\\ref{fig:phi_E}.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=12cm]{B_E_systematic.pdf}\n\\caption{Illustration of the $\\sim$1000-fold suppression of systematic errors associated with $\\omega^{\\E}$ provided by the $\\Nsw$ switch. Large values of $\\omega^{\\E}$ occur when there is a component of $\\B_z$ correlated with $\\Esw$, $\\B^\\E$. In previous eEDM experiments, this would have corresponded to a systematic error. In our experiment a much smaller shift in $\\omega^{\\mathcal{NE}}$ results from the small difference in magnetic moments between the two $\\Nsw$ levels. Error bars for the $\\omega^{\\E}$ data are significantly smaller than the data points. Data were taken with $\\E=142~\\mathrm{V}\/\\mathrm{cm}$ and the measured ratio of the slopes, $(\\partial\\omega^{\\mathcal{NE}}\/\\partial\\B^\\E)\/(\\partial\\omega^{\\E}\/\\partial\\B^\\E)=(2.8\\pm0.8)\\times 10^{-3}$ is consistent with the expected value $\\eta \\E\/g_1=(2.5\\pm0.1)\\times 10^{-3}$.}\n\\label{fig:phi_E}\n\\end{figure}\n\nThe intentionally applied $\\B^\\E$ was the only experimental parameter that was observed to produce a measurable shift in $\\omega^{\\E}$. Even large ($\\sim$20~mG) magnetic fields components along $\\hat{x}$ and $\\hat{y}$, which exaggerate the effect of motional magnetic fields, did not shift $\\omega^{\\E}$ (this is expected, since the large tensor Stark shift in $\\ket{H,J=1}$ dramatically suppresses the effect of motional magnetic fields \\cite{Player1970}). For our eEDM data set, $\\omega^{\\E}$ was consistent with zero. We included a contribution from $\\omega^{\\E}$ in our error budget for $\\omega^{\\mathcal{NE}}$ by multiplying the mean and uncertainty of the extracted $\\omega^{\\E}$ by our measured $|\\E|$-dependent suppression factors $\\eta\\E\/g_1$.\n\n\n\\subsection{\\texorpdfstring{$\\Nsw$}{N}-Correlated Laser Pointing}\n\\label{ssec:N_correlated_pointing}\nWe discovered a nonzero, time-dependent signal in $\\omega^{\\N}$ which was associated with an $\\Nsw$-correlated laser pointing, $\\hat{k}^{\\N}\\approx5~\\upmu$rad. An investigation into the mechanism behind this effect was inconclusive. We found that the pointing correlation appeared downstream of the AOMs that created the rapid polarisation switching and improved alignment was able to reduce the effect. We also found that the observed pointing was in some way correlated with the seed power and input angle of incidence into the high-power fiber amplifier immediately upstream of the polarisation switching, despite the fact that pointing out of the amplifier did not fluctuate. Since we used four different sets of AOMs to perform the $\\Nsw$ and $\\Psw$ switches before the amplifier, we observed laser pointing correlated with both of these switches. By matching the characteristics of these four beam paths we were able to suppress $\\hat{k}^{\\N}$ to $<1~\\upmu$rad.\n\nThe effect of $\\hat{k}^{\\N}$ on $\\omega^{\\N}$ was studied by exaggerating the former with piezoelectrically actuated mirrors. Examining $\\partial\\omega^{\\N}\/\\partial\\hat{k}^{\\N}$ showed significant fluctuations in its value. We were unable to identify the mechanism by which $\\hat{k}^\\N$ affected $\\omega^{\\N}$.\n\nWe had no evidence that the effect causing the observed variation in $\\omega^{\\N}$ also caused a systematic error in $\\omega^{\\mathcal{NE}}$, but to be cautious we included an associated systematic uncertainty in our systematic error budget (section \\ref{ssec:total_systematic_error_budget}). Assuming a linear relationship between $\\omega^{\\N\\E}$ and $\\omega^{\\N}$, we extracted $\\partial\\omega^{\\N\\E}\/\\partial\\omega^{N}$ from a combination of data taken under normal conditions and with an exaggerated $\\omega^{\\N}$ induced by an exaggerated $\\hat{k}^{\\N}$. We then placed an upper limit on a possible systematic error $\\omega^{\\mathcal{NE}}_{\\omega^{\\N}}$ based on the value of $\\omega^{\\N}$ obtained under normal running conditions.\nThe resulting systematic uncertainty was four times smaller than our statistical uncertainty.\n\n\\subsection{Laser Imperfections}\n\\label{ssec:laser_imperfections}\nOf the lasers used in our experiment, only the state preparation and readout lasers were known to produce possible systematic errors; imperfections in the rotational cooling, optical pumping or target ablation lasers simply resulted in a reduction in usable molecule flux. As part of our search for systematic errors, we intentionally exaggerated all known state preparation and readout laser imperfections possible without dismantling the apparatus (cf.\\ table~\\ref{tbl:syst_check}). In this section we describe this procedure and the resulting contributions to our systematic error budget.\n\n\\subsubsection{Laser Detuning}\n\\label{sssec:laser_detuning}\n\\hspace*{\\fill} \\\\\nThe correlated components of the state preparation and readout laser beam detunings are described in detail in section~\\ref{sec:state_prep_read}. Each detuning component was separately exaggerated and in some cases multiple components were simultaneously exaggerated. Most of the detuning terms in equation~\\ref{eq:detuningcorrelations} were exaggerated to $\\pm2\\pi\\times1$--2~MHz. No detuning or detuning correlation produced a significant shift in $\\omega^{\\mathcal{NE}}$ other than $\\Delta^{\\N\\E}$ caused by $\\E^{\\rm{nr}}$, discussed in section~\\ref{sssec:correlated_laser_parameters}. In some cases, shifts in other phase channels were induced, but all shifts were consistent with well-understood AC Stark shift and asymmetry models described in sections \\ref{sssec:AC_stark_shift_phases} and \\ref{ssec:asymmetry_effects}.\nFor example, the combination of nonzero $\\Delta^{\\N}$ and $\\Delta^{\\rm{nr}}$ coupled to the $\\B$-dependent component of the AC stark shift phase (equation~\\ref{eq:bdOmega}) induces a significant shift in $\\omega^{\\N\\B}$ (cf.\\ equation~\\ref{eq:Empirical_AC_Stark_Shift_Phase_Result}). \nAsymmetry correlations also resulted from these detuning correlations, but these were only manifested in channels odd with respect to $\\Psw$ and $\\Rsw$, and hence had no plausible effect on $\\omega^{\\mathcal{NE}}$. Because the YbF eEDM experiment \\cite{Kara2012} observed unexplained dependence of the measured eEDM value on state preparation microwave detuning, we included a systematic error contribution from all detuning imperfections in our systematic error budget.\n\n\\subsubsection{Laser Pointing and Intensity}\n\\label{sssec:laser_pointing_and_intensity}\n\\hspace*{\\fill} \\\\\nSimilar to detuning imperfections, the state preparation and readout lasers could have imperfect pointing and correlated intensities. Ideally the laser propagation direction, $\\hat{k}$, would have been parallel to the laboratory electric field. This would have diminished the amount of $\\hat{z}$ polarised light experienced by the molecules, which could drive unwanted off-resonant transitions, and prevented stray retroflection from the ITO field plate surfaces. Using this ITO retroflection as a guide, we aligned $\\hat{k}$ perpendicular to the field plate surface, and therefore parallel to $\\hat{\\E}$, to within $\\sim{3}$~mrad. To test for errors related to imperfect pointing, both the state preparation and readout pointing misalignments were exaggerated in the $x$-direction to $\\pm$10~mrad, as was the relative pointing of the $\\hat{X}$ and $\\hat{Y}$ state readout beams. The vacuum windows and $\\sim$3.8~cm wide holes in the magnetic shields prevented us from further misaligning the beams. To decouple pointing imperfections from detuning imperfections, the state preparation and readout laser frequencies were tuned to resonance after each pointing adjustment. No shift in $\\omega^{\\mathcal{NE}}$ was observed and no systematic error contribution from pointing imperfections was included. Pointing imperfections were only observed to affect the signal asymmetry, as previously discussed in section \\ref{ssec:asymmetry_effects}.\n\nUnlike laser pointing and detuning, there was no `ideal' value for laser intensity. The state preparation and readout laser intensities were chosen such that we were driving optical pumping to completion on the $H\\rightarrow C$ transition without producing unnecessary thermal stress on the field plates. We decreased each laser intensity by a factor of four to check that there was no variation in $\\omega^{\\mathcal{NE}}$. We observed a nonzero $\\Omega_{\\rm r}^{\\N}$ \ncaused by the $\\Nsw$-correlated seed power into the high-power fiber amplifiers and by Stark mixing between rotational levels in $H$ and $C$ as discussed in section \\ref{sssec:correlated_laser_parameters}. We exaggerated this imperfection by a factor of 20. Only $\\omega^{\\N\\B}$ was shifted, consistent with our understanding of the $\\B$-correlated AC Stark shift phase. These intensity systematic error checks were not included in the systematic error budget.\n\n\n\\subsection{Magnetic Field Imperfections}\n\\label{ssec:magnetic_field_imperfections}\nThe $H$ state is very insensitive to a magnetic field $\\B_z$ due to its small $g$-factor, as discussed in section~\\ref{sec:tho_molecule}. Sensitivity to the transverse fields is even further suppressed by the large size of the tensor Stark shift relative to the Zeeman interaction. Nevertheless, there are known mechanisms by which magnetic field imperfections can contribute to systematic errors: $\\B_z^{\\rm{nr}}$ can contribute to the $\\omega^{\\mathcal{NE}}_{\\Omega_{\\rm r}^{\\N\\E}}$ systematic error discussed in section~\\ref{sssec:correlated_laser_parameters}, and transverse fields $\\B_x^{\\rm{nr}}$ and $\\B_y^{\\rm{nr}}$ can lead to the geometric phase systematic errors \\cite{Vutha2010} discussed in section~\\ref{ssec:E_correlated_phase}. We designed the experiment to allow a wide variety of magnetic field tilts and gradients to be applied as described in section~\\ref{sec:bfields} and we directly looked for systematic errors resulting from these magnetic field imperfections.\n\nBoth $\\B$-correlated and uncorrelated imperfections were applied. We did not precisely measure the residual values of each of these parameters along the molecule beam line until we had studied all systematic errors and collected our published data set. Based on the projected ${\\sim}10^5$ magnetic shielding factor, we expected all stray magnetic fields and gradients to be on the order of 10~$\\upmu$G and 1 $\\upmu$G\/cm, respectively. For this reason we only exaggerated these imperfections to $\\sim$2~mG and $\\sim$0.5~mG\/cm. When we mapped out the magnetic field with a magnetometer inserted between the electric field plates as described in section \\ref{sec:bfields}, we discovered that several imperfections were much larger than we expected (e.g.\\ $\\B_y \\approx 0.5$~mG). This was caused by poor magnetic shielding due to insufficient shield degaussing. For this reason we gathered additional eEDM data with some magnetic field parameters exaggerated by an additional factor of five. $\\omega^{\\mathcal{NE}}$ and nearly all other frequency channels, apart from $\\omega^{\\rm{nr}}$ and $\\omega^{\\B}$ were not observed to be affected by any of these magnetic field parameters. \nBecause uncorrelated stray magnetic fields and magnetic field gradients caused unexpected eEDM offsets in the PbO eEDM experiment \\cite{Eckel2013}, we included contributions from all uncorrelated magnetic field imperfections in our systematic error budget described in section \\ref{ssec:total_systematic_error_budget}.\n\n\\subsection{Electric Field Imperfections}\n\\label{ssec:electric_field_imperfections}\nUnlike the magnetic field, we do not have the ability to control electric field gradients and stray electric fields, aside from the average value of $\\E^{\\rm{nr}}$. The field plates were located at the center of the experiment, inside the vacuum chamber and magnetic shields and coils, with no direct access available. To search for systematic errors related to the electric field, equal amounts of eEDM data were gathered with two different electric field magnitudes. The $\\omega^{\\mathcal{NE}}$ values from both field magnitudes were consistent with each other. The YbF eEDM experiment observed unexplained eEDM dependence on the voltage offset common to both field plates. For this reason we exaggerated this offset by a factor of 1000 (relative to its residual value of ${\\approx}5$~mV) and, even though it did not shift our eEDM measurement, included it in our systematic error budget. \n\n\\subsubsection{Molecule Beam} \\label{ssec:molecular_beam}\n\\hspace*{\\fill} \\\\\nThe molecule beam should have ideally travelled parallel to the electric field plates and well-centred between the plates. This minimizes Doppler shifts, protects the plates from being coated with ThO, and ensures that the molecules experience the most uniform electric field. The entire beam source vacuum chamber sat on a two axis ($yz$) translation stage. The exit aperture of the buffer gas cell was aligned to within 1~mm of the centre of the fixed collimators and electric field plates, using a theodolite. Geometric constraints only allowed us to exaggerate the cell misalignment by roughly a factor of three (up to 3~mm) before the molecules would have hit the sides of the field plates. We also \nvaried the transverse spatial and velocity distributions\nby using adjustable collimators between the beam source and spin-precession region to block half of the beam from the $\\pm\\hat{x},\\pm\\hat{z}$ directions. The value of $\\omega^{\\mathcal{NE}}$ was not observed to shift with any molecule beam parameter adjustment.\n\n\n\n\n\\subsection{Searching for Correlations in the eEDM Data Set} \\label{ssec:correlations_in_the_eEDM_data_set}\n\n\nIn addition to performing systematic error checks for possible variations of $\\omega^{\\N\\E}$ with various experimental parameters, we searched for statistically nonzero values within the set of 1536 possible correlations with the block and superblock switches. This analysis was performed for our primary measured quantities $\\omega$, $\\C$, and $\\F$ and for a wide range of auxiliary measurements such as laser powers, magnetic field, room temperature, etc. We also examined the switch-parity channels of $\\omega$, $\\C$, and $\\F$ as a function of time within the molecule beam pulse, and as a function of time within the polarisation switching cycle. We used the Pearson correlation coefficient to look for correlations between the aforementioned switch-parity channels and used the autocorrelation function to look for signs of time variation of the mean within those channels. Figure~\\ref{fig:pixel_plot} illustrates data from such a search with a subset of the previously described quantities. \nIn this search, we looked at 4390 quantities and we set the significance threshold at $4\\sigma$ which correponds to a probability of $p\\approx0.25$ that there will be one or more false positives above that threshold. We represented the significance of each of these quantities with a grayscale pixel. Each pixel that was significant at the $4\\sigma$ level is marked with a symbol corresponding to a known explanatory physical model, or a red dot if the signal is not yet explained. The fact that we understand most of the significant signals present in our experiment, combined with the fact that the statistical distribution of the remaining signals below the significance threshold is consistent with a normal distribution, gives us added confidence in our models of the experiment and our reported eEDM result.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{pixel_plot_new_jpeg.pdf}\n\\caption{Over 4,000 switch-parity channels (left) and correlations between switch parity channels (upper right) computed from the eEDM data set. The deviation of each quantity from zero in units of the statistical uncertainty is indicated by the grayscale shading. We set a significance threshold of $4\\sigma$ above which there is a probability of $p=25\\%$ of finding at least 1 false positive. We mark each significant channel\/correlation with a symbol corresponding to a model known to produce a signal in that channel. The quantities below this threshold exhibit a normal distribution, shown in the lower right.\n}\n\\label{fig:pixel_plot}\n\\end{figure}\n\nChannels\/correlations marked with symbols are significantly nonzero due to known mechanisms as follows:\n\\begin{itemize}\n    \\item Green stars: Correlations due to the nonzero and drifting signal in the $\\omega^{\\N}$ channel described in section~\\ref{ssec:N_correlated_pointing}.\n    \\item Light blue squares: Signals in $\\omega^{\\N\\E\\B}$ channels due to the $\\Bsw$-odd AC stark shift phase coupling to $\\Omega_{\\rm r}^{\\N\\E}$ as described in section~\\ref{sssec:correlated_laser_parameters}.\n    \\item Orange triangles: Correlations due to contrast or asymmetry coupling to $\\Omega_{\\rm r}^{\\N\\E}$. Contrast correlations arise simply because there is a linear dependence of total contrast on Rabi frequency, and the asymmetry correlation is described in section~\\ref{ssec:asymmetry_effects}.\n    \\item Brown diamond: Correlations in $\\C^{\\N}$ and related contrast channels due to nonzero Rabi frequency correlations $\\Omega_{\\rm r}^{\\N}$ and $\\Omega_{\\rm r}^{\\N\\P}$. These arise due to laser power correlations with the $\\Nsw$ and $\\Psw$ switches and due to Stark mixing between rotational levels in $H$ and $C$, which create $\\Nsw$- and $\\Psw$-correlated transition amplitudes on the $H\\rightarrow C$ transition as described in section~\\ref{sssec:laser_pointing_and_intensity}.\n    \\item Red dot: Signals above our significance threshold for which we have been unable to find a plausible explanation. Even if these quantities arise from real physical effects, they would need to couple to other correlated quantities to contribute to $\\omega^{\\N\\E}$ and there is no evidence for this in the eEDM dataset. \n\\end{itemize}\n\n\n\\subsection{Systematic Error Budget}\n\\label{ssec:total_systematic_error_budget}\n\n\n\n\n\nThe method used for construction of a systematic uncertainty varies from experiment to experiment (see for example \\cite{sinervo2003,barlow2002}), and it is ultimately a subjective quantity. Even if individual contributions are derived from objective measurements, their inclusion or exclusion in the systematic uncertainty is subjective. Furthermore, the systematic uncertainty cannot possibly be a measure of the uncertainty in all systematic errors in the experiment, but rather only those which were identified and searched for. Although we work hard to identify all significant systematic errors in the measurement, we cannot rule out the possibility that some were missed.\n\nOur criteria for including a given quantity in the systematic uncertainty consist of three classes of systematic errors in order of decreasing importance of inclusion:\n\n\\begin{enumerate}[(A)]\n\\item If we measured a nonzero correlation between $\\omega^{\\mathcal{NE}}$ and some parameter which had an ideal value in the experiment, we performed auxiliary measurements to evaluate the corresponding systematic error and subtract that error from $\\omega^{\\mathcal{NE}}$ to obtain $\\wNEt$. The statistical uncertainty in the shift made to $\\omega^{\\mathcal{NE}}$ contributed to the systematic uncertainty.\n\\item If we observed a signal in a channel that we deemed important to understand, and it was not understood, but was not observed to be correlated with $\\omega^{\\mathcal{NE}}$, we set an upper limit on the shift in $\\omega^{\\mathcal{NE}}$ due to a possible correlation between the two channels. Since such a signal represented a gap in our understanding of the experiment, we added this upper limit as a contribution to the systematic uncertainty.\n\\item If a similar experiment saw a nonzero, not understood correlation between their measurement channel and some parameter with an ideal experimental value, but we did not observe an analogous correlation, we set an upper limit on the shift in $\\omega^{\\mathcal{NE}}$ due to this imperfection. Since this signal may have signified a gap in our understanding of our experiment, we added this upper limit as a contribution to the systematic uncertainty.\n\\end{enumerate}\n\n\n\\begin{table}[tbp]\n\\centering\n\\caption{Systematic error shifts and uncertainties for $\\omega^{\\mathcal{NE}}$, in units of mrad\/s grouped by inclusion class (defined in the text). Total uncertainties are calculated by summing the individual contributions in quadrature. Note that $\\omega^{\\mathcal{NE}}\\approx1.3$~mrad\/s corresponds roughly to $1\\times10^{-29}~\\ecm$ for our experiment.}\n\\begin{tabular}{llcc}\n\\br\nClass & Parameter & Shift (mrad\/s) & Uncertainty (mrad\/s)\\\\\n\\mr\nA & $\\E^{\\rm{nr}}$ correction & $-0.81$ & $0.66$\\\\\nA & $\\Omega_{\\rm r}^{\\N\\E}$ correction & $-0.03$ & $1.58$\\\\\nA & $\\omega^{\\E}$ correlated effects & $-0.01$ & $0.01$\\\\\nB & $\\omega^{\\N}$ correlation &  & $1.25$\\\\\nC & Non-reversing $\\B$-field $\\left(\\B_{z}^{\\rm{nr}}\\right)$ &  & $0.86$\\\\\nC & Transverse $\\B$-fields $\\left(\\B_{x}^{\\rm{nr}},\\B_{y}^{\\rm{nr}}\\right)$ &  & $0.85$\\\\\nC & $\\B$-field gradients &  & $1.24$\\\\\nC & Prep.\/readout laser detunings &  & $1.31$\\\\\nC & $\\Nsw$ correlated detuning &  & $0.90$\\\\\nC & $\\E$-field ground offset &  & $0.16$\\\\\n\\mr\n  & Total Systematic & $-0.85$ & $3.24$\\\\\n\\mr\n  & Statistical Uncertainty &  & $4.80$\\\\\n\\mr\n  & Total Uncertainty &  & $5.79$\\\\\n\\br\n\\end{tabular}\n\\label{tbl:syst_error}\n\\end{table}\n\nTable \\ref{tbl:syst_error} contains a list of the contributions to our systematic error, grouped by inclusion class, with the corresponding shifts and\/or uncertainties. Accounting for class A systematic errors was obligatory, and the removal of these errors from $\\omega^{\\mathcal{NE}}$ can be viewed as a redefinition of the measurement channel to $\\wNEt$ which does not contain those unwanted effects. These systematic errors consisted of those that depended on the parameters $\\E^{\\rm{nr}}$, $\\Omega_{\\rm r}^{\\N\\E}$, and $\\omega^{\\E}$ as described in sections \\ref{sssec:correlated_laser_parameters} and \\ref{ssec:E_correlated_phase}, and as such our reported measurement of the $T$-odd spin precession frequency is defined as $\\wNEt=\\omega^{\\mathcal{NE}}-\\omega^{\\mathcal{NE}}_{\\E^{\\rm{nr}}}-\\omega^{\\mathcal{NE}}_{\\Omega_{\\rm r}^{\\N\\E}}-\\omega^{\\mathcal{NE}}_{\\omega^\\E}$. The class B and class C systematic errors were included in the systematic uncertainty to lend credance to our result despite unexplained signals and unexplained systematic errors in experiments similar to ours.\nAll uncertainties in the contributions to the systematic error were added in quadrature to obtain the systematic uncertainty.\n\nWith reference to the class B criterion, we deemed the following channels as important to understand: $\\omega^{\\N},$ $\\omega^{\\E},$ $\\omega^{\\E\\B}$, and $\\omega^{\\N\\E\\B}$. Signals were initially not expected in any of these channels and could be measured with the same precision as $\\omega^{\\mathcal{NE}}$. The $\\omega^{\\rm{nr}}$, $\\omega^{\\B}$ and $\\omega^{\\N\\B}$ channels were not included in our systematic error since the Zeeman spin precession signals present in these channels had non-stationary means and additional noise due to drift in the molecule beam velocity. Only one of these channels, $\\omega^{\\N}$, described in section \\ref{ssec:N_correlated_pointing}, met the class B inclusion criterion. \n\nWith reference to the class C criterion, we defined the set of experiments similar to ours to include other eEDM experiments performed in molecules: the YbF experiment \\cite{Hudson2011} and the PbO experiment \\cite{Eckel2013}. The PbO experiment observed unexplained systematic errors coupling to stray magnetic fields and magnetic field gradients (cf.\\  section~\\ref{ssec:magnetic_field_imperfections}), and the YbF experiment observed unexplained systematic errors proportional to detunings (cf.\\ section~\\ref{sssec:laser_detuning}) and a field plate ground voltage offset (cf.\\ section~\\ref{ssec:electric_field_imperfections}). Thus we included the systematic uncertainty associated with the aforementioned effects in our budget.\n\nAfter having accounted for the systematic errors and systematic uncertainty, we reported $\\wNEt$, the contribution to the channel $\\omega^{\\mathcal{NE}}$ induced by $T$-odd interactions present in the $H$ state of ThO, as\n\\begin{align}\n\\label{eq:wNEt_num}\n\\wNEt=&2.6 \\pm 4.8_{\\rm{stat}}\\pm 3.2_{\\rm{syst}}~\\rm{mrad}\/\\rm{s}\\\\\n     =&2.6 \\pm 5.8~\\rm{mrad}\/\\rm{s}, \\label{eq:wNEt_num_err_comb}\n\\end{align}\nwhere the combined uncertainty is defined as the quadrature sum of the statistical and systematic uncertainties, $\\sigma^2=\\sigma_{\\rm{stat}}^2+\\sigma_{\\rm{syst}}^2$. This result is consistent with zero within $1\\sigma$. Since $\\sigma_{\\rm{syst}}$ is to some extent a subjective quantity, its inclusion should be borne in mind when interpreting confidence intervals based on $\\sigma$. Nevertheless, this inclusion decision does not have a large impact on the meaning of the resulting confidence intervals since $\\sigma$ is only about 20\\% larger than $\\sigma_{\\rm{stat}}$.\n\n\n\n\n\\subsubsection{Overview}\n\\hspace*{\\fill} \\\\\nIn this section we provide an overview of our experimental procedure and the important components of our apparatus. The reader should consult subsequent subsections for further details. A schematic of the experimental apparatus is shown in figure~\\ref{fig:apparatus_overview}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.46]{apparatus_overview.pdf}\n\\caption{A schematic of the overall ACME experimental apparatus. A beam of ThO molecules was produced by a cryogenic buffer-gas-cooled source. After exiting the source, the molecules were rotationally cooled via optical pumping and microwave mixing and then collimated before entering a magnetically shielded spin-precession region where nominally uniform magnetic and electric fields were applied. Using optical pumping, the molecules were transferred into the eEDM-sensitive $H$ state and then a spin superposition state was prepared. The spin precessed for a distance of ${\\approx}22$~cm and was then read out via laser-induced fluorescence. The fluorescence photons were collected by lenses and passed out of the chamber for detection by photomultiplier tubes. See main text for further details.\\label{fig:apparatus_overview}}\n\\end{figure}\n\nThO molecules were produced via pulsed laser ablation of a ThO$_2$ ceramic target. This took place in a cryogenic neon buffer gas cell, held at a temperature of ${\\approx}16$~K, at a repetition rate of 50~Hz. The resulting molecular beam was collimated and had a forward velocity $v_{\\parallel}\\approx200$~m\/s. In the state readout region the molecular pulses had a temporal (spatial) length of around 2~ms (40~cm). The buffer gas beam source is described in detail in section~\\ref{sec:beamsource}.\n\nAfter leaving the buffer gas source, the molecules had a velocity distribution and rotational level populations consistent with a Maxwell-Boltzmann distribution at a temperature of ${\\approx}4$~K. This was lower than the cell temperature due to expansion cooling, which enhanced the number of usable ThO molecules in the relevant rotational state. Further rotational cooling was provided via optical pumping and microwave mixing (see section~\\ref{sec:rotcool}). The molecules then passed through adjustable horizontal and vertical collimators consisting of a double layer of razor blades affixed to linear translation vacuum feedthroughs. Under normal running conditions, these collimators were withdrawn so that they did not affect the profile of the molecule beam in the spin-precession region; however, they were used to modify the spatial profile of the molecule beam during systematic checks to investigate the effect of molecule beam position and pointing. Just before the field plates, 126~cm from the beam source, the molecules passed through a 1~cm square collimating aperture, which determined the beam profile in the spin-precession region and prevented particles in the beam from being deposited on the field plates.\n\nAs described in section~\\ref{sec:Measurement_scheme}, a spin precession measurement was performed where the precession angle provided a measure of the interaction energy of an eEDM with the effective electric field, $\\mathcal{E}_{\\rm eff}$, in the molecule. A pair of transparent, ITO-coated glass plates provided an electric field that polarised and aligned the molecules. Laser beams passed through these plates to perform state preparation and readout. Around the vacuum chamber were coils that provided a uniform magnetic field in the $+\\hat{z}$ direction, and five layers of magnetic shielding which shielded against environmental magnetic fields. The electric and magnetic fields are discussed in detail in sections~\\ref{ssec:efields} and \\ref{sec:bfields}. The fluorescence induced by the state readout laser beam was collected by a set of eight lenses and transferred out of the spin-precession region using fiber bundles and light pipes (see section \\ref{sec:fluorescence_collection}), where it was detected by photo-multiplier tubes\\footnote{Hamamatsu R8900U-20.}. \n\n\\subsubsection{Buffer Gas Beam Source}\n\\hspace*{\\fill} \\\\\n\\label{sec:beamsource}\nThe basic operation of our beam source \\cite{Maxwell2005,Petricka2007,Sushkov2008,Patterson2009,Campbell2009Review,Tu2009,Patterson2010,Hutzler2011,Barry2011,Lu2011,Skoff2011PRA,Skoff2011Thesis,Hutzler2012,HutzlerThesis,Hummon2013,Bulleid2013} is depicted in figure~\\ref{fig:beam_source_schematic}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=13.5cm]{beam_source_schematic.pdf}\n\\caption{A schematic of the buffer gas beam source. Neon buffer gas flowed into a cell at a temperature of 16~K where it served to thermalise the hot ThO molecules produced by laser ablation. The ThO was entrained in the buffer gas flow. The mixture exited the cell and its expansion cooled the ThO to $\\approx4$~K. The resulting beam passed through collimating apertures in the 4~K and 50~K radiation shields and exited the beam source into the high vacuum region of the experiment. Solid circles represent buffer gas atoms. Open circles represent ThO molecules being cooled (red to blue transition).}\n\\label{fig:beam_source_schematic}\n\\end{figure}\nNeon buffer gas was flowed at a rate of $\\approx30$~SCCM (standard cubic centimetres per minute) through a copper cell held at a $T\\approx16$~K. The inside of the cell was cylindrical with a diameter of 13~mm and a length of 75~mm. Within the cell ThO was introduced at high temperature via laser ablation: overlapped beams of light with wavelengths 532~nm and 1064~nm emitted by a pulsed Nd:YAG laser\\footnote{Litron Nano TRL 80-200.} were focussed onto a 1.9~cm diameter ${\\rm ThO}_2$ target fabricated from pressed and sintered powder \\cite{Balakrishna1988,KiggansPrivate}. The laser pulses had a duration of a few ns, a pulse energy up to approximately 100~mJ and a repetition rate of $50$~Hz. The resulting hot plume of ejected particles, which contained ThO along with various other ablation byproducts, was cooled by collisions with the neon buffer gas, became entrained, and then exited the cell. The cell temperature was maintained by a combination of a pulse tube refrigerator\\footnote{Cryomech PT415.} and a resistive heater.\n\nThe cell was surrounded by a 4~K copper shield that protected the cell from black-body radiation and cryopumped most of the neon emerging from the cell. This shield was also partially covered with activated charcoal that acted as a cryopump for residual helium in the neon buffer gas. We observed a background pressure of $10^{-7}$~Torr without any mechanical pumping of the beam source when cold and with no buffer gas flow. The 4~K shield had a stainless steel conical collimator with a circular aperture of diameter 6~mm, located 25~mm from the cell aperture, by which distance the expanding beam was sufficiently diffuse that intra-beam collisions were negligible and most trajectories were ballistic. This collimator thus functioned as a differential pumping aperture without affecting the beam's cooling, acceleration or expansion \\cite{Hutzler2011}. The collimator had a thermal standoff relative to the 4~K shield to which it was mounted so that it could be kept at a temperature above the freezing point of neon by a resistive heater to prevent ice buildup on the collimator adversely affecting the beam dynamics. Another layer of shielding surrounded the 4~K copper shield, constructed from aluminium and held at a temperature of 60~K. Both the 4~K and 60~K radiation shields were thermally connected to the pulse tube by heat links made of flexible copper rope.\n\nThe aluminium vacuum chamber that housed the buffer gas beam source\\footnote{Precision Cryogenic Systems Inc.} had windows on each side, providing optical access for both the ablation laser and spectroscopy lasers, the latter allowing characterisation and monitoring of beam properties. The ThO beam's forward velocity distribution was roughly Gaussian with mean $v_{\\parallel}\\approx200$~m\/s and standard deviation $\\sigma_{v_{\\parallel}}\\approx13$~m\/s, corresponding to a temperature of ${\\approx}5$~K. The rotational temperature was $T_{\\rm rot}\\approx4$~K (rotational constant $B_X\\approx0.33$~cm$^{-1}$), meaning that ${\\approx}90$\\% of the population was contained in the levels $J=0$--$3$. Upon exiting the cell, the beam had a FWHM angular spread of $\\approx45^{\\circ}$. Several stages of collimation were applied before reaching the spin-precession region. The final collimator subtended a solid angle of $\\approx6\\times10^{-5}~{\\rm sr}$, meaning 1 in ${\\sim}20,000$ molecules exiting the cell reached the spin-precession region, where the precession measurement was performed (see figure~\\ref{fig:apparatus_overview}).\n\nThO yields from a given ablation spot decreased significantly after ${\\sim}10^4--10^5$ YAG pulses (${\\sim}10$~mins), at which time the laser spot was moved to an un-depleted region via a motorised mirror to re-optimise the beam flux. Each target was found to provide acceptable levels of molecule flux for around 300~hours of continuous running (${\\approx}5\\times10^7$ shots) before requiring replacement.\n\n\\subsubsection{Rotational Cooling}\n\\hspace*{\\fill} \\\\\n\\label{sec:rotcool}\nWe observed that ${\\approx}2$~cm downstream of (further from) the buffer gas beam source cell aperture, $J$-changing collisions were `frozen out' \\cite{Hutzler2011}, and the distribution of rotational state populations was fairly well described by a Boltzmann distribution with temperature $T_{\\rm rot}\\approx4$~K. \nAt this temperature the resulting fractions of molecules in the $J=0$--3 levels were estimated to be 0.1, 0.3, 0.3 and 0.2 respectively.\n\nAs described in section~\\ref{sec:state_prep_read}, we sought to transfer as much of the initial ground state population as possible into $\\ket{H,J=1}$ via optical pumping. To enhance the population which was transferred, we accumulated population in a single rotational level of the ground state before state preparation. The scheme used to achieve this, which we refer to as rotational cooling, is illustrated schematically in figure~\\ref{fig:rotcool} and discussed in detail in \\cite{SpaunThesis}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.55]{rotcool.pdf}\n\\caption{Schematic of the rotational cooling process. Numbers label $J$ and $M_y$ (projection of total angular momentum along $y$) sublevels are unlabelled but are $-1$, 0, $+1$ from left to right. Population was first optically pumped out of the $J=2$ and $J=3$ levels ($C$-state $\\Omega$-doublet structure and $M_y$ sublevels omitted for clarity) in a nominally field-free region. Next, population was equilibrated between $\\ket{J=0}$ and $\\ket{J=1,M_y=0}$ via microwave pumping. An electric field of ${\\approx}40$~V\/cm along $\\hat{y}$ was empirically observed to lead to an increased population in $\\ket{X,J=1}$. Grey dots represent population before these pumping processes. The schematic on the right represents the populations inside the spin-precession region (after pumping).\\label{fig:rotcool}}\n\\end{figure}\nThe first stage of the process was the optical pumping of molecules out of $\\ket{X,J=2}$ ($\\ket{X,J=3}$), via $\\ket{C,J^{\\prime}=1}$ ($\\ket{C,J^{\\prime}=2}$) into $\\ket{X,J=0}$ ($\\ket{X,J=1}$) using laser light at 690~nm. The natural linewidth of the $X\\rightarrow C$ transition is ${\\approx}2\\pi\\times0.3$~MHz, however the usable molecules had a ${\\approx}\\pm0.7$~m\/s transverse velocity spread, corresponding to a $1\\sigma$ Doppler width of ${\\approx}2\\pi\\times1.5$~MHz at 690~nm. Because the lasers used had linewidths of ${\\lesssim}1$~MHz, to completely optically pump these molecules we relied on a combination of power broadening and extended interaction time. Optical pumping occured in a magnetically unshielded region where a background field $\\B\\approx500$~mG was present; however, the magnetic moment of $X$ ($C$) is ${\\sim}\\mu_{\\rm N}$ (${\\approx}\\mu_{\\rm B}\/J(J+1)$), the nuclear magneton, which led to a Zeeman shift of ${\\sim}2\\pi\\times400$~Hz (${\\lesssim}2\\pi\\times400$~kHz) such that the $M$ sublevels were not resolved by our lasers. The $\\ket{C,J=1}$ state has an $\\Omega$-doublet splitting of $\\Delta_{\\Omega,C,J=1}\\approx2\\pi\\times51$~MHz \\cite{Edvinsson1965}. This splitting scales as $\\Delta_{\\Omega,C,J}\\propto J(J+1)$, meaning we could spectroscopically resolve the $\\Omega$-doublets for all $\\ket{C,J}$. In addition, having no $\\E$-field present meant that the $M$ sublevels of $C$ and $X$ remained unresolved and the energy eigenstates remained parity eigenstates. The $X$ state is also insensitive to $\\E$-fields due to the lack of $\\Omega$-doublet substructure; opposite parity states are separated by ${\\sim}10$~GHz and were hence unmixed. Laser beams with linear polarisation alternating between $\\hat{x}$ and $\\hat{y}$ were used to ensure that all population in $\\ket{X,J=2,3}$ was addressed. This was achieved by directing around 10 passes of the beam, offset in $x$, through the vacuum chamber, passing through a quarter-wave plate twice in each pass, over a distance of around 2~cm. \n\nThe laser light for rotational cooling was derived from home-built extended cavity diode lasers (ECDLs). The lasers were frequency-stabilised using a scanning transfer cavity with a computer-controlled servo \\cite{YuliaThesis}. Frequency-doubled light at 1064~nm from a frequency-stabilised Nd:YAG laser, locked to a molecular iodine line via modulation transfer spectroscopy \\cite{FarkasThesis}, provided the reference for the transfer cavity.\n\nAfter this first stage of rotational cooling, there was significantly greater population in the $\\ket{X,J=0}$ state than in any of the $\\ket{X,J=1,M}$ sublevels. We obtained a ${\\approx}25~\\%$ increase in the $J=1$ population by applying a continuous microwave field, resonant with the $J=0\\rightarrow J=1$ transition; a sufficiently high microwave power combined with the inherent velocity dispersion of the molecule beam led to an equilibration of population between the coupled levels \\cite{SpaunThesis}. In this second stage of rotational cooling it was empirically observed that applying an electric field to lift the $M_y$ sublevel degeneracy was necessary to obtain the increased population in $\\ket{X,J=1}$. A pair of copper electric field plates (spacing $\\approx4$~cm) provided a field of ${\\approx}40$~V\/cm in the $\\hat{y}$ (vertical) direction. We applied microwaves resonant with the Stark-shifted $\\ket{J=0}\\rightarrow\\ket{J=1,M_y=0}$ transition at a frequency of $2\\pi\\times19.904521$~GHz from an \\emph{ex vacuo} horn. Between the rotational cooling and spin-precession regions of the experiment (see figure~\\ref{fig:apparatus_overview}) there was not a well-defined quantisation axis, and we observe that the populations of the $\\ket{J=1,M}$ magnetic sublevels were equalised by the time the molecules reached the state preparation region. \n\nOverall, we find that rotational cooling provided a factor of between 1.5 and 2.0 increase in the molecule fluorescence signal $F$ in the state readout region. This gain factor was observed to vary slowly over time, possibly due to variations in the rotational temperature of the molecule beam, with significant changes sometimes observed when the ablation target was changed.\n\n\\subsubsection{State Preparation and Readout}\n\\hspace*{\\fill} \\\\\n\\label{sec:state_prep_read}\nFollowing rotational cooling, the molecular beam passed into the spin-precession region, where the molecules experienced a nominally uniform electric field, $\\vec{\\E}$, which was nominally collinear with a magnetic field, $\\vec{\\B}$. Note that since neither of the states $X^1\\Sigma^+$ nor $A^3\\Pi_{0+}$ have $\\Omega$-doublet structure, parity remained a good quantum number for these levels for the small (${\\sim}100$~V\/cm) electric fields we applied. \n\nWe transferred the molecules into the $H$ electronic state via optical pumping, as illustrated in figure~\\ref{fig:op_sublevels}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=10cm]{op_sublevels.pdf}\n\\caption{Schematic of the optical pumping scheme used to populate the $H$ state. Spontaneous decay to the $H$ state (green arrows) led to an incoherent mixture of all indicated levels. See main text for detailed explanation.}\n\\label{fig:op_sublevels}\n\\end{figure}\nA 943~nm laser beam nominally propagating along $\\hat{z}$ excited molecules from the $\\ket{X,J=1}$ to $\\ket{A,J=0}$. The laser beam passed through a quarter-wave plate, was retroflected and offset in $x$, then passed again through the quarter-wave plate, such that the molecules were pumped by two spatially separated laser beams of orthogonal polarisations, allowing all population in both the $\\ket{X,J=1,M=\\pm1}$ levels to be excited. After excitation to $A$, the molecules could spontaneously decay into the $\\ket{H,J=1}$ manifold of states. We observed a transfer efficiency from $X$ to $H$ of ${\\approx}0.3$ \\cite{SpaunThesis}. In this decay, five out of the six sublevels were populated; 1\/6 of the population decayed to each of $\\ket{H,M=\\pm1,\\Nsw=\\pm 1}$ and 1\/3 to $\\ket{H,\\Psw=-1,M=0}$ (see sections~\\ref{sec:tho_molecule} and \\ref{sec:Measurement_scheme} for definitions of $\\Nsw$ and $\\Psw$); decay to $\\ket{H,\\Psw=+1,M=0}$ is forbidden. \nOf these five populated states, only one corresponded to the desired initial state described by equation~\\ref{eq:dark_state}, and only 1\/6 of the population in the $H$ state was in this desired state.\nWe estimated a total transfer efficiency from $\\ket{X,J=1,M=\\pm 1}$ to the state in equation~\\ref{eq:dark_state} of $30\\%\\times1\/6=5\\%$.\n\nThe 943~nm laser light was derived from a commercial ECDL and then amplified by a commercial tapered amplifier\\footnote{Toptica DL Pro and BoosTA.}, generating $\\approx400$~mW. As with the rotational cooling lasers, we verified that the power was sufficient to drive optical pumping to completion across the entire transverse velocity distribution of the molecular beam. This laser was also stabilised via the previously described (section~\\ref{sec:rotcool}) transfer cavity. The frequency of the laser light was monitored every 30--60 mins by scanning across the molecular resonances, allowing for independent fine-tuning and compensation of long-term frequency changes (${\\lesssim}2\\pi\\times100$~kHz per half hour) due to e.g. temperature drifts in the cavity.\n\nAround 1~cm downstream of the optical pumping laser beam that transferred population to $H$, we prepared the initial state of $H$ (equation~\\ref{eq:dark_state}) by driving the transition between $\\ket{H,M=\\pm1,\\Nsw}$ and $\\ket{C,\\Psw=+1}$ (see section~\\ref{sec:Measurement_scheme} for more details) using laser light at 1090~nm. A distance $L=22$~cm downstream of the preparation laser, a second 1090~nm laser beam was used to read out the molecule state via the same transition (but with the option to excite to either $\\Psw$ state). This laser light was also derived from a commercial ECDL. It was then amplified using a fiber amplifier\\footnote{Keopsys KPS-BT2-YFA-1083-SLM-PM-05-FA.}, increasing the power to ${\\approx}250$~mW. AOMs were then used to split and frequency shift the light to address both $\\Nsw$ states in the $H$ state, allowing spectroscopic selection of molecular alignment, and of both $\\Psw$ levels in the $C$ state. Switching between these frequencies was achieved with either RF switches\\footnote{Mini-Circuits ZYSWA-2-50DR.} or a DDS synthesizer\\footnote{Novatech 409B.}. Given the linear Stark shifts $D_1\\E\\approx2\\pi\\times146$~MHz ($2\\pi\\times37$~MHz) in $H$ with an applied electric field strength $|\\E|=141$~V\/cm (36~V\/cm), and the excited state $\\Omega$-doublet splitting $\\Delta_{\\Omega,C,J=1}\\approx50$~MHz in $C$, these transitions were spectroscopically well-resolved. We fixed the nominal frequency of the state preparation laser to only address $\\Psw=+1$, but periodically switched the state readout laser frequency to address $\\Psw=\\pm1$ (${\\sim}1$~min period). The transition frequencies of the state preparation and state readout laser beams were changed synchronously to always address the same $\\Nsw$ level, with a switch between $\\Nsw$ levels every 0.5~s. The state preparation and readout laser beams were then independently amplified with a pair of fiber amplifiers\\footnote{Nufern PSFA-1084-01-10W-1-3.}, providing ${\\sim}3$--4~W of power. Immediately before interrogating the molecules, the polarisation of the state readout laser beam was rapidly (100~kHz) switched between two orthogonal linear polarisations. The scheme for producing the $\\Nsw$ and $\\Psw$ switches, and this fast polarisation switch, together with the corresponding laser transitions, is shown in figure~\\ref{fig:HC_transitions_setup}. We now describe in detail how the appropriate frequency laser light was produced.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{HC_transitions_setup.pdf}\n\\caption{Top: transitions addressed during state preparation and readout (not to scale). The grey arrow represents the ECDL output frequency, $\\omega_0$, not resonant with any transition and referenced from halfway between the two $H$ state $\\Omega$-doublets. Bottom: simplified schematic of how we produced light at the appropriate frequencies. AOM-induced frequency shifts are denoted in the corresponding boxes. Bifurcation of grey lines represents light being split equally. Multiple lines represent different frequencies; only one frequency is used at once. Dashed grey lines represent a continuation of the optical path. AOMs to perform switching between $\\Nsw$ states; switching between $\\Psw$ states and adding relative detuning $\\Delta$; tuning Rabi frequency $\\Omega_{\\rm r}$; and performing polarisation switching are shown. The setup shown is used with $\\E=142$~V\/cm and changes slightly if a different value of $\\E$ is used. For a full description, consult the main text.}\n\\label{fig:HC_transitions_setup}\n\\end{figure}\n\\clearpage\n\nLight from the ECDL was amplified and split equally, passing to two AOMs which produced shifts $\\pm\\omega_{\\rm L}^{\\mathcal{N}}$ where $\\omega_{\\rm L}^{\\mathcal{N}}$ is half the splitting between the two $\\Nsw$ states; these AOMs were switched on and off to perform the $\\Nsw$ switch. The two frequency-shifted beams were combined and overlapped. For state preparation (lower branch of diagram), another AOM shifted the light by $+\\wLs{1}$, into resonance with the lower $\\Omega$-doublet in $C$ ($\\Psw=+1$). This light was then amplifed again and passed through an AOM to vary the power (used as a systematic check). For the state readout (upper branch of diagram), a single AOM switched frequency to produce shifts $+\\wLs{2,3}$ for the two $\\Psw$ states. A relative detuning between state preparation and readout laser beams (not shown) was also implemented with this AOM. (Shifts common to both beams were made by changing $\\omega_0$.) The light was then amplified again and passed through an AOM to vary the power. Finally, polarisation switching was achieved with two AOMs switched on and off at 100~kHz, $\\pi$ out of phase with each other; light not diffracted (and frequency shifted by $-\\wLs{\\rm PS}$) by the first AOM was diffracted (and also frequency shifted by $-\\wLs{\\rm PS}$) by the second AOM. The diffracted light from each path was combined on a polarising beam splitter such that the linear polarisation of the final output beam alternated. \n\nBased on the notation above we can now write the components of the frequencies of the state preparation and readout laser beams which do not reverse with any experimental switch as $\\omega_{\\rm L,prep}^{\\rm{nr}}=\\omega_{\\rm L,0}+\\omega_{\\rm L,1}$ and $\\omega_{\\rm L,read}^{\\rm{nr}}=\\omega_{\\rm L,0}+(\\omega_{\\rm L,2}+\\omega_{\\rm L,3})\/2-\\omega_{\\rm L,PS}$, respectively. We can also write the $\\Psw$-correlated frequency component of the state readout laser as $\\omega_{\\rm L,read}^{\\P}=(\\omega_{\\rm L,2}-\\omega_{\\rm L,3})\/2$. We then write the detuning components as $\\Delta_i=\\omega_{{\\rm L},i}-\\omega_{HC}$ where $i\\in\\left\\{\\mathrm{ prep},X,Y\\right\\}$ indexes the laser and $\\omega_{HC}$ is the transition frequency between the line centres of the $\\ket{H,J=1}$ and $\\ket{C,J=1}$ manifolds\\footnote{Note that this can in principle vary between different laser beams (denoted with the subscript $i$) if there is a relative pointing between them, which produces a relative Doppler shift, but we ignore this effect in our current treatment.}. We can rewrite this overall detuning in terms of various switch parity components:\n\\begin{align}\n\\Delta_{i}=&\\omega_{{\\rm L},i}-\\omega_{HC,i}\\\\\n=&\\left(\\omega_{{\\rm L},i}^{\\rm{nr}}+\\Nsw\\wL^{\\N}+\\Psw\\wL^{\\P}\\delta_{i,\\left\\{X,Y\\right\\} }\\right)-\\left(\\omega_{HC}^{\\rm{nr}}+\\Nsw D_1\\left|\\E(x_{i})\\tilde{\\E}+\\E^{\\rm{nr}}(x_{i})\\right|-\\frac{1}{2}\\Delta_{\\Omega,C,J=1}\\Psw\\delta_{i,\\left\\{ X,Y\\right\\} }\\right)\\\\\n=&\\Delta_{i}^{\\rm{nr}}+\\Nsw\\Delta_{i}^{\\N}+\\Nsw\\Esw\\Delta_{i}^{\\N\\E}+\\Psw\\Delta_{i}^{\\P}\\delta_{i,\\left\\{ X,Y\\right\\}}.\n\\label{eq:detuningcorrelations}\n\\end{align}\nIn the above equations we have defined detuning components of given switch parities --- we shall now explain each component in turn. $\\Delta_{i}^{\\N}=(\\wL^{\\N}-D_1\\E(x_{i}))$ is the mismatch between the Stark shift $D_1\\E(x_{i})$ and the AOM frequency $\\wL^{\\N}$ used to switch between resonantly addressing the two $\\Nsw$ states, where $x_i$ is the $x$ position of laser beam $i$. $\\Delta_{i}^{\\N\\E}=D_1\\E^{\\rm{nr}}(x_{i})$ is a detuning component correlated like an eEDM signal which is due to a non-reversing component of the applied electric field. To understand this relation, consider figure~\\ref{fig:Enr_wNE}. Recall that $\\Delta_{\\Omega,C,J=1}$ is the $\\Omega$-doublet splitting of the $C$ state.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=8cm]{Enr_wNE.pdf}\n\\caption{Illustration of $\\Delta^{\\N\\E}$ arising from a non-reversing electric field $\\E^{\\rm{nr}}$. Dashed lines show energy levels in the presence of $\\E^{\\rm{nr}}$. Colours indicate if the the laser shown in dark red is blue- or red-detuned from the transition.}\n\\label{fig:Enr_wNE}\n\\end{figure}\nFor a $\\E^{\\rm{nr}}\\ne0$, $|\\E|$, and hence the splitting between the $\\Nsw$ levels in $H$, depends on $\\Esw$. If the laser frequency for each $\\Nsw$ is set assuming $\\E^{\\rm{nr}}=0$, a nonzero $\\E^{\\rm{nr}}$ leads to blue or red detuning from resonance, correlated with $\\Esw$. Because the sign of the Stark shift is correlated with $\\Nsw$, the resulting detuning is also correlated with $\\Nsw$.\n\n$\\Delta_{i}^{\\P}=\\wL^{\\P}+\\Delta_{\\Omega,C,J=1}\/2$ is the mismatch between the excited state parity splitting and the AOM frequency, $\\wL^{\\P}=(\\omega_{\\rm L,3}-\\omega_{\\rm L,2})\/2$, used to switch between the two states ($\\delta_{i,\\left\\{ X,Y\\right\\}}$ is the Kronecker delta, 1 if $i=X$ or $i=Y$, zero else). We observed that $\\Delta^{\\N}$ ($\\Delta^{\\P}$) was typically less than $2\\pi\\times20$~kHz ($2\\pi\\times50$~kHz). Although we could measure $\\Delta^{\\N}$ with ${\\sim}2\\pi\\times1$~kHz precision, fluctuations in the Stark splitting, likely caused by thermally-induced fluctuations of the field plate spacing, limited our ability to zero out this correlated detuning.\n\nWe define $\\Delta^{\\rm{nr}}=(\\Delta^{\\rm{nr}}_{{\\rm prep}}+(1\/2)(\\Delta^{\\rm{nr}}_{X}+\\Delta^{\\rm{nr}}_{Y}))\/2$ as the average non-reversing detuning of the state preparation and readout laser beams; its value typically fluctuated by ${\\sim}2\\pi\\times0.1$~MHz over several hours. Every 30--60 minutes the value of $\\Delta^{\\rm{nr}}$ was scanned across the molecular resonance in the readout region using the $\\Delta$-tuning AOM (see figure~\\ref{fig:HC_transitions_setup}), as an auxiliary optimisation. $\\Delta^{\\rm{nr}}$ was set to the value where the fluorescence signal was maximum. This ensured that the average detuning of the state readout laser beams, $(\\Delta_X^{\\rm nr}+\\Delta_Y^{\\rm nr})\/2$, was zero, however, if the state preparation and readout laser beams were not exactly parallel, there could be a difference between $\\Delta^{\\rm{nr}}_i$ due to the resulting difference in Doppler shifts. The effect of a detuning difference between the two state readout polarisations $\\Delta^{XY}=(\\Delta^{\\rm{nr}}_{X}-\\Delta^{\\rm{nr}}_{Y})\/2$ is discussed in section~\\ref{ssec:asymmetry_effects}. Additionally, each day we scanned the frequency of the preparation laser across the molecule resonance while monitoring the contrast of our fluorescence signal to ensure $\\Delta^{\\rm{nr}}_{{\\rm prep}}$ was kept below $2\\pi\\times0.2$~MHz (an example scan is shown in figure~\\ref{fig:contrast}). The ways in which detuning components can contribute to systematic errors are discussed in detail in sections~\\ref{sssec:AC_stark_shift_phases} and \\ref{sssec:correlated_laser_parameters}.\n\nOther polarisation switches of the state preparation and readout laser beams ($\\Rsw$ and $\\Gsw$) were controlled independently via half-wave plates mounted in high resolution rotation stages\\footnote{Newport URS50BCC.}. These switches and their use in the experiment are described in detail in section~\\ref{sec:data_analysis}. Both beams were shaped using cylindrical lenses to be extended in $y$ so all molecules in the beam were addressed. The Gaussian standard deviations of the beam intensities were 1.1~mm and 7.5~mm in the $x$ and $y$ directions, respectively \\cite{SpaunThesis}. The preparation laser beam was temporally modulated at $50$~Hz with a chopper wheel, synchronous with the molecule beam pulses, to minimise the incident power on the field plates so as to reduce an important systematic error, described in sections~\\ref{sssec:AC_stark_shift_phases} and \\ref{sssec:polarization_gradients_from_thermal_stress_induced_birefringence}.\n\n\\subsubsection{Electric Field}\n\\hspace*{\\fill} \\\\\n\\label{ssec:efields}\nThe applied $\\E$-field was generated with a pair of 43~cm~$\\times$~23~cm parallel conducting plates composed of ${\\approx}1.25$~cm thick Borofloat glass, coated with a ${\\sim}200$~nm layer of indium tin oxide on the inner faces\\footnote{The plates were fabricated by Custom Scientific, Inc.}. The plates were transparent to the $X\\rightarrow A$ optical pumping laser (943~nm), the $H\\rightarrow C$ state preparation and readout lasers (1090~nm), and the $C\\rightarrow X$ molecule fluorescence (690~nm). The outside faces of the electric field plates were prepared with a broadband anti-reflection coating with a specified \\textless1\\% reflectivity at normal incidence from 600--1000~nm. The plates were made much larger than the precession region in order to minimise inhomogeneity of the field through which the molecules passed, and to enable large solid angle collection of fluorescence through the plates. One of the field plates was mounted in an aluminium frame fixed to the base of the vacuum chamber. The other field plate was secured a distance of 2.5~cm away in a kinematic aluminium frame. On the inward-facing surfaces, a frame of gold-plated copper clamped each field plate to the aluminium mounts and also functioned as a `guard ring' electrode, suppressing the effect of fringing fields near the edges of the plate. The field plates were protected from impinging molecular beam particles by a $1~\\mathrm{cm}\\times 1~\\mathrm{cm}$ square collimator fixed to the entrance of the assembly.\n\nThe applied electric field was controlled by a 20-bit DAC, amplified to produce up to $\\pm200$~V\\footnote{PA98A Power OpAmp.}. The field plate assembly was referenced to the vacuum chamber ground. Equal and opposite voltages, $\\pm V$, were applied to each side of the assembly. The direction of the field (the $\\Esw$ switch) was reversed every 1--2~s by reprogramming the output of the DAC channels to reverse their polarity. The configuration of the electrical connections between the amplified voltage and the field plates, denoted by $\\Lsw$, was reversed via a pair of mercury-wetted relays every 2.6~minutes\\footnote{Note that $\\Lsw$ constitutes a reversal of the supply voltages as well as a reversal of the leads connecting the power supply to the field plates, such that $\\Esw$ is unchanged.}. Data were also taken with two different values of $\\mathcal{E}=36$ and 141~V\/cm, varied on a ${\\sim}1$~day time scale.\n\nWe measured the homogoneity of the electric field in a number of ways which we shall describe in turn now. Firstly, an indirect measure was obtained by determining the spatial variation of the field plate separation $d$ using a `white light' Michelson interferometer \\cite{Patten1971}. A schematic of the setup is shown in figure~\\ref{fig:Interferometer_setup}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.4]{interferometer_setup_inset.pdf}\n\\caption{\\label{fig:Interferometer_setup}Schematic of the apparatus used to perform an interferometric measurement of the electric field plate separation. A spectrally broad light beam is reflected perpendicularly off the field plates and passes into a conventional Michelson interferometer setup with one fixed arm (length $L_2$) and one movable arm (length $L_1$). An example of a pair of beam paths of interest is shown as solid and dashed red lines. If the two paths are slightly tilted relative to each other, a spatial interference pattern (inset) is observed on the CCD detector when the path length difference between the two beams is less than the coherence length, e.g. $L_1+d-L_2<L_{\\rm c}$.}\n\\end{figure}\nWe directed a light beam at normal incidence through the electric field plates. This resulted in multiple reflected beams, but we restrict discussion to the reflections from the conducting surfaces as these are of primary interest and were efficiently experimentally isolated from all others. The reflected beams passed into a Michelson interferometer with one arm of fixed length ($L_2$) and one with length adjustable via a micrometer ($L_1$). Constructive (destructive) interference occured whenever the lengths of two reflected beam paths differed by an integer (odd half-integer) multiple of the wavelength of the light. This condition was restricted further by the use of a broadband superluminescent diode\\footnote{QPhotonics QSDM-680-2.} with a short coherence length $L_{\\rm c}$ (nominally $L_{\\rm c}\\approx15~\\upmu$m). Thus the interference was only substantial when the two beams differed in length by ${\\lesssim}L_{\\rm c}$. This occurred when $L_1=L_2$ (for reflections off the same surface) or when $L_1=L_2\\pm d$ (for reflections off surfaces spaced by $d$). The case where both beams reflected off the same surface was used as a reference to determine the position $L_1=L_2$. A measure of this interference was achieved by producing a spatial interference pattern (inset figure~\\ref{fig:Interferometer_setup}) through a slight tilting of the arms of the interferometer. Analysis of the spatial Fourier components of the resulting interference pattern provided a quantitative measure of the interference fringe contrast; a plot of contrast vs.\\ arm position $L_1$ yielded a peak with width $\\delta L_1\\approx L_{\\rm c}$. By performing this analysis while varying the path length $L_1$, the plate separation was deduced. This entire procedure was then performed over a range of transverse ($x,y$) positions on the field plates. The resulting data are shown in figure~\\ref{fig:Interferometer_data}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=14cm]{Interferometer_data.pdf}\n\\caption{Variation in the electric field plate separation as measured by the interferometric method. The left-hand plot shows the variation with $x$, the molecule beam direction, at two different values of $y$. The right-hand plot shows the variation with the $y$ (vertical) position at three different values of $x$. The coordinate origin is at the nominal centre of the plates. The shaded regions indicate the approximate extent of the molecular beam in the spin precession region. The change in separation is quoted relative to a common offset with an estimated error of $\\pm0.5~\\upmu$m. The mean separation over all $x$ is 25.00~mm.\\label{fig:Interferometer_data}}\n\\end{figure}\n\nThis measurement clearly showed a bowing of the electric field plates; the plate separation varied approximately quadratically with the position in $x$. This is shown in the left-hand plot of figure~\\ref{fig:Interferometer_data}. In the $\\hat{x}$ direction we observed a maximum variation in the plate separation of around 20~$\\upmu$m. We saw a roughly 80~$\\upmu$m variation in the $\\hat{y}$ (vertical) direction but note that the collimated molecular beam extended only over $\\pm5$~mm in $y$ so the biggest plate spacing variation at a given $x$ was ${\\approx}10~\\upmu$m. From these measurements and a typical applied voltage of $V=\\pm177$~V, we expected $\\E$ to vary by around 100~mV\/cm in the $\\hat{x}$ direction and ${\\lesssim}15$~mV\/cm in the $y$ direction in the region sampled by the molecules.\n\nThe indirect measurements of the spatial variation of the applied electric field provided by interferometric mapping of the field plate separation were later corroborated by direct measurements of $\\vec{\\E}(x)$. Spatial variation of $\\vec{\\E}$ could lead to the accumulation of geometric phases during the spin precession measurement \\cite{Vutha2009}. There are known mechanisms by which such phases can contribute to eEDM-like systematic errors, as described in section~\\ref{ssec:E_correlated_phase}, though simple estimates show that these effects are several orders of magnitude below the sensitivity of this measurement. However, additional $\\E$-field imperfections such as non-reversing fields, due to e.g.\\ variations in the ITO coating, which could produce patch potentials, are known to contribute to eEDM-like systematic errors and are only revealed by more direct measurements of the electric field, which we will now describe.\n\nWe can write the electric field present in the precession region in the following manner:\n\\begin{equation}\n\\vec{\\E}\\cdot\\hat{z}=\\E\\Esw+\\E^{\\rm{nr}}+\\E^{\\Ld}{\\Lsw}+\\E^{\\E\\Ld}\\tilde{\\E}{\\Lsw},\n\\end{equation}\nwhere, as usual, $\\Esw={\\rm sgn}(\\hat{z}\\cdot\\vec{\\E})$ is the direction of the field in the spin-precession region and $\\Lsw$ represents the binary state of the physical leads connecting the voltage supply to the field plates. The terms on the right-hand side are: $\\E\\Esw$, the intentionally applied electric field; $\\E^{\\rm{nr}}$, a non-reversing electric field; $\\E^{\\Ld}$, a non-reversing electric field component from the power supply that can be reversed by switching $\\Lsw$; and $\\E^{\\E\\Ld}\\tilde{\\E}{\\Lsw}$, a component of the applied field that is reversed by switching $\\Esw$ or $\\Lsw$.\n\nWe directly measured the components of $\\E$ using the molecules themselves, in three different ways. The first method used Raman spectroscopy, driving a two-photon Lambda-type transition between $\\Nsw$ levels in $\\ket{H,J=1}$ as shown in figure~\\ref{fig:Raman_transition}. The Raman transfer was performed at positions between, but close to, the state preparation and readout laser beams, where there was sufficient optical access. The procedure was as follows: first, an $\\hat{x}$-polarised state preparation laser beam depleted a superposition $\\ket{B(\\hat{x},\\Nsw=+1,\\Psw=+1)}$ (recall $\\ket{B}$ is the bright state as defined in section~\\ref{sec:Measurement_scheme}) by exciting it to the $C$ state. Next, at a point downstream, two co-propagating, $\\hat{x}$-polarised Raman beams were used to repopulate this depleted superposition by driving population from the other $\\Nsw$ state, via the transition $\\ket{B(\\hat{x},\\Nsw=-1,\\Psw=+1)}\\rightarrow \\ket{C,\\Psw=+1}\\rightarrow\\ket{B(\\hat{x},\\Nsw=+1,\\Psw=+1)}$. The frequencies of the two Raman beams were tuned with a pair of AOMs. The state readout laser then addressed the same transition as the preparation laser and excited the repopulated superposition to the $C$-state from which it spontaneously decayed back to $X$ and fluoresced at 690~nm.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=8cm]{Raman_transition.pdf}\n\\includegraphics[width=8cm]{raman_fits.pdf}\n\\caption{Left: Schematic of the Raman-type transition used to perform a measurement of the $\\E$-field in the spin-precession region. The pairs of red arrows represent the one-photon transitions driven by linearly polarised light, addressing superpositions of $M=\\pm1$. The single-photon detuning is given by $\\Delta+\\delta\/2$ and the two-photon detuning is given by $\\delta\/2$. $D_1|\\mathcal{E}|$ is the magnitude of the Stark shift due to the applied electric field. Right: Example scans for opposite $\\Esw$ states obtained by varying the two-photon detuning $\\delta\/2$ and observing fluorescence, with Gaussian fits to the data.\\label{fig:Raman_transition}}\n\\end{figure}\n\nEfficient transfer of population between the two $\\Nsw$ states occurred for zero two-photon detuning ($\\delta\/2$ in figure~\\ref{fig:Raman_transition}). This condition was indicated by a peak in fluorescence, giving a measure of the Stark shifted energy, and hence the absolute size of the applied field, $|\\E|$. This procedure was repeated for different positions of the Raman laser beams along the $\\hat{x}$ direction. The non-reversing component of the electric field was found by repeating the measurement after reversing the applied voltages. An example of such a pair of scans is shown on the right of figure~\\ref{fig:Raman_transition}.\n\nUsing this method we measured the electric field at $x$ positions where there was sufficient optical access, i.e.\\ near the state preparation and readout laser beams. The $\\Esw$-correlated two-photon detuning $\\delta^{\\E}=2\\pi\\times13$~kHz ($2\\pi\\times11$~kHz) allowed us to extract a value of the non-reversing electric field component, $\\E^{\\rm{nr}}=\\delta^{\\E}\/2D_1=-6.5\\pm0.3$~mV\/cm ($-5.5\\pm0.3$~mV\/cm), in the state preparation (readout) region. We did not observe any significant variation within the individual regions. We also observed that this non-reversing component did not vary with the size of the reversing electric field.\n\nThe second method used to measure the electric field had the greatest utility because it allowed for spatially resolved measurements along $x$ in the spin precession region with comparable precision to the Raman method without perturbing the experimental apparatus. This was achieved via microwave spectroscopy. A schematic of the experimental setup is shown in figure~\\ref{fig:microwave_setup}. \n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.65]{microwave_transitions.pdf}\n\\caption{The transition driven by microwaves during a measurement of the electric field. We used $\\hat{y}$-polarised microwaves of frequency $2\\pi\\times39$~GHz to drive a rotational transition between $\\ket{H,J=1}$ and $\\ket{H,J=2}$. The $M=0$ levels are labelled with their parity. We applied a moderate $\\E$-field such that $\\Delta_{\\Omega}\\ll D|\\E|\\ll B_H$ where $B_H=0.33~{\\rm cm}^{-1}$ is the rotational constant. The electric dipole moment of the $J=1$ state $D_1\\approx2\\pi\\times1~{\\rm MHz\/(V\/cm)}\\approx 3D_2$.\\label{fig:microwave_transitions}}\n\\end{figure}\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{microwave_setup_new.pdf}\n\\caption{Experimental setup for spatial measurement of $\\E$ via microwave spectroscopy. A molecular pulse (grey cloud) passed between the electric field plates (light blue). The optical pumping laser beam transferred population from $\\ket{X,J=1}$ to an incoherent mixture of states in $\\ket{H,J=1}$ as described in section~\\ref{sec:state_prep_read}. When the pulse was centred in the spin-precession region, a microwave $\\pi$-pulse was applied, driving population in $H$ from $J=1$ to $J=2$ when resonant (dark blue region). The depletion efficiency out of $J=1$ was subsequently read out by laser induced fluorescence as per the normal measurement scheme described in section~\\ref{sec:Measurement_scheme}. The time of arrival of the molecules in the state readout region encoded the position where they absorbed the microwaves.\n\\label{fig:microwave_setup}}\n\\end{figure}\n\nThe measurement procedure began with optical pumping of molecules into the $H$-state. The molecules travelled through the spin-precession region until it was entirely occupied by the molecule pulse. At this time, a $\\pi$-pulse of microwaves at $2\\pi\\times39$~GHz with nominal $\\hat{y}$ polarisation was applied counter-propagating to the molecule beam. When on resonance, this transferred population from $\\ket{B(\\hat{y},\\Nsw,\\Psw)}$ to $\\ket{H,J=2,M=0,\\Psw}$ (excitation to (from) either $\\Psw$ ($\\Nsw$) state was permitted) as shown in figure~\\ref{fig:microwave_transitions}. State readout was performed as usual (see section~\\ref{sec:Measurement_scheme}) by optically pumping with alternating polarisations $\\hat{x}$ and $\\hat{y}$. The measured asymmetry (as defined in equation~\\ref{eq:asymmetry}) served as a measure of the microwave transfer efficiency. The $x$ position of the molecules at the time of the microwave pulse was mapped onto their arrival time in the detection region and, with knowledge of the longitudinal molecular beam velocity, $v_{\\parallel}$, could be extracted. Thus, the spatial dependence of the resonant frequency, $\\omega_{\\rm MW}(x)$, was provided by the time-dependence of the asymmetry, $\\mathcal{A}(t)$. Due to the DC Stark shift, $\\omega_{\\rm MW}$ was linearly proportional to the electric field magnitude and $|\\E(x)|$ could be directly extracted.\n\n\nWe observed a resonance linewidth of ${\\approx}2\\pi\\times25~{\\rm kHz}\\approx2\\pi\/T$ which was limited by the microwave $\\pi$-pulse duration of $T=40~\\upmu\\rm{s}$. With our signal-to-noise, we were able to fit the resonance centre to a precision of ${\\sim}2\\pi\\times1$~kHz, typically using ${\\sim}50$ detuning values and averaging over ${\\sim}50$ molecule pulses per detuning value. Example data obtained via this method are shown in figure~\\ref{fig:asym_map}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=15cm]{asym_map_comp.pdf}\n\\caption{Colourmap: Plot of the asymmetry $\\A$ induced by a microwave pulse as the frequency of the microwaves was scanned. Red data points: Plot of the corresponding reversing component of the electric field obtained by extracting the centre of the resonance signal. The position is relative to the centre of the spin-precession region.\\label{fig:asym_map}}\n\\end{figure}\n\nIn these data, it is evident that the resonant frequency of the microwaves varied across the molecule pulse by around $2\\pi\\times60$~kHz. The position $x$ of the molecules at the time of the microwave pulse was assumed to be linearly related to the molecule arrival time in the state readout region. The observed spatial variation of $\\mathcal{E}$ was roughly consistent with expectations based on the measured variation of the plate spacing described above.\n\nBy switching $\\Nsw$ and $\\Esw$ between measurements of the $\\E$-field we were able to extract $\\E^{\\rm{nr}}$ from the $\\Nsw\\Esw$-correlated component of $\\omega_{\\rm MW}$. These measurements, shown in figure~\\ref{fig:enr}, were used to evaluate the corresponding systematic error in equation~\\ref{eq:Enr_systematic_error_value}. \n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.35]{enr_combined.pdf}\n\\caption{A plot of the spatial variation of $\\E^{\\rm{nr}}$. The black points are data obtained via microwave spectroscopy. The blue points are data obtained via Raman spectroscopy. The red data point was obtained by examining the variation of contrast with $\\Delta_{\\rm prep}$. The approximate position of the state preparation (state readout) laser beam is shown as a red dotted line on the left (right) of the figure. For the microwave spectroscopy data the uncertainty\/averaging range of the position is around 21~mm at the left-hand side of the plot and decreases to around 13~mm at the right-hand side --- see main text for details.\\label{fig:enr}}\n\\end{figure}\nWe clearly saw a non-uniform $\\E^{\\rm{nr}}$ across the spin precession region. The spatial variation shown in figure~\\ref{fig:enr} was reproducible for the period of several weeks over which these measurements of the electric field were taken. We are unsure as to the origin of the $\\E^{\\rm{nr}}$ but believe it may have been caused by patch potentials \\cite{Robertson2006} present on the electric field plates. We observed unexplained disagreement between the two measurement methods (Raman spectroscopy vs.\\ microwave spectroscopy), but note that both report non-reversing fields of a few mV\/cm with the same sign.\n\nThe mapping between arrival time in the detection region and $x$ position during the microwave pulse was approximate, suffering from spatial averaging due to a variety of effects. For example, velocity dispersion led to averaging of $dx\\times\\sigma_{v_{\\parallel}}\/v_{\\parallel}$, where $\\sigma_{v_{\\parallel}}$ is the longitudinal velocity spread of the molecular beam and $dx$ is the distance between microwave interrogation and state readout. This averaging distance was largest, ${\\approx}1.6~{\\rm cm}$, at the state preparation region. Spatial averaging also occurred across the ${\\approx}0.7$~cm distance traversed during the $T=40~\\upmu$s microwave pulse. Finally, there was averaging of the spatial position of the molecules due to the finite size of the state readout laser beam and the polarisation switching; molecules were optically pumped (with varying probability) throughout the ${\\approx}0.5$~cm wide laser beam.\n\nIn addition to spatial averaging, uncertainty in the mean longitudinal velocity also contributed an uncertainty in position. Changes of ${\\approx}10$~m\/s between molecule pulses were quite typical over the course of the $\\mathcal{E}$-field measurement, giving an estimated position uncertainty of ${\\lesssim}1$~cm.\n\n\nBy adding the above contributions in quadrature we concluded that the range of positions from which the microwave-induced signals could have originated increased from around ${\\approx}1.3$~cm at the state readout beam to ${\\approx}2.1$~cm at the optical pumping beam. These ranges are shown as horizontal error bars at the extrema of position in figure~\\ref{fig:enr}.\n\n\nWe used a third method to measure $\\mathcal{E}$ and $\\mathcal{E}^{\\mathrm{nr}}$ \\emph{in situ} throughout the eEDM dataset by performing `intentional parameter variation' tests with large $\\Delta_{{\\rm prep}}$ (denoted by `c' in figure~\\ref{fig:timing}). Detuning the state preparation laser resulted in a reduction in the measured contrast $|\\mathcal{C}|$ as shown in figure~\\ref{fig:contrast}~(B). Setting $\\Delta_{{\\rm prep}}\\approx\\pm2^{\\:}\\mathrm{MHz}$ gives $\\left|\\mathcal{C}\\right|\\approx0.5$, and the contrast was then approximately linearly proportional to $\\Delta_{\\rm prep}$ with a sensitivity\nof about $1\/\\gamma_{C}\\approx1\/(2\\pi\\times2~{\\rm MHz})$. Any variation in the electric field would change the Stark shift, and thus also $\\Delta_{\\rm prep}$, resulting in a change in contrast. Thus, using the previously described spin precession scheme, we indirectly measured parity components of the electric field from the appropriate parity components of the contrast:\n\\begin{align}\nD_1\\mathcal{E}^{\\mathrm{nr}}(x_{\\mathrm{prep}})\\approx&\\frac{\\partial\\Delta_{\\mathrm{prep}}}{\\partial\\mathcal{C}}\\mathcal{C}^{\\mathcal{NE}}\\\\\nD_1\\mathcal{E}(x_{\\mathrm{prep}})\\approx&\\omega_{\\rm L}^{\\mathcal{N}}+\\frac{\\partial\\Delta_{\\mathrm{prep}}}{\\partial\\mathcal{C}}\\mathcal{C}^{\\mathcal{N}}.\n\\end{align}\n\nWe looked for variation of $\\mathcal{E}$ or $\\mathcal{E}^{\\mathrm{nr}}$ every 3--4 hours. Measurements of $\\mathcal{E}^{\\mathrm{nr}}$ were consistent with the microwave measurements, with a constant value $\\mathcal{E}^{\\mathrm{nr}}(x_{\\mathrm{prep}})=-4.8\\pm0.9^{\\:}\\mathrm{mV\/cm}$. However, the mismatch $\\Delta^{\\mathcal{N}}=D_1\\mathcal{E}-\\omega_{\\rm L}^{\\mathcal{N}}$ between the Stark shift $D_1\\mathcal{E}$ and the $\\Nsw$-correlated laser frequency shift, $\\omega_{L}^{\\mathcal{N}}$, was found to drift significantly on the scale of around $2\\pi\\times20^{\\:}\\mathrm{kHz}\/\\mathrm{day}$. This drift of $\\Delta^{\\N}$ was servoed by tuning $\\omega^{\\N}_{\\rm L}$ after each measurement, ensuring $\\left|\\Delta^{\\mathcal{N}}\\right|<2\\pi\\times30$~kHz at all times \\cite{SpaunThesis}, see sections \\ref{sssec:correlated_laser_parameters} and \\ref{ssec:laser_imperfections} for more details. \n\n\\subsubsection{Magnetic Fields}\n\\hspace*{\\fill} \\\\\n\\label{sec:bfields}\nOur experimental scheme did not require the application of a magnetic field. This was not the case with some previous eEDM experiments, where the magnetic field was used to define a quantization axis \\cite{Commins1994,Regan2002}, or to cause the precession of spin to a direction associated with maximum sensitivity \\cite{Hudson2011,Kara2012}. Instead we used the electric field to define a quantization axis, and we used the relative polarisations of the state preparation and readout lasers to define the basis in which we read out the electron's spin precession with maximal sensitivity.\n\nHowever, we regularly applied a magnetic field $\\B$ in order to perform searches for systematic errors. The phase accumulation induced by an eEDM $\\delta d_{e}\\approx5\\times10^{-29}$~$\\ecm$ would have the same size as a Zeeman phase produced by a magnetic field of $\\mathcal{B}\\approx0.2~\\upmu{\\rm G}$, which is small compared to some of the magnetic field imperfections in the experiment. However, phases associated with magnetic-field-induced precession were distinguished from eEDM-induced precession by the use of the switches at our disposal (e.g.~electric field reversal). Nevertheless, it was important to investigate, quantify and minimize the effects of such magnetic fields, as they could have coupled with other experimental imperfections to give eEDM-like phases.\n\nUnder normal operating conditions we ran the experiment at three different magnetic field magnitudes, corresponding to a relative precession phase of $\\phi^{\\mathcal{B}}\\approx q\\frac{\\pi}{4}$ for $q=0,1,2$. The required $z$-component of the field was then $\\mathcal{B}_z=q\\mathcal{B}_0\\tilde{\\mathcal{B}}$, where $\\B_0=\\frac{\\pi}{4}\\frac{1}{g_1\\mu_{\\rm B}\\tau}\\approx 20~{\\rm mG}$. We also had the ability to apply transverse magnetic field components along $\\hat{x}$ and $\\hat{y}$, and all five linearly independent first-order gradients. The various coils that we used are illustrated in figure~\\ref{fig:coil_schematic}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.25]{coil_schematic.pdf}\n\\caption{A schematic of the magnetic field coils used. The main coils consisted of rectangular cosine coils (orange) wound on the surface of a cylindrical plastic frame together with additional end coils (red) to correct for the low aspect ratio (length\/diameter) in our system; a second set of these coils, mirrored in the $xy$ plane, is not shown. Also wrapped around this frame are a pair of circular auxiliary coils shown in yellow. The other auxiliary coils are shown in blue and green and consist of rectangular coils above and below the vacuum chamber. See the main text for descriptions of the functions of all of the coils.\\label{fig:coil_schematic}}\n\\end{figure}\n\nThe primary magnetic field, $\\mathcal{B}_z$, was produced by two sets of rectangular coils, shown in orange in figure~\\ref{fig:coil_schematic}. These were wound on the surface of two hemicylindrical plastic shells, on the $\\pm z$ sides of the spin-precession region. The coils were designed to maximize field uniformity and minimize distortion due to the boundary conditions imposed by the magnetic shielding. It was also possible to apply a $\\partial\\mathcal{B}_z\/\\partial z$ gradient with these coils. Two end coils (red in figure~\\ref{fig:coil_schematic}), located on the $\\pm x$ ends of the spin-precession region, enhanced the uniformity of the $\\B$-field along $x$ and enabled application of a $\\partial\\B_z\/\\partial x$. The main coils were powered by two separate commercial power supplies\\footnote{Krohn-Hite 521\/522}, and the end coils were powered by custom power supplies. The current flowing through these coils was continuously monitored throughout the course of the experiment by measuring with a digital multimeter the voltage dropped across precision resistors.\n\nWe used three sets of auxiliary magnetic field coils in systematic error searches. A pair of circular Helmholtz coils (yellow in figure~\\ref{fig:coil_schematic}) were wrapped around the same frame used for the main coils and were formed from ribbon cable. They provided a magnetic field in the $\\pm\\hat{x}$ directions and could also provide a $\\partial\\B_x\/\\partial x$. Above and below the spin-precession region chamber ($\\pm y$) there were four sets of rectangular coils (blue and green in figure~\\ref{fig:coil_schematic}). These allowed us to produce a field in the $\\pm\\hat{y}$ directions as well as all three associated first-order gradients. Note that the three first-order magnetic field gradients that we could not apply could be inferred from Maxwell's equations. A summary of the fields that we could apply is given in table~\\ref{tab:coils_table}.\n\\begin{centering}\n\\begin{table}\n\\caption{A summary of the magnetic fields and magnetic field gradients that we could produce. The coil colours refer to figure~\\ref{fig:coil_schematic}.\\label{tab:coils_table}}\n\\begin{tabular}{ccc}\n\\br\nCoil colour & Fields produced & Field gradients produced\\\\\n\\mr\nOrange & $\\B_z$ & $\\partial \\B_z\/\\partial z$\\\\\nRed & $\\B_z$ & $\\partial \\B_z\/\\partial x$, $\\partial \\B_z\/\\partial z$\\\\\nYellow & $\\B_x$ & $\\partial \\B_x\/\\partial x$\\\\\nBlue & $\\B_y$ & $\\partial \\B_y\/\\partial y$, $\\partial \\B_y\/\\partial z$\\\\\nGreen & $\\B_y$ & $\\partial \\B_y\/\\partial x$, $\\partial \\B_y\/\\partial y$\\\\\n\\br\n\\end{tabular}\n\\end{table}\n\\end{centering}\n\nSeveral measures were taken to minimize stray magnetic fields affecting the molecules. The simplest was to ensure no magnetized objects were placed within the spin-precession region. To ensure this, all components were fabricated from non-magnetic materials (e.g.\\ no stainless steel). The magnetization of all objects was also checked before installation by passing them across an AC-coupled magnetometer sensitive to 0.1~mG field variations.\n\nThe ambient $\\B$-field in the laboratory was dominated by that from the Earth's core (${\\sim}500~$mG approximately along $\\hat{x}+\\hat{y}$). To suppress this and other DC\/low-frequency fields, the spin-precession region was surrounded by a set of five concentric cylindrical magnetic shields constructed from ${\\approx}1.6$~mm thick mu-metal\\footnote{Amuneal Inc.}. Each layer of shielding should have provided around a factor of 10 reduction in the DC magnetic field \\cite{Vutha2010}; however, residual magnetisation of the mu-metal was found to limit the field components to $\\gtrsim20~\\upmu$G for $\\mathcal{B}_x$ and $\\mathcal{B}_y$, and $\\gtrsim500~\\upmu$G for $\\mathcal{B}_y$. \\footnote{We later found that the residual $\\mathcal{B}_y$ could be reduced to a level comparable to $\\mathcal{B}_x$ and $\\mathcal{B}_z$ by performing degaussing with a higher current.} Each shielding layer was divided into two half-cylinders and two end caps. The outermost (innermost) shield was 132~cm (86~cm) long and had a diameter of 107~cm (76~cm). These shields had holes to allow lasers to pass through in the $z$ direction, and to accommodate the molecule beam. There were also holes for the light pipes to extract molecule fluorescence, and some electric connections, in the $x$ direction. Measurements and simulations showed that these holes had a negligible impact on the shielding efficiency. The shielding factor remained approximately constant up to an AC frequency $\\sim2\\pi\\times3$~GHz for which the wavelength becomes comparable to the size of any apertures in the shields, ${\\sim}10$~cm, and the magnetic field noise starts to penetrate the shields. However, our measurement was only sensitive to magnetic field noise at frequencies up to roughly the inverse of the spin precession time $1\/\\tau\\approx2\\pi\\times1$~kHz \\cite{VuthaThesis}. The aluminium vacuum chamber also shielded AC magnetic noise above a frequency ${\\sim}1\/\\pi\\sigma t^2\\mu\\approx2\\pi\\times100$~Hz, where $\\sigma\\approx3.5\\times10^7$~S\/m is the electrical conductivity, $t\\approx1$~cm is the thickness and $\\mu$ is the permeability ${\\approx}\\mu_0$, the vacuum permeability \\cite{Mager1969,Sumner1987}.\n\nThe relatively large ($\\B\\sim10$~mG) fields applied by the $\\B_z$ coils caused the inner magnetic shields to become slightly magnetized, inducing a non-reversing magnetic field, $\\B^{\\rm{nr}}\\approx30~\\upmu{\\rm G}$. In order to suppress this remanent field we performed a degaussing procedure on the magnetic shields by passing a $200$~Hz sinusoidal current through sets of loosely wound ribbon cable coils which wrapped axially (in the $xy$ plane at $z=0$) between the shield layers. The maximum current amplitude was 1~A, sufficient to drive the mu-metal to saturation, and the amplitude was decreased with an exponential envelope over a period of 1~s. To fully degauss all layers of the magnetic shielding takes around 4~s. There was also a 1~s period of `dead time' during which the main magnetic field was turned back on and allowed to settle. This degaussing procedure was repeated every time the applied magnetic field was changed, which occured approximately every 40~s.\n\nVariations in the magnetic fields present were continuously measured throughout the experimental procedure. This was achieved using a set of four three-axis fluxgate magnetometers\\footnote{Bartington Mag-03.}, which were mounted in a tetrahedral configuration outside the spin-precession region vacuum chamber (but inside the magnetic shielding). We also used an additional fluxgate magnetometer which was positioned at a distance of around 1~m from the apparatus and outside of the magnetic shielding. By continuously recording the measurements provided by these magnetometers we were able to search for correlations of our data with the magnetic field present. In particular, we checked for the presence of a magnetic field correlated with the electric field, $\\mathcal{B}^{\\E}$, which would have been characteristic of a leakage current flowing between the electric field plates --- an effect known to contribute a significant systematic error in previous eEDM experiments \\cite{Regan2001,Kara2012}.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{probulator_data.pdf}\n\\caption{Magnetic field data taken with a flux-gate magnetometer passed along the molecular beam line. The left-hand plot shows the reversing components of field whilst a nominal $\\mathcal{B}_z$ was applied. The right-hand plot shows the corresponding non-reversing components. The data are fit by polynomial curves.\\label{fig:probulator_data}}\n\\end{figure}\nAdditional measurement of the magnetic fields was carried out by opening the vacuum system and passing a rotatable flux-gate magnetometer into the chamber. This allowed for measurement of the fields directly along the beam line. The freedom to rotate the magnetometers was crucial to distinguish between electronic offsets and $\\vec{\\B}^{\\rm nr}$ for fields ${\\lesssim}1$~mG. From these measurements we were able to directly characterise most of the magnetic fields and first-order field gradients, including non-reversing components. Example data obtained from these measurements are shown in Figure~\\ref{fig:probulator_data}. We saw that the applied fields were all flat to within 1~mG, and the non-reversing components, with the exception of $\\B^{\\rm nr}_y$, were less than 50~$\\upmu$G. Systematic uncertainty due to these fields is discussed in section~\\ref{ssec:magnetic_field_imperfections}.\n\n\\subsubsection{Fluorescence Collection and Detection}\n\\hspace*{\\fill} \\\\\n\\label{sec:fluorescence_collection}\nAs previously described, our experimental data consisted of laser-induced molecule fluorescence, emitted in all directions (with a well-defined angular distribution \\cite{Kirilov2013}) when the molecules were interrogated by the state readout laser beam. The apparatus for collecting this light is illustrated in figure~\\ref{fig:collection_optics}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=16cm]{collection_optics.pdf}\n\\caption{Fluorescence collection apparatus. Left: The mounted electric field plates are shown together with one of the two sets of four lens doublets. The mounting for the top-left doublet has been removed to show the lenses. The fiber bundles are shown schematically, fastened into the lens tubes behind the doublets. The lens assembly was mounted on rails and the entire assembly sat on a breadboard which was fastened to the vacuum chamber. The view on the right also shows the approximate position of the state readout laser beam as it passes through the apparatus.}\n\\label{fig:collection_optics}\n\\end{figure}\nThe fluorescence light passed through the transparent electric field plates, whose inner (outer) faces are ITO (anti-reflection [AR]) coated. Behind each field plate was a set of four AR-coated lens doublets, which collimated and then focussed the light. The optical axes of the doublets intersected a ray path from the centre of the fluorescing molecule region, accounting for refraction through the electric field plates. \nThe first (second) lens of each doublet was a 75~mm\\footnote{CVI Melles Griot LAG-75.0-50.0-C-SLMF-400-700.} (50~mm\\footnote{CVI Melles Griot LAG-50.0-35.0-C-SLMF-400-700.}) diameter spherical lens of focal length 50~mm (35~mm). On each side ($\\pm z$), each of the four lens doublets focussed light onto one of four sections of a `quadfurcated' fiber bundle\\footnote{Fiberoptic Systems.} whose input ends were 9~mm in diameter and fastened in lens tubes. The output of the fiber bundle was connected to a 19~mm diameter fused quartz light pipe with optical couplant gel\\footnote{Corning Q2-3067.} in between. The light pipe passed out of the spin-precession region vacuum chamber and magnetic shields and directed the light onto a PMT\\footnote{Hamamatsu R8900U-20.}. Bandpass filters\\footnote{Semrock FF01-689\/23-25-D.} were used to suppress backgrounds from e.g.\\ scattered light. Detailed tests of the light collection were carried out \\cite{SpaunThesis} which estimated that ${\\approx}14$~\\% of the fluorescence photons were collected. The major contributions to this efficiency were the finite solid angle subtended by the collection lenses (${\\approx}50~$\\%), finite coupling efficiency into fiber bundles (${\\approx}60~$\\%) and finite coupling efficiency between the fiber bundles and the light pipes (${\\approx}50~$\\%). In addition, the quantum efficiency of the PMT's was specified to be ${\\approx}10$~\\%, which further reduced the signal obtained.\n\n\\subsubsection{Data Acquisition}\n\\hspace*{\\fill} \\\\\n\\label{sec:data_acquisition}\nThe data acqusition system performed the following three functions: \n\\begin{enumerate}\n\\item Digital modulation of the experimental parameters necessary for acquiring the complete set of phase and contrast measurements required to extract the eEDM, as described in section~\\ref{sec:Measurement_scheme_more_detail}.\n\\item Rapid (5~MSa\/s) acquisition and storage of high-bandwidth fluorescence waveforms for the spin precession measurement.\n\\item Monitoring and logging of experimental parameters useful for checking the experimental state and for searching for systematic errors (e.g. magnetic fields, beam source temperatures).\n\\end{enumerate}\nAll functions were coordinated with a LabVIEW-based software system.\n\nData acquisition timing was controlled by a digital delay generator.\\footnote{SRS DG645.} Every 20~ms, a TTL signal was produced which triggered the ablation laser Q-switch, in turn creating a pulse of molecules. Molecule fluorescence signals, measured as a PMT photocurrent, were captured on a 20-bit digital oscilloscope\\footnote{National Instruments PXI-5922.}. The oscilloscope was triggered 6--7~ms after the ablation pulse, depending on the current molecule beam forward velocity, and recorded a 9~ms window of signal containing the entire molecule signal (1--2~ms) and several ms of background. The 100~kHz square wave that drove the fast polarisation switching of the state readout laser was synchronised with the 50~Hz Q-switch trigger so that the relative phase was fixed. The 5~MSa\/s data rate of the oscilloscope enabled resolution of the time-dependent structure within each 5~$\\upmu$s polarisation bin; this structure could vary on timescales as short as the C-state lifetime $1\/\\gamma_C\\approx500$~ns \\cite{Hess2014}.\n\nSignal waveforms, $S(t)$, were captured from two PMTs --- note that we were not counting individual photoelectrons, but instead amplified and read out a voltage proportional to the count rate. These waveforms were then transferred to the control PC where they were digitally averaged over 25 pulses to form one `trace'. The traces were then written to a hard drive. A file containing auxiliary measurements was recorded synchronously with each fluorescence trace. This file included the states of the experimental switches and other auxiliary measurements such as $\\E$-field voltages, $\\B$-field currents, laser power and polarisation, magnetic field measurements, molecular beam buffer gas flow rate, buffer gas cell temperature, and the temperature, pressure and humidity in our lab. This data proved useful in searching for systematic errors as described in section~\\ref{sec:systematics}.\n\\subsection{Experiment Switches}\nDuring the course of the experiment, we performed many parameter switches. Most of these switch parameter symbols are denoted by a superscript tilde $\\tilde{\\mathcal{X}}$, which indicates that that parameter takes on two\nvalues, $\\tilde{\\mathcal{X}}=\\pm1$.\n\\begin{description}\n\\item [{$\\Nsw$}] Used as a quantum number, $\\Nsw\\approx\\Esw\\rm{sgn}\\left(M\\Omega\\right)$,\nfor states of $\\left|M\\right|>0$, $\\left|\\Omega\\right|>0$, that\nrefers to states with opposite molecular alignment with respect to\nthe applied electric field. It is also used to refer to the experiment\nswitch between spectroscopically addressing states in $\\left|H,J=1\\right\\rangle $\nwith opposite values of $\\Nsw$.\n\\item [{$\\Esw$}] Denotes the alignment of the applied electric\nfield with respect to the laboratory $\\hat{z}$ axis, $\\Esw=\\rm{sgn}\\left(\\vec{\\mathcal{E}}\\cdot\\hat{z}\\right)$ where $\\vec{\\mathcal{E}}$ is the applied electric field.\n\\item [{$\\Bsw$}] Denotes the alignment of the applied magnetic\nfield with respect to the laboratory $\\hat{z}$ axis, $\\Bsw=\\rm{sgn}\\left(\\vec{\\mathcal{B}}\\cdot\\hat{z}\\right)$ where $\\vec{\\mathcal{B}}$ is the applied magnetic field.\n\\item [{$\\tilde{\\theta}$}] Denotes the state of the polarisation dither\nthat is used to extract the contrast in the spin precession measurement.\nIt refers to the direction of the offset angle in the $xy$ plane\nof the state readout polarisation basis $\\hat{X},\\hat{Y}$, relative\nto the average polarisation of these lasers.\n\\item [{$\\Psw$}] Used as a quantum number to denote the\nparity (eigenvalue of the parity operator $P$) of a given molecular\nstate of well-defined parity. It is also used to refer to the experiment\nswitch between spectroscopically addressing states in $\\left|C,J=1\\right\\rangle $\nwith opposite values of $\\Psw$ with the state readout\nlasers.\n\\item [{$\\Lsw$}] Denotes the state of the mapping between\nthe two output channels of the electric field voltage supply, and\nthe two electric field plates which can be either connected normally\n(+1), or inverted relative to normal (-1).\n\\item [{$\\Rsw$}] Denotes the state of an experimental switch\nof the state readout polarisation basis offset angle with respect\nto the $x$-axis by either 0 $\\left(+1\\right)$ or $\\pi\/2$ $\\left(-1\\right)$.\n\\item [{$\\Gsw$}] Denotes the state of an experimental switch\nof the global polarisation; the state preparation and state readout lasers are rotated synchronously by a common angle. This can be thought of as a redefinition\nof the $\\hat{x}$ and $\\hat{y}$ axes in the $xy$ plane.\n\\item [{$\\B_{z}$}] Denotes the magnitude of the magnetic field\nalong the $\\hat{z}$ direction in the laboratory, $\\B_{z}=|\\vec{\\B}\\cdot\\hat{z}|$.\nThis parameter is switched between three values differing by about\n$20^{\\:}\\mathrm{mG}$. In figure~\\ref{fig:pixel_plot}, channels $X$ that are `odd'\nwith respect to this parameter refer to the linear variation $\\partial X\/\\partial\\mathcal{B}_{z}$.\n\\item [{$\\E$}] Denotes the magnitude of the electric field, $\\E=|\\vec{\\E}|$.\nThis parameter is switched between two values.\n\\item [{$\\hat{k}\\cdot\\hat{z}$}] Denotes the orientation of both the state\npreparation and the state readout laser pointing directions with\nrespect to the laboratory $\\hat{z}$ axis. This is a binary switch,\n$\\hat{k}\\cdot\\hat{z}=\\pm1$, but we do not denote this switch with\na tilde as we do with the other binary switch parameters.\n\\end{description}\n\n\\subsection{Laser Parameters}\nThere are a variety of laser parameters which are used to describe\nthe state preparation laser that is denoted with a subscript `prep',\nor the state readout lasers that are denoted with a subscript `read'\nif the property applies to both state readout lasers, or with subscripts\n$X$ and $Y$, if the parameter can vary between the two readout lasers.\n\\begin{description}\n\\item [{$\\hat{k}$}] Laser pointing direction. In this paper, the pointing\ndirection is always nearly aligned or antialigned with respect to\nthe laboratory $\\hat{z}$ axis such that $\\hat{k}\\cdot\\hat{z}\\approx\\pm1$.\n\\item [{$\\vartheta_{k}$}] Defined in equation~\\ref{eq:pointing_imperfection}. Polar angle of deviation\nof the pointing $\\hat{k}$ from aligned or anti-aligned with the $\\hat{z}$\naxis.\n\\item [{$\\varphi_{k}$}] Defined in equation~\\ref{eq:pointing_imperfection}. Azimuthal angle denoting\nthe direction in the $xy$ plane, relative to the $x$-axis, of the deviation of the pointing\n$\\hat{k}$ from the $\\hat{z}$ axis.\n\\item [{$\\hat{\\epsilon}$}] Complex laser polarisation. The readout laser\npolarisations are also referred to as $\\hat{X}$ and $\\hat{Y}$ as\nan alternative to $\\hat{\\epsilon}_{X}$ and $\\hat{\\epsilon}_{Y}$\nat some points.\n\\item [{$\\hat{\\varepsilon}$}] Effective polarisation. Used to parameterize\nthe effect of experiment imperfections on the molecule state as the\npolarisation vector that would be required to obtain the same molecule\nstate in the absence of those experiment imperfections.\n\\item [{$\\theta$}] Defined in section~\\ref{sec:Measurement_scheme} and equation~\\ref{eq:polarization_parametrization} as the linear polarisation angle of the complex polarisation vector.\n\\item [{$\\Theta$}] Defined in section~\\ref{sec:Measurement_scheme} and equation~\\ref{eq:polarization_parametrization} as encoding the ellipticity of the complex polarisation vector.\n\\item [{$S$}] Defined in section~\\ref{sec:Measurement_scheme_more_detail} as the relative circular Stokes parameter,\n$S\\equiv S_{3}\/I=\\cos2\\Theta$.\n\\item [{$\\omega_{\\rm L}$}] Laser frequency.\n\\item [{$P$}] Laser power.\n\\item [{$\\Omega_{\\rm r}$}] Rabi frequency for a particular laser beam and transition. Defined as the transition dipole matrix element multiplied by the amplitude of the electric field associated with the laser beam.\n\\item [{$\\Gamma$}] Optical retardance for some birefrigent element along\nthe laser beam path.\n\\item [{$\\phi_{\\Gamma}$}] Angle in the $xy$ plane of the fast axis associated\nwith an optical retardance $\\Gamma$.\n\\end{description}\n\n\\subsection{Molecular States and Parameters}\nThese symbols are all used to describe the molecular energy level structure and the manner in which our laser light interacts with the molecules, in particular for the state preparation and readout processes.\n\\begin{description}\n\\item [{$J$}] Total angular momentum.\n\\item [{$M$}] Projection of $J$ onto the laboratory $\\hat{z}$-axis.\n\\item [{$\\Omega$}] Projection of $J$ onto the internuclear axis, $\\hat{n}$.\n\\item [{$B_H$}] Rotational constant of the $H$ state.\n\\item [{$\\Eeff$}] `Effective electric field' to which we consider the eEDM to be subjected.\n\\item [{$\\Delta_{\\Omega,1}$}] The $\\Omega$-doublet splitting of the $\\ket{H,J=1}$ state.\n\\item [{$D_1$}] Expectation value of the molecular electric dipole moment of the $\\ket{H,J=1}$ state.\n\\item [{$g_1$}] The $g$-factor of the $\\ket{H,J=1}$ state.\n\\item [{$\\eta$}] Defined in equation~\\ref{eq:eta_2}, it is proportional to the $g$-factor difference between the two $\\Nsw$ states.\n\\item [{$\\left|\\pm,\\Nsw\\right\\rangle $}] Sublevels within the $\\ket{H,J=1}$ (eEDM sensitive) manifold, labelled by their values of $M$ and $\\Nsw$.\n\\item [{$\\left|C,\\Psw\\right\\rangle $}] Sublevel to which molecules are excited during state preparation and readout. One of two sublevels in the $\\ket{C,J=1}$ manifold, with $M=0$ and parity $\\Psw=\\pm1$.\n\\item [{$\\left|B(\\hat{\\epsilon}),\\Nsw,\\Psw\\right\\rangle $}] Superposition of $M$ sublevels within the $\\ket{H,J=1,\\Nsw}$ manifold that is depleted during state preparation with a laser beam of polarisation $\\hat{\\epsilon}$, as defined in equation~\\ref{eq:bright_state}.\n\\item [{$\\left|D(\\hat{\\epsilon}),\\Nsw,\\Psw\\right\\rangle $}] Superposition of $M$ sublevels within the $\\ket{H,J=1,\\Nsw}$ manifold that remains after state preparation with a laser beam of polarisation $\\hat{\\epsilon}$, as defined in equation~\\ref{eq:dark_state}.\n\\item [{$\\left|B_{\\pm}(\\hat{\\epsilon}),\\Nsw,\\Psw\\right\\rangle$}] Instantaneous eigenvectors of the three-level system formed by $\\left|B(\\hat{\\epsilon}),\\Nsw,\\Psw\\right\\rangle$, $\\left|D(\\hat{\\epsilon}),\\Nsw,\\Psw\\right\\rangle$ and $\\left|C,\\Psw\\right\\rangle$, as defined in equation~\\ref{eq:inst_eigv}.\n\\item [{$\\Delta$}] One-photon detuning from resonance, discussed in section~\\ref{sec:state_prep_read} and defined in equation~\\ref{eq:detuningcorrelations}.\n\\item [{$\\gamma$}] Decay rate of the a given electronic state. The electronic state label is given in the subscript. In most of the paper, only $\\gamma_C$, the decay rate of the $C$ state, is relevant.\n\\item [{$\\Omega_{\\rm r}$}] Transition Rabi frequency, which is proportional\nto the square root of the laser intensity.\n\\item [{$E_{B\\pm},^{\\:}E_{D}$}] Instantaneous eigenenergies of the dressed three-level system, defined in equation~\\ref{eq:inst_eig}.\n\\item [{$\\dot{\\chi}$}] Complex polarisation rotation rate defined in section~\\ref{sssec:AC_stark_shift_phases}.\n\\item [{$\\Pi$}] Defined and discussed in section~\\ref{sssec:AC_stark_shift_phases} and equation~\\ref{eq:Pi_def}. This is a factor in the AC Stark shift phase that is independent of laser polarisation but depends on the laser detuning and Rabi frequency.\n\\item [{$v_{\\parallel}$}] The mean longitudinal velocity of the molecular beam.\n\\end{description}\n\n\\subsection{Measurement Quantities}\nThese symbols represent quantities related to the measurement of the accumulated phase and the way in which it is extracted during data analysis, as well as some related quantities pertaining to systematic studies.\n\\begin{description}\n\\item [{$N$}] Total number of measurments performed, equivalent to the number of detected photoelectrons.\n\\item [{$N_0$}] Number of molecules in the state readout region in the particular $\\Nsw$ level being addressed.\n\\item [{$f$}] Fraction of fluorescence photons emitted in the state readout region that are detected.\n\\item [{$S$}] Recorded photoelectron count rate measured on the photodetectors.\n\\item [{$F$}] Photoelectron count rate due to the molecule fluorescence.\n$F_{X,Y}$ is used to denote the molecular fluorescence induced by\nthe $X$ and $Y$ state readout lasers, respectively. $F_{\\mathrm{cut}}$\nis used to denote the fluorescence threshold above which data was\nincluded in the analysis.\n\\item [{$B$}] Background count rate primarily due to scattered\nlight from the state readout lasers. This background signal is subtracted\nfrom the raw photoelectron signal $S$ to obtain the fluorescence\nphotoelectron count rate, $F=S-B$.\n\\item [{$\\mathcal{A}$}] Signal asymmetry as defined in equation~\\ref{eq:Asymmetry}.\n\\item [{$\\mathcal{C}$}] Spin precession fringe contrast, as defined in\nequation~\\ref{eq:Contrast_Definition}, is the sensitivity of the asymmetry to molecular\nspin precession.\n\\item [{$\\phi$}] Actual spin precession phase of the molecules as defined in equation~\\ref{eq:total_phase}.\n\\item [{$\\Phi$}] Measured spin precession phase as described in section~\\ref{sec:Measurement_scheme_more_detail}, $\\Phi=\\mathcal{A}\/(2\\mathcal{C})$.\n\\item [{$\\tau$}] Measured spin precession time as described in sections~\\ref{sec:Measurement_scheme} and \\ref{sec:compute_phase}.\n\\item [{$\\omega$}] Measured spin precession frequency, as defined in equation~\\ref{eq:omega_def}, $\\omega=\\Phi\/\\tau$.\n\\item [{$\\chi^{2}$}] Reduced chi-squared statistic, $\\chi^2=\\frac{1}{N_{\\rm dof}}\\sum_i\\left(\\frac{x_i - f_i(\\{x\\})}{dx_i}\\right)^2$,\nwhere $N_{\\rm dof}$ is the number of degrees of freedom, $x_i$ are the data points, $dx_i$ are the uncertainties, and $f_i(\\{x\\})$ is a fit function that can depend on $i$ and the ensemble of all of the data, $\\{x\\}$. \nFor normally distributed data that fits well to the applied fit function, $\\chi^2$ should be consistent with 1.\n\\item [{$\\omega^{\\mathcal{NE}}$}] The measurement channel of interest, the spin precession frequency channel that is correlated with $\\Nsw$ and $\\Esw$. The expected eEDM signal should contribute to this channel.\n\\item [{$\\omega^{\\mathcal{NE}}_T$}] The contribution to spin precession frequency $\\omega^{\\mathcal{NE}}$ induced by $T$-odd spin precession effects in the $H$ state in ThO.\n\\item [{$\\omega^{\\mathcal{NE}}_P$}] A systematic error in the $\\omega^{\\mathcal{NE}}$ channel that is proportional to some parameter $P$.\n\\end{description}\n\n\\section{Introduction}\n\\input{Introduction_FINAL.tex} \n\n\\section{Atom and Molecule eEDM Experiments}\n\\subsection{Theory}\n\\input{theory_FINAL.tex}\n\n\\subsection{ThO Molecule}\n \\input{ThO_Molecule_FINAL.tex} \n\n\\section{ACME Experiment}\n\\subsection{Overview of Measurement Scheme}\n \\input{measurement_scheme_FINAL.tex}\n\n\\subsection{Apparatus}\n \\input{apparatus_FINAL.tex}\n\n\\section{Data Analysis}\n \\input{Data_Analysis_FINAL.tex}\n\n\\section{Systematic Errors}\n \\input{Systematics_FINAL.tex}\n\n\\section{Interpretation}\n \\input{Interpretation_FINAL.tex}\n\n\\section{Summary and Outlook}\n \\input{Conclusion_FINAL.tex}\n\n\n\\subsubsection{Basic Measurement Scheme}\n\\hspace*{\\fill} \\\\\nWe performed a spin precession measurement, resembling previous beam-based eEDM experiments \\cite{Hudson2011,Regan2002,Commins1994}, on $^{232}\\rm{Th}^{16}\\rm{O}$ molecules in a pulsed molecular beam generated by a cryogenic buffer gas beam source. Figure~\\ref{fig:meas_scheme_simple} shows a simplified schematic of the measurement. The molecules fly at velocity $v\\approx200$~m\/s into a magnetically shielded region with nominally uniform and parallel electric $\\vec{\\E}$ and magnetic $\\vec{\\B}$ fields. Molecule population is transfered from $|X^1\\Sigma^+,J=1,M=\\pm1\\rangle$ in the electronic ground state to the metastable $|H,J=1,M=\\pm1,\\Omega=\\tilde{\\N}\\tilde{\\E}M\\rangle\\equiv|\\pm,\\tilde{\\N}\\rangle$ state manifold (in the $\\ket{\\pm,\\Nsw}$ nomenclature we use $\\pm$ to refer to $M=\\pm1$) by optical pumping through the short-lived $|A^3\\Pi_{0^+},J=0,M=0\\rangle$ state with a 943~nm laser. This results in an even distribution of population in an incoherent mixture of the four $|\\pm,\\Nsw\\rangle$ states in $H$.\\footnote{A glossary of symbols used throughout this paper is provided in section~\\ref{sec:glossary}.} Figure~\\ref{fig:ThOlevels} shows the electronic states of ThO relevant to the eEDM measurement.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{ThO_Levels.pdf}\n\\caption{Levels and transitions in ThO used in our measurement of the eEDM, based on \\cite{Vutha2010,Edvinsson1985,Paulovic2003}. Solid arrows indicate transitions we address with lasers, wavy arrows indicate spontaneous decays of interest. For more details on how these transitions were used, see the main text.}\n\\label{fig:ThOlevels}\n\\end{figure}\n\nIn the absence of any experimental imperfections, we describe our system in terms of coordinate axes $+\\hat{z}$ along $+\\vec{\\E}$ (for a specified sign of applied field that we denote as positive, pointing approximately east to west in the lab) and $+\\hat{x}$ along the direction of the molecular beam (which travels approximately south to north) such that $+\\hat{y}$ is approximately aligned with gravity (cf.\\ figure~\\ref{fig:meas_scheme_simple}). Note that when we reverse the direction of the electric field, by construction the laboratory coordinate system does not change and the orientation of the electric field can be described by $\\Esw\\equiv{\\rm sgn}(\\hat{z}\\cdot\\vec{\\E})=\\pm1$. Analogously, we reverse the direction of the magnetic field between two $\\Bsw\\equiv{\\rm sgn}(\\hat{z}\\cdot\\vec{\\B})=\\pm1$ states. Since the directions of the fields are encoded by $\\Esw$ and $\\Bsw$, we define the magnitudes of the fields simply as $\\B_z\\equiv|\\B_z|$ and $\\E\\equiv|\\vec{\\E}|$.\n\nA superposition of the $M=\\pm1$ sublevels is prepared by optically pumping on the transition at 1090~nm between states $|\\pm,\\Nsw\\rangle$ and $|C^1\\Pi_1,J=1,M=0\\rangle(|\\Omega=+1\\rangle-\\Psw|\\Omega=-1\\rangle)\/\\sqrt{2}\\equiv|C,\\Psw\\rangle$, where $\\Psw=\\pm1$ is the excited state parity\\footnote{In this paper we follow the convention given in \\cite{Brown2003}.}, with laser light linearly polarised in the $xy$ plane. The resulting state corresponds to having the total angular momentum of the molecule aligned in the $xy$ plane. Because the $\\sigma$ electron's spin is aligned with $\\vec{J}$, by the Wigner-Eckart theorem this is equivalent to aligning the spin \\cite{Budker2008}, and we use this shorthand from here on. The state preparation laser frequency is tuned to spectroscopically select the molecule alignment $\\Nsw$, while the nearly degenerate $M=\\pm1$ states remain unresolved. The excited state $C$, which decays at a rate $\\gamma_C\\approx2\\pi\\times0.3$ MHz, decays primarily (${\\approx}75~\\%$ \\cite{Hess2014}) to the ground state so that one superposition of the two $|\\pm,\\Nsw\\rangle$ states is optically pumped out of $H$ and the remaining orthogonal superposition, which is `dark' to the preparation laser beam, is the prepared state. The linear polarisation of the state preparation laser beam, $\\hat{\\epsilon}_{{\\rm prep}}$, sets the relative coupling of each of the two $|\\pm,\\Nsw\\rangle$ states to $\\ket{C,\\Psw}$ and determines the spin alignment angle of the remaining state in the laboratory frame. The bright superposition $\\ket{B(\\hat{\\epsilon}_{\\rm prep})}$\nis pumped away, and the orthogonal dark superposition $\\ket{D(\\hat{\\epsilon}_{\\rm prep})}$ remains.\n\nFor the moment, we consider the specific case $\\Psw=+1$ and $\\hat{\\epsilon}_{{\\rm prep}}=\\hat{x}$, (the general case will be discussed in section \\ref{sec:Measurement_scheme_more_detail}). In this case, the prepared state\n\\begin{equation}\n\\ket{\\psi(t=0),\\Nsw}=\\frac{1}{\\sqrt{2}}\\left(\\ket{+,\\Nsw}-\\ket{-,\\Nsw}\\right)\n\\label{eq:initial_state}\n\\end{equation}\nhas the electron spin aligned along the $\\hat{y}$ axis. As the molecules traverse the spin precession region of length $L=22$ cm (which takes a time $\\tau\\approx1$~ms), the electric and magnetic fields exert torques on the electric and magnetic dipole moments, causing the spin to precess in the $xy$ plane by angle $2\\phi$; this corresponds to the state\n\\begin{equation}\n|\\psi(t=\\tau),\\Nsw\\rangle=\\frac{1}{\\sqrt{2}}\\left(e^{-i\\phi}|+,\\Nsw\\rangle-e^{+i\\phi}|-,\\Nsw\\rangle\\right),\n\\end{equation}\nwhere $\\phi$ is given approximately by the sum of the Zeeman and eEDM contributions to the spin precession angles,\n\\begin{equation}\n\\phi=-(\\Bsw g_1\\mu_{\\rm B}\\B_z+\\Nsw\\Esw d_e\\Eeff)\\tau.\n\\label{eq:simple_phase}\n\\end{equation}\nThe sign of the eEDM term, $\\Nsw\\Esw$, arises from the relative orientation between $\\vec{\\E}_{\\rm eff}$ and the electron spin as illustrated in figure~\\ref{fig:H-state}.\n\nAt the end of the spin precession region, we measure $\\phi$ by optically pumping on the same $H\\rightarrow C$ transition with the linearly polarised state readout laser beam. The polarisation alternates rapidly between two orthogonal linear polarisations $\\hat{X}$ and $\\hat{Y}$, such that each molecule is subject to excitation by both polarisations as it flies through the detection region, and we record the modulated fluorescence signals $F_X$ and $F_Y$ from the decay of $C$ to the ground state at 690 nm. This procedure amounts to a projective measurement of the spin onto $\\hat{X}$ and $\\hat{Y}$, which are defined such that $\\hat{X}$ is at an angle $\\theta$ with respect to $\\hat{x}$ in the $xy$ plane. To determine $\\phi$ we compute the asymmetry,\n\\begin{equation}\n\\mathcal{A}\\equiv\\frac{F_X-F_Y}{F_X+F_Y}\\propto\\cos{[2(\\phi-\\theta)]}.\n\\label{eq:asymmetry}\n\\end{equation}\nWe set $\\B_z$ and $\\theta$ such that $\\phi-\\theta\\approx(\\pi\/4)(2n+1)$ for integer $n$, so that the asymmetry is linearly proportional to small changes in $\\phi$ and maximally sensitive to the eEDM. A simplified schematic of the experimental procedure just described is shown in figure~\\ref{fig:meas_scheme_simple}.\n\\hspace*{\\fill} \\\\\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=16cm]{meas_scheme_simple.pdf}\n\\caption{Simplified schematic of the measurement scheme; numbers next to energy levels label $J$. \\textbf{1.} Molecules in the $\\ket{X,J=1}$ state are optically pumped via the $A$ state into $\\ket{H,J=1}$ by a retroflected (and offset in $x$) laser beam (blue arrows into\/out of page), polarised along $\\hat{x}$ and $\\hat{y}$ (blue arrows). \\textbf{2.} Molecules from one of the $\\Nsw$ states are then prepared in a superposition of $M$ sublevels ($M=-1,0,+1$ from left to right) by a linearly polarised laser beam (red) addressing the $H\\rightarrow C$ transition. This aligns the molecule's angular momentum, $\\vec{J}$, which in turn aligns the spin of the eEDM-sensitive $\\sigma$ electron, which is on average aligned with $\\vec{J}$. \\textbf{3.} The angular momentum (and hence electron spin) then precesses due to the electric and magnetic fields present (into the page) by an angle $\\phi$. This precession is dominated by the magnetic interaction but also includes a term linear in $d_e$ (see equation~\\ref{eq:simple_phase}). \\textbf{4.} The spin state is projected onto orthogonal superpositions of the $M$ sublevels by laser beams polarised along $\\hat{X},\\hat{Y}$ (red arrows). The resulting fluorescence is determined by the population in each superposition state and hence the precession angle $\\phi$.}\n\\label{fig:meas_scheme_simple}\n\\end{figure}\n\nBy repeating the measurement of $\\phi$ after having reversed any one of the signs $\\Nsw$, $\\Esw$ or $\\Bsw$, we may isolate the eEDM phase from the Zeeman phase. In practice, we repeat the phase measurement under all $2^3$ $(\\Nsw,\\Esw,\\Bsw)$ experiment states to reduce the sensitivity of the eEDM measurement to other spurious phases, and we extract the phase $\\phi^{\\N\\E}=-d_e\\Eeff\\tau=\\phi_{\\rm EDM}$. Here, we have introduced the notation $\\phi^u$, discussed in detail in the next section, which we use throughout this document to refer to the component of $\\phi$ that is odd under the set of switches listed in the superscript $u$, and implicitly even under those which are not listed (see section~\\ref{sec:Measurement_scheme_more_detail} and equation~\\ref{eq:general_parity} for a rigorous definition). A component which is even under all switches is considered to be `non-reversing' and is given an `nr' superscript.\n\n\\subsubsection{Measurement Scheme in Detail}\n\\label{sec:Measurement_scheme_more_detail}\n\\hspace*{\\fill} \\\\\nTo fully describe the method by which we extracted $d_e$ from the data in section \\ref{sec:data_analysis}, and to describe the systematic error models in section \\ref{sec:systematics}, we must introduce some additional formalism to describe the spin precession measurement to generalize the simple case described in the previous section. \n\nWe work in the regime in which the Stark shift in $H$ is approximately linear, $E_{\\rm Stark}\\approx-\\Nsw D_1\\E$, which holds when the Stark interaction energy is large compared to the $\\Omega$-doublet energy splitting $\\Delta_{\\Omega,1}$ but small compared to the rotational energy scale, described by the $H$-state rotational constant $B_H\\approx2\\pi\\times$~9.8 GHz, i.e. $\\Delta_{\\Omega,1}\\ll D_1\\E\\ll B_H$. In this regime, the molecular alignment is approximately related to $\\Omega$ by $\\Nsw=\\Esw M\\Omega$; this relation is assumed throughout this document. This is a good approximation, but it is notable that due to the Stark interaction at first order in perturbation theory, each $|M,\\Nsw\\rangle$ state is a superposition of all four $|H,J,M,\\Omega\\rangle$ states with $J=1,2$ and $\\Omega=\\pm1$. This effect is discussed further in sections~\\ref{sssec:correlated_laser_parameters} and \\ref{sssec:laser_pointing_and_intensity}.\n\nLet us consider the preparation of a spin-aligned state again. Starting from an incoherent mixture of the four $|\\pm,\\Nsw\\rangle$ states, we perform optical pumping on the electric dipole transition between $\\ket{\\pm,\\Nsw}$ and $\\ket{C,\\Psw}$, for a specific $\\Nsw$, with laser light of polarisation $\\hat{\\epsilon}_{{\\rm prep}}$ that is nominally linear in the $xy$ plane. This step depletes the bright superposition state (see e.g. \\cite{Bickman2009})\n\\begin{equation}\n\\ket{B(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw)}=(\\hat{\\epsilon}_{+1}^{*}\\cdot\\hat{\\epsilon}_{{\\rm prep}}^{*})\\ket{+,\\Nsw}-\\Psw(\\hat{\\epsilon}_{-1}^{*}\\cdot\\hat{\\epsilon}_{{\\rm prep}}^{*})\\ket{-,\\Nsw},\n\\label{eq:bright_state}\n\\end{equation}\nwhere $\\hat{\\epsilon}_{\\pm1}=\\mp\\left(\\hat{x}\\pm i\\hat{y}\\right)\/\\sqrt{2}$\nare unit vectors for circular polarisation. The corresponding dark state (with which the laser does not interact) is the orthogonal superposition\n\\begin{equation}\n\\ket{D(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw)}=(\\hat{\\epsilon}_{+1}^{*}\\cdot\\hat{\\epsilon}_{{\\rm prep}})\\ket{+,\\Nsw}+\\Psw(\\hat{\\epsilon}_{-1}^{*}\\cdot\\hat{\\epsilon}_{{\\rm prep}})\\ket{-,\\Nsw}.\n\\label{eq:dark_state}\n\\end{equation}\nThis dark state serves as the initial state, $|\\psi(0),\\Nsw\\rangle = |D(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw=+1)\\rangle$, for the spin-precession experiment, where we fixed the state preparation laser frequency to address the excited state with parity $\\Psw=+1$. The state preparation laser polarisation can be parameterised as\n\\begin{equation}\n\\hat{\\epsilon}_{{\\rm prep}}=-e^{-i\\theta_{{\\rm prep}}}\\cos\\Theta_{{\\rm prep}}\\hat{\\epsilon}_{+1}+e^{+i\\theta_{{\\rm prep}}}\\sin\\Theta_{{\\rm prep}}\\hat{\\epsilon}_{-1},\n\\label{eq:polarization_parametrization}\n\\end{equation}\nwhere $\\Theta_{{\\rm prep}}\\approx\\pi\/4$ defines the ellipticity Stokes parameter $(S_3\/I)_{{\\rm prep}}=\\cos2\\Theta_{{\\rm prep}}\\approx0$, and $\\theta_{{\\rm prep}}$ defines the linear polarisation angle with respect to $\\hat{x}$ in the $xy$ plane. From here on, we refer to the ellipticity Stokes parameter as $S\\equiv S_3\/I$. There is a one-to-one correspondence between the dark state superposition and the projection of the laser polarisation $\\hat{\\epsilon}_{{\\rm prep}}$ onto the $xy$ plane. If the laser polarisation does not lie entirely in the $xy$ plane, equations \\ref{eq:bright_state} and \\ref{eq:dark_state} are still appropriate, but require normalization. Note that if the laser is linearly polarised, switching the excited state parity $\\tilde{\\mathcal{P}}$ has the same effect on the dark state as rotating the laser polarisation angle by $\\pi\/2$.\n\nFollowing the initial state preparation, the molecules traverse the spin-precession region with their forward velocity nominally along $\\hat{x}$. In this region there are nominally uniform and parallel electric ($\\vec{\\E}$) and magnetic ($\\vec{\\B}$) fields, which produce energy shifts given by\n\\begin{equation}\nE(M,\\Nsw) =-|M|D_1\\E\\Nsw-Mg_1\\mu_{\\rm{B}}\\B_z\\tilde{\\B}-M\\eta\\mu_{\\rm{B}}\\E\\B_z\\Nsw\\tilde{\\B}-Md_e\\Eeff\\Nsw\\tilde{\\E},\n\\label{eq:Energy}\n\\vspace{10pt}\n\\end{equation}\nwhere $D_1$ is the electric dipole moment of $\\ket{H,J=1}$. Here $\\eta=0.79(1)$~nm\/V accounts for the $\\E$-dependent magnetic moment difference between the two sets of $\\Nsw$ levels in $\\ket{H,J=1}$ \\cite{Petrov2014}, as described in section~\\ref{sec:compute_phase}. The energy shift terms that depend on the sign of $M$ contribute to the spin precession angle $\\phi$, which is given by:\n\\begin{equation}\n\\phi=\\frac{1}{2}\\int_0^L(E(M=+1,\\Nsw)-E(M=-1,\\Nsw))\\frac{{\\rm d}x}{v}.\n\\label{eq:total_phase}\n\\end{equation}\nThis phase is dominated by the magnetic (Zeeman) interaction. The Stark shift, proportional to $|M|$, does not contribute. The state then evolves to: \n\\begin{equation}\n|\\psi(\\tau),\\Nsw\\rangle = \\left(e^{-i\\phi}|+,\\Nsw\\rangle \\langle +,\\Nsw| + e^{+i\\phi}|-,\\Nsw\\rangle \\langle -,\\Nsw|\\right)|\\psi(0),\\Nsw\\rangle,\n\\end{equation}\n(recall $\\ket{\\psi(0),\\Nsw}=\\ket{D(\\hat{\\epsilon}_{{\\rm prep}},\\Nsw,\\Psw=+1)}$ per equation~\\ref{eq:dark_state}) and molecules enter a detection region where the state is read out by optically pumping again between the $\\ket{H,J=1}$ and $\\ket{C,J=1}$ manifolds. This optical pumping is performed alternately by two laser beams with nominally orthogonal linear polarisations \n$\\hat{\\epsilon}_X$ and $\\hat{\\epsilon}_Y$.\\footnote{For convenience, the notation $\\hat{\\epsilon}_{X}$, $\\hat{\\epsilon}_{Y}$ is used interchangeably with the previously used notation $\\hat{X}$, $\\hat{Y}$.}\nThese beams excite the projection of $\\ket{\\psi(\\tau),\\Nsw}$ onto the bright states\n\\begin{equation}\n\\ket{B(\\hat{\\epsilon}_X,\\Nsw,\\Psw)}\\quad{\\rm and}\\quad\\ket{B(\\hat{\\epsilon}_Y,\\Nsw,\\Psw)},\n\\end{equation}\n(with the same $\\Nsw$ that was addressed in the state preparation optical pumping step, but with an independent choice of $\\Psw$) with probability $P_{X,Y}$ respectively. In the ideal case in which all laser polarisations are exactly linear, this probability is given by\n\\begin{equation}\n\\label{eq:xprojection}\nP_{X,Y}(\\phi,\\theta_{{\\rm prep}},\\theta_{ X,Y},\\Nsw,\\Psw)=\\left|\\braket{B(\\hat{\\epsilon}_{X,Y},\\Nsw,\\Psw)|\\psi(\\tau),\\Nsw}\\right|^2=\\left[1-\\Psw\\cos(2(\\theta_{{\\rm prep}}-\\theta_{X,Y}+\\phi))\\right]\/2,\n\\end{equation}\nwhere $\\theta_{X,Y}$ are the linear polarisation angles of the state readout beams, with respect to $\\hat{x}$. The result is a signal that varies sinusoidally with the precession angle $\\phi$. To measure these probabilities, we observe the associated modulated fluorescence signals, $F_{X,Y}=fN_0P_{X,Y}$, where $N_0$ is the number of molecules in the addressed $\\Nsw$ level at the state readout region, and $f$ is the fraction of total fluorescence photons that are detected. \n\nTo distinguish between molecule number fluctuations and phase variations, we normalize with respect to the former by rapidly switching the state readout laser between the two orthogonal polarisations, $\\hat{\\epsilon}_{X,Y}$, every 5~$\\upmu$s. This is significantly quicker than fluctuations in the molecule number and is sufficiently quick that every molecule is interrogated by both polarisations (see section~\\ref{sec:data_analysis} or \\cite{Kirilov2013} for more details). We then form an asymmetry $\\A$, which is immune to molecule number fluctuations, given by\n\\begin{equation}\n\\A=\\frac{F_X-F_Y}{F_X+F_Y}=\\Psw\\cos[2(\\phi-\\theta)],\n\\label{eq:Asymmetry}\n\\end{equation}\nwhere we have assumed that the readout polarisations are exactly orthogonal, given by $\\theta_X=\\theta_{\\rm{read}}$ and $\\theta_Y=\\theta_{\\rm{read}}+\\pi\/2$, and where we have defined $\\theta\\equiv\\theta_{\\rm{read}}-\\theta_{{\\rm prep}}$.\\footnote{Note that this reduces to equation~\\ref{eq:asymmetry} for $\\theta_{\\rm prep}=0$ (i.e. $\\hat{\\epsilon}_{\\rm prep}=\\hat{x}$) and $\\Psw=+1$.} In this equation and from now on unless otherwise noted, $\\Psw$ refers to the excited state parity that is addressed by the state readout laser, not to be confused with the excited state parity addressed by the state preparation laser, which is kept fixed.\n\nThe value of $\\mathcal{B}_z$ and the state preparation and readout laser beam polarisations are chosen so that $|\\phi-\\theta|\\approx\\pi\/4$. This corresponds to the linear part of the asymmetry fringe in equation~(\\ref{eq:Asymmetry}), where $\\A$ is most sensitive to, and linearly proportional to, small changes in $\\phi$ (cf.\\ figure~\\ref{fig:fringe}). A variety of effects including imperfect optical pumping, decay from $C$ back to $H$, elliptical laser polarisation and forward velocity dispersion, reduce the measurement sensitivity by a `contrast' factor\n\\begin{equation} \n\\C\\equiv-\\frac{1}{2}\\frac{\\partial\\A}{\\partial\\theta}\\approx \\frac{1}{2}\\frac{\\partial\\A}{\\partial\\phi},\n\\label{eq:Contrast_Definition}\n\\end{equation}\nwith $|\\C|\\le1$. We measure this parameter by dithering $\\theta=\\theta^{\\rm nr} + \\Delta\\theta\\tilde{\\theta}$ (where $\\theta^{\\rm nr}$ is the average or 'non-reversing' polarisation angle)\nbetween states of $\\tilde{\\theta}=\\pm 1$, with amplitude $\\Delta\\theta=0.05$~rad. \nWe found that typically $|\\C|\\approx0.94$. \nWe then extract the measured phase, $\\Phi=\\mathcal{A}\/(2\\mathcal{C})+q\\pi\/4$, by normalising the asymmetry measurements according to the measured contrast --- see section~\\ref{sec:data_analysis} for more details on the data analysis methods used to evaluate this quantity. In the ideal case, the measured phase matches closely with the precession phase, $\\Phi\\approx\\phi$. However, a variety effects that are investigated closely in section~\\ref{sec:systematics} lead to slight deviations between these two quantities, which can contribute to systematic errors in the measurement. Unless explicitly specified, $\\C$ is assumed to be an unsigned quantity from here on. In particular, when averaging over multiple states of the experiment, $|\\C|$ is used.\n\nTo isolate the eEDM term from other components of the energy shift in equation~(\\ref{eq:Energy}), the experiment is repeated under different conditions that are characterised by parameters whose sign is switched regularly during the experiment. The spin precession measurement is repeated for all $2^4$ experiment states defined by the four primary binary switch parameters: $\\Nsw$, the molecular orientation relative to the applied electric field (changed every 0.5~s); $\\Esw$, the direction of the applied electric field in the laboratory (2~s); $\\tilde{\\theta}$, the sign of the readout polarisation dither (10~s); and $\\tilde{\\mathcal{B}}$, the direction of the applied magnetic field in the laboratory (40~s). For each ($\\Nsw,\\Esw,\\tilde{\\mathcal{B}}$) state, the asymmetry $\\mathcal{A}(\\Nsw,\\Esw,\\tilde{\\mathcal{B}})$, contrast $\\mathcal{C}(\\Nsw,\\Esw,\\tilde{\\mathcal{B}})$, and measured phase $\\Phi(\\Nsw,\\Esw,\\tilde{\\mathcal{B}})$ are determined as described earlier. The data taken under all $2^4=16$ experimental states derived from these four binary switches constitutes a `block' of data.\n\nWe can write the phase $\\Phi(\\Nsw,\\Esw,\\tilde{\\mathcal{B}})$ in terms of components with particular parity with respect to the experimental switches:\n\\begin{align}\n\\Phi(\\tilde{\\mathcal{N}},\\tilde{\\mathcal{E}},\\tilde{\\mathcal{B}})=&\\Phi^{\\mathrm{nr}}+\\Phi^{\\mathcal{N}}\\tilde{\\mathcal{N}}+\\Phi^{\\mathcal{E}}\\tilde{\\mathcal{E}}+\\Phi^{\\mathcal{B}}\\tilde{\\mathcal{B}}\\nonumber\\\\+&\\Phi^{\\mathcal{NE}}\\tilde{\\mathcal{N}}\\tilde{\\mathcal{E}}+\\Phi^{\\mathcal{NB}}\\tilde{\\mathcal{N}}\\tilde{\\mathcal{B}}+\\Phi^{\\mathcal{EB}}\\tilde{\\mathcal{E}}\\tilde{\\mathcal{B}}+\\Phi^{\\mathcal{NEB}}\\tilde{\\mathcal{N}}\\tilde{\\mathcal{E}}\\tilde{\\mathcal{B}}.\n\\label{eq:phase_parity}\n\\end{align}\nWe refer to these components as `switch-parity channels'. A channel is said to be odd with respect to some subset of switches (labelled as superscripts) if it changes sign when any of those switches is performed. Thus it will also change sign if an odd number of those switches is performed. It is implicitly even under all other switches. We use this general notation throughout this document to refer to correlations of various measured quantities and experimental parameters with experiment switches. To generalize, if we have $k$ binary experiment switches $(\\tilde{\\mathcal{S}}_{1},\\tilde{\\mathcal{S}}_{2},\\dots,\\tilde{\\mathcal{S}}_{k})$ such that $\\tilde{\\mathcal{S}}_{i}=\\pm1$, and we perform a measurement of the parameter $X(\\tilde{\\mathcal{S}}_{1},\\tilde{\\mathcal{S}}_{2},\\dots,\\tilde{\\mathcal{S}}_{k})$ for a complete set of the $2^{k}$ switch states, then the component of $X$ that is odd under the product of switches $\\left[\\tilde{\\mathcal{S}}_{a}\\tilde{\\mathcal{S}}_{b}\\dots\\right]$ is given by\n\\begin{equation}\nX^{\\mathcal{S}_{a}\\mathcal{S}_{b}\\dots}\\equiv \\frac{1}{2^{k}}\\sum_{\\tilde{\\mathcal{S}}_{1}\\dots\\tilde{\\mathcal{S}}_{k}=\\pm1}\n\\left[\\tilde{\\mathcal{S}}_{a}\\tilde{\\mathcal{S}}_{b}\\dots\\right]X\\left(\\tilde{\\mathcal{S}}_{1},\\tilde{\\mathcal{S}}_{2},\\dots,\n\\tilde{\\mathcal{S}}_{k}\\right).\n\\label{eq:general_parity}\n\\end{equation}\nThe switch parity behavior of a given component is expressed in the superscript which lists the experimental switches with respect to which the component is odd. We order the switch labels in the superscripts such that the fastest switches are listed first and the slowest switches are listed last. Some components give particularly important physical quantities. Most notably, the eEDM precession phase is extracted from the $\\tilde{\\mathcal{N}}\\tilde{\\mathcal{E}}$-correlated component of the measured phase: that is, in the ideal case $\\Phi^{\\mathcal{NE}}=-d_{e}\\mathcal{E}_{\\mathrm{eff}}\\tau$. Additionally, the Zeeman precession phase is nominally given by $\\Phi^{\\B}=-\\mu_{\\rm B}g_1\\mathcal{B}_z\\tau$. Recall we label `non-reversing' components with an `nr' superscript. In a few cases, we drop the superscript parity because it is redundant. For example, we drop the superscript on the dominant components of the applied electric and magnetic fields, $\\mathcal{E}\\equiv\\mathcal{E}^{\\mathcal{E}}$ and $\\mathcal{B}_{z}\\equiv\\mathcal{B}_{z}^{\\mathcal{B}}$.\n\nMany other experimental parameters are also varied between blocks of data to suppress and monitor systematic errors (figure~\\ref{fig:timing}). These `superblock' switches include: excited-state parity addressed by the state readout laser beams, $\\Psw$ (chosen randomly after every block, with equal numbers of $\\Psw=\\pm1$); simultaneous change of the power supply polarity and interchange of leads connecting the electric field plates to their voltage supply, $\\Lsw$ (4~blocks); a rotation of the state readout polarisation basis by $\\theta_{\\rm{read}}\\rightarrow\\theta_{\\rm{read}}+\\pi\/2$ to interchange the roles of the $X$ and $Y$ beams, $\\Rsw$ (8~blocks); and a global polarisation rotation of both state preparation and readout lasers by $\\theta_{\\rm{read}}\\rightarrow\\theta_{\\rm{read}}+\\pi\/2$ and $\\theta_{{\\rm prep}}\\rightarrow\\theta_{{\\rm prep}}+\\pi\/2$, $\\Gsw$ (16~blocks).\n\nAdditionally, the magnitude of the magnetic field, $\\B_z$, was switched on the timescale of 64--128 blocks (${\\sim}1$~hour), and the magnitude of the applied electric field, $\\E$, and the laser propagation direction, $\\hat{k}\\cdot\\hat{z}$, were changed on timescales of ${\\sim}1$~day and ${\\sim}1$~week, respectively. \n\nOn these longer timescales, we also alternated between taking eEDM data under \\textit{Normal} conditions, for which all experiment parameters were set to their nominally ideal values, and taking data with \\textit{Intentional Parameter Variations} (IPVs), during which some experimental parameter was set to deviate from ideal so that we could monitor the size of the known systematic errors described in section \\ref{sssec:correlated_laser_parameters}. We took IPV data in which we varied (a) the non-reversing electric field $\\E^{\\rm{nr}}$ and (b) the $\\Nsw\\Esw$-correlated Rabi frequency, $\\Omega_{\\rm r}^{\\N\\E}$, to measure the sensitivity of the eEDM measurement to these parameters and we varied (c) the state preparation laser detuning $\\Delta_{{\\rm prep}}$ to monitor the size of the residual $\\E^{\\rm{nr}}$. These systematic errors are discussed in more detail in sections~\\ref{ssec:efields} and \\ref{sssec:correlated_laser_parameters}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=10cm]{Switch_Timescales.pdf}\n\\caption[Timescales of experimental parameter switches]{A schematic of the switches performed during our experiment and the associated timescales. See the main text for a description of each of the switch parameters and a description of the distinction between the \\textit{Normal} and IPV (\\emph{Intentional Parameter Variation}) data types. The 15-hour run time and $|\\E|$ switching timescale are approximate.}\n\\label{fig:timing}\n\\end{figure}\n\nThe details of the data analysis required to extract the eEDM-correlated phase $\\Phi^{\\mathcal{NE}}$ are described in section \\ref{sec:data_analysis}. A lower bound on the statistical uncertainty $\\delta\\Phi^{\\mathcal{NE}}$ of the eEDM-correlated phase is given by photoelectron shot noise to be $\\delta\\Phi^{\\mathcal{NE}}=1\/(2|\\C|\\sqrt{N})$ for $N$ detected photoelectrons \\cite{Khriplovich1997,VuthaThesis}. In the case where shot noise is the sole contribution, we can express the statistical uncertainty $\\delta d_e$ in our measurement of the eEDM as\n\\begin{equation}\n\\delta\\de=\\delta\\Phi^{\\N\\E}\\frac{1}{\\E_{\\rm eff}\\tau}=\\frac{1}{2|\\mathcal{C}|\\tau\\mathcal{E}_{\\rm eff}\\sqrt{\\dot{N}T}},\n\\end{equation}\nwhere $\\dot{N}\\approx f\\dot{N_0}$ is the measurement rate (equivalent to the photoelectron detection rate) and $T$ is the integration time (recall $f$ is the fraction of fluorescence photons detected and $N_0$ is the number of molecules in the addressed $\\Nsw$ level). Further discussion of the achieved statistical uncertainty is presented in section~\\ref{sec:data_analysis}.\n\\subsection{eEDM Interaction}\n\nTo make contact with common language in the literature about the eEDM in molecules, we first write the effective, nonrelativistic eEDM interaction in terms of an internal electric field $\\vec{\\mathcal{E}}_{\\rm int}$.  (As we will see, this is closely related, but not identical, to the effective field $\\vec{\\mathcal{E}}_{\\rm eff}$.)  We choose a convention where $\\vec{\\mathcal{E}}_{\\rm int} = -\\mathcal{E}_{\\rm int} \\hat{n}$. This means that the internal field vector is defined to be directed \\textit{opposite} to $\\hat{n}$, i.e., along the average direction of the electric field \\textit{inside} the molecule (here, from positive Th ion to negative O ion) when \n$\\mathcal{E}_{\\rm int}$ is positive.   We also adopt the convention that, in the $H$ state of ThO, there is an effective eEDM $\\vec{d}_e^{\\rm eff} = d_e\\vec{S}$ (where again $S=1$ to a fair approximation). This choice appears, at first glance, to contradict the discussion in section~\\ref{sec:theory}, where for a single electron we wrote $\\vec{d}_e=2d_e\\vec{s}$ (where $s=1\/2$). However, these two definitions are in fact consistent when taking into account that in the $H~^3\\Delta_1$ state of ThO only one of the two valence electrons (the one in the $\\sigma$ orbital) contributes significantly to the EDM energy shift, while both electrons contribute to the total spin $S=1$. Hence, in our formulation, the molecule-frame projection $\\vec{d}_e^{\\rm eff}\\cdot\\hat{n}$ can take extreme values $\\pm d_e$, as expected for a single contributing electron. (This `single contibuting electron' approximation is valid for all molecules used to date in searches for the eEDM.)\n\nWe then write the effective eEDM Hamiltonian $H_{\\rm EDM}^{\\rm eff}$ in the standard form for interaction of an electric dipole moment with the internal electric field:\n\\begin{equation}\nH_{\\rm EDM}^{\\rm eff}=-\\vec{d}_e^{\\rm eff}\\cdot\\vec{\\mathcal{E}}_{\\rm int}= +d_e \\mathcal{E}_{\\rm int} \\vec{S}\\cdot\\hat{n},\n\\label{eq:HEDMapp1a}\n\\end{equation}\nwhere the + sign in the final expression arises from the sign convention for $\\vec{\\mathcal{E}}_{\\rm int}$.\nIn eigenstates of $\\Omega$, the expectation value of $H_{\\rm EDM}^{\\rm eff}$---that is, the energy shift $\\Delta E_{\\rm EDM}$ due to the eEDM---can be written as\n\\begin{equation}\n\\Delta E_{\\rm EDM} = +d_e \\mathcal{E}_{\\rm int} \\left\\langle\\Sigma\\right\\rangle\n = +d_e \\left( \\mathcal{E}_{\\rm int} |\\!\\left\\langle\\Sigma\\right\\rangle\\! | \\right) \\mathrm{sgn}\\left(\\langle\\Sigma\\rangle\\right).\n\\label{eq:HEDMapp1b}\n\\end{equation}\n\nNow, we finally re-introduce the effective electric field $\\Eeff$ used throughout the main text of this paper. This is related to the internal field introduced above, via \n\\begin{equation}\n   \\Eeff \\equiv |\\!\\left\\langle\\Sigma\\right\\rangle\\! | \\mathcal{E}_{\\rm int},\n\\end{equation}\nWe can then use this notation to describe the effective nonrelativistic eEDM interaction, within a given electronic state and eigenstate of $\\Omega$ (and otherwise independent of molecular structure), as follows:\n\\begin{eqnarray}\n\\vec{\\mathcal{E}}_{\\rm eff} \\equiv -\\Eeff \\hat{n}; \\\\\n\\vec{d}_e \\equiv d_e \\vec{S}\/ | \\left\\langle \\Sigma \\right\\rangle |; \\\\\nH_{\\rm EDM}^{\\rm eff} = -\\vec{d}_e \\cdot \\vec{\\mathcal{E}}_{\\rm eff} = +d_e \\Eeff \\Sigma\/ | \\left\\langle \\Sigma \\right\\rangle |; \\\\\n\\Delta E_{\\rm EDM} = \\mathrm{sgn}(\\left\\langle \\Sigma \\right\\rangle) d_e \\Eeff,\n\\label{eq:HEDMapp1d}\n\\end{eqnarray}\nwhere the sign in the last expressions arises from the defined definitions of $\\Sigma$ (component of $\\vec{S}$ along $\\hat{n}$) and $\\vec{\\mathcal{E}}_{\\rm eff}$ (antiparallel to $\\hat{n}$). All relevant quantities are summarised pictorially in figure~\\ref{fig:ACME_signs}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=\\linewidth]{ACME_signs.pdf}\n\\caption{Summary of sign conventions used in the ACME experiment. All vectors depict expectation values of operators defined in the text, in the states $| H, J=1, \\tilde{\\N},M\\rangle$. Note the difference between\nscalar $\\Omega$ and vector $\\vec{\\Omega}$. The figure is drawn with a negative $g$-factor, i.e. the magnetic moment $\\vec{\\mu}$ opposes $\\vec{J}$, and with positive values of $d_e$ and $\\Eeff$. Energy levels are shown in the centre of the figure --- solid lines show the Stark-shifted levels ($M=0$ levels are unaffected), dashed lines include Zeeman shifts and dotted lines include a non-zero eEDM interaction. Figure inspired by \\cite{Lee2009}.}\n\\label{fig:ACME_signs}\n\\end{figure}\n\nIn most of the theoretical literature on this subject, this energy shift is written in the unambiguous form $\\Delta E_{\\rm EDM} = +d_e W_d \\Omega$.  However, there has been no consistent definition in the literature for the relation between $W_d$ and $\\Eeff$. In particular, both their relative signs and the dependence of their relative magnitude on the value of $|\\Omega |$ (encompassing both the case of one- and two-electron systems) are often defined differently, or imprecisely.  \nIn our notation, the expressions above imply a general relationship between $\\Eeff$ and $W_d$:\n\\begin{equation}\n\\Eeff = W_d \\Omega \\mathrm{sgn}(\\left\\langle \\Sigma \\right\\rangle).\n\\label{eq:genHEDMapp1}\n\\end{equation}\nThis relation is valid for systems with one or two valence electrons (in the `single contributing electron' approximation for the latter case), and regardless of the relative directions of $\\vec{\\Sigma}$ and $\\vec{\\Omega}$.\n\nNow we apply these general considerations to the specific case of the $H$ state of ThO. Here, since $\\left\\langle \\Sigma \\right\\rangle \\approx -\\Omega$, we find that $\\Eeff = -W_d$ with our conventions.  Thus, the energy shifts can be written for ThO as\n\\begin{equation}\n\\Delta E_{\\rm EDM} = -d_e\\Eeff \\Omega.\n\\label{eq:HEDMapp1c}\n\\end{equation}\nIn our experiment, this gives rise to energy shifts, for a given direction of the laboratory electric field $\\E$, given by\n\\begin{equation}\n\\langle H, J=1, \\tilde{\\N},M|H_{\\rm eEDM}^{\\rm eff}| H, J=1, \\tilde{\\N},M\\rangle=-d_e\\E_{\\rm eff}M\\tilde{\\N}\\tilde{\\E},\n\\label{eq:HEDMapp2}\n\\end{equation}\nsince in our notation $\\Omega=M\\tilde{\\N}\\tilde{\\E}$. \nThen, finally, the experimentally determined energy shift arising from the eEDM is \n\\begin{eqnarray}\n\\omega^{\\N\\E}_{\\rm EDM} = \\frac{1}{2}\\frac{1}{\\tilde{\\N}\\tilde{\\E}} \n\\left[ \\langle  H, J=1, \\tilde{\\N},M=+1|H_{\\rm eEDM}^{\\rm eff}| H, J=1, \\tilde{\\N},M=+1 \\rangle \\right. \\nonumber \\\\\n\\hphantom{H,J=1,\\N,} -\\left. \\langle H, J=1, \\tilde{\\N},M=-1|H_{\\rm eEDM}^{\\rm eff}| H, J=1, \\tilde{\\N},M=-1 \\rangle \\right] \\nonumber \\\\\n\\hphantom{\\omega^{\\N\\E}_{\\rm EDM}} = -d_e\\E_{\\rm eff} .\n\\label{eq:HEDMapp2a}\n\\end{eqnarray}\n\n\n\n\n\\subsection{Scalar-Pseudoscalar Nucleon-Electron Interaction}\n\nWe next turn to notation describing the $T$-violating scalar-pseudoscalar (SP) interaction between a nucleon and an electron.\nThe relativistic Hamiltonian for this interaction can be written as\n\\begin{equation}\nH_{\\rm SP}=i\\frac{G_{\\rm F}}{\\sqrt{2}}(ZC_{\\rm S,p}+NC_{\\rm S,n})\\gamma_0\\gamma_5\\rho_{\\rm N}(\\vec{r}),\n\\end{equation}\nwhere $G_F$ is the Fermi coupling constant, $\\gamma_i$ are Dirac matrices, $\\rho_{\\rm N}(\\vec{r})$ is the normalised nuclear density, $Z(N)$ is the proton (neutron) number, and $C_{\\rm S,p}$ and $C_{\\rm S,n}$ are dimensionless constants which describe the interaction strength (relative to that of the ordinary weak interaction) specifically for protons and neutrons, respectively. \nUsing the definition\n\\begin{equation}\nC_{\\rm S}=\\frac{Z}{A} C_{\\rm S,p} + \\frac{N}{A} C_{\\rm S,n} = \\frac{Z}{A} C_{\\rm S,p} + \\left( 1-\\frac{Z}{A}\\right) C_{\\rm S,n},\n\\end{equation}\nwhere $A=Z+N$, $C_\\mathrm{S}$ represents a weighted average of the couplings to protons and neutrons, and is different for every nuclear species. However, since the ratio $Z\/A$ is nearly the same for all heavy nuclei used in molecular and atomic EDM experiments (ranging only from $Z\/A=0.41$ for $^{133}$Cs to $Z\/A=0.39$ for $^{232}$Th), typically a common value for $C_{\\rm S}$ is assumed for all experiments of this type.\nThus we can write\n\\begin{equation}\nH_{\\rm SP}=i\\frac{G_{\\rm F}}{\\sqrt{2}} AC_{\\rm S} \\gamma_0\\gamma_5\\rho_{\\rm N}(\\vec{r}).\n\\label{eq:HPSapp1}\n\\end{equation}\nIn a given molecular electronic state, this gives rise to a non-relativistic, single-electron effective Hamiltonian of the form\n$ H_{\\rm SP}^{\\rm eff} = 2\\vec{s}\\cdot\\hat{n} C_{\\rm S} Y_{\\rm S}$; the factor of 2 is included so that the maximal energy shifts due to this term have the simple form $\\Delta E_\\mathrm{SP}^\\mathrm{max} = \\pm C_\\mathrm{S} Y_\\mathrm{S}$. \nBy analogy with our discussion of the eEDM Hamiltonian, in a molecular state with $S = 1$ and a `single contributing electron', as in the $H~^3\\Delta_1$ state of ThO, we rewrite this in the form\n\\begin{equation}\nH_{\\rm SP}^{\\rm eff} = \\vec{S}\\cdot\\hat{n} C_{\\rm S} Y_{\\rm S}.\n\\end{equation}\nHence, the energy shift due to this interaction can be written as\n\\begin{equation}\n\\Delta E_{\\rm SP} = \\langle \\vec{S}\\cdot\\hat{n} \\rangle C_{\\rm S} Y_{\\rm S} = Y_{\\rm S} \\left[ \\langle \\Sigma \\rangle \/ \\Omega \\right] \\Omega,\n\\end{equation}\nwhere the term in square brackets is a constant of the molecular state, determined by the fixed relative size and orientation of $\\vec{\\Sigma}$ and $\\vec{\\Omega}$, with value $\\approx -1$ in the $H~^3\\Delta_1$ state of ThO. In the literature on molecular eEDM systems, this energy shift is typically written in the simpler form\n\\begin{equation}\n\\Delta E_{\\rm SP} = C_{\\rm S} W_{\\rm S}\\Omega.\n\\end{equation}\nHere, in our notation, $W_{\\rm S}\\equiv Y_{\\rm S}\\left[\\langle\\Sigma\\rangle\/\\Omega\\right]$ ($\\approx -Y_{\\rm S}$ in ThO).\nHowever, quantities analogous to $Y_{\\rm S}$ (in terms of which the energy shifts depend explicitly on the spin direction) are rarely introduced in the literature; instead, only forms analogous to $W_{\\rm S}$ (where the energies depend only on $\\Omega$) are used.\n\n\nOur definition for $C_{\\rm S}$ was historically a standard notation used in the literature. However, in some recent papers (e.g. references \\cite{Skripnikov2015,Denis2016}) it is implicitly assumed that the neutron coupling $C_{\\rm S,n}$ vanishes.  In these papers, the factor $AC_{\\rm S}$ in equation \\ref{eq:HPSapp1} is replaced by $ZC_{\\rm S,p}$ (or its equivalent in a different notation),\\footnote{In reference \\cite{Skripnikov2015} our $C_{\\rm S,p}$ is denoted as $k_{\\rm T,P}$ and our $W_{\\rm S,p}$ as $W_{\\rm T,P}$; in Ref.\\ \\cite{Denis2016}, our $W_{\\rm S,p}$ is denoted simply as $W_{\\rm S}$. References \\cite{Dzuba2011,DzubaErratum2012} denote our $C_{\\rm S}$ as $C^{\\rm SP}$ and our $W_{\\rm S}$ as $W_{\\rm c}$.}\nand the energy shift is written in the analogous form \n$\\Delta E_{\\rm SP} = C_{\\rm S,p} W_{\\rm S,p}\\Omega$.  \nThese papers report values of $W_{\\rm S,p}$ in the $H$ state of ThO, based on sophisticated calculations of the molecular wavefunctions.  However, since there is no particular reason to expect this interaction to couple more strongly to protons than to neutrons, we prefer to report our results in terms of $C_{\\rm S}$. To do so, we use the relation $W_S = (A\/Z)W_{\\rm S,p}$ to determine $W_S$ from the reported values for $W_{\\rm S,p}$.  \n\nFinally, the experimentally determined energy shift arising from the nucleon-electron SP interaction is \n\\begin{equation}\n\\omega^{\\N\\E}_{\\rm SP} = C_{\\rm S} W_{\\rm S},\n\\label{eq:HPSapp3}\n\\end{equation}\nand the total T-violating energy shift is\n\\begin{equation}\n\\omega^{\\N\\E}_{\\rm T}= -d_e\\E_{\\rm eff} + C_{\\rm S} W_{\\rm S} = d_e W_d + C_{\\rm S} W_{\\rm S}.\n\\label{eq:HSP}\n\\end{equation}\nNote that the sign of the $C_{\\rm S}$ term is opposite to that used, incorrectly, in our original paper \\cite{Baron2014}.\n\n\\subsection{Relation to other notations in the literature}\n\nTable~\\ref{tab:sign_convs} shows some of the conventions used in the literature to describe the $T$-violating electron-nucleon interaction in molecular systems, and how they relate to our conventions. We note in particular three key differences between the (shared) conventions of references \\cite{Skripnikov2015,Denis2016}---which currently provide the most accurate values for $W_d$ and $W_S$---and ours.  First: these references define $\\hat{n}$ in the direction opposite to $\\vec{D}$, and hence opposite to ours.  This in turn means that their definition of $\\Omega$ has opposite sign to ours.  Hence, the same physical energy shifts (defined as $\\Delta E_{\\rm EDM} = W_d\\Omega$ both there and here) are obtained only if we take $W_d$ to have sign opposite to that of the reported $W_d$ in these papers. Second: these references define the eEDM energy shift as $\\Delta E_{\\rm EDM} = +d_e\\Eeff \\Omega$, while we have shown that in our notation $\\Delta E_{\\rm EDM} = -d_e\\Eeff \\Omega$.  Here there are two sign differences (one from the overall sign, one from the definition of $\\Omega$).  Hence, the same physical energy shifts are obtained when taking $\\Eeff$ to have the same sign as reported in these papers.  Third: these references formulate the scalar-pseudoscalar nucleon-electron interaction in terms of a quantity equivalent to our $W_{\\rm S,p}$ rather than our $W_{\\rm S}$.  Hence we must rescale these values as described above, using $W_S = (A\/Z)W_{\\rm S,p}$. In addition, the same physical energy shifts $\\Delta E_{\\rm SP} = W_S\\Omega$ are obtained only if we take $W_{\\rm S,p}$ to have sign opposite to that of the reported $W_{\\rm S,p}$ in these papers.\n\n\\begin{center}\n\\begin{threeparttable}\n\\centering\n\\begin{tabular}{C{2.9cm}C{1.1cm}C{0.6cm}C{4cm}C{4cm}}\n\\hline \n& $\\hat{n}$ & $\\vec{\\mathcal{E}}_{\\rm eff}$ & $\\Delta E_{\\rm EDM}$ & $\\Delta E_{\\rm SP}$ \\tabularnewline\n\\hline\nACME & \\includegraphics[width=10pt]{nup.pdf} & \\includegraphics[width=10pt]{adown.pdf} & $d_e W_d\\Omega=-d_e \\E_{\\rm eff} \\Omega = -\\vec{d}_e\\cdot\\vec{\\mathcal{E}}_{\\rm eff}$ & $C_{\\rm S}W_{\\rm S}\\Omega$ \\tabularnewline\nLee et al. \\cite{Lee2009} & \\includegraphics[width=10pt]{nup.pdf} & \\includegraphics[width=10pt]{adown.pdf} & $-\\vec{d}_e\\cdot\\vec{\\mathcal{E}}_{\\rm eff}$\\tnote{a} & \\tabularnewline\nYbF \\cite{Kara2012} & \\includegraphics[width=10pt]{nup.pdf} & \\includegraphics[width=10pt]{adown.pdf} & $-\\vec{d}_e\\cdot\\vec{\\mathcal{E}}_{\\rm eff}$\\tnote{b} & \\tabularnewline\nKozlov et al. \\cite{Kozlov1995,Kozlov2002} & \\includegraphics[width=10pt]{ndown.pdf} & & $+W_dd_e\\Omega$\\tnote{c} & \\tabularnewline\nSkripnikov et al. \\cite{Skripnikov2013,Skripnikov2015,Skripnikov2016} & \\includegraphics[width=10pt]{ndown.pdf} & & $+W_dd_e\\Omega=+d_e\\Eeff{\\rm sgn}(\\Omega)$\\tnote{d} & $+W_{T,P}k_{T,P}\\Omega$\\tnote{e}, where $k_{T,P}=AC_{\\rm S}\/Z$\\tnote{f} \\tabularnewline\nFleig et al. \\cite{Fleig2014,Fleig2013,Denis2016} & \\includegraphics[width=17.2pt]{bndown.pdf} & & $+W_dd_e\\Omega=+d_e\\Eeff[{\\rm sgn}(\\Omega)]$\\tnote{g} & $+W_{P,T}k_S\\Omega$\\tnote{h}, where $k_S=AC_S\/Z$ \\tabularnewline\nDzuba et al. \\cite{Dzuba2011a,DzubaErratum2012} & \\includegraphics[width=17.2pt]{bndown.pdf} & & $+W_dd_e[{\\rm sgn}(\\Omega)]=-d_e\\Eeff[{\\rm sgn}(\\Omega)]$\\tnote{i} & $+W_cC^{\\rm SP}[{\\rm sgn}(\\Omega)]$\\tnote{j} \\tabularnewline\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n    \\item[a] Reference \\cite{Lee2009}, p.\\ 2007\n    \\item[b] Reference \\cite{Kara2012}, p.\\ 3\n    \\item[c] Reference \\cite{Kozlov1995}, above equation 6.27\n    \\item[d] Reference \\cite{Skripnikov2015}, equation 1 and following\n    \\item[e] Reference \\cite{Skripnikov2015}, equation 4\n    \\item[f] Reference \\cite{Skripnikov2015}, equation 4 and \\cite{Dzuba2011a}, equation 25 and following\n    \\item[g] Reference \\cite{Fleig2014}, equation 1 and Reference \\cite{Fleig2013} equations 2--4\n    \\item[h] Reference \\cite{Denis2015}, equations 3 and 4\n    \\item[i] Reference \\cite{Dzuba2011a}, equation 24 and table IV\n    \\item[j] Reference \\cite{Dzuba2011a}, equation 25\n\\end{tablenotes}\n\\par\n\\protect\\caption{Summary of the different conventions used in some of the literature relating to eEDM measurements\/theory. Where entries are left blank the convention is not stated in the reference provided. Quantities in square brackets are not explicitly stated in the references but are implied. In some cases, nomenclature has been modified for consistency. Footnotes provide specific references for the equations shown.}\n\n\\label{tab:sign_convs}\n\\end{threeparttable}\n\\end{center}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nGraph Learning (GL), also referred to as Network Topology Inference or Graph Structure Learning, refers to the task of inferring graphs from data. Here we focus on learning undirected graphs in the supervised setting. GL has deep roots in statistics ~\\citep{dempster1972covariance} with significant contributions for probabilistic graphical model selection; see e.g.~\\citep{kolaczyk2009book,GLasso2008,drton2017structure}. Approaches dubbed `latent graph learning' have been used to learn interactions among coupled dynamical systems~\\citep{kipf2018icml}, or to obtain better task-driven representations of relational data for machine learning (ML) applications~\\citep{wang2019dynamicgraphcnn,kazi2020DGM,velickovic2020pgn}; for the related task of link-prediction see \\citep{hamilton2021book} .\nGraph signal processing (GSP) has been the source of recent advances using cardinal properties of network data such as smoothness~\\citep{dong2016learning,kalofolias2016learn} and graph stationarity~\\citep{segarra2017tsipn,pasdeloup2018tsipn}, exploiting models of network diffusion~\\citep{wasserman2022GDN, daneshmand2014estimating}, or taking a signal representation approach ~\\citep{dong2019learning,mateos2019spmag}. These works, referred to as Model-Based (MB) graph learning, use such data models to formulate the topology inference task as a (convex) optimization problem to be solved for each problem instance. When datasets are available, one can build on these MB approaches by unrolling an iterative solution procedure to produce a learned architecture optimized on the given data~\\citep{monga2021spmag}. Such Unrolling-Based (UB) methods offer several advantages over MB methods: they tend to be more expressive (MB methods are restricted to a small set of constraints expressible in a convex manner), require fewer layers than the corresponding iterative algorithm for a given performance requirement, and allow one to directly optimize for a (differentiable) metric of interest~\\citep{dong2019learning,wasserman2022GDN, shrivastava2019glad}. Thus by learning a distribution over such graphs by investing time upfront in training a model we can obtain better task-specific performance while avoiding expensively resolving each new problem instance.\n\n\nWhile the interdisciplinary history of GL has been a source of progress, it has also been a hindrance: these different fields use different tools (MATLAB\/Octave, R, Python, etc) in implementing their algorithms. This has slowed the spread of new algorithms and made comparing relative performance of different algorithms difficult. Importantly, there has been little effort to ensure implementations can leverage the increasing computation power of the GPU. Due to the reliance of GL algorithms on matrix\/vector operations, as well as the interest in handling ever increasing network sizes, GPUs offer an ideal environment for such computation to occur. \nThus the first aim of pyGSL is to provide standardized GPU-compatible implementations of fundamental GL methods, along with datasets to evaluate them on. \nThe second aim is lowering the barrier to use and develop such GL methods. As such, we provide a framework which makes it easy for researchers to extend - or build completely new - graph learning methods without needing to rebuild all the underlying software machinery. By consolidating wide-ranging GL methods into in a single GPU-compatible framework, pyGSL can be seen as the most  comprehensive Python library for GL to date.\n\n\n\\section{Architecture}\n\\label{sec:architecture}\n\nBoth MB and UB methods rely heavily on matrix\/vector operations, often require hyperparameter search capabilities, and are typically applied to problems with large graph sizes and\/or large sets of graphs; see Section \\ref{sec:gl_methods} for further details on said GL approaches. For these reasons, pyGSL implements all methods in PyTorch which has a rich ecosystem of software to interface with GPUs, run and visualize hyperparameter searches via W\\&B ~\\citep{wandb}, and abstract away the bug-prone training, validation and optimization logic required in gradient based training via PyTorch-Lightning ~\\citep{Falcon_PyTorch_Lightning_2019}.\n\nImplementations of UB methods share significant structure, motivating our novel \\texttt{UnrollingBase} class, which encapsulates this shared functionality; see Fig. \\ref{fig:pyGSL}. This minimizes repeated code, and makes implementing new UB methods as simple as defining the core layer-wise logic for that specific unrolling. By doing so, users can (i) cut their development time down significantly by relying on pretested methods; and (ii) automatically gain functionality such as intermediate output visualization and metric logging.\nSee Appendix \\ref{app:python_example} for a code snippet showing how pyGSL can be used.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=.3]{figures\/pyGSL_UML_model_horizontal.png}\n  \\caption{Class diagram of pyGSL's learned unrollings with a subset of variables and methods shown. \n  For a specific learned unrolling, say GDN, we would replace the LearnedUnrolling class with a GDN class which simply overwrites the setup(), shared\\textunderscore step(), and configure\\textunderscore optimizers() methods from its parent class pl.LightningModule. Since the GDN class also mixes in \\texttt{UnrollingBase}, it automatically inherits functionality common to all UB methods.} \n  \\label{fig:pyGSL}\n \n\\end{figure}\n\n\n\\section{A Unifying View of Graph Learning Methods}\n\\label{sec:gl_methods}\n\n\nThe GL problem is to infer a graph from data to uncover a latent complex information structure. Here, we focus on undirected and weighted graphs $\\mathcal{G}(\\mathcal{V},\\mathcal{E})$, where $\\mathcal{V}=\\{1,\\ldots, N\\}$ is the set of nodes (henceforth common to all graphs), and $\\mathcal{E}\\subseteq \\mathcal{V}\\times \\mathcal{V}$ collects the edges. A graph signal ${\\bm{x}} = [x_1, \\ldots, x_N] \\in \\mathbb{R}^N$ is a map ${\\bm{x}}: \\mathcal{V} \\rightarrow \\mathbb{R}$ which assigns a real value (say, a feature) to each vertex. We collect the $P$ graph signal observations together into data matrix ${\\bm{X}} = [{\\bm{x}}^{(1)}, \\cdots, {\\bm{x}}^{(P)}]$. A similarity function $S({\\bm{X}}): \\mathbb{R}^{N \\times P} \\mapsto \\mathbb{R}^{N \\times N}$ is chosen to compute the observed direct similarity between nodes. Common choices for $S$ include sample covariance\/correlation or Euclidean distance. \nWe can conceptualize the output of $S({\\bm{X}})$ as the symmetric adjacency matrix of an observed graph \nfrom which we would like to recover a latent graph with symmetric adjacency matrix denoted ${\\bm{A}}_L \\in \\mathbb{R}_+^{N\\times N}$. Below we discuss three popular approaches to tackle the GL task which exemplify the link between $S({\\bm{X}})$ and ${\\bm{A}}_L$; for discussions on Deep Learning (DL) methods for GL and tradeoffs between the three methods refer to \\ref{app:DL} and \\ref{app:tradeoffs}, respectively.\n\n\n\\begin{table}[t]\n\\caption{Model-Based methods with the associated Unrolling-Based method it inspires.}\n\\label{table:model_based_methods}\n\\begin{center}\n\\begin{tabular}{lll}\n\\multicolumn{1}{c}{\\bf Data Model}\n&\\multicolumn{1}{c}{\\bf MB Iterative Solution Procedure}\n&\\multicolumn{1}{c}{\\bf UB Model}\n\\\\ \\hline \\\\\nGaussian \n& Alternating Minimization\n& GLAD\\\\\nSmoothness       \n& Primal-Dual Splitting\n& L2G\\\\\nDiffusion \n& Proximal Gradient Descent\n& GDN\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\textbf{Model-Based Graph Learning.}\nThese GL approaches postulate some data model relating the observed data ${\\bm{X}}$ to the latent graph $\\mathcal{G}_L$ via ${\\bm{X}} \\sim \\mathcal{F}({\\bm{A}}_L)$.\nThis model can be the result of a network process, e.g. linear network diffusion, or statistical where ${\\bm{X}}$ follows a distribution determined by ${\\bm{A}}_L$\ne.g. in probabilistic graphical models. Thus the GL task reduces to an attempt to invert this relation $\\mathcal{F}^{-1}({\\bm{X}})$.\nTo do so, these techniques formulate a (convex) optimization problem which can be solved via iterative optimization methods.\nThese MB GL problems have the general form\n\\begin{align}\n\\label{prob:model-based-graph-learning}\n{\\bm{A}}^{*} \\in {}& \\underset{ {\\bm{A}} \\in \\mathcal{C} }  {\\text{ argmin}} \\: \\{\\mathcal{L}_{\\text{data}}({\\bm{A}},{\\bm{X}}) + \n\\mathcal{L}_{\\text{reg}}({\\bm{A}})\\},\n\\end{align}\nwhere $\\mathcal{L}_{\\text{data}}({\\bm{A}},{\\bm{X}})$ is the data fidelity term, $\\mathcal{L}_{\\text{reg}}({\\bm{A}})$ is the regularization term incorporating the structural priors (e.g., $\\norm{{\\bm{A}}}{1}$ for sparsity), and $\\mathcal{C}$ encodes a convex constraint on the optimization variable ${\\bm{A}}$, e.g., symmetry, non-negativity, or hollow diagonal. We inject our assumptions about the generative model into the objective of the problem. Note that we parameterize the canonical problem form with an adjacency ${\\bm{A}}$ for notational convenience, but many methods use other graph shift operators such as the Laplacian ${\\bm{L}} := \\text{diag}({\\bm{A}} \\mathbf{1}) - {\\bm{A}}$ or its normalized counterparts.\n\nTo make this clear, consider the following data models and resulting optimization formulations for graph recovery, summarized in Table \\ref{table:model_based_methods}. When the data ${\\bm{X}}$ are assumed to be Gaussian, the GL task is to estimate the precision matrix; the sparsity-regularized log-likelihood function for such a precision matrix is convex \n$\\arg\\min_{\\mathbf{A}\\succeq\\mathbf{0}}\\left\\{{\\textrm{Tr}(S({\\bm{X}}) \\mathbf{A})} {+\\alpha\\|\\mathbf{A}\\|_1 - \\beta\\log\\det\\mathbf{A}} \\right\\}$, where $S({\\bm{X}})$ is the sample covariance matrix~\\citep{GLasso2008}.\nWhen the data is assumed to be smooth on $\\mathcal{G}_L$, i.e., the total variation Tr$({\\bm{X}}^\\top {\\bm{L}} {\\bm{X}})$ of the signals on the graph is small, a standard GL formulation is $\\arg\\min_{\\mathbf{A} \\in \\mathcal{C}} \\left\\{ {\\norm{{\\bm{A}} \\odot S({\\bm{X}})}{1}} {- \\alpha \\mathbf{1}^\\top \\text{log}({\\bm{A}} \\mathbf{1}) + \\beta \\norm{{\\bm{A}}}{F}^2} \\right\\}$, where $S({\\bm{X}})$ is the Euclidean distance matrix~\\citep{kalofolias2016learn}. \nWhen the observed graph $S({\\bm{X}})$ satisfies a graph convolutional relationship with ${\\bm{A}}_L$, i.e. \n$S({\\bm{X}}) = \\sum_{i=0}^{K} \\alpha_i {\\bm{A}}_L^i$, as is the case in linear network diffusion where again $S({\\bm{X}})$ is the sample covariance matrix, we can pose the GL task as the non-convex problem\n$\\arg\\min_{\\mathbf{A} \\in \\mathcal{C}} \\left\\{ {\\norm{S({\\bm{X}})-\\sum_{i=0}^{K} \\alpha_i {\\bm{A}}_L^i}{F}^2} {+ \\beta\\norm{{\\bm{A}}}{1}} \\right\\}$~\\citep{wasserman2022GDN}.\n\nThe MB methods that admit iterative solutions take the generic form: ${\\bm{A}}[i+1] = h_{\\mathbf{\\theta}}({\\bm{A}}[i], S({\\bm{X}}))$ where ${\\bm{A}}[i]$ is output on the $i$-th iteration, $h_{\\mathbf{\\theta}}$ is the contractive function, and $\\mathbf{\\theta}$ are the regularization parameters. We implement the function $h_{\\mathbf{\\theta}}$ using GPU compatible operations in PyTorch and wrap it in a loop until uniform convergence is achieved. \n\n\n\\textbf{Unrolling-Based Graph Learning.}\nAlgorithm unrolling uses iterative algorithms, like those often used in signal processing, as an inductive bias in the design of the neural network architectures. The algorithm is unrolled into a deep network by associating each layer to a single iteration in the truncated algorithm, producing a finite number of stacked layers as shown in Figure \\ref{fig:generic_UB}. We transform the regularization parameters of the iterative algorithms into learnable parameters of the neural network. Using a dataset, such parameters can be optimized for a given task by choosing an appropriate loss function and backproprogating gradients; see ~\\citep{monga2021spmag} for more details.\n\nAll UB models share a significant amount of code in their construction, training, and evaluation. We consolidate this code into the novel \\texttt{UnrollingBase} class. To instantiate a new UB model, one simply inherits from \\texttt{UnrollingBase} and Pytorch-Lightning's \\texttt{LightningModule}, declares a layers' learnable parameters $\\theta^{i}$, and implements the $h_{\\mathbf{\\theta}^i}({\\bm{A}}[i], S({\\bm{X}}))$ \nfunction - denoted as \\texttt{shared\\_step} in the users class. \nRecall that each layer of the resulting unrolled network performs the same operation represented by $h_{\\mathbf{\\theta}}({\\bm{A}}[i], S({\\bm{X}}))$ here; thus specifiying the entire unrolled network reduces to providing a desired depth and specifying $h$ for a single layer.\nThe \\texttt{UnrollingBase} mixin takes care of stacking the layers together, feeding outputs of one layer as inputs of the next, and implements the required interface to allow the \\texttt{LightningModule} to automate the training and optimization.\n\n\\begin{comment}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.4in]{figures\/gdn_intermediate_outputs_adjust.png}\n\\end{center}\n\\caption{{Intermediate outputs of an Unrolling Method on a \\texttt{gl.data.diffuse} dataset.\nThis is an included plotting utility for all Unrolling-Based methods which use the \\texttt{UnrollingBase} mixin.}}\n\\end{figure}\n\\end{comment}\n\n\n\\begin{figure}[t]\n\\centering\n  \\begin{minipage}[b]{0.49\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/generic_unrolling_alter_SX.png}\n    \\caption{{Schematic of a generic UB method.}\n    \\label{fig:generic_UB}\n  \\end{minipage}\n \n  \\begin{minipage}[b]{0.49\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/gdn_intermediate_outputs_alter_SX.png}\n    \\caption{{Intermediate outputs plotted via pyGSL.}\n    \\label{fig:iterative_optim}\n  \\end{minipage}\n\\end{figure}\n\n\n\\section{Data}\n\\label{sec:data}\n\\textbf{Synthetics}\nAs discussed in Section \\ref{sec:gl_methods}, MB methods motivate the optimization problem through an assumed generative model on the observed data. Since UB methods are inspired by their MB counterparts, \nsynthetic datasets which match these assumed generative models are important for model validation across GL methods.\nWe thus provide (i) an efficient graph sampling interface using NetworkX ~\\citep{networkx} for a broad range of random graph ensembles;\nand (ii) an ability to generate signals conforming to \nsmoothness, \nlinear network diffusion (${\\bm{x}} = \\sum_{i=0}^p \\alpha_i A_L^{i}{\\bm{w}}$, where ${\\bm{w}}$ is typically white), or \nGaussianity (${\\bm{x}} \\sim \\mathcal{N}(\\mathbf{0}, {\\bm{A}}^{-1}$)). These datasets can be accessed via the \\texttt{smooth}, \\texttt{diffuse}, and \\texttt{gaussian} classes, respectively, in the \\texttt{gl.data} subdirectory.\n\n\n\\textbf{Real Data} We provide real datasets to evaluate the models on, including over $1000$ Structural Connectivity (SC) and Functional Connectivity (FC) pairs extracted from the HCP-YA neuroimaging dataset ~\\citep{glasser2016human}, as well as co-location and social network data\nfrom the Thiers13 dataset ~\\citep{Genois2018}. We also provide the ability to create `Pseudo-synthetic' data by using the graphs from the real datasets, and sampling synthetic signals on top of them that are e.g., smooth, diffused, or Gaussian. This provides a gentler transition to real data, which we found useful when developing and validating GL methods.\n\nSee Appendix \\ref{app:data} for a further discussion on the data and its associated class layout in pyGSL.\n\n\\begin{comment}\n\\section{Visualization, Logging, and Project Management}\nIn mixin already logging all metrics and loss values to WandB. Can be easily changed to any logger of choice due to the high level implimentation in pytorch lightning.\n\nShow unrolling image from GDN paper.\nShow WandB plots.\nEasy installation: Repository can be downloaded from GitHub link. Yaml files are included for simple conda environment setup to make project more agnostic to platform.\nCode quality: The code is includes unit-tests for fundamental shared functions, e.g. computing metrics, sampling graphs, etc, using pytest and hypothesis and, as of v0.3.1, test coverage is at 98. \n\\end{comment}\n\n\n\\section{Related Work}\nWhile there has been an explosion of software development in adjacent fields such as geometric DL~\\citep{pytorch_geom} and network-based modeling of complex systems~\\citep{deepgraph}, there has been little in the way of comprehensive GL packages.\n\\href{https:\/\/epfl-lts2.github.io\/gspbox-html\/doc\/learn_graph\/}{GSPBox} is (unmaintained) Matlab toolkit which focuses on traditional GSP tasks and offers a single function to perform GL on the smooth signal case ~\\citep{perraudin2016gspbox}.\nThere are many algorithms for Gaussian graphical model selection from a likelihood based ~\\citep{Yuan2007biometrika, GLasso2008}, regression based ~\\citep{meinshausen2006high, peng2009partial}, and constrained \n$\\ell 1$-minimization\n~\\citep{cai2011constrained} approaches; almost all implementations are in R or Matlab. Recently~\\citet{choi2021efficient} released an R-based GPU implementation of CONCORD-PCD, a regression based approach which uses parallelized coordinate descent for fast inference. Software in~\\citep{lassance2020mlsp} offers benchmarks for MB approaches.\nTo the best of the authors knowledge, no encompassing framework exists for UB methods, only standalone implementations released by the respective methods' authors.\n\n\n\n\\section{Concluding remarks}\nWe introduced the pyGSL framework for GPU-aware fast and scalable GL. We provide several synthetic and real datasets for model evaluation, as well as a novel class for UB architectures which dramatically lowers the time and expertise required to use and build such methods. It also ensures their compatibility as sub-modules in larger gradient-based learning systems. This is an active project, and we plan to add more benchmark datasets and the latest GL methods as they are published. We welcome more researchers and engineers to join, develop, maintain, and improve this toolkit to push forward the research and deployment of network topology inference algorithms.\n\n\n\n\n\\bibliographystyle{unsrtnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1} \\setcounter{equation}{0}\n\\setcounter{section}{1}\n\n\n\n\n\nA result of Hawking \\cite{Hawking} shows that a cross section of any connected component of the event horizon in a $4$-dimensional asymptotically flat stationary spacetime satisfying the dominated energy condition, has positive Euler characteristic, and hence must be topologically a $2$-sphere. The conclusion also holds without the stationarity condition provided one replaces a cross section of the event horizon with a stable apparent horizon. These results were generalized by Galloway and Schoen \\cite{GallowaySchoen} to show that a cross section of any connected component of the event horizon in an $n$-dimensional asymptotically flat stationary spacetime is an $(n-2)$-dimensional Riemannian manifold with positive Yamabe invariant. In dimension $5$ the additional hypothesis of bi-axial symmetry restricts the possible topologies further, so that the only admissible topologies are $S^3$, $S^1\\times S^2$, and $L(p,q)$ \\cite{HollandsYazadjiev}. Explicit examples of stationary vacuum bi-axisymmetric solutions\nwith horizon topology $S^3$ and $S^1\\times S^2$ have been constructed by Myers-Perry (sphere) \\cite{MyersPerry},\nEmparan-Reall (singly spinning ring) \\cite{EmparanReall}, and Pomeransky-Sen'kov (doubly spinning ring) \\cite{PomeranskySenkov}.\nIn particular, stationary vacuum black holes are not determined solely by their mass and angular momenta in higher dimensions. That is, the no-hair conjecture fails, as\nthere exist black ring solutions having the same\nmass and angular momenta as a Myers-Perry black hole.\nNonetheless the underlying result supporting the validity of the no-hair theorem in 4-dimensions, a uniqueness theorem for harmonic maps with prescribed singularities into a nonpositively curved target, still holds in higher dimensions. In particular any bi-axially symmetric stationary vacuum solution is determined by a finite set of parameters. It is the purpose of this paper to establish a partial converse: given any admissible set of parameters, there is a unique solution of the reduced equations. Whether this solution of the reduced equations then generates a physical spacetime solution then depends on the absence of conical singularities on the axes.\n\nThe axes correspond to the locus where a closed-orbit Killing field degenerates, and in the domain $\\mathbb{R}^3$ of the harmonic map these are identified by a number of intervals on the $z$-axis called \\emph{axis rods}. The axis rods are separated by intervals corresponding to horizons, and by points which are referred to as \\emph{corners}. Note that this precludes the case of degenerate horizons, in which horizons are represented by points instead of intervals. In addition, the end points of the horizon rods are named \\emph{poles}. Denote by $\\Gamma$ the $z$-axis with the interior of all the horizon rods removed, and let $\\{p_l\\}$ represent the corners and poles. Note that there are always two semi-infinite axes, labeled north and south. We assign a pair of relatively prime integers $(m_l,n_l)$ called the \\emph{rod structure} to each axis rod $\\Gamma_l$, such that the north and south semi-infinite axes are assigned the rod structures $(1,0)$ and $(0,1)$, respectively. This pair of numbers indicates which linear combination of rotational Killing fields vanishes on the associated rod.\nIf $(m_l,n_l)$ and $(m_{l+1},n_{l+1})$ are the rod structures assigned to two consecutive axis rods separated by a corner, then the \\emph{admissibility condition} \\cite{HollandsYazadjiev} is\n\\begin{equation} \\label{admissibility0}\n\\operatorname{det}\\begin{pmatrix} m_l & n_l \\\\ m_{l+1} & n_{l+1} \\end{pmatrix} = \\pm 1.\n\\end{equation}\nThis condition is to prevent orbifold singularities at the corners \\cite{Evslin}.\nHorizon rods are assigned the rod structure $(0,0)$. Finally, assign to each axis rod $\\Gamma_l$ a constant $\\mathbf{c}_l\\in\\mathbb{R}^2$, the \\emph{potential constant}. The difference between the values of these constants on two axes adjoining a horizon rod is proportional to the angular momenta of this horizon component, as calculated by Komar integrals. A \\textit{rod data set} $\\mathcal{D}$ consists of the corners and poles $\\{p_l\\}$, the rod structures $\\{(m_l,n_l)\\}$, and the potential constants $\\{\\mathbf{c}_l\\}$ which are assumed not to vary between two consecutive rods separated by a corner. This data determines uniquely the prescribed singularities of the harmonic map $\\varphi\\colon\\mathbb{R}^3\\setminus\\Gamma\\to SL(3,\\mathbb{R})\/SO(3)$ as described more precisely in the Section \\ref{sec4}, and will be referred to as admissible if it satisfies \\eqref{admissibility0} at each corner. For technical reasons an additional \\textit{compatibility condition} will be imposed to aid the existence result. This condition only applies when two consecutive corners are present. Let $p_{l-1}$ and $p_{l}$ be two consecutive corners with axis rods $\\Gamma_{l-1}$ above $p_{l-1}$, $\\Gamma_{l}$ between $p_{l-1}$ and $p_{l}$, and $\\Gamma_{l+1}$ below $p_{l}$.\nThen the compatibility condition states that the first component of the rod structures for $\\Gamma_{l-1}$ and $\\Gamma_{l+1}$ have opposite sign if both are nonzero\n\\begin{equation}\\label{compatibilitycondition}\nm_{l-1}m_{l+1}\\leq 0,\n\\end{equation}\nwhenever the determinants \\eqref{admissibility0} for the two corners $p_{l-1}$ and $p_{l}$ are both $+1$. Note that this latter requirement on the determinants may always be achieved by multiplying each component of the rod structures for $\\Gamma_{l-1}$ and $\\Gamma_{l}$ by $-1$ if necessary; this is an operation which does not change the properties of a rod.\n\n\nIn order to determine the physical relevance of a solution,\ndefine on each bounded axis rod $\\Gamma_l$ a function\n$b_l$ to be the logarithm of the limiting ratio between the length of the closed orbit of the Killing field degenerating on $\\Gamma_l$, and $2\\pi$ times the radius from $\\Gamma_l$ to this orbit. It turns out that $b_l$ is constant on $\\Gamma_l$. The absence of a conical singularity on $\\Gamma_l$ is the \\emph{balancing condition} $b_l=0$.\n\nAn asymptotically flat stationary vacuum spacetime will be referred to as \\textit{well-behaved} if the\norbits of the stationary Killing field are complete, the\ndomain of outer communication (DOC) is globally hyperbolic, and the DOC\ncontains an acausal spacelike connected hypersurface which is asymptotic to the canonical slice in the asymptotic end and whose boundary\nis a compact cross section of the horizon. These assumptions are consistent with those of \\cite{ChruscielCosta}, and are used for the reduction of the stationary vacuum equations. The main result may now be stated as follows.\n\n\n\\begin{theorem} \\label{main}\\par\n\\noindent\n\\begin{enumerate}[(i)]\n\\item\nA well-behaved 5-dimensional asymptotically flat, stationary, bi-axially symmetric solution of the vacuum Einstein equations without degenerate horizons gives rise to a harmonic map $\\varphi\\colon\\mathbb{R}^3\\setminus\\Gamma\\to SL(3,\\mathbb{R})\/SO(3)$ with prescribed singularities associated with an admissible rod data set $\\mathcal{D}$, and satisfying $b_l=0$ on all bounded axis rods.\n\\item\nConversely,\ngiven an admissible rod data set $\\mathcal{D}$ satisfying the compatibility condition \\eqref{compatibilitycondition}, there is a unique harmonic map $\\varphi\\colon\\mathbb{R}^3\\setminus\\Gamma\\to SL(3,\\mathbb{R})\/SO(3)$ with prescribed singularities on $\\Gamma$ corresponding to $\\mathcal{D}$.\n\\item\nA well-behaved 5-dimensional asymptotically flat, stationary, bi-axially symmetric solution of the vacuum Einstein equations without degenerate horizons can be constructed from $\\varphi$ if and only if the resulting metric coefficients are sufficiently smooth across $\\Gamma$ and $b_l=0$ on any bounded axis rod.\n\\end{enumerate}\n\\end{theorem}\n\nThe reduction of the Einstein vacuum equations to a harmonic map is well known \\cites{Harmark,Maison} and follows closely the 4-dimensional case. However, there are several new difficulties associated with the analysis of the resulting problem. First, even without angular momenta the problem is nonlinear, in contrast to the linear structure present in the static 4D setting. This makes the construction of a \\emph{model map} prescribing the singular behavior near $\\Gamma$\nmuch more delicate, whereas in the 4D case the superposition of Schwarzschild solutions is sufficient. Next, the target $SL(3,\\mathbb{R})\/SO(3)$ is a rank 2 symmetric space with nonpositive sectional curvature, rather than rank 1 with negative sectional curvature in 4D.  We recall that the theory of harmonic maps into rank 1 symmetric spaces, in particular real hyperbolic space $\\mathbb{H}^n$, has been extensively investigated e.g. \\cite{Schoen, LiTam}, yet comparatively little is known for the cases of higher rank targets.  These properties of the target hyperbolic space $\\mathbb{H}^2= SL(2,\\mathbb{R})\/SO(2)$ in dimension four played a central role in obtaining a priori estimates to prove existence, and without these properties in the 5D case new techniques must be developed. Furthermore, in higher dimensions there is an abundance of possible rod structures, and they must obey an admissibility condition \\eqref{admissibility0} not present in four dimensions. Finally, the study of conical singularities and their formulation as the balancing condition $b_i=0$, while similar to the 4D case, requires a more precise analysis.\n\nSeveral explicit solutions of these equations and related ones have previously been found. As mentioned above, the Myers-Perry black hole \\cite{MyersPerry} generalizes the Kerr black hole to 5-dimensions, and is a 3-parameter family of solutions with spherical $S^{3}$ horizon topology. Emparan and Reall \\cite{EmparanReall} found the first example with nontrivial topology, namely a family of black ring solutions with an $S^1\\times S^2$ horizon and one angular momentum. These were later generalized by Pomeransky-Sen'kov \\cite{PomeranskySenkov} to a full 3-parameter family with two angular momenta. A multiple horizon solution with two components consisting of an $S^3$ surrounded by an $S^1\\times S^2$, referred to as black saturn, was constructed by Elvang and Figueras \\cite{ElvangFigueras}. In this solution both the sphere and ring rotate only in one plane which is associated with the $S^1$ direction of the ring.\nFurther multiple horizon solutions include the dipole black rings (or di-rings) \\cites{EvslinKrishnan,IguchiMishima} consisting of two concentric singly spinning rings rotating in the same plane, and the bicycling black rings (or bi-rings) \\cites{ElvangRodriguez,Izumi} consisting of two singly spinning rings rotating in orthogonal planes. In the minimal supergravity setting, Kunduri and Lucietti \\cite{LuciettiKunduri} found the first examples of regular black holes having a lens space topology $\\mathbb{RP}^3=L(2,1)$. These were generalized by Tomizawa and Nozawa to more general lens topology $L(p,1)$ in \\cite{TomizawaNozawa}. Both of these black lens solutions are supersymmetric and hence extremal. It is an important open problem to\nfind regular vacuum black holes with lens topology. In this direction Chen and Teo \\cite{ChenTeo} found vacuum black lenses via the inverse scattering method, however their solutions either possess conical singularities or have a naked singularity.\nA disadvantage of the methods used to construct the above examples\nis that they cannot produce all possible regular solutions. In contrast, the PDE approach used here generates all candidates with an admissible\/compatible rod structure, where the only obstruction is the possibility of conical singularities on the bounded components of the axes. Furthermore, the variety of black holes that may be constructed from admissible rod data which also satisfy the compatibility condition is vast. In particular, multiple and single component black lenses $L(p,q)$ are possible, for arbitrary relatively prime $p$ and $q$, as is shown in Proposition \\ref{lensrod} of Section \\ref{sec4}.\n\nThe existence portion of Theorem \\ref{main} may be generalized by forgoing the admissibility condition \\eqref{admissibility0}. This requires instead of \\eqref{compatibilitycondition} a \\textit{generalized compatibility condition}\n\\begin{equation}\\label{gcompatibilitycondition0}\nm_{l-1}m_{l+1} \\operatorname{det}\\begin{pmatrix} m_{l-1} & n_{l-1} \\\\ m_{l} & n_{l} \\end{pmatrix}\n\\operatorname{det}\\begin{pmatrix} m_l & n_l \\\\ m_{l+1} & n_{l+1} \\end{pmatrix} \\leq 0,\n\\end{equation}\nwhich is used in the construction of a model map. Note that if \\eqref{admissibility0} is satisfied then \\eqref{gcompatibilitycondition0} reduces to \\eqref{compatibilitycondition}. However, without the admissibility condition orbifold singularities at corner points will be present.\n\n\\begin{theorem}\\label{main2}\nGiven a rod data set $\\mathcal{D}$ satisfying the generalized compatibility condition \\eqref{gcompatibilitycondition0}, there is a unique harmonic map $\\varphi\\colon\\mathbb{R}^3\\setminus\\Gamma\\to SL(3,\\mathbb{R})\/SO(3)$ with prescribed singularities on $\\Gamma$ corresponding to $\\mathcal{D}$. From this map a well-behaved 5-dimensional asymptotically flat, stationary, bi-axially symmetric solution of the vacuum Einstein equations without degenerate horizons can be constructed having orbifold singularities at the corners if and only if the resulting metric coefficients are sufficiently smooth across $\\Gamma$ and $b_l=0$ on any bounded axis rod.\n\\end{theorem}\n\nThis result has been generalized in \\cite{KhuriWeinsteinYamada}  to include the asymptotically Kaluza-Klein and asymptotically locally Euclidean cases, in which cross sections at infinity are $S^1 \\times S^2$ and quotients of $S^3$ respectively.\n\nThe organization of this paper is as follows.\nIn Section \\ref{sec2} we review the reduction of the Einstein vacuum equations, in the bi-axially symmetric stationary setting, to a harmonic map having the symmetric space $SL(3,\\mathbb{R})\/SO(3)$ as target. Relevant aspects of the geometry of this symmetric space are then discussed in Section \\ref{sec3}.\nIn Section \\ref{sec4} a detailed analysis of rod structures and the hypotheses associated with them is given. The model map is constructed in Section \\ref{sec5}, and existence and uniqueness for the harmonic map problem is proven in Section \\ref{sec7} using energy estimates established in Section \\ref{sec6}. Finally in Section \\ref{sec8} it is shown how the desired spacetime is produced from the harmonic map, and regularity issues are discussed. An appendix is included in order to give a topological characterization of corners.\n\n\n\n\n\n\\medskip\n\n\\textbf{Acknowledgements.}\nThe authors thank the Erwin Schr\\\"odinger International Institute for Mathematics and Physics and the organizers of its ``Geometry and Relativity'' program, where portions of this paper were written. The third author also thanks Koichi Kaizuka for useful conversations concerning the geometry of symmetric spaces.\n\n\\section{Dimensional Reduction of the Vacuum Einstein Equations} \\label{setup}\n\\label{sec2} \\setcounter{equation}{0}\n\\setcounter{section}{2}\n\n\n\n\nLet $\\mathcal{M}^5$ be the domain of outer communication for a well-behaved asymptotically flat, stationary vacuum, bi-axisymmetric spacetime. In particular its isometry group admits\n$\\mathbb{R}\\times U(1)^2$ as a subgroup in which the $\\mathbb{R}$-generator $\\xi$ (time translation) is timelike in the asymptotic region, and the $U(1)^2$-generators $\\eta^{(i)}$, $i=1,2$ yield spatial rotation.\nSince the three generators for the isometry subgroup commute, they may be expressed as coordinate vector fields $\\xi=\\partial_{t}$ and $\\eta^{(i)}=\\partial_{\\phi^{i}}$.\nMoreover by abusing notation so that the same symbols denote dual covectors it holds that\n\\begin{equation}\\label{aaa}\n\\star\\left(\\xi\\wedge\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\xi\\right)\n=\\star\\left(\\xi\\wedge\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\eta^{(1)}\\right)\n=\\star\\left(\\xi\\wedge\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\eta^{(2)}\\right)\n=0,\n\\end{equation}\nwhere $\\star$ denotes the Hodge star operation. To see this, observe that the vacuum\nequations imply that the exterior derivative of the three quantities in \\eqref{aaa} vanishes, and since these functions vanish on the axis in the asymptotically flat end they must vanish everywhere. Therefore the Frobenius theorem applies to show that the 2-plane distribution orthogonal to the three Killing vectors is integrable. We may then take coordinates on one of these resulting 2-dimensional orbit manifolds, and Lie drag them to get a system of coordinates such that the spacetime metric decomposes in the following way\n\\begin{equation}\ng=\\sum_{a,b=1}^{3}q_{ab}(x)dy^{a}dy^{b}+\\sum_{c,d=4}^{5}h_{cd}(x)dx^{c}dx^{d},\n\\end{equation}\nwhere $y=(\\phi^1,\\phi^2,t)$. The fiber metric may be expressed by\n\\begin{equation}\\label{fibermetric}\nq=f_{ij}(d\\phi^{i}+v^{i}dt)(d\\phi^{j}+v^{j}dt)-f^{-1}\\rho^2 dt^{2},\n\\end{equation}\nfor some functions $v^i$ where $f=\\operatorname{det} f_{ij}$ and $\\rho^2=-\\operatorname{det} q_{ab}$. It is shown in \\cites{Chrusciel,ChruscielCosta} that the determinant of the fiber metric is nonpositive, and\nthe vacuum equations imply that $\\rho$ is harmonic with respect to the metric $fh$, since\n\\begin{equation}\n\\Delta_{fh}\\rho=\\rho^{-1}R_{tt}-\\rho f^{-1}f^{ij} R_{ij}=0.\n\\end{equation}\nFrom this it can be shown \\cites{Chrusciel,ChruscielCosta} that $\\rho$ is a well-defined coordinate function on the quotient\n$\\mathcal{M}^{5}\/\\left[\\mathbb{R}\\times U(1)^2\\right]$ away from the poles, that is $|\\nabla\\rho|\\neq 0$. Since the orbit space is simply connected \\cite{HollandsYazadjiev1} there is a globally defined harmonic conjugate function $z$, which together with $\\rho$ yields an isothermal coordinate system so that\n\\begin{equation}\nfh=e^{2\\sigma}(d\\rho^2 +dz^2),\n\\end{equation}\nfor some function $\\sigma=\\sigma(\\rho,z)$. We now have the canonical Weyl-Papapetrou expression for the spacetime metric\n\\begin{equation} \\label{metric}\ng=f^{-1}e^{2\\sigma}(d\\rho^2+dz^2)-f^{-1}\\rho^2 dt^2\n+f_{ij}(d\\phi^{i}+v^{i}dt)(d\\phi^{j}+v^{j}dt).\n\\end{equation}\n\nLet\n\\begin{equation}\ng_{3}=e^{2\\sigma}(d\\rho^2+dz^2)-\\rho^2 dt^2,\\quad\\quad\\quad\nA^{(i)}=v^{i}dt,\n\\end{equation}\nthen\n\\begin{equation}\ng=f^{-1}g_{3}+f_{ij}(d\\phi^{i}+A^{(i)})(d\\phi^{j}+A^{(j)}).\n\\end{equation}\nThis represents a Kaluza-Klein reduction with 2-torus fibers. In this setting the vacuum Einstein equations yield a 3-dimensional version of Einstein-Maxwell theory, with the `Maxwell equations' given by\n\\begin{equation}\\label{maxwell}\nd(ff_{ij}\\star_{3} dA^{(j)})=0,\n\\end{equation}\nwhere $\\star_{3}$ is the Hodge star operation with respect to $g_{3}$. It follows that there exist globally defined (due to simple connectivity) twist potentials satisfying\n\\begin{equation}\\label{chi}\nd\\omega_{i}=2f f_{ij}\\star_{3}dA^{(j)}.\n\\end{equation}\nIn particular if $v^i$ are constant then the potentials $\\omega_i$ are constant, and vice versa. To explain the geometric meaning of the forms appearing on the right-hand side of \\eqref{chi}\nobserve that $\\eta^{(i)}=f_{ij}\\left(d\\phi^j+v^j dt\\right)$ is the dual 1-form to $\\partial_{\\phi^{i}}$, and according to Frobenius' theorem\nthe forms $\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\eta^{(i)}$\nmeasure the lack of integrability of the orthogonal complement distribution to the axisymmetric Killing fields. Moreover, it turns out that these forms are directly related to \\eqref{chi}. Indeed let $\\epsilon$, $\\epsilon_{3}$, and $\\star_{3}$ denote the volume forms with respect to $g$ and $g_{3}$, and the Hodge star operator with respect to $g_{3}$, respectively, then since\n\\begin{equation}\nd\\eta^{(i)}=f_{ij}d A^{(j)}+df_{ij}\\wedge\\left(f^{ja}\\eta^{(a)}\\right)\n\\end{equation}\nwe have\n\\begin{align}\\label{komar}\n\\begin{split}\n\\star(\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\eta^{(i)})=&\nf_{ij}\\star(\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge dA^{(j)})\\\\\n=&f_{ij}\\epsilon(\\text{ }\\!\\cdot\\text{ }\\!,\\partial_{\\phi^1},\\partial_{\\phi^2},\\partial_{l},\\partial_{k})\n\\left(dA^{(j)}\\right)^{lk}\\\\\n=&f^{-1}f_{ij}\\epsilon_{3}(\\text{ }\\!\\cdot\\text{ }\\!,\\partial_{l},\\partial_{k})\\left(dA^{(j)}\\right)^{lk}\\\\\n=&ff_{ij}\\star_{3} dA^{(j)}.\n\\end{split}\n\\end{align}\nNote also that since the spacetime is vacuum and $\\eta^{(i)}$ are dual to Killing fields,\nstandard computations along with Cartan's `magic' formula show that the 1-forms\n\\begin{equation}\\label{komar1}\n\\star(\\eta^{(1)}\\wedge\\eta^{(2)}\\wedge d\\eta^{(i)})=\\iota_{\\eta^{(1)}}\n\\iota_{\\eta^{(2)}}\\star d\\eta^{(i)}\n\\end{equation}\nare closed, where $\\iota$ denotes interior product. This yields an alternate proof of \\eqref{maxwell}, and confirms that the twist potentials $\\omega_i$ agree with those associated with the Komar expression for angular momentum.\n\n\nNext, following Maison \\cite{Maison} define the following $3\\times 3$ matrix\n\\begin{equation}\\label{bigmatrix}\n\\Phi=\n\\left(\n    \\begin{array}{ccc}\n      f^{-1} & - f^{-1} \\omega_1 & - f^{-1} \\omega_2 \\\\\n      -f^{-1} \\omega_1 & f_{11} + f^{-1} \\omega_1^2  & f_{12} + f^{-1} \\omega_1 \\omega_2  \\\\\n      -f^{-1} \\omega_2 & f_{12} + f^{-1} \\omega_1 \\omega_2  & f_{22} + f^{-1} \\omega_2^2\n    \\end{array}\n  \\right)\n\\end{equation}\nwhich is symmetric, positive definite, and has $\\operatorname{det}\\Phi=1$.\nThe inverse matrix is\n\\begin{equation}\n\\Phi^{-1}=\n\\left(\n    \\begin{array}{ccc}\n      f + f^{11} \\omega_1^2 + f^{22} \\omega_2^2 + 2 f^{12} \\omega_1 \\omega_2 & f^{11} \\omega_1 + f^{21} \\omega_2  & f^{12} \\omega_1 + f^{22} \\omega_2 \\\\\n      f^{11} \\omega_1 + f^{12} \\omega_2 & f^{11}  & f^{12}   \\\\\n      f^{21} \\omega_1 + f^{22} \\omega_2   &f^{21} & f^{22}    \\end{array}\n  \\right)\n .\n\\end{equation}\nThis allows for a simplified expression of the 3-dimensional reduced Einstein-Hilbert action\n\\begin{equation}\\label{action}\n\\mathcal{S}=\\int_{\\mathbb{R}\\times \\left(\\mathcal{M}^{5}\/[\\mathbb{R}\\times U(1)^2]\\right)}\nR^{(3)}\\star_{3}1+\\frac{1}{4}\\mathrm{Tr}\\left(\\Phi^{-1}d\\Phi\\wedge\\star_{3}\\Phi^{-1}d\\Phi\\right).\n\\end{equation}\nThe Einstein-harmonic map system arising from this action is\n\\begin{equation}\\label{einstein}\nR^{(3)}_{kl}-\\frac{1}{2}R^{(3)}(g_3)_{kl}=T_{kl},\\quad\\quad \\operatorname{div}_{\\mathbb{R}^3}\\left(\\Phi^{-1}\n\\nabla\\Phi\\right)=0,\n\\end{equation}\nwhere the stress-energy tensor for the harmonic map is\n\\begin{equation}\nT_{kl}=\\mathrm{Tr}\\left(J_{k}J_{l}\\right)\n-\\frac{1}{2}g_{3}^{mn}\\mathrm{Tr}\\left(J_{m}J_{n}\\right)\n(g_{3})_{kl}\n\\end{equation}\nwith the current\n\\begin{equation}\nJ_{l}=\\Phi^{-1}\\partial_{l}\\Phi.\n\\end{equation}\nNote that by taking a trace the Einstein equations may be reexpressed as\n\\begin{equation}\nR^{(3)}_{kl}=\\mathrm{Tr}\\left(J_{k}J_{l}\\right).\n\\end{equation}\nFurthermore, in the $\\Phi$ portion of the action cancelations occur so that $e^{2\\sigma}$ does not appear, and this results in the divergence of \\eqref{einstein} with respect to\nthe Euclidean metric\n\\begin{equation}\\label{flatmetric}\n\\delta=d\\rho^2+dz^2+\\rho^2 d\\phi^2,\n\\end{equation}\nwhere $\\phi$ is an auxiliary coordinate.\nThis also implies that the stress-energy tensor is divergence free with respect to the Euclidean metric\n\\begin{equation}\\label{divergence}\n0=\\left(\\operatorname{div}_{\\mathbb{R}^3}T\\right)(\\partial_{\\rho})\n=\\partial_{\\rho}(\\rho T_{\\rho\\rho})+\\partial_{z}(\\rho T_{\\rho z}),\\quad\\quad\n0=\\left(\\operatorname{div}_{\\mathbb{R}^3}T\\right)(\\partial_{z})\n=\\partial_{\\rho}(\\rho T_{\\rho z})+\\partial_{z}(\\rho T_{z z}).\n\\end{equation}\n\nThe divergence free property of $T$ follows from the harmonic map equations. To see this in a more general harmonic map setting, consider maps $\\varphi: (M,\\mathrm{g})\\rightarrow (N,\\mathrm{h})$ with harmonic energy\n\\begin{equation}\nE=\\frac{1}{2}\\int_{M}|d\\varphi|^2 dx_{\\mathrm{g}}=\\frac{1}{2}\\int_{M}\\mathrm{g}^{\\mathrm{ij}}\n\\mathrm{h}_{\\mathrm{lk}}\\partial_{\\mathrm{i}}\\varphi^{\\mathrm{l}}\n\\partial_{\\mathrm{j}}\\varphi^{\\mathrm{k}} dx_{\\mathrm{g}}.\n\\end{equation}\nThe first variation is given by\n\\begin{equation}\n\\frac{\\delta E}{\\delta \\mathrm{g}}=\\frac{1}{2}\\int_{M}\\delta \\mathrm{g}^{\\mathrm{ij}}\\left(\\mathrm{h}_{\\mathrm{lk}}\n\\partial_{\\mathrm{i}}\\varphi^{\\mathrm{l}}\\partial_{\\mathrm{j}}\\varphi^{\\mathrm{k}}\n-\\frac{1}{2}|d\\varphi|^2 \\mathrm{g}_{\\mathrm{ij}}\\right) dx_{\\mathrm{g}},\n\\end{equation}\nand the stress-energy tensor is\n\\begin{equation}\nT_{\\mathrm{ij}}=\\langle\\partial_{\\mathrm{i}}\\varphi,\\partial_{\\mathrm{j}}\n\\varphi\\rangle_{\\mathrm{h}}\n-\\frac{1}{2}|d\\varphi|^2 \\mathrm{g}_{\\mathrm{ij}}.\n\\end{equation}\nThe harmonic map equations\n\\begin{equation}\\label{tensiondef}\n\\tau(\\varphi)=\\hat{\\nabla}^{\\mathrm{i}}\\partial_{\\mathrm{i}}\\varphi=0\n\\end{equation}\nthen imply that the stress-energy tensor is divergence free\n\\begin{align}\n\\begin{split}\n\\nabla^{\\mathrm{i}}T_{\\mathrm{ij}}=\\langle\\hat{\\nabla}^{\\mathrm{i}}\n\\partial_{\\mathrm{i}}\\varphi,\\partial_{\\mathrm{j}}\\varphi\\rangle_{\\mathrm{h}}\n+\\langle\\partial_{\\mathrm{i}}\\varphi,\\hat{\\nabla}^{\\mathrm{i}}\n\\partial_{\\mathrm{j}}\\varphi\\rangle_{\\mathrm{h}}\n-\\mathrm{g}^{\\mathrm{lm}}\\langle\\hat{\\nabla}_{\\mathrm{j}}\n\\partial_{\\mathrm{l}}\\varphi,\\partial_{\\mathrm{m}}\\varphi\\rangle_{\\mathrm{h}}=0.\n\\end{split}\n\\end{align}\nHere $\\hat{\\nabla}$ is the induced connection on the bundle $T^{*}M\\otimes \\varphi^{-1} TN$, and $\\tau(\\varphi)$ denotes the tension field which is a section of the pullback bundle $\\varphi^{-1}TN$.\n\n\nThe Einstein equations of \\eqref{einstein} may be solved via quadrature. This may be shown by computing each equation in terms of metric components. Recall that\n\\begin{equation}\nR_{kl}^{(3)}=\\partial_{m}\\Gamma_{kl}^{m}-\\partial_{l}\\Gamma_{km}^{m}\n+\\Gamma_{kl}^{m}\\Gamma_{nm}^{n}-\\Gamma_{kn}^{m}\\Gamma_{lm}^{n},\n\\end{equation}\nand\n\\begin{equation}\nR^{(3)}=g_3^{kl}R_{kl}^{(3)}\n=-\\rho^{-2}R_{tt}^{(3)}+e^{-2\\sigma}\\left(R_{\\rho\\rho}^{(3)}\n+R_{zz}^{(3)}\\right).\n\\end{equation}\nThe Christoffel symbols are\n\\begin{equation}\n\\Gamma_{tt}^{l}=\\delta^{l\\rho}e^{-2\\sigma}\\rho,\\quad\\quad\n\\Gamma_{ti}^{l}=\\delta_{t}^{l}\\delta_{i}^{\\rho}\\rho^{-1},\\quad\\quad\n\\Gamma_{ij}^{l}=\\delta_{j}^{l}\\partial_{i}\\sigma+\\delta_{i}^{l}\\partial_{j}\\sigma\n-\\delta_{ij}\\delta^{lm}\\partial_{m}\\sigma\\text{ }\\text{ }\\text{ for }\\text{ }\\text{ }i,j\\neq t.\n\\end{equation}\nIt follows that\n\\begin{equation}\nR_{tt}^{(3)}=R_{ti}^{(3)}=0, \\quad i\\neq t,\\quad\nR_{\\rho\\rho}^{(3)}=-\\Delta_{\\mathbb{R}^2}\\sigma+\\frac{1}{\\rho}\\partial_{\\rho}\\sigma,\\quad\nR_{zz}^{(3)}=-\\Delta_{\\mathbb{R}^2}\\sigma-\\frac{1}{\\rho}\\partial_{\\rho}\\sigma,\\quad\nR_{\\rho z}^{(3)}=\\frac{1}{\\rho}\\partial_{z}\\sigma.\n\\end{equation}\nFrom this the quadrature equations for $\\sigma$ are found to be\n\\begin{equation}\n\\partial_{\\rho}\\sigma=\\frac{\\rho}{2}\\left(R_{\\rho\\rho}^{(3)}-R_{zz}^{(3)}\\right)\n=\\frac{\\rho}{2}\\left(\\mathrm{Tr}(J_{\\rho}J_{\\rho})-\\mathrm{Tr}(J_{z}J_{z})\\right)\n=\\rho T_{\\rho\\rho}=-\\rho T_{zz},\n\\end{equation}\n\\begin{equation}\n\\partial_{z}\\sigma=\\rho R_{\\rho z}^{(3)}=\\rho T_{\\rho z},\n\\end{equation}\nwhich may be rewritten more conveniently as\n\\begin{equation}\\label{sigma}\nd\\sigma=-\\iota_{\\eta}\\ast\\iota_{\\partial_{z}}T\n\\end{equation}\nwhere $\\ast$ is the Hodge star operation with respect to the metric $\\delta$ on $\\mathbb{R}^3$, and $\\eta=\\partial_{\\phi}$. To see this let $\\varepsilon$ denote the volume form for $\\delta$, then\n\\begin{equation}\n(\\ast\\iota_{\\partial_{z}}T)_{ij}=\\varepsilon_{ijl}T^{lz}\n\\end{equation}\nand hence\n\\begin{equation}\n(\\iota_{\\eta}\\ast\\iota_{\\partial_{z}}T)_{j}=\\varepsilon_{ijl}\\eta^{i}T^{lz}\n=\\varepsilon(\\partial_{\\phi},\\partial_{j},\\partial_{\\rho})T^{\\rho z}\n+\\varepsilon(\\partial_{\\phi},\\partial_{j},\\partial_{z})T^{zz}.\n\\end{equation}\nWe then have\n\\begin{equation}\\label{333}\n\\iota_{\\eta}\\ast\\iota_{\\partial_{z}}T\n=\\rho T_{zz} d\\rho -\\rho T_{\\rho z} dz,\n\\end{equation}\nwhich confirms \\eqref{sigma}. Moreover, for later use observe that this form is closed in light of the harmonic map equations\n\\begin{equation}\nd\\left(\\iota_{\\eta}\\ast\\iota_{\\partial_{z}}T\\right)\n=-\\left(\\operatorname{div}_{\\mathbb{R}^3}T\\right)(\\partial_{z})d\\rho\\wedge dz=0.\n\\end{equation}\nNote that we also have to show that $\\sigma$ obtained from quadrature is bi-axisymmetric. However this follows easily from \\eqref{333}, since\n\\begin{equation}\n\\iota_{\\eta^{(i)}}d\\sigma=\\iota_{\\eta^{(i)}}\\iota_{\\eta}\\ast\\iota_{\\partial_{z}}T=0.\n\\end{equation}\n\n\n\n\n\n\n\\section{The Riemannian Geometry of  $SL(3,\\mathbb{R})\/SO(3)$}\n\\label{sec3} \\setcounter{equation}{0}\n\\setcounter{section}{3}\n\n\n\n\n\nThe harmonic map arising from the dimensional reduction of the bi-axisymmetric stationary vacuum Einstein equations has as target space $SL(3,\\mathbb{R})\/SO(3)$. The geometry of this symmetric space plays an important role in the analysis of the harmonic map, and in this section the relevant aspects will be described.\n\nLet $G=SL(3,\\mathbb{R})$ then $K=SO(3)$ is a maximal compact subgroup. The quotient $\\mathbf{X}=G\/K$ is the space of equivalence classes $[A]$ in which\n\\begin{equation}\nA \\in SL(3, {\\mathbb{R}}) \\mbox{ and  } A \\sim A' \\Leftrightarrow A' = A B  \\mbox{ for some } B \\in SO(3).\n\\end{equation}\nIn other words $\\mathbf{X}$ is the space of left cosets of $K$ in $G$ and $G$ acts transitively on $\\mathbf{X}$ by\n\\begin{equation}\nA'K\\mapsto AA' K\\quad\\text{ }\\text{ for }\\text{ }\\quad A\\in G,\n\\end{equation}\nso that $K$ is the isotropy subgroup at $x_0=[\\mathrm{Id}]$.\nRecall now the construction of the canonical $G$-invariant Riemannian metric on the homogeneous space $\\mathbf{X}$, which yields a Riemannian symmetric space structure.\nThe Lie algebras will be denoted by\n\\begin{equation}\n\\mathfrak{g} = sl(3) = \\{Y \\in gl (3) \\,\\, | \\,\\, \\mathrm{Tr} Y = 0\\},\n\\end{equation}\nand\n\\begin{equation}\n\\mathfrak{k} = so(3) = \\{Y \\in gl(3) \\,\\, | \\,\\, Y^t = - Y\\}.\n\\end{equation}\nNote that $\\mathfrak{g}$ is semisimple since the Killing form $\\mathbf{B}:\\mathfrak{g}\\times\\mathfrak{g}\\rightarrow\\mathbb{R}$ given by\n\\begin{equation}\n\\mathbf{B}(Y, Z) = \\mathrm{Tr} (\\mathrm{ad} Y \\circ \\mathrm{ad} Z) = 6\\mathrm{Tr}(YZ)\n\\end{equation}\nis nondegenerate.  Let $\\mathfrak{p}$ be the orthogonal complement of $\\mathfrak{k}$ with respect to $\\mathbf{B}$, so that we have the Cartan decomposition\n\\begin{equation}\n\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}\n\\end{equation}\nwith\n\\begin{equation}\n\\mathfrak{p} = \\{Y \\in gl (3) \\,\\, | \\,\\, Y^t = Y, \\text{ }\\mathrm{Tr} Y = 0\\},\n\\end{equation}\nand satisfying the Cartan relations\n\\begin{equation}\n[\\mathfrak{k}, \\mathfrak{k}] \\subset \\mathfrak{k}, \\quad\\quad\n[\\mathfrak{p}, \\mathfrak{p} ] \\subset \\mathfrak{k}, \\quad\\quad\n[\\mathfrak{k},\\mathfrak{p}]\\subset\\mathfrak{p}.\n\\end{equation}\nThe Killing form $\\mathbf{B}$ is negative definite on $\\mathfrak{k}$ and positive definite on $\\mathfrak{p}$, in particular $\\mathbf{X}$ is of noncompact type.\n\nConsider the Cartan involution $\\theta: \\mathfrak{g} \\rightarrow \\mathfrak{g}$ with $\\theta|_{\\mathfrak{k}} = \\mathrm{id}, \\theta|_{\\mathfrak{p}} = - \\mathrm{id}$, where in our context $\\theta(Y) = - Y^t$. Then the quadratic form\n\\begin{equation}\n\\langle Y, Z\\rangle_{\\mathfrak{g}}=\n\\begin{cases}\n-\\frac{2}{3}\\mathbf{B}(Y,Z) & \\text{ if } Y,Z\\in\\mathfrak{k},\\\\\n- \\frac{2}{3}\\mathbf{B}(Y, \\theta(Z))& \\text{ if } Y,Z\\in\\mathfrak{p},\\\\\n0& \\text{ if } Y\\in\\mathfrak{k},\\text{ } Z\\in\\mathfrak{p},\n\\end{cases}\n\\end{equation}\nis positive definite and Ad $K$-invariant. From this the desired Riemannian metric at $x_0$ is obtained by restricting the quadratic form to $\\mathfrak{p}$ which is identified with $T_{x_0} \\mathbf{X}$, namely\n\\begin{equation}\n \\mathbf{g}_{x_0}(Y, Z) = 4 \\mathrm{Tr} (YZ^t) \\quad \\text{ for }\\quad Y, Z \\in \\mathfrak{p}.\n\\end{equation}\nThis in turn gives rise to the metric globally on $\\mathbf{X}$ via left translation. Let $L_B: \\mathbf{X} \\rightarrow \\mathbf{X}$ denote the left translation operator\n\\begin{equation}\nL_B(x) = L_B([A]) = [B A],\n\\end{equation}\nwhere $A,B\\in SL(3,\\mathbb{R})$ and $x=[A]$.\nSince $SL(3, {\\mathbb R})$ acts transitively on $\\mathbf{X}$, given $x \\in \\mathbf{X}$ there is a $B\\in SL(3, {\\mathbb R})$ such\nthat $L_B(x_0) = x$, and thus the $G$-invariant Riemannian metric at $x$ may be defined\nby pulling back the quadratic form at the identity\n\\begin{equation}\\label{symmetricspacemetric}\n\\mathbf{g}_x = L_{B^{-1}}^* \\mathbf{g}_{x_0}.\n\\end{equation}\nWith this metric $SL(3,\\mathbb{R})\/SO(3)$ becomes a symmetric space of noncompact type having rank 2 (see \\cite{BallmanGromovSchroeder}). In particular it has nonpositive curvature, with the sectional curvature of the plane spanned by orthonormal vectors $ Y, Z \\in \\mathfrak{p}$ given by $-\\parallel[Y,Z]\\parallel_{\\mathfrak{g}}^2$.\n\nIn order to connect the metric \\eqref{symmetricspacemetric} with the target space geometry associated to the harmonic map of the previous section, the following characterization of $\\mathbf{X}=SL(3,\\mathbb{R})\/SO(3)$ will be needed. Recall the polar decomposition for matrices, namely any $A\\in SL(3,\\mathbb{R})$ may be written uniquely as $A=PO$ where $O\\in SO(3)$ and $P\\in \\tilde{\\mathbf{X}}$ with\n\\begin{equation}\n\\tilde{\\mathbf{X}} = \\{A \\in SL(3, {\\mathbb R}) \\mid A \\text{ is symmetric and positive definite}\\}.\n\\end{equation}\nThis indicates that $\\mathbf{X}$ may be identified with $\\tilde{\\mathbf{X}}$, and in fact this is accomplished with the map $\\mathcal{I} : \\tilde{\\mathbf{X}} \\rightarrow \\mathbf{X}$ given by\n\\begin{equation}\n\\mathcal{I}(A) = [A^{1\/2}],\\quad\\quad\\quad \\mathcal{I}^{-1}([B])=BB^t.\n\\end{equation}\nObserve that $\\tilde{\\mathbf{X}}$ can be interpreted as the set of all ellipsoids in ${\\mathbb R}^3$ centered at the origin with unit volume, and is diffeomorphic to $\\mathbb{R}^5$ (hence the same is true of $\\mathbf{X}$). Moreover\n$SL(3,{\\mathbb{R}})$ acts transitively on $\\tilde{\\mathbf{X}}$ by the analogue of left translation $\\tilde{L}_B = \\mathcal{I}^{-1} \\circ L_B \\circ \\mathcal{I}$, that is\n\\begin{equation}\n\\tilde{L}_B (A) = B A B^t.\n\\end{equation}\n\n\n\nThe identification above naturally induces a pull-back metric $\\tilde{\\mathbf{g}}:= \\mathcal{I}^* \\mathbf{g}$ on $\\tilde{\\mathbf{X}}$. At the identity this is\n\\begin{equation}\\label{identitymetric}\n\\tilde{\\mathbf{g}}_{\\mathrm{Id}} (V, V) = \\mathbf{g}_{x_0} \\left(\\frac{V}{2}, \\frac{V}{2}\\right) =   \\mathrm{Tr}(VV^t),\n\\end{equation}\nfor\n\\begin{equation}\nV\\in T_{\\mathrm{Id}} \\tilde{\\mathbf{X}} = \\{W \\in Mat_{3 \\times 3} (\\mathbb{R}) \\,\\, | \\,\\, W^t = W, \\quad \\mathrm{Tr} W = 0 \\}.\n\\end{equation}\nAs for an arbitrary point $A\\in\\tilde{\\mathbf{X}}$ and $V\\in T_{A}\\tilde{\\mathbf{X}}$,\n\\begin{align}\n\\begin{split}\n\\tilde{\\mathbf{g}}_{A}(V,V)=&\\mathbf{g}_{\\mathcal{I}(A)}\\left(d\\mathcal{I}_{A}(V),\nd\\mathcal{I}_{A}(V)\\right)\\\\\n=&L_{A^{-1\/2}}^{*}\\mathbf{g}_{x_0}\\left(d\\mathcal{I}_{A}(V),d\\mathcal{I}_{A}(V)\\right)\\\\\n=&\\mathbf{g}_{x_0}\\left(d(L_{A^{-1\/2}}\\circ\\mathcal{I})_{A}(V),\nd(L_{A^{-1\/2}}\\circ\\mathcal{I})_{A}(V)\\right)\\\\\n=&\\mathrm{Tr}\\left([(d\\tilde{L}_{A^{-1\/2}})_{A}(V)][(d\\tilde{L}_{A^{-1\/2}})_{A}(V)]^{t}\\right).\n\\end{split}\n\\end{align}\nSince\n\\begin{equation}\n(d\\tilde{L}_{A^{-1\/2}})_{A}(V)=A^{-1\/2}V(A^{-1\/2})^t,\n\\end{equation}\nit follows that\n\\begin{align}\\label{targetmetric}\n\\begin{split}\n\\tilde{\\mathbf{g}}_{A}(V,V)=&\\mathrm{Tr}\\left(A^{-1\/2}V(A^{-1\/2})^t A^{-1\/2} V(A^{-1\/2})^t\\right)\\\\\n=&\\mathrm{Tr}\\left(A^{-1\/2}VA^{-1}V(A^{-1\/2})^t\\right)\\\\\n=&\\mathrm{Tr}\\left(A^{-1}V A^{-1}V\\right).\n\\end{split}\n\\end{align}\n\n\n\nRecall from the previous section that a given 5-dimensional bi-axisymmetric stationary vacuum spacetime yields a map $\\Phi:\\mathbb{R}^3\\setminus\\Gamma\\rightarrow\\tilde{\\mathbf{X}}$, where $\\mathbb{R}^3$ is parameterized by the Weyl-Papapetrou coordinates $(\\rho,z,\\phi)$, $\\Gamma$ denotes the $z$-axis, and $\\tilde{\\mathbf{X}}$ is parameterized by $(f_{ij},\\omega_{i})$. According to \\eqref{targetmetric} the pull-back metric is then given by\n\\begin{equation}\n\\Phi^* \\tilde{\\mathbf{g}} =  \\mathrm{Tr} ( \\Phi^{-1}d \\Phi \\, \\Phi^{-1} d \\Phi).\n\\end{equation}\nSince this agrees with the expression appearing in the reduced action \\eqref{action}, it follows that the bi-axisymmetric stationary vacuum Einstein equations reduce to a harmonic map problem with target space $SL(3,\\mathbb{R})\/SO(3)$.\n\n\n\n\n\n\n\\section{The Rod Structure}\n\\label{sec4} \\setcounter{equation}{0}\n\\setcounter{section}{4}\n\n\nA well-behaved asymptotically flat, stationary vacuum, bi-axisymmetric spacetime admits a global system of Weyl-Papapetrou coordinates in its domain of outer communication $\\mathcal{M}^5$, as described in Section \\ref{sec2}, in which the metric takes the form\n\\begin{equation}\\label{spacetimemetric}\ng=f^{-1}e^{2\\sigma}(d\\rho^2+dz^2)-f^{-1}\\rho^2 dt^2\n+f_{ij}(d\\phi^{i}+v^{i}dt)(d\\phi^{j}+v^{j}dt).\n\\end{equation}\nThe orbit space $\\mathcal{M}^{5}\/[\\mathbb{R}\\times U(1)^2]$ is diffeomorphic to the right-half plane $\\{(\\rho,z)\\mid \\rho>0\\}$ (see \\cite{HollandsYazadjiev1}), and\nits boundary $\\rho=0$ encodes nontrivial aspects of the topology. Let $q$ be the fiber metric \\eqref{fibermetric} consisting of the last two terms in \\eqref{spacetimemetric}. In order to avoid curvature singularities $\\mathrm{dim} \\left(\\mathrm{ker}\\text{ }\\! q(0,z)\\right)=1$ except at isolated points $p_{l}$, $l=1,\\ldots,L$ where the dimension of the kernel is 2 \\cites{Harmark,HollandsYazadjiev}. It follows that the $z$-axis is broken into $L+1$ intervals called rods\n\\begin{equation}\n\\Gamma_{1}=[z_{1},\\infty),\\text{ }\\Gamma_{2}=[z_2,z_1],\\text{ }\\ldots,\\text{ }\n\\Gamma_{L}=[z_{L},z_{L-1}],\\text{ }\\Gamma_{L+1}=(-\\infty,z_{L}],\n\\end{equation}\non which either $|\\partial_{t}+\\Omega_{1}\\partial_{\\phi^1}+\\Omega_{2}\\partial_{\\phi^2 }|$ vanishes (horizon rod) or $(f_{ij})$ fails to be of full rank (axis rod). Here $\\Omega_i$ denotes the angular velocity of the horizon and is given by $-v^i$ restricted to the rod. This must be a constant, and can be seen by solving for $dv^i$ from \\eqref{chi} and showing that it vanishes on the rod. The condition for an axis rod implies \\cite{HollandsYazadjiev} that for each such $\\Gamma_{l}$ there is a pair of relatively prime integers $(m_{l},n_{l})$ so that the Killing field\n\\begin{equation}\nm_{l}\\partial_{\\phi^1}+n_{l}\\partial_{\\phi^2}\n\\end{equation}\nvanishes on $\\Gamma_{l}$. Observe that $m_{l}$ and $n_{l}$ must be integers since elements of the isotropy subgroup at the axis are of the form $(e^{im_{l}\\phi},e^{in_{l}\\phi})$, $0\\leq\\phi<2\\pi$, and the isotropy subgroup forms a proper closed subgroup of $T^2=S^1\\times S^1$. That is, the isotropy subgroup yields a simple closed curve in the torus exactly when the slope of its winding is rational.\nThe pair $(m_{l},n_{l})$ is referred to as the rod structure for the rod $\\Gamma_{l}$, and $(0,0)$ serves as the rod structure for any horizon rod. Note that the rod structure is not unique in terms of the information that it encodes, although this type of uniqueness is valid when the rod structure is viewed as an element of $\\mathbb{RP}^1$.\n\nThe asymptotically flat condition is encoded by the rod structures of $\\Gamma_{1}$ and $\\Gamma_{L+1}$ by requiring them to be $(\\pm 1,0)$ and $(0,\\pm 1)$ or vice versa. This, of course, arises from the rod structure of Minkowski space $\\mathbb{R}^{4,1}$ which will now be described in order to motivate the definition of a `corner'. The Weyl-Papapetrou form of the Minkowski metric is derived from the polar coordinate expression with the help of Hopf coordinates $(\\theta,\\phi^1,\\phi^2)$, $\\phi^{i}\\in[0,2\\pi]$, $\\theta\\in[0,\\pi\/2]$ on the 3-sphere and a conformal mapping\n\\begin{align}\\label{Minkowski}\n\\begin{split}\n  g_{0} = & -dt^{2}+ dr^2 + r^2 d \\omega_{S^3}^2 \\\\\n    = & -dt^{2}+ dr^2 + r^2 \\left[d \\theta^2 + \\sin^2 \\theta (d \\phi^1)^2 + \\cos^2 \\theta (d \\phi^ 2)^2 \\right] \\\\\n    = & q_{0} +  dr^2 + r^2 d \\theta^2 \\\\\n    = & q_{0} + \\frac{1}{4\\sqrt{\\rho^2 + z^2 }} (d \\rho^2  + dz^2).\n\\end{split}\n\\end{align}\nHere the conformal map in the complex plane is given by\n\\begin{equation}\n\\zeta \\mapsto  \\zeta^2 \\quad :\\quad {\\mathbb R}_{\\geq 0} \\times  {\\mathbb R}_{\\geq 0} \\rightarrow \\mbox{$\\rho z$-half plane},\n\\end{equation}\nor rather\n\\begin{equation}\n\\rho =  r^2 \\sin 2 \\theta, \\quad\\quad z =  r^2 \\cos 2 \\theta .\n\\end{equation}\nIf $x^i$ denote cartesian coordinates then the Killing fields\n\\begin{equation}\n\\partial_{\\phi^1} = -x^2 \\partial_{x^1} + x^1 \\partial_{x^2},\\quad\\quad\\quad\n\\partial_{\\phi^2} = -x^4 \\partial_{x^3} + x^3 \\partial_{x^4},\n\\end{equation}\nvanish on the rods $\\Gamma_{1}=[0,\\infty)$ and $\\Gamma_{2}=(-\\infty,0]$, respectively. Therefore the rod structures for these two rods are $(1,0)$ and $(0,1)$. Moreover, because the origin $p_{1}$ in the $\\rho z$-plane corresponds to the vertex of the right-half quadrant under the inverse conformal map this is called a corner. For a general set of rod structures, a corner point $p_l$ is one which separates two axis rods, and a pole point is one which separates a horizon rod from an axis rod.\n\nPotential constants $\\mathbf{c}_{l}=(c_{l}^1,c_{l}^2)\\in\\mathbb{R}^2$ are prescribed on each axis rod $\\Gamma_{l}$, and are used as boundary conditions for the twist potentials\n$\\omega_{i}|_{\\Gamma_{l}}=c_{l}^{i}$. The constants may be chosen arbitrarily modulo the condition that they do not vary between adjacent rods separated by a corner. This is necessary for the construction of a model map in the next section, as well as a well-defined notion of angular momentum. In particular, the potential constants can only change after passing over a horizon rod, and this difference yields the angular momenta for each horizon component. Let $\\mathcal{S}$ denote the 3-dimensional horizon cross section component associated with a horizon rod $\\Gamma_{k}=[z_{k},z_{k-1}]$, then \\eqref{chi}, \\eqref{komar}, and \\eqref{komar1} may be used to compute the Komar angular momenta of this component by\n\\begin{equation}\n\\mathcal{J}_{i}=\\frac{1}{8\\pi}\\int_{\\mathcal{S}}\\star d\\eta^{(i)}\n=\\frac{\\pi}{2}\\int_{\\Gamma_{k}}\\iota_{\\eta^{(1)}}\\iota_{\\eta^{(2)}}\\star d\\eta^{(i)}\n=\\frac{\\pi}{4}\\int_{\\Gamma_{k}}d\\omega_{i}=\\frac{\\pi}{4}\\left[\\omega_{i}(p_{k-1})\n-\\omega_{i}(p_{k})\\right].\n\\end{equation}\nA rod data set $\\mathcal{D}$ consists of the collection of corners and poles $\\{p_l\\}$,\nrod structures $\\{(m_l,n_l)\\}$, and potential constants $\\{\\mathbf{c}_l\\}$.\n\nConsider now the topology of spacetime in a neighborhood of a corner point $p_l$ which separates axis rods $\\Gamma_{l}$ and $\\Gamma_{l+1}$ with rod structure $(m_l,n_l)$ and $(m_{l+1},n_{l+1})$. As is shown in the Appendix, new $2\\pi$-periodic coordinates $(\\bar{\\phi}^{1},\\bar{\\phi}^2)$ may be chosen so that the rod structures with respect to these coordinates are given by $(1,0)$ and $(q,p)$, $p\\neq 0$. That is, the Killing fields $\\partial_{\\bar{\\phi}^1}$ and $q\\partial_{\\bar{\\phi}^1}+p\\partial_{\\bar{\\phi}^2}$ vanish on $\\Gamma_{l}$ and $\\Gamma_{l+1}$, respectively. Next take any semicircle in the $\\rho z$-half plane (orbit space) centered at $p_l$ that connects a point on the interior of $\\Gamma_l$ to a point on the interior of $\\Gamma_{l+1}$. Note that each point on the interior of this semicircle represents a 2-torus in a constant time slice. By analyzing which 1-cycles collapse at the end points it follows that the semicircle represents a lens space $L(p,q)$. Recall that $L(1,q)\\cong S^3$, so that when $p=\\pm 1$ a neighborhood of the corner in a time slice is foliated by spheres, or rather a neighborhood of the corner in the spacetime is diffeomorphic to $\\mathbb{R}^5$. It turns out that $p=\\pm 1$ if and only if\n\\begin{equation} \\label{admissibility1}\n\\operatorname{det}\\begin{pmatrix} m_l & n_l \\\\ m_{l+1} & n_{l+1} \\end{pmatrix} = \\pm 1,\n\\end{equation}\nand therefore the spacetime has trivial topology in a neighborhood of the corner if and only if the admissibility condition \\eqref{admissibility1} holds, otherwise it has an orbifold singularity. The admissibility condition can be interpreted as stating that the\nintersection number of the two 1-cycles that degenerate on either side of the corner is equal to $\\pm 1$.\n\nIn addition to \\eqref{admissibility1}, the main results of this paper rely on what will be referred to as the compatibility condition. This supplementary requirement is only valid when two consecutive corners are present. As described above, let $p_l$ be a corner separating axis rods $\\Gamma_{l}$ and $\\Gamma_{l+1}$, and suppose that there is another corner $p_{l-1}$ at the top end of $\\Gamma_{l}$ connecting it to axis rod $\\Gamma_{l-1}$. Assuming that the admissibility condition \\eqref{admissibility1} holds at the two points $p_{l-1}$ and $p_{l}$, it may be arranged that these two determinants are $+1$ by multiplying each component of the rod structures by $-1$ if necessary. Observe that this operation on the rod structures does not change their properties, since the linear combinations of Killing fields that vanish at the rods is preserved.\nThe compatibility condition then states that the first component of the rod structures for $\\Gamma_{l-1}$ and $\\Gamma_{l+1}$ have opposite sign if both are nonzero\n\\begin{equation}\\label{compatibilitycondition1}\nm_{l-1}m_{l+1}\\leq 0.\n\\end{equation}\nThis technical condition is used only in the construction of the model map in the next section. Unlike the admissibility condition, it is not known whether Theorem \\ref{main} remains true without it. As mentioned in the introduction, if the admissibility condition is not assumed so that orbifold singularities are allowed then \\eqref{compatibilitycondition1} should be enhanced to the generalized compatibility condition\n\\begin{equation}\\label{gcompatibilitycondition}\nm_{l-1}m_{l+1} \\operatorname{det}\\begin{pmatrix} m_{l-1} & n_{l-1} \\\\ m_{l} & n_{l} \\end{pmatrix}\n\\operatorname{det}\\begin{pmatrix} m_l & n_l \\\\ m_{l+1} & n_{l+1} \\end{pmatrix} \\leq 0.\n\\end{equation}\nNote that the only way this quantity can vanish is if either $m_{l-1}=0$ or $m_{l+1}=0$, since for a corner the determinant is always nonzero.\n\nEach connected component cross section of the event horizon has one of the following topologies \\cite{HollandsYazadjiev1}: the sphere $S^3$, the ring $S^1\\times S^2$, or a lens space $L(p,q)$. These manifolds have a singular foliation whose leaves are 2-dimensional tori, and whose singular leaves are circles resulting from the degeneration of a 1-cycle in the torus.  This can be observed geometrically from the canonical metric on each manifold as follows. The round metric on $S^3$ in Hopf coordinates is given by\n\\begin{equation}\nd \\theta^2 + \\sin^2 \\theta (d \\phi^1)^2 + \\cos^2 \\theta (d \\phi^2)^2,\n\\end{equation}\nwhere $\\theta \\in [0, \\pi\/2]$, $\\phi^i \\in [0, 2 \\pi]$.  For $0 < \\theta < \\pi\/2$ the level set $\\{\\theta = \\mbox{const.} \\}$ is a flat 2-torus, and when $\\theta = 0, \\pi\/2$ the level sets degenerates to $S^1$.  These singular leaves are characterized by the fact that the Killing fields $\\partial_{\\phi^1}$ and $\\partial_{\\phi^2}$ vanish at $\\theta = 0, \\pi\/2$ respectively. Thus if $\\theta$ is viewed as parameterizing a horizon rod, then the rod structure at the two poles (end points) is\n$\\{(1,0),(0,1)\\}$. For the ring $S^1 \\times S^2$ the canonical product metric is\n\\begin{equation}\n[d \\theta^2 + \\sin^2 \\theta (d \\phi^1)^2] + (d \\phi^2)^2,\n\\end{equation}\nwhere $\\theta \\in [0, \\pi]$, $\\phi^i \\in [0, 2 \\pi]$.  The torus fibers are once again the level sets of $\\theta$, and the singular leaves occur when $\\theta = 0, \\pi$ and coincide with the vanishing of the Killing field $\\partial_{\\phi^1}$, while the other Killing field $\\partial_{\\phi^2}$ never degenerates. The associated rod structure at the poles is then $\\{(1,0),(1,0)\\}$.\n\n\n\n\\begin{figure}\n\\includegraphics[width=5cm]{lensidentification.pdf}\n\\caption{Identification Space}  \\label{toiletpaper}\n\\end{figure}\n\n\nConsider now the lens space $L(p, q)=S^{3}\/\\mathbb{Z}_{p}$ which inherits its canonical metric\n\\begin{equation}\nd \\theta^2 + \\sin^2 \\theta (d \\tilde{\\phi}^1)^2 + \\cos^2 \\theta (d \\tilde{\\phi}^2)^2\n\\end{equation}\nfrom the 3-sphere, where\n\\begin{equation}\n\\tilde{\\phi}^1 = \\phi^1 - \\frac{q}{p} \\phi^2, \\quad \\quad \\tilde{\\phi}^2 = \\frac1p \\phi^2,\n\\end{equation}\nwith $\\theta \\in [0, \\pi\/2]$, $\\phi^i \\in [0, 2 \\pi]$.\nSince $\\phi^2$ has period $2\\pi$, the following identifications are made\n\\begin{equation}\n\\tilde{\\phi}^1 \\sim \\tilde{\\phi}^1 + \\frac{2\\pi q}{p} , \\quad\\quad \\tilde{\\phi}^2 \\sim  \\tilde{\\phi}^2 +  \\frac{2\\pi}{p} .\n\\end{equation}\nThe singular leaves at $\\theta = 0, \\pi\/2$ are characterized by the vanishing of the Killing fields\n\\begin{equation}\n\\partial_{\\tilde{\\phi}^1}=\\partial_{\\phi^1},\\quad\\quad\n\\partial_{\\tilde{\\phi}^2}=q\\partial_{\\phi^1}+p\\partial_{\\phi^2},\n\\end{equation}\nrespectively, so that the associated rod structure at the poles is $\\{(1,0),(q,p)\\}$.\nRecall the model of the lens space as a quotient space of the unit sphere $S^3=\\{(z_1, z_2)\\in\\mathbb{C}^2 \\,\\, | \\,\\, |z_1|^2 + |z_2|^2 = 1 \\}$ via the equivalence relation\n\\begin{equation}\n(z_1, z_2 ) = (r_1 e^{ \\tilde{\\phi}^1 i}, r_2 e^{ \\tilde{\\phi}^2 i}) \\sim (r_1 e^{\\left(  \\tilde{\\phi}^1 + 2 \\pi q\/p\\right)i}, r_2 e^{\\left( \\tilde{\\phi}^2 + 2 \\pi\/p \\right)i}).\n\\end{equation}\nHere the pair of variables $(r_1, r_2)$ correspond to $(\\sin \\theta, \\cos \\theta)$ in the coordinates with which the lens space metric is written. A visualization of the lens space may be obtained by appropriately identifying the top, bottom, and sides of a solid cylinder as in Figure \\ref{toiletpaper}. Namely, first collapse the external cylinder $\\{\\theta =\\pi\/2\\}$ by identifying each vertical segment to a point, then identify the top and bottom discs via an orthogonal projection after performing a $2\\pi q\/p$ rotation of the top disc. The singular torus fibers occur where the action of the coordinate fields $\\partial_{\\tilde{\\phi}^1}$ and $\\partial_{\\tilde{\\phi}^2}$ degenerate, that is at $\\theta=0,\\pi\/2$.\n\n\n\nUsing a similar analysis the topology of arbitrary rod structures may be understood. In Figure \\ref{rod} four different rod structures for the orbit space are given, labeled by the topology of their horizons. Consider the first rod structure on the left in this diagram. The two semi-infinite rods are foliated by circle fibers none of which collapse, and hence they are 2-planes with an open disc removed. The finite rod has rod structure $(0,0)$ meaning that none of the rotational Killing fields vanish there. It is foliated by 2-tori such that each of the two 1-cycles generators in the torus degenerate on opposite poles. According to the description above, this yields an 3-sphere. Similarly, any simple curve in the $\\rho z$-plane connecting the two semi-infinite rods also produces an $S^3$. In the second and third rod structures of Figure \\ref{rod} it is clear that, by comparing with the singular foliations described above, these horizon rods represent a ring $S^1\\times S^2$ and a lens $L(p,1)$, respectively. In these two examples there is also a different type of rod not present in the first example, namely a finite rod bounded by a pole on top and a corner on the bottom. This type of rod is foliated by circles with a singular leaf at the corner, and thus it gives a topological disc. The last example in Figure \\ref{rod} has two horizon components in which the inner one is a lens $L(p,1)$ and the outer one is a ring $S^1\\times S^2$, and hence the name `Black Lens Saturn'.\n\n\n\\begin{figure}\n\\includegraphics[width=12cm]{rods.pdf}\n\\caption{Rod Strucures}  \\label{rod}\n\\end{figure}\n\nObserve that the rod structures of Figure \\ref{rod} satisfy the admissibility condition \\eqref{admissibility1} with $+1$ determinants, and the compatibility condition is vacuous. A natural question arises whether it is possible to produce a rod structure with a single horizon component having the general lens topology $L(p,q)$ without restricting to $q=1$, while at the same time satisfying the admissibility condition \\eqref{admissibility1} and compatibility condition \\eqref{compatibilitycondition1}. The following proposition answers this question affirmatively.\n\n\\begin{proposition}\\label{lensrod}\nLet $p$ and $q$ be integers satisfying $\\mathrm{gcd}(p,q)=1$ and $p>q\\geq 1$.\nThen there exists a rod structure appropriate for an asymptotically flat spacetime of the form\n\\begin{equation}\n\\{(1, 0), (0, 0), (q, p),(q_1, p_1), \\dots, (q_n, p_n), (0,\\pm 1)\\},\n\\end{equation}\nwhich has a single lens space horizon $L(p, q)$, satisfies the admissibility condition \\eqref{admissibility1} with positive determinants, and satisfies the compatibility condition \\eqref{compatibilitycondition1}.\n\\end{proposition}\n\nAs an example observe that the single lens horizon $L(9,7)$ is realized by the rod structures\n\\begin{equation}\n\\{(1,0), (0,0), (7,9), (-4,-5), (-3, -4), (1, 1), (0, 1)\\},\n\\end{equation}\nwhich clearly satisfy the admissibility condition with positive determinants as well as the compatibility condition. In order to prove Proposition \\ref{lensrod} we need a slightly modified version of Bezout's Lemma.\n\n\\begin{lemma}\\label{Bezout}\nLet $a \\neq 1$ and $b \\neq 1$ be relatively prime positive integers, then there exist integers $x$ and $y$ of the same sign such that\n\\begin{equation}\nax - by = 1,\n\\end{equation}\nwith $\\mathrm{gcd}(x, y)=1$ and $1\\leq |x| < b$, $1\\leq |y| < a$. Furthermore, if $a<b$ then $|x| \\geq |y|$.\n\\end{lemma}\n\n\\begin{proof}\nBy Bezout's Lemma there exist integers $\\overline{x}$, $\\overline{y}$ such that $a \\overline{x} + b \\overline{y} = 1$ with $|\\overline{x}|\\leq b$ and $|\\overline{y}|\\leq a$. Moreover, one of these may be an equality only if $a\\mid b$ or $b\\mid a$. Since $\\mathrm{gcd}(a,b)=1$ it must hold that $|\\overline{x}|< b$ and $|\\overline{y}|< a$.\nFurthermore, since $a, b>1$ we must have one of $\\overline{x}$, $\\overline{y}$ negative and the other positive.  Thus there are $\\tilde{x}>0$, $\\tilde{y}>0$ so that $a \\tilde{x} - b \\tilde{y} = \\pm 1$, with $\\tilde{x} < b$ and $\\tilde{y} < a$.  If $\\mathrm{gcd}(\\tilde{x},\\tilde{y})=c>1$ then $\\tilde{x}=c\\hat{x}$, $\\tilde{y}=c\\hat{y}$ and $c(a \\hat{x} - b \\hat{y}) = \\pm 1$. This, however, is impossible since $c>1$, and hence $\\mathrm{gcd}(\\tilde{x},\\tilde{y})=1$. If $a \\tilde{x} - b \\tilde{y} = 1$ then choose $(x, y)= (\\tilde{x}, \\tilde{y})$, and if  $a \\tilde{x} - b \\tilde{y} = -1$\nthen choose  $(x, y)=(-\\tilde{x}, -\\tilde{y})$. Lastly, neither $x$ nor $y$ may vanish as $a,b>1$.\n\nConsider now the case when $a < b$. It then follows from the equation $ax-by=1$ that either $x>y$ (when $x,y>0$) or $x\\leq y$ (when $x,y<0$). Hence $|x| \\geq |y|$ when $a < b$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{lensrod}]\nIf $q=1$ then append the rod structure $(0,1)$ after $(q,p)$ to solve the problem.\nAssume now that $p$ and $q$ are relatively prime with $p>q>1$. Apply Bezout's Lemma with\n$(a, b) = (q, p)$ to find a pair $(q_1, p_1)$ of relatively prime integers satisfying\n\\begin{equation}\nqp_1-pq_1 =1\n\\end{equation}\nas well as\n\\begin{equation}\n 1 \\leq |q_{1}| < q, \\quad\\quad\\quad 1 \\leq |p_{1}| < p.\n\\end{equation}\nIf  $|q_{1}|=1$, then by appending the rod structure $(0, \\pm 1)$ after $(q_1, p_1) = (\\pm1, p_{1})$ the desired result follows.\n\nConsider now the case when $|q_1|> 1$. Again apply Bezout's Lemma to find $(\\overline{q}_2, \\overline{p}_2)$ relatively prime and satisfying\n\\begin{equation}\n|q_1| \\overline{p}_2- |p_1| \\overline{q}_2 =1\n\\end{equation}\nas well as\n\\begin{equation}\n 1 \\leq |\\overline{q}_{2}| < |q_1|, \\quad\\quad\\quad 1 \\leq |\\overline{p}_{2}| < |p_1|.\n\\end{equation}\nNext define $(\\tilde{q}_2, \\tilde{p}_2)=\\pm (\\overline{q}_2, \\overline{p}_2)$ where the sign is chosen so that\n\\begin{equation}\nq_1 \\tilde{p}_2- p_1 \\tilde{q}_2 =1.\n\\end{equation}\n\nThe compatibility condition requires $q_0 q_2\\leq 0$, and since $q_0=q>0$ this can be achieved by setting $(q_2,p_2)=(\\tilde{q}_2,\\tilde{p}_2)$ if $\\tilde{q}_2<0$, and $(q_2,p_2)=(\\tilde{q}_2-|q_1|,\\tilde{p}_2-|p_1|)$ if $\\tilde{q}_2>0$. Clearly this also satisfies the admissibility condition\n\\begin{equation}\\label{DET}\nq_1 p_2- p_1 q_2 =1\n\\end{equation}\nas well as\n\\begin{equation}\n 1 \\leq |q_{2}| < |q_1|, \\quad\\quad\\quad 1 \\leq |p_{2}| < |p_1|,\n\\end{equation}\nand \\eqref{DET} implies that $q_2$ and $p_2$ are relatively prime.\nNote that if it were the case that $q_0<0$ then $(|q_1|,|p_1|)$ should be added in the last step, rather than subtracted, in order to satisfy the compatibility condition.\nThis iterative process may be continued until $|q_n|=1$. Then at that point, append the rod structure $(0,\\pm 1)$ after $(q_n,p_n)=(\\pm 1,p_n)$ in order to achieve the stated outcome.\n\\end{proof}\n\nWe end this section by noting an important property of the horizon rods, which corresponds to a well-known result in 4-dimensional spacetime \\cite{HawkingEllis}*{Proposition 9.3.1}.\nRecall that a horizon rod is defined as an interval on the $z$-axis where the matrix $(f_{ij})$ is invertible, so that the torus fibers are nondegenerate there. These fibers together with the horizon rod form a codimension 2 surface in the spacetime, which will be referred to as a horizon rod surface.\n\n\\begin{lemma}\nA horizon rod surface is a future apparent horizon, and within the $t=0$ slice it is a minimal surface.\n\\end{lemma}\n\n\\begin{proof}\nAt the beginning of this section we found that associated with a horizon rod there is a Killing field\n\\begin{equation}\n\\mathcal{K}=\\partial_{t}+\\Omega_1 \\partial_{\\phi^1}+\\Omega_2 \\partial_{\\phi^2},\\quad\\quad\\quad\n\\Omega_i\\in\\mathbb{R},\n\\end{equation}\nwhich is null on the horizon rod surface $S$. Since the tangent space to $S$ is spanned by the vector fields $\\partial_{z}$ and $\\partial_{\\phi^i}$, it easily follows from the structure of the spacetime metric \\eqref{spacetimemetric} and the values for $\\Omega_{i}$ that $\\mathcal{K}$ is normal to $S$. The second fundamental form of $S$ in the $\\mathcal{K}$-direction is then given by\n\\begin{equation}\nII_{ab}= g(\\nabla_{\\partial_{a}}\\mathcal{K},\\partial_{b}),\n\\end{equation}\nwhere $\\partial_{a}$ denotes a tangent vector to $S$. Since $\\mathcal{K}$ is Killing\n\\begin{equation}\ng(\\nabla_{\\partial_{a}}\\mathcal{K},\\partial_{b})\n=-g(\\nabla_{\\partial_{b}}\\mathcal{K},\\partial_{a}),\n\\end{equation}\nand hence $II_{ab}$ is antisymmetric. Let\n\\begin{equation}\n\\gamma=f^{-1}e^{2\\sigma}dz^2\n+f_{ij}d\\phi^{i}d\\phi^{j}\n\\end{equation}\nbe the induced metric on the horizon rod surface, then the future null expansion is\n\\begin{equation}\n\\theta_{+}=\\gamma^{ab}II_{ab}=0,\n\\end{equation}\nsince $\\gamma^{ab}$ is symmetric.\nBy definition, $S$ is then a future apparent horizon.\n\nIn order to show that $S$ is minimal within the $t=0$ slice, let\n\\begin{equation}\n\\nu=(\\nabla^a t)\\partial_{a}=g^{tt}\\partial_{t}+g^{t\\phi^i}\\partial_{\\phi^i}\n\\end{equation}\nbe the unnormalized normal to the slice. Then the second fundamental form of the slice is\ngiven by\n\\begin{equation}\n|\\nu|k_{cd}=g(\\nabla_{\\partial_{c}}\\nu,\\partial_d).\n\\end{equation}\nObserve that\n\\begin{equation}\n|\\nu|k(\\partial_{\\phi^i},\\partial_{\\phi^j})\n=g^{tt}g(\\nabla_{\\partial_{\\phi^i}}\\partial_{t},\\partial_{\\phi^j})\n+g^{t\\phi^l}g(\\nabla_{\\partial_{\\phi^i}}\\partial_{\\phi^l},\\partial_{\\phi^j})\n\\end{equation}\nis antisymmetric, and\n\\begin{equation}\n|\\nu|k(\\partial_{z},\\partial_{z})=g^{tt}g(\\nabla_{\\partial_{z}}\\partial_{t},\\partial_{z})\n+g^{t\\phi^l}g(\\nabla_{\\partial_{z}}\\partial_{\\phi^l},\\partial_{z})=0,\n\\end{equation}\nsince $\\partial_{t}$, $\\partial_{\\phi^i}$ are Killing. It follows that\n\\begin{equation}\n\\mathrm{Tr}_{S}k=\\gamma^{ab}k_{ab}=f e^{-2\\sigma}k(\\partial_{z},\\partial_{z})\n+f^{ij}k(\\partial_{\\phi^i},\\partial_{\\phi^j})=0.\n\\end{equation}\nLet $n$ denote the outward unit normal to $S$ within the $t=0$ slice, then\n$n+\\nu\/|\\nu|=\\psi\\mathcal{K}$ for some function $\\psi$ on $S$. We then have\n\\begin{equation}\n0=\\psi\\theta_{+}=H_{S}+\\mathrm{Tr}_{S}k=H_{S}\n\\end{equation}\nwhere $H_{S}$ denotes mean curvature, and therefore $S$ is a minimal surface within the slice.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Model Map}\n\\label{sec5} \\setcounter{equation}{0}\n\\setcounter{section}{5}\n\n\n\n\n\n\nIn this section a so called model map $\\Phi_0\\colon\\mathbb{R}^3\\setminus\\Gamma\\to \\tilde{\\mathbf{X}}\\cong SL(3,\\mathbb{R})\/SO(3)$ is constructed, which encodes the prescribed asymptotic behavior near the axis and at infinity for the desired harmonic map, and also has finite tension. It may be viewed as an approximate solution to the singular harmonic map problem near the axes and at infinity.\n\nThe construction bears some similarity to the one in \\cite{weinstein96}, but is more complex due to the abundance of rod structures, and the fact that even the non-rotating case is already nonlinear. We detail the construction in the case of a single component but the same approach works for all rod structures satisfying the compatibility condition. Where needed, we will point out differences required to make the approach work in the more general case.\n\n\nThe canonical Riemannian metric on $\\tilde{\\mathbf{X}}$ was constructed in Section \\ref{sec3}, and it was noted that this space is parameterized by a $2\\times 2$ symmetric positive definite matrix $F=(f_{ij})$ and a $2$-vector $\\omega=(\\omega_1,\\omega_2)^t$. If $f=\\operatorname{det} F$ then the metric in these coordinates \\cite{IdaIshibashiShiromizu} is given by\n\\begin{align}\n\\begin{split}\n\t\\tilde{\\mathbf{g}} =& \\frac14 \\frac{df^2}{f^2} +\\frac14 f^{ij}f^{kl}df_{ik}df_{jl}  + \\frac12 \\frac{f^{ij} d\\omega_i d\\omega_j}{f} \\\\\n\t=& \\frac14 [\\mathrm{Tr}(F^{-1}dF)]^2  +\\frac14 \\mathrm{Tr}(F^{-1}dF\\, F^{-1} dF)  + \\frac12 \\frac{d\\omega^t \\, F^{-1}\\, d\\omega}{f}.\n\\end{split}\n\\end{align}\nA computation shows that the components of the tension \\eqref{tensiondef} of a map $\\Phi_0=(F,\\omega)$ are\n\\begin{align}\\label{eulerlagrange}\n\\begin{split}\n\\tau^{f_{lj}}=&\\Delta f_{lj}-f^{km}\\nabla^{\\mu}f_{lm}\\nabla_{\\mu}f_{kj}\n+f^{-1}\\nabla^{\\mu}\\omega_{l}\\nabla_{\\mu}\\omega_{j},\\\\\n\\tau^{\\omega_{j}}=&\\Delta\\omega_{j}-f^{kl}\\nabla^{\\mu}f_{jl}\\nabla_{\\mu}\\omega_{k}\n-f^{lm}\\nabla^{\\mu}f_{lm}\\nabla_{\\mu}\\omega_{j},\n\\end{split}\n\\end{align}\nwhere $\\Delta$ is the Laplacian and $\\nabla$ the connection associated with the flat metric \\eqref{flatmetric} on $\\mathbb{R}^3$. This yields the harmonic map equations $\\tau=0$ in these coordinates. Let\n\\begin{equation}\nH=F^{-1}\\nabla F,\\quad\\quad G=f^{-1}F^{-1}\\left(\\nabla\\omega\\right)^2,\\quad\\quad\nK=f^{-1}F^{-1}\\nabla\\omega,\n\\end{equation}\nthat is\n\\begin{equation}\n  H_{\\mu}{}^i{}_j= f^{ik}\\nabla_\\mu f_{kj},\\quad\\quad\n  G^{i}_j=f^{-1} f^{ik} \\nabla_\\mu\\omega_k\\,\\nabla^\\mu\\omega_j,\\quad\\quad\n  K_\\mu{}^i=f^{-1}f^{ij}\\nabla_\\mu\\omega_j,\n\\end{equation}\nand observe that\n\\begin{equation}\n\\left(\\operatorname{div}H+G\\right)^{i}_j=f^{il}\\tau^{f_{lj}},\\quad\\quad\n\\left(\\operatorname{div}K\\right)^{i}=f^{-1}f^{ij}\\tau^{\\omega_{j}}.\n\\end{equation}\nWe then have\n\\begin{equation}\n  |\\tau|^2=\\frac14 \\left[ \\mathrm{Tr}(\\operatorname{div}H + G)\\right]^2 + \\frac14 \\mathrm{Tr} \\left[(\\operatorname{div}H + G)(\\operatorname{div}H + G) \\right]\n  + \\frac12 f (\\operatorname{div}K)^t F (\\operatorname{div}K).\n\\end{equation}\n\nIn order to state the main result of this section we will say that a map $\\Phi_0=(F,\\omega)$ \\textit{respects} a rod data set $\\mathcal{D}$, if $(m_l,n_l)$ is the rod structure and $\\mathbf{c}_l$ the potential constant within $\\mathcal{D}$ for an axis rod $\\Gamma_l$ then\n\\begin{equation}\n(m_l,n_l)\\in\\mathrm{ker \\text{ }} F|_{\\Gamma_l},\\quad\\quad\\quad \\omega|_{\\Gamma_l}=\\mathbf{c}_l.\n\\end{equation}\n\n\n\\begin{theorem} \\label{model}\nGiven a rod data set $\\mathcal{D}$ satisfying the generalized compatibility condition\n\\eqref{gcompatibilitycondition}, there exists a model map $\\Phi_0\\colon\\mathbb{R}^3\\setminus\\Gamma\\to \\tilde{X}$ with uniformly bounded tension having decay $|\\tau|=O(r^{-7\/2})$ which respects $\\mathcal{D}$.\n\\end{theorem}\n\n\\begin{proof}\nAs mentioned above, we give a detailed proof for the case of the rod configuration corresponding to a single lens horizon $L(p,1)$, see Figure~\\ref{domain}. However, we will indicate below the changes required for the general case.\n\n\\begin{figure}\n\\includegraphics[width=10cm]{domain.pdf}\n\\caption{Model Map Construction}  \\label{domain}\n\\end{figure}\n\n\nThe only requirement of the map $\\Phi_0$ within the white area in Figure~\\ref{domain} will be that it is a smooth extension of the map which will be defined explicitly in the gray region. This can easily be achieved since the white area remains a fixed distance away from the singular set $\\Gamma$, and this clearly implies that the tension is bounded within the white area.\n\nFor convenience, we define a pair of harmonic functions needed in the construction. For $a\\in\\mathbb{R}$ let $r_{a}$ be the Euclidean distance from the point $z=a$ on the $z$-axis, and let $\\theta_a$ be the polar angle about this center. Then set\n\\begin{equation}\n\tu_a =\\log(r_a-(z-a))=\\log\\bigl(2r_a\\sin^2(\\theta_a\/2)\\bigr), \\quad\\quad\nv_a = \\log(r_a+(z-a))=\\log\\bigl(2r_a\\cos^2(\\theta_a\/2)\\bigr).\n\\end{equation}\nIt is easy to check that these functions are harmonic. Furthermore $u_a$ behaves like $2\\log\\rho$ near the $z>a$ part of the $z$-axis and is locally bounded below on the $z<a$ part of the $z$-axis. Also, clearly $u_a(\\rho,z-a) = v_a(\\rho,-(z-a))$ and hence $v_a$ behaves like $2\\log\\rho$ on the $z<a$ part of the $z$-axis and is locally bounded below on the $z>a$ part of the $z$-axis.\n\nWe begin with the definition of $\\Phi_0$ outside a large ball. The map there is based on the Minkowski metric \\eqref{Minkowski} and is given by\n\\begin{equation}\n\tF=\\begin{pmatrix} e^{u_0-\\log2} & 0 \\\\ 0 & e^{v_0-\\log2} \\end{pmatrix}, \\qquad \\omega=\\omega(\\theta),\n\\end{equation}\nwhere $\\theta=\\theta_0$. The function $\\omega(\\theta)$ is smooth and chosen so that $\\omega$ is the appropriate constant on $[0,\\epsilon]\\cup[\\pi-\\epsilon,\\pi]$, with $0<\\epsilon<\\pi\/2$ fixed so that $\\omega$ is constant on the regions $\\mathcal N_0$ and $\\mathcal S_0$. Observe that this map is harmonic wherever $\\omega$ is constant, since $G=0$ and $\\div F^{-1}\\nabla F=0$. It will now be shown that the tension $|\\tau|$ decays like $O(r^{-7\/2})$, which as will be seen later is sufficient for the main existence and uniqueness arguments. Since the tension vanishes for $\\theta\\in[0,\\epsilon]\\cup[\\pi-\\epsilon,\\pi]$, we need only estimate $|\\tau|$ on the interval $[\\epsilon,\\pi-\\epsilon]$. An explicit calculation gives\n\\begin{align}\n\\begin{split}\n\tf (\\operatorname{div}K)^t F (\\operatorname{div}K) =&\n    \\frac{4\\csc^2\\theta \\sin^2(\\theta\/2)}{r^7}\n    \\left[\\csc^4(\\theta\/2) \\bigl(\\omega_1''-(\\csc\\theta+2\\cot\\theta) \\omega_1'\\bigr)^2 \\right. \\\\\n    &\\left. + 4 \\csc^2\\theta \\bigl(\\omega_2''+(\\csc\\theta-2\\cot\\theta) \\omega_2'\\bigr)^2\\right]\\\\\n    =& O(r^{-7}),\n\\end{split}\n\\end{align}\nand\n\\begin{equation}\n\tG=\\frac{\\csc^2(\\theta\/2)\\sec^2(\\theta\/2)}{r^5}\n\t\\begin{pmatrix}\n\t\t \\omega _1'^2\\csc^2(\\theta\/2) &\n\t\t \\omega _1'\\omega _2\\csc^2(\\theta\/2) \\\\[1ex]\n\t\t \\omega _1'\\omega _2 \\sec^2(\\theta\/2) &\n\t\t \\omega _2'^2 \\sec^2(\\theta\/2) &\n\t\\end{pmatrix}\n\t=O(r^{-5}).\n\\end{equation}\nSince $\\div H=0$, it follows that $|\\tau|=O(r^{-7\/2})$.\n\nIt remains to define the map inside the two tubular neighborhoods capped with hemispheres. Consider first the northern tubular neighborhood. Let $z=b$ indicate the location of the point $N$. Then in this region define\n\\begin{equation}\n\tF=\\begin{pmatrix} e^{u} & 0 \\\\ 0 & e^{v} \\end{pmatrix}, \\qquad \\omega=\\mathbf{c}_1,\n\\end{equation}\nwhere\n\\begin{equation}\nu=\\lambda (u_0-\\log2) + (1-\\lambda) u_b,\\quad\\quad v=\\lambda (v_0-\\log2),\n\\end{equation}\nand $\\lambda=\\lambda(z)$ is a smooth cut-off function with $\\lambda=1$ in $\\mathcal N_0$ and $\\lambda=0$ in $\\mathcal N_1$. This leads to the correct rod structure, and the definitions outside the large ball and in $\\mathcal N_0$ agree. Moreover\n\\begin{equation}\n\t\\div H = \\begin{pmatrix}  \\Delta [\\lambda(u_0-u_b)] & 0 \\\\ 0 & \\Delta [\\lambda v_0]\\end{pmatrix},\n\\end{equation}\nwhich is bounded. Indeed\n\\begin{equation}\n\t\\Delta[\\lambda(u_0-u_b)] = (u_0-u_b)\\Delta\\lambda + 2(\\partial_{z}\\lambda) \\partial_z(u_0-u_b),\n\\end{equation}\nand $\\partial_z u_a = 1\/r_a$ (on the $z$-axis) for $a=0, b$\nis clearly bounded in the transition region. Similarly $\\Delta[\\lambda v_0]$ is bounded since $\\partial_z v_0 = -1\/r_0=-1\/r$ (on the $z$-axis) is bounded. It follows that $|\\tau|$ is bounded in the northern region, as $G=0$ and $K=0$ due to the constancy of $\\omega$.\n\n\nConsider now the southern tubular neighborhood. The map in $\\mathcal S_0$ is defined exactly as in $\\mathcal N_0$, that is with the same $F$ but with $\\omega=\\mathbf{c}_2$. In fact $\\omega$ is set to be the constant $\\mathbf{c}_2$ in the entire southern tubular neighborhood. Next, let the south pole $S$ and corner point $C$ be located at $z=c$ and $z=0$,\nrespectively. Then in $\\mathcal S_1$ the remainder of the map is defined by\n\\begin{equation}\n\tF= hF_0h^t = h \\begin{pmatrix} e^{u} & 0 \\\\ 0 & e^{v} \\end{pmatrix}h^t,\n\\end{equation}\nwhere\n\\begin{equation} \\label{h}\n\th = \\begin{pmatrix}  1& -p \\\\ 0 & 1 \\end{pmatrix}\n\\end{equation}\nand $v=v_0-\\log2$, $u=u_0-u_c$. As before $\\div(F_0^{-1}\\nabla F_0)=0$ and hence \\begin{equation}\n\\div(F^{-1}\\nabla F) = h^{-t} \\div(F_0^{-1}\\nabla F_0) h^t = 0,\n\\end{equation}\nwhere for notational convenience $h^{-t}:=(h^t)^{-1}$.\nIt follows that $\\Phi_0$ is a harmonic map in $\\mathcal{S}_1$. In order to verify\nthat the rod structure is correct, observe that\n\\begin{equation} \\label{structure}\n\tF\\vector{1}{0} = \\vector{e^u+p^2e^v}{-pe^v}, \\quad\\quad F\\vector01 = \\vector{-pe^v}{e^v}, \\quad\\quad F\\vector1p = \\vector{e^u}{0}.\n\\end{equation}\nFrom this it is clear that the only direction which degenerates on the disk rod (between $S$ and $C$) is $(1,p)$, and the only direction that degenerates on the south rod (below $C$) is $(0,1)$. Furthermore, since $F_0$ is nonsingular on the horizon rod the same is true of $F$.\n\nLastly, the map will be defined on the southern transition region. Recall that $\\omega$ is constant. Moreover if $F$ defined in $\\mathcal{S}_1$ can be transitioned\nto a diagonal $F$ satisfying $\\div(F^{-1}\\nabla F)=0$, then we can complete the transition in the same manner as in the northern transition region. Thus it remains to demonstrate the transition to a diagonal $F$. Set\n\\begin{equation}\n\tF =  h(z) F_0 h(z)^t, \\qquad\\quad  h(z) = \\begin{pmatrix}  1& -p\\lambda(z) \\\\ 0 & 1 \\end{pmatrix},\n\\end{equation}\nwhere $F_0$ is as above, and $\\lambda(z)$ is a smooth cut-off function which is equal to $1$ near $\\mathcal S_1$ and equal to $0$ near $\\mathcal S_0$. To verify that $\\div(F^{-1}\\nabla F)$ is bounded in the transition region compute\n\\begin{equation}\n\tF^{-1} \\nabla F\n\t= (F_0h^t)^{-1} (h^{-1}\\nabla h) F_0h^t + h^{-t}  (F_0^{-1}\\nabla F_0) h^t + h^{-t}\\nabla h,\n\\end{equation}\nand\n\\begin{align}\n\\begin{split}\n\\label{divFdF}\n\t\\div(F^{-1} \\nabla F) = &[\\nabla(F_0h^t)^{-1}]\\cdot (h^{-1}\\nabla h) F_0h^t + (F_0h^t)^{-1} \\div (h^{-1}\\nabla h) F_0h^t\\\\\n\t&+ (F_0h^t)^{-1} (h^{-1}\\nabla h)\\cdot \\nabla (F_0h^t)\n+ (\\nabla h^{-t}) \\cdot (F_0^{-1}\\nabla F_0) h^t\\\\\n\t&+h^{-t}  \\div(F_0^{-1}\\nabla F_0) h^t + h^{-t}  (F_0^{-1}\\nabla F_0)\\cdot \\nabla h^t + \\div(h^{-t}\\nabla h).\n\\end{split}\n\\end{align}\nEach term may now be estimated individually. First note that\nthe fifth term vanishes and the seventh term is clearly bounded.\nFurthermore\n\\begin{equation}\n\tF_0^{-1} \\nabla F_0 = \\begin{pmatrix} \\nabla u & 0 \\\\ 0 & \\nabla v \\end{pmatrix},\n\\end{equation}\nand since $h$ depends only on $z$ we may replace $\\nabla u$ and $\\nabla v$ in \\eqref{divFdF} by $\\partial_z u$ and $\\partial_z v$, respectively. As explained above these $z$-derivatives are bounded, and since $h^t$, $h^{-t}$, $\\partial_{z}h^t$ and $\\partial_{z}h^{-t}$ are bounded it follows that the fourth and sixth terms are bounded. Next observe that the second term becomes\n\\begin{equation}\n\t (F_0h^t)^{-1} \\div (h^{-1}\\nabla h) F_0h^t= pe^{v-u}\\lambda''\n\t \\begin{pmatrix} p\\lambda & -1 \\\\ p^2\\lambda^2 & -p\\lambda \\end{pmatrix},\n\\end{equation}\nwhich is bounded. Furthermore the sum of the first and third terms is\n\\begin{align}\n\\begin{split}\n\t&[\\nabla(F_0h^t)^{-1}]\\cdot (h^{-1}\\nabla h) F_0h^t\n\t+ (F_0h^t)^{-1} (h^{-1}\\nabla h)\\cdot \\nabla (F_0h^t)\\\\\n =&\n\tpe^{v-u}\\lambda'\n\t\\begin{pmatrix} p\\bigl[\\lambda (\\partial_z v-\\partial_z u)+\\lambda'\\bigr] & \\partial_z u-\\partial_z v \\\\\n\tp^2\\lambda\\bigl[\\lambda(\\partial_z v-\\partial_z u)+2\\lambda'\\bigr] & -p\\bigl[\\lambda(\\partial_z v-\\partial_z u)+\\lambda'\\bigr] \\end{pmatrix},\n\\end{split}\n\\end{align}\nwhich again is bounded. It follows that $|\\tau|$ is bounded in the southern region, and this completes the proof for the rod data set associated with a single component lens horizon $L(p,1)$.\n\n\\begin{remark} \\label{integer}\nWe note that in the argument above showing that $\\div(F^{-1}\\nabla F)$ is bounded no use was made of the fact that $p$ is an integer. This is will be important in what follows.\n\\end{remark}\n\nConsider now the case of a general rod data set, in which consecutive corners may be present. In this situation the map will be defined inductively one corner at a time, with a transition region between any two consecutive corners, as well as a transition region on each of the two semi-infinite rods.  The only feature which remains to be treated is the case of two consecutive corners. Suppose then that consecutive corners occur at points $C_N$ and $C_S$ along the $z$-axis, with $z=\\mathrm{a}$ and $z=\\mathrm{b}$ at $C_N$ and $C_S$ respectively. Let there be rod structures $(m,n)$ above $C_N$, $(p,q)$ between $C_N$ and $C_S$, and $(r,s)$ below $C_S$. It will be assumed that $m\\neq 0$, $p\\neq 0$, $r\\neq 0$, and that the generalized compatibility condition is satisfied\n\\begin{equation}\\label{1256}\nmr(ps-rq)(mq-np)\\leq 0.\n\\end{equation}\nNote that this quantity is nonzero (and hence negative) since $ps-rq\\neq 0$ and $mq-np\\neq 0$ due to the fact that $C_N$ and $C_S$ are genuine corners.\n\nLet $v=u_{\\mathrm{b}}-u_{\\mathrm{a}}$ and $u=2\\log\\rho-v$ and set\n\\begin{equation}\n\tF_0=\\begin{pmatrix} e^u & 0 \\\\ 0 & e^v \\end{pmatrix},\n\\end{equation}\nso that $F_0$ gives rod structure $(1,0)$ above $C_N$ and below $C_S$, and $(0,1)$ between $C_N$ and $C_S$. Next define $F_N=h_NF_0h_N^t$ near $C_N$ and $F_S=h_SF_0h_S^t$ near $C_S$, where\n\\begin{equation} \\label{hnhs}\n\th_N=\\begin{pmatrix} -q\/p & -n\/m \\\\ 1 & 1 \\end{pmatrix}, \\qquad\n\th_S=\\begin{pmatrix} -q\/p & -s\/r \\\\ 1 & 1 \\end{pmatrix}.\n\\end{equation}\nIt is straightforward to check that the maps $F_N$ and $F_S$ yield the desired rod structures on each of the three rods in neighborhoods of $C_N$ and $C_S$ respectively, and that $(F_N,\\omega)$ and $(F_S,\\omega)$ are harmonic whenever $\\omega$ is constant. This latter property arises from the fact that although $F\\mapsto hFh^t$, $\\omega\\mapsto h\\omega$ is an isometry of $\\tilde{\\mathbf{X}}$ if and only if $\\operatorname{det} h=\\pm1$, this determinant condition is not required here for the harmonic map equations to be satisfied since $\\omega$ is constant. It remains to define $F$ in a transition region between $C_N$ and $C_S$. In order to do this first let $\\bar{F}_N=\\mathbf{k} F_0 \\mathbf{k}^t$ and $\\bar{F}_S=F_0$, where\n\\begin{equation}\\label{987}\n\t\\mathbf{k} = h_S^{-1}h_N=\\begin{pmatrix} 1 & \\frac{p(ms-nr)}{m(ps-qr)} \\\\[1ex]\n           0 & -\\frac{r(mq-np)}{m(ps-qr)} \\end{pmatrix}.\n\\end{equation}\nIf there is a smooth transition $\\mathbf{k}=\\mathbf{k}(z)$ from $h_S^{-1}h_N$ to\n\\begin{equation}\\label{jfh}\n\t \\begin{pmatrix} 1 & \\frac{p(ms-nr)}{m(ps-qr)} \\\\[1ex] 0 & 1 \\end{pmatrix},\n\\end{equation}\nthen by Remark~\\ref{integer}\nit is clear that we can further transition $\\mathbf{k}$ to the identity as in the arguments above the remark, since the only difference between \\eqref{jfh} and $h$ in \\eqref{h} is the fact that the off-diagonal element is an integer in the latter matrix.\nIt follows that $\\bar{F}$ would then be defined in the whole region encompassing both corners, having the property that it is equal to $\\bar{F}_N$ near $C_N$ and equal to $\\bar{F}_S$ near $C_S$. Finally, taking $F=h_S \\bar{F} h_S^t$ produces a map with finite tension which coincides with $F_N$ near $C_N$ and $F_S$ near $C_S$.\n\n\n\nIt remains to define the transition from \\eqref{987} to \\eqref{jfh}. Set\n\\begin{equation}\n\t\\mathbf{k}(z) = \\begin{pmatrix} 1 & \\varsigma \\\\ 0 & \\lambda(z) \\end{pmatrix},\\quad\\quad\\quad \\varsigma=\\frac{p(ms-nr)}{m(ps-qr)},\n\\end{equation}\nwhere $\\lambda(z)$ is a smooth cut-off function satisfying $\\lambda(z)=-\\frac{r(mq-np)}{m(ps-qr)}$ near $C_N$ and $\\lambda(z)=1$ for $z<(\\mathrm{a}+\\mathrm{b})\/2$.\nAccording to the generalized compatibility condition \\eqref{1256},\n$\\lambda(z)$ may be chosen strictly positive. The arguments following \\eqref{divFdF} may now be repeated to show that the tension remains bounded. In particular, the terms four through seven of \\eqref{divFdF} are bounded in the current setting. By denoting $F_{\\mathbf{k}}=F_0 \\mathbf{k}^t$ the second term becomes\n\\begin{equation}\n\tF_{\\mathbf{k}}^{-1} \\div (\\mathbf{k}^{-1}\\nabla \\mathbf{k})F_{\\mathbf{k}} =  \\varsigma e^{v-u}\\lambda'\n\t\\begin{pmatrix} -\\dfrac{\\varsigma}{\\lambda} & -1 \\\\[1.5ex]\n\t\\dfrac{\\varsigma^2+e^{u-v}}{\\lambda^2} & \\dfrac{\\varsigma^2e^{u-v}}{\\varsigma\\lambda}\n\t\\end{pmatrix},\n\\end{equation}\nand the sum of the first and third terms is\n\\begin{equation}\n\t\\nabla F_{\\mathbf{k}}^{-1} \\cdot(\\mathbf{k}^{-1}\\nabla \\mathbf{k}) F_{\\mathbf{k}} + F_{\\mathbf{k}}^{-1} (\\mathbf{k}^{-1}\\nabla \\mathbf{k})\\cdot \\nabla F_{\\mathbf{k}} = \\varsigma e^{v-u}\\lambda'\n\t\\begin{pmatrix} \\dfrac{\\varsigma(u_z-v_z)}{\\lambda} & u_z - v_z - \\dfrac{\\lambda'}{\\lambda} \\\\[1.5ex]\n\t\\dfrac{\\varsigma^2\\lambda(u_z-v_z)+(\\varsigma^2+e^{u-v}\\lambda')}{\\lambda^3} & \\dfrac{\\varsigma(v_z-u_z)}{\\lambda}\n\t\\end{pmatrix},\n\\end{equation}\nboth of which are bounded. Similar arguments may be used to treat the cases when one of $m$, $p$, $r$ is zero.\n\\end{proof}\n\n\n\n\\section{Energy Estimates}\n\\label{sec6} \\setcounter{equation}{0}\n\\setcounter{section}{6}\n\n\nIn the rank 1 case treated in \\cite{weinstein96}, a priori estimates for the singular harmonic map problem relied heavily on the uniformly strict negative curvature of the target spaces. In the current setting the target symmetric space $\\mathbf{X}=SL(3,\\mathbb{R})\/SO(3)$ is of rank 2, that is the dimension of a maximal flat subspace is 2. It follows that $X$ is of nonpositive curvature and the methods of \\cite{weinstein96} break down. In order to overcome this difficulty, we will employ a generalization of horospherical coordinates from hyperbolic space so that the flat directions as well as the coordinate planes of strict negative curvature are explicitly identified, and are thus more easily exploited. Coordinate systems of the symmetric space $\\mathbf{X}=SL(3,\\mathbb{R})\/SO(3)$ have been investigated previously, as in \\cite{MazzeoVasy}, yet what we need requires a different set of properties.\n\nConsider the Iwasawa decomposition \\cite{BallmanGromovSchroeder} of $G=SL(3,\\mathbb{R})$ given by $G=KAN$ where the three subgroups are $K=SO(3)$,\n\\begin{equation}\nA = \\{ \\mathrm{diag} (\\lambda_1, \\lambda_2, \\lambda_3) \\,\\, | \\,\\, \\lambda_i > 0,\\text{ for }i=1,2,3,\\text{ } \\lambda_1 \\lambda_2 \\lambda_3 =1 \\},\n\\end{equation}\nand\n\\begin{equation}\nN=\\{\\text{upper triangular matrices with 1's on the diagonal}\\}.\n\\end{equation}\nFor each $g\\in G$ there exist unique elements $k\\in K$, $a\\in A$, and $n\\in N$ such that\n$g=kan$. Moreover by taking inverses we have $G=NAK$, and hence $\\mathbf{X}=G\/K$ may be identified with the subgroup $NA$. Let $x_0=[Id]\\in \\mathbf{X}$ then the orbit $A\\cdot x_0$ represents a maximal flat so that it is a totally geodesic submanifold with vanishing curvature. The last property follows from the curvature formula in Section \\ref{sec3}, and the fact that the Lie algebra\n\\begin{equation}\n\\mathfrak{a} = \\{ \\mathrm{diag} (\\lambda_1, \\lambda_2, \\lambda_3) \\,\\, | \\,\\, \\sum \\lambda_i = 0\\}\n\\end{equation}\nassociated with $A$ is abelian ie. $[\\alpha_1,\\alpha_2]=0$ for all $\\alpha_1, \\alpha_2\\in \\mathfrak{a}$.\nOn the other hand, the orbit $N\\cdot x_0$ is a horocycle determined by the Weyl chamber\n\\begin{equation}\n\\mathfrak{a}^+ = \\{ \\mathrm{diag} (\\lambda_1, \\lambda_2, \\lambda_3) \\,\\, | \\,\\,\n\\lambda_1>\\lambda_2>\\lambda_3,\\text{ }\\sum \\lambda_i = 0\\}\\subset\\mathfrak{a}.\n\\end{equation}\nIt is a closed submanifold with the property that every flat which is asymptotic to the Weyl chamber at infinity\n\\begin{equation}\n\\mathrm{w}^{+}:=(A^{+}\\cdot x_0)(\\infty)=\\{\\gamma(\\infty)\\mid \\gamma(s)=\\mathrm{exp}(s\\alpha^{+})\\cdot x_0,\\text{ }\\alpha^{+}\\in\\mathfrak{a}^{+}\\},\n\\end{equation}\nintersects the horocycle orthogonally in exactly one point; recall that a flat $\\mathcal{F}$ is asymptotic to a Weyl chamber $\\mathrm{w}$ at infinity if $\\mathrm{w}\\subset\\mathcal{F}(\\infty)$.\nIn particular, the horocycle $N \\cdot x_0$ and flat $\\mathcal{F}_{x_0}:=A \\cdot x_0$\nintersect orthogonally at $x_0$, as can be seen from the orthogonality between the respective Lie algebras $\\mathfrak{n}$ (all upper triangular matrices with zeros on the diagonal) and $\\mathfrak{a}$ with respect to the Riemannian metric at $x_0$ given in Section \\ref{sec3}.\n\nA foliation by flats may be constructed \\cite{BallmanGromovSchroeder} from the action of $N$. More precisely\n\\begin{equation}\n\\mathbf{X} = \\bigcup_{n \\in N} n \\cdot \\mathcal{F}_{x_0},\n\\end{equation}\nwhere $n\\cdot \\mathcal{F}_{x_0} \\cap n' \\cdot \\mathcal{F}_{x_0} = \\emptyset$ for $n \\neq n'$ and each $n \\cdot\\mathcal{F}_{x_0}$ is asymptotic to the Weyl chamber $\\mathrm{w}^+$. Since each point $x \\in \\mathbf{X}$ can be uniquely written as\n$n a \\cdot x_0$, and $a \\cdot \\mathcal{F}_{x_0} = \\mathcal{F}_{x_0}$ as sets,\nthe assignment $x \\mapsto \\mathcal{F}_x=na\\cdot\\mathcal{F}_{x_0}$ defines a smooth foliation of $\\mathbf{X}$ whose leaves are the set of totally geodesic submanifolds $\\{ n \\cdot F_{x_0}\\}_{n \\in N}$, each of which is isometric to ${\\mathbb R}^2$. By homogeneity of $\\mathbf{X} = G\/K$,  the 3-dimensional horocycle $N \\cdot x$ and the 2-dimensional flat $\\mathcal{F}_x$ intersect orthogonally at (and only at) $x$.  In this sense, the pair $(a, n)$ gives a horocyclic orthogonal coordinate system for $\\mathbf{X}$.\n\nLet $\\gamma_{x_0}(s)$ be an arc-length parameterized geodesic satisfying $\\gamma_{x_0}(0)  = x_0$, and $\\gamma_{x_0}(\\infty) \\in \\mathrm{w}^+$. Equivalently $\\gamma_{x_0}'(0) \\in T_{x_0} X$ is an element of a Weyl chamber $\\mathfrak{a}^+$, so that $\\gamma_{x_0}$ is regular in the sense that it is contained in a unique 2-dimensional flat, namely $\\mathcal{F}_{x_0}$. Since the action by $na$ on $\\mathbf{X}$ is isometric and preserves the combinatorial structure of the Weyl chambers projected to $\\mathbf{X}(\\infty)$, it follows that $\\gamma_{x}(s):=na\\cdot \\gamma_{x_0}(s)$ is a regular geodesic contained in the flat $n\\cdot\\mathcal{F}_{x_0}$, and is asymptotic to $\\mathrm{w}^+$. In fact, the distance $d_{\\mathbf{X}}(n\\cdot \\gamma_{x_0}(s),\\gamma_{x_0}(s))$ decays exponentially and $d_{\\mathbf{X}}(na\\cdot \\gamma_{x_0}(s),\\gamma_{x_0}(s))\\rightarrow d_{\\mathbf{X}}(a\\cdot x_0, x_0)$.\n\n\nOn the flat $\\mathcal{F}_{x_0}$ there is a natural Euclidean coordinate system $r=(r_1, r_2)$, where the origin is identified with $x_0$, the $r_1$-axis coincides with the regular geodesic $\\gamma_{x_0}(s)$, and the $r_2$-axis is the orthogonal line to $\\gamma_{x_0}(s)$.  The $r_1$ axis is chosen to have the opposite orientation from that of $\\gamma_{x_0}$, so that $r_1\\rightarrow\\infty$ corresponds to $s\\rightarrow -\\infty$, and similarly for $r_2$.\nThe $(r_1, r_2)$ coordinate system may then be pushed forward to the flat $n \\cdot \\mathcal{F}_{x_0}$ where the origin is identified with $n \\cdot x_0$, the $r_1$-axis is the geodesic $\\gamma_{n \\cdot x_0}(s)$, and the $r_2$-axis is again the orthogonal line to $\\gamma_{n x_0}(s)$ in the flat.  Hence the the horocyclic coordinates $(a, n)$ may be represented by $(r, n)$. Moreover, for each $n'\\in N$ there is an isometry which preserves the $r$-coordinates and for each $r'$ there is a diffeomorphism which preserves the $n$-coordinates\n\\begin{equation}\n\\Xi_{n'} : (r_1, r_2, n) \\mapsto (r_1, r_2, n' n),\\quad\\quad\n\\Xi_{r'}: (r_1, r_2, n) \\mapsto (r_1+r_{1}', r_2+r_{2}', n).\n\\end{equation}\nThe $r$-translations map horocycles to horocylces, and thus if\n$\\theta=(\\theta^1,\\theta^2,\\theta^3)$ is a system of global coordinates on $N\\cdot x_0\\cong\\mathbb{R}^3$ then they may be pushed forward to all horocycles by the action of $\\Xi_{r'}$.\nIt follows that $(r,\\theta)$ form a system of global coordinates on $\\mathbf{X}$ with the property that the coordinate fields $\\partial_{r_i}$ and $\\partial_{\\theta^j}$ are orthogonal. By combining the observations above, the $G$-invariant Riemannian metric on $\\mathbf{X}$ can be expressed in these coordinates by\n\\begin{equation}\n\\mathbf{g}= dr^2+Q(d\\theta,d\\theta) = dr_{1}^2 +dr_{2}^2 + Q_{ij}d \\theta^{i} d\\theta^{j},\n\\end{equation}\nwhere the coefficients $Q_{ij}=Q_{ij}(r,\\theta)$ are smooth functions.\n\n\nAs a demonstration of this framework in the simpler setting of rank 1, consider the hyperbolic plane $\\mathbb{H}^2$. The half plane coordinates $(U,V)$, $U>0$ may be transformed to orthogonal horocyclic coordinates $(r,\\theta)$ by $r=\\log U$ and $\\theta=V$ to find\n\\begin{equation}\n\\mathbf{g}_{-1}=\\frac{dU^2 + dV^2}{U^2} = dr^2 + e^{-2r} d\\theta^2.\n\\end{equation}\nHere the flat $\\mathcal{F}_{x_0}$ in the upper half plane model with $x_0 =(0,1)$ is the positive $U$-axis $\\{V=0\\}$, and the horocycle $N\\cdot x_0$ is the horizontal line $\\{U=1\\}$.\n\n\n\nFor any unit tangent vector $Z \\in T_x \\mathbf{X}$ perpendicular to $\\mathcal{F}_x$, the sectional curvature\n\\begin{equation}\n\\mathcal{K}(Z,\\gamma_{x}'(0))=\\langle R(Z, \\gamma_{x}'(0))\\gamma_{x}'(0), Z \\rangle\n\\end{equation}\nis negative, since $\\mathcal{F}_x$ is a flat of maximal dimension. Moreover, such curvatures are uniformly negative (bounded away from zero) by compactness of the set of unit normal vectors to $\\mathcal{F}_x$ and the homogeneity of $\\mathbf{X}$.  The uniform (in $x$ as well as choice of 2-plane) upper and the lower bounds of these curvatures will be denoted by\n\\begin{equation}\\label{curvaturebounds}\n-c^2 \\leq  \\mathcal{K} \\leq -b^{2} < 0.\n\\end{equation}\n\n\\begin{lemma}\\label{lemmaimhof}\nLet $J$ be a Jacobi field perpendicular to the\nflat $\\mathcal{F}_x$ along an arc-length parameterized geodesic $\\gamma(s)\\in\\mathcal{F}_x$. Assume further that the Jacobi field is stable in that it is bounded as $s\\rightarrow -\\infty$, then\n\\begin{equation}\ne^{bs}|J(0)| \\leq|J(s)|\\leq e^{cs}|J(0)|.\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\nThis follows with slight modification from the proof of Theorem 2.4 in \\cite{HeintzeImhof}, which relies on Proposition 4.1 in \\cite{ImhofRuh}. The key observation is that the proof of Proposition 4.1 in \\cite{ImhofRuh} does not\nuse the bounds on all sectional curvatures, but rather only those appearing in \\eqref{curvaturebounds}.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lemmaQ}\nFor any vector $\\xi\\in\\mathbb{R}^3$ and $i=1,2$\n\\begin{equation} \\label{ineq:ab}\n2b Q(\\xi,\\xi) \\leq \\partial_{r_i} Q(\\xi,\\xi) \\leq 2c Q(\\xi,\\xi).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\psi:\\mathbb{R}^5\\rightarrow\\mathbf{X}$ denote the global coordinate patch constructed above, so that $\\psi^{-1}(x)=(r(x),\\theta(x))$. Consider the geodesics\n$\\gamma_{\\xi_0+\\varepsilon\\xi}: s\\mapsto \\psi(s,0 ,\\xi_0+\\varepsilon\\xi)$ where $\\varepsilon$ is a variation parameter and $\\xi_0\\in\\mathbb{R}^3$ is fixed. If $v=(0,\\xi)\\in\\mathbb{R}^5$ then $J_{\\xi}=d\\psi(v)$ is a Jacobi field along the geodesic $\\gamma_{\\xi_0}$. Moreover this Jacobi field is stable since\n$d_{\\mathbf{X}}(\\gamma_{\\xi_0+\\varepsilon\\xi}(s),\\gamma_{\\xi_0}(s))$ is bounded as $s\\rightarrow -\\infty$. Observe that\n\\begin{equation}\nQ_{x}(\\xi,\\xi)=\\mathbf{g}(d\\psi_{\\psi^{-1}(x)}(v),d\\psi_{\\psi^{-1}(x)}(v))\n=|J_{\\xi}(x)|^2,\n\\end{equation}\nso the inequalities \\eqref{ineq:ab} measure the logarithmic growth rate of stable Jacobi fields.\n\nIf $s \\leq t$ then Lemma \\ref{lemmaimhof} implies that\n\\begin{equation}\ne^{2b(t-s)}|J_{\\xi}(\\gamma_{\\xi_0}(s))|^2\n\\leq|J_{\\xi}(\\gamma_{\\xi_0}(t))|^2\\leq e^{2c(t-s)}|J_{\\xi}(\\gamma_{\\xi_0}(s))|^2.\n\\end{equation}\nThe desired result now follows for $i=1$ by taking logarithms, dividing by $t -s$, and letting $t\\rightarrow s$. Similar arguments hold for $i=2$.\n\\end{proof}\n\n\nConsider a smooth map $\\varphi:\\mathbb{R}^3\\setminus\\Gamma\\rightarrow \\mathbf{X}$ with Dirichlet energy density\n\\begin{equation}\n|d \\varphi|^2 = | \\nabla (r_1 \\circ \\varphi) |^2 + | \\nabla (r_2 \\circ \\varphi) |^2 + Q  \\Big( \\nabla (\\theta \\circ \\varphi), \\nabla (\\theta \\circ \\varphi) \\Big),\n\\end{equation}\nwhere the norms are computed with respect to the Euclidean metric $\\delta$ in \\eqref{flatmetric} and\n\\begin{equation}\nQ \\Big( \\nabla (\\theta \\circ \\varphi), \\nabla (\\theta \\circ \\varphi) \\Big)\n=Q_{ij}\\left(\\partial_{\\rho}\\theta^{i}\\partial_{\\rho}\\theta^{j}\n+\\partial_{z}\\theta^{i}\\partial_{z}\\theta^{j}\\right).\n\\end{equation}\nLet $\\overline{\\Omega}\\subset\\mathbb{R}^3\\setminus\\Gamma$ be the closure of a bounded domain situated away from the axis, and define the local Dirichlet energy\n\\begin{equation}\nE_{\\Omega}(\\varphi)=\\frac{1}{2}\\int_{\\Omega}|d\\varphi|^2.\n\\end{equation}\nTwo of the harmonic map equations associated with the Dirichlet energy are\n\\begin{equation}\\label{hmequations}\n\\Delta_\\delta r_i = \\partial_{r_i} Q(\\nabla \\theta, \\nabla \\theta),\\quad\\quad\\quad i= 1,2.\n\\end{equation}\nIt then follows from Lemma \\ref{lemmaQ} that each $r_i$ is subharmonic. Therefore if $\\overline{\\Omega}\\subset\\Omega'$ with $\\overline{\\Omega'}\\subset\\mathbb{R}^{3}\\setminus\\Gamma$ and $\\chi\\in C^{\\infty}_{c}(\\Omega')$ is a cut-off function with $\\chi=1$ on $\\Omega$, then multiplying by $\\chi^2 r_i$ and integrating by parts produces\n\\begin{equation}\\label{estimate1}\n\\int_{\\Omega'} \\chi^2 |\\nabla r_i|^2 \\leq 4 \\left(\\sup_{\\Omega'} r_i^2\\right)  \\int_{\\Omega'} | \\nabla \\chi |^2.\n\\end{equation}\n\nNext combine \\eqref{ineq:ab} with \\eqref{hmequations} to obtain\n\\begin{equation}\n\\Delta_{\\delta}r_{i}\\geq 2b Q(\\nabla\\theta,\\nabla\\theta).\n\\end{equation}\nThen multiplying by $\\chi^2$, integrating by parts, and applying \\eqref{estimate1} yields\n\\begin{equation}\\label{estimate2}\n\\int_{\\Omega'} \\chi^2 Q(\\nabla \\theta, \\nabla \\theta) \\leq \\frac{1}{b} \\int_{\\Omega'} \\chi \\nabla \\chi \\cdot \\nabla r \\leq\n\\frac{2}{b} \\left(\\sup_{\\Omega'} r_i\\right)  \\int_{\\Omega'}| \\nabla \\chi |^2.\n\\end{equation}\nTogether \\eqref{estimate1} and \\eqref{estimate2} give the desired local energy estimate\n\\begin{equation}\nE_{\\Omega}(\\varphi) \\leq \\Big[ 4 ( \\sup_{\\Omega'} r_1^2 +  \\sup_{\\Omega'} r_2^2 ) + \\frac{2}{b} (\\sup_{\\Omega'} r_1 + \\sup_{\\Omega'} r_2 )  \\Big] \\int_{\\Omega'}|\\nabla \\chi|^2.\n\\end{equation}\n\n\\begin{theorem}\\label{energybound}\nLet $\\varphi:\\mathbb{R}^3 \\setminus\\Gamma\\rightarrow\\mathbf{X}$ be a harmonic map and $\\Omega\\subset\\mathbb{R}^3 \\setminus\\Gamma$ be a bounded domain. If $\\varphi:\\Omega\\rightarrow B_{\\mathcal{R}}(x_0)$ then\n\\begin{equation}\nE_{\\Omega}(\\varphi)\\leq C,\n\\end{equation}\nwhere the constant $C$ depends only on the radius $\\mathcal{R}$ of the geodesic ball and $\\Omega$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Existence and Uniqueness} \\label{existence}\n\\label{sec7} \\setcounter{equation}{0}\n\\setcounter{section}{7}\n\n\n\n\nIn this section, we complete the proof of Theorem \\ref{main} and prove the existence and uniqueness of a harmonic map $\\varphi\\colon\\mathbb{R}^3\\setminus\\Gamma\\to \\mathbf{X}$ asymptotic to the model map $\\varphi_0$ constructed in Section \\ref{sec5}. Now that all the ingredients are in place, the proof is the same as in \\cite{weinstein96}. Nevertheless, we include it here for the sake of completeness.\nLet $\\varepsilon>0$ and define $\\Omega_\\varepsilon=\\{y\\in\\mathbb{R}^3\\colon d_{\\mathbb{R}^3}(y,\\Gamma)>\\varepsilon, \\text{ } y\\in B_{1\/\\varepsilon}(0)\\}$. Since the target $\\mathbf{X}$ is nonpositively curved, there is a smooth harmonic map $\\varphi_\\varepsilon\\colon\\Omega_\\varepsilon\\to\\mathbf{X}$ such that $\\varphi_\\varepsilon=\\varphi_0$ on $\\partial\\Omega_\\varepsilon$. We quote the following lemma from \\cite{weinstein96}, which essentially shows that the obstruction to a subharmonic distance function is given by the tension.\n\n\\begin{lemma}\nLet $\\varphi_1,\\varphi_2\\colon\\Omega\\to\\mathbf{X}$ be smooth maps into a nonpositively curved target. Then\n\\begin{equation}\n\t\\Delta\\left( \\sqrt{1 + d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)^2} \\right) \\geq\n\t-\\left( |\\tau(\\varphi_1)| + |\\tau(\\varphi_2)| \\right).\n\\end{equation}\n\\end{lemma}\n\nSet $\\varphi_1=\\varphi_\\varepsilon$ and $\\varphi_2=\\varphi_0$, and note that $\\tau(\\varphi_\\varepsilon)=0$. The remaining tension may be estimated by $\\Delta w\\leq -|\\tau(\\varphi_0)|$, where $w>0$ and $w\\rightarrow 0$ at infinity in $\\mathbb{R}^3$.\nThis is possible due to the boundedness and decay of $|\\tau(\\varphi_0)|$ as given in Theorem \\ref{model}. In particular we may take $w=c(1+r^2)^{-1\/4}$ so that\n\\begin{equation}\n\\Delta w\\leq -\\frac{c}{4}(1+r^2)^{-5\/4}\\leq -|\\tau(\\varphi_0)|,\n\\end{equation}\nif the constant $c>0$ is chosen sufficiently large. It follows that\n\\begin{equation}\n\t\\Delta\\left( \\sqrt{1 + d_{\\mathbf{X}}(\\varphi_{\\varepsilon},\\varphi_0)^2} - w \\right) \\geq 0, \\quad\\quad\\quad\\quad\n\t\\sqrt{1 + d_{\\mathbf{X}}(\\varphi_{\\varepsilon},\\varphi_0)^2} - w \\leq 1 \\text{ } \\text{ on }\\text{ }\\partial\\Omega_\\varepsilon.\n\\end{equation}\nThe maximum principle then yields a uniform $L^\\infty$ bound\n\\begin{equation}\\label{c0bound}\n\\sqrt{1 + \\operatorname{dist}_{\\mathbf{X}}(\\varphi_{\\varepsilon},\\varphi_0)^2} \\leq 1 + w \\text{ }\\text{ }\\text{ on }\\text{ }\\text{ } \\Omega_\\varepsilon.\n\\end{equation}\nFix a domain $\\Omega$ such that $\\overline{\\Omega}\\subset\\mathbb{R}^3\\setminus\\Gamma$ and take $\\varepsilon>0$  small enough to have $\\overline{\\Omega}\\subset\\Omega_\\varepsilon$. The $L^\\infty$ estimate combined with Theorem \\ref{energybound} produces an energy bound on $\\Omega$ independent of $\\varepsilon$. Furthermore, consider the Bochner identity\n\\begin{equation}\n\\Delta|d\\varphi_\\varepsilon|^2 = |\\hat{\\nabla} d\\varphi_\\varepsilon|^2\n\t- \\prescript{\\mathbf{X}}{}{\\operatorname{Riem}(d\\varphi_\\varepsilon,\nd\\varphi_\\varepsilon,d\\varphi_\\varepsilon,d\\varphi_\\varepsilon)}.\n\\end{equation}\nNonpositivity of the curvature shows that $|d\\varphi_{\\varepsilon}|^2$ is subharmonic. Thus a Moser iteration may be applied to find a uniform pointwise bound from the the energy estimate, namely\n\\begin{equation}\n\t\\sup_{\\Omega'} |d\\varphi_\\varepsilon|^2 \\leq C \\int_\\Omega |d\\varphi_\\varepsilon|^2\\leq C'\n\\end{equation}\nwhere $\\overline{\\Omega}'\\subset\\Omega$. Finally, using the harmonic map equations combined with the pointwise gradient and $L^{\\infty}$ bounds, we may now bootstrap to obtain uniform a priori estimates for all derivatives of $\\varphi_\\varepsilon$ on $\\Omega'$.\nBy letting $\\varepsilon\\rightarrow 0$, it follows that there exists a subsequence which converges together with any number of derivatives on $\\Omega'$. In the usual way, by choosing a sequence of exhausting domains and taking a diagonal subsequence, a sequence $\\varphi_{\\varepsilon_{i}}$ is produced which converges uniformly on compact subsets as $\\varepsilon_i\\rightarrow 0$. The limit $\\varphi$ is smooth and harmonic, and satisfies the $L^\\infty$ bound so that it is also asymptotic to $\\varphi_0$.\n\nThe proof of uniqueness is straightforward. If $\\varphi_1$ and $\\varphi_2$ are two harmonic maps asymptotic to $\\varphi_0$, then they are asymptotic to each other so that $d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)\\leq C$.\nMoreover\n\\begin{equation}\n\\Delta\\left(\\sqrt{1 + d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)^2} \\right)\\geq 0,\n\\end{equation}\nand since the set $\\Gamma$ on which $d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)$ may not be fully regular is of codimension 2, $\\sqrt{1 + d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)^2}$ is weakly subharmonic and the maximum principle applies \\cite{Weinstein}*{Lemma 8}.\nAs $\\sqrt{1 + d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)^2}\\to 1$ at infinity, it follows that $\\sqrt{1 + d_{\\mathbf{X}}(\\varphi_1,\\varphi_2)^2}\\leq 1$.\nConsequently $\\varphi_1=\\varphi_2$.\n\n\n\\subsection{Rod Data for the Harmonic Map}\n\n\n\n\nHaving constructed a harmonic map asymptotic to a prescribed model map, it remains to show that the rod data set arising from the harmonic map agrees with that of the model map. Let $\\Phi=(F,\\omega):\\mathbb{R}^3\\setminus\\Gamma\\rightarrow\\tilde{\\mathbf{X}}\\cong SL(3,\\mathbb{R})\/SO(3)$ denote the characterization of the harmonic map in the space of symmetric positive definite matrices, and let $\\Phi_0=(F_0,\\omega_0)$ denote the model map asymptotic to $\\Phi$. Recall that $F=(f_{ij})$ is a $2\\times 2$ symmetric positive definite matrix on $\\mathbb{R}^3\\setminus\\Gamma$ representing the fiber metric (associated with the rotational Killing directions) in a bi-axisymmetric stationary spacetime, and $\\omega=(\\omega_1,\\omega_2)^t$ are the twist potentials. The rod data associated with $\\Phi$ consists of the kernel of $F$ and the value of $\\omega$ on the axis.\n\n\\begin{theorem}\\label{rodstructharmonic}\nIf $\\Phi$ is asymptotic to $\\Phi_0$ then $\\mathrm{ker} \\text{ }F=\\mathrm{ker}\\text{ } F_0$ at each point of $\\Gamma$, and $\\omega=\\omega_0$ on each axis rod. In particular, the two maps respect the same rod data set. Furthermore, if $\\Phi$ is harmonic then $d_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)\\rightarrow 0$ at infinity in $\\mathbb{R}^3$.\n\\end{theorem}\n\nBefore proving this result we record several observations. Since the metric on $\\tilde{\\mathbf{X}}$ is $G$-invariant, the distance function is preserved under the action of left translation\n\\begin{equation}\nd_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)=\nd_{\\tilde{\\mathbf{X}}}(Id,\\tilde{L}_{B^{-1}}\\Phi),\n\\end{equation}\nwhere $B\\in SL(3,\\mathbb{R})$ satisfies $BB^{t}=\\Phi_0$. Note that\n\\begin{equation}\n\\tilde{L}_{B^{-1}}\\Phi=B^{-1}\n\\Phi(B^{-1})^{t}=e^{W}\n\\end{equation}\nfor some symmetric $W$ with $\\mathrm{Tr}\\text{ }W=0$. Since the Riemannian exponential map and the matrix exponential coincide for $\\tilde{\\mathbf{X}}$, Hadamard's theorem applies (using the fact that $\\tilde{\\mathbf{X}}$ is complete, simply connected, with nonpositive curvature) to show that the exponential map is a diffeomorphism, and the geodesic $\\gamma(t)= e^{tW}$ is minimizing. Therefore \\eqref{identitymetric} yields\n\\begin{equation}\nd_{\\tilde{\\mathbf{X}}}(Id,\\tilde{L}_{B^{-1}}\\Phi)=|\\gamma'(0)|=|W|\n=\\sqrt{\\mathrm{Tr}(WW^t)}\n=\\sqrt{\\mathrm{Tr}(W^2)}.\n\\end{equation}\n\nNow consider the function from the Mazur identity \\cite{IdaIshibashiShiromizu}, namely\n\\begin{align}\n\\begin{split}\n\\mathrm{Tr}\\left(\\Phi_{0}^{-1}\\Phi\\right)=&\n\\mathrm{Tr}\\left((B^{-1})^{t}B^{-1}\n\\Phi(B^{-1})^{t}B^{t}\\right)\\\\\n=&\\mathrm{Tr}\\left(B^{-1}\\Phi(B^{-1})^t\\right)\\\\\n=&\\mathrm{Tr}\\text{ } e^{W}.\n\\end{split}\n\\end{align}\nSince $e^{W}$ is symmetric and positive definite it may be diagonalized with positive eigenvalues\n$\\lambda_{i}$, $i=1,2,3$. We then have\n\\begin{equation}\n\\mathrm{Tr}\\text{ }e^{W}=\\lambda_{1}+\\lambda_{2}+\\lambda_{3},\\quad\\quad\\quad\n\\mathrm{Tr}(W^2)=(\\log\\lambda_{1})^2+(\\log\\lambda_{2})^2+(\\log\\lambda_{3})^2,\n\\end{equation}\nand since $W$ has zero trace\n\\begin{equation}\\label{jfjfj}\n\\log\\lambda_1+\\log\\lambda_2+\\log\\lambda_3=0.\n\\end{equation}\nIf $\\mathrm{Tr}\\text{ }e^{W}\\leq c$ then it is not difficult to see that \\eqref{jfjfj} implies $\\mathrm{Tr}(W^2)\\leq c_1$. Conversely if $\\mathrm{Tr}(W^2)\\leq c^2$\nthen each $|\\log\\lambda_{i}|\\leq c$, and it holds that $\\mathrm{Tr}\\text{ }e^{W}\\leq 3e^{c}$. We have thus proved the following.\n\n\\begin{lemma}\\label{lem1}\n$d_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)$ is uniformly bounded if and only if\nthe Mazur quantity $\\mathrm{Tr}\\left(\\Phi_{0}^{-1}\\Phi\\right)$\nis uniformly bounded.\n\\end{lemma}\n\n\\noindent\\textit{Proof of Theorem \\ref{rodstructharmonic}.}\nIf $\\Phi$ is asymptotic to $\\Phi_0$ then $d_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)\\leq c_0$, that is the distance is uniformly bounded, in particular near $\\Gamma$.\nBy Lemma \\ref{lem1} this implies that the Mazur function is also uniformly bounded\n\\begin{equation}\n\\mathrm{Tr}\\left(\\Phi_{0}^{-1}\\Phi\\right)\\leq c.\n\\end{equation}\nMoreover this quantity may be computed in terms of $F$, $F_0$, $\\omega$, and $\\omega_0$ as\n\\begin{equation}\\label{computationFF}\n\\mathrm{Tr}\\left(\\Phi_{0}^{-1}\\Phi\\right)\n=\\frac{f_0}{f}+\\mathrm{Tr}(FF_{0}^{-1})+\\frac{1}{f}(\\omega-\\omega_0)^{t}F_{0}^{-1}\n(\\omega-\\omega_0),\n\\end{equation}\nwhere $f=\\operatorname{det} F$ and $f_0=\\operatorname{det} F_0$. Since each of the terms on the right-hand side is nonnegative and the roles of $\\Phi$ and $\\Phi_0$ maybe reversed, we have\n\\begin{equation}\\label{potential}\n\\frac{f_0}{f}\\leq c,\\quad\\quad \\frac{f}{f_0}\\leq c,\\quad\\quad\n\\mathrm{Tr}(FF_{0}^{-1})\\leq c,\\quad \\quad \\frac{1}{f}(\\omega-\\omega_0)^{t}F_{0}^{-1}(\\omega-\\omega_{0})\\leq c.\n\\end{equation}\nIt follows that\n\\begin{equation}\nc^{-1}f_{0}\\leq f\\leq cf_{0}.\n\\end{equation}\n\nNext, since $F_0$ is symmetric it may be diagonalized with an orthogonal matrix $O$ so that $F_0=ODO^t$ where $D=\\mathrm{diag}(\\mu_1, \\mu_2)$. Working now at a point on an axis rod, the kernel of $F_0$ is 1-dimensional and so it may be assumed without loss of generality that $c^{-1}f_0\\leq \\mu_1\\leq cf_0$ and $0<c^{-1}\\leq \\mu_2\\leq c$. Let $\\tilde{F}=O^t FO$ then\n\\begin{equation}\n\\mathrm{Tr}(FF_{0}^{-1})=\\mathrm{Tr}(FO D^{-1}O^t)=\\mathrm{Tr}(O O^t FO D^{-1}O^t)\n=\\mathrm{Tr}(\\tilde{F}D^{-1})=\\tilde{f}_{11}\\mu_{1}^{-1}+\\tilde{f}_{22}\\mu_{2}^{-1}.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\tilde{f}_{11}\\mu_{2}+\\tilde{f}_{22}\\mu_{1}\\leq c\\mu_1 \\mu_2= cf_0,\n\\end{equation}\nso that\n\\begin{equation}\n\\tilde{f}_{11}\\leq cf_0,\\quad\\quad\\tilde{f}_{22}\\leq \\frac{cf_0}{\\mu_1}\\leq c_1.\n\\end{equation}\nFurthermore\n\\begin{equation}\nf=\\tilde{f}_{11}\\tilde{f}_{22}-\\tilde{f}_{12}^{2}\\leq\\tilde{f}_{11}\\tilde{f}_{22}\n\\leq cf_{0}\\tilde{f}_{22},\n\\end{equation}\nwhich produces the lower bound\n\\begin{equation}\n\\tilde{f}_{22}\\geq \\frac{c^{-1}f}{f_0}\\geq c_{1}^{-1}.\n\\end{equation}\nIn order to control the cross terms, observe that from the above\n\\begin{equation}\n\\tilde{f}_{12}^{2}=\\tilde{f}_{11}\\tilde{f}_{22}+f\\leq cf_0.\n\\end{equation}\nIn conclusion we obtain\n\\begin{equation}\n\\tilde{f}_{11}\\leq cf_0,\\quad\\quad |\\tilde{f}_{12}|\\leq c\\sqrt{f_0},\\quad\\quad\nc^{-1}\\leq \\tilde{f}_{22}\\leq c.\n\\end{equation}\nTherefore, on an axis rod both $D=O^t F_0 O$ and $\\tilde{F}=O^t FO$ have the same kernel, and thus $F_0$ and $F$ have the same kernel. Similar arguments hold for a horizon rod.\n\nLet us now show that the potentials agree on an axis rod. From \\eqref{potential} it holds that\n\\begin{equation}\n(\\tilde{\\omega}-\\tilde{\\omega}_0)^{t}D^{-1}(\\tilde{\\omega}-\\tilde{\\omega}_0)\n=(\\omega-\\omega_0)^{t}F_{0}^{-1}(\\omega-\\omega_0)\\leq cf\\leq c_1 f_0,\n\\end{equation}\nwhere\n\\begin{equation}\n(\\tilde{\\omega}-\\tilde{\\omega}_0)=O^t(\\omega-\\omega_0).\n\\end{equation}\nIt follows that\n\\begin{equation}\n\\mu_{1}^{-1}(\\tilde{\\omega}^1-\\tilde{\\omega}_{0}^1)^2\n+\\mu_{2}^{-1}(\\tilde{\\omega}^2-\\tilde{\\omega}_{0}^2)^2\\leq cf_0,\n\\end{equation}\nwhich implies\n\\begin{equation}\nf_{0}^{-1}(\\tilde{\\omega}^1-\\tilde{\\omega}_{0}^1)^2\n+(\\tilde{\\omega}^2-\\tilde{\\omega}_{0}^2)^2\\leq c_1 f_0.\n\\end{equation}\nWe then have\n\\begin{equation}\n|\\tilde{\\omega}-\\tilde{\\omega}_{0}|^2\\leq cf_0\\quad\\quad \\Rightarrow\n\\quad\\quad |\\omega-\\omega_0|^2\\leq cf_0,\n\\end{equation}\nshowing that $\\omega=\\omega_0$ on an axis rod.\n\nLastly if $\\Phi$ is harmonic then according to the $L^\\infty$ bound \\eqref{c0bound}, which holds globally for $\\varphi$ in place of $\\varphi_{\\varepsilon}$, it must hold that\n$d_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)\\rightarrow 0$ at infinity in $\\mathbb{R}^3$ since\n$w\\rightarrow 0$ in this limit. \\qed\n\n\n\n\n\n\\section{Reconstruction of the Spacetime Metric} \\label{reconstruct}\n\\label{sec8} \\setcounter{equation}{0}\n\\setcounter{section}{8}\n\n\n\n\n\nLet $\\Phi=(F,\\omega):\\mathbb{R}^3\\setminus\\Gamma\\rightarrow\\tilde{\\mathbf{X}}\\cong\nSL(3,\\mathbb{R})\/SO(3)$ be the harmonic map constructed in the previous section, represented in the space of symmetric positive definite matrices. Here we show how to build an asymptotically flat bi-axisymmetric stationary vacuum spacetime, which inherits the prescribed rod data set associated with the harmonic map. Note that the functions $F=(f_{ij})$ and $\\omega=(\\omega_1,\\omega_2)^t$ comprising the harmonic map are defined and smooth on the right-half plane $\\{(\\rho,z)\\mid \\rho>0\\}$, which will serve as the orbit space for the spacetime. The spacetime metric is given by \\eqref{metric}, and it suffices to show how each coefficient in \\eqref{metric} arises from $\\Phi$. The resulting spacetime will be asymptotically flat in light of the\ndecay of the model map $\\Phi_0$ and the fact that, by\nTheorem \\ref{rodstructharmonic}, $d_{\\tilde{\\mathbf{X}}}(\\Phi_0,\\Phi)\\rightarrow 0$ at infinity in $\\mathbb{R}^3$.\n\nFirst observe that $\\sigma$ is immediately obtained from \\eqref{sigma}, since the orbit space is simply connected and the form on the right-hand side is closed as a result of the harmonic map equations. It remains to find $A^{(i)}=v^{i}dt$, which will be derived from the harmonic map components $\\omega_{i}$. By solving for $dA^{(i)}$ in \\eqref{chi} we get\n\\begin{equation}\\label{0}\nd A^{(i)}=-\\frac{1}{2}f^{-1}f^{ij}\\star_{3} d\\omega_{j}.\n\\end{equation}\nObserve that from Cartan's magic formula and the fact that $\\partial_{t}$ is a Killing field\n\\begin{equation}\n\\iota_{\\partial_{t}} d A^{(i)}=-d \\iota_{\\partial_{t}}A^{(i)}=-dv^{i}.\n\\end{equation}\nIt follows that if\n\\begin{equation}\\label{1}\n\\iota_{\\partial_{t}}\\left(f^{-1}f^{ij}\\star_{3} d\\omega_{j}\\right)\n\\end{equation}\nis closed, then we may find $v^{i}$ by quadrature from the equation\n\\begin{equation}\ndv^{i}=\\frac{1}{2}\\iota_{\\partial_{t}}\\left(f^{-1}f^{ij}\\star_{3} d\\omega_{j}\\right).\n\\end{equation}\nIt turns out that showing \\eqref{1} is closed is equivalent to parts of the harmonic map equations. To see this, let $\\epsilon_{3}$ denote the volume form of $g_{3}$. Then\n\\begin{equation}\n(\\star_{3} d\\omega_{j})^{ab}=\\epsilon_{3}^{abc}\\partial_{c}\\omega_{j},\n\\end{equation}\nand\n\\begin{align}\\label{222}\n\\begin{split}\n\\iota_{\\partial_{t}}\\star_{3}d\\omega_{j}\n=&\\epsilon_{3}(\\partial_{t},\\partial_{\\rho},\\partial_{c})\n\\partial^{c}\\omega_{j} d\\rho\n+\\epsilon_{3}(\\partial_{t},\\partial_{z},\\partial_{c})\\partial^{c}\\omega_{j} dz\\\\\n=&\\rho e^{2\\sigma}\\partial^{z}\\omega_{j} d\\rho\n-\\rho e^{2\\sigma}\\partial^{\\rho}\\omega_{j} dz\\\\\n=&\\rho\\partial_{z}\\omega_{j} d\\rho-\\rho\\partial_{\\rho}\\omega_{j} dz.\n\\end{split}\n\\end{align}\nTherefore\n\\begin{align}\n\\begin{split}\nd\\left(f^{-1}f^{ij}\\iota_{\\partial_{t}}\\star_{3} d\\omega_{j}\\right)=&\nd\\left(\\rho f^{-1}f^{ij}\\partial_{z}\\omega_{j} d\\rho\n-\\rho f^{-1}f^{ij}\\partial_{\\rho}\\omega_{j} dz\\right)\\\\\n=&\\left[\\partial_{z}(\\rho f^{-1}f^{ij}\\partial_{z}\\omega_{j})+\n\\partial_{\\rho}(\\rho f^{-1}f^{ij}\\partial_{\\rho}\\omega_{j}) \\right]dz\\wedge d\\rho\\\\\n=&\\operatorname{div}_{\\mathbb{R}^3}\\left(f^{-1} f^{ij}\\nabla\\omega_{j}\\right)dz\\wedge d\\rho\\\\\n=&0,\n\\end{split}\n\\end{align}\nwhere the last equality arises from the second set of harmonic maps equations\nin \\eqref{eulerlagrange}.\nAnother way to obtain this calculation is to observe that\n\\begin{equation}\\label{2}\nf^{-1}f^{ij}\\iota_{\\partial_{t}}\\star_{3} d\\omega_{j}\n=\\ast\\left(f^{-1} f^{ij} d\\omega_{j}\\right)\n\\end{equation}\nand $\\operatorname{div}_{\\mathbb{R}^3}=\\ast d \\ast$,\nwhere $\\ast$ is the Hodge star operator with respect to $\\delta$ on $\\mathbb{R}^3$.\nLastly, it is clear from the equations involved that $\\sigma$ and $v^i$ are bi-axisymmetric.\n\n\n\n\n\\subsection{Regularity} \\label{conical}\n\n\n\nThe metric reconstructed above from a solution of the harmonic map problem is defined on $\\mathbb{R}\\times(\\mathbb{R}^3\\setminus\\Gamma)\\times U(1)$. In order to extend this metric across $\\Gamma$, two steps must be completed as described below.\n\n\\subsubsection{Analytic regularity}\n\nThe metric coefficients in~\\eqref{metric} must be shown to be smooth and even in $\\rho$ up to $\\Gamma$. This was achieved in the 4D case in~\\cite{weinstein90}, and then extended to the non-axially symmetric case in~\\cite{litian}. We believe that these methods are applicable to the 5D setting as well.\n\n\n\\subsubsection{Conical singularities}\n\nIn addition to the analytic regularity mentioned above, conical singularities on axis rods must be ruled out.  A conical singularity at a point on an axis rod $\\Gamma_{l}$ is measured by the angle deficiency $\\theta\\in(-\\infty,2\\pi)$ given by\n\\begin{equation}\n\\frac{2\\pi}{2\\pi-\\theta}=\\lim_{\\rho\\rightarrow 0}\\frac{2\\pi\\cdot\\mathrm{Radius}}\n{\\mathrm{Circumference}}=\\lim_{\\rho\\rightarrow 0}\n\\frac{\\int_{0}^{\\rho}\\sqrt{f^{-1}e^{2\\sigma}}}\n{\\sqrt{f_{ij}u^{i}u^{j}}}=\\lim_{\\rho\\rightarrow 0}\n\\sqrt{\\frac{\\rho^{2}f^{-1}e^{2\\sigma}}{f_{ij}u^{i}u^{j}}},\n\\end{equation}\nwhere $u=(u^1,u^2)=(m_{l},n_{l})$ is the associated rod structure so that $u$ is in the kernel of $F$ at $\\rho=0$. Absence of a conical singularity is characterized by a zero angle deficiency, that is when the right-hand side is 1; this is referred to as the balancing condition in Section \\ref{sec1}. By a standard change of coordinates from polar to Cartesian, it is straightforward to check that once analytic regularity has been established this condition is necessary and sufficient for the metric to be extendable across the axis.\n\nLet us denote by $b_l$ the value of $\\log\\left(\\frac{2\\pi}{2\\pi-\\theta}\\right)$ on the axis rod $\\Gamma_l$. Then, similarly to the 4D case, it can be shown from~\\eqref{sigma} that $b_l$ is constant on $\\Gamma_l$. Moreover asymptotic flatness implies that $b_l=0$ on the two semi-infinite axis rods, $l=1,L+1$. Thus it it remains to investigate the value of $b_l$ on the bounded axis rods. In the example from Figure~\\ref{domain}, to show regularity would only require showing that $b_3=0$ so that the angle deficit vanishes on the disk rod, between points $S$ and $C$.\n\nIn 4D very few cases have been worked out, see~\\cites{weinstein94,litian91}. In the current 5D setting, it is known that some configurations without any conical singularity do exist as mentioned in the introduction. We conjecture that many more such regular solutions can be found. These questions will be investigated in a future paper.\n\n\\begin{comment}\nSome preliminary observations are that we might have\n\\begin{equation}\n\\frac{\\rho^2}{f_{ij}v^{i}v^{j}}\\leq C\n\\end{equation}\nsince $F$ is asymptotic to the model map, and also\n\\begin{equation}\nf^{-1}e^{2\\sigma}\\leq C.\n\\end{equation}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{T}{he} 3D human digitization has a wide range of applications in industries such as film, animation, games, and virtual try-on. Existing approaches to obtain high-quality 3D human reconstruction often require expensive equipment such as multiple synchronized cameras~\\cite{textured_neural_avatars}, and RGB-D sensor~\\cite{selfportraits}, limiting their applications in practical scenarios.  On the other hand, for various 3D human reconstruction approaches~\\cite{PIFu,PIFuHD,Detailed_Human_Avatars,Video_avatars,MonoClothCap}, modeling complex geometric details such as hair, glasses, and cloth wrinkles of real humans remains a challenging problem.\n\n\\par\nIn this paper, we target to obtain photo-realistic 3D human avatars from monocular RGB videos, which are one of the most accessible video forms in daily life. Different from existing approaches based on pre-scanned human models~\\cite{AMAS} or parametric body models~\\cite{Detailed_Human_Avatars, MonoClothCap, Video_avatars}, our approach implicitly reconstructs the human geometry and appearance via generalizing Neural Radiance Fields (NeRF)~\\cite{NeRF}, which uses a neural network to encode color and density as a function of location and viewing angle, and generates photo-realistic images by volume rendering. \nNeRF~\\cite{NeRF} has shown impressive ability in reconstructing a static scene from multi-view images, and inspires many researchers in extending NeRF to scenes with severe lighting changes~\\cite{nerf-w} and non-rigid deformations~\\cite{nerfies,NGNeRF}. However, these approaches are uncontrollable and limited to nearly static scenes with small movements, failing to deal with human subjects with large movements.\n\nTo handle the dynamic human from monocular videos, we combine neural radiance fields with a parametric body model of SMPL~\\cite{SMPL}, which enables more precise human geometry and appearance modeling, and further makes the neural radiance fields controllable. \nIn particular, our approach extends NeRF via introducing the pose-guided deformation, which unwarps the observation space near the human body to a constant canonical space through the deformation of the SMPL.\nWe observe that even the state-of-the-art SMPL estimator from monocular videos cannot obtain accurate parameters, which inevitably leads to blurry results. To address this problem, we propose to jointly optimize the NeRF and SMPL parameters via analysis-by-synthesis, which not only obtains better results but also accelerates the convergence of training. \nWe demonstrate the superiority of the proposed method on multiple datasets, with both quantitative and qualitative results on novel view synthesis, 3D human reconstruction, and novel pose synthesis.\n\nIn summary, our work has the following contributions:\n\\begin{itemize}\n    \\item We propose a method explicitly deforming the points according to SMPL pose to reconstruct a canonical view NeRF model, relaxing the requirement of the static object and preserving details such as clothing and hair.\n    \\item We incorporate pose refinement into our analysis-by-synthesis approach to account for the inaccurate SMPL estimates, resulting in refined SMPL pose and greatly improved reconstruction quality.\n    \\item We achieve high-quality 3D human reconstruction from monocular RGB video, and can render photo-realistic images from novel views.\n    \\item Due to our controllable SMPL-based geometry deformation, we can synthesize novel pose images, showing that our learned canonical space NeRF model is animatable.\n\\end{itemize}\n\n\n\n\\section{Related work}\n\n\\textbf{3D Human Reconstruction}. Reconstructing 3D human has been more and more popular in recent years. \nVarious approaches attempt to digitize a human from a single-view image~\\cite{PIFu,PIFuHD,360degree,tex2shape,ARCH}, multi-view images~\\cite{PIFu,PIFuHD,dvv,PaMIR}, RGB videos~\\cite{Video_avatars, Detailed_Human_Avatars,lrp,MonoClothCap,DoubleFusion}, or RGB-D videos~\\cite{selfportraits,texmesh}. One stream of these approaches~\\cite{Detailed_Human_Avatars, Video_avatars, MonoClothCap} utilize a parametric body model such as SMPL~\\cite{SMPL} to represent a human body with cloth deformations, which produces an animatable 3D model with high-quality textures but struggles with limited expressive ability in complex geometries such as hair and dresses. On the other hand, PIFu~\\cite{PIFu} and PIFuHD~\\cite{PIFuHD} based methods use an implicit representation to reconstruct a 3D surface and achieves impressive results in handling people with complex poses, hairstyles and clothing, which however, suffers from a blurry appearance and requires further registration for animation. To handle more complex pose inputs, these methods~\\cite{ARCH,PaMIR,S3,ipnet} combine implicit representations and parametric models to obtain more robust results and are animatable.\n    \n\\textbf{Neural Representations}. Representing a scene with neural networks has achieved stunning success in recent years.\nSRN~\\cite{SRN} proposes an implicit neural representation that assigns feature vectors to 3D positions, and uses a differentiable ray marching algorithm for image generation.\nNeRF~\\cite{NeRF} establishes a static scene that maps 3D coordinates and viewing direction to density and color using a neural network. These methods~\\cite{SRN,NeRF,NSVF} can render very realistic images, but they are all limited to static scenes.\nDynamic NeRFs~\\cite{nerfies,D-NeRF,NSVF,NGNeRF,NSFF,HyperNeRF} extend NeRF to dynamic scenes by introducing the latent deformation field or scene flow fields.\nThese methods where the deformations are learned by networks allow to handle more general deformation, and synthesize novel poses by using interpolation in the latent space. \nHowever, it is difficult to implicitly control the complex non-rigid deformation of human body motion.\nThese works~\\cite{NerFACE,Neural_Body,A-NeRF,SCANimate,SMPLicit,LoopReg} combine scene representation network with parametric models~\\cite{3DMM,SMPL} to reconstruct dynamic humans.\nInstead of using latent codes or expression parameters as input, we use the human body model SMPL to explicitly deform over different poses and shapes and reconstruct the human body in the canonical pose.\nAt the same time, this explicit method allows us to fine-tune the parameters of the SMPL which is very practical in real scenarios.\nSimilar ideas with us have been used in recent works\\cite{Neural_Body, Anim-NeRF, Neural_Actor}, but these methods usually require multi-view images or accurate registered SMPL.\nIt is more challenging for monocular videos because of the difficulty of SMPL estimation.\n\n\\textbf{Human Motion Transfer}.\nHuman motion transfer aims to synthesize an image of a person with the appearance from a source human and the motion from a reference image. Recent advances using Generative Adversarial Networks (GAN) have shown convincing performance without recovering detailed 3D geometry.\nThese works~\\cite{monkey_net,eveybody_dance_now,LWGAN,LWGANPlusPlus} use image-to-image translation~\\cite{pix2pix,pix2pixhd} to map 2D skeleton images to rendering output. Due to the lack of 3D reasoning, the geometry of the generated humans is usually not consistent across multiple views and motions.\nTo better preserve the appearance of the source subject, these methods ~\\cite{StylePoseGAN,HumanGAN} use UV map to transform features from screen space to UV texture space to obtain the neural texture, then render the feature maps in the target pose by neural rendering network.\nIn addition to these general approaches between arbitrary subjects, there are other person-specific methods~\\cite{textured_neural_avatars,NHR,smplpix}.\nTextured Neural Avatar~\\cite{textured_neural_avatars} learns a uniform neural texture from different views and poses.\nSMPLpix\\cite{smplpix} and NHR\\cite{NHR} project the point clouds to 2D images and then feed them into an image-to-image translation network.\nHowever, these neural rendering methods fail to generate photorealistic results for novel poses that were not seen during training.\n\n\n\\begin{figure*}[ht]\n    \\begin{center}\n\t\\begin{tabular}{@{}c}\n\t\t\\includegraphics[width=0.8\\linewidth, ]{figures\/Overview.png} \n\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Overview of the proposed Animatable Neural Radiance Fields. Given a video sequence, we estimate the camera $K_t$ and SMPL parameters $M(\\theta_{t}, \\beta_{t})$ of the human subject for initialization. We use volume rendering to sample points $(x_t, y_t, z_t)$ along the camera ray in observation space, and transform these points to canonical space according to pose-guided deformation. Then we input these points $(x_t^0, y_t^0, z_t^0)$ into the neural radiance field to get densities $\\sigma$ and colors $\\mathbf{c}$. Then we use the integral equation Eq.~\\eqref{eq:rendering equation} to render the image, and jointly optimize the neural radiance field parameters $\\phi$ and SMPL parameters  $\\theta_t,\\beta_t$ by minimizing the error $\\mathcal{L}\\left(\\tilde{I}_t, I_t\\right)$ between the rendered image $\\tilde{I}_t$ and the ground truth image $I_t$ with the mask.} \\label{fig:framework}\n\\end{figure*}\n\n\\section{Method}\nIn this section, we will describe the method to create a human avatar from a single portrait video of a person as shown in Fig.~\\ref{fig:framework}.\nGiven a $n$-frame video sequence  $\\left\\{I_t\\right\\}_{t=1}^{n}$ of a single human subject turning around before the camera and holding an A-pose or T-pose, we estimate the SMPL~\\cite{SMPL,SMPL-X} parameters $M(\\theta_{t}, \\beta_{t})$ and camera intrinsics $K_{t}$ of each frame using existing human body shape and pose estimation models~\\cite{VIBE}. \nIn order to avoid the influence of background changes caused by camera movement, we first use a segmentation network~\\cite{RP-R-CNN} to obtain the foreground human mask and set the background color to white uniformly.\nOur animatable NeRF (Sec.~\\ref{Anim-NeRF}) can be decomposed into pose-guided deformation (Sec.~\\ref{Pose-Guided_deform}) and a neural radiance field (NeRF) defined in the canonical space. We can use the volume rendering (Sec.~\\ref{Volume_Render}) to render our neural radiance field. In order to avoid the negative effects of inaccurate SMPL parameters, we propose to jointly optimize the neural radiance field and SMPL parameters (Sec.~\\ref{Pose_Refine}). We also introduce background regularization and pose regularization to improve the robustness of optimization (Sec.~\\ref{Object_Fun}).\n\n\\subsection{Animatable Neural Radiance Fields}\\label{Anim-NeRF}\nTo model human appearance and geometry with complex non-rigid deformation, we introduce the parameterized human model SMPL~\\cite{SMPL} into the neural radiance field and present the animatable neural radiance fields (animatable NeRF) $F$ which maps the 3D position $\\mathbf{x}=(x, y, z)$, shape $\\mathbf{\\beta}_{t}$ and pose $\\mathbf{\\theta}_{t}$ into color $\\mathbf{c}=(r, g, b)$ and density $\\sigma$:\n\\begin{equation}\nF\\left(D\\left(\\mathbf{x}, \\mathbf{\\theta}_{t}, \\mathbf{\\beta}_{t} \\right)\\right) = \\left(\\mathbf{c},\\sigma\\right)\n\\end{equation}\nwhere $D\\left(\\mathbf{x}, \\mathbf{\\theta}_{t}, \\mathbf{\\beta}_{t} \\right)$ transforms the 3D position $\\mathbf{x}=(x, y, z)$ in the observation space to $\\mathbf{x}^{0}=(x^{0}, y^{0}, z^{0})$ in canonical space, aiming to handle human movements between different frames. The view dependence in NeRF is mainly for dealing with specular reflections of materials such as metal and glass. But the skin and clothes of humans are mainly diffuse reflective materials, so we remove the viewing direction from the input.  We will discuss the impact of viewing direction in Sec.~\\ref{impact_of_view_direction}.\n\n\\subsection{Pose-guided Deformation}\\label{Pose-Guided_deform}\nIn contrast to~\\cite{NGNeRF,nerfies} that implicitly control of the deformation of spatial points, we use the parametric body model - SMPL, to explicitly guide the deformation of spatial points. \nHere we define the observed image as the observation space and attempt to learn a template human in the canonical space.\nThe articulated SMPL model enables the explicit transformation of the spatial points (i.e. from observation space to canonical space), which facilitates the learning of a specified meaningful canonical space, and reduces the reliance of diverse input poses to generalize to unseen poses so that we can learn the NeRF space from dynamic scenes (containing moving person) and animate this person after training.\nThe template pose in the canonical space is defined as X-pose $\\theta^{0}$ (as shown in Fig.~\\ref{fig:framework}), due to its good visibility and separability of each body part.\nBy using the inverse transformation of the linear skinning of SMPL, the pose $\\theta_{t}$ in observation space can be transformed into the X-pose $\\theta^{0}$ in canonical space. Considering that the transformation functions are only defined on the surface vertices of the body mesh, we extend them to the space near the mesh surface based on the intuition that points in space near the mesh should move along with neighboring vertices. Following PaMIR~\\cite{PaMIR} we define the transformation of a point $\\mathbf{x}$ from observation space to canonical space as\n\n\n\n\n\n\\begin{equation}\n\\label{eq:transformation}\n\\begin{aligned}\n\\left[\\begin{matrix}\\mathbf{x}^{0} \\\\ 1\\end{matrix} \\right]&=\\mathbf{M}(\\mathbf{x}, \\beta_{t}, \\theta_{t}, \\theta^{0}) \\left[\\begin{matrix}\\mathbf{x} \\\\ 1\\end{matrix} \\right] \\\\\n\\mathbf{M}(\\mathbf{x}, \\beta_{t}, \\theta_{t}, \\theta^{0}) &=\\sum_{v_i \\in \\mathcal{N}\\left(\\mathbf{x}\\right)} \\frac{\\omega_{i}}{\\omega} \\mathbf{M}_{i}\\left(\\beta_{t}, \\theta^{0}\\right)\\left(\\mathbf{M}_{i}(\\beta_{t}, \\theta_{t})\\right)^{-1}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{N}\\left(\\mathbf{x}\\right)$ denotes the SMPL vertex set near $\\mathbf{x}$ in the observation space, and Eq.~\\eqref{eq:transformation} indicates that the movement of $\\mathbf{x}$ relies on the movement of neighboring vertices. \nThe transformation weight $\\omega_{i}$ that the vertex $v_i$ affects the point $\\mathbf{x}$ is defined as \n\n\\begin{equation}\n\\begin{aligned}\n\\omega_{i} &=\\exp \\left(-\\frac{\\left\\|\\mathbf{x}-v_{i}\\right\\| \\left\\|\\hat{\\mathbf{b}}-\\mathbf{b}_{i}\\right\\|}{2 \\sigma^{2}}\\right)  \\\\\n\\omega &=\\sum_{v_i \\in \\mathcal{N}(\\mathbf{x})} \\omega_{i}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{b}_{i}$ is the blend weight of $v_i$ and $\\hat{\\mathbf{b}}$ is the blend skinning weight of the nearest vertex, and $\\|\\mathbf{x}-v_{i}\\|$ computes the L2 distance between $\\mathbf{x}$ and $v_i$.\nConsider the fact that a point might be affected by different body parts, leading to ambiguous or even non-meaningful transformation,  we adopt the blend skinning weight which characterizes the movement patterns of a vertex along with the SMPL joints~\\cite{SMPL}, to strengthen the movement impact of the nearest neighbor. $\\omega$ is used for weight normalization.\n\nFollowing SMPL\\cite{SMPL}, the transformation matrix $\\mathbf{M}_{i}\\left(\\beta, \\theta\\right)$ of mesh vertex $v_i$ from rest pose to $\\theta$-pose is computed by\n\\begin{equation}\n\\mathbf{M}_{i}(\\beta, \\theta) = \\left(\\sum_{j=1}^{K}b_{i,j}\\mathbf{G}_j \\right)\n\\begin{bmatrix}\n  \\mathbf{I}  & \\mathbf{B}_{S,i}(\\beta)+\\mathbf{B}_{P, i}(\\theta) \\\\\n    \\mathbf{0}^T & 1\n\\end{bmatrix}\n\\end{equation}\nwhere $\\mathbf{G}_j\\in \\mathbb{R}^{4\\times4}$ is the world transformation of joint $j$, $b_{i,j}$ is the blend skinning weight representing how much the rotation of part $j$ affects vertex $v_i$, $\\mathbf{B}_{P, i}(\\theta)\\in \\mathbb{R}^{3}$ and $\\mathbf{B}_{S,i}(\\beta)\\in \\mathbb{R}^{3}$ are the pose blendshape and shape blendshape of vertex $v_i$ respectively. \n\n\\subsection{Volume Rendering}\\label{Volume_Render}\n\\label{sec:vr}\nWe use the same volume rendering techniques as in NeRF~\\cite{NeRF} to render the neural radiance field into a 2D image. For a given video frame $I_t$, we first convert the camera coordinate system to the SMPL coordinate system, that is, transform the SMPL global rotation and translation to the camera. Then the pixel colors are obtained by accumulating the colors and densities along the corresponding camera ray $\\mathbf{r}$. In practice, the continuous integration is approximated by sampling $N$ points $\\left\\{\\mathbf{x}_{k}\\right\\}_{k=1}^{N}$ between the near plane and the far plane along the camera ray $\\mathbf{r}$ as\n\\begin{equation}\n\\label{eq:rendering equation}\n\\begin{array}{c}\n\\tilde{C}_{t}(\\mathbf{r})=\\sum\\limits_{k=1}^{N}T_{k}\\left(1-\\exp \\left(\\eta_{t}(\\mathbf{x}_{k})\\sigma_{t}\\left(\\mathbf{x}_{k}\\right) \\delta_{k}\\right)\\right) \\mathbf{c}_{t}\\left(\\mathbf{x}_{k}\\right) \\\\\n\\tilde{D}_{t}(\\mathbf{r})=\\sum\\limits_{k=1}^{N}T_{k}\\left(1-\\exp \\left(\\eta_{t}(\\mathbf{x}_{k})\\sigma_{t}\\left(\\mathbf{x}_{k}\\right) \\delta_{k}\\right)\\right) \\\\\nT_{k}=\\exp \\left(-\\sum\\limits_{j=1}^{k-1} \\eta_{t}(\\mathbf{x}_{k})\\sigma_{t}\\left(\\mathbf{x}_{j}\\right) \\delta_{j}\\right)\n\\end{array}\n\\end{equation}\nwhere $\\delta_{k}=\\left\\|\\mathbf{x}_{k+1}-\\mathbf{x}_{k}\\right\\|_{2}$ is the distance between adjacent sampling points, and $\\eta_{t}(\\mathbf{x}_{k})$ is a prior 3D \\textit{mask} (detailed in the following) used to provide geometric prior guidance and deal with ambiguity during pose-guided deformation.\n\nSince we only focus on modeling a single human subject, here we introduce an assumption for learning a more accurate neural radiance field: The densities should be zeros for points far from the surface of human mesh;\n\\begin{equation}\n\\label{distance constraint}\n\\begin{aligned}\n\\eta_{t}(\\mathbf{x}_{k}) &=  d(\\mathbf{x}_{k}) \\leq \\delta\\\\\nd(\\mathbf{x}_{k}) &= \\sum_{v_i \\in \\mathcal{N}\\left(\\mathbf{x}_{k}\\right)} \\frac{\\omega_{j}}{\\omega} \\left\\|\\mathbf{x}_{k}-v_{i}\\right\\|\n\\end{aligned}\n\\end{equation}\nwhere $d(\\mathbf{x}_{k})$ is the weighted distance from point $\\mathbf{x_k}$ to the nearest neighbor vertices $\\mathcal{N}\\left(\\mathbf{x}_{k}\\right)$ in the observation space. $\\delta$ is the distance threshold limiting distance between the sample point to the SMPL surface in the observation space. In experiments we follow NeRF~\\cite{NeRF} to perform hierarchical volume sampling to obtain $\\tilde{C}_{t}^c(\\mathbf{r})$ and $\\tilde{C}_{t}^f(\\mathbf{r})$ with the coarse and fine networks, respectively.\n\n\\subsection{Pose Refinement via Analysis-by-Synthesis} \\label{Pose_Refine}\nOur proposed method learns an animatable NeRF for human subjects via explicitly deforming the observation space from different frames to a constant canonical space, under the guidance of SMPL transformations. Although the current state-of-the-art pose and shape estimation methods~\\cite{VIBE} is adopted to obtain more stable SMPL parameters, in experiments we observe that results estimated by these methods do not align well with the ground truth, especially in depth. The inaccurate human body estimation could easily lead to blurry results. To address this problem, we propose to fine-tune the SMPL parameters during training. \nSpecifically, we use VIBE~\\cite{VIBE} to estimate SMPL parameters $M(\\theta_{t}, \\beta_{t})$ for each frame $I_t$ as initialization of variables, which will be optimized during training.\nWe use the mean shape parameters $\\beta=\\frac{1}{n}\\sum_{t=1}^{n}\\beta_{t}$ for different frames. It turns out that the refined SMPL can better fit the input image, and helps to obtain clearer and sharper results as shown in Fig.~\\ref{fig:Comparision_with_different_method} and Fig.~\\ref{fig:3D_Rec_on_mutil_garment}.\n\n\n\\subsection{Objective Functions} \\label{Object_Fun}\nGiven a monocular video sequence, we learn the animatable NeRF by optimizing the following objective function\n\\begin{equation}\n\\mathcal{L}= \\mathcal{L}_{c} + \\mathcal{L}_{p} + \\lambda_{d} * \\mathcal{L}_{d}\n\\end{equation}\nwhere $\\mathcal{L}_{c}$, $\\mathcal{L}_{p}$, and $\\mathcal{L}_{d}$ are reconstruction loss, pose regularization, and background regularization, respectively. $\\lambda_{d}$ aims to balance the importance of background regularization.\n\n{\\bf Reconstruction Loss}. The reconstruction loss aims to minimize the error between the rendered images and the corresponding observed frames, which is defined as \n\\begin{equation}\n\\mathcal{L}_{c}=\\sum_{t}\\sum_{\\mathbf{r}}\\left\\|\\tilde{C}_{t}^{c}(\\mathbf{r})-C_{t}(\\mathbf{r})\\right\\|_{2}^{2} + \\left\\|\\tilde{C}_{t}^{f}(\\mathbf{r})-C_{t}(\\mathbf{r})\\right\\|_{2}^{2}\n\\end{equation}\nwhere $\\mathbf{r}$ is a camera ray passing through the image $I_{t}$.\n$C_t(\\mathbf{r})$ is the ground truth color of the pixel intersected by ray $\\mathbf{r}$ on the observed image $I_{t}$.\nAnd $\\tilde{C}_{t}^{c}(\\mathbf{r})$, $\\tilde{C}_{t}^{f}(\\mathbf{r})$ are the corresponding rendered colors from the coarse and fine networks respectively (see Sec.~\\ref{sec:vr}). \n\n\n{\\bf Pose Regularization}. \nTo obtain stable and smooth pose parameters, we add the following pose regularization term to encourage the optimized pose parameter to stay close to the initial pose, and the pose parameters between adjacent frames to be similar.\n\\begin{equation}\n\\mathcal{L}_{p}=\\lambda_1\\left\\|\\tilde{\\theta}_{t}-\\theta_{t}\\right\\|+\\lambda_2\\left\\|\\tilde{\\theta}_{t}-\\tilde{\\theta}_{t+1}\\right\\|\n\\end{equation}\nwhere $\\theta_{t}$ is the initial pose parameters, $\\tilde{\\theta}_{t}$ and $\\tilde{\\theta}_{t+1}$ are the optimized pose parameters of frame $t$ and $t+1$. $\\lambda_1$ and $\\lambda_2$ are the corresponding penalty weights.\n\n\n{\\bf Background Regularization}. \nWe only focus on reconstructing the human no matter what the background is, which means, ideally, density exists only inside the human. \nTo better achieve this goal, we first set the background color to white with the help of an off-the-shelf segmentation network.\nWe minimize the difference between the rendered integral density and the mask obtained by segmentation.\nSince the foreground (i.e. human) region is $1$ and the background region is $0$ in the mask, we are encouraging the human region's integral density to be 1 and encouraging the background region's integral density to be 0, resulting in a much cleaner empty space estimation and more solid and clearer person estimation in our canonical NeRF space.\nMathematically, our background regularization term is defined as follows,\n\\begin{equation}\n\\mathcal{L}_{d}=\\sum_{t}\\sum_{\\mathbf{r}}\\left\\|\\tilde{D}_{t}^{c}(\\mathbf{r})-D_{t}(\\mathbf{r})\\right\\| + \\left\\|\\tilde{D}_{t}^{f}(\\mathbf{r})-D_{t}(\\mathbf{r})\\right\\|\n\\end{equation}\nwhere $\\tilde{D}_{t}^{c}$ and $\\tilde{D}_{t}^{f}$ is the rendered integral density of the coarse and fine network for the camera ray $\\mathbf{r}$ from the image $I_{t}$, and $D_{t}(\\mathbf{r})$ is the corresponding segmentation mask and $D_{t}(\\mathbf{r})=1$ in the foreground region and $D_{t}(\\mathbf{r})=0$ in the background region.\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfigure[NeRF]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_nerf.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[SMPLpix]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_smplpix.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeuralBody]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_neuralbody.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeRF+U]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_nerf_u.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[Ours]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_ours.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[GT]{\n\t\\begin{minipage}[b]{0.14\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_gt.png}\n\t\\end{minipage}\n\t}\n    \\caption{Visual comparison of different methods about novel view synthesis on People-snapshot\\cite{Video_avatars}(1-2 rows) and iPER\\cite{LWGAN}(3-4 rows). \n    NeRF\\cite{NeRF} is struggling to handle dynamic scenes because the movement of the subject violates the multi-view consistency requirement. With the help of our proposed pose-guide deformation, NeRF+U (NeRF + Unpose) achieves much better results (row 1\\&2) if the estimated SMPL poses are accurate but still produces blurry results (row 3\\&4) if they are not.\n    Further adding pose refinement (ours) greatly improves the robustness as long as the estimated SMPL pose is reasonably good.\n    Compared with NeuralBody\\cite{Neural_Body} and SMPLpix\\cite{smplpix}, our approach can produce realistic images with well preserved identity and cloth details.\n   \n   \n    }\n    \\label{fig:Comparision_with_different_method}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.1116\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_iper_input.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 0]{\n\t\\begin{minipage}[b]{0.1116\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_iper_rec.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 1]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_iper_left.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 2]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_iper_right.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[Input]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_ps_input.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 0]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_ps_rec.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 1]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_ps_left.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[view 2]{\n\t\\begin{minipage}[b]{0.10\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_view_ps_right.png}\n\t\\end{minipage}\n\t}\n    \\caption{Results of Novel View Synthesis on iPER (a-d) and People-Snapshot (e-h). Our method can synthesize realistic and multi-view consistent results from different camera views while maintaining the subject pose fixed.}\n    \\label{fig:Novel_View}\n\\end{figure*}\n\n\n\\setlength{\\tabcolsep}{2.8pt}\n\\begin{table*}[ht]\n    \\caption{Quantitative comparison of novel view synthesis on People-Snapshot\\cite{Video_avatars} and iPER\\cite{LWGAN}.}\n\t\\label{Albation}\n\t\\begin{center}\n\t\t\\renewcommand{\\arraystretch}{1.1}\n\t\t\\begin{tabular}{|c|ccccc|ccccc|ccccc|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Subject ID} & \\multicolumn{5}{c|}  {PSNR$\\uparrow$} & \\multicolumn{5}{c|}{SSIM$\\uparrow$} &\n\t\t\t\\multicolumn{5}{c|}{LIPIS$\\downarrow$} \\\\\n\t\t\t\\cline{2-16}\n\t\t\t& NeRF & SMPLpix & NB & NeRF+U & OURS & NeRF &SMPLpix & NB & NeRF+U & OURS & NeRF &SMPLpix & NB & NeRF+U & OURS \\\\\n\t\t\t\\hline\n\t\t\tmale-3-casual & 20.64 & 23.74 & 24.94 & 23.88 & \\textbf{29.37}  & .8993 & .9229 & .9428 & .9329 & \\textbf{.9703} & .1008 & .0222 & .0326 & .0438 & \\textbf{.0168} \\\\\n\t\t\tmale-4-casual & 20.29 & 22.43 & 24.71 & 23.13 & \\textbf{28.37}  & .8803 & .9095 & .9469 & .9276 & \\textbf{.9605} & .1445 & .0305 & .0423 & .0554 & \\textbf{.0268} \\\\\n\t\t\tfemale-3-casual & 17.43 & 22.33 & 23.87 & 22.45 & \\textbf{28.91} & .8605 & .9288 & .9504  & .9413  & \\textbf{.9743} & .1696 & .0270 & .0346 & .0498 & \\textbf{.0215} \\\\\n\t\t\tfemale-4-casual & 17.63 & 23.35 & 24.37 & 23.13 & \\textbf{28.90} & .8578 & .9258 & .9451 & .9276 & \\textbf{.9678} & .1827 & .0239 & .0382 & .0556 & \\textbf{.0174} \\\\\n\t\t\t\\hline\n\t\t\tiper-009-4-1 & 19.54 & 20.25 & 25.46 & 21.56 & \\textbf{30.23} & .7870 & .9018 & .9378 & .8667 & \\textbf{.9466} & .2641 & \\textbf{.0293} & .0558 & .1197 & .0335 \\\\\n\t\t\tiper-023-1-1 & 17.41 & 19.48 & 25.44 & 20.25 & \\textbf{27.26} & .7623 & .8945 & .9330 & .8656 & \\textbf{.9457} & .2769 & .0442 & .0493 & .1109 & \\textbf{.0285} \\\\\n\t\t\tiper-002-1-1 & 16.01 & 19.64 & 23.06 & 18.75 & \\textbf{26.99} & .7500 & .8886 & .9394 & .8708 & \\textbf{.9502} & .3363 & .0392 &.0476 & .1205 & \\textbf{.0285} \\\\\n\t\t\tiper-026-1-1 & 17.09 & 19.03 & 23.77 & 18.48 & \\textbf{26.85} & .7580 & .8574 & .9351 & .8623 & \\textbf{.9542} & .2928 & .0494 & .0550 & .1282 & \\textbf{.0315} \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\\subsection{Applications}\nThe proposed approach learns an animatable NeRF, allowing us to reconstruct the implicit neural representation of the geometry and appearance of the human body, from a monocular video of a person turning around before a camera while holding the A-pose. For the original NeRF~\\cite{NeRF}, novel view images (Sec.~\\ref{NVS}) can be rendered through volume rendering, and the surface geometry (Sec.~\\ref{3DR}) of the scene can be extracted with the Marching Cubes algorithm~\\cite{MCubes}.\nSince we have explicitly incorporated the SMPL model into the NeRF training process, we can deform the neural radiance field to desired poses for rendering by our pose-guided deformation. This makes our NeRF \\textit{animatable}, and thus a new application that can demonstrate new poses or animating the reconstructed people (Sec.~\\ref{NPS}) is enabled as shown in Fig.~\\ref{fig:motion_transfer} and Table~\\ref{novel_pose_synthesis_with_neuralbody}.\n\n\n\n\\section{Experiments}\n\n\\subsection{Implementation Details}\nFollowing NeRF~\\cite{NeRF}, we use coarse and fine networks to represent the human body, and use $64$ coarse and $64+32$ fine rays samples for all experiments. \nFocusing on the foreground subject, we set $90\\%$ of the rays to be sampled from the foreground, and the remaining $10\\%$ to be sampled from the background. \nWe set the hyper-parameters as $\\left|\\mathcal{N}(i)\\right|=4$, $\\delta=0.2$, $\\lambda_1=0.001$, $\\lambda_2=0.01$ and $\\lambda_d=0.1$. \nWe use $512 \\times 512$ image in all experiments. \nFor training the model, we adopt the Adam optimizer~\\cite{Adam}, and it spends about 13 hours on 2 Nvidia GeForce RTX 3090 24GB GPUs.\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3d_rec_input.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeRF]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3d_rec_nerf.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeRF+L]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3D_rec_nerf_latent.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeRF+U]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3D_rec_nerf_unpose.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[Ours]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3d_rec_ours.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[NeRF+U(GT)]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3d_rec_nerf+unpose_gt.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[GT]{\n\t\\begin{minipage}[b]{0.12\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/3d_rec_gt.png}\n\t\\end{minipage}\n\t}\n    \\caption{Visualization of 3D reconstruction on Multi-Garment. NeRF\\cite{NeRF} and NeRF+U (NeRF + Unpose) fail to reconstruct 3D geometry due to the movement of the subject and the inaccurate SMPL. Compared with NeRF+L (NeRF + Latent) which produces over-smooth or under-smooth results, our results are more reasonable. As a reference, NeRF+U(GT) uses GT SMPL and learns geometry with very high precision, demonstrating the effectiveness of our pose-guided deformation and showing the importance of obtaining accurate SMPL for 3D reconstruction tasks.}\n    \\label{fig:3D_Rec_on_mutil_garment}\n\\end{figure*}\n\n\\setlength{\\tabcolsep}{2.8pt}\n\\begin{table*}[ht]\n    \\caption{Quantitative comparison of 3D reconstruction on Multi-Garment.}\n\t\\label{3D_Rec_on_mutil-garment}\n\t\\begin{center}\n\t\t\\renewcommand{\\arraystretch}{1.1}\n\t\t\\begin{tabular}{|c|ccccc|ccccc|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Subject ID} & \\multicolumn{5}{c|}  {P2S$\\downarrow$} & \\multicolumn{5}{c|}{Chamfer$\\downarrow$} \\\\\n\t\t\t\\cline{2-11}\n\t\t\t&NeRF &NeRF+L &NeRF+U &OURS &NeRF+U(GT) &NeRF &NeRF+L &NeRF+U &OURS &NeRF+U(GT) \\\\\n\t\t\t\\hline\n\t\t\tpeople1 & 65.53 & 13.57 & 33.51 & \\textbf{4.09} & 0.86  & 89.32 & 13.96 & 41.81 & \\textbf{4.25} & 0.25  \\\\\n\t\t\tpeople2 & 36.26 & 11.67 & 28.50 & \\textbf{1.55} & 0.85 & 34.95 & 10.78 & 28.86 & \\textbf{0.96} & 0.25 \\\\\n\t\t\tpeople3 & 34.78 & 16.01 & 36.40 & \\textbf{4.17} & 1.17 & 33.62 & 13.83 & 38.36 & \\textbf{3.30} & 0.43 \\\\\n\t\t\tpeople4 & 33.29 & 26.84 & 32.74 & \\textbf{3.53} & 1.06 & 33.70 & 26.59 & 32.08 & \\textbf{2.68} & 0.36 \\\\\n\t\t\t\\hline\n\t\t\tAverage & 42.46 & 17.02 & 33.28 & \\textbf{3.32} & 0.99 & 47.90 & 16.29 & 34.79 & \\textbf{2.80}  & 0.32 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\n\n\\subsection{Datasets and Evaluation}\n\n\\noindent \\textbf{Datasets}.\nTo evaluate the effectiveness of the proposed method, we conduct experiments on 3 different datasets, including People-Snapshot~\\cite{Video_avatars}, iPER~\\cite{LWGAN}, and Multi-Garment~\\cite{MGNet}.\nPeople-Snapshot\\cite{Video_avatars} and iPER\\cite{LWGAN} datasets both contain different monocular RGB videos captured in real-world scenes, where the subjects hold an A-pose and turn around before a fixed camera. \nIn addition, iPER dataset also contains videos of the same person with random motion sequences. Multi-Garment~\\cite{MGNet} dataset contains 3D scanned human body models and textures and the corresponding registered SMPLD models that can be used for animation.\nWe selected 4 human body models to synthesize the videos, according to motion sequences which the subjects rotate while holding an A-pose in People-Snapshot dataset.\nPeople-Snapshot and iPER datasets are mainly used for the evaluation of novel view synthesis and novel pose synthesis experiments.\nAnd the synthetic data from Multi-Garment dataset are used to evaluate the quality of the 3D reconstructions.\n\n\n\\noindent \\textbf{Evaluation}. \nIn our experiments, we use A-pose frames (2 circles) for training, and the remaining A-pose frames (1 circle) for testing novel view synthesis and random pose frames for testing novel pose synthesis. \nSince there are depth and scale ambiguities in optimizing the SMPL parameters for monocular videos, \nwe also optimize the SMPL parameters of the test frames for quantitative evaluation. \nNote that the parameters of the neural radiance field network remain fixed. \nFor quantitative evaluation, we evaluate our method for novel view synthesis and novel pose synthesis using the following metrics: peak signal-to-noise ratio (PSNR), structural similarity index (SSIM~\\cite{SSIM}), and learned perceptual image patch similarity (LPIPS~\\cite{lpips}). \nFor 3D reconstruction, we use point-to-surface Euclidean distance (P2S) and Chamfer distance~\\cite{Chamfer} (in cm) between the reconstructed and the ground truth surfaces. \nWe register our meshes to ground truth geometry for comparison in consideration of scale and depth ambiguities. \nThe datasets from the real scenes don't have the corresponding ground truth geometry, and we only provide qualitative results.\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\includegraphics[height=0.72\\linewidth, trim=0 0 0 0,clip]{figures\/3D_rec_with_smpld_input.png}\n\t}\n\t\\subfigure[Ours]{\n\t\\includegraphics[height=0.72\\linewidth, trim=10 0 20 0,clip]{figures\/3D_rec_with_smpld_ours.png}\n\t}\n\t\\subfigure[Video Avatars]{\n\t\\includegraphics[height=0.72\\linewidth, trim=20 0 0 0,clip]{figures\/3D_rec_with_smpld_video_avatar.png}\n\t}\n    \\caption{Comparisons of 3D reconstruction results on People-Snapshot with video avatars~\\cite{Video_avatars}. Compared with Video Avatars\\cite{Video_avatars}, our approach can generate more details such as hairs and clothes wrinkles.}\n    \\label{fig:3D_Rec_with_SMPLD}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/3D_rec.png}\n    \\caption{Visualization of our reconstructed geometry on iPER from different views.}\n    \\label{fig:3D_Rec_iPER}\n\\end{figure}\n\n\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/compare_with_neuralbody_on_unseen_pose_in_detail.png}\n    \\caption{Comparisons between NeuralBody\\cite{Neural_Body} (first row) and Ours (second row) on novel pose synthesis task. In contrast to NeuralBody, which fails to synthesize novel poses, our approach generalizes better on novel poses that are very different from the training poses.}\n    \\label{compare_with_neuralbody_on_unseen_pose_in_detail}\n\\end{figure}\n\n\\subsection{Novel view synthesis}\n\\label{NVS}\nLike the original NeRF~\\cite{NeRF}, our animatable NeRF can be rendered from arbitrary views (of the same pose). \nSince the monocular video does not have corresponding novel view images, we use the first part of the A-pose video frames to train our model and the remaining frames to test the rendered novel view images\\footnote{Technically, this is not a ``novel'' view apart from slightly different human pose}.\nTo compare against the original NeRF, we first transform the global rotation and translations of the SMPL estimation from every video frame to camera pose as if we are handling a collection of multi-view images of an almost static scene since the subject is holding an A-pose.\nIn order to verify the effectiveness of our pose fine-tuning strategy, we first use VIBE~\\cite{VIBE} to estimate the parameters of SMPL, and conduct comparative experiments between ``Ours'' (NeRF + Unpose + Pose Refinement) and ``NeRF+U'' (NeRF + Unpose) as shown in Fig.~\\ref{fig:Comparision_with_different_method}.\nWe also compare the proposed method with several state-of-the-art (SOTA) methods, including NeuralBody\\cite{Neural_Body}(NB) and SMPLpix\\cite{smplpix}.\nNeuralBody, which also combines SMPL and NeRF, is able to reconstruct dynamic human bodies from monocular video.\nSMPLpix takes SMPL pose as input to generate images via a neural rendering network.\nTable~\\ref{Albation} quantitatively compares the results of different approaches about novel view synthesis on the People-Snapshot and iPER datasets. \nAs described in the table, our proposed approach achieves higher PSNR and SSIM scores compared to other approaches. \nWe also provide qualitative comparisons in Fig.~\\ref{fig:Comparision_with_different_method} with the person examples drawn from the People-Snapshot the iPER datasets. \nWe can see that the proposed approach produces more realistic and reliable results.\nNeRF fails to handle such dynamic scenes since the movement of the subject violates the multi-view consistency requirement. \nExperiments in Fig.~\\ref{fig:Comparision_with_different_method}(d) and Table~\\ref{Albation} (see NeRF+U) also show that inaccurate SMPL parameters cause a very negative impact. \nIn contrast, after taking pose refinement into consideration, the quality of novel view synthesis has been significantly improved. \nAs shown in Fig.~\\ref{fig:Novel_View}(b)(c), SMPLpix and NeuralBody seem to overfit the training frames, while our results better preserve the details such as faces and hands.\nFig.~\\ref{fig:Novel_View} shows the realistic rendering results of more views of the proposed method on more persons with different dresses and hairstyles, indicating the applicability and robustness of the proposed method in real scenarios.\n\n\n\\subsection{3D human reconstruction}\n\\label{3DR}\nOn this task, we compare against the original NeRF~\\cite{NeRF} and NeRF+L baselines. NeRF+L extends NeRF to condition it on a (per-frame) learnable latent deformation code to handle dynamic scenes as shown in Fig~\\ref{fig:3D_Rec_on_mutil_garment}(c) and Table~\\ref{3D_Rec_on_mutil-garment}.\nFor synthetic data, we also show results using ground truth SMPL parameters (NeRF+U(GT)) for pose-guided deformation as the upper bound as shown in Fig~\\ref{fig:3D_Rec_on_mutil_garment}(f) and Table~\\ref{3D_Rec_on_mutil-garment}.\nQuantitative compassion of different strategies in 3D human reconstruction is shown in Table~\\ref{3D_Rec_on_mutil-garment}. \nWe can see that the proposed approach achieves much lower P2S and Chamfer distances, demonstrating the superiority of the proposed approach in reconstructing accurate 3D geometry. \nFig.~\\ref{fig:3D_Rec_on_mutil_garment} compares the qualitative results of 3D reconstruction. \nWe can see that NeRF fails to learn reasonable 3D geometry of the human subject with \\textit{small} movements. \nNeRF+U(see Sec. \\ref{NVS}) also produces messy results. Compared with NeRF+L, which produces over-smooth or under-smooth results, the proposed approaches better capture the geometric details such as cloth wrinkles, faces, and hairs. \nWith ground truth SMPL parameters, the P2S and Chamfer distance are much lower than all the approaches, which demonstrates the necessity of obtaining accurate poses and the effectiveness of our approximated pose-guided deformation. \nIn Fig.~\\ref{fig:3D_Rec_with_SMPLD}, we compare the reconstruction results with video avatar~\\cite{Video_avatars}, which deforms vertices of the SMPL model to fit the 2D human silhouettes over the video sequence. \nWe can see that the implicit learning of subject geometry with animatable NeRF generates a better quality of details, including cloth wrinkles, hair, and accessories. \nIn Fig.~\\ref{fig:3D_Rec_iPER} we show the reconstruction results of persons with varied clothes and hairstyles from iPER dataset. \nAlthough our pose-guided deformation does not take the deformation of clothes into account, the proposed method is capable of capturing the high-quality 3D geometry details, such as the hood (second line) and pigtail (third line), as long as the clothes do not have violent deformation.\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/motion_transfer.png}\n    \\caption{Novel pose synthesis on People-Snapshot\\cite{Video_avatars} and iPER\\cite{LWGAN}. We can feed novel SMPL pose parameters to the trained animatable NeRF to synthesize novel pose images. \n    Although trained only on A-pose images, our animatable NeRF has the capability to stably render new images containing complex poses.\n    }\n    \\label{fig:motion_transfer}\n\\end{figure*}\n\n\\subsection{Novel pose synthesis} \n\\label{NPS}\nDue to our explicit control of deformation via SMPL, our method can synthesize images under unseen poses even with only simple A-pose sequences as input. \nTo quantitatively evaluate the capability of our method on novel pose synthesis, we train the model using A-pose videos and test it using random pose videos of the same person. \nNeuralBody\\cite{Neural_Body} is the most similar work to ours in the sense that it also combines NeRF with SMPL. \nCompared to ours, it handles complex cloth geometry (which is not modeled by SMPL) better due to its use of latent code. \nHowever, each vertex's latent code would affect a much larger region after several sparse convolution layers, resulting in unpredictable artifacts for novel pose synthesis (see Fig. \\ref{compare_with_neuralbody_on_unseen_pose_in_detail}).\nTable~\\ref{novel_pose_synthesis_with_neuralbody} shows that our method achieves much better results than NeuralBody on novel pose synthesis. \nQualitative visualizations of novel pose synthesis on People-snapshot and iPER are provided in Fig.~\\ref{fig:motion_transfer}. \nSpecifically, different poses are fed into the trained animatable NeRF to obtain the aforementioned renderings. \nDespite the significant differences between the test novel poses and the training poses, the results show that our method can still produce realistic images with well preserved identity and cloth details of the subjects.\n\n\\setlength{\\tabcolsep}{2.8pt}\n\\begin{table}[ht]\n    \\caption{Quantitative comparison about novel pose synthesis with NeuralBody(NB)\\cite{Neural_Body} on the iPER dataset.}\n\t\\label{novel_pose_synthesis_with_neuralbody}\n\t\\begin{center}\n\t\t\\renewcommand{\\arraystretch}{1.1}\n\t\t\\begin{tabular}{|c|cc|cc|cc|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Subject ID} & \\multicolumn{2}{c|}  {PSNR$\\uparrow$} & \\multicolumn{2}{c|}  {SSIM$\\uparrow$} &\n\t\t\t\\multicolumn{2}{c|}  {LPIPS$\\downarrow$} \\\\\n\t\t\t\\cline{2-7}\n\t\t\t& NB & OURS & NB & OURS & NB & OURS \\\\\n\t\t\t\\hline\n\t\t\tiper-009-4-2 & 20.95 & \\textbf{24.11} & \\textbf{.9035} & .8927 & .0980 & \\textbf{.0782} \\\\\n\t\t\t\\hline\n\t\t\tiper-023-1-2 & 20.28 & \\textbf{21.98}  & \\textbf{.9009} & .8940 & .0870 & \\textbf{.0644} \\\\\n\t\t\t\\hline\n\t\t\tiper-026-1-2 & 17.42 & \\textbf{19.27} & \\textbf{.8795} & .8713 & .1192 & \\textbf{.0990} \\\\\n\t\t\t\\hline\n\t\t\tiper-002-1-2 & 19.07 & \\textbf{23.47} & .8957 & \\textbf{.9165} & .0749 & \\textbf{.0483} \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\n\\section{Discussion}\n\nIn the following, we will discuss the proposed approach in details from the following aspects:  analysis of pose refinement (Sec.~\\ref{analysis_of_pose_refinement}) and canonical poses (Sec.~\\ref{analysis_of_canonical_poses}), Impact of background regularization (Sec.~\\ref{impact_of_background_regularization}) and view direction (Sec.~\\ref{impact_of_view_direction}).\n\n\n\\subsection{Analysis of Pose Refinement}\n\\label{analysis_of_pose_refinement}\n\nHere we discuss the impact of pose refinement on our approach. \nOur method relies on SMPL parameters for explicit deformation, so inaccurate SMPL estimation may lead to catastrophic results as shown in Fig.~\\ref{fig:Comparision_with_different_method}(d).\nWe initialize the SMPL parameters with estimations from VIBE\\cite{VIBE}, which is the state-of-the-art pose and shape estimation method. \nHowever, pose estimation from monocular videos usually suffers from depth ambiguity.\nAs shown in Fig.~\\ref{Pose_Refine_Vis}(a), although our input video is a simple A-pose, the SMPL estimated by VIBE is usually misaligned at the foot joints.\nAfter pose refinement, the SMPL model is better fitting to the input image as shown in Fig.~\\ref{Pose_Refine_Vis}(b).\n\n\\begin{figure}[ht]\n    \\centering\n\t\\subfigure[VIBE est.]{\n\t\\begin{minipage}[b]{0.46\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/pose_refine_vibe.png}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[Ours]{\n\t\\begin{minipage}[b]{0.46\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/pose_refine_ours.png}\n\t\\end{minipage}\n\t} \n    \\caption{Visual comparison before and after pose refinement on iPER\\cite{LWGAN}. After pose refinement, the SMPL model is better aligned with the input image (e.g. foot joints).\n   \n    }\n\t\\label{Pose_Refine_Vis}\n\\end{figure}\n\n\\subsection{Analysis of Canonical Poses}\n\\label{analysis_of_canonical_poses}\n\nIn this section, we discuss the effect of using different canonical poses in canonical space on the reconstruction results. \nSince the pose-guided deformation is explicitly based on SMPL\\cite{SMPL}, we will get different canonical spaces with different canonical poses. \nTherefore, the choice of canonical poses has a crucial impact on the reconstruction and novel pose synthesis. Here we will discuss the reconstruction results of three different canonical poses: A-pose, T-pose, and X-pose. \nA-pose is the average pose of the body poses of the training frames, which is the closest to the poses in the training frames.  \nT-pose is the SMPL model's rest pose, where the arms are far away from the body, but the legs are closer to each other. \nIn comparison, our customized X-pose offers more spread body parts (see Fig. \\ref{Canonical_pose}(d)).\n\nAs shown in Fig. \\ref{Canonical_pose}, using A-pose as the canonical pose offers the best quality for canonical space NeRF learning, while using T-pose and X-pose as canonical poses result in some artifacts under the axilla and the thighs. \nIf a point is close to two different SMPL body parts (e.g. body and arm, two legs), it is hard to decide which part the point belongs to since SMPL models unclothed human body only without taking the offset of the clothes into consideration.\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.182\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/canonical_input.png}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[A-pose]{\n\t\\begin{minipage}[b]{0.182\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/canonical_A-pose.png}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[T-pose]{\n\t\\begin{minipage}[b]{0.24\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/canonical_T-pose.png}\n\t\\end{minipage}\n\t} \n    \\subfigure[X-pose]{\n\t\\begin{minipage}[b]{0.24\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/canonical_X-pose.png}\n\t\\end{minipage}\n\t} \n    \\caption{Visualization of different canonical NeRF spaces with different canonical poses during training on People-Snapshot\\cite{Video_avatars}.}\n\t\\label{Canonical_pose}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.18\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_pose_input.png}\n\t\\end{minipage}\n\t}\n\t\\subfigure[A-pose]{\n\t\\begin{minipage}[b]{0.21\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_pose_A-pose.png}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[T-pose]{\n\t\\begin{minipage}[b]{0.21\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_pose_T-pose.png}\n\t\\end{minipage}\n\t} \n    \\subfigure[X-pose]{\n\t\\begin{minipage}[b]{0.21\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/novel_pose_X-pose.png}\n\t\\end{minipage}\n\t} \n    \\caption{Novel pose synthesis with different canonical poses on People-Snapshot\\cite{Video_avatars}. X-pose is a better choice for novel pose synthesis compared to A-pose and T-pose, which produce unacceptable artifacts (e.g. multiple legs).}\n\t\\label{novel_pose_in_difference_canonical_pose}\n\\end{figure}\n\nFor reconstruction, A-pose is the best choice, but X-pose is more suitable for new pose synthesis. As shown in Fig. ~\\ref{novel_pose_in_difference_canonical_pose}. \nWhen using A-pose and T-pose as canonical poses for synthesizing a pose different from the poses in the training frames, there exist some unacceptable artifacts (e.g. multiple legs). This is because the different body parts are too close to each other in the canonical space of A-pose or T-pose, so that one body part (e.g. left leg) will be deformed by the transformation of another body part (e.g. right leg), resulting in multiple legs in Fig. ~\\ref{novel_pose_in_difference_canonical_pose}(b)(c). \n\n\n\\subsection{Impact of Background Regularization}\n\\label{impact_of_background_regularization}\n\nIn this section, we discuss the benefits of background regularization for our approach. Our approach focuses on the animatable NeRF of the human from a monocular video. \nTo avoid possible negative influence from the background, we set the background to white uniformly (with the help of an off-the-shelf segmentation network). \nHowever, it is common to appear some noisy density regions in the empty space (i.e. non-human space) after training.\nAs shown in Fig. \\ref{BG_Regularization}(b), although the background of the image is white, we notice some noisy non-zero density regions from the depth map. \nThis is because there is an ambiguity between the background (white in our case) and the cloth which happens to be the same color as the background. \nTo deal with this problem, we introduce background regularization to encourage the density of the background region to be zero. \nWith background regularization, the artifacts in the empty space are significantly reduced as shown in Fig.~\\ref{BG_Regularization}(c).\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.18\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_002_input.jpg}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[w\/o background reg.]{\n\t\\begin{minipage}[b]{0.35\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_002_without_BG.png}\n\t\\end{minipage}\n\t} \n\t\\subfigure[w\/ background reg.]{\n\t\\begin{minipage}[b]{0.36\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_002_with_BG.png}\n\t\\end{minipage}\n\t}   \n    \\caption{Impact of background regularization on iPER\\cite{LWGAN}. The background regularization can effectively reduce the artifacts in the background region.}\n\t\\label{BG_Regularization}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\begin{minipage}[b]{0.22\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_009_input.jpg}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[w\/ viewing direction]{\n\t\\begin{minipage}[b]{0.33\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_009_with_view.png}\n\t\\end{minipage}\n\t}   \n\t\\subfigure[w\/o viewing direction]{\n\t\\begin{minipage}[b]{0.33\\linewidth}\n\t\\includegraphics[width=1.0\\linewidth, trim=0 0 0 0,clip]{figures\/iper_009_without_view.png}\n\t\\end{minipage}\n\t} \n    \\caption{Impact of viewing direction on novel view synthesis on iPER. After removing viewing direction from the input, our model produces more consistent result across different views.}\n\t\\label{Impact_of_view_direction}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfigure[Input]{\n\t\\includegraphics[width=0.20\\linewidth]{figures\/complex_pose_input.png}\n\t}\n\t\\subfigure[w\/o pose refinement]{\n\t\\includegraphics[width=0.34\\linewidth]{figures\/complex_pose_without_pose_refinement.png}\n\t}\n\t\\subfigure[Ours]{\n\t\\includegraphics[width=0.35\\linewidth]{figures\/complex_pose_ours.png}\n\t}\n    \\caption{Visualization result of novel pose synthesis on a complex pose video 009-4-2 of the iPER\\cite{LWGAN}.}\n    \\label{complex_pose_result}\n\\end{figure}\n\n\\subsection{Impact of viewing direction}\n\\label{impact_of_view_direction}\n\nUnlike NeRF\\cite{NeRF}, which maps the 3D position and viewing direction \nto color and density, our approach excludes viewing direction from our input for robust dynamic reconstruction.\nNeRF's viewing direction is mainly used to handle specular reflections for materials such as glass and metal. \nFor dynamic scenes, it is very difficult to deal with changes of illumination, so we assume that the appearance of the subject is not view-dependent in our experiment.\nAlso, human skin and clothes are mainly diffuse reflective materials, and there exist very few specular reflections.\nAs shown in the Fig.~\\ref{Impact_of_view_direction}, the novel view synthesis results generated with viewing direction as a condition during training show unpredictable artifacts.\n\n\n\\section{Limitations}\n\n\nOur method reconstructs a detailed 3D human body model and renders realistic images from a monocular video. \nTypically, the training videos capture subjects turning around before the camera and holding an A-pose or T-pose.\nWhen trained on a video containing complex poses, our method still obtains reasonable results (Fig. \\ref{complex_pose_result}). But noticeable losses of details are observed compared to previous simple training videos.\nThe main reason is that it is more challenging to obtain accurate enough SMPL estimations for videos containing complex poses.\nAnother limitation is that it is difficult for our method to handle extremely loose clothes or complex non-rigid deformations of the garments, \nbecause our explicit pose-guided deformation associates spatial points to SMPL mesh, without explicit modeling of the garments.\nThus, the novel view synthesis results inevitably lose some details on the clothes. So, to the get best results, the performer should slowly turn around and hold a simple pose so that their clothes almost remain still relative to their body for high-quality rendering. \nLike all NeRF based methods trained for only one scene, Our method cannot reconstruct invisible parts, such as the underarms and the inner thighs, so the input video needs to cover the whole body of the human body as much as possible. \n\n\\section{Conclusion}\nIn this paper, we propose to learn an animatable neural radiance field from a monocular video, which allows us to perform photo-realistic novel-view synthesis, reconstruct the 3D geometry of the person with high-quality details, and animate the person with novel poses. \nTo achieve these goals, we extend the neural radiance field to dynamic scenes with human movements via introducing an explicit pose-guided deformation module and an analysis-by-synthesis pose refinement strategy. \nSpecifically, the pose-guided deformation attempts to deform the 3d position according to the neighboring SMPL vertices to learn a good and controllable human template in the canonical space, as well as to learn accurate 3d geometry.\nThe pose refinement strategy compensates for the negative impact of inaccurate pose estimation from existing approaches and provides more consistent\nguidance for learning better geometry (i.e. density) and appearance (i.e. RGB). \nExperiments on both synthetic data and real data demonstrate the effectiveness of the proposed approach. \n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe give some heuristics for counting elliptic curves with certain\nproperties. In particular, we re-derive the Brumer-McGuinness\nheuristic for the number of curves with positive\/negative discriminant\nup to~$X$, which is an application of lattice-point counting.\nWe then introduce heuristics (with refinements from random matrix theory)\nthat allow us to predict how often we expect an elliptic curve~$E$\nwith even parity to have $L(E,1)=0$. It turns out that we roughly expect\nthat a curve with even parity has $L(E,1)=0$ with probability proportional\nto the square root of its real period, and, since the real period\nis very roughly~$1\/\\Delta^{1\/12}$, this leads us to the\nprediction that almost all curves with even parity should have~$L(E,1)\\neq 0$.\nBy the conjecture of Birch and Swinnerton-Dyer, this says that\nalmost all such curves have rank~0.\n\nWe then make similar heuristics when enumerating by conductor.\nThe first task here is simply to count curves with conductor up to~$X$,\nand for this we use heuristics involving how often large powers of\nprimes divide the discriminant.\nUpon making this estimate, we are then able to imitate\nthe argument we made previously,\nand thus derive an asymptotic for the number of curves\nwith even parity and $L(E,1)=0$ under the ordering by conductor.\nWe again get the heuristic that almost all curves\nwith even parity should have~$L(E,1)\\neq 0$.\n\nWe then make a few remarks regarding how often curves should\nhave nontrivial isogenies and\/or torsion under different orderings,\nand then present some data regarding average ranks. We conclude by\ngiving data for Mordell-Weil lattice distribution for rank 2 curves,\nand speculating about symmetric power $L$-functions.\n\n\\section{The Brumer-McGuinness Heuristic}\n\nFirst we re-derive the Brumer-McGuinness\nheuristic~\\cite{brumer-mcguinness} for the number of elliptic\ncurves whose absolute discriminant is less than a given bound~$X$;\nthe technique here is essentially lattice-point counting, and we\nderive our estimates via the assumption that these counts are\nwell-approximated by the areas of the given regions.\n\n\\begin{conjecture}\\label{conj:bmcg}[Brumer-McGuinness]\nThe number $A_\\pm(X)$ of rational elliptic curves with a global\nminimal model (including at $\\infty$) and positive or negative\ndiscriminant whose absolute value is less than~$X$\nis asymptotically $A_\\pm(X)\\sim {\\alpha_{\\pm}\\over\\zeta(10)}X^{5\/6}$, where\n$\\alpha_{\\pm}={\\sqrt 3\\over 10}\\int_{\\pm1}^\\infty {dx\\over\\sqrt{x^3\\mp1}}$.\n\\end{conjecture}\n\nAs indicated by Brumer and McGuinness, the identity\n$\\alpha_{-}=\\sqrt{3}\\alpha_{+}$ was already known to Legendre,\nand is related to complex multiplication.\nThese constants can be expressed in terms of Beta integrals\n$B(u,v)=\\int_0^1 x^{u-1}(1-x)^{v-1}\\, dx=\n{\\Gamma(u)\\Gamma(v)\\over \\Gamma(u+v)}$ as\n$\\alpha_{+}={1\\over 3}{B}(1\/2,1\/6)$ and $\\alpha_{-}={B}(1\/2,1\/3)$.\n\nRecall that every rational elliptic curve has a unique integral\nminimal model $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$ with\n$a_1,a_3\\in\\{0,1\\}$ and $|a_2|\\le 1$.\nFix one of the 12 choices of~$(a_1,a_2,a_3)$. Since these are all\nbounded by~1 the discriminant is thus approximately~$-64a_4^3-432a_6^2$.\nSo we wish to count the number\nof $(a_4,a_6)$-lattice-points with~$|64a_4^3+432a_6^2|\\le X$,\nnoting that Brumer and McGuinness divide the curves according to the\nsign of the discriminant. The lattice-point count\nfor $a_1=a_2=a_3=0$ is given by\n$$\\mathop{\\sum\\sum}_{0<-64a_4^3-432a_6^2<X} 1\n\\quad+\\mathop{\\sum\\sum}_{-X<-64a_4^3-432a_6^2<0} 1.$$\nWe estimate this lattice-point count by the integral\n$\\mathop{\\int\\int}_{U} du_4 du_6$ for the region $U$ given by\n$|64u_4^3+432u_6^2|<X$. After splitting into two parts based\nupon the sign of the discriminant and performing the $u_4$-integration,\nwe get\n\\begin{align*}\n{2\\over (64)^{1\/3}}\n\\int_0^\\infty \\Bigl[(-432u_6^2)^{1\/3}&-(-X-432u_6^2)^{1\/3}\\Bigr]\\, du_6+\\\\\n&+{2\\over (64)^{1\/3}}\n\\int_0^\\infty \\Bigl[(X-432u_6^2)^{1\/3}-(-432u_6^2)^{1\/3}\\Bigr]\\, du_6,\n\\end{align*}\nwhere the factor of $2$ comes from the sign of~$u_6$.\nChanging variables $u_6=w\\sqrt{X\/432}$ and multiplying by~12\nfor the number of $(a_1,a_2,a_3)$-choices we get\n\\begin{align*}\n{24\\over (64)^{1\/3}}{X^{5\/6}\\over \\sqrt{432}}\n\\int_0^\\infty \\Bigl[(w^2+1)^{1\/3}&-(w^2)^{1\/3}\\Bigr]\\, dw+\\\\\n&+{24\\over (64)^{1\/3}}{X^{5\/6}\\over \\sqrt{432}}\n\\int_0^\\infty \\Bigl[(w^2)^{1\/3}-(w^2-1)^{1\/3}\\Bigr]\\, dw.\n\\end{align*}\nThese integrals are probably known, but I am unable to find a reference.\nThe first integral simplifies\\footnote{\n As N.~D.~Elkies indicated to us, we can write\n $I(a)=\\int_0^\\infty \\bigl[(t^2+a)^{1\/3}-(t^2)^{1\/3}\\bigr]\\, dt$,\n differentiate under the integral sign, then substitute~$t^2+a=ax^3$,\n and finally re-integrate to obtain~$I(1)$.}\nto ${3\\over 5}\\int_{1}^\\infty {dx\\over\\sqrt{x^3-1}}={1\\over 5}{B}(1\/2,1\/6)$,\nwhile the second becomes\n${3\\over 5}\\int_{-1}^\\infty {dx\\over\\sqrt{x^3+1}}={3\\over 5}{B}(1\/2,1\/3)$.\nThis counts all models of curves; if we eliminate non-minimal models,\nfor which we have $p^4|c_4$ and~$p^{12}|\\Delta$ for some prime~$p$,\nwe expect to accrue an extra factor\\footnote{\n Note that some choices of $(a_1,a_2,a_3)$ necessarily\n have odd discriminant, but the other choices compensate\n to give the proper Euler factors at 2 (and 3).}\nof~$\\zeta(10)$. From this, we get the conjecture of Brumer and McGuinness\nas stated above.\n\n\\section{Counting curves of even parity whose central $L$-value vanishes.}\n\nSee \\cite[\\S 15-16]{silverman} for definitions of the conductor $N$\nand $L$-function $L(E,s)$ of an elliptic curve~$E$.\nSince rational elliptic curves are modular, we have that the\ncompleted $L$-function $\\Lambda(E,s)=\\Gamma(s)(\\sqrt N\/2\\pi)^sL(E,s)$\nsatisfies $\\Lambda(E,s)=\\pm\\Lambda(E,2-s)$. When the plus sign occurs,\nwe say that $E$ has even parity.\n\nWe now try to count elliptic curves $E$ with\neven parity for which~$L(E,1)=0$.\nThroughout this section, $E$ shall be a curve with\neven parity, and we shall order curves by discriminant.\nVia the conjectural Parity Principle, we expect that,\nunder any reasonable ordering, half of the elliptic curves\nshould have even parity; in particular, we predict that there\nare asymptotically $A_{\\pm}(X)\/2$ curves with even parity and\npositive\/negative discriminant up to~$X$.\n\nOur main tool shall be random matrix theory, which\ngives a heuristic for predicting how often $L(E,1)$ is small.\nWe could alternatively derive a cruder heuristic by assuming the\nthe order of the Shafarevich-Tate group is a random square\ninteger in a given interval, but random matrix theory has the\nadvantage of being able to predict a more explicit asymptotic.\nOur principal heuristic is the following:\n\n\\begin{heuristic}\\label{conj:rank2disc}\nThe number $R(X)$ of rational elliptic curves $E$ with\neven parity and $L(E,1)=0$ and\nabsolute discriminant less than~$X$\nis given asymptotically by $R(X)\\sim cX^{19\/24}(\\log X)^{3\/8}$\nfor some computable constant~$c>0$.\n\\end{heuristic}\n\nIn particular, note that we get the prediction\nthat almost all curves with\neven parity have $L(E,1)\\neq 0$ under this ordering.\n\n\\subsection{Random matrix theory}\nOriginally developed in mathematical statistics by Wishart \\cite{wishart}\nin the 1920s and then in mathematical physics\n(especially the spectra of highly excited nuclei)\nby Wigner \\cite{wigner}, Dyson, Mehta, and others\n(particularly~\\cite{marcenko-pastur}), random matrix theory \\cite{mehta}\nhas now found some applications in number theory,\nthe earliest being the oft-told story of\nDyson's remark to Montgomery regarding the pair-correlation of zeros of\nthe Riemann $\\zeta$-function.\nBased on substantial numerical evidence, random matrix theory appears\nto give reasonable models for the distribution of $L$-values in families,\nthough the issue of what constitutes a proper family is a delicate one\n(see particularly \\cite[\\S 3]{CFKRS}, where the notion of family comes from\nthe ability to calculate moments of $L$-functions rather than\nfrom algebraic geometry).\n\nThe family of quadratic twists of a given elliptic curve\n$E:y^2=x^3+Ax+B$ is given by $E_d:y^2=x^3+Ad^2x+Bd^3$ for squarefree~$d$.\nThe work (most significantly a monodromy computation)\nof Katz and Sarnak \\cite{katz-sarnak} regarding families of\ncurves over function fields implies that when we restrict to quadratic twists\nwith even parity, we should expect that the $L$-functions are modelled\nby random matrices with even orthogonal symmetry.\nThough we have no function field analogue in our case, we brazenly\nassume (largely from looking at the sign in the functional equation)\nthat the symmetry type is again orthogonal with even parity.\nWhat this means is that we want to model properties of the $L$-function \nvia random matrices taken from ${\\rm SO}(2M)$ with respect to Haar measure.\nHere we wish the mean density of zeros\nof the \\hbox{$L$-functions} to match the mean density of eigenvalues\nof our matrices, and so, as in~\\cite{keating-snaith},\nwe should take $2M\\approx 2\\log N$. We suspect that the $L$-value\ndistribution is approximately given by the distribution of the evaluations\nat~$1$ of the characteristic polynomials of our random matrices.\nIn the large, this distribution is determined entirely by\nthe symmetry type, while finer considerations are distinguished\nvia arithmetic considerations.\n\nWith this assumption, via the moment conjectures of \\cite{keating-snaith}\nand then using Mellin inversion, as $t\\rightarrow 0$ we have\n(see (21) of \\cite{random-matrix-theory}) that\n\\begin{equation}\\label{RMTprob}\n{\\rm Prob}[L(E,1)\\le t]\\sim \\alpha(E) t^{1\/2}M^{3\/8}.\n\\end{equation}\nThis heuristic is stated for fixed $M\\approx\\log N$,\nbut we shall also allow $M\\rightarrow\\infty$.\nIt is not easy to understand this probability, as both the constant\n$\\alpha(E)$ and the matrix-size $M$ depend on~$E$.\nWe can take curves with $e^M\\le N\\le e^{M+1}$ to mollify the\nimpact of the conductor, but in order to average over a set of\ncurves, we need to understand how $\\alpha(E)$ varies.\nOne idea is that $\\alpha(E)$ separates into two parts, one of which\ndepends on local structure (Frobenius traces) of the curve,\nand the other of which depends only upon the size of the conductor~$N$.\nLetting $G$ be the Barnes $G$-function (such that $G(z+1)=\\Gamma(z)G(z)$\nwith $G(1)=1$) and $M=\\lfloor\\log N\\rfloor$ we have that\n$$\\alpha(E)=\\alpha_R(M)\\cdot \\alpha_A(E)$$\n$\\text{with}\\>\\>\\alpha_R(M)\n\\rightarrow\\hat\\alpha_R=2^{1\/8}G(1\/2)\\pi^{-1\/4}\\>\\>\n\\text{as}\\>\\> M\\rightarrow\\infty$\nand\n\\begin{equation}\\label{eqn:Fp}\n\\alpha_A(E)=\\prod_p F(p)=\n\\prod_p \\biggl(1-{1\\over p}\\biggr)^{3\/8} \\biggl({p\\over p+1}\\biggr)\n\\biggl({1\\over p}+{L_p(1\/p)^{-1\/2}\\over 2}+{L_p(-1\/p)^{-1\/2}\\over 2}\\biggr)\n\\end{equation}\nwhere $L_p(X)=(1-a_pX+pX^2)^{-1}$ when $p\\nmid\\Delta$\nand $L_p(X)=(1-a_pX)^{-1}$ otherwise;\nsee (10) of \\cite{random-matrix-theory} evaluated at~$k=-1\/2$,\nthough that equation is wrong at primes that\ndivide the discriminant --- see (20) of~\\cite{CPRW},\nwhere $Q$ should be taken to be~1.\nNote that the Sato-Tate conjecture \\cite{tate} implies that\n$a_p^2$ is $p$ on average, and this implies that the above\nEuler product converges.\n\n\\subsection{Discretisation of the $L$-value distribution}\nFor precise definitions of the Tamagawa numbers, torsion group,\nperiods, and Shafarevich-Tate group, see~\\cite{silverman},\nthough below we give a brief description of some of these.\nWe let $\\tau_p(E)$ be the Tamagawa number of~$E$ at the\n(possibly infinite) prime~$p$, and write $\\tau(E)=\\prod_p \\tau_p(E)$\nfor the Tamagawa product and $T(E)$ for the size of the torsion group.\nWe also write $\\Omega_{\\rm re}(E)$ for the real\nperiod and ${\\mbox{\\textcyr{Sh}}}_{\\rm an}(E)$ for the size of the Shafarevich-Tate group\nwhen $L(E,1)\\neq 0$, with ${\\mbox{\\textcyr{Sh}}}_{\\rm an}(E)=0$ when $L(E,1)=0$.\n\nWe wish to assert that sufficiently small values of~$L(E,1)$\nactually correspond to~$L(E,1)=0$.\nWe do this via the conjectural formula of Birch and\nSwinnerton-Dyer~\\cite{BSD}, which asserts that\n$$L(E,1)=\n\\Omega_{\\rm re}(E)\\cdot {\\tau(E)\\over T(E)^2}\\cdot {\\mbox{\\textcyr{Sh}}}_{\\rm an}(E).$$\nOur discretisation\\footnote\n{The precision of this discretisation might be the most-debatable\n methodology we use. Indeed, we are essentially taking a ``sharp cutoff'',\n while it might be better to have a more smooth transition function.\n For this reason, we do not specify the leading\n constant in our final heuristic.}\nwill be that\n$$L(E,1)<\\Omega_{\\rm re}(E)\\cdot {\\tau(E)\\over T(E)^2}\n\\quad\\text{implies}\\quad L(E,1)=0.$$\nNote that we are only using that ${\\mbox{\\textcyr{Sh}}}_{\\rm an}$ takes on integral\nvalues, and do not use the (conjectural) fact that it is square.\n\nUsing \\eqref{RMTprob}, we estimate the number of curves\nwith positive (for simplicity) discriminant less than~$X$\nand even parity and $L(E,1)=0$ via the lattice-point sum\n$$W(X)=\\mathop{\\sum\\sum}_{\\text{$c_4,c_6$ minimal}\\atop 0<c_4^3-c_6^2<1728X}\n\\alpha_R(M)\\alpha_A(E)\n\\cdot{\\sqrt{\\Omega_{\\rm re}(E)\\tau(E)\\over T(E)^2}}\\cdot M^{3\/8}.$$\nWe need to introduce congruence conditions on $c_4$ and $c_6$\nto make sure that they correspond to a minimal model of an elliptic curve.\nThe paper \\cite{stein-watkins} uses the work of Connell \\cite{connell}\nin a different context to get that\nthere are 288 classes of $(c_4\\>\\text{mod}\\>576,c_6\\>\\text{mod}\\>1728)$\nthat can give minimal models, and so we get a factor of~$288\/(576\\cdot 1728)$,\nassuming that each congruence class has the same impact\non all the entities in the sum. Indeed, this independence\n(on average) of various quantities with respect\nto $c_4$ and $c_6$ is critical in our estimation of~$W(X)$.\nThere is also the question of non-minimal models,\\footnote\n{At $p=2,3$, non-minimality occurs when $c_4\/p^4$ and $c_6\/p^6$\n satisfy the congruences.}\nfrom which we get a factor of~$1\/\\zeta(10)$.\n\n\\begin{guess}\\label{GUESS}\nThe lattice-point sum $W(X)$ can be approximated as $X\\rightarrow\\infty$ by\n$$\\hat W(X)={288\\over (576\\cdot 1728)}{1\\over\\zeta(10)}\\cdot\n\\hat\\alpha_R\\bar\\alpha_A\\beta_\\tau\\cdot\n\\mathop{\\int\\int}_{1\\le{u_4^3-u_6^2\\over 1728}<X}\n\\Omega_{\\rm re}(E)^{1\/2}\\cdot (\\log\\Delta)^{3\/8} du_4du_6.$$\nHere $\\hat\\alpha_R$ is the limit $2^{1\/8}G(1\/2)\\pi^{-1\/4}$ of $\\alpha_R(M)$\nas $M\\rightarrow\\infty$, while $\\bar\\alpha_A$ is a suitable average\nof the arithmetic factors $\\alpha_A(E)$, and $\\beta_\\tau$ is the\naverage of the square root of the Tamagawa product. We have also approximated\n$\\log N\\approx\\log\\Delta$ and assumed the torsion is trivial;\nbelow we will give these heuristic justification~(on average).\nNote that everything left in the integral is\na smooth function of $u_4$ and $u_6$.\n\\end{guess}\n\nWe shall first evaluate the integral in $\\hat W(X)$ given these suppositions,\nand then try to justify the various assumptions that are inherent in\nthis guess.\\footnote\n{Note that our methods do not readily generalise to higher rank, as there is\n no apparent way to model the heights of points (and thus the regulator).}\n\n\\subsection{Evaluation of the integral}\nWrite $E$ as $y^2=4x^3-(c_4\/12)x-c_6\/216$,\nand put $e_1>e_2>e_3$ for the roots of the cubic polynomial on the right side.\nWe have\n$$1\/\\Omega_{\\rm re}=\n{\\rm agm}\\bigl(\\sqrt{e_1-e_2},\\sqrt{e_1-e_3}\\bigr)\/\\pi.$$\nWe also have that $(e_1-e_2)(e_1-e_3)(e_2-e_3)=\\sqrt{\\Delta\/16}$\nfrom the formula for the discriminant.\nWe next write $e_1-e_2=\\Delta^{1\/6}\\lambda$ and $e_2-e_3=\\Delta^{1\/6}\\mu$\nso that we have $\\mu\\lambda(\\lambda+\\mu)=1\/4$,\nwhile $e_1={\\Delta^{1\/6}\\over 3}(\\mu+2\\lambda)$,\n$e_2={\\Delta^{1\/6}\\over 3}(\\mu-\\lambda)$,\nand $e_3=-{\\Delta^{1\/6}\\over 3}(2\\mu+\\lambda)$.\nThus we get\n$$-c_6\/864=-e_1e_2e_3=\n{\\Delta^{1\/2}\\over 27}(\\mu+2\\lambda)(\\mu-\\lambda)(2\\mu+\\lambda)$$\nand\n$$-c_4\/48=e_1e_2+e_1e_3+e_2e_3=\n-{\\Delta^{1\/3}\\over 3}(\\mu^2+\\lambda\\mu+\\lambda^2).$$\nChanging variables in the $\\hat W$-integral\ngives a Jacobian of $432\/\\Delta^{1\/6}\\sqrt{\\mu^4+\\mu}$\nso that\n$$\\hat W(X)=\\tilde c\\int_1^X\\int_0^\\infty {(\\log \\Delta)^{3\/8}\\over\n\\sqrt{\\Delta^{1\/12}\\,{\\rm agm}(\\sqrt\\lambda,\\sqrt{\\lambda+\\mu})}}\n{d\\mu\\,d\\Delta\\over\\Delta^{1\/6}\\sqrt{\\mu^4+\\mu}},$$\nwhere $\\lambda=(\\sqrt{\\mu^4+\\mu}-\\mu^2)\/2\\mu$.\nThus the variables are nicely separated, and since the $\\mu$-integral\nconverges, we do indeed get $\\hat W(X)\\sim cX^{19\/24}(\\log X)^{3\/8}$.\nA similar argument can be given for curves with negative discriminant.\nThis concludes our derivation of Heuristic~\\ref{conj:rank2disc},\nand now we turn to giving some reasons for our expectation that\nthe arithmetic factors can be mollified by taking their averages.\n\n\\subsection{Expectations for arithmetic factors on average}\nIn the next section we shall explain (among other things)\nwhy we expect that $\\log N\\approx \\log\\Delta$ for almost all curves,\nand in section~\\ref{section:torsion},\nwe shall recall the classical parametrisations of $X_1(N)$\ndue to Fricke to indicate why we expect the torsion size\nis 1 on average. Here we show how to compute the various averages\n(with respect to ordering by discriminant)\nof the square root of the Tamagawa product\nand the arithmetic factors~$\\alpha_A(E)$.\n\nFor both heuristics, we shall make the assumption that curves satisfying\nthe discriminant bound $|\\Delta|\\le X$ behave essentially the same as\nthose that satisfy $|c_4|\\le X^{1\/3}$ and $|c_6|\\le X^{1\/2}$.\nThat is, we approximate our region by a big box.\nWe write $D$ for the absolute value of~$\\Delta$.\nFirst we consider the Tamagawa product.\n\nWe wish to know how often a prime divides the discriminant to a high power.\nFix a prime~$p\\ge 5$ with $p$ a lot smaller than~$X^{1\/3}$.\nWe can estimate the probability that~$p^k|\\Delta$\nby considering all $p^{2k}$ choices of $c_4$ and $c_6$ modulo~$p^k$,\nthat is, by counting the number of solutions $S(p^k)$\nto $c_4^3-c_6^2=1728\\Delta\\equiv 0$ (mod~$p^k$).\nThis auxiliary curve $c_4^3=c_6^2$ is singular at $(0,0)$ over~${\\bf F}_p$,\nand has $(p-1)$ non-singular ${\\bf F}_p$-solutions\nwhich lift to $p^{k-1}(p-1)$ points modulo~$p^k$.\n\nFor $p^k$ sufficiently small, our $(c_4,c_6)$-region is so large that we can \nshow that the probability that $p^k|\\Delta$ is~$S(p^k)\/p^{2k}$.\nWe assume that big primes act (on average) in the same manner,\nwhile a similar heuristic can be given for~$p=2,3$.\nCurves with $p^4|c_4$ and $p^6|c_6$ will not be given by their\nminimal model; indeed, we want to exclude these curves, and\nthus will multiply our probabilities by $\\kappa_p=(1-1\/p^{10})^{-1}$\nto make them conditional on this criterion. For instance, the above\ncounting of points says that there is a probability of $(p^2-p)\/p^2$\nthat~$p\\nmid D$, and so upon conditioning upon minimal models we get\n$\\kappa_p(1-1\/p)$ for this probability.\n\nWhat is the probability $P_m(p,k)$ that a curve given by a minimal model\nhas multiplicative reduction at $p\\ge 5$ and $p^k\\|D$ for some~$k>0$?\nIn terms of Kodaira symbols, this is the case of I$_k$.\nFor multiplicative reduction we need that $p\\nmid c_4,c_6$.\nThese events are independent and each has a probability $(1-1\/p)$\nof occurring. Upon assuming these conditions and working modulo~$p^k$,\nthere are $(p^k-p^{k-1})$ such choices for each,\nand of the resulting $(c_4,c_6)$\npairs we noted above that $p^{k-1}(p-1)$ of them have~$p^k|D$.\nSo, given a curve with~$p\\nmid c_4,c_6$,\nwe have a probability of $1\/p^{k-1}(p-1)$ that~$p^k|D$,\nwhich gives $1\/p^k$ for the probability that~$p^k\\|D$.\nIn symbols, we have that (for~$p\\ge 5$ and $k\\ge 1$)\n$${\\rm Prob}\\Bigl[p^k\\|(c_4^3-c_6^2) \\Bigm| p\\nmid c_4,c_6\\Bigr]=1\/p^k.$$\nIncluding the conditional probability for minimal models, we get\n\\begin{equation}\\label{eqn:probm}\nP_m(p,k)=(1-1\/p^{10})^{-1}(1-1\/p)^2\/p^k\\quad\\text{(for $p\\ge 5$ and $k\\ge 1$).}\n\\end{equation}\nNote that summing this over $k\\ge 1$ gives $\\kappa_p(1-1\/p)\/p$ for the\nprobability for an elliptic curve to have multiplicative reduction at~$p$.\n\nWhat is the probability $P_a(p,k)$ that a curve given by a minimal\nmodel has additive reduction at $p\\ge 5$ and $p^k\\|D$ for some~$k>0$?\nWe shall temporarily ignore the factor\nof $\\kappa_p=(1-1\/p^{10})^{-1}$ from non-minimal models \nand include it at the end.\nWe must have that $p|c_4,c_6$, and thus get that~$k\\ge 2$.\nFor $k=2,3,4$, which correspond to Kodaira symbols II, III, and IV\nrespectively, the computation is not too bad:\nwe get that $p^2\\|D$ exactly when $p|c_4$ and~$p\\|c_6$,\nso that the probability is $(1\/p)\\cdot (1-1\/p)\/p=(1-1\/p)\/p^2$;\nfor $p^3\\|D$ we need $p\\|c_4$ and $p^2|c_6$\nand thus get $(1-1\/p)\/p\\cdot(1\/p^2)=(1-1\/p)\/p^3$ for the probability;\nand for $p^4\\|D$ we need $p^2|c_4$ and $p^2\\|c_6$ and so get\n$(1\/p^2)\\cdot(1-1\/p)\/p^2=(1-1\/p)\/p^4$ for the probability.\nNote that the case $k=5$ cannot occur.\nThus we have (for~$p\\ge 5$) the formula\n$P_a(p,k)=(1-1\/p^{10})^{-1}(1-1\/p)\/p^k$ for $k=2,3,4$.\n\nMore complications occur for $k\\ge 6$, where now we split into two\ncases depending upon whether additive reduction persists upon taking\nthe quadratic twist by~$p$. This occurs when $p^3|c_4$ and $p^4|c_6$,\nand we denote by $P_a^n(p,k)$ the probability that $p^k\\|D$ in this subcase.\nJust as above, we get that\n$P_a^n(p,k)=(1-1\/p^{10})^{-1}(1-1\/p)\/p^{k-1}$ for $k=8,9,10$.\nThese are respectively the cases of Kodaira symbols IV$^\\star$,\nIII$^\\star$, and~II$^\\star$.\nFor $k=11$ we have~$P_a^n(p,k)=0$, while for $k\\ge 12$ our\ncondition of minimality implies that we should take~$P_a^n(p,k)=0$.\n\nWe denote by $P_a^t(p,k)$ the probability that $p^6|D$ with either\n$p^2\\|c_4$ or $p^3\\|c_6$. First we consider curves for which $p^7|D$,\nand these have multiplicative reduction at~$p$ upon twisting.\nIn particular, these curves have $p^2\\|c_4$ and $p^3\\|c_6$,\nand the probability of this is $(1-1\/p)\/p^2\\cdot (1-1\/p)\/p^3$.\nConsider $k\\ge 7$. We then take $c_4\/p^2$ and $c_6\/p^3$ both modulo $p^{k-6}$,\nand get that $p^{k-6}\\|(D\/p^6)$ with probability $1\/p^{k-6}$\nin analogy with the above.\nSo we get that $P_a^t(p,k)=(1-1\/p^{10})^{-1}(1-1\/p)^2\/p^{k-1}$ for~$k\\ge 7$.\nThis corresponds to the case of I$_{k-6}^\\star$.\n\nFinally, for $p^6\\|D$ (which is the case I$_0^\\star$)\nwe get a probability of $(1\/p^2)\\cdot(1\/p^3)$ for the chance\nthat $p^2|c_4$ and $p^3|c_6$, and (since there are $p$ points mod~$p$\non the auxiliary curve $(c_4\/p^2)^3\\equiv(c_6\/p^3)^2 \\pmod{p}$)\na conditional probability of $(p^2-p)\/p^2$ that~$p^6\\|D$.\nSo we get that $P_a^t(p,6)=(1-1\/p^{10})^{-1}(1-1\/p)\/p^5$.\n\nWe now impose our current notation on the previous paragraphs,\nand naturally let $P_a^t(p,k)=0$ and $P_a^n(p,k)=P_a(p,k)$ for~$k\\le 5$.\nOur final result is that\n\\begin{equation}\\label{eqn:proba}\nP_a^n(p,k)=\n\\begin{cases}\n(1-1\/p^{10})^{-1}(1-1\/p)\/p^k & \\quad k=2,3,4\\\\\n(1-1\/p^{10})^{-1}(1-1\/p)\/p^{k-1} & \\quad k=8,9,10\n\\end{cases}\n\\end{equation}\n\\begin{equation}\\label{eqn:probd}\nP_a^t(p,k)=\n\\begin{cases}\n(1-1\/p^{10})^{-1}(1-1\/p)\/p^5 & \\quad k=6 \\\\\n(1-1\/p^{10})^{-1}(1-1\/p)^2\/p^{k-1} & \\quad k\\ge 7\n\\end{cases}\n\\end{equation}\nwith $P_a^n(p,k)$ and $P_a^t(p,k)$ equal to zero for other~$k$.\nWe conclude by defining $P_0(p,k)$ to be zero for $k>0$\nand to be the probability $(1-1\/p^{10})^{-1}(1-1\/p)$ that $p\\nmid D$ for~$k=0$.\nWe can easily check that we really do have the required probability relation\n$\\sum_{k=0}^\\infty \\bigl[P_m(p,k)+P_a^n(p,k)+P_a^t(p,k)+P_0(p,k)\\bigr]=1$,\nas: the cases of multiplicative reduction give $\\kappa_p(1-1\/p)\/p$;\nthe cases of Kodaira symbols II, III, and~IV give $\\kappa_p(1\/p^2-1\/p^5)$;\nthe cases of Kodaira symbols IV$^\\star$, III$^\\star$, and~II$^\\star$\ngive $\\kappa_p(1\/p^7-1\/p^{10})$; the cases of I$^\\star_k$ summed\nfor~$k\\ge 1$ give $\\kappa_p(1-1\/p)\/p^6$; the case of I$^\\star_0$\ngives $\\kappa_p(1-1\/p)\/p^5$; and the sum of these with\n$P_0(p,0)=\\kappa_p(1-1\/p)$ does indeed give us~1.\nWe could do a similar (more tedious) analysis for $p=2,3$,\nbut this would obscure our argument.\n\nGiven a curve of discriminant~$D$,\nwe can now compute the expectation for its Tamagawa number.\nWe consider primes~$p|D$ with $p\\ge 5$,\nand compute the local Tamagawa number~$t(p)$.\nWhen $E$ has multiplicative reduction at~$p$ and~$p^k\\|D$,\nthen $t(p)=k$ if $-c_6$ is square mod~$p$,\nand else $t(p)=1,2$ depending upon whether $k$ is odd or even.\nSo the average of $\\sqrt{t(p)}$ for this case is\n$\\epsilon_m(k)={1\\over 2}(1+\\sqrt k),{1\\over 2}(\\sqrt 2+\\sqrt k)$\nfor $k$ odd\/even respectively.\n\nWhen $E$ has potentially multiplicative reduction at~$p$ with~$p^k\\|D$,\nfor $k$ odd we have $t(p)=4,2$ depending on whether\n$(c_6\/p^3)\\cdot(\\Delta\/p^k)$ is square mod~$p$,\nand for $k$ even we have $t(p)=4,2$ depending on\nwhether $\\Delta\/p^k$ is square mod~$p$.\nIn both cases the average of $\\sqrt{t(p)}$ is~${1\\over 2}(\\sqrt 2+\\sqrt 4)$.\nIn the case of I$_0^\\star$ reduction where we have~$p^6\\|D$,\nwe have that $t(p)=1,2,4$ corresponding to whether the cubic\n$x^3-(27c_4\/p^2)x-(54c_6\/p^3)$ has $0,1,3$ roots modulo~$p$.\nSo the average of $\\sqrt{t(p)}$ is\n$${\\sqrt 1\\bigl((p-1)(p+1)\/3\\bigr)+\\sqrt 2\\bigl(p(p-1)\/2\\bigr)+\n\\sqrt 4\\bigl((p-1)(p-2)\/6\\bigr)\\over\n\\bigl((p-1)(p+1)\/3\\bigr)+\\bigl(p(p-1)\/2\\bigr)+\n\\bigl((p-1)(p-2)\/6\\bigr)}={2\\over 3}+{\\sqrt 2\\over 2}-{1\\over 3p}.$$\nin this case.\n\nFor the remaining cases,\nwhen $p^2\\|D$ or $p^{10}\\|D$ we have~$t(p)=1$,\nwhile when $p^3\\|D$ or $p^9\\|D$ we have~$t(p)=2$.\nFinally, when $p^4\\|D$ we have $t(p)=3,1$ depending\non whether $-6c_6\/p^2$ is square mod~$p$,\nand similarly when $p^8\\|D$ we have $t(p)=3,1$ depending\non whether $-6c_6\/p^4$ is square mod~$p$, so that the average\nof $\\sqrt{t(p)}$ in both cases is~${1\\over 2}(1+\\sqrt 3)$.\nWe get that $\\epsilon_a^n(k)=1,\\sqrt 2,{1\\over 2}(1+\\sqrt 3),\n{1\\over 2}(1+\\sqrt 3),\\sqrt 2,1$ for $k=2,3,4,8,9,10$, while\n\\begin{equation}\\label{eqn:epsilon}\n\\epsilon_m(k)=\n\\begin{cases}{1\\over 2}(1+\\sqrt k),&\\text{$k$ odd}\\\\\n{1\\over 2}(\\sqrt 2+\\sqrt k),&\\text{$k$ even}\\end{cases}\n\\quad\\text{and}\\quad\n\\epsilon_a^t(p,k)=\\begin{cases}\n{2\\over 3}+{\\sqrt 2\\over 2}-{1\\over 3p},&k=6\\\\\n{1\\over 2}(\\sqrt 2+\\sqrt 4),&k\\ge 7\\end{cases}\n\\end{equation}\nwith $\\epsilon_a^n(k)$ and $\\epsilon_a^t(p,k)$ equal to zero for other~$k$.\n\nWe define the expected square root of the Tamagawa number $K(p)$ at~$p$ by\n\\begin{equation}\\label{eqn:tama}\nK(p)=\\sum_{k=0}^\\infty\n\\bigl[\\epsilon_m(k) P_m(p,k)+\\epsilon_a^n(k) P_a^n(p,k)+\n\\epsilon_a^t(p,k) P_a^t(p,k)+P_0(p,k)\\bigr]\n\\end{equation}\nand the expected global\\footnote\n{Note that the Tamagawa number at infinity is 1 when $E$\n has negative discriminant and else is 2, the former\n occurring approximately $\\sqrt{3}\/(1+\\sqrt{3})\\approx 63.4\\%$ of the time.}\nTamagawa number to be~$\\beta_\\tau=\\prod_p K(p)$.\nThe convergence of this product follows from an analysis of the\ndominant $k=0,1,2$ terms of~\\eqref{eqn:tama}, which gives \na behaviour of~$1+O(1\/p^2)$.\nSo we get that the Tamagawa product is a constant on average,\nwhich we do not bother to compute explicitly (we would need to\nconsider $p=2,3$ more carefully to get a precise value).\n\nTo compute the average value of $\\alpha_A(E)=\\prod_p F(p)$\nin~\\eqref{eqn:Fp} we similarly assume\\footnote\n{This argumentative technique can also be used to bolster our assumption that\n using Connell's conditions should be independent of other considerations.}\nthat each prime acts independently; we then compute the\naverage value for each prime by calculating the distribution of $F(p)$\nwhen considering all the curves modulo~$p$\n(including those with singular reduction,\nand again making the slight adjustment for non-minimal models).\nThis gives some constant for the average $\\bar\\alpha_A$ of~$\\alpha_A(E)$,\nwhich we do not compute explicitly.\nNote that $\\prod_p F(p)$ converges if we assume the Sato-Tate\nconjecture \\cite{tate} since in this case we have that\n$a_p^2$ is $p$ on average.\n\n\\section{Relation between conductor and discriminant}\nWe now give heuristics for how often we expect the ratio\nbetween the absolute discriminant and the conductor to be large.\nThe main heuristic we derive is:\n\n\\begin{heuristic}\\label{conj:conductor}\nThe number $B(X)$ of rational elliptic curves whose conductor\nis less than~$X$ satisfies $B(X)\\sim cX^{5\/6}$ for an explicit\nconstant~$c>0$.\n\\end{heuristic}\n\nTo derive this heuristic, we estimate the proportion of curves with a given\nratio of (absolute) discriminant to conductor. Since the conductor\nis often the squarefree kernel of the discriminant, by way of\nexplanation we first consider the behaviour of $f(n)=n\/{\\rm sqfree}(n)$.\nThe probability that $f(n)=1$ is given by the probability that $n$\nis squarefree, which is classically known to be $1\/\\zeta(2)=6\/\\pi^2$.\nGiven a prime power~$p^m$, to have $f(n)=p^m$ says that $n=p^{m+1}u$\nwhere $u$ is squarefree and coprime to~$p$. The probability that\n$p^{m+1}\\|n$ is $(1-1\/p)\/p^{m+1}$, and given this, the conditional\nprobability that $\\bigl(n\/p^{m+1}\\bigr)$\nis squarefree is $(6\/\\pi^2)\\cdot(1-1\/p^2)^{-1}$.\nExtending this multiplicatively beyond prime powers, we get that\n$${\\rm Prob}\\bigl[n\/{\\rm sqfree}(n)=q\\bigr]=\n{6\\over\\pi^2}\\prod_{p^m\\|q} {1\/p^{(m+1)}\\over (1+1\/p)}=\n{6\\over\\pi^2}{1\\over q}\\prod_{p|q} {1\\over p+1}.$$\nIn particular, the average of $f(n)^\\gamma$ exists for~$\\gamma<1$;\nin our elliptic curve analogue, we will require such\nan average for~$\\gamma=5\/6$. We note that it appears to be\nan interesting open question to prove an asymptotic\nfor~$\\sum\\limits_{n\\le X} n\/{\\rm sqfree}(n)$.\n\n\\subsection{Derivation of the heuristic}\nWe keep the notation $D=|\\Delta|$ and wish to compute\nthe probability that $D\/N=q$ for a fixed positive integer~$q$.\nFor a prime power~$p^v$ with $p\\ge 5$, the probability that $p^v\\|(D\/N)$\nis given by: the probability that $E$ has multiplicative reduction\nat~$p$ and~$p^{v+1}\\|D$, that is~$P_m(p,v+1)$; plus the probability\nthat $E$ has additive reduction at~$p$ and~$p^{v+2}\\|D$, that is~$P_a(p,v+2)$;\nand the contribution from~$P_0(p,v)$, which is zero for $v>0$\nand for $v=0$ is the probability that $p$ does not divide~$D$.\nSo, writing $v=v_p(q)$, we get that\n(with a similar modified formula for~$p=2,3$)\n\\begin{equation}\\label{eqn:prob}\n{\\rm Prob}\\bigl[D\/N=q\\bigr]=\n\\prod_p E_p(v_p(q))=\\prod_p\\bigl[P_m(p,1+v)+P_a(p,2+v)+P_0(p,v)\\bigr].\n\\end{equation}\nIt should be emphasised that this probability is with respect to\n(as in the previous section) the ordering of the curves by discriminant.\nWe have\n\\begin{equation}\\label{eqn:sum}\n\\sum_{E: N_E\\le X} 1\n\\approx\\sum_{q=1}^\\infty\\sum_{E: D\\le qX} {\\rm Prob}\\bigl[D\/N=q\\bigr]\n\\sim\\sum_{q=1}^\\infty\\alpha (qX)^{5\/6}\\cdot{\\rm Prob}\\bigl[D\/N=q\\bigr],\n\\end{equation}\nwhere $\\alpha=\\alpha_++\\alpha_-$\nfrom the Brumer-McGuinness heuristic~\\ref{conj:bmcg}.\nIf this last sum converges, then we get Heuristic~\\ref{conj:conductor}.\n\nTo show the last sum in~\\eqref{eqn:sum} does indeed converge,\nwe upper-bound the probability in~\\eqref{eqn:prob}.\nWe have that $P_m(p,v+1)\\le 1\/p^{v+1}$ and~$P_a(p,v+2)\\le 2\/p^{v+1}$,\nwhich implies\n$$\\hat f(q)={\\rm Prob}\\bigl[D\/N=q\\bigr]=\n\\prod_p E_p(v_p(q))\\le {1\\over q}\\prod_{p|q} {3\\over p}.$$\nWe then estimate\n$$\\sum_{q=1}^\\infty q^{5\/6}\\hat f(q)\\le\n\\sum_{q=1}^\\infty {1\\over q^{1\/6}}\\prod_{p|q} {3\\over p}=\n\\prod_p\\biggl(1+\\sum_{l=1}^\\infty {3\/p\\over (p^l)^{1\/6}}\\biggr)\n\\le\\prod_p\\biggl(1+{3\/p\\over p^{1\/6}-1}\\biggr),$$\nand the last product is convergent upon comparison to~$\\zeta(7\/6)^3$.\nThus we shown that the last sum in~\\eqref{eqn:sum} converges,\nso that Heuristic~\\ref{conj:conductor} follows.\n\nWe can note that Fouvry, Nair, and Tenenbaum \\cite{FNT}\nhave shown that the number of minimal models with $D\\le X$\nis at least $cX^{5\/6}$, and that the number of curves with\n$D\\le X$ with Szpiro ratio ${\\log D\\over\\log N}\\ge \\kappa$\nis no more than $c_\\epsilon X^{1\/\\kappa+\\epsilon}$ for every~$\\epsilon>0$.\n\n\\subsection{Dependence of $D\/N$ and the Tamagawa product}\nWe expect that $D\/N$ should be independent of the real period,\nbut the Tamagawa product and $D\/N$ should be somewhat related.\\footnote\n{The size of the torsion subgroup should also be related to~$D\/N$,\n but in the next section we argue that curves with nontrivial\n torsion are sufficiently sparse so as to be ignored.}\nWe compute the expected square root of the Tamagawa product when~$D\/N=q$.\nAs with~\\eqref{eqn:prob} and using the $\\epsilon$ defined\nin~\\eqref{eqn:epsilon}, we find that this is given by\n$$\\eta(q)=\\prod_p\n{\\bigl[\\epsilon_m(v_1)P_m(p,v_1)+\\epsilon_a^n(v_2)P_a^n(p,v_2)+\n\\epsilon_a^t(p,v_2)P_a^t(p,v_2)+P_0(p,v)\\bigr]\n\\over \\bigl[P_m(p,v_1)+P_a(p,v_2)+P_0(p,v)\\bigr]},$$\nwhere $v_1=v+1$ and $v_2=v+2$ and $v=v_p(q)$.\n\n\\subsection{The comparison of $\\log\\Delta$ with $\\log N$}\nWe now want to compare $\\log\\Delta$ with~$\\log N$,\nand explicate the replacement therein in Guess~\\ref{GUESS}.\nIn order to bound the effect of curves with large~$D\/N$, we note that\n$${\\rm Prob}\\bigl[D\/N\\ge Y\\bigr]=\\sum_{q\\ge Y} \\hat f(q)\n\\le\\sum_{q\\ge Y} {1\\over q}\\prod_{p|q}{3\\over p},$$\nand use Rankin's trick, so that for any $0<\\alpha<1$ we have\n(using $p^\\alpha-1\\ge\\alpha\\log p$)\n\\begin{align*}\n{\\rm Prob}\\bigl[D\/N\\ge Y\\bigr]&\n\\le\\sum_{q=1}^\\infty\n\\biggl({q\\over Y}\\biggr)^{1-\\alpha}\\cdot {1\\over q}\\prod_{p|q}{3\\over p}\n={Y^\\alpha\\over Y}\\prod_p\\biggl(1+{3\\over p^{1+\\alpha}}+\n{3\\over p^{1+2\\alpha}}+\\cdots\\biggr)\\\\\n&={Y^\\alpha\\over Y}\\prod_p\\biggl(1+{3\/p\\over p^\\alpha-1}\\biggr)\n\\ll{Y^\\alpha\\over Y}\\exp\\biggl(\\sum_p{\\hat c\/p\\over \\alpha\\log p}\\biggr)\n\\ll {e^{c\\sqrt{\\log Y}}\\over Y}\n\\end{align*}\nfor some constants~$\\hat c,c$, by taking $\\alpha=1\/\\sqrt{\\log Y}$\n(this result is stronger than needed).\n\nHowever, a more pedantic derivation of Guess \\ref{GUESS} does not simply\nallow replacing $\\log N$ by~$\\log\\Delta$, but requires analysis\n(assuming $\\Omega_{\\rm re}(E)$ to be independent of~$q$) of\n$${\\hat\\alpha_R\\hat\\alpha_A\\over 3456\\,\\zeta(10)}\n\\cdot\\kern-10pt\\mathop{\\int\\int}_{\\sqrt X\\le {u_4^3-u_6^2\\over 1728}\\le X}\n\\kern-10pt\\Omega_{\\rm re}(E)\n\\cdot\\biggl[\\sum_{q<\\Delta} \\eta(q)(\\log \\Delta\/q)^{3\/8}\n\\cdot{\\rm Prob}\\bigl[D\/N=q\\bigr]\\biggr]\\, du_4\\,du_6.$$\nThe above estimate on the tail of the probability\nand a simple bound on $\\eta(q)$ in terms of the divisor function\nshows that we can truncate the $q$-sum at~$Y$\nwith an error of~$O(1\/Y^{8\/9})$, and choosing (say) $Y=e^{\\sqrt{\\log X}}$\ngives us that~$\\log(\\Delta\/q)\\sim\\log\\Delta$\n(note that we restricted to~$\\Delta>\\sqrt X$).\nSo the bracketed term becomes the desired\n$$\\sum_{q<Y} (\\log \\Delta)^{3\/8}\n\\eta(q)\\cdot{\\rm Prob}\\bigl[D\/N=q\\bigr]\\sim \\beta_\\tau(\\log\\Delta)^{3\/8},$$\nupon noting that the $q$-part of the sum converges to~$\\beta_\\tau$\nas~$Y\\rightarrow\\infty$.\n\n\\subsection{Counting curves with vanishing L-value}\nWe now estimate the number of elliptic curves~$E$ with even parity\nand $L(E,1)=0$ when ordered by conductor.\n\n\\begin{heuristic}\\label{conj:rank2cond}\nLet $\\tilde R(X)$ be the number of elliptic curves~$E$\nwith even parity and conductor less than~$X$ and~$L(E,1)=0$.\nThen $\\tilde R(X)\\sim cX^{19\/24}(\\log X)^{3\/8}$\nfor some explicit constant~$c>0$.\n\\end{heuristic}\n\nFrom Guess~\\ref{GUESS} we get that the number of even parity curves\nwith $0<\\Delta<qX$ and $D\/N=q$ and $L(E,1)=0$ is given by\n$$\\hat W(qX)\\cdot\\bigl(\\eta(q)\/\\beta_\\tau\\bigr)\n\\cdot{\\rm Prob}\\bigl[D\/N=q\\bigr],$$\nand we sum this over all~$q$.\nAs we argued above, the tail of the sum does not affect the asymptotic\n(and so we can take~$\\log\\Delta\\sim\\log N$ in~$\\hat W$),\nand again we get that the $q$-sum converges.\nThis then gives the desired asymptotic for the number of even parity\ncurves with conductor less than $X$ and vanishing central $L$-value\n(upon arguing similarly for curves with negative discriminant).\n\n\\section{Torsion and isogenies\\label{section:torsion}}\nWe can also count curves that have a given torsion group or isogeny\nstructure. For instance, an elliptic curve with a 2-torsion point\ncan be written as an integral model in the form $y^2=x^3+ax^2+bx$\nwhere $\\Delta=16b^2(a^2-4b)$; thus, by lattice-point counting,\nwe estimate about $\\sqrt X$ curves\nwith absolute discriminant less than~$X$. The effect on the conductor\ncan perhaps more easily be seen by using the Fricke parametrisation\n$c_4=(t+16)(t+64)T^2$ and $c_6=(t-8)(t+64)^2T^3$ of curves with\na rational \\hbox{$2$-isogeny,} and then substituting $t=p\/q$ and $V=T\/q$\nto get $c_4=(p+16q)(p+64q)V^2$ and $c_6=(p-8q)(p+64q)^2V^3$ so that\n$\\Delta=p(p+64q)^3q^2V^6$. The summation over the\ntwisting parameter~$V$ just multiplies our estimate by a constant,\nwhile ABC-estimates imply that there should be no more than\n$X^{2\/3+\\epsilon}$ coprime pairs $(p,q)$ with the squarefree kernel\nof $pq(p+64q)$ smaller than $X$ in absolute value.\nSo we get the heuristic that almost all curves have no $2$-torsion, \neven under ordering by conductor. Indeed, the exceptional set is\nso sparse that we can ignore it in our calculations.\nA similar argument applies for other isogenies, and more\ngenerally for splitting of division polynomials.\nAlso, the results of Duke~\\cite{duke} for exceptional\nprimes are applicable here, albeit with a different ordering.\n\n\\section{Experiments}\nWe wish to provide some experimental data for the above heuristics.\nIn particular, the two large datasets of\nBrumer-McGuinness \\cite{brumer-mcguinness}\nand Stein-Watkins \\cite{stein-watkins}\nfor curves of prime conductor\nup to $10^8$ and $10^{10}$ show little drop\nin the observed average (analytic) rank.\nBrumer and McGuinness considered about 310700 curves with prime conductor\nless than $10^8$ and found an average rank of about $0.978$,\nwhile Stein and Watkins extended this to over 11 million curves with\nprime conductor up to $10^{10}$ and found an average rank of about~$0.964$.\nBoth datasets are expected to be nearly exhaustive\\footnote\n{This is one reason to take prime conductor curves;\n we also have $|\\Delta|=N$ with few exceptions.}\namongst curves with prime conductor up to the given limit.\nThese results led some to speculate that\nthe average rank might be constant.\nTo test this, we chose a selection of curves with prime conductor\nof size~$10^{14}$. It is non-trivial to get a good dataset, since\nwe must account for congruence conditions on the elliptic curve\ncoefficients and the variation of the size of the real period.\n\n\\subsection\n{Average analytic rank for curves with prime conductor near $10^{14}$}\nAs in~\\cite{stein-watkins},\nwe divided the $(c_4,c_6)$ pairs into 288 congruence classes with\n$(\\tilde c_4,\\tilde c_6)=\n\\bigl(c_4\\>\\text{mod}\\>576,c_6\\>\\text{mod}\\>1728\\bigr)$.\nMany of these classes force the prime 2 to divide the discriminant,\nand thus do not produce any curves of prime conductor.\nFor each class $(\\tilde c_4,\\tilde c_6)$, we took the\n10000 parameter selections\n$$(c_4,c_6)=\\bigl(576(1000+i)+\\tilde c_4,1728(100000+j)+\\tilde c_6\\bigr)\n\\>\\text{for}\\> (i,j)\\in [1..10]\\times [1..1000],$$\nand then of these 2880000 curves,\ntook the 89913 models that had prime discriminant\n(note that all the discriminants are positive).\nThis gives us good distribution across congruence classes,\nand while the real period does not vary as much as possible,\nbelow we will attempt to understand how this affects the average rank.\n\nIt then took a few months to compute the (suspected) analytic ranks for\nthese curves. We got about $0.937$ for the average rank. We then\ndid a similar experiment for curves with negative discriminant given by\n$$(c_4,c_6)=\\bigl(576(-883+i)+\\tilde c_4,1728(100000+j)+\\tilde c_6\\bigr)\n\\>\\text{for}\\>(i,j)\\in [1..10]\\times [1..1000],$$\ntook the subset of 89749 curves with prime conductor,\nand found the average rank to be about~$0.869$.\nThis discrepancy between positive and negative discriminant is\nalso in the Brumer-McGuinness and Stein-Watkins datasets, and\nindeed was noted in \\cite{brumer-mcguinness}.\\footnote\n{``An interesting phenomenon was the systematic influence of the\n discriminant sign on all aspects of the arithmetic of the curve.''}\nWe do not average the results from positive and negative\ndiscriminant; the Brumer-McGuinness Conjecture \\ref{conj:bmcg}\nimplies that the split is not~50-50.\n\nIn any case, our results show a substantial drop in the average rank,\nwhich, at the very least, indicates that the average rank is not constant.\nThe alternative statistic of frequency of positive rank for curves with\neven parity also showed a significant drop. For prime positive discriminant\ncurves it was 44.1\\% for Brumer-McGuinness and 41.7\\% for Stein-Watkins,\nbut only 36.0\\% for our dataset --- for negative discriminant curves,\nthese numbers are 37.7\\%, 36.4\\%, and 31.3\\%.\n\n\\subsection{Variation of real period}\nOur random sampling of curves with prime conductor of size $10^{14}$\nmust account for various properties of the curves if our results\nare to possess legitimacy. Above we speculated that\nthe real period plays the most significant r\\^ole,\nand so we wish to understand how our choice has affected it.\n\nTo judge the effect that variation of the real period might have,\nwe did some comparisons with the Stein-Watkins database.\nFirst consider curves of positive prime discriminant,\nand write $E$ as $y^2=4x^3+b_2x^2+2b_4x+b_6$\nand $e_1>e_2>e_3$ for the real roots of the cubic.\nWe looked at curves with even parity and\nconsidered the frequency of positive rank as a function\nof the root quotient $t={e_1-e_2\\over e_1-e_3}$, noting that\\footnote\n{The calculation follows as in the previous sections;\n via calculus, we can compute that this function is maximised\n at $t\\approx .0388505246188$ with a maximum just below $4.414499094$.}\n$\\Omega_{\\rm re}\\Delta^{1\/12}={2^{1\/3}\\pi(t-t^2)^{1\/6}\\over{\\rm agm}(1,\\sqrt t)}$.\nThe curves we considered all had $0.617<t<0.629$.\n\nHowever, similar to when we considered curves ordered by conductor,\nbefore counting curves with extra rank,\nwe should first simply count curves. Figure~\\ref{fig:posdisc-rootquotient}\nindicates the distribution of root quotient $t$ for the\ncurves of prime (positive) discriminant and even parity\nfrom the Stein-Watkins database (more than 2 million curves\nmeet the criteria). The $x$-axis is divided up into bins\nof size $1\/1000$; there are more than 100 times as many curves with $t<0.001$\nas there with $0.500<t<0.501$, with the most extremal dots not\neven appearing on the graph.\n\n\\vskip0pt\\noindent\n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.97}{\\includegraphics{fig1.eps}}\n\\end{center}\n\\vskip-12pt\\noindent\n\\caption{$\\Delta>0$: Curve distribution as a function of~$t$\n\\label{fig:posdisc-rootquotient}}\n\\end{figure}\n\\vspace*{-1ex}\n\\begin{figure}[H]\n\\begin{center}\n\\scalebox{0.97}{\\includegraphics{fig2.eps}}\n\\end{center}\n\\vskip-12pt\\noindent\n\\caption\n{$\\Delta>0$:\\kern-0.5pt{} {}Positive rank frequency as a function of the\n\\newline\\mbox{}\\hspace{83pt}\nroot quotient~$t$, and $\\Omega_{\\rm re}\\Delta^{1\/12}$ as a function of~$t$.\n\\label{fig:posdisc-rankfreq}}\n\\end{figure}\n\nNext we plot the frequency of $L(E,1)=0$ as a function of the root quotient\nin Figure~\\ref{fig:posdisc-rankfreq}.\nSince there are only about 1000 curves in some of our bins, we do not\nget such a nice graph. Note that the left-most and especially the\nrightmost dots are much below their nearest neighbors,\nthe graph slopes down in general, and drops more at the end.\nWe see no evidence that our results should be overly biased.\nIn particular, the frequency of $L(E,1)=0$\nis 41.7\\% amongst all even parity curves of\nprime discriminant in the Stein-Watkins database,\nand is 42.8\\% for the 12324 such curves with $0.617<t<0.629$.\nThe function plotted (labelled on the right axis)\nin Figure~\\ref{fig:posdisc-rankfreq} is of\n$\\Omega_{\\rm re}\\Delta^{1\/12}={2^{1\/3}\\pi(t-t^2)^{1\/6}\\over{\\rm agm}(1,\\sqrt t)}$\nas a function of~$t$, and note that this goes to 0 as $t \\rightarrow 0,1$;\nthere is nothing canonical about the choice of our $t$-parameter,\nand we chose it more for convenience than anything else.\n\nSimilar computations can be made in the case of negative discriminant,\nwhich we briefly discuss for completeness (again restricting to curves\nwith even parity where appropriate). Let $r$ be the real root of\nthe cubic polynomial $4x^3+b_2x^2+2b_4x+b_6$,\nand $Z>0$ the imaginary part of the conjugate pair of nonreal roots.\nLetting $\\tilde r=r+b_2\/12$ and $c=\\tilde r\/Z$ we then have\\footnote\n{This is maximised at $c\\approx -33.58515148525$,\n with the maximum a bit less than $8.82921518$.}\n\\vskip-2pt\\noindent\n$$\\Omega_{\\rm re}|\\Delta|^{1\/12}=\n{\\pi\\sqrt{2}\\over (1+9c^2\/4)^{1\/12}\n{\\rm agm}\\Bigl(1,\\sqrt{{1\\over 2}+{3c\\over 4\\sqrt{1+9c^2\/4}}}\\Bigr)}.$$\nWe renormalise via taking~$C=1\/2+\\arctan(c)\/\\pi$,\nand graph the distribution of curves versus~$C$ in\nFigure~\\ref{fig:negdisc-rootquotient}.\nThe symmetry\\footnote\n{The blotches around 0.22-0.23 and 0.77-0.78 appear to come from\n the fact that curves with $a_4$ small (in particular~$\\pm 1$)\n tend to have $C$ in these ranges (for our discriminant range),\n and this causes instability in the counting function.}\nof the graph might indicate that the coordinate transform is reasonable.\nAll our curves have~$0.555<C<0.557$.\n\n\\vskip-8pt\\noindent\n\\begin{figure}[H]\n\\begin{center}\n\\scalebox{0.97}{\\includegraphics{fig3.eps}}\n\\end{center}\n\\vskip-18pt\\noindent\n\\caption{$\\Delta<0$: Distribution of curves as a function of $C$\n\\label{fig:negdisc-rootquotient}}\n\\vskip-6pt\n\\end{figure}\n\nNext we plot the frequency of $L(E,1)=0$ as a function of the root quotient\nin Figure~\\ref{fig:negdisc-rankfreq}. Again we also graph the function\n$\\Omega_{\\rm re}|\\Delta|^{1\/12}$ on the right axis.\nHere the drop-off is more pronounced\nthan with the curves of positive discriminant.\nNote the floating dot around $C=1\/2$.\nIndeed the 100 closest curves with $C<1\/2$ all have positive rank;\nthis breaks down when crossing the $1\/2$-barrier.\nThis is not particularly a mystery; these curves have $a_6=0$ and\/or $b_6=1$,\nand thus have an obvious rational point.\nRecall that $C=1\/2$ corresponds to $c=0=\\tilde r$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.97}{\\includegraphics{fig4.eps}}\n\\end{center}\n\\vskip-12pt\\noindent\n\\caption{$\\Delta<0$: Positive rank frequency as a function of~$C$,\n\\newline\\mbox{}\\hspace{82.5pt}\nand $\\Omega_{\\rm re}|\\Delta|^{1\/12}$ as a function of~$C$.\n\\label{fig:negdisc-rankfreq}}\n\\vskip-6pt\\noindent\n\\end{figure}\n\nWe again see no evidence that our results should be biased.\nIn particular, the frequency of $L(E,1)=0$ is 36.4\\% amongst\nall even parity curves of negative prime discriminant\nin the Stein-Watkins database, and is 37.0\\% for the\n4695 such curves with $0.555<C<0.557$.\n\n\\subsection{Other considerations}\nThe idea that the ``probability'' that an even parity curve possesses\npositive rank should be proportional to $\\sqrt{\\Omega_{\\rm re}}$\nis perhaps overly simplistic --- in particular, it is not borne out\ntoo precisely by the Stein-Watkins dataset.\nWe consider positive prime discriminant curves with even parity;\nfor those with $0.64<\\Omega_{\\rm re}<0.65$ we have 78784 curves of\nwhich 45.9\\% have positive rank, while of the 9872 with\n$0.32<\\Omega_{\\rm re}<0.325$ we have 36.0\\% with positive rank,\nfor a ratio of 1.28, which is not too close to~$\\sqrt 2$.\nOne consideration here is that we have placed a discriminant limit\non our curves, and there are curves with larger discriminant and\n$0.32<\\Omega_{\\rm re}<0.325$ that we have not considered.\nThis, however, is extra-particular to the idea that only the\nreal period should be of import.\n\nOne possibility is that curves with small discriminant and\/or\nlarge real period have \\textit{smaller} probability of $L(E,1)=0$\nthat our estimate of $c\\sqrt{\\Omega_{\\rm re}}$ would suggest --- indeed,\nit might be argued (maybe due to arithmetic considerations,\nor perhaps explicit formulae for the zeros of $L$-functions)\nthat curves with such small discriminant cannot realise their\nnominal expected frequency of positive rank. Unfortunately,\nwe cannot do much to quantify these musings, as the effect would\nlikely be in a secondary term, making it difficult to detect experimentally.\nNote also that a relative depression of rank for small discriminant\ncurves would give a reason for the near-constant average rank\nobserved by Brumer-McGuinness and Stein-Watkins.\n\n\\subsection{Mordell-Weil lattice distribution for rank 2 curves}\nWe have other evidence that curves of small discriminant might not\nbehave quite as expected. We undertook to compute generators for\nthe Mordell-Weil group for all 2143079 curves of (analytic) rank 2\nof prime conductor less than $10^{10}$ in the Stein-Watkins\ndatabase.\\footnote\n{We also computed the Mordell-Weil group for curves with higher ranks\n but do not describe the obtained data here.}\nJ. E. Cremona ran his \\verb+mwrank+ programme \\cite{mwrank} \non all these curves, and it was successful in provably finding\nthe Mordell-Weil group for 2114188 of these.\nFor about 2500 curves, the search region was\ntoo big to find the 2-covering quartics via invariant methods,\nwhile around 8500 curves had a generator of large height that\ncould not be found, and over 18000 had 2-Selmer rank greater than~2.\nWe then used the \\verb+FourDescent+ machinery of MAGMA \\cite{magma}\nwhich reduced the number of problematic curves to 54. Of these,\n19 have analytic {\\mbox{\\textcyr{Sh}}}\\ of 16.0 and we expect that either 3-descent\nor 8-descent \\cite{stamminger} will complete (assuming GRH to\ncompute the class group) the Mordell-Weil group verification;\nfor the 35 other curves, there is likely a generator\nof height more than 225 which we did not attempt to find.\\footnote\n{A bit more searching might resolve a few of the outstanding cases,\n but the extremal case of $[0,0,1,-237882589,-1412186639384]$ appears\n to have a generator of height more than 600, and thus other methods\n will likely be needed to try to find it. T.~A. Fisher has recently\n used 6-descent to find some of the missing points.}\n\nWe then looked at the distribution of the Mordell-Weil lattices\nobtained from the induced inner product from the height pairing;\nsince all of our curves have rank 2, we get 2-dimensional lattices.\nWe are not so interested in the size of the obtained lattices, but\nmore so in their shape. Via the use of lattice reduction (which\nreduces to continued fractions in this case), given any two generators\nwe can find the point $P$ of smallest positive height on the curve.\nBy normalising $P$ to be the unit vector, we then get a vector in the\nupper-half-plane corresponding to another generator~$Q$. Via the standard\nreduction algorithm, we can translate $Q$ so that it corresponds to a \npoint in the fundamental domain for the action of~${\\rm SL}_2({\\bf Z})$.\nFinally, by replacing $Q$ by $-Q$ if necessary, we can ensure that this\npoint is in the right half of the fundamental domain\n(in other words, we must choose an embedding for our Mordell-Weil lattice).\nIn this manner, for each rank 2 curve we associate a unique point\n$z=x+iy$ in the upper-half-plane with $x^2+y^2\\ge 1$ and $0\\le x\\le 1\/2$.\n\nWith no other guidance, we might expect that the obtained distribution\nfor the $z$ is given by\\footnote\n{Siegel \\cite{siegel} similarly uses Haar measure to put a natural\n measure on $n$-dimensional lattices of determinant~$1$.}\nthe Haar measure $(dx\\, dy)\/y^2$.\nWe find, however, that this is not borne out too well by experiment.\nIn particular, we should expect that\n${1\/2\\over \\pi\/6}\\approx 95.5\\%$ of the curves should have $y\\ge 1$,\nwhile the experimental result is about~93.5\\%. Furthermore, we should\nexpect that the proportion of curves with $y\\ge Y$ should die off like\n$1\/Y$ as $y\\rightarrow\\infty$; however, we get that 35.4\\% of the curves\nhave $y\\ge 2$, only 9.4\\% have $y\\ge 4$, while 1.7\\% have $y\\ge 8$\nand 0.2\\% have $y\\ge 16$. The validity of the vertical distribution\ndata might be arguable based upon concerns regarding the discriminant\ncutoff of our dataset, but the horizontal distribution is also skewed.\nIf we consider only curves with~$y\\ge 1$, then we should get uniform\ndistribution in the $x$-aspect; however, Table~\\ref{table:xdist} shows that\nwe do not have such uniformity.\n\n\\begin{table}[h]\n\\caption{Horiztonal distribution of rank 2 lattices with $y\\ge 1$\n\\label{table:xdist}}\n\\begin{center}\n\\begin{tabular}{*{4}{|@{\\hskip6pt}c@{\\hskip6pt}}|}\\hline\n$0.00\\le x<0.05$&9.0\\%&$0.25\\le x<0.30$&10.0\\%\\\\\n$0.05\\le x<0.10$&9.6\\%&$0.30\\le x<0.35$&10.2\\%\\\\\n$0.10\\le x<0.15$&9.8\\%&$0.35\\le x<0.40$&10.5\\%\\\\\n$0.15\\le x<0.20$&9.9\\%&$0.40\\le x<0.45$&10.6\\%\\\\\n$0.20\\le x<0.25$&10.0\\%&$0.45\\le x\\le 0.50$&10.5\\%\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe cannot say whether these unexpected results from the experimental data\nare artifacts of choosing curves with small discriminant; it is just as\nprobable that our Haar-measure hypothesis concerning the lattice distribution\nis simply incorrect.\n\n\\subsection{Symmetric power L-functions}\nSimilar to questions about the vanishing of $L(E,s)$, we can ask about\nthe vanishing of the symmetric power $L$-functions $L({\\rm Sym}^{2k-1} E,s)$.\nWe refer the reader to \\cite{martin-watkins} for more details about this,\nbut mention that, due to conjectures of Deligne and more generally\nBloch and Be\\u\\i linson \\cite{beilinson},\nwe expect that we should have a formula similar\nto that of Birch and Swinnerton-Dyer, stating that\n$L({\\rm Sym}^{2k-1} E,k)(2\\pi N)^{k\\choose 2}\/\n\\Omega_+^{k+1\\choose 2}\\Omega_-^{k\\choose 2}$\nshould be rational with small denominator. Here, for $k$ odd,\n$\\Omega_+$ is the real period and $\\Omega_-$ the imaginary period,\nwith this reversed for $k$ even. Ignoring the contribution\nfrom the conductor, and crudely estimating that\n$\\Omega_+\\approx\\Omega_-\\approx 1\/\\Delta^{1\/12}$, an application\nof discretisation as before gives that the probability\nthat $L({\\rm Sym}^{2k-1} E,s)$ has even parity and\n$L({\\rm Sym}^{2k-1} E,k)=0$ is bounded above (cf.~the ignoring of~$N$)\nby $c(\\log\\Delta)^{3\/8}\\cdot\\sqrt{1\/\\Delta^{k^2\/12}}$.\nAgain following the analogy of above, we can then upper-bound\nthe number of curves with conductor less than~$X$\nwith even-signed symmetric $(2k-1)$st power and $L({\\rm Sym}^{2k-1} E,k)=0$\nby $c_k(\\epsilon) X^{5\/6-k^2\/24+\\epsilon}$ for every $\\epsilon>0$.\nIt could be argued that\nwe should order curves according to the conductor of the symmetric\npower $L$-function rather than that of the curve, but we do not think\nsuch concerns are that relevant to our imprecise discussion.\nIn particular, the above estimate predicts that there are\nfinitely many curves with extra vanishing when~$k\\ge 5$. It should be\nsaid that this heuristic will likely mislead us about curves with\ncomplex multiplication, for which the symmetric power $L$-function\nfactors (it is imprimitive in the sense of the Selberg class),\nwith each factor having a 50\\% chance of having odd parity. However,\neven ignoring CM curves, the data of \\cite{martin-watkins} find a handful\nof curves for which the 9th, 11th and even the 13th symmetric powers appear\n(to 12 digits of precision)\nto have a central zero of order~2. We find this surprising, and casts some\ndoubt about the validity of our methodology of modelling of vanishings.\n\n\\subsection{Quadratic twists of higher symmetric powers}\nThe techniques we used earlier in this paper have also\nbeen used to model vanishings in quadratic twist families,\nand we can extend the analyses to symmetric powers.\n\n\\subsubsection{Non-CM curves}\nWe fix a non-CM curve~$E$ and let $E_d$ be its $d$th quadratic twist,\ntaking $d$ to be a fundamental discriminant. From an analogue of the\nBirch--Swinnerton-Dyer conjecture we expect to get a small-denominator\nrational from the quotient\\footnote\n{The contribution from the conductor actually comes from non-integral\n Tamagawa numbers from the Bloch-Kato exponential map, and in the case\n of quadratic twists, the twisting parameter~$d$ should not appear in\n the final expression.} \n$L({\\rm Sym}^3 E_d,2)(2\\pi N_E)\/\\Omega_{\\rm im}(E_d)^3\\Omega_{\\rm re}(E_d)$.\nWe have that $\\Omega_{\\rm im}(E_d)^3\\Omega_{\\rm re}(E_d)\n\\approx \\Omega_{\\rm im}(E)\/d^{3\/2}\\cdot \\Omega_{\\rm re}(E)\/d^{1\/2}$\nand so we expect the number of fundamental discriminants $|d|<D$\nsuch that $L({\\rm Sym}^3 E_d,s)$ has even parity with\n$L({\\rm Sym}^3 E_d,2)=0$ to be given crudely (up to log-factors)\nby $\\sum_{d<D} c\/\\sqrt{d^2}$.\nSo we expect about $(\\log D)^b$ quadratic twists with double zeros\nfor the 3rd symmetric power; generalising predicts finitely many\nextra vanishings for higher (odd) powers.\n\n\\begin{table}[h]\n\\caption\n{Fundamental $d$ with\n $\\mathop{\\rm ord}\\limits_{s=2} L({\\rm Sym}^3 E_d,s)\\ge 2$\\label{tbl:Ed}}\n\\vspace*{-12pt}\n\\begin{center}\n\\begin{tabular}{|r|l|}\\hline\n11a& $-40$ $-52$ $-563$ $-824$ $-1007$\n$-1239$ $-1460$ $-1668$ $-1799$ $-2207$\\\\\n&$-2595$ $-2724$ $-2980$ $-3108$ $-3592$ $-4164$ $-4215$ $-4351$ $-4399$\\\\\n&12 69 152 181 232 273 364 401 412 421 444 476 488 652 669 696\\\\\n&933 1101 1149 1401 1576 1596 1676 1884 1928 2348 2445 2616\\\\\n&2632 3228 3293 3404 $3720^\\star$ 3793 4060 4093 4161 4481 4665 4953\\\\\\hline\n14a&\n$-31$ $-52$ $-67$ $-87$ $-91$ $-111$ $-203$\n$-223$ $-255$ $-264$ $-271$ $-311$ $-327$\\\\\n&$-367$ $-535$ $-552$ $-651$ $-759$ $-804$\n$-831$ $-851$ $-852$ $-920$ $-1099$ $-1263$\\\\\n&$-1267$ $-1335$ $-1524$ $-1547$ $-1567$\n $-1623$ $-1679$ $-1707^\\star$ $-2047$ $-2235$\\\\\n&$-2280$ $-2407$ $-2443$ $-2563$ $-2824$\n $-2831$ $-3127$ $-3135$ $-3523$ $-4119$\\\\\n&$-4179$ $-4191$ $-4323$\\\\\n&137 $141^\\star$ 229 233 281 345 469 473\n 492 $497^\\star$ 697 901 1065 1068 1353 1457\\\\\n&1481 1513 1537 1793 1873 1905 2024 2093 2193 2265 2321 2589 2657 2668\\\\\n&2732 2921 2981 2993 3437 3473 3529 4001 4124 4389 4488 4661 4817\\\\\\hline\n15a&\n$-11$ $-51$ $-71$ $-164$ $-219$ $-232$ $-292$ $-295$ $-323^\\star$ $-340$\n$-356$ $-399$ $-519$\\\\\n&$-580$ $-583$ $-584$ $-671$ $-763$ $-804$ $-851$\n$-879$ $-943$ $-1012$ $-1060$ $-1151$\\\\\n&$-1199$ $-1284$ $-1288$ $-1363$\n$-1551$ $-1615$ $-1723$ $-1732$ $-2279$ $-2291$\\\\\n&$-2379$ $-2395$\n$-2407$ $-2571$ $-2632$ $-2635$ $-2756$ $-3396$ $-3588^\\star$ $-3832$\\\\\n&17 21 61 77 136 156 181 229 349 444 481 501 545 589 649 781 876 905\\\\\n&924 949 1009 1144 1249 1441 1501 1580 1621 1804 1861 1921 2041 2089\\\\\n&2109 2329 2581 2829 2840 2933 3001 3916\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe took the curves 11a:$[0,-1,1,0,0]$ and 14a:$[1,0,1,-1,0]$,\nand computed either $L({\\rm Sym}^3 E_d,2)$ or $L'({\\rm Sym}^3 E_d,2)$\nfor all fundamental discriminant $d$ with~$|d|<5000$.\nWe did the same for 15a:$[1,1,1,0,0]$ for~$|d|<4000$.\nWe then looked at the number of vanishings (to 9 digits of precision).\nFor 11a we found 58 double zeros and one triple zero\n(indicated by a star in Table~\\ref{tbl:Ed}) while for 14a\nwe found 88 double zeros and three triple zeros, and 15a yielded\n83 double zeros and two triple zeros.\n\n\\subsubsection{CM curves}\nNext we consider CM curves, for which we can compute significantly more data,\nbut the modelling of vanishings is slightly different.\nLet $E$ be a rational elliptic curve with~CM, and $\\psi$ its Hecke character.\nWe shall take~$\\psi$ to be ``twist-minimal'' --- this is not the same\nas the ``canonical'' character of Rohrlich~\\cite{rohrlich},\nbut rather we just take $E$ to be a minimal (quadratic) twist.\nIndeed, we shall only consider 11 different choices of~$E$,\ngiven (up to isogeny class) by 27a, 32a, 36a, 49a, 121a,\n256a, 256b, 361a, 1849a, 4489a, and~26569a, noting that 27a\/36a\nand 32a\/256b are respectively cubic and quartic twist-pairs.\nIn our tables, these can appear in a briefer format,\nsuch as $67^2$ for~4489a.\n\nWe normalise the Hecke $L$-function $L(\\psi,s)$ to have $s=1$\nbe the center of the critical strip.\nFor $d$ a fundamental discriminant, we let $\\psi_d$\nbe the Hecke Gr\\\"ossencharacter $\\psi$ twisted by the quadratic\nDirichlet character of conductor~$d$. Finally, note that\nthe symmetric powers $L({\\rm Sym}^k \\psi,s)$ are just~$L(\\psi^k,s)$,\nwhere we must take $\\psi^k$ to be the primitive underlying Gr\\\"ossencharacter.\n\nWe then expect $L(\\psi^3,2)(2\\pi)\/\\Omega_{\\rm im}(E)^3$\nto be rational with small denominator.\nWe can then use discretisation as before to count the expected number\nof fundamental discriminants $|d|<D$ for which the\n$L$-function $L(\\psi_d^3,s)$ has even parity\nbut vanishes at the central point --- since we have\n$\\Omega_{\\rm im}(E_d)^3\\approx \\Omega_{\\rm im}(E)\/d^{3\/2}$,\nwe expect the number of discriminants $d$ that\nyield even parity and $L(\\psi_d^3,2)=0$\nis crudely given by $\\sum_{d<D} 1\/\\sqrt{d^{3\/2}}$,\nso we should get about $D^{1\/4}$ such discriminants up to~$D$.\n\nFor higher symmetric powers, we expect that\n$L(\\psi^{2k-1},k)(2\\pi)^{k-1}\/\\Omega_+(E)^{2k-1}$\nis rational with small denominator, and thus get that\nthere should be finitely many quadratic twists of even parity\nwith vanishing central value.\n\nWe took the above eleven CM curves and took their (fundamental) quadratic\ntwists up to~$10^5$. We must be careful to exclude twists that are\nisogenous to other twists. In particular, we need to define a\n{\\bf primitive} discriminant for a curve with CM by an order\nof the field $K$ --- this is a fundamental discriminant~$d$\nsuch that ${\\rm disc}(K)$ does not divide~$d$,\nexpect for $K={\\bf Q}(i)$ when $d>0$ is additionally primitive when~$8\\|d$.\nNote also that 27a and 36a have the same symmetric cube $L$-function.\n\n\\vskip-12pt\\noindent\n\\begin{table}[h]\n\\caption{Counts of double order zeros for primitive twists\\label{tbl:overview}}\n\\vskip-8pt\\noindent\n\\begin{center}\n\\begin{tabular}{|c|ccccccccccc|}\\hline\n&27a&32a&36a&49a&121a&256a&256b&361a&1849a&4489a&26569a\\\\\\hline\n3rd&59&32&-&67&78&32&21&45&28&31&1\\\\\n5th&3&1&5&2&1&2&2&0&0&0&0\\\\\n7th&0&0&2&0&1&0&0&0&0&0&0\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\vskip-6pt\\noindent\n\nTable~\\ref{tbl:overview} lists the following results for counts\nof central double zeros (to 32 digits) for the $L$-functions\nof the 3rd, 5th, and 7th symmetric powers.\\footnote\n{We found no even twists with~$L(\\psi_d^9,5)=0$,\n and no triple zeros appeared in the data.}\nTables~\\ref{tbl:data} and \\ref{tbl:higher}\nlist the primitive discriminants that yield\nthe double zeros. The notable signedness can be\nexplained via the sign of the functional equation.\\footnote\n{The local signs at $p=2,3$ involve wild ramification are more complicated\n (see~\\cite{whitehouse,kobayashi,dm} for a theoretical description),\n and thus there is no complete correlation in some cases.}\nWe are unable to explain the paucity of double zeros for twists of~26569a;\nLiu and Xu have the latest results~\\cite{liu-xu} on the vanishing of\nsuch $L$-functions, but their bounds are far from the observed data.\nSimilarly, the last-listed double zero for 4489a at 67260 seems quite small.\n\nThere appear to be implications vis-a-vis higher vanishings in some cases;\nfor instance, except for 27a, in the thirteen cases that $L(\\psi_d^5,s)$ has\na double zero at $s=3$ then $L(\\psi_d,s)$ also has a double zero at~$s=1$.\nSimilarly, the 7th symmetric power for the 27365th twist of 121a has a\ndouble zero, as does the 3rd symmetric power, while the $L$-function\nof the twist itself has a triple zero.\nAlso, the 22909th twist of 36a has double zeros\nfor its first, third, and fifth powers (note that 36a does not appear\nin Table~\\ref{tbl:data} as the data are identical to that for~27a).\n\n\\begin{table}[H]\n\\vspace*{-18pt}\n\\caption{Primitive $d$ with\n         $\\mathop{\\rm ord\\,}\\limits_{s=2} L(\\psi_d^3,s)=2$\n\\label{tbl:data}}\n\\vspace*{-6pt}\n\\begin{center}\n\\begin{tabular}{|r|l|}\\hline\n27a&172 524 1292 1564 1793 3016 4169 4648 6508 9149 9452 9560 10636\\\\\n&11137 12040 13784 14284 15713 17485 17884 22841 22909 22936 25729\\\\\n&27065 27628 29165 30392 34220 35749 38636 40108 41756 44221 47260\\\\\n&51512 54385 57548 58933 58936 58984 59836 59996 62353 64268 70253\\\\\n&74305 77320 77672 78572 84616 86609 86812 87013 92057 95861 96556\\\\\n&97237 99817\\\\\\hline\n32a& $-395$ $-5115$ $-17803$ $-25987$ $-58123$\\\\\n& $-60347$ $-73635$ $-79779$ $-84651$ $-99619$\\\\\n& 257 1217 2201 2465 14585 26265 45201 82945\\\\\n&4632 5336 5720 7480 9560 30328 30360\\\\\n&31832 38936 45848 69784 71832 83512 92312\\\\\\hline\n49a&\n$-79$ $-311$ $-319$ $-516$ $-856$ $-1007$\n$-1039$ $-1243$ $-1391$ $-1507$ $-1795$\\\\\n&$-2024$ $-2392$ $-2756$ $-2923$ $-3527$\n$-3624$ $-4087$ $-4371$ $-4583$ $-4727$\\\\\n&$-5431$ $-5524$ $-5627$ $-6740$ $-7167$\n$-7871$ $-8095$ $-8283$ $-10391$ $-10628$\\\\\n& $-13407$ $-13656$ $-13780$ $-16980$ $-18091$\n$-22499$ $-27579$ $-28596$ $-30083$\\\\\n& $-30616$ $-32303$ $-32615$ $-36311$\n$-36399$ $-38643$ $-39127$ $-40127$ $-42324$\\\\\n& $-52863$ $-64031$ $-64399$ $-66091$ $-66776$\n $-66967$ $-69647$ $-70376$ $-71455$\\\\\n& $-72663$ $-73487$ $-73559$ $-77039$ $-84383$\n$-90667$ $-91171$ $-98655$ $-98927$\\\\\\hline\n$11^2$&\n12 140 632 1160 1208 1308 1704 1884 2072 2136 2380 2693 2716 3045\\\\\n&4120 4121 5052 5528 5673 5820 6572 7532 11053 11208 12277 12568\\\\\n&12949 13884 14844 15465 16136 18588 18885 19020 19884 24060 25788\\\\\n&27365 27597 28265 28668 29109 29573 32808 32828 35261 36552 37164\\\\\n&38121 38297 44232 44873 49512 49765 50945 52392 54732 55708 56076\\\\\n&56721 58460 59340 65564 66072 66833 71688 72968 79557 80040 80184\\\\\n&83388 84504 84620 84945 86997 87576 92460 95241\\\\\\hline\n\\kern-1.5pt 256a&\n401 497 2513 3036 3813 6933 6941 9596 9932 11436 14721 17133 17309\\\\\n&18469 21345 21749 26381 26933 28993 29973 30461 33740 51469 53084\\\\\n&62556 63980 67721 69513 73868 76241 81164 87697\\\\\\hline\n\\kern-1.5pt 256b&\n73 345 3521 5133 6693 7293 21752 25437 27113 34657 38485 41656\\\\\n&42433 44088 46045 75581 79205 83480 89737 93624 96193\\\\\\hline\n$19^2$&\n44 60 1429 1793 3297 3340 3532 3837 3880 4109 5228 5628 7761 8808\\\\\n&9080 9388 12280 12313 12545 13373 13516 13897 19164 22204 23241\\\\\n&25036 25653 41205 41480 42665 43429 44121 44285 44508 45660 48828\\\\\n&50584 52989 64037 74585 75324 76921 81885 85036 96220\\\\\\hline\n$43^2$&\n88 152 440 2044 4268 5852 6376 7880 8908 9880 14252\\\\\n&15681 17864 20085 20353 28492 29477 45368 55948 56172\\\\\n&57409 60177 68136 79916 84524 85580 86853 96216\\\\\\hline\n$67^2$&\n17 57 869 1612 1628 3260 6380 6385 7469 8328 11017 13772\\\\\n&14152 14268 14552 15901 22513 24605 24664 27992 29676 33541\\\\\n&33789 36344 36588 38028 40280 43041 49884 62353 67260 \\\\\\hline\n\\kern-1.5pt$163^2$&30720\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{-7.5pt}\n\\caption{Primitive $d$ with\n       $\\mathop{\\rm ord\\,}\\limits_{s=k} L(\\psi_d^{2k-1},s)=2$\n         for some $k\\ge 3$\\label{tbl:higher}}\n\\vspace*{-2pt}\n\\begin{center}\n\\begin{tabular}{|r|l|r|l|}\\hline\n27a&5th: $-13091$ 4040 18044&49a&5th: 437 19317\\\\\n32a&5th: 1704&121a&5th: $-183$\\quad 7th: 27365\\\\\n36a&5th: $-856$ $-2104$ $-31592$ $-88580$ 22909&256a&5th: $-79$ $-21252$\\\\\n36a&7th: $-95$ 2488&256b&5th: $-511$ 89320\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\goodbreak\n\n\\subsubsection{Comparison between the CM and non-CM cases}\nFor the twist computations for the symmetric powers,\nwe can go much further (about 20 times as far) in the CM case\nbecause the conductors do not grow as rapidly.\\footnote\n{In \\cite[\\S 8]{RVZ} Rodriguez Villegas and Zagier\n mention the possibility of a Waldspurger-type formula\n for the twists of the Hecke Gr\\\"ossencharacters,\n but it does not seem that such a formula has ever appeared.\n Similarly, one might hope to extend the work of Coates and Wiles~\\cite{cw}\n and\/or Gross and Zagier~\\cite{gz} to powers of Gr\\\"ossencharacters;\n there is some early work (among others) of Damerell \\cite{damerell}\n in this regard, while Guo \\cite{guo} shows partial results toward the\n Bloch-Kato conjecture.}\nFor the 3rd symmetric power, the crude prediction is that we should have\n(asymptotically) many more extra vanishings for twists in the CM case\nthan in the non-CM case, but this is not borne out by the data.\nAdditionally, we have no triple zeros in the CM case (where the\ndataset is almost 100 times as large),\nwhile we already have six for the non-CM curves.\nThis is directly antithetical to our suspicion that there should\nbe more extra vanishings in the CM case.\nAs before, this might cast some doubt on our methodology of modelling\nof vanishings.\n\n\\section{Acknowledgements}\nThe author was partially supported by\nEngineering and Physical Sciences Research Council (EPSRC)\ngrant GR\/T00658\/01 (United Kingdom).\nHe thanks N.~D.~Elkies, H.~A.~Helfgott, and A.~Venkatesh for useful comments,\nand N.~P.~Jones for the reference~\\cite{duke}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Intro}\n\nMaximum entropy models are an important class of statistical models for biology. \nFor instance, they have been found  to be a good model for protein folding \\cite{russ2005,Socolich2005},\nantibody diversity \\cite{Mora2010}, neural population activity \\cite{Schneidman2006,shlens2006,Tkacik2006,tang2008,bethge2008,yu2008,shlens2009}, and \nflock behavior \\cite{Bialek2012}.  In this paper we develop a general framework for studying maximum entropy distributions on weighted graphs, extending recent work of Chatterjee, Diaconis, and Sly~\\cite{Chatterjee}.  The development of this theory is partly motivated by the problem of sensory coding in neuroscience.\n\nIn the brain, information is represented by discrete electrical pulses, called \\textit{action potentials} or \\textit{spikes} \\cite{rieke1999}.  This includes neural representations of sensory stimuli which can take on a continuum of values.  For instance, large photoreceptor arrays in the retina respond to a range of light intensities in a visual environment, but the brain does not receive information from these photoreceptors directly.  Instead, retinal ganglion cells must convey this detailed input to the visual cortex using only a series of binary electrical signals. Continuous stimuli are therefore converted by networks of neurons to sequences of spike times. \n\nAn unresolved controversy in neuroscience is whether information is contained in the precise timings of these spikes or only in their ``rates\"  (i.e., counts of spikes in a window of time).  Early theoretical studies \\cite{mackay1952} suggest that information capacities of timing-based codes are superior to those that are rate-based (also see \\cite{hopfield1995} for an implementation in a simple model).  Moreover, a number of scientific articles have appeared suggesting that precise spike timing \\cite{abeles82, bair1996, neuenschwander1996, victor1996, liu2001variability, butts2007, Maldonado, nemenman2008, Desbordes2008, Kilavik} and synchrony \\cite{uhlhaas2009} are important for various computations in the brain.\\footnote{Although it is well-known that precise spike timing is used for time-disparity computation in animals \\cite{carr1993}, such as when owls track prey with binocular hearing or when electric fish use electric fields around their bodies for locating objects.}  Here, we briefly explain a possible scheme for encoding continuous vectors with spiking neurons that takes advantage of precise spike timing and the mathematics of maximum entropy distributions.\n\nConsider a network of $n$ neurons in one region of the brain which transmits a continuous vector $\\theta \\in \\mathbb{R}^n$ using sequences of spikes to a second receiver region.  We assume that this second region contains a number of coincidence detectors that measure the absolute difference in spike times between pairs of neurons projecting from the first region.  We imagine three scenarios for how information can be obtained by these detectors.  In the first, the detector is only measuring for synchrony between spikes; that is, either the detector assigns a 0 to a nonzero timing difference or a 1 to a coincidence of spikes.  In another scenario, timing differences between projecting neurons can assume an infinite but  countable number of possible values.  Finally, in the third scenario, we allow these differences to take on any nonnegative real values.  We further assume that neuronal output and thus spike times are stochastic variables.  A basic question now arises: How can the first region encode $\\theta$ so that it can be recovered robustly  by the second?  \n\nWe answer this question by first asking the one symmetric to this:  How can the second region recover a real vector transmitted by an unknown sender region from spike timing measurements?  We propose the following possible solution to this problem.  Fix one of the detector mechanics as described above, and set $a_{ij}$ to be the measurement of the absolute timing difference between spikes from projecting neurons $i$ and $j$.   We assume that the receiver population can compute the (local) sums $d_i = \\sum_{j \\neq i} a_{ij}$ efficiently.   The values $\\mathbf{a} = (a_{ij})$ represent a weighted graph $G$ on $n$ vertices, and we assume that $a_{ij}$ is randomly drawn from a distribution on timing measurements $(A_{ij})$.     Making no further assumptions, a principle of Jaynes \\cite{jaynes1957} suggests that the second region propose that the timing differences are drawn from the (unique) distribution over weighted graphs with the highest entropy \\cite{shannon48, coverthomas} having the vector $\\mathbf{d} = (d_1,\\ldots,d_n)$ for the expectations of the degree sums $\\sum_{j \\neq i} A_{ij}$.  Depending on which of the three scenarios described above is true for the coincidence detector, this prescription produces one of three different maximum entropy distributions.\n\nConsider the third scenario above (the other cases are also subsumed by our results).  As we shall see in Section~\\ref{Sec:Cont}, the distribution determined in this case is parameterized by a real vector $\\theta = (\\theta_1,\\ldots,\\theta_n)$, and finding the maximum likelihood estimator (MLE) for these parameters using $\\mathbf{d}$ as sufficient statistics boils down to solving the following set of  $n$ algebraic equations in the $n$ unknowns $\\hat{\\theta}_1,\\ldots,\\hat{\\theta}_n$:\n\\begin{equation}\\label{Eq:RetinaEqs}\nd_i = \\sum_{j \\neq i} \\frac{1}{\\hat{\\theta}_i+\\hat{\\theta}_j} \\quad \\text{ for } i = 1,\\dots,n.\n\\end{equation}\nGiven our motivation, we call the system of equations~\\eqref{Eq:RetinaEqs} the \\textit{retina equations} for theoretical neuroscience, and note that they have been studied in a more general context by Sanyal, Sturmfels, and Vinzant~\\cite{SturmEntDisc} using matroid theory and algebraic geometry. Remarkably, a solution $\\hat{\\theta}$ to~\\eqref{Eq:RetinaEqs} has the property that with high probability, it is arbitrarily close to the original parameters $\\theta$ for sufficiently large network sizes $n$ (in the scenario of binary measurements, this is a result of~\\cite{Chatterjee}).  In particular, it is possible for the receiver region to recover reliably a continuous vector $\\theta$ from a \\textit{single} cycle of neuronal firing emanating from the sender region.  \n\nWe now know how to answer our first question: \\textit{The sender region should arrange spike timing differences to come from a maximum entropy distribution}. We remark that this conclusion is consistent with modern paradigms in theoretical neuroscience and artificial intelligence, such as the concept of the Boltzmann machine \\cite{ackley1985},  a stochastic version of its (zero-temperature) deterministic limit, the Little-Hopfield network \\cite{little1974,hopfield1982}.\n\n\n\\paragraph{Organization.}\nThe organization of this paper is as follows.  In Section~\\ref{Sec:General}, we lay out the general theory of maximum entropy distributions on weighted graphs. In Section~\\ref{Sec:Specific}, we specialize the general theory to three classes of weighted graphs. For each class, we provide an explicit characterization of the maximum entropy distributions and prove a generalization of the Erd\\H{o}s-Gallai criterion for weighted graphical sequences. Furthermore, we also present the consistency property of the MLE of the vertex parameters from one graph sample. Section~\\ref{Sec:Proofs} provides the proofs of the main technical results presented in Section~\\ref{Sec:Specific}. Finally, in Section~\\ref{Sec:Discussion} we discuss the results in this paper and some future research directions.\n\n\n\\paragraph{Notation.}\nIn this paper we use the following notation. Let $\\mathbb{R}_+ = (0, \\infty)$, $\\mathbb{R}_0 = [0,\\infty)$, $\\mathbb{N} = \\{1,2,\\dots\\}$, and $\\mathbb{N}_0 = \\{0,1,2,\\dots\\}$. We write $\\sum_{\\{i,j\\}}$ and $\\prod_{\\{i,j\\}}$ for the summation and product, respectively, over all $\\binom{n}{2}$ pairs $\\{i,j\\}$ with $i \\neq j$. For a subset $C \\subseteq \\mathbb{R}^n$, $C^\\circ$ and $\\overline{C}$ denote the interior and closure of $C$ in $\\mathbb{R}^n$, respectively. For a vector $x = (x_1,\\dots,x_n) \\in \\mathbb{R}^n$, $\\|x\\|_1 = \\sum_{i=1}^n |x_i|$ and $\\|x\\|_\\infty = \\max_{1 \\leq i \\leq n} |x_i|$ denote the $\\ell_1$ and $\\ell_\\infty$ norms of $x$. For an $n \\times n$ matrix $J = (J_{ij})$, $\\|J\\|_\\infty$ denotes the matrix norm induced by the $\\|\\cdot\\|_\\infty$-norm on vectors in $\\mathbb{R}^n$, that is,\n\\begin{equation*}\n\\|J\\|_\\infty = \\max_{x \\neq 0} \\frac{\\|Jx\\|_\\infty}{\\|x\\|_\\infty} = \\max_{1 \\leq i \\leq n} \\sum_{j=1}^n |J_{ij}|.\n\\end{equation*}\n\n\n\n\n\n\\section{General theory via exponential family distributions}\n\\label{Sec:General}\n\nIn this section we develop the general machinery of maximum entropy distributions on graphs via the theory of exponential family distributions~\\cite{Jordan}, and in subsequent sections we specialize our analysis to some particular cases of weighted graphs.\n\n\nConsider an undirected graph $G$ on $n \\geq 3$ vertices with edge $(i,j)$ having weight $a_{ij} \\in S$, where $S \\subseteq \\mathbb{R}$ is the set of possible weight values. We will later consider the following specific cases:\n\\begin{enumerate}\n  \\item {\\em Finite discrete weighted graphs,} with edge weights in $S = \\{0,1,\\dots,r-1\\}$, $r \\geq 2$.\n  \\item {\\em Infinite discrete weighted graphs,} with edge weights in $S = \\mathbb{N}_0$.\n  \\item {\\em Continuous weighted graphs,} with edge weights in $S = \\mathbb{R}_0$.\n\\end{enumerate}\nA graph $G$ is fully specified by its \\textit{adjacency matrix} $\\mathbf{a} = (a_{ij})_{i,j=1}^n$, which is an $n \\times n$ symmetric matrix with zeros along its diagonal. For fixed $n$, a probability distribution over graphs $G$ corresponds to a distribution over adjacency matrices $\\mathbf{a} = (a_{ij}) \\in S^{\\binom{n}{2}}$. Given a graph with adjacency matrix $\\mathbf{a} = (a_{ij})$, let $\\deg_i(\\mathbf{a}) = \\sum_{j \\neq i} a_{ij}$ be the degree of vertex $i$, and let $\\deg(\\mathbf{a}) = (\\deg_1(\\mathbf{a}), \\dots, \\deg_n(\\mathbf{a}))$ be the degree sequence of $\\mathbf{a}$. \n\n\n\\subsection{Characterization of maximum entropy distribution}\n\nLet $\\mathcal{S}$ be a $\\sigma$-algebra over the set of weight values $S$, and assume there is a canonical $\\sigma$-finite probability measure $\\nu$ on $(S,\\mathcal{S})$. Let $\\nu^{\\binom{n}{2}}$ be the product measure on $S^{\\binom{n}{2}}$, and let $\\mathfrak{P}$ be the set of all probability distributions on $S^{\\binom{n}{2}}$ that are absolutely continuous with respect to $\\nu^{\\binom{n}{2}}$. Since $\\nu^{\\binom{n}{2}}$ is $\\sigma$-finite, these probability distributions can be characterized by their density functions, i.e.\\ the Radon-Nikodym derivatives with respect to $\\nu^{\\binom{n}{2}}$. Given a sequence $\\mathbf{d} = (d_1, \\dots, d_n) \\in \\mathbb{R}^n$, let $\\mathfrak{P}_\\mathbf{d}$ be the set of distributions in $\\mathfrak{P}$ whose expected degree sequence is equal to $\\mathbf{d}$,\n\\begin{equation*}\n\\mathfrak{P}_\\mathbf{d} = \\{ \\P \\in \\mathfrak{P} \\colon \\mathbb{E}_\\P[\\deg(A)] = \\mathbf{d}\\},\n\\end{equation*}\nwhere in the definition above, the random variable $A = (A_{ij}) \\in S^{\\binom{n}{2}}$ is drawn from the distribution $\\P$. Then the distribution $\\P^\\ast$ in $\\mathfrak{P}_\\mathbf{d}$ with maximum entropy is precisely the exponential family distribution with the degree sequence as sufficient statistics~\\cite[Chapter~3]{Jordan}. Specifically, the density of $\\P^\\ast$ at $\\mathbf{a} = (a_{ij}) \\in S^{\\binom{n}{2}}$ is given by\\footnote{We choose to use $-\\theta$ in the parameterization~\\eqref{Eq:MaxEntDist}, instead of the canonical parameterization $p^\\ast(\\mathbf{a}) \\propto \\exp(\\theta^\\top \\deg(\\mathbf{a}))$, because it simplifies the notation in our later presentation.}\n\\begin{equation}\\label{Eq:MaxEntDist}\np^\\ast(\\mathbf{a}) = \\exp \\big( -\\theta^\\top \\deg(\\mathbf{a}) - Z(\\theta) \\big),\n\\end{equation}\nwhere $Z(\\theta)$ is the \\emph{log-partition function},\n\\begin{equation*}\nZ(\\theta) = \\log \\int_{S^{\\binom{n}{2}}} \\exp\\big( -\\theta^\\top \\deg(\\mathbf{a}) \\big) \\; \\nu^{\\binom{n}{2}}(d\\mathbf{a}),\n\\end{equation*}\nand $\\theta = (\\theta_1, \\dots, \\theta_n)$ is a parameter that belongs to the \\emph{natural parameter space}\n\\begin{equation*}\n\\Theta = \\{\\mathbf{\\theta} \\in \\mathbb{R}^n \\colon Z(\\mathbf{\\theta}) < \\infty\\}.\n\\end{equation*}\nWe will also write $\\P^\\ast_\\theta$ if we need to emphasize the dependence of $\\P^\\ast$ on the parameter $\\theta$.\n\nUsing the definition $\\deg_i(\\mathbf{a}) = \\sum_{j \\neq i} a_{ij}$, we can write\n\\begin{equation*}\n\\exp \\big( -\\theta^\\top \\deg(\\mathbf{a}) \\big)\n= \\exp \\Big( -\\sum_{\\{i,j\\}} (\\theta_i+\\theta_j) a_{ij} \\Big)\n= \\prod_{\\{i,j\\}} \\exp \\big(-(\\theta_i+\\theta_j) a_{ij} \\big).\n\\end{equation*}\nHence, we can express the log-partition function as\n\\begin{equation}\\label{Eq:DecompositionLogPartition}\nZ(\\theta) = \\log \\prod_{\\{i,j\\}} \\int_S \\exp \\big(-(\\theta_i+\\theta_j) a_{ij} \\big) \\; \\nu(da_{ij}) = \\sum_{\\{i,j\\}} Z_1(\\theta_i+\\theta_j),\n\\end{equation}\nin which $Z_1(t)$ is the marginal log-partition function\n\\begin{equation*}\nZ_1(t) = \\log \\int_S \\exp (-ta) \\: \\nu(da).\n\\end{equation*}\nConsequently, the density in~\\eqref{Eq:MaxEntDist} can be written as\n\\begin{equation*}\np^\\ast(\\mathbf{a}) = \\prod_{\\{i,j\\}} \\exp \\big( -(\\theta_i + \\theta_j) a_{ij} - Z_1(\\theta_i + \\theta_j) \\big).\n\\end{equation*}\nThis means the edge weights $A_{ij}$ are independent random variables, with $A_{ij} \\in S$ having distribution $\\P_{ij}^\\ast$ with density\n\\begin{equation*}\np_{ij}^\\ast(a) = \\exp \\big( -(\\theta_i + \\theta_j) a - Z_1(\\theta_i + \\theta_j) \\big).\n\\end{equation*}\nIn particular, the edge weights $A_{ij}$ belong to the same exponential family distribution but with different parameters that depend on $\\theta_i$ and $\\theta_j$ (or rather, on their sum $\\theta_i + \\theta_j$). The parameters $\\theta_1, \\dots, \\theta_n$ can be interpreted as the potential at each vertex that determines how strongly the vertices are connected to each other. Furthermore, we can write the natural parameter space $\\Theta$ as\n\\begin{equation*}\n\\Theta = \\{\\theta \\in \\mathbb{R}^n \\colon Z_1(\\theta_i+\\theta_j) < \\infty \\; \\text{ for all } \\; i \\neq j \\}. \n\\end{equation*}\n\n\n\n\\subsection{Maximum likelihood estimator and moment-matching equation}\n\nUsing the characterization of $\\P^\\ast$ as the maximum entropy distribution in $\\mathfrak{P}_\\mathbf{d}$, the condition $\\P^\\ast \\in \\mathfrak{P}_\\mathbf{d}$ means we need to choose the parameter $\\theta$ for $\\P^\\ast_\\theta$ such that $\\mathbb{E}_\\theta[\\deg(A)] = \\mathbf{d}$.\\footnote{Here we write $\\mathbb{E}_\\theta$ in place of $\\mathbb{E}_{\\P^\\ast}$ to emphasize the dependence of the expectation on the parameter $\\theta$.} This is an instance of the moment-matching equation, which, in the case of exponential family distributions, is well-known to be equivalent to finding the maximum likelihood estimator (MLE) of $\\theta$ given an empirical degree sequence $\\mathbf{d} \\in \\mathbb{R}^n$.\n\nSpecifically, suppose we draw graph samples $G_1,\\dots,G_m$ i.i.d.\\ from the distribution $\\P^\\ast$ with parameter $\\theta^\\ast$, and we want to find the MLE $\\hat \\theta$ of $\\theta^\\ast$ based on the observations $G_1,\\dots,G_m$. Using the parametric form of the density~\\eqref{Eq:MaxEntDist}, this is equivalent to solving the maximization problem\n\\begin{equation*}\n\\max_{\\theta \\in \\Theta} \\: \\mathcal{F}(\\theta) \\equiv -\\theta^\\top \\mathbf{d} - Z(\\theta),\n\\end{equation*}\nwhere $\\mathbf{d}$ is the average of the degree sequences of $G_1,\\dots,G_m$. Setting the gradient of $\\mathcal{F}(\\theta)$ to zero reveals that the MLE $\\hat \\theta$ satisfies\n\\begin{equation}\\label{Eq:MLEEquation}\n-\\nabla Z(\\hat \\theta) = \\mathbf{d}.\n\\end{equation}\nRecall that the gradient of the log-partition function in an exponential family distribution is equal to the expected sufficient statistics. In our case, we have $-\\nabla Z(\\hat \\theta) = \\mathbb{E}_{\\hat \\theta}[\\deg(A)]$, so the MLE equation~\\eqref{Eq:MLEEquation} recovers the moment-matching equation $\\mathbb{E}_{\\hat \\theta}[\\deg(A)] = \\mathbf{d}$.\n\nIn Section~\\ref{Sec:Specific} we study the properties of the MLE of $\\theta$ from a {\\em single} sample $G \\sim \\P^\\ast_\\theta$. In the remainder of this section, we address the question of the existence and uniqueness of the MLE with a given empirical degree sequence $\\mathbf{d}$.\n\nDefine the \\emph{mean parameter space} $\\mathcal{M}$ to be the set of expected degree sequences from all distributions on $S^{\\binom{n}{2}}$ that are absolutely continuous with respect to $\\nu^{\\binom{n}{2}}$,\n\\begin{equation*}\n\\mathcal{M} = \\{ \\mathbb{E}_\\P[\\deg(A)] \\colon \\P \\in \\mathfrak{P} \\}.\n\\end{equation*}\nThe set $\\mathcal{M}$ is necessarily convex, since a convex combination of probability distributions in $\\mathfrak{P}$ is also a probability distribution in $\\mathfrak{P}$. Recall that an exponential family distribution is {\\em minimal} if there is no linear combination of the sufficient statistics that is constant almost surely with respect to the base distribution. This minimality property clearly holds for $\\P^\\ast$, for which the sufficient statistics are the degree sequence. We say that $\\P^\\ast$ is \\emph{regular} if the natural parameter space $\\Theta$ is open. By the general theory of exponential family distributions~\\cite[Theorem~3.3]{Jordan}, in a regular and minimal exponential family distribution, the gradient of the log-partition function maps the natural parameter space $\\Theta$ to the interior of the mean parameter space $\\mathcal{M}$, and this mapping\\footnote{The mapping is $-\\nabla Z$, instead of $\\nabla Z$, because of our choice of the parameterization in~\\eqref{Eq:MaxEntDist} using $-\\theta$.}\n\\begin{equation*}\n-\\nabla Z \\colon \\Theta \\to \\mathcal{M}^\\circ\n\\end{equation*}\nis bijective. We summarize the preceding discussion in the following result.\n\n\\begin{proposition}\\label{Prop:RegularMinimal}\nAssume $\\Theta$ is open. Then there exists a solution $\\theta \\in \\Theta$ to the MLE equation $\\mathbb{E}_{\\theta}[\\deg(A)] = \\mathbf{d}$ if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$, and if such a solution exists then it is unique.\n\\end{proposition}\n\nWe now characterize the mean parameter space $\\mathcal{M}$. We say that a sequence $\\mathbf{d} = (d_1, \\dots, d_n)$ is \\emph{graphic} (or a {\\em graphical sequence}) if $\\mathbf{d}$ is the degree sequence of a graph $G$ with edge weights in $S$, and in this case we say that $G$ \\emph{realizes} $\\mathbf{d}$. It is important to note that whether a sequence $\\mathbf{d}$ is graphic depends on the weight set $S$, which we consider fixed for now.\n\n\\begin{proposition}\\label{Prop:MConvW}\nLet $\\mathcal{W}$ be the set of all graphical sequences, and let $\\text{conv}(\\mathcal{W})$ be the convex hull of $\\mathcal{W}$. Then $\\mathcal{M} \\subseteq \\text{conv}(\\mathcal{W})$. Furthermore, if $\\mathfrak{P}$ contains the Dirac delta measures, then $\\mathcal{M} = \\text{conv}(\\mathcal{W})$.\n\\end{proposition}\n\\begin{proof}\nThe inclusion $\\mathcal{M} \\subseteq \\text{conv}(\\mathcal{W})$ is clear, since any element of $\\mathcal{M}$ is of the form $\\mathbb{E}_\\P[\\deg(A)]$ for some distribution $\\P$ and $\\deg(A) \\in \\mathcal{W}$ for every realization of the random variable $A$. Now suppose $\\mathfrak{P}$ contains the Dirac delta measures $\\delta_B$ for each $B \\in S^{\\binom{n}{2}}$. Given $\\mathbf{d} \\in \\mathcal{W}$, let $B$ be the adjacency matrix of the graph that realizes $\\mathbf{d}$. Then $\\mathbf{d} = \\mathbb{E}_{\\delta_B}[\\deg(A)] \\in \\mathcal{M}$, which means $\\mathcal{W} \\subseteq \\mathcal{M}$, and hence $\\text{conv}(\\mathcal{W}) \\subseteq \\mathcal{M}$ since $\\mathcal{M}$ is convex.\n\\end{proof}\n\nAs we shall see in Section~\\ref{Sec:Specific}, the result above allows us to conclude that $\\mathcal{M} = \\text{conv}(\\mathcal{W})$ for the case of discrete weighted graphs. On the other hand, for the case of continuous weighted graphs we need to prove $\\mathcal{M} = \\text{conv}(\\mathcal{W})$ directly since $\\mathfrak{P}$ in this case does not contain the Dirac measures.\n\n\\begin{remark}\nWe emphasize the distinction between a \\emph{valid} solution $\\theta \\in \\Theta$ and a \\emph{general} solution $\\theta \\in \\mathbb{R}^n$ to the MLE equation $\\mathbb{E}_\\theta[\\deg(A)] = \\mathbf{d}$. As we saw from Proposition~\\ref{Prop:RegularMinimal}, we have a precise characterization of the existence and uniqueness of the valid solution $\\theta \\in \\Theta$, but in general, there are multiple solutions $\\theta$ to the MLE equation. In this paper we shall be concerned only with the valid solution; Sanyal, Sturmfels, and Vinzant study some algebraic properties of general solutions~\\cite{SturmEntDisc}.\n\\end{remark}\n\nWe close this section by discussing the symmetry of the valid solution to the MLE equation. Recall the decomposition~\\eqref{Eq:DecompositionLogPartition} of the log-partition function $Z(\\theta)$ into the marginal log-partition functions $Z_1(\\theta_i+\\theta_j)$. Let $\\text{Dom}(Z_1) = \\{t \\in \\mathbb{R} \\colon Z_1(t) < \\infty\\}$, and let $\\mu \\colon \\text{Dom}(Z_1) \\to \\mathbb{R}$ denote the (marginal) \\emph{mean function}\n\\begin{equation*}\n\\mu(t) = \\int_S a \\: \\exp \\big(-ta - Z_1(t) \\big) \\: \\nu(da).\n\\end{equation*}\nObserving that we can write\n\\begin{equation*}\n\\mathbb{E}_\\theta[A_{ij}] = \\int_S a \\: \\exp \\big( -(\\theta_i + \\theta_j) a - Z_1(\\theta_i + \\theta_j) \\big) \\: \\nu(da) = \\mu(\\theta_i+\\theta_j),\n\\end{equation*}\nthe MLE equation $\\mathbb{E}_\\theta[\\deg(A)] = \\mathbf{d}$ then becomes\n\\begin{equation}\\label{Eq:MLE-Sym}\nd_i = \\sum_{j \\neq i} \\mu(\\theta_i+\\theta_j) \\quad \\text{ for } i = 1, \\dots, n.\n\\end{equation}\n\n\n\nIn the statement below, $\\text{sgn}$ denotes the sign function: $\\text{sgn}(t) = t\/|t|$ if $t \\neq 0$, and $\\text{sgn}(0) = 0$.\n\n\\begin{proposition}\\label{Prop:SymSoln}\nLet $\\mathbf{d} \\in \\mathcal{M}^\\circ$, and let $\\theta \\in \\Theta$ be the unique solution to the system of equations~\\eqref{Eq:MLE-Sym}. If $\\mu$ is strictly increasing, then\n\\begin{equation*}\n\\text{sgn}(d_i-d_j) = \\text{sgn}(\\theta_i-\\theta_j) \\quad \\text{ for all } i \\neq j,\n\\end{equation*}\nand similarly, if $\\mu$ is strictly decreasing, then\n\\begin{equation*}\n\\text{sgn}(d_i-d_j) = \\text{sgn}(\\theta_j-\\theta_i) \\quad \\text{ for all } i \\neq j.\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nGiven $i \\neq j$,\n\\begin{equation*}\n\\begin{split}\nd_i-d_j\n&= \\Big( \\mu(\\theta_i+\\theta_j) + \\sum_{k \\neq i,j} \\mu(\\theta_i+\\theta_k) \\Big)\n- \\Big( \\mu(\\theta_j+\\theta_i) + \\sum_{k \\neq i,j} \\mu(\\theta_j+\\theta_k) \\Big) \\\\\n&= \\sum_{k \\neq i,j} \\big( \\mu(\\theta_i+\\theta_k) - \\mu(\\theta_j+\\theta_k) \\big).\n\\end{split}\n\\end{equation*}\nIf $\\mu$ is strictly increasing, then $\\mu(\\theta_i+\\theta_k)-\\mu(\\theta_j+\\theta_k)$ has the same sign as $\\theta_i-\\theta_j$ for each $k \\neq i,j$, and thus $d_i-d_j$ also has the same sign as $\\theta_i-\\theta_j$. Similarly, if $\\mu$ is strictly decreasing, then $\\mu(\\theta_i+\\theta_k)-\\mu(\\theta_j+\\theta_k)$ has the opposite sign of $\\theta_i-\\theta_j$, and thus $d_i-d_j$ also has the opposite sign of $\\theta_i-\\theta_j$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Analysis for specific edge weights}\n\\label{Sec:Specific}\n\nIn this section we analyze the maximum entropy random graph distributions for several specific choices of the weight set $S$. For each case, we specify the distribution of the edge weights $A_{ij}$, the mean function $\\mu$, the natural parameter space $\\Theta$, and characterize the mean parameter space $\\mathcal{M}$. We also study the problem of finding the MLE $\\hat \\theta$ of $\\theta$ from one graph sample $G \\sim \\P^\\ast_\\theta$ and prove the existence, uniqueness, and consistency of the MLE. Along the way, we derive analogues of the Erd\\H{o}s-Gallai criterion of graphical sequences for weighted graphs. We defer the proofs of the results presented here to Section~\\ref{Sec:Proofs}.\n\n\n\n\n\\subsection{Finite discrete weighted graphs}\n\\label{Sec:FiniteDiscrete}\n\nWe first study weighted graphs with edge weights in the finite discrete set $S = \\{0,1,\\dots,r-1\\}$, where $r \\geq 2$. The case $r = 2$ corresponds to unweighted graphs, and our analysis in this section recovers the results of Chatterjee, Diaconis, and Sly~\\cite{Chatterjee}. The proofs of the results in this section are provided in Section~\\ref{Sec:ProofFiniteDisc}.\n\n\n\\subsubsection{Characterization of the distribution}\n\nWe take $\\nu$ to be the counting measure on $S$. Following the development in Section~\\ref{Sec:General}, the edge weights $A_{ij} \\in S$ are independent random variables with density\n\\begin{equation*}\np_{ij}^\\ast(a) = \\exp\\big(-(\\theta_i+\\theta_j)a - Z_1(\\theta_i+\\theta_j)\\big), \\quad 0 \\leq a \\leq r-1,\n\\end{equation*}\nwhere the marginal log-partition function $Z_1$ is given by\n\\begin{equation*}\nZ_1(t) = \\log \\sum_{a = 0}^{r-1} \\exp(-at) =\n\\begin{cases}\n\\log \\frac{1-\\exp(-rt)}{1-\\exp(-t)} \\quad & \\text{ if } t \\neq 0,\\\\\n\\log r &\\text{ if } t = 0.\n\\end{cases}\n\\end{equation*}\n\nSince $Z_1(t) < \\infty$ for all $t \\in \\mathbb{R}$, the natural parameter space $\\Theta = \\{\\theta \\in \\mathbb{R}^n \\colon Z_1(\\theta_i+\\theta_j) < \\infty, i \\neq j\\}$ is given by $\\Theta = \\mathbb{R}^n$. The mean function is given by\n\\begin{equation}\\label{Eq:MeanFuncFiniteDiscrete}\n\\mu(t) = \\sum_{a=0}^{r-1} a \\exp(-at - Z_1(t))\n= \\frac{\\sum_{a=0}^{r-1} a \\: \\exp(-at)}{\\sum_{a=0}^{r-1} \\exp(-at)}.\n\\end{equation}\nAt $t = 0$ the mean function takes the value\n\\begin{equation*}\n\\mu(0) = \\frac{\\sum_{a=0}^{r-1} a}{r} = \\frac{r-1}{2},\n\\end{equation*}\nwhile for $t \\neq 0$, the mean function simplifies to\n\\begin{equation}\\label{Eq:MeanFuncFiniteDiscreteAlt}\n\\mu(t)\n= -\\left(\\frac{1-\\exp(-t)}{1-\\exp(-rt)}\\right) \\cdot \\frac{d}{dt} \\sum_{a=0}^{r-1} \\exp(-at)\n= \\frac{1}{\\exp(t)-1}-\\frac{r}{\\exp(rt)-1}.\n\\end{equation}\nFigure~\\ref{Fig:FiniteDiscMean} shows the behavior of the mean function $\\mu(t)$ and its derivative $\\mu'(t)$ as $r$ varies.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\widgraph{0.42\\textwidth}{mu_plot}  &\n\\widgraph{0.42\\textwidth}{muprime_plot} \\\\\n(a) & (b)\n\\end{tabular}\n\\caption{Plot of the mean function $\\mu(t)$ (left) and its derivative $\\mu'(t)$ (right) as $r$ varies.}\n\\label{Fig:FiniteDiscMean}\n\\end{center}\n\\end{figure}\n\n\n\\begin{remark}\nFor $r = 2$, the edge weights $A_{ij}$ are independent Bernoulli random variables with\n\\begin{equation*}\n\\P^\\ast(A_{ij} = 1) = \\mu(\\theta_i+\\theta_j) = \\frac{\\exp(-\\theta_i-\\theta_j)}{1 + \\exp(-\\theta_i - \\theta_j)} = \\frac{1}{1+\\exp(\\theta_i+\\theta_j)}.\n\\end{equation*}\nAs noted above, this is the model recently studied by Chatterjee, Diaconis, and Sly~\\cite{Chatterjee} in the context of graph limits. When $\\theta_1 = \\theta_2 = \\dots = \\theta_n = t$, we recover the classical Erd\\H{o}s-R\\'enyi model with edge emission probability $p = 1\/(1+\\exp(2t))$.\n\\end{remark}\n\n\n\n\\subsubsection{Existence, uniqueness, and consistency of the MLE}\n\nConsider the problem of finding the MLE of $\\theta$ from one graph sample. Specifically, let $\\theta \\in \\Theta$ and suppose we draw a sample $G \\sim \\P^\\ast_\\theta$. Then, as we saw in Section~\\ref{Sec:General}, the MLE $\\hat \\theta$ of $\\theta$ is a solution to the moment-matching equation $\\mathbb{E}_{\\hat \\theta}[\\deg(A)] = \\mathbf{d}$, where $\\mathbf{d}$ is the degree sequence of the sample graph $G$. As in~\\eqref{Eq:MLE-Sym}, the moment-matching equation is equivalent to the following system of equations:\n\\begin{equation}\\label{Eq:MLEEqFiniteDisc}\nd_i = \\sum_{j \\neq i} \\mu(\\hat \\theta_i + \\hat \\theta_j), \\quad i = 1,\\dots,n.\n\\end{equation}\n\nSince the natural parameter space $\\Theta = \\mathbb{R}^n$ is open, Proposition~\\ref{Prop:RegularMinimal} tells us that the MLE $\\hat \\theta$ exists and is unique if and only if the empirical degree sequence $\\mathbf{d}$ belongs to the interior $\\mathcal{M}^\\circ$ of the mean parameter space $\\mathcal{M}$.\n\nWe also note that since $\\nu^{\\binom{n}{2}}$ is the counting measure on $S^{\\binom{n}{2}}$, all  distributions on $S^{\\binom{n}{2}}$ are absolutely continuous with respect to $\\nu^{\\binom{n}{2}}$, so $\\mathfrak{P}$ contains all probability distributions on $S^{\\binom{n}{2}}$. In particular, $\\mathfrak{P}$ contains the Dirac measures, and by Proposition~\\ref{Prop:MConvW}, this implies $\\mathcal{M} = \\text{conv}(\\mathcal{W})$, where $\\mathcal{W}$ is the set of all graphical sequences.\n\nThe following result characterizes when $\\mathbf{d}$ is a degree sequence of a weighted graph with edge weights in $S$; we also refer to such $\\mathbf{d}$ as a {\\em (finite discrete) graphical sequence}. The case $r = 2$ recovers the classical Erd\\H{o}s-Gallai criterion~\\cite{ErdosGallai}.\n\n\\begin{theorem}\\label{Thm:GraphicalFiniteDisc}\nA sequence $(d_1,\\dots,d_n) \\in \\mathbb{N}_0^n$ with $d_1 \\geq d_2 \\geq \\dots \\geq d_n$ is the degree sequence of a graph $G$ with edge weights in the set $S = \\{0,1,\\dots,r-1\\}$, if and only if $\\sum_{i=1}^n d_i$ is even and\n\\begin{equation}\\label{Eq:GraphicalFiniteDisc}\n\\sum_{i=1}^k d_i \\leq (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j, (r-1)k\\} \\quad \\text{ for } k = 1,\\dots,n.\n\\end{equation}\n\\end{theorem}\n\nAlthough the result above provides a precise characterization of the set of graphical sequences $\\mathcal{W}$, it is not immediately clear how to characterize the convex hull $\\text{conv}(\\mathcal{W})$, or how to decide whether a given $\\mathbf{d}$ belongs to $\\mathcal{M}^\\circ = \\text{conv}(\\mathcal{W})^\\circ$. Fortunately, in practice we can circumvent this issue by employing the following algorithm to compute the MLE. The case $r = 2$ recovers the iterative algorithm proposed by Chatterjee et al.~\\cite{Chatterjee} in the case of unweighted graphs.\n\n\\begin{theorem}\\label{Thm:MLEAlgFiniteDisc}\nGiven $\\mathbf{d} = (d_1, \\dots, d_n) \\in \\mathbb{R}_+^n$, define the function $\\varphi \\colon \\mathbb{R}^n \\to \\mathbb{R}^n$ by $\\varphi(\\mathbf{x}) = (\\varphi_1(\\mathbf{x}), \\dots, \\varphi_n(\\mathbf{x}))$, where\n\\begin{equation}\\label{Eq:VarphiFiniteDisc}\n\\varphi_i(\\mathbf{x}) = x_i + \\frac{1}{r-1} \\left( \\log \\sum_{j \\neq i} \\mu(x_i+x_j) -  \\log d_i\\right).\n\\end{equation}\nStarting from any $\\theta^{(0)} \\in \\mathbb{R}^n$, define\n\\begin{equation}\\label{Eq:MLEAlgFiniteDisc}\n\\theta^{(k+1)} = \\varphi(\\theta^{(k)}), \\quad k \\in \\mathbb{N}_0.\n\\end{equation}\nSuppose $\\mathbf{d} \\in \\text{conv}(\\mathcal{W})^\\circ$, so the MLE equation~\\eqref{Eq:MLEEqFiniteDisc} has a unique solution $\\hat \\theta$. Then $\\hat \\theta$ is a fixed point of the function $\\varphi$, and the iterates~\\eqref{Eq:MLEAlgFiniteDisc} converge to $\\hat \\theta$ geometrically fast: there exists a constant $\\beta \\in (0,1)$ that only depends on $(\\|\\hat \\theta\\|_\\infty, \\|\\theta^{(0)}\\|_\\infty)$, such that\n\\begin{equation}\\label{Eq:MLERateOfConvFiniteDisc}\n\\|\\theta^{(k)}-\\hat\\theta\\|_\\infty \\leq \\beta^{k-1} \\: \\|\\theta^{(0)}-\\hat\\theta\\|_\\infty, \\quad k \\in \\mathbb{N}_0.\n\\end{equation}\nConversely, if $\\mathbf{d} \\notin \\text{conv}(\\mathcal{W})^\\circ$, then the sequence $\\{\\theta^{(k)}\\}$ has a divergent subsequence.\n\\end{theorem}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\widgraph{0.38\\textwidth}{alg_conv}  &\n\\widgraph{0.28\\textwidth}{scatter_r2} &\n\\widgraph{0.28\\textwidth}{scatter_r5} \\\\\n(a) & (b) & (c)\n\\end{tabular}\n\\caption{(a) Plot of $\\log \\|\\theta^{(t)} - \\hat\\theta\\|_\\infty$ for various values of $r$, where $\\hat\\theta$ is the final value of $\\theta^{(t)}$ when the algorithm converges; (b) Scatter plot of the estimate $\\hat \\theta$ vs.\\ the true parameter $\\theta$ for $r = 2$; (c) Scatter plot for $r = 5$.}\n\\label{Fig:FiniteDisc}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{Fig:FiniteDisc} demonstrates the performance of the algorithm presented above. We set $n = 200$ and sample $\\theta \\in [-1,1]^n$ uniformly at random. Then for each $2 \\leq r \\leq 10$, we sample a graph from the distribution $\\P^\\ast_\\theta$, compute the empirical degree sequence $\\mathbf{d}$, and run the iterative algorithm starting with $\\theta^{(0)} = 0$ until convergence. The left panel (Figure~\\ref{Fig:FiniteDisc}(a)) shows the rate of convergence (on a logarithmic scale) of the algorithm for various values of $r$. We observe that the iterates $\\{\\theta^{(t)}\\}$ indeed converge geometrically fast to the MLE $\\hat \\theta$, but the rate of convergence decreases as $r$ increases. By examining the proof of Theorem~\\ref{Thm:MLEAlgFiniteDisc} in Section~\\ref{Sec:ProofFiniteDisc}, we see that the term $\\beta$ has the expression\n\\begin{equation*}\n\\beta^2 = 1-\\frac{1}{(r-1)^2} \\: \\left(\\min \\left\\{\\frac{\\exp(2K)-1}{\\exp(2rK)-1}, \\; -\\frac{\\mu'(2K)}{\\mu(-2K)} \\right\\}\\right)^2,\n\\end{equation*}\nwhere $K = 2\\|\\hat\\theta\\|_\\infty + \\|\\theta^{(0)}\\|_\\infty$. This shows that $\\beta$ is an increasing function of $r$, which explains the empirical decrease in the rate of convergence as $r$ increases.\n\n\nFigures~\\ref{Fig:FiniteDisc}(b) and (c) show the plots of the estimate $\\hat \\theta$ versus the true $\\theta$. Notice that the points lie close to the diagonal line, which suggests that the MLE $\\hat \\theta$ is very close to the true parameter $\\theta$. Indeed, the following result shows that $\\hat \\theta$ is a consistent estimator of $\\theta$. Recall that $\\hat \\theta$ is {\\em consistent} if $\\hat \\theta$ converges in probability to $\\theta$ as $n \\to \\infty$.\n\n\\begin{theorem}\\label{Thm:ConsistencyFiniteDisc}\nLet $M > 0$ and $k > 1$ be fixed. Given $\\theta \\in \\mathbb{R}^n$ with $\\|\\theta\\|_\\infty \\leq M$, consider the problem of finding the MLE $\\hat \\theta$ of $\\theta$ based on one graph sample $G \\sim \\P^\\ast_\\theta$. Then for sufficiently large $n$, with probability at least $1-2n^{-(k-1)}$ the MLE $\\hat \\theta$ exists and satisfies\n\\begin{equation*}\n\\|\\hat \\theta - \\theta\\|_\\infty \\leq C \\sqrt{\\frac{k \\log n}{n}},\n\\end{equation*}\nwhere $C$ is a constant that only depends on $M$.\n\\end{theorem}\n\n\n\n\\subsection{Continuous weighted graphs}\n\\label{Sec:Cont}\n\nIn this section we study weighted graphs with edge weights in $\\mathbb{R}_0$. The proofs of the results presented here are provided in Section~\\ref{Sec:ProofCont}.\n\n\n\\subsubsection{Characterization of the distribution}\n\nWe take $\\nu$ to be the Lebesgue measure on $\\mathbb{R}_0$. The marginal log-partition function is\n\\begin{equation*}\nZ_1(t) = \\log \\int_{\\mathbb{R}_0} \\exp(-ta) \\: da =\n\\begin{cases}\n\\log(1\/t) & \\text{ if } t > 0 \\\\\n\\infty \\quad & \\text{ if } t \\leq 0.\n\\end{cases}\n\\end{equation*}\nThus $\\text{Dom}(Z_1) = \\mathbb{R}_+$, and the natural parameter space is\n\\begin{equation*}\n\\Theta = \\{(\\theta_1, \\dots, \\theta_n) \\in \\mathbb{R}^n \\colon \\theta_i+\\theta_j > 0 \\text{ for } i \\neq j\\}.\n\\end{equation*}\nFor $\\theta \\in \\Theta$, the edge weights $A_{ij}$ are independent exponential random variables with density\n\\begin{equation*}\np_{ij}^\\ast(a) = (\\theta_i+\\theta_j) \\: \\exp\\big(-(\\theta_i+\\theta_j) \\: a\\big) \\quad \\text{ for } a \\in \\mathbb{R}_0\n\\end{equation*}\nand mean parameter $\\mathbb{E}_\\theta[A_{ij}] = 1\/(\\theta_i+\\theta_j)$. The corresponding mean function is given by\n\\begin{equation*}\n\\mu(t) = \\frac{1}{t}, \\quad t > 0.\n\\end{equation*}\n\n\n\\subsubsection{Existence, uniqueness, and consistency of the MLE}\n\nWe now consider the problem of finding the MLE of $\\theta$ from one graph sample $G \\sim \\P^\\ast_\\theta$. As we saw previously, the MLE $\\hat \\theta \\in \\Theta$ satisfies the moment-matching equation $\\mathbb{E}_{\\hat \\theta}[\\deg(A)] = \\mathbf{d}$, where $\\mathbf{d}$ is the degree sequence of the sample graph $G$. Equivalently, $\\hat \\theta \\in \\Theta$ is a solution to the system of equations\n\\begin{equation}\\label{Eq:MLEEqCont}\nd_i = \\sum_{j \\neq i} \\frac{1}{\\hat{\\theta}_i + \\hat{\\theta}_j}, \\quad i = 1,\\dots,n.\n\\end{equation}\n\n\\begin{remark}\nThe system~\\eqref{Eq:MLEEqCont} is a special case of a general class that Sanyal, Sturmfels, and Vinzant~\\cite{SturmEntDisc} study using algebraic geometry and matroid theory (extending the work of Proudfoot and Speyer~\\cite{proudfootspeyer}).  Define\n\\begin{equation*}\n\\chi(t) = \\sum_{k=0}^n \\left(\\stirlingtwo{n}{k} +n \\stirlingtwo{n-1}{k} \\right)(t-1)_k^{(2)},\n\\end{equation*}\nin which $\\stirlingtwo{n}{k}$ is the Stirling number of the second kind and $(x)_{k+1}^{(2)} = x(x-2)\\cdots (x-2k)$ is a generalized falling factorial.  Then, there is a polynomial $H(\\mathbf{d})$ in the $d_i$ such that for $\\mathbf{d} \\in \\mathbb R^n$ with $H(\\mathbf{d}) \\neq 0$, the number of solutions $\\theta \\in \\mathbb R^n$ to~\\eqref{Eq:MLEEqCont} is $(-1)^n \\chi(0)$.  Moreover, the polynomial $H(\\mathbf{d})$ has degree $2(-1)^n(n \\chi(0) + \\chi'(0))$ and characterizes those $\\mathbf{d}$ for which the equations above have multiple roots.  We refer to \\cite{SturmEntDisc} for more details.\n\\end{remark}\n\nSince the natural parameter space $\\Theta$ is open, Proposition~\\ref{Prop:RegularMinimal} tells us that the MLE $\\hat \\theta$ exists and is unique if and only if the empirical degree sequence $\\mathbf{d}$ belongs to the interior $\\mathcal{M}^\\circ$ of the mean parameter space $\\mathcal{M}$. We characterize the set of graphical sequences $\\mathcal{W}$ and determine its relation to the mean parameter space $\\mathcal{M}$.\n\nWe say $\\mathbf{d} = (d_1, \\dots, d_n)$ is a {\\em (continuous) graphical sequence} if there is a graph $G$ with edge weights in $\\mathbb{R}_0$ that realizes $\\mathbf{d}$. The finite discrete graphical sequences from Section~\\ref{Sec:FiniteDiscrete} have combinatorial constraints because there are only finitely many possible edge weights between any pair of vertices, and these constraints translate into a set of inequalities in the generalized Erd\\H{o}s-Gallai criterion in Theorem~\\ref{Thm:GraphicalFiniteDisc}. In the case of continuous weighted graphs, however, we do not have these constraints because every edge can have as much weight as possible. Therefore, the criterion for a continuous graphical sequence should be simpler than in Theorem~\\ref{Thm:GraphicalFiniteDisc}, as the following result shows.\n\n\\begin{theorem}\\label{Thm:GraphicalCont}\nA sequence $(d_1, \\dots, d_n) \\in \\mathbb{R}_0^n$ is graphic if and only if\n\\begin{equation}\\label{Eq:GraphicalCont}\n\\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i.\n\\end{equation}\n\\end{theorem}\n\nWe note that condition~\\eqref{Eq:GraphicalCont} is implied by the case $k = 1$ in the conditions~\\eqref{Eq:GraphicalFiniteDisc}. This is to be expected, since any finite discrete weighted graph is also a continuous weighted graph, so finite discrete graphical sequences are also continuous graphical sequences.\n\nGiven the criterion in Theorem~\\ref{Thm:GraphicalCont}, we can write the set $\\mathcal{W}$ of graphical sequences as\n\\begin{equation*}\n\\mathcal{W} = \\Big\\{(d_1, \\dots, d_n) \\in \\mathbb{R}_0^n \\colon \\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i \\Big\\}.\n\\end{equation*}\nMoreover, we can also show that the set of graphical sequences coincide with the mean parameter space.\n\n\\begin{lemma}\\label{Lem:W-Convex}\nThe set $\\mathcal{W}$ is convex, and $\\mathcal{M} = \\mathcal{W}$.\n\\end{lemma}\n\nThe result above, together with the result of Proposition~\\ref{Prop:RegularMinimal}, implies that the MLE $\\hat \\theta$ exists and is unique if and only if the empirical degree sequence $\\mathbf{d}$ belongs to the interior of the mean parameter space, which can be written explicitly as\n\\begin{equation*}\n\\mathcal{M}^\\circ = \\Big\\{(d_1', \\dots, d_n') \\in \\mathbb{R}_+^n \\colon \\max_{1 \\leq i \\leq n} d_i' < \\frac{1}{2} \\sum_{i=1}^n d_i' \\Big\\}.\n\\end{equation*}\n\n\n\\begin{example}\nLet $n = 3$ and $\\mathbf{d} = (d_1,d_2,d_3) \\in \\mathbb{R}^n$ with $d_1 \\geq d_2 \\geq d_3$. It is easy to see that the system of equations~\\eqref{Eq:MLEEqCont} gives us\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\hat{\\theta}_1+\\hat{\\theta}_2} &= \\frac{1}{2}(d_1+d_2-d_3), \\\\\n\\frac{1}{\\hat{\\theta}_1+\\hat{\\theta}_3} &= \\frac{1}{2}(d_1-d_2+d_3), \\\\\n\\frac{1}{\\hat{\\theta}_2+\\hat{\\theta}_3} &= \\frac{1}{2}(-d_1+d_2+d_3),\n\\end{split}\n\\end{equation*}\nfrom which we obtain a unique solution $\\hat{\\theta} = (\\hat{\\theta}_1,\\hat{\\theta}_2,\\hat{\\theta}_3)$. Recall that $\\hat{\\theta} \\in \\Theta$ means $\\hat{\\theta}_1+\\hat{\\theta}_2 > 0$, $\\hat{\\theta}_1+\\hat{\\theta}_3 > 0$, and $\\hat{\\theta}_2+\\hat{\\theta}_3 > 0$, so the equations above tell us that $\\hat{\\theta} \\in \\Theta$ if and only if $d_1 < d_2+d_3$. In particular, this also implies $d_3 > d_1-d_2 \\geq 0$, so $\\mathbf{d} \\in \\mathbb{R}_+^3$. Hence, there is a unique solution $\\hat{\\theta} \\in \\Theta$ to the system of equations~\\eqref{Eq:MLEEqCont} if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$, as claimed above.\n\\end{example}\n\nFinally, we prove that the MLE $\\hat \\theta$ is a consistent estimator of $\\theta$.\n\n\\begin{theorem}\\label{Thm:ConsistencyCont}\nLet $M \\geq L > 0$ and $k \\geq 1$ be fixed. Given $\\theta \\in \\Theta$ with $L \\leq \\theta_i + \\theta_j \\leq M$, $i \\neq j$, consider the problem of finding the MLE $\\hat \\theta \\in \\Theta$ of $\\theta$ from one graph sample $G \\sim \\P^\\ast_\\theta$. Then for sufficiently large $n$, with probability at least $1-2n^{-(k-1)}$ the MLE $\\hat \\theta \\in \\Theta$ exists and satisfies\n\\begin{equation*}\n\\|\\hat \\theta - \\theta\\|_\\infty \\leq \\frac{100M^2}{L} \\sqrt{\\frac{k \\log n}{\\gamma n}},\n\\end{equation*}\nwhere $\\gamma > 0$ is a universal constant.\n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Infinite discrete weighted graphs}\n\\label{Sec:InfiniteDisc}\n\nWe now turn our focus to weighted graphs with edge weights in $\\mathbb{N}_0$. The proofs of the results presented here can be found in Section~\\ref{Sec:ProofInfiniteDisc}.\n\n\n\\subsubsection{Characterization of the distribution}\n\nWe take $\\nu$ to be the counting measure on $\\mathbb{N}_0$. In this case the marginal log-partition function is given by\n\\begin{equation*}\nZ_1(t) = \\log \\sum_{a = 0}^\\infty \\exp(-at) = \n\\begin{cases}\n-\\log\\big( 1-\\exp(-t) \\big) & \\text{ if } t > 0, \\\\\n\\infty \\quad & \\text{ if } t \\leq 0.\n\\end{cases}\n\\end{equation*}\nThus, the domain of $Z_1$ is $\\text{Dom}(Z_1) = (0,\\infty)$, and the natural parameter space is\n\\begin{equation*}\n\\Theta = \\{(\\theta_1,\\dots,\\theta_n) \\in \\mathbb{R}^n \\colon \\theta_i+\\theta_j > 0 \\text{ for } i \\neq j\\},\n\\end{equation*}\nwhich is the same natural parameter space as in the case of continuous weighted graphs in the preceding section. Given $\\theta \\in \\Theta$, the edge weights $A_{ij}$ are independent geometric random variables with probability mass function\n\\begin{equation*}\n\\P^\\ast(A_{ij}=a) = \\big(1-\\exp(-\\theta_i-\\theta_j)\\big) \\: \\exp\\big( -(\\theta_i+\\theta_j) \\: a \\big), \\quad a \\in \\mathbb{N}_0.\n\\end{equation*}\nThe mean parameters are\n\\begin{equation*}\n\\mathbb{E}_{\\P^\\ast}[A_{ij}] = \\frac{\\exp(-\\theta_i-\\theta_j)}{1-\\exp(-\\theta_i-\\theta_j)} = \\frac{1}{\\exp(\\theta_i+\\theta_j)-1},\n\\end{equation*}\ninduced by the mean function\n\\begin{equation*}\n\\mu(t) = \\frac{1}{\\exp(t)-1}, \\quad t > 0.\n\\end{equation*}\n\n\n\n\\subsubsection{Existence, uniqueness, and consistency of the MLE}\n\nConsider the problem of finding the MLE of $\\theta$ from one graph sample $G \\sim \\P^\\ast_\\theta$. Let $\\mathbf{d}$ denote the degree sequence of $G$. Then the MLE $\\hat \\theta \\in \\Theta$, which satisfies the moment-matching equation $\\mathbb{E}_{\\hat \\theta}[\\deg(A)] = \\mathbf{d}$, is a solution to the system of equations\n\\begin{equation}\\label{Eq:MLEEqInfiniteDisc}\nd_i = \\sum_{j \\neq i} \\frac{1}{\\exp(\\hat{\\theta}_i + \\hat{\\theta}_j)-1}, \\quad i = 1,\\dots,n.\n\\end{equation}\n\nWe note that the natural parameter space $\\Theta$ is open, so by Proposition~\\ref{Prop:RegularMinimal}, the MLE $\\hat \\theta$ exists and is unique if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$, where $\\mathcal{M}$ is the mean parameter space. Since $\\nu^{\\binom{n}{2}}$ is the counting measure on $\\mathbb{N}_0^{\\binom{n}{2}}$, the set $\\mathfrak{P}$ contains all the Dirac measures, so we know $\\mathcal{M} = \\text{conv}(\\mathcal{W})$ from Proposition~\\ref{Prop:MConvW}. Here $\\mathcal{W}$ is the set of all {\\em (infinite discrete) graphical sequences}, namely, the set of degree sequences of weighted graphs with edge weights in $\\mathbb{N}_0$. The following result provides a precise criterion for such graphical sequences. Note that condition~\\eqref{Eq:GraphicalInfiniteDisc} below is implied by the limit $r \\to \\infty$ in Theorem~\\ref{Thm:GraphicalFiniteDisc}.\n\n\\begin{theorem}\\label{Thm:GraphicalInfiniteDisc}\nA sequence $(d_1, \\dots, d_n) \\in \\mathbb{N}_0^n$ is graphic if and only if $\\sum_{i=1}^n d_i$ is even and\n\\begin{equation}\\label{Eq:GraphicalInfiniteDisc}\n\\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i.\n\\end{equation}\n\\end{theorem}\n\nThe criterion in Theorem~\\ref{Thm:GraphicalInfiniteDisc} allows us to write an explicit form for the set of graphical sequences $\\mathcal{W}$,\n\\begin{equation*}\n\\mathcal{W} = \\Big\\{(d_1, \\dots, d_n) \\in \\mathbb{N}_0^n \\colon \\sum_{i=1}^n d_i \\text{ is even and } \\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i \\Big\\}.\n\\end{equation*}\nNow we need to characterize $\\text{conv}(\\mathcal{W})$. Let $\\mathcal{W}_1$ denote the set of all continuous graphical sequences from Theorem~\\ref{Thm:GraphicalCont}, when the edge weights are in $\\mathbb{R}_0$,\n\\begin{equation*}\n\\mathcal{W}_1 = \\Big\\{(d_1, \\dots, d_n) \\in \\mathbb{R}_0^n \\colon \\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i \\Big\\}.\n\\end{equation*}\nIt turns out that when we take the convex hull of $\\mathcal{W}$, we essentially recover $\\mathcal{W}_1$.\n\n\\begin{lemma}\\label{Lem:ConvWInfiniteDisc}\n$\\overline{\\text{conv}(\\mathcal{W})} = \\mathcal{W}_1$.\n\\end{lemma}\n\nRecalling that a convex set and its closure have the same interior points, the result above gives us\n\\begin{equation*}\n\\mathcal{M}^\\circ = \\text{conv}(\\mathcal{W})^\\circ = \\big(\\overline{\\text{conv}(\\mathcal{W})}\\big)^\\circ = \\mathcal{W}_1^\\circ =\n\\Big\\{(d_1, \\dots, d_n) \\in \\mathbb{R}_+^n \\colon \\max_{1 \\leq i \\leq n} d_i < \\frac{1}{2} \\sum_{i=1}^n d_i \\Big\\}.\n\\end{equation*}\n\n\\begin{example}\nLet $n = 3$ and $\\mathbf{d} = (d_1,d_2,d_3) \\in \\mathbb{R}^n$ with $d_1 \\geq d_2 \\geq d_3$. It can be easily verified that the system of equations~\\eqref{Eq:MLEEqInfiniteDisc} gives us\n\\begin{equation*}\n\\begin{split}\n\\hat{\\theta}_1+\\hat{\\theta}_2 &= \\log\\left( 1+\\frac{2}{d_1+d_2-d_3} \\right), \\\\\n\\hat{\\theta}_1+\\hat{\\theta}_3 &= \\log\\left( 1+\\frac{2}{d_1-d_2+d_3} \\right), \\\\\n\\hat{\\theta}_2+\\hat{\\theta}_3 &= \\log\\left( 1+\\frac{2}{-d_1+d_2+d_3} \\right),\n\\end{split}\n\\end{equation*}\nfrom which we can obtain a unique solution $\\hat{\\theta} = (\\hat{\\theta}_1,\\hat{\\theta}_2,\\hat{\\theta}_3)$. Recall that $\\hat{\\theta} \\in \\Theta$ means $\\hat{\\theta}_1+\\hat{\\theta}_2>0$, $\\hat{\\theta}_1+\\hat{\\theta}_3>0$, and $\\hat{\\theta}_2+\\hat{\\theta}_3>0$, so the equations above tell us that $\\hat\\theta \\in \\Theta$ if and only if $2\/(-d_1+d_2+d_3) > 0$, or equivalently, $d_1 < d_2+d_3$. This also implies $d_3 > d_1-d_2 \\geq 0$, so $\\mathbf{d} \\in \\mathbb{R}_+^3$. Thus, the system of equations~\\eqref{Eq:MLEEqInfiniteDisc} has a unique solution $\\hat{\\theta} \\in \\Theta$ if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$, as claimed above.\n\\end{example}\n\nFinally, we prove that with high probability the MLE $\\hat \\theta$ exists and converges to $\\theta$.\n\n\\begin{theorem}\\label{Thm:ConsistencyInfiniteDisc}\nLet $M \\geq L > 0$ and $k \\geq 1$ be fixed. Given $\\theta \\in \\Theta$ with $L \\leq \\theta_i + \\theta_j \\leq M$, $i \\neq j$, consider the problem of finding the MLE $\\hat \\theta \\in \\Theta$ of $\\theta$ from one graph sample $G \\sim \\P^\\ast_\\theta$. Then for sufficiently large $n$, with probability at least $1-3n^{-(k-1)}$ the MLE $\\hat \\theta \\in \\Theta$ exists and satisfies\n\\begin{equation*}\n\\|\\hat \\theta - \\theta\\|_\\infty \\leq \\frac{8 \\: \\exp(5M)}{L} \\: \\sqrt{\\frac{k \\log n}{\\gamma n}},\n\\end{equation*}\nwhere $\\gamma > 0$ is a universal constant.\n\\end{theorem}\n\n\n\n\n\\section{Proofs of main results}\n\\label{Sec:Proofs}\n\nIn this section we provide proofs for the technical results presented in Section~\\ref{Sec:Specific}. The proofs of the characterization of weighted graphical sequences (Theorems~\\ref{Thm:GraphicalFiniteDisc},~\\ref{Thm:GraphicalCont}, and~\\ref{Thm:GraphicalInfiniteDisc}) are inspired by the constructive proof of the classical Erd\\H{o}s-Gallai criterion by Choudum~\\cite{Choudum}.\n\n\n\\subsection{Preliminaries}\n\nWe begin by presenting several results that we will use in this section. We use the definition of sub-exponential random variables and the concentration inequality presented in~\\cite{vershynin}.\n\n\n\\subsubsection{Concentration inequality for sub-exponential random variables}\n\nWe say that a real-valued random variable $X$ is {\\em sub-exponential} with parameter $\\kappa > 0$ if\n\\begin{equation*}\n\\mathbb{E}[|X|^p]^{1\/p} \\leq \\kappa p \\quad \\text{ for all } p \\geq 1.\n\\end{equation*}\nNote that if $X$ is a $\\kappa $-sub-exponential random variable with finite first moment, then the centered random variable $X-\\mathbb{E}[X]$ is also sub-exponential with parameter $2 \\kappa $. This follows from the triangle inequality applied to the $p$-norm, followed by Jensen's inequality for $p \\geq 1$:\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}\\big[\\big|X-\\mathbb{E}[X]\\big|^p\\big]^{1\/p}\n&\\leq \\mathbb{E}[|X|^p]^{1\/p} + \\big|\\mathbb{E}[X]\\big|\n\\leq 2\\mathbb{E}[|X|^p]^{1\/p}.\n\\end{split}\n\\end{equation*}\nSub-exponential random variables satisfy the following concentration inequality.\n\n\\begin{theorem}[{\\cite[Corollary~5.17]{vershynin}}]\\label{Thm:ConcIneqSubExp}\nLet $X_1, \\dots, X_n$ be independent centered random variables, and suppose each $X_i$ is sub-exponential with parameter $\\kappa_i$. Let $\\kappa = \\max_{1 \\leq i \\leq n} \\kappa_i$. Then for every $\\epsilon \\geq 0$,\n\\begin{equation*}\n\\P\\left( \\left| \\frac{1}{n} \\sum_{i=1}^n X_i \\right| \\geq \\epsilon \\right) \\leq 2\\exp\\left[-\\gamma \\, n \\cdot \\min\\Big(\\frac{\\epsilon^2}{\\kappa^2}, \\: \\frac{\\epsilon}{\\kappa} \\Big) \\right],\n\\end{equation*}\nwhere $\\gamma > 0$ is an absolute constant.\n\\end{theorem}\n\nWe will apply the concentration inequality above to exponential and geometric random variables, which are the distributions of the edge weights of continuous weighted graphs (from Section~\\ref{Sec:Cont}) and infinite discrete weighted graphs (from Section~\\ref{Sec:InfiniteDisc}).\n\n\\begin{lemma}\\label{Lem:SubExp-Exp}\nLet $X$ be an exponential random variable with $\\mathbb{E}[X] = 1\/\\lambda$. Then $X$ is sub-exponential with parameter $1\/\\lambda$, and the centered random variable $X-1\/\\lambda$ is sub-exponential with parameter $2\/\\lambda$.\n\\end{lemma}\n\\begin{proof}\nFor any $p \\geq 1$, we can evaluate the moment of $X$ directly:\n\\begin{equation*}\n\\mathbb{E}[|X|^p] = \\int_0^\\infty x^p \\cdot \\lambda \\, \\exp(-\\lambda x) \\: dx\n= \\frac{1}{\\lambda^p} \\: \\int_0^\\infty y^p \\: \\exp(-y) \\: dy\n= \\frac{\\Gamma(p+1)}{\\lambda^p},\n\\end{equation*}\nwhere $\\Gamma$ is the gamma function, and in the computation above we have used the substitution $y = \\lambda x$. It can be easily verified that $\\Gamma(p+1) \\leq p^p$ for $p \\geq 1$, so\n\\begin{equation*}\n\\mathbb{E}[|X|^p]^{1\/p} = \\frac{\\big(\\Gamma(p+1)\\big)^{1\/p}}{\\lambda} \\leq \\frac{p}{\\lambda}.\n\\end{equation*}\nThis shows that $X$ is sub-exponential with parameter $1\/\\lambda$.\n\\end{proof}\n\n\\begin{lemma}\\label{Lem:SubExp-Geo}\nLet $X$ be a geometric random variable with parameter $q \\in (0,1)$, so\n\\begin{equation*}\n\\P(X = a) = (1-q)^a \\, q, \\quad a \\in \\mathbb{N}_0.\n\\end{equation*}\nThen $X$ is sub-exponential with parameter $-2\/\\log(1-q)$, and the centered random variable $X - (1-q)\/q$ is sub-exponential with parameter $-4\/\\log(1-q)$.\n\\end{lemma}\n\\begin{proof}\nFix $p \\geq 1$, and consider the function $f \\colon \\mathbb{R}_0 \\to \\mathbb{R}_0$, $f(x) = x^p (1-q)^x$. One can easily verify that $f$ is increasing for $0 \\leq x \\leq \\lambda$ and decreasing on $x \\geq \\lambda$, where $\\lambda = -p\/\\log(1-q)$. In particular, for all $x \\in \\mathbb{R}_0$ we have $f(x) \\leq f(\\lambda)$, and\n\\begin{equation*}\n\\begin{split}\nf(\\lambda) &= \\lambda^p (1-q)^\\lambda\n= \\left( \\frac{p}{-\\log (1-q)} \\cdot (1-q)^{-1\/\\log(1-q)} \\right)^p\n= \\left( \\frac{p}{-e \\cdot \\log (1-q)} \\right)^p.\n\\end{split}\n\\end{equation*}\n\nNow note that for $0 \\leq a \\leq \\lfloor \\lambda \\rfloor - 1$ we have $f(a) \\leq \\int_a^{a+1} f(x) \\: dx$, and for $a \\geq \\lceil \\lambda \\rceil + 1$ we have $f(a) \\leq \\int_{a-1}^a f(x) \\: dx$. Thus, we can bound\n\\begin{equation*}\n\\begin{split}\n\\sum_{a=0}^\\infty f(a)\n&= \\sum_{a=0}^{\\lfloor \\lambda \\rfloor - 1} f(a) + \\sum_{a=\\lfloor \\lambda \\rfloor}^{\\lceil \\lambda \\rceil} f(a) + \\sum_{a = \\lceil \\lambda \\rceil + 1}^\\infty f(a) \\\\\n&\\leq \\int_0^{\\lfloor \\lambda \\rfloor} f(x) \\: dx + 2f(\\lambda) + \\int_{\\lceil \\lambda \\rceil}^\\infty f(x) \\: dx \\\\\n&\\leq \\int_0^\\infty f(x) \\: dx + 2f(\\lambda).\n\\end{split}\n\\end{equation*}\nUsing the substitution $y = -x \\log(1-q)$, we can evaluate the integral to be\n\\begin{equation*}\n\\begin{split}\n\\int_0^\\infty f(x) \\: dx\n&= \\int_0^\\infty x^p \\exp\\left(x \\cdot \\log (1-q) \\right) \\: dx\n= \\frac{1}{(-\\log(1-q))^{p+1}} \\: \\int_0^\\infty y^p \\: \\exp(-y) \\: dy \\\\\n&= \\frac{\\Gamma(p+1)}{(-\\log(1-q))^{p+1}}\n\\leq \\frac{p^p}{(-\\log(1-q))^{p+1}},\n\\end{split}\n\\end{equation*}\nwhere in the last step we have again used the relation $\\Gamma(p+1) \\leq p^p$.\nWe use the result above, along with the expression of $f(\\lambda)$, to bound the moment of $X$:\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}[|X|^p] &= \\sum_{a=0}^\\infty a^p \\cdot (1-q)^a \\: q\n= q \\: \\sum_{a=0}^\\infty f(a) \\\\\n&\\leq q \\: \\int_0^\\infty f(x) \\: dx + 2q \\: f(\\lambda) \\\\\n&\\leq \\left(\\frac{q^{1\/p} \\: p}{(-\\log(1-q))^{1+1\/p}} \\right)^p + \\left( \\frac{(2q)^{1\/p} \\: p}{-e \\cdot \\log (1-q)} \\right)^p \\\\\n&\\leq \\left( \\frac{q^{1\/p} \\: p}{(-\\log(1-q))^{1+1\/p}} + \\frac{(2q)^{1\/p} \\: p}{-e \\cdot \\log (1-q)} \\right)^p,\n\\end{split}\n\\end{equation*}\nwhere in the last step we have used the fact that $x^p + y^p \\leq (x+y)^p$ for $x,y \\geq 0$ and $p \\geq 1$. This gives us\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{p} \\: \\mathbb{E}[|X|^p]^{1\/p}\n&\\leq \\frac{q^{1\/p}}{(-\\log(1-q))^{1+1\/p}} + \\frac{2^{1\/p} \\: q^{1\/p}}{-e \\cdot \\log (1-q)} \\\\\n&= \\frac{1}{-\\log(1-q)} \\: \\left(\\left(\\frac{q}{-\\log(1-q)}\\right)^{1\/p} + \\frac{2^{1\/p} \\: q^{1\/p}}{e}\\right).\n\\end{split}\n\\end{equation*}\nNow note that $q \\leq -\\log(1-q)$ for $0 < q < 1$, so $(-q\/\\log(1-q))^{1\/p} \\leq 1$. Moreover, $(2q)^{1\/p} \\leq 2^{1\/p} \\leq 2$. Therefore, for any $p \\geq 1$, we have\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{p} \\: \\mathbb{E}[|X|^p]^{1\/p}\n&\\leq \\frac{1}{-\\log(1-q)} \\: \\left(1 + \\frac{2}{e}\\right)\n< \\frac{2}{-\\log(1-q)}.\n\\end{split}\n\\end{equation*}\nThus, we conclude that $X$ is sub-exponential with parameter $-2\/\\log(1-q)$.\n\\end{proof}\n\n\n\n\\subsubsection{Bound on the inverses of diagonally-dominant matrices}\n\nAn $n\\times n$ real matrix $J$ is {\\em diagonally dominant} if\n\\begin{equation*}\n\\Delta_i(J) := |J_{ii}| - \\sum_{j \\neq i} |J_{ij}| \\geq 0, \\quad \\text{for } i = 1,\\dots,n.\n\\end{equation*}\nWe say that $J$ is {\\em diagonally balanced} if $\\Delta_i(J) = 0$ for $i = 1,\\dots,n$. We have the following bound from~\\cite{HLW} on the inverses of diagonally dominant matrices. This bound is independent of $\\Delta_i$, so it is also applicable to diagonally balanced matrices. We will use this result in the proofs of Theorems~\\ref{Thm:ConsistencyCont} and~\\ref{Thm:ConsistencyInfiniteDisc}.\n\n\\begin{theorem}[{\\cite[Theorem~1.1]{HLW}}]\\label{Thm:Main}\nLet $n \\geq 3$. For any symmetric diagonally dominant matrix $J$ with $J_{ij} \\geq \\ell > 0$, we have\n\\begin{equation*}\n\\|J^{-1}\\|_\\infty \\leq \\frac{3n-4}{2\\ell(n-2)(n-1)}.\n\\end{equation*}\n\\end{theorem}\n\n\n\n\n\n\\subsection{Proofs for the finite discrete weighted graphs}\n\\label{Sec:ProofFiniteDisc}\n\n\nIn this section we present the proofs of the results presented in Section~\\ref{Sec:FiniteDiscrete}.\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:GraphicalFiniteDisc}}\n\n\nWe first prove the necessity of~\\eqref{Eq:GraphicalFiniteDisc}. Suppose $\\mathbf{d} = (d_1,\\dots,d_n)$ is the degree sequence of a graph $G$ with edge weights $a_{ij} \\in S$. Then $\\sum_{i=1}^n d_i = 2\\sum_{(i,j)} a_{ij}$ is even. Moreover, for each $1 \\leq k \\leq n$, $\\sum_{i=1}^k d_i$ counts the total edge weights coming out from the vertices $1,\\dots,k$. The total edge weights from these $k$ vertices to themselves is at most $(r-1)k(k-1)$, and for each vertex $j \\notin \\{1,\\dots,k\\}$, the total edge weights from these $k$ vertices to vertex $j$ is at most $\\min\\{d_j,(r-1)k\\}$, so by summing over $j \\notin \\{1,\\dots,k\\}$ we get~\\eqref{Eq:GraphicalFiniteDisc}.\n\nTo prove the sufficiency of~\\eqref{Eq:GraphicalFiniteDisc} we use induction on $s := \\sum_{i=1}^n d_i$. The base case $s = 0$ is trivial. Assume the statement holds for $s-2$, and suppose we have a sequence $\\mathbf{d}$ with $d_1 \\geq d_2 \\geq \\dots \\geq d_n$ satisfying~\\eqref{Eq:GraphicalFiniteDisc} with $\\sum_{i=1}^n d_i = s$. Without loss of generality we may assume $d_n \\geq 1$, for otherwise we can proceed with only the nonzero elements of $\\mathbf{d}$. Let $1 \\leq t \\leq n-1$ be the smallest index such that $d_t > d_{t+1}$, with $t = n-1$ if $d_1 = \\cdots = d_n$. Define $\\mathbf{d}' = (d_1,\\dots,d_{t-1},d_t-1,d_{t+1},\\dots,d_{n-1},d_n-1)$, so we have $d_1' = \\cdots = d_{t-1}' > d_t' \\geq d_{t+1}' \\geq \\cdots \\geq d_{n-1}' > d_n'$ and $\\sum_{i=1}^n d_i' = s-2$.\n\nWe will show that $\\mathbf{d}'$ satisfies~\\eqref{Eq:GraphicalFiniteDisc}. By the inductive hypothesis, this means $\\mathbf {d}'$ is the degree sequence of a graph $G'$ with edge weights $a_{ij}' \\in \\{0,1,\\dots,r-1\\}$. We now attempt to modify $G'$ to obtain a graph $G$ whose degree sequence is equal to $\\mathbf{d}$. If the weight $a_{tn}'$ of the edge $(t,n)$ is less than $r-1$, then we can obtain $G$ by increasing $a_{tn}'$ by $1$, since the degree of vertex $t$ is now $d_t'+1 = d_t$, and the degree of vertex $n$ is now $d_n'+1 = d_n$. Otherwise, suppose $a_{tn}' = r-1$. Since $d_t' = d_1'-1$, there exists a vertex $u \\neq n$ such that $a_{tu}' < r-1$. Since $d_u' > d_n'$, there exists another vertex $v$ such that $a_{uv}' > a_{vn}'$. Then we can obtain the graph $G$ by increasing $a_{tu}'$ and $a_{vn}'$ by $1$ and reducing $a_{uv}'$ by $1$, so that now the degrees of vertices $t$ and $n$ are each increased by $1$, and the degrees of vertices $u$ and $v$ are preserved.\n\nIt now remains to show that $\\mathbf{d}'$ satisfies~\\eqref{Eq:GraphicalFiniteDisc}. We divide the proof into several cases for different values of $k$. We will repeatedly use the fact that $\\mathbf{d}$ satisfies~\\eqref{Eq:GraphicalFiniteDisc}, as well as the inequality $\\min\\{a,b\\}-1 \\leq \\min\\{a-1,b\\}$.\n\\begin{enumerate}\n  \\item For $k = n$:\n\\begin{equation*}\n\\sum_{i=1}^n d_i' = \\sum_{i=1}^n d_i-2 \\leq (r-1)n(n-1)-2 < (r-1)n(n-1).\n\\end{equation*}\n  \\item For $t \\leq k \\leq n-1$:\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^k d_i' &= \\sum_{i=1}^k d_i-1 \\leq (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j,(r-1)k\\} - 1 \\\\\n&\\leq (r-1)k(k-1) + \\sum_{j=k+1}^{n-1} \\min\\{d_j,(r-1)k\\} + \\min\\{d_n-1,(r-1)k\\} \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j',(r-1)k\\}.\n\\end{split}\n\\end{equation*}\n  \\item For $1 \\leq k \\leq t-1$: first suppose $d_n \\geq 1+(r-1)k$. Then for all $j$ we have\n\\begin{equation*}\n\\min\\{d_j',(r-1)k\\} = \\min\\{d_j,(r-1)k\\} = (r-1)k,\n\\end{equation*}  \nso\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^k d_i' &= \\sum_{i=1}^k d_i \\leq (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j,(r-1)k\\} \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j',(r-1)k\\}.\n\\end{split}\n\\end{equation*}\n  \\item For $1 \\leq k \\leq t-1$: suppose $d_1 \\geq 1 + (r-1)k$, and $d_n \\leq (r-1)k$. We claim that $\\mathbf{d}$ satisfies~\\eqref{Eq:GraphicalFiniteDisc} at $k$ with a strict inequality. If this claim is true, then, since $d_t = d_1$ and $\\min\\{d_t',(r-1)k\\} = \\min\\{d_t,(r-1)k\\} = (r-1)k$,\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^k d_i' &= \\sum_{i=1}^k d_i\n\\leq (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j,(r-1)k\\}-1 \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^{n-1} \\min\\{d_j',(r-1)k\\} + \\min\\{d_n,(r-1)k\\}-1 \\\\\n&\\leq (r-1)k(k-1) + \\sum_{j=k+1}^{n-1} \\min\\{d_j',(r-1)k\\} + \\min\\{d_n-1,(r-1)k\\} \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j',(r-1)k\\}.\n\\end{split}\n\\end{equation*}\nNow to prove the claim, suppose the contrary that $\\mathbf{d}$ satisfies~\\eqref{Eq:GraphicalFiniteDisc} at $k$ with equality. Let $t+1 \\leq u \\leq n$ be the smallest integer such that $d_u \\leq (r-1)k$. Then, from our assumption,\n\\begin{equation*}\n\\begin{split}\nk d_k &= \\sum_{i=1}^k d_i = (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j,(r-1)k\\} \\\\\n&\\geq (r-1)k(k-1) + (u-k-1)(r-1)k + \\sum_{j=u}^n d_j \\\\\n&= (r-1)k(u-2) + \\sum_{j=u}^n d_j.\n\\end{split}\n\\end{equation*}\nTherefore, since $d_{k+1} = d_k = d_1$,\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^{k+1} d_i &= (k+1)d_k\n\\geq (r-1)(k+1)(u-2) + \\frac{k+1}{k} \\sum_{j=u}^n d_j \\\\\n&> (r-1)(k+1)k + (r-1)(k+1)(u-k-2) + \\sum_{j=u}^n d_j \\\\\n&\\geq (r-1)(k+1)k + \\sum_{j=k+2}^n \\min\\{d_j, (r-1)(k+1)\\},\n\\end{split}\n\\end{equation*}\nwhich contradicts the fact that $\\mathbf{d}$ satisfies~\\eqref{Eq:GraphicalFiniteDisc} at $k+1$. Thus, we have proved that $\\mathbf{d}$ satisfies~\\eqref{Eq:GraphicalFiniteDisc} at $k$ with a strict inequality.\n  \\item For $1 \\leq k \\leq t-1$: suppose $d_1 \\leq (r-1)k$. In particular, we have $\\min\\{d_j,(r-1)k\\} = d_j$ and $\\min\\{d_j',(r-1)k\\} = d_j'$ for all $j$. First, if we have\n\\begin{equation}\\label{Eq:ErdosGallaiProofCond}\nd_{k+2} + \\cdots + d_n \\geq 2,\n\\end{equation}\nthen we are done, since  \n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^k d_i' &= \\sum_{i=1}^k d_i\n= (k-1)d_1 + d_{k+1} \\\\\n&\\leq (r-1)k(k-1) + d_{k+1} + d_{k+2} + \\cdots + d_n - 2 \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^n d_j' \\\\\n&= (r-1)k(k-1) + \\sum_{j=k+1}^n \\min\\{d_j',(r-1)k\\}.\n\\end{split}\n\\end{equation*}\nCondition~\\eqref{Eq:ErdosGallaiProofCond} is obvious if $d_n \\geq 2$ or $k+2 \\leq n-1$ (since there are $n-k-1$ terms in the summation and each term is at least $1$). Otherwise, assume $k+2 \\geq n$ and $d_n = 1$, so in particular, we have $k = n-2$ (since $k \\leq t-1 \\leq n-2$), $t = n-1$, and $d_1 \\leq (r-1)(n-2)$. Note that we cannot have $d_1 = (r-1)(n-2)$, for then $\\sum_{i=1}^n d_i = (n-1)d_1 + d_n = (r-1)(n-1)(n-2) + 1$ would be odd, so we must have $d_1 < (r-1)(n-2)$. Similarly, $n$ must be even, for otherwise $\\sum_{i=1}^n d_i = (n-1)d_1 + 1$ would be odd. Thus, since $1 \\leq d_1 < (r-1)(n-2)$ we must have $n \\geq 4$. Therefore,\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^k d_i' &= (n-2)d_1 = (n-3)d_1 + d_{n-1} \\\\\n&\\leq (r-1)(n-2)(n-3) - (n-3) + d_{n-1} \\\\\n&\\leq (r-1)(n-2)(n-3) + (d_{n-1}-1) + (d_n-1) \\\\\n&= (r-1)(n-2)(n-3) + \\sum_{j=k+1}^n \\min\\{d_j',(r-1)k\\}.\n\\end{split}\n\\end{equation*}\n\\end{enumerate}\nThis shows that $\\mathbf{d}'$ satisfies~\\eqref{Eq:GraphicalFiniteDisc} and finishes the proof of Theorem~\\ref{Thm:GraphicalFiniteDisc}.\n\n\n\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:MLEAlgFiniteDisc}}\n\nWe follow the outline of the proof of~\\cite[Theorem~1.5]{Chatterjee}. We first present the following properties of the mean function $\\mu(t)$ and the Jacobian matrix of the function $\\varphi$~\\eqref{Eq:VarphiFiniteDisc}. We then combine these results at the end of this section into a proof of Theorem~\\ref{Thm:MLEAlgFiniteDisc}.\n\n\n\\begin{lemma}\\label{Lem:PropertyMeanFunction}\nThe mean function $\\mu(t)$ is positive and strictly decreasing, with $\\mu(-t) + \\mu(t) = r-1$ for all $t \\in \\mathbb{R}$, and $\\mu(t) \\to 0$ as $t \\to \\infty$. Its derivative $\\mu'(t)$ is increasing for $t \\geq 0$, with the properties that $\\mu'(t) < 0$, $\\mu'(t) = \\mu'(-t)$ for all $t \\in \\mathbb{R}$, and $\\mu'(0) = -(r^2-1)\/12$.\n\\end{lemma}\n\\begin{proof}\nIt is clear from~\\eqref{Eq:MeanFuncFiniteDiscrete} that $\\mu(t)$ is positive. From the alternative representation~\\eqref{Eq:MeanFuncFiniteDiscreteAlt} it is easy to see that $\\mu(-t) + \\mu(t) = r-1$, and $\\mu(t) \\to 0$ as $t \\to \\infty$. Differentiating expression~\\eqref{Eq:MeanFuncFiniteDiscrete} yields the formula\n\\begin{equation*}\n\\mu'(t)\n= \\frac{-(\\sum_{a=0}^{r-1} a^2 \\, \\exp(-at))(\\sum_{a=0}^{r-1} \\exp(-at)) + (\\sum_{a=0}^{r-1} a \\: \\exp(-at))^2}{(\\sum_{a=0}^{r-1} \\exp(-at))^2},\n\\end{equation*}\nand substituting $t = 0$ gives us $\\mu'(0) = -(r^2-1)\/12$. The Cauchy-Schwarz inequality applied to the expression above tells us that $\\mu'(t) < 0$, where the inequality is strict because the vectors $(a^2 \\exp(-at))_{a=0}^{r-1}$ and $(\\exp(-at))_{a=0}^{r-1}$ are not linearly dependent. Thus, $\\mu(t)$ is strictly decreasing for all $t \\in \\mathbb{R}$.\n\nThe relation $\\mu(-t) + \\mu(t) = r-1$ gives us $\\mu'(-t) = \\mu'(t)$. Furthermore, by differentiating the expression~\\eqref{Eq:MeanFuncFiniteDiscreteAlt} twice, one can verify that $\\mu''(t) \\geq 0$ for $t \\geq 0$, which means $\\mu'(t)$ is increasing for $t \\geq 0$. See also Figure~\\ref{Fig:FiniteDiscMean} for the behavior of $\\mu(t)$ and $\\mu'(t)$ for different values of $r$.\n\\end{proof}\n\n\\begin{lemma}\\label{Lem:LowerBoundRatio}\nFor all $t \\in \\mathbb{R}$, we have\n\\begin{equation*}\n\\frac{\\mu'(t)}{\\mu(t)} \\geq -r+1 + \\frac{1}{\\sum_{a=0}^{r-1} \\exp(at)} > -r+1.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nMultiplying the numerator and denominator of~\\eqref{Eq:MeanFuncFiniteDiscrete} by $\\exp((r-1)t)$, we can write\n\\begin{equation*}\n\\mu(t)\n= \\frac{\\sum_{a=0}^{r-1} a \\: \\exp((r-1-a)t)}{\\sum_{a=0}^{r-1} \\exp((r-1-a)t)}\n= \\frac{\\sum_{a=0}^{r-1} (r-1-a) \\exp(at)}{\\sum_{a=0}^{r-1} \\exp(at)}.\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n\\begin{split}\n\\frac{\\mu'(t)}{\\mu(t)} &= \\frac{d}{dt} \\log \\mu(t)\n= \\frac{d}{dt} \\left( \\log \\sum_{a=0}^{r-1} (r-1-a)\\exp(at) - \\log \\sum_{a=0}^{r-1} \\exp(at) \\right) \\\\\n&= \\frac{\\sum_{a=0}^{r-1} a(r-1-a)\\exp(at)}{\\sum_{a=0}^{r-1} (r-1-a)\\exp(at)} - \\frac{\\sum_{a=0}^{r-1} a\\exp(at)}{\\sum_{a=0}^{r-1} \\exp(at)} \\\\\n&\\geq - \\frac{\\sum_{a=0}^{r-1} a\\exp(at)}{\\sum_{a=0}^{r-1} \\exp(at)} \\\\\n&= - \\frac{\\sum_{a=0}^{r-1} (r-1)\\exp(at) - \\sum_{a=0}^{r-1} (r-1-a)\\exp(at)}{\\sum_{a=0}^{r-1} \\exp(at)} \\\\\n&= -r+1 + \\frac{\\sum_{a=0}^{r-1} (r-1-a)\\exp(at)}{\\sum_{a=0}^{r-1} \\exp(at)} \\\\\n&\\geq -r+1+ \\frac{1}{\\sum_{a=0}^{r-1} \\exp(at)}.\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\nWe recall the following definition and result from~\\cite{Chatterjee}. Given $\\delta > 0$, let $\\mathcal{L}_n(\\delta)$ denote the set of $n \\times n$ matrices $A = (a_{ij})$ with $\\|A\\|_\\infty \\leq 1$, $a_{ii} \\geq \\delta$, and $a_{ij} \\leq -\\delta\/(n-1)$, for each $1 \\leq i \\neq j \\leq n$.\n\n\\begin{lemma}[{\\cite[Lemma~2.1]{Chatterjee}}]\\label{Lem:LnDelta}\nIf $A,B \\in \\mathcal{L}_n(\\delta)$, then\n\\begin{equation*}\n\\|AB\\|_\\infty \\leq 1-\\frac{2(n-2)\\delta^2}{(n-1)}.\n\\end{equation*}\nIn particular, for $n \\geq 3$,\n\\begin{equation*}\n\\|AB\\|_\\infty \\leq 1-\\delta^2.\n\\end{equation*}\n\\end{lemma}\n\nGiven $\\theta,\\theta' \\in \\mathbb{R}^n$, let $J(\\theta,\\theta')$ denote the $n \\times n$ matrix whose $(i,j)$-entry is\n\\begin{equation}\\label{Eq:JacobianMLEDef}\nJ_{ij}(\\theta,\\theta') = \\int_0^1 \\frac{\\partial \\varphi_i}{\\partial \\theta_j} (t\\theta + (1-t)\\theta') \\: dt.\n\\end{equation}\n\n\\begin{lemma}\\label{Lem:NormJInfinity}\nFor all $\\theta,\\theta' \\in \\mathbb{R}^n$, we have $\\|J(\\theta,\\theta')\\|_\\infty = 1$.\n\\end{lemma}\n\\begin{proof}\nThe partial derivatives of $\\varphi$~\\eqref{Eq:VarphiFiniteDisc} are\n\\begin{equation}\\label{Eq:PartialVarphi}\n\\frac{\\partial \\varphi_i(\\mathbf{x})}{\\partial x_i} = 1 + \\frac{1}{(r-1)}\\frac{\\sum_{j \\neq i} \\mu'(x_i+x_j)}{\\sum_{j \\neq i} \\mu(x_i+x_j)},\n\\end{equation}\nand for $i \\neq j$,\n\\begin{equation}\\label{Eq:PartialVarphiMixed}\n\\frac{\\partial \\varphi_i(\\mathbf{x})}{\\partial x_j} = \\frac{1}{(r-1)}\\frac{\\mu'(x_i+x_j)}{\\sum_{k \\neq i} \\mu(x_i+x_k)} < 0,\n\\end{equation}\nwhere the last inequality follows from $\\mu'(x_i + x_j) < 0$. Using the result of Lemma~\\ref{Lem:LowerBoundRatio} and the fact that $\\mu$ is positive, we also see that\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial \\varphi_i(\\mathbf{x})}{\\partial x_i}\n&= 1 + \\frac{1}{(r-1)} \\frac{\\sum_{j \\neq i} \\mu'(x_i+x_j)}{\\sum_{j \\neq i} \\mu(x_i+x_j)}\n> 1 + \\frac{1}{(r-1)} \\frac{\\sum_{j \\neq i} (-r+1) \\mu(x_i+x_j)}{\\sum_{j \\neq i} \\mu(x_i+x_j)}\n= 0.\n\\end{split}\n\\end{equation*}\nSetting $\\mathbf{x} = t\\theta + (1-t)\\theta'$ and integrating over $0 \\leq t \\leq 1$, we also get that $J_{ij}(\\theta,\\theta') < 0$ for $i \\neq j$, and $J_{ii}(\\theta,\\theta') = 1 + \\sum_{j \\neq i} J_{ij}(\\theta,\\theta') > 0$. This implies $\\|J(\\theta,\\theta')\\|_\\infty = 1$, as desired.\n\\end{proof}\n\n\n\\begin{lemma}\\label{Lem:JInLnDelta}\nLet $\\theta,\\theta' \\in \\mathbb{R}^n$ with $\\|\\theta\\|_\\infty \\leq K$ and $\\|\\theta'\\|_\\infty \\leq K$ for some $K > 0$. Then $J(\\theta,\\theta') \\in \\mathcal{L}_n(\\delta)$, where\n\\begin{equation}\\label{Eq:DeltaBoundJ}\n\\delta = \\frac{1}{(r-1)} \\: \\min \\left\\{\\frac{\\exp(2K)-1}{\\exp(2rK)-1}, \\; -\\frac{\\mu'(2K)}{\\mu(-2K)} \\right\\}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFrom Lemma~\\ref{Lem:NormJInfinity} we already know that $J \\equiv J(\\theta,\\theta')$ satisfies $\\|J\\|_\\infty = 1$, so to show that $J\\in \\mathcal{L}_n(\\delta)$ it remains to show that $J_{ii} \\geq \\delta$ and $J_{ij} \\leq -\\delta\/(n-1)$ for $i \\neq j$. In particular, it suffices to show that for each $0 \\leq t \\leq 1$ we have $\\partial \\varphi_i(\\mathbf{x})\/\\partial x_i \\geq \\delta$ and $\\partial \\varphi_i(\\mathbf{x})\/\\partial x_j \\leq -\\delta\/(n-1)$, where $\\mathbf{x} \\equiv \\mathbf{x}(t) = t\\theta + (1-t)\\theta'$.\n\nFix $0 \\leq t \\leq 1$. Since $\\|\\theta\\|_\\infty \\leq K$ and $\\|\\theta'\\|_\\infty \\leq K$, we also know that $\\|\\mathbf{x}\\|_\\infty \\leq K$, so $-2K \\leq x_i+x_j \\leq 2K$ for all $1 \\leq i,j \\leq n$. Using the properties of $\\mu$ and $\\mu'$ from Lemma~\\ref{Lem:PropertyMeanFunction}, we have\n\\begin{equation*}\n0 < \\mu(2K) \\leq \\mu(x_i+x_j) \\leq \\mu(-2K)\n\\end{equation*}\nand\n\\begin{equation*}\n\\mu'(0) \\leq \\mu'(x_i+x_j) \\leq \\mu'(2K) < 0.\n\\end{equation*}\nThen from~\\eqref{Eq:PartialVarphiMixed} and using the definition of $\\delta$,\n\\begin{equation*}\n\\frac{\\partial \\varphi_i(\\mathbf{x})}{\\partial x_j} \\leq \\frac{\\mu'(2K)}{(n-1)(r-1)\\mu(-2K)} \\leq -\\frac{\\delta}{n-1}.\n\\end{equation*}\nFurthermore, by Lemma~\\ref{Lem:LowerBoundRatio} we have\n\\begin{equation*}\n\\frac{\\mu'(x_i+x_j)}{\\mu(x_i+x_j)} \\geq -r+1 + \\frac{\\exp(x_i+x_j)-1}{\\exp(r(x_i+x_j))-1}\n\\geq -r+1 + \\frac{\\exp(2K)-1}{\\exp(2rK)-1}.\n\\end{equation*}\nSo from~\\eqref{Eq:PartialVarphi}, we also get\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial \\varphi_i(\\mathbf{x})}{\\partial x_i}\n&\\geq 1 + \\frac{1}{(r-1)} \\frac{\\sum_{j \\neq i} (-r+1+\\frac{\\exp(2K)-1}{\\exp(2rK)-1}) \\mu(x_i+x_j)}{\\sum_{j \\neq i} \\mu(x_i+x_j)}\n= \\frac{1}{(r-1)} \\left( \\frac{\\exp(2K)-1}{\\exp(2rK)-1} \\right)\n\\geq \\delta,\n\\end{split}\n\\end{equation*}\nas required.\n\\end{proof}\n\n\n\nWe are now ready to prove Theorem~\\ref{Thm:MLEAlgFiniteDisc}.\n\n\\begin{proof_of}{Theorem~\\ref{Thm:MLEAlgFiniteDisc}}\nBy the mean-value theorem for vector-valued functions~\\cite[p.~341]{Lang}, for any $\\theta,\\theta' \\in \\mathbb{R}^n$ we can write\n\\begin{equation*}\n\\varphi(\\theta) - \\varphi(\\theta') = J(\\theta, \\theta') (\\theta-\\theta'),\n\\end{equation*}\nwhere $J(\\theta,\\theta')$ is the Jacobian matrix defined in~\\eqref{Eq:JacobianMLEDef}. Since $\\|J(\\theta,\\theta')\\|_\\infty = 1$ (Lemma~\\ref{Lem:NormJInfinity}), this gives us\n\\begin{equation}\\label{Eq:FiniteDiscMVTBoundOneIter}\n\\|\\varphi(\\theta) - \\varphi(\\theta')\\|_\\infty \\leq \\|\\theta-\\theta'\\|_\\infty.\n\\end{equation}\n\nFirst suppose there is a solution $\\hat \\theta$ to the system of equations~\\eqref{Eq:MLEEqFiniteDisc}, so $\\hat \\theta$ is a fixed point of $\\varphi$. Then by setting $\\theta = \\theta^{(k)}$ and $\\theta' = \\hat \\theta$ to the inequality above, we obtain\n\\begin{equation}\\label{Eq:VarphiContraction}\n\\|\\theta^{(k+1)}-\\hat\\theta\\|_\\infty \\leq \\|\\theta^{(k)}-\\hat\\theta\\|_\\infty.\n\\end{equation}\nIn particular, this shows that $\\|\\theta^{(k)}\\|_\\infty \\leq K$ for all $k \\in \\mathbb{N}_0$, where $K := 2\\|\\hat\\theta\\|_\\infty + \\|\\theta^{(0)}\\|_\\infty$. By Lemma~\\ref{Lem:JInLnDelta}, this implies $J(\\theta^{(k)},\\hat \\theta) \\in \\mathcal{L}_n(\\delta)$ for all $k \\in \\mathbb{N}_0$, where $\\delta$ is given by~\\eqref{Eq:DeltaBoundJ}. Another application of the mean-value theorem gives us\n\\begin{equation*}\n\\begin{split}\n\\theta^{(k+2)}-\\hat \\theta \n&= J(\\theta^{(k+1)}, \\hat\\theta) \\: J(\\theta^{(k)}, \\hat\\theta) \\: (\\theta^{(k)} - \\hat\\theta),\n\\end{split}\n\\end{equation*}\nso by Lemma~\\ref{Lem:LnDelta},\n\\begin{equation*}\n\\begin{split}\n\\|\\theta^{(k+2)}-\\hat \\theta\\|_\\infty\n&\\leq \\|J(\\theta^{(k+1)}, \\hat\\theta) \\: J(\\theta^{(k)}, \\hat\\theta)\\|_\\infty \\: \\|\\theta^{(k)} - \\hat\\theta\\|_\\infty\n\\leq \\left(1-\\delta^2\\right) \\|\\theta^{(k)} - \\hat\\theta\\|_\\infty.\n\\end{split}\n\\end{equation*}\nUnrolling the recursive bound above and using~\\eqref{Eq:VarphiContraction} gives us\n\\begin{equation*}\n\\|\\theta^{(k)}-\\hat\\theta\\|_\\infty \\leq (1-\\delta^2)^{\\lfloor k\/2 \\rfloor} \\|\\theta^{(0)}-\\hat\\theta\\|_\\infty \\leq (1-\\delta^2)^{(k-1)\/2} \\|\\theta^{(0)}-\\hat\\theta\\|_\\infty,\n\\end{equation*}\nwhich proves~\\eqref{Eq:MLERateOfConvFiniteDisc} with $\\tau = \\sqrt{1-\\delta^2}$.\n\nNow suppose the system of equations~\\eqref{Eq:MLEEqFiniteDisc} does not have a solution, and suppose the contrary that the sequence $\\{\\theta^{(k)}\\}$ does not have a divergent subsequence. This means $\\{\\theta^{(k)}\\}$ is a bounded sequence, so there exists $K > 0$ such that $\\|\\theta^{(k)}\\|_\\infty \\leq K$ for all $k \\in \\mathbb{N}_0$. Then by Lemma~\\ref{Lem:JInLnDelta}, $J(\\theta^{(k)},\\theta^{(k+1)}) \\in \\mathcal{L}_n(\\delta)$ for all $k \\in \\mathbb{N}_0$, where $\\delta$ is given by~\\eqref{Eq:DeltaBoundJ}. In particular, by the mean value theorem and Lemma~\\ref{Lem:NormJInfinity}, we get for all $k \\in \\mathbb{N}_0$,\n\\begin{equation*}\n\\|\\theta^{(k+3)}-\\theta^{(k+2)}\\|_\\infty \\leq (1-\\delta^2) \\|\\theta^{(k+1)}-\\theta^{(k)}\\|_\\infty.\n\\end{equation*}\nThis implies $\\sum_{k=0}^\\infty \\|\\theta^{(k+1)}-\\theta^{(k)}\\|_\\infty < \\infty$, which means $\\{\\theta^{(k)}\\}$ is a Cauchy sequence. Thus, the sequence $\\{\\theta^{(k)}\\}$ converges to a limit, say $\\hat \\theta$, as $k \\to \\infty$. This limit $\\hat \\theta$ is necessarily a fixed point of $\\varphi$, as well as a solution to the system of equations~\\eqref{Eq:MLEEqFiniteDisc}, contradicting our assumption. Hence we conclude that $\\{\\theta^{(k)}\\}$ must have a divergent subsequence.\n\\end{proof_of}\n\n\nA little computation based on the proof above gives us the following result, which will be useful in the proof of Theorem~\\ref{Thm:ConsistencyFiniteDisc}.\n\n\\begin{proposition}\\label{Prop:FiniteDiscBoundFirstIter}\nAssume the same setting as in Theorem~\\ref{Thm:MLEAlgFiniteDisc}, and assume the MLE equation~\\ref{Eq:MLEEqFiniteDisc} has a unique solution $\\hat \\theta$. Then\n\\begin{equation*}\n\\|\\theta^{(0)}-\\hat\\theta\\|_\\infty \\leq \\frac{2}{\\delta^2} \\|\\theta^{(0)}-\\theta^{(1)}\\|_\\infty,\n\\end{equation*}\nwhere $\\delta$ is given by~\\eqref{Eq:DeltaBoundJ} with $K = 2\\|\\hat\\theta\\|_\\infty + \\|\\theta^{(0)}\\|_\\infty$.\n\\end{proposition}\n\\begin{proof}\nWith the same notation as in the proof of Theorem~\\ref{Thm:MLEAlgFiniteDisc}, by applying the mean-value theorem twice and using the bound in Lemma~\\ref{Lem:NormJInfinity}, for each $k \\geq 0$ we have\n\\begin{equation*}\n\\|\\theta^{(k+3)}-\\theta^{(k+2)}\\|_\\infty \\leq (1-\\delta^2) \\|\\theta^{(k+1)}-\\theta^{(k)}\\|_\\infty.\n\\end{equation*}\nTherefore, since $\\{\\theta^{(k)}\\}$ converges to $\\hat \\theta$,\n\\begin{equation*}\n\\begin{split}\n\\|\\theta^{(0)}-\\hat\\theta\\|_\\infty &\\leq \\sum_{k=0}^\\infty \\|\\theta^{(k)}-\\theta^{(k+1)}\\|_\\infty\n\\leq \\frac{1}{\\delta^2} \\big( \\|\\theta^{(0)}-\\theta^{(1)}\\|_\\infty + \\|\\theta^{(1)}-\\theta^{(2)}\\|_\\infty \\big)\n\\leq \\frac{2}{\\delta^2} \\|\\theta^{(0)}-\\theta^{(1)}\\|_\\infty,\n\\end{split}\n\\end{equation*}\nwhere the last inequality follows from~\\eqref{Eq:FiniteDiscMVTBoundOneIter}.\n\\end{proof}\n\n\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:ConsistencyFiniteDisc}}\n\nOur proof of Theorem~\\ref{Thm:ConsistencyFiniteDisc} follows the outline of the proof of Theorem~1.3 in~\\cite{Chatterjee}. Recall that $\\mathcal{W}$ is the set of graphical sequences, and the MLE equation~\\eqref{Eq:MLEEqFiniteDisc} has a unique solution $\\hat \\theta \\in \\mathbb{R}^n$ if and only if $\\mathbf{d} \\in \\text{conv}(\\mathcal{W})^\\circ$. We first present a few preliminary results. We will also use the properties of the mean function $\\mu$ as described in Lemma~\\ref{Lem:PropertyMeanFunction}.\n\nThe following property is based on~\\cite[Lemma~4.1]{Chatterjee}.\n\\begin{lemma}\\label{Lem:FiniteDiscBoundMLE}\nLet $\\mathbf{d} \\in \\text{conv}(\\mathcal{W})$ with the properties that\n\\begin{equation}\\label{Eq:LemFiniteDiscBoundMLE1}\nc_2 (r-1)(n-1) \\leq d_i \\leq c_1 (r-1)(n-1), \\quad i = 1,\\dots,n,\n\\end{equation}\nand\n\\begin{equation}\\label{Eq:LemFiniteDiscBoundMLE}\n\\min_{\\substack{B \\subseteq \\{1,\\dots,n\\},\\\\|B| \\geq c_2^2 (n-1)}} \\left\\{ \\sum_{j \\notin B} \\min\\{d_j, (r-1)|B|\\} + (r-1)|B| (|B|-1) - \\sum_{i \\in B} d_i \\right\\} \\geq c_3 n^2,\n\\end{equation}\nwhere $c_1, c_2 \\in (0,1)$ and $c_3 > 0$ are constants. Then the MLE equation~\\eqref{Eq:MLEEqFiniteDisc} has a solution $\\hat \\theta$ with the property that $\\|\\hat\\theta\\|_\\infty \\leq C$, where $C \\equiv C(c_1,c_2,c_3)$ is a constant that only depends on $c_1,c_2,c_3$.\n\\end{lemma}\n\\begin{proof}\nFirst assume $\\hat \\theta$ exists, so $\\hat \\theta$ and $\\mathbf{d}$ satisfy\n\\begin{equation*}\nd_i = \\sum_{j \\neq i} \\mu(\\hat \\theta_i + \\hat \\theta_j), \\quad i = 1,\\dots,n.\n\\end{equation*}\nLet\n\\begin{equation*}\nd_{\\max} = \\max_{1 \\leq i \\leq n} d_i, \\quad d_{\\min} = \\min_{1 \\leq i \\leq n} d_i, \\quad \\hat{\\theta}_{\\max} = \\max_{1 \\leq i \\leq n} \\hat \\theta_i, \\quad \\hat \\theta_{\\min} = \\min_{1 \\leq i \\leq n} \\hat \\theta_i,\n\\end{equation*}\nand let $i^*, j^* \\in \\{1,\\dots,n\\}$ be such that $\\hat \\theta_{i^*} = \\hat \\theta_{\\max}$ and $\\hat \\theta_{j^*} = \\hat \\theta_{\\min}$.\n\nWe begin by observing that since $\\mu$ is a decreasing function and we have the assumption~\\eqref{Eq:LemFiniteDiscBoundMLE1},\n\\begin{equation*}\nc_2(r-1) \\leq \\frac{d_{\\min}}{n-1} \\leq \\frac{d_{i^\\ast}}{n-1} = \\frac{1}{(n-1)} \\sum_{j \\neq i^\\ast} \\mu(\\hat \\theta_{\\max} + \\hat \\theta_j) \\leq \\mu(\\hat \\theta_{\\max} + \\hat \\theta_{\\min}),\n\\end{equation*}\nso\n\\begin{equation*}\n\\hat \\theta_{\\max} + \\hat \\theta_{\\min} \\leq \\mu^{-1}\\big(c_2(r-1)\\big).\n\\end{equation*}\nThus, if we have a lower bound on $\\hat \\theta_{\\min}$ by a constant, then we also get a constant upper bound on $\\hat \\theta_{\\max}$ and we are done.\n\nWe now proceed to prove the lower bound $\\hat \\theta_{\\min} \\geq -C$. If $\\theta_{\\min} \\geq 0$, then there is nothing to prove, so let us assume that $\\hat \\theta_{\\min} < 0$. We claim the following property.\n\n\\paragraph{Claim.} \\textit{If $\\hat \\theta_{\\min}$ satisfies $\\mu(\\hat \\theta_{\\min}\/2) \\geq c_1(r-1)$ and $\\mu(\\hat \\theta_{\\min}\/4) \\geq (r-1)\/(1+c_2)$, then the set $A = \\{i \\colon \\hat \\theta_i \\leq \\hat \\theta_{\\min}\/4\\}$ has $|A| \\geq c_2^2 (n-1)$.}\n\\vspace{2mm}\\\\\n\\begin{proof_of}{claim}\nLet $S = \\{i \\colon \\hat \\theta_i < -\\hat \\theta_{\\min}\/2\\}$ and $m = |S|$. Note that $j^\\ast \\in S$ since $\\hat \\theta_{j^*} = \\hat \\theta_{\\min} < 0$, so $|m| \\geq 1$. Then using the property that $\\mu$ is a decreasing function and the assumption on $\\mu(\\hat \\theta_{\\min}\/2)$, we obtain\n\\begin{equation*}\n\\begin{split}\nc_1(r-1)(n-1) &\\geq d_{\\max} \\geq d_{j^\\ast}\n= \\sum_{i \\neq j^\\ast} \\mu(\\hat \\theta_{\\min} + \\hat \\theta_i) \\\\\n&\\geq \\sum_{i \\in S \\setminus \\{j^*\\}} \\mu(\\hat \\theta_{\\min} + \\hat \\theta_i)\n> (m-1) \\: \\mu\\left(\\frac{\\hat \\theta_{\\min}}{2} \\right)\n\\geq c_1(r-1)(m-1).\n\\end{split}\n\\end{equation*}\nThis implies $m < n$, which means there exists $i \\notin S$, so $\\hat \\theta_i \\geq -\\hat \\theta_{\\min}\/2 > 0$.\n\nLet $S_i = \\{j \\colon j \\neq i, \\hat \\theta_j > -\\hat \\theta_i\/2\\}$, and let $m_i = |S_i|$. Then, using the properties that $\\mu$ is decreasing and bounded above by $r-1$, and using the assumption on $\\mu(\\hat \\theta_{\\min}\/4)$, we get\n\\begin{equation*}\n\\begin{split}\nc_2(r-1)(n-1) &\\leq d_{\\min} \\leq d_i\n= \\sum_{j \\in S_i} \\mu(\\hat \\theta_i + \\hat \\theta_j) + \\sum_{j \\notin S_i, j \\neq i} \\mu(\\hat \\theta_i + \\hat \\theta_j) \\\\\n&< m_i \\: \\mu\\left(\\frac{\\hat \\theta_i}{2}\\right) + (n-1-m_i)(r-1) \\\\\n&= (n-1)(r-1)-m_i \\left(r-1-\\mu\\left(\\frac{\\hat \\theta_i}{2}\\right) \\right) \\\\\n&= (n-1)(r-1)-m_i \\: \\mu\\left(-\\frac{\\hat \\theta_i}{2}\\right) \\\\\n&\\leq (n-1)(r-1) - m_i \\: \\mu\\left(\\frac{\\hat \\theta_{\\min}}{4}\\right) \\\\\n&\\leq (n-1)(r-1) - \\frac{m_i(r-1)}{1+c_2}.\n\\end{split}\n\\end{equation*}\nRearranging the last inequality above gives us $m_i \\leq (1-c_2^2)(n-1)$.\n\nNote that for every $j \\neq S_i$, $j \\neq i$, we have $\\hat \\theta_j \\leq -\\hat \\theta_i\/2 \\leq \\hat \\theta_{\\min}\/4$. Therefore, if $A = \\{j \\colon \\hat \\theta_j \\leq \\hat \\theta_{\\min} \/4\\}$, then we see that $S_i^c \\setminus \\{i\\} \\subseteq A$, so\n\\begin{equation*}\n|A| \\geq |S_i^c \\setminus \\{i\\}| = n-m_i-1 \\geq c_2^2 (n-1),\n\\end{equation*}\nas desired.\n\\end{proof_of}\n\\vspace{4mm}\n\n\nNow assume\n\\begin{equation*}\n\\hat \\theta_{\\min} \\leq \\min \\left\\{ 2\\mu^{-1}(c_1(r-1)), \\; 4\\mu^{-1}\\left(\\frac{r-1}{1+c_2}\\right), \\; -16\\right\\},\n\\end{equation*}\nfor otherwise we are done. Then $\\mu(\\hat \\theta_{\\min}\/2) \\geq c_1(r-1)$ and $\\mu(\\hat \\theta_{\\min}\/4) \\geq (r-1)\/(1+c_2)$, so by the claim above, the size of the set $A = \\{i \\colon \\hat \\theta_i \\leq \\hat \\theta_{\\min}\/4\\}$ is at least $c_2^2 (n-1)$. Let\n\\begin{equation*}\nh = \\sqrt{-\\hat \\theta_{\\min}} > 0,\n\\end{equation*}\nand for integers $0 \\leq k \\leq \\lceil h\/16 \\rceil - 1$, define the set\n\\begin{equation*}\nD_k = \\left\\{i \\colon -\\frac{1}{8} \\hat \\theta_{\\min} + kh \\leq \\hat \\theta_i < -\\frac{1}{8} \\hat \\theta_{\\min} + (k+1)h \\right\\}.\n\\end{equation*}\nSince the sets $\\{D_k\\}$ are disjoint, by the pigeonhole principle we can find an index $0 \\leq k^\\ast \\leq \\lceil h\/16 \\rceil - 1$ such that\n\\begin{equation*}\n|D_{k^\\ast}| \\leq \\frac{n}{\\lceil h\/16 \\rceil} \\leq \\frac{16n}{h}.\n\\end{equation*}\nFix $k^*$, and consider the set\n\\begin{equation*}\nB = \\left\\{ i \\colon \\hat \\theta_i \\leq \\frac{1}{8} \\hat \\theta_{\\min} - \\left(k^* + \\frac{1}{2} \\right) h \\right\\}.\n\\end{equation*}\nNote that $\\hat \\theta_{\\min}\/4 \\leq \\hat \\theta_{\\min}\/8 - (k^* + 1\/2)h$, which implies $A \\subseteq B$, so $|B| \\geq |A| \\geq c_2^2 (n-1)$. For $1 \\leq i \\leq n$, define\n\\begin{equation*}\nd_i^B = \\sum_{j \\in B\\setminus\\{i\\}} \\mu(\\hat \\theta_i + \\hat \\theta_j),\n\\end{equation*}\nand observe that\n\\begin{equation}\\label{Eq:DjBRelation}\n\\sum_{j \\notin B} d_j^B = \\sum_{j \\notin B} \\sum_{i \\in B} \\mu(\\hat \\theta_i + \\hat \\theta_j)\n= \\sum_{i \\in B} (d_i - d_i^B).\n\\end{equation}\n\nWe note that for $i \\in B$ we have $\\hat \\theta_i \\leq \\hat \\theta_{\\min}\/8$, so\n\\begin{equation}\\label{Eq:SumIinB}\n\\begin{split}\n(r-1)|B|(|B|-1) - \\sum_{i \\in B} d_i^B\n&= \\sum_{i \\in B} \\sum_{j \\in B \\setminus \\{i\\}} \\left(r-1-\\mu(\\hat \\theta_i + \\hat \\theta_j)\\right) \\\\\n&\\leq |B|(|B|-1)\\left(r-1-\\mu\\left(\\frac{\\hat\\theta_{\\min}}{4}\\right) \\right) \\\\\n&= |B|(|B|-1) \\: \\mu\\left(-\\frac{\\hat \\theta_{\\min}}{4}\\right)\n\\leq n^2 \\mu\\left(\\frac{h^2}{4}\\right),\n\\end{split}\n\\end{equation}\nwhere in the last inequality we have used the definition $h^2 = -\\hat \\theta_{\\min} > 0$.\nNow take $j \\notin B$, so $\\hat \\theta_j > \\hat \\theta_{\\min}\/8 - (k^*+1\/2)h$. We consider three cases:\n\\begin{enumerate}\n  \\item If $\\hat \\theta_j \\geq -\\hat \\theta_{\\min}\/8 + (k^*+1)h$, then for every $i \\notin B$, we have $\\hat \\theta_i + \\hat \\theta_j \\geq h\/2$, so\n\\begin{equation*}\n\\min\\{d_j, \\: (r-1)|B|\\} - d_j^B\n\\leq d_j - d_j^B\n= \\sum_{i \\notin B, i \\neq j} \\mu(\\hat \\theta_j + \\hat \\theta_i)\n\\leq n \\mu\\left(\\frac{h}{2}\\right).\n\\end{equation*}\n  \\item If $\\hat \\theta_j \\leq -\\hat \\theta_{\\min}\/8 + k^* h$, then for every $i \\in B$, we have $\\hat \\theta_i + \\hat \\theta_j \\leq -h\/2$, so\n\\begin{equation*}\n\\begin{split}\n\\min\\{d_j, \\: (r-1)|B|\\} - d_j^B\n&\\leq (r-1)|B| - \\sum_{i \\in B} \\mu(\\hat \\theta_j + \\hat \\theta_i) \\\\\n&\\leq (r-1)|B| - |B| \\: \\mu\\left(-\\frac{h}{2}\\right)\n= |B| \\: \\mu\\left(\\frac{h}{2}\\right)\n\\leq n \\mu\\left(\\frac{h}{2}\\right).\n\\end{split}\n\\end{equation*}\n  \\item If $-\\hat \\theta_{\\min}\/8 + k^* h \\leq \\hat \\theta_j \\leq -\\hat \\theta_{\\min}\/8 + (k^*+1)h$, then $j \\in D_{k^*}$, and in this case\n\\begin{equation*}\n\\min\\{d_j, \\: (r-1)|B|\\} - d_j^B \\leq (r-1)|B| \\leq n(r-1).\n\\end{equation*}  \n\\end{enumerate}\nThere are at most $n$ such indices $j$ in both the first and second cases above, and there are at most $|D_k^*| \\leq 16n\/h$ such indices $j$ in the third case. Therefore,\n\\begin{equation*}\n\\sum_{j \\notin B} \\big( \\min\\{d_j, \\: (r-1)|B|\\} - d_j^B \\big) \\leq n^2 \\mu\\left(\\frac{h}{2}\\right) + \\frac{16n^2(r-1)}{h}.\n\\end{equation*}\nCombining this bound with~\\eqref{Eq:SumIinB} and using~\\eqref{Eq:DjBRelation} give us\n\\begin{equation*}\n\\sum_{j \\notin B} \\min\\{d_j, (r-1)|B|\\} + (r-1)|B| (|B|-1) - \\sum_{i \\in B} d_i \\leq n^2 \\mu\\left(\\frac{h}{2}\\right) + \\frac{16n^2(r-1)}{h} + n^2 \\mu\\left(\\frac{h^2}{4}\\right).\n\\end{equation*}\nAssumption~\\eqref{Eq:LemFiniteDiscBoundMLE} tells us that the left hand side of the inequality above is bounded below by $c_3 n^2$, so we obtain\n\\begin{equation*}\n\\mu \\left(\\frac{h}{2}\\right) + \\frac{16(r-1)}{h} + \\mu\\left(\\frac{h^2}{4}\\right) \\geq c_3.\n\\end{equation*}\nThe left hand side is a decreasing function of $h > 0$, so the bound above tells us that $h \\leq C(c_3)$ for a constant $C(c_3)$ that only depends on $c_3$ (and $r$), and so $\\hat \\theta_{\\min} = -h^2 \\geq -C(c_3)^2$, as desired.\n\n\\paragraph{Showing existence of $\\hat \\theta$.}\nNow let $\\mathbf{d} \\in \\text{conv}(\\mathcal{W})$ satisfy~\\eqref{Eq:LemFiniteDiscBoundMLE1} and~\\eqref{Eq:LemFiniteDiscBoundMLE}. Let $\\{\\mathbf{d}^{(k)}\\}_{k \\geq 0}$ be a sequence of points in $\\text{conv}(\\mathcal{W})^\\circ$ converging to $\\mathbf{d}$, so by Proposition~\\ref{Prop:RegularMinimal}, for each $k \\geq 0$ there exists a solution $\\hat \\theta^{(k)} \\in \\mathbb{R}^n$ to the MLE equation~\\eqref{Eq:MLEEqFiniteDisc} with $\\mathbf{d}^{(k)}$ in place of $\\mathbf{d}$. Since $\\mathbf{d}$ satisfy~\\eqref{Eq:LemFiniteDiscBoundMLE1},~\\eqref{Eq:LemFiniteDiscBoundMLE}, and $\\mathbf{d}^{(k)} \\to \\mathbf{d}$, for all sufficiently large $k$, $\\mathbf{d}^{(k)}$ also satisfy~\\eqref{Eq:LemFiniteDiscBoundMLE1} and~\\eqref{Eq:LemFiniteDiscBoundMLE} with some constants $c_1', c_2', c_3'$ depending on $c_1, c_2, c_3$. The preceding analysis then shows that $\\|\\hat \\theta^{(k)}\\|_\\infty \\leq C$ for all sufficiently large $k$, where $C \\equiv C(c_1',c_2',c_3') = C(c_1,c_2,c_3)$ is a constant depending on $c_1, c_2, c_3$. This means $\\{\\hat \\theta^{(k)}\\}_{k \\geq 0}$ is a bounded sequence, so it contains a convergent subsequence $\\{\\hat \\theta^{(k_i)}\\}_{k_i \\geq 0}$, say $\\hat \\theta^{(k_i)} \\to \\hat \\theta$. Then $\\|\\hat \\theta\\|_\\infty \\leq C$, and since $\\hat \\theta^{(k_i)}$ is a solution to the MLE equation~\\eqref{Eq:MLEEqFiniteDisc} for $\\mathbf{d}^{(k_i)}$, $\\hat \\theta$ is necessarily a solution to~\\eqref{Eq:MLEEqFiniteDisc} for $\\mathbf{d}$, and we are done.\n\\end{proof}\n\n\n\nWe are now ready to prove Theorem~\\ref{Thm:ConsistencyFiniteDisc}.\n\n\\begin{proof_of}{Theorem~\\ref{Thm:ConsistencyFiniteDisc}}\nLet $\\mathbf{d}^\\ast = (d_1^*, \\dots, d_n^*)$ denote the expected degree sequence under $\\P^*_\\theta$, so $d_i^* = \\sum_{j \\neq i} \\mu(\\theta_i + \\theta_j)$. Since $-2M \\leq \\theta_i + \\theta_j \\leq 2M$ and $\\mu$ is a decreasing function, we see that \n\\begin{equation}\\label{Eq:ProofConsistencyFiniteDisc1}\n(n-1) \\: \\mu(2M) \\leq d_i^* \\leq (n-1) \\: \\mu(-2M), \\quad i = 1,\\dots,n.\n\\end{equation}\nFor $B \\subseteq \\{1,\\dots,n\\}$, let\n\\begin{equation*}\ng(\\mathbf{d}^*, B) = \\sum_{j \\notin B} \\min\\{d_j^*, \\: (r-1)|B|\\} + (r-1) |B| (|B|-1) - \\sum_{i \\in B} d_i^*,\n\\end{equation*}\nand similarly for $g(\\mathbf{d},B)$. Using the notation $(d_j^*)^B$ as introduced in the proof of Lemma~\\ref{Lem:FiniteDiscBoundMLE}, we notice that for $j \\notin B$,\n\\begin{equation*}\nd_j^* = \\sum_{i \\neq j} \\mu(\\theta_j + \\theta_i)\n\\geq \\sum_{i \\in B} \\mu(\\theta_j + \\theta_i)\n= (d_j^*)^B,\n\\end{equation*}\nand similarly,\n\\begin{equation*}\n(r-1)|B| \\geq \\sum_{i \\in B} \\mu(\\theta_j + \\theta_i) = (d_j^*)^B.\n\\end{equation*}\nTherefore, using the relation~\\eqref{Eq:DjBRelation}, we see that\n\\begin{equation*}\n\\begin{split}\ng(\\mathbf{d}^*, B)\n&\\geq \\sum_{j \\notin B} (d_j^*)^B + (r-1)|B|(|B|-1) - \\sum_{i \\in B} d_i^* \\\\\n&= (r-1)|B|(|B|-1) - \\sum_{i \\in B} (d_i^*)^B \\\\\n&= \\sum_{i \\in B} \\sum_{j \\in B \\setminus \\{i\\}} \\big(r-1-\\mu(\\theta_i + \\theta_j)) \\\\\n&\\geq |B|(|B|-1) \\: \\big(r-1-\\mu(-2M)\\big) \\\\\n&= |B|(|B|-1) \\: \\mu(2M).\n\\end{split}\n\\end{equation*}\n\nWe now recall that the edge weights $(A_{ij})$ are independent random variables taking values in $\\{0,1,\\dots,r-1\\}$, with $\\mathbb{E}_\\theta[A_{ij}] = \\mu(\\theta_i + \\theta_j)$. By Hoeffding's inequality~\\cite{hoeffding}, for each $i = 1,\\dots,n$ we have\n\\begin{equation*}\n\\begin{split}\n\\P\\left(|d_i-d_i^*| \\geq (r-1) \\sqrt{\\frac{k n \\log n}{2}} \\right)\n&\\leq \\P\\left(|d_i-d_i^*| \\geq (r-1) \\sqrt{\\frac{k (n-1) \\log n}{2}} \\right) \\\\\n&= \\P\\left(\\left|\\frac{1}{n-1} \\sum_{j \\neq i} (A_{ij} - \\mu(\\theta_i + \\theta_j)) \\right| \\geq (r-1) \\sqrt{\\frac{k \\log n}{2(n-1)}} \\right) \\\\\n&\\leq 2\\exp\\left( -\\frac{2(n-1)}{(r-1)^2} \\cdot \\frac{(r-1)^2 k \\log n}{2(n-1)} \\right) \\\\\n&= \\frac{2}{n^k}.\n\\end{split}\n\\end{equation*}\nTherefore, by union bound, with probability at least $1-2\/n^{k-1}$ we have $\\|\\mathbf{d}-\\mathbf{d}^*\\|_\\infty \\leq (r-1) \\sqrt{kn \\log n\/2}$. Assume we are in this situation. Then from~\\eqref{Eq:ProofConsistencyFiniteDisc1} we see that for all $i = 1,\\dots,n$,\n\\begin{equation*}\n(n-1) \\: \\mu(2M) - (r-1) \\sqrt{\\frac{k n \\log n}{2}} \\leq d_i \\leq (n-1) \\: \\mu(-2M) + (r-1) \\sqrt{\\frac{k n \\log n}{2}}.\n\\end{equation*}\nThus, for sufficiently large $n$, we have\n\\begin{equation*}\nc_2 (r-1) (n-1) \\leq d_i \\leq c_1(r-1)(n-1), \\quad i = 1,\\dots,n.\n\\end{equation*}\nwith\n\\begin{equation*}\nc_1 = \\frac{3\\mu(-2M)}{2(r-1)}, \\qquad\nc_2 = \\frac{\\mu(2M)}{2(r-1)}.\n\\end{equation*}\nMoreover, it is easy to see that for every $B \\subseteq \\{1,\\dots,n\\}$ we have $|g(\\mathbf{d},B)-g(\\mathbf{d}^*,B)| \\leq \\sum_{i=1}^n |d_i-d_i^*| \\leq n\\|\\mathbf{d}-\\mathbf{d}^*\\|_\\infty$. Since we already know that $g(\\mathbf{d}^*,B) \\geq |B|(|B|-1) \\: \\mu(2M)$, this gives us \n\\begin{equation*}\ng(\\mathbf{d},B) \\geq g(\\mathbf{d}^*,B) - n\\|\\mathbf{d}-\\mathbf{d}^*\\|_\\infty \\geq |B|(|B|-1) \\mu(2M) - (r-1) \\sqrt{\\frac{kn^3 \\log n}{2}}.\n\\end{equation*}\nThus, for $|B| \\geq c_2^2(n-1)$ and for sufficiently large $n$, we have $g(\\mathbf{d},B) \\geq c_3n^2$ with $c_3 = \\frac{1}{2} c_2^4 \\: \\mu(2M)$.\n\nWe have shown that $\\mathbf{d}$ satisfies the properties~\\eqref{Eq:LemFiniteDiscBoundMLE1} and~\\eqref{Eq:LemFiniteDiscBoundMLE}, so by Lemma~\\ref{Lem:FiniteDiscBoundMLE}, the MLE $\\hat \\theta$ exists and satisfies $\\|\\hat \\theta\\|_\\infty \\leq C$, where the constant $C$ only depends on $M$ (and $r$). Assume further that $C \\geq M$, so $\\|\\theta\\|_\\infty \\leq C$ as well.\n\nTo bound the deviation of $\\hat \\theta$ from $\\theta$, we use the convergence rate in the iterative algorithm to compute $\\hat \\theta$. Set $\\hat \\theta^{(0)} = \\theta$ in the algorithm in Theorem~\\ref{Thm:MLEAlgFiniteDisc}, so by Proposition~\\ref{Prop:FiniteDiscBoundFirstIter}, we have\n\\begin{equation}\\label{Eq:ProofConsistencyFiniteDisc2}\n\\|\\hat \\theta-\\theta\\|_\\infty \\leq \\frac{2}{\\delta^2} \\|\\theta-\\varphi(\\theta)\\|_\\infty,\n\\end{equation}\nwhere $\\delta$ is given by~\\eqref{Eq:DeltaBoundJ} with $K = 2\\|\\hat\\theta\\|_\\infty + \\|\\theta\\|_\\infty \\leq 3C$. From the definition of $\\varphi$~\\eqref{Eq:VarphiFiniteDisc}, we see that for each $1 \\leq i \\leq n$,\n\\begin{equation*}\n\\theta_i - \\varphi_i(\\theta) = \\frac{1}{r-1} \\left( \\log d_i - \\log \\sum_{j \\neq i} \\mu(\\theta_i + \\theta_j) \\right)\n= \\frac{1}{r-1} \\log \\frac{d_i}{d_i^*}.\n\\end{equation*}\nNoting that $(y-1)\/y \\leq \\log y \\leq y-1$ for $y > 0$, we have $|\\log (d_i\/d_i^*)| \\leq |d_i-d_i^*| \/ \\min\\{d_i, d_i^*\\}$. Using the bounds on $\\|\\mathbf{d}-\\mathbf{d}^*\\|_\\infty$ and $d_i, d_i^*$ that we have developed above, we get\n\\begin{equation*}\n\\begin{split}\n\\|\\theta-\\varphi(\\theta)\\|_\\infty\n&\\leq \\frac{\\|\\mathbf{d}-\\mathbf{d}^*\\|_\\infty}{\\min\\{ \\min_i d_i, \\; \\min_i d_i^*\\}}\n\\leq (r-1) \\sqrt{\\frac{kn \\log n}{2}} \\cdot \\frac{2}{\\mu(2M) \\: (n-1)}\n\\leq \\frac{2(r-1)}{\\mu(2M)} \\sqrt{\\frac{k \\log n}{n}}.\n\\end{split}\n\\end{equation*}\nPlugging this bound to~\\eqref{Eq:ProofConsistencyFiniteDisc2} gives us the desired result.\n\\end{proof_of}\n\n\n\n\n\n\n\n\\subsection{Proofs for the continuous weighted graphs}\n\\label{Sec:ProofCont}\n\nIn this section we present the proofs of the results presented in Section~\\ref{Sec:Cont}.\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:GraphicalCont}}\n\nClearly if $(d_1, \\dots, d_n) \\in \\mathbb{R}_0^n$ is a graphical sequence, then so is $(d_{\\pi(1)}, \\dots, d_{\\pi(n)})$, for any permutation $\\pi$ of $\\{1,\\dots,n\\}$. Thus, without loss of generality we can assume $d_1 \\geq d_2 \\geq \\cdots \\geq d_n$, and in this case condition~\\eqref{Eq:GraphicalCont} reduces to\n\\begin{equation}\\label{Eq:ErdosGallai-R2}\nd_1 \\leq \\sum_{i=2}^n d_i.\n\\end{equation}\n\nFirst suppose $(d_1, \\dots, d_n) \\in \\mathbb{R}_0^n$ is graphic, so it is the degree sequence of a graph with adjacency matrix $\\mathbf{a} = (a_{ij})$. Then condition~\\eqref{Eq:ErdosGallai-R2} is satisfied since\n\\begin{equation*}\nd_1 = \\sum_{i=2}^n a_{1i} \\leq \\sum_{i=2}^n \\sum_{j \\neq i} a_{ij} = \\sum_{i=2}^n d_i.\n\\end{equation*}\nFor the converse direction, we first note the following easy properties of weighted graphical sequences:\n\\begin{enumerate}[(i)]\n  \\item\\label{p:Cha} The sequence $(c, c, \\dots, c) \\in \\mathbb{R}_0^n$ is graphic for any $c \\in \\mathbb{R}_0$, realized by the ``cycle graph'' with weights $a_{i,i+1} = c\/2$ for $1 \\leq i \\leq n-1$, $a_{1n} = c\/2$, and $a_{ij} = 0$ otherwise.\n\n  \\item\\label{p:Eq} A sequence $\\mathbf{d} = (d_1, \\dots, d_n) \\in \\mathbb{R}_0^n$ satisfying~\\eqref{Eq:ErdosGallai-R2} with an equality is graphic, realized by the ``star graph'' with weights $a_{1i} = d_i$ for $2 \\leq i \\leq n$ and $a_{ij} = 0$ otherwise.\n\n  \\item\\label{p:Ext} If $\\mathbf{d} = (d_1, \\dots, d_n) \\in \\mathbb{R}_0^n$ is graphic, then so is $\\overline{\\mathbf{d}} = (d_1, \\dots, d_n, 0, \\dots, 0) \\in \\mathbb{R}_0^{n'}$ for any $n' \\geq n$, realized by inserting $n'-n$ isolated vertices to the graph that realizes $\\mathbf{d}$.\n  \n  \\item\\label{p:Sum} If $\\mathbf{d}^{(1)}, \\mathbf{d}^{(2)} \\in \\mathbb{R}_0^n$ are graphic, then so is $\\mathbf{d}^{(1)} + \\mathbf{d}^{(2)}$, realized by the graph whose edge weights are the sum of the edge weights of the graphs realizing $\\mathbf{d}^{(1)}$ and $\\mathbf{d}^{(2)}$.\n\\end{enumerate}\n\nWe now prove the converse direction by induction on $n$. For the base case $n = 3$, it is easy to verify that $(d_1,d_2,d_3)$ with $d_1 \\geq d_2 \\geq d_3 \\geq 0$ and $d_1 \\leq d_2 + d_3$ is the degree sequence of the graph $G$ with edge weights\n\\begin{equation*}\na_{12} = \\frac{1}{2}(d_1 + d_2 - d_3) \\geq 0, \\qquad\na_{13} = \\frac{1}{2}(d_1 - d_2 + d_3) \\geq 0, \\qquad\na_{23} = \\frac{1}{2}(-d_1 + d_2 + d_3) \\geq 0.\n\\end{equation*}\nAssume that the claim holds for $n-1$; we will prove it also holds for $n$. So suppose we have a sequence $\\mathbf{d} = (d_1, \\dots, d_n)$ with $d_1 \\geq d_2 \\geq \\cdots \\geq d_n \\geq 0$ satisfying~\\eqref{Eq:ErdosGallai-R2}, and let\n\\begin{equation*}\nK = \\frac{1}{n-2} \\left( \\sum_{i=2}^n d_i - d_1 \\right) \\geq 0\n\\end{equation*}\nIf $K = 0$ then~\\eqref{Eq:ErdosGallai-R2} is satisfied with an equality, and by property~\\eqref{p:Eq} we know that $\\mathbf{d}$ is graphic. Now assume $K > 0$. We consider two possibilities.\n\\begin{enumerate}\n  \\item Suppose $K \\geq d_n$. Then we can write $\\mathbf{d} = \\mathbf{d}^{(1)} + \\mathbf{d}^{(2)}$, where\n\\begin{equation*}\n\\mathbf{d}^{(1)} = (d_1-d_n, \\: d_2-d_n, \\: \\dots, \\: d_{n-1}-d_n, \\: 0) \\in \\mathbb{R}_0^n\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathbf{d}^{(2)} = (d_n, d_n, \\dots, d_n) \\in \\mathbb{R}_0^n.\n\\end{equation*}\nThe assumption $K \\geq d_n$ implies $d_1-d_n \\leq \\sum_{i=2}^{n-1} (d_i-d_n)$, so $(d_1-d_n, d_2-d_n, \\dots, d_{n-1}-d_n) \\in \\mathbb{R}_0^{n-1}$ is a graphical sequence by induction hypothesis. Thus, $\\mathbf{d}^{(1)}$ is also graphic by property~\\eqref{p:Ext}. Furthermore, $\\mathbf{d}^{(2)}$ is graphic by property~\\eqref{p:Cha}, so $\\mathbf{d} = \\mathbf{d}^{(1)} + \\mathbf{d}^{(2)}$ is also a graphical sequence by property~\\eqref{p:Sum}.\n\n  \\item Suppose $K < d_n$. Then write $\\mathbf{d} = \\mathbf{d}^{(3)} + \\mathbf{d}^{(4)}$, where\n\\begin{equation*}\n\\mathbf{d}^{(3)} = (d_1-K, \\: d_2-K, \\: \\dots, \\: d_n-K) \\in \\mathbb{R}_0^n\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathbf{d}^{(4)} = (K, K, \\dots, K) \\in \\mathbb{R}_0^n.\n\\end{equation*}\nBy construction, $\\mathbf{d}^{(3)}$ satisfies $d_1-K = \\sum_{i=2}^n (d_i-K)$, so $\\mathbf{d}^{(3)}$ is a graphical sequence by property~\\eqref{p:Eq}. Since $\\mathbf{d}^{(4)}$ is also graphic by property~\\eqref{p:Cha}, we conclude that $\\mathbf{d} = \\mathbf{d}^{(3)} + \\mathbf{d}^{(4)}$ is graphic by property~\\eqref{p:Sum}.\n\\end{enumerate}\nThis completes the induction step and finishes the proof of Theorem~\\ref{Thm:GraphicalCont}.\n\n\n\n\n\\subsubsection{Proof of Lemma~\\ref{Lem:W-Convex}}\n\nWe first prove that $\\mathcal{W}$ is convex. Given $\\mathbf{d} = (d_1, \\dots, d_n)$ and $\\mathbf{d}' = (d_1', \\dots, d_n')$ in $\\mathcal{W}$, and given $0 \\leq t \\leq 1$, we note that\n\\begin{equation*}\n\\begin{split}\n\\max_{1 \\leq i \\leq n} \\big( t d_i + (1-t) d_i' \\big)\n&\\leq t \\max_{1 \\leq i \\leq n} d_i + (1-t) \\max_{1 \\leq i \\leq n} d_i' \\\\\n&\\leq \\frac{1}{2} t \\sum_{i=1}^n d_i + \\frac{1}{2} (1-t) \\sum_{i=1}^n d_i' \\\\\n&= \\frac{1}{2} \\sum_{i=1}^n \\big( t d_i + (1-t) d_i' \\big),\n\\end{split}\n\\end{equation*}\nwhich means $t\\mathbf{d} + (1-t) \\mathbf{d}' \\in \\mathcal{W}$.\n\nNext, recall that we already have $\\mathcal{M} \\subseteq \\text{conv}(\\mathcal{W}) = \\mathcal{W}$ from Proposition~\\ref{Prop:MConvW}, so to conclude $\\mathcal{M} = \\mathcal{W}$ it remains to show that $\\mathcal{W} \\subseteq \\mathcal{M}$. Given $\\mathbf{d} \\in \\mathcal{W}$, let $G$ be a graph that realizes $\\mathbf{d}$ and let $\\mathbf{w} = (w_{ij})$ be the edge weights of $G$, so that $d_i = \\sum_{j \\neq i} w_{ij}$ for all $i = 1,\\dots,n$. Consider a distribution $\\P$ on $\\mathbb{R}_0^{\\binom{n}{2}}$ that assigns each edge weight $A_{ij}$ to be an independent exponential random variable with mean parameter $w_{ij}$, so $\\P$ has density\n\\begin{equation*}\np(\\mathbf{a}) = \\prod_{\\{i,j\\}} \\frac{1}{w_{ij}} \\exp\\left(-\\frac{a_{ij}}{w_{ij}}\\right),\n\\quad \\mathbf{a} = (a_{ij}) \\in \\mathbb{R}_0^{\\binom{n}{2}}.\n\\end{equation*}\nThen by construction, we have $\\mathbb{E}_\\P[A_{ij}] = w_{ij}$ and\n\\begin{equation*}\n\\mathbb{E}_\\P[\\deg_i(A)] = \\sum_{j \\neq i} \\mathbb{E}_\\P[A_{ij}] = \\sum_{j \\neq i} w_{ij} = d_i, \\quad i = 1, \\dots, n.\n\\end{equation*}\nThis shows that $\\mathbf{d} \\in \\mathcal{M}$, as desired.\n\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:ConsistencyCont}}\n\nWe first prove that the MLE $\\hat \\theta$ exists almost surely. Recall from the discussion in Section~\\ref{Sec:Cont} that $\\hat \\theta$ exists if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$. Clearly $\\mathbf{d} \\in \\mathcal{W}$ since $\\mathbf{d}$ is the degree sequence of the sampled graph $G$. Since $\\mathcal{M} = \\mathcal{W}$ (Lemma~\\ref{Lem:W-Convex}), we see that the MLE $\\hat \\theta$ does not exist if and only if $\\mathbf{d} \\in \\partial \\mathcal{M} = \\mathcal{M} \\setminus \\mathcal{M}^\\circ$, where\n\\begin{equation*}\n\\partial \\mathcal{M} = \\left\\{ \\mathbf{d}' \\in \\mathbb{R}_0^n \\colon \\min_{1 \\leq i \\leq n} d_i' = 0 \\; \\text{ or } \\; \\max_{1 \\leq i \\leq n} d_i' = \\frac{1}{2} \\sum_{i=1}^n d_i' \\right\\}.\n\\end{equation*}\nIn particular, note that $\\partial \\mathcal{M}$ has Lebesgue measure $0$. Since the distribution $\\P^\\ast$ on the edge weights $A = (A_{ij})$ is continuous (being a product of exponential distributions) and $\\mathbf{d}$ is a continuous function of $A$, we conclude that $\\P^\\ast(\\mathbf{d} \\in \\partial \\mathcal{M}) = 0$, as desired.\n\nWe now prove the consistency of $\\hat \\theta$. Recall that $\\theta$ is the true parameter that we wish to estimate, and that the MLE $\\hat \\theta$ satisfies $-Z(\\hat \\theta) = \\mathbf{d}$. Let $\\mathbf{d}^\\ast = -\\nabla Z(\\theta)$ denote the expected degree sequence of the maximum entropy distribution $\\P^\\ast_\\theta$. By the mean value theorem for vector-valued functions~\\cite[p.~341]{Lang}, we can write\n\\begin{equation}\\label{Eq:ConsistencyContMVT}\n\\mathbf{d} - \\mathbf{d}^\\ast = \\nabla Z(\\theta) - \\nabla Z(\\hat \\theta) = J(\\theta - \\hat \\theta).\n\\end{equation}\nHere $J$ is a matrix obtained by integrating (element-wise) the Hessian $\\nabla^2 Z$ of the log-partition function on intermediate points between $\\theta$ and $\\hat \\theta$:\n\\begin{equation*}\nJ = \\int_0^1 \\nabla^2 Z(t \\theta + (1-t) \\hat \\theta) \\: dt.\n\\end{equation*}\n\nRecalling that $-\\nabla Z(\\theta) = \\mathbb{E}_\\theta[\\deg(A)]$, at any intermediate point $\\xi \\equiv \\xi(t) = t \\theta + (1-t) \\hat \\theta$, we have\n\\begin{equation*}\n\\big(\\nabla Z(\\xi)\\big)_i = -\\sum_{j \\neq i} \\mu(\\xi_i + \\xi_j) = -\\sum_{j \\neq i} \\frac{1}{\\xi_i + \\xi_j}.\n\\end{equation*} \nTherefore, the Hessian $\\nabla^2 Z$ is given by\n\\begin{equation*}\n\\big( \\nabla^2 Z(\\xi) \\big)_{ij} = \\frac{1}{(\\xi_i + \\xi_j)^2} \\quad i \\neq j,\n\\end{equation*}\nand\n\\begin{equation*}\n\\big( \\nabla^2 Z(\\xi) \\big)_{ii} = \\sum_{j \\neq i} \\frac{1}{(\\xi_i+\\xi_j)^2} = \\sum_{j \\neq i} \\big( \\nabla^2 Z(\\xi) \\big)_{ij}.\n\\end{equation*}\nSince $\\theta,\\theta' \\in \\Theta$ and we assume $\\theta_i+\\theta_j \\leq M$, it follows that for $i \\neq j$,\n\\begin{equation*}\n0 < \\xi_i + \\xi_j \\leq \\max\\{\\theta_i+\\theta_j, \\: \\hat \\theta_i + \\hat \\theta_j\\} \\leq \\max\\{M, 2\\|\\hat\\theta\\|_\\infty\\} \\leq M + 2\\|\\hat \\theta\\|_\\infty.\n\\end{equation*}\nTherefore, the Hessian $\\nabla^2 Z$ is a {\\em diagonally balanced} matrix with off-diagonal entries bounded below by $1\/(M + 2\\|\\hat \\theta\\|_\\infty)^2$.\nIn particular, $J$ is also a symmetric, diagonally balanced matrix with off-diagonal entries bounded below by $1\/(M + 2\\|\\hat \\theta\\|_\\infty)^2$, being an average of such matrices. By Theorem~\\ref{Thm:Main}, $J$ is invertible and its inverse satisfies the bound\n\\begin{equation*}\n\\|J^{-1}\\|_\\infty \\leq \\frac{(M+2\\|\\hat\\theta\\|_\\infty)^2(3n-4)}{2(n-1)(n-2)} \\leq \\frac{2}{n} \\: (M + 2\\|\\hat \\theta\\|_\\infty)^2,\n\\end{equation*}\nwhere the last inequality holds for $n \\geq 7$. Inverting $J$ in~\\eqref{Eq:ConsistencyContMVT} and applying the bound on $\\|J^{-1}\\|_\\infty$ gives \n\\begin{equation}\\label{Eq:Consistency-R1}\n\\|\\theta-\\hat\\theta\\|_\\infty\n\\leq \\|J^{-1}\\|_\\infty \\: \\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty\n\\leq \\frac{2}{n} \\: (M + 2\\|\\hat \\theta\\|_\\infty)^2 \\: \\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty.\n\\end{equation}\n\nLet $A = (A_{ij})$ denote the edge weights of the sampled graph $G \\sim \\P^\\ast_\\theta$, so $d_i = \\sum_{j \\neq i} A_{ij}$ for $i = 1,\\dots,n$. Moreover, since $\\mathbf{d}^\\ast$ is the expected degree sequence from the distribution $\\P^\\ast_\\theta$, we also have $d_i^\\ast = \\sum_{j \\neq i} 1\/(\\theta_i+\\theta_j)$. Recall that $A_{ij}$ is an exponential random variable with rate $\\lambda = \\theta_i + \\theta_j \\geq L$, so by Lemma~\\ref{Lem:SubExp-Exp}, $A_{ij} - 1\/(\\theta_i + \\theta_j)$ is sub-exponential with parameter $2\/(\\theta_i+\\theta_j) \\leq 2\/L$. For each $i = 1,\\dots,n$, the random variables $(A_{ij} - 1\/(\\theta_i+\\theta_j), j \\neq i)$ are independent sub-exponential random variables, so we can apply the concentration inequality in Theorem~\\ref{Thm:ConcIneqSubExp} with $\\kappa = 2\/L$ and\n\\begin{equation*}\n\\epsilon = \\left(\\frac{4k \\log n}{\\gamma (n-1) L^2} \\right)^{1\/2}.\n\\end{equation*}\nAssume $n$ is sufficiently large such that $\\epsilon\/\\kappa = \\sqrt{k \\log n \/ \\gamma (n-1)} \\leq 1$. Then by Theorem~\\ref{Thm:ConcIneqSubExp}, for each $i = 1,\\dots,n$ we have\n\\begin{equation*}\n\\begin{split}\n\\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} \\right)\n&\\leq \\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{4 k (n-1) \\log n}{\\gamma L^2}} \\right) \\\\\n&= \\P\\left(\\Bigg|\\frac{1}{n-1}\\sum_{j \\neq i} \\left(A_{ij}-\\frac{1}{\\theta_i+\\theta_j}\\right)\\Bigg| \\geq \\sqrt{\\frac{4k \\log n}{\\gamma (n-1) L^2}} \\right) \\\\\n&\\leq 2\\exp\\left(-\\gamma \\: (n-1) \\cdot \\frac{L^2}{4} \\cdot \\frac{4k \\log n}{\\gamma (n-1) L^2}\\right) \\\\\n&= \\frac{2}{n^k}.\n\\end{split}\n\\end{equation*}\nBy the union bound,\n\\begin{equation*}\n\\begin{split}\n\\P\\Bigg(\\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty \\geq \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} \\; \\Bigg)\n&\\leq \\sum_{i=1}^n \\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}}\\right)\n\\leq \\frac{2}{n^{k-1}}.\n\\end{split}\n\\end{equation*}\n\nAssume for the rest of this proof that $\\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty \\leq \\sqrt{4kn \\log n\/(\\gamma L^2)}$, which happens with probability at least $1-2\/n^{k-1}$. From~\\eqref{Eq:Consistency-R1} and using the triangle inequality, we get\n\\begin{equation*}\n\\|\\hat \\theta\\|_\\infty\n\\leq \\|\\theta-\\hat\\theta\\|_\\infty + \\|\\theta\\|_\\infty\n\\leq \\frac{4}{L} \\: \\sqrt{\\frac{k \\log n}{\\gamma n}} \\: (M + 2\\|\\hat \\theta\\|_\\infty)^2 + M.\n\\end{equation*}\nWhat we have shown is that for sufficiently large $n$, $\\|\\hat \\theta\\|_\\infty$ satisfies the inequality $G_n(\\|\\hat \\theta\\|_\\infty) \\geq 0$, where $G_n(x)$ is the quadratic function\n\\begin{equation*}\nG_n(x) = \\frac{4}{L} \\: \\sqrt{\\frac{k \\log n}{\\gamma n}} \\: (M + 2x)^2 - x + M.\n\\end{equation*}\nIt is easy to see that for sufficiently large $n$ we have $G_n(2M) < 0$ and $G_n(\\log n) < 0$. Thus, $G_n(\\|\\hat \\theta\\|_\\infty) \\geq 0$ means either $\\|\\hat \\theta\\|_\\infty < 2M$ or $\\|\\hat \\theta\\|_\\infty > \\log n$. We claim that for sufficiently large $n$ we always have $\\|\\hat \\theta\\|_\\infty < 2M$. Suppose the contrary that there are infinitely many $n$ for which $\\|\\hat \\theta\\|_\\infty > \\log n$, and consider one such $n$. Since $\\hat \\theta \\in \\Theta$ we know that $\\hat \\theta_i + \\hat \\theta_j > 0$ for each $i \\neq j$, so there can be at most one index $i$ with $\\hat \\theta_i < 0$. We consider the following two cases:\n\\begin{enumerate}\n  \\item \\textbf{Case 1:} suppose $\\hat \\theta_i \\geq 0$ for all $i = 1,\\dots,n$. Let $i^\\ast$ be an index with $\\hat \\theta_{i^\\ast} = \\|\\hat \\theta\\|_\\infty > \\log n$. Then, using the fact that $\\hat \\theta$ satisfies the system of equations~\\eqref{Eq:MLEEqCont} and $\\hat \\theta_{i^\\ast} + \\hat \\theta_j \\geq \\hat \\theta_{i^\\ast}$ for $j \\neq i^\\ast$, we see that\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{M} &\\leq \\frac{1}{n-1} \\sum_{j \\neq i^\\ast} \\frac{1}{\\theta_{i^\\ast} + \\theta_j} \\\\\n&\\leq \\frac{1}{n-1} \\left| \\sum_{j \\neq i^\\ast} \\frac{1}{\\theta_{i^\\ast}+\\theta_j} - \\sum_{j \\neq i^\\ast} \\frac{1}{\\hat \\theta_{i^\\ast}+\\hat \\theta_j} \\right| + \\frac{1}{n-1} \\sum_{j \\neq i^\\ast} \\frac{1}{\\hat \\theta_{i^\\ast} + \\hat \\theta_j} \\\\\n&= \\frac{1}{n-1} \\left| d_i^\\ast - d_i \\right| + \\frac{1}{n-1} \\sum_{j \\neq i^\\ast} \\frac{1}{\\hat \\theta_{i^\\ast} + \\hat \\theta_j} \\\\\n&\\leq \\frac{1}{n-1} \\| \\mathbf{d}^\\ast - \\mathbf{d} \\|_\\infty + \\frac{1}{\\|\\hat \\theta\\|_\\infty} \\\\\n&\\leq \\frac{1}{n-1} \\: \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} + \\frac{1}{\\log n},\n\\end{split}\n\\end{equation*}\nwhich cannot hold for sufficiently large $n$, as the last expression tends to $0$ as $n \\to \\infty$.\n\n  \\item \\textbf{Case 2:} suppose $\\hat \\theta_i < 0$ for some $i = 1,\\dots,n$, so $\\hat \\theta_j > 0$ for $j \\neq i$ since $\\hat \\theta \\in \\Theta$. Without loss of generality assume $\\hat \\theta_1 < 0 < \\hat \\theta_2 \\leq \\cdots \\leq \\hat \\theta_n$, so $\\hat \\theta_n = \\|\\hat\\theta\\|_\\infty > \\log n$. Following the same chain of inequalities as in the previous case (with $i^\\ast = n$), we obtain\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{M} &\\leq \\frac{1}{n-1} \\| \\mathbf{d}^\\ast - \\mathbf{d} \\|_\\infty + \\frac{1}{n-1} \\left(\\frac{1}{\\hat \\theta_n + \\hat \\theta_1} + \\sum_{j = 2}^{n-1} \\frac{1}{\\hat \\theta_j + \\hat \\theta_n} \\right) \\\\\n&\\leq \\frac{1}{n-1} \\: \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} + \\frac{1}{(n-1)(\\hat \\theta_n + \\hat \\theta_1)} + \\frac{n-2}{(n-1)\\|\\hat \\theta\\|_\\infty} \\\\\n&\\leq \\frac{1}{n-1} \\: \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} + \\frac{1}{(n-1)(\\hat \\theta_n + \\hat \\theta_1)} + \\frac{1}{\\log n}.\n\\end{split}\n\\end{equation*}\nSo for sufficiently large $n$,\n\\begin{equation*}\n\\frac{1}{\\hat \\theta_1 + \\hat \\theta_n} \\geq (n-1)\\left(\\frac{1}{M} - \\frac{1}{n-1} \\: \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} - \\frac{1}{\\log n}\\right) \\geq \\frac{n}{2M},\n\\end{equation*}\nand thus $\\hat \\theta_1 + \\hat \\theta_i \\leq \\hat \\theta_1 + \\hat \\theta_n \\leq 2M\/n$ for each $i = 2,\\dots,n$. However, then\n\\begin{equation*}\n\\begin{split}\n\\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} &\\geq \\|\\mathbf{d}^\\ast - \\mathbf{d} \\|_\\infty\n\\geq |d_1^\\ast - d_1|\n\\geq -\\sum_{j=2}^n \\frac{1}{\\theta_1 + \\theta_j} +\\sum_{j=2}^n \\frac{1}{\\hat\\theta_1 + \\hat\\theta_j}\n\\geq - \\frac{(n-1)}{L} + \\frac{n(n-1)}{2M},\n\\end{split}\n\\end{equation*}\nwhich cannot hold for sufficiently large $n$, as the right hand side of the last expression tends to $\\infty$ faster than the left hand side.\n\\end{enumerate}\nThe analysis above shows that $\\|\\hat \\theta\\|_\\infty < 2M$ for all sufficiently large $n$. Plugging in this result to~\\eqref{Eq:Consistency-R1}, we conclude that for sufficiently large $n$, with probability at least $1-2n^{-(k-1)}$ we have the bound\n\\begin{equation*}\n\\|\\theta-\\hat\\theta\\|_\\infty \\leq \\frac{2}{n} \\: (5M)^2 \\: \\sqrt{\\frac{4k n \\log n}{\\gamma L^2}} = \\frac{100M^2}{L} \\sqrt{\\frac{k \\log n}{\\gamma n}},\n\\end{equation*}\nas desired.\n\n\n\n\n\n\n\n\\subsection{Proofs for the infinite discrete weighted graphs}\n\\label{Sec:ProofInfiniteDisc}\n\nIn this section we prove the results presented in Section~\\ref{Sec:InfiniteDisc}.\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:GraphicalInfiniteDisc}}\n\nWithout loss of generality we may assume $d_1 \\geq d_2 \\geq \\dots \\geq d_n$, so condition~\\eqref{Eq:GraphicalInfiniteDisc} becomes $d_1 \\leq \\sum_{i=2}^n d_i$. The necessary part is easy: if $(d_1,\\dots,d_n)$ is a degree sequence of a graph $G$ with edge weights $a_{ij} \\in \\mathbb{N}_0$, then $\\sum_{i=1}^n d_i = 2\\sum_{\\{i,j\\}} a_{ij}$ is even, and the total weight coming out of vertex $1$ is at most $\\sum_{i=2}^n d_i$. For the converse direction, we proceed by induction on $s = \\sum_{i=1}^n d_i$. The statement is clearly true for $s = 0$ and $s = 2$. Assume the statement is true for some even $s \\in \\mathbb{N}$, and suppose we are given $\\mathbf{d} = (d_1,\\dots,d_n) \\in \\mathbb{N}_0^n$ with $d_1 \\geq \\dots \\geq d_n$, $\\sum_{i=1}^n d_i = s+2$, and $d_1 \\leq \\sum_{i=2}^n d_i$. Without loss of generality we may assume $d_n \\geq 1$, for otherwise we can proceed with only the nonzero elements of $\\mathbf{d}$. Let $1 \\leq t \\leq n-1$ be the smallest index such that $d_t > d_{t+1}$, with $t = n-1$ if $d_1 = \\dots = d_n$, and let $\\mathbf{d}' = (d_1, \\dots, d_{t-1}, d_t-1, d_{t+1}, \\dots, d_n-1)$. We will show that $\\mathbf{d}'$ is graphic. This will imply that $\\mathbf{d}$ is graphic, because if $\\mathbf{d}'$ is realized by the graph $G'$ with edge weights $a_{ij}'$, then $\\mathbf{d}$ is realized by the graph $G$ with edge weights $a_{tn} = a_{tn}'+1$ and $a_{ij} = a_{ij}'$ otherwise.\n\nNow for $\\mathbf{d}' = (d_1', \\dots, d_n')$ given above, we have $d_1' \\geq \\dots \\geq d_n'$ and $\\sum_{i=1}^n d_i' = \\sum_{i=1}^n d_i-2 = s$ is even. So it suffices to show that $d_1' \\leq \\sum_{i=2}^n d_i'$, for then we can apply the induction hypothesis to conclude that $\\mathbf{d}'$ is graphic. If $t = 1$, then $d_1' = d_1-1 \\leq \\sum_{i=2}^n d_i -1 = \\sum_{i=2}^n d_i'$. If $t > 1$ then $d_1 = d_2$, so $d_1 < \\sum_{i=2}^n d_i$ since $d_n \\geq 1$. In particular, since $\\sum_{i=1}^n d_i$ is even, $\\sum_{i=2}^n d_i - d_1 = \\sum_{i=1}^n d_i - 2d_1$ is also even, hence $\\sum_{i=2}^n d_i - d_1 \\geq 2$. Therefore, $d_1' = d_1 \\leq \\sum_{i=2}^n d_i-2 = \\sum_{i=2}^n d_i'$. This finishes the proof of Theorem~\\ref{Thm:GraphicalInfiniteDisc}.\n\n\n\n\n\n\n\\subsubsection{Proof of Lemma~\\ref{Lem:ConvWInfiniteDisc}}\n\nClearly $\\mathcal{W} \\subseteq \\mathcal{W}_1$, so $\\overline{\\text{conv}(\\mathcal{W})} \\subseteq \\mathcal{W}_1$ since $\\mathcal{W}_1$ is closed and convex, by Lemma~\\ref{Lem:W-Convex}. Conversely, let $\\mathbb{Q}$ denote the set of rational numbers. We will first show that $\\mathcal{W}_1 \\cap \\mathbb{Q}^n \\subseteq \\text{conv}(\\mathcal{W})$ and then proceed by a limit argument. Let $\\mathbf{d} \\in \\mathcal{W}_1 \\cap \\mathbb{Q}^n$, so $\\mathbf{d} = (d_1, \\dots, d_n) \\in \\mathbb{Q}^n$ with $d_i \\geq 0$ and $\\max_{1 \\leq i \\leq n} d_i \\leq \\frac{1}{2} \\sum_{i=1}^n d_i$. Choose $K \\in \\mathbb{N}$ large enough such that $Kd_i \\in \\mathbb{N}_0$ for all $i = 1,\\dots,n$. Observe that $2K \\mathbf{d} = (2Kd_1, \\dots, 2Kd_n) \\in \\mathbb{N}_0^n$ has the property that $\\sum_{i=1}^n 2Kd_i \\in \\mathbb{N}_0$ is even and $\\max_{1 \\leq i \\leq n} 2Kd_i \\leq \\frac{1}{2} \\sum_{i=1}^n 2Kd_i$, so $2K\\mathbf{d} \\in \\mathcal{W}$ by definition. Since $0 = (0,\\dots,0) \\in \\mathcal{W}$ as well, all elements along the segment joining $0$ and $2K\\mathbf{d}$ lie in $\\text{conv}(\\mathcal{W})$, so in particular, $\\mathbf{d} = (2K\\mathbf{d})\/(2K) \\in \\text{conv}(\\mathcal{W})$. This shows that $\\mathcal{W}_1 \\cap \\mathbb{Q}^n \\subseteq \\text{conv}(\\mathcal{W})$, and hence $\\overline{\\mathcal{W}_1 \\cap \\mathbb{Q}^n} \\subseteq \\overline{\\text{conv}(\\mathcal{W})}$.\n\nTo finish the proof it remains to show that $\\overline{\\mathcal{W}_1 \\cap \\mathbb{Q}^n} = \\mathcal{W}_1$. On the one hand we have \n\\begin{equation*}\n\\overline{\\mathcal{W}_1 \\cap \\mathbb{Q}^n} \\subseteq \\overline{\\mathcal{W}_1} \\cap \\overline{\\mathbb{Q}^n} = \\mathcal{W}_1 \\cap \\mathbb{R}_0^n = \\mathcal{W}_1. \n\\end{equation*}\nFor the other direction, given $\\mathbf{d} \\in \\mathcal{W}_1$, choose $\\mathbf{d}_1, \\dots, \\mathbf{d}_n \\in W_1$ such that $\\mathbf{d}, \\mathbf{d}_1, \\dots, \\mathbf{d}_n$ are in general position, so that the convex hull $C$ of $\\{\\mathbf{d}, \\mathbf{d}_1, \\dots, \\mathbf{d}_n\\}$ is full dimensional. This can be done, for instance, by noting that the following $n+1$ points in $\\mathcal{W}_1$ are in general position:\n\\begin{equation*}\n\\{0, \\; \\mathbf{e}_1 + \\mathbf{e}_2, \\; \\mathbf{e}_1 + \\mathbf{e}_3, \\; \\cdots, \\; \\mathbf{e}_1 + \\mathbf{e}_n, \\; \\mathbf{e}_1 + \\mathbf{e}_2 + \\cdots + \\mathbf{e}_n\\},\n\\end{equation*}\nwhere $\\mathbf{e}_1, \\dots, \\mathbf{e}_n$ are the standard basis of $\\mathbb{R}^n$. For each $m \\in \\mathbb{N}$ and $i = 1, \\dots, n$, choose $\\mathbf{d}_i^{(m)}$ on the line segment between $\\mathbf{d}$ and $\\mathbf{d}_i$ such that the convex hull $C_m$ of $\\{\\mathbf{d}, \\mathbf{d}_1^{(m)}, \\dots, \\mathbf{d}_n^{(m)}\\}$ is full dimensional and has diameter at most $1\/m$. Since $C_m$ is full dimensional we can choose a rational point $\\mathbf{r}_m \\in C_m \\subseteq C \\subseteq \\mathcal{W}_1$. Thus we have constructed a sequence of rational points $(\\mathbf{r}_m)$ in $\\mathcal{W}_1$ converging to $\\mathbf{d}$, which shows that $\\mathcal{W}_1 \\subseteq \\overline{\\mathcal{W}_1 \\cap \\mathbb{Q}^n}$.\n\n\n\n\n\\subsubsection{Proof of Theorem~\\ref{Thm:ConsistencyInfiniteDisc}}\n\nWe first address the issue of the existence of $\\hat \\theta$. Recall from the discussion in Section~\\ref{Sec:InfiniteDisc} that the MLE $\\hat \\theta \\in \\Theta$ exists if and only if $\\mathbf{d} \\in \\mathcal{M}^\\circ$. Clearly $\\mathbf{d} \\in \\mathcal{W}$ since $\\mathbf{d}$ is the degree sequence of the sampled graph $G$, and $\\mathcal{W} \\subseteq \\text{conv}(\\mathcal{W}) = \\mathcal{M}$ from Proposition~\\ref{Prop:MConvW}. Therefore, the MLE $\\hat \\theta$ does not exist if and only if $\\mathbf{d} \\in \\partial \\mathcal{M} = \\mathcal{M} \\setminus \\mathcal{M}^\\circ$, where the boundary $\\partial \\mathcal{M}$ is explicitly given by\n\\begin{equation*}\n\\partial \\mathcal{M} = \\left\\{ \\mathbf{d}' \\in \\mathbb{R}_0^n \\colon \\min_{1 \\leq i \\leq n} d_i' = 0 \\; \\text{ or } \\; \\max_{1 \\leq i \\leq n} d_i' = \\frac{1}{2} \\sum_{i=1}^n d_i' \\right\\}.\n\\end{equation*}\nUsing union bound and the fact that the edge weights $A_{ij}$ are independent geometric random variables, we have\n\\begin{equation*}\n\\begin{split}\n\\P(d_i = 0 \\text{ for some } i)\n\\leq \\sum_{i=1}^n \\P(d_i = 0)\n&= \\sum_{i=1}^n \\P(A_{ij} = 0 \\text{ for all } j \\neq i) \\\\\n&= \\sum_{i=1}^n \\prod_{j \\neq i} \\left(1-\\exp(-\\theta_i-\\theta_j)\\right)\n\\leq n\\left(1-\\exp(-M)\\right)^{n-1}.\n\\end{split}\n\\end{equation*}\nFurthermore, again by union bound,\n\\begin{equation*}\n\\P\\left(\\max_{1 \\leq i \\leq n} d_i = \\frac{1}{2} \\sum_{i=1}^n d_i \\right)\n= \\P\\left(d_i = \\sum_{j \\neq i} d_j \\text{ for some } i \\right)\n\\leq \\sum_{i=1}^n \\P\\left(d_i = \\sum_{j \\neq i} d_j\\right).\n\\end{equation*}\nNote that we have $d_i = \\sum_{j \\neq i} d_j$ for some $i$ if and only if the edge weights $A_{jk} = 0$ for all $j,k \\neq i$. This occurs with probability\n\\begin{equation*}\n\\P\\left(A_{jk} = 0 \\text{ for } j,k \\neq i \\right)\n= \\prod_{\\substack{j,k \\neq i\\\\j \\neq k}}\\left(1-\\exp(-\\theta_j-\\theta_k)\\right)\n\\leq \\left(1-\\exp(-M)\\right)^{\\binom{n-1}{2}}.\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n\\begin{split}\n\\P(\\mathbf{d} \\in \\partial \\mathcal{M})\n&\\leq \\P(d_i = 0 \\text{ for some } i) + \\P\\left(\\max_{1 \\leq i \\leq n} d_i = \\frac{1}{2} \\sum_{i=1}^n d_i \\right) \\\\\n&\\leq n\\left(1-\\exp(-M)\\right)^{n-1} + n\\left(1-\\exp(-M)\\right)^{\\binom{n-1}{2}} \\\\\n&\\leq \\frac{1}{n^{k-1}},\n\\end{split}\n\\end{equation*}\nwhere the last inequality holds for sufficiently large $n$. This shows that for sufficiently large $n$, the MLE $\\hat \\theta$ exists with probability at least $1-1\/n^{k-1}$.\n\nWe now turn to proving the consistency of $\\hat \\theta$. For the rest of this proof, assume that the MLE $\\hat \\theta \\in \\Theta$ exists, which occurs with probability at least $1-1\/n^{k-1}$. The proof of the consistency of $\\hat \\theta$ follows the same outline as in the proof of Theorem~\\ref{Thm:ConsistencyCont}. Let $\\mathbf{d}^\\ast = -\\nabla Z(\\theta)$ denote the expected degree sequence of the distribution $\\P^\\ast_\\theta$, and recall that the MLE $\\hat \\theta$ satisfies $\\mathbf{d} = -\\nabla Z(\\hat \\theta)$. By the mean value theorem~\\cite[p.~341]{Lang}, we can write\n\\begin{equation}\\label{Eq:Consistency-MVT-Disc}\n\\mathbf{d} - \\mathbf{d}^\\ast = \\nabla Z(\\theta) - \\nabla Z(\\hat \\theta) = J(\\theta - \\hat\\theta),\n\\end{equation}\nwhere $J$ is the matrix obtained by integrating the Hessian of $Z$ between $\\theta$ and $\\hat \\theta$,\n\\begin{equation*}\nJ = \\int_0^1 \\nabla^2 Z(t\\theta + (1-t)\\hat \\theta) \\: dt.\n\\end{equation*}\n\nLet $0 \\leq t \\leq 1$, and note that at the point $\\xi = t\\theta + (1-t) \\hat \\theta$ the gradient $\\nabla Z$ is given by\n\\begin{equation*}\n\\big( \\nabla Z(\\xi) \\big)_i = -\\sum_{j \\neq i} \\frac{1}{\\exp(\\xi_i+\\xi_j)-1}.\n\\end{equation*}\nThus, the Hessian $\\nabla^2 Z$ is\n\\begin{equation*}\n\\big( \\nabla^2 Z(\\xi) \\big)_{ij} = \\frac{\\exp(\\xi_i+\\xi_j)}{(\\exp(\\xi_i+\\xi_j)-1)^2} \\quad i \\neq j,\n\\end{equation*}\nand\n\\begin{equation*}\n\\big( \\nabla^2 Z(\\xi) \\big)_{ii} = \\sum_{j \\neq i} \\frac{\\exp(\\xi_i+\\xi_j)}{(\\exp(\\xi_i+\\xi_j)-1)^2} = \\sum_{j \\neq i} \\big( \\nabla^2 Z(\\xi) \\big)_{ij}.\n\\end{equation*}\nSince $\\theta, \\hat \\theta \\in \\Theta$ and we assume $\\theta_i+\\theta_j \\leq M$, for $i \\neq j$ we have\n\\begin{equation*}\n0 < \\xi_i + \\xi_j \\leq \\max\\{\\theta_i + \\theta_j, \\; \\hat \\theta_i + \\hat \\theta_j \\} \\leq \\max\\{M, \\; 2\\|\\hat \\theta\\|_\\infty\\} \\leq M + 2\\|\\hat \\theta\\|_\\infty.\n\\end{equation*}\nThis means $J$ is a symmetric, diagonally dominant matrix with off-diagonal entries bounded below by $\\exp(M + 2\\|\\hat\\theta\\|_\\infty)\/(\\exp(M + 2\\|\\hat\\theta\\|_\\infty)-1)^2$, being an average of such matrices. Then by Theorem~\\ref{Thm:Main}, we have the bound\n\\begin{equation*}\n\\|J^{-1}\\|_\\infty\n\\leq \\frac{(3n-4)}{2(n-2)(n-1)} \\: \\frac{(\\exp(M + 2\\|\\hat\\theta\\|_\\infty)-1)^2}{\\exp(M + 2\\|\\hat\\theta\\|_\\infty)}\n\\leq \\frac{2}{n} \\: \\frac{(\\exp(M + 2\\|\\hat\\theta\\|_\\infty)-1)^2}{\\exp(M + 2\\|\\hat\\theta\\|_\\infty)},\n\\end{equation*}\nwhere the second inequality holds for $n \\geq 7$. By inverting $J$ in~\\eqref{Eq:Consistency-MVT-Disc} and applying the bound on $J^{-1}$ above, we obtain\n\\begin{equation}\\label{Eq:Consistency-R2}\n\\|\\theta-\\hat\\theta\\|_\\infty \\leq \\|J^{-1}\\|_\\infty \\: \\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty \\leq \\frac{2}{n} \\: \\frac{(\\exp(M + 2\\|\\hat\\theta\\|_\\infty)-1)^2}{\\exp(M + 2\\|\\hat\\theta\\|_\\infty)} \\: \\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty.\n\\end{equation}\n\nLet $A = (A_{ij})$ denote the edge weights of the sampled graph $G \\sim \\P^\\ast_\\theta$, so $d_i = \\sum_{j \\neq i} A_{ij}$ for $i = 1,\\dots,n$. Since $\\mathbf{d}^\\ast$ is the expected degree sequence from the distribution $\\P^\\ast_\\theta$, we also have $d_i^\\ast = \\sum_{j \\neq i} 1\/(\\exp(\\theta_i+\\theta_j)-1)$. Recall that $A_{ij}$ is a geometric random variable with emission probability\n\\begin{equation*}\nq = 1-\\exp(-\\theta_i-\\theta_j) \\geq 1-\\exp(-L),\n\\end{equation*}\nso by Lemma~\\ref{Lem:SubExp-Geo}, $A_{ij} - 1\/(\\exp(\\theta_i + \\theta_j)-1)$ is sub-exponential with parameter $-4\/\\log(1-q) \\leq 4\/L$. For each $i = 1,\\dots,n$, the random variables $(A_{ij} - 1\/(\\exp(\\theta_i+\\theta_j)-1), j \\neq i)$ are independent sub-exponential random variables, so we can apply the concentration inequality in Theorem~\\ref{Thm:ConcIneqSubExp} with $\\kappa = 4\/L$ and\n\\begin{equation*}\n\\epsilon = \\left(\\frac{16k \\log n}{\\gamma (n-1) L^2} \\right)^{1\/2}.\n\\end{equation*}\nAssume $n$ is sufficiently large such that $\\epsilon\/\\kappa = \\sqrt{k \\log n\/\\gamma (n-1)} \\leq 1$. Then by Theorem~\\ref{Thm:ConcIneqSubExp}, for each $i = 1,\\dots,n$ we have\n\\begin{equation*}\n\\begin{split}\n\\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} \\right)\n&\\leq \\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{16 k (n-1) \\log n}{\\gamma L^2}} \\right) \\\\\n&= \\P\\left(\\Bigg|\\frac{1}{n-1}\\sum_{j \\neq i} \\left(A_{ij}-\\frac{1}{\\exp(\\theta_i+\\theta_j)-1}\\right)\\Bigg| \\geq \\sqrt{\\frac{16 k \\log n}{\\gamma (n-1) L^2}} \\right) \\\\\n&\\leq 2\\exp\\left(-\\gamma \\: (n-1) \\cdot \\frac{L^2}{16} \\cdot \\frac{16 k \\log n}{\\gamma (n-1) L^2} \\right) \\\\\n&= \\frac{2}{n^k}.\n\\end{split}\n\\end{equation*}\nThe union bound then gives us\n\\begin{equation*}\n\\P\\left(\\|\\mathbf{d} - \\mathbf{d}^\\ast\\|_\\infty \\geq \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} \\right)\n\\leq \\sum_{i=1}^n \\P\\left(|d_i - d_i^\\ast| \\geq \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} \\right)\n\\leq \\frac{2}{n^{k-1}}.\n\\end{equation*}\n\nAssume now that $\\|\\mathbf{d} - \\hat{\\mathbf{d}}\\|_\\infty \\leq \\sqrt{16 kn \\log n\/(\\gamma L^2)}$, which happens with probability at least $1-2\/n^{k-1}$. From~\\eqref{Eq:Consistency-R2} and using the triangle inequality, we get\n\\begin{equation*}\n\\|\\hat \\theta\\|_\\infty\n\\leq \\|\\theta-\\hat\\theta\\|_\\infty + \\|\\theta\\|_\\infty\n\\leq \\frac{8}{L} \\: \\sqrt{\\frac{k\\log n}{\\gamma n}} \\: \\frac{(\\exp(M + 2\\|\\hat \\theta\\|_\\infty)-1)^2}{\\exp(M + 2\\|\\hat \\theta\\|_\\infty)} + M.\n\\end{equation*}\nThis means $\\|\\hat \\theta\\|_\\infty$ satisfies the inequality $H_n(\\|\\hat \\theta\\|_\\infty) \\geq 0$, where $H_n(x)$ is the function\n\\begin{equation*}\nH_n(x) = \\frac{8}{L} \\: \\sqrt{\\frac{k\\log n}{\\gamma n}} \\: \\frac{(\\exp(M + 2x)-1)^2}{\\exp(M + 2x)} - x + M.\n\\end{equation*}\nOne can easily verify that $H_n$ is a convex function, so $H_n$ assumes the value $0$ at most twice, and moreover, $H_n(x) \\to \\infty$ as $x \\to \\infty$. It is also easy to see that for all sufficiently large $n$, we have $H_n(2M) < 0$ and $H_n(\\frac{1}{4} \\log n) < 0$. Therefore, $H_n(\\|\\hat \\theta\\|_\\infty) \\geq 0$ implies either $\\|\\hat \\theta\\|_\\infty < 2M$ or $\\|\\hat \\theta\\|_\\infty > \\frac{1}{4} \\log n$. We claim that for sufficiently large $n$ we always have $\\|\\hat \\theta\\|_\\infty < 2M$. Suppose the contrary that there are infinitely many $n$ for which $\\|\\hat \\theta\\|_\\infty > \\frac{1}{4} \\log n$, and consider one such $n$. Since $\\hat \\theta_i + \\hat \\theta_j > 0$ for each $i \\neq j$, there can be at most one index $i$ with $\\hat \\theta_i < 0$. We consider the following two cases:\n\\begin{enumerate}\n  \\item \\textbf{Case 1:} suppose $\\hat \\theta_i \\geq 0$ for all $i = 1,\\dots,n$. Let $i^\\ast$ be an index with $\\hat \\theta_{i^\\ast} = \\|\\hat \\theta\\|_\\infty > \\frac{1}{4} \\log n$. Then, since $\\hat \\theta_{i^\\ast} + \\hat \\theta_j \\geq \\hat \\theta_{i^\\ast}$ for $j \\neq i^\\ast$,\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\exp(M)-1} &\\leq \\frac{1}{n-1} \\sum_{j \\neq i^\\ast} \\frac{1}{\\exp(\\theta_{i^\\ast} + \\theta_j)-1} \\\\\n&\\leq \\frac{1}{n-1} \\left| \\sum_{j \\neq i^\\ast} \\frac{1}{\\exp(\\theta_{i^\\ast}+\\theta_j)-1} - \\sum_{j \\neq i^\\ast} \\frac{1}{\\exp(\\hat \\theta_{i^\\ast}+\\hat \\theta_j)-1} \\right|\n + \\frac{1}{n-1} \\sum_{j \\neq i^\\ast} \\frac{1}{\\exp(\\hat \\theta_{i^\\ast} + \\hat \\theta_j)-1} \\\\\n&\\leq \\frac{1}{n-1} \\| \\mathbf{d} - \\mathbf{d}^\\ast \\|_\\infty + \\frac{1}{\\exp(\\|\\hat \\theta\\|_\\infty)-1} \\\\\n&\\leq \\frac{1}{n-1} \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} + \\frac{1}{n^{1\/4}-1},\n\\end{split}\n\\end{equation*}\nwhich cannot hold for sufficiently large $n$, as the last expression tends to $0$ as $n \\to \\infty$.\n\n  \\item \\textbf{Case 2:} suppose $\\hat \\theta_i < 0$ for some $i = 1,\\dots,n$, so $\\hat \\theta_j > 0$ for $j \\neq i$. Without loss of generality assume $\\hat \\theta_1 < 0 < \\hat \\theta_2 \\leq \\cdots \\leq \\hat \\theta_n$, so $\\hat \\theta_n = \\|\\hat \\theta\\|_\\infty > \\frac{1}{4} \\log n$. Following the same chain of inequalities as in the previous case (with $i^\\ast = n$), we obtain\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\exp(M)-1} &\\leq \\frac{1}{n-1} \\| \\mathbf{d} - \\mathbf{d}^\\ast \\|_\\infty + \\frac{1}{n-1} \\left(\\frac{1}{\\exp(\\hat \\theta_n + \\hat \\theta_1)-1} + \\sum_{j = 2}^{n-1} \\frac{1}{\\exp(\\hat \\theta_j + \\hat \\theta_n)-1} \\right) \\\\\n&\\leq \\frac{1}{n-1} \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} + \\frac{1}{(n-1)(\\exp(\\hat \\theta_n + \\hat \\theta_1)-1)} + \\frac{n-2}{(n-1)(\\exp(\\|\\hat \\theta\\|_\\infty)-1)} \\\\\n&\\leq \\frac{1}{n-1} \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} + \\frac{1}{(n-1)(\\exp(\\hat \\theta_n + \\hat \\theta_1)-1)} + \\frac{1}{n^{1\/4}-1}.\n\\end{split}\n\\end{equation*}\nThis implies\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\exp(\\hat \\theta_1 + \\hat \\theta_n)-1}\n&\\geq (n-1)\\left(\\frac{1}{\\exp(M)-1} - \\frac{1}{n-1} \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} - \\frac{1}{n^{1\/4}-1}\\right)\n\\geq \\frac{n}{2(\\exp(M)-1)},\n\\end{split}\n\\end{equation*}\nwhere the last inequality assumes $n$ is sufficiently large. Therefore, for $i = 2,\\dots,n$,\n\\begin{equation*}\n\\frac{1}{\\exp(\\hat \\theta_1 + \\hat \\theta_i)-1} \\geq \\frac{1}{\\exp(\\hat \\theta_1 + \\hat \\theta_n)-1} \\geq \\frac{n}{2(\\exp(M)-1)}.\n\\end{equation*}\nHowever, this implies\n\\begin{equation*}\n\\begin{split}\n\\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}} \\geq \\|\\mathbf{d} - \\mathbf{d}^\\ast \\|_\\infty\n\\geq |d_1-d_1^\\ast|\n&\\geq -\\sum_{j=2}^n \\frac{1}{\\exp(\\theta_1 + \\theta_j)-1} +\\sum_{j=2}^n \\frac{1}{\\exp(\\hat\\theta_1 + \\hat\\theta_n)-1} \\\\\n&\\geq - \\frac{(n-1)}{\\exp(L)-1} + \\frac{n(n-1)}{2(\\exp(M)-1)},\n\\end{split}\n\\end{equation*}\nwhich cannot hold for sufficiently large $n$, as the right hand side in the last expression grows faster than the left hand side on the first line.\n\\end{enumerate}\n\nThe analysis above shows that we have $\\|\\hat \\theta\\|_\\infty < 2M$ for all sufficiently large $n$. Plugging in this result to~\\eqref{Eq:Consistency-R2} gives us\n\\begin{equation*}\n\\begin{split}\n\\|\\theta-\\hat\\theta\\|_\\infty\n&\\leq \\frac{2}{n} \\: \\frac{(\\exp(5M)-1)^2}{\\exp(5M)} \\: \\sqrt{\\frac{16 kn \\log n}{\\gamma L^2}}\n\\leq \\frac{8 \\: \\exp(5M)}{L} \\: \\sqrt{\\frac{k \\log n}{\\gamma n}}.\n\\end{split}\n\\end{equation*}\nFinally, taking into account the issue of the existence of the MLE, we conclude that for sufficiently large $n$, with probability at least\n\\begin{equation*}\n\\left(1-\\frac{1}{n^{k-1}}\\right)\\left(1-\\frac{2}{n^{k-1}}\\right) \\geq 1-\\frac{3}{n^{k-1}},\n\\end{equation*}\nthe MLE $\\hat \\theta \\in \\Theta$ exists and satisfies\n\\begin{equation*}\n\\|\\theta-\\hat\\theta\\|_\\infty\n\\leq \\frac{8 \\: \\exp(5M)}{L} \\: \\sqrt{\\frac{k \\log n}{\\gamma n}},\n\\end{equation*}\nas desired. This finishes the proof of Theorem~\\ref{Thm:ConsistencyInfiniteDisc}.\n\n\n\n\n\n\\section{Discussion and future work}\n\\label{Sec:Discussion}\n\nIn this paper, we have studied the maximum entropy distribution on weighted graphs with a given expected degree sequence. In particular, we focused our study on three classes of weighted graphs: the finite discrete weighted graphs (with edge weights in the set $\\{0,1,\\dots,r-1\\}$, $r \\geq 2$), the infinite discrete weighted graphs (with edge weights in the set $\\mathbb{N}_0$), and the continuous weighted graphs (with edge weights in the set $\\mathbb{R}_0$). We have shown that the maximum entropy distributions are characterized by the edge weights being independent random variables having exponential family distributions parameterized by the vertex potentials. We also studied the problem of finding the MLE of the vertex potentials, and we proved the remarkable consistency property of the MLE from only one graph sample.\n\nIn the case of finite discrete weighted graphs, we also provided a fast, iterative algorithm for finding the MLE with a geometric rate of convergence. Finding the MLE in the case of continuous or infinite discrete weighted graphs can be performed via standard gradient-based methods, and the bounds that we proved on the inverse Hessian of the log-partition function can also be used to provide a rate of convergence for these methods. However, it would be interesting if we can develop an efficient iterative algorithm for computing the MLE, similar to the case of finite discrete weighted graphs.\n\nAnother interesting research direction is to explore the theory of maximum entropy distributions when we impose additional structures on the underlying graph. We can start with an arbitrary graph $G_0$ on $n$ vertices, for instance a lattice graph or a sparse graph, and consider the maximum entropy distributions on the subgraphs $G$ of $G_0$. By choosing different types of the underlying graphs $G_0$, we can incorporate additional prior information from the specific applications we are considering.\n\nFinally, given our initial motivation for this project, we would also like to apply the theory that we developed in this paper to applications in neuroscience, in particular, in modeling the early-stage computations that occur in the retina. There are also other problem domains where our theory are potentially useful, including applications in clustering, image segmentation, and modularity analysis.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction.}\n\nLong-period comets experience little solar exposure and heating, which means they can be used to reveal conditions that existed during the formation and early evolution of the solar system (Delsemme 1977; Lowry \\&{} Fitzsimmons 2005). The main evidence supporting this argument is that the post-perihelion activity decreases fast after the cut-off of water sublimation at 3 AU (Fern\\'andez 2005), and then the activity usually stops until the next apparition. A few exceptional comets, however, displayed activity far beyond 3 AU, which can be explained by the sublimation of CO, but the processes involved are not well understood (Mazzotta Epifani et al. 2007). Because of this, cometary activity at large heliocentric distances has raised a considerable interest recently, addressing the question of how intact the matter in comets is (Lowry et al. 1999). For example, dust activity throughout the entire orbit could result in a continuous resurfacing of the nucleus, while the surface composition could also be altered. The observed diversity of nucleus colors (Luu 1993; Jewitt 2002) may indicate that this is indeed the case. On the other hand, comets with disguised distant activity will have larger mass loss rate per orbit than we estimate, leading to overestimated comet lifetimes (Lowry et al. 1999; Mazzotta Epifani et al. 2007) and underestimated replenishment rate of the zodiacal dust (Liou et al. 1995).\n\nIn recent years, a number of studies reported on short-period comets active between 3 and 7 AU (e.g. Lowry et al. 1999; Lowry \\&{} Fitzsimmons 2001, 2005; Lowry \\&{} Wiessman 2003; Snodgrass et al. 2006, 2008; Mazzotta Epifani et al. 2006, 2007). These surveys aimed at the investigation of the bared nucleus, but a surprisingly high number of comets showed comae and even dust tails at large ($R>3)$ heliocentric distances, where volatile sublimation is expected to be low. The activity of some long period comets is similar, with ocassionally long dust tails (Szab\\'o et al. 2001, 2002), while 11 Centaur objects are also known with cometary activity (see e.g. Rousselot 2008 and references therein). Chiron is known to be active at solar distances between 8 and 14 AU (Meech et al. 1997), and was seen to display considerable outgassing near aphelion (at 17.8--18.8 AU) between 1969 and 1977 (Bus et al. 2001). Meech et al. (2004) found that the Oort-cloud comet C\/1987 H1 (Shoemaker) displayed an extensive tail at all distances between 5 and 18 AU, which is, as of this writing, the most distant example of cometary activity.\n\nDiscovered at 7.2 AU from the Sun, C\/1995 O1 (Hale--Bopp) has been a prime target for cometary studies. In prediscovery images it had a faint coma 0\\farcm4 in diameter and a total magnitude of $\\sim$18 (McNaught \\&{} Cass 1995), while the dust production rate was $\\approx 500$ kg\/s (Fulle et al. 1998) at a solar distance of 13.1 AU. At 7.0 AU, NIR absorption of water ice was detected (Davies et al. 1997). At that time the activity was driven by CO production (Biver et al. 1996, Jewitt et al. 1996), which switched to a water-driven activity at around 3 AU (Biver et al. 1997, Weaver et al. 1997). Approaching the 0.9 AU perihelion distance, the dust production rate was $2\\times 10^6$ kg\/s (Jewitt \\&{} Matthews, 1999). The size distribution of the dust, especially in the jets, showed a dominance of $\\lesssim0.5\\ \\mu$m grains, smaller than in any other comets. This was indicated by the unprecedentedly large superheat ($T_c \/ T_{bb}$ between 1.5--2 [Hayward et al. 2000] or 1.5--1.8 [Gr\\\"un et al. 2001]), and the scattering albedo and polarization (Hayward et al. 2000). The water production was $~10^{31}$ molecule\/s, the largest value ever observed. The dust to gas ratio was very high, between 5 and 10, regardless of the solar distance (Colom et al. 1997, Lisse et al. 1997, Weaver et al. 1999). The production rates observed post-perihelion were similar to those observed pre-perihelion (Capria et al. 2002), foreshadowing long-lasting distant activity. We have indeed detected evidence for cometary activity at 25.7 AU from the Sun; this Letter presents our major findings based on broadband imaging in late 2007.\n\n\\section{Observations}\n\n\nNew observations were taken with the 2.3 m ANU telescope at the Siding Spring Observatory on 2007 October 20, 21 and 22. The solar distance of the comet was 25.7 AU. We took 9$\\times$240 s exposures in Johnson-Cousins $VR_C$ filters with a 2$\\times$2 binned image scale of 0\\farcs67\/pixel. The seeing was 2\\farcs0--2\\farcs5 on the three nigths (see Table 1 for exposure data and ephemerides). \n\nThe images were corrected in a standard fashion, including bias and flat-field correction and fringing correction of the $R_C$ images. Every night we aligned and co-added the images by fitting a coordinate grid to the stars, yielding a ``star field'' image for photometric calibrations. The images were then re-aligned with respect to the proper motion of the comet, to get untrailed ``comet'' images. In this step, the MPC ephemerides at the time of each observation were used to match the individual frames. Fig. \\ref{zsaner} shows the ``comet'' image on 2007 October 21. The estimated size of the coma is 180$\\times 10^3$ km, slightly elongated north\/southward (Fig. \\ref{ME}).\n\n\\subsection{Photometry}\n\nA good proxy to the dust content inside the coma is its brightness, which we have measured by aperture photometry in a single aperture of 14\\arcsec{} across. On October 21, the comet and field stars were calibrated with all-sky photometry using the SA 98 field  of Landolt (1992), observed between airmasses 1.17 and 1.65 (Hale--Bopp was at $X=1.72$). Due to the slow apparent motion, we could use the same field stars as local standards on the other two nights as well. The measured brightnesses on October 21 were $V=20\\fm70\\pm0.1$, $R_C=20\\fm04\\pm0.1$. This corresponds to $Af\\rho=30~000$ cm, according to the definition by A'Hearn et al. (1984). For comparison, this value is twice as large as that for 29P\/Schwassmann-Wachmann 1 in outburst (Szab\\'o et al. 2002), and 3 times larger than for 174P\/Echeclus in outburst (Rousselot, 2008).\n\n\\subsection{Morphology}\n\nThe observed brightness can be converted to albedo $\\times$ dust surface, $a_RC$ (Eddington, 1910), which is the cross section of reflecting particles ($C$, in m$^2$) in the aperture, multiplied by the $a_R$ geometric albedo in the $R_C$ photometric band. It is calculated as \n\\begin{equation}\na_RC = {2.22\\times 10^{22} \\pi R^2 \\Delta^2 10^{0.4(m_{\\sun} - m_{\\rm comet})} \\over 10^{-0.4\\alpha\\beta}},\n\\end{equation} \nwhere $m_{\\sun}=-27\\fm11$, the apparent $R_C$ brightness of the Sun, and the $\\beta$ phase coefficient is usually assumed to be 0.04. Substituting the measured total brightness yields $a_RC\\approx 4300$ km$^2$. \nFor comparison, this cross section is 450 times larger than that of the dust cloud ejected in the Deep Impact experiment (e.g. Milani et al. 2007). After calibrating the image flux, the azimuthally averaged comet profile was determined from surface photometry. From this the local filling factor of the dust, $f$, can directly be expressed by replacing $m_{\\rm comet}$ with the $\\mu$ surface brightness:\n\\begin{equation}\na_Rf={1.34\\times 10^{17} R^2 10^{0.4(m_{\\sun} - \\mu)} \\over 10^{-0.4\\alpha\\beta}},\n\\end{equation}\nthat is the measured surface brightness relative to that of a reflecting surface with 1.0 albedo. We found that the surface brightness was 20\\fm3 in the inner coma, corresponding to $a_Rf \\approx 9\\times 10^{-6}$, which remained above $10^{-6}$ in the inner 70~000 km (Fig. 3). \n\n\\section{Discussion}\n\n\nAs we describe below, the observations are consistent with a CO-driven activity. Following Fern\\'andez (2005), the thermal equilibrium of the absorbed radiation, the emitted radiation and the latent heat lost by sublimation can be written as \n\\begin{equation}\n{F_{\\sun}\\over R^2}\\pi r^2 = 4 \\pi r^2 \\sigma T^4 + 4\\pi r^2 f \\zeta(T) l_s,\n\\end{equation} \nwhere $r$ is the radius of the nucleus, $T$ is the temperature, $f<1$ is the fraction of the active area, $\\zeta(T)$ is the gas production rate in molecules\/m$^2$\/s, and $l_s$ is the latent heat loss per one molecule. CO molecules deposited on CO ice (e.g. inclusions) and CO molecules deposited on H$_2$O can sublimate at large solar distances, due to their high volatility. By neglecting the CO ice inclusions, CO is condensed on water ice, for which $l_s=10^{-20}$ J\/molecule (Delsemme 1981), and $\\log\\zeta(T)\\approx 755.7\/T - 35.02$ (Mukai et al. 2001). The heat loss of such a nucleus is plotted in Fig. \\ref{sublimfig} for different values of $f$ and as a function of temperature.\n\nThe equivalent temperature of a freely sublimating ice globe ($f=1$) at 25.7 AU is 48.0 K, which is slightly less than 54.8 K for a blackbody ($f=0$), due to the sublimation of $2\\times 10^{19}$ molecules\/m$^2$\/s. This corresponds to $Q(CO)=4\\pi r^2\\zeta(T)=2.1\\times 10^{20} r^2$ molecule\/s. If the active area covers 1\\%{} of the surface, the equilibrium temperature is 53.1 K, $\\zeta(T)=6.2\\times 10^{20}$ molecule\/s\/m$^2$, $Q(CO)=8\\times 10^{19} r^2$ molecule\/s. The $Q(CO)$ production rate and the temperature just slightly depend on $f$, thus we can get a reasonable estimate assuming $f=0.01$. The thermal velocity of the gas is $u_g=\\sqrt{3kT\/m_{\\rm CO}} = 210$ m\/s, which is enough to carry off small dust grains ($m_{\\rm CO}$ is the mass of one CO molecule).\nThe acting drag force $F_D=\\pi a^2 u_g \\zeta(T) m_{\\rm CO},$ where $a$ is the size of the dust particle. Particles are carried off if the drag force exceeds the gravitation $F_D>F_G=(4 \\pi \/3)^2{G\\rho_n\\rho_p a^3 r},$ $\\rho_n$ and $\\rho_p$ are the density of the nucleus and the dust particle, respectively. Thus, the maximum radius of the escaping dust particles is \n\\begin{equation}\na_{\\rm max} = {9\\over 16 \\pi} {u_g \\zeta(T) m_{\\rm CO} \\over \\rho_n\\rho_p G r}.\n\\end{equation} \nFor an order-of-magnitude estimate we assumed $\\rho_n=1000$ kg\/m$^3$ (Capria et al. 2002), $\\rho_p=2500$ kg\/m$^3$, $r=15$ km (Meech et al. 2004)  or $r=30$ km (Fern\\'andez, 2000), leading to $a_{max}\\approx 100\\ \\mu$m, and $Q(CO)\\approx 1.7\\times 10^{28}\/$s$ = 790$ kg\/s in the case of the $r=15$ km nucleus and $Q(CO)\\approx 6.8\\times 10^{28}\/$s$ = 3160$ kg\/s if $r=30$ km.\n\n\nWith a more sophisticated model (the surface is dominated by water ice instead of CO on water, the majority of CO is present in inclusions, crystallization heats the nucleus, the nucleus rotates), Capria et al. (2002) predicted $Q(CO)=5\\times 10^{27}$ molecule\/s (equivalent to $230$ kg\/s) at 25 AU (see Fig. 4 in their paper). Assuming that the dust loading remained high ($m_{dust}\/m_{gas}=$1--10), the total dust production ranges from 230 to 2300 kg\/s. \nFor example, if 500 kg\/s dust is produced in the form of 1 $\\mu$m-sized particles, the projected area is 0.25 km$^2$. Assuming a 0.05 albedo results in a 12,000 m$^2$ excess of $a_RC$ every second. With these assumptions, the nucleus can produce the observed amount of matter in the coma within $\\sim$5 days. The measured radius of the coma and the time-scale of dust production needed gives a measure of dust ejection velocity as $90\\ 000$~km$\/5$ days$ \\approx 210$~m\/s, which is consistent with the thermal gas velocity derived at 25.7 AU.\n\n\nThis self-consistent picture is also supported by the brightness variation of Hale--Bopp between 10 AU and 26 AU. In Fig. \\ref{lcs}, we plot the observed brightness of Hale--Bopp (collected from ICQ and MPC bulletins) against the solar distance. For comparison, data for 6 dynamically young Oort-comets are also plotted (Meech et al. 2004). Hale--Bopp was consistently 3--5 magnitudes brighter than these Oort-comets at all distances $\\lesssim$15 AU. Beyond that, other comets quickly disappeared as opposed to Hale--Bopp, which kept its slow rate of fading. We also show a theoretical light curve predicted from the CO production curve by Capria et al. (2002), after scaling with $R^{-2}\\Delta^{-2}$ and a constant dust loading. The overall agreement indicates that the distant brightness change is consistent with a CO-driven activity.\n\nAn alternative explanation of the present activity of Hale--Bopp could be that the light halo is not a real coma but a preserved debris tail (e.g. Jenniskens et al. 1997; Sekanina et al., 2001).\nComets with periods over a few thousand years are unlikely to preserve dense debris tail, but, as Lyytinen \\& Jenniskens (2003) remarks, giant comets such as Hale--Bopp can be exception. At the current position, the orbital path of Hale--Bopp is almost parallel to the line of sight, with only 5$^\\circ$ inclination and a solar phase angle of 2.2$^\\circ$. Thus, a hypothetical thin, $\\sim$1--2 million km long debris tail would appear as a $\\sim$100--200 thousand km long ``tail'' in projection, which is approximately the diameter of the light halo around Hale--Bopp. However, this scenario  is not likely, because the optically detected dust trails are all very thin, only $\\sim$10--20$\\times 10^3$ km wide, and their projected images always appear as a thin feature pointing out of the nucleus (Ishuguro et al. 2007; Sarugaku et al. 2007). Our observation of a nearly spherical light halo, with the nucleus approximately in the center, is not compatible with such a dust trail. \n\n\n\\section{Conclusion}\nComet Hale--Bopp has been the single most significant comet encountered by modern astronomy and 11 years after perihelion it still displays fascinating phenomena. The detected activity can be well explained by theoretical models invoking CO sublimation at large heliocentric distances. Compared to other young Oort-cloud comets, the long-term behaviour of Hale--Bopp seems to suggest genuine differences beyond the difference in nucleus size.\n\nThe main results of this paper can be summarized as follows:\n\\begin{enumerate}\n\\item{} We detected cometary activity of Hale--Bopp at 25.7 AU, which is the most distant activity detection so far. \n\\item{} Our analysis indicates that the extrapolation of the Capria et al. (2002) model works very well for the distant Hale--Bopp, which confirms the physical assumptions of this model.\n\\item{} Further observations with 8 m-class telescopes can help constrain the presence of gas in the coma, effects of superheat due to small dust particles, and, ultimately, the cessation of mass loss processes in Hale--Bopp.\n\\end{enumerate}\n\n\\acknowledgments This research was supported by the ``Bolyai J\\'anos'' Research Fellowship of the Hungarian Academy of Sciences and a Group of Eight European Fellowship (Gy.M.Sz.), a University of Sydney Research Fellowship (L.L.K.) and a Hungarian State ``E\\\"otv\\\"os'' Fellowship (K.S.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Milky Way disc models}\nThe most elaborate model of the present day Milky Way is the Besan\\c{c}on Galaxy Model (BGM) including extinction, the bar, spiral arms and the warp. The most recent version of the BGM proposes an update of the initial mass function (IMF) and the star formation history (SFR) of the disc (Czekaj et al. 2014). Nevertheless there are still open issues to be solved concerning the degeneracy of the SFR and IMF and a consistent chemical abundance model. Our alternative local disc model (JJ-model) based on the kinematics of main sequence stars (Just \\& Jahrei{\\ss} 2010), the stellar content in the solar neighbourhood (Rybizki \\& Just 2015) and Sloan Digital Sky Survey (SDSS) star counts to the north Galactic pole (Just et al. 2011) has a significantly higher accuracy compared to the old BGM (Gao et al. 2013). In the JJ-model the local SFR, IMF, AVR (age--velocity dispersion relation) are determined self-consistently and it includes a simple chemical enrichment model. In order to extend the JJ-model over the full radial range of the disc, we have used the Jeans equation for the asymmetric drift to connect local dynamics with the radial scale-lengths of stellar sub-populations (Golubov et al. 2013). Based on RAVE (RAdial Velocity Experiment, Kordopatis et al. 2013) data we found an increasing scale-length with decreasing metallicity, which is consistent with a negative overall metallicity gradient of the disc. On the other hand Milky Way-like galaxies show a radial colour gradient of the disc to be bluer and younger in the outer part. Combining both observations immediately shows that the chemical enrichment in the inner disc must be faster\/larger compared to the outer disc. There are different paths of this inside-out growth of the disc. Models based on a Kennicutt-Schmidt law grow faster in the inner part of the disc due to higher densities. Alternatively, there may be a delay in star formation in the outer disc induced by the star formation threshold. It is a challenge to disentangle these different scenarios and to build a consistent disc model. Precise ages of main sequence stars (calibrated by astroseismology, e.g.) would be the silver bullet to solve this problem, but currently the more promising support comes from observations of abundances and abundance ratios of heavy elements for large stellar samples distributed over the full disc range. The $\\alpha$-enhancement is a good tracer for the enrichment timescale and can be used to infer the evolution history of the disc. It can also be used to disentangle the thin and thick disc without introducing a kinematic bias (see e.g. Lee et al. 2011).\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig1}\n\\caption{Normalised present day age distributions of main sequence stars with different lifetimes in the solar neighbourhood. For comparison the normalised SFR and the corresponding gas infall rate are shown.}\n\\label{just-fig1}\n\\end{figure}\n\nObservational data of stellar populations appear in two fundamentally different kinds leading to very different constraints on the models. Star counts (like luminosity functions, colour-magnitude diagrams, metallicity distribution functions (MDF), or velocity distribution functions) provide quantitative information about the stellar populations. They require complete datasets or a detailed understanding of incompleteness. On the other hand, correlations (like the AVR, the $\\alpha$-enhancement as function of metallicity, or the asymmetric drift--metallicity relation) tell us something about the dominating physical processes of the disc evolution. They require an unbiased selection of stars (or a grasp of the biases).\n\n\\section{Chemical enrichment}\n\nMany analytic chemical evolution models (see Matteucci 2012 for an overview) are still local, annuli in case of the disc, and rely at least partly on the instantaneous recycling approximation (IRA). In order to overcome these restrictions,  Sch\\\"onrich \\& Binney (2009) quantified the impact of radial mixing of stars and gas in an analytic model, whereas models based on numerical simulations start to incorporate a chemical evolution network to reproduce abundance distributions in detail (e.g. Minchev et al. 2014; Kubryk et al. 2015a,b).\n\nAs a first step to a consistent chemical enrichment in the framework of the JJ-model (default model A of Just \\& Jahrei{\\ss} 2010) we analyse the $\\alpha$-enhancement in a local one-zone model with gas infall. For comparison we use two local volume-complete datasets of main sequence stars, namely the Geneva-Copenhagen sample (GCS), where the $\\alpha$-enhancement was determined by Str\\\"omgren photometry (Casagrande 2011), and the Hipparcos sample, where the abundances were determined by high resolution spectra (Fuhrmann 2011). The chemical evolution depends on the SFR, the IMF and the gas infall rate. For comparison with the local sample, the local age distribution is needed additionally, since the dynamical heating given by the AVR results in an age-dependent vertical dilution factor measured by the vertical thickness of the corresponding sub-populations. \nThe SFR and the local age distribution for main sequence stars with different lifetimes, normalised to an average of 1\/Gyr, are shown in Fig.~\\ref{just-fig1} for the JJ-model. \nIn order to reproduce the age-selection for specific types of stars we feed the IMF and chemical enrichment of the JJ-model combined with a flat SFR into the {\\sc Galaxia} tool (Sharma et al. 2011) with PARSEC isochrones (Bressan et al. 2012) and create a large Mock sample (see Fig.~\\ref{just-fig2} for the CMD). Then we select the corresponding region in the CMD and weight the selected stars with the age distribution given by the JJ-model for the volume of interest (e.g. the yellow box in Fig.~\\ref{just-fig2} and the red line in Fig.~\\ref{just-fig1} to reproduce the local G dwarf sample).\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig2}\n\\caption{Model CMD with unweighted isochrones corresponding to a uniform age distribution. Red clump and lower main sequence selection boxes are shown in red and yellow, respectively.}\n\\label{just-fig2}\n\\end{figure}\n\nThere are two free parameters in the model. The high-mass slope of the IMF above 8\\,M$_{\\odot}$ was not determined in Rybizki \\& Just (2015), because there are no O- and early B-type stars in the solar neighbourhood. Secondly, the gas infall rate can be varied to reproduce the abundance distributions and the $\\alpha$-enhancement. The only boundary condition here is the present day surface density of gas. \n\nSome  $\\alpha$-elements (we use oxygen and magnesium) are predominantly produced by core collapse supernovae (SN2) of massive stars. Due to the short lifetime of the progenitor stars, the IRA is a good approximation and we can use the observed O- and Mg-abundance distributions to derive the gas infall rate. The result of our fiducial model (with high-mass slope -2.7 for the IMF (the Salpeter IMF has a slope of -2.35) and SN2 yields of Fran\\c{c}ois et al. (2004) is shown in Fig.~\\ref{just-fig1} with the same scaling as the SFR.  \n\nThe yields of O and Mg depend strongly on the high-mass slope of the IMF. Unfortunately, there are inconsistent SN2 yields published in the literature. This results in a degeneracy of the IMF slope and the yield set in the [O\/Mg] abundance ratio. An example, with the gas infall fixed to the fiducial case, is shown in Fig.~\\ref{just-fig3} with the empirically calibrated yields of Fran\\c{c}ois et al. (2004) and the theoretical yields of Chieffi \\& Limongi (2004). The Chieffi yields combined with an IMF slope of -2.7 (dashed lines) lead to the same [O\/Mg]$\\approx$0.12\\,dex (vertical offset in Fig.~\\ref{just-fig3}) as the Fran\\c{c}ois yields with an IMF slope of -2.3. On the other hand the oxygen yields depend much stronger on the IMF slope than the Mg yields. We conclude that it is required for a consistent disc model to have a more detailed look on the different $\\alpha$-elements (and clearly define, which elements are used to determine the $\\alpha$-enhancement) in order to fix the high-mass IMF slope and to determine the correct yields.\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig3}\n\\caption{The symbols show the $\\alpha$-enhancement for the datasets of Casagrande and Fuhrmann. The full lines (red for Mg, cyan for O) show the Fran\\c{c}ois yields combined with different IMF slopes (-2.3, -2.7, -3.0 from top to bottom). The dashed lines correspond to the Chieffi yields with an IMF slope of -2.7.}\n\\label{just-fig3}\n\\end{figure}\n\nAfter the determination of the gas infall rate and the IMF slope based on the $\\alpha$-element distribution, we derive in the next step the distribution in [Fe\/H]. The yields of supernovae type 1a (SN1a) are usually parametrised by the delay time distribution (DTD) and scaled by a number fraction of planetary nebulae (PN) exploding as SN1a. We have chosen a DTD with maximum at 1\\,Gyr and a decay timescale of 2.5\\,Gyr combined with a fraction of 0.2\\% exploding PNs. The resulting abundance distributions are smoothed by a rms scatter of 0.07\\,dex. In Fig.~\\ref{just-fig4} the [Mg\/H] and [Fe\/H] distributions are shown in comparison to the Casagrande and the Fuhrmann sample. There are still some issues to be solved (the low metallicity bump and the systematic shift at the high metallicity end), but the general element distributions and correlations can be reproduced in the framework of the local JJ-model.\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig4a}\n\\includegraphics[width=0.48\\textwidth]{just-fig4b}\n\\caption{Normalised [Mg\/H] and [Fe\/H] distributions compared to the datasets of Casagrande and Fuhrmann.}\n\\label{just-fig4}\n\\end{figure}\n\n\\section{Age distributions and vertical gradients}\n\nSince different types of stars have different age distributions due to different evolutionary stages, the resulting [Fe\/H] abundance distributions do also vary. The resulting [Fe\/H] distributions in the solar neighbourhood for lower main sequence, red clump, giant, and supergiant stars are shown in the top panel of Fig.~\\ref{just-fig5} as predicted by the JJ-model. Especially the red clump stars are very important, because they can be identified easily in the CMD and observed over large distances.\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig5a}\n\\includegraphics[width=0.48\\textwidth]{just-fig5b}\n\\includegraphics[width=0.48\\textwidth]{just-fig5c}\n\\caption{Top panel: [Fe\/H] distributions of different stellar types in the solar neighbourhood as predicted by the JJ-model.\nMiddle panel: Predicted [Fe\/H] distributions of red clump stars with increasing $|z|$ above the Galactic plane.\nBottom panel: Age distributions of red clump stars at different $|z|$.}\n\\label{just-fig5}\n\\end{figure}\nThe dynamical evolution of the thin disc results in vertical gradients of the stellar age distribution and the corresponding abundance distributions. The middle panel of Fig.~\\ref{just-fig5} shows the strong variation of the [Fe\/H] distribution of red clump stars with distance $|z|$ from the Galactic plane. The bottom panel of Fig.~\\ref{just-fig5} shows the corresponding age distributions. This demonstrates that the intrinsic structure of the thin disc alone leads to strong vertical gradients, e.g. a significant shift of the maximum by -0.2\\,dex at 900\\,pc. The increasing contribution of the thick disc with increasing distance to the mid-plane enhances the gradients additionally (but is not included here).\n\n\\section{Radial gradients and inside-out growth}\n\nThe extension of the chemo-dynamical disc model over the full radial distance range depends on a full 3-dimensional model and requires homogeneous datasets covering a large range of Galactocentric distances. With a Jeans analysis of the asymmetric drift of RAVE data we have shown that the radial scale-length depends on metallicity (Golubov et al. 2013). The radial scale-length decreases from 2.9\\,kpc at low metallicity to 1.6\\,kpc at super-solar metallicity  and is essentially independent of colour along the main sequence (see Fig.~\\ref{just-fig6}). Only in the low metallicity bin there is a decline with colour, which may originate from the contribution of the thick disc. A consistent extrapolation of these radial scale-lengths in the framework of a disc formation and enrichment model, where the SFR, AVR, and gas infall depend on Galactocentric distance, can be tested and constraint by using the recent spectroscopic surveys RAVE (Kordopatis et al. 2013), SDSS\/APOGEE (Hayden et al. 2014, Holtzman et al. 2015), and\nGaia-ESO (GES, Smiljanic et al. 2014; Mikolaitis et al. 2014). Radial and vertical metallicity gradients are well identified in these surveys and the meridional plane distribution of the $\\alpha$-enhancement and other abundance ratios can be derived. \nUltimately, adding the high-precision parallaxes and proper motions expected from the first full data release of the Gaia mission in summer 2017 will allow direct measurements of density profiles and kinematic properties over a couple of kpc in the disc.\nBased on these data we expect to be able to construct a self-consistent evolutionary thin disc model, which allows to determine the inside-out growth of the disc. An important task will also be to disentangle the thin and thick discs, and to quantify the impact of radial migration necessary to understand the chemical and dynamical properties of the stellar disc(s).\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{just-fig6} \n\\caption{Radial scale-lengths at different metallicity bins of RAVE data as determined by the Jeans analysis of the asymmetric drift (from Golubov et al. 2013). Data points are colour bins along the main sequence.}\n\\label{just-fig6}\n\\end{figure}\n\n\\acknowledgements\nThis work was supported by Sonderforschungsbereich SFB 881 \"The Milky Way System\" (sub-project A6) of the German Research Foundation (DFG). The work contains contributions of the undergraduate students Sarah Casura and Simon Sauer.\n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe AdS\/CFT duality conjecture \\cite{malda0,klebanov,witten} has passed\nan impressive number of consistency checks \\cite{review}. However, among these\ntests only few are not relying in one or another way on structures  enforced by supersymmetry and\/or conformal invariance. In this situation it appears worthwhile to further analyse any possible constraint set by the first principles of\nquantum field theory and to check, whether they are fulfilled by the corresponding dual partners in string theory\/supergravity.\n\nIn this sense the present letter is devoted to the concavity of the potential \nbetween static sources in a gauge theory. In the Euclidean formulation Osterwalder-Schrader reflection\npositivity \\cite{os} ensures this property for potentials derived from Wilson\nloops \\cite{seiler,bachas}. In the AdS\/CFT context the issue of concavity has been raised in ref.\\cite{olesen}. But the discussion so far has not taken into account the degree of freedom connected with the relative orientation of the static sources ($Q\\bar Q$) in internal space.\n\nWe will fill this gap by analysing in some detail the consequences of OS reflection positivity for potentials derived in standard manner from Wilson loops\nfor contours coupling both to the gauge bosons and to a set of scalar\nfields in the adjoint representation.  We take the Wilson loop in the form\nsuggested in \\cite{rey,malda} and analysed  in various ways in \\cite{gross}.\nFor the case where the gauge bosons and the scalars are just the bosonic\nfields of $D=4$, ${\\cal N}=4$ super-Yang-Mills theory it has been characterised\nas an object of BPS type \\cite{gross}.\n\nOur discussion closely follows \\cite{bachas}. The new input in our\npresentation is the handling of the contour parameter dependent coupling\nto the scalars, which is described by a curve on $S^5$. We also take care\nof the fact that the Wilson loop of refs.\\cite{malda,gross} is the trace of \na generically non-unitary matrix.\n\nThe virtue of the arising concavity condition lies in its inequality\nproperty. It has to be fulfilled both for the classical SUGRA approximation\nand for the expressions obtained by adding successive corrections. Therefore,\na violation at any level of approximation on the superstring side\nwould indicate a breakdown of the corresponding duality.\n\\section{Generalised concavity for potentials derived from BPS Wilson loops}\nWe start with the functional ($A\\dot x=A_{\\mu}\\dot x^{\\mu},~\\phi\\theta =\n\\phi _j\\theta ^j,~~~\\mu =0,..,3,~~j=4,..,9 $)\n\\begin{equation}\nU_{ab}[x,\\theta]=\\left (P\\exp\\int \\{iA(x(s))\\dot x (s)+\\phi (x(s))\\theta (s)\n\\vert \\dot x\\vert\\}ds \\right )_{ab}~.\n\\label{10}\n\\end{equation}\nThe expectation value of its trace for a closed path $x(s)$ yields the Wilson loop under investigation \\cite{malda,gross}. $\\theta (s)$ specifies the coupling to the scalars $\\phi $\nalong the contour $x(s)$.\n\nA reflection operation ${\\cal R}$ is defined by\n\\begin{eqnarray}\n({\\cal R} x)^1 (s)=-x^1 (s)~;~~~~({\\cal R} x)^{\\alpha}(s)=x^{\\alpha}(s),~~~ \\alpha\\neq 1\n\\nonumber\\\\\n{\\cal R} U_{ab}[x,\\theta ]=\\overline{U_{ab}[{\\cal R} x, \\theta]}~.\n\\label{11}\n\\end{eqnarray}\nIn addition, it is useful to define in connection with an isometry \n${\\cal I} \\in O(6)$ \nof $S^5$ acting on the  path $\\theta (s)$\n\\begin{equation}\n{\\cal I} U_{ab}[x,\\theta ]~=~U_{ab}[x, {\\cal I}\\theta]~.\n\\label{11a}\n\\end{equation} \nFor linear combinations of $U$'s for different contours we extend ${\\cal R} $\nand ${\\cal I}$ linearly.\n\nUsing the hermiticity of the matrices $A,~\\phi$ in the form\n$\\overline{A}=A^t,~~\\overline{\\phi}=\\phi ^t$ we can reformulate the\nr.h.s in the second line of (\\ref{11}) applying the following steps \n\\begin{eqnarray}\n\\overline{U_{ab}[x,\\theta]}&=&\\left (\nP\\exp\\int _{s_i}^{s_f}\\{-iA^t(x(s))\\dot x (s)+\\phi ^t(x(s))\\theta (s)\\vert \\dot x\\vert\\}ds\\right )_{ab}\\nonumber\\\\\n&=&\\left (\n\\hat{P}\\exp\\int _{s_i}^{s_f}\\{-iA(x(s))\\dot x (s)+\\phi (x(s))\\theta (s)\\vert \\dot x\\vert\\}ds\\right )_{ba}~.\n\\label{12}\n\\end{eqnarray}\nHere $P,~\\hat{P}$ denote ordering of matrices from right to left with\nincreasing\/decreasing argument $s$. $\\hat{P}$ applied to the path $x$ yields the\nsame result as $P$ applied to the backtracking path\n\\begin{equation}\n({\\cal B} x)(s)~=~x(s_f+s_i-s),~~~({\\cal B} \\theta )(s)~=~\\theta (s_f+s_i-s)~.\n\\label{13}\n\\end{equation}\nTherefore, we get\n\\begin{equation}\n\\overline{U_{ab}[x,\\theta]}~=~U_{ba}[{\\cal B} x,{\\cal B}\\theta]~.\n\\label{14}\n\\end{equation}\nThis, combined with (\\ref{11}),(\\ref{11a}) yields finally\n\\begin{equation}\n{\\cal R} {\\cal I} U_{ab}[x,\\theta ]~=~U_{ba}[{\\cal B}{\\cal R} x,{\\cal B}{\\cal I}\\theta]~.\n\\label{15}\n\\end{equation}\nIt is worth pointing out that for the result (\\ref{14}) the presence\/absence of the\nfactor $i$ in front of the $A$ and $\\phi $ term in $U$ is crucial. One could\nconsider this as another argument for the choice favoured by the investigations\nof ref. \\cite{gross}.\\\\\n\nWe now turn to a derivation of the basic Osterwalder-Schrader positivity\ncondition in a streamlined form within the continuum functional integral\nformulation. All steps can be made rigorously by a translation into a lattice\nversion with local and nearest neighbour interactions.\n\nLet denote $H_{\\pm}=\\{x^{\\mu}\\vert \\pm x^1>0\\},~~~H_{0}=\\{x^{\\mu}\\vert \nx^1=0\\}$. Then we consider for a functional of two paths \n$x^{(1)},x^{(2)}\\in H_+$\n\\begin{equation}\nf[x^{(1)},\\theta ^{(1)};x^{(2)},\\theta ^{(2)}]~=~U_{ab}[x^{(1)},\\theta ^{(1)}]\n~+~\\lambda U_{ab}[x^{(2)},\\theta ^{(2)}],~~~~\\lambda ~~\\mbox{real}~,\n\\label{15a}\n\\end{equation}\n\\begin{eqnarray}\n\\langle f[x,\\theta ]{\\cal R} {\\cal I} f[x,\\theta ]\\rangle&=&\\int {\\cal D} A{\\cal D} \\phi\nf[x,\\theta ]\\overline{f[{\\cal R} x,{\\cal I}\\theta ]}~e^{-S}\n\\\\\n&=&\\int {\\cal D} A^{(0)}{\\cal D} \\phi ^{(0)}~e^{-S_0}\\nonumber\\\\\n&\\cdot &\\int _{(b.c.)}{\\cal D} A^{(+)}{\\cal D} \\phi ^{(+)} f[x,\\theta ]~e^{-S_+}\\cdot\n\\int _{(b.c.)}{\\cal D} A ^{(-)}{\\cal D} \\phi ^{(-)}\\overline{f[{\\cal R} x,{\\cal I}\\theta ]}~e^{-S_-}~.\n\\nonumber\n\\label{15b}   \n\\end{eqnarray}\n$\\pm ,~0$ on the fields as well as on the action indicates that\nit refers to points in $H_{\\pm},~H_{0}$. The index for the two paths has been\ndropped, and the boundary condition $(b.c.)$ is\n$$A^{(\\pm )}\\vert _{\\partial H_{\\pm}}=A^{(0)},~~~\\phi ^{(\\pm )}\\vert\n_{\\partial H_{\\pm}}=\\phi ^{(0)}~.$$ \nWith the abbreviation\n\\begin{equation}\nh[A^{(0)},\\phi ^{(0)},x,\\theta ]~=~\\int _{(b.c.)}{\\cal D} A^{(+)}{\\cal D} \\phi ^{(+)}\nf[x,\\theta ]~e^{-S_+}~,\n\\label{15c}\n\\end{equation}\nthe standard reflection properties of the action imply\n\\begin{equation}\n\\langle f[x,\\theta ]{\\cal R} {\\cal I} f[x,\\theta ]\\rangle ~=~\\int {\\cal D} A^{(0)}{\\cal D} \\phi\n^{(0)}~e^{-S_0}~h[A^{(0)},\\phi ^{(0)},x,\\theta ]\\cdot \\overline{h[A^{(0)},\\phi\n  ^{(0)},x,{\\cal I}\\theta ] }~.\n\\label{15d}\n\\end{equation}\nFor ${\\cal I} = ${\\bf 1}$ $ the integrand of the final integration over the fields\nin the reflection hyperplane $H_0$ is non-negative, hence\n\\begin{equation}\n\\langle f[x,\\theta ]{\\cal R}  f[x,\\theta ]\\rangle ~\\geq ~0~.\n\\label{15e}\n\\end{equation}\n\nFor nontrivial ${\\cal I} $ the situation is by far more involved. If there would be\nno boundary condition, the result of the half-space functional integral in\n(\\ref{15c}) would be invariant with respect to \n$\\theta\\rightarrow{\\cal I}\\theta $. A given boundary configuration in general breaks\n$O(6)$ invariance on $S^5$. But due to the $O(6)$ invariance of the action,\nthe functional integration measure and the $\\phi\\theta$ coupling in $f$, we have instead\n\\begin{equation}\nh[A^{(0)},{\\cal I}\\phi ^{(0)},x,{\\cal I}\\theta ]~=~h[A^{(0)},\\phi ^{(0)},x,\\theta ]~.\n\\label{15g}\n\\end{equation}\nThis implies\n\\begin{eqnarray}\n\\langle f[x,\\theta ]{\\cal R} {\\cal I} f[x,\\theta ]\\rangle ~=~\\int {\\cal D} A^{(0)}{\\cal D} \\phi\n^{(0)}~e^{-S_0}\\label{15h} \\\\\n\\cdot~\\frac{1}{2}\\left (~h[A^{(0)},\\phi ^{(0)},x,\\theta ]~\\overline{\nh[A^{(0)},\\phi ^{(0)},x,{\\cal I}\\theta ] }\\right .&+&\\left . h[A^{(0)},\\phi ^{(0)},\nx,{\\cal I} ^{-1}\\theta ]~\\overline{h[A^{(0)},\\phi ^{(0)},x,\\theta ]}~\\right ) ~,\n\\nonumber\n\\end{eqnarray}\nwhich says us only ($R$ real numbers)\n\\begin{equation}\n\\langle f[x,\\theta ]{\\cal R} {\\cal I} f[x,\\theta ]\\rangle \\in R ~~~~\\mbox{for}~~\n{\\cal I} ^2={\\bf 1}~.\n\\label{15i}\n\\end{equation}   \nThe statements (\\ref{15e}) and (\\ref{15i}) are rigorous ones. \nBeyond them we found no real proof for sharpening (\\ref{15i}) to an inequality\nof the type (\\ref{15e}) for some nontrivial ${\\cal I} $. For later application\nto the estimate of rectangular Wilson loops we are in particular interested\nin nontrivial isometries keeping the, by assumption common, $S^5 $ position\nof the endpoints of the contours on $H_0 $ fixed. Then ${\\cal I} ={\\cal I}_{\\pi}$,\ndenoting a rotation around this fixpoint with angle $\\pi $, are the only\ncandidates.\n\nAt least for boundary fields $\\phi ^{(0)}$ in (\\ref{15c}), which as a map\n$R^3\\rightarrow S^5$ have a homogeneous distribution of their image points on\n$S^5$, we can expect that for contours of the type discussed in connection \nwith fig.1 below in the limit of large $T$ the orientation of $\\theta $\nrelative to $\\phi ^{(0)}$ becomes unimportant. Therefore, we conjecture for this special situation  \n\\begin{equation}\n\\langle f[x,\\theta ]{\\cal R} {\\cal I}_{\\pi} f[x,\\theta ]\\rangle ~\\geq ~0~.\n\\label{15j}\n\\end{equation}\nFrom (\\ref{15e}) and (\\ref{15j}) for any real $\\lambda $ in (\\ref{15a})\nwe get via the \nstandard derivation of Schwarz-type inequalities\n\\begin{eqnarray}\n\\langle U_{ab}[x^{(1)},\\theta ^{(1)} ]~{\\cal R}{\\cal I} U_{ab}[x^{(2)},\\theta ^{(2)} ]\n\\rangle ^2&\\leq &\\langle U_{ab}[x^{(1)},\\theta ^{(1)} ]~{\\cal R}{\\cal I}\nU_{ab}[x^{(1)},\\theta ^{(1)}]\\rangle \\label{15f}\\\\\n&&~~~~~~~~~~~~\\cdot\\langle U_{ab}[x^{(2)},\\theta ^{(2)} ]~{\\cal R}{\\cal I}\nU_{ab}[x^{(2)},\\theta ^{(2)}]\\rangle ~.\n\\nonumber\n\\end{eqnarray}\nThis is a rigorous result for ${\\cal I} ={\\bf 1}$ and a conjecture for ${\\cal I} ={\\cal I} _{\\pi}$. \n\\\\ \n\nLet us continue with the discussion of a Wilson loop for a\nclosed contour which crosses the reflection hyperplane twice and which\nis the result of going first along $x^-\\in H_-$ and then along $x^+\\in H_+$.\nIn addition we restrict to cases of coinciding $S^5$ position at the\nintersection points with $H_0$ and treat in parallel ${\\cal I} ={\\bf 1},{\\cal I} _{\\pi}$\n\\begin{eqnarray}\nW[x^+\\circ x^-,\\theta ^+\\circ \\theta ^-]~=~\\sum _{ab}\n\\langle U_{ab}[x^+,\\theta ^+]~U_{ba}[x^-,\\theta ^-]\\rangle\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\label{19}\\\\\n=~\\sum _{ab}\\langle U_{ab}[x^+,\\theta ^+]~{\\cal R}{\\cal I}\\overline{U_{ba}[{\\cal R} x^-,{\\cal I} ^{-1}\\theta ^-]}\\rangle ~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n\\leq \\sum _{ab}\\langle U_{ab}[x^+,\\theta ^+]~{\\cal R}{\\cal I} U_{ab}[x^+,\\theta^+]\\rangle ^{\\frac{1}{2}}\\langle\\overline{U_{ba}[{\\cal R} x^-,{\\cal I} ^{-1}\\theta ^-]}~\n{\\cal R}{\\cal I} \\overline{U_{ba}[{\\cal R} x^-,{\\cal I} ^{-1}\\theta ^-]}\\rangle ^{\\frac{1}{2}} \n\\nonumber\\\\\n\\leq (\n\\sum _{ab}\\langle U_{ab}[x^+,\\theta ^+]~{\\cal R}{\\cal I} U_{ab}[x^+,\\theta ^+]\\rangle\n)^{\\frac{1}{2}}\n~(\\sum _{cd}\\langle \\overline{U_{cd}[{\\cal R} x^-,{\\cal I} ^{-1}\\theta ^-]}~{\\cal R}{\\cal I} \n\\overline{U_{cd}[{\\cal R} x^-,{\\cal I} ^{-1}\\theta ^-]}\\rangle\n)^{\\frac{1}{2}}.\n\\nonumber\n\\end{eqnarray}\nWe have used (\\ref{11}), (\\ref{11a}), ${\\cal R}\\cR x=x$, (\\ref{15f}) and the usual Schwarz inequality\nin the last step. Now with (\\ref{14}),(\\ref{15}) we get\n\\begin{eqnarray}\nW[x^+\\circ x^-,\\theta ^+\\circ \\theta ^-]\n&\\leq &\\left (\nW[x^+\\circ{\\cal B}{\\cal R} x^+,\\theta ^+\\circ {\\cal B}{\\cal I}\\theta ^+]\n\\right )^{\\frac{1}{2}}\\nonumber\\\\ \n&&~~~~~~~\\cdot ~\\left (\nW[{\\cal B}{\\cal R} x^-\\circ x^-,{\\cal B} {\\cal I} ^{-1}\\theta ^-\\circ \\theta ^-]\\right )^\n{\\frac{1}{2}}~.\n\\label{20}\n\\end{eqnarray} \n\\begin{figure}\n\\begin{center}\n\\mbox{\\epsfig{file=fig1.eps, width=100mm}}\n\\end{center}\n\\noindent {\\bf Fig.1}\\ \\ {\\it From left to right the contours\n$x^+\\circ x^-,~~x^+\\circ {\\cal B}{\\cal R} x^+,~~{\\cal B}{\\cal R} x^-\\circ x^-$.}\n\\end{figure}\n\nTo evaluate the potential between two static sources ($Q\\bar Q $) separated\nby the distance $L$ and located at fixed $S^5 $-positions $\\theta _Q,~\\theta _\n{\\bar Q}$ we need Wilson loops for rectangular contours of extension $L\\times T$ in the large $T$-limit. We choose the $S^5$-position on the two $L$-sides\nlinearly interpolating between $\\theta _Q$ and $\\theta _{\\bar Q}$ on the corresponding great circle. For this restricted set of contours the Wilson loop becomes a  function of $L,~T$ and the angle between $\\theta _Q$ and $\\theta _{\\bar Q}$, called $\\Theta $. \n\nIn addition it is useful to restrict ourselves to contours which are situated \nin planes orthogonal to the reflection hyperplane and with $T$-sides running\nparallel to it in a distance $\\frac{L\\pm\\delta}{2}$, see fig.1. Then \n${\\cal I} ={\\cal I}_{\\pi}$ reflects $\\theta ^{\\pm}(s)$, \nwhich both lie on the great circle through $\\theta _Q$ and $\\theta _{\\bar Q}$,\nwith respect to the common $S^5$-position of the points $A$ and $B$, see \nfig.1. As a consequence, (\\ref{20}) implies\n\\begin{equation}\nW(L,T,\\Theta)~\\leq ~\\left (W(L-\\delta,T,\\frac{L-\\delta}{L}\\Theta )\\right )^{\\frac{1}{2}}~\\left (W(L+\\delta,T,\\frac{L+\\delta}{L}\\Theta )\\right )^{\\frac{1}{2}}~,\n\\label{21}\n\\end{equation}\nwhich by standard reasoning yields for the static potential \n\\begin{equation}\nV(L,\\Theta )~\\geq ~\\frac{1}{2}\\left (V(L-\\delta ,\\frac{L-\\delta}{L}\\Theta )~+~\nV(L+\\delta ,\\frac{L+\\delta}{L}\\Theta )\\right )~.\n\\label{22}\n\\end{equation}\nThe last inequality implies the local statement \n$\\frac{d^2}{d\\delta ^2}V(L+\\delta ,\\frac{L+\\delta }{L}\\Theta )\\leq 0$, i.e.\n\\begin{equation}\n\\left ( L^2~\\frac{\\partial ^2}{\\partial L^2}~+~2L\\Theta ~\\frac{\\partial ^2}{\\partial L\\partial \\Theta }~+~\\Theta ^2~\\frac{\\partial ^2}{\\partial \\Theta ^2}\\right  )~V(L,\\Theta )~\\leq 0~.\n\\label{24}\n\\end{equation}\nIt means concavity on each straight line across the origin, in the relevant\npart of the $(L,\\Theta )$-plane, $0<L<\\infty ,~~0<\\Theta\\leq\\pi $.\\\\\n\nBoth (\\ref{22}) and (\\ref{24}) rely on the conjecture (\\ref{15j}). \nFrom the rigorous point of view we are allowed to use (\\ref{20}) \nfor ${\\cal I} ={\\bf 1}$ only. Then the paths generated on the r.h.s. are, with\nrespect to their $S^5$ properties, no\nlonger of the type with which we started on the l.h.s. On the part of the\nspace time contour orthogonal to $H_0$ we go e.g. from $\\theta _{Q}$\nto the common $S^5$ position of the points A and B and then $back$ to $\\theta\n_{Q}$. Since in the large $T$-limit, relevant for the extraction of the\n$Q\\bar Q $-potential, only the behaviour on the large $T$-sides matters,\nwe get\n\\begin{equation}\nV(L,\\Theta )~\\geq ~\\frac{1}{2}\\left (V(L-\\delta ,0 )~+~\nV(L+\\delta ,0 )\\right )~.\n\\label{24a}\n\\end{equation}\nThis means standard concavity at $\\Theta =0$ $and$\n\\begin{equation}\nV(L,\\Theta )~\\geq ~V(L,0)~.\n\\label{24b}\n\\end{equation}  \nIf the same steps are repeated for rectangles with large $T$-sides still \nparallel to $H_0$, but spanning a plane no longer orthogonal to $H_0$\none finds\n\\begin{equation}\nV(L,\\Theta )~\\geq ~\\frac{1}{2}\\left (V(\\alpha (L-\\delta ),0 )~+~\nV(\\alpha (L+\\delta ) ,0 )\\right )~,~~~0\\leq \\alpha\\leq 1~.\n\\label{24c}\n\\end{equation}\nThe only new information gained from (\\ref{24c}) is that $V(L,0)$ is\nmonotonically non-decreasing in $L$.\n\\section{Test of the generalised concavity condition for potentials derived via AdS\/CFT duality}\nThe simplicity of the calculation recipe for Wilson loops in the classical\nSUGRA approximation via AdS\/CFT duality allows to make statements on universal\nproperties of the arising $Q\\bar Q$-potential for a large class of SUGRA\nbackgrounds \\cite{minahan, kinar}. We now enter a discussion of (\\ref{24})\nwithin this framework. The metric of the SUGRA background is assumed in the \nform\n\\begin{equation}\nG_{MN}dx^Mdx^N~=~G_{00}(u)dx^0dx^0~+~G_{\\vert\\vert}(u)dx^mdx^m~+~G_{uu}(u)dudu~+~G_{\\Omega}(u)d\\Omega _5^2~.\n\\label{25}\n\\end{equation}\nThen with\n\\begin{equation}\nf(u)~=~G_{00}G_{\\vert\\vert },~~~g(u)~=~G_{00}G_{uu},~~~j(u)=G_{00}G_{\\Omega }\n\\label{26}\n\\end{equation}\nwe get along the lines of \\cite{malda, minahan, kinar, dorn}\n\\begin{eqnarray}\nL^{(\\Lambda )}&=&2\\sqrt{f_0}\\sqrt{1-l^2}\\int_{u_0}^{\\Lambda}\\sqrt{\\frac{gj}{f}}\n\\frac{du}{\\sqrt{j(f-f_0)+(jf_0-j_0f)l^2}}~,\\nonumber\\\\\n\\Theta ^{(\\Lambda )}&=&2l\\sqrt{j_0}\\int_{u_0}^{\\Lambda}\\sqrt{\\frac{gf}{j}}\n\\frac{du}{\\sqrt{j(f-f_0)+(jf_0-j_0f)l^2}}~,\\nonumber\\\\\nV^{(\\Lambda )}&=& \\frac{1}{\\pi}\\int_{u_0}^{\\Lambda}\\sqrt{gfj}~\n\\frac{du}{\\sqrt{j(f-f_0)+(jf_0-j_0f)l^2}}~.\n\\label{27}\n\\end{eqnarray}\nWe defined $f_0=f(u_0)$ etc. $\\Lambda $ is a cutoff at large values of $u$.\nIn the following our discussion will be restricted to values of $L$ and\n$\\Theta $ for which all expressions appearing under square roots above are\npositive and where the inversion $u_0=u_0(L,\\Theta ),~~l=l(L,\\Theta )$ is well\ndefined. (\\ref{27}) implies\n\\begin{eqnarray}\nV^{(\\Lambda )}&=&\\frac{1}{\\pi}\\int_{u_0}^{\\Lambda}\\sqrt{\\frac{g}{fj}}\\sqrt{j(f-f_0)+(jf_0-j_0f)l^2}\\nonumber\\\\\n&+&\\frac{1}{2\\pi}\\sqrt{f_0}\\sqrt{1-l^2}~L^{(\\Lambda )}~+~\\frac{1}{2\\pi}\n\\sqrt{j_0}~l~\\Theta ^{(\\Lambda )}~.\n\\label{28}\n\\end{eqnarray}\nNow we differentiate with respect to $u_0$ and $l$. After this $\\Lambda $\ncan be sent to $\\infty$ ending up with a relation for the renormalised\npotential $V$:\n\\begin{eqnarray}\n\\frac{\\partial V}{\\partial u_0}&=&\\frac{1}{2\\pi}\\sqrt{f_0}\\sqrt{1-l^2}~\n\\frac{\\partial L}{\\partial u_0}~+~ \\frac{1}{2\\pi}\\sqrt{j_0}~l~\n\\frac{\\partial \\Theta}{\\partial u_0}~,\\nonumber\\\\\n\\frac{\\partial V}{\\partial l}&=&\\frac{1}{2\\pi}\\sqrt{f_0}\\sqrt{1-l^2}~\n\\frac{\\partial L}{\\partial l}~+~ \\frac{1}{2\\pi}\\sqrt{j_0}~l~\n\\frac{\\partial \\Theta}{\\partial l}~.\n\\label{29}\n\\end{eqnarray}\nFor $V$ defined by (\\ref{27}) implicitly as a function of $L$ and $\\Theta $\nthis means\n\\begin{equation}\n\\frac{\\partial V}{\\partial L}~=~\\frac{1}{2\\pi}\\sqrt{f_0}\\sqrt{1-l^2},~~~\n\\frac{\\partial V}{\\partial \\Theta}~=~\\frac{1}{2\\pi}\\sqrt{j_0}~l~,\n\\label{30}\n\\end{equation}\ni.e. $V$ is monotonically nondecreasing both in $L$ and $\\Theta $.\nThe monotony in $\\Theta $ is in agreement with our rigorous result\n(\\ref{24b}).\\\\\n\nCalculating now second derivatives one arrives at ($f'_0=\\frac{df(u_0)}{du_0}$ etc.)\n\\begin{eqnarray}\n\\left ( L^2~\\frac{\\partial ^2}{\\partial L^2}~+~2L\\Theta ~\\frac{\\partial ^2}{\\partial L\\partial \\Theta }\\right .&+&\\left .\\Theta ^2~\\frac{\\partial ^2}{\\partial \\Theta ^2}\\right  )~V(L,\\Theta )~=\\label{31}\\\\\n&=&\\frac{1}{4\\pi\\sqrt{f_0j_0}}~(Lf'_0\\sqrt{j_0(1-l^2)}+\n\\Theta j'_0\\sqrt{f_0} ~l)~(L\\frac{\\partial u_0}{\\partial L}+\\Theta\\frac{\\partial u_0}\n{\\partial \\Theta})\\nonumber\\\\\n&+&\\frac{1}{2\\pi\\sqrt{1-l^2}}~(\\Theta\\sqrt{j_0(1-l^2)}\n-L\\sqrt{f_0}~l)~(L\\frac{\\partial l}{\\partial L}+\\Theta\\frac{\\partial l}\n{\\partial \\Theta})~.\n\\nonumber\n\\end{eqnarray}\nNeglecting for a moment the issue of internal space dependence by\nrestricting oneselves to the case $\\Theta =l=0$, one finds usual concavity \nin $L$ from (\\ref{31}) if $f'_0~\\frac{\\partial u_0}{\\partial L}\\leq 0$.\nThe last inequality is for $f'>0$ guaranteed by theorem 1 of ref.\\cite{kinar}.\n\\footnote{Our $f$ and $g$ are called $f^2$ and $g^2$ in that paper.}\n\nTherefore, for $\\Theta =0$ standard concavity of $Q\\bar Q$-potentials with respect to the \ndistance in usual space is guaranteed for the wide class of SUGRA backgrounds\ncovered by theorem 1 of ref.\\cite{kinar}.\n\nHowever, due to the more complicated structure of the l.h.s. of (\\ref{31})\nfor $\\Theta\\neq 0$ we did not found a similar general statement in the\ngeneric case. We can only start checking (\\ref{24}) case by case.\\\\\n\nAs our first example we consider the original calculation of Maldacena\n\\cite{malda} for the $AdS_5\\times S^5$ background. The result was\n($R^2=\\sqrt{2g^2_{YM}N}$)\n\\begin{equation}\nV(L,\\Theta )~=~-~\\frac{2R^2}{\\pi}\\frac{F(\\Theta )}{L}~,\n\\label{32}\n\\end{equation}\nwith\n\\begin{eqnarray}\nF(\\Theta )&=&(1-l^2)^{\\frac{3}{2}}\\left ( \\int_1^{\\infty}\\frac{dy}{\ny^2\\sqrt{(y^2-1)(y^2+1-l^2)}}\\right )^2~,\\nonumber\\\\\n\\Theta &=&2l\\int _1^{\\infty}\\frac{dy}{\\sqrt{(y^2-1)(y^2+1-l^2)}}~.\n\\label{33}\n\\end{eqnarray}\nDue to this special structure ($L\\frac{\\partial V}{\\partial L}=-V,~L^2\\frac\n{\\partial ^2}{\\partial L^2}V=2V,~ \\frac{\\partial \\Theta}{\\partial u_0}=0$), (\\ref{24}) is equivalent to\n\\begin{equation}\n\\Theta ^3~\\frac{d^2}{d\\Theta ^2}\\left (\\frac{F}{\\Theta }\\right )~\\geq 0~.\n\\label{34}\n\\end{equation}\nA numerical calculation of $\\frac{F}{\\Theta }$ confirms (\\ref{34}) clearly,\nsee fig.2.\\\\\n\nNext we discuss the large $L$ confining potential including internal\nspace dependence and $\\alpha ^{\\prime} $ corrections of the background derived in\n\\cite{dorn}. It has the form ($\\gamma =\\frac{1}{8}\\zeta (3)R^{-6}$, $T$\ntemperature parameter)\n\\begin{equation}\nV(L,\\Theta )~=~\\frac{\\pi R^2T^2}{2}(1-\\frac{265}{8}\\gamma)\\cdot L~+~\n\\frac{R^2}{4\\pi}(1+\\frac{15}{8}\\gamma )~\\frac{\\Theta ^2}{L}~+~O(1\/L^3)~.\n\\label{35}\n\\end{equation}\n\\begin{figure}\n\\begin{center}\n\\mbox{\\epsfig{file=fig2.eps, width=80mm}}\n\\end{center}\n\\noindent {\\bf Fig.2}\\ \\ {\\it $\\frac{F}{\\Theta}$ as a function of $\\Theta$.\nUse has been made of the representation in terms of elliptic integrals given\nin \\cite{malda}.}\n\\end{figure}\nAlthough this potential for $\\Theta \\neq 0$ violates naive concavity\n$\\frac{\\partial ^2V}{\\partial L^2}\\leq 0$, there is $no$ conflict with\nthe correctly generalised concavity (\\ref{24}). Applied\nto (\\ref{35}) the differential operator just produces zero.\n\\section{Concluding remarks}\nThe $Q\\bar Q$-potential derived \\cite{malda} from the classical SUGRA approximation\nfor the type IIB string in $AdS_5\\times S^5$  fulfils our generalised\nconcavity condition at $\\Theta \\geq 0$. This adds another consistency check\nof this most studied case within the AdS\/CFT duality. \n\nPotentials have been almost completely studied only for $\\Theta =0$ in other\nbackgrounds. At least partly, this might\nbe due to the wisdom to approach in some way QCD, where after all there is no place for a parameter like this angle between different orientations in $S^5$.\nHowever, one has to keep in mind that this goal, in the approaches discussed so far, requires some additional limiting procedure. Before the limit the full 10-dimensionality inherited by the string is still present. Fluctuation determinants\nin all 10 directions have to be taken into account for quantum corrections\n\\cite{olesen,theisen} and the $\\Theta $-dependence of the potentials is of \ncourse not switched off. \n\nAlthough we proved in classical SUGRA approximation monotony in $L$ and\n$\\Theta $ as well as concavity at $\\Theta =0$\nfor a whole class of backgrounds, we were not able to get a similar general\nresult on concavity for $\\Theta >0$. Further work is needed to decide, whether at all\ngeneral statements for $\\Theta >0$ are possible. Alternatively one should\nperform case by case studies for backgrounds derived e.g. from\nrotating branes \\cite{russo}, type zero strings \\cite{tseytlin} or nonsupersymmetric\nsolutions of type IIB string theory \\cite{sfetsos}.\n\nOn the field theory side further work is necessary to really prove\nthe conjectered inequality (\\ref{15j}), otherwise the available set of\nrigorous constraints on the $L$ $and$ $\\Theta $ dependent potential,\nbeyond the standard concavity at $\\Theta =0$, would contain only the very mild\ncondition (\\ref{24b}).\n\\\\[10mm]\n{\\bf Acknowledgement}\\\\\nH.D. thanks G. Bali, H.-J. Otto and C. Preitschopf for\nuseful related discussions.\nThe work of V.D.P. was supported by GRACENAS grant, project No\n97-6.2-34; RFBR-DFG grant, project No 96-02-00180 and RFBR grant,\nproject No 99-02-16617.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and summary}\nThe pions ($\\pi^+,\\pi^0,\\pi^-$) are the Goldstone bosons of spontaneous chiral \nsymmetry breaking in QCD. Their strong interaction dynamics at low energies \ncan therefore be calculated systematically (and accurately) with chiral \nperturbation theory in form of a loop expansion based on an effective chiral\nLagrangian. The very accurate two-loop predictions \\cite{cola} for the S-wave \n$\\pi\\pi$-scattering lengths, $a_0=(0.220\\pm 0.005)m_\\pi^{-1}$ and $a_2=(-0.044\n\\pm 0.001)m_\\pi^{-1}$, have been confirmed experimentally by analyzing the \n$\\pi\\pi$ final-state interaction effects occurring in various (rare) charged \nkaon decay modes \\cite{bnl,batley,cusp}. Electromagnetic processes with pions \noffer further possibilities to test chiral perturbation theory. For example, \npion Compton scattering $\\pi^- \\gamma \\to\\pi^- \\gamma$ allows one to extract \nthe electric and magnetic polarizabilities ($\\alpha_\\pi$ and $\\beta_\\pi$) of the \ncharged pion. Chiral perturbation theory at two-loop order gives for the \ndominant pion polarizability difference the firm prediction $\\alpha_\\pi-\\beta_\\pi\n=(5.7\\pm1.0)\\cdot 10^{-4}\\,$fm$^3$ \\cite{gasser}. It is however in conflict with \nthe existing experimental results from Serpukhov $\\alpha_\\pi-\\beta_\\pi=(15.6\\pm \n7.8)\\cdot 10^{-4}\\,$fm$^3$ \\cite{serpukov} and MAMI $\\alpha_\\pi-\\beta_\\pi=(11.6\n\\pm 3.4)\\cdot 10^{-4}\\,$fm$^3$ \\cite{mainz} which amount to values more than \ntwice as large. Certainly, these existing experimental determinations of \n$\\alpha_\\pi-\\beta_\\pi$ raise doubts about their correctness since they violate\nthe chiral low-energy theorem notably by a factor 2. It is worth to note that a \nrecent dispersive analysis \\cite{mouss} of the Belle data for $\\gamma \\gamma \n\\to \\pi^+\\pi^-$ gives the fit value $\\alpha_\\pi-\\beta_\\pi=4.7\\cdot 10^{-4}\\,\n$fm$^3$, compatible with chiral perturbation theory.  \n\nIn that contradictory situation it is promising that the ongoing COMPASS \nexperiment \\cite{compass} at CERN aims at remeasuring the pion \npolarizabilities, $\\alpha_\\pi$ and $\\beta_\\pi$, with high statistics using the \nPrimakoff effect. The scattering of high-energy negative pions in the \nCoulomb-field of a heavy nucleus (of charge $Z$) gives access to cross \nsections for $\\pi^-\\gamma$ reactions through the equivalent photon method \n\\cite{pomer}. The consistent theoretical framework to extract the pion \npolarizabilities from the measured cross sections for (low-energy) pion \nCompton scattering $\\pi^- \\gamma \\to\\pi^- \\gamma$ or the primary pion-nucleus \nbremsstrahlung process $\\pi^- Z \\to\\pi^- Z \\gamma$ has been described (in\none-loop approximation) in refs.\\cite{picross,comptcor}. It has been stressed\nthat at the same order as the polarizability difference $\\alpha_\\pi-\\beta_\\pi$ \nthere exists a further (partly compensating) pion-structure effect in form of \na unique pion-loop correction (interpretable as photon scattering off the \n''pion-cloud around the pion''). In addition to these strong interaction \neffects, the QED radiative corrections to real and virtual pion Compton \nscattering $\\pi^-\\gamma^{(*)} \\to \\pi^- \\gamma$ have been calculated in \nrefs.\\cite{comptcor,bremscor}. The relative smallness of the \npion-structure effects in low-energy pion Compton scattering \\cite{picross} \nmakes it necessary to include such higher order electromagnetic corrections. The\nCOMPASS experiment is set up to detect simultaneously various (multi-particle) \nhadronic final-states which are produced in the Primakoff scattering of \nhigh-energy pions. The neutral pion production channel $\\pi^-\\gamma\\to \\pi^-\n\\pi^0$ serves as a test of the QCD chiral anomaly by measuring the $\\gamma \n3\\pi$ coupling constant $F_{\\gamma  3\\pi}= e\/(4\\pi^2 f_\\pi^3) = 9.72\\,$GeV$^{-3}$. For \nthe two-body process $\\pi^-\\gamma\\to \\pi^-\\pi^0$  the one-loop \\cite{picross,\nbijnens} and two-loop corrections \\cite{hannah} of  chiral perturbation theory \nas well as QED radiative corrections \\cite{ametller} have already been worked \nout. \n\nThe $\\pi^- \\gamma$ reaction with three charged pions in the final-state is used \nin the energy range above 1\\,GeV to study the spectroscopy of non-strange meson \nresonances \\cite{boris} and to search for so-called exotic meson resonances\n\\cite{exotic}. The very high statistics of the COMPASS experiment allows it to \ncontinue the event rates with three pions in the final-state even downward to \nthe threshold. The (differential) cross sections for the $\\pi^-\\gamma\\to 3\\pi$ \nreactions in the low-energy region offer new possibilities to test the strong \ninteraction dynamics of the pions as predicted by chiral perturbation theory. \nIn a recent work \\cite{dreipion} the production amplitudes for $\\pi^- \\gamma \n\\to \\pi^-\\pi^0 \\pi^0$ and $\\pi^- \\gamma \\to \\pi^+\\pi^-\\pi^-$ have been calculated\nanalytically at one-loop order in chiral perturbation theory. It has been \nfound that the next-to-leading order corrections from chiral loops and \ncounterterms enhance sizeably (by a factor $1.5 -1.8$) the total cross section \nfor neutral pion-pair production $\\pi^-\\gamma \\to\\pi^-\\pi^0\\pi^0$. By contrast\nthe total cross section for charged pion-pair production $\\pi^-\\gamma \\to\\pi^+\n\\pi^-\\pi^-$ remains almost unchanged in comparison to its tree-level result. \nThis different behavior can be understood from the varying influence of the\nchiral corrections on the pion-pion final-state interaction ($\\pi^+\\pi^- \\to\n\\pi^0 \\pi^0$ versus $\\pi^-\\pi^- \\to  \\pi^-\\pi^-$).  \n\nThe purpose of the present paper is to further improve the theoretical\ndescription of the $\\pi^- \\gamma \\to 3\\pi$ reactions by considering the\ncorresponding QED radiative corrections. We restrict ourselves here to the\nsimpler case of neutral pion-pair production $\\pi^- \\gamma \\to\\pi^-\\pi^0\\pi^0$, \nfor which the number of contributing one-photon loop diagrams is limited to \nabout a dozen. Another fortunate circumstance is that the (leading-order) \nchiral $\\pi^+\\pi^-\\to \\pi^0\\pi^0$ contact-vertex factors out of all photon-loop\ndiagrams and therefore the radiative corrections to $\\pi^- \\gamma\\to\\pi^-\\pi^0\n\\pi^0$ can be represented simply by a multiplicative correction factor $R\n\\sim \\alpha\/2\\pi$. Infrared finiteness of these virtual radiative corrections \nis achieved (in the standard way) by including soft photon radiation below an \nenergy cut-off $\\lambda$. Taking  $\\lambda=5\\,$MeV, we find that the radiative \ncorrections to the total cross section for $\\pi^- \\gamma\\to\\pi^-\\pi^0\\pi^0$ vary\nbetween $+2\\%$ and $-2\\%$ for center-of-mass energies from threshold up to \n$7m_\\pi$. An electromagnetic counterterm (necessary in order to cancel all \nultraviolet divergences generated by the photon-loops) gives an additional \nconstant contribution of about $1\\%$, however with a large uncertainty. The \nradiative corrections to the charged pion-pair production process $\\pi^- \\gamma\n\\to\\pi^+ \\pi^-\\pi^-$ can be roughly estimated to be a factor $2-4$ times larger,\narguing that in this case twice as many charged pions are involved in virtual \nphoton-loops and soft photon bremsstrahlung. \n\n\n\\section{Evaluation of one-photon loop diagrams}   \nIn this section we calculate analytically the radiative corrections to the \nneutral pion-pair photoproduction process: \n\\begin{equation}\\pi^-(p_1)+\\gamma(k,\\epsilon\\,) \\to\\pi^-(p_2) +\\pi^0(q_1)+ \n\\pi^0(q_2)\\,,\\end{equation}  \nas they arise from one-photon loop diagrams at order $\\alpha$. For a concise\npresentation of our analytical results it is convenient to introduce the \nfollowing dimensionless Mandelstam variables:\n\\begin{equation} s=(p_1+k)^2\/m_\\pi^2\\,, \\quad t=(p_1-p_2)^2\/m_\\pi^2\\,, \\quad \nu=(p_2-k)^2\/m_\\pi^2\\,,\\end{equation}\nwith $m_\\pi =139.570\\,$MeV the charged pion mass. In this (adapted) notation \n$\\sqrt{s}\\,m_\\pi$ is the total center-of-mass energy of the process. We will\nalso use  frequently the linear combination: \n\\begin{equation} \\Sigma = s+t+u-2 =(q_1+q_2)^2\/m_\\pi^2\\,,\\end{equation}\nrelated to the squared invariant mass of the produced neutral pion-pair.\nIn the physical region the following inequalities hold: $s>(1+2\\sqrt{r_0})^2$, \n$t<0$, $u<0\\,$\\footnote{The inequality $u<1$ follows immediately from the\ndefinition of $u$. In order to derive the sharper upper bound $u<0$, one uses\nthe relation for $u$ written in eq.(21) and inserts $y_{\\rm max}=1$ and \n$\\omega_{\\rm max} = (s+1-4r_0)\/2\\sqrt{s}$. In the end the condition $r_0>1\/4$ \nturns out to be crucial for $u$ to take on only negative values.} and \n$4r_0< \\Sigma <(\\sqrt{s}-1)^2$ where $r_0=(m_{\\pi^0}\/m_\\pi)^2=0.93526$ denotes\nthe  squared ratio between the neutral pion mass $m_{\\pi^0}=134.977\\, $MeV and \nthe charged pion mass $m_\\pi$.\n\nLet us recall the dynamical description of the process $\\pi^- \\gamma \\to \\pi^-\n\\pi^0\\pi^0$ at low energies \\cite{picross,dreipion}. When choosing for the \n(transversal) real photon $\\gamma(k,\\epsilon\\,)$ the Coulomb-gauge in the \ncenter-of-mass frame, the conditions $\\epsilon \\cdot p_1 =\\epsilon \\cdot k= 0$ \nimply that all diagrams for which the photon couples to the in-coming pion \n$\\pi^-(p_1)$ vanish identically. Furthermore, in the convenient parametrization\nof the special-unitary matrix-field $U =\\sqrt{1-\\vec  \\pi^{\\,2} \/f_\\pi^2}+ i\\vec \n\\tau \\cdot \\vec \\pi\/f_\\pi$ no $\\gamma 4\\pi$\nand $2\\gamma 4\\pi$ contact-vertices exist (at leading order). Under these \nassumptions one is left with one single $u$-channel pole diagram in which the \nchiral $\\pi^+\\pi^-\\to \\pi^0\\pi^0$ contact-vertex is followed by a photon-pion \ncoupling proportional to $\\epsilon\\cdot p_2$.\n\nThe virtual radiative corrections to $\\pi^- \\gamma \\to \\pi^-\\pi^0\\pi^0$ are\nobtained by dressing this tree diagram with a photon-loop in all possible ways \n(see Figs.\\,1-4). A fortunate circumstance is that the (leading order) chiral \n$\\pi^+\\pi^-\\to \\pi^0\\pi^0$ transition amplitude $[(q_1+q_2)^2-m_{\\pi^0}^2]\/f_\\pi^2$ \ndepends only on the $\\pi^0\\pi^0$ invariant mass and thus factors out of all \nphoton-loop diagrams. The Coulomb-gauge ($\\epsilon\\cdot p_1=\\epsilon\\cdot k=0$)\nleaves the scalar product $\\epsilon\\cdot p_2$ as the only possible coupling \nterm for the external real photon. As a consequence of these features the \nradiative corrections due to photon-loops can represented simply by a \nmultiplicative correction factor. Its real part which is only of relevance is \ndenoted by $R(s,t,u)$. We use dimensional regularization to treat both \nultraviolet and infrared divergences (where the latter are caused by the \nmasslessness of the photon). Divergent pieces of one-loop integrals show up in \nform of the composite constant:\n\\begin{equation} \\xi = {1\\over d-4}+{1\\over 2}(\\gamma_E-\\ln 4\\pi) +\\ln{m_\\pi\n\\over \\mu}\\,, \\end{equation}\ncontaining a simple pole at $d=4$ and $\\mu$ is an arbitrary mass scale.\nUltraviolet (UV) and infrared (IR) divergences are distinguished by the\nfeature of whether the condition for convergence of the $d$-dimensional\nintegral is $d<4$ or $d>4$. We discriminate them in the notation by putting\nappropriate subscripts, i.e. $\\xi_{UV}$ and $\\xi_{IR}$. In order to simplify\nall calculations we employ the Feynman gauge, where the photon propagator is \ndirectly proportional to the Minkowski metric tensor $g_{\\mu\\nu}$. We can now\nenumerate the analytical expressions for $R(s,t,u)$ as they emerge from the\neight classes of contributing one-photon loop diagrams.        \n\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[scale=1.,clip]{gam3pifig9.eps}\n\\end{center}\n\\vspace{-.5cm}\n\\caption{One-photon loop diagrams (I) and (II) for neutral pion-pair production \n$\\pi^-\\gamma \\to \\pi^-\\pi^0\\pi^0$. Arrows indicate out-going pions.}\n\\end{figure}\n\nThe two diagrams of class (I) shown in Fig.\\,1 introduce the wavefunction\nrenormalization factor $Z_2-1$ of the pion \\cite{comptcor}:   \n\\begin{equation}R^{(\\rm I)}={\\alpha \\over \\pi}\\big(\\xi_{IR}-\\xi_{UV}\\big) \\,. \n\\end{equation}\nDiagram (II) involves the once-subtracted (off-shell) selfenergy of the pion\nand leads to the result:\n\\begin{equation}R^{(\\rm II)}={\\alpha \\over \\pi}\\bigg[-\\xi_{UV}+1 -{u+1 \\over  2u}\n\\ln(1-u)\\bigg] \\,. \\end{equation}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.,clip]{gam3pifig10.eps}\n\\end{center}\n\\vspace{-.5cm}\n\\caption{One-photon loop diagrams (III), (IV) and (V).}\n\\end{figure}\nDiagram (III) shown in Fig.\\,2 gives rise a constant vertex correction:\n   \\begin{equation}R^{(\\rm III)}={\\alpha \\over 8\\pi}\\big(6\\xi_{UV}-7\\big) \\,, \n\\end{equation}\nwhile diagrams (IV) and (V) generate $u$-dependent vertex corrections:\n\\begin{equation}R^{(\\rm IV)}={\\alpha \\over 8\\pi}\\bigg[6\\xi_{UV}-6 -{1\\over u}+{u-1\n\\over u^2}(3u+1)\\ln(1-u)\\bigg] \\,, \\end{equation}\n\\begin{equation}R^{(\\rm V)}={\\alpha \\over 8\\pi}\\bigg[-4\\xi_{UV}+5 +{1\\over u}+{u^2\n+6u+1\\over u^2} \\ln(1-u)\\bigg] \\,. \\end{equation}\nIt is astonishing that the last four contributions $R^{(\\rm II)}+R^{(\\rm  III)}+\nR^{(\\rm  IV)}+R^{(\\rm V)}=0$ sum to zero.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.,clip]{gam3pifig11.eps}\n\\end{center}\n\\vspace{-.5cm}\n\\caption{One-photon loop diagrams (VI) and (VII).}\n\\end{figure}\n\nThe reducible $u$-channel pole diagram (VI) shown in Fig.\\,3 includes a \nphotonic vertex correction around the $2\\pi^0$ emission vertex. One finds for\nits contribution to the (real) $R$-factor the following result: \n\\begin{eqnarray} R^{(\\rm VI)} &=& {\\alpha \\over 2\\pi}\\bigg\\{-\\xi_{UV}+1+{1-u\\over\n2u} \\ln(1-u) +\\sqrt{\\Sigma-4\\over \\Sigma}\\ln{\\sqrt{\\Sigma-4}+\\sqrt{\\Sigma}\\over\n2} \\nonumber \\\\ &&+ \\bigg(s+t+{u-7\\over 2}\\bigg) -\\!\\!\\!\\!\\!\\!\\int_0^1 dx \n{\\ln|x^{-1}+\\Sigma(x-1)|-\\ln(1-u)\\over 1+(u-1)x+\\Sigma\\, x(x-1)} \\bigg\\} \\,.\n\\end{eqnarray}\nThe integrand of the principal-value integral $-\\!\\!\\!\\!\\!\\int_0^1 dx$ has \nsimple poles at $x_\\pm=[\\Sigma+1-u\\pm\\sqrt{(\\Sigma+1-u)^2-4\\Sigma}]\/2\\Sigma$,\nbut in the physical region $u<0$, $\\Sigma >4r_0$ only the pole at $x_-$ lies\ninside the unit-interval $0<x<1$. Due to this property an accurate numerical \ntreatment of the principal-value integral in eq.(10) (in combination with\nfurther three-body phase space integrations, see eq.(20)) is easily manageable.\nWe have also checked the Feynman parameter representation of the last loop \nintegral in eq.(10) against its  dispersion relation representation: \n\\begin{eqnarray}&&  -\\!\\!\\!\\!\\!\\!\\int_0^1\\!\\!dx{\\ln|x^{-1}+\\Sigma(x-1)|-\\ln(1-u) \n\\over 1+(u-1)x+\\Sigma\\, x(x-1)}=  -\\!\\!\\!\\!\\!\\!\\int_4^\\infty \\!\\! \\!{dx \\over x-\n\\Sigma }\\,{2 \\over \\sqrt{(x+1-u)^2-4x}} \\nonumber \\\\ && \\times \\ln{\\sqrt{x}\n(x-u-3)+\\sqrt{(x-4)[(x+1-u)^2-4x]} \\over 2(1-u)}\\,, \\end{eqnarray}\nwhere the imaginary part on the right hand side has been calculated via the \nCutkosky cutting rule. The analytical continuation of the last logarithmic \nterm in the  first line of eq.(10) for $0<\\Sigma <4$ is $\\sqrt{4\/\\Sigma-1}\n\\arcsin(\\sqrt{\\Sigma}\/2)$.\n\nIn the sum $R^{(\\rm I)}+R^{(\\rm VI)}$ obtained so far the ultraviolet divergences \n$\\xi_{UV}$ do not cancel and the remaining classes of diagrams (VII) and (VIII) \nare actually ultraviolet convergent. In a non-renormalizable effective field \ntheory like chiral perturbation theory such a behavior of the radiative\ncorrections is generic. In order to eliminate all ultraviolet divergences from \nphoton-loops additional electromagnetic counterterms have to be introduced \n\\cite{knecht}. The black square in the right tree diagram of Fig.\\,4 symbolizes\nthe pertinent electromagnetic counterterm for $\\pi^+\\pi^- \\to \\pi^0\\pi^0$ \nscattering. It gives rise to the following constant contribution to the \n$R$-factor:      \n\\begin{equation}R^{(\\rm ct)}={3\\alpha \\over 2\\pi}\\big(\\xi_{UV}+ \\bar k\\big) \\,, \n\\end{equation}\nwhere $\\bar k$ denotes the finite part of the electromagnetic counterterm\nwhich remains after canceling the ultraviolet divergence $\\xi_{UV}$. A \nnumerical estimate of $\\bar k$ will be given in section 4. For the sake of\ncompleteness we quote the expression for the on-shell $\\pi^+(q_+)+\\pi^-(q_-)\n\\to \\pi^0(q_1) +\\pi^0(q_2)$ scattering amplitude with inclusion of radiative\ncorrections \\cite{knecht}: \n\\begin{eqnarray} T_{+-,00} &=&{m_\\pi^2 \\over f_\\pi^2}(\\Sigma -r_0)\\Bigg\\{1+{\\alpha  \n\\over 2\\pi} \\bigg[ 2(\\xi_{IR}-\\xi_{UV}) + 3(\\xi_{UV}+\\bar k) \\nonumber \\\\ && \n-\\xi_{UV} +1 +\\sqrt{{\\Sigma -4\\over \\Sigma}} \\bigg(\\ln{\\sqrt{\\Sigma-4}+\\sqrt{\n\\Sigma}\\over 2} -{i \\, \\pi \\over 2}\\,\\theta(\\Sigma -4) \\bigg) \\nonumber \\\\ && \n+(\\Sigma -2) \\int_4^\\infty\\!\\!{dx\\over x-\\Sigma -i0^+} \\, {2\\xi_{IR}+\\ln(x-4) \\over\n\\sqrt{x^2-4x}} \\bigg] \\Bigg\\} \\,,  \\end{eqnarray}\nwhere $\\Sigma  = (q_1+q_2)^2\/m_\\pi^2$ and $f_\\pi = 92.4\\,$MeV denotes the pion\ndecay constant. In the order given the terms in the square bracket correspond \nto the pion wavefunction renormalization factor $Z_2-1$, the electromagnetic \ncounterterm, and the one-photon exchange contribution. Note that we have used \nthe (concise) spectral function representation for the infrared divergent \n(scalar) loop integral involving one photon and two pion propagators. If an\ninfinitesimal photon mass $m_\\gamma$ is introduced as an (alternative) infrared \nregulator the infrared divergence $\\xi_{IR}$ is to be identified with the\nlogarithm $\\ln(m_\\pi\/m_\\gamma)$.\n\n\nNext, we come to the irreducible $s$-channel diagrams of class (VII) shown in\nFig.\\,3. Since the contribution from the left diagram (involving the two-photon\ncontact-vertex) gets (partly) canceled by a term from the right diagram it is \nadvantageous to specify only their total contribution to the $R$-factor. After\nreducing the loop integrals with four propagators one obtains the following\nresult: \n\\begin{eqnarray} R^{(\\rm VII)} &=& {\\alpha \\over 2\\pi}(u-1)\\Bigg\\{{D(t)-D(\\Sigma)\n\\over 2(\\Sigma-t)} +\\int_0^1 dx {1-2x+2x^2 \\over 1-t\\, x(1-x)}  \\nonumber \\\\ &&\n\\times {\\ln|x^{-1}+\\Sigma(x-1)|-\\ln(s-1)\\over 1+(s-1)x+\\Sigma\\,x(x-1)} \\Bigg\\} \n\\,.\\end{eqnarray}\nNote that the last denominator has no poles in the physical region, since\n$s-1+\\Sigma(x-1) >s-1-\\Sigma>s-1-(\\sqrt{s}-1)^2=2(\\sqrt{s}-1)>0$. By taking\nthe absolute magnitude of the arguments of logarithms one gets directly a  \nsuitable representation of the only relevant real part. It is a fortunate\ncircumstance that the Feynman-parameter representation of loop functions leads\nto expressions which can be handled easily numerically in the physical region.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.,clip]{gam3pifig12.eps}\n\\end{center}\n\\vspace{-.5cm}\n\\caption{One-photon loop diagrams (VIII). The black square in the right tree \ndiagram (ct) symbolizes the electromagnetic counterterm for $\\pi^+\\pi^-\\to \n\\pi^0\\pi^0$ scattering.}\n\\end{figure}\n\nFinally, we come to the irreducible $u$-channel diagrams of class (VIII) shown \nin Fig.\\,4. The contribution from the left diagram gets completely absorbed by\na term from the right diagram. The resulting contribution to the (real) \n$R$-factor includes an infrared divergent term with a non-trivial \n$t$-dependence and after putting all pieces together it reads:   \n\\begin{eqnarray} R^{(\\rm VIII)} &=& {\\alpha \\over 2\\pi}\\Bigg\\{{1-u\\over  2}\\bigg[\n{D(t)-D(\\Sigma)\\over \\Sigma-t}- {1 \\over u}\\ln(1-u)  \\nonumber \\\\ && + \n-\\!\\!\\!\\!\\!\\!\\int_0^1 dx {1+4x^2-t\\,x(1+x)\\over 1-t\\,x(1-x)}\\,{\\ln|x^{-1}+\\Sigma\n(x-1)|-\\ln(1-u)\\over 1+(u-1)x+\\Sigma\\,x(x-1)} \\bigg]\\nonumber \\\\ &&+{t-2 \\over \n\\sqrt{t^2-4t}} \\bigg[4\\Big(\\xi_{IR}+\\ln(1-u)\\Big) \\ln{\\sqrt{4-t}+\\sqrt{-t}\\over\n2} +{\\rm Li}_2(w)\\nonumber \\\\ && -{\\rm Li}_2(1-w)+{1\\over 2}\\ln^2 w-{1\\over 2}\n\\ln^2(1-w) + {\\rm Li}_2(h_-)-{\\rm Li}_2(h_+)\\bigg]\\nonumber \\\\ &&  +(2-t)\n\\int_0^1 dx{\\ln|1+ \\Sigma\\, x(x-1)|\\over 1-t\\,x(1-x)} \\Bigg\\} \\,,\\end{eqnarray}\nwith the abbreviations\n\\begin{equation} w = {1\\over 2}\\Bigg(1-\\sqrt{-t\\over 4-t}\\,\\Bigg)\\,, \\qquad\n  h_\\pm = {1\\over 2}\\Big(t \\pm \\sqrt{t^2-4t}\\, \\Big)\\,. \\end{equation}\nOne observes that the term proportional to $D(t)-D(\\Sigma)$ drops out in the \nsum  $R^{(\\rm VII)} + R^{(\\rm VIII)}$ and therefore we do not need to specify it. \nLi$_2(w) = \\sum_{n=1}^\\infty n^{-2} w^n= w\\int_1^\\infty dx [x(x-w)]^{-1} \\ln x$ denotes \nthe conventional dilogarithmic function. Several of the results derived in \nsection 3 of ref.\\cite{bremscor} have been useful in order to obtain the \nexpression for $R^{(\\rm VIII)}$  written in eq.(15).\n  \n\\section{Infrared finiteness}\nIn the next step we have to consider the infrared divergent terms proportional\nto $\\xi_{IR}$ present in eqs.(5,15). At the level of the measurable cross \nsection these get eliminated by contributions from (undetected) soft photon \nbremsstrahlung. In its final effect, the (single) soft photon radiation off \nthe in- or out-going $\\pi^-$ multiplies the tree-level differential cross \nsection for $\\pi^-\\gamma \\to\\pi^- \\pi^0\\pi^0$ by a (universal) factor \n\\cite{comptcor,bremscor}:  \n\\begin{equation} \\delta_{\\rm soft}=2R_{\\rm soft}= \\alpha\\, \\mu^{4-d}\\!\\!\\int\\limits_{\n|\\vec l\\,|<\\lambda} \\!\\!{d^{d-1}l  \\over (2\\pi)^{d-2}\\, l_0} \\bigg\\{ {2p_1\\cdot p_2 \n\\over p_1 \\cdot l \\, p_2 \\cdot l} - {m_\\pi^2 \\over (p_1 \\cdot l)^2} - {m_\\pi^2 \n\\over (p_2 \\cdot l)^2} \\bigg\\} \\,, \\end{equation}\nwhich depends on a small energy cut-off $\\lambda$. Working out this momentum \nspace integral by the method of dimensional regularization (with $d>4$) one \nfinds the following contribution from soft photon emission to the $R$-factor: \n\\begin{eqnarray}R_{\\rm soft}^{(\\rm cm)}&=& {\\alpha \\over 4\\pi}\\Bigg\\{4\\bigg[1+{2 t-4 \n\\over\\sqrt{t^2-4t}} \\ln{\\sqrt{4-t}+\\sqrt{-t}\\over 2}\\bigg] \\bigg(\\ln{m_\\pi\\over\n2\\lambda} -\\xi_{IR}\\bigg) \\nonumber \\\\ && + {s+1 \\over s-1} \\ln s + {2\\omega\n\\over \\sqrt{\\omega^2-1}}\\ln\\Big(\\omega+\\sqrt{\\omega^2-1}\\,\\Big)+(t-2) \\nonumber \n\\\\ && \\times \\int_0^1 dx {s+1-\\Sigma\\,x\\over [1-t\\,x(1-x)] \\sqrt{W}} \\ln{ s+1\n-\\Sigma\\,x+\\sqrt{W} \\over s+1-\\Sigma\\,x-\\sqrt{W}}\\Bigg\\} \\,,\\end{eqnarray}\nwith the abbreviation $W= (s+1-\\Sigma\\,x)^2-4s[1-t\\,x(1-x)]$. In order to\nsimplify the last term in eq.(18) we have made use of the relation $\\Sigma= \ns+1-2 \\omega\\sqrt{s}$, where $\\omega$ denotes the center-of-mass energy of the\nout-going negative pion $\\pi^-(p_2)$ divided by $m_\\pi$. Note that the terms \nbeyond those proportional to $\\ln(m_\\pi\/2\\lambda) -\\xi_{IR}$ are specific for \nthe evaluation of the soft photon correction factor $R_{\\rm soft}$ in the \ncenter-of-mass frame with $\\lambda$ an infrared cut-off therein.  \n\nIn order to present a concrete example we have evaluated the complete radiative\ncorrection factor $R$ at the threshold in the isospin limit: $s_{\\rm th}=9$, \n$t_{\\rm  th}=-4\/3$,  $u_{\\rm th}=-5\/3$, $\\Sigma_{\\rm th}=4$,  $\\omega_{\\rm  th}=1$. In \nthis case one gets numerically:\n\\begin{equation}R_{\\rm th}={\\alpha \\over 2\\pi} \\bigg\\{11.093+3\\bar k+ \\bigg(2-\n{5\\over  2}\\ln 3\\bigg)\\ln{m_\\pi \\over 2\\lambda} -0.725\\bigg\\} \\,, \\end{equation} \nwhere the terms in the curly bracket correspond in the order written to virtual\nphoton-loops, the electromagnetic counterterm, the universal soft photon \ncontribution, and the  soft photon contribution specific for imposing an \ninfrared cut-off via $|\\vec l\\,|<\\lambda$ in the center-of-mass frame.  \n\\section{Results: radiative corrections to cross sections}\nAfter inclusion of radiative corrections the total cross section for neutral \npion-pair production $\\pi^-\\gamma \\to\\pi^-\\pi^0\\pi^0$ depends also on\nthe infrared cut-off $\\lambda$ for undetected soft photons. We multiply the \nsquared tree-level amplitude by $1+2R(s,t,u,\\lambda)$ and integrate over the \nthree-pion phase space. Applying the usual flux and symmetry factors the\ntotal cross  section reads:\n\\begin{eqnarray} \\sigma_{\\rm tot}(s,\\lambda) &=& {\\alpha\\,m_\\pi^2\\over 32\\pi^2f_\\pi^4\n(s-1)}\\int_{1}^{\\omega_{\\rm  max}}\\!\\!d\\omega\\,(\\omega^2-1)^{3\/2}\\sqrt{\\Sigma-4r_0 \\over \n\\Sigma }\\nonumber \\\\ && \\times\\int_{-1}^1 dy\\,(1-y^2) \\bigg({\\Sigma-r_0\\over\nu-1} \\bigg)^{\\!\\!2}\\,\\Big[1+2R(s,t,u,\\lambda)\\Big] \\,,\n\\end{eqnarray}\nwith $\\omega_{\\rm max} = (s+1-4r_0)\/2\\sqrt{s}$ the endpoint energy of the \nout-going $\\pi^-$ divided by $m_\\pi$. Using the relations $\\Sigma= s+t+u-2= \ns+1-2 \\omega \\sqrt{s}$ and:\n\\begin{equation} u= 1+{1-s \\over\\sqrt{s}}\\Big(\\omega-y  \\sqrt{\\omega^2-1}\\,\\Big)\n\\,, \\qquad t= 2-(s+1){\\omega \\over\\sqrt{s}}+{1-s \\over\\sqrt{s}} y \\sqrt{\n\\omega^2-1}\\,, \\end{equation} \nvalid in the center-of-mass frame the whole integrand in eq.(20) becomes a\nfunction of $\\omega$ and the directional cosine $y$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=.5,clip]{2piradcor.eps}\n\\end{center}\n\\vspace{-0.6cm}\n\\caption{Radiative corrections to the total cross section for neutral \npion-pair photoproduction $\\pi^- \\gamma \\to \\pi^-\\pi^0\\pi^0$ as a function of \nthe center-of-mass energy $\\sqrt{s}\\,m_\\pi$. The infrared cut-off for soft \nphoton emission has been set to the value $\\lambda = 5\\,$MeV.} \n\\end{figure}\n\n\nFig.\\,5 shows in percent the radiative corrections to the total cross section \nfor neutral pion-pair production $\\pi^-\\gamma \\to\\pi^-\\pi^0\\pi^0$ as a function \nof the center-of-mass energy $\\sqrt{s}\\,m_\\pi$. The dashed-dotted and dashed \ncurves display the separate contributions from soft photon bremsstrahlung and \nvirtual photon-loops. In each case the radiative correction is calculated as \nthe ratio of the shift in $\\sigma_{\\rm tot}(s,\\lambda)$ induces by the respective \n$R$-factor divided by the tree-level cross section. As in ref.\\cite{comptcor} \nthe infrared cut-off $\\lambda$ for undetected soft photons has been set to \n$\\lambda=5\\,$MeV, a value which seems appropriate for the COMPASS experiment. \nThe full line in Fig.\\,5 shows the complete radiative corrections. One observes \nan almost linear decrease which ranges from $+1.6\\%$ at threshold to $-1.6\\%$ \nat the center-of-mass energy $\\sqrt{s}\\,m_\\pi=7m_\\pi$. An interesting feature\nis that the positive radiative corrections from the virtual photon-loops get \ngradually reduced and turned into negative values by the soft photon\ncontributions.   \n    \nThe finite part of the electromagnetic counterterm $\\bar k$ shifts the \nradiative corrections (displayed by the full curve in Fig.\\,5) by a \nconstant amount of $3\\alpha\\bar k\/\\pi= 0.7\\%\\cdot \\bar k$. In order to give an\nestimate for $\\bar k$ we exploit the elaborate result of ref.\\cite{pionium}\nfor the pionium decay amplitude $(a_0-a_2)m_\\pi+\\varepsilon$. Guided by eq.(13) \nwe identify $(\\alpha\/ 2\\pi)(3\\bar k +1)$ with the ratio $\\varepsilon^{\\rm  elm}\/(\na_0-a_2)$, where $\\varepsilon^{\\rm  elm}$ is the electromagnetic correction to \nthe pionium decay amplitude. Subtracting from $\\varepsilon= (6.1\\pm 1.6)\\cdot \n10^{-3}$ the contribution $4.8\\cdot 10^{-3}$ due to the (charged and neutral)\npion mass difference (see eqs.(4.28,4.29) in ref.\\cite{pionium}) one gets \n$\\varepsilon^{\\rm elm}= (1.3\\pm 1.6)\\cdot 10^{-3}$. Together with the leading order \nexpression for the $\\pi\\pi$-scattering length difference $a_0-a_2 = 9m_\\pi\/\n(32\\pi f_\\pi^2) = 0.204 m_\\pi^{-1}$ one arrives at the estimate $\\bar k = 1.5\\pm \n2.2$ for the electromagnetic counterterm. Its central value implies a constant \nshift of the radiative corrections to $\\pi^-\\gamma \\to\\pi^-\\pi^0\\pi^0$ by\nabout $1.0\\%$. The large errorbar of $\\bar k = 1.5\\pm 2.2$ introduces at the\nsame time a wide errorband to the full curve in Fig.\\,5. Still an allowed\noption is to neglect to electromagnetic counterterm, setting $\\bar k=0$.     \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=.5,clip]{2pi0radspec.eps}\n\\end{center}\n\\vspace{-0.6cm}\n\\caption{Radiative corrections to the $\\pi^0\\pi^0$ mass spectra for neutral \npion-pair production $\\pi^- \\gamma \\to \\pi^-\\pi^0\\pi^0$ as a function of \nthe $\\pi^0\\pi^0$ invariant mass $\\sqrt{\\Sigma}\\,m_\\pi$. The numbers on the\ncurves correspond to $\\sqrt{s}$.} \n\\end{figure}\n\nFinally, we consider radiative corrections to more exclusive observables. An\nobvious candidate is the $\\pi^0\\pi^0$ mass spectrum $d\\sigma\/dm_{00}$ with $m_{00} \n=\\sqrt{\\Sigma}\\,m_\\pi$ the $\\pi^0\\pi^0$ invariant mass. The differential cross \nsection $d\\sigma\/dm_{00}$ is obtained by omitting the $d\\omega$-integration in \neq.(20) and applying the normalization factor $ m_\\pi^{-1}\\sqrt{\\Sigma\/s}$. \nFig.\\,6 shows in percent the radiative corrections to the $\\pi^0\\pi^0$ mass \nspectrum for neutral pion-pair production $\\pi^- \\gamma \\to \\pi^-\\pi^0\\pi^0$ as \na function of the $\\pi^0\\pi^0$ invariant mass $\\sqrt{\\Sigma}\\,m_\\pi$. The \nnumbers (4,\\,5,\\,6,\\,7) on the four rising curves correspond to $\\sqrt{s}$, the \ntotal center-of-mass energy divided by $m_\\pi$. The electromagnetic counterterm\n$\\bar k$ shifts again the whole pattern by a constant $3\\alpha \\bar k\/\\pi$. \n \nIn summary, we find that the radiative corrections to neutral pion-pair\nproduction $\\pi^- \\gamma \\to \\pi^-\\pi^0\\pi^0$ are comparable in size to those\nfor pion Compton scattering $\\pi^- \\gamma \\to\\pi^- \\gamma $ \\cite{comptcor}. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nPerceiving environments in three dimensions (3D) is important for locating objects, identifying free space, and motion planning in robotics and autonomous vehicles.\nAlthough these domains typically rely on 3D sensors to measure depth and identify free space (e.g., LiDAR \\cite{GaEtAl20} or RGBD cameras \\cite{FeLa19}), classifying and understanding raw 3D data is a challenging and ongoing area of research \\cite{KhEtAl19,SuSh19,LiuEtAl19,pointnet}.\nAlternatively, RGB cameras are less expensive and more ubiquitous than 3D sensors, and there are many more datasets and methods based on RGB images \\cite{ImageNet,maskrcnn,MSCOCO}.\nThus, even when 3D sensors are available, RGB images remain a critical modality for understanding data and identifying objects \\cite{FlCoGr19,WaEtAl19}.\n\n\nTo identify objects in a sequence of images, video object segmentation (VOS) addresses the problem of densely labeling target objects in video.\nVOS is a hotly studied area of video understanding, with frequent developments and improving performance on challenging VOS benchmark datasets \\cite{SegTrackv2,DAVIS,DAVIS17,SegTrack,YTVOS}.\nThese algorithmic advances in VOS support learning object class models~\\cite{OnReVeECCV2014,TaSuYaCVPR2013}, scene parsing~\\cite{LiHeCVPR2015,TiLaIJCV2012}, action recognition~\\cite{LuXuCoCVPR2015,SoIdShICCV2015,SoIdShCVPR2016}, and video editing applications~\\cite{ChChChACMM2012}. \n\n\n\\begin{figure} [t]\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{front_vos_1.jpg}\n\t\\caption{ \\textbf{Depth from Video Object Segmentation.}\n\t\tVideo object segmentation algorithms can densely segment target objects in a variety of settings (DAVIS \\cite{DAVIS}, \\textit{left}).\n\t\tGiven object segmentations and a measure of camera movement (e.g., from vehicle odometry or robot kinematics, \\textit{right}), our network can estimate an object's depth\n\t\n\t}\n\t\\label{fig:front_vos}\n\\end{figure}\n\nGiven that many VOS methods perform well in unstructured environments, in this work, we show that VOS can similarly support 3D perception for robots and autonomous vehicles.\nWe take inspiration from work in psychology that establishes how people perceive depth motion from the optical expansion or contraction of objects  \\cite{It51,SwGo86}, and we develop a deep network that learns object depth estimation from uncalibrated camera motion and video object segmentation (see Fig.~\\ref{fig:front_vos}).\nWe depict our optical expansion model in Fig.~\\ref{fig:optical_expansion}, which uses a moving pinhole camera and binary segmentation masks for an object in view.\nTo estimate an object's depth, we only need segmentations at two distances with an estimate of relative camera movement.\nNotably, most autonomous hardware platforms already measure movement, and even hand-held devices can track movement using an inertial measurement unit or GPS.\nFurthermore, although we do not study it here, if hardware-based measurements are not available, structure from motion is also plausible to recover camera motion \\cite{KaGaBa19,MuTa17,ScFr16}.\n\nIn recent work \\cite{GrFlCo20}, we use a similar model for VOS-based visual servo control, depth estimation, and mobile robot grasping.\nHowever, our previous analytic depth estimation method does not adequately account for segmentation errors.\nFor real-world objects in complicated scenes, segmentation quality can change among frames, with typical errors including: incomplete object segmentation, partial background inclusion, or segmenting the wrong object.\nThus, we develop and train a deep network that learns to accommodate segmentation errors and reduces object depth estimation error from \\cite{GrFlCo20} by as much as 59\\%.\n\nThe first contribution of our paper is developing a learning-based approach to object depth estimation using motion and segmentation,  \nwhich we experimentally evaluate in multiple domains.\nTo the best of our knowledge, this work is the first to use a learned, segmentation-based approach to depth estimation, which has many advantages.\nFirst, we use segmentation masks as input, so our network does not rely on application-specific visual characteristics and is useful in multiple domains.\nSecond, we process a series of observations simultaneously, thereby mitigating errors associated with any individual camera movement or segmentation mask.\nThird, our VOS implementation operates on streaming video and our method, using a single forward pass, runs in real-time.\nFourth, our approach only requires a single RGB camera and relative motion (no 3D sensors).\nFinally, our depth estimation accuracy will improve with future innovations in VOS.\n\n\n\\begin{figure} [t]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{optical_expansion_2.pdf}\n\t\\caption{ \\textbf{Optical Expansion and Depth.}\n\t\tAn object's projection onto the image plane scales inversely with the depth between the camera and object.\n\t\tWe determine an object's depth ($d$) using video object segmentation, relative camera movement, and corresponding changes in scale ($\\ell$).\n\t\tIn this example, $d_1 = \\frac{d_2}{2}$, $\\ell_1 = 2 \\ell_2$, and $d_1 \\ell_1 = d_2 \\ell_2$\n\t\n\t\n\t}\n\t\\label{fig:optical_expansion}\n\\end{figure}\n\nA second contribution of our paper is the \\textbf{O}bject \\textbf{D}epth via \\textbf{M}otion and \\textbf{S}egmentation (ODMS) dataset.\\footnote{Dataset and source code website: \\url{https:\/\/github.com\/griffbr\/ODMS}}\nThis is the first dataset for VOS-based depth estimation and enables learning-based algorithms to be leveraged in this problem space. \nODMS data consist of a series of object segmentation masks, camera movement distances, and ground truth object depth. \nDue to the high cost of data collection and user annotation  \\cite{DAVIS2018,ViGr09}, manually collecting training data would either be cost prohibitive or severely limit network complexity to avoid overfitting.\nInstead, we configure our dataset to continuously generate synthetic training data with random distances, object profiles, and even perturbations, so we can train networks of arbitrary complexity.\nFurthermore, because our network input consists simply of binary segmentation masks and distances, we show that domain transfer from synthetic training data to real-world applications is viable.\nFinally, as a benchmark evaluation, we create four ODMS validation and test sets with 15,650 examples in multiple domains, including robotics and driving.\n\n\n\\section{Related Work}\n\nWe use video object segmentation (VOS) to process raw input video and output the binary segmentation masks we use to estimate object depth in this work.\nUnsupervised VOS usually relies on generic object motion and appearance cues \\cite{NLC,GrCoWACV2019,KEY,FST,WeSz17,LIBSVX}, while semi-supervised VOS segments objects that are specified in user-annotated examples \\cite{CINM,PML,GrCo19,PREMVOS,RGMP,OSMN}.\nThus, semi-supervised VOS can learn a specific object's visual characteristics and reliably segment dynamic \\textit{or} static objects.\nTo segment objects in our robot experiments, we use One-Shot Video Object Segmentation (OSVOS) \\cite{OSVOS}.\nOSVOS is state-of-the-art in VOS, has influenced other leading methods \\cite{OSVOS-S,OnAVOS}, and does not require temporal consistency (OSVOS segments frames independently). \nDuring robot experiments, we apply OSVOS models that have been pre-trained with annotated examples of each object rather than annotating an example frame at inference time.\n\nWe take inspiration from many existing datasets in this work.\nVOS research has benefited from benchmark datasets like SegTrackv2 \\cite{SegTrack,SegTrackv2}, DAVIS \\cite{DAVIS,DAVIS17}, and YouTube-VOS \\cite{YTVOS}, which have provided increasing amounts of annotated training data.\nThe recently developed MannequinChallenge dataset \\cite{LiEtAl19} trained a network to predict dense depth maps from videos with people, with improved performance when given an additional human-mask input.\nAmong automotive datasets, Cityscapes \\cite{City} focuses on \\textit{semantic segmentation} (i.e., assigning class labels to all pixels), KITTI \\cite{KITTI} includes benchmarks separate from segmentation for single-image depth completion and prediction, and SYNTHIA \\cite{SYNTHIA} has driving sequences with simultaneous ground truth for semantic segmentation and depth images.\nIn this work, our ODMS dataset focuses on \\textbf{O}bject \\textbf{D}epth via \\textbf{M}otion and \\textbf{S}egmentation, establishing a new benchmark for segmentation-based 3D perception in robotics and driving.\nIn addition, ODMS is arbitrarily extensible, which makes learning-based methods feasible in this problem space.\n\n\n\\section{Optical Expansion Model}\n\\label{sec:solve}\n\nOur optical expansion model (Fig.~\\ref{fig:optical_expansion}) forms the theoretical underpinning for our learning-based approach in Section~\\ref{sec:learn} and ODMS dataset in Section~\\ref{sec:dataset}.\nIn this section, we derive the complete model and analytic solution for segmentation-based depth estimation.\nWe start by defining the inputs we use to estimate depth.\nAssume we are given a set of $n \\geq 2$ observations that consist of masks \n\\begin{align}\n\t\\mathsf{M} := \\{\\mathbf{M}_1, \\mathbf{M}_2, \\cdots, \\mathbf{M}_n\\}\n\n\t\\label{eq:masks}\n\\end{align}\nsegmenting an object and corresponding camera positions on the optical axis\n\\begin{align}\n\t\\mathbf{z} := \\{z_1, z_2, \\cdots, z_n\\}.\n\n\t\\label{eq:distances}\n\\end{align}\nEach binary mask image $\\mathbf{M}_i$ consists of pixel-level labels where 1 indicates a pixel belongs to a specific segmented object and 0 is background.\nFor the solutions in this work, the optical axis's origin and absolute position of  $\\mathbf{z}$ is inconsequential.\n\n\\subsection{Relating Depth and Scale}\n\nWe use changes in scale of an object's segmentation mask to estimate depth.\nAs depicted in Fig.~\\ref{fig:optical_expansion}, we relate depth and scale across observations using\n\\begin{align}\n\td_i \\ell_i = d_j \\ell_j \\implies \\frac{\\ell_j}{\\ell_i} = \\frac{d_i}{d_j},\n\t\\label{eq:elld}\n\\end{align}\nwhere $\\ell_i$ is the object's projected scale in $\\mathbf{M}_i$, $d_i$ is the distance on the optical axis from $z_i$ to the visible perimeter of the segmented object, and $\\frac{\\ell_j}{\\ell_i}$ is the object's change in scale between $\\mathbf{M}_i$ and $\\mathbf{M}_j$.\nNotably, it is more straightforward to track changes in scale using area (i.e., the sum of mask pixels) than length measurements.\nThus, we use Galileo Galilei's Square-cube law to modify \\eqref{eq:elld} as\n\\begin{align}\n\ta_j = a_i \\bigg(\\frac{\\ell_j}{\\ell_i}\\bigg)^2 \\implies\n\t\\frac{\\ell_j}{\\ell_i} = \\frac{\\sqrt{a_j}}{\\sqrt{a_i}} = \\frac{d_i}{d_j},\n\t\\label{eq:galileo}\n\\end{align}\nwhere $a_i$ is an object's projected area at $d_i$ and $\\frac{\\sqrt{a_j}}{\\sqrt{a_i}}$ is equal to the change in scale between $\\mathbf{M}_i$ and $\\mathbf{M}_j$.\nCombining \\eqref{eq:elld} and \\eqref{eq:galileo}, we relate observations as\n\\begin{align}\n\td_i \\sqrt{a_i} = d_j \\sqrt{a_j} = c,\n\t\\label{eq:c}\n\\end{align}\nwhere $c$ is a constant corresponding to an object's orthogonal surface area.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{optical_expansion_za.pdf}\n\n\t\\caption{ \\textbf{Calculating Object Depth.}\n\t\n\t\tFirst, we define $d_i$ in terms of its component parts $z_{\\text{object}}$ and $z_i$ \\eqref{eq:di}.\n\t\tSecond, we relate measured changes in camera pose $z_i$ and segmentation area $a_i$ across observations \\eqref{eq:cobj}.\n\t\tFinally, we solve for $z_{\\text{object}}$ using \\eqref{eq:zobj\n\t}\n\t\\label{fig:optical_expansion_za}\n\\end{figure}\n\n\\subsection{Object Depth Solution}\n\nTo find object depth $d_i$ in \\eqref{eq:c}, we first redefine $d_i$ in terms of its components as\n\\begin{align}\n\td_i := z_i - z_{\\text{object}},\n\t\\label{eq:di}\n\\end{align}\nwhere $z_{\\text{object}}$  is the object's static position on the optical axis and $\\dot{z}_{\\text{object}}=0$ (see Fig.~\\ref{fig:optical_expansion_za}).\nSubstituting \\eqref{eq:di} in \\eqref{eq:c}, we can now relate observations as\n\\begin{align}\n\t(z_i - z_{\\text{object}})  \\sqrt{a_i} = (z_j  - z_{\\text{object}}) \\sqrt{a_j} = c.\n\n\t\\label{eq:cobj}\n\\end{align}\nFrom \\eqref{eq:cobj}, we can solve $z_{\\text{object}}$ from any two unique observations ($z_i \\neq z_j$) as\n\\begin{align}\n\tz_{\\text{object}} = \\frac{z_i \\sqrt{a_i} - z_j \\sqrt{a_j}}{\\sqrt{a_i} - \\sqrt{a_j}} = \\frac{z_i  - z_j \\frac{\\sqrt{a_j}}{\\sqrt{a_i}}}{1 - \\frac{\\sqrt{a_j}}{\\sqrt{a_i}}}.\n\t\\label{eq:zobj}\n\\end{align}\nSubstituting $z_{\\text{object}}$ in \\eqref{eq:di}, we can now find object depth $d_i$ at any observation.\n\n\\section{Learning Object Depth from Camera Motion and Video Object Segmentation}\n\\label{sec:learn}\n\nUsing the optical expansion model from Section~\\ref{sec:solve}, we design a deep network, \\textbf{O}bject \\textbf{D}epth \\textbf{N}etwork (ODN),  \nthat learns to predict the depth of segmented objects given a series of binary masks $\\mathsf{M}$ \\eqref{eq:masks} and changes in camera position $\\mathbf{z}$ \\eqref{eq:distances}.\nTo keep ODN broadly applicable, we formulate a normalized relative distance input in Section~\\ref{sec:norm}.\nIn Sections~\\ref{sec:dist} and \\ref{sec:scale}, we derive three unique losses for learning depth estimation.\nAfter some remarks on using intermediate observations in Section~\\ref{sec:intermediate}, we detail our ODN architecture in Section~\\ref{sec:net}.\n\n\\subsection{Normalized Relative Distance Input}\n\\label{sec:norm}\n\nTo learn to estimate a segmented object's depth, we first derive a normalized relative distance input that increases generalization.\nAs in Section~\\ref{sec:solve}, assume we are given a set of $n$ segmentation masks $\\mathsf{M}$ with corresponding camera positions $\\mathbf{z}$.\nWe can use $\\mathsf{M}$ and $\\mathbf{z}$ as inputs to predict object depth, however, a direct $\\mathbf{z}$ input enables a learned prior based on absolute camera position, which limits applicability at inference. \nTo avoid this, we define a relative distance input\n\\begin{align}\n\t\\Delta  \\mathbf{z} := \\{z_2 - z_1, z_3 - z_1, \\cdots, z_n - z_1\\},\n\n\t\\label{eq:rel}\n\\end{align}\nwhere $z_1, z_2, \\cdots, z_n$ are the sorted $\\mathbf{z}$ positions with the minimum $z_1$ closest to the object (see Fig.~\\ref{fig:optical_expansion_za}) and $\\Delta  \\mathbf{z} \\in \\mathbb{R}^{n-1}$. \nAlthough $\\Delta  \\mathbf{z}$ consists only of relative changes in position, it still requires learning a specific SI unit of distance and enables a prior based on camera movement range.\nThus, we normalize \\eqref{eq:rel} as\n\\begin{align}\n\t\\mathbf{\\bar{z}} := \\Big\\{\\frac{z_i - z_1}{z_n - z_1} | z \\in \\mathbf{z}, 1 < i < n \\Big\\},\n\t\\label{eq:znorm}\t\n\\end{align}\nwhere $z_n - z_1$ is the camera move range, $\\frac{z_i - z_1}{z_n - z_1} \\in (0,1)$, and $\\mathbf{\\bar{z}} \\in \\mathbb{R}^{n-2}$.\n\nUsing $\\mathbf{\\bar{z}}$ as our camera motion input increases the general applicability of ODN.\nFirst, $\\mathbf{\\bar{z}}$ uses the relative difference formulation, so ODN does not learn to associate depth with an absolute camera position.\nSecond, $\\mathbf{\\bar{z}}$ is dimensionless, so our trained ODN can use camera movements on the scale of millimeters or kilometers (it makes no difference). \nFinally, $\\mathbf{\\bar{z}}$ is made a more compact motion input by removing the unnecessary constants $\\frac{z_1 - z_1}{z_n - z_1} = 0$ and $\\frac{z_n - z_1}{z_n - z_1} = 1$ in \\eqref{eq:znorm}.\n\n\n\\subsection{Normalized Relative Depth Loss}\n\\label{sec:dist}\n\nOur basic depth loss, given input masks $\\mathsf{M}$ \\eqref{eq:masks} and relative distances $\\Delta  \\mathbf{z}$ \\eqref{eq:rel}, is\n\\begin{align}\n\t\\mathcal{L}_d(\\textbf{W}) := d_1 - f_d(\\mathsf{M}, \\Delta  \\mathbf{z}, \\mathbf{W}),\n\t\\label{eq:depthloss}\n\\end{align}\nwhere $\\textbf{W}$ are the trainable network parameters, $d_1$ is the ground truth object depth at $z_1$ \\eqref{eq:di}, and $f_d \\in \\mathbb{R}$ is the predicted depth.\nTo use the normalized distance input $\\mathbf{\\bar{z}}$ \\eqref{eq:znorm}, we modify \\eqref{eq:depthloss} and define a normalized depth loss as\n\\begin{align}\n\n\n\n\n\n\n\t\\mathcal{L}_{\\bar{d}}(\\textbf{W}) := \\frac{d_1}{z_n - z_1} - f_{\\bar{d}}(\\mathsf{M}, \\mathbf{\\bar{z}}, \\mathbf{W}),\n\n\t\\label{eq:normloss}\n\\end{align} \nwhere $\\frac{d_1}{z_n - z_1}$ is the normalized object depth and $f_{\\bar{d}}$ is a dimensionless depth prediction that is in terms of the input camera movement range.\nTo use $f_{\\bar{d}}$ at inference, we multiply the normalized output $f_{\\bar{d}}$ by $(z_n - z_1)$ to find $d_1$.\n\n\\subsection{Relative Scale Loss}\n\\label{sec:scale}\n\nWe increase depth accuracy and simplify ODN's prediction by learning to estimate relative changes in segmentation scale.\nIn Section~\\ref{sec:dist}, we define loss functions that use a similar input-output paradigm to the analytic solution in Section~\\ref{sec:solve}.\nHowever, training ODN to directly predict depth requires learning many operations.\nAlternatively, if ODN only predicts the relative change in segmentation scale, we can finish calculating depth using \\eqref{eq:zobj}.\nThus, we define a loss for predicting the relative scale as\n\\begin{align}\n\t\\mathcal{L}_{\\ell}(\\textbf{W}) := \\frac{\\ell_n}{\\ell_1} - f_\\ell(\\mathsf{M}, \\mathbf{\\bar{z}}, \\mathbf{W}),\n\n\t\\label{eq:scaleloss}\n\\end{align}\nwhere $\\frac{\\ell_n}{\\ell_1}=\\frac{d_1}{d_n} \\in (0,1)$ \\eqref{eq:elld} is the ground truth distance-based change in scale between $\\mathbf{M}_n$ and $\\mathbf{M}_1$ and $f_\\ell$ is the predicted scale change.\nTo use $f_\\ell$ at inference, we output $f_\\ell \\approx \\frac{\\ell_n}{\\ell_1}$ and, using \\eqref{eq:galileo} to substitute $\\frac{\\ell_j}{\\ell_i}$ for $\\frac{\\sqrt{a_j}}{\\sqrt{a_i}}$ in \\eqref{eq:zobj}, find $z_{\\text{object}}$ as\n\\begin{align}\n\tz_{\\text{object}} =  \\frac{z_1  - z_n f_\\ell }{1 - f_\\ell } \\approx \\frac{z_1  - z_n \\big( \\frac{\\ell_n}{\\ell_1} \\big) }{1 - \\big( \\frac{\\ell_n}{\\ell_1} \\big) }.\n\n\t\\label{eq:zobjscale}\n\\end{align}\nAfter finding $z_{\\text{object}}$ in \\eqref{eq:zobjscale}, we use \\eqref{eq:di} to find object depth as $d_1 = z_1 - z_\\text{object}$.\n\n\\subsection{Remarks on using Intermediate Observations}\n\\label{sec:intermediate}\n\nAlthough the ground truth label $d_1$ in \\eqref{eq:depthloss}-\\eqref{eq:normloss} is determined only by camera position $z_1$ and label $\\frac{\\ell_n}{\\ell_1}$ in \\eqref{eq:scaleloss} is determined only by endpoint masks $\\mathbf{M}_n$, $\\mathbf{M}_1$, we emphasize that intermediate mask and distance inputs are still useful.\nConsider that, first, the ground truth mask scale monotonically decreases across all observations (i.e., $\\forall i, \\ell_{i+1} < \\ell_i$).\nSecond, the distance inputs make it possible to extrapolate $d_1$ and $\\frac{\\ell_n}{\\ell_1}$ from intermediate changes in scale.\nThird, if $z_1$, $z_n$, $\\mathbf{M}_1$, or $\\mathbf{M}_n$ have significant errors, intermediate observations provide the best prediction for $d_1$  or  $\\frac{\\ell_n}{\\ell_1}$.\nFinally, experiments in Section~\\ref{sec:exp_n} show that intermediate observations improve performance for networks trained on \\eqref{eq:depthloss}, \\eqref{eq:normloss}, or \\eqref{eq:scaleloss}.\n\n\\subsection{Object Depth Estimation Network Architecture}\n\\label{sec:net}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{DN_network_2.pdf}\n\t\\caption{ \\textbf{Object Depth Network Architecture} \n\t}\n\t\\label{fig:network}\n\\end{figure}\n\nOur ODN architecture is shown in Fig.~\\ref{fig:network}.\nThe input to the first convolution layer consists of $n$ 112$\\times$112 binary segmentation masks and, for three configurations in Section~\\ref{sec:radial}, a radial image.\nThe first convolution layer uses 14$\\times$14 kernels, and the remaining convolution layers use 3$\\times$3 kernels in four residual blocks \\cite{HeEtAl16}.\nAfter average pooling the last residual block, the relative camera position (e.g., $\\mathbf{ \\bar{z}}$) is included with the input to the first two fully-connected layers, which use ReLU activation and 20\\% dropout for all inputs during training \\cite{dropout14}.\nAfter the first two fully-connected layers, our ODN architecture ends with one last fully-connected neuron that, depending on chosen loss, is the output object depth $f_d(\\mathsf{M}, \\Delta \\mathbf{z}, \\mathbf{W}) \\in \\mathbb{R}$ using \\eqref{eq:depthloss}, normalized object depth $f_{\\bar{d}}(\\mathsf{M}, \\mathbf{\\bar{z}}, \\mathbf{W})$ using \\eqref{eq:normloss}, or relative scale $f_\\ell(\\mathsf{M}, \\mathbf{\\bar{z}}, \\mathbf{W})$ using \\eqref{eq:scaleloss}.\n\n\\section{ODMS Dataset}\n\\label{sec:dataset}\n\nTo train our object depth networks from Section~\\ref{sec:learn}, we introduce the \\textbf{O}bject \\textbf{D}epth via \\textbf{M}otion and \\textbf{S}egmentation dataset (ODMS).\nIn Section~\\ref{sec:gen_masks}, we explain how ODMS continuously generates new labeled training data, making learning-based techniques feasible in this problem space.\nIn Section~\\ref{sec:sets}, we describe the robotics-, driving-, and simulation-based test and validation sets we develop for evaluation.\nFinally, in Section~\\ref{sec:train}, we detail our ODMS training implementation.\n\n\\subsection{Generating Random Object Masks at Scale}\n\\label{sec:gen_masks}\n\n\\subsubsection{Camera Distance and Depth}\nWe generate new training data by, first, determining $n$ random camera distances (i.e., $\\mathbf{z}$ \\eqref{eq:distances}) for each training example.\nTo make ODMS configurable, assume we are given a minimum camera movement range ($\\Delta z_{\\text{min}}$) and minimum and maximum object depths ($d_{\\text{min}}$, $d_{\\text{max}}$). \nUsing these parameters, we define distributions for uniform random variables to find the endpoints\n\\begin{align}\n\t\\label{eq:z1}\n\tz_1 \\sim & ~\\mathcal{U}[d_{\\text{min}},d_{\\text{max}}-\\Delta z_{\\text{min}}], \\\\\n\tz_n \\sim & ~\\mathcal{U}[z_1 + \\Delta z_{\\text{min}},d_{\\text{max}}],\n\t\\label{eq:zn}\n\\end{align}\nand, for $1 < i < n$, the remaining intermediate camera positions \n\\begin{align}\n\tz_i \\sim \\mathcal{U}(z_1, z_n).\n\t\\label{eq:zi}\n\\end{align}\nUsing \\eqref{eq:z1}-\\eqref{eq:zi} to select $\\mathbf{z} = \\{z_1, \\cdots, z_n\\}$ ensures that the random camera movement range is independent of the number of observations $n$.\nFor the object depth label $d_1$, we choose an optical axis such that $z_\\text{object}=0$ and $d_1=z_1$ \\eqref{eq:di}.\nWe generate data in this work using $\\Delta z_\\text{min}=d_\\text{min}=0.1~\\textrm{m}$ and $d_\\text{max}=0.7~\\textrm{m}$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{data_gen.pdf}\n\t\\caption{ \\textbf{Generating Random Object Masks at Scale.} \n\t\tInitializing from a random number of points within a variable boundary (\\textit{left}), random curves complete the contour of each simulated object (\\textit{middle left}).\n\t\n\t\n\t\tThese contours are then scaled for each simulated distance and output as a filled binary mask (\\textit{right}).\n\t\tEach generated object is unique\n\t\n\t}\n\t\\label{fig:data_gen}\n\\end{figure}\n\n\\subsubsection{Random Object Contour and Binary Masks}\nAfter determining $\\mathbf{z}$, we generate a random object with $n$ binary masks (i.e., $\\mathsf{M}$ \\eqref{eq:masks}) scaled for each distance in $\\mathbf{z}$ (see Fig.~\\ref{fig:data_gen}).\nTo make each object unique, we randomly select parameters that change the object's size ($s_\\mathbf{p}$), number of contour points ($n_\\mathbf{p}$), and contour smoothness ($r_B$, $\\rho_B$).\nIn this work, we randomly select $s_\\mathbf{p}$ from $\\{100, 200, 300, 400\\}$ and $n_\\mathbf{p}$ from $\\{3, 4, \\cdots, 10\\}$.\nUsing $s_\\mathbf{p}$ and $n_\\mathbf{p}$, we select each of the random initial contour points, $\\mathbf{p}_i \\in \\mathbb{R}^2$ for $1 \\leq i \\leq n_\\mathbf{p}$, as\n\\begin{align}\n\t\\mathbf{p}_i= [ x_i,y_i ]',~\n\tx_i \\sim \\mathcal{U}[0,s_\\mathbf{p}],~ y_i \\sim \\mathcal{U}[0,s_\\mathbf{p}].\n\t\\label{eq:xy}\n\\end{align}\n\nTo complete the object's contour, we use cubic B\\'ezier curves with random smoothness to connect each set of adjacent coordinates $\\mathbf{p}_i$, $\\mathbf{p}_j$ from \\eqref{eq:xy}.\nEssentially, $r_B$ and $\\rho_B$ determine polar coordinates for the two intermediate B\\'ezier control points of each curve. \n$\\arctan (\\rho_B)$ is the rotation of a control point away from the line connecting $\\mathbf{p}_i$ and $\\mathbf{p}_j$,\nwhile $r_B $ is the relative radius of a control point away from $\\mathbf{p}_i$ (e.g., $r_B=1$ has a radius of $\\lVert \\mathbf{p}_i - \\mathbf{p}_j \\rVert$).\nIn this work, we randomly select $r_B$ from $\\{0.01, 0.05, 0.2, 0.5\\}$ and $\\rho_B$ from $\\{0.01, 0.05, 0.2\\}$ for each object.\nIn general, lower $r_B$ and $\\rho_B$ values result in a more straight-edged contour, while higher values result in a more curved and widespread contour.\nAs two illustrative examples in Fig.~\\ref{fig:data_gen}, the top ``straight-edged\" object uses $r_B=\\rho_B=0.01$ and the bottom ``curved\" object uses $r_B=0.5$ and $\\rho_B=0.2$.\n\nTo simulate object segmentation over multiple distances, we scale the generated contour to match each distance $z_i \\in \\mathbf{z}$ from \\eqref{eq:z1}-\\eqref{eq:zi} and output a set of binary masks $\\mathbf{M}_i \\in \\mathsf{M}$ \\eqref{eq:masks}.\nWe let the initial contour represent the object's image projection at $d_{\\text{min}}$, and designate this initial scale as $\\ell_{\\text{min}}=1$.\nHaving chosen an optical axis such that $z_\\text{object}=0$ in \\eqref{eq:di} (i.e., $d_i=z_i$), we modify \\eqref{eq:elld} to find the contour scale of each mask,  $\\ell_i $ for $1 \\leq i \\leq n$, as\n\\begin{align}\n\t\\ell_i = \\frac{d_\\text{min} \\ell_\\text{min}}{d_i}= \\frac{d_\\text{min}}{z_i}.\n\t\\label{eq:gen_scale}\n\\end{align}\nAfter finding $\\ell_i$, we scale, fill, and add the object contour to each mask $\\mathbf{M}_i$.\nIn this work, we position the contour by centering the scaled boundary ($\\ell_i s_\\mathbf{p}$) in a 480$\\times$640 mask.\nOur complete object-generating process is shown in Fig.~\\ref{fig:data_gen}.\n\n\n\\subsection{Robotics, Driving, and Simulation Validation and Test Sets}\n\\label{sec:sets}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{val_test_set.jpg}\n\t\\caption{ \\textbf{Robot Experiment Data.}\n\t\tHSR view of validation (yellow bin) and test set objects (blue bin) using head-mounted RGBD camera (\\textit{left}).\n\t\tUnfortunately, the depth image is missing many objects (\\textit{middle left}).\n\t\tHowever, using 4,400 robot-collected examples (\\textit{middle right}), we find that segmentation-based object depth works (\\textit{right})\n\t}\n\t\\label{fig:val_test}\n\\end{figure}\n\n\nWe test object depth estimation in a variety of settings using four ODMS validation and test sets.\nThese are based on robot experiments, driving, and simulated data with and without perturbations and provide a repeatable benchmark for ablation studies and future methods.\nAll examples include $n \\geq 10$ observations.\n\n\n\\subsubsection{Robot Validation and Test Set}\nOur robot experiment data provide an evaluation for object depth estimation from a physical platform using video object segmentation on real-world objects in a practical use case.\nWe collect data using a Toyota Human Support Robot (HSR), which has a 4-DOF manipulator arm with an end effector-mounted wide-angle grasp camera \\cite{UiYamaguchi2015,HSR_journal}.\nUsing HSR's prismatic torso, we collect 480$\\times$640 grasp-camera images as the end effector approaches an object of interest, with the intent that HSR can estimate the object's depth using motion and segmentation.\nWe use 16 custom household objects for our validation set and 24 YCB objects \\cite{YCB} for our test set (Fig.~\\ref{fig:val_test}, left).\nFor each object, we collect 30 images distanced 2~\\textrm{cm} apart of the object in isolation and, as an added challenge, 30 more images in a cluttered setting (see Fig.~\\ref{fig:val_test}, middle right).\nThe ground truth object depth ($d_1$) is manually measured at the closest camera position and propagated to the remaining images using HSR's kinematics and encoder values, which also measure camera positions ($\\mathbf{z}$).\nTo generate binary masks ($\\mathsf{M}$), we segment objects using OSVOS \\cite{OSVOS}, which we fine-tune on each object using three annotated images from outside of the validation and test sets.\nWe vary the input camera movement range between 18-58~\\textrm{cm} and object depth ($d_1$) between 11-60~\\textrm{cm} to generate 4,400 robot object depth estimation examples (1,760 validation and 2,640 test).\n\n\\subsubsection{Driving Validation and Test Set}\nOur driving data provide an evaluation for object depth estimation in a faster moving automotive domain with greater camera movement and depth distances.\nOur goal is to track driving obstacles using an RGB camera, segmentation, and vehicle odometry.\nChallenges include changing object perspectives, camera rotation from vehicle turning, and moving objects.\nWe collect data using the SYNTHIA Dataset \\cite{SYNTHIA}, which includes ground truth semantic segmentation, depth images, and vehicle odometry in a variety of urban scenes and weather conditions.\nTo generate binary masks ($\\mathsf{M}$), we use SYNTHIA's semantic segmentation over a series of 760$\\times$1280 frames for unique instances of pedestrians, bicycles, and cars (see Fig.~\\ref{fig:val_synthia}).\nFor each instance, the ground truth object depth ($d_1$) is the mean depth image values contained within corresponding mask  $\\mathbf{M}_1$.\nAs the vehicle moves, we track changes in camera position ($\\mathbf{z}$) along the optical axis of position $z_1$.\nWith an input camera movement range between 4.2-68~\\textrm{m} and object depth ($d_1$) between 1.5-62~\\textrm{m}, we generate 1,250 driving object depth estimation examples (500 validation and 750 test).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{synthia_val_test_set.jpg}\n\t\\caption{ \\textbf{Driving Test Set Examples and Results.}\n\t\tThe ODN$_\\ell$ object depth error is -6 and -23~\\textrm{cm} for the pedestrians, -10~\\textrm{cm} for the bicycle, and -4~\\textrm{cm} for the car\n\t\n\t\n\t}\n\t\\label{fig:val_synthia}\n\\end{figure}\n\n\\subsubsection{Simulation Validation and Test Sets}\n\\label{sec:sim_test}\nFinally, we generate a set of normal and perturbation-based data for simulated objects.\nThe normal set and the continuously-generated training data we use in Section~\\ref{sec:train} both use the same mask-generating procedure from Section~\\ref{sec:gen_masks}, so the normal set provides a consistent evaluation for the type of simulated objects we use during training.\n\nTo test robustness for segmentation errors, we also generate a set of simulated objects with random perturbations added to each mask, $\\mathbf{M}_i$ for $1 \\leq i \\leq n$, as\n\\begin{align}\n\t{p}_i \\sim  \\mathcal{N}(0,1), \n\t~\\mathbf{M}_{i,{p}} = \\begin{cases}\n\t\t\\text{dilate}(\\mathbf{M}_i, \\lfloor {p}_i + 0.5 \\rfloor) & \\text{if}  ~ {p}_i \\geq 0\\\\\n\t\t\\text{erode}(\\mathbf{M}_i, \\lfloor {p}_i + 0.5 \\rfloor) & \\text{if}  ~ {p}_i < 0\\\\\n\t\\end{cases},\n\t\\label{eq:perturb}\n\\end{align}\nwhere $\\mathcal{N}(0,1)$ is a Gaussian distribution with $\\mu=0$, $\\sigma^2=1$, ${p}_i$ randomly determines the perturbation type and magnitude, and $\\mathbf{M}_{i,{p}}$ is the perturbed version of initial mask $\\mathbf{M}_i$.\nNotably, the sign of ${p}_i$ determines a dilation or erosion perturbation, and the rounded magnitude of ${p}_i$ determines the number of iterations using a square connectivity equal to one.\nWhen generating perturbed masks $\\mathbf{M}_{i,{\\rm p}}$, we make no other changes to input data or ground truth labels.\n\nWe generate 5,000 object depth estimation examples (2,000 validation and 3,000 test) for both the normal and perturbation-based simulation sets.\n\n\n\\subsection{Training Object Depth Networks using ODMS}\n\\label{sec:train}\n\nUsing the architecture in Section~\\ref{sec:net}, we train networks for depth loss ${\\mathcal{L}_{d}}$ \\eqref{eq:depthloss}, normalized relative depth loss $\\mathcal{L}_{\\bar{d}}$  \\eqref{eq:normloss}, and relative scale loss $\\mathcal{L}_{\\ell}$ \\eqref{eq:scaleloss}.\nWe call these networks ODN$_d$, ODN$_{\\bar{d}}$, and ODN$_\\ell$ respectively.\nWe train each network with a batch size of 512 randomly-generated training examples using the framework in Section~\\ref{sec:gen_masks} with $n=10$ observations per prediction. We train each network for 5,000 iterations using the Adam Optimizer \\cite{adam14} with a $1\\times10^{-3}$ learning rate, which takes 2.6 days using a single GPU (GTX 1080 Ti).\nNotably, the primary time constraint for training is generating new masks, and we can train a similar configuration with $n=2$ for 5,000 iterations in 15 hours.\n\n\n\\section{Experimental Results}\n\\label{sec:results}\n\nOur primary experiments and analysis use the four ODMS test sets.\nFor each test set, the number of network training iterations is determined by the best validation performance, which we check at every ten training iterations.\nWe determine the effectiveness of each depth estimation method using the mean percent error for each test set, which is calculated for each example as\n\\begin{align}\n\t\\text{Percent Error} = \\left| \\frac{d_1 - \\hat{d}_1}{d_1} \\right| \\times 100 \\%,\n\t\\label{eq:percent}\n\\end{align}\nwhere $d_1$ and $\\hat{d}_1$ are ground truth and  predicted object depth at final pose $z_1$.\n\n\\subsection{ODMS Test Results}\n\n\\setlength{\\tabcolsep}{6.75pt}\n\\begin{table} [t]\n\t\\centering\n\t\\small\n\t\\caption{\\textbf{ODMS Test Set Results}\t\n\t}\n\t\\begin{tabular}{| l | c | c | c| c | c | c | c |}\n\t\t\\hline\t\\multicolumn{1}{|c|}{} & Object &\t$n$\t&\t\\multicolumn{5}{c|}{ Mean Percent Error  \\eqref{eq:percent} }  \\\\\t\\cline{4-8}\n\t\t\\multicolumn{1}{|c|}{Config.}\t& Depth &  Input  & \\multicolumn{1}{c|}{Robot} & \\multicolumn{1}{c|}{Driving} & \\multicolumn{2}{c|}{Simulated Objects} & All \\\\ \\cline{6-7}\n\t\t\\multicolumn{1}{|c|}{ID} & Method  &\tMasks  &\t\\multicolumn{1}{c|}{Objects}  & \\multicolumn{1}{c|}{Objects}\t&\t\\multicolumn{1}{c|}{Normal} & \\multicolumn{1}{c|}{Perturb} & Sets \\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_\\ell$\t&\t$\\mathcal{L}_{\\ell}$ \\eqref{eq:scaleloss}\t&\t10\t&\t19.3\t& \\bf\t30.1\t&\t8.3\t&\t18.2\t& \\bf\t19.0\t\\\\\t\n\t\tODN$_{\\bar{d}}$\t&\t$\\mathcal{L}_{\\bar{d}}$  \\eqref{eq:normloss} \t&\t10\t&\t18.5\t&\t30.9\t&\t8.2\t&\t18.5\t&\t19.0\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_d$\t&\t${\\mathcal{L}_{d}}$ \\eqref{eq:depthloss}\t&\t10\t& \\bf\t18.1\t&\t47.5\t& \\bf\t5.1\t& \\bf\t11.2\t&\t20.5\t\\\\\t\n\t\tVOS-DE \t&\t\\cite{GrFlCo20}\t&\t10\t&\t32.6\t&\t36.0\t&\t7.9\t&\t33.6\t&\t27.5\t\\\\\t\\hline\n\t\n\t\\end{tabular}\n\t\\label{tab:main}\n\\end{table}\n\nObject depth estimation results for all four ODMS test sets are provided in Table~\\ref{tab:main} for our three ODN configurations and VOS-DE \\cite{GrFlCo20}.\nWe use $n=10$ observations, and ``All Sets'' is an aggregate score across all test sets.\nNotably, VOS-DE uses only the largest connected region of each mask to reduce noise.\n\nThe relative scale-based ODN$_\\ell$ performs best on the Driving set and overall.\nWe show a few quantitative depth estimation examples for ODN$_\\ell$ in Fig.~\\ref{fig:val_test} and Fig.~\\ref{fig:val_synthia}.\nNormalized depth-based ODN$_{\\bar{d}}$ comes in second overall, and depth-based ODN$_{d}$ performs best in three categories but worst in driving.\nBasically, ODN$_{d}$ gets a performance boost from a camera movement range- and depth-based prior (i.e., $\\Delta \\mathbf{z}$ and $f_d$ in \\eqref{eq:depthloss}) at the cost of applicability to other domains where the scale of camera input and depth will vary.\nOn the other hand, the generalization of ODN$_{\\bar{d}}$ and ODN$_\\ell$ from small distances in training to large distances in Driving is highly encouraging.\nVOS-DE performs the worst overall, particularly on test sets with segmentation errors or moving objects.\nHowever, VOS-DE does perform well on normal simulated objects, which only have mask discretization errors. \n\n\n\\subsubsection{Results on Changing the Number of Observations}\n\\label{sec:exp_n}\n\nObject depth estimation results for varied number of observations are provided in Table~\\ref{tab:num}.\nWe repeat training and validation for each new configuration to learn depth estimation with less observations.\nAs $n$ changes, each test set example uses the same endpoint observations (i.e., $\\mathbf{M}_1,\\mathbf{M}_n,z_1,z_n$).\nHowever, the $n-2$ intermediate observations are evenly distributed and do change (e.g., $n=2$ has none).\nNotably, at $n=2$, VOS-DE is equivalent to \\eqref{eq:zobj} and $\\mathbf{\\bar{z}} \\in \\mathbb{R}^{n-2}$ \\eqref{eq:znorm} gives no input to ODN$_{\\bar{d}}$, ODN$_\\ell$.\n\nODN$_\\ell$ has the most consistent and best performance for all $n$ settings, aside from a second place to ODN$_{\\bar{d}}$ at $n=5$.\nODN$_\\ell$ also requires the fewest training iterations for all $n$.\nIn general, ODN$_{\\bar{d}}$ and ODN$_{d}$ performance starts to decrease for $n \\leq 3$. \nVOS-DE performance decreases most significantly at $n=2$, having 2.5 times the error of ODN$_\\ell$ at $n=2$.\nAmazingly, all $n=2$ ODN configurations outperform $n=10$ VOS-DE.\nThus, even with significantly less input data, our learning-based approach outperforms prior work.\n\n\\setlength{\\tabcolsep}{4pt} \n\\begin{table} [t]\n\t\\centering\n\t\\small\n\t\\caption{\\textbf{ODMS Test Set Results vs. Number of Observations}\t\n\t}\n\t\\begin{tabular}{| l | c | c | c| c | c | c | c | c | c |}\n\t\t\\hline\n\t\t\\multicolumn{1}{|c|}{Config.}\t& Depth &  \t\\multicolumn{4}{c|}{ Overall Mean Percent Error }  & \\multicolumn{4}{c|}{ Average Training Iterations}\\\\\t\\cline{3-10}\n\t\t\\multicolumn{1}{|c|}{ID} & Method  &\t$n=2$  &\t$n=3$ &\t$n=5$ &\t$n=10$   & $n=2$  &\t$n=3$ &\t$n=5$ &\t$n=10$ \\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_\\ell$\t&\t$\\mathcal{L}_{\\ell}$ \\eqref{eq:scaleloss}\t& \\bf\t20.4\t& \\bf\t19.9\t&\t20.0\t& \\bf\t19.0\t& \\bf\t2,590\t& \\bf\t3,460\t& \\bf\t3,060\t& \\bf\t3,138\t\\\\\t\n\t\tODN$_{\\bar{d}}$\t&\t$\\mathcal{L}_{\\bar{d}}$  \\eqref{eq:normloss} \t&\t22.7\t&\t20.9\t& \\bf\t19.9\t&\t19.0\t&\t3,993\t&\t4,330\t&\t3,265\t&\t3,588\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_d$\t&\t${\\mathcal{L}_{d}}$ \\eqref{eq:depthloss}\t&\t21.6\t&\t21.2\t&\t20.5\t&\t20.5\t&\t4,138\t&\t4,378\t&\t4,725\t&\t3,300\t\\\\\t\n\t\tVOS-DE\t&\t \\cite{GrFlCo20}\t&\t50.3\t&\t29.7\t&\t27.6\t&\t27.5\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\n\t\\end{tabular}\n\t\\label{tab:num}\n\\end{table}\n\n\n\n\\setlength{\\tabcolsep}{4.7pt} \n\\begin{table} [t]\n\t\\centering\n\t\\small\n\t\\caption{\\textbf{Test Results with Perturb Training Data and Radial Input Image}\t\n\t\n\t}\n\t\\begin{tabular}{| l | c | c | c | c| c | c | c | c |}\n\t\t\\hline\t\\multicolumn{1}{|c|}{} & Object &  Radial& Type of &\t\\multicolumn{5}{c|}{ Mean Percent Error  \\eqref{eq:percent} }  \\\\\t\\cline{5-9}\n\t\t\\multicolumn{1}{|c|}{Config.}\t& Depth &  Input & Training &  \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} & \\multicolumn{2}{c|}{Simulated} & All \\\\ \\cline{7-8}\n\t\t\\multicolumn{1}{|c|}{ID} & Method  &  Image & Data &\t\\multicolumn{1}{c|}{Robot}  & \\multicolumn{1}{c|}{Driving}\t&\t\\multicolumn{1}{c|}{Normal} & \\multicolumn{1}{c|}{Perturb} & Sets \\\\\t\\hline \n\t\t\\hline \\multicolumn{9}{|c|}{Perturb Training Data}\\\\ \\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell p}$\t&\t$\\mathcal{L}_{\\ell}$ \\eqref{eq:scaleloss}\t&\tNo\t&\tPerturb\t&\t22.2\t& \\bf\t29.0\t&\t11.1\t&\t13.0\t&\t18.8\t\\\\\t\n\t\tODN$_{\\bar{d} p}$\t&\t$\\mathcal{L}_{\\bar{d}}$ \\eqref{eq:normloss} \t&\tNo\t&\tPerturb\t&\t25.8\t&\t31.4\t&\t11.1\t&\t13.2\t&\t20.4\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d p}$\t&\t${\\mathcal{L}_{d}}$ \\eqref{eq:depthloss}\t&\tNo\t&\tPerturb\t&\t20.1\t&\t60.9\t&\t7.3\t& \\bf\t8.2\t&\t24.1\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Radial Input Image}\\\\ \\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell r}$\t&\t$\\mathcal{L}_{\\ell}$ \\eqref{eq:scaleloss}\t&\tYes\t&\tNormal\t& \\bf\t13.1\t&\t31.7\t&\t8.6\t&\t17.9\t& \\bf\t17.8\t\\\\\t\n\t\tODN$_{\\bar{d} r}$\t&\t$\\mathcal{L}_{\\bar{d}}$  \\eqref{eq:normloss} \t&\tYes\t&\tNormal\t&\t15.2\t&\t30.9\t&\t8.4\t&\t18.5\t&\t18.3\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d r}$\t&\t${\\mathcal{L}_{d}}$ \\eqref{eq:depthloss}\t&\tYes\t&\tNormal\t&\t13.4\t&\t48.6\t& \\bf\t5.6\t&\t11.2\t&\t19.7\t\\\\\t\\hline\n\t\n\t\n\t\n\t\n\t\\end{tabular}\n\t\\label{tab:rad}\n\\end{table}\n\n\\subsubsection{Results with Perturbation Training Data}\nWe train each $n=10$ ODN on continuously-generated perturbation data \\eqref{eq:perturb} from Section~\\ref{sec:sim_test}.\nAs shown in Table~\\ref{tab:rad}, this improves performance for each ODN on the Perturb test set, and demonstrates that we can learn robust depth estimation for specific errors.\nThe perturbed ODN$_\\ell$ configuration, ODN$_{\\ell p}$, improves performance overall and has the best Driving result of any method.\n\n\\subsubsection{Results with Radial Input Image}\n\\label{sec:radial}\nFor our final ODMS results in Table~\\ref{tab:rad}, we train each $n=10$ ODN with an added input radial image for convolution.\nPixel values $\\in [0,1]$ are scaled radially from 1 at the center to 0 at each corner (see Fig.~\\ref{fig:network}).\nThis serves a similar purpose to coordinate convolution \\cite{LiEtAl18} but simply focuses on how centered segmentation mask regions are.\nThis improves overall performance for each ODN, particularly on the Robot test, where objects are generally centered for grasping and peripheral segmentation errors can be ignored.\nNotably, ODN$_{\\ell r}$ has the best Robot and overall result of any method.\n\n\n\\subsection{Robot Object Depth Estimation and Grasping Experiments}\n\\label{sec:robot_exp}\n\nAs a live robotics demonstration, we use ODN$_{\\ell r}$ to locate objects for grasping.\nExperiments start with HSR's grasp camera approaching an object while generating segmentation masks at 1~\\textrm{cm} increments using pre-trained OSVOS.\nOnce ten masks are available, ODN$_{\\ell r}$ starts predicting depth as HSR continues approaching and generating masks. \nBecause ODN$_{\\ell r}$'s prediction speed is negligible compared to HSR's data-collection speed, we use the median depth estimate of multiple permutations of collected data to improve robustness against segmentation errors.\nOnce ODN$_{\\ell r}$ estimates the object to be within 20~\\textrm{cm} of grasping, HSR stops collecting data and grasps the object at that depth.\nUsing this active depth estimation process, we are able to successfully locate and grasp consecutive objects at varied heights in a variety of settings, including placing laundry in a basket and clearing garbage off a table (see Fig.~\\ref{fig:front_vos}).\nWe show these robot experiments in our Supplementary Video at: \\url{https:\/\/youtu.be\/c90Fg_whjpI}.\n\n\n\\section{Conclusions}\n\nWe introduce the \\textbf{O}bject \\textbf{D}epth via \\textbf{M}otion and \\textbf{S}egmentation (ODMS) dataset, which continuously generates synthetic training data with random camera motion, objects, and even perturbations.\nUsing the ODMS dataset, we train the first deep network to estimate object depth from motion and segmentation, leading to as much as a 59\\% reduction in error over previous work.\nBy using ODMS's simple binary mask- and distance-based input, our network's performance transfers across sim-to-real and diverse application domains, as demonstrated by our results on the robotics-, driving-, and simulation-based ODMS test sets.\nFinally, we use our network to perform object depth estimation in real-time robot grasping experiments, demonstrating how our segmentation-based approach to depth estimation is a viable tool for real-world applications requiring 3D perception from a single RGB camera.\n\n\\subsection*{Acknowledgements}\nWe thank Madan Ravi Ganesh, Parker Koch, and Luowei Zhou for various discussions throughout this work. Toyota Research Institute (``TRI\") provided funds to assist the authors with their research but this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity.\n\n\\section*{Appendix}\n\n\\subsection*{Least-squares Solution for Object Depth}\n\nIn previous work \\cite{GrFlCo20}, we propose a least-squares object depth solution (VOS-DE) that uses more than two observations to add robustness for camera position and segmentation errors. \nWe include this solution here for reference.\nThe VOS-DE formulation derives an alternative form of \\eqref{eq:cobj} as\n\\begin{align}\n\tz_{\\text{object}} \\sqrt{a_i}   + c = z_i  \\sqrt{a_i},\n\\end{align}\nwhich over $n$ observations in $\\mathbf{A}\\mathbf{x}=\\mathbf{b}$ form yields\n\\begin{align}\n\t\\begin{bmatrix}\n\t\t\\sqrt{a_1} & 1 \\\\ \\sqrt{a_2} & 1 \\\\ \\vdots & \\vdots \\\\ \\sqrt{a_n} & 1\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\t\\hat{z}_{\\text{object}} \\\\ \\hat{c}\\\\ \n\t\\end{bmatrix}\n\t=\n\t\\begin{bmatrix}\n\t\tz_{1} \\sqrt{a_1} \\\\ z_{2} \\sqrt{a_2} \\\\ \\vdots \\\\ z_{n} \\sqrt{a_n}\n\t\\end{bmatrix}.\n\t\\label{eq:axb}\n\\end{align}\nSolving \\eqref{eq:axb} for $\\hat{z}_{\\text{object}}$ does provide a more robust depth estimate than the two-observation solution \\eqref{eq:zobj}. \nHowever, our learning-based approach from Section~\\ref{sec:learn} outperforms both analytic solutions in experiments.\n\n\\subsection*{ODMS Random Object Mask Examples}\n\nWe provide a few random object mask examples using ODMS's data-generation framework from Section~\\ref{sec:gen_masks}.\nThese synthetic object examples are shown in Fig.~\\ref{fig:synth_obj} and demonstrate the B\\'ezier curve behaviors associated with changing parameters $r_B$ and $\\rho_B$.\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{synth_object.jpg}\n\t\\caption{ \\textbf{ODMS Random Object Mask Examples.}\n\t\tAll examples use $s_\\mathbf{p}=400$, $n_\\mathbf{p}=5$, and $\\ell=\\ell_\\text{min}=1$.  \n\t\t$r_B$ values are 0.01, 0.05, 0.2, and 0.5 (\\textit{from left to right}) and $\\rho_B$ values are 0.01, 0.05, and 0.2 (\\textit{from top to bottom}).\n\t\tEach generated object is unique\n\t}\n\t\\label{fig:synth_obj}\n\\end{figure}\n\n\\subsection*{ODMS Validation Results}\n\nAs mentioned in Section~\\ref{sec:results}, the number of network training iterations is determined by the best validation performance, which we check at every ten training iterations.\nIn Table~\\ref{tab:all}, we provide the ODMS validation results and corresponding number of training iterations for all configurations.\nIn general, the relative performance of each configuration is consistent between the ODMS validation and test sets.\n\n\\setlength{\\tabcolsep}{1.75pt} \n\\setlength{\\tabcolsep}{4pt} \n\\begin{table}\n\t\\centering\n\t\\scriptsize\n\t\\caption{\\textbf{Complete ODMS Validation and Test Set Results}\t\n\t}\n\t\\begin{tabular}{| l | c | c | c | c | c | c | c | c |}\n\t\t\\hline\n\t\t\\multicolumn{1}{|c|}{Config.}\t & \t\\multicolumn{4}{c|}{ Mean Percent Error (Validation\/Test) }  & \\multicolumn{4}{c|}{ Training Iterations}\\\\\t\\cline{2-9}\n\t\t\\multicolumn{1}{|c|}{ID}   &\tRobot  &\tDriving &\tNormal  &\tPerturb   & Robot  &\tDriving &\tNormal  &\tPerturb \\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Standard Configuration}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_\\ell$\t&\t21.6\/19.3\t&\t29.4\/30.1\t&\t8.2\/8.3\t&\t18.4\/18.2\t&\t2390\t&\t1920\t&\t3370\t&\t4870\t\\\\\t\n\t\tODN$_{\\bar{d}}$ \t&\t19.6\/18.5\t&\t32.0\/30.9\t&\t7.9\/8.2\t&\t18.4\/18.5\t&\t4140\t&\t2990\t&\t3690\t&\t3530\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_d$ \t&\t19.9\/18.1\t&\t48.1\/47.5\t& \\bf\t4.9\/5.1\t&\t11.5\/11.2\t&\t2380\t&\t1650\t&\t4740\t&\t4430\t\\\\\t\n\t\t\\scriptsize VOS-DE \t&\t27.4\/32.6\t&\t35.9\/36.0\t&\t7.9\/7.9\t&\t34.1\/33.6\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=5$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell}$ \t&\t23.4\/20.5\t&\t31.5\/30.5\t&\t8.4\/8.6\t&\t20.2\/20.4\t&\t1000\t&\t1850\t&\t4870\t&\t4520\t\\\\\t\n\t\tODN$_{\\bar{d}}$ \t&\t22.8\/19.5\t&\t34.2\/31.1\t&\t8.4\/8.4\t&\t20.5\/20.6\t&\t1510\t&\t3450\t&\t3770\t&\t4330\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d}$\t&\t21.0\/19.4\t&\t44.6\/44.2\t&\t5.4\/5.5\t&\t13.4\/12.9\t&\t4690\t&\t4260\t&\t4980\t&\t4970\t\\\\\t\n\t\t\\scriptsize VOS-DE\t&\t29.5\/35.1\t&\t34.8\/34.6\t&\t7.8\/7.9\t&\t32.8\/32.6\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=3$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell}$\t&\t20.3\/18.6\t&\t31.8\/31.1\t&\t8.4\/8.4\t&\t21.9\/21.6\t&\t1820\t&\t2750\t&\t4890\t&\t4380\t\\\\\t\n\t\tODN$_{\\bar{d}}$ \t&\t19.9\/20.6\t&\t34.7\/33.1\t&\t8.4\/8.4\t&\t21.6\/21.5\t&\t4130\t&\t4320\t&\t4620\t&\t4250\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d}$ \t&\t24.0\/21.8\t&\t45.1\/44.5\t&\t5.4\/5.6\t&\t13.8\/12.9\t&\t4800\t&\t3040\t&\t4990\t&\t4680\t\\\\\t\n\t\t\\scriptsize VOS-DE \t&\t33.7\/41.2\t&\t45.2\/34.0\t&\t8.0\/8.1\t&\t37.0\/35.7\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=2$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell}$\t&\t21.3\/19.2\t&\t30.4\/31.4\t&\t8.7\/8.9\t&\t22.0\/22.0\t&\t1140\t&\t1010\t&\t3910\t&\t4300\t\\\\\t\n\t\tODN$_{\\bar{d}}$ \t&\t29.1\/24.2\t&\t39.6\/35.9\t&\t8.6\/8.9\t&\t21.8\/21.8\t&\t3410\t&\t4570\t&\t3370\t&\t4620\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d}$\t&\t23.3\/21.1\t&\t45.3\/44.8\t&\t5.8\/6.0\t&\t14.9\/14.4\t&\t2850\t&\t4120\t&\t4610\t&\t4970\t\\\\\t\n\t\t\\scriptsize VOS-DE \t&\t95.8\/65.5\t&\t55.0\/41.1\t&\t8.2\/8.3\t&\t90.6\/86.2\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Perturb Training Data}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell p}$\t&\t21.4\/22.2\t& \\bf\t28.6\/29.0\t&\t10.7\/11.1\t&\t12.8\/13.0\t&\t100\t&\t140\t&\t5000\t&\t5000\t\\\\\t\n\t\tODN$_{\\bar{d} p}$\t&\t25.6\/25.8\t&\t31.4\/31.4\t&\t11.0\/11.1\t&\t13.1\/13.2\t&\t420\t&\t2760\t&\t2730\t&\t4270\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d p}$\t&\t20.5\/20.1\t&\t59.4\/60.9\t&\t7.0\/7.3\t& \\bf\t8.1\/8.2\t&\t50\t&\t330\t&\t4860\t&\t4780\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Radial Input Image}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell r}$\t& \\bf\t13.8\/13.1\t&\t31.6\/31.7\t&\t8.4\/8.6\t&\t18.2\/17.9\t&\t1710\t&\t870\t&\t4940\t&\t3940\t\\\\\t\n\t\tODN$_{\\bar{d} r}$\t&\t16.6\/15.2\t&\t30.7\/30.9\t&\t8.3\/8.4\t&\t18.6\/18.5\t&\t2010\t&\t4200\t&\t4990\t&\t4440\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d r}$\t&\t14.1\/13.4\t&\t49.0\/48.6\t&\t5.5\/5.6\t&\t11.7\/11.2\t&\t2210\t&\t460\t&\t4870\t&\t4710\t\\\\\t\\hline\n\t\\end{tabular}\n\t\\label{tab:all}\n\\end{table}\n\n\\subsection*{ODMS Absolute Error Results}\n\nIn Table~\\ref{tab:abs}, we provide ODMS test results for the mean absolute error, which is calculated for each example as\n\\begin{align}\n\t\\text{Absolute Error} = \\left| d_1 - \\hat{d}_1 \\right|,\n\t\\label{eq:abs}\n\\end{align}\nwhere $d_1$ and $\\hat{d}_1$ are ground truth and  predicted object depth at final pose $z_1$.\nNotably, our motivation to use percent error \\eqref{eq:percent} in the paper is to provide a consistent comparison across domains with markedly different object depth distances.\nFor example, the 6 \\textrm{cm} absolute error from Fig.~\\ref{fig:val_synthia} is much better for the driving domain than it would be for robot grasping.\n\n\\setlength{\\tabcolsep}{1.75pt} \n\\setlength{\\tabcolsep}{4pt} \n\\begin{table}\n\t\\centering\n\t\\scriptsize\n\t\\caption{\\textbf{Complete ODMS Validation and Test Set Results (Absolute Error)}\t\n\t}\n\t\\begin{tabular}{| l | c | c | c | c | c | c | c | c |}\n\t\t\\hline\n\t\t\\multicolumn{1}{|c|}{}\t & \t\\multicolumn{4}{c|}{ Mean Absolute Error (Validation\/Test) }  & \\multicolumn{4}{c|}{ Training Iterations}\\\\\t\\cline{2-9}\n\t\t\\multicolumn{1}{|c|}{Config.}   &\tRobot  &\tDriving &\tNormal  &\tPerturb   &   &\t &\t  &\t \\\\\t\n\t\t\\multicolumn{1}{|c|}{ID}   &\t\\textrm{(cm)}  &\t\\textrm{(m)}  &\t\\textrm{(cm)}   &\t\\textrm{(cm)}    & Robot  &\tDriving &\tNormal  &\tPerturb \\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Standard Configuration}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_\\ell$\t&\t7.2\/6.6\t&\t3.8\/4.3\t&\t3.4\/3.4\t&\t7.3\/7.2\t&\t2390\t&\t1920\t&\t3370\t&\t4870\t\\\\\t\n\t\tODN$_{\\bar{d}}$ \t&\t6.4\/6.0\t&\t4.1\/4.4\t&\t3.1\/3.1\t&\t7.4\/7.3\t&\t4140\t&\t2990\t&\t3690\t&\t3530\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_d$ \t&\t6.8\/6.3\t&\t7.1\/7.8\t& \\bf\t1.8\/1.8\t&\t3.9\/3.7\t&\t2380\t&\t1650\t&\t4740\t&\t4430\t\\\\\t\n\t\t\\scriptsize VOS-DE \t&\t8.8\/10.0\t&\t5.0\/5.4\t&\t2.8\/2.8\t&\t15.3\/14.9\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=5$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell 5}$ \t&\t7.8\/7.0\t&\t3.9\/4.3\t&\t3.4\/3.5\t&\t8.2\/8.1\t&\t1000\t&\t1850\t&\t4870\t&\t4520\t\\\\\t\n\t\tODN$_{\\bar{d}5}$ \t&\t7.0\/6.1\t&\t4.4\/4.6\t&\t3.3\/3.3\t&\t8.1\/8.0\t&\t1510\t&\t3450\t&\t3770\t&\t4330\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d 5}$\t&\t7.3\/6.8\t&\t6.5\/7.2\t&\t1.9\/2.0\t&\t4.9\/4.7\t&\t4690\t&\t4260\t&\t4980\t&\t4970\t\\\\\t\n\t\t\\scriptsize VOS-DE$_5$\t&\t9.6\/10.8\t&\t5.0\/5.2\t&\t2.9\/2.9\t&\t14.3\/14.2\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=3$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell 3}$\t&\t6.8\/6.3\t&\t4.1\/4.5\t&\t3.4\/3.4\t&\t8.8\/8.6\t&\t1820\t&\t2750\t&\t4890\t&\t4380\t\\\\\t\n\t\tODN$_{\\bar{d}3}$ \t&\t6.8\/7.0\t&\t4.4\/4.7\t&\t3.3\/3.3\t&\t8.6\/8.4\t&\t4130\t&\t4320\t&\t4620\t&\t4250\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d 3}$ \t&\t7.9\/7.3\t&\t6.6\/7.3\t&\t1.9\/1.9\t&\t4.8\/4.4\t&\t4800\t&\t3040\t&\t4990\t&\t4680\t\\\\\t\n\t\t\\scriptsize VOS-DE$_3$ \t&\t11.2\/12.6\t&\t6.3\/5.0\t&\t2.9\/2.9\t&\t15.7\/15.3\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{$n=2$ Observations}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell 2}$\t&\t7.0\/6.4\t&\t3.7\/4.3\t&\t3.5\/3.6\t&\t8.5\/8.4\t&\t1140\t&\t1010\t&\t3910\t&\t4300\t\\\\\t\n\t\tODN$_{\\bar{d}2}$ \t&\t9.2\/7.8\t&\t4.8\/5.0\t&\t3.5\/3.5\t&\t8.6\/8.4\t&\t3410\t&\t4570\t&\t3370\t&\t4620\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d 2}$\t&\t8.0\/7.2\t&\t6.8\/7.5\t&\t2.0\/2.1\t&\t5.4\/5.1\t&\t2850\t&\t4120\t&\t4610\t&\t4970\t\\\\\t\n\t\t\\scriptsize VOS-DE$_2$ \t&\t36.2\/21.9\t&\t8.5\/6.7\t&\t3.0\/3.0\t&\t42.1\/39.7\t&\tN\/A\t&\tN\/A\t&\tN\/A\t&\tN\/A\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Perturb Training Data}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell p}$\t&\t7.0\/6.9\t& \\bf\t3.5\/4.1\t&\t4.3\/4.5\t&\t5.2\/5.2\t&\t100\t&\t140\t&\t5000\t&\t5000\t\\\\\t\n\t\tODN$_{\\bar{d} p}$\t&\t8.4\/8.5\t&\t4.0\/4.4\t&\t4.4\/4.4\t&\t5.2\/5.1\t&\t420\t&\t2760\t&\t2730\t&\t4270\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d p}$\t&\t6.7\/5.8\t&\t8.9\/9.9\t&\t2.4\/2.5\t&\t\\bf 2.8\/2.8\t&\t50\t&\t330\t&\t4860\t&\t4780\t\\\\\t\\hline\n\t\t\\hline \\multicolumn{9}{|c|}{Radial Input Image}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\t\\hline\n\t\t\\rowcolor{rowgray}\tODN$_{\\ell r}$\t& \\bf\t4.4\/4.3\t&\t4.0\/4.5\t&\t3.5\/3.5\t&\t7.4\/7.2\t&\t1710\t&\t870\t&\t4940\t&\t3940\t\\\\\t\n\t\tODN$_{\\bar{d} r}$\t&\t5.6\/5.0\t&\t3.8\/4.3\t&\t3.3\/3.4\t&\t7.5\/7.4\t&\t2010\t&\t4200\t&\t4990\t&\t4440\t\\\\\t\n\t\t\\rowcolor{rowgray}\tODN$_{d r}$\t&\t4.4\/4.4\t&\t7.2\/8.0\t&\t1.9\/1.9\t&\t4.3\/4.0\t&\t2210\t&\t460\t&\t4870\t&\t4710\t\\\\\t\\hline\n\t\\end{tabular}\n\t\\label{tab:abs}\n\\end{table}\n\n\\subsection*{ODMS Robot Test Set Segmentation Examples}\n\nFor the ODMS Robot test set, we intentionally choose challenging objects, spanning from a single die to the 470~\\textrm{mm} long pan.\nNot surprising, segmenting diverse objects presents varied challenges.\nTo illustrate this point, in Fig.~\\ref{fig:test_obj} we show the closest and farthest Robot test set segmentations for the die and pan.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.975\\textwidth]{test_objects.pdf}\n\t\\caption{ \\textbf{ODMS Robot Test Set Segmentation Examples.}\n\t\tThe small die segmentation (\\textit{top}) has fragments of other objects in the closest view (\\textit{left}) and completely misses the die in the farthest view (\\textit{right}).\n\t\tOn the other hand, the larger pan segmentation (\\textit{bottom}) misses parts of the handle that are out of the image in the closest view (\\textit{left}) but is fairly accurate in the farthest view (\\textit{right})\n\t}\n\t\\label{fig:test_obj}\n\\end{figure}\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe introduce the task of canonicalized-alignment, a universal first step for multi-purpose garment manipulation pipelines.\nBy funneling a diverse array of crumpled clothing into a small set of high-visibility configurations, this task addresses much of the complexity associated with cloth's infinite DOF state space and severe self-occlusion and thus simplifies downstream tasks. After training in simulation with our novel factorized reward formulation for canonicalized alignment our learned policy generalizes to the real-world robot system and can be directly used for garment ironing and folding.\nDue to imperfect cloth simulators, we hypothesize that canonicalized-alignment performance can be improved if a real-world supervision signal could be derived to enable real-world finetuning, and believe this is an interesting direction for future work.\n\\section{Introduction}\n\n\n\n\n\nWhy is everyday cloth manipulation conceptually simple for humans?\nTo us, there are only a few meaningful states shirts could\n\nIn-the-wild cloth manipulation is made easy if the cloth is setup mindful of the downstream manipulation.\nConcretely, this involves canonicalizing the garment, which unfolds the garment into a standard configuration thus exposing all keypoints, while ensuring the downstream task manipulation can kinematically reach task-relevant parts of the cloth.\nRigid-goal conditioned canonicalizing are a fundamental skill for robotic cloth manipulation systems, because this task-agnostic skill \\emph{funnels} severely self-occluded and crumpled garment configurations found in-the-wild into highly structured configurations, allowing for simple task-specific policies to predictably and robustly manipulate the garments.\n\n\n\n\\iffalse\n   \n   \n   \n    Consider how we go through clothes pile after laundry.\n   \n    One-by-one, we would align the garments at a specific pose, with a large swing then a small adjustment or two, before executing a routine folding algorithm.\n    All the severely self-occluded and crumple states clothes could assume are turned into a standard configuration.\n    This task-agnostic canonicalization step enables simple (i.e., hardcoded) task-specific routines to robustly and predictably manipulate the garments.\n   \n\n   \n   \n   \n    In-the-wild cloth manipulation is made easy through canonicalization:\n   \n    by first bringing severely self-occluded crumpled garments into their standard canonical configuration [at a specific pose], downstream tasks such as ironing and folding becomes easy.\n\n   \n    Every day cloth manipulation\n    effortlessly\n    because conceptually involves only two different modes,\n    first is canonicalizing garment into a standard state from the infinite number of configs\n    then downstream cloth manipulation relies on highly regular repeatable actions (e.g. a folding algorithm).\n\n    From these observations, we ask the following question:\n    ``How can we define a unified learning framework over multiple primitives and cloths''\n\n\n\\fi\n\nMany prior works have designed\nWhile prior works have tackled visual goal conditioned manipulation, we argue their problem formulation is too general for typical cloth manipulation tasks.\nOur key observation is goal-conditioned cloth manipulation is much simpler from a canonical state, given\n\nIn summary, our approach has three main contributions:\n\\begin{enumerate}\n    \\item \\textbf{A canonicalization objective factorization}, which not only give a better landscape for learning and greedy interaction policies\\footnote{don't like this word, too vague, also need to back up by experiments later.} but also enables rigid goal-conditioning (i.e., SE2 transforms) of the canonical pose by design.\n    \\item \\textbf{A multi-primitive, multi-arm setup for canonicalization}, which enables efficient unfolding using high velocity flings while reliably reaching garments' canonical configurations with high precision\\footnote{high precision means different things in different contexts, ideally swap out for something else} pick and places.\n    \\item \\textbf{A real-world shirt ironing and folding system}, which demonstrates our key observation that cloth manipulation in the wild is made tractable given a robust and efficient rigid-goal conditioned canonicalization.\n\\end{enumerate}\n<<<<<<< HEAD\n\n\n\n\n\\section{Introduction}\n\n\nWhy hasn't the garments manufacturing received the same level of success in automation as robotic manufacturing in other industries?\nDue to cloths' infinite degrees of freedom, severe self-occlusion, and the possible crumpled states they can assume during many manufacturing stages, defining simple, repetitive controllers with low failure rates (which characterizes non-deformable manufacturing pipelines) is a challenging feat.\nIn this work, we ask: ``how \\emph{should} category-level cloth manipulation pipelines be designed, which enables maximum flexibility to arbitrary initial configurations (\\textit{i}.\\textit{e}.  crumpled initial states) and manipulation task?''\n\nRecently, real-world cloth manipulation has received a lot of attention.\nSome of the earliest cloth manipulation works designed elaborate state machines for core cloth manipulation tasks in manufacturing, such as cloth smoothing~\\cite{ken2022speedfolding} and folding~\\cite{zhou2021folding}, but only for a specific instance, configuration, and task.\nMore recently, fully self-supervised approaches learn to unfold~\\cite{ha2021flingbot} and tackle visual goal-conditioned manipulation of a single cloth instance~\\cite{lee2020learning} from arbitrary initial configurations.\nOn the other hand, approaches which uses human demonstrations for smoothing~\\cite{ken2022speedfolding}, folding~\\cite{zhou2021folding}, but require costly human demonstrations\/annotations.\n\n\nOur main observation is that if given appropriately-aligned and canonicalized garments and a garment category-level keypoint detector, keypoint-based cloth-manipulation state machines are sufficient for many manufacturing pipelines.\nIn other words, instead of learning arbitrary cloth manipulation tasks from arbitrary initial configurations, we hypothesize that it is sufficient to learn a robust canonicalization and alignment policy.\nThis is because it \\emph{funnels} unstructured and self-occluded cloth configurations into highly structured states with clearly visible keypoints, enabling simple heuristics to work with high success rate.\n\nWe argue that learning a robust, task-agnostic, garment-category level canonicalization and alignment policy is an enabler for simple downstream task-specific state machines.\nTo this end, we propose a canonicalization and alignment objective, which factorizes into two terms, one which accounts for the deformation cost and another for the planar rigid transform cost of the garment.\nWe demonstrate that this objective function enables better canonicalization and alignment compared to a naive unfactorized per-vertex cost objective function.\nWe implement a self-supervised multi-primitive, dual-arm system for efficient unfolding using high velocity flings and fine-grained adjustments using quasi-static pick and places.\nWe directly deploy our simulation-trained policy in the real world, where we demonstrate its effectiveness in a steam-ironing and folding pipeline.\nThese downstream applications exploit the highly structured configurations from our policy \n\nIn summary, our approach has three main contributions:\n\\begin{enumerate}\n   \n    \\item \\textbf{A canonicalization and alignment objective factorization}, which we show emperically correlates better with downstream cloth manipulation success.\n    \\item \\textbf{A multi-primitive, multi-arm setup for canonicalization and alignment}, which uses dynamic flings and quasi-static pick \\& places.\n    \\item \\textbf{A real-world shirt steam-ironing and folding system}, which demonstrates our key observation that garment-category level cloth manipulation from arbitrary initial states is made tractable given a robust canonicalization and alignment.\n\\end{enumerate}\n\\section{Introduction}\n\nWhy has garment manipulation proved more difficult to automate than more typical rigid and articulated objects? We argue that two key factors are severe self-occlusion, which is present in the large set of possible crumpled states, and the infinite degrees of freedom inherent to clothing. As a result, it is impractical to manually define manipulation policies which result in reliable manipulation \u2014 a cornerstone of current automated non-deformable manufacturing pipelines.\n\nIn this work, we explore bridging the gap between existing approaches to automation and the challenging domain of clothing. We show that when a robot first learns to robustly manipulate arbitrarily configured clothing items into a specific pose, downstream manipulation skills can be made to work significantly more reliably on deformable clothing.\n\n\n\nRecently, real-world cloth manipulation has received significant attention.\n\\todo{need to cite earlier works, like Peiter Abbeel's folding with  PR2}Some of the earliest cloth manipulation work employed elaborate state machines for core cloth manipulation tasks in manufacturing, such as cloth smoothing~\\cite{ken2022speedfolding} and folding~\\cite{zhou2021folding}, but only for a specific clothing instance, configuration, and task.\n\nMore recently, learning-based approaches have shown success in creating more general cloth manipulation behavior. One such approach consists of supervised methods that use human demonstrations for smoothing~\\cite{ken2022speedfolding} and folding~\\cite{zhou2021folding} behaviors but require costly human demonstrations\/annotations. Another recent line of work employs fully self-supervised learning and has shown success in learning to unfold~\\cite{ha2021flingbot} and in tackling visual goal-conditioned manipulation of a single cloth instance~\\cite{lee2020learning} from arbitrary initial configurations.\n\n\nOur main observation is that if given appropriately-aligned and canonicalized garments\\todo{should define this, first time use the term}, keypoint-based task-specific state machines are sufficient for many garment manipulation tasks.\nInstead of learning arbitrary cloth manipulation tasks from arbitrary initial configurations, we hypothesize that it is sufficient to learn a robust canonicalization and alignment policy from which other manipulation skills may be chained. This is because such a policy \\emph{funnels} unstructured and self-occluded cloth configurations into highly structured states with clearly visible key points, enabling simple heuristics to work with a high success rate.\n\nTo this end, we define a new ``canonicalized-alignment'' task for garment manipulation, where the goal is to transform a garment from its arbitrary initial state into a canonical shape (defined by its category) and align it with a particular rigid pose defined by rotation and translation in a 2D plane.\n\nThe end result is a decomposition of garment manipulation into two factorized pieces. The first of which funnels a chaotic and diverse set of cloth configurations into a narrow distribution in a structured region of state space while the second piece consists of downstream behaviors that are able to rely on friendly initial configurations and high observability.\n\n\n\n\n\\todo{feels redundant}Enabled by canonicalized-alignment, we propose \\textbf{CASSIE}, a recipe for multi-purpose garment manipulation, which includes \\underline{c}anonicalization, \\underline{a}lignment, and \\underline{s}tate machines.\nBy learning to canonicalize, our approach can turn arbitrarily crumpled initial configurations into standard configurations where all the garment's key-points are visible.\nBy simultaneously learning alignment, the canonicalized garment can be appropriately positioned and oriented for the downstream state machine's kinematic constraints.\nLastly, by using a mature off-the-shelf key-point detector to extract a key-point garment representation, our proposed recipe enables the usage of key-point-based task-specific state machines.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/teaser.png} \\vspace{-4mm}\n    \\caption{Transformed-canonicalization funnels a wide distribution of possible initial configurations into a small set of well structured and highly observable states.} \\vspace{-5mm}\n    \\label{fig:task}\n\\end{figure}\n\nOur primary contribution is the introduction of the ``canonicalized-alignment'' task for garment manipulation, which can be serve as \\textit{universal} starting point for many downstream garment manipulation applications. To achieve this, we propose a learned multi-arm, multi-primitive manipulation policy that strategically choose between dynamic flings and quasi-static pick\\&place actions to efficiently and precisely transform the garment into its canonicalized and aligned configuration. \nThis policy is based on a novel factorized objective function that effectively guided the learning and avoid ambiguities and local minima which plague the generic goal-reaching formulations by decoupling shape and pose.\nWe evaluate our approach's effectiveness by applying CASSIE to multiple downstream garment manipulation tasks in the real-world on a physical robot, including category-level garment folding and ironing and we show empirically that incorporation of canonicalized-alignment significantly improves the reliability of downstream tasks. These results suggest that robust canonicalized-alignment provides a practical step forward toward general-purpose garment manipulation from arbitrary states for diverse tasks by significantly reducing the difficulty of learning performant downstream manipulation behavior.\n\\section{Introduction}\n\nWhy has garment manipulation proved more difficult to automate than more typical rigid and articulated objects?\nWe argue that two key factors are severe self occlusion, which is present in the large set of possible crumpled states, and the infinite degrees of freedom inherent to clothing. \nAs a result, it is impractical to manually define manipulation policies that achieve reliable manipulation \u2014 a cornerstone of current automated non-deformable manufacturing pipelines.\nIn this work, we explore bridging the gap between existing approaches to automation and the challenging domain of clothing.\nWe show that when a robot first manipulates arbitrarily configured clothing items into a predefined configuration (\\textit{i}.\\textit{e}.  canonicalization) at an appropriate pose (\\textit{i}.\\textit{e}.  alignment), downstream manipulation skills work significantly more reliably.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/teaser.png} \\vspace{-6mm}\n    \\caption{Canonicalized-Alignment \\emph{funnels} the large space of possible cloth configurations into a much smaller and better structured set of highly-visible states that greatly simplifies downstream tasks such as ironing or folding.} \n    \\vspace{-5mm}\n    \\label{fig:teaser}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/method_overview_v2.jpg}\n    \\vspace{-7mm}\n    \\caption{\\textbf{Approach Overview.} a) A batch of scaled and rotated observations are created from a top-down RGB image of the workspace and then concatenated with a scale-invariant coordinate map. b) The batch of inputs is fed through the factorized network architecture, producing a batch of rotated and scaled value maps for each primitive. c) All primitive batches are concatenated and the maximum value pixel parameterizes the action to be executed.}\n    \\vspace{-7mm}\n    \\label{fig:pipeline}\n\\end{figure*}\n\nRecently, real-world cloth manipulation has received significant attention.\nSome of the earliest cloth manipulation work explored manually designed heuristics which worked well for specific clothing types, configurations, and tasks, such as cloth unfolding~\\cite{cusumano2011bringing,maitin2010cloth,triantafyllou2016geometric,osawa2007unfolding}, smoothing~\\cite{Sun2013AHA,willimon2011model}, folding~\\cite{zhou2021folding,maitin2010cloth,osawa2006clothes,miller2011parametrized}, \nbut their strong assumptions initial states, fiducial markers, specialized tools, or cloth type\/shape do not generalize.\nMore recently, learning-based approaches have shown success in more general cloth manipulation behavior. \nOne line of work has explored supervised-learning from human demonstrations for smoothing~\\cite{ken2022speedfolding} and folding~\\cite{zhou2021folding}, but those methods required costly human demonstrations\/annotations. \nAnother recent line of work employs fully self-supervised learning and has shown success in learning to unfold~\\cite{ha2021flingbot} (but doesn't generalize to other tasks) and in tackling visual goal-conditioned manipulation of a single square cloth instance~\\cite{lee2020learning}.\n\nInstead of learning arbitrary monolithic cloth manipulation tasks, we hypothesize that it is more efficient to learn a robust task-agnostic canonicalization and alignment policy from which other task-specific manipulation skills may be chained.\nThis is because such a policy \\emph{funnels} unstructured and self-occluded cloth configurations into structured states with clearly visible key points (Fig.~\\ref{fig:teaser}, middle), reducing the complexity of the task-specific downstream policy, and enabling even simple heuristics to work with a high success rate.\n\nTo this end, we define a new ``canonicalized-alignment'' task for garment manipulation, where the goal is to transform a garment from its arbitrary initial state into a canonical shape (defined by its category) and align it with a particular 2D translation and rotation.\nThe end result is a decomposition of garment manipulation into two factorized parts.\nThe first part funnels a diverse set of cloth configurations into SE-2 transforms of a small set of states with high visibility. \nThe second part consists of downstream behaviors which relies on kinematically feasible transforms of structured initial configurations and full keypoint observability to achieve high task success rates.\n\n\nOur primary contribution is the introduction of ``canonicalized-alignment'', a garment manipulation task which serves as a cloth funnel for reducing general-purpose garment manipulation complexity.\nWe achieve this by the following technical contributions: \n\\begin{itemize}\n    \\item  We propose a learned multi-arm, multi-primitive manipulation policy that strategically chooses between dynamic flings and quasi-static pick\\&place actions to efficiently and precisely transform the garment into its canonicalized and aligned configuration.\n    \n    \\item To train the policy, we proposed a novel factorized reward function that avoids adverse local minima which plague the generic goal-reaching formulations by decoupling deformable shape and rigid pose.\n    \n    \\item We evaluate our approach in multiple downstream garment manipulation tasks in the real-world on a physical robot, including folding and ironing.\n\\end{itemize}\nOur experiments show that incorporation of canonicalized-alignment significantly reduces the complexity of downstream applications, suggesting that robust canonicalized-alignment provides a practical step forward toward multi-purpose garment manipulation from arbitrary states for diverse tasks.\n\n\n\\section{Introduction}\n\n\n\\textcolor{red}{ALPER:} \\textcolor{red}{I'm struggling to begin this introduction}\n\nUnfolding and folding laundry, aligning sheets of carbon-fiber on a mold at a factory, and positioning  sheets on a person during a surgical operation are only some of the tasks that require goal conditioned cloth manipulation.\n\n\\textcolor{red}{ALPER:} \\textcolor{red}{Big jump but I do want to mention this before introducing it in challenges}\nAlthough many prior works have studied folding of wearable garments, most of them make the assumption that the cloth begins in a perfectly \\textit{canonicalized}\\cite{huang2022mesh} state. In reality, clothes in natural environments are crumpled and self-occluded. What's the point of having a robot fold your clothes if you have to flatten them first? \n\nDespite the importance of cloth manipulation to home and industrial settings, making a general and robust robotic manipulation framework has not been fully addressed due to the the following challenges:\n\n\\begin{enumerate}\n\n    \\item \\textbf{Generality}\n    \n    Previous works that have investigated the task of cloth folding typically rely on strong assumptions about the cloth, such as full keypoint visibility, canonicalized initial state, or a single\/square cloth instance. Since clothes come in vastly different categories, shapes, and sizes, as well as unstructured and self-occluded states, these methods cannot generalize to most natural cloth configurations.\n    \n    \\item \\textbf{Precision-Efficiency Trade-off}\n    \n  \n    Much of the existing cloth manipulation literature has been successful in using quasi-static action primitives to achieve folding, smoothing, canonicalization, and unfolding. Although single-arm quasi-static action primitives (e.g. pick and place, pick and drag) are precise and predictable, they are inefficient against crumpled and self-occluded clothes. As an alternative, recent works have shown that dynamic action primitives (e.g. flinging, blowing-air) can be extremely efficient at unfolding; however, these dynamic actions lack the precision required for more general goal-conditioned general cloth manipulation.\n    \n    \\textcolor{red}{ALPER:} \\textcolor{red}{This is sort of what I previously had, I like some of the words in it but it doesn't fit the \"challenges\" section but it's more like \"what we need\":}\n      Efficient cloth manipulation requires both coarse-grained and fine-grained actions. Without large, coarse grained actions such as a fling, the task becomes difficult to solve, and self-occlusion becomes an issue. Without fine-grained actions, the model cannot quite reach the full goal(e.g. arms still stick out in random directions, corners folded in, etc.)\n        \n         \n    \\item \\textbf{Large State Space and Loss of Reward Equivariances}\n    \n    Cloth manipulation presents a learning challenge because a single piece of cloth can deform into practically infinite configurations, not to mention that clothes come in different shapes and sizes. Prior unfolding works in cloth unfolding \\cite{ha2021flingbot} \\cite{huang2022mesh}, have taken advantage of equivariant reward definitions, but equivariances lose their validity in the goal-conditioned setting. Given that we want to goal-condition a cloth, the supervision must depend on both the position and the rotation of the cloth relative to the goal.\n    \n\\end{enumerate}\n\nTo address the challenges posed above, we propose the follownig contributions:\n\n\\begin{enumerate}\n  \\item \\textbf{Factorization of Goal Conditioning}\n  \n  We assert that all goal-conditioned deformable object manipulation tasks can be factorized into the \\textit{deformable goal} and the \\textit{rigid goal}. Abstractly, the deformable goal concerns the shape and the deformation of the cloth, and it is invariant to translations and rotations. The \\textit{rigid goal} complements the deformable goal by specifying the position and the rotation of the cloth.\n  \n  Our work is centered around the following insights: 1. Majority useful goal-states for garments in the real world are rigid transformations of a very few deformable goals. 2. Many of these deformable goals can be achieved starting from the canonicalized deformable goal. 3. Since cloth canonicalization eliminates self-occlusion and enforces lots of structure, getting from the canonicalized state to a useful deformable goal can be achieved using demonstrations or a heuristic. \n  \n  \\textcolor{red}{ALPER:} \\textcolor{red}{Remark: I'm not sure if we want to include this but I think it's interesting that canonicalization is 1. universal and has a clear objective 2. deals with unstructured states, whereas folding 1. has no clear objective function, it has many local optima demonstrated by a wide variety of existing folding techniques 2. requires structure. Therefore it makes more sense to \\textbf{learn how to canonicalize} and \\textbf{demonstrate\/hardcode how to fold. }}\n  \n  Therefore, we demonstrate that we can accomplish category-level goal-conditioning by first training a model to canonicalize a garment category at a given rigid goal, thereby funneling all instances of that category in a structured state, and then using a simple keypoint detection model and a heuristic to fold the garment.\n  \n  \\item  \\textbf{Combinatorial Action Primitives}\n  \n   We propose a simple yet efficient universal multi-primitive cloth manipulation framework that unites both dynamic and quasi-static action primitives under the same reward \\textcolor{red}{ALPER:} \\textcolor{red}{Unification under the same reward was mentioned by Huy, is this something that's novel or worth emphasizing?}. The resulting model outperforms all single-primitive baselines in terms of both speed and performance. \\textcolor{red}{ALPER:} \\textcolor{red}{Is it worth highlighting that it's easy to add new primitives}\n  \n  \\item  \\textbf{Factorization Reward and Architecture}\n  \n  Unfolding the arm of a shirt near the goal and away from the goal must result in vastly different supervision under the rigid goal-conditioned reward; however, we can still learn from the commonality of unfolding the arm in both examples. To accomplish this, we factorize our reward into rigid and deformable components, and we similarly factorize our networks to learn each reward separately. Doing so disentangles the supervision and results in a more equivariant reward formulation. \\textcolor{red}{ALPER:} \\textcolor{red}{Not saying anything about the inherent value of the reward itself or the fact that it's more aligned with human preferences yet, I will confirm these things as my multi-shirt abelations conclude, what I wrote is the worst case scenario}\n\n\\end{enumerate}\n\n\\section{Method}\n\nOur method is divided into two distinct stages:\n\n\\begin{enumerate}\n    \\item The rigid goal-conditioned canonicalization stage, for which we train a self-supervised model inside the PyFlex cloth simulator \\cite{lin2020softgym}. The goal of rigid goal-conditioned canonicalization stage is to take an arbitrarily crumpled cloth, and to execute a combination of dynamic and quasi-static actions that will canonicalize the cloth at the goal rotation and translation, ensuring keypoint visibility and satisfying the kinematic constraints for downstream tasks.\n    \\item A simple, user-defined follow-up folding stage parametrized by the cloth keypoints to demonstrate the viability of heuristic folding policies after RGC.\n\\end{enumerate}\n\n\n\n\\subsection{Action Primitive Definitions}\n\nTo achieve both efficient and precise rigid goal-conditioned canonicalization in the first stage, our approach combines Pick \\& Fling and Pick \\& Place primitives.\n\nThe Fling \\& Place primitive is largely influenced by FlingBot \\cite{ha2021flingbot} with a few modifications suited to our task.\nThe key difference is that our primitive is not in a fixed direction and position. The direction of the fling is determined by the rotation of the-goal, and the position of the fling depends on the translation of the rigid-goal. This ensures that we can move our model's viewport around for rigid-goal conditioning.\n\nThe Fling \\& Place primitive involves a sequence of picking, lifting, stretching, and flinging actions. First, picking primitive performs a bimanual top-down grasp on the $\\textbf{x}_{\\text{1}}, \\textbf{x}_{\\text{2}}\\in \\mathbb{R}^3$ coordinates, which correspond to left and right grasp locations. Second, the arms lift up the cloth until there's a $5 cm$ margin with the ground, as opposed to a fixed fling height, and the height of the cloth from the ground is recorded. This is to ensure that smaller clothes do not fold back onto themselves during flinging. Third, the cloth is stretched as much as possible, with a maximum stretch width of 0.7m. Finally, the arms move forward $0.70m$ and they move linearly move down back onto the workspace surface at $1.4ms^{-1}$ by $0.70m + \\frac{\\text{Cloth Height}}{2}$ to ensure that the cloth is centered on the viewport.\n\nThe Pick \\& Place primitive involves a sequence of picking, lifting, dragging, and placing actions. Given $\\mathbf{x}_{\\text{1}}, \\mathbf{x}_{\\text{2}} \\in \\mathbb{R}^3$, the cloth is top-down pinch grasped at $\\mathbf{x}_{\\text{1}}$, lifted by $0.1 m$, then dragged to the xy coordinate $\\mathbf{x}_{\\text{2}}$ and placed back down onto the board surface.\n\n\\subsection{Reward Formulation For Rigid Goal-Conditioned Canonicalization}\n\nFor each cloth instance, the goal is determined by the ground-truth canonicalized vertices of the same cloth mesh. Obtaining the canonicalized vertices is an extremely simple task, since cloth meshes in the CLOTH3D dataset, as well as many other meshes on the internet, come in a canonicalized pose. All we do is to create goal configurations is to flatten the cloth at the center of the workspace.\n\n\nLet the flattened two-dimensional vertices of a cloth's configuration be defined as $\\{\\mathbf{v}_i\\}$ and the corresponding flattened goal vertices be defined as $\\{ \\mathbf{g}_i\\}$\n\nA naive way to define a goal is to use take the mean L2 distance of all vertices with respect to the corresponding goal vertices. Since the magnitude of this goal differs based on cloth size, we can attempt to normalize it by dividing by $\\sqrt{HW}$ where $H$ and $W$ are the height and the width of the canonicalized cloth respectively. This gives us the baseline unfactorized reward:\n\n\\begin{equation}\n    R_{\\textrm{Unf}} = -\\frac{\n        \\sum_i \\lvert\\lvert \\textbf{g}_i - \\textbf{v}_i\\rvert\\rvert\n    }{\\sqrt{HW}}\n    \\label{eq:unfactorized}\n\\end{equation}\n\nDespite being a viable dense reward, L2 reward has the following deficiencies: \\newline\n\\begin{enumerate}\n    \\item\\textbf{Entangled supervision:} Using the naive L2 reward, adjusting the arm of a shirt at the corner of the board results in a vastly different supervision than pulling the adjusting the arm at the center of the board. How is our model supposed to learn how to adjust the arms of a shirt?\n    \\item\\textbf{Rough reward landscape:} Actions that end up shifting most of the cloth body result in extremely sharp changes in reward, whereas smaller actions become insignificant. This makes it make to land final adjustments to the cloth.\n\\end{enumerate}\n\nTo address these problems, we propose the \\textit{factorized reward}, which is the sum of the \\textit{rigid reward}, which represents the distance of a cloth's overall translation and rotation away from a goal position, and the \\textit{deformable reward}, which represents the distance of a cloth away from its canonicalized configuration in terms of shape or deformation.\n\nTo calculate the two components, we introduce a new point cloud $\\{\\mathbf{g'}_{i}\\}$. $\\{\\mathbf{g'}_{i}\\}$, which is obtained superimposing the goal vertices $\\{\\mathbf{g}_{i}\\}$ onto the cloth's workspace vertices using the ICP (iterative closest point) algorithm. Note that $\\{\\mathbf{g'}_{i}\\}$ is the combination of a planar translation and rotation of $\\{\\mathbf{g}_{i}\\}$. Now, we define the factorized reward as follows:\n\n\\begin{align} \n    \\label{eq:factorized}\n    R_{\\textrm{Fac}} & = (1-\\alpha) R_{\\textrm{Rigid}} + \\alpha R_{\\textrm{Deform}}\n\\end{align}\n\nwhere the rigid cost is given by \n\\begin{equation}\n    R_{\\textrm{Rigid}} = \n    -\\frac{\\sum_i \\lvert\\lvert \\mathbf{v}_i - \\mathbf{g'}_i\\rvert\\rvert}\n    {\\sqrt{HW}}\n    \\label{eq:rigid}\n\\end{equation}\nand the deformable cost is given by \n\\begin{equation}\n    R_{\\textrm{Deform}} = \n    -\\frac{\\sum_i \\lvert\\lvert \\mathbf{g'}_i - \\mathbf{g}_i\\rvert\\rvert}{\n        \\sqrt{HW}\n    }\n    \\label{eq:deform}\n\\end{equation}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/method_reward_v1.pdf}\n    \\caption{\\textbf{Reward Computation.} TBD: Awaiting review.}\n    \\label{fig:task}\n\\end{figure}\n\nThe first term in the sum, the deformable distance,  represents \"Given that the cloth is perfectly positioned and rotated, how bad is the shape?\". The second term is the rigid distance represents \"If the cloth were perfectly canonicalized in place, how off would it be from one that's canonicalized at the goal?\"\n\n$\\alpha$ is a tunable parameter which determines how much emphasis is put onto the deformable reward.\nOur factorized architecture formulation, which we introduce in the next subsection, predicts each of these from a separate network head, so $\\alpha$ can be tuned at test time to optimize a desired metric. In our experiments, we found that $\\alpha=0.6$ maximizes the final IoU performance for long sleeve shirts.\n\n\\textcolor{red}{ALPER:} \\textcolor{red}{I'm not sure where to properly introduce how we actually goal condition}\n\n\\subsection{Learning Reward-Maximizing Actions}\n\nThe goal of our policy is to take a top-down RGB image of the workspace, and then execute the action that will maximize the single step delta-reward that it receives. To to unify the primitive parametrizations, enforce constraints easily, and equip our model with strong inductive bias with respect to scale, translation, and rotations, we use a spatial action map approach inspired by FlingBot.\n\n\n\\textcolor{red}{ALPER:} \\textcolor{red}{A couple sentences introducing spatial action maps}.\n\nTo extend the spatial action maps to multiple primitives, we propose a network architecture with one common encoder and $N$ decoder heads, where $N$ is the number of primitives.\n\nFor model inference, the top-down square $128 \\times 128$ RGB image of the viewport is captured, concatenated with a scale invariant positional encoding, and then a batch of scaled and rotated observations are generated from this image, resulting in $B$ \\textit{transformed} observations. In our experiments we use the cross product of scales $[0.75, 1.0, 1.5, 2.0, 2.5, 3.0]$ and 16 rotations spanning $360$ degrees, resulting in $B= 6 \\times 16 = 96$ observations.\n\nAfter a $B \\times 128 \\times 128$ value map is generated by each of the $N$ primitive decoders, pixels that result in an invalid action are filtered out, and the maximum value pixel among the $N \\times B \\times 128 \\times 128$ value maps are chosen, and the chosen pixel coordinate, the chosen action primitive, and the delta rewards are recorded.\n\n\nGiven the pixel coordinate $(x,y)$ executed on some transformed observation, $(x,y+10)$ and $(x,y-10)$ are selected as the 2D pixel corresponding to the action primitive parameters $\\mathbf{x}_1$ and $\\mathbf{x}_2$. These pixel coordinates are transformed back into the world space coordinates, and the top-down depth of the coordinate is appended as the $z$ coordinate to obtain the three dimensional locations that parametrize our action primitives.\n\nThe specified action is executed, and the step statistics are saved into a replay buffer for off-policy training.\n\nDuring the off-policy training stage, a batch of $(S_i, A_i, R_i)_{i=1, \\cdots , N}$ tuples are sampled for each action primitive $i$, where $S_i$ represents the top-down network observation, $A_i$ represents the chosen pixel coordinate $(x, y)$, and $R_i$ represents the delta-reward received by executing the action primitive $i$ on the pixel $A_i$ with respect to the scaled and rotated reference frame of the observation $S_i$.\n\n\n\nThe loss is accumulated by sequentially forwarding each observation $S_i$ through the network, decoding the observation with the $i$ th head into a value map $V_i$, and then adding to the total loss the L2 distance between $V_i[x,y]$ and $R_i$. Once the loss is accumulated for each primitive, the gradient step is taken. In our approach, we use Adam optimizer with lr=1e-4.\n\nIn addition to predicting delta-rewards using a network, we also investigate learning rigid and deformable rewards using two separate networks, with one encoder and $N$ decoders each. For inference, the $N$ value maps from each network are added together by the weight $\\alpha$ according to factorized reward formulation [\\ref{eq:factorized}], and the maximum pixel is chosen for execution. The training procedure is identical to the one mentioned above, where one network is supervised on delta-rigid-rewards and the other one on delta-deformable-rewards.\nFor more details, we refer to readers the code [Link]\n\nWe train our model using the aformentioned setup until convergence, for 12,500 episodes with 8 steps per episode. This process takes about 48 hours on 4 RTX3090 GPUs.\n\n\n\\subsection{Follow-up Folding Heuristic}\n\nUnlike unfolding, folding a cloth is a structured, step-by-step task that does not involve self-occlusion. In addition, folding methods vary based on habits and preferences, so it does not make sense to train an entire model to fold a cloth in one specific way.\n\nIn our approach, we use a simple heuristic that identifies cloth keypoints using a UNet, and then executes a user-defined closed-loop policy based on these keypoints.\n\nTo train the UNet keypoint model for a given cloth category, we take about 100 128x128 pixel images of high-coverage observations from a training replay buffer, and hand-label them using a custom GUI. Using a heavily augmented training pipeline, we are able to train a model that can robustly identify the keypoints of a canonicalized cloth with high sim to real generalizability. The canonicalized pose makes the keypoint identification almost trivial.\n\nAfter identifying the cloth keypoints, we execute a user-defined pick-and-place policy parametrized by the keypoint locations.\n\n\\section{Method}\n\n\\subsection{A Two-staged Recipe for Garment Processing}\n\nCommonly cited reasons for the challenges of cloth manipulation are their infinite degrees-of-freedom (DOF) and severely crumpled configurations, which implies a large state space and partial observability, respectively.\nThus, the same simple, low-failure-rate state machines which has dominated non-deformable manufacturing pipelines does not trivially extend to garment pipelines.\nThis is because such human-designed state machines typically assumes a low-dimensional state space and highly-structured configurations, which requires limited-to-no sensing.\nAre we forced to take a fully end-to-end learning approach for each cloth manipulation task we care about then?\n\nWe propose to factorize a cloth-manipulation pipeline into two stages\\todo{consider putting two part factorization into teaser figure}.\nIn the first stage, the robot system execute a learned \\emph{task-agnostic} canonicalization \\& alignment policy, which leaves the garment is in a standard pose with all keypoints are visible.\nIn the second stage, the robot will execute a \\emph{task-specific} human-specified state machine, which operates purely from key-points.\nWe argue that this factorization comes with three benefits:\n\\begin{enumerate}\n    \\item \\textbf{Arbitrary initial configuration via canonicalization}:\n    Since cloths readily deform and crumple during any stage of the manufacturing pipeline, simple manually-designed state machine do not have the structure they require to perform their reliable manipulation.\n   \n   \n    In contrast, our first stage performs canonicalization, which \\emph{funnels} the large space of cloth configurations into the high-structured, fully-observable configurations which such state machines require, which enables our pipeline to work on arbitrary initial configurations.\n    \\item \\textbf{Downstream task-awareness via alignment}:\n    Manufacturing pipelines may involve many different tools (\\textit{e}.\\textit{g}.  iron, sewing machines) placed on different sides of a workspace.\n    Thus, depending on the downstream task, it is insufficient for canonicalized cloths to be at any position and orientation on the workspace~\\cite{huang2022mesh}\n    In contrast, our first stage also performs garment alignment, which enables our system to canonicalize the cloth at a specified planar pose which is kinematically most suitable for downstream tasks.\n    \\item \\textbf{Garment-level generalization via Key-point-based State Machines}:\n    Engineering and debugging a task-specific state-machine operating on key-points to achieve low failure rates is much more straight forward and, thus, less costly than that for a behaviour-cloning, imitation learning, or reinforcement learning approach.\n    This is because the only data-driven part of this second-stage builds upon the maturity of off-the-shelf key-point detectors and the existing user-friendly frameworks (which abstracts away the semi-supervised, data-augmentation machinery which makes them work so robustly).\n    While a key-point-based cloth representation contains less information than an image~\\cite{ha2021flingbot} or mesh~\\cite{}, we argue that this is a feature, not a bug.\n    This is because such key-point representations effectively reduces the infinite DOF down to a handful meaninful ones, which makes manually writing a state machine for a new garment manipulation task simple.\n    With a good canonicalization and alignment, we hypothesize that our key-point-based state machines would achieve much better intra-garment-class generalization than data-driven approaches\\todo{ideally, we have an approach which compares behaviour cloning with state machine based approaches}.\n\\end{enumerate}\nWe name our two-staged garment manufacturing pipeline recipe, which includes \\underline{C}anonicalization, \\underline{A}lignment, and a \\underline{S}tate machine, \\textbf{CASPAr}.\nNext, we will discuss how we formulate the canonicalization \\& alignment objective (Sec.~\\ref{sec:method:can_align_task}), how we learn the task (Sec.~\\ref{sec:spatial_policy}), and how we implement the system and state-machines (Sec.~\\ref{sec:method:heuristics}).\n\n\n\n\\subsection{The Canonicalization \\& Alignment Task}\n\\label{sec:method:can_align_task}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/method_reward_v2.pdf}\n    \\caption{\\textbf{Reward Computation.} TBD: Awaiting review.}\n    \\label{fig:task}\n\\end{figure}\n\n\\mypara{Problem Statement.}\nFix a garment class $c \\in \\{\\textrm{shirt},\\textrm{pants},\\textrm{skirts}\\}$.\nGiven a garment instance in class $c$ with $N$ vertices, let $\\mathbf{v}= \\{v_i\\}_{i\\in N}$ denote the local configuration of the cloth, which is the set of vertex planar positions $ v_i \\in \\mathbb{R}^2$ in the cloth's local frame $T = (x,y,\\theta)$.\nHere, $(x,y)\\in \\mathbb{R}^2$ is the origin and $\\theta \\in [-180^\\circ,180^\\circ]$ is the orientation of the cloth's frame with respect to the global (workspace) framework.\nDenote by $ \\mathbf{g}  = \\{ g_i \\}_{i\\in N}$ the local canonical configuration of the cloth, with each $g_i \\in \\mathbb{R}^2$ in the cloth's local frame.\nHere, $\\mathbf{g}$ for each garment instance the conventional configuration for class $c$, such as the T-shaped pose when $c=\\textrm{shirt}$ and the upside-down V-shaped pose $c=\\textrm{pants}$.\nLet the target planar pose $T^\\prime = (x^\\prime,y^\\prime, \\theta^\\prime)$ describe the global planar position $(x^\\prime,y^\\prime) \\in \\mathbb{R}^2$ and rotation $\\theta^\\prime\\in [-180^\\circ,180^\\circ]$ at which the garment instance's pose $T$ should be located.\nUsing only top-down views of the garment instance, the goal of canonicalization is to manipulate the garment instance such that  $\\mathbf{v} = \\mathbf{g}$.\nSimilarly, from only $I$, the goal of alignment is to ensure that  $T = T^\\prime$.\n\n\\mypara{Naive Reward Formulation.}\nClearly, a global canonical configuration $\\mathbf{g}^\\prime$ can be derived from  $\\mathbf{g}$ and a specified target planar pose $T^\\prime$, and similarly from the current local configuration $\\mathbf{v}$ to the global canonical configuration $\\mathbf{v}^\\prime$.\nThis leads to the following straight-forward objective for canonicalization and alignment\n\\begin{equation}\n    R_{\\textrm{Unf}}  = -\\lvert\\lvert \\mathbf{g}^\\prime - \\mathbf{v}^\\prime \\rvert\\rvert_{2}\n    \\label{eq:unfactorized}\n\\end{equation}\nBy summing up the $L_2$ distances between all vertices, this objective formulation is applicable to generic cloth manipulation since $\\mathbf{g}^\\prime$ could be any cloth configuration of interest.\nHowever, we argue that this leads to two downsides:\n\\begin{enumerate}\n    \\item \\textbf{Entangled supervision.}\n    Reaching a canonicalized configuration at the center versus the corner of the workspace could give vastly different $R_{\\textrm{Unf}}$ values depending on what the target planar pose $T^\\prime$ is.\n    This makes it difficult for the policy to tell whether it should make a planar transformation of the cloth configuration (such as shifting entire cloth to the right) or a deformable adjustment (such as flipping shirt's sleeve outwards).\n    \\item \\textbf{Over-emphasis on cloth pose.}\n    Actions that shift the cloth result in sharp and large changes in $R_{\\textrm{Unf}}$, while actions of smaller magnitudes become insignificant.\n    Since such small adjustment actions are required to bring a poorly canonicalized cloth to a well-canonicalized one,  $R_{\\textrm{Unf}}$ fails to put enough emphasis on the canonicalization subtask.\n\\end{enumerate}\n\n\\mypara{Factorized Reward Formulation.}\nIn contrast, we propose a reward factorization (Fig.~\\ref{fig:reward_formulation}), which retains the \\underline{c}anonicalization and \\underline{a}lignment nature of the task,\n\\begin{align} \n    \\label{eq:factorized}\n    R_{\\textrm{CA}} & = (1-\\alpha) R_{\\textrm{C}} + \\alpha R_{\\textrm{A}}\n\\end{align}\nwhere $\\alpha \\in (0,1)$ is a weighing hyperparameter, the canonicalization reward is given by \n\\begin{equation}\n    R_{\\textrm{C}} = \n    - \\lvert\\lvert \\mathbf{v} - \\mathbf{g}\\rvert\\rvert_2\n    \\label{eq:rigid}\n\\end{equation}\nand the alignment reward is given by \n\\begin{equation}\n    R_{\\textrm{A}} = \n    -\\lvert\\lvert \\mathbf{g}^\\prime - \\mathbf{g}\\rvert\\rvert_2\n    \\label{eq:deform}\n\\end{equation}\nHere, $R_{\\textrm{C}}$ encodes ``if the cloth is perfectly positioned and oriented, how bad is the deformable shape?'', while $R_{\\textrm{A}}$ asks, ``if the cloth was perfectly canonicalized, how bad is the planar pose offset?''.\nWith this factorization, we can provide separate supervision $R_{\\textrm{C}}$ and $R_{\\textrm{A}}$ during training, while acting with respect to $R_{\\textrm{CA}}$ during data-collection\/inference time.\nThis helps the policy distinguish how actions separately affect the cloth shape or planar pose.\nFurther, with a tunable $\\alpha$, we can emphasize $R_{\\textrm{C}}$ more than $R_{\\textrm{A}}$, which we have observed quantitatively correlates better with downstream task success and qualitatively aligns with a better canonicalization.\n\n\\mypara{Objective Implementation.}\nIn our experiments, we observed that $\\alpha = 0.6$ performs best.\nTo account for different cloth sizes, we normalize all $R_{\\textrm{C}}$, $R_{\\textrm{A}}$ and $R_{\\textrm{Unf}}$ by the geometric mean of the cloth's height and width in a canonicalized configuration.\nWe define a cloth's planar pose $T$ as the planar transformation for which $\\lvert\\lvert\\mathbf{v}^\\prime - \\mathbf{g}^\\prime \\rvert \\rvert$ is minimized, and approximate this using a modified version of ICP that utilizes the ground-truth correspondence between the current mesh and the goal mesh. In this modified method, vertices that are closer to their goal coordinates below a certain threshold are used to superimpose the current vertices onto the goal vertices, and this process iterates until convergence.\n\n\n\\subsection{Spatial Action Maps for Canonicalization and Alignment}\n\\label{sec:spatial_policy}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/pipeline.png}\n    \\caption{\\textbf{Approach Overview.} }\n    \\label{fig:pipeline}\n\\end{figure*}\n\nCoarse-grain, high-velocity multi-arm flings can efficiently unfold and align garments from crumpled initial states~\\cite{ha2021flingbot}, but is insufficient for fine-grained adjustments to achieve good canonicalization.\nInstead, a multi-arm, multi-primitive system which combines quasi-static and dynamic actions can achieve the best of both worlds.\nTo unify the primitive parametrizations, easily enforce constraints, and equip our model with strong inductive bias with respect to scale, translation, and rotations, we use a spatial action maps policy.\n\n\\mypara{Spatial Action Maps} is a convolutional neural network~\\cite{lecun2010convolutional} (CNN) based policy for learning value maps~\\cite{Wu_2020} where actions are defined on the pixel grid.\nThrough its simple and practical exploitation of these translational, rotational, and scale equivariances, spatial action maps is a popular framework for learning robotic policies~\\cite{}.\nWe refer the reader to \\citet{ha2021flingbot} for more details.\n\nWe extend \\citeauthor{ha2021flingbot}'s spatial action maps approach as follows.\nGiven a top-down view $I\\in \\mathbb{R}^{H\\times W \\times 3}$ of the workspace (Fig.~\\ref{fig:pipeline}a), we concatenate a $H \\times W \\times 2$ normalized coordinate map to $I$ to help the network reason about the cloth's alignment.\nThe resulting map is transformed to 16 rotations spanning $360^\\circ$ and scales in $[0.75, 1.0, 1.5, 2.0, 2.5, 3.0]$, which effectively enforces the arm-cross-over and grasp-width safety constraints.\nTo enable multiple primitives, we propose a factorized network architecture (Fig.~\\ref{fig:pipeline}b) with two common encoder, one for each task's reward ($R_{\\textrm{C}}$, $R_{\\textrm{A}}$) and multiple decoder heads, one for each primitive.\nThe decoded value map for each primitive from each encoder is combined using \\eqref{eq:factorized}, and the highest value action (over all action parameters and primitives) is chosen (Fig.~\\ref{fig:pipeline}c), with $\\epsilon$-greedy exploration.\n\n\\mypara{Implementation.}\nIn our experiments, we consider two primitives, quasi-static pick\\&place and dynamic flings.\nWe use $(H,W) = (128, 128)$, and a $\\epsilon_v$ value exploration and $\\epsilon_a$ action primitive exploration schedule with halflife $\\epsilon_v=2500$ steps and $\\epsilon_a=5000$ steps.\nOur value networks' predictions are supervised to the delta-reward values -- which is the difference in $R_{\\textrm{CA}}$ before and after an action is taken -- using the MSE loss and the Adam~\\cite{kingma2014adam} optimizer with a learning rate of $1e-4$.\nWe train our model until convergence, for 12,500 episodes with 8 optimization steps per episode, which takes about 48 hours on 4 NVIDIA RTX3090s.\n\n\n\\subsection{Task-specific State Machines for Ironing and Unfolding}\n\\label{sec:method:heuristics}\n\n[OUTLINE]\n\nSetup for state machines:\n\\begin{itemize}\n    \\item Run canonicalization \\& alignment policy, described in previous section, with a specified target pose $T^\\prime$\n    \\item Use DeepLabv3 to detect keypoints on the garment\n    \\item Execute the task-specific key-point based state machine\n    \\item Key benefit is that the highly structured canonicalized states enables the usage of state machines, which are simple to define and debug for engineers.\n\\end{itemize}\n\nDeepLabv3\n\\begin{itemize}\n    \\item Emphasize that this is mature technology, used in many other fields, and has also been used in robotics (e.g. IRP).\n    \\item We collected XXX data points for each garment class, and trained one model for each garment class.\n    \\item Qualitatively, we observed that this key-point model generalized to novel garment instances.\n    \\item We provide quantitative numbers for DeepLabv3 results on held-out cloth meshes in supplementary material.\n\\end{itemize}\n\nIroning state machine for shirts:\n\\begin{itemize}\n    \\item The ironing state machine is the simplest in that it does not require any perception. A third arm holding an iron moves back and forth between two way points.\n    \\item Our policy performs 4 canonicalization \\& alignment, one for each orientation, such that the iron covers multiple sides of the garment\n    \\item Ironing Score: The binary mask overlay between the iron's path and the garment.\n\\end{itemize}\n\nFolding state machine for shirts:\n\\begin{itemize}\n    \\item Folding consists of 2 steps:\n        \\begin{enumerate}\n            \\item Grasp wrist keypoints, then bring them to the far waist keypoints.\n            \\item Grasp shoulder keypoints, then bring them to the waist keypoints.\n        \\end{enumerate}\n    \\item Folding Score: composed of three scores:\n    \\begin{enumerate}\n        \\item Shirt Body IoU score: get the shirt's body mask with the target shirt body mask, computed before step 1 of the folding state machine.\n        \\item Shirt Arm IoU score: similarly for shirt arms, computed between step 1 and step 2 of the folding state machine.\n        \\item Final folded IoU, computed after step 2 of the folding state machine.\n    \\end{enumerate}\n    The final folding score is the mean of these three numbers.\n\\end{itemize}\n\\section{Method}\n\\label{sec:method}\n\n\\subsection{A Multi-Purpose Garment Manipulation Pipeline}\n\\label{sec:method:recipe}\n\n\nWe propose a factorized approach to multi-purpose garment manipulation from arbitrary states that decomposes the process into two steps.\nFirst, the robot executes a learned \\emph{task-agnostic} canonicalized-alignment policy, which leaves the garment in a known configuration predefined for the clothing category at a specified 2D rotation and translation.\nSecond, the robot executes a \\emph{task-specific} keypoint based policy, which could be as simple as a manually-designed heuristic.\nThis approach confers three primary benefits:\n\\begin{itemize}\n    \\item \\textbf{Arbitrary initial configuration}:\n    Canonicalization \\emph{funnels} the large space of possible cloth configurations into a narrow distribution of highly-structured fully-observable configurations from which downstream policies can more easily operate.\n    \\item \\textbf{Downstream task-awareness}:\n    Flexible goal-conditioned alignment allows the canonicalized cloths to be placed at specified positions and orientations that are kinematically appropriate for particular downstream tasks.\n    \\item \\textbf{Clothing category generalization}:\n    A keypoint-based cloth representation effectively reduces the observation space from having to represent the infinite DoF down to a few meaningful keypoints.\n    Further, cloths are always in a known canonicalized configuration.\n    These two properties combined not only simplifies learning downstream task-specific manipulation policies, but also makes it possible to engineer heuristics that work reliably for a clothing category.\n\\end{itemize}\nNext, we will discuss how the canonicalized-alignment task is formulated (Sec.~\\ref{sec:method:can_align_task}), learned (Sec.~\\ref{sec:spatial_policy}), and implemented alongside the several downstream task policies (Sec.~\\ref{sec:method:heuristics}).\n\n\n\n\\subsection{The Canonicalization \\& Alignment Task}\n\\label{sec:method:can_align_task}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/method_reward_v2.png}\n    \\vspace{-6mm}\n    \\caption{\\textbf{Reward Computation.}\n    From the goal configuration $\\mathbf{g}$ (green) and current configuration $\\mathbf{v}$ (magenta), we compute a best-alignment configuration $\\mathbf{g}^\\prime$ (orange).\n   \n    Then, the average vertex distance between $\\mathbf{g}^\\prime$ and $\\mathbf{g}$ (where only rigid transforms matter) gives the alignment reward $R_{\\textrm{A}}$, while that between $\\mathbf{g}$ and $\\mathbf{v}$ (where only deformation matters) gives the canonicalization reward $R_{\\textrm{C}}$.\n   \n    Our factorized reward $R_{\\textrm{CA}}$ is a weighed sum between $R_{\\textrm{A}}$ and  $R_{\\textrm{C}}$.\n   \n    }\n    \\vspace{-6mm}\n    \\label{fig:reward_formulation}\n\\end{figure}\n\n\\mypara{Problem Formulation.}\nGiven a clothing item in some clothing category in an arbitrary initial configuration, the goal of canonicalization is to reach the human-defined standard deformable configuration for that clothing category, such as a T-shaped configuration for shirts and the upside-down V-shaped configuration for pants.\nNote that this only accounts for the garment's shape, but not its pose in the workspace.\nThis means canonicalization alone can't ensure that downstream tasks are kinematically feasible.\nTo address this shortcoming, the goal of ``canonicalized-alignment'' is to reach canonicalization at a specific planar position and rotation in the workspace.\n\n\\mypara{Naive Reward Formulation.}\nGiven a simulated cloth instance with $N$ vertices, let  $\\mathbf{v} = \\{v_{i}\\}_{i\\in[N]}$ denote the current configuration of the cloth (Fig.~\\ref{fig:reward_formulation}, magenta), where each $v_i \\in \\mathbb{R}^3$ is the position of the $i$th cloth vertex.\nGiven a goal configuration $\\mathbf{g} = \\{g_{i}\\}_{i\\in[N]}$, the average per-vertex distance between $\\mathbf{g}$ and $\\mathbf{v}$ gives a generic goal-conditioned cloth manipulation cost.\nIn the specific case where $\\mathbf{g}$ is a canonicalized configuration of the cloth at the goal position and rotation (Fig.~\\ref{fig:reward_formulation}, green), we have the following straightforward canonicalized-alignment reward \n\\begin{equation}\n    R_{\\textrm{Unf}}  = -\\lvert\\lvert \\mathbf{g} - \\mathbf{v} \\rvert\\rvert_{2}\n    \\label{eq:unfactorized}\n\\end{equation}\nClearly,  $R_{\\textrm{Unf}}$ is consistent, in that a policy which achieves $R_{\\textrm{Unf}}=0$ achieves perfect canonicalized-alignment.\nHowever, this formulation has two primary downsides:\n\\begin{enumerate}\n    \\item \\textbf{Entangled supervision.}\n    When $R_{\\textrm{Unf}}$ is low, it can be difficult for the policy to tell whether it should make a planar transformation of the cloth configuration (such as shifting entire cloth to the right) or a deformable adjustment (such as flipping a shirt's sleeve outwards).\n    \\item \\textbf{Over-emphasis on cloth pose.}\n    Actions that shift the cloth result in sharp and large changes in $R_{\\textrm{Unf}}$, while actions of smaller magnitudes become insignificant.\n    Since such small adjustment actions are required to bring a poorly canonicalized cloth to a well-canonicalized one,  $R_{\\textrm{Unf}}$ fails to put enough emphasis on the canonicalization subtask, and leads to a problematic local minima in policy learning.\n\\end{enumerate}\n\n\n\\mypara{Factorized Reward Formulation.}\nTo alleviate these shortcomings, we propose a reward factorization, that expresses the canonicalization $R_{\\textrm{C}}$ and alignment $R_{\\textrm{A}}$ aspects of the task separately:\n\\begin{align} \n    \\label{eq:factorized}\n    R_{\\textrm{CA}} & = (1-\\alpha) R_{\\textrm{C}} + \\alpha R_{\\textrm{A}}\n\\end{align}\nwhere $\\alpha \\in (0,1)$ is a hyperparameter.\nWith this factorization, we can provide separate supervision $R_{\\textrm{C}}$ and $R_{\\textrm{A}}$ during training, while acting with respect to $R_{\\textrm{CA}}$ during data-collection and inference.\nThis helps the policy distinguish how actions separately affect the cloth shape or planar pose. \nFurther, with a tunable $\\alpha$, we can emphasize $R_{\\textrm{C}}$ more than $R_{\\textrm{A}}$, which leads to a better canonicalization.\n\nTo factorize the reward, we propose to compute a transform $T$, which transforms $\\mathbf{g}$ into a best-aligned goal configuration $\\mathbf{g}^\\prime$ (Fig.~\\ref{fig:reward_formulation}, orange).\nGiven such a $\\mathbf{g}^\\prime$, its distance to $\\mathbf{v}$ accounts only for their deformable shape mismatch, which serves as the canonicalization reward,\n\\begin{equation}\n    R_{\\textrm{C}} = \n    - \\lvert\\lvert \\mathbf{v} - \\mathbf{g}^\\prime\\rvert\\rvert_2\n    \\label{eq:rigid}\n\\end{equation}\nMeanwhile, by $T$'s definition, the distance between $\\mathbf{g}^\\prime$ and $ \\mathbf{g}$ accounts only for the mismatch in planar position and rotation, which serves as the alignment reward,\n\\begin{equation}\n    R_{\\textrm{A}} = \n    -\\lvert\\lvert \\mathbf{g}^\\prime - \\mathbf{g}\\rvert\\rvert_2.\n    \\label{eq:deform}\n\\end{equation}\n\n\n\\mypara{Factorization Implementation.}\nTo find $\\mathbf{g}^\\prime$, we have observed that naively minimizing the average per-vertex distance between $\\mathbf{g}$ and $\\mathbf{v}$ is extremely sensitive to outliers, so does not give us the best alignment.\nSuch outliers arise due to mismatches in $\\mathbf{g}$'s and $\\mathbf{v}$'s deformable shapes where small protrusions with large offsets (\\textit{e}.\\textit{g}.  a shirt's arm folded inwards) could significantly shift the minimum distance configuration.\nTo filter out such outliers, we optimize the transform $T$ which minimizes this distance for only a subset of points, where point $i$ is included if $\\lvert\\lvert g_i - v_i\\rvert\\rvert_2 \\leq \\tau$ for some scalar threshold $\\tau$ then apply $T$ to $\\mathbf{g}$ to get $\\mathbf{g}^\\prime$.\nWe repeat this minimization and filter procedure in iterations, using the previous iteration's $\\mathbf{g}^\\prime$ as the current $\\mathbf{g}$, until convergence.\n\nIn our experiments, we observed that $\\alpha = 0.6$ and $\\tau=0.3$ performs best.\nTo account for different cloth sizes, we normalize all $R_{\\textrm{C}}$, $R_{\\textrm{A}}$, $R_{\\textrm{Unf}}$, and $\\tau$ by the geometric mean of the cloth's height and width in a canonicalized configuration.\nSince most garments are mirror-symmetric, we select the highest reward from either the goal configuration or its mirror-flip in the goal's local frame.\n\n\n\\subsection{Multi-Primitive Spatial Action Maps Policy}\n\\label{sec:spatial_policy}\n\nCoarse-grain dynamic multi-arm flings can efficiently unfold and align garments from crumpled states~\\cite{ha2021flingbot}, but are insufficient for the fine-grained adjustments required to achieve canonicalization.\nTo overcome this challenge, we propose a multi-arm, multi-primitive system that combines quasi-static and dynamic actions, which enables both efficient \\emph{and} fine-grained manipulation. To unify the primitive parametrizations and easily enforce constraints, we use a spatial action maps policy.\n\n\\mypara{Spatial Action Maps} is a convolutional neural network~\\cite{lecun2010convolutional} (CNN) policy for learning value maps~\\cite{Wu_2020} where actions are defined on a pixel grid.\nThrough its simple and effective exploitation of translational, rotational, and scale equivariances, spatial action maps is a popular framework for learning robotic policies~\\cite{wu2020spatial,ha2021flingbot,xu2022dextairity}.\n\nWe extend FlingBot~\\cite{ha2021flingbot}'s spatial action maps approach as follows.\nGiven a $H\\times W \\times 3$ top-down view of the workspace (Fig.~\\ref{fig:pipeline}a), we rotate and scale it to form a stack of transformed observations of shape $K \\times H\\times W \\times 3$.\nTo help the network reason about the cloth's alignment, we concatenate a $K \\times H \\times W \\times 2$ scale-invariant, normalized (between -1 and 1) positional encoding to the transformed observation stack.\nTo enable multiple primitives, we propose a factorized network architecture (Fig.~\\ref{fig:pipeline}b) with two encoders, one for each task's reward ($R_{\\textrm{C}}$, $R_{\\textrm{A}}$), where each encoder has two decoder heads, one for each primitive.\nThe encoders take in the transformed observation stack, and the decoders output value maps, one for each reward-primitive pair.\nThe value maps are combined using \\eqref{eq:factorized}, and the highest value action (over all action parameters and primitives) is chosen (Fig.~\\ref{fig:pipeline}c).\n\n\\mypara{System Implementation.} \nIn our experiments, we consider two primitives, quasi-static pick\\&place and dynamic flings.\nWe use $(H,W) = (128, 128)$\nand a decaying $\\epsilon$-greedy with $\\epsilon = 1$ for exploration of\n1) action primitives (\\textit{i}.\\textit{e}.  fling v.s. pick\\&place) with half life 5000, and \n2) action parameters within each primitive with half life 2500.\nBy constraining the observation's transforms to 16 rotations spanning $360^\\circ$ and scales in $\\{0.75, 1.0, 1.5, 2.0, 2.5, 3.0\\}$ (giving $K=96$), we can ensure that arms neither collide nor cross-over each other. \nOur value networks' predictions are supervised using the delta-reward values -- which is the difference in $R_{\\textrm{CA}}$ before and after an action is taken -- using the MSE loss and the Adam~\\cite{kingma2014adam} optimizer with a learning rate of $1e-4$.\nWe train our model for 12,500 episodes which takes 2 days on 4 NVIDIA RTX3090s.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/qualitative_categories_v1.jpg}\n    \\vspace{-3mm}\n    \\caption{\n    \\textbf{Canonicalized-Alignment of Multiple Categories.}\n    In each row, we demonstrate a sequence of 5 actions taken by the model corresponding to a clothing category in simulation.}\n    \\vspace{-6mm}\n    \\label{fig:canal:sim}\n\\end{figure}\n\n\n\\subsection{Keypoint-based Task Heuristics}\n\\label{sec:method:heuristics}\n\nCompared to learning-based approaches, heuristics are highly interpretable and thus simple to define.\nHere, we demonstrate that it's possible to use heuristics for shirt ironing with no keypoints and folding with a small set of keypoints.\n\n\\mypara{Keypoint Detection.}\nWe collect 200 cloth configurations from simulation with coverage at least $60\\%$ and trained a DeepLabv3~\\cite{chen2019rethinking} detector for each garment class.\nUsing a random 80\/20 training\/evaluation split, we observed that this detector generalizes well to novel garment instances with average error of 5\/128 pixels.\nSetting up a keypoint detector model for a new clothing category takes roughly 1 hour.\nAfter detection, each keypoint is depth-projected into 3D points and transformed into the workspace frame of reference.\nBy representing cloths as a set keypoints, we sidestep  their infinite DoF by using a few meaningful keypoints as the representation, which makes it simpler to define heuristics over them.\nFor instance, long sleeve shirts have six keypoints for two sleeves, shoulders, and waists.\n\n\\mypara{Ironing Heuristic.}\nFor garment manipulation pipelines, specialized tools are placed at a fixed location in the workspace.\nFor ironing, the extra tools involved are the ironing board and the arm holding the iron (Fig.~\\ref{fig:downstream:real} left).\nGiven a well canonicalized and aligned shirt, an open-loop ironing primitive where the end effector moves from one end of the ironing board and back without any perception can be sufficient.\nIn our setup, we use two transforms such that the left and right side of the shirt is on the ironing board respectively.\n\n\\mypara{Folding Heuristic.}\nFirst, the sleeves are folded towards the waist using a dual-arm pick and place action.\nHere, the pick point is the sleeve keypoint, while the place point is the quarter and three-quarter point along the waist line (computed from the waist keypoints).\nSince not all shirt arms are long enough to reach the waist points, the place points are constrained to be an arm's length distance away from the shoulder keypoints.\nThe arm length can be computed from keypoints as the minimum distance between the sleeve and the shoulder keypoints over the left and right arms.\nIn the second step, with the arms folded in, the shoulder keypoints are picked and placed at the waist keypoints (Fig.~\\ref{fig:downstream:real} right).\n\\section{Related work}\n\nPrior works have explored the topics of goal-conditionining and manipulation within the domain of deformable objects. Not only are these systems limited in their manipulation capabilities (i.e. only using quasi-static or only using dynamic actions), but they also make strong assumptions about the cloth that's being manipulated (e.g. square cloth, full visibility, access to cloth image under goal configuration).   \n\n{\\color{red} I'm not sure how to group these approaches. 1 and 2 are related, but 3 and 4 are about a completely different part of the task}\n\n\\label{sec:citations}\n\n\n\\textbf{Reinforcement Learning for Cloth Manipulation}\n\\cite{wu2019withoutdemonstrations}\\cite{jangir2019dynamic}\n\\cite{tsurumine2019deeprl}\nSimplifying assumptions about square cloths or physically marked landmarks on fixed shirt dot dot dot.\n\n\n\n\\textbf{Cloth Manipulation From Expert Demonstrations} \\newline\nSince defining the goal by itself is a challenge in cloth manipulation, prior reinforcement learning approaches \\cite{seita2019imitation, ganapathi2020learning} have leveraged expert demonstrations. While expert demonstrations simultaneously sidestep the exploration problem and the goal conditioning problem, they either rely on simplified instances such as a square cloth \\cite{seita2019imitation} or expensive human input \\cite{ganapathi2020learning}.\n\n\n\\textbf{Single Primitive Cloth Manipulation} \\newline\n Bypassing the dependency on an expert, self-supervised cloth manipulation has been demonstrated in unfolding with a factorized pick \\& place action space by \\cite{wu2019withoutdemonstrations} and goal conditioned folding for simplified cloth instances (square cloth, rope) {\\color{red} they only have unfolding btw }\n using spatial action maps~\\cite{Wu_2020,zeng2019tossingbot,zeng2018learning} by \\cite{lee2020learning}.\nHowever, all these approaches operate entirely in quasi-static action spaces. A recent approach \\cite{ha2021flingbot} has demonstrated the effectiveness of pairing dynamic actions with spatial action maps. Although dynamic actions are useful for unfolding a cloth and reducing self-occlusion, by themselves they are not sufficient for goal-conditioned cloth manipulation. \n\n\n\n\\textbf{Instance-level Goal Conditioning} \\newline\n A learning framework developed by Hoque et. al. \\cite{hoque2020robotic} first learns a cloth's visual dynamics model from random interactions, and then is able to take a visual goal image and take actions towards the goal state; however, this method assumes manipulation a square cloth. Even if visual dynamics are replaced by a more generalized framework such as optical flow \\cite{weng2021fabricflownet}, which allows visual goal-conditioning on arbitrary cloth instances, it assumes that we have access to an image of a cloth in a specific goal state, which is impossible to obtain for novel cloth instances under self-occlusion.\n \n \\textbf{Reward Engineering For Goal-conditioned Cloth Manipulation?}\n \n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\\textbf{Heuristic-based cloth manipulation.}\nHeuristic-based manipulation pipelines -- where action selection and planning is manually designed -- can produce impressive results.\nHowever, the generality and robustness of these approaches is limited due to strong assumptions regarding pre-canonicalized initial state~\\cite{li2019learning,miller2011parametrized}, fiducial markers~\\cite{bersch2011bimanual}, specialized tools~\\cite{osawa2006clothes}, and cloth type and shape~\\citep{Sun2013AHA,doumanoglou2016folding,maitin2010cloth,tanaka2007origami,balkcom2008robotic,willimon2011model,cusumano2011bringing,stria2014garment,osawa2007unfolding}.\n\n\n\n\n\\textbf{Learning-based cloth unfolding.}\nLearning-based methods can self-discover the best policies for a distribution of cloths using real-world self-supervision~\\cite{ha2021flingbot,xu2022dextairity} or simulator states~\\cite{huang2022mesh,lin2022learning}. \nWhile these approaches have been successfully applied to cloth unfolding~\\cite{ha2021flingbot,xu2022dextairity} or canonicalization~\\cite{huang2022mesh}, they do not consider canonicalized-alignment.\nThis limits their applicability since heuristic-based pipelines cope poorly with unmet cloth assumptions or kinematic constraints. \n\n\n\n\\textbf{Goal-conditioned cloth manipulation.}\nTowards generic goal-conditioned cloth manipulation, prior works have investigated reinforcement learning~\\cite{matas2018sim,hietala2021closing,jangir2020dynamic,tsurumine2019deep}, real-world self-supervised learning~\\cite{lee2020learning} and imitation learning~\\cite{seita2020deep}.\nHowever, these methods often struggle to bridge the sim2real gap~\\cite{jangir2020dynamic}, generalize across cloth instances~\\cite{lee2020learning,weng2022fabricflownet,seita2020deep} or generalize between garment types~\\cite{tanaka2018emd, hietala2021closing, tsurumine2019deep}.\nFurthermore, all goal-conditioned works do not address how goal vertices\/key points\/images can be obtained for a completely novel cloth instance.\nInstead, our proposed approach can accommodate different garment categories and generalize to a variety of novel real-world garment instances from simulation training.\n\n\n\n\n\n\\section{Results}\n\n\\textbf{In simulation}, we conduct ablation studies of reward formulation (Sec.~\\ref{sec:eval:reward_formulation}) and action primitives (Sec.~\\ref{sec:eval:combined_primitives}).\nNext, we demonstrated our approach on five garment categories from Cloth3D~\\cite{bertiche2020cloth3d} (Sec.~\\ref{sec:eval:multicategory}) and a folding downstream task (Sec.~\\ref{sec:eval:downstream}).\n\\textbf{In the real world}, we include primitive and reward comparisons for the long-sleeve shirt category on canonicalized-alignment (Tab.~\\ref{tab:primitive:real}, Fig.~\\ref{fig:canal:real}), folding and ironing. \n\n\n\n\n\\begin{figure}\n    \\centering\n    \\vspace{-2mm}\n    \\includegraphics[width=0.98\\linewidth]{figures\/new_comparison_v3.png}\n    \\vspace{-3mm}\n    \\caption{\\textbf{Reward Comparisons. }\n    We qualitatively compare the final configuration achieved by policies trained with our factorized and the unfactorized reward formulation.\n    For the qualitative comparisons, the IoU of the final frame of various rollouts are shown in top-left corner of each square.\n   \n   \n    }\n    \\label{fig:reward_comparison}\n    \\vspace{-5mm}\n\\end{figure}\n\n\\subsection{Metrics}\nAfter running each policy for 10 steps, we evaluate the final \n$R_{\\textrm{Unf}}$~\\eqref{eq:unfactorized}, $R_{\\textrm{A}}$~\\eqref{eq:rigid}, and $R_{\\textrm{C}}$~\\eqref{eq:deform} in the episode.\nWhile these rewards account for the full configuration of the garment, they measure distance based on effectively ground-truth vertex correspondence.\nThis means they give larger distances for radially symmetric garments, like skirts and dresses, and they can't readily be computed on real world garments.\nTo address these shortcomings, we also evaluate the IoU and percent coverage from the current cloth binary image mask and the goal cloth binary image mask.\nIn all tables, $\\downarrow$ indicates that lower is better while $\\uparrow$ indicates that larger values are preferred.\n\n\\subsection{Task Generation}\n\nOur task datasets contain randomized initial configurations of a filtered\\footnote{\n    Since CLOTH3D meshes are automatically generated, we manually filter unrealistic mesh examples (\\textit{e}.\\textit{g}.  arms as wide as cloth body), and we ensured all cloth meshes are shorter than 0.7m. \n} subset of meshes from Cloth3D~\\cite{bertiche2020cloth3d}, whose configurations are generated as follows:\n\\begin{enumerate}\n    \\item \\textbf{Hard Tasks} have low coverage and severe self-occlusion.\n    They are generated by randomly rotating the cloth, picking a random point on the cloth, dropping it from a random height in $[0.5, 1.5]\\mathrm{m}$, and then translating the cloth by a random distance in $[0.0, 0.2]\\mathrm{m}$. \n    \\item \\textbf{Easy Tasks} have high coverage to test policies' abilities to perform small adjustments crucial to canonicalization. \n    They are generated by starting with the canonicalized configuration, and dragging a random point on the cloth by an angle uniformly sampled from $[0, 360]$ degrees by a distance uniformly sampled in $[0.5, 1]\\mathrm{m}$.\n\\end{enumerate}\nEach garment category has 2000 training and 400 testing tasks with unseen meshes, with a 75-25 and 50-50 split between hard and easy tasks respectively.\n\n\\subsection{Reward Ablation}\n\\label{sec:eval:reward_formulation}\nWe compare the canonicalized-alignment performance between the unfactorized~\\eqref{eq:unfactorized} and our factorized reward formulation~\\eqref{eq:factorized} on the long sleeve category.\nWe observe that our approach consistently does best on all metrics (Tab.~\\ref{tab:reward_formulation}), \nreflected in qualitatively more consistent canonicalized-alignment (Fig.~\\ref{fig:reward_comparison}).\nWe hypothesize that the $R_{\\textrm{Unf}}$ baseline struggles to canonicalize properly due to faint supervision on small deformable adjustments.\nMeanwhile, our approach can emphasize canonicalization more with $\\alpha = 0.6$.\n\n\\begin{table}\n\\vspace{-2mm}\n\\caption{ \n        \\footnotesize\n        \\centering\n        \\textbf{Reward Ablation on Hard Tasks}\n       \n    }\n    \\centering\n    \\vspace{-2mm}\n    \\begin{tabular}{l|ccccc}\n        \\toprule\n        \\textbf{Metric}            &  $\\mathbf{R_{\\textrm{Unf}}} \\downarrow$   & $\\mathbf{R_{\\textrm{A}}}\\downarrow$ &  $\\mathbf{R_{\\textrm{C}}} \\downarrow$ & \\textbf{IoU} $\\uparrow$  & \\textbf{Cov.}$\\uparrow$ \\\\\n        \\midrule\n         $\\mathbf{R_{\\textrm{Unf}}}$       & 0.093          & 0.069          & 0.064               & 0.684          & 0.879             \\\\\n         $\\mathbf{R_{\\textrm{CA}}}$  ($\\alpha=0.6$) & \\textbf{0.075} & \\textbf{0.051} & \\textbf{0.052}      & \\textbf{0.728} & \\textbf{0.887}             \\\\\n        \\bottomrule\n    \\end{tabular}\n    \n    \\vspace{-3mm}\n    \\label{tab:reward_formulation}\n\\end{table}\n\n\n\n\\subsection{Effectiveness of Combined Primitives}\n\\label{sec:eval:combined_primitives}\n\nWhile high-velocity dynamic actions enable efficient unfolding, precise quasi-static actions are necessary for fine-grained adjustments involved in canonicalization.\nWe compare two single primitive systems, only Aligned-Fling (Flingbot~\\cite{ha2021flingbot}'s fling with fling direction and location specified by the target alignment) and only Pick \\& Place (P\\&P), with our combined primitive system on canonicalized-alignment of long sleeve shirts.\n\n\\input{tables\/primitive_comparison_sim_v1.tex}\n\nOn easy tasks in simulation (Tab.~\\ref{tab:primitive:sim:easy}, top), P\\&P and our combined primitive system perform similarly, confirming that quasi-static actions alone are effective for small adjustments.\nIn contrast, the fling-only system performs poorly because the imprecise flinging actions are ill-suited for cases where fine-grained manipulation is required (\\textit{e}.\\textit{g}.  Fig.~\\ref{fig:pipeline}).\n\nFor hard tasks (Tab.~\\ref{tab:primitive:sim:easy} bottom in simulation, Tab.~\\ref{tab:primitive:real} in real), we observe that both single primitive baselines perform poorly.\nNotably, Fling achieves a similar performance between easy and hard tasks in simulation, suggesting that the effect of coarse grain high-velocity actions can be effective for unfolding but is not versatile enough for canonicalization.\nMeanwhile, the P\\&P baseline performed significantly worse on hard tasks in simulation, confirming the findings reported in FlingBot~\\cite{ha2021flingbot}.\nOur combined primitive system achieves the best performance, demonstrating the synergy between coarse-grain high-velocity flings and fine-grain quasi-static actions for canonicalized-alignment.\n\nDue to imperfect cloth simulators, deformable dynamics such as arms twisting (instance \\#2, last column in Fig.~\\ref{fig:canal:real}) and stretching (instance \\#3, last column in Fig.~\\ref{fig:canal:real}) are never observed in simulation.\nDespite this sim2real gap, our real world qualitative results (Fig.~\\ref{fig:canal:real}) confirmed our simulation findings.\nSpecifically, we found that on average, our combined primitive approach achieved an IoU of 0.65, while other single primitive baselines\/ablations reached 0.56 or less.\n\n\\subsection{Canonicalized-Alignment on Garment Category}\n\\label{sec:eval:multicategory}\n\nUsing the same learning and system configuration, we trained new models on 4 other garment categories, one model per garment category.\nFrom the quantitative evaluation in Tab.~\\ref{tab:multicategory}, our approach achieves around 70 IoU for all categories, which demonstrates the generality of our problem formulation with respect to garment categories.\nFurther, from Fig.~\\ref{fig:canal:sim}, our policy has learned that while flinging is crucial for quickly unfolding crumpled garments, a few pick\\&places are usually required to achieve good canonicalization.\n\n\\input{tables\/multicategory_v1.tex}\n\n\n\\subsection{Application in Downstream Cloth Manipulation}\n\\label{sec:eval:downstream}\nA primary motivation for this work is the improvement of downstream tasks; we hypothesize that effective canonicalized-alignment will significantly reduce the complexity of subsequent manipulation skills.\nTo this end, we study two garment manipulation tasks, ironing and folding (Sec.~\\ref{sec:method:heuristics}).\nFor folding, we measured the final $R_{\\textrm{Unf}}$ achieved by each approach when the goal configuration is the ground-truth folded configuration at a specified alignment.\nWe also include a folding success rate which is a thresholded $R_{\\textrm{Unf}}$, where the boundary is chosen by qualitatively.\n\n\n\n\\textbf{Canonicalized-Alignment Improves Downstream Tasks.}\nFrom Tab.~\\ref{tab:folding_sim}, we observe that manually-designed folding heuristic completely fails at the task if starting from random initial configurations.\nWhile success rate improves with FlingBot~\\cite{ha2021flingbot}'s unfolded configurations, achieving high-coverage configurations (the goal of unfolding) does not always give structured, high-visibility configurations required by heuristics (Fig.~\\ref{fig:downstream:folding}), so FlingBot~\\cite{ha2021flingbot} still performs poorly.\nWith canonicalized-alignment, policies trained with $R_{\\textrm{Unf}}$ and $R_{\\textrm{CA}}$ achieve success rates of 84.9 and 87.8 respectively, demonstrating the importance of canonicalized-alignment reducing downstream task complexity, such that even simple manually designed task heuristics can work well.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/sim_folding_v1.png}\\vspace{-3mm}\n    \\caption{\\textbf{Simulation Folding Qualitative Comparison.}  We compare our folding heuristic on our method's canonicalization results vs. FlingBot's unfolding results. We qualitatively (images) and quantitatively (histogram) demonstrate that not all high-coverage configurations ensure folding success.}\n    \\label{fig:downstream:folding}\n   \n\\end{figure}\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/realworld_canal_v1.jpg} \\vspace{-3mm}\n    \\caption{\n        \\textbf{Real-world Canonicalized-Alignment.}\n        High-coverage configurations achieved by Flingbot aren't always aligned, which is improved with our Aligned-Fling.\n        However, using only coarse-grained Aligned-Fling fails to fully canonicalize the shirt, and using only fine-grained Pick \\& Place fails to fully-unfold the shirt as can be seen by the average of the final cloth masks (bottom row).\n    }\n    \\label{fig:canal:real}  \\vspace{-3mm}\n\\end{figure}\n\n\\input{tables\/primitive_comparison_v1}\n\n\\input{tables\/folding_performance_v1.tex}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.98\\linewidth]{figures\/real_downstream.jpg} \\vspace{-3mm}\n    \\caption{\n        \\textbf{Real-world Ironing \\& Folding.}\n        Reliable canonicalized-alignment not only gives high-visibility starting configurations, which reduces complexity for downstream tasks like folding (step 1), but can also be called multiple times with different target alignments, which is useful in ironing (step 2 \\& 3).\n        More video results on the \\href{https:\/\/clothfunnels.cs.columbia.edu}{project website}.\n    }\n    \\label{fig:downstream:real}  \\vspace{-3mm}\n\\end{figure}\n\n\n\\label{sec:result}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe OPERA experiment \\cite{1}, located at the Gran Sasso laboratory (LNGS), aims at observing the $\\nu_{\\mu}\\rightarrow\\nu_{\\tau}$ neutrino oscillation in the direct appearance mode in the CERN neutrino to Gran Sasso (CNGS) \\cite{3a,3b} beam by detecting the decay of the $\\tau$ produced in charged current (CC) interactions. A detailed description of the detector can be found in \\cite{1,2a,2b,2c,2d,2e,2f}. The OPERA detector consists of two identical  Super Modules (SM), each of them consisting of a target area and a muon spectrometer, as shown in Fig. \\ref{strauss-fig1}. The target area consists of alternating layers of scintillator strip planes and target walls. The muon spectrometer is used to reconstruct and identify muons from $\\nu_{\\mu}$-CC interactions and estimate their momentum and charge. \n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig01-eps-converted-to.pdf}}}\n\\caption{Picture of the OPERA detector, with a view of a reconstructed neutrino interaction occurring in the 2nd Super Module.}\n\\label{strauss-fig1}\n\\end{myfigure}\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig14-eps-converted-to.pdf}}}\n\\caption{The CNGS neutrino beamline. Figure from \\cite{6}.}\n\\label{author-fig4}\n\\end{myfigure}\n\nThe target walls are trays in which target units of $10\\times12.5\\mathrm{ cm^2}$ and a depth of 10\\,$X_0$ in lead (7.9\\,cm)\t with a mass of around 10~kg each are stored: they are also refered to as bricks. A brick is formed by alternating layers of lead plates and emulsion films (2 emulsion layers separated by a plastic base) building an emulsion cloud chamber (ECC). This provides high granularity and high mass, which is ideal for $\\nu_{\\tau}$ interaction detection. Fig. \\ref{strauss-fig3} left shows an image of the unwrapped ECC brick. On the right, the arrangement of the scintillator strip planes (Target Tracker, TT) and the ECC is shown. Note an extra pair of emulsion films in a removable box called a changeable sheet (CS), shown in blue. The total mass of each target area is about 625\\,tons, leading to a target mass of 1.25\\,ktons for 145'000 bricks.\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig03-eps-converted-to.pdf}\\includegraphics{thomasstrauss_2012_0n_fig05-eps-converted-to.pdf}}}\n\\caption{$\\tau$ detection principle in OPERA.}\n\\label{strauss-fig3}\n\\end{myfigure}\n\n\\section{DAQ and analysis}\nThe information from the reconstructed event, recorded by the electronic detectors, is used to predict the most probable ECC for the neutrino interaction vertex \\cite{4}. A display of a reconstructed event is indicated in Fig. \\ref{strauss-fig1}. Fig. \\ref{strauss-fig2} shows the procedure for localizing the event vertex in the ECC, by extrapolating the reconstructed tracks from the electronic detector to the CS emulsion films. \n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig02-eps-converted-to.pdf}}}\n\\caption{Event detection principle in the OPERA experiment.  The candidate interaction brick is determined from the prediction of the electronic detector (blue). The changeable sheet (CS) is used to confirm the prediction. From the CS result, the tracks are followed up to the interaction point inside the ECC.}\n\\label{strauss-fig2}\n\\end{myfigure}\n\nThe signal recorded in the CS films will confirm the prediction from the electronic detector, or will act as a veto and trigger a search in neighboring bricks to find the correct ECC in which the neutrino interaction is contained. After a positive CS result, the ECC will be unpacked and the emulsion films are developed and sent to one of the various scanning stations in Japan and Europe. Dedicated automatic scanning systems allow us to follow the tracks from the CS prediction up to their stopping point. Around these stopping points, a volume of 1\\,cm$^2$ times 15\\,emulsion films will be scanned to find the interaction vertex. As illustrated in Fig. \\ref{strauss-fig2}, only track segments in the active emulsion volume are visible and a reconstruction of the event is needed to find tracks and vertices. A dedicated procedure, called a ``decay search\" is used to search for possibly interesting topologies, like the $\\tau$-decay pictured in Fig. \\ref{strauss-fig2}. The accuracy of the track reconstruction goes from cm in the electronic detector, down to mm for the CS analysis and to micrometric precision in the final vertex reconstruction (after aligning the ECC emulsion plates with passing-through cosmic-ray tracks).\n\\newline\n\\subsection{Tau detection} $\\tau$ detection is only possible due to the micrometric resolution of the emulsion films, as it allows us to separate the primary neutrino interaction vertex from the decay vertex of the $\\tau$ particle. The most prominent background for $\\tau$ decay is either hadron scattering or charged charm decays. The background from hadron scattering can be controlled by cuts applied to the event kinematics. The background due to charm can be reduced by identifying the muon at the primary vertex, as charm will occur primarily in $\\nu_{\\mu}$-CC interactions (for further details see \\cite{2e,4}). After topological and kinematical cuts are applied, the number of background events in the nominal events sample is anticipated to be 0.7 at the end of the experiment.\n\nIn 2010 the first $\\nu_{\\tau}$ candidate was reconstructed inside the OPERA emulsion. It was recorded in an event classified as a neutral current, as no muon was identified in the electronic detector. To crosscheck the $\\tau$ hypothesis, all tracks were followed downstream of the vertex until their stopping or their re-interaction point. They were all attributed to be hadrons, and no soft muon ($E<2$\\,GeV) was found. In 2010, the expected total number of $\\tau$ candidates was 0.9, while the expected background was less than 0.1 events. More details on the analysis are presented in \\cite{2e}. In Fig. \\ref{strauss-fig8} shows a picture of this event. Track number 4, labeled as the parent, is the $\\tau$, decaying into one charged daughter track. The two showers are most likely connected to the decay vertex of the $\\tau$, rather than activity from the primary interaction. Thus the decay is compatible with: $\\tau\\rightarrow\\eta^-(\\pi^-+\\pi_0)\\nu_{\\tau}$. \n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig08-eps-converted-to.pdf}}}\n\\caption{Display of the 2010 $\\tau^-$ candidate event. Top left: view transverse to the neutrino\ndirection. Top right: same view zoomed on the vertices. Bottom: longitudinal view. Figure from \\cite{2e}.}\n\\label{strauss-fig8}\n\\end{myfigure}\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig09-eps-converted-to.pdf}}}\n\\caption{MC distribution of: a) the kink angle for $\\tau$ decays, b) the path length of the $\\tau$, c) the momentum of the decay daughter, d) the total transverse momentum $P_T$ of the detected daughter particles of $\\tau$ decays with respect to the parent track. The red band shows the 68.3\\,\\% of values allowed for the candidate event, and the dark red line indicates the\nmost probable value. The dark shaded area represents the excluded region corresponding to the a priori $\\tau$ selection cuts. Figure from \\cite{2e}.}\n\\label{strauss-fig9}\n\\end{myfigure}\n\nFig. \\ref{strauss-fig9} shows the cuts used for the selection criteria defined at the time of the proposal and the kinematic variables of the $\\tau$ decay observed by the OPERA experiment. At the time of this conference, the number of expected $\\tau$ candidates was 1.7, with 0.5 events expected in the single-prong channel. The expected background for the analyzed event sample corresponding to $4.9\\times10^{19}$\\,protons on target (pot) was $0.16\\pm0.05$ events. \n\nAt the time of writing these proceedings, a second $\\tau$ candidate appeared \\cite{7a}.\n\n\\subsection{Physics run performance and data analysis status}\nSince 2007, the OPERA experiment has collected a total of $18.5\\times10^{19}$\\,pot. This corresponds to about 15'000 interactions in the target areas of the experiment. Fig. \\ref{strauss-fig6} shows from top to bottom, as a function of time, the integrated number of events occurring in the target (showing the CNGS shutdown periods),  with their vertex reconstructed by the electronic detectors, for which at least one brick has been extracted, for which at least one CS has been analysed, for which this analysis has been positive (track stubs corresponding to the event have been found), for which the brick has been analysed, for which the vertex has been located, for which the decay search has been completed.\n\nThe efficiency of the analysis of the most probable CS is rather low, about 65\\,\\%, and is significantly lower for NC events, among which $\\nu_{\\tau}$ interactions are the most likely to be found. To recover this loss, multiple brick extraction is performed; this brings the final efficiency of observing tracks of the event in the CS to about 74\\,\\%. The efficiency for locating an event seen on the CS in the brick is about 70\\,\\%. At the time of this conference, a total of 4611 events has been localised in the bricks and the decay search has been completed for 4126 of them.\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig07-eps-converted-to.pdf}}}\n\\caption{Events recorded and analyzed in the OPERA experiment from 2008 until 2012.}\n\\label{strauss-fig6}\n\\end{myfigure}\nAfter an ECC has been identified to be the most likely interaction brick, the ECC is developed and sent to one of the scanning laboratories, where it is scanned within a short time. The efficiency for locating an event within the ECC is about 70\\,\\% with respect to the number of positive CS results. After the event has been located, a dedicated decay search is performed to obtain a data sample which can be compared to MC and which provides uniform data quality from all laboratories. This decay search includes the search for decay daughters and a reconstruction of the kinematics of the particles at the vertex. At the time of this conference, 4611 events have been localized in the ECC, with a total of 4126 CC and NC events having completed the decay search.\n\n\\subsection{Charm decay topologies in neutrino interactions}\nIn about 5\\,\\% of the $\\nu_{\\mu}$-CC interactions, the production of a charmed particle at the primary vertex takes place. Charmed particles have lifetimes similar to the $\\tau$ and similar decay channels. Thus charm events provide a subsample of decay topologies similar to $\\tau$ decay, for which the detection efficiency can be estimated based on MC simulation. A study of a high purity selection of charm events in  2008 and 2009 shows agreement with the data \\cite{5}. One-prong charm decay candidates are retained if the charged daughter particle has a momentum larger than 1 GeV\/c. This leads to an efficiency of $\\epsilon_{\\mbox{short}} = 0.31 \\pm 0.02 (\\mbox{stat.})\\pm 0.03 (\\mbox{syst.})$ for short and $\\epsilon_{\\mbox{long}} = 0.61 \\pm 0.05 (\\mbox{stat.}) \\pm 0.06 (\\mbox{syst.})$ for long charm decays, wherein `long' means the production and the decay vertices are not located in the same lead plate. The number of events for which the charm search is complete is 2167 CC interactions. In these we expect $51\\pm7.5$ charm candidates, with a background of $5.3\\pm2.3$ events. The number of observed candidates is 49, which is in agreement with expectations. \n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig10-eps-converted-to.pdf}}}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig11-eps-converted-to.pdf}}}\n\\caption{Display of one of the charm events. Top: View of the reconstructed event in the emulsion. Bottom: Zoom in the vertex region of the primary and secondary vertex. }\n\\label{strauss-fig10}\n\\end{myfigure}\n\nFig. \\ref{strauss-fig10} shows a charm decay detected in the OPERA experiment. In the electronic detector reconstruction, two muons were observed, one charged positively, the other negatively. The $\\mu^-$ is attached to the primary vertex, while the $\\mu^+$ is connected to the decay vertex. This topology corresponds with a charged charm decaying into a muon and the measured kinematic parameters are a flight length of 1330\\,$\\mu$m and a kink angle of 209\\,mrad. The impact parameter (IP) of the $\\mu^+$ with respect to the primary vertex is 262\\,$\\mu$m, and its momentum is measured as 2.2\\,GeV\/c. This accounts for a transverse momentum ($P_T$) of 0.46\\,GeV\/c.\n\n\\section{$\\nu$-velocity measurement}\nDue to the time structure of the CERN SPS beam, the OPERA experiment is able to trigger on the proton spill hitting the CNGS target. As a result, the electronic detector provides a time signal of the recorded events, which can be used to measure the neutrino velocity in the CNGS beam. One needs to measure with precision of some ns the flight time (time of flight - TOF) between CERN and LNGS, and the distance between reference points in the detector and the CNGS. The concept of the neutrino time of flight measurement is illustrated in Fig. \\ref{strauss-fig15}. \n\n\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig15-eps-converted-to.pdf}}}\n\\caption{Scheme of the time of flight measurement. Figure from \\cite{6}.}\n\\label{strauss-fig15}\n\\end{myfigure}\nThe procedures are explained in great detail in \\cite{6}. Since the time of the conference, an instrumental mistake has been identified that makes the results presented at this conference obsolete. Updated results taken from \\cite{7b} are presented below.\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig16-eps-converted-to.pdf}}}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig17-eps-converted-to.pdf}}}\n\\caption{Top: Scheme of the timing sytem at CERN. Bottom: Scheme of the timing system at LNGS. Figures from \\cite{6}.}\n\\label{strauss-fig16}\n\\end{myfigure}\n\nFig. \\ref{strauss-fig16} shows the timing systems both at CERN and LNGS, which allowed time calibration between both sites within accuracy of $\\pm4$\\,ns.\n\nThe distance between CERN and the OPERA detector was measured via GPS geodesy and extrapolation down to the location of both the CNGS target and the OPERA detector with terrestrial traverse methods. The effective baseline is measured as $731278.0\\pm0.2$\\,m.\n\nThe proton wave form for each SPS extraction was measured with a beam current transformer (BCT). The sum of the wave forms restricted to those associated to a neutrino interaction in OPERA was used as PDF for the time distribution of the events within the extraction. The maximum likelihood method was used to extract the time shift between the two distributions, i.e. the neutrino time of flight. Internal NC and CC interactions in the OPERA target and external CC interactions occurring in the upstream rock from the 2009, 2010 and 2011 CNG runs were used for this analysis. As shown in Fig. \\ref{strauss-fig20}, it is measured to be: $$\\delta t = \\mbox{TOF}_c - \\mbox{TOF}_\\nu = (6.5\\pm7.4(\\mbox{stat.})\\pm ^{+8.3}_{-8.0}(\\mbox{syst.}))\\,\\mbox{ns.}$$ Modifying the analysis by using each neutrino interaction waveform as PDF instead of their sum gives a comparable result of\n$$\\delta t = (3.5\\pm5.6(\\mbox{stat.})\\pm ^{+9.4}_{-9.1}(\\mbox{syst.}))\\,\\mbox{ns.}$$ No energy dependence was observed.\n\n\\begin{myfigure}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig19-eps-converted-to.pdf}}}\n\\centerline{\\resizebox{70mm}{!}{\\includegraphics{thomasstrauss_2012_0n_fig20-eps-converted-to.pdf}}}\n\\caption{Top: Comparison of the measured neutrino interaction time distributions (data points) and the proton PDF (red and blue line) for the two SPS extractions resulting from the maximum likelihood analysis. Bottom: Blow-up of the leading edge (left plot) and the trailing edge (right plot) of the measured neutrino interaction time distributions (data points) and the proton PDF (red line) for the first SPS extraction after correcting for $\\delta t=6.5$\\,ns. Within errors, this second extraction is equal to the first one. Figures from \\cite{6}.}\n\\label{strauss-fig20}\n\\end{myfigure}\n\nTo cross-check for systematic effects, a dedicated bunched beam run was performed, where the SPS proton delivery is split into 3\\,ns long spills, separated by 524\\,ns in time during autumn 2011, and a similar mode of 3\\,ns with 100\\,ns separation in spring 2012. \n\nThe value of $\\delta t$ obtained in 2011 by using timing information provided by the target tracker is $1.9\\pm3.7$\\,ns; it is $0.8\\pm3.5$\\,ns when based on the spectrometer data \\cite{6}. For the 2012 run, the corresponding values of $\\delta t$ are $\\delta t =(-1.6\\pm1.1(stat.)^{+6.1}_{-3.7})$\\,ns \\cite{7b}. All results are in agreement with the measurement from standard CNGS beam operation.\n\n\\section{Conclusions}\nThe OPERA experiment detected two $\\tau$ neutrino events appearing in the CNGS beam. Further, we measured the neutrino velocity to be in agreement with the speed of light in a vacuum to the O($10^{-6}$). Other short decay topologies like $\\nu_e$ or charm decays can also be detected and are in agreement with MC expectations, thus providing a benchmark for validating the $\\tau$ efficiency expectations.\n\n\n\\thanks\nFirstly, I thank the organizers of the workshop and the OPERA PTB for the possibility to join this workshop. The OPERA collaboration thanks CERN, INFN and LNGS for their support and work. In addition OPERA is grateful for funding from the following national agencies: \n\nFonds de la Recherche Scientique - FNRS and Institut Interuniversitaire des Sciences Nucl\\'eaires for Belgium; MoSES for Croatia; CNRS and IN2P3 for France; BMBF for Germany; INFN for Italy; JSPS (Japan Society for the Promotion of Science), MEXT (Ministry of Education, Culture, Sports, Science and Technology), QFPU (Global COE program of Nagoya University, ``Quest for Fundamental Principles in the Universe\", supported by JSPS and MEXT) and Promotion and Mutual Aid Corporation for Private Schools of Japan for Japan; The Swiss National Science Foundation (SNF), the University of Bern and ETH Zurich for Switzerland; the Russian Foundation for Basic Research (grant 09-02-00300 a), the Programs of the Presidium of the Russian Academy of Sciences ``Neutrino Physics\" and ``Experimental and theoretical researches of fundamental interactions connected with work on the accelerator of CERN\", the support programs of leading schools (grant 3517.2010.2), and the Ministry of Education and Science of the Russian Federation for Russia; the Korea Research Foundation Grant (KRF-2008-313-C00201) for Korea; and TUBITAK ``Scientific and Technological Research Council of Turkey\", for Turkey. In addition the OPERA collaboration thanks the technical collaborators and the IN2P3 Computing Centre (CC-IN2P3).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn equilibrium, it is not possible to get a steady current in a periodic potential  \nwithout the application of external forces. This is in conformity with the second law of \nthermodynamics. However, in recent times it has become a popular theme to obtain such \ncurrents in nonequilibrium situations. The currents so obtained in asymmetric but \nperiodic potentials with no imposed directional preferences in the presence of \nthermal noise are called thermal ratchet currents\\cite{feyn, reim}. The usual \nmethods one generally employs to realise such currents are: Rock the system periodically \nalternately with constant forces $|F|$ and $-|F|$ for equal short intervals \\cite{magn}, \nor flash the potential either periodically or randomly with two unequal (for example, \nzero and nonzero) amplitudes \\cite{prost}. However, the method of obtaining current in \nsymmetric periodic potentials subject to asymmetric periodic forcings with zero mean per \nperiod has also been established \\cite{mcm1}. In most of the cases the particle motion is \nconsidered overdamped which is  more suitable to model particle (such \nas a molecular motor) transport in biological systems where large friction plays important \nrole \\cite{magn, prost, svob}. There is yet another model wherein the friction is \nconsidered nonuniform in space and in particular periodically varying similar to the \npotential function. The ratchet current could be obtained, in this inhomogeneous case, \neven in symmetric periodic potentials driven periodically and symmetrically in \ntime\\cite{pareek}. So far such currents are obtained in overdamped\\cite{dan1} \ninhomogeneous systems and the possibility of obtaining such currents in underdamped \nsystems\\cite{mcm2} had only been indicated earlier. This model is very appropriate to \ndescribe Josephson junctions \\cite{falco} (where the frictional inhomogeneity bears an \nexact analogy to the coupling between concomitant Cooper pair\nand the quasiparticle tunnelings), and motion of adatoms on \ncrystal surfaces\\cite{wahn}. In this part (I) of the work we present detailed \ncharacteristics of the ratchet current in inhomogeneous (nonuniform friction) inertial \nsystems when driven adiabatically. And in the next part (II) we shall present results \nfor similar systems when driven at finite frequency.\n\nAs will be described in the next section, the steady state mean velocity $\\bar v$ of a\nparticle under the action of a constant force $F$ is calculated using the matrix\ncontinued fraction method (MCFM). The algebraic sum \n$\\Sigma \\bar v$ ($=\\bar v(|F|)+\\bar v(-|F|)$)\n of these mean velocities $\\bar v$ with applied forces $|F|$ and $-|F|$ is termed here \nas the ratchet current in the adiabatic limit. Since the phase difference $\\phi (\\ne 0,\nn\\pi)$ between the sinusoidal potential $V(x)$ and the friction coefficient $\\gamma(x)$\nis the only parameter in the model that breaks the directional symmetry of the problem, \nthe ratchet current is expected to be small. However, $\\Sigma \\bar v$ turns out to be an \ninteresting complicated function of $F$, the temperature $T$, and the mean friction \ncoefficient $\\gamma_0$. This is not surprising and can be understood quite simply on \nphysical grounds as discussed elsewhere\\cite{wanda}.\n\nIn the same work\\cite{wanda} the variation of the ratchet current $\\Sigma \\bar v$ as a \nfunction of applied force $|F|$ for various values of the average friction coefficient \n$\\gamma_0$ at temperature $T=0.4$ has been presented. (The same choice of dimensionless\nparameters, $T$, etc, are elaborated in Sec. II below.) It shows that at the smallest\nexplored $\\gamma_0 (=0.035)$, $\\Sigma \\bar v$ peaks, with value $\\simeq -0.08$, close to\n$F=0.09 \\ll F_c(=1)$, the critical field. The current is thus thermally assisted.  The\nvariation of $\\Sigma \\bar v$ is very sharp: Appreciable current could be seen only in\nthe range $0.05\\leq |F|\\leq 0.12$. The current strength reduces almost to zero for\n$F=0.12$. Beyond this $F$ no meaningful $\\bar v$ could be obtained from MCFM. As\n$\\gamma_0$ is increased the range of $F$, where appreciable $\\Sigma \\bar v$ results,\nwidens monotonically. However, the peak current becomes maximum for \n$\\gamma_0 \\simeq 0.4$, with\nmaximum current $\\simeq -0.28$ at $|F|\\simeq 0.75$, and then gradually decreases as\n$\\gamma_0$ is increased further. By a close observation of the reported figure\n(3, ref. \\cite{wanda}) one can conclude that $\\Sigma \\bar v$ at $T=0.4$ will show\nnonmonotonic variation with $\\gamma_0$ at carefully chosen $F$ values, which as we \npresent in the following is, indeed, the case.\n\nFor a given applied force $F$, and the temperature $T$ the current $\\Sigma \\bar v$\ninitially increases, attains a peak, and then gradually decreases as $\\gamma_0$ is \nincreased. The result is reminiscent of the well-known result\\cite{kramers} obtained \nby Kramers, however, in an entirely different context of particle flux across a \npotential barrier. In all situations $\\Sigma \\bar v$ increases with $|F|$, attains a \npeak, and gradually decreases. With appropriately chosen parameters $\\gamma_0$ and $T$,  \n  $\\Sigma \\bar v$ shows current reversal as well and consequently as $|F|$ \nis increased further $\\Sigma \\bar v$ changes its direction and increases in magnitude \neventually again decreases in magnitude after attaining a second peak. Interestingly, \n$\\Sigma \\bar v$ shows very similar behaviour also as a function of temperature $T$ in \nsome parts of the parameter space $(\\gamma_0, F)$.\n\nAs the value (and direction) of $\\Sigma \\bar v$ changes one naturally asks about how it  \nis correlated with the changes in energies of the system. It turns out that the average \nkinetic energy of the particle increases monotonically with $|F|$, whereas the average\npotential energy shows a nonmonotonic behaviour. Again, in this case too we consider\nthe mean of the average kinetic (as also of the potential) energies for $F=\\pm|F|$. The\naverage kinetic energy overwhelms the potential energy as $F$ is increased forcing \nthe average total energy also to increase monotonically. However, the contribution of the \naverage potential energy of the particle is reflected in the derivative of the average\ntotal energy with respect to $F$. These derivatives show distinctive behaviour where\n$\\Sigma \\bar v$ peaks as a function of $F$.\n\nIn order to be useful for practical purposes the system must have good performance \ncharacteristics. We calculate the efficiency of energy transduction of this inhomogeneous\ninertial system driven adiabatically and symmetrically and moving in a symmetric \nperiodic potential. The system yields small current and hence the work done against an \napplied load is also small if at all it is possible to obtain. Naturally, the \nthermodynamic efficiency of the system is also small and it is feasible to obtain only \nin a very small region of parameter space. However, even without an applied load the \nsystem continuously performs work against the frictional force and in order to take it \ninto account a quantity  called Stokes efficiency has been \ndefined\\cite{wang, seki, parr}. We calculate the Stokes efficiency of our system. \nThis efficiency also has a complex behaviour and, in particular, it is nonmonotonic. \nIt is also possible, interestingly, to observe increasing Stokes efficiency \nwith temperature contrary to what is observed in usual macroscopic heat engines. This \nkind of behaviour has been shown to occur in overdamped inhomogeneous but asymmetrically \ndriven systems\\cite{raish}.\n\nAdmittedly, the quantitative effect of the inhomogeneity is not very large. Nevertheless, \nour investigation  shows that many important results can be obtained, some even \ncounterintuitive, albeit at a small but nonnegligible scale, using system inhomogeneity. \nThe ratchet current, the efficiency of performance, and so on, can be improved by orders \nof magnitude, if zero mean asymmetric periodic drive is employed instead of a symmetric \ndrive. In this case, however, inhomogeneity does not play a decisive role. Similar \nresults can be obtained without system inhomogeneity as the asymmetric drive plays the \nmajor role.\n\nThe above mentioned results will be elaborated upon in section III. The paper will be \nconcluded with a discussion in section IV.\n\n\\section{The matrix continued fraction method for inhomogeneous inertial systems}\nThe motion of a particle of mass $m$, moving in a periodic potential $V(x)=-V_0 sin(kx)$\nin a medium with friction coefficient $\\gamma (x)=\\gamma_0(1-\\lambda sin(kx+\\phi))$\n($0<\\lambda<1$) and subjected to a constant uniform force $F$, is described by the \nLangevin equation\\cite{pareek, amj1},\n\\begin{equation}\nm\\frac{d^{2}x}{dt^{2}}=-\\gamma (x)\\frac{dx}{dt}-\\frac{\\partial{V(x)}}{\\partial\nx}+F+\\sqrt{\\gamma(x)T}\\xi(t).\n\\end{equation}\nHere $T$ is the temperature in units of the Boltzmann constant $k_B$. The Gaussian \ndistributed fluctuating forces $\\xi (t)$ satisfy the statistics: $<\\xi (t)>=0$,\nand $<\\xi (t)\\xi(t^{'})>=2\\delta(t-t^{'})$. For covenience, we write down Eq.(2.1) in \ndimensionless units by setting $m=1$, $V_0=1$, $k=1$ so that, for example, the scaled\n$\\gamma_0$ is expressed in terms of the old $\\sqrt{mV_0}k$, etc. The reduced Langevin \nequation, with reduced variables denoted again by the same symbols, is written as\n\\begin{equation}\n\\frac{d^{2}x}{dt^{2}}=-\\gamma(x)\\frac{dx}{dt}\n+cos x +F+\\sqrt{\\gamma(x) T}\\xi(t),\n\\end{equation}\nwhere $\\gamma(x)=\\gamma_0(1-\\lambda sin(x+\\phi))$. Thus the periodicity of the potential\n$V(x)$ and also the friction coefficient $\\gamma$ is $2\\pi$. The barrier height between\nany two consecutive wells of $V(x)$ persists for all $F<1$ and it just disappears at \nthe critical field value $F_c=1$. The noise variable, with the same symbol $\\xi$, \nsatisfies exactly similar statistics as earlier. The Fokker-Planck equation corresponding \nto Eq.(2.2) is given as\n\\begin{equation}\n\\frac{\\partial W(x,v,t)}{\\partial t} = {\\cal{L}}_{FP}W(x,v,t),\n\\end{equation}\nwhere the Fokker-Planck operator\n\\begin{eqnarray}\n{\\cal{L}}_{FP} & = & -v\\frac{\\partial}{\\partial x}+\\gamma_0(1-\\lambda sin(x+\\phi))\n\\frac{\\partial}\n{\\partial v}v - (cosx+F)\\frac{\\partial}{\\partial v}+ \\nonumber \\\\\n      &    &\\gamma_0T(1-\\lambda sin (x+\\phi))\\frac{\\partial^{2}}{\\partial v^{2}},\n\\end{eqnarray}\nand $W(x,v,t)$ is the probablity distribution of finding the particle at position $x$\nwith velocity $v=\\frac{dx}{dt}$ at time $t$.\n\nIt is hard to solve the equation in the general case. However, our aim is to solve Eqs.\n (2.3-4) in the steady state case. We apply MCFM developed by Risken, \net. al. \\cite{risk}, and used by others to suit various circumstances \\cite{ferra}. The \nbasic idea behind the method is to expand the distribution function $W(x,v,t)$ in series \nin terms of products of (Hermite) functions $\\psi_n(v)$, which are explicit functions of \nvelocity $v$, and $C_n(x,t)$ (as coefficients) which are explicit functions of position \n$x$ and time $t$:\n\\begin{equation}\nW(x,v,t)=(2\\pi T)^{\\frac{-1}{4}}e{^\\frac{-v^{2}}{4T}}\\sum_{n=0}^\n{\\infty}C_n(x,t)\\psi_n(v),\n\\end{equation}\nwith\n\\begin{equation}\n\\psi_n=\\frac{(b^{\\dagger})^n\\psi_0}{\\sqrt{n!}}, \n\\end{equation}\nand\n\\begin{equation}\n\\psi_0=(2\\pi)^\\frac{-1}{4}T^\\frac{-1}{4}e^\\frac{-v^2}{4T}.\n\\end{equation}\nIt turns out that $\\psi_n(v)$ $(n=0, 1, 2, ...)$ are the eigenfunctions of the operator\n$-\\gamma(x)b^\\dagger b$:\n\\begin{equation}\n-\\gamma(x)b^\\dagger b \\psi_n(v) = -\\gamma(x) n \\psi_n(v),\n\\end{equation}\nwhere $b$ and $b^\\dagger$ are the bosonic-like  annihilation and creation operators \nsatisfying\n\\begin{equation}\nbb^\\dagger-b^\\dagger b=1,\n\\end{equation}\nand defined as\n\\begin{equation}\nb=\\sqrt{T}\\frac{\\partial}{\\partial v}+\\frac{1}{2}\\frac{v}{\\sqrt{T}},\n\\end{equation}\n\\begin{equation}\nb^\\dagger=-\\sqrt{T}\\frac{\\partial}{\\partial v}+\\frac{1}{2}\\frac{v}{\\sqrt{T}}.\n\\end{equation}\n(Note: $b^\\dagger \\psi_n(v)=\\sqrt{n+1}\\psi_{n+1}(v)$ and \n$b \\psi_n(v)=\\sqrt{n}\\psi_{n-1}(v)$.)\n  \nSubstituting (2.5) in the Fokker-Planck equation (2.3) and multiplying from left by \n$e^\\frac{v^2}{4T}$ on both sides of Eq. (2.3) and on simplifying, we get,\n\\begin{equation}\n\\sum_{m=0}^{\\infty}\\frac{\\partial C_m(x,t)}{\\partial t}\\psi_m =\n\\bar{\\cal{L}}_{FP}\\sum_{m=0}^{\\infty}C_m(x,t)\\psi_m(v),\n\\end{equation}\nwhere \n\\begin{equation}\n\\bar{\\cal{L}}_{FP}=e^{\\frac{v^2}{4T}}{\\cal{L}}_{FP}e^{\\frac{-v^2}{4T}}.\n\\end{equation}\nThe operator $\\bar{\\cal{L}}_{FP}$ is given by\n\\begin{equation}\n\\bar{\\cal{L}}_{FP}=-\\gamma(x)b^{\\dagger} b -bD-b^{\\dagger} \\hat D,\n\\end{equation}\nwhere the operators $D$ and $\\hat D$ are defined as,\n\\begin{equation}\nD=\\sqrt{T}\\frac{\\partial}{\\partial x},\n\\end{equation}\n\\begin{equation}\n\\hat D=\\sqrt{T}\\frac{\\partial}{\\partial x} + \\frac{V^{'}-F}{\\sqrt{T}},\n\\end{equation}\nwith the prime on the potential $V$ denoting differentiation with respect to $x$.\n\nMultiplying both sides of Eq.(2.12) from left by $\\psi_n$ and integrating over the\nvelocity variable and using the orthonormality properties of the functions $\\psi_n$, we \nget the equation on $C_n(x,t)$:\n\\begin{equation}\n\\frac{\\partial C_n(x,t)}{\\partial t}+\nn\\gamma(x)C_n(x,t)+\\sqrt{n+1}DC_{n+1}(x,t)+\\sqrt{n}\\hat{D}C_{n-1}(x,t)=0.\n\\end{equation}\nIn the stationary (steady) state case, we thus have,\n\\begin{equation}\nn\\gamma(x)C_n(x)+\\sqrt{n+1}DC_{n+1}(x)+\\sqrt{n}\\hat{D}C_{n-1}(x)=0\n\\end{equation}\ngiving a recursion relation on $C_n(x)$ ($n=0, 1, 2, ...$). The first ($n=0$) of the\nrecursion relations reveals that $C_1$ is a constant.\n\nIn this steady state case, the normalization condition is given by\n\\begin{equation}\n\\int_{0}^{2\\pi}\\int_{-\\infty}^{\\infty}W(x,v)dx dv = \\int_{0}^{2\\pi}C_0 dx =1,\n\\end{equation}\ngiving a condition on the coefficient $C_0(x)$. The mean velocity $\\bar v(F)$, the \naverage potential energy $E_{pot}$, and the average kinetic energy $E_{kin}$, are\nobtained, respectively, as,\n\\begin{equation}\n\\bar v = \\int_{0}^{2\\pi} \\int_{-\\infty}^{\\infty}v W(x,v)dx dv = 2\\pi \\sqrt{T}C_1,\n\\end{equation}\n\\begin{equation}\nE_{pot}=\\int_{-\\pi}^{\\pi}V(x)C_0(x) dx,\n\\end{equation}\nand\n\\begin{equation}\nE_{kin}=<\\frac{v^2}{2}>=\\frac{1}{2}+\\frac{1}{\\sqrt{2}}\\int_{0}^{2\\pi}C_2(x)dx.\n\\end{equation}\nThus, all the physical quantities of our interest can be calculated provided $C_0$, $C_1$, \nand $C_2$ are known from Eq.(2.18).\n\nWriting $C_n(x) =\\sum_{q}C_n^qe^{iqx}$, $V(x)=\\sum_{q}V^qe^{iqx}$, and\n$\\gamma (x)=\\sum_{q}\\gamma ^q e^{iq(x+\\phi)}$, where $i=\\sqrt{-1}$ and $q$'s as integer\nindices the recurrence relation (2.18) becomes,\n\\begin{eqnarray}\n\\lefteqn{0=} \\nonumber \\\\\n & & n\\sum_{p}\\gamma^{q-p}e^{i(q-p)\\phi}C_n^p \n+ \\sqrt{n+1}\\sqrt{T}\\sum_{p}(ip)\\delta_{q,p}C_{n+1}^p \n+\\sqrt{n}\\sqrt{T}\\sum_{p}(ip)\\delta_{q,p}C_{n-1}^p \\nonumber \\\\ \n& & -\\frac{\\sqrt{n}}{\\sqrt{T}}\\sum_{p}F\\delta_{q,p}C_{n-1}^p\n+\\frac{\\sqrt{n}}{\\sqrt{T}}\\sum_{p}i(q-p)V^{q-p}C_{n-1}^p.\n\\end{eqnarray}\n\nDefining the matrices $\\mathbf{\\gamma}, \\mathbf{D} , \\mathbf{1}$, and $\\mathbf{V^{'}}$ \nwith components\n\\begin{eqnarray}\n(\\mathbf{\\gamma})^{q,p}=\\gamma^{q-p}e^{i(q-p)\\phi},\\\\\n(\\mathbf{D})^{q,p}=ip\\delta_{q,p},\\\\\n(\\mathbf{1})^{q,p}=\\delta_{q,p},\\\\\n(\\mathbf{V^{'}})^{q,p}=i(q-p)V^{q-p},\n\\end{eqnarray}\nand the column matrices $\\mathbf{C_n}, n=0,1,2, ..$ with components\n\\begin{equation}\n(\\mathbf{C_n})^q=C_n^q,\n\\end{equation}\nthe recurrence relation can be written as,\n\\begin{equation}\nn\\mathbf{\\gamma C_n} + \\sqrt{n+1}\\sqrt{T}\\mathbf{DC_{n+1}} +\n\\sqrt{n}(\\sqrt{T}\\mathbf{D}+\\frac{1}{\\sqrt{T}}[\\mathbf{V^{'}}-F\\mathbf{1}])\\mathbf{\nC_{n-1}}=0.\n\\end{equation}\nIntroducing the matrices $\\mathbf{M_n}, (n=0, 1, 2, ..)$ such that\n\\begin{equation}\n\\sqrt{n}\\mathbf{DC_n}=\\mathbf{M_nC_{n-1}}\n\\end{equation}\nthe recursion relation (2.29) for $n=2$ becomes\n\\begin{equation}\n\\mathbf{M_2C_1}(=\\sqrt{2}\\mathbf{DC_2})=\n-\\mathbf{D}(\\mathbf{\\gamma}+\\frac{\\sqrt{T}}{2}\\mathbf{M_3})^{-1}\n(\\sqrt{T}\\mathbf{D}+\\frac{1}{\\sqrt{T}}[\\mathbf{V^{'}}-F\\mathbf{1}])\\mathbf{C_1}.\n\\end{equation}\nHence, for nonzero $\\mathbf{C_1}$, $\\mathbf{M_2}$ can be given in terms of $\\mathbf{M_3}$\nas\n\\begin{equation}\n\\mathbf{M_2}=\\mathbf{D}(\\mathbf{\\gamma}+\\frac{\\sqrt{T}}{2}\\mathbf{M_3})^{-1}\n(\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}} ] - \\sqrt{T}\\mathbf{D}).\n\\end{equation}\nand similarly, one can write down for $\\mathbf{M_3}$, $\\mathbf{M_4}$, and so on. And, in \ngeneral,\n\\begin{equation}\n\\mathbf{M_n}=\\mathbf{D}(\\mathbf{\\gamma}+\\frac{\\sqrt{T}}{n}\\mathbf{M_{n+1}})^{-1}\n(\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}} ] - \\sqrt{T}\\mathbf{D}),\n\\end{equation}\nfor $2\\leq n\\leq N-1$, if the series of recursion relations is terminated at $n=N$ \n($\\mathbf{C_n}=\\mathbf{0}$ for $n\\geq N+1$). Thus, the last ($n=N$) relation becomes,\n\\begin{equation}\n\\mathbf{M_N}=\\mathbf{D}\\mathbf{\\gamma}^{-1}(\n\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}}]-\\sqrt{T}\\mathbf{D}),\n\\end{equation}\nfor $N\\geq 2$. Hence, $\\mathbf{M_{N-1}}$, $\\mathbf{M_{N-2}}$, ... $\\mathbf{M_2}$ can be\ngiven entirely in terms of $\\mathbf{\\gamma}$, $\\mathbf{V^{'}}$, and $F\\mathbf{1}$.\nUsing $\\sqrt{2}\\mathbf{DC_2}=\\mathbf{M_2C_1}$, the recursion relation $(n=1)$\\\\ \n$\\mathbf{\\gamma C_1}+\\sqrt{2}\\sqrt{T}\\mathbf{DC_2}+\n(\\sqrt{T}\\mathbf{D}+\\frac{1}{\\sqrt{T}}[\\mathbf{V^{'}}-F\\mathbf{1}])\\mathbf{C_0}=0$ \\\\\nyields\n\\begin{equation}\n\\mathbf{C_0}=\\mathbf{HC_1},\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{H}=(\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}}]-\\sqrt{T}\\mathbf{D})^{-1}(\n\\mathbf\\gamma + \\sqrt{T}\\mathbf{M_2}).\n\\end{equation}\nSince $C_1$ is a constant,\n\\begin{equation}\nC_0^q=\\sum_{p}(\\mathbf{H})^{q,p}(\\mathbf{C_1})^p = (\\mathbf{H})^{q,0}C_1.\n\\end{equation}\nFrom the normalization condition (2.19), we obtain $2\\pi (\\mathbf{C_0})^0 =1$, and hence \nfrom Eq.(2.36),\n\\begin{equation}\nC_1=\\frac{1}{2\\pi(\\mathbf{H})^{0,0}}.\n\\end{equation}\nTherefore, once $\\mathbf{H}$ is calculated $C_1$ is known and hence all the components\n$C_0^q=(\\mathbf{C_0})^q$ are evaluated.\n\n\n\\section{Numerical results}\nIn order to calculate the physical quantities of interest, namely, the mean velocity\n$\\bar v$, the average kinetic and potential energies for given values of $F$, $\\gamma_0$,\nand temperature $T$ one needs to evaluate the matrix $\\mathbf{H}$. For our special case, \nthe potential $V(x)$ and $\\gamma(x)$ each has only two Fourier components. Therefore, the \ndimension of $\\mathbf{H}$ depends on the number $Q$ of Fourier components to be kept for\nthe coefficients $C_n(x)$. Also, the degree of complication increases with the number\n$N$ of recursion relations one requires to be considered.  The acceptable solution will\nrequire the matrix $\\mathbf{H}$ to remain essentially unchanged as $Q$ and $N$ are \nchanged. All these numbers depend on the values of $F$, $\\gamma_0$, and the temperature \n$T$. Typically, in the worst case, it is sufficient if $Q<30$ and $N<2000$ is taken for \nsmall $\\gamma_0 \\ll 1$ in the appropriate parameter space range of ($F, T$). Outside this \nrange no \nacceptable solution is found. It turned out that the paramater space $(\\gamma_0, F, T)$ \nrange where solutions could be extracted was just the region that yielded finite ratchet \ncurrent $\\Sigma \\bar v$. In all our calculations for which the results are described in \nthe following we have set $\\phi=0.35$ and $\\lambda=0.9$.\n\n\\subsection{The ratchet current}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig1.eps}}\n\\caption{Shows the ratchet current $\\Sigma{v}$ as a function of $\\gamma_{0}$ for two \nvalues of $F$. The inset shows the variation of current vs $\\gamma_{0}$ for $F =$ 0.1.}\n\\label{fig.1}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig2.eps}}\n\\caption{Ratchet current versus F is shown for $\\gamma_0=0.1$ and two values of temperature. \nThe inset shows variation of $\\Sigma{\\bar v}$ for smaller values of $F$ at $ T = $0.6. The \ndotted line for zero current is shown just to guide the eye.}\n\\label{fig.2}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig3.eps}}\n\\caption{Shows the variation of $\\mu(|F|)$ and $\\mu(-|F|)$ with $F$ for $\\gamma_0 =$ 0.5. \nThe little vertical marking ($|$) indicates the position of current maximum.}\n\\label{fig.3}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig4.eps}}\n\\caption{The variation of ratchet current is shown for $\\gamma_0 =$ 0.035 for different \nvalues of $F$. $\\Sigma \\bar v$ decreases slowly with $T(>4)$. The inset shows two current \nreversals for $\\gamma_0 =$ 0.1, $F= 0.1$. The zero $\\Sigma \\bar v$ line is shown to \nguide the eye.}\n\\label{fig.4}\n\\end{figure}\n\nFigure 1 shows how $\\Sigma \\bar v$ changes with $\\gamma_0$ at $T=0.4$. Contrary to usual \nperception current increases as the system becomes more damped. This trend is seen for \nlower $\\gamma_0$ values. However, in the higher range of $\\gamma_0$, as expected, the \ncurrent diminishes with damping. $\\bar v(|F|)$ and $\\bar v(-|F|)$ are both monotonically \ndecreasing in magnitude with $\\gamma_0$. It is only their algebraic sum $\\Sigma \\bar v$\nthat shows the nonmonotonic behaviour. It is purely an effect brought about by the \nsymmetry breaking due to nonzero phase difference $\\phi$ and any analogy with potential \nbarrier crossing phenomenon at low damping described by Kramers is superfluous.\n\nIn Fig.2, the variation of $\\Sigma \\bar v$ as a function of $|F|$ for $\\gamma_0=0.1$ is\nshown for two values of temperature $T=0.6$ and $2.0$. It is to be noted that $T=2.0$\ncorresponds to an energy exactly equal to the potential barrier height at $F=0$. One \ncan see that in the scale of the figure the current $\\Sigma \\bar v$ remains negative for \n$T=0.6$, whereas at $T=2.0$ the current is positive at low $|F|$ but it changes sign and \nremains negative at higher $|F|$ values for the same phase lag $\\phi$. This current \nreversal, however, is neither confined to higher $T$ nor to\nhigher $\\gamma_0$ values. The inset of the figure shows that at smaller $|F|$ region \n$\\Sigma \\bar v$ is positive and then it becomes negative as $|F|$ is increased. Of \ncourse, in this case the positive current is an order of magnitude smaller and hence is \nnot visible in the larger figure. A similar result is obtained for $\\gamma_0=0.035$ where\nagain the negative current at larger $|F|$ dominate but positve currents at low $|F|$\nregion are also appreciable. Here again $\\bar v(|F|)$ and $\\bar v(-|F|)$ both increase\nmonotonically with $|F|$ only their relative values change giving rise to nonmonotonic \nbehaviour of $\\Sigma \\bar v$, including its direction reversal. At \nhigher values of $|F|$ the current should saturate with a small value since the mobility \n$\\mu = |\\frac{dv}{dF}|$ for both $\\pm F$ tend to $\\gamma_0^{-1}$. The variation of $\\mu$ \nas a function of $|F|$ is shown in Fig.3. It shows that at the position of the current \nmaximum $\\mu(|F|)=\\mu(-|F|)$ as it should be.\n\nThe variation of $\\Sigma \\bar v$ as a function of temperature $T$ at various values of\napplied force $|F|$, as shown in Fig.4, reveals that current can also increase with \ntemperature. Moreover, one can achieve current reversals as well. In Fig.4, plots are given \nfor $\\gamma_0=0.035$ and $|F|=0.025$, and 0.075. $|F|=0.075$ yields large currents and shows one current reversal. For larger $|F|$ the current remains negative for all $T$. This \nis not shown here to preserve clarity of the figure. However, in order to highlight the \nexistence of two current reversals we plot the graph for $\\gamma_0=0.1$, and $|F|=0.1$ \nin the inset of the figure. For low values of $|F|$, for example $|F|=0.025$, the \ncurrent always remains positive, though small, which is consistent with the observation \nmade with reference to Fig.2. It is noteworthy that the magnitude of current obtained \nhere is quite large compared to what has been reported\\cite{raish} for overdamped \nsystems.\n\n\\subsection{The energetics}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig5.eps}}\n\\caption{Shows the variation of the average kinetic, potential and total energies at \n$T= 0.4$ for $\\gamma_0 = 0.4$ as a function of $F$.}\n\\label{fig.5}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=15cm\n\\centerline{\\epsfbox{fig6.eps}}\n\\caption{In fig.$[a]$ the slopes of the average total energies are shown for various \nvalues of $\\gamma_0 < 1.0$. The graphs for $\\gamma_0 =$ 0.2, 0.4, 0.5 and 0.75 \nare scaled by factors of 10, 20, 25 and 30, respectively, to make them visible in the\nfigure with the graph for $\\gamma_0 =$0.1. Fig.$[b]$ shows plots for $\\gamma_0 > 1.0$. \nThe slopes for $\\gamma_0 =$ 2.0, 4.0 and 5.0 are blown up by 4, 5 and 5 times, \nrespectively, to scale with the curve for $\\gamma_0 =$1.0. The little vertical markings \non the curves show positions of the current maxima.}\n\\label{fig.6}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig7.eps}}\n\\caption{$T-\\gamma_{0}$ plane showing two disjointed regions with possibility of negative curvatures of the total energy $E_{tot} (|F|)$ separated by a region without the negative curvature of  $E_{tot} (|F|)$.}\n\\label{fig.7}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig8.eps}}\n\\caption{Stokes efficiency versus F for different values of $\\gamma_0$. The inset shows \n$\\eta_{Stokes}$ and $\\Sigma{v}$ plotted for $\\gamma_0 =$ 0.75.}\n\\label{fig.8}\n\\end{figure}\n\\begin{figure}[]\n\\newpage\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig9.eps}}\n\\caption{Shows the variation of Stokes efficiency with $T$ for various values of  $F$ \nfor $\\gamma_0 =$ 0.4. The little vertical markings on the efficiency curves indicate the \npositions of the current maxima. The inset shows efficiency and current for $F =$ 0.7}.\n\\label{fig.9}\n\\end{figure}\n\nThe average potential and kinetic energies can be easily calculated. The expressions\nare given in the Appendix. Fig.5 shows the variation of the average kinetic ($E_{kin}$),\npotential ($E_{pot}$), and total ($E_{tot}$) energies at $T=0.4$ for $\\gamma_0=0.4$ as a\nfunction of $|F|$. Monotonically increasing nature of $E_{kin}$, and $E_{tot}$ and\nnonmonotonic behaviour of $E_{pot}$ are seen for all (explored) $\\gamma_0$ and $T$ values.\nInterestingly, the slope of $E_{tot}$ shows a shoulder peak which after reaching a minimum\nrises again, Fig. 6. For a given temperature the shoulder peak occurs for small $\\gamma_0$ \nand large $\\gamma_0$ values. However, there is an intermediate range of $\\gamma_0$ \nvalues where the slope does not show this peaking behaviour; the total energy does not\nacquire negative curvature in this range of $\\gamma_0$. The slope gives valuable \ninformation about the variation of the ratchet current $\\Sigma \\bar v$ as shown in Fig.6.\nFor small $\\gamma_0 < 1$ values the slope peaks at an intermediate value of $|F|$, and \nthis is precisely where the current maximum occurs. For $\\gamma_0 > 1$ the positions of \nthe current maximum and that of the slope of the total energy are clearly well separated.\nIn fact, the current maximum occurs, for these large $\\gamma_0$ values, even beyond the \nposition of the minimum (after the peak) of the slope of the total energy. At this point \nit is worth mentioning that for not too large temperatures for $\\gamma_0 < 1$ the current \npeak occurs at $|F|<1$ whereas for all $\\gamma_0 > 1$ the current peaks invariably occur \nat $|F|>1$. Recall that $|F|=1$ is the critical field at which the potential barrier to \nmotion just disappears.\nA careful observation of the results at various temperatures $T$, thus, allows one to\nconstruct a diagram in the ($T-\\gamma_0$) plane, Fig.7, with two disjointed regions with \npossibility of negative curvatures of $E_{tot}(|F_0|)$ separated by a region with only \npositive curvatures. In the \"negative curvature\" region with $\\gamma_0<1$ the slope of \n$E_{pot}$ peaks close to where $\\Sigma \\bar v$ becomes maximum. Whereas in the other \nnegative curvature region with $\\gamma_0>1$ current maximum occurs far away from the \nposition ($F_0$) where the slope of $E_{tot}(|F_0|)$ peaks. It shows that one can \nclearly separate low damping ($\\gamma_0<1$) region from the high damping ($\\gamma_0>1$) \ncase with each showing qualitatively different characteristic behaviour of particle motion. \n\n\\subsection{The efficiency of performance}\nThe currents obtained by application of symmetric drives $\\pm|F|$ are small and not \nsustainable against applied loads of perceptible magnitude. Therefore it is not feasible \nto calculate work done by the system against the load and hence it is difficult to study \nits behaviour as the other parameters vary. However, the particle keeps moving, essentially \nwith average velocity $\\Sigma \\bar v$, against the frictional force and hence performs \nsome work, though not useful practically. How efficiently the system does this work is \nobtained by comparing the magnitude of this work ($=\\gamma_0(\\sum \\bar v)^2$) with the \ninput energy supplied to it. A measure of this quantity called the Stokes efficiency is \ndefined and given (in our case) as\\cite{wang, seki, parr, raish},\n\\begin{equation}\n\\eta_{S}=\\frac{\\gamma_0(\\sum \\bar v)^2}{|F||(\\bar v(|F|)-\\bar v(-|F|))|}.\n\\end{equation}    \nIn Fig.8, we plot $\\eta_{S}$ as a function of $|F|$ for various $\\gamma_0$ values at \n$T=0.4$. $\\eta_{S}$ behaves in a very similar fashion as the corresponding current does. \nHowever, $\\eta_{S}$ and current do not peak at the same value of F as will be clear from \nthe inset of the figure where $\\eta_{S}$ and $\\Sigma \\bar v$ are plotted together for \n$\\gamma_0=0.75$. It is remarkable that with such a modest approach one can find regions \nin the parameter space $(\\gamma_0,|F|)$ where the efficiency of performance exceeds two \nper cent. It shows that exploiting system inhomogeneity to obtain ratchet current in \ninertial systems may be useful for practical purposes too.\n\nThe variation of Stokes efficiency $\\eta_{S}$ with $T$ is shown in Fig.9 for $|F|=\n0.66, 0.70$, and 0.80 for which  the efficiency and current are appreciable. It is \nnoteworthy that at low temperatures the performance of the system improves with rise in \ntemperature and only after attaining a peak it declines with further rise in temperature. \nIn this case, however, the efficiency and current seem to peak at the same temperature as \ncan be seen from the inset of the figure. For $F=0.66$ we could not see the low temperature\nbehaviour because we do not get sensible results in this region from MCFM.\n\n\\subsection{The asymmetrically driven ratchet}\n\\begin{figure}[]\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig10.eps}}\n\\caption{Stokes efficiency as a function of the asymmetry parameter $\\epsilon$ for \ndifferent combinations of values of $F$ and $\\gamma_0$.}\n\\label{fig.10}\n\\end{figure}\n\\begin{figure}[]\n\\epsfxsize=9cm\n\\centerline{\\epsfbox{fig11.eps}}\n\\caption{The thermodynamic efficiency  $\\eta_{th}$  versus load for two values of $F$ with \nfixed  $\\epsilon =$ 0.5, $\\gamma_0 =$ 0.035 and $T =$ 0.4.}\n\\label{fig.11}\n\\end{figure}\n\nIt is clear that in general the efficiency of performance of the inhomogeneous \ninertial ratchets is small except in some special sections of the parameter space\n$(\\gamma_0, |F|, T)$. This efficiency can be dramatically enhanced if the ratchet is \ndriven asymmetrically with an asymmetry parameter $\\epsilon (0\\leq \\epsilon<1)$. Here the \ndriving force in the positive and in the negative directions are not same: The force\n$F=\\frac{1+\\epsilon}{1-\\epsilon}|F|$ is applied in the positive direction for a fraction\nof time $\\tau=\\frac{1-\\epsilon}{2}$ whereas $F=-|F|$ is applied for a fraction of time\n$1-\\tau=\\frac{1+\\epsilon}{2}$. The ratchet current now will be given by\n\\begin{equation}\n\\Sigma \\bar v=\\bar v^{+}+\\bar v^{-}\n\\end{equation}\nwith\n\\begin{equation}\n\\bar v^{+}=\\frac{1-\\epsilon}{2}\\bar v(\\frac{1+\\epsilon}{1-\\epsilon}|F|),\n\\end{equation}\n\\begin{equation}\n\\bar v^{-}= \\frac{1+\\epsilon}{2}\\bar v(-|F|)\n\\end{equation}\nThe Stokes efficiency of the thermal ratchet motion is similarly defined as in Eq. (3.1):\n\\begin{equation}\n\\eta_{S}=\\frac{\\gamma_0 (\\sum \\bar v)^2}{\n|F||\\bar v^{+}(\\frac{1+\\epsilon}{1-\\epsilon})-\\bar v^{-}|}.\n\\end{equation}\nIn this case the ratchet current, in a typical situation, is quite large. \nCorrespondingly, the Stokes efficiency also become large. Fig.10 shows the variation of \n$\\eta_{S}$ as a function of the asymmetry parameter $\\epsilon$ for various values of\n (a) $|F|=$0.017, $\\gamma_{0}=$0.035; (b) $|F|=$0.05, $\\gamma_{0}=$0.1 and (c) $|F|=$0.27, $\\gamma_{0}=$0.4  at  $T=0.4$. Clearly, more than 10 per cent of ratchet operation \nefficiency becomes achievable. Since the currents are large \none can also drive these ratchet currents against applied loads, $L$. Hence, the \nthermodynamic efficiency $\\eta_{th}$, defined in this case as \\cite{seki, parr, raish},\n\\begin{equation}\n\\eta_{th}=\\frac{L\\sum \\bar v}{|F||\\bar v^{+}(\\frac{1+\\epsilon}\n{1-\\epsilon})-\\bar v^{-}|} \n\\end{equation}\ncan be calculated. The variation of $\\eta_{th}$ as a function of $L$ for $\\gamma_0=0.035$\nat $T=0.4$ is shown in Fig.11 for $|F|=0.03$, and 0.053. From the figure one can see\nthat thermodynamic efficiency can be substantial. One can find an optimum load, $L$, and \nan optimum asymmetry parameter, $\\epsilon$, of field drive where the ratchet performs most\nefficiently. It should, however, be noted that the effect of asymmetric drive is \noverwhelming and the system inhomogeneity plays only a secondary role in this case. One \ncan achieve almost similar results without invoking system inhomogeneity as hs been seen \nin the overdamped case\\cite{raish}. \n  \n\\section{Discussion and Conclusion}\n\nAll the results of inertial particle motion, in a symmetric periodic potential and driven\nsymmetrically ( and of course adiabatically), such as ratchet current and its \nnonmonotonic behaviour (including its direction reversals) as a function of $|F|, T$, \nand $\\gamma_0$ are possible here only because the friction was considered space-dependent. \nThe periodic variation of the friction in space is just a model and we have not provided \nany microscopic basis for it. However, it has been shown earlier that in order to describe\nthe motion of an adsorbed particle on a crystalline surface (of same atoms as the \nadsorbed one) by a Fokker-Planck equation suitable for one-dimensional particle motion,\none necessarily have to consider space-dependent friction\\cite{wahn}. It turned out from\ntheir microscopic theory that the space dependence has to be periodic as the average\npotential created by the crystalline (surface) substrate for the adatom and in antiphase\n($\\phi=\\pi$) with it. In other words, the friction will  be large where the substrate \npotential is small. The possiblity of obtaining current due to diffusion \ninhomogeneity\\cite{landa} and frictional inhomogeneity\\cite{amj1} has been argued \nearlier. The phase difference $\\phi\\ne n\\pi (n=0, 1, 2, ...)$ is necessary to break the \nspatial symmetry in order to get ratchet current. We have found that the ratchet current \ndepends on the value of $\\phi$. For instance, taking $\\phi=0.4$ gives different and \nlarger current from what we have reported here for $\\phi=0.35$ at the same point in the \nparameter space of ($|F|, T, \\gamma_0$). \n\nThe increase of ratchet current and also the efficiency of the ratchet as a function of\n$T$ at small temperature range is because the particle has a large potential to surmount \nat small $|F|$ and increasing temperature helps in achieving that. However at the larger\ntemperature range obtaining a second current peak as shown in Fig.4 cannot be explained\nas simply and it arises because of the competition between the temperature and friction\neffect as argued elsewhere\\cite{wanda}.\n\nThe behaviour of the slope of the total energy vis-a-vis the position of ratchet current\npeak shows a clear demarcation occurring around  $\\gamma_0=1$, in the ($T-\\gamma_0$) \nplane. The investigation of time scales of running and locked states of the particle \nmotion may provide a clue to this remarkable behaviour. We are carrying out numerical \nLangevin dynamic simulation on this aspect of the problem and the results will be \nreported elsewhere. We conclude by reiterating that particle motion in inhomogeneous \nsystems has the potential of a rich field of research.  \n\\section*{ACKNOWLEDGEMENT}\n\nThe authors thank BRNS, Department of Atomic Energy, Govt. of India, for partial\nfinancial support. MCM thanks A.M. Jayannavar for discussion, and Abdus Salam ICTP,\nTrieste, Italy for providing opportunity to visit where the paper was written.\n\n\\section*{APPENDIX}\nUsing Eqs. (2.36) and (2.37), the potential energy expression (2.21) is given by\n\\begin{displaymath}\nE_{pot}=\\sum_{q}V^{(-q)}\\frac{(\\mathbf H)^{q,0}}{(\\mathbf H)^{0,0}}.\n\\end{displaymath}\nThe kinetic energy expression (2.22) can be written as\n\\begin{equation}\nE_{kin}=\\frac{1}{2}+{\\sqrt{2}}\\pi(\\mathbf {C_2})^0. \\nonumber\n\\end{equation}\nFrom Eqs. (2.29) and (2.32) one can write down a general equation relating \n$\\mathbf{C_n}$ and $\\mathbf{C_{n-1}} (n\\geq 1)$ as\n\\begin{displaymath}\n\\mathbf{C_n}=\\frac{1}{\\sqrt{n}}\\mathbf{A_nC_{n-1}},\n\\end{displaymath}\nwhere the matrices $\\mathbf{A_m} (m=0, 1, 2, ...)$ are given by\n\\begin{displaymath}\n\\mathbf{A_m}=(\\gamma+\\sqrt{T}(m+1)^{-1}\\mathbf{DA_{m+1}})^{-1}(\n\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}}]-\\sqrt{T}\\mathbf{D}).\n\\end{displaymath}\nIf the recursion is terminated at $m=N$ $(\\mathbf{A_{N+1}}=\\mathbf{0})$\n\\begin{displaymath}\n\\mathbf{A_N}=\\gamma^{-1}(\\frac{1}{\\sqrt{T}}[F\\mathbf{1}-\\mathbf{V^{'}}]\n-\\sqrt{T}\\mathbf{D}),\n\\end{displaymath}\nfor $N\\geq 2$. $\\mathbf{C_2}$ is thus calculated as\n\\begin{displaymath}\n\\mathbf{C_2}=\\frac{1}{\\sqrt{2}}\\mathbf{A_1A_0C_0}.\n\\end{displaymath}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGiven the wide spread use of deep networks \\cite{gan} across various applications and their black box nature, explainability in artificial intelligence (XAI) has been one of the problems at the forefront in AI research recently \\cite{montavon2017methods,patternet,lime,CEM,simple}. Darpa's call for creating interpretable solutions \\cite{xai} and the General Data Protection Regulation (GDPR) passed in Europe \\cite{gdpr} which requires businesses to provide understandable justifications to their users for decisions that may affect them has made this need even more acute.\n\nThere have been many (posthoc) interpretability methods proposed to interpret decisions of neural networks \\cite{Ormas,patternet,saliency,bach2015pixel,CEM,simple} which assume complete access to the model. Locally interpretable model-agnostic explanation method (LIME) \\cite{lime} is amongst the few that can provide local explanations for any model with just query access.\nThis is an extremely attractive feature as it can be used in settings where the model owner may not want to expose the inner details of the model but may desire local explanations using say a remote service. Another application is in interpreting decisions not just of neural networks but other models such as random forests, boosted trees and ensembles of heterogeneous models which are known to perform quite well in many domains that use structured data \\cite{kaggle}.\n\nIn this paper, we thus propose the model agnostic contrastive explanations method (MACEM) that requires only query access to the classification model with particular focus on structured data. Structured data can be composed of real and categorical features, and we provide a principled way of creating contrastive explanations for such data. Contrastive explanations are a rich form of explanation where one conveys not only what is (minimally) sufficient to justify the class of an input i.e. pertinent positives (PPs), but also what should be (minimally) necessarily absent to maintain the original classification i.e. pertinent negatives (PNs) \\cite{CEM}. Such explanations are commonly used in social settings as well as in domains such as medicine and criminology \\cite{pertneg}. For example, a patient with symptoms of cough, cold and fever (PPs) could have flu or pneumonia. However, the absence of chills or mucous (PNs) would indicate that the person has flu rather than pneumonia. Thus, in addition to the symptoms that were present, the symptoms that are absent are also critical in arriving at a decision. As such, these type of explanations are also sought after in the financial industry where in the recently completed FICO explainability challenge \\cite{FICO} it was explicitly stated that if a loan was rejected it would be highly desirable for an explanation to elucidate what changes to the loan application (i.e. input) would have led to its acceptance. Not to mention the data in the challenge was in structured format.\n\nAdditionally, working with experts across multiple industries (finance, healthcare, manufacturing, utility) we have found that they want explanation methods to satisfy two main criteria: a) be model agnostic so that one can explain a model on their private cloud through just query access and b) be trustworthy in that the method closely captures what the model is trying to do. For b) they are moving away from proxy model approaches such as LIME, since it does NOT meet their regulatory standards.\n\\begin{wrapfigure}{r}{0.55\\textwidth}\n\\vspace{-0.5cm}\n\\begin{center}\n    \\includegraphics[width=0.55\\textwidth]{PPPNEg.png}  \n  \\caption{Above we see an example explanation for a loan application from the German Credit dataset that was rejected by a black box model which was a tree. We depict the important features for PPs and PNs. Our PPs convey that even if the person didn't need a co-applicant and had lower credit card debt the application would still be rejected. In contrast, our PNs inform us that if the persons checking amount had more money, the loan installment rate was lower and there were no people that he\/she was responsible for then the loan would have been accepted.}\n  \\label{VisualPPPN}\n  \\end{center}\n\\vspace{-0.5cm}\n\\end{wrapfigure}\nBeing contrastive is also extremely useful as it helps them understand sensitivities of the model decisions. Given this we believe our contribution is timely and significant.\n\nIn previous works \\cite{CEM}, a method to produce such explanations was proposed. However, the method was restricted to differentiable models such as deep neural networks and strong (implicit) assumptions were made in terms of the semantics of the data used to train the models and obtain explanations. In particular, following are the key differences between our current and the prior work:\\\\\n     \\textbf{i) Gradients not available:} In this work we want to create contrastive explanations with only query access to any classification model. This is a significant step given that the prior work a) assumed complete access to the model and b) could be used only for differentiable models like deep neural networks.\\\\% Our method can be used for any classification model (viz. decision trees, forests, ensembles), where we estimate gradients (with theoretically bounded bias) using only oracle access.\\\\\n     \\textbf{ii) Using (and estimating) base values:} To compute PPs and PNs one needs to know what it means for a feature to be absent i.e., what value for a feature indicates there is no signal or is essentially the least interesting value for that feature. We refer to such values as \\emph{base values}. In the prior work on contrastive explanations \\cite{CEM} the value 0 for a feature was considered as the base value, which can be unrealistic.\n     Ideally, the user should provide us with these values. In this paper we adapt our methods to utilize such base values and also propose ways to estimate them in situations that they are not provided. It is important to note here that existence base values is implicitly assumed in most of explainability research. For example, explanations for images involve highlighting\/selecting pixels \\cite{saliency,bach2015pixel,lime} which implicitly assumes that a blank image is zero information although it too may be classified in a class with high confidence.\\\\\n     \\textbf{iii) Handling categorical features:} In the prior work all features were considered to be real valued and no special consideration was given to handle categorical features. However, in this work we remain cognizant to the fact that categorical features are fundamentally different than real valued ones and propose a principled as well as scalable approach to handle them for our explanations.\\\\\n     \\textbf{iv) Computing PPs and PNs:} Given the above differences we propose new ways of computing PPs and PNs that are consistent with their intuitive definitions mentioned before. \\emph{As such, we define a PP for an input $\\mathbf{x}$ as the sparsest example (w.r.t. base values) whose feature values are no farther from the base values than those of $\\mathbf{x}$, with it lying in the same class as $\\mathbf{x}$. Consequently, a PN for $\\mathbf{x}$ is defined as an example that is closest to $\\mathbf{x}$ but whose feature values are at least as far away from the base values as those of $\\mathbf{x}$ with it lying in a different class}. Important features for an example PP and PN for a loan application in the German Credit dataset are depicted in figure \\ref{VisualPPPN}.\n    \n   \n\n\n \n \n \n \n\n\\vspace{-0.3cm}\n\\section{Related Work}\n\\vspace{-0.3cm}\nTrust and transparency of AI systems has received a lot of attention recently \\cite{xai}. Explainability is considered to be one of the cornerstones for building trustworthy systems and has been of particular focus in the research community \\cite{lipton2016mythos,tip}. Researchers are trying to build better performing interpretable models \\cite{decl,twl,Caruana:2015,irt,bastani2017interpreting,simple} as well as improved methods to understand black box models such as deep neural networks \\cite{lime,bach2015pixel,CEM}.\n\nThe survey \\cite{montavon2017methods} which is mainly focused on deep learning explainability methods looks broadly at i) prototype selection methods \\cite{nguyen2016synthesizing,nguyen2016multifaceted} to explain a particular class and ii) methods that highlight relevant features for a given input \\cite{bach2015pixel,patternet,lime,unifiedPI}. There are other works that fall under (i) such as \\cite{l2c,proto} as well as those that fall under (ii) for vision \\cite{selvaraju2016grad,saliency,Ormas} and NLP applications \\cite{lei2016rationalizing}. Most of these works though do not provide contrastive explanations in a model agnostic setting. There are also interesting works which try to quantify interpretability \\cite{QEval,tip} and suggest methods for doing so.\n \nTwo of the most relevant recent works besides \\cite{CEM} which we have already contrasted with are \\cite{anchors,symbolic}. In \\cite{anchors}, the authors try to find sufficient conditions to justify a classification that are global in nature. For example, the presence of the word \"bad\" in a sentence would automatically indicate negative sentiment irrespective of the other words. As such, they do not find input specific minimally sufficient values that would maintain the classification or minimal values that would change classification. Such global anchors also may not always be present in the data that one is interested in. The other work \\cite{symbolic} tries to provide (stable) suggestions more than local explanations for decisions based on a neural network. Moreover, the approach is restricted to neural networks using rectified linear units and is feasible primarily for smallish to medium sized neural networks in asymmetric binary settings, where suggestions are sought for a specific class (viz. loan rejected) and not the other (viz. loan accepted).\n\n\\vspace{-0.3cm}\n\\section{MACEM Method}\n\\vspace{-0.3cm}\n Let $\\mathcal{X}$ denote the feasible data space and let $(\\mathbf{x}_0,t_0)$ denote an input example $\\mathbf{x}_0 \\in \\mathcal{X}$ and its inferred class label $t_0$ obtained from a black-box classification model. The modified example $\\mathbf{x} \\in \\mathcal{X}$ based on $\\mathbf{x}_0$ is defined as $\\mathbf{x}=\\mathbf{x}_0 + \\boldsymbol{\\delta}$, where $\\boldsymbol{\\delta}$ is a perturbation applied to $\\mathbf{x}_0$. Our method of finding pertinent positives\/negatives is formulated as an optimization problem over the perturbation variable $\\boldsymbol{\\delta}$ that is used to explain the model's prediction results. We denote the prediction of the model on the example $\\mathbf{x}$ by $\\zeta(\\mathbf{x})$, where $\\zeta(\\cdot)$ is any function that outputs a vector of confidence scores over all classes, such as the log value of prediction probability. Let $c,~\\beta,~\\gamma$ be non-negative regularization parameters.\n\n\\begin{algorithm}[htbp]\n    \\caption{Model Agnostic Contrastive Explanations Method (MACEM)}\n    \\label{cem}\n\\begin{algorithmic}\n\\STATE \\textbf{Input:} Black box model $\\mathcal{M}$, base values $\\mathbf{b}$, allowed space $\\mathcal{X}$, example $(\\mathbf{x}_0,t_0)$,  and (estimate of) input probability distribution $p(x)$ (optional).\\\\\n\\STATE 1) Find PPs $\\boldsymbol{\\delta}^{\\textnormal{pos}}$ by solving equation \\ref{eqn_per_pos} and PNs $\\boldsymbol{\\delta}^{\\textnormal{neg}}$ by solving equation \\ref{eqn_per_neg}.\n\n \\STATE 2) Return $\\boldsymbol{\\delta}^{\\textnormal{pos}}$ and $\\boldsymbol{\\delta}^{\\textnormal{neg}}$. \\COMMENT{Explanation: The input $\\mathbf{x}_0$ would still be classified into class $t_0$ even if it were (closer to base values) $\\mathbf{x}_0+\\boldsymbol{\\delta}^{\\textnormal{pos}}$. However, its class would change if it were perturbed (away from original values) by $\\boldsymbol{\\delta}^{\\textnormal{neg}}$, i.e., if it became $\\mathbf{x}_0+\\boldsymbol{\\delta}^{\\textnormal{neg}}$.\n}\n\\end{algorithmic}\n\\end{algorithm}\n\\vspace{-0.5cm}\n\\subsection{Computing Pertinent Positives}\n\\vspace{-0.2cm}\nAssume an example $\\mathbf{x}_0$ has $d$ features each with base values $\\{b_i\\}_{i=1}^d$. Let $\\Delta_{PP}$ denote the space $\\{\\boldsymbol{\\delta}:  | \\mathbf{x}_0+\\boldsymbol{\\delta} - \\mathbf{b} | \t\\preceq   | \\mathbf{x}_0 - \\mathbf{b} |  \\textnormal{~and~} \\mathbf{x}_0 + \\boldsymbol{\\delta} \\in \\mathcal{X}  \\}$, where $\\mathbf{b}=[b_1,\\ldots,b_d]$, and $|\\cdot|$ and $\\preceq$ implies element-wise absolute value and inequality, respectively. To solve for PP, we propose the following:\n\n\\begin{align}\n\\label{eqn_per_pos}\n\\boldsymbol{\\delta}^{\\textnormal{pos}} \\leftarrow \\argmin_{\\boldsymbol{\\delta} \\in \\Delta_{PP}}~&c \\cdot \\max \\{  \\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i - [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0}, - \\kappa   \\}+ \\beta \\|\\mathbf{x}_0 + \\boldsymbol{\\delta} - \\mathbf{b} \\|_1 \\nonumber\\\\ &+ \\|\\mathbf{x}_0 + \\boldsymbol{\\delta} - \\mathbf{b}\\|_2^2-\\gamma p(\\mathbf{x}_0 + \\boldsymbol{\\delta}).\n\\end{align}\n The first term\n is a designed loss function that encourages the modified example $\\mathbf{x}=\\mathbf{x}_0+\\boldsymbol{\\delta}$ relative to the base value vector $\\mathbf{b}$, defined as  $\\mathbf{x}-\\mathbf{b}$,\n to be predicted as the same class as the original label $t_0=\\arg \\max_i [\\zeta(\\mathbf{x}_0)]_i$\nThe loss function\nis a hinge-like loss and\nthe term $\\kappa \\geq 0$  controls the gap between $[\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0}$ and the other most probable class. In particular, the loss attains its minimal value when $[\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0}$ is $\\kappa$ larger than $\\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i$. The parameter $c \\geq 0$ is the regularization coefficient associated with the first term\nThe second and third terms in \\eqref{eqn_per_pos} are jointly called the elastic-net regularizer \\cite{zou2005regularization}, which aids in selecting a set of highly relevant features from $\\mathbf{x}-\\mathbf{b}$, and the parameter $\\beta \\geq 0$ controls the sparsity of the vector $\\mathbf{x}-\\mathbf{b}$\n\nOptionally, the input distribution $p(\\mathbf{x})$ also maybe estimated from the data\nwhich could be used to further direct the search so that we produce realistic or high probability $\\mathbf{x}$.\n\n\n\\subsection{Computing Pertinent Negatives}\nAnalogous to PP, for PN let $\\Delta_{PN}$ denote the space $\\{\\boldsymbol{\\delta}:  | \\mathbf{x}_0+\\boldsymbol{\\delta} - \\mathbf{b} | \t\\succ   | \\mathbf{x}_0 - \\mathbf{b} |  \\textnormal{~and~} \\mathbf{x}_0 + \\boldsymbol{\\delta} \\in \\mathcal{X}    \\}$. To solve for PN, we propose the following problem formulation:\n\\begin{align}\n\\label{eqn_per_neg}\n\\boldsymbol{\\delta}^{\\textnormal{neg}}\\leftarrow\\argmin_{\\boldsymbol{\\delta} \\in \\Delta_{PN}}  ~c \\cdot\\max \\{ [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0} - \\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i, - \\kappa   \\}+\\beta \\|\\boldsymbol{\\delta}  \\|_1 + \\|\\boldsymbol{\\delta} \\|_2^2- \\gamma p(\\mathbf{x}_0 + \\boldsymbol{\\delta})\n\\end{align}\nIn other words, for PN, we aim to find the least modified changes in $\\boldsymbol{\\delta} \\in \\Delta_{PN}$, evaluated by the elastic-net loss on $\\boldsymbol{\\delta}$,\nsuch that its addition to $\\mathbf{x}_0$ leads to a different prediction from $t_0$, \n\n\\subsection{Method Details}\n\n\nWe now describe the details of how the optimization of the above objectives is implemented along with estimation and modeling of certain key aspects\n\n\\subsubsection{Optimization Procedure}\n\n\n\nHere we first illustrate how FISTA solves for PP and PN, assuming the gradient is available. This is very similar to previous work \\cite{CEM} with the main difference lying in the projection operators $\\Delta_{PN}$ and $\\Delta_{PP}$.\n FISTA is an efficient solver for optimization problems involving $L_1$ regularization. Take pertinent negative as an example, let $g (\\boldsymbol{\\delta}) = \\max \\{ [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0} - \\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i, - \\kappa   \\} + \\|\\boldsymbol{\\delta}\\|_2^2 -  \\gamma p(\\mathbf{x}_0 + \\boldsymbol{\\delta}) $ denote the objective function of (\\ref{eqn_per_neg}) without the $L_1$ regularization term.\nGiven the initial iterate $\\boldsymbol{\\delta}^{(0)}=\\mathbf{0}$,\nprojected FISTA iteratively updates the perturbation $I$ times by\n\\begin{equation}\n\\boldsymbol{\\delta}^{(k+1)}=\\Pi_{\\Delta_{PN}} \\{ S_{\\beta}(\\mathbf{y}^{(k)}-\\alpha_k \\nabla g(\\mathbf{y}^{(k)})) \\};~~~\n \\mathbf{y}^{(k+1)}=\\Pi_{\\Delta_{PN}} \\{ \\boldsymbol{\\delta}^{(k+1)}+\\frac{k}{k+3} (\\boldsymbol{\\delta}^{(k+1)} - \\boldsymbol{\\delta}^{(k)}) \\},\n\\end{equation}\nwhere $\\Pi_{\\Delta_{PN}}$ denotes the vector projection onto the set $\\Delta_{PN}$,\n$\\alpha_k$ is the step size, $\\mathbf{y}^{(k)}$ is a slack variable accounting for momentum acceleration with $\\mathbf{y}^{(0)}=\\boldsymbol{\\delta}^{(0)}$, and $S_{\\beta}: \\mathbb{R}^{p} \\mapsto \\mathbb{R}^{p}$ is an element-wise shrinkage-thresholding function which is 0 if $\\forall i \\in \\{1,\\ldots,d\\}$ $|\\mathbf{z}_i| \\leq \\beta$, else takes the values $\\mathbf{z}_i - \\beta$ or $\\mathbf{z}_i + \\beta$ for $\\mathbf{z}_i  > \\beta$ or $\\mathbf{z}_i  < -\\beta$ respectively.\nThe final perturbation $\\boldsymbol{\\delta}^{(k^*)}$ for pertinent negative analysis is selected from the set $\\{ \\boldsymbol{\\delta}^{(k)}\\}_{k=1}^I$ such that $f^{\\textnormal{neg}}_{\\kappa}(\\mathbf{x}_0,\\boldsymbol{\\delta}^{(k^*)})=0$ and  $k^* = \\arg \\min_{k \\in \\{1,\\ldots, I\\}} \\beta \\| \\boldsymbol{\\delta}\\|_1 + \\| \\boldsymbol{\\delta}\\|_2^2 $.\nA similar projected FISTA approach is applied to PP analysis. \n\n\n\\subsubsection{Gradient Estimation}\nIn the black-box setting, in order to balance the model query complexity and algorithmic convergence rate using zeroth-order optimization, in this paper we use a two-point evaluation based gradient estimator averaged over $q$ different random directions \\cite{duchi2015optimal,liu2017zeroth,liu2018zeroth}. Specifically, given a scalar function $f(\\cdot)$, its gradient at a point $\\mathbf{x} \\in \\mathbb{R}^d$ is estimated by \n\\begin{align}\n\\label{eqn_grad_est}\n\\widehat{\\nabla} f (\\mathbf{x}) = \\frac{d}{q \\mu} \\sum_{j=1}^q \\frac{f(\\mathbf{x}+\\mu \\mathbf{u}_j) - f(\\mathbf{x})}{\\mu} \\cdot \\mathbf{u}_j,\n\\end{align}\nwhere $\\{\\mathbf{u}_j\\}_{j=1}^q$ is a set of i.i.d. random directions drawn uniformly from a unit sphere, and $\\mu>0$ is a smoothing parameter.\n\nThe estimation error between $\\widehat{\\nabla} f (\\mathbf{x})$ and the true gradient $\\nabla f(\\mathbf{x})$ can be analyzed through a smoothed function $f_\\mu (\\mathbf{x})=\\mathbb{E}_{\\mathbf{u} \\in U_b} [f(\\mathbf{x}+\\mu \\mathbf{u})]$, where $U_b$ is a uniform distribution over the unit Euclidean ball. Assume $f$ is an $L$-smooth function, that is, its gradient $\\nabla f$ is $L$-Lipschitz continuous.\nIt has been shown in \\cite{liu2018zeroth} that $\\widehat{\\nabla} f (\\mathbf{x})$ is an unbiased estimator of the gradient $\\nabla f_\\mu (\\mathbf{x})$, i.e., $\\mathbb{E}_{\\mathbf{u}} [\\widehat{\\nabla} f (\\mathbf{x})]=\\nabla f_{\\mu} (\\mathbf{x})$. Moreover, using the bounded error between $f$ and $f_\\mu$, one can show that  the mean squared error between $\\widehat{\\nabla} f (\\mathbf{x})$ and  $\\nabla f (\\mathbf{x})$ is upper bounded by \n\\begin{align}\n\\label{eqn_bound}\n\\mathbb{E}_{\\mathbf{u}} [ \\| \\widehat{\\nabla} f (\\mathbf{x})- \\nabla f (\\mathbf{x})\\|_2^2 ]\\leq O \\left(\\frac{q+d}{q} \\right) \\|\\nabla f (\\mathbf{x})\\|_2^2 + O(\\mu^2 L^2 d^2).\n\\end{align}\n\n\n\n\\subsubsection{Determining Base Values}\nAs mentioned before, ideally, we would want base values as well as allowed ranges or limits for all features be specified by the user. This should in all likelihood provide the most useful explanations. However, this may not always be feasible given the dimensionality of the data and the level of expertise of the user. In such situations we compute base values using our best judgment.\n\nFor real valued features, we set the base value to be the median value of the feature. This possibly is the least interesting value for that feature as well as being robust to outliers\nMedians also make intuitive sense where for sparse features $0$ would rightly be chosen as the base value as opposed to some other value which would be the case for means.\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n  \\centering  \n      \\includegraphics[width=0.5\\textwidth]{FMA.png}  \n  \\caption{Above we see a categorical feature taking three values A, B and C with frequencies 11, 6 and 1 respectively as indicated on the vertical axis. Our mapping function in equation 11 for FMA maps these frequencies and hence the categorical values to 0, 0.5 and 1 in the $[0,1]$ interval. The red horizontal lines depict the function $h(.)$ showcasing the range of values that map back to either A, B or C.}\n  \\label{FMA}\n\\vspace{-0.75cm}\n\\end{wrapfigure}\nFor categorical features, we set the base value to be the mode for that feature. Here we use the fact that the mode is the least informative value for a categorical feature \\cite{inftheory} and rarer values are likely to be more interesting to the user. For example in a dataset containing health records most people will probably not have cancer and so having cancer is something that should stand out as it indicates a state away from the norm. Such states or behaviors we believe carry information that is more likely to surprise the user and draw attention, which could be a prelude to further actions.\n\n\\subsubsection{Modeling Categorical Features}\nGiven that categorical features do not impose an explicit ordering like real features do along with the fact that only the observed values have semantic meaning, there is a need to be model them differently when obtaining explanations. We now present a strategy that accomplishes this in an efficient manner compared to one-hot encoding where one could have an explosion in the feature space if there are many categorical features with each having many possible discrete values. Both these strategies are described next\n\n\n\n \n \n \n\n\n\\noindent{\\textbf{Frequency Map Approach (FMA):}} In this approach we want to directly leverage the optimization procedure described above where we need to define an ordered set\/interval in which to find the perturbations $\\delta$ for both PPs and PNs.\n\n\\noindent\\textit{Mapping:} As described above for categorical features we set the base value to be the mode of the values that occur. Given this a natural ordering can be created based on the frequencies of the different values. Thus, the least frequent value would be considered to be the farthest from the base value. Based on this we can map the $k$ discrete values $v_{i1}, ..., v_{ik}$  of the $i^{th}$ feature occurring with frequencies\/counts $c_{i1}, ..., c_{ik}$ to real values ${r}_{i1}, ..., {r}_{ik}$ respectively in the $[0,1]$ interval using the following mapping for any $j\\in \\{1, ..., k\\}$:\n   $ {r}_{ij}=\\frac{c_{\\text{max}}-c_{ij}}{c_{\\text{max}}-1}$\nwhere $c_{\\text{max}}=\\max\\limits_{j\\in \\{1, ..., k\\}} c_{ij}$. This maps the discrete value with the highest frequency to 0 making it the base value, while all other values with decreasing frequencies lie monotonically away from 0. Every candidate value has to have a frequency of at least 1 and so every discrete value gets mapped to the $[0,1]$ interval. We divide by $c_{\\text{max}}-1$, rather than $c_{\\text{max}}-c_{\\text{min}}$, where $c_{\\text{min}}=\\min\\limits_{j\\in \\{1, ..., k\\}} c_{ij}$ since, we do not want values that occur with almost equal frequency to be pushed to the edges of the interval as based on our modeling they are of similar interest.\n\n\\noindent\\textit{Method and Interpretation:} Based on the equation for $r_{ij}$, we run our algorithm for categorical features in the interval $[0,1]$, where every time we query the model we round the $\\mathbf{x}_0+\\boldsymbol{\\delta}$ to the closest $r_{ij}$ so that a \\emph{valid} categorical value $v_{ij}$ can be mapped back to and sent as part of the query input.\n\nThe question now is what are we exactly doing in the mathematical sense. It turns out that rather than optimizing $f^{\\textnormal{neg}}_{\\kappa}(\\mathbf{x}_0,\\boldsymbol{\\delta})=\\max \\{ [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0} - \\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i, - \\kappa   \\}$ or $f^{\\textnormal{pos}}_{\\kappa}(\\mathbf{x}_0,\\boldsymbol{\\delta})=\\max \\{  \\max_{i \\neq t_0} [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_i - [\\zeta(\\mathbf{x}_0+\\boldsymbol{\\delta})]_{t_0}, - \\kappa   \\}$, we are optimizing $f^{\\textnormal{neg}}_{\\kappa}(h(\\mathbf{x}_0,\\boldsymbol{\\delta}))$ or $f^{\\textnormal{pos}}_{\\kappa}(h(\\mathbf{x}_0,\\boldsymbol{\\delta}))$ respectively, where $h(.)$ is the identity map for real features,\nbut a step function defined over the $[0,1]$ interval for categorical features. Let $h_i(.)$ denote the application of the function $h(.)$ to the categorical feature $i$. If $\\mathbf{x} = \\mathbf{x}_0+\\boldsymbol{\\delta}$ and $\\mathbf{x}_i$ denotes the value of the feature in the mapped $[0,1]$ interval then,\n    $h_i(\\mathbf{x}_0,\\boldsymbol{\\delta}) = v_{ij}$, if $|\\mathbf{x}_i-r_{ij}| \\le |\\mathbf{x}_i-r_{im}|$ $\\forall m\\in\\{1, ..., k\\}$ and $m\\neq j$\nwhere $|.|$ denotes absolute value.\nAn example function $h(.)$ is depicted in figure \\ref{FMA}, where we see how real values are mapped back to valid categorical values by rounding to the closest $r_{ij}$.\n\n\\noindent\\textbf{Simplex Sampling Approach (SSA):} In this method of handling categorical variables, we will assume that a one-hot encoding of the input. Let $\\mathbf{x}=[\\mathbf{x}_C~\\mathbf{x}_R]$ be the input feature vector where $\\mathbf{x}_C$ denotes the categorical part while $\\mathbf{x}_R$ denotes the set of real features. Let there be $C$ categorical features in total. Then for all $c \\in [1:C], x_c \\in [1:d_c]$ where $x_c$ is the $c$-th categorical variable and it takes one of $d_c$ values.\n Note that we imply no ordering amongst the $d_c$ values. Generically they are assumed to have distinct integer values from $1$ till $d_c$. \nWe assume that input $\\mathbf{x}$ is processed into $\\mathbf{\\tilde{x}} = [\\mathbf{\\tilde{x}}_C ~ \\mathbf{\\tilde{x}}_R] $ where $\\mathbf{\\tilde{x}}_R = \\mathbf{x}_R$ while $\\mathbf{\\tilde{x}}_C \\in \\mathbb{R}^{1 \\times \\prod_{c \\in C} d_c }$. Each component $\\mathbf{\\tilde{x}}_c \\in \\mathbb{R}^{1 \\times d_c} $ is set to $\\mathbf{e}_i$, the canonical unit vector with $1$ in the $i$-th coordinate, if $x_c$ takes the value $i$. \n\n\\textit{Interpretation:} Now, we provide an interpretation when every categorical component $c$ lies in the $d_c$ dimensional simplex, i.e. when $\\mathbf{\\tilde{x}}_c \\in \\Delta_{d_c} $. Here, $\\Delta_N$ denotes the $N$-dimensional simplex. The actual function can be evaluated only on the inputs where each categorical component takes values from one of the corner points on the simplex, namely $\\mathbf{e}_i, ~ i \\in [1:d_c]$. Therefore, we interpolate the function when $\\mathbf{\\tilde{x}}_c$ is assigned a real vector in the simplex. \n\nLet $f(\\cdot)$ capture the soft output of the classifier when the one-hot encoded categorical variables take the values at the corner points of their respective simplices. Now, we extend the definition of $f$ as follows:\n\n$f([\\mathbf{\\tilde{x}}_C~\\mathbf{x}_R]) = \n\\mathbb{E}_{\\mathbf{e}_{i_c} \\sim \\mathbf{\\tilde{x}}_c,~\\forall c \\in [1:C] } \\left[ f(\\mathbf{e}_{i_1}, \\ldots \\mathbf{e}_{i_c}, \\ldots \\mathbf{e}_{i_{C}}, \\mathbf{x}_R)\\right].$\n\nEssentially, we sample the $c$-th unit vector from the distribution represented by $\\mathbf{\\tilde{x}_c} \\in \\Delta_{d_c}$ on the simplex independent of other categorical variables. The function value is the expected value of the functional evaluated on unit vectors obtained from this product distribution along with the fixed real coordinates $\\mathbf{x}_R$.\n\nWhen we perform the gradient descent as a part of algorithm \\ref{cem}, we actually do a projected gradient descent for $\\mathbf{\\tilde{x}}_C $ in the product of simplices $\\Delta_{d_1} \\times \\ldots \\Delta_{d_c}$. We cannot evaluate the function exactly, hence we can average over a certain number of samples drawn from the product distribution for every function evaluation on a candidate $\\mathbf{\\tilde{x}}$.\\\\\n\n\\noindent\\textbf{FMA vs SSA Tradeoffs:} As such the SSA strategy has stronger theoretical grounding, however from a practical standpoint it requires a lot of additional averaging through sampling for every function evaluation along with repeated projections to the simplices defined for every categorical feature during gradient descent to optimize the objective in Algorithm \\ref{cem}. \nFMA can also take more general format of inputs as they don't need to be one-hot-encoded which guards against further explosion of the feature space which could potentially result from categorical features having many distinct values.\n\n\\section{Experiments}\nWe now empirically evaluate our approach on 5 public datasets covering diverse domains. The datasets along with their domains are as follows. Finance: German Credit \\cite{uci} and FICO \\cite{FICO}, Astronomy: Digital Sky Survey \\cite{kaggleskysurvey}, Health care: Vertebral Column \\cite{uci} and Neuroscience: Olfaction \\cite{olfs}. A summary of the datasets is given in table \\ref{tab:data}. All the datasets except olfaction have class labels. However, as done in previous studies \\cite{tip}, we use \\emph{Pleasantness} as the target which is binarized using a threshold of 50. That is molecules with a rating $>$ 50 (on a 100 point scale) are categorized as being pleasant to smell, while the rest are deemed as unpleasant.\n\n\\begin{wraptable}{r}{0.6\\textwidth}\n\\vspace{-0.5cm}\n\\centering\n\\caption{Dataset characteristics, where $N$ denotes dataset size and $d$ is the dimensionality.}\n \\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n    \\small{Dataset} & \\small{$N$} & \\small{$d$} & \\small{\\# of} & \\small{Domain} \\\\\n    &&  & \\small{Classes}& \\\\\n    \\hline\n     \\small{German Credit} & \\small{1000} & \\small{20} & \\small{2} & \\small{Finance} \\\\\n     \\hline\n    \\small{FICO} & \\small{10459} & \\small{24} & \\small{2} & \\small{Finance} \\\\\n \\hline\n   \\small{Sky Survey} & \\small{10000} & \\small{17} & \\small{3} & \\small{Astronomy} \\\\\n   \\hline\n \\small{Vertibral Column} & \\small{310} & \\small{6} & \\small{3} & \\small{Health care} \\\\\n \\hline\n\\small{Olfaction} & \\small{476} & \\small{4869} & \\small{2} & \\small{Neuroscience} \\\\\n \\hline\n \\end{tabular}\n \\label{tab:data}\n\\vspace{-0.25cm}\n\\end{wraptable}\nWe test the methods for two (black box) classification models namely, CART decision trees (depth $\\le$ 5) and random forest (size 100). In all cases we use a random 75\\% of the dataset for training and a remaining 25\\% as test. We repeat this 10 times and average the results. The explanations are generated for the test points and correspondingly evaluated. Given the generality and efficiency of the FMA approach as stated above, we use that in our implementation of MACEM. For all datasets and all features the ranges were set based on the maximum and minimum values seen in the datasets. The base values for the German Credit, Sky Survey, Vertibral  Column dataset and FICO were set to median for the real features and mode for the categorical ones. For FICO the special values (viz. -9, -7, -8) were made 0. The base values for all features in the olfaction dataset were set at 0. We do not learn the underlying distribution so $\\gamma$ is set to 0 and the other parameters are found using cross-validation. We generate 50 random samples for estimating gradients at each step and run our search for 100 steps. We compare with LIME (\\url{https:\/\/github.com\/marcotcr\/lime}) which is arguably the most popular method to generate local explanations in a model agnostic fashion for (especially non-neural network) classifiers trained on structured data. We quantitatively evaluate our results on all 5 datasets as described next. We also provide qualitative evaluations for German Credit and the Olfaction dataset based on studying specific cases along with obtaining expert feedback.\n\n \n  \n   \n   \n    \n    \n   \n\n \n  \n\n\n\n\n\n\\vspace{-0.2mm}\n\\subsection{Quantitative Evaluation Metrics}\n\\vspace{-0.2mm}\nWe define three quantitative metrics: Correct Classification Percentage (CCP), Correct Feature Ranking (CFR) and Correct Feature Importance Percentage (CFIP) where we evaluate feature importances as well as how accurate the explanations are in predicting the same or different classes for PPs and PNs respectively.\n\nFor LIME we create proxies for PPs and PNs that are intuitively similar to ours for a fair comparison. The PP proxy for LIME is created by replacing all the negatively correlated features by base values in the original example, while maintaining the positively correlated feature values. For PNs, we create a proxy by setting all the positively correlated feature values to base values, while maintaining the negatively correlated features.\nFor our PPs, we compute feature importance by taking the absolute difference of each feature value to the corresponding base value and dividing by the features standard deviation. For PNs, we take the absolute value of the change of the perturbed feature and divide again by standard deviation. For LIME, we use the absolute value of the coefficients of the PP\/PN proxies.\n\n\\noindent\\textbf{Correct Classification Percentage (CCP):} For this metric we compute what percentage of our PPs actually lie in the same (predicted) class as the original example and for PNs what percentage of the new examples lie in a different class than the predicted one of the original example. Formally, if $(x_1,t_1), ..., (x_n,t_n)$ denote $n$ examples with $t_i$ being the predicted class label for $x_i$ and $PP_i$, $PN_i$ being the respective pertinent positives and negatives for it, then if $\\lambda(.)$ denotes an indicator function which is 1 if the condition inside it is true and 0 otherwise, we have (higher values better)\n\\begin{align}\n\\label{ccp}\n    CCP_{PP}=\\sum_i\\frac{\\lambda\\left(\\max [\\zeta(PP_i)]=t_i\\right)}{n}\\times 100, ~~~\n    CCP_{PN}=\\sum_i\\frac{\\lambda\\left(\\max [\\zeta(PN_i)]\\neq t_i\\right)}{n}\\times 100\n\\end{align}\n\n\\noindent\\textbf{Correct Feature Ranking (CFR):} For this metric we want to evaluate how good a particular explanation method's feature ranking is. For PPs\/PNs we independently set the top-k features to base\/original values and in each case note the class probability of classifying that input into the black box models predicted class. We then rank the features based on these probabilities in descending order and denote this ranking by $r^*_{PP}$ (or $r^*_{PN}$). Our CFR metric is then a correlation between the explanation models ranking of the features ($r_{PP}$ or $r_{PN}$) and this ranking. Higher the CFR better the method. If $\\rho(.,.)$ indicates the correlation between two lists then,\n\\begin{equation}\nCFR_{PP} = \\rho(r_{PP},r^*_{PP}),~~~CFR_{PN} = \\rho(r_{PN},r^*_{PN})\n\\end{equation}\nThe intuition is that most important features if eliminated should produce the most drop in predicting the input in the desired class. This is similar in spirit to ablation studies for images \\cite{Ormas} or the faithfulness metric proposed in \\cite{selfEx}.\n\n\\noindent\\textbf{Correct Feature Importance Percentage (CFIP):} Here we want to validate if the features that are actually important for an ideal PP and PN are the ones we identify.\nOf course, we do not know the globally optimal PP or PN for each input. So we use proxies of the ideal to compare with, which are training inputs that are closest to the base values and satisfy the PP\/PN criteria as defined before.\nIf our (correct) PPs and PNs are closer to the base values than these proxies, then we use them as the golden standard. We compute the CFIP score as follows: Let $f^*_{PP}(x)$ and $f^*_{PN}(x)$ denote the set of features in the tree corresponding to the ideal (proxy) PPs and PNs for an input $\\mathbf{x}$ as described before. Let $f_{PP}(x)$ and $f_{PN}(x)$ denote the top-k important features (assuming $k$ is tree path length) based on our method or LIME then,\n\\begin{align}\n\\label{cfi}\n    CFIP_{PP}=\\frac{100}{n}\\sum_i\\frac{|f_{PP}(x_i)\\cap f^*_{PP}(x_i)|_{\\text{card}}}{k}, ~~~\n    CFIP_{PN}=\\frac{100}{n}\\sum_i\\frac{|f_{PN}(x_i)\\cap f^*_{PN}(x_i)|_{\\text{card}}}{k}\n\\end{align}\nwhere, $|.|_{\\text{card}}$ denotes cardinality of the set. Here too higher values for both $CFIP_{PP}$ and $CFIP_{PN}$ are desirable.\n\\vspace{-0.3cm}\n\\begin{table*}[htbp]\n\\centering\n\\caption{Below we see the quantitative results for CCP metric. The statistically significant best results are presented in bold based on paired t-test.}\n \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n  \\hline\n    \\multirow{2}{*}{\\small{Dataset}} & \\multicolumn{2}{c|}{\\small{$CCP_{PP}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CCP_{PN}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CCP_{PP}$ (Forest)}} & \\multicolumn{2}{c|}{\\small{$CCP_{PN}$ (Forest)}} \\\\\n     & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME}  & \\small{MACEM} & \\small{LIME}\\\\\n     \\hline\n     \\small{German Credit} & \\small{\\textbf{100}} & \\small{96.0} & \\small{\\textbf{100}} & \\small{10.2} & \\small{\\textbf{100}} & \\small{89.6} & \\small{\\textbf{100}} & \\small{9.0} \\\\\n \\hline\n    \\small{FICO} & \\small{\\textbf{100}} & \\small{91.45} & \\small{\\textbf{100}}& \\small{49.31} & \\small{\\textbf{100}} & \\small{46.47} & \\small{\\textbf{100}} & \\small{40.82}\\\\\n \\hline\n  \\small{Sky Survey}& \\small{\\textbf{100}} & \\small{58.48}& \\small{\\textbf{100}}& \\small{25.01} & \\small{\\textbf{100}}& \\small{47.32} & \\small{\\textbf{100}} & \\small{39.78}\\\\\n  \\hline\n \\small{Vertibral Column} & \\small{\\textbf{100}} & \\small{33.33} & \\small{\\textbf{100}}& \\small{44.87} & \\small{\\textbf{100}} & \\small{91.02}& \\small{\\textbf{100}}& \\small{19.23}\\\\\n  \\hline\n\\small{Olfaction} & \\small{\\textbf{100}} & \\small{79.22} & \\small{\\textbf{100}}& \\small{13.22} & \\small{\\textbf{100}} & \\small{74.19}& \\small{\\textbf{100}}& \\small{19.21}\\\\\n \\hline\n \\end{tabular}\n \\label{tab:quant}\n\\vspace{-0.1cm}\n\\end{table*}\n\\vspace{-0.3cm}\n\\begin{table*}[htbp]\n\\centering\n\\caption{Below we see the quantitative results for CFR metric. The statistically significant best results are presented in bold based on paired t-test.}\n \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n  \\hline\n    \\multirow{2}{*}{\\small{Dataset}} & \\multicolumn{2}{c|}{\\small{$CFR_{PP}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CFR_{PN}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CFR_{PP}$ (Forest)}} & \\multicolumn{2}{c|}{\\small{$CFR_{PN}$ (Forest)}} \\\\\n     & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME}  & \\small{MACEM} & \\small{LIME}\\\\\n     \\hline\n     \\small{German Credit} & \\small{\\textbf{0.68}} & \\small{0.64} & \\small{\\textbf{0.70}} & \\small{0.68} & \\small{\\textbf{0.43}} & \\small{0.35} & \\small{\\textbf{0.48}} & \\small{\\textbf{0.46}} \\\\\n \\hline\n    \\small{FICO} & \\small{\\textbf{0.68}} & \\small{0.49} & \\small{\\textbf{0.74}}& \\small{0.70} & \\small{\\textbf{0.29}} & \\small{0.09} & \\small{\\textbf{0.54}} & \\small{0.31}\\\\\n \\hline\n  \\small{Sky Survey}& \\small{\\textbf{0.55}} & \\small{0.48}& \\small{\\textbf{0.81}}& \\small{0.53} & \\small{\\textbf{0.42}}& \\small{0.35} & \\small{\\textbf{0.54}} & \\small{0.36}\\\\\n  \\hline\n \\small{Vertibral Column} & \\small{\\textbf{0.63}} & \\small{\\textbf{0.64}} & \\small{\\textbf{0.75}}& \\small{0.66} & \\small{\\textbf{0.20}} & \\small{-0.02}& \\small{\\textbf{0.33}}& \\small{0.23}\\\\\n  \\hline\n\\small{Olfaction} & \\small{\\textbf{0.71}} & \\small{0.58} & \\small{\\textbf{0.78}}& \\small{0.59} & \\small{\\textbf{0.73}} & \\small{0.62}& \\small{\\textbf{0.82}}& \\small{0.65}\\\\\n \\hline\n \\end{tabular}\n \\label{tab:quant2}\n\\vspace{-0.1cm}\n\\end{table*}\n\n\\begin{table*}[htbp]\n\\centering\n\\caption{Below we see the quantitative results for CFIP metric. The statistically significant best results are presented in bold based on paired t-test.}\n \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n  \\hline\n    \\multirow{2}{*}{\\small{Dataset}} & \\multicolumn{2}{c|}{\\small{$CFIP_{PP}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CFIP_{PN}$ (Tree)}} & \\multicolumn{2}{c|}{\\small{$CFIP_{PP}$ (Forest)}} & \\multicolumn{2}{c|}{\\small{$CFIP_{PN}$ (Forest)}} \\\\\n     & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME} & \\small{MACEM} & \\small{LIME}  & \\small{MACEM} & \\small{LIME}\\\\\n     \\hline\n     \\small{German Credit} & \\small{\\textbf{83.73}} & \\small{68.98} & \\small{\\textbf{62.32}} & \\small{48.28} & \\small{\\textbf{71.26}} & \\small{30.65} & \\small{\\textbf{30.13}} & \\small{13.98} \\\\\n \\hline\n    \\small{FICO} & \\small{\\textbf{79.13}} & \\small{78.96} & \\small{\\textbf{82.92}}& \\small{58.76} & \\small{\\textbf{89.92}} & \\small{33.56} & \\small{\\textbf{54.98}} & \\small{33.25}\\\\\n \\hline\n  \\small{Sky Survey}& \\small{\\textbf{98.05}} & \\small{76.55}& \\small{\\textbf{93.52}}& \\small{68.55} & \\small{\\textbf{78.96}}& \\small{\\textbf{80.18}} & \\small{\\textbf{89.54}} & \\small{\\textbf{88.26}}\\\\\n  \\hline\n \\small{Vertibral Column} & \\small{\\textbf{85.95}} & \\small{68.24} & \\small{\\textbf{77.38}}& \\small{64.98} & \\small{\\textbf{94.23}} & \\small{86.12}& \\small{\\textbf{100.00}}& \\small{89.45}\\\\\n  \\hline\n\\small{Olfaction} & \\small{\\textbf{87.19}} & \\small{83.92} & \\small{\\textbf{72.82}}& \\small{47.96} & \\small{\\textbf{82.23}} & \\small{77.56}& \\small{\\textbf{74.72}}& \\small{52.28}\\\\\n \\hline\n \\end{tabular}\n \\label{tab:quant3}\n\\vspace{-0.1cm}\n\\end{table*}\n\n\n\\vspace{-0.2cm}\n\\subsection{Quantitative Evaluation}\n\\vspace{-0.1cm}\nFirst looking at Table \\ref{tab:quant} we observe that our method MACEM as designed, on all datasets produces PPs and PNs that lie in the same or different class as the original input respectively. This is indicated by metrics $CCP_{PP}$ and $CCP_{PN}$ where we are 100\\% accurate. This observation is reassuring as it means that whenever we return a PP or PN for an input it is valid. Although LIME has reasonable performance for PPs (much worse for PNs) no such promise can be made.\n\\begin{wrapfigure}{r}{0.49\\textwidth}\n  \\centering  \n      \\includegraphics[width=0.49\\textwidth]{GermanCreditExample.png}\\\\  \n      \\includegraphics[width=0.49\\textwidth]{OlfactionExample.png}\n      \\vspace{-0.5cm}\n  \\caption{Above we compare the actual tree path (blue arrows) and the corresponding PP\/PN important features for an input in the a) German Credit dataset and b) Olfaction dataset. The PP columns (center) and the PN columns (right) list the top 3 features highlighted by MACEM for the corresponding PPs and PNs. The PP feature importance reduces top to bottom, while the PN feature importance reduces bottom to top. The red arrows indicate PP and PN features that match the features in the tree path for the respective inputs. }\n  \\label{GCeg}\n\\vspace{-1cm}\n\\end{wrapfigure}\nLooking at Table \\ref{tab:quant2} we observe that the feature ranking obtained by our method seems to be more indicative of the true feature importances than LIME, as it correlates better with the ablation based ranking in almost all cases as conveyed by the higher CFR score. We are the best performer in most cases with high correlation values depicting that we accurately capture (relative) feature importances.\n\nWe now look at how efficient the different methods are in picking the correct features for PPs and PNs. This is reflected in the $CFIP_{PP}$ and $CFIP_{PN}$ metrics in Table \\ref{tab:quant3}. We are still better than LIME in both cases. Our PP results are generally better than PN. A reason for this maybe that when we run our algorithm (based on FISTA) most initial movements, especially for inputs in the interior of the decision boundary, will result in not changing the class, so the candidate PNs to choose from eventually (where we output the sparsest) will be much smaller than the candidate PPs for many inputs. This leads to a much richer choice from which to choose the (sparsest) PP resulting in them being more accurate\n\n\\vspace{-0.2cm}\n\\subsection{Qualitative Evaluation}\n\\vspace{-0.1cm}\n\nWe now look at specific inputs from two of the datasets and garner expert feedback relating to the insights conveyed by the models and our explanations for them.\n\nThe smallest change to alter the class will probably occur by changing values of the lower level features in the tree path, which is what we observe. Hence, both PPs and PNs seem to capture complementary information and together seem to be important in providing a holistic explanation.\n\\noindent\\textbf{Comparing with Tree Paths:} In figure \\ref{GCeg}, we see two examples the left from German Credit dataset and the right from Olfaction dataset. In both these examples we compare the top features highlighted by our method with actual tree paths for those inputs. In figure \\ref{GCeg}a, we see that our top-2 PP features coincide with the top-2 features in the tree path. While the bottom-2 features in the tree path correspond to the most and $3^{rd}$ most important PN feature. A qualitatively similar result is seen in figure \\ref{GCeg}b for the olfaction dataset. We observe this for over 80\\% inputs in these datasets. What this suggests is that our PP features tend to capture more global or higher level features, while our PNs capture more detailed information. This makes intuitive sense since for PPs we are searching for the sparsest input that lies in a particular class and although the input (absolute) feature values have to upper bound the PPs feature values, this is a reasonably weak constraint leading to many inputs in a class having the same PPs, thus giving it a global feel. On the other hand, a PN is very much input dependent as we are searching for minimal (addition only) perturbations that will change the class. \n\n\\noindent\\textbf{Human Expert Evaluations:}\nWe asked an expert in Finance and another in Neuroscience to validate our decision trees (so that their intuition is consistent with the model) and explanations. Each expert was given 50 randomly chosen explanations from the test set and they were blinded to which method produced these explanations. They were then given the binary task of categorizing if each explanation was reasonable or not.\n\nThe expert in Finance considered the features selected by our decision tree to be predictive. He felt that 41 of our PPs made sense, while around 34 PPs from LIME were reasonable. Regarding PNs, he thought 39 of ours would really constitute a class change, while 15 from LIME had any such indication. An expert from Neuroscience who works on olfaction also considered our tree model to be reasonable as the features it selected from the Dragon input features \\cite{olfs} had relations to molecular size, paclitaxel and citronellyl phenylacetate, which are indicative of pleasantness. In this case, 43 of our PPs and 34 of LIMEs were reasonable. Again there was a big gap in PN quality, where he considered 40 of our PNs to be reasonable versus 18 of LIMEs. For random forests the numbers were qualitatively similar. For the finance dataset, 44 of ours and 27 of LIMEs PPs made sense to the expert. While 38 of our PNs and 19 of LIMEs were reasonable. For the olfaction dataset, 41 of our PPs and 32 of LIMEs were reasonable, while 39 of our PNs and 20 of LIMEs made sense.\n\n\\vspace{-0.4cm}\n\\section{Discussion}\n\\vspace{-0.2cm}\nIn this paper we provided a model agnostic black box contrastive explanation method specifically tailored for structured data that is able to handle real as well as categorical features in a meaningful and scalable manner. We saw that our method is quantitatively as well as qualitatively superior to LIME and provides more complete explanations.\n\nIn the future, we would like to extend our approach here to be applicable to also unstructured data. In a certain sense, the current approach could be applied to such data if it is already vectorized or the text is embedded in a feature space. In such cases, although minimally changing a sentence to another lying in a different class would be hard, one could identify important words or phrases which a language model or human could use as the basis for creating a valid sentence\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Voronoi diagram is one of the most useful representations for tessellations in computational geometry. The fundamental concepts, generalizations, and applications have been studied widely, e.g., \\cite{BAurenhammer,BOkabe}. Here we are concerned with weighted Voronoi diagrams called Laguerre Voronoi diagrams, which are also known as a power diagrams. This class of Voronoi diagrams are important because the boundaries are linear instead of curved.\n\nThe Laguerre Voronoi diagram was introduced by \\cite{Imi,Aurenhammer}. Briefly, for a set $S$ of $n$ spheres $s_i=(\\textbf{x}_i, r_i)$ in $\\mathbb{R}^d$, where $\\textbf{x}_i$ is the position of the center and $r_i$ is the radius, which is equivalent to the generator weight, the Laguerre distance of $\\textbf{x}\\in \\mathbb{R}^d$ from $s_i$ is \n\\begin{equation}\\label{LagDisOr}\nd_{\\text{L}}(\\textbf{x},s_i)=\\|\\textbf{x}-\\textbf{x}_i\\|^2-r_i^2.\n\\end{equation}\nIn \\cite{Aurenhammer2,Aurenhammer3} the correspondence between the Laguerre Voronoi diagram in $\\mathbb{R}^d$ and a polyhedron in $\\mathbb{R}^{d+1}$ was investigated. Then using the correspondence, an algorithm for constructing the Laguerre Voronoi diagram was presented in \\cite{Imi,Aurenhammer}, and a robust version was presented in \\cite{Sugihara1}. The Laguerre Voronoi diagram can be extended for tessellations on a sphere as defined in \\cite{Sugihara2}. Applications of the spherical Laguerre Voronoi diagram were also studied in \\cite{Mach,Chaidee3} for modeling objects with spheres.\n\nIt is sometimes necessary to consider the inverse of the above problem: the determination of whether or not a given tessellation is the Voronoi diagram. If it is, we can recover the corresponding generators. This problem is known as the Voronoi recognition problem, and was studied in \\cite{BLoeb,JAsh,JAurenhammer,JHartvigsen,JSchoenberg,JAloupis}. On the other hand, if the tessellation cannot be represented by a Voronoi diagram,  we approximate it with the best fitting Voronoi diagram; this is called the Voronoi approximation problem. Examples of the Voronoi approximation problems are found in \\cite{JHonda1,JHonda2,JSuzuki,JEvan,Chaidee1}, for unweighted Voronoi diagrams. These approximations have a number of useful applications, for instance, if we have a tessellation that is found in the real world, and we can approximate the tessellation with a Voronoi diagram, we can use this Voronoi diagram as a model for understanding the pattern's formation.\n\nThe inverse problems were widely studied for the case of ordinary Voronoi diagrams, whereas relatively little work has been done for Laguerre diagrams. Duan et al. \\cite{Duan} considered the inverse problem for the planar Laguerre Voronoi diagram, and presented an algorithm for recovering the generators and their weights from a tessellation. Lautensack \\cite{Lau} and Lyckegaard et al. \\cite{Lyckegaard} used Laguerre Voronoi diagrams to study the relation between structures and their physical properties. Recently, Spettl et al. \\cite{Duan2} fitted Laguerre Voronoi diagrams to tomographic image data. In the case of the spherical Laguerre Voronoi diagram, Chaidee and Sugihara \\cite{Chaidee3} provided a framework for approximating the weights of spherical Laguerre Voronoi diagrams when the location of the Voronoi generators are known.\n\nIn this paper, we focus on the spherical Laguerre Voronoi diagram recognition problem. Our goal is to judge whether or not a given spherical tessellation is a spherical Laguerre Voronoi diagram. Remark that in the case of the ordinary Voronoi diagram recognition problem, the generator positions are unique whereas there are many sets of the generating circles generating the same Laguerre Voronoi diagram. With this reason, the Laguerre Voronoi diagram recognition problem is more difficult than the ordinary Voronoi diagram recognition problem. By the nonuniqueness property of generating circles, for each spherical Laguerre Voronoi diagram, there is a class of polyhedra whose projections coincide with the spherical Laguerre Voronoi diagram. Therefore, for a given tessellation, if we find these polyhedra, we can judge it is a  spherical Laguerre Voronoi diagram, and can recover the generators and their weights. Otherwise, we judge that the given tessellation is not a spherical Laguerre Voronoi diagram.\n\nThis paper is organized as follows. In Section 2, we provide the definitions and theorems which are necessary for our study. The recognition problem is also mathematically defined in this section. In Section 3, we focus on the properties of projective transformations which transform a polyhedron associated with a spherical Laguerre Voronoi diagram to other polyhedra in the same class. In Section 4, we give algorithms for constructing a polyhedron from a given tessellation and for judging whether a given tessellation is a spherical Laguerre Voronoi diagram. Finally, we summarize our research in Section 5, and give suggestions for future work.\n\n\n\\section{Preliminaries}\n\nIn this section, we provide the necessary definitions and theories on the spherical Laguerre Voronoi diagram. Then we state the problem considered in this paper.\n\nWe first present some fundamental definitions from spherical geometry.\n\nLet $U$ be a unit sphere whose center is located at the origin $O(0, 0, 0)$ of a Cartesian coordinate system. \n\nFor two distinct points $p, p_i \\in U$ with position vectors $\\textbf{x}, \\textbf{x}_i$, respectively, the \\textit{geodesic arc} $\\wideparen{e}_{p,p_i}$ of $p, p_i$ is defined as the shortest arc between $p$ and $p_i$ of the great circle passing through $p$ and $p_i$. The \\textit{geodesic arc length}, also called the \\textit{geodesic distance}, is given as follows,\n\\begin{equation}\n\\tilde{d}(p, p_i) = \\arccos(\\textbf{x}^{\\text{T}}\\textbf{x}_i)\\leq \\pi. \\label{GeoDist}\n\\end{equation}\n\nIn this paper, we focus on spherical tessellations. We define the spherical polygon as follows.\n\nLet $(q_1, ..., q_m)$ be a sequence of distinct vertices on the sphere such that the geodesic arcs $\\wideparen{e}_{q_i,q_i+1}$ ($i=1, ..., m$; $q_{m+1}$ is read as $q_1$) do not intersect except at the vertices. The left area enclosed by the collection of these geodesic arcs is called the \\textit{spherical polygon} $Q(q_1, ..., q_m)$. $Q(q_1, ..., q_m)$ is abbreviated as $Q$ hereafter.\n\nThe spherical polygon $Q$ is said to be \\textit{convex} if and only if no geodesic arc joining the two points in $Q$ goes outside of $Q$.\n\n$\\mathcal{T}$ is said to be a \\textit{spherical tessellation} if $\\mathcal{T}$ is a decomposition of $U$ into a countable number of spherical polygons whose interiors are pairwise disjoint. $\\mathcal{T}$ is said to be \\textit{convex} if all of these spherical polygons are convex.\n\nNext, we consider a special case of a spherical tessellation. Let $G=\\{p_1, ..., p_n\\}$ be a set of points on the sphere $U$. Assignment of each point on $U$ to the nearest point in $G$ with respect to the geodesic distance forms a tessellation, which is called the spherical Voronoi diagram on the sphere $U$. Algorithms for constructing the spherical Voronoi diagram were provided in \\cite{Renka,Sugihara2}.\n\nWe can generalize the spherical Voronoi diagram to the spherical Laguerre Voronoi diagram. The necessary definitions and theorems were originally presented in \\cite{Sugihara1,Sugihara2}. We briefly introduce those definitions and theorems as follows.\n\nFor a given sphere $U$ and point $p_i \\in G$, the spherical circle $\\tilde{c}_i$ centered at $p_i$ is defined by\n\\begin{equation}\\label{SC}\n\\tilde{c}_i=\\{p\\in U|\\tilde{d}(p_i, p)=r_i\\},\n\\end{equation}\nwhere $0\\leq r_i <\\pi\/2$. $r_i$ is called the radius of the spherical circle $\\tilde{c}_i$.\n\nFollowing \\cite{Sugihara2}, we define the Laguerre proximity, the distance measured from an arbitrary point $p$ on the sphere $U$ to a spherical circle  $\\tilde{c}_i$, as follows,\n\\begin{equation}\\label{RealCircle}\n\\tilde{d}_L(p, \\tilde{c}_i)=\\frac{\\cos \\left(\\tilde{d}(p_i, p)\\right)}{\\cos \\left(r_i\\right)}.\n\\end{equation}\nLet $\\tilde{c}_i$ and $\\tilde{c}_j$ be two circles. The Laguerre bisector of $\\tilde{c}_i$ and $\\tilde{c}_j$ is defined by\n\\begin{equation}\nB_L(\\tilde{c}_i, \\tilde{c}_j)=\\{p\\in U|\\tilde{d}_L(p, \\tilde{c}_i)=\\tilde{d}_L(p, \\tilde{c}_j)\\}. \\label{LagBi}\n\\end{equation}\nFor a set $\\tilde{G}=\\{\\tilde{c}_1, ..., \\tilde{c}_n\\}$ of $n$ spherical circles on $U$, we define the region\n\\begin{equation}\n\\tilde{R}(\\tilde{G}, \\tilde{c}_i)=\\{p\\in U|\\tilde{d}_L(p, \\tilde{c}_i)<\\tilde{d}_L(p, \\tilde{c}_j), j\\neq i\\}.\n\\end{equation}\nThe regions $\\tilde{R}(\\tilde{G}, \\tilde{c}_1), ..., \\tilde{R}(\\tilde{G}, \\tilde{c}_n)$, together with their boundaries constitute a tessellation, which is called the spherical Laguerre Voronoi diagram of $U$. Figure \\ref{SLVD} shows an example of the spherical Laguerre Voronoi diagram. This is a stereo diagram. If the right diagram is viewed with the left eye and the left diagram with the right eye, the sphere can be seen; the upper pair represents the front hemisphere and the lower pair represents the rear hemisphere.\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.7]{slv50}\n\t\t\\caption{Stereographic images of a spherical Laguerre Voronoi diagram.} \\label{SLVD}\n\t\\end{center}\n\\end{figure}\n\nIn \\cite{Sugihara1,Sugihara2}, an algorithm for constructing the spherical Laguerre Voronoi diagram and its dual, i.e., the spherical Laguerre Delaunay diagram, was proposed. For a spherical circle  $\\tilde{c}_i$ of $U$, let $\\pi(\\tilde{c}_i)$ be the plane passing through $\\tilde{c}_i$, and let $H(\\tilde{c}_i)$ be the halfspace bounded by $\\pi(\\tilde{c}_i)$ and containing $O$. Let $\\ell_{i,j}$ be the line of intersection of $\\pi(\\tilde{c}_i)$ and $\\pi(\\tilde{c}_j)$.\n\nThe Laguerre bisector shown in (\\ref{LagBi}) is characterized by the following theorems.\n\n\\begin{theorem}[\\cite{Sugihara2}]\\label{SugiThm1}\n\tThe Laguerre bisector $B_L(\\tilde{c}_i,\\tilde{c}_j)$ is a great circle, and it crosses the geodesic arc connecting the two centers $p_i$ and $p_j$ at right angles.\n\\end{theorem}\n\n\\begin{theorem}[\\cite{Sugihara2}]\\label{SugiThm2}\n\tThe bisector $B_L(\\tilde{c}_i,\\tilde{c}_j)$ is the intersection of $U$ and the plane containing $\\ell_{i,j}$ and $O$.\n\\end{theorem}\n\nFor a set $\\tilde{G}$ of spherical circles, the spherical Laguerre Voronoi diagram is constructed by the following process. For circles $\\tilde{c}_i$, we construct the planes $\\pi(\\tilde{c}_i)$ and the half spaces $H(\\tilde{c}_i)$, and the intersection of all halfspaces. We finally project the edge of the resulting polyhedron onto $U$, with the center of the projection at $O$. Then we have the spherical Laguerre Voronoi diagram.\n\nThroughout this paper, the term \\textit{spherical polygon} is abbreviated as \\textit{polygon}. Let $\\mathcal{T}$ be the given convex spherical tessellation on the unit sphere $U$ having $n$ polygons. We assume that all vertices of $\\mathcal{T}$ are of degree 3.\n\nThe main concern of this paper is to judge whether or not a given tessellation $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram.\n\n\\section{Polyhedron Transformation}\n\nFrom the definition and theorems of the spherical Laguerre Voronoi diagram in the previous section, the following proposition state the correspondence between the spherical Laguerre Voronoi diagram and a polyhedron.\n\\begin{proposition}\\label{twoWay}\n\t$\\mathcal{L}$ is a spherical Laguerre Voronoi diagram if and only if there is a convex polyhedron $\\mathcal{P}$ containing the center of the sphere whose central projection coincides with $\\mathcal{L}$.\n\\end{proposition}\n\\begin{proof}\n\tWe firstly prove the necessity of the condition. Let $\\mathcal{L}$ be a spherical Laguerre Voronoi diagram. Hence, the generating circles exist. Therefore, we get the intersection of halfspaces with boundary planes passing through the spherical circles including the center of the sphere. The projection of this polyhedron onto $U$ coincides with $\\mathcal{L}$ because of Theorem \\ref{SugiThm2}.\n\t\n\tInversely, let $\\mathcal{L}$ be a given spherical tessellation and $\\mathcal{P}$ be a convex polyhedron containing the center of the sphere such that its central projection coincides with $\\mathcal{L}$. If there exists a plane $P_i$ such that $P_i$ does not intersect $U$, shrink the polyhedron $\\mathcal{P}$ so that $P_i$ intersects $U$ for all $i$. Let $\\tilde{c}_i$ be the intersection of the plane containing the face $P_i$ with $U$ for all $i$. Then, by Theorem \\ref{SugiThm2}, $\\mathcal{L}$ is the spherical Laguerre Voronoi diagram for $\\tilde{c}_i$'s as the generators.\n\t\n\\end{proof}\n\nTherefore, for each spherical Laguerre Voronoi diagram $\\mathcal{L}$, there is a class of polyhedra whose central projections coincide with $\\mathcal{L}$. To solve the spherical Laguerre Voronoi recognition problem, we will construct an algorithm to find those polyhedra.\n\nTo find these polyhedra, we study the transformation of the polyhedron that preserves the projection onto the sphere $U$. Let $P^3(\\mathbb{R})$ be the three-dimensional projective space. The following properties are the requirements for the transformation.\n\n\\begin{definition}(Projection Preservation Property)\\label{PPP}\n\tLet $f$ be a transformation from $P^3(\\mathbb{R})$ to $P^3(\\mathbb{R})$. $f$ is said to be the \\textit{projection preserving mapping with respect to the origin $O$} if $f$ satisfies the following properties:\n\t\\begin{enumerate}\n\t\t\\item $f(O)=O$;\n\t\t\\item For any point $v\\in P^3(\\mathbb{R})$, $v$ and $f(v)$ are on the same line passing through $O$.\n\t\\end{enumerate}\n\\end{definition}\n\nLet $\\textbf{v}_a=(t_a, x_a, y_a, z_a) \\in P^3(\\mathbb{R})$ be a homogeneous coordinate representation of a vertex of the polyhedron $\\mathcal{P}$ in the projective space. We define a map $f:P^3(\\mathbb{R})\\rightarrow P^3(\\mathbb{R})$ by\n\\begin{equation}\\label{map}f(\\textbf{v}_a)=\\begin{pmatrix}\n\\alpha & \\beta & \\gamma & \\delta \\\\ \n0 & \\eta & 0 & 0 \\\\ \n0 & 0 & \\eta & 0 \\\\ \n0 & 0 & 0 & \\eta\n\\end{pmatrix}\\textbf{v}_a \\end{equation}\nwhere $\\alpha, \\beta, \\gamma, \\delta, \\eta \\in \\mathbb{R}$ and $\\alpha\\neq 0, \\eta\\neq 0$.\n\n\\begin{theorem}\\label{transMapThm}\n\tThe mapping $f$ defined by equation (\\ref{map}) is a projection preserving mapping.\n\\end{theorem}\n\\begin{proof}\n\t\n\tWe now prove that the map $f$ satisfies the conditions in Definition \\ref{PPP}. It is easy to verify that $f(O)=O$. It remains to be shown that $f(\\textbf{v}_a)=\\textbf{v}_a'$, where $\\textbf{v}_a'$ lies on the line passing through the origin $L_a$ of $\\textbf{v}_a$.\n\t\n\tFor $\\textbf{v}_a=(t_a, x_a, y_a, z_a) \\in P^3(\\mathbb{R})$, we have\n\t\\begin{equation}\n\tf(\\textbf{v}_a)=\\begin{pmatrix}\n\t\\alpha & \\beta & \\gamma & \\delta \\\\ \n\t0 & \\eta & 0 & 0 \\\\ \n\t0 & 0 & \\eta & 0 \\\\ \n\t0 & 0 & 0 & \\eta\n\t\\end{pmatrix}\\begin{pmatrix}\n\tt_a \\\\ \n\tx_a \\\\ \n\ty_a \\\\ \n\tz_a\n\t\\end{pmatrix} = \\begin{pmatrix}\n\t\\varLambda \\\\ \n\t\\eta x_a \\\\ \n\t\\eta y_a \\\\ \n\t\\eta z_a\n\t\\end{pmatrix} \\label{transMap}\n\t\\end{equation}\n\twhere $\\varLambda=\\alpha t_a + \\beta x_a + \\gamma y_a + \\delta z_a$.\n\t\n\tNote that for the transformed point $\\textbf{v}_a'=(\\varLambda, \\eta x_a, \\eta y_a, \\eta z_a)$ in the projective space $P^3(\\mathbb{R})$, \n\t$$\\left(\\frac{\\eta x_a}{\\varLambda}, \\frac{\\eta y_a}{\\varLambda}, \\frac{\\eta z_a}{\\varLambda}\\right)= \\frac{\\eta t_a}{\\varLambda}\\left(\\frac{x_a}{t_a}, \\frac{y_a}{t_a}, \\frac{z_a}{t_a}\\right)$$\n\tis a point in the space $\\mathbb{R}^3$ which implies that $\\textbf{v}_a'$ lies on the same line as $\\textbf{v}_a$, which concludes the proof.\n\\end{proof}\n\nSince the transformation is a projective map, it preserves the planarity of faces of a polyhedron. Also, by the reason that $f$ satisfies the projection preservation property, the transformed point $f(v)$ is on the same line as $v$ passing through $O$. However, the transformation as defined in (\\ref{map}) does not guarantee the convexity of the projected polyhedron because some vertices may be mapped to the other side of the origin. Note that if $\\alpha=\\eta=1$ and $\\beta=\\gamma=\\delta=0$, the map $f$ is an identity map. The set of the transformations of the form (\\ref{map}) form a continuous group, and hence for each convex polyhedron $\\mathcal{P}$, there exists a positive constant $\\epsilon$ such that for any $-\\epsilon \\leq \\alpha-1, \\eta-1, \\beta, \\gamma, \\delta \\leq \\epsilon$, the transformation (\\ref{map}) maps $\\mathcal{P}$ to a convex polyhedron.\n\nNote that the transformation is uniquely determined if we fix the five parameters $\\alpha, \\beta, \\gamma, \\delta$, and $\\eta$. However, the homogeneous coordinate representation itself has one degree of freedom to represent each point. Hence, the choice of the polyhedron transformed from $\\mathcal{P}$ by (\\ref{map}) has four degrees of freedom.\n\n\\section{Algorithms}\n\n\\subsection{Recognition procedure}\n\nIn the previous section, the existence of the class of polyhedra whose projections coincide with the given spherical Laguerre Voronoi diagram is proved by Theorem \\ref{transMapThm}. In this section, we propose an algorithm for constructing a polyhedron with respect to a given tessellation.  Let $\\mathcal{T}$ be a given spherical tessellation. If a polygon $i$ is adjacent to a polygon $j$, then there exists a geodesic arc $\\wideparen{e}_{i,j}$ which is a tessellation edge partitioning polygons $i$ and $j$. The tessellation vertex that is the intersection of edges $\\wideparen{e}_{i,j}$, $\\wideparen{e}_{j,k}$, and $\\wideparen{e}_{i,k}$ is denoted by $v_{i,j,k}$, as shown in Figure \\ref{polygons}. During the algorithm, $\\ell_{i,j}$ is defined as the line intersecting $\\pi(\\tilde{c}_i)$ and $\\pi(\\tilde{c}_j)$.\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\t\\definecolor{uququq}{rgb}{0.25098039215686274,0.25098039215686274,0.25098039215686274}\n\t\t\\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]\n\t\t\\clip(2.8160640470324694,3.5067554088250974) rectangle (9.718902215139163,6.965008986114204);\n\t\t\\draw [shift={(6.342008502719588,3.617125273275333)}] plot[domain=0.035538821603734644:3.1108262356690495,variable=\\t]({1.*3.168612589062205*cos(\\t r)+0.*3.168612589062205*sin(\\t r)},{0.*3.168612589062205*cos(\\t r)+1.*3.168612589062205*sin(\\t r)});\n\t\t\\draw [shift={(11.743458897258435,5.290836630761076)}] plot[domain=2.8644792148819374:3.229205627611263,variable=\\t]({1.*5.449456835313343*cos(\\t r)+0.*5.449456835313343*sin(\\t r)},{0.*5.449456835313343*cos(\\t r)+1.*5.449456835313343*sin(\\t r)});\n\t\t\\draw [shift={(8.476872026504891,7.202718578526532)}] plot[domain=3.9767765086173488:4.666125717960333,variable=\\t]({1.*3.2218106032764178*cos(\\t r)+0.*3.2218106032764178*sin(\\t r)},{0.*3.2218106032764178*cos(\\t r)+1.*3.2218106032764178*sin(\\t r)});\n\t\t\\draw [shift={(5.341385632169549,8.590198620619178)}] plot[domain=4.419859512679194:4.9646990170726895,variable=\\t]({1.*3.902136057996358*cos(\\t r)+0.*3.902136057996358*sin(\\t r)},{0.*3.902136057996358*cos(\\t r)+1.*3.902136057996358*sin(\\t r)});\n\t\t\\draw (5.814606881363096,4.746316083031945) node[anchor=north west] {$v_{i,j,k}$};\n\t\t\\draw (6.479352421394042,6.623284324326742) node[anchor=north west] {$\\wideparen{e}_{i,j}$};\n\t\t\\draw (4.497349383026773,6.021848919580811) node[anchor=north west] {$\\text{polygon }i$};\n\t\t\\draw (7.268554244414403,5.447751487777876) node[anchor=north west] {$\\text{polygon }j$};\n\t\t\\draw (5.659213233104137,4.067183854156534) node[anchor=north west] {$\\text{polygon }k$};\n\t\t\\draw (4.647708234213256,5.14703378540491) node[anchor=north west] {$\\wideparen{e}_{i,k}$};\n\t\t\\draw (7.559202352642416,4.5182604077159825) node[anchor=north west] {$\\wideparen{e}_{j,k}$};\n\t\t\\begin{scriptsize}\n\t\t\\draw [fill=uququq] (6.314903792454812,4.81400408777157) circle (1.5pt);\n\t\t\\end{scriptsize}\n\t\t\\end{tikzpicture}\n\t\t\\caption{Three adjacent polygons corresponding to a tessellation vertex.} \\label{polygons}\n\t\\end{center}\n\\end{figure}\n\nThe overview of the recognition procedure is to try to construct a set of planes composing a polyhedron whose central projection coincides with $\\mathcal{T}$. The planes are chosen sequentially. Suppose that the chosen sequence of the first three planes of polygons is $(i, j, k)$. The choice of the first two planes of polygons $i, j$ has the degrees of freedom, while the third plane of the polygon $k$ can be constructed uniquely up to the first two planes of polygons $i, j$.\n\nThe following algorithm is for the construction of the first three planes of the polyhedron corresponding to the three polygons $i, j, k$.\n\\\\\n\\\\\n\\textbf{Algorithm 1: Plane Construction with Three Adjacent Sites}\n\\\\ \\textbf{Input:} Tessellation edges $\\wideparen{e}_{i,j}$, $\\wideparen{e}_{j,k}$, $\\wideparen{e}_{i,k}$, and degree-three tessellation vertex $v_{i,j,k}$.\n\\\\ \\textbf{Output:} The three planes $P_i, P_j, P_k$ with respect to polygons $i, j, k$.\n\\\\ \\textbf{Procedure:} \n\\begin{enumerate}\n\t\\item select a spherical circle  $\\tilde{c}_i$ with center $p_i\\in U$ and radius $r_i$ in polygon $i$.\n\t\\item construct a plane $P_i:=\\pi(\\tilde{c}_i)$ of the spherical circle  $\\tilde{c}_i$;\n\t\\item construct a plane $P_{i,j}$, $P_{i, k}$, $P_{j,k}$ passing through $\\wideparen{e}_{i,j}$, $\\wideparen{e}_{i,k}$, and $\\wideparen{e}_{j,k}$, respectively;\n\t\\item find the line $\\ell_{i,j}$ by intersecting $P_i$ and $P_{i,j}$, and $\\ell_{i,k}$ by intersecting $P_i$ and $P_{i,k}$;\n\t\\item construct a geodesic arc  $\\wideparen{e}_{i,j}^c$ such that $\\wideparen{e}_{i,j}^c$ passes through $p_i$ and is perpendicular to  $\\wideparen{e}_{i,j}$;\n\t\\item choose a point $q_j$ in polygon $j$ on the arc $\\wideparen{e}_{i,j}^c$;\n\t\\item construct the plane $P_j$ passing through $\\ell_{i,j}$ and $q_j$;\n\t\\item find the line $\\ell_{j,k}$ by intersecting the planes $P_j$ and $P_{j,k}$;\n\t\\item construct the plane $P_k$ passing through the line $\\ell_{i,k}$ and $\\ell_{j,k}$.\n\\end{enumerate}\n\\textbf{end Procedure}\n\\\\\n\\\\\nIn step 8 of Algorithm 1, $\\ell_{j,k}$ is constructed from the intersection of $P_j$ and $P_{j,k}$ which is used in step 9 of the Algorithm 1. The following lemma guarantees the co-planarity of $\\ell_{i,k}$ and $\\ell_{j,k}$.\n\n\\begin{lemma}\n\t$\\ell_{i,k}$ generated in step 4 of Algorithm 1 and $\\ell_{j,k}$ generated in step 8 are co-planar.\n\\end{lemma}\n\\begin{proof}\n\tFrom Algorithm 1, suppose that the construction sequence is $(i, j, k)$. Then there exists an intersection between planes $P_i$ and $P_{i,k}$ and $P_{i, j}$, written as $\\ell_{i,k}$ and $\\ell_{i,j}$, respectively which are obviously coplanar. Since $P_{i, k}$ and $P_{i, j}$ pass through $\\wideparen{e}_{i,k}$ and $\\wideparen{e}_{i,j}$ such that $\\wideparen{e}_{i,k}$ intersects $\\wideparen{e}_{i,k}$ at $v_{i, j, k}$, there exists a line $\\ell_{i, j, k}$ which is the intersection of $P_{i, k}$ and $P_{i,j}$. Therefore, there exists the intersection point of $\\ell_{i, k}, \\ell_{i,j}$ and $\\ell_{i,j,k}$, says $V_{i,j,k}$.\n\t\n\tBy step 7 of Algorithm 1, the plane $P_j$ is constructed through $\\ell_{i,j}$, and also $V_{i, j, k}$. Then $\\ell_{j, k}$ is constructed through the intersection of $P_j$ and $P_{j, k}$. Hence, $V_{i, j, k}$ is laid in the line $\\ell_{j, k}$. Since $V_{i,j,k}\\in \\ell_{i,k}$ and $V_{i,j,k}\\in \\ell_{j,k}$, there exists the unique plane passing through $\\ell_{i,k}$ and $\\ell_{j,k}$ which implies that $\\ell_{i,k}$ and $\\ell_{j,k}$ are co-planar. \n\\end{proof}\n\n\nWe next extend this process to construct all planes $P_1, ..., P_n$ corresponding to the polygons $1, ..., n$ of the given tessellation. \n\nLet $\\mathcal{V}_i$ be the set of vertices of the $i$-th spherical polygon. Note that $\\mathcal{V}_i$ is written as $\\mathcal{V}_i=\\{v_{i,j_1,k_1},...,v_{i,j_m,k_m}\\}$ where $m$ is the number of vertices of the $i$-th spherical polygon. The set of spherical tessellation vertices is denoted by $\\mathcal{V}=\\cup_{i=1}^n \\mathcal{V}_i$. \n\\\\\n\\\\\n\\textbf{Algorithm 2: Construction of $n$ planes }\n\\\\ \\textbf{Input:} Spherical tessellation $\\mathcal{T}$ where all vertices are of degree 3, and the set $\\mathcal{V}$ of tessellation vertices.\n\\\\ \\textbf{Output:} The planes $P_1, ..., P_n$ with respect to the polygons $1, ..., n$, and $\\text{Mark}(v)\\in \\{0, 1\\}$ for $v\\in \\mathcal{V}$.\n\\\\ \\textbf{Comment:} $\\mathbb{P}$ is the set of planes constructed in the  procedure.\n\\\\ \\textbf{Procedure:} \n\\begin{enumerate}\n\t\\item make $\\mathbb{P}$ empty;\n\t\\item set $\\text{Mark}(v)=0$ for all $v \\in \\mathcal{V}$;\n\t\\item choose an arbitrary vertex $v_{i, j, k}\\in \\mathcal{V}$ and employ Algorithm 1 to construct planes $P_i, P_j, P_k$;\n\t\\item add the planes $P_i, P_j, P_k$ to $\\mathbb{P}$;\n\t\\item set $\\text{Mark}(v_{i, j, k})=1$;\n\t\\item \\textbf{while} there exists vertex $v_{p,q,l}\\in\\mathcal{V}$ such that $\\text{Mark}(v_{p,q,l})=0$ and exactly two planes $P_p, P_q$ are included in $\\mathbb{P}$,\\\\\n\t\\textbf{do} \\\\\n\t\\text{\\hspace{0.4cm}} apply steps 3, 4 of Algorithm 1, where $(i, j, k)$ are read as $(p, q, l)$, to find $\\ell_{p,q}$ and $\\ell_{p,l}$;\\\\\n\t\\text{\\hspace{0.4cm}} compute $\\ell_{q,l}$ from the intersection of $P_q$ and $P_{q,l}$;\\\\\n\t\\text{\\hspace{0.4cm}} compute $\\ell_{p, q, l}$ from the intersection of $P_{q, l}$ and  $P_{p, l}$;\\\\\n\t\\text{\\hspace{0.4cm}} compute $V_{p, q, l}$ from the intersection of $P_p$ and $\\ell_{p, q, l}$;\\\\\n\t\\text{\\hspace{0.4cm}} choose a point $v'_{q, l}$ on the line $\\ell_{q, l}$;\\\\\n\t\\text{\\hspace{0.4cm}} construct a plane $P_l$ from the point $v'_{q, l}, V_{p, q, l}$ and line $\\ell_{p,l}$.\\\\\n\t\\text{\\hspace{0.4cm}} add $P_l$ to $\\mathbb{P}$;\\\\\n\t\\text{\\hspace{0.4cm}} set $\\text{Mark}(v_{p, q, l})=1$;\\\\\n\t\\textbf{end while}\n\\end{enumerate}\n\\textbf{end Procedure}\n\\\\\n\\\\\nIn step 6 of Algorithm 2, for any arbitrary planes $P_p, P_q\\in \\mathbb{P}$ to construct a plane $P_l$, the choice of the point $v'_{q, l}$ from the line $\\ell_{q, l}$ affects the uniqueness of the plane. That is, it takes a degree of freedom for each $v'_{q,l}$. However, if the given tessellation $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram, then a point $v'_{q, l}$ in step 6 of Algorithm 2 is arbitrarily chosen to obtain the unique plane $P_l$ passing through $\\ell_{p, l}$ and $\\ell_{q, l}$, which means that there is no degree of freedom in the choice of $v'_{q,l}$. For that purpose, we claim that for any point $v'_{q, l}$ on the line $\\ell_{q, l}$ and $\\ell_{p, l}$ are co-planar, which is proved by the following lemma.\n\n\\begin{lemma}\\label{Coplanar2}\n\tIf $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram, then the plane $P_l$ of the construction sequence $(p, q, l)$ constructed in the step 6 of Algorithm 2 does not depend on the choices of $v'_{q,l}$.\n\\end{lemma}\n\\begin{proof}\n\tLet $\\mathcal{T}$ be a spherical Laguerre Voronoi diagram. We prove $V_{p,q,l}\\in\\ell_{q,l}$ to imply that any choices of $v'_{q,l}$ on $\\ell_{q,l}$ gives the same plane $P_l$ passing through $V_{p,q,l}, \\ell_{p,l}$ and $\\ell_{q,l}$.\n\t\n\tIn the step 6 of Algorithm 2, suppose that $v_{p, q, l}$ is the tessellation vertex of the adjacent polygons $p, q, l$. Suppose that there are planes $P_p, P_q \\in \\mathbb{P}$. By Theorem \\ref{SugiThm2}, $\\ell_{p,q}$ of the intersection of $P_p$ and $P_q$ is laid on the plane $P_{p,q}$ passing through the geodesic arc $\\wideparen{e}_{p,q}$. Note that the line $\\ell_{p,q,l}$ of the intersection of planes $P_{p, q}$ and $P_{q, l}$ is also included in $P_{p, l}$, and the intersection $V_{p, q, l}$ of $P_p$ and $\\ell_{p, q, l}$ is on the lines $\\ell_{p, q}$ and $\\ell_{p, l}$.\n\t\n\tUsing the fact that $\\ell_{p,q}$ lays on the plane $P_{p,q}$ of $\\wideparen{e}_{p,q}$ and Theorem \\ref{SugiThm2}, it is implied that $\\ell_{p,l}$ ,the intersection of $P_p$ and $P_{p,l}$, and $\\ell_{q,l}$, the intersection of $P_q$ and $P_{q,l}$, intersect at $V_{p,q,l}$, which means that $V_{p,q,l}\\in\\ell_{q,l}$. Therefore, there exist a unique plane $P_l$ independent from the choice of $v'_{q,l}$.\n\\end{proof}\n\nIn Algorithm 1, we arbitrarily choose the first spherical circle  $\\tilde{c}_i$ (i.e., with generator position $p_i$ and radius $r_i$), and the generator position with the choice of $q_j$ of the adjacent polygon $j$ which lies on the geodesic arc $\\wideparen{e}_{i,j}^c$. This means that we have four degrees of freedom pursuing in Algorithm 1. This reflects the freedom in the choice of the polyhedron for representing the spherical Laguerre Voronoi diagram as shown in the following theorem.\n\n\\begin{theorem}\\label{4degs}\n\tThere are exactly four degrees of freedom in the choice of a polyhedron $\\mathcal{P}$ with respect to the given spherical Laguerre Voronoi diagram.\n\\end{theorem}\n\n\\begin{proof}\n\tTo prove this theorem, we will derive the lower and upper bounds for the degrees of freedom in the choice of the polyhedron.\n\t\n\tThe lower bound is four as we have seen in Theorem \\ref{transMapThm} and the discussion immediately after the theorem.\n\t\n\tFor the upper bound, we show that if we have two polyhedra $\\mathcal{P}$ and $\\mathcal{P}'$ whose projections give the same spherical Laguerre Voronoi diagram, $\\mathcal{P}$ is transformed to $\\mathcal{P}'$ by the transformation (\\ref{map}).\n\t\n\tLet $\\mathcal{P}$ and $\\mathcal{P}'$ be two different polyhedra whose central projections give the same spherical Laguerre Voronoi diagram. Assume that $P_i$ and $P_j$ are planes containing mutually adjacent faces $F_i, F_j$ of $\\mathcal{P}$. Then there are vertices $v_{i,j,k_1}, v_{i,j,k_2}\\in F_i \\cap F_j$ for some $k_1, k_2$. Without loss of generality, we choose vertices $v_{i, m_1, m_2}\\in F_i$ and $v_{j, n_1, n_2}\\in F_j$ of the polyhedron $\\mathcal{P}$. Note that the remaining polyhedron vertices are uniquely constructed by Algorithm 2 and Lemma \\ref{Coplanar2}.\n\t\n\tOn the polyhedron $\\mathcal{P}'$, we consider the polyhedron vertices $v'_{i,j,k_1}, v'_{i,j,k_2}\\in F'_i \\cap F'_j$ and $v'_{i, m_1, m_2}\\in F'_i$ and $v'_{j, n_1, n_2}\\in F'_j$. Then there exists a transformation $f$ which transforms the vertices $v_{i,j,k_1}$, $v_{i,j,k_2}$, $v_{i, m_1, m_2}$, $v_{j, n_1, n_2}$ to the vertices $v'_{i,j,k_1}$, $v'_{i,j,k_2}$, $v'_{i, m_1, m_2}$, $v'_{j, n_1, n_2}$, respectively.\n\t\n\tLet $\\mathcal{P}''$ be the polyhedron transformed by $f$ from $\\mathcal{P}$. Since the first four points of polyhedron $\\mathcal{P}''$ are $v'_{i,j,k_1}=f(v_{i,j,k_1})$ , $v'_{i,j,k_2}=f(v_{i,j,k_2})$, $v'_{i, m_1, m_2}=f(v_{i, m_1, m_2})$, $v'_{j, n_1, n_2}=f(v_{j, n_1, n_2})$, and all the other vertices are determined by Algorithm 2, $\\mathcal{P}''$ is the same polyhedron as $\\mathcal{P}'$. This means that the upper bound of the degrees of freedom is four.\n\t\n\tThus, the degrees of freedom for choosing the polyhedron are exactly four.\n\\end{proof}\n\nNote that in Algorithm 1, we choose 4 parameters arbitrarily, two for $p_i$, one for $r_i$ in step 1 and one for $q_j$ in step 6. Therefore, if the given tessellation $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram, the projection of the arrangement of the planes constructed by Algorithm 2 on the sphere gives $\\mathcal{T}$ for any spherical circle $\\tilde{c}_i$ and the point $q_j$ is chosen in Algorithm 1. \n\nWe have the following corollary which is directly implied from Lemma \\ref{Coplanar2}, Algorithm 2, and Theorem \\ref{4degs}.\n\n\\begin{corollary}\\label{uniquePoly}\n\tIf $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram, then Algorithm 2 gives the unique polyhedron up to the choice of the first four degrees of freedom.\t\n\\end{corollary}\n\nWe use the contraposition of Corollary \\ref{uniquePoly} to verify that, if the constructed planes composing a polyhedron with respect to the tessellation $\\mathcal{T}$ using Algorithm 2 are not uniquely constructed from the first choice of four degrees of freedom, then the given tessellation is not the spherical Laguerre Voronoi diagram.\n\nThe following lemma characterizes the properties of the polyhedron vertices and the given tessellation.\n\n\\begin{lemma}\\label{ConsCheckLemma}\n\tLet $\\mathcal{T}$ be a given tessellation  and $\\mathbb{P}$ a set of planes constructed by Algorithm 2. $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram if and only if for all vertices $v_{i, j, k}\\in \\mathcal{V}$,\n\t\\begin{enumerate}\n\t\t\\item there exists the unique point $V_{i,j,k}$ of the intersection of the plane $P_i, P_j, P_k \\in \\mathbb{P}$; and\n\t\t\\item there exists $t\\in\\mathbb{R}\\backslash\\{0\\}$ such that $V_{i,j,k}=tv_{i,j,k}$.\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tFirstly, let $\\mathcal{T}$ be a spherical Laguerre Voronoi diagram and $v_{i,j,k}\\in \\mathcal{V}$. Without loss of generality, assume that $P_i, P_j$ are constructed sequentially. By Lemma \\ref{Coplanar2}, $P_k$ is uniquely determined from $P_i$ and $P_j$ which is constructed from $\\ell_{i,k}$ and $\\ell_{j,k}$. Hence, $V_{i,j,k}\\in P_i\\cap P_j \\cap P_k$ uniquely. In addition, since $v_{i,j,k}\\in \\wideparen{e}_{i,j}\\cap \\wideparen{e}_{i,k} \\cap \\wideparen{e}_{j,k}$ and $\\ell_{i,j,k}:=P_{i,j}\\cap P_{j,k}\\cap P_{i,k}$ is a line passing through $O$ and $v_{i,j,k}$,  Algorithm 2 and Lemma \\ref{Coplanar2} implies that $V_{i,j,k}$ is laid on the line $\\ell_{i,j,k}$. That is, the condition 1 and 2 of the lemma are satisfied.\n\t\n\tConversely, suppose that the conditions 1 and 2 of the lemma hold. Since $V_{i,j,k}$'s are formulated from the halfspaces intersection of the planes in $\\mathbb{P}$ which are constructed from Algorithm 2, and the central projection is preserved from the condition 2, Proposition \\ref{twoWay} implies that $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram.\n\\end{proof}\n\nLemma \\ref{ConsCheckLemma} means that from the constructed polyhedron, the vertices selected in Algorithm 2, i.e., the vertices $v$ with $\\text{Mark}(v)=1$ set in Algorithm 2, are located on the lines emanating from the origin and passing through the associated tessellation vertices. Therefore, the given tessellation $\\mathcal{T}$ is not the spherical Laguerre Voronoi diagram if other vertices are not necessarily on the radial lines emanating from the origin and passing through the associated tessellation vertices.\n\nAs mentioned before, the uniqueness of the plane $P_l$ constructed from $P_p$ and $P_q$ is up to the choice of the point $v'_{q,l}$ in the step 6 of Algorithm 2. Lemma \\ref{Coplanar2}, Corollary \\ref{uniquePoly} together with Lemma \\ref{ConsCheckLemma} lead to Algorithm 3 for checking whether the given tessellation is a spherical Laguerre Voronoi diagram or not.\n\nFrom Algorithm 2, the set of tessellation vertices $\\mathcal{V}$ is divided into two groups: marked vertices and unmarked vertices. The unmarked vertices are vertices $v\\in\\mathcal{V}$ such that $\\text{Mark}(v)=0$. Remark that for the marked vertex $v_{p,q,l}$, the plane $P_l$ constructed from $P_p$ and $P_q$ in Algorithm 2 passes through the intersection point $V_{p,q,l}$ on the line $\\ell_{p,q,l}$ passing through $v_{p,q,l}$. Therefore, it is sufficient to check the consistency in Algorithm 3 among the unmarked vertices.\n\\\\\n\\\\\n\\textbf{Algorithm 3: Spherical Laguerre Voronoi Diagram Recognition}\n\\\\ \\textbf{Input:} The tessellation $\\mathcal{T}$, the set $\\mathbb{P}$ and $\\text{Mark}(v), v\\in \\mathcal{V}$ constructed from Algorithm 2.\n\\\\ \\textbf{Output:} ``true'' or ``false''.\n\\\\ \\textbf{Comment:} ``true'' means that $\\mathcal{T}$ is a spherical Laguerre Voronoi diagram.\n\\\\ \\textbf{Procedure:} \n\\begin{enumerate}\n\t\\item choose an unmarked vertex $v_{p,q,l}$;\n\t\\item \\textbf{if} compute the intersection $V_{p,q,l}$ of planes $P_p, P_q, P_l\\in\\mathbb{P}$, there exists $t\\in \\mathbb{R}$ such that $V_{p,q,l}=tv_{p,q,l}$ \\textbf{then}\\\\\n\t\\text{\\hspace{0.4cm}} $\\text{Mark}(v_{p, q, l})=1$;\\\\\n\t\\text{\\hspace{0.0cm}} \\textbf{else}\\\\\n\t\\text{\\hspace{0.4cm}} report ``false'' and terminate the process;\\\\\n\t\\textbf{end if};\n\t\\item \\textbf{if} $\\text{Mark}(v)=1$ for all $v\\in \\mathcal{V}$ \\textbf{then}\\\\\n\t\\text{\\hspace{0.4cm}} report ``true'';\\\\\n\t\\text{\\hspace{0.0cm}} \\textbf{else}\\\\\n\t\\text{\\hspace{0.4cm}} go to step 1;\\\\\n\t\\textbf{end if}\n\\end{enumerate}\n\\textbf{end Procedure}\n\n\\subsection{Algorithm analysis}\n\nIn this section, we analyze the proposed algorithm. In Algorithm 1, the main operations are for the plane construction and the intersection of two planes, each of whose complexity is $O(1)$. Since Algorithm 1 is related to the first three polygons, the complexity of Algorithm 1 is $O(1)$.\n\nThe spherical Laguerre Voronoi diagram recognition problem is mainly considered by Algorithms 2 and 3. The following theorem shows the complexity of the problem.\n\n\\begin{theorem}\n\tFor an $n$ cells tessellation, we can judge whether the given tessellation is a spherical Laguerre Voronoi diagram in $O(n \\log n)$ time.\n\\end{theorem}\n\n\\begin{proof}\n\t\n\tWe now consider the complexity of Algorithm 2. In Algorithm 2, the time complexity of steps 1 to 5 is $O(1)$. Most of the computation time is spent on step 6. We use a priority queue with a heap structure whose nodes contain vertices as follows.\n\t\n\tFor each tessellation vertex $v\\in \\mathcal{V}$, we define the key function $k(v)=c$, where $c$ is the number of planes constructed around vertex $v$. Firstly, let $k(v)=0$ for all $v\\in \\mathcal{V}$. During the course of processing, the key value increases and $0 \\leq k(v) \\leq 3$. At the end of step 3 of Algorithm 2, we have $k(v_{i,j,k})=3$ and for the three vertices $v_{i,j,a_1}, v_{i,a_2,k}$ and $v_{a_3,j,k}$ adjacent to $v_{i,j,k}$, $k(v_{i,j,a_1})=k(v_{i,a_2,k})=$ $k(v_{a_3,j,k})=2$. \n\t\n\tWe store vertices with key values less than or equal to 2 in the heap according to the key function so that the root node has the largest key value. For each repetition of step 6 in Algorithm 2, the root node is deleted from the heap, and the vertex contained within is set to $v_{p,q,l}$. After we construct the plane, we update the key function. Then we remove nodes which have a key value of 3. The other nodes whose key values have been changed are relocated in the heap. We repeat this process until all nodes have been removed from the heap. Adding and removing a node from the heap requires $O(\\log n)$ time. Since the spherical tessellation is planar, the number of vertices is $O(n)$. Hence, the complexity of step 6 in Algorithm 2 is $O(n \\log n)$. \n\t\n\tAfter we obtain the planes $P_1,..., P_n$ from Algorithm 2, the construction of the polyhedron can be performed using the intersection of halfspaces, including the origin, which requires $O(n\\log n)$ time.\n\t\n\tIn Algorithm 3, we have to check unmarked vertices. For each vertex, we pick the planes of the corresponding polygons and find the intersection of three planes at the vertex which take the time complexity $O(1)$. Therefore, the complexity of Algorithm 3 is $O(n)$ through the unmarked vertices.\n\t\n\tSummarizing the above considerations, we have proved that the complexity of the spherical Laguerre Voronoi  diagram recognition is $O(n \\log n)$.\n\t\n\\end{proof}\n\n\\subsection{Interpretation of the generators}\n\nOnce the tessellation $\\mathcal{T}$ is judged as a spherical Laguerre Voronoi diagram, by Algorithm 1, 2, 3, we can construct the associated set of generators in the following way.\n\nFirst, shrink the polyhedron constructed by Algorithm 2 so that all the planes intersect with $U$. Next, collect all the intersections, i.e., circles, between the planes and $U$. The resulting set of circles is the set of generators. Note that the generator set is not unique. The set of circles obtained by the above procedure is an example of the generator set.\n\n\\section{Concluding Remarks}\n\nWe proposed an algorithm for determining whether a spherical tessellation is a spherical Laguerre Voronoi diagram. The algorithm utilizes the convex polyhedron corresponding to a given tessellation. The criterion for recognizating a spherical Laguerre Voronoi diagram is obtained from this polyhedron. If the given tessellation is a spherical Laguerre Voronoi diagram, we can recover the generating spherical circles.\n\nThe properties of the polyhedron presented in the algorithm can be applied to the spherical Laguerre Voronoi approximation problem. If we have a spherical tessellation which does not correspond exactly with a spherical Laguerre Voronoi diagram, employing Algorithm 2 can give us an approximate spherical Laguerre Voronoi diagram. An interesting problem is the determination of the spherical Laguerre Voronoi diagram which provides the best approximation to the spherical tessellation.\n\nFrom the perspective of practical applications, we \\cite{Chaidee3} recently presented an algorithm for finding the spherical Laguerre Voronoi diagram which most closely approximates a tessellation, using an example of a planar tessellation extracted from a photo taken from a curved surface with generators. For the case that a given spherical tessellation does not contain generators, we may apply the proposed algorithms for approximating the generators and their weights. One of our future areas for study is the application of the algorithms presented here to the analysis of polygonal patterns found in the real world.\n\n\\subparagraph*{Acknowledgements}\n\nThe first author acknowledges the support of the MIMS Ph.D. Program of the Meiji Institute for the Advanced Study of Mathematical Sciences, Meiji University, and the Development and Promotion of Science and Technology Talents Project (DPST) of the Institute for the Promotion of Teaching Science and Technology (IPST), Ministry of Education, Thailand. The authors also thank the reviewers who gave useful comments for improving the manuscript. This research is supported in part by the Grant-in-Aid for Basic Research [24360039]; and Exploratory Research [151512067] of MEXT.\n\n\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\chapter{Natural Language Processing for Policymaking}\n\n\\chapterauthor{Zhijing Jin}{jinzhi@ethz.ch}{Max Planck Institute \\& ETH Z\u00fcrich}\n\n\\chapterauthor{Rada Mihalcea}{mihalcea@umich.edu}{University of Michigan}\n\n\\begin{abstract}\nLanguage is the medium for many political activities, from campaigns to news reports. Natural language processing (NLP) uses computational tools to parse text into key information that is needed for policymaking. In this chapter, we introduce common methods of NLP, including text classification, topic modeling, event extraction, and text scaling. We then overview how these methods can be used for policymaking through four major applications including data collection for evidence-based policymaking, interpretation of political decisions, policy communication, and investigation of policy effects.\nFinally, we highlight some potential limitations and ethical concerns when using NLP for policymaking.\n\n\\end{abstract}\n\n\\keywords{Natural Language Processing, Text Analysis, Policymaking, Artificial Intelligence, Machine Learning}\n\n\n\\thispagestyle{alim}\n\n\n\\input{main_content}\n\n\\bibliographystyle{acl_natbib}\n\n\\section{Introduction}\\label{sim:sec:intro}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{fig\/img_intro.pdf}\n    \\caption{Overview of NLP for policymaking.}\n    \\label{fig:intro}\n\\end{figure}\n\nLanguage is an important form of data in politics.  Constituents express their stances and needs in text such as social media and survey responses. Politicians conduct campaigns through debates, statements of policy positions, and social media. Government staff needs to compile information from various documents to assist in decision-making. Textual data is also prevalent through the documents and debates in the legislation process, negotiations and treaties to resolve international conflicts, and media such as news reports, social media, party platforms, and manifestos.\n\nNatural language processing (NLP) is the study of computational methods to automatically analyze text and extract meaningful information for subsequent analysis.\nThe importance of NLP for policymaking has been highlighted since the last century \\citep{gigley-1993-projected}. With the recent success of NLP and its versatility over tasks such as classification, information extraction, summarization, and translation \\citep{devlin2019bert,brown2020language}, there is a rising trend to integrate NLP into the policy decisions and public administrations \\citep{misuraca2020use,engstrom2020government,vanroy2021ai}. Main applications include extracting useful, condensed information from free-form text \\citep{engstrom2020government}, and \nanalyzing sentiment and citizen feedback by NLP \\citep{biran2022policycloud} as in many projects funded by EU Horizon projects \\citep{european2011proposal}.\nDriven by the broad applications of NLP \\citep{jin-etal-2021-good,gonzalez2022good}, the research community also starts to connect NLP with various social applications in\nthe fields of computational social science \\citep{lazer2009computational,shah2015big,engel2021handbook,luz2022computational} and political science in particular \\citep{grimmer2013text,glavas-etal-2019-computational}.\n\n\n\nWe show an overview of NLP for policymaking in Figure~\\ref{fig:intro}. According to this overview, the chapter will consist of three parts. First, we introduce in Section~\\ref{sec:nlp_tools} NLP methods that are applicable to political science, including text classification, topic modeling, event extraction, and score prediction. Next, we cover a variety of cases where NLP can be applied to policymaking in Section~\\ref{sec:nlp4policy}. Specifically, we cover four stages: analyzing data for evidence-based policymaking, improving policy communication with the public, investigating policy effects, and interpreting political phenomena to the public. Finally, we will discuss limitations and ethical considerations when using NLP for policymaking in Section~\\ref{sec:ethics}.\n\n\n\\section{NLP for Text Analysis}\\label{sec:nlp_tools}\n\nNLP brings powerful computational tools to analyze textual data \\citep{jurafsky2000speech}.\nAccording to the type of information that we want to extract from the text, we introduce four different NLP tools to analyze text data: text classification (by which the extracted information is the \\textit{category} of the text), topic modeling (by which the extracted information is the \\textit{key topics} in the text), event extraction (by which the extracted information is the list of \\textit{events} mentioned in the text), and score prediction (where the extracted information is a \\textit{score} of the text). Table~\\ref{tab:nlp_tasks} lists each method with the type of information it can extract and some example application scenarios, which we will detail in the following subsections.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n\\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{llll}\n\\toprule\n\\textbf{NLP Method} & \\textbf{Information to Extract} & \\textbf{Example Applications} \\\\ \\midrule\nText classification & Category of text & Identify the sentiment, stance, etc.\n\\\\\nTopic modeling & Key topics in text & Summarize topics in political agenda\n\\\\\nEvent extraction & List of events & Extract news events, international conflicts\n\\\\\nScore prediction & Score & Text scaling\n\\\\\n\\bottomrule\n    \\end{tabular}\n}\n    \\caption{Four common NLP methods, the type of information extracted by each of them, and example applications.}\n    \\label{tab:nlp_tasks}\n\\end{table}\n\n\\subsection{Text Classification}\nAs one of the most common types of text analysis methods, text classification reads in a piece of text and predicts its category using an NLP text classification model, as in Figure~\\ref{fig:classification}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9 \\textwidth]{fig\/img_classification.pdf}\n    \\caption{The usage and example applications of text classification on political text.}\n    \\label{fig:classification}\n\\end{figure} \nThere are many off-the-shelf existing tools for text classification \\citep{yin-etal-2019-benchmarking,brown2020language,loria2018textblob} such as {the implementation}\\footnote{\\url{https:\/\/discuss.huggingface.co\/t\/new-pipeline-for-zero-shot-text-classification\/681}} using the Python package  \\texttt{transformers} \\citep{wolf-etal-2020-transformers}. \nA well-known subtask of text classification is sentiment classification (also known as sentiment analysis, or opinion mining), which aims to distinguish the subjective information in the text, such as positive or negative sentiment \\citep{pang2007opinion}.\nHowever, the existing tools only do well in categories that are easy to predict. If the categorization is customized and very specific to a study context, then there are two common solutions. One is to use dictionary-based methods, by a list of frequent keywords that correspond to a certain category \\citep{albaugh2013automated} or using general linguistic dictionaries such as the Linguistic Inquiry and Word Count (LIWC) dictionary \\citep{pennebaker2001linguistic}.\nThe second way is to adopt the data-driven pipeline, which requires human hand coding of documents into a predetermined set of categories, then train an NLP model to learn the text classification task \\citep{sun2019finetune}, and verify the performance of the NLP model on a held-out subset of the data, as introduced in \\citet{grimmer2013text}. An example of adapting the state-of-the-art NLP models on a customized dataset is demonstrated in {this guide}.\\footnote{\\url{https:\/\/skimai.com\/fine-tuning-bert-for-sentiment-analysis\/}}\n\nUsing the text classification method, we can automate many types of analyses in political science. As listed in the examples in Figure~\\ref{fig:classification}, researchers can detect political perspective of news articles \\citep{huguet-cabot-etal-2020-pragmatics}, the stance in media on a certain topic \\citep{luo-etal-2020-detecting}, whether campaigns use positive or negative sentiment \\citep{ansolabehere1995going}, which issue area is the legislation about \\citep{adler2011congressional},\ntopics in parliament speech \\citep{albaugh2013automated,osnabrugge2021cross}, congressional bills \\citep{hillard2008computer,collingwood2012tradeoffs} and political agenda \\citep{karan-etal-2016-analysis},\nwhether the international statement is peaceful or belligerent \\citep{schrodt2000pattern}, whether a speech contains positive or negative sentiment \\citep{schumacher2016euspeech}, and whether a U.S. Circuit Courts case decision is conservative or liberal \\citep{hausladen2020text}.\nMoreover, text classification can also be used to categorize the type of language devices that politicians use, such as what type of framing the text uses \\citep{huguet-cabot-etal-2020-pragmatics}, and \nwhether a tweet uses political parody \\citep{maronikolakis-etal-2020-analyzing}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Topic Modeling}\nTopic modeling is a method to uncover a list of frequent topics in a corpus of text. For example, news articles that are against vaccination might frequently mention the topic ``autism,'' whereas news articles supporting vaccination will be more likely to mention ``immune'' and ``protective.''\nOne of the most widely used models is the Latent Dirichlet Allocation (LDA) \\citep{blei2001latent} which is available in the Python packages \\texttt{NLTK} and \\texttt{Gensim}, as in {this guide}.\\footnote{\\url{https:\/\/skimai.com\/fine-tuning-bert-for-sentiment-analysis\/}} \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{fig\/img_topic_model.pdf}\n    \\caption{Given a collection of text documents, topic modeling generates a list of topic clusters.}\n    \\label{fig:topic_model}\n\\end{figure}\n\nSpecifically, LDA is a probabilistic model that models each topic as a mixture of words, and each textual document can be represented as a mixture of topics. As in Figure~\\ref{fig:topic_model}, given a collection of textual documents, LDA topic modeling generates a list of topic clusters, for which the number $N$ of topics can be customized by the analyst.  In addition, if needed, LDA can also produce a representation of each document as a weighted list of topics. While often the number of topics is predetermined by the analyst, this number can also be dynamically determined by measuring the perplexity of the resulting topics. In addition to LDA, other topic modeling algorithms have been used extensively, such as those based on principal component analysis (PCA) \\citep{Chung2008-fo}.\n\nTopic modeling, as described in this section, can facilitate various studies on political text.\nPrevious studies analyzed the topics of legislative speech \\citep{quinn2010analyze,quinn2006automated}, Senate press releases \\citep{grimmer2010bayesian}, and electorial manifestos \\citep{menini-etal-2017-topic}.\n\n\n\\subsection{Event Extraction}\nEvent extraction is the task of extracting a list of events from a given text. It\nis a subtask of a larger domain of NLP called information extraction \\citep{manning2008introduction}.\nFor example, the sentence ``Israel bombs Hamas sites in Gaza'' expresses an event ``\\textit{Israel $\\xrightarrow[]{\\text{bombs}}$ Hamas sites}'' with the location ``\\textit{Gaza}.'' Event extraction usually incorporates both entity extraction (e.g., Israel, Hamas sites, and Gaza in the previous example) and relation extraction (e.g., ``bombs'' in the previous example).\n\n\nEvent extraction is a handy tool to monitor events automatically, such as detecting news events \\citep{walker2006ace,mitamura2017events}, \nand detecting international conflicts \\citep{azar1980conflict,trappl2006programming}. To foster research on event extraction, there are tremendous efforts into textual data collection \\citep{mcclelland1976world,schrodt2006twenty,merritt1993international,raleigh_introducing_2010,sundberg2013introducing}, event coding schemes to accommodate different political events \\citep{goldstein1992conflict,bond1997mapping,gerner2002conflict}, and dataset validity assessment \\citep{schrodt1994validity}.\n\n\n\n\n\n\n\n\n\nAs for event extraction models, similar to text classification models, there are off-the-shelf tools such as the Python packages \\texttt{stanza} \\citep{qi2020stanza} and \\texttt{spaCy} \\citep{honnibal2020spacy}. In case of customized sets of event types, researchers can also train NLP models on a collection of textual documents with event annotations \\citep[\\textit{inter alia}]{hogenboom2011overview,liu2020extracting}.\n\n\\subsection{Score Prediction}\nNLP can also be used to predict a score given input text. A useful application is political text scaling, which aims to predict a score (e.g., left-to-right ideology, emotionality, and different attitudes towards the European integration process) for a given piece of text (e.g., political speeches, party manifestos, and social media posts) \\citep[\\textit{inter alia}]{laver2003extracting,lowe2011scaling,slapin2008scaling,gennaro2021emotion}.\n\nTraditional models for text scaling include Wordscores \\citep{laver2003extracting} and WordFish \\citep{slapin2008scaling,lowe2011scaling}. Recent NLP models represent the text by high-dimensional vectors learned by neural networks to predict the scores \\citep{glavas-etal-2017-unsupervised,nanni2019political}. One way to use the NLP models is to apply off-the-shelf general-purpose models such as InstructGPT \\citep{ouyang2022instructGPT} and design a prompt to specify the type of the scaling to the API,\\footnote{\\url{https:\/\/beta.openai.com\/docs\/introduction}}, or borrow existing, trained NLP models if the same type of scaling has been studied by previous researchers. Another way is to collect a dataset of text with hand-coded scales, and train NLP models to learn to predict the scale, similar to the practice in \\citet{slapin2008scaling,gennaro2021emotion}, \\textit{inter alia}.\n\n\n\n\n\\section{Using NLP for Policymaking}\\label{sec:nlp4policy}\n\nIn the political domain, there are large amounts of textual data to analyze \\citep{neuendorf2015content}, such as parliament debates \\citep{van2017debates}, speeches \\citep{schumacher2016euspeech}, legislative text \\citep{baumgartner2006comparative,bevan2017gone}, database of political parties worldwide \\citep{doring2019party}, and expert survey data \\citep{bakker2015measuring}. Since it is tedious to hand-code all textual data, NLP provides a low-cost tool to automatically analyze such massive text. \n\nIn this section, we will introduce how NLP can facilitate four major areas to help policymaking: before policies are made, researchers can use NLP to analyze data and extract key information for evidence-based policymaking (Section~\\ref{sec:use1}); after policies are made, researchers can interpret the priorities among and reasons behind political decisions (Section~\\ref{sec:use2}); researchers can also analyze features in the language of politicians when communicating the policies to the public (Section~\\ref{sec:use3}); finally, after the policies have taken effect, researchers can investigate the effectiveness of the policies (Section~\\ref{sec:use4}).\n\n\n\n\n\n\\subsection{Analyzing Data for Evidence-Based Policymaking}\\label{sec:use1}\n\nA major use of NLP is to extract information from large collections of text. This function can be very useful for analyzing the views and needs of constituents, so that policymakers can make decisions accordingly.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{fig\/img_use_survey.pdf}\n    \\caption{NLP to analyze data for evidence-based policymaking.}\n    \\label{fig:use_survey}\n\\end{figure}\n\nAs in Figure~\\ref{fig:use_survey}, we will explain how NLP can be used to analyze data for evidence-based policymaking from three aspects: data, information to extract, and political usage.\n\n\\myparagraph{Data.}\nData is the basis of such analyses. Large amounts of textual data can reveal information about constituents, media outlets, and influential figures. The data can come from a variety of sources, including social media such as Twitter and Facebook, survey responses, and news articles.\n\n\\myparagraph{Information to Extract.}\nBased on the large textual corpora, NLP models can be used to extract information that are useful for political decision-making, ranging from information about people, such as sentiment \\citep{thelwall2011sentiment,rosenthal2015semeval}, stance \\citep{thomas-etal-2006-get,gottipati-etal-2013-learning,stefanov-etal-2020-predicting,luo-etal-2020-detecting}, ideology \\citep{hirst2010party,iyyer-etal-2014-political,preotiuc-pietro-etal-2017-beyond}, and reasoning on certain topics \\citep{egami2018make,demszky-etal-2019-analyzing,camp2021thin}, to factual information, such as main topics \\citep{gottipati-etal-2013-learning}, events \\citep{trappl2006programming,mitamura2017events,ding-riloff-2018-human,ding-etal-2019-improving}, and needs \\citep{sarol-etal-2020-empirical,crayton2020narratives,paul-frank-2019-ranking} expressed in the data. \nThe extracted information cannot only be about people, but also about political entities, such as the left-right political scales of parties and political actors \\citep{slapin2008scaling,glavas-etal-2017-unsupervised}, which claims are raised by which politicians \\citep{blessing-etal-2019-environment,pado-etal-2019-sides}, and the legislative body's vote breakdown for state bills by backgrounds such as gender, rural-urban and ideological splits \\citet{davoodi-etal-2020-understanding}.\n\nTo extract such information from text, we can often utilize the main NLP tools introduced in Section~\\ref{sec:nlp_tools}, including text classification, topic modeling, event extraction and score prediction (especially text scaling to predict left-to-right ideology). In NLP literature, social media, such as Twitter, is a popular source of textual data to collect public opinions  \\citep{thelwall2011sentiment,paltoglou2012twitter,pak-paroubek-2010-twitter,arunachalam-sarkar-2013-new,rosenthal2015semeval}.\n\n\\myparagraph{Political Usage.}\nSuch information extracted from data is highly valuable for political usage. For example, voters' sentiment, stance, and ideology are important supplementary for traditional polls and surveys to gather information about the constituents' political leaning. Identifying the needs expressed by people is another important survey target, which helps politicians understand what needs they should take care of, and match the needs and availabilities of resources \\citep{hiware-etal-2020-narmada}.\n\nAmong more specific political uses is to understand the public opinion on parties\/president, as well as on certain topics. The public sentiment towards parties \\citep{pla-hurtado-2014-political} and President \\citep{marchetti-bowick-chambers-2012-learning} can serve as a supplementary for the traditional approval rating survey, and stances towards certain topics \\citep{gottipati-etal-2013-learning,stefanov-etal-2020-predicting,luo-etal-2020-detecting} can be important information for legislators to make decisions on debatable issues such as abortion, taxes, and legalization of same-sex marriage.\nMany existing studies use NLP on social media text to predict election results \\citep{oconnor2010tweets,beverungen2011evaluating,unankard2014predicting,mohammad2015sentiment,tjong-kim-sang-bos-2012-predicting}.\nIn general, big-data-driven analyses can facilitate decision-makers to collect more feedback from people and society, enabling policymakers to be closer to citizens, and increase transparency and engagement in political issues \\citep{arunachalam-sarkar-2013-new}.\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{Interpreting Political Decisions}\\label{sec:use2}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/img_use_interpret.pdf}\n    \\caption{NLP to interpret political decisions.}\n    \\label{fig:use_interpret}\n\\end{figure}\n\nAfter policies are made, political scientists and social scientists can use textual data to interpret political decisions. \nAs in Figure~\\ref{fig:use_interpret}, there are two major use cases: mining political agendas, and discovering policy responsiveness.\n\n\\myparagraph{Mining Political Agendas.}\nResearchers can use textual data to infer a political agenda, including the topics that politicians prioritize, political events, and different political actors' stances on certain topics.\nSuch data can come from press releases, legislation, and electoral campaigns. \nExample of previous studies to analyze the topics and prioritization of political bodies include the research on the prioritization each Senator assigns to topics using press releases \\citep{grimmer2010representational}, topics in different parties' electoral manifestos \\citep{glavas-etal-2017-cross}, topics in EU parliament speeches \\citep{lauscher2016entities} and other various types of text \\citep{king2003automated,hopkins2010method,grimmer2010bayesian,roberts2014structural}, as well as political event detection from congressional text and news \\citep{nanni2017building}.\n\nResearch on politicians' stances include identifying policy positions of politicians \\citep[\\textit{inter alia}]{winter1977content,laver2003extracting,slapin2008scaling,lowe2011scaling},\nhow different politicians agree or disagree on certain topics in electoral campaigns \\citep{menini-tonelli-2016-agreement}, \nand assessment of political personalities \\citep{immelman1993assessment}.\n\n\n\n\nFurther studies look into how political interests affect legislative behavior. Legislators tend to show strong personal interest in the issues that come before their committees \\citep{fenno1973congressmen}, and \\citet{mayhew2004congress} identifies that Senators replying on appropriations secured for their state have a strong incentive to support legislations that allow them to secure particularistic goods.\n\n\n\\myparagraph{Discovering Policy Responsiveness.}\nPolicy responsiveness is the study of how policies respond to different factors, such as how changes in public opinion lead to responses in public policy \\citep{stimson1995dynamic}. One major direction is that politicians tend to make policies that align with the expectations of their constituents, in order to run for successful re-election in the next term \\citep{canes2002out}.\nStudies show that policy preferences of the state public can be a predictor of future state policies \\citep{caughey2018policy}.\nFor example, \\citet{lax2009gay} show that more LGBT tolerance leads to more pro-gay legislation in response.\n\n\n\nA recent study by \\citet{jin-etal-2021-mining-cause} uses NLP to analyze over 10 million COVID-19-related tweets targeted at US governors; using classification models to obtain the public sentiment, they study how public sentiment leads to political decisions of COVID-19 policies made by US governors. Such use of NLP on massive textual data contrasts with the traditional studies of policy responsiveness which span over several decades and use manually collected survey results \\citep{caughey2018policy,lax2009gay,lax2012democratic}. \n\n\n\\subsection{Improving Policy Communication with the Public}\\label{sec:use3}\n\nPolicy communication is the study to understand how politicians present the policies to their constituents. As in Figure~\\ref{fig:use_comm}, common research questions in policy communication include how politicians establish their images \\citep{fenno1978home} such as campaign strategies \\citep{petrocik1996issue,simon2002winning,sigelman2004avoidance}, how constituents allocate credit, what receives attention in Congress \\citep{sulkin2005issue}, and what receives attention in news articles \\citep{semetko2000framing,mccombs2004setting,armstrong2006whose}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/img_use_comm.pdf}\n    \\caption{NLP to analyze policy communication.}\n    \\label{fig:use_comm}\n\\end{figure}\n\nBased on data from press releases, political statements, electoral campaigns and news articles,\\footnote{Other data sources used in policy communication research include surveys of Senate staffers \\citep{cook1988press}, newsletters that legislators send to constituents \\citep{lipinski2009congressional} and so on.} researchers usually analyze two types of information: the language techniques politicians use, and the contents such as topics and underlying moral foundations in these textual documents. \n\n\\myparagraph{Language Techniques.}\nPolicy communication largely focuses on the types of languages that politicians use. Researchers are interested in first analyzing the language techniques in political texts, and then, based on these techniques, researchers can dive into the questions of why politicians use them, and what are the effects of such usage. \n\nFor example, previous studies analyze what portions of political texts are position-taking versus credit-claiming \\citep{grimmer2012words,grimmer2013appropriators}, whether the claims are vague or concrete \\citep{baerg2018central,eichorst2019resist}, the frequency of credit-claiming messages versus the actual amount of contributions\n\\citep{grimmer2012words}, and whether politicians tend to make credible or dishonorable promises \\citep{grimmer2010representational}. Within the political statements, it is also interesting to check the ideological proportions \\citep{sim-etal-2013-measuring}, and how politicians make use of dialectal variations and code-mixing \\citep{sravani-etal-2021-political}.\n\nThe representation styles usually affect the effectiveness of policy communication, such as \nthe role of language\nambiguity in framing the political agenda \\citep{page1976theory,campbell1983ambiguity}, and the effect of credit-claiming messages on constituents' allocation of credit \\citep{grimmer2012words}.\n\n\n\n\n\n\\myparagraph{Contents.}\nThe contents of policy communication include the topics in the political statements, such as what Senators discuss in floor statements \\citep{hill2002symbolic}, and what Presidents address in daily speeches \\citep{lee2008dividers}, and also the moral foundations used by politicians underlying their political tweets \\citep{johnson-goldwasser-2018-classification}.\n\nUsing the extracted content information, researchers can explore further questions such as whether competing politicians or political elites emphasize the same issues  \\citep{petrocik1996issue,gabel2007estimating}, and how the priorities politicians articulate co-vary with the issues discussed in\nthe media \\citep{bartels1996politicians}. Another open research direction is to analyze the interaction between newspapers and politicians' messages, such as how often newspapers cover a certain politician's message and in what way, and how such coverage affects incumbency advantage.\n\n\n\n\n\\myparagraph{Meaningful Future Work.}\nApart from analyzing the language of existing political texts that aims to maximize political interests, an advanced question that is more meaningful to society is how to improve policy communication to steer towards a more beneficial future for society as a whole. There is relatively little research on this, and we welcome future work on this meaningful topic.\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{Investigating Policy Effects}\\label{sec:use4}\n\nAfter policies are taken into effect, it is important to collect feedback or evaluate the effectiveness of policies. Existing studies evaluate the effects of policies along different dimensions: one dimension is the change in public sentiment, which can be analyzed by comparing the sentiment classification results before and after policies, following a similar paradigm in Section~\\ref{sec:use1}.\nThere are also studies on how policies affect the crowd's perception of the democratic process \\citep{miller1990voters}.\n\nAnother dimension is how policies result in economic changes. \\citet{calvo2018winners} investigate the negative consequences of policy volatility that harm long-term economic growth. Specifically, to measure policy volatility, they first obtain main topics by topic modeling on presidential speeches, and then analyze how the significance of topics changes over time.\n\n\n\n\\section{Limitations and Ethical Considerations} \\label{sec:ethics}\n\nThere are several limitations that researchers and policymakers need to take into consideration when using NLP for policymaking, due to the data-driven and black-box nature of modern NLP. First, the effectiveness of the computational models relies on the quality and comprehensiveness of the data. Although many political discourses are public, including data sources such as news, press releases, legislation, and campaigns, when it comes to surveying public opinions, social media might be a biased representation of the whole population. Therefore, when making important policy decisions, the traditional polls and surveys can provide more comprehensive coverage. Note that in the case of traditional polls, NLP can still be helpful in expediting the processing of survey answers. \n\nThe second concern is the black-box nature of modern NLP models. We do not encourage decision-making systems to depend fully on NLP, but suggest that NLP can assist human decision-makers. Hence, all the applications introduced in this chapter use NLP to compile information that is necessary for policymaking instead of directly suggesting a policy. Nonetheless, some of the models are hard to interpret or explain, such as text classification using deep learning models \\citep{yin-etal-2019-benchmarking,brown2020language}, which could be vulnerable to adversarial attacks by small paraphrasing of the text input \\citep{jin2020bert}. In practical applications, it is important to ensure the trustworthiness of the usage of AI. There could be a preference for transparent machine learning models if they can do the work well (e.g., LDA topic models, and traditional classification methods using dictionaries or linguistic rules), or tasks with well-controlled outputs such as event extraction to select spans of the given text that mention events. In cases where only the deep learning models can provide good performance, there should be more detailed performance analysis (e.g., a study to check the correlation of the model decisions and human judgments), error analysis (e.g., different types of errors, failure modes, and potential bias towards certain groups), and studies about the interpretability of the model (e.g., feature attribution of the model, visualization of the internal states of the model).\n\nApart from the limitations of the technical methodology, there are also ethical considerations arising from the use of NLP. Among the use cases introduced in this chapter, some applications of NLP are relatively safe as they mainly involve analyzing public political documents and fact-based evidence or effects of policies. However, others could be concerning and vulnerable to misuse. For example, although effective, truthful policy communication is beneficial for society, it might be tempting to overdo policy communication and by all means optimize the votes. As it is highly important for government and politicians to gain positive public perception, overly optimizing policy communication might lead to propaganda, intrusion of data privacy to collect more user preferences, and, in more severe cases, surveillance and violation of human rights.\nHence, there is a strong need for policies to regulate the use of technologies that influence public opinions and pose a challenge to democracy.\n\n\\section{Conclusions}\n\nThis chapter provided a brief overview of current research directions in NLP that provide support for policymaking. We first introduced four main NLP tasks that are commonly used in text analysis: text classification, topic modeling, event extraction, and text scaling. We then showed how these methods can be used in policymaking for applications such as data collection for evidence-based policymaking, interpretation of political decisions, policy communication, and investigation of policy effects. We also discussed potential limitations and ethical considerations of which researchers and policymakers should be aware.\n\nNLP holds significant promise for enabling data-driven policymaking. In addition to the tasks overviewed in this chapter, we foresee that other NLP applications, such as text summarization (e.g., to condense information from large documents), question answering (e.g., for reasoning about policies), and culturally-adjusted machine translation (e.g., to facilitate international communications), will soon find use in policymaking. The field of NLP is quickly advancing, and close collaborations between NLP experts and public policy experts will be key to the successful use and deployment of NLP tools in public policy. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpectropolarimetric data in the optical\/UV range put important\nconstraints on the emission and scattering geometry of AGNs (see\ne.g. \\cite{antonucci2002}). For the interpretation of\nspectropolarimetric data detailed modeling tools are important. Here,\nwe use a new, publicly available radiative transfer code, {\\sc Stokes}\n(\\cite[Goosmann \\& Gaskell\\ 2007]{goosmann2007}), to model the\nobscuring torus of AGNs. The code is based on the Monte-Carlo method\nand allows the simulation of various emission and scattering\ngeometries. Polarization due to Thomson and dust (Mie-)scattering is\nincluded. Moreover, the code computes wavelength-dependent time delays\nand can thus be used to study polarization reverberation (\\cite[Shoji,\nGaskell, \\& Goosmann\\ 2005]{shoji2005}).\n\n\n\\section{Modeling an obscuring torus}\n\nWe consider a torus with an elliptical cross-section and centered on a\npoint source. The source isotropically emits a flat continuum spectrum\nbetween 1600~\\AA \\ and 8000~\\AA. The torus half-opening angle is set to\n$\\theta_0 = 30^\\circ$. The inner and outer radii of the torus are fixed at\n0.25 pc and 100 pc respectively. The radial optical depth in the equatorial\nplane is 750 for the V-band. The dust models (table~\\ref{tab}) assume\na mixture of graphite and ``astronomical silicate'' and a grain radii\ndistribution $n(a) \\propto a^\\alpha_s$ between $a_{\\rm min}$ and\n$a_{\\rm max}$.\n\n\\begin{table}\\def~{\\hphantom{0}}\n  \\begin{center}\n    \\caption{Parameterization of the dust models}\n    {\\tiny\n    \\label{tab}\n    \\begin{tabular}{lccccc}\\hline\n      Type & Graphite & Silicate & $a_{\\rm min}$ & $a_{\\rm max}$ &\n      $\\alpha_s$\\\\\\hline\n      Galactic & 62.5\\% & 37.5\\% & 0.005$\\, \\mu{\\rm m}$ & 0.250$\\, \\mu{\\rm m}$\n               & $-3.5$\\\\\n      AGN      &  85\\%  &  15\\%  & 0.005$\\, \\mu{\\rm m}$ & 0.200$\\, \\mu{\\rm m}$\n               & $-2.05$\\\\\\hline\n    \\end{tabular}\n    }\n  \\end{center}\n\\end{table}\n\nThe ``Galactic dust'' model reproduces the interstellar extinction\nfor $R_{\\rm V} = 3.1$ whilst the ``AGN dust'' parameterization is\nobtained from quasar extinction curves derived by \\cite[Gaskell\n\\etal\\ 2004]{gaskell2004}. This latter dust type favors larger grain\nsizes.\n\n\n\\section{Results and discussion}\n\nThe angular distributions of the polarization and flux spectra are\nshown in figure~\\ref{fig}. In a face-on view, the central source is\nseen directly and the obtained polarization is low. At obscured\ninclinations $i>\\theta_0$, the scattering properties of both dust\ntypes lead to different results. For ``AGN dust'', the polarization is\nlower in the UV but rises quickly toward longer wavelengths.  Detailed\nspectropolarimetric observations of dust reflection thus effectively\nconstrain the dust composition. For both dust types the polarization\nposition angle is oriented perpendicularly to the projected symmetry\naxis of the object.\n\n\\begin{figure}\n  \\vskip +0.4cm\n  \\includegraphics[width=0.48\\textwidth]{S238-goosmann-poster2-fig1.eps}\n  \\hfill\n  \\includegraphics[width=0.48\\textwidth]{S238-goosmann-poster2-fig2.eps}\n  \\caption{Polarization (top) and flux (bottom) spectra for a\n  centrally-illuminated torus filled with ``Galactic dust'' (left) or\n  ``AGN dust'' (right). The flux spectra are normalized to the\n  illuminating flux and the inclination angle $i$ is measured from the\n  symmetry axis.}\n  \\label{fig}\n\\end{figure}\n\nIf the torus is filled with ``AGN dust'' the total flux spectra are\nwavelength-independent above 2500~\\AA \\ at all possible\ninclinations. For the ``Galactic dust'' model the scattered flux rises\ngradually toward longer wavelengths. The albedo of both dust\ncompositions changes significantly below 2500~\\AA \\ so that less\nradiation is scattered in the UV.\n\nSpectropolarimetric data from AGNs contain not only information\nabout the obscuring torus, but also include other scattering\nregions. To obtain a more detailed picture of AGN polarization it is\ntherefore necessary to model several, radiatively coupled scattering\nregions self-consistently. While this is beyond the scope of this\nproceedings note, {\\sc Stokes} is capable of solving such problems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe are interested in using climate models to produce objective probabilistic forecasts of future climate that\ninclude parameter uncertainty. The word `objective' is used here in a technical\nstatistical sense that means that the prior distribution for the parameters is determined by a rule,\nrather than from intuition.\nIn statistics the most widely discussed rule is the Jeffreys' rule~\\citep{jeffreys}.\nJeffreys' rule could, in principal, be applied to a climate model directly from the definition.\nThis would translate into using numerical methods to differentiate the predicted probabilities from\ninitial condition ensembles and to take expectations over all simulated climate states.\nThat could, however, be computationally demanding.\nAs an alternative one could fit distributions to the output from the model, and differentiate the estimated parameters of\nthe distributions instead, which is likely to be considerably easier.\nThe most obvious distribution to fit is then the multivariate normal distribution.\nIn previous work we have considered two special cases.\nIn~\\citet{jp1} we considered the case where the predicted variables are independent, but both the mean and the variance of initial condition\nensembles from the model are allowed to vary as a function of the model parameters.\nIn~\\citet{jp2} we considered the complementary case where the predicted variables can be correlated, but with a covariance matrix that is constant\nas a function of the model parameters.\nWe now consider the general case, with correlated predicted variables \\emph{and} a covariance matrix that can vary as a function of the\nmodel parameters.\nThis general case contains the two special cases.\nThe algebra in this case is not quite as elementary as the two special cases, in that we have\nto take expectations of a vector quadratic form, and differentiate the determinants, traces and inverses of matrices.\nPedagogically, therefore, we consider the derivations used in the two simpler cases as still being useful, especially\nas climate modelling practice is perhaps unlikely to reach the stage where it would be necessary to consider the covariance\nmatrix varying as a function of parameters for quite some time (there are many other more important challenges to be dealt with first).\n\nDetailed explanations of Jeffreys' Prior, and the motivation for its use, are given in~\\citet{jp1} and will not be repeated here.\nIn section~\\ref{s2} below we give the expressions for Jeffreys' Prior and the multivariate normal density.\nIn section~\\ref{s3} we then derive the expression for Jeffreys' Prior for the multivariate normal distribution where the climate model\nhas just a single parameter.\nIn section~\\ref{s4} we derive the same expression, but considering multiple parameters.\nIn both section~\\ref{s3} and section~\\ref{s4} we also consider four special cases:\nindependence (taking us back to the results in~\\citet{jp1}),\nconstant covariance (taking us back to the results in~\\citet{jp2}),\nconstant correlation and\nconstant variance.\n\n\\section{Jeffreys' Prior and the Multivariate Normal Density}\\label{s2}\n\nWe start with definitions of the Jeffreys' Prior and the multivariate normal density.\n\n\\subsection{Jeffreys' Prior}\n\nJeffreys' Prior is given by:\n\n\\begin{equation}\\label{eq1}\n    p(\\theta)=\\sqrt{\\mbox{det}\\left[-\\mbox{E}\\left(\\frac{\\partial^2 \\log p(x|\\theta)}{\\partial \\theta_j \\partial \\theta_k}\\right)\\right]}\\\\\n    \\label{jp_def}\n\\end{equation}\n\nDetailed explanations of this equation, which can at first be rather difficult to understand, are given in~\\citet{jp1} and~\\citet{jp2},\nas well as many statistics textbooks such as~\\citet{peterlee}.\n\n\\subsection{The Multivariate Normal Density}\n\nProbability densities from the multivariate normal distribution are given by:\n\\begin{equation}\np(x|\\theta)=\\frac{1}{(2\\pi)^\\frac{n}{2}} \\frac{1}{D^{\\frac{1}{2}}} \\mbox{exp}\\left(-\\frac{1}{2}(x-\\mu)^T S(x-\\mu)\\right)\n\\end{equation}\n\nwhere, in our application:\n\\begin{itemize}\n  \\item $x$ is a vector of those variables predicted by the climate model that are to be compared\n  with observations\n  \\item $\\theta$ is a vector for the underlying parameters in the climate model\n  \\item $\\mu$=$\\mu(\\theta)$ is a vector of the mean response of the model (in other words, the ensemble mean\n  of $x$ for an infinite-sized initial condition ensemble for fixed parameters $\\theta$)\n  \\item $\\Sigma=\\Sigma(\\theta)$ is the covariance matrix of the response of the model\n   (in other words, the ensemble covariance matrix of $x$ for an infinite-sized initial condition ensemble for fixed parameters $\\theta$)\n  \\item $S=\\Sigma^{-1}=S(\\theta)$ is the inverse of the covariance matrix\n  \\item $D=\\mbox{det}(\\Sigma)=D(\\theta)$ is the determinant of the covariance matrix\n\\end{itemize}\n\nThis gives:\n\\begin{eqnarray}\n\\ln p(x|\\theta)\n&=&-\\frac{n}{2}\\ln 2\\pi -\\frac{1}{2} \\ln D -\\frac{1}{2}(x-\\mu)^T S(x-\\mu)\\\\\n&=&-\\frac{n}{2}\\ln 2\\pi -\\frac{1}{2} \\ln D -\\frac{1}{2}(x^T S x-x^TS \\mu-\\mu^T S x+\\mu^T S \\mu)\n\\end{eqnarray}\n\nSince $S$ is symmetric we have $x^TS\\mu=\\mu^T S x$\nwhich means that the above expression for $\\ln p(x|\\theta)$ simplifies a little to:\n\\begin{eqnarray}\n\\ln p(x|\\theta)\\label{eq1}\n&=&-\\frac{n}{2}\\ln 2\\pi -\\frac{1}{2} \\ln D -\\frac{1}{2}(x^T S x-2x^TS \\mu+\\mu^T S \\mu)\n\\end{eqnarray}\n\nWe now consider two cases: one parameter and multiple parameters.\n\n\\section{One parameter}\\label{s3}\n\nThe first case we consider is where there is just a single parameter in the climate model.\nWe mainly consider this case as a warm-up for the multiple parameter case, although it would also be\nrelevant if one only wanted to model the uncertainty due to a single parameter, which might be\na good approximation to the overall parameter uncertainty if that single parameter dominates the uncertainty.\n\nIf we consider $\\theta$ to be this single (scalar) parameter, then differentiating equation~\\ref{eq1} by $\\theta$ gives:\n\\begin{eqnarray}\n\\frac{\\partial \\ln p(x|\\theta)}{\\partial \\theta}\n&=&-\\frac{1}{2} \\frac{\\partial \\ln D}{\\partial \\theta}\n   -\\frac{1}{2}x^T \\frac{\\partial S}{\\partial \\theta} x\n   +x^T\\frac{\\partial (S \\mu)}{\\partial \\theta}\n   -\\frac{1}{2}\\frac{\\partial (\\mu^T S \\mu)}{\\partial \\theta}\n\\end{eqnarray}\n\nExpanding the third and fourth terms:\n\\begin{eqnarray}\n\\frac{\\partial \\ln p(x|\\theta)}{\\partial \\theta}\n&=&-\\frac{1}{2} \\frac{\\partial \\ln D}{\\partial \\theta}\n   -\\frac{1}{2}x^T \\frac{\\partial S}{\\partial \\theta} x\n   +x^T \\left(\\frac{\\partial S}{\\partial \\theta}\\mu+S\\frac{\\partial \\mu}{\\partial \\theta}\\right)\n   -\\frac{1}{2}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu+\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu+\\mu^T S \\frac{\\partial \\mu}{\\partial \\theta}\\right)\\\\\n&=&-\\frac{1}{2} \\frac{\\partial \\ln D}{\\partial \\theta}\n   -\\frac{1}{2}x^T \\frac{\\partial S}{\\partial \\theta} x\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\mu+x^T S\\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu-\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu-\\frac{1}{2}\\mu^T S \\frac{\\partial \\mu}{\\partial \\theta}\n\\end{eqnarray}\n\nSince $S$ is symmetric $\\mu^T S \\frac{\\partial \\mu}{\\partial \\theta}=\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu$ and so\nthe 5th and 7th terms combine, giving:\n\n\\begin{eqnarray}\\label{eq10}\n\\frac{\\partial \\ln p(x|\\theta)}{\\partial \\theta}\n&=&-\\frac{1}{2} \\frac{\\partial \\ln D}{\\partial \\theta}\n   -\\frac{1}{2}x^T \\frac{\\partial S}{\\partial \\theta} x\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\mu\n   +x^T S\\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu\n\\end{eqnarray}\n\nDifferentiating again wrt $\\theta$ gives:\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\n   +x^T \\frac{\\partial}{\\partial \\theta}\\left(\\frac{\\partial S}{\\partial \\theta}\\mu\\right)\n   +x^T \\frac{\\partial}{\\partial \\theta}\\left(S\\frac{\\partial \\mu}{\\partial \\theta}\\right)\\\\\n&& -\\frac{\\partial}{\\partial \\theta}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu\\right)\n   -\\frac{1}{2}\\frac{\\partial}{\\partial \\theta}\\left(\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu\\right)\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\n   +x^T \\left(\\frac{\\partial^2 S}{\\partial \\theta^2}\\mu+\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\right)\n   +x^T \\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}+S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\\right)\\\\\n&&   -\\left(\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n                    +\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n                    +\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\right)\n   -\\frac{1}{2}\\left(\\frac{\\partial \\mu^T}{\\partial \\theta}\\frac{\\partial S }{\\partial \\theta}\\mu\n         +\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\n         +\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\right)\\\\\n&&\\mbox{(where we have expanded the derivatives of products)}\\nonumber\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\n   +x^T \\frac{\\partial^2 S}{\\partial \\theta^2}\\mu\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\n   +x^T S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\\\\\n&&   -\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T}{\\partial \\theta}\\frac{\\partial S }{\\partial \\theta}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\n   +x^T \\frac{\\partial^2 S}{\\partial \\theta^2}\\mu\n   +2x^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\n   +x^T S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\\\\\n&&   -\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n     -2\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n     -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\\\\\n&&\\mbox{(where we have combined the 4th and 5th, and 8th, 10th and 12th terms)}\\nonumber\n\\end{eqnarray}\n\nTaking expectations:\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\\right)\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2}E\\left(x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\\right)\n   +E(x^T) \\frac{\\partial^2 S}{\\partial \\theta^2}\\mu\n   +2E(x^T) \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&&   +E(x^T) S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\n    -\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n     -2\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\n\\end{eqnarray}\n\nBut $E(x)=\\mu$, and, from appendix 2,\n\\begin{equation}\nE\\left(x^T \\frac{\\partial^2 S}{\\partial \\theta^2} x\\right)=\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)+\\mu^T \\frac{\\partial^2 S}{\\partial \\theta^2} \\mu\n\\end{equation}\n\nand so:\n\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\\right)\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2} \\left(\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma \\right)+\\mu^T \\frac{\\partial^2 S}{\\partial \\theta^2} \\mu\\right)\n   +\\mu^T \\frac{\\partial^2 S}{\\partial \\theta^2}\\mu\n   +2\\mu^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&&   +\\mu^T S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\n   -\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n     -2\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n   -\\frac{1}{2} \\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)\n   -\\frac{1}{2} \\mu^T \\frac{\\partial^2 S}{\\partial \\theta^2} \\mu\n   +\\mu^T \\frac{\\partial^2 S}{\\partial \\theta^2}\\mu\n   +2\\mu^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&&   +\\mu^T S\\frac{\\partial^2 \\mu}{\\partial \\theta^2}\n   -\\frac{\\partial^2 \\mu^T }{\\partial \\theta^2}S \\mu\n     -2\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\theta} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta^2}\\mu\\\\\\label{eq2}\n&=&-\\frac{1}{2} \\left[\\frac{\\partial^2 \\ln D}{\\partial \\theta^2}+\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)\\right]\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&&\\mbox{(where we have cancelled various terms)}\\nonumber\n\\end{eqnarray}\n\nBut\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{\\partial \\log D}{\\partial \\theta}\\right)\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{1}{D}\\frac{\\partial D}{\\partial \\theta}\\right)\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{1}{D}\\left(D\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\right)\\right)\\\\\n&&\\mbox{(using a standard result for the derivative of a determinant known as Jacobi's formula)}\\nonumber\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial}{\\partial \\theta}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\right)\\\\\n&&\\mbox{(using a standard result for the derivative of a trace)}\\nonumber\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}+S\\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)+\\mbox{tr}\\left( S \\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}\\right)\\\\\n&&\\mbox{(using the linearity of the trace operator)}\\nonumber\n\\end{eqnarray}\n\nNow note that\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial \\theta}(S \\Sigma)&=&S \\frac{\\partial \\Sigma}{\\partial \\theta}+\\frac{\\partial S}{\\partial \\theta}\\Sigma\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\frac{\\partial^2}{\\partial \\theta^2}( S \\Sigma)\n&=&S \\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}+\\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma+2\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\n\\end{eqnarray}\n\nBut $S\\Sigma=I$ and so $\\frac{\\partial^2}{\\partial \\theta^2}( S \\Sigma)=0$, implying that\n\\begin{equation}\nS\\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}=-\\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma -2\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\n\\end{equation}\n\nand so\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta^2}\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)+\\mbox{tr}\\left( S \\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)+\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta^2}-2\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\Sigma\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)-\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)-\\mbox{tr}\\left(2\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\\\\n&=&-\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma \\right)-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n\\end{eqnarray}\n\nGiving:\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta^2}+\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)\n&=&\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)+\\mbox{tr}\\left(S \\frac{\\partial^2 \\Sigma}{\\partial \\theta^2}\\right)\\\\\\label{eq3}\n&=&-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n\\end{eqnarray}\n\nReturning to equation~\\ref{eq2} and substituting in the expression given in equation~\\ref{eq3} gives:\n\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\\right)\n&=&-\\frac{1}{2} \\left[\\frac{\\partial^2 \\ln D}{\\partial \\theta^2}+\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta^2}\\Sigma\\right)\\right]\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&=&-\\frac{1}{2} \\left[-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\right]\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&=&\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\\\\n&=&-\\frac{1}{2}\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta}S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\\\\\n   &&\\mbox{(using the standard result that $\\frac{\\partial A^{-1}}{\\partial \\theta}=-A^{-1}\\frac{\\partial A}{\\partial \\theta}A^{-1}$)}\n\\end{eqnarray}\n\nThis gives the prior:\n\\begin{eqnarray}\np(\\theta)\n&=&\\sqrt{-E\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\\right)}\\\\\n&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n-\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)}\\label{eq100}\\\\\n&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n+\\frac{1}{2}\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta}S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)}\\label{eq100b}\n\\end{eqnarray}\n\nIf there are $n$ observations then:\n\\begin{itemize}\n  \\item $\\mu$ is an $n$ x $1$ vector\n  \\item $\\frac{\\partial \\mu}{\\partial \\theta}$ is an $n$ by $1$ vector\n  \\item $\\mu^T$ is a $1$ x $n$ vector\n  \\item $\\frac{\\partial \\mu^T}{\\partial \\theta}$ is a $1$ by $n$ vector\n  \\item $S$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}$ is a scalar\n  \\item $\\frac{\\partial S}{\\partial \\theta}$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial \\Sigma}{\\partial \\theta}$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}$ is an $n$ by $n$ matrix\n  \\item tr$\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)$ is a scalar\n\\end{itemize}\n\nWe now consider various special cases of this general formula, starting with the two cases described in~\\citet{jp1} and~\\citet{jp2}.\n\n\\subsection{Independence}\n\nIf the observations are modelled as independent then:\n\\begin{eqnarray}\n\\Sigma\n&=&\\mbox{diag}\\left(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2\\right)\\\\\nS\n&=&\\mbox{diag}\\left(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2}\\right)\\\\\n\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n&=&\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\mu_i}{\\partial \\theta}\\right)^2\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=&\\mbox{diag}\\left(2\\sigma_1 \\frac{\\partial \\sigma_1}{\\partial \\theta},2\\sigma_2\\frac{\\partial \\sigma_2}{\\partial \\theta},...,2\\sigma_n\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta}\n&=&\\mbox{diag}\\left(-2\\sigma_1^{-3} \\frac{\\partial \\sigma_1}{\\partial \\theta},-2\\sigma_2^{-3}\\frac{\\partial \\sigma_2}{\\partial \\theta},...,-2\\sigma_n^{-3}\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=&\\mbox{diag}\\left(-4\\sigma_1^{-2} \\frac{\\partial \\sigma_1}{\\partial \\theta},-4\\sigma_2^{-2}\\frac{\\partial \\sigma_2}{\\partial \\theta},...,-4\\sigma_n^{-2}\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)\\\\\n&=&-4\\mbox{diag}\\left(\\frac{1}{\\sigma_1^2} \\frac{\\partial \\sigma_1}{\\partial \\theta},\\frac{1}{\\sigma_2^2}\\frac{\\partial \\sigma_2}{\\partial \\theta},...,\\frac{1}{\\sigma_n^2}\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)\\\\\n-\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n&=&2\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\sigma_i}{\\partial \\theta}\\right)^2\n\\end{eqnarray}\n\nand so the prior is\n\\begin{eqnarray}\np(\\theta)&=&\\sqrt{\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\mu_i}{\\partial \\theta}\\right)^2\n               +2\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\sigma_i}{\\partial \\theta}\\right)^2}\n\\end{eqnarray}\n\nwhich agrees with equation 19 in~\\citet{jp1}.\n\n\\subsection{Constant Covariance}\n\nIf the covariance $\\Sigma$ is constant then equation~\\ref{eq100} reduces immediately to\n\\begin{eqnarray}\np(\\theta)&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}}\n\\end{eqnarray}\n\nwhich agrees with equation 19 in~\\citet{jp2}.\n\n\\subsection{Constant Correlation}\n\nIf the correlations are modelled as constant (but the variances are allowed to vary) then:\n\n\\begin{eqnarray}\n\\Sigma&=&VCV\\\\\n\\mbox{where }V&=&\\mbox{diag}(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2)\\\\\n\\mbox{and }C&=&\\mbox{the correlation matrix}\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=& \\frac{\\partial V}{\\partial \\theta}CV+VC\\frac{\\partial V}{\\partial \\theta}\\\\\n&=& 2\\frac{\\partial V}{\\partial \\theta}CV\\\\\n\\mbox{where } \\frac{\\partial V}{\\partial \\theta}\n&=&2\\mbox{diag}(\\sigma_1\\frac{\\partial \\sigma_1}{\\partial \\theta},\\sigma_2\\frac{\\partial \\sigma_2}{\\partial \\theta},...,\\sigma_n\\frac{\\partial \\sigma_n}{\\partial \\theta})\\\\\nS&=&V^{-1} C^{-1} V^{-1}\\\\\n\\mbox{where }V^{-1}&=&\\mbox{diag}(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2})\\\\\n\\mbox{and }C^{-1}&=&\\mbox{the inverse correlation matrix}\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=& \\frac{\\partial V^{-1}}{\\partial \\theta}C^{-1}V^{-1}+V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta}\\\\\n&=& 2V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta}\\\\\n\\mbox{where } \\frac{\\partial V^{-1}}{\\partial \\theta}\n&=&-2\\mbox{diag}\\left(\\frac{1}{\\sigma_1^2}\\frac{\\partial \\sigma_1}{\\partial \\theta},\\frac{1}{\\sigma_2^2}\\frac{\\partial \\sigma_2}{\\partial \\theta},...,\\frac{1}{\\sigma_n^2}\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=&4 V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta}\\frac{\\partial V}{\\partial \\theta}CV\\\\\n\\frac{\\partial V^{-1}}{\\partial \\theta}\\frac{\\partial V}{\\partial \\theta}\n&=&-4\\mbox{diag}\\left(\\frac{1}{\\sigma_1}\\left(\\frac{\\partial \\sigma_1}{\\partial \\theta}\\right)^2,\\frac{1}{\\sigma_2}\\left(\\frac{\\partial \\sigma_2}{\\partial \\theta}\\right)^2,...,\\frac{1}{\\sigma_n}\\left(\\frac{\\partial \\sigma_n}{\\partial \\theta}\\right)^2\\right)\n\\end{eqnarray}\n\nand the prior is:\n\\begin{eqnarray}\np(\\theta)\n&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n-\\frac{1}{2}\\mbox{tr}\\left(4 V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta}\\frac{\\partial V}{\\partial \\theta}CV\\right)}\n\\end{eqnarray}\n\n\\subsection{Constant Variance}\n\nIf the variances are modelled as constant (but the correlations are allowed to vary) then:\n\\begin{eqnarray}\n\\Sigma&=&VCV\\\\\n\\mbox{where }V&=&\\mbox{diag}(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2)\\\\\n\\mbox{and }C&=&\\mbox{the correlation matrix}\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta}&=& V \\frac{\\partial C}{\\partial \\theta}V\\\\\nS&=&V^{-1} C^{-1} V^{-1}\\\\\n\\mbox{where }V^{-1}&=&\\mbox{diag}(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2})\\\\\n\\mbox{and }C^{-1}&=&\\mbox{the inverse correlation matrix}\\\\\n\\frac{\\partial S}{\\partial \\theta}&=&V^{-1} \\frac{\\partial C^{-1}}{\\partial \\theta}V^{-1}\\\\\n\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\theta}\n&=&V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta}V^{-1}V\\frac{\\partial C}{\\partial \\theta}V\\\\\n&=&V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta}\\frac{\\partial C}{\\partial \\theta}V\\\\\n&=&-V^{-1}C^{-1}\\frac{\\partial C}{\\partial \\theta}C^{-1}\\frac{\\partial C}{\\partial \\theta}V\n\\end{eqnarray}\n\nand so the prior is:\n\\begin{eqnarray}\np(\\theta)\n&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n-\\frac{1}{2}\\mbox{tr}\\left(V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta}\\frac{\\partial C}{\\partial \\theta}V\\right)}\\label{eq101}\\\\\n&=&\\sqrt{\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\theta}\n+\\frac{1}{2}\\mbox{tr}\\left(V^{-1}C^{-1}\\frac{\\partial C}{\\partial \\theta}C^{-1}\\frac{\\partial C}{\\partial \\theta}V\\right)}\\label{eq101b}\n\\end{eqnarray}\n\n\\section{Multiple Parameters}\\label{s4}\n\nWe now consider the multiparameter case.\nWe start with two parameters and generalise to multiple parameters later.\nThe derivations are only slightly more complex than those for the single parameter case.\nStarting from equation~\\ref{eq10}, which was:\n\n\\begin{eqnarray}\n\\frac{\\partial \\ln p(x|\\theta)}{\\partial \\theta}\n&=&-\\frac{1}{2} \\frac{\\partial \\ln D}{\\partial \\theta}\n   -\\frac{1}{2}x^T \\frac{\\partial S}{\\partial \\theta} x\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\mu\n   +x^T S\\frac{\\partial \\mu}{\\partial \\theta}\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu\n\\end{eqnarray}\nwe now take the derivative wrt a second parameter $\\phi$:\n\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta \\partial \\phi}\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} x\n   +x^T \\frac{\\partial}{\\partial \\phi}\\left(\\frac{\\partial S}{\\partial \\theta}\\mu\\right)\n   +x^T \\frac{\\partial}{\\partial \\phi}\\left(S\\frac{\\partial \\mu}{\\partial \\theta}\\right)\\\\\n&& -\\frac{\\partial}{\\partial \\phi}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta}S \\mu\\right)\n   -\\frac{1}{2}\\frac{\\partial}{\\partial \\phi}\\left(\\mu^T\\frac{\\partial S }{\\partial \\theta}\\mu\\right)\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} x\n   +x^T \\left(\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\mu+\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\right)\n   +x^T \\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\mu}{\\partial \\theta}+S\\frac{\\partial^2 \\mu}{\\partial \\theta \\partial \\phi}\\right)\\\\\n&&   -\\left(\\frac{\\partial^2 \\mu^T }{\\partial \\theta \\partial \\phi}S \\mu\n                    +\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\phi} \\mu\n                    +\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\\right)\n   -\\frac{1}{2}\\left(\\frac{\\partial \\mu^T}{\\partial \\phi}\\frac{\\partial S }{\\partial \\theta}\\mu\n         +\\mu^T\\frac{\\partial^2 S }{\\partial \\theta \\partial \\phi}\\mu\n         +\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\right)\\nonumber\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} x\n   +x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\mu\n   +x^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\n   +x^T \\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\mu}{\\partial \\theta}\n   +x^T S\\frac{\\partial^2 \\mu}{\\partial \\theta \\partial \\phi}\\\\\n&&   -\\frac{\\partial^2 \\mu^T }{\\partial \\theta \\partial \\phi}S \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\phi} \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T}{\\partial \\phi}\\frac{\\partial S }{\\partial \\theta}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta \\partial \\phi}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\n\\end{eqnarray}\n\nTaking expectations:\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta^2}\\right)\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}E\\left(x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} x\\right)\n   +E(x^T) \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\mu\n   +E(x^T) \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&& +E(x^T) \\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\mu}{\\partial \\theta}\n   +E(x^T) S\\frac{\\partial^2 \\mu}{\\partial \\theta \\partial \\phi}\n     -\\frac{\\partial^2 \\mu^T }{\\partial \\theta \\partial \\phi}S \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\phi} \\mu\\\\\n&&   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T}{\\partial \\phi}\\frac{\\partial S }{\\partial \\theta}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta \\partial \\phi}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\n\\end{eqnarray}\n\nBut $E(x)=\\mu$, and, from appendix 2,\n\\begin{equation}\nE\\left(x^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} x\\right)\n=\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma \\right)+\\mu^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} \\mu\n\\end{equation}\n\nand so:\n\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta \\partial \\phi}\\right)\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}\\left(\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)+\\mu^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} \\mu\\right)\n   +\\mu^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\mu\n   +\\mu^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&& +\\mu^T \\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\mu}{\\partial \\theta}\n   +\\mu^T S\\frac{\\partial^2 \\mu}{\\partial \\theta \\partial \\phi}\n     -\\frac{\\partial^2 \\mu^T }{\\partial \\theta \\partial \\phi}S \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\phi} \\mu\\\\\n&&   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T}{\\partial \\phi}\\frac{\\partial S }{\\partial \\theta}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta \\partial \\phi}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&=&-\\frac{1}{2} \\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   -\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma \\right)\n   -\\frac{1}{2}\\mu^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi} \\mu\n   +\\mu^T \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\mu\n   +\\mu^T \\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&& +\\mu^T \\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\mu}{\\partial \\theta}\n   +\\mu^T S\\frac{\\partial^2 \\mu}{\\partial \\theta \\partial \\phi}\n     -\\frac{\\partial^2 \\mu^T }{\\partial \\theta \\partial \\phi}S \\mu\n     -\\frac{\\partial \\mu^T }{\\partial \\theta}\\frac{\\partial S}{\\partial \\phi} \\mu\\\\\n&&   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\n   -\\frac{1}{2}\\frac{\\partial \\mu^T}{\\partial \\phi}\\frac{\\partial S }{\\partial \\theta}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial^2 S }{\\partial \\theta \\partial \\phi}\\mu\n   -\\frac{1}{2}\\mu^T\\frac{\\partial S }{\\partial \\theta}\\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&=&-\\frac{1}{2} \\left[\\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   +\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)\\right]\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\\label{eq4}\n\\end{eqnarray}\n\nBut\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{\\partial \\log D}{\\partial \\phi}\\right)\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{1}{D}\\frac{\\partial D}{\\partial \\phi}\\right)\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\frac{1}{D}\\left(D\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\\right)\\right)\\\\\n&=&\\frac{\\partial}{\\partial \\theta}\\left(\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial}{\\partial \\theta}\\left(S\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}+S\\frac{\\partial^2 \\Sigma}{\\partial \\theta \\partial \\phi}\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)+\\mbox{tr}\\left( S \\frac{\\partial^2 \\Sigma}{\\partial \\theta \\partial \\phi}\\right)\n\\end{eqnarray}\n\nHowever\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial \\theta}\\left(S \\Sigma\\right)&=&S \\frac{\\partial \\Sigma}{\\partial \\theta}+\\frac{\\partial S}{\\partial \\theta}\\Sigma\\\\\n\\frac{\\partial^2}{\\partial \\theta \\partial \\phi}\\left( S \\Sigma\\right)\n&=&S \\frac{\\partial^2 \\Sigma}{\\partial \\theta \\partial \\phi}\n    +\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\n    +\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\n    +\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\n\\end{eqnarray}\n\nBut $S\\Sigma=I$ and so $\\frac{\\partial^2}{\\partial \\theta \\partial \\phi}( S \\Sigma)=0$, implying that\n\\begin{equation}\nS\\frac{\\partial^2 \\Sigma}{\\partial \\theta \\partial \\phi}=- \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\n-\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\n-\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\n\\end{equation}\n\nand so\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)+\\mbox{tr}\\left( S \\frac{\\partial^2 \\Sigma}{\\partial \\theta \\partial \\phi}\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\n+\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\n -\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\n -\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\\\\n&=&\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\n+\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)\n-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta}\\frac{\\partial \\Sigma}{\\partial \\phi}\\right)\n-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\\\\n&=&\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)\n-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n\\end{eqnarray}\n\nGiving:\n\\begin{eqnarray}\n\\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}+\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma \\right)\n&=&\\mbox{tr}\\left(- \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)\n-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n+\\mbox{tr}\\left(\\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma \\right)\\\\\n&=&-\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\\label{eq5}\n\\end{eqnarray}\n\nSubstituting expression~\\ref{eq5} into equation~\\ref{eq4} gives:\n\n\\begin{eqnarray}\nE\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta \\partial \\phi}\\right)\n&=&-\\frac{1}{2} \\left[\\frac{\\partial^2 \\ln D}{\\partial \\theta \\partial \\phi}\n   +\\mbox{tr}\\left( \\frac{\\partial^2 S}{\\partial \\theta \\partial \\phi}\\Sigma\\right)\\right]\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&=&\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\phi}\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\\\\\n&=&-\\frac{1}{2}\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\phi}S\\frac{\\partial \\Sigma}{\\partial \\theta}\\right)\n   -\\frac{\\partial \\mu^T }{\\partial \\theta}S \\frac{\\partial \\mu}{\\partial \\phi}\n\\end{eqnarray}\n\nWe now generalize from two to multiple parameters. We change the notation so that $\\theta$ is the vector\nof all parameters. The prior is then:\n\\begin{eqnarray}\np(\\theta)\n&=&\\sqrt{-\\mbox{det}E\\left(\\frac{\\partial^2 \\ln p(x|\\theta)}{\\partial \\theta_j \\partial \\theta_k}\\right)}\\\\\n&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n-\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\\right)\\right)}\\label{eq101}\\\\\n&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n+\\frac{1}{2}\\mbox{tr}\\left(S\\frac{\\partial \\Sigma}{\\partial \\theta_j}S\\frac{\\partial \\Sigma}{\\partial \\theta_k}\\right)\\right)}\\label{eq101b}\n\\end{eqnarray}\n\nIf there are $n$ observations and $m$ parameters then:\n\\begin{itemize}\n  \\item $\\mu$ is an $n$ x $1$ vector\n  \\item $\\frac{\\partial \\mu}{\\partial \\theta_j}$ is an $n$ by $1$ vector\n  \\item $\\mu^T$ is a $1$ x $n$ vector\n  \\item $\\frac{\\partial \\mu^T}{\\partial \\theta_k}$ is a $1$ by $n$ vector\n  \\item $S$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}$ is a scalar\n  (which is the $(j,k)$th element of a matrix)\n  \\item $\\frac{\\partial S}{\\partial \\theta_j}$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial \\Sigma}{\\partial \\theta_k}$ is an $n$ by $n$ matrix\n  \\item $\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}$ is an $n$ by $n$ matrix\n  \\item tr$\\left(\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\\right)$ is a scalar\n    (which is the $(j,k)$th element of a matrix)\n\\end{itemize}\n\nWe now again consider various special cases of this general formula, starting with the two cases described in~\\citet{jp1} and~\\citet{jp2}.\n\n\\subsection{Independence}\n\nIf the observations are modelled as independent then:\n\\begin{eqnarray}\n\\Sigma\n&=&\\mbox{diag}\\left(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2\\right)\\\\\nS\n&=&\\mbox{diag}\\left(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2}\\right)\\\\\n\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n&=&\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\mu_i}{\\partial \\theta_j}\\right)\\left(\\frac{\\partial \\mu_i}{\\partial \\theta_k}\\right)\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta_k}\n&=&\\mbox{diag}\\left(2\\sigma_1 \\frac{\\partial \\sigma_1}{\\partial \\theta_k},2\\sigma_2\\frac{\\partial \\sigma_2}{\\partial \\theta_k},...,2\\sigma_n\\frac{\\partial \\sigma_n}{\\partial \\theta_k}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta_j}\n&=&\\mbox{diag}\\left(-2\\sigma_1^{-3} \\frac{\\partial \\sigma_1}{\\partial \\theta_j},-2\\sigma_2^{-3}\\frac{\\partial \\sigma_2}{\\partial \\theta_j},...,-2\\sigma_n^{-3}\\frac{\\partial \\sigma_n}{\\partial \\theta_j}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\n&=&\\mbox{diag}\\left(-4\\sigma_1^{-2} \\frac{\\partial \\sigma_1}{\\partial \\theta_j}\\frac{\\partial \\sigma_1}{\\partial \\theta_k},-4\\sigma_2^{-2}\\frac{\\partial \\sigma_2}{\\partial \\theta_j}\\frac{\\partial \\sigma_2}{\\partial \\theta_k},...,-4\\sigma_n^{-2}\\frac{\\partial \\sigma_n}{\\partial \\theta_j}\\frac{\\partial \\sigma_n}{\\partial \\theta_k}\\right)\\\\\n&=&-4\\mbox{diag}\\left(\\frac{1}{\\sigma_1^2} \\frac{\\partial \\sigma_1}{\\partial \\theta_j}\\frac{\\partial \\sigma_1}{\\partial \\theta_k},\\frac{1}{\\sigma_2^2}\\frac{\\partial \\sigma_2}{\\partial \\theta_j}\\frac{\\partial \\sigma_2}{\\partial \\theta_k},...,\\frac{1}{\\sigma_n^2}\\frac{\\partial \\sigma_n}{\\partial \\theta_j}\\frac{\\partial \\sigma_n}{\\partial \\theta_k}\\right)\\\\\n-\\frac{1}{2}\\mbox{tr}\\left(\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\\right)\n&=&2\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\frac{\\partial \\sigma_i}{\\partial \\theta_j}\\frac{\\partial \\sigma_i}{\\partial \\theta_k}\n\\end{eqnarray}\n\nand so\n\\begin{eqnarray}\np(\\theta)&=&\\sqrt{\\mbox{det}\\left(\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\mu_i}{\\partial \\theta_j}\\right)\\left(\\frac{\\partial \\mu_i}{\\partial \\theta_k}\\right)\n               +2\\sum_{i=1}^{n}\\frac{1}{\\sigma_i^2}\\left(\\frac{\\partial \\sigma_i}{\\partial \\theta_j}\\right)\\left(\\frac{\\partial \\sigma_i}{\\partial \\theta_k}\\right)\\right)}\n\\end{eqnarray}\n\nwhich agrees with equation 36 in~\\citet{jp1}.\n\n\\subsection{Constant Covariance}\n\nIf the covariance $\\Sigma$ is constant then equation~\\ref{eq101} reduces immediately to\n\\begin{eqnarray}\np(\\theta)&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\\right)}\n\\end{eqnarray}\n\nwhich agrees with equation 28 in~\\citet{jp2}.\n\n\\subsection{Constant Correlation}\n\nIf the correlations are modelled as constant (but the variances are allowed to vary) then:\n\n\\begin{eqnarray}\n\\Sigma&=&VCV\\\\\n\\mbox{where }V&=&\\mbox{diag}(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2)\\\\\n\\mbox{and }C&=&\\mbox{the correlation matrix}\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta_k}\n&=& \\frac{\\partial V}{\\partial \\theta_k}CV+VC\\frac{\\partial V}{\\partial \\theta_k}\\\\\n&=& 2\\frac{\\partial V}{\\partial \\theta_k}CV\\\\\n\\mbox{where } \\frac{\\partial V}{\\partial \\theta_k}\n&=&2\\mbox{diag}(\\sigma_1\\frac{\\partial \\sigma_1}{\\partial \\theta_k},\\sigma_2\\frac{\\partial \\sigma_2}{\\partial \\theta_k},...,\\sigma_n\\frac{\\partial \\sigma_n}{\\partial \\theta_k})\\\\\nS&=&V^{-1} C^{-1} V^{-1}\\\\\n\\mbox{where }V^{-1}&=&\\mbox{diag}(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2})\\\\\n\\mbox{and }C^{-1}&=&\\mbox{the inverse correlation matrix}\\\\\n\\frac{\\partial S}{\\partial \\theta_j}\n&=& \\frac{\\partial V^{-1}}{\\partial \\theta_j}C^{-1}V^{-1}+V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta_j}\\\\\n&=& 2V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta_j}\\\\\n\\mbox{where } \\frac{\\partial V^{-1}}{\\partial \\theta_j}\n&=&-2\\mbox{diag}\\left(\\frac{1}{\\sigma_1^2}\\frac{\\partial \\sigma_1}{\\partial \\theta_j},\\frac{1}{\\sigma_2^2}\\frac{\\partial \\sigma_2}{\\partial \\theta_j},...,\\frac{1}{\\sigma_n^2}\\frac{\\partial \\sigma_n}{\\partial \\theta_j}\\right)\\\\\n\\frac{\\partial V^{-1}}{\\partial \\theta_j}\\frac{\\partial V}{\\partial \\theta_k}\n&=&-4\\mbox{diag}\\left(\\frac{1}{\\sigma_1}\\frac{\\partial \\sigma_1}{\\partial \\theta_j}\\frac{\\partial \\sigma_1}{\\partial \\theta_k},\\frac{1}{\\sigma_2}\\frac{\\partial \\sigma_2}{\\partial \\theta_j}\\frac{\\partial \\sigma_2}{\\partial \\theta_k},...,\\frac{1}{\\sigma_n}\\frac{\\partial \\sigma_n}{\\partial \\theta_j}\\frac{\\partial \\sigma_n}{\\partial \\theta_k}\\right)\\\\\n\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\n&=&4 V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta_j}\\frac{\\partial V}{\\partial \\theta_k}CV\n\\end{eqnarray}\n\nand the prior is:\n\\begin{eqnarray}\np(\\theta)\n&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n-\\frac{1}{2}\\mbox{tr}\\left(4 V^{-1}C^{-1}\\frac{\\partial V^{-1}}{\\partial \\theta_j}\\frac{\\partial V}{\\partial \\theta_k}CV\\right)\\right)}\n\\end{eqnarray}\n\n\\subsection{Constant Variance}\n\nIf the variances are modelled as constant (but the correlations are allowed to vary) then:\n\\begin{eqnarray}\n\\Sigma&=&VCV\\\\\n\\mbox{where }V&=&\\mbox{diag}(\\sigma_1^2,\\sigma_2^2,...,\\sigma_n^2)\\\\\n\\mbox{and }C&=&\\mbox{the correlation matrix}\\\\\n\\frac{\\partial \\Sigma}{\\partial \\theta_k}&=& V \\frac{\\partial C}{\\partial \\theta}V\\\\\nS&=&V^{-1} C^{-1} V^{-1}\\\\\n\\mbox{where }V&=&\\mbox{diag}(\\sigma_1^{-2},\\sigma_2^{-2},...,\\sigma_n^{-2})\\\\\n\\mbox{and }C{-1}&=&\\mbox{the inverse correlation matrix}\\\\\n\\frac{\\partial S}{\\partial \\theta_j}&=&V^{-1} \\frac{\\partial C^{-1}}{\\partial \\theta_j}V^{-1}\\\\\n\\frac{\\partial S}{\\partial \\theta_j}\\frac{\\partial \\Sigma}{\\partial \\theta_k}\n&=&V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta_j}VV^{-1}\\frac{\\partial C}{\\partial \\theta_k}V\\\\\n&=&V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta_j}\\frac{\\partial C}{\\partial \\theta_k}V\\\\\n&=&-V^{-1}C^{-1}\\frac{\\partial C}{\\partial \\theta_j}C^{-1}\\frac{\\partial C}{\\partial \\theta_k}V\n\\end{eqnarray}\n\nand the prior is:\n\\begin{eqnarray}\np(y|x)\n&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n-\\frac{1}{2}\\mbox{tr}\\left(V^{-1}\\frac{\\partial C^{-1}}{\\partial \\theta_j}\\frac{\\partial C}{\\partial \\theta_k}V\\right)\\right)}\\\\\n&=&\\sqrt{\\mbox{det}\\left(\\frac{\\partial \\mu^T }{\\partial \\theta_j}S \\frac{\\partial \\mu}{\\partial \\theta_k}\n+\\frac{1}{2}\\mbox{tr}\\left(V^{-1}C^{-1}\\frac{\\partial C}{\\partial \\theta_j}C^{-1}\\frac{\\partial C}{\\partial \\theta_k}V\\right)\\right)}\n\\end{eqnarray}\n\n\\section{Summary}\\label{s5}\n\nClimate models are statistical models, in that they produce probabilistic predictions\n(when run as initial condition ensembles) and have certain parameters that can only\nbe determined by comparison of model results with observations.\nAs a result, the Bayesian framework, in which probabilistic predictions made from models with different parameter values are\ncombined together to make a single best probabilistic prediction, can be applied.\nWithin that framework one has to specify a prior, and one must choose between a prior based on intuition or a prior\nbased on a rule. The former is known as subjective Bayesian statistics, and the latter, objective Bayesian statistics.\nThe authors are pursuing a research programme that is exploring methods by which objective Bayesian statistics can be applied in\nclimate modelling. In this article we have discussed the application of the most standard objective prior, known as\nJeffreys' Prior.\n\nClimate models are complex and the relationship between the parameters\nand the predicted distributions is also complex. However, the form of the predicted distributions themselves can often\nbe rather simple, and for many groups of variables a multivariate normal may be a good approximation.\nWe have shown that, by making this approximation, the calculation of Jeffreys' Prior can be reduced to\ndifferentiating the parameters of the multivariate normal by the parameters of the underlying climate model.\nWe derive expressions for Jeffreys' Prior in this situation.\n\nThe results from our two previous articles on this topic (\\citet{jp1} and~\\citet{jp2}) are special cases of the general results shown here.\n\nIn all this work we have expressed Jeffreys' Prior in terms of the true, rather than the estimated, parameters of the distributions from climate model predictions. In other words, we have assumed infinite rather than finite size initial condition ensembles.\nA further challenge is to rederive the expressions given above but incorporating estimation uncertainty.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe physical motivation for the development of supermanifolds stems from quantum field theory in its functional integral formulation, which describes fermionic particles by anticommuting fields. In the 1970s, pioneering work by Berezin strongly suggested that commuting and anticommuting variables should be treated on equal footing. Several theories of supermanifolds have been advocated, among which the definition of Berezin, Kostant, and Leites is one of the most commonly used in mathematics. \n\nOur motivation for the study of supermanifolds comes from the study of certain nonlinear $\\sigma$-models with supersymmetry. Indeed, it is known from the work of the third named author \\cite{zirnbauer-rsss} that Riemannian symmetric superspaces occur naturally in the large $N$ limit of certain random matrix ensembles, which correspond to Cartan's ten infinite series of symmetric spaces. In spite of their importance in physics, the mathematical theory of these superspaces is virtually non-existent. (But compare \\cite{duflo-petracci-formalss,zirnbauer-superbos,goertsches-diss}.) We intend to initiate the systematic study of Riemannian symmetric superspaces, in order to obtain a good understanding of, in particular, the invariant differential operators, the spherical functions, and the related harmonic analysis. \nThe present work lays an important foundation for this endeavour: the generalisation of Chevalley's restriction theorem to the super setting. \n\nTo describe our results in detail, let us make our assumptions more precise. Let $\\mathfrak g$ be a complex Lie superalgebra with even centre such that  $\\mathfrak g_0$ is reductive in $\\mathfrak g$ and $\\mathfrak g$ carries an even invariant supersymmetric form. Let $\\theta$ be an involutive automorphism of $\\mathfrak g$, and denote by $\\mathfrak g=\\mathfrak k\\oplus\\mathfrak p$ the decomposition into $\\theta$-eigenspaces. We say that $(\\mathfrak g,\\mathfrak k)$ is a \\emph{reductive superpair}, and it is of \\emph{even type} if there exists an even Cartan subspace $\\mathfrak a\\subset\\mathfrak p_0$. \n\nAssume that $(\\mathfrak g,\\mathfrak k)$ is a reductive symmetric superpair of even type. Let $\\bar\\Sigma_1^+$ denote the set of positive roots of $\\mathfrak g_1:\\mathfrak a$ such that  $\\lambda,2\\lambda$ are no roots of $\\mathfrak g_0:\\mathfrak a$. To each $\\lambda\\in\\bar\\Sigma_1^+$, one associates a set $\\mathcal R_\\lambda$ of differential operators with rational coefficients on $\\mathfrak a$. \n\nOur main results are as follows. \n\n\\begin{Th*}[A]\n\tLet $I(\\mathfrak a^*)$ be the image of the restriction map $S(\\mathfrak p^*)^{\\mathfrak k}\\to S(\\mathfrak a^*)$ (which is injective). Then $I(\\mathfrak a^*)$ is the set of $W$-invariant polynomials on $\\mathfrak a$ which lie in the common domain of all operators in $\\mathcal R_\\lambda$, $\\lambda\\in\\bar\\Sigma_1^+$. Here, $W$ is the Weyl group of $\\mathfrak g_0:\\mathfrak a$.\n\\end{Th*}\n\nFor $\\lambda\\in\\bar\\Sigma_1^+$, let $A_\\lambda\\in\\mathfrak a$ be the corresponding coroot, and denote by $\\partial(A_\\lambda)$ the directional derivative operator in the direction of $A_\\lambda$. Then the image $I(\\mathfrak a^*)$ can be characterised in more explicit terms, as follows.\n\n\\begin{Th*}[B]\n\tWe have $I(\\mathfrak a^*)=\\bigcap_{\\lambda\\in\\bar\\Sigma_1^+}S(\\mathfrak a^*)^W\\cap I_\\lambda$ where\n\t\\[\n\t\tI_\\lambda=\\textstyle\\bigcap_{j=1}^{\\frac12m_{1,\\lambda}}\\dom\\lambda^{-j}\\partial(A_\\lambda)^j\\mathtxt{if}\\lambda(A_\\lambda)=0\\ ,\n\t\\]\n\tand if $\\lambda(A_\\lambda)\\neq0$, then $I_\\lambda$ consists of those $p\\in\\cplxs[\\mathfrak a]$ such that \n\t\\[\n\t\t\\partial(A_\\lambda)^kp|_{\\ker\\lambda}=0\\mathfa[odd integers] k\\ ,\\ 1\\sle k\\sle m_{1,\\lambda}-1\\ .\n\t\\]\n\tHere, $m_{1,\\lambda}$ denotes the multiplicity of $\\lambda$ in $\\mathfrak g_1$ (and is an even integer).\n\\end{Th*}\n\nIf the symmetric pair $(\\mathfrak g,\\mathfrak k)$ is of \\emph{group type}, \\emph{i.e.}~$\\mathfrak g=\\mathfrak k\\oplus\\mathfrak k$ with the flip involution, then for all  $\\lambda\\in\\bar\\Sigma_1^+$, $\\lambda(A_\\lambda)=0$, and the multiplicity $m_{1,\\lambda}=2$. In this case, Theorem (B) reduces to $I(\\mathfrak a^*)=\\bigcap_{\\lambda\\in\\bar\\Sigma_1^+}S(\\mathfrak a^*)^W\\cap\\dom\\lambda^{-1}\\partial(A_\\lambda)$. The situation where $\\lambda(A_\\lambda)\\neq0$ for some  $\\lambda\\in\\bar\\Sigma_1^+$ occurs if and only if $\\mathfrak g$ contains symmetric subalgebras $\\mathfrak s\\cong C(2)=\\mathfrak{osp}(2|2)$ where $\\mathfrak s_0\\cap\\mathfrak k=\\sll(2,\\cplxs)$. This is case for $\\mathfrak g=C(q+1)$ with a special involution, and in this case, the invariant algebra $I(\\mathfrak a^*)$ defines the singular curve $z^{2q+1}=w^2$ (\\thmref{Cor}{singular}). \n\nLet us place our result in the context of the literature. The Theorems (A) and (B) apply to the case of classical Lie superalgebras with non-degenerate invariant even form (equivalently, finite-dimensional contragredient Lie superalgebras), considered as symmetric superspaces of group type. In this case, the result is due to Sergeev \\cite{sergeev-invpol}, Kac \\cite{kac-laplace}, and Gorelik \\cite{gorelik-kacconstruction}, and we simply furnish a new (and elementary) proof. (The results of Sergeev are also valid for basic Lie superalgebras which are not contragredient.) For some particular cases, there are earlier results by Berezin \\cite{berezin}.\n\nSergeev's original proof involves case-by-case calculations. The proof by Gorelik---which carries out in detail ideas due to Kac in the context of Kac--Moody algebras---is classification-free, and uses so-called Shapovalov determinants. Moreover, the result of Kac and Gorelik actually characterises the image of the Harish-Chandra homomorphism rather than the image of the restriction map on the symmetric algebra, and is therefore more fundamental than our result.\n\nStill in the case of symmetric superpairs of group type, Kac \\cite{kac-typical} and Santos \\cite{santos-zuckerman} describe the image of the restriction morphism in terms of supercharacters of certain (cohomologically) induced modules (instead of a characterisation in terms of a system of differential equations). This approach cannot carry over to the case of symmetric pairs, as is known in the even case from the work of Helgason \\cite{helgason-fundamental}. \n\nOur result also applies in the context of Riemannian symmetric superspaces, where one has an even non-degenerate $\\mathcal G$-invariant supersymmetric form on $\\mathcal G\/\\mathcal K$ whose restriction to the base $G\/K$ is Riemannian. In this setting, it is to our knowledge completely new and not covered by earlier results. We point out that a particular case was proved in the PhD thesis of Fuchs \\cite{fuchs-diss}, in the framework of the `supermatrix model', using a technique due to Berezin. \n\nIn the context of harmonic analysis of even Riemannian symmetric spaces $G\/K$, Chevalley's restriction theorem enters crucially, since it determines the image of the Harish-Chandra homomorphism, and thereby, the spectrum of the algebra $\\mathbb D(G\/K)$ of $G$-invariant differential operators on $G\/K$. It is an important ingredient in the proof of Harish-Chandra's integral formula for the spherical functions. In a series of forthcoming papers, we will apply our generalisation of Chevalley's restriction theorem to obtain analogous results in the context of Riemannian symmetric superspaces. \n\n\\medskip\\noindent\nLet us give a brief overview of the contents of our paper. We review some basic facts on root decompositions in sections 2.1-2.2. In section 2.3, we introduce our main tool in the proof of Theorem (A), a certain twisted action $u_z$ on the supersymmetric algebra $S(\\mathfrak p)$. In section 3.1, we define the `radial component' map $\\gamma_z$ via the twisted action $u_z$. The proofs of Theorems (A) and (B) are contained in sections 3.2 and 3.3, respectively. The former comes down to a study of the singularities of $\\gamma_z$ as a function of the semi-simple $z\\in\\mathfrak p_0$, whereas the latter consists in an elementary and explicit discussion of the radial components of certain differential operators. In sections 4.1 and 4.2, we discuss the generality of the `even type' condition, and study an extreme example in some detail. \n\nThe first named author wishes to thank C.~Torossian (Paris VII) for his enlightening remarks on a talk given on an earlier version of this paper. The first and second named author wish to thank M.~Duflo (Paris VII) for helpful discussions, comments, and references. The second named author wishes to thank K.~Nishiyama (Kyoto) for several discussions on the topic. Last, not least, we wish to thank an anonymous but diligent referee whose suggestions greatly improved the presentation of our main technical device. \n\nThis research was partly funded by the IRTG ``Geometry and Analysis of Symmetries'', supported by Deutsche Forschungsgemeinschaft (DFG), Minist\\`ere de l'\\'Education Nationale (MENESR), and Deutsch-Franz\\\"osische Hochschule (DFH-UFA), and by the SFB\/Transregio 12 ``Symmetry and Universality in Mesoscopic Systems'', supported by Deutsche Forschungsgemeinschaft (DFG).\n\n\\section{Some basic facts and definitions}\n\n\\begin{Par*}\n\tIn this section, we mostly collect some basic facts concerning (restricted) root decompositions of Lie superalgebras, and the (super-) symmetric algebra, along with some definitions which we find useful to formulate our main results. As general references for matters super, we refer the reader to \\cite{kostant-supergeom,deligne-morgan-susy,kac-liesuperalgs,scheunert-liesuperalgs}\n\\end{Par*}\n\n\\subsection{Roots of a basic quadratic Lie superalgebra}\n\n\\begin{Def}\n\tLet $\\mathfrak g=\\mathfrak g_0\\oplus\\mathfrak g_1$ be a Lie superalgebra over $\\cplxs$ and $b$ a bilinear form $b$. Recall that $b$ is \\emph{supersymmetric} if $b(u,v)=(-1)^{\\Abs0u\\Abs0v}b(v,u)$ for all  homogeneous $u,v$. We shall call $(\\mathfrak g,b)$ \\emph{quadratic} if $b$ is a non-degenerate, $\\mathfrak g$-invariant, even and supersymmetric form on $\\mathfrak g$. We shall say that $\\mathfrak g$ is \\emph{basic} if $\\mathfrak g_0$ is reductive in $\\mathfrak g$ (\\emph{i.e.}~$\\mathfrak g$ is a semi-simple $\\mathfrak g_0$-module) and $\\mathfrak z(\\mathfrak g)\\subset\\mathfrak g_0$ where $\\mathfrak z(\\mathfrak g)$ denotes the centre of $\\mathfrak g$.\n\\end{Def}\n\n\\begin{Par}\n\tLet $(\\mathfrak g,b)$ be a basic quadratic Lie superalgebra, and $\\mathfrak b$ be a Cartan subalgebra of $\\mathfrak g_0$. \n\t\n\tAs usual \\cite[Chapter II, \\S~4.6]{scheunert-liesuperalgs}, we define\n  \\[\n  V^\\alpha=\\Set1{x\\in V}{\\exists\\,n\\in\\nats\\,:\\,(h-\\alpha(h))^n(x)=0\\ \\textfor all  h\\in\\mathfrak b}\\quad,\\quad\\alpha\\in\\mathfrak b^*\n  \\]\n  for any $\\mathfrak b$-module $V$. Further, the sets of even resp.~odd roots for $\\mathfrak b$ are  \n  \\[\n  \\Delta_0(\\mathfrak g:\\mathfrak b)\n  =\\Set1{\\alpha\\in\\mathfrak b^*\\setminus0}{\\mathfrak g_0^\\alpha\\neq0}\n  \\mathtxt\\AND\\Delta_1(\\mathfrak g:\\mathfrak b)\n  =\\Set1{\\alpha\\in\\mathfrak b^*}{\\mathfrak g_1^\\alpha\\neq0}\\ .\n  \\]\n  We also write $\\Delta_j=\\Delta_j(\\mathfrak g:\\mathfrak b)$. Let $\\Delta=\\Delta(\\mathfrak g:\\mathfrak b)=\\Delta_0\\cup\\Delta_1$. The elements of $\\Delta$ are called \\emph{roots}. We have \n  \\[\n  \\mathfrak g=\\mathfrak b\\oplus\\textstyle\\bigoplus_{\\alpha\\in\\Delta}\\mathfrak g^\\alpha\n  =\\mathfrak b\\oplus\\bigoplus_{\\alpha\\in\\Delta_0}\\mathfrak g_0^\\alpha\n  \t\\oplus\\bigoplus_{\\alpha\\in\\Delta_1}\\mathfrak g_1^\\alpha\\ .\n  \\]\n  It is obvious that $\\Delta_0=\\Delta(\\mathfrak g_0:\\mathfrak b)$, so in particular, it is a reduced abstract root system in its real linear span. Also, since $\\mathfrak g_0$ is reductive in $\\mathfrak g$, the root spaces $\\mathfrak g_i^\\alpha$ are the joint eigenspaces of $\\ad h$, $h\\in\\mathfrak b$ (and not only generalised ones).\n  \n  We collect the basic statements about $\\mathfrak b$-roots. The results are known (\\emph{e.g.}~\\cite{scheunert-liesuperalgs,benayadi-root}), so we omit their proofs.\n\\end{Par}\n\n\\begin{Prop}[superroot]\nLet $\\mathfrak g$ be a basic quadratic Lie superalgebra with invariant form $b$, and $\\mathfrak b$ a Cartan subalgebra of $\\mathfrak g_0$.\n\\begin{enumerate}\n\\item For $\\alpha,\\beta\\in\\Delta\\cup0$, we have $b(\\mathfrak g_j^\\alpha,\\mathfrak g_k^\\beta)=0$ unless $j=k$ and $\\alpha=-\\beta$.\n\\item The form $b$ induces a non-degenerate pairing $\\mathfrak g_j^\\alpha\\times\\mathfrak g_j^{-\\alpha}\\to\\cplxs$. In particular, we have $\\dim\\mathfrak g_j^\\alpha=\\dim\\mathfrak g_j^{-\\alpha}$ and $\\Delta_j=-\\Delta_j$ for $j\\in\\ints\/2\\ints$. \n\\item The form $b$ is non-degenerate on $\\mathfrak b$, so for any $\\lambda\\in\\mathfrak b^*$, there exists a unique $h_\\lambda\\in\\mathfrak b$ such that  $b(h_\\lambda,h)=\\lambda(h)$ for all  $h\\in\\mathfrak b$. \n\\item If $\\alpha(h_\\alpha)\\neq0$, $\\alpha\\in\\Delta_1$, then $2\\alpha\\in\\Delta_0$. In particular, $\\Delta_0\\cap\\Delta_1=\\vvoid$. \n\\item We have $\\mathfrak g_1^0=\\mathfrak z_1(\\mathfrak g)=\\{x\\in\\mathfrak g_1\\mid[x,\\mathfrak g]=0\\}=0$, so $0\\not\\in\\Delta_1$.\n\\item All root spaces $\\mathfrak g^\\alpha$, $\\alpha\\in\\Delta$, $\\alpha(h_\\alpha)\\neq0$, are one-dimensional. \n\\end{enumerate}\n\\end{Prop}\n\n\\subsection{Restricted roots of a reductive symmetric superpair}\n\n\\begin{Def}\nLet $(\\mathfrak g,b)$ be a complex quadratic Lie superalgebra, and $\\theta:\\mathfrak g\\to\\mathfrak g$ an involutive automorphism leaving the form $b$ invariant. If $\\mathfrak g=\\mathfrak k\\oplus\\mathfrak p$ is the $\\theta$-eigenspace decomposition, then we shall call $(\\mathfrak g,\\mathfrak k)$ a \\emph{symmetric superpair}.  We shall say that $(\\mathfrak g,\\mathfrak k)$ is \\emph{reductive} if, moreover, $\\mathfrak g$ is basic.\n\nNote that for any symmetric superpair $(\\mathfrak g,\\mathfrak k)$, $\\mathfrak k$ and $\\mathfrak p$ are $b$-orthogonal and non-degenerate. It is also useful to consider the form $b^\\theta(x,y)=b(x,\\theta y)$ which is even, supersymmetric, non-degenerate and $\\mathfrak k$-invariant.\n\nLet $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair. For arbitrary subspaces $\\mathfrak c,\\mathfrak d\\subset\\mathfrak g$, let $\\mathfrak z_{\\mathfrak d}(\\mathfrak c)=\\Set0{d\\in\\mathfrak d}{[d,\\mathfrak c]=0}$ denote the centraliser of $\\mathfrak c$ in $\\mathfrak d$. Any linear subspace $\\mathfrak a=\\mathfrak z_{\\mathfrak p}(\\mathfrak a)\\subset\\mathfrak p_0$ consisting of semi-simple elements of $\\mathfrak g_0$ is called an \\emph{even Cartan subspace}. If an even Cartan subspace exists, then we say that $(\\mathfrak g,\\mathfrak k)$ is of \\emph{even type}. \n\\end{Def}\n\n\\begin{Par*}\n\tWe state some generalities on even Cartan subspaces. These are known and straightforward to deduce from standard texts such as \\cite{dixmier-envalg,borel-rss}.\n\\end{Par*}\n\n\\begin{Lem}\nLet $\\mathfrak a\\subset\\mathfrak g$ be an even Cartan subspace.\n\\begin{enumerate}\n\\item $\\mathfrak a$ is reductive in $\\mathfrak g$, \\emph{i.e.}~$\\mathfrak g$ is a semi-simple $\\mathfrak a$-module.\n\\item $\\mathfrak z_{\\mathfrak g_0}(\\mathfrak a)$ and $\\mathfrak z_{\\mathfrak g_1}(\\mathfrak a)$ are $b$-non-degenerate.\n\\item $\\mathfrak z_{\\mathfrak g_0}(\\mathfrak a)=\\mathfrak m_0\\oplus\\mathfrak a$ and $\\mathfrak z_{\\mathfrak g_1}(\\mathfrak a)=\\mathfrak m_1$ where $\\mathfrak m_i=\\mathfrak z_{\\mathfrak k_i}(\\mathfrak a)$, and the sum is $b$-orthogonal.\n\\item $\\mathfrak m_0$, $\\mathfrak m_1$, and $\\mathfrak a$ are $b$-non-degenerate.  \n\\item There exists a $\\theta$-stable Cartan subalgebra $\\mathfrak b$ of $\\mathfrak g_0$ containing $\\mathfrak a$. \n\\end{enumerate}\n\\end{Lem}\n\n\\begin{Par}[grouptype]\n\tLet $\\mathfrak k$ be a classical Lie superalgebra with a non-degenerate invariant even form $B$ \\cite{kac-repnclassical}. Then $\\mathfrak k_0$ is reductive in $\\mathfrak k$, and $\\mathfrak z(\\mathfrak k)$ is even. We may define $\\mathfrak g=\\mathfrak k\\oplus\\mathfrak k$, and $b(x,y,x',y')=B(x,x')+B(y,y')$. Then $(\\mathfrak g,b)$ is basic quadratic. The flip involution $\\theta(x,y)=(y,x)$ turns $(\\mathfrak g,\\mathfrak k)$ into a reductive symmetric superpair (where $\\mathfrak k$ is, as is customary, identified with the diagonal in $\\mathfrak g$). We call such a pair of \\emph{group type}. \n\t\n\tMoreover, any Cartan subalgebra $\\mathfrak a$ of $\\mathfrak k_0$ yields an even Cartan subspace for the superpair $(\\mathfrak g,\\mathfrak k)$. Indeed, $\\mathfrak p=\\Set1{(x,-x)}{x\\in\\mathfrak k}$, and the assertion follows from \\thmref{Prop}{superroot}~(v).\n\\end{Par}\n\n\\begin{Par}\n\tIn what follows, let $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair of even type, $\\mathfrak a\\subset\\mathfrak p$ an even Cartan subspace, and $\\mathfrak b\\subset\\mathfrak g_0$ a $\\theta$-stable Cartan subalgebra containing $\\mathfrak a$. The involution $\\theta$ acts on $\\mathfrak b^*$ by $\\theta\\alpha=\\alpha\\circ\\theta$ for all  $\\alpha\\in\\mathfrak b^*$. Let $\\alpha_\\pm=\\frac12(1\\pm\\theta)\\alpha$ for all  $\\alpha\\in\\mathfrak b^*$, and set\n\t\\[\n\t\t\\Sigma_j=\\Sigma_j(\\mathfrak g:\\mathfrak a)=\\Set1{\\alpha_-}{\\alpha\\in\\Delta_j\\ ,\\ \\alpha\\neq\\theta\\alpha}\\ ,\\ \\Sigma=\\Sigma(\\mathfrak g:\\mathfrak a)=\\Sigma_0\\cup\\Sigma_1\\ .\n\t\\]\n\t(The union might not be disjoint.) Identifying $\\mathfrak a^*$ with the annihilator of $\\mathfrak b\\cap\\mathfrak k$ in $\\mathfrak b^*$, these may be considered as subsets of $\\mathfrak a^*$. The elements of $\\Sigma_0$, $\\Sigma_1$, and $\\Sigma$ are called \\emph{even restricted roots}, \\emph{odd restricted roots}, and \\emph{restricted roots}, respectively. For $\\lambda\\in\\Sigma$, let \n\t\\[\n\t\t\\Sigma_j(\\lambda)=\\Set1{\\alpha\\in\\Delta_j}{\\lambda=\\alpha_-}\\ ,\\ \\Sigma(\\lambda)=\\Sigma_0(\\lambda)\\cup\\Sigma_1(\\lambda)\\ .\n\t\\]\n\tIn the following lemma, observe that $\\lambda\\in\\Sigma_j(\\lambda)$ means that $\\lambda\\in\\Delta_j$. We omit the simple proof, which is exactly the same as in the even case \\cite[Chapter 1.1, Appendix 2, Lemma 1]{warner-vol1}.\n\\end{Par}\n\n\\begin{Lem}[rootrestr-fibre-even]\n\tLet $\\lambda\\in\\Sigma_j$, $j=0,1$. The map $\\alpha\\mapsto-\\theta\\alpha$ is a fixed point free involution of $\\Sigma_j(\\lambda)\\setminus\\lambda$. In particular, the cardinality of this set is even. \n\\end{Lem}\n\n\\begin{Par}\n\tFor $\\lambda\\in\\Sigma$, let \n\t\\[\n\t\t\\mathfrak g_{j,\\mathfrak a}^\\lambda=\\Set1{x\\in\\mathfrak g_j}{\\forall h\\in\\mathfrak a\\,:\\,[h,x]=\\lambda(h)\\cdot x}\\ ,\\ \\mathfrak g_{\\mathfrak a}^\\lambda=\\mathfrak g_{0,\\mathfrak a}^\\lambda\\oplus\\mathfrak g_{1,\\mathfrak a}^\\lambda\\ ,\n\t\\]\n\tand $m_{j,\\lambda}=\\dim_\\cplxs\\mathfrak g^\\lambda_{j,\\mathfrak a}$, the \\emph{even} or \\emph{odd multiplicity} of $\\lambda$, according to whether $j=0$ or $j=1$. It is clear that\n\t\\[\n\t\t\\textstyle\\mathfrak g_{j,\\mathfrak a}^\\lambda=\\bigoplus_{\\alpha\\in\\Sigma_j(\\lambda)}\\mathfrak g^\\alpha_j\\ ,\\ m_{j,\\lambda}=\\sum_{\\alpha\\in\\Sigma_j(\\lambda)}\\dim_\\cplxs\\mathfrak g_j^\\alpha\\ ,\\ \\text{\\AND}\\ \\mathfrak g=\\mathfrak z_{\\mathfrak g}(\\mathfrak a)\\oplus\\bigoplus_{\\lambda\\in\\Sigma}\\mathfrak g^\\lambda_{\\mathfrak a}\\ .\n\t\\]\n\\end{Par}\n\n\\begin{Par*}\n\tThe following facts are certainly well-known. Lacking a reference, we give the short proof. \n\\end{Par*}\n\n\\begin{Prop}[res-superroot]\nLet $\\alpha,\\beta\\in\\Delta$, $\\lambda\\in\\Sigma$, and $j,k\\in\\{0,1\\}$.\n\\begin{enumerate}\n\t\\item The form $b^\\theta$ is zero on $\\mathfrak g^\\alpha_j\\times\\mathfrak g_k^\\beta$, unless $j=k$ and $\\alpha=-\\theta\\beta$, in which case it gives a non-degenerate pairing.\n\t\\item There exists a unique $A_\\lambda\\in\\mathfrak a$ such that  $b(A_\\lambda,h)=\\lambda(h)$ for all  $h\\in\\mathfrak a$. \n\t\\item We have $\\dim_\\cplxs\\mathfrak g_j^\\alpha=\\dim_\\cplxs\\mathfrak g_j^{-\\theta\\alpha}$. \n\t\\item The subspace $\\mathfrak g_j(\\lambda)=\\mathfrak g_{j,\\mathfrak a}^\\lambda\\oplus\\mathfrak g_{j,\\mathfrak a}^{-\\lambda}$ is $\\theta$-invariant and decomposes into $\\theta$-eigenspaces as $\\mathfrak g_j(\\lambda)=\\mathfrak k_j^\\lambda\\oplus\\mathfrak p_j^\\lambda$.\n\t\\item The odd multiplicity $m_{1,\\lambda}$ is even, and $b^\\theta$ defines a symplectic form on both $\\mathfrak k_1^\\lambda$ and $\\mathfrak p_1^\\lambda$. \n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n\tThe form $b^\\theta$ is even, so $b^\\theta(\\mathfrak g_0,\\mathfrak g_1)=0$. For $x\\in\\mathfrak g^\\alpha_j$, $y\\in\\mathfrak g^\\beta_j$, we compute, for all  $h\\in\\mathfrak b$, \n\t\\begin{align*}\n\t\t(\\alpha+\\theta\\beta)(h)b^\\theta(x,y)&=b^\\theta([h,x],y)+b^\\theta(x,[\\theta h,y])\\\\\n\t\t&=b^\\theta([h,x]+[x,h],y)=0\\ .\n\t\\end{align*}\n\tHence, $b^\\theta(x,y)=0$ if $\\alpha\\neq-\\theta\\beta$. Since $b^\\theta$ is non-degenerate and $\\mathfrak g\/\\mathfrak b$ is the sum of root spaces, $b^\\theta$ induces a non-degenerate pairing of $\\mathfrak g_j^\\alpha$ and $\\mathfrak g_j^{-\\theta\\alpha}$. We also know already that $\\mathfrak a$ is non-degenerate for $b^\\theta$, and (i)-(iii) follow. Statement (iv) is immediate. \n\t\n\tWe have \n\t\\[\n\t\t\\mathfrak g_{1,\\mathfrak a}^\\lambda\/\\mathfrak g_1^\\lambda\\cong\\textstyle\\bigoplus_{\\alpha\\in\\Sigma_j(\\lambda)\\setminus\\lambda}\\mathfrak g_1^\\alpha\\ .\n\t\\]\n\tBy (iii) and \\thmref{Lem}{rootrestr-fibre-even}, this space is even-dimensional. But $\\lambda$ is a root if and only if $\\lambda=-\\theta\\lambda$. Then $b^\\theta$ defines a symplectic form on $\\mathfrak g_1^\\lambda$ by (i), and this space is even-dimensional. Thus, $m_{1,\\lambda}$ is even, and again by (i), $\\mathfrak g_{1,\\mathfrak a}^\\lambda$ is $b^\\theta$-non-degenerate. It is clear that $\\mathfrak k_1^\\lambda$ and $\\mathfrak p_1^{\\lambda}$ are $b^\\theta$-non-degenerate because $\\mathfrak g_{1,\\mathfrak a}^\\lambda$ and $\\mathfrak g_{1,\\mathfrak a}^{-\\lambda}$ are. Hence, we obtain assertion (v).\n\\end{proof}\n\n\\begin{Rem}\n\tUnlike the case of unrestricted roots, there may exist $\\lambda\\in\\Sigma_1$ such that  $2\\lambda\\not\\in\\Sigma$ but $\\lambda$ is still anisotropic, \\emph{i.e.}~$\\lambda(A_\\lambda)\\neq0$. Indeed, consider $\\mathfrak g=\\mathfrak{osp}(2|2,\\cplxs)$ ($\\cong\\sll(2|1,\\cplxs)$). Then $\\mathfrak g_0=\\mathfrak o(2,\\cplxs)\\oplus\\mathfrak{sp}(2,\\cplxs)=\\mathfrak{gl}(2,\\cplxs)$ and $\\mathfrak g_1$ is the sum of the fundamental representation of $\\mathfrak g_0$ and its dual. \n\t\n\tDefine the involution $\\theta$ to be conjugation by the element $\\begin{Matrix}0\\sigma&0\\\\0&1_2\\end{Matrix}$ where $\\sigma=\\begin{Matrix}00&1\\\\1&0\\end{Matrix}$. One finds that $\\mathfrak k_0=\\sll(2,\\cplxs)$ and $\\mathfrak p_0=\\mathfrak a=\\mathfrak z(\\mathfrak g_0)$ which is one-dimensional and non-degenerate for the supertrace form $b$. On the other hand, $\\mathfrak g_1=\\mathfrak g_1(\\lambda)$ is the sum of the root spaces for certain odd roots $\\pm\\alpha$, $\\pm\\theta\\alpha$ which restrict to $\\pm\\lambda$. Clearly, there are no even roots, so $2\\lambda$ is not a restricted root. Since $A_\\lambda$ generates $\\mathfrak a$, it is a $b$-anisotropic vector. We discuss this issue at some length in section 4.2. \n\t\n\tWe point out that it is also not hard to prove that any such root $\\lambda$ occurs in this setup. \\emph{I.e.}, given a reductive symmetric superpair $(\\mathfrak g,\\mathfrak k)$, for any $\\lambda\\in\\Sigma_1$, $2\\lambda\\not\\in\\Sigma$, $\\lambda(A_\\lambda)\\neq0$, there exists a $b$-non-degenerate $\\theta$-invariant subalgebra $\\mathfrak s\\cong\\mathfrak{osp}(2|2,\\cplxs)$ such that  $\\mathfrak p\\cap\\mathfrak s_0=\\cplxs A_\\lambda=\\mathfrak z(\\mathfrak s_0)$ (the centre of $\\mathfrak s_0$), and $\\dim\\mathfrak s\\cap\\mathfrak g_1(\\lambda)=4$.\n\t\n\tThis phenomenon, of course, cannot occur if the symmetric superpair $(\\mathfrak g,\\mathfrak k)$ is of group type. This reflects the fact that the conditions characterising the invariant algebra may be different in the general case than one might expect from the knowledge of the group case (\\emph{i.e.}~the theorems of Sergeev and Kac, Gorelik).\n\\end{Rem}\n\n\\subsection{The twisted action on the supersymmetric algebra}\n\n\\begin{Par}\n\tLet $V=V_0\\oplus V_1$ be a finite-dimensional super-vector space over $\\cplxs$. We define the supersymmetric algebra  $S(V)=S(V_0)\\otimes\\bigwedge(V_1)$. It is $\\ints$-graded by total degree, as follows: $S^{k,\\mathrm{tot}}(V)=\\bigoplus_{p+q=k}S^p(V_0)\\otimes\\bigwedge^q(V_1)$. This grading is not compatible with the $\\ints_2$-grading, but will of be of use to us nonetheless. \n\t\n\tLet $U$ be another finite-dimensional super-vector space, and moreover, let $b:U\\times V\\to\\cplxs$ be a bilinear form. Then $b$ extends to a bilinear form $S(U)\\times S(V)\\to\\cplxs$: It is defined on linear generators by \n\t\\[\n\t\tb(x_1\\dotsm x_m,y_1\\dotsm y_n)=\\delta_{mn}\\cdot\\sum\\nolimits_{\\sigma\\in\\mathfrak S_n}\\alpha^\\sigma_{x_1,\\dotsc,x_n}\\cdot b(x_{\\sigma(1)},y_1)\\dotsm b(x_{\\sigma(n)},y_n)\t\n\t\\]\n\tfor all  $x_1,\\dotsc,x_m\\in U$, $y_1,\\dotsc,y_n\\in V$ where $\\alpha=\\alpha^\\sigma_{x_1,\\dotsc,x_n}=\\pm1$ is determined by the requirement that $\\alpha\\cdot x_{\\sigma(1)}\\dotsm x_{\\sigma(n)}=x_1\\dotsm x_n$ in $S(V)$. If $b$ is even (resp.~odd, resp.~non-degenerate), then so is its extension. Here, recall that a bilinear form has degree $i$ if $b(V_j,V_k)=0$ whenever $i+j+k\\equiv1\\ (2)$.\n\t\n\tIn particular, the natural pairing of $V$ and $V^*$ extends to a non-de\\-ge\\-ne\\-rate even pairing $\\Cdual0\\cdot\\cdot$ of $S(V)$ and $S(V^*)$. By this token, $S(V)$ embeds injectively as a subsuperspace in $\\widehat S(V)=S(V^*)^*$. Its image coincides with the graded dual $S(V^*)^{*\\mathrm{gr}}$ whose elements are the linear forms vanishing on $S^{k,\\mathrm{tot}}(V^*)$ for $k\\gg1$. \n\t\n\tWe define a superalgebra homomorphism $\\partial:S(V)\\to\\End0{\\widehat S(V^*)}$ by \n\t\\[\n\t\t\\Cdual0p{\\partial(q)\\pi}=\\Cdual0{pq}\\pi\\mathfa p,q\\in S(V)\\,,\\,\\pi\\in S(V)^*\n\t\\]\n\twhere $\\widehat S(V^*)=S(V)^*$. Clearly, $\\partial(q)$ leaves $S(V^*)$ invariant. \n\\end{Par}\n\n\\begin{Par}\n\tIf $U$ is an even finite-dimensional vector space over $\\cplxs$, then we have the well-known isomorphism $S(U^*)\\cong\\cplxs[U]$ as algebras, where $\\cplxs[U]$ is the set of polynomial mappings $U\\to\\cplxs$. We recall that the isomorphism can be written down as follows. \n\t\n\tThe pairing $\\Cdual0\\cdot\\cdot$ of $S(U)$ and $S(U^*)$ extends to $\\widehat S(U)\\times S(U^*)$. For any $d\\in S(U)$, the exponential $e^d=\\sum_{n=0}^\\infty\\frac{d^n}{n!}$ makes sense as an element of the algebra $\\widehat S(U)=\\prod_{n=0}^\\infty S^n(U)$. Now, define a map $S(U^*)\\to\\cplxs[U]:p\\mapsto P$ by \n\t\\[\n\t\tP(z)=\\Cdual0{e^z}p=\\textstyle\\sum_{n=0}^\\infty\\frac1{n!}\\Cdual0{z^n}{p}=\\sum_{n=0}^\\infty\\frac1{n!}\\Cdual01{\\partial(z)^np}\\ .\n\t\\]\n\tObserve \n\t\\[\n\t\t\\tfrac d{dt}P(z_0+tz)\\big|_{t=0}=\\tfrac d{dt}\\Cdual0{e^{tz}e^{z_0}}p\\big|_{t=0}=\\Cdual0{ze^{z_0}}p\\ .\n\t\\]\n\tIterating this formula, we obtain $\\Cdual0{z_1\\dotsm z_n}p$ for any $z_j\\in U$ as a repeated directional derivative of $P$, and the map is injective. Since it preserves the grading by total degree, it is bijective because of identities of dimension in every degree. \n\\end{Par}\n\n\\begin{Par}\n\tLet $V=V_0\\oplus V_1$ be a finite-dimensional super-vector space. We apply the above to define an isomorphism $\\phi:S(V^*)\\to\\Hom[_{S(V_0)}]0{S(V),\\cplxs[V_0]}$. Here, $S(V_0)$ acts on $S(V)$ by left multiplication, and it acts on $\\cplxs[V_0]$ by natural extension of the action of $V_0$ by directional derivatives:\n\t\\[\n\t\t(\\partial_zP)(z_0)=\\tfrac d{dt}P(z_0+tz)\\big|_{t=0}\\mathfa P\\in\\cplxs[V_0]\\,,\\,z,z_0\\in V_0\\ .\n\t\\]\n\tThe isomorphism $\\phi$ is given by the following prescription for $P=\\phi(p)$:\n\t\\[\n\t\tP(d;z)=(-1)^{\\Abs0d\\Abs0p}\\Cdual0{e^z}{\\partial(d)p}\\mathfa p\\in S(V^*)\\,,\\,z\\in V_0\\,,\\,d\\in S(V)\\ .\n\t\\]\n\tHere, note that $\\widehat S(V_0)\\subset\\widehat S(V)$ since $S(V^*_0)$ is a direct summand of $S(V^*)$, $S(V^*)=S(V_0^*)\\oplus S(V_0^*)\\otimes\\bigwedge^+(V_1^*)$, where $\\bigwedge^+=\\bigoplus_{k\\sge1}\\bigwedge^k$. Hence, $e^z$ may be considered as an element of $\\widehat S(V)$. \t\n\t\n\tThe map $\\phi$ is an isomorphism as the composition of the isomorphisms\n\t\\begin{align*}\n\t\t\\Hom[_{S(V_0)}]0{S(V),\\cplxs[V_0]}&\\cong\\Hom[_{S(V_0)}]0{S(V_0)\\otimes\\textstyle\\bigwedge V_1,S(V_0^*)}\\\\\n\t\t&\\cong\\textstyle S(V_0^*)\\otimes\\bigwedge V_1^*\\cong S(V^*)\\ .\n\t\\end{align*}\n\\end{Par}\n\n\\begin{Def}[restrhom]\n\tLet $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair of even type, and $\\mathfrak a\\subset\\mathfrak p$ an even Cartan subspace. We apply the isomorphism $\\phi$ for $V=\\mathfrak p$ to define natural \\emph{restriction homomorphisms}\n\t\\[\n\tS(\\mathfrak p^*)\\to S(\\mathfrak p_0^*):p\\mapsto\\bar p\\mathtxt\\AND S(\\mathfrak p^*)\\to S(\\mathfrak a^*):p\\mapsto\\bar p\\ .\n\t\\]\n\tHere, $\\bar p\\in S(\\mathfrak p_0^*)$ (resp.~$\\bar p\\in S(\\mathfrak a^*)$) is defined via its associated polynomial $\\bar P\\in\\cplxs[\\mathfrak p_0]$ (resp.~$\\bar P\\in\\cplxs[\\mathfrak a]$) where \n\t\\[\n\t\t\\bar P(z)=P(1;z)\\mathtxt\\AND P=\\phi(p)\\ .\n\t\\]\n\tThis is a convention we will adhere to in all that follows. \n\t\n\tSince $\\mathfrak p_0$ is complemented by $\\mathfrak p_1$ in $\\mathfrak p$, and $\\mathfrak a$ is complemented in $\\mathfrak p_0$ by $\\bigoplus_{\\lambda\\in\\Sigma_0}\\mathfrak p_0^\\lambda$, we will in the sequel consider $\\mathfrak p_0^*\\subset\\mathfrak p^*$ and $\\mathfrak a^*\\subset\\mathfrak p_0^*$.\n\\end{Def}\n\n\\begin{Par}\n\tLet $K$ be a connected Lie group with Lie algebra $\\mathfrak k_0$ such that  the restricted adjoint representation $\\ad:\\mathfrak k_0\\to\\End0{\\mathfrak g}$ lifts to a homomorphism $\\Ad:K\\to\\GL(\\mathfrak g)$. (For instance, one might take $K$ simply connected.) Then $\\mathfrak k$ (resp.~$K$) acts on $S(\\mathfrak p)$, $S(\\mathfrak p^*)$, $\\widehat S(\\mathfrak p)$, $\\widehat S(\\mathfrak p^*)$ by suitable extensions of $\\ad$ and $\\ad^*$ (resp.~$\\Ad$ and $\\Ad^*$) which we denote by the same symbols. Here, the sign convention for $\\ad^*$ is\n\t\\[\n\t\t\\Cdual0y{\\ad^*(x)\\eta}=\\Cdual0{[y,x]}\\eta=-(-1)^{\\Abs0x\\Abs0y}\\Cdual0{\\ad(x)(y)}\\eta\n\t\\]\n\tfor all  $x,y\\in\\mathfrak g$, $\\eta\\in\\mathfrak g^*$. \n\t\n\tLet $z\\in\\mathfrak p_0$. We have $e^z=\\sum_{k=0}^\\infty\\frac{z^k}{k!}\\in\\widehat S(\\mathfrak p)$, and this element is invertible with inverse $e^{-z}$. Define \n\t\\[\n\tu_z(x)d=\\ad(x)(de^z)e^{-z}\\mathfa x\\in\\mathfrak k\\,,\\,d\\in\\widehat S(\\mathfrak p)\\ .\n\t\\]\n\tObserve that  \n\t\\[\n\t\\ad(x)(e^z)=\\textstyle\\sum_{n=0}^\\infty\\frac1{n!}\\ad(x)(z^n)=\\sum_{n=1}^\\infty\\frac n{n!}[x,z]z^{n-1}=[x,z]e^z\\ ,\n\t\\]\t\n\tbecause $z$ is even. Hence, \n\t\\[\n\t\tu_z(x)d=\\ad(x)(de^z)e^{-z}=[x,z]d+\\ad(x)(d)\\ .\n\t\\]\n\tIn particular, $u_z(x)$ leaves $S(\\mathfrak p)\\subset\\widehat S(\\mathfrak p)$ invariant. \n\\end{Par}\n\n\\begin{Lem}[twistedaction-cov]\n\tLet $z\\in\\mathfrak p_0$. Then $u_z$ defines a $\\mathfrak k$-module structure on $S(\\mathfrak p)$, and for all  $x\\in\\mathfrak k$, $k\\in K$, we have \n\t\\[\n\t\t\\Ad(k)\\circ u_z(x)=u_{\\Ad(k)(z)}(\\Ad(k)(x))\\circ\\Ad(k)\\ .\n\t\\]\n\\end{Lem}\n\n\\begin{proof}\n\tWe clearly have \n\t\\[\n\t\tu_z(x)u_z(y)d=(\\ad(x)\\ad(y)(de^z))e^{-z}\\ .\n\t\\]\n\tNow $u_z$ is a $\\mathfrak k$-action because $\\ad$ is a homomorphism. Similarly,\n\t\\begin{align*}\n\t\t\\Ad(k)(u_z(x)d)&=\\ad(\\Ad(k)(x))(\\Ad(k)(d)e^{\\Ad(k)(z)})e^{-\\Ad(k)(z)}\\\\\n\t\t&=u_{\\Ad(k)(z)}(\\Ad(k)(x))\\Ad(k)(d)\\ ,\n\t\\end{align*}\n\twhich manifestly gives the second assertion. \n\\end{proof}\n\n\\begin{Par}[twisted-action-def]\n\tLet $u_z$ also denote the natural extension of $u_z$ to $\\Uenv0{\\mathfrak k}$. Then we may define an action $\\ell$ of $\\Uenv0{\\mathfrak k}$ on $\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]}$ via \n\t\\[\n\t\t(\\ell_vP)(d;z)=(-1)^{\\Abs0v\\Abs0P}P(u_z(S(v))d;z)\n\t\\]\n\tfor all  $P\\in\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]}$, $v\\in\\Uenv0{\\mathfrak k}$, $d\\in S(\\mathfrak p)$, $z\\in\\mathfrak p_0$. Here, we denote by $S:\\Uenv0{\\mathfrak g}\\to\\Uenv0{\\mathfrak g}$ the unique linear map such that  $S(1)=1$, $S(x)=-x$ for all  $x\\in\\mathfrak g$, and $S(uv)=(-1)^{\\Abs0u\\Abs0v}S(v)S(u)$ for all  homogeneous $u,v\\in\\Uenv0{\\mathfrak g}$ (\\emph{i.e.}~the principal anti-automorphism). Compare \\cite{koszul-superaction} for a similar definition in the context of the action of a supergroup on its algebra of superfunctions. \n\t\n\tWe also define \n\t\\[\n\t\t(L_kP)(d;z)=P(\\Ad(k^{-1})(d);\\Ad(k^{-1})(z))\n\t\\]\n\tfor all  $P\\in\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]}$, $k\\in K$, $d\\in S(\\mathfrak p)$, $z\\in\\mathfrak p_0$.\n\\end{Par}\n\n\\begin{Lem}[flat-radial-fmla]\n\tThe map $\\ell$ (resp.~$L$) defines on $\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]}$ the structure of a module over $\\mathfrak k$ (resp.~$K$) making the isomorphism $\\phi$ equivariant for $\\mathfrak k$ (resp.~$K$). \n\\end{Lem}\n\n\\begin{proof}\n\tLet $P=\\phi(p)$. Then\n\t\\begin{align*}\n\t\t(\\ell_xP)(d;z)&=-(-1)^{\\Abs0x\\Abs0p}P(u_z(x)d;z)=-(-1)^{\\Abs0d\\Abs0p}\\Cdual1{\\ad(x)(e^zd)}p\\\\\n\t\t&=(-1)^{\\Abs0d(\\Abs0x+\\Abs0p)}\\Cdual1{e^zd}{\\ad^*(x)(p)}=\\phi\\Parens1{\\ad^*(x)(p)}(d;z)\\ .\n\t\\end{align*}\n\t\n\tSimilarly, we check that\n\t\\begin{align*}\n\t\t(L_kP)(d;z)&=P(\\Ad(k^{-1})(d);\\Ad(k^{-1})(z))\\\\\n\t\t&=(-1)^{\\Abs0d\\Abs0p}\\Cdual1{e^{\\Ad(k^{-1})(z)}\\Ad(k^{-1})(d)}p\\\\\n\t\t&=(-1)^{\\Abs0d\\Abs0p}\\Cdual1{\\Ad(k^{-1})(e^zd)}p=\\phi\\Parens1{\\Ad^*(k)(p)}(z;d)\\ .\n\t\\end{align*}\n\tThis proves our assertion.\n\\end{proof}\n\n\\section{Chevalley's restriction theorem}\n\n\\subsection{The map $\\gamma_z$}\n\n\\begin{Par*}\n\tFrom now on, let $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair of even type, and let $\\mathfrak a\\subset\\mathfrak p_0$ be an even Cartan subspace. \n\\end{Par*}\n\n\\begin{Def}\n\tAn element $z\\in\\mathfrak p_0$ is called \\emph{oddly regular} whenever the map $\\ad(z):\\mathfrak k_1\\to\\mathfrak p_1$ is surjective. \tRecall that $z\\in\\mathfrak p_0$ is called \\emph{regular} if $\\dim\\mathfrak z_{\\mathfrak k_0}(z)=\\dim\\mathfrak z_{\\mathfrak k_0}(\\mathfrak a)$. We shall call $z$ \\emph{super-regular} if it is both regular and oddly regular. \n\t\t\n\tFix an even Cartan subspace $\\mathfrak a$, and let $\\Sigma$ be the set of (both odd and even) restricted roots. Let $\\Sigma^+\\subset\\Sigma$ be any subset such that $\\Sigma$ is the disjoint union of $\\pm\\Sigma^+$. Define $\\Sigma_j^\\pm=\\Sigma_j\\cap\\Sigma^\\pm$ for $j\\in\\ints\/2\\ints$. Let $\\bar\\Sigma_1$ be the set of $\\lambda\\in\\Sigma_1$ such that  $m\\lambda\\not\\in\\Sigma_0$ for $m=1,2$. Denote $\\bar\\Sigma_1^+=\\bar\\Sigma_1\\cap\\Sigma^+$. Note that $\\Pi_1\\in S(\\mathfrak a^*)^W$ where $\\Pi_1(h)=\\prod_{\\lambda\\in\\Sigma_1}\\lambda(h)$, and $W$ is the Weyl group of $\\Sigma_0$. \n\t\n\tBy Chevalley's restriction theorem, restriction $S(\\mathfrak p_0^*)^{\\mathfrak k_0}\\to S(\\mathfrak a^*)^W$ is a bijective map. Let $\\Pi_1$ also denote the unique extension to $S(\\mathfrak p^*_0)^{\\mathfrak k_0}$ of $\\Pi_1$. \n\\end{Def}\n\n\\begin{Rem}\n\tThe space $\\mathfrak p_0$ contains non-semi-simple elements, and the definitions we have given above work in this generality. However, it will suffice for our purposes to consider the set of \\emph{semi-simple} super-regular elements in $\\mathfrak p_0$, by the following reasoning. \n\t\n\tFirst, the set of semi-simple elements in $\\mathfrak p_0$ is Zariski dense (a linear endomorphism is semi-simple if and only if its minimal polynomial has only simple zeros). Second, the set of semi-simple elements in $\\mathfrak p_0$ equals $\\Ad(K)(\\mathfrak a)$ \\cite[Chapter III, Proposition~4.16]{helgason-geoman}. Thus, given any \\emph{semi-simple} $z\\in\\mathfrak p_0$, $z$ is oddly regular (super-regular) if and only if $\\lambda(\\Ad(k)(z))\\neq0$ for all  $\\lambda\\in\\Sigma_1$ ($\\lambda\\in\\Sigma$), and for some (any) $k\\in K$ such that  $\\Ad(k)(z)\\in\\mathfrak a$. In particular, the set of super-regular elements of $\\mathfrak a$ is the complement of a finite union of hyperplanes. Hence, the set of semi-simple super-regular elements of $\\mathfrak p_0$ is non-void and therefore Zariski dense; in particular, this holds for the set of semi-simple oddly regular elements. \n\\end{Rem}\n\n\\begin{Lem}[centnondegen]\n\tIf $z\\in\\mathfrak p_0$ is semi-simple, then $\\mathfrak k_i=\\mathfrak z_{\\mathfrak k_i}(z)\\oplus[z,\\mathfrak p_i]$, and the subspaces $\\mathfrak z_{\\mathfrak k_i}(z)$ and $[z,\\mathfrak p_i]$ are $b$-non-degenerate. \n\\end{Lem}\n\n\\begin{proof}\n\tSince $\\ad z$ is a semi-simple endomorphism of $\\mathfrak g$ ($\\mathfrak g$ is a semi-simple $\\mathfrak g_0$-module and $z$ is semi-simple), we have $\\mathfrak g_i=\\mathfrak z_{\\mathfrak g_i}(z)\\oplus[z,\\mathfrak g_i]$. Taking $\\theta$-fixed parts, we deduce $\\mathfrak k_i=\\mathfrak z_{\\mathfrak k_i}(z)\\oplus\\mathfrak[z,\\mathfrak p_i]$. The summands, being $b$-orthogonal, are non-degenerate. \n\\end{proof}\n\n\\begin{Par}\n\tLet $z\\in\\mathfrak p_0$ be semi-simple and oddly regular. Let $\\beta:S(\\mathfrak g)\\to\\Uenv0{\\mathfrak g}$ be the supersymmetrisation map. Let\n\t\\[\n\t\t\\Gamma_z:\\textstyle\\bigwedge(\\mathfrak p_1)\\otimes S(\\mathfrak p_0)\\to S(\\mathfrak p):q\\otimes p\\mapsto u_z\\Parens1{\\beta([z,q])}p\n\t\\]\n\ton elementary tensors and extend linearly. \n\\end{Par}\n\n\\begin{Prop}[radial-part]\nLet $z$ be oddly regular and semi-simple. Then $\\Gamma_z$ is bijective, and $\\gamma_z=(\\eps\\otimes 1)\\circ\\Gamma_z^{-1}:S(\\mathfrak p)\\to S(\\mathfrak p_0)$ satisfies \n\\[\n\\gamma_{\\Ad(k)(z)}\\circ\\Ad(k)=\\Ad(k)\\circ\\gamma_z\\mathfa k\\in K\\ .\n\\] \nHere $\\eps:\\bigwedge(\\mathfrak p_1)\\to\\cplxs$ is the unique unital algebra homomorphism.\n\nMoreover, on $S^{m,\\mathrm{tot}}(\\mathfrak p)$, $\\Pi_1(z)^m\\gamma_z$ is polynomial in $z$, \\emph{i.e.}~it extends to an element $\\Pi_1(\\cdot)^m\\gamma_\\cdot$ of the space $\\cplxs[\\mathfrak p_0]\\otimes\\Hom0{S^{m,\\mathrm{tot}}(\\mathfrak p),S(\\mathfrak p_0)}$. \n\\end{Prop}\n\n\\begin{proof}\n\tBy the assumption on $z$, $\\ad z:\\mathfrak p_1\\to[z,\\mathfrak p_1]$ is bijective. Moreover, $\\Gamma_z$ respects the filtrations by total degree, and the degrees of these filtrations are equidimensional by the assumption. Hence, $\\Gamma_z$ will be bijective once it is surjective. In degree zero, $\\Gamma_z$ is the identity. We proceed to prove the surjectivity in higher degrees by induction.\n\n\tBy assumption, $\\ad z:[z,\\mathfrak p_1]\\to\\mathfrak p_1$ is also bijective (since its kernel is $\\mathfrak z_{\\mathfrak k_1}(z)\\cap[z,\\mathfrak p_1]$, which is $0$ by \\thmref{Lem}{centnondegen}). Let $y_1,\\dotsc,y_m\\in\\mathfrak p_1$, $y_1',\\dotsc,y_n'\\in\\mathfrak p_0$. Let $x_j\\in\\mathfrak p_1$ such that  $[[z,x_j],z]=y_j$. We find\n\t\\[\n\t\t\\Gamma_z(x_1\\dotsm x_m\\otimes y_1'\\dotsm y_n')\\equiv y_1\\dotsm y_my_1'\\dotsm y_n'\\quad\\Parens1{\\textstyle\\bigoplus\\nolimits_{k<m+n}S^{k,\\mathrm{tot}}(\\mathfrak p)}\\ ,\n\t\\]\n\tso the first assertion follows by induction. \n\t\n\tAs to the covariance property, observe first that we have the identity $\\Ad(k)([z,\\mathfrak p_1])=[\\Ad(k)(z),\\mathfrak p_1]$. Moreover, \n\t\\begin{align*}\n\t\t(\\Ad(k)\\circ\\gamma_z)(\\Gamma_z(v\\otimes d))&=\\eps(v)\\Ad(k)(d)=\\eps(\\Ad(k)(v))\\Ad(k)(d)\\\\\n\t\t&=\\gamma_{\\Ad(k)(z)}\\Parens1{\\Gamma_{\\Ad(k)(z)}(\\Ad(k)(v)\\otimes\\Ad(k)(d))}\\\\\n\t\t&=\\gamma_{\\Ad(k)(z)}\\Parens1{u_{\\Ad(k)(z)}(\\Ad(k)(\\beta([z,v])))\\Ad(k)(d)}\\\\\n\t\t&=\\gamma_{\\Ad(k)(z)}\\Parens1{\\Ad(k)(u_z(\\beta([z,v]))(d))}\\\\\n\t\t&=(\\gamma_{\\Ad(k)(z)}\\circ\\Ad(k))(\\Gamma_z(v\\otimes d))\n\t\\end{align*}\n\tfor all  $v\\in\\bigwedge(\\mathfrak p_1)$ and $d\\in S(\\mathfrak p_0)$, by \\thmref{Lem}{twistedaction-cov}.\n\t\n\tTo show that $\\Pi_1(z)^m\\gamma_z:S^{m,\\mathrm{tot}}(\\mathfrak p)\\to S(\\mathfrak p_0)$ is given by the restriction of a polynomial function, we remark that its domain of definition---the set $U$ of semi-simple oddly regular elements in $\\mathfrak p_0$---is Zariski dense. We need only prove that $f:U\\to\\Hom0{\\mathfrak p_1,\\mathfrak k_1}$, $f(z)=\\Pi_1(z)(\\ad z)^{-1}$, is polynomial in $z$, where we consider $(\\ad z)^{-1}:\\mathfrak p_1\\to[z,\\mathfrak p_1]$ as a linear map $\\mathfrak p_1\\to\\mathfrak k_1$. \n\t\t\n\tThus, let $z\\in\\mathfrak p_0$ be semi-simple and oddly regular. It is contained in some even Cartan subspace $\\mathfrak a$ (say). We have $\\mathfrak z_{\\mathfrak k_1}(\\mathfrak a)=\\mathfrak m_1=\\mathfrak k_1\\cap[z,\\mathfrak p_1]^\\perp$ by \\thmref{Lem}{centnondegen} and $(\\mathfrak k_1\\cap\\mathfrak m_1^\\perp)\\oplus\\mathfrak p_1=\\bigoplus_{\\lambda\\in\\Sigma_1^+}\\mathfrak g_{1,\\mathfrak a}^\\lambda$. If $x=u+v\\in\\mathfrak g^\\lambda_{1,\\mathfrak a}$, and $u\\in\\mathfrak k_1$, $v\\in\\mathfrak p_1$, then $[z,u]=\\lambda(z)v$. It follows that $\\Pi_1(z)(\\ad z)^{-1}$ depends polynomially on $z$, proving our claim. \n\\end{proof}\n\n\\begin{Prop}[flat-radial]\n\tLet $p\\in S(\\mathfrak p^*)^{\\mathfrak k}$. Then $P(d;z)=P(\\gamma_z(d);z)$ for all  oddly regular and semi-simple $z\\in\\mathfrak p_0$ and $d\\in S(\\mathfrak p)$. \n\\end{Prop}\n\n\\begin{proof}\n\tFix an oddly regular $z\\in\\mathfrak p_0$, and let $x_1,\\dotsc,x_n\\in\\mathfrak p_1$. By \\thmref{Lem}{flat-radial-fmla}, we find for $n>0$ \n\t\\[\n\t\tP\\Parens1{\\Gamma_z(x_1\\dotsm x_n\\otimes q);z}=(\\ell_{\\beta([z,x_1\\dotsm x_n])}P)(q;z)=0\\ .\n\t\\]\n\tSince $d-\\gamma_z(d)\\in\\Gamma_z(\\bigwedge^+(\\mathfrak p_1)\\otimes S(\\mathfrak p_0))$, where $\\bigwedge^+(\\mathfrak p_1)$ denotes the kernel of $\\eps:\\bigwedge(\\mathfrak p_1)\\to\\cplxs$ (\\emph{i.e.}, the set of elements without constant term), the assertion follows immediately.  \n\\end{proof}\n\n\\begin{Cor}[res-inj]\n\tLet $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair of even type. The algebra homomorphism $p\\mapsto\\bar p:I(\\mathfrak p^*)=S(\\mathfrak p^*)^{\\mathfrak k}\\to S(\\mathfrak p_0^*)$ is injective. In particular, $I(\\mathfrak p^*)$ is commutative and purely even. \n\\end{Cor}\n\n\\begin{proof}\n\tLet $p\\in I(\\mathfrak p^*)$. Assume that $\\bar p=0$. Let $d\\in S(\\mathfrak p)$. For all $z\\in\\mathfrak p_0$ which are oddly regular and semi-simple, \n\t\\[\n\t\tP(d;z)=P(\\gamma_z(d);z)=[\\partial_{\\gamma_z(d)}\\bar P](z)=0\\ ,\n\t\\]\n\tby \\thmref{Prop}{flat-radial}. It follows that $P(d;-)=0$ on $\\mathfrak p_0$, since it is a polynomial. Since $d$ was arbitrary, we have established our contention. \n\\end{proof}\n\n\\begin{Rem}\n\tThe statement of the Corollary can, of course, be deduced by applying the inverse function theorem for supermanifolds, as in \\cite[Proposition 1.1]{sergeev-invpol}. Nonetheless, we find it instructive to give the above proof based on the map $\\gamma_z$, as it illustrates the approach we will take to determine the image of the restriction map. \n\\end{Rem}\n\n\\subsection{Proof of Theorem (A)} \n\n\\begin{Par}\n\tLet $(\\mathfrak g,\\mathfrak k)$ be a reductive symmetric superpair of even type, and let $\\mathfrak a$ be an even Cartan subspace. We denote by $\\mathfrak a'$ the set of super-regular elements of $\\mathfrak a$. Let $\\mathcal R$ be the algebra of differential operators on $\\mathfrak a$ with rational coefficients which are non-singular on $\\mathfrak a'$. For any $z\\in\\mathfrak a'$ and any $D\\in\\mathcal R$, let $D(z)$ be the local expression of $D$ at $z$. This is defined by the requirement that $D(z)$ be a differential operator with constant coefficients, and \n\t\\[\n\t\t(Df)(z)=(D(z)f)(z)\\mathfa z\\in\\mathfrak a'\\ ,\n\t\\]\n\tand all regular functions $f$. \n\t\n\tWe associate to $\\Sigma\\subset\\mathfrak a^*$, the restricted root system of $\\mathfrak g:\\mathfrak a$, the subset $\\mathcal R_\\Sigma=\\bigcup_{\\lambda\\in\\bar\\Sigma_1^+}\\mathcal R_\\lambda\\subset\\mathcal R$ where\n\t\\[\n\t\t\\mathcal R_\\lambda=\\Set1{D\\in\\mathcal R}{\\exists\\,d\\in S(\\mathfrak p_1^\\lambda)\\colon D(z)=\\gamma_z(d)\\text{ for all }z\\in\\mathfrak a'}\\ .\n\t\\]\n\t\\emph{I.e.}, $\\mathcal R_\\Sigma$ consists of those differential operators which are given as radial parts of operators with constant coefficients on the $\\mathfrak p$-projections $\\mathfrak p_1^\\lambda$ of the restricted root spaces for the $\\lambda\\in\\bar\\Sigma_1^+$. For any $D\\in\\mathcal R$, let the \\emph{domain} $\\dom D$ be the set of all $p\\in\\cplxs[\\mathfrak a]$ such that  $Dp\\in\\cplxs[\\mathfrak a]$. \n\t\n\tAs we shall see, the image of the restriction map is the set of $W$-invariant polynomials in the common domain of $\\mathcal R_\\Sigma$. We will subsequently determine $\\mathcal R_\\Sigma$ in order to describe this common domain in more explicit terms. \n\\end{Par}\n\n\\begin{Th}[super-chevalley]\n\tThe restriction homomorphism $I(\\mathfrak p^*)\\to S(\\mathfrak a^*)$ from \\thmref{Def}{restrhom} is a bijection onto the subspace $I(\\mathfrak a^*)=S(\\mathfrak a^*)^W\\cap\\bigcap_{D\\in\\mathcal R_\\Sigma}\\dom D$. \n\\end{Th}\n\n\\begin{Par*}\n\tThe \\emph{proof} of the Theorem requires a little preparation. \n\\end{Par*}\n\n\\begin{Lem}[doubleprojection-trick]\n\tLet $q\\in S(\\mathfrak p_0^*)^K$, $Q=\\phi(q)$, and $z\\in\\mathfrak p_0$ be super-regular and semi-simple. For all  $x\\in\\mathfrak k$, and $w\\in S(\\mathfrak p)$, we have\n\t\\[\n\t\tQ\\Parens1{\\gamma_z(u_z(x)w);z}=0\\ .\n\t\\]\n\\end{Lem}\n\n\\begin{proof}\n\tThere is no restriction to generality in supposing $z\\in\\mathfrak a'$, so that $\\mathfrak z_{\\mathfrak k}(z)=\\mathfrak z_{\\mathfrak k}(\\mathfrak a)=\\mathfrak m$ and $\\mathfrak z_{\\mathfrak k_0}(z)=\\mathfrak z_{\\mathfrak k_0}(\\mathfrak a)=\\mathfrak m_0$. We define linear maps\n\t\\[\n\t\t\\gamma_z':S(\\mathfrak p_0)\\to S(\\mathfrak a)\\mathtxt\\AND\\gamma_z'':S(\\mathfrak p)\\to S(\\mathfrak a)\n\t\\]\n\tby the requirements that $v-\\gamma_z'(v)\\in u_z(\\mathfrak m_0^\\perp\\cap\\mathfrak k_0)(S(\\mathfrak p_0))$ for all  $v\\in S(\\mathfrak p_0)$ and $w-\\gamma_z''(w)\\in u_z(\\mathfrak m^\\perp\\cap\\mathfrak k)(S(\\mathfrak p))$ for all  $w\\in S(\\mathfrak p)$. (That such maps exist and are uniquely defined by these properties follows in exactly the same way as for \\thmref{Prop}{radial-part}. We remark that $[z,\\mathfrak p_i]=\\mathfrak k_i\\cap\\mathfrak m_i^\\perp$ by \\thmref{Lem}{centnondegen}.) Then \n\t\\begin{align*}\n\t\tw&-\\gamma_z'(\\gamma_z(w))=w-\\gamma_z(w)+\\gamma_z(w)-\\gamma_z'(\\gamma_z(w))\\\\\n\t\t&\\in u_z(\\mathfrak m_1^\\perp\\cap\\mathfrak k_1)(S(\\mathfrak p))+u_z(\\mathfrak m_0^\\perp\\cap\\mathfrak k_0)(S(\\mathfrak p_0))\\subset u_z(\\mathfrak m^\\perp\\cap\\mathfrak k)(S(\\mathfrak p))\n\t\\end{align*}\n\tfor all  $w\\in S(\\mathfrak p)$, where $\\mathfrak m_1=\\mathfrak z_{\\mathfrak k_1}(\\mathfrak a)$. This shows that $\\gamma_z''=\\gamma_z'\\circ\\gamma_z$. \n\t\n\tMoreover, by the $K$-invariance of $q$, we have $Q(v;z)=Q(\\gamma_z'(v);z)$ for all  $v\\in S(\\mathfrak p_0)$. We infer  \n\t\\[\n\t\tQ\\Parens1{\\gamma_z(u_z(x)w);z}=Q\\Parens1{\\gamma_z''(u_z(x)w);z}=0\\mathfa x\\in\\mathfrak m^\\perp\\cap\\mathfrak k\\ ,\\ w\\in S(\\mathfrak p)\n\t\\]\n\tsince $u_z(x)w\\in u_z(\\mathfrak m^\\perp\\cap\\mathfrak k)(S(\\mathfrak p))$ belongs to $\\ker\\gamma_z''$. \n\t\n\tNext, we need to consider the case of $x\\in\\mathfrak m$. Then $\\ad(x):S(\\mathfrak p)\\to S(\\mathfrak p)$ annihilates the subspace $S(\\mathfrak a)$, and moreover, $\\ad(x)(e^z)=0$. From this we find for all  $y\\in\\mathfrak m^\\perp\\cap\\mathfrak k$, $d\\in S(\\mathfrak p)$\n\t\\begin{align*}\n\t\t\\ad(x)\\Parens1{u_z(y)(d)}&=(\\ad(x)\\ad(y)(de^z))e^{-z}\\\\\n\t\t&=(\\ad([x,y])(de^z))e^{-z}+(-1)^{\\Abs0x\\Abs0y}\\ad(y)(\\ad(x)(d)e^z)e^{-z}\\\\\n\t\t&=u_z([x,y])d+(-1)^{\\Abs0x\\Abs0y}u_z(y)\\ad(x)(d)\\ .\n\t\\end{align*}\n\tSince $\\mathfrak m$ is a subalgebra and $b$ is $\\mathfrak k$-invariant, $\\mathfrak m^\\perp\\cap\\mathfrak k$ is $\\mathfrak m$-invariant. Hence, the above formula shows that $\\ker\\gamma_z''=u_z(\\mathfrak m^\\perp\\cap k)(S(\\mathfrak p))$ is $\\ad(x)$-invariant.\n\t\n\tBy the definition of $\\gamma_z''$, we find that \n\t\\[\n\t\t\\gamma_z''(\\ad(x)d)=\\ad(x)\\gamma_z''(d)=0\\mathfa x\\in\\mathfrak m\\,,\\,d\\in S(\\mathfrak p)\\ .\n\t\\]\n\tReasoning as above, we see that \n\t\\[\n\tQ(\\gamma_z(u_z(x)d);z)=Q(\\gamma_z(\\ad(x)d);z)=0\\mathfa x\\in\\mathfrak m\\,,\\,d\\in S(\\mathfrak  p)\\ .\n\t\\]\n\tSince $\\mathfrak k=\\mathfrak m\\oplus(\\mathfrak m^\\perp\\cap\\mathfrak k)$, this proves the lemma. \n\\end{proof}\n\n\\begin{Par*}\n\tLet $\\mathfrak p_0'$ be the set of semi-simple super-regular elements in $\\mathfrak p_0$. Recall the polynomial $\\Pi_1$, and consider the localisation $\\cplxs[\\mathfrak p_0]_{\\Pi_1}$. Let $q\\in S(\\mathfrak p_0^*)^K$, $Q=\\phi(q)$, and define \n\t\\[\n\t\tP(v;z)=Q(\\gamma_z(v);z)\\mathfa v\\in S(\\mathfrak p)\\ ,\\ z\\in\\mathfrak p_0'\\ .\n\t\\]\n\tBy \\thmref{Prop}{radial-part}, $P\\in\\Hom0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]_{\\Pi_1}}$. We remark that the $\\mathfrak k$-action $\\ell$ defined in \\thmref{Par}{twisted-action-def} extends to $\\Hom0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]_{\\Pi_1}}$, by the same formula. \n\\end{Par*}\n\n\\begin{Lem}[sp0-linear]\n\tRetain the above assumptions. Then $P$ is $S(\\mathfrak p_0)$-linear and $\\mathfrak k$-invariant, \\emph{i.e.}~$P\\in\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]_{\\Pi_1}}^{\\mathfrak k}$.\n\\end{Lem}\n\n\\begin{proof}\n\tBy \\thmref{Lem}{doubleprojection-trick}, $P$ is $\\mathfrak k$-invariant. It remains to prove that $P$ is $S(\\mathfrak p_0)$-linear. To that end, we first establish that $P$ is $K$-equivariant as linear map $S(\\mathfrak p)\\to\\cplxs[\\mathfrak p_0]_{\\Pi_1}$. Since $q$ is $K$-invariant, \n\t\\begin{align*}\n\t\tP\\Parens1{\\Ad(k)(v);\\Ad(k)(z)}&=Q\\Parens1{\\gamma_{\\Ad(k)(z)}(\\Ad(k)(v));\\Ad(k)(z)}\\\\\n\t\t&=Q\\Parens1{\\Ad(k)(\\gamma_z(v));\\Ad(k)(z)}\\\\\n\t\t&=Q(\\gamma_z(v);z)=P(v;z)\\ .\n\t\\end{align*}\n\n\tNext, fix $z\\in\\mathfrak p_0'$. Then $S(\\mathfrak p)=S(\\mathfrak p_0)\\oplus u_z(\\mathfrak z_{\\mathfrak k_1}(z)^\\perp\\cap\\mathfrak k_1)(S(\\mathfrak p))$ where the second summand equals $\\ker\\gamma_z$. We may check the $S(\\mathfrak p_0)$-linearity on each summand separately. \n\n\tFor $v\\in S(\\mathfrak p_0)$, we have $P(v;z)=Q(v;z)$, so for any $y\\in\\mathfrak p_0$ \n\t\\[\n\t\t[\\partial_yP(v;-)](z)=[\\partial_yQ(v;-)](z)=Q(yv;z)=P(yv;z)\\ .\n\t\\]\n\t\n\tWe are reduced to considering $v=u_z(x)v'$ where $x\\in\\mathfrak z_{\\mathfrak k_1}(z)^\\perp\\cap\\mathfrak k_1$ and $v'\\in S(\\mathfrak p)$. We may assume w.l.o.g.~$z\\in\\mathfrak a$ (since $z$ is semi-simple), so that $\\mathfrak z_{\\mathfrak k_1}(z)=\\mathfrak z_{\\mathfrak k_1}(\\mathfrak a)=\\mathfrak m_1$. By our assumption on $z$, $\\mathfrak p_0=\\mathfrak a\\oplus[\\mathfrak k_0,z]$, and we may consider $y$ in each of the two summands separately. \n\t\n\tLet $y\\in\\mathfrak a$. For sufficiently small $t$, we have $z+ty\\in\\mathfrak a'=\\mathfrak a\\cap\\mathfrak p_0'$, so that $\\mathfrak z_{\\mathfrak k_1}(z+ty)=\\mathfrak m_1=\\mathfrak z_{\\mathfrak k_1}(z)$. Hence, $\\gamma_{z+ty}(u_{z+ty}(x)v')=0$. By the chain rule,  \n\t\\[\n\t\t0=\\tfrac d{dt}\\gamma_{z+ty}(u_{z+ty}(x)v')\\big|_{t=0}=d\\gamma_\\cdot(v)_z(y)+\\gamma_z\\Parens1{\\tfrac d{dt}u_{z+ty}(x)v'\\big|_{t=0}}\\ ,\n\t\\]\n\tSince $\\tfrac d{dt}u_{z+ty}(x)v'\\big|_{t=0}=[x,y]v'$, we have \n\t\\[\n\t\td\\gamma_\\cdot(v)_z(y)=-\\gamma_z(\\tfrac d{dt}u_{z+ty}(x)v'\\big|_{t=0})=\\gamma_z([y,x]v')\\ .\n\t\\]\n\t\n\tMoreover, as operators on $S(\\mathfrak p)$, \n\t\\[\n\t\t[y,u_z(x)]=y[x,z]+y\\ad(x)-[x,z]y-\\ad(x)y=[y,x]\\ ,\n\t\\]\n\tand thus $yv=yu_z(x)v'\\equiv[y,x]v'$ modulo $\\ker\\gamma_z$. We conclude  \n\t\\[\n\t\td\\gamma_\\cdot(v)_z(y)=\\gamma_z([y,x]v')=\\gamma_z(yv)=\\gamma_z(yv)-y\\gamma_z(v)\n\t\\]\n\tsince $\\gamma_z(v)=0$. \tHence, \n\t\\[\n\t\t[\\partial_yP(v;-)](z)=Q\\Parens1{d\\gamma_\\cdot(v)_z(y)+y\\gamma_z(v);z}=Q\\Parens1{\\gamma_z(yv);z}=P(yv;z)\\ .\n\t\\]\n\n\tNow let $y=[u,z]$ where $u\\in\\mathfrak k_0$. We may assume that $u\\perp\\mathfrak z_{\\mathfrak k_0}(z)$. Define $k_t=\\exp tu$. Then by the $K$-invariance of $P$, \n\t\\begin{align*}\n\t\t[\\partial_yP(v;-)](z)&=\\tfrac d{dt}P\\Parens1{v;\\Ad(k_t)(z)}\\big|_{t=0}\n\t\t=\\tfrac d{dt}P\\Parens1{\\Ad(k_t^{-1})(v);z}\\big|_{t=0}\\\\\n\t\t&=-P\\Parens1{\\ad(u)(v);z}=P(yv;z)-P\\Parens1{u_z(u)v;z}=P(yv;z)\n\t\\end{align*}\n\twhere in the last step, we have used \\thmref{Lem}{doubleprojection-trick}. \n\\end{proof}\n\n\\begin{proof}[\\protect{Proof of \\thmref{Th}{super-chevalley}}]\n\tThe restriction map is injective by \\thmref{Cor}{res-inj} and Chevalley's restriction theorem for $\\mathfrak g_0$. By the latter, the image lies in the set of $W$-invariants. Let $\\bar p\\in S(\\mathfrak a^*)$ be the restriction of $p\\in I(\\mathfrak p^*)$, and $P=\\phi(p)$. For any $d\\in S(\\mathfrak p)$, and $D\\in\\mathcal R_\\Sigma$ given by $D(z)=\\gamma_z(d)$, we have by \\thmref{Prop}{flat-radial}\n\t\\[\n\t\t(D\\bar p)(z)=(\\partial_{\\gamma_z(d)}\\bar P)(z)=P(\\gamma_z(d);z)=P(d;z)\\mathfa z\\in\\mathfrak a'\\ .\n\t\\]\n\tThe result is clearly polynomial in $z$, so $\\bar p\\in\\dom D$. This shows that the image of the restriction map lies in $I(\\mathfrak a^*)$. \n\t\n\tLet $r\\in I(\\mathfrak a^*)$. By Chevalley's restriction theorem, there exists a unique $q\\in I(\\mathfrak p_0^*)=S(\\mathfrak p_0^*)^K$ such that  $Q(h)=R(h)$ for all  $h\\in\\mathfrak a$. \n\t\n\tNext, recall that for $d\\in S(\\mathfrak p)$ and $z\\in\\mathfrak p_0'$: \n\t\\[\n\t\tP(d;z)=Q(\\gamma_z(d);z)\\ .\n\t\\]\n\tBy \\thmref{Lem}{sp0-linear}, $P\\in\\Hom[_{S(\\mathfrak p_0)}]0{S(\\mathfrak p),\\cplxs[\\mathfrak p_0]_{\\Pi_1}}^{\\mathfrak k}$. Hence, $P$ will define an element $p\\in I(\\mathfrak p^*)$ by virtue of the isomorphism $\\phi$, as soon as it is clear that, as a linear map $S(\\mathfrak p)\\to\\cplxs[\\mathfrak p_0]_{\\Pi_1}$, it takes its values in $\\cplxs[\\mathfrak p_0]$. \n\t\n\tWe only have to consider $z$ in the Zariski dense set $\\mathfrak p_0'$. The function $\\Pi_1(z)^k\\cdot P(d;z)$ depends polynomially on $z$, where we assume $d\\in S^{\\sle k,\\mathrm{tot}}(\\mathfrak p)$.  To prove that $P$ has polynomial values, it will suffice (by the removable singularity theorem and the conjugacy of Cartan subspaces) to prove that $P(d;h)$ is bounded as $h\\in\\mathfrak a'=\\mathfrak a\\cap\\mathfrak p_0'$ approaches one of the hyperplanes $\\lambda^{-1}(0)$ where $\\lambda\\in\\Sigma_1^+$ is arbitrary. Since $r$ is $W$-invariant, $r-r_0$ (where $r_0$ is the constant term of $r$) vanishes on $\\lambda^{-1}(0)$ if a multiple of $\\lambda$ belongs to $\\Sigma_0^+$. Such a multiple could only be $\\pm\\lambda,\\pm2\\lambda$. Hence, it will suffice to consider $\\lambda\\in\\bar\\Sigma_1^+$. By definition, $2\\lambda\\not\\in\\Sigma$. \n\t\n\tConsider $P(d;h)$ as a map linear in $d$, and let $N_h=\\ker P(-;h)$. Let $d\\in S^{\\sle k,\\mathrm{tot}}(\\mathfrak p)$. Assume that $d=zd'$ where $z$ is defined by $x=y+z$, $y\\in\\mathfrak k$, $z\\in\\mathfrak p$, for some  $x\\in\\mathfrak g^\\mu_{\\mathfrak a}$ and $\\mu\\in\\Sigma^+$, $\\mu\\neq\\lambda$. Then, modulo $N_h$, \n\t\\[\n\t\td=zd'\\equiv zd'+\\frac{u_h(y)d'}{\\mu(h)}=zd'+\\frac{[y,h]d'}{\\mu(h)}+\\frac{\\ad(y)(d')}{\\mu(h)}=\\frac{\\ad(y)(d')}{\\mu(h)}\\ .\n\t\\]\n\tThe root $\\mu$ is not proportional to $\\lambda$ and the total degree of $\\ad(y)(d')$ is strictly less than that of $d$. By induction, modulo $N_h$, \n\t\\[\n\t\td\\equiv\\frac{\\tilde d}{\\Pi_{\\mu\\in\\Sigma^+\\setminus\\lambda}\\,\\mu(h)^k}\n\t\\]\n\tfor some  $\\tilde d$ which lies in the subalgebra of $S(\\mathfrak p)$ generated by $\\mathfrak a\\oplus\\mathfrak p_1^\\lambda$, and depends polynomially on $h$ and linearly on $d\\in S^{\\sle k,\\mathrm{tot}}(\\mathfrak p)$. \n\t\n\tHence, the problem of showing that $P(d;h)$ remains bounded as $h$ approaches $\\lambda^{-1}(0)$ is reduced to the case of $d\\in S(\\mathfrak a\\oplus\\mathfrak p_1^\\lambda)$. For $d\\in S(\\mathfrak p_1^\\lambda)$, the polynomiality of $P(d;-)$ immediately follows from the assumption on $r$. \nIf $d=d'd''$ where $d'\\in S(\\mathfrak a)$ and $d''\\in S(\\mathfrak p_1^\\lambda)$, then $P(d;z)=[\\partial(d')P(d'';-)](z)$ since $P$ is $S(\\mathfrak p_0)$-linear. But $P(d'';-)\\in\\cplxs[\\mathfrak p_0]$ and this space is $S(\\mathfrak p_0)$-invariant, so $P(d;-)\\in\\cplxs[\\mathfrak p_0]$. \n\n\tTherefore, there exists $p\\in I(\\mathfrak p^*)$ such that  $P=\\phi(p)$. By its definition, it is clear that $p$ restricts to $r$, so we have proved the theorem.\n\\end{proof}\n\n\\subsection{Proof of Theorem (B)}\n\n\\begin{Par}[symplbasis]\n\tIn order to give a complete description of the image of the restriction map, we need to compute the radial parts $\\gamma_h(d)$ for $d\\in S(\\mathfrak p_1^\\lambda)$ and $h\\in\\mathfrak a'$ explicitly. First, let us choose bases of the spaces $S(\\mathfrak p_1^\\lambda)$. \n\n\tLet $\\lambda\\in\\Sigma^+_1$. By \\thmref{Prop}{res-superroot}~(v) we may choose $b^\\theta$-symplectic bases $y_i,\\tilde y_i\\in\\mathfrak k_1^\\lambda$, $z_i,\\tilde z_i\\in\\mathfrak p_1^\\lambda$, $i=1,\\dotsc,\\frac12m_{1,\\lambda}$, $m_{1,\\lambda}=\\dim\\mathfrak g_{1,\\mathfrak a}^\\lambda$. \\emph{I.e.},\n\t\\[\n\t\tb(y_i,\\tilde y_j)=b(\\tilde z_j,z_i)=\\delta_{ij}\\,,\\,b(y_i,y_j)=b(\\tilde y_i,\\tilde y_j)=b(z_i,z_j)=b(\\tilde z_i,\\tilde z_j)=0\\ .\n\t\\]\n\tWe may impose the conditions $x_i=y_i+z_i,\\tilde x_i=\\tilde y_i+\\tilde z_i\\in\\mathfrak g_{1,\\mathfrak a}^\\lambda$, so that  \n\t\\[\t\n\t\t[h,y_i]=\\lambda(h)z_i\\,,\\,[h,\\tilde y_i]=\\lambda(h)\\tilde z_i\\,,\\,[h,z_i]=\\lambda(h)y_i\\,,\\,[h,\\tilde z_i]=\\lambda(h)\\tilde y_i\n\t\\]\n\tfor all  $h\\in\\mathfrak a$. (Compare \\thmref{Prop}{res-superroot}~(iv).) \n\t\n\tGiven partitions $I=(i_1<\\dotsm<i_k)$, $J=(j_1<\\dotsm<j_\\ell)$, we define monomials $z_I\\tilde z_J=z_{i_1}\\dotsm z_{i_k}\\tilde z_{j_1}\\dotsm\\tilde z_{j_\\ell}$ in $S(\\mathfrak p_1^\\lambda)=\\bigwedge(\\mathfrak p_1^\\lambda)$. They form a basis of $S(\\mathfrak p_1^\\lambda)$. \n\\end{Par}\n\n\\begin{Lem}[radial-rank1-odd]\n\tFix $\\lambda\\in\\bar\\Sigma_1^+$. Let $h\\in\\mathfrak a$ be oddly regular, $I,J$ be multi-indices where $k=\\Abs0I$, $\\ell=\\Abs0J$, and let $m$ be a non-negative integer. Modulo $\\ker\\gamma_h$,\n\t\\[\n\t\tz_I\\tilde z_JA_\\lambda^m\\equiv\\begin{cases}0&I\\neq J\\ ,\\\\\n\t\tA_\\lambda^m & I=J=\\vvoid\\ ,\\\\\n\t\t(-1)^kz_{I'}\\tilde z_{I'}\\textstyle\\sum_{j=0}^m(-1)^j\\tfrac{\\lambda(A_\\lambda)^j}{\\lambda(h)^{j+1}}(m)_jA_\\lambda^{m+1-j}&I=J=(i<I')\\ ,\\end{cases}\n\t\\]\n\twhere $(m)_j$ is the falling factorial $m(m-1)\\dotsm(m-j+1)$, and $(m)_0=1$.\n\t\\end{Lem}\n\n\\begin{proof}\n\t For $k=\\ell=0$, there is nothing to prove. We assume that $k>0$ or $\\ell>0$, and write $I=(i<I')$ if $k>0$, $J=(j<J')$ if $\\ell>0$. We claim that modulo $\\ker\\gamma_h$,\n\t\\[\n\t\tz_I\\tilde z_JA_\\lambda^m\\equiv\\begin{cases}0&k\\neq\\ell\\text{ or }i\\neq j\\ ,\\\\\n\t\t(-1)^kz_{I'}\\tilde z_{J'}\\textstyle\\sum_{n=0}^m(-1)^n\\tfrac{\\lambda(A_\\lambda)^n}{\\lambda(h)^{n+1}}(m)_nA_\\lambda^{m+1-n}&i=j\\ .\\end{cases}\n\t\\]\n\tWe argue by induction on $\\max(k,\\ell)$. There will also be a sub-induction on the integer $m$. First, we assume that $k>0$, and compute \n\t\\[\n\t\tz_I\\tilde z_JA_\\lambda^m\\equiv z_iz_{I'}\\tilde z_JA_\\lambda^m+\\tfrac1{\\lambda(h)}u_h(y_i)(z_{I'}\\tilde z_JA_\\lambda^m)=\\tfrac1{\\lambda(h)}\\ad(y_i)(z_{I'}\\tilde z_JA_\\lambda^m)\\ .\n\t\\]\n\t\n\tFor any $q$, we have \n\t\\[\n\t\tb\\Parens1{[y_i,z_q],h'}=-\\lambda(h')b(y_i,y_q)=0\\mathfa h'\\in\\mathfrak a\\ ,\n\t\\]\n\tso $b([y_i,z_q],\\mathfrak a)=0$, and $[y_i,z_q]\\in\\mathfrak p_0$. Hence $[y_i,z_q]\\in\\mathfrak g_{0,\\mathfrak a}^{2\\lambda}\\oplus\\mathfrak g_{0,\\mathfrak a}^{-2\\lambda}=0$. Similarly, for $i\\neq q$, we have $[y_i,\\tilde z_q]=0$. Now, assume that $i\\sle J$. Then \n\t\\begin{align*}\n\tz_I\\tilde z_JA_\\lambda^m&\\equiv(-1)^{k-1}\\tfrac1{\\lambda(h)}z_{I'}\\ad(y_i)(\\tilde z_JA_\\lambda^m)\\\\\n\t&=(-1)^{k-1}\\tfrac1{\\lambda(h)}[y_i,\\tilde z_j]z_{I'}\\tilde z_{J'}A_\\lambda^m-m\\tfrac{\\lambda(A_\\lambda)}{\\lambda(h)}z_I\\tilde z_JA_\\lambda^{m-1}\\tag{$*$}\n\t\\end{align*}\n\tsince $[y_i,A_\\lambda^m]=-m\\lambda(A_\\lambda)z_iA_\\lambda^{m-1}$. As it stands, equation ($*$) only holds for $\\ell>0$, but if we take the first summand to be $0$ if $\\ell=0$, then it is also true in the latter case.\n\t\n\tIf $\\ell>0$ and $i<J$, then the first summand also vanishes, and arguing by induction on $m$, we find \n\t\\[\n\tz_I\\tilde z_JA_\\lambda^m\\equiv(-1)^mm!\\tfrac{\\lambda(A_\\lambda)^m}{\\lambda(h)^m}z_I\\tilde z_J=(-1)^{m+k-1}m!\\tfrac{\\lambda(A_\\lambda)^m}{\\lambda(h)^{m+1}}[y_i,\\tilde z_j]z_{I'}\\tilde z_J=0\\ .\n\t\\]\n\t Virtually the same reasoning goes through for $\\ell=0$. In particular, whenever $\\gamma_h(z_I\\tilde z_JA_\\lambda^m)\\neq0$ and $k>0$, then $i\\sle J$ implies $\\ell>0$ and $i=j$.\n\t\t\n\tIf $\\ell>0$ and $j\\sle I$, then we observe that $z_I\\tilde z_J=(-1)^{k\\ell}\\tilde z_Jz_I$. Formally exchanging the letters $z_s$ and $\\tilde z_s$ in the above equations, and reordering all terms in the appropriate fashion, we obtain\n\t\\[\\tag{$**$}\n\t\tz_I\\tilde z_JA_\\lambda^m\\equiv(-1)^k\\tfrac1{\\lambda(h)}[\\tilde y_j,z_i]z_{I'}\\tilde z_{J'}A_\\lambda^m-m\\tfrac{\\lambda(A_\\lambda)}{\\lambda(h)}z_I\\tilde z_JA_\\lambda^{m-1}\\ ,\n\t \\]\n\t because $k\\ell+\\ell-1+(k-1)(\\ell-1)=k(2\\ell-1)\\equiv k\\ (2)$. Arguing as above, the right hand side of equation ($**$) is equivalent to $0$ modulo $\\ker\\gamma_h$ if $k=0$ or $j<I$. Therefore, $\\gamma_h(z_I\\tilde z_JA_\\lambda^m)$ vanishes unless $k,\\ell>0$ and $i=j$. \n\t \n\t We consider the case of $k,\\ell>0$ and $i=j$. Since $[y_i,\\tilde z_i]-[\\tilde y_i,z_i]=-2A_\\lambda$ by standard arguments, we find, by adding equations ($*$) and ($**$), \n\t\\[\n\t\tz_I\\tilde z_JA_\\lambda^m\\equiv(-1)^k\\tfrac1{\\lambda(h)}z_{I'}\\tilde z_{J'}A_\\lambda^{m+1}-m\\tfrac{\\lambda(A_\\lambda)}{\\lambda(h)}z_I\\tilde z_JA_\\lambda^{m-1}\\ .\n\t\\]\n\tWe may now apply this formula recursively to the second summand, to conclude\n\t\\[\n\tz_I\\tilde z_JA_\\lambda^m\\equiv(-1)^kz_{I'}\\tilde z_{J'}\\textstyle\\sum_{n=0}^m(-1)^n\\tfrac{\\lambda(A_\\lambda)^n}{\\lambda(h)^{n+1}}(m)_nA_\\lambda^{m+1-n}\\ .\n\t\\]\n\n\tBy induction on $\\max(k,\\ell)$, the right hand side belongs to $\\ker\\gamma_h$ unless $k=\\ell$. We have proved our claim, and thus, we arrive at the assertion of the lemma. \n\\end{proof}\n\n\\begin{Par}\t\n\tFix $\\lambda\\in\\bar\\Sigma_1^+$ and $h\\in\\mathfrak a'$. Let $I=(i_1<\\dotsm<i_k)$ and $1\\sle\\ell\\sle k$. Set $I'=(i_{\\ell+1}<\\dotsm<i_k)$. Let \n\t\\[\n\t\t\\eps^k_\\ell=(-1)^{\\sum_{j=k-\\ell+1}^kj}=(-1)^{\\frac\\ell2(2k-\\ell+1)}\\ .\n\t\\]\n\tWe claim that there are $b_{s\\ell}\\in\\nats$, $s<\\ell$, $b_{01}=1$, such that, modulo $\\ker\\gamma_h$, \n\t\\[\n\t\tz_I\\tilde z_I\\equiv\\eps^k_\\ell z_{I'}\\tilde z_{I'}\\textstyle\\sum_{j=0}^{\\ell-1} b_{j\\ell}\\tfrac{(-\\lambda(A_\\lambda))^j}{\\lambda(h)^{\\ell+j}}A_\\lambda^{\\ell-j}\\ .\\tag{$***$}\n\t\\]\n\t\n\tThe case $\\ell=1$ has already been established. To prove the inductive step, let $I''=(i_\\ell,\\dotsc,i_k)=(i_\\ell<I')$, and $J=(i_0<I)$. We compute\n\t\\begin{align*}\n\t\tz_J\\tilde z_J&\\equiv\\eps_\\ell^{k+1}z_{I''}\\tilde z_{I''}\\textstyle\\sum_{j=0}^{\\ell-1}b_{j\\ell}\\tfrac{(-\\lambda(A_\\lambda))^j}{\\lambda(h)^{\\ell+j}}A_\\lambda^{\\ell-j}\\\\\n\t&\\equiv(-1)^{k-\\ell+1}\\eps_\\ell^{k+1}z_{I'}\\tilde z_{I'}\\textstyle\\sum_{s=0}^\\ell\\sum_{j=0}^{\\min(s,\\ell-1)}(\\ell-j)_{s-j}b_{j\\ell}\\tfrac{(-\\lambda(A_\\lambda))^s}{\\lambda(h)^{\\ell+1+s}}A_\\lambda^{\\ell+1-s}\\ ,\n\t\\end{align*}\n\tso \n\t\\[\n\t\tb_{s,\\ell+1}=\\textstyle\\sum_{j=0}^{\\min(s,\\ell-1)}(\\ell-j)_{s-j}b_{j\\ell}=\\frac1{(\\ell-s)!}\\sum_{j=0}^{\\min(s,\\ell-1)}(\\ell-j)!b_{j\\ell}\\ .\n\t\\]\n\tThis proves our claim, where the constants $b_{s\\ell}$ obey the recursion relation set out above. \n\t\n\tTo solve this recursion, we claim that \n\t\\[\n\t\tb_{s\\ell}=\\frac{(\\ell-1+s)!}{2^s(\\ell-1-s)!s!}\\mathfa 0\\sle s<\\ell\\ .\n\t\\]\n\tThis is certainly the case for $\\ell=1$. By induction, for all  $0\\sle s\\sle\\ell$, $\\ell\\sge1$, \n\t\\[\n\t\tb_{s,\\ell+1}=\\tfrac1{(\\ell-s)!}\\textstyle\\sum_{j=0}^{\\min(s,\\ell-1)}(\\ell-j)\\tfrac{(\\ell-1+j)!}{2^jj!}\\ .\n\t\\]\n\tAs is easy to show by induction, $\\sum_{j=0}^N(\\ell-j)\\tfrac{(\\ell-1+j)!}{2^jj!}=\\tfrac{(\\ell+N)!}{2^NN!}$. Hence, \n\t\\[\n\t\tb_{s,\\ell+1}=\\begin{cases}\\frac{(\\ell+s)!}{2^s(\\ell-s)!s!}&0\\sle s<\\ell\\\\\\frac{(2\\ell-1)!}{2^{\\ell-1}(\\ell-1)!}=\\frac{(2\\ell)!}{2^\\ell\\ell!} & s=\\ell\\end{cases}\n\t\\]\n\twhich establishes the claim. \n\t\n\tSetting $\\ell=k=\\Abs0I$ in $(***)$, we obtain the following lemma.\n\\end{Par}\n\n\\begin{Lem}[radialfmla]\n\tFix $\\lambda\\in\\bar\\Sigma_1^+$. Let $h\\in\\mathfrak a$ be oddly regular, $I$ be a multi-index where $k=\\Abs0I$. Then \n\t\\[\n\t\t\\gamma_h(z_I\\tilde z_I)=(-1)^{\\frac{k(k+1)}2}\\textstyle\\sum_{j=0}^{k-1}\\tfrac{(k-1+j)!}{2^j(k-1-j)!j!}\\tfrac{(-\\lambda(A_\\lambda))^j}{\\lambda(h)^{k+j}}A_\\lambda^{k-j}\\ .\n\t\\]\n\\end{Lem}\n\n\\begin{Rem}\n\tIn passing, note that $b_{k-2,k}=b_{k-1,k}=\\frac{(2k-2)!}{2^{k-1}(k-1)!}$. We remark also that $\\theta_n(z)=\\sum_{j=0}^nb_{j,n+1}z^{n-j}$ are so-called \\emph{Bessel polynomials} \\cite{grosswald-besselpol}, \\cite[A001498]{OEIS}.\n\\end{Rem}\n\n\\begin{Par}\n\tLet $\\lambda\\in\\bar\\Sigma_1^+$, $\\lambda(A_\\lambda)=0$. By \\thmref{Lem}{radialfmla}, we find for all  $I$, $\\Abs0I=k$, that $\\gamma_h(z_I\\tilde z_I)=(-1)^{\\frac12k(k+1)}\\lambda(h)^{-k}A_\\lambda^k$ ($h\\in\\mathfrak a'$). Hence, \n\t\\[\n\t\t\\bigcap\\nolimits_{D\\in\\mathcal R_\\lambda}\\dom D=\\bigcap\\nolimits_{k=1}^{\\frac12m_{1,\\lambda}}\\dom\\lambda^{-k}\\partial(A_\\lambda)^k\\ .\n\t\\]\n\t\n\tThe situation in the case $\\lambda(A_\\lambda)\\neq0$ is different and requires a more detailed study.\n\\end{Par}\n\t\n\\begin{Par}[radial-action]\n\t\tLet $\\lambda\\in\\bar\\Sigma_1^+$, $\\lambda(A_\\lambda)\\neq0$. Then $\\cplxs[\\mathfrak a]\\cong R[\\lambda]$ where $R=\\cplxs[\\ker\\lambda]$. This isomorphism is equivariant for $S(\\cplxs A_\\lambda)$ if we define an action $\\partial$ on $R[\\lambda]$ by requiring that $\\partial(A_\\lambda)$ be the unique $R$-derivation for which $\\partial(A_\\lambda)\\lambda=\\lambda(A_\\lambda)$. \n\t\t\n\t\tNow, let $R$ be an arbitrary commutative unital $\\cplxs$-algebra. We define an action $\\partial$ of $S(\\cplxs A_\\lambda)$ on $R[\\lambda,\\lambda^{-1}]$ by requiring that $\\partial(A_\\lambda)$ be the unique $R$-derivation such that  $\\partial(A_\\lambda)=\\lambda(A_\\lambda)$ and $\\partial(A_\\lambda)\\lambda^{-1}=-\\lambda(A_\\lambda)\\lambda^{-2}$. The action $\\partial$ is faithful, because $\\lambda(A_\\lambda)\\neq0$.\n\t\t\n\t\tLet $\\mathcal D_\\lambda$ be the subalgebra of $\\End[_\\cplxs]0{R[\\lambda,\\lambda^{-1}]}$ generated by $\\partial(S(\\cplxs A_\\lambda))$ and $\\cplxs[\\lambda,\\lambda^{-1}]$. In particular, we may embed $\\mathcal R_\\lambda\\subset\\mathcal D_\\lambda$. We consider the action of $D\\in\\mathcal R_\\lambda$, $D(h)=\\gamma_h(z_I\\tilde z_I)$, $\\Abs0I=k$, on $p=\\sum_{j=0}^Na_j\\lambda^j\\in R[\\lambda]$, \n\t\t\\[\n\t\t\tDp=(-1)^{\\frac{k(k+1)}2}\\textstyle\\sum_{j=1}^Na_j\\lambda(A_\\lambda)^k\\lambda^{j-2k}\\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}\\in R[\\lambda,\\lambda^{-1}]\\ .\n\t\t\\]\n\t\tSince $\\lambda(A_\\lambda)\\neq0$, we have $Dp\\in R[\\lambda]$ if and only if  \n\t\t\\[\n\t\t\ta_j\\textstyle\\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}=0\\mathfa j=1,\\dotsc,2k-1\\ .\n\t\t\\]\n\t\t\n\t\tWe need to determine when the number\n\t\t\\begin{equation}\\label{eq:ajk-def}\n\t\ta_{jk}=\\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}=\\sum_{i=(k-j)_+}^{k-1}\\Parens2{-\\frac12}^i(j)_{k-i}\\frac{(k-1+i)!}{(k-1-i)!i!}\n\t\t\\end{equation}\n\t\tis non-zero. \n\\end{Par}\n\t\n\\begin{Par}\n\tFix $k\\sge1$. For $x\\in\\reals$ and $1\\sle j\\sle k$, let\n\t\\[\n\t\ta_{2k-j,k}(x)=\\sum_{i=0}^{k-1}x^i(2k-j)_{k-i}\\frac{(k-1+i)!}{(k-1-i)!i!}\\ .\n\t\\]\n\tWe claim that \n\t\\begin{equation}\\label{eq:coeffx}\n\t\ta_{2k-j,k}(x)=\\textstyle\\frac{(j-1)!(2k-j)!}{(k-1)!}\\sum_{\\ell=0}^{j-1}\\binom{k-1}\\ell\\binom{k-1}{j-1-\\ell}x^\\ell(1+x)^{k-1-\\ell}\\ .\n\t\\end{equation}\n\tTo that end, we rewrite \n\t\\[\n\t\ta_{2k-j,k}(x)=\\frac{(j-1)!(2k-j)!}{(k-1)!}\\sum_{i=0}^{k-1}\\binom{k-1}i\\binom{k+i-1}{j-1}x^i\\ .\n\t\\]\n\tThen, for fixed $x\\in\\reals$, we form the generating function\n\t\\[\n\t\tf(z)=\\sum_{j=1}^\\infty z^{j-1}\\sum_{i=0}^{k-1}\\binom{k-1}i\\binom{k+i-1}{j-1}x^i\\ .\n\t\\]\n\tIt is easy to see \n\t\\begin{align*}\n\t\tf(z)&=\\sum_{i=0}^{k-1}\\binom{k-1}ix^i\\sum_{j=1}^{k+i}\\binom{k+i-1}{j-1}z^{j-1}\\\\\n\t\t&=(1+z)^{2k-2}\\sum_{i=0}^{k-1}\\binom{k-1}ix^i\\Parens2{\\frac1{1+z}}^{k-1-i}\\\\\n\t\t&=(1+z)^{k-1}((1+z)x+1)^{k-1}\\ .\n\t\\end{align*}\n\t\n\tOn the other hand, we may form the generating function for the right hand side of (\\ref{eq:coeffx}), \n\t\\[\n\t\tg(z)=\\sum_{j=1}^\\infty z^{j-1}\\sum_{\\ell=0}^{j-1}\\binom{k-1}\\ell\\binom{k-1}{j-1-\\ell}x^\\ell(1+x)^{k-1-\\ell}\\ .\n\t\\]\n\tThen\n\t\\begin{align*}\n\t\tg(z)&=\\sum_{\\ell=0}^{k-1}\\binom{k-1}\\ell x^\\ell(1+x)^{k-1-\\ell}\\sum_{j=\\ell+1}^{k+\\ell}\\binom{k-1}{j-1-\\ell}z^{j-1}\\\\\n\t\t&=\\sum_{\\ell=0}^{k-1}\\binom{k-1}\\ell(xz)^\\ell(1+x)^{k-1-\\ell}\\sum_{j=0}^{k-1}\\binom{k-1}jz^j\\\\\n\t\t&=(xz+x+1)^{k-1}(1+z)^{k-1}=f(z)\\ .\n\t\\end{align*}\t\n\tSince the generating functions coincide, we have proved (\\ref{eq:coeffx}). \n\\end{Par}\n\n\\begin{Par}[evencoeffcase1]\n\tWe notice that for $k\\sge 1$ and $j=1,\\dotsc,k$, $k-(2k-j)=j-k\\sle0$, so $a_{2k-j,k}=a_{2k-j,k}\\Parens1{-\\tfrac12}$ by (\\ref{eq:ajk-def}). By (\\ref{eq:coeffx}), we obtain\n\t\\[\n\t\ta_{2k-j,k}=\\frac{(j-1)!(2k-j)!}{2^{k-1}(k-1)!}\\sum_{\\ell=0}^{j-1}(-1)^\\ell\\binom{k-1}\\ell\\binom{k-1}{j-1-\\ell}\n\t\\]\n\tFor $j=1$, one gets\n\t\\[\n\t\ta_{2k-1,k}=\\frac{(2k-1)!}{2^{k-1}(k-1)!}\\neq0\\ .\n\t\\]\n\tNow, let $j=2n$ where $1\\sle n\\sle\\lfloor\\tfrac k2\\rfloor$. Then $\\ell\\mapsto(-1)^\\ell\\binom{k-1}\\ell\\binom{k-1}{2n-1-\\ell}$ is odd under the permutation $\\ell\\mapsto2n-1-\\ell$ of $\\{0,\\dotsc,2n-1\\}$, so \n\t\\[\n\t\ta_{jk}=0\\mathfa j=k,\\dotsc,2k-2\\ ,\\ j\\equiv0\\ (2)\\ .\n\t\\]\n\\end{Par}\n\n\\begin{Par}[evencoeffcase2]\n\tNext, we study the behaviour of $a_{k-j,k}$ for $k\\sge1$ and $j=1,\\dotsc,k-1$, by a similar scheme. To that end, write \n\t\\begin{align*}\n\t\ta_{k-j,k}&=\\sum_{i=j}^{k-1}\\frac{(k-j)!(k-1+i)!}{(i-j)!(k-1-i)!i!}\\Parens2{-\\frac12}^i\\\\\n\t\t&=\\frac{(k-1+j)!(k-j)!}{(k-1)!}\\sum_{i=j}^{k-1}\\binom{k-1}i\\binom{k-1+i}{k-1+j}\\Parens2{-\\frac12}^i\\ .\n\t\\end{align*}\n \tObserve that we may sum over $i=0,\\dotsc,k-1$ since the second binomal coefficient vanishes for $i<j$. \n\t\n\tNow, we fix $x\\in\\reals$ and define $f(z)=\\sum_{j=1}^{k-1}a_{k-j,k}(x)z^{k+j-1}\\in\\cplxs[z]$ where\n\t\\[\n\t\ta_{k-j,k}(x)=\\sum_{i=0}^{k-1}\\binom{k-1}i\\binom{k-1+i}{k-1+j}x^i\\ .\n\t\\]\n\tWe wish to study the coefficients of the polynomial $f$. Observe that the lowest power of $z$ occuring in $f(z)$ is $z^k$. Thus, we compute, modulo $\\cplxs[z]_{<k}$, \n\t\\begin{align*}\n\t\tf(z)&=\\sum_{i=0}^{k-1}\\binom{k-1}ix^i\\sum_{j=1}^i\\binom{k-1+i}{k-1+j}z^{k+j-1}\\\\\n\t\t&=\\sum_{i=0}^{k-1}\\binom{k-1}ix^i\\sum_{j=k}^{k-1+i}\\binom{k-1+i}jz^j\\\\\n\t\t&\\equiv(1+z)^{k-1}\\sum_{i=0}^{k-1}\\binom{k-1}i(x(1+z))^i=(1+z)^{k-1}(1+x(1+z))^{k-1}\\ .\n\t\\end{align*}\t\n\t\n\tFor $j=k,\\dotsc,2k-2$, $a_{2k-j-1,k}(x)$ is the coefficient of $z^j$ in $f(z)$. Since \n\t\\[\n\t\t(1+z)^{k-1}(1+x(1+z))^{k-1}=\\sum_{j=0}^{2k-2}z^j\\sum_{i=0}^j\\binom{k-1}{j-i}\\binom{k-1}i(1+x)^{k-1-i}x^i\\ ,\n\t\\]\n\twe find, for $j=k,\\dotsc,2k-2$,\n\t\\begin{align*}\n\t\ta_{2k-j-1,k}(x)&=\\sum_{i=0}^j\\binom{k-1}{j-i}\\binom{k-1}i(1+x)^{k-1-i}x^i\\\\\n\t\t&=(1+x)^{k-1}\\sum_{i=j-k+1}^{k-1}\\binom{k-1}{j-i}\\binom{k-1}i\\Parens2{\\frac x{1+x}}^i\\ .\n\t\\end{align*}\n\t\n\tIn particular,  \n\t\\[\n\t\ta_{2k-j-1,k}\\Parens1{-\\tfrac12}=2^{1-k}\\sum_{i=j-k+1}^{k-1}(-1)^i\\binom{k-1}{j-i}\\binom{k-1}i\\ .\n\t\\]\n\tNotice that the function $i\\mapsto(-1)^i\\binom{k-1}{j-i}\\binom{k-1}i$ has parity $j$ with respect to the permutation $i\\mapsto j-i$ of $\\{j-k+1,\\dotsc,k-1\\}$. Since $2k-j-1$ is even and only if $j$ is odd, this implies\n\t\\[\n\t\ta_{jk}=0\\mathfa j=2,\\dotsc,k-1\\ ,\\ j\\equiv0\\ (2)\\ .\n\t\\]\n\t\n\tWe summarise the above considerations in the following proposition.\n\\end{Par}\n\n\\begin{Prop}[commondom]\n\tLet $R$ be a commutative unital $\\cplxs$-algebra, and $\\lambda\\in\\bar\\Sigma_1^+$ such that  $\\lambda(A_\\lambda)\\neq0$. Let $m\\sge1$ be an integer, and for $k=1,\\dotsc,m$, define\n\t\\[\n\t\tD_k=(-1)^{\\frac{k(k+1)}2}\\textstyle\\sum_{j=0}^{k-1}\\frac{(k-1+j)!}{2^j(k-1-j)!j!}\\frac{(-\\lambda(A_\\lambda))^j}{\\lambda^{k+j}}A_\\lambda^{k-j}\\in \\mathcal D_\\lambda\\ .\n\t\\]\n\tLet $p=\\sum_{j=0}^Na_j\\lambda^j\\in R[\\lambda]$. Then $D_kp\\in R[\\lambda]$ for all  $k=1,\\dotsc,m$ if and only \n\t$a_j=0$ for all $j=1,\\dotsc,2m-1$, $j\\equiv1\\ (2)$. \n\\end{Prop}\n\n\\begin{proof}\n\tLet $1\\sle k\\sle m$. We have $a_{2k-1}a_{2k-1,k}=0$ and $a_{2k-1,k}\\neq0$, so $a_{2k-1}=0$. Conversely, there are no further conditions, since $a_{km}=0$ for even $k$, $1<k<2m$. \n\\end{proof}\n\n\\begin{Par}\n\tTo apply \\thmref{Prop}{commondom} to the determination of the image of the restriction map, let $\\lambda\\in\\bar\\Sigma_1^+$, $\\lambda(A_\\lambda)\\neq0$. Note that $\\cplxs[\\mathfrak a]=\\cplxs[\\ker\\lambda][\\lambda]$. Then for all  $p\\in\\cplxs[\\mathfrak a]$, \n\t\\[\n\t\tp=\\textstyle\\sum_{j=0}^\\infty(j!)^{-1}\\partial(A_\\lambda)^jp|_{\\ker\\lambda}\\Parens1{\\frac\\lambda{\\lambda(A_\\lambda)}}^j\\ .\n\t\\]\n\t\\emph{I.e.}, if we take $R=\\cplxs[\\ker\\lambda]$, then $p=\\sum_ja_j\\lambda^j$ where the coefficients are given by $a_j=\\frac1{\\lambda(A_\\lambda)^jj!}\\partial(A_\\lambda)^jp|_{\\ker\\lambda}\\in R$. Also, $\\partial(A_\\lambda)^ip|_{\\ker\\lambda}=0$ for all  $i=1,\\dotsc,j$ if and only if $p\\in\\cplxs\\oplus\\lambda^{j+1}\\cplxs[\\mathfrak a]$. Together with \\thmref{Th}{super-chevalley}, we immediately obtain our main result, as follows. \n\\end{Par}\n\n\\begin{Th}[chevrestrb]\n\tThe restriction homomorphism $I(\\mathfrak p^*)\\to S(\\mathfrak a^*)$ is a bijection onto the subspace $I(\\mathfrak a^*)=\\bigcap_{\\lambda\\in\\bar\\Sigma_1^+}S(\\mathfrak a^*)^W\\cap I_\\lambda$ where\n\t\\[\n\t\tI_\\lambda=\\textstyle\\bigcap_{j=1}^{\\frac12m_{1,\\lambda}}\\dom\\lambda^{-j}\\partial(A_\\lambda)^j\\mathtxt{if}\\lambda(A_\\lambda)=0\n\t\\]\n\tand if $\\lambda(A_\\lambda)\\neq0$, then $I_\\lambda$ consists of those $p\\in\\cplxs[\\mathfrak a]$ such that \n\t\\[\n\t\t\\partial(A_\\lambda)^kp|_{\\ker\\lambda}=0\\mathfa[odd integers] k\\ ,\\ 1\\sle k\\sle m_{1,\\lambda}-1\\ .\n\t\\]\n\\end{Th}\n\n\\section{Examples}\n\n\\subsection{Scope of the theory}\n\n\\begin{Par}\n\tAs remarked in \\thmref{Par}{grouptype}, \\thmref{Th}{chevrestrb} applies to a symmetric superpair of group type where $\\mathfrak k$ is classical and carries a non-degenerate invariant even form. The assumptions are still fulfilled if we add to $\\mathfrak k$ an even reductive ideal. Hence, $\\mathfrak k$ may be a direct sum of a reductive Lie algebra, and copies of any of the following Lie superalgebras \\cite{kac-liesuperalgs}: \n\t\\begin{gather*}\n\t\t\\mathfrak{gl}(p|q,\\cplxs)\\ ,\\ \\sll(p|q,\\cplxs)\\ (p\\neq q)\\ ,\\ \\sll(p|p,\\cplxs)\/\\cplxs\\ ,\\\\\n\t\t\\mathfrak{osp}(p|2q,\\cplxs)\\ ,\\ D(1,2;\\alpha)\\ ,\\ F(4)\\ ,\\ G(3)\\ .\n\t\\end{gather*}\n\tAs follows from \\thmref{Prop}{superroot} (iv), in this situation one has $\\lambda(A_\\lambda)=0$ for all  $\\lambda\\in\\bar\\Sigma_1^+$.\n\\end{Par}\n\n\\begin{Par}[glexample]\n\tIf we take $(\\mathfrak g,\\mathfrak k)$ to be an arbitrary reductive symmetric superpair, then the assumption of \\emph{even type} amounts to an additional condition. \n\t\n\tAs an example, we consider $\\mathfrak g=\\mathfrak{gl}(p+q|r+s,\\cplxs)$, $p,q,r,s\\sge0$, where $\\theta$ is given by conjugation with the diagonal matrix whose diagonal entries are the matrix blocks $1_p$, $-1_q$, $1_r$, $-1_s$. Let $\\mathfrak a\\subset\\mathfrak p_0$ be the maximal Abelian subalgebra of all matrices\n\t\\[\n\t\t\\begin{Matrix}10&A&0&0\\\\-A^t&0&0&0\\\\0&0&0&B\\\\0&0&-B^t&0\\end{Matrix}\\in\\cplxs^{(p+q+r+s)\\times(p+q+r+s)}\n\t\\]\n\twhere $A=(D,0)$ or $A=\\begin{Matrix}0D\\\\0\\end{Matrix}$ for a diagonal matrix $D\\in\\cplxs^{\\min(p,q)\\times\\min(p,q)}$, and similarly for $B$. Let $x_j$, $j=1,\\dotsc,\\min(p,q)$, and $y_\\ell$, $\\ell=1,\\dotsc,\\min(r,s)$, be the linear forms on $\\mathfrak a$ given by the entries of the diagonal blocks of $A,B$. \n\t\n\tConsider the $\\mathfrak a$-module $\\mathfrak g_1$. Then the non-zero weights are \n\t\\[\n\t\t\\pm(x_j\\pm y_\\ell)\\ (2)\\ ,\\ \\pm x_j\\ (2\\Abs0{r-s})\\ ,\\ \\pm y_\\ell\\ (2\\Abs0{p-q})\n\t\\]\n\twith multiplicities given in parentheses \\cite{schmittner-zirnbauer}. The sum $U\\subset\\mathfrak g_1$ of the non-zero weight spaces therefore has dimension\n\t\\begin{align*}\n\t\t8\\min(p,q)\\min(r,s)&+4\\Abs0{r-s}\\min(p,q)+4\\Abs0{p-q}\\min(r,s)\\\\\n\t\t&=2\\Parens1{(p+q)(r+s)-\\Abs0{p-q}\\Abs0{r-s}}\\ .\n\t\\end{align*}\n\t(The equation follows by applying the formula $2\\min(a,b)=a+b-\\Abs0{a-b}$.)\n\t\n\tWe have that $U$ is $\\theta$-stable, and the action of a generic $h\\in\\mathfrak a$ induces an automorphism of $U$. Hence, we have $\\dim U_{\\mathfrak k}=\\dim U_{\\mathfrak p}=\\frac12\\dim U$ where $U_{\\mathfrak k}$ and $U_{\\mathfrak p}$ are the projections of $U$ onto $\\mathfrak k_1$ and $\\mathfrak p_1$, respectively. It follows that $\\dim U_{\\mathfrak p}=(p+q)(r+s)-\\Abs0{p-q}\\Abs0{r-s}$. On the other hand, \n\t\\[\n\t\t\\dim\\mathfrak p_1=2(ps+rq)=(p+q)(r+s)-(p-q)(r-s)\\ .\n\t\\]\n\tHence, $\\mathfrak z_{\\mathfrak p_1}(\\mathfrak a)=0$ if and only if $(p-q)(r-s)\\sge0$, and $(\\mathfrak g,\\mathfrak k)$ is of even type  if and only if this condition holds. \n\t\n\tWe remark that in this case, the set $\\bar\\Sigma_1^+$ consists of the weights $x_j\\pm y_\\ell$ (for a suitably chosen positive system). For each $\\lambda\\in\\bar\\Sigma_1^+$, one has $\\lambda(A_\\lambda)=0$. \n\\end{Par}\n\n\\begin{Par}\n\tA similar example arises by restricting the involution from \\thmref{Par}{glexample} to the subalgebra $\\mathfrak g=\\mathfrak{osp}(p+q|r+s,\\cplxs)$, where we now assume $r$ and $s$ to be even. We realise $\\mathfrak g$ by taking the direct sum of the standard non-degenerate symmetric forms on $\\cplxs^p\\oplus\\cplxs^q$, and the direct sum of the standard symplectic forms on $\\cplxs^r\\oplus\\cplxs^s$. \n\t\n\tFor $k$ even, denote by $J_k\\in\\cplxs^{k\\times k}$ the matrix representing the standard symplectic form. Let $\\mathfrak a\\subset\\mathfrak p_0$ be the maximal Abelian subalgebra of all matrices\n\t\\[\n\t\t\\begin{Matrix}10&A&0&0\\\\-A^t&0&0&0\\\\0&0&0&B\\\\0&0&J_sB^tJ_r&0\\end{Matrix}\\in\\cplxs^{(p+q+r+s)\\times(p+q+r+s)}\n\t\\]\n\twhere $A=(D,0)$ or $A=\\begin{Matrix}0D\\\\0\\end{Matrix}$ for a diagonal matrix $D\\in\\cplxs^{\\min(p,q)\\times\\min(p,q)}$, and $B=(D',0)$ or $B=\\begin{Matrix}0D'\\\\0\\end{Matrix}$ for a diagonal matrix $D'\\in\\cplxs^{\\frac12\\min(r,s)\\times\\frac12\\min(r,s)}$. \n\t\n\tBy restriction, we obtain the following non-zero $\\mathfrak a$-weights in $\\mathfrak g_1$, \n\t\\[\n\t\t\\pm(x_j\\pm y_\\ell)\\ (2)\\ ,\\ \\pm x_j\\ (\\Abs0{r-s})\\ ,\\ \\pm y_\\ell\\ (2\\Abs0{p-q})\\ ,\n\t\\]\n\twhere now $j=1,\\dotsc,\\min(p,q)\\,,\\,\\ell=1,\\dotsc,\\tfrac12\\min(r,s)$, and the multiplicities are given in parentheses \\cite{schmittner-zirnbauer}.\n\t\n\tLet $U$ be the sum of all weight spaces for non-zero weights of the $\\mathfrak a$-module $\\mathfrak g_1$. Then the dimension of $U$ is \n\t\\begin{align*}\n\t\t4\\min(p,q)\\min(r,s)&+2\\Abs0{r-s}\\min(p,q)+2\\Abs0{p-q}\\min(r,s)\\\\\n\t\t&=(p+q)(r+s)-\\Abs0{p-q}\\Abs0{r-s}\\ .\n\t\\end{align*}\n\tIf $U_{\\mathfrak p}$ is the projection of $U$ onto $\\mathfrak p_1$, then by the same argument as in \\thmref{Par}{glexample}, $\\dim U_{\\mathfrak p}=\\frac12\\dim U$. We have \n\t\\[\n\t\\dim\\mathfrak p_1=pq+rs=\\tfrac12\\Parens1{(p+q)(r+s)-(p-q)(r-s)}\\ ,\n\t\\]\n\tso, as above, $(\\mathfrak g,\\mathfrak k)$ is of even type if and only if $(p-q)(r-s)\\sge0$. In this case, as in \\thmref{Par}{glexample}, the set $\\bar\\Sigma_1^+$ consists of the weights $x_j\\pm y_\\ell$ (for a suitable choice of positive system), and again we have $\\lambda(A_\\lambda)=0$ for all  $\\lambda\\in\\bar\\Sigma_1^+$. \n\\end{Par}\n\n\\subsection{An extremal class: $\\mathfrak g=C(q+1)=\\mathfrak{osp}(2|2q,\\cplxs)$, $\\mathfrak k_0=\\mathfrak{sp}(2q,\\cplxs)$}\n\n\\begin{Par}\n\tConsider the Lie superalgebra $\\mathfrak g=C(q+1)=\\mathfrak{osp}(2|2q,\\cplxs)$ where $q\\sge1$ is arbitrary. Let $I=\\begin{Matrix}00&1\\\\1&0\\end{Matrix}\\in\\cplxs^{2\\times2}$ and $J=\\begin{Matrix}00&1\\\\-1&0\\end{Matrix}\\in\\cplxs^{2q\\times2q}$. If we realise $\\mathfrak g$ with respect to the orthosymplectic form $I\\oplus J$, it consists of the matrices\n\t\\[\n\t\tx=\\begin{Matrix}1a&0&-w^{\\prime t}&z^{\\prime t}\\\\0&-a&-w^t&z^t\\\\ z&z'&A&B\\\\ w&w'&C&-A^t\\end{Matrix}\t\n\t\\]\n\twhere $a\\in\\cplxs$, $z,z',w,w'\\in\\cplxs^q$, $A,B=B^t,C=C^t\\in\\cplxs^{q\\times q}$.\n\n\tThe matrix $g=\\begin{Matrix}0I&0\\\\0&1\\end{Matrix}\\in\\cplxs^{(2+2q)\\times(2+2q)}$ represents an even automorphism of the super-vector space $\\cplxs^{2|2q}$, of order $2$. Since $g$ leaves the orthosymplectic form invariant, $\\theta(x)=gxg$ defines an involutive automorphism of $\\mathfrak g$. Moreover, since $g^2=1$, the supertrace form $b(x,y)=\\str(xy)$ on $\\mathfrak g$ is $\\theta$-invariant. Hence, $(\\mathfrak g,\\mathfrak k)$, where $\\mathfrak k=\\mathfrak g_\\theta$, is a reductive symmetric superpair.\n\t\n\tWe compute\n\t\\[\n\t\t\\theta(x)=\\begin{Matrix}1-a&0&-w^t&z^t\\\\0&a&-w^{\\prime t}&z^{\\prime t}\\\\z'&z&A&B\\\\w'&w&C&-A^t\\end{Matrix}\n\t\\]\n\twhen $x\\in\\mathfrak g$ is written as above. Hence, the general elements of $\\mathfrak k$ and $\\mathfrak p$ are respectively of the form \n\t\\[\n\t\tx=\\begin{Matrix}10&0&-w^t&z^t\\\\0&0&-w^t&z^t\\\\z&z&A&B\\\\w&w&C&-A^t\\end{Matrix}\\mathtxt\\AND\n\t\tx=\\begin{Matrix}1a&0&w^t&-z^t\\\\0&-a&-w^t&z^t\\\\z&-z&0&0\\\\w&-w&0&0\\end{Matrix}\\ .\n\t\\]\n\t\n\tIt is immediate that the one-dimensional space $\\mathfrak a=\\mathfrak p_0$ is self-centralising in $\\mathfrak p_0$. In particular, any non-zero element of $\\mathfrak a$ is $b$-anisotropic (since $\\mathfrak p_0$ is non-degenerate). The bracket relation for the general element of $[\\mathfrak a,\\mathfrak g_1]$\n\t\\[\n\t\t\\left[\\begin{Matrix}10&a&0&0\\\\-a&0&0&0\\\\0&0&0&0\\\\0&0&0&0\\end{Matrix},\\begin{Matrix}10&0&-w^{\\prime t}&z^{\\prime t}\\\\0&0&-w^t&z^t\\\\z&z'&0&0\\\\w&w'&0&0\\end{Matrix}\\right]=\\begin{Matrix}10&0&-aw^{\\prime t}&az^{\\prime t}\\\\0&0&aw^t&-az^t\\\\-az&az'&0&0\\\\-aw&aw'&0&0\\end{Matrix}\n\t\\]\n\timplies in particular that $\\mathfrak z_{\\mathfrak p_1}(\\mathfrak a)=0$. Hence, $\\mathfrak a$ is an even Cartan subspace, and $(\\mathfrak g,\\mathfrak k)$ is of even type.\n\t\n\tAlso, there are only two restricted roots, $\\pm\\lambda$, where $\\lambda$ maps $x\\in\\mathfrak a$ (as above) to $a$. Necessarily, $\\lambda$ is odd, so $2\\lambda\\not\\in\\Sigma=\\{\\pm\\lambda\\}$, and $W=W(\\Sigma_0)=1$. Since $A_\\lambda$ is $b$-anisotropic, we have $\\lambda(A_\\lambda)\\neq0$.\n\t\n\tMoreover, we must have $\\mathfrak p_1=\\mathfrak p_1^\\lambda$, and this space has dimension $2q$, so $m_{1,\\lambda}=2q$. From \\thmref{Th}{chevrestrb}, we obtain the following result.\n\\end{Par}\n\n\\begin{Prop}[specialosp22q]\n\tLet $\\mathfrak g=\\mathfrak{osp}(2|2q,\\cplxs)$, with the involution defined above. The image of the restriction map $S(\\mathfrak p^*)^{\\mathfrak k}\\to S(\\mathfrak a^*)=\\cplxs[\\lambda]$ is \n\t\\[\n\t\tI(\\mathfrak a^*)=\\Set1{p=\\textstyle\\sum_ja_j\\lambda^j}{a_{2j-1}=0\\ \\forall j=1,\\dotsc,q}\\ .\n\t\\]\n\t\n\tIn particular, the algebra $I(\\mathfrak a^*)$ is isomorphic to the commutative unital $\\cplxs$-algebra defined by the generators $\\lambda_2$, $\\lambda_{2q+1}$, and the relation\n\t\\[\n\t\t(\\lambda_2)^{2q+1}=(\\lambda_{2q+1})^2\\ .\n\t\\]\n\\end{Prop}\n\n\\begin{proof}\n\tWe only need to prove the presentation of $I(\\mathfrak a^*)$. Let $A$ be the unital commutative $\\cplxs$-algebra defined by the above generators and relations. It is clear that there is a surjective algebra homomorphism from $\\phi:A\\to I(\\mathfrak a^*)$, defined by $\\phi(\\lambda_n)=\\lambda^n$. \n\t\t\n\tConsider on $I(\\mathfrak a^*)$ the grading induced by $\\cplxs[\\lambda]$. For any multiindex $\\alpha=(\\alpha_2,\\alpha_{2q+1})$, define $\\lambda_\\alpha=(\\lambda_2)^{\\alpha_2}(\\lambda_{2q+1})^{\\alpha_{2q+1}}$ in the free commutative algebra $\\cplxs[\\lambda_2,\\lambda_{2q+1}]$. The latter is graded via $\\Abs0{\\lambda_\\alpha}=\\Abs0\\alpha=2\\alpha_2+(2q+1)\\alpha_{2q+1}$. The relation defining $A$ is homogeneous for this grading, so that $A$ inherits a grading. \n\n\tBy definition, $\\phi$ respects the grading, and in fact, it is surjective in each degree of the induced filtration (and hence, in each degree of the grading). The relation of $A$ ensures that the image of $\\lambda_\\alpha$ in $A$, for any $\\alpha$, depends only on $\\Abs0\\alpha$. Hence, $\\dim A_j\\sle1$ for all  $j$. This proves that $\\phi$ is injective. \n\\end{proof}\n\n\\begin{Cor}[singular]\n\tUnder the assumptions of \\thmref{Prop}{specialosp22q}, the algebra $I(\\mathfrak a^*)$ defines the singular curve in $\\cplxs^2$ given by the equation $z^{2q+1}=w^2$. \n\\end{Cor}\n\n\\begin{Par}\n\tWe substantiate the above by some explicit computations. We have \n\t\\[\n\t\t\\str\\begin{Matrix}1a&0&w^t&-z^t\\\\0&-a&-w^t&z^t\\\\z&-z&0&0\\\\w&-w&0&0\\end{Matrix}\\begin{Matrix}1a'&0&w'^t&-z'^t\\\\0&-a'&-w'^t&z'^t\\\\z'&-z'&0&0\\\\w'&-w'&0&0\\end{Matrix}=2aa'+4(w^tz'-z^tw')\n\t\\]\n\tfor the trace form $b$ on $\\mathfrak p=\\mathfrak a\\oplus\\mathfrak p_1^\\lambda$. In particular, \n\t\\[\n\t\tA_\\lambda=\\tfrac12\\left(\\begin{Matrix}[0]01&0&0&0\\\\0&-1&0&0\\\\0&0&0&0\\\\0&0&0&0\\end{Matrix}\\right)\\ ,\\ \\lambda(A_\\lambda)=\\tfrac12\\ .\n\t\\]\n\tSetting \n\t\\begin{gather*}\n\t\tz_i=\\tfrac12\\left(\\begin{Matrix}[0]00&0&0&-e_i^t\\\\0&0&0&e_i^t\\\\e_i&-e_i&0&0\\\\0&0&0&0\\end{Matrix}\\right)\\ ,\\ \\tilde z_i=\\tfrac12\\left(\\begin{Matrix}[0]00&0&e_i^t&0\\\\0&0&-e_i^t&0\\\\0&0&0&0\\\\e_i&-e_i&0&0\\end{Matrix}\\right)\\ ,\\\\\n\t\ty_i=\\tfrac12\\left(\\begin{Matrix}[0]00&0&0&-e_i^t\\\\0&0&0&-e_i^t\\\\-e_i&-e_i&0&0\\\\0&0&0&0\\end{Matrix}\\right)\\ ,\\ \\tilde y_i=\\tfrac12\\left(\\begin{Matrix}[0]00&0&e_i^t&0\\\\0&0&e_i^t&0\\\\0&0&0&0\\\\-e_i&-e_i&0&0\\end{Matrix}\\right)\\ ,\n\t\\end{gather*}\n\tone verifies the conditions from \\thmref{Par}{symplbasis}, namely\n\t\\begin{gather*}\n\t\ty_i,\\tilde y_i\\in\\mathfrak k_1\\,,\\,z_i,\\tilde z_i\\in\\mathfrak p_1\\,,\\,y_i+z_i,\\tilde y_i+\\tilde z_i\\in\\mathfrak g_1^\\lambda\\,,\\,b(y_i,\\tilde y_j)=b(\\tilde z_j,z_i)=\\delta_{ij}\\,,\\\\\n\t\tb(y_i,y_j)=b(\\tilde y_i,\\tilde y_j)=b(z_i,z_j)=b(\\tilde z_i,\\tilde z_j)=0\\ .\n\t\\end{gather*}\n\tThen one computes\n\t\\begin{gather*}\n\t\t[y_i,z_j]=[\\tilde y_i,\\tilde z_j]=0\\ ,\\ [y_i,\\tilde z_j]=-\\delta_{ij}A_\\lambda\\ ,\\ [\\tilde y_i,z_j]=\\delta_{ij}A_\\lambda\\ ,\\\\\n\t\t[A_\\lambda,y_i]=\\tfrac12z_i\\ ,\\ [A_\\lambda,z_i]=\\tfrac12y_i\\ ,\\ [A_\\lambda,\\tilde y_i]=\\tfrac12\\tilde z_i\\ ,\\ [A_\\lambda,\\tilde z_i]=\\tfrac12\\tilde y_i\\ .\n\t\\end{gather*}\n\t\n\tLet $\\zeta_i$, $\\tilde\\zeta_i$, $i=1,\\dotsc,q$, be the basis of $\\mathfrak p_1^*$, dual to $z_i$, $\\tilde z_i$, $i=1,\\dotsc,q$, so \n\t\\[\n\t\t\\Cdual0{\\tilde z_j}{\\zeta_i}=-\\Cdual0{z_i}{\\tilde\\zeta_j}=\\delta_{ij}\\ ,\\ \\Cdual0{z_j}{\\zeta_i}=\\Cdual0{\\tilde z_i}{\\tilde\\zeta_j}=0\\ .\n\t\\]\n\tThen $\\Cdual0z{\\zeta_i}=b(z,z_i)$, $\\Cdual0z{\\tilde\\zeta_i}=b(z,\\tilde z_i)$, and one has\n\t\\begin{gather*}\n\t\t\\ad^*(y_i)\\zeta_j=\\ad^*(\\tilde y_i)\\tilde\\zeta_j=0\\ ,\\ -\\!\\ad^*(y_i)\\tilde\\zeta_j=\\ad^*(\\tilde y_i)\\zeta_j=\\delta_{ij}\\lambda\\ ,\\\\\n\t\t\\ad^*(y_i)\\lambda=-\\tfrac12\\zeta_i\\ ,\\ \\ad^*(\\tilde y_i)\\lambda=-\\tfrac12\\tilde\\zeta_i\\ .\n\t\\end{gather*}\n\tAlso, we observe $\\Cdual0{z_I\\tilde z_Jh^\\nu}{\\zeta_K\\tilde\\zeta_L\\lambda^\\mu}=\\delta_{IL}\\delta_{JK}\\delta_{\\nu\\mu}(-1)^{\\Abs0I\\Abs0J}\\nu!\\lambda(h)^\\nu$. \n\t\n\tThe preimages $p_2,p_{2q+1}$ of the generators $\\lambda^2,\\lambda^{2q+1}$ in $S(\\mathfrak p^*)^{\\mathfrak k}$ under the restriction map can be deduced from \\thmref{Par}{radial-action}, because $\\mathfrak p=\\mathfrak a\\oplus\\mathfrak p_1^\\lambda$. Indeed, let $P=\\phi(p_N)$ where $N=2$ or $N=2q+1$. By the formulae from \\thmref{Par}{radial-action}, for $q\\sge\\Abs0I=k>0$ and $h\\in\\mathfrak a'$, \n\t\\begin{align*}\n\t\t\\textstyle\\sum_{\\nu=0}^\\infty\\tfrac1{\\nu!}\\Cdual0{z_I\\tilde z_Jh^\\nu}{p_N}&=P(z_I\\tilde z_J;h)=(\\partial_{\\gamma_h(z_I\\tilde z_J)}\\lambda^N)(h)\\\\\n\t\t&=\\delta_{IJ}(-1)^{\\tfrac12k(k+1)}2^{-k}a_{Nk}\\lambda(h)^{N-2k}\n\t\\end{align*}\n\twhere\n\t\\[\n\t\ta_{Nk}=\\textstyle\\sum_{i=(k-N)_+}^{k-1}\\Parens1{-\\tfrac12}^i(N)_{k-i}\\frac{(k-1+i)!}{(k-1-i)!i!}\\ .\n\t\\]\n\tThus, \n\t\\[\n\t\tp_N=\\lambda^N+\\textstyle\\sum_{k=1}^{\\min(N,q)}(-1)^{\\tfrac12k(k+3)}2^{-k}a_{Nk}\\lambda^{N-2k}\\sum_{\\Abs0I=k}\\zeta_I\\tilde\\zeta_I\\ .\n\t\\]\n\t\n\tWhen $N=2$ and $k\\sge2$, then $a_{Nk}=0$ by \\thmref{Par}{evencoeffcase1} and \\thmref{Par}{evencoeffcase2}. On the other hand, $a_{21}=2$. Hence,\n\t\\[\n\t\tp_2=\\lambda^2+\\textstyle\\sum_{i=1}^q\\zeta_i\\tilde\\zeta_i\n\t\\]\n\tis the super-Laplacian, and \n\t\\[\n\t\tp_{2q+1}=\\lambda^{2q+1}+\\textstyle\\sum_{k=1}^q(-1)^{\\tfrac12k(k+3)}2^{-k}a_{2q+1,k}\\lambda^{2(q-k)+1}\\sum_{\\Abs0I=k}\\zeta_I\\tilde\\zeta_I\\ .\n\t\\]\n\tThese elements are clearly subject to the relation $p_2^{2q+1}=p_{2q+1}^2$. \n\t\n\tOne readily checks\n\t\\[\n\t\t\\ad^*(y_i)p_2=-\\lambda\\zeta_i+\\zeta_i\\lambda=0\\mathtxt\\AND\\ad^*(\\tilde y_i)p_2=-\\lambda\\tilde\\zeta_i+\\lambda\\tilde\\zeta_i=0\\ .\n\t\\]\n\tIn case $q=1$, one has $p_3=\\lambda^3+\\frac32\\lambda\\zeta_1\\tilde\\zeta_1$, and\n\t\\begin{align*}\n\t\t\\ad^*(y_1)p_3&=-\\tfrac32\\lambda^2\\zeta_1-\\tfrac32\\lambda\\zeta_1\\ad^*(y_1)\\tilde\\zeta_1=-\\tfrac32\\lambda^2\\zeta_1+\\tfrac32\\lambda\\zeta_1\\lambda=0\\ ,\\\\\n\t\t\\ad^*(\\tilde y_1)p_3&=-\\tfrac32\\lambda^2\\tilde\\zeta_1+\\tfrac32\\lambda\\ad^*(\\tilde y_1)(\\zeta_1)\\tilde\\zeta_1=-\\tfrac32\\lambda^2\\tilde\\zeta_1+\\tfrac32\\lambda^2\\tilde\\zeta_1=0\\ .\n\t\\end{align*}\n\t\n\tTo verify the $\\mathfrak k_0$-invariance, let \n\t\\[\n\t\tx=\\left(\\begin{Matrix}[0]00&0&0&0\\\\0&0&0&0\\\\0&0&A&B\\\\0&0&C&-A^t\\end{Matrix}\\right)\\in\\mathfrak k_0=\\mathfrak{sp}(2q,\\cplxs)\\ .\n\t\\]\n\tThen \n\t\\[\n\t\t\\ad^*(x)\\zeta_i=\\textstyle\\sum_{j=1}^q\\Parens1{A_{ji}\\zeta_j+C_{ji}\\tilde\\zeta_j}\\mathtxt\\AND\\ad^*(x)\\tilde\\zeta_i=\\sum_{j=1}^q\\Parens1{B_{ji}\\zeta_j+A_{ji}\\tilde\\zeta_j}\\ .\n\t\\]\n\tThis implies\n\t\\[\n\t\t\\ad^*(x)(\\zeta_i\\tilde\\zeta_i)=\\textstyle\\sum_{j\\neq i}\\Parens1{C_{ji}\\tilde\\zeta_j\\tilde\\zeta_i-B_{ji}\\zeta_i\\zeta_j}\\ .\n\t\\]\n\tSince $B=B^t$, $C=C^t$, we deduce $\\sum_{i=1}^q\\ad^*(x)(\\zeta_i\\tilde\\zeta_i)=0$. Since $\\mathfrak a=\\mathfrak z(\\mathfrak g_0)$ and thus $\\ad^*(\\mathfrak k_0)\\lambda=0$, this implies that $p_2$ (for general $q$) and $p_3$ (for $q=1$) are $\\mathfrak k$-invariant. \n\\end{Par}\n\n\\bibliographystyle{alpha}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGravitational wave observations from the merger of compact objects like black holes and neutron stars\nprovide a new powerful tool to learn about the regime of strong gravity and thus about\nGeneral Relativity (GR) and alternative theories of gravity\n\\cite{TheLIGOScientific:2016src,Abbott:2018lct,LIGOScientific:2019fpa,Will:2014kxa,Berti:2015itd,Barack:2018yly}.\nAmong the plethora of gravity theories, in particular theories with higher curvature corrections, as motivated\nby quantum gravity considerations, have received much attention in recent years.\nAn attractive class of such theories is Einstein-scalar-Gauss-Bonnet (EsGB) theory,\nsince they are ghost-free and lead to second order equations of motion\n\\cite{Horndeski:1974wa,Charmousis:2011bf,Kobayashi:2011nu}.\n\nThe coupling function $f(\\varphi)$ of the scalar field to the Gauss-Bonnet (GB) term\nhas a decisive influence on the properties of the resulting EsGB black holes and neutron stars.\nEffective low energy string theories feature exponential coupling functions,\nwith the scalar field representing the dilaton.\nThese theories possess black holes with scalar hair,\nwhose properties have been investigated in great detail\n\\cite{Kanti:1995vq,Torii:1996yi,Guo:2008hf,Pani:2009wy,Pani:2011gy,Kleihaus:2011tg,Ayzenberg:2013wua,Ayzenberg:2014aka,Maselli:2015tta,Kleihaus:2014lba,Kleihaus:2015aje,Blazquez-Salcedo:2016enn,Cunha:2016wzk,Zhang:2017unx,Blazquez-Salcedo:2017txk,Konoplya:2019hml,Zinhailo:2019rwd}.\nThe GB term allows to circumvent  the no-hair theorems of GR\n(see e.g. \\cite{Chrusciel:2012jk,Herdeiro:2015waa}),\nbut these dilatonic theories do not allow for the black hole solutions of GR,\nthe Schwarzschild solutions and the Kerr solutions.\n\nAn interesting rather new development has led to the insight,\nthat for a whole class of coupling functions $f(\\varphi)$ the black hole solutions\nof GR remain solutions of the respective EsGB theory,\nbut become unstable at certain\nvalues of the GB coupling, where branches of scalarized black holes emerge\n\\cite{Doneva:2017bvd,Antoniou:2017acq,Silva:2017uqg}.\nThis phenomenon is referred to as curvature-induced spontaneous scalarization,\nand is akin to the well-known matter-induced spontaneous scalarization\nin scalar-tensor theories in neutron stars \\cite{Damour:1993hw}.\nBy now, a variety of coupling functions has been employed to\nstudy such spontaneously scalarized black holes and their properties\n\\cite{Doneva:2017bvd,Antoniou:2017acq,Silva:2017uqg,Antoniou:2017hxj,Blazquez-Salcedo:2018jnn,Doneva:2018rou,Minamitsuji:2018xde,Silva:2018qhn,Brihaye:2018grv,Doneva:2019vuh,Myung:2019wvb,Cunha:2019dwb,Macedo:2019sem,Hod:2019pmb,Collodel:2019kkx,Bakopoulos:2020dfg}.\n\nA central question concerning these scalarized black holes is of course their stability.\nIf physically relevant, these black holes should be stable, at least on astrophysical timescales.\n{A coupling function leading to potentially stable spontaneously scalarized black holes\nfor a wide range of parameters has been proposed in \\cite{Doneva:2017bvd}.}\nFor a fixed value of the GB coupling constant,\nthere is a critical value $r_{\\rm B}$ of the horizon size of the Schwarzschild black hole,\nwhere the fundamental branch of spontaneously scalarized black holes\nemerges and continues to exist for all $r_{\\rm H}<r_{\\rm B}$.\nIn fact, there is a whole sequence of bifurcation points at smaller values of $r_{\\rm H}$,\nwhere radially excited scalarized black holes emerge.\nThe black holes on the fundamental branch are thermodynamically\n{preferred} for the coupling function \\cite{Doneva:2017bvd}.\nHowever, their dynamical stability has only partly been investigated up to now.\n\nTo determine the dynamical stability, a study of the\nlinear mode stability of the solutions is a first important step.   \nIn perturbation theory the modes of the spherically symmetric black holes\ncan be studied separately according to their parity.\nThe polar modes have positive parity, while the axial modes have negative parity.\nRadial perturbations represent a special subset of the polar modes,\nwhich often reveal already instabilities of the solutions.\nWe have therefore first considered radial perturbations\nfor the spontaneously scalarized black holes \\cite{Blazquez-Salcedo:2018jnn}.\nThese have indeed revealed radial instabilities of the Schwarzschild black holes\nand of the radially excited scalarized black holes.\nHowever, the spontaneously scalarized black holes on the fundamental branch\ndo not have unstable radial modes,\nwhen their horizon size $r_{\\rm H}$ is in the interval $r_{\\rm S1} < r_{\\rm H} < r_{\\rm B}$.\nAt the critical value $r_{\\rm S1}$ the perturbation equations lose hyperbolicity,\nand the employed formalism is no longer well-defined.\n\nTo show linear mode stability, we have\nnext considered axial perturbations \\cite{Blazquez-Salcedo:2020rhf}.\nHere the scalar field decouples and only gravitational quadrupole\n(and higher multipole) modes are present.\nHyperbolicity of the axial perturbation equations is lost as well,\nbut at a slightly larger value of the horizon size $r_{\\rm S2} > r_{\\rm S1}$,\nfor  fixed coupling.\nWithin the interval $r_{\\rm S2} < r_{\\rm H} < r_{\\rm B}$,\nhowever, the black holes are mode stable with respect to axial perturbations.\n\nIn this paper we complete the analysis of linear mode stability\nby considering also the non-radial polar modes.\nThe quadrupole modes now come in two kinds, the scalar-led modes\nand the gravitational-led modes. The names imply, that in the\nlimit of vanishing backgound scalar field, the modes correspond\nto the Schwarzschild modes, which are purely scalar\nand purely gravitational. For a finite background scalar field, of course\nmixing between these channels arises.\nThe presence of the scalar background field then destroys the\nisospectrality present for Schwarzschild black holes.\nIn addition to the quadrupole modes, the scalarized black holes\npossess also dipole modes and scalar modes, which involve both\nthe scalar field and the gravitational field.\nWe will see, that all of these modes are stable\nin the relevant range $r_{\\rm S2} < r_{\\rm H} < r_{\\rm B}$.\n\nThis paper is organized as follows.\nIn section II we recall the action and the background solutions.\nIn section III we discuss the equations for the polar perturbations,\nincluding the asymptotic behavior and the numerical method.\nWe present our numerical results for the polar modes of the\nspontaneously scalarized black holes \nin section IV, and we conclude in section V.\n\n\n\n\\section{Action and background}\n\nThe action in Einstein-scalar-Gauss-Bonnet theory is given by\n\\begin{eqnarray}\nS=&&\\frac{1}{16\\pi}\\int d^4x \\sqrt{-g}\n\\Big[R - 2\\nabla_\\mu \\varphi \\nabla^\\mu \\varphi\n+ \\lambda^2 f(\\varphi){\\cal R}^2_{GB} \\Big] \\ ,\\label{eq:quadratic}\n\\end{eqnarray}\nwhere the spacetime metric is $g_{\\mu\\nu}$ with Ricci scalar $R$,\n$\\varphi$ is the scalar field with coupling function $f(\\varphi)$, and $\\lambda$  is the GB coupling constant\nwith dimension of $length$.\nThe GB invariant ${\\cal R}^2_{GB}$ is defined as\n${\\cal R}^2_{GB}=R^2 - 4 R_{\\mu\\nu} R^{\\mu\\nu}\n+ R_{\\mu\\nu\\alpha\\beta}R^{\\mu\\nu\\alpha\\beta}$\nwith Ricci tensor $R_{\\mu\\nu}$\nand Riemann tensor $R_{\\mu\\nu\\alpha\\beta}$.\n\nThe field equations that result from this action are\n\\begin{eqnarray}\\label{FE1}\n&&R_{\\mu\\nu}- \\frac{1}{2}R g_{\\mu\\nu} + \\Gamma_{\\mu\\nu}= 2\\nabla_\\mu\\varphi\\nabla_\\nu\\varphi -  g_{\\mu\\nu} \\nabla_\\alpha\\varphi \\nabla^\\alpha\\varphi \\ ,\\\\\n&&\\nabla_\\alpha\\nabla^\\alpha\\varphi=  -  \\frac{\\lambda^2}{4} \\frac{df(\\varphi)}{d\\varphi} {\\cal R}^2_{GB} \\ , \\label{FE2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Gamma_{\\mu\\nu}&=& - R(\\nabla_\\mu\\Psi_{\\nu} + \\nabla_\\nu\\Psi_{\\mu} ) - 4\\nabla^\\alpha\\Psi_{\\alpha}\\left(R_{\\mu\\nu} - \\frac{1}{2}R g_{\\mu\\nu}\\right) +\n4R_{\\mu\\alpha}\\nabla^\\alpha\\Psi_{\\nu} + 4R_{\\nu\\alpha}\\nabla^\\alpha\\Psi_{\\mu} \\nonumber \\\\\n&& - 4 g_{\\mu\\nu} R^{\\alpha\\beta}\\nabla_\\alpha\\Psi_{\\beta}\n+ \\,  4 R^{\\beta}_{\\;\\mu\\alpha\\nu}\\nabla^\\alpha\\Psi_{\\beta} \\ , \\\\\n\\Psi_{\\mu}&=& \\lambda^2 \\frac{df(\\varphi)}{d\\varphi}\\nabla_\\mu\\varphi \\ .\n\\end{eqnarray}\n\nIn the present paper we will use the coupling function\nintroduced in \\cite{Doneva:2017bvd}\n\\begin{equation} \\label{eq:coupling_function}\nf(\\varphi)=  \\frac{1}{12} \\left(1- e^{-6\\varphi^2}\\right) \\ .\n\\end{equation}\nNote that the coupling function $f(\\varphi)$ satisfies\nthe conditions\n$\\frac{df}{d\\varphi}(0)=0$ and $b^2=\\frac{d^2f}{d\\varphi^2}(0)>0$,\nand hence we have curvature induced spontaneous scalarization\nof black holes in the theory,\nwhen we fix the cosmological value of the scalar field to zero,\n$ \\varphi_{\\infty}=0$\n\\cite{Doneva:2017bvd,Antoniou:2017acq,Silva:2017uqg}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_vs_Ah_for_polar}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_vs_Qd_for_polar}\\\\[2ex]\n\t\\caption{(\\textit{left}) Scaled horizon area $A_{\\rm H}\/\\lambda^2$\nvs. scaled total mass $M\/\\lambda$, and\n(\\textit{right}) Scaled scalar charge $Q_{\\rm D}\/\\lambda$ vs. scaled total mass\nfor {the fundamental branch of EsGB black holes }(blue and red) and the Schwarzschild black holes (grey).\nThe bifurcation point $r_{\\rm B}$ is marked in blue,\nthe points $r_{\\rm S1}$ and $r_{\\rm S2}$, where hyperbolicity is lost\nfor the radial and axial perturbation equations, are marked in green and red, respectively.}\n\t\\label{fig:static}\n\\end{figure}\n\n\nSpherically symmetric solutions\nof the field equations can be obtained with the Ansatz for the metric\n\\begin{eqnarray}\\label{eq:metric_BG}\nds^2= - f(r)dt^2 + \\frac{1}{1-\\frac{2m(r)}{r}} dr^2\n+ r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2 ) \\ ,\n\\end{eqnarray}\nwhere the metric functions $f(r)$ and $m(r)$ depend only on the radial coordinate,\nand the scalar field $\\varphi(r)$ is likewise only a function of $r$.\nBy solving the set of field equations for these functions\nsubject to the conditions of asymptotic flatness and regularity at and outside the horizon,\nthe domain of existence of spontaneously scalarized black holes solutions has been\nmapped out in \\cite{Doneva:2017bvd}.\nIn fact, one can work with $\\lambda=1$ without loss of generality,\nsince all dimensionful quantities can be rescaled with respect to $\\lambda$.\nThe black holes can then be parametrized by the value\nof the horizon radius $r_{\\rm H}$.\n\nHere we focus only on the fundamental branch of the scalarized black holes,\nsince this should be the physically most relevant branch of solutions.\nThis branch is shown together with the Schwarzschild branch\nin Fig.~\\ref{fig:static},\nwhere the scaled horizon area $A_{\\rm H}\/\\lambda^2$ (left)\nand the scaled scalar charge $Q_{\\rm D}\/\\lambda$ (right) are exhibited\nas functions of the scaled mass $M\/\\lambda$.\nNote, that the scalar charge $Q_{\\rm D}$ is defined as the coefficient of the dominant\nterm in the asymptotic expansion of the scalar field.\nThe fundamental branch bifurcates from the Schwarzschild branch\nat the horizon radius $r_{\\rm B}=1.173944$ and continues to exist for all\nradii $r_{\\rm H}<r_{\\rm B}$.\nThe fundamental branch is marked in blue and red in the figures,\nwhile the Schwarzschild branch is marked in grey.\n\nAnalysis of the radial perturbation equations for the solutions on the fundamental\nbranch shows, that hyperbolicity of the equations is lost at\na critical horizon radius $r_{\\rm S1}=0.191605$.\nHowever, in the interval $r_{\\rm S1} < r_{\\rm H}< r_{\\rm B}$\nall solutions are stable with respect to radial perturbations \\cite{Blazquez-Salcedo:2018jnn} .\nSimilarly, analysis of the axial perturbation equations\nreveals loss of hyperbolicity at $r_{\\rm S2}=0.19994$,\nbut stability with respect to axial perturbations\nin the interval $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$ \\cite{Blazquez-Salcedo:2020rhf}.\nThe critical points $r_{\\rm S1}$ and $r_{\\rm S2}$ ($r_{\\rm S1} < r_{\\rm S2}$)\nare marked in Fig.~\\ref{fig:static} in green and red, respectively.\n\nWe conclude, that there is a potentially physically interesting stable range\n$r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$ of scalarized black holes.\nTo show mode stability of these black holes we still have to\ninvestigate their non-radial polar modes.\nThis will be done in the following {sections}.\n\n\n\n\\section{Polar Perturbations}\n\n\\subsection{Ansatz}\n\nIntroducing a perturbation control parameter $\\epsilon$, and the spherical harmonics $Y_{lm}(\\theta,\\phi)$ with angular numbers $l$ and $m$, we  adopt for the polar perturbation {of the metric the following Ansatz in the Regge-Wheeler gauge}\n\\cite{Regge:1957td}\n\\begin{eqnarray}\n\\label{eq:pert_metric}\nds^2=-f(r)\\left[1+\\epsilon e^{-i\\omega t}H_0(r)Y_{lm}(\\theta,\\phi)\\right]dt^2+\\frac{1}{1-\\frac{2m(r)}{r}}\\left[1+\\epsilon e^{-i\\omega t}{H_2}(r) Y_{lm}(\\theta,\\phi)\\right]dr^2 \\nonumber \\\\ +r^2\\left[1+\\epsilon e^{-i\\omega t}{T}(r) Y_{lm}(\\theta,\\phi)\\right](d\\theta^2 + \\sin^2\\theta d\\phi^2 ) - 2\\epsilon  e^{-i\\omega t}H_1(r)Y_{lm}(\\theta,\\phi)drdt \\ ,\n\\end{eqnarray}\nand for the scalar field\n\\begin{eqnarray}\n\\label{eq:pert_scalar}\n\\varphi= \\varphi_0(r) \\left[1+\\epsilon e^{-i\\omega t}\\varphi_1(r)Y_{lm}(\\theta,\\phi)\\right] \\ .\n\\end{eqnarray}\n\nThe resulting set of equations is given in the Appendix. {Due to the spherical symmetry of the background solution the radial perturbation equations do not depend on the value of $m$.}\n{They can be reduced to} a second order differential equation for the scalar field perturbation function $\\varphi_1$ coupled to two first order differential equations for the metric perturbation functions $H_1$ and $T$. {After solving this system the remaining functions $H_0$ and $H_2$ can be obtained by means of two relations, which represent them by means of the rest of the perturbation fields.}\n\nThe coupled system can be expressed in the schematic first order form\n\\begin{equation}\n\\frac{d}{dr}{{\\Psi}}+{V}{\\Psi}= 0\n\\label{eq:perturbations}\n\\end{equation}\nwhere ${\\Psi}$ is a column vector with components $(H_1,T,\\varphi_1,\\varphi_1')$.\nThe matrix ${V}$ contains the coupling between the background functions and the perturbation functions.\nThe set of coupled first order equations can also be cast\ninto a set of coupled second-order Schr\\\"odinger-like equations.\n\nLet us address here what happens in the Schwarzschild limit.\nAt the bifurcation point $r_{\\rm H}=r_{\\rm B}$, the scalar field of the solution vanishes, and the exterior of the black hole is described by the Schwarzschild solution, with $f(r)=1-\\frac{2M}{r}$, $m(r)=M$ and $\\phi_0(r)=0$. In practice, the polar perturbations decouple into two sets of second order equations: one set for $(H_1,T)$, that can be written like the Zerilli equation using the standard definitions we shall not repeat here. The second set for $(\\varphi_1,\\varphi_1')$ results in the following equation for the scalar field perturbations\n\\begin{eqnarray}\n\\frac{d^2 \\varphi_1}{dr^2}=-\\frac{2(r-M)}{r\\left(r-2M\\right)}\\frac{d \\varphi_1}{dr}+\\left[\n\\frac{l(l+1)}{r(r-2M)}\n-\\frac{12\\lambda^2M^2}{r^5(r-2M)}\n-\\frac{r^2\\omega^2}{(r-2M)^2}\n\\right] \\varphi_1 \\, .\n\\end{eqnarray}\nNote that this equation is not exactly the one for a minimally coupled scalar field, since it has a source term proportional to the Gauss-Bonnet invariant and the coupling constant $\\lambda^2$. In practice, this means that the scalar modes at the bifurcation point will be shifted with respect to the standard scalar modes of Schwarzschild that one obtains in the $\\lambda=0$ coupling limit.\n\n\\subsection{Asymptotic behaviour and numerical method}\n\nFor the study of the asymptotic behaviour of the perturbations, we introduce the tortoise coordinate\n\\begin{eqnarray}\n\\frac{dR^*}{dr} = \\frac{1}{\\sqrt{f(r)\\left(1-\\frac{2m(r)}{r}\\right)}} ,\n\\end{eqnarray}\nand express the spatial derivatives in terms of this coordinate.\nThe quasinormal modes are then obtained by solving the set of equations\n\\eqref{eq:perturbations} subject to the proper boundary conditions,\nwhich correspond to purely outgoing waves at infinity\nand purely ingoing waves at the black hole horizon\n\\begin{equation} \\Psi \\propto\n\\left\\{ \\begin{array}{ll} e^{-i\\omega (t+R_*)}\\,,\n&r\\to r_{\\rm H} \\,,\\\\\ne^{-i\\omega (t-R_*)}\\,,\n&r\\to\\infty\\,. \\end{array} \\right.\n\\end{equation}\n\nThe far-field behavior of the perturbations is determined by the expansions\n\\begin{eqnarray}\nT &=& e^{i\\omega R^*} \\left[ A_g^+ \\left(1 + \\frac{1}{r^2}\\left(\\frac{Q_D^2}{2}-\\frac{3iM}{2\\omega}+\\frac{l(l-1)(l+2)}{8\\omega^2}\\right)+  O(r^{-3}) \\right) + \\right. \\nonumber \\\\\n&&\\left. + A_s^+\\left(\\frac{-2i Q_D}{\\omega r^3} +  \\frac{Q_D(l^2+l-3)}{\\omega^2 r^4} +  O(r^{-5}) \\right) \\right] , \\\\\nH_1 &=& r \\omega e^{i\\omega R^*} \\left[ A_g^+ \\left(1 + \\frac{1}{r}\\left(2M+\\frac{i(l-1)(l+2)}{2\\omega}\\right)+  O(r^{-2}) \\right) + \\right. \\nonumber \\\\\n&&\\left. + A_s^+\\left(\\frac{-Q_D}{\\omega^2 r^4} +  \\frac{Q_D}{r^5}\\left(\\frac{24M\\lambda^2}{5}+\\frac{2M}{5\\omega^2}-\\frac{i(l-2)(l+3)}{2\\omega^3}\\right) +  O(r^{-6}) \\right) \\right] ,\n\\end{eqnarray}\n\\begin{eqnarray}\n \\varphi_1 &=& \\frac{1}{r} e^{i\\omega R^*} \\left[ A_g^+ \\left(\\frac{-MQ_D}{r}-\\frac{Q_D}{r^2}\\left(2M^2+\\frac{iM(l+2)(l-1)}{2\\omega}\\right)+  O(r^{-3}) \\right) + \\right. \\nonumber \\\\\n &&\\left. + A_s^+\\left(1+\\frac{il(l+1)}{2\\omega r} + O(r^{-2}) \\right) \\right] ,\n\\end{eqnarray}\nwhere $A_s^+$ and $A_g^+$ are the scalar  and\nspace-time perturbation amplitudes, respectively, characterizing the expansions.\nThe far-field expansion is written in terms of the global parameters of the background solution, the total mass $M$ and the scalar charge $Q_D$.\nThe near horizon behavior on the other hand is given by the expansions\n\\begin{eqnarray}\nT &=&  e^{-i\\omega R^*} \\left[ A_g^- + O(r-r_{\\rm H}) \\right] \\,  \\\\\n\\varphi_1 &=& \\frac{1}{r} e^{-i\\omega R^*} \\left[ A_s^- + O(r-r_{\\rm H}) \\right]\\, \\\\\nH_1 &=& \\frac{1}{r-r_{\\rm H}} e^{-i\\omega R^*} \\left[ A_s^-\\left(\\frac{-4\\lambda^2 \\varphi_{\\rm H}}{r_{\\rm H}^3e^{6 \\varphi_{\\rm H}^2}} +  O(r-r_{\\rm H}) \\right) +\\right.  \\\\\n &&\\hskip -1cm \\left. + A_g^- \\left(\\frac{4\\omega^2r_{\\rm H}+2i\\omega\\sqrt{f'_{\\rm H} r_{\\rm H}(1-2m'_{\\rm H})(l^2+l+1)}-f'_{\\rm H}(1-2m'_{\\rm H})l(l+1)}{r_{\\rm H}^2(f'_{\\rm H}(1-2m'_{\\rm H})+4r_{\\rm H}\\omega^2)} +  O(r-r_{\\rm H}) \\right) \\right] \\, \\nonumber\n\\end{eqnarray}\nwhere $A_s^-$ and $A_g^-$ are again the scalar and space-time perturbation amplitudes.\nIn the near horizon expansion we have defined the horizon parameters $ \\varphi_{\\rm H}= \\varphi_0(r_{\\rm H})$, $ \\varphi'_{\\rm H}=\\frac{d \\varphi_0}{dr}|_{r_{\\rm H}}=$, $m'_{\\rm H}=\\frac{dm}{dr}|_{r_{\\rm H}}$ and $f'_{\\rm H}=\\frac{df}{dr}|_{r_{\\rm H}}$, which can be obtained from the background solution. Not all of these background near-horizon parameters are free, since in particular the horizon expansion at zero order imposes the relations $(2\\lambda^2 \\varphi_{\\rm H} \\varphi'_{\\rm H}+r_{\\rm H}e^{6 \\varphi_{\\rm H}^2})m'_{\\rm H}=\\lambda^2 \\varphi_{\\rm H} \\varphi'_{\\rm H}$ and $\\lambda^2r_{\\rm H} \\varphi_{\\rm H} \\varphi'_{\\rm H}=\\sqrt{r_{\\rm H}^4e^{12 \\varphi_{\\rm H}^2}-24\\lambda^2 \\varphi_{\\rm H}}-r_{\\rm H}e^{6 \\varphi_{\\rm H}^2}$.\n\nThe eigenvalue $\\omega=\\omega_R+i\\omega_I$ consists of a real part $\\omega_R$,\ncorresponding to the frequency of the mode, and of an imaginary part $\\omega_R$,\ncorresponding to the damping rate, as long as it is positive.\nA negative imaginary part, however, implies an instability.\nThus a criterion for linearized mode stability of the black holes is\nthat all quasinormal mode frequencies have a positive imaginary part.\n\nTo calculate the quasinormal mode frequencies, we employ a shooting method,\nthat is generating solutions that satisfy the near-horizon expansion, and solutions that satisfy the far-field expansion, with different values of the scalar and space-time amplitudes $A_s^\\pm$ and $A_g^\\pm$.\nSubsequently we match them at an intermediate point.\nContinuity of the functions and their derivatives then determine the values of $\\omega$,\nthat define the quasinormal modes. We always calibrate and cross-check the numerics with the different limits that we can continuously study (in this case the GR modes when $\\lambda=0$, or the Schwarzschild modes when $ \\varphi_0(r)=0$).\nFurther details on the numerical procedure can, for instance, be found in \\cite{Blazquez-Salcedo:2018pxo}.\n\n\\section{Results}\n\nIn the following, we will present our results\nfor the polar quasinormal modes of the spontaneously scalarized EsGB black holes\non the fundamental branch in the physically interesting range\n$r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nconcluding the demonstration of their mode stability in this range.\nWe note, that we have extended the calculations of the presented modes\ninto the range $ r_{\\rm H} < r_{\\rm S2}$,\nwithout finding any pecularity there\nindicating a further instability or another breakdown of hyperbolicity.\n\nWe will discuss the modes in the order of their angular number $l$, comprising\nthe monopole case $l=0$, the dipole case $l=1$, and the quadrupole case $l=2$.\nIn the latter case we have to distinguish between two families of modes \\cite{Blazquez-Salcedo:2016enn}:\n\\begin{itemize}\n\\item[i.]\nModes connected with purely gravitational perturbations\nof the Schwarzschild solution are referred to as gravitational-led (grav-led) modes.\nThese have dominant amplitude $A_g^{\\pm}$.\n\\item[ii.]\nModes connected with purely scalar perturbations\nof the Schwarzschild solution are referred to as scalar-led modes.\nThese have dominant amplitude $A_s^{\\pm}$.\n\\end{itemize}\nWe will also demonstrate the breaking of isospectrality.\n\n\\subsection{$l=0$}\n\n\\begin{figure}\n\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaR_w_lambda_l0}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaI_w_lambda_l0}\n\t\\caption{\nScaled polar $l=0$ eigenvalue $\\lambda \\omega$\nvs. scaled total mass $M\/\\lambda$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nwhere hyperbolicity is lost at $r_{\\rm S2}$, and\n$r_{\\rm B}$ is the bifurcation point from the Schwarzschild solution.\nFor comparison also the Schwarzschild mode is shown (solid grey).}\n\t\\label{fig:plotl0}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaR_l0}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaI_l0}\n\t\\caption{\nScaled polar $l=0$ eigenvalue $M \\omega$\nvs. scalar charge $Q_D\/M$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$.\nHyperbolicity is lost at $r_H=r_{\\rm S2}$, the maximum value of $Q_D\/M$ shown.\n$Q_D\/M=0$ corresponds to the bifurcation point $r_{\\rm B}$\nfrom the Schwarzschild solution.}\n\t\\label{fig:plotl0_Qd}\n\\end{figure}\n\nLet us now consider the results for the monopole ($l=0$) modes.\nThey are obtained by starting at the bifurcation point $r_{\\rm H} = r_{\\rm B}$,\nwhere a corresponding mode of the Schwarzschild solution\nrefers to an independent scalar field in  the Schwarzschild background.\nThis mode is given by $M\\omega=0.114-i0.090$.\nFrom $ r_{\\rm B}$ the mode is then tracked along the fundamental branch.\nThe $l=0$ mode is exhibited in Fig.~\\ref{fig:plotl0},\nwhere the scaled eigenvalue $\\lambda \\omega$ is shown\nversus the scaled total mass $M\/\\lambda$.\nThe left plot exhibits the real part $\\omega_R$,\nrepresenting the frequency,\nand the right plot exhibits the imaginary part $\\omega_I$,\nrepresenting the inverse damping time.\nFor comparison also the Schwarzschild mode is shown (solid grey).\nWe observe that the scaled frequency of the scalarized black holes\nof a given mass follows closely the corresponding frequency\nof the Schwarzschild black holes.\nThe damping rate of the scalarized black holes\nis always smaller than its Schwarzschild counterpart,\nthus the damping time is larger for the scalarized solutions.\n\nIn Fig.~\\ref{fig:plotl0_Qd} we illustrate the dependence of the scaled eigenvalue\n$M \\omega$ on the scaled scalar charge $Q_D\/M$ of the black holes.\nThe corresponding Schwarzschild values are read off at $Q_D\/M=0$.\nThe scaled frequency varies by less than 20\\%, whereas\nthe damping rate decreases strongly for larger values of the scaled scalar charge.\nWe emphasize that we do not find unstable modes.\n\n\\subsection{$l=1$}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaR_w_lambda_l1}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaI_w_lambda_l1}\n\t\\caption{\nScaled polar $l=1$ eigenvalue $\\lambda \\omega$\nvs. scaled total mass $M\/\\lambda$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nwhere hyperbolicity is lost at $r_{\\rm S2}$, and\n$r_{\\rm B}$ is the bifurcation point from the Schwarzschild solution.\nFor comparison also the Schwarzschild mode is shown (solid grey).}\n\t\\label{fig:plotl1}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaR_l1}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaI_l1}\n\t\\caption{\nScaled polar $l=1$ eigenvalue $M \\omega$\nvs. scalar charge $Q_D\/M$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$.\nHyperbolicity is lost at $r_H=r_{\\rm S2}$, the maximum value of $Q_D\/M$ shown.\n$Q_D\/M=0$ corresponds to the bifurcation point $r_{\\rm B}$\nfrom the Schwarzschild solution.}\n\t\\label{fig:plotl1_Qd}\n\\end{figure}\n\nTo obtain the dipole ($l=1$) modes,  we proceed analogously to the $l=0$ case.\nWe start again at the bifurcation point $r_{\\rm H} = r_{\\rm B}$,\nwith the corresponding mode of the Schwarzschild solution\n$M\\omega=0.293-i0.097$,\nand then track the mode along the fundamental branch.\nThe $l=1$ mode is exhibited in Fig.~\\ref{fig:plotl1},\nwhere again $\\lambda \\omega$ is shown versus $M\/\\lambda$,\nwith frequency $\\omega_R$ on the left side and damping rate $\\omega_I$ on the right.\nFor comparison again the Schwarzschild mode is shown (solid grey).\nWe observe that the  frequency and the damping time of the scalarized black holes\nfollow more or less the corresponding frequency and damping  time\nof the Schwarzschild black holes, being mostly slightly larger than\ntheir Schwarzschild counterparts.\nWhen scaled with the mass and considered as a function of the scalar charge,\nthe deviations from the Schwarzschild values become more pronounced\nas seen in Fig.~\\ref{fig:plotl1_Qd}.\nAgain, we emphasize that we do not find unstable modes.\n\n\n\\subsection{$l=2$}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaR_w_lambda_l2_grav}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaI_w_lambda_l2_grav}\n\t\\caption{\nScaled polar grav-led $l=2$ eigenvalue $\\lambda \\omega$\nvs. scaled total mass $M\/\\lambda$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nwhere hyperbolicity is lost at $r_{\\rm S2}$, and\n$r_{\\rm B}$ is the bifurcation point from the Schwarzschild solution.\nFor comparison also the Schwarzschild mode is shown (solid grey).}\n\t\\label{fig:plotl2_grav}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaR_w_lambda_l2_scalar}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaI_w_lambda_l2_scalar}\n\t\\caption{\nScaled polar scalar-led $l=2$ eigenvalue $\\lambda \\omega$\nvs. scaled total mass $M\/\\lambda$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nwhere hyperbolicity is lost at $r_{\\rm S2}$, and\n$r_{\\rm B}$ is the bifurcation point from the Schwarzschild solution.\nFor comparison also the Schwarzschild mode is shown (solid grey).}\n\t\\label{fig:plotl2_scalar}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaR_l2_grav}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaI_l2_grav}\n\t\\caption{\nScaled polar grav-led $l=2$ eigenvalue $M \\omega$\nvs. scalar charge $Q_D\/M$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$.\nHyperbolicity is lost at $r_H=r_{\\rm S2}$, the maximum value of $Q_D\/M$ shown.\n$Q_D\/M=0$ corresponds to the bifurcation point $r_{\\rm B}$\nfrom the Schwarzschild solution.}\n\t\\label{fig:plotl2_grav_Qd}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaR_l2_scalar}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_Qd_omegaI_l2_scalar}\n\t\\caption{\nScaled polar scalar-led $l=2$ eigenvalue $M \\omega$\nvs. scalar charge $Q_D\/M$: real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$.\nHyperbolicity is lost at $r_H=r_{\\rm S2}$, the maximum value of $Q_D\/M$ shown.\n$Q_D\/M=0$ corresponds to the bifurcation point $r_{\\rm B}$\nfrom the Schwarzschild solution.}\n\t\\label{fig:plotl2_scalar_Qd}\n\\end{figure}\n\nIn the case of the quadrupole ($l=2$) modes we now have to distinguish between the\ngrav-led modes and the scalar-led modes, since quadrupole modes\nare also present in the absence of a scalar field.\nFor both families of modes we start again\nat the  bifurcation point $r_{\\rm H} = r_{\\rm B}$,\nwhere the modes of the Schwarzschild solution are\n$M\\omega=0.3737-i0.08895$ and $M\\omega=0.481-i0.0894$\nfor the gravitational mode and the scalar mode, respectively.\nThen we track these modes again along the fundamental branch.\n\nThe grav-led $l=2$ mode and the scalar-led $l=2$ mode\nare exhibited in Fig.~\\ref{fig:plotl2_grav} and Fig.~\\ref{fig:plotl2_scalar}, respectively,\nwhere again $\\lambda \\omega$ is shown versus $M\/\\lambda$,\nwith frequency $\\omega_R$ on the left and damping rate $\\omega_I$ on the right.\nAgain, the Schwarzschild modes are also shown (solid grey).\nThe frequency of the grav-led modes is larger than the frequency\nof the Schwarzschild modes, with the difference increasing\nalong the branch. The damping rate, in contrast, is smaller\nthan the Schwarzschild damping rate,\nbecoming even very small in some intermediate range, while increasing again towards the\nSchwarzschild value as the critical point $r_{\\rm S2}$ is approached.\n\nThe scalar-led mode exhibits less strong deviations from Schwarzschild.\nThe frequency is first slightly smaller than the Schwarzschild value,\nbut then starts to increase more rapidly towards smaller $r_{\\rm H}$.\nThe damping rate, on the other hand, is always\nlarger than the Schwarzschild value.\nFig.~\\ref{fig:plotl2_grav_Qd} and Fig.~\\ref{fig:plotl2_scalar_Qd}\nshow the frequency and damping time of the grav-led and\nscalar-led mode again versus the scalar charge.\nAs before, these diagrams enhance the deviations from Schwarzschild.\nAlso for the $l=2$ modes, we emphasize that we do not find unstable modes.\n\n\\subsection{Isospectrality breaking}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaR_w_lambda_l2_grav_iso}\n\t\\includegraphics[width=0.34\\linewidth,angle=-90]{plot_M_omegaI_w_lambda_l2_grav_iso}\n\t\\caption{\nScaled $l=2$ eigenvalue $\\lambda \\omega$\nvs. scaled total mass $M\/\\lambda$\nfor the polar grav-led mode (blue),\nand the axial mode (red): real part\/frequency $\\omega_R$\n(\\textit{left}) and imaginary part\/inverse damping time $\\omega_I$ (\\textit{right}),\ndepicted in the range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$,\nwhere hyperbolicity is lost at $r_{\\rm S2}$, and\n$r_{\\rm B}$ is the bifurcation point from the Schwarzschild solution.\nFor comparison also the Schwarzschild mode is shown (solid grey).}\n\t\\label{fig:plotl2_iso}\n\\end{figure}\n\nThe modes of the Schwarzschild black holes are known to exhibit\nisospectrality,\nmeaning that the polar and the axial mode spectra coincide.\nPreviously it has been observed, that isospectrality\nis broken in the presence of a dilatonic coupling \\cite{Blazquez-Salcedo:2016enn}.\nThis is not surprising,\nsince the scalar field enters the polar equations, while it does not\nenter the axial equations.\nTherefore breaking of isospectrality is also expected for EsGB\nblack holes. \n\nWe demonstrate the breaking of isospectrality in Fig.~\\ref{fig:plotl2_iso},\nwhere we show the modes $\\lambda \\omega$ versus $M\/\\lambda$\n(with frequency $\\omega_R$ on the left and damping rate $\\omega_I$ on the right),\ncomparing the polar grav-led mode (blue),\nwith the axial mode (red) and also with\nthe isospectral Schwarzschild modes (solid grey).\nWe note, that both the polar grav-led and axial frequencies follow the behavior\nof the Schwarzschild frequency, but exceed it the more, {the smaller the black hole is},\nwith the polar mode {frequency} increasing faster with decreasing black hole size.\nThe  decay rate, on the other hand,  follows for the axial case the Schwarzschild decay rate\nto some extent, whereas the polar case features a completely different decay rate,\ndeviating significantly from the Schwarzschild decay rate except at the boundaries\nof the relevant range $r_{\\rm S2} < r_{\\rm H}< r_{\\rm B}$.\n\n\\section{Conclusions}\n\nWe have studied the polar quasinormal modes on the fundamental branch\nof spontaneously scalarized black hole solutions of the EsGB  theory\nproposed and studied in \\cite{Doneva:2017bvd}.\nThese scalarized black holes are thermodynamically {preferred over the GR one}s.\nHowever, they should also be dynamically stable\nor at least sufficiently long-lived\nto be of astrophysical interest. Here the analysis\nof their quasinormal modes is an important tool,\nsince it reveals either linear (mode) stability or instability\nof the black holes, and in case {that} the black holes are unstable the analysis\nalso provides the time-scale of the instability.\n\nThe quasinormal modes consist of polar and axial modes, where the\npolar modes are composed of radial and non-radial modes.\nThe radial modes were investigated before,\nwhere the analysis was focused on the detection of potential instabilities\n\\cite{Blazquez-Salcedo:2018jnn}.\n{This} analysis revealed the loss of hyperbolicity of\nthe perturbation equations at a critical upper size $r_{\\rm S1}$\n(for a given coupling constant) for the\nscalarized black holes. However, for scalarized black holes\nwith sizes in the range between this critical value\nand the bifurcation point $r_{\\rm B}$ from the Schwarzschild branch,\nno instabilities were detected,\nmaking scalarized black holes in this range\npotentially stable physically interesting objects.\n\nSubsequent investigation of the axial modes of these black holes\nrevealed the loss of hyperbolicity of\nthe perturbation equations at a critical upper size $r_{\\rm S2}$,\nslightly larger than $r_{\\rm S1}$.\nAgain, no instabilities were detected in the range\n$r_{\\rm S2} < r_{\\rm H} < r_{\\rm B}$,\nretaining a slightly smaller interval\nof potentially stable physically interesting scalarized black holes\n \\cite{Blazquez-Salcedo:2020rhf}.\n\nIn the present study we have investigated the non-radial polar modes\nof the black holes in this interval $r_{\\rm S2} < r_{\\rm H} < r_{\\rm B}$.\nWe have calculated the monopole ($l=0$) modes,\nthe dipole ($l=1$) modes and the quadrupole ($l=2$) modes,\nwhere the quadrupole modes consist of grav-led and scalar-led modes.\nSince we did not find any instabilities, we conclude\nthat the scalarized black holes on  the fundamental branch\nare linearly (mode) stable in the range $r_{\\rm S2} < r_{\\rm H} < r_{\\rm B}$\n(for fixed coupling).\nWe note, that we have also continued the analysis of the modes\ninto the range below $r_{\\rm S2}$, where we did not find\nany further pathologies.\n\nWe have compared the frequencies and the damping times of the\nmodes of the scalarized black hole with their Schwarzschild counterparts.\nWe have not seen a general trend for the deviations of the various\nmultipolar modes, which all feature more or less substantial deviations\nin the frequencies and damping times.\nAn expected outcome of our investigations has been the breaking of isospectrality,\ni.e., of the degeneracy of the axial and polar Schwarzschild modes.\nIndeed, in the EsGB black holes we have observed a splitting between the axial modes\nand the polar grav-led modes as soon as scalarization arises.\n\nThis can be relevant with regard to gravitational wave astronomy. With the improvement of sensitivities of the detectors, it could be possible for the LIGO-VIRGO collaboration to measure the ringdown frequencies. In this paper we have quantitatively explored the smoking gun that characterizes the ringdown phase of scalarized black holes in EsGB theory as compared to GR: the richer quasinormal mode spectrum that appears, first, because of the breaking of isospectrality between the grav-led modes in the axial and polar channels, and second, because of the existence of scalar-led modes, that appear at any multipole, and in particular, at $l=0$. Of course, only stable solutions describing astrophysically conceivable objects are relevant candidates in this respect.\n\n\nWhile our study has concluded our linear stability analysis, a non-linear\nstability analysis would be a natural next objective.\nOn the other hand, investigation of the instabilities of\nrotating spontaneously scalarized black holes in this theory should also be\nan  interesting and challenging step forward.\n\n\n\n\\section*{Acknowledgements}\nJLBS, SK, JK and PN gratefully acknowledge support by the DFG funded\nResearch Training Group 1620 ``Models of Gravity''.\nJLBS would like to acknowledge support from the DFG project BL 1553.\nDD acknowledges financial support via an Emmy Noether Research Group\nfunded by the German Research Foundation (DFG) under grant no. DO 1771\/1-1.\nDD is indebted to the Baden-W\\\"urttemberg Stiftung for the financial support\nof this research project by the Eliteprogramme for Postdocs.\nSY would like to thank the University of T\\\"ubingen for the financial support\nand acknowledges the partial support by the Bulgarian NSF Grant KP-06-H28\/7.\nP.N. is partially supported by the Bulgarian NSF Grant  KP-06-H38\/2.\nThe authors would also like to acknowledge networking support by the\nCOST Actions CA16104 and CA15117.\n\n\\section*{Appendix}\n\nIn this appendix we will present the perturbation equations. Using for the metric the Ansatz (\\ref{eq:pert_metric}) and for the scalar {field} the Ansatz (\\ref{eq:pert_scalar}), we get a system of equations for { the polar parity perturbation functions $H_0,\\,H_1,\\,H_2,\\,T$ and $ \\varphi_1$}. For simplicity we introduce the function $K(r)=\\frac{1}{1-2m(r)\/r}$. Also in the following equations the prime symbol denotes the derivative with respect to $r$: $K'=\\frac{dK}{dr}$, etc...\n\nWe start with the Einstein equations.\nFrom the $(t,t)$-component we get\n\\begin{eqnarray}\n\\label{pert_tt}\n\t&{T''}&=\\,{\\left(\\frac {4\\left( K-1 \\right) {\\lambda}^\n\t\t\t{2}}{r \\left( -rK  {{ e}^{{6{ \\varphi_0}}^{2}}}\n\t\t\t+4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }\\right) \\varphi_0\\varphi_{1}''}+\\,{\\left(\\frac {\n\t\t\t-4\\,K{ \\varphi_0}\\,{ \\varphi_0'}\\, \\left( K-3 \\right) {\\lambda}^{2}-2\n\t\t\t\\,{K}^{2} {{ e}^{{6{ \\varphi_0}}^{2}}} r  {\n\t\t}}{2rK \\left( -rK {{ e}^{{6{ \\varphi_0}}^{2}}} +4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }\\right) H_{2}''} \\nonumber \\\\\n\t&+&\\,{\\left( \\frac { \\left(\n\t\t\t-8\\,K{ \\varphi_0}\\,{ \\varphi_0''}\\,r+12\\,{ \\varphi_0}\\,{ \\varphi_0'}\\,{ K'}\n\t\t\t\\,r+8\\,K{ \\varphi_0'}\\, \\left( 12\\,{{ \\varphi_0}}^{2}{ \\varphi_0'}\\,r-{ \\varphi_0'}\\,r\n\t\t\t-2\\,{ \\varphi_0} \\right)  \\right) {\\lambda}^{2}}{2rK \\left( -r\n\t\t\tK {{ e}^{{6{ \\varphi_0}}^{2}}}+4\\,{\\lambda}^{2}{\n\t\t\t\t\\varphi_0'}\\,{ \\varphi_0} \\right)} \\right. } \\nonumber \\\\\n\t&+&{\\left. \\frac{-K{ K'}\\,{\n\t\t\t\t{ e}^{{6{ \\varphi_0}}^{2}}}{r}^{2}+6\\,{K}^{2} {\n\t\t\t\t{ e}^{{6{ \\varphi_0}}^{2}}}r }{2rK \\left( -r\n\t\t\tK {{ e}^{{6{ \\varphi_0}}^{2}}}+4\\,{\\lambda}^{2}{\n\t\t\t\t\\varphi_0'}\\,{ \\varphi_0} \\right) } \\right) T'} \\nonumber \\\\\n\t&+&\\,{\\left(\\frac { \\left(  -4\\,{\\lambda}^{2}{ \\varphi_0}\\,\n\t\t\t\\left( K-3 \\right) { K'}-16\\,K{ \\varphi_0'}\\, \\left( 12\\,{{ \\varphi_0}}^{2\n\t\t\t}-1 \\right)  \\left( K-1 \\right)  \\right) {\\lambda}^{2}+4\\,{K}^{2}{\n\t\t\t\t\\varphi_0'}\\, {{ e}^{{6{ \\varphi_0}}^{2}}}{r}^{2}  }{2rK \\left( -rK {{ e}^{{6{ \\varphi_0}}^{2}}}\n\t\t\t+4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }\\right) \\varphi_{1}'} \\nonumber \\\\\n\t&+& {\\left(\\frac {4\\,{\\lambda}^{2} \\left( K-2 \\right) \\varphi_0\\varphi_{0}''}{r \\left( rK\n\t\t\t{{ e}^{{6{ \\varphi_0}}^{2}}}-4\\,{\\lambda}^{2}{\n\t\t\t\t\\varphi_0'}\\,{ \\varphi_0} \\right) }\\right. }-{\\frac {  \\left(rK {{ e}^{{6{ \\varphi_0\n\t\t\t\t\t}}^{2}}} +2\\,K{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0}-12\\,{\n\t\t\t\t\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0}\\right) K' }{rK \\left( rK\n\t\t\t{{ e}^{{6{ \\varphi_0}}^{2}}}-4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{\n\t\t\t\t\\varphi_0} \\right) } } \\nonumber \\\\\n\t&+&\\, { \\frac {2\\,K{{ \\varphi_0'}}^{2} {{ e}^{{6\n\t\t\t\t\t\t{ \\varphi_0}}^{2}}} {r}^{3}+{K}^{2} {{ e}^{{6{ \\varphi_0}\n\t\t\t\t\t}^{2}}} {l}^{2}r+{K}^{2} {{ e}^{{6{ \\varphi_0}}^{2}}}\n\t\t\tlr-96\\,K{{ \\varphi_0}}^{2}{{ \\varphi_0'}}^{2}{\\lambda}^{2}r+2\\,rK\n\t\t\t{{ e}^{{6{ \\varphi_0}}^{2}}} -4\\,K{ \\varphi_0}\\,{ \\varphi_0'\n\t\t\t}\\,{l}^{2}{\\lambda}^{2}} {2{r}^{2} \\left( rK {\n\t\t\t\t{ e}^{{6{ \\varphi_0}}^{2}}} -4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{\n\t\t\t\t\\varphi_0} \\right) }  } \\nonumber \\\\\n\t&+& \\left. \\frac{192\\,{{ \\varphi_0}}^{2}{{ \\varphi_0'}}^{2}{\\lambda}^{2\n\t\t}r-4\\,K{ \\varphi_0}\\,{ \\varphi_0'}\\,l{\\lambda}^{2}+8\\,K{{ \\varphi_0'}}^{2}{\\lambda\n\t\t}^{2}r-16\\,{{ \\varphi_0'}}^{2}{\\lambda}^{2}r}{2{r}^{2} \\left( rK {\n\t\t\t{ e}^{{6{ \\varphi_0}}^{2}}} -4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{\n\t\t\t\\varphi_0} \\right)} \\right) H_2 \\nonumber \\\\\n\t&+& \\left(\\frac { \\left( 8\\,K{\n\t\t\t\\varphi_0}\\, \\left( K-1 \\right) { \\varphi_0''}-4\\,{ \\varphi_0}\\,{ \\varphi_0'}\\,\n\t\t\\left( K-3 \\right) { K'}-8\\,K{{ \\varphi_0'}}^{2} \\left( 12\\,{{ \\varphi_0}}\n\t\t^{2}-1 \\right)  \\left( K-1 \\right)  \\right) {\\lambda}^{2}}{2rK \\left( -rK {{ e}^{{6{ \\varphi_0\n\t\t\t\t}}^{2}}}+4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right)} \\right.  \\nonumber \\\\\n\t&-&\\left.\\frac{2\\,K{ K'}\n\t\t\\,  {{ e}^{{6{ \\varphi_0}}^{2}}}r+2\\,{K}^{2}  {\n\t\t\t{ e}^{{6{ \\varphi_0}}^{2}}} \\left( -{{ \\varphi_0'}}^{2}{r}^{2}+\n\t\tK-1 \\right)  }{2rK \\left( -rK {{ e}^{{6{ \\varphi_0\n\t\t\t\t}}^{2}}}+4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right)}\\right) H_0 \\nonumber \\\\\n\t&+&\\left( -\\,{\\frac {2K{ \\varphi_0}\\,{\\lambda}^{2} \\left( l+2 \\right)\n\t\t\t\\left( l-1 \\right) { \\varphi_0''}}{r \\left( rK {{ e}^{{6{ \\varphi_0}\n\t\t\t\t\t}^{2}}} -4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }}+{\n\t\t\\frac {{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0}\\, \\left( l+2 \\right)  \\left( l\n\t\t\t-1 \\right) { K'}}{r \\left( rK {{ e}^{{6{ \\varphi_0}}^{2}}}\n\t\t\t-4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }}\\right) T \\nonumber \\\\\n\t&+& \\left( \\frac {K \\left( K {{ e}^{{6{ \\varphi_0}}^{2}}}+48\\,{\n\t\t\t{ \\varphi_0}}^{2}{{ \\varphi_0'}}^{2}{\\lambda}^{2}-4\\,{{ \\varphi_0'}}^{2}{\\lambda}^\n\t\t{2} \\right)  \\left( l+2 \\right)  \\left( l-1 \\right)}{2r \\left( rK\n\t\t{{ e}^{{6{ \\varphi_0}}^{2}}}-4\\,{\\lambda}^{2}{\n\t\t\t\\varphi_0'}\\,{ \\varphi_0} \\right) }\\right)  T \\nonumber \\\\\n\t&+& \\left(\\,{\\frac {4{\\lambda}^{2}\n\t\t\t\\left( 12\\,{{ \\varphi_0}}^{2}-1 \\right)  \\left( K-1 \\right) { \\varphi_0''}}{r\n\t\t\t\\left( rK  {{ e}^{{6{ \\varphi_0}}^{2}}}-4\\,{\\lambda}\n\t\t\t^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }}\\right) \\varphi_1 \\nonumber \\\\\n\t&-&\\,\\left({\\frac {2{\\lambda}^{2} \\left( 12\n\t\t\t\\,K{{ \\varphi_0}}^{2}{ \\varphi_0'}\\,r-K{ \\varphi_0}\\,l(l+1)-36\\,{{ \\varphi_0}}^{2}{\n\t\t\t\t\\varphi_0'}\\,r-K{ \\varphi_0'}\\,r+3\\,{ \\varphi_0'}\\,r \\right) { K'\n\t\t}}{K{r}^{2} \\left( rK {{ e}^{{6{ \\varphi_0}}^{2}}}-4\n\t\t\t\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0} \\right) }}\\right) \\varphi_1 \\nonumber \\\\\n\t&-&\\left(\\,{\\frac {144{ \\varphi_0}\\,{\n\t\t\t\t{ \\varphi_0'}}^{2}{\\lambda}^{2} \\left( 2\\,{ \\varphi_0}-1 \\right)  \\left( 2\\,{\n\t\t\t\t\\varphi_0}+1 \\right)  \\left( K-1 \\right) }{r \\left( rK  {{ e}^{6{\n\t\t\t\t\t\t{ \\varphi_0}}^{2}}} -4\\,{\\lambda}^{2}{ \\varphi_0'}\\,{ \\varphi_0}\n\t\t\t\\right) }} \\right) { \\varphi_1} \n\\end{eqnarray}\n\nFrom the $(t,r)$-component we get\n\\begin{eqnarray}\n&{ T'}&=\\,{\\left(\\frac {4\\, \\left( K-1 \\right) {\\lambda}^{\n\t\t\t2}}{r \\left( -rK  {{ e}^{{6{  \\varphi_0}}^{2}}}+4\\,{\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) } \\right) \\varphi_{0} \\varphi_{1}'}+\\,{\\left(\\frac {\n\t\t-4\\,f{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\\lambda}^{2}-2\n\t\t\\,  {{ e}^{{6{  \\varphi_0}}^{2}}} fKr\n\t}{2rf \\left( -rK  {{ e}^{{6{  \\varphi_0}}^{2}}} +4\\,{\n\t\t\t\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) }\\right) H_2} \\nonumber \\\\\n&+&\\,\\left({\\frac {2\\,  {{ e}^{{6{  \\varphi_0}}^{2}}} K{{\n\t\t\t\t \\varphi_{0}'}}^{2}{r}^{3}+  {{ e}^{{6{  \\varphi_0}}^{2}}}{K}^{2\n\t\t}(l+2)(l-1))r-2\\,\n\t\t{{ e}^{{6{  \\varphi_0}}^{2}}}{K}^{2}r+2\\,rK  {\n\t\t\t{ e}^{{6{  \\varphi_0}}^{2}}} - 96K{{  \\varphi_0}}^{2}{{  \\varphi_{0}'}}^\n\t\t{2}{\\lambda}^{2}r }{2K{r}^{2} \\left( rK  {{ e}^{{6{  \\varphi_0}}^{\n\t\t\t\t\t2}}} -4\\,{\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) }\n} \\right. \\nonumber \\\\\n&+& \\,{\\frac {96\\,{{\n\t\t\t\t \\varphi_0}}^{2}{{  \\varphi_{0}'}}^{2}{\\lambda}^{2}r+8\\,{ \\varphi_{0}''}\\,(K-1){  \\varphi_0}\\,{\n\t\t\t\\lambda}^{2}r-4\\,K{  \\varphi_0}\\,{  \\varphi_{0}'}\\,{\\lambda}^{2}l(l+1)-8\\,{{  \\varphi_{0}'\n\t\t}}^{2}{\\lambda}^{2}r(1-K)}{2K{r}^{2} \\left( rK  {{ e}^{{6{  \\varphi_0}}^{\n\t\t\t\t\t2}}} -4\\,{\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0} \\right)}}  \\nonumber \\\\\n&-&\\left. {\\frac { \\left( rK {{ e}^{{6{  \\varphi_0}}^{2}}} +2\\,K{\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0}-6\\,{\\lambda}^{2}{  \\varphi_{0}'}\\,{\n\t\t\t \\varphi_0} \\right) { K'}}{r \\left( rK  {{ e}^{{6{  \\varphi_0}}^{2}}\n\t\t} -4\\,{\\lambda}^{2}{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) {K}^{2}}} \\right) \\frac{H_{1}}{\\omega} \\nonumber \\\\\n&+&\\,{\\left(\\frac {  -4\\, \\left( K-1\n\t\t\\right)  \\left( 24\\,f{{  \\varphi_0}}^{2}{  \\varphi_{0}'}+{ f'}\\,{  \\varphi_0}-2\\,f{\n\t\t\t \\varphi_{0}'} \\right) {\\lambda}^{2}+4\\,  {{ e}^{{6{  \\varphi_0}}^{2}}}\n\t\tfK{  \\varphi_{0}'}\\,{r}^{2} }{2rf \\left( -rK\n\t\t{{ e}^{{6{  \\varphi_0}}^{2}}} +4\\,{\\lambda}^{2}{\n\t\t\t \\varphi_{0}'}\\,{  \\varphi_0} \\right) } \\right)  \\varphi_1} \\nonumber \\\\\n&+& { \\left( \\frac {  4\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( {\n\t\t\tf'}\\,r-2\\,f \\right) {\\lambda}^{2}-rK  {{ e}^{{6{  \\varphi_0}}^{2}}}\n\t\t\\left( { f'}\\,r-2\\,f \\right)  }{2rf \\left( -r\n\t\tK  {{ e}^{{6{  \\varphi_0}}^{2}}} +4\\,{\\lambda}^{2}{\n\t\t\t \\varphi_{0}'}\\,{  \\varphi_0} \\right) } \\right) T} \\, .\n\\end{eqnarray}\n\nFrom the $(t, \\varphi)$-component we get\n\\begin{eqnarray}\n\\label{pert_tphi}\n{H_1'}&=&K\\omega H_2 -{\\frac {4{\\lambda}^{2}{ \\varphi_0}{K'}}{4 {\\lambda}^{2}{ \\varphi_0}{ \\varphi_0'}-rK {{e}^{6{{ \\varphi_0}}^{2}}}}}\\omega \\varphi_1 -{\\frac { -{f'}  {{e}^{6{{ \\varphi_0}}^{2}}}{K}^{2}{r}^{2}+{K'}  {{e}^{6{{ \\varphi_0}}^{2}}}f{r}^{2}K }{2f Kr \\left(4 {\\lambda}^{2}{ \\varphi_0}{ \\varphi_0'}-rK {{e}^{6{{ \\varphi_0}}^{2}}} \\right)}}{H_1} \\nonumber \\\\\n&-&{\\frac {\\left(2 { \\varphi_0} { \\varphi_0'} {K'} r -4 K{ \\varphi_0} { \\varphi_0''} r+4 K{{ \\varphi_0'}}^{2}r \\left( 12 {{ \\varphi_0}}^{2}-1 \\right) \\right) {\\lambda}^{2}+{K}^{2} {{e}^{6{{ \\varphi_0}}^{2}}}r}{4 {\\lambda}^{2}{ \\varphi_0}{ \\varphi_0'}-rK {{e}^{6{{ \\varphi_0}}^{2}}}}} \\omega T \\nonumber  \\\\\n&-&{\\frac { 4 f K{ \\varphi_0} { \\varphi_0''} r-6 f{ \\varphi_0} { \\varphi_0'} {K'} r+2 K{ \\varphi_0'}  \\left(2 f{ \\varphi_0'} r(1-12 {{ \\varphi_0}}^{2})+{ \\varphi_0}({f'}r-2 f) \\right) }{f Kr \\left(4 {\\lambda}^{2}{ \\varphi_0}{ \\varphi_0'} -rK {{e}^{6{{ \\varphi_0}}^{2}}} \\right)}}{\\lambda}^{2}{H_1}  \\, .\n\\end{eqnarray}\n\nFrom the $(r,r)$-component we get\n\\begin{eqnarray}\n\\label{pert_rr}\n&{ H_{0}'}&= \\left({\\frac { \\left( 12\\,{  \\varphi_0}\\,{  \\varphi_{0}'\n\t\t}\\,{\\lambda}^{2}{r}-{{ e}^{6\\,{{  \\varphi_0}}^{2}}}K{r}^{2} \\right)\n\t\t{ f'}}{2f \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\n\t\t\t\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }}-{\\frac {rK{\n\t\t\t{ e}^{6\\,{{  \\varphi_0}}^{2}}}}{2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3\n\t\t\\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}}} \\right) {\n\tT'} \\nonumber \\\\\n&+& \\left( \\,{\\frac {-2{  \\varphi_0}\\, \\left( K-3 \\right) {\\lambda}^{2}{\n\t\t\tf'}}{f \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\\lambda\n\t\t}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }}+\\,{\\frac {2{{ e}^{\n\t\t\t\t6\\,{{  \\varphi_0}}^{2}}}K{  \\varphi_{0}'}\\,{r}^{2}}{2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,\n\t\t\\left( K-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}}}\n\\right) {  \\varphi_{1}'} \\nonumber \\\\\n&+& \\left( \\,{\\frac {-6{\\lambda}^{2}{ f'}\\,{  \\varphi_{0}'\n\t\t}\\,{  \\varphi_0}}{f \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\n\t\t\t\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }}+{\\frac {{\n\t\t\t{ e}^{6\\,{{  \\varphi_0}}^{2}}}{K}^{2}}{2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K\n\t\t-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}}} \\right) {\n\tH_2} \\nonumber \\\\\n&+&\\,{\\left(\\frac {  8\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,r \\left( K-3\n\t\t\\right) {\\lambda}^{2}+4\\,{{ e}^{6\\,{{  \\varphi_0}}^{2}}}K{r}^{2}\n\t}{2rf \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K\n\t\t-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) } \\right) \\omega H_1} \\nonumber \\\\\n&+&\\,{\\left(\\frac { -4\\,fK{  \\varphi_0}\\,{  \\varphi_{0}'}\\,l \\left( l+1 \\right) {\n\t\t\t\\lambda}^{2}+{{ e}^{6\\,{{  \\varphi_0}}^{2}}}f{K}^{2}lr \\left( l+1\n\t\t\\right)   }{2rf \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left(\n\t\tK-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }\\right) H_0} \\nonumber \\\\\n&+&\\left( -{\\frac {K{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( l+2 \\right)  \\left( l-1\n\t\t\\right) {\\lambda}^{2}{ f'}}{f \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,\n\t\t\\left( K-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}\n\t\t\\right) }}+\\,{\\frac { \\left( 8\\,K{  \\varphi_0}\\,{  \\varphi_{0}'}\\,{\\lambda}^{2\n\t\t}{r}^{2}-2\\,{{ e}^{6\\,{{  \\varphi_0}}^{2}}}{K}^{2}{r}^{3} \\right) {\n\t\t\t\\omega}^{2}}{2rf \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\n\t\t\t\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }} \\right. \\nonumber \\\\\n&+& \\,\\left.{\\frac {{\n\t\t\t{ e}^{6\\,{{  \\varphi_0}}^{2}}}{K}^{2} \\left( l+2 \\right)  \\left( l-1\n\t\t\\right) }{4\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\\lambda}^{2}+2r\n\t\tK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}}} \\right) T \\nonumber \\\\\n&+& \\left( \\,{\\frac {\n\t\t\\left( 48\\,K{{  \\varphi_0}}^{2}{  \\varphi_{0}'}\\,r-4\\,K{  \\varphi_0}\\,{l}^{2}-144\\,{{\n\t\t\t\t \\varphi_0}}^{2}{  \\varphi_{0}'}\\,r-4\\,K{  \\varphi_{0}'}\\,r-4\\,K{  \\varphi_0}\\,l+12\\,{  \\varphi_{0}'\n\t\t}\\,r \\right) {\\lambda}^{2}{ f'}}{2rf \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,\n\t\t\\left( K-3 \\right) {\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}}\n\t\t\\right) }} \\right. \\nonumber \\\\\n&-&\\left. \\,{\\frac {4K{  \\varphi_0}\\, \\left( K-1 \\right) {\\lambda}^{2}{\n\t\t\t\\omega}^{2}}{f \\left( 2\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( K-3 \\right) {\n\t\t\t\\lambda}^{2}+rK{{ e}^{6\\,{{  \\varphi_0}}^{2}}} \\right) }} \\right) {\n\t \\varphi_1}   \\, .\n\\end{eqnarray}\n\nFrom the $(r,\\theta)$-component we get\n\\begin{eqnarray}\n\\label{pert_rtheta}\n&{T'}&={\\left( \\frac {  2\\, {{ e}^{{6{  \\varphi_0}}\n\t\t\t\t^{2}}} fK{r}^{2}-8\\,f{  \\varphi_0}\\,{  \\varphi_{0}'}\\,{\\lambda}^{2}r\n\t}{2{r}^{2} \\left( -K {{ e}^{{6{  \\varphi_0}}^{2}}\n\t\t}f +2\\,{\\lambda}^{2}{ f'}\\,{  \\varphi_{0}'}\\,{  \\varphi_0} \\right)\n\t} \\right) H_{0}'}-{\\left( \\frac {4{ f'}\\,{\\lambda}^{2}}{r \\left( -K\n\t\t{{ e}^{{6{  \\varphi_0}}^{2}}} f+2\\,{\\lambda}^{2}{\n\t\t\tf'}\\,{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) }\\right)  \\varphi_0 \\varphi_{1}' } \\nonumber \\\\\n&+& {\\left( \\frac {  12\\,{ f'}\n\t\t\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,{\\lambda}^{2}r- {{ e}^{{6{  \\varphi_0}}^{2}\n\t\t}} Kr \\left( { f'}\\,r+2\\,f \\right)  }{2{\n\t\t\tr}^{2} \\left( -K  {{ e}^{{6{  \\varphi_0}}^{2}}} f+2\\,{\n\t\t\t\\lambda}^{2}{ f'}\\,{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) } \\right) H_2 }+{\\left(\\frac {\n\t\t-2\\, \\left( {{ e}^{{{  \\varphi_0}}^{2}}} \\right) ^{6}K{r}^{2}+8\n\t\t\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\,{\\lambda}^{2}r  }{2{r}^{2\n\t\t} \\left( -K \\left( {{ e}^{{{  \\varphi_0}}^{2}}} \\right) ^{6}f+2\\,{\n\t\t\t\\lambda}^{2}{ f'}\\,{  \\varphi_{0}'}\\,{  \\varphi_0} \\right) }\\right) \\omega H_1} \\nonumber \\\\\n&+&{\\left( \\frac {\n\t\t-4\\,{  \\varphi_0}\\,{  \\varphi_{0}'}\\, \\left( { f'}\\,r-2\\,f \\right) {\n\t\t\t\\lambda}^{2}+rK \\left( {{ e}^{{{  \\varphi_0}}^{2}}} \\right) ^{6} \\left(\n\t\t{ f'}\\,r-2\\,f \\right)  }{2{r}^{2} \\left( -K  {\n\t\t\t{ e}^{{6{  \\varphi_0}}^{2}}} f+2\\,{\\lambda}^{2}{ f'}\\,{\n\t\t\t \\varphi_{0}'}\\,{  \\varphi_0} \\right) } \\right) H_0} \\nonumber \\\\\n&+&{ \\left( \\frac {  8\\,{ f'}\\,\n\t\t\\left( 12\\,{{  \\varphi_0}}^{2}{  \\varphi_{0}'}\\,r-{  \\varphi_{0}'}\\,r+{  \\varphi_0} \\right) {\n\t\t\t\\lambda}^{2}+8\\, {{ e}^{{6{  \\varphi_0}}^{2}}} fK{\n\t\t\t \\varphi_{0}'}\\,{r}^{2}}{2{r}^{2} \\left( -K {{ e}^{{6{\n\t\t\t\t\t\t \\varphi_0}}^{2}}} f+2\\,{\\lambda}^{2}{ f'}\\,{  \\varphi_{0}'}\\,{\n\t\t\t \\varphi_0} \\right) } \\right)  \\varphi_1}  \\, .\n\\end{eqnarray}\n\n\nFrom the $(\\theta, \\phi)$-component we get\n\\begin{eqnarray}\n\\label{pert_tphi}\n&{H_2}&={\\left(\\frac {f \\left(  \\left( -4\\,K{  \\varphi_0}\\,{ \\varphi_0''}+2\n\t\t\\,{  \\varphi_0}\\,{ K'}\\,{  \\varphi_0'}+4\\,K{{  \\varphi_0'}}^{2} \\left( 12\\,{{\n\t\t\t\t \\varphi_0}}^{2}-1 \\right)  \\right) {\\lambda}^{2}+{K}^{2}  {{e}^{6{{\n\t\t\t\t\t\t \\varphi_0}}^{2}}}  \\right)}{ \\left( -K  {\n\t\t\t{e}^{{6{  \\varphi_0}}^{2}}}  f+2\\,{\\lambda}^{2}{ f'}\\,{\n\t\t\t \\varphi_0'}\\,{  \\varphi_0} \\right) K}\\right) H_0} \\nonumber \\\\\n&+&\\,{\\left(\\frac {2{\\lambda}^{2}\n\t\t\\left( { f'}\\,f{ K'}-K \\left( 2\\,f{ f''}-{{ f'}}^{2}\n\t\t\\right)  \\right) }{ \\left( -K  {{e}^{{6{  \\varphi_0}}^{2}\n\t\t}} f+2\\,{\\lambda}^{2}{ f'}\\,{  \\varphi_0'}\\,{  \\varphi_0}\n\t\t\\right) fK}\\right)  \\varphi_0 \\varphi_1}  \\, .\n\\end{eqnarray}\n\nFinally, from the scalar field equation we get\n\\begin{eqnarray}\n& \\varphi_1''&=\n\\left( -{\\frac {{f'}{ \\varphi_0} \\left( K-3 \\right) {\\lambda}^{2}}{2fK{r}^{2} \\left( {{e}^{6{{ \\varphi_0}}^{2}}}\\right)}}+\\frac{ \\varphi_0'}{2} \\right) {H_2'} +{\\left(\\frac {{ \\varphi_0} f\\left( K-1 \\right){\\lambda}^{2}}{fK{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}\\right)}{H_0''}\n-{\\left(\\frac {{ \\varphi_0} {f'}r{\\lambda}^{2}}{fK{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}\\right)} {T''}\n\\nonumber \\\\\n&+& \\left( {\\frac {{K'}}{2K}}-{\\frac {{f'}r+4f}{2fr}} \\right) { \\varphi_1'}\n+ \\left({\\frac {Kl \\left( l+1\\right) }{{r}^{2}}} -{\\frac {K{\\omega}^{2}}{f}}\\right)  \\varphi_1\n\\nonumber \\\\\n&-&{\\left(\\frac {2{ \\varphi_0}\\left( K-1 \\right) {\\lambda}^{2}\\omega}{fK{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}\\right)}{H_1'}+ \\left(  \\left( -{\\frac {{ \\varphi_0} \\left( K-3 \\right) {K'}}{2{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right){K}^{2}}}+{\\frac {{f'}{ \\varphi_0}\\left( K-1 \\right) }{fK{r}^{2} \\left( {{\\rm e}^{{{ \\varphi_0}}^{2}}}\\right) ^{6}}} \\right) {\\lambda}^{2}-\\frac{ \\varphi_0'}{2} \\right) {H_0'} \\nonumber  \\\\\n&+&\\left(  \\left( {\\frac {3{f'}{ \\varphi_0}{K'}}{2fr \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right){K}^{2}}}+{\\frac {{ \\varphi_0} \\left( -2{f''}fr+{{f'}}^{2}r-4{f'}f \\right) }{2K{r}^{2}{f}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}}\\right) {\\lambda}^{2}-{ \\varphi_0'} \\right) {T' } \\nonumber  \\\\\n&+&\\left( {\\frac {{ \\varphi_0} \\left( K-1 \\right) {\\lambda}^{2}{\\omega}^{2}}{f{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}}+{ \\varphi_0''}-{\\frac {{ \\varphi_0'}{K'}}{2K}}+{\\frac {{ \\varphi_0'} \\left( {f'}r+4f \\right) }{2fr}} \\right) {H_2} \\nonumber  \\\\\n&+&\\left( \\left( {\\frac {{f'}{ \\varphi_0} \\left( K-6 \\right) {K'}}{2f{r}^{2}\\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right){K}^{2}}}+{\\frac {{ \\varphi_0} \\left( {f'}{l(l+1)}fK+(2{f''}fr-{{f'}}^{2} r)(2-K) \\right) }{2K{f}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right){r}^{3}}} \\right) {\\lambda}^{2} \\right) {H_2} \\nonumber \\\\\n&+& \\left( {\\frac {{ \\varphi_0} \\left( K-3 \\right) {K'}{\\lambda}^{2}}{f{r}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right){K}^{2}}}+{\\frac {{ \\varphi_0'}}{f}} \\right) \\omega H_1-{\\left(\\frac {{ \\varphi_0}l \\left( l+1 \\right) {K'}{\\lambda}^{2}}{2\\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)K{r}^{3}}\\right)} H_0 \\nonumber  \\\\\n&+& \\left( {\\frac {{K'}{ \\varphi_0}{\\lambda}^{2}{\\omega}^{2}}{rKf \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}}+ \\left({\\frac {{f'}{ \\varphi_0} \\left( l+2 \\right)  \\left( 1-l \\right) {K'}}{4fK{r}^{2} \\left( {{e}^{6{{ \\varphi_0}}^{2}}} \\right)}}-{\\frac{{ \\varphi_0} \\left({{f'}}^{2}-2{f''}f \\right)  \\left( l+2\\right)  \\left( l-1 \\right) }{4{r}^{2}{f}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}} \\right) {\\lambda}^{2} \\right)T \\nonumber  \\\\\n&+& \\left( \\left({\\frac {{f'} \\left( 12{{ \\varphi_0}}^{2}-1 \\right)  \\left( K-3 \\right) {K'}}{2f{r}^{2}\\left({{e}^{{6{ \\varphi_0}}^{2}}} \\right){K}^{2}}}+{\\frac { \\left( 12{{ \\varphi_0}}^{2}-1 \\right)  \\left( K-1 \\right)  \\left({{f'}}^{2}-2{f''}f \\right) }{2K{r}^{2}{f}^{2} \\left( {{e}^{{6{ \\varphi_0}}^{2}}} \\right)}} \\right) {\\lambda}^{2}\\right)  \\varphi_1  \\, .\n\\label{pert_scalar}\n\\end{eqnarray}\n\nThe remaining components of the Einstein equations are zero or can be written in terms of the previous components.\n\n{This system of equations can be further simplified. After some algebra we can derive an equation which relates the function $H_0$ to the rest of the metric perturbation functions, as well as the perturbation of the scalar field $\\varphi_1$ and its derivative. Using this relation and the algebraic relation $(\\ref{pert_tphi})$ we can eliminate the functions $H_0$ and $H_2$, and reduce the perturbation equations to a coupled system of three differential equations for the remaining metric perturbation functions $H_1$ and $T$ and the perturbation of the scalar field $\\varphi_1$. It can be proven that these equations together with the two relations representing the functions $H_0$ and $H_2$ in terms of the rest of the perturbation functions are equivalent to the initial set of perturbation equations. }\n\n{The system of equations for the perturbations functions  $H_1$, $T$ and $\\varphi_1$ can be represented schematically in the form}\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n H_1' \\\\\n T' \\\\\n  \\varphi_1' \\\\\n  \\varphi_1''\n\\end{array}\\right)\n+\\left(\\begin{array}{c}\nV_{11} \\;\\; V_{12} \\;\\; V_{13} \\;\\; V_{14} \\\\\nV_{21}\\;\\; V_{22}\\;\\; V_{23} \\;\\; V_{24} \\\\\nV_{31}\\;\\; V_{32}\\;\\; V_{33}\\;\\; V_{34} \\\\\nV_{41}\\;\\; V_{42}\\;\\; V_{43} \\;\\;V_{44}\n\\end{array}\\right)\\left(\n\\begin{array}{c}\n H_1 \\\\\n T \\\\\n  \\varphi _1\\\\\n  \\varphi _1'\n\\end{array}\\right)\n=0 ,\n\\label{eq:expanper}\n\\end{equation}\n{where $V_{3k}=0$ for $k\\neq 4$, and $V_{34}=-1$.} \nDue to the complexity of the components of \n$V_p$, we shall not show them here. However, by making the dependence in $\\omega$ explicit, we can write the previous equation like\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\tH_1'\/\\omega \\\\\n\tT' \\\\\n\t \\varphi _1' \\\\\n\t \\varphi _1''\n\\end{array}\\right)\n+\n\\left[\n\\left(\\begin{array}{c}\n\tV_{11}^{(1)} \\;\\; V_{12}^{(1)} \\;\\; V_{13}^{(1)} \\;\\; V_{14}^{(1)} \\\\\n\tV_{21}^{(1)}\\;\\; V_{22}^{(1)}\\;\\; V_{23}^{(1)} \\;\\; V_{24}^{(1)} \\\\\n\t0\\;\\;\\;\\;\\;\\; 0\\;\\;\\;\\;\\;\\; 0\\;\\;\\;\\;\\;\\; -1 \\\\\n\tV_{41}^{(1)}\\;\\; V_{42}^{(1)}\\;\\; V_{43}^{(1)} \\;\\;V_{44}^{(1)}\n\\end{array}\\right)\n+\\omega^2\\left(\\begin{array}{c}\n\tV_{11}^{(2)} \\;\\; V_{12}^{(2)} \\;\\; V_{13}^{(2)} \\;\\;\\; 0 \\\\\n\tV_{21}^{(2)}\\;\\; V_{22}^{(2)}\\;\\; V_{23}^{(2)} \\;\\;\\; 0 \\\\\n\t0\\;\\;\\;\\;\\;\\;\\; 0\\;\\;\\;\\;\\;\\;\\; 0\\;\\;\\;\\;\\;\\;\\; 0 \\\\\n\tV_{41}^{(2)}\\;\\; V_{42}^{(2)}\\;\\; V_{43}^{(2)} \\;\\;\\;0\n\\end{array}\\right)\n\\right]\n\\left(\n\\begin{array}{c}\n\tH_1\/\\omega \\\\\n\tT \\\\\n\t \\varphi _1\\\\\n\t \\varphi _1'\n\\end{array}\\right)\n=0 ,\n\\label{eq:expanper}\n\\end{equation}\nwhere the functions $V_{ij}^{(1,2)}$ only depend on $f(r)$, $m(r)$, $ \\varphi_0(r)$, $\\lambda$ and $l$.\n\n\n\nThe boundary conditions for the perturbations can be found with the help of the expansions of the background metric and the scalar field. Because of the natural divergence at spatial infinity, one has to consider a high-order expansion of the perturbations at infinity for the quasinormal modes.\n\nFor the lowest values $l=0,1$, simpler gauge choices can be chosen~\\cite{Zerilli:1971wd}. We note that for $l=1$ we can choose a gauge in which $T$ vanishes identically. Therefore, we can use the first two equations in~\\eqref{eq:expanper} to eliminate $H_1'$ and $H_1$ in favor of $ \\varphi_1$, $ \\varphi_1'$. In the case of $l=0$ we can pick a gauge such that both $T$ and $H_1$ vanish. The equations are then again reduced to a second order equation for the scalar perturbation. This is possible since some harmonics in the expansion are identically zero when $l=0$ or $l=1$.\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nTesting planarity of graphs with additional constraints is a popular theme in\nthe area of graph visualizations.\nOne of most the prominent such planarity variants, c-planarity, raised in 1995 by Feng, Cohen and Eades~\\cite{FCEa95,FCEb95} \nasks for a given planar graph $G$ equipped with a hierarchical structure on its vertex \nset, i.e., clusters, to decide if a planar embedding $G$ with the following property exists:\nthe vertices in each cluster are drawn inside a disc so that the discs\nform a laminar set family corresponding to the given hierarchical structure\nand the embedding has the least possible number of edge-crossings with the boundaries of the discs.\nShortly after, several groups of researchers tried to settle\nthe main open problem formulated by Feng et al. asking to decide its complexity\nstatus, i.e., either provide\na polynomial\/sub-exponential-time  algorithm for c-planarity or show its \\hbox{\\rm \\sffamily NP}-hardness.\nFirst, Biedl~\\cite{B98} gave\n a polynomial-time algorithm for c-planarity with two clusters. A different approach\nfor two clusters was considered by Hong and Nagamochi~\\cite{HN16}\nand quite recently in~\\cite{FKMP15}.\n The result also follows from a work by Gutwenger et al.~\\cite{GJL+02}.\nBeyond two clusters a polynomial time algorithm for c-planarity was obtained only in special cases,\ne.g.,~\\cite{BFPP08,GLS05,GJL+02,JJK+09,JKK+09}, and most recently in~\\cite{BR14+,CBFK14+}. Cortese et al.~\\cite{CDPP05} shows that c-planarity is solvable in polynomial\ntime if the underlying  graph is a cycle and the number of clusters\nis at most three.\n\n In the present work we generalize the result of Cortese et al. to the class of all planar graphs\nwith a given combinatorial embedding. In a recent pre-print~\\cite{F14+} we established \na strengthening for trees, where we do not fix the embedding.\nIn the general case (including already the case of three clusters) of so-called flat clustered graphs a similar result was obtained\nonly in very limited cases. Specifically, either when every face of $G$ is incident\nto at most five vertices~\\cite{BF07,FKMP15}, or when there exist at most two vertices of a cluster incident to a single face~\\cite{CBFK14+}.\nWe remark that the techniques of the previously mentioned papers do not give\na polynomial-time algorithm for the case of three clusters, and also do not seem to be adaptable\nto this setting. Our result and the technique used to achieve it suggest that, for a fairly general class of clustered graphs, c-planarity could be tractable\/solvable in  sub-exponential time at least with a fixed combinatorial embedding. \n\n {\\bf Notation.}\nLet $G=(V,E)$ denote a connected planar graph possibly with multi-edges.\nFor  standard graph theoretical definitions such as path, cycle, walk etc.,\nwe refer reader to~\\cite[Section 1]{D05}. \nA \\emph{drawing} of $G$ is a representation of $G$ in the plane where every vertex\n in $V$ is represented by a unique point and every\nedge $e=uv$ in $E$ is represented by a Jordan arc joining the two points that represent $u$ and $v$. \nWe assume that in a drawing no edge passes through a vertex,\nno two edges touch and every pair of edges cross in finitely many points.\nAn \\emph{embedding} of $G$ is an  edge-crossing free drawing.\nIf it leads to no confusion, we do not distinguish between\na vertex or an edge and its representation in the drawing and we use the words ``vertex'' and ``edge'' in both\n contexts.\nA  \\emph{face} in an embedding is a connected component of the complement of the embedding \nof $G$ (as a topological space) in the plane.\n The \\emph{facial walk} of $f$ is the closed walk in $G$ with a fixed orientation that we obtain by traversing the boundary of $f$ counter-clockwise.\nIn order to simplify the notation we sometimes denote the facial walk of a face $f$ by $f$. \n  A pair of consecutive edges $e$ and $e'$ in a facial walk $f$ creates a \\emph{wedge} incident to $f$ at their common vertex.\n  A vertex or an edge is \\emph{incident} to a face $f$, if it appears on its facial walk.\n\nThe \\emph{rotation} at a vertex is the counter-clockwise cyclic order of the end pieces of its incident edges\nin a drawing of $G$.\nAn embedding of $G$ is up to an isotopy and the choice of an \\emph{outer} (unbounded) face described by the rotations at its vertices. We call such a description of an embedding of $G$ a \\emph{combinatorial embedding}. Remaining faces are \\emph{inner faces}.\nThe \\emph{interior} and \\emph{exterior} of a cycle in an embedded graph is the bounded and unbounded, respectively, connected component\nof its complement in the plane. \nSimilarly, the \\emph{interior} and \\emph{exterior} of an inner face in an embedded graph is the bounded and unbounded, respectively, connected component\nof the complement of its facial walk in the plane, and vice-versa for the outer face.\nWhen talking  about interior\/exterior or area of a cycle  \nin a graph $G$ with a combinatorial embedding and a \\emph{designated} outer face  we mean it with respect to an embedding in the isotopy class that $G$ defines.\nFor $V'\\subseteq V$ we denote by $G[V']$ the sub-graph of $G$ induced by $V'$. \n\n\n\n\n\n\n\n \n\n\nA \\emph{flat clustered graph}, shortly  \\emph{c-graph}, is a pair $(G,T)$, where $G=(V,E)$ is a graph and $T=\\{V_0, \\ldots, V_{c-1}\\}$, $\\biguplus_i V_i=V$, is a partition of the\nvertex set into \\emph{clusters}. See Figure~\\ref{fig:treeEx} for an illustration.\nA  c-graph $(G,T)$ is \\emph{clustered planar} (or briefly \\emph{c-planar}) if $G$ has an\n embedding in the plane such that (i)\nfor every $V_i\\in T$ there is a topological disc $D(V_i)$, where $\\mathrm{interior}(D(V_i))\\cap \\mathrm{interior} (D(V_j))=\\emptyset$, if $i\\not=j$,\n containing all the vertices of $V_i$ in its interior, and (ii)\n every edge of $G$ intersects the boundary of $D(V_i)$ at most once for every $D(V_i)$.\nA c-graph  $(G,T)$ with a given combinatorial embedding of $G$ is \\emph{c-planar} \nif additionally the embedding is combinatorially described as given.\n A \\emph{clustered drawing and embedding} of a flat clustered graph $(G,T)$ is a drawing and embedding, respectively,\n of $G$ satisfying (i) and (ii).\nIn 1995\n Feng, Cohen and Eades~\\cite{FCEa95,FCEb95} introduced the notion of clustered planarity for clustered graphs, shortly c-planarity, (using, a more general, hierarchical clustering)\nas a natural generalization of graph planarity. (Under a different name\nLengauer~\\cite{L89} studied a similar concept in 1989.)\n\n\n\\begin{wrapfigure}{r}{.5\\textwidth}\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=0.4]{treeEx}\n    \t}\n\\subfloat[]{\n\\includegraphics[scale=0.4]{treeEx+}\n\t\t}\n\\caption{A c-graph that is not c-planar (left); and a c-planar c-graph (right).}\n\\label{fig:treeEx}\n\\end{wrapfigure}\n\nBy slightly abusing the notation for the rest of the paper $G$ denotes  a flat c-graph $(G,T)=(V_0 \\uplus V_1 \\uplus \\ldots \\uplus V_{c-1}, E)$ with $c$ clusters\n$V_0,V_1,\\ldots $ and $V_{c-1}$, and a given combinatorial embedding, and we assume that $G$ is \\emph{cyclic}~\\cite[Section 6]{FKMP15}. Thus, every $e=uv$ of $G$ is such that\n$u\\in V_i$ and $v\\in V_j$ where $j-i \\mod c \\le 1$ and for every\n$i$ there exists an edge in $G$ between $V_i$ and $V_{i+1 \\mod c}$.\nIn the case of three clusters, the first condition is redundant.\nIf the second condition is violated, the problem was essentially solved for three clusters as discussed in\nSection~\\ref{sec:wnwn}.\nWe assume that $G$ is connected, since in the problem that we are studying, the connected components of $G$ can be treated separately. Indeed, \\iflong as we show in Section~\\ref{sec:fan} \\fi without loss of generality  we  assume throughout the paper that in a clustered embedding of $G$ the clusters are unbounded wedges defined by pairs of rays emanating from the origin (see Figure~\\ref{fig:wedges}) that is disjoint from all the edges (see Appendix). We call such a clustered drawing  a \\emph{fan drawing}. \\\\\n\n\n\n\n\n\n\n\nThus, a connected component in a clustered embedding \ncan be drawn so that it is disjoint from a ball $B$ centered at the origin of radius $\\epsilon>0$\nfor any $\\epsilon$. The rest of the graph is then embedded inductively inside $B$.\nThe aim of the present work is to prove the following.\n\n\\begin{theorem}\nThere exists a quadratic-time algorithm in $|V(G)|$ to test if a cyclic c-graph $(G,T)$ is c-planar.\n\\end{theorem}\n\n\n\n\n{\\bf Further research directions.}\nWe think that our technique should be extendable by means of Euler's formula to resolve the c-planarity in more general situations than the one treated in the present paper. In particular, we suspect that\nthe technique should yield a generalization of the characterization of strip planar clustered graphs~\\cite[Section 5]{F14+}. \nThat would allow us to work with graphs without a fixed embedding. We mention that the tractability in a special case \nof our problem known as cyclic level planarity, when the embedding is not fixed, follows from a recent work of Angelini et al.~\\cite{angelini2015beyond}.\n\n{\\bf Organization.} In Section~\\ref{sec:pre} we introduce concepts used in the proof of\nour result. We give an outline of our approach in Section~\\ref{sec:out}.\nA more detailed description and a proof of correctness of our algorithm is in Section~\\ref{sec:alg}.\n\n\n \n\\section{Preliminaries}\n\\label{sec:pre}\n\n\\iflong\n\\subsection{Fan drawings}\n\\label{sec:fan}\nWe show that the clusters can be drawn as regions, each bounded by a pair of rays emanating from the origin.\nSuppose that $G=(V_0\\uplus \\ldots \\uplus V_{c-1},E) $ is given by a clustered embedding\nliving in the $xy$ plane of $\\mathbb{R}^3$.\nWe assume that boundries of discs representing clusters do not touch.\nConsider a stereographic projection from the north pole of a two-dimensional sphere $S$ \nsitting at the origin of $\\mathbb{R}^3$.\nLet $D$ be a stereographical pre-image of the embedding of $G$ on $S$.\nLet $S'$ denote the union of $G$ (as a topological space) with the boundaries of the clusters in $D$.\nLet $R_n$ and $R_s$ be a connected component of the complement of $S'$ in $S$, respectively, containing the north pole and south pole.\nIf necessary, we apply an isotopy  to $D$ (a continuous deformation keeping $D$ to be a clustered embedding all the time)  so that in the resulting embedding $D$ of $G$ on $S$ every boundary of a cluster intersects (in fact touches) the closure of $R_n$ and  the closure of $R_s$. \n\nWe show that a desired isotopy exists. We contract every cluster to a point thereby\ntreating clusters as vertices in an embedding $D'$ of a cycle $C$ of length $c$ having multi-edges. \nFormally, this can be viewed as a quotient $S\/\\sim$, where $x\\sim y$ iff\n$x$ and $y$ belong to the same cluster.\nIn $D'$ there must be a pair of faces $f$ and $f'$ whose facial walk is $C$ since any cycle in the corresponding multi-graph is obtained as a symmetric difference of facial walks. Apply an isotopy to $D'$ such that $f$ contains\nthe north pole in its interior and $f'$ contains the south pole in its interior. Finally, we decontract clusters in the end. The above procedure can be easily turned into an isotopy of $D$. \n\nBy projecting the resulting spherical embedding back to the plan we can also assume that we have a clustered embedding of $G$\nsuch that clusters are represented by small discs of diameter $\\epsilon>0$ each drawn in a close vicinity\nof a different vertex of a regular convex $c$-gon with the center at the origin, and the edges\nbetween clusters $V_i$ and $V_{i+1 \\mod c}$, for every $i$, are closely following the edge\nof the $c$-gon between the corresponding pair of vertices. \nThe desired rays bounding clusters are those from the origin orthogonal to the sides of the $c$-gon. \\\\\n\\fi\n\n\n\n\n\\subsection{Outline of the approach}\n\\label{sec:out}\nBy~\\cite[Theorem 1]{FCEb95} deciding c-planarity of instances $G$ in which all $G[V_i]$'s are connected amounts to \nchecking if an outer face of $G$ can be chosen so that every $V_i$ is embedded in the outer face \nof $G[V\\setminus V_i]$. On the other hand, once we have a clustered embedding of $G$ we can augment $G$ by adding edges drawn inside clusters without creating an edge-crossing so that clusters become connected.\nThese observations suggest that c-planarity of $G$ could be viewed as a connectivity augmentation problem, for example as in~\\cite{CBFK14+,FKMP15},\nin which we want to decide if it is possible to make clusters connected  while maintaining the planarity of $G$.\nOne minor problem with this viewpoint is the fact that if $G$ is c-planar we do not allow\na cluster $V_i$ to induce a cycle such that clusters $V_j$ and $V_{j'}$, $i\\not=j,j'$, are drawn on its opposite \nsides. However, this cannot happen  if $G$ is cyclic.\nFollowing the above line of thought our algorithm tries to augment $G$ by subdividing its faces with\npaths and edges. We proceed in two steps. In the first step, Section~\\ref{sec:norm}, we either \ndetect that $G$ is not c-planar or similarly as in~\\cite{ADDF13} and~\\cite{F14+} by \nturning clusters into independent sets and adding certain paths we normalize the instance. In the second step, Section~\\ref{sec:const}, we decide if the normalized instance can be further augmented by  edges as desired.\n\nIn order to prove the correctness of the second step of the algorithm\nwe use the notion of the \\emph{winding number} $\\mathrm{wn}(W)\\in \\mathbb{Z}$ of a walk $W$ of $G$, as defined\nin Section~\\ref{sec:wnwn}. The parameter $\\mathrm{wn}(W)$ says how many times and in which sense \na walk $W$ of $G$ winds around the origin in a clustered drawing of $G$.\nThus, $G$ is not c-planar if there exists a face $f$ such that for its facial walk $|\\mathrm{wn}(f)|>1$ or\nif there exists at least two inner faces $f$ with $|\\mathrm{wn}(f)|>0$.\nHowever, it can be easily seen that this necessary condition of c-planarity is not sufficient\nexcept when $G$ is a cycle~\\cite{CDPP05}.\nThe  necessary condition allows us to reduce the c-planarity testing problem of a normalized instance to the problem\nof finding a perfect matching in an auxiliary face-vertex incidence graph which is polynomially solvable.\nThe novelty of our work lies in the use of the winding number in the context of connectivity augmentation guided\nby the flow and matching in the auxiliary face-vertex incidence graph \\`a la~\\cite{ADDF13} and~\\cite{F14+}, respectively. \n\nWe remark that the approach of~\\cite{ADDF13} via a variant of upward embeddings\nfor directed graphs in our settings has several problems that seem quite hard to overcome,\nthe main one being the fact that the result of Bertolazzi et al.~\\cite{BBLM94} does not extend, at least not in a natural way, to the drawings on the rolling cylinder, see e.g.,~Auer et al.\\cite{Auer201536} for the definition of these drawings.\nWe are not aware of a polynomial-time algorithm for the corresponding  problem,\nnor a corresponding \\hbox{\\rm \\sffamily NP}-hardness result, and\nfind the corresponding algorithmic question interesting and related to our problem.\n\n\\begin{figure}\n  \\centering\n\\centering\n\\subfloat[]{\n\\label{fig:wedges}\n\\includegraphics[scale=0.5]{3ex}\n    \t}\n    \t\\hspace{10px}\n    \t\\subfloat[]{\n    \t\\label{fig:semiSimple}\n    \t\\includegraphics[scale=0.5]{semiSimple}\n    \t}\n\\caption{(a) A clustered graph $G=(V_0 \\uplus V_1 \\uplus V_2,E)$ with clusters represented by wedges bounded by rays meeting at the origin. The highlighted wedge at $u$ is concave and at $v$ convex. (b) A semi-simple face $f$\nand the outer face $f_o$ with an incident concave wedge.}\n\\end{figure}\n\n\n\\subsection{Winding number} \n\\label{sec:wnwn1}\nWe define the winding number $\\mathrm{wn}(W)$ of a closed oriented walk $W$ in a drawing disjoint from the origin of a graph $G$ (possibly with crossings). In what follows  facial walks are understood with the orientations as in\n  an embedding of $G$ with the given rotations and a face $f_o$ being a designated outer face.\nBy viewing a closed walk $W$ in the drawing  as a continuous  function $w$ from the unit circle $S^1$ to $\\mathbb{R}^2\\setminus {\\bf 0}$,\nthe winding number $\\mathrm{wn}(W)\\in \\mathbb{Z}$ corresponds to the element of the fundamental group of  $S^1$~\\cite[Chapter 1.1]{Hatch02}  represented by $\\frac{w(x)}{||w(x)||_2}$.\nLet $W_1$ and $W_2$ denote a pair of oriented closed walks meeting in a vertex $v$.\nLet $W$ denote the closed oriented walk from $v$ to $v$ obtained by concatenating $W_1$ and $W_2$.\nBy the definition of $\\mathrm{wn}$ we have $\\mathrm{wn}(W)=\\mathrm{wn}(W_1)+\\mathrm{wn}(W_2)$.\nLet $f_1$ and $f_2$, $f_o\\not = f_1, f_2$, denote a pair of faces of $G$ whose walks intersect in a single walk.\nLet $G'$ denote a graph we get from $G$ by deleting edges incident to both $f_1$ and $f_2$.\nLet $f$ denote the new face thereby obtained. Since $f_1$ and $f_2$ intersect in a single walk, the boundary of $f$ is connected.\nIn the drawing of $G'$ inherited from the drawing of $G$ we have $\\mathrm{wn}(f)=\\mathrm{wn}(f_1)+\\mathrm{wn}(f_2)$, since common edges of $f_1$ and $f_2$ are traversed\nin opposite directions by $f_1$ and $f_2$.\nA face  or a vertex is in the interior of a closed walk $W$ in $G$ if it is in the interior of \na cycle induced by the edges of $W$ in an embedding of $G$ with the given rotations and  $f_o$ as the outer face.\nThe previous observation is easily generalized by a simple inductive argument \nas follows \\\\\n$ {\\bf (*)} \\ \\ \\ \\   \\sum_f \\mathrm{wn}(f)=\\mathrm{wn}(W)$ \\\\\n where we sum over all  faces~$f$ of $G$ in the interior of the\nclosed walk $W$ in $G$. In particular, $\\sum_f \\mathrm{wn}(f)=\\mathrm{wn}(f_o)$, where we sum over all \n faces  $f\\not=f_o$ of $G$.\n\n\\subsection{Labeling vertices} \n\\label{sec:wnwn}\nLet $\\gamma:V \\rightarrow \\{0,1,\\ldots c-1\\}$ be a labeling of the vertices $V$ by integers\nsuch that $\\gamma(v) = i$ if $v\\in V_i$.\n Let $W$ denote an oriented closed  walk in a clustered drawing of  $G$. We put $\\mathrm{height}(W)=\n\\sum_{{v'u'}\\in E(W)} g(\\gamma(u')-\\gamma(v'))$,\n where $g(0)=0, \\ g(1)=g(1-c)=1$ and $g(c-1)=g(-1)=-1$.\nWe have the following.\n\n\\begin{lemma}\n\\label{lemma:wn}\nFor a walk $W$ in a fan drawing of $G$ we have $\\mathrm{wn}(W)=\\mathrm{height}(W)\/c$.\n\\end{lemma}\n\n\\begin{proof}\nThe number of times\nthe walk $W$ crosses the ray between $V_i$ and $V_{i+1 \\mod c}$ from right\nto left w.r.t. to the direction of the ray is $\\mathrm{wn}_i^+(W)=\\sum_{{v'u'}} g(\\gamma(u')-\\gamma(v'))$, \nwhere we sum over the edges $v'u'$ in the walk $W$, where\n $v'\\in V_i$ immediately precedes $u'\\in V_{i+1 \\mod c}$ in the walk.\n Similarly, we define \\\\  $\\mathrm{wn}_i^-(W)=\\sum_{{v'u'}} g(\\gamma(u')-\\gamma(v'))$, \nwhere we sum over the edges $v'u'$ in $W$, where\n $v'\\in V_{i+1 \\mod c}$ immediately precedes $u'\\in V_{i}$ in the walk.\nWe have, $\\mathrm{wn}(W) = \\mathrm{wn}_i^+(W) + \\mathrm{wn}_i^-(W)$\nwhich in turn implies $c\\cdot\\mathrm{wn}(W) = \\sum_{i} (\\mathrm{wn}_i^+(W) + \\mathrm{wn}_i^-(W))=\\mathrm{height}(W) $.\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n \nBy the previous lemma $\\mathrm{wn}(W)$ is determined already by the c-graph $G$ and is the same in all  clustered drawings of $G$, and hence, putting $\\mathrm{wn}(W):=\\mathrm{height}(W)\/c$, for a walk $W$ with a fixed orientation, allows us to speak about $\\mathrm{wn}(W)$ without\nreferring to a particular drawing of $G$.\n Thus, $\\mathrm{wn}(W)$ tells us the winding number of $W$ in any clustered \ndrawing.\nBy Jordan-Sch\\\"onflies theorem $G$ the following holds. \n\n\\begin{lemma}\n\\label{lem:2}\n$G$ is not c-planar if there exists a face $f$ such that $|\\mathrm{wn}(f)|>1$ or if there exists more than one inner face $f'$ with $|\\mathrm{wn}(f')|=1$.\n\\end{lemma}\n\\begin{proof}\nIn a crossing free drawing  $|\\mathrm{wn}(f)|\\le 1$ for every face $f$.\nIf $|\\mathrm{wn}(f')|=1$ the origin ${\\bf 0}$ lies in the interior of $f'$ since\notherwise the facial walk is null-homotopic, i.e., homotopic to a constant map, in $\\mathbb{R}^2\\setminus {\\bf 0}$ (contradiction). However, interiors of faces are disjoint.\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n If $\\mathrm{wn}(f)=0$ for all faces $f$,~\\cite[Lemma 1.2]{F14+} extends easily to this case,\nreducing the problem to the work of Angelini et al.~\\cite{ADDF13}.\nThus, by Lemma~\\ref{lem:2} and for the sake of simplicity of the presentation, throughout the paper we assume that there exists a pair  of faces $f_o,f_o'$, $\\mathrm{wn}(f_o)=\\mathrm{wn}(f_o')\\not=0$  (by\n~$(*)$ there cannot be just one such face) one of which, let's say $f_o$,\nwe designate as an \\emph{outer face}.  The roles of $f_o$ and $f_o'$ are, in fact, interchangeable.\nAlso such a restriction is by no means crucial in our problem, and alternatively, it is always possible\nto choose and subdivide the outer face in the normalized instance (defined later) by a path so that the restriction is satisfied.\n\n\nViewing a facial walk $f$ as a sequence of vertices and edges $w_0e_0w_1e_2\\ldots e_mw_m$, where $e_{i-1}=w_{i-1}w_i$,\nlet $V_f$ be the set $\\{w_0,\\ldots, w_m\\}$ of \\emph{vertex occurrences} along~$f$.\nWe treat $V_f$ also as a multi-set of vertices, and thus, $\\gamma$ is defined on its elements.\nLet $\\gamma_f:V_f \\rightarrow \\mathbb{N}$, for $f\\not=f_o,f_o'$, be a labeling of the elements of $V_f$ by integers defined as follows.\nWe mark all the vertex occurrences  in $V_f$ as unprocessed.\nWe pick an arbitrary vertex occurrence  $v\\in V_f$, set $\\gamma_f(v):=\\gamma(v)$\nand mark $v$ as processed.\nWe repeatedly pick an unprocessed vertex occurrence  $u\\in V_f$ that has its predecessor or successor $v$ along the boundary walk of $f$ in $V_f$  processed.\n We put  $\\gamma_f(u):=\\gamma_f(v)+g(\\gamma(u)-\\gamma(v))$.\n Intuitively, $\\gamma_f$ records \n the distance  in terms of ``winding around origin'' of vertex occurrences \n along the boundary walk of $f$ from a single chosen vertex occurrence.\n Since $\\mathrm{wn}(f)=0$ the function  $\\gamma_f(u)$ is completely determined by\nthe choice of the  first occurrence of a vertex we processed. \nThis choice is irrelevant for our use of $\\gamma_f$ as we see later.\nAlso notice that $\\gamma(v) = \\gamma_f(v) \\mod c$ for all vertices incident to $f$.\n\nA normalized instance allows only the faces of the types defined next.\nAn element $v$ in $V_f$ is a \\emph{local minimum} (\\emph{maximum}) of a face $f$ if in the  facial walk  $f$ the value of $\\gamma(v)$ is not bigger (not smaller)\nwith respect to the  relation $0<1<\\ldots <c-1<0$  than the value of its successor and predecessor.\nA walk $W$ in $G$ is \\emph{(strictly) monotone with respect to $\\gamma$} if the labels of the occurrences of vertices on $W$ form a (strictly) monotone sequence\nwith respect to the  relation $0<1<\\ldots <c-1<0$ when ordered\nin the correspondence with their appearance on  $W$.\nThe  face $f$ is \\emph{simple} if $f$ has at most one local minimum. It follows \nthat a simple face $f$ has also at most one local maximum.\nThe inner face   $f\\not=f_o'$ is \\emph{semi-simple}  if $f$ has exactly two local minima and maxima and these minima and maxima, respectively,  have the same $\\gamma_f$ value.\n\n\n\\section{Algorithm}\n \\label{sec:alg}\n\nA cyclic c-graph $G$ is \\emph{normalized } if \n\n\\noindent\n(i) $G$ is connected; \\\\\n(ii) each cluster $V_i$ induces an independent set; and \\\\\n(iii) each face of $G$ is simple or semi-simple, and $f_o$ and $f_o'$ are both simple. \n\nSuppose that (i)--(iii) are satisfied. \nBy~(ii) we  put directions on all the edges in $G$ as follows.\nLet $\\overrightarrow{G}$ denote the directed c-graph obtained from $G$ by orienting every edge\n$uv$ from the vertex with the smaller label $\\min (\\gamma(u), \\gamma(v))$ to the vertex with the bigger label $\\max (\\gamma(u), \\gamma(v))$ with respect to the relation $0<1<\\ldots < c-1<0$.\nA \\emph{sink} and \\emph{source} of $\\overrightarrow{G}$ is\na vertex with no outgoing and incoming, respectively, edges.\n\nLet $e$ denote an edge of $G$ not contained in a single \ncluster.\nGiven a clustered embedding $\\mathcal{D}$ of $G$ let ${\\bf p_{e}}:={\\bf p_{e}}(\\mathcal{D})$ denote the intersection point of $e$ with a ray separating a pair of clusters.\nLet $e_0,\\ldots,e_{k-1}$ be  the  edges\n incident to a sink or source $u$.\nBy Jordan curve theorem it is not hard to see that (i)--(iii) imply that a clustered embedding $\\mathcal{D}$ of $G$ is ``combinatorially'' determined once we order the set \n  of intersection points\n ${\\bf p_{e_0}}, \\ldots , {\\bf p_{e_{k-1}}}$ along\n rays separating clusters for every sink and sources $u$ in $G$. Moreover, the set of intersection points corresponding to a sink or source $u$ admits in\n an embedding only orders that are cyclic shifts of\n one another, since we have the rotations at vertices \n of $G$ fixed.\n The wedge in $\\mathcal{D}$ formed by a pair of edges $e_i$ and $e_{i+1}$ incident to a face $f$ at its local extreme $u$ is \\emph{concave} (see Figure~\\ref{fig:wedges} \n for an illustration) if \n$u$ is a sink or source of $\\overrightarrow{G}$ and\nthe line segment ${\\bf p_{e_i}}{\\bf p_{e_{i+1}}}$ contains all the other points ${\\bf p_{e_j}}$ or\nin other words the order of intersection points\ncorresponding to $u$ is ${\\bf p_{e_{i+1}},p_{e_{i+2}}, \\ldots, p_{e_{k-1}},p_{e_{0}}, \\ldots , p_{e_i}}$.\nA non-concave wedge is \\emph{convex}.\\iflong\\footnote{The terminology comes from the fact that in an embedding that is ``combinatorially the same'' with straight line edges, wedges become convex and concave in the usual sense.}\\fi\nNote that in $\\mathcal{D}$ every sink or source is\nincident to exactly one concave wedge that in\nturn determines the order of intersection points.\nThus, combinatorially $\\mathcal{D}$ is also determined\nby a prescription of concave wedges at sink and sources.\n\nLet  $S$ be the set of sinks and sources of $\\overrightarrow{G}$. Let $F$ denote the union of the set of \nsemi-simple faces of $G$ with a subset of $\\{f_o,f_o'\\}$ containing faces incident to a sink and a source.\nWe construct a planar bipartite graph $I=(S\\cup F, E(I))$ with parts $S$ and $F$,\n where $s\\in S$ and $f\\in F$\nis joined by an edge if $s$ is incident to $f$. \nGiven that (i)--(iii) are satisfied, the existence of a perfect matching $M$ in $I$ is a necessary condition for $G$ being c-planar. Indeed, as we just said,\nin a clustered embedding, each source or sink has exactly one of its wedges concave.\nOn the other hand, by Jordan curve theorem  it  can be easily checked that in the clustered embedding  \\\\\n{\\bf (A)} every semi-simple face is incident to exactly one concave wedge \\\\\n{\\bf (B)} faces $f_o$ and $f_o'$ are incident to one concave wedge if they are incident to a sink and source, and \\\\ {\\bf (C)} all the other faces are not incident to any concave wedges at the minimum and maximum. \\\\\nThis is fairly easy to see if $G$ is vertex two-connected, see Figure~\\ref{fig:semiSimple} for an illustration.\nThe cycle $C$ \\emph{corresponding to a closed walk} is obtained\nby traversing the walk and introducing a new vertex for each\nvertex occurrence in the walk.\nFor a face $f$ incident to cut-vertices, {\\bf (A)--(C)} follows by considering the cycle corresponding to the facial walk of $f$ (treated as a face) embedded in a close vicinity of the boundary of $f$. \n Thus, a desired matching $M$ is obtained by matching each source or sink with\nthe face incident to its concave wedge.\n\n \nWe show in Section~\\ref{sec:const}  that if $M$ exists $G$  is c-planar by augmenting \n$G$ with edges as described in Section~\\ref{sec:out}.\nTesting the existence, but even counting perfect matchings in a planar bipartite graph can be carried out in a polynomial time~\\cite[Section 8]{L09}.\n\n\nThe running time of our algorithm is $O(|V|^2)$ since finding the perfect matching \ncan be done in $O(|V|^2)$  time, due to $|E(I)|=O(|V|)$, and the pre-processing step including the construction of $I$ and the normalization will be easily seen to have this time complexity. Also computing the winding number for all the faces can be performed in a linear time by Lemma~\\ref{lemma:wn}.\nFirst, we explain and prove the correctness of the algorithm for\n instances satisfying~(i)--(iii). In Section~\\ref{sec:norm},\n we show a polynomial-time reduction of the general case  to instances  satisfying~(i)--(iii). We often use Jordan--Sch\\\"onflies theorem without\n explicitly mentioning~it.\n\n\n\\iflong\\else\n\\fi\n\\subsection{Constructing a clustered embedding}\n \\label{sec:const}\n \n\\begin{wrapfigure}{r}{.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.65]{fuk}\n\\caption{Subdividing a semi-simple face   (left). Subdividing a simple face $f_o'$ (right).}\n\n\\label{fig:figfig}\n\\end{wrapfigure}\n\n \n Given a normalized instance $G$ and a matching $M$ between sources and sinks in $S$, and  faces in $F$ of $G$ we construct a  clustered embedding of $G$ as follows. Recall that we assume that $G$ does not have a face \n $f$ with $|\\mathrm{wn}(f)|>0$ besides $f_o$ and $f_o'$.\n  We start with $\\overrightarrow{G}$ defined\nabove and add edges to it thereby eliminating all the sinks and sources, see Figure~\\ref{fig:figfig}.\nLet $u\\in S$ be a source matched in $M$ with $f$. If $f$ is a semi-simple inner face\nlet $u'$ denote another local minimum  incident to $f$. We add to  $\\overrightarrow{G}$\n  an edge $\\overrightarrow{u'u}$ embedded in the interior of $f$. \n  If $f=f_o$ or $f=f_o'$ we join $u$ by $\\overrightarrow{u'u}$ with the vertex in the same cluster $u'$\nso that we subdivide $f$ into two simple faces $f'$ and $f''$ such that\n$\\mathrm{wn}(f')=0$ and $\\mathrm{wn}(f'')=\\mathrm{wn}(f)$. If $f=f_o$  face $f''$ is the new outer face. By Lemma~\\ref{lemma:wn}, such a vertex $u'$ exists and it is unique.\n\n\n\nWe proceed with $u\\in S$ that are sinks analogously thereby eliminating all the sinks and source in the resulting graph $\\overrightarrow{G'}$, where by $G'$ we denote its underlying undirected graph.\nBy Lemma~\\ref{lemma:wn}, there still exists exactly one inner face $f_o'$ with a non-zero winding number in the resulting graph $G'$.\n\n\n\n\\begin{lemma}\n\\label{lemma:keyFact0}\n$G'$ has exactly one inner face $f_o'$ such that $|\\mathrm{wn}(f_o')|=1$.\n\\end{lemma}\n\nSince $\\gamma(v) = \\gamma_f(v) \\mod c$ for every face $f\\not=f_o,f_o'$ and $v$ incident to $f$, \nevery edge we added joins a pair of vertices in the same cluster.\n\n\n    \\begin{lemma}\n\\label{lemma:keyFact1}\nThe induced sub-graph $G'[V_i]$ of (undirected) $G'$ does not contain a cycle for $i=0,1,\\ldots, c-1$.\n\\end{lemma}\n\\begin{proof}\nFor the sake of contradiction suppose that a cycle $C$ is contained in $G'[V_{j'}]$.\nLet us choose $C$ such that the area of its interior is minimized.\nSince $G[V_{j'}]$ is an independent set all the edges of $C$ are newly added.\nThus, by looking at the rotation of an arbitrary vertex $v'$ of $C$ we see that $v'$ is incident to a vertex \n $v$ from  $V_j$, $j\\not=j'$,  in the interior of $C$. Indeed, no two edges of $C$ subdivide the same face of $G$.\n \n \n\n\n\n \nUsing the fact that $\\overrightarrow{G'}$ does not contain any source or sink, we show that \na vertex $w$ in the interior of $C$ belongs to an oriented cycle $C'$ (by chance also directed in $\\overrightarrow{G'}$), whose interior is contained in the interior of $C$   \nsuch that $\\mathrm{wn}(C')>0$.\nThe cycle $C'$ is obtained by following a directed path in $\\overrightarrow{G'}$ (from which it inherits its orientation)\n passing through  $v$.\nEither both ends of the path meet each other, they both meet $C$, or the path meet itself in the interior. \nIn the first two cases we can take $w:=v$ in the last case it can happen that the directed path gives rise to a cycle $C'$ not containing $v$. However, $C'$ is not induced by a single cluster by the choice of $C$, and thus, $\\mathrm{wn}(C')>0$ by Lemma~\\ref{lemma:wn} and $C'$ contains a vertex $w$ from $V_j$.\n Let $F'$ denote the set of faces\nin the interior of $C$ and not in the interior of $C'$. \nIn all cases it can be seen   by Lemma~\\ref{lemma:wn} that $\\mathrm{wn}(C')>0$.\n\nIndeed, as we proved in  the proof of Lemma~\\ref{lemma:wn} \n$\\mathrm{wn}(C') = \\mathrm{wn}_j^+(C') + \\mathrm{wn}_j^-(C')$.\nSince $C'$ follows a directed path and is not induced by a single cluster we have $\\mathrm{wn}_j^+(W)>0$ and $\\mathrm{wn}_j^-(W)=0$.\nHence, $\\mathrm{wn}(C') = \\mathrm{wn}_j^+(C') + \\mathrm{wn}_j^-(C')>0$. \n\n\n\n By~(*) it follows that $C'$ contains the unique\ninner face with a non-zero winding number in its interior.\nThen Lemma~\\ref{lemma:keyFact0} with~(*) yields the following  contradiction\n$$0=\\mathrm{wn}(C)=\\mathrm{wn}(C')+\\sum_{f\\in F'} \\mathrm{wn}(f)=\\mathrm{wn}(C')\\not=0$$\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\nLet $E'\\subseteq \\bigcup_i{V_i \\choose 2}\\setminus E(G')$ such that each edge in $E'$ can be added to \nthe embedding of $G'$ without creating a crossing or increasing the number of inner faces with a non-zero winding number.\nWe do not put any direction on the edges in $E'$.\nSince every inner face $\\not=f_o'$ in $G'$ is simple, and its outer face and the face $f_o'$ are not adjacent\nto a source or sink, all the edges in $E'$ can be \nintroduced simultaneously without creating a crossing. \nIn particular, no edge of $E'$ subdivides $f_o'$ or the outer face.\nLet $E''$ denote a maximal subset of $E'$\nthat does not introduce a cycle in $(G'\\cup E'')[V_i]$ for every $i=0,1,\\ldots, c-1$ (see Figure~\\ref{fig:simpleFace}), where $G' \\cup E'' = (V(G'), E(G') \\cup E'')$.\nBy Lemma~\\ref{lemma:keyFact1}, $E''$ is well-defined.\n\n\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.65]{simple_face}\n\\caption{A simple face $f$ of $G'$ (left). The face $f$ subdivided with edges of $E''$ (right). Labels at vertices are their $\\gamma$ values (or indices of their clusters).}\n\\label{fig:simpleFace}\n\\end{figure}\n\n\n\n\n\n\\begin{lemma}\n\\label{lemma:keyFact2}\n  $(G'\\cup E'')[V_i]$  is a tree for $i=0,1,\\ldots, c-1$.\n  \\end{lemma}\n\\begin{proof}\nSuppose for the sake of contradiction that $(G'\\cup E'')[V_i]$ for some $i$ is not a tree, and thus, it is just a  forest with more than one connected component.\nIt follows that either (1) there exists a cycle in $(G'\\cup E'')[V\\setminus V_i ]$\ncontaining a vertex $v$ of $V_i$ in its interior\nor (2) a pair of vertices of $V_i$ in different\nconnected components of $(G'\\cup E'')[V_i]$  are incident to the same face of $(G'\\cup E'')$.  \nThe claim (1) or (2) implies that there exists a cycle\n$C$ in $(G'\\cup E')[V\\setminus V_i]$ containing a vertex $w$ of $V_i$ in its interior.\nSimilarly as in the proof of Lemma~\\ref{lemma:keyFact1}, by following\na directed path through $v$ we obtain \nan oriented cycle $C'$ (this time not necessarily directed) in $G$,  whose interior is contained in the interior of $C$ with $\\mathrm{wn}(C')>0$ yielding a contradiction.\n\nIndeed, as we proved in  the proof of Lemma~\\ref{lemma:wn} \n$\\mathrm{wn}(C') = \\mathrm{wn}_i^+(C') + \\mathrm{wn}_i^-(C')$.\nSince $C'$ is not induced by a single cluster and follows in the interior of $C$ a directed path, and $C$ does not have any vertex in $V_i$ we have\n $\\mathrm{wn}_i^+(W)>0$ and $\\mathrm{wn}_i^-(W)=0$.\nHence, $\\mathrm{wn}(C') = \\mathrm{wn}_i^+(C') + \\mathrm{wn}_i^-(C')>0$. \n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n\n\n\n\n\nBy Lemma~\\ref{lemma:keyFact2}, every $F_i$ is a tree. Taking a close neighborhood of each such $F_i$ as a disc representing\nthe cluster $V_i$ we obtain a desired clustered embedding of $(G'\\cup E'')$. In the obtained embedding we just delete edges not belonging to $G$ and that concludes the proof of the correctness of our algorithm. \\\\\n\n\n\n\n\n\n\\subsection{Normalization}\n \\label{sec:norm}\n\n \n \n In the present section we normalize the instance so that~(i)-(iii) are satisfied.\nWe argued the connectedness in Introduction, and hence, (i) is taken care of.\n\\iflong \n\nA \\emph{contraction} of an  edge $e=uv$ in a topological graph is an operation that turns\n$e$ into a vertex\nby moving $v$ along $e$ towards $u$ while dragging all the other edges incident to $v$ along $e$.\nBy a contraction we can introduce multi-edges or loops at the vertices.\nWe will also  use the following operation which can be thought of as the inverse operation of the edge contraction\nin a topological graph.\nA \\emph{vertex split} in a drawing of a graph $G$ is an operation that replaces a vertex $v$ by two vertices $v'$ and $v''$\ndrawn in a small neighborhood of $v$ joined by a short crossing free edge so that the neighbors of $v$ are partitioned into two parts\naccording to whether they are joined with $v'$ or $v''$ in the resulting drawing, the rotations at $v'$ and $v''$ are inherited from the\nrotation at $v$, and the new edges are drawn in the small  neighborhood of the edges they correspond to in $G$.\n\n\n\nRegarding~(ii), by a series of successive edge contractions we contract each connected component of $G[V_i]$'s to a vertex.\nWe delete any created loop.\nIf a loop at a vertex from $V_i$ contains a vertex from a different cluster $V_j$, $j\\not=i$, in its interior we know that the \ninstance is not c-planar, since for every $j$ all the vertices in $V_j$ must be contained in the outer face of $G[V\\setminus V_j]$ in a positive instance. This all can be easily checked in  polynomial time.\nOtherwise, a contraction preserves c-planarity of $G$, since\ndeleted empty loops can be introduced in a c-planar embedding of the reduced graph,\nand contracted edges recovered via vertex splits.\nFrom now on we assume that clusters of $G$ form independent sets. \nIt remains to satisfy (iii).\n\\else\nTo achieve~(ii) is fairly standard by contracting components induced by clusters to vertices. \\fi\nThus, it remains to satisfy (iii).\n\n\nWe want to sub-divide a non-simple face $f$  into \na pair of faces one of which is\nsemi-simple by a monotone path $P'$ w.r.t. $\\gamma$.\nLet $uPv$ denote an oriented monotone  sub-walk of $f$ w.r.t. $\\gamma$ joining a local minimum $u$ and maximum $v$ of $f$ minimizing  $|\\mathrm{height}(P)|$.\nLet $vQv'$ denote the oriented  monotone walk with $|\\mathrm{height}(P)|=|\\mathrm{height}(Q)|$ immediately following $P$ on the facial walk of $f$, and let $u'Q'u$ be such walk  immediately preceding $P$ on the facial walk of $f$. Note that $Q$ and $Q'$ exists due to the minimality of $P$ and that we have\n$\\mathrm{height}(Q)=\\mathrm{height}(Q') = - \\mathrm{height}(P)$.\nSimilarly as in~\\cite{F14+} we subdivide $f$ into two faces $f'$ and $f''$\nby a strictly monotone path $v'P'u'$  w.r.t. $\\gamma$. Hence,  $\\mathrm{height}(P) = \\mathrm{height}(P')$.\nWe have $\\mathrm{height}(Q)=\\mathrm{height}(Q') = - \\mathrm{height}(P)= -\\mathrm{height}(P')$.\nThus, by Lemma~\\ref{lemma:wn} if $f$ with $\\mathrm{wn}(f)\\not=0$ is semi-simple we obtain a simple face $f'$ with \n$\\mathrm{wn}(f')\\not=0$ and a semi-simple face $f''$ with $\\mathrm{wn}(f'')=0$ as desired.\nIndeed, $\\mathrm{wn}(f'') = \\mathrm{height}(P') +\\mathrm{height}(Q') + \\mathrm{height}(P)+\\mathrm{height}(Q)=0$\nand $c\\cdot\\mathrm{wn}(f) = \\mathrm{height}(v'P''u') + \\mathrm{height}(Q')+\\mathrm{height}(P) + \\mathrm{height}(Q) =\\mathrm{height}(v'P''u') - \\mathrm{height}(P')= c\\cdot\\mathrm{wn}(f')$. It remains to show the following lemma, since both $f'$ and $f''$ \nare incident to less local minima and maxima than $f$ if $f$ is not semi-simple.\nHence, after $O(|V|)$ facial subdivisions we obtain a desired instance, since $|E(I)|=O(|V|)$.\n\n\\begin{lemma}\n\\label{lemma:norm}\nIf the c-graph $G$ is c-planar then by subdividing $f$ of $G$ by $P'$ into\na pair of faces $f'$ and $f''$, where $f''$ is semi-simple we obtain a c-planar c-graph. Moreover, \n$\\mathrm{wn}(f')=\\mathrm{wn}(f)$ and $\\mathrm{wn}(f'')=0$.\n\\end{lemma}\n\\begin{proof}\nThe second statement is proved above.\nHence, we deal just with the first one.\nLet $e_{u}$ and $e_{u}'$ denote the first edge on $P$ and the last edge on $Q'$, respectively. Let $e_{v}$ and $e_v'$\ndenote the last edge on $P$ and the first edge on $Q$, respectively.\nLet $e_{v'}$ and $e_{u'}$ denote the last edge on $Q$ and the first edge on $Q'$.\nLet  ${\\bf p_{u}=p_{e_{u}}}, \\iflong {\\bf p_{u}'=p_{e_{u}'}}, {\\bf p_{u'}=p_{e_{u'}}}, {\\bf p_{v}}={\\bf p_{e_v}},{\\bf p_{v}'}={\\bf p_{e_{v}'}}\\fi$  and ${\\bf p_{v'}=p_{e_{v'}}}$\n denote the intersection of the edges $e_u,\\iflong e_{u}',e_{u'},e_v,e_v'\\fi$ and $e_{v'}$, respectively,  with a ray separating a pair of clusters.\nLet $\\omega_u$ and $\\omega_v$ denote the wedge between $e_u,e_u'$ and $e_v,e_v'$, respectively, in $f$.\n\n\nWe presently show that subdividing $f$ with $P'$  preserves c-planarity, since a clustered\nembedding without $P'$ can be deformed so that $P'$ can be added to a clustered planar embedding without creating a crossing, while keeping the embedding clustered. \nThis is not hard to see if, let's say $\\omega_v$, is convex and the line segment ${\\bf p_up_{v'}}$ is not crossed by an edge. Since $\\omega_v$ is\nconvex, the relative interior of ${\\bf p_up_{v'}}$ is contained in the interior of $f$. Note that $u'Q'PQv'$ is a sub-walk of $f$ since $f$ is not simple. We draw a curve $C$ joining $u'$ with $v'$ following the walk $u'Q'PQv'$ in its small neighborhood in the interior $f$; we cut $C$ at its (two) intersection points with ${\\bf p_up_{v'}}$ and reconnected the severed ends on both sides by a curve following ${\\bf p_up_{v'}}$ in its small neighborhood thereby obtaining a closed curve, and a curve $C'$ joining  $v'$ and $u'$. Finally, $C'$ can be subdivided by vertices thereby \nyielding a desired embedding of $G\\cup P'$.\nOtherwise, if $\\omega_v$ is concave\nor ${\\bf p_up_{v'}}$ is crossed by an edge of $G$ we need to deform the clustered\nembedding of $G$ so that this is not longer the case.\n\n \\begin{figure}[h]\n  \\centering\n\\centering\\textbf{\u2022}\n{\n\\includegraphics[scale=0.6]{stork}\\textbf{\u2022}\n    \t}\n\n\\caption{A pair of deformations of the clustered embedding of $G$ so that $f$ can be subdivided by $P'$. For the sake of clarity clusters are drawn as horiznotal strips rather than wedges.}\n \\label{fig:stork}\n\\end{figure}\n\n\nBy a \\emph{spur} with the \\emph{tip} $u$ we understand a closed curve  obtained  as  a concatenation of a line segment contained in a ray separating clusters\nand a curve contained in the boundary of $f$ passing through exactly one extreme $u$\nof $f$ such that the curve is longest possible. The \\emph{length} is the spur\nis one plus the number of its crossings with rays separating clusters divided by two.\n If $\\omega_u$ is concave, the vertex $u$ is a tip of a spur whose length is the distance of $u$ to a closest other extreme along the face. Note that both $P$ and $Q'$ must be paths in this case.\nThe rough idea in the omitted part of the proof is that shortest spurs have room around them to be deformed while maintaining the embedding\nclustered such that $P'$ can be added.\nSpurs  are deformed  as illustrated in Fig.~\\ref{fig:stork}. (see Appendix for the rest of the proof)\n\\iflong\n\n\n \\begin{figure}[h]\n  \\centering\n\\centering\n{\n\\includegraphics[scale=0.65]{deform1}\n    \t}\n\n\\caption{Deformation in the case when both $u$ and $v$ have concave wedges incident to $f$ that is indicated by grey.\nThe dashed curve represents the path $P'$ subdividing $f$.\nOn the left,  point ${\\bf p_v'}\\in{\\bf p_{v}p_{u'}}$. In the middle, \n   point ${\\bf p_v'}\\not\\in{\\bf p_{v}p_{u'}}$. On the right, the corresponding\n deformation.}\n\\label{fig:deform1}\n\\end{figure}\n\n\n First, we suppose that $\\omega_u$ is concave.\n W.l.o.g. we assume that ${\\bf p_{v}'}\\not\\in{\\bf p_{v}p_{u'}}$. This holds when $\\omega_v$ is convex,\n Figure~\\ref{fig:deform2}. Otherwise,\nwe exchange the roles of $u$ and $v$, see Figure~\\ref{fig:deform1}. Combinatorially, there are two\ncases depending on whether $v$ is concave, but we treat them\nsimultaneously.\nWe isolate a part of the embedding of $G$ inside a spur represented\nby a topological disc $D$. In order to get a desired deformed clustered embedding of $G$ we define a homeomorphism from $D$ that we use to redraw the corresponding part of $G$  thereby disconnecting some edges\nthat are reconnected in the end.\n Let $D_0$ denote the topological disc bounded by the closed curve\nobtained by concatenating the line segment ${\\bf p_{v}}{\\bf p_{u'}}$ with the parts\nof $P$ and $Q'$ connecting endpoints of ${\\bf p_{v}}{\\bf p_{u'}}$  with $u$. \nWe assume that $v'\\not\\in D_0$ which holds automatically when $\\omega_v$ is concave due to ${\\bf p_{v}'}\\not\\in{\\bf p_{v}p_{u'}}$. \n\n \n \n\n \\begin{figure}[h]\n  \\centering\n\\centering\n\\subfloat[]{\n\\label{fig:deform3}\n\\includegraphics[scale=0.65]{deform3}\n    \t}\n    \t\\hspace{10px}\n\\subfloat[]{\n\\label{fig:deform5}\n\\includegraphics[scale=0.65]{deform5}\t\n    \t}\n\\caption{(a) A ``covering map'' $g$ from a disc onto a spur. (b) Identifying parts of the boundary of the discs $D'$ \naccording to $D$.}\n\n\\end{figure}\n \n\n  If $v$ does not have a concave wedge incident to $f$ it could happen that the boundary of $D_0$ crosses itself. As we will see later due to this reason we cannot simply put $D:=D_0$.\nWe use a ``covering map'' $g$, see Figure~\\ref{fig:deform3}.\nIn the light of Jordan-Sch\\\"onflies theorem,\n  let a topological disc $D$ be a pre-image of a continuous  map  $g:D \\rightarrow D_0$ such that the map $g$ is injective when restricted (in the target) to the embedding of $G$;\n$g$ maps the boundary of $D$ to the concatenation of ${\\bf p_{v}}{\\bf p_{u'}}$\nwith the parts of $P$ and $Q'$ connecting endpoints of ${\\bf p_{v}}{\\bf p_{u'}}$  with $u$;\nand the pre-image (of  parts) of the rays separating clusters\n consists of a union of a connected part of the boundary of $D$ \n (contained in the pre-image of ${\\bf p_{v}}{\\bf p_{u'}}$) and\na set of pairwise disjoint diagonals of $D$.\nTreating $G$ as a topological space let $G|_D:=g^{-1}(G)$.  \n\n If $G$ contains cut-vertices $D$ can have pairs of boundary points identified, see Figure~\\ref{fig:deform5}.\nLet $\\ell_0'$ denote  a line segment contained inside the intersection of the interior of $f$ with a ray separating clusters containing ${\\bf p_v'}$, whose end vertex is very close to ${\\bf p_v'}$, and $\\ell_0'\\subset{\\bf p_v'p_v}$ if and only if $\\omega_v$ is convex. Let $D'$ denote a disc bounded by $\\ell_0'$ and a curve \njoining the endpoints of $\\ell_0'$ following $Q$ towards $v'$ and back in its small neighborhood in the interior of $f$.\nLet $\\ell_0$ be the connected component on the boundary of $D$ in $g^{-1}({\\bf p_vp_{u'}})$.\nLet $\\ell_i$ for $0<i<|\\mathrm{height}(P)|$, denote a connected component of a pre-image by $g$ of (a part of) a ray separating clusters. The segments \n$\\ell_i$'s are indexed by the order of appearance of their endpoints along the boundary of $D$.\nA line segment $\\ell_i$ is just a point if it joins identified boundary points in $D$.\nSimilarly, let $\\ell_i'$ for $0<i<|\\mathrm{height}(P)|$, denote a connected component of the \nintersection of $D'$ with rays separating clusters.\nWe contract $\\ell_i'$ to a point iff $\\ell_i$ is a single point\nand we  contract  interior parts of $D'$ in the correspondence with $D$ and $\\ell_i$'s as illustrated in Figure~\\ref{fig:deform5}. Since $v'\\not\\in D_0$, we have $(g(D)=D_0)\\cap D' = \\emptyset$.\n\n \n \n\n \\begin{figure}[h]\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=0.65]{deform2}\n\\label{fig:deform2}    \t}\n    \t\\hspace{3px}\n\\subfloat[]{\n\\includegraphics[scale=0.65]{deform4}\n\\label{fig:deform4}    \t}\n\n\\caption{(a) Deformation in the case when only $u$ has a concave wedge incident to $f$ that is indicated by grey.\nThe dashed curve represents the desired path subdividing $f$.\nOn the left, the line segment ${\\bf p_{v'}p_{u}}$ crosses  edges of $G$. \nOn the right, the corresponding deformation. (b) The wedge at $u$\nin $f$, that is indicated by grey, is convex.  A sink $s$  in the interior of a spur having the vertex $u$\nas the tip.}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n \\centering\n\\includegraphics[scale=0.7]{surgery}\n\\caption{Re-routing $Q'$ past $u$ along the boundary of $f$.}\n\\label{fig:deform8} \n\\end{figure}\n\nWe map by a homeomorphism $h$ the disc $D$ to $D'$ so that $\\ell_i$\nis mapped to $\\ell_i'$ for all $0\\leq i<|\\mathrm{height}(P)|$\nand so that the endpoint ${\\bf x}$ of $\\ell_0$ for which $g({\\bf x})$ is closer to $\\ell_0'$ is mapped to the point of $\\ell_0'$ closest to (farthest from) \n$g({\\bf x})$ if $\\omega_v$ is concave (convex), see Figure~\\ref{fig:stork} for an illustration.\nWe alter the embedding of $G$ by deleting $g(G|_D)$ and replacing it\nby $h(G|_D)$.\nFinally, we  reconnect the severed end pieces of edges intersecting ${\\bf p_vp_{u'}}$ by curves inside the cluster containing $v$ without\ncreating any edge crossing. \n\n\n\nIf $v'\\in D_0$, we redraw the portion of $G$ contained in the disc $D$ (we override the previous $D$) bounded by the line segment ${\\bf p_{v'}p_u}$\nand the parts of $P$ and $Q$ joining end points of ${\\bf p_{v'}p_u}$  with $v$, see Figure~\\ref{fig:deform6}.\nNote that the boundary of the disc $D$ is non-self intersecting.\nIf $Q'$ does not cross ${\\bf p_{v'}p_u}$ we make ${\\bf p_{v'}p_u}$ crossing free by mapping\nthe part of $G$ in $D$ to a long skinny disc $D'$  (we override the previous $D'$) in the vicinity of $Q'$ and reconnecting the severed edges similarly \nas above.\n\n \\begin{figure}\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=1]{3rdcaseT}\n\\label{fig:deform6}  }\n    \t\\hspace{1px}\n\\subfloat[]{\n\\includegraphics[scale=1]{3rdcase2T}\n\\label{fig:deform7}}\n\\caption{The face $f$ is indicated by grey. (a) The vertex  $v'\\in D_0$ corresponding to the spur with the tip $u$. (b) Corresponding  deformation involving $Q'$ and the dashed path $P'$ subdividing $f$.}\n\\end{figure}\n\n\n\nIf $Q'$ crosses ${\\bf p_{v'}p_u}$ we cannot apply the previous argument since $D \\cap D' \\not=\\emptyset$,\nand thus, we need a different approach.\nWe temporarily add to $G$ a path $u'Q''w$ starting at $u'$, following closely $Q'$ in the interior of $f$\nand ending in a vertex $w$. Let ${\\bf p_w}$ denote the crossing point of the last edge on $Q''$\nseparating clusters with a ray separating clusters. \nWe cut edges at ${\\bf p_{v'}p_u}$ by removing a small $\\epsilon>0$ neighborhood of their crossing points.\nWe reroute the paths $Q'$  (see Figure~\\ref{fig:deform8}) and $Q''$ without crossing an edge of $G$ from their severed ends outside of $D$ past $u$ and along $Q'$ in the interior of $f$ so that we closely follow the boundary of $f$.\nWe  easily avoid creating crossings since we cut edges at ${\\bf p_{v'}p_u}$.\nLet $D''$  be the disc bounded by rerouted parts of $Q'$ and $Q''$ from their first intersections on the way from $u$ and $w$, respectively, with ${\\bf p_wp_{u}'}$ to their common vertex $u'$; and by $\\ell_0''\\subset {\\bf p_wp_{u}'}$ connecting the ends of those parts.\nLet $\\ell_i''$, $0\\leq i < |\\mathrm{height}(P)|-c$, denote the line segments in the intersection of $D''$ with rays separating clusters listed in the order of appearance along the boundary of $D''$.\nWe deform the embedding of $G$ by a mapping $g$ from $D\\setminus (P \\cup Q)$, see Figure~\\ref{fig:deform7} for an illustration, such that $g$ maps the relative interior of $\\ell_0={\\bf p_{v'}p_u}$ to the relative interior of  ${\\bf p_wp_{u}'}$, the parts of $Q'$ and $Q''$ in $D$ to their rerouted counterparts, and for the open line segments $\\ell_i$, $1\\leq i <|\\mathrm{height}(P)|-c$, in the intersection of $D$ with the rays separating clusters we have $g(\\ell_{i+c})\\subset \\ell_i'' \\subset g(\\ell_i)$.\nThe mapping $g$ is then extended to the whole $D\\setminus (P \\cup Q)$ such that (i) $g$ is a homeomorphism when restricted to the interior\nof the slab between $\\ell_i$ and $\\ell_{i+1}$, for all $0\\le i<|\\mathrm{height}(P)|$, where $\\ell_{|\\mathrm{height}(P)|}$ is the \nfinal piece of the boundary of $D''$, and (ii) no edge crossings are introduced, i.e., $g(D|_{G})$ is injective\nand $g(D|_{G}) \\cap G =\\emptyset$. It is not hard to see that $g$ exists. Finally, we reconnect the severed end pieces of edges inside the cluster\ncontaining $u$ and remove $Q''$.\nThe previous deformation is perhaps easier seen as follows. The clustered embedding of $G|_{D}$ can be made  arbitrarily skinny. Thus, for the purpose of  deformation, we can picture that $G|_{D}$ consists just of the part of $Q'$ and $Q''$ in $D$ and a strictly monotone \npath starting at  $u'$ as in Figure\\ref{fig:deform6}. We map $G|_{D}$ as indicated in Figure~\\ref{fig:deform7}.\n \n   \nSecond, if both $\\omega_u$ and $\\omega_v$ are convex we can subdivide $f$ by $P'$ unless ${\\bf p_{v}p_{u'}}$ is intersected by edge(s) of $G$ (we still assume that  ${\\bf p_v'}\\not\\in{\\bf p_{v}p_{u'}}$).\n   However, if this is the case $D_0$ (defined above) contains a sink or source $s$ in its interior, see Figure~\\ref{fig:deform4}. Note that we can assume that $s$ is also a tip of a shortest spur of $f$, and hence, the previous case applies.\\fi\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n\n\t\n\n\n\n\n\n\n\n\n{\\bf Acknowledgment} I would like to thank Jan Kyn\\v{c}l and D\\\"om\\\"ot\\\"or P\\'alv\\\"olgyi for many comments and suggestions that helped to improve the presentation of the result.\n    \n\t\n\t\n\t\n\n\n\n\n \n\n\n \n \n\n \n   \n\n\n\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nAt a high energy hadron or ionized nucleus collider, the dynamics of the collision is overwhelmingly dominated by quantum chromodynamics (QCD).  Because of the Landau pole and confinement of QCD at energies comparable to hadron masses, the momentum exchanged in these hadronic collisions is typically small compared to center-of-mass energies, leading to a production of a large number of observed particles with momenta not far above the QCD scale.  With the high luminosities of the experiments at the Large Hadron Collider (LHC) or the Relativistic Heavy Ion Collider (RHIC), it is not currently technically feasible to record all collision events for later analysis so triggers are employed that flag an event as interesting.  While passing triggers does introduce some bias in the events that are recorded, consideration of all events that pass the various triggers provides an ensemble of events that, as close as possible, probes low-energy QCD in a high-energy collision environment.  Developing an understanding of these so-called minimum bias events has been an active area of research for the past several decades \\cite{Sjostrand:1987su,Durand:1988ax,Butterworth:1996zw,Borozan:2002fk,Ryskin:2009tj}.\n\n\nDespite significant research efforts, a first-principles understanding of minimum bias events is lacking precisely because it exists at the scale for which QCD becomes strongly interacting.  Thus, minimum bias events are typically described by models with many parameters, like that used in modern general-purpose event simulation programs like Pythia \\cite{Sjostrand:2006za} or Herwig \\cite{Bahr:2008dy}.  A hydrodynamics limit is often used to model and interpret collisions of heavy ions which result in extremely high particle multiplicities  \\cite{Kolb:2003dz,Romatschke:2009im,deSouza:2015ena},  providing evidence for production of the quark-gluon plasma (QGP) \\cite{BraunMunzinger:2007zz,Shuryak:2008eq,Nouicer:2015jrf,Pasechnik:2016wkt}.  However, with a model, experimental results can only be interpreted within the context of that model and without a more general framework may obscure other consistent descriptions of data.  What is more, such a description relies on unphysical or unmeasurable parameters (like temperature or chemical potential), and properties of collisions are often described in terms of the unmeasurable impact parameter or overlap of the nuclei. In this paper our philosophy is to propose and explore a framework in which certain collider observables may be captured in a model independent fashion, in terms of purely physical\/measurable quantities.\n\nOne instance where not being tied to a particular model(s) may be particularly useful is in the study of the relatively newly discovered phenomena of collective behavior in small systems such as pp collisions at the LHC and pA collisions at RHIC (see, e.g., \\Ref{doi:10.1146\/annurev-nucl-101916-123209} for a review). For instance, in order to disentangle evidence of the production of small `droplets' of QGP from other physics that leads to particle correlations, it would be desirable to have a framework that can interpolate between different system sizes, and collision energies.\n\nThe basic idea of the model independent approach we explore is to work directly with the measured particles, and  constrain the $S$-matrix of the minimum bias events. This philosophy  aligns with a bootstrap approach to QCD which is actively being pursued for low-multiplicity processes in quantum field theory \\cite{Paulos:2016fap,Paulos:2016but,Paulos:2017fhb,Cordova:2018uop,Mazac:2018mdx,Mazac:2018ycv,Cordova:2019lot,Karateev:2019ymz,Correia:2020xtr, Homrich:2019cbt, Komatsu:2020sag,Caron-Huot:2020adz,Guerrieri:2021tak}. The nature and high-multiplicity of minimum bias events offers some simplifications by appealing to the principle of ergodicity: a given particle is representative of any particle in the event. Further, events with $N$ and $N+1$ particles have approximately equal descriptions, and thus we can expect to be able to expand in the multiplicity parameter $N$.\nIn the absence of a first-principles understanding, we look for a detector-level effective  description of minimum bias for which there exists concrete power counting and symmetries that guide the description and has parameters that can be fit to data that are then ultimately described in QCD. \n\nAt a hadron collider, there are always particles that are unobservable as they go straight down the beampipe. We show that  integrating out these particles from the description of events has the effect of smearing the phase space of the observed particles in the detector in a universal fashion, reproducing the flat-in-(pseudo)rapidity distribution argued by Feynman~\\cite{Feynman:1969ej}. The expansion of the matrix element of the observed particles then provides fluctuations about this flat distribution. \n\nImportantly, the approach is applicable to observables that are binned in $N$. That is, it does not provide a first-principles description of  fluctuations in multiplicity. Thus the kind of statements that can be made are, for example, how {\\it normalized} distributions, binned in $N$, may change as a function of $N$. We also assume a fixed collider energy, $Q$, when taking the large $N$ limit, in the sense that we do not consider a `t Hooft coupling-like limit of taking $Q,N\\to\\infty$ at fixed $Q\/N$; statements about the scaling with $Q$, however, are obtained.\n\nOur aim is to explore concrete, quantitative predictions that must follow from consistency of the above-described approach to minimum bias.  We first define the power counting scheme for minimum bias events and identify its symmetries exclusively in terms of measurable quantities, just like the starting point of any effective field theory (EFT).  Through this procedure, the relevant expansion parameter we identify is novel and unlike familiar EFTs.  We formally take the number of observed particles $N\\gg 1$, which is similar to a hydrodynamics limit.  From the general expression for the cross section for production of $N$ final state particles in minimum bias events, we expand the squared matrix element in symmetric polynomials of the phase space coordinates, ordered in powers of $1\/N$.  Among predictions that we validate in collider data include:\n\\begin{itemize}\n\n\\item In the $N\\to\\infty$ \\ limit the symmetries of minimum bias events and the central limit theorem require that the squared matrix element is exclusively a function of the squared total energy of the observed final state particles.\n\n\\item Inspired by the soft and collinear singularities of perturbative QCD, we show that the distribution of particle transverse momentum is universal and depends on a single parameter in the large-$N$ limit.  Further, the transverse momentum distribution implies that particles in minimum bias events have a fractional dispersion relation $\\omega \\propto k_\\perp^{2\/3}$.\n\n\\item By positivity of the squared matrix element, we demonstrate that all pairwise particle azimuthal correlations necessarily vanish as $N\\to \\infty$ at fixed center-of-mass collision energy. \n\n\\item Long-distance pairwise particle pseudorapidity correlations are a consequence of factorization of the squared matrix element as $N\\to\\infty$.\n\n\\item Non-trivial pairwise particle azimuthal correlations are highly suppressed in minimum bias events in $e^+e^-$ collisions because of the full Lorentz invariance of the observed final state particles.\n\n\\end{itemize}\nThese results and others suggest that development of an effective theory for minimum bias would describe a wide range of phenomena and could be matched to perturbative QCD or hydrodynamics.\n\nThis paper is organized as follows.  In \\Sec{sec:ppmin}, we establish the power counting for minimum bias in identical hadron or heavy ion collisions and the corresponding symmetries that those events enjoy.  We show how to construct the squared matrix element in terms of symmetric polynomials of the phase space coordinates.  In \\Sec{sec:datacomp}, we use this effective form of the minimum bias cross section to interpret collider data.  We show that this large-$N$ expansion can provide quantitative and precise predictions that agree with data in its realm of applicability.  In \\Sec{sec:other}, we describe the power counting and symmetries for different collider environments.  Because minimum bias events in $e^+e^-$ collisions enjoy extensive symmetry, we show that these symmetries strongly suppress pairwise particle azimuthal correlations in the large-$N$ limit.  In \\Sec{sec:concs}, we summarize our findings and discuss future avenues to further develop an effective theory of minimum bias.\n\n\\section{Power Counting and Symmetries of pp\/AA Collision Min-Bias}\\label{sec:ppmin}\n\nWe first establish the power counting and symmetries we employ to describe minimum bias events.  This will correspondingly precisely define what we theoretically mean by ``minimum bias'', which is distinct from the experimental definition.  We will focus on minimum bias events at a hadron collider, in particular, proton-proton (pp) or identical heavy ion (AA) collisions.  We assume that the lab frame is also the center-of-mass frame of the collisions.  The center-of-mass collision energy $Q$ is assumed to be much larger than the masses of the initial, colliding particles.  Throughout this paper, we assume that $Q$ is an arbitrary, but fixed, energy, but will occasionally comment on the dependence of distributions of observables as a function of $Q$.  Additionally, we assume that the experimental detector cannot measure all particles and that there is an unmeasureable beam region.  Concretely, we assume that there is a maximum pseudorapidity $\\eta_{\\max}$ of the detector and particles with pseudorapidity $|\\eta|>\\eta_{\\max}$ are lost down the beampipe.  Finally, we assume that our detector can only measure particle momenta, but has perfect angular and energy resolution.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14cm]{.\/detector}\n\\caption{Schematic illustration of detector (cylinder), its pseudorapidity range $\\eta_{\\max}$, the beam regions around $\\eta = \\pm\\infty$, and the characteristic transverse momentum of detected particles.\n\\label{fig:detect}\n}\n\\end{center}\n\\end{figure}\n\nWith these assumptions, we can establish power counting that defines minimum bias events and the corresponding symmetries of our collision experiment and of the detected final state particles.  The power counting that we take is as follows.\n\\begin{enumerate}\n\n\\item The beam is a small angular region outside the detection apparatus and we restrict our description of the event to far from the beam region, where detected particle pseudorapidity satisfies $|\\eta|\\sim 1\\ll \\eta_{\\max}$.\n\n\\item  We assume that the mass of the particles is irrelevant and so detected particle transverse momentum $p_\\perp$ is parametrically larger than the QCD scale or pion mass, $p_\\perp \\gg m_\\pi$. \n\n\\item The momentum lost down the beam region is an order-1 fraction of the center-of-mass energy $Q$. \n\n\\item The number of detected particles $N$ for which their pseudorapidity $|\\eta|\\ll\\eta_{\\max}$ is large: $N\\gg1$.\n\n\\item We assume that the mean transverse momentum of the detected particles is representative of all particles' momenta and so the mean and the root mean square momenta are comparable: $\\langle p_\\perp\\rangle\\sim \\sqrt{\\langle p_\\perp^2\\rangle}$.\n\\end{enumerate}\nWe will see below how these power counting assumptions then imply that our expansion parameter is $1\/N \\sim p_\\perp\/Q\\ll 1$.\\footnote{Actually, as we will see in the explicit examples below, there is a (dominant) additional fixed factor of $(Q\/p_{\\perp\\text{cut}})^\\frac{1}{2}$ in this scaling relation, with $p_{\\perp\\text{cut}}$ the experimental cut on transverse momentum.  This factor is fixed over an ensemble of events and we absorb it into `$\\sim$' here and in the following section when evaluating the scaling in $1\/N$ of terms in the expansion.} Additionally, these power counting assumptions imply that the minimum bias events we consider satisfy something like an ergodic hypothesis.  That is, we assume that any individual particle in an event is representative of all particles in an event.  Thus, averages over particles in an event are equivalent to averages of events over an ensemble.  This ergodic hypothesis will have important implications for the structure of the squared matrix element in the $N\\to\\infty$ limit.  An illustration of the physical configuration established by these power counting assumptions is provided in \\Fig{fig:detect}.\n\nHadron collision events that satisfy these power counting requirements then enjoy the following symmetries:\n\\begin{enumerate}\n\n\\item O(2) rotation and reflection symmetry  about the beam,\n\n\\item reflection of the beam $\\eta\\to-\\eta$ because of the identical colliding particles,\n\n\\item $S_N$ permutation symmetry in all $N$ detected particles, and\n\n\\item translation symmetry in pseudorapidity, $\\eta\\to \\eta+\\Delta \\eta$.\n\\end{enumerate}\nMost of these symmetries should be apparent as a consequence of the familiar cylindrical shape of particle detectors.  Because we assume only particle momenta are measured, no distinguishing information of particles (like electric charge) is collected, and hence there is complete permutation symmetry. By our power counting, we assume that the beam region is defined by $\\eta_{\\max}\\gg1$.  Taking the formal limit $\\eta_{\\max}\\to\\infty$, a translation in $\\eta$ by a finite amount $\\Delta \\eta$ can never move particles out of the detected region into the beam region, or vice-versa.  \n\nWe then define minimum bias events in hadron collisions as those that satisfy the power counting assumptions, and therefore inherit the symmetries.  Experimentally, minimum bias events are typically defined as those that pass a minimal trigger threshold for, say, activity in a forward calorimeter.  As the name suggests, passing a trigger does bias the event selection somewhat, but this bias can potentially be reduced by performing other cuts on these events.  For example, to force an event into the regime described by our power counting, one could require that a fixed fraction of particles have a transverse momentum within some range about the mean transverse momentum.  Or, one could require that an event does not have any identified jets with transverse momentum larger than some fraction of the center-of-mass energy.  We will return to jets below, and explicitly demonstrate that the existence of hard jets in the final state violates our power counting.\n\nBecause of the high luminosity of modern hadron colliders, truly zero bias events can collected by exploiting pile-up, or secondary hadron collisions per bunch crossing.  A hard hadron collision that produces jets or high-energy leptons, for example, would pass the triggers, but then secondary hadron collision events in the bunch crossing would have no requirements placed on them, and still be measured.  The challenge with extracting these zero bias events is then pushed to the ability to distinguish the point of collision where different particles were produced.  Any number of our power counting assumptions can be relaxed to provide a more realistic description of these events; however, we will find that even these strongly constraining assumptions will be able to explain and understand a wide breadth of data in various collider environments.\n \n\\subsection{Effective Form of the Cross Section}\n\nWe will use the power counting and symmetries to provide an effective description of the scattering cross section for these minimum bias events exclusively in terms of properties of the measured particles.  To derive this description, we start from the completely general formula for the cross section, considering $2\\to N+N_{B_1}+N_{B_2}$ scattering events, where $N_{B_1}+N_{B_2}$ particles are beam remnants; i.e., $N_{B_1}$ particles have pseudorapidity $\\eta > \\eta_{\\max}$, $N_{B_2}$ particles have $-\\eta > \\eta_{\\max}$, and $N$ particles have $|\\eta|< \\eta_{\\max}$.  The cross section $\\sigma$ for this scattering can be expressed in generality as\n\\begin{align}\n\\sigma &\\sim \\int d\\Pi_{N+N_{B_1}+N_{B_2}}\\, |{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N+N_{B_1}+1,\\dotsc,N+N_{B_1}+N_{B_2})|^2 \\nonumber\\\\\n&\\sim  \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, \\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,\n\\nonumber\\\\\n&\n\\hspace{0.5cm}\\times|{\\cal M}(1,\\dotsc,N+N_{B_1}+N_{B_2})|^2\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^-- \\sum_{k=1}^{N_{B_2}} k_k^-\n\\right)\\nonumber\\\\\n&\\hspace{0.5cm}\n\\times \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^+- \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}+\\sum_{j=1}^{N_{B_1}} \\vec p_{\\perp j}+\\sum_{k=1}^{N_{B_2}} \\vec p_{\\perp k}\n\\right)\\,.\n\\end{align}\nIn writing this expression, we have ignored overall factors of the center-of-mass energy $Q$ and powers of $2\\pi$.  In the first line, we implicitly write the cross section as the integral of the squared matrix element over $N+N_{B_1}+N_{B_2}$-body phase space, and then in the remaining lines, expand out the massless Lorentz-invariant integration measure and momentum-conserving $\\delta$-functions.  $k^+_i$ and $k^-_i$ are the plus and minus components of particle $i$'s lightcone momentum with respect to the beam axis (the $z$ axis).  The momentum-conserving $\\delta$-functions fix the lab frame to be the center-of-mass frame of the collisions.\n\nNow, with the assumption that $\\eta_{\\max}\\gg 1$, we will formally take $\\eta_{\\max} \\to\\infty$, so the beam remnants live at $\\pm \\infty$ pseudorapidity.  With that assumption, their transverse momenta are 0 and one component of their lightcone momenta is also 0, depending on whether the particles are in beam 1 or beam 2.  These assumptions simplify the momentum conserving $\\delta$-functions and the cross section becomes  \n\\begin{align}\n\\sigma &\\sim   \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, \\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,|{\\cal M}(1,\\dotsc, N+N_{B_1}+N_{B_2})|^2\\nonumber \\\\\n&\n\\hspace{0.5cm}\\times \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^-\n\\right)\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i} - \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,.\n\\end{align}\nIt may seem like the assumption that the beam remnants carry no net transverse momentum is too constraining.  However, assuming that the number of detected particles $N$ is large and the correlations between the detected particles are relatively weak, the net transverse momentum of the detected particles is necessarily small.  The detected particles in an individual event can be imagined to randomly walk in the transverse momentum plane and the average net transverse momentum in an ensemble of events will be 0.  As a random walk, the standard deviation of the net transverse momentum in this ensemble of events scales like $1\/\\sqrt{N}$ at fixed collision energy $Q$.  Then, to leading approximation in the large-$N$ limit, we assume that the net transverse momentum of the detected particles is 0.  It would be interesting to determine the consequences of relaxing this assumption, but we leave that to future work.\n\nBecause we do not measure the particles in the beam region, we want to integrate over them.  With the assumption of permutation symmetry of the particles and the fact that we assume that all beam remnants travel in the exact same direction down either pipe, the only quantity that we can be sensitive to in experiment is their total momentum.  That is, the squared matrix element can only depend on a function of the total plus or minus lightcone momentum that goes down the beampipes:\n\\begin{align}\n&|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\\\\n&\n\\hspace{1cm}\\sim f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, \\frac{|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2}{f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)}\\nonumber\\\\\n&\\hspace{1cm}\\equiv f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\nonumber\\,.\n\\end{align}\nHere, $f$ is some function of the total lightcone momentum that goes in either beam direction.  In the final line, we have expressed squared matrix element with dependence on the measured particles $1,\\dotsc, N$ and dependence on the total lightcone momentum in the beam regions is left implicit.  We will return to the explicit form of this matrix element later. The $\\eta\\to -\\eta$ symmetry of the event requires that the function $f$ is symmetric in its arguments: $f(x,y) = f(y,x)$.  Further, by the $\\eta\\to \\eta+\\Delta \\eta$ translation symmetry, the argument of the function $f$ must be the product of the total plus and minus lightcone momenta, because lightcone momentum components transform homogeneously under boosts along the $z$ axis.  Then, the squared matrix element can be expressed as\n\\begin{align}\n|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\sim f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\,.\n\\end{align}\n\nBy momentum conservation and choosing the collision frame to be the center-of-mass frame, we also know that the total lightcone momentum lost down the beam regions is related to the lightcone momentum of the detected particles.  In particular, \n\\begin{align}\n& \\sum_{j=1}^{N_{B_1}} k_j^- = Q-\\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\\,, &\\sum_{k=1}^{N_{B_2}} k_k^+ = Q-\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\\,.\n\\end{align}\nThus, we can equivalently express the function $f$ exclusively in terms of directly measurable quantities.  We define\n\\begin{align}\n&k^-\\equiv \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\\,, &k^+\\equiv\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\\,,\n\\end{align}\nso that the argument of the function $f$ is just the product $k^+k^-$, by the same reasoning above.  That is, the matrix element can be expressed as\n\\begin{align}\n|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\sim f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\,,\n\\end{align}\nfor some function $f(k^+k^-)$ and our power counting assumes that the momentum lost down the beam regions (or, the $z$-axis boost of the detected particles), is of a comparable size to the center-of-mass energy, $ Q^2$.  \n\nThen, with the power counting and symmetries enforced on the form of the matrix element, in the cross section we can integrate over the momentum lost down the beams.  The cross section can then be expressed as\n\\begin{align}\\label{eq:xsecmaster}\n\\sigma &\\sim \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\,\\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,f\\left(\nk^+k^-\n\\right)\\nonumber \\\\\n&\n\\hspace{1cm}\\times  |{\\cal M}(1,2,\\dotsc,N)|^2\\, \\delta\\left(\nQ-k^- -\\sum_{j=1}^{N_{B_1}} k_j^-\n\\right)\\, \\delta\\left(\nQ-k^+ - \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\n\\nonumber\\\\\n&\\hspace{1cm}\n\\times \\delta\\left(\nk^--\\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\,\\delta\\left(\nk^+-\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\nonumber\n\\\\\n&\\sim  \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2 \\nonumber\\\\\n&\n\\hspace{1cm}\\times \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,.\n\\end{align}\nIn doing the integrals over the unobservable beam remnant particles, we have ignored overall factors of the center-of-mass energy $Q$, and additional dependence on the product $k^+k^-$ has been absorbed in the definition of $f(k^+k^-)$. Because $f(k^+k^-)$ is a physical squared matrix element, we make the reasonable assumption that it is finite and analytic on its domain.  It can therefore be expanded in an appropriate basis of orthonormal polynomials, and this sum can be truncated for approximation to fit data.\n\nIn analysis that we will present later, it will be useful to know the volume of this $N$-body phase space smeared by a boost along the beam axis.  For a center-of-mass energy $Q$, the volume of $N$-body phase space is\n\\begin{equation}\\label{eq:psvol}\n\\int d\\Pi_N = (2\\pi)^{4-3N}Q^{2N-4}\\frac{2\\pi^{N-1}}{(N-1)!(N-2)!}\\,.\n\\end{equation}\nSetting the function $f(k^+k^-)$ and the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$ to unity, the volume of this smeared phase space is therefore\n\\begin{align}\n&\\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\,\\\\\n&\n\\hspace{2cm}\\times \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\\\\n&\\hspace{1cm}=(2\\pi)^{4-3N}\\frac{2\\pi^{N-1}}{(N-1)!(N-2)!}\\int_0^Q dk^+ \\int_0^Q dk^-\\, (k^+k^-)^{N-2}\\nonumber\\\\\n&\\hspace{1cm}=(2\\pi)^{4-3N}Q^{2N-2}\\frac{2\\pi^{N-1}}{(N-1)^2(N-1)!(N-2)!}\\nonumber\\,.\n\\end{align}\nWe also note that the topology of the smeared phase space is that of a $(3N-2)$-ball, found by integrating over two of the dimensions of the $N$-body phase space manifold \\cite{Larkoski:2020thc}.\n\n\\subsection{Expansion of the Squared Matrix Element}\n\nHaving established the form of the cross section for minimum bias events expressed exclusively in terms of observable particle momenta, we now use the power counting and symmetries to expand the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$.  The natural expansion parameter in which to do this according to our power counting is the number of detected particles $N\\gg 1$.  Namely, we would want to determine the momentum dependence of the squared matrix element systematically at every power of $1\/N$, with undetermined parameters that can be fit to data.  We will do this in a few steps.  First, we will present an expansion of the squared matrix element in inverse powers of the center-of-mass energy $Q$, for which terms can be established order-by-order through use of momentum conservation.  From this expansion, we can identify the scaling with $N$ for each of the terms and reorganize the expansion according to our power counting rules.\n\nWe first note that, with this construction, the leading approximation of the squared matrix element is that it is constant.  As we can fix overall normalization later, the leading approximation of squared matrix element in the $1\/Q$ expansion is\n\\begin{equation}\n|{\\cal M}(1,2,\\dotsc,N)|^2 = 1 + {\\cal O}(Q^{-1})\\,.\n\\end{equation}\nA linearly-independent set of momentum-dependent terms at higher powers in $1\/Q$ can be established using momentum conservation.  The transverse and longitudinal momentum conservation equations of the detected particles are:\n\\begin{align}\\label{eq:momconserv}\n0&=\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)^2 = \\sum_{i=1}^N p_{\\perp i}^2 + \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cos(\\phi_i-\\phi_j)\\,,\\\\\nk^+k^-&=\\left(\n\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\left(\n\\sum_{j=1}^N p_{\\perp j}e^{\\eta_j}\n\\right) = \\sum_{i=1}^N p_{\\perp i}^2 + \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cosh(\\eta_i-\\eta_j)\\,.\\nonumber\n\\end{align}\nOn the right, we have expanded the equalities in terms of permutation-symmetric polynomials of the phase space coordinates.  The symmetries of minimum bias events forbid terms at order $1\/Q$ (and actually any odd power of this) and these fundamental momentum conservation equations define a linearly-independent set of terms at order $1\/Q^2$.  We can then write\n\\begin{equation}\n|{\\cal M}(1,2,\\dotsc,N)|^2 = 1+\\frac{c_1^{(2)}}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2+{\\cal O}(Q^{-4})\\,,\n\\end{equation}\nwhere $c_1^{(2)}$ is some dimensionless, numerical coefficient.  We will discuss the potential dependence of $c_1^{(2)}$ on the number of particles $N$ later.\n\nTo construct linear relationships between terms at higher mass dimension, we can multiply the momentum conservation equations by symmetric polynomials of the minimum bias phase space coordinates.  For example, two identities that can be used to establish terms at order $1\/Q^4$ are:\n\\begin{align}\n&\\sum_{i=1}^N p_{\\perp i}^2\\left(\n\\sum_{j=1}^N p_{\\perp j}^2 + \\sum_{j\\neq k}^N p_{\\perp j}p_{\\perp k}\\cos(\\phi_j-\\phi_k)\n\\right) = 0 \\\\\n&\n\\hspace{1cm}=\\sum_{i=1}^N p_{\\perp i}^4 +  \\sum_{j\\neq k}^N p_{\\perp j}^3p_{\\perp k}\\cos(\\phi_j-\\phi_k)+\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cos(\\phi_j-\\phi_k)\\,,\\nonumber\\\\\n&\\sum_{i=1}^N p_{\\perp i}^2\\left(\nk^+k^--\\sum_{j=1}^N p_{\\perp j}^2 - \\sum_{j\\neq k}^N p_{\\perp j}p_{\\perp k}\\cosh(\\eta_j-\\eta_k)\n\\right) = 0 \\nonumber\\\\\n&\\hspace{1cm}=k^+k^-\\sum_{i=1}^N p_{\\perp i}^2-\\sum_{i=1}^N p_{\\perp i}^4 - \\sum_{j\\neq k}^N p_{\\perp j}^3p_{\\perp k}\\cosh(\\eta_j-\\eta_k)-\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_j-\\eta_k)\\,.\\nonumber\n\\end{align}\nAgain, on the right, we have expanded the identities in terms of independent symmetric polynomials.  Then, through the first few orders, the squared matrix element can be expanded as:\n\\begin{align}\n\\hspace{-0.2cm}|{\\cal M}(1,2,\\dotsc,N)|^2 &= 1+\\frac{c_1^{(2)}}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2+\\frac{c_1^{(4)}}{Q^4}k^+k^-\\sum_{i=1}^N p_{\\perp i}^2+\\frac{c_2^{(4)}}{Q^4}\\sum_{i=1}^N p_{\\perp i}^4+\\frac{c_3^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\\\\n&\n\\hspace{0.5cm}+\\frac{c_4^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^3p_{\\perp j}\\cosh(\\eta_i-\\eta_j)+\\frac{c_5^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\cosh(2(\\eta_i-\\eta_j))\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_6^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_i-\\eta_j)\\cosh(\\eta_i-\\eta_k)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_7^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}\\cos(\\phi_i-\\phi_j)+\\frac{c_8^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\cos(2(\\phi_i-\\phi_j))\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_9^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cos(\\phi_i-\\phi_j)\\cos(\\phi_i-\\phi_k)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_{10}^{(4)}}{Q^4}\\sum_{i\\neq  j}^N p_{\\perp i}^2p_{\\perp j}^2\\cosh(\\eta_i-\\eta_j)\\cos(\\phi_i-\\phi_j)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_{11}^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_i-\\eta_j)\\cos(\\phi_i-\\phi_k) + {\\cal O}(Q^{-6})\\,.\n\\nonumber\n\\end{align}\nThe $c_i^{(4)}$ are dimensionless numerical coefficients.  Linear relationships from conservation of momentum have already been accounted for and (hyperbolic) trigonometric identities have been used. Note also that any terms that exclusively depend on the product $k^+k^-$ can be absorbed into the function $f(k^+k^-)$ and are not included.\n\nThe numerical coefficients $c_i^{(n)}$ in general depend on the number of detected particles $N$.  For the most part, we will be agnostic as to what their precise scaling with $N$ is, but we will make the following weak, but constraining, assumption.  We assume that, in the $N\\to\\infty$ limit, the dependence of the coefficients $c_i^{(n)}$ on $N$ is such that the entire corresponding term in the squared matrix element remains finite.  For example, consider the term at ${\\cal O}(Q^{-2})$.  With our power counting assumption that $p_\\perp \\sim Q\/N$ and ergodicity, the kinematic dependence of this term scales like\n\\begin{equation}\n\\frac{1}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2 \\sim \\frac{1}{N}\\,,\n\\end{equation}\nbecause there are $N$ terms in the sum and each term scales like $1\/N^2$.  Thus, the coefficient $c_1^{(2)}$ can scale at worst linear in $N$ as $N\\to\\infty$ and the whole term will still be finite in the squared matrix element.\\footnote{In matching this expansion to a short-distance description from QCD, the coefficients $c_i^{(n)}$ can generically also depend on the ratio of collision energy $Q$ to the QCD scale $\\Lambda_\\text{QCD}$ or pion mass $m_\\pi$.  Explicitly doing the matching or resumming large logarithms of this ratio requires an effective field theory for minimum bias, which we do not construct here and leave to future work.}\n\nFurther, the assumption of ergodicity implies that in the limit that $N\\to\\infty$, the entire matrix element becomes a constant, independent of the event's dynamics.  Our ergodic assumption implies that the symmetric sums over terms that appear in the squared matrix element reduce to the corresponding mean values for $N\\to\\infty$.  For example, consider the term at ${\\cal O}(Q^{-2})$ again.  The sum of the squared transverse momentum of all particles in the event returns the mean square transverse momentum $\\langle p_\\perp^2\\rangle$ times $N$, as $N\\to\\infty$.  Because the sum consists of $N$ terms, the variance of the sum is also of order $N$ and so the whole expression takes the form\n\\begin{equation}\n\\lim_{N\\to\\infty}\\sum_{i=1}^N p_{\\perp i}^2 \\to N\\langle p_\\perp^2\\rangle + {\\cal O}\\left(\\sqrt{N}\\langle p_\\perp^2\\rangle\\right)\\,.\n\\end{equation}\nIn the strict $N\\to\\infty$ limit, the variance of the sum vanishes, reducing to a constant value on all of phase space.  Every term in the squared matrix element reduces to its mean as $N\\to\\infty$ by ergodicity, and so the whole squared matrix element itself becomes a constant on the $N$-body phase space as $N\\to\\infty$.  We will exploit this limit of the matrix element in our predictions of the following section.\n\nWhile constructive, this $1\/Q$ expansion isn't exactly the expansion that is natural with our power counting.  Terms at a fixed order in $1\/Q$ do not in general have a homogeneous scaling with $1\/N$.  For example, at ${\\cal O}(Q^{-4})$, the term\n\\begin{equation}\n\\frac{1}{Q^4}k^+k^-\\sum_{i=1}^N p_{\\perp i}^2 \\sim \\frac{1}{N}\\,,\n\\end{equation}\nwhile the term\n\\begin{equation}\n\\frac{1}{Q^4}\\sum_{i=1}^N p_{\\perp i}^4 \\sim \\frac{1}{N^3}\\,,\n\\end{equation}\nas $N\\to\\infty$.  The particular form of higher-order terms in $1\/N$ is subtle to determine because, in general, terms that arise in the $1\/Q$ expansion don't have a homogeneous scaling in $1\/N$.  For example, assuming that $k^+k^-$ is homogeneous in powers of $N$ with $k^+k^-\\sim N^0$, then from \\Eq{eq:momconserv} the term with hyperbolic cosine is not homogeneous in $N$:\n\\begin{equation}\n\\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cosh(\\eta_i-\\eta_j)\\sim {\\cal O}(N^0) + {\\cal O}(N^{-1})\\,.\n\\end{equation}\nThis follows because of the assumption that $\\eta_i\\sim 1$ and so $\\cosh(\\eta_i-\\eta_j)\\sim 1$, the transverse momentum $p_{\\perp i}\\sim 1\/N$ and that the sum consists of $N^2-N$ terms.  We do not attempt to determine the general form of the $1\/N$ expansion of the squared matrix element here, although a systematic approach could be pursued along the lines of \\Refs{Henning:2015daa,Henning:2015alf,Henning:2017fpj,Henning:2019enq,Henning:2019mcv,Graf:2020yxt,Melia:2020pzd}.  However, what we will do in \\Sec{sec:datacomp}, when using the power counting and symmetries to understand collider data, is identify the $1\/N$ expansion for particular subsets of terms in the squared matrix element that are relevant for the specific measurements of interest that we consider in this work.\n\nOne important point to make at this stage is that we are considering fixed, but large, number of particles $N$.  That is, our expansion can make predictions for observables that survive in the $N\\to\\infty$ limit or for observables that are conditioned on $N$.  We will see a number of examples of such observables in \\Sec{sec:datacomp}.  However, with our present formulation of the cross section for minimum bias, we cannot make predictions for which the relative probability of different numbers of detected particles is important.  Perhaps the simplest such observable is just the multiplicity $N$ of detected particles itself.  We leave an understanding of particle multiplicity within this framework to future work.\n\n\\subsection{Where are Jets?}\n\nJets, while the most ubiquitous phenomena of high-energy QCD, are not described within the framework of our minimum bias expansion of the cross section.  From the perspective of our goals with this expansion, this is a good thing, as it implies that the expansion focuses on the physics of interest, namely the low-energy dynamics of QCD.  Specifically, the presence of high-energy jets in the detected final state particles violates the power counting assumption that the mean particle transverse momentum, $\\langle p_\\perp\\rangle$, is representative of all particles' transverse momentum.  Jets are collimated streams of particles which carry an order-1 fraction of the jet's energy.  Very low energy particles fill in the regions at wide angles from the jets.  The distribution of particle transverse momentum then has two dominant populations corresponding to high energy or very low energy particles.  Of course, for any distribution of $N$ particles with total energy $Q$, the mean transverse momentum $\\langle p_\\perp\\rangle \\sim Q\/N$.  However, when jets are present in the final state, the root mean square transverse momentum $\\sqrt{\\langle p_\\perp^2\\rangle}$ will be dominated by the high energy particles of the jets, and so we find the hierarchy that $\\sqrt{\\langle p_\\perp^2\\rangle}\\gg \\langle p_\\perp\\rangle \\sim Q\/N$.  This then means that jets are not described by any low-order truncation of either the $1\/Q$ or $1\/N$ expansion of the squared matrix element. \n\nOf course, this is not surprising as jets arise because of the approximate scale invariance of QCD at high energies, in addition to the small coupling $\\alpha_s$.  Even approximate scale invariance is not a symmetry of our definition of minimum bias, and so is not exploited in the expansion of the squared matrix element.  Further, the smallness of $\\alpha_s$ for controlling jet dynamics means that jets can be described by a perturbative effective field theory, known as soft-collinear effective theory \\cite{Bauer:2000yr,Bauer:2001ct,Bauer:2001yt,Bauer:2002nz}.  For minimum bias, because the only relevant energy scale is the mean transverse momentum $\\langle p_\\perp\\rangle \\sim Q\/N\\ll Q$, we cannot assume that $\\alpha_s$ is small so as to formulate a perturbative effective field theory.  \n\nNevertheless, even in the presence of collimated jets, the low energy particles that fill the detector away from the jets should exhibit many of the properties we have already discussed.  Namely, as long as the number $N$ of low energy particles not in the jets is large, we still expect their net transverse momentum to be approximately 0, up to the relative $1\/\\sqrt{N}$ standard deviation of a random walk.  However, jets in the final state introduce new, relevant directions that spontaneously break the symmetries of minimum bias as well as introduce correlations that would manifest themselves in the form of the squared matrix element.  As discussed, such correlations would not be captured to low orders in our expansion of the cross section. \n\n\\section{Interpretation of Data}\\label{sec:datacomp}\n\nIn this section, we exploit this expansion of the cross section for minimum bias events to understand and re-interpret collider data from this perspective.  Because the expansion as we have developed it thus far can describe any identical hadron or nucleus scattering, we apply it to understand data from both pp and heavy ion collisions.  Particularly in the case of heavy ion collisions, experimental data are very often interpreted or expressed in terms of strictly unobservable quantities as established in some model of the collision, like the centrality or the number of participating nucleons.  By contrast, our expansion is expressed exclusively in terms of directly observable quantities, like the total number of detected particles or the total observed final state energy. \n\n\n\\subsection{Pseudorapidity Distributions}\n\nThe first observable that we consider is the pseudorapidity distribution of observed final state particles.  This can be calculated directly from the form of the minimum bias cross section we established in \\Eq{eq:xsecmaster}.  From the permutation symmetry of the particles, every particle has the same pseudorapidity distribution $p(\\eta)$, so we can just fix the particle of interest to be particle 1.  Then, we have\n\\begin{align}\np(\\eta)&\\sim \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2 \\\\\n&\n\\hspace{1cm}\\times \\delta(\\eta-\\eta_1)\\, \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\,.\n\\end{align}\nTo proceed, we will make a number of simplifying assumptions.  First, we will work in the $N\\to\\infty$ limit in which the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$ reduces to a constant on phase space, as discussed earlier.  Because this constant is just an overall scaling, we can take it to be 1, for simplicity.  With this assumption, we can then determine the pseudorapidity distribution on flat phase space $p_\\text{flat}(\\eta)$, in the large-$N$ limit.  We have\n\\begin{align}\\label{eq:flatmaster}\np_\\text{flat}(\\eta) &\\sim \\lim_{N\\to\\infty}\\int \\prod_{i=1}^N \\left[\np_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}\n\\right]\\, \\delta(\\eta - \\eta_1)\\\\\n&\n\\hspace{2cm}\n\\times \\delta\\left(\nk^--\\sum_i p_{i\\perp}e^{\\eta_i}\n\\right)\\delta\\left(\nk^+-\\sum_i p_{i\\perp}e^{-\\eta_i}\n\\right)\\, \\delta^{(2)}\\left(\n\\sum_i \\vec p_{i\\perp}\n\\right)\\nonumber\\\\\n&\\propto \\lim_{N\\to\\infty}\\int dp_{\\perp 1}\\, p_{\\perp 1}\\, d\\eta_1\\, \\delta(\\eta - \\eta_1)\\left[\n(k^+-p_{\\perp 1}e^{-\\eta_1})(k^--p_{\\perp 1}e^{\\eta_1})-p^2_{\\perp 1}\n\\right]^N\\nonumber\\\\\n&\\propto\\lim_{N\\to\\infty}\\int  dp_\\perp\\,p_\\perp \\left(\n1-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,p_\\perp\n\\right)^N\\nonumber\\\\\n&=\\int_0^\\infty  dp_\\perp\\,p_\\perp \\, e^{-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,Np_\\perp}\\nonumber\\\\\n&=\\frac{(k^+k^-)^2}{N^2}\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\,.\\nonumber\n\\end{align}\nTo get this expression, we used the large-$N$ limit of the volume of phase space from \\Eq{eq:psvol} and ignore overall factors.  The normalized probability distribution of particle pseudorapidity on flat phase space is then\n\\begin{equation}\\label{eq:flateta}\np_\\text{flat}(\\eta) = 2k^+k^-\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\,.\n\\end{equation}\nIn the large-$N$ limit, this can also be directly derived from assuming that all particles have a distribution flat in $\\cos\\theta$, the polar angle from the beam.  Note that this large-$N$ limit ignores momentum conservation in transforming to the exponential integrand.\n\nWe can then use this result to determine the observed pseudorapidity distribution $p(\\eta)$ smeared against the function $f(k^+k^-)$.  We have\n\\begin{align}\\label{eq:etadistsmear}\np(\\eta)&=\\frac{1}{Q^2}\\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\n\\, p_\\text{flat}(\\eta) \\\\\n&= \\frac{1}{Q^2}\\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\n\\, 2k^+k^-\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\nonumber\\\\\n&=\\int_0^1 dx\\, f(x)\\, \\frac{1-x^2}{1+x^2+2x \\cosh(2\\eta)}\n\\,.\\nonumber\n\\end{align}\nIn the final line, we have set $xQ^2=k^+k^-$ and integrated over the pseudorapidity of the system of final state particles in the lab frame.  Written in this way, $f(x)$ is itself a probability distribution whose normalization is inherited from its expression in terms of $k^+$ and $k^-$:\n\\begin{align}\n1&=\\frac{1}{Q^2}\\int_0^Q dk^+ \\int_0^Q dk^-\\, f(k^+k^-) = \\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\,.\n\\end{align}\nWe also note that because $f(x)\\geq 0$ on $x\\in[0,1]$, the pseudorapidity distribution $p(\\eta)$ monotonically decreases as $|\\eta|$ increases away from $\\eta=0$.  The integral in the final line of \\Eq{eq:etadistsmear} is also dominated by the region for $0\\leq x\\lesssim 1\/\\cosh(2\\eta)$.\n\nTo go further, we need an explicit form for the function $f(x)$.  As we noted earlier, $f(x)$ should be analytic on $x\\in[0,1]$ because it is a physical squared matrix element.  Also, because of the collinear singularity of QCD, we expect a very flat distribution of pseudorapidity over a wide range.  From our expression for the pseudorapidity distribution $p(\\eta)$ in \\Eq{eq:etadistsmear}, we note that if $f(x) = \\delta(x)$, then $p(\\eta)$ is flat, but not normalizable.  So, combining these observations suggests that $f(x)$ should be highly-peaked at $x = 0$ and analytic.  This motivates the following form:\n\\begin{align}\\label{eq:cutffunc}\nf(k^+k^-) = \\frac{n+1}{H_{n+1}}\\left(1-\\frac{k^+k^-}{Q^2}\\right)^n\\simeq \\frac{n}{\\gamma_E+\\text{log}\\, n}\\,e^{-n\\frac{k^+k^-}{Q^2}}\\,,\n\\end{align}\nwhere $n\\gg 1$ ensures strong peaking at $x=0$, $H_n$ is the harmonic number and $\\gamma_E$ is the Euler-Mascheroni constant.\\footnote{We have currently described the function $f(k^+k^-)$ as a function that corresponds to smearing the flat phase space distribution.  However, it can equivalently be interpreted as a squared matrix element $|{\\cal M}|^2$ for the $N$ detected particles using the relationship from \\Eq{eq:momconserv}.  With this interpretation, the squared matrix element is\n\\begin{equation}\n|{\\cal M}|^2 \\sim e^{-n\\frac{k^+k^-}{Q^2}} \\sim \\exp\\left[\n-n\\sum_{i\\neq j}^N \\frac{p_{\\perp i}p_{\\perp j}}{Q^2}\\cosh(\\eta_i-\\eta_j)\n\\right]\\,,\n\\end{equation}\nto leading order in $1\/N$ in the exponent.  Of course, regardless of the interpretation, we find the same predictions for the desired measured quantities.} We discuss the independence of our results on the precise form of this function shortly.\n\nTo determine the value $n$, we must use data.  Pseudorapidity distributions have been measured extensively in the ATLAS \\cite{ATLAS:2011ag,Aad:2012mfa,Aad:2015wga,Aad:2016mok}, CMS \\cite{Khachatryan:2010xs,Khachatryan:2010us,Chatrchyan:2011av,Chatrchyan:2014qka,Khachatryan:2015jna,Chatrchyan:2011pb,Sirunyan:2019cgy} and ALICE \\cite{Aamodt:2009aa,Abbas:2013bpa,Adam:2015kda,Adam:2015pza,Adam:2016ddh,Acharya:2018hhy,Aamodt:2010my,Abelev:2013ala,ALICE:2015qqj,Acharya:2018qsh,Acharya:2018eaq} experiments at the LHC, in both pp and heavy ion collisions.  From those data, we can determine the parameter $n$ by noting the following.  The mean value of $x$ from \\Eq{eq:cutffunc} is roughly $1\/n$, which is the region that dominates the integral.  We call $\\eta_{1\/2}$ where the pseudorapidity distribution is approximately half of its value at $\\eta = 0$ which occurs when $2x \\cosh(2\\eta_{1\/2}) \\sim 1$; or when\n\\begin{equation}\\label{eq:etamaxn}\nn = 2\\cosh(2\\eta_{1\/2})\\,.\n\\end{equation}\nFrom the 8 TeV LHC pp experiment data that follows, for instance, the value of $\\eta_{1\/2}\\sim 6$, and so $n \\sim 1.6\\times 10^5$. The parameter $n$ has implicit $Q$ dependence, at least by the connection to maximum value of $\\eta$.\n\nWe emphasize that in what follows that the fit above to the tail around $|\\eta|\\gtrsim \\eta_{1\/2}$ is not important for the observables that are captured in our approach; that is, any step-like function with a cutoff around $\\eta_{1\/2}$ would suffice.  Kinematic observables binned in $N$ and  at fixed $Q$ in the region of validity of this effective framework are insensitive as $n\\to\\infty$; we will see this explicitly below with the single particle $p_\\perp$ distribution, where we also give a quantitative estimate for the validity condition. In order to correctly capture scalings with $Q$, the value of $n$ is important in that it captures the value of $\\eta_{1\/2}$, which scales with $Q$. Any other step-like function, however, would capture the same scaling and, at least at large enough $Q$, this scaling should be independent of the precise form of the tail.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{.\/eta_large_dist}\n\\caption{Comparison of particle density in pseudorapidity between 8 TeV pp collider data from the CMS and TOTEM experiments \\cite{Chatrchyan:2014qka} to our prediction of \\Eq{eq:etadistsmear} (solid) and the flat phase space distribution of \\Eq{eq:flateta} (dashed).  The data only include charged particles with transverse momentum greater than 40 MeV and our prediction uses the form of smearing function $f(x)$ in \\Eq{eq:cutffunc} with $n = 1.6\\times 10^5$. \n\\label{fig:etacomp}\n}\n\\end{center}\n\\end{figure}\n\nThe above established value of $n$ can then be used to compare our predicted pseudorapidity distribution to data.  This is done in \\Fig{fig:etacomp} where we compare our prediction to data from the CMS and TOTEM experiments \\cite{Chatrchyan:2014qka} of the charged particle number density as a function of pseudorapidity, $dN\/d\\eta$, in 8 TeV pp collisions.  Our prediction for the pseudorapidity distribution is just a normalized probability distribution, so we multiply by a factor to fit data.  As mentioned earlier, our present formulation of the expansion of the minimum bias cross section does not enable us to predict the multiplicity distribution, so we can't predict the overall normalization here.  In general, we find good agreement between our prediction and data, with a noticeable lack of a dip in our prediction near $\\eta = 0$.  Hadrons are of course massive particles and there is a distinction between their rapidity and pseudorapidity which is manifest as the dip in the pseudorapidity distribution.  We assumed that the transverse momentum of particles is much larger than their mass, so we ignore the pseudo\/rapidity distinction.  However, the data include particles with transverse momentum above 40 MeV which includes hadrons with transverse momentum comparable to their mass.\n\n\\subsection{Transverse Momentum Distributions}\n\nWe now turn to understanding transverse momentum distributions in pp collisions.  The set-up for this analysis will be the same as that for pseudorapidity.  We take the squared matrix element for the detected particles $|{\\cal M}|^2 = 1$ and take the large-$N$ limit of phase space.  The first steps are therefore very similar to that for pseudorapidity, so we won't repeat them here.  From \\Eq{eq:flatmaster}, the flat phase space distribution of the transverse momentum is\n\\begin{equation}\np_\\text{flat}(p_\\perp) \\propto p_\\perp \\int_{-\\infty}^\\infty d\\eta\\, e^{-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,Np_\\perp}= p_\\perp \\, K_0\\left(\n\\frac{2N p_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\,,\n\\end{equation}\nwhere $K_0(z)$ is a modified Bessel function.  The unit normalized distribution is\n\\begin{equation}\\label{eq:flatpspt}\np_\\text{flat}(p_\\perp) = \\frac{4N^2 p_\\perp}{k^+k^-}\\, K_0\\left(\n\\frac{2Np_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\,.\n\\end{equation}\nWith this result, the distribution smeared with the function $f(k^+k^-)$ is, in general,\n\\begin{align}\np(p_\\perp)&=\\frac{1}{Q^2} \\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\\, p_\\text{flat}(p_\\perp)\\\\\n&=\\frac{1}{Q^2} \\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\\, \\frac{4N^2 p_\\perp}{k^+k^-}\\, K_0\\left(\n\\frac{2Np_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\nonumber\\\\\n&=\n\\frac{4N^2 p_\\perp}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\n\\nonumber\\,.\n\\end{align}\n\nUnlike the pseudorapidity distribution, the transverse momentum distribution depends explicitly on the number of detected particles $N$.  This number fluctuates event-by-event and we do not predict the multiplicity distribution. So, instead, we will re-write this distribution in terms of the mean transverse momentum, which we can calculate and is unique over an ensemble of collision events.  This mean is\n\\begin{align}\n\\langle p_\\perp \\rangle &= \\int_0^\\infty dp_\\perp\\, p_\\perp \\, p(p_\\perp) =\\frac{4N^2}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x} \\int_0^\\infty dp_\\perp \\,p_\\perp^2\\,  K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\\\\\n&=\\frac{\\pi}{4}\\frac{Q}{N}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\,f(x)\\, \\sqrt{x} \\equiv \\frac{\\pi}{4}\\frac{Q}{N} \\langle \\sqrt{x}\\rangle\n\\nonumber\\,.\n\\end{align}\nSubstituting this expression for $N$, the transverse momentum distribution is then\n\\begin{align}\np(p_\\perp)&=\\frac{\\pi^2 \\langle \\sqrt{x}\\rangle^2}{4\\langle p_\\perp\\rangle^2 }\\,p_\\perp\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{\\pi \\langle \\sqrt{x}\\rangle}{2\\langle p_\\perp\\rangle\\sqrt{x}}\\, p_\\perp\n\\right)\n\\nonumber\\,.\n\\end{align}\nWith $f(x)$ from \\Eq{eq:cutffunc}, the mean transverse momentum is\n\\begin{equation}\\label{eq:meanpt}\n\\langle p_\\perp\\rangle = \\frac{\\pi}{4}\\frac{Q}{N} \\langle \\sqrt{x}\\rangle \\simeq \\frac{\\pi^{3\/2}Q}{8\\sqrt{n}N}\\,,\n\\end{equation}\nin the $n\\to\\infty$ limit. The $p_\\perp$ distribution with this form of $f(x)$ is\n\\begin{align}\\label{eq:perpdistfit}\np(p_\\perp)&=\n\\frac{\\pi^3 p_\\perp}{16 \\langle p_\\perp\\rangle^2}\\frac{1}{\\gamma_E+\\text{log}\\,n }\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, e^{-nx}\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{\\pi^{3\/2}p_\\perp}{4\\langle p_\\perp\\rangle\\sqrt{x n}}\n\\right)\\,.\n\\end{align}\nData of the transverse momentum distribution are often displayed as a number density per phase space volume, and so what we will plot is actually\n\\begin{equation}\n\\frac{1}{2\\pi p_\\perp}\\frac{dN}{dp_\\perp} \\propto \\frac{p(p_\\perp)}{p_\\perp}\\,.\n\\end{equation}\nNote that once we determine $f(x)$ from pseudorapidity data, the prediction of the transverse momentum distribution only depends on a single parameter, $\\langle p_\\perp\\rangle$.  Additionally, a cut on the maximum pseudorapidity of particles that contribute to this distribution can be incorporated, as such a cut is always imposed in data.  However, any such cut that is relevant experimentally has an exceedingly small effect on the transverse momentum distribution, so we will not include it in what follows.\n\nThe reason for effective independence on a pseudorapidity cut is as follows.  In the large $n$ limit, the transverse momentum distribution of \\Eq{eq:perpdistfit} is itself independent of $n$.  The Bessel function has an asymptotic form of\n\\begin{equation}\nK_0(z) \\to \\sqrt{\\frac{\\pi}{2z}}\\,e^{-z}\\,,\n\\end{equation}\nfor $z\\gg1$.  With this approximation, the distribution is\n\\begin{align}\np(p_\\perp)&\\sim\n\\sqrt{p_\\perp} \\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, \\frac{e^{-nx-\\frac{\\pi^{3\/2}p_\\perp}{4\\langle p_\\perp\\rangle\\sqrt{x n}}}}{x^{3\/4}}\\,,\n\\end{align}\nignoring overall constant factors.  Now, with $n\\gg1$, we can saddle-point approximate the exponential.  The value of $x$ for which the exponent factor is minimized is\n\\begin{equation}\nx_{\\min} = \\frac{\\pi}{4n}\\frac{p_\\perp^{2\/3}}{\\langle p_\\perp\\rangle^{2\/3}}\\,.\n\\end{equation}\nJust setting $x$ in the integrand equal to this minimum value, taking $n\\to\\infty$ and ignoring non-exponential factors, we have\n\\begin{align}\\label{eq:asymppt}\np(p_\\perp)&\\sim e^{-\\frac{3\\pi}{4}\\frac{p_\\perp^{2\/3}}{\\langle p_\\perp\\rangle^{2\/3}}}\\,.\n\\end{align}\nAs the value of $n$ in turn determines the maximum value of pseudorapidity according to \\Eq{eq:etamaxn}, as long as $n$ is large enough, any dependence on a pseudorapidity cut is eliminated.\n\n\nThis probability distribution is like the Boltzmann factor for the ``gas'' of $N$ detected particles.  Hence, they have a dispersion relation of $\\omega \\propto k_\\perp^{2\/3}$.  Fractional dispersion relations are very strange, but can arise from integrating out a gapless subsystem of a larger system \\cite{Watanabe:2014zza}.  This is essentially what we did here: the beam remnants were gapless, but we exclusively wanted an effective description of the detected particles.  Integrating out gapless modes introduces non-locality in the effective theory, which is manifested as a non-analytic dispersion relation.  In particular, note that the non-analyticity is not present in the flat phase space distribution of \\Eq{eq:flatpspt}.  The asymptotic form of the Bessel function implies that\n\\begin{equation}\np_\\text{flat}(p_\\perp)\\sim e^{-\\frac{\\pi}{2}\\frac{p_\\perp}{\\langle p_\\perp\\rangle}}\\,,\n\\end{equation}\nignoring overall non-exponential factors.  No gapless modes are integrated out on flat phase space, hence the dispersion relation $\\omega \\propto k_\\perp$ is analytic.  The particular form of the dispersion relation implied by \\Eq{eq:asymppt} may provide {a hint towards the construction of an effective field theory for minimum bias based on spontaneously broken symmetry breaking patterns, but we leave that to future work.}\n\nWe can also verify the consistency of our transverse momentum prediction, within the framework of our assumed power counting.  The second moment of the transverse momentum is\n\\begin{align}\n\\langle p_\\perp^2 \\rangle &= \\int_0^\\infty dp_\\perp\\, p_\\perp^2 \\, p(p_\\perp) =\\frac{4N^2}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x} \\int_0^\\infty dp_\\perp \\,p_\\perp^3\\,  K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\\\\\n&=\\frac{Q^2}{N^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\,f(x)\\,x\n= \\frac{Q^2}{n N^2}\\,,\\nonumber\n\\end{align}\nwhere the final equality is the large $n$ limit of the second moment of the distribution of \\Eq{eq:perpdistfit}.  Then, the root mean square and expectation value of the transverse momentum are related by\n\\begin{equation}\n\\sqrt{\\langle p_\\perp^2 \\rangle} = \\frac{8}{\\pi^{3\/2}}\\langle p_\\perp \\rangle \\simeq 1.44 \\langle p_\\perp \\rangle\\,,\n\\end{equation}\nsatisfying our power counting of $\\sqrt{\\langle p_\\perp^2 \\rangle}\\sim\\langle p_\\perp \\rangle$ and demonstrating consistency of our prediction.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/pt_dist}\n\\caption{Transverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=7$ TeV pp collisions at CMS from \\Ref{Khachatryan:2010us} compared to our prediction.  The data have an $|\\eta|< 2.4$ cut on the charged particles.  Our prediction from the asymptotic expression of \\Eq{eq:asymppt} (solid) uses $\\langle p_\\perp\\rangle = 0.65$ GeV, and also plotted is the distribution on flat phase space of \\Eq{eq:flatpspt} (dashed).\n\\label{fig:ptpred}}\n\\end{center}\n\\end{figure}\n\nWith this prediction for the transverse momentum distribution, we can compare it to collider data.  \\Fig{fig:ptpred} shows the transverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=7$ TeV pp collisions at CMS from \\Ref{Khachatryan:2010us}.  A pseudorapidity cut of $|\\eta|<2.4$ is also imposed, but, as mentioned earlier, that will not matter for our prediction.  This figure also plots the asymptotic form of our prediction from \\Eq{eq:asymppt} where we have set the parameter $\\langle p_\\perp\\rangle = 0.65$ GeV, as well as the transverse momentum distribution on flat $N$-body phase space.  Our prediction is in remarkably good agreement with the data over the plotted range of transverse momentum.  We want to emphasize that once the form of the smearing function $f(x)$ is determined from pseudorapidity data, our prediction for the transverse momentum distribution depends on only one parameter, the mean value of the transverse momentum.\n\nWhile we are unable to directly predict the multiplicity distribution from this framework, we can nevertheless demonstrate consistency between parameters in the expansion of the cross section with measured values of the mean multiplicity.  From \\Eq{eq:meanpt}, the expected value of $N$ with our choice for the form of the function $f(x)$ is\n\\begin{equation}\\label{eq:expmult}\nN  \\simeq \\frac{\\pi^{3\/2}Q}{8\\sqrt{n}\\langle p_\\perp\\rangle}\\,.\n\\end{equation}\nFitting the 7 TeV data from CMS in \\Fig{fig:ptpred} fixes $\\langle p_\\perp\\rangle = 0.65$ GeV, and from the 8 TeV CMS+TOTEM data of the pseudorapidity distribution from \\Fig{fig:etacomp}, we fit $n=1.6\\times 10^5$.  While these data are from different collision energies and so are perhaps not directly comparable and interpretable from one to the other, we only anticipate logarithmic dependence on center-of-mass collision energy, so their distinction should be minimal when applied to our (rather coarse) scaling predictions.  Taking these values along with $Q = 8$ TeV, the expected value of the detected multiplicity from \\Eq{eq:expmult} is then\n\\begin{equation}\nN \\simeq 21\\,.\n\\end{equation}\nWhile this is a very simple and crude prediction, it is nevertheless in the same order-of-magnitude as the number of observed charged particles from \\Fig{fig:etacomp}, for example.  In that figure, the number density of charged particles with transverse momentum greater than 40 MeV is roughly 6.5 per unit pseudorapidity for $|\\eta| \\lesssim 5.5$.  So, there are roughly 72 charged particles in each event.  Our prediction is about a factor of 3 smaller, which is likely accounted for by the transverse momentum cut.  40 MeV is less than the mass of the pion, and so violates an assumption of our power counting.  Increasing this cut would correspondingly decrease the number of detected particles, while not affecting our fit value for $\\langle p_\\perp\\rangle$.\n\nFurther, the expression for multiplicity $N$ from \\Eq{eq:expmult} implies a non-trivial dependence on the center-of-mass collision energy.  First, note that the value of $n$ in the form of the function $f(x)$ is related to the collision energy $Q$ through \\Eq{eq:etamaxn}:\n\\begin{equation}\n\\eta_{\\max}\\simeq \\text{log}\\frac{Q}{p_{\\perp\\text{cut}}} \\simeq \\text{log} \\,n\\,.\n\\end{equation}\nHere, $p_{\\perp\\text{cut}}$ is the experimental lower bound on detected particle transverse momentum.  Then, as long as the dependence of the mean transverse momentum $\\langle p_\\perp\\rangle$ on the collision energy $Q$ is relatively weak, the multiplicity scales with a fractional power of $Q$:\n\\begin{equation}\nN \\sim Q^{1\/2} = s^{1\/4}\\,.\n\\end{equation}\nThis particular fractional power scaling is a prediction of the Fermi-Landau model \\cite{Fermi:1950jd,Landau:1953gs,Belenkij:1955pgn,Wong:2008ex}.  In the context of our analysis here, we note that it is a consequence of our particular choice of the function $f(x)$ in the smeared cross section.  Additionally, the inclusion of a squared matrix element with non-trivial dependence on the detected particles will affect this scaling.  Data prefer a slightly smaller power-law scaling of the multiplicity, e.g., \\Ref{ALICE:2015qqj} in which a power law of $s^{0.11}$ fits the charged particle multiplicity over decades of collision energies.  Nevertheless, this simple result within the context of our large-$N$ expansion suggests that an appropriate squared matrix element could fit the data.  We leave a more detailed analysis of the collision energy dependence of the multiplicity to future work.\n\nData of transverse momentum distributions in minimum bias are often compared to Tsallis distributions \\cite{Tsallis:1987eu,Wilk:2008ue,Biro:2008hz} that assume there are fluctuations in the $N$ particle final state that are quantified by a non-extensive form of entropy.  This is an intriguing interpretation and the success of such models may point to fundamental fractal-like structure of particles produced in minimum bias collisions.  While not inconsistent with this interpretation, our expansion of the minimum bias cross section has a more mundane understanding as a consequence of the symmetries of these collision events.  Further, the Tsallis distribution reduces to a power law at large transverse momentum, while our smeared prediction is exponential, though dependent on a fractional power of transverse momentum.  A detailed study to distinguish the consequences of these two (or other) models of minimum bias dynamics may reveal the microscopic description of these events and lead to an effective field theory in which precision calculations can be performed.\n\n\\subsubsection{Limit of Large-$N$ Expansion}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/highpt_dist}\n\\caption{\nTransverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=2.76$ TeV pp collisions at ATLAS from \\Ref{Aad:2015wga} compared to our asymptotic prediction and a Tsallis distribution.  The data have an $|\\eta|< 2$ cut on the charged particles.  In our prediction of \\Eq{eq:asymppt}, we set the parameter $\\langle p_\\perp\\rangle = 0.5$ GeV and the parameters of the Tsallis distribution are $n=6.6$ and $T = 0.12$ GeV.\n\\label{fig:highptpred}}\n\\end{center}\n\\end{figure}\n\nWith data that extends to higher values of transverse momentum, we can see the limit of our prediction, with the assumptions we have made thus far.  In particular, we have assumed that the squared matrix element is just 1, which is its expression at lowest order in the $1\/N$ expansion.  However, at higher transverse momentum, higher order terms in the squared matrix element become more important and may be necessary to describe the distribution.  In Fig.~\\ref{fig:highptpred}, we compare charged particle transverse momentum distribution data from $\\sqrt{s}=2.76$ TeV pp collisions to our asymptotic prediction of \\Eq{eq:asymppt} and a Tsallis distribution.  This Tsallis distribution takes the form\n\\begin{equation}\n\\frac{1}{2\\pi p_\\perp}\\frac{d\\sigma}{dp_\\perp}\\propto \\left(\n1+\\frac{p_\\perp}{nT}\n\\right)^{-n}\\,,\n\\end{equation}\nfor parameters $n$ and $T$. In the plot, we have set the parameters in our prediction and the Tsallis distribution to be $\\langle p_\\perp\\rangle = 0.5$ GeV, $n=6.6$ and $T = 0.12$ GeV.  At low transverse momentum, both our prediction and the Tsallis distribution follow the data extremely well, but around 5 GeV, our prediction exponentially drops, while the Tsallis distribution largely follows the power-law distribution at high transverse momentum.  It would be interesting to study the effect of maintaining momentum conservation in our prediction and non-trivial terms in the matrix element to see if they can reproduce the high-$p_\\perp$ tail, but we leave that to future work.  \n\nHowever, with the approximations that we have made thus far, we can estimate where our description of the transverse momentum distribution should break down.  In the derivation of our results, an important approximation that we made in the large-$N$ limit was that momentum conservation could be ignored, which in turn resulted in an exponential appearing in the calculation of distributions on flat phase space in \\Eq{eq:flatmaster}.  By carefully analyzing this approximation, we can identify where it  breaks down.  First, the large-$N$ flat phase space factor can be written as\n\\begin{align}\n\\left(\n1-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp\n\\right)^N &= \\exp\\left[\nN\\,\\text{log}\\left(\n1-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp\n\\right)\n\\right]\\\\\n&=\\exp\\left[\nN\\left(\n-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp-\\frac{1}{2}\\frac{(k^+e^\\eta  + k^-e^{-\\eta})^2}{(k^+k^-)^2}p^2_\\perp + \\cdots\n\\right)\n\\right]\n\\nonumber\\,.\n\\end{align}\nSo far, this is exact, but higher-order terms in the exponent can be safely ignored only if\n\\begin{equation}\np_\\perp \\ll \\frac{k^+k^-}{k^+e^\\eta  + k^-e^{-\\eta}} \\,.\n\\end{equation}\nRecall that $k^+k^-$ is the total squared energy of the detected particles.  In coordinates that we used earlier, we expressed\n\\begin{align}\n&x = \\frac{k^+k^-}{Q^2}\\,, &\\Delta\\eta =\\frac{1}{2}\\, \\text{log}\\frac{k^-}{k^+}\\,,\n\\end{align}\nand so the limit on transverse momentum is\n\\begin{equation}\np_\\perp \\ll \\frac{k^+k^-}{k^+e^\\eta  + k^-e^{-\\eta}} = \\frac{\\sqrt{x}\\,Q}{2\\cosh(\\eta - \\Delta\\eta)}\\,,\n\\end{equation}\nwhere $x\\in[0,1]$ and \n\\begin{equation}\n\\frac{1}{2}\\,\\text{log}\\, x \\leq \\Delta\\eta \\leq \\frac{1}{2}\\,\\text{log}\\frac{1}{x}\\,.\n\\end{equation}\n\nTo set an upper bound for the transverse momentum, we first maximize over $x$ and $\\Delta\\eta$, for which $x = 1$ and $\\Delta \\eta = 0$:\n\\begin{equation}\np_\\perp \\ll \\frac{Q}{\\cosh \\eta}\\,.\n\\end{equation}\nThat is, the energy of an individual particle must be parametrically less than the collision energy.  We had found that for pp collisions with energy of several TeV, the maximum pseudorapidity was about $\\eta_{\\max}\\sim 6$.  For $Q = 2.76$ TeV from \\Fig{fig:highptpred}, the transverse momentum then must be smaller than\n\\begin{align}\np_\\perp \\ll \\frac{Q}{\\cosh \\eta_{\\max}} \\sim 14\\text{ GeV}\\,.\n\\end{align}\nThis estimate of the limit agrees well with \\Fig{fig:highptpred}, in which our prediction diverges from data well before 14 GeV.  Conversely, for a fixed transverse momentum $p_\\perp$, the maximum pseudorapidity is logarithmically related to the collision energy $Q$, which is well known \\cite{Feynman:1969ej,Wilson:1970zzb}.\n\n\n\\subsection{Azimuthal Correlations}\n\nIn this section, we discuss azimuthal correlations between pairs of particles produced in pp or heavy ion collisions.  We will focus on the ellipticity and the long pseudorapidity-distance correlations or ``ridge'' phenomena later, but we will first determine the form of the probability distribution of the pairwise azimuthal angle difference within the context of our minimum-bias expansion.  Non-trivial azimuthal correlations require a non-trivial squared matrix element.  In general, the form of the terms in the squared matrix element relevant for azimuthal correlations are\n\\begin{align}\n|{\\cal M}|^2&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{(\\vec p_{\\perp i}\\cdot \\vec p_{\\perp j})^n}{Q^{2n}} \\\\\n&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{p_{\\perp i}^n p_{\\perp j}^n}{Q^{2n}} \\cos(n(\\phi_i-\\phi_j))\\nonumber\\,,\n\\end{align}\nfor some coefficient functions $g_n(k^+k^-,N)$.  On the second line, we have absorbed multiplicative factors in the expansion of $(\\vec p_{\\perp i}\\cdot \\vec p_{\\perp j})^n$ into a redefinition of $g_n(k^+k^-,N)$.  All other terms in the squared matrix element that are independent of azimuthal differences would just contribute to the constant ``1'' term and would therefore just be an overall normalization. With our power counting assumption that $\\langle p_\\perp\\rangle \\sim \\sqrt{\\langle p_\\perp^2\\rangle}\\sim Q\/N$, we replace $p_\\perp \\sim Q\/N$ to establish the scaling with the number of observed particles $N$.  Our squared matrix element then becomes\n\\begin{align}\n|{\\cal M}|^2&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{p_{\\perp i}^n p_{\\perp j}^n}{Q^{2n}} \\cos(n(\\phi_i-\\phi_j))\\\\\n&\\sim1+\\sum_{n=1}^\\infty \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N  \\cos(n(\\phi_i-\\phi_j))\\nonumber\\,.\n\\end{align}\n\nThe maximal scaling with $N$ of the coefficient functions $g_n(k^+k^-,N)$ can be established by demanding that the squared matrix element is non-negative.  To do this, we note that, in the $N\\to\\infty$ limit, transverse momentum conservation is trivially satisfied and so azimuthal angles are uncorrelated and uniformly distributed on flat phase space:\n\\begin{equation}\n\\int d\\Pi_N \\to \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}\\,.\n\\end{equation}\nThus, on flat phase space, the mean of the sum over the cosine factors of the difference of azimuthal angles is 0:\n\\begin{equation}\n\\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}  \\sum_{j\\neq k}^N  \\cos(n(\\phi_j-\\phi_k))=0\\,.\n\\end{equation}\nOn the other hand, the central limit theorem states that the variance $\\sigma^2$ of the sum of cosine factors is\n\\begin{align}\n\\sigma^2 \\equiv \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}  \\left[\\sum_{j\\neq k}^N  \\cos(n(\\phi_j-\\phi_k))\\right]^2 = N^2 \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi} \\, \\cos^2(n(\\phi_1-\\phi_2)) = \\frac{N^2}{2}\\,,\n\\end{align}\nas there are $N^2$ terms in the sum, in the $N\\to\\infty$ limit.  To ensure that the squared matrix element is non-negative, the sum over cosine factors at each value of $n$ must not be significantly negative to overwhelm the constant ``1'' term.  Therefore, we must enforce that the bulk of the possible values of the sum over cosines is less than 1:\n\\begin{align}\n1\\gtrsim \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N  \\cos(n(\\phi_i-\\phi_j)) \\sim \\frac{g_{n}(k^+k^-,N)}{N^{2n}} \\, \\sigma\\sim \\frac{g_{n}(k^+k^-,N)}{N^{2n-1}}\\,.\n\\end{align}\nThat is, the coefficient functions $g_{n}(k^+k^-,N)$ are required to scale with $N$ no greater than\n\\begin{align}\ng_n(k^+k^-,N) \\lesssim N^{2n-1}\\,.\n\\end{align} \nThis property will have important consequences for the large-$N$ predictions of azimuthal correlations.\\footnote{Few general results are known about conditions for positivity of Fourier transforms.  One sufficient condition for positivity of the continuous Fourier transform of a function $u(x)$ for $x>0$ is that it is decreasing and concave-up for all $x>0$ \\cite{tuck_2006,Giraud:2014sba}.  Our simple result based on scaling of terms in the Fourier series is consistent with this result.}\n\nContinuing, we can integrate over the smeared phase space to establish the probability distribution for the pairwise azimuthal angle difference, $\\Delta\\phi$.  We have\n\\begin{align}\\label{eq:azidist}\np(\\Delta \\phi) &\\sim\\frac{1}{Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-)\\, \\int d\\Pi_N |{\\cal M}|^2\\, \\delta(\\Delta\\phi -(\\phi_1-\\phi_2))\\\\\n&\\sim \\frac{1}{Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-) \\int_0^{2\\pi} \\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}\\, \\delta(\\Delta\\phi -(\\phi_1-\\phi_2))\\nonumber\\\\\n&\n\\hspace{1cm}\\times\\left(\n1+\\sum_{n=1}^\\infty \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N \\cos(n(\\phi_i-\\phi_j))\n\\right)\\nonumber\\\\\n&=\\frac{1}{2\\pi}+\\frac{1}{\\pi}\\sum_{n=1}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,.\\nonumber\n\\end{align}\nHere, we have used the permutation symmetry of particles to just define $\\Delta \\phi$ as the difference of the azimuthal angles of particles 1 and 2.  The Fourier coefficients $d_n(N)$ are defined to be\n\\begin{equation}\\label{eq:fcoeffsazi}\nd_n(N) =  \\frac{1}{2Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-)\\, g_{n}(k^+k^-,N)\\,.\n\\end{equation}\nNote that the maximal $N$ scaling of the coefficients $d_n(N)$ is inherited from the form of $g_n(k^+k^-,N)$ established above by positivity; that is,\n\\begin{equation}\nd_n(N) \\lesssim N^{2n-1}\\,.\n\\end{equation}\nThis scaling means that the Fourier coefficients at each $n$ necessarily vanish in the $N\\to\\infty$ limit:\n\\begin{equation}\\label{eq:fourcoefflimit}\n\\lim_{N\\to\\infty}\\frac{d_n(N)}{N^{2n}} = 0\\,.\n\\end{equation}\nThe distribution on the final line of \\Eq{eq:azidist} is normalized on $\\Delta\\phi \\in[0,2\\pi)$.\\footnote{Recall that we are working at fixed center-of-mass energy $Q$.  It is known that azimuthal correlations have a finite, non-zero value at fixed centrality as both $Q$ and $N$ increase (see, e.g., \\cite{Aamodt:2010pa}).  This is not inconsistent with this analysis because the Fourier coefficients defined in \\Eq{eq:fcoeffsazi} have implicit dependence on the center-of-mass energy $Q$ and other mass scales in the system, like the pion mass $m_\\pi$.}\n\nThe particular scaling of the coefficient $d_1(N)$ with $N$ can also be established by momentum conservation.  Transverse momentum conservation states that\n\\begin{align}\n0=\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)^2=\\sum_{i=1}^N p_{\\perp i}^2+\\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cos(\\phi_i-\\phi_j)\\,.\n\\end{align}\nNow, with $p_{\\perp}\\sim Q\/N$, this relationship implies the scaling\n\\begin{equation}\n\\frac{1}{N^2}\\sum_{i\\neq j}^N \\cos(\\phi_i-\\phi_j) \\sim -\\frac{1}{N}\\,.\n\\end{equation}\nWith our ergodic assumption, the term on the left is just the mean value of $\\cos(\\Delta\\phi)$ in the large-$N$ limit and so\n\\begin{equation}\n\\langle \\cos(\\Delta \\phi)\\rangle \\sim -\\frac{1}{N}\\,.\n\\end{equation}\nThis mean value can also be calculated from the probability distribution $p(\\Delta\\phi)$.  We have\n\\begin{equation}\n\\langle \\cos(\\Delta \\phi)\\rangle = \\int_0^{2\\pi} d\\Delta\\phi\\, p(\\Delta\\phi)\\, \\cos(\\Delta\\phi) = \\frac{d_1(N)}{N^{2}}\\,,\n\\end{equation}\nfrom \\Eq{eq:azidist}.  If this is to scale like $-1\/N$, the coefficient $d_1(N)$ must scale like\n\\begin{equation}\nd_1(N) \\sim -N = -N^{2\\cdot 1-1}\\,,\n\\end{equation}\nexactly as predicted from positivity.\\footnote{Yet another way to prove this scaling with $N$ follows from demanding that the squared matrix element is finite in the $N\\to\\infty$ limit.}  Then, the distribution of the azimuthal angle difference can be expressed as\n\\begin{align}\\label{eq:delphidistfinal}\np(\\Delta\\phi)  =\\frac{1}{2\\pi}-\\frac{d_1}{\\pi}\\frac{1}{N}\\cos(\\Delta\\phi) +\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,,\n\\end{align}\nwhere now $d_1>0$ is some constant value in the $N\\to\\infty$ limit.  A similar relationship for the $n=1$ coefficient of the Fourier expansion using momentum conservation was established in \\Ref{Luzum:2010fb}, but to the best of our knowledge, the scaling with the number of particles $N$ is novel.\n\n\\subsubsection{Ellipticity}\n\n\n\nEspecially in collisions of heavy ions, azimuthal correlations amongst particles are used as evidence for collective flow phenomena and the production of exotic states of QCD matter.  The first non-trivial azimuthal correlation is referred to as elliptic flow and quantifies particle correlations with respect to the reaction plane of the collision, the plane about which particle production is maximized in the plane and minimized orthogonal to it.  As a proxy for the elliptic flow, a pairwise azimuthal correlation moment is often measured instead.  Here, we will focus on one such moment, the average over all $N$ particles in an event denoted as $\\langle 2\\rangle$ and defined to be\n\\begin{align}\n\\langle 2\\rangle=\\frac{1}{N^2}\\sum_{j,k=1}^N e^{2i(\\phi_j-\\phi_k)}&=\\frac{1}{N}+\\frac{1}{N^2}\\sum_{j\\neq k}^N\\cos\\left(2(\\phi_j-\\phi_k)\\right)\\,,\n\\end{align}\nwhere $\\phi_j$ is the azimuthal angle of particle $j$.  As above, the ergodic assumption establishes that the sum over cosine factors is just the mean value of cosine of the azimuthal angle difference in the large $N$ limit, and so\n\\begin{equation}\\label{eq:azcorrdef}\n\\langle 2\\rangle=\\frac{1}{N}+\\langle\\cos\\left(2\\Delta \\phi\\right)\\rangle\\,.\n\\end{equation}\nWhen further averaged over an ensemble of events, this is often denoted as $\\langle\\langle 2\\rangle\\rangle$ or $c_2(2)$.  The notation $c_2(2)$ is common in the literature with the parenthetical 2 referencing the pairwise azimuthal correlation and the subscript 2 referencing the 2 in the argument of cosine in \\Eq{eq:azcorrdef}.\n\nAzimuthal correlations have been extensively measured in heavy ion collisions for the past few decades, in experiments at ATLAS \\cite{Aad:2015gqa,Aad:2014eoa,ATLAS:2011ah}, CMS \\cite{Sirunyan:2017fts,Chatrchyan:2012ta,Chatrchyan:2013nka,Sirunyan:2017uyl,Sirunyan:2017gyb,Sirunyan:2017pan,CMS:2013bza,Chatrchyan:2013kba,Chatrchyan:2012vqa,Chatrchyan:2012xq,Chatrchyan:2012wg}, and ALICE \\cite{Acharya:2019vdf,Adam:2016izf,Abelev:2014mda,Adam:2015eta,Aamodt:2010pa} at the LHC and at STAR \\cite{Adamczyk:2012ku,Agakishiev:2011id,Abelev:2008ae,Adler:2002pu,Ackermann:2000tr,Adam:2019woz,Adamczyk:2017ird,Adamczyk:2015obl,Abdelwahab:2014sge,Adams:2004bi,Adams:2004wz} and PHENIX \\cite{Adare:2018zkb,Adare:2010ux} at RHIC.  However, outside of an explicit hydrodynamic model, measurements of azimuthal correlations are typically challenging to interpret.  Azimuthal correlations are often binned in unphysical or unobservable quantities that are fit from models of the ion collisions, like the number of nucleons participating in the collision or the centrality.\\footnote{Experimentally, the centrality is measurable as it is defined as a quantile of the multiplicity distribution for a given collision event data set.  However, the centrality is not defined for an individual event in isolation and ``centrality'' connotes the overlap of the ion nuclei in collision, which is not measurable.}  Few of the listed references plot azimuthal correlations as a function of directly observable quantities, such as the number of charged particles in the event \\cite{Aad:2015gqa,Chatrchyan:2013nka,Sirunyan:2017uyl,Acharya:2019vdf,Abelev:2014mda,Adamczyk:2015obl}.  In what follows we will show how  the features of the observables can in fact be elucidated purely in terms of observable quantities, by appealing to the expansion of the minimum bias cross section derived above.\n\nAs mentioned earlier, our expansion of the minimum bias cross section is defined for a fixed collision energy $Q$ and fixed number $N$ of observed particles.  With our current formulation, we are not able to predict the dependence of the azimuthal correlations either as a function of $Q$ or $N$.  Thus, we will focus on predictions for fixed $Q$ binned in the number of observed particles.  Nevertheless, we will be able to make a number of concrete predictions.  For fixed $N$ and $Q$, the average azimuthal correlation $c_2(2)$ in terms of our expansion of minimum bias established in \\Eq{eq:delphidistfinal} is\n\\begin{align}\nc_2(2) &=\\frac{1}{N}+\\int_0^{2\\pi}d\\Delta\\phi\\, p(\\Delta\\phi)\\, \\cos(2\\Delta\\phi) = \\frac{1}{N} + \\frac{d_2(N)}{N^4}\\,.\n\\end{align}\nThis form immediately identifies two distinct contributions.  First, the explicit $1\/N$ term is completely independent of any of the dynamics of the collision (i.e., the squared matrix element and smearing).  Such a contribution is sometimes called ``non-flow'' in the literature.  The second term, by contrast, is only non-zero if there are non-trivial azimuthal correlations; otherwise, the integral over phase space of the sinusoidal function vanishes.  Such a contribution is sometimes called ``flow''.  Whatever its short-distance interpretation, we can make concrete predictions of the flow contribution within our expansion.\n\nFirst, the azimuthal correlation $c_2(2)$ vanishes in the $N\\to\\infty$ limit.  This follows directly from the results established in \\Eq{eq:fourcoefflimit}.  Additionally, this large-$N$ limit is also the limit of small centrality (the quantile of highest multiplicity in an ensemble of collision events).  Thus, azimuthal correlations should also vanish in the limit of small centrality.  In the nucleus-overlap model of heavy ion collisions, it is also predicted that azimuthal correlations vanish in the low centrality limit because a head-on, perfectly overlapping nucleus collision is completely rotationally symmetric and has no preferred particle production axis.  However, we stress that this prediction in our formulation makes no reference to the unobservable nucleus overlap and relies instead exclusively on our power counting of the relevant observable quantities.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/ellipN}\n\\caption{Data of the azimuthal correlation $c_2(2)$ as a function of charged particle multiplicity $N_\\text{ch}$ from the ALICE experiment \\cite{Abelev:2014mda}.  Lead ions are collided with a nucleon-nucleon center of mass of $\\sqrt{s_\\text{NN}} = 2.76$ TeV.  Only charged particles with pseudorapidity $|\\eta|<1$ and transverse momentum $0.2 < p_\\perp <3.0$ GeV contribute.  The open circle data include all charged particles, and the open triangle, diamond, and square data only include those pairs of charged particles with a pseudorapidity difference of at least 0.4, 1.0, and 1.4, respectively.\n\\label{fig:ellipNch}}\n\\end{center}\n\\end{figure}\n\nThis prediction is observed in data from heavy ion collisions.  In \\Fig{fig:ellipNch}, we plot data of the azimuthal correlation $c_2(2)$ from PbPb collisions at the ALICE experiment, as a function of the number of charged particles $N_\\text{ch}$ in the pseudorapidity range $|\\eta| < 1.0$ and transverse momentum in the range $0.2 < p_\\perp < 3.0$ GeV \\cite{Abelev:2014mda}.  Only charged particles are used to compute $c_2(2)$, but as discussed earlier, as long as the number of particles is large, we still expect our expansion to accurately describe these events.  For significantly large $N_\\text{ch}$ the azimuthal correlation decreases toward 0, as our expansion predicts.  Though we do not show a plot here, data of the azimuthal correlation binned in centrality also vanishes as centrality decreases (see, e.g., \\Ref{Adam:2016izf}), corresponding to the highest multiplicity events in an ensemble.\n\nThe plot presented in \\Fig{fig:ellipNch} also illustrates another feature described in our expansion.  The four data sets in that plot correspond to the azimuthal correlation of pairs of particles with a minimal pseudorapidity difference, $|\\Delta\\eta|$.  As the pairwise pseudorapidity difference increases, the azimuthal correlation at small multiplicity $N_\\text{ch}$ significantly decreases.  At small multiplicities, momentum conservation induces significant correlations between pairs of particles that is imprinted on $c_2(2)$, especially when further restricted to long-range pseudorapidity correlations.  However, at large values of $N_\\text{ch}$ the data with different pseudorapidity differences converge onto a universal curve.  This is expected within our expansion.  At large $N_\\text{ch}$, correlations induced by momentum conservation are eliminated.  Further, the pseudorapidity distribution of particles is quite flat over the acceptance region of the detector, and we argued that the transverse momentum distribution of particles effectively independent of a pseudorapidity cut.  So, for sufficiently large number of particles, there is nothing special about the particular pseudorapidity of a particle and its correlation to other particles.  Therefore, in the large $N_\\text{ch}$ limit, neighboring particles and far away particles in pseudorapidity will exhibit the same azimuthal correlations.\n\n\n\n\\subsubsection{Two-Particle Angular Correlations; the ``Ridge''}\n\nA more general analysis of two-particle correlations can be studied through measurement of the cross section differential in both pairwise azimuthal differences $\\Delta\\phi$ as well as pairwise pseudorapidity differences $\\Delta\\eta$.  Such double differential cross sections have been measured at RHIC \\cite{Abelev:2009af,Alver:2009id} and other experiments before the LHC, but became exciting evidence of potential collective phenomena in pp collisions in measurements at CMS \\cite{Khachatryan:2010gv} in 2010.  Since then, CMS \\cite{Chatrchyan:2011eka,Chatrchyan:2012wg,Chatrchyan:2013nka,CMS:2013bza,Khachatryan:2015lva,Khachatryan:2016txc}, ATLAS \\cite{ATLAS:2012ap,Aad:2015gqa,Aaboud:2016yar}, and ALICE \\cite{Aamodt:2011by,Adam:2015gda} have measured pairwise particle correlations in pp and PbPb ion collisions.  In heavy ion collisions, it is well-established that there are long range in $\\Delta \\eta$ correlations, which is interpreted as collective phenomena due to the high-density medium produced in collision.  Extending out to $\\Delta\\eta\\sim 4$, the distribution of azimuthal correlations in $\\Delta\\phi$ is nearly independent of $\\Delta\\eta$ and exhibit an over-density in the distribution around $\\Delta\\phi = 0$, the so-called ``ridge''.  A long distance factorization of the $\\Delta\\eta$ and $\\Delta \\phi$ distributions and the ridge are challenging to observe in pp collisions because of the overwhelming dominance of short-distance correlations, or jet phenomena.  However, with significantly restrictive cuts at high particle multiplicity, the CMS analysis from 2010 and later measurements identified a small, but non-negligible ridge in pp collisions suggesting that a dense QCD medium was produced.\n\nIn this section, we will analyze these angular correlations and the ridge from the perspective of our expansion of high multiplicity, minimum bias events.  First, the quantity that is typically measured and plotted to identify the ridge is the cross section ratio\n\\begin{align}\n\\frac{d^2 N^\\text{pair}}{d\\Delta\\eta \\, d\\Delta\\phi} = B(0,0)\\times \\frac{S(\\Delta\\eta,\\Delta\\phi)}{B(\\Delta\\eta,\\Delta\\phi)}\\,.\n\\end{align}\nHere, $N^\\text{pair}$ is the number of pairs of particles from which their pseudorapidity difference $\\Delta\\eta$ and azimuthal angle difference $\\Delta \\phi$ are measured.  $S(\\Delta\\eta,\\Delta\\phi)$ is the double differential cross section for ``signal'' pairs of particles drawn from the same event.  To eliminate trivial correlations to some extent, this is divided by the ``background'' distribution $B(\\Delta\\eta,\\Delta\\phi)$ which consists of pairs of particles drawn from different events.  Thus, the paired particles that contribute to $B(\\Delta\\eta,\\Delta\\phi)$ are truly uncorrelated.  Finally, $B(0,0)$ is a convenient normalization factor.\n\nStarting with the signal distribution $S(\\Delta\\eta,\\Delta\\phi)$, it is defined to be\n\\begin{align}\nS(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{same}}{d\\Delta\\eta \\, d\\Delta\\phi}\\,,\n\\end{align}\nwhere $N^\\text{same}$ is the number of pairs of particles drawn from the same event.  In the limit that the number of detected particles in the event $N\\to\\infty$, correlations between pairs of particles due to momentum conservation are eliminated.  Further, we expect that the pseudorapidity and azimuthal angle differences are uncorrelated, or only very weakly correlated, because the squared matrix element approaches a constant in this limit.  This suggests that the signal distribution factorizes to leading power in $1\/N$ as\n\\begin{align}\nS(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{same}}{d\\Delta\\eta \\, d\\Delta\\phi} \\sim \\frac{dN^\\text{same}}{d\\Delta\\eta}\\times\\frac{1}{N^\\text{same}}\\frac{dN^\\text{same}}{d\\Delta\\phi} = \\frac{dN^\\text{pair}}{d\\Delta\\eta}\\, p(\\Delta\\phi)\\,.\n\\end{align}\nHere, $p(\\Delta\\phi)$ is the probability distribution for azimuthal angle differences established in \\Eq{eq:delphidistfinal}, and $dN^\\text{pair}\/d\\Delta\\eta$ is the pairwise $\\Delta\\eta$ distribution.\n\nContinuing to the background distribution $B(\\Delta\\eta,\\Delta\\phi)$, we expect no correlation between $\\Delta\\eta$ and $\\Delta\\phi$ for pairs of particles from different events, so this distribution would naturally factorize.  Further, the azimuthal difference should be flat because the experiment is symmetric about the colliding beams and each event spontaneously breaks this rotational symmetry, but independent of all other events.  We also expect that the pseudorapidity difference distribution of the background is identical to that of signal because in the large $N$ limit, correlations between pairs of particles in the same event vanish.  Thus, the pseudorapidity difference distribution for same-event or distinct-event pairs is just determined by drawing from the single particle pseudorapidity distribution twice, to leading power in $1\/N$.  That is, the background distribution factorizes to leading power in $1\/N$ as\n\\begin{align}\nB(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{mix}}{d\\Delta\\eta \\, d\\Delta\\phi} \\sim \\frac{dN^\\text{mix}}{d\\Delta\\eta}\\times\\frac{1}{N^\\text{mix}}\\frac{dN^\\text{mix}}{d\\Delta\\phi} = \\frac{1}{2\\pi}\\frac{dN^\\text{pair}}{d\\Delta\\eta}\\,,\n\\end{align}\nwhere $N^\\text{mix}$ is the number of pairs of particles from distinct or ``mixed'' events.\n\nWith these assumptions, our prediction for the double differential pairwise correlation distribution is independent of $\\Delta\\eta$ and just determined by the azimuthal difference distribution:\n\\begin{align}\n\\frac{d^2 N^\\text{pair}}{d\\Delta\\eta \\, d\\Delta\\phi} \\propto p(\\Delta\\phi)\\,.\n\\end{align}\nThe large-$N$ expansion of the minimum bias cross section immediately implies that there are long range in $\\Delta\\eta$ correlations, through the interpretation that the $\\Delta\\phi$ distribution at disparate values of $\\Delta\\eta$ are, to first order, the same.  While these long-range correlations are observed in data and are a requirement for existence of a ridge, it doesn't by itself predict an over-density of pairs of particles with $\\Delta\\phi = 0$.  That requires the distribution $p(\\Delta\\phi)$ to have a maximum at $\\Delta\\phi = 0$.  One can perform an analysis of the coefficients of the Fourier expansion of $p(\\Delta\\phi)$, and compare to data.  Recall that our large-$N$ expansion of minimum bias predicts the form\n\\begin{equation}\np(\\Delta\\phi)  =\\frac{1}{2\\pi}-\\frac{d_1}{\\pi}\\frac{1}{N}\\cos(\\Delta\\phi) +\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,,\n\\end{equation}\nwhere $d_1>0$ and the $d_n(N)$ coefficients scale at worst like $N^{2n-1}$ in the large $N$ limit.  Additionally, from ellipticity, we know that $d_2(N)>0$, at least for minimum bias events in PbPb collisions. \n\nFor existence of a ridge, we would additionally need that \n\\begin{align}\n&\\left.\\frac{dp(\\Delta\\phi)}{d\\Delta\\phi}\\right|_{\\Delta\\phi = 0}=0\\,,\n&\\left.\\frac{d^2p(\\Delta\\phi)}{d\\Delta\\phi^2}\\right|_{\\Delta\\phi = 0}<0\\,.\n\\end{align}\nThe vanishing of the first derivative at $\\Delta\\phi=0$ is guaranteed by the symmetries of minimum bias collisions.  The concave down requirement constrains the Fourier coefficients, but without a short-distance theory from which to predict the $d_n(N)$ values, we can't say much more within the context of our minimum bias expansion.  However, the fact that the distribution $p(\\Delta\\phi)$ is physical highly constrains the coefficients $d_n(N)$.  Assuming that a physical distribution is infinitely differentiable and the Fourier series of every derivative is absolutely convergent, the $d_n(N)$ must vanish as $n\\to\\infty$ faster than any power of $n$.  For example, the Fourier series could actually just be finite for which there is a maximum $n$ at which $d_n(N)$ is non-zero, or $d_n(N)$ might vanish exponentially fast, like $d_n(N)\\sim e^{-n}$.  Infinite differentiability implies that there are only a finite number of terms in the Fourier expansion that can have an appreciable effect on the first and second derivatives of $p(\\Delta \\phi)$ at $\\Delta \\phi=0$.  This seems to be borne out in data, e.g., figure 6 of \\Ref{Chatrchyan:2011eka}, in which the $d_2(N)$, $d_3(N)$, and $d_4(N)$ coefficients would be positive, but $d_5(N)$ appears to be vanishingly small.  Of course, with any finite dataset, measuring higher Fourier modes is problematic, but the trend is consistent with expectations from differentiability.\n\n\n\\section{Min-Bias in Other Colliders}\\label{sec:other}\n\nWhile our focus in this paper has been the application of the large-$N$ expansion of minimum bias events to understand identical hadron or identical heavy ion collisions, the power counting and symmetries can be appropriately modified to describe nearly any collision environment.  In this section, we will just briefly describe extension to minimum bias at other colliders.  Minimum bias in electron-positron collisions will be significantly different that we will provide more details about its symmetries and predictions.\n\n\\subsection{pA Collisions and Electron-Ion Collisions}\n\nFor different types of colliders, we must modify our power counting and symmetry assumptions as established in \\Sec{sec:ppmin}.  Here, we will just discuss how the power counting and symmetries are modified for proton-ion (pA) and electron-ion collisions, leaving a detailed analysis of unique predictions from our minimum bias expansion in those environments to future work.  First, for pA collisions, as the initial state still consists of hadronic matter, there are still unmeasurable beam regions in which all we can determine is the total momentum lost down the beam.  Thus, all power counting assumptions for minimum bias in pA collisions remain identical to that of pp\/AA collisions.  The only change is to the symmetries, in which pA collisions lack the beam reflection $\\eta\\to -\\eta$ symmetry of pp\/AA collisions.  As a consequence, the expansion of the corresponding matrix element is not reflection symmetric about the beams and so can have terms that consist of the hyperbolic sine of pseudorapidity differences, as well as hyperbolic cosine.  For example, the squared matrix element for the $N$ detected particles of pA collisions can now have terms of the form\n\\begin{equation}\n|{\\cal M}|^2\\supset \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\sinh(\\eta_i-\\eta_j)\\,.\n\\end{equation}\n\nFor electron-ion colliders, the power counting and symmetries are even more different.  Now, because the electron is not hadronic, there is only one unmeasurable beam region, in the direction of the momentum of the initial ion.  This isn't necessarily to say that there is perfect experimental resolution in the direction of the electron's momentum, it is instead that because the electron is not hadronic, its direction of momentum is not special for establishing beam remnants that take away an order-1 energy fraction at very high pseudorapidity.  As there is only one unmeasurable beam region, we only smear over one component of lightcone momentum, $k^+$, say.  This also means that the pseudorapidity translation symmetry $\\eta\\to \\eta+\\Delta\\eta$ is broken because we can effectively measure all particles at arbitrary pseudorapidities in the direction of the electron's momentum.  Additionally, as the colliding particles are not identical, there is no beam reflection symmetry, $\\eta\\to-\\eta$.  Thus in this case, the cross section for minimum bias events takes the form:\n\\begin{align}\n\\sigma &\\sim \\frac{1}{Q}\\int_0^Q dk^+\\,f(k^+) \\int d\\Pi_N\\, |{\\cal M}|^2\\\\\n&\n\\hspace{2cm}\\times\\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\,.\n\\end{align}\nHere, $f(k^+)$ is some function purely of the total $k^+$ momentum lost down the beampipe in the direction of the initial ion's momentum and $|{\\cal M}|^2$ is the squared matrix element of the $N$ detected particles.  Decreasing the constraining symmetries makes the expansion of the squared matrix element significantly more involved, so we leave predictions for electron-ion collisions to future work.\n\n\\subsection{Electron-Positron Collisions}\n\nThe description of hadronic minimum bias events in electron-positron ($e^+e^-$) collisions is sufficiently constraining and simple that we will explicitly construct the large-$N$ expansion and make some predictions.  First and foremost, because electrons are color-singlet, fundamental particles, there are no unmeasurable beam regions whatsoever, so (in principle) all final state particles can be detected.  Consequently, because the initial state is the hadronic vacuum, we assume full Lorentz invariance of the final state, so natural phase space coordinates are the energies of particles and their location on the celestial sphere.  Further, as with our pp\/AA collision analysis, we assume that only momenta are measured; no information on electric charge, etc., is recorded for particles.\n\nWith these assumptions, the power counting for hadronic minimum bias events in $e^+e^-$ collisions are:\n\\begin{enumerate}\n\n\\item All $N$ particles produced in collision can in principle be detected.  Only their four-momenta are measured; no charge information is determined.\n\n\\item  We assume that the mass of the particles is irrelevant and so detected particle energy $E$ is parametrically larger than the QCD scale or pion mass, $E \\gg m_\\pi$.\n\n\\item The number of detected particles $N$ is large: $N\\gg1$.\n\n\\item We assume that the mean energy of the detected particles is representative of all particles' energies and so the mean and the root mean square energies are comparable: $\\langle E\\rangle\\sim \\sqrt{\\langle E^2\\rangle} \\sim Q\/N$.\n\\end{enumerate}\nThese power counting assumptions then imply that these minimum bias events enjoy the following symmetries:\n\\begin{enumerate}\n\n\\item Complete Lorentz invariance O$(3,1)$ and parity symmetry of the final state,\n\n\\item reflection of the electron-positron beams because no charge information is measured, and\n\n\\item $S_N$ permutation symmetry in all $N$ detected particles.\n\\end{enumerate}\nThe reflection symmetry of the beams is already accounted for with the O(3,1) Lorentz invariance, but we mention it explicitly because it would not be true if charge information were retained.\n\nWith these symmetries, the cross section for minimum bias events can just be expressed in the textbook form of Fermi's Golden Rule with four-momentum conservation in the center-of-mass frame:\n\\begin{align}\n\\sigma &\\sim \\int \\Pi_N\\, |{\\cal M}|^2\\sim \\int \\prod_{i=1}^N\\frac{d^3 p_i}{2E_i}\\, |{\\cal M}|^2\\, \\delta^{(4)}\\left(\nQ-\\sum_{i=1}^N p_i\n\\right)\\\\\n&\\sim \\int \\prod_{i=1}^N \\left[E_i\\, d E_i\\, d\\cos\\theta_i\\, \\frac{d\\phi_i}{2\\pi}\\right]\\, |{\\cal M}|^2\\, \\delta\\left(\nQ-\\sum_{i=1}^N E_i\n\\right)\\, \\delta\\left(\n\\sum_{i=1}^N E_i\\cos\\theta_i\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,,\n\\nonumber\n\\end{align}\nwhere $|{\\cal M}|^2$ is the Lorentz-invariant squared matrix element for $N$ final state particles, $E_i$ is the energy of particle $i$, and $\\theta_i$ and $\\phi_i$ are its polar and azimuthal angle coordinates on the celestial sphere.  As it is fully Lorentz invariant, the squared matrix element can be expanded in symmetric polynomials of the Mandelstam invariants $s_{ij} = 2p_i\\cdot p_j$, for two particles' four-momenta $p_i$ and $p_j$.  Up through mass dimension 4 terms, the squared matrix element can be expanded in symmetric polynomials as:\n\\begin{align}\n|{\\cal M}|^2 &=1+\\frac{c_1^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k\\neq l}^N s_{ij}s_{kl}+\\frac{c_2^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^Ns_{ij}s_{ik}+\\frac{c_3^{(4)}}{Q^4}\\sum_{i\\neq j}^Ns_{ij}^2+{\\cal O}(Q^{-6})\n\\end{align}\nThe $c_i^{(n)}$ factors are constants that may in general have dependence on the number of particles $N$.  The only constraint we impose on them is that in the $N\\to\\infty$ limit, their scaling with $N$ produces a finite contribution to the squared matrix element and that they are not sufficiently negative to make the squared matrix element negative.  For example, assuming that the energy of particle $i$ scales like $E_i\\sim Q\/N$ and so the second non-trivial term scales with $N$ as\n\\begin{equation}\n\\sum_{i\\neq j\\neq k}^Ns_{ij}s_{ik} \\sim N^3\\frac{Q^4}{N^4}\\sim \\frac{Q^4}{N}\\,,\n\\end{equation}\nbecause there are $N^3$ terms in the sum in the $N\\to\\infty$ limit.  Therefore, for finiteness and positivity, the coefficient $c_2^{(4)}$ is bounded as\n\\begin{equation}\n-N \\lesssim c_2^{(4)}\\lesssim N\\,.\n\\end{equation}\n\nAs discussed in the expansion of the squared matrix element for pp\/AA collisions, in the $N\\to\\infty$ limit, the various terms in the squared matrix element relax to their ensemble mean with 0 variance, with our assumption of ergodicity and the central limit theorem.  So, in the $N\\to\\infty$ limit, our minimum bias power counting implies that the matrix element is just a constant, corresponding to $N$ free final state particles only constrained by momentum conservation.\n\n\\subsubsection{Suppression of a Ridge}\n\nWithin this hadronic minimum bias expansion for $e^+e^-$ collision events, we will concretely demonstrate that this predicts a strong suppression of possible ridge phenomena in azimuthal correlations of pairs of particles.  A ridge in two-particle correlations was searched for in $e^+e^-$ collisions from archived ALEPH data recently \\cite{Badea:2019vey}, but no evidence for such a ridge was found.  Lack of azimuthal correlations over a long distance in pseudorapidity in $e^+e^-$ collisions is perhaps not surprising because the initial state is the QCD vacuum and there are no special directions in which particles are produced at arbitrarily high pseudorapidities.  Nevertheless, it is satisfying to see that the power counting and symmetries of $e^+e^-$ collisions immediately predict that such correlations, if they do exist, are very suppressed in the large-$N$ limit.  \n\nTo demonstrate this suppression of azimuthal correlations, we will focus on terms in the squared matrix element that directly correlate pairs of particles:\n\\begin{align}\n|{\\cal M}|^2 &\\supset 1+\\sum_{n=2}^\\infty c_n\\sum_{i\\neq j}^N \\frac{s_{ij}^n}{Q^{2n}} = 1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{Q^{2n}}\\sum_{i\\neq j}^N p_{\\perp i}^np_{\\perp j}^n\\left(\n\\cosh \\Delta \\eta_{ij} - \\cos\\Delta \\phi_{ij}\n\\right)^n\\,.\n\\end{align}\nHere, the $c_n$ are some constants that possibly depend on $N$ but are constrained by finiteness and on the right, we have expanded the Mandelstam invariant $s_{ij}$ in terms of cylindrical detector coordinates.  Now, we would like to simplify this expression, focusing on the scaling with $N$ and the azimuthal correlations explicitly, so we will make some replacements.  First, the transverse momentum $p_\\perp = E\\sin\\theta$ and on flat phase space, most particles are located around the ``equator'' of the celestial sphere, where $\\theta = \\pi\/2$.  So, as $E\\sim Q\/N$, so too does $p_\\perp \\sim Q\/N$ in $e^+e^-$ collisions.  Further, because most particles lie near the equator, the hyperbolic cosine of the pairwise pseudorapidity difference is close to 1, so we can take the scaling $\\cosh \\Delta \\eta \\sim 1$.  With these scaling assumptions, the squared matrix element takes the form:\n\\begin{align}\n|{\\cal M}|^2 &\\supset  1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{Q^{2n}}\\sum_{i\\neq j}^N p_{\\perp i}^np_{\\perp j}^n\\left(\n\\cosh \\Delta \\eta_{ij} - \\cos\\Delta \\phi_{ij}\n\\right)^n\\\\\n&\\sim1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\sum_{i\\neq j}^N \\left(\n1 - \\cos\\Delta \\phi_{ij}\n\\right)^n\\nonumber\\,.\n\\end{align}\n\nNow, with this squared matrix element, we can determine the probability distribution of the pairwise azimuthal angle difference $\\Delta\\phi$, in the large-$N$ limit.  As in our analysis of pp\/AA collisions, correlations from momentum conservation become trivial in the $N\\to\\infty$ limit, and the probability distribution can be calculated from\n\\begin{align}\np(\\Delta\\phi)&\\propto \\int_0^{2\\pi}\\prod_{k=1}^N\\frac{d\\phi_k}{2\\pi}\\, |{\\cal M}|^2 \\, \\delta(\\Delta \\phi-(\\phi_1-\\phi_2))\\\\\n&= \\int_0^{2\\pi}\\prod_{k=1}^N\\frac{d\\phi_k}{2\\pi}\\,\\left(\n1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\sum_{i\\neq j}^N \\left(\n1 - \\cos\\Delta \\phi_{ij}\n\\right)^n\n\\right)\\, \\delta(\\Delta \\phi-(\\phi_1-\\phi_2))\\nonumber\\\\\n&=\\frac{1}{2\\pi}+\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\left[\n\\frac{2^{n-1}\\Gamma\\left(\\frac{1}{2}+n\\right)}{\\sqrt{\\pi}\\Gamma(1+n)} \\,(N+1)(N-2)+\\left(\n1 - \\cos\\Delta \\phi\n\\right)^n\n\\right]\n\\nonumber\n\\end{align}\nWe have used permutation symmetry to define $\\Delta\\phi = \\phi_1-\\phi_2$, and the result in the final line is the exact integral over azimuthal angles of the second line.  In this final form, it is obvious that any azimuthal dependence is suppressed by a relative factor of $N^2$ in the large-$N$ limit.  This explicitly follows from Lorentz invariance of the final state, enforcing that the transverse and longitudinal momentum components to the beam must appear in the expansion of the squared matrix element with the same coefficient.  \n\n\\section{Outlook}\\label{sec:concs}\n\nIn this paper, we elucidated a power counting scheme to describe minimum bias events at colliders. The main tenets of this approach are a principle of particle ergodicity---each particle is representative of any other particle and averages over particles in an event are equivalent to averages of events over an ensemble---and the assumption of large particle multiplicity, $N$, in an event which provides the expansion parameter $1\/N$. Under these conditions, the {\\it variance} of the QCD squared matrix element vanishes as $1\/N$, and as $N\\to \\infty$, this effectively reduces all physical observables to their flat phase space limits, with a smearing factor in p or A collisions to account for beam losses.\n\nWe showed that the above assumptions do indeed lead to a remarkably good description of minimum bias within a realm of validity that we quantified, performing an explicit comparison to minimum bias collider data. Notably, the (smeared) flat phase space values reproduce kinematic distributions well. We also studied observable features that can be described by an expansion in higher order kinematic harmonics about flat phase space, focusing in particular on azimuthal correlations, specifically elliptic flow in heavy ion collisions and the ridge phenomena in pp collisions. We showed that the conditions of positivity and momentum conservation fix the sign and scaling with $N$ of the leading harmonic of two particle azimuthal difference $\\Delta \\phi$ beyond the flat limit, and argued that the different symmetries of $e^+e^-$ collisions alone imply additional suppression $1\/N^2$ of any ridge that could be observed there. \n\nThe results of the above study are promising, in that they provide confidence that the $S$-matrix framework to describe minimum bias can be developed. Benefits of this approach are that it equally applies to small and large collision system sizes at high and low energies, and it does not depend on any underlying model or unphysical parameters. This could be utilized to elucidate the nature of small scale collective phenomena of QCD that were newly discovered at the LHC, and which are the subject of some debate. In particular, it would be interesting to study the emergence of jets in our picture, and see whether this can address the question of jet quenching in small (potentially QGP phase) systems. Another avenue would be exploring the use of this formalism at  low-energy heavy ion beam scans that aim to hit  the QCD critical point.\n\nTheoretically, there are a number of interesting directions to pursue. A systematic study of the harmonic expansion of the phase space manifold could be undertaken; power spectra of minimum bias collisions can be envisaged from this, and could be an efficient way of determining flow vs.~non-flow physics. It would be interesting and potentially very natural to incorporate detector resolution into this language via, e.g., a maximum harmonic sensitivity.\n\nWe formally took a limit $N\\to\\infty$ while working at fixed $Q$, while ignoring the mass of the QCD pions. This approximation agrees with data well, giving confidence that it is a good approximation to low-energy QCD dynamics in the minimum bias regime. However, strictly this limit does not exist, a maximum multiplicity being ensured by the explicit breaking of chiral symmetry and the pion mass, $N_{\\text{max}}=Q\/m_{\\pi}$. The obvious connection between the limit we worked in and broader studies of strongly-coupled conformal field theories at colliders~\\cite{Hofman:2008ar} would be interesting to pursue. It might be possible to address whether  the $Q$ and $N$ dependence of the coefficients the appeared in our expansion be fixed by, e.g., consistency conditions that arise from a non-trivial limit of a distribution as $N\\to\\infty$.\n\nFinally, by working purely with the  $S$-matrix of final particles, the hydrodynamic description of the unobservable QGP was sidestepped. Nevertheless one might wonder how hallmarks of transport coefficients and, e.g., the theoretical lower bound on specific shear viscosity may show up in our coarse-grained approach. \n\n\\acknowledgments\n\nWe thank Bryan Webber for valuable feedback regarding our approach. T.M. is supported by the World Premier International Research Center Initiative (WPI) MEXT, Japan, and by JSPS KAKENHI grants JP19H05810, JP20H01896, and JP20H00153. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{{Introduction}}\n\nIn recent years a significant attention has been devoted to\ninvestigation of low-dimensional asymmetric simple exclusion\nprocesses (ASEPs)\n\\cite{derrida98,zia95,schutz,chowdhury00,derrida93}. They play  a\ncritical role for understanding  fundamental properties of\nnon-equilibrium phenomena in Chemistry, Physics and Biology.\nASEPs have been widely utilized for description of traffic\nphenomena \\cite{chowdhury00}, kinetics of biopolymerization\n\\cite{macdonald68}, protein synthesis\n\\cite{shaw03,chou04,shaw04,dong07}, and biological transport of\nmotor proteins \\cite{lipowsky01,parmeggiani03}. The advantage of\nusing asymmetric exclusion processes for studying mechanisms of\nnon-equilibrium phenomena is due to the fact that some\nhomogeneous versions of ASEPs can be solved exactly via\nmatrix-product approach and related methods\n\\cite{derrida98,derrida93,blythe07}. In addition, understanding\nof processes in ASEPs can be achieved by utilizing a\nphenomenological domain wall approach \\cite{DW}. In order to have\na more realistic description of different non-equilibrium\nphenomena ASEPs with inhomogeneous distribution of rates are\nrequired. However, there is a limited number of  studies dealing\nwith ASEPs with disorder in the transition rates at sites (static impurities)\n\\cite{chou04,shaw04,janowsky92,schutz93,schutz93PRE,janowsky94,kolomeisky98,tripathy97,barma98,kolwanker,ha,stinchcombe,derrida04,juhasz05,barma06,pierobon06,foulaadvand07,foulad07,dongPRE,greulich1,greulich2}\nand with disorder associated to particles's hopping rates (moving impurities) \\cite{speers,mallick,evans,kim,fouladvand99}.\nIn this case exact solutions are not obtained, and extensive\nMonte Carlo computer simulations and approximate theories are\nutilized in order to understand particle dynamics. Disorder has a\nstrong effect on the behavior of ASEPs. Even a single defect bond\nfar away from the boundaries lead to dramatic effects in the\nstationary properties both in closed\n\\cite{janowsky92,janowsky94}  and open boundary conditions\n\\cite{kolomeisky98}.  It was shown recently that the dynamics of\nASEPs is also influenced by several defects that are close to\neach other \\cite{chou04,dong07,greulich1}, although the mechanism\nof this phenomenon is not well understood. This interaction\nbetween  defects is important for understanding several\nbiological transport phenomena \\cite{chou04,dong07}.  Recently\nthe particular case of two defects has been extensively\ninvestigated by Monte Carlo simulations \\cite{dong07}. It has been\nshown that the system current exhibits a notable dependence on the\ndistance between defects with equal hopping rates. Moreover, it\nwas found that the density profile is linear between defects\nwhich marks the existence of wandering shock between defects\n\\cite{dong07}. The case of two defective sites with equal rate\nhas been generalized to include extended objects \\cite{dongPRE}.\nTheoretical efforts to analyze ASEPs with disorder have been\nmostly directed to the cases with a single or few defects\n\\cite{kolomeisky98,chou04,greulich1}. In Ref. \\cite{kolomeisky98}\nASEP with open boundaries  and with a local inhomogeneity in\nthe bulk  has been investigated by arguing that the defect bond\ndivides the system into two coupled homogeneous ASEPs. This\ntheoretical approach can be called a {\\it defect mean-field}\n(DMF) because the mean-field assumptions are made only at the\nposition of  local inhomogeneity.  Although a good agreement with\ncomputer simulations has been found, there were significant\ndeviations in statistical properties of the phase with the\nmaximal current that was attributed to the neglect of\ncorrelations at the defect bond in the proposed theory\n\\cite{kolomeisky98}. A related approach called {\\it interacting\nsubsystem approximation} (ISA) has been proposed in Ref.\n\\cite{greulich1} for ASEPs with a single defect or several\nconsecutive defects (bottleneck). Here it was suggested that due\nto the defect bonds there are 3 segments in the system: two\nhomogeneous ASEPs are coupled by a segment that includes all\nsites that surround defect bonds. Explicit results have been\nused  inside the segments, and  mean-field assumptions have been\nutilized for particle dynamics between the segments. A better\nagreement with Monte Carlo computer simulations has been found,\nand the method was also successfully applied to describe\ninteractions of defects with boundaries. It was argued that ISA\ncan be used for analyzing properties of general ASEPs with\ndisorder \\cite{greulich1,greulich2}. However, ISA has not been\napplied for the systems with 2 defects at finite distances from\neach other, and because of this observation it is difficult to\napply ISA for understanding mechanisms of more complex\ninhomogeneous asymmetric exclusion processes. A slightly\ndifferent method of calculations has been proposed by Chou and\nLakatos \\cite{chou04}, who applied a {\\it finite segment\nmean-field theory} (FSMFT). According to this approach, the\nsegment of finite length $n$ that covers the defect and\nsurrounding sites is considered, and its dynamics is fully\ndescribed by solving explicitly for eigenvectors of the\ncorresponding transition rate matrix. The segment is then coupled\nin the mean-field fashion to the rest of the system. However,\nthis approach becomes numerically quite involved for cluster\nsizes  larger than $\\approx$20, and it also limits its\napplicability. Different studies of asymmetric exclusion\nprocesses with disorder point out to importance of correlations\nin the system. It is reasonable to expect that correlations are\nstronger near the slow defect sites. However, it is not clear how\nfar from the local inhomogeneity and how fast these correlations\ndecay. In addition, it is also unclear  how correlations from two\nclose defects affect each other. The goal of this paper is to\ninvestigate the role of correlations in dynamics of ASEPs with\ndisorder. By analyzing several analytical approaches in\ncombination with extensive Monte Carlo computer simulations it\nwill be shown that a successful description of disordered driven\ndiffusive systems can be achieved by properly accounting for\ncorrelations near the defect bonds.\n\n\n\\section{Model and Theoretical Description}\n\nWe investigate a totally asymmetric simple exclusion processes\nwith disorder. In the one-dimensional lattice\nthe particle at the site $i$ can jump forward with the rate\n$p_{i}$ if the next site $i+1$ is unoccupied, otherwise it stays\nat the same place. The particle can enter the system with the\nrate $\\alpha$ if the site is empty, and it can also exit the\nlattice with the rate $\\beta$. When all $p_{i}=1$ we have a\nhomogeneous ASEP for which dynamic properties are known\nexplicitly from exact solutions \\cite{derrida98,schutz}.\nASEPs with disorder correspond to the situation when there is\ninhomogeneities in the transition rates, and $p_{i}$ are drawn\nfrom arbitrary distributions. Numerous theoretical and\ncomputational studies indicate that in the limit of large times\nthe dynamics in the system can be determined by comparing\nentrance rate, exit rates and the transition rate at the slowest\ndefect bond \\cite{kolomeisky98,greulich2}. This observation has a\nsignificant consequence for properties of ASEPs with disorder,\nyielding a generic  phase diagram with 3 phases.  When the\nentrance is a rate-limiting process the system can be found in\nthe low-density phase, while for slow exiting the high-density\nphase governs the system. If the rate-limiting process is the\ntransition via the slowest defect bond the system is in the\nmaximal-current phase. In this maximal-current phase, a segregation\nof density profile into macroscopic high and low regions occurs at the\nlocation of slowest defect bond. Other defects only perturb the density\nprofile on a local scale. However, when the number slowest defect bonds exceeds\ntwo or more the above picture needs modification. Furthermore, the previous studies\non disordered ASEPs lack investigations on correlation effects induced by defects.\nTo address these questions, we analyze the simplest model with 2 identical defects\nin the bulk of the system far away from the boundaries. It was shown earlier\n\\cite{greulich1} that positioning of the slow defects close to the boundaries leads\nonly to rescaling of the effective entrance and\/or exit rates, and we will not\nconsider this possibility in this paper. Note that in this paper we are using\nterms of defect bonds and defect sites. To clarify, we define the defect site as\nthe site $i$ from which the particle hopes to the site $i+1$ with the rate $q<1$.\nCorrespondingly, the bond connecting sites $i$ and $i+1$ is a defect one.\n\n\n\\subsection{ Defect Mean-Field Theory}\n\nConsider a totally asymmetric exclusion processes with open\nboundaries and with 2 slow defective sites at $i=d_1$ and $i=d_2$ at a distance $d$\nwith $d_2-d_1=d$ (separated by $d-1$ normal sites), as shown in Fig. 1.\nAt the defects the particle jump to the right with the rate $q < 1$, in all\nother sites the hopping rate is equal to one. It can be seen that two defects\ndivide the system into three segments.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig1.eps}\n\\caption{ Fig.1: Schematic of ASEP with two defective sites at $i=d_1$ and $i=d_2$ separated by $d-1$ normal sites\ni.e., $d_2-d_1=d$. The reduced hopping rates at each defect is equal to $q$.\nIn normal sites the hoping rates are one. } \\label{fig:bz2}\n\\end{figure}\n\nThe particle dynamics inside each segment can be calculated exactly,\nhowever, it is assumed that there are no correlations between the segments.\nIf entrance to the lattice is the slowest process then the system\ncan be found in low-density (LD) phase with stationary current\nand bulk densities given by\n\n\\begin{equation}\nJ=\\alpha(1-\\alpha), \\quad \\rho_{bulk}=\\alpha.\n\\end{equation}\n\nSimilarly, when the exit becomes a bottleneck process the system\nis in high-density (HD) phase with\n\n\\begin{equation}\nJ=\\beta(1-\\beta), \\quad \\rho_{bulk}=1-\\beta.\n\\end{equation}\n\nNote that in both phases particle densities near the defect bonds\nwill deviate from the bulk values. The more interesting case is\nwhen the dynamics in the system is governed by transitions via\nlocal inhomogeneities. In this phase, which has the maximal current, we expect\nto have density phase segregation similar to the case a single defect ASEPs.\nWe emphasize that all our investigation in this paper is on this maximal-current phase.\nFirst consider a lattice segment after the second defect.\nThe dynamics in this part of the system is controlled by the entrance\nof particle via the defect, then it has a low-density profile with unknown\nbulk density $\\rho^{*}<1\/2$. Similar arguments can be used to analyze the density\nprofile in the segment before the first defect. Here the flux is\nlimited by the exit rate via the local inhomogeneity, leading to\nthe  high-density phase. Since at stationary-state condition\nthe flux through any segment should be the same,\n$J=\\rho^{*}(1-\\rho^{*})$, the bulk density in this segment is\nequal to $1-\\rho^{*}$. The region between two defects can be\nviewed as asymmetric exclusion process on a finite lattice with\n$d$ sites. The effective entrance and exit rates to this segment\ncan be easily evaluated using our mean-field assumptions,\n\n\\begin{equation}\\label{boundary_rates}\n\\alpha_{eff}=\\beta_{eff}=q(1-\\rho^{*}).\n\\end{equation}\n\nThe stationary properties of the lattice segment with $d$ sites\nbetween the defects can be evaluated explicitly by utilizing\nexact results for finite-size ASEPs \\cite{derrida93}.\nSpecifically, the particle flux is given by\n\n\\begin{equation}\\label{J0}\nJ_{0}(\\alpha,\\beta,d)=\\frac{R_{d-1}(1\/\\beta)-R_{d-1}(1\/\\alpha)}{R_{d}(1\/\\beta)-R_{d}(1\/\\alpha)},\n\\end{equation}\n\nwhere the function $R_{d}(x)$ is defined as\n\n\\begin{equation}\\label{R_eq}\nR_{d}(x)=\\sum_{p=2}^{d+1} \\frac{(p-1)(2d-p)!}{d!(d+1-p)!} x^{p}.\n\\end{equation}\n\nTo understand the density profile in the segment we can use a\ndomain-wall picture of asymmetric exclusion processes \\cite{DW}.\nSince the entrance and exit rates are the same [see Eq.\n(\\ref{boundary_rates})], the domain wall that separates\nhigh-density and low-density blocks performs an unbiased random,\nleading to a linear density profile with a positive slope.\nExplicit expressions for particle densities can also be found in\nRef. \\cite{derrida93}. The full description of dynamics in ASEPs\nwith two defects is obtained by solving for the unknown parameter\n$\\rho^{*}$. It can be done by applying the condition of\nstationarity in the particle flux,\n\n\\begin{equation}\\label{relation}\nJ=\\rho^{*}(1-\\rho^{*})=J_{0}(\\alpha_{eff},\\beta_{eff},d).\n\\end{equation}\n\nThis equation can always be solved analytically or numerically\nexactly for any number of sites between local inhomogeneities,\nleading to stationary particle currents and density profiles. it\nis important to note that there is a particle-hole symmetry in\nthe system because defects are far away from the boundaries. To\nillustrate our approach let us calculate dynamic properties of\nASEPs with two defects for several values of the parameter $d$.\nFirst, let us analyze the simplest case of $d=1$ with consecutive\ndefects in the bulk. It can be shown that for this system\n\n\\begin{equation}\nJ_{0}(\\alpha,\\beta,d=1)=\\frac{\\alpha \\beta}{\\alpha + \\beta}.\n\\end{equation}\n\nThen Eq. (\\ref{relation}) can be written as\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=q(1-\\rho^{*})\/2,\n\\end{equation}\n\nwhich produces simple expressions for the bulk density and the\nparticle current,\n\n\\begin{equation}\n\\rho^{*}=q\/2, \\quad J=q(2-q)\/4.\n\\end{equation}\n\nThe density $l$ at the site between the defects can also be found\nfrom the condition that the flux via this site, $J=q l\n(1-\\rho^{*})$, should be equal to the flux through other\nsegments, and this yields\n\n\\begin{equation}\nl=\\rho^{*}\/q=1\/2.\n\\end{equation}\n\nThis result could also be obtained from the particle-hole\nsymmetry arguments. Note that for $q=1$ we obtain $\\rho^{*}=1\/2$\nand  $J=1\/4$ as expected for homogeneous ASEPs in the\nmaximal-current phase. For $d=2$ there are two lattice sites\nbetween the defects, and stationary properties of this system can\nalso be obtained analytically. From Eq. (\\ref{R_eq}) one can\neasily derive\n\n\\begin{equation}\nR_{1}(x)=x^{2}, \\quad R_{2}(x)=x^{2}+x^{3},\n\\end{equation}\n\nwhich produces the following expression for the current in the\nsegment between the defects\n\n\\begin{equation}\\label{eq_2}\nJ_{0}(\\alpha,\\beta,d=2)=\\frac{\\frac{1}{\\alpha}+\n\\frac{1}{\\beta}}{\\frac{1}{\\alpha}+\n\\frac{1}{\\beta}+\\frac{1}{\\alpha^{2}}+\n\\frac{1}{\\beta^{2}}+\\frac{1}{\\alpha \\beta}}.\n\\end{equation}\n\nUsing the expression for the effective entrance and exit rates\nfor the segment between the inhomogeneities [see Eq.\n(\\ref{boundary_rates})], the condition for the stationary current\nleads to\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=\\frac{2q(1-\\rho^{*})}{3+2q(1-\\rho^{*})}.\n\\end{equation}\n\nThis quadratic equation can be solved, and taking the physically\nreasonable root we obtain\n\n\\begin{equation}\n\\rho^{*}=\\frac{2q+3-\\sqrt{9+12q-12q^{2}}}{4q};\n\\end{equation}\n\\begin{equation}\nJ=\\frac{8q^{2}-6q-9 + 3\\sqrt{9+12q-12q^{2}}}{8q^{2}}.\n\\end{equation}\n\nIt can be checked  that for $q=1$ these equations reduce to\nexpected relations  $\\rho^{*}=1\/2$ and  $J=1\/4$. We can also\ncalculate the densities $l_{1}$ and $l_{2}$ at the sites between\nthe defects. Because of the particle-hole symmetry one can argue\nthat\n\n\\begin{equation}\nl_{2}=1-l_{1},\n\\end{equation}\n\nand the density at the first site  can be found by analyzing the\ncurrent via the first defect,\n\n\\begin{equation}\nJ=q(1-\\rho^{*})(1-l_{1})=\\rho^{*}(1-\\rho^{*}).\n\\end{equation}\nThen we have\n\\begin{equation}\nl_{1}=1-\\frac{\\rho^{*}}{q}=\\frac{4q^{2}-2q-3+\\sqrt{9+12q-12q^{2}}}{4q^{2}}.\n\\end{equation}\n\nWe have solved equation (6) for the case $d=3$. In this case we\nhave:\n\n\\begin{equation}\nR_{3}(x)=2x^{2}+2x^{3}+x^{4},\n\\end{equation}\n\nAfter some lengthy but straightforward algebra we arrive at the\nfollowing cubic equation for $\\rho^{*}$:\n\n\\begin{equation}\n4q^2(\\rho^{*})^{3} -(8q^2+6q)(\\rho^{*})^{2} +\n(6q^2+6q+4)\\rho^{*}-q(3+2q)=0.\n\\end{equation}\n\nFor brevity we avoid writing the answer explicitly. Analytical\nresults for ASEP with 2 defects can also be obtained in the limit\nof very large distances between the inhomogeneities ($d \\gg 1$).\nIn this case the segment between the defects can be viewed as a\nhomogeneous ASEP in the state of the phase transition between\nhigh-density and low-density phases ($\\alpha_{eff}=\\beta_{eff}$).\nThis corresponds to a linear density profile for the segment\nbetween the defects. Then it leads to the following expression\nfor the current\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=q(1-\\rho^{*})\\left[1-q(1-\\rho^{*})\\right],\n\\end{equation}\nand finally we obtain\n\\begin{equation}\n\\rho^{*}=\\frac{q}{1+q}, \\quad J=\\frac{q}{(1+q)^{2}}.\n\\end{equation}\n\nThese results are identical to stationary properties of ASEP with\nonly one local inhomogeneity far away from the boundaries\nobtained using DMF approximation \\cite{kolomeisky98}, suggesting\nthat 2 defects at large distances do not affect each other\n\\cite{chou04}. For a general $d$ equation (6) leads to a polynomial equation\nof order $d$ for the unknown $\\rho^*$. For $d>3$ this equation can be solved numerically to\nfind the acceptable answer. In figure (2) we have sketched the behavior of current\n$J$ as a function of $q$ for $d=1,2$ and $3$ and have compared them to the results\nobtained via Monte Carlo simulations. As expected $J$ is an\nincreasing function of both $q$ and $d$. DMF notably\nunderestimates the current in comparison to the MC simulation\nespecially in the intermediate values of $q$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig3-colour.eps}\n\\caption{ Fig.2: (Colour online) $J$ vs $q$ for\n$d=1,2,3$ obtained by DMF method and MC simulation.} \\label{fig:bz2}\n\\end{figure}\n\n\\subsection{Interacting Subsystem Approximation}\n\nInteracting subsystem approximation (ISA) is another method of\ncalculating stationary properties of ASEPs with a single defect\nor a single bottleneck developed by Greulich and Schadschneider\n\\cite{greulich1}.  It can be easily extended to the case of\nasymmetric exclusion processes with 2 defects separated by $d$\nlattice sites. Similarly to DMF this method divides the lattice\ninto several segments. Particle dynamics inside the segments is\ntreated exactly, while between the segments mean-field\nassumptions are made. ISA differs from DMF in the defining of\nsegments. In DMF the position of defects separates different\nparts, and there is always 3 segments in the system. In ISA the\nsites that are connected by the defect bond are put together in\none segment, as shown in Fig. 3.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig2.eps}\n\\caption{ Fig.2: (Colour online) Fig.3: (Colour online) interacting subsystems connected via mean-field assumption.\nFor d$>$2 the system is divided into five segments. } \\label{fig:bz2}\n\\end{figure}\n\nFor $d=1$ there are also 3 parts in the lattice, and the middle segment has 3 sites.\nFor $d=2$ there are 4 segments and 2 middle segments (with 2 lattice sites\neach) border each other.  For any larger distance between local\ninhomogeneities ISA assumes 5 segments: see Fig. 3. Note that the\nsize of the middle segment is equal to $d-2$.\n\nLet us consider a general case of 5 segments ($d>2$) for\ncomputation of stationary properties of ASEPs with 2 defects. As\nwas argued above, the system can be found in one of three phases:\nLD, HD or HD\/LD (maximal-current). Since the derivation of\nproperties in HD and LD phases is the same as for DMF approach,\nwe concentrate on description of the maximal-current phase. As\nbefore we assume that the bulk density in the segments 1 and 5\nare $1-\\rho^{*}$ and $\\rho^{*}$ correspondingly. Let us define\n$l_{1}$ and $l_{2}$ as the probabilities to find the particles at\nthe corresponding sites of the segment around the first defect.\nSimilarly, $l_{3}$ and $l_{4}$ describe densities in the segment\naround the second defect bond. For the middle segment with $d-2$\nlattice sites we define $x_{i}$ for $i=1,\\cdots,d-2$ as the\nparticle density at $i$-th site of this segment. As for DMF\napproach, the existing particle-hole symmetry simplifies\ncalculations significantly. Specifically, it suggests that\n\n\\begin{equation}\nl_{4}=1-l_{1},\\quad l_{3}=1-l_{2}, \\quad x_{i}=1-x_{d+1-i}.\n\\end{equation}\n\nThe overall particle current in the system can be written as\n\n\\begin{equation}\nJ=\\rho^{*}(1-\\rho^{*}),\n\\end{equation}\n\nwhile due to the mean-field assumptions the current between the\nfirst and the second segments is equal to\n\n\\begin{equation}\nJ_{12}=(1-l_{1})(1-\\rho^{*})=\\alpha_{2}(1-\\rho^{*}),\n\\end{equation}\n\nwhere $\\alpha_{2}$ is the effective rate to enter the second\nsegment.  At large times we expect to find the system in the\nstationary state, i.e., $J=J_{12}$, yielding\n\n\\begin{equation}\n1-\\alpha_{2}= l_{1}=(1-\\rho^{*}).\n\\end{equation}\n\nThe current between segments 2 and 3 can be presented in the\nseveral ways,\n\n\\begin{equation}\nJ_{23}=l_{2}(1-x_{1})=\\beta_{2} l_{2}=\\alpha_{3}(1-x_{1}),\n\\end{equation}\n\nwith $\\beta_{2}$ being the effective exit rate from the segment\n2, while $\\alpha_{3}$ is the effective rate to enter the segment\n3. When the system reaches stationary phase, $J=J_{23}$, and we\nobtain\n\n\\begin{equation}\n\\beta_{2}=1-x_{1}, \\quad\n\\alpha_{3}=\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}}.\n\\end{equation}\n\nBecause of the particle-hole symmetry the effective entrance and\nexit rates from the segment 3 are the same,\n$\\alpha_{3}=\\beta_{3}$. The particle current via the ASEP segment\nwith $N$ sites and with entrance and exit rates $\\alpha$ and\n$\\beta$, respectively, $J(\\alpha,\\beta, N)$, can be calculated\nexplicitly \\cite{derrida93}. Then to obtain stationary properties\nof ASEP  with 2 defects in the maximal-current phase the\nfollowing system of equations should be solved,\n\n\n$$\\rho^{*}(1-\\rho^{*}) = qJ(\\frac{1-\\rho^{*}}{q},\\frac{1-x_{1}}{q},2); $$\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*}) =\nJ(\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}},\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}},d-2).\n\\end{equation}\n\n\nwhere $\\rho^{*}$ and $x_{1}$ are 2 unknown variables. The\nexpression on the right side of the first equation describes the\ncurrent inside the segment 2 and 4. Because the hopping rate is\n$q<1$, the effective entrance and exit rates must be rescaled by\nthe same factor. The right side of the second equation gives the\ncurrent inside the segment 3. The application of ISA for $d=1$\nand $d=2$ cases is different. In the case of 2 consecutive defect\nbonds the system is divided  only in 3 segments. The middle\nsegment has 3 sites that surround defect bonds. In the HD\/LD\nphase the effective entrance  rate is $\\alpha_{2}=1-\\rho^{*}$,\nand the stationary properties can be obtained by solving only one\nequation\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=qJ(\\frac{1-\\rho^{*}}{q},\\frac{1-x_{1}}{q},3).\n\\end{equation}\n\nUsing Eqs. (\\ref{J0}) and (\\ref{R_eq}) for the middle segment\nwith equal entrance and exit rates gives us\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=\\frac{q(1-\\rho^{*})\\left[2(1-\\rho^{*})+3q\\right]}{2\\left[2(1-\\rho^{*})^{2}+3q(1-\\rho^{*})+2q^{2}\\right]},\n\\end{equation}\n\nwhich can be simplified into the following expression,\n\\begin{equation}\n4(\\rho^{*})^{3}-2(3q+4)(\\rho^{*})^{2}+4(q+1)^{2}\\rho^{*}-q(3q+2)=0.\n\\end{equation}\n\nThis cubic equation can be solved explicitly, yielding\n\\begin{equation}\n\\rho^{*}=\\left[3q+4+(4-3q^{2})\/D+D\\right]\/6,\n\\end{equation}\nwhere\n\\begin{equation}\nD=\\left[-8+9q^{2}-27q^{3}+3\\sqrt{3}\n\\sqrt{16q^{3}-q^{4}-18q^{5}+28q^{6}} \\right]^{1\/3}.\n\\end{equation}\n\nISA also works differently in the case of $d=2$. There are 4\nsegments in the system, and because of the neglect of correlations\nbetween segments 2 and 3  we have\n\n\\begin{equation}\nl_{2}(1-l_{3})=l_{2}^{2}=\\rho^{*}(1-\\rho^{*}).\n\\end{equation}\n\nThen the effective entrance rate to the segment 2 is\n$\\alpha_{2}=1-\\rho^{*}$, and the effective exit rate is equal to\n$\\beta_{2}=l_{2}=\\sqrt{\\rho^{*}(1-\\rho^{*})}$. The unknown\nparameter $\\rho^{*}$ is determined from the equation for the\nstationary current,\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=qJ(\\frac{\\alpha_{2}}{q},\\frac{\\beta_{2}}{q},2).\n\\end{equation}\n\nSubstituting the values of the effective boundary rates and\nutilizing Eq. (\\ref{eq_2}) we obtain\n\n\\begin{equation}\n\\rho^{*} \\sqrt{1-\\rho^{*}}+(\\rho^{*})^{3\/2}=q \\sqrt{1-\\rho^{*}}.\n\\end{equation}\n\nwhich can be recast in the form of a cubic equation:\n\n\n\\begin{equation}\n 2(\\rho^{*})^{3} - (2q+1)(\\rho^{*})^{2} + (q^2+2q)\\rho^{*}-q^2=0,\n\\end{equation}\n\nThis equation can be solved analytically but for brevity we do not\nwrite the solution. It can also be shown that in the limit of $d\n\\gg 1$ ISA method with 2 defects produces the stationary current\nand bulk densities which are indistinguishable form the situation\nwith only one defect \\cite{greulich1},\n$$\n\\rho^{*}=\\left[3q+2-\\sqrt{9q^{2}-4q+4} \\right]\/4,\n$$\n\\begin{equation}\nJ=\\left[2q-9q^2+3q\\sqrt{9q^{2}-4q+4} \\right]\/8.\n\\end{equation}\n\nLet us now exhibit the dependence of $J$ on q in the ISA method.\nIn figure (4) we have drawn $J$ vs $q$ for $d=1,2$ and have\ncompared the results to those obtained by DMF method and MC\nsimulations.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig4-colour.eps}\n\\caption{ Fig.4: (Color online) Current vs q for\n$d=1,2$ and a large $d$ obtained within ISA method and MC simulation.\nIn simulations we have taken $\\alpha=\\beta=0.6$. } \\label{fig:bz2}\n\\end{figure}\n\nIn general, ISA method gives a better estimation of current\ncompared to DMF at least for small values of $d$ we have\nconsidered.\n\n\n\\section{Correlations near defect}\n\n\\subsection{Monte Carlo Simulations}\n\nIn this section we aim to investigate correlations in the\nvicinity of defects. We restrict ourselves to adjacent two-point\ncorrelations and will present our results for the general two-point and\nmulti-point correlations in a future work. Let us now introduce the\nnormalized connected two-point correlation function $C_i$ between the neighbouring\nsites $i$ and $i+1$. This quantity is defined as follows:\n\n\\begin{eqnarray}\nC_i= \\frac{\\langle \\tau_i\\tau_{i+1} \\rangle - \\langle \\tau_i\n\\rangle \\langle \\tau_{i+1} \\rangle}{ \\sqrt{\\langle \\tau_i^2\n\\rangle- \\langle \\tau_{i} \\rangle ^2}\\sqrt{\\langle \\tau_{i+1}^2\n\\rangle- \\langle \\tau_{i+1} \\rangle ^2}} ~ i=1,\\cdots,L-1.\n\\end{eqnarray}\n\nThe function $C_i$ measures the correlation and it lies between\n$-1$ and $1$. Negative values correspond to anti-correlation\nbetween neighboring sites whereas a positive value signifies\ncorrelation. The values near zero are regarded as uncorrelated.\nFig. (5) depicts the simulated profiles of correlation at $d=10$\nand 100 each for three values of $q$. The system size is $L=500$ and we have\ntaken $\\alpha=\\beta=0.6$ in all our simulation results unless stated otherwise.\nThe system has been updated for $T$ Monte Carlo steps. Each step consists of $L$ moves.\nIn each move, we randomly choose a site and update its status according\nto ASEP rules described above. We discard the first $\\frac{T}{5}$ steps to\nensure reaching steady state, and we have accumulated data\nseparated by 10 MC steps to avoid any possible temporal\ncorrelations. The value of $T$ is taken $10^8$ in our simulations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig9-colour.eps}\n\\includegraphics[width=7cm]{fig12-colour.eps}\n\\caption{ Fig.5: (Color online) Profile of normalized correlation function at $d=10$ (top)\nand $d=100$ (bottom) for $q=0.1,0.3,0.5$. } \\label{fig:bz2}\n\\end{figure}\n\n\nTwo defects are symmetrically placed with respect to chain mid\npoint. We observed that correlations are large in sites\nbetween the defects. There is a rather strong anti-correlation in\nthe sites immediately after the first defect and before the\nsecond defect. The correlations are growing up for middle sites\nwhere the maximum value is achieved. It can be seen that\ncorrelations are greater when $d$ is increased. This is unexpected\nand counterintuitive because increasing the distance between the defects\nreduces their interaction. It has been observed via MC\nsimulations that when $d$ is increased the current reaches\nasymptotically to its mean-field value\n$J_{MF}=\\frac{q}{(1+q)^2}$  \\cite{dong07,dongPRE}. Therefore, one\nexpects the correlations to exhibit a reducing behavior with\nrespect to distance $d$ but this is not observed in our\nsimulations. To have a deeper understanding, we have sketched the\nbehavior of correlation profile upon varying the distance $d$ for\n$q=0.1$ and $q=0.3$ in Fig. 6.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig10-colour.eps}\n\\includegraphics[width=7cm]{fig13-colour.eps}\n\\caption{ Fig.6: (Color online) Profile of\nnormalized correlation functions at various values of $d$ for\n$q=0.1$ (top) and $q=0.3$ (bottom). } \\label{fig:bz2}\n\\end{figure}\n\n\nFor fixed values of $q$, increasing the distance $d$ between the\ndefects gives rise to enhancement of\ncorrelations\/antocorrelations. For instance, the correlation value\nin the middle point rises from roughly 0.5 at small $d \\sim 10$\nto 0.65 for $d \\sim 100$. It can be observed that there is no\nnotable difference in correlation values for $d$ larger than\n$100$. Moreover, the correlations are always greater than\nanti-correlations. By increasing $q$, the correlations\/anti-correlations\nare notably reduced in values. This is expected since in the limit of\nhomogeneous ASEP where $q\\rightarrow 1$ the correlation functions become very small.\nHere we wish to make a pause and have a discussion on correlations in normal ASEPs.\nIn fact the middle segment between two defects can be regarded as an ASEP chain with\nlength $d$ with equal entrance and exit rates. To the best of our knowledge, correlations\nin ASEP with random sequential update, has only been analytically discussed by\nDerrida and Evans who obtained exact analytical expression for a general two-point function\nand made a conjecture to generalize their findings to $n$-point function \\cite{evans93}. Their\nstudy was restricted to the special case $\\alpha=\\beta=1$ and they found that\nlong range correlations persist in the bulk which was attributed as a boundary effect.\nIn order to see if the large value of the connected two-point function survives in\nthe normal ASEP with equal entrance and exit rates, we performed MC simulations. Our\nresults show that when $\\alpha=\\beta$, the profile of $C_i$ reaches a small constant (almost zero) in the bulk.\nThe correlations become large near boundaries. This boundary behaviour depends on whether $\\alpha=\\beta <0.5$\nor $\\alpha=\\beta >0.5$. Figure (7) illustrates this aspect.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig15-colour.eps}\n\\caption{ Fig.7: (Color online) Profile of normalized correlation functions in a normal ASEP chain with $\\alpha=\\beta$ .\n} \\label{fig:bz2}\n\\end{figure}\n\n\nWe recall that correlations in other types of update such as parallel updating has been discussed in\n\\cite{schutz93,schutz93PRE}. It is worthwhile to examine the behavior of density profile between\ndefects. Dong et al have recently shown via extensive MC simulations that\nthe density profile takes a linear shape between defects \\cite{dong07}. This behavior remains\nunchanged in ASEP with extended objects \\cite{dongPRE}. For the\nsake of completeness, we show some typical density profiles in Fig. 8.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig11-colour.eps}\n\\caption{ Fig.8: (Color online) Profile of density at various values of $d$ for $q=0.1$.\nThe profile exhibits a linear behaviour with positive slope. This behaviour is associated\nto the existence of wandering shock in the region between left and right defects. } \\label{fig:bz2}\n\\end{figure}\n\n\nThe interesting point is the absence of boundary layer in this\nphase-segregated regime. It would be illustrative\nto look at the dependence of two-point correlation\nfunctions at some particular sites on values of $q$ and $d$.\nThese results are sketched in Fig. 9 where correlations at the\nfirst defect site ($d_1$), its rightmost site ($d_1+1$) and in\nthe middle site of the chain are considered.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig8-colour.eps}\n\\includegraphics[width=7cm]{fig14-colour.eps}\n\\caption{ Fig.9: (Color online) Dependence of normalized correlation functions at three\nselected sites on $q$ (top) and on $d$ (bottom). } \\label{fig:bz2}\n\\end{figure}\n\n\nNote that all correlations\/anti-correlations approach zero when\n$q$ tends to one. Moreover, increasing defect's separation $d$\nincreases the correlations. The dependence on $d$ is more\ninteresting. While the values of correlation functions reach an\nasymptotic value at large d, the behavior is not monotonous.\nEspecially for $C_{d_1+1}$ correlation increases up to a maximum\nand then begin to decrease smoothly towards its asymptotic value.\nThe value of $d$ where $C_{d_1+1}$ is maximized does not show a\nsignificant dependence on $q$. In can be concluded that varying\n$d$ can dramatically affect the system characteristics as far as\ncorrelations are considered.\n\n\n\\subsection{Analytical theory }\n\nOur simulation findings in the preceding section confirms that in\nbetween the defects the correlations are notably higher than other\nsites. In this section we try to develop a theoretical framework\nto capture this feature. Suppose we have two slow defects located\nin the bulk at sites $k$ and $l$ ($k<l$) respectively both with rate\n$q$. The mean occupation at\nsite $i$ in the steady state is denoted by $n_i=\\langle \\tau_i\n\\rangle$, in which $\\tau_i=0,1$ is the occupation number at site\n$i$. We assume that a simple MF assumption, i.e., $\\langle\n\\tau_i\\tau_{i+1}\\rangle =\\langle \\tau_i \\rangle \\langle\n\\tau_{i+1}\\rangle =n_in_{i+1}$ holds for all sites except\n$i=k,\\cdots,l$ i.e.; defective sites themselves and all the sites between them.\nAt these sites the correlations are strong enough to violate the simple\nmean-field assumption. Let us introduce two-point correlation\nfunctions $m_i$ in the following way,\n\n\\begin{eqnarray}\nm_i=\\langle \\tau_{i}\\tau_{i+1}\\rangle.\n\\end{eqnarray}\n\nThere are $L+l-k+2$ unknowns, namely, $\nn_1,n_2,\\cdots,n_L,m_{k},m_{k+1},\\cdots,m_{l}$ and lastly the\ncurrent $J$. In the stationary state there exists $L+1$ equations among these unknowns.\nLet us label them by $A_0$ to $A_{L}$. These\nequations can be obtained by expressing the current $J$ in terms\nof the mean site densities and two point correlators. The first\nequation, $A_0$, is $J=\\alpha(1-n_1)$. The equations $A_i\n$ for $~i=1,\\cdots,k-1$ and $i=l+1,\\cdots,L-1$ have the following form,\n\n\\begin{eqnarray}\nJ= n_i(1-n_{i+1}).\n\\end{eqnarray}\n\nEquation $A_k$ is:\n\\begin{eqnarray}\nJ=q \\langle \\tau_k(1-\\tau_{k+1}) \\rangle =q(n_k-m_k).\n\\end{eqnarray}\nSimilarly, equation $A_l$ is given below\n\\begin{eqnarray}\nJ=q \\langle \\tau_{l}(1-\\tau_{l+1}) \\rangle =q(n_l-m_l).\n\\end{eqnarray}\n\nEquations $A_i~ (i=k+1,\\cdots,l-1) $ have the following form\n\n\\begin{eqnarray}\nJ= \\langle \\tau_{i}(1-\\tau_{i+1}) \\rangle =(n_i-m_{i+1}).\n\\end{eqnarray}\n\nlastly equation $A_L$ is as follows,\n\n\\begin{eqnarray}\nJ=\\beta \\langle \\tau_L\\rangle=\\beta n_L.\n\\end{eqnarray}\n\nWe do not intend to add more equations. Unfortunately the above\nequations are nonlinear and it would be a formidable task to solve\nthem analytically. Alternatively, we shall utilize a numerical\napproach to solve the system of equations by exploiting their recursive structure.\nThis approach was originally introduced in \\cite{foulaadvand07} in the context of\ndisordered ASEP and was later applied to the problem of two\nintersecting ASEP chains \\cite{foulad07}. According to this\nnumerical scheme, we assign a value to $J$. Having $J$, it is possible\nto iterate equations (in forward direction) and obtain $n_1$ up to $n_k$. Then we\nproceed to find $m_k$ by incorporating the relation $J=q(n_k\n-m_k)$. At this stage it is not possible to proceed further because both\n$n_{k+1}$ and $m_{k+1}$ are unknown and we have only one relation\nbetween them : $J=n_{k+1} -m_{k+1}$. In order to proceed, we\napproximate $n_{k+1}$ in the following way. Consider the rate equation for\nthe 2-point function $\\langle \\tau_{k} \\tau_{k+1} \\rangle$ which is governed by\nthe following master equation:\n\n\\begin{eqnarray}\n\\frac{d\\langle \\tau_{k} \\tau_{k+1} \\rangle }{dt}= \\langle\n\\tau_{k-1}(1-\\tau_k)\\tau_{k+1}\\rangle - \\langle\n\\tau_{k}\\tau_{k+1}(1-\\tau_{k+2})\\rangle.\n\\end{eqnarray}\n\nIn the steady state, the left hand side becomes zero, and\ntherefore two terms on the right hand side will be equal. To\nproceed further we have to approximate 3-point functions. This is\nachieved by utilizing the cluster mean-field assumption \\cite{chowdhury02}.\nAccording to this assumption we replace any three point function\nby the product of 2-point functions as follows,\n\n\\begin{eqnarray}\n\\langle n_in_jn_k\\rangle=\\frac{\\langle n_in_j\\rangle\\langle\nn_jn_k\\rangle}{\\langle n_j\\rangle}.\n\\end{eqnarray}\n\nWe then replace all 2-point functions by the product of 1-point\nfunctions except $m_k=\\langle \\tau_{k}\\tau_{k+1} \\rangle$.\nThen it is possible to express $1-n_{k+2}$ in\nterms of $n_{k-1},m_1$ and $n_{k+1}$. After some algebra a quadratic\nequation for $n_{k+1}$ is obtained:\n\n\\begin{eqnarray} n_{k-1}n_{k+1}^2 - m_kn_{k-1}n_{k+1}\n-m_k J=0.\n\\end{eqnarray}\n\nThe physically reasonable solution for this equation is given by\n\n\\begin{eqnarray}\nn_{k+1}= \\frac{ m_k  + \\sqrt{ m_k^2 + \\frac{4m_kJ}{n_{k-1}} }\n}{2}.\n\\end{eqnarray}\n\nNow it is possible to find $m_{k+1}$ via equation $J=n_{k+1} -m_{k+1}$. Analogous to the above\nprocedure we can find $n_{k+2}$ as follows:\n\n\\begin{eqnarray}\nn_{k+2}= \\frac{ qm_km_{k+1}  + \\sqrt{ (qm_km_{k+1})^2 + 4qJn_kn_{k+1}^2m_{k+1} }\n}{2qn_kn_{k+1}}.\n\\end{eqnarray}\n\n\nAfter having $n_{k+2}$ we simply obtain $m_{k+2}$ via equation $J=n_{k+2} -m_{k+2}$. Now it would be possible\nto proceed iteratively after taking into account some approximation. To this end we recall the equality\n\n\\begin{eqnarray}\n\\frac{d\\langle \\tau_{i} \\tau_{i+1} \\rangle }{dt}= \\langle\n\\tau_{i-1}(1-\\tau_i)\\tau_{i+1}\\rangle - \\langle\n\\tau_{i}\\tau_{i+1}(1-\\tau_{i+2})\\rangle.\n\\end{eqnarray}\n\nPutting the left hand side equal to zero in the steady state, utilizing cluster mean-field in three point functions and finally\nsubstituting $m_{i+1}$ by $m_{i+1}=n_{i+1} -J$ we arrive at the following equation:\n\n\\begin{eqnarray}\nn_{i+1}= \\frac{ m_{i-1}m_{i}  + \\sqrt{ (m_{i-1}m_{i})^2 + 4Jn_{i-1}n_{i}^2m_{i} }\n}{2n_{i-1}n_{i}}.\n\\end{eqnarray}\n\nNote that we have approximated $\\langle \\tau_{i-1} \\tau_{i+1} \\rangle$ by the mean-field relation $\\langle \\tau_{i-1} \\rangle \\langle \\tau_{i+1} \\rangle$.\nWe can iterate equation (53) together with $m_{i+1}=n_{i+1}-J$ from $i=k+2$ to $l-2$ to find the corresponding $n_i$ and $m_i$ up to $i=l-1$.\nThe site $i=l$ needs to be treated separately. Following the same strategy we easily find:\n\n\\begin{eqnarray}\nn_{l}= \\frac{ m_{l-2}m_{l-1}  + \\sqrt{ (m_{l-2}m_{l-1})^2 + 4Jn_{l-2}n_{l-1}^2m_{l-1} }\n}{2n_{l-2}n_{l-1}}.\n\\end{eqnarray}\n\n\nFrom which one can compute $m_l$. In a similar fashion, we can obtain $n_{l+1}$. We only should\nshift up all the subscripts in equation (54) by one. Now it is possible again to proceed iteratively to the end of the chain and\nevaluate $n_L$ which gives us the output current $J^{out}$. If the given value of input $J$ were correct,\nthe output current $J^{out}$, which is $\\beta n_L$, should be the same, up to a given precision, as the input value of $J$.\nBy systematically increasing the input $J$ in an small amount $\\delta J$, we can determine the correct $J$ and\ncorrespondingly the mean densities $n_1,\\cdots,n_L$ together with correlators $m_k,\\cdots,m_l$.\nIn Fig. (10) the dependence $J$ on $q$ obtained by the numeric scheme devised above\nis sketched. For the sake of comparison, we have augmented the\nfigure with the analogous graphs obtained by MF, DMF, ISA and MC\nmethods.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig5-colour.eps}\n\\caption{ Fig.10: (Color online) Current vs q for $d=2$ obtained by various analytical and\nnumerical methods and MC simulation. $J$ approaches to 0.25 when $q$ tends to one in accordance to normal ASEP.\n} \\label{fig:bz2}\n\\end{figure}\n\n\nThe result of the numerical scheme is almost identical to ISA\nmethod. They both are in very good agreement with Monte Carlo\nsimulations. However, the numerical scheme has an advantage over\nISA method in the sense that it can easily be implemented for any\n$d$, whereas finding the solution of the ISA nonlinear set of\nequations, i.e., Eq. (29) is not an easy task even by employing\nadvanced numerical methods. Finally in figure (11) we have sketched the dependence of $J$ on $d$ obtained\nfrom MC simulation and the numeric scheme.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig7-colour.eps}\n\\caption{ Fig.11: (Color online) Current vs $d$ for $q=0.1,0.3$ and $0.5$ obtained by the\nnumeric method and MC simulation. } \\label{fig:bz2}\n\\end{figure}\n\nThe results of the numeric method are in rather good agreement\nwith those obtained by MC simulations. $J$ is an increasing\nfunction of $d$ and becomes saturated after some short\n$q$-dependent distance. The results confirms the earlier finding\nin \\cite{dong07}. Note that the length scale on which $J$\nrecovers its single-defect value is of the same order of\nmagnitude of the correlation length in the density profile near\nboundaries. Finally we would like to add that our numerical scheme\nis not capable of reproducing the profile of correlators obtained\nvia MC simulations. Figure (12) depicts the profile of\nunnormalised adjacent two-point correlation function $\\langle\n\\tau_{i+1}\\tau_i \\rangle - \\langle \\tau_{i+1} \\rangle \\langle \\tau_i\n\\rangle$ for two methods of simulation and numerical scheme.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig16-colour.eps}\n\\caption{ Fig.12: (Color online) Profile of corrrelators  for $d=100$ and $q=0.3$. } \\label{fig:bz2}\n\\end{figure}\n\nWe see that within the numerical framework both the value and the extension of correlators are small in comparison to\nthe simulation results. The reason lies in the implementation of a series of approximation in several places in this\nnumerical algorithm.\n\n\n\\section{Summary and Conclusions}\n\nAn open ASEP chain with two defective sites with reduced hopping rates $q<1$ has been investigated. The system current and mean site densities\nat defective sites and their vicinities have been obtained by two analytical methods namely defect mean-field (DMF) and interacting\nsubsystem approximation (ISA). Both methods combine mean-field approach near defects with known exact solutions. Our results are accomplished\nby extensive Monte Carlo simulations. We focus on the phase-segregated phase in which defects globally affect the system properties and\nthe system is not input\/output rate-limited but rather defect-limited. MC simulations have revealed strong short ranged correlations at the\ndefective sites and at all the site between them. Additionally, the profile of density between defects takes a linear form which marks the\nexistence wandering shock in this intermediate region. In order to take into account these correlations, we have introduced a numerical approach\nwhich utilizes a cluster mean-field assumption. Comparison of the three methods show that ISA and the numeric approach give a current value which\nis in a good agreement with MC simulations. DMF method however, only gives a good results compared to MC and other two methods for small $q$ less\nthan $0.1$ which is due to strong correlations. Furthermore, we have obtained the profile of neighbouring two-point correlation function throughout\nthe chain. It is shown that these two point correlators exhibit a rather strong anti correlation at the first defective site then they grow to the\nmiddle of the defects and after that they start diminishing. In general, two point correlators will tend to a tiny value when $q$ approaches one.\nHowever, the dependence of two point correlators for a fixed $q$ as a function of distance between defects is not monotonous. Despite reaching\nto an asymptotic value for large distances, one observes a peak at short distances for the two point correlators near defects. Our theoretical\nanalysis indicates that correlations are critically important for dynamics of particles in disordered ASEPs. It also shows that it is possible\nto devise an approximate method that can take into account these correlations, providing a satisfactory description of stationary properties.\n\n\\section*{Acknowledgments}\n\nABK acknowledges the support from the Welch Foundation (under Grant\nNo. C-1559), and from the US National Science Foundation (grants\nCHE-0237105 and NIRT ECCS-0708765). MEF expresses his gratitude to\nM. F. miri for their useful help.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Solutions}%\n}\n\n\n\n\\section{Introduction} \nThe increasing demand for wireless services and higher data volume per area has brought to the research forefront of both the scientific community (e.g., the METIS project~\\cite{METIS}) and the standardization bodies  a new technology known as massive multiple-input-multiple-output (MIMO) networks~\\cite{Marzetta2010,Rusek2013}. The keystone of this concept, currently considered for the design of  next generation wireless networks~\\cite{5g,Larsson2014,Osseiran2014}, is based on that the numbers of base station (BS) antennas and users are scaled up by 1-2 orders of magnitude under the same time-frequency resources.  This scientific area is motivated by the resulting gains stemming from large channel matrices, i.e., the arising advantages due to the asymptotics of random matrix theory~\\cite{Couillet2011}. Specifically, by  allowing the number of BS antennas tend to infinity,  small-scale fading and thermal noise average out, as well as the analysis is simplified. \n\n\nNotably, the channel state information (CSI)  is  pivotal to the performance of  such systems that take into account for multi-user MIMO (MU-MIMO)~\\cite{Gesbert2007}. Thus, recognizing the need for a realistic study with imperfect CSI, we focus on an important factor-phenomenon, being present in time-varying channels, that deteriorates the quality of CSI available at the BS in \ntime-division-duplex (TDD) systems. Our consideration includes TDD systems and not frequency-division-duplex (FDD) systems, since the  application of the latter in massive MIMO  systems is meaningful only when  \nthe channel matrices are sparse~\\cite{Bjoernson2015}\\footnote{The hypothesis of sparsity is still an open question requiring further research and channel measurements.}. This phenomenon, known as {\\em  channel aging}, describes the mismatch appearing between the current  and the estimated channel due to the relative movement between the users and the BS antennas in addition to any processing delays. In other words, it models the divergence occurring between the channel estimate used in the precoder\/detector, and the channel over which the data transmission actually  takes place. \n\nAlthough the significance of  channel aging due to user mobility, its effect on the performance of massive MIMO systems has not been addressed adequately as  few studies  on this area exist. In fact, only the user mobility has been considered as the only channel aging cause~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC,Papazafeiropoulos2015a,Papazafeiropoulos2015,PapazafeiropoulosPIMRC, Kong2015,Kong2015a}. Basically, channel aging  due to only user mobility was the focal point in~\\cite{Truong2013}, where an application of the deterministic equivalent (DE) analysis was presented by considering linear techniques in the uplink and downlink in terms of maximum ratio combining (MRC) detector and  maximum ratio transmission (MRT) precoder, respectively. Further investigations have been performed in~\\cite{Papazafeiropoulos2015a}, where more sophisticated linear techniques are employed, i.e., minimum-mean-square-error (MMSE) receivers (uplink) and regularized zero-forcing precoders (downlink) are considered, and a comparison between the various linear transceiver techniques is performed. Moreover, in~\\cite{Papazafeiropoulos2015}, the optimal linear receiver in the case of cellular massive MIMO  systems has been derived  by exploiting the correlation between the channel estimates and the interference from other cells, while in~\\cite{PapazafeiropoulosPIMRC}, the uplink analysis of a cellular network with zero-forcing (ZF) receivers that holds for any finite as well as infinite number of BS antennas has been provided.\n\nIn general, hardware undergoes several types of impairments such as high power amplifier nonlinearities and I\/Q imbalance~\\cite{Schenk2008,Studer2010,Qi2012,Mehrpouyan2012,Goransson2008,Bjornson2012Optimal,Qi2010,Bjoernson2013,Zhang2015,Pitarokoilis2015,Bjornson2015,Krishnan2015,Bjornson2014,Demir2000,Bittner2007,Wu2004,Petrovic2007,Krishnan2014,Pitarokoilis2013}\n\\footnote{The arising hardware mismatch between the uplink and the downlink has a negative effect on the  channels reciprocity. For example, a recent insightful work (see \\cite{Zhang2015} and  references therein) has presented the performance analysis in the large number of antennas regime as well as pre-precoding and post-precoding calibration schemes. Herein, focusing on the fundamentals of channel aging, where we want to quantify the individual contributory time-dependent factors (Doppler shift and phase noise), we neglect any hardware mismatch. However, future work can\ninvestigate this mismatch in the channel aging.}.  These hardware impairments can be classified into two categories. In the first category, the variable, describing the hardware impairments, is multiplied with the channel vector, and it might cause channel attenuations and phase shifts. In the case of slow variation of these impairments, they can be characterized as sufficiently static, and thus, can be assimilated by the channel vector by an appropriate scaling of its covariance matrix or due to the property of circular symmetry of the channel distribution.      Note that this factor cannot be incorporated by the channel vector by an appropriate scaling of its covariance matrix or due to the property of circular symmetry of the channel distribution, when it changes faster than the channel. Focusing on this scenario, we include in our analysis the phase noise in the case which results to its accumulation within the channel coherence period~~\\cite{Mehrpouyan2012,Bjornson2015,Pitarokoilis2015,Krishnan2015}. Given that additive transceiver impairments are not time-dependent, we have not included them in our analysis~\\cite{Bjoernson2013,Bjornson2014,Bjornson2015}.\n\nHence, deeper consideration reveals that  user mobility is not the only source of channel aging. Interestingly, {the unavoidable \\em phase noise}, being one of the most severe transceiver impairments~\\cite{Demir2000}, contributes to the further degradation of the system performance due to channel aging because it accumulates over time. Ideally, local oscillators (LOs), used during the conversion of the baseband signal to bandpass and vice-versa, should output a sinusoid with stable amplitude, frequency, and phase. Unfortunately, the presence of phase noise  is inevitable due to the inherent imperfections in the circuitry of  LOs that bring up to the sinusoid a random phase drift.  A lot of research has been conducted regarding phase noise in conventional MIMO systems.  In fact, a big part of the literature has dealt with multi-carrier MIMO systems that employ orthogonal frequency-division multiplexing (OFDM)~\\cite{Bittner2007,Wu2004,Petrovic2007,Krishnan2014}. For example, in~\\cite{Wu2004}, the signal-to-interference-and-noise-ratio (SINR) degradation in OFDM is studied, and a method to mitigate the effect of phase noise is proposed. In~\\cite{Mehrpouyan2012}, a method to jointly estimate the channel coefficients and the phase noise in a point-to-point MIMO system is presented. Not to mention that the degradation appears to be more significant in coherent communications such as massive MIMO  systems.  Specifically, although massive MIMO technology is attractive (cost-efficient) for network deployment, only if the antenna elements consist of inexpensive hardware (more imperfections), most of the research contributions are based on the strong assumption of perfect hardware without phase noise, which is quite idealistic in practice. Reasonably, the introduction of a larger number of antennas will have a more severe contribution to the degradation provoked by phase noise, which could be partially avoided in conventional MIMO  systems, where more expensive hardware with less induced phase is affordable. Hence, the  larger the number of antennas and the corresponding employed LOs, the larger the impact of phase noise. Thus, in the case of massive MIMO  systems, it is a dire necessity to study the impact of channel aging induced by phase noise. Furthermore, motivated by the need of employing low-quality RF elements (e.g., oscillators) for cost-effective construction of large BS antenna arrays, it is  expected that the phase noise will be more severe in a massive MIMO setting.  In particular, the large number of antennas regime has been partially     investigated~\\cite{Pitarokoilis2013,Pitarokoilis2015,Bjornson2015,Krishnan2015}. Specifically, the phase noise has been taken into account for single carrier  uplink  massive MIMO by considering ZF and time-reversal MRC  in~\\cite{Pitarokoilis2013} and~\\cite{Pitarokoilis2015}, respectively. \nRegarding the downlink,  an effort to address the effect of phase noise, being of great importance in massive MIMO systems,  has taken place only in~\\cite{Krishnan2015}. However, the analysis therein does not account for fast time-varying conditions due to the mobility of the users.\n\nIn this paper, we provide a novel realistic CSI model that allows studying channel aging  thoroughly by means of the simultaneous impact of    user mobility and phase noise at the LOs. Taking into account that   next generation wireless systems need to be cost-efficient, implying that real systems may include low-quality RF elements with more imperfections, makes the significance of the proposed model in describing the ineluctable time-varying nature of realistic scenarios remarkable.  Under these conditions, we analyze the downlink performance of a massive MIMO system with imperfect CSI using linear precoders (MRT and RZF). To this end, we invoke random matrix theory (RMT) tools, and we derive the DEs (asymptotically tight approximations as system's dimensions increase). Actually, these approximations agree with simulation results as has been  already shown in the literature~\\cite{Couillet2011,Truong2013,Papazafeiropoulos2015a}. As a result, these deterministic approximations make any lengthy simulations\nunnecessary. Furthermore, as far as phase noise is concerned, we address two system arrangements: a) all BS antennas are connected to a common LO (CLO), and b) each BS antenna includes its own separate LO (SLOs setup). The contributions of this paper can be summarized as follows:\n\\begin{itemize}\n \\item Given a single cell MU-MIMO realistic system model with path loss and shadowing, we present a joint channel-phase noise estimate describing the impact of channel aging by incorporating the time-varying effects of Doppler shift and phase noise at the BS and user elements (UEs).\n \\item Using tools of large RMT, we obtain the DEs of the downlink SINRs, when the BS applies MRT and RZF precoders under imperfect CSI due to uplink training and time variation of the channel (generalized channel aging). Moreover, we account for special cases of the channel covariance, corresponding to various interesting effects  described by the large-scale component, that simplify a lot the resulting expressions.\n \\item We demonstrate that the square root power-law, generally holding in imperfect CSI scenarios, stands for the case of generalized channel aging as well. Hence, the transmit power per user can be cut down by scaling the number of BS antennas to the inverse of their square root.\n \\item We corroborate our analysis concerning deterministic approximations by performing simulations coinciding with the analytical expressions.  \n \\item We elaborate on insightful conclusions, drawn from our analysis. For example, Doppler shift has a far more severe impact than phase noise even for low vehicle velocities. \nIn addition, massive MIMO  systems outperform conventional MIMO even in generalized channel aging conditions with RZF precoder behaving better than MRC for various Doppler shifts and phase noise severities.  \n \\end{itemize}\n\nThe remainder of this paper is organized as follows. Section II introduces the system  and signal models accounting for both the effects of user mobility and phase noise, which allow a more complete characterization of channel aging. Next, Section III presents the downlink transmission, in terms of the received signal, the applied precoders, and the achievable user rate.  In Section IV, we proceed with the main results that include the derivation of the DEs of the downlink SINRs, as well as the investigation of certain simpler cases such as the scenario where the large-scale fading component is absent. Section V reports a set of numerical results that confirms the validity of our analytical results, and sheds light on the behavior of channel aging for various mobility environments, phase noise settings, and a number of antennas in a massive MIMO system. We summarize our results and observations in Section VI. In Appendix A, some useful results from the literature are presented, while the following appendices provide the proofs for the main analytical results.\n\n\\textit{Notation:} Vectors and matrices are represented by boldface lower and upper case symbols. $(\\cdot)^{\\scriptscriptstyle\\mathsf{T}}$, $(\\cdot)^\\H$, and $\\mathop{\\mathrm{tr}}\\nolimits{\\cdot}$ denote the transpose, Hermitian  transpose, and trace operators, respectively. The expectation and variance operators, as well as the spectral norm of a matrix are denoted by $\\mathbb{E}\\left[\\cdot\\right]$ and $\\mathrm{var}\\left[\\cdot\\right]$, as well as $\\| \\cdot \\|$,\nrespectively, while $\\mathrm{Re}\\{\\cdot\\}$  is the real part of a complex number. The  $\\mathrm{diag}\\{\\cdot\\}$ operator generates a diagonal matrix from a given vector, and the symbol $\\triangleq$ declares definition. The notations $\\mathcal{C}^{M}$ and $\\mathcal{C}^{M\\times N}$ refer to complex $M$-dimensional vectors and  $M\\times N$ matrices, respectively. Finally, ${\\mathbf{b}} \\sim {\\cal C}{\\cal N}{({\\mathbf{0}},\\mathbf{\\Sigma})}$  or ${\\mathbf{b}} \\sim {\\cal N}{({\\mathbf{0}},\\mathbf{\\Sigma})}$ denote a circularly symmetric complex Gaussian  or a real Gaussian vector ${\\mathbf{b}}$ with zero-mean and covariance matrix $\\mathbf{\\Sigma}$.\n\n\n\n\n\n\n\n\\section{System Model}\nThis work considers a single-cell system, where a BS communicates with a number of single-antenna non-cooperative UEs belonging to the set $\\mathcal{K}$ with cardinality $K=|\\mathcal{K}|$ being the number of UEs. In particular,  each UE is assigned an index $k$ in $\\mathcal{K}$, while the BS is deployed with an array of $M$ antennas\\footnote{The antennas can be co-located at the BS or can be distributed as fully-coordinated, multiple, smaller BSs~\\cite{Bjornson2015}.}. Since each UE has a single antenna, the terms user and user element, are used interchangeably in the following analysis. Our interest focuses on large-scale topologies, where both number of antennas $M$ and users $K$ grow infinitely large while keeping a finite ratio $\\beta$, i.e., $M, K\\to \\infty$ with $K\/M=\\beta$ fixed such that $\\mathrm{lim\\,sup}_{M} K\/M < \\infty$\\footnote{ In the following, in order to simplify notation, $M\\to \\infty$ will be used instead of $M, K\\to \\infty$, unless  stated otherwise.}.\n\nDuring the transmission  of the $n$th symbol, the channel gain vector between the BS and user $k$, denoted by ${\\mathbf{h}}_{k,n}  \\in {\\mathbb{C}}^{M}$ exhibits flat-fading, and it is assumed constant for the symbol period, while it may vary slowly from symbol to symbol. Basically, given the time-frequency resources of the system, the symbol duration is assumed smaller or equal to the coherence time of all the users. Moreover, in order to account for certain inevitable effects such as path loss and lognormal shadowing, which are dependent on the distance of the users from the BS,  we express the channel between user $k$ and the BS as\n\\begin{align}\n {\\mathbf{h}}_{k} = {\\mathbf{R}}^{1\/2}_{k}{\\mathbf{w}}_{k},\n\\end{align}\nwhere ${\\mathbf{R}}_{k} =\\mathbb{E}\\left[ {\\mathbf{h}}_{k,n} {\\mathbf{h}}_{k,n}^{\\H}\\right]\\in {\\mathbb{C}}^{M \\times M}$ is a deterministic Hermitian-symmetric positive-definite matrix representing the aforementioned effects\\footnote{The independence of ${\\mathbf{R}}_{k}$ from the time index, $n$, originates from the assumption that the  shadowing vary with time in a longer slower pace  than the coherence time~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}. Interestingly, this model is quite versatile, since  ${\\mathbf{R}}_{k}$ can also describe the antenna correlation due to either insufficient antenna spacing or a lack of scattering.However, the following results focus only on diagonal ${\\mathbf{R}}_{k}$ matrices describing only large-scale fading, and ${\\mathbf{w}}_{k,n} \\in {\\mathbb{C}}^{M}$ is an uncorrelated fast-fading Gaussian channel vector drawn as  a realization from a zero-mean circularly symmetric complex Gaussian distribution, i.e., ${\\mathbf{w}}_{k,n} \\sim {\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{I}}_{M})$.}. \n\n\\subsection{Phase Noise Model}\nIn practice, both  the transmitter and the receiver are impaired by phase noise induced during the up-conversion of the baseband signal to passband and vice-versa\\footnote{The conversion takes place by multiplying the signal with the LO's output.}. As a result, the received signal is distorted during the reception processing, while at the transmitter side, a mismatch appears between the signal that  is intended to be transmitted and the generated signal.\n\nIn our analysis regarding the $n$th time slot, we take into account for non-synchronous operation at the BS, i.e., the BS antennas have independent phase noise processes $\\phi_{m,n}, m=1,\\ldots,M$ with $\\phi_{m,n}$ being the phase noise process at the $m$th antenna. Note that the  phase noise processes are considered as mutually independent, since each antenna has its own oscillator, i.e., separate LOs (SLOs) at each antenna~\\cite{Pitarokoilis2015,Krishnan2015}.  In case that all the antennas have a common LO (CLO), the phase noise processes $\\phi_{m,n}$ are identical for all $m = 1, \\ldots, M$ (synchronous operation). Similarly, we denote $\\varphi_{k,n}$, $k=1,\\ldots, K$ the phase noise process at the single-antenna user $k$. \n\n\nPhase noise during the $n$th symbol can be described by a discrete-time  independent Wiener process, i.e., the phase noises at the  LOs of the $m$th antenna of the BS and $k$th user are modeled as\n\\begin{align}\n \\phi_{m,n}&=\\phi_{m,n-1}+\\delta^{\\phi_{m}}_{n}\\label{phaseNoiseBS}\\\\\n \\varphi_{k,n}&=\\varphi_{k,n-1}+\\delta^{\\varphi_{k}}_{n},\\label{phaseNoiseuser}\n\\end{align}\nwhere $\\delta^{\\phi_{m}}_{n}\\sim {\\cal N}(0,\\sigma_{\\phi_{m}}^{2}) $ and $\\delta^{\\varphi_{k}}_{n}\\sim {\\cal N}(0,\\sigma_{\\varphi_{k}}^{2})$~\\cite{Demir2000,Pitarokoilis2015,Krishnan2015}. Note that $\\sigma_{i}^{2}=4\\pi^{2}f_{\\mathrm{c}} c_{i}T_{\\mathrm{s}}$, $i=\\phi_{m}, \\varphi_{k}$ describes the phase noise increment variance with $T_{\\mathrm{s}}$, $c_{{i}}$, and $f_{\\mathrm{c}}$ being the  symbol interval, a constant dependent on the oscillator, and the carrier frequency, respectively. \n\nThe phase noise, being the distortion in the phase due to the random phase drift  in the signal coming from the LOs of the BS and user $k$,  is expressed as a multiplicative factor to the channel vector as \n\\begin{align}\n{\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n},\n\\end{align}\nwhere ${\\boldsymbol{\\Theta}}_{k,n}\\!\\triangleq\\!\\mathrm{diag}\\!\\left\\{ e^{j \\theta_{k,n}^{(1)}}, \\ldots, e^{j \\theta_{k,n}^{(M)}} \\!\\right\\}=e^{j \\varphi_{k,n}}{\\boldsymbol{\\Phi}}_{n}\\in \\mathbb{C}^{M\\times M}$ with ${\\boldsymbol{\\Phi}}_{n}\\!\\triangleq\\!\\mathrm{diag}\\!\\left\\{\\! e^{j \\phi_{1,n}}, \\ldots, e^{j \\phi_{M,n}} \\!\\right\\}$ being the phase noise sample matrix  at time $n$ because of the imperfections in the LOs of the BS,  while $e^{j \\varphi_{k,n}}$ corresponds to the phase noise contribution by the LO of user $k$ \\cite{Pitarokoilis2015,Krishnan2015}.\n\\subsection{Channel Estimation}\nIn realistic conditions,  where no perfect CSI is available, the BS needs to estimate the uplink channel~\\cite{Papazafeiropoulos2015a,Ngo2013,Hoydis2013}. We take advantage of the TDD operation scheme, where  an uplink training phase during each transmission block (coherence period) exists by means of transmit pilot symbols. Since in TDD the number of required pilots scales with the number of terminals $K$, but not the number of BS antennas $M$, not only the analysis of systems with large number of BS antennas is  possible because of the finite duration of the coherence time\\footnote{TDD is the only viable solution in fast time-varying channel conditions with massive MIMO systems, where the limitations for the coherence time are more stringent, while FDD can be considered only when there is some kind of channel sparsity~\\cite{Bjoernson2015}.}, but also the knowledge of the downlink channel is considered known due to the property of reciprocity.   \n\n\nBy assuming that the channel estimation takes place at time $0$, we now derive the joint channel-phase noise linear minimum mean-square\nerror detector  (LMMSE) estimate of the effective channel ${\\mathbf{g}}_{k,0}={\\boldsymbol{\\Theta}}_{k,0}{\\mathbf{h}}_{k,0}$ in the presence of phase noise, small-scale fading, and channel impairments such as path loss. During this phase, we neglect the channel aging effect by assuming that both the channel and phase noise remain constant~\\cite{Truong2014}. This is a valid assumption since the duration of the training phase is small, and the consequent time-variation of the channel is unnoticeable.  However, in the data transmission phase, taking place for ($T_{\\mathrm{c}} -\\tau$ ) symbols, the channel is supposed to vary from symbol to symbol, where $\\tau$ and $T_{\\mathrm{c}}$ are  the duration of the  training sequence during the training phase and   the channel coherence time, respectively.\n\\begin{proposition}\\label{LMMSE} \nThe LMMSE estimator of ${\\mathbf{g}}_{k,0}$, obtained during the training phase, is\n \\begin{align}\\label{estimatedChannel}\n \\hat{{\\mathbf{g}}}_{k,0}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{\\mathrm{b}}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}\\tilde{{\\mathbf{y}}}_{k,0}^{{\\mathrm{p}}},\n\\end{align}\nwhere $\\sigma_{\\mathrm{b}}^{2}$ is the variance of the post-processed noise at\nbase station, $\\tilde{{\\mathbf{y}}}_{,k,0}^{{\\mathrm{p}}}$ is a noisy observation of the effective channel from user $k$ to the BS, and $p_{\\mathrm{p}}=\\tau p_{\\mathrm{u}}$  with $p_{\\mathrm{u}}$   being the power per user in the uplink data transmission phase.\n\\end{proposition}\n\\proof: The proof of Proposition~\\ref{LMMSE} is given in Appendix~\\ref{proposition1}.\\endproof\n\n\\subsection{Channel Aging}\nThe relative movement between the users and the BS inflicts a phenomenon, known as channel aging in the literature~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}. Specifically, the  Doppler shift arising from this movement causes the channel to vary between when it is learned by estimation, and when the estimate is applied for detection\/precoding. Especially, channel aging is more severe in massive MIMO systems, where  narrow transmit beams appear due to the high angular resolution. Thus, the precoders that are used for achieving this spatial focus, bring a critical performance loss, which worsens as the users move with increasing velocity~\\cite{Ying2014}. An autoregressive model of order $1$ enables  the  joint channel-phase noise process ${\\mathbf{g}}_{k,n}$ between the BS and the $k$th user at time $n$ (current time) to be approximated as\\footnote{Contrary to other works~\\cite{Vu2007,Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}, ${\\mathbf{A}}_{n}$, being the factor of ${\\mathbf{g}}_{k,0}$,   is described by a matrix  instead of a scalar to account for the setting of SLOs. Moreover, in this work, ${\\mathbf{A}}_{n}$ is time-dependent by the  delay  $n$ in terms of the number of symbols transmitted, in order to include the accumulative channel aging effect~\\cite{Kong2015a}.  Therefore, the longer the delay, the smaller the ${\\mathbf{A}}_{n}$, which in turn results in less accurate CSI. Hence, the channel estimated at time $0$, is used at time $1$, $2$, and so on till $n$. It is intuitive that the CSI will be much less reliable at time $n$, which results to a more realistic model. We assume that\nit can be embodied in the channel vector ${\\mathbf{g}}_{k,0}$ by employing the circular symmetry of the channel distribution, or mainly, by the inclusion of an appropriate scaling of the covariance matrix.}\n\\begin{align}\n{\\mathbf{g}}_{k,n}  = {\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0} + {\\mathbf{e}}_{k,n},\\label{eq:GaussMarkoModel}\n\\end{align}\nwhere, initially, ${\\mathbf{A}}_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n) \\mat{\\mathrm{I}}_{M}$ is supposed to model 2-D isotropic scattering~\\cite{WCJr1974}\\footnote{Normally, the BS antennas are deployed in a fixed space instead of considering  distantly distributed small BSs constituting one massive MIMO BS. In such case, ${\\mathbf{A}}_{n}$ is a scalar, since it can be  reasonably assumed that all the BS antennas have the same relative movement comparing to the user.}, ${\\mathbf{g}}_{k,0}$ is the channel vector received during the training phase, and ${\\mathbf{e}}_{k,n} \\in {\\mathbb{C}}^{M}$ is the uncorrelated channel error vector due to the channel variation modelled as a stationary Gaussian random process with independent and identically distributed (i.i.d.)~entries and distribution ${\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{R}}_{k}-{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n})$. \n\nThis model can be enriched by incorporating other interesting effects. By introducing the effect of channel estimation to the autoregressive model of~\\eqref{eq:GaussMarkoModel}, the BS  takes into account for both estimated and delayed CSI, in order to   design the detector ${\\mathbf{W}}_{n}\\in \\mathbb{C}^{M\\times M}$ or the precoder ${\\mathbf{F}}_{n}\\in \\mathbb{C}^{M\\times M}$ at time instance $n$ for the uplink and downlink, respectively. Hence, the effective channel at time $n$ is expressed by\n\\begin{align}\n{\\mathbf{g}}_{k,n} \n= & {\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0} + {\\mathbf{e}}_{k,n}\\\\\n= & {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} + \\tilde{{\\mathbf{e}}}_{k,n},\\label{eq:GaussMarkov2}\\end{align} \nwhere $\\hat{{\\mathbf{g}}}_{k,0}\\sim {\\cal C}{\\cal N}\\left({\\mathbf{0}},{\\mathbf{D}}_{k}\\right)$ with ${\\mathbf{D}}_{k}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{b}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}{\\mathbf{R}}_{k}$ and  $\\tilde{{\\mathbf{e}}}_{k,n}\\triangleq {\\mathbf{A}}_{n} \\tilde{{\\mathbf{g}}}_{k,n} +  {\\mathbf{e}}_{k,n}\\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}{\\mathbf{D}}_{k}{\\mathbf{A}}_{n})$  are mutually independent, while ${\\mathbf{A}}_{n}$, is assumed to be known to the BS, now incorporates both the effects of 2-D isotropic scattering, user mobility,  and phase noise, as shown below by Theorem~\\ref{LMMSE}. In other words, the combined error $\\tilde{{\\mathbf{e}}}_{k,n}$ depends on both the imperfect and the general delayed CSI effects, allowing the elicitation of interesting outcomes during the ensuing analysis in Section~\\ref{Deterministic}.  \n  \nWe turn our attention to the derivation of ${\\mathbf{A}}_{n}$ by accounting for user mobility, phase noise, and the general channel effects modelled by  ${\\mathbf{R}}_{k}$.\n\\begin{Theorem}\\label{LMMSE}\n The  effective joint channel-phase noise  vector at time $n$  is expressed as\n \\begin{align}\n  {{\\mathbf{g}}}_{k,n}^{\\mathrm{Joint}}={\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0}\n \\end{align}\nwith \n\\begin{align}\n{\\mathbf{A}}_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}}{2}n}\\Delta{\\boldsymbol{\\Phi}}_{n}, \\label{agingParameter}\n\\end{align}\nwhere $\\Delta{\\boldsymbol{\\Phi}}_{n}=\\mathrm{diag}\\left\\{e^{-\\frac{\\sigma_{\\phi_{1}}^{2}}{2}n} ,\\ldots,e^{-\\frac{\\sigma_{\\phi_{M}}^{2}}{2}n}\\right\\}$.\n \\end{Theorem}\n\\proof\nThe total mean square error (MSE), which is actually  given by means of the error covariance matrix, is $\\mathrm{MSE}=\\mathop{\\mathrm{tr}}\\nolimits {{\\mathbf{E}}}_{k,n}$, where ${{\\mathbf{E}}}_{k,n}$ is the error covariance written as\n\\begin{align}\n&\\!\\!{{\\mathbf{E}}}_{k,n}=\\mathbb{E}\\left[ \\left( {\\mathbf{g}}_{k,n}-{\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0} \\right) \\left( {\\mathbf{g}}_{k,n}-{\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0} \\right)^{\\H}\\right]\\nonumber\\\\\n&\\!\\!=\\mathbb{E}| {\\mathbf{g}}_{k,n}|^{2}+{\\mathbf{A}}_{n} \\mathbb{E}|{{\\mathbf{g}}}_{k,0}|^{2}{\\mathbf{A}}_{n}^{\\H}-2\\mathbb{E}\\left[ \\mathrm{Re}\\left\\{{\\mathbf{A}}_{n}  {{\\mathbf{g}}}_{k,0}{\\mathbf{g}}_{k,n}^{\\H} \\right\\} \\right]\\nonumber\\\\\n&\\!\\!={\\mathbf{R}}_{k}+{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n}^{\\H}-2\\mathbb{E}\\left[ \\mathrm{Re}\\left\\{{\\mathbf{A}}_{n}  {{\\mathbf{g}}}_{k,0}{\\mathbf{g}}_{k,n}^{\\H} \\right\\} \\right]\\nonumber\\\\\n&\\!\\!\\!\\!={\\mathbf{R}}_{k}\\!+\\!{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n}^{\\H}\\!-\\!2 {\\mathrm{J}_{0}}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}}{2}n}\\!{\\mathbf{A}}_{n} {\\mathbf{R}}_{k}\\Delta{\\boldsymbol{\\Phi}}_{n}.\\label{error_Cov_mat}\n\\end{align}\nNote that the expectation is taken over all channel and phase noise  realizations. Also,  we have used that ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n}$ with ${\\boldsymbol{\\Theta}}_{k,n}$ and ${\\mathbf{h}}_{k,n}$ being uncorrelated. Also, we have denoted $\\Delta{\\boldsymbol{\\Phi}}_{n}=\\mathrm{diag}\\left\\{e^{-\\frac{\\sigma_{\\phi_{1}}^{2}}{2}n} ,\\ldots,e^{-\\frac{\\sigma_{\\phi_{M}}^{2}}{2}n}\\right\\}$, and  \n\\begin{align}\n\\mathbb{E} [{{\\mathbf{h}}}_{k,0}{{\\mathbf{h}}}_{k,n}^{\\H}] = \\mathrm{J}_{0}(2 \\pi f_{D}T_{s}n){\\mathbf{R}}_{k}.\\label{eq:autoCorrelation2}\n\\end{align}\nThe multiplcative parameter matrix ${\\mathbf{A}}_{n}$ is determined by minimizing the MSE in~\\eqref{error_Cov_mat}. Thus, if we differentiate~\\eqref{error_Cov_mat} with respect to ${\\mathbf{A}}_{n}$, and equate the resulting expression to zero,  we obtain the Hermitian  matrix ${\\mathbf{A}}_{n}$ as in~\\eqref{agingParameter}.\n\\endproof\n\\begin{remark}\n During the data transmission phase, the available CSI at time $n$ is obtained in terms of the estimated CSI at time 0. The parameter matrix ${\\mathbf{A}}_{n}$  has a physical meaning, since it describes the effects of 2-D isotropic scattering, user mobility, and phase noise impairments at  both the UE and the BS. \n\\end{remark}\n\\begin{remark}\nNotably, phase noise induces an extra loss due to the additional deviation between the actual channel at time n, and the channel estimate from time 0 because of the symbol-by-symbol random phase drift. In other words, the phase noise contributes to the channel aging phenomenon that  imposes a further challenge to  investigate the realistic potentials of massive MIMO systems.\n\\end{remark}\n\n\n\\begin{corollary}\n In the case where  all the BS antennas are connected to the same oscillator or  all the oscillators are identical, ${\\mathbf{A}}_{n}$ degenerates to a scalar $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi_{k}}^{2}}{2}n}$ or a scaled identity matrix $\\alpha_{n}\\mat{\\mathrm{I}}_{M}$, respectively. \n\\end{corollary}\n\n\n\n\\section{Downlink Transmission}\nThe BS transmits data simultaneously to all users in its cell by employing space-division multiple access (SDMA). Applying reciprocity, the downlink channel is written as the Hermitian  transpose of the uplink channel. The coherence time,  the received signal $y_{k,n}\\in{\\mathbb{C}}$ by the $k$th UE during the data transmission phase ($n=\\tau+1,\\ldots, T_{\\mathrm{c}}$) is\n\\begin{align}\n\\!\\!y_{k,n}=\\sqrt{p_{\\mathrm{d}}}{\\mathbf{h}}^\\H_{k,n}{\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{s}}_{n}+z_{k,n}\\label{eq:DLreceivedSignal}\n\\end{align}\nwhere  ${\\mathbf{s}}_{n}=\\sqrt{\\lambda}{\\mathbf{F}}_{n}{\\mathbf{x}}_{n}$ denotes the signal vector transmitted by the BS with $\\lambda$, ${\\mathbf{F}}_{n} \\in {\\mathbb{C}}^{M \\times K}$ and  ${\\mathbf{x}}_{n} = \\big[x_{1,n},~x_{2,n},\\cdots,~x_{K,n}\\big]^{\\scriptscriptstyle\\mathsf{T}} \\in {\\mathbb{C}}^{K}\\sim {\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{I}}_{K})$ being a normalization parameter, the linear precoding matrix,  and the data symbol vector to its $K$ served users, respectively. Moreover, we have assumed that the BS transmits data to all the users with the same power $p_{\\mathrm{d}}$, and $z_{k,n} \\sim {\\cal C}{\\cal N}(0,\\sigma_{k}^{2})$ is complex Gaussian noise at user $k$.  The need for constraining the transmit power per user to $p_{\\mathrm{d}}$, i.e., $\\mathbb{E}\\left[\\frac{p_{\\mathrm{d}}}{K}{\\mathbf{s}}_{n}^{\\H}{\\mathbf{s}}_{n}\\right]=p_{\\mathrm{d}}$, provides the normalization parameter as\n\\begin{align}\n\\lambda=\\frac{1}{\\mathbb{E} \\left[\\frac{1}{K} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{F}}_{n}{\\mathbf{F}}^\\H_{n}\\right]}. \\label{eq:lamda} \n\\end{align}\n\nHenceforth, by taking into account that ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n}$, we proceed with the following presentation of the received signal, obtained by~\\eqref{eq:DLreceivedSignal}.  Thus, we have\n\\begin{align}\n&\\!\\!y_{k,n}\\!\\!=\\!\\!{\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\!{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}x_{k,n}\n\\!\\!+\\!\\!\\sum_{i \\neq k}\\!\\! {\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\!{\\boldsymbol{\\Theta}}_{k,n}^{2} { {\\mathbf{f}}}_{i,n}x_{i,n}\\!+\\!z_{k,n}\\label{eq:DLreceivedSignal1}\\\\\n\\!\\!&=\\!\\!\\underbrace{{ \\sqrt{\\!\\lambda p_{\\mathrm{d}}}}\\mathbb{E}\\!\\!\\left[{\\mathbf{g}}^\\H_{k,n} \\!{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}\\right]\\!x_{k,n}}_{\\mathrm{desired~signal}}\\!+\\!\\underbrace{z_{k,n}}_{\\mathrm{noise}}\n\\!\\!+\\!\\!\\sum_{i \\neq k} \\!{\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\! {\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{i,n}x_{i,n}\\nonumber\\\\\n\\!\\!&+\\!{ \\sqrt{\\!\\lambda p_{\\mathrm{d}}}}\\left({\\mathbf{g}}^\\H_{k,n}{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}x_{k,n}\\!-\\!\\mathbb{E}\\!\\!\\left[{\\mathbf{g}}^\\H_{k,n} {\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}\\right]\\!x_{k,n}\\right)\\!.\\label{eq:DLreceivedSignalgeneral}\n\\end{align}\n\nIn~\\eqref{eq:DLreceivedSignal1}, we have  applied a similar technique to~\\cite{Medard2000} because UEs do not have access to instantaneous CSI during the downlink phase, but we can  assume that, in particular, UE $k$  is aware of only $\\mathbb{E}[{\\mathbf{g}}^\\H_{k,n} {\\mathbf{f}}_{jm} ]$.\n\nGiven that the input symbols are Gaussian, we provide the achievable user rate based on the analysis in~\\cite[Theorem $1$]{Hassibi2003}. Specifically, the worst case uncorrelated additive noise is also zero-mean with variance equal to the variance of interference plus noise. In addition, since ${\\mathbf{g}}_{k,n}$ is circularly symmetric, we let ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,0}^{2}{\\mathbf{g}}_{k,n}$. Thus, by denoting $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=\\Delta{{\\boldsymbol{\\Theta}}}_{k,n}$,  where $\\Delta{{\\boldsymbol{\\Theta}}}_{k,n}=\\left( {\\boldsymbol{\\Theta}}_{k,n}{{\\boldsymbol{\\Theta}}_{k,0}}^{\\!\\!\\!\\H} \\right)^{2}=\\mathrm{diag}\\left\\{ e^{2 j \\left(\\theta_{k,n}^{(1)}-\\theta_{k,0}^{(1)}  \\right)}, \\ldots, e^{2 j \\left(\\theta_{k,n}^{(1)}-\\theta_{k,0}^{(1)}  \\right)} \\right\\}$, we may consider a SISO model with the desired signal power and the interference plus noise power at UE $k$ given by \n \\begin{align}\nS_{k,n}\n=  \\lambda\\Big|\\mathbb{E}\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{f}}}_{k,n}\\right]  \\Big|^{2},\\label{eq:DLgenSignalPower}\n\\end{align}\nand \n\\begin{align}\nI_{k,n}\n&\\!=\\!   \\lambda{\\mathrm{var}}\\!\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{f}}_{k,n}\\right] \\! +  \\! \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}} \\!+\\!    \\sum_{i\\ne k}   \\!  \\lambda\\mathbb{E}\\bigg[\\Big|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{f}}_{i,n}\\Big|^{2}\\bigg] .   \\label{eq:DLgenIntfPower}\n\\end{align}\n\nSince the available CSI at the BS at time $n$ is ${\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}$,   as can be seen from~\\eqref{eq:GaussMarkov2}, the expression corresponding to the MRT precoder is\n\\begin{align}\n{\\mathbf{F}}_{n}&=  {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}. \\label{eq:precoderMRT}\n\\end{align}\n\nIn a similar way, \nthe BS  designs its RZF precoder as\\! \\cite{Hoydis2013}\n\\begin{align}\n{\\mathbf{F}}_{n}\n&= \\left({\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^\\H {\\mathbf{A}}_{n} \\!+\\! {\\mathbf{Z}} + M  a \\mat{\\mathrm{I}}_M\\right)^{-1}{\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}\\nonumber\\\\\n&= {{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}, \\label{eq:precoderRZF}\n\\end{align}\nwhere we define ${{\\mathbf{\\Sigma}}}\\triangleq\\left({{\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H}{\\mathbf{A}}_{n}} + {\\mathbf{Z}} +   a M \\mat{\\mathrm{I}}_M\\right)^{-1}$ with ${\\mathbf{Z}} \\in {\\mathbb{C}}^{M \\times M}$ being an arbitrary Hermitian\nnonnegative definite matrix and $a$ being a regularization scaled by $M$, in order to converge to a constant, as $M$, $K\\to \\infty$. Note that both ${\\mathbf{Z}}$ and $a$ could be optimized, but this is outside the\nscope of this paper and left to future work. One possible common choise for these parameters is provided in~\\cite{Hoydis2013}.\n\n\nFollowing a similar approach to~\\cite{Bjornson2015,Pitarokoilis2015}, we compute the achievable rate for each time instance of the data transmission phase. Thus, the downlink ergodic achievable rate of user $k$, is given by\n\\begin{align}\nR_{k}& = \\frac{1}{T_{c}}\\sum_{n=1}^{T_{c}-\\tau}R_{k,n}\\nonumber\\\\\n &=\\frac{1}{T_{c}}\\sum_{n=1}^{T_{c}-\\tau}\\mathrm{log}_{2}\\left(1+ \\gamma_{k,n} \\right),\n\\label{eq:DLrate}\n\\end{align}\nwhere $\\gamma_{k,n}=\\frac{S_{k,n} }{I_{k,n}}$ is the instantaneous downlink SINR at time $n$.  In other words, the mutual information between the received signal and the transmitted symbols is lower bounded by the achievable rate per user provided in~\\eqref{eq:DLrate}. \n\n\n\\section{Deterministic Equivalent Analysis}\\label{Deterministic}\nIn this section, we provide the derivations of the downlink SINRs after applying  MRT and RZF precoders in the presence of imperfect  and delayed CSI due to user mobility and phase noise. Use of RMT in terms of DEs enables the extraction of asymptotic expressions as $K,M \\rightarrow \\infty$, while keeping their ratio ${K}\/{M}=\\beta$ finite. Given that the theory of  DEs provides tight approximations  even for moderate system dimensions, the significance of our  results is of great importance. Confirmation of this tightness is provided in Section~\\ref{results} by  simulations. Interestingly, the results mirror the dependence of the harmful effects described by channel aging.\n\nThe DE of the SINR $\\gamma_{k,n}$ is such that $\\gamma_{k,n}-\\bar{\\gamma}_{k,n}\\xrightarrow[M \\rightarrow \\infty]{\\mbox{a.s.}}0$\\footnote{Note that $\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}$ denotes almost sure convergence, and  $a_n\\asymp b_n$ expresses the equivalence relation $a_n - b_n  \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}  0$ with $a_n$  and $b_n$  being two infinite sequences.}, while the deterministic rate of user $k$ is obtained by the dominated convergence~\\cite{Billingsley2008} and the continuous mapping theorem~\\cite{Vaart2000} by means of~\\eqref{eq:DLrate} \n\\begin{align}\nR_{k}-\\frac{1}{T_{\\mathrm{c}}}\\sum_{n=1}^{T_{\\mathrm{c}}-\\tau}\\log_{2}(1 + \\bar{\\gamma}_{k,n}) \\xrightarrow[ N \\rightarrow \\infty]{\\mbox{a.s.}}0.\\label{DeterministicSumrate}\n\\end{align}\n\n\n\n\n\nThe DE downlink achievable  user rates corresponding to MRT and RZF can be obtained by means of the following theorems. Hereafter, for the sake of exposition and without loss of generality, we assume that in the case of  SLOs, the phase noises obey to identical statistics. However, in Appendix~\\ref{theorem2}, we scrutinize the general setting, where  phase noises across different LOs  are independent, but not identically distributed. First, we consider MRT precoding. \n\\subsection{MRT}\nThis section presents the general DE regarding the SINR with MRT precoding. Then, this asymptotic result particularizes to a number of cases of interest. In additions, it is investigated the dependence of the transmit power per user with the number of BS antennas (power scaling law), in order to support a specific desired rate.\n\\begin{Theorem}\\label{theorem:DLagedCSIMRT}\nThe downlink DE of the SINR of user $k$ at time $n$ with MRT precoding , accounting for imperfect CSI and  delayed CSI due to phase noise and user mobility, is given by \n\\begin{align}\n\\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n} {\\delta}_{k}^{2}}{{\\frac{1}{M}{\\delta}_{k}^{'}}+  \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}  \\bar{\\lambda}M}\n + \\sum_{i\\neq k}\\frac{1}{M}\\delta_{i}^{''}},\\label{eq:DLdelayedCSIetaMRT1}\n \\end{align}\n with \n  \\begin{align}\n  \\bar{\\lambda}&=\\left( \\frac1K\\sum_{i=1}^{K}\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)^{-1},\\nonumber\n  \\end{align}\n${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2} {\\mathbf{D}}_{k} \\left( {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)$, and $\\delta_{i}^{''}=\\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{R}}_{k}$.\n\\end{Theorem}\n\\proof: The proof of Theorem~\\ref{theorem:DLagedCSIMRT} is given in Appendix~\\ref{theorem2}.\\endproof\n\\begin{corollary}\\label{NoCorrelation}\nIn  case that ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$ (no large-scale component), and ${\\mathbf{A}}_{n}$ degenerates to a scalar $\\alpha_{n}$ or a $\\alpha_{n}\\mat{\\mathrm{I}}_{M}$, i.e., the BS employs a CLO or a SLOs setting with identical LOs, respectively, we obtain\n \\begin{align}\n  \\bar{\\gamma}_{k,n} =\\frac{\\alpha_{n}^{2} d  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n} }{  \\frac{1-\\alpha_{n}^{2} d}{M}+  \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}} M}\n +{  \\frac{ K-1 }{M} }},\\label{eq:DLdelayedCSIetaMRT3}\n \\end{align}\n where $d=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}$ and $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}}{2}n}$.\n \\end{corollary}\n\\proof Using ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$ leads to ${\\mathbf{D}}_{k}=d\\,\\mat{\\mathrm{I}}_{M}=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}\\mat{\\mathrm{I}}_{M}$. Moreover, if ${\\mathbf{A}}_{n}$ is replaced by the scalar $\\alpha_{n}$ or the scaled identity matrix $\\alpha_{n}\\mat{\\mathrm{I}}_M$,   Theorem~\\ref{theorem:DLagedCSIMRT} allows to simplify enough the deterministic SINR. Hence, by substituting  $\\bar{\\lambda}=\\left(\\alpha_{n}^{2} d\\right)^{-1}$, ${\\delta}=\\alpha_{n}^{2} d$, ${\\delta}^{'}=\\alpha_{n}^{2} d \\left( 1 - \\alpha_{n}^{2} d \\right)$, and $\\delta^{''}=\\alpha_{n}^{2} d$, we obtain $\\bar{\\gamma}_{k,n}$ as in~\\eqref{eq:DLdelayedCSIetaMRT3}.\n\\endproof\n\nNevertheless, it is interesting to investigate the effect of the combined impairments of phase noise and user mobility  on the asymptotic power scaling law. For the sake of exposition, we focus on MRT precoding, however similar results can be obtained in the case of RZF precoding as well.\n\\begin{proposition}\\label{powerLaw}\n For the identical SLOs or CLO setting, the downlink achievable SINR of user $k$ at time $n$ with MRT precoding  subject to imperfect CSI, delayed CSI due to phase noise  and user mobility, and collocated  BS antennas  becomes\n \\begin{align}\n  \\gamma_{k,n}=\\frac{\\tau E_{\\mathrm{d}}E_{\\mathrm{u}}}{\\sigma^{4}M^{2 q-1}}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-3\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{R}}_{k}^{2}]_{mm},\\label{scalingPower}\n \\end{align}\nwhere the transmit uplink and downlink powers are scaled proportionally to $1\/M^{q}$, i.e.,  $p_{\\mathrm{u}} = E_{\\mathrm{u}}\/M^{q}$ and $p_{\\mathrm{d}}= E_{\\mathrm{d}}\/M^{q}$\nfor fixed $E_{\\mathrm{u}}$ and $E_{\\mathrm{d}}$, and $q>0$.\n\\end{proposition}\n\\proof\nLet us first substitute the MRT precoder, given by~\\eqref{eq:precoderMRT},  in~\\eqref{eq:DLreceivedSignalgeneral}. Then, we divide both the desired and the interference parts by $\\lambda$ and $1\/M^{2}$. The desired signal power is written as\n\\begin{align}\nS_{k,n}^{\\mathrm{MRT}}\n&=  \\frac{1}{M^{2}}\\Big|{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n} {\\hat{{\\mathbf{g}}}}_{k,0}\\Big|^{2}\\nonumber\\\\\n&=  \\frac{1}{M^{2}}\\Big|\\hat{{\\mathbf{g}}}^\\H_{k,0}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n}^{2} {\\hat{{\\mathbf{g}}}}_{k,0}\\Big|^{2}\\label{desiredMid}\\\\\n&=\\frac{1}{M^{2}}\\mathrm{J}_{0}^4{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-4 \\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}^{2},\\label{desired}\n\\end{align}\nwhere $[{\\mathbf{D}}_{k}]_{mm}$ is the $m$th diagonal element of the matrix  ${\\mathbf{D}}_{k}$ expressing the variance of the $m$th element. Given that both $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ and $  {\\mathbf{A}}_{n}$ are diagonal matrices,   we have taken into account in~\\eqref{desired} that they can commute. Moreover, in the last step of~\\eqref{desired}, we have used the law of large numbers for large $M$ in the case of collocated  BS antennas, as well as that ${\\mathbf{A}}_{n}$ is a scaled identity matrix  or a scalar in the case of identical SLOs or CLO, respectively. In other words, $ {\\hat{{\\mathbf{g}}}}_{k,0}$ has i.i.d. elements with variance $[{\\mathbf{D}}_{k}]_{mm}$. As far as the interference is concerned, the first and third terms of~\\eqref{eq:DLgenIntfPower} vanish to zero as $M \\to \\infty$ by means of the same law. Thus, we have\n\\begin{align}\nI_{k,n}^{\\mathrm{MRT}}&=\\frac{\\sigma^{2}_{k}}{M^{2}p_{\\mathrm{d}} \\lambda}\\nonumber\\\\\n&=\\frac{\\sigma^{2}_{k}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}}{M p_{\\mathrm{d}} }, \n\\end{align}\nwhere  we have applied the law of large numbers to obtain $ \\lambda\\!=\\!\\left(\\frac{1}{M}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\left( {\\sigma_{\\varphi_{k}}^{2}\\!+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}\\! \\right)^{-1}\\!$ from~\\eqref{eq:lamda}, and we have substituted  ${\\mathbf{D}}_{k}=\\frac{\\tau E_{\\mathrm{u}}}{M^{q}\\sigma^{2}_{k}}{\\mathbf{R}}_{k}^{2}$, since  $p_{\\mathrm{p}}$ depends on $p_{\\mathrm{u}}$, the result in~\\eqref{scalingPower} is obtained. \\endproof\n\nProposition~\\ref{powerLaw} reveals that the selection of $q$ affects heavily the achievable rate per user. In fact, proper selection allows maintaining the same performance, even by further scaling down  the transmit power of each user. Specifically, if $q$ is set less than $1\/2$, $\\gamma_{k,n}$ is unbounded, i.e., it tends to infinity.  On the contrary, the SINR of user $k$ diminishes to zero, if $q>1\/2$. This clearly indicates that the transmit powers of each user during the training and downlink phases have been reduced over the required value. Most importantly, in the special case that $q=1\/2$, the SINR approaches a non-zero limit given by the following corollary.\n\\begin{corollary}\nIn the presence of phase noise and user mobility, when the transmit uplink and downlink powers are scaled down by $p_{\\mathrm{u}} = E_{\\mathrm{u}}\/\\sqrt{M}$ and $p_{\\mathrm{d}} = E_{\\mathrm{d}}\/\\sqrt{M}$ for fixed $E_{\\mathrm{u}}$ and $E_{\\mathrm{d}}$, the downlink SINR per user with MRT can be finite  as\n\\begin{align}\n   \\gamma_{k,n}=\\frac{\\tau^{2} E_{\\mathrm{d}} E_{\\mathrm{u}}}{\\sigma_{k}^{4}}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-3\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{R}}_{k}^{2}]_{mm}.\n\\end{align}\n\\end{corollary}\n\nEvidently, phase noise and user mobility reduce the SINR, but the power scaling law is not affected. \n\\subsection{RZF}\nAs far as RZF is concerned, the analysis is more complex, as shown below. First we present the DE of the SINR in general channel aging conditions by means of the following theorem.\n\\begin{Theorem}\\label{theorem:DLagedCSIRZF}\nThe downlink DE of the SINR user $k$ at time $n$ with RZF precoding, accounting for imperfect CSI and delayed CSI due to phase noise and user mobility, is given by \n\\begin{align}\n\\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {\\delta}_{k}^{2}}{{\\frac{1}{M}{\\delta}_{k}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta_{k} \\right)^{2}}{p_{\\mathrm{d}}  \\bar{\\lambda}}\n + \\sum_{i\\neq k}{ \\frac{{Q}_{ik}\\left( 1+\\delta_{k} \\right)^{2}}{M\\left(1+{\\delta_{i}}\\right)^{2}} }},\\label{eq:DLdelayedCSIetaRZF}\n \\end{align}\n with \n  \\begin{align}\n  \\bar{\\lambda}&=\\frac{K}{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} +a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)}\\nonumber\\\\\n{Q}_{ik}&\\asymp \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}{\\mathbf{C}}^{'''}\n+\\frac{\\left|{\\delta_{k}}\\right|^{2}\\delta_{k}^{''}}{\\left( 1+\\delta_{k} \\right)^{2}}-2\\mathrm{Re}\\left\\{  \\frac{\\delta_{k}\\delta_{k}^{''} }{\\left( 1+\\delta_{k} \\right)}\\right\\},\\nonumber\n  \\end{align}\n  ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}},\n {\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{{\\mathbf{C}}}^{'}, \\mathrm{and}~\n \\delta_{k}^{''}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}$,\n where\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\ast$}\n\\item ${\\mathbf{T}}={\\mathbf{T}}(a)$ and ${\\hbox{\\boldmath$\\delta$}}=[{\\delta}_{1},\\cdots,{\\delta}_{K}]^{\\scriptscriptstyle\\mathsf{T}}={\\hbox{\\boldmath$\\delta$}}(a)={\\vect{e}}(a)$ are given by Theorem~\\ref{th:detequ} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}={\\mathbf{T}}^{'}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}=\\mat{\\mathrm{I}}_M$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{K}}=\\mat{\\mathrm{I}}_M$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}^{'}={\\mathbf{T}}^{''}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{K}}= {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}^{''}={\\mathbf{T}}^{'''}(a )$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}_j\/M$, ${\\mathbf{K}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}.$\n\\item  ${\\mathbf{C}}^{'''}={\\mathbf{T}}^{''''}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$, ${\\mathbf{S}}={\\mathbf{Z}}_j\/M$, ${\\mathbf{K}}={\\mathbf{R}}_{k}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}.$\n\\end{itemize} \n\\end{Theorem}\n\\proof: The proof of Theorem~\\ref{theorem:DLagedCSIRZF} is given in Appendix~\\ref{theorem3}.\\endproof\n\\begin{corollary}\\label{equal_correlation}\n Let ${\\mathbf{R}}_{k}={\\mathbf{R}}$, i.e., the large-scale effects (path loss and shadowing) affect the same all users, then ${\\mathbf{D}}_{k}={\\mathbf{D}}$. In such case, the deterministic SINR of Theorem~\\ref{theorem:DLagedCSIRZF}  $\\bar{\\gamma}_{k,n}$ can be simplified to\n \\begin{align}\n  \\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}    {{\\delta}}^{2}}{  {\\frac{1}{M}{\\delta}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta \\right)^{2}}{p_{\\mathrm{d}} \\bar{\\lambda}}\n + \\left( K-1 \\right){  \\frac{{Q}}{M} }},\\label{eq:DLdelayedCSIetaRZF2}\n \\end{align}\nwith  ${\\delta}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{\\mathbf{T}}\\triangleq e,\n {\\delta} ^{'}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{{\\mathbf{C}}}^{'},  \\delta ^{''}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{\\mathbf{C}}^{''}$,  $ {\\mathbf{T}} = ({{\\mathbf{A}}_{n}^{2}{\\mathbf{D}} }\/{\\beta\\left( 1+\\delta \\right)}$  $  +{\\mathbf{Z}}\/M + a\\mat{\\mathrm{I}}_M)^{-1}$, $\\bar{\\lambda}={K}\/{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} +a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)}$, and\n \\begin{align}\n  {Q}\\asymp \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}} {\\mathbf{C}}^{''}+\\frac{\\left|{\\delta }\\right|^{2}\\delta ^{''}}{\\left( 1+\\delta  \\right)^{2}}-2\\mathrm{Re}\\left\\{  \\frac{\\delta \\delta ^{''} }{\\left( 1+\\delta  \\right)}\\right\\},\\nonumber\n \\end{align}\nwhile the general expression of ${\\mathbf{T}}^{'}$ is given by \n \\begin{align}\n{\\mathbf{T}}^{'}={\\mathbf{T}}{\\mathbf{K}}{\\mathbf{T}}+e^{'}_{{\\mathbf{K}}}\/{\\beta\\left(1+\\delta\\right)^2}{\\mathbf{T}}{{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}  }{\\mathbf{T}},  \\nonumber\n \\end{align}\nwhere  $e^{'}_{{\\mathbf{K}}}=\\beta \\left( 1+\\delta \\right)^{2}e_{111}^{{\\mathbf{K}}}\/\\left( \\beta-e_{201}^{{\\mathbf{K}}} \\right)$  with  $e_{ijm}^{{\\mathbf{K}}}=1\/\\left( 1+\\delta \\right)^{j+m}\\frac1M \\mathop{\\mathrm{tr}}\\nolimits\\left( {\\mathbf{A}}_{n}^{2}{\\mathbf{D}} \\right)^{i}{\\mathbf{T}}{\\mathbf{K}}^{j}{\\mathbf{T}}^{m}$.\n \\end{corollary}\n \\begin{corollary}\\label{NoCorrelation1}\nIf no large-scale component $({\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$) are assumed, and the phase noise  from all the oscillators is considered identical or the BS has only a CLO, i.e., ${\\mathbf{A}}_{n}$ becomes $\\alpha_{n}\\mat{\\mathrm{I}}_M$ or a scalar $\\alpha_{n}$, we obtain\n \\begin{align}\n  \\bar{\\gamma}_{k,n} = \\frac{    e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {{\\delta}}^{2}}{  {\\frac{1}{M}{\\delta}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta \\right)^{2}}{p_{\\mathrm{d}} \\bar{\\lambda}}\n + \\left( K-1 \\right){  \\frac{{Q}}{M} }},\\label{eq:DLdelayedCSIetaRZF3}\n \\end{align}\n where $d=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}$ and $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}}{2}n}$.\n \\end{corollary}\n\\proof When ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$, ${\\mathbf{D}}_{k}$  becomes a scaled identity matrix, i.e., ${\\mathbf{D}}_{k}=d \\mat{\\mathrm{I}}_{M}=\\left( \\frac{p_{{\\mathrm{p}}}\\tau}{p_{{\\mathrm{p}}}\\tau+\\sigma_{b}^{2}}\\right)\\mat{\\mathrm{I}}_{M}$. In addition, replacing ${\\mathbf{A}}_{n}$ by the scalar $\\alpha_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi_{k}}^{2}}{2}n}$, we obtain $\\delta=\\alpha_{n}^{2} d  t$, \n ${\\delta} ^{'}=\\alpha_{n}^{2}dt^{'}_1$,  $\\delta ^{''}=\\alpha_{n}^{2} d t^{'}_{2}$,   $\\bar{\\lambda}=\\frac{K}{ \\left(t -t^{'}_{3}a- \\frac{ t^{'}_{3}z}{M}  \\right)}$, \n \\begin{align}\n  {Q}&\\asymp\\frac{\\beta d^4 \\delta^2 \\left(\\delta^2+\\delta+1\\right)}{\\beta (\\delta+1)^2-d^4 \\delta^2}.\\nonumber\n  \\end{align}\n By considering the unique positive root of the quadratic equation  $\\delta=\\alpha_{n}^{2}d t$, we obtain~\\eqref{delta}.\n \\begin{figure*}\n \\begin{align}\n \\delta =\\frac{\\sqrt{4 \\alpha_{n}^2 \\beta^2 d (a +z)+\\left(\\alpha_{n}^2 d (1-\\beta )+\\beta (a +z)\\right)^2}+\\alpha_{n}^2 (\\beta-1) d+\\alpha_{n} \\beta-\\beta (a +z)}{2 \\beta (a +z)}.\\label{delta}\n \\end{align}\n \\line(1,0){470}\n\\end{figure*}\nNote that  the general expression of $t^{'}_{i}$ for $i=1,2,3$ is given by \n \\begin{align}\nt^{'}_{i}=\\frac{\\beta d^2 \\delta^{2}(\\delta+1)^2 k_i}{a^4 \\left(\\beta (\\delta+1)^2-d^4 \\delta^2\\right)}, \\nonumber\n\\end{align}\nsince  $e^{'}_{k_{i}}=e^{'}_{{\\mathbf{K}}}=\\beta d \\delta^2 k_i\/(\\alpha_{n}^2 (\\beta - (d^2 \\delta^2)\/(1 + \\delta)^2)) $. Here, $k_{1}=(1  - \\alpha_{n}^{2}d)$, $k_{2}=\\alpha_{n}^{2} d$, and $k_{3}=1$. \n\\endproof\n\n\n\n\n\\section{Numerical Results}\\label{results}\nThe purpose of this section is to present some representative numerical examples enabling the study of the time variation of the channel due to the effects of phase noise and  user mobility on the performance of massive MIMO systems with MRT and RZF precoders. The metrics under study are the achievable sum-rates and the required transmit power achieving a certain user rate. Note that the achievable sum-rates are described by means of the SINRs, given by Theorems~\\ref{theorem:DLagedCSIMRT} and~\\ref{theorem:DLagedCSIRZF}. In addition, the correctness of the proposed results is validated by Monte-Carlo simulations. Notably, despite that the analytical results are obtained by  assuming that $M,K \\to \\infty$, they coincide with the simulations even for finite values of $M$ and $K$. Actually, this is a known observation in the literature concerning the DEs~\\cite{Couillet2011,Hoydis2013,Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}.\n\\subsection{Simulation Setup}\nGiven that the interest of this work is based  on the investigation of the impact of practical channel impairments, the setting of our scenario includes long-term evolution (LTE) system specifications~\\cite{Marzetta2010}. Specifically,  the simulation setup considers a single cell with a radius of $R = 1000$ meters, and  a guard range of $r_{0} = 100$ meters specifying the distance between the nearest user and the BS. The BS, comprised of  $M$ antennas, broadcasts to $K$ users that are  uniformly distributed within the cell. According to the system model, the channel vector between the BS and the UE $k$ in the $n$th time slot describes the large-scale fading and path loss or spatial correlation by means of ${\\mathbf{R}}_{k}$. Herein,  ${\\mathbf{R}}_{k}$ describes  large-scale fading  modelled as ${\\mathbf{R}}_{k}=l_{k}\\mat{\\mathrm{I}}_{M}$ with $l_{k}=q_{k}\/\\left( r_{k}\/r_{0} \\right)^{\\upsilon}$. Especially, $q_k$ is a log-normal random variable with standard deviation $\\sigma$ ($\\sigma = 8$ dB) expressing the shadow-fading effect, $r_{k}$ denotes the distance between UE $k$ and the BS, and $\\upsilon$ ($\\upsilon = 3.8$) is the path loss exponent. The uplink and downlink powers are $p_{\\mathrm{u}}=p_{\\mathrm{d}}=46$dBm, while the thermal noise density is $-174$ dBm\/Hz. The length of the training duration is $\\tau=K$ symbols, and the phase noises  at the  BS and user LOs are simulated as discrete Wiener processes given by~\\eqref{phaseNoiseBS} and~\\eqref{phaseNoiseuser}, with increment variances in the interval $\\operatorname{0\\deg-2\\deg}$~\\cite{Colavolpe2005}. The coherence time is $T_{\\mathrm{c}}=1\/4 f_{\\mathrm{D}}=1~\\mathrm{ms}$, where $f_{\\mathrm{D}}=250~\\mathrm{Hz}$ is the Doppler spread corresponding to a relative velocity of  $135$ km\/h between the BS and the users, if  the  center frequency is assumed to be $f_\\mathrm{c}= 2~\\mathrm{GHz}$. Moreover, given that the bandwidth for LTE-A is $\\mathrm{W}=20\\mathrm{MHz}$, the symbol time is $T_s=1\/(2\\mathrm{W})=0.025~\\mathrm{\\mu s}$. In order to account for fast varying channels, where high mobility occurs, the coherence block is assumed to include $T=196$ channel uses corresponding to the coherence bandwidth ${B}_{\\mathrm{c}}=196~\\mathrm{KHz}$.\n\nOne useful metric, providing the means for study the considered system, is\n\\begin{align} \\label{eq num 1}\n \\mathcal{S} \\triangleq\\sum_{k=1}^K\n \\bar{R}_{k},\n\\end{align}\nwhere $\\mathcal{S}$ is the sum-rate and $\\bar{R}_{k}$ is the DE rate of UE $k$ given by~\\eqref{DeterministicSumrate}, and \nTheorems~\\eqref{theorem:DLagedCSIMRT} and~\\eqref{theorem:DLagedCSIRZF} for the cases of MRT and RZF, respectively.\n\nIn the following figures, the red and black line patterns correspond to the rates with RZF and MRT precoding for various phase noise nominal values, respectively. Thus, the ``solid'', ``dash'', and ``dot'' lines designate the analytical results with no channel aging, $\\phi_{m,n}=0\\deg,\n \\varphi_{k,n}=2\\deg$, and $\\phi_{m,n}= \\varphi_{k,n}=2\\deg$, respectively.   The bullets represent the simulation results. Finally,  the green ``solid'' and ``dot'' lines, where applicable, depict a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively.\n\\subsection{Rate Comparison between MRT and RZF}\nThe rate comparison between MRT and RZF considers the impact of both Doppler shift and phase noise. Clearly, the RZF outperforms the MRT for the scenarios considered. Especially, Fig.~\\ref{fig:1} depicts the achievable rates by varying the number of BS antennas for different values of phase noise, when the users are assumed to be static ($v=0~\\mathrm{Km\/h}$), or equivalently, in the case of no Doppler shift ($f_{\\mathrm{D}}=0~\\mathrm{Hz}$). By increasing the hardware imperfection in terms of phase noise, the sum-rate decreases. As far as the RZF is concerned, when the phase noise at the user side has a variance of $\\sigma_{\\varphi_{k}}^{2}=2\\deg$ the rate loss is $75\\%$ with the BS having $M=30$ antennas, but when $M$ reaches $300$, the loss is smaller, i.e.,  $52\\%$.  \nMoreover, when the phase noise increases, the sum-rate saturates faster. Similar conclusions can be made in the case of MRT precoding. The perfect match between Monte Carlo simulations and analytical results validates the analytical results.\n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsAntennas.pdf}}\n    \\caption{Simulated and DE downlink sum-rates with MRT and RZF precoders in a static environment versus the number of BS antennas for  various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the  black bullets refer to the simulation results.}\n    \\label{fig:1}\n\\end{figure}\n\nOn the other hand, Fig.~\\ref{fig:2} provides the variation of $\\mathcal{S}$ versus the normalized Doppler shift $f_{\\mathrm{D}}T_{\\mathrm{s}}$ when $M=60$ for different values of phase noise starting from perfect LOs (no phase noise) to high phase noise ($\\sigma_{\\phi_{k}}^{2}=\\sigma_{\\varphi_{k}}^{2}=2\\deg$). It is revealed that the effect of the Doppler shift on the achievable rate is more detrimental than that of phase noise. In fact, for low velocities in the order of $30~\\mathrm{km\/h}$ equivalent to $f_{\\mathrm{D}}T_{\\mathrm{s}}\\approx 0.2$, the degradation due to phase noise starts to become  insignificant, but then the achievable rate can become so low that it is inadequate for practical applications, i.e., it is not worthy to investigate the impact of phase noise in high mobility environments. Hence, the higher the mobility, the less important role the phase noise plays. In the same figure, the straight lines illustrate the sum-rate with imperfect CSI, but with no user mobility and with perfect LOs at both BS and user ends.  Furthermore, it is evident that as phase noise increases, the performance worsens. However, the loss due to phase noise is prominent in static environments. \n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsDoppler.pdf}}\n    \\caption{Simulated and DE downlink sum-rates with MRT and RZF precoders when $M=60$ as a function of  the normalized Doppler shift for  various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the black bullets refer to the  simulation results. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:2}\n\\end{figure}\n\\subsection{Required Transmit Power Comparison between MRT and RZF}\nRegarding the required transmit power achieving a specific rate per user equal to $1$ bit\/s\/Hz, it is quite insightful to investigate how it changes by varying the number of BS antennas, the amount of Doppler shift, and  the severity of phase noise.\n\nHence, Fig.~\\ref{fig:3} illustrates the variation of the transmit power $p_{\\mathrm{d}}$ versus the number of BS antennas in a static environment ($f_{\\mathrm{D}}T_{\\mathrm{s}}=0$). Specifically,  $p_{\\mathrm{d}}$ decreases considerably when we increase $M$. Especially, a closer observation shows a reduction in the transmit power by approximately 1.5dB  after doubling the number of BS antennas, which agrees with previously known results e.g.,~\\cite{Ngo2013}. Notably, the more severe is the phase noise at the user, the more the required transmit power should be. \n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingPowerVsAntennas.pdf}}\n    \\caption{Required transmit power  to achieve $1$ bit\/s\/Hz per user  with MRT and RZF precoders in a static environment versus the number of BS antennas and various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the black bullets refer to the simulation results.}\n    \\label{fig:3}\n\\end{figure}\n\nFig.~\\ref{fig:4} depicts $p_{\\mathrm{d}}$ versus the varying normalized Doppler shift $f_{\\mathrm{D}}T_{\\mathrm{s}}$ and phase noise,  when $M=60$. In particular, the straight lines represent the required transmit power with no channel aging. As a result, these do not depend on $f_{\\mathrm{D}}T_{\\mathrm{s}}$ or phase noise. However, as the Doppler shift increases,  $p_{\\mathrm{d}}$ becomes higher, and very soon (low velocities), it saturates to a constant. In other words, after a specific value of  $f_{\\mathrm{D}}T_{\\mathrm{s}}$, any increase of  $p_{\\mathrm{d}}$ cannot achieve any benefit. Furthermore, the higher the phase noise, the faster the saturation ensues.\n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingPowerVsDoppler.pdf}}\n    \\caption{Required transmit power  to achieve $1$ bit\/s\/Hz per user with MRT and RZF precoders  when $M=60$ as a function of the normalized Doppler shift for  various values of phase noise.  Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:4}\n\\end{figure}\n\\subsection{Extension to Multi-Cell Large MIMO Systems}\nHerein, we focus on the impact of channel aging in the practical case of a hexagonal cellular system. Especially, we consider the downlink of a cellular MIMO system with $L$ cells\noperating under the same frequency band, i.e., the system is impaired by pilot contamination. Actully, each cell can be assumed as an instance of the single-cell setting provided in Section II. The received signal at the $k$th user, located at the $i$th cell, is given by\n\\begin{align}\n\\!\\!y_{ik,n}=\\sqrt{p_{\\mathrm{d}}}\\sum_{l=1}^{L}{\\mathbf{h}}^\\H_{lik,n}{\\boldsymbol{\\Theta}}_{lk,n}{\\mathbf{s}}_{l,n}+z_{ik,n}\\label{eq:DLreceivedSignal}\n\\end{align}\nwhere  ${\\mathbf{s}}_{l,n}$ denotes the unit-norm signal vector transmitted by the BS. Moreover, we have assumed that the BS transmits data to all the users with the same power $p_{\\mathrm{d}}$, and $z_{ik,n} \\sim {\\cal C}{\\cal N}(0,\\sigma_{k}^{2})$ is \ncomplex Gaussian noise at user $k$, found in the $i$th cell.\n\nThe LMMSE estimator of ${\\mathbf{g}}_{ik,0}$, obtained during the training phase, is\n \\begin{align}\\label{estimatedChannel}\n \\hat{{\\mathbf{g}}}_{ik,0}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{\\mathrm{b}}^{2}}{p_{{\\mathrm{p}}}}\\sum_{l}{\\mathbf{R}}_{lik}^{-1}\\right)^{-1}\\tilde{{\\mathbf{y}}}_{iik,0}^{{\\mathrm{p}}},\n\\end{align}\nwhere $\\sigma_{\\mathrm{b}}^{2}$ is the variance of the post-processed noise at\nbase station, $\\tilde{{\\mathbf{y}}}_{ik,0}^{{\\mathrm{p}}}$ is a noisy observation of the effective channel from user $k$ to the $i$th BS, and $p_{\\mathrm{p}}=\\tau p_{\\mathrm{u}}$  with $p_{\\mathrm{u}}$   being the power per user in the uplink data transmission phase.\n\nGiven that the focal point of this work is to conduct an investigation of a generalized channel aging model, and  shed deeper light on channel aging as well as reveal new properties, we present only simulation results and not the corresponding deterministic equivalent analysis, which is straightforward and will distract the reader from the main objective of this part. In particular, we consider the same design parameters with the setup of the single cell, but now we employ $L=7$ cells. Moreover, the average sum-rate is going to be investigated for the central cell, while it has to be mentioned that in the multi-cell setting the large-scale fading depends on the distance of not only the distance of $k$th user from its associated BS, but also from the neighbor BSs. In such scenario, the equivalent SISO model for UE $k$ in the central cell has \ndesired signal power and interference plus noise power given by \n \\begin{align}\nS_{ik,n}\n=  \\lambda\\Big|\\mathbb{E}\\left[{\\mathbf{g}}^\\H_{iik,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n}{{\\mathbf{f}}}_{ik,n}\\right]  \\Big|^{2}\\label{eq:DLgenSignalPower5}\n\\end{align}\nand \n\\begin{align}\nI_{ik,n}\n&=   \\lambda{\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{iik,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n} {\\mathbf{f}}_{ik,n}\\right]  +   \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}} \\nonumber\\\\\n&+    \\sum_{(l,m \\ne i,k)}     \\lambda\\mathbb{E}\\bigg[\\Big|{\\mathbf{g}}^\\H_{lim,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{lm,n} {\\mathbf{f}}_{lm,n}\\Big|^{2}\\bigg],   \\label{eq:DLgenIntfPower5}\n\\end{align}\nwhere $\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n}=\\Delta{{\\boldsymbol{\\Theta}}}_{ik,n}$ and ${{\\mathbf{f}}}_{ik,n}$ represent the accumulated phase noise and the precoder applied by the $i$th BS.\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsDopplerCells.pdf}}\n    \\caption{Simulated downlink sum-rates with MRT and RZF precoders in a cellular setting with $L=7$ when $M=60$ as a function of  the normalized Doppler shift for  various values of phase noise. Red and black lines correspond to the simulated sum-rates with RZF and MRT precoding, respectively. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:2}\n\\end{figure}\n\nObviously, the difference between the single-cell and multi-cell analyses is limited to the term describing the inter-cell interference, or in other words, the pilot contamination. Fig.~5 illustrates the sum-rate of the $i$th (central) cell versus the normalized Doppler shift. As can be seen, the impact of channel aging on the inter-cell interference is similar to the intra-cell interference, and results only in an extra degradation of the  system performance. In addition, the system is more sensitive to the normalized Doppler shift. In other words, the more severe the channel aging, and basically the user mobility, the higher the degradation of the achievable  sum-rate because it affects the inter-cell interference term.\n\n\\section{Conclusions}\nIn this work, we  modeled channel aging by incorporating both effects of Doppler shift coming from user mobility and phase noise due to circuitry imperfections of LOs, since both the effects contribute to the time variation of the effective channel multiplying\/affecting the transmitted data. Given this novel integrated framework, we provided a joint channel-phase noise estimate. Next, the new CSI model was exploited to construct the MRT  and the RZF precoders, and derived the DEs of the corresponding downlink SINRs. As attested by Monte Carlo simulations, these DEs are tight approximations for the rate performance of the studied system. In addition, the numerical results showed that the degradation due to Doppler shift dominates against phase noise. As a result, the detrimental effect of phase noise is meaningful only in low mobility conditions. \nNotably, the use of massive MIMO systems should be preferred even in general channel aging conditions,  where, in addition, RZF behaves better than MRT as expected. Finally, we showed that in the case of MRT precoding, the required transmit power per user to achieve a certain rate can be scaled down at most by the inverse of the square root of the number of antennas, while a similar result should hold for RZF precoding as well.\n\n\\begin{appendices}\n\\section{Useful Lemmas}\n\\begin{lemma}[Matrix inversion lemma (I) {\\cite[Eq.~2.2]{Bai1}}]\\label{lemma:inversion}\n\\\\Let ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  invertible. Then, for any vector ${\\mathbf{x}}\\in\\mathbb{C}^{M}$, and any scalar $\\tau\\in\\mathbb{C}^{M}$ such that ${\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H$ is invertible,\n\\begin{align}\n{\\mathbf{x}}^\\H({\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H)^{-1}=\\frac{{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}}{1+\\tau{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}{\\mathbf{x}}}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\\begin{lemma}[Matrix inversion lemma (II) {\\cite[Lemma~2]{Hoydis2013}}]\\label{lemma:inversion2}\n\\\\Let ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  invertible. Then, for any vector ${\\mathbf{x}}\\in\\mathbb{C}^{M}$, and any scalar $\\tau\\in\\mathbb{C}^{M}$ such that ${\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H$ is invertible,\n\\begin{align}\n({\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H)^{-1}={\\mathbf{B}}-\\frac{{\\mathbf{B}}^{-1}\\tau{\\mathbf{x}}\\bx^\\H{\\mathbf{B}}^{-1}}{1+\\tau{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}{\\mathbf{x}}}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\n\\begin{lemma}[Rank-1 perturbation lemma {\\cite[Lemma~2.1]{Bai2}}]\n\\\\Let $z\\in<0$, ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$, ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ with ${\\mathbf{B}}$ Hermitian  nonnegative-definite, and ${\\mathbf{x}}\\in\\mathbb{C}^{M}$. Then,\n\\begin{align}\n|\\mathop{\\mathrm{tr}}\\nolimits\\left(({\\mathbf{B}}-z\\mat{\\mathrm{I}}_M)^{-1} -({\\mathbf{B}}+{\\mathbf{x}}\\bx^\\H-z\\mat{\\mathrm{I}}_M)^{-1}{\\mathbf{B}}\\right)|\\leq\\frac{\\|{\\mathbf{B}}\\|}{|z|}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\n\\begin{lemma}[{\\cite[Thm. 3.7]{Hoydis2013}},{\\cite[Lem. 1]{Truong2013}}]\\label{lemma:asymptoticLimits}\nLet ${\\mathbf{B}} \\in {\\mathbb{C}}^{M \\times M}$ with uniformly bounded spectral norm (with respect to $M$). Consider ${\\mathbf{x}}$ and ${\\mathbf{y}}$, where ${\\mathbf{x}}, {\\mathbf{y}} \\in {\\mathbb{C}}^{M}$, ${\\mathbf{x}} \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\boldsymbol{\\Phi}}_{x})$ and ${\\mathbf{y}}  \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\boldsymbol{\\Phi}}_{y})$, are mutually independent and independent of ${\\mathbf{B}}$. Then, we have\n\\begin{align}\n\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{x}} - \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}}{\\boldsymbol{\\Phi}}_{x} & \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0 \\label{eq:oneVector}\\\\\n\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{y}} & \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0 \\label{eq:twoVector}\\\\\n\\mathbb{E}\\!\\!\\left[\\left|\\left(\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{x}}\\right)^{2}\\!\\! - \\!\\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} {\\boldsymbol{\\Phi}}_{x} \\right)^{2} \\right|\\right] \\!\\!&\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}  0\\label{eq:squared}\\\\\n\\frac{1}{M^{2}} |{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{y}}|^{2} - \\frac{1}{M^{2}} \\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} {\\boldsymbol{\\Phi}}_{x} {\\mathbf{B}}^{\\H} {\\boldsymbol{\\Phi}}_{y}  & \\xrightarrow[ M\\rightarrow \\infty]{\\mbox{a.s.}} 0. \\label{eq:twoVectorGeneral}\n\\end{align}\n\\end{lemma}\n\n\\begin{lemma}[{\\cite[p. 207]{Tao2012}}]\\label{lemma:asymptoticproduct}\nLet ${\\mathbf{A}}\\mathrm{,}~{\\mathbf{B}}\\in \\mathcal{C}^{M\\times M}$ be freely independent random\nmatrices  with uniformly bounded spectral norm for all $M$. Further, let all the moments of the entries of ${\\mathbf{A}}\\mathrm{,}~{\\mathbf{B}}$ be finite. Then,\n\\begin{align}\n\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}{\\mathbf{B}}-\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}} \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} \\xrightarrow[ M\\rightarrow \\infty]{\\mbox{a.s.}} 0.\n\\end{align}\n\\end{lemma}\n\n\n\\begin{Theorem}[{\\cite[Theorem 1]{Wagner2012}}]\\label{th:detequ}\nLet ${\\mathbf{L}}\\in\\mathbb{C}^{M\\times M}$ and ${\\mathbf{S}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  nonnegative definite matrices, and let ${\\mathbf{H}}\\in\\mathbb{C}^{M\\times K}$ be a random matrix with columns ${\\mathbf{v}}_k\\sim {\\cal C} {\\cal N}\\left(0,\\frac{1}{M}{\\mathbf{R}}_k\\right)$. Assume that ${\\mathbf{L}}$ and the matrices ${\\mathbf{R}}_k$, $k=1,\\dots,K$, have uniformly bounded spectral norms (with respect to $M$). Then, for any $\\rho>0$,\n\\begin{align*}\n\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}\\left({\\mathbf{H}}\\bH^\\H +{\\mathbf{S}} +\\rho\\mat{\\mathrm{I}}_M\\right)^{-1} - \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}{\\mathbf{T}}(\\rho) \\xrightarrow[ M_{{\\mathrm{t}}} \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align*}\nwhere ${\\mathbf{T}}(\\rho)\\in\\mathbb{C}^{M\\times M}$ is defined as\n$$ {\\mathbf{T}}(\\rho) = \\left(\\frac1M\\sum_{k=1}^K\\frac{{\\mathbf{R}}_k}{1+e_k(\\rho)}  +{\\mathbf{S}} + \\rho\\mat{\\mathrm{I}}_M\\right)^{-1},$$\nand the elements of $\\boldsymbol{e}(\\rho)=\\left[e_1(\\rho)\\cdots e_K(\\rho)\\right]^{\\scriptscriptstyle\\mathsf{T}}$ are defined as $ e_k(\\rho)  = \\lim_{t \\to \\infty}e_k^{(t)}(\\rho),$ where for $t=1,2,\\ldots$\n\\begin{align}\n\\!\\!\\!e_k^{(t)}(\\rho)\\!=\\!\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k\\!\\left(\\!\\!\\frac1M\\!\\!\\sum_{j=1}^{K}\\!\\!\\frac{{\\mathbf{R}}_j}{1+e_j^{(t-1)}(\\rho)}+{\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\!\\!\\right)\\!\\!^{-1}\n\\end{align}\nwith initial values $e_k^{(0)}(\\rho)=\\frac{1}{\\rho}$ for all $k$.\n\\end{Theorem}\n\\vspace{5pt}\n\n\\begin{Theorem}[{\\cite[Theorem 2]{Hoydis2013}}]\\label{th:detequder}\nLet ${\\boldsymbol{\\Theta}}\\in\\mathbb{C}^{M\\times M}$ be a Hermitian  nonnegative definite matrix with uniformly bounded spectral norm (with respect to $M$). Under the same conditions as in Theorem~\\ref{th:detequ}, we have  \n\\begin{align*}\n\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}\\left({\\mathbf{H}}\\bH^\\H + {\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\right)^{-1}&{\\mathbf{K}}\\left({\\mathbf{H}}\\bH^\\H  +{\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\right)^{-1}\\nonumber\\\\\n&- \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}{\\mathbf{T}}^{'}(\\rho) \\xrightarrow[]{\\text{a.s.}} 0,\n\\end{align*}\nwhere ${\\mathbf{T}}^{'}(\\rho)\\in\\mathbb{C}^{M\\times M}$ is defined as\n$$ {\\mathbf{T}}^{'}(\\rho) = {\\mathbf{T}}(\\rho){\\mathbf{K}} {\\mathbf{T}}(\\rho) + {\\mathbf{T}}(\\rho)\\frac1M\\sum_{k=1}^K\\frac{{\\mathbf{R}}_k e^{'}_k(\\rho) }{\\left(1+e_k(\\rho)\\right)^2}{\\mathbf{T}}(\\rho)$$\nwith ${\\mathbf{T}}(\\rho)$ and $\\boldsymbol{e}_k(\\rho)$ as defined in Theorem~\\ref{th:detequ} and $\\boldsymbol{e}^{'}(\\rho) = \\left[e^{'}_1(\\rho)\\cdots e^{'}_K(\\rho)\\right]^{\\scriptscriptstyle\\mathsf{T}}$ given by\n\\begin{align}\n \\boldsymbol{e}^{'}(\\rho) &= \\left(\\mat{\\mathrm{I}}_K - {\\mathbf{J}}(\\rho)\\right)^{-1}{\\mathbf{v}}(\\rho).\n\\end{align}\nThe elements of ${\\mathbf{J}}(\\rho)\\in\\mathbb{C}^{K\\times K}$ and ${\\mathbf{v}}(\\rho)\\in\\mathbb{C}^{K}$ are defined as\n\\begin{align}\n[{\\mathbf{J}}(\\rho)]_{kl} = \\frac{\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k{\\mathbf{T}}(\\rho){\\mathbf{R}}_l{\\bf T}(\\rho)}{M\\left(1+e_k(\\rho)\\right)^2}\\nonumber\n\\end{align}\nand\n\\begin{align}\n[{\\mathbf{v}}(\\rho)]_k = \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k{\\bf T}(\\rho){\\mathbf{K}}{\\bf T}(\\rho).\\nonumber\n\\end{align}\n\n\\end{Theorem}\n\n\n\n\n\\section{Proof of Proposition~\\ref{LMMSE}}\\label{proposition1} \nDuring the training phase,  users transmit mutually orthogonal training sequences consisting of $\\tau$ symbols, and it is assumed that the channel remains  constant during this phase.  In particular,  the pilot sequences can be represented by  ${\\boldsymbol{\\Psi}} = [{\\boldsymbol{\\psi}}_{1}; \\cdots; {\\boldsymbol{\\psi}}_{K}] \\in {\\mathbb{C}}^{K \\times \\tau}$ with ${\\boldsymbol{\\Psi}}$ normalized, i.e., ${\\boldsymbol{\\Psi}}\\bPsi^\\H = {\\mathbf{I}}_K$. Given that the channel estimation takes place at time $0$, the received signal at the BS is written as\n\\begin{align}\\label{eq:Ypt}\n{\\mathbf{Y}}_{{\\mathrm{p}},0}= & \\sqrt{p_{{\\mathrm{p}}}}{\\boldsymbol{\\Theta}}_{k,0} {\\mathbf{H}}_{0}{\\boldsymbol{\\Psi}} + {\\mathbf{Z}}_{{\\mathrm{p}},0},\n\\end{align}\nwhere $p_{{\\mathrm{p}}}$ is the common average transmit power for all users, ${\\boldsymbol{\\Theta}}_{k,0}=\\mathrm{diag}\\left\\{ e^{j \\theta_{k,0}^{(1)}}, \\ldots, e^{j \\theta_{k,0}^{(M)}} \\right\\}$ is the phase noise because of the BS and user $k$ LOs at time $0$, and ${\\mathbf{Z}}_{{\\mathrm{p}},0} \\in {\\mathbb{C}}^{M \\times \\tau}$ is the spatially white additive Gaussian noise matrix at the  BS   during the training phase. Note that $ \\theta_{k,0}^{(m)}=\\phi_{m,0}+\\varphi_{k,0},~m=1,\\ldots, M$. By correlating the received signal with the training sequence $\\frac{1}{\\sqrt{p_{{\\mathrm{p}}}}}{\\boldsymbol{\\psi}}^\\H_k$ of user $k$, and by substituting ${\\mathbf{g}}_{k,0}={\\boldsymbol{\\Theta}}_{k,0} {{\\mathbf{h}}_{k,0}}$ the BS obtains\n\\begin{align}\n\\tilde{{\\mathbf{y}}}_{k,0}^{{\\mathrm{p}}}\n= {\\mathbf{g}}_{k,0}  + {\\frac{1}{\\sqrt{p_{{\\mathrm{p}}}}}\\tilde{{\\mathbf{z}}}_{{\\mathrm{p}},0}},\\label{eq:Ypt3}\n\\end{align}\nwhere $\\tilde{{\\mathbf{z}}}_{{\\mathrm{p}},0}\\triangleq {\\mathbf{Z}}_{{\\mathrm{p}},0}{\\boldsymbol{\\psi}}^\\H_{k}\\sim {\\cal C}{\\cal N}({\\mathbf{0}},\\sigma_{\\mathrm{b}}^{2}{\\mathbf{I}}_{M})$.  After applying the MMSE estimation method~\\cite{Kay}, the  effective channel estimated at the BS  can be written as~\\eqref{estimatedChannel}.\nEmploying the orthogonality principle, the  channel decomposes as\n\\begin{align}\n{\\mathbf{g}}_{k,0} = \\hat{{\\mathbf{g}}}_{k,0} + \\tilde{{\\mathbf{g}}}_{k,0},\\label{eq:MMSEorthogonality}\n\\end{align}\nwhere $\\hat{{\\mathbf{g}}}_{k,0}$ is distributed as ${\\cal C}{\\cal N}\\left({\\mathbf{0}},{\\mathbf{D}}_{k}\\right)$ with ${\\mathbf{D}}_{k}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{b}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}{\\mathbf{R}}_{k}$, and $\\tilde{{\\mathbf{g}}}_{k,0} \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\mathbf{R}}_{k} - {\\mathbf{D}}_{k})$ is the channel estimation error vector. Note that $\\hat{{\\mathbf{g}}}_{k}$ and $\\tilde{{\\mathbf{g}}}_{k}$ are statistically independent because they are uncorrelated and jointly Gaussian.\n \\section{Proof of Theorem~\\ref{theorem:DLagedCSIMRT}}\\label{theorem2}\n We consider the desired signal power \n\\begin{align}\nS_{k,n}\n&= \\frac{\\lambda}{M}  \\Big|\\mathbb{E}\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}   {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\!\\!\\Big|^{2}.\\label{normalizedMRT}\n\\end{align}\nFirst, we  derive the DE of the normalization parameter $\\lambda$. Hence, the normalization parameter can be written by means of~\\eqref{eq:lamda} and~\\eqref{eq:precoderMRT} as\n\\begin{align}\n\\lambda&=\n\\frac{K}{\\mathbb{E}\\Big[\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H} {\\mathbf{A}}_{n}   \\Big]} \\nonumber\\\\\n&= \\frac{K}{\\mathbb{E}\\Big[\\sum_{i=1}^{K}\\frac{1}{M}{\\mathbf{A}}_{n}^{2}  \\hat{{\\mathbf{g}}}_{k,0} \\hat{{\\mathbf{g}}}_{k,0}^{\\H}  \\Big]}\\nonumber\\\\\n&\\asymp \\left( \\frac1K\\sum_{i=1}^{K}\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)^{-1},\\label{desired1MRT}\n\\end{align}\nwhere we have applied Lemma~\\ref{lemma:asymptoticLimits}.\nAs far as the other term in~\\eqref{normalizedMRT} is concerned, we obtain\n\\begin{align}\n \\frac{1}{M}\\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\n &\\!=\\! \\frac{1}{M}\\mathbb{E}\\!\\left[(\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}  \\!+ \\!\\tilde{{\\mathbf{e}}}^\\H_{k,n} ) \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\nonumber\\\\\n &=\\frac{1}{M}\\mathbb{E}\\Big[{\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} }\\Big],\\nonumber\\\\\n &\\asymp\\frac{1}{M}\\mathbb{E}\\left[\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n}{\\mathbf{D}}_{k}\\label{desired4MRT}\\right]\\\\\n &\\asymp \\frac{e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k},\\label{desired2MRT}\n\\end{align}\nwhere in~\\eqref{desired4MRT} and~\\eqref{desired2MRT}, we have applied Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct}, respectively. Note that ${\\mathbf{A}}_{n}$ and $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ commute because both are diagonal matrices. In particular, regarding the CLO setting, it holds that $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{ 2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} +\\phi_{k, n}-\\phi_{k, 0}\\right)}$, while in the case of the SLOs setup, we obtain $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} \\right)-\\sigma_{\\phi}^{2}n}$ or  $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} \\right)}\\frac{1}{M}\\prod_{i=1}^{M}e^{-\\sigma_{\\phi_{i}}^{2}n}$, when the LOs obey to identical or non-identical statistics, respectively. Herein, we focus on the former scenario regarding the BS antennas. Especially, in such case their increment variance is $\\sigma_{\\phi}^{2}$. Application of the expectation operator to  $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ gives~$e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}$ for both the  CLO and SLOs settings. Finally, after simple algebraic maipulations, we lead to ~\\eqref{desired2MRT}. By denoting $\\bar{S}_{k,n} = \\lim_{M \\rightarrow \\infty} S_{k,n}$ the DE signal power, and by using~\\eqref{desired1MRT} and~\\eqref{desired2MRT}, we have\n\\begin{align}\n\\bar{S}_{k,n}\\asymp  \\bar{\\lambda}e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {\\delta}_{k}^{2},\\label{eq:theorem4.5MRT}\n\\end{align}\nwhere ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$.\nNow, we proceed with the derivation of each term in~\\eqref{eq:DLgenIntfPower}. The first term on the right hand side can be written as \n\\begin{align}\n\\frac{1}{M} {\\mathrm{var}}&\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right]\\nonumber\\\\\n&-\\frac{1}{M^{2}} \\mathbb{E}\\bigg[\\Big|{\\tilde{{\\mathbf{e}}}_{k,n}^{\\H}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n}\\hatvg\n_{k,0} }\\Big|^2\\bigg]\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align}\nwhere the property of the variance operator $\\mathrm{var}\\left[ x \\right]=\\mathbb{E}[x^{2}]- \\mathbb{E}^{2}[x] $ together with~\\eqref{eq:GaussMarkov2} have been used. Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct} enable us to derive this DE as\n\\begin{align}\n \\frac{\\lambda}{M} {\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right] \n\\asymp&  \\frac{\\bar{\\lambda}}{M}{\\delta}_{k}^{'}, \\label{eq:theorem4.7MRT}\n\\end{align}\nwhere ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\left( {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)$.\nSimilarly, use of Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct} in the last term of~\\eqref{eq:DLgenIntfPower} completes the proof. Thus, we have\n\\begin{align}\n  \\frac{\\lambda}{M^{2}}\\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n&\\asymp \\frac{\\bar{\\lambda}}{M^{2}} \\mathop{\\mathrm{tr}}\\nolimits\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{R}}_{k},\\nonumber\\\\\n&=\\frac{\\bar{\\lambda}}{M}\\delta_{i}^{''},\\label{lasttermMRT}\n\\end{align}\nsince ${\\mathbf{g}}_{k,n} $ and $\\hatvg_{i,0} $ are mutually independent. Note here that $\\delta_{i}^{''}=\\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{R}}_{k}$. \n\n \\section{Proof of Theorem~\\ref{theorem:DLagedCSIRZF}}\\label{theorem3}\nFirst, we obtain  the DE of the normalization parameter $\\lambda$. Taking into account for~\\eqref{eq:lamda} and~\\eqref{eq:precoderRZF}, and making a simple algebraic manipulation, we lead to\n\\begin{align}\n\\lambda&=\n\\frac{K}{\\mathbb{E}\\Big[\\mathop{\\mathrm{tr}}\\nolimits{ {\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H} {\\mathbf{A}}_{n} {{\\mathbf{\\Sigma}}} \\Big]} \\label{eq:theorem4.6}\\\\\n&= \\frac{K}{\\mathbb{E}\\Big[\\mathop{\\mathrm{tr}}\\nolimits {{\\mathbf{\\Sigma}}}  - \\mathop{\\mathrm{tr}}\\nolimits \\left( {\\bf Z}+ a M\\mat{\\mathrm{I}}_M\\right)  {{\\mathbf{\\Sigma}}}^{2} \\Big]},\n\\end{align}\nwhile its DE is \n\\begin{align}\n \\bar{\\lambda}=\\frac{K}{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} + a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)},\\label{desired1}\n\\end{align}\nwhere we have applied Theorems 1 and 2 for ${\\mathbf{K}}=\\mat{\\mathrm{I}}_M$.\nRegarding the other term of the desired signal power, we have\n\\begin{align}\n  &\\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\n\\!=\\! \\mathbb{E}\\!\\left[(\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\!  + \\!\\tilde{{\\mathbf{e}}}^\\H_{k,n} )\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} \\right]\\label{desiredsig1}\\nonumber\\\\\n &=\\mathbb{E}\\Big[\\frac{\\hat{{\\mathbf{g}}}^\\H_{k,0} {\\mathbf{A}}_{n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_k{\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} }{1+{\\hatvg}^\\H_{k,0}{\\mathbf{A}}_{n}  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_k \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}{\\hatvg}_{k,0} }\\Big],\n\\end{align}\nwhere Lemmas~\\ref{lemma:inversion} and~\\ref{lemma:asymptoticLimits} have been applied in \\eqref{desiredsig1}, while $ {{\\mathbf{\\Sigma}}}_k $ is defined as\n\\begin{align}\n {{\\mathbf{\\Sigma}}}_k\\! &=\\!\\left(\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}   \\hat{{\\mathbf{G}}}_{0} \\hat{{\\mathbf{G}}}^\\H_{0} {\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} \\!-\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,n}\\hat{{\\mathbf{g}}}^\\H_{k,n}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\n \\right.\\nonumber\\\\\n & \\left.+ {\\mathbf{Z}} + M  a\\mat{\\mathrm{I}}_M\\right)^{-1}.\\nonumber\n\\end{align}\nExploiting Lemmas~\\ref{lemma:asymptoticLimits},~\\ref{lemma:asymptoticproduct}, and Theorem~\\ref{th:detequ} gives\n\\begin{align}\n   \\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\asymp \\mathbb{E}\\!\\left[\\frac{\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{T}}}{1+\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}}}\\right].\\label{desired2}\n\\end{align}\n\nHence, use of~\\eqref{desired1} and~\\eqref{desired2} provides the DE signal power ${\\bar{S}_{k,n}}$ as\n\\begin{align}\n\\bar{S}_{k,n}\\asymp  \\bar{\\lambda}  \\bigg(\\frac{e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}{\\delta}_{k}}{1+{\\delta}_{k}}\\bigg)^{2},\\label{eq:theorem4.5}\n\\end{align}\nwhere ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}}$. Note that the manipulation regarding $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ follows a similar analysis to~\\eqref{desired2MRT}.\nWe continue with the derivation of each term of~\\eqref{eq:DLgenIntfPower}. Specifically, after applying  Lemmas~1 and~\\ref{lemma:asymptoticLimits}, as well as~\\eqref{eq:GaussMarkov2} to the first term in~\\eqref{eq:DLgenIntfPower}, we have\n\\begin{align}\n &{\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{k,0} \\right]\\nonumber\\\\\n & - \\mathbb{E}\\bigg[\\Big|\\frac{\\tilde{{\\mathbf{e}}}_{k,n}^{\\H}   \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_k {\\mathbf{A}}_{n}\\hatvg\n_{k,0} }{1+\\hatvg^\\H_{k,0}  {\\mathbf{A}}_{n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_k \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n} \\hatvg_{k,0} }\\Big|^2\\bigg]\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align}\nwhich yields\n\\begin{align}\n \\lambda {\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right] \n\\asymp \\bar{\\lambda} \\frac{\\frac{1}{M}{\\delta}_{k}^{'}}{\\left(1+{\\delta}_{k}\\right)^2}, \\label{eq:theorem4.7}\n\\end{align}\nwhere we have used Theorems~\\ref{th:detequ} and~\\ref{th:detequder} as well as Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct}. Note that ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{{\\bf T}}^{'}$ and ${\\mathbf{K}}= {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$.\nNext, we focus on the last term of~\\eqref{eq:DLgenIntfPower}, where we make use of Theorems~\\ref{th:detequ} and~\\ref{th:detequder} as well as Lemmas~1 and~\\ref{lemma:asymptoticLimits} as before. In particular, if $i \\neq k$, we have\n\\begin{align}\n& \\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n= \\mathbb{E}\\bigg[\\left|\\frac{{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{i,0}  }{1+\\hatvg^\\H_{i,0} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hatvg_{i,0} }\\right|^{2}\\bigg]\\nonumber\\\\\n&= \\mathbb{E}\\bigg[\\frac{ {\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hatvg_{i,0} \\hatvg_{i,0}^{\\H}{\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{g}}_{k,n}}{\\left( 1+\\hatvg^\\H_{i,0} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right)^{2} }\\bigg]\\nonumber\\\\\n&\\asymp \\mathbb{E}\\bigg[\\frac{ {\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n}{\\mathbf{D}}_{i} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{g}}_{k,n}}{\\left( 1+\\delta_{i}\\right)^{2} }\\bigg],\\label{lastterm}\n\\end{align}\nwhere we have taken into consideration that ${\\mathbf{g}}_{k,n} $ and $\\hatvg_{i,0} $ are mutually independent. Unfortunately, upon inspecting~\\eqref{lastterm}, we observe that ${\\mathbf{\\Sigma}}_i$ is not independent of $\\hat{{\\mathbf{g}}}_{k,0}$. For this reason, we use  Lemma~\\ref{lemma:inversion2}, which gives\n\\begin{align}\n{\\mathbf{\\Sigma}}_i={{\\mathbf{\\Sigma}}}_{ik}-\\frac{{{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}\\hatvg_{k,0}^{\\H}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik}}{1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n} \\hatvg_{k,0} },\\label{eq:theorem2.I.51}\n\\end{align}which introduces a new matrix ${{\\mathbf{\\Sigma}}}_{ik}$ to~\\eqref{lastterm} defined as\n\\begin{align}\n{{\\mathbf{\\Sigma}}}_{ik}\\!&=\\!\\left(\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{G}}}_{0}\\hat{{\\mathbf{G}}}_{0}^\\H{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\\! -\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{i,0}\\hat{{\\mathbf{g}}}^\\H_{i,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} \\right.\\nonumber\\\\\n&\\left.\\!-\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\\!+\\! {\\mathbf{Z}}\\! + \\!M a\\mat{\\mathrm{I}}_M\\right)^{\\!-\\!1}.\n\\end{align}\nBy substituting~\\eqref{eq:theorem2.I.51} into~\\eqref{lastterm}, we obtain\n\\begin{align}\n  \\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n&=\\frac{{Q}_{ik}}{M\\left(1+{\\delta_{i}}\\right)^{2}},\\label{eq:theorem2.I.6}\n\\end{align}           \nwhere ${Q}_{ik}$ is given in~\\eqref{eq:theorem2.I.mu1} with $\\widetilde{{\\mathbf{\\Sigma}}}_{ik}=\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_{ik}$.\n\\begin{figure*}\n\\begin{align}\n{Q}_{ik}&= {\\mathbf{g}}^\\H_{k,n}\n\\widetilde{{\\mathbf{\\Sigma}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{ {\\mathbf{\\Sigma}}}_{ik}^{\\H}  {\\mathbf{g}}_{k,n}\\!+\\!\\frac{\\left|  {\\mathbf{g}}^\\H_{k,n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0}\\right|^{2}\\hat{{\\mathbf{g}}}^\\H_{k,0} {\\mathbf{A}}_{n} \\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}}{\\left( 1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0} \\right)^{2}}\\nonumber\\\\\n&-2\\mathrm{Re}\\left\\{  \\frac{\\hatvg^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H} {\\mathbf{g}}_{k,n}{\\mathbf{g}}_{k,n}^{\\H}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}}{1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}}\\right\\}.\n \\label{eq:theorem2.I.mu1}\n\\end{align}\n\\line(1,0){470}\n\\end{figure*}\nThe DE of each term in~\\eqref{eq:theorem2.I.mu1} is obtained as\n\\begin{align}\n \\!\\!\\!\\!{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{\\Sigma}}_{ik}  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{g}}_{k,n}&\\asymp  \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{R}}_{k}{\\mathbf{C}}^{''}\n \\end{align}\n\\begin{align}\n \\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}{{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}\n &\\asymp  \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}\\nonumber\\\\\n &=\\frac{ \\delta_{k}^{''}}{M}\n\\end{align}\n\\begin{align}\n{\\mathbf{g}}_{k,n}^{\\H} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}&{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}\\nonumber\\\\\n&\\asymp \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}\\nonumber\\\\\n&=\\frac{ \\delta_{k}^{''}}{M}\n\\end{align}\n\\begin{align}\n\\!\\!{Q}_{ik}\\!\\asymp\\! \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_{k}{\\mathbf{C}}^{''}\\!\\!+\\!\\frac{\\left|{\\delta_{k}}\\right|^{2}\\delta_{k}^{''}}{M\\left( 1\\!+\\!\\delta_{k} \\right)^{2}}\\!-\\!2\\mathrm{Re}\\left\\{ \\! \\frac{\\delta_{k}\\delta_{k}^{''} }{M\\left( 1\\!+\\!\\delta_{k} \\right)}\\!\\right\\}\\!,\n \\label{eq:theorem2.I.mu}\n\\end{align}\nwhere ${\\mathbf{K}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$ with  $\\delta_{k}^{''}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}$.\nThis concludes the proof for the derivation of   $\\bar{\\gamma}_{k,n}$.\n\n\n\\end{appendices}\n\\section*{Acknowledgement}\nThe author would like to express his gratitude to Dr. Rajet Krishnan for his help and support in making this work possible.\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\n\nQuantum entanglement is  a vital resource for quantum information\nprocessing \\cite{r1}. However, isolating a quantum system completely\nfrom its environment is plainly an impossible task and each quantum\nsystem will inevitably interact with its environment. Therefore, it\nis important to investigate the behavior of an entangled quantum\nsystem under the influence of its environment. Recently, Yu and\nEberly \\cite{r7,r8}  investigated the dynamics of two-qubit\nentangled states undergoing various modes of decoherence. They found\nthat it takes an infinite time to complete decoherence locally, the\nglobal entanglement may be lost in  a finite time, and the decay of\na single-qubit coherence can be slower than the decay of a two-qubit\nentanglement. The abrupt disappearance of entanglement in a finite\ntime was named ''entanglement sudden death''(ESD). A geometric\ninterpretation of the phenomenon is given in Ref.\\cite{Geometric}.\nIn addition, experimental evidences of ESD have been reported for\noptical setups \\cite{Optical1,Optical2}  and atomic ensembles\n\\cite{Atomic}.  Clearly, ESD can seriously affect the applications\nof entangled states in  a practical quantum information processing.\nRecently,  dynamics of entanglement has received increasing\nattention \\cite{nomar,zhaopra1,zhaopra2}.\n\n\n\nESD in  finite-dimensional systems is not limited only to two-qubit\nsystems. It may be occurs in a composite quantum system with a\nlarger dimension  and a multiqubit system  as well\n\\cite{Large1,Large2,r9,r10,mul1,mul2,mul3}.  The dissipative\ndynamics for a specific one-parameter class of states in a\nqubit-qutrit ($2\\otimes3$) system interacting with dephasing and\ndepolarizing channels was studied by Ann \\emph{et al.} \\cite{r9} and\nKhan \\cite{r10}, respectively. Ann \\emph{et al.} \\cite{r9}\nconjectured that ESD exists in all bipartite quantum systems. Khan\n\\cite{r10} showed that no ESD happens in any density matrix of a\nqubit-qutrit system when only the qubit is coupled to its local\ndepolarizing channel but the re-birth of entanglement occurs in\nparticular initial states. However, for general qubit-qutrit states\nand other common noise channels, the dissipative dynamics of a\nhybrid qubit-qutrit is not presented.\n\n\n\n\n\nIn this paper, we devote to investigate the behavior of entanglement\nfor a two-parameter class of states in a qubit-qutrit system under\nthe influence of both two independent (multi-local) and only one\n(local) various noise channels, such as dephasing, phase-flip,\nbit-(trit-) flip, bit-(trit-) phase-flip, and depolarizing channels.\nUsing negativity for quantifying entanglement, some analytical or\nnumerical results are presented. We find that ESD is a general\nphenomenon in a  qubit-qutrit system undergoing all these noise\nchannels, not only the case with dephasing and depolarizing channels\nobserved by others \\cite{r9}.  It is interesting to show that ESD\nalways takes place in any density matrix when each subsystem couples\nto its depolarizing channel or only the qutrit couples to its\ntrit-flip or trit-phase-flip channels. ESD can only be avoided in\nsome initial states undergoing particular noise channels. For\nexample, no ESD occurs when the system under the influence of\nmulti-local (local) dephasing, multi-local (local) phase-flip, local\nbit-flip, and local bit-phase-flip channels if it is initially in\nthe state shown in Eq.(\\ref{eq.1}) with the parameter $b=0$.  Our\nresults show that the noise channels affect the entanglement and the\ncoherence of a hybrid qubit-qutrit system in very different ways.\nFor local or multi-local dephasing, phase-flip, and depolarizing\nnoise channels, a time scale of disentanglement is usually shorter\nthan the decay of the off-diagonal dynamics, and coherence\ndisappears in an infinite-time limit. For multi-local and local\nbit-flip and bit-phase-flip channels, disentanglement  occurs in an\ninfinite time, but coherence does not disappear even though\n$t\\mapsto\\propto$.\n\n\n\n\nThis paper is organized as follows. In Sec.\\ref{sec2}, we motivate\nthe choice of a two-parameter class of states in a qubit-qutrit\nsystem, and the physical model  are introduced. In Sec.\\ref{sec3},\nentanglement dynamics of a two-parameter class of states in a\nqubit-qutrit system under the influence of  local and multi-local\ndephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing noise channels are discussed, respectively. Discussion\nand summary are shown in Sec.\\ref{sec4}.\n\n\n\n\n\\section{ Initial states and noise model }\\label{sec2}\n\n\nA two-parameter  class of states with real parameters in a hybrid\nqubit-qutrit ($2 \\otimes 3$) quantum system \\cite{r15} can be\ndescribed as\n\\begin{eqnarray}\n\\rho_{bc}(0)&=&\na\\left(|02\\rangle\\langle02|+|12\\rangle\\langle12|\\right)+\nb(|\\phi^{+}\\rangle\\langle\\phi^{+}|+ \\nonumber\n\\\\&&|\\phi^{-}\\rangle\\langle\\phi^{-}|+|\\psi^{+}\\rangle\\langle\\psi^{+}|)+c|\\psi^{-}\\rangle\\langle\\psi^{-}|, \\label{eq.1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n|\\phi^{\\pm}\\rangle &=& \\frac{1}{\\sqrt{2}}(|00\\rangle\\pm|11\\rangle),\\nonumber\\\\\n|\\psi^{\\pm}\\rangle &=& \\frac{1}{\\sqrt{2}}(|01\\rangle\\pm|10\\rangle),\n\\label{eq.2}\n\\end{eqnarray}\nand  $a$, $b$, and $c$ are three real parameters, and they satisfy\nthe relation $2a+3b+c=1$. $\\vert 0\\rangle$ and $\\vert 1\\rangle$ are\nthe two eigenstates of a two-level quantum system (qubit) or the\neigenstates of a three-level quantum system (qutrit) with the other\neigenstate $\\vert 2\\rangle$. The two-parameter class of states\n$\\rho_{bc}(0)$ can be obtained from an arbitrary state of a $2\n\\otimes 3$ quantum system by means of local quantum operations and\nclassical communication \\cite{r15}.\n\n\n\n\n\nFor an arbitrary mixed state $\\rho^{AB}$ in a $2\\otimes2$ or a\n$2\\otimes3$ system, its entanglement can well be characterized and\nquantified by its\nnegativity,\\textsuperscript{\\cite{negativity1,negativity2}}\n\\begin{eqnarray}                                               \\label{eq.3}\nN(\\rho^{AB})=\\parallel\\rho^{T_{B}}\\parallel_{1}-1,\n\\end{eqnarray}\nwhich corresponds to the absolute value of the sum of negative\neigenvalues of $\\rho^{T_{B}}$(the partial transpose $\\rho^{T_{B}}$\nassociated with an arbitrary product orthonormal basis $f_i \\otimes\nf_j$ is defined by the matrix elements:\n$\\rho_{m\\mu,n\\nu}^{T_B}\\equiv \\langle f_m \\otimes\nf_\\mu|\\rho^{T_B}|f_n \\otimes f_\\nu\\rangle=\\rho_{m\\nu,n\\mu}$), i.e.,\n\\begin{eqnarray}                                        \\label{eq.4}\nN(\\rho^{AB})=2\\max\\{0,-\\lambda_{S}\\},\n\\end{eqnarray}\nwhere $\\lambda_{S}$ represents the sum of all negative eigenvalues\nof $\\rho^{T_{B}}$. $N(\\rho^{AB})=0$ for an unentangled state.\nTherefore, from Eq.(\\ref{eq.1}) one can obtain the range of\nparameters as $3b<c\\leq1-3b$, i.e., $b\\in[0,1\/6)$  for the initial\nentangled states.\n\n\n\n\n\n\n\n\nIn our physical model of noise for a qubit-qutrit system (composed\nof a two-level subsystem $A$ and a three-level subsystem $B$), the\ntwo subsystems interact with their  environments independently. The\nevolved states of the initial density matrix of such a system when\nit is influenced by multi-local environments can be given compactly\nby\n\\begin{eqnarray}                                        \\label{eq.5}\n\\rho_{bc}^{AB}(t)=\\sum_{i=1}^{2}\\sum_{j=1}^{3}F_{j}^{B}E_{i}^{A}\\rho^{AB}_{bc}(0)E_{i}^{A\\dagger}F_{j}^{B\n\\dagger}.\n\\end{eqnarray}\nHere, the operators $E_{i}^{A}$ and $F_{j}^{B}$ are the Kraus\noperators which are used to describe the noise channels acting on\nthe qubit $A$ and the qutrit $B$, respectively. They satisfy the\ncompleteness relations $E_{i}^{A\\dagger}E_{i}^{A}=I$ and $F_{j}^{B\n\\dagger}F_{j}^{B}=I$ for all $t$.\n\n\n\n\\bigskip\n\\section{Dynamics of entanglement under decoherence} \\label{sec3\n\nIt is important to consider the possible degradation of any\ninitially prepared entanglement due to decoherence. In this section\nwe investigate what happens to the entanglement in a qubit-qutrit\nsystem under common noise channels for qubit (qutrit): dephasing,\nphase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing channels. The two specific environment noise situations\nwill be considered: (i) local and (ii) multi-local. In the case (i),\nonly one part of a qubit-qutrit system ($S$) interacts with its\nenvironment. In the case (ii), both the two parts of $S$ interact\nwith their local environments, independently.\n\n\n\n\\subsection{Dephasing channels} \\label{sec3.1\n\nThe set of Kraus operators for a single qubit $A$ and a single\nqutrit $B$ that reproduce the effect of a dephasing channel are\ngiven  by\n\\begin{eqnarray}                                                   \\label{eq.6}\nE_{1}^{A}&=&\\left(\\begin{array}{cc}\n1&0\\\\\n0&\\sqrt{1-\\gamma_{A}}\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\left(\\begin{array}{cc}\n0&0\\\\\n0&\\sqrt{\\gamma_{A}}\\\\\n\\end{array}\n\\right)\\otimes I_{3},\\\\                       \\label{eq.7}\nF_{1}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&\\sqrt{1-\\gamma_{B}}&0\\\\\n0&0&\\sqrt{1-\\gamma_{B}}\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n0&0&0\\\\\n0&\\sqrt{\\gamma_{B}}&0\\\\\n0&0&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n0&0&0\\\\\n0&0&0\\\\\n0&0&\\sqrt{\\gamma_{B}}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\nThe time-dependent parameters are defined as\n$\\gamma_{A}=1-e^{-t\\Gamma_{A}}$ and $\\gamma_{B}=1-e^{-t\\Gamma_{B}}$.\nHere $\\gamma_{A}$, $\\gamma_{B}\\in[0,1]$. $\\Gamma_{A}$ ($\\Gamma_{B}$)\ndenotes the decay rate of the subsystem $A$ ($B$).\n\n\n\nAccording to Eq.(\\ref{eq.5}), the time-dependent evolved density\noperator $\\rho^{AB}(t)$ of a hybrid qubit-qutrit system, which is\ninitially in the entangled state $\\rho^{AB}(0)$, is given by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.8}\n\\rho^{AB}(t)=\\left(\\begin{array}{cccccc}\nb&0&0&0&0&0\\\\\n0&\\frac{b+c}{2}&0&\\frac{(b-c)\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}}{2}&0&0\\\\\n0&0&\\frac{1-c-3b}{2}&0&0&0\\\\\n0&\\frac{(b-c)\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}}{2}&0&\\frac{b+c}{2}&0&0\\\\\n0&0&0&0&b&0\\\\\n0&0&0&0&0&\\frac{1-c-3b}{2}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nIn order to characterize the dynamics of evolution for  the density\nmatrix and consider the entanglement of this system quantified by\nnegativity, we should calculate the eigenvalues of the partial\ntranspose of the time-evolved density matrix $\\rho^{AB}(t)$ and\ndetermine the potential negative eigenvalues.\n\n\n\n\\textsl{(1) Multi-local dephasing channel}. The eigenvalue which can\npotentially be negative is\n\\begin{eqnarray}                                                   \\label{eq.9}\n\\lambda=b-\\frac{c-b}{2}\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}.\n\\end{eqnarray}\nThe entanglement of the qubit-qutrit system under a multi-local\ndephasing channel  is\n\\begin{eqnarray}                                                   \\label{eq.10}\nN^{mul-loc}(\\rho^{AB})=2\\max\\{0,\n\\frac{c-b}{2}\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}-b\\}.\\nonumber\\\\\n\\end{eqnarray}\nIt is easy to obtain that all the states which are initially\nentangled ($3b<c\\leq1-3b$) become separable when\n$(1-\\gamma_{A})(1-\\gamma_{B})\\leq\\left(\\frac{2b}{c-b}\\right)^2$ and\n$b\\neq0$.\n\n\n\n\\textsl{(2) Qubit dephasing channel only}. If the qutrit field is\nturned off (i.e., $\\gamma_{B}=0$), that is, only a dephasing noise\nacts on qubit $A$ alone, the  entanglement of the qubit-qutrit\nsystem is\n\\begin{eqnarray}                                                     \\label{eq.11}\nN^{bit}(\\rho^{AB})=2\\max\\{0, \\frac{c-b}{2}\\sqrt{1-\\gamma_{A}}-b\\}.\n\\end{eqnarray}\nAll the states which are initially entangled ($3b<c\\leq1-3b$) become\nseparable as soon as $\\gamma_{A}\\geq\n1-\\left(\\frac{2b}{c-b}\\right)^2$ and $b\\neq0$.\n\n\n\n\\textsl{(3) Qutrit dephasing channel only}. By the same argument as\nthat made in the qubit dephasing noise, for a dephasing noise acting\non qutrit $B$ alone, the  entanglement is\n\\begin{eqnarray}                                                   \\label{eq.12}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{c-b}{2}\\sqrt{1-\\gamma_{B}}-b\\}.\n\\end{eqnarray}\nA single-qutrit dephasing channel will induce ESD in the\nqubit-qutrit system when $\\gamma_{B}\\geq\n1-\\left(\\frac{2b}{c-b}\\right)^2$ and $b\\neq 0$.\n\n\nTogether with these pieces of results, one can see that ESD takes\nplace under local and multi-local dephasing channels in a general\nqubit-qutrit system if and only if the system is initially in the\nstate shown in Eq.(\\ref{eq.1}) with the parameter $b\\neq0$.\nMoreover, the smaller $c\/b$, the longer the death time range.\nHowever, if $b=0$,\nno ESD occurs.\\\\\n\n\n\n\n\n\\subsection{ Phase-flip channels}\\label{sec3.2}\n\nThe Kraus operators describing the phase-flip channel for a single\nqubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.13}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&-1\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.14}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{-i2\\pi\/3}&0\\\\\n0&0&e^{i2\\pi\/3}\\\\\n\\end{array}\n\\right),  \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$. $\\Gamma_{A}$ ($\\Gamma_{B}$)\nrepresents the decay rate of the subsystem $A$ ($B$).\n\n\n\nWe can obtain the time-evolved density-matrix dynamics\n$\\rho^{AB}(t)$ of the qubit-qutrit system under a phase-flip\nchannel, according to Eq.(\\ref{eq.5}). That is,\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.15}\n\\rho^{AB}(t)=\\left(\\begin{array}{cccccc}\nb&0&0&0&0&0\\\\\n0&\\frac{b+c}{2}&0&\\frac{(b-c)(1-\\gamma_{A})(1-\\gamma_{B})}{2}&0&0\\\\\n0&0&\\frac{1-c-3b}{2}&0&0&0\\\\\n0&\\frac{(b-c)(1-\\gamma_{A})(1-\\gamma_{B})}{2}&0&\\frac{b+c}{2}&0&0\\\\\n0&0&0&0&b&0\\\\\n0&0&0&0&0&\\frac{1-c-3b}{2}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nWith the same argument as that made in dephasing channels, some\nresults can be presented as follows.\n\n\n\n\n\\textsl{(1) Multi-local phase-flip channel}. The entanglement of the\nqubit-qutrit system under a multi-local phase-flip channel is\ncalculated as\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.16}\n &&N^{mul-loc}(\\rho^{AB})= 2\\max\\left\\{0,\n\\frac{(c-3b)+(c-b)((1-\\gamma_{A})(1-\\gamma_{B})-1)}{2}\\right\\}.\\nonumber\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nThat is,  all the states which are initially entangled\n($3b<c\\leq1-3b$) become separable for all values of\n$(1-\\gamma_{A})(1-\\gamma_{B})\\leq \\frac{2b}{c-b}$ and $b\\neq0$.\n\n\n\n\\textsl{(2) Qubit phase-flip channel only}. The entanglement of the\nsystem is\n\\begin{eqnarray}                                                   \\label{eq.17}\nN^{bit}(\\rho^{AB})=2\\max\\{0,\\frac{c-3b-\\gamma_{A}(c-b)}{2}\\}.\n\\end{eqnarray}\nAll the states which are initially entangled ($3b<c\\leq1-3b$) become\nseparable as soon as $\\gamma_{A}\\geq \\frac{c-3b}{c-b}$  and $b \\neq\n0$.\n\n\n\n\\textsl{(3) Qutrit phase-flip channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) is given\nas\n\\begin{eqnarray}                                                   \\label{eq.18}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{c-3b-\\gamma_{B}(c-b)}{2}\\}.\n\\end{eqnarray}\nAll the states which are initially entangled become separable when\n$\\gamma_{B}\\geq \\frac{c-3b}{c-b}$ and $b\\neq 0$.\n\n\n\nSimilar to the case with a dephasing noise, ESD  also takes place\nwhen the system undergos a phase-flip noise environment but no ESD\noccurs when the parameter $b=0$. Difference from the dephasing\nchannel, the decay time range  under the multi-local phase-flip\nchannel is shorter than that in the local channel if $3b<c<5b$, and\nlonger for $5b<c\\leq 1-3b$.\n\n\n\n\n\\bigskip\n\n\\subsection{ Bit-(Trit-) flip channels} \\label{sec3.3\n\n\n\nThe Kraus operators describing the bit-flip channel for a single\nqubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.19}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n0&1\\\\\n1&0\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.20}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n0&0&1\\\\\n1&0&0\\\\\n0&1&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n0&1&0\\\\\n0&0&1\\\\\n1&0&0\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$.\n\n\n\n\nThe matrix elements of $\\rho$ after the interaction time $t$ under a\nmulti-local bit-(trit-) flip channel are given by\n\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.21}\n\\rho_{11}(t)&=\\rho_{55}(t)=&\\frac{1}{12}(12b+3(b-c)\\gamma_{A}(\\gamma_{B}-1)\\nonumber\\\\&&+2(1-6b)\\gamma_{B}),\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)\\gamma_{A}(\\gamma_{B}-1)\\nonumber\\\\&&+(2-6b-6c)\\gamma_{B}),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}\\left(3(1-3b-c)+(9b+3c-2)\\gamma_{B}\\right),\\nonumber\\\\\n\\rho_{15}(t)&=\\rho_{51}(t)=&\\frac{1}{12}(b-c)\\gamma_{A}(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{35}(t)&=\\rho_{53}(t)=&\\rho_{16}(t)=\\rho_{61}(t)=\\frac{1}{12}(b-c)(2-\\gamma_{A})\\gamma_{B},\\nonumber\\\\\n\\rho_{26}(t)&=\\rho_{62}(t)=&\\rho_{34}(t)=\\rho_{43}(t)=\\frac{1}{12}(b-c)\\gamma_{A}\\gamma_{B},\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{12}(b-c)(2-\\gamma_{A})(3-2\\gamma_{B}),\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero. Here\n$\\rho=(\\rho_{ij})$  and $i,j=1,\\ldots,6$.\n\n\n\n\n\\textsl{(1) Qubit bit-flip channel only}. We obtain the same results\nas the case with a phase-flip channel.\n\n\n\n\\textsl{(2) Qutrit trit-flip channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) can be\ngiven by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                        \\label{eq.22}\nN^{trit}(\\rho^{AB})=2\\max\\left\\{0,\\frac{3b-9c-(1-8b+2c)\\gamma_{B}}{6}\\right\\}.\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nThe states  become separable once $\\gamma_{B}\\geq\n\\frac{3c-9b}{1-8b+2c}$.\n\n\n\n\\textsl{(3) Multi-local bit-(trit-) flip channel}. For simplicity,\nwe choose the local asympotic bit-(trit-) flip rates\n$\\Gamma_{A}=\\Gamma_{B}=\\Gamma$ to discuss the effect of this noise\non entanglement. Unfortunately, it is difficult to calculate the\nanalytical eigenvalues of the partial transpose of the time-evolved\ndensity matrix. Therefore, we take the numerical calculation for\nvarious initial states as examples to investigate the dynamics of\nentanglement of the qubit-qutrit system under a multi-local\nbit-(trit-) flip channel.\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig1.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit- (trit-) flip noise with $a=0$. $N$ represents the\nnegativity of a hybrid qubit-qutrit system. $\\gamma$ is the\ntime-dependent parameter and\n$\\gamma_A=\\gamma_B=\\gamma=1-e^{-t\\Gamma}$. The solid, dashed,\nshort-dashed, dashed-dotted, and dotted lines correspond to $b=0$,\n$b=1\/30$, $b=2\/30$, $b=3\/30$, and $b=4\/30$, respectively.}\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig2.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) flip noise with parameter $b=0$. The solid,\ndashed,  dashed-dotted, and dotted lines correspond to $c=1$,\n$c=3\/4$, $c=1\/2$, and $c=1\/4$, respectively.}\\label{Fig2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig3.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) flip noise with $b=1\/20$. The solid, dashed,\ndashed-dotted, and dotted lines correspond to $c=16\/20$, $c=12\/20$,\n$c=8\/20$, and $c=4\/20$, respectively.}\\label{Fig3}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\nAs the first example, we consider the initial entangled states shown\nin Eq.(\\ref{eq.1}) with the parameter $a=0$ which are equivalent to\nWerner states \\cite{r17}  in a $2 \\otimes 2$ systems but with\ndifferent noise channels. Their dynamics of entanglement is shown in\nFig.\\ref{Fig1}. We note that the time-evolved density matrix for the\ninitial state with the parameters $a=b=0$ is just the case for the\nmaximally entangled Bell state $\\vert \\psi^-\\rangle$ \\cite{r18},\nand the system does not suffer from ESD under a multi-local\nbit-(trit-) flip channel.\n\n\n\nAs a second example, we consider the initial entangled states with\nthe parameter $b=0$. Their dynamics of entanglement is shown in\nFig.\\ref{Fig2}.\n\n\n\n\nFinally, we consider the initial states with $a, b, c >0$. The\ndynamics of entanglement is plotted in Fig.\\ref{Fig3} for a\nparticular value of the parameter $b=0.05$.\n\n\n\n\nFrom the above analysis and numerical results, we conjecture that\nESD in a hybrid qubit-qutrit system under local and multi-local bit-\n(trit-) flip channels is a general phenomenon. No ESD takes place\nunder a qubit-flip channel alone if and only if the qubit-qutrit\nsystem is initially in the states shown in Eq.(\\ref{eq.1}) with the\nparameter $b=0$. However, ESD always occurs when the system\nundergoes a qutrit-flip channel alone.\n\n\n\n\\bigskip\n\n\n\\subsection{Bit-(Trit-) phase-flip channels}\\label{sec3.4}\n\n\n\nThe Kraus operators describing the bit-phase flip channel for a\nsingle qubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.23}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n0&-i\\\\\ni&0\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.24}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&0&e^{i2\\pi\/3}\\\\\n1&0&0\\\\\n0&e^{-i2\\pi\/3}&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&0&e^{-i2\\pi\/3}\\\\\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n\\end{array}\n\\right),\\nonumber\\\\\nF_{4}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&e^{-i2\\pi\/3}&0\\\\\n0&0&e^{i2\\pi\/3}\\\\\n1&0&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{5}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n1&0&0\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$.\n\n\n\n\nThe  elements of the density matrix $\\rho$ after the interaction\ntime $t$ under multi-local bit-(trit-) phase-flip channels are given\nby\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.25}\n\\rho_{11}(t)&=\\rho_{55}(t)=&b+\\frac{1}{4}(b-c)\\gamma_{A}(\\gamma_{B}-1)+(\\frac{1}{6}-b)\\gamma_{B},\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)\\gamma_{A}(\\gamma_{B}-1)+(2-6b-6c)\\gamma_{B}),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}(3(1-3b-c)+(9b+3c-2)\\gamma_{B}),\\nonumber\\\\\n\\rho_{15}(t)&=\\rho_{51}(t)=&-\\frac{1}{12}(b-c)\\gamma_{A}(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{16}(t)&=\\rho_{61}(t)=&\\rho_{35}(t)=\\rho_{53}(t)=\\frac{1}{24}(b-c)(\\gamma_{A}-2)\\gamma_{B},\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{12}(b-c)(2-\\gamma_{A})(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{26}(t)&=\\rho_{62}(t)=&\\rho_{34}(t)=\\rho_{43}(t)=\\frac{1}{12}(b-c)\\gamma_{A}\\gamma_{B},\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero.\n\n\n\n\\textsl{(1) Local bit-(trit-) phase-flip channel only}. We obtain\nthe same results as the case with a bit-(trit-) flip channel.\n\n\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig4.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) phase-flip noise with  the parameter $a=0$.\nThe solid, dashed, short-dashed, dashed-dotted, and dotted lines\ncorrespond to $b=0$, $b=1\/30$, $b=2\/30$, $b=3\/30$, and $b=4\/30$,\nrespectively.}\\label{Fig4}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig5.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) phase-flip noise with the parameter $b=0$.\nThe solid, dashed, dashed-dotted, and dotted lines correspond to\n$c=1$, $c=3\/4$, $c=1\/2$, and $c=1\/4$, respectively.}\\label{Fig5}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig6.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(tri-t) phase-flip noise with the parameter\n$b=1\/20$. The solid, dashed, dashed-dotted, and dotted lines\ncorrespond to $c=16\/20$, $c=12\/20$, $c=8\/20$, and $c=4\/20$,\nrespectively. }\\label{Fig6}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\textsl{(2) Multi-local bit-(trit-) phase-flip channel}. For\nsimplicity, the time dependent parameters are also defined as\n$\\gamma_{A}=\\gamma_{B}=\\gamma$. It is  difficult to obtain the\nanalytical results. The similar work is made as that in the case\nwith a bit-(trit-) flip channel, and the dynamics of entanglement of\nthe initial states with the parameter $a=0$ is displayed in\nFig.\\ref{Fig4}. Different from a multi-local bit-(trit-) flip\nchannel, one can see that there exists ESD for the maximally\nentangled Bell state (solid line). The dynamics of entanglement of\nthe initial states shown in Eq.(\\ref{eq.1}) with the parameters\n$b=0$ and  $a, b, c\n> 0$ are displayed in Fig.\\ref{Fig5} and Fig.\\ref{Fig6}, respectively.\n\n\n\n\n\n\n\\subsection{Depolarizing channels}\\label{sec3.5\n\n\n\n\nA depolarizing channel represents the process in which the density\nmatrix is dynamically replaced by the maximally mixed state $I\/d$.\nHere $I$ is the identity matrix of a single qudit. The set of Kraus\noperators that reproduces the effect of the depolarizing channel for\na single qubit $A$  are given by\n\\begin{eqnarray}                                                   \\label{eq.26}\nE_{1}^{A} &=& \\sqrt{1-\\frac{3\\gamma_{A}}{4}}I_{6},\\nonumber\\\\\nE_{2}^{A} &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{1}\\otimes I_{3},\\nonumber\\\\\nE_{3}^{A} &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{2}\\otimes\nI_{3},\\nonumber\\\\\nE_{4}^{A}  &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{3}\\otimes I_{3},\n\\end{eqnarray}\nwhere $\\sigma_i$ ($i=1,2,3$) are the three Pauli matrices. The Kraus\noperators describing a single-qutrit depolarizing noise are given\nby\\textsuperscript{\\cite{19}}\n\\begin{eqnarray}                                                   \\label{eq.27}\nF_{1}^{B} &=& I_{2}\\otimes\\sqrt{1-\\frac{8\\gamma_{B}}{9}}I_{3},\\nonumber \\\\\nF_{2}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y,\\nonumber \\\\\nF_{3}^{B} &=& I_{2}\\otimes\n\\frac{\\sqrt{\\gamma_{B}}}{3}Z,\\nonumber \\\\\nF_{4}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2},\\nonumber \\\\\nF_{5}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}YZ,\\nonumber \\\\\nF_{6}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2}Z,\\nonumber \\\\\nF_{7}^{B} &=& I_{2}\\otimes\\frac{\\sqrt{\\gamma_{B}}}{3}YZ^{2},\\nonumber \\\\\nF_{8}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2}Z^{2},\\nonumber \\\\\nF_{9}^{B} &=& I_{2}\\otimes\\frac{\\sqrt{\\gamma_{B}}}{3}Z^{2},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}                                                   \\label{eq.28}\nY &=& \\left(\\begin{array}{ccc}\n0&1&0\\\\\n0&0&1\\\\\n1&0&0\\\\\n\\end{array}\n\\right), \\nonumber\\\\ Z &=& \\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nand $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, $\\gamma_{A}, \\gamma_{B}\\in[0,1]$.\n\n\n\n\n\n\nThe matrix elements of $\\rho$ after the interaction time $t$ under a\nmulti-local depolarizing channel are given by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.29}\n\\rho_{11}(t)&=\\rho_{55}(t)=&\\frac{1}{12}(12b+3(b-c)(\\gamma_B-1)\\gamma_{A} + 2(1-6b)\\gamma_{B}),\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)(\\gamma_B-1)\\gamma_A+ (2-6b-6c)\\gamma_B),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}(3(1-3b-c)+(9b+3c-2)\\gamma_B),\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{2}(b-c)(1-\\gamma_{A})(-\\gamma_{B}),\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero.\n\n\n\n\n\n\n\n\\textsl{(1) Multi-local depolarizing channel}. The negativity for\nthe composite system with initial entangled states ($3b<c\\leq1-3b$)\nis given by\n\\begin{eqnarray}                                                   \\label{eq.30}\nN^{mul-loc}(\\rho^{AB})=2\\max\\{0,-\\lambda\\},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\lambda=\\frac{9(b-c)\\gamma_{A}(\\gamma_{B}-1)+2\\gamma_{B}(1-9b+3c)+18b-6c}{12}.\\nonumber\\\\\n\\label{eq.31}\n\\end{eqnarray}\nIt is easy to obtain the result that all the states, which are\ninitially entangled ones, become separable if and only if\n$9(b-c)\\gamma_{A}(\\gamma_{B}-1)+2\\gamma_{B}(1-9b+3c)\\geq 6(c-3b)$.\n\n\n\n\n\\textsl{(2) Qubit depolarizing channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) is given\nby\n\\begin{eqnarray}                                                   \\label{eq.32}\nN^{bit}(\\rho^{AB})=2\\max\\left\\{0,\\frac{2c-6b-3\\gamma_{A}(c-b)}{4}\\right\\}.\n\\end{eqnarray}\nThe states become separable for all values of $\\gamma_{A}\\geq\n\\frac{2c-6b}{3(c-b)}$.\n\n\n\n\n\\textsl{(3) Qutrit depolarizing channel only}. The negativity for\nall the states which are initially entangled ($3b<c\\leq1-3b$) can be\nwritten as\n\\begin{eqnarray}                                                   \\label{eq.33}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{3c-9b-(1-9b+3c)\\gamma_{B}}{6}\\}.\n\\end{eqnarray}\nThe states become separable for all values of  $\\gamma_{B}\\geq\n\\frac{3c-9b}{1-9b+3c}$.\n\n\n\n\\begin{widetext}\n\\begin{center}\n\\begin{table}[!h]\n\\tabcolsep 0pt \\caption{ESD in $2\\otimes3$ systems under certain\nnoises. }\n\\begin{center}\n\\def\\temptablewidth{0.7\\textwidth}\n{\\rule{\\temptablewidth}{1pt}}\n\\begin{tabular*}{\\temptablewidth}{@{\\extracolsep{\\fill}}cccc}\n                              & multi-local           & qubit noise only           & qutrit noise only  \\\\\\hline\n      dephasing               & ESD with $b\\neq 0$    & ESD with $b\\neq 0$         & ESD with $b\\neq 0$   \\\\\n       phase-flip             & ESD with $b\\neq 0$    & ESD with $b\\neq 0$         & ESD with $b\\neq 0$    \\\\\n       bit-(trit-) flip       & exist ESD             & ESD with $b\\neq0$          & ESD  \\\\\n      bit-(trit-) phase-flip  & exist ESD             &ESD with $b\\neq0$           & ESD  \\\\\n      depolarizing            & exist ESD             &ESD                         & ESD\n       \\end{tabular*}\n       {\\rule{\\temptablewidth}{1pt}}\n       \\end{center}\\label{tab1}\n       \\end{table}\n       \\end{center}\n\\end{widetext}\n\n\n\n\n\\section{Discussion and summary}\\label{sec4\n\n\n\n\nPutting all the pieces of our results together, one can see that ESD\nis a general phenomenon in a qubit-qutrit system undergoing various\nindependent noise channels and we show the outcomes explicitly in\nTable.\\ref{tab1}. ``ESD with $b\\neq 0$'' denotes  ESD takes places\nin a qubit-qutrit system if and only if $b\\neq 0$, that is, if b=0,\nno ESD occurs.``exist ESD'' denotes ESD may occurs in a qubit-qutrit\nsystem, but not necessary. ``ESD'' denotes ESD always occurs.\n\n\nDecoherence which is characterized by the decay of the off-diagonal\nelements of the density matrix describing the system\n\\cite{Decoherence},  results from the unwanted interactions of a\nquantum system with its environment. According to\nEqs.(\\ref{eq.8},\\ref{eq.15},\\ref{eq.21},\\ref{eq.25},\\ref{eq.29}),\none can see that the coherence of a qubit-qutrit system can be\ndestroyed completely when the system undergos the multi-local or\nlocal\n dephasing, phase-flip and depolarizing channels, while\nthe disappearance of coherence does not occur even though\n$\\gamma_{A}, \\gamma_{B}\\mapsto 1$ for multi-local or local the\nbit-(trit-)flip and bit-(trit-)phase-flip\n channels.\n\n\n\n\nUsing the negativity criterion for quantifying entanglement, it is\nshown that the density matrix of a hybrid qubit-qutrit system always\nsuffers from ESD and decoherence under both local and multi-local\ndepolarizing channels, which is different from the result that no\nESD in any density matrix of a qubit-qutrit system occurs when only\nthe qubit is coupled to its depolarizing channel presented in Ref.\n\\cite{r10}.\n\n\n\n\n\n\n\n\n\n\nThe  entanglement dynamics of a specific one-parameter class of\nstates interacting with a local and multi-lcoal dephasing noise in\nhybrid qubit-qutrit systems has been studied by Ann \\emph{et al.}\n\\cite{r9}  in 2008. We have generalized their study to a more\ngeneral case, that is, a two-parameter class of states under the\ninfluence of local and multi-local dephasing, phase-flip,\nbit-(trit-) flip, bit-(trit-) phase-flip, and depolarizing channels.\nWith analytical and numerical analysis, we have shown that ESD is a\ngeneral phenomenon in a qubit-qutrit system under various noise\nchannels,\n not only the case with dephasing and\ndepolarizing ones \\cite{r9}.  It can only be avoided in some initial\nstates undergoing particular noise channels.\n\n\n\n\n\n\nIn summary, our results show that the  environment, which causes\ndephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing, affects the entanglement and the coherence of  a\nhybrid qubit-qutrit system in a two-parameter class of entangled\nstates in very different ways. ESD is a general phenomenon in a\nqubit-qutrit system undergoing various independent noise channels\nand it can only be avoided in some initial states undergoing\nparticular noise channels. Moreover, we can divide those noise\nchannels into two groups. For multi-local and local dephasing,\nphase-flip, and depolarizing noise channels, a time scale of\ndisentanglement is usually shorter than the decay of the\noff-diagonal dynamics, and coherence disappears in an infinite-time\nlimit $t\\mapsto\\propto$ in which $\\gamma_{A}, \\gamma_{B}\\mapsto 1$.\nFor multi-local or local bit-(trit-) flip and bit-(trit-) phase-flip\nchannels, disentanglement may occur in the infinite-time limit, but\nthe disappearance of coherence does not occur even though\n$\\gamma_{A}, \\gamma_{B}\\mapsto 1$.\n\n\n\n\n\n\\section*{ACKNOWLEDGEMENTS}\n\n\nThis work is supported by the National Natural Science Foundation of\nChina under Grant Nos. 10974020 and 11174039,  NCET-11-0031, and the\nFundamental Research Funds for the Central Universities.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe algebra of symbols of classical pseudodifferential operators $\\Psi \\mathrm{DO}_{cl}(X,E)$ on \na closed manifold $X$ acting on sections of a vector bundle $E$ can be defined as the \nquotient of $\\Psi \\mathrm{DO}_{cl}(X,E)$ by the ideal of smoothing operators. Since pseudodifferential  \noperators are smooth off the diagonal the symbol algebra is localized on the diagonal and \nit therefore can also be defined locally, using the product expansion formula and the change \nof charts formula for pseudodifferential operators. That the local heat kernel coefficients \nand the index of elliptic pseudodifferential operators are locally computable relies on the \nfact that the index and asymptotic spectral properties of pseudodifferential operators depend \nonly on their class in the symbol algebra. Note that the principal symbol of a \npseudodifferential operator is a section of the bundle of endomorphisms of $\\pi^* E$, \nwhere $\\pi: T^* X \\to X$ is the canonical projection.\n\nThe bundle of endomorphisms of a complex hermitian vector bundle is a bundle of simple \nmatrix algebras with $*$-structure. However, not all bundles of simple matrix algebras with \n$*$-structure, so-called Azumaya bundles, are isomorphic to endomorphism bundles of hermitian \nvector bundles. The obstruction is the so-called Dixmier-Douady class in $H^3(X,\\mathbb{Z})$. \nGiven an Azumaya bundle $\\mathcal{A}$ it is possible to construct algebras of symbols whose \nprincipal symbols takes values in the space of sections of the Azumaya bundle \n$\\pi^*\\mathcal{A}$ (see for example \\cite{MMS05} and the discussion in \\cite{MMS06}). \nFollowing \\cite{MMS05} we refer to such a symbol algebra as the algebra of symbols of \nprojective pseudodifferential operators. For such symbol algebras one can define an index \nand Mathai, Melrose and Singer \\cite{MMS06} proved an index formula for projective \npseudodifferential operators, analogous to the Atiyah-Singer index formula. The topological \nindex in this case is a map from twisted $K$-theory to $\\mathbb{R}$. It has also been shown \nin \\cite{MMS06} that any oriented manifold admits a projective Dirac operator even if the \nmanifold does not admit a spin structure. In this case its index may fail to be an integer.\n\nAnother important quantity that depends only on the class of the symbol of a \npseudodifferential operator is the so-called Wodzicki residue or noncommutative residue. \nUp to a factor it is the \nunique trace on the algebra of pseudodifferential operators. The Wodzicki residue appeared \nfirst as a residue of a zeta function measuring spectral asymmetry (\\cite{APS76,Wo84}). \nWodzicki showed that the regularity of the $\\eta$-function of a Dirac-type operator at \nzero -- a necessary ingredient to define the $\\eta$-invariant -- follows as a special case \nfrom the vanishing of the Wodzicki residue on pseudodifferential projections (as remarked \nby Br\\\"uning and Lesch \\cite{BL99} the regularity of the $\\eta$-function at zero for any \nDirac type operator and the vanishing of the Wodzicki residue on pseudodifferential \nprojections are actually equivalent). The regularity of the $\\eta$-function was proved by \nAtiyah, Patodi and Singer in \\cite{APS76} in the case when $X$ is odd dimensional and \nlater by Gilkey (\\cite{G81}) in the general case using $K$-theoretic arguments. Note that \nwhereas the Wodzicki residue can be locally computed its vanishing on pseudodifferential \nprojections is not a local phenomenon. Gilkey \\cite{G79} constructed a pseudodifferential \nprojection whose residue density is non-vanishing but integrates to zero. \n\nIn our paper we show that the Wodzicki residue can also be defined for projective \npseudodifferential operators (this has already been observed in \\cite{MMS06}) and show \nthat it vanishes on projections in case the dimension of the manifold is odd. \nOur proof is based on the  Leray-Hirsch theorem in twisted $K$-theory and a purely \nalgebraic result on `residue-traces' in filtered rings. \n\nIf $L$ is a filtered ring then we call a linear functional $\\tau: L \\to \\mathbb{C}$ a \nresidue trace if $\\tau(L^{-N})=\\{0\\}$ for $N$ large enough. We prove that the value of \n$\\tau$ on projections depends only on their class in $L^{(0)}:=L\/L^{-1}$. Thus, if the \nmap $K^0_{alg}(L) \\to K^0_{alg}(L^{(0)})$ is surjective the map $\\tau$ descends to a map \nfrom the algebraic $K$-theory of $L^{(0)}$ to $\\mathbb{C}$. This result can be applied \nto the Wodzicki residue showing that it descends to a map from twisted $K$-theory  \n$K^0(S^*X,\\pi^*\\mathcal{A})$ to $\\mathbb{C}$. We then use the Leray-Hirsch theorem to \nshow that this map actually vanishes. We reduce the \nproblem to positive spectral projections of generalized Dirac operators for which it \nis known \\cite{BG92} that the residue density vanishes.\n\n\\section{Convolution bundles and Azumaja bundles}\n\nPseudodifferential operators on a smooth closed Riemannian manifold $X$ acting on sections of a vector \nbundle $E$ can be understood as co-normal distributional sections in the vector bundle \n$E \\boxtimes E^*$\\footnote{$E \\boxtimes E^*$ denotes the external tensor product of $E$ \nand its dual bundle $E^*$, i.e., the fibre over a point $(x,y)$ is $E_x\\otimes E^*_y$.} \nover the space $X \\times X$, by identifying the operators with their distributional kernel. \nThe bundle $E \\boxtimes E^*$ has the following structure that allows to convolve kernels of \nintegral operators: any element in the fibre over ${(x,y)}$ may be multiplied by an element \nin the fibre over ${(y,z)}$ to give an element in the fibre over ${(x,z)}$. \nMoreover, this multiplication satisfies natural conditions such as associativity. \nIn order to define projective pseudodifferential operators it is \nconvenient to formalize this structure, as we shall do in this section.\n\n\\subsection{Convolution bundles}\n\nLet $\\mathcal{U}$ denote an open neighborhood of the diagonal $\\Delta(X)$ \nin $X \\times X$ which is symmetric under the reflection map $s:(x,y)\\mapsto(y,x)$. \nLet $p_{ik}:X \\times X \\times X \\to X \\times X$ be defined by \n$p_{ik}(x_1,x_2,x_3)=(x_i,x_k)$ and set \n$\\widetilde{\\mathcal{U}}\n   :=p_{12}^{-1}(\\mathcal{U}) \\cap p_{23}^{-1}(\\mathcal{U})  \n   \\cap p_{13}^{-1}(\\mathcal{U})$. \nDenote by $\\widetilde p_{ik}$ the restriction of the map $p_{ik}$ to $\\widetilde{\\mathcal{U}}$.\n\n\\begin{definition}\\label{def:bundle}\nLet $\\pi: F \\to \\mathcal{U}$ be a locally trivial vector bundle with typical fibre \n$\\mathrm{Mat}(k)$, the complex $k\\times k$-matrices. We call $F$ a \\emph{convolution bundle} \nif there exists a homomorphism of vector bundles \n$m: \\widetilde{p}_{12}^* F \\otimes \\widetilde{p}_{23}^* F \\to F$ \nsuch that the following conditions are satisfied:\n\\begin{itemize}\n \\item[(i)] The following diagram is commutative: \n   \\begin{equation*\n    \\xymatrix@1@=1.2cm@M=5pt{{\\widetilde{p}_{12}^* F \\otimes \\widetilde{p}_{23}^* F} \n    \\ar[d] \\ar[r]^{\\qquad m}  & F \\ar[d] \\\\\n    \\widetilde{\\mathcal{U}}   \\ar[r]^{\\widetilde{p}_{13}} & \\mathcal{U}}\n   \\end{equation*}\n \\item[(ii)] $m$ is associative, i.e., whenever $f_{ij}$ belong to the fibre \n   $F_{(x_i,x_j)}$ then \n   \\begin{align*}\n    m \\left( m(f_{12}\\otimes f_{23}) \\otimes f_{34}\\right)= \n    m \\left( f_{12}\\otimes m(f_{23} \\otimes f_{34})\\right).\\footnotemark  \n   \\end{align*} \n \\item[(iii)] There\\footnotetext{Here we implicitely assume that $(x_1,x_2,x_3)$, \n   $(x_1,x_3,x_4)$, and $(x_1,x_2,x_4)$ belong to $\\widetilde{\\mathcal{U}}$.} \n   is an atlas $\\{\\mathcal{O}_\\alpha\\}$ of $\\mathcal{U}$ together with local trivializations \n    $$\\phi_\\alpha: \\pi^{-1}\\mathcal{O}_\\alpha \\to \\mathcal{O}_\\alpha \\times \n      \\mathrm{Mat}(k),$$ \n   such that\n    $$\\phi_\\alpha(m(f_{12} \\otimes f_{23})) = \\phi_\\alpha(f_{12}) \\cdot \\phi_\\alpha(f_{23})$$\n   whenever $f_{ij}\\in F_{(x_i,x_j)}$ with \n   $(x_1,x_2,x_3)\\in \\widetilde{p}_{12}^{-1}(\\mathcal{O}_\\alpha)\n   \\cap\\widetilde{p}_{23}^{-1}(\\mathcal{O}_\\alpha)$. \n \\end{itemize}\n\\end{definition}\n\n\\begin{definition}\\label{def:star}\nA \\emph{$*$-structure} on $F$ is a conjugate linear map $*: F \\to F$ of vector bundles \nsuch that \n\\begin{gather*}\n \\xymatrix@1@=1.2cm@M=5pt{\n     F  \\ar[d] \\ar[r]^{*} & F \\ar[d] \\\\\n    \\mathcal{U} \\ar[r]^{s} & \\mathcal{U} \n }\n\\end{gather*}\ncommutes, such that\n$\n (m(f \\otimes g))^*=m(g^* \\otimes f^*), \n$\nand such that the above local trivializations additionally satisfy  \n$$\n \\forall f \\in \\pi^{-1}(\\mathcal{O}_\\alpha \\cap s(\\mathcal{O}_\\alpha)): \\qquad \\phi_\\alpha(f^*)=\\phi_\\alpha(f)^*, \n$$\nwhere the star on the right hand side denotes the hermitian conjugation of matrices.\nWe will refer to a convolution bundle with $*$-structure as a \\emph{$*$-convolution bundle}.\n\\end{definition}\n\nNote that $E \\boxtimes E^*$ is a particular example for a $*$-convolution bundle; in this \ncase we can choose $\\mathcal{U}=X \\times X$. The restriction of a $*$-convolution \nbundle $F$ to the diagonal in $X \\times X$ is a bundle $\\mathcal{A}$ of finite dimensional \nsimple $C^*$-algebras. Following the literature \nwe refer to such bundles of matrix algebras as Azumaja bundles. \n\nAs shown in \\cite{MMS05} any Azumaja bundle $\\mathcal{A}$ \non $X$ gives rise to a convolution bundle near the diagonal in the following way, using an \natlas of local trivializations with respect to a good \ncover\\footnote{A cover is good if finite intersections of elements therein are either empty \nor contractible.} \n$\\{{U}_\\alpha\\}$ of $X$: The transition functions $\\sigma_{\\alpha \\beta}$ are smooth functions \non ${U}_{\\alpha \\beta}={U}_{\\alpha} \\cap {U}_{\\beta}$ with values in the automorphisms of \n$\\mathrm{Mat}(k)$. Since all automorphims are inner we \ncan choose local functions $\\varphi_{\\alpha \\beta}: {U}_{\\alpha \\beta} \\to SU(k)$\nthat implement $\\sigma_{\\alpha \\beta}$, i.e., \n$\\sigma_{\\alpha \\beta}(x)(A) = \\varphi_{\\alpha \\beta}^{-1}(x) A \\varphi_{\\alpha \\beta}(x)$. \nIn general, the functions $\\varphi_{\\alpha \\beta}$ may violate the co-cycle condition \nand therefore are not the transition functions of a vector bundle. The cocycle condition \nfor the $\\sigma_{\\alpha\\beta}$ together with the condition that the $\\varphi_{\\alpha \\beta}$ \nare chosen in $SU(k)$ show that any \n$\\varphi_{\\alpha \\beta} \\varphi_{\\beta \\gamma} \\varphi_{\\gamma \\alpha}$\nmust be a constant function on ${U}_\\alpha \\cap {U}_\\beta \\cap {U}_\\gamma$, \nequal to an $k$-th root of unity times the identity \nmatrix.\\footnote{On different triple intersections, the resulting unit-root can be different. \nThis induces a torsion element in $H^3(X,\\mathbb{Z})$, the Dixmier-Douady class.} \nThen we obtain a convolution bundle $F$ with typical fibre $\\mathrm{Mat}(k)$ \non a neighborhood of the diagonal by choosing the transition functions\n $$\\phi_{\\alpha\\beta}(x,y)(A)=\\varphi_{\\alpha \\beta}(x) A\\, \\varphi_{\\alpha \\beta}(y)^{-1},\n   \\qquad A\\in \\mathrm{Mat}(k),$$ \non ${U}_{\\alpha\\beta} \\times {U}_{\\alpha\\beta}$. \nThere are also other possible extensions of $\\mathcal{A}$, cf.\\ \\cite{MMS06}, and Proposition \n\\ref{prop:transition}, below. \n\n\\begin{remark}\\label{rem:atlas}\nIn the sequel it will be occasionally convenient to choose an atlas for $F$ consisting of sets \n$\\mathcal{O}_\\alpha:=U_\\alpha\\times U_\\alpha$, where $\\{U_\\alpha\\}$ is a good cover of $X$; \nthe corresponding trivialisations we shall denote by $\\phi_\\alpha$ $($so we use the same notation \nas in Definition {\\rm\\ref{def:bundle}.(iii)} above, but possibly have changed the atlas$)$. \n\\end{remark}\n\\subsection{Transition functions} \n\nIn the previous section we have seen how an Azumaja bundle leads to a convolution bundle by \nchoosing certain transition functions. Let us now have a closer look to the transition functions \nof an arbitrary $*$-convolution bundle. Fix an atlas as explained in Remark \\ref{rem:atlas} and let \n$\\phi_{\\alpha\\beta}:\\mathcal{O}_\\alpha\\cap \\mathcal{O}_\\beta=:\\mathcal{O}_{\\alpha\\beta}\n\\longrightarrow GL(\\mathrm{Mat}(k))$\nbe the transition functions defined by \n\\begin{equation*}\\label{eq:trans0}\n \\phi_\\beta\\circ\\phi_\\alpha^{-1}\\big((x,y),A\\big)=\\big((x,y),\\phi_{\\alpha\\beta}(x,y)(A)\\big),\n\\end{equation*}\nThen condition iii$)$ of Definition \\ref{def:bundle} is equivalent to \n $$\\phi_{\\alpha\\beta}(x,y)(A)\\phi_{\\alpha\\beta}(y,z)(B)=\\phi_{\\alpha\\beta}(x,z)(AB).$$\nIn particular, \n\\begin{equation}\\label{eq:trans1}\n (x,x)\\mapsto\\phi_{\\alpha\\beta}(x,x):\\mathcal{O}_{\\alpha\\beta}\\cap\\Delta(X)\n \\longrightarrow\\mathrm{Aut}(\\mathrm{Mat}(k)).\n\\end{equation} \nMoreover, Definition \\ref{def:star} on the level of the transition functions means that \n\\begin{equation}\\label{eq:trans2a}\n \\phi_{\\alpha\\beta}(x,y)(A^*)=\\phi_{\\alpha\\beta}(y,x)(A)^*.\n\\end{equation}\n\n\\begin{proposition}\\label{prop:transition}\nLet $F$ be a $*$-convolution bundle with transition functions $\\phi_{\\alpha\\beta}$ as described above. \nThen\n\\begin{equation}\\label{eq:trans2}\n \\phi_{\\alpha\\beta}(x,y)(A)\n =\\lambda_{\\alpha\\beta}(x,y)\\varphi_{\\alpha\\beta}(x)A\\,\\varphi_{\\alpha\\beta}(y)^{-1}\n\\end{equation}\nwith mappings \n $$\\varphi_{\\alpha\\beta}:\\mathcal{O}_{\\alpha\\beta}\\longrightarrow SU(k), \\qquad \n   \\lambda_{\\alpha\\beta}:\\mathcal{O}_{\\alpha\\beta}\\longrightarrow {\\mathbb C},$$\nsatifying  \n $$\\lambda_{\\alpha\\beta}(x,x)=1,\\qquad \n   \\lambda_{\\alpha\\beta}(x,y)\\lambda_{\\alpha\\beta}(y,z)=\\lambda_{\\alpha\\beta}(x,z),\\qquad\n   \\lambda_{\\alpha\\beta}(x,y)=\\overline{\\lambda_{\\alpha\\beta}(y,x)},$$\nand such that all $\\varphi_{\\alpha\\beta}\\varphi_{\\beta\\gamma}\\varphi_{\\gamma\\alpha}$  \nare constant functions on their domain of definition, equal to a $k$-th root of unity times \nthe identity matrix.\n\\end{proposition} \n\\begin{proof}\nCombining \\eqref{eq:trans1} with \\eqref{eq:trans2a} we find $\\varphi_{\\alpha\\beta}$ with \n\\begin{equation*}\n \\phi_{\\alpha\\beta}(x,x)(A)=\\varphi_{\\alpha\\beta}(x)A\\,\\varphi_{\\alpha\\beta}(x)^{-1}, \n\\end{equation*}\nsince all automorphisms of $\\mathrm{Mat}(k)$ are inner. \nNow let us define\n $$\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\n   =\\varphi_{\\alpha\\beta}(x)^{-1}\\phi_{\\alpha\\beta}(x,y)(A)\\varphi_{\\alpha\\beta}(y).$$\nWe then have \n $$\\phi_{\\alpha\\beta}^\\prime(x,x)(A)=A,\\qquad \n   \\phi_{\\alpha\\beta}^\\prime(A)(x,y)\\phi_{\\alpha\\beta}^\\prime(y,z)(B)=\\phi_{\\alpha\\beta}^\\prime(x,z)(AB).$$\nIt follows that \n$\\phi_{\\alpha\\beta}^\\prime(x,y)(AB)=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\\phi_{\\alpha\\beta}^\\prime(y,y)(B)\n=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)B$\nand, analogously, $\\phi_{\\alpha\\beta}^\\prime(x,y)(AB)=A\\phi_{\\alpha\\beta}^\\prime(x,y)(B)$. \nTherefore, for all matrices $A$, \n $$\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1})A=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\n   =A\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1}),$$\nwhere $\\mathbf{1}$ is the identity matrix. \nThis shows $\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1})$ is a multiple of the identity matrix. \nDenoting the corresponding factor by $\\lambda_{\\alpha\\beta}(x,y)$, the claim follows. \n\\end{proof}\n\n\\section{Projective Pseudodifferential Operators}\n\nProjective pseudodifferential operators have been defined in \\cite{MMS05}. We adapt this definition to fit \nin our setting of convolution bundles. \n\n\n\\subsection{Pseudodifferential operators}\n\nTo clarify notation let us briefly recall the definition of classical (or polyhomogeneous) \npseudodifferential operators on an open subset $\\Omega$ of ${\\mathbb R}^n$. Let $V\\cong{\\mathbb C}^k$ be a \n$k$-dimensional vector space.\n\nA symbol of order $m\\in{\\mathbb R}$ is a smooth function \n$a:\\Omega\\times\\Omega\\times{\\mathbb R}^n\\to\\mathrm{End}(V)=V\\otimes V^*$ satisfying estimates \n $$\\left\\|\\partial^\\alpha_\\xi\\partial^\\beta_{(x,y)} a(x,y,\\xi)\\right\\|\\le \n   C_{\\alpha\\beta K}(1+|\\xi|)^{m-|\\alpha|}$$\nfor any multi-indices $\\alpha,\\beta$ and any compact subset $K$ of $\\Omega\\times\\Omega$, \nand having an asymptotic expansion $a\\sim\\sum\\limits_{j=0}^\\infty \\chi a_{m-j}$ with a\nzero-excision function $\\chi=\\chi(\\xi)$ and homogeneous components $a_{m-j}$, i.e., \n $$a_{m-j}(x,y,t\\xi)=t^{m-j}a_{m-j}(x,y,\\xi)$$\nfor all $(x,\\xi)$ with $\\xi\\not=0$ and all $t>0$. The pseudodifferential operator \n$\\mathrm{op}(a):C^\\infty_{0}(\\Omega,V)\\to C^\\infty(\\Omega,V)$ associated with $a$ is \n $$[\\mathrm{op}(a)\\varphi](x)=\\iint e^{i(x-y)\\xi}a(x,y,\\xi)\\varphi(y)\\,dyd\\hspace*{-0.08em}\\bar{}\\hspace*{0.1em}\\xi,\n   \\qquad \\varphi\\in C^\\infty_0(\\Omega,V).$$ \nAn operator $R:C^\\infty_{0}(\\Omega,V)\\to C^\\infty(\\Omega,V)$ is called smoothing if it has \na smooth integral kernel $k\\in C^\\infty(\\Omega\\times\\Omega,\\mathrm{End}(V))$, i.e., \n $$(R\\varphi)(x)=\\int_\\Omega k(x,y)\\varphi(y)\\,dy,\\qquad \\varphi\\in C^\\infty_0(\\Omega,V).$$\nA pseudodifferential operator of order $m\\in{\\mathbb R}$ on $\\Omega$ is an operator of the form \n$A=\\mathrm{op}(a)+C$, where $a$ is a symbol of order $m$ and $R$ is smoothing. \n\nAny pseudodifferential operator $A=\\mathrm{op}(a)+R$ of order $m$ can be represented in the form \n$\\mathrm{op}(a_L)+R^\\prime$, where $a_L(x,\\xi)$ is a $y$-independent `left-symbol' of order $m$; up to \norder $-\\infty$ the left-symbol is uniquely determined by the asymptotic expansion \n $$a_L(x,\\xi)\\sim \\sum_{|\\alpha|=0}^\\infty\\frac{1}{\\alpha!}\\partial^\\alpha_\\xi D^\\alpha_y \n   a(x,y,\\xi)\\Big|_{x=y}.$$\nThe homogeneous components of $A$ are by definition those of $a_L$,\n $$\\sigma_{m-j}(A)(x,\\xi):=(a_L)_{m-j}(x,\\xi).$$  \nBy the Schwarz kernel theorem, we can identify $A$ with its distributional \nkernel \n $$K_A\\in \\mathscr{D}^\\prime(\\Omega\\times\\Omega,V\\otimes V^*),$$\nthe topological dual of $C^\\infty_0(\\Omega\\times\\Omega;V^*\\otimes V)$. \nIt is uniquely defined by the relation \n $$\\skp{K_A}{\\psi\\otimes\\varphi}=\\skp{\\psi}{A\\varphi},\\qquad \n   \\psi\\in C^\\infty_0(\\Omega,V^*),\\quad\n   \\varphi\\in C^\\infty_0(\\Omega,V).$$\nDenoting by $\\mathrm{tr}:V^*\\otimes V\\to{\\mathbb C}$ the canonical contraction map, we have explicitly \n $$\\skp{K_A}{u}=\\int_\\Omega \\mathrm{tr}[A u(x,\\cdot)](x)\\,dx,\\qquad \n   u\\in C^\\infty_0(\\Omega\\times\\Omega;V^*\\otimes V).$$\nBy pseudo-locality, \n$K_A\\in C^\\infty(\\Omega\\times\\Omega\\setminus\\Delta(\\Omega),V\\otimes V^*)$. \n\nIf $U\\subset X$ is a coordinate neighborhood, we can pull-back the local operators \nunder the coordinate map. The resulting space of operators we shall denote by \n$\\Psi \\mathrm{DO}^m_{cl}(U;\\mathrm{End}(V))$, the subspace of smoothing operators by  \n$\\Psi \\mathrm{DO}^{-\\infty}(U;\\mathrm{End}(V))$. \n\n\\subsection{Projective pseudodifferential operators}\n\nIn the following choose an atlas as explained in Remark \\ref{rem:atlas}. \n\n\\begin{definition}\\label{def:pseudo}\n Let $F$ be a $*$-convolution bundle over $\\mathcal{U}$. A distribution \n $A \\in \\mathcal{D}^\\prime(\\mathcal{U}, F)$ \n is called a projective pseudodifferential operator of order $m\\in{\\mathbb R}$ if \n \\begin{itemize}\n  \\item[(i)] $A$ is smooth outside the diagonal, \n  \\item[(ii)] for any $\\alpha$ the distribution \n   $\\left( \\phi_\\alpha^{-1} \\right)^* A\\big|_{U_\\alpha\\times U_\\alpha}$ is the distributional kernel of a \n   pseudodifferential operator $A_\\alpha\\in\\Psi \\mathrm{DO}^m_{cl}(U_\\alpha;\\mathrm{End}({\\mathbb C}^{k}))$. \n \\end{itemize}\n We denote the vector space of $m$-th order projective pseudodifferential \n operators by $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$, the subspace of smoothing elements by \n $\\Psi \\mathrm{DO}^{-\\infty}(\\mathcal{U};F)$.  \n\\end{definition}\n\nThe subspace $\\mathrm{Diff}^m(\\mathcal{U};F)$ of projective differential operators consists \nof all projective pseudodifferential operators which are supported on the diagonal. \n\n\\begin{remark}\nIf $\\mathcal{U} = X \\times X$ and $F=E\\boxtimes E^*$ for a bundle $E$ over $X$ \nthen $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$ coincides with $\\Psi \\mathrm{DO}^m_{cl}(X;E,E)$, the \npseudodifferential operators of order $m$ acting on sections into $E$. \n\\end{remark}\n\nThough projective pseudodifferential operators, in general, are not operators in the \nusual sense (i.e., acting between sections of vector bundles) all elements of \nthe standard calculus can be generalized to this setting. In particular, \nthe $*$-structure gives rise to a conjugation on $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$, \ndefined by $A^*(x,y):=(A(y,x))^*$ in the distributional sense. \n\nLet $A$ projective pseudodifferential operator with local representatives $A_\\alpha$ and $A_\\beta$, \ncf.\\ Definition \\ref{def:pseudo}, where $\\mathcal{O}_{\\alpha\\beta}$ is not empty. \nBy passing to local coordinates on $U_\\alpha\\cap U_\\beta$, we can associate with $A_\\alpha$ \nand $A_\\beta$ local symbols $a_\\alpha(x,\\xi)$ and $a_\\beta(x,\\xi)$, respectively. These symbols are \nthen related by \n\\begin{align}\\label{eq:asymptotic}\n\\begin{split}\n a_\\beta(x,\\xi)\n &=\\sum_{|\\gamma|=0}^\\infty \n   \\frac{1}{\\gamma!}\\partial^\\gamma_\\xi D^\\gamma_y\\Big|_{y=x}\n   \\phi_{\\alpha\\beta}(x,y)\\big(a_\\alpha(x,\\xi)\\big)\\\\ \n &=\\sum_{|\\gamma|=0}^\\infty \n   \\frac{1}{\\gamma!}\\partial^\\gamma_\\xi D^\\gamma_y\\Big|_{y=x}\n   \\big[\\lambda_{\\alpha\\beta}(x,y)\\varphi_{\\alpha\\beta}(x)a_\\alpha(x,\\xi)\\varphi_{\\alpha\\beta}(y)^{-1}\\big],\n\\end{split}\n\\end{align}\nwhere the transition function $\\phi_{\\alpha\\beta}$ is as described in \\eqref{eq:trans0} and \nProposition \\ref{prop:transition}. Note that this behaviour, in general, differs from the standard case, \ndue the factor $\\lambda_{\\alpha\\beta}(x,y)$. \nHowever, \\eqref{eq:asymptotic} together with $\\lambda_{\\alpha\\beta}(x,x)=1$ shows that with $A$ we can \nassociate a well-defined homogeneous principal symbol \n $$\\sigma_m(A)(x,\\xi)\\in C^\\infty(S^*X,\\pi^*\\mathcal{A}),$$\nwhere $\\pi:S^*X\\to X$ is the canonical co-sphere bundle over $X$. Vice versa, any given such \nsection can be realized as the principal symbol of a projective pseudodifferential operator. \n\nIf the projective pseudodifferential operators $A_1$ and $A_2$ are supported in a \nsufficiently small neighborhood of the diagonal in $\\mathcal{U}$ their usual composition \n$$ \n (A_1 \\circ A_2)(x,z)= \\int_{X} m\\left(A_1(x,y) \\otimes A_2(y,z)\\right)\\, dy\n$$\nis a distribution. By passing to local coordinates and using the composition theorems for \npseudodifferential operators one can see that $A_1 \\circ A_2$ is a projective pseudodifferential \noperator. The homogeneous principal symbol behaves multiplicative under composition. Of course, \nany projective pseudodifferential operator can be written as a sum of two operators, where one \nis smoothing and the other is supported near the diagonal. Summarizing, the coset space \n\\begin{equation}\\label{eq:coset}\n L^*_{cl}(\\mathcal{U},F):=\\Psi \\mathrm{DO}^*_{cl}(\\mathcal{U},F) \/ \\Psi \\mathrm{DO}^{-\\infty}(\\mathcal{U},F)\n\\end{equation}\nis a filtered $*$-algebra. As in the standard case, asymptotic summations of sequences \nof projective operators of one-step decreasing orders are possible and parametrices \n(i.e., inverses modulo smoothing remainders) to elliptic elements can be constructed.  \n\n\\begin{theorem}\\label{wres}\n Let $F$ be a $*$-convolution bundle and let $A$ be a projective pseudodifferential \n operator. For $x\\in X$ define\n $$\n  \\mathrm{WRes}_x(A) := \\int_{S^*_xX} \\mathrm{tr}\\, a_{-n}(x,\\xi) \\,d\\sigma(\\xi),\n $$ \n where $a_{-n}(x,\\xi)$, $n=\\mathrm{dim}\\,X$, is the homogeneous component of \n order $-n$ of a symbol of a local representative $A_\\alpha$ with $x\\in\\mathcal{O}_\\alpha$, \n cf. Definition {\\rm\\ref{def:pseudo}}. Then  $\\mathrm{WRes}_x(A)$ is well-defined and \n defines a global density on $X$. Moreover,\n $$ \n  \\mathrm{WRes}(A) := \\int_X \\mathrm{WRes}_x(A) \\,dx\n $$\n defines a trace functional on the algebra $L^*_{cl}(\\mathcal{U},F)$, the so-called \n \\emph{noncommutative residue} or Wodzicki residue.\n\\end{theorem}\n\\begin{proof}\n Let $A_\\beta$ be another local representative and $x\\in\\mathcal{O}_\\beta$. Fixing local coordinates on \n $\\mathcal{O}_\\alpha\\cap\\mathcal{O}_\\beta$, the local symbols $a_\\alpha$ and $a_\\beta$ are related by \n the asymptotic expansion \\eqref{eq:asymptotic}. Following the proof in \\cite{FGLS} terms containing \n a derivative $\\partial_\\xi^\\gamma$, $|\\gamma|\\ge1$, vanish under integration. We thus obtain the same \n value for $\\mathrm{WRes}_x(A)$ using either $a_\\alpha(x,\\xi)$ or $a_\\beta(x,\\xi)$. \n That $\\mathrm{WRes}_x(A)$ transforms as density under changes of \n coordinates is seen as in the standard case, cf.\\ \\cite{FGLS}. \n \n To see that the integral of the residue density defines a  trace functional we need to show that it \n vanishes on commutators $[A,B]$. To this end fix a cover $\\{U_\\sigma^\\prime\\}$ of $X$ by coordinate maps \n together with a subordinate partition of unity, such that $U_\\sigma^\\prime\\cup U_\\rho^\\prime$ is contained \n in some $U_\\alpha$ whenever $U_\\sigma^\\prime\\cap U_\\rho^\\prime$ is not empty. We then \n can write $A= \\sum_\\alpha A_{\\sigma}$ and $B= \\sum_\\alpha B_{\\sigma}$ modulo smoothing operators, \n where the $A_\\sigma$ and $B_\\sigma$ are supported in \n $\\mathcal{O}_{\\sigma}^\\prime:=U_\\sigma^\\prime \\times U_\\sigma^\\prime$. Then the commutator\n $[A,B]$ can be written as a sum of terms $[A_\\sigma,B_\\rho]$. Such a commutator is smoothing \n if $\\mathcal{O}_{\\sigma}^\\prime\\cap\\mathcal{O}_{\\rho}^\\prime$ is empty. Otherwise it \n is contained in some set $\\mathcal{O}_\\alpha$. Therefore the calculation reduces to a local one, which \n is not different from the one for usual pseudodifferential operators that can be found in \\cite{FGLS}. \n\\end{proof}\n\nFor purposes below let us establish the following result: \n\n\\begin{proposition}\\label{projection}\nLet $F$ be a $*$-convolution bundle and $\\mathcal{A}$ be the Azumaja \nbundle obtained by restricting $F$ to the diagonal. Moreover, let \n$p\\in C^\\infty(S^*X,\\pi^*\\mathcal{A})$ with $p^2=p$. Then there exists a \nprojective pseudodifferential operator $P\\in L^0_{cl}(\\mathcal{U},F)$ which is a projection, \ni.e., $P^2=P$, and which has $p$ as its principal symbol. If, additionally, $p^*=p$ \nthen $P$ can be chosen such that $P^*=P$. \n\\end{proposition}\n\\begin{proof}\nThe proof follows \\cite{Schu01}. Using local coordinates together with a partition of \nunity we can construct a $P_1\\in L^0_{cl}(\\mathcal{U},F)$ having $p$ as principal symbol. \nIf $V:={\\mathbb C}\\setminus\\{0,1\\}$ then $\\lambda-P_1$ is elliptic for any \n$\\lambda\\in V$. Thus there is a parametrix $Q(\\lambda)$, depending holomorphically \non the parameter $\\lambda\\in V$ (i.e., the local symbols depend holomorphically on $\\lambda$). \nThen one takes   \n $$P=\\int_\\mathscr{C} Q(\\lambda)\\,d\\lambda,$$\nwhere $\\mathscr{C}=\\partial U_{1\/2}(1)$ is the counter-clockwise oriented boundary of the disc of \nradius $1\/2$ centred in $1$ $($more precisely, we decompose $Q(\\lambda)$ in a sum of \nlocal terms and integrate each of these terms seperately over $\\mathscr{C})$. \nIf also $p^*=p$ we repeat the above construction, replacing $P_1$ by $P^*P$. \n\\end{proof}\n\n\\section{The noncommutative residue in twisted $K$-theory}\n\n\\subsection{Twisted K-theory}\n\nSuppose that $\\mathcal{A}$ is an Azumaja bundle over a compact manifold $X$.\nThe twisted K-theory is defined to be the K-theory of the $C^*$-algebra\nof continuous sections $C(X; \\mathcal{A})$ of $\\mathcal{A}$.\n\nIf $Y \\subset X$ is a closed subset then the set of sections $C(X,Y;\\mathcal{A})$\nvanishing on $Y$ is a closed two-sided ideal in $C(X,Y;\\mathcal{A})$\nand the quotient by this ideal can be identified with the space of continuous\nsections $C(Y;\\mathcal{A})$ of the Azumaja bundle $\\mathcal{A}|_Y$. We therefore have\nthe six term exact sequence as a consequence of the six term exact sequence in the theory\nof $C^*$-algebras.\n\\begin{gather*}\n  \\xymatrix@1@=1.2cm@M=2pt{\n     K^0(X,Y;\\mathcal{A}) \\ar[r] & K^0(X;\\mathcal{A}) \\ar[r] & K^0(Y;\\mathcal{A}) \\ar[d] \\\\\n    K^1(Y;\\mathcal{A})  \\ar[u] \\ar[r] & K^1(X;\\mathcal{A}) \\ar[r] & K^1(X,Y;\\mathcal{A}) }\n\\end{gather*}\nwhere the relative $K$-groups $K^*(X,Y;\\mathcal{A})$ are defined as\n$K_*(C(X,Y;\\mathcal{A}))$.\n\nThere is a natural map\n\\begin{gather*}\n K_*(C(X,Y;\\mathcal{A})) \\otimes_\\mathbb{Z} K_*(C(X)) \\mapsto K_*(C(X,Y;\\mathcal{A}) \\hat \\otimes C(X)).\n\\end{gather*}\nHere $\\hat \\otimes$ is the tensor product of $C^*$-algebras which is well defined in this case as $C(X)$\nis nuclear.\nThe usual multiplication\n\\begin{gather*}\n C(X,Y;\\mathcal{A}) \\hat \\otimes C(X) \\to C(X,Y;\\mathcal{A})\n\\end{gather*}\ninduces a map $K_*(C(X,\\mathcal{A}) \\hat \\otimes C(X)) \\to K_*(C(X,\\mathcal{A}))$.\nThe composition of these two maps makes $K^*(X,Y;\\mathcal{A})$\na module over the $\\mathbb{Z}_2$-graded ring $K^*(X)$.\nChoosing $Y=\\emptyset$ defines a $K^*(X)$ module structure\non $K^*(X;\\mathcal{A})$.  Note that the morphisms in the six term exact sequence are module\nhomomorphisms.\n\nThese observations can be used to prove the following Leray-Hirsch theorem: \n\n\\begin{theorem}\\label{leray}\n Let $R$ be a commutative torsion-free ring.\n Suppose that $\\pi: M \\xrightarrow{F} X$ is a compact smooth fibre bundle with fibre $F$ \n over $X$ and let $\\mathcal{A}$ be an Azumaja bundle over $X$. Assume that $K^*(F) \\otimes_{{\\mathbb Z}} R$ is a \n free $R$-module and suppose there exist elements\n $c_1,\\ldots,c_N \\in K^*(M) \\otimes_{{\\mathbb Z}} R$ such that the $c_j|_{M_x}$ form a basis \n for $K^*(M_x)\\otimes_{{\\mathbb Z}} R$ for every $x \\in X$. Then the following map is an isomorphism$:$\n \\begin{equation*}\\label{eq:leray}\n   K^*(X;\\mathcal{A})\\otimes_{{\\mathbb Z}}  R^N \\longrightarrow K^*(M,\\pi^*(\\mathcal{A})) \\otimes_{{\\mathbb Z}} R,\\qquad \n   (p,\\alpha) \\mapsto \\sum_{j=1}^N \\alpha_j \\pi^*(p) \\cdot c_j. \n \\end{equation*}\n\\end{theorem}\n\nIndeed, the usual proof  of the Leray-Hirsch theorem in topological $K$-theory (see e.g. \\cite{hatcher}, Theorem 2.25)\ncan be adapted to our setting in the following way.\nIf $Y \\subset X$ is a closed subset of $X$, we have the following diagram:\n\\begin{gather*}\n  \\xymatrix@=0.5cm@1@=0.5cm@=0.5cm@M=2pt{\n     \\ar[r] & K^*(X,Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R^N \\ar[d]^\\Phi  \\ar[r] &  K^*(X;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R^N  \n     \\ar[d]^\\Phi \\ar[r] & K^*(Y;\\mathcal{A})\\otimes_{{\\mathbb Z}} R^N   \\ar[d]^\\Phi  \\ar[r] &  \\\\\n     \\ar[r] &K^*(\\pi^{-1}X,\\pi^{-1}Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R \\ar[r] & K^*(\\pi^{-1}X;\\mathcal{A})\\otimes_{{\\mathbb Z}} R \\ar[r] & K^*(\\pi^{-1}Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R\\ar[r] & }\n\\end{gather*}\nHere  $\\Phi$ is defined as in the theorem,\n$\\Phi(p,\\alpha) = \\sum\\limits_{j=1}^N \\alpha_j \\pi^*(p) \\cdot c_j$. The rows of this diagram are exact\nsince tensoring with $R^N$ and $R$ is an exact functor. All maps in the six term exact sequence are natural\nand therefore the pull back $\\pi^*$ commutes with them. Moreover, the maps in the six term exact sequence for\nthe pair $(\\pi^{-1}X,\\pi^{-1}Y)$ are $K^*(M)$ module homomorphisms. Thus, the diagram commutes.\nSince $X$ is a finite cell complex one can proceed in the usual way using the $5$-lemma and induction\nin the number of cells and the dimension  to prove the theorem.\n\n\\subsection{Residue-traces on filtered rings}\n\nLet $L$ be a ring with filtration, i.e., $L=L^0\\supset L^{-1}\\supset L^{-2}\\supset\\ldots$ \nwith sub-rings $L^{-j}$ and the multiplication induces maps $L^{-i}\\times L^{-j}\\to L^{-i-j}$ \nfor any choice of $i,j$. \n\nA trace functional on $L$ is a map $\\tau:L\\to V$ for some vector space $V$ having the following properties: \n\\begin{itemize}\n \\item[$(1)$] $\\tau$ is linear, $\\tau(A+B)=\\tau(A)+\\tau(B)$ for all $A,B\\in L$, \n \\item[$(2)$] $\\tau$ vanishes on commutators, $\\tau([A,B])=\\tau(AB-BA)=0$ for all $A,B\\in L$.\n\\end{itemize}\nWe call $\\tau$ a residue-trace if, additionally,    \n\\begin{itemize}\n \\item[$(3)$] there exists a $N$ such that $\\tau(A)=0$ for all $A\\in L^{-N}$. \n\\end{itemize}\nWe shall now show that a residue-trace restricted to the set of projections in $L$ is insensible \nfor lower order terms. The proof is elementary and purely algebraic. \n\n\\begin{theorem}\\label{prop1}\nLet $\\tau$ be a residue-trace on $L$ and $P,\\widetilde{P}\\in L$ be two projections, i.e., \n$P^2=P$ and $\\widetilde{P}^2=\\widetilde{P}$. If $P-\\widetilde{P}\\in L^{-1}$ then $\\tau(P)=\\tau(\\widetilde{P})$. \n\\end{theorem}\n\\begin{proof}\nSet $R=\\widetilde{P}-P$ and then define  \n $$A=P R P,\\quad B=P R(1-P),\\quad C=(1-P)R P,\\quad D=(1-P)R(1-P).$$\nObviously then $\\widetilde{P}=P+R$ and $R=A+B+C+D$. Using that $P(1-P)=(1-P)P=0$ we obtain \n\\begin{align*}\n (P+R)(P+R)=&P+2A+B+C+A^2+AB+BC+ \\\\\n &+BD+CA+CB+DC+D^2.\n\\end{align*}\nOn the other hand, using that $\\widetilde{P}$ is a projection, \n $$(P+R)(P+R)=(P+R)=P+A+B+C+D.$$\nEquating these two expressions and rearranging of terms yields \n $$A^2+A+BC+D^2-D+CB+AB+BD+CA+DC= 0.$$\nMultiplying this identity from the left and the right with $P$ and $1-P$, respectively, \nyields \n\\begin{equation*}\n A^2+A+BC = 0,\\qquad D^2-D+CB =0.\n\\end{equation*}\nLet us rewrite these equations as \n\\begin{equation*}\n A(1+A)=-BC,\\qquad (-D)\\big(1+(-D)\\big)=-CB.  \n\\end{equation*}\nMultiplying the first equation by $(1-A)$ yields $A\\equiv -BC$ modulo $L^{-3}$. Multiplying it \nwith $(1-A+A^2)$ then yields $A\\equiv-BC-(BC)^2$ modulo $L^{-4}$. Proceeding by induction we obtain \n $$A\\equiv \\sum_{k=1}^\\ell c_{k\\ell}(BC)^{k}\\mod L^{-(2+\\ell)}$$\nfor any $\\ell\\in{\\mathbb N}$ with suitable constants $c_{k\\ell}$. In the same way, with the same \nconstants $c_{k\\ell}$,  \n $$-D\\equiv \\sum_{k=1}^\\ell c_{k\\ell}(CB)^{k}\\mod L^{-(2+\\ell)}.$$\nTherefore we have \n $$A+D=\\sum_{k=1}^\\ell c_{k\\ell}\\left[B,(CB)^{k-1}C\\right]\\mod L^{-(2+\\ell)}.$$\nChoosing $\\ell$ large enough, we deduce that $\\tau(A)+\\tau(D)=0$. Furthermore, \n $$\\tau(B)=\\tau(P R(1-P))=\\tau((1-P)P R)=0$$ \nand, analogously, $\\mathrm{res}(C)=0$. Altogether we obtain \n $$\\tau(\\widetilde{P})=\\tau(P)+\\tau(A)+\\tau(D)+\\tau(B)+\\tau(C)=\\tau(P)$$\nwhich is the claim we wanted to prove. \n\\end{proof}\n\n\\subsection{The noncommutative residue}\n\nWe shall show that the noncommutative residue induces a map on twisted $K$-theory. \n\n\\begin{proposition}\nLet $\\mathcal{A}$ be the Azumaja bundle obtained by restricting a $*$-con\\-vo\\-lution bundle \n$F$ to the diagonal. The noncommutative residue from Theorem {\\rm\\ref{wres}} descends to a \ngroup homomorphism\n\\begin{equation}\\label{eq:wres}\n \\mathrm{WRes}:K^0(S^*X,\\pi^*\\mathcal{A}) \\to \\mathbb{C},\n\\end{equation}\nwhere $\\pi:S^*X\\to X$ denotes the co-sphere bundle over $X$. \n\\end{proposition}\n\\begin{proof}\nA typical element in $K^0(S^*X,\\pi^*\\mathcal{A})$ can be represented by a section  \n$p\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_N(\\mathcal{A}))$ which is $($pointwise$)$ a projection. \nThis is possible, since the natural inclusion of the $K$-theory of the local $C^*$-algebra \n$C^\\infty(S^*X,\\pi^*\\mathcal{A})$ into that of $C(S^*X,\\pi^*\\mathcal{A})$ is an isomorphism, \ncf. \\cite{Bl98}. By Proposition \\ref{projection} each such section is the principal symbol \nof a projective \npseudodifferential operator $P\\in L^0(X;\\mathrm{Mat}_N(F))$ which is a projection. \nThe noncommutative residue of the $K$-group element is then defined as $\\mathrm{WRes}(P)$ \nin the sense of Theorem \\ref{wres}. \n\nWe have to show that this map is well-defined. So let \n$\\widetilde{p}\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_M(\\mathcal{A}))$ \nrepresent the same element as $p$ does. Let $\\widetilde{P}\\in L^0(X;\\mathrm{Mat}_M(F))$ \nbe associated with $\\widetilde{p}$. \nSince $p$ and $\\widetilde{p}$ are equivalent there exists a unitary \n$u\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_{M+N}(\\mathcal{A}))$ such that \n$u(p\\oplus 0_{M})u^{-1}$ coincides with $0_{N}\\oplus\\widetilde{p}$. \nLet $U\\in L^0(X;\\mathrm{Mat}_{M+N}(F))$ have $u$ as its principal symbol. Then \n $$\\mathrm{WRes}(U(P\\oplus 0_{M})U^{-1})=\\mathrm{WRes}(P\\oplus 0_{M})=\\mathrm{WRes}(P).$$\nOn the other hand $U(P\\oplus 0_{M})U^{-1}$ is a projection having the same principal \nsymbol as $0_{N}\\oplus\\widetilde{P}$. Thus, by Theorem \\ref{prop1},  \n $$\\mathrm{WRes}(U(P\\oplus 0_{M})U^{-1}) =\\mathrm{WRes}(0_{N}\\oplus\\widetilde{P})=\\mathrm{WRes}(\\widetilde{P}).$$\nThis shows that the noncommutative residue is independent of the choice of the representative. \n\\end{proof}\n\n\n\\section{Twisted Dirac operators and connections}\n\nLet $\\mathcal{A}$ be the Azumaja bundle obtained by restricting a $*$-convolution bundle \n$F$ to the diagonal. \n\n\\begin{definition}\\label{def:connection}\nA projective connection $\\nabla=\\nabla^F$ on $F$ is a linear map \n $$Y\\mapsto\\nabla_Y:\\quad C^\\infty(X;TX)\\longrightarrow \\mathrm{Diff}^1(\\mathcal{U};F)$$\nsatisfying, for any vector field $Y\\in C^\\infty(X,TX)$ and any function \n$f\\in C^\\infty(X)$,  \n\\begin{itemize}\n \\item[$(1)$] $\\nabla_{fY}=f\\nabla_Y$, \n \\item[$(2)$] $[ \\nabla_Y , f] = Yf$ for any $f\\in C^\\infty(X)$.\n\\end{itemize}\nIt is called a hermitian connection if additionally \n\\begin{itemize}\n \\item[$(3)$] $\\nabla_Y^* + \\nabla_Y + \\mathrm{div}\\, Y =0$\n\\end{itemize}\n$($here, $f$ and $\\mathrm{div}\\,Y$ are considered as elements of $\\mathrm{Diff}^0(\\mathcal{U};F))$. \n\\end{definition}\n\nNote that in case $\\mathcal{U}=X\\times X$ and $F=E \\boxtimes E^*$ for a vector bundle \n$E$ over $X$ we just recover a usual hermitian connection on $E$. One can always construct \na projective hermitian connection from local hermitian connections \nby gluing with a partition of unity.\n\nIf $\\nabla=\\nabla^F$ is a projective connection and $\\phi_\\alpha$ is a local trivialization\nof $F$ over $U_\\alpha\\times U_\\alpha$ as described in Remark \\ref{rem:atlas}, the corresponding \nlocal differential operator \n $$\\nabla^\\alpha_Y\\in\\mathrm{Diff}^m(U_\\alpha,\\mathrm{End}({\\mathbb C}^k))$$ \nis of the form  \n $$\\nabla^\\alpha_Y=Y+\\Gamma_Y^\\alpha(x),\\qquad \n   \\Gamma_Y^\\alpha\\in C^\\infty(U_\\alpha,\\mathrm{Mat}(k)).$$ \nIf we use another trivialisation $\\phi_\\beta$ of $F$ on $U_\\beta\\times U_\\beta$, \nwe have the relation \n $$\\Gamma_Y^\\beta(x)=\\phi_{\\alpha\\beta}(x,x)(\\Gamma_Y^\\alpha(x))\n   +Y_y\\phi_{\\alpha\\beta}(x,y)(\\mathbf{1})\\big|_{y=x},\\qquad x\\in U_\\alpha\\cap U_\\beta,$$\nwhere $\\mathbf{1}$ is the identity matrix. Thus, analogous to the theory of standard connections, \nwe may describe projective connections by `connection matrices' $\\Gamma_Y^\\alpha$ associated to \na covering $X=\\mathop{\\mbox{\\large$\\cup$}}_\\alpha U_\\alpha$ satisfying the above compatibility \nrelations. For a hermitian connection the connection matrices also have to be skew-symmetric, \n$\\Gamma_Y^\\alpha(x)^*=-\\Gamma_Y^\\alpha(x)$.\n\nSuppose now $S$ is a Clifford module over $X$ and let $\\gamma$ denote the Clifford \nmultiplication. Moreover, let $\\nabla^{S}$ be a connection on $S$ which is compatible \nwith the Clifford structure. Writing $\\widetilde{F}:=S\\boxtimes S^*$ it is easy to see that \n$F\\otimes\\widetilde{F}$ is a $*$-convolution bundle over $\\mathcal{U}$ and we can define \nthe projective hermitian connection \n $$\\nabla:=\\nabla^F\\otimes 1+1\\otimes\\nabla^{S}$$ \nby choosing the corresponding connection matrices as  \n $$\\Gamma_Y^{F,\\alpha}(x)\\otimes 1+1\\otimes \\Gamma_Y^{S,\\alpha}(x),\\qquad x\\in U_\\alpha,$$\nwhere the $U_\\alpha$ are chosen in such a way that both $F$ and $\\widetilde{F}$ are locally trivial \nover $U_\\alpha\\times U_\\alpha$. \nThen we can define the twisted Dirac operator \n $$D:=(1\\otimes\\gamma)\\circ \\nabla\\in \\mathrm{Diff}^1(\\mathcal{U};F\\otimes\\widetilde{F});$$\nin fact, in each local trivialisation $\\nabla$ is a usual hermitean connection and we can compose \nit locally with $1\\otimes\\gamma$. \n\n\\section{Vanishing of the Wodzicki residue}\n\n\\begin{theorem}\\label{thm:vanish}\n If $X$ is an odd dimensional oriented manifold \n the map $\\mathrm{WRes}$ of \\eqref{eq:wres} vanishes identically.\n\\end{theorem}\n\nAs a direct consequence, $\\mathrm{WRes}(P)=0$ for any projection \n$P\\in L^0(\\mathcal{U},F)$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:vanish}]\nSuppose the dimension of $X$ is $n=2 \\ell -1$.\nLet $S=\\mathop{\\mbox{$\\oplus$}}\\limits_{k\\text{ even}}\\Lambda^k(T^*X)$ denote the bundle of \neven-degree forms over $X$ and let $*: \\Lambda^k(T^*X) \\to \\Lambda^{n-k}(T^*X)$ the Hodge star\noperator and denote by $d$ and $\\delta$ the exterior differential and the co-differential respectively.\nDefine the operator $D^S$ acting on sections of $S$ as\n $$D^S=i^{\\ell} * (\\delta+(-1)^{k+1}d)\\quad\\text{on $k$-forms}.$$\nThen by Proposition 1.22 and 2.8 in \\cite{BGV04} this is a generalized Dirac operator, \nwhere the Clifford action on $S$ is given by \n$\\gamma(\\xi)=i^{\\ell} * \\left(\\mathrm{int}(\\xi)+(-1)^{k+1}\\mathrm{ext}(\\xi)\\right)$ for $\\xi\\in T^*X$\nand the compatible connection is the Levi-Civita connection.\nThe principal symbol of $D^S$ restricted to the co-sphere bundle is a self-adjoint involution and the\nprojection $\\sigma_+(D^S)=\\frac{1}{2}(\\sigma(D^S)+1)$ onto its $+1$ eigenspace defines an element\nin $K^0(S^*X)$. It is well known (see for instance \\cite{APS76}) that restriction of this element to each\nco-sphere $S^*_x X$ equals $2^\\ell$ times the Bott element on $S^{n-1}$ which together with the class of the trivial\nline bundle generates $K^0(S^{n-1})$.\n\nFor notational convenience denote by $K_{{\\mathbb R}}^*(X)$ the groups $K^*(X) \\otimes_{{\\mathbb Z}} {\\mathbb R}$.\nBy Theorem \\ref{leray} applied to the co-sphere bundle of $X$, any element of \n$K^0_{{\\mathbb R}}(S^*X,\\pi^*\\mathcal{A})$ can be represented in the form \n $$\\alpha_0\\pi^*(p)\\cdot [\\mathbf{1}]+\\alpha_1\\pi^*(p)\\cdot [\\sigma_+(D^S)]$$\nfor some $\\alpha_0,\\alpha_1 \\in {\\mathbb R}$ and some $p\\in K^*(X,\\mathcal{A})$. Here both\nthe class $[\\mathbf{1}]$ of the trivial line bundle and the class $[\\sigma_+(D^S)]$ are understood\nas elements in $K^0_{\\mathbb R}(S^*X)$.\nThe elements in $\\alpha_0\\pi^*(p)\\cdot [\\mathbf{1}]$ can be represented by projections in \n$C^\\infty(X;\\mathrm{Mat}_N(\\mathcal{A}))$.\nTherefore, the noncommutative residue of these elements vanishes. \nIt remains to show that this is also true for the second summand. \n\nTo this end let $p$ be a projection in $\\mathrm{Mat}_N(C^\\infty(X;\\mathcal{A}))$. \nLet us define the new convolution bundle $F_p$ \nhaving fibre $p(x)\\mathrm{Mat}_N(F)_{(x,y)}p(y)\\subset \\mathrm{Mat}_N(F)_{(x,y)}$ in $(x,y)$. \nWe now apply the above construction and build a twisted Dirac operator $D_p$ with respect to \n$F_p\\otimes\\widetilde{F}$, $\\widetilde{F}=S\\boxtimes S^*$. Then $\\sigma_+(D_p)$ represents the class \n$\\pi^*([p])\\cdot[\\sigma_+( D^S)]$ in $K^0(S^*X,\\pi^*\\mathcal{A})$. \n\nThe projective differential operator $D_p$ can now be used to construct a certain \nprojection $Q \\in L^0_{cl}(X,F_p)$  which has principal symbol \nas $\\sigma_+(D_p)$ on $S^* X$.\nIn the case of a Dirac type operator $D$ acting on a vector bundle the projection would just be the\noperator $\\frac{1}{2}(|D|^{-1} D+1)$. The symbol of this projection can be constructed from the a parametrix\nof $D$ and this construction is local modulo smoothing operators. That is the full symbol of $\\frac{1}{2}(|D|^{-1} D+1)$\nmodulo smoothing terms in local coordinates depends only on the full symbol of $D$ in these local coordinates.\nThus, the construction can be repeated for the operator $D_p$ to yield an element in $L^0_{cl}(X,F_p)$\nwhich we denote by $Q$ or formally $\\frac{1}{2}(|D_p|^{-1} D_p+1)$. By construction $[\\sigma(Q)] \\in K^0_{\\mathbb R}(S^*X;\\mathcal{A})$ \nis equal to  $\\pi^*([p]) \\cdot [\\sigma_+(D^S)]$.\n\nIn  \\cite{BG92} (Theorem 3.4) Branson and Gilkey have used invariant theory to show that the residue \\emph{density} \nof the positive spectral projection for any generalized Dirac operator vanishes identically. \nLocally, $D_p$ is a generalized Dirac operator\nand since the construction of the residue density is local the residue density of  $Q$\nvanishes as well.\nSo we can conclude that the noncommutative residue of  $Q$ vanishes which\ncompletes our proof. \n\\end{proof}\n\n\\section{Discussion of even dimensional manifolds}\n\nThe vanishing of the Wodzicki residue on projective pseudodifferential projections is very likely\nto hold also in even dimensions.\nTo compute the residue as a map from $K^0(S^* X;\\pi^* \\mathcal{A})$ it is enough to calculate it\non the generators of $$K^0_{{\\mathbb R}}(S^* X;\\pi^* \\mathcal{A}) \/ \\pi^* K^0_{\\mathbb R}(X;\\mathcal{A})$$ which by\nthe Leray-Hirsch theorem is isomorphic to $K^1_{\\mathbb R}(X;\\mathcal{A})$. Of course the result then also holds\nif $K^1_{\\mathbb R}(X;\\mathcal{A})=\\{0\\}$.\n\n\\noindent{\\bf Acknowledgements.} The authors would like to thank\nThomas Schick for comments and for pointing out a gap in an earlier version\nof this paper. \n\n\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\\label{sec:conclusion}\n\\noindent\nWe provide a systematic study of error correction techniques that address leakage faults where qubit states leave the computational space. We developed a simple model that captures realistic elements of leakage phenomena. Using this model we studied the performance of various circuits that suppress leakage in the toric code, ranging from a simple circuit that only adds one $CNOT$ gate per syndrome extraction, to a scheme that reduces leakage after the application of every gate. To accurately compare the performance of these circuits, we considered two scenarios, one in which measurements distinguish leaked qubits from qubits in the computational state, and another in which they do not. For each of these cases we designed an optimized decoding algorithm and recorded its success rate in Monte Carlo simulations. Our results show that the Partial-LRU and Quick circuits are effective at reducing leakage and reduce the accuracy threshold of the toric code by less than a factor of $4$ in a realistic scenario where leakage was added to depolarizing noise. We see a factor of $2$ threshold improvement over Standard decoders for the HL Decoder that uses information about leakage detection events. Finally, there is evidence that the Full-LRU, which reduces leakage after every gate, may achieve the lowest logical error rates of all of the circuits when the physical error rate is less than $2\\times 10^{-4}$. Future work could explore the logical error rates of these circuits at higher distances and at error rates that are far below threshold, perhaps by applying the splitting method \\cite{bravyi13}.\n\n\\nonumsection{Acknowledgements}\n\\noindent\nWe thank Sergey Bravyi, Ken Brown, Oliver Dial, David DiVincenzo, Easwar Magesen, Graeme Smith, and John Smolin for helpful discussions, Chris Lirakis and Mark Ritter for their support, and Karen Bard for making Blue Gene available to us. AWC and JMG acknowledge support from ARO under contract {\\em W911NF-14-1-0124}.\n\n\\section{Heralded Leakage Reduction}\\label{sec:heralded}\n\\noindent\n\nUp to this point in the discussion, we have assumed that measurements cannot distinguish a leaked qubit from a qubit in the state $|1\\rangle$, and therefore there was no mechanism to detect leakage faults. We now assume that measurements produce a third outcome ``L'' given a leaked qubit and optimistically assume that they output ``L'' if and only if the input qubit is leaked. Taking the same circuits considered in Sec.~\\ref{sec:reduction} (specifically the Partial-LRU and Quick circuits), we modify the decoders to use this additional information. This approach is relatively simple, requiring no change to the quantum circuits, and leads to significantly improved logical error rates and threshold (see Sec.~\\ref{sec:results}). A 3-outcome measurement is not equivalent to a simple erasure model because the precise space-time location of the leakage fault is not available, and leakage faults induce regular errors that depend on the leakage fault location.\n\nThe \\emph{Heralded Leakage (HL) decoder} ingests results from 3-outcome measurements. Using these 3-outcome measurements provides an advantage because in our model leakage is likely to produce regular errors on specific locations around the leaked qubit, and the decoder is then able to match defects using correct prior probabilities in the space-time neighborhood of the ``L'' event in the decoding graph. Such events do not indicate exactly which gate suffered a leakage fault, since ``L'' events can only be detected by measuring a qubit, which only happens in an LRU or when measuring a syndrome. Therefore, upon registering an ``L'' event, the decoder only ``knows'' that the measured qubit leaked at some location between its initialization and measurement.\n\nConditioned on the observed ``L'' events, the HL decoder generates a new decoding graph from the Standard decoding graph ${\\cal G}$ whose edge weights are given by Table~\\ref{table:weights}. The ``L'' events are independent in our model, so suppose that a set of these events ${\\cal L}$ occurs. Because our leakage model independently depolarizes qubits when they interact with a leaked qubit, we can construct a conditional decoding graph ${\\cal G}_{\\cal L}$, for bit-flips say, whose edges $\\varepsilon$ again correspond to independent faults and whose edge weights are computed from conditional probabilities $p(\\varepsilon|{\\cal L})$. These conditional probabilities are computed in the same way as explained in Section~\\ref{sec:reduction}. ${\\cal G}_{\\cal L}$ is generally not translation invariant and contains additional edges not present in Fig.~\\ref{fig:cube}. There will be edges in ${\\cal G}_{\\cal L}$ with very low weight associated to highly probable errors on all of the qubits that interacted with leaked qubits.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=quickCircuit.pdf, width=.88\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:quickCircuit} (Color online) A measurement detects a leakage event in the Quick circuit at time $t+1$. This circuit depicts all of the gates in error-correction cycles $t$, $t+1$, and $t+2$. Each qubit that interacts with the potentially leaked qubit may be depolarized; subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $p_1$ in Fig.~6 (b)) where an ancilla leaks and introduces an $X$ error on qubit $U$ that propagates to other qubits. The notation ``$X\\mapsto I$'' indicates a cancellation of errors.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=quickKey,width=.23\\textwidth}\n   \\label{subfig:quickKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=quickXCube,width=.36\\textwidth}\n   \\label{subfig:quickXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=quickZCube,width=.27\\textwidth}\n   \\label{subfig:quickZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:quick} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Quick circuit. (b,c) Upon leakage detection at time $t$ on the plaquette ancilla *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nIf an ``L'' event occurs when measuring a syndrome, the corresponding syndrome bit is absent from the syndrome history and the usual approach for computing defect locations is somewhat modified. Suppose that ``L'' events occur for some check operator at $m$ contiguous times $t+1$, $\\dots$, $t+m$. In this case, each of the corresponding ``$a$'' edges $e_{t+1}$, $\\dots$, $e_{t+m}$ in the decoding graph are assigned weight $0$. Let $s_t$ be value of the check operator's syndrome bit at time $t$. If $s_t$ differs from $s_{t+m+1}$, we place a defect on the earliest vertex in $\\partial\\{e_{t+1}\\}$.\n\nTo compute the edge weights in the conditional decoding graph, we proceed as follows, examining each ``L'' event independently. When an ``L'' event occurs on some qubit $q$, we consider each of the $n$ fault locations between and including $q$'s initialization and measurement. Since each gate suffers leakage faults independently and with equal probability, the probability that $q$ is leaked when interacting at the $i$-th location is approximately $i\/n$. An $X$ and $Z$ error each occurs on the qubit that interacts with $q$ at location $i$ with probability $p_i = i\/2n$. We consider each of these errors separately, find the defect pair they cause, and modify the weight of the edge connecting this defect pair. Since $p_i$ can be large, we take any probability $p_0$ previously associated to the edge and set the new probability equal to $p_0+p_1-2p_0p_i$.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=circuitAncillaCircuit.pdf, width=.67\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitAncillaCircuit} (Color online) A plaquette ancilla measurement detects a leakage event at time $t$ in the \\emph{Partial-LRU} circuit. This circuit depicts all of the gates on this set of qubits in the $t$th and $t+1$th error-correction cycle. Each data qubit that interacts with the ancilla may be depolarized, and subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $p_3$ in Fig.~8 (c)) where an ancilla leaks and introduces a $Z$ error on qubit $R$ that propagates to other qubits.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=circuitAncillaKey,width=.23\\textwidth}\n   \\label{subfig:circuitAncillaKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitAncillaXCube,width=.36\\textwidth}\n   \\label{subfig:circuitAncillaXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitAncillaZCube,width=.27\\textwidth}\n   \\label{subfig:circuitAncillaZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitAncilla} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Partial-LRU scheme. (bc) Upon leakage detection at time $t$ on the plaquette ancilla *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nThe added edges and their weights are obviously specific to each leakage reduction circuit, so we now describe how to construct conditional decoding graphs in greater detail for the Quick and Partial-LRU circuits.\n\nFig.~\\ref{fig:quickCircuit} shows three rounds of plaquette measurements using the Quick circuit. Leakage is detected at time $t+1$ on the ancilla qubit labeled with $*$. Fig.~\\ref{fig:quick} shows the relative position of the qubits in the circuit, and the low weight edges that are added to or modified in the $X$- and $Z$-error conditional decoding graphs. The edges are shown relative to the position of the leaked qubit.  Consider the following example in which we show that the long edge labelled $p_1$ in Fig.~\\ref{subfig:quickXCube} is associated to an event that occurs with probability $\\approx 1\/22$ conditioned on the ``L'' event shown in Fig.~\\ref{fig:quickCircuit}. This edge is not present in the standard decoding graph and is added to the conditional decoding graph for $X$ errors. The error labels in Fig.~\\ref{fig:quickCircuit} show the events responsible for addition of the edge. A total of $11$ gates act on the leaked qubit between its initialization and measurement, and the qubit is leaked directly after initialization with probability $1\/11$. Therefore, an $X$ error spreads to qubit $U$ at time $t$ with probability $1\/2 \\times 1\/11$. This $X$ error spreads to qubit $\\alpha$ at time $t$ and continues to be detected by the $\\alpha$ plaquette from time $t$ onward. An $X$ error also spreads to the ancilla that replaces the ancilla $*$ at time $t+1$, which in turn spreads to qubit $\\delta$ starting at time $t+2$, as indicated. Therefore, this error introduces the long edge labeled $p_1$ connecting the $\\alpha$ and $\\delta$ qubits in the $X$ error decoding graph. Other edges in the figure are obtained by considering other errors.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=circuitHDataCircuit.pdf, width=.61\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitHDataCircuit} (Color online) A data qubit's trailing LRU detects a leakage event at time $t$. This circuit depicts all of the gates that act on the data qubit in the $t$th error-correction cycle, together with the preceding LRU. Each ancillary qubit that interacts with the data may be depolarized, as well as the outgoing qubit of the trailing LRU, and subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $P_3$ in Fig.~10 (c)) where the data leaks and introduces a $Z$ error on an ancilla qubit.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=circuitHDataKey,width=.23\\textwidth}\n   \\label{subfig:circuitHDataKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitHDataXCube,width=.36\\textwidth}\n   \\label{subfig:circuitHDataXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitHDataZCube,width=.30\\textwidth}\n   \\label{subfig:circuitHDataZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitHData} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Partial-LRU scheme. (bc) Upon leakage detection at time $t$ on the data qubit *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nUnlike the Quick circuit, the Partial-LRU circuit can detect leakage on ancillas, which are directly measured, and data qubits, which are measured by LRUs. First consider the ancillas. Two rounds of plaquette measurements are shown in Fig.~\\ref{fig:circuitAncillaCircuit} with an ``L'' event at the ancilla $*$ at time $t$. The modified edges of the decoding graph, conditioned on the ``L'' event, are shown in Fig.~\\ref{fig:circuitAncilla}. The error labels in Fig.~\\ref{fig:circuitAncillaCircuit} show the events responsible for addition of the edge labeled $p_3$ in Fig.~\\ref{subfig:circuitAncillaZCube}. The probability that the ancilla is leaked after the second $CNOT$ gate is $3\/5$ and a $Z$ error spreads to qubit $R$ with probability $p_3=3\/10$. The $Z$ error spreads to the qubits labeled $2$ and $4$ at time $t$ and $t+1$, and hence we modify the weight of the diagonal edge labeled $p_3$ connecting these two qubits in the decoding graph. Next, consider Fig.~\\ref{fig:circuitHDataCircuit}, which shows LRUs acting on a data qubit. If the second LRU detects leakage, the edges shown in Fig.~\\ref{fig:circuitHData} are added or modified to construct the conditional decoding graphs. The error labels in Fig.~\\ref{fig:circuitHDataCircuit} correspond to the error responsible for addition of edge labeled $p_3$ in the $Z$ error decoding graph.\n\n\n\\section{Introduction}\n\nLarge-scale quantum computers require a fault-tolerant architecture based on quantum error-correcting codes. While there are many approaches to quantum fault-tolerance, topological codes~\\cite{kitaev03,bravyi98} stand out due to their many favorable properties, including local check operators, simple syndrome extraction circuits~\\cite{dennis02}, and flexible fault-tolerant logic based on transversal gates, code deformation~\\cite{rh07,raussendorf07}, or lattice surgery~\\cite{horsman12}. These properties endow topological codes with a high accuracy threshold. The value of the surface code's accuracy threshold depends on assumptions regarding the noise model and error-correction approach, but estimates vary from $0.67\\%$ \\cite{stephens14} to above $1\\%$ \\cite{wang11}.\n\nThese threshold estimates assume a depolarizing noise model that approximates realistic noise, but the model does not include so-called leakage faults. Leakage faults map quantum states out of the 2-dimensional qubit subspace and into a higher-dimensional Hilbert space. Many physical realizations of qubits are multilevel systems that we expect to suffer from leakage faults. Leakage errors can be mitigated by constructing quantum circuits with suitable properties. Broadly speaking, such gadgets convert leakage errors into ``regular'' errors and may or may not simultaneously raise a flag to indicate the location of the leakage error. Gadgets that detect leakage convert it into a located loss error that is easier to correct~\\cite{gottesman97,preskill98,knill98}. Leakage reduction units (LRUs) convert leakage errors into regular errors but do not give an indication that leakage has occurred~\\cite{aliferis07, fong11}. Leakage mitigation with LRUs was rigorously analyzed by Aliferis and Terhal~\\cite{aliferis07}, who showed existence of a threshold to leakage in concatenated codes. Subsequent work has shown that circuits can be further simplified while remaining fault-tolerant \\cite{fortescue14}.\n\nMost studies of quantum codes do not consider errors due to leakage, and the cost of leakage mitigation has not been thoroughly understood, particularly in the surface code. When leakage errors are converted to detected loss, topological codes are known to have a 50\\% loss threshold in an idealized model~\\cite{stace09,fujii12}. However, the strategy of~\\cite{stace09} measures high weight ``super'' check operators. While this may be natural for measurement-based quantum computing \\cite{barrett10}, it is more difficult with a fixed planar array of qubits, as is the case with some superconducting architectures \\cite{chow14}. Furthermore, if leakage errors instead remain undetected, or are detected but could have originated from multiple space-time points in the circuit, they may lead to additional error spread. Leakage of this form has been studied for a quantum repetition code~\\cite{fowler13} and for specific types of gates applied to superconducting qubits~\\cite{ghosh13, ghosh14}. Despite this work, the impact of leakage on the accuracy threshold of topological codes remains unclear. To address this problem, we systematically study several leakage reducing circuits and develop a new syndrome processing strategy that further enhances the benefits of these circuits.\n\nThe paper is organized as follows. Section~\\ref{sec:toriccode} reviews the toric code and fault-tolerant error-correction using bare stabilizer measurements. Section~\\ref{sec:leakagemodel} introduces a new model of leakage faults that is amenable to simulation within the stabilizer formalism, much like the standard depolarizing noise model. Section~\\ref{sec:reduction} reviews circuit-based leakage reduction techniques, introduces the leakage mitigation circuits we study and why they work, and explains how matching-based syndrome processing algorithms are modified for these circuits. Section~\\ref{sec:heralded} then proposes a model where we are given 3-outcome projective measurements for detecting leaked qubits and shows how to further modify syndrome processing algorithms to take advantage of this extra information. Section~\\ref{sec:simulation} explains our simulation approach and relevant details of the implementation. Finally, Section~\\ref{sec:results} presents our numerical results and Section~\\ref{sec:conclusion} concludes.\n\n\n\\section{Leakage Errors}\\label{sec:leakagemodel}\n\nLeakage errors are possible because the two-dimensional Hilbert space of a qubit is a subspace of a larger physical Hilbert space and interactions with the system may populate states in that larger physical space. The two-dimensional Hilbert space ${\\cal H}_{C[j]}$ of the $j$th qubit is encoded into a Hilbert space ${\\cal H}_{S[j]}={\\cal H}_{C[j]}\\oplus {\\cal H}_{L[j]}$ where ${\\cal H}_{L[j]}$ is an auxiliary leakage space associated to the qubit. We say that the $j$th qubit is {\\em leaked} (resp. {\\em contained}) if its state is supported entirely on ${\\cal H}_{L[j]}$ (resp. ${\\cal H}_{C[j]}$). The total Hilbert space of the system is ${\\cal H}=\\bigotimes_j{\\cal H}_{S[j]}={\\cal H}_S\\oplus {\\cal H}_L$ where ${\\cal H}_S=\\bigotimes_j{\\cal H}_{C[j]}$ is the computational subspace and ${\\cal H}_L$ is the remaining leakage subspace. States supported on ${\\cal H}_L$ have at least one leaked qubit.\n\nConsider a quantum computation implemented by one- and two-subsystem unitary gates that act on the total Hilbert space. For convenience we refer to these as one- and two-qubit gates. A {\\em sealed single qubit gate} maps states in ${\\cal H}_{C[j]}$ back to ${\\cal H}_{C[j]}$ and therefore has the form $U_j = U_{{\\cal H}_{C[j]}}\\oplus U_{{\\cal H}_{L[j]}}$, i.e. it does not leak. Similarly, a {\\em sealed two-qubit gate} acting on states in ${\\cal H}_{S[i]}\\otimes {\\cal H}_{S[j]}$ is a direct sum of unitary gates\n\\begin{equation}\nU = U_{{\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}} \\oplus\nU_{{\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}} \\oplus\nU_{{\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}} \\oplus\nU_{{\\cal H}_{L[i]}\\otimes {\\cal H}_{L[j]}}.\\label{eq:idealgate}\n\\end{equation}\nIt is clear that the subspaces corresponding to each block are invariant under these gates. In this sense, sealed gates do not propagate leakage errors. In many physical settings, where gates are applied by a resonant or off-resonant drive or by tuning subsystems into resonance \\cite{chow11,strauch13}, all blocks except ${\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}$ are approximately diagonal in the eigenbasis of the undriven system Hamiltonian and hence the physical gates are approximately sealed.\n\nLeakage errors are generally coherent operations that are challenging to simulate for large quantum codes. Physical leakage noise may not be accurately captured by a Markovian noise model and generally needs to be treated as a coherent process \\cite{aliferis07}. For simplicity and tractability, we define and simulate a stochastic leakage model, analogous to the standard approach of using depolarizing faults as a proxy for realistic decoherence and control faults. Although such a model does not accurately capture all conceivable leakage phenomena, it captures some aspects and yet remains amenable to stabilizer simulation.\n\nOur stochastic model assumes ideal gates are sealed and that each subsystem is a 3-level system (a qutrit). We add three new elements to define faulty gates. The first two elements are rather straightforward. First, discrete leakage events occur independently with probability $p_\\uparrow$ on each output qutrit of an ideal gate. These events are described by the map\n\\begin{equation}\n{\\cal E}_\\uparrow(\\rho_j)=(1-p_\\uparrow)\\rho_j + p_\\uparrow |2\\rangle\\langle 2|\n\\end{equation}\nwhere $\\rho_j$ is the state of the $j$th qutrit. Second, there is a relaxation process with probability $p_\\downarrow$, analogous to amplitude damping, that acts independently on each output qutrit of an ideal gate. The stochastic map for this process is given by\n\\begin{equation}\n{\\cal E}_\\downarrow(\\rho_j)=(1-p_\\downarrow)\\rho_j+p_\\downarrow {\\cal E}_{\\mathrm{decay}}(\\rho_j)\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal E}_{\\mathrm{decay}}(\\rho_j)=A\\rho_j A^\\dagger + \\sum_{k=0,1} A_{k}\\rho A_{k}^\\dagger\n\\end{equation}\nis a decay map with elements\n\\begin{equation}\nA = |0\\rangle\\langle 0| + |1\\rangle\\langle 1|,\\ A_{k} = \\frac{1}{\\sqrt{2}}|k\\rangle\\langle 2|.\n\\end{equation}\nThese elements are chosen so that when a leaked qubit decays, it is replaced by a maximally mixed contained qubit. Let $Z^{(2)}=|0\\rangle\\langle 0|+|1\\rangle\\langle 1|-|2\\rangle\\langle 2|$ and suppose that an initial state $\\rho_0$ is a fixed point of\n\\begin{equation}\n\\Lambda(\\rho) = \\frac{1}{2}\\left( \\rho + Z^{(2)}\\rho Z^{(2)} \\right).\n\\end{equation}\nThen it is clear that any sequence of operations ${\\cal E}_\\uparrow$, ${\\cal E}_\\downarrow$, and sealed single qubit gates applied to $\\rho_0$ produces a state that is also a fixed point of $\\Lambda$, i.e. states are mixtures where qubits are leaked or contained. For convenience we call such mixtures on any number of qubits $|2\\rangle$-{\\em dephased}.\n\nThe final element of our model for faulty gates concerns two-qubit gates. Sealed two-qubit gates apply the intended operation to the ${\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}$ block and act arbitrarily on the remaining blocks. The ${\\cal H}_{L[i]}\\otimes {\\cal H}_{L[j]}$ block consists of a single matrix element that applies what amounts to a harmless global phase in our model. The ${\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}$ and ${\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}$ blocks of a sealed gate merely apply a single qubit gate to qubit $i$ or $j$, respectively. As a final simplifying assumption, these blocks each implement Haar distributed random unitary gates, so that the resulting process completely depolarizes the contained qubit. Noisy two-qubit gates are followed by independent excitation and relaxation maps on each output qutrit. This is motivated by the assumption that matrix elements coupling computational states to $|22\\rangle$ are significantly smaller than other matrix elements of the drive Hamiltonian. Like before, $|2\\rangle$-dephased states remain so under the action of these two-qubit gates.\n\nIt is important to observe that this two-qubit gate model destroys correlations that could otherwise arise from interacting with leaked qubits. When multiple contained qubits interact with a leaked qubit, those contained qubits will experience independent depolarizing noise rather than some collective process. This would not be the case if, for example, the ${\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}$ and ${\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}$ blocks are fixed unknown single qubit unitaries. In this sense, our model is not a worst-case stochastic model. However, for topological codes, each leaked qubit interacts with a small set of neighbors and persists for at most a few syndrome extraction cycles before being reset by leakage reduction circuits. This would limit the impact of even a worst-case stochastic model but quantifying that impact we leave to future work.\n\nStabilizer circuits constructed from one- and two- qubit gates under the influence of our noise model can be simulated within the stabilizer formalism. The main observation is that $|2\\rangle$-dephased states remain $|2\\rangle$-dephased in our model, so we can introduce a classical {\\em leakage indicator bit} for each qubit whose value indicates if the qubit is contained or leaked. We assume all qubits are initially contained. Quantum operations now act on an $n$-qubit stabilizer state and an $n$-bit leakage indicator array in the following way. The map ${\\cal E}_\\uparrow$ acts as identity with probability $1-p_\\uparrow$ and otherwise traces out the corresponding qubit and sets the leakage indicator bit. Likewise ${\\cal E}_{\\mathrm{decay}}$ resets the leakage indicator bit and replaces the corresponding qubit, setting its state to be completely mixed. How sealed gates act on the $n$-qubit stabilizer state depends on the value of the leakage indicator array. Roughly speaking, the future quantum circuit is rewritten depending on the state of the leakage indicators. For our model, sealed one-qubit gates act as intended on contained qubits and as identity on leaked qubits. Likewise, sealed two-qubit gates also act as intended when both inputs are contained. However, a sealed two-qubit gate depolarizes one of the input qubits when the other input is leaked. When both inputs are leaked, a sealed two-qubit gate acts as identity. Taken together, and given a suitable definition of measurement for leaked qubits, one can sample from the output distribution of faulty stabilizer circuits built from these elements.\n\nOften a full-blown stabilizer simulation is not needed for the study of fault-tolerant quantum error-correction and it suffices to track the Pauli error for each qubit. This works because we assume that the code qubits are always in an eigenstate of the stabilizer, and we can use the error operator to compute the eigenvalues of the stabilizers and normalizers and to predict deviations from ideal measurement outcomes without maintaining a description of the current quantum state as a set of stabilizer generators. For our leakage model, can we similarly divest ourselves of the full stabilizer framework and simply propagate for each qubit a Pauli error operator and a leakage indicator bit? The main concern is that the stabilizer circuit we simulate may not always measure operators in the current stabilizer when one or more leakage indicators are set. For example, leaked code qubits cause subsequent syndrome extraction circuits to measure check operators that are supported only on contained qubits. Likewise, leaked syndrome qubits ``turn off'' the associated check operator measurement. Without knowing the full stabilizer, we cannot predict how the measurement outcomes are correlated through parity constraints enforced by the stabilizer. The key observation that enables us merely to work with Pauli error and leakage indicator labels is that our model completely depolarizes syndrome qubits when code qubits are leaked, destroying any correlations that might otherwise have been present. Furthermore, when leaked qubits relax or pass through circuits to reset them, they are replaced by completely depolarized qubits. This guarantees that past syndrome history is uncorrelated with the particular state of the reset qubit (although it is correlated with the fact that the qubit has leaked). Reassured by this rough argument, we do not retain a complete set of stabilizer generators for the quantum state in the simulation. Instead, we simulate the model by maintaining and propagating a label ($I$, $X$, $Y$, $Z$ or $L$) for each qubit, where ``L'' denotes that the leakage indicator bit is set.\n\nNow that we have explained our model and justified our simulation approach, here follows a summary of the details of the model. The model has parameters $p$ (depolarizing error probability), $q$ (syndrome measurement error probability, we set $q=p$), $p_\\uparrow$ (probability of excitation outside of the computational state), and $p_\\downarrow$ (probability of relaxation back to the computational state). For convenience, we also define the relative excitation rate $r = p_\\uparrow \/ p$ and relaxation rate $s = p_\\downarrow \/ p$. Elementary quantum gates behave as follows. Idle qubits are subject to depolarizing noise with probability $p$, where we apply one of $X$, $Y$, or $Z$ uniformly at random. They undergo free evolution and do not leak. However, a leaked qubit may relax to the computational space with probability $p_\\downarrow$. The $|0\\rangle$ or $|+\\rangle$ state preparation succeeds with probability $1-p$, otherwise the orthogonal state is prepared. With probability $p_\\uparrow$ the freshly prepared qubit leaks. If the input qubit is in the computational space, measurements report the incorrect outcome with probability $p$ and otherwise report the correct outcome. When the input qubit is leaked, we consider two scenarios: either the measurement cannot distinguish higher levels and reports ``1'', or the measurement has a third outcome ``L''. Noisy CNOT gates are subject to joint depolarizing noise with probability $p$, where we apply one of the non-identity two-qubit Pauli errors uniformly at random. In addition, if any input to the CNOT is leaked, all non-leaked qubits are completely depolarized, i.e. one of the $4$ single-qubit Pauli operators is applied uniformly to each non-leaked qubit. Finally, each output qubit leaks with probability $p_\\uparrow$ and relaxes with probability $p_\\downarrow$.\n\nUnlike a similar model in~\\cite{fowler13}, we assume that qubits reach a steady state leakage error rate at the beginning of any simulation, which more faithfully models the conditions of a long computation with the code. Consider the error-correction circuits in the standard toric code. Unlike ancillas, the data qubits are never re-initialized, and therefore they gradually accumulate leakage according to the transition probabilities $p_\\uparrow$ and $p_\\downarrow$ until an equilibrium is reached. It is easy to show from direct calculation of eigenstates of the transition matrix that the equilibrium distribution for data qubits in the toric code is given by $p_{eq}\\approx\\frac{4p_\\uparrow}{4p_\\uparrow+6p_\\downarrow}$ where the factor $4$ is due to the four $CNOT$ gates, and the factor $6$ is due to the four $CNOT$s and two idle time steps. The second eigenvalue of the transition matrix is $(1-p_\\downarrow)^6(1-p_\\uparrow)^4$, so the non-equilibrium component of the distribution for each qubit decays as $\\mathrm{exp}(-\\zeta n)$ where $\\zeta=-6\\ln\\left[(1-p_\\downarrow)(1-p_\\uparrow)\\right]$. This suggests that starting from an equilibrium distribution may not be an overly pessimistic assumption. Under the reasonable assumptions $p_\\uparrow \\approx p_\\downarrow$, the fraction of leaked data qubits reaches $40\\%$ at which point the threshold is far exceeded and computation is clearly impossible (without leakage reduction, of course).\n\n\n\\section{Leakage Reduction}\\label{sec:reduction}\n\\noindent\n\nLeakage reduction is a process that converts leakage errors into regular errors that can be later corrected by quantum codes. Prompt leakage reduction is desirable as a single leaked qubit can cause many other errors (e.g., a leaked data qubit can damage ancillas which then spread more errors to other data qubits). A circuit that converts leakage on a single qubit into a regular error is a leakage reduction unit (LRU). LRUs satisfy two properties \\cite{aliferis07}: (a) if their input is in the computational space then the identity operation is performed, and (b) if their input is in the leakage space then their output is some state in the computational space. We use LRUs as building blocks to construct several possible leakage suppressing error-correction circuits in the toric code.\n\n\\begin{figure}[b]\n\\vspace*{13pt}\n\\centering\n\\epsfig{file=LRU,width=.19\\textwidth}\n\\fcaption{\\label{fig:lru} A leakage reduction unit (LRU) implemented by one-bit teleportation.}\n\\end{figure}\n\nAn LRU based on one-bit teleportation \\cite{zhou00} is shown in Fig.~\\ref{fig:lru}. This LRU differs from canonical LRUs based on full quantum teleportation \\cite{mochon04,aliferis07}. It does not satisfy the properties of an LRU if ideal two-qubit gates can output a pair of leaked qubits when one of the inputs is leaked. In our model, ideal two-qubit gates are sealed and therefore the simpler one-bit teleportation circuit is an LRU and uses fewer resources than full quantum teleportation. Note that reinitializing qubits with most of their population in the computational space is essential to implementing any LRU in practice.\n\nWe use this LRU to build several error-correction circuits that represent a tradeoff between the circuit complexity and effectiveness of leakage reduction. We list the circuits from the most complicated one with the best leakage suppression capability to the simplest one with the fewest extra ancillas:\n\n\\begin{enumerate}\n\\item \\textbf{Full-LRU}: We call an error-correction circuit that replaces each gate by a rectangle containing the gate followed by LRUs on each output qubit (as in \\cite{aliferis07}) the \\emph{Full-LRU} circuit; see Fig~\\ref{fig:LRcircuits}~(a). This circuit removes leakage immediately after each gate but uses a large number of additional gates and qubits per cycle. The Full-LRU circuit uses $16$ additional gates (in $4$ LRUs) per data and ancilla qubit per cycle. No LRU is applied to ancillas prior to measurement.\n\n\\item \\textbf{Partial-LRU}: The \\emph{Partial-LRU} error-correction circuit uses LRUs less frequently than the Full-LRU circuit. Once per error correction cycle, LRUs act on each data qubit while the ancillas are measured, as shown in Fig.~\\ref{fig:LRcircuits}~(b). No additional effort is made for the ancillas, which are periodically measured and reinitialized anyway. While this strategy applies LRUs less frequently, which leads to greater spread of errors relative to the \\emph{Full-LRU}, it only uses $4$ additional gates (in $1$ LRU) per data qubit per syndrome extraction and no additional gates applied to the ancillas.\n\n\\item \\textbf{Quick}: The \\emph{Quick} error-correction circuit further reduces the frequency of leakage reduction; see Fig.~\\ref{fig:LRcircuits}~(c). At the end of each cycle, the circuit swaps each data qubit with an ancilla. A similar construction was proposed in~\\cite{ghosh14} and shown to reduce leakage for a particular model relevant to superconducting qubits. In our case we choose to swap the ancilla with the data qubit $d_D$ immediately below the ancilla. Three $CNOT$ gates implement a $SWAP$ and, due to gate cancellation, there is only one additional $CNOT$ gate per data qubit compared to the standard circuit. After the swap, the physical qubits representing data and ancilla have traded roles. Each physical qubit is measured and reset every other cycle, so qubits do not remain leaked for many cycles.\n\n\\item \\textbf{No LRU}: The last and simplest circuit we study is the standard syndrome extraction circuit depicted in Fig.~\\ref{fig:toric}~(b,c), which we call \\emph{No LRU} for convenience. The circuit is not suitable for error correction in the presence of leakage since data qubits are never reset. This circuit provides a point of reference from which to assess the effectiveness of the other circuits.\n\\end{enumerate}\n\nWhile the Partial- and Full-LRU circuits are relatively straightforward to understand since they incorporate LRUs directly, the Quick circuit does not use LRUs. If there are no leakage errors, it is clear that the Quick circuit is functionally equivalent to the standard circuit. However, if an input ancilla or data qubit has leaked, one can calculate the effective circuit within our leakage model. Referring to Fig.~\\ref{fig:LRcircuits}~(c), a leaked ancilla $a_z$ depolarizes each data qubit in the corresponding check, and the leaked qubit takes the role of $d_D$. The original $d_D$ is depolarized and measured, producing a corrupted syndrome bit. On the other hand, a leaked data qubit, which we take to be $d_D$, depolarizes the ancilla $a_z$. The data and ancilla exchange roles and the leaked qubit is measured. Meanwhile, $d_U$, $d_L$, and $d_R$ have interacted with $a_z$ before it was depolarized. One can check that these gates merely re-encode the plaquette operators and do not produce additional errors.\n\nThe realization of the Quick circuit in the toric code does not require extra ancillas. In the surface code, which does not have periodic boundary conditions, one could alternate swapping the ancillas with the $D$ and $U$ qubits in every odd and even error-correction cycle respectively. This approach only requires $O(d)$ additional qubits at the boundary of the lattice.\n\n\\begin{figure}[t]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=fullLRU,width=.37\\textwidth}\n   \\label{subfig:full}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitModel,width=.27\\textwidth}\n   \\label{subfig:partial}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=quickModel,width=.32\\textwidth}\n   \\label{subfig:quick}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:LRcircuits} (a) The Full-LRU circuit replaces each gate with the gate followed by LRUs on each output qubit. (b) In the partial-LRU scheme, we insert an LRU on each data qubit at the end of each error-correction cycle. (c) The Quick scheme uses a circuit that is equivalent to those in Fig.~\\ref{fig:toric}. The final CNOT gate acting on the ancilla and the $d_U$ data qubit has been replaced by a CNOT followed by a SWAP. This simplifies to a pair of CNOT gates.}\n\\end{figure}\n\nTo decode syndromes produced by the leakage reducing circuits, we adjust the edge weights in the decoding graph so that the corresponding decoders again find a correction from among the most likely errors consistent with the syndrome. The decoder paired with the standard error-correction circuit (our No LRU circuit) uses a decoding graph depicted in Fig.~\\ref{fig:cube}. The prior probability of an edge $\\varepsilon$ (restricted to $\\{a,b,c,d,e,f\\}$ due to translation invariance) is taken as $p(\\varepsilon)=\\sum_j p_j(\\varepsilon)$ where $p_j(\\varepsilon)$ is the probability that a fault occurs at location $j$ and yields the syndrome $\\partial\\{\\varepsilon\\}$. The sum is taken over all locations in the circuit although not all locations contribute positive probability. If the physical error rates are sufficiently small, $p(\\varepsilon)$ is approximately proportional to the probability that $\\varepsilon$ carries an error. The edge weights are then given by $-\\log p(\\varepsilon)$. Edge weights for the standard circuit were approximated in this way in~\\cite{wang11} by counting single-location faults, and we use an analogous approach to calculate the participating edges of the decoding graph and their weights for the other circuits introduced in this section. Specifically, we iterate over regular faults placed at single fault locations $j$, compute the corresponding syndrome and associated pair of defects, identify an edge $\\varepsilon$ with the same pair of defects $\\partial\\{\\varepsilon\\}$, and add the probability of the fault $p_j(\\varepsilon)$ to $p(\\varepsilon)$.\n\n\\begin{table}[t]\n\\tcaption{Edges weights for each leakage reduction circuit}\n\\centerline{\\footnotesize\\smalllineskip\n\\begin{tabular}{r c c c c c c}\\\\\n\\hline\n{} & $\\mathrm{exp}(-w_a)$ & $\\mathrm{exp}(-w_b)$ & $\\mathrm{exp}(-w_c)$ & $\\mathrm{exp}(-w_d)$ & $\\mathrm{exp}(-w_e)$ & $\\mathrm{exp}(-w_f)$ \\\\\n\\hline\n{No LRU} & $31\/15p+q$ & $28\/15p$ & $16\/15p$ & $52\/15p$ & $8\/15p$ & $8\/15p$ \\\\\n{Quick} & $7\/3p+q$ & $32\/15p$ & $4\/3p$ & $4p$ & $8\/15p$ & $8\/15p$ \\\\\n{Full-LRU} & $103\/15p+q$ & $52\/15p$ & $88\/15p$ & $172\/15p$ & $32\/15p$ & $32\/15p$ \\\\\n{Partial-LRU} & $31\/15p+q$ & $52\/15p$ & $16\/15p$ & $76\/15p$ & $8\/15p$ & $8\/15p$ \\\\\n\\hline\\\\\n\\end{tabular}}\n\\label{table:weights}\n\\end{table}\n\nThe calculated edge weights in the decoding graph are summarized in Table~\\ref{table:weights}. Here $p$ is the probability of gate, initialization, and idle error and $q$ is the probability of measurement error, which we set equal to $p$. The edge labels correspond to labels shown in Fig.~\\ref{fig:cube}. The symmetry between the weights of edge pairs $b$-$d$ and $c$-$e$ is broken because the stabilizer measurements are not atomic operations but involve $CNOT$ gates applied in a particular order (see \\cite{wang11}). Although the leakage reducing circuits differ from the No LRU circuit, there are no new edges in the decoding graphs of these circuits. One can see that, for example, the weights for the Full-LRU circuit on edges $e$ and $f$ are $4$ times the corresponding weights of the other circuits due to the $4$ locations in the LRU. The edge weights for the No LRU circuit agree with the weights in~\\cite{wang11} except for edge $a$. Our probability for edge $a$ is higher because our error-correction circuits include ancilla initializations. We call a decoder that uses these edge weights the \\emph{Standard decoder}. Note that the Standard decoder does not account for leakage events, and hence we expect it will be suboptimal.\n\n\\section{Results}\\label{sec:results}\n\\noindent\n\nA systematic exploration of the accuracy thresholds for the Standard and HL decoders is depicted in Fig.~\\ref{fig:thresholds}~(a). The plot shows the accuracy threshold as a function of the amount of leakage (the relative excitation rate $r$). The plot was obtained by recording the failure rates for code distance $d=7$ and $d=9$ and by varying $r$ in increments of $0.1$ and $p$ in increments of $0.01\\%$. We chose a relative relaxation rate $s = 1$. For each parameter choice we found the crossover of the failure rates for the two code distances, which approximates the threshold.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=allThresholds,width=.595\\textwidth}\n   \\label{subfig:all}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=pdExploration,width=.365\\textwidth}\n   \\label{subfig:pdexplore}\n}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:thresholds} (Color online) (a) Summary of thresholds for the HL and Standard decoders. The HL decoder offers a substantially improved threshold. (b) The threshold, show here for the Quick circuit with HL decoder, does not change with the relative relaxation rate $s$.}\n\\end{figure}\n\nBased on Fig.~\\ref{fig:thresholds}~(a) we make the following observations. In the regime with no leakage ($r = 0$), simpler circuits, i.e. circuit with fewer locations per syndrome extraction, have higher thresholds than more complicated circuits, in agreement with expectations. With no leakage, the No LRU circuit has a threshold of about $0.70\\%$ whereas the Full-LRU circuit, which is the most complicated circuit we studied, has a threshold of about $0.22\\%$. We also observe that the Quick and Partial-LRU circuits give an almost identical threshold, differing visibly only for $r<0.3$. The Quick circuit benefits from having fewer gates and qubits, whereas the Partial-LRU benefits from more frequent qubit reinitializations and therefore more effective leakage suppression. These features appear to have comparable effects on the threshold. The HL decoder significantly improves the threshold compared to the Standard decoder when $r$ exceeds about $0.3$.\n\nAs the $r$ increases, the threshold decreases monotonically for all circuits and decoders. To understand this, consider an idealized setting where we assume that leaked qubits are immediately replaced by completely depolarized qubits, causing errors $X$, $Y$, $Z$, and $I$ each with equal probability. This makes a leakage event equivalent to a regular depolarizing error with probability $\\beta=3\/4$. Suppose that physical errors occur at an effective rate $\\tilde{p}=(1+\\beta r)p$ that is the sum of two terms where the first one is due to regular depolarizing errors and the second one due to leakage. Let's also assume that there is some minimum number of faults $m$ that can produce a logical error. The logical error rate is then given by\n\\begin{equation}\np' = A\\tilde{p}^m+O(\\tilde{p}^{m+1}) = A(1+\\beta r)^{m}p^m + O(p^{m+1}).\n\\end{equation}\nNeglecting the higher order terms and solving for the fixed point gives us a very rough approximation for the threshold as\n\\begin{equation}\np_{th}^{(r)} \\approx \\left[A(1+\\beta r)^m\\right]^{-1\/(m-1)}.\n\\end{equation}\nWe are interested in the ratio of thresholds when $r=0$ versus $r>0$, which is\n\\begin{equation}\n\\frac{p_{th}^{(r)}}{p_{th}^{(0)}} = (1+\\beta r)^{-m\/(m-1)} \\approx \\frac{1}{1+\\beta r}.\n\\end{equation}\n\n\\begin{figure} [b!]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=threshCombined.pdf,width=.585\\textwidth}\n   \\label{subfig:linear}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=threshCombinedLog.pdf,width=.375\\textwidth}\n   \\label{subfig:log}\n}\n\\fcaption{\\label{fig:thCombined} (Color online) Comparisons of decoding success probabilities for the Full-LRU circuit and the Quick circuit with the Standard and HL decoder. (a) Linear scale on the left, (b) log scale on the right.}\n\\end{figure}\n\nThe threshold then decays as $\\frac{\\alpha}{1+\\beta r}$ where $\\alpha$ is a constant equal to the numeric value of the threshold in the regime without leakage. Assuming $\\beta=3\/4$ and choosing $\\alpha$ such that the threshold at $r=0$ intersects the threshold of the Quick circuit with HL decoder, we plot the threshold of the ``idealized decoder'' in Fig.~\\ref{fig:thresholds}~(a). A Standard decoder's threshold will not exceed that of the idealized one we just described, but a decoder that uses results from three-outcome measurements could in principle. Table~\\ref{table:fitparameters} lists some extracted values of $\\alpha$ and $\\beta$ from the data in Fig.~\\ref{fig:thresholds}~(a).\n\nFig.~\\ref{fig:thresholds}~(b) shows that the threshold does not depend much on the relaxation rate $s$, likely because relaxation is much slower than the leakage reduction occuring in the circuits. This plot was obtained by recording thresholds for the Quick circuit with the HL decoder using the same technique as for Fig.~\\ref{fig:thresholds}.\n\nIn our last set of simulations, we chose $r = s = 1$ and recorded the failure rate of the decoders as a function of $p$. Recall that $p_\\uparrow = r p$ and $p_\\downarrow = s p$. In Fig.~\\ref{fig:thCombined} (a) we show failure rates for the Full-LRU circuit and the Quick circuit with both the Standard and HL Decoder. The thresholds of each circuit\/decoder are visible as crossing points of the curves. The Full-LRU circuit has the lowest threshold, followed by the Quick circuit with the Standard decoder and the Quick circuit with the HL decoder. The success rate for the Partial-LRU circuit is not plotted because it was indistinguishable from the Quick circuit.\n\nThe same plot on the log scale in Fig.~\\ref{fig:thCombined} (b) shows that when $p$ is well below threshold, the success rate of the Full-LRU circuit improves faster than for the Quick circuit with Standard decoder, which suggests that for low enough physical error rates the Full-LRU circuit will be better than the other circuits. We have a sufficient number of samples to observe at least $100$ failures at each data point shown in the figure. Table~\\ref{table:fitparameters} presents the degree of error suppression $\\gamma$ for each circuit based on fitting the logarithm of the logical error rate. For the HL decoder, we give the fit for each distance $d=7$, $9$, and $11$ since we appear to be observing the effects of higher order terms in the logical error rate and have not reached a sufficiently low physical error rate to capture the lowest order term in the polynomial. Since numeric simulations at extremely low physical error rates are inefficient and may require different techniques \\cite{bravyi13}, we leave further exploration of this question for future work.\n\n\\begin{table}[t]\n\\tcaption{Threshold $\\sim \\frac{\\alpha}{1+\\beta r}$ and sub-threshold logical error rate $\\sim Ap^{\\gamma d}$}\n\\centerline{\\footnotesize\\smalllineskip\n\\begin{tabular}{r c c c}\\\\\n\\hline\n{} & $\\alpha$ & $\\beta$ & $\\gamma$ \\\\\n\\hline\n{idealized} & $0.65$ & $3\/4$ & n.a. \\\\\n{Full-LRU} & $0.22$ & $1.72$ & $~0.67$\\\\\n{Quick} & $0.65$ & $3.59$ & $~0.45$ \\\\\n{Partial-LRU} & $0.55$ & $2.55$ & $~0.46$ \\\\\n{Quick (HL)} & $0.62$ & $1.23$ & $~\\{0.45,0.52,0.64\\}$ \\\\\n{Partial-LRU (HL)} & $0.55$ & $0.92$ & $~\\{0.48,0.53,0.65\\}$ \\\\\n\\hline\\\\\n\\end{tabular}}\n\\label{table:fitparameters}\n\\end{table} \n\\section{Simulation Methods}\\label{sec:simulation}\n\\noindent\n\nWe implemented simulations of the No LRU, Partial-LRU, Full-LRU, and Quick leakage reduction strategies. We evaluate both the Standard and HL decoders for the Partial-LRU and Quick strategies, and the Standard decoder for the No LRU and Full-LRU strategies. The simulation propagates labels $I$, $X$, $Y$, $Z$ or $L$ for each ancilla and data qubit in a toric code of distance $d$. We simulate the propagation of regular and leakage errors according to the error model described in Section~\\ref{sec:leakagemodel}. Fault-tolerant error-correction in the toric code of distance $d$ requires $O(d)$ rounds of syndrome measurements so we simulate $d$ rounds. In addition, we include a round of perfect syndrome measurement at the end of each simulation where leaked qubits are replaced by completely depolarized qubits and the syndromes are measured without errors. All $d+1$ syndromes are processed together, a correction is found, and a failure is registered if the state of the code qubits after correction anticommutes with a logical operator of the code. Syndrome processing is performed such that the final state of the code qubits commutes with the stabilizer.\n\nWe assume that the $d$ rounds of fault-tolerant error-correction take place in the course of a longer computation. Therefore, each qubit is initially assigned the $L$ label with probability that is calculated based on the parameters of the model ($p_\\uparrow$ and $p_\\downarrow$) and on the frequency of the qubit re-initializations for the particular circuit. Specifically, persistent qubits that are never re-initialized, such as data qubits in the No LRU circuit, start in the $L$ state with the equilibrium probability derived in Section~\\ref{sec:leakagemodel}.\n\nThe Standard decoding graph is built using the unit cell in Fig.~\\ref{fig:cube}. The translation invariance of the decoding graph allows us to calculate the distance of two defects directly from their $(x, y, t)$ coordinates where the $x$ and $y$ specify the location of the defect on the torus and $t$ is the location in the time dimension. The decoding graph of the HL decoder is not translation invariant due to the presence of the low weight edges introduced in Section~\\ref{sec:heralded}. Since calculating distance of the defects based on their coordinates would be difficult, we build the entire decoding graph and use a shortest path algorithm to find the distance between defects.\n\nThe implementation uses the Boost~\\cite{Boost} libraries to find shortest paths, and the Blossom V~\\cite{kolmogorov09} library to perform minimum weight perfect matching. Our results were generated by Monte Carlo simulations which were repeated until we reached at least $10,000$ iterations and at least $1,000$ failures for each configuration. These simulations were run on an IBM Blue Gene\/Q~\\cite{BGTeam11} using about 30,000 CPU-hours.\n\\section{Toric Code}\\label{sec:toriccode}\n\nThe toric codes \\cite{kitaev03} are the prototypical example of topological stabilizer codes. Each toric code is defined on a $d$ by $d$ square array whose left-right and top-bottom boundaries are associated. The vertices of the array are connected to form a graph with $d^2$ vertices, $2d^2$ edges, and $d^2$ faces. Each of the $n=2d^2$ edges carries a physical qubit of the code, called a code or data qubit, and each vertex and face carries an ancillary qubit used for error-correction, called a syndrome or ancilla qubit (see Fig.~\\ref{subfig:lattice}). The stabilizer $S$ of the quantum code is generated by a set of check operators $\\{A_v\\}$ and $\\{B_f\\}$ that belong to the $n$-qubit Pauli group and are attached to each vertex $v$ and face $f$ of the graph. These operators are tensor products of single qubit Pauli operators\n\n\\begin{figure}[t]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=lattice,width=.31\\textwidth}\n   \\label{subfig:lattice}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=ZZZZMeas,width=.33\\textwidth}\n   \\label{subfig:ZZZZ}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=XXXXMeas,width=.33\\textwidth}\n   \\label{subfig:XXXX}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:toric} (Color online) (a) The toric code has a natural two-dimensional layout on the surface of a torus. Qubits function as either data qubits, used to store the encoded quantum state, or ancilla qubits, used to measure check operators (stabilizers) of the quantum code. Z-type check operators associate to faces (plaquettes) and X-type check operators associate to vertices (stars). Each check operator involves four data qubits and is measured using an ancilla qubit. The torus encodes a pair of qubits whose representative logical Pauli operators $X_1$, $Z_1$, $X_2$, and $Z_2$ are shown. (b,c) These circuits measure (b) plaquette and (c) star operators using four CNOT gates together with preparations and measurements. In each circuit, data qubits $d_{D,R,L,U}$ interact with an ancilla qubit $a_{z\/x}$ that is prepared in the state $|0\\rangle$ or $|+\\rangle=\\frac{1}{\\sqrt{2}}\\left(|0\\rangle+|1\\rangle\\right)$. This ancilla is then measured in the basis of eigenstates of Z or X, respectively.}\n\\end{figure}\n\n\\begin{equation}\nX = \\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right)\\ \\textrm{and}\\ Z = \\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right).\n\\end{equation}\nThe vertex (or star) operators\n\\begin{equation}\nA_v =\\bigotimes_{\\varepsilon\\in N(v)}X_\\varepsilon\n\\end{equation}\nare $X$-type checks that apply a Pauli $X$ operator to the qubit on each edge $\\varepsilon$ in the neighborhood $N(v)$ of vertex $v$. The neighborhood $N(v)$ of a vertex is the set of edges incident to $v$. The face (or plaquette) operators\n\\begin{equation}\nB_f =\\bigotimes_{\\varepsilon\\in N(f)}Z_\\varepsilon\n\\end{equation}\nare $Z$-type checks that apply a Pauli $Z$ operator to the qubit on each $\\varepsilon$ in the neighborhood $N(f)$ of a face $f$. The neighborhood $N(f)$ of a face is the set of edges on the boundary of $f$. Each check operator involves only four code qubits since each vertex is incident to four edges and each face is bounded by four edges. The set of generators clearly commutes since any star shares zero or two edges with any face. Due to the periodic boundary conditions, any set of $d^2-1$ faces and $d^2-1$ vertices associate to an independent set of check operators that generate the stabilizer of the toric code. This implies that the toric code encodes a pair of logical qubits. Representatives for each class of logical Pauli $X$ and $Z$ operators are shown in Fig.~\\ref{subfig:lattice}.\n\nThe overcomplete set of $2d^2$ check operators are measured simultaneously \\cite{dennis02} using the circuits shown in Fig.~\\ref{subfig:ZZZZ} and Fig.~\\ref{subfig:XXXX}. In the first step, we prepare all plaquette ancillas in $|0\\rangle$ and all site ancillas in $|+\\rangle\\propto |0\\rangle+|1\\rangle$. In the next four steps, CNOT gates act between each ancilla and the data qubit above, left, right, and below the ancilla, in that order. This corresponds to the gate order used in \\cite{wang11,stephens14}. Finally, we measure each plaquette ancilla in the $Z$ eigenbasis and each star ancilla in the $X$ eigenbasis. These six steps constitute an error-correction cycle. One cycle produces a single noisy syndrome given by $2d^2$ bits.\n\nA single error-correction cycle cannot be fault-tolerant in the toric code since local errors in the syndrome can lead to macroscopic errors after error-correction. However, $O(d)$ cycles suffice to improve confidence in the syndrome so that error-correction becomes fault-tolerant. A processing algorithm ingests these $O(d)$ syndromes and infers a corrective Pauli operator. The classic approach \\cite{dennis02}, which we follow in this work, processes plaquette and site syndromes independently to find bit and phase error corrections. The problem of inferring the most probable error given the observed syndrome is mapped to a minimum weight perfect matching problem that can be solved with Edmond's algorithm \\cite{edmonds65}.\n\n\\begin{figure} [b]\n\\vspace*{13pt}\n\\centerline{\\epsfig{file=noLeakCube.pdf, width=.30\\textwidth}}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:cube} (Color online) The unit cell of the decoding graph has six distinctly-weighted edges shown here. Edges $b$ and $d$ correspond to qubit errors, edge $a$ to measurement error, edges $c$ and $e$ to correlated qubit-measurement errors, and edge $f$ to correlated qubit-qubit-measurement errors. The particular edges that appear are determined by the error-correction circuits.}\n\\end{figure}\n\nThe concept of a decoding graph ${\\cal G}=({\\cal V},{\\cal E})$ is a useful abstraction for relating errors and corresponding syndromes \\cite{bravyi13}. There is a separate decoding graph for bit flip ($X$) and phase flip ($Z$) errors. With the exception of Sec.~\\ref{sec:heralded}, the graphs we consider are translation invariant, generated by translating the unit cell shown in Fig.~\\ref{fig:cube} in each of three orthogonal directions. The edges ${\\cal E}$ correspond to error events in the error-correction circuits, and, with the exception of diagonal edges $c$, $e$, and $f$ that correspond to correlated errors, are identified with qubits (horizontal edges ``$b$'' and ``$d$'') and measurement outcomes (vertical edges ``$a$'') for each error-correction cycle. An error on an edge $\\varepsilon$ of ${\\cal G}$ creates a pair of defects at the two vertices incident to $\\varepsilon$. Errors on a subset of edges $E\\subseteq {\\cal E}$, called an error chain, create defects on $\\partial E$ where $\\partial E\\subseteq {\\cal V}$ is the set of vertices with an odd number of edges incident from $E$. Operationally, syndrome bit outcomes label the vertical ``$a$'' edges, and a defect occurs on vertex $v$ if the two incident ``$a$'' edges to $v$ have different syndrome labels.\n\nThe syndrome processing algorithm, or decoder, uses the minimum weight matching algorithm to match pairs of defects on the decoding graph. The error is then corrected by applying the $X$ or $Z$ correction on an error chain connecting each pair of matched defects. A chain $E$ is closed if $\\partial E$ is empty, which occurs if the chain commutes with the stabilizer. A closed chain is contractible if the corresponding error belongs to the stabilizer. All error chains with the same $\\partial E$ are equivalent up to closed chains, which include harmless contractible chains but also include non-contractible chains that are logical operators. The weight of each edge in the decoding graph is chosen to approximate the negative logarithm of the probability of the corresponding error. Therefore, independently for $X$ and $Z$ errors, a minimum weight matching decoder finds a correction from among the most likely errors and fails when the product of the correction and the error chain is not in the stabilizer.\n\nIn this work we restrict our attention to fault-tolerant quantum error-correction but our results apply to both active memory and computation \\cite{rh07,horsman12}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nWater masers arising from thin disks around massive black holes provide \nhigh brightness temperature non-thermal dynamical tracers of gas in Keplerian \norbits.  As such, water maser disks viewed edge-on provide tracers of the\nKeplerian potential and enable measurement of the black hole mass, provided a distance \nis known in order to translate the apparent angular disk size into a physical \nsize \\citep[e.g.,][]{miyoshi1995}.\nMaser accelerations and proper motions can also be observed, and because\nthe circular velocity is known from Doppler shifts, the geometric distance \nto the black holes (and host galaxies) can be determined \\citep[e.g.,][]{herrnstein1998}.  \nGeometric distances obtained from water masers provide a crucial independent measurement of the Hubble\nconstant and can be used to calibrate other distance indicators such as \nthe period-luminosity relation of Cepheids \\citep{riess2016}.\n\nThese measurements require maser disks that are viewed within a few degrees of edge-on:\notherwise, maser beaming directs emission away from the observer because masers \npropagate along velocity-coherent paths through the disk.  For thin disks, this propagation \noccurs along the radial path along the line of sight toward the black hole and along the disk \ntangent points.  For warped disks, such as that found in NGC 4258, the picture is\nmore nuanced because a warped disk provides numerous sightlines and inclinations that intersect velocity-coherent \nparts of the disk \\citep{humphreys2013}.  Nonetheless, inclined maser disks are generally not seen in water maser\nsurveys.  Or are they?\n\n\\begin{figure*}[ht!]\n\\epsscale{1.1}\n\\plotone{lensed_masers.eps}       \n\\caption\nMaser amplification pathways for Keplerian disks orbiting massive black holes.  These diagrams are schematic and not to scale.\nLeft:  the geometry of an egde-on rotating disk showing systemic velocity masers (green) and high-velocity redshifted (red) and blueshifted\n(blue) masers at the tangent points of the disk.  \nCenter:  in the black hole rest frame, all disk motion is perpendicular to the radial direction, providing velocity-coherent amplification \npathways at the systemic velocity (or rest frame).  In-going (and out-going) radial masers will populate the disk in this reference frame.\nRight:  In-going masers may be gravitationally lensed\/deflected by the central black hole into the sightline of an observer who would \nnot otherwise see maser\nemission from an inclined disk.   Only the systemic velocity masers would be seen by this observer, and they will appear to \narise from the location of the black hole.  High-resolution imaging might reveal a ring or arc of water maser emission, \nproviding a black hole mass measurement.  \n\\label{fig:schematic}}\n\\end{figure*}\n\nIt seems likely that inclined water maser disks have already been detected by single dish surveys, \nbut they have been discarded because they show no high velocity lines that have canonically been\nused to identify maser disks.  \nIn this paper, we propose a mechanism to produce detectable maser emission from inclined disks \nthat may also be used to obtain black hole masses\n(Section \\ref{sec:thought}), we present a method to detect inclined maser disks based on extant surveys \n(Section \\ref{sec:candidates}), we present Karl G. Jansky Very Large Array \n(VLA)\\footnote{The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.} observations of candidate inclined disks \n(Sections \\ref{sec:obs} and \\ref{sec:results}), and we \nprovide a list of inclined disk candidates for further study (Sections \\ref{sec:analysis} and \\ref{sec:discussion}).  \nIn what follows, we assume a flat cosmology with parameters  $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$, $\\Omega_M = 0.27$, and \n$\\Omega_\\Lambda = 0.73$, and we calculate distances from redshifts in the CMB rest frame.\n\n\n\\section{A Thought Experiment} \\label{sec:thought}\n\nConsider a typical 22 GHz water maser thin disk in Keplerian orbit around a massive black hole.\nThis thin disk shows systemic velocity masers at the center of the observed disk, along a radial amplification path, \nand it shows high-velocity red- and blue-shifted masers at the tangent points of the disk, along azimuthal paths, \namplifying spontaneous emission and background host galaxy continuum (Figure \\ref{fig:schematic}, left).  \nIf a thin disk is inclined by more than a few degrees, the masers are \nno longer beamed toward the observer and the disk would not be detected.  \nThe known water maser disks show an inclination typically within $\\sim$5$^\\circ$ of edge-on \\citep[e.g.,][]{kuo2011}, \nsuggesting that an order of magnitude additional maser disks exist, and they are simply beamed in directions we cannot observe.  \nNote that the famous water maser disk in NGC 4258 \\citep[e.g.,][]{miyoshi1995,herrnstein1998} is atypical, with \na disk inclination of 72$^\\circ$, but the disk warp provides sightlines that are nearly edge-on, and \nthis is where the masers are seen \\citep{humphreys2013}.\n\nFor the following discussion we will use fiducial parameters similar to those of many of the known maser disks \n\\citep{kuo2011,greene2016}.  We assume a distance\nof 10 Mpc, a black hole mass of $M_{BH} = 10^7\\ M_\\odot$ (and therefore a Schwarzschild radius of $R_s\\sim1\\ \\mu$pc), and\na circular rotation speed of 1000 km s$^{-1}$ in the maser-active part of the disk located 0.5 pc from the black hole and \nspanning 0.2 pc.  We \nassume a disk thickness equal to that of NGC 4258: $5.5\\times10^{14}$ cm or 0.18 mpc \\citep[1$\\sigma$;][]{argon2007}. \n\nConsider how this system appears to an observer at the location of the black hole:\nthe disk describes a plane in the sky.  Water masers reaching this observer must be purely radial (propagating inward) but will have a\ncoherent amplification path in all radial directions (Figure \\ref{fig:schematic}, center).  All Keplerian Doppler shifts\nwill be across the line of sight, so all maser emission reaching this location will have zero velocity in this reference frame (ignoring\nthe transverse Doppler shift, \nwhich is negligible at $\\sim$2 km~s$^{-1}$).  The black hole-frame observer will thus see a ring of maser light \nor maser spots with low velocity spread.  The maser-active part of the disk would subtend $\\sim$2.5 arcminutes (1$\\sigma$). \nThe expected maser opening angle would be similar, $\\sim$3 arcminutes, assuming a maser path-to-size ratio of \n$\\sim$0.2 pc\/0.2 mpc $\\simeq 1000$, implying maser light spanning $\\sim$500 $R_s$ at the black hole.   \n \nThe black hole Kerr metric would thus be bathed in maser light with a continuous distribution of impact parameters, \nimplying gravitational deflection of maser light \ninto nearly all angles (the deflection at 100 $R_s$ is 1.1$^\\circ$ and grows inversely with impact parameter $R$ as $\\theta = 2 R_s\/R$;\n\\citet{einstein1915}).  \nOne would therefore\nexpect incoming maser light to be scattered into many angles, representing a gravitational lensing de-focusing of a given incoming\nbeam.  But portions of multiple maser beams from around the disk may be deflected into a given direction, making the \nnet flux as a function of angular direction uncertain and in need of numerical modeling. \n\nNGC 4258 has a warped disk \\citep[e.g.,][]{humphreys2013}, which means that the in-going radial masers can be misaligned and thus be\nbeamed above or below the dynamical center of the disk, and that the continuous distribution of disk inclination angles will \nfully populate the black hole metric volume with beams from multiple angles, \nfurther enhancing the sampling of impact parameters and deflection angles. \nIf warped disks are common, \nthen maser disks with high velocity lines will be detected more often than one would assume based on random inclination alone \n(in other words, parts of many \ndisks will have inclinations appropriate for detectable maser amplification), and the black hole will be illuminated by in-going \nmasers from above and below\nthe nominal disk plane, producing a range of incoming maser impact parameters and angles with respect to the disk,\nthereby enhancing the gravitational deflection probability toward any given observer.  \nRegardless of whether disks are warped or not, maser light will be \nscattered away from the maser disk plane, making inclined disks potentially detectable, but likely faint compared to edge-on disk\nmasers.\n\nMasers amplify seed photons, and these seed photons can be the portion of a radio continuum that matches the maser line \nfrequency, appropriately Doppler-shifted, or they can be spontaneously emitted maser line photons.  In known maser disks, the\nsystemic radial masers typically amplify the AGN radio continuum, and the high-velocity tangential masers amplify spontaneous \nemission or possibly radio continuum from the host galaxy itself.  In-going radial masers do not have access to AGN seed photons\nand must amplify either continuum from the host galaxy or inward-directed spontaneous emission from the outer parts of the disk.  \nIn either case, it is reasonable to expect in-going masers to be weaker than the out-going systemic masers from \nedge-on disks.  It is unclear how the amplification of in-going masers would compare to the amplification of the \nhigh-velocity masers in edge-on disks because \nthe tangential amplification pathway may or may not have a physically longer or higher column density \nvelocity-coherent path for amplification.  \n\nIf one were to observe a water maser disk from a direction other than edge-on, would water maser emission be seen?  \nThe thought experiment \ndescribed here suggests that it would, and the inclination of the disk with respect to the observer would select the deflection angle\n(or equivalently the impact parameter) of the observed maser light.  For example, deflection by 10$^\\circ$ would allow observation of masers\nfrom the back side of a disk with an inclination of 80$^\\circ$, requiring an impact parameter of 12 $R_s$.  Other parts of the disk could be viewed if light reaches smaller impact parameters.  If, as expected, there is a continuous range of impact parameters, then a continuous distribution of maser light from an extended portion of the disk will contribute to a spectrally narrow maser line complex seen by the observer at nearly any inclination (but perhaps with higher likelihood and intensity at higher inclinations).  \n\n\\subsection{Observable Signatures}  \n \nThe observational signatures of an inclined water maser disk would be:\\\\\n\\indent 1.  A narrow line or line complex\\\\\n\\indent 2.  at the systemic velocity\\\\\n\\indent 3.  at the apparent black hole location.\\\\\nThe black hole location would be indicated by the radio continuum core, ideally observed at the same frequency as the water\nmaser.\nThe observational signatures of an inclined water maser disk, however, may also arise from other mechanisms.\nWater masers can be produced in radio jet-molecular cloud interactions \\citep[e.g.,][]{gallimore1996,claussen1998,peck2003,henkel2005}, in \nstar-forming regions \n\\citep[e.g.,][]{tarchi2002a,tarchi2002b,henkel2005,hofner2006, darling2008,brogan2010,darling2011,tarchi2011,amiri2016}, \nand in outflows \\citep[e.g.,][but note that some objects in the latter survey may be \ncandidates for inclined disk masers]{greenhill2003b,kondratko2005,tarchi2011a}.\nThese are likely to be the main contaminant among an inclined disk\nsurvey sample.  VLBI identification of the maser with an AGN via spatial coincidence with \nthe core radio continuum --- identified by the spectral index --- can resolve the ambiguity (Section \\ref{sec:results}).\n\n\nIn contrast to edge-on disk systemic masers, it is unclear whether inclined disk maser lines will show proper motion or acceleration.  This may depend on the clumpiness of maser-emitting regions, on the maser beam sizes, on the physical extent of the \ndisk that is sampled by the observed line, and on the deflection angle to the observer.\nOne might expect lensed masers to show little or no time variability, but the substructure seen in known water maser disks \nand the natural variability of water masers suggests that this may be a bad assumption.  \n\nMany extragalactic water masers detected in previous surveys meet some or all of the above observational criteria:  \ninclined water maser disks may have already been detected!  In most cases, when \na single systemic velocity line is detected in water maser surveys, there is no interferometric follow-up\nbecause distance or black hole mass measurements (traditionally) require edge-on disks, the signature of which are \nthe high velocity lines emitted from the tangent points of the disk.  \nHigh resolution observations may also be frustrated by weak or variable water masers.\n\n\nIn Section \\ref{sec:obs} we present interferometric mapping of a sample of narrow-line systemic velocity water masers that appear to be inclined disk maser candidates.  If observations show that the masers remain unresolved and are centered at the location of the \ncentral massive black hole (as identified by simultaneous radio continuum observations at 20 GHz), then they remain candidates and should be mapped with VLBI.\n\n\n\n\n\\subsection{Black Hole Masses}\n\nThe Einstein radius of a strong gravitational lens is \n\\begin{equation}\n   \\theta_E = \\sqrt{2 R_s {D_{LS}\\over D_L D_S}} \n                 = 9.2 \\sqrt{{R_s \\over {\\rm mpc}} \\, {D_{LS} \\over {\\rm pc}}} \\left( D_L \\over {\\rm Mpc} \\right)^{-1} {\\rm mas}, \n\\end{equation}\nwhere $D_L$, $D_S$, and $D_{LS}$ are the angular diameter distances to the lens, to the source, and between the lens and the source,\nrespectively, and $D_L \\simeq D_S$ for the black hole-maser disk configuration \\citep[after][]{einstein1936}.\nFor the fiducial parameters listed above ($R_s = 1$ $\\mu$pc, $D_{LS} = 0.5$ pc, and $D_L = 10$ Mpc),\n$2 \\theta_E = 0.041$ mas. \nSince the angular resolution (HPBW) of the Very Long Baseline Array (VLBA) is 0.3 milliarcseconds at 22.2 GHz (which should be \ncompared to twice the Einstein radius), \nthe Einstein radius for the fiducial maser disk and black hole would require space-based VLBI to resolve.  \nOn the other hand, terrestrial VLBI could resolve \na water maser Einstein ring for a more massive black hole with a physically larger maser disk:  for \n$R_s = 100$ $\\mu$pc ($M_{BH} = 10^9\\ M_\\odot$), $D_{LS} = 2$ pc, and $D_L = 10$ Mpc,\n$2 \\theta_E = 0.82$ mas.  Unfortunately, water maser disks have yet to be identified orbiting $10^9\\ M_\\odot$ black holes, \nbut it is unclear whether is this a selection effect or a consequence of physics \\citep{vandenbosch2016}.  The most \nmassive black hole measured using a water maser disk to date is in NGC 1194 with $M_{BH} = 10^{7.85\\pm0.05}\\ M_\\odot$ \\citep{kuo2011,greene2016}.\n\nWere an Einstein ring observable from a back side in-going maser in an edge-on disk or from an in-going maser from \nan edge-on portion of an inclined warped disk, then one can infer a black hole mass from the angular size of the ring.  \nOne does need independent measurements of the luminosity distance to the black hole ($D_L$) and the size of the \nmaser-emitting disk ($D_{LS}$).  $D_L$ can be obtained from an assumed cosmology and the cosmological redshift, but \n$D_{LS}$ may be more difficult to measure.  The precision of black hole masses obtained from this method may be limited\nby our ability to measure, model, or estimate the radius of the maser-emitting part of the disk.  \n\nEinstein rings require linear alignment between the source, the lens, and the observer, which is not germane to the inclined\nmaser disk geometry (for unwarped disks). \nInstead, one would na\\\"{i}vely expect to see multiple images of the same maser, which can also be related to the black hole mass.  This\nexpectation, which is correct for isotropic emitters, is probably incorrect for masers.\n\nAn important difference between maser emission and the standard treatment of gravitational lensing is maser \nbeaming:  while there may be sightlines in a gravitational lens geometry that land on the emitter, emission \nmay not be seen if light is not beamed along that sightline.  For a general pointlike isotropic light source offset from \nthe lens by angle $\\theta_S$, there are two solutions to the lens equation:\n\\begin{equation}\n  \\theta_\\pm = {1\\over2} \\left( \\theta_S \\pm \\sqrt{\\theta_S^2 + 4 \\theta_E^2}\\right).\n\\end{equation}\n$\\theta_+$ represents the angle between the lens and the source image appearing outside $\\theta_E$, and \n$\\theta_-$ represents the angle of the image appearing inside $\\theta_E$\n\\citep[][Equation 24, Figures 5 and 7]{narayan1995}.\nFor an inclined maser disk configuration, no emission is directed \nsignificantly out of the disk plane, so no maser emission would be seen in the $\\theta_+$ direction.\nOn the other hand, $\\theta_-$ may be small enough that this sightline is included in the maser beam \npassing very close to the central black hole.  In this case, the source-observer deflection angle $\\alpha_-$ nearly matches the \ncomplement of the disk inclination:  $\\alpha_- \\simeq \\pi\/2 - i$ (Figure \\ref{fig:schematic}, right).  \n\nIn contrast to canonical lensing, \nwe expect that the maser beaming will produce only one maser spot image, where light is deflected in the \nmanner shown schematically in Figure \\ref{fig:schematic} (right).  This maser image will lie inside the Einstein radius.\nFor isotropic extended emitters, this image would also be demagnified, but masers are not isotropic emitters and \ncan be exceptionally compact (equivalently, they demonstrate high brightness temperatures).  The degree of demagnification\nis therefore unclear and requires numerical ray-tracing to assess.\n\nFor the fiducial parameters above, and assuming an inclination of $80^\\circ$,\n$\\theta_S = (R_{\\rm maser}\/D_S) \\sin\\alpha_- = 1.8$ mas, $\\theta_E = 0.021$ mas, and therefore $|\\theta_-| = 0.2$ $\\mu$as, \nwhich is equivalent to 11 $R_s$.  The maser beam spans this distance from \nthe black hole, so the maser emission can be lensed toward the observer in this case. \nFor the $10^9\\ M_\\odot$ black hole with a larger maser disk described above, \n$\\theta_S = (R_{\\rm maser}\/D_S) \\sin\\alpha_- = 7.2$ mas,\n$\\theta_E = 0.41$ mas, and therefore $|\\theta_-| = 24$ $\\mu$as, \nwhich is again equivalent to 11 $R_s$ (the impact parameter determines the \ndeflection angle, which is determined by the inclination).  \n\nOne would therefore expect lensed masers from inclined disks to be faint and appear to arise from \nthe black hole location ($\\theta_- \\ll 1$ mas).   \nThis treatment assumes a single pointlike maser rather than an extended continuous disk of masers or a set of distributed maser \nspots.  In this more realistic scenario, numerical ray-tracing is required to connect the observable lensed maser image to the black \nhole mass and maser disk configuration.  \n\nLensed masers from inclined disks may appear to be pointlike or they may describe arcs.\nIn-going masers from the far side of an edge-on disk or in-going masers from an inclined but warped disk may produce Einstein\nrings.  \nSingle-epoch VLBI maps can therefore provide black hole masses, but space-based VLBI may be required\nfor the typical $\\sim10^7\\ M_\\odot$ black hole associated with water maser disks.   The mass measurement precision will be limited by \nuncertainty about the maser disk size rather than by the distance to the object\n (the maser will provide the systemic redshift, which can be converted into a distance, and\nthe distance uncertainty will be dominated by peculiar velocity departures from the Hubble flow).\n\n\n\n\\subsection{Back-Side Masers in Edge-on Disks}  \n\nIf there are in-going radial water masers in maser disks (and there is no compelling reason to think otherwise), \nthen there should be systemic water masers seen in edge-on disks from the back side of the disk, \nboth lensed and unlensed by the black hole.  \nAre such masers in extant data?  \n\nIf there are also front-side systemic masers in an edge-on disk, then they will be orders of magnitude brighter \nthan back-side masers, even in the presence of strong lensing, simply due to amplification considerations.  \nThe front-side masers can amplify AGN radio continuum, whereas the back-side masers will either be driven by \nstimulated emission (but with an amplification pathway equal to the front-side masers) or by host galaxy continuum, \nwhich is substantially weaker than AGN continuum at 22 GHz in many cases.   The back-side maser contribution may therefore\nbe confused by the front-side emission, particularly since both types of maser emission will occur at the systemic velocity.\n\nIn rare cases, a back-side maser might be distinguishable from the front-side emission either by a position or a velocity offset\n(there is some spread to systemic velocity masers, e.g. \\citet{gao2016}).  The observational signature of a back-side maser would be an acceleration \nin the opposite sense of the front-side systemic maser acceleration (i.e., negative acceleration under the convention that \npositive velocities are redshifted).   This is not seen in published systemic maser acceleration measurements \n\\citep[e.g.][]{greenhill1995,nakai1995,braatz2010,kuo2015,gao2016}, but such a signal could be lost amid the brighter and numerous front-side masers.\n\nIf a back-side maser is lensed into an arc or Einstein ring by the central black hole, then it might be extended in VLBI maps.  Resolved \nsystemic maser emission would therefore be an additional observable signature of back-side masers.  \n\n\n\\section{Candidate Selection}\\label{sec:candidates}\n\nIf inclined water maser disks can be detected via gravitational lensing or deflection of in-going masers by massive black holes, then \nthey have likely already been detected in surveys for maser disks.  But they were rejected as disk candidates because they \nlacked high-velocity emission.  Inclined maser disks will appear to have maser\nemission only at the systemic velocity of the galaxy or AGN.  We therefore use extant water maser surveys to select inclined disk \ncandidates.\n\nUsing extant water maser surveys, most of which favor Seyfert 2 AGN, we examined single-dish spectra \ncompiled by the Megamaser Cosmology Project\\footnote{\\label{footnote:maser_url}\\url{https:\/\/safe.nrao.edu\/wiki\/bin\/view\/Main\/PrivateWaterMaserList}}\nto select objects showing\na narrow systemic velocity maser or maser complex.  We also imposed a 30 mJy line flux limit and excluded objects south of $-20^\\circ$ declination.  \nThis process identified 16 inclined maser disk candidates (Table \\ref{tab:obs}), and most candidates (14) have only been observed\nwith a single dish.  Those that do have interferometric maps are NGC 3556, which was mapped using the VLA in CnB and DnA configurations \\citep{tarchi2011} with no 22 GHz continuum detected (1$\\sigma$ rms noise of $\\sim$0.5 mJy beam$^{-1}$),  \nand NGC 3735,which was mapped using A-array \\citep{greenhill1997}, but no 22 GHz continuum was detected.  \n\n\nSince the inclinations of known maser disks are nearly edge-on, the number of inclined maser disks must be large, \nroughly an order of magnitude larger than the number of detected maser disks.\nHowever, the gravitational lensing or deflection of detectable in-going radial\nmasers adds substantial uncertainty to the detection expectations.   We do not know how common observable lensed inclined maser\ndisks are in the universe because we do not know the opening angle of the masers, the \nsize of the maser spots with respect to the black hole's Schwarzschild radius, the brightness of in-going masers, which will \ndepend on the 22 GHz seed photons from the host galaxy, or the degree of gravitational lensing demagnification.  \n\n\n\n\\section{Observations and Data Reduction}\\label{sec:obs}\n\nWe observed the 22.23508 GHz $6_{16}-5_{23}$ ortho water maser line and 20 GHz radio continuum \ntoward 16 candidate inclined maser disks \n(see Section \\ref{sec:candidates} and Table \\ref{tab:obs}) using the \nVLA in A configuration (the highest angular resolution configuration).  Observations \nof program 15A-297 spanned June 19 2015 through September 26 2015 in five sessions.  The \nfifth session occurred during the A-array to D-array reconfiguration.   Each session included visits to \nflux and bandpass calibrators, and observations of each target object were interleaved with nearby \ncomplex gain calibrators with a $\\sim$4 minute switching cadence.  \n\nSpectral line and continuum observations were simultaneous.  The spectral line observations were centered on the redshift of the\nhost galaxy, had 1.1--1.9 km s$^{-1}$ spectral resolution, used 1536 channels to span 128 MHz (1700--2900 km s$^{-1}$), \nand used dual circular polarization\nand 8-bit sampling.  Continuum observations spanned 4 GHz, 18--22 GHz, using 32 spectral windows spanning 128 MHz each \nusing 128 channels in dual circular polarization and 3-bit sampling.  \n\nTable \\ref{tab:obs} lists the details of the observations and the rms noise in the line cubes and continuum maps.  Typical beam sizes were\n80--100 milliarcseconds. Noise was about 3 mJy beam$^{-1}$ in 1.2 km~s$^{-1}$ channels in the spectral line cubes and \nabout 15--20 $\\mu$Jy beam$^{-1}$ in the continuum.  The exception was the $z\\simeq0.66$ water maser J0804+3607 \n\\citep{barvainis2005}\nthat was redshifted to 13.4 GHz, in Ku band, which necessarily had lower angular and spectral resolution and lower rms noise\nin the line but higher rms continuum noise.  In this case, the continuum was centered on 14 GHz and spanned 12--16 GHz.\n\nThe observing session on September 15 2015 had poor 22 GHz weather for A configuration and could not be calibrated or imaged.  \nNGC 3359, NGC 3556, and NGC 3735\nare therefore only listed in Table \\ref{tab:obs} and are not discussed further or included in any analysis of the remaining 13 objects.\n\nAll data reduction and analysis was performed using the Common Astronomy Software Applications package \\citep[CASA;][]{mcmullin2007}.  Calibration and flagging \nused a modified CASA pipeline plus additional manual flagging.  Imaging used Briggs weighting with robustness 0.5.  \nSpectra were extracted from spectral line cubes using a maser-centered beam, and integrated line maps were restricted to \nline-emitting channels.  All spectra use the optical velocity definition in the Barycentric reference frame.  \n\n\n\\floattable\n\\begin{deluxetable}{cccCrCccc}\n\\tablecaption{Journal of Observations \\label{tab:obs}}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{UT Date} &\n\\colhead{Integration} & \n\\multicolumn{3}{c}{Beam\\tablenotemark{a}} & \n\\multicolumn{2}{c}{Line} &\n\\colhead{Continuum} \\\\\n\\cline{4-6}\n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{Angular} & \n\\colhead{PA} & \n\\colhead{Physical} & \n\\colhead{rms} & \n\\colhead{$\\Delta v$} & \n\\colhead{rms} \\\\\n\\colhead{} & \n\\colhead{} & \n\\colhead{(m)} & \n\\colhead{(milliarcsec)} & \n\\colhead{($^\\circ$)} & \n\\colhead{(pc)} & \n\\colhead{(mJy bm$^{-1}$)} &\n\\colhead{(km s$^{-1}$)} &\n\\colhead{($\\mu$Jy bm$^{-1}$)}  \n}\n\\startdata\nNGC 291          & 2015-06-21 & 23.3  &  128\\times84 & 20.4 & 50\\times33 & 2.6 & 1.2 & 17  \\\\\nNGC 520b        & 2015-06-21 & 23.3  &  102\\times89 & 27.8 & 14\\times12 & 3.1 & 1.1 & 20 \\\\\nJ0350$-$0127 & 2015-06-21 & 23.3  &  100\\times87 & 20.5 & 81\\times70 & 3.1 & 1.2 & 15  \\\\\nIC 485              & 2015-09-26 & 23.4  &   84\\times82 & 19.6 & 47\\times46 & 2.2 & 1.2 & 18  \\\\\nJ0804+3607 & 2015-09-26 & 23.3 & 151\\times130 & $-$81.3 & 1065\\times917 & 0.9 & 1.9 & 47   \\\\\nCGCG 120$-$039 & 2015-09-26 & 23.3  & 85\\times81 & 33.7 & 44\\times42 & 2.1 & 1.2 & 20  \\\\\nJ0912+2304    & 2015-09-06 & 23.3  & 92\\times81 & 65.6 & 67\\times59 & 2.9 & 1.2 & 16  \\\\\nJ1011$-$1926 & 2015-09-06 & 23.4  & 173\\times86 & $-$20.8 & 98\\times49 & 5.3 & 1.2 & 19  \\\\\nNGC 3359\\tablenotemark{b}       & 2015-09-15 & 23.3  & 112\\times72 & 48.6 & 9\\times6 & \\nodata & 1.1 & \\nodata \\\\\nNGC 3556\\tablenotemark{b}       & 2015-09-15 & 23.3  & 104\\times72 & 45.0 & 6\\times4 & \\nodata & 1.1 & \\nodata  \\\\\nNGC 3735\\tablenotemark{b}       & 2015-09-15 & 23.3  & 112\\times68 & 36.8 & 21\\times13 & \\nodata & 1.2 & \\nodata  \\\\\nUGC 7016       & 2015-09-06 & 23.3  & 114\\times91 & $-$74.8 & 54\\times43 & 3.8 & 1.2 & 16  \\\\\nNGC 5256       & 2015-06-19 & 23.3  & 96\\times84 & $-$73.8 & 55\\times48 & 3.0 & 1.2 & 17 \\\\\nNGC 5691       & 2015-06-19 & 23.4  & 131\\times85 & 39.3 & 15\\times9 & 3.7 & 1.1 & 17 \\\\\nCGCG 168$-$018 & 2015-06-19 & 23.3 & 86\\times82 & 50.0 & 63\\times60 & 2.7 & 1.2 & 15  \\\\\nJ1939$-$0124 & 2015-06-19 & 25.0 & 96\\times87 & 37.7 & 41\\times38 & 3.0 & 1.2 & 16 \\\\\n\\enddata\n\\tablenotetext{a}{Synthesized beam properties for the spectral line observations.  The continuum maps are slightly different.}\n\\tablenotetext{b}{Objects observed on 2015 September 15 could not be calibrated or imaged due to poor weather.  }\n\\end{deluxetable}\n\n\n\n\\section{Results}\\label{sec:results}\n\nWater masers were detected in 9 out of 13 objects, and 20 GHz continuum emission was detected in 7 out of 13 objects.  \nOnly five objects show both maser and continuum emission, and two objects were detected in neither line nor continuum.  \nFigures \\ref{fig:NGC291}--\\ref{fig:2MASX1939} show a 1\\arcsec\\ square field of view of the first moment maser maps and\ncontinuum contours and they show spectra of the detected objects (in line, continuum, or both).  Table \\ref{tab:positions} lists the \nmaser and continuum centroids based on two-dimensional Gaussian fits.  For the spectral lines, these fits were made\nto the integrated line maps.  The maser emission was universally unresolved, but the continuum emission was formally resolved \nwhen deconvolved from the beam in all but two objects, IC 485 and CGCG 168$-$018.  \n\nTable \\ref{tab:positions} also lists the maser-continuum offsets in angular and physical units. Offsets for four of the five objects \ndetected in both maser and continuum are non-significant with $1\\sigma$ uncertainties ranging from about 1 to 40 pc.  The only object\nshowing a significant offset between the maser and continuum centroids is CGCG 168$-$018, with a $21.6\\pm2.7$ pc offset (Figure \\ref{fig:CGCG168-018}).  \n\nTable \\ref{tab:masers} lists the measured and derived water maser properties:  peak and integrated flux densities, luminosity distance, \nisotropic line luminosity, the range of velocities spanned by the line emission 3$\\sigma$ above the noise, the velocity of peak emission, and \nthe adopted systemic velocity.  For J0804+3607, we list redshifts rather than velocities.  The detected maser velocities are consistent with previous \nobservations, although the masers can be substantially offset from the systemic velocities, which are obtained from optical and HI 21 cm \nlines.  It is unclear whether these velocity offsets are physical (i.e., due to different line-emitting regions genuinely having different velocities, as\nis seen in shock-induced maser emission), due to obscuration (optical vs. radio lines), or due to measurement error, particularly \nin optical redshifts.  We therefore do not rely on the velocity offset between the maser emission and the adopted systemic  \nvelocity as a criterion for assessing the likelihood of a maser arising from an inclined disk.  \nIsotropic maser luminosities range from kilomaser values ($3.05 \\pm 0.26\\ L_\\odot$ in NGC 520b) to the exceptionally luminous, \n$L_{iso} = (1.8 \\pm 0.1) \\times 10^4\\ L_\\odot$ in J0804+3607 (Section \\ref{sec:discussion}).  \n\nTable \\ref{tab:continuum} shows the 20 GHz radio continuum properties of the seven detected objects.  We include the peak flux density, \nthe integrated flux density, the spectral index derived solely from the 18--22 GHz bandpass, and the deconvolved angular size.  \nAmong the six continuum sources with enough signal-to-noise to derive a significant spectral index, four are steep spectrum \n($\\alpha = -1$ to $-2$) and two are flat ($\\alpha \\simeq -0.2 \\pm 0.2$).  One of the latter, CGCG 120$-$039, is only marginally resolved.\\\nIC 485, which does not have a spectral index measurement, but which shows both maser and continuum emission, does not \nhave a resolved continuum.\n\n\n\\floattable\n\\begin{deluxetable}{cllllcc}\n\\tablecaption{Maser and Radio Continuum Positions \\label{tab:positions}}\n\\tablecolumns{7}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\multicolumn{2}{c}{Maser Centroid\\tablenotemark{a}} &\n\\multicolumn{2}{c}{Continuum Centroid\\tablenotemark{b}} &\n\\multicolumn{2}{c}{Offset} \\\\\n\\colhead{} & \n\\colhead{RA} & \n\\colhead{Dec} & \n\\colhead{RA} & \n\\colhead{Dec} & \n\\colhead{Angular} & \n\\colhead{Physical}\\\\\n\\colhead{} & \n\\colhead{(hms)} & \n\\colhead{(dms)} & \n\\colhead{(hms)} & \n\\colhead{(dms)} & \n\\colhead{(milliarcsec)} & \n\\colhead{(pc)} \n}\n\\startdata\nNGC 291          & \\nodata & \\nodata & 00:53:29.9101(11) & $-$08.46.03.740(14) & \\nodata & \\nodata \\\\\nNGC 520b        & 01:24:34.91412(14) & +03.47.29.7864(22) & 01:24:34.9099(28) & +03.47.29.7783(92) & 64(42) & 8.6(5.6) \\\\\nJ0350$-$0127 & 03:50:00.352168(29) & $-$01.27.57.39574(53) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nIC 485              & 08:00:19.752486(74) & +26.42.05.0526(10) & 08:00:19.7515(14) & +26.42.05.050(14) & 13(19) & 7.6(10.5)\\\\\nJ0804+3607 &  08:04:31.01144(40) & +36.07.18.1937(52) & 08:04:31.01138(12) & +36.07.18.19927(91) & 5.6(5.3) & 39.6(37.2) \\\\\nCGCG 120$-$039 & 08:49:14.07078(16) & +23.22.48.9408(20) &  08:49:14.07097(13) & +23.22.48.9346(17) & 6.7(2.7) & 3.5(1.4) \\\\\nJ0912+2304    & 09:12:46.36659(33) & +23.04.27.2421(28) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nJ1011$-$1926 & 10:11:50.56731(17) & $-$19.26.43.9645(59) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nUGC 7016       & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nNGC 5256\\tablenotemark{c} & \\nodata & \\nodata & 13:38:17.79219(15) & +48.16.41.1389(19) & \\nodata & \\nodata \\\\\n                       & \\nodata & \\nodata & 13:38:17.24843(76) & +48.16.32.2095(92) & \\nodata & \\nodata \\\\\nNGC 5691       & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nCGCG 168$-$018 &  16:30:40.90329(19) & +30.29.19.7066(29) &  16:30:40.90134(21) & +30.29.19.7216(20) & 29.3(3.6) & 21.6(2.7) \\\\\nJ1939$-$0124 & 19:39:38.91545(38) & $-$01.24.33.2553(39) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n\\enddata\n\\tablenotetext{a}{All detected maser emission was unresolved.  See Table \\ref{tab:obs} for beam sizes.}\n\\tablenotetext{b}{All detected continuum emission was resolved except for IC 485 and CGCG 168$-$018.  \n  See Table \\ref{tab:continuum} for continuum measurements.}\n\\tablenotetext{c}{NGC 5256 shows two widely-separated continuum components (both are listed).}\n\\tablecomments{Coordinates are epoch J2000, and parenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n \n\n\n\\floattable\n\\begin{deluxetable}{crrrrcrrc}\n\\tablecaption{Water Maser Properties and Redshifts \\label{tab:masers}}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{$S_{\\rm peak}$\\tablenotemark{a}} &\n\\colhead{$S_{\\rm int}$\\tablenotemark{b}} &\n\\colhead{$D_L$} & \n\\colhead{$L_{\\rm iso}$\\tablenotemark{c}} & \n\\colhead{$v_{\\rm range}$\\tablenotemark{d}} & \n\\colhead{$v_{\\rm peak}$} & \n\\colhead{$v_{\\rm sys}$} & \n\\colhead{Ref.\\tablenotemark{e}} \\\\\n\\colhead{} & \n\\colhead{(mJy)} & \n\\colhead{(mJy km s$^{-1}$)} & \n\\colhead{(Mpc)} & \n\\colhead{($L_\\odot$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{} \n}\n\\startdata\nNGC 520b        & 35(3) & 166(14) & 28 & 3.05(26) & 2270--2273 & 2272(1) & 2288(8) & 1 \\\\\nJ0350$-$0127 &  347(4) & 7200(150)  & 180 & 5181(108) & 12336--12393\\tablenotemark{f} & 12369(1) & 12322(18) & 2 \\\\\nIC 485              & 78(2) & 2470(130) & 125 & 868(46) & 8307--8387 & 8356(1) & 8338(10) & 3 \\\\ \nJ0804+3607\\tablenotemark{g} & 9(1) & 80(6) & 3997 & $1.8(1)\\times10^4$ & 0.66038--0.66051 & 0.66045(1) & 0.65654(37)  & 4 \\\\\nCGCG 120$-$039 & 82(2) & 438(46) & 116 &  133(14) & 7559--7565 & 7565(1) & 7684(26) & 5\\\\ \nJ0912+2304    & 16(3) & 252(44) & 164 & 151(26) & 10855--10878 & 10855(1) & 10861(26) &  6\\\\\nJ1011$-$1926 & 44(4) & 624(82) & 123 & 213(28) & 8043--8065 & 8048(1) & 8065(31) & 7 \\\\\nCGCG 168$-$018 & 25(2) & 279(36) & 162 & 163(21) & 11134--11164 & 11140(1) & 11015(29) & 5 \\\\ \nJ1939$-$0124 & 30(2) & 519(80) & 93 & 102(16) & 6170--6206 & 6198(1) & 6226(20)  & 8 \\\\\n\\enddata\n\\tablenotetext{a}{The peak flux density was obtained from a spectrum formed from a single beam centered \non the peak (unresolved) maser emission.}\n\\tablenotetext{b}{The integrated flux density of the water maser complex was \nobtained from fitting a single two-dimensional Gaussian to the velocity-integrated spectral line cube.}\n\\tablenotetext{c}{The isotropic luminosity is computed from the integrated line flux density $S_{int}$ via \n  $L_{iso} = 23.1\\ L_\\odot \\times S_{\\rm int} ({\\rm mJy\\ km\\ s}^{-1}) \\times D_L ({\\rm Gpc})^2\/(1+z)$, where $D_L$ is the luminosity distance and \n  $z$ is the cosmological redshift.}\n\\tablenotetext{d}{$v_{\\rm range}$ is the velocity range over which the water maser exceeds 3$\\sigma$ significance.}\n\\tablenotetext{e}{References for the systemic velocities:\n1 -- \\citet{springob2005}; \n2 -- \\citet{huchra2012};\n3 -- \\citet{RC3};\n4 -- \\citet{hewett2010};\n5 -- \\citet{SDSSDR5};\n6 -- \\citet{SDSSDR6}; \n7 -- \\citet{6dFDR3};\n8 -- \\citet{theureau2007}.}\n\\tablenotetext{f}{There appears to be an additional narrow maser line at 12422 km s$^{-1}$ (Figure \\ref{fig:2MASX0350}).}\n\\tablenotetext{g}{We list redshifts instead of velocities for $v_{\\rm range}$, $v_{\\rm peak}$, and $v_{\\rm sys}$ for J0804+3607.}\n\\tablecomments{Only objects with detected water maser emission are listed (see Table \\ref{tab:obs} for rms noise values for nondetections).\nParenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n\n\\floattable\n\\begin{deluxetable}{ccccCcC}\n\\tablecaption{20 GHz Radio Continuum Properties \\label{tab:continuum}}\n\\tablecolumns{7}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{$S_{\\rm peak}$\\tablenotemark{a}} &\n\\colhead{$S_{\\rm int}$\\tablenotemark{a}} &\n\\colhead{$\\alpha$\\tablenotemark{b}} &\n\\multicolumn{3}{c}{Size\\tablenotemark{a}} \\\\\n\\cline{5-7}\n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{Angular} & \n\\colhead{PA} & \n\\colhead{Physical}\\\\\n\\colhead{} & \n\\colhead{($\\mu$Jy bm$^{-1}$)} & \n\\colhead{(mJy)} & \n\\colhead{} & \n\\colhead{(milliarcsec)} & \n\\colhead{($^\\circ$)} & \n\\colhead{(pc)} \n}\n\\startdata\nNGC 291          & 263(24) & 1.93(20) & $-$2.0(3) & 451(52)\\times146(19) & 51.8(3.4) & 177(20)\\times57(7) \\\\\nNGC 520b        & 272(31) & 4.31(52) & $-$0.23(19) & 775(100)\\times164(24) & 95.6(2.3) & 105(14)\\times22(3) \\\\\nIC 485              &  77(15)  &0.180(48)& \\nodata & < 93\\times88            & $-$69.5 &    < 52\\times49 \\\\\nJ0804+3607 & 4247(83) & 4.71(16) & $-$1.5(3) & 62(12)\\times23(13) & 71(17) & 437(85)\\times162(92) \\\\\nCGCG 120$-$039 & 482(20) & 0.555(39) & $-$0.27(16) & 44(8)\\times23(14) & 133(33) & 23(4)\\times12(7) \\\\\nNGC 5256\\tablenotemark{c} & 633(28) & 1.543(68) & $-$2.0(2) & 127(6)\\times96(8) & 148.7(9.6) & 72(3)\\times55(5)\\\\\n                      & 78(17) & 0.48(11) & \\nodata & 433(23)\\times68(58) & 163.1(3.1) & 247(13)\\times39(33) \\\\\nCGCG 168$-$018 &  267(16) & 0.308(30) & $-$0.95(17) & < 87\\times85 & 77.8 & < 64\\times63\\\\\n\\enddata\n\\tablenotetext{a}{The peak flux density, integrated flux density, and the radio source size and orientation were \nobtained from fitting a single two-dimensional Gaussian to the source image and \ndeconvolving the source from the beam. Upper limits indicate unresolved detections and list the continuum beam parameters.}\n\\tablenotetext{b}{The spectral index $\\alpha$ was measured at the peak of the continuum emission solely from within the 18--22 GHz bandpass\nand follows the convention $S_\\nu \\propto\\nu^\\alpha$. It is not listed for objects with inadequate signal-to-noise in the 20 GHz continuum. }\n\\tablenotetext{c}{NGC 5256 shows two widely-separated continuum components (both are listed).}\n\\tablecomments{Only objects with detected 20 GHz continuum emission are listed (see Table \\ref{tab:obs} for rms noise values for nondetections).  \nParenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.535\\textwidth]{NGC291_overlay.eps}\n\\includegraphics[width=0.46\\textwidth]{NGC291_spec.eps}\n\\caption{Left:  NGC 291 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 393 pc.  \nRight:  water maser nondetection spectrum at the continuum peak with the systemic velocity and its uncertainty indicated by the vertical bars\n\\citep[$5705\\pm4$ km s$^{-1}$;][]{SDSSDR2}.\nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:NGC291}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{NGC520b_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{NGC520b_spec.eps}\n\\includegraphics[width=1.0\\textwidth,trim={0 0 0 0},clip]{NGC520b_overlay_m0_wide.eps}\n\\caption{Top Left:  NGC 520b integrated water maser (image) and 20 GHz radio continuum (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 135 pc.  \nTop Right:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars  (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered\non the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \nBottom: a 5\\arcsec$\\times$2\\arcsec\\ (675 pc $\\times$ 170 pc) field of view to show the full extent of the 20 GHz continuum.\n\\label{fig:NGC520b}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.52\\textwidth]{2MASX0350_overlay.eps}\n\\includegraphics[width=0.475\\textwidth]{2MASX0350_spec.eps}\n\\caption{Left:  J0350$-$0127  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 805 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX0350}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.525\\textwidth]{IC485_overlay.eps}\n\\includegraphics[width=0.47\\textwidth]{IC485_spec.eps}\n\\caption{Left: IC 485 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 3 and 4 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 562 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}). \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:IC485}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{SDSSJ0804_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{SDSSJ0804_spec.eps}\n\\caption{Left:  J0804+3607 integrated water maser (image) and 14 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 7.055 kpc.  \nRight:  water maser spectrum centered on the peak maser emission at $z=0.66045$, plotted in the object rest frame (Table \\ref{tab:masers}).  \n\\label{fig:SDSSJ0804}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{CGCG120-039_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{CGCG120_039_spec.eps}\n\\caption{Left:  CGCG 120$-$039 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 519 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:CGCG120-039}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.52\\textwidth]{2MASX0912_overlay.eps}\n\\includegraphics[width=0.475\\textwidth]{2MASX0912_spec.eps}\n\\caption{Left:  J0912+2304  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 723 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars  (Table \\ref{tab:masers}).  \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX0912}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{2MASX1011_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{2MASX1011_spec.eps}\n\\caption{Left: J1011$-$1926  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 564 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX1011}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.51\\textwidth]{NGC5256_overlay_c1.eps}\n\\includegraphics[width=0.485\\textwidth]{NGC5256_c1_spec.eps}\n\\includegraphics[width=0.51\\textwidth]{NGC5256_overlay_c2.eps}\n\\includegraphics[width=0.485\\textwidth]{NGC5256_c2_spec.eps}\n\\caption{Left:  NGC 5256 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, 16, and 32 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 570 pc.  \nRight:  water maser nondetection spectra at the continuum peaks with the systemic velocity and its uncertainty indicated by the vertical bars\n\\citep[$8353\\pm13$ km s$^{-1}$;][]{RC3}.\nThe spectra are roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \nTwo continuum sources were detected, but no water maser emission was detected toward either continuum source or in the \nlarger field of view.\n\\label{fig:NGC5256}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{CGCG168-018_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{CGCG168_018_spec.eps}\n\\caption{Left:  CGCG 168$-$018 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 737 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:CGCG168-018}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{2MASX1939_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{2MASX1939_spec.eps}\n\\caption{Left:  J1939$-$0124  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 432 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX1939}}\n\\end{figure*}\n\n\n\\section{Analysis} \\label{sec:analysis}\n\nFigure \\ref{fig:offsets} shows the projected physical offset between the water maser line and continuum centroids for the \nfive objects detected in both line and continuum (Table \\ref{tab:positions}).  Only CGCG 168$-$018 shows a significant offset\nof $29.3\\pm3.6$ milliarcseconds or $21.6\\pm2.7$ pc, but based \non its 18--22 GHz  continuum spectral index of $-0.95\\pm0.17$ (Table \\ref{tab:continuum}), \nthe continuum could be jet emission and not the core of the AGN.  \nAll other maser-continuum\noffsets are not significant and are therefore consistent with\ninclined maser disk expectations. \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{offsets.eps}\n\\caption{Maser-continuum offsets for those objects detected in both.  Only CGCG 168$-$018 shows a significant offset.  Error bars are \nplotted symmetrically.\n\\label{fig:offsets}}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nAmong the 13 water maser hosts, we did not detect water masers in four objects:\nNGC 291, UGC 7016, NGC 5256, and NGC 5691.\nThe water maser in NGC 520b is likely associated with star formation and is rejected \nas an inclined maser disk candidate (see \\ref{subsubsec:NGC520b}).  \nObjects with unresolved masers but not detected in the 20 GHz continuum remain ambiguous \n(these are J0350$-$0127, J0912+2304, J1011$-$1926, and J1939$-$0124; note that none of these\nhave other sub-arcsecond radio continuum observations), as does \nCGCG 168$-$018, the only object to show a significant maser-continuum offset (see \\ref{subsubsec:CGCG168-018}).\nIC 485, J0804+3607, and CGCG 120$-$039 remain inclined maser disk candidates:  they show no \nsignificant maser-continuum offset and no high velocity lines (by selection).  While the maser\nemission from IC 485 is broad and multi-component, this type of structure spanning $\\sim$100 km s$^{-1}$ is seen in the systemic masers\nin many disk systems \\citep[e.g.,][]{kuo2011}.  J0808+3607 and CGCG 120$-$039 have narrow maser lines and are particularly \ngood inclined disk candidates.  \n\n\\subsection{Individual Objects}\n \nWe discuss the previous water maser observations, general characteristics, and our results for each \nindividual object below. \n\n\n\\subsubsection{NGC 291}   \nNGC 291 is a barred spiral galaxy with a Seyfert 2 nucleus \\citep{kewley2001,nair2010}.\nThe maser was detected as a narrow $\\sim$60 mJy line in 2006 by GBT program \nGBT06A-009,\\textsuperscript{\\ref{footnote:maser_url}}\nbut we did not detect it (2.6 mJy beam$^{-1}$ rms per 1.2 km s$^{-1}$ \nchannel; Table \\ref{tab:obs} and Figure \\ref{fig:NGC291}) nor did \\citet{kondratko2006} in 2002 (15 mJy rms per 1.3 km s$^{-1}$ \nchannel).  \nWe detect extended 20 GHz continuum with 18--22 GHz spectral index $-2.0\\pm0.3$\n(Table \\ref{tab:continuum}). \n\n\n\\subsubsection{NGC 520b}    \\label{subsubsec:NGC520b}\nNGC 520b is one galaxy in the colliding pair in NGC 520 \\citep[e.g.,][]{stanford1990}.  \n\\citet{castangia2008} detected a maser in 2005, measuring a $\\sim40$ mJy peak with the VLA \nand $L_{iso} \\sim 1\\  L_\\odot$.  Our VLA detection shows a $35\\pm3$ mJy peak and\n$L_{iso} = 3.05\\pm0.26\\ L_\\odot$ (Figure \\ref{fig:NGC520b} and Table \\ref{tab:masers}).  \nOur VLA map shows extended 20 GHz radio continuum emission (Figure \\ref{fig:NGC520b}, bottom) with a \nflat 18--22 GHz spectral index of $-0.23\\pm0.19$ (Table \\ref{tab:continuum}) and  a similar east-west \nmorphology to the \\citet{castangia2008} 14.9 GHz map.  In our map the maser is slightly north of the \\citet{castangia2008}\nmaser position.  There is a significant maser-continuum peak offset, although the single-component centroid \nfits listed in Table \\ref{tab:positions} are consistent because the radio emission is extended.  \\citet{castangia2008}\nfavor a star formation origin for the water maser in NGC 520b, but cannot rule out a low luminosity AGN.  \nThe preponderance of evidence suggests that NGC 520b is a poor candidate for an inclined maser disk.  \n\n\n\\subsubsection{J0350$-$0127}  \nJ0350$-$0127 is an almost otherwise unknown spiral or possibly irregular galaxy included in the 2MASS Redshift Survey \\citep{huchra2012}\nand detected in water maser emission by the GBT programs GBT09-051\\textsuperscript{\\ref{footnote:maser_url}} in 2010 and \nGBT10C-019\\textsuperscript{\\ref{footnote:maser_url}} in 2011.  \nThe peak flux density was $\\sim$350 mJy in 2011, in good agreement with our VLA measurement of $347\\pm4$ mJy (Table \\ref{tab:masers} and \nFigure \\ref{fig:2MASX0350}).  \nThis is a broad and luminous maser ($L_{\\rm iso} = 5181\\pm108\\ L_\\odot$) with no associated 20 GHz continuum, down to an rms noise of 15 $\\mu$Jy beam$^{-1}$\n(Table \\ref{tab:obs}).  \nWithout a continuum detection, the nature of this maser remains ambiguous, although it is almost certainly associated with an AGN.\n\n\n\\subsubsection{IC 485}  \nIC 485 is a spiral galaxy detected in the water maser line in 2006 in GBT program\nGBT06C-035.\\textsuperscript{\\ref{footnote:maser_url}}  The maser has a \n$\\sim$80 mJy peak and shows a broad profile.  \\citet{zhu2011} lists this object as \na maser nondetection.  Our VLA observations show a \nbroad, multi-component maser with a similar peak ($78\\pm2$ mJy) \nand a high luminosity, $L_{iso} = 868\\pm46\\ L_\\odot$ (Figure \\ref{fig:IC485} and Table \\ref{tab:masers}).  The 20 GHz continuum \nis detected but unresolved and faint  ($77\\pm15$ $\\mu$Jy beam$^{-1}$ peak flux density; Table \\ref{tab:continuum}).\nThis galaxy was also detected at 1.4 GHz (4.4 mJy), and its dominant radio energy source was classified as star formation \nby \\citet{condon2002}.  This classification does not exclude the presence of an AGN: \\citet{liu2011} classify the optical nucleus \nof IC 485 as a LINER.  \nThe maser-continuum offset is not significant --- less than 10.5 pc (1$\\sigma$; Table \\ref{tab:positions}) --- so this maser remains an inclined disk candidate.\n\n\n\\subsubsection{J0804+3607}   \nThis is a type 2 quasar showing luminous water maser emission at $z\\simeq0.66$ \\citep{zakamska2003,barvainis2005}.  \nThe isotropic maser luminosity was $2.31\\pm0.46\\times10^4\\ L_\\odot$ in 2005 \n\\citep[][error from the quoted 20\\% calibration uncertainty]{barvainis2005} and $1.8\\pm0.1\\times10^4\\ L_\\odot$ in 2015 (this work),\nconsistent with no variation.   The maser and 20 GHz continuum \nemission are coincident to within 40 pc (1$\\sigma$; Table \\ref{tab:positions} and Figure \\ref{fig:SDSSJ0804}), \nwhich is less precise than the other objects due to the much larger distance and lower observing frequency.  The 12--16 GHz spectral \nindex is $-1.5\\pm0.3$ (Table \\ref{tab:continuum}).  This object remains an inclined maser disk candidate. \n\n\n\\subsubsection{CGCG 120$-$039}\nThe water maser in this little-studied galaxy was detected in 2013 in GBT program\n GBT13A-236.\\textsuperscript{\\ref{footnote:maser_url}}\nIt had a peak flux density of $\\sim$210 mJy, was blueshifted from the systemic velocity, and \nshowed several components.  \nWe detect the maser --- now at $82\\pm2$ mJy peak, but still showing multiple components (Figure \\ref{fig:CGCG120-039})\n--- and the 20 GHz continuum at $0.56\\pm0.04$ mJy, which are spatially coincident to within a remarkably small $3.5\\pm1.4$ pc \n(Table \\ref{tab:positions}).  \nThe 18--22 GHz spectral index is flat:  $\\alpha = -0.27\\pm0.16$ (Table \\ref{tab:continuum}).\nThis object is a good inclined maser disk candidate.  \n\n\\subsubsection{J0912+2304}  \nThe water maser in the galaxy J0912+2304 was detected in 2008 by GBT program GBT07A-034,\\textsuperscript{\\ref{footnote:maser_url}}\nshowing a $\\sim$30 mJy peak.  \\citet{zhu2011} list this as a water maser nondetection.  The VLA observations show a $16\\pm3$ mJy \npeak, but no 20 GHz continuum down to an rms noise of 16 $\\mu$Jy beam$^{-1}$ (Figure \\ref{fig:2MASX0912} and Tables \\ref{tab:obs} and\n\\ref{tab:masers}).  The provenance of the maser remains ambiguous.\n\n\\subsubsection{J1011$-$1926}  \nThe maser in this almost unknown galaxy was detected in GBT program GBT07A-066\\textsuperscript{\\ref{footnote:maser_url}}\nin 2008.  \\citep{zhu2011} list this object as a water maser nondetection.  \nThe GBT detection shows a broad line with a $\\sim$80 mJy peak.  The VLA detection shows a broad multi-component maser with a \n$44\\pm4$ mJy peak in good agreement with the systemic velocity (Figure \\ref{fig:2MASX1011} and Table \\ref{tab:masers}).  \nWe do not detect the 20 GHz continuum.  The nature of this maser therefore remains ambiguous.\n\n\\subsubsection{UGC 7016}   \nThe water maser in this barred spiral \\citep{RC3} was \ndetected in 2007 by GBT program GBT07A-034,\\textsuperscript{\\ref{footnote:maser_url}} but \\citet{zhu2011} list this as a nondetection.\nThe GBT detection spectrum had a peak flux density of $\\sim$55 mJy, but we did not detect the maser or the 20 GHz continuum emission.\nVLA rms noise values were 3.8 mJy beam$^{-1}$ per 1.2 km s$^{-1}$ channel in the spectral line cube and \n16 $\\mu$Jy beam$^{-1}$ in the continuum map (Table \\ref{tab:obs}).\n\n\n\\subsubsection{NGC 5256}    \n\\citet{braatz2004} discovered the water maser in this merging luminous infrared galaxy in 2003.  \nIt had a peak flux density of 99 mJy ($L_{iso} = 30\\ L_\\odot$)  and was redshifted from the systemic\nvelocity by $\\sim$300 km s$^{-1}$.  \\citet{braatz2004} claim that all maser emission originates \nfrom the southern nucleus \\citep[a Sey 2 nucleus according to, e.g.,][]{mazzarella1993}.\nWe detect two 20 GHz continuum sources that are consistent with the positions of the two nuclei \n\\citep[Tables \\ref{tab:positions} and \\ref{tab:continuum};][]{brown2014},\nbut neither location (nor the larger region that includes the overlap region between the two nuclei)\nshows maser emission down to an rms noise of 3.0 mJy beam$^{-1}$ in 1.2 km s$^{-1}$ channels\n(Table \\ref{tab:obs} and Figure \\ref{fig:NGC5256}).\nThe spectral index of the northern nuclear continuum is $-2.0\\pm0.2$ in the 18--22 GHz band  (Table \\ref{tab:continuum}).  \n\n\n\\subsubsection{NGC 5691}   \nThe water maser in NGC 5691, a barred non-Seyfert spiral galaxy \\citep{mulchaey1997}, was detected in \n2009 by GBT program AGBT08C-035\\textsuperscript{\\ref{footnote:maser_url}}\nwith a $\\sim$45 mJy peak flux density.  It was listed by \\citet{zhu2011} as a maser nondetection.  \nWe did not detect the maser or any 20 GHz continuum in this galaxy, with rms noise levels of 3.7 mJy beam$^{-1}$ per 1.1 km s$^{-1}$ channel\nand 17 $\\mu$Jy beam$^{-1}$, respectively (Table \\ref{tab:obs}).\n\n\\subsubsection{CGCG 168$-$018}  \\label{subsubsec:CGCG168-018}\nCGCG 168$-$018 is a little-studied galaxy classified as an AGN by \\citet{schawinski2010} and\nlisted as a water maser nondetection by \\citet{zhu2011}.\nWater maser emission was detected by GBT program GBT07A-066\\textsuperscript{\\ref{footnote:maser_url}}\nin 2008.  The GBT water maser spectrum shows \na $\\sim$50 mJy peak, detected by this work at $25\\pm2$ mJy (Table \\ref{tab:masers}).  \nThe $0.31\\pm0.03$ mJy continuum shows a 18-22 GHz spectral index of $-0.95\\pm0.17$ (Table \\ref{tab:continuum}).   \nThe maser emission and 20 GHz continuum\nare unresolved but show a significant relative offset of $21.6\\pm2.7$ pc (Table \\ref{tab:positions} and Figure \\ref{fig:CGCG168-018}).  \nThis is the only object in the sample\nthat shows a significant offset between the maser and continuum emission (Figure \\ref{fig:offsets}).\nWhile the offset suggests that the maser emission is not deflected from an inclined maser disk, the continuum \nspectral index suggests that the continuum may arise from a jet, and the radio core is not detected.  The provenance \nof the maser therefore remains ambiguous.\n\n\\subsubsection{J1939$-$0124}   \n\\citet{greenhill2003} detected the water maser in this spiral galaxy hosting a Sey 2 nucleus in 2002 using the Tidbinbilla antenna.  \nThe maser showed a peak flux density of $\\sim$28 mJy, and \\citet{henkel2005} report $L_{iso} \\simeq 160\\ L_\\odot$. \nNo 22 GHz continuum was detected by \\citet{greenhill2003} using the VLA ($< 2.8$ mJy).  \nWe likewise detect no continuum, with rms noise of 16 $\\mu$Jy beam$^{-1}$ (Table \\ref{tab:obs}), but we do detect the \nmaser emission with peak $30\\pm2$ mJy ($L_{iso} = 102\\pm16 \\ L_\\odot$) although with a substantially different maser profile \n(Figure \\ref{fig:2MASX1939} and Table \\ref{tab:masers}).  \nThe 6 and 20 cm continua were detected at $5.4\\pm0.4$ mJy  and $15.5\\pm1.0$ mJy, respectively, by \\citet{vader1993}, and the \n6 cm continuum position agrees with the maser position to within $\\sim$1\\arcsec ($\\sim$430 pc).  \nThe nature of this maser remains ambiguous.  \n\n\n\n\\section{Conclusions}\n\nThis paper presents a physical mechanism that may enable detection of inclined water maser disks orbiting massive black holes\nvia the lensing\/deflection of in-going systemic masers.  The observational signature of an inclined disk \nis a maser line or line complex with limited Doppler extent that appears to arise at the location of the black \nhole, as identified by its radio continuum core.  With enough angular resolution, it may be possible to \nmeasure the black hole mass if the maser emission forms a lensing arc or Einstein ring, but the mass\nprecision will be limited by one's ability to measure or estimate the size of the maser-emitting portion of the disk.  \n\nWe suggest that if inclined maser disks can be detected at all, then they have probably already been detected\nin single-dish surveys but discarded for interferometric follow-up because they did not show high-velocity lines.\nWe present original 0.07--0.17\\arcsec\\ (4--100 pc) resolution VLA observations of inclined maser disk candidates with the goal\nof identifying systems where the maser emission is unresolved and is coincident with the 20 GHz continuum \nemission.  \n\nOf the 16 masers observed with the VLA, we obtained useful data for 13, and among these, five were \ndetected in both 22 GHz maser line emission and in 20 GHz continuum.  Of these five, one maser is most likely\nassociated with star formation (NGC 520b), and one shows a significant spatial offset between the maser emission \nand the continuum (CGCG 168$-$018, but it could still host an inclined maser disk --- this case is ambiguous).  \nThree objects are good inclined maser disk candidates that merit further study with VLBI:  \nIC 485, J0804+3607, and CGCG 120$-$039.\nFive maser hosts remain ambiguous, based either on non-detected or offset continua:\nJ0350$-$0127,  J0912+2304, J1011$-$1926, CGCG 168$-$018, and J1939$-$0124.  \n\nMore straightforward methods for measuring black hole masses from molecular lines may be in the offing.  For example,\n\\citet{davis2013} and \\citet{barth2016a,barth2016b} have used carbon monoxide kinematics in thin disks that approach or are\nwithin the black hole gravitational sphere of influence to obtain black hole mass measurements.  \\citet{barth2016a}, in particular, \ndemonstrate the ability of ALMA to measure black hole masses with $\\sim$10\\% uncertainty.   Although they lack the \nintrinsic brightness of masers that enables VLBI mapping, \nthermal molecular lines have the advantage of being observable at any disk inclination.\n\n\n\\acknowledgments\n\nWe thank A. Hamilton and M. Eracleous for helpful discussion and B. Butler for assistance with \nmetadata repair of bespoke observing configurations.  \nWe also thank the anonymous referee for important feedback.\nThis research has made use of NASA's Astrophysics Data System Bibliographic Services and\nthe NASA\/IPAC Extragalactic Database (NED),\nwhich is operated by the Jet Propulsion Laboratory, California Institute of Technology,\nunder contract with the National Aeronautics and Space Administration.\n\n \\facility{VLA}\n\n\\software{CASA}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Compact representation of cuts}\n\\label{sec:compact}\n\\newcommand{{\\mathcal F}}{{\\mathcal F}}\n\nThis part serves as a ``communication language'' for various components in our data structure.\nSince a cut can have up to $\\Omega(m)$ edges, the data structure cannot afford to describe it explicitly. We will use a compact representation of cuts~\\cite{ChekuriQ17}, which allows us to describe any $(1+\\epsilon)$-approximate solution in a given weighted graph using $\\tilde{O}(1)$ bits; notice that, in the MWU framework, we only care about (punishing) near-optimal solutions, so it is sufficient for us that we are able to concisely describe such cuts.\n\n\n\nFormally, we say that a family ${\\mathcal F}$ of  subsets of edges is $\\epsilon$-canonical for $(G,\\textbf{w})$ if (i) $|{\\mathcal F}| \\leq \\widetilde{O}(|E|)$, (ii) any $(1+\\epsilon)$-approximate minimum cut of $(G,\\textbf{w})$ is a disjoint union of at most $\\tilde{O}(1)$ sets in ${\\mathcal F}$, (iii) any set $S \\in {\\mathcal F}$ can be described concisely by $\\tilde{O}(1)$ bits, and (iv) every edge in the graph belongs to $\\tilde{O}(1)$ sets in ${\\mathcal F}$.\nIt follows that any $(1+\\epsilon)$-approximate cut admits a short description.\nDenote by $[[S]]$ a short description of cut $S \\in {\\mathcal F}$, and for each $(1+\\epsilon)$ approximate cut $C$, $[[C]]$ a short description of $C$.\n\n\n\n\n\\begin{lemma}[implicit in \\cite{ChekuriQ17}] \\label{lem:canonical cuts}\nThere exists a randomized data structure that, on input $(G,\\textbf{w})$, can be initialized in near-linear time,  (w.h.p) constructs an $\\epsilon$-canonical family ${\\mathcal F} \\subseteq 2^{E(G)}$, and handles the following queries:\n\\begin{itemize}\n\n    \\item Given a description $[[C]]$ of a $(1+\\epsilon)$-approximate cut, output a list of $\\tilde{O}(1)$ subsets in ${\\mathcal F}$ such that $C$ is a disjoint union of those subsets in $\\tilde{O}(1)$ time.\n\n    %\n\n    \\item Given a description of $[[S]]$, $S \\in {\\mathcal F}$, output a list of edges in $S$ in $\\widetilde{O}(|S|)$ time.\n\n    %\n     %\n   \n\\end{itemize}\n\\end{lemma}\n\n\n\n\\subsection{Range Cut-listing Data Structure}\n\\label{sec:cut listing}\n\n\n\nThe cut listing data structure is encapsulated in the following theorem.\n\n\\begin{theorem} [Range Cut-listing Data Structure \\cite{ChekuriQ17}] \\label{thm:cut listing}\nThe cut-listing data structure, denoted by $\\mathcal{D}$, maintains dynamically changing weighted graph $(G,\\widehat{\\textbf{w}})$ and supports the following operations.\n\\begin{itemize}\n\\item $\\mathcal{D}$.\\textsc{Init}$(G,\\vecw^{\\operatorname{init}},\\lambda, \\epsilon)$ where $G$ is a graph, $\\widehat{\\textbf{w}}$ is an initial weight function, and ${\\sf mincut}_{\\widehat{\\textbf{w}}} \\geq \\lambda$: initialize the data structure and  the weight $\\widehat{\\textbf{w}} \\leftarrow \\vecw^{\\operatorname{init}}$ in $\\tilde{O}(m)$ time.\n\n\n     \\item $\\mathcal{D}$.\\textsc{FindCut}$():$ output either a short description of a $(1+O(\\epsilon))$-approximate mincut $[[C]]$ or $\\emptyset$ (when ${\\sf mincut}_{\\widehat{\\textbf{w}}} > (1+\\epsilon) \\lambda$). The operation takes amortized $\\tilde{O}(1)$ time.\n\n    \\item $\\mathcal{D}$.\\textsc{Increment$(\\Delta)$} where $\\Delta = \\{(e,\\delta_e)\\}$ is the set of increments (defined by a pair of an edge $e \\in E$ and a value $\\delta_e \\in \\mathbb{R}_{\\geq 0}$): For each $(e,\\delta_e) \\in \\Delta$, $\\widehat{\\textbf{w}}(e) \\gets \\widehat{\\textbf{w}}(e) + \\delta_e$.   The operation takes $\\tilde{O}(|\\Delta|)$ time (note that $|\\Delta|$ corresponds to the number of increments).\n\\end{itemize}\n\\end{theorem}\n\n\nAs outlined earlier, the cut listing data structure will be invoked with $\\widehat{\\textbf{w}} = \\textbf{w}_{\\rho}$.\n\n\n\n\n\n\n\\subsection{Truncated Lazy MWU Increment}\n\\label{sec:lazy weight}\n\nThe data structure is formally summarized by the definition below.\n\n\n\n\n\n\n\n\n\\begin{definition}[Truncated Lazy MWU Increment]\n A truncated lazy MWU increment denoted by $\\mathcal{L}$ maintains the approximate weight function $\\textbf{w}$ explicitly, and exact weight $\\vecw^{\\operatorname{mwu}}$ implicitly and  supports the following operations:\\footnote{This is implicit in the sense that $w$ is divided into parts and they are internally stored in different memory segments. Whenever needed, the real weight can be constructed from the memory content in near-linear time.}\n\n \\begin{itemize}\n     \\item   $\\mathcal{L}.\\textsc{Init}(G,\\vecw^{\\operatorname{init}},\\rho)$ where $G$ is a graph, $\\vecw^{\\operatorname{init}}$ is the initial weight function, $\\rho \\in \\mathbb{R}_{>0}$: Intialize the data structure, and set $\\textbf{w} \\leftarrow \\vecw^{\\operatorname{init}}$.\n\n     \\item $\\mathcal{L}.\\textsc{Punish}([[C]])$ where $C$ is a cut: Internally punish the free cut $(C,F)$ for some $F$ (to be made precise later) and output a list of increment $\\Delta = \\{(e,\\delta_e)\\}$ so that for each $e \\in E$, $\\vecw^{\\operatorname{init}}(e)$ plus the total increment over $e$ is $\\textbf{w}_{\\rho}(e)$.\n\n     \\item  $\\mathcal{L}.\\textsc{Flush}()$: Return the exact weight $\\vecw^{\\operatorname{mwu}}$.\n \\end{itemize}\n\\end{definition}\n\nRemark that the output list of increments returned by {\\sc Punish} is mainly for the purpose of syncing with the cut listing data structure (so it aims at maintaining $\\textbf{w}_{\\rho}$ instead of $\\textbf{w}$).  Also, in the {\\sc Punish} operation, the data structure must compute the set $F \\subseteq C$ of free edges efficiently (these are the edges whose weights would not be increased).\n This is one of the reasons for which we cannot use the lazy update data structure in~\\cite{ChekuriQ17} as a blackbox. Section~\\ref{sec:dataStructures} will be devoted to proving the following theorem.\n\n\\begin{theorem} \\label{thm:tlmi}\nThere exists a lazy MWU increment with the following time complexity: (i) init operation takes $\\tilde{O}(m)$ time, (ii)  \\textsc{Punish} takes $\\tilde{O}(K) +\\widetilde{O}\\left(\\sum_e \\log \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)$ time in total where $K$ is the number of calls to $\\textsc{Punish}$ and outputs at most $\\widetilde{O}\\left(\\sum_e \\log \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)$ increments, and (iii) flush takes $\\tilde{O}(m)$ time.   Moreover, the Invariant~\\ref{inv:new approx weight} is maintained throughout the execution.\n\\end{theorem}\n\n\n\n %\n\n  %\n\n\n\\subsection{A Fast Range Punisher for Normalized Free Cut Problem}\n\n\n\n\\label{sec:implementationPunisher}\n\n\\newcommand{\\mathcal{T}}{\\mathcal{T}}\n\\newcommand{\\mathcal{D}}{\\mathcal{D}}\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\n\n\n\nNow we have all necessary ingredients to prove~\\Cref{lem:fast range punisher}. The algorithm is very simple and described in~\\Cref{alg:fast range punisher}. We initialize the cut-listing data structure ${\\mathcal D}$ so that it maintains the truncated weight $\\textbf{w}_{\\rho}$ and the lazy weight data structure $\\mathcal{L}$.  \nWe iteratively use ${\\mathcal D}$ to find a cheap cut in $(G,\\textbf{w}_{\\rho})$ until no such cut exists. Due to our mapping theorem, such a cut found can be used for our problem, and the data structure $\\mathcal{L}$ is responsible for punishing the weights (Line 8) and returns the list of edges to be updated (this is for the cut-listing ${\\mathcal D}$ to maintain its weight function $\\textbf{w}_{\\rho}$).\n\n\n\n\\subsubsection*{Algorithm}\n\\begin{algorithm}[H]\n\\KwIn{$G,\\vecw^{\\operatorname{init}},\\lambda , \\epsilon$ such that ${\\sf OPT}_{\\vecw^{\\operatorname{init}}}\\geq \\lambda$.}\n\\KwOut{a correct weight function $\\textbf{w} = \\vecw^{\\operatorname{mwu}}$ such that ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$.}\n\\BlankLine\n$\\textbf{w} \\gets \\vecw^{\\operatorname{init}}$ and $\\rho \\gets (1+\\epsilon)\\lambda$\\;\nLet $\\textbf{w}_{\\rho}$ be the truncated weight function of $\\textbf{w}$.\\;\n\\lIf{ ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$}{ \\Return{$\\textbf{w}$.}} \\label{line:early return}\n  Let $\\mathcal{D}$ and $\\mathcal{L}$ be cut listing data structure, and truncated lazy MWU increment. \\;\n  $\\mathcal{D}.\\textsc{Init}(G,\\textbf{w}_{\\rho}, k\\rho\/(1+\\epsilon), \\epsilon)$\\;\n  $\\mathcal{L}.\\textsc{Init}(G,\\textbf{w},\\rho,\\epsilon)$\\;\n  \\While{$\\mathcal{D}.\\textsc{FindCut}()$ \\normalfont{returns} $[[C]]$}\n  {\n    $\\Delta \\gets \\mathcal{L}.\\textsc{Punish}([[C]])$\\;\n    $\\mathcal{D}.\\textsc{Increment}(\\Delta)$\\;\n  }\n  $\\textbf{w} \\gets \\mathcal{L}.\\textsc{Flush}()$\\;\n\n  \\Return{$\\textbf{w}$.} \\label{line:late return}\n\\caption{\\textsc{FastRangePunisher}($G,\\textbf{w},\\lambda$)}\n\\label{alg:fast range punisher}\n\\end{algorithm}\n\n\\subsubsection*{Analysis}\nBy input assumption, we have ${\\sf OPT}_w \\geq \\lambda$. If $\\textbf{w}$ is returned at line \\ref{line:early return}, then ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$. By \\Cref{thm:approx mapping}(\\ref{item:approx mapping1}), ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq {\\sf OPT}_{\\textbf{w}} \\geq \\rho= (1+\\epsilon)\\lambda$, and we are done (since minimum cut can be computed in near-linear time). Now, we assume that $\\textbf{w}$ is returned at the last line. The following three claims imply \\Cref{lem:fast range punisher}.\n\n\\begin{claim} For every cut $[[C]]$ returned by the range cut listing data structure during the execution of \\Cref{alg:fast range punisher}, we have that $(C,H_{\\textbf{w},\\rho} \\cap C)$ is a $(1+O(\\epsilon))$-approximation to ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}}$ at the time $[[C]]$ is returned.\n\\end{claim}\n\nWe remark that it is important that our cut punished must be approximately optimal w.r.t. the actual MWU weight.\n\n\\begin{proof}\n  By definition of $\\mathcal{L}.\\textsc{Flush}()$ operation, we always have that the exact weight function and approximate weight function are identical at the beginning of the loop.  By definition of $\\mathcal{L}.\\textsc{Punish}([[C]])$, the total increment plus the initial weight at the beginning of the loop for every edge $e$ is $\\textbf{w}_{\\rho}(e)$ and \\Cref{inv:new approx weight} holds. Therefore, by definition of $\\mathcal{D}.\\textsc{Increment}(\\Delta)$, the range cut-listing data structure maintains the weight function $\\textbf{w}_{\\rho}$ internally. We now bound the approximation of each cut $[[C]]$ that $\\mathcal{D}.\\textsc{FindCut}()$ returned. Let $F =  H_{\\textbf{w},\\rho}\\cap C$. By definition of $\\textsc{FindCut}()$, we have that $\\textbf{w}_{\\rho}(C) < k\\rho$. By \\Cref{thm:approx mapping}(\\ref{item:approx mapping2}), ${\\sf val}_{\\textbf{w}}(C, F) < (1+\\epsilon)\\lambda$. By \\Cref{inv:new approx weight}, we have that ${\\sf val}_{\\vecw^{\\operatorname{mwu}}}(C, \\tilde F) < (1+O(\\epsilon))\\lambda$.  Since ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq {\\sf OPT}_{w^{\\operatorname{init}}} \\geq \\lambda$, we have $(C, F)$ is a $(1+O(\\epsilon))$-approximation to ${\\sf OPT}_{w}$.\n\\end{proof}\n\n\\begin{claim}\nAt the end of \\Cref{alg:fast range punisher},  we have ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq (1+\\epsilon)\\lambda.$\n\\end{claim}\n\\begin{proof}\n  Consider the time when $\\mathcal{D}.\\textsc{FindCut}()$ outputs $\\emptyset$. The fact that this procedure terminates means that ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$. Therefore,  \\Cref{thm:approx mapping}(\\ref{item:approx mapping1})  implies that  ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$. Let $(C^*, F^*)$ be an optimal normalized free cut with respect to $\\vecw^{\\operatorname{mwu}}$. We have\n\n  \\begin{align*}\n    {\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} &= {\\sf val}_{\\vecw^{\\operatorname{mwu}}}(C^*,F^*) \\\\\n    &\\geq {\\sf val}_{\\textbf{w}}(C^*,F^*) \\\\\n    & \\geq {\\sf OPT}_{\\textbf{w}} \\\\\n    &\\geq (1+\\epsilon)\\lambda\n  \\end{align*}\n\n  \\noindent where the first inequality follows from \\Cref{inv:new approx weight}.\n  %\n\\end{proof}\n\n\n\\begin{claim}\n  \\Cref{alg:fast range punisher} terminates in $\\tilde{O}(m+K+\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)}))$ time where $K$ is the number of \\textsc{Punish} operations.\n\\end{claim}\n\n\\begin{proof}\nWe first bound the running time due to truncated lazy MWU increment. By \\Cref{thm:tlmi}, the total running time due to $\\mathcal{L}$ (i.e.,  $\\mathcal{L}.\\textsc{Init}, \\mathcal{L}.\\textsc{Punish}, \\mathcal{L}.\\textsc{Flush}$)  is  $\\tilde{O}(m+K+\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)}))$ time where $K$ is the number of \\textsc{Punish} operations.  We bound the running time due to cut-listing data structure. Observe that the number of cuts listed equals the number of calls of \\textsc{Punish} operations, and the total number of edge increments in $\\mathcal{D}$ is $\\widetilde{O}\\left(\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)})\\right)$. By \\Cref{thm:cut listing}, the total running time due to $\\mathcal{D}$ (i.e, $\\mathcal{D}.\\textsc{Init}, \\mathcal{D}.\\textsc{FindCut}(), \\mathcal{D}.\\textsc{Increment}(\\Delta)$) is as desired.\n\\end{proof}\n\n\n\\subsection{Additive Accuracy}\n\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\n\nNotice that, at any time, we always have $\\vecw^{\\operatorname{mwu}}(e)= \\frac{1}{c(e)} \\cdot\n\\exp \\left( \\vecv^{\\operatorname{mwu}}(e) s(e)\\right)$ for some positive real numbers $\\vecv^{\\operatorname{mwu}}(e)$\nand $s(e) = \\frac{\\epsilon}{c(e)}$.\nFor the true vector $\\vecv^{\\operatorname{mwu}}$, the update rule for $(C, F)$ becomes the\nfollowing: $\\vecv^{\\operatorname{mwu}}(e) \\leftarrow \\vecv^{\\operatorname{mwu}}(e) + c_{\\min}$ for all $e \\in C\n\\setminus F$. This update causes all\nedges in $C \\setminus F$ to increase their $\\vecv^{\\operatorname{mwu}}(e)$ by the same amount\nof $c_{\\min}$. By \\Cref{thm:approx mapping}(2), it is enough to use\n$F$ to be always $H_{\\rho, \\textbf{w}} \\cap C$, i.e., the set of heavy edges with respect to $\\textbf{w}$ inside\n$C$.  From now on, we always use $H_{\\textbf{w},\\rho} \\cap C$ as a free\nedge set whenever we punish $C$.  \n\nInstead of maintaining the approximate vector $\\textbf{w}$ for the real\nvector $\\vecw^{\\operatorname{mwu}}$, we instead work\nwith the additive form of the approximate vector $\\textbf{v}$ for the real vector $\\vecv^{\\operatorname{mwu}}$,\nand we bound the additive error:\n\\begin{align} \\label{eq:additive error small}\n  \\forall e \\in E, \\vecv^{\\operatorname{mwu}}(e) - \\eta\/s(e) \\leq \\textbf{v}(e) \\leq \\vecv^{\\operatorname{mwu}}(e) \n\\end{align}\nNext, we show that it is enough to work on $\\vecv^{\\operatorname{mwu}}$ with additive errors. \n\\begin{proposition}\n  If \\Cref{eq:additive error small} holds, then $\\forall e \\in E, \\vecw^{\\operatorname{mwu}}(e)\n  (1-\\eta) \\leq \\textbf{w}(e) \\leq \\vecw^{\\operatorname{mwu}}(e)$. \n\\end{proposition}\n\\begin{proof}\n  Fix an arbitrary edge $e \\in E$, we have $\\textbf{w}(e) \\leq\n  \\vecw^{\\operatorname{mwu}}(e)$. Moreover,\n$$ \\textbf{w}(e) \\geq \\frac{1}{c(e)} \\exp( (\\vecv^{\\operatorname{mwu}}(e) - \\eta\/s(e)) s(e)) =\n\\frac{1}{c(e)} \\cdot \\frac{\\exp(\\vecv^{\\operatorname{mwu}}(e)s(e))}{\\exp(\\eta)} \\geq (1-\\eta) \\vecw^{\\operatorname{mwu}}(e).$$\n\\end{proof}\n\\subsection{Local Bookkeeping} \\label{sec:local book}\n\n\\newcommand{\\operatorname{q}}{\\operatorname{q}}\n\\newcommand{\\operatorname{ref}}{\\operatorname{ref}}\n\\newcommand{\\operatorname{last}}{\\operatorname{last}}\n\\newcommand{\\operatorname{diff}}{\\operatorname{diff}}\n\\newcommand{\\operatorname{priority}}{\\operatorname{priority}}\n\nWe describe the set of variables to maintain in order to support\n$\\textsc{Punish}$ operation efficiently. Let $\\mathcal{F}$ be a\n$\\epsilon$-canonical family of subsets of edges (as defined in\n\\Cref{lem:canonical cuts}). We call each subset of edges in $\\mathcal{F}$ as a canonical cut.  Let $\\bar E = E \\setminus H_{\\textbf{w}, \\rho}\n$ be the set of non-heavy edges where $H_{\\rho, \\textbf{w}} = \\{e \\in E\\colon \\textbf{w}(e) \\geq \\rho\\}$ is the set of heavy edges. We define a bipartite graph $\\mathcal{B} =\n(\\mathcal{F}, \\bar E, E_{\\mathcal{B}})$ where the first vertex partition  is the set\nof canonical cuts $\\mathcal{F}$ , the\nsecond vertex partition is $\\bar E$, and for each $S \\in \\mathcal{F}$ and for\neach $e \\in \\bar E$, we add an edge  $(S,e)$ to $E_{\\mathcal{B}}$ if and\nonly if $e \\in S$.  Let $\\operatorname{q}(\\mathcal{B}) =$ the maximum degree of\nnodes in $\\bar E$ in graph $\\mathcal{B}$.  Since $\\mathcal{F}$ is\n$\\epsilon$-canonical, $\\operatorname{q}(\\mathcal{B}) = \\tilde{O}(1)$. By \\Cref{lem:canonical\n  cuts}, given a description\n$[[C]]$ of 1 or 2$-$repsecting cut, we can compute a list of at most\n$\\tilde{O}(1)$ canonical cuts in $\\mathcal{F}$ in $\\tilde{O}(1)$ time.\n\nWe maintain the following variables:\n\\begin{enumerate}\n\\item For each canonical cut $S \\in \\mathcal{F}$,\n  \\begin{enumerate} \n     \\item we have a non-negative\n  real number $\\operatorname{ref}(S)$ representing the reference point for\n  the total increase in $S$ so far.  %\n  \\item Also, we create a min priority queue\n  $Q_{S}$ containing the set of neighbors $N_{\\mathcal{B}}(S)$ (which is the\n  set of edges in $\\bar E$ that $S$ contains). %\n %\n  \\item Also, we define $c_{\\mathcal{B}}(S) = \\min_{e \\in N_{\\mathcal{B}}(S)}c(e)$ for the purpose of\n  computing $c_{\\min}$  which is the minimum capacity $c(e)$ for all\n  edge $e$ in the cut (excluding  heavy edges) that we want to punish. %\n  \\end{enumerate}\n\\item For each edge $(S,e) \\in E_{\\mathcal{B}}$, we  have a number\n  $\\operatorname{last}(S,e)$ representing the last update point for $e$ in $S$.\n \\item For each edge $e \\in E$, we maintain $\\textbf{v}(e)$.\n\\end{enumerate}\n\nFor each edge $(S,e) \\in E_{\\mathcal{B}}$, we define $\\operatorname{diff}(S,e) =\n\\operatorname{ref}(S) - \\operatorname{last}(S,e) \\geq 0$.  This difference represents the total\nslack from the exact weight of $e$ on $S$ (we will ensure that the\nslack is non-negative by being ``lazy''). When summing over all canonical cuts that contains $e$, we\nensure that $\\sum_{S \\ni  e}\\operatorname{diff}(S,e) =\n\\vecv^{\\operatorname{mwu}}(e) - \\textbf{v}(e)$.  More formally, we maintain the following\ninvariants throughout the execution of the truncated lazy increment.\n\n\\begin{invariant}  \\label{inv:lazy update}Let $\\eta' = \\eta\/\\operatorname{q}({\\mathcal B})$. %\n  \\begin{enumerate}[(a)]\n   \\item \\label{item:total diff not far} for all $e \\in \\bar E$, $ \\frac{\\eta}{s(e)}  \\geq \\sum_{S:\n       (S,e) \\in E_{\\mathcal{B}}}\\operatorname{diff}(S,e) = \\vecv^{\\operatorname{mwu}}(e) - \\textbf{v}(e)$,  \n   \\item \\label{item:priority set correctly} for all $(S,e) \\in E_{\\mathcal{B}}$, $Q_S.\\operatorname{priority}(e) =\n     \\operatorname{last}(S,e) +  \\frac{\\eta'}{s(e)}$, and\n   \\item  \\label{item:heavy edges exact}for all $e \\in H_{\\textbf{w},\\rho}$, $\\vecv^{\\operatorname{mwu}}(e) = \\textbf{v}(e)$.\n %\n  %\n   \\end{enumerate}\n\\end{invariant}\n\nIntuitively, the first invariant means for each $e \\in \\bar E$, the total difference over all\n$S \\ni e$ is bounded. The second invariant ensures that $\\operatorname{ref}(S)\n\\leq Q_S.\\operatorname{priority}(e)$ if and only if \n$\\operatorname{diff}(S,e)$ is small, and we can apply extract min operations on $Q_S$ to detect all\nedges whose priority exceeds the reference point efficiently.  The\nthird invariant means we restore the exact value for all heavy\nedges.\n\n\\begin{proposition}\n\\Cref{inv:lazy update}\\ref{item:total diff not far} implies \\Cref{eq:additive error small}.\n\\end{proposition}\n\nAlso, this invariant allows us to ``reset'' $\\textbf{v}$ to be $\\vecv^{\\operatorname{mwu}}$\nefficiently. \n\n\\begin{algorithm}[H]\n  \\BlankLine\n  $\\textbf{v}(e) \\gets \\textbf{v}(e) + \\sum_{S: (S,e) \\in E_{\\mathcal{B}}} \\operatorname{diff}(S,e)$ \\; %\n  \\For{\\normalfont{each} $S: (S,e) \\in E_{\\mathcal{B}}$}{\n  $\\operatorname{last}(S,e) \\gets \\operatorname{ref}(S)$ \\;\n   $Q_S.\\operatorname{priority}(e) \\gets \\operatorname{last}(S,e) + \\frac{\\eta'}{s(e)}$ \\;\n }\n\\caption{\\textsc{Reset}($e$)}\n\\label{alg:reset}\n\\end{algorithm}\n\nSince priority queue supports the change of priority in $O(\\log m)$ time, we\nhave:\n\\begin{proposition}\n  The procedure \\textsc{Reset} can be implemented in time $O( \\operatorname{q}(\\mathcal{B}) \\cdot \\log m) = \\tilde{O}(1)$.\n\\end{proposition}\n\n\\subsection{Init}\nDefine $\\textbf{v} = v_0$ where $v_0$ is the additive form of $w_0$. We construct the bipartite graph $\\mathcal{B} = (\\mathcal{F},\n\\bar E, E_{\\mathcal{B}})$ as defined in \\Cref{sec:local book}. We use \\Cref{lem:canonical cuts} to construct\n$\\mathcal{B}$ in $\\tilde{O}(m)$ time. For each $S \\in \\mathcal{F}$, we create a min priority\nqueue $Q_S$ containing all the elements in $N_{\\mathcal{B}}(S)$ where for\neach $e \\in N_{\\mathcal{B}}(S)$, we set $Q_s.\\operatorname{priority}(e) =\n\\eta'\/s(e)$. We also define $\\operatorname{ref}(S) = 0$ for all $S \\in \\mathcal{F}$, and\n$\\operatorname{last}(S,e) = 0$ for all $(S,e) \\in E_{\\mathcal{B}}$. By design, the\ninvariants are satisfied. The total running time of this step is\n$\\tilde{O}(m)$.  \n\\subsection{Punish}\n\nGiven a short description of 1 or 2-respecting cut $[[C]]$, we apply\n\\Cref{lem:canonical cuts} to obtain a set $\\mathcal{S} \\subseteq \\mathcal{F}$ of $\\tilde{O}(1)$ canonical\ncuts whose disjoint union is $C$ in $\\tilde{O}(1)$ time.  Recall that the\nupdate increases $\\vecv^{\\operatorname{mwu}}(e)$ by $c_{\\min}$ for each $e \\in C -\nH_{\\textbf{w},\\rho}$ where $c_{\\min} = \\min_{e \\in C - H_{\\textbf{w},\\rho}} c(e)$.\n\\begin{claim}\nWe can compute $c_{\\min}$ in $\\tilde{O}(1)$ time.\n\\end{claim}\n\\begin{proof}\nBy definition of $c_{\\mathcal{B}}(S)$,   $\\min_{S \\in \\mathcal{S}}\nc_{\\mathcal{B}}(S) = \\min_{S \\in \\mathcal{S}} \\min_{e \\in N_{\\mathcal{B}}(S)}c(e) =  \\min_{e \\in C -\n  H_{\\textbf{w},\\rho}} c(e) = c_{\\min}$. The claim follows because there are $\\tilde{O}(1)$ canonical\ncuts in $\\mathcal{S}$ and we maintain the value $c_{\\mathcal{B}}(S)$ for every $S \\in \\mathcal{F}$.%\n\\end{proof}\n\nIn the first step, for each $S \\in \\mathcal{S}$, we set $\\operatorname{ref}(S) \\gets\n\\operatorname{ref}(S) + c_{\\min}$. This takes $\\tilde{O}(1)$ time because $|\\mathcal{S}| =\n\\tilde{O}(1)$ and potentially causes a violation to \\Cref{inv:lazy\n  update}\\ref{item:total diff not far}.\n\nIn the second step, we check and fix the invariant violation as follows.  For\neach $S \\in \\mathcal{S}$, let $W_S = \\{ e \\in S \\setminus H_{\\textbf{w},\\rho} \\colon \\operatorname{ref}(S) > Q_S.\\operatorname{priority}(e)\\}$ be the set of all edges\nin $S \\setminus H_{\\textbf{w},\\rho}$ whose priority in $Q_S$ is smaller\nthan the reference point $\\operatorname{ref}(S)$. For each $e \\in W_S$, we call\nthe procedure $\\textsc{Reset}(e)$. This take times $O(r \\cdot q(\\mathcal{B})\n\\log m) = \\tilde{O}(r)$ where $r$ is the number of calls to $\\textsc{Reset}$\nprocedure. There will be new heavy edges after this step, which means\nwe need to update $\\mathcal{B}$ to correct the set $\\bar E$. \n %\n\nIn the third step, we identify new heavy edges from the set of edges\nthat we called \\textsc{Reset} procedure in the second step, then we\nremove each edge in the set from the associated priority queues and\nfrom the graph $\\mathcal{B}$ as follows. Let $U = \\bigcup_{S \\in \\mathcal{S}}W_S$. Define $U_{H} = \\{ e \\in U \\colon \\textbf{w}(e) \\geq\n\\rho\\}$.  For each $e \\in U_H$, for all $D \\in N_{\\mathcal{B}}(e)$, remove $e$ from the priority queue\n$Q_D$ and update the value $c_{\\mathcal{B}}(D)$ (to get a new minimum after\nremoving $e$). Finally, delete all nodes in $U_H$ from $\\mathcal{B}$. The third step\ntakes $O( |U| + |U_H| \\operatorname{q}(\\mathcal{B}) \\log m + |U_H| \\operatorname{q}(\\mathcal{B}))\n= \\tilde{O}(r)$ time. The running time follows because the $|U| = r$ and $|U_H| \\leq |U|$.\n\nFinally, we output $\\Delta$ where $\\Delta$ is constructed as\nfollows. For each $e \\in U$, let $\\textbf{w}'(e)$ be the weight of $e$\nbefore \\textsc{Reset}$(e)$ is invoked. If $e \\not \\in U_H$, then we define $\\delta_e =\n\\textbf{w}(e) - \\textbf{w}'(e)$. Otherwise, we define $\\delta_e = \\rho -\n\\textbf{w}'(e)$. Then, we add $(e,\\delta_e)$ to $\\Delta$. \n\n\n\\begin{lemma}\n  If \\Cref{inv:lazy update} holds before calling $\\textsc{Punish}([[C]])$, then\n  \\Cref{inv:lazy update} holds afterwards.\n\\end{lemma}\n\\begin{proof}\n  In the first step, we have $ \\bigcup_{S \\ni \\mathcal{S}} N_{\\mathcal{B}}(S) = C\n  \\setminus H_{\\textbf{w},\\rho}$, and thus the violation to \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} can only happen due to some edge $e \\in C\n\\setminus H_{\\textbf{w},\\rho}$.  Because the unions are over disjoint sets, for each\nedge $e \\in C \\setminus H_{\\textbf{w},\\rho}$, there is a unique\ncanonical cut $S_e \\in \\mathcal{S}$ such that $N_{\\mathcal{B}}(S_e) \\ni e$.\n\n\\begin{claim} \n  If there is a violation to \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} due to an edge $e \\in C \\setminus H_{\\textbf{w},\\rho}$, then \\textsc{Reset}$(e)$ is invoked in the second step.\n\\end{claim}\n\\begin{proof}\n    Since  \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} is violated due to an edge $e$, we have $\\sum_{S':\n  (S',e) \\in E_{\\mathcal{B}}} \\operatorname{diff}(S',e) > \\eta\/s(e)$. By averaging argument,\nthere is a canonical cut $S^*$ such that $\\operatorname{diff}(S^*,e) >\n\\frac{\\eta}{s(e)} \\cdot q(\\mathcal{B})$. Since $\\operatorname{diff}(S_e,e)$ is the\nonly term in the summation that is increased, we have $S^* = S_e$. Therefore, we have %\n$$ \\frac{\\eta}{s(e)}\\cdot q(\\mathcal{B}) < \\operatorname{diff}(S_e,e) = \\operatorname{ref}(S_e)\n- \\operatorname{last}(S_e,e) \\overset{\\ref{item:priority set correctly}}{=} \\operatorname{ref}(S_e) - Q_{S_e}.\\operatorname{priority}(e) +\n\\frac{\\eta'}{s(e)}.$$  Therefore, $\\operatorname{ref}(S_e) >\nQ_{S_e}.\\operatorname{priority}(e)$, and so $e \\in W_S$ as defined in the\nsecond step. Hence, \\textsc{Reset}$(e)$ is invoked. \n\\end{proof}\n\nSince \\textsc{Reset}$(e)$ is invoked for every violation, we have that \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} is maintained. By design, the second invariant is\ntrivially maintained whenever \\textsc{Reset} is invoked, and also the last\ninvariant is automatically maintained by the third step. This completes the proof.\n\\end{proof}\n\n\\subsection{Flush}\nFor each $e \\in \\bar E$, we call the procedure\n$\\textsc{Reset}(e)$. Then, we output $\\textbf{w}$ which is the same as\n$\\vecw^{\\operatorname{mwu}}$.  The total running time is $O(q(\\mathcal{B}) |\\bar{E}|)\n= \\tilde{O}( m )$.  \n \n\\subsection{Total Running Time}\n\nThe initialization takes $\\tilde{O}(m)$. Let $K$ be the number of calls to\n\\textsc{Punish}$([[C]])$ and let $I$ be the number of calls to\n\\textsc{Reset}$(e)$ before calling \\textsc{Flush}(). The total running time\ndue to the first step is $O(K\\log^2 n) = \\tilde{O}(K)$, and total running\ntime due to the second and third steps is $\\tilde{O}(I)$. It remains to\nbound $I$, the total number of calls to \\textsc{Reset}$(e)$. Since\neach \\textsc{Reset}$(e)$ increases of weight $\\vecw^{\\operatorname{mwu}}(e)$ by a factor of\n$1+\\eta'$, the total number of resets is\n\n\\begin{align*}\n  O(\\sum_{i \\in [n]}  \\log_{1+O(\\eta')} \\left(\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)) &= O\\left(\\frac{q({\\mathcal B})}{\\eta} \\cdot  \\sum_{i \\in [n]} \n                                                 \\log ( \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})  \\right) \\\\\n  &=  \\tilde{O} \\left(\\frac{1}{\\eta} \\cdot  \\sum_{i \\in [n]} \n  \\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})  \\right).\n\\end{align*}\n\n\n\n\n\n\n\\section{Introduction}\n\n\nIn the $k$-Edge-Connected Spanning Subgraph problem ($k$ECSS), we are given an undirected $n$-node $m$-edge graph $G=(V,E)$ together with edge costs, and want to find a minimum-cost $k$-edge connected spanning subgraph.\\footnote{\\label{foot:intro:sub-multigraph}Note that this problem should not be confused with a variant that allows to pick the same edge multiple time, which is sometimes also called $k$ECSS (e.g.,~\\cite{ChekuriQ17}). We follow the convention in \\cite{CzumajL07} and call the latter variant minimum-cost $k$-edge connected spanning sub-multigraph ($k$ECSSM) problem. (See also the work by Pritchard~\\cite{Pritchard10}.)}\nFor $k=1$, this is simply the minimum spanning tree problem, and thus can be solved in $O(m)$ time~\\cite{karger1995randomized}.\nFor $k\\geq 2$, the problem is a classical NP-hard problem whose first approximation algorithm was given almost four decades ago, where Frederickson and Jaja \\cite{FredericksonJ81} gave a $3$-approximation algorithm that runs in $O(n^2)$ time for the case of $k=2$.\nThe approximation ratio was later improved to $2$ by an\n$\\tilde O(mnk)$-time\nalgorithm of Khuller and Vishkin~\\cite{KhullerV94}.\\footnote{$\\tilde O$ hides $\\operatorname{polylog}(n)$ factor.}\nThis approximation factor of $2$  has remained the best for more than 30 years, even for a special case of $2$ECSS called the weighted tree augmentation problem.\nWhen the running time is of the main concern, the fastest known algorithm takes  $O(n^2)$  time at the cost of a significantly higher $(2k-1)$-approximation guarantee, due to Gabow, Goemans, and Williamson~\\cite{GabowGW98}.\n\nThis above state-of-the-arts leave a big gap between\nalgorithms achieving the best approximation ratio and the best time complexity. This gap exists even for $k=2$.\nIn this paper, we improve the running time of both aforementioned algorithms of~\\cite{KhullerV94,GabowGW98} while keeping the approximation ratio arbitrarily close to two.\nOur main contribution is a near-linear time algorithm that $(1+\\epsilon)$-approximates the optimal {\\em fractional} solution.\n\n\\begin{theorem}\\label{thm:intro:fractional $k$ECSS}\nFor any $\\epsilon > 0$, there is a randomized $\\widetilde{O}(m\/\\epsilon^2)$-time algorithm that outputs a $(1+\\epsilon)$-approximate {\\em fractional} solution for $k$ECSS.\n\\end{theorem}\n\nFollowing, in the high-level, the arguments of Chekuri and Quanrud \\cite{ChekuriQuanrud18} (i.e. solving the minimum-weight $k$ disjoint arborescences in the style of \\cite{KhullerV94} on the support of the sparsified fractional solution), the above fractional solution can be turned into a fast $(2+\\epsilon)$-approximation algorithm for the integral version of $k$ECSS.\n\n\\begin{corollary}\\label{thm:intro:integral $k$ECSS}\nFor any $\\epsilon>0$, there exist\n\\begin{itemize}\n    \\item a randomized $\\widetilde{O}(m\/\\epsilon^2)$-time algorithm that estimates the {\\em value} of the optimal solution for $k$ECSS to within a factor $(2+\\epsilon)$, and\n\n    \\item a randomized $\\tilde O\\left(\\frac{m}{\\epsilon^2} + \\frac{k^2n^{1.5}}{\\epsilon^2}\\right)$-time algorithm\nthat produces a feasible $k$ECSS solution of cost at most $(2+\\epsilon)$ times the optimal value.\n\\end{itemize}\n\\end{corollary}\n    %\n    %\n    %\n    %\n    %\n\nWe remark that the term $\\tilde{O}(k^2 n^{1.5})$ is in fact ``tight'' up to the state-of-the-art algorithm for finding minimum-weight $k$ disjoint arborescences.\\footnote{More formally, if a minimum-weight union of $k$ edge-disjoint arborescences can be found in time $T(k,m,n)$, then our algorithm would run in time $T(k, kn, n)$. The term $O(k^2 n^{1.5})$ came from Gabow's algorithm~\\cite{gabow1995matroid} that runs in time $O(km \\sqrt{n} \\log (n c_{\\max}))$.}\n\nPrior to our results, a sub-quadratic time algorithm was not known even for special cases of $k$ECSS, called\n{\\em $k$-Edge-Connected Augmentation} ($k$ECA). In this problem, we are given a $(k-1)$-edge-connected subgraph $H$ of a graph $G$, and we want to minimize the total cost of adding edges in $G$ to $H$ so that $H$ becomes $k$-edge connected. It is not hard to see that if we can $\\alpha$-approximates $k$ECSS, then we can $\\alpha$-approximates $k$ECA by assigning cost 0 to all edges in $H$. This problem previously admits a $O(kn^2)$-time $2$-approximation algorithm for any even integer $k$~\\cite{khuller1993approximation}\\footnote{In Khuller and Vishkin~\\cite{khuller1993approximation}, the $k$ECA problem aims at augmenting the connectivity from $k$ to $(k+1)$ (but for us it is from $(k-1)$ to $k$.)}. The approximation ratio of 2 remains the best even for 2ECA. Our result in \\Cref{thm:intro:integral $k$ECSS} improves the previously best time complexity by a $\\tilde{\\Theta}(\\sqrt{n})$ factor.\n\n\\textbf{Perspective.} The gap between algorithms with best approximation ratio and best time complexity in fact reflects a general lack of understanding on {\\em fast approximation algorithms}. While polynomial-time algorithms were perceived by many as efficient, it is not a reality in the current era of large data, where it is nearly impossible to take $O(n^3)$ time to process a graph with millions or billions of nodes. Research along this line includes algorithms for sparsest cut \\cite{KhandekarRV09,KhandekarKOV2007cut,Sherman09,Madry10-jtree}, multi-commodity flow \\cite{GargK07,Fleischer00,Madry10}, and travelling salesman problem \\cite{ChekuriQ17,ChekuriQuanrud18}. Some of these algorithms have led to exciting applications such as fast algorithms for max-flow \\cite{Sherman13}, dynamic connectivity \\cite{NanongkaiSW17,ChuzhoyGLNPS19,SaranurakW19,Wulff-Nilsen17,NanongkaiS17dynamic}, vertex connectivity \\cite{LiNPSY21} and maximum matching \\cite{BrandLNPSSSW20}.\n\nThe $k$ECSS problem belongs to the class of survivable network design problems (SNDPs), where the goal is to find a subgraph ensuring that every pair of nodes $(u,v)$ are $\\kappa(u,v)$-edge-connected for a given function $\\kappa$. ($k$ECSS is the {\\em uniform} version of SNDP where $\\kappa(u,v)=k$ for every pair $(u,v)$.)\nThese problems typically focus on building a network that is resilient against device failures (e.g. links or nodes), and are arguably among the most fundamental problems in combinatorial optimization.\nResearch in this area has generated a large number of beautiful algorithmic techniques during the 1990s, culminating in the result of Jain~\\cite{Jain01} which gives a $2$-approximation algorithm for the whole class of SNDPs. Thus, achieving a {\\em fast} $2$-approximation algorithm for SNDPs is a very natural goal.\n\n\nTowards this goal and towards developing fast approximation algorithms in general, there are two common difficulties:\n\\begin{enumerate}\n    \\item Many approximation algorithms inherently rely on solving a {\\em linear program (LP)} to find a fractional solution, before performing  rounding steps. However, the state-of-the-art general-purpose linear program solvers are still quite slow, especially for $k$ECSS and SNDP where the corresponding LPs are {\\em implicit}.%\n\n    In the context of SNDP, the state-of-the-art (approximate) LP solvers still require at least quadratic time: Fleischer~\\cite{fleischer2004fast} designs an $\\tilde{O}(m n k)$ for solving $k$ECSS LP, and more generally for SNDP and its generalization~\\cite{fleischer2004fast,FeldmannKPS16} with at least $\\Theta(m \\min\\{n,k_{\\max}\\})$ iterations  of minimum cost flow's computation  are the best known running time where $k_{\\max}$ is the maximum connectivity requirements.\n\n    \\item Most existing techniques that round fractional solutions to integral ones are not ``friendly'' for the design of fast algorithms. For instance, Jain's celebrated iterative rounding~\\cite{Jain01} requires solving the LP  $\\Omega(m)$ times.  Moreover, most LP-based network design algorithms are fine-tuned to optimize approximation factors, while designing near-linear time LP rounding algorithms requires limiting ourselves to a relatively small set of tools, about which we currently have very limited understanding.\n\n\\end{enumerate}\n\n\n\n\nThis paper completely resolves the first challenge for $k$ECSS  and manages to identify a fundamental bottleneck of the second challenge.\n\n\n\\textbf{Challenges for LP Solvers.} Our main challenge is handling the so-called {\\em box constraints} in the LPs. To be concrete, below is the LP relaxation of $k$ECSS on graph $G=(V,E)$.\n\\begin{align}\n\\min \\{\\sum_{e \\in E} c_e x_e: \\sum_{e \\in \\delta_G(S)} x_e \\geq k\\ (\\forall S \\subseteq V), x \\in [0,1]^E \\} \\label{eq:intro:$k$ECSS LP}\n\\end{align}\nwhere $\\delta_G(S)$ is the set of edges between nodes in $S$ and $V\\setminus S$.\nThe box constraints refer to the constraints  $x \\in [0,1]^E$.\nWithout these constraints, we can select the same edge multiple times in the solution; this problem is called $k$ECSSM in \\cite{CzumajL07} (see \\Cref{foot:intro:sub-multigraph}). Removing the box constraints often make the problem significantly easier. For example, the min-cost $st$-flow problem without the box constraints become computing the shortest $st$-path, which admits a much faster algorithm.\n\nFor $k$ECSS, it can be shown that solving \\eqref{eq:intro:$k$ECSS LP} without the box constraints can be reduced to solving \\eqref{eq:intro:$k$ECSS LP} {\\em with $k=1$} and multiplying all $x_e$ with $k$. In other words, without the box constraints, fractional $k$ECSS is equivalent to fractional 1ECSS.%\n This \\emph{fractional} 1ECSS can be $(1+\\epsilon)$-approximated in near-linear time by plugging in the dynamic minimum cut data structure of Chekuri and Quanrud ~\\cite{ChekuriQ17} to the multiplicative weight update framework (MWU).\n\nHowever, with the presence of box constraints, to use the MWU framework we would need a dynamic data structure for a much more complicated cut problem, that we call, the \\textit{minimum normalized free cut} problem (roughly, this is a certain normalization of the minimum cut problem where the costs of up to $k$ heaviest edges in the cut are ignored.)\nFor our problem, the best algorithm in the static setting we are aware of (prior to this work) is to use Zenklusen's $\\tilde O(mn^4)$-time algorithm \\cite{Zenklusen14} for the {\\em connectivity interdiction} problem.\\footnote{In the connectivity interdiction problem, we are given $G=(V,E)$ and $k \\in {\\mathbb N}$, our goal is to compute $F \\subseteq E$ to delete from $G$ in order to minimize the minimum cut in the resulting graph.}\nThis results in an $\\tilde O(kmn^4)$-time static algorithm.  Speeding up and dynamizing this algorithm seems very challenging.\nOur main technical contribution is an efficient dynamic data structure (in the MWU framework) for the $(1+\\epsilon)$-approximate minimum normalized free cut problem.\nWe explain the high-level overview of our techniques in Section~\\ref{sec:overview}.\n\n\n\n\n\n\n\\textbf{Further Related Works.} %\nThe $k$ECSS and its special cases have been studied extensively. For all $k \\geq 2$, the $k$ECSS problem is known to be APX-hard~\\cite{fernandes1998better} even on bounded-degree graphs~\\cite{csaba2002approximability} and when the edge costs are $0$ or $1$~\\cite{Pritchard10}.  Although a factor $2$ approximation for $k$ECSS has not been improved for almost $3$ decades, various special cases of $k$ECSS admit better approximation ratios (see for instance~\\cite{grandoni2018improved,fiorini2018approximating,adjiashvili2018beating}).\nFor instance, the unit-cost $k$ECSS ($c_e=1$ for all $e \\in E$) behaves very differently, admitting a $(1+ O(1\/k))$ approximation algorithm~\\cite{gabow2009approximating,laekhanukit2012rounding}.\nFor the $2$ECA problem, one can get a better than $2$ approximation when the edge costs are bounded~\\cite{adjiashvili2018beating,fiorini2018approximating}. Otherwise, for general edge costs, the factor of $2$ has remained the best known approximation ratio even for the $2$ECA problem.\n\n\nThe $k$ECSS problem in special graph classes have also received a lot of attention.\nIn Euclidean setting, a series of papers by Czumaj and Lingas led to a near-linear time approximation schemes for constant $k$~\\cite{czumaj2000fast,czumaj1999approximability}.\nThe problem is solvable in near-linear time when $k$ and treewidth are constant~\\cite{berger2007minimum,chalermsook2018survivable}. In planar graphs, 2ECSS, 2ECSSM and 3ECSSM admit a PTAS~\\cite{czumaj2004approximation,borradaile2014polynomial}.\n\n\n\\textbf{Organization.} We provide a high-level overview of our proofs in Section~\\ref{sec:overview}.\nIn Section~\\ref{sec:prelim}, we explain the background on Multiplicative Weight Updates (MWU) for completeness (although this paper is written in a way that one can treat MWU as a black box). In~\\Cref{sec: range map}, we prove our main technical component. In~\\Cref{sec:fast LP solver}, we present our LP solver. In~\\Cref{sec: rounding}, we show how to round the fractional solution obtained from the LP solver.\nDue to space limitations, many proofs are deferred to Appendix.\n\n\n\\section{Overview of Techniques}\n\\input{overview.tex}\n\\section{Preliminaries}\n\\input{prelim.tex}\n\n\\section{Range Mapping Theorem} %\n\\input{rangemapping}\n\\section{Fast Approximate LP Solver} \n\\input{algo} \n\\section{LP Rounding for $k$ECSS (Proof of \\Cref{thm: fast rounding})}\\label{sec: rounding}\n\\input{rounding.tex}\n\n\n\\section{Omitted Proofs} \n\n\n\\subsection{Polynomially Bounded Cost in Proof of \\Cref{thm: warmup}} \\label{sec:polynomially bounded cost}\nLet us assume that the costs $c_e$ are integers (but they can be exponentially large in values).\nKarger's sampling~\\cite{Karger00mincut} gives a near-linear time algorithm to create a skeleton graph $H$ so that all  cuts in $H$ are approximately preserved, and the minimum cut value is $O(\\log |E|)$.\nIt only requires an easy modification of Karger's arguments to show that we can create a skeleton $H$ such that all $k$-free minimum cuts are approximately preserved, and that the value of the minimum $k$-free cuts is $\\Theta(k \\log |E|)$.\nWe will run our static algorithm in graph $H$ instead.\nAs outlined in Karger's paper~\\cite{DBLP:journals\/mor\/Karger99}, the assumption that we do not know the value of the optimal can be resolved by enumerating them in the geometric scales, and the sampling will guarantee that the running time would not blow up by more than a constant factor.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:mwu}}  \\label{sec:mwu proof}\n\nThe proof is done via duality. The primal and dual solutions will be maintained and updated, until the point where one can argue that their values converge to each other; this implies that both the primal and dual solutions are approximately optimal.  \nRecall the primal LP is the covering LP: \n\\[\\min \\{c^T x: A x \\geq 1, x \\geq 0\\} \\]\nThe dual LP is the following packing LP: \n\\[\\max \\{ y^T \\mathbf{1}: y^T A \\leq c^T, y \\geq 0  \\} \\]  \nFor the primal LP, we maintain vectors $\\textbf{w}^{(t)} \\in {\\mathbb R}^n$, where $\\textbf{w}^{(0)}_i = 1\/c_i$ for each $i \\in [n]$. The tentative primal solution on day $t$ is $\\bar{\\textbf{w}}^{(t)} = \\textbf{w}^{(t)}\/\\textsc{MinRow}(A, \\textbf{w}^{(t)})$.\nFor the dual packing LP, we maintain vectors $\\textbf{f}^{(t)} \\in {\\mathbb R}^m$ where $\\textbf{f}^{(0)} = \\mathbf{0}$. \nThe tentative dual solution on day $t$ is defined as  $\\bar{\\textbf{f}}^{(t)} = f^{(t)}\/{\\sf cong}(\\textbf{f}^{(t)})$, where ${\\sf cong}(\\textbf{f})$ is the maximum ratio of violated constraints by $\\textbf{f}$, that is, \\begin{align} \\label{def:congestion} {\\sf cong}(\\textbf{f}) = \\max_{i \\in [n]} \\frac{(\\textbf{f}^T A)_i}{c_i}.\\end{align} Notice that \n$\\bar{\\textbf{f}}^{(t)}$ is a feasible dual solution on each day.  \n\nNow we explain the update rules on each day. Let $j(t)$ be the row that achieves $A_{j(t)} \\textbf{w}^{(t-1)} \\leq (1+\\epsilon) \\textsc{MinRow}(A,\\textbf{w}^{(t-1)})$.  \n\\begin{itemize} \n\\item Update $\\textbf{f}^{(t)}_{j(t)} \\leftarrow \\textbf{f}^{(t-1)}_{j(t)} + \\delta(t)$ where $\\delta(t) = \\min_{i \\in [n]} \\frac{c_i}{A_{j(t), i}}$ is the ``increment'' on day $t$. \n\n\\item Update $\\textbf{w}^{(t)}_i \\leftarrow \\textbf{w}^{(t-1)}_i \\exp\\left(\\epsilon \\cdot \\frac{\\delta(t) A_{j(t),i}}{c_i} \\right)$ for each $i \\in [n]$.  \n\\end{itemize}  \n\n\nDenote the primal value at time $t$ by $P(t)= c^T \\bar{\\textbf{w}}^{(t)}$ and the dual by $D(t) =  ||\\bar{\\textbf{f}}^{(t)}||_{1}$; so we have $P(t) \\geq D(t)$ for all $t$. \n\\begin{theorem}\n\\label{thm:potential}  \nLet $t^*$ be the day $t$ for which $P(t)$ is minimized and $N = \\Omega(\\frac{n}{\\epsilon^2} \\ln n)$ be the total number of days. \nThen we have that $P(t^*) \\leq (1+O(\\epsilon)) D(N)$. In particular, $\\bar{\\textbf{w}}^{(t^*)}$ and $\\bar{\\textbf{f}}^{(N)}$ are near-optimal primal and dual solutions.   \n\\end{theorem} \n\nOur proof relies on the estimates of a potential function defined as $\\Phi^{(t)} = c^T \\textbf{w}^{(t)}= \\sum_{i \\in [n]} c_i \\textbf{w}^{(t)}_i$. \n\n\\begin{lemma} \nWe have, on each day $t$, $$\\exp(\\epsilon \\cdot {\\sf cong}(\\textbf{f}^{(t)}))  \\leq \\Phi^{(t)} \\leq n \\cdot \\exp  \\left( \\epsilon (1+3\\epsilon) \\sum_{0 < t' \\leq t} \\frac{\\delta(t')}{P(t'-1)} \\right).$$\n\\end{lemma}  \n\n\n\n\\begin{proof} \nFirst we show the lower bound of $\\Phi^{(t)}$. Fix column $i \\in [n]$ such that $\\frac{ ((\\textbf{f}^{(t)})^T A)_i  }{c_i} = {\\sf cong}(\\textbf{f}^{(t)})$. Notice that the value of $c_i \\textbf{w}^{(t)}_i$ is equal to: \n\\[\\exp\\left( \\frac{\\epsilon}{c_i} \\cdot \\sum_{t' \\leq t} \\delta(t') A_{j(t'), i} \\right). \\]\n The term $\\delta(t') A_{j(t), i}$ is exactly the increase in $((\\textbf{f}^{(t)})^T A)_i$ at time $t$, so we have that \n$$c_i \\textbf{w}^{(t)}_i \\geq  \\exp\\left( \\frac{\\epsilon}{c_i} \\cdot ((\\textbf{f}^{(t)})^T A)_i  \\right) = \\exp(\\epsilon \\cdot {\\sf cong}(\\textbf{f}^{(t)})),  $$ as desired. \n\nNext, we prove the upper bound on the potential function. \nObserve that\\footnote{In particular, we use the inequality $e^\\gamma \\leq 1 + \\gamma + \\gamma^2$ for $\\gamma \\in [0,1)$ and the fact that the ratio $\\delta(t) A_{j(t),i}\/c_i$ is at most $1$.} $\\textbf{w}^{(t)}_i \\leq \\textbf{w}_i^{(t-1)} (1+ \\epsilon (1+\\epsilon)\\cdot \\frac{\\delta(t) A_{j(t),i } }{c_i})$. \nThis formula shows the increase of potential at time $t$ to be at most \n$$\\Phi^{(t)} \\leq \\Phi^{(t-1)} + \\sum_{i \\in [n]} \\epsilon (1+\\epsilon) \\cdot \\delta(t) A_{j(t), i}\\textbf{w}^{(t-1)}_i \\leq \\Phi^{(t-1)} \\exp \\left( \\frac{\\epsilon(1+\\epsilon) \\delta(t)}{\\Phi^{(t-1)}} \\cdot \\sum_{i \\in [n]}  A_{j(t), i} \\textbf{w}^{(t-1)}_i\\right)$$ \nNotice that $\\sum_{i \\in [n]} A_{j(t),i} \\textbf{w}^{(t-1)}_i = (A_{j(t)} \\textbf{w}^{(t-1)})$ is at most $(1+\\epsilon) \\textsc{MinRow}(A, \\textbf{w}^{(t-1)})$ by the choice of the update rules. The term reduces further to: \n\\[\\Phi^{(t)} \\leq \\Phi^{(t-1)} \\exp \\left(\\frac{\\epsilon(1+\\epsilon)^2 \\delta(t)}{P(t-1)} \\right)\\leq \\Phi^{(t-1)} \\exp\\left( \\frac{\\epsilon(1+3 \\epsilon) \\delta(t)}{P(t-1)}. \\right)\\]\nBy applying the fact that $\\Phi^{(0)} = n$ and the above fact iteratively, we get the desired bound. %\n\\end{proof} \n\nFinally, we argue that the lemma implies Theorem~\\ref{thm:potential}. Consider the last day $N$. Taking logarithms on both sides gives us: \n\\[{\\sf cong}(\\textbf{f}^{(N)} ) \\leq \\frac{\\ln n}{\\epsilon} + (1+3\\epsilon) \\sum_{0 < t' \\leq N} \\frac{\\delta(t')}{P(t'-1)} \\leq \\frac{\\ln n}{\\epsilon} + (1+3\\epsilon) \\frac{||\\textbf{f}^{(N)}||_1}{P(t^*)}\\]  \nThe second inequality uses the fact that $||f^{(N)}||_1 = \\sum_{t'} \\delta(t')$ and that $P(t^*) \\leq P(t)$ for all $t$. \n \n \n\\begin{claim} \n${\\sf cong}(\\textbf{f}^{(N)}) \\geq N\/n$, so this implies that ${\\sf cong}(\\textbf{f}^{(N)}) \\geq \\ln n\/ \\epsilon^2$ when $N \\geq n \\ln n\/ \\epsilon^2$.  \n\\end{claim} \n\\begin{proof}\nWe will argue that $\\sum_{i \\in [n]} \\frac{\\textbf{f}^{(t)}A}{c_i}$ increases by at least one on each day. Since this sum is at most $n {\\sf cong}(\\textbf{f}^{(t)})$, we have the desired result. \nTo see the increase, let $i$ be the column that defines $\\delta(t)$, that is $i = \\arg \\min_{i \\in [n]} c_i\/ A_{j(t), i}$. Notice that $((\\textbf{f}^{(t+1)})^T A)_i = ((\\textbf{f}^{(t)})^T A)_i+  \\delta(t) A_{j(t), i} \\geq  ((\\textbf{f}^{(t)})^T A)_i+ c_{i}$. This shows an increase of one in the above sum. \n\\end{proof} \n\nPlugging in this term, we have that: \n\\[{\\sf cong}(\\textbf{f}^{(N)}) \\leq \\epsilon {\\sf cong}(\\textbf{f}^{(N)}) + (1+3\\epsilon) \\frac{||\\textbf{f}^{(N)}||_1}{P(t^*)} \\]\nThis implies that $P(t^*) \\leq (1+6 \\epsilon) D(N)$. \n\n\\subsection{Proof of \\Cref{lem:KC for box}}\n\nLet $x$ be a feasible solution $A^{kc} x \\geq 1$. Consider $x'_i = \\min (x_i,1)$ for each $i \\in [n]$. \nWe claim that $x'$ satisfies $A x' \\geq \\kappa$. Consider the constraint $A_j x' \\geq \\kappa_j$. Let $F =\\{i \\in \\textsf{supp}(A_j): x_i > 1\\}$. If $|F| \\geq \\kappa_j$, it would imply that $A_j x' \\geq \\kappa_j$ and we are done. Otherwise, we have $|F| \\leq \\kappa_j-1$, and the KC constraints guarantee that \n\\[\\sum_{i \\in \\textsf{supp}(A_j)} x'_i = \\sum_{i \\in \\textsf{supp}(A_j) \\setminus F} x_i + |F| \\geq \\kappa_j  \\] \n\n\nConversely, let $x$ be a feasible solution $A x \\geq \\kappa, x \\in [0,1]^n$. \nConsider any KC constraint: For any $j \\in [m]$ and $F \\subseteq \\textsf{supp}(A_j), |F| \\leq \\kappa_j-1$\n\\[\\sum_{i \\in \\textsf{supp}(A_j)\\setminus F} x_i = \\sum_{i \\in \\textsf{supp}(A_j)} x_i - \\sum_{i \\in F} x_i \\geq \\kappa_j -|F|\\] \nThis implies that $x$ itself is feasible for $A^{kc} x \\geq 1$. \n\n\n\\subsection{Proof of \\Cref{thm:fast LP solver}} \\label{sec:fast lp solver full}\n\n\nBy \\Cref{lem:KC for box}, it is enough to solve kECSS LP with KC inequalities.\n\n\\subsubsection{Interpretation of MWU Framework}  \\label{sec:interpretation}\n\nWe interpret the analysis in \\Cref{sec:mwu proof} in the language of graphs. An interesting feature is that the dual variables are only used in the analysis; it is not used in the implementation at all.\n\n\nWe use $\\vecw^{\\operatorname{mwu}}$ to be the weights that the primal LP maintains. Let $\\{(C^{(t)},F^{(t)}, c_{\\min}^{(t)})\\}_{t \\leq T}$ a sequence of normalized free cuts $(C^{(t)},F^{(t)})$ and the value $c_{\\min}^{(t)} = \\min_{e \\in C^{(t)}\\setminus F^{(t)}} c(e)$ obtained by the MWU algorithm up to day $T$. For each edge $e$, we define congestion ${\\sf cong}(e) = \\frac{1}{c(e)} \\cdot \\sum_{t \\leq T \\colon e \\in C^{(t)} \\setminus F^{(t)}} c_{\\min}^{(t)}$.  The congestion of the graph is denoted as ${\\sf cong}(G) = \\max_{e \\in E} {\\sf cong}(e)$. Note that ${\\sf cong}(G)$ is precisely the same as ${\\sf cong}$ in \\Cref{def:congestion} when we restrict the LP instance to kECSS LP. Furthermore,  by definition, we have\n\n\\begin{align} \\label{eq:we and cong}\n\\forall e,  \\vecw^{\\operatorname{mwu}}(e) \\leq \\frac{1}{c(e)} \\cdot \\exp(\\epsilon {\\sf cong}(G)) \n\\end{align}\n \nSince the running time of the Range Punisher depends on the change of weights, we need to ensure that the total change (the sum-of-log (SOL) terms) is at most near-linear. We bound the SOL term using a slightly different stopping criteria: Observe that the  analysis rely crucially on the fact that congestion ${\\sf cong}(G) \\geq \\frac{1}{\\epsilon^2} \\ln m$. We could also use ${\\sf cong}(G) \\geq \\frac{1}{\\epsilon^2} \\ln m$ as a stopping condition (instead of running up to $O(\\frac{1}{\\epsilon^2} m \\log m)$ days), and the stopping condition implies the number of days is at most $O(\\frac{1}{\\epsilon^2} m \\log m)$.\n\nWe can infer ${\\sf cong}(G)$ from the weight function $\\vecw^{\\operatorname{mwu}}$ by the following.  Let $ \\phi^{\\operatorname{mwu}}(e) :=  \\frac{1}{\\epsilon} \\cdot \\ln (c(e) \\cdot \\vecw^{\\operatorname{mwu}}(e))$ for all $e \\in E$. By definition of ${\\sf cong}(e)$, we have $\\vecw^{\\operatorname{mwu}}(e) = \\frac{1}{c(e)} \\cdot \\exp (\\epsilon {\\sf cong}(e))$, and so $\\phi^{\\operatorname{mwu}}(e) = {\\sf cong}(e)$. Therefore, we have\n\\begin{align}\\label{eq:relation to congestion}\n\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = {\\sf cong}(G).\n\\end{align}\n\n\n\n\n\n\n\\subsubsection{Algorithm}\n\n\nFor the implementation, recall that we denote  $\\vecw^{\\operatorname{mwu}}$ to be the real weights on MWU framework, and $\\textbf{w}$ to be the approximate weight that the data structure maintains. \n\nWe describe extra bookkeeping from \\textsc{RangePunisher} to construct to the final solution. First, it outputs a pair of weight function $(\\vecw^{\\operatorname{mwu}},\\textbf{w}^{\\operatorname{sol}})$ where $\\vecw^{\\operatorname{mwu}}$ is the weights at the end of $\\textsc{RangePunisher}$ and $\\textbf{w}^{\\operatorname{sol}}= \\frac{\\vecw^{\\operatorname{init}}}{{\\sf val}_{\\vecw^{\\operatorname{init}}}(C,F)}$ where $\\vecw^{\\operatorname{init}}$ is the initial weight function for \\textsc{RangePunisher}, and $(C,F)$ is the first normalized mincut obtained during the range punisher.\n\n Since the range punisher maintains approximate weights, we next explain how to detect the stopping condition using approximate weights. We want to stop as soon as $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} > \\frac{1}{\\epsilon^2}  \\cdot \\ln m$. Since we can only keep the approximate weights, we can only detect the approximate value with $O(1\/\\epsilon)$-additive error as follows.  First, it keeps track of $\\phi(e) :=  \\frac{1}{\\epsilon} \\cdot \\ln (c(e) \\cdot \\vecw(e))$ for all $e \\in E$, and early stop as soon as $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon} \\cdot \\ln m$. Since $\\textbf{w}$ is $(1+\\epsilon)$-approximation to the real weight $\\vecw^{\\operatorname{mwu}}$, it implies that with respect to weight right before the stopping day, $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} \\leq \\frac{1}{\\epsilon} \\cdot \\ln m + O(\\epsilon^{-1}) = O( \\frac{1}{\\epsilon} \\ln m)$. \n\n\nThe algorithm for LP solver is described in \\Cref{alg:fast LP solver}.\n\n\n\\begin{algorithm}[H]\n\\KwIn{An undirected graph $G = (V,E)$, a cost function $c$, $\\epsilon \\in (0,1)$}\n\\KwOut{A fractional solution $\\vecw^{\\operatorname{sol}}$.}\n\\BlankLine\n\n$\\forall e \\in E, \\vecw^{\\operatorname{mwu}}(e) \\gets \\frac{1}{c(e)}$ \\;\nLet $\\tilde \\lambda$ be an $(1+\\epsilon)$-approximation to ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}}$\\;\n$\\lambda \\gets \\frac{\\tilde \\lambda }{1+\\epsilon}$\\;\n$\\vecw^{\\operatorname{best}} \\gets \\frac{\\vecw^{\\operatorname{mwu}}}{\\tilde \\lambda}$ \\; \n\\Repeat{$\\exists$ \\normalfont{a} day such that $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon^2} \\cdot \\ln m$ (and early terminate)}\n{\n  $(\\vecw^{\\operatorname{mwu}}, \\vecw^{\\operatorname{sol}}) \\gets \\textsc{RangePunish}(G,\\vecw^{\\operatorname{mwu}},\\lambda)$\\; \n  $\\lambda \\gets \\lambda (1+\\epsilon)$\\;\n \\lIf{$c^T \\vecw^{\\operatorname{best}} > c^T\\vecw^{\\operatorname{sol}}$}{ $\\vecw^{\\operatorname{best}} \\gets \\vecw^{\\operatorname{sol}}$.}\n}\n\n  \\Return{$\\textbf{w}^{\\operatorname{best}}$.}\n\\caption{\\textsc{kECSSLPSolver}($G,c,\\epsilon$)}\n\\label{alg:fast LP solver}\n\\end{algorithm}\n\n\n\\subsubsection*{Correctness}\nWe first show that \\Cref{alg:fast LP solver} punish a sequence of $(1+O(\\epsilon))$-approximate normalized free cuts with respect to $\\vecw^{\\operatorname{mwu}}$ where the weight update rule is defined in the \\textsc{PunishMin} operations. Initially, $\\vecw^{\\operatorname{mwu}}(e) = \\frac{1}{c(e)}$ for all $e \\in E$. By definition, ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\tilde \\lambda\/(1+\\epsilon), \\tilde \\lambda)$ and thus ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$. For each iteration where ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$, the range punisher (\\Cref{lem:fast range punisher}) keeps punishing $(1+O(\\epsilon))$-approximate normalized free cuts until ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq (1+\\epsilon)\\lambda$.\n\nBy discussion in \\Cref{sec:interpretation}, and \\Cref{thm:potential}, there must be a day $t^*$ such that in some range such that $\\frac{w ^{(t^*)}}{{\\sf val}_{w^{(t^*)}}(C^{(t^*)},F^{(t^*)})}$ is $(1+O(\\epsilon))$-approximation to the LP solution where $w^{(t^*)}$ is $\\vecw^{\\operatorname{mwu}}$ at day $t^*$.  Since each normalized cut value is within $(1+\\epsilon)$ factor from any other cut inside the same range, we can easily show that the first cut in the range is $(1+\\epsilon)$-competitive with \\textit{any} cut in the range. Therefore, \\Cref{alg:fast LP solver} outputs $(1+O(\\epsilon))$-approximate solution to kECSS LP. \n\n\\subsubsection*{Running Time} \n By \\Cref{cor:normalized mincut}, the running time for computing the value $\\tilde \\lambda$ is $\\tilde{O}(\\frac{1}{\\epsilon}\\cdot m)$. \n By \\Cref{lem:fast range punisher}, the total running time is $$ \\widetilde{O}(m \\ell + K+ \\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})), $$ where $\\ell $ is the number of iterations, and $K$ is the total number of normalized free cuts punished (including all iterations),  $\\vecw^{\\operatorname{mwu}}$ is the final weight at the end of the algorithm, and $\\vecw^{\\operatorname{init}}(e) = 1\/c(e)$ for all $e$.\n\n Since we early stop as soon as $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon^2} \\cdot \\ln m$,  it means that the day right before we stop we have $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = O(\\frac{1}{\\epsilon^2} \\cdot \\ln m)$. By the stopping condition,\n\n \\begin{align} \\label{eq:stopping cg}\n {\\sf cong}(G) \\overset{(\\ref{eq:relation to congestion})}{=} \\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = O(\\frac{1}{\\epsilon^2} \\cdot    \\ln m). \n \\end{align}\n\n\n The following three claims finish the proof.\n \n \\begin{claim}\n    $\\ell = O(\\frac{1}{\\epsilon^2} \\log m)$.\n  \\end{claim}\n  \\begin{proof}\n    Initially, we have ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$.  By \\Cref{eq:we and cong}, we have $\\vecw^{\\operatorname{mwu}}(e) \\leq \\frac{1}{c(e)} \\cdot \\exp(\\epsilon {\\sf cong}(G)) \\overset{(\\ref{eq:stopping cg})}{=} O(\\frac{1}{c(e)} \\cdot m^{O(\\frac{1}{\\epsilon})})$ for all $e \\in E$.  Let $(C^{(0)}, F^{(0)})$ be the first normalized free cut that we punish. Let $\\lambda_0$ be the value of that cut. We have that each edge is increase by at most a factor of $m^{O(\\frac{1}{\\epsilon})}$, and thus the cut at day right before the stopping happens must be smaller than $\\lambda_0 \\cdot m^{O(\\frac{1}{\\epsilon})}$. Therefore, the number of ranges is $\\log_{1+\\epsilon}(m^{O(\\frac{1}{\\epsilon})}) = O(\\frac{1}{\\epsilon^2} \\log m)$.  \n  \\end{proof}\n\n  \\begin{claim}\n   $K = O( \\frac{1}{\\epsilon^2} m\\log m)$.\n \\end{claim}\n \\begin{proof}\n  Observe that for each normalized free cut $(C,F)$ that we punish there exists a bottleneck edge $e \\in C \\setminus F$ whose $c(e)$ is minimum. By the weight update rule, the congestion is this edge is increased by exactly $1$.  Therefore, the number of normalized free cuts is at most $O(m\\cdot {\\sf cong}(G)) \\overset{(\\ref{eq:stopping cg})}{=} O(\\frac{1}{\\epsilon^2} m\\log m)$.  \n \\end{proof}\n\n   \\begin{claim}\n   For each $e$, $\\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})) = O(\\frac{1}{\\epsilon} \\log m)$. \n \\end{claim}\n \\begin{proof}\n   Recall that the initial weight $\\textbf{w}^{\\text{init}}(e) = 1\/c(e)$ for all $e$. Therefore,\n   $$ \\forall e \\in E,   \\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}))  \\overset{(\\ref{eq:we and cong})}{\\leq}  \\epsilon {\\sf cong}(G) \\overset{(\\ref{eq:stopping cg})}{\\leq} O(\\frac{1}{\\epsilon} \\cdot \\log m).$$\n \\end{proof}\n\n\\subsection{Proof of~\\Cref{lem:spase graph}} \\label{sec:spase graph}\n\nIt suffices to prove the following lemma. \n\n\\begin{lemma} \\label{lem:sparsex}\nGiven a feasible solution $x$ to kECSS, and a non-negative cost function $  : E \\rightarrow \\mathbb{R}_{\\geq 0}$, and $\\epsilon > 0$, there is an algorithm that runs in $\\tilde{O}(m)$ time, and w.h.p., outputs another feasible solution $y$ to kECSS such that\n\\begin{itemize}%\n\\item $\\sum_{e \\in E} c_ey_e \\leq (1+\\epsilon) \\sum_{e \\in E} c_ex_e$.\n\\item $\\operatorname{support}(y) \\subseteq \\operatorname{support}(x)$.\n\\item $|\\operatorname{support} (y)| = O\\left(\\frac{k n \\log n}{\\epsilon^2}\\right)$.\n\\end{itemize}\n\\end{lemma}\n\n\nWe devote the rest of this subsection to proving~\\Cref{lem:sparsex}.\n\n\nLet $x$ be a near-optimal $k$ECSS fractional solution obtained by Theorem~\\ref{thm:intro:fractional $k$ECSS}. \nCompute the solution $y$ using Lemma~\\ref{lem:sparsex}. \nCreate a graph $G'$ by keeping only edges in the support of $y$. \n\n\nBefore proving the lemma, we first develop an extension to the sparsification theorem from the paper of Benczur and Karger.\n\n\nWe follow the definitions by \\cite{BenczurK15, ChekuriQuanrud18}.\n\\begin{definition} [Edge stength]  Let $G = (V,E,w)$ be a weighted undirected graph.\n\n\\begin{itemize}%\n\\item $G$ is $k$-\\textit{connected} if every cut in $G$ has weight at least $k$.\n\\item A $k$-\\textit{strong component} is a maximal non-empty $k$-connected vertex-induced subgraph of $G$.\n\\item The \\textit{strength} of an edge $e$, denoted as $\\kappa_e$ is the maximum $k$ such that both endpoints of $e$  belong to some  $k$-strong component.\n\\end{itemize}\n\\end{definition}\n\n\\begin{lemma}[\\cite{BenczurK15}]\n$$ \\sum_{e \\in E} \\frac{w_e}{\\kappa_e} \\leq n - 1 $$\n\\end{lemma}\n\n\\begin{lemma}[\\cite{BenczurK15}] \\label{lem:approx-strength}\nIn $\\tilde{O}(m)$ time, we can compute approximate stength $\\tilde \\kappa_e$ for each edge $e \\in E$ such that $\\tilde \\kappa_e \\leq \\kappa_e$ and   $\\sum_{e \\in E} \\frac{w_e}{\\tilde \\kappa_e} = O(n) $\n\\end{lemma}\n\n\nGiven a cut $C$ and a subset $S\\subseteq C$ of \nits edges, where $|S|\\leq k-1$, we say $C\\S$ \nis a \\emph{constrained cut}. The next theorem states that all constrained cuts would have \ntheir weights closed to their original weights \nafter the sampling.\n\n\n\\begin{theorem} [Extension to Compression Theorem \\cite{BenczurK15}] \\label{thm:compression-thm}\nGiven $G = (V,E,w)$, let $p : E \\rightarrow [0,1]$ be a probability function over edges of $G$. We construct a random weighted graph  $H = (V,E_H,w')$ as follows.  For each edge $e \\in E$, we independently add edge $e$ into $E_H$ with weight $w'_e = w_e\/p_e $, with probability $p_e$.  For $\\delta \\geq \\Omega( k d \\log n)$, if $p_e \\geq \\min\\{1, \\delta \\frac{w_e}{\\kappa_e} \\}$ for all $e \\in E$,\nthen with high probability $\\left(\\text{over } 1 - \\frac{1}{n^d}\\right)$,  every constrained cut in $H$ has weight between $(1-\\epsilon)$ and $(1+\\epsilon)$ times its value in $G$.\n\\end{theorem}\n\n\\begin{proof} \n\nThis theorem follows almost closely the proof of Benczur-Karger.\nWe sketch here the part where we need a minor modification.\n\nThe proof of Benczur-Karger roughly has two components. The first reduces the analysis for general case to the ``weighted sum'' of the ``uniform'' cases where the minimum cut is large, i.e. edge weights are at most $1$ and minimum cut at least $D=  \\Omega\\left(kd\\log n\\right)$.\nThis first component works exactly the same in our case.\n\nNow in each uniform instance which is the second component of Benczur-Karger,  the probabilistic arguments can be made in the following way: For each cut $C$, since edges are sampled independently, we can use Chernoff bound to upper bound the probability that each cut $C$ deviates more than $(1+\\epsilon)$ factor (after sampling).\nLet $\\mu_C$ denote this probability.\nTherefore, the bad event that there is a cut deviating too much is upper bounded by $\\sum_{C} \\mu_C$.\n\nBenczur-Karger analyzes this probability by constructing an auxiliary experiment: Imagine each edge is deleted with probability $p$, then the sum is exactly the expected number of ``empty cuts'' in the resulting graph.\nThey upper bound this by using the term ${\\mathbb E}[2^R]$ where $R$ is the (random) number of connected components in the resulting graph.\nThey show (using a coupling argument) that ${\\mathbb E}[2^R] = O\\left(n^2 p^D\\right)$, which vanishes whenever $D = \\Omega\\left(d\\log n\\right)$.\nHere is where we need to slightly change the proof. The bad even that we need to bound is not just all the cuts $\\left(\\displaystyle \\sum_{C} \\mu_C\\right)$, but also all the constraint cuts. Let $\\mu_{C \\setminus S}$ be the probability of the bad event that the constraint cut $C \\setminus S$ is deviating too much. We want to bound\n$$\n    \\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}.\n$$\n\n\nWe will create, by enumerating, $m \\choose k$ different graphs $H$ so that each $H$ has at most $k$ edges removed from $G$. Note that all constrained cuts are now defined in these graphs $H$. In the original sampling, if an edge $G$ \nis removed, then we remove it similarly in all graphs $H$ (ignoring it is present in $H$ or not).  \n\n\nGiven that there are $R$ connected components in $H$, there are $O(2^R)$ empty cuts.\nWe consider $m \\choose k$ different graphs derived from $H$ by exhaustively remove\na subset $S \\subseteq E$ of $k$ edges. Some edges in $S$ might already be removed in $H$, so some configurations will be identical.\nWe now count the empty cuts in these $m \\choose k$ graphs. \nTo upper bound $\\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}$, we just need to compute the total number of ``empty cuts'' in all these graphs $H$. \n\nIn each $H$, there are at most $R + k$ connected components. Hence, each graph\nhas at most $2^{R + k}$ empty cuts. Sum up this number among all the graphs,\nwe get that $$\n    \\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}\n    \\leq \\mathbb E \\left[{m \\choose k} 2^{R+k}\\right].\n$$\n\n\nSince ${\\mathbb E}[2^R] = O(n^2 p^D)$, we get that\n$$\n\\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k} \\mu_{C \\setminus S} = O\\left(\n{m \\choose k} 2^k n^2 p^D\\right) = O\\left( {\\left ( \\frac{2em}{k} \\right )}^k n^2 p^D\\right),\n$$\n\n which again vanishes if $D$ is large enough (at least $\\Omega(k d \\log n)$).\n\n\n\\end{proof}\n\nWe are now ready to prove \\Cref{lem:sparsex}. In fact the same proof in \\cite{ChekuriQuanrud18} can be applied once we have \\Cref{thm:compression-thm}. \n\n\\begin{proof}[Proof of \\Cref{lem:sparsex}]\nWe first use \\Cref{lem:approx-strength} to compute approximate edge strength $\\tilde \\kappa_e$ for each edge $e \\in E$ so that $\\tilde \\kappa_e \\leq \\kappa$ and  $\\sum_{e \\in E} \\frac{w_e}{\\tilde \\kappa_e} = O(n) $ in $\\tilde{O} (m)$ time.\nLet $\\delta = \\Theta(k d\\log n)$ for some large constant $d$.  Let $\\text{cost}(x) = \\sum_{e \\in E} c_e x_e\\ $.  For each edge $e \\in E$ let $p_e = \\min \\{1, \\frac{ \\delta x_e}{\\epsilon^2 \\tilde \\kappa_e} \\}$, and $q_e = \\min \\{1, \\frac{\\delta c_e x_e}{\\epsilon^2 \\text{cost}(x) } \\} $, and define $r_e = \\max (p_e,q_e)$.\n\nWe will focus on $x$ from the perspective of kECSS LP with knapsack constraints.\n\nWe construct a random graph $H = (V, E', x')$ using $r$ as a probability function over edges of $G$ and we $x$ as weight function of the graph as follows. For each edge $e \\in E$, we independently sample edge $e$ into $E'$ with weight $x'_e = x_e\/r_e $ with probability $r_e$.    Since\n$$\n    r_e =  \\max (p_e,q_e) \\geq p_e =  \\min \\{1, \\frac{ \\delta x_e}{\\epsilon^2 \\tilde \\kappa_e} \\} \\geq  \\min\\{1, \\delta \\frac{x_e}{\\kappa_e} \\}\n$$\nfor sufficiently large constant $d$, by \\Cref{thm:compression-thm},\nwe get w.h.p.,\n$$\n    \\forall C \\in \\mathcal{C} \\forall S \\in C, |S| \\leq k-1, \\sum_{e\\in C \\setminus S} x'_e \\geq (1-\\epsilon) \\sum_{e\\in C \\setminus S} x_e \\geq (1-\\epsilon)(k - |S|).\n$$\n\nObserve that $$ \\sum_{e \\in E} r_e \\leq \\sum_{e \\in E} p_e + \\sum_{e \\in E} q_e = O( \\frac{n\\delta}{\\epsilon^2}   + \\frac{\\delta}{\\epsilon^2}) = O(\\frac{n\\delta}{\\epsilon^2} )$$\n\nBy Chernoff bound, we have  $$ P(\\sum_{e\\in E} c_e x'_e \\geq (1+\\epsilon) \\sum_{e\\in E} c_e x_e ) \\leq \\exp(-\\Omega(\\delta)) $$\nand,\n$$ P( |E'| \\geq (1+\\epsilon) O( \\frac{n\\delta}{\\epsilon^2})) \\leq \\exp(-\\delta\/\\epsilon^2) $$\nBy the union bound, we have the followings  w.h.p.\n\\begin{align*}\n\\sum_{e\\in C \\setminus S} x'_e \\geq   (1-\\epsilon) (k - |S|), \\quad  \\forall C \\in \\mathcal{C}, \\forall S \\in C, |S| \\leq k-1, \\\\\n  %\n  |\\operatorname{support}(x')| \\leq O( \\frac{n\\delta}{\\epsilon^2})\\quad \\text{ and } \n\\sum_{e\\in E} c_e x'_e \\leq (1+\\epsilon)  \\sum_{e\\in E} c_e x_e\n\\end{align*}\n\nTherefore, $y' = (1+\\epsilon)x'$ is a feasible solution to kECSS. Also,  $ |\\operatorname{support}(y')| \\leq O( \\frac{n\\delta}{\\epsilon^2})$, and $\\sum_{e\\in E} c_e y'_e \\leq (1+\\epsilon)^2 \\sum_{e\\in E} c_e x_e$. \nFinally, we can get $(1+\\epsilon') \\sum_{e\\in E} c_e x_e$ by a proper  scaling factor for $\\epsilon$.\n\n\n\n\\end{proof}\n\n\n\\subsection{Proof of \\Cref{thm: KV}} \\label{sec: proof KV}\n\nFor the first part of the theorem, \nlet $x$ denote the optimal solution in the relaxed LP of $k$ECSS of graph $H$. \nWe create a fractional solution $z$ in $D[H]$\nas follows: for every edge $e \\in E$ in $H$, \nif $e_1$ and $e_2$ are the two opposite directed edges in $D[H]$ derived from $e$, we set \n$z_{e_1}=z_{e_2}=x_e$. It is clear that \n$c(z) = 2c(x)$. We just need to argue that \n$z$ is feasible in the relaxed $k$-arboresences \nproblem. Consider a cut $C \\in \\mathcal{C}$ \n(where $r \\in C$ and $C \\neq V$). As $x(C) \\geq k$, $\\sum_{e \\in \\delta^{+}(C)} z_e \\geq k$. \nFurthermore, by Lemma~\\ref{lem:KC for box}, $x$ satisfies \nthe boxing constraint, that is, \n$0 \\leq x_e \\leq 1$ for all edges $e \\in H$, \nimplying that $0 \\leq z_e \\leq 1$ for all directed edges $e \\in D[H]$. This shows that $z$ is feasible and the first part of the theorem is proved. \n\nFor the second part, consider a feasible solution for $k$-arborescences in $D[H]$. If any of the two opposite directed edges is part of the $k$-arborescences, we include its corresponding \nundirected edge in $H$ as part of our induced solution. \nClearly, the cost of the induced solution cannot be higher and it is a feasible $k$ECSS solution, since it guarantees that the cut value is at least $k$ for all cuts. \n\n\n\\subsection{Polynomially bounded costs}\n\\label{subsec:bound_cmax}\n\nSince Gabow's algorithm for arborescences has the running time depending on $c_{\\max}$, the maximum cost of the edges, we discuss here how to ensure that $c_{\\max}$ is polynomially bounded.\n\nLet $x$ be the LP solution obtained from our LP solver. Denote by $C^* = \\sum_{e \\in E} c_e x_e$, so we have that $C^*$ is between ${\\sf OPT}\/2$ and ${\\sf OPT}$, where ${\\sf OPT}$ is the optimal integral value.\n\nFirst, whenever we see an edge $e \\in E$ with $c_e > 2 C^*$, we remove such an edge from the graph $G$.\nFor each remaining edge $e \\in E$, we round the capacity $c_e$ up to the next multiple of $M = \\lceil \\epsilon C^*\/|E| \\rceil$.\nSo, after this rounding up, we have the capacities in $\\{M, 2M, \\ldots, C^*\\}$, and we can then scale them down by a factor of $M$ so that the resulting capacities $c'_e$ are between $1$ and $O(|E|\/\\epsilon)$.\nIt is an easy exercise to verify that any $\\alpha$-approximation algorithm for $(G,c')$ can be turned into an $\\alpha(1+\\epsilon)$-approximation algorithm for $(G,c)$.\n\n\n\\subsection{Step 1: Static Algorithm}\nWe show that the minimum normalized free cut problem can be solved efficiently in the static setting. For  convenience, we often use the term cut to refer to a set of edges instead of a set of vertices.\n\nDefine the objective function of our problem as, for any cut $C$,\n$${\\sf val}_{\\textbf{w}}(C) = \\min_{F\\subseteq C: |F| \\leq k-1} \\frac{\\textbf{w}(C \\setminus F)}{k-|F|}.$$\nFor any weight function $\\textbf{w}$, denote by ${\\sf OPT}_{\\textbf{w}} = \\min_C {\\sf val}_{\\textbf{w}}(C)$. In this paper, the graph $G$ is always fixed, while $\\textbf{w}$ is updated dynamically by the algorithm (so we omit the dependence on $G$ from the notation ${\\sf val}$ and ${\\sf OPT}$).\nWhen $\\textbf{w}$ is clear from context, we sometimes omit the subscript $\\textbf{w}$.\n\nWe show that the truncation technique can be used to establish a connection between our problem and minimum cut.\n\n\n\n\n\\begin{lemma}\n\\label{lem: baby mapping}\nWe are given a graph $G=(V,E)$, weight function $\\textbf{w}$,  integer $k$, and $\\epsilon >0$.\nFor any threshold $\\rho \\in ({\\sf OPT}_{\\textbf{w}}, (1+\\epsilon){\\sf OPT}_{\\textbf{w}}]$,\n\\begin{itemize}\n    \\item any optimal normalized free cut in $(G,\\textbf{w})$ is a $(1+\\epsilon)$-approximate minimum cut in $(G,\\textbf{w}_{\\rho})$, and\n\n    \\item any minimum cut $C^*$ in $(G,\\textbf{w}_{\\rho})$ is a $(1+\\epsilon)$-approximation for the minimum normalized free cut.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFirst, consider any cut $C$ with ${\\sf val}(C) = {\\sf OPT}$.\nLet $F \\subseteq C$ be an optimal set of  free edges for $C$, so we have $\\textbf{w}_{\\rho}(C \\setminus F) \\leq \\textbf{w}(C \\setminus F) = (k-|F|) {\\sf OPT}$. Moreover, $\\textbf{w}_{\\rho}(F) \\leq |F| \\rho$. This implies that\n\\begin{equation}\n\\textbf{w}_{\\rho}(C) = \\textbf{w}_{\\rho}(C \\setminus F) + \\textbf{w}_{\\rho}(F) < k \\rho\n\\label{eq:exact map upper}\n\\end{equation}\nNext, we prove that any cut in $(G,\\textbf{w}_{\\rho})$ is of value at least $k{\\sf OPT}$ (so the cut $C$ is a $(1+\\epsilon)$ approximate minimum cut).\nAssume for contradiction that there is a cut $C'$ such that $\\textbf{w}_{\\rho}(C') < k{\\sf OPT}$. Let $F' \\subseteq C'$ be the set of $\\rho$-heavy edges.\nObserve that $|F'| \\leq k-1$ since otherwise the total weight $\\textbf{w}_{\\rho}(C')$ would have already exceeded $k{\\sf OPT}$.\nThis implies that $\\textbf{w}(C' \\setminus F') = \\textbf{w}_{\\rho}(C' \\setminus F') < (k-|F'|) {\\sf OPT}$ and that\n\t$${\\sf val}(C') \\leq \\frac{\\textbf{w}(C'\\setminus F') }{(k-|F'|)} < {\\sf OPT}$$\n\twhich is a contradiction. Altogether, we have proved the first part of the lemma.\n\n\nTo prove the second part of the lemma, consider a minimum cut $C^*$ in $(G,\\textbf{w}_{\\rho})$, we have that $\\textbf{w}_{\\rho}(C^*) < \\textbf{w}_{\\rho}(C) < k \\rho$ (from~\\Cref{eq:exact map upper}). Again, the set of heavy edges $F^* \\subseteq C^*$ can contain at most $k-1$ edges, so we must have $\\textbf{w}(C^* \\setminus F^*) < (k-|F^*|) \\rho \\leq (k-|F^*|)(1+\\epsilon) {\\sf OPT}$, implying that ${\\sf val}(C^*) < (1+\\epsilon) {\\sf OPT}$.\n\\end{proof}\n\nWe remark that this reduction from the minimum normalized free cut problem to the minimum cut problem does not give an exact correspondence, in the sense that a minimum cut in $(G,\\textbf{w}_{\\rho})$ cannot be turned into a minimum normalized free cut in $(G,\\textbf{w})$. In other words, the approximation factor of $(1+\\epsilon)$\nis unavoidable.\n\n\n\\begin{theorem}\n\\label{thm: warmup}\nGiven a graph $G=(V,E)$ with weight function $\\textbf{w}$ and integer $k$, the minimum normalized free cut problem can be $(1+\\epsilon)$ approximated by using $O( \\frac{1}{\\epsilon} \\cdot \\log n)$ calls to the exact minimum cut algorithm.\n\\end{theorem}\n\n\\begin{proof}\nWe assume that the minimum normalized free cut of $G$ is upper bounded by some value $M$ which is polynomial in $n = |V(G)|$ (we show how to remove this assumption in \\Cref{sec:polynomially bounded cost}). %\nFor each $i$ such that $(1+\\epsilon)^i \\leq M$, we compute the minimum cut $C_i$ in $(G,\\textbf{w}_{\\rho_i})$\nwhere $\\rho_i = (1+\\epsilon)^i$ and return one with minimum value ${\\sf val}(C_i)$. Notice that there must be some $i^*$ such that $\\rho_{i^*} \\in ({\\sf OPT}_{\\textbf{w}}, (1+\\epsilon){\\sf OPT}_{\\textbf{w}}]$ and by the lemma, we must have that $C_{i^*}$ is a $(1+\\epsilon)$-approximate solution for the normalized free cut problem.\n\\end{proof}\nBy using any  near-linear time minimum cut algorithm e.g.,~\\cite{Karger00mincut}, the collorary follows.\n\n\\begin{corollary} \\label{cor:normalized mincut}\nThere exists a $(1+\\epsilon)$ approximation algorithm for the minimum normalized free cut problem that runs in time $\\tilde{O}(|E|\/\\epsilon)$.\n\\end{corollary}\n\n\n\n\\subsection{Step 2: Dynamic Algorithm}\n\n\nThe next idea we use is from Chekuri and Quanrud~\\cite{ChekuriQ17}. One of the key concepts there is that it is sufficient to solve a ``range punishing'' problem in near-linear time; for completeness we  prove this sufficiency in Appendix.\nIn particular, the following proposition is a consequence of their work:\n\n\\begin{definition}\nA \\textbf{range punisher}\\footnote{Our range punisher corresponds to an algorithm of\nChekuri and Quanrud~\\cite{ChekuriQuanrud18} in one epoch.} is an algorithm that, on any input graph  $G$, initial weight function $\\textbf{w} = \\vecw^{\\operatorname{init}}$, real numbers $\\epsilon$, and $\\lambda \\leq {\\sf OPT}_{\\vecw^{\\operatorname{init}}}$, iteratively applies {\\sc PunishMin} on $(G,\\textbf{w})$  until the optimal becomes at least ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$.\n\\end{definition}\n\nThe following proposition connects a fast range punisher to a fast LP solver.\n\n\n\\begin{proposition}\n\\label{prop: CQ epoch}\nIf there exists a range punisher running in time $$\\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)})\\right)$$ where $K$ is the number of cuts punished, then, there exists a fast dynamic punisher, and consequently the $k$ECSS LP can be solved in near-linear time.\n\\end{proposition}\n\n\n\nThis proposition applies generally in the MWU framework independent of problems. That is, for our purpose of solving $k$ECSS LP, we need a fast range punisher for the minimum normalized free cut problem. For Chekuri and Quanrud~\\cite{ChekuriQ17}, they need such algorithm for the minimum cut problem (therefore a fast LP solver for the Held-Karp bound).\n\n\n\\begin{theorem}[\\cite{ChekuriQ17}, informal]\n\\label{thm: CQ range punisher}\nThere exists a fast range punisher for the minimum cut problem.\n\\end{theorem}\n\n\nOur key technical tool in this paper is a more robust reduction from the range punishing of normalized free cuts to the one for minimum cuts. This reduction works for all edge weights and is suitable for the dynamic setting.\nThat is, it is a strengthened version of Lemma~\\ref{lem: baby mapping} and is summarized below (see its proof in \\Cref{sec: range map}).\n\n\\begin{theorem}[Range Mapping Theorem]\n\\label{lem: full mapping}\nLet $(G=(V,E),\\textbf{w})$ be a weighted graph.\nLet $\\lambda > 0$ and $\\rho = (1+\\gamma)\\lambda$.\n\n\n\\begin{enumerate}\n\n\n\n  %\n\n    \\item If the value of optimal normalized free cut is in $[\\lambda, (1+\\gamma)\\lambda)$, then the value of minimum cut in $(G,\\textbf{w}_{\\rho})$ lies in $[k\\rho\/(1+\\gamma), k \\rho)$.\n\n    %\n    \\item For any cut $C$ where $\\textbf{w}_{\\rho}(C) <k \\rho$, then\n    $\\frac{\\textbf{w}(C \\setminus F)}{k-|F|} < (1+\\gamma) \\lambda$ where $F$ contains all $\\rho$-heavy edges in $C$. In particular, ${\\sf val}(C) < (1+\\gamma) \\lambda$.\n    %\n\\end{enumerate}\n\\end{theorem}\n\nGiven the above reduction, we can implement range punisher fast. We present its full proof in \\Cref{sec:fast LP solver} and sketch the argument below.\n\n\\begin{theorem}\n\\label{thm: free cut range punisher}\nThere exists a fast range punisher for the minimum normalized free cut problem.\n\\end{theorem}\n\\begin{proof}(sketch)\nWe are given $\\lambda$ and weighted graph $(G,\\textbf{w}): \\textbf{w}=\\vecw^{\\operatorname{init}}$ such that ${\\sf OPT}_{\\vecw^{\\operatorname{init}}} \\geq \\lambda$.  Our goal is to punish the normalized free cuts until the optimal value in $(G,\\textbf{w})$ becomes at least $(1+\\epsilon)\\lambda$. We first invoke \\Cref{thm: warmup} to get a $(1+\\epsilon)$-approximate cut, and if the solution is already greater than $(1+\\epsilon)^2\\lambda$, we are immediately done (this means ${\\sf OPT} > (1+\\epsilon)\\lambda$).\n\nNow, we know that ${\\sf OPT} \\leq (1+\\epsilon)^2 \\lambda \\leq (1+3\\epsilon) \\lambda$.\nWe invoke Lemma~\\ref{lem: full mapping}(1) with $\\gamma = 3\\epsilon$.\nThe minimum cut in $(G,\\textbf{w}_{\\rho})$ has size in the range $[k\\rho\/ (1+3\\epsilon), k \\rho)$. We invoke (one iteration of) Theorem~\\ref{thm: CQ range punisher} with $\\lambda' = k \\rho (1+3\/\\epsilon)$ to obtain a cut $C$ whose size is less than $k \\rho$ and therefore, by Lemma~\\ref{lem: full mapping}(1), ${\\sf val}(C) < (1+3\\epsilon) \\lambda$. This is a cut that our algorithm can punish (we ignore the detail of how we actually punish it -- we would need to do that implicitly since the cut itself may contain up to $m$ edges). We repeat this process until all cuts whose values are relevant have been punished, that is, we continue this process until the returned cut $C$ has size at least $k \\rho$.\n\nThe running time of this algorithm is\n$$\\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}_{\\rho}(e)}{\\vecw^{\\operatorname{init}}_{\\rho}(e)})\\right) \\leq \\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w} (e)}{\\vecw^{\\operatorname{init}} (e)})\\right)$$\nNotice that we rely crucially on the property of our reduction using truncated weights.\n\\end{proof}\n\nWe remark that in the actual proof of Theorem~\\ref{thm: free cut range punisher}, there are quite a few technical complications (e.g., how to find optimal free edges for a returned cut $C$?), and we cannot invoke Theorem~\\ref{thm: CQ range punisher} in a blackbox manner. We refer to \\Cref{sec:fast LP solver} for the details. %\n\n\n\n\\subsection{LP Rounding for $k$ECSS}\\label{subsec: LP rounding}\n\nMost known techniques for $k$ECSS (e.g.~\\cite{GabowGW98,laekhanukit2012rounding}) rely on iterative LP rounding, which is computationally expensive.\nWe achieve fast running time by making use of the 2-approximation algorithm of Khuller and Vishkin~\\cite{khuller1993approximation}.\n\nRoughly speaking, this algorithm creates a directed graph $H$ from the original graph $G$ and then compute on $H$ the minimum-weight $k$ disjoint arboresences. The latter can be found by Gabow's algorithms, in either\n$\\tilde{O}(|E||V| k)$ or $\\tilde{O}(k|E| \\sqrt{|V|} \\log c_{\\max})$\ntime.\n\nTo use their algorithm, we will construct $H$ based on the support\nof the fractional solution $x$ computed by the LP solver. By the integrality of the arborescence polytope~\\cite{schrijver2003combinatorial}, an integral\nsolution is as good as the fractional solution. However, the support of $x$ can be potentially large, which causes Gabow's algorithm to take longer time. Here our idea is a sparsification of the support, by extending the celebrated sparsification theorem of Benzcur and Karger~\\cite{BenczurK15} to handle our problem, i.e., we prove the following (see~\\Cref{sec: rounding} for the proofs):\n\n\\begin{theorem}\n\\label{thm: sparsification}\nLet $G$ be a graph and $c_G$ its capacities. There exists a capacitated graph $(H, c_H)$ on the same set of vertices that can be computed in\n$\\tilde{O}(m)$\nsuch that (i) $|E(H)| = \\tilde{O}(n k )$, and (ii) for every cut $S$ and $F \\subseteq S: |F| \\leq (k-1)$, we have $c_G(S~\\setminus~F)~=~(1~\\pm~\\epsilon)~c_H(S~\\setminus~F)$.\n\\end{theorem}\n\nBenzcur and Karger's theorem corresponds to this theorem when $k = 1$.  We believe that this theorem might have further applications, e.g., for providing a fast algorithm for the connectivity interdiction problem.\nOur result implies the following (see Section~\\ref{sec: rounding} for the proof):\n\n\\begin{theorem} \\label{thm: fast rounding}\nAssume that there exists an algorithm that finds a minimum-weight $k$-arborescences in an $m$-edge $n$-node graph in time  $T_k(m,n)$. Then there exists a $(2+\\epsilon)$ approximation algorithm for $k$ECSS running in time $\\tilde{O}(m\/\\epsilon^2 + T_k(kn\/\\epsilon^2,n))$\n\\end{theorem}\n\nApplying \\Cref{thm: fast rounding} with the Gabow's algorithm (see~\\Cref{thm:fast-karbor} in \\Cref{sec: rounding}), we obtain  \\Cref{thm:intro:integral $k$ECSS}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we define a new modular functor based on Kac-Wakimoto admissible\nrepresentations\nover $\\widehat{\\mbox{\\g sl}}_{2}$.\nThe modular functor introduced by Segal \\cite{seg} assigns a finite-dimensional\nvector space\nto the data consisting of a punctured curve, a rank 2 vector bundle and a\ncollection of integral\ndominant highest weights attached to the punctures. Our modular functor does\nthe same for\nthe Segal's data (with  integral\ndominant highest weights replaced with admissible highest weights) extended by\nthe lines in the\nfibers over the punctures. As the data `` surface, vector bundle, punctures,\nlines in fibers over punctures'' evolve, so does the corresponding finite\ndimensional vector space.\nThis leads to a new $\\mbox{${\\cal D}$}-$module on the moduli space of rank 2 vector bundles\nwith  parabolic\nstructure (fixed lines in certain fibers). The main feature of this\n$\\mbox{${\\cal D}$}-$module, as opposed\nto the standard one (see Tsuchiya-Ueno-Yamada \\cite{ts_u_ya},\n or Beilinson-Feigin-Mazur \\cite{beil_feig_maz}), or Moore-Seiberg\n\\cite{moorseib}) is that\n it is singular over a certain set\nof exceptional vector bundles. The latter is closely related to the Hitchin's\nglobal nilpotent\ncone.\n\nWe also prove that our $D-$module has (in a proper sense) regular singularities\nat infinity and that dimension of the generic fiber can be calculated by the\nusual combinatorial algorithm: by pinching the surface the problem is reduced\nto the case of a  sphere with $\\geq 3$ punctures and further to a collection of\nspheres with 3 punctures. Dimension of the space attached to the datum ``3\nmodules sitting at 3 points on a sphere'' is calculated explicitly. It is a\npure\nlinear algebra calculation of dimension of the space of coinvariants of a\ncertain infnite dimensional algebra with coefficients in a certain infinite\ndimensional representation. As the  result is amusing we will record it here.\n\nFirst of all, and it is important, in the genus zero case, one can work with\nmodules at a generic level, as opposed to admissible representations which only\nexist when the  level is rational. It is in complete analogy with the usual\nWZW model, where the famous theory of Knizhnik-Zamolodchikov equations arises\nfrom a collection of the so-called Weyl modules sitting on a sphere\n(terminology is borrowed from  \\cite{kazh_luszt}). The family of Weyl modules\nis good for the purpose of studying integrable representations because each\nintegrable representation is a quotient of some Weyl module.\nThis is no longer the case as far as admissible representations are concerned.\nA family of modules suitable for our needs is that of what we call {\\em\ngeneralized Weyl modules}; the latter is defined to be  a Verma module\n quotiented out by a singular vector.\n\nGeneralized Weyl modules are naturally parametrized by the symbols\n$(V_{r}^{\\epsilon},V_{s})\\;\\; r,s\\geq 0,\\;\\epsilon\\in\\nz\/2\\nz$. Here $V_{r}$ is\nto be thought of as the $r+1-$dimensional irreducible $\\mbox{\\g sl}_{2}-$module;\nmeaning\nof $V_{r}^{\\epsilon}$ will be explained soon. It is appropriate to keep in mind\nthat the conventional Weyl module is defined to be the module induced from\n$V_{r}$. Therefore usually Weyl modules are labelled by $\\mbox{\\g sl}_{2}-$modules. In\nour\nsituation Weyl modules are those related to symbols $(V_{0}^{0},V_{s})$.\n\nAccording to Verlinde, dimensions of the spaces associated to 3 modules on a\nsphere are structure constants of Verlinde algebra. Result of calculation of\nVerlinde algebra in our situation is as follows:\n\n\\begin{eqnarray}\n\\label{intr_ourforgen}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})&=&\\\\\n(V_{r_{1}+r_{2}}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&+&\n(V_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1},V_{s_{1}}\\otimes V_{s_{2}})+\n(V_{r_{1}+r_{2}-2}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})+\\cdots\n+\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&.&\\nonumber\n\\end{eqnarray}\nRecall that the usual Verlinde algebra built on Weyl modules is as follows:\n\n\\[V_{s_{1}}\\circ V_{s_{2}}=V_{s_{1}}\\otimes V_{s_{2}},\\]\ni.e. it is the Grothendieck ring if the category of finite dimensional\nrepresentations of $\\mbox{\\g sl}_{2}$. Observe that our formula agrees with the latter\none on Weyl modules.\n\nThe first component of the right hand side of our formula is  equally easy to\ninterpret. It is known that the symbols $V_{r}^{\\epsilon}$ naturally\nparametrize\nfinite dimensional representations of the simplest rank 1 superalgebra\n$\\mbox{\\g osp}(1|2)$. The category of finite dimensional $\\mbox{\\g osp}(1|2)-$modules\nis a tensor category and (\\ref{intr_ourforgen}) reads as follows: Verlinde\nalgebra is isomorphic to the product of Grothendieck rings of the categories of\nfinite dimensional representations of $\\mbox{\\g osp}(1|2)$ and $\\mbox{\\g sl}_{2}$.\n\nIt is known in principle what to do when passing from modules  to their\nquotients, in our case\nfrom generalized Weyl modules at a generic level to admissible representations\nat a rational level: one has to replace Lie algebras with quantized universal\nenveloping algebras at roots of unity and consider Grothendieck rings\nof the corresponding semi-simple ``quotient categories''. Examples: Verlinde\nalgebra built on integrable $\\mbox{\\g sl}_{2}-$modules has to do with $\\mbox{\\g sl}_{2}$ in\nthis way, and Verlinde algebra built on minimal representations of Virasoro\nalgebra in this way has to do with 2 copies of $\\mbox{\\g sl}_{2}$. It appears that\nVerlinde algebra built on admissible representations is related to the pair\n$(\\mbox{\\g osp}(1|2),\\mbox{\\g sl}_{2})$ in exactly the same way as $Vir-$Verlinde algebra\nis related to the pair of $\\mbox{\\g sl}_{2}$'s.\n\nInterest in admissible representation  originates in the fact that the\ncharacters of admissible representations representations at a fixed level give\na\nrepresentation of the modular group. However realization of this fact\nimmediately gave rise to two puzzles:\n\n(i) Given a representation of the modular group, Verlinde formula  produces\nstructure constants of Verlinde algebra; in the case of admissible\nrepresentations some of the structure constants are negative. This does not\nmake much sense as they are supposed to count dimensions.\n\n(ii) Quantum Drinfeld-Sokolov reduction provides a functor from the category\nof $\\widehat{\\mbox{\\g sl}}_{2}-$modules to the category of $Vir-$modules, which sends\nadmissible representations to minimal representations. It should give\nan epimorphism (or some weakened version of it) of a suitably defined Verlinde\nalgebra for  $\\widehat{\\mbox{\\g sl}}_{2}$ on the well-known Verlinde algebra for\n$Vir$.\n\nWe are able to give an answer to (ii), and a partial answer to (i).\n\nAs far as (ii) is concerned, let us for simplicity step aside and consider\n$Vir-$modules at a generic (not necessarily rational) level. Then there is an\nanalogue of a generalized Weyl module -- Verma module quotiented out by a\nsingular vector -- and these are naturally parametrized by the symbols\n$(V_{r},V_{s})$.  The desired  epimorphism is given by:\n\\[(V_{r}^{\\epsilon},V_{s})\\mapsto (V_{r},V_{s})+(V_{r-1},V_{s}).\\]\n\nThis map is naturally related to the  Drinfeld-Sokolov reduction in the\nfollowing way.\nAs we have fixed the category of representations, we have triangular\ndecomposition of $\\mbox{\\g sl}_{2}$; in particular we have 2 opposite nilpotent\nsubalgebras, $\\mbox{${\\bf C}$} e$, $\\mbox{${\\bf C}$} f$. Therefore there are in fact 2 Drinfeld-Sokolov\nfunctors, $\\phi_{e}$, $\\phi_{f}$. It happens that the map above is induced\nby the direct sum $\\phi_{e}\\oplus\\phi_{f}$.\n\nAs to (i), the situation is as follows. The structure constants naturally\narrange in a tensor $\\{c_{ij}^{r}\\}$, the indices running through a set of\nrepresentations in question. Let us compare the set  $\\{c_{ij}^{r}\\}$ of the\nstructure\ncoefficients of our algebra and the set $\\{b_{ij}^{r}\\}$ of structure\ncoefficients of\nthe algebra calculated by Verlinde formula:\n\n\n\n If our $c_{ij}^{r}=0$, then $b_{ij}^{r}=0$. If $c_{ij}^{r}\\neq 0$, then\n$b_{ij}^{r}$ is ``most certainly'' zero, however in some exceptional cases it\nis\nnon-zero. The latter cases in our situation are interpreted in the following\nway. Recall that we have not only 3 modules, $i,j,r,$ but also 3 Borel\nsubalgebras, $\\mbox{\\g b}_{i},\\mbox{\\g b}_{j},\\mbox{\\g b}_{r}$, which vary. Now as $c_{ij}^{r}\\neq\n0$, the fiber of our $D-$module is $\\neq 0$ (in fact it is 1-dimensional), if\nthe\n3 Borel subalgebras are pairwise different. If however 2 of them meet, the\nfiber\nusually vanishes, but sometimes survives. It survives if and only if\n$b_{ij}^{r}\\neq 0$. If non-zero, $b_{ij}^{r}$ can be $\\pm 1$. There is no doubt\nthat $b_{ij}^{r}$ is a result of some cohomological calculation related to the\n$D-$module. Unfortunately we cannot make it more precise at the moment.\n\n\n\n\n\n\n\n\nJust as in the usual case Weyl modules on a sphere produce a trivial vector\nbundle with the flat (Knizhnik-Zamolodchikov)connection,\n in our case\nwe get a bundle with a flat connection on a space of the 2 times greater\ndimension. The extra\ncoordinates come from the flag manifold, recall that we are dealing  with\nmoduli of vector bundles\nwith parabolic structure. Horizontal sections of this connection satisfy a\nsystem of differential equations; we get twice as many equations as\nthere are KZ equations: half of them are indeed KZ equations and the other half\ncomes from singular vectors in  Verma modules over $\\widehat{\\mbox{\\g sl}}_{2}$. The\nlatter is but natural -- it is exactly one of the lessons of the pioneering\nwork \\cite{bpz}.\n This allows to put the integral formulas for solutions of\nKnizhnik-Zamolodchikov equations, which we wrote in \\cite{feig_mal}, in a\nproper context: they\ngive horizontal sections of this new connection. We conjecture that our\nmethods, in fact, provide\nall horizontal sections. The relation of our formulas to those in\n\\cite{sch_varch} is that the\nlatter are necessarily polynomials as functions on the flag manifold while ours\nare not.\n\nWe wish to acknowledge that there has been a number of works approaching\nWZW model for admissible representation from different points of view, see for\nexample \\cite{awata,feld,f_g_p_p,peters,ramg}. It would be interesting to\nrelate\n our integral formulas with those in \\cite{peters} and the new Hopf algebra of\n\\cite{ramg} to the above mentioned ``$\\mbox{\\g osp}(1|2)\\times\\mbox{\\g sl}_{2}$'' at roots of\nunity. To the best of our knowledge, Verlinde algebras proposed in these work\ndo not solve (ii) above -- those algebras are rather trivial when compared to\nthe $Vir-$analogue.  Our starting point, see  \\cite{feig_mal1}, was the work\n\\cite{awata}, where Verlinde algebra for admissible representations was\nfirst calculated (in the form equivalent but much less illuminating than the\none  described above), using\nthe language which left completely open the problem of existence of a\n$D-$module, such that dimension of the  fiber is calculated through this\nalgebra.\n\n{\\bf Acknowledgments.} Parts of this work were reported\nat the AMS meeting in Hartford, March, 1995, and at Service de Physique\nTheorique at Saclay, in November, 1994. We are grateful to J.-B. Zuber for\ninvitation and warm hospitality. Considerable part of this work was done over\nthe 2 years one of us spent at Yale. Inspiring and friendly atmosphere at the\nDepartment of Mathematics contributed a lot -- and so did the discussions\nwith Igor Frenkel, Ian Grojnowski, Gregg Zuckerman. We are grateful to Itzhak\nBars for bringing to our attention the paper \\cite{peters}, to Sanjaye\nRamgoolam\nfor sending his work, and to David Kazhdan for an interesting conversation at\nHarvard.\n\n\n\n\n\n\\section{{\\bf Notations and known results}}\n\\label{notat_known_res}\n\n\\subsection{ }\n Some notations from  commutative algebra are as follows:\n\n$\\mbox{${\\bf C}$} [t]$ is a polynomial ring, $\\mbox{${\\bf C}$} [[t]]$ is its completion by\npositive powers of $t$; $\\mbox{${\\bf C}$}[t,t^{-1}]$ is a ring of Laurent\npolynomials and $\\mbox{${\\bf C}$} ((t))$ is its completion by positive powers of $t$.\n\nBy functions on the formal (punctured) neighborhood of a non-singular point on\na curve we\nwill mean a ring isomorphic to $\\mbox{${\\bf C}$} [[t]]$ ($\\mbox{${\\bf C}$} ((t))$ resp.); to specify\nsuch an isomorphism means to pick a local coordinate $t$. The analogous meaning\nwill\nbe given to the phrase `` sections of a vector bundle\non the formal (punctured) neighborhood of a non-singular point on a curve''.\n\n\\subsection{ }\\label{defofalgebrandmodusu}\nSet $\\mbox{\\g g} = \\mbox{\\g sl}_{2}$, $\\mbox{$\\hat{\\gtg}$}=\\widehat{\\mbox{\\g sl}}_{2}=\\mbox{\\g sl}_{2}\n\\otimes\\mbox{${\\bf C}$} [z,z^{-1}]\\oplus\\mbox{${\\bf C}$} c$. Choose a basis $e,\\,h,\\,f$ of\n$\\mbox{\\g g}$\nsatisfying the standard relations $[h,e]=2e,\\;[h,f]=-2f,\\;[e,f]=h$.\nWe say that\n\n$\\mbox{\\g g}_{\\geq}=\\mbox{${\\bf C}$} e\\oplus \\mbox{${\\bf C}$} h$\nand\n$\\mbox{$\\hat{\\gtg}$}_{\\geq} =\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g b}\\oplus\\mbox{${\\bf C}$} c$ are standard Borel\nsubalgebras of $\\mbox{\\g g}$ and $\\mbox{$\\hat{\\gtg}$}$ resp;\n\n$\\mbox{\\g g}_{>}=\\mbox{${\\bf C}$} e$\nand\n$\\mbox{$\\hat{\\gtg}$}_{>} =\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g g}_{>}$ are standard ``maximal\nnilpotent subalgebras''\nof $\\mbox{\\g g}$ and $\\mbox{$\\hat{\\gtg}$}$ resp.;\n\n$\\mbox{${\\bf C}$} h$ and $\\mbox{${\\bf C}$} h\\oplus\\mbox{${\\bf C}$} c$ are standard Cartan subalgebras of $\\mbox{\\g g}$ and\n$\\mbox{$\\hat{\\gtg}$}$ resp.\n\nThe Verma module $M_{\\lambda,k}$ is a module induced from the\ncharacter\nof $\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g b}\\oplus\\mbox{${\\bf C}$} c$ annihilating\n$\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[z]\\oplus \\mbox{${\\bf C}$} e$ and sending $ h$ and $c$ to\n$\\lambda$ and $k$ resp. $k$ is often referred to as\na level. Generator of $M_{\\lambda,k}$ is usually denoted by\n$v_{\\lambda,k}$. A quotient of a Verma module is called highest weight module.\n\nThe algebra $\\mbox{$\\hat{\\gtg}$}$ is $\\nz^{2}_{+}-$graded by assigning $f\\otimes\nz^{n}\\mapsto\n(1,-n),\\;e\\otimes z^{n}\\mapsto (-1,-n)$ and so is a Verma module ( as well as\nits quotients):\n$M_{\\lambda,k}=\\oplus\n_{i,j}M_{\\lambda,k}^{i,j}$.\n\nThere is a canonical antiinvolution $\\omega :\\mbox{$\\hat{\\gtg}$}\\rightarrow\\mbox{$\\hat{\\gtg}$}$\ninterchanging $\\mbox{$\\hat{\\gtg}$}_{>}$ and\n$\\mbox{$\\hat{\\gtg}$}_{<}$ and constant on the Cartan subalgebra. For any highest weight\nmodule $V$ denote by\n$V^{c}$ and call {\\em contragredient} the module equal to the\nrestricted dual $V^{\\ast}$ as a vector space with the following action of\n$\\mbox{$\\hat{\\gtg}$}$:\n\\[<gx,y>=<x,\\omega(g)y>,\\;g\\in\\mbox{$\\hat{\\gtg}$},x\\in V^{\\ast},y\\in V.\\]\n\nIf a highest weight module $V$ is irreducible then it is isomorphic to $V^{c}$.\nA morphism of\nhighest weight modules $V_{1}\\rightarrow V_{2}$ naturally induces the morphism\nof the\ncorresponding contragredient modules: $V_{2}^{c}\\rightarrow V_{1}^{c}$.\n\nA morphism of Verma modules $M_{\\lambda,k}\\rightarrow M_{\\mu,k}$ is\ndetermined by the image of $v_{\\lambda,k}$. The image can be written as\n$Sv_{\\mu,k}$ for a uniquely determined element $S$ of the universal\nenveloping algebra of  $\\mbox{\\g g}\\otimes z^{-1}\\mbox{${\\bf C}$}[z^{-1}]\\oplus\\mbox{${\\bf C}$} f$.\nIf non-zero, the vector $Sv_{\\mu,k}$, or even $S$ for this matter, is called\n{\\em singular}.\nThe singular vector can be equivalently defined as an eigenvector of the Cartan\nsubalgebra\n of $\\mbox{$\\hat{\\gtg}$}$\n annihilated by $\\mbox{$\\hat{\\gtg}$}_{>}$. In this form definition applies to an arbitrary\n$\\mbox{$\\hat{\\gtg}$}-$module.\n\n\\subsection{Singular vector formula}\n\\label{Singular_vector_formula}\nIt follows from Kac-Kazhdan\ndeterminant formula that a singular vector generically appears in the\nhomogeneous\ncomponents\nof degree either $n(-1,m),\\;m> 0,\\,n>0$ or $n(1,m),\\;m\\geq 0,n> 0$.\nDenote\nthe corresponding singular vectors by $S_{n,m}^{1}$ and\n$S_{n,m}^{0}$ resp.\n\n\n Singular vectors $S_{nm}^{i}$ were found in ~\\cite{malff}\nin an unconventional form containing  non-integral powers of\nelements\nof $\\mbox{$\\hat{\\gtg}$}$ ( see also ~\\cite{ba_soch} for another approach):\n\n\\begin{equation}\nS_{nm}^{1}=(e\\otimes z^{-1})^{n+mt}f^{n+(m-1)t}\n(e\\otimes z^{-1})^{n+(m-2)t}\\cdots (e\\otimes z^{-1})^{n-mt}\n\\label{s_v_1},\n\\end{equation}\n\\begin{equation}\nS_{nm}^{0}=f^{n+mt}(e\\otimes z^{-1})^{n+(m-1)t}\nf^{n+(m-2)t}\\cdots f^{n-mt},\n\\label{s_v_2}\n\\end{equation}\nwhere $t=k+2$.\n\nThis form is not always convenient to calculate a singular vector. It is,\nhowever,\n a useful tool to derive properties of a singular vector. For example,\ndenoting\nby\n$\\pi:\\mbox{$\\hat{\\gtg}$}\\rightarrow\\mbox{\\g g},\\;g\\otimes z^{n}\\mapsto \\mbox{\\g g}$ the evaluation\nmap,\none uses (~\\ref{s_v_1},~\\ref{s_v_2}) to derive that (see ~\\cite{fuchs}, also\n{}~\\cite{mal} for\nthe proof in a more general quantum case):\n\n\\begin{eqnarray}\n\\pi S_{nm}^{1}=(\\prod_{i=1}^{m}\\prod_{j=1}^{N} P(-it-j))e^{N}\n\\label{p_s_v_1}\\\\\n\\pi S_{nm}^{0}=(\\prod_{i=1}^{m}\\prod_{j=0}^{N-1} P(it+j))f^{N}\n\\label{p_s_v_2},\n\\end{eqnarray}\nwhere $P(t)=ef-(t+1)h-t(t+1)$.\n\n\n\\subsection{Generalized Weyl modules and admissible representations}\n\nThe  structure of Verma modules over $\\mbox{$\\hat{\\gtg}$}$ is known in full detail\n(\\cite{mal_2}).\nOutside the critical level ($k=-2$) a Verma module\nis generically irreducible. $M_{\\lambda,k}$ happens\nto be reducible if and only if it contains a singular vector.\nIf $M_{\\lambda,k}$ is reducible then the following 2 cases arise:\n\n(i) $k$ is generic (not rational) and $M_{\\lambda,k}$  contains only one\nsingular vector;\n\n(ii)$k+2=p\/q>0$ is a ratio of 2 positive integers and $M_{\\lambda,k}$ contains\ninfinitely\nmany singular vectors.\n\nIt can of course happen that $k+2=p\/q<0$. We will not be interested\nin this case and confine to mentioning that here the situation is  in a sense\ndual to (ii).\n\n\\subsubsection{ Case (i)}\n\\label{Case_(i)}\n$M_{\\lambda,k}$ contains a unique proper submodule $M$\n generated by the singular vector. $M$ is, in fact, a Verma module.\n\n{\\bf Definition.}The irreducible quotient $V_{\\lambda,k}$\nis called {\\em generalized\nWeyl module}. $\\Box$\n\n\\bigskip\n\n\nThere arises the exact sequence\n\\begin{equation}\n\\label{exactseqforweyl}\n0\\rightarrow M\\rightarrow M_{\\lambda,k}\\rightarrow V_{\\lambda,k}\\rightarrow 0.\n\\end{equation}\n\nA simple property of Kac-Kazhdan equations \\cite{kac_kazhd} is that, given\n(\\ref{exactseqforweyl}),\n the module\n$M$ is irreducible and does not project on any generalized Weyl module. Note\nthat\nif the composition series of a $\\mbox{$\\hat{\\gtg}$}-$module only consist of generalized Weyl\nmodules\nthen this module\nbreaks into a direct sum of its components. (This can be proved by methods of\nDeodhar-Gabber-Kac\n\\cite{deodgabbkac}.)\n\nIt is an exercise on Kac-Kazhdan equations to derive that the highest weight\n$(\\lambda,k)$\n of a generalized Weyl module $V_{\\lambda.k}$ belongs to either the line\n\\begin{equation}\n\\label{param_eq_line_1}\n\\lambda=-it+j-1,\\;k=t-2,\n\\end{equation}\nfor some $i\\geq 0,j\\geq 1$, or\n  to the line\n\\begin{equation}\n\\label{param_eq_line_2}\n\\lambda=it-j-1,\\;k=t-2,\n\\end{equation}\nfor some $i,j\\geq 1$; in both cases $t$ is regarded as a parameter. Formula\n(\\ref{param_eq_line_1}) cooresponds to the case when $V_{\\lambda,k}$ is\nobtained from $M_{\\lambda,k}$ by quotienting out the singular vector\n$S^{0}_{i,j}$;\nanalogously, (\\ref{param_eq_line_2}) cooresponds to the case when\n$V_{\\lambda,k}$ is obtained from $M_{\\lambda,k}$ by quotienting out the\nsingular vector $S^{1}_{i,j}$.\n\n\n\nWe see that for a fixed level $k$  generalized Weyl modules are parametrized by\nthe\ntriples consisting of a pair of nonnegative numbers, $i$, $j$ in the formulas\nabove, and an element taking one of the 2 values needed to distinguish between\n(\\ref{param_eq_line_1}) and (\\ref{param_eq_line_2}).\nTo be more\nprecise, denote by $V_{i}$ the $i+1-$dimensional irreducible representation of\n$\\mbox{\\g g}$.\n\n\n{\\bf Notation.}\nAssign\nto $V_{\\lambda,k}$ either the symbol $(V_{i}^{0},V_{j-1}),\\; i\\geq 0,j\\geq 1$\nif $(\\lambda,k)$ satisfies (\\ref{param_eq_line_1}), or the symbol\n$(V_{i-1}^{1},V_{j-1}),\\; i,j\\geq 1$ if $(\\lambda,k)$ satisfies\n(\\ref{param_eq_line_2}). $\\Box$\n\n\\bigskip\n\n\nThis gives us a one-to-one correspondence between the set of generalized Weyl\nmodules at a fixed generic level\nand the set of symbols $(V_{i}^{\\epsilon},V_{j})$, where\n$\\epsilon$ is understood as an element of $\\nz\/2\\nz$.\n\n\n\n Observe that the conventional Weyl module of the level $k$\n is  defined to be the induced\nrepresentation\n\\[\\mbox{Ind}_{\\mbox{\\g g}[[z]]\\oplus\\mbox{${\\bf C}$} c}^{\\mbox{$\\hat{\\gtg}$}}V_{n},\\]\nwhere $\\mbox{\\g g}[[z]]$ operates on $V_{n}$ via the evaluation map\n$\\mbox{\\g g}[[z]]\\rightarrow\\mbox{\\g g}$\nand $c\\mapsto k$. From our point of view the Weyl module is a quotient of the\nVerma module\n$M_{n,k}$ by the submodule generated by the singular vector\n$f^{n+1}v_{\\lambda,k}$. In other\nwords, Weyl modules are associated to the symbols $(V_{0}^{0},V_{n})$.\nThis   partially explains appearance of $\\mbox{\\g g}-$modules in our notations.\n\n\n\n\\subsubsection{Case (ii)}\n\\label{Case_(ii)}\n A Verma module\ncontains\ninfinitely many singular vectors and is embedded in finitely many\nother\nVerma modules. Among all singular vectors in\n $M_{\\lambda,k}$ there are 2 independent ones and these\ngenerate\nthe maximal proper submodule.\nAlthough formally all such Verma modules look alike a special role is\nplayed\nby those which can only embed (non-trivially) in themselves. Highest\nweights of such modules were called by Kac and Wakimoto\n{\\em admissible} (~\\cite{kac_wak}) and are described as follows.\n\nLet $k+2=p\/q$, where $p,q$ are relatively prime positive integers.\nThe set of admissible highest weights at the level $k=p\/q-2$ is given\nby\n\\[\\Lambda_{k}=\\{\\lambda(m,n)=m\\frac{p}{q}-n-1\\, :\\;0<m\\leq q,\\,\n0\\leq n\\leq p-1\\}.\\]\n\n\nWhat is said above about the structure of Verma modules implies that\nany\nVerma module appears in the exact sequence of the form\n\\begin{equation}\n\\label{bgg_res}\n0\\leftarrow L_{\\lambda_{0},k}\\leftarrow\nM_{\\lambda_{0},k}\\stackrel{d_{0}}{\\leftarrow} M_{\\lambda_{1},k}\\oplus\nM_{\\mu_{1},k}\n\\stackrel{d_{1}}{\\leftarrow} M_{\\lambda_{2},k}\\oplus\nM_{\\mu_{2},k}\\stackrel{d_{2}}{\\leftarrow}\\cdots,\n\\end{equation}\nwhere $\\lambda_{0}$ is an admissible weight at the level $k$ and\n$L_{\\lambda_{0},k}$ is\nthe corresponding irreducible module.\n$L_{\\lambda_{0},k}$ is also called {\\em admissible}.\n The exact sequence\n(\\ref{bgg_res}) is called Bernstein - Gel`fand - Gel'fand ( BGG ) resolution.\n\nAgain cohomological arguments show (see e.g. \\cite{kac_wak}) that if the\ncomposition series of\na $\\mbox{$\\hat{\\gtg}$}-$module only consists of admissible representations then the module is\ncompletely\nreducible.\n\nThe parametrization of the set of admissible representations we are going to\nuse is as follows.\nTwo different generalized Weyl modules project onto one and the same\nadmissible representation: formula (\\ref{bgg_res}) implies that the the two\nmodules\nprojecting onto $L_{\\lambda_{0},k}$ are $M_{\\lambda_{0},k}\/M_{\\lambda_{1},k}$\nand\n$M_{\\lambda_{0},k}\/M_{\\mu_{1},k}$.\nTherefore\ntwo different triples $(V_{m}^{\\epsilon},V_{n})$\nare related to the same admissible represenation.\n Introduce the equivalence relation $\\approx$ by\n$(V_{m}^{\\epsilon},V_{n})\\approx (V_{q-1-m}^{\\epsilon+1},V_{p-2-n}),\\; 0\\leq\nm\\leq q-1,\\, 0\\leq n\\leq p-2$.\n Denote\nby $(V_{m}^{\\epsilon},V_{n})^{\\sim}$ the equivalence class of\n$(V_{m}^{\\epsilon},V_{n})$.\n\nIt easy to check that\nadmissible representations are parametrized by the equivalence  classes of the\ntriples:\n\\begin{equation}\n\\label{parametrofadmreprformula}\n\\{\\mbox{ admissible representations }\\}\\Longleftrightarrow\n\\{(V_{m}^{\\epsilon},V_{n})^{\\sim}\\}.\n\\end{equation}\n\n\n\n\\subsection{ }\n\\label{whatcanbesaidhighrank}\nConsiderable part of the above carries over to the arbitrary Kac-Moody algebra\ncase.\nHere,\nfor example, is the definition of an admissible representation.\nDrop the condition that $\\mbox{\\g g}=\\mbox{\\g sl}_{2}$, let $M_{\\lambda,k}$\nbe a Verma module over $\\mbox{$\\hat{\\gtg}$}$ and $L_{\\lambda,k}$ be its irreducible quotient.\nCall $(\\lambda,k)$\nadmissible if $M_{\\lambda,k}$ satisfies the following projectivity condition:\nif composition\nseries of a $\\mbox{$\\hat{\\gtg}$}-$module $W$ contains $L_{\\lambda,k}$ then $M_{\\lambda,k}$\nnon-trivially maps\nin $W$.\n\n Unfortunately we do not have a reasonable definition of a generalized Weyl\nmodule\nin the higher rank case. This is one of the reasons\nfor which we have to  confine mostly to the $\\mbox{\\g sl}_{2}-$case.\n\n\n \\subsection{Loop modules }\n\\label{Loop_modules}\n\n We will also be using $\\mbox{$\\hat{\\gtg}$}-$modules different from Verma modules\nor corresponding irreducible ones.\n\nDenote by $\\mbox{${\\cal F}$}_{\\alpha \\beta} $ a $\\mbox{\\g g}-$ module with the basis\n$F_{i},\\;i\\in\\nz$\nand the action given by\n\\[eF_{i}=-(\\alpha+i-\\beta)F_{i+1},\\;hF_{i}=(2\\alpha+2i-\\beta)F_{i},\\;\nfF_{i}=(-\\alpha-i)F_{i-1}.\\]\nThe space $\\mbox{${\\cal F}$}_{\\alpha \\beta} ^{\\mbox{${\\bf C}$}^{\\ast}}=\\mbox{${\\cal F}$}_{\\alpha \\beta}\\otimes\\mbox{${\\bf C}$}\n[z,z^{-1}]$ is endowed with\nthe natural $\\mbox{$\\hat{\\gtg}$}-$module structure. The elements $F_{ij}=\nF_{i}\\otimes z^{j},\\;i,j\\in\\nz$\nserve as a natural basis in it.\n\nRecall (see \\ref{Singular_vector_formula}) that   $S^{1}_{nm},\\;S^{0}_{nm}$\nstand for a singular\nvector of degree $n(-1,m)$ or $n(1,m)$ resp. in a Verma module. The following\nformulas are proved\nby using (\\ref{p_s_v_1},\\ref{p_s_v_2}):\n\n\\begin{eqnarray}\n\\label{actonloops1}\nS^{1}_{nm}F_{n,nm}=\\{&\\prod_{i=1}^{m}\\prod_{j=1}^{n}&(-it-j-\\alpha+\\beta)(-it-j-\\alpha)\\}\n\\{\\prod_{s=1}^{n}\n(\\alpha+i)\\}F_{00}\\\\\nS^{2}_{nm}F_{-n,nm}=\\{&\\prod_{i=1}^{m}\\prod_{j=1}^{n}&(it+j-\\alpha+\\beta)(-it-j-\\alpha)\\}\n\\{\\prod_{s=1}^{n}\n(\\alpha-\\beta-i)\\}F_{00},\\label{actonloops2}\n\\end{eqnarray}\nwhere $t=k+2$.\n\n\n\n\\section{Construction of the modular functor.}\n\nAlthough most of our results have to do with $\\mbox{\\g sl}_{2}$,\n up to some point it is no extra effort to work in greater\ngenerality. So until sect.\\ref{The_spaces_of_coinvariants}, $\\mbox{\\g g}$ will stand\nfor $\\mbox{\\g sl}_{n}$ unless\notherwise stated.\n\\subsection{ Algebra $\\mbox{$\\hat{\\gtg}$}^{A}$ and categories of $\\mbox{$\\hat{\\gtg}$}^{A}$-modules }\n\\subsubsection{ }\n\\label{algebra_hgta}\nLet $\\mbox{${\\cal C}$}$ be a smooth compact algebraic curve and $\\rho:\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}$ be\na rank\n$n$ vector bundle  with a flat connection. The connection relates to a section\n$s$ of any\nbundle $\\mbox{${\\cal A}$}$ associated with $\\mbox{${\\cal E}$}$ the section $dc$ of $\\Omega\\otimes\\mbox{${\\cal A}$}$ where\n$\\Omega$\nis the sheaf of differential forms over $\\mbox{${\\cal C}$}$. A typical example of $\\mbox{${\\cal A}$}$ is\nthe bundle\n$\\mbox{End}\\mbox{${\\cal E}$}$ of fiberwise endomorphisms of $\\mbox{${\\cal E}$}$. The sheaf of sections of\n$\\mbox{End}\\mbox{${\\cal E}$}$\nis naturally a sheaf of Lie algebras over $\\mbox{${\\cal C}$}$.\n\n\nFor a point $P\\in \\mbox{${\\cal C}$}$ let $\\mbox{\\g g}^{P}$ be the algebra of sections of\n$\\mbox{End}\\mbox{${\\cal E}$}$ over the formal\nneighborhood of $P$. For a finite subset $\\bar{A}=\\{ P_{1},P_{2},\\ldots ,\nP_{m}\\}\n\\subset \\mbox{${\\cal C}$}$ set $\\mbox{\\g g}^{\\bar{A}}= \\oplus_{i=1}^{m}\\mbox{\\g g}^{P_{i}}$. Define\n$\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}$ to\nbe the central extension of $\\mbox{\\g g}^{\\bar{A}}$ by the cocycle\n \\[ <x,y>=\\sum_{i=1}^{m}\\mbox{Res}_{P_{i}}\\mbox{Tr}dx\\cdot y.\\]\nIn particular, we obtain the splitting\n\\begin{equation}\n\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}=\\mbox{\\g g}^{\\bar{A}}\\oplus\\mbox{${\\bf C}$}\\cdot c.\n\\label{splitting}\n\\end{equation}\n\n\n\nConsider a finite set $A=\\{(P_{1},\\mbox{\\g b}_{1}),\\ldots ,(P_{m},\\mbox{\\g b}_{m})\\}$\n  where $P_{i}\\in\\mbox{${\\cal C}$}$ are pairwise different and $\\mbox{\\g b}_{i}$ is a Borel\nsubalgebra\nof the algebra of  traceless linear transformations of the fiber\n$\\rho^{-1}P_{i}$\n ($1\\leq i\\leq m$). Let $\\bar{A}$ be the projection of $A$ on $\\mbox{${\\cal C}$}$. Set\n $\\mbox{$\\hat{\\gtg}$}^{A}=\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}$.\n\n\\subsubsection{ }\n\\label{defofocat}\nGiven $A$ as above, set $\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$.\nDenote by $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$ the subalgebra\nconsisting of sections $x(.)$ such that $x(P_{i})\\in \\mbox{\\g n}_{i},\\;1\\leq i\\leq m,$\nand by $\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}$  the subalgebra spanned by the space\n of sections $x(.)$ such that $x(P_{i})\\in \\mbox{\\g b}_{i},\\;1\\leq i\\leq m,$\n and the central\nelement $c$.\nThese are analogues of the maximal ``nilpotent'' and maximal ``solvable''\nsubalgebras for $\\mbox{$\\hat{\\gtg}$}^{A}$, c.f.\\ref{defofalgebrandmodusu}.\n\n\nDenote by $\\mbox{${\\cal O}$}^{A}_{k},\\;k\\in\\mbox{${\\bf C}$}$, the category of\nfinitely generated $\\mbox{$\\hat{\\gtg}$}^{A}$-modules satisfying\n the conditions:\n\n(i) $c$ acts as  multiplication by $k$;\n\n(ii) the action of the subalgebra $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$ is locally finite.\n\nIn much the same way as in \\ref{defofalgebrandmodusu} one defines\nVerma and generalized Weyl modules over $\\mbox{$\\hat{\\gtg}$}^{A}$:\n\n{\\bf Definition.}\n\n(i) We will say that $(\\lambda,k)$ is a highest weight of $\\mbox{$\\hat{\\gtg}$}^{A}$ if\n$\\lambda$ is a functional on $\\oplus _{i}\\mbox{\\g b}_{i}\/\\mbox{\\g n}_{i}$ and $k$ is a\nnumber.\n\n(ii) A highest weight $(\\lambda,k)$ naturally determines a character of\n$\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}$\nsending $c$ to $k$ and\n annihilating $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$. Denote by $\\mbox{${\\bf C}$}_{\\lambda,k}$\nthe corresponding 1-dimensional representation.\n\n(iii)  Define the Verma module\n$M^{A}_{\\lambda ,k}$ to be the induced representation\n\\[\\mbox{Ind}_{\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}}^{\\mbox{$\\hat{\\gtg}$}^{A}}\\mbox{${\\bf C}$}_{\\lambda}. \\; \\Box\\]\n\n\n\\bigskip\n\nThere is an isomorphism\n\\[M^{A}_{\\lambda ,k}\\approx\n \\otimes _{i=1}^{m} M^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}.\\]\n\nSuppose now that each $M^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}$ has at least one\nsingular vector. If\n$k\\in \\mbox{${\\bf C}$} \\setminus \\mbox{${\\bf Q}$}$ then this\nsingular vector is unique for each $i$. Quotienting out all of\nthem one obtains the\n {\\em generalized Weyl module} $V_{\\lambda,k}^{A}$. As above there is an\nisomorphism\n\\[V_{\\lambda,k}^{A}\\approx\n\\otimes _{i=1}^{m} V^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}.\\]\n\nIf $k$ is not a rational number then any generalized Weyl module is\nirreducible. Denote by $\\mbox{$\\tilde{\\co}$}_{k}$ the full subcategory of $\\mbox{${\\cal O}$}_{k}$ consisting\nof all $\\mbox{$\\hat{\\gtg}$}^{A}-$modules whose composition series consist of generalized\nWeyl modules. Again if $k$ is not a rational number then $\\mbox{$\\tilde{\\co}$}_{k}$ is\nsemisimple.\n\nIf $k$ is rational then there arises the admissible representation\n$L_{\\lambda,k}^{A}$ if $(\\lambda,k)$ is admissible. If the composition series\nof a module $V^{A}$ consists only of admissible representations, then $V^{A}$\nis\ncompletely reducible.\n\n\\begin{remark}\n\\label{functorwrtA}\nThere is a canonical isomorphism $\\mbox{\\g b}_{1}\/\\mbox{\\g n}_{1}\\approx\\mbox{\\g b}_{2}\/\\mbox{\\g n}_{2}$\nfor any 2 Borel subalgebras $\\mbox{\\g b}_{1},\\mbox{\\g b}_{2}$. Therefore if Borel subalgebras\nappearing in $A$ evolve, so does the projectivization of the module\n$M_{\\lambda,k}^A$\nor its quotients.\n\n  \\end{remark}\n\n\n\\subsubsection{ }\n\\label{algebra_gofA}\nLet $A$ be as in \\ref{algebra_hgta}. Let $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ be the Lie algebra\nof meromorphic sections of\n $\\mbox{End}\\mbox{${\\cal E}$}$  holomorphic outside $\\bar{A}$.\n The  maps of restriction to formal neighborhoods give rise to the Lie algebra\nmorphism\n\\begin{equation}\n\\label{globalfunct_local}\n\\mbox{\\g g}(\\mbox{${\\cal C}$},A)\\rightarrow \\mbox{\\g g}^{A}\n\\end{equation}\nThe splitting (\\ref{splitting}) provides us with the section\n$s_{A}:\\; \\mbox{\\g g}^{A}\\rightarrow \\mbox{$\\hat{\\gtg}$}^{A}$. Composition of\n(\\ref{globalfunct_local}) with $s_{A}$\ngives the linear morphism\n\\begin{equation}\n\\label{globalfunct_affine}\n\\mbox{\\g g}(\\mbox{${\\cal C}$},A)\\rightarrow\\mbox{$\\hat{\\gtg}$}^{A}.\n\\end{equation}\nThe residue theorem implies that (\\ref{globalfunct_affine}) is a Lie algebra\nmorphism\n(even though $s_{A}$ is not!).\n\nBy (\\ref{globalfunct_affine}), the standard\npullback makes each object of $M^{A}\\in\\mbox{${\\cal O}$}_{k}^{A}$ into a\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$-module.\nHence there arises the space of coinvariants\n\\[(M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}=M\/\\mbox{\\g g}(\\mbox{${\\cal C}$},A)M.\\]\n\n\\subsection{ Localization of $\\mbox{$\\hat{\\gtg}$}^{A}$-modules }\n\n\\subsubsection{ }\n\\label{whi1jetparam}\n  let us\nrecall that with an $n-$dimensional vector space $W$ one associates the flag\nmanifold $F(W)=\nGL(n,\\mbox{${\\bf C}$})\/B$ and {\\em the base affine space} $Base(W)= GL(n,\\mbox{${\\bf C}$})\/N$, where $B$\nis a Borel\nsubgroup and $N$ unipotent subgroup of $B$. The natural map $Base(W)\\rightarrow\nF(W)$ is a\nprincipal $(\\mbox{${\\bf C}$}^{\\ast})^{\\times n}$-bundle.\n\nNow return to a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $V^{A}$ and suppose for simplicity that $A$\nconsists of\n1 element $(P,\\mbox{\\g b})$. Consider a family of the data\n$\\{P, \\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}\\}$ -- let us not care about Borel subalgebras for the\nmoment. One expects that the corresponding family of vector spaces arranges\nthen in a locally trivial vector bundle. An obstacle to get this is that we\nhave defined\n$V^{P}$ up to an isomorphism but have not specified any such isomorphism. For\nexample, an attempt to\nchoose a basis in $V^{P}$ requires to choose\n(in particular) a local coordinate $z$ at $P$, such that $z(P)=0$.\nDifferent choices of $z$ are essentially different as the group\n$\\mbox{Diff}(P)$\nof diffeomorphisms of the formal neighborhood of $P$ does not\nin general act on $V^{P}$. However the subgroup $\\mbox{Diff}(P)_{1}\\subset\n\\mbox{Diff}(P)$\nof diffeomorphisms preserving the 1-jet of parameter does act on $V^{P}$. We\nsee that $V^{P}$,\nin fact, depends on the 1-jet of parameter at $P$.\n\nTo take care of Borel subalgebras,  let us\nrecall that with an $n-$dimensional vector space $W$ one associates the flag\nmanifold $F(W)=\nGL(n,\\mbox{${\\bf C}$})\/B$ and {\\em the base affine space} $Base(W)= GL(n,\\mbox{${\\bf C}$})\/N$, where $B$\nis a Borel\nsubgroup and $N$ unipotent subgroup of $B$. The natural map $Base(W)\\rightarrow\nF(W)$ is a\nprincipal $(\\mbox{${\\bf C}$}^{\\ast})^{\\times n}$-bundle.\n\n\nSimilar arguments applied to $\\mbox{\\g b}$ show that\n\nthe module $V^{A}=V^{P,\\mbox{\\g b}}$ depends on the quadruple $(P,\\mbox{\\g b},j,x)$ such that\n$j$ is a 1-jet\nof parameter at $P$ and $x\\in Base(\\mbox{${\\bf C}$}^{n})$ belongs to the preimage of $\\mbox{\\g b}$.\n\nOne concludes that we do get a locally trivial vector bundle after pull-back to\nthe space of pairs ``1-jet of parameter at $P$, element of the maximal torus of\nthe\nBorel group related to $\\mbox{\\g b}$''. Let us be more precise now.\n\n\n\n\n\\subsubsection{ }\n\\label{maingenerresults}\n\nLet $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ be a family of smooth projective curves\nand $\\rho_{S}:\\; \\mbox{${\\cal E}$}_{S}\\rightarrow \\mbox{${\\cal C}$}_{S}$ be a rank $n$ vector bundle. There\narise 2 more\nbundles:\n\n(i) the  bundle $Base(\\rho_{S}):\\; Base(\\mbox{${\\cal E}$}_{S})\\rightarrow \\mbox{${\\cal C}$}_{S}$ with the\nfiber\nover any $x\\in\\mbox{${\\cal C}$}_{S}$  equal to the base affine space\nof the vector space $\\rho_{S}^{-1}x$;\n\n(ii) the $\\mbox{${\\bf C}$}^{\\ast}-$bundle $J^{(1)}(\\mbox{${\\cal C}$}_{S})\\rightarrow \\mbox{${\\cal C}$}_{S}$ of  1-jets\nof coordinates along\nfibers of $\\bar{\\pi}$.\n\n Consider the fibered product\n$Base(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})$ and the natural map\n\\[\\pi:\\; Base(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})\\rightarrow S.\\]\n\nPick a  non empty  finite set $A_{S}$ of sections of $\\pi$   satisfying the\ncondition:\n\nfor any $s\\in S$ the natural projection of\n the set $A_{S}(s)=\\{a(s),\\;a\\in A_{S}\\}$\n on  $\\bar{\\pi}^{-1}(s)$\nis an injection.\n\nPick  an arbitrary curve, say $\\mbox{${\\cal C}$}_{s_{0}}$, from our family. Consider a\nhighest weight\nmodule  $M^{A }$ over $\\mbox{$\\hat{\\gtg}$}^{A}$, where we write $A$ instead of the lengthy\n$A_{S}(s_{0})$; what follows is obviously independent of the choice of $s_{0}$.\n\nBy \\ref{defofocat}, remark  \\ref{functorwrtA}, and \\ref{whi1jetparam}, we get a\n$\\mbox{$\\hat{\\gtg}$}^{A_{S}(s)}-$module $M^{A_{S}(s)}$ for any $s\\in S$ and the collection\n $\\{M^{A_{S}(s)},\\; s\\in S\\}$ arranges in a locally trivial vector bundle.\nWith\neach $s\\in S$ we can further associate a vector space, that is the space of\ncoinvariants\n\\[(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s))},\\]\nsee \\ref{algebra_gofA}.\n\n\n\\begin{theorem}\n\\label{existofDmod}\nSuppose the collection $\\psi=(M^{A},\\pi,A_{S})$, satisfying the conditions\nimposed above, is given.\n Then\nthere is a twisted $ \\mbox{${\\cal D}$}-$module (that is\na sheaf of modules over a certain algebra of twisted differential operators) on\n$S$ such\nthat its fiber over $s\\in S$ is $(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s)}$.\n\\end{theorem}\n\nThis theorem is an immediate consequence of \\cite{beil_feig_maz} and\n\\cite{bern_beil,bryl_kash} . Briefly the construction is as follows. Take a\nvector field  $\\xi$ on $U\\subset S$. It lifts to a meromorphic vector field\non $\\mbox{${\\cal C}$}_{S} - A_{S}(S)$ over $U$, and\nfurther to a meromorphic vector field on $\\pi^{-1}(U)\\subset\nBase(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})$; denote this vector field by\n$\\xi^{\\ast}$.  Trivializing  the infinitesimal neighborhood of\n$A_{S}(U)\\subset\\mbox{${\\cal C}$}_{S}$ by chosing, locally with respect to $U\\subset S$,\ncoordinates in the fibers, one gets vertical components\n$\\{\\xi^{\\ast}_{vert;i}\\}$, so that $\\xi^{\\ast}_{vert;i}$ is the vertical\ncomponent in the formal neighborhood of the $i-$ section. Projecting\n$\\xi^{\\ast}_{vert;i}$ on $Base(\\mbox{${\\cal E}$}_{S})$ one gets some element of $U(\\mbox{\\g g})$,\nsay\n$u_{i}$;  projecting $\\xi^{\\ast}_{vert;i}$ on $J^{(1)}(\\mbox{${\\cal C}$}_{S})$ one gets some\nvector field, say $v_{i}$. Both $u_{i},\\; v_{i}$ act on our $\\mbox{$\\hat{\\gtg}$}^{A}$-module\n$M^{A}$:\n$u_{i}$ naturally, $v_{i}$ by means of the Sugawara construction. Going over\ndefinitions one gets that this well defines a twisted $\\mbox{${\\cal D}$}-$module with the\nfiber as in the theorem. $\\Box$\n\n\\bigskip\n\n\nDenote the constructed $\\mbox{${\\cal D}$}-$module by $\\Delta_{\\psi}(M^{A})$.\n\nIn the case when $M^{A}$ is an admissible representation the following result\nis valid.\n\n\\begin{theorem}\n\\label{smoothinadm}\nIf $n=2$ and $M^{A}$ is an admissible $\\mbox{$\\hat{\\gtg}$}^{A}-$module then\n$\\Delta_{\\psi}(M^{A})$ is {\\em holonomic}\nfor almost any vector bundle $\\mbox{${\\cal E}$}_{S}$\n(i.e.as a sheaf $\\Delta_{\\psi}(M^{A})$ is isomorphic to a sheaf of sections\n of a certain finite rank  vector bundle over some open set in $S$ ).\n\\end{theorem}\n\n{\\bf Proof.}\n\nTo prove this theorem essentially  means to show that the spaces\n $(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s)},\\; s\\in S,$ are all finite\ndimensional. That will be done\nin \\ref{finofcoinvhighgensubs},  Proposition \\ref{proofoflisse} in the higher\ngenus case\nand in \\ref{finofcoinvsferesubs}, Proposition \\ref{fincinvforcpadmweyl} for\n$\\cp$. . We will also give there a precise meaning\nto the phrase ``almost any vector bundle'' in Theorem\\ref{smoothinadm}. $\\Box$\n\n\\bigskip\n\nResults of \\ref{Quadratic_degeneration} will show that the standard\ncombinatorial algorithm can be used to calculate the dimension of the fiber of\nour\n$\\mbox{${\\cal D}$}-$module using the dimensions of the spaces of coinvariants on a sphere\nwith 3 punctures.\nThe latter dimensions will be calculated in \\ref{The_rational_leve_case}.\n\n\\section{The spaces of coinvariants}\n\\label{The_spaces_of_coinvariants}\n In this section we will be concerned\nwith the space of coinvariants $(M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}$\n(or  spaces closely related to it ) in the case when $M^{A}$ is either\na generalized Weyl module or an admissible representation. The standard tool to\nget finiteness results\nabout coinvariants is the notion of\n {\\em singular support}.\n\n\\subsection{Singular support and coinvariants}\n\\label{singsuppandcoinvvvvv}\n\nLet $\\mbox{\\g a}$ be a Lie algebra. Universal enveloping algebra $U\\mbox{\\g a}$ is filtered\nin\n the standard way so that the associated graded algebra is $S\\mbox{\\g a}$. One says\nthat a filtration of a finitely generated $\\mbox{\\g a}-$module $V$ is good if (i) it\nis\ncompatible with the filtration of $U\\mbox{\\g a}$, and (ii) the associated graded\nmodule $Gr\\, V$ is finitely generated as an $S\\mbox{\\g a}$-module.\n\n{\\bf Definition} Singular support, $SS V$, of $V$ is the zero set of the\nvanishing ideal of the $S\\mbox{\\g a}$-module $Gr\\, V$. $\\Box$\n\n\\bigskip\n\nObviously, $SS V$ is a conical subset of $\\mbox{\\g a}^{\\ast}$.\n\nFor a subalgebra $\\mbox{\\g n}\\subset \\mbox{\\g a}$, call  $V$ an $(\\mbox{\\g a},\\mbox{\\g n})$-module if it is\nan $\\mbox{\\g a}-$module and\n$\\mbox{\\g n}$ acts on $V$ locally nilpotently. Typical example: any module from the\n$\\mbox{${\\cal O}$}-$category is a $(\\mbox{$\\hat{\\gtg}$},\\mbox{$\\hat{\\gtg}$}_{>})-$module.\n\n\n\\begin{lemma} (\\mbox{ see \\cite{beil_feig_maz}})\n\\label{singsupp-finitenessofcoinv}\nLet $\\mbox{\\g a}$ be a Lie algebra and $\\mbox{\\g p}\\subset\\mbox{\\g a}$ be its subalgebra. Denote by\n$\\mbox{\\g p}^{\\perp}$ the\nannihilator of $\\mbox{\\g p}$ in $\\mbox{\\g a}^{\\ast}$\nLet $V$ be an $(\\mbox{\\g a},\\mbox{\\g n})$-module.\nIf $SSM\\cap\\mbox{\\g p}^{\\perp}=\\{0\\}$ and $dim\\, \\mbox{\\g a}\/\\mbox{\\g n}\\oplus\\mbox{\\g p}<\\infty$ then\n$\\mbox{dim}M_{\\mbox{\\g p}}<\\infty$.\n\\end{lemma}\n\nRecall that from now on $\\mbox{\\g g}=\\mbox{\\g sl}_{2}$ unless otherwise stated.\n\n\\subsection{ Singular support of $\\mbox{$\\hat{\\gtg}$}^{A}-$modules}\n\\label{singsuppppofhgta}\nObserve that there is an involution $\\sigma$ of $\\mbox{$\\hat{\\gtg}$}$ sending $f$ to\n$e\\otimes z^{-1}$ and\n$e\\otimes z^{-1}$ to $f$, see\n\\ref{defofalgebrandmodusu}\nfor notations. There arises the involution, also denoted by $\\sigma$, acting\non the algebras $\\mbox{$\\hat{\\gtg}$}^{A}$ and their duals. This involution is not canonical\nbut we do not have\nto care as our considerations here are purely local.\n\nDenote by $\\Omega^{A}$ the space of $\\mbox{\\g g}-$ valued differential\nforms on the formal neighborhoods of the points from $A$. There is a natural\nembedding\n$\\Omega^{A}\\hookrightarrow (\\mbox{\\g g}^{A})^{\\ast}$ (``take the traces and\n then sum up all the residues!'')\n\nWe will make use of 2 subspaces of $\\Omega^{A}$: $\\Omega^{A}_{reg}$ is all\nregular forms and\n$\\Omega^{A}_{nilp}$ is all forms with values in the nilpotent\ncone.\n\\begin{theorem}\n\\label{theoronsingsupp}\n\n(i) If $M^{A}$ is a generalized Weyl module then\n$SSM^{A}=\\Omega^{A}_{reg}\\cup\\sigma\\Omega^{A}_{reg}$.\n\n(ii) (E.Frenkel, B.F.) If $M^{A}$ is an admissible representation then\n$SSM^{A}=\\Omega^{A}_{nilp}\\cup\\sigma\\Omega^{A}_{nilp}$.\n\\end{theorem}\n\n\\begin{remark}\nIt is easy to see that although $\\sigma$ is not determined uniquely the\nspaces $\\sigma\\Omega^{A}_{reg}$, $\\sigma\\Omega^{A}_{nilp}$ are canonical.\nFor example  $\\sigma\\Omega^{A}_{reg}$ is the space of forms such that:\n\nthey have at most order 1 pole at $\\bar{A}$;\n\ntheir residue at each $P_{i}\\in \\bar{A}$ belongs to $\\mbox{\\g n}_{i}$;\n\nat each $P_{i}\\in\\bar{A}$ their constant term  belongs to $\\mbox{\\g b}_{i}$.\n\\end{remark}\n\n\\subsection{Finiteness of coinvariants --  the higher genus case}\n\\label{Finiteness_of_coinvariants-hghergen}\n\n\n\n\n\n\\subsubsection{ Hitchin's theorem.}\n\\label{hitchinstheorem}\nFirst recall a well-known result of  Hitchin, \\cite{hitch}.\nWith a vector bundle $\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}$ associate the map\n\\begin{eqnarray}\n\\label{def-fhitchmap}\nH(\\mbox{${\\cal E}$}):\\;H^{0}(\\mbox{${\\cal C}$},\\Omega\\otimes\\mbox{End}\\mbox{${\\cal E}$})\\rightarrow\n \\oplus_{i=2}^{n}H^{0}(\\mbox{${\\cal C}$},\\Omega^{\\otimes i}),\\\\\nX\\mapsto \\mbox{Tr}X^{i}\\nonumber\n\\end{eqnarray}\nCall a bundle $\\mbox{${\\cal E}$}$ {\\em exceptional} if $\\mbox{ker}H(\\mbox{${\\cal E}$})\\neq 0$. Obviously\n$\\mbox{ker}H(\\mbox{${\\cal E}$})$ is exactly the space of global differential forms\nwith values in nilpotent endomorphisms of the vector bundle $\\mbox{${\\cal E}$}$.\n\n\\begin{theorem} (\\mbox{Hitchin \\cite{hitch}})\n\\label{hitchi}\n Zero set of the map (\\ref{def-fhitchmap}) is a maximal Lagrangian\nsubmanifold in the cotangent bundle of the moduli space of vector bundles over\n$\\mbox{${\\cal C}$}$. In particular,\nexceptional\nvector bundles form a positive codimension algebraic subset of the moduli space\nof vector bundles.\n\\end{theorem}\n\nFor us, importance of Theorem \\ref{hitchi} is in  that generically a vector\nbundle does not allow\na non-trivial global differential form with coefficients in nilpotent\nendomorphisms of the bundle.\n\n\\subsubsection { Subtracting lines from rank 2 vector bundles.}\n\\label{subtrlinesfromvectbundles}\n\n An analogue of subtracting a point\nfrom a line bundle (or, better to say, from  its divisor) is an operation of\nsubtracting a line from a rank 2  vector bundle.\n\nTo a  rank 2 vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}$ one can associate a  module over\nthe sheaf of regular functions -- the\nsheaf of sections of $\\mbox{${\\cal E}$}$.; denote this sheaf by $Sect(\\mbox{${\\cal E}$})$. This establishes\na one-to-one correspondence between rank 2 vector bundles and rank 2 locally\nfree modules over the sheaf of regular functions.\n Now fix a line, $l$, in a fiber of $\\mbox{${\\cal E}$}$ over some point $P\\in\\mbox{${\\cal C}$}$. Denote by\n$S (l)$ a sheaf such that:\n\n(i) $S(l)|_{U}=Sect(\\mbox{${\\cal E}$})|_{U}$ if $P$ does not belong to $U$;\n\n(ii)  $S(l)|_{U}, P\\in U,$ is the space of meromorphic sections of $\\mbox{${\\cal E}$}$ over\n$U$ regular outside $P$, having at most order 1 pole at $P$ and such that their\nresidue at $P$ belongs to the fixed line $l$.\n\nIt is obvious that $S(l)$ is a rank 2 locally free module. Therefore it\ndefines a rank 2  vector bundle. Denote this vector bundle by $\\mbox{${\\cal E}$}(l)$.\nIf a collection of lines -- $l_{1},l_{2},...,l_{m}$ -- is subtracted, then\ndenote\nthe corresponding vector bundle by $\\mbox{${\\cal E}$}(l_{1}+\\cdots + l_{m})$.\n\nSuppose we have a moduli space of rank 2 vector bundles with parabolic\nstructure with fixed determinant. Elements of such a space are isomorphism\nclasses of the data\n(vector bundle $\\mbox{${\\cal E}$}$, fixed lines $l_{1},...,l_{m}$ in some fibers.) It is\nrather\nclear  that the map $(\\mbox{${\\cal E}$},\\; l_{1},...,l_{m})\\mapsto (\\mbox{${\\cal E}$}(l_{1}+\\cdots\n+l_{m}),\\; l_{1},...,l_{m})$ is a homeomorphism of 2 moduli spaces with\ndifferent determinants.\n\n{\\bf Definition.}  Call the data $(\\mbox{${\\cal E}$},\\; l_{1},...,l_{m})$ generic if\n $\\mbox{${\\cal E}$}(l_{i_{1}}+\\cdots +l_{i_{s}})$ is not  exceptional for\nany subset $\\{i_{1},...,i_{s}\\}\\subset\\{1,2,..., m\\}$. $\\Box$\n\n\\bigskip\n\nIt follows from Theorem \\ref{hitchi} that the set of generic vector bundles is\nopen and everywhere dense.\n\n\n\n\n\\subsubsection { Finiteness of coinvariants.}\n\\label{finofcoinvhighgensubs}\n\nSuppose we are in the situation of \\ref{defofocat}: we have an admissible\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module\n$M^{A}$ on the curve $\\mbox{${\\cal C}$}$ with a vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}$.\nAs $A$ is a collection of borel subalgebras $\\mbox{\\g b}_{1},...,\\mbox{\\g b}_{m}$ operating\nin fixed fibers, we have\nparabolic structure -- lines $l_{1},...,l_{m}$ in the corresponding fibers\npreserved by the $\\mbox{\\g b}_{i}$'s.\nCall the data $(\\mbox{${\\cal E}$},A)$  generic if the data $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$ is\ngeneric\nin the sense of \\ref{subtrlinesfromvectbundles} above.\n\n\n\n\n\n\nRecall that we are interested in the space of coinvariants\n$M^{A}_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}$, where $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is an algebra of endomorphisms of\nthe bundle $\\mbox{${\\cal E}$}$ regular outside points from the corresponding $\\bar{A}$, see\n\\ref{algebra_gofA} and \\ref{maingenerresults}, Theorem \\ref{existofDmod}.\n\n\n\\begin{proposition}\n\\label{proofoflisse}\n Let $(\\mbox{${\\cal E}$}, A)$ be generic. Then\n\\[\\mbox{dim} M^{A}_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}<\\infty.\\]\n\\end{proposition}\n\n{\\bf Proof.}  One extracts from definitions that the annihilator\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}$ of the algebra $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is the space\n$\\Omega_{\\mbox{${\\cal C}$},A}(\\mbox{${\\cal E}$})$ of global meromorphic $End(\\mbox{${\\cal E}$})-$valued differential\nforms regular outside\n$\\bar{A}\\subset\\mbox{${\\cal C}$}$.\n\nBy Theorem \\ref{theoronsingsupp}(ii) we get that\n$SSM^{A}\\cap\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}=\\Omega_{nilp}(\\mbox{${\\cal E}$})\\cup\\sigma\\Omega_{nilp}(\\mbox{${\\cal E}$})$, where\n$\\Omega _{nilp}(\\mbox{${\\cal E}$})$ is the space global nilpotent transformations of $\\mbox{${\\cal E}$}$,\nand $\\sigma$ is the twist introduced in \\ref{singsuppppofhgta}.\n\nGenericity condition means that $\\Omega_{nilp}(\\mbox{${\\cal E}$})=0$, see\n\\ref{hitchinstheorem} and  \\ref{subtrlinesfromvectbundles}.\n\nOn the other hand it is easy to see that the operation of subtracting a line\ngenerates the twist $\\sigma$ on endomorphisms. (In fact one has to compose\nsubtracting of a line\nwith a reflection in the fiber, but this does not change the isomorphism class\nof the bundle.) Therefore genericity condition also implies that\n$\\sigma\\Omega_{nilp}(\\mbox{${\\cal E}$})=0$.\n\nHence we get that $SSM^{A}\\cap\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}=0$. And as the space\n$\\mbox{$\\hat{\\gtg}$}^{A}_{>}+\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is  of finite codimension in $\\mbox{$\\hat{\\gtg}$}^{A}$,\napplication\nof Lemma \\ref{singsupp-finitenessofcoinv}, see \\ref{singsuppandcoinvvvvv},\ncompletes the proof. $\\Box$\n\n\\bigskip\n\n\n\\bigskip\n\nIn order to study quadratic degenerations we will need the following\nstronger finiteness result. Along with the set\n$A=\\{(P_{1},\\mbox{\\g b}_{1}),...,(P_{m},\\mbox{\\g b}_{m})\\}$, consider the set $ A_{2} =\\,\n\\{(P_{m+1},\\mbox{\\g b}_{m+1}), (P_{m+2},\\mbox{\\g b}_{m+2})\\}$ such that the points\n$P_{1},...,P_{m+2}\\in\\mbox{${\\cal C}$}$ are different. Denote by $\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})$ the\nsubalgebra of $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ consisting of functions taking values in\n$\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$ at point $P_{i}$, $i=m+1,m+2$.\n\n\\begin{proposition}\n\\label{highgen_finit_afterpionching}\n\nIf $(\\mbox{${\\cal E}$}, A\\bigsqcup A_{2})$ is generic and $M^{A}$ is admissible, then\n\\[dim\\, (M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})}<\\infty.\\]\n\n\\end{proposition}\n\n{\\bf Proof.} We are again going to apply Lemma\n\\ref{singsupp-finitenessofcoinv}. Observe that\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}$ consists of meromorphic forms on $\\mbox{${\\cal C}$}$ with values\nin $End(\\mbox{${\\cal E}$})$,\nregular outside $\\{P_{1},...,P_{m+2}\\}\\subset\\mbox{${\\cal C}$}$, having at most order 1 poles\nat $P_{m+1},P_{m+2}$, their residues at the latter points lying in $\\mbox{\\g b}_{1}$\n( $\\mbox{\\g b}_{2}$ resp.).\n\nBy Theorem \\ref{theoronsingsupp}(ii), $\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}\\cap\\, SSM^{A}$\nconsists of forms with values in nilpotent endomorphisms, satisfying the above\nlisted global conditions. This implies, in particular, that actually residues\nof our forms\nbelong to $\\mbox{\\g n}_{m+1},\\mbox{\\g n}_{m+2}$ at $P_{m+!}, P_{m+2}$ resp..\n\nGiven an element $\\omega\\in\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}\\cap\\, SSM^{A}$, subtract\nsome lines from $\\mbox{${\\cal E}$}$ so as to make $\\omega$ be everywhere regular. Genericity\ncondition implies then that $\\omega=0$, and application of   Lemma\n\\ref{singsupp-finitenessofcoinv} completes the proof. $\\Box$\n\n\\bigskip\n\n\n\n\n\n\n\n\n\n\\subsection{Finiteness of coinvariants -- the case of $\\cp$}\n\\label{Finiteness_of_coinvariants}\n\n\\subsubsection{Generic vector bundles on $\\cp$}\n\\label{genervectbumdonsph}\n\nLet $O(n)$ be the degree $n$ line bundle over $\\cp$. It is known, e.g.\n\\cite{pr_seg}, that any rank 2 vector bundle over $\\cp$ is a direct sum\n$O(r)\\oplus O(s)$ for some $r$, $s$.\n\nAs there are no moduli, it is hard to speak about generic vector bundles.\nNevertheless we will call $O(r)\\oplus O(s)$ {\\em exceptional} if $|r-s|>1$.\nHere is\na justification.\n\n\\begin{lemma}\n\\label{whyexceptional}\n\nLet $\\mbox{${\\cal E}$}= O(r)\\oplus O(s)$ and $(\\mbox{${\\cal E}$},l_{1},...,l_{m})$, $m\\geq |r-s|$, a vector\nbundle with parabolic structure. Then generically with respect to\n$l_{1},...,l_{m}$ the bundle $\\mbox{${\\cal E}$}(l_{1}+\\cdots +l_{m})$ is not exceptional:\n\n\\[\\mbox{${\\cal E}$}(l_{1}+\\cdots +l_{m})=\\left\\{\\begin{array}{lll}\nO(p+1)\\oplus O(p)&\\mbox{ if } r+s-m=2p+1\\\\\n O(p)\\oplus O(p)&\\mbox{ if } r+s-m=2p.\n\\end{array}\n\\right.\\]\n\\end{lemma}\n\nLemma \\ref{whyexceptional} seems to be common knowledge, although we failed\nto find a reference with its proof.\n\nProceed just like we did in \\ref{subtrlinesfromvectbundles}:  call  $(\\mbox{${\\cal E}$},\\;\nl_{1},...,l_{m})$ generic if\n $\\mbox{${\\cal E}$}(l_{i_{1}}+\\cdots +l_{i_{s}})$ is not  exceptional for\nany subset $\\{i_{1},...,i_{s}\\}\\subset\\{1,2,..., m\\}$.\n\n\\subsubsection{Finiteness of coinvariants}\n\\label{finofcoinvsferesubs}\n\nA specific feature of the genus zero case is that we do not necessarily have\nto consider admissible representations -- generalized Weyl modules, see\n\\ref{Case_(i)},\n will also do.\n\nLet us again consider a vector bundle $\\mbox{${\\cal E}$}$ over $\\cp$ and a $\\mbox{$\\hat{\\gtg}$}^{A}-$module\n$M^{A}$. As in \\ref{finofcoinvhighgensubs}, $A$ determines a parabolic\nstructure\non $\\mbox{${\\cal E}$}$, say $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$. Call the data $(\\mbox{${\\cal E}$},A)$ generic\nif $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$ is also.\n\n\n\\begin{proposition}\n\\label{fincinvforcpadmweyl}\nIf  $(\\mbox{${\\cal E}$},A)$ is generic and $M^{A}$ is either admissible or generalized Weyl\nmodule, then\n\n\\[\\mbox{dim } (M^{A})_{\\mbox{\\g g}(\\cp,A)}<\\infty.\\]\n\n\\end{proposition}\n\n{\\bf Proof}  is a simplified version of the proof of Proposition\n\\ref{proofoflisse} in \\ref{finofcoinvhighgensubs}. The new features are as\nfollows: to include generalized Weyl modules one uses Theorem\n\\ref{theoronsingsupp}(i) in addition to Theorem \\ref{theoronsingsupp}(ii);\ninstead of the Hitchin's theorem one uses the ``observation'' that $O(n)$ has\nno\nnon-zero global sections if $n<0$. $\\Box$\n\n\\begin{corollary}\n\\label{existofdmodforweyl}\n If $M^{A}$ is a generalized Weyl module then\nthere is a holonomic\ntwisted $D$-module living in the space $(J^{(1)}(\\cp)\\times\nJ^{(1)}(\\cp))^{\\times m}$ with the fiber $M^{A}_{\\mbox{\\g g}(\\cp,A)})$.\n\\end{corollary}\n\n{\\bf Proof.} Repeating word for word proof of  Theorem\\ref{smoothinadm} one\nderives\n from Proposition \\ref{fincinvforcpadmweyl} existence of a twisted $D-$module\non\nthe space $(Base(\\mbox{${\\bf C}$}^{2})\\times J^{(1)}(\\cp))^{\\times m}$. But for $\\mbox{\\g sl}_{2}$,\n the flag manifold is $\\cp$ and\nthe base affine space $(Base(\\mbox{${\\bf C}$}^{2})$ is  also the space of 1-jets of\nparameter $ J^{(1)}(\\cp))^{\\times m}$. $\\Box$\n\n\\bigskip\n\nAs in \\ref{finofcoinvhighgensubs}, we want to prove a generalization of\nProposition \\ref{fincinvforcpadmweyl} in order to prepare grounds for studying\nquadratic degeneration.\n\nAlong with $A=\\{(P_{1},\\mbox{\\g b}_{1}),\\ldots ,(P_{m},\\mbox{\\g b}_{m})\\}$ consider 2 sets\n$A_{1}=\\{(P_{m+1},\\mbox{\\g b}_{m+1})\\}$ and $A_{2}=\\{(P_{m+1},\\mbox{\\g b}_{m+1}),\n(P_{m+2},\\mbox{\\g b}_{m+2})\\}$\nsuch that $P_{1},...,P_{m+2}$ are different points in $\\mbox{${\\cal C}$}$.\n\nWith $A_{1}$ and $A_{2}$ associate the following 2 subalgebras of\n$\\mbox{\\g g}(\\cp,A)$:\n$\\mbox{\\g g}(\\cp,A,A_{1})$ consists of all functions taking values in\n$\\mbox{\\g n}_{m+1}=[\\mbox{\\g b}_{m+1},\\mbox{\\g b}_{m+1}]$\nat the point $P_{m+1}$;\n$\\mbox{\\g g}(\\cp,A,A_{2})$ consists of all functions taking values in\n$\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$\nat the point $P_{i}$, $i=m+1,\\;m+2$.\n\n\\begin{proposition}\n\\label{finofcoinv_general}\n Let   $(\\mbox{${\\cal E}$}, A_{2})$ be\ngeneric. Then\n\n(i) If $M^{A}$ is a generalized Weyl module over $\\mbox{$\\hat{\\gtg}$}^{A}$, then\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}<\\infty$;\n\n(ii)  If $M^{A}$ is an admissible representation of $\\mbox{$\\hat{\\gtg}$}^{A}$, then\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$.\n\\end{proposition}\n\n \\bigskip\n\n{\\bf Proof.} of (ii) repeats almost word for word that of Proposition\n\\ref{highgen_finit_afterpionching} in \\ref{finofcoinvhighgensubs} with\nsimplifications analogous to those\nindicated in the proof of Proposition  \\ref{fincinvforcpadmweyl}.\n\nAs to (i), its proof is again application of the same technique in a slightly\ndifferent\nform: one has to take a form $\\omega\\in\\mbox{\\g g}(\\cp,A,A_{2})^{\\perp}\\cap SSM^{A}$\nand to subtract lines from $\\mbox{${\\cal E}$}$ so as to make $\\omega$ into\na form with either one pole (at $P_{m+1}$) or 2 poles (one of them is\nagain at $P_{m+1}$) in such a way that the bundle obtained is $O(n)\\oplus\nO(n)$.\nThe 2 cases are of course distinguished by the parity of the difference\nbetween the degrees of the determinant of $\\mbox{${\\cal E}$}$ and $\\omega$. In both cases\nit is easy to prove that $\\omega=0$ using the fact that any differential form\nwith trivial coefficients has at least 2 poles. $\\Box$\n\n\\subsubsection{Holonomic $D$-module on $(\\mbox{${\\bf C}$}\\times\\mbox{${\\bf C}$})^{\\times m}$}\n\\label{holondmodoncc}\n\n We will now get rid of twisted differential operators in Corollary\n\\ref{existofdmodforweyl} under the assumption that the vector bundle\n$\\mbox{${\\cal E}$}\\rightarrow \\cp$ is trivial.\nConsider the set $A'=A\\bigsqcup (P_{\\infty},\\mbox{\\g b}_{\\infty})$. Attach to the\npoint $(P_{\\infty},\\mbox{\\g b}_{\\infty})$  the module $(V^{0}_{0},V_{0})$ known\nas the vacuum representation, see \\ref{Case_(i)} for notations.\n$(P_{\\infty},\\mbox{\\g b}_{\\infty})$ can be redefined as the module induced from the\ntrivial representation (see also \\ref{Case_(i)}) and therefore there is an\nisomorphism $ M^{A}_{\\mbox{\\g g}(\\cp,A)}\\approx M^{A'}_{\\mbox{\\g g}(\\cp,A')}$. Now consider\nthe twisted $D-$module with fiber $ M^{A'}_{\\mbox{\\g g}(\\cp,A')}$ on the space\n$(J^{(1)}(\\cp)\\times J^{(1)}(\\cp))^{\\times m+1}$. Restrict it to the space\n $(J^{(1)}(\\cp)\\times J^{(1)}(\\cp))^{\\times m}$ by having  the point\n$(P_{\\infty},\\mbox{\\g b}_{\\infty})$ fixed. The result of this operation is that the\nbundles in question trivialize: $\\cp - \\mbox{\\g b}_{\\infty}=\\cp - P_{\\infty}=\\mbox{${\\bf C}$}$ and\n$J^{(1)}(\\mbox{${\\bf C}$})=\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}$. Further pushing forward by ``integrating\nalong $\\mbox{${\\bf C}$}^{\\ast}$'' one gets a $D-$module over the space $(\\mbox{${\\bf C}$}\\times\n\\mbox{${\\bf C}$})^{m}$.\nObserving that it is appearance of the bundle $J^{(1)}(\\cp)\\rightarrow\\cp$\nwhich\nwas responsible for the twisting of the $D-$module, one argues that we get\na usual holonomic $D$-module on $(\\mbox{${\\bf C}$}\\times \\mbox{${\\bf C}$})^{m}$ with fiber $\nM^{A}_{\\mbox{\\g g}(\\cp,A)}$. In particular, we get a bundle with flat connection\nover an open subset of $(\\mbox{${\\bf C}$}\\times \\mbox{${\\bf C}$})^{m}$.\n\n\n{\\bf Notation.} Denote the constructed in this way bundle with flat connection\nby $\\Delta(M^{A})$. $\\Box$\n\n\\bigskip\n\n\n\n  We are unable to describe this open\nsubset explicitly at present. It follows from the requirement that\n$(\\mbox{${\\cal E}$}, A)$ be generic in all our finiteness results however that the diagonals\nshould\nbe thrown away meaning that $P_{i}\\neq P_{j}$ and $\\mbox{\\g b}_{i}\\neq\n\\mbox{\\g b}_{i}\\neq\\mbox{\\g b}_{j}$ for all $i\\neq j$.\n\nOne may want to write down differential equations satisfied by horizontal\nsections of this bundle. We will show in \\ref{diffeqsatbycorrfuntttt} that\nhorizontal sections satisfy a system of $2m$ differential equations of\nwhich\n$m$equations are Knizhnik-Zamolodchikov equations and the other $m$ are\nobtained from singular vectors of the Verma module projecting onto $M^{A}$.\n\nEverything said here holds true for an admissible representation. It is easy to\nsee\nthat the bundle associated with an admissible representation is a quotient of\nthe just constructed bundle for the corresponding generalized Weyl module.\n\n\n\n\n\n\n\\subsection{Calculation of the dimensions of coinvariants. Fusion algebra}\nLet $\\mbox{${\\cal E}$}\\rightarrow\\cp$ be the rank 2 trivial vector bundle and $M^{A}$ be a\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module.\nHere we will calculate the dimension of the space\n$(M^{A})_{\\mbox{\\g g}(\\cp,A)},\\;\\sharp A=3$, in the\nfollowing 2 cases:\n(i) the level $k$ is not\n rational and $M^{A}$ is a generalized Weyl module; (ii) $k+2=p\/q$, $p\\mbox{\nand }q$\n being positive integers,\nand $M^{A}$ is an admissible representation. Without loss of generality we can:\n\nfix a coordinate $z$ on $\\cp$; assume that\n$A=\\{(0,\\mbox{\\g b}_{0}),\\;(1,\\mbox{\\g b}_{1}),\\; (\\infty,\\mbox{\\g b}_{\\infty})\\}$, where\n$\\mbox{\\g b}_{0}=\\mbox{${\\bf C}$} e\\oplus\\mbox{${\\bf C}$} h,\\;\\mbox{\\g b}_{\\infty}=\\mbox{${\\bf C}$} f\\oplus\\mbox{${\\bf C}$} h$ and\n$\\mbox{\\g b}_{1}=\\mbox{${\\bf C}$}(e-h-f)\\oplus\n\\mbox{${\\bf C}$}(h+2f)$.\n\n(In fact, for any $\\mbox{\\g b}_{0}\\neq\\mbox{\\g b}_{\\infty}$  we can always choose a basis of\n$\\mbox{\\g g}$ so that\n$\\mbox{\\g b}_{0},\\mbox{\\g b}_{\\infty}$ are as above. As to $\\mbox{\\g b}_{1}$, there really is some\nfreedom but it is\neasy to see that all the calculations below are independent of the choice. We\nhave set\n$\\mbox{\\g b}_{1}=(\\exp{f})\\mbox{\\g b}_{0}(\\exp{-f})$.)\n\n\\subsubsection{ The generic level case}\n\\label{Thegenericlevelcase}\nSo by \\ref{defofocat}  we are given three irreducible generalized Weyl\n modules $V^{0}_{\\lambda_{0},k},\nV^{1}_{\\lambda_{1},k},V^{\\infty}_{\\lambda_{\\infty},k}$.\nRecall, see \\ref{Case_(i)},\nthat generalized Weyl modules are parametrized by symbols\n$(V_{m}^{\\epsilon},V_{n})$, where\n$m,n$ are nonnegative integers,\n$\\epsilon\\in\\nz\/2\\nz$ and $V_{m}$ is an $m+1$-dimensional $\\mbox{\\g g}-$module.\nTherefore we\ncan and will assume that we have\n\\[(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}),\\;i=0,1,\\infty .\\]\n\nIt is convenient to interpret the result of calculation of $\\mbox{dim }\n(\\otimes_{i}(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ in terms of\nthe\n {\\em fusion algebra}.\nThe latter is defined as follows.   Suppose\n that for any pair of generalized Weyl modules, say\n$(V_{r_{i}}^{\\alpha_{i}},V_{s_{i}}),\\;i=0,1$, there is only finite number of\n$(V_{r_{\\infty}}^{\\alpha_{\\infty}},V_{s_{\\infty}})$ such that\n\\[\\mbox{dim }\n(\\otimes_{i=0,1,\\infty}(V_{r_{i}}^{\\alpha_{i}},V_{s_{i}}))_{\\mbox{\\g g}(\\cp,A)}\\neq\n0.\\]\nNow view the symbols $(V_{m}^{\\epsilon},V_{n})$ as generators of a free abelian\ngroup. Then\nthere naturally arises an algebra (over $\\nz$) with the operation of\n  multiplication $\\circ$ defined by\n\\[(V^{\\alpha_{0}}_{r_{0}},V_{s_{0}})\\circ (V^{\\alpha_{1}}_{r_{1}},V_{s_{1}})=\n\\sum_{(r_{\\infty},s_{\\infty},\\alpha_{\\infty})}\\mbox{dim }\\{\n(\\otimes_{i=0,1,\\infty}(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}\\}\n(V^{\\alpha_{\\infty}}_{r_{\\infty}},\nV_{s_{\\infty}}).\\]\nThe algebra defined in this way is called {\\em fusion\nalgebra}. Of course structure constants of the fusion\nalgebra determine the dimensions of the spaces of coinvariants.\n\n\nOne last piece  of notation:\n in the following\ntheorem  we  formally set $(X\\oplus Y,Z)=(X,Z)+(Y,Z)$\nand $(X,Y\\oplus Z)=(X,Y)+(X,Z)$. Recall also that in the category of\n$\\mbox{\\g g}-$modules\none has\n\\[V_{r}\\otimes V_{s}\\approx V_{r+s}\\oplus V_{r+s-2}\\oplus\\cdots\\oplus\nV_{|r-s|}.\\]\n\n\\begin{theorem}\n\\label{fusalggencase}\n(i) For any triple of generalized Weyl modules\nthe space $(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ is finite\ndimensional.\n\n(ii) The fusion algebra is well-defined, multiplication being given by the\nfollowing formula\n\\begin{eqnarray}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})&=&\\nonumber\\\\\n(V_{r_{1}+r_{2}}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&+&\n(V_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1},V_{s_{1}}\\otimes V_{s_{2}})+\n(V_{r_{1}+r_{2}-2}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})+\\cdots\n+\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&.&\\nonumber\n\\end{eqnarray}\n\\end{theorem}\n\n\\subsubsection{ }\n{\\bf Proof of Theorem\\ref{fusalggencase}.}\n\nThroughout the proof $A$ will stand for $\\{(0,\\mbox{\\g b}_{0}),(1,\\mbox{\\g b}_{1})\\}$,\n$A_{1}$ -- for\n$\\{(\\infty,\\mbox{\\g b}_{\\infty})\\}$. Along with the algebras\n$\\mbox{\\g g}(\\cp,A),\\; \\mbox{\\g g}(\\cp,A,A_{1})$ (see \\ref{Finiteness_of_coinvariants})\nintroduce the\nalgebra $\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\\subset \\mbox{\\g g}(\\cp,A)$ consisting of all\nfunctions\ntaking values in $\\mbox{\\g b}_{\\infty}$ at the point $\\infty$.\n\n Of course\n$\\mbox{\\g g}(\\cp,A,A_{1})\\subset\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})$ is an ideal and\n$\\mbox{dim }\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\/\\mbox{\\g g}(\\cp,A,A_{1})=1$. Define\n$\\bar{h}_{\\infty}$\nto be a basis element of $\\mbox{dim\n}\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\/\\mbox{\\g g}(\\cp,A,A_{1})$.\nIt is a standard (and simple) fact of Lie algebra cohomology theory that\n$\\bar{h}_{\\infty}$ acts on $(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}$.\n\n\\begin{lemma}\n\\label{cohom_for_Weyl}\n\n\n Let $M^{A}$ be a generalized Weyl module.  The element $\\bar{h}_{\\infty}$\n has a simple spectrum as an operator acting on\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})} $. Further, if\n$M^{A}=(V_{r_{1}}^{\\alpha},V_{s_{1}})\\otimes (V_{r_{2}}^{\\beta},V_{s_{2}})$\nthen the\nset of eigenvalues\nof $\\bar{h}_{\\infty}$ is the set of the highest weights of the modules\nappearing in\nthe right-hand side of Theorem\\ref{fusalggencase}(ii).\n\\end{lemma}\n\nProof of this lemma is essentially the same as that of Theorem 4.4 in\n\\cite{feig_mal1} and\nmostly consists of solving a system of 2 equations related to 2 singular\nvectors -- one\nin $(V_{r_{1}}^{\\alpha},V_{s_{1}})$, another in\n$(V_{r_{2}}^{\\beta},V_{s_{2}})$.\nWe will discuss it in \\ref{prooflemma49}. Derivation\nof Theorem \\ref{fusalggencase} from Lemma \\ref{cohom_for_Weyl} is again very\nsimilar\nto that of Theorem 3.2 from Theorem 4.4 in {\\em loc. cit}\nand uses Verma modules as follows.\n\n\\begin{lemma}\n\\label{Berma-Weyl-gener}\n\n(i) Let $(M^{A})^{\\mu}_{\\mbox{\\g g}(\\cp,A,A_{1})}\\in (M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})} $\n be the eigenspace related to the eigenvalue\n$\\mu$ of $\\bar{h}_{\\infty}$. Then $(M^{A})^{\\mu}_{\\mbox{\\g g}(\\cp,A,A_{1})}\\approx\n(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}$.\n\n(ii) Projection of a Verma module $M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}$ onto a\ngeneralized Weyl module $ W$ induces an isomorphism\nof the coinvariants\n\\[(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup\nA_{1})}\\approx\n(M^{A}\\otimes W)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}.\\]\n\\end{lemma}\n\n{\\bf Proof of Lemma\\ref{Berma-Weyl-gener}}\n\n(i) A Verma module sitting at a point is induced from the 1-dimensional\nrepresentation of the\n algebra\nof functions on the formal disk whose value at the point belong to the\ncorresponding Borel\nsubalgebra. Therefore (i) follows from Frobenius duality.\n\n\n\n\n\n(ii)  Consider the resolution of $W$ by Verma modules (see \\ref {Case_(i)},\nformula\n(\\ref{exactseqforweyl}) ):\n\\[0\\rightarrow M\\rightarrow M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}\\rightarrow\nW\\rightarrow 0\\]\nand tensor it with $M^{A}$. There arises the long exact sequence of homology\ngroups of which\nwe consider the following part:\n\n\\[(M^{A}\\otimes M)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}\\rightarrow\n(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup\nA_{1})}\\rightarrow\n(M^{A}\\otimes W)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}\\rightarrow 0.\\]\n\nSince $M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}$ projects onto a Weyl module, the Verma\nmodule $M$\ndoes not, see \\ref{Case_(i)}. Lemma\\ref{cohom_for_Weyl} and  now give that\n$(M^{A}\\otimes M)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}=\\{0\\}.$ $\\Box$\n\n\\bigskip\n\nTo complete the proof of Therorem\\ref{fusalggencase} observe that\nLemma \\ref{cohom_for_Weyl}  and Lemma \\ref{Berma-Weyl-gener} together is a\nreformulation of\nTherorem\\ref{fusalggencase}. $\\Box$\n\\bigskip\n\n\\begin{corollary} Let $\\sharp A=1$ and let $A_{1}$ and $A_{2}$\nbe as in \\ref{Finiteness_of_coinvariants}. The following conditions are\nequivalent\n\\label{oncompseries-gener}\n\n(i) $M^{A}$ is a direct sum of generalized Weyl module;\n\n(ii)$SSM^{A}=\\Omega^{A}_{reg}\\cup\\sigma\\Omega^{A}_{reg}$;\n\n(iii) For any Verma module $W^{A_{1}}$\n$\\mbox{dim}(M^{A}\\otimes W^{A_{1}})_{\\mbox{\\g g}(\\cp,A\\cup A_{1},A_{2})}<\\infty$.\n\\end{corollary}\n\n\\subsubsection{ }\n\\label{prooflemma49}\nHere we sketch the proof of Lemma \\ref{cohom_for_Weyl}. First of all replace\n$M^{A}$ with\nthe corresponding Verma module -- $\\bar{M}^{A}$. Then\npass from the space\n$(\\bar{M}^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}$ to its dual, that is to the space\nof $\\mbox{\\g g}(\\cp,A,A_{1})-$invariant functionals on $\\bar{M}^{A}$.\nChoose $h\\otimes (1-z^{-1})$ to be a representative of $\\bar{h}_{\\infty}$. Let\n$\\Psi$ be the\n eigenvector of $h\\otimes (1-z^{-1})$. By definition $\\Psi$ is a linear\nfunctional on\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}\\otimes M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$. It is\nan excersise\non Frobenius duality to show that such a functional exists and unique.\n\n Define $F$ to be the\nfollowing linear\nfunctional on $M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$: $F(w)=\\Psi(w\\otimes\nv_{\\lambda_{1}})$, where,\nas usual, $v_{\\lambda_{1}})$ is the vacuum vector of\n$M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$. As\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$ is $\\nz_{+}\\times\\nz_{+}-$graded (see\n\\ref{defofalgebrandmodusu}),\nwe denote by $F_{ij}$ the restriction of $F$ to the $(i,j)-$component. Direct\ncalculations show that\nwith respect to the natural action of $\\mbox{$\\hat{\\gtg}$}$ on\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$:\n\n\\begin{eqnarray}\n\\label{strofloopmodact}\n& &\\oplus_{i,j\\in\\nz}\\mbox{${\\bf C}$} F_{ij}\\approx \\mbox{${\\cal F}$}_{\\alpha\\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\\\\n& &\\mbox{where }\\alpha=\\frac{\\lambda_{\\infty}-\\lambda_{1}-\\lambda_{0}-2}{2},\\;\n\\beta=\\lambda_{1}\n\\end{eqnarray}\n\nThe functional $F$ factors through the projection\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}\\rightarrow V_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$ if\nand only if\nit vanishes on the singular vector of $M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$. In\nother words, if this\nsingular vector, say $S$, has degree $(i,j)$ then the following equation holds\n\\[ S F_{ij}=0.\\]\nThe latter equation can be  written down and solved explicitly using formulas\n(\\ref{actonloops1} or\n\\ref{actonloops2}). Similar arguments go through for the module\n$M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$\ngiving another equation, say\n\\[ S' F_{i'j'}=0.\\]\n\nSimultaneous solutions to these 2 equations give the desired result.\nBy the way, as (\\ref{actonloops1},\n\\ref{actonloops2}) show, each of the expressions $ S F_{ij},\\; S' F_{i'j'}$\nsplits in a product of linear factors; therefore geometrically the solution is\na collection of intersection points of 2 families of lines in the plane. $\\Box$\n\n\n\n\n\\subsubsection {The rational level case}\n\\label{The_rational_leve_case}\nSuppose $k+2=p\/q$, $p\\mbox{ and }q$\n being positive integers.\nNow instead of 3 generalized Weyl modules sitting at 3 points in $\\cp$ we are\ngiven\n3 admissible representations sitting at 3 points on $\\cp$.\nRecall, see \\ref{Case_(ii)}, that  admissible representations\nare parametrized by symbols $(V_{m}^{\\epsilon},V_{n}),\\;0\\leq m\\leq q-1,0\\leq\nn\\leq p-2$ modulo the relation\n$(V_{m}^{\\epsilon},V_{n})=(V_{q-1-m}^{\\epsilon+1},V_{p-2-n})$. Denote\nby $(V_{m}^{\\epsilon},V_{n})^{\\sim }$  an equivalence class\nof $(V_{m}^{\\epsilon},V_{n})$. We assume that $(V_{m}^{\\epsilon},V_{n})^{\\sim}$\nsatifies\nthe same bilinear condition  $(V_{m}^{\\epsilon},V_{n})$ in\nTheorem\\ref{fusalggencase} does.\n\nThe definition of the fusion (Verlinde) algebra in this case repeats\nword for word that in \\ref{Thegenericlevelcase}.\n\n\nRecall finally that  Kazhdan-Lusztig fusion functor \\cite{kazh_luszt} gives\n\\[V_{r}\\dot{\\otimes}_{k}V_{s}= V_{|m-n|}\n\\oplus V_{|m-n|+2}\\cdots \\oplus V_{\\mbox{min}\\{2k-r-s,r+s\\}}.\\]\nThe following theorem was proved in \\cite{feig_mal1}\nin an equivalent but much less illuminating form.\n\n\\begin{theorem}\n\\label{fusalgratcase}\n(i) For any triple of admissible representations\nthe space $(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ is finite\ndimensional.\n\n(ii) The fusion algebra is well-defined, multiplication being given by the\nfollowing formula\n\\begin{eqnarray}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})^{\\sim }\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})^{\\sim\n}&=&\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2} V_{s_{2}})^{\\sim\n}&+&\n(V_{|r_{1}-r_{2}|+1}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2}\nV_{s_{2}})^{\\sim }+\n(V_{|r_{1}-r_{2}|+2}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2}\nV_{s_{2}})^{\\sim }+\\cdots +\n\\nonumber\\\\\n(V_{N}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2} V_{s_{2}})^{\\sim }&\n&\\nonumber,\n\\end{eqnarray}\nwhere $N=\\mbox{min}\\{2q-2-r-s,r+s\\}$.\n\\end{theorem}\n It is an easy exercise to derive this theorem from Theorem\n\\ref{fusalggencase}. For future\npurposes, however, we now sketch its original proof.  Set\n$A=\\{(\\infty,\\mbox{\\g b}_{\\infty}\\}$,\n$A_{2}=\\{(0,\\mbox{\\g b}_{0}),(1,\\mbox{\\g b}_{1})\\}$. In addition to the algebras\n$\\mbox{\\g g}(\\cp,A),\\mbox{\\g g}(\\cp,A,A_{2})$\nas in \\ref{Finiteness_of_coinvariants}, we introduce an algebra\n$\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\\subset\\mbox{\\g g}(\\cp,A)$. The latter consists of all\nfunctions whose values\nat the points $0$ ($1$ resp.) belong to $\\mbox{\\g b}_{0}$ ($\\mbox{\\g b}_{1}$ resp.).\nObviously\n$\\mbox{\\g g}(\\cp,A,A_{2})\\subset\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})$ is an ideal and the quotient\nalgebra\n$\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\/\\mbox{\\g g}(\\cp,A,A_{2})$ is commutative and 2-dimensional.\nThis algebra naturally operates on the space\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}$. Let $\\bar{h}_{0},\n\\bar{h}_{1}$ be a basis of $\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\/\\mbox{\\g g}(\\cp,A,A_{2})$.\n\n\\begin{lemma}(\\mbox{\\cite{feig_mal1}})\n\\label{cohom_for_admiss}\n\n (i) $\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$ if and only if $M^{A}$ is\n  an admissible representation.\n\n(ii) Let $M^{A}$ be an admissible representation. The elements\n$\\bar{h}_{0},\n\\bar{h}_{1}$\n have simple spectra\nas  operators acting on\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}$. Their eigenvalues recover the structure\nconstants of the\nfusion algebra.\n\\end{lemma}\n\n``Inserting'' Verma modules and using BGG resolution one derives Theorem\n\\ref{fusalgratcase}\nfrom Lemma\\ref{cohom_for_admiss} in  a way similar to that we used in\n\\ref{Thegenericlevelcase}.\n\nAnother important corollary of Lemma\\ref{cohom_for_admiss} is as follows.\n\n\\begin{corollary} Let $\\sharp A=1$ The following conditions are equivalent\n\\label{oncompseries-ratin}\n(i) $M^{A}$ is a sum of admissible representations;\n\n(ii)$SSM^{A}=\\Omega^{A}_{nilp}\\cup\\sigma\\Omega^{A}_{nilp}$;\n\n(iii)\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$.\n\\end{corollary}\n\n\\subsubsection {Classical and quantum $\\mbox{\\g osp}(1|2)$. Fusion algebra as a\nGrothendieck ring}\n\\label{fusalgasgrotring}\n\n{\\bf A. } $\\mbox{\\g osp}(1|2)$ is a rank 1 superalgebra -- one of the superanalogues\nof $\\mbox{\\g sl}_{2}$. It can be defined as an algebra on 2 odd generators,\n$x_{+},x_{-}$, one even generator, $h$, and relations\n\\[[x_{+},x_{-}]=h,\\; [h,x_{\\pm}]=\\pm x_{\\pm}.\\]\n\nEven part of this algebra is $\\mbox{\\g sl}_{2}$ and is generated by $x_{\\pm}^{2}$;\nodd part is $V_{1}$ as an $\\mbox{\\g sl}_{2}-$ module, its basis is $x_{+},\\; x_{-}$.\n\n From this it is easy to obtain the following classification of all simple\nfinite dimensional $\\mbox{\\g osp}(1|2)-$modules. (It is even simpler to do this in the\nway\nmodelling the $\\mbox{\\g sl}_{2}-$case -- by starting with Verma modules and then\nquotienting out a singular vector; for details see \\cite{kul_resh}). Each\n$\\mbox{\\g osp}(1|2)-$module  $W$is a sum of an even and odd part\n$W=^{even}W\\oplus^{odd}W$;\neach $^{\\cdot}W$ is an $\\mbox{\\g sl}_{2}-$module, i.e. direct sum of $V_{n}$'s. These\nare generalities. But in reality each irreducible $\\mbox{\\g osp}(1|2)-$module is of\none of the 2 following\ntypes:\n\\[ V_{n}^{0} \\mbox{ such that } ^{even}V_{n}^{0}=V_{n},\\;\n^{odd}V_{n}^{0}=V_{n-1};\\]\n\n\\[ V_{n}^{1} \\mbox{ such that } ^{even}V_{n}^{0}=V_{n-1},\\;\n^{odd}V_{n}^{0}=V_{n}.\\]\n\nThe fact that the dimensions of the even and odd parts are different by 1 is a\nconsequence  of the fact  that odd part of the algebra is $V_{1}$.\n\nWe see that each irreducible $\\mbox{\\g osp}(1|2)-$module is odd-dimesional; further\n$V_{n}^{0}$ and $V_{n}^{1}$ are isomorphic as modules and obtained from each\nother by the change of parity. This is the category of finite dimensional\nrepresentations of $\\mbox{\\g osp}(1|2)$; denote it $Rep(\\mbox{\\g osp}(1|2))$. As in the\n$\\mbox{\\g sl}_{2}-$case, one proves that $Rep(\\mbox{\\g osp}(1|2))$ is semisimple.\n\nThe universal enveloping algebra $U\\mbox{\\g osp}(1|2)$ is in fact a Hopf algebra, for\nexample the comultiplication is given by the standard formula $g\\mapsto\ng\\otimes 1+1\\otimes g,\\; g\\in\\mbox{\\g osp}(1|2)$. This makes $Rep(\\mbox{\\g osp}(1|2))$ a\ntensor category: $Rep(\\mbox{\\g osp}(1|2))\\times Rep(\\mbox{\\g osp}(1|2))\\rightarrow\nRep(\\mbox{\\g osp}(1|2))$, $A,B\\mapsto A\\otimes B$, where the $\\mbox{\\g osp}(1|2)-$module\nstructure on $A\\otimes B$ is determined through the comultiplication (and the\nrule of sign!). Decomposing the tensor product of 2 irreducible modules one\ngets the Grothendieck ring of $Rep(\\mbox{\\g osp}(1|2))$.\n\n\\begin{lemma}\n\\label{decomposptensprod}\n\\[ V_{r_{1}}^{\\alpha}\\otimes V_{r_{2}}^{\\beta}=\nV_{r_{1}+r_{2}}^{\\alpha+\\beta}+\nV_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1}+\nV_{r_{1}+r_{2}-2}^{\\alpha+\\beta}+\\cdots +\nV_{|r_{1}-r_{2}|}^{\\alpha+\\beta}.\\]\n\\end{lemma}\n{\\bf Proof.} Direct calculations show that $ V_{r_{1}}^{\\alpha}\\otimes\nV_{r_{2}}^{\\beta}$ contains one and only one singular (annihilated by $x_{+}$)\nvector of each weight from $|r_{1}-r_{2}|$ to $r_{1}+r_{2}$ and that the\nsubmodules generated by these vectors are irreducible. Proof is completed by\ncounting dimensions. $\\Box$\n\n\\bigskip\n\n Theorems \\ref{fusalggencase} and \\ref{fusalgratcase} provide us with 2\ncommutative algebras. Here\nwe interprete these algebras as Grothendieck rings of certain categories. Start\nwith the algebra\nof Theorem \\ref{fusalggencase} and denote it $\\mbox{${\\cal A}$}^{gen}$. Obviously\n$\\mbox{${\\cal A}$}^{gen}=\\mbox{${\\cal A}$}_{0}\\otimes\\mbox{${\\cal A}$}$, where $\\mbox{${\\cal A}$}$ is the Grothendieck ring of the\ncategory of\nfinite-dimensional representations of $\\mbox{\\g g}$ (its multiplication law is defined\nby the formula\npreceding Theorem \\ref{fusalggencase}) and $\\mbox{${\\cal A}$}_{0}$ is the algebra with basis\n$V_{i}^{\\alpha},\\;i\\geq 0,\\alpha\\in\\nz\/2\\nz$, multiplication being given by\n\\begin{equation}\n\\label{reprospalg}\nV_{r_{1}}^{\\alpha}\\circ V_{r_{2}}^{\\beta}=\nV_{r_{1}+r_{2}}^{\\alpha+\\beta}+\nV_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1}+\nV_{r_{1}+r_{2}-2}^{\\alpha+\\beta}+\\cdots +\nV_{|r_{1}-r_{2}|}^{\\alpha+\\beta}.\n\\end{equation}\n\nComparing (\\ref{reprospalg}) with Lemma \\ref{decomposptensprod} we get the\nfollowing.\n\n\\begin{proposition}\n\\label{interptofgenlev}\n$\\mbox{${\\cal A}$}_{0}$ is the Grothendieck ring of the category of finite-dimensional\nrepresentations\nof the superalgebra $ \\mbox{\\g osp}(1|2)$.\n\\end{proposition}\n\nAppearance of $\\mbox{\\g osp}(1|2)$ here, although artificial as it may seem to be, has\ndeep reasons  behind it. To see this we will analyze the rational level case\nusing quantized enveloping algebras.\n\n\\begin{remark}\nIt follows from Lemma \\ref{decomposptensprod} that the functor\n$Rep(\\mbox{\\g osp}(1|2))\\rightarrow Rep(\\mbox{\\g sl}_{2})$, $V_{m}^{\\alpha}\\mapsto\nV_{m}\\oplus V_{m-1}$ induces  an epimorhpism of the Grothendieck rings.\n\\label{renonmorh[pring}\n\\end{remark}\n\n \\bigskip\n\n\\bigskip\n\n{\\bf B.} Both $U\\mbox{\\g sl}_{2}$ and $U\\mbox{\\g osp}(1|2)$ admit quantization,\n$U_{t}\\mbox{\\g sl}_{2}$ and $U_{t}\\mbox{\\g osp}(1|2)$ resp.. Let us remind the relevant\nformulas.\nThe Drinfeld-Jimbo (see \\cite{drinf,jimbo}) algebra $U_{t}\\mbox{\\g sl}_{2},\\; t\\in\\mbox{${\\bf C}$}$\nis defined to be an associative\nalgebra on generators $E,F,K^{\\pm 1}$ and relations\n\n\\[EF-FE=\\frac{K-K^{-1}}{t-t^{-1}},\\; KEK^{-1}=t^{2}E,\\; KFK^{-1}=t^{-2}F.\\]\n\n$U_{t}\\mbox{\\g osp}(1|2)$ is similarly defined \\cite{kul_resh} as an associative\nalgebra on generators $X_{+}, X_{-}, K^{\\pm 1}$ and relations\n\n\\[X_{+}X_{-}+X_{-}X_{+}=\\frac{K-K^{-1}}{t-t^{-1}},\\; KX_{\\pm}K^{-1}=t^{\\pm\n1}X_{\\pm}.\\]\n\nThe representation theory of $\\mbox{\\g sl}_{2}$ and $\\mbox{\\g osp}(1|2)$ ``deforms to'' the\nrepresentation theory of $U_{t}\\mbox{\\g sl}_{2}$ and  $U_{t}\\mbox{\\g osp}(1|2)$ resp. We will\n continue denoting by $V_{m}$ the $m+1-$dimensional module over\n$U_{t}\\mbox{\\g sl}_{2}$, and by $V^{0}_{m},\\; V^{1}_{m}$ the 2 $(2m+1)-$ dimensional\nmodules over $U_{t}\\mbox{\\g osp}(1|2)$. For generic $t$ these modules are irreducible,\nthe categories of finite dimensional representations, $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$, generated by these modules are semisimple.\n\nThe deformations $U_{t}\\mbox{\\g sl}_{2}$ and $U_{t}\\mbox{\\g osp}(1|2)$ are especially\nremarkable in that they afford simultaneous deformation of the Hopf algebra\nstructure. We get 2 tensor categories $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$. What has been said implies that the Grothendieck rings\nof $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$ are isomorphic to the Grothendieck rings of the\ncorresponding classical objects.\n\nIf however $t$ is a root of unity, things change dramatically. Suppose for\nsimplicity that $t$ is a primitive $l$-th root of unity, $l$ being odd. Then\n\n\\begin{equation}\n\\mbox{(i) $V_{m}$ is irreducible if and only if $m< l$;}\n\\label{irratroots1}\n\\end{equation}\n\\begin{equation}\n\\label{irratroots2}\n\\mbox{(ii) $V_{m}^{\\epsilon}$ is irreducible if and only if $m< l$.}\n\\end{equation}\n\n(Both statements are proved by direct computations.)\n\nWhat is even more important is that the categories $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$ are no longer semisimple. For example, tensor product\nof 2 irreducible representations is not semisimple. Things, however, are still\nvery\nmuch under control.\n\n\\begin{lemma}\n\\label{decompprodatroots}\nLet $t$ be a primitive $l$-th root of unity, $l$ being odd, $m,n<l$. Then\n\n\\[\\mbox{ (i) } V_{m}\\otimes V_{n}= V_{|m-n |}\n\\oplus V_{|m-n|+2}\\cdots \\oplus V_{\\mbox{min}\\{2(l-1)-m-n,m+n\\}}\\oplus W,\\]\nwhere $W$ is not semisimple.\n\\[\\mbox{ (ii) } V_{m}^{\\alpha}\\otimes V_{n}^{\\beta}= V_{|m-n |}^{\\alpha+\\beta}\n\\oplus V_{|m-n|+1}^{\\alpha+\\beta+1}\\cdots \\oplus\nV_{\\mbox{min}\\{2(l-1)-m-n,m+n\\}}^{\\alpha+\\beta}\\oplus W,\\]\nwhere $W$ is not semisimple.\n\n\n\\end{lemma}\n\n{\\bf Sketch of Proof.} (i) is well-known, see \\cite{resh_tur}. We will however\nreview both cases as at our level of brevity there will no difference between\nthem. First, direct calculations as in the proof of Lemma\n\\ref{decomposptensprod} show that  regardless of $t$ at each weight space there\ncan always be only one singular vector. Now decomposition of Lemma\n\\ref{decomposptensprod}, statements (\\ref{irratroots1}, \\ref{irratroots2})\nand this uniqueness result show that the submodules $V_{l-1+i}$ and $V_{l-1-i}$\n(or $V_{l-1+i}^{\\epsilon}$ and $V_{l-1-i}^{\\epsilon}$), $i\\leq m+n-l+1$ are\nnon-trivially tangled. Other $V_{j}$ coming from generic $t$ are still\nirreducible and appear as direct summands. $\\Box$\n\n\\bigskip\n\n{\\bf Definition.}\n\n(i)  Define $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$ to be subcategories of $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$resp. consisting of direct sums of irreducible modules\n$V_{m}$ (or $V_{m}^{\\alpha}$ resp.), $m< l$.\n\n(ii) Define functors\n\\[Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}\\times Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}\\rightarrow\nRep(U_{t}\\mbox{\\g sl}_{2})^{(l)},\\; A,B\\mapsto A\\dot{\\otimes}B,\\]\n\n\\[Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}\\times Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}\\rightarrow\nRep(U_{t}\\mbox{\\g osp}(1|2))^{(l)},\\; A,B\\mapsto A\\dot{\\otimes}B,\\]\nby taking the usual tensor product and then throwing away $W$ in the right hand\nside of formulas in Lemma \\ref{decompprodatroots}. $\\Box$\n\n\\bigskip\n\nWe get tensor categories $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$.\n\n\\bigskip\n\n\\bigskip\n\n{\\bf C.} It is easy now to interpret the  fusion algebra at the rational level\nin terms of the Grothendieck rings of $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$.   In view of Lemma \\ref{decompprodatroots} and\nDefinition\nabove, Theorem \\ref{fusalgratcase} reads as follows.\n\n\\begin{proposition}\n\\label{fusalgeasgrotratlev}\nFusion algebra at the level $k+2=p\/q$ is a quotient of the tensor product\nof the Grothendieck rings of the categories  $Rep(U_{t_{1}}\\mbox{\\g sl}_{2})^{(p-1)}$\nand\n$Rep(U_{t_{2}}\\mbox{\\g osp}(1|2))^{(q)}$. Further, the fusion algebra always contains\nthe Grothendieck ring of $Rep(U_{t_{2}}\\mbox{\\g osp}(1|2))^{(q)}$ via the classes\nof symbols $V_{m}^{\\alpha}, V_{0}$.\n\\end{proposition}\n\n\n\n\n\n\\subsubsection{Kac-Moody vs. Virasoro}\n\\label{kacmoodyvsvir}\nVirasoro algebra, $Vir$, is defined to be a vector space with basis\n$\\{ L_{i},z, \\; i\\in\\nz\\}$ and bracket\n\\[[L_{i},L_{j}]=(i-j)L_{i+j}+\\delta_{i,-j}\\frac{i^{3}-i}{12}z.\\]\nRepresentation theory of Virasoro algebra is to a great extent parallel\nto that of $\\mbox{$\\hat{\\gtg}$}$. We will confine to essentials, making reference to\n\\cite{fei_fuchs}.\n\nOne defines the Verma module $M_{h,c}$, where $(h,c)$ is a highest weight, i.e.\neigenvalues of $L_{0}\\;, z$ resp. determined by the vacuum vector; $c$ is\nsometimes referred to as level. A Verma module\nis reducible if and only if it contains a singular vector.  $M_{h,c}$\ngenerically has no singular vectors.By the Kac determinant formula, there is a\nfamily of hyperbolas labelled by pairs of positive integers $m,n$ in the plane\nwith coordinates $(h,c)$ such that if $M_{h,c}$ contains a singular vector,\nthen $(h,c)$ belongs to one of  these hyperbolas; generically along hyperbolas\nthe singular vector is unique.   Denote the singular vector arising in\n$M_{h,c}$ as\n$(h,c)$ gets on hyperbola with the label $m,n$ by $S_{mn}$. There arises\nthe $Vir$-analogue of the generalized Weyl module $M_{h,c}\/<S_{mn}=0>$.\nAttach to        $M_{h,c}\/<S_{mn}=0>$ the symbol $(V_{n-1},V_{m-1})$. Further,\nfor $c$ fixed there arises a one-to-one correspondence between the\n$Vir$-analogues of generalized Weyl modules and symbols $(V_{n-1},V_{m-1})$.\nThis has all been in precise analogy with \\ref{Case_(i)}.\n\nIt has hardly been written anywhere, but is nevertheless known that the $Vir-$\nanalogue\nof the fusion algebra from \\ref{Thegenericlevelcase}, i.e. at a generic level,\nis as follows:\n\\begin{equation}\n\\label{fusalgforvir}\n(V_{n_{1}},V_{m_{1}})\\circ (V_{n_{2}},V_{m_{2}})=(V_{n_{1}}\\otimes\nV_{n_{2}},V_{m_{1}}\\otimes V_{m_{2}}).\n\\end{equation}\n(The interested reader can prove this result using methods of\n\\cite{feig_fuchs_2}; our treatment of the $\\mbox{$\\hat{\\gtg}$}$-fusion algebra in\n\\ref{Thegenericlevelcase} is also a direct analogue of these.)\n\n\nThere is a functor sending $\\mbox{$\\hat{\\gtg}$}-$modules to $Vir-$modules -- quantum\nDrinfeld-Sokolov reduction. One of the prerequisites for it is a choice of\na nilpotent subalgebra of $\\mbox{\\g sl}_{2}$. The two obvious possibilities are\n$\\mbox{${\\bf C}$} e$ and $\\mbox{${\\bf C}$} f$. Denote the corresponding functors $\\phi_{e}$ and\n$\\phi_{f}$. It can be extracted from \\cite{feig_fr} that  both functors send\ngeneralized Weyl modules to generalized Weyl modules. In our terminology one\ngets\n\n\\[\\phi_{e}:\\begin{array}{lll}\n(V_{m}^{0},V_{n})&\\mapsto& (V_{m},V_{n})\\\\\n(V_{m}^{1},V_{n})&\\mapsto& (V_{m-1},V_{n}),\n\\end{array}\\]\n\n\\[\\phi_{f}:\\begin{array}{lll}\n(V_{m}^{0},V_{n})&\\mapsto& (V_{m-1},V_{n})\\\\\n(V_{m}^{1},V_{n})&\\mapsto& (V_{m},V_{n}),\n\\end{array}\\]\nwhere the symbol $V_{-1}$, if arises, is understood as zero.\n\nThe $Vir-$analogue of admissible representations is the celebrated minimal\nrepresentations. The latter can be defined as quotients of generalized Weyl\nmodules\nby repeating word for word definition of  admissible representations from\n\\ref{Case_(ii)}. It is known that minimal representations\narise only when\n \\[c=c_{pq}=1-\\frac{6(p-q)^{2}}{pq},\\]\nwhere $p,q$ are relatively prime positive integers.\n There are again 2 generalized Weyl modules projecting on\na given minimal representation. Therefore minimal representations are labelled\nby equivalence classes of symbols $(V_{m}, V_{n})$. It can be shown that the\nequivalence relation is as follows: $(V_{m},V_{n})\\approx (V_{q-2-m},\nV_{p-2-n})$ for $c=c_{pq}$. From this and (\\ref{fusalgforvir}) one can easily\ncalculate the fusion algebra. We will not write down the relevant formulas here\nand confine to mentionaing that the algebra is related to the product of\nGrothendieck rings of 2 quantum $U_{t}(\\mbox{\\g sl}_{2})$ at appropriate roots of\nunity in much the same way as the fusion algebra for $\\mbox{$\\hat{\\gtg}$}$ is related to the\nproduct\nof Grothendieck rings of  $U_{t}(\\mbox{\\g osp}(1|2))$ and $U_{t}(\\mbox{\\g sl}_{2})$. Recall\nalso that the $Vir$-fusion algebra  was calculated in \\cite{bpz};\nmathematically acceptable exposition can be found in \\cite{feig_fuchs_2}.\n\n\n\nAnother property of the Drinfeld-Sokolov reduction is that both  $\\phi_{e}$ and\n$\\phi_{f}$ send admissible representations at the level $k=2-p\/q$ of $\\mbox{$\\hat{\\gtg}$}$ to\nminimal representations\nof $Vir$ at the level $c_{pq}$, see \\cite{fkw} .\n\n\\begin{proposition}\n\\label{homosl2vir}\n The functor $\\phi_{e}\\oplus\\phi_{f}$ determines an epimorphism of the\n$\\mbox{$\\hat{\\gtg}$}-$fusion algebra onto the $Vir-$fusion algebra at both generic and\nrational levels.\n\\end{proposition}\n\n{\\bf Proof.} The generic level case follows from Remark \\ref{renonmorh[pring}\nand formula (\\ref{fusalgforvir}) above. In the rational level case, the\nstatement follows from the fact that both, $\\mbox{$\\hat{\\gtg}$}-$ and $Vir$-, fusion algebras\nare obtained from their generic level counterparts by imposing the equivalence\nrelations and the 2 equivalence relations agree with each other.  $\\Box$\n\n\\subsection{Fusion functor.}\nThis part is an announcement, proofs will appear elsewhere\n\nSuppose we have a trivial vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\cp$,\n$A=\\{(P_{1},\\mbox{\\g b}_{1}),(P_{2},\\mbox{\\g b}_{2})\\}$, $B=\\{(P_{3},\\mbox{\\g b}_{3})\\}$, so that\n$(\\mbox{${\\cal E}$},A\\bigsqcup B)$ is generic. There is a construction which to a\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module associates a $\\mbox{$\\hat{\\gtg}$}^{B}-$module. This construction is a\nnatural adjustment of the Kazhdan-Lusztig tensoring \\cite{kazh_luszt} to our\nneeds.\n\nDenote by $\\mbox{\\g g}(\\cp,A,B)$ the subalgebra of $\\mbox{\\g g}(\\cp,A)$ consisting of\nfunctions taking values in $\\mbox{\\g n}_{3}=[\\mbox{\\g b}_{3},\\mbox{\\g b}_{3}]$ -- just like we did\nin \\ref{finofcoinvsferesubs}. For a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $M^{A}$, denote by\n$M^{A}_{N}$ the subspace of $(M^{A})^{\\ast}$ annihilated by\n$\\mbox{\\g g}(\\cp,A,B)^{N}$. Obviously  $M^{A}_{N}\\subset M^{A}_{N+1}$, $N\\geq 1$. Set\n\n\\[F^{A\\rightarrow B}(M^{A})=\\cup_{N\\geq 1} M^{A}_{N}.\\]\n\nOne can show that the vector space\n $F^{A\\rightarrow B}(M^{A})$ affords in a natural way a structure of an\n$\\mbox{$\\hat{\\gtg}$}^{B}-$module at the same level; this is easy to show in the spirit of\n\\cite{kazh_luszt, beil_feig_maz}. Using our methods one can show that\n\n(i) if $M^{A}$ is from the $\\mbox{${\\cal O}$}-$category, or further a generalized Weyl\nmodule, or further an admissible representation, then $F^{A\\rightarrow\nB}(M^{A})$ is also as a $\\mbox{$\\hat{\\gtg}$}^{B}-$module;\n\n(ii) the arising in this way Grothendieck rings coincide with those in Theorem\n\\ref{fusalggencase} or Theorem\n\\ref{fusalgratcase} if the level is generic or rational resp..\n\nThis generalizes the statement for the integrable representations, see\n\\cite{fink}.\n\n{\\bf Problem.} Describe the arising tensoring in the spirit of Kazhdan-Lusztig.\n\n\n \\subsection{Quadratic degeneration}\n\\label{Quadratic_degeneration}\n\n\\subsubsection { }\n\\label{setupquadrdeg}\nThe setup here will the following version of  \\ref{maingenerresults}:\n\n(i) $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ be a family of  curves over  a formal\ndisk  $S$, such that the fiber over the generic point of $S$ (``outside\norigin'')\nis a smooth projective curve, and over the origin, $O$, the fiber is a curve\n$\\mbox{${\\cal C}$}_{O}$.\nwith exactly one quadratic singularity;\n\n(ii) $\\rho_{S}:\\; \\mbox{${\\cal E}$}_{S}\\rightarrow \\mbox{${\\cal C}$}_{S}$ is a rank $2$ vector bundle.\n\nAs in \\ref{maingenerresults}, we complete these data to the localization data\nwith logarithmic singularities, say\n$\\tilde{\\psi}$.\nIn the standard way, Theorem \\ref{existofDmod} rewrites to give a $D-$module\nover $S$ with logarithmic singularities at $O$; call it\n$\\Delta_{\\tilde{\\psi}}(M^{A})$. This is because $Spec(S)$ is $\\mbox{${\\bf C}$}[[t]]$ and\nvector fields\nvanishing at $q=0$ are exactly those which can be lifted to $\\mbox{${\\cal C}$}_{S}$.\n\n\nAlong with the family $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ consider the family\n$\\bar{\\pi}^{\\vee}:\\;\\mbox{${\\cal C}$}_{S}^{\\vee}\\rightarrow S$, obtained from\n$\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ by replacing the singular fiber $\\mbox{${\\cal C}$}_{O}$\nwith\nits normalization $\\mbox{${\\cal C}$}_{O}^{\\vee}$ (i.e. be tearing $\\mbox{${\\cal C}$}_{O}$ apart at the\nself-intersection point). There is a projection $\\mbox{${\\cal C}$}_{O}^{\\vee}\\rightarrow\n\\mbox{${\\cal C}$}_{O}$\nand the preimage of the self-intersection point $a\\in C_{O}$ consists\nof 2 points $a_{0}, a_{\\infty}\\in C_{O}^{\\vee}$.\n\nIt is obvious that the datum $\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}_{S}$\nis equivalent to the data  ``$\\rho^{\\vee}_{S}: \\mbox{${\\cal E}$}_{S}^{\\vee}\\rightarrow\n\\mbox{${\\cal C}$}_{S}^{\\vee}$, equivalence $(\\rho^{\\vee}_{S})^{-1}(a_{0})\\approx\n(\\rho^{\\vee}_{S})^{-1}(a_{\\infty})$''. The localization data with logarithmic\nsingularities $\\tilde{\\psi}$ rewrites to give a ``normalized'' localization\ndata\n$\\psi^{\\vee}$.\n\nIn addition fix 2 different lines $l_{0},l_{\\infty}$ in the fiber of $\\mbox{${\\cal E}$}_{S}$\nover the point $a\\in\\mbox{${\\cal C}$}_{O}$. This determines 2 Borel subalgebras,\n$\\mbox{\\g b}_{0},\\mbox{\\g b}_{\\infty}$ operating in the fiber over $a$.\n\nAfter normalization these additional data determine the line $l_{0}$ and the\nBorel subalgebra $\\mbox{\\g b}_{0}$ operating in the fiber of $\\mbox{${\\cal E}$}_{S}^{\\vee}$ over\n$a_{0}$, as well as the line $l_{\\infty}$ and the Borel subalgebra\n$\\mbox{\\g b}_{\\infty}$ operating in the fiber over $a_{\\infty}$. We also get a\ndistinguished Cartan subalgebra\n$\\mbox{\\g h}=\\mbox{\\g b}_{0}\\cap\\mbox{\\g b}_{\\infty}$. Set\n$A^{\\vee}=A\\bigsqcup\\{(a_{0},\\mbox{\\g b}_{0}),(a_{\\infty},\\mbox{\\g b}_{\\infty})\\}$.\n\nNow with a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $M^{A}$ at the level $k$ and an admissible weight\n  $\\lambda\\in\\mbox{\\g h}^{\\ast}$ we associate the $\\mbox{$\\hat{\\gtg}$}^{A^{\\vee}}-$module\n$M^{A}\\otimes L^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}$.  We get a $D-$module for the\n``normalized''localization data:\n\n \\[\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\n\\begin{proposition}\n\\label{statemonquadrgen}\nGenerically with respect to $l_{0},l_{\\infty}$,\nif $\\Delta_{\\tilde{\\psi}}(M^{A})$ is smooth then\n$\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k})$ is also and there is an isomorphism\nof $D-$modules\n\\[\\Delta_{\\tilde{\\psi}}(M^{A})\\approx\n\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\\end{proposition}\n\n\n\n\\subsubsection { Proof}\n\n(i) Begin with the genus zero case.Observe that the algebra of regular\nfunctions\non the neighborhood of the point $a$ is\n$\\mbox{${\\bf C}$}[t_{0},t_{\\infty}][[t]]\/<t_{0}t_{\\infty}=t>$ where $t$ is a coordinate on\n$S$; $\\mbox{${\\cal C}$}_{O}^{\\vee}$\nin this case is just a union of 2 spheres. Therefore the set $A$ splits in two:\n$A'$ and $A''$, each of which has to do with one of the spheres.\n\nHence the algebra $\\mbox{\\g g}(\\bar{\\pi}^{-1}(s),A)$ can be degenerated into the\nfollowing one\nas $s$ ``approaches'' $O$:\n\n \\[(\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))+\\mbox{\\g h})\\oplus_{\\mbox{\\g h}}\n(\\mbox{\\g h}+\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty})).\\]\n\nMeaning of the last expression is as follows: recall, see\n\\ref{Finiteness_of_coinvariants}, that $\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))$ consists\nof functions regular outside $\\bar{A}$ and sending $P_{0}$ to $\\mbox{\\g n}_{0}$;\n$\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))$ is defined similarly with\n$P_{0},\\mbox{\\g n}_{0}$ replaced with $P_{\\infty},\\mbox{\\g n}_{\\infty}$; further the algebra\n$\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))+\\mbox{\\g h}$ is the algebra of functions sending\n$P_{0}$ to $\\mbox{\\g b}_{0}$, the same is true for\n$\\mbox{\\g h}+\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))$; finally ``$\\oplus_{\\mbox{\\g h}}$''\nmeans\ndirect product over $\\mbox{\\g h}$.\n\nTherefore the coinvariants degenerate into the space\n\\[((M^{A'})_{\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))}\\otimes\n(M^{A''})_{\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))})_{\\mbox{\\g h}},\\]\nwhere $\\mbox{\\g h}$ acts by means of the diagonal embedding; this makes sense as the\nfibers are identified.\n\nBy Proposition \\ref{finofcoinv_general}, the space\n\\[(M^{A'})_{\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))}\\otimes\n(M^{A''})_{\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))}\\]\nis finite dimensional. It is easy to extract from Lemma \\ref{cohom_for_Weyl}\nthat as an $\\mbox{\\g h}$-module this space is semisimple and therefore is\nisomorphic to\n\\[\\oplus_{\\lambda}((M^{A}\\otimes M_{\\lambda,k}^{P_{0},\\mbox{\\g b}_{0}}\\otimes\nM_{\\lambda,k}^{P_{\\infty},\\mbox{\\g b}_{\\infty}})_{\\mbox{\\g g}(\\mbox{${\\cal C}$}_{O}^{\\vee},A^{\\vee})}.\\]\nBy Lemma \\ref{cohom_for_admiss}, in the last formula $\\lambda$ can be chosen to\nbe admissible and the Verma modules can be replaced with the corresponding\nadmissible representations.\n\n\n\n\n\nThis proves that $\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k})$ is smooth and gives a morphism\n\n\\[\\Delta_{\\tilde{\\psi}}(M^{A})\\rightarrow\n\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\nThat this is an isomorphism can be shown in the standard way constructing the\ninverse map using the formal character of $L_{\\lambda,k}$, see\n\\cite{beil_feig_maz}.\n\n(ii) The higher genus case is not much different. For example, pinching makes a\ntorus into  a sphere. Therefore in this case proof is literally the same. It\nalso proves  an analogue of Lemma \\ref{cohom_for_admiss} for a torus. This\nprovides a basis for induction.\n\nIn genus $\\geq 2$\nat an appropriate place instead\nof Proposition \\ref{finofcoinv_general} one has to make reference to\nProposition\n\\ref{highgen_finit_afterpionching} and then use induction. $\\Box$\n\n\\subsubsection{Remarks}\n\\label{remonquadrdeg}\n\n(i) Meaning of Proposition \\ref{statemonquadrgen} is that the dimension of the\ngeneric\nfiber of the $D-$module $\\Delta_{\\psi}(M^{A})$ can be calculated by the usual\ncombinatorial algorithm: by pinching the surface and further inserting\ninserting all possible representations the problem is reduced to the case of a\nsphere with three punctures and in the latter case the complete results are\navailable.\n\n(ii) In the genus 0 case the  analogue of Proposition \\ref{statemonquadrgen}\nfor generalized Weyl modules\nis valid. To see this it is enough to examine  part (i) of the proof and\nconvince oneself that the only requirement on $M^{A}$ used there was that\n$M^{A}$ be generalized Weyl module; in fact at an appropriate place instead of\nLemma \\ref{cohom_for_admiss} one has to use Lemma \\ref{Berma-Weyl-gener}.\n\n(iii) Quadratic degeneration for generalized Weyl modules on the sphere allows\nto write horizontal sections of the corresponding bundle as a product of vertex\noperators. This will be explained in  sect.\\ref{screenopcorrfuncttt}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{{\\bf Screening operators and correlation functions}}\n\\label{screenopcorrfuncttt}\n\nIn this section we will study in detail the situation described in\n\\ref{holondmodoncc}: we have the trivial rank 2 bundle $\\mbox{${\\cal E}$}\\rightarrow\\cp$, a\ngeneralized Weyl module $M^{A}$, and\na holonomic $D-$module $\\Delta(M^{A})$ on the space $\\mbox{${\\bf C}$}^{m}\\times\\mbox{${\\bf C}$}^{m}$ with\nfiber\n$(M^{A})_{\\mbox{\\g g}(\\cp,A)}$. For the reasons which will become clear later\nwe replace this bundle with the dual one, its fiber being\n$((M^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. Denote the corresponding $D-$module by\n$\\Delta(M^{A})^{\\ast}$. Using our results on quadratic degeneration we rewrite\nhorizontal sections of the corresponding bundle with flat connection as  matrix\nelements of vertex operators, which serves the two-fold purpose: we find that\nthe differential equations satisfied by horizontal sections are provided by the\nsingular vectors of the corresponding Verma module and write down integral\nrepresentations for solutions to these differential equations.\n\n\n\n\n\\subsection{\\bf Vertex operators and corelation functions}\n\\label{Vertex_operators_and_corelation_functions}\nAn alternative to the language of coinvariants in the genus zero case is the\nlanguage of {\\em vertex operators}.\n\n{\\bf Definition.} A vertex operator is a $\\mbox{$\\hat{\\gtg}$}-$morphism\n\\begin{equation}\n\\label{defvertoper}\nY:\\;\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{1}\\rightarrow V_{2},\n\\end{equation}\nwhere $\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}$ is a loop module (see\n\\ref{Loop_modules}) and\n $V_{1},V_{2}\\in\\mbox{${\\cal O}$}_{k}$ are highest weight modules. $\\Box$\n\n\\bigskip\n\nIn other words, a vertex operator is an embedding\n $\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\hookrightarrow\n\\mbox{Hom}_{\\mbox{${\\bf C}$}}V_{1}\\rightarrow V_{2}$. The space\n$\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}$ has the basis $\\{F_{ij}=F_{i}\\otimes\nz^{j},\\;i,j\\in\\nz\\}$,\n where $\\{F_{i},\\;i\\in \\nz\\}$ is a basis in $\\mbox{${\\cal F}$}_{\\alpha \\beta}$, see\n\\ref{Loop_modules}.\n Given a vertex operator $Y$, consider the generating function\n\\[Y(x,z)=x^{\\Delta_{1}}z^{\\Delta_{2}}\n\\sum_{i,j=-\\infty}^{\\infty}F_{ij}x^{-i}z^{-j},\\]\nthe ``monodromy\ncoefficients'' $\\Delta_{1},\\Delta_{2}$ are defined by:\n\\[\\Delta_{1}=\\frac{-\\lambda_{2}+\\lambda_{1}+\\beta}{2},\\;\n\\Delta_{2}=\\frac{-C(\\lambda_{2})+C(\\lambda_{1})+C(\\beta)}{2},\\]\nwhere $\\lambda_{i}$ is the highest weight of $V_{i}$ and\n$C(\\lambda)=\\lambda(\\lambda+2)\/2$.\n($\\Delta_{1},\\Delta_{2}$ will later appear as genuine monodromy coefficients of\na certain flat\nconnection.)\n\n The formal series\n$Y(x,z)$ is, of course, an element of $Hom_{\\mbox{${\\bf C}$}}(V_{1},V_{2}\\otimes\n x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},z^{\\pm 1}]])$. Further, for any\n$g\\in\\mbox{\\g g}$ the\ncommutator $[g\\otimes z^{n},Y(x,z)]$ is also a well-defined element of\n$Hom_{\\mbox{${\\bf C}$}}(V_{1},V_{2}\\otimes\n x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},z^{\\pm 1}]])$. For the standard\nbasis of $\\mbox{\\g g}$, see\n\n                                           \\ref{defofalgebrandmodusu}, one\nderives from the definition of a vertex operator that\n\n\\begin{equation}\n[e\\otimes z^{n},Y(x,z)]=z^{n}(-x^{2}\\frac{\\partial}{\\partial x}+\\beta x)Y(x,z),\n\\label{comm_vert_op_1}\n\\end{equation}\n\\begin{equation}\n[f\\otimes z^{n},Y(x,z)]=z^{n}\\frac{\\partial}{\\partial x}Y(x,z),\n\\label{comm_vert_op_2}\n\\end{equation}\n\\begin{equation}\n[h\\otimes z^{n},Y(x,z)]=z^{n}(2x^{2}\\frac{\\partial}{\\partial x}-\\beta x)Y(x,z).\n\\label{comm_vert_op_3}\n\\end{equation}\nWe conclude that for any $g\\in\\mbox{\\g g}$ there is a differential operator $D_{g}(x)$\nin $x$ such that\n\\begin{equation}\n[g\\otimes z^{n},Y(x,z)]=z^{n}D_{g}(x)Y(x,z),\n\\label{comm_vert_op_4}\n\\end{equation}\n\n\nSuppose now we are given a collection of vertex operators\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nThe product of the corresponding generating functions\n$Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})$ is a\nwell-defined element of\n$Hom_{\\mbox{${\\bf C}$}}(V_{1\/2},V_{m+1\/2}\\otimes\n\\prod_{i}x_{i}^{\\Delta_{i,1}}z_{i}^{\\Delta_{i,2}}\n\\mbox{${\\bf C}$}[[x_{1}^{\\pm 1},\\ldots x_{m}^{\\pm 1},z_{1}^{\\pm 1},\n\\ldots z_{m}^{\\pm 1}]])$. The matrix element\n\\[ <v^{\\ast},Y_{m}(x_{m},z_{m}\\cdots\nY_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>,\\;v\\in V_{1\/2},\nv^{\\ast}\\in V_{m+1\/2}^{\\ast}\\]\nis, therefore, a formal Laurent series in $x_{i},z_{i},\\;1\\leq i\\leq m$.\n\n{\\bf Definition} Suppose $Y_{i}(x_{i},z_{i}),\\;1\\leq i\\leq m$  are as above.\nThen the matrix\nelement\n\\begin{equation}\n\\label{defcorrfun}\n\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>\n\\end{equation}\nis called {\\em correlation function} if $V_{1\/2},\\ldots,V_{m+1\/2}$ are\nirreducible generalized\nWeyl modules, $V_{1\/2}$ is the vacuum module,\n$v$ is the highest weight vector of $V_{1\/2}$ and $v^{\\ast}$ is the dual to the\nhighest weight vector of $V_{m+1\/2}$. (The latter condition is meaningful in\nview of the weight\nspace decomposition of a highest weight module.) $\\Box$\n\n\\bigskip\n\nA correlation function has been understood as a formal power series. We will\nshow that, in fact,\nit is a holomorphic function satisfying a certain holonomic system of partial\ndifferential\nequations. In order to do that we  will interpret  vertex opeartors as\nhorizontal sections\nof a line bundle with a flat connection provided by three modules on $\\cp\\times\n\\cp$.\n\n\\subsection {{\\bf From coinvariants to vertex operator algebra}}\n\n\\subsubsection{ }\n\\label{coinv_corrfunsn}\n\nWe return to the setup of \\ref{Thegenericlevelcase}. In the cartesian product\n$\\cp\\times\\cp$\nfix coordinate system $(x,z)$. Attach to the point $x$ in the first factor the\nBorel subalgebra\n$\\mbox{\\g b}_{x}$ spanned by the vectors $e_{x}=e-xh-x^{2}f,\\; h_{x}=h+2xf$. This\nmeans, in particular,\nthat $\\mbox{\\g b}_{0}$ is the standard Borel subalgebra $\\mbox{${\\bf C}$} e\\oplus\\mbox{${\\bf C}$} h$\n(see  \\ref{defofalgebrandmodusu} ) and $\\mbox{\\g b}_{\\infty}$ is the opposite one. Set\n$A=\n\\{(0,0),(x,z),(\\infty,\\infty)\\}$. Let $V^{A}=V^{\\mbox{\\g b}_{0},0}_{0}\\otimes\nV^{\\mbox{\\g b}_{x},z}_{1}\n\\otimes V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty}$ be a generalized Weyl module over\n$\\mbox{$\\hat{\\gtg}$}^{A}$.\nConsider the space of invariants $((V^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. By Theorem\n\\ref{fusalggencase}\n this\nspace is either 0- or 1-dimensional. Suppose the latter possibility is the\ncase. Then by\nTheorem \\ref{smoothinadm} we get a line bundle with flat connection over\n$\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}^{\\ast}$ whose fiber over the point $(x,z)\\in\n\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}^{\\ast}$ is\n$((V^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. There arises an embedding\n\\[V^{\\mbox{\\g b}_{x},z}_{1}\\hookrightarrow \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0}\\otimes\n V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty},\\mbox{${\\bf C}$}).\\]\n\nThe dual space $(V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty})^{\\ast}$ as a $\\mbox{$\\hat{\\gtg}$}-$module\nis isomorphic\nto the contragredient module $(V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty})^{c}$,\nsee \\ref{defofalgebrandmodusu}. As the level is generic,\nthe latter module is irreducible and is, therefore, isomorphic to a certain\ngeneralized Weyl\nmodule\n$V^{\\mbox{\\g b}_{0},0}_{\\infty}$. Hence we get an embedding\n\\[V^{\\mbox{\\g b}_{x},z}_{1}\\hookrightarrow \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},\nz^{\\pm 1}]]),\\]\nwhere $\\Delta_{1},\\Delta_{2}$ are monodromy coefficients of the flat\nconnection. We conclude that\nany $w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ can be looked upon as a certain generating\nfunction\n$w(x,z)=x^{\\Delta_{1}-n}z^{\\Delta_{2}-l}\\sum_{i,j\\in\\nz}w_{ij}x^{-i}z^{-j}$ of\na family of\noperators $\\{w_{ij}\\subset \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty})$, where $(n,l)$ is a bidegree of $w$ as an element of\n$V^{\\mbox{\\g b}_{x},z}_{1}$.\n\n\\begin{lemma}\n\\label{vacuum_vertoper}\nSuppose $v_{1}\\in V^{\\mbox{\\g b}_{x},z}_{1}$ is the highest weight vector. Then\n\n(i) $v_{1}(x,z)$ is a generating function of a certain vertex operator as in\n\\ref{Vertex_operators_and_corelation_functions};\n\n(ii) any vertex operator is obtained in this way.\n\\end{lemma}\n\n{\\bf Proof} is a direct and simple  calculation using definitions, see also\n \\ref{prooflemma49} formula  (\\ref{strofloopmodact}). $\\Box$\n\n\\bigskip\n\n\n\nLet us now relate correlation functions to horizontal sections of the bundle\nbuilt on the generalized Weyl module $M^{A}$, $\\Delta(M^{A})^{\\ast}$, see\nbeginning of sect.\\ref{screenopcorrfuncttt} for notations.\nSuppose that $M^{A}$ is the tensor product of ``individual'' generalized Weyl\nmodules \\[\\otimes_{i=1}^{m} V_{i}^{z_{i},\\mbox{\\g b}_{x_{i}}}.\\] Consider all possible\ncorrelation functions\n\\[<v^{\\ast}, v_{m}(x_{m},z_{m})\\cdots v_{1}(x_{1},z_{1})v>,\\]\nwhere $v_{i}(x_{i},z_{i})$ is a generating function of a vertex operator\nrelated\nto the highest weight vector $v_{i}\\in V_{i}^{z_{i},\\mbox{\\g b}_{x_{i}}}$.\n\\begin{corollary}\n\\label{betwcorrfunandhorizsecttt}\nLet $M^{A}$ be as above.\nOver a suitable  open contractible subset $U$ of $\\mbox{${\\bf C}$}^{m}\\times\\mbox{${\\bf C}$}^{m}$, there\nis\nan isomorphism between the space of horizontal sections of the  bundle\n$\\Delta(M^{A})^{\\ast}$ and the space of correlation functions\n\\[<v^{\\ast}, v_{m}(x_{m},z_{m})\\cdots v_{1}(x_{1},z_{1})v>.\\]\n\\end{corollary}\n\n{\\bf Proof.} Intertwining properties of vertex operators imply a correlation\nfunction is a horizontal section of $\\Delta(M^{A})^{\\ast}$ in a formal sense.\nThis give a map in one direction. A map in the opposite direction in provided\nby\nquadratic degeneration, see Proposition \\ref{statemonquadrgen}. $\\Box$\n\n\n\n\\subsubsection{ }\n\\label{vertop_opervertalg}\nBy Lemma \\ref{vacuum_vertoper} coinvariants recover vertex operators. In fact\nthey give us much more: the\ncollection of generating functions $w(x,z),\\;w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ affords a\nkind of\n{\\em vertex operator algebra} structure. We will not discuss the latter in\ndetail\n(see \\cite{fren_lep_meur}) and only explain how one can get exlicit formulas\nfor\n$w(x,z),\\;w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ in terms of the vertex operator $v_{1}(x,z)$\nrelated to the\nhighest weight vector $v_{1}$.\n\nFor any $g\\in\n\\mbox{\\g g}$ set $g(i)=g\\otimes z^{i}\\in\\mbox{$\\hat{\\gtg}$}$. Define the {\\em current} $g(z)$ to be\n$g(z)=\\sum_{i\\in\\nz}g(i)z^{-1-i}\\in\\mbox{$\\hat{\\gtg}$}\\otimes\\mbox{${\\bf C}$}[[z^{\\pm 1}]]$. Define\n$g(z)^{(l)}$ to be\nthe $l-$th (formal) derivative of $g(z)$ with respect to $z$.\n For any  $g(z)^{(l)}$ set\n\\[(g(z)^{(l)})_{+}=(\\frac{d}{dz})^{l}\\sum_{i=0}^{\\infty}g_{-i-1}z^{i},\\;\n(g(z)^{(l)}_{-}=g(z)^{(l)}-(g(z)^{(l)})_{+}.\\]\nObserve that for any $w(x,z)\\in \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$\nand any $g\\in\\mbox{\\g g}$, the\nproducts $(g(z)^{(l)})_{-}w(x,z)$, $w(x,z)(g(z)^{(l)})_{+}$ are also\nwell-defined elements of\n$\\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$.\n\n\n\nDefine for any $g\\in\\mbox{\\g g},\\;w(x,z)\\in \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$\n\\begin{equation}\n\\label{normalordering}\n:g(z)^{(k)}w(x,z):=(g(z)^{(k)})_{-}w(x,z)+w(x,z)(g(z)^{(k)})_{+}.\n\\end{equation}\n\n\\begin{lemma}\n\\label{descendants}\n Let $g\\in\\mbox{\\g g}$, $w\\in V^{\\mbox{\\g b}_{x},z}_{1}$, $w(x,z)$ the corresponding element\nof $\\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$. Then\n\n(i) $(g\\cdot w)(x,z)=[g,w(x,z)]$;\n\n(ii) $(g(-l)\\cdot w)(x,z)=(1\/(l-1)!):g(z)^{(l-1)}w(x,z):,\\;l>0$;\n\\end{lemma}\n\n{\\bf Proof} is a direct calculation of matrix elements of the operator\n$(g(-l)\\cdot w)(x,z)$\nbased on the definition of the space of coinvariants. $\\Box$\n\n\\subsection{ Differential equations satisfied by correlation functions}\n\\label{diffeqsatbycorrfuntttt}\nWe return to the setup of \\ref{Vertex_operators_and_corelation_functions} and\nconsider\na correlation function\n\\[\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>,\\]\ncoming from the  product of vertex operators\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nUsing Lemma \\ref{vacuum_vertoper} we assume that there are generalized Weyl\nmodules\n$V_{i},\\;1\\leq i\\leq m$ with highest weight vectors $v_{i},\\;1\\leq i\\leq m$\nsuch that\n$Y_{i}(x,z)=v_{i}(x,z)$. An advantage of this point of view is that for any\ncollection\nof elements $w_{i}\\in V_{i},\\;1\\leq i\\leq m$ we can consider the matrix element\n\\[<v^{\\ast},w_{m}(x_{m},z_{m})\\cdots w_{2}(x_{2},z_{2})w_{1}(x_{1},z_{1})v>.\\]\n\n\\begin{lemma}\n\\label{fromprimtodesc}\nFor any\n $w_{i}\\in V_{i},\\;1\\leq i\\leq m$\n\n\\[<v^{\\ast},w_{m}(x_{m},z_{m})\\cdots w_{2}(x_{2},z_{2})w_{1}(x_{1},z_{1})v>=\nD\\cdot \\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m}),\\]\nwhere $D$ is a differential operator in $x'$s with coefficients in rational\nfunctions in $z'$s.\n\\end{lemma}\n\n{\\bf Proof.} Start  with the function\n\\[<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(g(-l)v_{i})(x_{i},z_{i})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;l>0.\\]\nBy Lemma \\ref{descendants} (ii) it rewrites as\n\\[\n<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(g(z)^{(l-1)}_{-}v_{i}(x_{i},z_{i})-\nv_{i}(x_{i},z_{i})g(z)^{(l-1)}_{+})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;l>0.\\]\nThen commute all $g_{i},\\; i<0$ through to the right and all $g_{i},\\; i\\geq 0$\nto the left in\na standard way, c.f.\\cite{fren_resh}  and use commutation relations\n(\\ref{comm_vert_op_1},\\ref{comm_vert_op_2},\\ref{comm_vert_op_3}). The case\n$l=0$ is treated in\n a similar\nand\nsimpler way using Lemma \\ref{descendants} (i). Further argue by induction using\nagain\nLemma \\ref{descendants}. $\\Box$\n\n\\bigskip\n\nBy definition each $V_{i}$ is a quotient of a Verma module and therefore there\nare elements, singular vectors in the corresponding Verma module (see\n\\ref{defofalgebrandmodusu})\n$S_{i}\\in U(\\mbox{$\\hat{\\gtg}$})$ such that $S_{i}v_{i}=0,\\;1\\leq i\\leq m$. On the other\nhand,\nby Lemma \\ref{fromprimtodesc} there are differential operators $D_{i},\\;1\\leq\ni\\leq m$ such that\n\\begin{eqnarray}\n& &D_{i}<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{2}(x_{2},z_{2})v_{1}(x_{1},z_{1})v>=\\nonumber\\\\\n& &<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(S_{i}v_{i})(x_{i},z_{i})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;1\\leq i\\leq m.\\nonumber\n\\end{eqnarray}\nWe arrive to the following result.\n\n\\begin{lemma}\n\\label{firsthalfequat}\nThe correlation function\n\\[\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots v_{2}(x_{2},z_{2})v_{1}(x_{1},z_{1})v>\\]\nsatisfies the system of equations\n\\begin{equation}\n\\label{firsthalfequat_eq}\nD_{i}\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=0,\\;1\\leq i\\leq m.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\bigskip\n\nObserve that, although there are in general no explicit formulas for $D_{i}$,\nthe fact that\n$[D_{i},D_{j}]=0$ is an obvious consequence of the definition.\n\nWe have obtained $m$ equations our function of $2m$ variables satisfies. The\nrest is, of course,\n the Knizhnik-Zamolodchikov equations. Let us write them down explicitly.\n Recall that we can look upon\n$\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ as a function of $z_{1},\\ldots\nz_{m}$ with\ncoefficients in a completed tensor product of $m$ $\\mbox{\\g g}-$modules.\n(The variables $x_{1}\\ldots x_{m}$\nare responsible for that, see\n(\\ref{comm_vert_op_1},\\ref{comm_vert_op_2},\\ref{comm_vert_op_3}).)\n For any $A=\\sum_{s}a_{i}\\otimes b_{s}\n\\in\\mbox{\\g g}\\otimes\\mbox{\\g g}$ denote by $A_{ij},1\\leq i,j\\leq m$\nan operator  acting on the $m-$fold tensor product of $\\mbox{\\g g}$-modules by the\nformula\n\\[A_{ij}\\cdot w_{1}\\otimes\\cdots w_{m}=\\sum_{s}w_{1}\\otimes\na_{s}w_{i}\\otimes\\cdots b_{s}w_{j}\n\\otimes\\cdots w_{m}.\\]\nThe formula (\\ref{comm_vert_op_4}) implies that $A_{ij}$ is a differential\noperator\n in $x_{i},x_{j}$.\nSet $\\Omega=ef+fe+h^{2}\/2$.\n\n\\begin{lemma} (\\cite{knizh_zam})\n\\label{knzameq}\n\nThe correlation function\n$\\Psi=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ satisfies the  system of\nKnizhnik-Zamolodchikov\n equations\n\\begin{equation}\n\\label{knzameq_eq}\n(k+2)\\frac{\\partial}{\\partial z_{i}}\\Psi=\\sum_{j\\neq\ni}\\frac{\\Omega_{ij}}{z_{i}-z_{j}}\\Psi,\\;\n1\\leq i\\leq m.\n\\end{equation}\n\\end{lemma}\n\nThere is no need to prove this lemma here as one can repeat\nword for word the known proofs.\n However we point out that if one considers a highest weight module $V$ as a\nmodule\nover the semi-direct product of $\\mbox{$\\hat{\\gtg}$}$ and the Virasoro algebra $Vir$ then $V$\nis annihilated\nby the element $d\/dz-L_{-1}$, where $L_{-1}$ is one of the Sugawara elements.\nOne then shows that\n  the singular vectors\n$(d\/dz-L_{-1})v_{i}$, where $v$ is a highest weight vector of $V_{i}$, give\nrise to\nthe equations (\\ref{knzameq_eq}) in exactly the same way the singular vectors\n$S_{i}$ gave\nrise to the equations (\\ref{firsthalfequat_eq}). An immediate consequence of\nthis proof is\nthat the system of equations (\\ref{knzameq_eq},\\ref{firsthalfequat_eq}) is\nconsistent.\n\n\n\n\n\\subsection{Screening operators and integral representations of correlation\nfunctions}\n\\label{scr_and_int_repr}\n\\subsubsection{ }\n\\label{motivation}\nSuppose a  function $\\Psi_{old}=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$\n is the matrix element of the product\nof vertex operators\n\\[\\Psi_{old}=<v^{\\ast},\\pi\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{o}>,\\]\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m,\\]\nsatisfying the same conditions as the expression  in (\\ref{defcorrfun}),\nsee\\ref{Vertex_operators_and_corelation_functions}, except\nthat instead of assuming that $V_{m+1\/2}$ is a generalized Weyl module we\nassume that\n$V_{m+1\/2}$ is a contragredient Verma module, see \\ref{defofalgebrandmodusu}.\n(Why ``old'' will become clear in a moment.) It is easy to see that\n $\\Psi_{old}=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ satisfies the same\nsystem of equations\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). Suppose in addition that there is a\nprojection\n $\\pi:\\; V_{m+1\/2}\\rightarrow W$ onto another contragredient Verma module $W$.\nDenoting\nby $w^{\\ast}$ an element dual to the highest weight vector $w\\in W$ one can\nconsider the\nmatrix element\n\\[\\Psi_{new}=<w^{\\ast},\\pi\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>.\\]\nWe again observe that $\\Psi_{old}$ is a solution to the same system\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). This new solution can be calculated\n as follows.\n\nThere arises the dual map $\\pi^{\\ast}:\\; W^{\\ast}\\rightarrow V_{m+1\/2}^{\\ast}$\nand by definition\n   there is an element $S$ of $U(\\mbox{$\\hat{\\gtg}$}_{>})$\nsuch that $\\pi^{\\ast}(w^{\\ast})= Sv^{\\ast}$. We now take the definition of\n$\\Psi_{new}$, replace\nin it $\\pi^{\\ast}(w^{\\ast})$ with $Sv^{\\ast}$ and get\n\n\\begin{equation}\n\\label{replachwvect}\n\\Psi_{new}=<S\\cdot v^{\\ast},\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>.\n\\end{equation}\n\n\n Then we  commute $S$ through to the right.\n The intertwining properties of vertex operators tell us\nthat\n\\begin{equation}\n\\label{fromoldtonew}\n\\Psi_{new}=S^{t}\\cdot\\Psi_{old},\n\\end{equation}\nwhere $^{t}$ signifies the canonical\nantiinvolution an a Lie algebra\n($g_{1}g_{2}\\cdots g_{n}\\rightarrow g_{n}g_{n-1}\\cdots g_{1}$)\nand the action  is determined by the following condition: if $g\\in\\mbox{\\g g}$ then\n\\[(g\\otimes\nz^{n})\\cdot\\Psi_{old}=\\sum_{i=1}^{m}D_{g}(x_{i})z_{i}^{n}\\Psi_{old},\\]\nsee (\\ref{comm_vert_op_4}).\n\nWe intend to use (\\ref{fromoldtonew}) in the case when $\\pi$ and therefore $S$\ndo not exist!\n\n\\subsubsection{Screening operators }\nLet $V_{\\lambda_{\\infty},k}$ be a highest weight module and\n$v\\in V_{\\lambda_{\\infty},k}$ a highest weight vector.\n If the obvious integrality conditions are satisfied\nthen the vectors $f^{\\lambda_{\\infty}+1}v,\\;( e\\otimes\nz^{-1})^{k-\\lambda_{\\infty}+1}v$ are\nsingular and give rise to embeddings of the type $W\\hookrightarrow\nV_{\\lambda_{\\infty},k}$.\nNow take 3 highest weight modules $V_{\\lambda_{i},k},\\; i=0,1,\\infty$ attach\nthem\nto 3 point in $\\cp$ and consider the space of coinvariants\n\\[(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}.\\]\nOf course an embedding $W\\hookrightarrow V_{\\lambda_{\\infty},k}$ gives rise to\na map\n\\[(W^{\\mbox{\\g b}_{\\infty},\\infty}\\otimes_{i=0,1}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}\n\\hookrightarrow\n(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}.\\]\nIt is remarkable that even if the embedding $W\\hookrightarrow\nV_{\\lambda_{\\infty},k}$ does not\nexist the last map still does. In the language of vertex operators this\nphenomenon was explained\nin great detail in \\cite{feig_mal}.\n\nTherefore with each of the formal singular vectors --\n$f^{\\lambda_{\\infty}+1}v\\mbox{ or }( e\\otimes z^{-1})^{k-\\lambda_{\\infty}+1}v$\n-- we have\nassociated an operator acting on coinvariants. Call these operators {\\em\nscreenings} and denote\nthem $R_{1}$ and $R_{0}$ respectively.\n\nLet us calculate the action of the  screenings explicitly. By definition\n\\newline\n$R_{j}(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}$ only\ndepends on $V_{\\lambda_{\\infty},k}$ so we will be simply writing\n$R_{j}(V_{\\lambda_{\\infty},k})$.\nNow formulas for the related singular vectors\n ($f^{\\lambda_{\\infty}+1}v,\\;( e\\otimes z^{-1})^{k-\\lambda_{\\infty}+1}v$) and a\nvery simple\ncalculation using the formulas (\\ref{param_eq_line_1},\\ref{param_eq_line_2})\ngive the following\nresult:\n\\begin{eqnarray}\n\\label{act_scr_vert_op_1}\nR_{1}((V_{m}^{0},V_{n}))=(V_{m-1}^{1},V_{n})\\\\\n\\label{act_scr_vert_op_2}\nR_{1}((V_{m}^{1},V_{n}))=(V_{m+1}^{0},V_{n})\\\\\n\\label{act_scr_vert_op_3}\nR_{0}((V_{m}^{0},V_{n}))=(V_{m+1}^{1},V_{n})\\\\\n\\label{act_scr_vert_op_4}\nR_{0}((V_{m}^{1},V_{n}))=(V_{m-1}^{0},V_{n})\n\\end{eqnarray}\n\nSuppose we are given 2 generalized Weyl modules and a vertex operator acting\nbetween them.\nSuppose in addition that this vertex operator is related to a highest weight in\nthe third\ngeneralized Weyl module, say $(V_{m}^{\\epsilon},V_{n})$.\nTheorem \\ref{fusalggencase} tells us that given such a vertex operator\nour screenings give us all the others of the type $(V_{i}^{\\alpha},V_{n})$ --\nwe cannot only\nchange the value of $n$.\nBut then there is the standard screening operator -- $S$ -- which takes care of\n$n$, see\ne.g. \\cite{feig_fr_1}. So these three -- $R_{1},R_{2},S$ -- provide us with all\nvertex operators.\n This has\nan important application to the calculation of correlation functions.\n\nStart with a simple correlation function given by the product of vertex\noperators, each of which\nis characterized by the condition $m=0$. Then applying $S$ an appropriate\nnumber\n of times one gets all\nvertex operators and, hence,\nall correlation functions in spirit of Varchenko-Schechtman, see\\cite{awtsyam}.\n\nNow take a Varchenko-Schechtman\ncorrelation function $\\Psi_{old}$. It comes from a product of vertex operators:\n\\[\\Psi_{old}=<v^{\\ast}, Y(x_{m},z_{m})\\cdots Y_{1}(x_{1},z_{1})v_{o}>,\\]\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nLet $W_{i},\\;0\\leq i\\leq m$, be words on 2 letters $R_{1}$ and $R_{2}$.\nReplacing $V_{i+1\/2}$\nwith $W_{i}(V_{i+1\/2})$ we get a new correlation function $\\Psi_{new}$. Doing\nthis with all\n$\\Psi_{old}$ and sufficiently many $W_{i},\\;0\\leq i\\leq m$ we get all solutions\nto\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). In principle all these solutions\ncan be written\ndown explicitly. It is especially simple to do so in the case when we keep\n$V_{i+1\/2},\\;0\\leq i\\leq m-1,$ and only change $V_{m+1\/2}$.\n\nSo assume that $\\Psi_{old}$ is as above and replace $V_{m+1\/2}$ with\n$R_{j}(V_{m+1\/2}),\\; j=0,1$.\nThen by (\\ref{replachwvect}) one is to expect that\n\\[\\Psi_{new}=<X^{\\alpha}\\cdot v^{\\ast},\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>,\\]\nwhere $X$ is either $e$ or $f\\otimes z$ if $j=1$ or 0 resp., and $\\alpha$ is\neither $\\lambda+2$\nor $k-\\lambda+2$ resp., where $\\lambda$ is\n the highest weight of $V_{m+1\/2}$.\n\nOf course if $\\alpha$ is not a nonnegative integer  then the last formula does\nnot make much sense.\nNevertheless using it and (\\ref{fromoldtonew}) as a motivation we arrive to\n\\[\\Psi_{new}=X^{\\alpha}\\Psi_{old}.\\]\nNow the left-hand side of the last equality does make sense:\n$X$ is a first order differential operator,\nsee \\ref{motivation}, therefore  we can set in a rather straightforward manner\n\\[X^{\\alpha}\\Psi_{old}=\\int t^{-\\alpha-1}\\{\\exp(-Xt)\\Psi_{old}\\}dt\\]\nand get a nice integral operator, for details see \\cite{feig_mal}.\n\nThis procedure can be easily iterated to provide the functions\n\\begin{equation}\n\\label{our_solut}\n\\int \\prod_{i=1}^{n}t_{i}^{-\\alpha_{i}-1}\\{\\exp(-X_{1}t_{1})\\exp(-X_{2}t_{2})\n\\cdots \\exp(-X_{n}t_{n})\\Psi_{old}\\}\\prod_{i=1}^{n}dt{i},\n\\end{equation}\nwhere $X_{1},X_{2},...$ is either $e,f\\otimes z,e,...$ or $f\\otimes\nz,e,f\\otimes z,...$.\n\n\\begin{lemma}\n\\label{our_sol_stae}\nFunctions (\\ref{our_solut}) are solutions to\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}).\n\\end{lemma}\n\n{\\bf Proof} is  same as the proof of the analogous statement in\n\\cite{feig_mal1}. In fact it is an easy exrcise to make the heuristic arguments\nwhich have lead us to the formula (\\ref{our_solut}) into a precise proof.\n$\\Box$\n\n\\bigskip\n\nIntegrating functions (\\ref{our_solut}) with respect to $x'$s\n(or doing something similar but more esoteric) one is supposed to get the\nDotsenko-Fateev correlation functions for the Virasoro algebra. It would be\ninteresting to do\nthis explicitly and compare the result with the calculations in \\cite{f_g_p_p}.\n\n\\begin{conjecture}\n\n(i) Formulas (\\ref{our_solut}) provide all solutions to the system\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}).\n\n(ii) If the level $k$ is rational, then there arises a subbundle of  the bundle\nin question, the one with fiber $((L^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A}$, where\n$L^{A}$ is the corresponding admissible representation. We conjecture that in\nthis case formulas (\\ref{our_solut}) actually give horizontal sections of the\nlatter bundle.\n\n\n \\end{conjecture}\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMobile robotic networks (MRNs) have received considerable attention in recent years. \nThanks to the mobility, flexibility, and distributed fashion, MRNs are widely deployed in numerous applications, e.g., surveillance, reconnaissance, search and environmental monitoring \\cite{bullo2009distributed}. \nAmong these applications, formation control is a fundamental technique to enhance the cooperation performance by maintaining a preset geometric shape \\cite{olfati2004consensus,oh2015survey,10.1145\/3366696,10.1145\/3366701,kamel2020formation}. \n\n\n\nIt has been reported that MRNs under formation control may suffer severe security vulnerabilities from both cyber and physical spaces \\cite{choi2018detecting,10.1145\/3366698}. \nExisting works mainly focus on the cybersecurity of MRNs, and various defense techniques are developed to protect the information availability, integrity and confidentiality. \nIn the literature, the attacker is generally assumed with strong knowledge about the system (e.g., network structure, dynamics model) or powerful capabilities (e.g., access to the system), lacking for deep investigation on the capabilities and modes of attackers in more realistic situations. \nThis paper demonstrates that an external attacker without strong prior knowledge can observe the state evolution of the MRN to obtain data, learn the interaction rules from the data analysis, and further leverage the rules to implement attacks that cannot be handled by traditional security countermeasures. \nSeveral prior works have made some progress in this direction and support our demonstration. \nFor example, it is possible to infer whether two robots are connected \\cite{8985069} or learn how a single robot avoids an obstacle \\cite{li2019learning}. \nHowever, considering what exact interaction rules can be learned and under what conditions to obtain them, how to further destroy the cooperation performance by leveraging the rules and where is the attack impact boundary, they are all still open issues and need further exploration. \n\nMotivated by above discussions, we design a sneak attack against MRNs under formation control, where the attack robot has only partial observation and aims to replace one robot therein, obtaining partial control over the MRN and destroying the cooperation performance. \nThis attack is inspired by that i) the shape-forming process essentially reflects the internal interaction structure among cooperating robots \\cite{olfati2007consensus}; \nii) the obstacle-avoidance behavior embodies how the robots interact with the environment \\cite{pandey2017mobile}; \niii) both the interaction rules are limited by physical distance in practical applications, either by communication or sensing. \nThese universal interaction characteristics indicate the possibility of learning them to implement advanced attacks, making our work of both theoretical and practical significance. \nFrom the attack view, it enlightens the feasibility of revealing the interaction rules that support the MRN, and guides for more advanced attack design (e.g., invade an MRN). \nIn turn, from the defense view, it provides deeper insights into the trade-off between the system security and observability, and helps the countermeasure design by understanding the possible attack capabilities and modes. \n\n\n\n\\textbf{Challenges and contributions}. \nThe major challenges of achieving the sneak attack lie in two aspects. \nFirst, the attacker can only observe the measurable states of a partial MRN (e.g., position, velocity), and the states are influenced by both the model-unknown interaction rules and the unobservable part of the MRN. \nIt is difficult to split the coupled influences of the multiple factors and approximate the interaction rules. \nSecond, even if the interaction rules are obtained, their mutual impacts will closely intersect with the influence of the attack dynamics, making it further hard to design feasible and efficient strategies for the attacker to achieve maximum attack rewards. \nTo overcome above issues, we holistically formulate the entire observe (excite)-attack process as different stages. \nBy exploiting the independence and correlation characteristics of the interaction rules in different stages, the whole attack mode is designed in an active-knowledge-growth form and accommodates the coupled dynamics of the attacker and the MRN. \n\n\nThe differences between this paper and its conference version \\cite{lys} include i) a systematic intelligent attack design is introduced rather than mere topology inference, \nii) the learnability of both the internal interaction structure within the MRN and the external interaction mechanism with the environment is explored, \nand iii) the sneak attack is introduced with comprehensive feasibility and performance analysis. \nThe main contributions of this paper are summarized as follows. \n\n\\begin{itemize}\n\\item We contribute to the existing body of research by understanding what interaction rules in MRNs under formation can be separately revealed by observing over the state evolution, which is a mutual consequence of multiple factors. \nNo strong prior information like dynamics model or access to the MRN is required. \n\n\\item We find a novel sneak attack where an attacker with weak knowledge can learn the interaction rules and replace a victim robot to obtain partial control over the MRN. \nThe observation and excitation oriented ideas endow the ``ignorant'' attacker with growing intelligence to achieve maximum attack impacts, and this kind of attack is hard to be tackled by traditional defense techniques. \n\n\\item We develop an in-depth framework to handle different interaction rules of MRNs. \nWe prove the learnability of the interaction rules by extracting their universal characteristics. \nUtilizing the proposed node controllability, we design an ECR strategy and establish the sufficient conditions for the attack. \nPerformance study by simulation demonstrates the effectiveness of the sneak attack. \n\\end{itemize}\n\nIn a larger sense, this paper promotes to explore the security vulnerabilities in the interaction process, and beckons further research to address the new physical observation and excitation oriented attacks on generic networked systems. \n\n\\textbf{Organization and Notation}. \nThe remainder of this paper is organized as follows. \nSection \\ref{r-work} presents related literatures. \nSection \\ref{preliminary} gives the modeling for the interaction process of MRNs and formulates the attack process. \nSection \\ref{revealing} studies how to reveal the interaction rules. \nSection \\ref{attack} develops how to design control strategies for the final sneak. \nSimulation results are shown in Section \\ref{simulation}, \nfollowed by the concluding remarks and further research issues in Section \\ref{conclusion}. \nAll the proofs of the Lemmas and Theorems are provided in the Appendix. \n\nThroughout the paper, the set variable is expressed in Euclid font, and the compact all-dimension state of a robot is expressed in bold font. \nUnless otherwise noted, the formulas with non-bold font state variable apply to every dimension of the actual state space. \n$\\mathcal{A} \\backslash \\mathcal{B}$ represents the elements in $\\mathcal{A}$ that are not in $\\mathcal{B}$. \nThe superscripts ${[i,:]}$, ${[:,i]}$ of a matrix denote its $i$-th row vector and $i$-th column vector, respectively. \nThe scripts $\\tilde \\cdot $ and  $\\hat \\cdot$ right above a variable indicate its corresponding observation and estimation value, respectively. \nThe basic notation definitions are given in Table \\ref{tab:test}. \n\n\n\n\n\\section{Related Work}\\label{r-work}\n\\textit{Formation control in MRNs}. \nThe fundamental rules for formation control were first introduced by the famous Reynolds' Rules \\cite{reynolds1987flocks}: separation, alignment, and cohesion. \nBased on the rules, numerous methods have been proposed to achieve the desired performance, and consensus-based algorithms become the mainstream, e.g., \\cite{sun2016optimal,zhao2018affine,alonso2019distributed,xu2020affine}. \nThe key idea of consensus-based algorithms is that the formation is modeled as a graph, and every robot implements information interaction (like positions and velocities) with their neighbors and compute their control inputs. \nTherefore, the interaction structure lays critical support for effective formation control and is mostly based on network communication. \nIn recent years, communication-free formation control \\cite{deghat2014localization,cheng2017event,trinh2018bearing} was also developed and attracts great research interests, thanks to the emergent development of sensor technology. \nCommunication-free interaction avoids information delays and network bandwidth consumption, and even enables stealth modes of operation \\cite{kan2011network}. \nFor instance, formation control with only bearing measurements by vision sensors was investigated in \\cite{zhao2019bearing}. \n\nAlthough the Reynolds' Rules lay a solid foundation for formation control, they do not take into account some unanticipated faults (e.g., interaction failure, formation splitting), which are easily incurred when performing tasks in the real environment. \nMany advanced interaction rules and techniques were developed to better handle those accidental and unpremeditated faults \\cite{feng2016robust,khalili2018distributed,liu2018hierarchical}. \nFor instance, a consensus-based splitting\/merging algorithm was designed to tackle complex obstacle distributions in \\cite{zhu2019distributed}. \nA topology recovery method was designed to reconnect a broken link based on historical data in \\cite{li2019optimal}. \n\\cite{Zhe2015A,liu2017self} proposed a gradient-based self-repairing algorithm to restore the topology of formation and further synchronize the formation motion. \nThe motion synchronization methods were investigated to form desired\/predefined formation configuration in the presence of robot malfunctions \\cite{liu2018failure} or undesirable formation variation \\cite{dong2017time}. \n\\cite{yang2019fault} gave a comprehensive survey for more types of fault-tolerant design about formation control. \nNote that by either way, the interaction capability is limited by distance due to the energy constraint in real-world applications, and this simple yet crucial point is utilized in our attack design. \n\n\n\n\\textit{The security of MRNs}. \nEffective formation control in MRNs is supported by physical sense and network interaction, which constitute the feedback loops for control and also incur security vulnerabilities, and an attack on these components may cause disastrous consequences. \nMRN under formation control a typical cyber-physical-system (CPS), while the existing work mainly focused on the defense design to cyber attacks, e.g., DoS, false data injection, and replay attacks \\cite{pasqualetti2013attack,mo2015performance,sandberg2015cyberphysical,sanchez2019bibliographical}. \nThe attacker is generally assumed with strong prior knowledge about the MRN (like knowing the system structure or node states \\cite{pasqualetti2012consensus}, corrupting exchanged information \\cite{silvestre2017stochastic}), and formulated with a powerful attack model. \nRecently, a few works emerged which are dedicated to attacks against formation from physical space. \nFor instance, a noise-generating attack was proposed in \\cite{son2015rocking} to alter gyroscopic sensor data, leading to drone crashes. \n\\cite{tippenhauer2011requirements,bianchin2019secure} proposed a spoofing attack to disturb the GPS sensor readings, gradually causing trajectory deviations. \nThese attacks are against a specific transducer by utilizing its sensing mechanism, and are hard to be generalized to other scenarios. \n\\cite{li2019learning} developed a trial-and-learning based method to infer the obstacle-avoidance mechanism of mobile robots by disguising the attack robot as an obstacle, which requires little prior knowledge about mobile robots. \n\nIn summary, concerning what attack capability is feasible for an ``ignorant'' attacker from external observation and what is the upper limit of the attack impacts on MRNs, it still remains an open yet critical issue that motivates this work. \n\n\n\\section{Preliminaries and Problem Formulation}\\label{preliminary}\n\\subsection{Graph Basics}\nLet $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ be a directed graph that models an MRN, where $\\mathcal{V}=\\{1, \\cdots, {N}\\}$ is the finite set of nodes (i.e., robots) and $\\mathcal{E}\\subseteq \\mathcal{V}\\times \\mathcal{V}$ is the set of interaction edges. \nAn edge $(i,j)\\in \\mathcal{E}$ indicates that $i$ will use information from $j$. \nThe adjacency matrix $A=[a_{ij}]_{N \\times N}$ of $\\mathcal{G}$ is defined such that ${a}_{ij}\\!>\\!0$ if $(i,j)$ exists, and ${a}_{ij}\\!=\\!0$ otherwise. \nDenote ${\\mathcal{N}_i^{in}}=\\{j\\in \\mathcal{V}:a_{ij}>0\\}$ and ${\\mathcal{N}_i^{out}}=\\{j\\in \\mathcal{V}:a_{ji}>0\\}$ as the in-neighbor and  out-neighbor sets of $i$, respectively. \nA directed path is a sequence of nodes $\\left \\{1, 2, \\cdots, j\\right\\}$ such that $({i+1},i)\\!\\in\\!\\mathcal{E}$, $i=1, 2, \\cdots, j-1$. \nA directed graph has a (directed) spanning tree if there exists at least a node having a directed path to all other nodes. \n$\\mathcal{G}$ must have a spanning tree to guarantee the information flow in robot formation. \n\n\nDefine $L=\\text{diag}\\{A \\bm{1}\\}-A$ as the Laplacian matrix of $\\mathcal{G}$, where $\\bm{1}$ is a vector of all ones. \nThen, $L\\bm{1}=0$ holds. \nLet $L=\\Gamma \\Lambda \\Gamma^{-1}$ be the Jordan decomposition of $L$, where $\\Gamma=[q_1 \\cdots q_{\\sss N}]$ is the transformation matrix, $\\Gamma^{-1}=[p_1 \\cdots p_{\\sss N}]$, and $\\Lambda  \\!=\\! \\text{diag}\\{\\bm{\\lambda_i}\\}$ with $\\bm{\\lambda_i}$ being the Jordan block of eigenvalue $\\lambda_i$.\nThe eigenvalues and eigenvectors satisfy\n\\begin{equation}\\label{f1}\n({\\lambda _i}I - L){q_i} = 0,~p_i^\\mathsf{T}({\\lambda _i}I - L) = 0,~p_i^\\mathsf{T}q_j\\!=\\!0~(i\\!\\ne \\!j).\n\\end{equation}\nSpecially, take $q_1=\\bm{1}$ and ${p_1 = {[{p_{\\sss 11}} \\cdots {p_{\\sss 1N}}]^\\mathsf{T}}}$. \nAll the right and left eigenvector are normalized such that ${p_i^\\mathsf{T}}{q_i} = 1$. \n\n\n\\begin{table}[t]\n\\small\n \\caption{\\label{tab:test}Notation Definitions} \n \\begin{tabular}{cl}\n \\toprule \n  Symbol  & Definition  \\\\ \n  \\midrule\n   $z_i$ & the state of robot $i$ ($r_i$) in one dimension\\\\\n  $z_a$ & the state of attack robot ($r_a$) in one dimension\\\\\n  $z_v$ & the state of victim robot ($r_v$) in one dimension\\\\\n  $\\bm{z}_{\\omega}$ & the compact state of a robot in all dimensions\\\\\n  $g$ & the external interaction mechanism with environment\\\\\n  $W$ & the internal interaction matrix among robots\\\\\n  $R_{\\omega}$ & \\makecell[t{p{7cm}}]{ the radius indicating different interaction range, where $\\omega=c,o,s,f$ represent the interaction, obstacle avoidance, safety, and observation radius, respectively. }\\\\\n  $\\mathcal{D}_{\\omega}$ & \\makecell[t{p{7cm}}]{ the dataset collected in different stages, where $\\omega=c,~s,~e,~a$ represent the convergence, stabilization, excitation and attack processes, respectively.}\\\\\n   $\\mathcal{V}_{i}^{p}$ & the $r_i$-vertexed convex polygon covering $\\mathcal{N}_{i}^{out}$\\\\  \n $\\mathcal{N}_{i}^{\\omega}$ & the in\/out-neighbor set ($\\omega=in\/out$) of robot $i$ \\\\ \n $\\mathcal{V}_{\\sss F}$ & the formation under partial observation \\\\  \n $\\Omega$ & the rule set of formation interaction\\\\ \n   \\bottomrule \n \\end{tabular} \n\\end{table}\n\\subsection{Distance Constraints and Shape Specification}\nThe robots are assumed to be set in short distance interaction with their neighbors (by communication or sensing) and need to detect obstacles. \nThe interaction radius $R_c$, the obstacle detection radius $R_o$, and the safe radius $R_s$ indicating danger range are supposed to satisfy \n\\begin{equation}\nR_c>R_o>R_s.\n\\end{equation}\n\nLet $a_{ij}^0$ be the preset interaction weight of $(i,j)$ initially, and $z_i$ ($\\bm{z_i}$) be the position state of robot $i$ in one (all) dimension(s). \nBased on the well-known disk model that considers the boundedness of $R_c$ \\cite{bullo2009distributed}, $a_{ij}$ is assumed to satisfy\n\\begin{equation}\\label{eq-12}\n{a_{ij}} = a_{ij}^0 \\cdot \\text{sign}({R_c} - {\\left\\| {\\bm{z_j} - \\bm{z_i}} \\right\\|_2}), \n\\end{equation}\nwhere the indicative function $\\text{sign}(x)\\!=\\!1$ if $x\\!>\\!0$, or $0$ otherwise. \nAccordingly, define the $R$-disk of arbitrary $\\bm{z}$ as \n\\begin{equation}\n\\mathcal{P}(\\bm{z}, R)= \\left\\{ { \\bm{z_{\\omega}} : {{\\left\\| { \\bm{z_{\\omega}} - \\bm{z} } \\right\\|}_2} < R} \\right\\}. \n\\end{equation}\n\n\n\n\n\n\n\n\\subsection{Interaction Modeling of Formation Control} \\label{s2-c}\nThe following modeling is given to describe how to form the preset shape through internal interaction, and deal with the environmental interferences through external interaction. \n\n\\textit{Forming formation shape via internal interactions}. \nTo describe the predefined geometric shape under formation control, a group of parameters $\\{h_i\\}(i\\in\\mathcal{V})$ is introduced, where $h_i$ is the desired relative deviation between robot $i$ (abbreviated to $r_i$ hereafter) and a common reference point. \nNote that once the formation shape is specified, the choice of the reference point will make no difference as $h_{ij}=h_j-h_i$ remains unchanged. \nTo achieve this pattern, a commonly used first-order consensus-based formation control algorithm and its global form are given by \\cite{olfati2007consensus}\n\\begin{equation}\\label{eq-1}\n{{\\dot z}_i} \\!\\!=\\!\\!\\!\\sum\\limits_{j \\in \\mathcal{N}_i^{in}} \\!\\!{{a_{ij}}(z_j\\!-\\!z_i\\!-\\!h_{ij})},~~\\dot z(t) =- Lz(t)+Lh,\n\\end{equation}\nwhere $h_{ij}=h_j-h_i$, $z = {\\left[ { {{z_1}}, \\cdots, {{z_{\\sss N}}}} \\right]^\\mathsf{T}}$, and $h = {\\left[ { {{h_1}}, \\cdots, {{h_{\\sss N}}}} \\right]^\\mathsf{T}}$. \nGenerally, to dynamically guide the formation motion, one robot will be specified as the leader with an extra velocity input. \nFor simplicity and without loss of generality, $r_{\\sss N}$ is taken as the leader and reference node, and suppose that it runs in a constant velocity $c$. \nDenote $u_0=[0,\\cdots,0, c]^\\mathsf{T}$ and $u=Lh+u_0$. \nThen, the global dynamics in (\\ref{eq-1}) is rewritten as \n\\begin{equation}\\label{eq-22}\n\\dot z(t)=- Lz(t)+u. \n\\end{equation}\nAccordingly, the solution of (\\ref{eq-22}) is given by \n\\begin{equation}\\label{ll}\nz(t) = {e^{ - Lt}}z(0) + \\int_0^t {{e^{ - L(t - \\tau )}}} ud\\tau. \n\\end{equation}\n\nNote that (\\ref{ll}) guarantees zero deviation for static formation and bounded tracking error for dynamic formation \\cite{lewis2013cooperative}. \nTo eliminate the error, solutions contain imposing a virtual leader to all robots (which is impractical when the scale of the MRN is large), or adopting a second-order controller which further utilizes the velocities of its neighbors. \nThe global form by second-order controllers is the same as that by first-order ones, except that $z_i$ is augmented with the velocity state. \nConsequently, the internal interaction structure is related to both the positions and velocities of neighbors. \nTo avoid fussy expressions and simplify the analysis, we adopt the first-order case in this paper. \n\\begin{remark}\nThe second-order controller approximates the real leader speed, and can be seen as a nonlinear first-order model \n$\\dot z(t)=- Lz(t)+Lh+u_s(t)$, where $u_s(t)$ is determined by the second-order controller design and satisfies $\\mathop {\\lim }\\limits_{t\\to \\infty } u_s(t)= c\\bm{1}$ \\cite{lewis2013cooperative}. \nIn Section \\ref{simulation}, we will illustrate that our linear approximation method also applies to this nonlinear model if only using first-order states. \n\\end{remark}\n\n\\textit{Obstacle avoidance via external interaction interface}. \nThe obstacle-avoidance mechanism is the major interface for MRNs to interact with the physical environment. \nLet $z_{i*}$ be the desired goal of $r_i$,\n$z_{ob}$ and $v_{ob}$ be the state and velocity of the obstacle, respectively.\nThe obstacle-avoidance behavior is mainly determined by the relative positions between a robot and the obstacle, and their velocities. \nTherefore, regardless of the detail design of different algorithms, we formulate the universal avoidance behavior as a general mechanism\n\\begin{equation}\\label{local}\n{{\\dot z}_i} = g({z_{ob}}-{z_i},z_{i*}-{z_i},v_{ob},{v_i}). \n\\end{equation}\nNote that in multi-robot cases, $z_{i*}$ is usually time-varying and determined by ${\\mathcal{N}_i^{in}}$, \nand numerous obstacle-avoidance mechanisms do not consider the mobility of obstacles, i.e., the last parameter in $g(\\cdot)$ may not be used (e.g., in artificial potential method and genetic approach). \nBesides, the influence of $(z_{i*} - z_i )$ is negligible when $\\left\\| \\bm{z_a}-\\bm{z_i}\\right \\|_2 \\to R_s$, and the effect of the obstacle is dominant. \nTypically, the boundedness of $g(\\cdot)$ is described as follows\n\\begin{equation}\n\\left\\{ {\\begin{aligned}\n&g( \\cdot ) = 0,&&~\\text{if}~\\left\\| \\bm{z_{ob}}- \\bm{z_i} \\right \\|_2> R_o,\\\\\n&{0 \\le |g( \\cdot ) |\\le b},&&~\\text{if}~R_s\\le\\left\\| \\bm{z_{ob}}-\\bm{z_i} \\right \\|_2\\le R_o,\\\\\n&{|g( \\cdot )|=b},&&~\\text{if}~\\left\\| \\bm{z_{ob}}-\\bm{z_i} \\right \\|_2<R_s.\n\\end{aligned}} \\right.\n\\end{equation}\n\n\n\\textit{Formation maintenance}. \nTo deal with possible faults and hazardous interference in the environment, many methods have been developed to support autonomy and self-organization for interaction maintenance, e.g., restoring the interaction topology of splitting robot groups \\cite{liu2017self,li2019optimal}. \nIn line with these considerations, we extract the main characteristics and formulate them as the following ruleset $\\Omega=\\{\\Omega_R,\\Omega_A\\}$. \n\\begin{itemize}\n\\item Neighbor-connection restoration rule $\\Omega_R$. \nFor $r_i$, if $a_{ij}(t)\\ne a_{ij}^0$, $\\forall j\\in\\mathcal{N}_i^{in}$ at time $t$, then $r_i$ will try to move back to its desired position $z_{i*}$ and reconnect with $r_j$, $\\forall j\\in\\mathcal{N}_i^{in}$. \nThe necessary condition of successful restoration is that $z_{i*}$ is available. \n\\item Neighbor-replacement authentication rule $\\Omega_A$. \nGiven $r_a\\notin \\mathcal{V}$, $i\\in \\mathcal{V}$ and a time slot $t_l$, $j\\in \\mathcal{N}_i^{in}$ can be replaced by $r_a$ at $(t_0+t_l)$ if and only if $\\forall t \\!\\in\\! [{t_0},t_0+t_l)$, \n\\begin{align}\na_{ij}(t)\\ne a_{ij}^0,~~\\bm{z_a}(t) \\in \\mathcal{P}(  \\bm{z_i}(t), R_c),\\\\~~{\\left\\| {  \\bm{z_a}(t) - \\bm{z_{j*}}(t) } \\right\\|_2} \\!<\\! {\\left\\| {  \\bm{z_j}(t) \\!- \\! \\bm{z_{j*}}(t)} \\right\\|_2}. \n\\end{align}\n\\end{itemize}\n\n\\begin{remark}\nThe reasonability and generality of the $\\Omega$ lie in that it \ni) considers the prespecified neighborship configuration, instead of merely proximity among robots as in many existing works \\cite{feng2016robust,zavlanos2009hybrid,kan2018distributed}; \nii) works for situations where cyber authentication is not available, e.g., the interaction is purely based on sensors \\cite{cheng2017event,aranda2015coordinate}. \n\\end{remark}\nThe existence of the internal interaction structure, the external interaction mechanism, the organization ruleset $\\Omega$ and their distance-constrained characteristic constitute all the weak knowledge of the attacker about the MRN. \nThe knowledge is natural for an external attacker and provides no explicit information about system structure, model or parameters. \n\n\\begin{figure}[t]\n\\centering\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\includegraphics[width=0.45\\textwidth]{pic\/range6.png}\n\\caption{Illustration of different observation ranges. \nThe blue circle area enclosing $\\mathcal{V}_{\\sss H}$ is with radius $R_h$, and larger circle area enclosing $\\mathcal{V}_{\\sss F}$ is with radius $R_f$.}\n\\label{fig-range}\n\\vspace*{-8pt}\n\\end{figure}\n\n\n\\subsection{Partial Observation Ability of The Attacker}\nDenote the sampling moment by $k$ and the discrete time form of (\\ref{ll}) is given by \n\\begin{equation} \\label{basic-formation}\nz^{k+1} = (I - \\varepsilon_{\\sss T} L)z^{k} + \\varepsilon_{\\sss T} u^{k} = Wz^{k} + \\varepsilon_{\\sss T} u^{k},\n\\end{equation}\nwhere the sampling interval $\\varepsilon_{\\sss T}$ is small enough, and $W$ is the interaction structure matrix. \nNote that $W$ equivalently reflects the internal interaction structure and effect as $L$, since they are all linear transformation of the adjacent matrix $A$. \nTherefore, we directly seek to obtain $W$ for revealing the structure in the following sections. \n\n \nSuppose the observation of the attacker is with \\textit{i.i.d.} Gaussian noise $\\xi_0^{k}\\sim N(0,{\\sigma ^2}I)$, and it is given by \n\\begin{equation}\\label{eq-3}\n\\tilde z^{k}=z^{k}+\\xi_o^{k}=Wz^{k-1}+\\varepsilon_{\\sss T} u^{k}+\\xi_o^{k}. \n\\end{equation}\nFor every two consecutive observations, we obtain \n\\begin{equation}\\label{eq-4}\n\\begin{aligned}\n\\tilde z^{k}&=W(\\tilde z^{k-1} -\\xi_o^{k-1})+\\varepsilon_{\\sss T} u^{k}+\\xi_o^{k}\\\\\n&=W\\tilde z^{k-1}+\\varepsilon_{\\sss T} u^{k}+\\xi^{k},\n\\end{aligned}\n\\end{equation}\nwhere $\\xi=\\xi_o- W\\xi_o$. \nNote that (\\ref{eq-4}) only illustrates the quantitative relationship between two consecutive measurements and does not represent the essential dynamic process. \n\n\nNext, denote the partially observable part of $\\mathcal{V}$ by $\\mathcal{V}_{\\sss F}$, and use the smallest enclosing circle covering $\\mathcal{V}_{\\sss F}$ with radius $R_f$. \nFor convenience, we assume $R_f\\ge2R_c$. \nSince a robot will only use information from their neighbors within its interaction range, \nfor the robots near the observation boundary, their neighbors may not all be in $\\mathcal{V}_{\\sss F}$, making it hard to infer the interaction structure in $\\mathcal{V}_{\\sss F}$ with high accuracy. \nTo deal with this issue, we narrow down the inferring range and use a concentric circle to cover the feasible subset $\\mathcal{V}_{\\sss H} \\!\\subseteq\\! \\mathcal{V}_{\\sss F}$ with radius $R_h$, as shown in Fig.~\\ref{fig-range}. \n\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.9\\textwidth]{pic\/framework4.png}\n\\caption{The whole scheme of our proposed sneaking attack. \nFirst, at Stage 1 and 2, $r_a$ leverages $\\mathcal{D}_c$ and $\\mathcal{D}_s$ to infer the steady pattern parameters and $W$. \nAt Stage 3, $r_a$ makes tentative excitation on $r_v$ and and regresses $g$ based on the observed response dataset $\\mathcal{D}_e$. \nMeanwhile, the interaction radius $R_c$ is estimated and used as feedback to further optimize $W$. \nFinally, at Stage 4, $r_a$ begins its sneak with the support of the learned rules. And the response dataset $\\mathcal{D}_s$ can also be used to expand the data for regressing $g$. \n}\n\\label{frame}\n\\end{figure*}\n\n\\subsection{Problem of Interest}\\label{problemof}\nAn MRN $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ of $N$ robots is deployed to execute mission in physical environment, while running to a goal position $z_{g}$ with specified formation shape configuration $\\{h_{ij}\\}$.  \nA malicious attack robot (denoted as $r_a$) who has limited observation over $\\mathcal{V}_{\\sss F}\\subseteq \\mathcal{V}$, aims to sneak into $\\mathcal{V}$ by replacing a victim robot (denoted as $r_v$). \n\n\n\n\n\n\nTo achieve above purpose, $r_a$ needs to acquire sufficient interaction knowledge about $\\mathcal{V}$. \nNote that what is most regular to $r_a$ is the steady formation shape, \ntherefore, $r_a$ can first identity the steady pattern and then determine the convergence process, which reflects the internal interaction structure in $\\mathcal{V}$. \nSince a robot in $\\mathcal{V}$ has to react to the obstacles nearby, $r_a$ can actively make excitation on the robot to learn the external interaction rule. \nWith the interaction rules obtained, $r_a$ begins the sneak attack efficiently. \nAccordingly, we divide the whole process into four stages: formation shape convergence (observe), steady formation maintenance (observe), tentatively excitation (explore), and attack launching (sneak). \nThe observations on the four stages are constructed as four datasets: $\\mathcal{D}_c$, $\\mathcal{D}_s$, $\\mathcal{D}_e$ and $\\mathcal{D}_a$. \nOverall, to achieve the sneak attack, we need to solve the following four problems. \n\n\n\n\\begin{itemize}\n\\item Stage 1: given $\\mathcal{D}_s$, infer the steady pattern $\\hat h$, $\\hat c$.  \n\\item Stage 2: given $\\mathcal{D}_c$, $\\hat h$ and $\\hat c$, find the mapping relation $\\phi$ that embodies $W$ by solving \n\\begin{equation}\n\\begin{aligned}\n\\mathop {\\min }\\limits_{\\phi (\\mathcal{D}_c) \\mapsto W} {\\| {W - \\hat W} \\|_{\\sss \\text{Frob}}}.\n\\end{aligned}\n\\end{equation}\n\\item Stage 3: given $\\mathcal{D}_e$, find the mapping relation $\\varphi$ that embodies $g$ by solving \n\\begin{equation}\n\\begin{aligned}\n\\mathop {\\min }\\limits_{\\varphi (\\mathcal{D}_e) \\mapsto g} {\\| {g - \\hat g} \\|_2}. \n\\end{aligned}\n\\end{equation}\n\\item Stage 4: given $\\hat W$ and $\\hat g$, design a control strategy $\\bm{u}_{a,0:H}=\\{u_a^0,u_a^1,\\cdots,u_a^H\\}$ such that\n\\begin{equation}\n\\mathcal{V}(t_{\\sss H})=(\\mathcal{V}(t_0)\\backslash\\{r_v\\})\\cup\\{r_a\\}.\n\\end{equation}\n\\end{itemize}\nBy solving the problems sequentially, $r_a$ is enabled to learn the interaction rules and leverage them to implement sneak attack. \nThe whole framework with details is shown in Fig.~\\ref{frame}. \n\n\\begin{figure}[t]\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.45\\textwidth]{pic\/approximation.png}\n\\caption{Illustration of the approximation of the interaction rules. The formation state evolution is governed by the interaction rules, and the observed state at instant $k$ is a resulting output of the interaction rules taking the observed state at instant $k-1$ as input (possibly with preprocessing). }\n\\label{fig-app}\n\\vspace*{-5pt}\n\\end{figure}\n\n\n\n\\section{Revealing the Interaction Rules}\\label{revealing}\nIn this section, we first analyze the feasibility of revealing the internal and external interaction rules of an MRN, and then highlight the methods addressing the closely related problems in Stage 1-3, respectively. \nNote that the interaction range $R_c$ is estimated by active excitation in Stage 3 and is used to approximate a more informative internal interaction structure than the one obtained in Stage 2. \nThe obtained results are leveraged to constitute the cornerstone for the sneak attack. \n\n\\subsection{Feasibility Analysis}\nTo reveal the interaction rules from the observations, there are two foremost issues that need to be addressed. \nFirst, under what condition the observations are meaningful and reflect the interaction rules. \nSecond, how large the observations amount is sufficient to approximate the interaction rules with tolerable errors. \nIf these issues are settled, the problem is then converted to parameter identification of a given system model or model fitting from a statistic perspective, and different methods are adopted to approximate the two kinds of interaction rules, as shown in Fig.~\\ref{fig-app}. \nThe details are given below. \n\n\n\nFor the internal interaction rule, what matters most is the structure among the robots. \nTherefore, the classical linear state space equation is sufficient to approximate the internal interaction rule, where the structure is explicitly represented as the state transition matrix. \nUnder this model, the widely used least square method (LSM) provides an efficient way to obtain the optimal parameter estimation in the sense of minimizing the observation errors \\cite{dobson2008introduction}. \nNote that we can only approximate $W$ locally due to the partial observation nature. \n\nFor the external interaction rule, the obstacle-avoidance mechanism is fundamentally needed for a robot, and the avoidance behavior is inevitable if an obstacle is involved. \nOur major focus is to exploit how the robot will react to a close obstacle by $g$, not necessarily seeking the explicit analytical expression of $g$, especially considering its wide varieties. \nTherefore, we adopt generic learning-based methods to approximate the external interaction rule. \nBesides, due to the distance-constrained interaction, the interaction radius $R_c$ can be inferred through active excitation, and this result is utilized to expand the approximation range of a local $W$. \n\n\\subsection{Steady Pattern Identification}\nThe steady pattern of the MRN reflects the formation shape and moving speed of the MRN. \n\nAssume that $r_a$ begins its observation as the MRN begins forming the formation shape from an unstable pattern, and $\\mathcal{V}_{\\sss H}$ is initially determined by a small $R_h$ (e.g., $R_h=0.3R_f$). \nWhen the formation pattern is reached, its state dynamics are illustrated by the following result.  \n\n\n\\begin{theorem}\\label{th01}\nFor the global dynamic model (\\ref{ll}) with $u=Lh+[0\\cdots0 \\;c]^\\mathsf{T}$, we have \n\\begin{equation}\n\\mathop {\\lim }\\limits_{t \\to \\infty } \\left\\|z(t)-{c}t\\! \\cdot \\!\\bm{1} -s \\right\\|_2=0, \n\\end{equation}\nwhere the offset vector $s\\!=\\!{q_1}{p_1^\\mathsf{T}}z(0)\\!+\\! (I-{q_1}{p_1^\\mathsf{T}}) h\\!+\\!\\sum\\limits_{i = 2}^N { \\frac{  cp_{\\sss iN} }{{{\\lambda _i}}}{q_i}}$. \n\\end{theorem}\n\n\nTheorem \\ref{th01} illustrates that when the formation reaches the steady pattern, all robots are running at a common speed with fixed relative state deviations. \nThis provides a theoretical foundation to identify the steady pattern parameters. \nBefore that, let $\\Delta z_i^k={z_i^k} \\!-\\! {z_i^{k-1}}$, $d_{ij}^k=\\|{z_j^k}-{z_i^k}\\|_2$, and we define the $l$-period window as $[k,k+l]$. \nThe second-order state difference accumulation of $r_i$ during $l$-period is given by\n\\begin{equation}\n{  \\Delta S_{i}^{k_0:k_0+l} } \\!=\\! \\sum\\limits_{k = k_0+1}^{k_0+l-1} {{\\| \\Delta z_i^{k+1} - \\Delta z_i^{k} \\|}_2} .\n\\end{equation}\nThen, based on the observations, the steady velocity is obtained by solving \n\\begin{equation} \\label{c0}\n\\hat c(k_0,l) = \\mathop {\\arg \\min }\\limits_c \\sum\\limits_{k = {k_0}}^{{k_0} + l} {{{\\left\\| {\\tilde z_{\\sss F}^k} - (c{\\varepsilon _T}k + {b_0})\\bm{1} \\right\\|}_2}},\n\\end{equation}\nwhere $b_0$ is an auxiliary constant. \nNote that (\\ref{c0}) can be easily solved by generic LSM. \nNext, define the real time when the steady pattern is reached as $k_s$, and we present the following theorem to show how to identify the formation leader and the performance of estimated $\\hat c$. \n\\begin{corollary}\\label{thooo}\nGiven  $\\mathcal{V}_{\\sss F}$ and $k_s$, if the leader $r_{\\sss N}\\in \\mathcal{V}_{\\sss F}$, then it is identified by \n\\begin{equation}\n\\hat r_{\\sss N}= {\\arg \\mathop {\\min }\\limits_{i} \\left\\{ \\Delta  S_{i}^{2:k_s}: \\Delta S_{i}^{2:k_s}<\\epsilon, i\\in\\mathcal{V}_{\\sss F} \\right\\}},  \\label{ww1}\n\\end{equation}\nwhere $\\epsilon$ is an arbitrarily small positive constant. \n$\\forall k_0>k_s$, we have \n$\\mathop {\\lim }\\limits_{l \\to \\infty } \\hat c({k_0},l) = c. $\n\\end{corollary}\n\nCorollary \\ref{thooo} is straightforward due to the uniqueness of $r_{\\sss N}$ and the steady pattern characteristic. Thus the proof is omitted here. \nNote that smaller $\\Delta S_{i}^{k_0:k_0+l} $ means less velocity variation in $r_i$, and it illustrates how to identify the leader and estimate the stable velocity. \nBased on it, given preset $l$ and $\\epsilon$, the following criterion is designed to estimate $k_s$, i.e.,\n\\begin{equation}\\label{time}\n{\\hat k_s} \\!=\\! \\inf \\left\\{ {{k_0} \\!:\\left({  \\sum\\nolimits_{i \\in {\\mathcal{V}_{\\sss F}}}  \\Delta \\tilde S_{i}^{k_0:k_0+l} } \\right)\\!\\le\\! \\epsilon } \\right\\}.\n\\end{equation}\nOnce $r_a$ obtains $\\hat k_s$, the datasets $\\mathcal{D}_c$ are $\\mathcal{D}_s$ are constructed by \n\\begin{equation}\\label{data}\n\\mathcal{D}_c=\\{ \\tilde z_{\\sss F}^k:k \\le \\hat k_s \\},~\\mathcal{D}_s=\\{ \\tilde z_{\\sss F}^k:k > \\hat k_s \\},\n\\end{equation}\nwhere $\\tilde z_{\\sss F}^k=[\\tilde z_{i_1}^k,\\tilde z_{i_2}^k,\\cdots, \\tilde z_{i_{|{\\sss{ \\mathcal{V}_{\\sss F} }}|}}^k]^\\mathsf{T}$. \nMeanwhile, the formation shape parameters $\\{h_i\\}$ are calculated by \n\\begin{equation}\n\\hat h = \\hat s - {{\\hat s}_{j}} \\bm{1}, \n\\end{equation}\nwhere $\\hat s = {  \\sum\\nolimits_{k = \\hat k_s+1}^{\\hat k_s+l} {(\\tilde z_{\\sss F}^k - \\hat c{\\varepsilon _{\\sss T}} k\\!\\cdot\\! \\bm{1})} }\/{l}$.\n\n\n\n\n\n\\subsection{Internal Interaction Structure Approximation}\nAfter identifying the steady pattern, we then try to extract the interaction structure among $\\mathcal{V}_{\\sss F}$. \n\n\nDenote the unobserved robot set as $\\mathcal{V}_{\\sss F'}=\\mathcal{V}\\backslash\\mathcal{V}_{\\sss F}$, and define the indicative leader vector \n\\begin{equation}\n\\mathbb{I}_{\\sss F}^{[i]}=\\left\\{ {\\begin{aligned}\n&1, &&\\text{if}~\\exists i\\in \\mathcal{V}_{\\sss F}, i=\\hat r_{\\sss N},\\\\\n&0, &&\\text{otherwise}. \n\\end{aligned}} \\right.\n\\end{equation}\nThen, the global dynamics (\\ref{eq-4}) is divided into \n\\begin{equation}\n\\left[ {\\begin{aligned}\n{z}_{\\sss F}^{k+1}\\\\\n{z}_{\\sss F'}^{k+1}\n\\end{aligned}} \\right] \\!=\\! \\left[ {\\begin{aligned}\nW_{\\sss FF}~W_{\\sss FF'}\\\\\nW_{\\sss F'F}~W_{\\sss F'F'}\n\\end{aligned}} \\right]\\left[ {\\begin{aligned}\n{z}_{\\sss F}^{k}\\\\\n{z}_{\\sss F'}^{k}\n\\end{aligned}} \\right] \\!+ \\!{\\varepsilon _{\\sss T}}\\left[ {\\begin{aligned}\nu_{\\sss F}^{k}\\\\\nu_{\\sss F'}^{k}\n\\end{aligned}} \\right],\n\\end{equation}\nwhere $u_{\\sss F}=h_{\\sss F}+ c \\mathbb{I}_{\\sss F}$ and $u_{\\sss F'}=h_{\\sss F'}+c \\mathbb{I}_{\\sss F'}$. \nAccordingly, the observations of $r_a$ over $\\mathcal{V}_{\\sss F}$ is given by\n\\begin{equation} \\label{eq-6}\n{{\\tilde z}_{\\sss F}}^{k+1} = {W_{\\sss FF}}{{\\tilde z}_{\\sss F}}^{k} + {W_{\\sss FF'}}{{\\tilde z}_{\\sss F'}}^{k} + \\varepsilon _{\\sss T}\\hat u_{\\sss F}^{k}+\\xi_{\\sss F}^{k}.\n\\end{equation}\nBased on (\\ref{eq-6}), there are two points that need to be noticed: \n\\begin{itemize}\n\\item For $r_a$, it can only observe ${z}_{\\sss F}$, unaware of ${W_{\\sss FF'}}$ and ${z}_{\\sss F'}$;\n\\item The evolution of ${z}_{\\sss F}$ is a coupled consequence of ${z}_{\\sss F}$ itself and the unobservable ${ z}_{\\sss F'}$.\n\\end{itemize}\nTherefore, it is quite difficult to directly solve ${W_{\\sss FF}}$ by (\\ref{eq-6}). \n\n\nTo avoid the above issue, we narrow down the inference range from $\\mathcal{V}_{\\sss F}$ to a smaller $\\mathcal{V}_{\\sss H}$. \nLet $\\mathcal{V}_{\\sss H'}=\\mathcal{V}_{\\sss F}\\backslash{\\mathcal{V}_{\\sss H}}$, \n$W_{\\sss HF}=[W_{\\sss HH}~W_{\\sss HH'}]$, $y_{\\sss H}^{k}=\\tilde z_{\\sss H}^{k} -\\hat h_{\\sss H} - {\\varepsilon _{\\sss T}}\\hat c\\mathbb{I}_{\\sss H} $ and \n$y_{\\sss F}^{k} = {[(\\tilde z_{\\sss H}^{k} -\\hat h_{\\sss H} )^\\mathsf{T},(\\tilde z_{\\sss H'}^{k})^\\mathsf{T}]^\\mathsf{T}}$. \nWe obtain the following result of approximating the structure. \n\\begin{theorem}\\label{th-topo}\nGiven $\\mathcal{D}_c$ and ${\\hat k_s}$, if $\\mathcal{D}_c$ is linearly modeled by $r_a$, then in the sense of expectation, the observations satisfy \n\\begin{equation}\\label{tt}\n\\bm{E}(y_{\\sss H}^{k+1}) = {W_{\\sss HF}}\\bm{E}(y_{\\sss F}^{k}).\n\\end{equation}\nIf $|\\mathcal{V}_{\\sss F}| \\! + \\! 1\\! \\le |\\mathcal{D}_s|$,\nthen the optimal estimation of ${\\phi (\\mathcal{D}_c)\\mapsto W}$ in sense of least square is\n\\begin{equation}\\label{solution1}\n\\phi (\\mathcal{D}_c):~\\hat W_{\\sss HF}=\\left(  ({Y_{\\sss F}} {Y_{\\sss F}^\\mathsf{T}})^{^{-1}} {Y_{\\sss F}}  {Y_{\\sss H}^\\mathsf{T} }   \\right)^\\mathsf{T}, \n\\end{equation}\nwhere ${Y_{\\sss H}} = [y_{\\sss H}^{2},y_{\\sss H}^{3},\\cdots,y_{\\sss H}^{l}]$ and \n${Y_{\\sss F}} = [y_{\\sss F}^{1},y_{\\sss F}^{2},\\cdots,y_{\\sss F}^{l-1}]$.\n\\end{theorem}\n\nTheorem \\ref{th-topo} gives the least square solution of $W_{\\sss HF}$ under the linear approximation model. \nAlthough the number of feasible observations is limited in practice, Theorem \\ref{th-topo} nevertheless can be used as the basis for approximating $W_{\\sss HF}$, especially to obtain a robust connection structure even the real model is nonlinear and with different noise variance, which will be verified in Section \\ref{simulation}. \n\n\\subsection{External Interaction Rule Approximation}\\label{estimate_external}\nNext, we present the tentative excitation based method to illustrate how to infer the interaction radius $R_c$ and construct dataset $\\mathcal{D}_e$ for learning the external interaction rule $g$. \n\n\n\n\\begin{definition}\\label{direct}\n(\\textbf{Direct controllability}) \nA node is directly controllable if one can control it to reach any given state $z_c^*$ in finite steps by direct external excitations.\n\\end{definition}\n\\begin{lemma}\\label{attack1}\nIf $g$ and $z_{i*}$ is known, $\\forall i \\in \\mathcal{V}_{\\sss F}$, $r_i$ is directly controllable by $r_a$. \n\\end{lemma}\n\n\n\nLemma \\ref{attack1} shows that $r_i$ can be steered to any state by utilizing the characteristic of $g$. \nGiven the input configuration of $g$, the avoidance behavior is unique. \nTherefore, as in \\cite{li2019learning}, we make $r_a$ actively excite on $r_i$ and observe its external interaction response to learn $g$. \n\nFirst, based on the analysis in last section, for $i\\in \\mathcal{V}_{\\sss F}$, its desired position of $z_{i*}$ is estimated by \n\\begin{equation}\\label{predict}\n\\hat z_{i*}^{k+1}\\!=\\! \n\\left\\{ \n\\begin{aligned}\n&\\hat c\\cdot k+\\hat s^{[i]},~~~~~\\text{if}~\\mathcal{N}_i^{in}~\\text{is~in~steady~pattern}; \\\\\n&\\sum\\limits_{j \\in\\mathcal{V}_{\\sss F} } {{{\\hat a}_{ij}}({\\tilde z_j^k} - {\\tilde z_i^k}-  h_{\\sss F}^{[j]}+ h_{\\sss F}^{[i]} )},~\\text{otherwise},\n\\end{aligned}\\right.\n\\end{equation} \nwhere ${\\hat a_{ij}} =\\hat w_{ij}\/{\\varepsilon _{\\sss T}}~(i\\neq j)$. \nThe next implicit issue is the obstacle detection radius $R_o$, which can be directly inferred by approaching $r_i$ from a distant position. \n\n\n\n\nThen, the excitation strategy is designed considering two possible situations. \nIf there $\\exists i\\in\\mathcal{V}_{\\sss H}$, $|\\mathcal{N}_i^{in}|>1$, we randomly select a $j\\in\\mathcal{N}_i^{in}$ as the direct excitation target. \nNote that the excitation is adopted when $\\mathcal{N}_j^{in}$ is in steady pattern, thus $z_{j*}$ is available. \nThe excitation input is designed such that $u_e$ is orthogonal to the ideal moving direction of $r_j$, $\\overrightarrow{dir}(\\bm{z_{j*}})$, i.e., \n\\begin{equation}\n\\!\\{ {\\bm{u_e}}\\!:{\\bm{u_e}} \\!=\\! \\lambda \\overrightarrow{dir}^{\\perp}(\\bm{z_{j*}}),\\|\\bm{u_e}\\|_2\\!\\le\\! b, {\\left\\| {{\\bm{z_a}}({\\bm{u_e}})\\! - \\!{\\bm{z_{j*}}}} \\right\\|_2} \\!<\\! {\\hat R_o}\\},\\!\\! \n\\end{equation}\nwhere $\\lambda>0$ and ${\\perp}$ means the orthogonal vector. \nSince $r_i$ is an out-neighbor of $r_j$, when within in the interaction range, its state is predicted by \n\\begin{equation}\n{{\\hat z}_i}^{k+1} = {  \\hat W_{\\sss HF}^{[i,:]} }{{\\tilde z}_{\\sss F}}^{k} +\\varepsilon _{\\sss T}\\hat u_i^{k}. \n\\end{equation}\nOnce the continuous excitation of $r_a$ makes $r_i$ lose connections with any one in $\\mathcal{N}_i^{in}$, \nthe deviation $\\| \\bm{\\hat{z}}_{i}^{k+1}- \\bm{\\tilde{z}}_{i}^{k+1}\\|_2$ will be significant large, which inspires us to design the following criterion to estimate $R_c$, given by\n\\begin{equation}\\label{aa}\n\\left\\{ \n\\begin{aligned}\n    k_2&=\\min\\{k:\\left\\|  \\bm{\\hat z_i}^{k+1}- \\bm{\\tilde z_i}^{k+1} \\right\\|_2>\\beta\\}, \\\\\n    \\hat R_c&=\\max\\{\\| \\bm{\\tilde z_i}^{k_2}-  \\bm{\\tilde z_j}^{k_2}\\|_2:~j\\in\\mathcal{N}_i^{in}\\}, \n\\end{aligned}\\right.\n\\end{equation}\nwhere $\\beta>0$ is a given threshold. \nTechnically, a larger $\\beta$ will guarantee a more conservative $\\hat R_c$. \n\nIf there $\\forall i\\in\\mathcal{V}_{\\sss H}$, $|\\mathcal{N}_i^{in}|\\le1$, we randomly select a $r_i$ with $|\\mathcal{N}_i^{in}|=1$ as the direct excitation target. \nIn this case, we introduce a time window $k_l$ to design an intermittent excitation strategy. \nAt each iteration $k$ satisfying $[k\\!\\!\\mod\\! k_l=0]$, $ \\bm{u_e}(k)$ is randomly selected from \n\\begin{equation}\n\\!\\{ \\bm{u_e}\\!:\\bm{{z_a}({u_e}}) \\!=\\! \\alpha \\bm{z_{i*}} \\!+\\! (1 \\!-\\! \\alpha )\\bm{z_i},{\\left\\| { \\bm{{z_a}({u_e}}) \\!-\\! \\bm{z_{i*}} } \\right\\|_2} \\!<\\! {\\hat R_o}\\},\\!\n\\end{equation}\nand $r_a$ stays still at other iterations. \nLikewise, the same judging criterion (\\ref{aa}) is used to estimate $R_c$ at iteration $k$ satisfying $[k\\mod k_l=k_l-1]$. \nOnce a $\\hat R_c$ is estimated, $r_a$ does reverse excitation to steer the robot to form the original steady pattern, and observes more obstacle-avoidance behaviors. \n\n\nFinally, with $\\hat R_c$ determined, $\\mathcal{V}_{\\sss H}$ is re-specified by setting $R_h=R_f-\\hat R_c$, and a more informative $\\hat W_{\\sss HF}$ is further approximated by (\\ref{solution1}). \nDuring the whole process of this stage, the avoidance behavior corresponding to every active excitation is recorded as the following input-output pair\n\\begin{equation}\\label{input1}\n\\left\\{ \n\\begin{aligned}\n&Q_{in}^{k}\\!=\\![\\tilde z_v^{k}-\\tilde z_a^{k}, \\tilde z_{v*}^{k}-\\tilde z_v^{k}, \\Delta \\tilde z_v^{k}\/\\varepsilon_{\\sss T},\\Delta \\tilde z_a^{k}\/\\varepsilon_{\\sss T}],\\\\\n&Q_{out}^{k}=\\Delta \\tilde z_v^{k+1}.\n\\end{aligned}\\right.\n\\end{equation}\nFurther, construct the dataset $\\mathcal{D}_e = \\left\\{ {\\mathop  \\cup \\limits \\{Q_{in}^{k},Q_{out}^{k}\\} } \\right\\}$, and $g$ can be learned using many mature learning methods by solving \n\\begin{equation}\\label{ques}\n\\hat g = \\mathop {\\arg \\min }\\limits_{g: Q_{in}\\mapsto Q_{out}} \\sum\\nolimits_{k = 1}^{ |\\mathcal{D}_e | } {{{\\left\\| Q_{out}^{k}-g(Q_{in}^{k}) \\right\\|}_2}}.\n\\end{equation}\nSpecifically, we adopt SVR method, which has good performance on nonlinear approximation and strong generalization ability when the amount of data is not vast \\cite{cristianini2000introduction}. \n\nAt the end of this stage, $\\hat R_c$, $\\hat W_{\\sss HF}$ and $\\hat g$ are all obtained, providing support to design efficient sneak strategies. \n\n\n\n\n\n\\section{Sneak Attack Design}\\label{attack}\nIn this section, we first analyze the attack feasibility by introducing the definitions of 1-hop convex formation and the indirect controllability of a robot. \nThen, the sufficient conditions for the attack are established. \nFinally, we propose the ECR strategy to dynamically replace $r_v$ by $r_a$, completing sneak.\n\n\n\\subsection{Feasibility Analysis}\nIn this part, we give the theoretical guarantees for the existence of attack positions and the controllability of robots. \n\nNote that the success of the attack lies in two parts: i) there exists a feasible attack position and ii) $r_v$ is controllable such that the attack condition is satisfied. \nAccordingly, we introduce the following definitions. \n\\begin{definition} \\label{polygon}\n(\\textbf{1-hop convex pattern}) \nGiven $r_i$, if $\\exists \\mathcal{V}_{i}^{p}\\subseteq\\{ i \\cup \\mathcal{N}_i^{out} \\}$ such that i) the nodes in $\\mathcal{V}_{i}^{p}$ constitutes a convex polygon; \nii) the polygon covers $\\{ i \\cup \\mathcal{N}_i^{out} \\}$; \niii) $i$ is one vertex of the polygon, then $\\mathcal{V}_{i}^{p}$ is the 1-hop convex pattern of $r_i$. \n\\end{definition} \n\\begin{definition}\n(\\textbf{Indirect controllability}) \nA node is indirectly controllable if one can control another node, and a let a chain reaction make the tagged node reach any $z_c^*$ in finite steps. \n\\end{definition}\n\\begin{theorem}\\label{property}\nGiven $r_i$, if $\\mathcal{V}_{i}^{p}$ exists, then\n\\begin{enumerate}[i)]\n\\item $\\mathcal{V}_{i}^{p}$ is unique. \n\\item $\\mathcal{Z}_i^{f}={ \\mathop \\bigcap \\limits_{\\mathclap{j \\in \\mathcal{N}_i^{out} }} {\\mathcal{P}}( \\bm{z_j} , d_{ij})}\\neq\\emptyset$, and ${\\left\\| { \\bm{z} - \\bm{z_j} } \\right\\|_2} < {\\left\\| { \\bm{z_i} - \\bm{z_j} } \\right\\|_2}$, $\\forall \\bm{z}\\in \\mathcal{Z}_i^{f}$, $ j \\in \\mathcal{N}_i^{out}$. \n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{property} shows the nice distance properties brought by $\\mathcal{V}_{i}^{p}$, which helps to the sneak strategy design. \nBesides, in its proof in Appendix, we also provide a method to find the positions that satisfy the property. \nNext, we illustrate under what conditions the robots are indirectly controllable. \n\nSuppose that $i\\in\\mathcal{V}$ is under the excitation of $r_a$, and it can be seen as another leader. \nThen, $z_i$ is directly determined by $u_e=g(\\cdot)$. \nIn this situation, denote the new adjacent matrix as $A^e$ where \n$a^e_{ij}=0,\\forall j\\in\\mathcal{N}_i^{in}$, and other elements are the same as in $A$. \nAccordingly, its Laplacian matrix is $L^e=\\text{diag}\\{A^e \\bm{1}\\}-A^e$, and $p^e_1\\!=\\![p^e_{\\sss 11},\\!\\cdots\\!,p^e_{\\sss 1N}]^\\mathsf{T}$ is the corresponding left eigenvector for $\\lambda_1$ of $L^e$.\n\\begin{lemma}\\label{lh44}\nGiven desired state $z_c^*$ and initial state $z_i^0$, $r_i$ is indirectly controllable by $r_j$ iff the following conditions hold\n\\begin{equation}\n\\left\\{ \n\\begin{aligned}\n&u_e u_c>0,&\\text{if}~(z_c^*-z_i^0)u_c>0, \\\\\n&|p^e_{\\sss 1j}{u_e}| >|p^e_{\\sss 1N}{u_c}|,&\\text{if}~(z_c^*-z_i^0)u_c<0. \n\\end{aligned}\\right.\n\\end{equation}\n\\end{lemma}\n\n\nLemma \\ref{lh44} points out the necessary and sufficient condition for indirect controllability from the perspective of global connectivity. \nHowever, under partial observation, the eigenvectors of the interaction matrix are hard to obtain, making it impossible to verify the conditions. \nTo overcome this issue, we leverage the idea of Lemma \\ref{lh44}, and obtain the local conditions that guarantee the direct controllability. \n\\begin{theorem}\\label{th44}\nGiven desired state $z_c^*$ and initial state $z_i^0$, $r_i$ is indirectly controllable by $r_j$ when $u_e,u_c$ satisfy \n\\begin{equation}\\label{str}\n\\left\\{ \n\\begin{aligned}\n&u_e u_c>0,&\\text{if}~(z_c^*-z_i^0)u_c>0, \\\\\n&|a_{ij}{u_e}| >|\\bar{a}_{ij}{u_c}|,&\\text{if}~(z_c^*-z_i^0)u_c<0,\n\\end{aligned}\\right.\n\\end{equation}\nwhere $\\bar{a}_{ij}=\\!\\!\\!\\!\\! \\sum\\limits_{j' \\in \\{ \\mathcal{N}_i^{in}\\backslash{j} \\}}{\\!\\!\\!\\!\\!  a_{ij'}}$. \n\\end{theorem}\nWithout knowing the global interaction structure in $\\mathcal{V}$, Theorem \\ref{th44} provides a sufficient criterion to find the robots satisfying the indirect controllability by considering the local interaction with their in-neighbors. \nThis condition will be used in the following strategy design. \n\n\n\\subsection{Sneak Strategy: Evaluate-Cut-Restore}\nTo achieve maximum attack rewards, a feasible $r_v$ needs to be selected based on its significance evaluation in the formation first, and then $r_a$ is controlled to replace $r_v$. \nBesides, the strategy design should be flexible for different local geometric patterns of $r_v$. \nTo meet the neighbor-replacement rule $\\Omega_A$, the key point is to ensure the existence of $\\mathcal{V}_{v}^{p}$. \nBriefly, the ideas of our proposed sneak strategy are summarized as follows. \n\\begin{itemize}\n\\item \\textbf{Evaluate} the significance of the robots based on their potential indirect controllability and the state-propagation effects. \nSelect the most valuable one as $r_v$. \n\\item \\textbf{Cut} the connections between $r_v$ and $\\mathcal{N}_v^{in}$. \nIf the 1-hop convex pattern $\\mathcal{V}_{v}^{p}$ does not exist, $r_a$ needs to make $\\mathcal{V}_{v}^{p}$ form first. \n\\item \\textbf{Restore} the original steady pattern, by making $r_a$ be regarded as the in-neighbor of $\\mathcal{N}_v^{out}$ based on ruleset $\\Omega$ and replacing $r_v$ by $r_a$.\n\\end{itemize}\n\nDuring the sneak stage, the estimated state of $r_v$ under attack and the state of $r_a$ are respectively given by  \n\\begin{equation}\\label{state1}\n\\hat z_v^{k + 1} (u_a^k)= \\tilde z_v^k + \\hat g({u_a^k}), \n\\end{equation}\n\\begin{equation}\\label{state2}\nz_a^{k+1}=z_a^k+u_a^k,\n\\end{equation}\nwhere $\\hat g({u_a^k})$ is simplified for $\\hat g({z_a^k}-{\\tilde z_i^k},{z_{i*}^k}-{\\tilde z_i^k},{v_a^k}-{v_i^k})$. \nThe details of ECR strategy are given below. \n\n\\textbf{Evaluate phase}. \nWith $\\hat W_{\\sss HF}$ obtained, we need to find a victim robot to achieve the maximum rewards. \nNote that for a robot in the MRN, a larger out-degree means it has the potential capability to control more robots while a smaller in-degree indicates it is less affected by others. \nFrom this perspective, the evaluation criteria is designed as \n\\begin{subequations}\\label{node-value0}\n\\begin{align}\n\\mathop {\\max }\\limits_{i}  ~~~~~~&( |\\mathcal{N}_i^{out}|+ {\\| {W_{\\sss\\! HF}^{[:,i]}} \\|_1} - |\\mathcal{N}_i^{in}|-{\\| {W_{\\sss HF}^{[i,:]}}\\|_1})\\\\\n\\text{s.t.}~~~~~~&i \\in \\mathcal{V}_{\\sss H}, ~\\mathcal{N}_i^{out}|\\ge1, |\\mathcal{N}_i^{in}|\\le \\alpha_1,\n\\end{align}\n\\end{subequations}\nwhere ${W_{\\sss HF}^{[:,i]}}$, ${W_{\\sss HF}^{[i,:]}}$ represent the $i$-th column and row of $W_{\\sss HF}$, respectively, and $\\alpha_1$ a preset possible integer. \nThe solution of (\\ref{node-value0}) is selected as $r_v$. \nDue to the small scale of the observed formation, (\\ref{node-value0}) can be easily solved by a common traverse search method with algorithm complexity of $O(|\\mathcal {H}| |\\mathcal {F}|)$.\n\n\n\\textbf{Cut phase}. \nIn this core phase, $r_a$ aims to make the connections between $r_v$ and its in-neighbors break, i.e., $a_{vj}(t)\\ne a_{vj}^0$. \nIf $\\mathcal{V}_{i}^{p}$ exists and $r_v$ is readily to be excited directly (i.e., not many other robots are distributed around), the following strategy is adopted to break the connections,\n\\begin{subequations}\\label{eq-pro}\n\\begin{align}\n\\mathop{\\max }\\limits_{\\bm{u_a}^k}~&{\\alpha_2}{\\| {\\bm{\\hat z_v}^{k + 1}(\\bm{u_a}^k) \\!-\\! \\bm{\\hat z_{v*}}^{k + 1}}\\|_2} + {\\alpha _3}\\!\\sum\\limits_{j \\in \\mathcal{N}_v^{in}} \\!\\!\\!{{{\\| {\\bm{\\hat z_j}^{k + 1} \\!-\\! \\bm{\\hat z_v}^{k + 1} \\!-\\! {\\bm{\\tilde h_{jv}} }} \\|}_2}} ,\\label{eq-pro-a}\\\\\n{\\rm{s.t.}}~& (\\ref{predict}),~(\\ref{state1})~\\text{for}~r_v,~\\text{and}~(\\ref{state2}),\\\\ \n&\\|\\bm{u_a}^k \\|_2 < b.  \n\\end{align}\n\\end{subequations}\nwhere $\\alpha_2,\\alpha_3>0$ are the weight parameters to balance the offsets of $r_v$ with its ideal position and in-neighbors. \nAn efficient way to solve this cut-edge problem is to adopt a greedy heuristic by sampling from the feasible solution space. \nThe attack iteration under (\\ref{eq-pro}) is terminated by the same criterion (\\ref{aa}) for estimating $R_c$ in Section \\ref{estimate_external}. \n\nIf $\\mathcal{V}_{v}^{p}$ does not exist or the $r_v$ can not be attacked readily, we utilize the indirect controllability to attack an in-neighbor of $r_v$ first, such that the formation pattern $\\mathcal{V}_{i}^{p}$ is reached and $r_v$ is easier to approach. \nLet $r_{l_{1}}, r_{l_{2}}$ be the two robots in $\\mathcal{N}_i^{out}$ that are closest to $r_j$, \nand $\\overrightarrow{\\bm{z}_{l_1} \\bm{z}_{l_2} }$ be the vector from $\\bm{z}_{l_1}$ to $\\bm{z}_{l_2}$. \nThen, the following strategy is adopted\n\\begin{align}\\label{chain}\n\\bm{u_a}^k = \\mathop {\\arg }\\limits_{\\bm{u_a}} \\max \\{ &{{\\| {\\bm{\\hat z_j}^{k + 1}(\\bm{u_a}) - \\bm{\\hat z_{j*}}^{k + 1}} \\|}_2}: \\bm{u_a}\\in \\mathcal{U}_{\\sss F}, \\nonumber \\\\\n&j\\in\\mathcal{N}_i^{in} ~\\text{and}~ r_j~\\text{satisfies}~(\\ref{str})  \\},\n\\end{align}\nwhere $\\mathcal{U}_{\\sss F}\\!=\\!\\{\\bm{u_a}: { {\\left\\| {\\bm{u_a}} \\right\\|}_2} < b,\\bm{u_a} = \\lambda \\overrightarrow{\\bm{z}_{l_1} \\bm{z}_{l_2} }^{\\perp} ,\\forall \\lambda>0 \\} $. \nIteratively, $r_a$ updates the control inputs until $\\mathcal{V}_{i}^{p}$ is formed, then the control mode is turned to (\\ref{eq-pro}). \n\n\n\n\\textbf{Restore phase}. \nAfter the $\\mathcal{V}_{v}^{p}$ is formed and the connections between $r_v$ and $\\mathcal{N}_v^{in}$ are cut, $r_a$ only needs to be in a position that is close to arbitrary $j\\in\\mathcal{N}_v^{in}$. \nUtilizing the properties in Theorem \\ref{property}, $r_a$ is controlled by \n\\begin{align}\\label{chain2}\n\\bm{u_a}^k = \\mathop {\\arg }\\limits_{\\bm{u_a}} \\max \\left\\{ {{\\left\\| {\\bm{\\hat z_v}^{k + 1}(\\bm{u_a}) - \\bm{\\hat z_{v*}}^{k + 1}} \\right\\|}_2}: \\bm{z_a}^{k+1}\\in \\mathcal{Z}_v^{f}  \\right\\}.\n\\end{align}\nBy continuously implementing the procedure, $r_a$ takes the ideal position that belongs to $r_v$, and will be regarded as the real $r_v$ by $r_v$'s out-neighbors at certain time according to rule $\\Omega_A$. \nAfter that, the original $r_v$ becomes an outlier and $r_a$ is controlled to move towards $\\hat z_{v*}$ and $\\bm{z_a} \\in \\mathcal{P}(\\bm{z_j}, R_c)$. \nFinally, $r_a$ is recognized as a member and obtains partial control over $\\mathcal{V}$. \nThe sneak attack is completed. \n\nDuring this attack-implementing stage, $r_a$ also collects dataset $\\mathcal{D}_a=\\{Q_{in}^{k},Q_{out}^{k}\\}$ and uses it to update $\\hat g$ online. \nAfter sneaking into the MRN, $r_a$ is able to implement further attacks, for example, infer the interaction structure in $\\mathcal{V}_{\\sss H'}$, or intentionally split the MRN by misleading $\\mathcal{N}_v^{out}$ and the out-neighbors of them. \nThese interesting problems are not the focus of this paper and are omitted here. \n\nTo sum up, the whole procedures of the sneak attack consisting of four stages are summarized in Algorithm \\ref{algo-all}. \n\n\\begin{algorithm}[t]\n \\small\n    \\caption{Sneak Attack against MRNs}\n    \\label{algo-all}\n    \\begin{algorithmic}[1]\n    \\REQUIRE{Partial observation range $R_f$, moving ability bound $b$.}\\;\n    \\ENSURE{Interaction rules $W_{\\sss HF}$, $g$.}\\;\n    \\STATE Observe the dynamic process of $\\mathcal{V}_{\\sss F}$ converging to the steady formation geometric pattern.\\;\n    \\STATE Judge when the steady pattern is reached by (\\ref{time}).\\;\n    \\STATE Construct datasets $\\mathcal{D}_c$, $\\mathcal{D}_s$, and estimate the parameters of steady pattern by (\\ref{data}).\\;\n    \\STATE Obtain an initial approximation of $W_{\\sss HF}$ based on $\\mathcal{D}_c$ by (\\ref{solution1}).\\;\n    \\STATE Leveraging direct controllability, make tentative excitation on $r_j$ continuously to infer $R_o$, $R_c$ by (\\ref{aa}).\\;\n    \\STATE Expand the approximation range of $W_{\\sss HF}$ utilizing $\\hat R_c$.\\;\n    \\STATE Construct dataset $\\mathcal{D}_e= \\left\\{ {\\mathop  \\cup \\limits \\{Q_{in}^{k},Q_{out}^{k}\\} } \\right\\}$ during the excitation process, and learn the external interaction mechanism $g$ by (\\ref{ques}).\\;\n    \\STATE Leveraging indirect controllability, adopt ECR (Evaluate-Cut-Restore) attack strategy to replace a selected $r_v$ by $r_a$.\\;\n \t\\STATE Construct dataset $\\mathcal{D}_a= \\left\\{ {\\mathop  \\cup \\limits \\{Q_{in}^{k},Q_{out}^{k}\\} } \\right\\}$ during the sneak process, and update $g$ online  by (\\ref{ques}).\\;\n \t\\STATE Obtain partial control over $\\mathcal{V}_{\\sss F}$ and support further attacks.\\;\n    \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\section{Simulation}\\label{simulation}\nIn this section, we conduct extensive simulations to demonstrate the feasibility of revealing the interaction rules from external observations and excitation, and also validate the theoretical analysis and the proposed sneak attack strategy. \n\n\n\n\\subsection{Simulation Setting}\n\\textbf{MRN specification}. \nAn MRN consisting of 17 robots is considered, with two kinds of interaction structure and common preset formation shape, as shown in Fig.~\\ref{se_example}. \nSpecifically, robot 1 is set as the leader, and the moving speed in stable stage is set as $0.2m\/s$. \nThe maximum speed of $r_a$ is $1m\/s$, and the radii are setting as $R_c=7m$, $R_o=2m$ and $R_s=0.5m$. \n\n\\begin{figure}[t]\n \n  \\setlength{\\abovecaptionskip}{0.1cm}\n    \\subfigure[]{ \n    \\label{sae1}\n    \\includegraphics[width=0.23\\textwidth]{pic\/example22} \n  }\n  \\hspace{-5mm}\n  \\subfigure[]{ \n    \\label{sae2}\n    \\includegraphics[width=0.23\\textwidth]{pic\/example33} \n  } \n  \\caption{An MRN of 17 robots with two different interaction structure. The formation shape is set the same, while the intersection structure is different. \n  The detailed interaction weights are denoted in red font. } \n  \\label{se_example}\n  \\vspace*{-13pt}\n\\end{figure}\n\n\\begin{figure*}[ht]\n\\setlength{\\abovecaptionskip}{0.1cm}\n  \\centering \n    \\subfigure[]{ \n    \\label{se0}\n    \\includegraphics[width=0.3\\textwidth]{pic\/speed} \n    }\n    \\subfigure[]{ \n    \\label{se1}\n    \\includegraphics[width=0.3\\textwidth]{pic\/error_sample_model1} \n  } \n  \\subfigure[]{ \n    \\label{se2}\n   \n    \\includegraphics[width=0.3\\textwidth]{pic\/error_sample_model2} \n   } \n  \\caption{ The approximation results based on calculated $\\hat k_s$. (a). Formation velocity estimation. (b) and (c) are the approximation results with increasing data amount of $\\mathcal{D}_c $ structure case 1 and 2, respectively.} \n  \\label{error_sample}\n  \\vspace*{-8pt}\n\\end{figure*}\n\n\\textbf{Comparisons under different models and parameters}. \nFor the interaction structure $\\hat W_{\\sss HF}$, we present the approximation results from three perspectives. \nFirst, we adopt the liner model $\\dot z(t) =- Lz(t)+Lh+u_0$ and the nonlinear model $\\dot z(t) =- Lz(t)+Lh+u_s(t)$, respectively. \nSecond, we consider the influence of different sample amounts and different observation noise variance. \nAdditionally, $\\hat W_{\\sss HF}$ can be further optimized by considering the feedback of $\\hat R_c$, i.e., \n\\begin{subequations} \\label{further}\n\\begin{align}\n\\mathop {\\min }\\limits_{{W_{\\sss HF}}} ~& \\| Y_{\\sss H} - {W_{\\sss HF}}Y_{\\sss F}  \\|_{\\sss \\text{Frob}}  \\\\\n\\text{s.t.}~~&W_{\\sss HF}^{[i,j]}=0,\\text{if}~\\|\\tilde z_i-\\tilde  z_j\\|_2>\\hat R_c, i\\in\\mathcal{V}_{\\sss H},j\\in\\mathcal{V}_{\\sss F}.\n\\end{align}\n\\end{subequations}\nAs a typical constrained linear least square problem, (\\ref{further}) can be solved by many mature optimization techniques, e.g., Newton interior-point method \\cite{boyd2004convex}. \nFor the external interaction mechanism $\\hat g$, we present the approximation results under different sample amounts and different noise variance. \n\n\n\\textbf{Metric of evaluation}. \nTo evaluate $\\hat W_{\\sss HF}$, we use the following two indexes \n\\begin{align}\n{\\varepsilon _1} &=({{{\\| {\\text{sign}({\\hat W}_{\\sss HF}) - \\text{sign}(W_{\\sss HF}}) \\|}_0}})\/  {( |\\mathcal{V}_{\\sss H}| |\\mathcal{V}_{\\sss F}| )}, \\\\\n{\\varepsilon _2} &= ({\\| {{\\hat W}_{\\sss HF} - W_{\\sss HF}} \\|_{\\text{Frob}} }) \/ {\\| W_{\\sss HF}\\|_{\\text{Frob}}},\n\\end{align}\nwhere ${\\varepsilon _1}$ evaluates the structure correctness in terms of whether two robots have an interaction connection, and ${\\varepsilon _2}$ represent the magnitude correctness in terms of how larger is the connection weight between two robots. \nTo evaluate $\\hat g$, we adopt mean directional accuracy (MDA), root mean square error (RMSE) and mean absolute error (MAE), which are respectively calculated by  \n\\begin{align}\n{\\rm{MDA}} &= \\frac{1}{m}\\sum\\limits_{i = 1}^m {  { \\text{sign}  }({y_i} - {{y'}_i}} ),\\\\\n{\\rm{RMSE}} &= \\sqrt {\\frac{1}{m}\\sum\\limits_{i = 1}^m {{{({y_i} - {{y'}_i})}^2}} } ,\\\\\n {\\rm{MAE}} &= \\sum\\limits_{i = 1}^m {\\frac{{\\left| {{y_i} - {{y'}_i}} \\right|}}{{{m}}}},\n\\end{align}\nwhere $y_i$, $y'_i$ are the actual and predicted values, $m$ is the amount of samples. \n\n\\subsection{Results and Analysis}\nWe mainly illustrate the results from the perspectives of the four stages described in Section \\ref{problemof}. \n\n\\textbf{Stage 1}. \nRecall that the time instant $\\hat k_s$ is computed by (\\ref{time}) to identify when the MRN has reached the steady pattern. \nUsing different threshold $\\epsilon$ to compute $\\hat k_s$ will result in different sample number of $\\mathcal{D}_c $. \nAs shown in Fig.~\\ref{se0}, when the MRN reaches steady state, the velocity estimation remains stable. \nFig.~\\ref{se1}-\\ref{se2} illustrate the sample number mainly affects the accuracy of $\\varepsilon_2$, and when the amount becomes larger, both $\\varepsilon_1$ and $\\varepsilon_2$ become stable. \nThis is because that the latter observations in $\\mathcal{D}_c $ are observed from the stage that is close to the steady process, only providing redundant information to approximate ${W}_{\\sss HF}$. \nIn fact, this demonstrates that an accurate calculation of $\\hat k_s$ is not important under both linear and nonlinear models, especially for the structure error. \nNevertheless, more data are still helpful to obtain ${W}_{\\sss HF}$ with smaller magnitude errors. \n\n\n\\textbf{Stage 2}. \nHereafter, we present the results based on the interaction structure given in Fig.~\\ref{sae1}. \nFig.~\\ref{error_w} shows the approximation errors of the internal interaction structure under different variances of observation noises. \nThe variance of observation noise varies from 0 to 0.5. \nFig.~\\ref{error_w_a} presents the approximation evaluation of the MRN under linear formation law. \nAs it shows, the structure error $\\varepsilon_1$ is smaller than $\\varepsilon_2$, and when noises are involved, the approximation error is generally stable under different noise variances. \nSpecifically, when the interaction radius feedback is used to further optimize ${W}_{\\sss HF}$, i.e., solving (\\ref{further}), the approximation errors are reduced significantly. \nThe details in Fig.~\\ref{error_w_b} is likewise, and under the same conditions other than the nonlinear formation control, the approximation errors are basically larger than that under linear formation control, which is not hard to understand. \nHowever, we observe that even though the approximation errors in the nonlinear case is higher, the difference is not significant. \nThe errors remain small and are sufficient to get a feasible ${W}_{\\sss HF}$ to support the following attack, especially in terms of the connection structure. \nThese results verify that the accuracy of the proposed linear approximation for ${W}_{\\sss HF}$. \n\\begin{figure}[t]\n  \\centering \n  \\setlength{\\abovecaptionskip}{0.1cm}\n    \\subfigure[]{ \n    \\label{error_w_a}\n    \\includegraphics[width=0.24\\textwidth]{pic\/linearmodel} \n  } \n  \\hspace{-6mm}\n  \\subfigure[]{ \n    \\label{error_w_b}\n    \\includegraphics[width=0.24\\textwidth]{pic\/nonlinearmodel} \n  } \n  \\caption{The approximation result comparison of ${\\hat W}_{\\sss HF}$ with and without the feedback of $\\hat R_c$. (a) under linear formation control. (b) under nonlinear formation control.} \n  \\label{error_w}\n  \\vspace*{-10pt}\n\\end{figure}\n\\begin{figure*}[t]\n\\centering\n\\setlength{\\abovecaptionskip}{0.05cm}\n\\includegraphics[width=0.9\\textwidth]{pic\/noise.png}\n\\caption{Obstacle-avoidance mechanism regression results based on SVR method. \nUnder different numbers of the samples, 3 groups of experiments are conducted for each sample number. }\n\\label{regression_noise}\n\\vspace*{-10pt}\n\\end{figure*}\n\n\\textbf{Stage 3}. \nTo verify the approximation performance of the obstacle-avoidance mechanism (here we use classical artificial potential method \\cite{khatib1986real}), SVM with a Gaussian kernel function is used for regression, and the parameters are optimized by 10-fold grid search cross-validation. \nWe conduct 4 groups of regression with 3 repeated procedures based on $\\mathcal{D}_e$ collected from the tentative excitation process. The $\\sigma=0.1$ here. \nWe randomly select 25, 50, 100, 200 samples to train the SVR model in each group, respectively. \nA fixed set of 120 samples are used for testing. \nThe testing results are shown in Fig.~\\ref{regression_noise}, \nwith the statistic results given in Table ~\\ref{regression_table}. \nBy comparing the results of the test samples in each group, it is obvious that the fitting performance is improved with more training data. \nMeanwhile, the index does not vary much with more samples, which is illustrated by comparing the results of repeated tests. \nThe MDA result shows a rising trend from 25-sample-training to 200-sample-training, which indicates that more training samples lead to high accuracy. \nNevertheless, we observe that even though a larger number of samples bring more accurate results, the improvement of the accuracy is not significant. \nEven with a small number of samples such as the 25-samples, the method is still feasible, especially in terms of MDA (above 90\\%). \nIt is worth mentioning that RMSE and MAE fluctuate due to observation noises. \nSpecifically, RMSE is no larger than 0.6, corresponding to the famous $3\\sigma$-principle \\cite{pukelsheim1994three}. \nOverall, these show the effectiveness of the proposed method to learn the external interaction rule. \n\n\n\n\n\\begin{table}[]\n\\setlength{\\abovecaptionskip}{0.3cm}\n\\caption{Statistic obstacle-avoidance mechanism regression results based on SVR method.  }\n\t\\begin{tabular}{cccclccc}\n\t\t\\toprule[1pt]\n\t\t& \\multicolumn{3}{c}{25 samples}  &  & \\multicolumn{3}{c}{50 samples}  \\\\ \\cline{2-4} \\cline{6-8} \n\t\tIndex & MDA       & RMSE     & MAE      &  & MDA       & RMSE     & MAE      \\\\ \\cline{2-8} \n\t\tTraining         & 0.880        & 0.253    & 0.154    &  & 0.913      & 0.217    & 0.113    \\\\\n\t\tTesting          & 0.933      & 0.601    & 0.404    &  & 0.933      & 0.581    & 0.300    \\\\ \\hline\n\t\t& \\multicolumn{3}{c}{100 samples} &  & \\multicolumn{3}{c}{200 samples} \\\\ \\cline{2-4} \\cline{6-8} \n\t\tIndex & MDA       & RMSE     & MAE      &  & MDA       & RMSE     & MAE      \\\\ \\cline{2-8} \n\t\tTraining         & 0.910        & 0.333    & 0.146    &  & 0.923      & 0.426    & 0.206    \\\\\n\t\tTesting          & 0.956      & 0.541    & 0.291    &  & 0.967      & 0.496    & 0.264    \\\\ \n\t\t\\bottomrule[1pt]\n\t\\end{tabular}\n\t\t\\label{regression_table}\n\t\t\\vspace*{-10pt}\n\\end{table}\n\n\n\n\\begin{figure}[t]\n  \\centering \n \n    \\subfigure[]{ \n    \\label{stage_a}\n    \\includegraphics[width=0.4\\textwidth]{pic\/all_error5} \n  } \n  \\subfigure[]{ \n    \\label{stage_b}\n    \\includegraphics[width=0.4\\textwidth]{pic\/neibour_error2} \n  } \n  \\caption{The illustration of the sneak process. (a). The position errors between the real and the desired positions of the robots, and $r_a$ takes the $z_5^*$ as its desired position. (b) The distance deviations  $({\\left\\| {z_a - \\tilde z_j} \\right\\|_2} - {\\left\\| {{\\tilde z_i} - \\tilde z_j} \\right\\|_2}) $, here $i=5$, $j=6,7,8$.  } \n  \\label{attacker_w}\n  \\vspace{-15pt}\n\\end{figure}\n\n\\textbf{Stage 4}. \nFirst, at the Evaluate phase, robot 5 is selected as $r_v$ by the evaluation criteria (\\ref{node-value0}), which has only one in-neighbor and three out-neighbors, and its corresponding 1 hop convex pattern $\\mathcal{V}_{5}^{p}$ exists. \nThen, during the Cut phase, $r_a$ utilizes the learned $\\hat g$ and makes consequent excitations on $r_v$ by solving (\\ref{eq-pro}), such that $r_v$ is gradually pulled out of the interaction range of its in-neighbors, shown as the phase before the break time in Fig.~\\ref{stage_a}. \nAfter the connections of $r_v$ with its in-neighbors break, $r_a$ in the Restore phase is controlled by solving $(\\ref{chain2})$ dynamically to move within $\\mathcal{Z}_i^{f}={ \\mathop \\bigcap \\limits_{\\mathclap{j \\in \\mathcal{N}_i^{out} }} {\\mathcal{P}}(z_j, d_{ij})}$ ($i=5$ here), such that ${\\left\\| {z_a - z_j} \\right\\|_2} < {\\left\\| {{z_i} - z_j} \\right\\|_2}$, $\\forall j \\in \\mathcal{N}_i^{out}$ (i.e., $j=6,7,8$ here). \nThis phase is illustrated as the process between the break time and sneaking start time in Fig.~\\ref{stage_a}, with detailed presentation of distance deviations $({\\left\\| {z_a - \\tilde z_j} \\right\\|_2} - {\\left\\| {{\\tilde z_i} - \\tilde z_j} \\right\\|_2}) $ in Fig.~\\ref{stage_b}, where the deviations are all negative. \nBased on the neighbor-replacement authentication rule $\\Omega_A$, the robots $\\mathcal{N}_5^{out}$ will take $r_a$ as the new neighbor replace robot 5. \nFinally, $r_a$ here just moves towards $z_{5*}$ and its out-neighbors begin following it, and the formation will reach steady state again. \nNote that the position fluctuation of $\\mathcal{N}_5^{out}$ before beginning sneaking in Fig.~\\ref{stage_a} also verifies the indirect controllability in Theorem \\ref{th44}. \nLikewise, this can be used to intentionally split the MRN formation shape, and the results are omitted here due to space limits. \n \n\n\n\n\n\n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, we investigated the security of MRNs under formation control by designing a sneak attack. \nWe demonstrated that the interaction rules in MRNs are learnable even the attacker is only with partial observation over the MRN, without any strong prior knowledge like the system dynamics model or structure. \nFirst, we proposed an excitation-based method to approximate the internal interaction structure within the MRN and the obstacle-avoidance mechanism with the environment. \nThen, we designed the ECR strategies, which make the attacker replace a victim robot and gain optimal local control over the MRN. \nFinally, comprehensive theoretical analysis and numerical results illustrated the effectiveness of the proposed attack. \nThis work reveals the possibility of learning the interaction rules of an MRN from physical observations and excitations, and shows how to utilize them to design an intelligent attack, even though the interaction rules are model agnostic to the attacker. \n\nIn a broader sense, our work broadens the horizons of the security of CPSs by exploiting the system vulnerabilities from physical observations and interactions. \nFurthermore, we emphasize that the system interaction nature can be leveraged by malicious attackers to cause disastrous consequences on the system. \nConcerning the CPS security from this perspective, a lot of issues remain to be further addressed, including i) how to design efficient detection methods to identify the potential threats; \nii) how to reliably secure the interaction protocols that only leak confusing states on the external observer; \nand iii) how to construct robust and dynamic interaction defensive strategies with the physical world. \n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe quantization of simple second-class constrained mechanical systems has usually been \naccomplished by adopting as canonical commutation relations those given by the so-called \nDirac algorithm. This method, although successful in many situations, can potentially lead \nto wrong results, and additional strategies or physical considerations are needed. This, \nwhich is not a serious problem when the system under consideration contains ``known \nphysics'', as in the case of the free particle constrained to move on a sphere surface \n\\cite{properdirac, circle, stuecksphere}, could become dramatic if we want to unfold new physical features.\n\nIn general, canonical quantization makes extensive use of Hamiltonian formulation of \nclassical mechanics, trying to put the classical theory in a form in which the quantization \nwould be straightforward and\/or as unambiguous as possible. However, it should be \nrecalled once again that it is the quantum system what contains the physical entity,\nthe classical theory being a mere approximation. \n\nThis paper is devoted to show how, in the case of the partial-trace non-linear sigma model (NLSM) at least, a proper quantum theory can be obtained straightforwardly if one leaves aside \nthe idea that the classical theory must lead to the quantum one through a process of  \n\\textit{quantization}, which is inherently ambiguous. Here we suggest that the \n\\textit{quantized system} should be \\textit{directly} obtained through a deeper knowledge \nof the corresponding dynamical symmetry transformations, i.e. those that are able to span the whole space of physical states \n(and\/or parameterize the classical solution manifold by means of Noether invariants). We aim at obtaining the quantum theory not from an appropriate treatment of canonical classical \nquantities, but from a clear algorithm based in the complete symmetry of the system, \nwhich naturally selects the basic quantum operators. \nThis method, named Group Approach to Quantization (GAQ) \\cite{23}, has proved to be \nsuccessful for many physical systems (see, for instance \\cite{oscilrelativ} and \\cite{revisited}).\n\n\nAfter introducing the general setting of the problem we shall be explicitly concerned with the example of $SU(2)$.\n\n\n\\section{Sigma Model-type systems}\n\\label{4}\n\nGenerally speaking, the non-linear sigma model (NLSM) consists of a set of coupled \nscalar fields \n$\\epsilon^i(x^\\mu), i=1,\\dots,d,$ in a $D$-dimensional spacetime \n$M, \\mu=0,1,2,\\dots,D-1$, with Lagrangian (density)\n\\begin{equation}\n\\mathcal L_\\Sigma=\\frac{1}{2}\\kappa \ng_{ij}(\\epsilon)\\partial^\\mu\\epsilon^i\\partial_\\mu\\epsilon^j, \n\\label{nlsmaction}\\end{equation} \nwhere $\\partial^\\mu=\\eta^{\\mu\\nu}\\partial_\\nu, \n\\partial_\\nu=\\partial\/\\partial x^\\nu$, $\\eta$ is the spacetime  \nmetric and $\\kappa$ a constant. The field theory \n(\\ref{nlsmaction}) is called the NLSM with metric $g_{ij}(\\epsilon)$ \n(usually a positive-definite field-dependent matrix). The fields \n$\\epsilon^i$ themselves can also be considered as the coordinates of an \ninternal (pseudo-)Riemannian manifold $\\Sigma$ with metric $g_{ij}$. \n\nIn this letter we shall restrict ourselves to the quantum mechanical case $D=1$, so that \nthe fields $\\epsilon^i(x^\\mu)$ are just curves $\\epsilon^i(t)$ on $\\Sigma$, which we shall take as a \n(semisimple, linear) Lie group manifold $G$, or a given coset $G\/G_\\lambda$ (see later). Let us denote by $g=e^{\\epsilon^iT_i}$ \nan element of $G$, where $T_i, i=1,\\dots,{\\rm dim}(G)$, stands for the Lie algebra generators, with commutation relations $[T_i,T_j]=C^{\\phantom{ij}k}_{ij} T_k$.  For the sake of simplicity, we shall choose $T_i$ in the adjoint representation, whose matrix elements are $(T_i)^k_j=C^{\\phantom{ij}k}_{ij}$. Then the Killing form, used to raise and lower indices, is given by $K_{ij}\\equiv C^{\\phantom{il}k}_{il} C^{\\phantom{jk}l}_{jk}={\\rm Tr}_G(T_iT_j)$. With this notation, the NLSM Lagrangian acquires the simple algebraic form:\n\\begin{equation}\n \\mathcal{L}_{G}=\\frac{1}{2} \\kappa{\\rm Tr}_{G}(\\theta{\\theta})\\,,\n\\label{trazatoa}\n\\end{equation}\nwhere $\\theta\\equiv g^{-1}\\dot g$ and here $\\kappa$ is intended to have the dimensions of an inertia moment.\n\nIt is well-known the difficulty found by canonical quantization in dealing \nwith non-linear systems. Even symmetry-based quantization techniques face the \nimpossibility of parameterizing the solution manifold by a finite dimensional Lie group. \nThis is essentially because $g^{-1} \\dot g$ is not a total derivative, except for Abelian\ngroups. However, this obstruction disappears when the manifold $\\Sigma$ is considered to be \na coset $G\/G_\\lambda$ of $G$, $G_\\lambda$ being the isotropy subgroup of a given Lie algebra element $\\lambda$ under the adjoint action $\\lambda \\to g \\lambda g^{-1}$. To be precise $\\lambda$ should have been defined as an element of the dual of the Lie algebra, which is equivalent to the Lie algebra since $G$ is semisimple. In this sense $\\lambda$, which can be seen as an ordinary vector, may be endowed with length dimensions so that $\\kappa$ would appear as a mass $m$ rather than an inertia moment.\n\nIn this case, the (total-trace) NLSM Lagrangian (\\ref{trazatoa}) takes \nthe (partial-trace) form:      \n\\begin{equation}\n \\mathcal{L}_{G\/G_\\lambda}=\\frac{1}{2} m {\\rm Tr}_{G\/G_\\lambda}(\\theta \\theta)\n \\equiv\\frac{1}{2} m {\\rm Tr}_{G}(\\theta_\\lambda \\theta_\\lambda), \\label{trazapar}\n\\end{equation} \n where we have defined \n\\begin{equation}\n\\theta_\\lambda\\equiv[\\theta,\\lambda]\\,.\n\\end{equation}\n\nIt can be realized that, defining\n\\[\nS \\equiv g\\lambda g^{-1},\\,\\,g\\in G\\,,\n\\]\nwe have an alternative way of writing (\\ref{trazapar}) as \n\\begin{equation}\n\\mathcal{L}_{G\/G_\\lambda}=\\frac{1}{2} m {\\rm Tr}_{G}(\\dot S \\dot S)=\\frac{1}{2} m K_{ij}\\dot S^i\\dot S^j, \\label{nesfera}.\n\\end{equation} \nwhere $S^i \\equiv {\\rm Tr}_{G}(T^i S)$. \nNote that this Lagrangian is singular due to the existence of constraints like, for example, ${\\rm Tr}_{G}(S^2)=\n{\\rm Tr}_{G}(\\lambda^2)\\equiv r^2$. We shall not deal with constraints \nat this stage. They will be naturally addressed inside our quantization procedure below.\n \n \n\n\nWe can try to find two sets of generators mimicking the basic symmetry of the Galilean\nparticle (i.e., translations and boosts). For the first set, we choose the generators of the group itself: \n\\begin{equation}\n   X_i = C^{\\phantom{ij}k}_{ij} S^j \\frac{\\partial}{\\partial S^k} + C^{\\phantom{ij}k}_{ij} \\dot S^j\n    \\frac{\\partial}{\\partial \\dot S^k}\\,.\n   \\label{xsinpunto}\n\\end{equation}\nThe Lagrangian (\\ref{nesfera}) is \\textit{strictly} invariant under the action of these generators, i.e.:\n\n\\[\n   L_{X_i}\\mathcal{L}_{G\/G_\\lambda} = 0\\,,  \\qquad  \\forall i=1,\\dots,{\\rm dim}(G)\\ ,\n\\]\nwhere  $L_{X}$ stands for the Lie derivative with respect to a generator $X$. In this computation, the fact that the product $K_{lm} C^{\\phantom{ij}l}_{ij}\\equiv C_{ijm}$ is fully antisymmetric has been used.\n\nAs far as the second set of symmetries is concerned (those playing the role of boosts), we propose the following one:\n\n\n\\begin{equation}\n  X'_i = \\frac{\\partial}{\\partial \\dot S^i}\\,.\n  \\label{xpunto}\n\\end{equation}\n\nThese generators leave the Lagrangian \\textit{semi}-invariant in the sense that they give a total derivative (thus leaving the action strictly invariant), much in the same way the generators of boosts do in the free Galilean particle, that is,  \n\\begin{equation}\n   L_{X'_i}\\mathcal L_{G\/G_\\lambda} = m \\dot{S}_i , \\qquad  \\forall i=1,\\dots,{\\rm dim}(G) \\ .\n   \\label{semiinv}\n\\end{equation}\n\nThe generators (\\ref{xsinpunto}) and (\\ref{xpunto}) close a finite-dimensional Lie algebra with commutation relations\n\\begin{equation}\n\\left[X_i, X_j\\right]= - C_{ij}^{\\phantom{ij}k} X_{k},\\;\\; \n\\left[X_i, X'_j\\right] = - C_{ij}^{\\phantom{ij}k}  X'_k,\\;\\; \n\\left[X'_i, X'_j\\right] = 0.\n\\label{g1}\n\\end{equation}\nThe corresponding symmetry group is the (co-)tangent group of $G$ and will be denoted by $G^{(1)}$ (see \\cite{jetgaugegroup} as regards gauge theory).\n\n\nWe shall assume this algebra (in fact, a central extension of it) as the basic symmetry of the quantum  particle constrained to move on the manifold $G\/G_\\lambda$.\nIn fact, the semi-invariance (\\ref{semiinv}) of the Lagrangian suggests the presence of a central extension $\\tilde G^{(1)}$ of the group $G^{(1)}$, as happens in the quantum Galilean particle \\cite{centralext}. At the Lie algebra level, this central extension only affects the second commutator in (\\ref{g1}), which now reads\n\n\\begin{equation}\n\\left[\\tilde X_i, \\tilde X'_j\\right] = - C_{ij}^{\\phantom{ij}k} \\tilde X'_k \n- C_{ij}^{\\phantom{ij}k} \\lambda_k \\frac{m}{\\hbar} \\Xi\n\\label{gtilde1}\n\\end{equation}\nwhere $\\Xi$ denotes the central generator.\n\nThis centrally extended group $\\tilde G^{(1)}$ is the group of \\textit{strict} invariance of the system, and contains the necessary information to obtain the quantum theory. Although this central extension is trivial from a strict mathematical point of view, in the sense that a redefinition of a generator eliminates the central generator from the r.h.s. of the Lie algebra commutators, physically it behaves as a non-trivial one, since under an In\\\"on\\\"u-Wigner contraction (limit process) leads to a non-trivial extension of the contracted group \\cite{saletan} (see also \\cite{refinement} and references therein).\n\nIt should be remarked that the commutation relations providing the central term on the right hand side generalize those of the Heisemberg-Weyl algebra, where $\\tilde X_i$ play the role of (non-Abelian) ``translation'' generators, and $\\tilde X'_j$ the role of ``boost'' generators, although restricted to the coset space  $G\/G_\\lambda$. This point will be further clarified in the example of the $SU(2)$ group.\n\n\n\\section{Quantum theory}\n\\label{3}\n\nThe leading idea is to obtain an irreducible and unitary representation (unirep) of the basic group suggested in the previous section, i.e. the (centrally extended) $\\tilde G^{(1)}$ . The central extension will select a specific representation, associated with a given coadjoint orbit.\n\nThis can be performed in any desired way. Here we shall choose a specific algorithm, GAQ, developed by some of the authors.\n\n\\subsection{Group Approach to Quantization}\n\nThe basic idea of GAQ consists in taking advantage of having {\\it two} mutually {\\it commuting} \ncopies of the Lie algebra $\\tilde{\\cal G}$ of a group $\\tilde{G}$ ($G$ centrally extendended by $U(1)$) of {\\it strict symmetry} of a given physical system, that is,\n\n\\[{\\cal X}^L(\\tilde{G})\\approx\\tilde{\\cal G}\\approx{\\cal \nX}^R(\\tilde{G})\\]\n\n\\noindent  in such a way that one copy, let us say \n${\\cal X}^R(\\tilde{G})$, plays the role of {\\it pre-Quantum \nOperators} acting (by usual derivation) on complex (wave) functions on \n$\\tilde{G}$, whereas the other,\n${\\cal X}^L(\\tilde{G})$, is used to {\\it reduce} the \nrepresentation in a manner {\\it compatible} with the action of the operators, thus providing the {\\it true \nquantization}.\n\nIn fact, from the group law $g''=g'*g$ of any group $\\tilde{G}$, we can read \ntwo different actions:\n\\begin{eqnarray}\ng''&=&g'*g\\equiv L_{g'}g\\nonumber\\\\\n&&\\nonumber\\\\\ng''&=&g'*g\\equiv R_{g}g'\\nonumber\n\\end{eqnarray}\n\n\\noindent The two actions commute and so do the generators $\\tilde{X}^R_a$  and \n$\\tilde{X}^L_b$ of the {\\it left} and {\\it right} actions respectively, \ni.e.\n\\[[\\tilde{X}^L_a,\\;\\tilde{X}^R_b]=0\\;\\;\\forall a,b\\;.\\]\n\n\\noindent The generators $\\tilde{X}^R_a$ are right-invariant vector fields \nclosing a Lie algebra, ${\\cal X}^R(\\tilde{G})$, isomorphic to the tangent space\nto $\\tilde{G}$ at the identity, $\\tilde{\\cal G}$. The same, changing $L\\leftrightarrow R$, applies\nto $\\tilde{X}^L\\in{\\cal X}^L(\\tilde{G})$. \n\n\nWe consider the space of complex functions $\\Psi$ on the whole group \n$\\tilde{G}$ and restrict them to only $U(1)$-functions, that is, those \nwhich are homogeneous of degree one on the argument $\\zeta\\equiv e^{i\\phi}\\in \nU(1)$; that is,\n\\[ \\tilde{X}^L_\\phi\\Psi= i \\Psi\\,, \\]\n\n\\noindent where $\\tilde{X}^L_\\phi$ is the (central) generator of the $U(1)$ subgroup. On these \nfunctions the right-invariant vector fields act as {\\it pre-Quantum Operators} by ordinary derivation. \nHowever, this action is not \\textit{irreducible} since there is a set of non-trivial operators \ncommuting with this representation. \nIn fact, all the left-invariant vector fields do commute with the right-invariant ones, i.e. the (pre-quantum) operators. According to Schur's Lemma\nthose operators must be ``trivialized'' in order to turn the right-invariant vector fields into true \\textit{quantum operators}.\n\nWe have seen that the action of the central generator (which is, in particular, left-invariant) is fixed to be non-zero by the $U(1)$-function condition. Thus, not every left-invariant vector field can be nullified in a compatible way with this condition. That is, if \n\n\\[\n   \\left[ \\tilde X_{a}^L,\\tilde X_{b}^L\\right]=\\tilde{X}^L_\\phi,\n\\]\nthen $X_{a}^L \\Psi=0, \\; X_{b}^L \\Psi=0$ is not compatible with $\\tilde{X}^L_\\phi\\Psi= i \\Psi$. Of course, \nthis null condition on $\\Psi$ can be imposed by those generators that never produce a central term by\nconmutation; they constitute the \\textit{Characteristic sub-algebra} $\\mathcal G_{\\Theta}$. But also half\nof the rest of left-generators can be joined to $\\mathcal G_{\\Theta}$ to constitute a \\textit{Polarization} subalgebra ${\\cal P}$.\n\nThen, the role of a Polarization is\nthat of {\\it reducing} the representation, which now constitutes a true {\\it Quantization}. Therefore we impose that \\textit{wave functions} satisfy the Polarization condition:  \n\\[ \\tilde{X}^L_b\\Psi=0\\,, \\ \\ \\forall \\tilde{X}^L_b\\in {\\cal P}\\, .\\]\nWe refer the reader to Ref. \\cite{23} for further details.\n\n\n\\subsection{The case of a particle on the sphere}\n\n\nThe procedure presented above is quite simple when applied to the case of a particle moving on the coset space $\\mathbb S^2\\equiv SU(2)\/U(1)$. We begin with a specific realization of the basic group $\\tilde G^{(1)}$, which in this case turns out to be a centrally extended Euclidean Group. In it, the fields $X_{\\epsilon^i}\\,, i=1,2,3$ generate ordinary rotations, parameterized by vectors $\\vec\\epsilon$ whose direction determines the axis of rotation and its modulus the angle of rotation by $|\\vec\\epsilon|=2\\sin \\frac{\\chi}{2} $. In the same way, the fields $X_{\\theta^i}\\,, i=1,2,3$ correspond to the (co-)tangent subgroup, parameterized by $\\theta^i$. In terms of these variables the group law is:\n \n \\begin{eqnarray*}\n    R(\\vec\\epsilon\\,'') &=& R(\\vec\\epsilon\\,') R(\\vec\\epsilon\\,) \\\\\n    \\vec \\theta\\,'' &=& R^{-1}(\\vec\\epsilon\\,) \\vec \\theta\\,' +  \\vec \\theta\\\\\n    \\zeta'' &=& \\zeta' \\zeta \\exp \\left\\lbrace \n    i\\frac{m r}{\\hbar} \\vec\\lambda \\cdot \\left( \\vec \\theta\\,''-\\vec \\theta\\,'-\\vec \\theta\\right)\\right\\rbrace =\n    \\zeta' \\zeta \\exp \\left\\lbrace i\\frac{m r}{\\hbar} \\vec\\lambda \\cdot \\left(R^{-1}(\\vec\\epsilon) \\vec \\theta\\,'\n    - \\vec \\theta\\,'\\right)\\right\\rbrace \\\\\n    \\phantom{\\zeta''} & \\phantom{=} & ( R(\\vec\\epsilon\\,)^i_j \\equiv\n                                      (1-\\frac{{\\vec\\epsilon}^{\\;2}}{2})\\delta^i_j-\n                                      \\sqrt{1-\\frac{{\\vec\\epsilon}^{\\;2}}{4}} \\eta^i_{\\cdot jk}\\epsilon^k +\n                                      \\frac{1}{2} \\epsilon^i \\epsilon_j \\;\n                                      ),\n \\end{eqnarray*}\n\n\n\\noindent where $\\vec\\lambda$ is an arbitrary, constant vector in the (co-)algebra with modulus $r$, and $\\zeta\\equiv e^{i\\phi}$ is the quantum mechanical phase. The $\\hbar$ constant has been introduced to keep the exponent dimensionless.\n\nWe can immediately calculate the corresponding infinitesimal generators of the left action, which are the  right-invariant vector fields:\n\n\\begin{eqnarray*}\n    \\tilde X^{R}_{\\vec\\epsilon} &=& X^{R\\;(SU(2))}_{\\vec\\epsilon} \\\\\n    \\tilde X^{R}_{\\vec \\theta}       &=& R^{-1} \\frac{\\partial}{\\partial \\vec \\theta} \n                                    + \\frac{m r}{\\hbar}(R \\vec\\lambda - \\vec\\lambda)^t\n                                     \\Xi\\\\\n    \\tilde X^{R}_{\\phi}        &=& \\hbox{Re}\\left(i\\zeta \n                                    \\frac{\\partial}{\\partial\\zeta}\\right)\n                                    \\equiv \\Xi.\n\\end{eqnarray*}\n\n\n\\noindent Needless to say that this set of generators closes (a particular case of) the Lie algebra that we found in the previous section:\n\n\\begin{eqnarray*}\n\\left[\\tilde X^{R}_{\\epsilon^{i}}, \\tilde X^{R}_{\\epsilon^{j}}\\right] \n&=& \n- \\eta_{ij\\cdot}^{\\phantom{ij}k} \\tilde X^{R}_{\\epsilon^{k}}\n\\\\\n\\left[\\tilde X^{R}_{\\epsilon^{i}}, \\tilde X^{R}_{\\theta^{j}}\\right] \n&=&\n- \\eta_{ij\\cdot}^{\\phantom{ij}k} \\left( \\tilde X^{R}_{\\theta^{k}} +\\frac{m r}{\\hbar} \\lambda_{k} \\Xi \\right) \n\\\\\n\\left[\\tilde X^{R}_{\\theta^{i}}, \\tilde X^{R}_{\\theta^{j}}\\right]\n&=& \n0\\;.\n\\end{eqnarray*}\n\n\nWe can compute as well the infinitesimal generators of the right action, which are the  left-invariant vector fields:\n\n\\begin{eqnarray*}\n    \\tilde X^{L}_{\\vec\\epsilon} &=& X^{L\\;(SU(2))}_{\\vec\\epsilon} - \n                                  \\vec \\theta \\wedge \\frac{\\partial}{\\partial \\vec \\theta}\n                                  -\\frac{m r}{\\hbar}\\vec \\theta \\wedge \\vec \\lambda \\;\\Xi \\\\\n    \\tilde X^{L}_{\\vec \\theta}       &=& \\frac{\\partial}{\\partial \\vec \\theta} \\\\\n    \\tilde X^{L}_{\\phi}        &=& \\hbox{Re}\\left(i\\zeta \n                                   \\frac{\\partial}{\\partial \\zeta}\\right)\\equiv \\Xi,\n\\end{eqnarray*}\n\n\\noindent closing the same Lie algebra but with opposite structure constants.\n\nThe characteristic module, i.e., the sub-algebra generated by those vector fields which do not produce a central term under conmutation, and therefore without dynamical content, is generated by two fields:\n\n\\[\n   \\mathcal G_{\\Theta}= \\langle \\vec\\lambda \\cdot \\tilde X^{L}_{\\vec\\epsilon},\\, \n   \\vec\\lambda \\cdot \\tilde X^{L}_{\\vec \\theta}\\rangle,\n\\]\n\nA polarization sub-algebra is given by the characteristic module together with half of the conjugated pairs: \n\n\\[\n   \\mathcal P = \\langle \\vec\\lambda \\cdot \\tilde X^{L}_{\\vec\\epsilon},\\, \\tilde X^{L}_{\\vec \\theta}\\rangle.\n\\]\n\nThis must not be interpreted as constraint conditions, since they preserve the action of the right-invariant vector fields, i.e., those which will be the physical operators. An alternative treatment, making explicit use of constraints can be seen, for instance, in \\cite{coherent}.\n\nAn irreducible representation of the group is given by the action of the right-invariant vector fields of the group on the complex functions valued over the group manifold, provided that these functions are polarized and satisfy the condition of U(1)-function:\n \n \\[\n    \\mathcal P \\Psi = 0, \\qquad \\Xi \\Psi = i \\Psi.\n \\]\n\nIt can be easily checked that such functions (now true \\textit{wave functions}) are of the form\n\n\\[\n   \\Psi \\left(R(\\vec\\epsilon\\,)\\vec\\lambda,\\vec \\theta, \\zeta\\right)=\\zeta \n    \\Phi \\left(R(\\vec\\epsilon\\,)\\vec\\lambda\\right),\n\\]\n\\noindent where $\\Phi$ is an arbitrary function, or equivalently,  in terms of the variable $\\vec S \\equiv R(\\vec\\epsilon\\,)\\vec\\lambda$ (the positions on the surface of a sphere of radius $r$):\n\n\\[\n   \\Psi (\\vec S,\\vec \\theta, \\zeta)=\\zeta \\Phi (\\vec S).\n\\]\n\nThe explicit action of the right-invariant vector fields on these wave functions is computed to give:\n\n\n\\begin{eqnarray*} \n   \\tilde X^{R}_{\\vec\\epsilon} \\Psi &=& \\zeta \n           \\left(\\vec S \\wedge \\vec \\nabla_{S} \\Phi \\right)= \\vec S \\wedge \\vec \\nabla_{S} \\Psi  \\\\\n   \\tilde X^{R}_{\\vec \\theta} \\Psi       &=& i\\frac{m r}{\\hbar} \\left( \\vec S - \\vec \\lambda \\right) \\Psi.\n\\end{eqnarray*}\n\nIt is possible to redefine the vector fields to obtain the actual quantum operators, acting only on the arbitrary part of the wave functions.\nThus, we end up with the explicit representation over the wave-functions $\\Phi$ depending only in the variable $\\vec S = R\\vec\\lambda$,\n\n\\begin{eqnarray*}\n   \\hat{\\vec L} \\Phi (\\vec S) &\\equiv& i \\hbar \\vec S \\wedge \\vec \\nabla \\Phi(\\vec S)\\\\\n   \\hat{\\vec S} \\Phi (\\vec S) &\\equiv& \\vec S \\Phi(\\vec S).\n\\end{eqnarray*}\n\n\\noindent It becomes evident at this point that the domain of the wave functions have been naturally selected without imposing any constraint condition as such.\n\nAs was previously emphasized, the $\\hat{\\vec L}$ operators now play the role of ``generators of translations'' on the surface of the sphere, $\\hat{\\vec S}$ playing that of a ``position operator'' on the sphere surface\\footnote{Note that, in this representation of the Euclidean Group, the relation $\\hat{\\vec S}\\cdot \\hat{\\vec L}=0$ is fulfilled. However, the Euclidean Group admits another family of central extensions, where $\\hat{\\vec S}\\cdot \\hat{\\vec L}\\neq 0$, and the corresponding representations can realize a magnetic monopole in the center of the sphere. They will be studied elsewhere.}. \n\nFinally, to obtain a Hamiltonian, we proceed as in the free particle: it is defined as the generator of ``translations'' squared, so that\n\n\\[\n   \\hat H = \\frac{\\left.\\hat{\\vec L}\\right.^{2}}{2 m} = \\frac{\\hbar^{2}}{2 m} \\nabla^{2}.\n\\]\n\nThere is no ambiguity in this expression, since it corresponds to the squared action of $\\hat{\\vec L}$ as a ``basic operator''. $\\hat{H}$ thus  provides the energy spectrum\n\n\\[\n   E_l = \\frac{\\hbar^{2}}{2 m} l (l+1)\n\\]\n\n\\noindent with no extra terms. Note that $\\hat H$ coincides with the Casimir\nof $SU(2)$ restricted to $\\mathbb{S}^2$. In the general case, the energy operator \nwill be the quadratic Casimir of $G$ restricted to $G\/G_\\lambda$, and this turns out\nto be the Laplace-Beltrami operator on  $G\/G_\\lambda$.\n\n\n\n\n\n\\section{Conclusions}\n\n\nAs we have shown with the particular example of the free particle in the sphere $\\mathbb S^2$, the quantization of a non-linear system can be achieved if the complete (quantum) symmetry of the system is identified. As a remarkable advantage over other quantization methods, the followed one, a Group Approach to Quantization, is coordinate-free, in such a way that the quantum operators are automatically selected. In particular, this implies that there is no need to impose any constraint. This is specially well suited for non-linear systems, where normal-ordering problems arise in canonical quantization.\n\nWe would like to point out that a similar strategy can be explicitly adopted in dealing with particles moving on coadjoint orbits of semi-simple groups, non-necessarily spheres, as shown in our general scheme.\n\n\n\n\\subsection*{Acknowledgments}\n\nWork partially supported by the Spanish MCYT, Junta de Andaluc\\'\\i a and Fundaci\\'on\nS\\'eneca under projects FIS2005-05736-C03-01, P06-FQM-01951 and 03100\/PI\/05. F. F. \nL\\'opez Ruiz would like to thank the C.S.I.C. for an I3P grant.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries }\n \\setcounter{equation}{0}\n Let us recall the theorem of H.Bohr \\cite{bohr3:gnus} in 1914.\n\\begin{theorem} If a power series\n\\begin{eqnarray}\nf(z_1)=\\sum\\limits_{k=0}^{\\infty}c_kz_1^k\n\\end{eqnarray}\n converges in the\nunit disk $U_1$ and its sum has modulus\n less than $1$, then\n \\begin{eqnarray}\n \\sum\\limits_{k=0}^{\\infty }\\vert c_k z_1^k\\vert <1,\n \\end{eqnarray}\n if $ \\vert z_1\\vert <{1\\over 3}$. Moreover,\n the constant ${1\\over 3}$ cannot be improved.\n \\end{theorem}\n For reasons of convenience we write the inequality $(1.2)$ in the\n following equivalent form\n \\begin{eqnarray*}\n\\sum\\limits_{k=1}^{\\infty }\\vert c_k z_1^k\\vert <1-\\vert c_0\\vert .\n\\end{eqnarray*}\nGeneralizations and modifications  of this result can be found in\n\\cite{aiz2:gnus}, \\cite{aiz4:gnus}, \\cite{aizad1:gnus} --\n\\cite{aizg:gnus}, \\cite{aizvi:gnus}, \\cite{ben:gnus},\n{\\cite{dra:gnus}, \\cite{kres:gnus},  \\cite{pau:gnus},\n\\cite{sid:gnus}, \\cite{tom:gnus} . On the other hand, formulation\nof Bohr theorem in several complex variables appeared very\nrecently. Given a complete Reinhardt domain ${\\mathcal{D}}\\subset\n\\mathbb{C}^n$, we denote by $R( {\\mathcal D})$ the largest\nnon-negative number $r$ with the property that if the power series\n \\begin{eqnarray}\n f(z)=\\sum\\limits_{\\vert \\alpha \\vert \\geq 0}c_{\\alpha}z^{\\alpha},\n \\; z\\in {\\mathcal D},\n \\end{eqnarray}\n where as usual, $\\alpha=(\\alpha_1,\\dots, \\alpha_n)\\in \\mathbb{N}^n_0$,\n $z^{\\alpha}=z_1^{\\alpha_1}\\dots z_n^{\\alpha_n}$, $\\vert \\alpha\n \\vert =\\alpha_1+\\dots \\alpha_n $, converges in ${\\mathcal D}$ and the\n modulus of its sum is less than $1$, then\n \\begin{eqnarray*}\n \\sum\\limits_{\\vert \\alpha \\vert \\geq 1}\\vert c_{\\alpha}z^{\\alpha}\n \\vert < 1-\\vert c_0\\vert\n \\end{eqnarray*}\n in the homothetic domain $\\mathcal{D}_r=r\\mathcal{D}$. Here\n $c_0=c_{0,\\dots, 0}$. A variety of results, related to this\n particular $R(\\mathcal{D})$ or other multidimensional generalizations of Bohr radius are found in \\cite{aiz1:gnus} - \\cite{aizvi:gnus}, \\cite{ben:gnus} - \\cite{ boha:gnus},\n  \\cite{deff:gnus} -- \\cite{dra:gnus}.\n Key results, used by us in the present paper, were obtained in\n \\cite{aiz4:gnus}, \\cite{aize:gnus}, \\cite{bal:gnus}.\\\\\n Let $\\widetilde G $ be a convex hull of the domain $G\\subset \\mathbb{C}$. For a a complete Reinhardt domain  $\\mathcal{D}$ in $\\mathbb{C}^n$, denote by\n $R(\\mathcal{D},G)$\n the largest $r\\geq 0 $ such that if the function $f(z)$ from $(1.3)$ is\n holomorphic in $\\mathcal{D}$, $f(\\mathcal{D})\\subset G$ and  $\\widetilde G \\not=\\mathbb{C}$,\n then\n \\begin{eqnarray*}\n \\sum\\limits_{\\vert \\alpha \\vert \\geq 1}\\vert\n c_{\\alpha}z^{\\alpha}\\vert <dist (c_0,\\partial\\widetilde G)\n \\end{eqnarray*}\nin homothety $\\mathcal{D}_r$. A point $p\\in \\partial G $ is called\n{\\it{a point of convexity}} if $p\\in \\partial\\widetilde G$. A\npoint of convexity $p$ is called {\\it {regular}} if there exists a\ndisk $U\\subset G $\nso that $p\\in\\partial U$. \\\\\nThe following result is a consequence of a more general result,\nproved \\cite{aiz4:gnus}. We state it in a suitable for us form for\nthe purposes of the present article.\n\\begin{theorem}\nIf $\\widetilde G\\not=\\mathbb{C}$, then $R(\\mathcal{D},G)$ is not\nsmaller than $R(\\mathcal{D},U_1)$. If $\\partial G $ contains at\nleast one point of regular convexity , then\n\\begin{eqnarray*}\nR(\\mathcal{D},G)=R(\\mathcal{D},U_1).\n\\end{eqnarray*}\n\\end{theorem}\n\\begin{corollary}\nIf the domain $G$ is convex and $G\\not= \\mathbb{C}$, then\n$R(\\mathcal{D}, G)$ is independent of the choice of $G$.\n\\end{corollary}\nBesides the Bohr radius we will use the radius of Rogosinski, whose\nclassic result of 1923, \\cite{ rog:gnus, szego:gnus,laga:gnus}, is\ndescribed in the following statement:\n\\begin{theorem}\nIf the function $f(z_1)$ from $(1.1)$ is holomorphic in the unit\ndisk $U_1$ and $\\vert f(z_1)\\vert <1 $ in $U_1$, then all of its\npartial sums are less than $1$ in the disk of radius ${1\\over 2}$,\nthat is:\n\\begin{eqnarray*}\n\\vert \\sum\\limits_{k=0}^{m}c_kz_1^k\\vert <1\n\\end{eqnarray*}\nfor $\\vert z_1\\vert <{1\\over 2}$ and this radius is sharp.\n\\end{theorem}\nThe following result is the consequence of more general fact\nproved in \\cite{aize:gnus}. Again, we are going to use its\nparticular, convenient for us, formulation: let $\\mathcal{A}$ be a\nlattice in $\\mathbb{N}_0^n$, which is represented by\n\\begin{eqnarray*}\n\\mathcal{A}=\\{\\alpha \\in \\mathbb{N}_0^n:\\;\nm_1\\alpha_1+\\dots+m_n\\alpha_n\\leq m\\},\n\\end{eqnarray*}\nwhere all the numbers $m_1,\\dots, m_n, m$ belong to $\\mathbb{N}_0$\nand have no common divisor.\n\\begin{theorem}\nLet the function $f(z)$ in $(1.3)$ be holomorphic in the Reinhardt\ndomain $\\mathcal{D}$ and $f(\\mathcal{D})\\subset G$, where $G$ is a\nconvex domain in $\\mathbb{C}$ so that $G\\not=\\mathbb{C}$. Then the\npolynomial (partial sum)\n\\begin{eqnarray*}\n\\sum\\limits_{\\alpha \\in \\mathcal{ A}}c_{\\alpha}z^{\\alpha}\n\\end{eqnarray*}\nmaps $\\mathcal{D}_{\\mathcal{A}}$ into $G$, where\n\\begin{eqnarray*}\n\\mathcal{D}_{\\mathcal{A}}=\\{z\\in\\mathbb{C}^n:\\;({z_1\\over\nr_m^{m_1}},\\dots,{z_n\\over r_m^{m_n}})\\in \\mathcal{D}\\},\n\\end{eqnarray*}\ndoes not depend on $G$, $r_1={1\\over 2}$, $r_2={\\sqrt 3\\over 8}$\nand for $l\\geq 3$ the number  $r_l$ is the unique positive\nsolution of the equation\n\\begin{eqnarray*}\n1-r-2r^{l+1}=0.\n\\end{eqnarray*}\n\\end{theorem}\nIn the present article we investigate the Dirichlet series\n\\begin{eqnarray}\nf(s)=\\sum\\limits_{n=1}^{\\infty}{a_n\\over n^s},\n\\end{eqnarray}\nconverging in the right half-plane $\\Pi=\\{s\\in \\mathbb{C}:\\;\\Re s\n=\\sigma >0\\}$ and $f(s)\\in H^{\\infty}(\\Pi )$, that is\n$\\|f\\|=\\sup\\limits_{s\\in \\Pi}\\vert f(s)\\vert <\\infty $. Following\n\\cite{bal:gnus}, we call {\\it{isometric Bohr abscissa}} ${\\bf{b}}$\nthe non-negative real number defined as the infimum of those\n$\\sigma \\geq 0 $ such that for all $f\\in H^{\\infty}(\\Pi) $ which\ncan be expressed as a Dirichlet series $(1.4)$, the following\nholds\n\\begin{eqnarray}\n\\sum\\limits_{n=1}^{\\infty}{\\vert a_n\\vert \\over n^{\\sigma}}<\\|f\\|.\n\\end{eqnarray}\n For reasons of convenience, we write $(1.5)$ as\n \\begin{eqnarray*}\n\\sum\\limits_{n=2}^{\\infty} {\\vert a_n\\vert \\over n^{\\sigma}}\\leq\n\\|f\\|-a_1.\n\\end{eqnarray*}\nThe next result was obtained in \\cite{bal:gnus}.\n\\begin{theorem}\nFor the isometric Bohr abscissa the following estimates are valid\n\\begin{eqnarray*}\n1.5850\\dots={\\log 3\\over \\log 2}\\leq {\\bf{b}}<1.8154\n\\end{eqnarray*}\nIf $f\\in H^{\\infty}(\\Pi )$ and $\\|f\\| =1 $, then\n\\begin{eqnarray*}\n\\vert a_1\\vert ^2+\\sum\\limits_{n=2}^{\\infty} {\\vert a_n\\vert \\over\nn^{\\sigma}}\\leq 1,\n\\end{eqnarray*}\nwhere $\\sigma <1.7287 $.\n\\end{theorem}\n Analogously, to the Bohr radius\n$R(\\mathcal{D},G)$, {\\it{the isometric Bohr abscissa}}\n${\\bf{b}}(G)$, where $G$ is a domain in $\\mathbb{C}$, $\\widetilde\nG\\not=\\mathbb{C}$, is defined as the infimum of those $\\sigma \\geq\n0 $ such that if $f\\in H^{\\infty}(\\Pi)$, $f(s)$ is like in $(1.4)$\nand $f(\\Pi)\\subset G$, then\n\\begin{eqnarray}\n\\sum\\limits_{n=2}^{\\infty} {\\vert a_n\\vert \\over\nn^{\\sigma}}<dist(a_1,\\partial \\widetilde G).\n\\end{eqnarray}\nFurthermore, we define the {\\it{ Rogosinski abscissa\n${\\bf{r}}(G)$}} for functions $f\\in H^{\\infty}(\\Pi)$,\n$f(\\Pi)\\subset G$, as the infimum of $\\sigma ^{\\prime}\\geq 0 $,\nsuch that for all partial sums of the series $(1.4)$\n\\begin{eqnarray}\n\\sum\\limits_{n=1}^{k}{a_n\\over n^s}=P_k(s)\n\\end{eqnarray}\nthe inclusion $P_k(\\Pi _{\\sigma ^{\\prime}})\\subset G$ is valid,\nwhere $\\Pi_{\\sigma^{\\prime}}=\\{s\\in \\mathbb{C}: \\;\\sigma\n>\\sigma^{\\prime}\\}$.\n\\section{The main results}\n\\setcounter{equation}{0}\n\\begin{theorem}\nLet $G$ be a be a bounded domain in $\\mathbb{C}$. Then the\nisometric Bohr abscissa ${\\bf{b}}(G)$ is not larger then the\nisometric Bohr abscissa ${\\bf{b}}(U_1)$. If $\\partial G$ contains\nat least one regular point of convexity, then\n${\\bf{b}}(G)={\\bf{b}}(U_1)$.\n\\end{theorem}\n{\\bf{Proof:}} We are going to exploit the connection between\nclassical Dirichlet series and the power series in the infinite\ndimensional polydisc found by H.Bohr in \\cite{bohr2:gnus}.\nConsider the series $(1.4)$. Every natural number $n$ is product\nof its prime factors\n\\begin{eqnarray*}\nn=p_1^{\\alpha_1}\\dots p_m^{\\alpha_m}.\n\\end{eqnarray*}\nWe set $z=(p_1^{-s}, p_2^{-s},\\dots )$. Then\n\\begin{eqnarray}\nf(s)=\\sum\\limits_{n=1}^{\\infty}a_n(p_1^{-s})^{\\alpha_1}\\dots\n(p_m^{-s})^{\\alpha_m}=\\sum\\limits_{n=1}^{\\infty}a_nz_1^{\\alpha\n_1}\\dots z_m^{\\alpha _m},\n\\end{eqnarray}\nwhere $m=m(n)$. If the power series $(1.4)$ converges absolutely\nfor $\\sigma =\\sigma_0$, then the set of values of $f(s)$ on the\nvertical line $\\{s\\in \\mathbb{C}:\\;\\Re s=\\sigma _0\\}$ is\neverywhere dense in the set of the values of the series $(2.1)$ on\nthe set $\\{z:\\; \\vert z_1\\vert ={1\\over\np_1^{\\sigma_0}},\\dots,\\vert z_m\\vert ={1\\over p_m^{\\sigma_0}}\\}$,\n\\cite{bohr2:gnus}. We will be using this result in the particular\ncase when the series $(1.4)$ is just a Dirichlet polynomial. The\nsame conclusion is easily deduced from a theorem of Kronecker,\n\\cite{bay:gnus}. Denote by $\\sigma _u$ the abscissa of uniform\nconvergence of the series $(1.4)$, that is, the infimum of those\n$\\sigma $, so that the series $(1.4)$ converges uniformly in the\nhalf-plane $\\Pi_{\\sigma}=\\{s\\in \\mathbb{C}:\\;\\Re s>\\sigma \\}$. Let\n$\\sigma _{b}$ be infimum of those $\\sigma $ for which the series\n$(1.4)$ is bounded (and naturally holomorphic) in the half-plane\n$\\Pi_{\\sigma}$.  A non-trivial result of Bohr states that\n$\\sigma_u=\\sigma_{b}$, \\cite{bohr1:gnus},\\cite{bay:gnus}.\nTherefore, if $f(\\Pi)\\subset G$, where $G$ is a bounded domain in\n$\\mathbb{C}$, then in every half-plane $\\Pi_{\\sigma}$, $\\sigma >0$\nthe series $(1.4)$ converges uniformly. Hence, for $\\sigma\n$-fixed, we can find for every $\\epsilon >0$ an index $k\\in\n\\mathbb{ N}$ so that $\\vert f(s)-P_k(s)\\vert <\\epsilon $, where\nthe polynomial $P_k(s)$ is taken from $(1.7)$.We remark that\n$P_k(\\Pi_{\\sigma})\\subset G_{\\epsilon}$, where $G_{\\epsilon}$ is\nan $\\epsilon $-neighborhood of $G$. Therefore the polynomial\n\\begin{eqnarray*}\n\\widetilde P_k(z)=\\sum\\limits_{n=1}^ka_nz_1^{\\alpha_1}\\dots\nz_m^{\\alpha_m}\n\\end{eqnarray*}\nin the corresponding polydisc $\\mathcal{D}$ is such that\n$\\widetilde P_k(\\mathcal{D})\\subset G_{\\epsilon }$. From Theorem\n1.2 then follows  that $R(\\mathcal{D},G_{\\epsilon})\\geq\nR(\\mathcal{D}, U_1)$ and therefore the claim\n\\begin{eqnarray}\n\\sum\\limits_{n=2}^k \\vert a_n\\vert {1\\over p_1^{\\sigma}}\\dots\n{1\\over p_m^{\\sigma}}<dist(a_1,\\partial\\widetilde G_{\\epsilon})\n\\end{eqnarray}\nfor $\\sigma \\leq {\\bf{b}}(U_1)$ is valid. Taking the limit in\n$(2.2)$ for $k\\longrightarrow \\infty$ (this means $\\epsilon\n\\longrightarrow 0$), we obtain that for such  a $\\sigma $ the\nrelation $(1.6)$ holds, but instead of the proper inequality $< $,\nthe inequality $\\leq $ is now possible. But if we had equality\nthere, then by taking smaller $\\sigma $ we would have had obtained\nthe converse inequality, which is impossible. Therefore\n${\\bf{b}}(G)\\leq {\\bf{b}}(U_1)$.\\par\n Let us assume now that on\nthe boundary $\\partial G$ there exists at least one regular point\nof convexity. Denote it by $p_0$. Then there exists a disk\n$U\\subset G$ so that $p_0\\in\\partial U\\cap\\partial G\\cap \\partial\n\\widetilde G$. Remark that homothety and  parallel translation for\nthe disk $U_1$ (that is all $a_n$ are replaced by $ra_n$ and after\nthis $ra_1$ is replaced by $ra_1+c$, and therefore the new disc\n$U$ is $rU_1+ c$) do not alter the condition $(1.6)$ for the disc.\nHence ${\\bf{b}}(U)={\\bf{b}}(U_1)$. From here, it follows that\n${\\bf{b}}(G)\\geq {\\bf{b}}(U_1)={\\bf{b}}(U)$ since the\ncorresponding set of holomorphic functions, which are represented\nby Dirichlet series in $\\Pi $ and are satisfying\n$f(\\Pi) \\subset G$, is larger than the set of functions satisfying $f(\\Pi)\\subset U$. Thus, in this case we obtain ${\\bf{b}}(G)={\\bf{b}}(U_1)$.$\\diamondsuit $.\\\\\n\\begin{corollary}\nIf $G$ is a bounded, convex domain in $\\mathbb{C}$ then the\nisometric Bohr abscissa ${\\bf{b}}(G)$ does not depend from the\nchoice of the domain $G$.\n\\end{corollary}\nWe point out that in the proof of the Theorem 2.1, we used the\nboundedness of the domain $G$ only to be able to approximate the\nDirichlet series by a Dirichlet polynomial, in order to use the\nlater's  uniform convergence in the corresponding half-plane and\nthus to be able to apply Bohr result  about the everywhere density\nof its values. Instead, one could demand that the series $(1.4)$\nconverges absolutely in the half plane $\\Pi $. Thus, one is able\nto formulate the following statement, where instead of isometric\nBohr abscissa ${\\bf{b}}(G)$ one considers ${\\bf{b}}_{a}(G)$,\ndefined in the same manner as ${\\bf{b}}(G)$, but only for\nabsolutely converging Dirichlet series in the half plane $\\Pi$.\n\\begin{theorem}\nLet $G$ be a domain in $\\mathbb{C}$, $\\widetilde G\n\\not=\\mathbb{C}$. Then the isometric Bohr abscissa\n${\\bf{b}}_{a}(G)$ is not greater ${\\bf{b}}(U_1)$. If $\\partial G$\ncontain at least one regular point of convexity, then\n${\\bf{b}}_{a}(G)={\\bf{b}}(U_1)$.\n\\end{theorem}\n\\begin{theorem}\nLet $G$ be a bounded convex domain in $\\mathbb{C}$. Then the\nfollowing Rogosinski abscissas are equal: ${\\bf {r}}(G)={\\bf\n{r}}(U_1)$.\n\\end{theorem}\n{\\bf{Proof:}} The proof  repeats the steps of the proof of the\nTheorem 2.1, but instead of Theorem 1.2 one uses the Theorem 1.4\nand the following remark: the Dirichlet polynomial $(1.7)$ can be\ndescribed via the inequality\n\\begin{eqnarray}\n\\alpha_1\\log 2+\\alpha_2\\log 3+\\dots +\\alpha_m\\log m\\leq \\log k,\n\\end{eqnarray}\nwhere the prime numbers are all prime numbers that appear in the\nprime decomposition of the integers $2,3,4,\\dots, k$.\nApproximating the logarithms in $(2.3)$ with fractions so that no\nnew integers appear and the existing ones remain, one can obtain\nthe lattice\n\\begin{eqnarray}\n\\alpha_1d+\\alpha_2d_2+\\dots +\\alpha_md_m\\leq d,\n\\end{eqnarray}\ninstead of the lattice $(2.3)$, where $d, d_i\\in \\mathbb{Q}$,\n$i=1,\\dots, m $. Crucial fact here is that the new lattice is\ndescribed by the same $m$-tuples $(\\alpha_1,\\dots, \\alpha _m)$ as\nin $(2.3)$. For the final step it is enough to obtain a common\ndenominator for the rational numbers $d, d_i$, $i=1,\\dots, m $ and\nget the lattice $\\mathcal {A}$ from the Theorem\n1.4.$\\diamondsuit$\\par\n We remark that defining the Rogosinski abscissa ${\\bf {r}}_a(G)$ in\nthe same manner as ${\\bf {r}}(G)$, but only for absolutely\nconvergent Dirichlet series in the half-plane, one obtains\n\\begin{theorem}\nIf the domain $G$ is convex then ${\\bf {r}}_a(G)={\\bf {r}}(U_1)$.\n\\end{theorem}\nIt is a well know fact that the Rogosinski radius is not smaller\nthan the Bohr radius for power series in one and several complex\nvariables (that is, the Rogosinski condition is satisfied in a\ndisc (ball) of larger radius than the Bohr condition). Therefore,\none might expect that the Rogosinski abscissa is not greater that\nthe isometric Bohr abscissa for ordinary Dirichlet series. This is\nthe content of the next result.\n\\begin{theorem}\nLet $G$ be a convex bounded domain in $\\mathbb{C}$. Then ${\\bf\n{r}}(G)\\leq {\\bf {b}}(G)$. If the domain $G\\not=\\mathbb{C}$ is\njust convex, then ${\\bf {r}}_a(G)\\leq {\\bf {b}}_a(G)$.\n\\end{theorem}\n{\\bf{Proof:}} Actually, if the domain $G$ is bounded  and $\\sigma\n>{\\bf{b}}(G)$, then\n\\begin{eqnarray*}\n\\vert P_k(s)-a_1\\vert =\\vert \\sum\\limits_{n=2}^k{a_n\\over\nn^s}\\vert \\leq  \\sum\\limits_{n=2}^k\\vert {a_n\\over n^s}\\vert\\leq\n\\sum\\limits_{n=2}^{\\infty}\\vert {a_n\\over n^s}\\vert\n<dist(a_1,\\partial G).\n\\end{eqnarray*}\nBut the obtained inequality\n\\begin{eqnarray*}\n\\vert P_k(s)-a_1\\vert \\leq dist(a_1,\\partial G)\n\\end{eqnarray*}\nmeans geometrically that $P_k(\\Pi _{\\sigma})\\subset G$. Thus, if\n$\\sigma >{\\bf {b}}(G)$, then $\\sigma >{\\bf {r}}(G)$ also.\nTherefore ${\\bf {r}}(G)\\leq {\\bf {b}}(G)$. The case when the\nboundedness of the convex domain $G$ is not required is considered\nanalogously. $\\diamondsuit $.\n\\begin{remark}\nWe point out the following open problems:\\\\\n1. Is it possible to remove the condition on the boundedness of\nthe domain $G$ in the statements of the Theorems 2.1, 2.4 and the\nCorollary 2.2?\\\\\n2) Can one prove the second parts of the  Theorems 2.1, 2.3\nwithout the assumption on the existence of at least one regular\npoint of convexity?\n\\end{remark}\n\\section{Estimates for the Bohr and Rogosinski abscissas}\n\\setcounter{equation}{0}\n\\begin{theorem}\nLet $G$ be a convex bounded domain in $\\mathbb{C}$. Then the\nisometric Bohr abscissa satisfies\n\\begin{eqnarray}\n{\\bf{b}}(G)\\leq 1.7267.\n\\end{eqnarray}\nIf $G$ is a domain in $\\mathbb{C}$, $\\widetilde G\\not=\n\\mathbb{C}$, then the isometric Bohr abscissa ${\\bf{b}}_a(G)$\nsatisfies the same estimate.\n\\end{theorem}\n{\\bf{Proof:}} Denote by $\\Omega (n)$ the number of the prime\ndivisors (counted with their multiplicity ) of the natural number\n$n$. Pivotal for us is the following result (\\cite{bal:gnus},\nProp.2.1): let $f$ be like in $(1.4)$ and $\\|f\\|=1 $, then for\n$k\\geq 1$ one has\n\\begin{eqnarray}\n(\\sum\\limits_{\\Omega (n)=k, n\\geq 2}\\vert a_n\\vert^2)^{{1\\over\n2}}\\leq 1-\\vert a_1\\vert ^2.\n\\end{eqnarray}\nFurthermore\n\\begin{eqnarray*}\n\\sum\\limits_{k=1}^{\\infty}{\\vert a_k\\vert\\over n^{\\sigma}}&\\leq\n&\\vert a_1\\vert+\\sum\\limits_{k=1}^{\\infty}(\\sum\\limits_{{\\Omega\n(n)=k,\\atop n\\geq 2}}\\vert a_n\\vert^2)^{{1\\over\n2}}(\\sum\\limits_{{\\Omega (n)=k,\\atop n\\geq 2}}{1\\over\nn^{2\\sigma}})^{{1\\over\n2}}\\\\\n&\\leq & \\vert a_1\\vert+ (1-\\vert\na_1\\vert^2)\\sum\\limits_{k=1}^{\\infty}(\\sum\\limits_{{\\Omega\n(n)=k,\\atop n\\geq 2}}{1\\over n^{2\\sigma}})^{{1\\over\n2}}\\\\\n\\end{eqnarray*}\nConsider now the equation\n\\begin{eqnarray}\n\\sum\\limits_{k=1}^{\\infty}(\\sum\\limits_{{\\Omega (n)=k,\\atop n\\geq\n2}}{1\\over n^{2\\sigma}})^{{1\\over 2}}={1\\over 2}.\n\\end{eqnarray}\nIf $\\sigma _0$ is the unique solution of the equation $(3.3)$,\nthen\n\\begin{eqnarray*}\n\\sum\\limits_{k=1}^{\\infty}{\\vert a_k\\vert\\over n^{\\sigma _0}}\\leq\n\\vert a_1\\vert+ (1-\\vert a_1\\vert ^2){1\\over 2}\\leq 1,\n\\end{eqnarray*}\nsince $\\vert a_1\\vert \\leq 1$. Therefore ${\\bf{b}}(U_1)\\leq \\sigma\n_0$. The solution of the equation $(3.3)$ was obtained\nnumerically, using Maple. Thus the estimate $\\sigma_0<1.7267$ was\nobtained. $\\diamondsuit $\\\\\n\\begin{corollary}\nLet $G$ be a bounded domain in $\\mathbb{C}$. Assume also that\n$\\partial G$ contains at least one regular point of convexity.\nThen\n\\begin{eqnarray*}\n1.5850\\dots ={\\log 3\\over \\log 2}\\leq {\\bf{b}}(G)<1.7267.\n\\end{eqnarray*}\nIf one does not require the boundedness of the domain\n$G\\not=\\mathbb{C}$, then ${\\bf{b}}_a(G)$ satisfies the same\nestimates.\n\\end{corollary}\n\\begin{theorem}\nLet $G$ be a convex bounded domain in $\\mathbb{C}$. Then, if the\nfunction $f$ is from $(1.5)$ and $\\|f\\|=1$, then for every such\n$f$\n\\begin{eqnarray}\n\\vert a_1\\vert^2+\\sum\\limits_{k=2}^{\\infty}{\\vert a_k\\vert\\over\nn^{\\sigma }}\\leq 1,\n\\end{eqnarray}\nwhere $1\\leq \\sigma < 1.2061$.\n\\end{theorem}\n{\\bf{Proof:}} From $(3.2)$, we obtain as before, that the left\nhand-side in $(3.4)$ is not larger than\n\\begin{eqnarray*}\n\\vert a_1\\vert ^2+ (1-\\vert a_1\\vert\n^2)\\sum\\limits_{k=1}^{\\infty}(\\sum\\limits_{{\\Omega (n)=k,\\atop\nn\\geq 2}}{1\\over n^{2\\sigma}})^{{1\\over 2}}.\n\\end{eqnarray*}\nConsider the equation\n\\begin{eqnarray}\n\\sum\\limits_{k=1}^{\\infty}(\\sum\\limits_{{\\Omega (n)=k,\\atop n\\geq\n2}}{1\\over n^{2\\sigma}})^{{1\\over 2}}=1.\n\\end{eqnarray}\nIf $\\sigma _0 $ is the root of the equation $(3.5)$, then for\n$\\sigma =\\sigma _0$ the relation $(3.4)$ holds. The equation\n$(3.5)$ was solved numerically by using Maple to obtain the\nestimate\n$\\sigma_0<1.2061$.\\\\\nOn the other hand, a particular case of the Dirichlet series\n$(1.4)$ is the series\n\\begin{eqnarray}\nf(s)=\\sum\\limits_{n=0}^{\\infty} {a_n\\over 2^{ns}},\n\\end{eqnarray}\nwhich is a power series relatively to the variable ${1\\over 2^s}$.\nIt is known (\\cite{pau:gnus}) that the best value for the  radius\n$r$, for which the relation\n\\begin{eqnarray*}\n\\vert c_0\\vert ^2+\\sum\\limits_{k=1}^{\\infty}\\vert c_k\\vert r^k\n\\leq 1\n\\end{eqnarray*}\nis valid for every power series $(1.1)$ satisfying $\\vert\nf(z_1)\\vert <1 $ in the unit disk $U_1$, is ${1\\over 2}$. Then for\nthe Dirichlet series $(3.6)$ we obtain\n\\begin{eqnarray*}\n{1\\over 2^{\\sigma_0}}={1\\over 2},\n\\end{eqnarray*}\nthat is $\\sigma _0=1 $. Since the series $(3.6)$ is a particular\ncase of the series $(1.4)$, we deduce that $\\sigma $ from Theorem\n3.3 is greater or equal to 1.$\\diamondsuit$\\\\\nAnalogously one can prove the following\n\\begin{theorem}\nLet $G$ be a convex bounded domain in $\\mathbb{C}$. Then the\nRogosinski abscissa ${\\bf{r}}(G)\\geq 1 $. If the domain $G\\not\n=\\mathbb{C}$ is not bounded then the Rogosinski abscissa\n${\\bf{r}}_{a}(G)$ satisfies the same inequality.\n\\end{theorem}\n\\begin{remark}\nWe conclude the present paper by by stating the following\nhypothesis: for every convex bounded domain $G\\subset \\mathbb{C}$\nthe equality ${\\bf{r}}(G)=1 $ is true. If the domain $G\\not\n=\\mathbb{C}$ is convex, but not necessarily bounded then also\n${\\bf{r}}_{a}(G)=1$.\n\\end{remark}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Algorithm}\\label{sec:alg.}\n\\subsection{Follow the perturbed leader}\nThe agent is in an MDP without a specified reward function. Instead, the agent is presented with expert policy and features $\\pi_E$ and $\\mu_E$. The mapping between the policy and features $\\phi$ is also known. This expert policy could be acquired by human demonstration or learned from an MDP that has all properties of the current MDP and a known reward function. We assume that we have access to an RL algorithm that can learn the optimal policy given a full MDP, so that we have the expert policy and thus the features. Recall the max-min goal as in equation \\ref{def:max-min}, we will learn a roughly max-min solution using FPL. Our algorithm to learn such a policy $\\tilde \\pi$ is as follows:\n\\todo{Say adaptive adversary.}\n\nUpon termination, the algorithm returns ${\\tilde{\\mu}} = 1\/T \\sum_{i=1}^T \\mu_i $ and the corresponding ${\\tilde {\\pi}} $\n\n\\section*{Supplementary Material}\n\n\\section{Missing Proofs from Section~\\ref{sec:alg}}\\label{sec:missingproof}\n\n\\paragraph{The Ellipsoid Method} The following theorem, reworded from~\\cite{Khachiyan79,GrotschelLS81, KarpP82}, states that given a separation oracle of a convex polytope, the ellipsoid method can optimize any linear function over the convex polytope in polynomial time.\n\n\\begin{theorem}[Ellipsoid Method]\\label{thm:ellipsoid}(\\cite{Khachiyan79,GrotschelLS81,KarpP82}) Let $P$ be a $d$-dimensional closed, convex subset of $\\mathbb{R}^d$ defined as the intersection of finitely many halfspaces, and $SO$ be a {poly-time} separation oracle for $P$. Then it is possible to\nfind an element in $\\text{argmax}_{{x} \\in P} \\{{c} \\cdot {x}\\}$ for any ${c} \\in \\mathbb{R}^d$ (i.e. solve linear programs) in time polynomial in $d$ and $\\langle P \\rangle$ using the ellipsoid method, if $P$ can be described implicitly\nusing $\\langle P\\rangle$ bits.~\\footnote{We say a polytope $P$ can be described implicitly using $\\ell$ bits if there exists a description of the polytope $P$ such that all constraints only use coefficients with bit complexity $\\ell$.\n\n\\end{theorem}\n\n\n\\begin{prevproof}{Lemma}{lem:SO_f}\n\tLemma~\\ref{lem:bit complexity of P_F}  shows that $P_F$ can be implicitly described using $\\langle P_F\\rangle = \\text{poly}(\\langle M\\setminus R \\rangle,\\langle \\phi \\rangle,k)$ bits. Maximizing any linear function $w\\cdot \\mu$ can be solved by querying $\\textsc{ALG}$ on MDP $M\\setminus R$ with reward function $w\\cdot \\phi(\\cdot)$. Since MDP $\\left (M\\setminus R, w\\cdot \\phi(\\cdot)\\right)$ has bit complexity polynomial in $\\langle M\\setminus R \\rangle$, $\\langle \\phi \\rangle$, $k$, and $\\langle w \\rangle$,  we can solve the linear optimization problem in time $\\text{poly}(\\langle P_F\\rangle,k, \\langle w \\rangle)$. By Lemma~\\ref{lem:sep-opt}, we can solve the separation problem in time $\\text{poly}(\\langle P_F\\rangle, k, \\langle y \\rangle)$ on any input $y\\in \\mathbb{R}^k$. Hence, we can design a polynomial time separation oracle.\n\\end{prevproof}\n\n\\begin{lemma}\\label{lem:bit complexity of P_F}\nPolytope $P_F$ for any MDP without reward function $M\\setminus R$ can be implicitly described using $\\text{poly}(\\langle M\\setminus R \\rangle,\\langle \\phi \\rangle,k)$ bits.\n\\end{lemma}\n\\begin{proof} \nThe following constraints explicitly describe all $\\mu \\in P_F$, where $x_{sa}$s correspond to the occupancy measure of some policy $\\pi$. \n\\begin{align*}\n\t&\\mu = \\sum_{s} \\phi(s) \\sum_{a} x_{s a} &\\\\\n& \\sum_a x_{sa} = \\text{Pr}(s_0 = s) + \\gamma \\sum_{s',a}x_{s'a} P_{sa} \\qquad \\forall s &\\\\\t\n& x_{sa} \\geq 0 &\n\\end{align*}\nOur statement follows from the fact that all the coefficients in these constraints have bit complexity $\\langle M\\setminus R \\rangle$ or $\\langle \\phi \\rangle$.\n\\end{proof}\n\n\n\\paragraph{Intuition behind Lemma~\\ref{lem:convhull}} The intuition behind Lemma~\\ref{lem:convhull} is that the separation oracle $SO_F$ tries to search over all possible weights $w$ to find one to separate the query point $\\mu$ from  $P_F$ using the ellipsoid method. Along the way, it queries a set of weights (this is our set $C$) on $\\textsc{ALG}$ trying to find a separating weight $w$ such that $\\mu\\cdot w>\\mu_w \\cdot w$. If such a separating weight is found, $SO_F$ terminates immediately and outputs ``\\textsc{No}'' together with the corresponding separating hyperplane. The $SO_F$ says ``\\textsc{Yes}'' only when it has searched over a polynomial number of weights and concludes that there is no possible weight to separate $\\mu$. The reason that $SO_F$ can draw such a conclusion is due to the ellipsoid method. In particular, when $SO_F$ says ``\\textsc{Yes}'',  the correctness of the ellipsoid algorithm implies that $\\mu$ is in the convex hull of all the extreme points of $P_F$ that have been outputted by the $\\textsc{ALG}$.\n\n\\paragraph{Follow-the-Perturbed-Leader} Kalai and Vempala~\\cite{KalaiV05} proposed the FPL algorithm and showed that in expectation, the regret is small against any oblivious adversary.~\\cite{HutterP05} showed that the same regret bound extends to settings with adaptive adversary. To obtain a high probability bound, one can construct a martingale to connect the actual reward and the expected reward obtained by the agent, then apply the Hoeffding-Azuma inequality.\n\n\\begin{theorem}[Follow-the-Perturbed-Leader]~\\cite{KalaiV05,HutterP05, CL}\\label{thm:FPL}\n\tLet $d_1,\\ldots,d_T$ be a sequence of decisions. Let $s_1,\\ldots, s_T$ be a state sequence chosen by an adaptive adversary, that is, $s_t$ can be selected based on all the previous states $s_1,\\ldots,s_{t-1}$ and all the previous decisions $d_1,\\ldots, d_{t-1}$ for every $t\\leq T$. If we let $d_t$ be $\\text{argmax}_{d\\in D} d\\cdot \\left( \\sum_{i=1}^{t-1} s_{i}+p_t\\right)$, where $p_t$ is drawn uniformly from $[0,1\/\\delta]^n$ for some $\\delta>0$,  then \n\t$$\\mathbb{E}\\left[\\sum_{t=1}^T d_t\\cdot s_t-\\max_{d\\in D} \\sum_{t=1}^T d\\cdot s_t\\right]\\geq -\\delta\\cdot C_1C_2 T-\\frac{2C_3}{\\delta}.$$ $C_1$ is an upper bound of $||s||_1$ for all $s\\in \\mathbb{S}$, $C_2$ is an upper bound of $|d\\cdot s|$ for all $d\\in D$ and $s\\in \\mathbb{S}$, and $C_3$ is an upper bound of $||d||_1$ for all $d\\in D$. Moreover, for all $\\xi\\geq 0$, with probability at least $1-\\xi$, the actual accumulative reward under any adaptive adversary satisfies, $$\\sum_{t=1}^T d_t\\cdot s_t-\\max_{d\\in D} \\sum_{t=1}^T d\\cdot s_t\\geq -\\delta\\cdot C_1C_2 T-\\frac{2C_3}{\\delta}-2C_2\\sqrt{T\\ln \\frac{1}{\\xi}}.$$ \n\\end{theorem}\n\n\nUsing Theorem~\\ref{thm:FPL}, we are ready to prove Theorem~\\ref{thm:FPL maxmin}.\n\n\\begin{prevproof}{Theorem}{thm:FPL maxmin}\n\tWe use $P$ to denote the sequence $p_1,\\ldots, p_T$ and $Q$ to denote the sequence $q_1,\\ldots, q_T$. First, notice that every realization of $Q$ defines a deterministic adaptive adversary for the agent. In the setting of Algorithm~\\ref{alg:FPL maxmin}, we can take $C_1$ to be $k$, $C_2$ to be $k^2$, and $C_3$ to be $k$. By Theorem~\\ref{thm:FPL} (Section~\\ref{sec:missingproof} of the supplementary material), we know that for all $\\xi\\geq 0$, $\\text{Pr}_{P\\sim U[0,1\/\\delta]^{kT}} [\\sum_{t=1}^T \\mu_t\\cdot w_t-\\max_{\\mu\\in P_F} \\sum_{t=1}^T \\mu\\cdot w_t \\geq -k^2\\sqrt{T}(3+2\\sqrt{\\ln 1\/\\xi}) | Q]\\geq 1-\\xi$ for every realization of $Q$. Similarly, every realization of $P$ also defines a deterministic adaptive adversary for the designer, and by Theorem~\\ref{thm:FPL} , we know that $\\text{Pr}_{Q\\sim U[0,1\/\\delta]^{kT}} [-\\sum_{t=1}^T \\mu_t\\cdot w_t+\\min_{w\\in P_R} \\sum_{t=1}^T \\mu_t\\cdot w \\geq -k^2\\sqrt{T}(3+2\\sqrt{\\ln 1\/\\xi})| P]\\geq 1-\\xi$ for any realization of $P$. Let $B= k^2\\sqrt{T}\\left(3+2\\sqrt{\\ln 1\/\\xi}\\right)$. By the union bound, with probability at least $1-2\\xi$ over the randomness of $P$ and $Q$  \\begin{equation}\\label{eq:agent regret}\n\\sum_{t=1}^T \\mu_t\\cdot w_t-\\max_{\\mu\\in P_F} \\sum_{t=1}^T \\mu\\cdot w_t \\geq -B\n  \\end{equation}\n  and \n\\begin{equation}\\label{eq:designer regret}\n-\\sum_{t=1}^T \\mu_t\\cdot w_t+\\min_{w\\in P_R} \\sum_{t=1}^T \\mu_t\\cdot w \\geq -B\n\\end{equation}\n\t\n\tNext, we argue that $\\frac{1}{T}\\cdot \\sum_{t=1}^T \\pi_t$ is an approximate maxmin policy.\n\t\\begin{align*}\n\t\\min_{w\\in P_R}\\sum_{t=1}^T \\mu_t\\cdot w&\\geq \\sum_{t=1}^T \\mu_t\\cdot w_t\t-B & (\\text{Eq.~\\eqref{eq:designer regret}})\\\\\n\t&\\geq \\max_{\\mu\\in P_F} \\sum_{t=1}^T \\mu\\cdot w_t-2B & (\\text{Eq.~\\eqref{eq:agent regret}})\\\\\n\t&\\geq T\\cdot \\max_{\\mu\\in P_F}\\min_{w\\in P_R} \\mu\\cdot w-2B &\n\t\\end{align*}\nThe last inequality is because that on the LHS (line 2) the designer is choosing a fixed strategy $\\frac{1}{T}\\cdot\\sum_{t=1}^T w_t$, while on the RHS (line 3) the designer can choose the worst possible strategy for the agent. Therefore, if the agent uses policy $\\frac{1}{T}\\cdot \\sum_{t=1}^T \\pi_t$, it guarantees expected reward $\\max_{\\mu\\in P_F}\\min_{w\\in P_R} \\mu\\cdot w-2B\/T$.\n\nFinally, in every iteration $t$, we query $\\textsc{ALG}$ once to compute $\\pi_t$ and $\\mu_t$, and we use the ellipsoid method to find $w_t$ using $O(k^2 (\\langle M\\backslash R \\rangle^2+(\\log T)^2))$ queries to $SO_R$ and poly$(k, \\langle M\\backslash R \\rangle, \\log T)$ regular computation steps. During each query, $SO_R$ calls $\\textsc{ALG}$. Thus, our result is a reduction from the maxmin learning problem to simply solving an MDP under given weights. Any improvement on $\\textsc{ALG}$ will also improve the running time of Algorithm~\\ref{alg:FPL maxmin}. We discuss the empirical running time in section~\\ref{sec:suppexp} of the supplementary material.\n\\end{prevproof}\n\n\n\\input{model_free_approx_solver}\n\\section{Experiment Details}\\label{sec:suppexp}\nIn every iteration of Algorithm~\\ref{alg:FPL maxmin} and Algorithm~\\ref{alg:eps FPL maxmin}, step 5 computes a minimizing weight in $P_R$. Instead of using the ellipsoid method to solve the LP, we use the analytic center cutting-plane method (see~\\cite{BoydV07} for a brief overview) throughout our experiments. The method combines good practical performance with reasonable simplicity. \n\n\\subsection{Gridworld}\nThe domain contains five types of terrain. Four terrain types are used in the demonstration gridworld where we construct the expert policy. We select the rewards for these four terrain types uniformly from $[-0.5,0]$, and the target has a reward of 10. The reward of each terrain type is deterministic. The demonstration MDP is uniformly composed of four terrain types, 25\\% each type. The fifth terrain type (red colored as in Figure~\\ref{fig:gw}) is not present in the demonstration gridworld. The agent is trained in a \"real-world\" MDP that is composed uniformly of all five terrain types, 20\\% each. We select maps that are feasible, such that for all rewards in the consistent reward polytope, value iteration has a solution for the agent to reach the goal. We use feature vectors that indicate the terrain type of each state, choose a discount factor of 0.95, and use value iteration throughout the experiment. The consistency between the expert policy and the reward function is defined with $\\epsilon = 0.5$. \n\n\\paragraph{Deterministic transition model} In an MDP with deterministic transition model, the agent moves in exactly the direction chosen by the agent. We run FPL for $5000$ iterations and use the average of policies output by the last $2500$ iterations as the maxmin policy. Figure~\\ref{fig:dtgw_occu} shows that our maxmin policy is much safer than a baseline. The baseline policy is computed in an MDP whose reward weights are the same as the demonstration MDP for the first four terrain types and the fifth terrain weight is chosen uniformly at random from [-1,0]. The expert policy for the displayed results is constructed by computing the optimal policy in an demonstration MDP with rewards for the first four terrain type set as $[-0.5, -0.2, -0.4, -0.1]$. The results are accumulated from 100 individual runs using the same expert policy. Examples of the baseline trajectories are shown in Figure~\\ref{fig:gw-1}. \n\n\\begin{figure}[ht]\n\\begin{center}\n\t\\includegraphics[width=0.2\\textwidth]{figures\/alternative_1}\\hfill\n\\end{center}\n\\caption{Trajectories chosen by policies generated using weights randomly assigned to the red-colored unknown feature. Although this feature may have negative side effects, the random agent may still go through it. }\n\\label{fig:gw-1}\n\\end{figure}\n\n\n\\paragraph{Stochastic transition model} At each state, there is 10\\% chance that the agent will go in a random direction regardless of the action chosen by the agent. The agent will receive rewards based on the state it actually lands in. We show in Figure~\\ref{fig:gw-stoc-1} that to mitigate the higher risk of traversing the unknown terrain type, our maxmin policy appears to be more conservative than the deterministic case. Although Figure~\\ref{fig:stocgw_occu} shows that it cannot absolutely avoid the unknown terrain type due of the stochastic nature of the model, the percentage is much lower than the baseline. The baseline was computed with the same reward weights as in the deterministic case.\n\n \\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.2\\textwidth]{figures\/stoch0p1_agent_2}\\hfill\n\\end{center}\n\\caption{At each state, there is $10\\%$ chance that the agent will go in a random direction irrespective of the action chosen by the agent,  our maxmin method is still valid. Comparing to Figure~\\ref{fig:gw} ({\\bf right}), the maxmin policy also avoids going to the peripheral of the red-colored unknown feature.  }\n\\label{fig:gw-stoc-1}\n\\end{figure}\n\n \\begin{figure}\\centering\n\t\\includegraphics[width = 0.5 \\textwidth]{figures\/gw_mmvsrand_stoch}\n\t\\caption{In the gridworld with stochastic transition model, our maxmin policy has a small chance of traversing the unknown terrain type disregard of being more conservative than the maxmin policy in the deterministic case. The percentage is much lower than the baseline.  }\n\t\\label{fig:stocgw_occu}\n\\end{figure}\n\n\\paragraph{Computation Performance}\nIn our grid world experiment, the worst case running time of ALG is $O(n^2)$, but experiments show a more benign runtime of $O(n^{1.5})$. For a $50\\times50$ grid world with 25 features, Algorithm~\\ref{alg:FPL maxmin} appears to converge after 325 iterations of FPL with total runtime of 3324 seconds (average of 20 trials, ordinary desktop computer). \nInstead of using the ellipsoid method, we used analytic center cutting-plane method, and the running time appears to scale in the order of $O(k^2)$. \n\n\n\n\n \n\n \\subsection{CartPole}\n We modify the classic CartPole task in the OpenAI Gym environment by adding features that may incur additional rewards. This is represented by the question blocks in Figure~\\ref{fig:cartpole_env}. The two question blocks correspond to feature indicators for the agent's horizontal position in the range of $[-1.2, 0)$ and $[0.6, 1.8)$. We keep the same episode termination criterions for the pole angle and cart position as the original environment. An episode is considered ending without failing if the pole angel and cart position meet the criterion and the episode length is greater than 500. The agent receives a reward of $+1$ for surviving every step. \n \n We use longer episodes than the original problem to allow more diverse movement, while it also makes the task more challenging. During validation of a policy, we consider the task solved as getting a target average reward over 100 consecutive episodes with less than five failed episodes. The target average reward depends on the reward we assign for passing the question blocks. If each step spent at question block $i$ incurs reward of $r_i$, the target average reward is set to be $450+25\\sum r_i$. For example, in $scenario \\ A$, only the blue question block exists and it incurs reward of $-2$, our expert policy $Expert \\ A$ passes the validation criterion by getting average reward higher than 400 over 100 consecutive episodes with less than 5 failed episodes. Indeed, our $Expert \\ A$ policy performs quite well by getting a reward greater than 450 in $scenario \\ A$. In $scenario \\ B$, only the yellow question block is present and it incurs reward of $+2$. $Expert \\ B$ passes the validation criterion with reward greater than 1000. \n \n The agent is in an MDP with both blue and yellow question blocks whose reward polytopes are implicitly defined by the expert policy. We use Q-learning and apply updates using minibatches of stored samples as the MDP solver. Notice that for this problem, our MDP solver is not necessarily optimal. We computed maxmin policies when provided with different expert policies. The results in Figure~\\ref{fig:cp_maxmin} are from testing the maxmin policy for $2000$ episodes.\n\\section{Discussion}\n\nIn this paper, we provided a theoretical treatment of the problem of reinforcement learning in the presence of mis-specifications of the agent's reward function, by leveraging data provided by experts.  The posed optimization can be solved exactly in polynomial-time by using the ellipsoid methods, but a more practical solution is provided by an algorithm which takes a follow-the-perturbed-leader approach. Our experiments illustrate the fact that this approach can successfully learn robust policies from imperfect expert data, in both discrete and continuous environments. It will be interesting to see whether our maxmin formulation can be combined with other methods in RL such as hierarchical learning to produce robust solutions in larger problems.\n\n\\section{Experiments}\\label{sec:exp.}\n\n\\paragraph{Gridworld}\\label{sec:exp_gw}\nWe use gridworlds in the first set of experiments. Each grid may have a different \"terrain\" type such that passing the grid will incur certain reward. For each grid,  a feature vector $\\phi(s)$ denotes the terrain type, and the true reward can be expressed as $R^* =  w^* \\cdot \\phi(s)$. The agent's goal is to move to a {goal} grid with {\\it maximal} reward under the {\\it worst possible} weights that are consistent with the expert.\n  In other words, the maxmin policy is a safe policy, as it avoids possible negative side effects \\cite{AmodeiOSCSM16}. In the experiments, we construct the expert policy $\\pi_E$ in a small (10$\\times$10) demonstration gridworld that contains a subset of the terrain types. One expert policy is provide, and the number of trajectories that we need to estimate the expert policy's cumulative feature follows the sample complexity analysis as in~\\cite{SyedS08}. In the following experiment we set $\\epsilon=0.5$ which defines $P_R$ and captures how close to optimal the expert is.\n\n\nAn example behavior is shown in Figure \\ref{fig:gw}. There are 5 possible terrain types. The expert policy in Figure~\\ref{fig:gw} (left) has only seen 4 terrain types. We compute the maxmin policy in the \"real-world\" MDP of a much larger size (50$\\times$50) with all 5 terrain types using Algorithm~\\ref{alg:FPL maxmin} with the reward polytope $P_R$ implicitly specified by the expert policy. Figure~\\ref{fig:gw} (middle) shows that our maxmin policy avoids the red-colored terrain that was missing from the demonstration MDP.  To facilitate observation, Figure~\\ref{fig:gw} (right) shows the same behavior by an agent trained in a smaller MDP. Figure~\\ref{fig:dtgw_occu} compares the maxmin policy to a baseline. The baseline policy is computed in an MDP whose reward weights are the same as the demonstration MDP for the first four terrain types and the fifth terrain weight is chosen at random. Our maxmin policy is much safer than the baseline as it completely avoids the fifth terrain type. It also imitates the expert's behavior by favoring the same terrain types.\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\t\\vspace{-10pt}\n\t\\centering\n\t\\includegraphics[width = 0.45 \\textwidth]{figures\/gw_mmvsrand}\n\t\\caption{Experiment results comparing our maxmin policy to a baseline. The baseline was computed with a random reward for the fifth terrain and the other four terrain rewards set the same as the demonstration MDP. Our maxmin policy is much safer than the baseline as it completely avoids traversing the fifth (unknown) terrain type. It should also be noticed that the maxmin policy learns from the expert policy while achieving the goal of avoiding potential negative side effects, as the fraction of trajectory of each terrain type closely resemble the expert.  }\n\t\\label{fig:dtgw_occu}\n\\end{wrapfigure}\n\nWe also implemented the maxmin method in gridworlds with a stochastic transition model. The maxmin policy (see Figure~\\ref{fig:gw-stoc-1} in Section~\\ref{sec:suppexp} of the supplementary material) is more conservative comparing to the deterministic model, and chooses paths that are further away from any unknown terrains. More details and computation time can be found in the supplementary material.\n\nIt should be noted that the training and testing MDPs are different. More specifically, the red terrain type is missing from the expert demonstration, and the testing MDP is of a larger size. As discussed in the Introduction, our formulation allows the testing MDP in which the agent operates to be different from the training MDP in which the expert demonstrates, as long as the two MDPs share the same feature space. All of our experiments have this property. To the limit of our knowledge, apprenticeship learning requires the training and testing MDPs to be the same, thus a direct comparison is not possible. For example, in the gridworld experiments, one has to explicitly assign a reward to the \"unknown\" feature in order to apply apprenticeship learning, which may cause the problem of reward misspecification and negative side effects. Our maxmin solution is robust to such issues. \n\n\\begin{figure}[h]\n\\centering\n\t\\hspace*{\\fill}%\n\t\\begin{subfigure}\\centering\n\t\t\\includegraphics[width=0.22\\textwidth]{figures\/expert_1}\\hfill\n\t\\end{subfigure}\n\t\\begin{subfigure}\\centering\n\t\t\\includegraphics[width=0.22\\textwidth]{figures\/agent50}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}\\centering\n\t\t\\includegraphics[width=0.22\\textwidth]{figures\/agent_1}\n\t\\end{subfigure}\n\t\\hspace*{\\fill}%\n\t\\caption{An example of maxmin policy in gridworlds.  {\\bf Left}: an expert policy in the small demonstration MDP, where 4 of 5 terrain types were used and their weights were randomly chosen. The expert policy guides moving towards the yellow goal grid while preferring the terrains with higher rewards (light blue and light green).  {\\bf Middle}: when faced with terrain types (red-colored) that the expert policy never experienced, maxmin policy avoids such terrains and the accompanying negative side effects. The agent learns to operate in a larger (50$\\times$50) grid world. \n\t{\\bf Right}: an agent in a smaller MDP to facilitate observation\n\tThe maxmin policy generates two possible trajectories.  }\n\n\\label{fig:gw}\n\\end{figure}\n\n\\begin{wrapfigure}{r}{0.4\\textwidth}\n\n\t\\centering\n\t\\includegraphics[width = 0.3\\textwidth]{figures\/CartPole}\n\t\\caption{Modified cartpole task with two additional features -- questions blocks to either side of the center. The rewards associated with passing these blocks are not provided to the agent.}\n\n\t\\label{fig:cartpole_env}\n\\end{wrapfigure}\n\n\\paragraph{CartPole}Our next experiments are based on the classic control task of cartpole and the environment provided by OpenAI Gym \\cite{BrockmanCPSSTZ16}.  While we can only solve the problem approximately using model-free learning methods, our experiments show that our FPL-based algorithm can learn a safe policy efficiently for a continuous task. Moreover, if provided with more expert policies, our maxmin learning method can easily accomodate and learn from multiple experts. \n\n\\begin{figure}\n\n\t\\centering\n\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{figures\/cp_basic}\n\t\t\\caption{Behavior examples of different policies. Occupancy is defined as the number of steps appearing at a location divided by the total steps. {\\bf top}: In the default setting without any question blocks, the travel range is relatively symmetric around the center of the field. {\\bf mid}: In the presence of the blue question block to the left, an expert policy $A$ guides movements to the right. {\\bf bottom}: In {\\it scenario B} where only the yellow question block is present, expert policy $B$ also guides movement to the right. }\n\t\t\\label{fig:cp_basic}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{figures\/cp_maxmin}\n\t\t\\caption{Maxmin policy learnt with different expert policies. {\\bf top}: Given {\\it Expert A} policy only, the agent learns to stay within a  narrow range near slightly right to the center to avoid both question blocks. Because the agent has no knowledge about the yellow block, a maxmin policy avoids it.  {\\bf bottom}: When given both {\\it Expert A} and {\\it Expert B} policies, the agent learns that it is safe to pass the yellow block, so the range is wider and extends more to the right comparing to the maxmin policy learnt from {\\it Expert A} alone.   }\n\t\t\\label{fig:cp_maxmin}\n\t\\end{minipage}\n\t\\vspace{-10pt}\n\\end{figure}\n\nWe modify the cartpole problem by adding two possible features to the environment as the two question blocks shown in Figure~\\ref{fig:cartpole_env}, and more details in the supplementary material. The agent has no idea of what consequences passing these two blocks may have. Instead of knowing the rewards associated with these two blocks, we have expert policies from two other related scenarios. The first expert policy ({\\it Expert A}) performs well in {\\it scenario A} where only the blue block to the left of the center is present, and the second expert policy ({\\it Expert B}) performs well in {\\it scenario B} where only the yellow block to the right of the center is present.  The behavior of expert policies in a default scenario (without any question blocks),  and scenarios $A$ and $B$ are shown in Figure~\\ref{fig:cp_basic}. It is obvious that comparing with the default scenario, the expert policies in the other two scenarios prefer to travel to the right side. Intuitively, it seems that the blue block incurs negative effects while the yellow block is either neutral or positive.\n\n\nNow we train the agent in the presence of both question blocks. First, we provide the agent with {\\it Expert A} policy alone, and learn a maxmin policy. The maxmin policy's behavior is shown in Figure~\\ref{fig:cp_maxmin} ({\\bf top}). It tries to avoid both question blocks since it observes that Expert A avoids the blue block and it has no knowledge of the yellow block. Then, we provide both $Expert \\ A$ and $Expert \\ B$ to the agent, and the resulting maxmin policy guides movement in a wider range extending to the right of the field as shown in Figure~\\ref{fig:cp_maxmin} ({\\bf bottom}). This time, our maxmin policy also learns from {\\it Expert B} that the yellow block is not harmful. \n\n\nThe experiment demonstrates that our maxmin method works well with complex reinforcement learning tasks where only approximate MDP solvers are available. \n\n\n\n\n\n\n\n\\section{Introduction}\n\nIn Reinforcement Learning (RL), agent behavior is driven by an objective function defined through the specification of rewards. Misspecified rewards may lead to {\\it negative side effects} \\cite{AmodeiOSCSM16}, when the agent acts unpredictably responding to the aspects of the environment that the designer overlooked, and potentially causes harms to the environment or itself. As the environment gets richer and more complex, it becomes more challenging to specify and balance rewards for every one of its aspects. Yet if we want to have some type of safety guarantees in terms of the behavior of an agent learned by RL once it is deployed in the real world, it is crucial to have a learning algorithm that is robust to mis-specifications.\n\nWe assume that the agent has some knowledge about the reward function either through past experience or demonstrations from experts. The goal is to choose a robust\/safe policy that achieves high reward with respect to any reward function that is consistent with the agent's knowledge~\\footnote{{Note that the safety as used here is more in the context of AI safety, and a policy is safe because it is robust to misspecified rewards and the consequent negative side effects. }}.We formulate this as a \\emph{maxmin learning} problem where the agent chooses a policy and an adversary chooses a reward function that is consistent with the agent's current knowledge and minimizes the agent's reward. The goal of the agent is to learn a policy that maximizes the worst possible reward. \n\n\n We assume that the reward functions are linear in some feature space. Our formulation has two appealing properties: (1) it allows us to combine demonstrations from multiple experts even though they may disagree with each other; and (2) the training environment\/MDP in which the experts operate need not be the same as the testing environment\/MDP where the agent will be deployed, our results hold as long as the testing and training MDPs share the same feature space. As an application, our algorithm can learn a maxmin robust policy in a new environment that contains a few features that are not present in the training environment. See our gridworld experiment in Section~\\ref{sec:exp_gw}. \n \n Our first result (Theorem~\\ref{thm:exact maxmin}) shows that given any algorithm that can find the optimal policy for an MDP in polynomial time, we can solve the maxmin learning problem exactly in polynomial time. Our algorithm is based on a seminal result from combinatorial optimization -- the equivalence between separation and optimization~\\cite{GrotschelLS81,KarpP82} -- and the ellipsoid method. To understand the difficulty of our problem, {it is useful to think of maxmin learning as a two-player zero-sum game between the agent and the adversary.} The deterministic policies correspond to the pure strategies of the agent. The consistent reward functions we define in Section~\\ref{sec:reward SO} form a convex set and the adversary's pure strategies are the extreme points of this convex set. Unfortunately, both the agent and the adversary may have exponentially many pure strategies, which are hard to describe explicitly. This makes solving the two-player zero-sum game challenging.  Using tools from combinatorial optimization, we manage to construct separation oracles for both the agent's and the adversary's set of policies using the MDP solver as a subroutine. With the separation oracles, we can solve the maxmin learning problem in polynomial time using the ellipsoid method.\n\n \n\n \n  Theorem~\\ref{thm:exact maxmin} provides a polynomial time algorithm, but as it heavily relies on the ellipsoid method, it is computationally expensive to run in practice. We propose another algorithm (Algorithm~\\ref{alg:FPL maxmin}) based on the online learning algorithm -- \\emph{followed-the-perturbed-leader (FPL)}, and show that after $T$ iterations the algorithm computes a policy that is at most $O\\left(\\nicefrac{1}{\\sqrt{T}}\\right)$ away from the true maxmin policy (Theorem~\\ref{thm:FPL maxmin}). Moreover, each iteration of our algorithm is polynomial time. Notice that many other low-regret learning algorithms, such as the multiplicative weights update method (MWU), are not suitable for our problem. The MWU requires explicitly maintaining a weight for every pure strategy\n   and updates them in every iteration, resulting in an exponential time algorithm for our problem. Furthermore, we show that Algorithm~\\ref{alg:FPL maxmin} still has similar performance when we only have a fully polynomial time approximation scheme (FPTAS) for solving the MDP. The formal statement and proof are postponed to the supplemental material due to space limit\n\n\\subsection{Related Work}\\label{sec:related}\nIn the sense of using expert demonstrations, our work is related to inverse reinforcement learning (IRL) and apprenticeship learning~\\cite{NgR00,AbbeelN04,SyedS08,SyedBS08}. In particular, the apprenticeship learning problem {aims to learn a policy that performs at least as well as the expert's policy on all basis rewards}, and can also be formulated as a maxmin problem~\\cite{SyedS08,SyedBS08}. Despite the seemingly similarity, our maxmin learning problem aims to solve a completely different problem than apprenticeship learning. Here is a simple example: {consider in a gridworld, there are two basis rewards, $w_1$ and $w_2$, and there are only two routes\/policies -- $top$ and $bottom$. The expert takes the $top$ route, getting $100$ under $w_1$ and $70$ under $w_2$. Alternatively, taking the $bottom$ route gets $90$ under both $w_1$ and $w_2$. Apprenticeship learning will return the { $top$ route}, because taking the alternative route performs worse than the expert under $w_1$ and violates the requirement. What is our solution? Assuming $\\epsilon=25$~\\footnote{See Section~\\ref{sec:reward SO} for the formal definition of consistent rewards. Intuitively, it means that the expert's policy yields a reward that is within $\\epsilon$ of the optimal possible reward.}, both $w_1$ and $w_2$ (or any convex combination of them) are consistent with the expert demonstration. If we choose the top route, the worst case performance (under $w_2$ in this case) is 70, while the correct maxmin solution to our problem is to take the {$bottom$ route} so that its worst performance is $90$.}\\notshow{assume the expert is consistent with two reward functions $w_1$ and $w_2$ with $\\epsilon = 25$~\\footnote{See Section~\\ref{sec:reward SO} for the formal definition of consistent rewards. Intuitively, it means that the expert's policy yields a reward that is within $\\epsilon$ of the optimal possible reward.}. The expert gets $100$ under $w_1$ and $70$ under $w_2$, and the only alternative policy gives $90$ and $90$. Apprenticeship learning will return the expert policy while ours will return the latter solution.} In the worst case (under $w_2$), our maxmin policy has better guarantees and thus is more robust.\nUnlike apprenticeship learning\/IRL, we do not want to mimic the experts or infer their rewards, but we want to produce a policy with robustness guarantees by leveraging their data. As a consequence, our results are applicable to settings where the training and testing environments are different (as discussed in the Introduction). Moreover, our formulation allows us to combine multiple expert demonstrations. \n\n\\notshow{\\paragraph{Apprenticeship Learning} Similarity: maxmin formulation, expert guidance. Difference: apprenticeship hopes to imitate the expert's behavior. We just want to infer consistent rewards from the expert and then do maxmin planning. Give example. Different application: we can accommodate multiple experts and do not need the training domain to be the same as testing domain. See experiment.~\\cite{SyedBS08,SyedS08}}\n\nInverse reward design~\\cite{HadfieldMARD17}  uses a proxy reward and infers the true reward by estimating its posterior. Then it uses risk-averse planning together with samples from the posterior in the testing environment to achieve safe exploration. Our approach achieves a similar goal without assuming any distribution over the rewards and is arguably more robust. We apply a single reward function to the whole MDP while they apply (maybe too pessimistically) per step\/trajectory maxmin planning. Furthermore, our algorithm is guaranteed to find the maxmin solution in polynomial time, and can naturally accommodate multiple experts. \n\n{In repeated IRL~\\cite{AminJS17}, the agent acts on the behalf of a human expert in a variety of tasks, and the human expert corrects the agent when the agent's policy is far from the optimum. The goal is to minimize the number of corrections from the expert, and they provide an upper bound on the number of corrections by reducing the problem to the ellipsoid method. Their model requires continuous interaction with an expert while our model only assumes the availability of one or a couple expert policies prior to training. Furthermore, we aim to find a maxmin optimal policy, while their paper focuses on minimizing the number of corrections needed.}\n\nRobust Markov Decision Processes  \\cite{NilimEG05,Iyengar05}  have addressed the problem of performing dynamic programming-style optimization environments in which the transition probability matrix is uncertain. Lim, Xu \\& Mannor \\cite{LimXM13} have extended this idea to reinforcement learning methods.  This body of work also uses min-max optimization, but because the optimization is with respect to worst-case transitions, this line of work results in very pessimistic policies.  Our algorithmic approach and flavor of results are also different.\n\\cite{MorimotoD05} have addressed a similar adversarial setup, but in which the environment designs a worst-case disturbance to the dynamics of the agent, and have addressed this setup using $H_\\infty$ control. \n\n\n\n\n\\paragraph{Paper Organization:} We introduce the notations and  define the maxmin learning problem in Section~\\ref{sec:prelim}.\nWe provide three different ways to  \ndefine the set of consistent reward functions in Section~\\ref{sec:reward SO}, and present the ellipsoid-based exact algorithm and its analysis in Section~\\ref{sec:exact alg}. The FPL-based algorithm and its analysis are in Section~\\ref{sec:exact FPL}, followed by experimental results in Section~\\ref{sec:exp.}.\n\n\\section{Model-Based Learning using an Exact MDP Solver}~\\label{sec:model based}\n\\subsection{Exact max-min solution with dual methods for MDPs}\n\nGiven a policy $\\pi$, we can calculate the occupancy measure $x^\\pi$ for all state and action pair as\n\\begin{equation}\n\tx^\\pi_{sa} = E [\\sum_{t=0}^\\infty \\gamma^t \\mathbbm{1}_{s_t = s, a_t = a} | \\pi , P_{sa}].\n\\end{equation}\nThis measures the expected and discounted number of visits to each state-action pair following the policy $\\pi$. It requires the knowledge of the transition probabilities. The expectation has also taken into account the initial state distribution.The policy's expected reward can be represented as a linear function over the occupancy measure \n\\begin{equation}\n\tV(\\pi) := \\sum_{s, a}R_{sa} x^\\pi_{s a}.\n\\end{equation}\nSimilarly, we can express $\\mu(\\pi)$ as:\n\\begin{equation}\n\t\\Psi(\\pi) := \\sum_{s} \\phi(s) \\sum_{a} x^\\pi_{s a}\n\\end{equation}\nThus, we can replace the constraints on $\\mu$ in our max-min problem with the {\\it Bellman flow constraints} \\cite{Puterman14}, and rewrite the max-min LP as following.\n\n\\begin{align*}\n &\\max z &\\\\\n &z \\leq \\mu \\cdot w,  \\qquad \\forall w \\in P_R &\\\\\n&\\mu = \\sum_{s} \\phi(s) \\sum_{a} x_{s a} &\\\\\n& \\sum_a x_{sa} = \\text{Pr}(s_0 = s) + \\gamma \\sum_{s',a}x_{s'a} P_{sa} \\qquad \\forall s &\\\\\n& x_{sa} \\geq 0 &\\\\\n\\end{align*}\n\n\\section{Maxmin Learning using an Approximate MDP Solver}\\label{sec:approx FPL}\n\n\n\nIn the previous sections, we assume that we have access to an MDP solver $\\textsc{ALG}$ that solves any MDP $M$ optimally in time polynomial in $\\langle M\\rangle$. However, in practice, solving large-size MDPs, e.g. continuous control problems, exactly could be computationally expensive or infeasible. Our FPL-based algorithm also works in cases where we can only solve MDPs approximately. \n\n\n{Suppose we are given access to an additive FPTAS $\\widehat{\\textsc{ALG}}$ for solving MDPS. More specifically, $\\widehat{\\textsc{ALG}}$ finds in time polynomial in $\\langle M\\rangle, 1\/\\eta$ a solution $(\\pi^*_{\\eta}, \\mu^*_{\\eta})$, such that $\\mathbb{E}_{s_0\\sim D}[V^{\\pi^*_{\\eta}}(s_0) | M]\\geq \\max_\\pi \\mathbb{E}_{s_0 \\sim D}[V^\\pi(s_0) | M] - \\eta $. Notice that  the weights of $M$'s reward function have $L_1$-norm $L$.}\n\nWe face two challenges when we replace $\\textsc{ALG}$ with $\\widehat{\\textsc{ALG}}$: (i) we can no longer find the best policy $\\mu_t$ with respect to all the previous weights plus the perturbation in every iteration, and (ii) we no longer have a separation oracle for $P_R$, as the $SO_R$ (Algorithm~\\ref{alg:SO}) relies on the MDP solver when $P_R$ is implicitly specified by the expert's policy. It turns out (i) is not hard to deal with, as the FPL algorithm is robust enough to work with only an approximate leader. (ii) is much more subtle. We design a new algorithm and use it as a proxy for the polytope $P_R$. We call this new algorithm a weird separation oracle (following the terminology in~\\cite{CaiDW13a}) as the points it may accept do not necessarily form a convex set, even though it does accept all points in $P_R$. It may seem at first not clear at all why such a weird separation oracle can help us. However, we manage to prove that just with this weird separation oracle, we can still compute an approximate minimizing weight vector $w_t$ in $P_R$ in every iteration (Step 5 of Algorithm~\\ref{alg:FPL maxmin}). Combining this with our solution for challenge (i), we can still compute an approximately maxmin policy with essentially the same performance as in Algorithm~\\ref{alg:FPL maxmin}. \n\n\\begin{theorem}\\label{thm: approx FPL maxmin}\n\n{If we replace the exact MDP solver $\\textsc{ALG}$ with an approximate solver $\\widehat{\\textsc{ALG}}$ in step 4 of Algorithm ~\\ref{alg:FPL maxmin},} then for any $\\xi\\in (0,1\/2)$ and any $c>0$, with probability at least $1-2\\xi$, Algorithm~\\ref{alg:FPL maxmin} finds a policy $\\pi$ after $T$ rounds of iterations such that its expected reward under any weight from $P_R$ is at least $\\max_{\\mu\\in P_F}\\min_{w\\in P_R} \\mu\\cdot w-\\frac{k^2\\left(6+4\\sqrt{\\ln 1\/\\xi}\\right)}{\\sqrt{T}} - 2c $. In every iteration, Algorithm~\\ref{alg:FPL maxmin} makes one query to $\\widehat{\\textsc{ALG}}$ and a polynomial number of queries to $SO_R$. In particular, for every query to $\\widehat{\\textsc{ALG}}$, we first divide the input by $2T$ then feed it to $\\widehat{\\textsc{ALG}}$ and ask for a policy that is at most $c\/2T$ worse than the optimal one.\n\\end{theorem}\n\n\n\n\nThe proof of Theorem~\\ref{thm: approx FPL maxmin} is similar to the proof of Theorem~\\ref{thm:FPL maxmin}. We use the bounds provided by Lemma~\\ref{lem:approximateFPL} instead of Theorem~\\ref{thm:FPL}, and change the RHS in Equation~\\eqref{eq:agent regret} from $-B$ to $-k^2\\sqrt{T}\\left(3+2\\sqrt{\\ln 1\/\\xi}\\right) -2cT$ accordingly. The rest of the proof remains the same. \n\nAssume we have a procedure $M_\\eta$ for $\\eta$-approximating linear programs over the decision set $D$ such that for all $s\\in \\mathbb{R}^k$, \n\\[\ts \\cdot M_\\eta(s) \\geq \\text{argmax}_{d\\in D} s \\cdot d - \\eta.\\]\n\n\\begin{lemma}[Follow the Approximate Perturbed Leader]~\\cite{Ben-TalHKM15}\\label{lem:approximateFPL} Let $d_1,\\ldots,d_T$ be a sequence of decision made by an $\\eta$-approximating procedure $M_\\eta$ such that $d_t = M_\\eta(\\sum_{i=1}^{t-1} s_i + p_t)$. Then\n$$\\mathbb{E}\\left[\\sum_{t=1}^T d_t\\cdot s_t-\\max_{d\\in D} \\sum_{t=1}^T d\\cdot s_t\\right]\\geq -\\delta\\cdot C_1C_2 T-\\frac{2C_3}{\\delta} - 2\\eta T .$$ The definition of constants $C_1$, $C_2$ and $C_3$ are the same as in Theorem~\\ref{thm:FPL}.\nMoreover, for all $\\xi\\geq 0$, with probability at least $1-\\xi$, the actual accumulative reward under any adaptive adversary satisfies, \\begin{align*}\n&\\sum_{t=1}^T d_t\\cdot s_t-\\max_{d\\in D} \\sum_{t=1}^T d\\cdot s_t\\geq\\\\\n &~~~~~~~~~~~~~~~~~~ -\\delta\\cdot C_1C_2 T-\\frac{2C_3}{\\delta}-2C_2\\sqrt{T\\ln \\frac{1}{\\xi}} - 2\\eta T .\n \\end{align*}\n \\label{lemma:approx FPL}\n\\end{lemma}\n\nAn astute reader may have noticed that in the analysis above, we used the same separation oracle $SO_R$ as in section~\\ref{sec:reward SO}. However, in the case when the separation oracle for the reward polytope is implicitly specified by an expert policy, $SO_R$ queries the MDP solver in step 1 of algorithm~\\ref{alg:SO}. If we do not have an exact MDP solver $\\textsc{ALG}$, it is not clear how we can define a separation oracle for polytope $P_R$. We use Algorithm~\\ref{alg:WSOR} as an proxy to polytope $P_R$.\n\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\STATE Let $\\mu_{w'}^{(\\eta)}:=\\widehat{\\textsc{ALG}}(w',\\eta) $. It is the feature vector of the policy computed by the approximate MDP solver $\\widehat{\\textsc{ALG}}$ with accuracy $\\eta$, and $\\mu_{w'}^{(\\eta)}\\cdot w' \\geq \\max_{\\mu\\in P_R} \\mu\\cdot w' - \\eta$. \n\\IF {$\\mu_{w'}^{(\\eta)}\\cdot w'> \\mu_E \\cdot w' + \\epsilon$} \\STATE output ``NO'' , and \n\t $ \\left(\\mu_E- \\mu_{w'}^{(\\eta)} \\right)\\cdot w + \\epsilon\\geq 0$ as the separating hyperplane, \n\t since for all $w \\in P_R, \\mu_E\\cdot w \\geq \\mu_{w'} ^{(\\eta)}\\cdot w -\\epsilon$.\n\\ELSE \\STATE output ``YES''.\n\\ENDIF\n\\end{algorithmic}\n\\caption{{\\sf Weird Separation Oracle $WSO^\\eta_R$ for the reward polytope $P_R$}}\n\\label{alg:WSOR}\n\\end{algorithm}\n\nWe call $WSO^\\eta_R$ a weird separation oracle for for the reward polytope for $P_R$, because the set of $w'$ that it will accept is not necessarily convex. For example, the following may happen. First, we query two points $w_1$ and $w_2$ that are close to each other. Both are accepted by $WSO_R^\\eta$, and it happens to be the case that $\\widehat{\\textsc{ALG}}(w_1,\\eta)$ and $\\textsc{ALG}(w_2,\\eta)$ are both $\\eta$ away from the optimal solutions. Now we query $w_3 = (w_1+w_2)\/2$, and run $WSO^\\eta_R$. Luckily (or unfortunately) $\\widehat{\\textsc{ALG}}(w_3,\\eta)$ is close to optimal, and $w_3$ is rejected.  \n\n\\begin{lemma}\\label{lem:WSOR}\n\tFor any linear optimization problem, we can construct a polynomial time algorithm based on the ellipsoid-method that queries $WSO_R^{\\eta}$, such that it finds a solution that is at least as good as the best solution in polytope ${P = \\{w | w \\cdot \\mu_E \\geq w\\cdot  \\widehat{\\textsc{ALG}}(w',\\eta)-\\epsilon, \\forall w' , \\|w'\\|_1 } \\leq L\\}$, although our solution does not necessarily lie in $P$. \n\t\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:WSOR}\nWe only sketch the proof here. Solving a linear optimization can be converted into solving a sequence of feasibility problems by doing  binary search on the objective value. We show that for any objective value $\\alpha$, as long as there is a solution $x\\in P$ whose objective value $c\\cdot x\\geq \\alpha$, our algorithm also finds a solution $x'$ such that $c\\cdot x'\\geq \\alpha$. First, imagine we have a separation oracle for $P$, and the ellipsoid method needs to run $N$ iterations to determine whether there is a solution in $P$ whose objective value is at least $\\alpha$. The correctness of ellipsoid method guarantees that if it hasn't found any solution after $N$ iterations, then the intersection of the  halfspace $c\\cdot x\\geq \\alpha$ and $P$ is empty. The reason is that if the intersection is not empty it must have volume at least $r$, and the ellipsoid method maintains an ellipsoid that contains the intersection of the halfspace $c\\cdot x\\geq \\alpha$ and $P$ and shrinks the volume of the ellipsoid in every iteration. After $N$ iterations the ellipsoid already has volume less than $r$\n\nOur algorithm also runs the ellipsoid method for $N$ iterations. In each iteration, we first check the constraint $c\\cdot x\\geq \\alpha$, if not satisfied, we output this constraint as the separating hyperplane. If it is satisfied, instead of querying the real separation oracle for $P$, we query $WSO^\\eta_R$. If the answer is ``YES\", we have found a solution $x$ such that $c\\cdot x\\geq \\alpha$. If the answer is ``NO\", clearly this query point is not in $P$, and the outputted separating hyperplane contains the intersection of the halfspace $c\\cdot x\\geq \\alpha$ and $P$. Therefore, whenever our algorithm accepts a point, it must have objective value higher than $\\alpha$. Otherwise, the shrinking ellipsoid still contains the intersection of the halfspace $c\\cdot x\\geq \\alpha$ and $P$. If our algorithm terminates after $N$ iterations without accepting point, we know that the intersection between the  halfspace $c\\cdot x\\geq \\alpha$ and $P$ is empty as the volume of the ellipsoid after $N$ iterations is already too small.\n\n\\end{prevproof}\n\n\nConsider the following three polytopes:\n\n\\noindent (i) $P_R := \\left\\{w\\ |\\ w\\cdot\\mu_E \\geq w\\cdot \\mu -\\epsilon,\\ \\forall \\mu \\in P_F\\right \\}$\n\n\\vspace{.1in}\n\\noindent (ii) $P = \\{{w|w \\cdot \\mu_E \\geq w\\cdot \\widehat{\\textsc{ALG}}(w',\\eta) -\\epsilon, \\forall w' , \\|w'\\|_1 \\leq L}\\}$\n\n\\vspace{.1in}\n\\noindent (iii) $P_R^{(\\epsilon + \\eta)}:= \\{w\\ |\\ w\\cdot\\mu_E \\geq w\\cdot \\mu -\\epsilon-\\eta,  \\forall \\mu \\in P_F\\}$. \n\n\\begin{fact}\n\t$P_R\\subseteq P$.\n\t\\label{fact:PRinP}\n\\end{fact}\n\\begin{fact}\n\t$WSO_R^\\eta$ only accepts points that are in $P_R^{(\\epsilon + \\eta)}$.\n\t\\label{fact:PRee}\n\\end{fact}\n\\begin{proof}\n\tSuppose $w\\notin P_R^{(\\epsilon+\\eta)}$, then clearly $w\\cdot \\widehat{\\textsc{ALG}}(w,\\eta)\\geq \\max_{\\mu\\in P_F} w\\cdot \\mu -\\epsilon-\\eta>w\\cdot \\mu_E$. Hence, $WSO_R^{\\eta}$ will not accept $w$.\n\\end{proof}\n\\begin{lemma}\n\tFor all $w$ in $P_R^{(\\epsilon+\\eta)}$,  $w\\cdot\\frac{\\epsilon}{\\epsilon+\\eta}$ is in $P_R$.\n\t\\label{lemma:shrinkw}\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lemma:shrinkw}\nFrom the definition of $P_R^{(\\epsilon+\\eta)}$, multiply both side of the inequality with $\\frac{\\epsilon}{\\epsilon+\\eta}$, and let $w' = w\\cdot\\frac{\\epsilon}{\\epsilon+\\eta}$, $w'$ is in $P_R$.\n\\end{prevproof}\n\n\\begin{theorem}\nFor any $c>0$ and $\\xi\\in (0,1\/2)$, with probability at least $1-2\\xi$, Algorithm~\\ref{alg:FPL maxmin} finds a policy $\\pi$ after $T$ rounds of iterations such that its expected reward under any weight from $P_R$ is at least $\\max_{\\mu\\in P_F}\\min_{w\\in P_R} \\mu\\cdot w-\\frac{k^2\\left(6+4\\sqrt{\\ln 1\/\\xi}\\right)}{\\sqrt{T}} - 4c$. In every iteration, Algorithm~\\ref{alg:eps FPL maxmin} makes one query to $\\textsc{ALG}_\\eta$ and a polynomial number of queries to Algorithm~\\ref{alg:WSOR}.\n\t\\label{thm:FPL-WSOR}\n\\end{theorem}\n\nNow, we are ready to describe the algorithm using only access to $WSO_R^{\\eta}$.\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\INPUT $T$: the number of iterations\n\\STATE Set $\\delta:=\\frac{1}{k\\sqrt{T}}$, where $||w||_1\\leq L$ for all $w\\in P_R$. Set $\\eta_1:=\\frac{c}{2T}$ and $\\eta_2:=\\frac{c\\epsilon}{2k^2T-c}$.\n\\STATE Arbitrarily pick some policy $\\pi_1$ and compute $\\mu_1 \\in P_F$. Arbitrarily pick some reward weights $w_1$, and set $t = 1$.\n\\WHILE{$t\\leq T$}{\n\\STATE Let policy $\\pi_t$ and $\\mu_t = \\Psi(\\pi_t)$ be the output of $\\widehat{\\textsc{ALG}}\\left( \\left(\\sum_{i=1}^{t-1} w_i+p_t\\right )\/T, \\eta_1 \\right)$\n, where $p_t$ is drawn uniformly from $[0,1\/\\delta]^k$.\n\\STATE Use our algorithm in Lemma~\\ref{lem:WSOR} with $WSO_R^{\\eta_2}$ to solve $\\mathop{\\min} w^T (\\sum_{i=1}^{t-1}\\mu_t + q_t)$, where $q_t$ is drawn uniformly from $[0,1\/\\delta]^k$. Let $w'_t$ be the solution and set $w_t$ to be $w'_t\\cdot\\frac{\\epsilon}{\\epsilon+\\eta_2}$.\n\\STATE $t:= t+1$.}\n\\ENDWHILE\n\\STATE Output the randomized policy $\\frac{1}{T}\\cdot \\sum_{t=1}^T \\pi_t$.\n\\end{algorithmic}\n\\caption{{\\sf Finding the Maxmin Policy using Follow-the-Perturbed-Leader (FPL)}}\n\\label{alg:eps FPL maxmin}\n\\end{algorithm}\n\n\n\n\\begin{prevproof}{Theorem}{thm:FPL-WSOR}\nAt each time step $t$, using $WSO_R$,  Algorithm~\\ref{alg:eps FPL maxmin} step 5 outputs a $w_t$. By Lemma~\\ref{lem:WSOR} and Fact~\\ref{fact:PRinP}, \n$$w'_t\\cdot\\left(\\sum_{i=1}^{t-1}\\mu_i + q_t\\right) \\leq \\min_{w \\in P_R}w\\cdot\\left(\\sum_{i=1}^{t-1}\\mu_i +q_t\\right).$$\nBy Lemma~\\ref{lemma:shrinkw} and Fact~\\ref{fact:PRee}, \n\\begin{align*}\n\t&w_t \\cdot\\left(\\sum_{i=1}^{t-1}\\mu_i + q_t\\right) = \\frac{\\epsilon}{\\epsilon+\\eta_2}\\cdot w_t' \\cdot\\left(\\sum_{i=1}^{t-1}\\mu_i + q_t\\right)\\\\\n\t \\leq & \\min_{w \\in P_R }w\\cdot\\left(\\sum_{i=1}^{t-1}\\mu_i +q_t\\right)\n+ \\frac{2k^2T}{\\epsilon+\\eta_2},\n\\end{align*}  where we used the fact that $$ -w_t\\left(\\sum_{i=1}^{t-1}\\mu_i + q_t\\right) \\leq  2k^2T.$$\nSince $c = \\frac{2\\eta_2 k^2T}{\\epsilon+\\eta_2}$, we can use Lemma~\\ref{lemma:approx FPL} and replace the RHS in Equation~\\eqref{eq:designer regret} that was used in the proof of Theorem~\\ref{thm:FPL maxmin} to  $-k^2\\sqrt{T}\\left(3+2\\sqrt{\\ln 1\/\\xi}\\right) - 2cT$. The analysis for $\\mu_t$ remains the same as in Thoerem~\\ref{thm: approx FPL maxmin}.\n\\end{prevproof}\n\n\n\n\n \n\n\n\\section{Maxmin Learning using an Exact MDP Solver}\\label{sec:alg}\nIn this section, we show how to design maxmin learning algorithms. Our algorithm only interacts with the MDP through the MDP solver, which can be either model-based or model-free. Our first algorithm solves the maxmin learning problem exactly using the ellipsoid method. Despite the fact that the ellipsoid method has provable worst-case polynomial running time, it is known to be inefficient sometimes in practice. Our second algorithm is an efficient iterative method based on the online learning algorithm -- \\emph{follow-the-perturbed-leader (FPL)}.\n\n\\subsection{Ellipsoid-Method-Based Solution}\\label{sec:exact alg}\n\\begin{theorem}\\label{thm:exact maxmin}\n\tGiven a polynomial time separation oracle $SO_R$ for the consistent reward polytope $P_R$ and an exact polynomial time MDP solver $\\textsc{ALG}$, we have a polynomial time algorithm such that for any MDP without the reward function $M\\backslash R$, the algorithm computes the maxmin policy $\\pi^*$ with respect to $M\\backslash R$ and $P_R$. \\end{theorem}\n\t\n\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\\centering\n\\colorbox{MyGray}{\n\\begin{minipage}{0.45\\textwidth} \n\\vspace{-10pt}\n\\begin{equation*}\n\\begin{split}\n\\max\\quad \\displaystyle &z \\\\\n \\text{subject to}\\quad \\displaystyle &z \\leq \\mu \\cdot w, ~~~ \\forall w \\in P_R\\\\\n&\\mu \\in P_F\n\\end{split}\t\n\\end{equation*}\n\\end{minipage}}\n\\caption{Maxmin Learning LP.}\n\\label{fig:maxmin LP}\n\\vspace{-10pt}\n\\end{wrapfigure}\nThe plan is to first solve the maxmin learning problem in the feature space then convert it back to the policy space. Solving the maxmin learning problem in the feature space is equivalent to solving the linear program in Figure~\\ref{fig:maxmin LP}.\n\n\nThe challenges for solving the LP are that (i) it is not clear how to check whether $\\mu$ lies in the polytope $P_F$, and (ii) there are seemingly infinitely many constraints of the type $z\\leq \\mu\\cdot w$ as there are infinitely many $w\\in P_R$. Next, we show that given an exact MDP solver $\\textsc{ALG}$, we can design a polynomial time separation oracle for the set of feasible variables $(\\mu,z)$ of LP~\\ref{fig:maxmin LP}. With this separation oracle, we can apply the ellipsoid method (see Theorem~\\ref{thm:ellipsoid} in the supplementary material) to solve LP~\\ref{fig:maxmin LP} in polynomial time.\n\n\nFirst, we design a separation oracle for polytope $P_F$ by invoking a seminal result from optimization -- the equivalence between separation and optimization.\n\n\\begin{lemma}[Separation $\\equiv$ Optimization]~\\cite{GrotschelLS81,KarpP82}\\label{lem:sep-opt}\nConsider any convex polytope $P = \\{x : Ax \\leq b\\}\\in \\mathbb{R}^d$ and the following two problems:\n\\begin{itemize}\n\t\\item \\textbf{Linear Optimization:} given a linear objective $c \\in \\mathbb{R}^d$, compute $x^* \\in \\text{argmax}_{x\\in P} c\\cdot x$\n\t\\item \\textbf{Separation:} given a point $y \\in\\mathbb{R}^d$, decide that $y \\in P$, or else find $h \\in \\mathbb{R}^d$ s.t. $h\\cdot x < h\\cdot y$, $\\forall x \\in P$.\n\\end{itemize}\nIf $P$ can be described implicitly\nusing $\\langle P\\rangle$ bits, then the separation problem is solvable in $\\text{poly}(\\langle P\\rangle, d,\\langle y\\rangle)$ time for P  if and\nonly if the linear optimization problem is solvable in $\\text{poly}(\\langle P\\rangle, d,\\langle c\\rangle)$\ntime.\\end{lemma}\n\t\n\t It is not hard to see that if one can solve the separation problem, one can construct a separation oracle in polynomial time and apply the ellipsoid method to solve the linear optimization problem. The less obvious direction in the result above states that if one can solve the linear optimization problem, one can also use it to construct a separation oracle. The equivalence between these two problems  turns out to have profound implications in combinatorial optimization and has enabled numerous polynomial time algorithms for many problems that are difficult to solve otherwise.\n\t\\begin{algorithm}[H]\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\INPUT $(\\mu',z')\\in\\mathbb{R}^{k+1}$\n\t\t\t\\STATE Query $SO_F(\\mu')$.\n\t\t\t\\IF{$\\mu' \\notin P_F$} \\STATE{output ``\\textsc{No}'' and output the same separating hyperplane as outputted by $SO_F(\\mu')$.}\n\t\t\t\\ELSE\\STATE Let $w^* \\in \\mathop{\\text{argmin}}_{w \\in P_R} \\mu' \\cdot w$ and $V=\\mu' \\cdot w^*$. This requires solving a linear optimization problem over $P_R$ using the ellipsoid method with the separation oracle $SO_R$.\n\n\t\t\t\\IF{$z' \\leq V$}\\STATE {output ``\\textsc{Yes}''}\n\t\t\t\\ELSE \\STATE output ``\\textsc{No}'', and a separating hyperplane $z \\leq \\mu \\cdot w^*$, as $z'> \\mu'\\cdot w^*$ and all feasible solutions of LP~\\ref{fig:maxmin LP} respect this constraint.\n\t\t\t\\ENDIF\n\t\t\t\\ENDIF\n\t\t\\end{algorithmic}\n\t\t\\caption{{\\sf Separation Oracle for the feasible $(\\mu, z)$ in LP~\\ref{fig:maxmin LP}}}\n\t\t\\label{alg:SO maxmin}\n\t\\end{algorithm}\n\n\n\nOur goal is to design a polynomial time separation oracle for the polytope $P_{F}$. The key observation is that the linear optimization problem over polytope $P_{F}$: $\\max_{\\mu\\in P_F} w\\cdot \\mu$ is exactly the same as solving the MDP with reward function $R(\\cdot)=w\\cdot \\phi(\\cdot)$. Therefore, we can use the MDP solver to design a polynomial time separation oracle for $P_F$.\n\n\\begin{lemma}\\label{lem:SO_f}\n\tGiven access to an MDP solver $\\textsc{ALG}$ that solves any MDP $M$ in time polynomial in $\\langle M\\rangle$, we can design a separation oracle $SO_F$ for $P_F$ that runs in time polynomial in $\\langle M\\backslash R \\rangle$, $\\langle \\phi \\rangle$, $k$, and the bit complexity of the input~\\footnote{Note that  $SO_F$ only depends on the bit complexity of $M\\backslash R$, but not the actual model of $M\\backslash R$ such as the distributions of the initial states or  the transition probabilities. We only require access to $ALG$ and an upper bound of $\\langle M\\backslash R \\rangle$.}.\n\\end{lemma}\n\n\n\n\n\n\\notshow{\t\t\\begin{figure}[h!]\n\t\\colorbox{MyGray}{\n\t\\begin{minipage}{0.46\\textwidth} {\t\n\t\t$WSO(\\vec{\\pi})=$\n\t\\begin{itemize}\t\n\t\\item ``\\textbf{Yes}'' if the ellipsoid algorithm with $N$ iterations\\footnote{The appropriate choice of $N$ for our use of $WSO$ is provided in Corollary~\\ref{cor:N} of Section~\\ref{sec:runtime}. The only place that requires an appropriate choice of $N$ is the proof of Lemma~\\ref{lem:convhull}.} outputs ``infeasible'' on the following problem:\n\t\t\t \n\t\t\t  \\underline{\\textbf{variables:}} $\\vec{w}, t$;\n\t\t\t \n\t\t\t \\underline{\\textbf{constraints:}}\n\t\t\t\t\\begin{itemize}\n\t\t \t\t\\item $\\vec{w} \\in [-1,1]^d;$\n\t\t \t\t\\item $t - \\vec{\\pi} \\cdot \\vec{w} \\leq -\\delta$;\\footnote{The appropriate choice of $\\delta$ for our use of $WSO$ is provided in Lemma~\\ref{lem:delta} of Section~\\ref{sec:runtime}. The only place that requires an appropriate choice of $\\delta$ is the proof of Lemma~\\ref{lem:convhull}.}\n\t\t \t\t\\item $\\widehat{WSO}(\\vec{w},t) = $\n\t\t \t\t\\begin{itemize}\n\t\t \t\t\\item ``yes'' if $t \\geq \\mathcal{A}(\\vec{w}) \\cdot \\vec{w}$;\\footnote{Notice that the set $\\{(\\vec{w},t)|\\widehat{WSO}(\\vec{w},t) =$ ``Yes''$\\}$ is not necessarily convex or even connected.}\n\t\t \t\t\\item the violated hyperplane $t' \\geq \\mathcal{A}(\\vec{w})\\cdot \\vec{w}'$ otherwise.\n\t\t \t\t\\end{itemize}\n\t\t\t\t \t\t\\end{itemize}\n\t\\item If a feasible point $(t^*,\\vec{w}^*)$ is found, output the violated hyperplane $\\vec{w}^* \\cdot \\vec{\\pi}' \\leq t^*$.\n\t\\end{itemize}\t\t}\n\t\t\t\\end{minipage}} \\caption{A ``weird'' separation oracle.}\\label{fig:WSO}\n\t\\end{figure}\n}\n\n\n\nWith $SO_F$, we first design a polynomial time separation oracle for checking the feasible $(z,\\mu)$ pairs in LP~\\ref{fig:maxmin LP} (Algorithm~\\ref{alg:SO maxmin}). With the separation oracle, we can solve LP~\\ref{fig:maxmin LP} using the ellipsoid method. The last difficulty is that the optimal solution only gives us the maxmin feature vector instead of the corresponding maxmin policy. We use the following nice property of $SO_F$ to convert the optimal solution in the feature space to the policy space. See Section~\\ref{sec:missingproof} in the supplementary material for intuition behind Lemma~\\ref{lem:convhull}.\n\n\\begin{lemma}~\\cite{GrotschelLS81,KarpP82,CaiDW13a}\\label{lem:convhull}\n If $SO_F(\\mu) = $ ``\\textsc{Yes}'', there exists a set, $C$, of weights ${w}\\in \\mathbb{R}^k$ such that $SO_F$ has queried the MDP solver $\\textsc{ALG}$ on reward function $w\\cdot \\phi(\\cdot)$ for every $w\\in C$. Let $(\\pi_w,\\mu_w)$ be the output of $\\textsc{ALG}$ on weight $w$, then $\\mu$ lies in the convex hull of  $\\{\\mu_w | {w} \\in C\\}$. \n\t\\end{lemma}\n\t\n\n\n\n\n\\notshow{\\begin{enumerate}\n\\item Check if $\\mu' \\in P_F$ by querying the separation oracle of $P_F$, if not, directly output ``NO'' and output the same separating hyperplane as output by $P_F$.\n\\item Let $w^* = \\mathop{\\text{argmin}}_{w \\in P_R} \\mu' \\cdot w$ and $V=\\mu' \\cdot w^*$. \n\\item If $z' \\leq V$, output ``YES'';\n\\item Else output ``NO'', and a separating hyperplane $z \\leq \\mu \\cdot w^*$.\n\\end{enumerate}}\n\n\n\\begin{prevproof}{Theorem}{thm:exact maxmin}\nIt is not hard to see that Algorithm~\\ref{alg:SO maxmin} is a valid polynomial time separation oracle for the feasible $(\\mu,z)$ pairs in LP~\\ref{fig:maxmin LP}. Hence, we can solve LP~\\ref{fig:maxmin LP} in polynomial time with the ellipsoid method with access to Algorithm~\\ref{alg:SO maxmin}. Next, we show how to convert the optimal solution $\\mu^*$ of LP~\\ref{fig:maxmin LP} to the corresponding maxmin optimal policy $\\pi^*$. Here, we invoke Lemma~\\ref{lem:convhull}. We query $SO_F$ on $\\mu^*$ and we record all weights $w$ that $SO_F$ has queried the MDP solver $\\textsc{ALG}$ on. Let $C=\\{w_1,\\ldots, w_\\ell\\}$ be all the queried weights. As $SO_F$ is a polynomial time algorithm, $\\ell$ is also polynomial in the input size. By Lemma~\\ref{lem:convhull}, we know that $\\mu$ is in the convex hull of $\\left(\\{\\mu_w | {w} \\in C\\}\\right)$, which means there exists a set of nonnegative numbers $p_1,\\ldots, p_\\ell$, such that $\\sum_{i=1}^\\ell p_i =1$ and $\\mu^*=\\sum_{i=1}^\\ell p_i\\cdot \\mu_{w_i}$. Clearly, the discounted accumulated feature value of the randomized policy $\\sum_{i=1}^\\ell p_i\\cdot \\pi_{w_i}$ equals to $\\sum_{i=1}^\\ell p_i\\cdot \\Psi(\\pi_{w_i})=\\sum_{i=1}^{\\ell} p_i\\cdot\\mu_{w_i}=\\mu^*$. We can compute the $p_i$s in poly-time via linear programming and  $\\sum_{i=1}^\\ell p_i\\cdot \\pi_{w_i}$ is the maxmin policy.\n\\end{prevproof}\n\n\n\n\n\n\\subsection {Finding the Maxmin Policy using Follow the Perturbed Leader}\\label{sec:exact FPL}\nThe exact algorithm of Theorem~\\ref{thm:exact maxmin} may be computationally expensive to run, as the separation oracle $SO_F$ requires running the ellipsoid method to answer every query, and on top of that we need to run the ellipsoid method with queries to $SO_F$. In this section, we propose a simpler and faster algorithm that is based on the algorithm -- \\emph{follow-the-perturbed-leader (FPL)}~\\cite{KalaiV05}.\n\n\\begin{theorem}\\label{thm:FPL maxmin}\n\n\tFor any $\\xi\\in (0,1\/2)$, with probability at least $1-2\\xi$, Algorithm~\\ref{alg:FPL maxmin} finds a policy $\\pi$ after $T$ rounds of iterations such that its expected reward under any weight from $P_R$ is at least $\\max_{\\mu\\in P_F}\\min_{w\\in P_R} \\mu\\cdot w-\\frac{k^2\\left(6+4\\sqrt{\\ln 1\/\\xi}\\right)}{\\sqrt{T}}$. In every iteration, Algorithm~\\ref{alg:FPL maxmin} makes one query to $\\textsc{ALG}$ and $O\\left(k^2 \\left((\\log k)^2+\\left((b+\\langle\\phi \\rangle) (|\\mathcal{A}||\\mathcal{S}|+k)+\\log T \\right)^2\\right)\\right)$ queries to $SO_R$, where $b$ is an upper bound on the number of bits needed to specify the transition probability\t$P_{sa}$ for any state $s$ and action $a$\n\\end{theorem}\nFPL is a classical online learning algorithm that solves a problem where a series of decisions $d_1, d_2, ...$ need to be made. Each $d_i$ is from a possibly infinite set $D \\subseteq \\mathbb{R}^n$. The state $s_t \\in \\mathbb{S} \\subseteq \\mathbb{R}^n$ at step $t$ is observed after the decision $d_t$. The goal is to have the total reward $\\sum_t d_t \\cdot s_t$ not far from the best expert's reward with hindsight, that is $\\max_{d\\in D} \\sum_t{d \\cdot s_t}$. The FPL algorithm guarantees that after $T$ rounds, the regret $\\sum_t d_t \\cdot s_t-\\max_{d\\in D} \\sum_t{d \\cdot s_t}$ scales linearly in $\\sqrt{T}$. This guarantee holds for both oblivious and adaptive adversary, and the bound holds both in expectation and with high probability (see Theorem~\\ref{thm:FPL} in Section~\\ref{sec:missingproof} of the supplementary material for the formal statement). \n\n\n\t\\begin{algorithm}[H]\n\t\t\\begin{algorithmic}[1]\n\\INPUT $T$: the number of iterations\n\\STATE Set $\\delta:=\\nicefrac{1}{k\\sqrt{T}}$\n\\STATE Arbitrarily pick some policy $\\pi_1$, compute $\\mu_1 \\in P_F$. Arbitrarily pick some reward weights $w_1$, and set $t = 1$.\n\\WHILE{$t\\leq T$}{\n\\STATE Use $\\textsc{ALG}$ to compute the optimal policy $\\pi_t$ and $\\mu_t = \\Psi(\\pi_t)$ that maximizes the expected reward under reward function $ \\left(\\sum_{i=1}^{t-1} w_i+p_t\\right )\\cdot \\phi(\\cdot)$, where $p_t$ is drawn uniformly from $[0,1\/\\delta]^k$.\n\\STATE Let $w_t := \\mathop{\\text{argmin}}_{w\\in P_R} w^T (\\sum_{i=1}^{t-1}\\mu_t + q_t)$, where $q_t$ is drawn uniformly from $[0,1\/\\delta]^k$.\n\\STATE $t:= t+1$.}\n\\ENDWHILE\n\\STATE Output the randomized policy $\\frac{1}{T}\\cdot \\sum_{t=1}^T \\pi_t$.\n\\end{algorithmic}\n\\caption{{\\sf FPL Maxmin Learning}}\n\\label{alg:FPL maxmin}\n\\end{algorithm}\n\n\nFPL falls into a large class of algorithms that are called low-regret algorithms, as the regret grows sub-linearly in $T$. It is well known that low-regret algorithms can be used to solve two-player zero-sum games approximately. The maxmin problem we face here can also be modeled as a two-player zero-sum games. One player is the agent whose strategy is a policy $\\pi$, and the other player is the reward designer whose strategy is a weight $w\\in P_R$. The agent's payoff is the reward that it collects using policy $\\pi$, which is $\\Psi(\\pi)\\cdot w$, and the designer's payoff is $-\\Psi(\\pi)\\cdot w$. Finding the maxmin strategy for the agent is equivalent to finding the maxmin policy. One challenge here is that the numbers of strategies for both players are infinite. Even if we only consider the pure strategies which correspond to the extreme points of $P_F$ and $P_R$, there are still exponentially many of them.\nMany low-regret algorithms such as multiplicative-weights-update requires explicitly maintaining a distribution over the pure strategies, and update it in every iteration. In our case, these algorithms will take exponential time to finish just a single iteration. This is the reason why we favor the FPL algorithm, as the FPL algorithm only requires finding the best policy giving the past weights, which can be done by the MDP solver $\\textsc{ALG}$. We also show that a similar result holds even if we replace the exact MDP solver with an additive FPTAS $\\widehat{\\textsc{ALG}}$.\n{The proof of Theorem~\\ref{thm:FPL maxmin} can be found in Section~\\ref{sec:missingproof} in the supplementary material. Our generalization to cases where we only have access to $\\widehat{\\textsc{ALG}}$ is postponed to Section~\\ref{sec:approx FPL} in the supplementary material.} \n\n\n\n\n\\notshow{The algorithm works as follows. In the first round, we choose arbitrary $\\mu_1 \\in P_F$, and $w_1 \\in P_R$. Then for every $t > 1$,  choose $\\mu_t$ using FPL given all $w_{1:t-1}$. That is, \n\n\\begin{equation}\\label{def:FTPL}\n\\mathop{\\text{argmax}}_{\\mu \\in P_F} \\mu\\cdot (\\mathop{\\sum}_{j=1}^{t-1} w_j + p_t), \n\\end{equation}\n\nwhere $p_t$ is drawn from a distribution that is defined by the FPL. Once the agent has chosen the policy and thus the features, the reward weights are chosen adversarially such that $w_t= \\text{argmin}_{w \\in P_R} w_t^T \\mu_t$. The game will go on for $T$ rounds and we claim that the time average $1\/T \\sum_{t=1}^T \\mu_t$ is roughly the max-min solution, and proof is as follows:\n\n\\begin{align*}\n\\mathop {\\min}_{w\\in P_R} \\frac{1}{T} \\sum_{t=1}^T w^T \\mu_t & \\geq  \\frac{1}{T} \\sum_{t=1}^T w_t^T\\mu_t &&\\\\\n& \\geq \\frac{1}{T}  \\mathop{\\max}_{\\mu \\in P_F} \\sum_{t=1}^T w_t^T\\mu - \\text{regret} && (\\text {regret bound of FPL})\\\\\n& \\geq \\mathop{\\min}_{w \\in P_R} \\mathop{\\max}_{\\mu \\in P_F} w^T \\mu - \\text {regret} && \\\\\n& = \\mathop{\\max}_{\\mu \\in P_F} \\mathop{\\min}_{w \\in P_R} w^T \\mu - \\text{regret}\n\\end{align*}\n\nThe first inequality holds because each $w_t$ was chosen to minimize the reward with respect to each $\\mu_t$. Thus the sum of rewards on the right hand side is smaller or equal to the sum of minimal rewards achieved by a common set of weights that is on the left hand side.\n\nThe regret bound of FPL is \n\n\\begin{equation}\n\\sum_{t=1}^T w_t^T\\mu_t  \\geq  \\mathop{\\max}_{\\mu \\in P_F} \\sum_{t=1}^T w_t^T\\mu - \\text{regret} \\cdot T,\n\\end{equation}\n\nand\n\\begin{equation}\n\\text{regret} = \\sqrt{\\frac{RAD}{T}},\n\\end{equation}\n\nwhere $R$ is the largest reward possible in the MDP, $A$ is the $L_1$ norm of any possible weights, and  $D$ is the $L_1$ diameter of the feature polytope $P_F$. The (average) regret decreases as the total time step $T$ increases, $\\text{regret} \\approx O(1\/\\sqrt{T})$. Thus the time average $\\sum_{t=1}^T \\mu_t$ is roughly the max-min solution. We can adjust the total time step to achieve the desired accuracy. The perturbation term $p_t$ is chosen randomly according to an exponential distribution independently for each coordinate, that is to choose from a standard exponential distribution $\\epsilon_2 \\exp(-\\epsilon_2 x)$, with $\\epsilon_2= \\sqrt{D\/RAT} $.}\n\n\\section{Preliminary}\\label{sec:prelim}\n\nAn MDP is a tuple $M = (\\mathcal{S}, \\mathcal{A}, P_{sa}, \\gamma, D, R)$, including a finite set of states, $\\mathcal{S}$, a set of actions, $\\mathcal{A}$, and transition probabilities, $P_{sa}$.  $\\gamma$ is a discount factor, and $D$ is the distribution of initial states.  The reward function $R$ instructs the learning process. We assume that the reward is a linear function of some vector of features $\\phi$: $ \\mathcal{S} \\to  [0, 1] ^ k$  over states. That is $R(s) = w \\cdot \\phi(s)$ for every state $s\\in \\mathcal{S}$, where $w\\in\\mathbb{R}^k$ is the \\emph{reward weights} of the MDP. The true reward weights $w^*$ is assumed to be unknown to the agent. We use $\\langle\\cdot\\rangle$ to denote the bit complexity of an object. In particular, we use $\\langle M\\rangle$ to denote the bit complexity of $M$, which is the number of bits required to represent the distribution of initial states, transition probabilities, the discount factor $\\gamma$, and the rewards at all the states. We use the notation $M\\backslash R$ to denote a MDP without the reward function, and $\\langle M\\backslash R \\rangle$ is its bit complexity. We further assume that $\\phi(s)$ can be represented using at most $\\langle \\phi \\rangle$ bits for any state $s\\in \\mathcal{S}$.\n\nAn agent selects the action according to a policy $\\pi$. The value of a policy under rewards $w$ is $ \\mathbb{E}_{s_0 \\sim D}[V^\\pi(s_0) | M] = w \\cdot \\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^t\\phi(s_t) | M,\\pi]$. It is expressed as the weights multiplied by the accumulated discounted feature value given a policy, which we define as $\\Psi(\\pi)=\\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^t\\phi(s_t) | M,\\pi]$.\n\n\\paragraph{MDP solver}\nWe assume that there is a RL algorithm $\\textsc{ALG}$ that takes an MDP as input and outputs an optimal policy and its corresponding representation in the feature space. In particular, $\\textsc{ALG}(M)$ outputs $(\\pi^*,\\mu^*)$ such that $\\mathbb{E}_{s_0\\sim D}[V^{\\pi^*}(s_0) | M]=\\max_\\pi  \\mathbb{E}_{s_0 \\sim D}[V^\\pi(s_0) | M]$ and $\\mu^*=\\Psi(\\pi^*)$\n\n\\paragraph{Maxmin Learning}\nAll weights that are consistent with the agent's knowledge form a set $P_R$. We will discuss several formal ways to define this set in Section~\\ref{sec:reward SO}. The goal of the agent is to learn a policy that maximizes the reward for any reward function that could be induced by weights in $P_R$ and adversarially chosen. More specifically, the max-min learning problem is $\\boldsymbol{\\mathop{\\max}_{\\mu\\in P_F} \\mathop{\\min}_{w\\in P_R} w^T\\mu}$,\nwhere $P_F$ is the polytope that contains the representations of all policies in the feature space, i.e. $P_F=\\{\\mu\\ | \\ \\mu =\\Psi(\\pi) \\text{ for some policy $\\pi$ } \\}$. WLOG, we assume that all weights lie in $[-1,1]^k$.\n\n\n\n\\notshow{\\subsection {Ellipsoid}\n\nThe Ellipsoid method can solve an LP in polynomial time. Consider an input LP that has the form of:\n\n\\begin {equation}\nK = { Ax \\leq b : x \\geq 0}, \n\\end{equation}\n\nwhere $A \\subseteq \\mathbb{R} ^ n$. If $n=1$, we can solve the LP directly. Without loosing generality, we asume $n>1$. The Ellipsoid algorithm maintains an ellipsoid $E$ that completely contains the convex set $K$, if $K$ is nonempty.  We check if the center of the ellipsoid is in $K$. If it is, we are done and we find a point in K. If it is not, we find a hyperplane $a^Tx \\leq b$ that is satisfied by all points in $K$ but not the center of the ellipsoid. The Ellipsoid algorithm is as follows:\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE {An input set $K \\subseteq \\mathbb{R}^n$.}\n\\STATE Initialize an ellipsoid $E_o$ that contains $K$, $k = 0$.\n\\IF {the center $c_k$ of $E_k$ is in $K$}\n\t\\STATE Terminate and report $c_k$\n\\ELSE \\STATE Get a separating hyperplane through $c_k$, i.e., $a^T x = a^T c_k $ such that the set K is completely contained in the half-ellipsoid formed by the intersection of the half space and the current ellipsoid. \n\t\\STATE Take the smallest volume ellipsoid containing the half-ellipsoid that contains $K$, set $k=k+1$ and repeat the iteration\n\\ENDIF\n\\end{algorithmic}\n\\caption{{\\sf Ellipsoid Algorithm}}\n\\label{alg:Ellipsoid}\n\\end{algorithm}\n\nThe ellipsoids constructed by the algorithm shrink in volume as the algorithm proceeds. In the end, we either find a point in the input set (solve the LP) or determine that the input set is empty. The algorithm is polynomial-time. In order to use the ellipsoid method, to solve LPs in our problem, we need separation oracles as access to the polytopes of interest, which we discuss later.}\n\n\\paragraph{Separation Oracles} To perform maxmin learning, we often need to optimize linear functions over convex sets that are intersections of exponentially many halfspaces. Such optimization problem is usually intractable, but if the convex set permits a polynomial time \\emph{separation oracle}, then there exists polynomial time algorithms (e.g. ellipsoid method) that optimize linear functions over it.\n\n\n\\begin{definition}(\\textbf{Separation Oracle}) Let $P$ be a closed, convex subset of Euclidean space $\\mathbb{R}^d$. Then a \\emph{Separation Oracle} for $P$ is an algorithm that takes as input a point ${x}\\in\\mathbb{R}^d$ and outputs ``\\textsc{Yes}'' if ${x} \\in P$, or a hyperplane $({w},c)$ such that $w\\cdot {y} \\leq c$ for all ${y} \\in P$, but $w\\cdot{x} > c$. Note that because $P$ is closed and convex, such a hyperplane always exists whenever ${x} \\notin P$.\n\n\n\\end{definition}\n\n\\section{Related Work}\\label{sec:related}\n\nIn the sense of using expert demonstrations, our work is related to inverse reinforcement learning and apprenticeship learning~\\cite{NgR00,AbbeelN04,SyedS08,SyedBS08}. In particular, the apprenticeship learning problem was formulated as a maxmin problem in~\\cite{SyedS08,SyedBS08}. Despite the seemingly similarity, our maxmin learning problem aims to solve a completely different problem than apprenticeship learning. Here is a simple example: the expert is consistent with two reward functions $w_1$ and $w_2$ with $\\epsilon = 25$. Say the expert gets 100 under $w_1$ and 70 under $w_2$, and the only alternative policy gives 90 and 90. Apprenticeship learning will return the expert policy while ours will return the latter solution. In the worst case (under $w_2$), our maxmin policy has better guarantees and thus is more robust.\nWe do not infer a specific reward function or imitate expert trajectories directly, but rather use them to provide information about the potential reward function.\n\n\\notshow{\\paragraph{Apprenticeship Learning} Similarity: maxmin formulation, expert guidance. Difference: apprenticeship hopes to imitate the expert's behavior. We just want to infer consistent rewards from the expert and then do maxmin planning. Give example. Different application: we can accommodate multiple experts and do not need the training domain to be the same as testing domain. See experiment.~\\cite{SyedBS08,SyedS08}}\n\nInverse reward design~\\cite{HadfieldMARD17}  uses a proxy reward and infers the true reward by estimating its posterior. Then it uses risk-averse planning together with samples from the posterior in the testing environment to achieve safe exploration. Our approach achieves a similar goal without assuming any distribution over the rewards and is arguably more robust. We apply a single reward function to the whole MDP while they apply (maybe too pessimistically) per step\/trajectory maxmin planning. Furthermore, our algorithm is guaranteed to find the maxmin solution in poly time, and it naturally accommodates multiple experts. \n\nRobust Markov Decision Processes  \\cite{NilimEG05,Iyengar05}  have addressed the problem of performing dynamic programming-style optimization environments in which the transition probability matrix is uncertain. Lim, Xu \\& Mannor \\cite{LimXM13} have extended this idea to reinforcement learning methods.  This body of work also uses min-max optimization, but because the optimization is with respect to worst-case transitions, this line of work results in very pessimistic policies.  Our algorithmic approach and flavor of results are also different.\n\\cite{MorimotoD05} have addressed a similar adversarial setup, but in which the environment designs a worst-case disturbance to the dynamics of the agent, and have addressed this setup using $H_\\infty$ control. \n\n\n\\section{Consistent Reward Polytope}\\label{sec:reward SO}\nIn this section, we discuss several ways to define the consistent reward polytope $P_R$.\n\n\\paragraph{Explicit Description} We assume that the agent knows that the weights satisfy a set of explicitly defined linear inequalities of the form $c\\cdot w \\geq b$. For example, such an inequality can be learned by observing that a particular policy yields a reward that is larger or smaller than a certain threshold.~\\footnote{Note that with a polynomial number of trajectories, one can apply standard Chernoff bounds to derive such inequalities that hold with high probability. It is often the case that the probability is so close to $1$ that the inequality can be treated as true always for any practical purposes.}\n\n\\paragraph{Implicitly Specified by an Expert Policy} Usually, it may not be easy to obtain many explicit inequalities about the weights. Instead, we may have observed a policy $\\pi_E$ used by an expert. We further assume that the expert's policy has a reasonably good performance under the true rewards $w^*$. Namely, $\\pi_E$'s expected reward is only $\\epsilon$ less than the optimal one. Let the expert's feature vector $\\mu_E = \\Psi(\\pi_E)$. The set $P_R$ therefore contains all $w$ such that $\\mu_E \\cdot w \\geq \\mu^T \\cdot w - \\epsilon, \\forall \\mu \\in P_F$. It is not hard to verify that under this definition $P_R$ is a convex set. Even though explicitly specifying $P_R$ is extremely expensive as there are infinitely many $\\mu\\in P_F$, we can construct a polynomial time separation oracle $SO_R$ (Algorithm~\\ref{alg:SO}). An alternative way to define $P_R$ is to assume that the expert policy can achieve $(1-\\epsilon)$ of the optimal reward (assuming the final reward is positive). We can again design a polynomial time separation oracle similar to Algorithm~\\ref{alg:SO}. \n\n\n\t\t\\begin{algorithm}[H]\n\t\t\t\\begin{algorithmic}[1]\n\t\t\t\t\\INPUT {$w' \\in \\mathbb{R}^{k}$}\n\t\t\t\t\\STATE Let $\\mu_{w'}:=\\mathop{\\text{argmax}}_{\\mu\\in P_F} \\mu \\cdot w' $. Notice that $\\mu_{w'}$ is the feature vector of the optimal policy under reward weights $w'$. Hence, it can be computed by our MDP solver $\\textsc{ALG}$.\n\t\t\t\t\\IF {$\\mu_{w'}\\cdot w'> \\mu_E \\cdot w' + \\epsilon$} \\STATE output ``NO'' , and \n\t \t\t\t\t$ \\left(\\mu_E- \\mu_{w'} \\right)\\cdot w + \\epsilon\\geq 0$ as the separating hyperplane, \n\t \t\t\t\tsince for all $w \\in P_R, \\mu_E\\cdot w \\geq \\mu_{w'} \\cdot w -\\epsilon$.\n\t\t\t\t\\ELSE \\STATE output ``YES''.\n\t\t\t\t\\ENDIF\n\t\t\t\\end{algorithmic}\n\t\t\\caption{{\\sf Separation Oracle $SO_R$ for the reward polytope $P_R$}}\n\t\t\\label{alg:SO}\n\t\t\\end{algorithm}\n\n\n\n\\paragraph{Combining Multiple Experts} \nHow can we combine demonstrations from experts operating in drastically different environments? Here is our model. For each environment $i$, there is a separate MDP $M_i$, and all the MDPs share the same underlying weights as they are all about completing the same task although in different environments. The $i$-th expert's policy is nearly optimal in $M_i$. More specifically, for expert $i$, her policy $\\pi_{E_i}$ is at most $\\epsilon_i$ less than the optimal policy in $M_i$. Therefore, each expert $i$ provides a set of constraints that any consistent reward needs to satisfy, and $P_R$ is the set of rewards that satisfy all constraints imposed by the experts. For each expert $i$, we can design a separation oracle $SO^{(i)}_R$ (similar to Algorithm~\\ref{alg:SO}) accepting weights that respect the constraints given by expert $i$'s policy. We can easily design a separation oracle for $P_R$ that only accepts weights that will be accepted by all separation oracles $SO^{(i)}_R$.\n\nFrom now on, we will not distinguish between different ways to define and access the consistent reward polytope $P_R$, but simply assume that we have a polynomial time separation oracle for it. All the algorithms we design in this paper only require access to this separation oracle. In Section~\\ref{sec:exp.}, we will specify how the $P_R$ is defined for each experiment.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nThe effect of gravitation on electromagnetic phenomena is of great interest due to its applications in both particle and astrophysics. To the astrophysicist, such study sheds light on phenomena occuring around black holes and other strong gravitational sources. Of particular interest to the authors of this paper are the closed form fundamental solutions for electric potential that can be found for these situations. Such solutions, when they exist, are particularly interesting as they provide the basis for research on, amongst others, particle self-interaction [eg. \\cite{linet1984}, \\cite{leaute_self-interaction_1985}, \\cite{Piazzese1991} \\cite{drivas2011}] and electromagnetic phenomena around wormholes [eg. \\cite{Beskin2011}, \\cite{Khusnutdinov2010}, \\cite{Khusnutdinov2005}, \\cite{bezerra2009}]. See also \\cite{Dehghani2006}, \\cite{Hendi2008} and \\cite{Hendi2012}. This paper investivates the stability of a class of electrovac Brans-Dicke spacetimes linearly perturbed by a static electric point charge. \n\nIn 1927, Whittaker \\citep{Whittaker1927b} investigated electric phenomena in gravitational fields including the study of the electrostatic potential generated by a static electric point charge in a Schwarzschild and a quasi-uniform gravitational background. Using the method of separation of variables, he was able to find an infinite series solution describing the former and a closed form solution for the latter. \n\nShortly after in 1928, Copson \\cite{Copson1928} used Hadamard's \\citep{hadamard1923} theory  of ``elementary solutions\" (referred to in recent literature as fundamental solutions) of partial differential equations to not only reproduce Whittaker's original expression for the quasiuniform field but to go on and derive an exact closed form expression for the potential generated by a static electric point charge in a Schwarzschild spacetime written in isotropic coordinates. Copson noted his result differed from Whittaker's infinite series solution by a zeroth order term (see Section \\ref{surfintandbc} for further discussion on Whittaker's and Copson's solutions).\n\nIndependently in the 1970s, Cohen and Wald \\cite{cohen_point_1971} and Hanni and Ruffini \\cite{hanni_lines_1973} used the method of separation of variables to express the electrostatic potential generated by a static electric point charge in a Schwarzschild background as an infinite series which concurred with Whittaker's earlier result.\n\nCopson's result was ammended by Linet \\cite{Linet1976} who applied the boundary condition at infinity and an asymptotic expansion of Copson's solution to prove that it was for not one but two charges, the second residing within the horizon. Linet resolved this issue by excising the second charge, his result coinciding with those found using multipole expansions by Cohen and Wald \\cite{cohen_point_1971}, Hanni and Ruffini \\cite{hanni_lines_1973} and Whittaker \\cite{whittaker_course_1927}. Linet was also able to transform Copson's fundamental solution from isotropic coordinates into the usual Schwarzschild coordinates. \n\nUsing the Copson-Hadamard method Linet went on to derive expressions for the potential of a static electric point charge in the Reissner-Nordstr\\\"{o}m field with Leaute \\citep{leaute_electrostatics_1976} and in a Brans-Dicke-Schwarzschild field with Teyssandier  \\citep{Linet1979}. Copson revisited his method of solution in 1978 \\citep{Copson1978}, developing a closed form solution for the potential generated by a static electric point charge (what he terms an ``electron\") in a Reissner-Nordstr\\\"{o}m background field. His result again differed to that obtained by Leaute and Linet \\citep{leaute_electrostatics_1976} due to a different choice of boundary conditions, which will be discussed in Section \\ref{surfintandbc}.\n\nThe Copson method for solving for the electric potential involves identifying a new independent variable that converts the governing Maxwell equation into a linear second order ordinary differential equation. Linet and co-authors adopted the form of Copson's independent variable to consider the Reissner-Nordstr\\\"{o}m and Brans-Dicke-Schwarzschild cases and were able to find the corresponding governing equations for the above mentioned cases as second order ordinary differential equations (see also \\citep{Linet2005}). The results obtained by these authors were on a case-by-case basis. Here we introduce a method by which one is able to extract the form of the new independent variable and obtain the general second order ordinary differential equation for all the electrovac spherically symmetric Brans type solutions which are reducible to the Schwarzschild and Reissner-Nordstr\\\"{o}m black hole solutions in the Einstein theory.\n\nAs Copson's method involves solving a linear second order ordinary differential equation which would naturally produce two linearly independent solutions, it is necessary to impose appropriate boundary conditions that would allow one to determine the relationship between the two constant coefficients of the general solution. As mentioned earlier, it was due to a different choice of boundary condition that Copson's result differed from that found by Linet and co-authors, thus it is clear to see how the interpretation of the solution hinges upon the choice of boundary condition. As a result of his choice of boundary conditions Copson's solution exhibited two charges which was in contradiction to Hadamard's theory of ``elementary\" solutions that stipulates that there must exist only one singular point. Here we impose a boundary condition such that a Gauss' Law type integral over any closed surface in space not enclosing the perturbing charge must vanish even if that region contains a surface of inversion which exists in all Brans type I solutions which are reducible to the Schwarzschild and Reissner-Nordstr\\\"{o}m black hole cases (see Theorem 1 in Section \\ref{STFT}). This boundary condition proves to be sufficient in determining the relationship between the constant coefficients such that they are in agreement with known multipole solutions found using the method of separation of variables \\citep{whittaker_course_1927}, \\citep{cohen_point_1971}, \\citep{hanni_lines_1973} and those found using Hadamard's definition of ``elementary\" solutions \\citep{Linet1976}, \\citep{leaute_electrostatics_1976}, \\citep{Linet1979}. \n\nIn Section \\ref{STFT} we give a detailed overview of the Brans-Dicke Reissner-Nordstr\\\"{o}m background which is the exact solution for the gravitational field generated by a point charge in a scalar-tensor field. The general Brans-Dicke electrovac solutions has six constants of integration, two of which can be determined by scaling the coordinates $r$ and $t$. Luke and Szamosi \\cite{Luke1972} show that the remaining four contants of integration can be constrained such that the solution reduces to the Reissner-Nordstr\\\"{o}m solution in Einstein's theory. The salient features of these Brans-Dicke metrics is that in isotropic coordinates a surface of inversion separates the solution into two regions. This results in a double-covering of the spacetime region corresponding to the exterior of a non-rotating black hole in General Relativity. It is also important to detail the choice of constants and their subsequent physical interpretations as this will influence our choice of boundary condition (see Appendix A for more detail). As demonstrated by Arnowitt, Deser and Misner \\citep{Arnowitt1960}, \\citep{Arnowitt1960a}, the interpretations of the source terms of a spherically symmetric spacetime in isotropic coordinates requires careful analysis. The results in this section, together with the analysis in Appendix A, extends some of the results in \\citep{Arnowitt1960} and \\citep{Arnowitt1960a} done using the ADM technique on Schwarzschild and Reissner-Nordstr\\\"{o}m spacetimes. We also state the appropriate choice of parameter values for the static spherically symetric Brans-Dicke electrovac solutions as required by the weak-field approximation. We briefly discuss observational constraints on $\\omega$ as this is of particular interest due to the fact that scalar fields and the variable cosmological ``constant\" have become two of the most popular candidates for dark energy \\citep{Bergmann1968}, \\citep{Wagoner1970}, \\citep{Berman1989}, \\citep{Nakamura2006}, \\citep{Thushari2010a}.\n\nIn Section \\ref{EP}, we briefly outline Hadamard's theory of fundamental solutions of curved space Laplace equations containing first order terms. In Section \\ref{exten} we extend Copson's method to find the first four terms of the fundamental solution describing the potential generated by an static electric point charge placed outside the surface of inversion in a Brans-Dicke Reissner-Nordstr\\\"{o}m background. We then develop a method of identifying the new independent variable using the Brans-Dicke field equations as outlined in Appendix A. \n\nIn Section \\ref{solving}, we solve the linear second order differential equation to give us a closed-form solution, which can be used to construct the fundamental solution that represents the electric potential generated by a point charge residing outside the surface of inversion in a Brans-Dicke-Reissner-Nordst\\\"{o}m spacetime. It is important to note that due to the nature of the background metric in isotropic coordinates, the region interior to the surface of inversion is an exact copy of the exterior. Therefore the closed-form solution obtained has an additional singular point at the inversion point of the perturbing static electric point charge. \n\nIn Section \\ref{surfintandbc}, we introduce a boundary condition that will allow one to determine the relationship between the two constant coefficients and essentially eliminate the singularity that creeps into the spacetime region that lies beyond the inversion\nsurface and which is not covered by the isotropic coordinates. Hence we obtain a process to derive the fundamental solution for a class of curved space Laplace equations containing first order terms thus making it unnecessary to compare with multipole expansion solutions. \n\nLastly, in Section \\ref{specificcases} we show how our method also yields the fundamental solutions of the three known cases (Schwarzschild, Reissner-Nordstr\\\"{o}m, Brans-Dicke Type I) which are summarized in Table \\ref{tab:tab}.\n\nIn Appendix A, details of how the scalar field, the metric functions and the electrostatic potential are all essentially a combination of the metric variable $r^2 \\phi e^{\\alpha + \\beta}$ are discussed. Appendix A also outlines how the Brans-Dicke Reissner-Nordstr\\\"{o}m background metric can be determined. The surface integral inner boundary condition is outlined in Appendix B for the Brans-Dicke Reissner-Nordstr\\\"{o}m background .\n\n\n\n\\section{Scalar-Tensor Field Theory}\\label{STFT}\nThe field equations in the Brans-Dicke theory are\n\n\\begin{eqnarray} \\label{fieldone} \\nonumber\nR_{a b}-\\frac{1}{2}g_{a b}R &=& \\frac{8 \\pi T_{a b}}{c^{4} \\phi} + \\frac{1}{\\phi} (\\nabla_{a} \\partial_{b} \\phi - g_{a b} \\square \\phi) \n\\\\ && + \\frac{\\omega}{\\phi^{2}}(\\partial_{a} \\phi \\partial_{b} \\phi - \\frac{1}{2}g_{a b} g^{c d} \\partial_{c} \\phi \\partial_{d} \\phi) \n\\end{eqnarray}\n\\begin{eqnarray}\\label{fieldtwo}\n\\square \\phi= \\frac{8 \\pi T}{(2 \\omega + 3)c^{4}}, \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\square \\phi &:=& \\nabla_{b} (g^{ab} \\partial_{a} \\phi) = \\frac{1}{\\sqrt{-g}} \\partial_{b} (\\sqrt{-g}g^{ab} \\partial_{a} \\phi)\n\\end{eqnarray}\nand $\\square$ is the scalar wave operator. \n\n\nHere the notations have their usual meaning. The contribution of the electromagnetic field, encoded in the Faraday tensor $F_{ab}$, to the energy-momentum tensor is\n\\begin{eqnarray}\nT_{ab} = F_{ac}F^{c}_{b} - \\frac{1}{4}g_{ab}F_{cd}F^{cd} \\; , \\: T^{a}_{a} = 0,\n\\end{eqnarray}\nwhere $F_{ab}$ satisfies the source-free Maxwell equations\n\\begin{eqnarray} \\label{maxwell}\n\\nabla_b F^{a b}= 0 \\: , \\: \\nabla_{[c} F_{ab]} =0.\n\\end{eqnarray}\n\nFollowing the method of Luke and Szamozi \\citep{Luke1972} while at the same time conforming to the choice of boundary conditions in Brans \\citep{Brans1962} (see equations (\\ref{weakfieldone}) to (\\ref{weakfieldthree})) one can verify that an electrically charged Brans-Dicke field that reduces to the Reisnner-Nordstr\\\"{o}m solution in isotropic coordinates when the long-range field equals the reciprocal of the gravitational constant, i.e. $\\phi = (G_{0})^{-1}$, can be summarized as follows (see Appendix A for a brief derivation).\n\n\n\\textbf{Theorem 1}\n\\textit{A static spherically symmetric electrically charged Brans-Dicke-Reissner-Nordstr\\\"{o}m (BDRN) solution of equations (\\ref{fieldone}), (\\ref{fieldtwo}) and (\\ref{maxwell}) in isotropic coordinates $(t,r,\\theta, \\phi)$ is given by the line element}\n\\begin{eqnarray} \\label{lineelement}\nds^{2}=-c^{2}e^{2\\alpha(r)}dt^{2}+e^{2\\beta(r)}[dr^{2}+r^{2}(d\\theta^{2}+\\sin\\ \\theta d\\phi^{2})],\n\\end{eqnarray}\n\\textit{where the static electric potential $V_0(r)$, the Faraday tensor $F_{ab}$ and the corresponding energy-momentum tensor $T^{a}_{b}$ are:}\n\\begin{eqnarray}\nV_0(r) &=& Q \\int_{\\infty}^{r} \\frac{e^{\\alpha(r)-\\beta(r)}}{r^{2}}dr, \\label{V}\\\\\nF_{ab}&=& -cV'_0(r) \\begin{pmatrix}\n0 & 1 & 0 & 0 \\\\ \n-1 & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & 0\n\\end{pmatrix}, \\label{Fab}\\\\\nT^{a}_{b}&=& -\\frac{e^{4\\beta(r)}Q^{2}}{2r^{4}} \\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & -1 & 0 \\\\ \n0 & 0 & 0 & -1\n\\end{pmatrix}. \\label{Tab}\n\\end{eqnarray}\n\n\n\\textit{The reduced long-range scalar field wave equation derived from equation (\\ref{fieldone}) is }\n\\begin{eqnarray} \\label{scalarfield}\n[r^2 \\exp(-\\alpha(r) + \\beta(r)) V'_0(r)]'=0.\n\\end{eqnarray}\n\n\\textit{The metric functions $e^{2 \\alpha(r)}$ and $e^{2 \\beta(r)}$ are}\n\\begin{eqnarray} \ne^{2\\alpha(r)} &=& \\frac{e^{2\\alpha_0} \\left| \\frac{r-B}{r+B} \\right|^{\\frac{2}{\\lambda}}}{\\left( p_{+}^{2}-p_{-}^{2}\\left| \\frac{r-B}{r+B} \\right|^{\\frac{C+2}{\\lambda}} \\right)^{2}}, \\label{e2alpha}\\\\ \\nonumber\ne^{2\\beta(r)} &=&e^{2\\beta_0}  \\left(1 + \\frac{B}{r} \\right)^4  \n\\left| \\frac{r-B}{r+B}\\right| ^{2 \\left( \\frac{\\lambda-C-1}{\\lambda} \\right)} \\\\&& \\times \\left( p_{+}^{2}-p_{-}^{2}\\left| \\frac{r-B}{r+B} \\right|^{\\frac{C+2}{\\lambda}} \\right)^{2} , \\label{e2beta}\n\\end{eqnarray} \n\\textit{and the long range scalar field $\\phi(r)$ is}\n\\begin{eqnarray}\n\\phi = \\phi_{0} \\left| \\frac{r-B}{r+B} \\right|^{\\frac{C}{\\lambda}}. \\label{phi}\n\\end{eqnarray}\n\n\\noindent \\textit{The functions $V_0(r)$, $e^{2\\alpha(r)}$, $e^{2\\beta(r)}$ and $\\phi(r)$ are defined for all non-negative $r$ except at $r=B$. The boundary values $e^{\\alpha_0}$ and $e^{\\beta_0}$ can be rescaled to unity by scaling the $t$ and $r$ coordinates respectively. The nine parameters $Q, B, p_{+}^{2}, p_{-}^{2}, \\lambda, C, \\phi_{0}, e^{2\\alpha_0}$ and $e^{2\\beta_0}$ in equations (\\ref{e2alpha}) to (\\ref{phi}) are related via }\n\n\\begin{equation} \\label{lambda}\n4\\lambda^{2}= (2 \\omega +3) C^2 + (C+2)^2, \\rule{0.5cm}{0cm} \\omega \\geq -\\frac{3}{2},\n\\end{equation}\n\n\\begin{equation}\\label{B}\nB = \\frac{1}{2} \\sqrt{m_{B}^{2}-q_{B}^{2}} ,\n\\end{equation}\n\n\\begin{equation} \\label{mnq}\nm_{B}:= \\frac{2M}{c^{2}\\phi_{0}} \\frac{e^{\\beta_0 }\\lambda}{C+2}, \\rule{0.5cm}{0cm} q_{B}= 2\\sqrt{ \\frac{4 \\pi}{\\phi_{0}}} \\frac{Q}{c^{2}} \\frac{e^{\\beta_0 }\\lambda}{C+2},\n\\end{equation}\n\n\\begin{equation} \np_{\\pm}^{2} = \\frac{m_{B} \\pm \\sqrt{m_{B}^{2}-q_{B}^{2}} }{2 \\sqrt{m_{B}^{2}-q_{B}^{2}}}  \\label{pnp},\n\\end{equation}\n\n\\begin{equation} \\label{p-}\np_{+}^{2} - p_{-}^{2} =1,\n\\end{equation}\n\n\\noindent \\textit{where $M$ and $Q$ are non-negative real constants and are identified, respectively, as mass measured in conventional units (kg) and charge measured in electrostatic units (e.s.u), which has the dimensions of $[mass]^{\\frac{1}{2}}[length]^{\\frac{3}{2}}[time]^{-1}$. Here $\\omega$ is the coupling constant that couples the scalar field to matter, while $c$ is the speed of light in a vacuum.}\n\nTaking into account equations (\\ref{lambda}) to (\\ref{mnq}), there remain only four essential parameters in the BDRN solution. We adopt the independent parameter set ${M, Q, C, \\phi_0}$.\n\\begin{enumerate}\n \\item The choice of the physical parameters of mass, $M$, and charge, $Q$, in the characterization of the BDRN metric is natural.\n \\item As opposite charges neutralize one another, in most astrophysical applications it is reasonable to assume $M \\geq \\sqrt{4 \\pi \\phi_{0}}Q \\geq 0$, and hence the parameter $B$ in equation (\\ref{B}) is non-negative.\n \\item $\\phi_0$ is the value of the long range scalar field at spatial infinity. It has the dimensions of $[mass][length]^{-3}[time]^{2}$.\n \\item \nThe parameter $C$ is dimensionless and relates to the local strength of the long-range scalar field $\\phi(r)$. Equation (\\ref{lambda}) gives $\\lambda^{2}$ as a quadratic expression in $C$ with the discriminant $\\bigtriangleup= -(2\\omega +3)$. Thus when $\\omega >- \\frac{3}{2}$, $C$ is real and $\\sqrt{\\frac{2\\omega+3}{2\\omega+4}}<\\vert\\lambda \\vert< \\infty$. Constraining the BDRN solutions to conform with the weak-field approximation (see \\citep{Brans1961}) we expand the metric functions and the scalar field, equations (\\ref{e2alpha}), (\\ref{e2beta}) and (\\ref{phi}), to the order of $1\/r$, and obtain the following restrictions on the parameters:\n\\begin{eqnarray} \\label{weakfieldone}\n\\alpha_0&=&\\beta_0=0,\\\\\n\\phi_{0} &=& \\frac{1}{G_{0}} \\left( \\frac{2 \\omega+4}{2 \\omega +3} \\right),\\\\ \\label{weakfieldtwo}\n\\lambda &\\cong& \\sqrt{\\frac{2\\omega+3}{2\\omega+4}}\\\\\nC &\\cong& -\\frac{1}{\\omega+2}, \\label{C} \\\\\nm_B &\\cong& \\frac{M}{c^4\\phi_0} \\sqrt{\\frac{2\\omega+4}{2\\omega+3}}, \\\\\nq_B &\\cong& \\sqrt{\\frac{4\\pi}{\\phi_0}} \\frac{Q}{c^2} \\sqrt{\\frac{2\\omega+4}{2\\omega+3}}, \\label{weakfieldthree}\n\\end{eqnarray}\nwhere $G_0$ is defined as the gravitational constant (for the BDRN spacetimes), while $G$ denotes Newton's universal constant of gravity (see Case 2 and Case 3 below).\n\n\\item  \\label{obs} Observational constraints put even stronger requirements on the values of $\\omega$. The latest results obtained from the Cassini-Huygen experiment \\citep{Bertotti2003} put the value of $\\omega$ at over $40000$. The coupling constant $\\omega$ represents the strength of the coupling between the scalar field and the gravitational field. Therefore its value is of great importance in any discussion regarding (a) the existence and properties of Brans-Dicke black holes, and (b) candidates for dark energy. \n\n\\item When an inversion is applied, that is, transforming from $r$ to $r* = \\frac{B^{2}}{r}$, the region $B <r < \\infty$ is mapped one to one onto the region $0 < r < B$. Under such a reflection at the sphere of $r_{BDRN}=B$, the functions $V_0(r)$, $e^{2\\alpha(r)}$, $\\phi(r)$ and the line element (\\ref{lineelement}) remain invariant while the metric function $e^{2\\beta(r)}$ is transformed into $e^{2\\beta(r^{*})}=\\frac{r^{4}}{B^{4}}e^{2\\beta(r)}$ and the flat 3-metric $d\\ell^{2}:=[dr^{2}+r^{2}(d\\theta^{2} + \\sin^{2} \\theta d \\phi^{2})]$ is mapped conformally onto the flat metric $d(\\ell^{*})^{2}=\\frac{r^{4}}{B^{4}}d\\ell^{2}$. Therefore the spherical surface at $r_{BDRN}:= B =\\frac{1}{2}\\sqrt{m_{B}^{2}-q_{B}^{2}}$ is an inversion surface in the sense that the BDRN solution in isotropic coordinates is invariant under the geometric inversion transformations $r(r*)=B^{2}$. The two copies of the BDRN spacetime, one exterior to and the other interior to the inversion surface, are identical. At the spherical surface of inversion $r_{BDRN}=B$, the line element (\\ref{lineelement}) is singular. It is the inaccessible boundary of the two identical copies of the BDRN spacetime in isotropic coordinates. Throughout this article we use the exterior copy where $B< r< \\infty$, unless stated otherwise. This will have important consequences (see below) on how to interpret the Copson-Hadamard method \\citep{Copson1928} in the construction of the fundamental solution to the Laplace equation of a perturbed electrostatic potential in a BDRN background solution. \n\n\\item \\label{bhadra} An investigation by \\citep{Bhadra2005} found that Brans Type I solutions may represent an external gravitational field for nonsingular spherically symmetric matter sources. They concluded however, that Brans-Dicke black holes cannot exist, as a condition equivalent to equation (\\ref{C}) (that is, the weak-field approximation) would require that $-2>\\omega>-(2+\\frac{1}{\\sqrt{3}})$; a requirement which clearly violates observational constraints \\citep{Bertotti2003}. \n \\end{enumerate}\n\nBy choosing various combinations of the four independent parameters $M, Q, C$ and $\\phi_0$ to vanish, we obtain the following limiting solutions:\n\n\\textbf{Case 1:} Brans Type I (BS) metric in isotropic coordinates\n\nBy setting the charge parameter $Q$ to zero, it implies that $B= \\frac{m_{B}}{2}= \\frac{M}{2c^{2}\\phi_{0}}, p_{+}^{2}=1$ and $p_{-}^{2}=0$. We recover the Brans Type I metric \\citep{Brans1962} of the Brans-Dicke theory:\n\\begin{eqnarray}\n\\phi(r)&=& \\phi_{0} \\left| \\frac{r-B}{r+B} \\right|^{\\frac{C}{\\lambda}},\\\\\ne^{2\\alpha(r)} &=& e^{2 \\alpha_0} \\left|\\frac{r-B}{r+B} \\right|^{\\frac{2}{\\lambda}},\\\\\ne^{2\\beta(r)} &=&  e^{2 \\beta_0} \\left(1+\\frac{B}{r}\\right)^4  \\left| \\frac{r-B}{r+B} \\right|^{2 \\left( \\frac{\\lambda-C-1}{\\lambda} \\right)},\\\\\n\\end{eqnarray}\nwhere $B < r < \\infty$ and the inversion spherical surface is at $r_{BS} = B = \\frac{M}{2c^{2}\\phi_{0}}$ and is a curvature singularity.\n\n\\textbf{Case 2:} Reissner-Nordstr\\\"{o}m (RN) metric in isotropic coordinates\n\nBy setting the parameters $C=\\alpha_0=\\beta_0=0$, it implies that $\\lambda^{2}=1, \\phi_{0}=(G)^{-1}, B= \\frac{1}{2}\\sqrt{m^{2}-q^{2}}, p_{+}^{2}= \\frac{m+\\sqrt{m^{2}-q^{2}}}{2\\sqrt{m^{2}-q^{2}}}$ and $p_{-}^{2}= \\frac{m-\\sqrt{m^{2}-q^{2}}}{2\\sqrt{m^{2}-q^{2}}}$, where $m:=\\frac{GM}{c^{2}}$ and $q:= \\frac{\\sqrt{4 \\pi G}Q}{c^{2}}$ are respectively the mass and the electric charge measured in grativtational units. The metric functions reduce to the usual Reissner-Nordstr\\\"{o}m solution in isotropic coordinates:\n\\begin{eqnarray} \\label{RN1}\ne^{2\\alpha(r)}&=& \\frac{\\left(r- \\frac{\\sqrt{m^2-q^2}}{2} \\right)^2 \\left(r+ \\frac{\\sqrt{m^2-q^2}}{2}\\right)^2}{ \\left(r+\\frac{m-q}{2}\\right)^2 \\left(r+ \\frac{m+q}{2}\\right)^2}, \\\\ \\label{RN2}\ne^{2\\beta(r)}&=& \\frac{\\left(r+\\frac{m-q}{2}\\right)^2 \\left(r+ \\frac{m+q}{2}\\right)^2}{r^4},\n\\end{eqnarray}\nwhere $\\frac{1}{2}\\sqrt{m^{2}-q^{2}} < r< \\infty$, and the inversion spherical surface is at $r_{H+}=\\frac{1}{2}\\sqrt{m^{2}-q^{2}}$, which is also the outer event horizon of the RN spacetime in isotropic coordinates.\n\nThe Reisnner-Nordstr\\\"{o}m metric in isotropic coordinates was first derived in the form given in equations (\\ref{RN1}) and (\\ref{RN2}) above using the ADM technique (see \\citep{Arnowitt1960} and \\citep{Arnowitt1960a}).\n\n\\textbf{Case 3:} Schwarzschild (S) metric in isotropic coordinates\n\nBy setting the parameters $Q= C=\\alpha_0=\\beta_0=0$, it implies that $\\lambda^{2}=1, \\phi_{0}=(G)^{-1}, B= \\frac{m}{2}= \\frac{GM}{2c^{2}}, p_{+}^{2}=1$ and $ p_{-}^{2}=0$ where $m= \\frac{GM}{c^{2}}$ is the mass in gravitational units. The metric functions reduce to the well known Schwarzschild solution in isotropic coordinates:\n\\begin{eqnarray}\ne^{2\\alpha(r)}= \\left( \\frac{1- \\frac{m}{2r}}{1+ \\frac{m}{2r}} \\right)^2\t\\:, \\: e^{2\\beta(r)}= \\left(1+ \\frac{m}{2r} \\right)^4,\n\\end{eqnarray}\nwhere $\\frac{1}{2} m< r< \\infty$ and the inversion spherical surface is at $r_{H}=\\frac{1}{2}m$, which is also the event horizon of the Schwarzschild spacetime in isotropic coordinates.\n\n\\section{Electrostatics and the Hadamard Method} \\label{EP}\nWe now consider the electrostatic potential due to a ``small\" static electric charge $-\\epsilon_0 (\\vert \\epsilon_{0} \\vert \\ll m_{B})$ situated outside the spherical surface of inversion $B$. \n\nWe let $V(r,\\theta,\\phi)$ denote the linearly perturbated electrostatic potential so that the perturbed Faraday tensor $F_{ab}$ takes the form\n\\begin{eqnarray} \\label{max}\nF_{0i}=-F_{i0}= -c \\partial_i V(r,\\theta,\\phi)\\\\\\nonumber\n F_{ij}=0 \\: ; \\: i,j=1,2,3 \\cdots.\n\\end{eqnarray}\n\nThe perturbed Maxwell equations $\\nabla_{[a}F_{bc]}=0$ is automatically satisfied by equation ({\\ref{max}). \n\nWithout loss of generality, the perturbed Maxwell equations due to a single electrostatic charge yields\n\\begin{eqnarray}\n\\frac{1}{\\sqrt{-g}} \\partial_b (\\sqrt{-g}F^{ab})=J^0\n\\end{eqnarray}\nwhich implies\n\\begin{eqnarray} \\label{maxwell1.5}\n\\nabla^2 V(r,\\theta, \\phi) - (\\alpha'(r) - \\beta'(r)) \\frac{\\partial V(r,\\theta, \\phi)}{\\partial r}\\\\=ce^{2(\\alpha(r) +  \\beta(r))}J^0\n\\end{eqnarray}\nwhere the current density $J^0=-\\frac{4 \\pi \\epsilon_0}{cr^2} e^{-2\\alpha(r)-3 \\beta} \\delta(r-b) \\delta(\\cos \\theta- \\cos \\theta_0)$. Here $\\alpha(r)$ and $\\beta(r)$ are given by equations (\\ref{e2alpha}) and (\\ref{e2beta}) respectively, $\\nabla^2=\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}$ is the 3-dimensional Euclidean space Laplacian with $x=r\\sin \\theta \\cos \\phi$, $y=r \\sin \\theta \\sin\\phi$ and $z=r\\cos \\theta$. Note that $\\partial_r= \\frac{x}{r} \\partial_x +\\frac{y}{r} \\partial_y+\\frac{z}{r} \\partial_z$. We define\n\\begin{eqnarray} \\label{Gamma} \n\\Gamma(r, \\theta)=r^2+b^2-2br\\cos \\theta\n\\end{eqnarray}\nwhich is equal to the square of the ``radial\" distance from the charged particle at $z=b$. \n\nA brief overview of Hadamard's theory of fundamental solutions \\citep{hadamard1923} is neccessary to fully understand Copson's construction (\\cite{Copson1928}, \\cite{Copson1978}). We adapt Hadamard's result that includes equation (\\ref{maxwell1.5}) as a particular case as follows:\n\n\n\\textbf{Theorem 2 (Hadamard's Theorem):}\n\\noindent \\textit{Consider a second order linear partial differential equation of the form \n\\begin{eqnarray}  \\label{hadamardspde}\n\\Im(u) =  \\sum_{i,j =1}^{3} \\delta^{ij} \\frac{\\partial^2 u}{\\partial x^i\\partial x^j} + \\sum_{i=1}^3 h(r) \\frac{x^i}{r} \\frac{\\partial u}{\\partial x^i}  =0 \n\\end{eqnarray}\nwhere $\\delta^{ij}$ is the Kronecker tensor, $h(r)$ is a differentaible function of $r=\\delta_{ij} x^{i}x^{j}$. The fundamental solution of equation (\\ref{hadamardspde}) is continuous and differentiable everywhere except at the singular point $(r,\\theta, \\phi)=(b, \\theta_0, \\phi_0)$ and can be written as \n\\begin{eqnarray}\nu=   \\frac{U(r,\\theta,\\phi)}{\\Gamma^ \\frac{1}{2}} \n\\end{eqnarray}\nwhere $\\Gamma$ is given by equation (\\ref{Gamma}) and the function $U(r,\\theta, \\phi)$ is real analytic everywhere in the domain of definition of equation (\\ref{hadamardspde}) including the singular point $(r,\\theta, \\phi)=(b, \\theta_0, \\phi_0)$ . $U(r)$ is expandable as a convergent power series in $\\Gamma$ such that\n\\begin{eqnarray} \\label{gam}\nU(r, \\theta, \\phi) =   U_0(r) + U_1(r) \\Gamma + U_2(r) \\Gamma^2 + \\cdots,\n\\end{eqnarray}\nwhere $U_n$ is given by the reccurent formula\n\\begin{eqnarray} \\label{Un}\nU_n(r)&=& \\frac{U_0}{4 (n-\\frac{1}{2}) s} \\int^s_0 \\frac{s^{n-1}}{U_0} \\Im(U_{n-1})ds \\\\ \\nonumber \n n&=&1,2,3 \\cdots \\\\\ns &=& \\sqrt{r^2+b^2-2rb \\cos \\theta} \\nonumber\n\\end{eqnarray}\nand \n\\begin{eqnarray} \\label{U0}\nU_0(r)= \\exp \\left( - \\int^r_b h(r) d r \\right).\n\\end{eqnarray}}\n\nIn the case of the BDRN metric, the coefficient $h(r)$ in equations (\\ref{hadamardspde}) and (\\ref{U0}), is given by\n\\begin{eqnarray}\nh(r) = - \\alpha'(r) + \\beta'(r).\n\\end{eqnarray} \n\n\n\n\\section{Extension of the Copson Construction} \\label{exten}\n\nEquation (\\ref{maxwell1.5}) for the BDRN metric can be expressed in the form\n\\begin{eqnarray} \\nonumber\n   \\nabla^2 V+ \\frac{2B}{r^2-B^2} \\left(    2k \\left[ \\frac{\\eta^\\ast(r)}{\\eta(r)} \\right] + \\frac{B}{r} \\right) \\frac{\\partial V}{\\partial r} \\\\ \\label{maxwell2}\n= ce^{2(\\alpha(r) +  \\beta(r))}J^0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{k}\nk= \\frac{C+2}{2\\lambda} \\quad ; \\quad  \\eta(r)= p_{+}^{2}-p_{-}^{2}\\left(\\frac{r-B}{r+B} \\right)^{2k},\\\\\n\\eta^\\ast(r) =  - p_{+}^{2}-p_{-}^{2}\\left(\\frac{r-B}{r+B} \\right)^{2k}.\n\\end{eqnarray}\n\n\n\n\nInstead of using the formal expression in equation (\\ref{Un}) we follow Copson \\citep{Copson1928} by substituting equation (\\ref{gam}) into (\\ref{maxwell2}). After some algebra we obtain the first four terms of the recurrent series of the Brans-Dicke-Reissner-Nordstr\\\"{o}m metric\n\\begin{eqnarray}\nU_0(r) &=& \\frac{r}{b} \\frac{\\eta_0}{\\eta(r)} \\frac{(r-B)^{k - \\frac{1}{2}}} {(r+B)^{k + \\frac{1}{2}}} \\frac{(b+B)^{k + \\frac{1}{2}}} {(b-B)^{k - \\frac{1}{2}}}, \\\\\nU_1 (r)&=& \\frac{3B^2 \\left( 1+\\frac{4}{3}(1-k^2) \\right)}{2(r^2-B^2)(b^2-B^2)}U_0,\\\\\nU_2 (r)&=& \\frac{B^2  \n\\left( -5+\\frac{4}{3}(1-k^2)\\right)}{4(r^2-B^2)(b^2-B^2)}U_1,\\\\\nU_3 (r)&=& \\frac{B^2  \\left( -7+\\frac{4}{3}(1-k^2)  \\right)}{10(r^2-B^2)(b^2-B^2)}U_2,\n\\end{eqnarray}\nwhere $\\eta_0=\\eta(b)$.\nSee Table \\ref{tab:tab} for the three degenerate cases. \n\nWe introduce the method by which the substitution can be determined for any background with a line element of the form given by equation (\\ref{lineelement}) which satisfies the Brans-Dicke electrovac field equations (see Appendix A for the governing equations).\nLike Copson, from the first few terms given above we find that the ratio of the ($n+1$)th term to the $n$th term of the power series (\\ref{Un}) is proportional to \n\\begin{eqnarray}\n\\frac{B^2}{b^2-B^2}\\frac{\\Gamma}{r^2-B^2},\n\\end{eqnarray}\nwhere $r^2-B^2$ is proportional to $\\phi r^2 e^{\\alpha+ \\beta}$ (see equation (\\ref{key}) in Appendix A).\n\nFurthermore, the first term of the infinite series, $\\frac{U_0}{\\Gamma^{1\/2}}$, given by equation (\\ref{U0}) is proportional to\n\\begin{eqnarray}\n\\frac{e^{\\frac{1}{2}( \\alpha - \\beta)}}{\\Gamma^{1\/2}}= \\frac{1}{\\gamma^{1\/2} \\sqrt{\\phi} r e^{\\beta}}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{potential}\n\\gamma(r, \\theta)=  \\frac{B^2}{b^2-B^2} \\frac{\\Gamma(r,\\theta)}{\\phi r^{2} e^{\\alpha(r)+\\beta(r)}}.\n\\end{eqnarray}\n\nNow we introduce a new dependent variable $F(\\gamma)$ such that the perturbed electrostatic potential takes the form\n\\begin{eqnarray} \\label{epsub}\nV(r, \\theta, \\phi)= \\frac{F(\\gamma)}{\\sqrt{\\phi} r e^{\\beta(r)}}.\n\\end{eqnarray}\n\n\nFor the Brans-Dicke Reissner-Nordstr\\\"{o}m background, equation (\\ref{epsub}) and (\\ref{potential}) become, respectively,\n\\begin{eqnarray} \\label{elec pot}\nV(r,\\theta)&=& \\frac{r \\phi_{0}}{\\eta(r) (r+B)^{2}} \\left( \\frac{r-B}{r+B} \\right)^{k-1} F(\\gamma)\\\\ \\label{gamma}\n\\gamma(r, \\theta)&=&  \\frac{B^2}{b^2-B^2}\\frac{\\Gamma(r, \\theta)}{r^{2} - B^{2}}.\n\\end{eqnarray}\n\nSubstituting equations (\\ref{elec pot}) and (\\ref{gamma}) into (\\ref{maxwell2}) gives us a second order linear differential equation on $F(\\gamma)$. \n\\begin{eqnarray} \\nonumber\n\\gamma(\\gamma+1)F''(\\gamma)+ \\frac{3}{2}(2\\gamma +1) F'(\\gamma) + (1-k^2)F(\\gamma) = 0.  \\\\  \\label{fdiff} \n\\end{eqnarray}\nWe have allowed the right-hand side of the above equation to vanish as we are only interested in regions away from the point cource where the right-hand side of equation (\\ref{fdiff}) is zero. We later use our boundary condition to verify that the delta-function source term is satisfied and to also determine the constants of integration of the solution to equation (\\ref{fdiff}).\n\n.\n\n\n\\section{Fundamental Solutions} \\label{solving}\n\nEquation (\\ref{fdiff}) can be solved if we transform the independent variable $\\gamma$ as\n\\begin{eqnarray} \\label{subs}\n\\gamma &=& \\sinh^{2} \\frac{\\zeta}{2},\n\\end{eqnarray}\nwhich implies \n\\begin{eqnarray}\n\\gamma+1 = \\cosh^{2} \\frac{\\zeta}{2},\n\\end{eqnarray}\nand we write the dependent variable $F(\\gamma)$ as\n\\begin{eqnarray}\nF(\\gamma)&=& \\Phi(\\zeta). \\label{subs2}\n\\end{eqnarray}\n\nUsing equations (\\ref{subs}) and (\\ref{subs2}), equation (\\ref{fdiff}) can be written in terms of the new variables as follows \n\\begin{eqnarray}\n\\Phi''(\\zeta) + 2 \\coth \\zeta \\: \\Phi'(\\zeta) + (1-k^2) \\Phi(\\zeta) = 0\n\\end{eqnarray}\nwhich has the closed-form solution (see \\cite{kamke1971})\n\\begin{eqnarray} \n\\Phi(\\zeta)= \\frac{k}{\\sinh \\zeta} (\\hat{W}_{1}e^{k\\zeta}-\\hat{W}_{2}e^{-k\\zeta})\n\\end{eqnarray}\nwhere $\\hat{W}_{1}$ and $\\hat{W}_{2}$ are integration constants. The solution in terms of $\\gamma$ is therefore\n\\begin{eqnarray} \\nonumber\nF(\\gamma)&=& \\frac{k}{2\\sqrt{\\gamma}\\sqrt{\\gamma+1}} [\\hat{W}_{1}(\\sqrt{\\gamma+1}+\\sqrt{\\gamma})^{2k} \\\\\n&& \\: \\: \\: - \\hat{W}_{2}(\\sqrt{\\gamma+1}-\\sqrt{\\gamma})^{2k}]. \\label{fgamma}\n\\end{eqnarray}\n\nSubstituting equation (\\ref{fgamma}) into equation (\\ref{elec pot}) gives the electrostatic potential $V(r,\\theta)$ as follows\n\\begin{eqnarray} \\label{solution} \\nonumber\nV(r,\\theta)&=& \\frac{k}{2 \\eta(r) \\sqrt{\\gamma}\\sqrt{\\gamma+1}} \\frac{r}{(r+B)^{2}} \\left( \\frac{r-B}{r+B} \\right)^{k-1}  \\\\\n&& \\times [\\hat{W}_{1}(\\sqrt{\\gamma+1}+\\sqrt{\\gamma})^{2k} \\nonumber \\\\ \n&& \\: \\: \\: \\: - \\hat{W}_{2}(\\sqrt{\\gamma+1}-\\sqrt{\\gamma})^{2k} ].\n\\end{eqnarray}\n\nConsider the inversion point of the static electric point charge $(0,0,(b*))$, where $(b*)= \\frac{B^2}{b}$. Let\n\\begin{eqnarray}\n\\gamma*= \\frac{B^2}{B^2-(b*)^2} \\frac{\\Gamma*(r,\\theta)}{B^2-r^2}\\\\\n\\Gamma*= r^2 +(b*)^2 -2(b*)r \\cos \\theta.\n\\end{eqnarray}\nThus $\\Gamma*$ is equal to the square of the ``radial\" distance from the inversion point at $z=(b*)$. It is straightforward to verify that\n\\begin{eqnarray}\n\\gamma + 1 = \\gamma*.\n\\end{eqnarray}\n\nThe electrostatic potential $V(r,\\theta)$ in equation (\\ref{solution}) is therefore singular at the point charge $z=b$ and also at its inversion point $z=(b*)$. One can also verify that as the field point $r$ approaches the inversion surface $r=B$, the potential approaches a finite limit value provided that $C>-2$.\n\nFinally, to determine the fundamental solution for the electrostatic potential, which allows only one free parameter to arise from the presence of the perturbing electrostatic charge, it is necessary to establish the relationship between the two arbritrary constants in equation (\\ref{solution}).\n\n\n\n\\section{Determination of Integration Constants} \\label{surfintandbc}\n\n\nIn 1927, Whittaker, using the method of separation of variables in the usual Schwarzschild coordinates, found the solution expressing the electrostatic potential of a charge in a Schwarzschild background as an infinite series \\citep{Whittaker1927b}. His result was later confirmed by Cohen and Wald \\citep{cohen_point_1971} in 1971 and Hanni and Ruffini \\citep{hanni_lines_1973} in 1973. A commonality of these works is the use of a boundary condition stating that a charge should not arise inside the horizon as a result of the presence of the perturbing electric charge situated outside the horizon.\n\nThis boundary condition was not implemented by Copson in his determination of integration constants in \\citep{Copson1928} and \\citep{Copson1978} due to the fact that the region inside the horizon is excised in the isotropic coordinates. Instead Copson chose values for the integration constants such that the overall solution would be symmetric in interchanging the position of the field point $r$ with the position  of the perturbing charge $b$. As a result his solution, as he pointed out himself, was in contradiction to Whittaker's solution by the existence of a non-vanishing zero-order term. Linet \\citep{Linet1976}, using the boundary condition at infinity and Gauss' theorem, found this second charge which was necessarily excised to give a result which was in accordance to those given by \\citep{Whittaker1927b}, \\citep{hanni_lines_1973}, and \\citep{cohen_point_1971}.\n\nIn \\citep{Linet1979}, Linet and Teyssandier found a single closed-form solution describing the electrostatic potential generated by a perturbing charge in a Brans-Dicke background. They expressed the fundamental solution as a sum of this solution and legendre functions before performing a multipole expansion and writing the fundamental solution completely in terms of legendre functions. By expressing the solution as a multipole expansion they are then able to impose boundary conditions at infinity to get a meaningful solution upon which the Gauss' theorem can then be implemented to yield their final closed-form solution.\n\nHere we introduce a method of determining the integration constants of equation (\\ref{solution}) which does not require one to expand the closed-form solution into an infinite series and which is even more stringent than those set by \\citep{Whittaker1927b}, \\citep{cohen_point_1971} and \\citep{Hanni1973}. We impose the condition that any integration over a closed spatial region not containing the perturbing charge must be exactly zero even if that area contains a surface of inversion. Naturally, an integration over an area containing the perturbing charged particle must therefore equal exactly $4 \\pi \\epsilon$, where $\\epsilon$ is the charge of the particle. From Appendix B we know that for the Brans-Dicke Reissner-Nordstr\\\"{o}m background, the generalized Gauss's theorem can be written as the following\n\n\\begin{widetext}\n\\begin{eqnarray} \\label{int}\n\\int_{\\Re} J^{0} d \\upsilon =\\int_{0}^{2\\pi}  \\int_{-\\pi}^{\\pi}  \\eta(r)^{2} (r+B)^{2}  \\left(\\frac{r-B}{r+B} \\right)^{\\frac{\\lambda - C - 2}{\\lambda}} \n \\frac{\\partial V(r)}{\\partial r} \\sin \\theta d\\theta d\\phi. \n\\end{eqnarray}\n\\end{widetext}\n\nHere, $\\Re$ is a region of 3-dimensional space residing in a hypersurface and $\\partial \\Re$ is its closed 2-dimensional boundary. Again, $d \\upsilon$ is an element of spatial proper volume in $\\Re$. \nIn order to integrate the above we convert equation (\\ref{solution}) into a function of $\\sinh \\zeta$ where $\\gamma = \\sinh^{2} \\frac{\\zeta}{2}$. We find that the only term that requires integration is the term containing the integration constants, the integral of which is \n\\begin{eqnarray} \\nonumber\n\\int_{-\\pi}^\\pi \\frac{\\hat{W}_1 e^{k\\zeta} - \\hat{W}_2 e^{-k\\zeta}}{\\sinh \\zeta} \\sin \\theta d \\theta =\\\\ \\nonumber \\frac{2brB^2[(b+B)^{2k} - (b-B)^{2k}]}{k (b^2-B^2)^{k-1}(r^2-b^2)^{k-1}} \\\\ \\label{no charge} \\times[\\hat{W}_1 (r+B)^{2k} - \\hat{W}_2 (r-B)^{2k}] \n\\end{eqnarray}\nfor $B<r<b$ and\n\\begin{eqnarray}\\nonumber\n\\int_{-\\pi}^\\pi \\frac{\\hat{W}_1 e^{k\\zeta} - \\hat{W}_2 e^{-k\\zeta}}{\\sinh \\zeta} \\sin \\theta d \\theta =\\\\ \\nonumber \\frac{2brB^2[(r+B)^{2k} - (r-B)^{2k}]}{k (b^2-B^2)^{k-1}(r^2-b^2)^{k-1}} \\\\ \\label{charge} \\times[\\hat{W}_1 (b+B)^{2k} - \\hat{W}_2 (b-B)^{2k}] \n\\end{eqnarray}\nfor $B<b<r$.\n\nWhen we return equation (\\ref{no charge}) into equation (\\ref{int}) it is fairly straightforward to see that for the electrostatic potential to vanish for the region not containing a charge the integration constants must be chosen as the following\n\\begin{eqnarray}\n\\hat{W}_1= p^2_+ \\hat{W}\\\\\n\\hat{W}_2=p^2_- \\hat{W},\n\\end{eqnarray}\nwhere $\\hat{W}$ is a constant yet to be determined. \n\nBy returning equation (\\ref{charge}) into equation (\\ref{int}) and under the condition that for this region ($B<b<r$) the equation (\\ref{int}) must equal $-4 \\pi \\epsilon_0$ we can quickly solve for $\\hat{W}$ giving \n\\begin{eqnarray}\n\\hat{W} = \\frac{bB (b^2-B^2)^{k-1} \\sqrt{\\phi_0} \\epsilon_0}{k [p_+^2 (b+B)^{2k} - p_-^2 (b-B)^{2k}]}.\n\\end{eqnarray}\n\n\n\n\\section{Degenerate Cases} \\label{specificcases}\n\n\n\nThe relationship between the four cases and the process by which one reduces to the other is made obvious in Table \\ref{tab:tab}. It is straightforward to convert the equations in the Reissner-Nordstr\\\"{o}m and Schwarzschild spacetimes into their more familiar form when one conducts the transformations given in Section \\ref{STFT} of this paper. When the transformations are made we find that the solutions are in agreement with the closed-form solutions given by Linet \\cite{Linet1976}, Leaute and Linet  \\citep{leaute_electrostatics_1976} and Linet and Teyssandier \\citep{Linet1979} and with multipole expansions given by Hanni and Ruffini \\citep{hanni_lines_1973} and Cohen and Wald \\citep{cohen_point_1971}.\n\n\n\\section{Conclusion}\nIn this paper we have established an ansatz to solve the perturbed Maxwell equations due to an electrostatic charge in a Brans-type spacetime in isotropic coordinates which is reducible to the Schwarzschild and Reissner-Nordstr\\\"{o}m black hole solutions by extending Copson's method. As Copson's solution is based on Hadamard's theory of fundamental solutions of general Laplace equations it would be interesting to see whether Hadamard's infinite series converges to give Copson's closed-form result.\n\nBy finding the coefficients to $U_0, U_1, U_2, \\cdots$ through the direct substitution of  Hadamard's infinite series into the field equations one is able to compare them with the coefficients given in this paper using Copson's method. \n\nIn a seperate paper, a formal proof of Hadamard's fundamental solution equation (\\ref{hadamardspde}) is given. We find that Copson's results in \\citep{Copson1928} are in fact exactly equal to those found using Hadamard's method and go on to investigate how the Hadamard method relates to the results obtained by Linet in \\citep{Linet2005}. We also find that the discrepancy between Copson and Hadamard with those from the literature including Whittaker, Hanni and Ruffini and Cohen and Wald lies in the domain of definition of fundamental solutions in the presence of a surface of inversion when considering the situation in isotropic coordinates.\n\nFurthermore, we investigate the scope of applicability of Hadamard's theorem including its application to more general Scalar-Tensor-Vector theories and $f(R)$ theory and in particular to other branches of the Brans-Dicke theory such as the Barker and Schwinger cases (see also \\citep{Singh1984}). For detailed discussions on the scope of applicability of the Copson method in higher dimensions see \\citep{Linet2005}.\n\nIn a separate paper, we convert the results given in this paper from isotropic coordinates to the usual Schwarzschild coordinates using Linet's transformation (outlined in \\citep{Linet1976}) and plot equipotential surfaces in both coordinate systems. As alluded to in Item \\ref{bhadra} of Section \\ref{STFT}, \\citep{Bhadra2005} found that Brans-Dicke black holes cannot exist if the weak-field approximation is to be upheld. In our next paper we postulate that the weak-field approximation need not be satisfied \\citep{Barcelo2000}, \\citep{McInnes2002}, \\citep{Wu1986}, \\citep{Hawking1975}. Thus we find it worthwhile to plot the results of this paper in the usual coordinates to gain better insight into the behavior of the scalar field inside the horizon and thereby shed light on the physical possibility of Brans-Dicke black holes. \n\n\n\n\\section*{Acknowledgements}\nOne of the authors of this paper (M.W) would like to thank the Australian Government for the International Postgraduate Research Scholarship and Monash University for the Monash Graduate Scholarship.\n\nThe authors would also like to thank S. Deser for his helpful advice and an anonymous referee for their useful suggestions.\n\n\n\n\\begin{sidewaystable*}\n\\vspace*{8cm} \n\\hspace*{-2.5cm}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline \n  & Brans-Dicke-Reissner-Nordstr\\\"{o}m & Brans-Dicke & Reissner-Nordstr\\\"{o}m & Schwarzschild \\\\ \n\\hline \n\n\n{ \\rule{0cm}{0.7cm} $e^{2\\alpha}$  } \n& {$\\frac{e^{2\\alpha_0} \\left|\\frac{r-B}{r+B} \\right|^{\\frac{2}{\\lambda}}}{\\eta(r)^{2}}$  } \n& { $e^{2\\alpha_0}\\left|\\frac{r-B}{r+B} \\right|^{\\frac{2}{\\lambda}}$ } \n& { $\\frac{\\left(\\frac{r-B}{r+B} \\right)^2}{\\eta(r)^{2}}$ }  \n& { $\\left(\\frac{r-B}{r+B} \\right)^2$  }  \\\\ \n\\hline \n\n\n{ \\rule{0cm}{0.7cm} $e^{2\\beta}$ } \n& {$  e^{2\\beta_0} \\eta(r)^2  \\left( 1 + \\frac{B}{r} \\right)^{4} \\left| \\frac{r-B}{r+B} \\right|^{2 \\left( \\frac{\\lambda-C-1}{\\lambda} \\right)} $ }  \n& {$ e^{2\\beta_0} \\left( 1 + \\frac{B}{r} \\right)^{4} \\left| \\frac{r-B}{r+B} \\right|^{2 \\left( \\frac{\\lambda-C-1}{\\lambda} \\right)}$ } \n& {$ \\eta(r)^2 \\left( 1 + \\frac{B}{r} \\right)^{4} $ } \n& {$\\left( 1 + \\frac{B}{r} \\right)^{4}$ }  \\\\ \n\\hline \n\n\n{ \\rule{0cm}{0.7cm} $\\eta(r)$  } \n& { $ p_{+}^{2}-p_{-}^{2}\\left|\\frac{r-B}{r+B} \\right|^{2k}$ } \n& { $1$  }  \n& {$ p_{+}^{2}-p_{-}^{2}\\left(\\frac{r-B}{r+B} \\right)^{2}$ } \n& {$1 $ }  \\\\ \n\\hline \n\n\n{ \\rule{0cm}{0.7cm} $\\phi$ } \n& {$\\phi_{0} \\left| \\frac{r-B}{r+B} \\right|^{\\frac{C}{\\lambda}}$   } \n& {$\\phi_{0} \\left| \\frac{r-B}{r+B} \\right|^{\\frac{C}{\\lambda}}$  } \n& { $\\phi_0$  } \n& { $\\phi_0$ }  \\\\\n\\hline \n\n\n{\\rule{0cm}{0.7cm} $U_{0}$   } \n& {$\\frac{r}{b} \\frac{\\eta_0}{\\eta(r)} \\frac{(r-B)^{k - \\frac{1}{2}}} {(r+B)^{k + \\frac{1}{2}}} \\frac{(b+B)^{k + \\frac{1}{2}}} {(b-B)^{k - \\frac{1}{2}}}$  }\n& { $\\frac{r}{b} \\frac{(r-B)^{k - \\frac{1}{2}}}{(r+B)^{k + \\frac{1}{2}}} \\frac{(b+B)^{k + \\frac{1}{2}}}{(b-B)^{k - \\frac{1}{2}}} $  } \n& { $\\frac{r}{b} \\frac{\\eta_0}{\\eta(r)} \\frac{(r-B)^{\\frac{1}{2}}}{(r+B)^{\\frac{3}{2}}} \\frac{(b+B)^{\\frac{3}{2}}}{(b-B)^{\\frac{1}{2}}} $  } \n& { $\\frac{r}{b}  \\frac{(r-B)^{\\frac{1}{2}}}{(r+B)^{\\frac{3}{2}}} \\frac{(b+B)^{\\frac{3}{2}}}{(b-B)^{\\frac{1}{2}}} $  }  \\\\\n\\hline \n\n{\\rule{0cm}{0.7cm} $U_{1}$   } \n& { $\\frac{3B^2 \\left( 1+\\frac{4}{3}(1-k^2) \\right)}{2(r^2-B^2)(b^2-B^2)}U_0$ } \n& { $\\frac{3B^2 \\left( 1+\\frac{4}{3}(1-k^2) \\right)}{2(r^2-B^2)(b^2-B^2)}U_0$  } \n& { $\\frac{3B^2}{2(r^2-B^2)(b^2-B^2)}U_0$   } \n& { $\\frac{3B^2}{2(r^2-B^2)(b^2-B^2)}U_0$   }  \\\\\n\\hline \n\n{\\rule{0cm}{0.7cm} $U_{2}$   } \n& {$\\frac{B^2 \\left( 1+\\frac{4}{3}(1-k^2) \\right) \n\\left( -5+\\frac{4}{3}(1-k^2)\\right)}{8(r^2-B^2)^2(b^2-B^2)^2}U_0$   } \n&{$\\frac{B^2 \\left( 1+\\frac{4}{3}(1-k^2) \\right) \n\\left( -5+\\frac{4}{3}(1-k^2)\\right)}{8(r^2-B^2)^2(b^2-B^2)^2}U_0$   } \n& { $-\\frac{5B^4}{8(r^2-B^2)^2(b^2-B^2)^2}U_0$ } \n& { $-\\frac{5B^4}{8(r^2-B^2)^2(b^2-B^2)^2}U_0$  }  \\\\\n\\hline \n\n{\\rule{0cm}{0.7cm} $U_{3}$  } \n& {$\\frac{B^6 \\left( 1+\\frac{4}{3}(1-k^2)  \\right) \\left( -5+\\frac{4}{3}(1-k^2) \\right) \\left( -7+\\frac{4}{3}(1-k^2)  \\right)}{80(r^2-B^2)^3(b^2-B^2)^3}U_0$  } \n& { $\\frac{B^6 \\left( 1+\\frac{4}{3}(1-k^2)  \\right) \\left( -5+\\frac{4}{3}(1-k^2) \\right) \\left( -7+\\frac{4}{3}(1-k^2)  \\right)}{80(r^2-B^2)^3(b^2-B^2)^3}U_0$ } \n& {$-\\frac{7B^6}{80(r^2-B^2)^3(b^2-B^2)^3}U_0$  } \n& {$-\\frac{7B^6}{80(r^2-B^2)^3(b^2-B^2)^3}U_0$  }  \\\\\n\\hline \n\n{\\rule{0cm}{0.7cm} Eq. (\\ref{epsub})  }  \n& { $\\frac{r \\phi_{0}}{\\eta(r)(r+B)^{2}} \\left( \\frac{r-B}{r+B} \\right)^{k-1} F(\\gamma)$   } \n& {$\\frac{r}{(r+B)^{2}} \\left( \\frac{r-B}{r+B} \\right)^{k-1} F(\\gamma)$   } \n& { $\\frac{r \\phi_{0}}{\\eta(r)(r+B)^{2}}  F(\\gamma)$   } \n& { $\\frac{r \\phi_0}{(r+B)^{2}} F(\\gamma)$ }  \\\\ \n\\hline \n\n\n{\\rule{0cm}{0.7cm} $V(r,\\theta)$ }\n\n& \\parbox[t]{6cm}{$\\frac{\\epsilon_0 r}{\\eta(r) (r^2-B^2)} \\left[ \\frac{r-B}{r+B} \\right]^{k}  \\frac{bB (b^2-B^2)^{k-1}}{[p_+^2(b+B)^{2k}-p_-^2(b-B)^{2k}]} $ \n  \\\\  $\\times \\frac{p_+^2(\\sqrt{\\gamma+ 1}+ \\sqrt{\\gamma})^{2k} - p_-^2(\\sqrt{\\gamma+ 1}- \\sqrt{\\gamma})^{2k}  }{2\\sqrt{\\gamma}\\sqrt{\\gamma+1}}$ }\n\n&  \\parbox[t]{5cm}{$  \\frac{\\epsilon_0 r}{ r^2-B^2} \\left[ \\frac{r-B}{r+B} \\right]^{k}  \\frac{bB (b^2-B^2)^{k-1}}{(b+B)^{2k}}$\n \\\\ \n $ \\times\\frac{(\\sqrt{\\gamma+ 1}+ \\sqrt{\\gamma})^{2k} }{2\\sqrt{\\gamma}\\sqrt{\\gamma+1}}$ }\n  \n&  \\parbox[t]{5cm}{$ \\frac{\\epsilon_0 r}{\\eta(r) (r+B)^2}   \\frac{bB }{[p_+^2(b+B)^{2}-p_-^2(b-B)^{2}]}$\n \\\\ \n $ \\times \\frac{p_+^2(\\sqrt{\\gamma+ 1}+ \\sqrt{\\gamma})^{2} - p_-^2(\\sqrt{\\gamma+ 1}- \\sqrt{\\gamma})^{2}  }{2\\sqrt{\\gamma}\\sqrt{\\gamma+1}}$ \n}  \n& \\parbox[t]{2cm}{$ \\frac{\\epsilon_0 r}{ (r+B)^2}   \\frac{bB }{(b+B)^{2}} $\n \\\\ \n$ \\times  \\frac{(\\sqrt{\\gamma+ 1}+ \\sqrt{\\gamma})^{2} }{2\\sqrt{\\gamma}\\sqrt{\\gamma+1}}$ \n }\\\\\n\\hline \n\\end{tabular}\n\\caption{The four cases and their solutions for the electrostatic potential generated by a charged particle at $r=b$ where $k=\\frac{C+2}{2 \\lambda}$}\n\\label{tab:tab}\n\\end{sidewaystable*} \n\\cleardoublepage\n\\pagebreak\n\\newpage\n\n\n\n\\section*{Appendix A: Brans-Dicke Reissner-Nordstr\\\"{o}m Background} \\label{appendixa}\n\nThe Brans-Dicke electrovac equations (\\ref{fieldone}), (\\ref{fieldtwo}) and (\\ref{maxwell}) arising from the static spherically symmetric line element (\\ref{lineelement}) in isotropic coordinates can be simplified when the following substitutions are introduced \n\\begin{eqnarray} \\label{tildeA}\n\\tilde{A}(r) &:=& \\alpha(r) + \\frac{1}{2}[\\ln \\phi(r)],\\\\ \\label{tildeB}\n\\tilde{B}(r) &:=& \\beta(r) + \\frac{1}{2}[\\ln \\phi(r)].\n\\end{eqnarray}\nThe electrovac equations ($e^{2\\beta}G^t_t$, $e^{2\\beta}G^r_r$ and $e^{2\\beta}(G^r_r +G^\\theta_\\theta)$) from equation (\\ref{fieldone}) can then be written as\n\\begin{eqnarray} \\label{gtt}\n  2 \\tilde{B}''(r) + \\tilde{B}'(r)^2 + \\frac{4}{r} \\tilde{B}'(r) + \\frac{4 \\pi Q^2 e^{-2\\tilde{B}(r)}}{c^4 } \\rule{1.5cm}{0cm} \\nonumber \\\\ + \\frac{2\\omega+3}{4} ([\\ln \\phi(r)]')^2 =0, \\rule{0.5cm}{0cm}\\\\ \\label{grr}\n  \\tilde{B}'(r)^2 + 2\\tilde{A}'(r) \\tilde{B}'(r) + \\frac{2}{r} (\\tilde{A}'(r) + \\tilde{B}'(r)) \n\\rule{1.8cm}{0cm} \\nonumber \\\\ + \\frac{4 \\pi Q^2 e^{-2\\tilde{B}(r)}}{c^4}  - \\frac{2\\omega+3}{4} ([\\ln \\phi(r)]')^2= 0, \\rule{0.5cm}{0cm}  \\\\\n\\label{grrp}\n   \\tilde{A}''(r) + \\tilde{B}''(r) + ((\\tilde{A}(r)+ \\tilde{B})')^2  \\rule{3cm}{0cm} \\nonumber \\\\ + \\frac{3}{r} (\\tilde{A}'(r) +\\tilde{B}'(r))= 0, \\rule{0.5cm}{0cm} \n\\end{eqnarray}\n\nThe above three equations are not linearly independent, but instead are related via the following\n\\begin{eqnarray}\n- \\tilde{A}'(r) e^{2\\beta} G^t_t + \\left( \\frac{d}{dr}+ \\left(\\tilde{A}'(r)+ 2\\tilde{B}'(r) + \\frac{4}{r} \\right) \\right) e^{2\\beta}G^r_r \\nonumber \\\\  - 2 \\left(\\tilde{B}'(r) + \\frac{1}{r} \\right) e^{2\\beta} (G^r_r + G^\\theta_\\theta) =0. \\rule{1cm}{0cm}\n\\end{eqnarray}\n\nWe point out here that the integrations below are carried out formally without taking ito account the signature or actual boundary values of $\\tilde{A}_b$, $\\tilde{B}_b$, $\\tilde{A}_{b}'+\\tilde{B}_{b}'$, $\\phi_b$ and $\\phi_{b}'$ where the former are the corresponding values of $\\tilde{A}(r)$, $\\tilde{B}(r)$, $\\tilde{A}(r)'+\\tilde{B}(r)'$, $\\phi(r)$ and $\\phi(r)'$ at the boundary point at infinity. \n\nEquation (\\ref{grrp}) can be expressed as a Cauchy-Euler equation\n\\begin{eqnarray} \\label{grr2}\n(e^{\\tilde{A}(r)+\\tilde{B}(r)})'' + \\frac{3}{r} (e^{\\tilde{A}(r)+\\tilde{B}(r)})' =0,\n\\end{eqnarray}\nwhich can be solved to give \n\\begin{eqnarray} \\label{key}\ne^{\\tilde{A}(r)+\\tilde{B}(r)} = e^{\\tilde{A}_b+\\tilde{B}_b} \\left(1- \\frac{\\varepsilon^2 B^2}{r^2} \\right),\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\lim_{r \\rightarrow \\infty}r^3 (\\tilde{A}'(r) + \\tilde{B}'(r))= 2 \\varepsilon^2 B^2,\\\\\n\\varepsilon^2 \\in \\lbrace -1, +1 \\rbrace. \\rule{1.7cm}{0cm}\n\\end{eqnarray}\n\n\n\nThe reduced long-range scalar field wave equation (equation (\\ref{scalarfield})) can be written in terms of $\\tilde{A}$ and $\\tilde{B}$ as\n\\begin{eqnarray} \\label{scalarwave}\n\\left(  \\frac{r^2 e^{\\tilde{A}+\\tilde{B}}\\phi'(r)} {\\phi} \\right)'=0.\n\\end{eqnarray}\nBy integrating equation (\\ref{scalarwave}) twice from $r$ to infinity we obtain\n\\begin{eqnarray} \\label{36}\n\\phi= \\phi_0 \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{\\frac{C}{\\varepsilon \\lambda}},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\lambda^2= \\frac{\\varepsilon^2}{4} \\left((2 \\omega+3) C^2+(C+2)^2  \\right) >0.\n\\end{equation}\n\n\n\nWe rewrite the modified field equation (\\ref{grr}) into the following form\n\\begin{eqnarray} \\label{40}\n\\tilde{A}'(r)^2= (\\tilde{A}+ \\tilde{B})' \\left(\\tilde{A}' + \\tilde{B}' + \\frac{2}{r} \\right)  \\rule{1 cm}{0cm} \\nonumber \\\\ + \\frac{4 \\pi Q^2 e^{-2\\tilde{B}(r)}}{c^4 } - \\frac{2\\omega+3}{4} ([\\ln \\phi(r)]')^2 \\rule{0.25cm}{0cm}\n\\end{eqnarray}\n\nUsing equations (\\ref{key}) and (\\ref{36}), after some algebra we obtain a first order second degree separable differential equation\n\\begin{widetext}\n\\begin{eqnarray}\n\\left( \\frac{d}{dr} \\left( e^{-\\tilde{A}(r)} \\right)  \\right)^2&=& \\frac{4 \\pi Q^2 e^{-2(\\tilde{A}_b + \\tilde{B}_b)}}{c^4 (r^2- \\varepsilon^2 B^2)^2} \\left[ \\left( \\frac{c^4 B^2 e^{2(\\tilde{A}_b + \\tilde{B}_b)} (C+2)^2}{4\\pi Q^2 \\lambda^2}  \\right) e^{-2\\tilde{A}(r)} + 1 \\right].\n\\end{eqnarray}\nSince $e^{\\tilde{A}_b}=\\sqrt{\\phi_0}e^{\\alpha_b}$ and $e^{\\tilde{B}_b}=\\sqrt{\\phi_0}e^{\\beta_b}$ the solution to this equation gives\n\\begin{eqnarray} \\label{alphafinal}\ne^{-\\alpha(r)} =  e^{-\\alpha_b} \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{\\frac{C}{2 \\varepsilon \\lambda}} \\left( p_{+}^2 \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{\\frac{C+2}{2 \\varepsilon \\lambda}} - p_{-}^2  \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{-\\frac{C}{2 \\varepsilon \\lambda}} \\right),\n\\end{eqnarray}\nwhere $p_+$ and $p_-$ are given in equation (\\ref{pnp}).\n\\begin{eqnarray} \\label{betafinal}\ne^{\\beta(r)} = e^{\\beta_b} \\left( 1+ \\frac{\\varepsilon B^2}{r^2} \\right) \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{-\\frac{C}{2 \\varepsilon \\lambda}} \\left( p_{+}^2 \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{\\frac{C+2}{2 \\varepsilon \\lambda}} - p_{-}^2  \\left( \\frac{r-\\varepsilon B}{r+ \\varepsilon B} \\right)^{-\\frac{C}{2 \\varepsilon \\lambda}} \\right).\n\\end{eqnarray}\n\\end{widetext}\nWhen $\\varepsilon^2=+1$ the above coincides with the BDRN metric given in Theorem 1 in Section (\\ref{STFT}) above. The solutions corresponding to the Brans Type II, and Type III and IV solutions are given by setting, respectively, $\\epsilon^2=-1$ and taking the limit when $\\epsilon\\rightarrow 0$  (using L'Hopital's rule) on equations (\\ref{36}), (\\ref{alphafinal}) and (\\ref{betafinal}).\n\n\\section*{Appendix B:  Gauss' Theorem} \\label{appendixc}\n\nIn order to determine the integration constants in equation (\\ref{solution}) we use Gauss's theorem; a brief overview of which is given here.\nLet $\\Re$ be a region of 3-dimensional space residing in a hypersurface $\\varrho$ and let $\\partial \\Re$ be its closed 2-dimensional boundary. Gauss' theorem states that for the electric field $E^a$ (and indeed for any given vector field, see Wald \\citep{Wald})\n\\begin{eqnarray}\n\\int_{\\Re} \\nabla_a E^a d\\upsilon = \\int_{\\partial \\Re} E^a \\cdot n_a dS \\label{surfint}\n\\end{eqnarray}\nwhere $d\\upsilon$ is an element of spatial proper volume in $\\Re$, $n_a$ is the outward facing unit vector orthogonal to the closed 2-dimensional boundary $\\partial \\Re$ and $dS$ is the usual surface element $dS=r^{2}\\sin \\theta d\\theta d\\phi$.\n\nWe know that the electric field is related to the Faraday tensor by the following \n\\begin{eqnarray}\nE^a = F^{ab} n_b.\n\\end{eqnarray}  \nUsing the above and equation (\\ref{max}) we find that the electric field is indeed equal to the gradient of the electrostatic potential $V(r,\\theta,\\phi)$ and therefore the right-hand side of equation (\\ref{surfint}) can be written as \n\\begin{eqnarray}\n\\int_{d\\Re} E^a \\cdot n_a dS = \\int_{d\\Re} \\nabla V \\cdot \\hat{n} dS.\n\\end{eqnarray}\n\nFrom Maxwell's equations the left-hand side of equation (\\ref{surfint}) can be written as $\\int_{\\Re} J^{0} d \\upsilon$ where \n\\begin{equation}\nJ^0=-\\frac{4 \\pi \\epsilon_0}{cr^2} e^{-2\\alpha(r)-3 \\beta} \\delta(r-b) \\delta(\\cos \\theta- \\cos \\theta_0)\n\\end{equation}\nis the charge density. \n\nIt follows that for the region $B<b<r$ containing the point charge $-\\epsilon_{0}$ positioned at $r=b, \\theta=0$ the left-hand side of equation (\\ref{surfint}) becomes $-4 \\pi \\epsilon_{0}$ and for any region not containing the charge, i.e. $B<r<b$, the left-hand side vanishes. \n\nFor the Brans-Dicke Reisnner-Nordstr\\\"{o}m spacetime as $n^a$ is orthogonal to $\\partial \\Re$ the only term that remains is the $r$ term and equation (\\ref{surfint}) becomes\n\\begin{eqnarray} \\nonumber\n\\int_{\\Re} J^{0} d \\upsilon = \\eta(r)^{2} (r+B)^{2} \\left(\\frac{r-B}{r+B} \\right)^{1-2k}   \\\\ \\times \\int_{0}^{2\\pi}  \\int_{-\\pi}^{\\pi}  \\frac{\\partial V(r, \\theta)}{\\partial r} \\sin \\theta d\\theta d\\phi. \\label{intre}\n\\end{eqnarray}\n\n\n\n\n$\\rule{0cm}{1cm}$ The left-hand side of equation ({\\ref{intre}) is determined by whether or not the region $\\Re$ contains the singular point at $r=b$. In particular for the purposes of this investigation, the theorem determines the choice of integration constants in $V(r, \\theta)$ as can be seen in the main section of this article. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe familiar generalized hypergeometric function $_{p}F_{q}$ is defined as\nfollows (see,\\cite{Foxc, Rainville}): \n\\begin{equation}  \\label{eqn-1-hyper}\n_{p}F_{q}\\left[ \n\\begin{array}{r}\n\\left( a _{p}\\right) ; \\\\ \n\\left( c _{q}\\right) \n\\end{array}\nx\\right] =\\sum\\limits_{n=0}^{\\infty }\\frac{\\Pi _{j=1}^{p}\\left( a_{j}\\right)\n_{n}}{\\Pi _{j=1}^{q}\\left( c _{j}\\right) _{n}}\\frac{x^{n}}{n! },\n\\end{equation}\n\\begin{equation*}\n(p\\leq q,\\, x \\in \\mathbb{C};\\,\\, p=q+1,\\, \\left\\vert x\\right\\vert <1),\n\\end{equation*}\nwhere $(a)_{n}$ denoted by Pochhammer symbol given by\n$$(a)_{n}=a(a+1)_{n-1}; (a)_{0}=1.$$\n\nIn particular, if $p=2$ and $q=1$,  \\eqref{eqn-1-hyper} reduced to Gaussian hypergeometric function,\n\\begin{equation}\\label{2F1}\n{}_2F_{1}(a, b; c; x)=\\sum_{k=0}^{\\infty}\\frac{(a)_{n}(b)_{n}}{(c)_{n}}\\frac{x^{n}}{n!}.\n\\end{equation}\n\nThe polynomial $M_{n}^{\\left( p,q\\right) }\\left( x\\right) $ is the solution\nof the differential equatio\n\\begin{equation}\nx\\left( x+1\\right) y_{n}^{\\prime \\prime }\\left( x\\right) +\\left( 2-p\\right)\nx+\\left( 1+q\\right) y_{n}^{\\prime }\\left( x\\right) -n\\left( n-1+p\\right)\ny_{n}\\left( x\\right) =0,  \\label{Eq1}\n\\end{equation\nis defined b\n\\begin{equation}\nM_{n}^{\\left( p,q\\right) }\\left( x\\right) =\\left( -1\\right)\n^{n}n!\\sum_{k=0}^{n}\\left( \n\\begin{array}{c}\np-n-1 \\\\ \n\n\\end{array\n\\right) \\left( \n\\begin{array}{c}\nq+n \\\\ \nn-\n\\end{array\n\\right) \\left( -x\\right) ^{k}.  \\label{Eq2}\n\\end{equation\nThese polynomials are orthogonal on $\\left[ 0,\\infty \\right] $ with respect\nto the weight function $w_{p,q}\\left( x\\right) =x^{q}\\left( 1+x\\right)\n^{-\\left( p+q\\right) }$ if and only if $p>2n+1$ and $q>-1.$\n\nThe relation between $M_{n}^{\\left( p,q\\right) }\\left( x\\right) $ and\n\\eqref{2F1} can be expressed as\n\\begin{equation}\nM_{n}^{\\left( p,q\\right) }\\left( x\\right) =\\left( -1\\right) ^{n}n!\\left( \n\\begin{array}{c}\nq+n \\\\ \n\n\\end{array\n\\right) \\text{ }_{2}F_{1}\\left( -n,n+1-p;q+1;-x\\right).   \\label{Eq3}\n\\end{equation\nThe relation between Jacobi polynomial $P_{n}^{\\left( \\alpha ,\\beta \\right) }\n$ and $M_{n}^{\\left( p,q\\right) }$ is given by \n\\begin{equation}\nM_{n}^{\\left( p,q\\right) }\\left( x\\right) =\\left( -1\\right)\n^{n}n!P_{n}^{\\left( a,-p-q\\right) }\\left( 2x+1\\right),   \\label{Eq4}\n\\end{equation\nwhich can be expressed as \n\\begin{equation}\nP_{n}^{\\left( a,-p-q\\right) }\\left( x\\right) =\\frac{\\left( -1\\right) ^{n}}{n\n}M_{n}^{\\left( -p-q,p\\right) }\\left( \\frac{x-1}{2}\\right).   \\label{Eq5}\n\\end{equation\nFor more details about Jacobi polynomials and related results, one can refer \\cite{Malik, Malik1, Malik2, Malik3}.\n\n\n\n\\vskip3mm \nThen the generalized fractional integral operators involving the Appell\nfunctions $F_{3}$ are defined for $\\delta $, $\\delta ^{\\prime }$, $\\mu $, $\\mu ^{\\prime }$, \n\\epsilon \\in \\mathbb{C}$ with $\\Re (\\epsilon )>0$ and $x\\in \\mathbb{R}^{+}$as follows:\n\\begin{equation}\n\\left( I_{0{+}}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}f\\right) (x)=\\frac{x^{-\\delta }}{\\Gamma (\\epsilon )}\\int_{0}^{x}(x-t)^\n\\epsilon -1}t^{-\\delta ^{\\prime }}F_{3}\\left( \\delta ,\\delta ^{\\prime },\\mu\n,\\mu ^{\\prime };\\epsilon ;1-\\frac{t}{x},1-\\frac{x}{t}\\right) f(t)\\,\\mathrm{d}t,\n\\label{Int-1}\n\\end{equation\nand \n\\begin{equation}\n\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}f\\right) (x)=\\frac{x^{-\\delta ^{\\prime }}}{\\Gamma (\\epsilon )\n\\int_{x}^{\\infty }(t-x)^{\\epsilon -1}t^{-\\delta }F_{3}\\left( \\delta ,\\delta\n^{\\prime },\\mu ,\\mu ^{\\prime };\\epsilon ;1-\\frac{t}{x},1-\\frac{x}{t}\\right)\nf(t)\\,\\mathrm{d}t.  \\label{Int-2}\n\\end{equation\nThe integral operators of the types \\eqref{Int-1} and\n\\eqref{Int-2} have been introduced by Marichev \\cite{Marichev} and later\nextended and studied by Sagio and Maeda \\cite{Saigo}. Recently, many researchers (see, \\cite{Baleanu, Kumar, SRN, Purohit}) have studied the image formulas for MSM fractional integral operators involving various special functions.\n\nThe corresponding fractional differential operators have their respective\nforms: \n\\begin{equation}\n\\left( D_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}f\\right) \\left( x\\right) =\\left( \\frac{\\mathrm{d}}{\\mathrm{d}x}\\right) ^\n\\left[ \\Re \\left( \\epsilon \\right) \\right] +1}\\left( I_{0+}^{-\\delta ^{\\prime\n},-\\delta ,-\\mu ^{\\prime }+\\left[ \\Re \\left( \\epsilon \\right) \\right] +1,-\\mu\n,-\\epsilon +\\left[ \\Re \\left( \\epsilon \\right) \\right] +1}f\\right) \\left(\nx\\right)  \\label{Dif-1}\n\\end{equation\nand \n\\begin{equation}\n\\left( D_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}f\\right) \\left( x\\right) =\\left( -\\frac{\\mathrm{d}}{\\mathrm{d}x}\\right) ^\n\\left[ \\Re \\left( \\epsilon \\right) \\right] +1}\\left( I_{-}^{-\\delta ^{\\prime\n},-\\delta ,-\\mu ^{\\prime },-\\mu +\\left[ \\Re \\left( \\epsilon \\right) \\right]\n+1,-\\epsilon +\\left[ \\Re \\left( \\epsilon \\right) \\right] +1}f\\right) \\left(\nx\\right) .  \\label{Dif-2}\n\\end{equation\n\nHere, we recall the following results (see \\cit\n{Kilbas-itsf, Saigo}):\n\n\\begin{lemma}\n\\label{lem-1} Let $\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n,\\tau \\in \\mathbb{C}$ be such that $Re{(\\epsilon)}>0$ and \n\\begin{equation*}\n\\Re{(\\tau)}>\\max \\{0,\\Re{(\\delta-\\delta^{\\prime }-\\mu-\\epsilon)},\\Re\n(\\delta^{\\prime }-\\mu^{\\prime })}\\}.\n\\end{equation*\nthen there exists the relation \n\\begin{equation}\n\\left( I_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( \\tau \\right) \\Gamma \\left(\n\\tau +\\epsilon -\\delta -\\delta ^{\\prime }-\\mu \\right) \\Gamma \\left( \\tau\n+\\mu {^{\\prime }}-\\delta {^{\\prime }}\\right) }{\\Gamma \\left( \\tau +\\mu \n^{\\prime }}\\right) \\Gamma \\left( \\tau +\\epsilon -\\delta -\\delta {^{\\prime }\n\\right) \\Gamma \\left( \\tau +\\epsilon -\\delta {^{\\prime }}-\\mu \\right) \nx^{\\tau -\\delta -\\delta ^{\\prime }+\\epsilon -1}\n\\end{equation\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem-2} Let $\\delta $, $\\delta ^{\\prime }$, $\\mu $, $\\mu ^{\\prime }\n, $\\epsilon $, $\\tau \\in \\mathbb{C}$ such that $\\Re (\\epsilon )>0$ and \n\\begin{equation*}\n\\Re {(\\tau )}>\\max \\{\\Re {(\\mu )},\\,\\Re {(-\\delta -\\delta ^{\\prime\n}+\\epsilon )},\\,\\Re {(-\\delta -\\mu ^{\\prime }+\\epsilon )}\\},\n\\end{equation*\nthen  \n\\begin{equation}\n\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau }\\right) (x)  \\label{Lem2-Eq2} \\\\\n=\\frac{\\Gamma \\left( -\\mu +\\tau \\right) \\Gamma \\left( \\delta +\\delta\n^{\\prime }-\\epsilon +\\tau \\right) \\Gamma \\left( \\delta +\\mu {\\ ^{\\prime }\n-\\epsilon +\\tau \\right) }{\\Gamma \\left( \\tau \\right) \\Gamma \\left( \\delta\n-\\mu +\\tau \\right) \\Gamma \\left( \\delta +\\delta {^{\\prime }+\\mu \n^{^{\\prime }}-\\epsilon +\\tau \\right) }x^{-\\delta -\\delta ^{\\prime }+\\epsilon\n-\\tau }.  \\notag\n\\end{equation}\n\\end{lemma}\nAlso, we need the following lemmas \\cite{Kilbas-itsf}:\n\\begin{lemma}\\label{lem-3}\nLet $\\delta, \\mu, \\epsilon \\in \\mathbb{C}$ be such that $\\Re(\\delta)>0, \\Re(\\tau)>max[0, \\Re(\\mu-\\epsilon)]$\nthen\n\\begin{equation}\\label{lem3-eqn1}\n\\left( I_{0+}^{\\delta ,\\mu,\\epsilon\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( \\tau \\right) \\Gamma \\left(\n\\tau +\\epsilon -\\mu \\right)}{\\Gamma \\left( \\tau -\\mu\\right) \\Gamma \\left( \\tau +\\epsilon +\\delta\n\\right)\nx^{\\tau -\\mu -1}.\n\\end{equation\n\\end{lemma}\n\nIn particular,\n\\begin{equation}\\label{lem3-eqn2}\n\\left( I_{0+}^{\\delta\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( \\tau \\right)}{\\Gamma \\left( \\tau+\\delta\n\\right)\nx^{\\tau+ \\delta-1}, \\Re(\\delta)>0, \\Re(\\tau)>0,\n\\end{equation\n\n\\begin{equation}\\label{lem3-eqn3}\n\\left( I_{\\epsilon,\\delta}^{+\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( \\tau+\\epsilon \\right)}{\\Gamma \\left( \\tau+\\delta+\\epsilon\n\\right)\nx^{\\tau+ \\delta-1}, \\Re(\\delta)>0, \\Re(\\tau)>\\Re(\\epsilon).\n\\end{equation\n\n\\begin{lemma}\\label{lem-4}\nLet $\\delta, \\mu, \\epsilon \\in \\mathbb{C}$ be such that $\\Re(\\delta)>0, \\Re(\\tau)<1+min[\\Re(\\mu), \\Re(\\epsilon)]$\nthen \n\\begin{equation}\\label{lem4-eqn1}\n\\left( I_{-}^{\\delta ,\\mu,\\epsilon\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( \\mu-\\tau+1 \\right) \\Gamma \\left(\n\\epsilon -\\tau+1 \\right)}{\\Gamma \\left( 1-\\tau\\right) \\Gamma \\left( \\delta+\\mu+\\epsilon-\\tau+1\n\\right)\nx^{\\tau -\\mu -1}.\n\\end{equation\n\\end{lemma}\n\nIn particular,\n\\begin{equation}\\label{lem4-eqn2}\n\\left( I_{-}^{\\delta\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left( 1-\\delta-\\tau \\right)}{\\Gamma \\left(1- \\tau\n\\right)\nx^{\\tau+ \\delta-1}, 0<\\Re(\\delta)<1-\\Re(\\tau),\n\\end{equation\n\n\\begin{equation}\\label{lem4-eqn3}\n\\left( K_{\\epsilon,\\delta}^{+\n}\\;t^{\\tau -1}\\right) (x)=\\frac{\\Gamma \\left(\\epsilon-\\tau+1 \\right)}{\\Gamma \\left(\\delta+\\epsilon-\\tau+1\n\\right)\nx^{\\tau-1}, \\Re(\\tau)<1+\\Re(\\epsilon).\n\\end{equation\n\nThe main aim of this paper is to apply the generalized operators of fractional\ncalculus for the Jacobi type orthogonal polynomials in order to get certain new\nimage formulas. The basic definitions and results of fractional calculus,\none may refer to \\cite{Kim, Kiryakova, Miller, Srivastava2}.\n\n\\section{Fractional integrals of Jacobi type orthogonal polynomials}\nIn this section, we derive the following theorems,\n\\begin{theorem} \\label{Th1}\nLet $\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon ,\\tau\n\\in \\mathbb{C}$ be such that $Re{(\\epsilon )}>0$ and \n\\begin{equation*}\n\\Re {(\\tau )}>\\max \\{0,\\Re {(\\delta -\\delta ^{\\prime }-\\mu -\\epsilon )\n,\\Re {(\\delta ^{\\prime }-\\mu ^{\\prime })}\\}.\n\\end{equation*\nthen \n\\begin{eqnarray*}\n&&\\left( I_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{\\tau -1}~M_{n}^{\\left( p,q\\right) }\\left( t\\right) \\right) (x) \\\\\n&=&x^{\\tau -\\delta -\\delta ^{^{\\prime }}+\\epsilon -1}\\frac{\\left( -1\\right)\n^{n}\\Gamma \\left( q+n+1\\right)\\Gamma(\\tau)\\Gamma(\\tau+\\delta-\\delta'-\\mu)\\Gamma(\\tau+\\mu'-\\delta') }{ \\Gamma \\left(q+1\\right)\\Gamma(\\tau+\\mu')\\Gamma(\\tau+\\epsilon-\\delta-\\delta')\\Gamma(\\tau+\\epsilon-\\delta'-\\mu) } \\\\\n&&\\times _{5}F _{4}\\left[ \n\\begin{array}{c}\n1+n-p, -n, \\tau, \\tau+\\delta-\\delta'-\\mu, \\tau+\\mu'-\\delta', \\\\ \n1+q, \\tau+\\mu', \\tau+\\epsilon-\\delta-\\delta', \\tau+\\epsilon-\\delta'-\\mu\n\\end{array\n\\Big|-x\\right] .\n\\end{eqnarray*}\n\\end{theorem}\n\n\\begin{proof}\n\\label{Pf1} Applying the definition of $M_{n}^{\\left( p,q\\right) }\\left(\nx\\right) $ given in \\eqref{Eq2} and denote the left hand side by \\emph{I}$_{1}$, we\nget \n\\begin{eqnarray*}\n\\emph{I}_{1} &=&\\left( I_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime\n},\\epsilon }t^{\\tau -1}~M_{n}^{\\left( p,q\\right) }\\left( t\\right) \\right) (x),\n\\\\\n&=&\\left( I_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{\\tau -1}\\left( -1\\right) ^{n}n!\\sum\\limits_{k=0}^{\\infty }\\left( \n\\begin{array}{c}\np-n-1 \\\\ \n\n\\end{array\n\\right) \n\\left( \n\\begin{array}{c}\nq+n \\\\ \nn-\n\\end{array\n\\right) \\text{ }\\left( -t\\right) ^{k}\\right) \\left( x\\right) ,\n\\end{eqnarray*\nOn interchanging the integration and summation, we obtain \n\\begin{equation*}\n\\emph{I}_{1}=\\left( -1\\right) ^{n}n!\\sum_{k=0}^{\\infty }\n\\frac{\\Gamma(p-n)}{k!\\Gamma(p-n-k)}\\frac{\\Gamma(q+n+1)(-1)^{k}}{\\Gamma(n-k+1)\\Gamma(q+k+1)}\\left( I_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}(t^{\\tau+k-1})\\right)(x)\n\\end{equation*\nNow, for any $k=0,1,2,...$.Since  \n\\begin{equation*}\n\\Re \\left( \\tau +k\\right) \\geq \\Re \\left( \\tau \\right) >\\max \\left[ 0,\\Re\n\\left( \\delta -\\delta ^{^{\\prime }}-\\mu -\\epsilon \\right) ,\\Re \\left(\n\\delta ^{^{\\prime }}-\\mu ^{^{\\prime }}\\right) \\right] ,\n\\end{equation*\nand applying Lemma \\ref{lem-1} and after some calculations, we ge\n\\begin{eqnarray*}\n\\emph{I}_{1} &=&\\sum_{k=0}^{\\infty }\\frac{\\left( -1\\right) ^{n}\\Gamma \\left( q+n+1\\right) (n-p)_{k}(-n)_{k}(-1)^{k}}{k\n\\Gamma \\left(q+1+k\\right)} \\\\\n&&\\times \\frac{\\Gamma \\left( \\tau +k\\right) \\Gamma \\left( \\tau +k+\\delta\n-\\delta ^{^{\\prime }}-\\mu \\right) \\Gamma \\left( \\tau +k+\\mu ^{\\prime\n}-\\delta ^{^{\\prime }}\\right) }{\\Gamma \\left( \\tau +k+\\mu ^{^{\\prime\n}}\\right) \\Gamma \\left( \\tau +k+\\epsilon -\\delta -\\delta ^{^{\\prime\n}}\\right) \\Gamma \\left( \\tau +k+\\epsilon -\\delta ^{\\prime }-\\mu \\right) } \\\\\n&&\\times x^{\\tau +k-\\delta -\\delta ^{^{\\prime }}+\\epsilon -1}.\n\\end{eqnarray*\nIn view of \\eqref{eqn-1-hyper} and using the well known relation \n\\begin{equation}\\label{GR}\n\\Gamma(x+k)=(x)_{k}\\Gamma(x),\n\\end{equation} \nwe arrive desired result.\n\\end{proof}\nIf we set $\\delta', \\mu'=0, \\delta=\\delta+\\mu, \\mu=-\\epsilon$ and $\\epsilon=\\delta$ in theorem \\ref{Th1}, then we get the following result,\n\\begin{corollary}\\label{Cor1} \nLet $\\delta, \\mu, \\epsilon \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)>max[0, \\Re(\\mu-\\epsilon)]$, then\n\\begin{eqnarray*}\n\\left( I_{0+}^{\\delta ,\\mu ,\\epsilon }t^{\\tau -1}~M_{n}^{\\left( p,q\\right)\n}\\left( t\\right) \\right) (x)\n=\\frac{\\left( -1\\right) ^{n}\\Gamma \\left( \\tau \\right) \\Gamma \\left( \\tau\n+\\epsilon -\\mu \\right) \\Gamma \\left( q+n+1\\right) }{\\Gamma \\left( 1+q\\right)\n\\Gamma \\left( \\tau -\\mu \\right) \\Gamma \\left( \\tau +\\delta +\\epsilon \\right) \nx^{\\tau -\\mu -1} \\\\\n\\times _{4}F_{3}\\left[ \n\\begin{array}{c}\n1+n-p,-n,\\tau ,\\tau +\\epsilon -\\mu,  \\\\ \n1+q,\\tau-\\mu ,\\tau+\\epsilon +\\delta, \n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\n\\end{array\n\\right. \\right] .\n\\end{eqnarray*}\nwhich is theorem 2.1 of \\cite{Malik}.\n\\end{corollary}\n\nSubstituting $\\mu=-\\delta$ in corollary \\ref{Cor1} and using \\eqref{Eq2}, we get ,\n\\begin{corollary}\\label{Cor2} \nLet $\\delta, \\tau \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)>0$, then\n\\begin{eqnarray*}\n\\left( I_{0+}^{\\delta}t^{\\tau-1}~M_{n}^{\\left( p,q\\right)\n}\\left( t\\right) \\right) (x)\n=\\frac{\\left( -1\\right) ^{n}\\Gamma(1+q+n)\\Gamma \\left( \\tau \\right)}{\\Gamma \\left( 1+q\\right)\n\\Gamma \\left( \\tau +\\delta\\right) \nx^{\\tau+\\delta-1} \\\\\n\\times _{3}F_{2}\\left[ \n\\begin{array}{c}\n1+n-p,-n,\\tau, \\\\ \n1+q,\\tau+\\delta, \n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\n\\end{array\n\\right. \\right] \n\\end{eqnarray*}\n\\end{corollary}\n\n\nSubstituting $\\mu=0$ in corollary \\ref{Cor1}, we have,\n\\begin{corollary}\\label{Cor3} \nLet $\\delta, \\epsilon, \\tau \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)>\\Re(\\epsilon)$, then\n\\begin{eqnarray*}\n\\left( I_{\\epsilon, \\delta}^{+}t^{\\tau-1}~M_{n}^{\\left( p,q\\right)\n}\\left( t\\right) \\right) (x)\n=\\frac{\\left( -1\\right) ^{n}\\Gamma(1+q+n)\\Gamma \\left( \\tau+\\epsilon \\right)}{\\Gamma \\left( 1+q\\right)\n\\Gamma \\left( \\tau +\\delta+\\epsilon\\right) \nx^{\\tau-1} \\\\\n\\times _{3}F_{2}\\left[ \n\\begin{array}{c}\n1+n-p,-n,\\tau+\\epsilon,  \\\\ \n1+q,\\tau+\\delta+\\epsilon, \n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\n\\end{array\n\\right. \\right] \n\\end{eqnarray*}\n\\end{corollary}\n\n\\begin{theorem}\n\\label{Th2} Let $\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n,\\tau\\in \\mathbb{C}$,   and \n\\begin{equation*}\n\\Re(\\epsilon)>0, \\Re {(\\tau )}>\\max \\{\\Re(\\mu), \\Re {(-\\delta -\\delta ^{\\prime }+\\epsilon^{\\prime } )},\\Re \n(-\\delta-\\mu ^{\\prime }+\\epsilon )},\n\\end{equation*\nthen \n\\begin{eqnarray*}\n&&\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau }~M_{n}^{\\left( p,q\\right) }\\left( \\frac{1}{t}\\right) \\right) (x) \\\\\n&=&x^{-\\delta -\\delta ^{^{\\prime }}+\\epsilon -\\tau}\\frac{\\left( -1\\right)\n^{n}\\Gamma \\left( q+n+1\\right)\\Gamma(\\tau-\\mu)\\Gamma(\\delta+\\delta^{\\prime}-\\epsilon+\\tau)\\Gamma(\\delta+\\mu^{\\prime}-\\epsilon+\\tau) }{\\Gamma \\left( \\tau\\right) \\Gamma \\left(q+1\\right)\\Gamma(\\delta-\\mu+\\tau)\\Gamma(\\delta+\\delta^{\\prime}+\\mu^{\\prime}-\\epsilon+\\tau) } \\\\\n&&\\times _{5}\\Psi _{4}\\left[ \n\\begin{array}{c}\n1+n-p, -n, -\\mu+\\tau, \\delta+\\delta^{\\prime}-\\epsilon+\\tau, \\delta+\\mu^{\\prime}-\\epsilon+\\tau , \\\\ \n1+q, \\tau, \\delta-\\mu+\\tau, \\delta+\\delta^{\\prime}+\\mu^{\\prime}-\\epsilon+\\tau \n\\end{array\n\\Big|-\\frac{1}{x}\\right] .\n\\end{eqnarray*}\n\\end{theorem}\n\n\\begin{proof}\\label{Pf2}  \n Applying the definition of $M_{n}^{\\left( p,q\\right) }\\left(\nx\\right) $ in \\ref{Eq2} and denote the left hand side by \\emph{I}$_{2}$, we\nget \n\\begin{eqnarray*}\n\\emph{I}_{2} &=&\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau }~M_{n}^{\\left( p,q\\right) }\\left( \\frac{1}{t}\\right) \\right) (x)\n\\\\\n&=&\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau }\\left( -1\\right) ^{n}n!\\sum\\limits_{k=0}^{\\infty }\\left( \n\\begin{array}{c}\np-n-1 \\\\ \n\n\\end{array\n\\right) \n\\left( \n\\begin{array}{c}\nq+n \\\\ \nn-\n\\end{array\n\\right) \\text{ }\\left( -\\frac{1}{t}\\right) ^{k}\\right) \\left( x\\right) ,\n\\end{eqnarray*\nOn interchanging the integration and summation, we obtain \n\\begin{equation*}\n\\emph{I}_{2}=\\left( -1\\right) ^{n}n!\\sum_{k=0}^{\\infty }\n\\frac{\\Gamma(p-n)}{k!\\Gamma(p-n-k)}\\frac{\\Gamma(q+n+1)(-1)^{k}}{\\Gamma(n-k+1)\\Gamma(q+k+1)}\\left( I_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}(t^{-\\tau-k})\\right)(x)\n\\end{equation*\nNow, for any $k=0,1,2,...$.Since  \n\\begin{equation*}\n\\Re \\left( \\tau+k \\right) >\\max \\left[ \\Re(\\mu),\\Re\n\\left(-\\delta -\\delta ^{^{\\prime }}-\\epsilon'\\right) ,\\Re \\left(\n-\\delta-\\mu^{\\prime }+\\epsilon\\right) \\right] ,\n\\end{equation*\nand applying Lemma \\ref{lem-2} and some simple calculations, we ge\n\\begin{eqnarray*}\n\\emph{I}_{2} &=&\\sum_{k=0}^{\\infty }\\frac{\\left( -1\\right) ^{n}\\Gamma \\left( q+n+1\\right) (n-p)_{k}(-n)_{k}(-1)^{k}}{k\n\\Gamma \\left(q+1+k\\right)} \\\\\n&&\\times \\frac{\\Gamma \\left( -\\mu+\\tau +k\\right) \\Gamma \\left( \\tau+\\delta\n+\\delta ^{^{\\prime }}-\\epsilon+k \\right) \\Gamma \\left( \\tau+\\mu ^{\\prime\n}+\\delta-\\epsilon+k\\right) }{\\Gamma \\left( \\tau +k\\right) \\Gamma \\left( \\tau +k+\\delta-\\mu\\right) \\Gamma \\left( \\delta+\\delta^{\\prime}+\\mu^{\\prime}-\\epsilon+\\tau+k \\right) } \\\\\n&&\\times x^{-\\tau-k-\\delta -\\delta ^{^{\\prime }}+\\epsilon}.\n\\end{eqnarray*\nIn view of \\eqref{eqn-1-hyper} and \\eqref{GR}, we reach the required result.\n\\end{proof}\n\nIf we set $\\delta', \\mu'=0, \\delta=\\delta+\\mu, \\mu=-\\epsilon$ and $\\epsilon=\\delta$ in theorem \\ref{Th2}, then we get the following result,\n\\begin{corollary}\\label{Cor4} \nLet $\\delta, \\mu, \\epsilon \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)>1+min[\\Re(\\mu), \\Re(\\epsilon)]$, then\n\\begin{eqnarray}\n\\left( I_{-}^{\\delta ,\\mu ,\\epsilon }t^{\\tau -1}~M_{n}^{\\left( p,q\\right)\n}\\left( \\frac{1}{t}\\right) \\right) (x)\n=\\frac{\\Gamma(1+q+n)\\Gamma(\\mu-\\tau+1)\\Gamma(\\epsilon-\\tau+1)}{\\Gamma(1+q)\\Gamma(1-\\tau)\\Gamma(\\delta+\\mu+\\epsilon-\\tau+1)}x^{\\tau-\\mu-1} \\notag\\\\\n\\times _{4}F_{3}\\left[ \n\\begin{array}{c}\n1+n-p,-n,\\mu-\\tau+1, \\epsilon-\\tau+1,  \\\\ \n1+q,1-\\tau ,\\delta+\\mu+\\epsilon-\\tau+1,\n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\\frac{1}{x\n\\end{array\n\\right. \\right] \n\\end{eqnarray}\n\\end{corollary}\nwhich is theorem 2.2 of \\cite{Malik}.\n\n\nSubstituting $\\mu=-\\delta$ in corollary \\ref{Cor4} and using \\eqref{Eq2}, we get,\n\\begin{corollary}\\label{Cor5} \nLet $\\delta, \\epsilon, \\tau \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)<1+min[\\Re(-\\delta), \\Re(\\epsilon)]$ and $\\tau+\\delta \\neq 1,2,...$, then\n\\begin{eqnarray*}\n\\left( I_{-}^{\\delta}t^{\\tau-1}~M_{n}^{\\left( p,q\\right)\n}\\left( \\frac{1}{t}\\right) \\right) (x)\n=\\frac{\\left( -1\\right) ^{n}\\Gamma(1+q+n)\\Gamma \\left(-\\delta- \\tau+1 \\right)}{\\Gamma \\left( 1+q\\right)\n\\Gamma \\left(1-\\tau\\right) \nx^{\\tau+\\delta-1} \\\\\n\\times _{3}F_{2}\\left[ \n\\begin{array}{c}\n1+n-p,-n,-\\delta-\\tau+1,  \\\\ \n1+q,1-\\tau, \n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\\frac{1}{x\n\\end{array\n\\right. \\right] \n\\end{eqnarray*}\n\\end{corollary}\n\n\nSubstituting $\\mu=0$ in corollary \\ref{Cor4}, we have,\n\\begin{corollary}\\label{Cor6} \nLet $\\delta, \\epsilon, \\tau \\in \\mathbb{C}$ such that $\\Re(\\delta)>0, \\Re(\\tau)<1+min[0, \\Re(\\epsilon)]$ and let $\\tau-\\epsilon \\neq 1,2,...$, then\n\\begin{eqnarray*}\n\\left( K_{\\epsilon, \\delta}^{-}t^{\\tau-1}~M_{n}^{\\left( p,q\\right)\n}\\left( \\frac{1}{t}\\right) \\right) (x)\n=\\frac{\\left( -1\\right) ^{n}\\Gamma(1+q+n)\\Gamma \\left(\\epsilon-\\tau+1 \\right)}{\\Gamma \\left( 1+q\\right)\n\\Gamma \\left(\\delta+\\epsilon-\\tau+1\\right) \nx^{\\tau-1} \\\\\n\\times _{3}F_{2}\\left[ \n\\begin{array}{c}\n1+n-p,-n,\\epsilon-\\tau+1,  \\\\ \n1+q,\\delta+\\epsilon-\\tau+1, \n\\end{array\n\\left\\vert \n\\begin{array}{c}\n-\\frac{1}{x\n\\end{array\n\\right. \\right] \n\\end{eqnarray*}\n\\end{corollary}\n\n\\section{Fractional differentials of Jacobi type polynomials}\n\n\\label{Sec3}\n\nIn this section devoted to derive the MSM fractional differentiation of \\eqref{Eq2}. We recall the following lemmas\n(see \\cite{Kataria}).\n\n\\vskip 3mm\n\n\\begin{lemma}\n\\label{lem-5} Let $\\delta $, $\\delta ^{\\prime },$ $\\mu ,$ $\\mu ^{\\prime }\n, $\\epsilon $, $\\tau \\in \\mathbb{C}$ such that \n\\begin{equation*}\n\\Re \\left( \\tau \\right) >\\max \\left\\{ 0,\\Re \\left( -\\delta +\\mu \\right)\n,\\Re \\left( -\\delta -\\delta ^{\\prime }-\\mu +\\epsilon \\right)\n\\right\\} .\n\\end{equation*\nThen \n\\begin{eqnarray}\n&&\\left( D_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{\\tau -1}\\right) \\left( x\\right)  \\label{D1} \\\\\n&=&\\frac{\\Gamma \\left( \\tau \\right) \\Gamma \\left( -\\mu +\\delta +\\tau\n\\right) \\Gamma \\left( \\delta +\\delta ^{\\prime }+\\mu ^{^{\\prime }}-\\epsilon\n+\\tau \\right) }{\\Gamma \\left( -\\mu +\\tau \\right) \\Gamma \\left( \\delta\n+\\delta ^{^{\\prime }}-\\epsilon +\\tau \\right) \\Gamma \\left( \\delta +{\\mu \n^{^{\\prime }}-\\epsilon +\\tau \\right) }x^{\\delta +\\delta ^{\\prime }-\\epsilon\n+\\tau -1}.  \\notag\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem-6} Let $\\delta $, $\\delta ^{\\prime }$, $\\mu $, $\\mu ^{\\prime }\n, $\\epsilon $, $\\tau \\in \\mathbb{C}$ such that \n\\begin{equation*}\n\\Re \\left( \\tau \\right) >\\max \\left\\{ \\Re \\left( -\\mu ^{\\prime }\\right) ,\\Re\n\\left( \\delta ^{\\prime }+\\mu -\\epsilon \\right) ,\\Re \\left( \\delta +\\delta\n^{^{\\prime }}-\\epsilon \\right) +\\left[ \\Re \\left( \\epsilon \\right) \\right]\n+1\\right\\} .\n\\end{equation*\nThen the following formula holds true: \n\\begin{eqnarray}\n&&\\left( D_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau }\\right) \\left( x\\right)  \\label{D2} \\\\\n&=&\\frac{\\Gamma \\left( \\mu ^{\\prime }+\\tau \\right) \\Gamma \\left( -\\delta\n-\\delta ^{\\prime }+\\epsilon +\\tau \\right) \\Gamma \\left( -\\delta ^{\\prime\n}-\\mu +\\epsilon +\\tau \\right) }{\\Gamma \\left( \\tau \\right) \\Gamma \\left(\n-\\delta ^{\\prime }+\\mu ^{\\prime }+\\tau \\right) \\Gamma \\left( -\\delta\n-\\delta ^{\\prime }-\\mu +\\epsilon +\\tau \\right) }x^{\\delta +\\delta ^{\\prime\n}-\\epsilon -\\tau }.  \\notag\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{theorem}\n\\label{Th3} Let $\\delta, \\delta^{\\prime}, \\mu, \\mu^{\\prime}, \\epsilon, \\tau \\in \\mathbb{C}$ such that\\\\\n $\\Re(\\tau+k)>max\\{0, \\Re(-\\delta+\\mu), \\Re(-\\delta-\\delta^{\\prime}-\\mu+\\epsilon)\\}$, \nthen the following formula hold true: \n\\begin{eqnarray*}\n&&\\left( D_{0+}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{\\tau -1}~M_{n}^{(p, q)}(t)\\right)\n\\left( x\\right) \\\\\n&=&\\frac{(-1)^{n}\\Gamma(q+n+1)\\Gamma(\\tau)\\Gamma(\\tau+\\delta-\\mu)\\Gamma(\\tau+\\delta+\\delta^{\\prime}+\\mu^{\\prime}-\\epsilon)}{\\Gamma(q+1)\\Gamma(\\tau-\\mu)\\Gamma(\\tau+\\delta+\\delta^{\\prime}-\\epsilon)\\Gamma(\\tau+\\delta+\\mu^{\\prime}-\\epsilon)}x^{\\delta +\\delta ^{^{\\prime }}-\\epsilon +\\tau-1} \\\\\n&&\\times _{5}F _{4}\\left[ \n\\begin{array}{c}\n1+n-p, -n, \\tau, -\\mu+\\delta+\\tau, \\delta+\\delta^{\\prime}+\\mu^{\\prime}-\\epsilon+\\tau , \\\\ \nq+1, -\\mu^{\\prime}+\\tau, \\delta+\\delta^{\\prime}-\\epsilon+\\tau, \\delta+\\mu^{\\prime}-\\epsilon+\\tau \n\\end{array\n\\Big|-x\\right],\n\\end{eqnarray*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{Th4} Let $\\delta, \\delta^{\\prime}, \\mu, \\mu^{\\prime}, \\epsilon, \\tau \\in \\mathbb{C}$ such that\\\\\n $\\Re(\\tau+k)>max\\{0, \\Re(-\\mu'), \\Re(\\delta^{\\prime}+\\mu-\\epsilon), \\Re(\\delta+\\delta^{\\prime}-\\epsilon)+[\\Re(\\epsilon)]+1\\}$, then the following formula hold true: \n\\begin{eqnarray*}\n&&\\left( D_{-}^{\\delta ,\\delta ^{\\prime },\\mu ,\\mu ^{\\prime },\\epsilon\n}t^{-\\tau}~M_{n}^{(p,q)}\\left( \\frac{1}{t\n\\right) \\right) \\left( x\\right) \\\\\n&=&\\frac{(-1)^{n}\\Gamma(q+n+1)\\Gamma(\\mu^{\\prime }+\\tau)\\Gamma(-\\delta-\\delta^{\\prime }+\\epsilon+\\tau)\\Gamma(-\\delta^{\\prime }-\\mu+\\epsilon+\\tau)}{\\Gamma(q+1)\\Gamma(\\tau)\\Gamma(-\\delta^{\\prime }+\\mu^{\\prime }+\\tau)\\Gamma(-\\delta-\\delta^{\\prime }-\\mu+\\epsilon+\\tau)} x^{\\delta+\\delta'-\\epsilon-\\tau}\\\\\n&&\\times _{5}F_{4}\\left[ \n\\begin{array}{c}\n1+n-p, -n, \\tau+\\mu^{\\prime }, \\tau-\\delta-\\delta^{\\prime }+\\epsilon, \\tau-\\delta^{\\prime }-\\mu+\\epsilon , \\\\ \n\\tau, \\tau-\\delta^{\\prime }+\\mu^{\\prime }, \\tau+\\epsilon-\\delta-\\delta^{\\prime }-\\mu, q+1,\n\\end{array\n\\Big|-\\frac{1}{x}\\right],\n\\end{eqnarray*}\nprovided both the sides exists.\n\\end{theorem}\n\n\\section{Conclusion}\nThe generalized fractional integration and differentiation of of Jacobi type orthogonal polynomials are derived in this paper. Many known results (see \\cite{Malik}, \\cite{Kilbas-itsf}) reduced as the particular case of the main theorems. This concludes that the results obtained here are general in nature and can easily obtain various known results.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn relativistic heavy ion collisions, large amount of pions can be\ncreated in small space-time regions. Theoretically, it is possible\nfor these pions to condensate into the zero momentum and form\nBose-Einstein condensation(BEC) at certain temperatures and\ndensities. For this problem, there are some discussions in the\nliterature, for example, the BEC of pions in the non-equilibrium\nprocess~\\cite{a,b}, the pion condensation in thermal equilibrium\nincorporating boundary effects~\\cite{c}, and the search for the\n``cold spots\" in multipion system~\\cite{d,e}, etc. These studies\nshow a great interest on this issue. On the other hand, in high\nenergy physics, the chiral symmetry which is spontaneously broken\nat low temperature is expected to be restored at high temperature\nand\/or density~\\cite{f}. Long before, Kapusta and Haber had\ndiscussed the relation between BEC and the spontaneous symmetry\nbreaking by using different scalar models~\\cite{g,h}. Their main\ninterest, however, was focused on the physics of the electroweak\nsystem. Here in the strong interacting pion system, one may wonder\nwhether there are some relations between the chiral symmetry\nrestoration and BEC.\n\nOur motivation in this paper is to discuss the relation between\nthese two phenomena in the pion system and find the phase diagram\nof them through a model study. As far as we know, this topic has\nnot been much discussed in the literature. In~\\cite{i}, the chiral\nsymmetry restoration is discussed in the interaction pion gas\nthrough virial expansion, where BEC is only discussed in the case\nof free pion gas. Here we will discuss BEC and the chiral phase\ntransition in the interacting pion system with the linear sigma\nmodel. The linear sigma model has been extensively used to discuss\nthe chiral symmetry restoration at finite\ntemperature~\\cite{j,k,l}. In our discussion, to address BEC and\nthe chiral phase transition simultaneously, we will consider a\nfinite pion density as well as the temperature. The pion chemical\npotential can be introduced in different ways. Theoretically, at\ngiven conditions when the pion number can be regarded as being\nfixed~\\cite{m}, one could introduce the pion number density.\nHowever, generally speaking, pion number is not a conserved\nquantity and one should find a certain conserved charge of the\nsystem. For the linear sigma model which has the isospin\ninvariance, one can introduce a chemical potential associated with\nthe isospin charge. For example, in~\\cite{n}, the pion isospin\nchemical potential is introduced by the third component of the\nisospin charge and applied to the discussions of the thermal pion\nmass within the chiral perturbation theory. In our discussion, the\nisospin chemical potential of pions is introduced to the linear\nsigma model in the same way.\n\nThe organization of the present paper is as follows. In section 2\nthe effective potential for the linear sigma model is calculated\nwithin the Cornwall, Jackiw, Tomboulis resummation scheme at\nfinite temperature and finite pion density in the Hartree\napproximation. The gap equations are derived by the stationary\ncondition, which form the basis of our numerical calculation. In\nsection 3, the BEC and the chiral phase transition are studied by\na numerical calculation. We discuss their $\\mu-T$ phase diagrams\nand give the physical analysis. The last section comprises a\nsummary of our results.\n\n\\section{CJT formalism and effective Lagrangian}\nIn studies of phase transitions the effective potential is an\nimportant and popular theoretical tool. It is defined through an\neffective action which is the generating functional of the one\nparticle irreducible graphs. A generalized version is the\neffective potential for composite operators introduced by\nCornwall, Jackiw and Tomboulis(CJT)~\\cite{o}. This formalism\nassumes that the effective potential $V(\\phi, G)$ depends not only\non $\\phi(x)$, a possible expectation value of the quantum field\n$\\Phi(x)$, but also on $G(x,y)$, a possible expectation value of\nthe time-ordered product $T\\Phi(x)\\Phi(y)$. Physical solutions\ndemand minimization of the effective potential with respect to\nboth $\\phi$ and $G$, which means \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} \\0{dV(\\phi,G)}{d\\phi}=0, \\ \\\n\\ \\ \\ \\ \\0{dV(\\phi,G)}{dG}=0. \\eea This formalism was originally\nwritten at zero temperature. Then it has been extended to finite\ntemperature occasion by Amelino-Camelia and Pi for investigations\nof the effective potential of the $\\l\\phi^4$ theory~\\cite{p}. It\nhas been also used in the linear sigma model for discussing the\nchiral phase transition at finite temperature~\\cite{j}. At finite\ndensity as well as finite temperature, we will employ this\nformalism to discuss both BEC and the chiral phase transition with\nthe linear sigma model.\n\nFor the linear sigma model, here we will ignore the fermion\nsector. The Lagrangian can be written as \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} {\\cal\nL}=\\012(\\pl\\bf\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho)^2&+&\\012(\\pl\\vec\\pi)^2-\\012m^2\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2-\\012m^2\\vec\\pi^2\n\\nonumber\n\\\\ &-&\\0\\l{24}(\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2+{\\vec\\pi}^2)^2-\\e\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho , \\eea\nwhere $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$ and ${\\vec\\pi}$ are the sigma field and the three pion\nfields ($\\pi_1, \\pi_2, \\pi_3$) respectively. The explicit chiral\nsymmetry breaking term is $\\e=f_\\pi m^2_\\pi$, where $f_\\pi=93MeV$\nis the pion decay constant. The coupling constant $\\l$ and\nnegative mass parameter $m^2$ of the model are chosen to be\n$\\l=3(m^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho-m^2_\\pi)\/f^2_\\pi$ and $-m^2=(m^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho-3m^2_\\pi)\/2>0$.\nIn our discussion, we only consider the case in the chiral limit\n$\\e=0$, then at zero temperature the pion mass is $m_\\pi=0$ and\nthe sigma mass is taken as $m_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\approx 600MeV$.\n\nThe conventional treatment of spontaneous chiral symmetry breaking\nof the linear sigma model is to shift the sigma field as\n$\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\to\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\phi$, where $\\phi$ is expectation value of the sigma\nfield and the order parameter of the chiral phase transition. Then\nthe Lagrangian results as \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} {\\cal\nL}=\\012(\\pl\\vec\\pi)^2&+&\\012(\\pl\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho)^2-\\012(m^2+\\0\\l2\\phi^2)\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2\n-\\012(m^2+\\0\\l6\\phi^2)\\vec\\pi^2 \\nonumber \\\\ &-&\\012m^2\\phi^2\n-\\0\\l{24}\\phi^4\n-\\0\\l{24}(\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2+\\vec\\pi^2)^2-\\0\\l6\\phi\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^3-\\0\\l6\\phi\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\pi^2. \\eea\nIn the following discussion, we will adopt Hartree approximation\nwhich means only the bubble diagrams with four-point vertex need\nto be calculated. So the last two terms in the Lagrangian are\nneglected~\\cite{j}.\n\nThe chemical potential is introduced through the third component\nof isospin charge. The Lagrangian (3) is invariant under the\nisospin transformation, therefore according to Noether's theorem\nthe conserved current is $\\vec V_\\mu =\\vec\\pi\\times\\pl_\\mu\\vec\\pi$\nand the conserved isospin charge is $\\vec V_0\n=\\vec\\pi\\times\\pl_0\\vec\\pi.$ Then the partition function of the\nsystem can be written as \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} Z=\\int\n[d\\Pi_{\\vec\\pi}][d\\Pi_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho][d\\vec\\pi][d\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho]exp\\left[i\\int d^4x({\\cal\nL}+\\vec\\mu_I\\cdot\\vec V_0)\\right], \\eea where $\\Pi_{\\vec\\pi}$ and\n$\\Pi_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$ are the conjugate momenta of $\\vec\\pi$ and $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$. The\nisospin chemical potential $\\vec\\mu_I=(0, 0, \\mu)$, in which $\\mu$\nis the third component of isospin chemical potential, is\nintroduced associated with $\\vec V_0$~\\cite{n}, thus \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\vec\\mu_I\\cdot\\vec V_0=\\mu (\\pi_1\\pl_0\\pi_2-\\pi_2\\pl_0\\pi_1). \\eea\nHere it is more convenient to work with fields possessing\nwell-defined charges: \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} \\pi_-\\equiv\\01{\\sqrt2}(\\pi_1+i\\pi_2), \\\n\\ \\pi_+\\equiv\\01{\\sqrt2}(\\pi_1-i\\pi_2),\\ \\ \\pi_0\\equiv\\pi_3. \\eea\nIntegrating out the conjugate momenta in equation (4) and using\nthe redefined pion fields, one obtains \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nZ=\\int[d\\pi_+][d\\pi_-][d\\pi_0][d\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho]exp\\left[i\\int d^4x{\\cal\nL}_{eff}\\right], \\eea where ${\\cal L}_{eff}$ is the effective\nLagrangian of the system and can be written as, \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} &{\\cal\nL}&_{eff}=|\\pl_I\\pi|^2-(m^2+\\0\\l6\\phi^2)|\\pi|^2+\\012(\\pl\\pi_0)^2+\\012(\\pl\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho)^2\n \\nonumber \\\\\n&-&\\012(m^2+\\0\\l6\\phi^2)\\pi^2_0-\\012(m^2+\\0\\l2\\phi^2)\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2-\\012m^2\\phi^2-\\0\\l{24}\\phi^4\n\\nonumber \\\\\n&-&\\0\\l{24}\\lbrack\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^4+4|\\pi|^4+\\pi_0^4+4\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2|\\pi|^2+2\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho^2\\pi_0^2+4|\\pi|^2\\pi_0^2\\rbrack\n\\eea with $|\\pl_I\\pi|^2=(\\pl_\\mu+i\\mu\\delta_{\\mu 0}\n)\\pi_+(\\pl^\\mu-i\\mu\\delta^{\\mu 0})\\pi_-$ and $|\\pi|^2=\\pi_+\\pi_-$.\nIt should be noted that the chemical potential $\\mu$ has entered\nthe lagrangian in $|\\pl_I\\pi|^2$. The following calculations and\ndiscussions will be based on this effective Lagrangian.\n\n\\section{Gap equations and thermodynamic potential}\nWithin the imaginary time formalism of finite temperature field\ntheory, we have \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\int\\0{d^4k}{(2\\pi)^4}f(k)\\to\\01{\\b}\\sum_n\\int\\0{d^3\\bf\nk}{(2\\pi)^3}f(i\\o_n,{\\bf k})  \\equiv\\int_\\b f(i\\o_n, {\\bf k}),\n\\eea where $\\b$ is the inverse temperature, $\\b=1\/T$; the\nintegration over the time component $k_0$ has been replaced by a\nsummation over discrete frequencies. For boson there are\n$\\o_n=2\\pi nT$ and $n=0, \\pm1, \\pm2, \\cdots$.  For the sake of\nsimplicity in what follows, a shorthand notation is used to denote\nthe integration and the summation.\n\nAccording to the Lagrangian (8), the tree level inverse\npropagators of $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$, $\\pi_0$ and $\\pi_\\pm$ can be written\nrespectively as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray} D^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho&=&\\o^2_n+{\\bf k}^2+m^2+\\0\\l2\\phi^2,  \\\\\nD^{-1}_0&=&\\o^2_n+{\\bf k}^2+m^2+\\0\\l6\\phi^2, \\\\\nD^{-1}&=&(\\o_n+i\\mu)^2+{\\bf k}^2+m^2+\\0\\l6\\phi^2. \\eea The\nequation (12) represents the inverse propagator of $\\pi_+$ and\n$\\pi_-$. As the summation in equation (9) is symmetric over $n$\nfrom $-\\infty$ to $+\\infty$, $\\o_n+i\\mu$ and $\\o_n-i\\mu$ are\nequivalent in describing the propagators of $\\pi_+$ and $\\pi_-$.\nHere we just choose $\\o_n+i\\mu$ to describe both $\\pi_+$ and\n$\\pi_-$, and write the two propagators in the same form as in\nequation (12). According to \\cite{j}, we can write down the\ncorresponding effective potential at finite temperature as \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nV(&\\phi&, G)=\\012m^2\\phi^2+\\0\\l{24}\\phi^4+\\012\\int_\\b\\ln G^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\n+\\012\\int_\\b \\left[D^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho-1\\right]+\\012\\int_\\b\\ln G^{-1}_0\n\\nonumber \\\\ &+&\\012\\int_\\b\\left[D^{-1}_0G_0-1\\right] +\\int_\\b\\ln\nG^{-1} +\\int_\\b\\left[D^{-1}G-1\\right]+V_2(\\phi, G), \\eea where\n$G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho, G_0$ and $G$ are the full propagators of $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho, \\pi_0$ and\n$\\pi_\\pm$ respectively. They are determined by the stationary\ncondition (1). The chemical potential is included in the full\npropagator. $V_2(\\phi,G)$ represents the infinite sum of the two\nparticle irreducible vacuum graphs. However, in CJT formalism at\nthe Hartree approximation we need only to calculate the ``double\nbubble\" diagrams with four-point vertex and treat each loop line\nas the full propagator~\\cite{j,o}. Therefore, $V_2$ can be written\nas \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} V_2(\\phi, G)&=&\\0\\l8\\left[\\int_\\b\nG_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\right]^2+\\0\\l3\\left[\\int_\\b\nG\\right]^2+\\0\\l8\\left[\\int_\\b G_0\\right]^2 \\nonumber \\\\\n&+&\\0\\l6\\int_\\b G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\int_\\b G+\\0\\l{12}\\int_\\b G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho\\int_\\b G_0\n+\\0\\l6\\int_\\b G\\int_\\b G_0. \\eea\n\nBy minimizing the effective potential $V(\\phi, G)$ with respect to\nthe full propagators, we get the following set of nonlinear gap\nequations, \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} G^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho&=&D^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l2\\int_\\b\nG_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l3\\int_\\b G+\\0\\l6\\int_\\b G_0, \\\\\nG^{-1}_0&=&D^{-1}_0+\\0\\l2\\int_\\b G_0+\\0\\l6\\int_\\b\nG_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l3\\int_\\b G,  \\\\ G^{-1}&=&D^{-1}+\\0{2\\l}3\\int_\\b\nG+\\0\\l6\\int_\\b G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l6\\int_\\b G_0 . \\eea Furthermore we take\nthe similar ansatz for the full propagators as in \\cite{j},\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray} G^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho&=&\\o^2_n+{\\bf k}^2+M^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho, \\\\\nG^{-1}_0&=&\\o^2_n+{\\bf k}^2+M^2_0, \\\\ G^{-1}&=&(\\o_n+i\\mu)^2+{\\bf\nk}^2+M^2, \\eea where the effective masses $M_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho, M_0$ and $M$ have\nbeen introduced for $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho, \\pi_0$ and $\\pi_\\pm$ respectively.\nSubstituting them into the above gap equations, we obtain a set of\neffective mass gap equations, \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nM^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho&=&m^2+\\0\\l2\\phi^2+\\0\\l2\\int_\\b\nG_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l3\\int_\\b G+\\0\\l6\\int_\\b G_0, \\\\\nM^2_0&=&m^2+\\0\\l6\\phi^2+\\0\\l2\\int_\\b G_0+\\0\\l6\\int_\\b\nG_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l3\\int_\\b G, \\\\ M^2&=&m^2+\\0\\l6\\phi^2+\\0{2\\l}3\\int_\\b\nG+\\0\\l6\\int_\\b G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0\\l6\\int_\\b G_0. \\eea Accordingly the\neffective potential can be written as \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} V(\\phi,\nM)&=&\\012m^2\\phi^2+\\0\\l{24}\\phi^4+\\012\\int_\\b\\ln\nG^{-1}_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho \\nonumber \\\\\n&-&\\012\\int_\\b(M^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho-m^2-\\0\\l2\\phi^2)G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\012\\int_\\b\\ln G^{-1}_0\n\\nonumber \\\\ &-&\\012\\int_\\b(M^2_0-m^2-\\0\\l6\\phi^2)G_0+\\int_\\b\\ln G^{-1}\n\\nonumber \\\\ &-&\\int_\\b(M^2-m^2-\\0\\l6\\phi^2)G+V_2(\\phi, M). \\eea By\nminimizing the potential with respect to the order parameter\n$\\phi$, we obtain one more equation, \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nm^2\\phi+\\0\\l6\\phi^3+\\0{\\l\\phi}2\\int_\\b G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho+\\0{\\l\\phi}6\\int_\\b\nG_0+\\0{\\l\\phi}3\\int_\\b G=0. \\eea From equations (21)---(23) and\n(25), the effective masses and order parameter can be solved\nself-consistently at given temperature and chemical potential.\nThere are some discussions concerning the renormalization on the\nCJT formalism in different models~\\cite{p,q,r}. Recent\ninvestigation about the renormalization of the O(N) linear sigma\nmodel has been addressed in \\cite{k}. In our discussion, the\ndivergent parts of the integrals are neglected as is done in\n\\cite{j}. And we take the finite parts of the integrals as\nfollows:  \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} &&\\int_\\b\\ln G^{-1}=\\int_\\b\\ln[(\\o_n+i\\mu)^2+{\\bf\nk}^2+M^2] \\nonumber\n\\\\\n&=&\\01{\\b}\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\left[\\ln(1-e^{-\\b(\\o+\\mu)})+\\ln(1-e^{-\\b(\\o-\\mu)})\\right],\n\\\\ &&\\int_\\b G=\\int_\\b\\01{(\\o_n+i\\mu)^2+{\\bf k}^2+M^2} \\nonumber \\\\ &=&\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\01{2\\o}\\left[\\01{e^{\\b(\\o+\\mu)}-1}+\\01{e^{\\b(\\o-\\mu)}-1}\\right],\n\\eea where $\\o=\\sqrt{{\\bf k}^2+M^2}$. The integration over $G_0$\nand $G_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$ will be dealt with in the same way. With the chiral\nlimit ($\\e=0$), the corresponding coupling constant and the\nnegative mass parameter can be taken as $\\l\\approx 125$ and\n$-m^2\\approx 1.8\\times 10^5MeV^2$.\n\nIn the discussion of the thermodynamic system, the effective\npotential $V$ is equivalent to the thermodynamic potential\n$\\Omega$. Thus we get \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Omega&=&\\012m^2\\phi^2+\\0\\l{24}\\phi^4+T\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\ln(1-e^{-\\b\\o_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho}) \\nonumber \\\\\n&-&\\012(M^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho-m^2-\\0\\l2\\phi^2)\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\01{\\o_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho}\\01{e^{\\b\\o_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho}-1} \\nonumber \\\\\n&+&T\\int\\0{d^3{\\bf k}}{(2\\pi)^3}\\ln(1-e^{-\\b\\o_0})\n-\\012(M^2_0-m^2-\\0\\l6\\phi^2) \\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\01{\\o_0}\\01{e^{\\b\\o_0}-1} \\nonumber \\\\ &+&T\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\left[\\ln(1-e^{-\\b(\\o+\\mu)})+\\ln(1-e^{-\\b(\\o-\\mu)})\\right]\n\\nonumber \\\\ &-&(M^2-m^2-\\0\\l6\\phi^2)\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\01{2\\o}\n\\left[\\01{e^{\\b(\\o+\\mu)}-1}+\\01{e^{\\b(\\o-\\mu)}-1}\\right]+\\Omega_2,\n\\eea where $\\Omega_2=V_2$, $\\o_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho=\\sqrt{{\\bf k}^2+M^2_\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho},\n\\o_0=\\sqrt{{\\bf k}^2+M^2_0}$ and $\\o=\\sqrt{{\\bf k}^2+M^2}$.\nAccording to the relation \\begin{eqnarray}} \\def\\eea{\\end{eqnarray} \\rho=-\\0{\\pl\\Omega}{\\pl\\mu}, \\eea\nand equation (23), we can get the net charge density \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\r=\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\left[\\01{e^{\\b(\\o-\\mu)}-1}-\\01{e^{\\b(\\o+\\mu)}-1}\\right].\n\\eea This expression of density seems very similar to that of\nideal gas, but actually they are different. Because here the\neffective mass $M$ in $\\o$ is a function of temperature and will\nbe determined self-consistently by the gap equations.\n\n\\section{BEC and chiral phase transition}\nNow we are in a position to discuss BEC and the chiral phase\ntransition. We will study BEC first. It is known that when\n$\\mu=M$, BEC occurs. The equation (30) should be written as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray} \\r=\\r_0+\\r^*(\\b,\\mu=M), \\nonumber \\\\\n\\r^*(\\b,\\mu=M)=\\int\\0{d^3{\\bf\nk}}{(2\\pi)^3}\\left[\\01{e^{\\b(\\o-M)}-1}-\\01{e^{\\b(\\o+M)}-1}\\right],\n\\eea where $\\r_0$ represents the charge density of the\nzero-momentum state~\\cite{p1,s}. Let us determine the point at\nwhich $\\mu=M$ is reached. From equation (30), when $\\r$ is fixed,\nby solving the gap equations (21)---(23) and equation (25), we\nfind both $\\mu$ and $M$ are functions of $T$. It can be plotted in\nfigure 1. We can see that with $T$ decreasing, $M$ decreases first\nand then increases, while $\\mu$ keeps increasing quickly and\napproaches $M$. At certain temperature, $\\mu$ catches up with $M$\nand BEC happens. From equation (31), we know the critical\ntemperature $T_c$ is determined implicitly by the equation \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\r=\\r^*(\\b_c,\\mu=M). \\eea When $T\\leq T_c$, the system goes into\nBEC phase. The equation (31) will be solved together with the gap\nequations with $\\r$ fixed. We find $\\mu$ and $M$ are still\nfunctions of $T$ and $\\mu(T)=M(T)$. They both decrease with\ntemperature decreasing, which is indicated in figure 1.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f1}\n\\end{center}\n\\caption{$\\mu$ and $M$ as functions of $T$ at the fixed total\ncharge density $\\r=0.06fm^{-3}$. BEC happens at $T_c=80MeV$.}\n\\end{figure}\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f2}\n\\end{center}\n\\caption{The ratio $\\r_0\/\\r$ as a function of the ratio $T\/T_c$ at\n$\\r=0.06fm^{-3}$ and $T_c=80MeV$.}\n\\end{figure}\n\nFrom equation (31) which is solved together with the gap equations\nat fixed $\\r$, we can get $\\r_0$ as a function of $T$. This is\nshown in figure 2. It is clear that when $\\r$ is fixed, at\n$T>T_c$, $\\r_0=0$; at $T\\leq T_c$, $\\r_0$ starts to increase\nquickly with temperature decreasing, which means large amount of\nthe charged pions will reside to the zero momentum state and form\nBEC.\n\n\nFor different fixed total density $\\r$, there will be different\nvalues of critical temperature $T_c$ and chemical potential\n$\\mu(T_c)$, so one can study the $\\mu-T$ phase diagram of BEC. In\nsolving the self-consistent equations, we observe that $T_c$ and\n$\\mu(T_c)$ depend on the order parameter $\\phi$. In low\ntemperature regime, the order parameter $\\phi$ is nonzero and\ndetermined by equation (25). When it is solved together with the\ngap equations (21)---(23) and equation (32), the critical\ntemperature of BEC at a given density can be determined. The\n$\\mu-T$ phase diagram for BEC can be given accordingly as shown in\nfigure 3. The dividing line which separates BEC and the normal\nphase at $\\phi\\neq 0$ is curve AC. At $T>T_C$ (the temperature at\nthe point C), there is no solution with $\\phi\\neq 0$ for the onset\nof BEC. Because in the high temperature regime the chiral symmetry\nis restored and $\\phi=0$. At $\\phi=0$ the phases dividing line is\ncurve DF as shown in figure 3. We can see that BEC happens in\nchiral symmetry broken state and restored state at low temperature\nand high temperature respectively.\n\nIn the regime of $T_B<T<T_C$, it seems that BEC can happen at both\n$\\phi\\neq 0$ and $\\phi=0$. However, if we calculate thermodynamic\npotential from equation (28), we find under the same temperature\nthe thermodynamic potential of $\\phi\\neq 0$ has lower values than\nthat of $\\phi=0$, which means at $T_B<T<T_C$, BEC would happen\nalong the curve BC. Then the final $\\mu-T$ phase diagram of BEC is\ndetermined and shown in figure 4. In low temperature regime BEC\nphase and the normal phase are divided by curve AC, while in high\ntemperature regime, the phases are divided by curve EF. The whole\nphase plane has been divided into two phase areas. The area above\nthe curve belongs to the BEC phase, while the area below the curve\nrepresents the normal phase.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f3}\n\\end{center}\n\\caption{The phase diagram of $\\mu$ versus $T$ for BEC. (AC and DF\nare the dividing lines which separate BEC and normal phases in low\ntemperature regime and high temperature regime respectively.)}\n\\end{figure}\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f4}\n\\end{center}\n\\caption{The $\\mu-T$ phase diagram for BEC with no ambiguity.}\n\\end{figure}\n\nNow we will discuss the chiral phase transition in this system. By\nusing the linear sigma model, there are many discussions on the\nchiral phase transition at finite temperature~\\cite{j,k,l}. As we\nhave considered finite pion density as well as finite temperature,\nthe calculation becomes more involved. However, the chiral phase\ntransition is still characterized by the order parameter $\\phi$.\nIn our discussion, $\\phi$ is not only temperature dependent but\nalso chemical potential dependent. In \\cite{j}, the chiral phase\ntransition has been discussed at finite temperature with the\nlinear sigma model. By using CJT resummation scheme in Hartree\napproximation, the transition is found to be first order in the\nchiral limit. If we take $\\mu=0$, we can reproduce their result.\nIn the case of $\\mu\\neq 0$, we find the chiral phase transition is\nstill first order. This can be seen if we take a look at $\\Omega$\nas a function of $\\phi$. As the gap equations (21)---(23) can be\nsolved at certain $T$ and $\\mu$, from equation (28), the\nthermodynamical potential $\\Omega$ can be determined. The $\\phi$\ndependence of $\\Omega$ can be plotted as shown in figure 5. In the\ntransition, it can be seen that there are two minima of $\\Omega$\nwhich correspond to the stable and meta-stable states at certain\n$T$ and $\\mu$. If $T$ is fixed, by increasing $\\mu$, the lower\nminimum of $\\phi\\neq 0$, which stays in a stable state, will be\nlifted up. When it becomes higher than the $\\phi=0$ minimum, it\nchanges to a meta-stable state from the stable state. In other\nwords, the $\\phi=0$ minimum becomes a stable state which means\nchiral symmetry is restored. It is clear that the chiral phase\ntransition is first order. The transition is usually defined at\nthe point where the two minima become degenerate. In figure 5,\nwhen $T=160MeV$ and $T=180MeV$, we can see that the transitions\nhappen at $\\mu=237MeV$ and $\\mu=100MeV$ respectively. According to\nthis way, the $\\mu-T$ phase diagram for the chiral phase\ntransition can be plotted as shown in figure 6.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f5}\n\\end{center}\n\\caption{$\\Omega$ as functions of $\\phi$. Solid lines stand for\n$T=160MeV$, and the chemical potentials for different lines are\n$260MeV, 237MeV$ and $220MeV$ respectively (from top to bottom).\nDashed lines stand for $T=180MeV$, and the chemical potentials are\n$140MeV, 100MeV$ and $60MeV$ respectively (also from top to\nbottom).}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f6}\n\\end{center}\n\\caption{The phase diagram of $\\mu$ versus $T$ for the chiral\nphase transition. When $T>144MeV$, the area above the curve is the\nrestored phase; the area below the curve is the broken phase.}\n\\end{figure}\n\nFrom figure 6, we can see that the transition temperature is\n$184MeV$ for $\\mu=0$, which was obtained in \\cite{j}. Here when\n$\\mu\\ne 0$, the transition temperature will decrease with the\nchemical potential increasing. At certain temperature ($T\\approx\n144MeV$), the phase curve of the chiral phase transition will\nterminate, because at this time $\\mu=M$ and BEC will happen. As a\nresult the chiral phase transition can not take place further at\nlower temperatures. When $T>144MeV$, the area above the curve is\nthe chiral symmetry restored phase. However, the phase diagram is\nnot complete, because we do not know what the phase is when\n$T<144MeV$. Now let us recall the $\\mu-T$ phase diagram of BEC\n(figure 4). If we combine the two phase diagrams (figure 4 and\nfigure 6) into one figure, we can see the different phase areas\nclearly. This is shown in figure 7. When the chiral phase\ntransition terminates, it means that the system will go into BEC\nphase. Furthermore, the normal phase area in figure 4 is divided\ninto the chiral symmetry broken phase and the chiral symmetry\nrestored phase. When the system is in a chiral symmetry broken\nphase, if we decrease temperature with density fixed, BEC will\noccur at ceratin temperature; if we increase temperature, the\nsystem may go into a chiral symmetry restored phase. In the\nrestored phase, at relatively low densities, if we decrease\ntemperature, the chiral phase transition may take place; at\nrelatively high densities, if we decrease temperature with density\nfixed, BEC will take place.\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=220pt,height=130pt]{f7}\n\\end{center}\n\\caption{The combined phase diagram of $\\mu$ versus $T$ for BEC\nand the chiral phase transition. The CSB and CSR stand for chiral\nsymmetry broken phase and chiral symmetry restored phase\nrespectively.}\n\\end{figure}\n\nWe noted that in \\cite{h}, BEC and the spontaneous symmetry\nbreaking in interacting Bose system have been discussed with the\nO(N) model. In their discussion, a chemical potential $\\mu$\nassociated with O(2) symmetry is introduced when N is even. There\nare $N\/2$ such chemical potentials for $N$ scalars. For only one\nchemical potential and the large-N limit, their result shows that\nthe transition to broken symmetry phase is just the onset of BEC\nin the the ultra-relativistic regime. This is because the scalar\nfield associated with $\\mu$ acquires a nonzero vacuum expectation\nvalue at the point that $\\mu=M$ is reached when temperature is\nlowered. Thus the symmetry, which is actually the O(2) symmetry,\nis broken at this time. In our discussion, we have studied the\nchiral symmetry restoration and BEC in the O(4) linear sigma\nmodel. In this model, the four scalar fields have real physical\nmeaning, which are three pion fields and one sigma field. The\nchiral symmetry breaking in the vacuum is chosen along the $\\sigma}  \\def\\b{\\beta} \\def\\p{\\psi} \\def\\r{\\rho$\ndirection. The chiral phase transition is characterized by the\norder parameter $\\phi$. The chemical potential here is associated\nwith the pion fields. The BEC means the BEC of the charged pions.\nThe relation between BEC and the chiral phase transition can be\nillustrated by the $\\mu-T$ phase diagram as indicated in figure 7.\n\n\\section{Conclusion}\nIn summary, we have discussed Bose-Einstein condensation and the\nchiral phase transition for the linear sigma model in the chiral\nlimit. The effective potential of the linear sigma model is\ncalculated within the CJT resummation scheme in the Hartree\napproximation at finite temperature and finite pion density. By\nsolving the self-consistent equations we have discussed BEC in the\ninteracting pion system. The $\\mu-T$ phase diagram of BEC is\npresented. At relatively low temperature, BEC happens in chiral\nsymmetry broken state, while at relatively high temperature, BEC\nhappens in chiral symmetry restored state. The phase plane has\nbeen divided into the BEC phase and the normal phase. Then we have\ndiscussed the chiral phase transition and also plotted the $\\mu-T$\nphase digram of the chiral phase transition. By comparing it with\nthe BEC phase diagram, we find the chiral phase transition happens\nin the normal phase and further divided the normal phase into the\nchiral symmetry broken phase and restored phase. So the whole\nphase plane has been divided into three phase areas: the BEC phase\narea, the chiral symmetry broken phase area and the chiral\nsymmetry restored phase area. BEC can happen either from the\nchiral symmetry broken phase or from the chiral symmetry restored\nphase. When the system goes into BEC phase, there is no chance for\nthe onset of the chiral phase transition.\n\n\n\n\n\n\\ack We would like to thank Professor Ji-Sheng Chen and Professor\nDe-Fu Hou for their help. This work was supported in part by the\nNational Natural Science Foundation of China with No. 90303007.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{in}\n\\vskip 0.8cm\nTo present date the hypothesis that the tau lepton and its neutrino are a\nstandard lepton doublet is confirmed by all existing data. Beyond this the\nstructure of the weak interaction of the $\\tau$ is still poorly determined\nexperimentally. From the theoretical point of view the charged leptons could\nhave physical properties substantially different from those predicted by the\nstandard model.\nlifetimes, rare lepton and\/or CP violating decay modes, weak universality\nviolating interactions, {\\sl etc...} Therefore the $\\tau$ lepton is an\nextremely clean laboratory to look for new physics through the study of its\ndecay modes. Two complementary approaches can be followed. One can $precisely$\nmeasure the parameters characterizing the well-known leptonic and hadronic\n$\\tau$ decays and look for possible deviations of their expected ``standard\"\nvalues. Alternatively, one can look for $rare$ $\\tau$ decays, not expected or\nexplicitly forbidden in the context of the standard model.\n\nBoth precision measurements and rare decay studies will require not only very\ngood statistics, but also very good detectors providing excellent particle\nidentification, $\\pi^0$ and neutral hadron recognition and hermiticity. These\nrequirements bring naturally the idea of a {\\sl particle factory}\n($\\tau-$charm factory, B factory, $Z$ factory) as the ideal place to perform\nthese studies. These facilities can provide very large samples (of the order\nof $10^7$ $\\sim$ $10^8$  $\\tau$ pairs per year) and project state-of-the-art\ndetectors. The physics prospects, as well as the challenging detector and\nmachine requirements for these factories have been extensively discussed in\nthe literature \\cite{tcf1,bf1,lep1}.\n\nIn this paper we study the sensitivity to possible phenomena involving ``new\nphysics\" that can be achieved by the study of exotic $\\tau$ decays in both the\nB factory (BF) and the $\\tau-$charm factory (tcF). We will study ``typical\"\nrare $\\tau$ decays such as $\\tau \\rightarrow e(\\mu) \\gamma$ and $\\tau\n\\rightarrow e(\\mu) \\pi^0$, three body decays of the type $\\tau \\rightarrow lll\n{}~(l=e,\\mu)$ as well as two body decays involving the emission of light weakly\ninteracting spin zero bosons, for example $\\tau \\rightarrow e(\\mu) J$, where\n$J$ denotes the majoron.\n\nSuch decays are predicted in a wide class of extensions of the Standard Model\n\\cite{perga}. In Section two we present some models where the previous decays\ncan have relative large branching ratios. The experimental aspects of $\\tau$\nphysics at the tcF and BF factories and details of the Monte Carlo simulation\nused for this study are given in Section three. In Section four we study the\nlimits on the different rare $\\tau$ decays. Finally in Section five we present\nour conclusions.\n\n\\section{Theoretical Overview}\n\nAlthough in the standard model all leptons flavours are exactly conserved\nnothing is sacred about these conservation laws when one considers models\nbeyond the standard model. Therefore looking for rare lepton flavour violating\n(LFV) decays is a way to search for so far unknown interactions. There are\nseveral motivations and ways to extend the standard electroweak theory leading\nto such rare tau decays. If we restrict ourselves to models with the standard\ngauge structure, lepton flavour violation can arise either from a) the mixing\nof heavy leptons in the leptonic charged and neutral current or b) due to the\nexistence of LFV Yukawa couplings to new scalar particles.\n\nIn general models of type a) include neutrino masses and, although model\ndependent, the expected LFV rates will generally tend to be very much\nrestricted due to the smallness of neutrino masses that follows from\nlaboratory, astrophysics and cosmology \\cite{perga}. However it has been\nrecently noted that flavour violation can occur even when neutrinos  are\n``protected'' from acquiring a mass by some symmetry such as the conservation\nof the  {\\sl total} lepton number (this leaves the door open to the possible\nviolation of {\\sl partial} lepton flavours \\cite{BER}). This is the case, for\nexample if neutral heavy leptons (NHL) exist in nature \\cite{BER,CERN,MPLA}.\nRare decays processes associated to the existence of NHL include those shown\nin table \\ref{NHLtable}.\n\nSupersymmetry, both in its conventional realization, as well as in the\nalternative scenarios without R parity conservation \\cite{RPP,MASI-ROMA} are\nexamples of models of type b). It offers a possible origin for new signals in\n$\\tau$ and $\\mu$ decays. One example would be the emission of very light\nsneutrinos \\cite{KANE}, but this possibility is now ruled out in the minimal\nsupersymmetric standard model. Moreover, sneutrinos are not expected to be\nlight in this case. However, the emergence of a massless sneutrino is a\ngeneric feature of a class of $SU(2)\\otimes U(1)$ supersymmetric models where\nthe $R_p$ symmetry is broken spontaneously \\cite{ARCA}. In these models one\ncombination of the sneutrinos remains massless because there must be a\nphysical Nambu-Goldstone boson associated with the spontaneous violation of\nlepton number symmetry. This is called the majoron and denoted by $J$. Recent\nprecision measurements of the $Z$ width at LEP \\cite{LEP1} have excluded\n\\cite{concha} the simplest version of this model \\cite{ARCA} but certainly do\nnot rule out the idea of a massless isosinglet majoron \\cite{MASI-ROMA}. As a\nconsequence majoron emission in many weak decays such as those in table\n\\ref{Jtable} is experimentally viable and can, in fact, produce sizeable\neffects \\cite{Nuria}.\n\n\\subsection{Rare Tau Decays and NHLS}\n\nNeutral isosinglet heavy leptons arise in many extensions of the electroweak\ntheory. Being isosinglets, they couple to the standard gauge bosons only\nthrough mixing with ordinary isodoublet neutrinos. The resulting charged\ncurrent (CC) leptonic weak interaction  has the form \\cite{2227}\n\\begin{equation}\n\\L=\\frac{g}{\\sqrt{2}} W_\\mu\n\\sum_{ij}\n\\overline l^-_{Li}  \\gamma^\\mu (K_{Lij}\\nu_{Lj}\n+K_{Hij}N_{Lj}) + h.c.\n\\end{equation}\nSimilarly the neutral current  (NC) Lagrangian for the neutral\nleptons is given by\n\\begin{equation}\n\\L=\\frac{g}{\\cos\\theta_W}Z_\\mu\n\\sum_{ij}\n[\\overline \\nu_{Li}  \\gamma^\\mu P_{LLij}\\nu_{Lj}+\n\\overline\\nu_{Li} \\gamma^\\mu P_{LHij} N_{Lj} +\n\\overline N_{Li} \\gamma^\\mu P_{HLij} \\nu_{Lj} +\n\\overline N_{Li} \\gamma^\\mu P_{HHij} N_{Lj}]\n\\end{equation}\nwhere $P_{AB}=K_A^\\dagger K_B$. The admixture in the CC and NC leads in\ngeneral to violations of universality which limit the attainable values of\nthe $K_H$ matrix elements.\n\nThese neutral heavy leptons can engender several LFV decays of the charged\nleptons, such as those in table \\ref{NHLtable}. These are one loop processes\nwhere there is at least one virtual NHL in the loop. For detailed expressions\nof the different decay widths see Ref \\cite{MPLA}. The widths for the\ndifferent processes can be written in the general form\n\\begin{equation}\n\\Gamma_p = C_p \\times m_\\tau^n |\\sum_a K_{H\\tau a}^* F_p(M_a) K_{H a j} |^2\nPS\n\\end{equation}\nwhere $C_p$  is a constant that depends on the process, $n=3,5$ also depending\non the process. The index $j$ labels the flavour of the final lepton and\n$F_p(M_a)$ is the form factor from the integration of the corresponding loop\ndiagram with a NHL of mass  $M_a$ in some internal leg. Finally PS is the\nphase space factor which includes the effect of the masses in the final state.\nTherefore the size of the attainable branching ratio will depend on the NHL\nmass and of course on the allowed values for the couplings $K_H$. In some\nmodels the $K_H$ are severely constrained by the limits on neutrino masses.\nHowever, it is possible to avoid altogether these constraints. In fact flavour\nviolation can occur with strictly massless neutrinos. In this case the\nconstraints on $K_H$ are only those that follow from weak universality,\nleading to branching ratios as large as those in Fig. \\ref{NHLFig}. We see\nthat present constraints allow decay branching ratios as large as\n$O(10^{-6})$. A detailed discussion of this is given in ref.\n\\cite{CERN,perga}.\n\n\\subsection{Rare Decays and Supersymmetry}\n\nSupersymmetry can produce rare tau decays with detectable rates. This may\noccur in simple extensions of the minimal supersymmetric standard model\n\\cite{SUSYLFV}.\n\nA particularly attractive case correspond to the situation where supersymmetry\nis realized in the context of the standard model gauge group in such a way\nthat R parity is broken spontaneously at the $TeV$ scale. As pointed out these\nmodels contain a massless Nambu-Goldstone, the majoron. In the minimal SUSY\nmodel with broken R parity \\cite{ARCA} the existence of the majoron is\naccompanied by another light scalar particle, denoted $\\rho$, which receives a\nmass of order $v_L$, the scale characterizing the spontaneous violation of R\nparity. The value of this scale $v_{L}$ is severely constrained by the\nastrophysics of stellar cooling and evolution \\cite{KIM} to be $v_L < O\n\\:(30\\:KeV)$. As a result there is a new kinematically possible decay mode for\nthe neutral gauge boson $Z \\rightarrow \\rho + J $. This leads to an additional\ncontribution to the invisible $Z$ width. The precision measurements of the $Z$\nwidth at LEP \\cite{LEP1} is sufficient to exclude this model \\cite{concha}. It\nis however possible to formulate the R parity breaking model in such a way\nthat the majoron is replaced by an $SU(2) \\otimes U(1)$ $singlet$ so that it\ndoes not couple to the $Z$ \\cite{MASI-ROMA}. The scale characterizing R parity\nbreaking is now large, {\\it i.e.,} $v_{R} = O(1)$ TeV.\n\nIn these models the spectrum of the tau decay is modified by single majoron\nemission. This can be described in terms of the effective Lagrangian\ninteraction of the majoron with charged leptons which can given as\n\\begin{equation}\ni J \\bar{l}_j \\{ \\eta_k {\\rm L} A_{kj} - \\eta_j {\\rm R} A_{jk} \\} l_k\n\\label{lag}\n\\end{equation}\nwhere ${\\rm L}$ and ${\\rm R}$ are the chiral projectors and $\\eta_i$ and\n$\\eta_j$ are sign factors related to the relative CP parities of the leptons.\nThe coupling matrix $A$ depends on unknown model parameters and are\nconstrained by different sets of experimental data as detailed in Ref\n\\cite{Nuria}.\n\nThe corresponding charged lepton decay width is given as\n\\begin{equation}\n\\Gamma = \\frac {m_k} {32 \\pi} [A_{kj}^2 + A_{jk}^2]\n\\end{equation}\nand from this the single majoron emission branching ratio\nis easily determined.\n\nAs an example we show in Fig. \\ref{JFig} the attainable values of the single\nmajoron emission tau decay branching ratios $\\tau\\rightarrow \\mu J$ as a\nfunction of the tau neutrino mass. This indicate that the flavour-violating\n$\\tau$ decay processes with single majoron emission could lead to observable\neffects for a wide range of the allowed parameters, even at a level similar to\nthe present experimental sensitivities \\cite{SINGLE}.\n\n\\section {Tau Physics at the tcF and the BF Factories}\n\nBoth tau and B factories may provide complementary information on several\nissues on tau physics, including studies of rare tau decay branching ratios.\n\n\\subsection{Tau Studies at tcF}\n\nThe distinctive feature that  makes the tcF extremely well suited for almost\nall $\\tau$ physics, is its ability to take data at several operating points\n(see Figs. \\ref{tauer} and \\ref{taucr}) with unique properties:\n\\begin{itemize}\n\n\\item\n$3.56 GeV$. Right below $\\tau^+\\tau^-$ threshold. Taking data at this point\none $measures$ experimentally all non-$\\tau$ backgrounds.\n\\item\n$3.57 GeV$.  At this point the $\\tau$'s are produced practically at rest. Due\nto the Coulomb interaction between the $\\tau^+$ and the $\\tau^-$, the\n$\\tau^+\\tau^-$ cross-section has a finite value of 0.24 nb, ignoring\ninitial-state radiation \\cite{volo}. With a collision spread of about 1 MeV\nthe $\\tau^+\\tau^-$ cross-section is 0.47 nb and $\\beta_\\tau~=~0.033$. Then,\nthe two-body $\\tau$ decays, such as $\\tau\\rightarrow\\pi\\nu_\\tau$ and\n$\\tau\\rightarrow K\\nu_\\tau $ (and also possible exotic decays of the type\n$\\tau\\rightarrow e (\\mu) J$, see section two), give rise to monochromatic\nsecondaries, as illustrated in Fig. \\ref{mono}.a. The consequences are clean\nsignatures for event selection, as well as $kinematic$ separation of the\ndifferent decay modes. If a monochromator scheme at the interaction point is\nimplemented \\cite{monocrom} the spread in the collision energy could be\nreduced to 0.1 MeV and thus $\\beta_\\tau~\\sim~0.01$, giving rise to still\nsharper monochromatic distributions.\n\\item\n$3.67 GeV$.  This energy provides the highest $\\tau^+\\tau^-$ cross-section\nbelow the  $\\psi'(3.69)$ and $D\\bar{D}$(3.73) thresholds. Tau decays can be\ncharacterized taking advantage of the fact that they are the only source of\nprompt leptons and neutrinos.\n\\item\n$3.69 GeV$.  The highest $\\tau^+\\tau^-$ cross-section occurs at this energy.\nExperiments with a very clean signal (such as the neutrinoless $\\tau$ decays\nthat we will discuss later) can be made here. Notice that, while the standard\noptics will result in an observed cross-section of about 4.5 nb, the\nmonochromator scheme would raise it to 10 nb.\n\\item\n$4.25 GeV$. This is near the maximum energy accessible by tcF. the\n$\\tau^+\\tau^-$ cross section is large and the $\\tau$'s boost is moderate,\n$\\beta_\\tau~=~0.54.$ This point is well suited for measurements of the\nenergy-energy correlations between $\\tau$ decay products.\n\\end{itemize}\n\nThe fact that $\\tau$ decays are the only source of prompt leptons and\nneutrinos at energies below charm threshold makes possible to $single-tag$\n$\\tau^+\\tau^-$ events. Single tagging requires a clean signature from a single\n$\\tau$ decay. The absence of heavy flavor contamination allows several\ndistinct signatures: $e+E_{miss},\\mu+E_{miss}$ and $\\pi+E_{miss}$ (at 3.57\nGeV).  Studies \\cite{jasp1,jjlfv} indicate backgrounds of between $10^{-3}$\n(at 3.57 GeV) and $10^{-4}$ (at 3.67 GeV) in the case of the $e+E_{miss}$\ntrigger, which has a $\\tau^+\\tau^-$ event detection efficiency of 24 \\%.  %\n\n\n\\subsection{Tau studies at BF}\n\nAt the BF, the $\\tau$'s are produced with a large boost, $\\beta_\\tau=0.93$ at\n10 GeV. This results in a topological separation of the $\\tau$ decay products.\nThe signature of a $\\tau$ pair event is two small-multiplicity (up to five\ncharged tracks) back-to-back jets. The event is also characterized by missing\nenergy and transverse momentum due to the undetected neutrinos. $\\tau$ physics\nis more difficult here than at the tcF. Backgrounds are higher and must be\nestimated from Monte Carlo calculation. Single-tagging of $\\tau$'s is not\npossible. While this is a serious drawback for studies requiring a very\naccurate control of systematics (such as the limits on the $\\tau$ neutrino\nmass or the precision measurement of the $\\tau$ topological branching ratios)\nthere is still a large potential for $\\tau$ physics. In particular, the large\nboost allows a measurement of the $\\tau$ lifetime. Most exotic decays can also\nbe well studied at the BF.\n\nIn fact, one can obtain a good idea of what kind of $\\tau$ physics results can\nbe produced by the BF by looking at the CLEO-II\/CESR experiment, with more\nthan two million $\\tau$ pair events recorded.  Recent results from CLEO\n\\cite{richard} are the measurement of the $\\tau$ lifetime (with a relative\nerror of 2.6 \\%, about the same that LEP experiments), the measurement of the\nelectronic branching ratio (where they achieve a relative error of 3.7 \\%\nagain on the same range than LEP experiments) and the measurement of branching\nratios to several hadronic final states containing one or more $\\pi^0$'s.\n\n\n\\subsection{ The Detector for tcF and BF}\n\nBoth the tcF and the BF project state-of-the-art detectors \\cite{jasp2,bf1}.\nTheir main features are very similar. Again, the CLEO-II \\cite{cldet} detector\nis a good example of an operating detector on the line of the requirements of\nthe tcF and BF. This requirements can be summarized as follows.\n\\begin{itemize}\n\\item\nPrecise charged particle resolution: This is achieved by a precise tracker\n(drift chamber) embedded in a high magnetic field (around 1-1.5 T).\n\\item\nMaximum possible solid angle coverage (90-95 \\% of the solid angle) for the\nelectromagnetic calorimetry.\n\\item\nHigh resolution electromagnetic calorimetry. Crystals like CsI(Tl) and CsI(Na)\nare being studied.\n\\item\nHermetic calorimetry; an outer hadronic calorimeter to detect neutrons and\n$K^0_{L}$'s and a careful design to avoid blind regions of the detector.\n\\item\nGood particle identification for $\\pi K p$ and $e\\mu$ separation; the different\nmethods being investigated include the use of TOF, dE\/dx and RICH.\n\n\\end{itemize}\n\nAn illustration of these detectors can be found in references\n\\cite{jasp2,bf1}.\n\n\\subsection{Monte Carlo simulation}\n\nSince the general features of the detectors for tcF and BF are similar, it is\nan interesting exercise to compare the possibilities offered by both machines\nfor the study of the $\\tau$ exotic decays by assuming the same detector\nparameters. In table \\ref {pdet}  the basic parameters  of  the  projected tcF\ndetector are show. We have developed a simplified simulation of this detector\nincluding track momentum smearing in the tracker, electron and photon energy\nsmearing in the electromagnetic calorimeter, $\\gamma-\\pi^0$ separation and\ndetector inefficiencies. Notice that to understand the possibilities offered\nby both the tcF and the BF in the search for new physics through the study of\nexotic $\\tau$ decays a very detailed simulation is not essential, since\nstraight consequences can be drawn based on very simple facts like the\nkinematical separation of $\\tau\\rightarrow e\\nu_e\\nu_\\tau,\\tau\\rightarrow\n\\mu\\nu_\\mu \\nu_\\tau$ and $\\tau\\rightarrow\\pi\\nu_\\tau$ in the tcF or the\nexcellent mass resolution of both BF and tcF that will allow very easy\nidentification of $\\tau$ neutrinoless decays.\n\nThe $\\tau^+ \\tau^-$ events were generated using the KORALB program.  This\ngenerator includes all effects due to finite mass of the $\\tau$, longitudinal\nand transverse spin effects of $\\tau$'s and e's and order $\\alpha^3$ QED\nradiative corrections including single hard bremsstrahlung \\cite{koralb}. We\nhave generated and passed through the simulated detector large sample of\nevents at  two  different center of mass energies;  $\\sqrt s~=~3.67$GeV, tcF\nand $\\sqrt s~=~10$GeV, BF. Table \\ref{ntaus} shows the expected number of\n$\\tau$ pairs per year run at these energy points. A peak luminosity of\n$10^{33}$ $cm^{-2} s^{-1}$ is assumed for the tcF. For the BF we assume a\n``pessimistic\" peak luminosity of  $10^{33}$ $cm^{-2} s^{-1}$ and an\n``optimistic\" peak luminosity of $10^{34}$ $cm^{-2} s^{-1}.$\n\n\\section{Limits on Tau Exotic Decays}\n\\subsection {$\\tau$ selection at the tcF and BF}\n\nAlthough the selection of $\\tau$ events depend to some extent on the physical\nchannel that one is studying, the general philosophy is clear in both tcF and\nBF.\n\\begin{itemize}\n\\item\n$tcF.$   A $\\tau$ event is $single~tagged$ with lepton+$E_{miss}$ (50 $\\%$\nefficiency) and $\\pi+ E_{miss}$ ( at 3.57 GeV with 10 $\\%$ efficiency). These\ntags provide $\\tau$ samples virtually background-free below the charm\nthreshold and with very low background  above.\n \\item\n$BF.$  A $\\tau$ event is characterized by requiring a two back-to-back jets\nwith $1.vs.1$ or $1.vs.3$ charged tracks topology.  Events where both tracks\nare identified as electrons or muons are rejected to prevent Bhabha and dimuon\nbackgrounds. In addition, missing energy and\/or missing transverse momentum\ncan be used as a distinctive feature of tau decays on the one-prong side.\nTypically one would require the momentum of the one prong side to be smaller\nthan a fraction of the center of mass energy (i.e., $0.5E_{cms}$).  The\nefficiency of these selection criteria is high, around $90\\%$, and the\nbackground moderately small,  $3-5\\%$.\n\\end{itemize}\n\n\n\\subsection {$\\tau \\rightarrow e(\\mu) J$}\n\nThe two-body decay $\\tau\\rightarrow e (\\mu) J,$ where  $J$  is a massless,\nweakly interacting, spin zero Goldstone boson is a very clear example of the\nadvantage  of being able to produce the $\\tau'$s at rest. When $\\beta_\\tau\n\\rightarrow 0,$ this decay has the striking signature of a monochromatic\nlepton right at the end-point of the standard $\\tau\\rightarrow l \\bar{\\nu_l}\n\\nu_\\tau$ momentum distribution. At higher energies, the sensitivity is much\nweaker, since the lepton spectrum from the $e(\\mu) J$ decay is broad,\nspreading over the full spectrum of the standard $\\tau\\rightarrow l \\bar{\\nu_l}\n\\nu_\\tau$ decay. See Fig. \\ref{mono} (the $\\pi$ spectrum from the decay\n$\\tau\\rightarrow\\pi\\nu_\\tau$ is identical to the lepton spectrum from  the\ndecay $\\tau\\rightarrow e (\\mu) J$).\n\nThe procedure to extract a limit on the branching ratio of a possible exotic\ndecay $\\tau\\rightarrow e (\\mu) J$ at the tcF uses the fact that one knows the\nexact shape of both the exotic decay $\\tau\\rightarrow e (\\mu) J$ being\nsearched and the standard leptonic decay $\\tau\\rightarrow l \\bar{\\nu_l}\n\\nu_\\tau$  distributions; this is illustrated in Fig. \\ref{exotcf} where\nsimulated data in which the standard leptonic decay $\\tau\\rightarrow l\n\\bar{\\nu_l} \\nu_\\tau$ (18 $\\%$ branching ratio)  is mixed  with 1 $\\%$\nbranching ratio exotic decay $\\tau\\rightarrow e (\\mu) J$ is shown. The center\nof mass energy is $3.57$ GeV (standard optics is assumed). The region of\ninterest is $E\/E_{max}~>~0.95$. In this region, the shape of the new decay is\nknown from the data by fitting the $\\tau\\rightarrow\\pi\\nu_\\tau$ distribution\n(in the case of $\\tau \\rightarrow e J$ one needs to correct for the stronger\neffect of the radiative corrections) and also the shape of the standard signal\nis known (being simply a straight line). Thus, one can perform a maximum\nlikelihood fit. This is customarily done by minimizing the quantity\n\\begin{equation}\n -log( L(\\alpha))=-\\sum_{i=1}^n log[(1-\\alpha)U_s(x_i)~+~\\alpha U_e(x_i)],\n\\end{equation}\nwhere the $n$ is the total number of events passing the energy cut,\n$x~=~E\/E_{max}$ and the parameter $\\alpha$ gives the mixing between the\nstandard and the exotic decay, whose shapes are described by the functions\n$U_s$ and $U_e$ respectively.\n\nAt $\\sqrt s=~3.57$ GeV with standard optics we obtain a branching ratio\nsensitivity to the exotic decay $\\tau\\rightarrow e (\\mu) J$ of $10^{-5}$,\nassuming that one can separate the lepton produced in the decay\n$\\tau\\rightarrow e (\\mu) J$ and the $\\pi$ produced in the decay\n$\\tau\\rightarrow\\pi\\nu_\\tau$. The required $\\pi$\/l rejection is $10^{-4}.$\nThis can be  achieved in the case of electrons, where a $e\/\\pi$ rejection  of\n$0.1\\%$  (for 0.9 GeV particles) should come from calorimetry and, a further\nrejection smaller of 10 $\\%$ from TOF and dE\/dx. A RICH \\cite{ribe} would\nfurther improve the $e\/\\pi$ rejection by at least a factor $10^{-2}.$ On the\nother hand, $\\mu\/\\pi$ separation is limited to the order of 1 $\\%$ unless a\nRICH  is used. Thus, one could obtain the limits, $B(\\tau \\rightarrow e\nJ)~<~10^{-5}$ (standard optics), $B(\\tau \\rightarrow e J)~<~10^{-6}$\n(monochromator); and $B(\\tau \\rightarrow \\mu J)~<~10^{-3}$  (limited by\n$\\mu\/\\pi$ separation, it can be improved by an order of magnitude with a\nRICH).\n\nThe situation is far more difficult at the BF. Here, the leptons arising from\nthe standard and the exotic decay are mixed up in the full spectrum.\nCompare Figs. \\ref{exotcf} and \\ref{exobf}. The mixing of\nexotic\/standard decays is the same as above, but now the center of mass energy\nis $10.0$ GeV. The best limit that we obtain in this case is\n$B(\\tau\\rightarrow e (\\mu) J)~=~5 \\cdot 10^{-3}$.\n\n\\subsection{Neutrinoless Decays}\n\nThese kind of rare $\\tau$ decays are characterized by $zero$ missing energy,\nand the possibility of fully reconstructing the $\\tau$ mass. As a consequence\nthe event signature is straightforward. One $\\tau$ is required to decay to a\nclean ``tag\" ( i.e. $\\tau\\rightarrow l \\bar{\\nu_l} \\nu_\\tau$ giving a prompt\nlepton and missing transverse momentum) and the other to decay to the rare\ndecay under consideration. This means in general the identification of all or\nsame of the final state particles, and the reconstruction of the $\\tau$ mass.\nSince the mass resolution of the tcF or BF detector is very good, the signal\nis very clear and the backgrounds small,  in both tcF and BF. As an example we\ndiscuss here two ``typical\" rare decays.  A two-body, neutrinoless decay $\\tau\n\\rightarrow \\mu \\gamma$ and a three body neutrinoless decay $\\tau \\rightarrow\n\\mu\\mu\\mu.$ Clearly many rare decays can searched at both the tcF and the BF\nbut the general lines will be similar to the ones we discuss here. On the\nother hand, the fact that both tcF and BF can reach $10^8$ $\\tau$ pairs, and\nthus, limits $a~priori$ of $10^{-8}$ for the rare decays that we are\nconsidering, makes it interesting to consider whether backgrounds can be\nsuppressed to this very low level. We will consider two energies for our\nstudy; $\\sqrt s=3.67 $ GeV (tcF  below charm threshold ), and $\\sqrt s=10.0$\nGeV (BF). The main difference between this two cases is the possible impact of\nthe backgrounds due to heavy quarks. At both points, with ``standard\"\nassumptions one gets at least $10^7$ $\\sim$ $10^8$ $\\tau$ pairs a year, so a\ntotal yield of $10^8$ $\\tau$ pairs can be assumed.\n\nTo gain a feeling of the rejection to different backgrounds  we compute the\nrejection factor for each cut independently. Then, we make the naive approach\nthat the total rejection factor is the product of the rejection factors for\neach cut, ignoring correlations between them. This is necessary in order to\nkeep the Monte Carlo generation of background events down to some sensible\nlevel and we have checked that the  results are consistent with an exact\ncalculation using $10^5$ hadronic events.\n\n\\subsubsection {Tagging of tau neutrinoless events}\n\nTau decays have a characteristic signature that allows to tell them apart from\nbackgrounds very efficiently at both BF and tcF. For the examples that we are\nconsidering (neutrinoless decays with at least a $\\mu$ on the final state) we\nwill select the events by imposing one $\\tau$ to decay via the complementary\nlepton tag, i.e. the decay $\\tau\\rightarrow e\\nu_e\\nu_\\tau$. For the\nneutrinoless decays with electrons on final state one uses $\\tau\\rightarrow\n\\mu\\nu_\\mu \\nu_\\tau$ as a tag. Of course, others tags are possible (it also\ndepends on the specific final state that one is considering) and can result in\nincreased efficiency but its sensitivity to backgrounds could also be higher\n(i.e., tagging the decay $\\tau \\rightarrow \\mu \\gamma$ with the decay\n$\\tau\\rightarrow e\\nu_e\\nu_\\tau$ biases against radiative electron and muon\npairs). Thus, for the shake of clarity we take the `gold plated' case and\nstudy whether one can reject backgrounds to the very low level required in\nthis case.\n\nThe event is then accepted if:\n\\begin{itemize}\n\\item An energetic electron\n(more than 0.4GeV at  tcF and more than 0.8GeV\n      at  BF) is identified in the event.\n\\item The event has missing energy (more than 0.8GeV at tcF, more\n      than 1.0GeV at BF).\n\\end{itemize}\n\nFigs. \\ref{tage} and \\ref{tagmis} illustrate the effectiveness of the tags. In\nFig. \\ref{tage} the energy spectrum of the identified electron is shown for\nboth the signal (a tau neutrinoless decay with the other tau decaying to an\nelectron) and the most important background, the multihadronic events. Notice\nthe logarithmic scale. A cut $p_c$ in the electron momentum of 0.4 $GeV$ at\nthe tcF (1 GeV at BF) keeps about $70\\%$ of the signal while suppressing the\nbackground by more than one order of magnitude. In Fig. \\ref{tagmis} the\nmissing energy spectrum  {\\it when an electron has been identified with\n$p>p_c$} is shown for signal and background. Cutting at $p_{miss}$ $>$ 0.8 GeV\nat the tcF ($p_{miss}$ $>$ 1 GeV BF) again suppresses very efficiently the\nbackground while keeping a high efficiency for the signal.\n\n\n\\subsubsection {$\\tau \\rightarrow \\mu \\gamma$}\n\nThe signal for  the two energies considered will be a tagged event with only\none additional track and only one photon of at least 100 MeV (multiplicity\ncut). This track must be identified as a muon (neutrinoless decay particle\nidentification cut). Additionally, the total energy of the muon and the photon\nmust be  within the resolution of the beam energy, labeled $E_{beam}$. The\nassumed detector energy  resolutions are 20 MeV for the tcF and 50 MeV for the\nBF \\cite{jasp2,cinabro}.  Finally the invariant mass of the $\\mu-\\gamma$\nsystem must be within two $\\sigma_m$ (where $\\sigma_m$ is the mass resolution\nof the tcF or BF, typically 20 MeV) of  the $\\tau$ mass (neutrinoless mass\ncut) as illustrated in Fig. \\ref{masstcf} and in Fig. \\ref{massbf}.\n\nThe more important backgrounds to consider are multihadronic background, and\ninternal $\\tau$ background ($\\tau$ events faking the $\\tau \\rightarrow \\mu\n\\gamma$ decay). We will consider also $two-\\gamma$ events (QED events where\ntwo quasi-real photons scatter to produce a reaction of the type  $e^+e^-\n\\rightarrow e^+e^-ll (\\gamma)$ with $l$ being either an electron or a muon)\nalthough the cross section of the relevant processes (i.e, $\\gamma \\gamma\n\\rightarrow ee\\tau\\tau$ is the only relevant background for the $\\tau\n\\rightarrow \\mu\\gamma$ decay) is small. Other potentially dangerous\nbackgrounds like radiative electron or muon pair events are very effectively\nsuppressed by explicitly biasing against them as explained above   and will\nnot be considered here, but they must be carefully studied if other additional\ntags ($\\tau\\rightarrow \\mu\\nu_\\mu \\nu_\\tau\n,\\tau\\rightarrow\\pi\\nu_\\tau,\\tau\\rightarrow \\rho\\nu_\\tau$)  are used.\n\\footnotemark\n\\footnotetext{There are preliminary results from the CLEO collaboration that\nare\nexploring a wider tagging of exclusive decays.}\n\nOur results are show in table \\ref{tmu} and \\ref{bmu} for the different\nenergies considered. As one could expect the most serious background is the\ninternal $\\tau$ background that may be the limiting factor in the sensitivity\nfor this searches in the limit of very high statistics. The limits on the\nbranching ratios that can be achieved in a year run  will be: $B(\\tau\n\\rightarrow \\gamma\\mu) \\sim 10^{-7}$ for the tcF and $10^{-6}$ for the BF.\n\n\\subsubsection {$\\tau \\rightarrow \\mu\\mu\\mu$}\n\nThis study is very similar to the study described above. The signal for  the\ntwo energies considered will be a tagged event with exactly three additional\ntracks (multiplicity cut). At least one  track must be identified as a muon\n(neutrinoless decay particle identification cut). The energy of the three\ntracks must be close to the beam energy. Finally the invariant mass of the 3\ntrack system must be close to the $\\tau$ mass (neutrinoless mass cut) as\nillustrated in Fig. \\ref{masstcf2} and in Fig. \\ref{massbf2}.\n\nOur results are show in table \\ref{tele} and \\ref{bele} for the different\nenergies considered. The only limiting factor is the size of the data sample.\nTherefore the attainable limits for the branching ratio  for a year run are\n$B(\\tau \\rightarrow \\mu\\mu\\mu) \\sim$ $10^{-7}$ for the Tau Charm and the B\nFactory.\n\n\\section{Conclusions}\n\nWe have investigated the potential of $\\tau$ and B factories in searching for\nnew physics through the study of rare $\\tau$ decays. Some of these decays can\nhave substantial branching fractions even in situations where the associated\ntau neutrino is very or even massless. Therefore the physics of rare tau\ndecays is $complementary$ to the efforts of directly improving the neutrino\nmass limits, as has been widely advertised can be done at a tau factory.\n\nOur results can be summarized in table \\ref{conclu}. In the light of our\nresults a tcF is preferred for certain rare decays, such as those involving\nmajoron emission. In this case the attainable branching ratio is limited by\nthe $e(\\mu)\/\\pi$ separation, for which special attention should be given in\nthe detector design.  For neutrinoless decays both machines will do very well\nand the only limiting factor is basically the size of the data sample.\n\n\\acknowledgments\nThis work was supported by CICYT and the Ministerio de Educaci\\'on y  Ciencia\n(Spain) and the Department of Energy (USA). We thank J. Kirkby for useful\ndiscussions during this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe notion of Heisenberg algebra and of Heisenberg group appear since long time in the description of  quantum dynamical systems and sophisticated techniques of algebra, topology and functional analysis are often used, in order to understand the behaviour of elementary particles or, more in general, of some quantum systems. When it is possible to detect algebraic structures like nilpotent Lie algebras, we are in a very good position to recognize symmetries and conservation laws of significant interest in physics. This is exactly the content of the well known Noether's Theorem, \\cite{gold}. The reference of Snobl and others  \\cite{snobl} show some recent techniques in the context of Lie algebras and theoretical physics, but here we will focus on a specific classification, due to the so called ``Berkovich--Zhou invariant'' (or ``corank'') of nilpotent finite dimensional Lie algebras (over a field of characteristic $0$). \n\n\nWe will refer to the recent works \\cite{nr1, nr2, nr3}, which generalize \\cite{bat, es, es2}, but, at the same time, the present contribution continues the efforts we made in \\cite{bagrus2018}, where we looked at dynamical systems involving pseudo-bosons in terms of finite dimensional nilpotent Lie algebras for values of the dimension smaller than five.\n\n\n\n \n During our discussions, we will list a first classification \n of finite dimensional nilpotent Lie algebras by the size of\n their dimension, and, specifically, when the dimension is at\n most $5$ (see \\cite{degraaf, morozov, turk} and Theorem\n \\ref{classification} below). Then we will use  the notion of\n \\textit{Schur multiplier} of a  Lie algebra, referring to \\cite{weibel} for its classical approach via homological algebra.  Schur \\cite{schur} introduced this concept in 1907 in group theory, but his approach was subject to generalizations of various nature for different algebraic and geometric structures. \n \n \n \n The Schur multiplier does not seem to be related with the classification of finite dimensional nilpotent Lie algebras by dimension, but this is only apparently true. Indeed we\n may define a non-negative integer (which turns out to be a\n numerical invariant), called ``corank''  and due to\n Berkovich and Zhou \\cite{berkovich, zhou}. This number\n measures the size of the Schur multiplier of a nilpotent Lie\n algebra and we will see that the corank allows us to get a\n second classification of nilpotent Lie algebras, but this\n time the classification gives information on the way in\n which our Lie algebra may be written as the sum of two\n smaller abelian Lie algebras. \n \n \n This second classification is more useful if we\n want to focus on the structure of a general Lie algebra.\n Moreover,  this new classification presents again nilpotent\n Lie algebras of dimension $\\le 5$ when the value of the\n corank is smaller than $6$ (see Theorem \\ref{tl} below\n for more details).\n \n  \n While the parallel approach of classification via corank and\n via dimension was known from a pure algebraic--geometric perspective (see\n \\cite{bat, es, es2, mon, nr1, nr2}), one of the\n interesting result of the present paper is that explicit\n physical examples of dynamical systems can be related to\n this classification. This point is not known in literature,\n to the best of our knowledge.\n \n  In fact we will show that all finite dimensional nilpotent\n Lie algebras, over the field of complex numbers may be\n realized  in terms of (pseudo-) bosonic operators  when\n their corank is at most $6$ (see Theorem \\ref{main2}\n below). Moreover we will provide examples of significant\n interest in physics when the value of the corank is\n arbitrary big (see Theorem \\ref{main1}).  \n \n\n\n\n\n\n\n\n\n\n\n\n \n\n\nSection 2 recalls some well known results and definitions of algebraic nature. They can be found in \\cite{beltita,snobl,weibel} and are integrated by some recent material in \\cite{nr1, nr2, nr3}. \n We refer to \\cite[Section 3]{bagrus2018} for the terminology and notations which will be involved in the proofs of our main results, where a framework of functional analysis and of theory of the operators is applied. Finally, Section 3 illustrates  the main results with corresponding proofs. \n\n\n\\section{Some results of low dimensional cohomology}\n\nIn order to focus on the main topics, without dispersion of energies in additional details, we do not report the classical notion of \\textit{Chevalley-Eilenberg complex} for a Lie algebra, since it can be found in \\cite[Definition 7.7.1]{weibel}. Analogously, we will assume that the reader is familiar with the second cohomology Lie algebra with complex coefficients. This Lie algebra is widely described in \\cite[Corollary 7.7.3 and Exercise 7.7.5]{weibel} and  we will call it  \\textit{ Schur multiplier} of the Lie algebra $\\mathfrak{l}$, using the notation $M(\\mathfrak{l})$. In fact we will work only with finite dimensional  Lie algebras over the field of complex numbers.\n\n\nThe size of the Schur multiplier describes how many extensions we can actually form, beginning from an abelian Lie algebra with another one acting on it. Furthemore, we will see  that it is possible to introduce a numerical invariant, which we will call  ``corank'', or  ``Berkovich--Zhou invariant'', and this measures the size of the Schur multiplier from some ideal cases for which we known essentially all. The main results of this paper will indeed show that it is possible to find concrete examples for any possible value of this corank and, to the best of our knowledge, it is the first time that these examples can be found in physical contexts.\n\n\n We begin with the following notion, available in \\cite{weibel}.\n\n\n\n\\begin{defn}\\label{extensions}\nA Lie algebra $\\mathfrak{l}$ is an \\textit{extension} of an abelian Lie algebra $\\mathfrak{a}$ (called \\textit{abelian kernel}) by another  Lie algebra $\\mathfrak{b}$, if there is a short exact sequence\n$$0 \\ \\longrightarrow \\  \\mathfrak{a} {\\overset{\\mu}{\\longrightarrow}} \\ \\mathfrak{l} \\ {\\overset{\\varepsilon}{\\longrightarrow}} \\mathfrak{b} \\ \\longrightarrow \\ 0$$\nsuch that  $\\mathfrak{a}$ is an abelian ideal of $\\mathfrak{l}$ and $\\mathfrak{l}\/\\mathfrak{a} \\simeq \\mathfrak{b}$ is the Lie algebra quotient. Here $\\mu$ is a monomorphism and $\\varepsilon$ an epimorphism of Lie algebras such that $\\ker \\varepsilon = \\mathrm{Im} \\ \\mu $. If in addition, $\\mathfrak{a}$ is contained in the center \\[Z(\\mathfrak{l})=\\{ a \\in \\mathfrak{l}  \\ | \\ [a,b]=0  \\ \\forall b \\in \\mathfrak{l}\\} \\]\n of $\\mathfrak{l}$, the Lie algebra $\\mathfrak{l}$ is called \\textit{central extension} of $\\mathfrak{a}$ by $\\mathfrak{b}$.\n\\end{defn}\n\nDefinition \\ref{extensions} is reported from \\cite[Definition 7.6.1]{weibel}, where it is noted that $\\mathfrak{a}$ may be endowed with the structure of $\\mathfrak{b}$-module  defining $ba$ to be the product $[\\widetilde{b},a]$ in $\\mathfrak{l}$, where $\\varepsilon(\\widetilde{b})=b$ for all $a \\in \\mathfrak{a}$ and $b \\in \\mathfrak{b}$. Following \\cite[Extension Problem 7.6.2]{weibel}, two extensions  $0 \\ \\longrightarrow \\  \\mathfrak{a} {\\overset{\\mu_1}{\\longrightarrow}} \\ \\mathfrak{l}_1 \\ {\\overset{\\varepsilon_1}{\\longrightarrow}} \\mathfrak{b} \\ \\longrightarrow \\ 0$ and $0 \\ \\longrightarrow \\  \\mathfrak{a} {\\overset{\\mu_2}{\\longrightarrow}} \\ \\mathfrak{l}_2 \\ {\\overset{\\varepsilon_2}{\\longrightarrow}} \\mathfrak{b} \\ \\longrightarrow \\ 0$ are equivalent if there is an isomorphism $\\varphi$ such that the following diagram \n$$\\begin{CD}\n0  @>>>   \\mathfrak{a} @>\\mu_1>> \\mathfrak{l}_1 @>\\varepsilon_1>>  \\mathfrak{b} @>>> 0\\\\ \n @. @| @VV\\varphi V @| @. \\\\ \n0  @>>>   \\mathfrak{a} @>\\mu_2>> \\mathfrak{l}_2 @>\\varepsilon_2>>  \\mathfrak{b} @>>> 0\n\\end{CD}\n$$\nis commutative. Then we can define the set $\\mathrm{Ext}(\\mathfrak{b},\\mathfrak{a})$ of equivalence classes of the extensions of  $\\mathfrak{a} $ by  $\\mathfrak{b}$ realizing $\\mathfrak{l}$. The problem of the description of $\\mathrm{Ext}(\\mathfrak{b},\\mathfrak{a})$ is known as Extension Problem and a  classical result \\cite[Classification Theorem 7.6.3]{ weibel} shows an important relation with the Schur multiplier.\n\n\n\\begin{lem}[See \\cite{weibel}, Chapter 7]\\label{h2}Given an extension $0 \\ \\rightarrow \\  \\mathfrak{a} \\rightarrow \\ \\mathfrak{l} \\rightarrow \\mathfrak{b} \\ \\rightarrow \\ 0$ of a Lie algebra $\\mathfrak{l}$ as in Definition \\ref{extensions}:\n\\begin{itemize}\n\\item[(i)]There is a 1-1 correspondence between $M(\\mathfrak{l})$ and $\\mathrm{Ext}(\\mathfrak{b},\\mathfrak{a})$;\n\\item[(ii)] If $M(\\mathfrak{l})=0$, then $\\mathfrak{l}$ cannot be realized as  extension with abelian kernel.\n\\end{itemize}\n\\end{lem}\n\nAs indicated in Lemma \\ref{h2} (ii),  the size of the Schur multiplier actually measures how many extensions are possible. Further important examples of extensions are constructed below.\n\n\\begin{defn}\\label{semidirect}\nA Lie algebra $\\mathfrak{l}$ is the {\\em semidirect sum} of two of its Lie subalgebras $\\mathfrak{a}$ and $\\mathfrak{b}$ if  the following conditions are satisfied:\n\\begin{itemize}\n\\item[(i)]$\\mathfrak{a}$  is an ideal of $\\mathfrak{l}$,\n\\item[(ii)] $\\mathfrak{l}= \\mathfrak{a} + \\mathfrak{b}$,\n\\item[(iii)] $\\mathfrak{a} \\cap \\mathfrak{b} = 0$.\n\\end{itemize}\nOf course, if $\\mathfrak{a} \\subseteq Z(\\mathfrak{l})$, then $\\mathfrak{l}$ is a central extension of $\\mathfrak{a}$ by $\\mathfrak{b}$.\n\\end{defn}\n\nHaving in mind Lemma \\ref{h2} (i),  Definition \\ref{semidirect} and Definition \\ref{extensions}, it is clear that semidirect sums are special cases of extensions with abelian kernel, and so the Schur multiplier gives information of structure of a Lie algebra. In this perspective, we recall another classical result. A semisimple Lie algebra is a sum of simple Lie algebras. A well known theorem of Levi illustrates the importance of the semidirect sum in the structure of the finite dimensional Lie algebras.\n\n\n\\begin{thm}[See \\cite{weibel}, Chapter 7, Levi Decomposition]\nAny finite dimensional Lie algebra $\\mathfrak{l}$ over $\\mathbb{C}$  is semidirect sum of a solvable Lie algebra $\\mathfrak{a}$ by a semisimple Lie algebra $\\mathfrak{b}$.\n\\end{thm}\n\n\n Note that a solvable Lie algebra is a Lie algebra which\n contains a finite series of ideals in which each factor is\n an abelian Lie algebra. While the role of the Schur\n multiplier of solvable Lie algebras is quite difficult to\n compute, the  result above is classical and may help for the\n case of semisimple Lie algebras. An important consequence is\n the following.\n\n\n\\begin{cor}[See \\cite{weibel}, Chapter 7]\\label{semisimple} A semisimple Lie algebra $\\mathfrak{l}$ over $\\mathbb{C}$ of finite dimension has $M(\\mathfrak{l})=0$.\n\\end{cor}\n\n\n\n\n\nFor a series of motivations, which will be clear later on, we need to investigate Schur multipliers of Lie algebras in which we may iterate the notion of center.\n\n\\begin{defn}\\label{center}If $\\mathfrak{l}$ is a Lie algebra, the \\textit{derived} Lie subalgebra of $\\mathfrak{l}$ is defined as the smallest Lie ideal containing all the commutators  $[a,b]=ab-ba$  of $\\mathfrak{l}$, that is,\n\\[[\\mathfrak{l},\\mathfrak{l}]=\\langle [a,b]  \\ | \\ a,b \\in \\mathfrak{l} \\rangle. \\]\nNote that the Lie algebra quotient $\\mathfrak{l}\/[\\mathfrak{l},\\mathfrak{l}]$ is always abelian,\n \\end{defn}\n\n\n\n\nBoth  $Z(\\mathfrak{l})$ and $[\\mathfrak{l},\\mathfrak{l}]$ are ideals of $\\mathfrak{l}$ and so the size of their quotients may  measure  how far we are from having an abelian Lie algebra. Since we do not know whether $\\mathfrak{l}\/Z(\\mathfrak{l})$ is abelian or not in general,  there is a way to generalize the above notions via appropriate series (see \\cite{ weibel}).\n\n\n\\begin{defn}\\label{uppercentralseries}The \\textit{upper central series} of $\\mathfrak{l}$ is the series\n\\[0=Z_0(\\mathfrak{l}) \\leq Z_1(\\mathfrak{l})=Z(\\mathfrak{l}) \\leq Z_2(\\mathfrak{l}) \\leq \\ldots \\leq Z_i(\\mathfrak{l}) \\le Z_{i+1}(\\mathfrak{l}) \\leq \\ldots, \\]\nwhere each $Z_i(\\mathfrak{l})$ turns out to be an ideal of $\\mathfrak{l}$ (called $i$th $center$ of $\\mathfrak{l}$) and \n\\[ \\frac{Z_1(\\mathfrak{l})}{Z_0(\\mathfrak{l})}=Z(\\mathfrak{l}), \\frac{Z_2(\\mathfrak{l})}{Z_1(\\mathfrak{l})}=Z\\left(\\frac{\\mathfrak{l}}{Z(\\mathfrak{l})}\\right),   \\ldots, \\frac{Z_{i+1}(\\mathfrak{l})}{Z_i(\\mathfrak{l})}=Z\\left(\\frac{\\mathfrak{l}}{Z_i(\\mathfrak{l})}\\right), \\ldots \\]\nis an abelian quotient Lie algebra. We say that $\\mathfrak{l}$ is  \\textit{nilpotent of class $c$} if $\\mathfrak{l}=Z_c(\\mathfrak{l})$, that is, if the upper central series of $\\mathfrak{l}$ ends after a finite number $c \\ge 0$ of steps.\n\nBy analogy, the \\textit{lower central series} of $\\mathfrak{l}$ is defined  inductively by\n\\[\\gamma_1(\\mathfrak{l})=\\mathfrak{l} \\geq \\gamma_2(\\mathfrak{l})=[\\mathfrak{l},\\mathfrak{l}]  \\geq  \\gamma_3(\\mathfrak{l})=[[\\mathfrak{l},\\mathfrak{l}],\\mathfrak{l}]  \\geq \\ldots \\geq \\gamma_i(\\mathfrak{l}) \\geq \\gamma_{i+1}(\\mathfrak{l}) \\geq \\ldots, \\]\nwhere each $\\gamma_i(\\mathfrak{l})$ turns out to be an ideal of $\\mathfrak{l}$ (called $i$th $derived$ of $\\mathfrak{l}$).\n\\end{defn}\n\n\nThe presence of finite dimensional nilpotent Lie algebras plays a fundamental role in mathematical physics, because these structures have been observed in several quantum dynamical systems (see \\cite{beltita}).  Probably one of the first nilpotent Lie algebras, which was used for the description of the behaviour of complex dynamical systems, in which bosons are involved, is due to Heisenberg and we report the formal definition here.\n\n\n\\begin{defn}\\label{Heisenberg}  A finite dimensional Lie algebra  $\\mathfrak{l}$ is called $Heisenberg$ provided that $[\\mathfrak{l},\\mathfrak{l}]=Z(\\mathfrak{l})$ and $\\mathrm{dim}([\\mathfrak{l},\\mathfrak{l}]) = 1$. Such algebras are odd dimensional with\nbasis $v_1, \\ldots , v_{2m}, v$ and the only non--zero commutator between basis elements is $[v_{2i-1}, v_{2i}] = -\n[v_{2i}, v_{2i-1}]= v$ for $i = 1,2, \\ldots ,m$. The symbol $\\mathfrak{h}(m)$\ndenotes the Heisenberg algebra of dimension $2m + 1$.\n\\end{defn}\n\n\n\n One can see easily that $\\mathfrak{h}(m)$ is a nilpotent\n Lie algebra of class $2$, that is, such that\n $Z_2(\\mathfrak{h}(m)))=\\mathfrak{h}(m)$, of\n arbitrary dimension of the form $2m+1$,  and such that the\n Lie algebra quotient\n $\\mathfrak{h}(m)\/Z(\\mathfrak{h}(m))$ is abelian,\n where $Z(\\mathfrak{h}(m))$ a 1-dimensional Lie algebra.\n It turns out that the structure of finite dimensional\n nilpotent Lie algebras of low dimension involves strongly\n Heisenberg algebras and may be described in terms of\n generators and relations in the following way.\n \n\n\n\n\\begin{thm}[Classification of Finite Dimensional Nilpotent Lie Algebras of Dimension 3, 4 and  5]\\label{classification}\nLet $\\mathfrak{l}$ be a finite dimensional nilpotent Lie algebra (over any field of characteristic $\\neq 2$) and $\\mathfrak{i}$ an abelian Lie algebra of dimension 1. Then\n\\begin{itemize}\n\\item[(1)] $\\mathrm{dim} \\ \\mathfrak{l} =3$ if and only if $\\mathfrak{l}$ is isomorphic to one of the following Lie algebras:\n \\begin{itemize}\n\\item[-] $\\mathfrak{l}_{3,1}$ abelian Lie algebra of dimension 3,\n\\item[-] $\\mathfrak{l}_{3,2}  \\simeq \\mathfrak{h}(1)$.\n\\end{itemize}\n\\item[(2)]$\\mathrm{dim} \\ \\mathfrak{l} =4$ if and only if    $\\mathfrak{l}$ is isomorphic to one of the following Lie algebras:\n \\begin{itemize}\n\\item[-] $\\mathfrak{l}_{4,1} =  \\mathfrak{l}_{3,1} \\oplus \\mathfrak{i}$,\n\\item[-] $\\mathfrak{l}_{4,2} =  \\mathfrak{l}_{3,2} \\oplus \\mathfrak{i}$\n\\item[-] $\\mathfrak{l}_{4,3} = \\langle v_1, v_2, v_3, v_4 \\ | \\ [v_1,v_2]=v_3, [v_1,v_3]=v_4 \\rangle$\n\\end{itemize}\n\\item[(3)]$\\mathrm{dim} \\ \\mathfrak{l} =5$ if and only if $\\mathfrak{l}$ is isomorphic to one of the following Lie algebras:\n\\begin{itemize}\n\\item[-] $\\mathfrak{l}_{5,1} = \\mathfrak{l}_{4,1} \\oplus \\mathfrak{i} $,\n\\item[-] $\\mathfrak{l}_{5,2} = \\mathfrak{l}_{4,2} \\oplus \\mathfrak{i} \\simeq \\mathfrak{h}(1) \\oplus \\mathfrak{i} \\oplus \\mathfrak{i}$,\n\\item[-] $\\mathfrak{l}_{5,3} = \\mathfrak{l}_{4,3} \\oplus \\mathfrak{i} $,\n\\item[-] $\\mathfrak{l}_{5,4} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=[v_3,v_4]=v_5 \\rangle \\simeq \\mathfrak{h}(2) $,\n\\item[-] $\\mathfrak{l}_{5,5} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=v_3, [v_1,v_3]=[v_2,v_4]=v_5 \\rangle$,\n\\item[-] $\\mathfrak{l}_{5,6} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=v_3, [v_1,v_3]=v_4,$ $$[v_1,v_4]=[v_2,,v_3]=v_5  \\rangle,$$\n\\item[-] $\\mathfrak{l}_{5,7} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=v_3, [v_1,v_3]=v_4, [v_1,v_4]=v_5 \\rangle$,\n\\item[-] $\\mathfrak{l}_{5,8} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=v_4, [v_1,v_3]=v_5 \\rangle$,\n\\item[-] $\\mathfrak{l}_{5,9} =  \\langle v_1, v_2, v_3, v_4, v_5 \\ | \\ [v_1,v_2]=v_3, [v_1,v_3]=v_4, [v_2,v_3]=v_5 \\rangle$.\n\\end{itemize}\n\\end{itemize}\n\\end{thm}\n\nAn important observation can be done here: this is designed for our purposes.\n\n\\begin{lem}\\label{technical}\nThe nilpotent Lie algebras $\\mathfrak{l}_{4,3}$, $\\mathfrak{l}_{5,4}$, $\\mathfrak{l}_{5,5}$,  $\\mathfrak{l}_{5,6}$, $\\mathfrak{l}_{5,7}$, $\\mathfrak{l}_{5,8}$ and $\\mathfrak{l}_{5,9}$ may be realized as central extensions.\n\\end{lem}\n\n\\begin{proof} We begin to consider $\\mathfrak{l}_{4,3}$, which is  described at \\cite[Page 3530]{har1}. We note that  $\\mathfrak{l}_{4,3}$ is nilpotent of class $2$ and is realized as central extension of a  1-dimensional ideal $\\mathfrak{i}=\\mathfrak{a} \\subseteq Z(\\mathfrak{l}_{4,3})$ by the  quotient  $\\mathfrak{b}=\\mathfrak{l}_{4,3}\/\\mathfrak{a}=\\mathfrak{h}(1)$.  Definition \\ref{extensions} is satisfied. On the other hand, it is clear that the Heisenberg algebra of dimension 5 is realized as central extension, just looking at the way in which it is defined. Then we look at \\cite[Pages 3531--3532, Case 2]{har1} and see that $\\mathfrak{l}_{5,5}$ is also nilpotent of class $2$ and  realized as central extension of a  1-dimensional ideal $\\mathfrak{i}=\\mathfrak{a} \\subseteq Z(\\mathfrak{l}_{5,5})$ by the Lie algebra quotient  $\\mathfrak{b}=\\mathfrak{l}_{5,5}\/\\mathfrak{a}=\\mathfrak{h}(1)+\\mathfrak{i}$. Again Definition \\ref{extensions} is satisfied. On the other hand, we may look at \\cite[Pages 3531--3532]{har1} and see that $\\mathfrak{l}_{5,8}$ may be  realized as central extension of a  1-dimensional ideal $\\mathfrak{i}=\\mathfrak{a} \\subseteq Z(\\mathfrak{l}_{5,5})$ by the Lie algebra quotient  $\\mathfrak{b}=\\mathfrak{l}_{5,5}\/\\mathfrak{a}=\\mathfrak{h}(1)+\\mathfrak{i}$. Again Definition \\ref{extensions} is satisfied.  Then we consider $\\mathfrak{l}_{5,6}$. We  look  at \\cite[Page 4206, Case 2.1]{har} and see that it is nilpotent of class $2$ and realized as central extension of a 1-dimensional ideal  $\\mathfrak{i}=\\mathfrak{a} \\subseteq \\mathfrak{l}_{5,6}=\\mathfrak{l}_{4,3}$. Definition \\ref{extensions} is satisfied and, in the same way, we can construct  $\\mathfrak{l}_{5,7}$ and $\\mathfrak{l}_{5,9}$ as central extensions with central ideal of dimension one. These three central extensions of a central ideal of dimension one by $\\mathfrak{l}_{4,3}$ are not equivalent, as indicated by the computations at \\cite[Page 4206, Case 2.1]{har}. The result follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n We will quote now some recent results on Schur multipliers of Lie algebras, in order to understand numerically the size of $M(\\mathfrak{l})$\n\n\n\n\n\\begin{thm}[See \\cite{nr1}, Lemmas 2.4 and 2.5] \\label{ab} \\\n\n\\begin{itemize}\n\\item[(i)]$\\mathrm{dim} \\ M(\\mathfrak{h}(1))=2$.\n\\item[(ii)]$\\mathrm{dim} \\ M(\\mathfrak{h}(m)) =2m^2-m-1$ for all $m\\geq 2$.\n\\item[(iii)]A nilpotent finite dimensional Lie algebra $\\mathfrak{l}$ of dimension $n$ has\n$$\\mathrm{dim} \\ M(\\mathfrak{l}) \\le \\frac{1}{2}n(n-1)$$\nand the bound is exactly $\\frac{1}{2}n(n-1)\n$ if and only if $\\mathfrak{l}$ is abelian.\n\\end{itemize}\n\\end{thm}\n\n\n\nThe condition (iii) above is due to Moneyhun \\cite{mon} and allows us to introduce the following notion, which has been studied in \\cite{nr3}.\n\n\\begin{defn}\\label{corank} Given a nilpotent Lie algebra $\\mathfrak{l}$ of dimension $n$, there exists a (unique) integer  $t(\\mathfrak{l})\\geq 0$ such that\n\\begin{equation}\nt(\\mathfrak{l})=\\frac{1}{2}n(n - 1) - \\mathrm{dim}\\ M(\\mathfrak{l}).\n\\end{equation}\nThis number is called \\textit{corank} of $\\mathfrak{l}$ (in the sense of Berkovich--Zhou).\n\\end{defn}\nThe  invariant of Definition \\ref{corank} is originally due to Berkovich \\cite{berkovich} and Zhou \\cite{zhou}, who  introduced it for finite $p$-groups. It clearly gives information on the gap between the size of $M(\\mathfrak{l})$ and the critical value $\\frac{1}{2}n(n - 1)$ in which we find abelian Lie algebras only (see Theorem \\ref{ab}). The same information can be rephrased in the context of groups as well. Of course, the case of Lie algebra is more rich than the case of finite $p$-groups and we have more instruments than the group case; this has allowed to describe the structure of all nilpotent Lie algebras in \\cite{har} and \\cite{har1} with $t(\\mathfrak{l}) = 3, 4, 5, 6, 7 ,8$.\n\n\n \\begin{thm}[Classification of Finite Dimensional Nilpotent Lie algebras via $t(\\mathfrak{l}) \\le 6$]\\label{tl} Let $\\mathfrak{l}$ be an $n$-dimensional nilpotent Lie algebra. Then\n\\begin{itemize}\n\\item[(i)]$t(\\mathfrak{l})=0$ if and only if $\\mathfrak{l}$ is abelian;\n\\item[(ii)]$t(\\mathfrak{l})=1$ if and only if $ \\mathfrak{l} \\simeq \\mathfrak{h}(1)$;\n\\item[(iii)]$t(\\mathfrak{l})=2$ if and only if $ \\mathfrak{l}  \\simeq \\mathfrak{l}_{4,2} \\simeq \\mathfrak{h}(1) \\oplus \\mathfrak{i}$;\n\\item[(iv)]$t(\\mathfrak{l})=3$ if and only if $ \\mathfrak{l} \\simeq \\mathfrak{l}_{5,2} \\simeq \\mathfrak{l}_{4,2} \\oplus \\mathfrak{i}$;\n\\item[(v)]$t(\\mathfrak{l})=4$ if and only if either $ \\mathfrak{l} \\simeq \\mathfrak{l}_{5,2} \\oplus \\mathfrak{i}$, or $ \\mathfrak{l} \\simeq  \\mathfrak{l}_{4,3}$ or $ \\mathfrak{l} \\simeq \\mathfrak{l}_{5,8}$;\n\\item[(vi)] $t(\\mathfrak{l})=5$ if and only if $ \\mathfrak{l} \\simeq \\mathfrak{l}_{7,2} \\simeq \\mathfrak{l}_{5,2}\\oplus \\mathfrak{i} \\oplus \\mathfrak{i}$ or $ \\mathfrak{l} \\simeq \\mathfrak{h}(2)$.\n\\item[(vii)] $t(\\mathfrak{l})=6$ if and only if either  $ \\mathfrak{l} \\simeq \\mathfrak{l}_{4,2}\\oplus \\mathfrak{i}$, or $\\mathfrak{l} \\simeq \\mathfrak{l}_{5,5}$, or $ \\mathfrak{l} \\simeq \\mathfrak{h}(2) \\oplus \\mathfrak{i}$, or $\\mathfrak{l} \\simeq \\mathfrak{l}_{5,8} \\oplus \\mathfrak{i}$, or $\\mathfrak{l} \\simeq \\mathfrak{l}_{8,2} \\simeq \\mathfrak{l}_{7,2} \\oplus \\mathfrak{i}$;\n\\end{itemize}\n\\end{thm}\n\n\n\n\nThere are a series of open problems which prevent us to have a more general approach for higher dimensions. \n\n\\begin{op}The classification for $t ( \\mathfrak{l} ) \\ge 9$ is not known in general. There are some partial classification, imposing restrictions of various nature on $\\mathfrak{l}$. \n\\end{op}\n\nOn the other hand, \\cite{har} shows that for  $t ( \\mathfrak{l} ) \\in \\{7,8\\}$ we have nilpotent Lie algebras of dimension higher than $5$ for which an argument as in  Lemma \\ref{technical} is not available. Therefore the main results of this paper will show that it is possible to realize physically the above nilpotent Lie algebras using concepts of quantum physics. This is the first time that the classification above may be interpreted in terms of concrete dynamical systems, and we will see that this approach is so sophisticated that may cover in fact all the Lie algebras of Theorem \\ref{tl}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Main results}\nIn the present section we mainly refer to pseudo-bosons, considering bosons as a particular case of those and assume the reader is familiar with the terminology which is used in \\cite{baginbagbook, a2, bagBS, bagnew1, bagnew2, barnett, gold, schu}. \n\nIn particular, one needs to have clear the notions of  \n\n\\begin{itemize}\n\\item[-] bounded operators in Hilbert spaces (see \\cite[Definition 3.1]{bagrus2018})  \n\\item[-]Riesz basis (see \\cite{schu}), \n\\item[-] $\\mathcal{G}$-quasi basis in \\cite[Definition 3.3]{bagrus2018}, \n\\item[-]canonical commuting relations (see \\cite{baginbagbook}), \n\\item[-] distribution (see \\cite[Eq. (3.2)]{bagrus2018}), \n\\item[-]Assumption $\\mathcal{D}$-pb 1,  Assumption $\\mathcal{D}$-pb 2, Assumption $\\mathcal{D}$-pb 3, Assumption $\\mathcal{D}$-pbw 3, Assumption $\\mathcal{D}$-pbs 3, reported in \\cite[Section 3]{bagrus2018}. \n\\end{itemize}\n\n\nThe above concepts are strictly related to the field of the functional analysis and allow us to construct the framework  in which pseudo-bosons may form the structure of a Lie algebra, which is described in the previous section in abstract.\n \n\n\n\n\n\\begin{thm}\\label{main1}There is at least one family of  pseudo-bosonic operators and a nilpotent Lie algebra $\\mathfrak{l}$ of finite dimension for arbitrary values of $t(\\mathfrak{l})$.\n\\end{thm}\n\n\\begin{proof} About the notations and the terminology for pseudo-bosonic operators, we will use the same in \\cite[Section 3]{bagrus2018}. We begin to consider the nilpotent Lie algebra \\begin{equation}\n\\mathfrak{l} = \\mathfrak{h}(m) \\oplus \\underbrace{\\mathfrak{i} \\oplus \\ldots \\oplus \\mathfrak{i}}_{k-\\mbox{times}}.\n\\end{equation}\nThis Lie algebra has dimension $(2m+1)+k$, where $m \\ge 1$ and $k \\ge 0$ are arbitrary integers.\n\n\n\nIf $k=0$, then $\\mathfrak{l} \\simeq \\mathfrak{h}(m)$. In this case a possible example  can be constructed with $m$-dimensional bosonic operators, and with their pseudo-bosonic versions. \n\n\nLet $x_j$, $j=1,2,\\ldots,m$ be $m$ position operators, and let $p_j=-i\\frac{d}{dx_j}$ the related momentum operator. Then the generators of $\\mathfrak{h}(m)$ can be written as follows:\n$$\nv_{2j-1}=\\frac{1}{\\sqrt{2}}(x_j+ip_j), \\qquad v_{2j}=\\frac{1}{\\sqrt{2}}(x_j-ip_j),\n$$\n$j=1,2,\\ldots,m$, with $v_{2m+1}=\\mathbb{I}$, the identity operator. There are just lowering and raising operators associated to an $m$-dimensional quantum harmonic oscillator, or (alternatively) to $m$ modes of bosons. \n\n\nA different representation can also be constructed easily in terms of pseudo-bosonic operators, simply by considering a shifted version of the $v_j$, that is,  \n\n\n\n$$w_{2j-1}=v_{2j-1}+\\alpha_j \\mathbb{I}, \\qquad  w_{2j}=v_{2j}+\\beta_j \\mathbb{I},$$  \n$j=1,2,\\ldots,m,$, for any scalar $\\alpha_j, \\beta_j \\in \\mathbb{C}$ with $\\alpha_j\\neq\\overline{\\beta_j}.$ \n\nOn the other hand,  we may consider the genuine abelian case $\\mathfrak{l}$ of dimension $k$ putting $m=0$. Therefore\n\\begin{equation}\\label{heisenberg+abelian}\n\\mathfrak{l}=\\langle v_1, v_2, \\ldots, v_{2m}, v_{2m+1}, \\ldots, v_{2m+k+1} \\ | \\ [v_{2j-1},v_{2j}]=\\mathbb{I} \\  \\ \\forall j =1, 2, \\ldots, m,\n\\end{equation}\n$$ \\mbox{and} \\ v_{2m+1}= v_{2m+2}= \\ldots = v_{2m+k+1}=\\mathbb{I} \\rangle. $$\n\n\nThis means that there is no loss of generality in assuming both $m \\ge 0$ and $k\\ge 0$. \n\n\nFrom \\cite[Theorem 4.2, Page 44]{nr3}, we can see that\n\\begin{equation}\nt(\\mathfrak{l}) = \\mathrm{dim} \\ \\mathfrak{l} = 2m+k+1,\n\\end{equation}\nwhere $m$ and $k$ can grow arbitrarily.\nThis allows us to conclude that $\\mathfrak{l}$ is always realized by pseudo-bosons and its corank may be arbitrary big, so the result follows.\n\n\\end{proof}\n\n\nNow we provide concrete examples for the Lie algebras with $t(\\mathfrak{l}) \\le 6$.\n\n\n\n\\begin{thm}\\label{main2}All the nilpotent Lie algebras of Theorem \\ref{tl} are realized by pseudo-bosonic operators.\n\\end{thm}\n\n\n\\begin{proof}\nWe begin to consider $t(\\mathfrak{l})$ for the values $0,1,2,3,4$. Looking at Theorem \\ref{tl} (i), (ii), (iii), (iv) and (v), we identify just two cases which do not fit into the form of the Lie algebra \\eqref{heisenberg+abelian}. These are $\\mathfrak{l}_{4,3}$ and $\\mathfrak{l}_{5,8}$. We are going to prove that both of them may be described in terms of pseudo-bosonic operators. About the notations and the terminology for pseudo-bosonic operators, we will use again that  in \\cite[Section 3]{bagrus2018}.\n\n\nLet $c$ be a bosonic operator such that $[c,c^\\dagger]=cc^\\dagger-c^\\dagger\\,c=\\mathbb{I}$.  We define\n\\begin{equation}\\label{add1}\nv_1=c,\\quad v_2=\\frac{1}{2}\\,(c^\\dagger)^2,\\quad v_3=c^\\dagger,\\quad v_4=\\mathbb{I}.\n\\end{equation}\nIt is easy to check that all the commutators are zero, except that\n$$ [v_1,v_2]=v_3, \\qquad [v_1,v_3]=v_4.$$\nHence we get $\\mathfrak{l}_{4,3}$. These operators can be used in quantum optics to construct two-photons Hamiltonians which are relevant, for instance, in connection with squeezed states, \\cite{barnett}. The Hamiltonian is $H_0=\\omega c^\\dagger c+i(g{c^\\dagger}^2-\\overline{g}c^2)$ which, in terms of $v_j$ above can be written as\n$$\nH=\\omega v_3v_1+i\\left(2gv_2-\\overline{g}v_1^2\\right).\n$$\nHere $\\omega$ is a real quantity related to the free energy of the bosons, while the complex $g$ {\\em measures} the degree of squeezing of the system, \\cite{barnett}. \n\n\n\n\nAnother example of the same algebra can be constructed using pseudo-bosonic operators $a$ and $b$ satisfying $[a,b]=\\mathbb{I}$. In this case we put\n$$\nv_1=a,\\quad v_2=\\frac{1}{2}\\,b^2,\\quad v_3=b,\\quad v_4=\\mathbb{I}.\n$$ They also describe $\\mathfrak{l}_{4,3}$.\n\nOf course, the same Hamiltonian $H$ as above could be used simply replacing $v_j$ with their pseudo-bosonic counterparts. This could be relevant in the analysis of bi-squeezed states, which extend squeezed states to a non hermitian context, as bi-coherent states do for coherent states.\n\nNow we pass to describe $\\mathfrak{l}_{5,8}$.\nIn this case one single bosonic or pseudo-bosonic operator is not enough. For this reason, let us consider two bosonic operators $c_1$ and $c_2$ such that $$\n[c_k,c_l^\\dagger]=\\delta_{k,l}\\mathbb{I}, \\qquad [c_k,c_l]=0,\n$$\nwhere $\\delta_{k,l}$ denotes the well known symbol of Kronecker. Then, defining\n$$\nv_1=c_1,\\quad v_2=c_1^\\dagger,\\quad v_3=c_1^\\dagger\\,c_2,\\quad v_4=\\mathbb{I}, \\quad v_5=c_2,\n$$\nthe only non zero commutators are the following:\n$$\n[v_1,v_2]=v_4, \\qquad [v_1,v_3]=v_5.$$\nHence we recover $\\mathfrak{l}_{5,8}$. The same result can be found if we use two pseudo-bosonic operators $[a_1,b_1]=[a_2,b_2]=\\mathbb{I}$, all the other commutators being zero. In this case we have to define\n$$ v_1=a_1,\\quad v_2=b_1,\\quad v_3=b_1\\,a_2,\\quad v_4=\\mathbb{I}, \\quad v_5=a_2.\n$$\nAs for a physical appareance of these (bosonic or pseudo-bosonic) operators we could consider the operator\n $$\n H=\\lambda(v_3+v_2v_5),\n $$\nfor real $\\lambda$. $H$ is manifestly not self-adjoint either if we use bosonic or pseudo-bosonic operators. The meaning of $H$ is simple: it describes interactions between the two modes of particles. Notice that no kinetic term is present in $H$. This is not strange, in concrete models, since quite often the most relevant part of an Hamiltonian is that term which describes the interactions. \n\n\n\nThen we pass to consider the case of $t(\\mathfrak{l})=5$, looking at Theorem \\ref{tl} (vi). Again here we may realize pseudo-bosonic operators via the Lie algebra \\eqref{heisenberg+abelian}.\n\n\nPassing to the case $t(\\mathfrak{l})=6$,  Theorem \\ref{tl} (vii) shows that $\\mathfrak{l}_{5,5}$ and $\\mathfrak{l}_{5,8} \\oplus \\mathfrak{i}$ are the cases not involved in \\eqref{heisenberg+abelian}. While $\\mathfrak{l}_{5,8} \\oplus \\mathfrak{i}$ may be written easily in terms of pseudo-bosonic operators adding an operator which commutes with all those of $\\mathfrak{l}_{5,8}$, an explicit construction for $\\mathfrak{l}_{5,5}$  must be given.\n\n\n\nFor that, we can invoke Lemma \\ref{technical} and see that it is possible to construct this Lie algebra with a family of pseudo-bosonic operators realizing  the Lie algebra $\\mathfrak{h}(1) \\oplus \\mathfrak{i}$ and a central ideal $\\mathfrak{a}=\\mathfrak{i}$. Therefore $\\mathfrak{l}_{5,5}$ is the central extension of $\\mathfrak{a}$ by $\\mathfrak{b}=\\mathfrak{l}_{5,5}\/\\mathfrak{a}$. This is a first possible abstract approach.\n\n\nA different realization of $\\mathfrak{l}_{5,5}$ can be constructed using again two modes of bosonic (or pseudo-bosonic) operators $c_1$ and $c_2$ as for $\\mathfrak{l}_{5,8}$. Now we define\n$$\nv_1=c_1,\\quad v_2=c_2+\\frac{1}{2}(c_1^\\dagger)^2,\\quad v_3=c_1^\\dagger,\\quad v_4=c_2^\\dagger,\\quad v_5=\\mathbb{I}.\n$$\nThe only non zero commutators are $[v_1,v_2]=v_3$, and $[v_1,v_3]=[v_2,v_4]=v_5$. These commutators can also be recovered using pseudo-bosonic operators $(a_1,b_1)$ and $(a_2,b_2)$. In this case we define\n$$\nv_1=a_1,\\quad v_2=a_2+\\frac{1}{2}b_1^2,\\quad v_3=b_1,\\quad v_4=b_2,\\quad v_5=\\mathbb{I}.\n$$\nA possible Hamiltonian constructed out of these last operators is the following:\n$$\nH=\\omega v_3v_1+\\lambda v_4v_2,\n$$\nwhich is manifestly non self-adjoint, and describes a sort of \"always increasing\"\n interaction between two modes, without any decreasing.\n\\end{proof}\n\n\n\n\n\nAs we have seen in the proof of the previous theorem, the notion of central extension in Definition \\ref{extensions} is appropriate to describe nilpotent Lie algebras in terms of pseudo-bosons. In fact we can describe all those of dimension at most five.\n\n\\begin{cor}\\label{dimension}\nAll the nilpotent Lie algebras of Theorem \\ref{classification} may be realized by central extensions of families of pseudo-bosonic operators.\n\\end{cor}\n\n\\begin{proof}\nUsing Lemma \\ref{technical} and the fact that \\eqref{heisenberg+abelian} is described by pseudo-bosons, the result follows.\n\\end{proof}\n\n\n \n The conclusions of the above corollary may be rephrased in a\n more significant way, showing the the theory of the\n pseudo-boson operators is really a powerful tool both in\n mathematics and physics.\n \n \n \\begin{cor}\\label{final}\n If there exists a finite dimensional nilpotent Lie algebras\n which cannot be realized by central extensions of families\n of pseudo-bosonic operators, then its dimension must be\n $\\ge 6$.\n \\end{cor}\n \n\\section*{Acknowledgements}\nF.B. acknowledges partial support from Palermo University and from the Gruppo Nazionale di Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM). \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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C}}\n\n\n\n\\newcommand{{\\mathbb Z}}{{\\mathbb Z}}\n\\newcommand{{\\mathbb R}}{{\\mathbb R}}\n\\newcommand{{\\mathbb N}}{{\\mathbb N}}\n\\newcommand{{\\mathbb C}}{{\\mathbb C}}\n\\newcommand{{\\cal M}}{{\\cal M}}\n\\newcommand{{\\mathbb Q}}{{\\mathbb Q}}\n\\newcommand{\\widetilde{X}}{\\widetilde{X}}\n\\newcommand{\\Omega}{\\Omega}\n\\newcommand{{\\mathbb J}}{{\\mathbb J}}\n\\def{\\rm Tr}{{\\rm Tr}}\n\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newcommand{\\beta}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{equation*}}{\\begin{equation*}}\n\\newcommand{\\end{equation*}}{\\end{equation*}}\n\\newcommand{\\begin{eqnarray*}}{\\begin{eqnarray*}}\n\\newcommand{\\end{eqnarray*}}{\\end{eqnarray*}}\n\\newcommand{\\begin{pmatrix}}{\\begin{pmatrix}}\n\\newcommand{\\end{pmatrix}}{\\end{pmatrix}}\n\\newcommand{\\begin{align}}{\\begin{align}}\n\\newcommand{\\end{align}}{\\end{align}}\n\\newcommand{\\eq}[1]{(\\ref{#1})}\n\\newcommand\\Top{\\rule{0pt}{2.6ex}}\n\\newcommand\\Bottom{\\rule[-1.2ex]{0pt}{0pt}}\n\\renewcommand{\\labelenumii}{\\arabic{enumi}-\\roman{enumii}}\n\\def\\alpha{\\alpha}\n\\def\\beta{\\beta}\n\\def\\gamma{\\gamma}\n\\def\\delta{\\delta}\n\\def\\d_{D}{\\delta_{D}}\n\\def\\epsilon{\\epsilon}\n\\def\\f{\\phi}              \n\\def\\varphi{\\varphi}\n\\def\\gamma{\\gamma}\n\\def\\eta{\\eta}\n\\def\\iota{\\iota}\n\\def\\mathrm{Im}{\\mathrm{Im}}\n\\def\\infty{\\infty}\n\\def\\psi{\\psi}\n\\def\\k{\\kappa}            \n\\def\\lambda{\\lambda}\n\\def\\mu{\\mu}\n\\def\\nu{\\nu}\n\\def\\nabla{\\nabla}\n\\def\\omega}  \\def\\omega{\\omega{\\omega}  \\def\\omega{\\omega}\n\\def\\partial{\\partial}\n\\def\\re{\\mathrm{Re}}               \n\\def\\r{\\rho}                                    \n\\def\\sigma{\\sigma}                                  \n\\def\\tau{\\tau}\n\\def\\theta{\\theta}\n\\def\\tilde{\\tilde}\n\\def\\upsilon{\\upsilon}\n\\def\\varphi{\\varphi}\n\\def\\xi{\\xi}\n\\def\\wedge{\\wedge}\n\\def\\zeta{\\zeta}\n\\def\\overline{z}{\\overline{z}}\n\\def\\Delta{\\Delta}\n\\def\\Phi{\\Phi}\n\\def\\Gamma{\\Gamma}\n\\def\\Psi{\\Psi}\n\\def\\Lambda{\\Lambda}\n\\def\\Omega{\\Omega}\n\\def\\Pi{\\Pi}\n\\def\\Theta{\\Theta}\n\\def\\Upsilon{\\Upsilon}\n\\def\\Xi{\\Xi}\n\\def\\tau{\\tau}\n\\def\\bar{\\tau}{\\overline{\\tau}}\n\\def\\bar{\\rho}{\\overline{\\rho}}\n\n\\def\\v^{\\text{stand\\vphantom{p}}}{\\varphi^{\\text{stand\\vphantom{p}}}}\n\\def\\Phi^{\\rm stand}{\\Phi^{\\rm stand}}\n\\def\\mathfrak{F}_P#1{\\mathcal A_{#1}}\n\\def\\BP#1{\\mathcal B_{#1}}\n\\def\\v^{\\text{opt}}{\\varphi^{\\text{opt}}}\n\\def\\Phi^{\\rm opt}{\\Phi^{\\rm opt}}\n\\def\\mathcal V{\\mathcal V}\n\n\\def\\Delta{\\Delta}\n\n\\def\\CE{{\\cal E }}\n\\def\\REM#1{{\\bf[#1]}}\n\\def\\vartheta{\\vartheta}\n \n \n \n\n\\title{Quantum Black Holes, Wall Crossing, and Mock Modular Forms}\n\n\\thispagestyle{empty}\n\n\\preprint{}\n\\author{Atish Dabholkar$^{1, 2}$, Sameer Murthy$^3$, and Don Zagier$^{4, 5}$\\\\\n\n\\vspace{0.2cm}\n\n\\it $^{1}${Theory Division, CERN, PH-TH Case C01600,  CH-1211,  23 Geneva, Switzerland}\n\\vspace{0.2cm}\n\n\\it $^2${Sorbonne Universit\\'es, UPMC Univ Paris 06\\\\\n\\it  \\hspace{0.15cm}UMR 7589, LPTHE, F-75005, Paris, France}\n\\vspace{0.2cm}\n\n\\it \\hspace{0.05cm} {CNRS, UMR 7589, LPTHE, F-75005, Paris, France}\n\\vspace{0.2cm}\n\n\n\\noindent \\it$^3${NIKHEF theory group, Science Park 105}, \\it {1098 XG Amsterdam, The Netherlands} \n\\vspace{0.2cm}\n\n\\noindent \\it $^4${Max-Planck-Institut f\\\"ur Mathematik},\\it{Vivatsgasse 7, 53111 Bonn, Germany}\n\\vspace{0.2cm}\n\n\\noindent \\it $^5${Coll\\`ege de France},  \n\\it{3 Rue d'Ulm, 75005 Paris, France}\n\n\\vspace{4mm}\n\\hspace{-5mm}\n\\leftline{Emails: \n\\tt{atish@lpthe.jussieu.fr,  murthy.sameer@gmail.com, don.zagier@mpim-bonn.mpg.de}}\n}\n\n\n\\abstract{\nWe show that the meromorphic Jacobi form that counts the  quarter-BPS states  in ${\\cal N}=4$ \nstring theories can be  canonically decomposed as a sum of  a \\textit{mock Jacobi form}  and an \n\\textit{Appell-Lerch sum}. The  quantum degeneracies of single-centered black holes are Fourier \ncoefficients of this mock Jacobi form, while the Appell-Lerch sum captures  the degeneracies of \nmulti-centered black holes which decay upon wall-crossing. The \\textit{completion} of the mock \nJacobi form  restores the modular symmetries expected from $AdS_3\/CFT_2$  holography but  has \na holomorphic anomaly reflecting the non-compactness of the microscopic CFT.   For every positive \nintegral value $m$ of  the magnetic charge invariant  of the black hole, our analysis leads \nto a  special mock Jacobi form of weight two  and  index $m$, which we characterize uniquely \nup to a  Jacobi cusp form. This family  of special forms and another closely related family \nof weight-one forms contain almost all the known mock modular forms  including the mock theta \nfunctions of Ramanujan, the generating function of Hurwitz-Kronecker class numbers, the mock modular forms \nappearing in the Mathieu and Umbral moonshine, as well as an infinite number of new examples.  \n}\n\\keywords{black holes, mock modular forms, superstrings, wall-crossing}\n\n\n\n\\setcounter{page}{1}\n\n\n\\begin{document}\n\n\n\\textit{``My dream is that I will live to see the day when our young physicists, struggling \nto bring the predictions of superstring theory into correspondence with the facts of nature, \nwill be led to enlarge their analytic machinery to include not only theta-functions but mock \ntheta-functions\\,\\dots\\; But before this can happen, the purely mathematical exploration of the \nmock-modular forms and their mock-symmetries must be carried a great deal further.''}\n\n\\vspace{2mm}\n\\centerline{ \\hspace{60mm} \\textsl{ Freeman Dyson (1987 Ramanujan Centenary Conference)}}\n\n\n\n\n\\section{Introduction}\n\nThe quantum degeneracies associated with a black hole horizon are of central importance in \nquantum gravity. The counting function for these degeneracies for  a large class of black holes\nin string theory is expected to be  modular from the perspective of holography. However, \nin situations with wall-crossing, there is an apparent loss of modularity and   it is far \nfrom clear if and how such a counting function can be modular.   In the context of quarter-BPS \nblack holes in ${\\cal N}=4$ supersymmetric theories, we develop the  required analytic machinery \nthat provides a complete answer to this question which leads naturally to the mathematics of\nmock modular forms. We  present a number of new mathematical results motivated by but \nindependent of these physical considerations.  \n\nSince this paper is of possible interest to both theoretical physicists (especially\nstring theorists) and theoretical mathematicians (especially number theorists),\nwe give two introductions in their respective dialects.\n\n\\subsection{Introduction for mathematicians}\n\nIn the quantum theory of black holes in the context of string theory, the physical\nproblem of counting the dimensions of certain eigenspaces (``the number of\nquarter-BPS dyonic states of a given charge'') has led to the study of Fourier\ncoefficients of certain {\\it meromorphic} Siegel modular forms and to the\nquestion of the modular nature of the corresponding generating functions.\nUsing and refining results of S. Zwegers \\cite{Zwegers:2002}, we show that\nthese generating  functions belong to the recently discovered class of functions\ncalled {\\it mock modular forms}.\n\nSince this notion is still not widely known, it will be reviewed in some detail\n(in \\S\\ref{basicdefs}). Very roughly, a mock modular form of weight $k$ \n(more precisely, ``pure'' mock modular forms; we will also introduce\na somewhat more general notion of ``mixed'' mock modular forms in~\\S\\ref{MMMF}) \nis a holomorphic function~$f$ in the upper half plane to which is associated a \nholomorphic  modular form~$g$ of weight $2-k$, called the ``shadow'' of~$f$, such that the \nsum of~$f$ and a suitable non-holomorphic integral of~$g$ transforms like \na holomorphic modular form of weight~$k$. Functions of this type occur \nin several contexts in mathematics and mathematical physics: as certain $q$-hypergeometric series \n(like Ramanujan's original mock theta functions), as generating functions of \nclass numbers of imaginary quadratic fields~\\cite{Zagier:1975}, or as characters of extended \nsuperconformal algebras~\\cite{Ooguri:1989fd}, with special cases of the last class being conjecturally \nrelated to the Mathieu group~$M_{24}$~\\cite{Eguchi:2010ej}. They also arise, as was shown \nby Zwegers in his thesis~\\cite{Zwegers:2002},  as the Fourier coefficients of \nmeromorphic Jacobi forms. It is this last occurrence which is  at the origin of the \nconnection to black hole physics, because the Fourier coefficients of meromorphic Jacobi forms have  \nthe same wall-crossing behavior as that exhibited by the degeneracies of BPS states. \n\nThe specific meromorphic Jacobi forms that are of interest for the black hole counting problem  \nare the Fourier-Jacobi coefficients $\\psi_m (\\tau,z)$ of  the meromorphic Siegel modular form\n  \\beta\\label{reciproigusa} \\frac 1{\\Phi_{10}(\\Omega)} \\= \\sum_{m=-1}^{\\infty} \\psi_m (\\tau,z) \\, p^m  \\,, \\qquad \n    \\Bigl(\\Omega = \\left( \\begin{array}{cc} \\tau & z \\\\ z & \\sigma \\\\ \\end{array} \\right),\\;\\, p=e^{2\\pi i \\sigma} \\Bigr) \\,, \\end{equation}\nthe reciprocal of the Igusa cusp form of weight $10$, which arises as the partition function of \nquarter-BPS dyons in the type~II compactification on the product of a $K3$ surface and an elliptic curve \\cite{Dijkgraaf:1996it,Shih:2005uc,David:2006yn}.\nThese coefficients, after multiplication by the discriminant function $\\Delta(\\tau)$, are meromorphic \nJacobi forms of weight~$2$ with a double pole at $z=0$ and no others (up to translation by the period lattice). \n\nThe new mathematical results of the paper are contained in Sections~\\ref{MockfromJacobi}, \\ref{flyJac}\nand \\ref{structure}. In~\\S\\ref{MockfromJacobi}, \\-extending the results of~\\cite{Zwegers:2002}, we show \nthat any meromorphic Jacobi form $\\varphi(\\tau,z)$ having poles only at torsion points $z=\\alpha\\tau+\\beta$ ($\\alpha,\\,\\beta\\in{\\mathbb Q}$)\nhas a canonical decomposition into two pieces~$\\varphi^{\\rm F}(\\tau,z)$ and~$\\varphi^{\\rm P}(\\tau,z)$, called its ``finite\" part and  \n``polar\" part, respectively, of which the first is a finite linear combination of classical theta series\nwith mock modular forms as coefficients, and the second is an elementary expression that is\ndetermined completely by the poles of~$\\varphi$. Again using the results of\nZwegers, we give explicit formulas for the polar part~$\\varphi^P$ for all~$\\varphi$ having only simple or double poles.  \nIn the particular case of~$\\psi_{m}$, the polar part is given by the formula \n  \\beta \\label{simfun} \\psi_{m}^{\\rm P}(\\tau,z) \\= \\frac{p_{24}(m+1)}{\\Delta(\\tau)} \\; \\mathfrak{F}_P{2,m}(\\tau,z) \\qquad \n  \\big(q=e^{2\\pi i \\tau} \\,, \\;  y=e^{2\\pi i z} \\big) \\, , \\end{equation}\nwhere $\\Delta(\\tau)$ is the Ramanujan discriminant function, $p_{24}(m+1)$ the coefficient \nof~$q^{m}$ in $\\Delta(\\tau)^{-1}$, and $\\mathfrak{F}_P{2,m}(\\tau,z)$ the elementary function (Appell-Lerch sum) \n  \\beta \\label{defBP} \\mathfrak{F}_P{2,m}(\\tau,z)\\= \\sum_{s\\in{\\mathbb Z}} \\frac{q^{ms^2 +s}y^{2ms+1}}{(1 -q^s y)^2} \\,. \\end{equation}\nNote that $\\mathfrak{F}_P{2,m}$ exhibits wall-crossing: for  $0<\\Im(z) < \\Im(\\tau)$ it has the Fourier expansion \n  \\beta \\label{AL2} \\mathfrak{F}_P{2,m}(\\tau,z) \\= \\sum_{r\\ge\\ell>0 \\atop r\\,\\equiv\\ell\\; \\mod \\; {2m}} \\ell\\,q^{\\frac{r^{2}-\\ell^{2}}{4m}}\\,y^r\\,, \\end{equation}\nbut the Fourier expansions are different in other strips $n<\\Im(z)\/\\Im(\\tau)<n+1$.  \nThis wall-crossing is at the heart of both the mathematical and the physical theories. On the mathematical side\nit explains the ``mockness\" of the finite part $\\psi_m^{\\rm F}$ of~$\\psi_m$. On the physical side it has\nan interpretation in terms of counting two-centered black holes, with the integer $\\ell$ in \\eqref{AL2} being the  \ndimension   of the $SU(2)$ multiplet with angular momentum $(\\ell -1)\/2$ contained in the electromagnetic \nfield produced by the two centers.\n\n\nStarting in \\S\\ref{flyJac} we focus attention on two particular classes of functions\n$\\{\\varphi_{1,m}\\}$ and $\\{\\varphi_{2,m}\\}$, where $\\varphi_{k,m}$ is a meromorphic Jacobi form \nof weight $k\\in\\{1,\\,2\\}$ and index~$m\\in{\\mathbb N}$ having singular part $(2\\pi iz)^{-k}\\+\\text O(1)$ \nas $z\\to0$ and no other poles except the translates of this one by the period lattice.  \nThese functions in the case~$k=2$ are related to the Fourier coefficients~$\\psi_{m}$ defined \nin \\eqref{reciproigusa} by \n\\beta \\label{psifromphim}\n\\psi_{m}(\\tau,z) \\= \\frac{p_{24}(m+1)}{\\Delta(\\tau)} \\; \\varphi_{2,m}(\\tau,z) \\+ \\text{$($weakly holomorphic Jacobi form$)$}.\n\\end{equation}\nThe polar part of\n$\\varphi_{2,m}$ is the function~$\\mathfrak{F}_P{2,m}$ defined above, and that of $\\varphi_{1,m}$ an Appell-Lerch sum $\\mathfrak{F}_P{1,m}$ \nwith a similar definition (eq.~\\eqref{AAm}), but the functions themselves are not unique, since we \ncan add to them any (weak) Jacobi forms of weight~$k$ and index~$m$ without changing the defining property.\nThis will change the mock Jacobi form $\\Phi_{k,m}=\\varphi_{k,m}^{\\rm F}$ in the corresponding way.  Much of our\nanalysis concerns finding ``optimal\" choices, meaning choices for which the functions $\\Phi_{k,m}$ have \npoles of as small an order as possible at infinity and consequently Fourier coefficients whose asymptotic \ngrowth is as small as possible.  Our main results concern the case~$k=2$, and are as follows:\n\n\\begin{myenumerate}\n  \\item If $m$ is a prime power, then the functions $\\Phi_{2,m}$ can be chosen to be strongly \nholomorphic (i.e., with Fourier expansions containing only monomials $q^ny^r$ with $4nm-r^2\\ge0$).\nTheir Fourier coefficients then have polynomial growth and are given by explicit linear\ncombinations of class numbers of imaginary quadratic fields.\n \\item More generally, for arbitrary $m$, the function $\\Phi_{2,m}$ can be chosen to be the sum\nof the images under appropriate ``Hecke-like operators\" of special mock Jacobi forms ${\\cal Q }_M$,\nwhere~$M$ ranges over all divisors of~$m$ having an even number of distinct prime factors. \nThese summands are the eigencomponents of~$\\Phi_{2,m}$ with respect to the Atkin-Lehner-like operators\n$W_{m_1}$ ($m_1|m$, $(m_1,m\/m_1)=1$) acting on the space of mock Jacobi forms of index~$m$.\n  \\item The mock Jacobi form~${\\cal Q }_1$ is the generating function of class numbers of imaginary quadratic fields,\nand is strongly holomorphic. The other special mock Jacobi forms~${\\cal Q }_M$ $(M=6$, 10, 14, \\dots) \ncan be chosen to have ``optimal growth\" (meaning that their Fourier expansions contain only monomials \n$q^ny^r$ with $4nm-r^2\\ge-1$). Their\nFourier coefficients then grow like $e^{\\pi\\sqrt{\\Delta}\/m}$ as $\\Delta:=4nm-r^2$ tends to infinity.\n  \\item One can also choose $\\Phi_{2,m}(\\tau,z)$ for arbitrary $m\\ge1$ to be of the form \n$\\sum\\Phi_{2,m\/d^2}^0(\\tau,dz)$, where $d$ ranges over positive integers with $d^2|m$ and each~$\\Phi_{2,m\/d^2}^0$\nhas optimal growth.\n  \\item There are explicit formulas for the polar coefficients (those with $\\Delta<0$) of $\\Phi_{2,m}^0$ \nand~${\\cal Q }_M$.  In particular, the coefficient of $q^ny^r$ in~${\\cal Q }_M$ vanishes if $4nM-r^2=0$ and equals\n$\\pm\\varphi(M)\/12$ if $4nM-r^2=-1$, where $\\varphi(M)=\\prod_{p|M}(p-1)$ is the Euler $\\varphi$-function of~$M$.\n\\end{myenumerate}\nThe proofs of these properties are contained in~\\S\\ref{structure}, which gives a detailed description \n(based on difficult results on Jacobi forms proved in~\\cite{SkoruppaZagier2, SkoruppaZagier}) of the way that the space of\nholomorphic Jacobi forms is contained in the spaces of weak or of weakly holomorphic Jacobi forms.\nThis analysis contains several surprises, such as the result (Theorem~\\ref{chooseKm}) that for\nall $m\\ge1$ the space of holomorphic Jacobi forms of index~$m$ has codimension exactly~1 in\nthe space of Jacobi forms of index~$m$ with optimal growth. \n\nFor $\\varphi_{1,m}$ the description is much less complete.  There is still a result like~2., with $M$ now \nranging over the divisors of~$m$ with an~{\\it odd}\nnumber of distinct prime factors, but it is no longer possible in general to choose the~${\\cal Q }_M$ to have\noptimal growth.  The few cases where this is possible turn out to be related to mock theta functions\nthat have played a prominent role in the past. Thus ${\\cal Q }_{30}$ and ${\\cal Q }_{42}$ are essentially equal\nto the mock theta functions of ``order~5\" and ``order~7\" in Ramanujan's original letter to Hardy, and \n${\\cal Q }_2$ and several of the other ${\\cal Q }_M$ are related to the ``Mathieu moonshine\" and \n``Umbral moonshine'' story~\\cite{Eguchi:2010ej, Cheng:2012tq}.\nThe mock Jacobi forms~${\\cal Q }_M$ for the $k=2$ case are also related to special mock theta functions, \nnow of weight~$3\/2$, e.g.~${\\cal Q }_6$ is the mock theta function with shadow~$\\eta(\\tau)$ given in \\cite{Zagier:2007}. \n\nThe applications of the results (for the $k=2$ case, via the fact that $\\psi_m$ is the sum of\n$p_{24}(m+1)\\varphi_{2,m}\/\\Delta$ and a weakly holomorphic Jacobi form of weight~$-10$ and index~$m$) to the\noriginal physics problem are explained in~\\S\\ref{MockforDyons} and in the following ``second introduction.\"\n\n\\subsection{Introduction for physicists}\n\nThe microscopic quantum description of supersymmetric black holes in string\ntheory usually starts with a brane configuration of given\ncharges and mass at weak coupling, which is localized at a single point in the\nnoncompact spacetime. One then computes an appropriate indexed partition\nfunction in the world-volume theory of the branes, which from the perspective of enumerative geometry \ncomputes topological invariants such as the  Donaldson-Thomas invariants. At strong coupling, the brane\nconfiguration gravitates and the indexed partition function is expected to count the\nmicrostates of the corresponding macroscopic gravitational configurations.\nAssuming that the gravitational configuration is a single-centered black hole then\ngives a way to obtain a statistical understanding of the entropy of the black hole in\nterms of its microstates, in accordance with the Boltzmann relation\\footnote{Under certain \nconditions the index equals the absolute number \\cite{Sen:2008yk,Sen:2008vm, Dabholkar:2010rm}.}.\n\nOne problem that one often encounters is that the macroscopic configurations are no longer localized \nat a point and include not only a single-centered black hole of interest but also several multi-centered \nones \\cite{Denef:2000nb,Denef:2002ru,Bates:2003vx,Denef:2007vg}. Moreover, the indexed degeneracy of \nthe multi-centered configurations typically jumps upon crossing walls of marginal stability in the \nmoduli space where the multi-centered configuration breaks up into its single-centered constituents. \nThese jumps are  referred to  as the  `wall-crossing phenomenon'.  \n\nIf one is interested in the physics of the horizon or the microstates of a single black\nhole, the multi-centered configurations and the associated wall-crossings are thus something of a \nnuisance. It is  desirable to have a mathematical characterization that isolates the single-centered \nblack holes directly at the microscopic level. One  distinguishing feature of single-centered \nblack holes is that they are `\\textit{immortal}' in that they exist as stable quantum states for \nall values of the moduli and hence their degeneracy does not exhibit the wall-crossing phenomenon. \nWe will use this property later to   define the counting function for the  immortal black holes.\n\nThe wall-crossing phenomenon raises important conceptual questions regarding\nthe proper holographic formulation in this context. In many cases, the black hole can be viewed \nas an excitation of a black string. The near  horizon geometry of a black string is $AdS_3$ which \nis expected to be holographically dual to a two-dimensional conformal field theory $CFT_2$.  \nThe conformal boundary of  Euclidean $AdS_3$ is a 2-torus with a complex structure parameter $\\tau$,\n and the physical partition function of $AdS_3$ and of the boundary $CFT_2$ is a function of $\\tau$. \nThe  $SL(2, {\\mathbb Z})$ transformations of $\\tau$ can be identified geometrically with global diffeomorphisms \nof the boundary of  $AdS_3$ space. The partition function is expected to have good modular properties\nunder this geometric  symmetry. This symmetry is crucial for the Rademacher-type expansions of the \nblack hole degeneracies for understanding the quantum entropy of these black holes via holography \\cite{Sen:2008yk,Sen:2008vm,Dabholkar:2011ec,Dijkgraaf:2000fq,deBoer:2006vg,Manschot:2007ha,Strominger:1998yg, \nMurthy:2009dq,Manschot:2009ia,DGMKloos}. Implementing the modular symmetries  presents several subtleties in situations when there is wall-crossing. \n\nThe wall-crossing phenomenon has another important physical implication for the invariance of the \nspectrum under large gauge transformations of the antisymmetric tensor field.  Large gauge transformations \nlead to the `spectral flow symmetry' of the partition function of the black string \\cite{deBoer:2006vg}. \nSince these transformations act both on the charges and the moduli, degeneracies of states with a \ncharge vector $\\Gamma$ at some point $\\phi$ in the moduli space  get mapped to the degeneracies of states with charge\nvector $\\Gamma^{\\prime}$ at some other point $\\phi^{\\prime}$ in the moduli space. Typically, there are many \nwalls separating the point $\\phi^{\\prime}$  and the original point $\\phi$. As a result, the degeneracies \nextracted from the black string at a \\textit{given} point $\\phi$ in the moduli space do not exhibit the \nspectral-flow symmetry.  On the other hand, the spectrum of immortal black holes is independent of \nasymptotic moduli and hence must exhibit the  spectral-flow symmetry.  This raises the question as to \nhow to make the spectral-flow symmetry manifest for the degeneracies of immortal black holes in the \ngeneric situation when there is wall-crossing.\n\nWith these motivations, our objective will be to isolate the partition functions of the black string \nassociated with immortal black holes and investigate their transformation properties  under the\nboundary modular group and large gauge transformations. More\nprecisely, we would like to investigate the following four questions.\n\\begin{myenumerate}\n  \\item Can one define a \\textit{microscopic}  counting function that cleanly\nisolates  the microstates of immortal black holes from  those of  the multi-centered black configurations?\n  \\item What are the modular properties of  the counting function of  immortal black holes when the \nasymptotic spectrum exhibits the wall-crossing phenomenon?\n  \\item Can this counting function be related to a quantity that is properly modular as\nmight be expected from the perspective of near-horizon $AdS_3\/CFT_2$   holography?\n  \\item Can one define a partition function of the immortal black holes that \nmanifestly exhibits the spectral-flow symmetry resulting from large gauge transformations?\n\\end{myenumerate}\n\nThe main difficulties in answering these questions stem from the complicated  moduli dependence\nof the black hole spectrum which is often extremely hard to compute. To address the central conceptual\nissues in a tractable context, we consider  the compactification of Type-II on $K3 \\times T^2$\nwith ${\\cal N} =4$ supersymmetry in four dimensions. The spectrum of  quarter-BPS dyonic states in this\nmodel is exactly computable \\cite{Dijkgraaf:1996it,Gaiotto:2005gf,Shih:2005uc,Shih:2005he,LopesCardoso:2004xf,David:2006yn} \nand by now is well understood  at \\textit{all} points in the moduli space \\cite{Sen:2007vb,Dabholkar:2007vk,\nCheng:2007ch} and for \\textit{all} possible duality orbits \\cite{Dabholkar:2007vk,\nBanerjee:2008pv,Banerjee:2008pu,Banerjee:2008ri,Dabholkar:2008zy}. \nMoreover, as we will see, this particular model exhibits  almost all of the essential issues\nthat we wish to address.  The  ${\\cal N}=4$ black holes have the remarkable property that even though\ntheir spectrum is moduli-dependent, the partition function itself is moduli-independent. The  entire moduli\ndependence of the black hole degeneracy is captured by the moduli dependence of\nthe choice of the Fourier contour \\cite{Sen:2007vb,Dabholkar:2007vk, Cheng:2007ch}.\nMoreover,  the only  multi-centered configurations that contribute to the supersymmetric index of quarter-BPS states\nare the ones with only two centers,  each of which is half-BPS \\cite{Dabholkar:2009dq}.  Consequently,  \nthe only way the index can jump at a wall is by the decay of  a two-centered configuration  into its half-BPS \nconstituents \\cite{Sen:2007vb,Dabholkar:2007vk, Cheng:2007ch}; this  is a considerable simplification compared \nto the general ${\\cal N}=2$ case where more complicated multi-centered decays  are possible. These features  \nmake the ${\\cal N}=4$ case  much more tractable.  \n\nThe number of microstates of quarter-BPS  dyonic states for the above-mentioned compactification is given by a Fourier\ncoefficient of a meromorphic Jacobi form $\\psi_{m}(\\tau, z)$   with a moduli-dependent contour. \nThe partition function (\\ref{reciproigusa}) referred to earlier is the generating\nfunction for these meromorphic Jacobi forms. Using this simplicity of the moduli\ndependence and the knowledge of the exact spectrum, it is possible to give very\nprecise answers to the above questions in the ${\\cal N} =4$ framework, which  turn out\nto naturally involve mock modular forms.\n\\begin{myenumerate}\n  \\item One can define a holomorphic  function for counting the\nmicrostates of immortal black holes\\footnote{We will use the terms  `immortal' and `single-centered' interchangeably.  \nIn general, the moduli-independent `immortal' degeneracies  can receive contributions not only from single black \nholes but also from scaling solutions \\cite{Denef:2007vg}. They are not expected to contribute to the ${\\cal N}=4$ \nindex that we consider \\cite{Dabholkar:2009dq}. In addition, there can be  `hair' degrees of freedom \\cite{Banerjee:2009uk, Jatkar:2009yd}, \nwhich are degrees of freedom localized outside the  black hole horizon that carry part of the charge of the black hole. \nIn frames where the black hole is represented entirely in terms of D-branes, such hair modes are expected to be absent.} as a \nFourier coefficient of the partition function of the black string for a specific choice of the Fourier \ncontour \\cite{Sen:2007vb,Dabholkar:2007vk,Cheng:2007ch}. The contour corresponds to\nchoosing the asymptotic moduli of the theory in the attractor region of the single-centered black hole.\n  \\item Because the asymptotic counting function  is a \\textit{meromorphic} Jacobi form, the\nnear horizon counting function of immortal black holes is a \\textit{mock} modular form in that it fails to be modular\nbut in a very specific way.  The failure of modularity is governed by a\n\\textit{shadow}, which is given in terms of another holomorphic modular form. \n  \\item Given a mock modular form and its shadow, one can define its\n\\textit{completion} which is a non-holomorphic modular form. The failure of\nholomorphy can be viewed as a `holomorphic anomaly' which is also governed by the\nshadow.\n\\item The partition function of  immortal black holes with manifest spectral-flow invariance \nis a mock Jacobi form -- a new mathematical object  defined and  elaborated upon in \\S{\\ref{MockJacobi}}. \n\\end{myenumerate}\n\nThe main physical payoff of the mathematics of mock modular forms in this context\nis the guarantee that one can still define a completion as in (3) which \\textit{is} modular albeit non-holomorphic. \nAs mentioned earlier, the modular\ntransformations on the $\\tau$ parameter can be identified with global\ndiffeomorphisms of the boundary of the near horizon $AdS_3$. This connection\nmakes the mathematics of mock modular forms physically very relevant for\n$AdS_3\/CFT_2$ holography in the presence of wall-crossing and holomorphic\nanomalies. The required mathematical results concerning mock modular forms  \nare developed in sections \\S{\\ref{Mock}}, \\S\\ref{MockfromJacobi}, \n\\S\\ref{flyJac}, and \\S\\ref{structure}. \n\nTo orient the physics reader, we summarize the essential conclusions of this mathematical investigation from \nthe perspective of the questions posed above. \n\n\n\\begin{enumerate}\n\n\\item  \\textbf{Decomposition}: Given an asymptotic counting function,  the degeneracies of single-centered  \nblack holes can be isolated using the results in \\S\\ref{MockfromJacobi}. Applying Theorem~\\ref{Thmdouble} to the \n meromorphic Jacobi form  $\\psi_{m}(\\tau, z)$ gives a unique decomposition\n\\beta\\label{decomp} \\psi_{m}(\\tau, z)=\\psi^{\\rm F}_{m}(\\tau, z)\\+\\psi^{\\rm P}_{m}(\\tau, z)\\,, \\end{equation}\nsuch that $\\psi^{\\rm P}_{m}(\\tau, z)$ is a simple function \\eqref{simfun} with  the same pole structure in $z$ \nas $\\psi_{m}(\\tau, z)$ and $\\psi^{\\rm F}_{m}(\\tau, z)$ has no poles. \nThe elegant decomposition \\eqref{decomp} is motivated partly by the choice of `attractor contour' for \nsingle-centered black holes and and has a direct physical interpretation:  $\\psi_{m}(\\tau, z)$ is the counting \nfunction of all asymptotic states including both single and multi-centered configurations, $\\psi^{\\rm F}_{m}(\\tau, z)$ \nis the counting function of immortal black holes, whereas $\\psi^{\\rm P}_{m}(\\tau, z)$ is the counting function of multi-centered black holes. \n\n\\hspace{6mm} Since both $\\psi_{m}(\\tau, z)$ and $\\psi^{\\rm P}_{m}(\\tau, z)$ have poles in $z$,  their Fourier \ncoefficients depend on the choice of the contour which in turn depends on the moduli. On the other hand,  \nthe Fourier coefficients of $\\psi^{\\rm F}_{m}(\\tau, z)$ are unambiguously defined without any contour or \nmoduli dependence. This is what is expected for immortal black holes. \n\n\\item \\textbf{Modular Completion}: \nThe immortal counting function $\\psi^{\\rm F}_{m}(\\tau, z)$  defined by the decomposition \\eqref{decomp} is \nnot modular.  However, theorem 8.3 ensures that by adding a specific nonholomorphic function  to $\\psi^{\\rm F}_{m}(\\tau, z)$,  \none can obtain its nonholomorphic completion $\\widehat \\psi^{\\rm F}_{m}(\\tau, z)$  which \\textit{is} modular and transforms as \na Jabobi form.  The failure of holomorphy of  the completion   $\\widehat \\psi^{\\rm F}_{m}(\\tau, z)$ is given by the equation\n\\beta\\label{holanom1}\n \\tau_2^{3\/2} \\; \\frac{\\partial} {\\partial \\overline{\\tau}}   \\, \\widehat \\psi_{m}^{\\rm F}(\\tau,z) \\= \n\\sqrt{\\frac{m}{8 \\pi i}} \\; \\frac{ p_{24}(m+1)}{ \\Delta(\\tau)} \\,  \n \\sum_{\\ell \\, {\\rm mod} \\, (2m)}  {\\overline{\\vartheta_{m,\\ell}(\\tau,0)}} \\, \\vartheta_{m,\\ell} (\\tau,z) \\, .\n\\end{equation}\nHence the counting function   of immortal black holes has a hidden modular symmetry and more specifically is\na mock Jacobi form as defined  in \\S\\ref{MockJacobi}. This is one of our main physics results and is described \nin more detail in \\S\\ref{MockforDyons}.\n\n\n\\item  \\textbf{Holomorphic anomaly}:  The completion is a natural object to be identified with the indexed \npartition function of the superconformal field theory $SCFT_{2}$ dual to a single-centered $AdS_{3}$, which is expected  to be modular. \n\n\\hspace{6mm} {}From this perspective,  the equation \\eqref{holanom1} can be viewed  as a holomorphic anomaly and \ncan in fact be taken as a defining property of a mock Jacobi form for physics applications. \n Naively, an indexed partition function is expected to be holomorphic because of a   cancellation between \nright-moving bosons and fermions as for the elliptic genus \\cite{Witten:1986bf}. However, if the $SCFT_{2}$ \nis noncompact, then the spectrum is continuous and this naive reasoning may fail leading  to an `anomaly'. \nThe holomorphic anomaly can then arise as a consequence of the fact that for the right-movers in a noncompact \n$SCFT$, the density of states of bosons and fermions may be slightly different \\cite{Troost:2010ud, Ashok:2011cy} \nand may not precisely cancel.  The detailed connection between the holomorphic anomaly and the noncompactness\nof the $SCFT_{2}$ in this context needs to be understood better from a path-integral perspective.\n\n\n\\item \n \\textbf{Optimality}:  \n The Fourier coefficients of $\\psi^{\\rm F}_{m}(\\tau, z)$ grow exponentially rapidly as expected for a counting \nfunction of black hole degeneracies. It is clear from the anomaly equation \\eqref{holanom1} that if  we add  \na holomorphic  true Jacobi form to $\\psi^{\\rm F}_{m}(\\tau, z)$ with the same weight and index, it will still \nadmit a modular completion satisfying the same anomaly equation. This raises the question whether for a \ngiven holomorphic anomaly there is an `optimal'  mock  Jacobi form whose Fourier coefficients grow as slowly as possible.  \nThe  answer to this question (for the functions~$\\varphi_{2,m}$ related to~$\\psi_{m}$ by \\eqref{psifromphim}) \nis  in the affirmative but is subtle and requires several new results in the theory of Jacobi forms \ndeveloped in~\\S\\ref{structure}, motivated by and explaining the numerical experiments and \nobservations described in~\\S\\ref{flyJac}. \n\n\n\\hspace{6mm} A practical implication of such an optimal choice is that the leading contribution to the black \nhole entropy is  then determined essentially by a Fourier coefficient of a true Jacobi form.  One can thus \napply the  familiar Cardy formula  and  the Rademacher expansion of the Fourier coefficients of true modular \nforms for the leading answer. There will be exponentially subleading corrections to this leading answer\ncoming from the optimal mock Jacobi form. A Rademacher expansion for these corrections requires  a   \ngeneralization~\\cite{BringOno, BringMahlburg} applicable for mock rather than true modular forms.\n\n\\item  \\textbf{Examples}:  Modular forms with slowly growing  Fourier coefficients are mathematically \nparticularly interesting and many of the best-known examples of mock modular forms share this property.   \nAn `optimal' choice thus enables us in many  cases  to obtain a very  explicit expression for the immortal \ncounting function $\\psi^{\\rm F}_{m}(\\tau, z)$ in terms of these known  mock modular forms. For example, \nfor all $m$ prime, the optimal mock part  of $\\psi^{\\rm F}_{m}(\\tau, z)$  can be expressed in terms  \nof the generating function of Hurwitz-Kronecker class numbers (see \\S\\ref{choosephi}).\nOn the other hand, for a nonprime $m$ our analysis leads to  new  mock modular forms with very \nslowly growing Fourier coefficients.\n\n\n\\hspace{6mm} The functions $\\psi_{m}(\\tau, z)$ that arise in the black hole problem have a double \npole at~$z=0$ and its translates, but our methods can also be applied to a related class of functions \nwith just a single pole at $z=0$. This leads to a second infinite family of mock modular forms, this time \nof weight~1\/2. (In the double pole case, the mock Jacobi forms had weight~2 and their coefficients \nwere mock modular forms of weight~3\/2. In both cases, the coefficients are in fact mock theta functions, i.e., \nmock modular forms whose shadows are unary theta series.) Unlike the first case, where we found that \nthe mock modular forms occurring can always be chosen to have at most simple poles at the cusps, \nthe pole order here is much greater in general, and there are only a handful of examples having only \nsimple poles. It is remarkable that these include essentially all the most prominent examples of mock theta\nfunctions, including the original ones of Ramanujan, the mock theta function conjecturally related \nto the Mathieu group $M_{24}$~\\cite{Eguchi:2010ej} and the functions arising in the umbral moonshine \nconjecture~\\cite{Cheng:2010pq}. \n\n\\end{enumerate}\n\nModular symmetries are very powerful in physics applications because they relate\nstrong coupling to weak coupling, or high temperature to low temperature.   The  hidden modular symmetry \nof mock modular forms is therefore expected to be  useful in  diverse physics contexts.  As mentioned above,\nmock modularity of the counting function in the present context of black holes is a consequence of meromorphy \nof the asymptotic counting function which  in turn is a consequence of noncompactness of the target space \nof the microscopic SCFT. Now, conformal field theories with a  noncompact target space occur naturally \nin several physics contexts. For example, a general class of four-dimensional BPS black holes is obtained \nas a supersymmetric D-brane configuration in Type-II compactification on a Calabi-Yau three-fold $X_6$. \nIn the M-theory limit, these black holes can be viewed as excitations of the MSW black \nstring \\cite{Maldacena:1997de,Minasian:1999qn}. The microscopic theory describing the low energy \nexcitations of the MSW string is the $(0, 4)$ MSW SCFT. The target space of this SCFT does not necessarily \nhave to be compact in which case the considerations of this paper will apply. Very similar \nobjects \\cite{Zagier:1975, Hirzebruch:1976} have already made their appearance in the context of topological supersymmetric\nYang-Mills theory on $\\mathbb{CP}{^2}$~\\cite{Vafa:1994tf}. Other examples include \nthe theory of multiple M5-branes~\\cite{Alim:2010cf} quantum Liouville theory and E-strings~\\cite{Minahan:1998vr}, \nand the $SL(2,\\relax{\\rm I\\kern-.18em R})\/U(1)$ SCFT~\\cite{Eguchi:2004yi,Troost:2010ud,Ashok:2011cy} where the CFT is noncompact. \nThe appearance of a holomorphic anomaly in the regularized Poincar\\'e series for the elliptic genus of $\\rm CFT_{2}$\nwas noted in \\cite{Manschot:2007ha} in the context of the $\\rm AdS_{3}\/CFT_{2}$ correspondence. \n\nWe expect that the general framework of mock modular forms and Jacobi forms developed in this paper is likely \nto have varied physical applications in  the context of  non-compact conformal field theories, wall-crossings \nin enumerative geometry, and recently formulated  Mathieu and umbral moonshine conjectures~\\cite{Eguchi:2010ej, Cheng:2010pq}.\n\n\n\\subsection{Organization of the paper}\n\nIn \\S{\\ref{ReviewK3}}, we review the physics\nbackground concerning the string compactification on $K3 \\times T^2$ and the\nclassification of BPS states corresponding to the supersymmetric black holes in\nthis theory. In sections \\S{\\ref{Modular}}, \\S{\\ref{Jacobi}}, and  \\S{\\ref{Siegel}}, we review the basic mathematical\ndefinitions of various types of classical modular forms  (elliptic, Jacobi, Siegel)\nand illustrate an application to the physics of quantum black holes in each\ncase by means of an example. In \\S{\\ref{Contours}}, we review the moduli\ndependence of the Fourier contour prescription for extracting the degeneracies of quarter-BPS black holes \nin the ${\\cal N}=4$ theory from the partition function which is a meromorphic Siegel modular form. \nIn \\S{\\ref{MockfromJacobi}}, we refine results due to Zwegers to show that \nany meromorphic Jacobi form with poles only at the sub-multiples of the period lattice\ncan be decomposed canonically into two pieces, one of which is a mock Jacobi form. We give explicit \nformulas for this decomposition in the case when the poles have at most order~2,\nand again give several examples. \nIn \\S\\ref{flyJac} we give a detailed description of the experimental results for the \nFourier coefficients of the two families of mock Jacobi forms~$\\{\\Phi_{2,m}\\}$ and~$\\{\\Phi_{1,m}\\}$, \nand formulate the main mathematical results concerning them. \nThe proofs of these results are given in \\S\\ref{structure}, after we have formulated \nand proved a number of structural results about holomorphic and weakly holomorphic \nJacobi forms that are new and may be of independent interest. In \\S{\\ref{MockforDyons}, we apply these results \nin the physical context to determine the mock Jacobi form that counts the  degeneracies of single-centered black holes\nand discuss the implications for $AdS_{2}\/CFT_{1}$ and $AdS_{3}\/CFT_{2}$ holography. \n\n\n\n\n\\section{Review of Type-II superstring theory on $K3 \\times T^2$  \\label{ReviewK3}}\n\nSuperstring theories are naturally formulated in ten-dimensional Lorentzian\nspacetime $ {\\cal M}_{10}$. A `compactification' to four-dimensions is obtained by\ntaking ${\\cal M}_{10}$ to be a product manifold $ {\\mathbb R}^{1, 3} \\times X_6$ where\n$X_6$ is a compact Calabi-Yau threefold and ${\\mathbb R}^{1, 3}$ is the noncompact\nMinkowski spacetime. We will focus in this paper on a compactification of Type-II\nsuperstring theory when $X_6$ is itself the product $X_6 = K3 \\times T^2$. A\nhighly nontrivial and surprising result from the 90s is the statement that this\ncompactification is quantum equivalent or `dual' to a compactification of\nheterotic string theory on $T^4 \\times T^2$ where $T^4$ is a four-dimensional\ntorus \\cite{Hull:1994ys, Witten:1995ex}. One can thus describe the theory either\nin the Type-II frame or the heterotic frame.\n\nThe four-dimensional theory in ${\\mathbb R}^{1,3}$ resulting from this\ncompactification has $ \\mathcal{N}=4$ supersymmetry\\footnote{This supersymmetry is a super Lie algebra\ncontaining $ISO(1, 3) \\times SU(4)$ as the even subalgebra where $ISO(1,3)$ is the Poincar\\'e symmetry\nof the ${\\mathbb R}^{1, 3}$ spacetime and $SU(4)$ is an internal symmetry usually referred to as R-symmetry. \nThe odd generators of the superalgebra are called supercharges. With $ {\\cal N}=4$ supersymmetry,\nthere are eight complex supercharges which transform as a spinor of $ISO(1,3)$ and a fundamental \nof~$SU(4)$.}. The massless fields in the theory consist of $22$ vector multiplets in addition to the\nsupergravity multiplet. The massless moduli fields consist of the $S$-modulus $\\lambda$ taking values in the coset\n  \\beta\\label{Smoduli} SL(2,{\\mathbb Z})\\backslash SL(2, {\\mathbb R})\/ O(2, {\\mathbb R}), \\end{equation}\nand the $T$-moduli $\\mu$ taking values in the coset\n  \\beta\\label{Narainmoduli} O(22, 6, {\\mathbb Z}) \\backslash O(22, 6, {\\mathbb R}) \/O(22, {\\mathbb R}) \\times O(6, {\\mathbb R}). \\end{equation}\nThe group of discrete identifications $SL(2,{\\mathbb Z})$ is called the $S$-duality group. In the heterotic frame,\nit is the electro-magnetic duality group \\cite{Sen:1994yi, Sen:1994fa}, whereas in the\ntype-II frame, it is simply the group of area-preserving global diffeomorphisms\nof the $T^2$ factor. The group of discrete identifications $O(22, 6, \\mathbb{Z})$ \nis called the $T$-duality group. Part of the $T$-duality group $O(19, 3, \\mathbb{Z})$ \ncan be recognized as the group of geometric identifications on the\nmoduli space of K3; the other elements are stringy in origin and have to do with mirror symmetry.\n\n\nAt each point in the moduli space of the internal manifold $K3 \\times T^2$, one\nhas a distinct four-dimensional theory. One would like to know the spectrum of\nparticle states in this theory. Particle states are unitary irreducible\nrepresentations, or supermultiplets, of the ${\\cal N} =4$ superalgebra. The\nsupermultiplets are of three types which have different dimensions in the rest\nframe. A long multiplet is $256$-dimensional, an intermediate multiplet is\n$64$-dimensional, and a short multiplet is $16$-dimensional. A short multiplet\npreserves half of the eight supersymmetries ({\\it i.e.} it is annihilated by four\nsupercharges) and is called a half-BPS state; an intermediate multiplet preserves\none quarter of the supersymmetry ({\\it i.e.} it is annihilated by two\nsupercharges), and is called a quarter-BPS state; and a long multiplet does not\npreserve any supersymmetry and is called a non-BPS state. One consequence of the\nBPS property is that the spectrum of these states is `topological' in that it\ndoes not change as the moduli are varied, except for jumps at certain walls in\nthe moduli space \\cite{Witten:1978mh}.\n\n\nAn important property of a BPS states that follows from the superalgebra is\nthat its mass is determined by its charges and the moduli \\cite{Witten:1978mh}.\nThus, to specify a BPS state at a given point in the moduli space, it suffices to\nspecify its charges.  The charge vector in this theory transforms in the vector\nrepresentation of the $T$-duality group $O(22, 6, \\mathbb{Z})$ and in the\nfundamental representation of the $S$-duality group $SL(2, \\mathbb{Z})$. It is thus\ngiven by a vector $\\Gamma^{\\alpha I}$  with integer entries\n  \\beta\\label{chargevector} \\Gamma^{\\alpha I} \\= \\left( \\begin{array}{c} N^I \\\\ M^I \\\\ \\end{array} \\right)\n   \\qquad \\text{where} \\qquad {  \\alpha = 1,2\\, ; \\quad {I=1,2, \\ldots 28}} \\, \\,  \\end{equation}\ntransforming  in the $(2, 28)$ representation of $SL(2, \\mathbb{Z}) \\times O(22, 6, \\mathbb{Z})$.\nThe vectors $N$ and $M$ can be regarded as the quantized electric and magnetic\ncharge vectors of the state respectively. They both belong to an even, integral,\nself-dual lattice $\\Pi^{22, 6}$. We will assume in what follows that $ \\Gamma =\n(N, M)$ in (\\ref{chargevector}) is primitive in that it cannot be written as an\ninteger multiple of $ (N_0, M_0)$ for $N_0$ and $M_0$ belonging to $\\Pi^{22, 6}$.\nA state is called purely electric if only $N$ is non-zero, purely magnetic if\nonly $M$ is non- zero, and dyonic if both $M$ and $N$ are non-zero.\n\nTo define $S$-duality transformations, it is convenient to represent the $S$-modulus\nas a complex field $S$ taking values in the upper half plane. An $S$-duality transformation\n  \\beta\\label{Sgroup} \\gamma \\equiv\\left( \\begin{array}{cc} a&b\\\\ c&d \\end{array} \\right) \\in SL(2;{\\mathbb Z}) \\end{equation}\nacts simultaneously on the charges and the $S$-modulus by\n  \\beta\\label{stransform}\\left(\\begin{array}{c} N \\\\ M \\\\ \\end{array} \\right) \\rightarrow\n    \\left( \\begin{array}{cc} a&b\\\\ c&d \\end{array} \\right)\n    \\left( \\begin{array}{c} N \\\\ M \\\\ \\end{array} \\right)\\;, \\qquad S\\to\\frac{a S+b}{c S+d}\\;. \\end{equation}\n\nTo define $T$-duality transformations, it is convenient to represent the $T$-moduli\nby a $28 \\times 28$ matrix $\\mu^A_{\\,\\,I}$ satisfying \n  \\beta \\mu^{t} \\, L \\, \\mu = L \\end{equation} \nwith the identification that $\\mu\\sim k\\mu$ for every $k\\in O(22;{\\mathbb R})\\times O(6;{\\mathbb R})$. Here $L$ is the $28\\times28$ matrix\n  \\beta\\label{lorentzian}  L_{IJ} = \\left( \\begin{array}{ccc} - \\textbf{C}_{16} & \\textbf{0} & \\textbf{0}\\\\\n    \\textbf{0} & \\textbf{0} & \\textbf{\\rm I}_6  \\\\ \\textbf{0} &   \\textbf{\\rm I}_6 & \\textbf{0}  \\\\ \\end{array} \\right), \\end{equation}\nwith $\\textbf{\\rm I}_s$ the $s\\times s$ identity matrix and $ \\textbf{C}_{16}$ is the Cartan matrix of\n$E_8\\times E_8$ . The $T$-moduli are then represented by the matrix \n  \\beta {\\cal M} \\=\\mu^t \\mu \\end{equation}\nwhich satisifies\n  \\beta {\\cal M}^t \\= {\\cal M}\\,, \\qquad {\\cal M}^t L {\\cal M} = L\\,. \\end{equation}\nIn this basis, a $T$-duality transformation can then be represented by a $28\\times 28 $ matrix $R$ with integer entries satisfying\n  \\beta\\label{defRtduality}  R^t L R \\= L\\,, \\end{equation}\nwhich acts simultaneously on the charges and the $T$-moduli by\n  \\beta\\label{ttransform}  N \\rightarrow R N; \\quad M \\rightarrow R M\\,; \\quad \\mu \\rightarrow \\mu R^{-1} \\end{equation}\n\nGiven the matrix $\\mu^{A}_I$, one obtains an embedding $ \\Lambda^{22, 6}\\subset\n\\mathbb{R}^{22, 6}$ of $\\Pi^{22, 6}$ which allows us to define the\nmoduli-dependent charge vectors $Q$ and $P$ by\n  \\beta\\label{physcharges} Q^A =  \\mu^A_I N_I\\,,   \\qquad P^{A} =  \\mu^A_I M_I \\, . \\end{equation}\nThe matrix $L$ has a $22$-dimensional eigensubspace with eigenvalue $-1$ and a\n$6$- dimensional eigensubspace with eigenvalue $ +1$. Given $Q$ and $P$, one can\ndefine the `right-moving' and `left-moving' charges\\footnote{The right-moving charges couple to\nthe graviphoton vector fields associated with the right-moving chiral currents in\nthe conformal field theory of the dual heterotic string.} $Q_{R,L}$ and $P_{L,R}$ \nas the projections\n\\beta \nQ_{R, L} \\=\\frac{(1 \\pm L)}2\\,Q \\, ; \\qquad P_{R, L} \\= \\frac{(1 \\pm L)}2\\,P \\, . \n\\end{equation}\n\nIf the vectors $N$ and $M$ are nonparallel, then the state is quarter-BPS. On the\nother hand, if $N= p N_0$ and $ M =q N_0$ for some $N_0 \\in \\Pi^{22, 6}$ with $p$\nand $q$ relatively prime integers, then the state is half-BPS.\n\nAn important piece of nonperturbative information about the dynamics of the\ntheory is the exact spectrum of all possible dyonic BPS-states at all points in\nthe moduli space. More specifically, one would like to compute the number\n$d(\\Gamma)|_{S, \\mu}$ of dyons of a given charge $\\Gamma$ at a specific point\n$(S, \\mu)$ in the moduli space. Computation of these numbers is of course a very\ncomplicated dynamical problem. In fact, for a string compactification on a\ngeneral Calabi-Yau threefold, the answer is not known. One main reason for\nfocusing on this particular compactification on $K3 \\times T^2$ is that in this\ncase the dynamical problem has been essentially solved and the exact spectrum of\ndyons is now known. Furthermore, the results are easy to summarize and the\nnumbers $d(\\Gamma)|_{S, \\mu}$ are given in terms of Fourier coefficients of\nvarious modular forms.\n\nIn view of the duality symmetries, it is useful to classify the inequivalent\nduality orbits labeled by various duality invariants. This leads to an\ninteresting problem in number theory of classification of inequivalent duality\norbits of various duality groups such as $SL(2, \\mathbb{Z}) \\times O(22, 6; {\\mathbb Z})$\nin our case and more exotic groups like $E_{7,7} ({\\mathbb Z})$ for other choices of\ncompactification manifold $X_6$. It is important to remember though that a\nduality transformation acts simultaneously on charges and the moduli. Thus, it\nmaps a state with charge $\\Gamma$ at a point in the moduli space $ (S, \\mu)$ to\na state with charge $ \\Gamma'$ but at some other point in the moduli space $(S',\n\\mu')$.  In this respect, the half-BPS and quarter-BPS dyons behave differently.\n\\begin{itemize}\n  \\item For half-BPS states, the spectrum does not depend on the moduli. Hence\n$d(\\Gamma)|_{S', \\mu'} = d(\\Gamma)|_{S, \\mu}$. Furthermore, by an $S$-duality\ntransformation one can choose a frame where the charges are purely electric with\n$M=0$ and $N \\neq 0$. Single-particle states have $N$ primitive and the number of\nstates depends only on the $T$-duality invariant integer $n \\equiv N^2\/2$. We can\nthus denote the degeneracy of half-BPS states $d(\\Gamma)|_{S', \\mu'}$ simply by $d(n)$.\n  \\item For quarter-BPS states, the spectrum does depend on the moduli, and\n$d(\\Gamma)|_{S', \\mu'} \\neq d(\\Gamma)|_{S, \\mu}$. However, the partition function\nturns out to be independent of moduli and hence it is enough to classify the\ninequivalent duality orbits to label the partition functions. For the specific\nduality group $SL(2, \\mathbb{Z}) \\times O(22, 6; {\\mathbb Z})$ the partition functions are\nessentially labeled by a single discrete invariant \\cite{Dabholkar:2007vk,Banerjee:2007sr,Banerjee:2008ri}.\n      \\beta\\label{gcd}  I \\= \\gcd (N \\wedge M) \\,,       \\end{equation}\nThe degeneracies themselves are Fourier coefficients of the partition\nfunction. For a given value of $I$, they depend only on\\footnote{There is an additional \ndependence on  arithmetic $T$-duality invariants but the degeneracies for states with nontrivial\nvalues of these $T$-duality invariants can be obtained from the degeneracies discussed here\nby demanding $S$-duality invariance \\cite{Banerjee:2008ri}. } the moduli and the three\n$T$-duality invariants $(m, n, \\ell) \\equiv (M^2\/2, N^2\/2, N \\cdot M)$. Integrality\nof $ (m, n, \\ell)$ follows from the fact that both $N$ and $M$ belong to $\\Pi^{22, 6}$. \nWe can thus denote the degeneracy of these quarter-BPS states $d(\\Gamma)|_{S, \\mu}$ simply\nby $d(m, n, l)|_{S, \\mu}$.  For simplicity, we consider only $I=1$ in this paper.\n\\end{itemize}\n\n\nGiven this classification, it is useful to choose a representative set of charges\nthat can sample all possible values of the three $T$-duality invariants. For this\npurpose, we choose a point in the moduli space where the torus $T^2$ is a product\nof two circles $S^1 \\times {\\tilde S}^1$ and choose the following charges in a Type-IIB frame.\n\\begin{myitemize}\n  \\item For electric charges, we take $n$ units of momentum along the circle $S^1$, \nand $\\tilde K$  Kaluza- Klein monopoles associated with the circle ${\\tilde S}^1$.\n  \\item For magnetic charges, we take $Q_1$ units of D1-brane charge wrapping $S^1$, $Q_5$\nD5-brane wrapping $K3 \\times S^1$ and $l$ units of momentum along the ${\\tilde S}^1$ circle.\n\\end{myitemize}\nWe can thus write \n\\beta\\label{final charges} \\Gamma \\= \\left[ \\begin{array}{c}  N \\\\ M \\end{array} \\right] \n   \\= \\left[ \\begin{array}{cccc} 0,& n; & 0, & \\tilde K \\\\ Q_1, & \\tilde n; & Q_5, & 0\\\\ \\end{array} \\right] \\,. \\end{equation}\nThe $T$-duality quadratic invariants can be computed using a restriction of the matrix\n(\\ref{lorentzian}) to a $ \\Lambda^{(2,2)}$ Narain lattice of the form\n  \\beta\\label{lorentzintwo}  L \\= \\left( \\begin{array}{cc} \\textbf{0} & \\textbf{\\rm I}_2  \\\\ \n                  \\textbf{\\rm I}_2 & \\textbf{0} \\\\ \\end{array} \\right) , \\end{equation}\nto obtain \n  \\beta\\label{invts} M^2\/2 \\=  Q_1Q_5\\,, \\quad N^2\/2 \\= n \\tilde K\\,, \\quad N \\cdot M =\\tilde n \\tilde K\\,. \\end{equation}\nWe can simply the notation further by choosing $\\tilde K=Q_5=1$, $Q_1= m$, $\\tilde n = l$ to obtain\n  \\beta\\label{invts}  M^2\/2 \\= m, \\quad N^2\/2 \\= n, \\quad N \\cdot M =   l\\,. \\end{equation}\n\nFor this set of charges,  we can focus our attention on a subset of $T$-moduli associated with the torus $T^2$ parametrized by\n  \\beta\\label{moduli-matrix} {\\cal M} \\=  \\left( \\begin{array}{ccc} G^{-1}\\,\\quad  &G^{-1}B \\\\\n   -BG^{-1}\\,\\quad &G -B G^{-1} B\\end{array}\\right) \\,,\\end{equation}\nwhere $G_{ij}$ is the metric on the torus and $B_{ij} $ is the antisymmetric tensor field. Let $U = U_1 + i U_2$ be the\ncomplex structure parameter, $A$ be the area,  and $\\epsilon_{ij}$ be the Levi-Civita symbol with $\\epsilon_{12} = -\\epsilon_{21} =1$, then\n  \\beta\\label{metric-B} G_{ij} = \\frac{A}{U_2} \\left(\\begin{array}{cc} 1& U_1 \\\\ U_1 & |U|^2 \\\\ \\end{array}\\right)\n   \\quad \\text{and} \\quad B_{ij} = A B \\epsilon_{ij} \\,, \\end{equation}\nand the complexified K\\\"ahler modulus $U = U_{1 }+ iU_2$ is defined as $U:= B + i A$. The $S$-modulus $S = S_1 + S_2$\nis defined as \n\\beta\\label{Sdef}\nS:= a + i\\exp{(-2\\phi)}\n\\end{equation}\n where $a$ is the axion and $\\phi$ is the dilaton field in the four dimensional\nheterotic frame.  the relevant moduli can be parametrized by three complex scalars $S, T, U$ which define the so-called\n`STU' model in ${\\cal N} =2$ supergravity.  Note that these moduli are labeled naturally in the heterotic frame which \nare related to the $S_{B}$, $T_{B}$, and $U_{B}$ moduli in the Type-IIB frame by\n  \\beta\\label{hetBrel} S \\= U_B, \\quad T = S_B,  \\quad U = T_B \\,. \\end{equation}\n\n\n\n\n\\section{Modular forms in one variable \\label{Modular}}\n\nBefore discussing mock modular forms, it is useful to recall the variety of\nmodular objects that have already made their appearance in the context of\ncounting black holes. In the following sections we give the basic\ndefinitions of modular forms, Jacobi forms, and Siegel forms, using the notations\nthat are standard in the mathematics literature, and then in each case illustrate\na physics application to counting quantum black holes by means of an example.\n\n\nIn the physics context, these modular forms arise as generating functions for\ncounting various quantum black holes in string theory. The structure of poles of\nthe counting function is of particular importance in physics, since it determines\nthe asymptotic growth of the Fourier coefficients as well as the contour\ndependence of the Fourier coefficients which corresponds to the wall crossing\nphenomenon. These examples will also be relevant later in \\S{\\ref{MockforDyons}} in\nconnection with mock modular forms. We suggest chapters I and III of\n\\cite{Mod123} respectively as a good general reference for classical and Siegel\nmodular forms and \\cite{Eichler:1985ja} for Jacobi modular forms.\n\n\\subsection{Basic definitions and properties} \\label{modularbasic}\nLet $\\mathbb{H}$ be the upper half plane, \\textit{i.e.}, the set of complex numbers $\\tau$\nwhose imaginary part satisfies $\\Im(\\tau)>0$. Let $SL(2, \\mathbb{Z})$ be the group of matrices  $\\sm abcd$\n with integer entries such that $ad - bc =1$.\n\n\nA \\emph{modular form} $f (\\tau)$ of weight $k$ on $SL(2,{\\mathbb Z})$ is a holomorphic\nfunction on $\\relax{\\rm I\\kern-.18em H}$, that transforms as\n  \\beta\\label{modtransform0} f(\\frac{a\\tau + b }{c\\tau + d}) = (c\\tau + d)^k\\,f(\\tau) \\qquad \\forall \\;\\left(\n       \\begin{array}{cc} a & b \\\\ c & d \\\\  \\end{array} \\right) \\in SL(2,{\\mathbb Z})\\,, \\end{equation}\nfor an integer $k$ (necessarily even if $f(0) \\neq 0$). It follows from the definition\nthat $f(\\tau)$ is periodic under $\\tau \\to \\tau +1$ and can be written as a Fourier series\n  \\beta\\label{holmod} f(\\tau) \\= \\sum_{n= -\\infty}^\\infty a(n)\\,q^n   \\qquad \\bigl(q :=  e^{2 \\pi i \\tau}\\bigr)\\,,  \\end{equation}\nand is bounded as $\\rm{Im}(\\tau) \\to \\infty$.\nIf $a(0) =0$, then the modular form vanishes at infinity and is called a \\emph{cusp form}.\nConversely, one may weaken the growth condition at $\\infty$ to $f(\\tau) = {\\cal O}(q^{-N}) $\nrather than $ {\\cal O}(1)$ for some $N \\ge 0$; then the Fourier coefficients of $f$ have the behavior\n$a(n)=0$ for $n < -N$. Such a function is called a \\emph{weakly holomorphic modular form}.\n\nThe vector space over $\\mathbb{C}$ of holomorphic modular forms of weight $k$ is\nusually denoted by~$M_k$. Similarly, the space of cusp forms of weight $k$ and\nthe space of weakly holomorphic modular forms of weight $k$ are denoted by $S_k$\nand $M^{\\,!}_k$ respectively. We thus have the inclusion\n  \\beta\\label{in}  S_k \\; \\subset M_k \\; \\subset \\;M^{\\,!}_k \\;.  \\end{equation}\nThe Fourier coefficients of the modular forms in these spaces have different growth properties:\n\\begin{enumerate}\n\\item $f \\in S_{k}\\;\\Rightarrow \\; a_{n} \\= {\\cal O}(n^{k\/2})$ \\ as $n \\to \\infty\\,$;\n\\item $f \\in M_{k}\\; \\Rightarrow \\; a_{n} \\= {\\cal O}(n^{k-1})$ \\ as $n \\to \\infty\\,$;\n\\item $f \\in M_{k}^{\\,!}\\;\\Rightarrow\\; a_{n} \\= {\\cal O}(e^{C \\sqrt{n}})$ \\ as $n \\to \\infty$ for some $C>0\\,$.\n\\end{enumerate}\n\nSome important modular forms on $SL(2,{\\mathbb Z})$ are:\n\\begin{enumerate}\n\\item The {\\it Eisenstein series} $E_{k} \\in M_{k}$ ($k \\ge 4$).  The first two of these are\n\\begin{eqnarray}\\label{eisen}\n   E_4 (\\tau) &\\=& 1 \\+ 240\\,\\sum_{n=1}^\\infty \\frac{n^3q^n}{1- q^n} \\= 1+ 240 q + 2160q^2 + \\cdots \\, , \\\\\n   E_6 (\\tau) &\\=& 1 \\,-\\, 504\\,\\sum_{n=1}^\\infty \\frac{n^5q^n}{1- q^n} \\= 1 -504q -16632q^2 - \\cdots \\,.\n\\end{eqnarray}\n\\item The {\\it discriminant function} $\\Delta$. It is given by the product expansion\n  \\beta\\label{discrim}  \\Delta(\\tau)  \\=  q\\,\\prod_{n=1}^\\infty {(1 - q^n)^{24}} \\= q - 24 q^2 + 252 q^3 + ... \\end{equation}\nor by the formula $\\Delta =  \\left( E_4^3 - E_6^2 \\right)\/1728$.  We mention for later use that the \nfunction $E_2(\\tau)=\\dfrac1{2\\pi i}\\dfrac{\\Delta'(\\tau)}{\\Delta(\\tau)}=1-24\\sum\\limits_{n=1}^\\infty\\dfrac{nq^n}{1-q^n}$ is also an Eisenstein series, \nbut is not modular. (It is a so-called {\\it quasimodular form}, meaning in this case that the non-holomorphic function \n$\\widehat E_2(\\tau)=E_2(\\tau)-\\dfrac3{\\pi\\,\\Im(\\tau)}$ transforms like a modular form of weight~2.) This function can be used to form the {\\it Ramanujan-Serre derivative}\n  \\beta\\label{RSderiv} \\dRS{k}\\,:\\,M_k\\to M_{k+2}\\,, \\qquad  \\dRS{k}f(\\tau)\\,:=\\,\\frac1{2\\pi i}\\,f'(\\tau) \\,-\\,\\frac k{12}\\,E_2(\\tau)\\,f(\\tau)\\,. \\end{equation}\n\\end{enumerate}\n\nThe ring of modular forms on $SL(2,{\\mathbb Z})$ is generated freely by $E_4$ and $E_6$, so any modular form of weight $k$ can be \nwritten (uniquely) as a sum of monomials $E_4^\\alpha E_6 ^\\beta$ with $4\\alpha+6\\beta =k$.  We also have $M_{k}={\\mathbb C}\\cdot E_k\\oplus S_k$ \nand $S_{k} = \\Delta \\cdot M_{k-12}$, so any $f \\in M_{k}$ also has a unique expansion as $\\sum_{n=0}^{[k\/12]} \\alpha_n\\, E_{k-12n} \\,\\Delta^{n}$ \n(with $E_0=1$ and $E_2$ replaced by~0). From either representation, we see that a modular form is uniquely determined by its weight and first few Fourier coefficients.\n\n\nGiven two modular forms $(f,g)$ of weight $(k,l)$, one can produce a sequence of modular forms of \nweight $k+l+2n, \\, n \\ge 0$ using the {\\it Rankin-Cohen bracket} \n\\beta\\label{RCbrac}   [f,g]_{n} \\= [f,g]^{(k,l)}_{n} \\= \n   \\sum_{r+s=n} (-1)^s \\left( {k+n-1 \\atop r} \\right) \\left( {\\ell+n-1 \\atop s} \\right) f^{(s)}(\\tau) g^{(r)}(\\tau) \\end{equation}\nwhere $f^{(m)} := \\left(\\frac 1{2 \\pi i} \\frac{d}{d\\tau} \\right)^{m} f$. For $n=0$, this is simply the product \nof the two forms, while for $n>0$ we always have $[f,g]_n \\in S_{k+l+2n}$.  The first two non-trivial examples are \n  \\beta\\label{rcbraceg} [E_4, E_6]_1 \\= -3456 \\,\\Delta \\ , \\qquad [E_4, E_4]_2 = 4800 \\, \\Delta \\ . \\end{equation}\n\nAs well as modular forms on the full modular group $SL(2,{\\mathbb Z})$, \none can also consider modular forms on subgroups of finite index, with the same transformation law\n\\eqref{modtransform0} and suitable conditions on the Fourier coefficients to define the notions of \nholomorphic, weakly holomorphic and cusp forms. The weight $k$ now need no longer be even, \nbut can be odd or even half integral, the easiest way to state the transformation property when \n$k \\in {\\mathbb Z} + {\\frac12}$ being to say that $f(\\tau)\/\\theta(\\tau)^{2k}$ is invariant under some congruence subgroup\nof $SL(2,{\\mathbb Z})$, where $\\theta(\\tau)=\\sum_{n\\in{\\mathbb Z}}q^{n^2}$. The graded vector space of modular forms \non a fixed subgroup $\\Gamma \\subset SL(2,{\\mathbb Z})$ is finite dimensional in each weight, finitely generated \nas an algebra, and closed under Rankin-Cohen brackets.  Important examples of modular forms of half-integral\nweight are the {\\it unary theta series}, i.e., theta series associated to a quadratic form in one variable. \nThey come in two types:\n\\beta\\label{unary1}  \n  \\sum_{n\\in{\\mathbb Z}}\\varepsilon(n)\\,q^{\\lambda n^2} \n  \\qquad\\text{for some $\\lambda\\in{\\mathbb Q}_+$ and some even periodic function $\\varepsilon$} \n\\end{equation}\nand\n\\beta\\label{unary2} \n \\sum_{n\\in{\\mathbb Z}}n\\,\\varepsilon(n)\\,q^{\\lambda n^2} \n \\qquad\\text{for some $\\lambda\\in{\\mathbb Q}_+$ and some odd periodic function $\\varepsilon$}\\,, \n\\end{equation}\nthe former being a modular form of weight 1\/2 and the latter a cusp form of weight 3\/2.  \nA theorem of Serre and Stark says that in fact every modular form of weight 1\/2 is a linear \ncombination of form of the type \\eqref{unary1}, a simple example\nbeing the identity\n\\beta\\label{defeta}\n  \\eta(\\tau)\\;:=\\; q^{1\/24} \\prod_{n=1}^{\\infty} (1-q^{n}) \n  \\= \\sum_{n=1}^{\\infty} \\chi_{12}(n)\\,q^{n^2\/24}\\,,\n\\end{equation}\nproved by Euler for the so-called {\\it Dedekind eta function} $\\eta(t)=\\Delta(\\tau)^{1\/24}$. \nHere $\\chi_{12}$ is the function of period 12 defined by\n\\beta\\label{defchi12} \n \\chi_{12}(n) \\=  \\left\\{ \\begin{array}{lll} + 1 \\quad & {\\rm if} \\; n \\equiv \\pm 1 \\, ({\\rm mod} \\, 12) \\\\  \n  - 1 \\quad & {\\rm if} \\; n \\equiv \\pm 5 \\, ({\\rm mod} \\, 12) \\\\  0 \\quad & {\\rm if}  \\; (n,12)>1 \\ . \\end{array} \\right. \n\\end{equation}\n\n\nFinally, we recall the definition of the Petersson scalar product. If $f(\\tau)$ and $g(\\tau)$\nare two modular forms of the same weight $k$ and the same ``multiplier system'' on some subgroup $\\Gamma$ of \nfinite index of $SL_{2}({\\mathbb Z})$ (this means that the quotient $f\/g$ is invariant under $\\Gamma$), and if either $k<1$ \nor else at least one of $f$ and $g$ is a cusp form, then we can define the (normalized) \n\\emph{Petersson scalar product}  of $f$ and $g$ by \n\\beta \\label{Peterkabeta}\n\\big(f,g\\big) \\=  \\int_{{\\cal F}_{\\Gamma}} \\, f(\\tau) \\, \\overline{g(\\tau)} \\, \\tau_{2}^{k} \\, d \\mu(\\tau) \n\\left\/  \\int_{{\\cal F}_{\\Gamma}} \\, d \\mu(\\tau) \\right.\n  \\qquad\\Bigl(\\,d \\mu(\\tau)\\,:=\\, \\frac{d\\tau_{1} d\\tau_{2}}{\\tau_{2}^{2}} \\Bigr)\\, , \n\\end{equation}\nwhere ${\\cal F}_{\\Gamma}$ is a fundamental domain for $\\Gamma$. \nThis definition is independent of the choice of the subgroup $\\Gamma$ and the fundamental \ndomain ${\\cal F}_{\\Gamma}$.  By the Rankin-Selberg formula (see, for example,~\\cite{Zagier:1982}),\nwe have that if $f=\\sum_\\lambda a_\\lambda q^\\lambda$ and $g=\\sum_\\lambda b_\\lambda q^\\lambda$ (where $\\lambda\\ge0$ may have a denominator), then \n\\beta \\label{RS} (f,g)\\=\\frac{\\Gamma(k)}{(4\\pi)^k}\\;\n\\text{Res}_{s=k}\\biggl(\\,\\sum_{\\lambda>0}\\frac{a_\\lambda \\, \\overline{b_\\lambda}}{\\lambda^s}\\biggr)\\,,\\end{equation}\na formula that will be used later. For instance, for $f=g=\\eta$ we have $k=1\/2$ and \n$\\sum_{\\lambda} a_{\\lambda}  b_{\\lambda}  \\lambda^{-s} = 24^{s}(1-2^{-2s})(1-3^{-2s}) \\, \\zeta(2s)$, and hence  \n$(\\eta, \\eta) = 1\/\\sqrt{6}$.\n\n\n\n\\subsection{Quantum black holes and modular forms}\n\nModular forms occur naturally in the context of counting the Dabholkar-Harvey  (DH)\nstates \\cite{Dabholkar:1989jt,Dabholkar:1990yf}, which are states in the string\nHilbert space that are dual to perturbative BPS states. The spacetime helicity\nsupertrace counting the degeneracies reduces to the partition function of a\nchiral conformal field theory on a genus-one worldsheet.  The $\\tau $ parameter\nabove becomes the modular parameter of the genus one Riemann surface. The\ndegeneracies are given by the Fourier coefficients of the partition function.\n\nA well-known simple example is the partition function $Z(\\tau) $ which counts the\nhalf-BPS DH states for the Type-II compactification on $K3 \\times T^2$ considered\nhere. In the notation of (\\ref{chargevector}) these states have zero magnetic\ncharge $M=0$, but nonzero electric charge $N$ with the $T$-duality invariant $N^2\n=2n$, which can be realized for example by setting $Q_1 =Q_5=l =0$ in (\\ref{final\ncharges}). They are thus purely electric and perturbative in the heterotic\nframe\\footnote{Not all DH states are half-BPS. For example, the states that are\nperturbative in the Type-II frame correspond to a Type-II string winding and\ncarrying momentum along a cycle in $T^2$. For such states both $M$ and $N$ are\nnonzero and nonparallel, and hence the state is quarter- BPS.}. The partition\nfunction is given by the partition function of the chiral conformal field theory\nof $24$ left-moving transverse bosons of the heterotic string. The Hilbert space\n$\\cal H$ of this theory is a unitary Fock space representation of the commutation algebra\n\\beta\\label{osci}  [a_{in},\\,a_{j m}^\\dagger]  \\= \\delta_{ij}\\,\\delta_{n+m,0} \n  \\qquad (i,\\,j = 1,\\ldots,24\\,, \\quad n,\\,m = 1,\\,2, \\ldots,\\infty)  \\end{equation}\nof harmonic modes of oscillations of the string in $24$ different directions. The Hamiltonian is\n  \\beta\\label{ham}  H \\= \\sum_{i=1}^{24} n\\,a_{in}^\\dagger\\,a_{in} \\;-\\;1\\,, \\end{equation}\nand the partition function is\n  \\beta\\label{defpart}  Z(\\tau) \\= {\\rm Tr}_{\\cal H} (q^H) \\,. \\end{equation}\nThis can be readily evaluated since each oscillator mode of energy $n$ contributes to the trace\n  \\beta\\label{harm}  1 + q^n + q^{2n} + \\ldots  \\= \\frac 1{1-q^n} \\,. \\end{equation}\nThe partition function then becomes\n  \\beta\\label{partsmall}  Z(\\tau) \\=  \\frac 1{\\Delta(\\tau)}\\,,  \\end{equation}\nwhere $\\Delta$ is the cusp form (\\ref{discrim}). Since $\\Delta$\nhas a simple zero at $q=0$, the partition function itself has a pole at $q=0$,\nbut has no other poles in $\\mathbb{H}$. Hence, $Z(\\tau)$ is a weakly holomorphic\nmodular form of weight $-12$. This property is essential in the present physical\ncontext since it determines the asymptotic growth of the Fourier coefficients.\n\nThe degeneracy $d(n)$ of the state with electric charge $N$ depends only on the\n$T$-duality invariant integer $n$ and is given by\n  \\beta\\label{partsmall2}  Z(\\tau) \\=   \\sum_{n=-1}^\\infty d(n)\\,q^n  \\,. \\end{equation}\nFor the Fourier integral\n  \\beta\\label{dn}  d(n)\\ = \\int_C e^{-2\\pi i\\tau n} Z(\\tau) d\\tau  \\,,  \\end{equation}\none can choose the contour ${\\cal C}$ in $\\mathbb{H}$ to be\n  \\beta\\label{cont1}  0 \\leq \\Re(\\tau) <1 \\,,  \\end{equation}\nfor a fixed imaginary part $\\Im(\\tau) $. Since the partition function has no poles in\n$ \\mathbb{H}$ except at $q =0$, smooth deformations of the contour do not change\nthe Fourier coefficients and consequently the degeneracies $d(n)$ are uniquely\ndetermined from the partition function. This reflects the fact that the half-BPS states are\nimmortal and do not decay anywhere in the moduli space. As a result, there is no wall crossing\nphenomenon, and no jumps in the degeneracy.\n\nIn number theory, the partition function above is well-known in the context of\nthe problem of partitions of integers. We can therefore identify\n  \\beta\\label{colorpartition}  d(n) \\= p_{24}(n +1)  \\qquad (n \\geq 0)\\,.  \\end{equation}\nwhere $p_{24}(I)$ is the number of colored partitions of a positive integer $I$ using\nintegers of $24$ different colors.\n\nThese states have a dual description in the Type-II frame where they can be\nviewed as bound states of $Q_1$ number of D1-branes and $Q_5$ number of D5-branes\nwith $M^2\/2 = Q_1 Q_5 \\equiv m $. This corresponds to setting $n= {\\tilde K} = l =0$ in\n(\\ref{final charges}). In this description, the number of such bound states\n$d(m)$ equals the orbifold Euler character $ \\chi(\\textrm{Sym}^{m+1}(K3))$ of the\nsymmetric product of $(m+1)$ copies of $K3$-surface \\cite{Vafa:1994tf}. The\ngenerating function for the orbifold Euler character\n  \\beta\\label{Zeuler} \\hat Z(\\sigma) = \\sum_{m=-1}^\\infty  \\chi(\\textrm{Sym}^{m+1}(K3))\\, p^m\n  \\qquad \\bigl(p:= e^{2\\pi i\\sigma}\\bigr)\\end{equation}\ncan be evaluated \\cite{Goettsche:1990go} to obtain\n  \\beta\\label{Zeuler2} \\hat Z(\\sigma) = \\frac 1{\\Delta(\\sigma)} \\,. \\end{equation}\nDuality requires that the number of immortal BPS-states of a given charge must equal the\nnumber of BPS-states with the dual charge. The equality of the two partition functions\n(\\ref{partsmall}) and (\\ref{Zeuler2}) coming from two very different counting\nproblems is consistent with this expectation. This fact was indeed one of the\nearly indications of a possible duality between heterotic and Type-II strings\n\\cite{Vafa:1994tf}.\n\nThe DH-states correspond to the microstates of a small black hole \n\\cite{Sen:1995in,Dabholkar:2004yr,Dabholkar:2004dq} for large $n$. The macroscopic entropy\n$S(n)$ of these black holes should equal the asymptotic growth of the degeneracy\nby the Boltzmann relation \\beta\\label{bolt} S(n) = \\log d(n); \\quad n\n\\gg 1 \\,. \\end{equation} In the present context, the macroscopic entropy can be\nevaluated from the supergravity solution of small black holes \n\\cite{LopesCardoso:1998wt,LopesCardoso:1999ur,LopesCardoso:1999cv,\nLopesCardoso:1999xn,Dabholkar:2004yr,Dabholkar:2004dq}. \nThe asymptotic growth of the microscopic degeneracy can\nbe evaluated using the Hardy-Ramanujan expansion (Cardy formula). There is a\nbeautiful agreement between the two results \\cite{Dabholkar:2004yr,Kraus:2005vz}\n  \\beta\\label{bolt2} S(n) \\= \\log d(n) \\sim 4 \\pi \\sqrt{n}\\quad n \\gg 1 \\,.   \\end{equation}\nGiven the growth properties of the Fourier coefficients mentioned above, it is clear\nthat, for a black hole whose entropy scales as a power of $n$ and not as\n$\\log(n)$, the partition function counting its microstates can be only weakly\nholomorphic and not holomorphic.\n\nThese considerations generalize in a straightforward way to congruence subgroups\nof $SL(2, \\mathbb{Z})$ which are relevant for counting the DH-states in various\norbifold compactifications with ${\\cal N}=4$ or $ {\\cal N}=2$ supersymmetry\n\\cite{Dabholkar:2005by,Sen:2005ch,Dabholkar:2005dt}.\n\n\\section{Jacobi forms \\label{Jacobi}}\n\n\\subsection{Basic definitions}\n\nConsider a holomorphic function $\\varphi(\\tau, z)$ from $\\mathbb{H} \\times{\\mathbb C}$ to ${\\mathbb C}$ which \nis ``modular in $\\tau$ and elliptic in $z $'' in the sense that it transforms under the modular group as\n  \\beta\\label{modtransform}  \\varphi\\Bigl(\\frac{a\\tau+b}{c\\tau+d},\\frac{z}{c\\tau+d}\\Bigr) \\= \n   (c\\tau+d)^k\\,e^{\\frac{2\\pi imc z^2}{c\\tau+d}}\\,\\varphi(\\tau,z)  \\qquad \\forall \\quad\n   \\Bigl(\\begin{array}{cc} a&b\\\\ c&d \\end{array} \\Bigr) \\in SL(2; {\\mathbb Z}) \\end{equation}\nand under the translations of $z$ by $\\mathbb{Z} \\tau + \\mathbb{Z}$ as\n  \\beta\\label{elliptic}  \\varphi(\\tau, z+\\lambda\\tau+\\mu)\\= e^{-2\\pi i m(\\lambda^2 \\tau + 2 \\lambda z)} \\varphi(\\tau, z)\n  \\qquad \\forall \\quad \\lambda,\\,\\mu \\in {\\mathbb Z} \\, , \\end{equation}\nwhere $k$ is an integer and $m$ is a positive integer.\n\nThese equations include the periodicities $\\varphi(\\tau+1,z) = \\varphi(\\tau,z)$ and $\\varphi(\\tau,z+1) = \\varphi(\\tau,z)$, so $\\varphi$ has a Fourier expansion\n  \\beta\\label{fourierjacobi} \\varphi(\\tau,z) \\= \\sum_{n, r} c(n, r)\\,q^n\\,y^r\\,, \\qquad\\qquad\n   (q :=e^{2\\pi i \\tau}, \\; y := e^{2 \\pi i z}) \\ . \\end{equation}\nEquation \\eqref{elliptic} is then equivalent to the periodicity property\n  \\beta\\label{cnrprop}  c(n, r) \\= C(4 n m - r^2 , \\, r) \\ ,\n  \\qquad \\mbox{where} \\; C(\\Delta, r) \\; \\mbox{depends only on} \\; r \\; \\mod \\; {2m} \\ . \\end{equation}\n(We will sometimes denote $c(n,r)$ and $C(\\Delta,r)$ by $c_\\varphi(n,r)$ and $C_{\\varphi}(\\Delta,r)$ \nor by $c(\\varphi;n,r)$ and $C(\\varphi\\, ;\\Delta,r)$ when this is required for emphasis or clarity.) \nThe function $\\varphi(\\tau, z)$ is called a \\emph{holomorphic Jacobi form} (or simply a \\emph{Jacobi form})\nof weight $k$ and index $m$ if the coefficients $C(\\Delta,r)$ vanish for $\\Delta<0$, {\\it i.e.} if \n  \\beta\\label{holjacobi}  c(n, r) \\= 0 \\qquad \\textrm{unless} \\qquad 4mn \\ge r^2\\,. \\end{equation}\nIt is called a \\emph{Jacobi cusp form} if it satisfies the stronger condition that\n$C(\\Delta,r)$ vanishes unless $\\Delta$ is strictly positive, {\\it i.e.}\n  \\beta\\label{cuspjacobi}  c(n, r) = 0 \\qquad \\textrm{unless} \\qquad 4mn > r^2 \\ , \\end{equation}\nand it is called a \\emph{weak Jacobi form} if it satisfies the weaker condition\n  \\beta\\label{weakjacobi} c(n, r) \\= 0\\qquad   \\textrm{unless}  \\qquad n \\geq 0 \\, \\end{equation}\nrather than \\eqref{holjacobi}, whereas a merely \\emph{weakly holomorphic Jacobi form} satisfies only the yet weaker condition\nthat $c(n,r)=0$ unless $n\\ge n_0$ for some possibly negative integer~$n_0$ (or equivalently $C(\\Delta,r)=0$ unless $\\Delta\\ge\\Delta_0$\nfor some possibly negative integer~$\\Delta_0$).  The space of all holomorphic (resp.~cuspidal, weak, or weakly holomorphic) Jacobi\nforms of weight~$k$ and index~$m$ will be denoted by $J_{k,m}$ (resp.~$J_{k,m}^0$,  $\\widetilde J_{k,m}$, or $\\widetilde J^{\\,!}_{k,m}$).\n\nFinally, the quantity $\\Delta=4mn-r^2$, which by virtue of the above discussion is the crucial invariant of a monomial\n$q^ny^r$ occurring in the Fourier expansion of~$\\varphi$, will be referred to as its {\\it discriminant}.  \n(It would be mathematically more correct to use this word for the quantity~$-\\Delta$, but $\\Delta$ is usually positive \nand it is more convenient to work with positive numbers.)\n\n\\subsection{Theta expansion and Taylor expansion \\label{ThetaTaylorExp}}\n\nA Jacobi form has two important representations, the {\\it theta expansion} and the {\\it Taylor expansion}. \nIn this subsection, we explain both of these and the relation between them. \n\nIf $\\varphi(\\tau, z)$ is a Jacobi form, then the transformation property (\\ref{elliptic}) implies its\nFourier expansion with respect to $z$ has the form\n  \\beta\\label{jacobi-Fourier} \\varphi(\\tau, z) \\= \\sum_{\\ell\\,\\in\\, {\\mathbb Z}} \\;q^{\\ell^2\/4m}\\;h_\\ell(\\tau) \\; e^{2\\pi i\\ell z} \\end{equation}\nwhere $h_\\ell(\\tau)$ is periodic in $\\ell$ with period $2m$.  In terms of the coefficients \\eqref{cnrprop} we have\n  \\beta\\label{defhltau}  h_{\\ell}(\\tau) \\= \\sum_{\\Delta} C(\\Delta,\\ell) \\,  q^{\\Delta\/4m} \\, \\qquad \\qquad (\\ell \\,\\in\\, {\\mathbb Z}\/2m {\\mathbb Z})\\;.  \\end{equation}\nBecause of the periodicity property, equation \\eqref{jacobi-Fourier} can be rewritten in the form \n  \\beta\\label{jacobi-theta} \\varphi(\\tau,z) = \\sum_{\\ell\\,\\in\\, {\\mathbb Z}\/2m{\\mathbb Z}} h_\\ell(\\tau) \\, \\vartheta_{m,\\ell}(\\tau, z)\\,, \\end{equation}\nwhere $\\vartheta_{m,\\ell}(\\tau,z)$ denotes the standard index $m$ theta function \n  \\begin{eqnarray} \\label{thetadef} \\vartheta_{m,\\ell}(\\tau, z) \n   \\;:=\\; \\sum_{{r\\,\\in\\,{\\mathbb Z}} \\atop {r\\,\\equiv\\,\\ell\\,({\\rm mod}\\,2m)}} q^{r^2\/4m} \\, y^r \\, \\,\n   \\= \\sum_{n\\,\\in\\,{\\mathbb Z}} \\,q^{(\\ell+2mn)^2\/4m} \\,y^{\\ell+2mn}  \\end{eqnarray}\n(which is a Jacobi form of weight ${\\frac12}$ and index $m$ on some subgroup of\n$SL(2, \\mathbb{Z})$). This is the theta expansion of $\\varphi$.  The coefficiens $h_{\\ell}(\\tau)$ are  modular forms of weight $k-\\frac12$\nand are weakly holomorphic, holomorphic or cuspidal if $\\varphi$ is a weak Jacobi form, a Jacobi form or a Jacobi cusp form, respectively. \nMore precisely, the vector $h := ( h_1, \\ldots, h_{2m})$ transforms like a modular form of weight $k-\\frac 12$ under $SL(2,{\\mathbb Z})$.\n\nThe theta decomposition \\eqref{jacobi-theta} leads to the definition of a differential operator on Jacobi forms as follows.  Let\n$$ {\\cal L}_m \\= \\frac{4m}{2\\pi i}\\,\\frac\\partial{\\partial\\tau} \\,-\\,\\frac1{(2\\pi i)^2}\\,\\frac{\\partial^2}{\\partial z^2}  $$\nbe the index $m$ heat operator, which sends $\\varphi=\\sum c(n,r)q^ny^r$ to $\\sum(4nm-r^2)c(n,r)q^ny^r$. Here the Fourier coefficients\nhave the same periodicity property~\\eqref{cnrprop} as for~$\\varphi$, so ${\\cal L}_m\\varphi$ has the same elliptic transformation properties as a Jacobi\nform of index~$m$. Moreover, since ${\\cal L}_m$ annihilates all the $\\vartheta_{m,\\ell}$, we have \n${\\cal L}_m(\\sum h_\\ell\\vartheta_{m,\\ell})=4m\\sum h'_\\ell\\vartheta_{m,\\ell}$, so the modified heat operator\n\\beta\\label{LRS} {\\cal L}_{k,m} = {\\cal L}_m -\\frac{m(k-\\frac12)}3\\,E_2  \\;\\,:\\;\\;\n   \\sum_{\\ell} h_\\ell(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z) \\;\\mapsto\\; 4m\\,\\sum_\\ell \\dRS{k-\\frac12}h_\\ell(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z) \\,, \\end{equation}\n\nwhere $\\dRS{*}$ is the Ramanujan-Serre derivative defined in~\\eqref{RSderiv}, sends $J_{k,m}$ to $J_{k+2,m}$ \n(and also $\\widetilde J_{k,m}$ to $\\widetilde J_{k+2,m}$ and $J^{\\,!}_{k,m}$ to $J^{\\,!}_{k+2,m}$).  These operators will be used later.\n\nA Jacobi form $\\varphi \\in \\widetilde J_{k,m}$ also has a Taylor expansion in $z$ which for $k$ even takes the form \n  \\beta\\label{defxin}  \\varphi(\\tau,z) \\= \\xi_0(\\tau) \\+ \\left(\\frac{\\xi_1(\\tau)}2 + \\frac{ m \\xi'_0(\\tau)}{k} \\right) (2\\pi iz)^2   \n  \\+ \\left(\\frac{\\xi_2(\\tau)}{24} + \\frac{m \\xi'_1(\\tau) }{2\\,(k+2)}+ \\frac{m^2  \\xi^{''}_0 (\\tau)}{2 k(k+1)} \\right) \\, (2 \\pi i z)^4 + \\cdots \\end{equation}\nwith $\\xi_{\\nu} \\in M_{k+2 \\nu} (SL(2,{\\mathbb Z}))$ and the prime denotes $\\frac 1{2 \\pi i} \\frac{d}{d \\tau} $ as before. \nIn terms of the Fourier coefficients of $\\varphi$, the modular form $\\xi_\\nu$ is given by\n  \\beta\\label{xicnrsrel}  \\frac{(k+2\\nu -2)!}{(k+ \\nu -2)!} \\, \\xi_{\\nu} (\\tau) \n  \\= \\sum_{n=0}^{\\infty} \\left( \\sum_{r} P_{\\nu,k} (nm,r^2) c(n,r) \\right) q^{n}\\,,  \\end{equation}\nwhere $P_{\\nu,k}$ is a homogeneous polynomial of degree $\\nu$ in $r^2$ and $n$ \nwith coefficients depending on $k$ and $m$, the first few being\n  \\begin{eqnarray}\\label{Pnu012}  P_{0,k} & \\= &  1 \\ , \\cr   P_{1,k} & \\= &   k r^2 \\,-\\, 2nm  \\ , \\cr\n  P_{2,k} & \\= &  (k+1)(k+2)r^4 \\,-\\, 12 (k+1) r^2 m n  \\+ 12 m^2  n^2 \\ .  \\end{eqnarray}\nThe Jacobi form $\\varphi$ is determined by the first $m+1$ coefficients $\\xi_{\\nu}$, and the map   $\\varphi \\mapsto (\\xi_0,\\dots,\\xi_m)$\nis an isomorphism from $\\widetilde J_{k,m}$ to $M_{k} \\oplus M_{k+2} \\oplus \\cdots \\oplus M_{k+2m}$.  \nFor $k$ odd, the story is similar except that \\eqref{defxin} must be replaced by \n  \\beta\\label{defxinodd}\n  \\varphi(\\tau,z) \\= \\xi_0(\\tau)\\;(2\\pi iz) \\+ \\left(\\frac{\\xi_1(\\tau)}6+\\frac{m \\xi'_0(\\tau)}{k+2} \\right) \\, (2\\pi iz)^3 \\+ \\cdots     \\end{equation}\nwith $\\xi_{\\nu} \\in M_{k+2\\nu+1}(SL(2,{\\mathbb Z}))$, and the map $\\varphi \\mapsto (\\xi_0, \\dots, \\xi_{m-2})$ gives an \nisomorphism from $\\widetilde J_{k,m}$ to $M_{k+1} \\oplus M_{k+3} \\oplus \\cdots \\oplus M_{k+2m-3}$. \n\n\nWe also observe that even if $\\varphi$ is weak, so that the individual coefficients $c(n,r)$ grow like $C^{\\sqrt{4nm - r^2}}$, \nthe coefficients  $ \\sum_{r} P_{\\nu,k} (nm,r^2) \\, c(n,r) $ of $\\xi_{\\nu}$ still have only polynomial growth. We thus have the\nfollowing descriptions (analogous to those given for classical modular forms in \\S\\ref{modularbasic}) \nof holomorphic and weak Jacobi forms in terms of their asymptotic properties: \\\\\n\n\\vspace{0.2cm}\n\n\\begin{tabular}{ccl}\n  Holomorphic Jacobi & $\\Longleftrightarrow$ &  $c(n,r) =0$ for $4mn-r^2 < 0$ \\\\\n   & $\\Longleftrightarrow$ & the function $q^{m\\alpha^2} \\varphi(\\tau,\\alpha \\tau + \\beta)$ (which is a modular form of weight $k$  \\\\\n   && and some level) is bounded as $\\tau_2 \\to \\infty$ for every $\\alpha,\\beta \\in {\\mathbb Q}$\\\\\n   & $\\Longleftrightarrow$ &  all $h_{j}(\\tau)$ in \\eqref{jacobi-theta} are bounded as $\\tau_2 \\to \\infty$ \\\\\n   & $\\Longleftrightarrow$ &  $c(n,r)$ have polynomial growth. \\\\   &&\\\\\n  Weak Jacobi & $\\Longleftrightarrow$ &  $c(n,r) =0$ for $n < 0$ \\\\\n   & $\\Longleftrightarrow$ &   $\\varphi(\\tau,\\alpha \\tau + \\beta)$ is bounded as $\\tau_2 \\to \\infty$ for any fixed $z \\in {\\mathbb C}$ \\\\\n   & $\\Longleftrightarrow$ & all $h_{j}(\\tau) = O(q^{-j^2\/4m})$ as $\\tau_2 \\to \\infty$ \\\\\n   & $\\Longleftrightarrow$ &  $\\sum_{r} P_{\\nu,k} (nm,r^2) c(n,r)$ have polynomial growth. \n\\end{tabular}\n\nFinally, the relation between the Taylor expansion \\eqref{defxin} of a Jacobi form and its theta expansion \\eqref{jacobi-theta} \nis given by   \n  \\beta\\label{xinhrel} \\xi_\\nu(\\tau) \\= (4m)^\\nu \\left({k+2\\nu -2 \\atop \\nu }\\right)^{-1}  \\,\n   \\sum_{\\ell \\, ({\\rm mod} \\, 2m)} \\bigl[ h_{\\ell}(\\tau), \\, \\vartheta^0_{m,\\ell}(\\tau) \\bigr]_\\nu \\ , \\end{equation}\nwhere $[\\; , \\; ]_{\\nu} = [\\; , \\; ]_{\\nu}^{(k-\\frac 12,\\frac 12)}$ denotes the Rankin-Cohen bracket \n(which, as we mentioned above, also works in half-integral weight), and $\\vartheta^0_{m,\\ell}(\\tau) = \\vartheta_{m,\\ell}(\\tau,0)$ (Thetanullwerte).\nThere is a similar formula in the odd case, but with $\\vartheta^0_{m,\\ell}(\\tau)$ replaced by \n\\beta\\label{defth1}\n\\vartheta^1_{m,\\ell}(\\tau) \\= \\frac 1{2 \\pi i} \\, \\frac{\\pi}{\\pi z} \\vartheta_{m,\\ell}(\\tau,z) \\Big|_{z=0} \\= \\sum_{r\\equiv \\ell \\, ({\\rm mod} \\; 2m)} r\\,q^{r^2\/4m} \\, . \n\\end{equation} \n\n\n\\subsection{Example: Jacobi forms of index $1$} \\label{MJF1}\n\nIf $m=1$, \\eqref{cnrprop} reduces to $c(n,r) = C(4n-r^2)$ where $C(\\Delta)$ is a\nfunction of a single argument, because $4n - r^2$ determines the value of $r \\, ({\\rm mod} \\, 2)$. \nSo any $\\varphi \\in J_{k,1}^{\\textrm{weak}} $ has an expansion of the form \n  \\beta\\label{phiC}   \\varphi(\\tau, z) \\= \\sum_{n, r \\in {\\mathbb Z}} C(4n-r^2) \\, q^n\\,y^r \\,.  \\end{equation}\nIt also follows that $k$ must be even, since in general, $C(\\Delta,-r) = (-1)^{k} C(\\Delta,r)$.\n\nOne has the isomorphisms $J_{k,1} \\cong M_{k} \\oplus S_{k+2}$ and $J_{k,1}^{\\textrm{weak}} \\cong M_{k} \\oplus M_{k+2}$. If\n$\\varphi \\in J_{k,1}^{\\textrm{weak}}$ with an expansion as in  \\eqref{phiC}, then \n  \\beta\\label{phiAB} \\varphi(\\tau, 0) \\=  \\sum_{n=0}^\\infty a(n)\\,q^n\\,, \\qquad  \n  \\frac1{2(2\\pi i)^2} \\varphi''(\\tau, 0) \\= \\sum_{n=0}^\\infty b(n)\\,q^n\\,,  \\end{equation}\nwhere \n  \\beta\\label{defAB}  a(n) \\= \\sum_{r\\in{\\mathbb Z}} C(4n-r^2)\\,, \\qquad  b(n) \\= \\sum_{r>0} r^2\\,C(4n-r^2)\\,, \\end{equation}\nand the isomorphisms are given (if $k>0$) by the map $\\varphi \\mapsto ({\\cal A}, {\\cal B})$ with \n \\beta\\label{ABmod} {\\cal A}(\\tau) \\= \\sum a(n)\\,q^n \\in M_{k}\\,, \\qquad {\\cal B}(\\tau) \\=  \\sum\\bigl(kb(n)-na(n)\\bigr)\\,q^{n} \\in M_{k+2}\\,.\\end{equation}\nFor $J_{k,1}$ one also has the isomorphism $J_{k,1} \\cong M_{k-\\frac 12}^{+}\\left( \\Gamma_0(4)\\right)$ given by \n\\beta\\label{phitog} \\varphi(\\tau,z) \\leftrightarrow g(\\tau) \\= \\sum_{{\\Delta\\ge 0} \\atop \\Delta \\equiv 0,\\,3 \\, {\\rm mod}4} \nC(\\Delta)\\,q^{\\Delta}\\,. \\end{equation}\n\nWe have four particularly interesting examples $ \\varphi_{k, 1}$\n\\beta\\label{phik} \\varphi_{k, 1}(\\tau, z) \\= \\sum_{n,\\,r\\in{\\mathbb Z}} C_k(4n -r^2)\\,q^n\\,y^r\\,,\\quad k=-2,\\,0,\\,10,\\,12\\,, \\end{equation}\nwhich have the properties (defining them uniquely up to multiplication by scalars)\n\\begin{myitemize}\n  \\item $\\varphi_{10,1}$ and $\\varphi_{12,1}$ are the two index $1$ Jacobi cusp forms of smallest weight;\n  \\item $\\varphi_{-2,1}$ and $\\varphi_{0,1}$ are the unique weak Jacobi forms of index $1$ and weight $ \\le 0$;\n  \\item $\\varphi_{-2,1}$ and $\\varphi_{0,1}$ generate the ring of weak Jacobi forms of even weight freely over the ring of\n      modular forms of level~$1$, so that\n   \\beta\\label{jacovermod}  J^{\\textrm{weak}}_{k,m} \\= \\bigoplus_{j=0}^{m} M_{k+2j}\\left(SL(2,{\\mathbb Z}) \\right) \n      \\cdot\\varphi_{-2,1}^j\\, \\varphi_{0,1}^{m-j}\\qquad\\text{($k$ even)}\\,; \\end{equation}\n \\item $\\varphi_{-2,1} = \\varphi_{10,1}\/\\Delta$, $\\varphi_{0,1} = \\varphi_{12,1}\/ \\Delta$,\n  and the quotient $ \\varphi_{0,1}\/ \\varphi_{-2,1} = \\varphi_{12,1}\/ \\varphi_{10,1}$ is a multiple of the Weierstrass ${\\cal P}$ function.\n\\end{myitemize}\nThe Fourier coefficients of these functions can be computed from the above recursions, since the pairs $({\\cal A},{\\cal B})$ for $\\varphi=\\varphi_{-2,1}$, \n$\\varphi_{0,1}$, $\\varphi_{10,1}$ and $\\varphi_{12,1}$ are proportional to $(0,1)$, $(1,0)$, $(0,\\Delta)$ and $(\\Delta,0)$, respectively\\footnote{For $k=0$, the second formula in~\\eqref{ABmod} must be modified, and the function\n$\\sum b(n) q^{n}$ for $\\varphi_{0,1}$ is in fact~$E_{2}(\\tau)$.}.  \nThe results for \nthe first few Fourier coefficients are given in Table \\ref{tablefcoeffs} below. In particular, the Fourier expansions of $\\varphi_{-2,1}$ and $\\varphi_{0,1}$ begin\n  \\begin{eqnarray}\\label{Fourirphi02}  \\varphi_{-2,1} & \\= & \\frac{(y-1)^2}{y} \\,-\\, 2\\,\\frac{(y-1)^4}{y^2}\\,q \\+  \\frac{(y-1)^4(y^2-8y+1)}{y^3}\\,q^2 \\+ \\cdots \\ , \\\\\n  \\varphi_{0,1} & \\= & \\frac{y^2 + 10 y +1}{y} \\+ 2 \\frac{(y-1)^2\\,(5 y^2 - 22 y + 5)}{y^2}\\,q \\+ \\cdots  \\ .   \\end{eqnarray}\n\n\n\\begin{table}[h]     \\centering\n   \\begin{tabular}{|c|ccccccccc|ccccc}   \\hline   \n   $k$ &  $C_{k}(-1)$ & $C_{k}(0)$ &$C_{k}(3) $& $C_{k}(4)$& $C_{k}(7)$ & $C_{k}(8)$ &$C_{k}(11)$& $C_{k}(12)$& $C_{k}(15)$ \\\\\n   \\hline $-2$ & 1 & $-2$ &8 &$-12$&39&$-56$&152&$-208$&513\\\\   $0$ &  1  & 10 &$-64$&108&$-513$&808&$-2752$&4016&$-11775$\\\\\n     $10$ & 0 & 0 &1&$-2$&$-16$&36&99&$-272$&$-240$\\\\   $12$ & 0 & 0 &1&10&$-88$&$-132$&1275&736&$-8040$\\\\  \n   \\hline \\end{tabular} \\label{tablefcoeffs} \\captionof{table}{ \\small {Some Fourier coefficients}}   \\end{table}\n\\smallskip\n\nThe functions $\\varphi_{k,1}$ ($k=10,\\,0,\\,-2$) can be expressed in terms of the Dedekind eta function \n\\eqref{defeta} and the Jacobi theta functions $\\vartheta_1,\\,\\vartheta_2,\\,\\vartheta_3,\\,\\vartheta_4$ by the formlas\n  \\beta\\label{phi10} \\varphi_{10, 1}(\\tau, z) \\= \\eta^{18}(\\tau) \\, {\\vartheta_1^2(\\tau, z)}\\,, \\end{equation}\n   \\beta\\label{phiminus2}  \\varphi_{-2,1}(\\tau, z) \\= \\frac{\\vartheta_1^2(\\tau, z)}{\\eta^6(\\tau)}\n          \\= \\frac{\\varphi_{10, 1}(\\tau, z)} {\\Delta(\\tau)} \\;. \\end{equation}\n \\beta\\label{phizero}  \\varphi_{0, 1} (\\tau, z) \\= 4 \\left( \\frac{\\vartheta_2(\\tau, z)^2}{\\vartheta_2(\\tau)^2} +\n    \\frac{\\vartheta_3(\\tau, z)^2}{\\vartheta_3(\\tau)^2} +\\frac{\\vartheta_4(\\tau, z)^2}{\\vartheta_4(\\tau)^2} \\right) \\, , \\end{equation}\n\nFinally, we say a few words about Jacobi forms of odd weight.  Such a form cannot have index~1, as we saw.\nIn index~2, the isomorphisms $J_{k,2} \\cong S_{k+1}$ and $J_{k,2}^{\\textrm{weak}} \\cong M_{k+1}$ show\nthat the first examples of holomorphic and weak Jacobi forms occur in weights 11 and $-1$, respectively,\nand are related by $\\varphi_{-1,2}=\\varphi_{11,2}\/\\Delta$. The function $\\varphi_{-1,2}$ is given explicitly by\n \\beta\\label{phiminus1}\\varphi_{-1,2}(\\tau, z) \\= \\frac{\\vartheta_1(\\tau,2z)}{\\eta^3(\\tau)} \\;, \\end{equation}\nwith Fourier expansion beginning\n \\beta  \\label{Fourirphi1}  \\varphi_{-1,2} \\= \\frac{y^2-1}{y} \\,-\\, \\frac{(y^2-1)^3}{y^3}\\,q \\,-\\,3\\,\\frac{(y^2-1)^3}{y^3}\\,q^2 \\+ \\cdots \\,,  \\end{equation}\nand its square is related to the index~1 Jacobi forms defined above by \n\\beta \\label{weierstrass}\n  432\\,\\varphi_{-1,2}^2 \\= \\varphi_{-2,1}\\,\\bigl(\\varphi_{0,1}^3 \\,-\\, 3\\,E_4\\,\\varphi_{-2,1}^2\\,\\varphi_{0,1} \\+ 2\\,E_6\\,\\varphi_{-2,1}^3\\bigr)\\;. \n\\end{equation}\n(In fact, $\\varphi_{-1,2}\/\\varphi_{-2,1}^2$ is a multiple of ${\\cal P}'(\\tau,z)$ and this equation, divided by $\\varphi_{-2,1}^4$, is just the\nusual equation expressing $\\,{{\\cal P}'}^{\\,2}\\,$ as a cubic polynomial in~${\\cal P}$.)  It is convenient to introduce abbreviations\n  \\beta\\label{ABC}  A\\=\\varphi_{-2,1}\\,,  \\qquad B\\=\\varphi_{0,1}\\,,\\qquad  C\\=\\varphi_{-1,2}\\;. \\end{equation}\nWith these notations, the structure of the full bigraded ring of weak Jacobi forms is given by\n  \\beta\\label{WJF}  J_{*,*}^{\\textrm{weak}} \\= {\\mathbb C}[E_4,E_6,A,B,C]\\,\\bigl\/\\,(432C^2-AB^3 + 3E_4A^3B - 2E_6A^4)\\;.  \\end{equation}\n\n\n\\subsection{Hecke-like operators \\label{hecke}}\n\nIn~\\cite{Eichler:1985ja} Hecke operators $T_{\\ell}$ acting on $J_{k,m}$ were introduced, but also various ``Hecke-like'' operators, \nagain defined by the action of certain combinations of elements of $GL(2,{\\mathbb Q}) \\rtimes {\\mathbb Q}^2$, which send Jacobi forms \nto Jacobi forms, but now possibly changing the index. We describe these operators here.  \nThis subsection is more technical than the preceding three, but will be essential later. \n\n\nThe first is the very simple operator $U_{s}$ ($s \\ge 1$) which sends $\\varphi(\\tau,z)$ to $\\varphi(\\tau,s z)$, i.e.,\n  \\beta\\label{defUl} U_s\\,: \\quad \\sum_{n,r} c(n,r)\\,q^n\\,y^r  \\quad \\mapsto  \\quad \\sum_{n,r} c(n,r)\\,q^n\\,y^{sr} \\ ,  \\end{equation}\nThis operator maps $J_{k,m}$ to $J_{k,s^2m}$. \n\nThe second operator, $V_{k,t}$ ($t \\ge 1$), sends $J_{k,m}$ to $J_{k,t m}$. It is given in terms of its action on Fourier coefficients by \n\\beta\\label{defVl}\nc(\\varphi|V_{k,t}\\,; n,r) \\= \\sum_{d|(n,r,t)} d^{k-1} c\\Big(\\varphi \\,; \\frac{nt}{d^2},\\frac{r}{d}\\Big) \\, , \\qquad \nC(\\varphi|V_{k,t}\\,; \\Delta,r) \\= \\sum_{d|(\\frac{r^{2}+\\Delta}{4mt},r,t)} d^{k-1} C\\Big(\\varphi \\,; \\frac{\\Delta}{d^2},\\frac{r}{d}\\Big) \\, .\n\\end{equation}\nIn~\\S\\ref{flyJac}, we will also need the modified operator ${\\cal V }_{k,t}^{(m)}:J_{k,m} \\mapsto J_{k,tm}$ defined by \n\\beta\\label{Vkt1}  {\\cal V }_{k,t}^{(m)}\\= \\sum_{s^2|t \\atop (s,m)=1}\\mu(s)\\,s^{k-1}\\,V_{k,t\/s^2}\\,U_s\\, ,  \\end{equation}\nso that, for instance, ${\\cal V }_{k,t}^{(m)}=V_{k,t}$ if~$t$ is square-free, but~${\\cal V }_{k,4}^{(m)}=V_{k,4}-2^{k-1}U_{2}$ if $m$ is odd.  \n\n\n\nNext, for each positive integer $m_{1}$ with $m_{1} \\| m$ (this means that $m=m_1 m_2$ with $m_{1}$\nand~$m_{2}$ coprime) we have an involution $W_{m_1}$ on $J_{k,m}$,  which we will \ncall an \\emph{Atkin-Lehner involution,}\\footnote{This terminology was not used in~\\cite{Eichler:1985ja}, \nwhere these involutions were introduced, but is justified by the results of~\\cite{SkoruppaZagier}\nwhere it is shown that they correspond\nin a precise sense to the Atkin-Lehner involutions on the space of modular forms of  weight~$2k-2$ and \nlevel~$m$.} that is defined in terms of the theta expansion of Jacobi forms by \n  \\beta\\label{defW} W_{m_1}\\,: \\quad  \\sum_{\\ell \\; ({\\rm mod} \\, 2m)} h_{\\ell} (\\tau) \\, \\vartheta_{m,\\ell} (\\tau,z)  \\quad \\mapsto  \\quad   \n  \\sum_{\\ell \\; ({\\rm mod} \\, 2m)} h_{\\ell^*} (\\tau) \\, \\vartheta_{m,\\ell} (\\tau,z) \\end{equation}\n(or equivalently by $C(\\Delta,r) \\mapsto  C(\\Delta,r^*)$), where the involution $\\ell \\mapsto \\ell^*$ on ${\\mathbb Z}\/2m{\\mathbb Z}$ is defined by \n  \\beta\\label{definvol}  \\ell^* \\equiv - \\ell \\, ({\\rm mod} \\, 2m_1) , \\qquad  \\ell^* \\equiv   + \\ell \\, ({\\rm mod} \\, 2m_2) \\ . \\end{equation}\nThese operators commute and satisfy $W_{m\/m_1} = (-1)^{k} W_{m_1}$, so that we get an eigenspace decomposition \n  \\beta\\label{Jkmeigen} J_{k,m} \\= \\bigoplus_{\\pmb{\\ve}=(\\varepsilon_1,\\dots ,\\varepsilon_t) \\,\\in\\,\\{ \\pm 1\\}^t  \\atop \\varepsilon_1 \\cdots \\varepsilon_t \\= (-1)^k}  J_{k,m}^{\\pmb{\\ve}} \\ ,  \\end{equation}\nwhere $m=p_1^{r_1} \\cdots p_{t}^{r_{t}}$ is the prime power decomposition of $m$ and $\\varepsilon_i$ is the eigenvalue of $W_{p_{i}^{r_{i}}}$. \nOne can also introduce the ``$p$-adic Atkin-Lehner involution'' $W_{p^{\\infty}}$, which acts on any~$J_{k,m}$\nas~$W_{p^{\\nu}}$ where~$p^{\\nu}||m$. (In particular, it acts as the identity if~$p \\nmid m$.) This operator \nsimply changes the second index~$r$ in~$C(\\Delta,r)$ to~$r^{*}$, where~$r$ is equal to the negative \nof~$r^{*}$ $p$-adically and to~$r^{*}$ $p'$-adically for any~$p' \\neq p$. \nIn particular,~$W_{p^{\\infty}}$ commutes with all~$U_{s}$, $V_{k,t}$, and~${\\cal V }_{k,t}^{m}$. \n\n\nWe now describe a further Hecke-like operator, which was not given in \\cite{Eichler:1985ja} but is mentioned \nbriefly in \\cite{SkoruppaZagier}, p.~139, and  \nwhich we will also need in the sequel. If $t$ is a positive integer whose square divides $m$, then we \ndefine $u_{t}: \\; J_{k,m} \\; \\longrightarrow J_{k,m\/t^{2}}$ by \n\\beta\\label{Ucech}\nu_t\\,: \\quad \\sum_{n,r} c(n,r)\\,q^n\\,y^r \\quad \\mapsto \\quad \n    \\sum_{n,r} \\left( \\sum_{a \\; \\mod \\; {t}} c \\bigl( n+ ar+\\frac{m}{t^{2}} a^{2}, \\, tr + \\frac{2ma}{t}  \\bigr)\\right) q^n\\,y^r \n\\end{equation}\nor equivalently by \n\\beta\\label{Ucech2}\nC_{{\\varphi|u_{t}}}(\\Delta,\\,\\ell \\; \\mod \\; {2m\/t^{2}}) \n\\= \\sum_{r \\; \\mod \\; {2m\/t} \\atop r \\equiv \\ell \\; \\mod \\; {2m\/t^{2}}} C_{\\varphi}(\\Delta t^{2}, \\, rt  \\; \\mod \\; {2m} ) \\; .\n\\end{equation}\nIt is easily checked that this map does indeed send Jacobi forms to Jacobi forms of the index stated, \nand that \nfor any integers $m,\\,t >0$, the composite map $J_{k,m\/t^{2}} \\stackrel{U_{t}}{\\longrightarrow} \nJ_{k,m} \\stackrel{u_{t}}{\\longrightarrow} J_{k,m\/t^{2}}$ is simply multiplication by~$t$ \n(because, if $\\varphi = \\varphi_{1} \\! \\mid \\! {U_{t}}$ with $\\varphi_{1}$ of index $m\/t^{2}$, then each term \non the right of \\eqref{Ucech2} equals $C_{\\varphi_{1}}(\\Delta,\\ell)$). It follows that we have \n\\beta\\label{primdecomp}\nJ_{k,m} \\= \\bigoplus_{t^{2}\\mid m} J^{\\rm prim}_{k,\\frac{m}{t^{2}}} \\mid {U_{t}} \\, ,\n\\end{equation}\nwhere\n\\beta\\label{primdecomp2}\nJ^{\\rm prim}_{k,m}  \\=  \\bigcap_{t^{2}\\mid m \\atop t >1} \\ker \\big(J_{k,m} \\stackrel{u_{t}}{\\longrightarrow} J_{k,m\/t^{2}} \\big) \n\\= \\bigcap_{p^{2}\\mid m \\atop \\text{$p$ prime}} \\ker \\big(J_{k,m} \\stackrel{u_{p}}{\\longrightarrow} J_{k,m\/p^{2}} \\big) \n\\end{equation}\n(the equivalence of the two definitions follows because $u_{t} \\, u_{t'} = u_{tt'}$ for any $t,\\,t'$),\nwith the projection map~$\\pi^{\\rm prim}: J_{k,m} \\mapsto J_{k,m}^{\\rm prim}$ onto the first factor in~\\eqref{primdecomp} \ngiven by\n\\beta \\label{primproj}\n\\pi^{\\rm prim}(\\varphi) \\= \\sum_{t^{2}|m} \\frac{\\mu(t)}{t} \\, \\varphi \\! \\mid \\!  u_{t}\\! \\mid \\! U_{t} \\, . \n\\end{equation}\n\n\n\n\n\n\n\nWe will sometimes find it useful to work, not with the space of Jacobi forms, but \nwith the larger space of holomorphic functions on $\\mathbb{H} \\times{\\mathbb C}$\nthat obey the elliptic transformation law \\eqref{elliptic} of Jacobi forms, but are not required to \nobey the modular transformation property \\eqref{modtransform}. \nWe shall call such functions \\emph{elliptic forms}.  (The word \\emph{Jacobi-like forms} has been\nused in the literature to denote functions which, conversely, satisfy \\eqref{modtransform} \nbut not \\eqref{elliptic}.) Any elliptic form has a Fourier expansion as in \\eqref{fourierjacobi}\nwith the periodicity condition \\eqref{cnrprop}. We will denote by $\\CE_{m}$, $\\CE^{0}_{m}$, $\\widetilde \\CE_{m}$ \nthe spaces of all elliptic forms of index $m$ satisfying the growth conditions \n\\eqref{holjacobi}, \\eqref{cuspjacobi}, and \\eqref{weakjacobi}, respectively, and we will call them\n\\emph{holomorphic, cuspidal}, and \\emph{weak} respectively. \nWe also denote by \n$\\CE_{\\pm,m}$, $\\CE^{0}_{\\pm,m}$, $\\widetilde \\CE_{\\pm,m}$ the $(\\pm 1)-$eigenspaces \nof theta functions under the involution $\\varphi(\\tau,z) \\mapsto \\varphi(\\tau,-z)$, i.e., those functions \nwhose Fourier coefficients $c(n,r)$ satisfy the parity condition $c(n,-r) = \\pm c(n,r)$. \nBecause any Jacobi form of weight $k$ satisfies this parity condition with $\\pm 1 = (-1)^{k}$\n(by \\eqref{modtransform} applied to $-{\\bf 1}_{2}$), we have \n\\beta \\label{PrimParts}\nJ_{k,m} \\subset \\CE_{\\pm,m} \\, , \\quad J^{0}_{k,m} \\subset \\CE^{0}_{\\pm,m} \\, , \\quad  \n\\widetilde J_{k,m} \\subset \\widetilde \\CE_{\\pm,m} \\,  \\qquad \\big(\\, (-1)^{k} = \\pm 1 \\, \\big) \\, . \n\\end{equation}\nIt is now clear from the definitions above that all of the Hecke-like operators \n$V_{t}$, $U_{t}$, $W_{m_1}$ for $(m_1,m\/m_1)=1$ and $u_t$ for $t^2|m$ defined \nin this section extend from the space of Jacobi forms to the space of elliptic forms, \nand that the decompositions \\eqref{Jkmeigen} and \\eqref{primdecomp} remain true if we replace all the \n$J_{k}$s by $\\CE_{+}$s.\n \nFinally, we should warn the reader explicitly that, although the operators $U_{t}$ and $V_{t}$ also act on\nthe spaces $\\widetilde J_{k,*}$ and $\\widetilde \\CE_{\\pm,*}$ of weak Jacobi or weak elliptic forms, the operators \n$W_{m_1}$ and $u_{t}$ do not: the condition \\eqref{weakjacobi} is equivalent to saying that $C(\\Delta,\\ell)$ \nfor $|\\ell| \\le m$ vanishes whenever $\\Delta < -\\ell^{2}$, and this condition is preserved neither under \nthe permutation $\\; \\ell \\; \\mod \\; {2m} \\mapsto \\ell^{*}  \\; \\mod \\; {2m}$ defining $W_{m_{1}}$,\nnor by the summation conditions in  \\eqref{Ucech2}.  As a concrete example, the 7-dimensional space $\\widetilde J_{2,6}$\ncontains a 6-dimensional subspace of forms invariant under the involution $W_2$, but no anti-invariant form,\nand its image under the projection operator $\\pi_2^-=\\frac12(1-W_2)$ to anti-invariant forms is the 1-dimensional\nspace spanned by \n\\beta \\label{phi26min}  \\varphi_{2,6}^-(\\tau,z) \\= j'(\\tau)\\,\\frac{\\vartheta_{1}(\\tau,4z)}{\\vartheta_{1}(\\tau,2z)} \\= \n \\sum_{n,\\,r\\in{\\mathbb Z} } \\chi_{12}(r)\\,C_{2,6}(24n-r^2)\\;q^n\\,y^r \\,,\n\\end{equation}\nwith \n$$\n\\begin{tabular}{|c|ccccccc|}  \\hline     \n$\\Delta$ & $\\;\\,\\le-49$ & $-25$& $-1$& $23$ & 47 & 71 & $\\cdots$ \\\\\n       \\hline  $C_{2,6}(\\Delta)$ & \\;0 & $-1$ & $-1$ & 196882 & 43184401 & 2636281193 & $\\cdots$ \\\\\n\\hline  \\end{tabular}  \\quad ,\n$$\nwhich is a weakly holomorphic but not a weak Jacobi form.  (Specifically, for any $\\varphi\\in\\widetilde J_{2,6}$ we have\n$\\pi^-_2(\\varphi)=6\\,c_\\varphi(0,5)\\,\\varphi_{2,6}^-\\,$.)\n\n\n\n\n\n\n\n\n\n\n\\subsection{Quantum black holes and  Jacobi forms}\\label{BHandJF}\n\nJacobi forms usually arise in string theory as elliptic genera of two-dimensional superconformal field theories (SCFT) \nwith $(2, 2)$ or more worldsheet supersymmetry\\footnote{An SCFT with $(r, s) $ supersymmetries has $r$  \nleft-moving and $s$ right-moving supersymmetries.}.  We denote the superconformal field theory by $\\sigma({\\cal M})$ when it corresponds\nto a sigma model with a target manifold $\\mathcal{M}$.  Let $H$ be the Hamiltonian in the Ramond sector, and $J$ be the left-moving\n$U(1)$ $R$-charge. The elliptic genus $\\chi(\\tau, z; \\mathcal{M})$ is then defined as \\cite{Witten:1986bf,Alvarez:1987wg, Ochanine:1987}\na trace over the Hilbert space $\\mathcal{H}_R$ in the Ramond sector\n  \\beta\\label{ell}  \\chi(\\tau, z; \\mathcal{M}) \\= {\\rm Tr}_{\\mathcal{H}_R} \\bigl( (-1)^F q^H y^J  \\bigr) \\,, \\end{equation}\nwhere $F$ is the fermion number.\n\nAn elliptic genus so defined satisfies the modular transformation property\n(\\ref{modtransform}) as a consequence of modular invariance of the path integral.\nSimilarly, it satisfies the elliptic transformation property (\\ref{elliptic}) as\na consequence of spectral flow.  Furthermore, in a unitary SCFT, the positivity\nof the Hamiltonian implies that the elliptic genus is a weak Jacobi form. The\ndecomposition (\\ref{jacobi-theta}) follows from bosonizing the $U(1)$ current in\nthe standard way so that the contribution to the trace from the momentum modes of\nthe boson can be separated into the theta function (\\ref{thetadef}). See, for\nexample, \\cite{Kawai:1993jk,Moore:2004fg} for a discussion. This notion of the\nelliptic genus can be generalized to a $(0, 2)$ theory using a left-moving $U(1)$\ncharge $J$ which may not be an R- charge. In this case spectral flow is imposed\nas an additional constraint and follows from gauge invariance under large gauge\ntransformations \\cite{deBoer:2006vg,Gaiotto:2006wm,Kraus:2006nb,Denef:2007vg}.\n\nA particularly useful example in the present context is $ \\sigma(K3)$, which is a $(4,4)$ SCFT whose target space\nis a $K3$ surface. The elliptic genus is a topological invariant and is independent of the moduli of the $K3$. Hence, it can\nbe computed at some convenient point in the $K3$ moduli space, for example, at the orbifold point where the $K3$ is the Kummer\nsurface. At this point, the $\\sigma(K3)$ SCFT can be regarded as a $\\mathbb{Z}_2$ orbifold of the $\\sigma(T^4)$ SCFT, which \nis an SCFT with a torus $T^4$ as the target space. A simple computation using standard techniques of orbifold conformal field theory\n\\cite{Dixon:1985jw, Ginsparg:1988ui} yields \n  \\beta\\label{k3}  {\\chi}(\\tau,z; K3) \\= 2\\,\\varphi_{0,1}(\\tau,z) \\= 2 \\sum \\, C_0(4n- l^2)\\, q^n\\,y^l \\,.  \\end{equation}\nNote that for $z=0$, the trace (\\ref{ell}) reduces to the Witten index of the SCFT and correspondingly the elliptic genus reduces to\nthe Euler character of the target space manifold. In our case, one can readily verify from (\\ref{k3}) and (\\ref{phizero}) that\n$\\chi(\\tau, 0; K3)$ equals $24$, which is the Euler character of $K3$.\n\nA well-known physical application of Jacobi forms is in the context of the five-dimensional Strominger-Vafa black \nhole\\cite{Strominger:1996sh}, which is a bound state of $Q_1$ D1-branes, $Q_5$ D5-branes, $n$ units of momentum and $l $\nunits of five-dimensional angular momentum \\cite{Breckenridge:1996is}. The degeneracies $d_m(n,l)$ of such black holes depend \nonly on $m =Q_1 Q_5$. They are given by the Fourier coefficients $c(n, l)$ of the elliptic genus ${\\chi}(\\tau, z;\n\\textrm{Sym}^{m+1} (K3))$ of symmetric product of $(m+1)$ copies of $K3$-surface.\n\nLet us denote the generating function for the elliptic genera of symmetric products of K3 by\n\\beta\\label{defZ5d}  \\widehat Z(\\sigma,\\tau,z) := \\sum_{m=-1}^\\infty \\chi(\\tau,z; \\textrm{Sym}^{m+1}(K3))\\,p^m \\end{equation}\nwhere $\\chi_m(\\tau,z)$ is the elliptic genus of $ \\textrm{Sym}^m(K3)$. A standard orbifold computation \\cite{Dijkgraaf:1996xw} gives\n  \\beta\\label{ellprzero}  \\widehat Z(\\sigma,\\tau,z) = \\frac 1{p}\\prod_{r>0,\\; s\\ge0,\\; t} \\frac 1 {(1- q^s y^t p^r)^{2C_0(rs, t)}} \\end{equation}\nin terms of the Fourier coefficients $2 C_o$ of the elliptic genus of a single copy of $K3$.\n\nFor $z=0$, it can be checked that,  as expected, the generating function (\\ref{ellprzero}) for elliptic genera\nreduces to the generating function (\\ref{Zeuler2}) for Euler characters\n  \\beta\\label{Ztoeta}  \\widehat Z(\\sigma,\\tau, 0) = \\hat Z(\\sigma) = \\frac 1{\\Delta(\\sigma)} \\, .  \\end{equation}\n\n\\section{Siegel modular forms \\label{Siegel}}\n\n\\subsection{Definitions and examples of Siegel modular forms}\n\nLet $Sp(2, \\mathbb{Z})$ be the group of $(4\\times 4)$ matrices $g$ with integer entries satisfying $g J g^t =J$ where\n  \\beta    J \\equiv \\left( \\begin{array}{cc} 0 & -I_2 \\\\ I_2 & 0 \\\\ \\end{array} \\right)  \\end{equation}\nis the symplectic form. We can write the element $g$ in block form as\n  \\beta\\label{sp}  \\left(  \\begin{array}{cc}  A & B \\\\ C & D \\\\ \\end{array} \\right),\\end{equation}\nwhere $A, B, C, D$ are all $(2\\times 2)$ matrices with integer entries.\nThen the condition $g J g^t =J$ implies\n  \\beta\\label{cond}  AB^t=BA^t, \\qquad CD^t=DC^t, \\qquad AD^t-BC^t= \\mathbf 1\\, ,  \\end{equation}\nLet $\\mathbb{H}_2$ be the (genus two) Siegel upper half plane, defined as the set\nof $(2\\times 2)$ symmetric matrix $\\Omega$ with complex entries\n  \\beta\\label{period}   \\Omega = \\left( \\begin{array}{cc} \\tau & z \\\\  z & \\sigma \\\\ \\end{array} \\right) \\end{equation}\nsatisfying\n  \\beta\\label{cond1}   \\textrm{Im} (\\tau) > 0, \\quad \\textrm{Im} (\\sigma) > 0, \\quad \\det(\\Im(\\Omega)) > 0 \\, .  \\end{equation}\nAn element $g \\in Sp(2, \\mathbb{Z})$ of the form (\\ref{sp}) has a natural action on $\\mathbb{H}_2$ under which it is stable:\n  \\beta\\label{trans} \\Omega \\to (A \\Omega + B )(C\\Omega + D ) ^{-1}.  \\end{equation}\nThe matrix $\\Omega$ can be thought of as the period matrix of a genus two \nRiemann surface\\footnote{See \\cite{Gaiotto:2005hc,Dabholkar:2006xa,Banerjee:2008yu} for a\ndiscussion of the connection with genus-two Riemann surfaces.} on which there is a natural symplectic action of $Sp(2, \\mathbb{Z})$.\n\nA Siegel form $F(\\Omega)$ of weight $k$ is a holomorphic function $ \\mathbb{H}_2\\rightarrow \\mathbb{C}$ satisfying\n  \\beta\\label{trans2}   F\\bigl((A\\Omega + B)(C\\Omega + D) ^{-1}\\bigr) \\=  \\det(C\\Omega + D)^k \\, F (\\Omega)\\,.  \\end{equation}\nA Siegel modular form can be written in terms of its Fourier series\n  \\beta\\label{siegelfourier}   F(\\Omega) \\= \\sum_{n,\\,r,\\,m\\,\\in\\,{\\mathbb Z} \\atop r^2\\le4mn} a(n,r, m)  \\, q^n\\,y^r\\,p^m \\,.  \\end{equation}\nIf one writes this as \n  \\beta\\label{siegeljacobi}    F(\\Omega) = \\sum_{m=0}^{\\infty}  \\varphi^{\\rm F}_{m} (\\tau, z)  \\, p^m   \\end{equation}\nwith \n  \\beta\\label{siegeljacfourier}   \\varphi^F_m(\\tau, z) \\= \\sum_{n,\\,r} a(n,r, m)  \\, q^n\\,y^r  \\, ,  \\end{equation}\nthen each $\\varphi^{\\rm F}_{m} (m \\geq 0)$ is a Jacobi form of weight $k$ and index $m$.\n\nAn important special class of Siegel forms were studied by Maass which he called the\n{\\it Spezialschar}.  They have the property that $a(n,r,m)$ depends only on the\ndiscriminant $4 mn -r^2$ if $ (n,r,m)$ are coprime, and more generally\n  \\beta\\label{Cnrm}  a(n, r, m) \\= \\sum_{d \\mid (n,r,m),\\;\\,d>0} d^{k-1} C\\Bigl(\\frac{4 mn - r^2}{d^2}\\Bigr)  \\end{equation} \nfor some coefficients $C(N)$. Specializing to $m=1$, we can see that these numbers are simply the coefficients associated \nvia \\eqref{phiC} to the Jacobi form $\\varphi = \\varphi^{\\rm F}_1 \\in J_{k,1}$, and that \\eqref{Cnrm} says precisely that the \nother Fourier-Jacobi coefficients of $F$ are given by $\\varphi_{m}^{\\rm F} = \\varphi_1^{\\rm F}|V_{m}$ with $V_{m}$ as in \\eqref{defVl}. \nConversely, if $\\varphi$ is any Jacobi form of weight $k$ and index $1$ with Fourier expansion \\eqref{phiC}, then the function \n$F(\\Omega)$ defined by \\eqref{siegelfourier} and \\eqref{Cnrm} or by $F(\\Omega) = \\sum_{m=0}^{\\infty} \\bigl( \\varphi|V_{m} \\bigr)(\\tau,z)\\,p^{m}$\nis a Siegel modular form of weight $k$ with $\\varphi_1^{\\rm F} = \\varphi$.   The resulting map from $J_{k,1}$ to the Spezialschar is called the \nSaito-Kurokawa lift or \\textit{additive lift} since it naturally gives the sum representation of a Siegel form using the Fourier \ncoefficients of a Jacobi form as the input. (More information about the additive lift can be found in \\cite{Eichler:1985ja}.)\n\nThe example of interest to us is the Igusa cusp form $\\Phi_{10}$ (the unique cusp form of weight $10$) which is the Saito-Kurokawa\nlift of the Jacobi form $ \\varphi_{10, 1}$ introduced earlier, so that\n\\beta\\label{igusa-additive}  \n  \\Phi_{10}(\\Omega) \\= \\sum_{m=1}^{\\infty} \\bigl(\\varphi_{10,1}|V_{m}\\bigr) (\\tau,z) \\, p^{m}   \n  \\= \\sum_{n,\\,r,\\,m} a_{10}(n,r,m) \\,q^n\\,y^r\\,p^m \\,, \n \\end{equation}\nwhere $a_{10}$ is defined by (\\ref{Cnrm}) with $k=10$ in terms of the coefficients $C_{10}(d)$ given in Table \\ref{tablefcoeffs}.\n\n\n\nA Siegel modular form sometimes also admits a product representation, and can be obtained as\nBorcherds lift or {\\it multiplicative lift} of a weak Jacobi form of weight zero\nand index one. This procedure is in a sense an exponentiation of the additive\nlift and naturally results in the product representation of the Siegel form using\nthe Fourier coefficients of a Jacobi form as the input. Several examples of\nSiegel forms that admit product representation are known but at present there is\nno general theory to determine under what conditions a given Siegel form admits a product representation. \n\nFor the Igusa cusp form $\\Phi_{10}$, a product representation does exist. It was obtained by Gritsenko and Nikulin \n\\cite{GritNik1, GritNik2}\nas a multiplicative lift of the elliptic genus $ \\chi(\\tau,z; K3) =2\\varphi_{0,1}(\\tau,z) $ and is given by\n  \\beta \\label{final2}  \\Phi_{10}(\\Omega) \\= q y p \\prod_{(r, s, t) >0}\\bigl( 1 - q^s y^t p^r\\bigr)^{2C_0(4rs-t^2)}, \\end{equation}\nin terms of $C_0$ given by (\\ref{phizero}, \\ref{phik}).  Here the notation $(r,s,t)>0$ means that $r,\\,s,\\,t\\,\\in{\\mathbb Z}$ with\neither $r>0$ or $r=0$, $s >0$, or $r=s=0$, $t<0$.  In terms of the $V_{m}$, this can be rewritten in the \nform $\\; \\Phi_{10}= p \\, \\varphi_{10,1} \\, \\exp\\bigl(-2\\sum_{m=1}^{\\infty} (\\varphi_{0,1}|V_{m}) \\, p^{m}\\bigr)$. \n\n\n\n\\subsection{Quantum black holes and Siegel modular forms}\n\nSiegel forms occur naturally in the context of counting of quarter-BPS dyons.  The partition function for these dyons depends on three\n(complexified) chemical potentials $(\\sigma, \\tau, z)$, conjugate to the three $T$-duality invariant integers $ (m, n, \\ell)$ respectively \nand is given by\n\\beta\\label{igusa} \n \\textrm{Z}(\\Omega) \\= \\frac 1{\\Phi_{10}(\\Omega)} \\ .  \n\\end{equation}\n\nThe product representation of the Igusa form is particularly useful for the\nphysics application because it is closely related to the generating function for\nthe elliptic genera of symmetric products of $K3$ introduced earlier.  This is a\nconsequence of the fact that the multiplicative lift of the Igusa form is\nobtained starting with the elliptic genus of $K3$ as the input. Comparing the\nproduct representation for the Igusa form (\\ref{final2}) with \\eqref{ellprzero}, we get the relation:\n  \\beta\\label{4d5drel}  \\textrm{Z}(\\tau,z,\\sigma) \\= \\frac 1{\\Phi_{10}(\\tau,z,\\sigma)} \\= \\frac{\\widehat Z (\\tau,z,\\sigma)} {\\varphi_{10,1}(\\tau,z)} \\ .  \\end{equation}\n\nThis relation to the elliptic genera of symmetric products of $K3$ has a deeper\nphysical significance based on what is known as the 4d-5d lift\n\\cite{Gaiotto:2005gf}. The main idea is to use the fact that the geometry of the\nKaluza-Klein monopole in the charge configuration (\\ref{final charges}) reduces\nto five-dimensional flat Minkowski spacetime in the limit when the radius of the\ncircle ${\\tilde S}^1$ goes to infinity. In this limit, the charge $l$ corresponding to\nthe momentum around this circle gets identified with the angular momentum $l$ in\nfive dimensions. Our charge configuration (\\ref{final charges}) then reduces\nessentially to the Strominger-Vafa black hole \\cite{Strominger:1996sh} with\nangular momentum \\cite{Breckenridge:1996is} discussed in the previous subsection.\nAssuming that the dyon partition function does not depend on the moduli, we thus\nessentially relate $Z(\\Omega)$ to $\\hat Z(\\Omega)$. The additional factor in\n(\\ref{4d5drel}) involving $\\Phi_{10} (\\sigma,\\tau,z)$ comes from bound states of\nmomentum $n$ with the Kaluza-Klein monopole and from the center of mass motion of\nthe Strominger-Vafa black hole in the Kaluza-Klein geometry \\cite{Gaiotto:2005hc,David:2006yn}.\n\nThe Igusa cusp form has double zeros at $z=0$ and its $Sp(2,{\\mathbb Z})$ images. The partition\nfunction is therefore a \\textit{meromorphic} Siegel form (\\ref{trans2}) of weight\n$-10$ with double poles at these divisors. This fact is responsible for much of\nthe interesting physics of wall-crossings in this context as we explain in the next section.\n\nUsing  \\eqref{igusa-additive} or \\eqref{final2}, one can compute the coefficients $\\psi_{m}$ \nin the Fourier-Jacobi expansion~\\eqref{reciproigusa}. The first few, multiplied by $\\Delta$ and a suitable integer, are given by \n\\begin{eqnarray}\\label{firstpsim}\n\\Delta\\,\\psi_{-1} & \\= & A^{-1}\\;, \\nonumber \\\\ \n\\Delta\\,\\psi_{0\\;}  & \\= & 2\\,A^{-1}B\\;, \\nonumber \\\\\n4\\,\\Delta \\,\\psi_{1\\;} &\\=& 9\\,A^{-1}B^2 \\+ 3 E_4 A\\;, \\\\\n27 \\, \\Delta \\, \\psi_{2\\;}  & \\= & 50 \\, A^{-1} B^3 \\+  48 E_4 AB \\+ 10 E_6 A^2\\;,  \\nonumber  \\\\\n384 \\, \\Delta \\, \\psi_{3\\;}  & \\= & 475 \\,  A^{-1} B^4 \\+ 886 E_4 AB^2 \\+ 360 E_6 A^2 B \n\\+ 199 E_4^2 A^3\\;, \\nonumber \\\\\n72 \\, \\Delta \\, \\psi_{4\\;}  & \\= & 51 \\,  A^{-1} B^5 \\+ 155 E_4 A B^3 \\+  93 E_6 A^2 B^2 \n\\+ 102 E_4^2 A^3 B \\+ 31 E_4 E_6 A^4\\;. \\nonumber \n\\end{eqnarray}\n(Here $E_4$, $E_6$, $\\Delta$ are as in \\S\\ref{Modular} and $A=\\varphi_{-2,1}$, $B=\\varphi_{0,1}$ as in \\S\\ref{Jacobi}.) \nThe double zero of $\\Phi_{10}$ at $z=0$ is reflected by the denominator $A$ in the $A^{-1} B^{m+1}$ terms in these formulas. \nNote that, by \\eqref{4d5drel}, the right-hand side of \\eqref{firstpsim} multiplied by $A$ and divided by \na suitable denominator are equal to the elliptic genus of symmetric products of a $K3$ surface. For example,\n\\beta\n\\chi({\\rm Sym}^{4} K3; \\tau,z) \\= \\frac{1}{384} \\big(475 \\,  B^4 \\+ 886 E_4 A^{2} B^2 \\+ 360 E_6 A^3 B \n\\+ 199 E_4^2 A^4\\; \\big) \\, .\n\\end{equation}\n\n\n\\section{Walls and contours \\label{Contours}}\n\nGiven the partition function \\eqref{igusa}, one can extract the black hole\ndegeneracies from the Fourier coefficients. The three quadratic $T$-duality\ninvariants of a given dyonic state can be organized as a $2 \\times 2$ symmetric matrix\n\\beta\\label{matrix_charge_vector}  \n\\Lambda \\=\\begin{pmatrix} N\\!\\cdot\\! N & N\\!\\cdot\\! M \\\\  M\\!\\cdot\\!N & M\\!\\cdot\\! M \\end{pmatrix} \\= \\begin{pmatrix} 2n& \\ell \\\\ \\ell & 2 m \\end{pmatrix} \\; , \n\\end{equation}\nwhere the dot products are defined using the $O(22, 6; \\mathbb{Z})$ invariant metric $L$. The matrix $ \\Omega$ in (\\ref{igusa}) \nand (\\ref{period}) can be viewed as the matrix of complex chemical potentials conjugate to the charge matrix $\\Lambda$.\nThe charge matrix $\\Lambda$ is manifestly $T$-duality invariant. Under an $S$-duality transformation (\\ref{Sgroup}),  it transforms as\n\\beta\\label{lambdatransform} \n \\Lambda \\;\\to\\; \\gamma \\Lambda \\gamma^t  \n\\end{equation}\nThere is a natural embedding of this physical $S$-duality group $SL(2,{\\mathbb Z})$ into $Sp(2, \\mathbb{Z})$:\n\\beta\\label{sembed} \n \\left( \\begin{array}{cc}  A & B \\\\ C & D \\\\ \\end{array} \\right)\n    \\= \\left( \\begin{array}{cc} (\\gamma^t)^{-1} & \\textbf{0} \\\\ \\textbf{0} & \\gamma   \\\\ \\end{array} \\right)  \n    \\= \\left( \\begin{array}{cccc} d & -c & 0 & 0 \\\\ -b & a & 0 & 0 \\\\ 0 & 0 & a & b \\\\ 0 & 0 & c & d \\\\ \n    \\end{array} \\right) \\; \\in\\, Sp(2, \\mathbb{Z})\\,.  \n\\end{equation}\nThe embedding is chosen so that $\\Omega \\to (\\gamma^T)^{-1} \\Omega \\gamma^{-1}$ and ${\\rm Tr}(\\Omega \\cdot \\Lambda)$ in the Fourier integral \nis invariant. This choice of the embedding ensures that the physical degeneracies extracted from the Fourier\nintegral are $S$-duality invariant if we appropriately transform the moduli at the same time as we explain below.\n\nTo specify the contours, it is useful to define the\nfollowing moduli-dependent quantities. One can define the matrix of right-moving $T$-duality invariants\n  \\beta\\label{rightmatrix_charge_vector} \\Lambda_{R}  =\\begin{pmatrix} Q_R\\cdot Q_R & Q_R\\cdot P_R \\\\ P_R\\cdot Q_R & P_R\\cdot P_R \\end{pmatrix} \\;, \\end{equation}\nwhich depends both on the integral charge vectors $N, M$ as well as the $T$-moduli $\\mu$.  One can then define \ntwo matrices naturally associated to the $S$-moduli $S = S_1+iS_2$ and the $T$-moduli $\\mu$ respectively by\n  \\beta\\label{ST_moduli} {\\cal S} = \\frac 1{S_2} \\begin{pmatrix} |S|^2 & S_1\\\\\\ S_1& 1 \\end{pmatrix}\\quad, \\quad {\\cal T} \n  \\= \\frac{ \\Lambda_{R}}{{|\\det (\\Lambda_{R})|^\\frac 12}}\\;.  \\end{equation}\nBoth matrices are normalized to have unit determinant. In terms of them, we\ncan construct the moduli-dependent `central charge matrix'\n  \\beta\\label{def_Z_vec} {\\mathcal Z} = {{|\\det(\\Lambda_{R})|^\\frac 14}}\\, \\big({\\cal S}+{\\cal T} \\big)\\,, \\end{equation}\nwhose determinant equals the BPS mass\n  \\beta M_{Q,P} \\= |\\det{\\mathcal Z}|\\;.  \\end{equation}\nWe define \n  \\beta \\widetilde \\Omega \\equiv \\left( \\begin{array}{cc} \\sigma &  -z \\\\  -z & \\tau \\\\ \\end{array} \\right)\\;. \\end{equation}\nThis is related to $\\Omega$ by an $SL(2,{\\mathbb Z})$ transformation\n  \\beta  \\tilde\\Omega \\=  \\widehat S \\Omega  \\widehat S^{-1}  \\quad \\text{where} \\quad \n   \\widehat S \\=  \\left( \\begin{array}{cc} 0 &  1 \\\\  -1 & 0 \\\\ \\end{array} \\right)\\,, \\end{equation}\nso that, under a general $S$-duality transformation $\\gamma$, we have the transformation $\\tilde\\Omega\\rightarrow \\gamma\\widetilde\\Omega \\gamma^T$\nas  $\\Omega \\to (\\gamma^T)^{-1} \\Omega \\gamma^{-1}$.\n\n\nWith these definitions,  $\\Lambda, \\Lambda_{R}, {\\cal Z}$ and $\\tilde \\Omega$ all transform as $X \\to \\gamma X \\gamma^T$ under an \n$S$-duality transformation (\\ref{Sgroup}) and are invariant under $T$-duality transformations. \nThe moduli-dependent Fourier contour can then be  specified in a duality-invariant fashion by\\cite{Cheng:2007ch}\n  \\beta\\label{contour}  {\\cal C} = \\{\\mathrm{Im}  (\\widetilde \\Omega) = \\varepsilon^{-1} {\\cal Z};\\quad 0 \\leq \\Re(\\tau), \\, \\Re(\\sigma), \\, \\Re(z) < 1 \\},   \\end{equation}\nwhere $\\varepsilon \\rightarrow 0^+$. For a given set of charges, the contour depends on the moduli $S, \\mu$ through the definition of the central\ncharge vector (\\ref{def_Z_vec}). The degeneracies $d(n, \\ell,m)\\lvert_{S, \\mu}$ of states with the $T$-duality invariants $(n,\\ell,m)$\nat a given point $(S,\\mu)$ in the moduli space are then given by\\footnote{The physical degeneracies have an additional multiplicative \nfactor of  $(-1)^{\\ell +1}$ which we omit here for simplicity of notation in later chapters.} \n  \\beta\\label{inverse}  d(n, \\ell, m)\\lvert_{S, \\mu} = \\int_{{\\cal C}} \\, e^{-i\\pi {\\rm Tr} (\\Omega \\cdot \\Lambda)}\\, \\, {\\textrm{Z}(\\Omega)} \\, d^3 \\Omega \\, . \\end{equation}\n\nThis contour prescription thus specifies how to extract the degeneracies from the partition \nfunction for a given set of charges and in any given region of the moduli space. In particular, \nit also completely summarizes all wall-crossings as one moves around in the moduli space for a fixed set of charges. \nEven though the indexed partition function has the same functional form throughout the moduli space, the spectrum is moduli\ndependent because of the moduli dependence of the contours of Fourier integration and the pole structure of the partition function. \nSince the degeneracies depend on the moduli \\textit{only} through the dependence of the contour ${\\cal C}$, moving around in the \nmoduli space corresponds to deforming the Fourier contour. This does not change the degeneracy except when\none encounters a pole of the partition function. Crossing a pole corresponds to crossing a wall in the moduli \nspace. The moduli space is thus divided up into domains separated by ``walls of marginal stability.'' In each domain the degeneracy\nis constant but it jumps upon crossing a wall as one goes from one domain to the other. The jump in the degeneracy has a nice mathematical \ncharacterization. It is simply given by the residue at the pole that is crossed while deforming the Fourier contour in going from one domain to the other.\n\nWe now turn to the degeneracies of single-centered black holes. Given the\n$T$-duality invariants $\\Lambda$, a single centered black hole solution is known to\nexist in all regions of the moduli space as long as $\\det (\\Lambda)$ is large and\npositive. The moduli fields can take any values $(\\lambda_ \\infty, \\mu_ \\infty)$ at\nasymptotic infinity far away from the black hole but the vary in the black hole geometry.  Because of the attractor phenomenon\n\\cite{Ferrara:1995ih,Strominger:1996kf}, the moduli adjust themselves so that near the horizon of the black hole of charge\n$\\Lambda$ they get attracted to the values $(\\lambda_*(\\Lambda), \\mu_*(\\Lambda))$ which are determined by the requirement that the\ncentral charge $ \\mathcal Z_*(\\Lambda)$ evaluated using these moduli becomes proportional to $\\Lambda$. These attractor values are\nindependent of the asymptotic values and depend only on the charge of black hole. We call these moduli the\nattractor moduli. This enables us to define the \\textit{attractor contour} for a\ngiven charge $\\Lambda$ by fixing the asymptotic moduli to the attractor values corresponding to this charge. In this case\n  \\beta\\label{att}  {\\cal Z }(\\lambda_\\infty, \\mu_\\infty) \\=  {\\cal Z }(\\lambda_*(\\Lambda),\\mu_*(\\Lambda)) \\sim \\Lambda \\end{equation}\nand we have the attractor contour\n  \\beta\\label{contourattract} {\\cal C}_* \\= \\{\\mathrm{Im} (\\widetilde \\Omega) \\= \\varepsilon^{-1} {\\Lambda};\\quad 0 \\leq \\Re(\\tau), \\, \\Re(\\sigma), \\, \\Re(z) < 1 \\} \\end{equation}\nwhich depends only on the integral charges and not on the moduli. The significance of the attractor\nmoduli in our context stems from the fact if the asymptotic moduli are tuned to these values for given\n$(n,\\ell,m)$, then only single-centered black hole solution exists.  The degeneracies $d^*(n,\\ell,m)$ obtained using the attractor contour\n  \\beta\\label{inverse3}   d^*(n, \\ell,m) = \\int_{ {\\cal C}_*}  \\, e^{-i\\pi {\\rm Tr} (\\Omega \\cdot \\Lambda)}\\, \\, {\\textrm{Z}(\\Omega)} \\, d^3 \\Omega \\end{equation}\nare therefore expected to be the degeneracies of the immortal single-centered black holes.\n\n\n\\section{Mock modular forms \\label{Mock}}\n\nMock modular forms are a relatively new class of modular objects, although individual examples had been\nknown for some time. They were first isolated explicitly by S. Zwegers in his thesis \\cite{Zwegers:2002}\nas the explanation of the ``mock theta functions'' introduced by Ramanujan in his famous last letter to\nHardy.  An expository account of this work can be found in \\cite{Zagier:2007}. \n\nRamanujan's mock theta functions are examples of what we will call ``pure'' mock modular forms. \nBy this we mean a holomorphic function in the upper half plane which transforms under\nmodular transformations almost, but not quite, as a modular form. The\nnon-modularity is of a very special nature and is governed by another holomorphic\nfunction called the \\textit{shadow} which is itself an ordinary modular form. We will describe this \nmore precisely in \\S\\ref{basicdefs}  and give a number of examples.  In \\S{\\ref{MockJacobi}}, \nwe introduce a notion of mock Jacobi forms (essentially, holomorphic functions of  $\\tau$ and $z$ \nwith theta expansions like that of usual Jacobi forms, but in which the coefficients $h_{\\ell}(\\tau)$ \nare mock modular forms) and show how all the examples given in the first subsection \noccur naturally as pieces of mock Jacobi forms. Finally in \\S\\ref{MMMF} we define a more \ngeneral notion of mock modular forms in which the shadow is replaced by a sum of products \nof holomorphic and anti-holomorphic modular forms. This has the advantage that there are now \nmany examples of strongly holomorphic mock modular forms, whereas pure mock modular forms \nalmost always have a negative power of $q$ at at least one cusp, and also that almost all of the examples \noccurring naturally in number theory, algebraic geometry, or physics are of this more general \n``mixed'' type, with the pure mock modular forms arising as quotients of mixed mock modular forms \nby eta products. \n\n\n\\subsection{Pure mock modular forms \\label{basicdefs}}\n\nWe define a (weakly holomorphic) {\\it pure mock modular form} of weight $k\\in \\frac12 {\\mathbb Z}$ as the first member \nof a pair $(h,g)$, where\n\\begin{enumerate}\n\\item $h$ is a holomorphic function in $\\relax{\\rm I\\kern-.18em H}$ with at most exponential growth at all cusps, \n\\item the function $g(\\tau)$, called the \\textit{shadow} of $h$, is a holomorphic\\footnote{One can also consider the case where the shadow is allowed\nto be a weakly holomorphic modular form, but we do not do this since none of our examples will be of this type.} modular form of weight $2-k\\,$, \nand \n\\item the sum $\\widehat h := h\\+g^*$, called the {\\it completion} of $h$, transforms like a holomorphic modular form of weight $k$,\n   {\\it i.e.} $\\widehat h (\\tau)\/\\theta (\\tau)^{2k}$ is invariant under $\\tau \\to \\gamma \\tau$  for all $ \\tau \\in \\relax{\\rm I\\kern-.18em H} $ and for all\n   $\\gamma$ in some congruence subgroup of $SL(2,{\\mathbb Z})$. \n\\end{enumerate} \nHere $g^*(\\tau)$, called the {\\it non-holomorphic Eichler integral}, is a solution of the differential equation\n\\beta\\label{starinv} (4\\pi\\tau_2)^k\\,\\frac{\\partial g^*(\\tau)}{\\partial \\overline{\\tau}} \\= -2\\pi i\\;\\overline{g(\\tau)} \\, . \\end{equation}\nIf $g$ has the Fourier expansion $g(\\tau)=\\sum_{n\\ge0}\\;b_n\\,q^n$, we fix the choice of $g^*$ by setting \n\\beta\\label{defstar} \n  g^*(\\tau)  \\=  \\overline b_0\\,\\frac{(4\\pi\\tau_2)^{-k+1}}{k-1} \\+ \\sum_{n>0} \\;n^{k-1}\\,\\overline b_n \\;\\Gamma(1-k,4\\pi n\\tau_2)\\;q^{-n}\\, , \n\\end{equation}\nwhere $\\tau_2 = \\rm{Im}(\\tau)$ and $\\Gamma(1-k,x) = \\int_x^{\\infty} t^{-k}\\,e^{-t}\\,dt$ denotes the incomplete gamma function, and where \nthe first term must be replaced by $ -\\overline b_0\\,\\log(4\\pi\\tau_2)\\,$ if $k=1$. Note that the series in~\\eqref{defstar} converges despite \nthe exponentially large factor $q^{-n}$ because $\\Gamma(1-k,x)= O(x^{-k}e^{-x})\\,$. If we assume either that $k>1$ or that $b_{0}=0$,\nthen we can define $g^*$ alternatively by the integral  \n\\beta\\label{Lkintrep} g^*(\\tau) = \\biggl(\\frac i{2\\pi}\\Bigr)^{k-1} \\int_{-\\overline{\\tau}}^{\\infty} (z+ \\tau)^{-k} \\ \\overline{g(-\\overline{z})}\\; dz \\;. \\end{equation}\n(The integral is independent of the path chosen because the integrand is holomorphic in $z$.) \nSince $h$ is holomorphic, \\eqref{starinv} implies that the completion of $h$ is related to its shadow by\n    \\beta\\label{ddtbarh}   (4\\pi\\tau_2)^k\\,\\,\\frac{\\partial \\widehat h(\\tau)}{\\partial \\overline{\\tau}} \\= -2\\pi i\\;\\overline{g(\\tau)}\\;.  \\end{equation}\n\nWe denote by $ {\\mathbb M}_{k|0}^{\\,!}$ the space of weakly holomorphic pure mock modular forms\nof weight $k$ and arbitrary level. (The ``0'' in the notation corresponds to the word ``pure'' \nand will be replaced by an arbitrary integer or half-integer in \\S\\ref{MMMF}.)  \nThis space clearly contains the space $M_{k}^{\\,!}$ of ordinary weakly holomorphic\nmodular forms (the special case $g=0$, $h=\\widehat h$) and we have an exact sequence  \n  \\beta\\label{exseqmmf} 0 \\longrightarrow M_{k}^{\\,!} \\longrightarrow {\\mathbb M}_{k|0}^{\\,!} \\stackrel{{\\cal S}}{\\longrightarrow} \\overline {M_{2-k}}  \\; \\, ,\\end{equation}\nwhere the {\\it shadow map} ${\\cal S}$ sends $ h$ to $\\overline{g} $.\\footnote{We will use the word ``shadow\"  to denote either $g(\\tau)$\nor $\\overline{g(\\tau)}$, but the shadow {\\it map}, which should be linear over ${\\mathbb C}$, always sends $h$ to $\\overline g$, the complex conjugate \nof its holomorphic shadow.  We will also often be sloppy and say that the shadow of a certain mock modular form ``is'' some modular\nform $g$ when in fact it is merely proportional to $g$, since the constants occurring are arbitrary and are often messy.} \n\n\nIn the special case when the shadow $g$ is a unary theta series as in \\eqref{unary1} or \\eqref{unary2} (which can only\nhapen if $k$ equals 3\/2 or 1\/2, respectively), the mock modular form $h$ is called a {\\it mock theta function}.\nAll of Ramanujan's examples, and all of ours in this paper,  are of this type.   In these cases the incomplete gamma functions\nin~\\eqref{defstar} can be expressed in terms of the complementary error function $\\,\\text{erfc}(x)=2\\pi^{-1\/2}\\int_x^\\infty e^{-t^2}dt\\;$:\n\\beta \\label{erfc}  \\Gamma\\bigl(-\\frac12,\\,x\\bigr)\\= \\frac2{\\sqrt x}\\,e^{-x}\\;-\\;2\\,\\sqrt\\pi\\;\\text{erfc}\\bigl(\\sqrt x\\bigr)\\;, \\qquad\n     \\Gamma\\bigl(\\frac12,\\,x\\bigr)\\= \\sqrt\\pi\\;\\text{erfc}\\bigl(\\sqrt x\\bigr)\\;.  \\end{equation}\n\n\n\nA very simple, though somewhat artificial, example of a mock modular form is the weight~2 Eisenstein series $E_2(\\tau)$ \nmentioned in~\\S\\ref{modularbasic}, which is also a quasimodular form. Here the shadow $g(\\tau)$ is a constant and its \nnon-holomorphic Eichler integral $g^*(\\tau)$ simply a multiple of~$1\/\\tau_2\\,$.  \nThis example, however, is atypical, since most quasimodular forms (like $E_2(\\tau)^2$) are not mock modular forms.\nWe end this subsection by giving a number of less trivial examples. Many more will occur later. \n\n\\smallskip \n\\noindent {\\bf Example 1.}  In Ramanujan's famous last letter to Hardy in 1920, he gives 17 examples of mock theta functions, though\nwithout giving any complete definition of this term.  All of them have weight $1\/2$ and are given as $q$-hypergeometric\nseries. A typical example (Ramanujan's second mock theta function of ``order 7\"\\,---\\,a notion that he also does not define) is \n  \\beta\\label{Ramaeg}  {\\cal F}_{7,2}(\\tau) = - q^{-25\/168}  \\, \\sum_{n=1}^{\\infty} \\frac{q^{n^2}}{(1-q^{n}) \\cdots (1 - q^{2n-1})}\n  \\= - q^{143\/168} \\, \\bigl( 1 + q + q^2 + 2 q^3 + \\cdots \\bigr) \\ .  \\end{equation}\nThis is a mock theta function of weight $1\/2$ on $\\Gamma_0(4) \\cap \\Gamma(7) $ with shadow the unary theta series \n  \\beta\\label{nequiv2} \\sum_{n\\equiv 2 \\, ({\\rm mod}\\;7)}  \\chi_{12}(n)\\,n\\,q^{n^2\/168}\\,, \\end{equation}\nwith $\\chi_{12}(n)$ as in \\eqref{defchi12}.  The product $\\eta(\\tau){\\cal F}_{7,2}(\\tau)$ is a strongly holomorphic mixed mock modular\nform of weight $(1,1\/2)$, and by an identity of Hickerson~\\cite{Hickerson} is equal to an indefinite theta series:\n  \\beta\\label{Hickerson} \\eta(\\tau) \\, {\\cal F}_{7,2} (\\tau) \\= \\sum_{r,\\,s\\,\\in\\,{\\mathbb Z} + \\frac5{14}} \n  \\frac12\\bigl({\\rm sgn}(r) + {\\rm sgn}(s)\\bigr)\\, (-1)^{r-s} \\, q^{(3r^2 +8rs+3s^2)\/2} \\ . \\end{equation}\n\n\\smallskip\n\\noindent {\\bf Example 2.} \nThe second example is the generating function of the Hurwitz-Kronecker class numbers $H(N)$. \nThese numbers are defined for $N>0$ as the number of $PSL(2,{\\mathbb Z})$-equivalence classes of \nintegral binary quadratic forms of discriminant $-N$, weighted by the reciprocal of the number of \ntheir automorphisms (if $-N$ is the discriminant of an imaginary quadratic field $K$ other than ${\\mathbb Q}(i)$ or \n${\\mathbb Q}(\\sqrt{-3})$, this is just the class number of $K$), and for other values of $N$ by \n$H(0) = -1\/12$ and $H(N)=0$ for $N<0$. It was shown in  \\cite{Zagier:1975} that the function \n  \\beta\\label{classnum}  {\\bf H}(\\tau) \\;:=\\; \\sum_{N=0}^{\\infty} H(N)\\,q^{N} \n \\= -\\frac 1{12} \\+ \\frac13 q^3 \\+ \\frac12 q^4 \\+ q^7 \\+ q^8 \\+ q^{11} \\+ \\cdots  \\end{equation}\nis a mock  modular form of weight $3\/2$ on $\\Gamma_0(4)$, with shadow the classical theta function $\\theta (\\tau)  = \\sum q^{n^2}$. \nHere ${\\bf H}(\\tau)$ is strongly holomorphic, but this is very exceptional. In fact, up to minor variations it is essentially the {\\it only} \nknown non-trivial example of a strongly holomorphic pure mock modular form, which is why we will introduce mixed mock modular forms below.\nAs mentioned earlier, this mock modular form has appeared in the physics literature in the context of topologically twisted supersymmetric\nYang-Mills theory on $\\mathbb{CP}{^2}$ \\cite{Vafa:1994tf}.\n\n\n\\smallskip\n\\noindent {\\bf Example 3. }\nThis example is taken from \\cite{Zagier:2007}.  Set\n  \\beta\\label{defF62}  F_2^{(6)}(\\tau) \\= - \\sum_{r>s>0} \\chi_{12}(r^2-s^2) \\, s \\, q^{rs\/6} \\= q + 2 q^2 +  q^3 + 2 q^4 - q^5 + \\cdots \\end{equation}\nwith $\\chi_{12}$ as in \\eqref{defchi12}, and let $E_2(\\tau)$ be the quasimodular Eisenstein series of weight~2 on $SL(2,{\\mathbb Z})$ as defined\nin \\S\\ref{modularbasic}.  Then the function \n \n \n   \\beta\\label{defh6}  h^{(6)}(\\tau) \\= \\frac{12 F_2^{(6)}(\\tau) - E_2(\\tau)}{\\eta(\\tau)} \n   \\= q^{-1\/24} \\, \\bigl(-1 + 35 q + 130q^2 + 273q^3 + 595q^4 + \\cdots \\bigr) \\end{equation}\nis a weakly holomorphic mock modular form of weight $3\/2$ on $SL(2,{\\mathbb Z})$ with shadow proportional to $\\eta(\\tau)$. More generally, if we define \n  \\beta\\label{defF6k}  F_k^{(6)}(\\tau) \\=  - \\sum_{r>s>0} \\chi_{12}(r^2-s^2) \\, s^{k-1} q^{rs\/6}  \\qquad (\\text{$k>2$, $k$ even})\\,,  \\end{equation}\nthen for all $\\nu \\ge 0$ we have \n  \\beta\\label{hetaRC6} \\frac{24^{\\nu}}{\\left({2\\nu \\atop \\nu }\\right)}\\,[h^{(6)},\\eta]_\\nu \n   \\=12 F_k^{(6)} \\,-\\, E_k \\+ {\\rm cusp \\; form \\; of \\; weight \\;} k \\ , \\quad k = 2\\nu +2 \\ , \\end{equation}\nwhere $ [h^{(6)}, \\eta ]_{\\nu}$ denotes the $\\nu$th Rankin-Cohen bracket of the mock modular form $h^{(6)}$ and the modular \nform $\\eta$ in weights $(3\/2,1\/2)$. This statement, and the similar statements for other mock modular forms which come later, \nare proved by the method of holomorphic projection, which we do not explain here, and are intimately connected with \nthe mock Jacobi forms introduced in the next subsection. \nThat connection will also explain the superscript ``6\" in~\\eqref{defF62}--\\eqref{hetaRC6}.\n\n\\smallskip\n\\noindent {\\bf Example 4.}  Our last example is very similar to the preceding one. Set \n\\beta\\label{defF22}  F_2^{(2)}(\\tau) \\=  \\sum_{r>s>0 \\atop r-s \\, \\rm odd} (-1)^r\\,s\\,q^{rs\/2} \\= q + q^2 -  q^3 +  q^4 - q^5 + \\cdots \\, . \\end{equation}\nThen the function\n  \\beta\\label{defh2}  h^{(2)}(\\tau) \\= \\frac{24 F_2^{(2)}(\\tau) - E_2(\\tau)}{\\eta(\\tau)^3}  \n    \\= q^{-1\/8} \\, \\bigl(-1 + 45 q + 231q^2 + 770q^3 + 2277q^4  + \\cdots \\bigr)\\end{equation}\nis a weakly holomorphic mock modular form of weight $1\/2$ on $SL(2,{\\mathbb Z})$ with shadow proportional to $\\eta(\\tau)^3$ and, as in Example~3, if we set  \n  \\beta\\label{defF2k}   F_k^{(2)}(\\tau) \\=  - \\sum_{r>s>0 \\atop r-s \\, \\rm odd} (-1)^{r} \\, s^{k-1} \\, q^{rs\/2}  \\qquad (\\text{$k>2$, $k$ even})\\,,\\end{equation}\nthen for all $\\nu \\ge 0$ we have \n  \\beta\\label{hetaRC2} \\frac{8^\\nu}{\\left({2\\nu \\atop \\nu }\\right)}\\,[h^{(2)},\\eta^3]_{\\nu} \n   \\= 24 F^{(2)}_k \\,-\\, E_k \\+ {\\rm cusp \\; form \\; of \\; weight \\;} k \\ , \\quad k = 2\\nu +2 \\ , \\end{equation}\nwhere $[h^{(2)}, \\eta^3]_\\nu$ denotes the Rankin-Cohen bracket in weights $(1\/2,3\/2)$. \n\n\nIn fact, the mock modular form $h^{(2)}$, with the completely different definition \n\\beta\\label{oogfor}\nh^{(2)}(\\tau) \\= \\frac{\\vartheta_2(\\tau)^4 - \\vartheta_4(\\tau)^4}{\\eta(\\tau)^3} \\, - \\, \\frac{24}{\\vartheta_3(\\tau)} \\, \n\\sum_{n\\in{\\mathbb Z}} \\frac{q^{n^2\/2-1\/8}}{1+q^{n-1\/2}} \\, , \n\\end{equation}\narose from the works of Eguchi-Taormina \\cite{Eguchi:1988af} and \nEguchi--Ooguri--Taormina--Yang \\cite{EOTY:1989} in the late 1980's \nin connection with characters of the ${\\cal N}=4$ superconformal algebra in two dimensions and the elliptic genus of $K3$ surfaces, and\nappears explicitly in the work of Ooguri \\cite{Ooguri:1989fd} (it is $q^{-1\/8}(-1+F(\\tau)\/2)$ in Ooguri's notation) and\nWendland~\\cite{Wendland:2000}. It has recently aroused considerable interest because of the ``Mathieu moonshine\" discovery by Eguchi, \nOoguri and Tachikawa (see~\\cite{Eguchi:2010ej, Cheng:2010pq}) that the coefficients 45, 231, 770, \\dots\\ in~\\eqref{defh2} are \nthe dimensions of certain representations of the Mathieu group~$M_{24}\\,$.  \nThe equality of the right-hand sides of \\eqref{defh2} and \\eqref{oogfor}, which is not a priori \nobvious, follows because both expressions define mock modular forms whose shadow is the \nsame multiple of $\\eta(\\tau)^{3}$, and the first few Fourier coefficients agree. \n\n\n\\subsection{Mock Jacobi forms \\label{MockJacobi}}\n\nBy a (pure) {\\it mock Jacobi form}\\footnote{We mention that there are related constructions and definitions in the \nliterature. The weak Maass-Jacobi forms of~\\cite{BringRicht}\ninclude the completions of our mock Jacobi forms, but in general are non-holomorphic in $z$ as \nwell as in $\\tau$. See also~\\cite{Choie} and~\\cite{BringmannRaum}. The functions discussed in this paper\nwill be holomorphic in~$z$.} \n(resp.~{\\it weak mock Jacobi form}) of weight $k$ and index $m$  we will mean a \nholomorphic function $\\varphi$ on $\\relax{\\rm I\\kern-.18em H} \\times {\\mathbb C}$ that satisfies the elliptic transformation property~\\eqref{elliptic},\nand hence has a Fourier expansion as in~\\eqref{fourierjacobi} with the periodicity property~\\eqref{cnrprop}\nand a theta expansion as in~\\eqref{jacobi-theta}, and that also satisfies the same cusp conditions~\\eqref{holjacobi}\n(resp.~\\eqref{weakjacobi}) as in the classical case, but in which the modularity property with respect to the\naction of $SL(2,{\\mathbb Z})$ on $\\relax{\\rm I\\kern-.18em H}\\times{\\mathbb Z}$ is weakened: the coefficients $h_{\\ell}(\\tau)$ in \\eqref{jacobi-theta}\nare now mock modular forms rather than modular forms of weight $k-\\frac12$, and the modularity property of\n$\\varphi$ is that the {\\it completed} function \n  \\beta\\label{defphihat} \\widehat \\varphi(\\tau,z) \\;=\\!\\sum_{\\ell\\,\\in\\,{\\mathbb Z}\/2m{\\mathbb Z}} \\widehat h_\\ell(\\tau) \\, \\vartheta_{m,\\ell}(\\tau,z)\\,,  \\end{equation}\nrather than $\\varphi$ itself, transforms according to \\eqref{modtransform}.  If $g_\\ell$ denotes the shadow of $h_l$, then we have\n  $$ \\widehat\\varphi(\\tau,z) \\;= \\! \\varphi(\\tau,z)\\+\\sum_{\\ell\\,\\in\\,{\\mathbb Z}\/2m{\\mathbb Z}} g^*_\\ell(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z)$$\nwith $g^*_\\ell$ as in \\eqref{defstar} and hence, by \\eqref{starinv},   \n \\beta \\psi(\\tau,z)\\;:=\\; \\tau_2^{k-1\/2}\\,\\frac{\\partial}{\\partial\\overline\\tau}\\widehat\\varphi(\\tau,z)\\;\\doteq\\;\\sum_{\\ell\\,\\in\\,{\\mathbb Z}\/2m{\\mathbb Z}} \\overline{g_\\ell(\\tau)}\\,\\vartheta_{m,\\ell}(\\tau,z)\\,. \\end{equation}\n(Here $\\doteq$ indicates an omitted constant.)  The function $\\psi(\\tau,z)$ is holomorphic in $z$, satisfies the same elliptic\ntransformation property~\\eqref{elliptic} as $\\varphi$ does (because each $\\vartheta_{m,\\ell}$ satisfies this), satisfies the heat equation\n$\\,\\bigl(8\\pi im\\,\\frac{\\partial}{\\partial\\tau}-\\frac{\\partial^2}{\\partial z^2}\\bigr)\\psi=0$  (again, because each $\\vartheta_{m,\\ell}$ does), and, by virtue\nof the modular invariance property of $\\widehat\\varphi(\\tau,z)$, also satisfies the transformation property\n \\beta\\label{modtransformpsi}  \\psi(\\frac{a\\tau+b}{c\\tau+d}, \\frac{z}{c\\tau+d}) \\=   |c\\tau+d|\\,(c\\overline\\tau+d)^{2-k}\\,e^{\\frac{2\\pi i m c z^2}{c\\tau +d}}\n  \\, \\psi(\\tau,z)  \\qquad \\forall\\;\\left(\\begin{array}{cc} a&b\\\\ c&d \\end{array} \\right) \\,\\in\\, SL(2;{\\mathbb Z}) \\end{equation}\nwith respect to the action of the modular group.  These properties say precisely that $\\psi$ is a {\\it skew-holomorphic Jacobi form} of weight~$3-k$\nand index~$m$ in the sense of Skoruppa \\cite{Skoruppa1,Skoruppa2}, \nand the above discussion can be summarized by saying that we have an exact sequence\n  \\beta\\label{exseqJmf} 0 \\longrightarrow J_{k,m}^{\\text{weak}} \\longrightarrow {\\mathbb J}_{k|0,m}^{\\text{weak}}\n          \\stackrel{{\\cal S}}{\\longrightarrow} J_{3-k,m}^{\\textrm{skew}} \\end{equation}\n(and similarly with the word ``weak\" omitted), where ${\\mathbb J}_{k|0,m}^{\\textrm{weak}}$ (resp.~${\\mathbb J}_{k|0,m}$) denotes the space of weak \n(resp.~strong) pure mock Jacobi forms and the ``shadow map\" ${\\cal S}$ \nsends\\footnote{This shadow map was introduced in the context of weak Maass-Jacobi forms in~\\cite{BringRicht}.}\n $\\varphi$ to $\\psi$.\n\nIt turns out that most of the classical examples of mock theta functions occur as the components of a\nvector-valued mock modular form which gives the coefficients in the theta series expansion of a mock Jacobi\nform.  We illustrate this for the four examples introduced in the previous subsection. \n\n\n\n\\smallskip\n\\noindent {\\bf Example 1.}   The function ${\\cal F}_{7,2}(\\tau)$ in the first example of \\S\\ref{basicdefs} is actually one of three \n``order~7 mock theta functions\" $\\{{\\cal F}_{7,j}\\}_{j=1,2,3}$ defined by Ramanujan, each given by a \n$q$-hypergeometric formula like \\eqref{Ramaeg}, each having a shadow  $\\Theta_{7,j}$ like in \\eqref{nequiv2}  \n(but now with the summation over $n \\equiv j$ rather than $n \\equiv 2$ modulo~7), and each  satisfying an indefinite \ntheta series identity like \\eqref{Hickerson}.  We extend $\\{{\\cal F}_{7,j}\\}$ to all $j$ by defining it to be an odd periodic\nfunction of $j$ of period $7$, so that the shadow of ${\\cal F}_{7,j}$ equals $\\Theta_{7,j}$ for all~$j\\in{\\mathbb Z}$. Then the function \n  \\beta {\\cal F}_{42}(\\tau,z) \\= \\sum_{\\ell\\;({\\rm mod}\\;84)} \\chi_{12}(\\ell) \\, {\\cal F}_{7,\\ell} (\\tau) \\, \\vartheta_{42,\\ell} (\\tau,z) \\ \\end{equation}\nbelongs to ${\\mathbb J}_{1,42}^{\\text{weak}}$. The Taylor coefficients $\\xi_{\\nu}$ as defined in equation \\eqref{defxinodd} are \nproportional to $\\sum_{j=1}^3 \\bigl[{\\cal F}_{7,j},\\Theta_{7,j}\\bigr]_{\\nu}$ and have the property\nthat their completions  $\\widehat \\xi_{\\nu} = \\sum_{j=1}^3 \\bigl[\\widehat {\\cal F}_{7,j},\\Theta_{7,j}\\bigr]_{\\nu}$ \ntransform like modular forms of weight $2 \\nu +2$ on the full modular group $SL(2,{\\mathbb Z})$. \n\n \n\\smallskip\n\\noindent {\\bf Example 2.}  Set ${\\cal H}_0(\\tau)=\\sum_{n=0}^\\infty H(4n)q^n$ and \n${\\cal H}_1(\\tau)=\\sum_{n=1}^\\infty H(4n-1)q^{n-\\frac14}$ so that ${\\cal H}_{0}(\\tau)+{\\cal H}_{1}(t) = {\\bf H}(\\tau\/4)$. \nThen the function\n  \\beta\\label{classH}  {\\cal H}(\\tau,z) \\= {\\cal H}_0(\\tau)\\vartheta_{1,0}(\\tau,z)\\+{\\cal H}_1(\\tau)\\vartheta_{1,1}(\\tau,z) \n   \\= \\sum_{\\substack{n,\\,r\\in{\\mathbb Z} \\\\ 4 n - r^2 \\ge 0}}  H(4n-r^2) \\, q^n\\,y^r\\,,  \\end{equation} \nis a mock Jacobi form of weight $2$ and index $1$ with shadow $\\overline{\\vartheta_{1,0}(\\tau,0)}\\,\\vartheta_{1,0}(\\tau,z)\\+\n\\overline{\\vartheta_{1,1}(\\tau,0)}\\,\\vartheta_{1,1}(\\tau,z)$.  The $\\nu$th Taylor coefficient $\\xi_\\nu$ of ${\\cal H}$ is given by\n  \\beta\\label{thetaHRC} \\frac{4^\\nu}{\\binom{2\\nu}\\nu}\\,\\sum_{j=0}^1 [\\vartheta_{1,j},{\\cal H}_j]_\\nu\n  \\= \\delta_{k,2}\\,E_{k} \\,-\\,F_k^{(1)}\\+ \\text{(cusp form of weight $k$ on $SL(2,{\\mathbb Z})$)}\\,, \\end{equation}\nwhere $k=2\\nu+2$ and\n  \\begin{eqnarray}\\label{defF1k} F_k^{(1)}(\\tau) \\;:=\\; \\sum_{n>0}\\Bigl(\\sum_{d|n} \\min\\bigl(d,\\frac nd\\bigr)^{k-1}\\Bigr) q^{n}\\qquad\\quad \\text{($k$ even, $k\\ge2$)}\\,. \\end{eqnarray} \nIn fact the cusp form appearing in \\eqref{thetaHRC} is a very important one, namely (up to a factor $-2$) the sum of the\nnormalized Hecke eigenforms in $S_k(SL(2,{\\mathbb Z}))$, and equation~\\eqref{thetaHRC} is equivalent to the famous formula of Eichler\nand Selberg expressing the traces of Hecke operators on~$S_k(SL(2,{\\mathbb Z}))$ in terms of class numbers of imaginary quadratic fields.\n\n\n\n\\smallskip\n\\noindent {\\bf Example 3.}  Write the function $h^{(6)}$ defined in~\\eqref{defh6} as \n\\beta \\label{defC6}\nh^{(6)} (\\tau) \\= \\sum_{\\substack{\\Delta \\ge - 1 \\\\ \\Delta \\equiv -1 \\; ({\\rm mod} \\; 24)}} C^{(6)}(\\Delta) \\, q^{\\Delta\/24}\n\\end{equation}\nwith the first few coefficients $C^{(6)}(\\Delta)$ given by\n$$\n\\begin{tabular}{|c|ccccccccccc|c}  \\hline     $\\Delta$ & $-1$& $23$& $47$ & $71$ &$95$& $119$& $143$ & 167 & 191 & 215 & 239 \\\\\n       \\hline  $C^{(6)}(\\Delta)$ &$-1$ & 35 & 130 & 273 & 595 & 1001 & 1885 & 2925 & 4886 & 7410 & 11466 \\\\\n\\hline         \\end{tabular}  \\quad .\n$$\nThen the function \n\\beta \\label{defCF6} {\\cal F}_6(\\tau,z) \\= \\sum_{\\substack{n,\\,r\\in{\\mathbb Z} \\\\ 24n-r^2 \\ge -1}} \\chi_{12}(r)\\,C^{(6)}(24n-r^2)\\;q^n\\,y^r \\end{equation}\nis a mock Jacobi form of index~6 (explaining the notation $h^{(6)}$). Note that,  \nsurprisingly, this is even {\\it simpler} than the expansion of the index~1 mock Jacobi form just discussed,  \nbecause its twelve Fourier coefficients $h_{\\ell}$ are all multiples of just one of them, while the two Fourier \ncoefficients $h_{\\ell}$ of ${\\cal H}(\\tau,z)$ were not proportional. (This is related to the fact that the shadow\n$\\eta(\\tau)$ of $h^{(6)}(\\tau)$ is a modular form on the full modular group, \nwhile the shadow $\\theta(\\tau)$ of ${\\cal H}(\\tau)$ is a modular form on a congruence subgroup.)\nSpecifically, we have $h_{\\ell}(\\tau) = \\chi_{12}(\\ell) h^{(6)}(\\tau)$ for all $\\ell\\in{\\mathbb Z}\/12{\\mathbb Z}$,\nwhere $\\chi_{12}$ is the character defined in~\\eqref{defchi12}, so that ${\\cal F}_6$ has the factorization \n\\beta\\label{F6splits} \n {\\cal F}_6(\\tau,z) \\= h^{(6)}(\\tau)\\,\\sum_{\\ell\\;({\\rm mod} 12)} \\chi_{12}(\\ell)\\,\\vartheta_{6,\\ell}(\\tau,z) \\= \\eta(\\tau)\\,h^{(6)}(\\tau)\\,\\frac{\\vartheta_{1}(\\tau,4z)}{\\vartheta_{1}(\\tau,2z)} \\, .  \n\\end{equation}\nCombining this with \\eqref{xinhrel} and noting that $\\vartheta^{0}_{6,1}-\\vartheta^{0}_{6,5}=\\vartheta^{0}_{6,11}-\\vartheta^{0}_{6,7}=\\eta$,\nwe see that the functions described in~\\eqref{hetaRC6} are proportional to the Taylor coefficients $\\xi_{\\nu}(\\tau)$ of ${\\cal F}_6(\\tau,z)$. \n\n\n\\smallskip\n\\noindent {\\bf Example 4.} The fourth example is very similar. Write the mock modular form~\\eqref{defh2} as \n\\beta\nh^{(2)} (\\tau) \\=  \\sum_{\\substack{\\Delta \\ge - 1  \\\\ \\Delta \\equiv -1 \\; ({\\rm mod} \\; 8)}} C^{(2)}(\\Delta) \\, q^{\\Delta\/8} \n\\end{equation}\nwith initial coefficients given by\n$$\n\\begin{tabular}{|c|cccccccccc|cc}   \\hline    $\\Delta$ & $-1$& 7& 15 & 23 & 31& 39 & 47 & 55 & 63 & 71  \\\\\n       \\hline  $C^{(2)}(\\Delta)$ &$-1$ & 45 & 231 & 770 & 2277 & 5796 & 13915 & 30843 & 65550 & 132825 \\\\\n \\hline        \\end{tabular}  \\quad . \n$$\nThen the function \n\\begin{eqnarray}\\label{defCF2}\n{\\cal F}_2(\\tau,z) & \\= & \\sum_{\\substack{n,\\,r\\in{\\mathbb Z} \\\\ 8 n - r^2 \\ge -1}} \\chi_4(r) \\, C^{(2)}(8n-r^2)\\,q^n \\, y^r \n\\= h^{(2)}(\\tau)\\,\\bigl(\\vartheta_{2,1}(\\tau,z) - \\vartheta_{2,3}(\\tau,z)\\bigr) \\cr\n& \\= &  h^{(2)}(\\tau)\\,\\vartheta_ 1(\\tau,2z) =  \\eta(\\tau)^{3} \\, h^{(2)}(\\tau) \\, C(\\tau,z)\n\\end{eqnarray}\n(where $\\chi_4(r) = \\pm 1$ for $r \\equiv \\pm 1 \\, ({\\rm mod} \\; 4)$ and $\\chi_4(r) =0$ for $r$ even) \nis a mock Jacobi form of index~2. (Here $C$ is the Jacobi form of weight $-1$ and index 2 \nintroduced in \\S\\ref{MJF1}.)  \nAs in Example 3, the functions given in~\\eqref{hetaRC2} are proportional to the Taylor coefficients\n$\\xi_{\\nu}$ of ${\\cal F}_2$, because $\\vartheta_{2,1}^ 1 = - \\vartheta_{2,3}^ 1 = \\eta^3$, where $\\vartheta_{2,\\ell}^ 1$ is defined by~\\eqref{defth1}.\n\n\n\n\\subsection{Mock modular forms: the general case \\label{MMMF}}\n\nIf we replace the condition ``exponential growth'' in $1.$ of~\\ref{basicdefs} by ``polynomial growth,\"  we get\nthe class of {\\it strongly holomorphic mock modular forms}, which we can denote ${\\mathbb M}_{k|0}$, and an exact \nsequence $0 \\to M_{k} \\to {\\mathbb M}_{k|0} \\to \\overline {M_{2-k}} $. This is not very useful, however, because there are \nalmost no examples of  ``pure'' mock modular forms that are strongly holomorphic, essentially the only ones being \nthe function ${\\cal H}$ of Example~2 of \\S\\ref{basicdefs} and its variants. It becomes useful if we generalize to {\\it mixed \nmock modular forms} of mixed weight $k|\\ell$. (Here $k$ and $\\ell$ can be integers or half-integers.) These are\nholomorphic functions $h(\\tau)$, with polynomial growth at the cusps, that have completions $\\widehat h$ of the\nform $\\widehat h = h + \\sum_j f_j\\,g^*_j$ with $f_j \\in M_{\\ell}$, $g_j \\in M_{2-k+\\ell}$ \nthat transform like modular\nforms of weight $k$. The space ${\\mathbb M}_{k|\\ell}$ of such forms thus fits into an exact sequence \n  \\beta\\label{exseqmmf2} 0 \\longrightarrow M_{k} \\longrightarrow {\\mathbb M}_{k|\\ell}  \\stackrel{{\\cal S}}{\\longrightarrow} \n  M_{\\ell}    \\otimes    \\overline {M_{2-k+\\ell}} \\,, \\end{equation}\nwhere the shadow map ${\\cal S}$ now sends $h$ to $\\sum_{j} f_{j}\\,\\overline g_{j}$. If $\\ell = 0$ this reduces to the \nprevious definition, since $M_0={\\mathbb C}$, but with the more general notion of mock modular forms \nthere are now plenty of strongly holomorphic examples, and, just as for ordinary modular forms, they have \nmuch nicer properties (notably, polynomial growth of their Fourier coefficients) than the weakly holomorphic ones. \nIf the shadow of a mixed mock modular form $h \\in {\\mathbb M}_{k|\\ell}$ happens to contain only one term \n$f(\\tau) \\overline{g(\\tau)}$, and if $f(\\tau)$ has no zeros in the upper half-plane, then $f^{-1} \\, h $ is a weakly holomorphic \npure mock modular form of weight $k-\\ell$ (and in fact, all weakly holomorphic pure mock modular forms arise in this way). \nFor functions in the larger space ${\\mathbb M}_{k|\\ell}^{\\,!}$ of weakly holomorphic mixed mock modular forms, defined in the \nobvious way, this always happens, so ${\\mathbb M}_{k|\\ell}^{\\,!}$  simply coincides with $M_{\\ell}\\otimes{\\mathbb M}_{k-\\ell}^{\\,!}$ and does\nnot give anything new, but the smaller space ${\\mathbb M}_{k|\\ell}$ does not have such a decomposition and is more interesting.\nMoreover, there are many examples.  In fact, apart from Ramanujan's original examples of mock theta functions, which \nwere defined as $q$-hypergeometric series and were weakly holomorphic pure mock modular forms of weight 1\/2, most of\nthe examples of mock modular forms occurring ``in nature,'' such as Appell-Lerch sums or indefinite theta series,\nare strongly holomorphic mixed modular forms.\n\nWe can also define ``even more mixed\" mock modular forms by replacing ${\\mathbb M}_{k|\\ell}$ by the space ${\\mathbb M}_{k}=\\sum_{\\ell}{\\mathbb M}_{k|\\ell}$,\ni.e., by allowing functions whose shadow is a finite sum of products $f_j(\\tau)\\overline{g_j(\\tau)}$ with the $f_j$ of\nvarying weights~$\\ell_j$ and $g_j$ of weight $2-k+\\ell_j$. This space fits into an exact sequence \n  \\beta\\label{exseqmmf3} 0 \\longrightarrow M_{k} \\longrightarrow {\\mathbb M}_{k}  \\stackrel{{\\cal S}}{\\longrightarrow} \n \\bigoplus_\\ell M_\\ell \\otimes \\overline {M_{2-k+\\ell}} \\;. \\end{equation}\nIn fact, the functions in this space are the most natural objects, and from now on we will use ``mock modular forms'' to\nrefer to this more general class, including the adjectives ``mixed'' and ``pure'' only when needed for emphasis. To justify this\nclaim, we consider the space $\\mathfrak M_{k}$ of all {\\it real-analytic modular forms of weight~$k$}, i.e.,  \nreal-analytic functions that transform like modular forms of weight~$k$ and have polynomial \ngrowth at all cusps. More generally, one can consider the space $\\mathfrak M_{k,\\ell}$ of real-analytic functions \nof polynomial growth that transform according to $F(\\gamma \\tau) = (c\\tau+d)^{k} (c \\overline \\tau + d)^{\\ell} F(\\tau)$ for all \nmatrices $\\gamma = \\sm abcd$ belonging to some subgroup of finite index of $SL(2,{\\mathbb Z})$. Since the function $\\tau_2=\\Im (\\tau)$\ntransforms by $(\\gamma \\tau)_2 = \\tau_2\/|c\\tau+d|^2$, the spaces  $\\mathfrak M_{k,\\ell}$ and $\\tau_2^{r} \\, \\mathfrak M_{k+r,\\ell+r}$ \ncoincide for all~$r$. (Here, $k$, $\\ell$ and $r$ can each be integral or half-integral.) Obviously the space $\\mathfrak M_{k,\\ell}$\ncontains the space $M_{k} \\otimes \\overline{M_{\\ell}}$, so by the comment just made, it also contains the direct sum \n$\\bigoplus_{r \\in {\\mathbb Z}} \\tau_2^{r} \\,  M_{k+r} \\otimes \\overline{M_{\\ell+r}}$. Elements of this subspace will be called {\\em decomposable}. \nNow, if a function $F \\in \\mathfrak M_{k} = \\mathfrak M_{k,0}$ has the property that its $\\overline \\tau$-derivative (which automatically belongs to \n$\\mathfrak M_{k,2}\\,$) is decomposable, say as $\\sum_{j} {\\tau_2}^{r_{j}} \\, f_j \\,\\overline{g_j}$ (where the weights of $f_{j}$ and $g_{j}$ \nare necessarily equal to $k+r_{j}$ and $2+r_{j}$, respectively), then $F$ is the sum of a holomorphic function \n$h$ and a multiple of $\\sum_{j} f_{j} g^*_{j}$ and the function $h$ is a (mixed) mock modular form in the sense just explained. \nConversely, a holomorphic function $h$ is a (mixed) mock modular form of weight $k$ if and only if there exist numbers $r_{j}$ and \nmodular forms $f_{j} \\in M_{k+r_{j}}$ and $g_{j} \\in M_{2 + r_{j}}$ such that the sum $\\widehat h:= h+ \\sum_{j} f_{j} g^*_{j}$ \nbelongs to~$\\mathfrak M_{k}$. We can thus summarize the entire discussion by the following very simple definition: \n\\begin{quote}\n{\\it A mock modular form is the holomorphic part of a real-analytic modular form whose $\\overline \\tau$-derivative is decomposable.}  \n\\end{quote}\n\nAs examples of this more general class we mention the functions $F_2^{(6)}$ and $F_2^{(2)}$ defined in equations~\\eqref{defF62}\nand~\\eqref{defF22}, which are both (strong) mixed mock modular forms of total weight~2 and trivial character on the full modular group $SL(2,{\\mathbb Z})$,\nbut with different mixed weights: $12F_2^{(6)}$ is the sum of the functions $E_2(\\tau)$ and $\\eta(\\tau)\\,h^{(6)}(\\tau)$ of weights $2|0$ and\n$\\frac32|\\frac12$, while $24F_2^{(2)}$ is the sum of the functions $E_2(\\tau)$ and $\\eta(\\tau)^3h^{(2)}(\\tau)$ of weights $2|0$ and $\\frac12|\\frac32$.\n\n\nWe remark that in the pure case we have $\\tau_2^{k}\\cdot\\pi_{\\overline \\tau}F \\,\\in\\,\\overline{M_{2-k}}=\\textrm{Ker}\\bigl(\\pi_\\tau\\!:\n\\mathfrak M_{0,2-k} \\to \\mathfrak M_{2,2-k} \\bigr)$,  so that the \nsubspace ${\\mathbb M}_{k,0}$ of ${\\mathbb M}_{k}$ can be characterized by a {\\it single} second-order differential equation $\\Delta_k \\widehat h=0$,\nwhere $\\Delta_k=\\tau_2^{2-k}  \\pi_{\\tau}  \\tau_2^{k}  \\pi_{\\overline \\tau}$ is the Laplacian in weight~$k$. For this reason, (weak) pure \nmock modular forms can be, and often are, viewed as the holomorphic parts of (weak) harmonic Maass forms, i.e., of functions \nin $\\mathfrak M_{k}^{\\,!}$ annihilated by~$\\Delta_{k} \\,$.  But in the general case, there is no connection between mock modular \nforms or their completions and any second-order operators.  Only the Cauchy-Riemann operator~$\\pi\/\\pi\\overline \\tau$ is needed,\nand the new definition of mock modular forms as given above is both more flexible and more readily generalizable than the\nmore familiar definition of pure mock modular forms in terms of harmonic Maass forms. \n\nFinally, we can define mixed mock Jacobi forms in exactly the same way as we defined pure ones (i.e., they have\ntheta expansions in which the coefficients are mixed mock modular forms, with completions that transform like ordinary Jacobi forms).\nWe will see in the next section that such objects arise very naturally as the ``finite parts\" of meromorphic Jacobi forms.\n\n\n\n\\subsection{Superconformal characters and mock modular forms\\label{FinitePart}} \n\nAs already mentioned, mock modular forms first made their appearance in physics some ten years before the work of Zwegers, \nin connection with the characters of superconformal algebas~\\cite{Eguchi:1988af, EOTY:1989}.  \nThe representations of the $\\mathcal{N}=4$ superconformal algeba with central charge~$c=6k$ are labeled by two quantum numbers, the\nenergy~$h$ and spin~$l$ of the lowest weight state~$|\\Omega\\rangle$, which satisfy $h>k\/4$, $l\\in\\{1\/2,\\,1,\\dots,k\/2\\}$  for the non-BPS \n(massive) case and $h=k\/4$, $l\\in\\{0,1\/2,\\dots,k\/2\\}$ for the BPS (massless) case. The corresponding character, defined by\n $$ \\text{ch}^{\\tilde R}_{k,h,l}(\\tau,z) \\= \\text{Tr}\\bigl(q^{L_0-c\/24}\\,e^{4\\pi izJ_0}\\bigr)\n     \\qquad\\bigl(\\,L_0|\\Omega\\rangle=h|\\Omega\\rangle,\\quad J_0|\\Omega\\rangle=l|\\Omega\\rangle\\bigr)\\,, $$\nturns out to be, up to a power of $q$, a Jacobi form in the non-BPS case but a ``mock\" object in the BPS case.  In \nparticular, for $k=1$ the non-BPS characters are given by\n  \\beta\\label{nBPSchar}\\text{ch}^{\\tilde R}_{1,h,\\frac12}(\\tau,z) \\= -\\,q^{h-\\frac38}\\,\\frac{\\vartheta_1(\\tau,z)^2}{\\eta(\\tau)^3}   \n     \\= -\\,q^{h-\\frac38}\\,\\eta(\\tau)^3\\,\\varphi_{-2,1}(\\tau,z)   \\qquad\\bigl(h>\\frac14\\bigr) \\end{equation}\nand the BPS characters by\n  \\beta\\label{BPSchar} \\text{ch}^{\\tilde R}_{1,\\frac14,0}(\\tau,z) \\,=\\, -\\,\\frac{\\vartheta_1(\\tau,z)^2}{\\eta(\\tau)^3}\\,\\mu(\\tau,z)\\,,  \n    \\;\\quad  \\text{ch}^{\\tilde R}_{1,\\frac14,\\frac12}(z,\\tau)+2\\text{ch}^{\\tilde R}_{1,\\frac14,0}(\\tau,z) \n   \\,=\\, -\\, q^{-\\frac18}\\,\\frac{\\vartheta_1(\\tau,z)^2}{\\eta(\\tau)^3}\\,, \n\\end{equation}\nwhere $\\mu(\\tau,z)$ is the function defined by\n  \\beta\\label{mudef}  \\mu(\\tau,z)\\=\\frac{e^{\\pi iz}}{\\vartheta_1(z,\\tau)}\\;\\sum_{n\\in{\\mathbb Z}}\\frac{(-1)^nq^{\\frac{n^2+n}2}e^{2\\pi inz}}{1\\,-\\,q^n\\,e^{2\\pi iz}}\n   \\qquad(q=e^{2\\pi i\\tau},\\;y=e^{2\\pi iz})\\,, \\end{equation}\nan ``Appell-Lerch sum\" of a type that also played a central role in Zwegers's work\\footnote{More precisely, the function $\\mu(\\tau,z)$\nis the specialization to $u=v=z$ of the 2-variable function $\\mu(u,v;\\tau)$ considered by Zwegers.} and that we will discuss in the next \nsection.  The ``mock\" aspect is the fact that $\\mu(\\tau,z)$ must be completed by an appropriate non-holomorphic correction \nterm in order to transform like a Jacobi form.\n\nThe mock modular form $h^{(2)}$ arises in connection with the decomposition of the elliptic genus of a $K3$ surface as a linear\ncombination of these characters.  This elliptic genus is, as discussed in \\ref{BHandJF}, equal to the weak Jacobi form $2B=2\\varphi_{0,1}$.\nIt can be represented as a specialization of the partition  function (= generating function of states) of superstrings compactified\non a $K3$ surface~\\cite{EOTY:1989} and as such can be decomposed into pieces corresponding to the representations of the $N=4$ superconformal \nalgebra with central charge~$6k=6$.  The decomposition has the form \n\\cite{Ooguri:1989fd, Wendland:2000, EST:2007}\n  \\beta\\label{SCA} 2\\varphi_{0,1}(\\tau,z)\\=20\\,\\text{ch}^{\\tilde R}_{1,\\frac14,0}(z,\\tau)\n    \\+A_0\\,\\text{ch}^{\\tilde R}_{1,\\frac14,\\frac12}(z,\\tau)\n     \\,+\\, \\sum_{n=1}^\\infty A_n\\, \\text{ch}^{\\tilde R}_{1,n+\\frac14,\\frac12}(z,\\tau) \\,, \\end{equation}\nwhere $A_0=-2$, $A_1=90$, $A_2=462$, \\dots\\ are the Fourier coefficients $2C^{(2)}(8n-1)$ of the mock modular form $2h^{(2)}(\\tau)$.\nInserting~\\eqref{nBPSchar} and~\\eqref{BPSchar}, we can rewrite this as\n \\beta \\label{K3decomp} \\eta(\\tau)^{-3}\\,\\frac{\\varphi_{0,1}(\\tau,z)}{\\varphi_{-2,1}(\\tau,z)} \\= -\\,12\\,\\mu(\\tau,z) \\,-\\, h^{(2)}(\\tau)\\,. \\end{equation}\n\nIn the next section we will see how to interpret this equation as an example of a certain canonical \ndecomposition of meromorphic Jacobi forms \\eqref{CoverA}. \n\n\n\n\\section{From meromorphic Jacobi forms to mock modular forms \\label{MockfromJacobi}}\n\nIn this section we consider Jacobi forms $\\varphi(\\tau,z)$ that are meromorphic with respect to the variable~$z$.  It was discovered\nby Zwegers~\\cite{Zwegers:2002} that such forms, assuming that their poles occur only at points $z\\in{\\mathbb Q}\\tau+{\\mathbb Q}\\,$ (i.e., at torsion\npoints on the elliptic curve ${\\mathbb C}\/{\\mathbb Z}\\tau+{\\mathbb Z}\\,$),  have a modified theta expansion related to mock modular forms.  \nOur treatment is based on his, but the presentation is quite different and the results go further in one key respect. \nWe show that $\\varphi$ decomposes canonically into two pieces, one constructed directly from its poles and consisting of a finite\nlinear combination of Appell-Lerch sums with modular forms as coefficients, and the other  being a mock Jacobi form in the sense\nintroduced in the preceding section.  Each piece separately\ntransforms like a Jacobi form with respect to elliptic transformations.  Neither piece separately transforms like a Jacobi form with\nrespect to modular transformations, but each can be completed by the addition of an explicit and elementary non-holomorphic correction\nterm so that it does transform correctly with respect to the modular group.\n\nIn \\S\\ref{FinitePart} we explain how to modify the Fourier coefficients $h_\\ell$ defined in \\eqref{jacobi-Fourier}\nwhen $\\varphi$ has poles, and use these to define a ``finite part\" of~$\\varphi$ by the theta decomposition~\\eqref{jacobi-theta}. \nIn \\S\\ref{PolarPart} we define (in the case when $\\varphi$ has simple poles only) a ``polar part\" of $\\varphi$ as a finite linear \ncombination of standard Appell-Lerch sums times modular forms arising as the residues of~$\\varphi$ at its poles, and show that\n$\\varphi$ decomposes as the sum of its finite part and its polar part.  Subsection \\ref{MockDecomposition} gives the proof\nthat the finite part of~$\\varphi$ is a mock Jacobi form and a description of the non-holomorphic correction term needed\nto make it transform like a Jacobi form.  This subsection also contains a summary in tabular form of the various functions that \nhave been introduced and the relations between them.  In \\S\\ref{double} we describe the modifications needed in the case of double poles\n(the case actually needed in this paper) and in \\S\\ref{MJExamples}\nwe present a few examples to illustrate the theory.  Among the ``mock\" parts of these are two of the most interesting mock\nJacobi forms from \\S\\ref{Mock} (the one related to class numbers of imaginary quadratic fields and the one conjecturally\nrelated to representations of the Mathieu group~$M_{24}$).  Many other examples will be given in \\S\\ref{flyJac}.\n\nThroughout the following sections, we use the convenient notation $\\mathbf{e}(x) := e^{2\\pi i x}$.\n\n\\subsection{The Fourier coefficients of a meromorphic Jacobi form\\label{FinitePart}} \nAs indicated above, the main problem we face is to find an analogue of the\ntheta decomposition~\\eqref{jacobi-theta} of holomorphic Jacobi forms in the meromorphic case.  We will approach this\nproblem from two sides: computing the Fourier coefficients of $\\varphi(\\tau,z)$ with respect to~$z$, and computing the \ncontribution from the poles.  In this subsection we treat the first of these questions.\n\nConsider a meromorphic Jacobi form $\\varphi(\\tau, z)$ of weight $k$ and index $m$. We assume\nthat $\\varphi(\\tau, z)$ for each $\\tau \\in \\relax{\\rm I\\kern-.18em H}$ is a meromorphic function of $z$ which has poles only at points $z = \\alpha \\tau + \\beta$ wit\\\n$\\alpha$ and $\\beta$ rational. In the case when $\\varphi$ was holomorphic, we could write its Fourier expansion in the form \\eqref{jacobi-Fourier}.\nBy Cauchy's theorem, the coefficient $h_\\ell(\\tau)$ in that expansion could also be given by the integral formula\n \\beta\\label{hAl} h^{(z_0)}_{\\ell}(\\tau) \\= q^{-\\ell^2\/ 4m}  \\int_{z_0}^{z_0+1} \\varphi(\\tau,z)\\;\\mathbf{e}(-\\ell z)\\;dz\\,, \\end{equation}\nwhere $z_0$ is an arbitrary point of ${\\mathbb C}$.  From the holomorphy and transformation properties of~$\\varphi$ it follows that the value\nof this integral is independent of the choice of $z_0$ and of the path of integration and depends only on $\\ell$ modulo~$2m$\n(implying that we have the theta expansion \\eqref{jacobi-theta}) and that each $h_\\ell$ is a modular form of weight~$k-\\frac12$.\nHere each of these properties fails: the integral \\eqref{hAl} is not independent of the path of integration (it jumps when\nthe path crosses a pole); it is not independent of the choice of the initial point $z_0$; it is not periodic in $\\ell$\n(changing $\\ell$ by $2m$ corresponds to changing $z_0$ by $\\tau$); it is not modular; and of course the expansion \\eqref{jacobi-theta}\ncannot possibly hold since the right-hand-side has no poles in $z$. \n\nTo take care of the first of these problems, we specify the path of integration in \\eqref{hAl} as the horizontal line\nfrom $z_0$ to $z_0+1$.  If there are poles of $\\varphi(\\tau,z)$ along this line, this does not make sense; in that case, we define\nthe value of the integral as the average of the integral over a path deformed to pass just above the poles and the\nintegral over a path just below them.  (We do not allow the initial point~$z_0$ to be a pole of $\\varphi$, so this makes sense.)\nTo take care of the second and third difficulties, the dependence on~$z_0$ and the non-periodicity in~$\\ell$, we play one of\nthese problems off against the other.  From the elliptic transformation property~\\eqref{elliptic} we find that\n\\begin{eqnarray*} &h_\\ell^{(z_0+\\tau)}(\\tau) &\\= q^{-\\ell^2\/4m}\\,\\int_{z_0}^{z_0+1}\\varphi(\\tau,\\,z+\\tau)\\,\\mathbf{e}(-\\ell\\,(z+\\tau))\\;dz \\\\ \n  & &\\=   q^{-(\\ell+2m)^2\/4m}\\,\\int_{z_0}^{z_0+1}\\varphi(\\tau,\\,z)\\,\\mathbf{e}(-(\\ell+2m)\\,z)\\;dz  \\= h_{\\ell+2m}^{(z_0)}(\\tau)\\,,  \\end{eqnarray*}\ni.e., changing $z_0$ by $\\tau$ corresponds to changing $\\ell$ by $2m$, as already mentioned.  It follows that if we\nchoose $z_0$ to be $-\\ell\\tau\/2m$ (or $-\\ell\\tau\/2m+B$ for any~$B\\in{\\mathbb R}$, since it is clear that the value of the integral~\\eqref{hAl}\ndepends only on the height of the path of integration and not on the initial point on this line), then the quantity\n\\beta\\label{Defofhl}  h_\\ell(\\tau)\\;:=\\;h_\\ell^{(-\\ell\\tau\/2m)}(\\tau)\\,,  \\end{equation}\nwhich we will call the $\\ell$th {\\bf canonical Fourier coefficient} of~$\\varphi$, depends only on the value of $\\ell$ (mod $2m$).\nThis in turn implies that the sum \n  \\beta\\label{phiF}  \\varphi^F(\\tau,z)\\;:=\\; \\sum_{\\ell\\,\\in\\, {\\mathbb Z}\/2m{\\mathbb Z}} h_\\ell(\\tau) \\, \\vartheta_{m,\\ell}(\\tau, z)\\,, \\end{equation}\nwhich we will call the {\\bf finite} (or {\\bf Fourier}) {\\bf part} of $\\varphi$, is well defined.  If $\\varphi$ is holomorphic, then\nof course $\\varphi^F=\\varphi$, by virtue of \\eqref{jacobi-theta}.\nHence if we define the {\\bf polar part}~$\\varphi^{P}$ of~$\\varphi$ to be~$\\varphi-\\varphi^{F}$, then~$\\varphi^{P}$ only depends\non the principal part of~$\\varphi$ at its singularities.  We will give explicit formulas in \\S\\ref{PolarPart} and \\S\\ref{double}\nin the cases when all poles are single or double. \n\n\nNote that the definiton of $h_\\ell(\\tau)$ can also be written\n\\beta\\label{Defofhl2} h_\\ell(\\tau) \\= q^{\\ell^2\/4m}\\,\\int_{{\\mathbb R}\/{\\mathbb Z}} \\varphi(\\tau,z-\\ell\\tau\/2m)\\;\\mathbf{e}(-\\ell z)\\;dz\\,, \\end{equation}\nwith the same convention as above if $\\varphi(\\tau,z-\\ell\\tau\/2m)$ has poles on the real line.\n\nFinally, we remark that none of these definitions have anything to do with the modular transformation\nproperty \\eqref{modtransform}, and would also work if our initial function $\\varphi(\\tau,z)$ satisfied only the elliptic \ntransformation property \\eqref{elliptic}. (In that case, however, the functions $h_{\\ell}$ in general need not \nbe modular, and in fact will be mock modular when $\\varphi(\\tau,z)$ is a meromorphic Jacobi form.)\nWhat we have said so far can be summarized by a commutative diagram \n\\begin{eqnarray} \n{\\cal E }^{\\rm Mer}_{\\pm,m}  & \\stackrel{F}{\\longrightarrow} & {\\cal E }_{\\pm,m}  \\cr \n\\cup \\; \\; & & \\;  \\cup  \\qquad , \\cr\nJ^{\\rm Mer}_{k,m}  & \\stackrel{F}{\\longrightarrow} & {\\mathbb J}_{k,m} \n\\end{eqnarray}\nwhere $J^{\\rm Mer}_{k,m}$ (resp.~${\\cal E }^{\\rm Mer}_{\\pm,m}$) denote the subspace of meromorphic\nJacobi (resp.~elliptic) forms having poles only at torsion points $z \\in {\\mathbb Q} \\tau + {\\mathbb Q}$, and the map~$F$\nsends $\\varphi \\mapsto \\varphi^{\\rm F}$. \n\n\n\n\n\n\\subsection{The polar part of $\\varphi$ (case of simple poles) \\label{PolarPart}}\nWe now consider the contribution from the poles. To present the results we first need to\nfix notations for the positions and residues of the poles of our meromorphic function~$\\varphi$.  \nWe assume for now that the poles are all simple.\n\nBy assumption, $\\varphi(\\tau,z)$ has poles only at points of the form $z=z_s=\\alpha\\tau+\\beta$ for $s=(\\alpha,\\beta)$ belonging to some subset $S$ \nof~${\\mathbb Q}^2$.  The double periodicity property~\\eqref{elliptic} implies that $S$ is invariant under translation by~${\\mathbb Z}^2$, and\nof course $S\/{\\mathbb Z}^2$ must be finite.  The modular transformation property~\\eqref{modtransform} of $\\varphi$ implies that  $S$ is $SL(2,{\\mathbb Z})$-invariant.\nFor each $s=(\\alpha,\\beta) \\in S$, we set\n  \\beta\\label{defDs} D_s(\\tau) \\= 2\\pi i\\;\\mathbf{e}(m\\alpha z_s)\\, {\\rm Res}_{z=z_s}\\bigl(\\varphi(\\tau,z)\\bigr) \\qquad  (s=(\\alpha,\\beta)\\in S,\\;z_s=\\alpha\\tau+\\beta)\\;.  \\end{equation}\nThe functions $D_{s}(\\tau)$ are holomorphic modular forms of weight $k-1$ and some level, and only finitely many of them are distinct. More precisely, they satisfy\n  \\begin{eqnarray}\\label{Dell} \\bullet && \\;\\, D_{(\\alpha+\\lambda,\\beta+\\mu)} (\\tau)\\;\\=\\;\\mathbf{e}(m(\\mu\\alpha-\\lambda\\beta))\\,D_{(\\alpha,\\beta)}(\\tau) \\quad {\\rm for} \\quad(\\lambda,\\mu)\\,\\in\\,{\\mathbb Z}^2\\,, \\\\\n   \\label{Dmod}  \\bullet && D_s \\Bigl(\\dfrac{a\\tau+b}{c\\tau+d}\\Bigr) \\= (c\\tau+d)^{k-1}\\,D_{s\\gamma}(\\tau) \\quad {\\rm for} \\quad \n      \\gamma = \\bigl(\\begin{array}{cc} a & b \\\\ c & d \\\\ \\end{array} \\bigr) \\,\\in\\, SL(2,{\\mathbb Z}) \\;, \\end{eqnarray}\nas one sees from the transformation properties of $\\varphi$.  (The calculation for equation~\\eqref{Dmod} is given in \\S\\ref{double}.)  It is precisely to obtain the simple\ntransformation equation~\\eqref{Dmod} that we included the factor $\\mathbf{e}(m\\alpha z_s)$ in~\\eqref{defDs}.  Since we are assuming for the time\nbeing that there are no higher-order poles, all of the information about the non-holomorphy of $\\varphi$ is contained in these functions.\n\nThe strategy is to define a ``polar part\" of $\\varphi$ by taking the poles $z_s$ in some fundamental parallelogram for the action of\nthe lattice ${\\mathbb Z}\\tau+{\\mathbb Z}$ on ${\\mathbb C}$ (i.e., for $s=(\\alpha,\\beta)$ in  the intersection of $S$ with some square box  $[A,A+1)\\times[B,B+1)\\,$)\nand then averaging the residues at these poles over all translations by the lattice.  But we must be careful to do this in just\nthe right way to get the desired invariance properties.  For each $m\\in\\mathbb{N}$ we introduce the {\\it averaging operator}\n\\beta \\label{defAvm}  \\Av m{F(y)} \\;:=\\; \\sum_{\\lambda} \\,q^{m\\lambda^2}y^{2m\\lambda}F(q^\\lambda y)  \\end{equation}\nwhich sends any function of $y$ (= ${\\mathbb Z}$-invariant function of $z$) of polynomial growth in~$y$ to a function of $z$\ntransforming like an index~$m$ Jacobi form under translations by the full lattice ${\\mathbb Z}\\tau+{\\mathbb Z}$.  For example, we have\n\\beta \\label{ThetAv} q^{\\ell^2\/4m}\\,\\Av m{y^\\ell}\\= \\sum_{\\lambda\\,\\in\\,{\\mathbb Z}} \\,q^{(\\ell+2m\\lambda)^2\/4m}\\,y^{\\ell+2m\\lambda} \\= \\vartheta_{m,\\ell}(\\tau, z) \\end{equation}\nfor any $\\ell\\in{\\mathbb Z}\\,$.  If $F(y)$ itself is given as the average\n\\beta\\label{defAvZ} F(y)\\=\\text{Av}_\\IZ\\bigl[f(z)\\bigr] \\;:=\\; \\sum_{\\mu\\in{\\mathbb Z}} f(z+\\mu)   \\qquad(z\\in{\\mathbb C},\\;\\, y=\\mathbf{e}(z)) \\end{equation}\nof a function $f(z)$ in ${\\mathbb C}$ (of sufficiently rapid decay at infinity), then we have\n\\beta\\label{defAvL} \\Av m{F(y)} \\=  \\text{Av}^{(m)}_{\\IZ\\tau+\\IZ}\\bigl[f(z)\\bigr] \\;:=\\; \\sum_{\\lambda,\\,\\mu\\in{\\mathbb Z}} e^{2\\pi im(\\lambda^2\\tau+2\\lambda z)}\\,f(z+\\lambda\\tau+\\mu)\\;. \\end{equation}\n\nWe want to apply the averaging operator \\eqref{defAvm} to the product of the function $D_s(\\tau)$ with a standard rational\nfunction of $y$ having a simple pole of residue~1 at $y=y_s=\\mathbf{e}(z_s)$, but the choice of this rational function is not obvious.\nThe right choice turns out to be $\\mathcal R_{-2m\\alpha}(y)$, where $\\mathcal R_c(y)$ for $c\\in{\\mathbb R}$ is defined by\n\\beta\\label{defRc}   \\mathcal R_c(y)\\=\\begin{cases} \\quad\\dfrac12\\,y^c\\,\\dfrac{y+1}{y-1} & \\text{if $c\\in{\\mathbb Z}$\\,,} \\\\\n    \\quad\\, y^{\\ceil c}\\,\\dfrac1{y-1} &\\text{if $c\\in{\\mathbb R}\\smallsetminus{\\mathbb Z}$\\,.} \\end{cases}  \\end{equation}\n(Here $\\ceil c$ denotes the ``ceiling\" of $c$, i.e., the smallest integer $\\ge c$. The right-hand side can also be written  \nmore uniformly as $\\dfrac12\\;\\dfrac{y^{\\flo c+1}+y^{\\ceil c}}{y-1}\\,$, where $\\flo c=-\\ceil{-c}$ denotes the ``floor\" of~$c$,\ni.e., the largest integer $\\le c$.)  This function looks artificial, but is in fact quite natural.  First of all, by\nexpanding the right-hand side of~\\eqref{defRc} in a geometric series  we find\n\\beta\\label{Rcseries}  \\mathcal R_c(y)\\=\\begin{cases} -\\sum^*_{\\ell\\ge c} y^\\ell &\\text{if $|y|<1$,} \\\\\n                            \\phantom{-}\\sum^*_{\\ell\\le c} y^\\ell &\\text{if $|y|>1$,}\\end{cases}\\end{equation}\nwhere the asterisk on the summation sign means that the term $\\ell=c$ is to be counted with multiplicity 1\/2 when it occurs \n(which happens only for $c\\in{\\mathbb Z}$, explaining the case distinction in~\\eqref{defRc}).  This formula, which can be seen as the \nprototypical example of wall-crossing, can also be written in terms of $z$ as a Fourier expansion (convergent for all $z\\in{\\mathbb C}\\smallsetminus{\\mathbb R}$)\n\\beta\\label{Rcseries2} \\mathcal R_c(\\mathbf{e}(z))\\=-\\sum_{\\ell\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(\\ell-c)\\+\\mbox{sgn}(z_2)}2\\;\\mathbf{e}(\\ell z) \\qquad(y=\\mathbf{e}(z),\\;\\, z_2=\\Im(z)\\ne0)\\,, \\end{equation}\nwithout any case distinction.  Secondly, $R_c(y)$ can be expressed in a  natural way as an average:\n\\begin{proposition}\\label{RcAsAv} For $c\\in{\\mathbb R}$ and $z\\in{\\mathbb C}\\smallsetminus{\\mathbb Z}$  we have  \n\\beta  \\mathcal R_c(\\mathbf{e}(z))\\=\\text{Av}_\\IZ\\Bigl[\\frac{\\mathbf{e}(cz)}{2\\pi iz}\\Bigr]\\;.\\end{equation}\n\\end{proposition} \n\\noindent {\\it Proof:} If $c\\in{\\mathbb Z}$, then\n$$ \\text{Av}_\\IZ\\Bigl[\\frac{\\mathbf{e}(cz)}{2\\pi iz}\\Bigr]\\=\\frac{y^c}{2\\pi i}\\;\\sum_{n\\in{\\mathbb Z}}\\frac1{z-n}\n   \\=\\frac{y^c}{2\\pi i}\\;\\frac\\pi{\\tan\\pi z}\\=\\frac{y^c}2\\;\\frac{y+1}{y-1}$$\nby a standard formula of Euler.  If $c\\notin{\\mathbb Z}$ then the Poisson summation formula and \\eqref{Rcseries2} give\n\\begin{align*} \\text{Av}_\\IZ\\Bigl[\\frac{\\mathbf{e}(cz)}{2\\pi iz}\\Bigr] &\\=\\sum_{n\\in{\\mathbb Z}}\\,\\frac{\\mathbf{e}(c(z+n))}{2\\pi i(z+n)} \n     \\=\\sum_{\\ell\\in{\\mathbb Z}}\\,\\biggl(\\int_{iz_2-\\infty}^{iz_2+\\infty}\\frac{\\mathbf{e}(c(z+u))}{2\\pi i(z+u)}\\,\\mathbf{e}(-\\ell u)\\,du\\biggr)\\,\\mathbf{e}(\\ell z) \\\\\n   &\\= -\\,\\sum_{\\ell\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(\\ell-c)\\+\\mbox{sgn}(z_2)}2\\;\\mathbf{e}(\\ell z)\\=\\mathcal R_c(\\mathbf{e}(z))  \\end{align*}\nif $z_2\\ne0$, and the formula remains true for $z_2=0$ by continuity. An alternative proof can be obtained by noting\nthat $\\mathbf{e}(-cz)\\mathcal R_c(\\mathbf{e}(z))$ is periodic of period~1 with respect to~$c$ and expanding it as a Fourier series in~$c$,\nagain by the Poisson summation formula.\n\n\\medskip\n\nFor $m\\in{\\mathbb N}$ and $s=(\\alpha,\\beta)\\in{\\mathbb Q}^2$ we now define a {\\it universal Appell-Lerch sum} $\\mathcal{A}_m^s(\\tau,z)$ by\n  \\beta \\label{defFms} \\mathcal{A}_m^s(\\tau,z)\\= \\mathbf{e}(-m\\alpha z_s)\\,\\Av m{\\mathcal R_{-2m\\alpha}(y\/y_s)}\\qquad(y_s=\\mathbf{e}(z_s)=\\mathbf{e}(\\beta)q^\\alpha)\\,. \\end{equation}\nIt is easy to check that this function satisfies\n  \\beta \\mathcal{A}_m^{(\\alpha+\\lambda,\\beta+\\mu)}\\= \\mathbf{e}(-m(\\mu\\alpha-\\lambda\\beta))\\,\\mathcal{A}_m^{(\\alpha,\\beta)} \\qquad(\\lambda,\\,\\mu\\in{\\mathbb Z}) \\end{equation}\nand hence, in view of the corresponding property \\eqref{Dell} of $D_s\\,$, that the product $D_s(\\tau)\\mathcal{A}_m^s(\\tau,z)$ depends\nonly on the class of~$s$ in $S\/{\\mathbb Z}^2$.  We can therefore define \n\\beta \\label{phiP}  \\varphi^P(\\tau,z)\\;:=\\;\\sum_{s\\in S\/{\\mathbb Z}^2}D_s(\\tau)\\,\\mathcal{A}_m^s(\\tau,z)\\,, \\end{equation}\nand it is obvious from the above discussion that this function, which we will call the {\\bf polar part} of $\\varphi$,\n satisfies the index~$m$ elliptic transformation property~\\eqref{elliptic}\nand has the same poles and residues as~$\\varphi$, so that the difference $\\varphi-\\varphi^P$ is holomorphic and has a theta expansion. In fact, we have:\n\n\\begin{thm}\\label{Thmsingle}  Let $\\varphi(\\tau,z)$ be a meromorphic Jacobi form with simple poles at $z=z_s=\\alpha\\tau+\\beta$ for\n$s=(\\alpha,\\beta)\\in S\\subset{\\mathbb Q}^2$, with Fourier coefficients $h_\\ell(\\tau)$ defined by \\eqref{hAl} and \\eqref{Defofhl} or by~\\eqref{Defofhl2}\nand residues $D_s(\\tau)$ defined by~\\eqref{defDs}.  Then $\\varphi$ has the decomposition\n\\beta\\label{FplusP}  \\varphi(\\tau,z)\\=\\varphi^F(\\tau,z)\\+\\varphi^P(\\tau,z)\\,, \\end{equation}\nwhere $\\varphi^F$ and $\\varphi^P$ are defined by equations \\eqref{phiF} and \\eqref{phiP}, respectively.\n\\end{thm}\n\n\\noindent {\\it Proof:} Fix a point $P=A\\tau+B\\in{\\mathbb C}$ with $(A,B)\\,\\in\\,{\\mathbb R}^2\\smallsetminus S$.  Since $\\varphi$, $\\varphi^F$ and $\\varphi^P$ are meromorphic, it\nsuffices to prove the decomposition~\\eqref{FplusP} on the horizontal line $\\,\\Im(z)=\\Im(P)=A\\tau_2$.  On this line we have the Fourier expansion\n$$ \\varphi(\\tau,z)\\=\\sum_{\\ell\\,\\in\\,{\\mathbb Z}} q^{\\ell^2\/4m}\\,h_\\ell^{(P)}(\\tau)\\,y^\\ell  \\,, $$\nwhere the coefficients $h_\\ell^{(P)}$ are defined by~\\eqref{hAl} (modified as explained in the text there if $A=\\alpha$ for\nany $(\\alpha,\\beta)\\in S$, but for simplicity we will simply assume that this is not the case, since we are free to choose~$A$ any\nway we want).  Comparing this with~\\eqref{phiF} gives\n\\beta\\label{v-vF} \\varphi(\\tau,z)\\,-\\,\\varphi^F(\\tau,z) \\= \\sum_{\\ell\\,\\in\\,{\\mathbb Z}}\\bigl(h_\\ell^{(P)}(\\tau)-h_\\ell(\\tau)\\bigr)\\,q^{\\ell^2\/4m}\\,y^\\ell \\qquad(\\Im(z)=\\Im(P))\\,.\\end{equation}\nBut $q^{\\ell^2\/4m}(h_\\ell^{(P)}(\\tau)-h_\\ell(\\tau))$ is just $2\\pi i$ times the sum of the residues of $\\varphi(\\tau,z)\\mathbf{e}(-\\ell z)$ in the parallelogram\nwith width~1 and horizontal sides at heights $A\\tau_2$ and $-\\ell\\tau_2\/2m$, with the residues of any poles on the latter line being\ncounted with multiplicity~1\/2 because of the way we defined $h_\\ell$ in that case, so\n\\begin{align}  q^{\\ell^2\/4m}\\bigl(h_\\ell^{(P)}(\\tau)-h_\\ell(\\tau)\\bigr) \n  &\\=2\\pi i\\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}} \\frac{\\mbox{sgn}(\\alpha-A)-\\mbox{sgn}(\\alpha+\\ell\/2m)}2\\;\\text{Res}_{z=z_s}\\bigl(\\varphi(\\tau,z)\\mathbf{e}(-\\ell z)\\bigr) \\notag \\\\\n    &\\= \\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}} \\frac{\\mbox{sgn}(\\alpha-A)-\\mbox{sgn}(\\ell+2m\\alpha)}2\\;D_s(\\tau)\\,\\mathbf{e}(-(\\ell+m\\alpha)z_s)\\,. \\notag  \\end{align} \n(Here ``$(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}$\" means that we consider all $\\alpha$, but $\\beta$ only modulo~1, which is the same by periodicity as\nconsidering only the $(\\alpha,\\beta)$ with $B\\le \\beta<B+1$.)  Inserting this formula into~\\eqref{v-vF} and using~\\eqref{Rcseries2}, we find\n\\begin{align}   \\varphi(\\tau,z)\\,-\\,\\varphi^F(\\tau,z)\n   &\\= -\\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}} \\mathbf{e}(-m\\alpha z_s)\\,D_s(\\tau)\\sum_{\\ell\\in{\\mathbb Z}} \\frac{\\mbox{sgn}(\\Im(z-z_s))+\\mbox{sgn}(\\ell+2m\\alpha)}2\\,\\bigr(\\frac y{y_s}\\bigl)^\\ell \\notag\\\\ \n  &\\= \\;\\,\\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}} \\,\\mathbf{e}(-m\\alpha z_s)\\;D_s(\\tau)\\;\\mathcal R_{-2m\\alpha}(y\/y_s) \\notag\\\\\n  &\\= \\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}^2}\\,\\sum_{\\lambda\\in{\\mathbb Z}}\\mathbf{e}\\bigl(-m(\\alpha-\\lambda)(z_s-\\lambda\\tau)\\bigr)\\,D_{(\\alpha-\\lambda,\\beta)}(\\tau)\\;\\mathcal R_{-2m(\\alpha-\\lambda)}(q^\\lambda y\/y_s) \\notag \\\\\n   &\\= \\sum_{s=(\\alpha,\\beta)\\,\\in\\, S\/{\\mathbb Z}^2}\\,D_s(\\tau)\\,\\mathbf{e}(-m\\alpha z_s)\\;\\sum_{\\lambda\\in{\\mathbb Z}}q^{m\\lambda^2}\\,y^{2m\\lambda}\\,\\mathcal R_{-2m\\alpha}(q^\\lambda y\/y_s)\\,, \\notag  \\end{align}\nwhere in the last line we have used the periodicity property~\\eqref{Dell} of $D_s(\\tau)$ together with the obvious periodicity property\n$\\mathcal R_{c+n}(y)=y^n\\mathcal R_c(y)$ of $\\mathcal R_c(y)$.  But the inner sum in the last expression is just $\\Av m{\\mathcal R_{-2m\\alpha}(y\/y_s)}$, so from the\ndefinition~\\eqref{defFms} we see that this agrees with $\\varphi^P(\\tau,z)$, as claimed.\n\n\n\\subsection{Mock modularity of the Fourier coefficients \\label{MockDecomposition}}\n\nIn subsections \\S\\ref{FinitePart} and \\S\\ref{PolarPart} we introduced a canonical splitting of a meromorphic Jacobi\nform $\\varphi$ into a finite part $\\varphi^F$ and a polar part $\\varphi^P$, but there is no reason yet (apart from the simplicity of\nequation~\\eqref{Defofhl}) to believe that the choice we have made is the ``right\" one:  we could have defined\nperiodic Fourier coefficients $h_\\ell(\\tau)$ in many other ways (for instance, by taking $P=P_0-\\ell\/2m\\tau$ with any\nfixed $P_0\\in{\\mathbb C}$ or more generally $P=P_\\ell-\\ell\\tau\/2m$ where $P_\\ell$ depends only on $\\ell$ modulo~$2m$)\nand obtained other functions $\\varphi^F$ and~$\\varphi^P$. What makes the chosen decomposition special is that, as we will now\nshow, the Fourier coefficients defined in~\\eqref{Defofhl} are (mixed) mock modular forms and the function $\\varphi^F$\ntherefore a (mixed) mock Jacobi form in the sense of \\S\\ref{MMMF}.  The corresponding shadows will involve\ntheta series that we now introduce.\n\nFor  $m\\in{\\mathbb N}$, $\\,\\ell\\in{\\mathbb Z}\/2m{\\mathbb Z}$ and $s=(\\alpha,\\beta)\\in{\\mathbb Q}^2$  we define the unary theta series\n\\beta\\label{deftheta3\/2} \n  \\Theta^s_{m,\\ell}(\\tau) \\= \\mathbf{e}(-m\\alpha\\beta)\\,\\sum_{\\lambda\\,\\in\\,{\\mathbb Z}+\\alpha+\\ell\/2m}\\lambda\\,\\mathbf{e}(2m\\beta\\lambda)\\,q^{m\\lambda^2}  \\end{equation}\nof weight 3\/2 and its Eichler integral\\footnote{Strictly speaking, the Eichler integral as defined by equation~\\eqref{defstar}\nwith $k=1\/2$ would be this multiplied by $2\\sqrt{\\pi\/m}$, but this normalization will lead to simpler\nformulas and, as already mentioned, there is no good universal normalization for the shadows of mock modular forms.}   \n\\beta\\label{defEichler3\/2} \n \\Theta^{s\\,*}_{m,\\ell}(\\tau) \\= \\frac{\\mathbf{e}(m\\alpha\\beta)}2 \\,\\sum_{\\lambda\\,\\in\\,{\\mathbb Z}+\\alpha+\\ell\/2m}\n      \\mbox{sgn}(\\lambda)\\,\\mathbf{e}(-2m\\beta\\lambda)\\,\\text{erfc}\\bigl(2|\\lambda|\\sqrt{\\pi m\\tau_2}\\bigr)\\,q^{-m\\lambda^2}  \\end{equation}\n(cf.~\\eqref{defstar} and \\eqref{erfc}). One checks easily that these functions transform by\n\\begin{eqnarray}    \\Theta^{(\\alpha+\\lambda,\\beta+\\mu)}_{m,\\ell}(\\tau) \\=& \\mathbf{e}(m(\\mu\\alpha-\\lambda\\beta))\\,\\Theta^{(\\alpha,\\beta)}_{m,\\ell}(\\tau)  &\\qquad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,, \\\\\n     \\Theta^{(\\alpha+\\lambda,\\beta+\\mu)\\,*}_{m,\\ell}(\\tau) \\=& \\mathbf{e}(-m(\\mu\\alpha-\\lambda\\beta))\\,\\Theta^{(\\alpha,\\beta)\\,*}_{m,\\ell}(\\tau) &\\qquad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,. \\end{eqnarray}\nwith respect to translations of $s$ by elements of ${\\mathbb Z}^2$.\nFrom this and \\eqref{Dell} it follows that the products $D_s\\overline{\\Theta^s_{m,\\ell}}$ and $D_s\\Theta^{s\\,*}_{m,\\ell}$ \ndepend only on the class of $s$ in $S\/{\\mathbb Z}^2$, so that the sums over~$s$ occurring in the following theorem make sense.\n\n\\begin{thm}\\label{Thmmock} Let $\\varphi$, $h_\\ell$ and $\\varphi^F$ be as in Theorem~\\ref{Thmsingle}.  Then each $h_\\ell$ is a mixed\nmock modular form of weight $k-\\frac12\\,|\\,k-1$, with shadow $\\sum_{s\\in S\/{\\mathbb Z}^2} D_s(\\tau)\\,\\overline{\\Theta^s_{m,\\ell}(\\tau)}$, and the\nfunction $\\varphi^F$ is a mixed mock Jacobi form.  More precisely, for each $\\,\\ell\\in{\\mathbb Z}\/2m{\\mathbb Z}\\,$ the completion of $h_\\ell$ defined by\n\\beta\\label{defhcompleted} \\widehat h_\\ell(\\tau)\\;:=\\; h_\\ell(\\tau)\\; - \\; \\sum_{s\\in S\/{\\mathbb Z}^2} D_s(\\tau)\\,\\Theta^{s\\,*}_{m,\\ell}(\\tau)\\,, \\end{equation}\nwith $\\Theta^{s\\,*}_{m,\\ell}$ as in~\\eqref{defEichler3\/2}, tranforms like a modular form of weight $k-1\/2$ with\nrespect to some congruence subgroup of $SL(2,{\\mathbb Z})$, and the completion of $\\varphi^F$ defined by\n\\beta\\label{defphiFhat}  \\widehat\\varphi^F(\\tau,z)\\;:=\\;   \\sum_{\\ell\\; \\mod \\; {2m}} \\widehat h_\\ell(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z) \\end{equation}\ntransforms like a Jacobi form of weight~$k$ and index~$m$ with respect to the full modular group.\n\\end{thm}\n\nThe key property needed to prove this theorem is the following proposition, essentially due to Zwegers, which says that the\nfunctions $\\mathcal{A}_m^s(\\tau,z)$ defined in \\S\\ref{PolarPart} are (meromorphic) mock Jacobi forms of weight 1 and index~$m$, with shadow \n$\\sum_{\\ell\\;\\text{(mod $2m$)}}\\overline{\\Theta^s_{m,\\ell}(\\tau)}\\,\\vartheta_{m,\\ell}(\\tau,z)$ (more precisely, that each $\\mathcal{A}_m^s$ is\na meromorphic mock Jacobi form of this weight, index and shadow with respect to some congruence subgroup of $SL(2,{\\mathbb Z})$ \ndepending on~$s$ and that the collection of all of them is a vector-valued mock Jacobi form on the full modular group):  \n\\begin{proposition}\\label{Sander} For $m\\in{\\mathbb N}$ and $s\\in{\\mathbb Q}^2$ the completion $\\widehat \\mathcal{A}_m^s$ of $\\mathcal{A}_m^s$ defined by\n  \\beta \\label{defFsmhat} \\widehat \\mathcal{A}^s_m(\\tau,z) \\;:=\\; \\mathcal{A}^s_m(\\tau,z) \\; + \\sum_{\\text{\\rm $\\ell$ (mod $2m$)}} \\Theta^{s\\,*}_{m,\\ell}(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z)\\;. \\end{equation}\nsatisfies\n\\begin{align}\\label{FsmPeriodic}    \\widehat\\mathcal{A}^{(\\alpha+\\lambda,\\beta+\\mu)}_m(\\tau,z) &\\=\\mathbf{e}(-m(\\mu\\alpha-\\lambda\\beta))\\,\\widehat\\mathcal{A}^{(\\alpha,\\beta)}_m(\\tau)\\qquad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,,   \\\\\n  \\label{FsmElliptic}       \\widehat \\mathcal{A}^s_m(\\tau,z+\\lambda\\tau+\\mu) &\\= \\mathbf{e}(-m(\\lambda^2\\tau+2\\lambda z))\\,\\widehat \\mathcal{A}^s_m(\\tau) \\qquad\\quad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,, \\\\\n  \\label{FsmModular}  \\widehat \\mathcal{A}^s_m\\Bigl(\\frac{a\\tau+b}{c\\tau+d},\\,\\frac z{c\\tau+d}\\Bigr)\\!&\\= \n     (c\\tau+d)\\;\\mathbf{e}\\Bigl(\\frac{mcz^2}{c\\tau+d}\\Bigr)\\,\\widehat\\mathcal{A}^{s\\gamma}_m(\\tau,z) \\quad\\;\n    \\Bigl(\\gamma=\\Bigl(\\begin{matrix} a&b\\\\gamma&d\\end{matrix}\\Bigr)\\,\\in\\, SL(2,{\\mathbb Z})\\Bigl)\\,.      \\end{align}\n\\end{proposition}\n\\noindent {\\it Proof:} The first two properties are easy to check because they hold for each term in \\eqref{defFsmhat} separately.\nThe modular transformation property, considerably less obvious, is essentially the content of Proposition~3.5 of~\\cite{Zwegers:2002},\nbut the functions he studies are different from ours and we need a small calculation to relate them.  Zwegers defines two\nfunctions $f_u^{(m)}(z;\\tau)$ and $\\tilde f_u^{(m)}(z;\\tau)$ ($m\\in{\\mathbb N}$, $\\tau\\in\\mathbb H$, $\\,z,\\,u\\in{\\mathbb C}$) by\n$$ f_u^{(m)}(z;\\tau)= \\AvB m{\\frac1{1-y\/\\mathbf{e}(u)}},\\quad\\;\n   \\tilde f_u^{(m)}(z;\\tau) =  f_u^{(m)}(z;\\tau)-\\frac12\\sum_{\\text{$\\ell$ (mod $2m$)}} R_{m,\\ell}(u;\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z) $$\n(here we have rewritten Definition 3.2 of~\\cite{Zwegers:2002} in our notation), where \n\\beta\\label{defRml}  R_{m,\\ell}(u;\\tau) \\= \\sum_{r\\in\\ell+2m{\\mathbb Z}}\\Bigl\\{\\mbox{sgn}\\bigl(r+\\frac12\\bigr)\n   \\,-\\,\\text{erf}\\Bigl(\\sqrt\\pi\\,\\frac{r\\tau_2+2mu_2}{\\sqrt{m\\tau_2}}\\Bigr)\\Bigr\\}\\;q^{-r^2\/4m}\\;\\mathbf{e}(-ru)\\,, \\end{equation}\nand shows (Proposition 3.5) that $\\tilde f_u^{(m)}$ satisfies\nthe modular transformation property \n\\beta\\label{ZwegersMod}  \\tilde f_{u\/(c\\tau+d)}^{(m)}\\Bigl(\\frac z{c\\tau+d};\\,\\frac{a\\tau+b}{c\\tau+d}\\Bigr)\n   \\= (c\\tau+d)\\;\\mathbf{e}\\Bigl(\\frac{mc(z^2-u^2)}{c\\tau+d}\\Bigr)\\,\\tilde f_u^{(m)}(z;\\tau)  \\end{equation}\nfor all $\\gamma= \\Bigl(\\begin{matrix} a&b\\\\gamma&d\\end{matrix}\\Bigr)\\in SL(2,{\\mathbb Z})$.  Noting that $\\,\\text{erf}(x)=\\mbox{sgn}(x)(1-\\text{erfc}(|x|))$, we find that \n$$  \\frac12\\,R_{m,\\ell}(z_s;\\tau)\\;=\\sum_{r\\equiv\\ell\\text{ (mod $2m$)}} \\frac{\\mbox{sgn}(r+\\frac12)-\\mbox{sgn}(r+2m\\alpha)}2\\,q^{-r^2\/4m}\\,y_s^{-r}\n    \\+ \\mathbf{e}(m\\alpha z_s)\\,\\Theta^{s\\,*}_{m,l}(\\tau) $$\nin our notation.  On the other hand, from~\\eqref{defRc} we have\n\\beta  \\mathcal R_{-2m\\alpha}(y)\\=\\frac1{y-1} \\+\\sum_{r\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(r+\\frac12)-\\mbox{sgn}(r+2m\\alpha)}2\\,y^r \\end{equation}\n(note that the sum here is finite).  Replacing $y$ by $y\/y_s$ and applying the operator $\\text{Av}^{(m)}$, we find (using~\\eqref{ThetAv})\n$$ \\mathbf{e}(m\\alpha z_s)\\,\\mathcal{A}_m^s(\\tau,z)\\=-f^{(m)}_{z_s}(z;\\tau)\\+ \\sum_{r\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(r+\\frac12)-\\mbox{sgn}(r+2m\\alpha)}2\\,q^{-r^2\/4m}\\,y_s^{-r}\\,\\vartheta_{m,r}(\\tau,z)\\,. $$\nCombining these two equations and rearranging, we obtain \n$$ \\widehat \\mathcal{A}_m^s(\\tau,z) \\= -\\,\\mathbf{e}(-m\\alpha z_s)\\,\\tilde f^{(m)}_{z_s}(z;\\tau)\\,,$$\nand the modularity property~\\eqref{FsmModular} then follows from~\\eqref{ZwegersMod} after a short calculation.\n\n\\smallskip\nThe proof of Theorem~\\ref{Thmmock} follows easily from Proposition~\\ref{Sander}. We define the completion of the function $\\varphi^P$ studied in~\\S\\ref{PolarPart} by\n \\beta\\label{defPolarhat} \\widehat\\varphi^P(\\tau,z)\\;:=\\; \\sum_{s\\in S\/{\\mathbb Z}^2} D_s(\\tau)\\;\\widehat \\mathcal{A}^s_m(\\tau,z)\\,. \\end{equation}\nThe sum makes sense by \\eqref{FsmPeriodic}, and from the transformation equations \\eqref{FsmElliptic}--\\eqref{FsmModular}\ntogether with the corresponding properties \\eqref{Dell}--\\eqref{Dmod} of the residue functions $D_s(\\tau)$ it follows that \n$\\widehat\\varphi^P(\\tau,z)$ transforms like a Jacobi form of weight~$k$ and index~$m$ with respect to the full modular group.\nComparing equations~\\eqref{defPolarhat} and~\\eqref{defFsmhat} with equations~\\eqref{defphiFhat} and~\\eqref{defhcompleted}, we find that\n$$ \\widehat\\varphi^P(\\tau,z) \\,-\\, \\varphi^P(\\tau,z) \n    \\= \\sum_{\\ell\\,\\in\\,{\\mathbb Z}\/2m{\\mathbb Z}}\\sum_{s\\,\\in\\, S\/{\\mathbb Z}^2} D_s(\\tau)\\;\\Theta^{s\\,*}_{m,\\ell}(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z) \n    \\=  \\varphi^F(\\tau,z)\\,-\\,\\widehat\\varphi^F(\\tau,z)   $$\nand hence, using Theorem~\\ref{Thmsingle}, that\n$$  \\widehat\\varphi^F(\\tau,z) \\+ \\widehat\\varphi^P(\\tau,z) \\= \\varphi^F(\\tau,z)\\+\\varphi^P(\\tau,z) \\= \\varphi(\\tau,z)\\,. $$\nSince both $\\varphi(\\tau,z)$ and $\\widehat\\varphi^P(\\tau,z)$ transform like Jacobi forms of weight~$k$ and index~$m$, it follows that $\\widehat\\varphi^F(\\tau,z)$\nalso does, and then the fact that each $\\widehat h_\\ell$ transforms like a modular form of weight $k-1\/2$ (and hence that each $h_\\ell$ is \na mixed mock modular form with the weight and shadow given in the theorem) follows by the same argument that proves the modularity of\nthe coefficients $h_\\ell$ in the theta expansion~\\eqref{jacobi-theta} in the classical case.\n\n\n\\bigskip\n {\\bf Summary.}  For the reader's convenience, we give a brief summary of the results given up to now.  We have the following six\nfunctions of two variables $(\\tau,z)\\in\\mathbb{H}\\times{\\mathbb C}\\,$:\n\\begin{myitemize}\n\\item  $\\varphi(\\tau,z)$, a meromorphic Jacobi form of weight $k$ and index~$m$, assumed to have only simple poles\n   at $z=z_s=\\alpha\\tau+\\beta$ for $s=(\\alpha,\\beta)$ in some discrete subset $S\\subset{\\mathbb Q}^2\\,$;\n\\item $\\varphi^F(\\tau,z)$, the {\\bf finite} part of $\\varphi$, defined by the theta expansion $\\sum_{\\text{$\\ell$ (mod $2m$)}}h_\\ell(\\tau)\\vartheta_{m,\\ell}(\\tau,z)$\n  where $h_\\ell(\\tau)$ is $q^{\\ell^2\/4m}$ times the $\\ell$th Fourier coefficient of $\\varphi(\\tau,z-\\ell\\tau\/2m)$ on the real line;\n\\item $\\varphi^P(\\tau,z)$, the {\\bf polar} part of $\\varphi$, defined as $\\sum_{s\\in S\/{\\mathbb Z}^2} D_s(\\tau)\\mathcal{A}_m^s(\\tau,z)$, where $\\mathcal{A}_m^s$ is an \n   explicit Appell-Lerch   sum having simple poles at $z\\in z_s+{\\mathbb Z}\\tau+{\\mathbb Z}\\,$;\n\\item $\\varphi^{C}(\\tau,z)$, a non-holomorphic {\\bf correction} term, defined as $\\sum_s\\sum_\\ell D_s(\\tau)\\Theta_{m,\\ell}^{s\\,*}(\\tau)\\vartheta_{m,\\ell}(\\tau,z)$ where\n  the $\\Theta_{m,\\ell}^{s\\,*}$ are the Eichler integrals of explicit unary theta series of weight $3\/2\\,$;\n\\item $\\widehat \\varphi^F(\\tau,z)$, the completed finite part, defined as $\\varphi^F(\\tau,z)-\\varphi^{C}(\\tau,z)\\,$;\n\\item $\\widehat \\varphi^P(\\tau,z)$, the completed polar part, defined as $\\varphi^P(\\tau,z)+\\varphi^{C}(\\tau,z)\\,$.\n\\end{myitemize}\nThese six functions are related by\n\\beta\\label{relations}  \\varphi^F\\+\\varphi^P \\=\\varphi\\= \\widehat\\varphi^F\\+\\widehat\\varphi^P\\;,\\qquad \\varphi^F\\,-\\,\\widehat\\varphi^F\\=\\varphi^{C}\\=\\widehat\\varphi_P\\,-\\,\\varphi_P\\;.  \\end{equation}\nEach of them is real-analytic in $\\tau$, meromorphic in~$z$, satisfies the elliptic transformation property~\\eqref{elliptic} with respect to~$z$,\n and has precisely two of four further desirable properties of such a function (note that $6=\\bigl({4\\atop2}\\bigr)$), as shown in the following table\n   \\begin{center}     \\begin{tabular}{r|c|c|c|c|c|c|}\n   \\multicolumn 1{c}{} & \\multicolumn 1{c}{$\\;\\;\\varphi\\;\\;$} & \\multicolumn 1{c}{$\\;\\varphi^F\\;$} \n      & \\multicolumn 1{c}{$\\;\\varphi^{\\rm P}\\;$} & \\multicolumn 1{c}{$\\;\\,\\varphi^{C}\\;\\,$} & \\multicolumn 1{c}{$\\;\\widehat\\varphi^{\\rm F}\\;$} & \\multicolumn 1{c}{$\\;\\widehat\\varphi^{\\rm P}\\;$}\\\\\n      \\cline{2-7}        holomorphic in $\\tau\\,?\\;$ & \\checkmark & \\checkmark & \\checkmark & $-$  & $-$  & $-$ \\\\\n               transforms like a Jacobi form\\,?\\; & \\checkmark & $-$  & $-$  & $-$  & \\checkmark & \\checkmark\\\\\n                   holomorphic in $z\\,?\\;$  & $-$ & \\checkmark & $-$  & \\checkmark & \\checkmark& $-$ \\\\\n     determined by the poles of $\\varphi\\,?\\;$ & $-$  & $-$ & \\checkmark  & \\checkmark  & $-$ & \\checkmark  \\\\\n   \\cline{2-7}      \\end{tabular} \\qquad\\qquad\\qquad   \\end{center}   \\bigskip\nin which the three checked entries in each row  correspond to one of the equations~\\eqref{relations}.  Each Fourier\ncoefficient $h_\\ell$ of $\\varphi$ is a mixed mock modular form of weight $(k-1,1\/2)\\,$, and the finite part $\\varphi^F$ is a mixed mock Jacobi form. \nIn the holomorphic case, the functions $\\varphi$, $\\varphi^F$ and $\\widehat\\varphi^F$ coincide and the functions $\\varphi^P$, $\\varphi^{C}$ and~$\\widehat\\varphi^P$ vanish.  \n\nWe end by mentioning one further property of the canonical decomposition $\\varphi=\\varphi^F+\\varphi^P$ that seems of interest.  The finite part $\\varphi^P$\nof~$\\varphi$ is a linear combination of terms $h_\\ell(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z)$ where $h_\\ell(\\tau)$ is a Laurent power series in~$q$ and\n$\\vartheta_{m,\\ell}(\\tau,z)$ a linear combination of terms $q^{r^2\/4m}y^r$, so it satisfies the same growth condition as a weak Jacobi form,\nviz., that it contains only monomials $q^ny^r$ with discriminant $4mn-r^2$ bounded from below.  But the Appell-Lerch sums $\\mathcal{A}_m^s(\\tau,z)$\noccurring in $\\varphi^P(\\tau,z)$ have the opposite property: substituting the Fourier expansion~\\eqref{Rcseries2} of $\\mathcal R_c$ into the\ndefinition~\\eqref{defFms}, we find that they contain only terms $q^ny^r$ with $4mn-r^2=-\\delta^2$ where the numbers $\\delta$ occurring\nare rational (more precisely, $\\delta\\in r+2m({\\mathbb Z}+\\alpha)$) and unbounded (more precisely, $\\delta$ lies between~0 and~$r+\\text O(1)$).\n\n\n\\subsection{The case of double poles \\label{double}}\n\nIn this subsection we extend our considerations to the case when $\\varphi$ is allowed to have double poles,\nagain assumed to be at points $z=z_s=\\alpha\\tau+\\beta$ for $s=(\\alpha,\\beta)$ belonging to some discrete subset $S$ of~${\\mathbb Q}^2$.\nThe first thing we need to do is to generalize the definition~\\eqref{defDs} to this case. For $s\\in S$ we\ndefine functions $E_s$ and $D_s$ on $\\mathbb H$ by the Laurent expansion\n  \\beta\\label{defDE}    \\mathbf{e}(m\\alpha z_s)\\,\\varphi(\\tau,z_s+\\varepsilon) \n    \\= \\frac{E_s(\\tau)}{(2\\pi i\\varepsilon)^2} \\+ \\frac{D_s(\\tau)-2m\\alpha\\,E_s(\\tau)}{2\\pi i\\varepsilon} \\+ O(1) \\qquad\\text{as $\\varepsilon\\to 0$.}  \\end{equation}\n(Notice that $D_s(\\tau)$ is the same function as in \\eqref{defDs} when the pole is simple.)\nIt is easily checked that the behavior of these functions under translations of $s$ by elements of~${\\mathbb Z}^2$ is given by\nequation~\\eqref{Dell} and its analogue for $E_s$.  For the modular behavior, we have:\n\\begin{proposition}  The functions $E_s(\\tau)$ and $D_s(\\tau)$ defined by~\\eqref{defDE} are modular forms of weight $k-2$ and $k-1$,\nrespectively. More respectively, for all $s\\in S$ and $\\sm abcd\\in SL(2,{\\mathbb Z})$ we have\n  \\beta\\label{modEDdoub}   E_s\\Bigl(\\frac{a\\tau+b}{c\\tau+d}\\Bigr)\\=(c\\tau+d)^{k-2}E_{s\\gamma}(\\tau)\\,, \n     \\qquad D_s\\Bigl(\\frac{a\\tau+b}{c\\tau+d}\\Bigr)\\=(c\\tau+d)^{k-1}D_{s\\gamma}(\\tau)\\,.   \\end{equation}\n\\end{proposition}\n\\noindent {\\it Proof:} We rewrite~\\eqref{defDE} as\n $$\\mathbf{e}(m\\alpha(z_s+2\\varepsilon))\\,\\varphi(\\tau,z_s+\\varepsilon) \\=  \\frac{E_s(\\tau)}{(2\\pi i\\varepsilon)^2} \\+ \\frac{D_s(\\tau)}{2\\pi i\\varepsilon} \\+ \\text O(1)\\,,$$\nand also write $\\alpha_s$ and $z_s(\\tau)$ instead of just $\\alpha$ and $z_s$. Then using the easily checked identities\n$$  z_s(\\gamma\\tau)\\=\\frac{z_{s\\gamma}(\\tau)}{c\\tau+d}\\,,\\qquad \\alpha_{s\\gamma}\\,z_{s\\gamma}(\\tau)\\,-\\,\\alpha_s\\,z_s(\\gamma\\tau)\\=\\frac{c\\,z_{s\\gamma}(\\tau)^2}{c\\tau+d}  \n     \\qquad\\Bigl(\\gamma\\tau:=\\frac{a\\tau+b}{c\\tau+d}\\;\\Bigr)\\,, $$\nand the modular transformation equation~\\eqref{modtransform}, we find \n\\begin{align}   \\frac{(c\\tau+d)^2\\,E_s(\\gamma\\tau)}{(2\\pi i\\varepsilon)^2} &\\+ \\frac{(c\\tau+d)\\,D_s(\\gamma\\tau)}{2\\pi i\\varepsilon}  \n    \\equiv\\; \\mathbf{e}\\Bigl(m\\alpha\\Bigl(z_s(\\gamma\\tau)+\\frac{2\\varepsilon}{c\\tau+d}\\Bigr)\\Bigr)\\,\\varphi\\Bigl(\\gamma\\tau,\\,z_s(\\gamma\\tau)+\\frac\\varepsilon{c\\tau+d}\\Bigr)    \\notag \\\\\n  &\\qquad\\equiv\\;  \\mathbf{e}\\Bigl(m\\alpha\\frac{z_{s\\gamma}(\\tau)+2\\varepsilon}{c\\tau+d}\\Bigr)\\,\\varphi\\Bigl(\\gamma\\tau,\\,\\frac{z_{s\\gamma}(\\tau)+\\varepsilon}{c\\tau+d}\\Bigr) \\notag \\\\ \n  &\\qquad\\equiv\\;(c\\tau+d)^k\\,\\mathbf{e}\\Bigl(m\\alpha\\frac{z_{s\\gamma}+2\\varepsilon}{c\\tau+d}+\\frac{mc(z_{s\\gamma}+\\varepsilon)^2}{c\\tau+d}\\Bigr)\\,\\varphi\\Bigl(\\tau,\\,z_{s\\gamma}+\\varepsilon\\Bigr) \\notag\\\\\n  &\\qquad\\equiv\\;(c\\tau+d)^k\\,\\mathbf{e}(m\\alpha_{s\\gamma}(z_{s\\gamma}+2\\varepsilon))\\,\\varphi(\\tau,\\,z_{s\\gamma}+\\varepsilon) \\notag\\\\\n  &\\qquad\\equiv\\;(c\\tau+d)^k\\,\\Bigl[\\frac{E_{s\\gamma}(\\tau)}{(2\\pi i\\varepsilon)^2} \\+ \\frac{D_{s\\gamma}(\\tau)}{2\\pi i\\varepsilon} \\Bigr]\\;,\\notag  \\end{align}\nwhere ``$\\,\\equiv\\,$\" means equality modulo a quantity that is bounded as $\\varepsilon\\to0$. The claim follows.\n\\medskip\n\nNext, we must define a standard Appell-Lerch sum with a double pole at $z=z_s$.  We begin by defining a rational\nfunction $\\mathcal R^{(2)}_c(y)$ with a double pole at $y=1$ for each $c\\in{\\mathbb R}$. Motivated by Proposition~\\ref{RcAsAv}, we require\n\\beta\\label{RRcAsAv}   \\mathcal R^{(2)}_c(\\mathbf{e}(z))\\= \\text{Av}_\\IZ\\biggl[\\frac{\\mathbf{e}(cz)}{(2\\pi iz)^2}\\biggr] \n   \\= \\frac1{(2\\pi i)^2}\\,\\sum_{n\\in{\\mathbb Z}} \\frac{\\mathbf{e}(c(z-n))}{(z-n)^2}\\,. \\end{equation}\nTo calculate this explicitly as a rational function, we could imitate the proof of Proposition~\\ref{RcAsAv}, but it is\neasier to note that $\\mathcal R^{(2)}_c= \\bigl(-\\dfrac1{2\\pi i}\\dfrac d{dz}\\+c\\bigr)\\mathcal R_c= \\bigl(-y\\dfrac d{dy}\\+c\\bigr)\\mathcal R_c$ and hence\nfrom equations~\\eqref{Rcseries2}, \\eqref{Rcseries} and~\\eqref{defRc} we get three alternative definitions\n\\begin{align}\\label{defRRseries} \\mathcal R^{(2)}_c(y) &\\=  \\sum_{\\ell\\in{\\mathbb Z}}\\frac{|\\ell-c| \\+ \\mbox{sgn}(z_2)\\,(\\ell-c)}2\\;y^\\ell  \\\\\n \\label{defRRc2}  &\\= \\begin{cases} \\sum_{\\ell\\ge c} (\\ell-c)\\,y^\\ell &\\text{if $|y|<1$} \\\\\n                            \\sum_{\\ell\\le c} (c-\\ell)\\,y^\\ell &\\text{if $|y|>1$}\\end{cases}  \\\\\n          &\\=    y^{\\flo c+1}\\,\\biggl(\\frac 1{(y-1)^2} \\+ \\frac{c\\,-\\,\\flo c}{y-1}\\biggr)\\,. \\end{align}\nof $\\mathcal R^{(2)}_c(y)$. (Notice that in these equations neither the asterisk on the summation sign nor the case distinction for $c\\in{\\mathbb Z}$ and $c\\notin{\\mathbb Z}$\nare needed as before, and that the function $\\mathcal R^{(2)}_c(y)$, unlike $\\mathcal R_c(y)$, is continuous in~$c$.)  For $s=(\\alpha,\\beta)\\in{\\mathbb Q}^2$ and $m\\in{\\mathbb N}$ we set \n  \\beta \\label{defGms} \\mathcal{B}_m^s(\\tau,z)\\= \\mathbf{e}(-m\\alpha z_s)\\,\\Av m{\\mathcal R^{(2)}_{-2m\\alpha}(y\/y_s)}\\,, \\end{equation}\nin analogy with \\eqref{defFms}.  If we then define the polar part $\\varphi^P$ of $\\varphi$ by\n\\beta \\label{phiPdoub}  \\varphi^P(\\tau,z)\\= \\sum_{s\\in S\/{\\mathbb Z}^2}\\Bigl( D_s(\\tau)\\,\\mathcal{A}_m^s(\\tau,z)\\+ E_s(\\tau)\\,\\mathcal{B}_m^s(\\tau,z) \\Bigr)\\,; \\end{equation}\nthen the definitions of the functions $D_s$, $E_s$, $\\mathcal{A}_m^s$ and $\\mathcal{B}_m^s$ immediately imply that $\\varphi^P$ has the same\nsingularities as~$\\varphi$, so that the difference \n  \\beta\\label{NewFinitePart}  \\varphi^F(\\tau,z) \\= \\varphi(\\tau,z)\\,-\\,\\varphi^P(\\tau,z) \\end{equation}\nis a holomorphic function of $z$.\n\nAs before, the key property of the Appell-Lerch sums is that they are again mock Jacobi forms, of a somewhat\nmore complicated type than before.  We introduce the unary theta series\n\\beta\\label{deftheta1\/2} \n  \\vartheta^s_{m,\\ell}(\\tau) \\= \\mathbf{e}(-m\\alpha\\beta)\\,\\sum_{\\lambda\\,\\in\\,{\\mathbb Z}+\\alpha+\\ell\/2m}\\mathbf{e}(2m\\beta\\lambda)\\,q^{m\\lambda^2}  \\end{equation}\nof weight 1\/2 and its (again slightly renormalized) Eichler integral \n\\beta\\label{defEichler1\/2} \n  \\vartheta^{s\\,*}_{m,\\ell}(\\tau) \\= \\frac{\\overline{\\vartheta^s_{m,\\ell}(\\tau)}}{2\\pi\\sqrt{m\\tau_2}} \\,-\\, \n   \\mathbf{e}(m\\alpha\\beta)\\!\\sum_{\\lambda\\,\\in\\,{\\mathbb Z}+\\alpha+\\ell\/2m} |\\lambda|\\,\\mathbf{e}(-2m\\beta\\lambda)\\,\\text{erfc}\\bigl(2|\\lambda|\\sqrt{\\pi m\\tau_2}\\bigr)\\,q^{-m\\lambda^2}  \\end{equation}\n(cf.~\\eqref{defstar} and \\eqref{erfc}).  Then we can define the completion $\\widehat\\mathcal{B}_m^s$ of $\\mathcal{B}_m^s$ by \n\\beta \\label{defGsmhat} \\widehat\\mathcal{B}^s_m(\\tau,z)\\;:=\\;\\mathcal{B}^s_m(\\tau,z)\\;+\\,m\\,\\sum_{\\text{\\rm $\\ell$ (mod $2m$)}}\n\\vartheta^{s\\,*}_{m,\\ell}(\\tau)\\,\\vartheta_{m,\\ell}(\\tau,z)\\;.\\end{equation}\n\n\\begin{proposition}\\label{Sander2} For $m\\in{\\mathbb N}$ and $s\\in{\\mathbb Q}^2$ the completion $\\widehat \\mathcal{B}_m^s$ of $\\mathcal{B}_m^s$ defined\nby \\eqref{defGsmhat} satisfies\n\\begin{align}\\label{GsmPeriodic}    \\widehat\\mathcal{B}^{(\\alpha+\\lambda,\\beta+\\mu)}_m(\\tau,z) &\\=\\mathbf{e}(-m(\\mu\\alpha-\\lambda\\beta+\\lambda\\mu))\\,\\widehat\\mathcal{B}^{(\\alpha,\\beta)}_m(\\tau)\\qquad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,,   \\\\\n  \\label{GsmElliptic}       \\widehat\\mathcal{B}^s_m(\\tau,z+\\lambda\\tau+\\mu) &\\= \\mathbf{e}(-m(\\lambda^2\\tau+2\\lambda z))\\,\\widehat\\mathcal{B}^s_m(\\tau) \\qquad\\quad(\\lambda,\\,\\mu\\in{\\mathbb Z})\\,, \\\\\n  \\label{GsmModular}  \\widehat\\mathcal{B}^s_m\\Bigl(\\frac{a\\tau+b}{c\\tau+d},\\,\\frac z{c\\tau+d}\\Bigr)\\!&\\= \n     (c\\tau+d)^2\\;\\mathbf{e}\\Bigl(\\frac{mcz^2}{c\\tau+d}\\Bigr)\\,\\widehat\\mathcal{B}^{s\\gamma}_m(\\tau,z) \\quad\\;\n    \\bigl(\\gamma=\\Bigl(\\begin{matrix} a&b\\\\gamma&d\\end{matrix}\\Bigr)\\in SL(2,{\\mathbb Z})\\bigl)\\,.      \\end{align}\n\\end{proposition}\n\nThe proof is exactly similar to that of~\\ref{Sander}.  We define functions $g_u^{(m)}(z;\\tau)$ and $\\tilde g_u^{(m)}(z;\\tau)$ by\napplying the operator $\\,\\dfrac1{2\\pi i}\\dfrac\\partial{\\partial u}-2m\\dfrac{u_2}{\\tau_2}\\,$ to $f_u^{(m)}(z;\\tau)$ and $\\tilde f_u^{(m)}(z;\\tau)$;\nthen the transformation equation~\\eqref{ZwegersMod} of   $\\tilde f_u^{(m)}$ implies the same transformation equation for  $\\tilde g_u^{(m)}$ but with\nthe initial factor $(c\\tau+d)$ replaced by $(c\\tau+d)^2$, and a calculation exactly similar to the one given before shows that $\\mathbf{e}(m\\alpha z_s)\\mathcal{B}_m^s(\\tau,z)$\ndiffers from $g_{z_s}^{(m)}(z;\\tau)$ by a finite linear combination of functions $\\vartheta_{m,r}(\\tau,z)$ and that \n$\\mathbf{e}(m\\alpha z_s)\\widehat\\mathcal{B}_m^s(\\tau,z)=\\tilde g_{z_s}^{(m)}(z;\\tau)$.  We omit the details.\n\n\\begin{thm}\\label{Thmdouble}  \n Let $\\varphi$ be as above, with singularities at $z=z_s$ $(s\\in S\\subset{\\mathbb Q}^2)$ given by~\\eqref{defDE}.  Then the finite part $\\varphi^F$\nas defined by~\\eqref{NewFinitePart} coincides with the finite part defined by the theta expansion~\\eqref{phiF}, the coefficients $h_\\ell(\\tau)$\nin this expansion are mixed mock modular forms, with completion given by\n $$  \\widehat h_\\ell(\\tau) \\=   h_\\ell(\\tau)\\+\\sum_{s\\in S\/{\\mathbb Z}^2} \\Bigl(D_s(\\tau)\\,\\Theta^{s\\,*}_{m,\\ell}(\\tau) \\+ E_s(\\tau)\\,\\vartheta^{s\\,*}_{m,\\ell}(\\tau)\\Bigr)\\,,$$\nand the completion $\\widehat\\varphi^F$ defined by~\\eqref{defphiFhat} transforms like a Jacobi form of weight~$k$ and index~$m$.\\end{thm}\n\n\n\\smallskip\nThe proof follows the same lines as before: the equivalence of~\\eqref{phiF} and~\\eqref{NewFinitePart} is proved by expanding $\\varphi(\\tau,z)$ as a Fourier\nseries along the horizontal line $\\Im(z)=\\Im(P)$ for some generic point $P\\in{\\mathbb C}$ and calculating the difference $\\varphi-\\varphi^F$ as\na sum of residues, and the mock modularity is proved by decomposing $\\varphi$ as $\\widehat\\varphi^F+\\widehat\\varphi^P$ with $\\widehat\\varphi^F$ as in~\\eqref{defphiFhat}  and\n$\\widehat\\varphi^P=\\sum_{s\\in S\/{\\mathbb Z}^2}\\bigl(D_s\\widehat \\mathcal{A}_m^s+E_s\\widehat\\mathcal{B}_m^s\\bigr)$, which transforms like a Jacobi form by virtue of Proposition~\\ref{Sander2}.\nAgain the details are left to the reader.  Note that here the mock modular forms $h_\\ell(\\tau)$ are of the ``even more mixed\" variety mentioned\nat the end of \\S\\ref{MockJacobi}, since they now have a shadow that is a linear combination of two terms $\\sum_sD_s\\,\\overline{\\Theta_{m,l}^s}$ and\n$\\sum_sE_s\\,\\overline{\\vartheta_{m,l}^s}$ belonging to two different tensor products $M_{k-1}\\otimes\\overline{M_{3\/2}}$ and $M_{k-2}\\otimes\\overline{M_{1\/2}}$\nand hence two different mixed weights $k-\\frac12\\,|\\,k-1$ and $k-\\frac12\\,|\\,k-2$ rather than a single mixed weight $k-\\frac12\\,|\\,k-1$\nas in the case of simple poles.\n\n\n\\subsection{Examples \\label{MJExamples}}  We end this section by giving five examples of meromorphic Jacobi forms\nand their canonical decompositions into a mock Jacobi form and a finite linear combination of Appell-Lerch sums.  We  systematically\nuse the notations $A=\\varphi_{-2,1}$, $B=\\varphi_{0,1}$, $C=\\varphi_{-1,2}$ for the three basic generators of the ring of weak Jacobi forms\nas described in equations~\\eqref{Fourirphi02}--\\eqref{WJF}.\n \n\\medskip\n\\noindent{\\bf Example 1: Weight 1, index 1, simple pole at $z=0$.}  As our first example we take the Jacobi form $\\varphi=C\/A\\in J_{1,1}^{\\text{mer}}$,\nwhich has a simple pole of residue $1\/\\pi i$ at $z=0$ and a Fourier expansion beginning\n$$\\frac{y+1}{y-1}\\,-\\,(y^2-y^{-2})\\,q\\,-\\,2(y^3-y^{-3})\\,q^2\\,-\\,2(y^4-y^{-4})\\,q^3 \\,-\\,(2y^5+y^4-y^{-4}-2y^{-5})\\,q^4\\,-\\,\\cdots\\;.$$  \nCalculating the Fourier expansion of the polar part $\\varphi^P=\\Av1{\\frac{y+1}{y-1}}$, we find that it begins the same way, and indeed, we must have\n$\\varphi=\\varphi^P$ because the Fourier coefficients $h_\\ell$ all vanish (we have $h_{-\\ell}=-h_\\ell$ because the weight is odd and $h_{\\ell+2}=h_\\ell$\nbecause the index is~1).  So here there is no mock Jacobi form at all, but only the polar correction term given by the Appell-Lerch sum, a\nkind of a ``Cheshire cat\" example which is all smile and no cat.\n\n\\medskip \\noindent\n{\\bf Example 2: Weight 1, index 2, simple pole at $z=0$.} As a second example take $\\varphi=BC\/A\\in J_{1,2}^{\\text{mer}}$.  Here we find\n\\begin{eqnarray*} \\varphi^P&\\=&12\\,\\AvB2{\\frac{y+1}{y-1}}\\=12\\,\\frac{y+1}{y-1}\\,-\\,12(y^4-y^{-4})q^2\\+24(y^5-y^{-5})\\,q^3\\+\\cdots \\\\\n\\varphi^F&\\=&(y-y^{-1})\\,-\\,(y^3+45y-45y^{-1}-y^{-3})q\\+(45y^3-231y+231y^{-1}-45y^{-3})q^2\\+\\cdots \\, , \\end{eqnarray*}\nand by comparing the shadows and the first few Fourier coefficients (cf.~eq.~\\eqref{defCF2}), we see \nthat $\\varphi^F$ is the negative of the mock Jacobi form ${\\cal F}_2(\\tau,z)=h^{(2)}(\\tau)\\,\\vartheta_ 1(\\tau,2z)$ \ndiscussed in Example~4 of \\S\\ref{MockJacobi} that is conjecturally related to representations of~$M_{24}\\,$. \nUsing~\\eqref{phiminus1} and dividing by $\\vartheta_1(\\tau,2z)$, we can rewrite the identity $\\varphi=\\varphi^P+\\varphi^F$ as\n  \\beta \\label{CoverA} \n  \\eta(\\tau)^{-3}\\,\\frac{\\varphi_{0,1}(\\tau,z)}{\\varphi_{-2,1}(\\tau,z)} \\= -\\,\\frac{12}{\\vartheta_1(\\tau,2z)}\\,\\AvB2{\\frac{1+y}{1-y}}\\,-\\,h^{(2)}(\\tau)\\,. \\end{equation}\nOne can check that $\\vartheta_1(\\tau,2z)\\mu(\\tau,z)=\\Av2{\\frac{1+y}{1-y}}$, so this agrees with the splitting~\\eqref{K3decomp} coming \nfrom the decomposition of the elliptic genus of $K3$ into superconformal characters.\n\n\\medskip \\noindent\n{\\bf Example 3: Weight 1, index 1\/2, simple pole at $z=0$.} We can also multiply~\\eqref{K3decomp} by $\\vartheta_1(\\tau,z)$ and write it in the form\n  \\beta \\label{BoversqrtA}\n  \\frac B{\\sqrt A} \\= -12\\,\\AvB{\\frac12,-}{\\frac{\\sqrt y}{1-y}} \\;-\\;h^{(2)}(\\tau)\\,\\vartheta_1(\\tau,z)\\,, \\end{equation}\nwhere $\\,\\text{Av}^{(\\frac12,-)}[F(y)]:=\\sum_{n\\in{\\mathbb Z}}(-1)^nq^{n^2\/2}\\,y^n\\,F(q^ny)$.  This decomposition also fits into the scheme explained in\nthis section, but slightly modified because the index here is half-integral.\n\n\n\\medskip \\noindent\n{\\bf Example 4: Weight 2, index 0, double pole at $z=0$.} Now take $\\varphi=B\/A$, which, as we saw in \\S\\ref{MJF1}, is just \na multiple of the Weierstrass ${\\cal P}$-function.This example does not quite fit in with \nthe discussion given so far, since we did not allow $m=0$ in the definition~\\eqref{defAvm} of the averaging operator \n($m<0$ doesn't work at all, because the sum defining the average diverges, and $m=0$ is less interesting since a form\nof index~0 is clearly determined up to a function of $\\tau$ alone by its singularities, so that in our discussion we \nexcluded that case too), but nevertheless it works.  The decomposition $\\varphi=\\varphi^P+\\varphi^F$ takes the form\n  \\beta \\frac{\\varphi_{0,1}(\\tau,z)}{\\varphi_{-2,1}(\\tau,z)} \\= 12\\,\\AvB0{\\frac{y}{(1-y)^2}}\\+ E_2(\\tau)\\,,\\end{equation}\nwith the finite part being simply the quasimodular Eisenstein series $E_2(\\tau)$, which\nis also a mock modular form. (It is automatically independent of~$z$ since the index is~0.)\n\n\n\\medskip \\noindent\n{\\bf Example 5: Weight 2, index 1, double pole at $z=0$.}  The next example is a special case of an infinite family\nthat will be treated in detail in \\S\\ref{flyJac}.  Take $\\varphi=B^2\/A\\in J_{2,1}^{\\text{mer}}$. Then we find\n\\begin{eqnarray*}\n\\varphi &\\=& \\Bigl(\\frac{144\\,y}{(1-y)^2}\\+y+22+y^{-1}\\Bigr) \\+ \\Bigl(22y^2+152y-636+152y^{-1}+ 22y^{-2}\\Bigr)\\,q \\\\\n&&  \\+\\Bigl(145y^3-636y^2+3831y-7544+3831y^{-1}-636y^{-2}+145y^{-3}\\Bigr)\\,q^2 \\+\\cdots \\, , \\\\ \n\\varphi^{\\rm P} &\\=& \\AvB 1{\\frac{144\\,y}{(1-y)^2}} \\= \\frac{144\\,y}{(1-y)^2} \\+ 0\\,q \\+ 144\\,(y^3 + y^{-3})q^2 \\+ 288\\,(y^4+y^{-4})q^3 \\+ \\cdots \n\\end{eqnarray*}\nand hence that $\\varphi^F=\\varphi-\\varphi^P=\\sum\\limits_{4n-r^2\\ge-1} C(4n-r^2)\\,q^n\\,y^r$ with the first few $C(\\Delta)$ given by \n$$ \\begin{tabular}{|c|ccccccccc|}   \\hline    $\\Delta$ & $-1$& 0&3 & 4 & 7& 8 & 11 & 12 & 15  \\\\\n       \\hline  $C(\\Delta)$ &1 & 22 & 152 & $-636$ & 3831& $-7544$ & 33224 & $-53392$ & 191937\\\\  \\hline  \n  \\end{tabular}  \\;\\, . \n$$\nOn the other hand, we have the weak Jacobi form $E_4(\\tau)\\varphi_{-2,1}(\\tau,z)=\\sum C^*(4n-r^2) q^n y^r$ with \n$$\n\\begin{tabular}{|c|ccccccccc|}   \\hline  $\\Delta$ & $-1$& 0&3 & 4 & 7& 8 & 11 & 12 & 15 \\\\ \n \\hline  $C^*(\\Delta)$ &1 & $-2$ & 248 & $-492$ & 4119 & $-7256$ & 33512& $-53008$ & 192513  \\\\  \\hline  \n\\end{tabular}  \\;\\, , \n$$\nWe see that $C(\\Delta)$ and $C^*(\\Delta)$ are very close to each other (actually, asymptotically the same) \nand that their difference is precisely $-288$ times the Hurwitz-Kronecker \nclass number $H(d)$. We thus have $\\varphi^F= E_4A -288{\\cal H}$, and we have found~${\\cal H}(\\tau,z)$, the simplest of\nall mock Jacobi forms, as the finite part of the meromorphic Jacobi form $(E_4A^2-B^2)\/288A$.  We\nremark that the numbers $-C^*(\\Delta)$ have an interesting interpretation as ``traces of singular moduli\"~\\cite{Zagier:2002}.\n\n\\medskip \\noindent\n{\\bf Example 6: Weight $-$5, index 1, simple poles at the 2-torsion points.} \nFinally, we give an example with more than one pole. Take $\\varphi=A^3\/C\\in J_{1,-5}$.  This function has three poles, all\nsimple, at the three non-trivial 2-torsion points on the torus ${\\mathbb C}\/({\\mathbb Z}\\tau+{\\mathbb Z})$.  The three corresponding modular forms,\neach of weight~$-6$, are given by \n$$ D_{(0,\\frac12)}(\\tau) \\= 16\\,\\frac{\\eta(2\\tau)^{12}}{\\eta(\\tau)^{24}}\\,, \\quad\n   D_{(\\frac12,0)}(\\tau) \\= -\\frac14\\,\\frac{\\eta\\bigl(\\frac\\t2\\bigr)^{12}}{\\eta(\\tau)^{24}}\\,, \\quad\n   D_{(\\frac12,\\frac12)}(\\tau) \\= \\frac i4\\,\\frac{\\eta\\bigl(\\frac{\\tau+1}2\\bigr)^{12}}{\\eta(\\tau)^{24}} \\, , $$\nand one finds \n\\begin{eqnarray*} \n&&\\varphi \\= \\varphi^{\\rm P} \\= D_{(0,\\frac12)}(\\tau)\\,\\AvB1{\\frac12 \\, \\frac{y-1}{y+1}}\n  \\+ q^{1\/4}\\, D_{(\\frac12,0)}(\\tau)\\,\\AvB1{\\frac 1{2y} \\, \\frac{y+\\sqrt q}{y-\\sqrt q}} \\\\\n &&\\qquad\\qquad\\quad  \\+ q^{1\/4}\\, D_{(\\frac12,\\frac12)}(\\tau)\\,\\AvB1{\\frac 1{2y} \\, \\frac{y-\\sqrt q}{y+\\sqrt q}} \\,,\\qquad \\varphi^F\\;\\equiv\\;0\\,, \\end{eqnarray*}\nanother ``Cheshire cat\" example (of necessity, for the same reason as in Example~1,\nsince again $m=1$ and $k$ is odd). \n\n\n\\section{Special mock Jacobi forms \\label{flyJac}}\n\nWe now turn to the study of certain families of meromorphic Jacobi forms of weights 1 or 2 and arbitrary \npositive index and of their related mock Jacobi forms, the weight 2 case being the one relevant for the application to \nblack hole microstate counting. In this section, we introduce these families and formulate a nunmber of properties\nof them that were found experimentally and that we will prove in~\\S\\ref{structure}.  In particular, it will turn\nout that all of these functions can be expressed,\nusing the Hecke-like operators introduced in \\S\\ref{hecke}, in terms of a collection of mock Jacobi forms ${\\cal Q }_{m}$\ndefined (though not quite uniquely: there is some choice involved starting at $m = 91$) for all square-free~$m$, \nof weight either~1 or 2 depending on whether~$m$ has an odd or even number of prime factors.  These functions, which\nseem to have particularly nice arithmetic properties, include all of the examples studied in \\S\\ref{Mock}, \nand several other special mock modular forms that have appeared in the literature.  \n\n\n\\subsection{The simplest meromorphic Jacobi forms \\label{simplest}}\n\nThe two simplest situations for the theory described in \\S\\ref{MockfromJacobi} are when the meromorphic \nJacobi form $\\varphi=\\varphi_{k,m}$ satisfies either \n\\newline\\noindent\\null\\quad\n$\\bullet$ $k=1$ and $\\varphi(\\tau,z)$ has a simple pole at $z=0$ and no other poles in $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}\/({\\mathbb Z}\\tau + {\\mathbb Z})$ or \n\\newline\\noindent\\null\\quad\n$\\bullet$ $k=2$ and $\\varphi(\\tau,z)$ has a double pole at $z=0$ and  no other poles in $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}\/({\\mathbb Z}\\tau + {\\mathbb Z})$, \\\\\nsince in these two cases  the modular forms $D_{s}(\\tau)$ (resp.~$E_{s}(\\tau)$) defined in \\S\\ref{PolarPart} (resp.~\\S\\ref{double}) \nare simply constants and the canonical Fourier coefficients $h_{\\ell}$ of $\\varphi$ are pure, rather than mixed, mock modular forms.\nIf we normalize $\\varphi$ by $\\varphi(\\tau,z) \\sim (2\\pi i z)^{-1}$ in the first case and $\\varphi(\\tau,z) \\sim (2\\pi i z)^{-2}$ in the second case \nas $z \\to 0$, then any two choices of $\\varphi$ differ by a (weakly) holomorphic Jacobi form whose Fourier coefficients \nare true modular forms and are therefore well understood. It therefore suffices in principle to make some specific\nchoice of $\\varphi_{1,m}$ and $\\varphi_{2,m}$ for each $m$ and to study their finite parts $\\varphi_{k,m}^{\\rm F}$, but, as we shall\nsee, the problem of choosing a good representative $\\varphi$ is very interesting and has several subtleties. \nFor any choice of $\\varphi_{k,m}$, we will use the notation~$\\Phi_{k,m}:=\\varphi_{k,m}^{\\rm F}$, with decorations like \n$\\Phi_{k,m}^{\\rm stand}$ or $\\Phi_{k,m}^{\\rm opt}$ corresponding to the choice of $\\varphi_{k,m}$. \nThe shadow of the mock Jacobi form $\\Phi_{k,m}$  is independent of the choice of representative and is a multiple of \n$\\sum_{\\ell\\; \\mod \\; {2m}}\\overline{\\vartheta_{m,\\ell}^{2-k}(\\tau)}\\,\\vartheta_{m,\\ell}(\\tau,z)$,\nwhere $\\vartheta_{m,\\ell}^1$ and $\\vartheta_{m,\\ell}^0$ are defined as at the end of \\S\\ref{ThetaTaylorExp}.\n\nThe Jacobi forms $\\varphi_{2,m}$ are the ones related to the Fourier coefficients $\\psi_m$ of \n$\\Phi_{10}(\\Omega)^{-1}$ that appear in the application to black holes (eq.~\\eqref{reciproigusa}). \nIndeed, since the principal part of the pole of $\\psi_{m}(\\tau,z)$ at $z=0$ equals $p_{24}(m+1)\\,\\Delta(\\tau)^{-1}\\,(2\\pi iz)^{-2}$,\nwe can write $\\Delta(\\tau)\\psi_m(\\tau,z)$ as the sum of $p_{24}(m+1)\\,\\varphi_{2,m}$ and a weak Jacobi form.  \nIf we make the simplest choice for $\\varphi_{2,m}$, namely \n  \\beta\\label{defphi2m}  \\v^{\\text{stand\\vphantom{p}}}_{2,m} \\= \\frac{B^{m+1}}{12^{m+1}A} \\,, \\end{equation}\nwhere $ A = \\varphi_{-2,1}$, $B = \\varphi_{0,1}$ are the special Jacobi forms introduced in \\S\\ref{Jacobi} \n(which has the correct principal part because $A(\\tau,z) \\sim (2\\pi i z)^2$ and $B(\\tau,z) \\to 12$ \nas $z \\to 0$), then the first few values of  $\\Delta(\\tau)\\,\\psi_{m}(\\tau,z)-p_{24}(m+1)\\,\\v^{\\text{stand\\vphantom{p}}}_{2,m}(\\tau,z)$ can be read off from~\\eqref{firstpsim}. For instance, we have $p_{24}(3)=3200$ and the fourth equation of~\\eqref{firstpsim} says that \n$\\Delta\\psi_2=3200\\,\\v^{\\text{stand\\vphantom{p}}}_{2,2}+\\frac{16}9E_4AB+\\frac{10}{27}E_6A^2$.\n\nIn the weight one case we can also make a simple standard choice of $\\varphi$.  It has to be divisible\nby $C$ because the weight is odd, and since $C\\sim4\\pi iz$ for $z\\to0$ we can choose\n  \\beta\\label{defphi1m}  \\v^{\\text{stand\\vphantom{p}}}_{1,m} \\= \\frac{B^{m-1}\\,C}{2\\cdot 12^{m-1}\\,A}\\= \\frac C2 \\, \\varphi^{\\rm stand}_{2,m-2}\\;.   \\end{equation}\nas our standard functions. However, we emphasize that \\eqref{defphi2m} and \\eqref{defphi1m} are only the starting points \nfor our discussion and will be replaced by better choices as we go along.\n\nWe will apply the decomposition of \\S\\ref{MockfromJacobi} to the functions $\\varphi$.  The polar part is independent of the \nchoice of $\\varphi$ and is given by $\\varphi_{k,m}^{\\rm P}=\\mathfrak{F}_P{k,m}$, \nwhere $\\mathfrak{F}_P{k,m}$ is given\n(in the notation of equations~\\eqref{defFms} and~\\eqref{defGms}) by\n  \\beta\\label{AP1AP2}  \\mathfrak{F}_P{1,m} \\= \\mathcal{A}_{1,m}^{(0,0)} \\= \\AvB m{\\frac12 \\, \\frac{y+1}{y-1} } \\, , \\qquad   \\mathfrak{F}_P{2,m} \\= \\mathcal{A}_{2,m}^{(0,0)}\n  \\= \\AvB m{ \\frac{y}{(y-1)^2} }  \\end{equation}\nor---written out explicitly---by\n  \\beta\\label{AP1AP2again}\n  \\mathfrak{F}_P{1,m}(\\tau,z) \\=  -\\frac12 \\, \\sum_{s \\in \\mathbb{Z}} q^{ms^2\\,}y^{2ms} \\, \\frac{1 +q^s y}{1 -q^s y} \\, ,  \\qquad \n  \\mathfrak{F}_P{2,m}(\\tau,z) \\=  \\sum_{s \\in \\mathbb{Z}} \\frac{q^{ms^2 +s\\,}y^{2ms+1}}{(1 -q^s y)^2} \\, .  \\end{equation}\nAlso independent of the choice of $\\varphi$ is the correction term which must be added to the finite \npart and subtracted from the polar part of $\\varphi_{2,m}$ in order to make these functions transform\nlike Jacobi forms. In particular, for the double pole case, we find from \\eqref{defGsmhat} that the completion \n$\\widehat \\Phi_{2,m}$ of $\\Phi_{2,m} = \\varphi^{\\rm F}_{2,m}$ for any choice of $\\varphi_{2,m}$ is given by \n\\beta\\label{phi2mhat}\n\\widehat \\Phi_{2,m}(\\tau,z) \\= \\Phi_{2,m}(\\tau,z) \\, - \\, \\sqrt{\\frac{m}{4\\pi}} \\, \\sum_{\\ell \\; \\mod \\; {2m}} \n\\big( \\vartheta_{m,\\ell}^{0} \\big)^{*} (\\tau)\\, \\vartheta_{m,\\ell} (\\tau,z) \\, , \n\\end{equation}\nwhere $\\vartheta_{m,\\ell}^{0}(\\tau) = \\vartheta_{m,\\ell}(\\tau,0)$ and  \n$(\\vartheta_{m,\\ell}^{0})^{*}$ is normalized\\footnote{Note that the function $\\vartheta_{m,\\ell}^{(0,0)\\,*}$ \nas defined in \\eqref{defEichler1\/2} is equal to $1\/\\sqrt{4\\pi m}$ times $(\\vartheta_{m,\\ell}^{0})^{*}$.}\nas in equations~\\eqref{defstar} and~\\eqref{erfc}.\n\n\n\nThe functions $\\mathfrak{F}_P{1,m}$, $\\mathfrak{F}_P{2,m}$ have wall-crossings, so their Fourier expansions \ndepend on the value of $z_2 = \\Im(z)$. In the range $0 < z_2 < \\tau_2$, or $|q| < |y| < 1$, they are given by \n\\begin{eqnarray}  \\label{AAm}\n\\mathfrak{F}_P{1,m} \\=  \\biggl(- \\sum_{s \\ge 0}  \\sideset{}{^{*}} \\sum_{\\ell \\ge 0} \\+  \\sum_{s < 0} \\sideset{}{^{*}} \\sum_{\\ell \\le 0} \\biggr) \\, \nq^{ms^2 + \\ell s} \\, y^{2ms+\\ell} \\, , \\\\ \\label{BBm}\n\\mathfrak{F}_P{2,m} \\=  \\biggl(\\sum_{s \\ge 0}  \\sideset{}{^{*}} \\sum_{\\ell \\ge 0} \\, - \\, \\sum_{s < 0}  \\sideset{}{^{*}} \\sum_{\\ell \\le 0} \\biggr) \\, \n\\ell \\, q^{ms^2 + \\ell s} \\, y^{2ms+\\ell} \\, , \n\\end{eqnarray}\nwhere the asterisk on the summation sign has the usual meaning (count the term $\\ell=0$ with multiplicity $1\/2$).\nHence the coefficient of $q^{n} y^{r}$ is zero unless $\\Delta=4mn-r^2$ is the negative of the square of an \ninteger congruent to $r$ modulo $2m$, in which case it is $0$, $\\pm \\frac12$, $\\pm1$ or $\\pm 2$ if $k=1$ and is\n0, $\\pm \\sqrt{-\\Delta}$ or $\\pm 2 \\sqrt{-\\Delta}$ if $k=2$. (Compare the remark at the end of \\S\\ref{MockDecomposition}.)\nIf we expand $\\mathfrak{F}_P{1,m}$ and $\\mathfrak{F}_P{2,m}$ in Fourier series for $z_2$ in some other interval $(N,N+1)\\tau_2$, then the formulas~\\eqref{AAm}\nand~\\eqref{BBm} are unchanged except for a shift in the inequalities defining the summation.  In summary, the Fourier coefficients\nof the polar part of $\\varphi$, although not quite like the coefficients of a usual Jacobi form---they do not only depend \non $\\Delta$ and on $r$ mod $2m$, they are non-zero for infinitely many negative $\\Delta$, and there is wall-crossing---are\nextremely simple and have very slow growth (at most like $\\sqrt{|\\Delta|}$). \n\nThe interesting part of the Fourier expansion of $\\varphi_{k,m}$ is therefore the mock part~$\\Phi_{k,m}$. Here the choice \nof $\\varphi_{k,m}$ makes a big difference. For instance, Examples 1--4 of  \\S\\ref{MJExamples} say that the mock Jacobi forms $\\Phi^{\\rm stand}_{k,m}$\ncorresponding to the standard choices~\\eqref{defphi2m} and~\\eqref{defphi1m} for small~$m$ are\n\\beta\n\\Phi^{\\rm stand}_{1,1} \\= 0, \\quad\\; \\Phi^{\\rm stand}_{1,2} \\= \\frac1{24} \\, {\\cal F}_2 \\,, \\quad\\; \\Phi^{\\rm stand}_{2,0} \\= \\frac 1{12} \\, E_2 \\, ,\n\\quad\\; \\Phi^{\\rm stand}_{2,1} \\= \\frac 1{144} \\, E_4\\,A \\,-\\, 2\\,{\\cal H} \\,,\n\\end{equation}\n and we see that this choice is a good one in the first three cases (indeed, in these cases there {\\it is} no choice!) \nbut that, as we have already observed, a better choice of $\\varphi$ in the fourth case would have been the function \n$\\v^{\\text{opt}}_{2,1}=\\v^{\\text{stand\\vphantom{p}}}_{2,1} - \\frac 1{144} E_4 A$, with mock part $\\Phi^{\\rm opt}_{2,1}=-2 {\\cal H}$,\nbecause it has no terms $q^{n} y^{r}$ with discriminant $\\Delta=4n-r^2$ smaller than zero and (consequently) \nhas Fourier coefficients that grow polynomially in~$\\Delta$ (actually, like $\\Delta^{1\/2+\\varepsilon}$) rather than\nexponentially in~$\\sqrt{\\Delta}$. Our next main objective, to which we devote the next subsection, \nis therefore to replace our initial choice of $\\varphi_{k,m}$\nas given by \\eqref{defphi2m} or \\eqref{defphi1m} by an optimal choice $\\v^{\\text{opt}}_{k,m}$, where\n``optimal'' means ``with the negative discriminants $d=4mn-r^2$ that occur being as little negative \nas possible'' or, equivalently, ``with Fourier coefficients growing as slowly as possible.'' \nThis choice turns out  to have Fourier coefficients of very much \nsmaller growth than those of our initial choice $\\v^{\\text{stand\\vphantom{p}}}_{k,m}$ in all cases, and to be unique in many cases. \n\n\n\n\\subsection{Choosing the function $\\varphi$ for weight 2 and small index \\label{choosephi}}\nWe now consider in more detail the meromorphic Jacobi forms of weight 2 and index $m$\nwith a double pole at the origin introduced in~\\S\\ref{simplest}. Any such form will have the form \n\\beta\n\\varphi_{2,m} \\= \\frac{\\psi_{0,m+1}}{12^{m+1}A}\\,, \\qquad\n\\psi_{0,m+1} \\;\\, \\in \\;\\,  B^{m+1} \\+ A\\cdot \\widetilde J_{2,m} \\;\\,\\subset\\;\\, \\widetilde J_{0,m+1} \\;.\n\\end{equation}\nFor each $m$, we want to choose a meromorphic Jacobi form $\\v^{\\text{opt}}_{2,m}$ differing from $\\v^{\\text{stand\\vphantom{p}}}_{2,m}$\nby a weak Jacobi form of weight~2 and index~$m$ whose coefficients grow as slowly as possible.  \nWe are interested both in the form of the corresponding~$\\psi_{0,m+1}=\\psi_{0,m+1}^{\\rm opt}$ as a \npolynomial in $A$ and $B$ with modular form coefficients, and in the nature of the finite \npart~$\\Phi_{2,m}^{\\rm opt} = \\v^{\\text{opt}}_{2,m} - \\mathfrak{F}_P{2,m}$ as a mock Jacobi form. \nIn this subsection we will look at the optimal choice for $0\\le m \\le 11$ to see what \nphenomena occur, while in the next subsection, we report on the calculations for higher values up to\nabout~250 and formulate the general results. \n\nFor {\\bf m=0}, we have to choose \n  $$ \\v^{\\text{opt}}_{2,0} \\= \\v^{\\text{stand\\vphantom{p}}}_{2,0} \\= \\frac{B}{12A} \\; , \\qquad \\psi^{\\rm opt}_{0,1} \\= B  $$\nbecause there are no weak Jacobi forms of weight~2 and index~0. The mock part $\\Phi^{\\rm opt}_{2,0}$\nhere is $E_2(\\tau)\/12$, as we have already seen (Example~3 of \\S\\ref{MJExamples}). \n\nFor {\\bf m=1} the only freedom is the addition of a multiple of $E_4 A$ and (as recalled in \\S\\ref{simplest}) \nthe optimal choice is \n\\beta\\label{PHI21} \n \\v^{\\text{opt}}_{2,1} \\= \\v^{\\text{stand\\vphantom{p}}}_{2,1}\\,-\\,\\frac{E_4 A}{144}\\,,\\qquad \\psi^{\\rm opt}_{0,2}\\=B^2\\,-\\,E_4 A^2\\,, \n\\end{equation}\nbecause then $\\Phi^{\\rm opt}_{2,1}=-2{\\cal H}$ has no terms $q^{n} y^{r}$ with $4n-r^2 <0$  and its Fourier coefficients \n$-2 H(4n-r^2)$ have only polynomial growth as $4n-r^2 \\to \\infty$. \n\nNow we look at {\\bf m=2}. Here we can modify $12^3\\,\\Phi^{\\rm stand}_{2,2} = B^3\/A - 12^3 \\mathfrak{F}_P{2,2}$ \nby adding to it any linear combination of the two weak \nJacobi forms $E_4 A B$ and $E_6 A^2$. Each of these three functions has Fourier coefficients of the form \n$c(n,r) = C_{\\varphi}(8n-r^2)$  (because the $(n,r)$-coefficient of a Jacobi form of prime or prime power index $m$ and \neven weight always depends only on the discriminant $4mn-r^2$), with the first few given by \n  \\begin{center} \\begin{tabular}{|c|ccccccccc|ccc}  \n\\hline\n       $\\Delta$ & $-4$& $-1$& $0$ & $4$ &$7$& $8$& $12$ & $15$ &$16$ \\\\  \\hline\n       $C(12^3\\Phi^{\\rm stand}_{2,2};\\Delta)$ &1&32& $366$ & $528$ &1056 &$-10144$&$-14088$&$92832$&$-181068$\\\\\n      $C(E_4AB;\\Delta)$ &1 &8& $-18$  & 144 &$2232$&$-4768$&$-22536$&96264&$-151884$\\\\\n      $C(E_6A^2;\\Delta)$ &1 & $-4$ & 6  & $-480$ &$1956$&$-2944$&$-28488$&96252&$-139452$\\\\\n\\hline\n       \\end{tabular} \\end{center}\n\n\\noindent It follows from this table that the linear combination \n\\beta\\label{defX2} \n\\v^{\\text{opt}}_{2,2} \\=  \\v^{\\text{stand\\vphantom{p}}}_{2,2} \\, - \\, \\frac 1{576} \\, E_4 A B \\+ \\frac 1{864} E_6 A^2 \n \\= \\frac{B^3 \\, - \\, 3 E_4  A^2 B \\+ 2 E_6 A^3 }{12^3 \\, A} \n\\end{equation}\nhas a mock Jacobi part  $ \\Phi^{\\rm opt}_{2,2} =  \\v^{\\text{opt}}_{2,2} - \\mathfrak{F}_P{2,2}$\nwith no negative terms and much smaller coefficients:\n\\begin{center}\n\\begin{tabular}{c|cccccccccccc}  \n       $d$ & $-4$& $-1$& $0$ & $4$ &$7$& $8$& $12$ & $15$ &$16$ \\\\\n       \\hline       $C(\\Phi^{\\rm opt}_{2,2};\\Delta)$ &0&0& 1\/4 & $-1\/2$ &$-1$ &$-1$&$-2$& $-2$& $-5\/2$ \\\\\n        \\end{tabular}  \n\\end{center}\n\\noindent On inspection of these numbers and the next few values, we recognize them as simple linear combinations of \nHurwitz class numbers, namely:\n  $$  C(\\Phi^{\\rm opt}_{2,2}; \\Delta) =  \\begin{cases} - H(\\Delta) \\quad  &  {\\rm if} \\, \\Delta \\equiv 7 \\; \\mod \\;  8 \\, ,  \\\\ \n  -  H(\\Delta) - 2 H(\\Delta\/4)  \\quad &   {\\rm if} \\, \\Delta \\equiv 0 \\; \\mod \\;  4 \\, ,  \\end{cases}   $$\nwhich says that $\\Phi^{\\rm opt}_{2,2}$ is simply $-{\\cal H}|V_2$, with ${\\cal H}$ as in \\eqref{classH}  and $V_2$ as in \\eqref{defVl}.\n(Note, by the way, the similarity  between \\eqref{defX2} and \\eqref{weierstrass}. Together they say that \n$\\v^{\\text{opt}}_{2,2} = (C\/2A)^2$.)  \n\n\nThe cases {\\bf m=3}, {\\bf m=4}, {\\bf m=5}, {\\bf m=7}, {\\bf m=8}, {\\bf m=9} and {\\bf m=11} are similar to the case $m=2$. For each\nof these values of~$m$ the coefficient of $q^ny^r$ in $\\Phi^{\\rm stand}_{2,m}$ depends only on $\\Delta=4nm-r^2$, the number of negative $\\Delta$ for\nwhich this coefficient is non-zero equals the dimension of the space of weak Jacobi forms of weight~2 and index~$m$, and we\ncan subtract a unique weak Jacobi form from $\\v^{\\text{stand\\vphantom{p}}}_{2,m}$ to obtain a meromorphic Jacobi form $\\v^{\\text{opt}}_{2,m}$ whose finite part is\na strongly holomorphic mock Jacobi form with very small Fourier coefficients.  For example, for $m=3$ the optimal choice of\n$\\v^{\\text{opt}}_{2,m}$ is $\\v^{\\text{opt}}_{2,3}=\\psi^{\\rm opt}_{0,4}\/12^4A$ with \n  \\beta\\label{PSIF4} \\psi^{\\rm opt}_{0,4}   \\= B^4\\,-\\, 6 E_4 A^2 B^2 \\+ 8 E_6 A^3 B \\,-\\, 3 E_4^2 A^4 \\end{equation}\nand the first few Fourier coefficients of $12^4\\,\\Phi^{\\rm stand}_{2,3}$ and $\\Phi^{\\rm opt}_{2,3}$ are given by\n  \\begin{center} \\begin{tabular}{|c|ccccccccc|ccc}   \\hline\n       $\\Delta$ & $-9$& $-4$& $-1$& $0$ & $3$ &$8 $& $11$& $12$ & $15$  \\\\  \\hline\n       $12^4\\,C(\\Phi^{\\rm stand}_{2,3};\\Delta)$ & 1 & 42 & 687 & 5452 & 1104 & 8088 & $-1488$ & $-139908$ & $-19629$\\\\\n      $C(\\Phi^{\\rm opt}_{2,3};\\Delta)$ & 0 & 0 & 0  & $1\/3$ & $-1\/3 $& $-1$  & $-1$ & $-4\/3$ & $-2$ \\\\\n       \\hline \\end{tabular} \\end{center}\nAgain we recognize the coefficients $C(\\Phi^{\\rm opt}_{2,3};\\Delta)$ as $-H(\\Delta)$ if $9\\!\\nmid\\!\\Delta$ and $-H(\\Delta)-3H(\\Delta\/9)$ \nif $9\\!\\mid\\!\\Delta$, so that\njust as for $m=2$ we find that the mock part of $\\v^{\\text{opt}}_{2,3}$ equals $-{\\cal F}_ 1|V_3$.  The same thing also happens for $m=5$, 7 and 11, \nsuggesting that for prime values of $m$ we can always choose\n \\beta\\label{PHI2p}    \\Phi^{\\rm opt}_{2,p} \\= -\\,{\\cal H}|V_p  \\qquad \\text{($p$ prime).} \\end{equation}\nThis will be proved in \\S\\ref{structure}.  \nFor the prime-power values $m=4$, 8 and 9 this formula no longer holds, but the\ncoefficients of $\\Phi^{\\rm opt}_{2,m}$ are still simple linear combinations of Hurwitz-Kronecker class numbers and we find the expressions\n$$  \\Phi^{\\rm opt}_{2,4}=-{\\cal H}|V_4 + 2 {\\cal H}|U_2\\,, \\quad \\Phi^{\\rm opt}_{2,8}=-{\\cal H}|V_8 + 2 {\\cal H}|U_2V_2\\,,  \\quad \\Phi^{\\rm opt}_{2,9}=-{\\cal H}|V_9 + 3 {\\cal H}|U_3\\,, $$\nwhere $U_t$ is the Hecke-like operator defined in \\S\\ref{hecke} that multiplies the index of a Jacobi form (or mock Jacobi form) by~$t^2$.\nThis suggests that for prime power values of $m$ we can choose\n \\beta\\label{PHI2pp}    \\Phi^{\\rm opt}_{2,m} \\= -2\\,{\\cal H}|\\,{\\cal V }^{(1)}_{2,m}  \\qquad \\text{($m$ a prime power)} \\end{equation}\nwhere ${\\cal V }^{(1)}_{k,t}$ for $t\\in{\\mathbb N}$ is the Hecke-like operator from Jacobi (or weakly holomorphic Jacobi, or mock Jacobi) forms of weight~$k$ and index~$1$ to forms of weight~$k$ and index~$t$ defined \nin \\eqref{Vkt1}. \nFormula \\eqref{PHI2pp} will also be proved in \\S\\ref{structure}. \n\nThe data for the values discussed so far ($0\\le m\\le11$, $m\\ne6,\\,10$) is summarized in the table below, which expresses\nthe weak Jacobi form $\\psi^{\\rm opt}_{0,m+1}=12^{m+1}A\\,\\v^{\\text{opt}}_{2,m}$ for each of these values of~$m$ as a polynomial in~$A$ and~$B$.\nIn this table we notice many regularities, e.g., the coefficients of $E_4A^2B^{m-2}$, $E_6A^3B^{m-3}$ and $E_4^2A^4B^{m-4}$  \nin $\\psi^{\\rm opt}_{0,m}$ are always given by $-\\bigl({m\\atop2}\\bigr)$, $2\\bigl({m\\atop3}\\bigr)$ and $-3\\bigl({m\\atop4}\\bigr)$, respectively,  \nand more generally, if we write $\\psi^{\\rm opt}_{0,m}$ as $\\sum_{i=0}^m f_{m,i} \\, A^{i} \\, B^{m-i}$ with \n$f_{m,i}=f_{m,i}(\\tau) \\in M_{2i}$, $f_{m,0}=1$, then the constant terms $f_{m,i}(\\infty)$ of the modular forms $f_{m,i}$ (which \ncan be read off from the table by setting $E_4=E_6=1$ and $D=0$) are given in each case by \n$f_{m,i}(\\infty)=(-1)^{i-1}(i-1)\\bigl({m\\atop i}\\bigr)$ or equivalently by the generating function   \n  \\beta\\label{coeffCm} \\sum_{i=0}^m  f_{m,i}(\\infty) \\, X^{m-i}\\=  (X-1)^m \\+m \\, (X-1)^{m-1}  \\= (X-1)^{m-1}(X+m-1)\\;, \\end{equation}\ne.g.~$X^8 - 28X^6 + 112X^5 - 210X^4 + 224X^3 - 140X^2 + 48X - 7 = (X-1)^7(X+7)$.  This formula has a simple interpretation: the\nconstant term with respect to~$q$ of $\\sum f_{m,i}\\,A^{i}\\,B^{m-i}$ is $\\sum f_{m,i}(\\infty)(y^{-1}-2+y)^i(y^{-1}+10+y)^{m-i}$\nand this Laurent polynomial---subject to the condition $f_{m,1}(\\infty)=0$, which comes from the fact that there are no modular forms of weight~2 on $SL(2,{\\mathbb Z})$---has minimal growth $\\,\\text O\\bigl(y+y^{-1}\\bigr)\\,$ precisely for the coefficients defined by~\\eqref{coeffCm}. \n\n\\vspace{0.2cm}\n\n\\begin{minipage}{\\linewidth}\n    \\vspace{5pt}\n  \\begin{centering}\n    \\begin{tabular}{!{\\vrule width 1pt}   r  !{\\vrule width 1pt}   r   r   r   r   r   r   r   r   r  r  !{\\vrule width 1pt}      }\n       \\noalign{\\hrule height 1pt}\n      $m$ & $1$  & $2$ & $3$  & $4$ & $5$& $6$ & $8$ & $9$ & $10$ & $12$ \\\\\n       \\noalign{\\hrule height 1pt}\n      $B^m$ & 1 & $1$  & $1$ & $1$  & $1$ & $1$& $1$ &$1$& $1$ & $1$\\\\\n       \\hline\n      $E_4 \\cdot A^2 B^{m-2}$ & 0 & $-1$  & $-3$ & $-6$  & $-10$ & $-15$& $-28$& $-36$ &$-45$ & $-66$\\\\\n       \\hline\n      $E_6 \\cdot A^3 B^{m-3} $ & 0 & $0$  & $2$ & $8$  & $20$ & $40$ &$112$& $168$ &$240$ & $440$\\\\\n       \\hline\n      $E_4^2  \\cdot  A^4 B^{m-4}$ & 0 & $0$  & $0$ & $-3$  & $-15$ & $-45$& $-210$& $-378$ &$-630$ & $-1485$\\\\\n       \\hline\n      $E_4 E_6 \\cdot A^5 B^{m-5} $ & 0 & $0$  & $0$ & $0$  & $4$ & $24$& $224$& $504$ &$1008$ & $3168$\\\\\n       \\hline\n      $E_4^3  \\cdot A^6 B^{m-6} $ & 0 & $0$  & $0$ & $0$  & $0$ & $-5$&  $-140$& $-420$ &$-1050$&$-4620$\\\\\n      $D  \\cdot A^6 B^{m-6} $ & 0 & $0$  & $0$ & $0$  & $0$ & $1$& $10$& $30$ &$48$ & $222$\\\\\n       \\hline\n      $E_4^2 E_6 \\cdot A^7 B^{m-7} $ & 0 & $0$  & $0$ & $0$  & $0$ & $0$ &$48$& $216$ &$720$ & $4752$\\\\\n       \\hline\n      $E_4^4 \\cdot  A^8 B^{m-8}$ & 0 & $0$  & $0$ & $0$  & $0$ &  $0$ &$-7$& $-63$ &$-315$&$-3465$\\\\\n      $E_4 D \\cdot A^8 B^{m-8} $ & 0 & $0$  & $0$ & $0$  & $0$ &  $0$ &$-4$& $-36$ &$-18$&$-360$\\\\\n       \\hline\n      $E_4^3 E_6 \\cdot  A^9 B^{m-9}$ & 0 & $0$  & $0$ & $0$   & $0$& $0$ &$0$& $8$ & $80$ & $1760$\\\\\n      $ E_6 D \\cdot A^9 B^{m-9} $ & 0 & $0$  & $0$ & $0$  & $0$& $0$ &$0$& $20$ & $-16$ & $80$\\\\\n       \\hline\n      $E_4^5  \\cdot A^{10} B^{m-10} $ & $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$-9$ & $-594$\\\\\n      $E_4^2 D \\cdot A^{10} B^{m-10} $ & $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$18$&$702$\\\\\n       \\hline\n      $E_4^4 E_6 \\cdot  A^{11} B^{m-11}$ & $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$0$ & $120$\\\\\n      $E_4 E_6 D \\cdot A^{11} B^{m-11} $ &  $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$0$ & $-672$\\\\\n       \\hline\n      $E_4^6 \\cdot  A^{12} B^{m-12}$ & $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$0$ & $-11$\\\\\n      $E_4^3 D \\cdot A^{12} B^{m-12} $ &  $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$0$ & $188$\\\\\n      $D^2 \\cdot A^{12} B^{m-12} $ &  $0$  & $0$ & $0$  & $0$ & $0$& $0$ &$0$& $0$ &$0$ & $-8$\\\\\n       \\noalign{\\hrule height 1pt}\n     \\end{tabular}\n   \\end{centering}\n    \\captionof{table}{\n            \\small Coefficients of $\\psi^{\\rm opt}_{0,m}$ in terms of the standard basis. Here $D := 2^{11} 3^3 \\Delta$.}\n \\label{Cmcoeffs}\n\\end{minipage}\n \n\\vspace{0.6cm}\n \n\n\nFor {\\bf m=6} several new things happen, all \nconnected with the fact that~6 is the first value of~$m$ having more than one prime factor.\nFirst of all, the formula \\eqref{PHI2pp}, which worked for prime powers, now fails: \n$-2 {\\cal H}|{\\cal V }_{2,6}^{(1)}$\nis not a possible choice for $\\Phi_{2,6}$. The reason is that, as we will see later in general, the form\n${\\cal H}|{\\cal V }_{2,m}^{(1)}$ (for any~$m$) is invariant under all Atkin-Lehner involutions, but $\\Phi_{2,m}$ is \nnot when $m$ is not a prime power. The expression $-2{\\cal H}|{\\cal V }_{2,6}^{(1)}$   \n\ndoes, however, give the optimal choice of the eigencomponent $\\Phi_{2,6}^{++}$ of $\\Phi_{2,6}$ with eigenvalues \n$W_{2} = W_{3} = +1$. The other eigencomponent, corresponding to $W_{2}=W_{3}=-1$, is obtained by subtracting a \nmultiple of the non-weak Jacobi form $\\varphi_{2,6}^{-}$ given in \\eqref{phi26min} from $\\pi_{2}^{-}(\\Phi^{\\rm opt}_{2,6})$ \n(which is weakly holomorphic but not weak), and turns out to be simply a multiple of the mock Jacobi form \n${\\cal F}_{6}$ defined in \\S\\ref{MockJacobi} (Example~3), giving as our first choice of the ``optimal'' mock modular form \n\\beta\\label{PhiA26}\n\\Phi^{\\rm I}_{2,6} \\= -2 \\, {\\cal H}\\,|\\,{\\cal V }_{2,6}^{(1)} \\,-\\, \\frac{1}{24}\\,{\\cal F}_{6} \\; .\n\\end{equation}\n\n\nThe second new phenomenon for $m=6$ is that requiring the order of the poles to be as small as possible \nno longer fixes a unique optimal choice. The function~$\\Phi_{2,6}^{\\rm I}$ defined above has an even \npart that is strongly holomorphic (because ${\\cal H}$ is) and an odd part that has discriminant~$\\Delta \\ge -1$ \n(because~$C^{(6)}(\\Delta)=0$ for $\\Delta < -1$), but we can change this choice without disturbing \nthe discriminant bound $\\Delta \\ge -1$ by adding to it an arbitrary multiple of the function\n\\beta {\\cal K}_{6} = \\frac{E_4A B^5-5E_6A^2B^4+10E_4^2A^3B^3-10E_4E_6A^4B^2+(5E_4^3-\\frac14D)A^5B-E_4^2E_6A^5}{12^5} \\end{equation}\nwhich is (up to a scalar factor, fixed here by requiring ${\\cal K}_{6} = y -2 +y^{-1} + \\text O(q)$) \nthe unique Jacobi form of weight 2 and index 6 satisfying the same discriminant bound. \n\nTwo particularly nice choices besides the function $\\Phi_{2,6}^{\\rm I}$ are the functions \n\\beta \\Phi_{2,6}^{\\rm II} \\=  \\Phi_{2,6}^{\\rm I} \\,-\\, \\frac1{24} \\, {\\cal K}_{6} \\, \\qquad  \\Phi_{2,6}^{\\rm III} \\= \\Phi_{2,6}^{\\rm I} \\+ \\frac14 \\, {\\cal K}_{6} \\end{equation}\n  \n \n \nThe special properties of these two choices are that $\\Phi_{2,6}^{\\rm II}$ is the unique choice with $c(\\Phi;0,1)=0$\nand (hence) has the form $12^{-7} \\sum_{i=0}^7 f_{7,i} A^{i-1} B^{7-i}$ with $f_{7,i}$ satisfying \n\\eqref{coeffCm}, while $\\Phi_{2,6}^{\\rm III}$ is the unique choice for which $c(n,r)=0$\nwhenever $24n=r^{2}$.  (This last choice is from some points of view the most canonical one.)\nEach of these three special choices, as we shall see below, has analogues for all~$m$, although in general\nnone of them is canonical, since for large~$m$ we can change them by adding cusp forms without \naffecting their defining properties and we do not know any criterion that selects a unique form.\n\nThe following table, which gives the first few Fourier coefficients $C(\\Phi;\\Delta,\\,\\text{$r\\;$(mod 12)})$ for the Jacobi form ${\\cal K}_6$ \nand each of the five mock Jacobi forms $-2{\\cal H}|{\\cal V }_{2,6}^{(1)}$, ${\\cal F}_6$, $\\Phi_{2,6}^{\\rm I}$, $12\\Phi_{2,6}^{\\rm II}$ and \n$\\Phi_{2,6}^{\\rm III}$, summarizes the above discussion. In this table we have set in boldface the Fourier coefficients~$C(\\Delta,r)$\nwhose special properties determine the choice of $\\Phi$ (namely, $C(-1,\\pm1)+C(-1,\\pm5)=0$ for~$\\Phi^{\\rm I}$, $C(-1,\\pm1)=0$ \nfor $\\Phi^{\\rm II}$, and $C(0,0)=0$ for $\\Phi^{\\rm III}$).\n\\begin{table}[h]\n \\begin{center} \\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}cc|c|c|c|c|c|cc|c|c|c|c!{\\vrule width 1pt}}  \n \\noalign{\\hrule height 1pt}\n       $\\Delta$ & $-1$& $-1$& $0$& $8$ & $12$ & $15$ & $20$ & $23$ & $23$ & 24 & 32 & 36  & 39 \\\\  \n       $r\\!\\pmod{12}$ & $\\pm1$ & $\\pm5$ & $0$ & $\\pm4$ & $6$ & $\\pm3$ & $\\pm2$ & $\\pm1$ & $\\pm5$ & $0$ & $\\pm4$ & $6$ & $\\pm3$ \\\\  \\noalign{\\hrule height 1pt}\n       ${\\cal K}_6$ & $1$ & $1$ & $-2$ & $10$ & $28$ & $10$ & $-12$ & $13$ & $13$ & $-44$ & $58$  & $104$ & $2$ \\\\ \n       $-2{\\cal H}\\,|\\,{\\cal V }_{2,6}$ & $0$& $0$& $\\fr12$ & $-\\fr12$ & $-1$ & $-1$ & $-1$ & $-\\fr32$ & $-\\fr32$ & $-1$ & $-\\fr52$ & $-2$ & $-2$ \\\\  \n       ${\\cal F}_6$ & $-1$ & $1$& $0$& $0$ & $0$ &$0$& $0$& $35$ & $-35$ & $0$ & $0$  & $0$ & $0$ \\\\        \\hline\n     $\\Phi_{2,6}^{\\rm I}$ & $\\fr{\\bold 1}{\\bold 2\\bold 4}$ & $\\bold{-}\\fr{\\bold 1}{\\bold 2\\bold 4}$ \n          & $\\fr12$ & $-\\fr12$ & $-1$ & $-1$ & $-1$ & $-\\fr{71}{24}$ & $-\\fr1{24}$ & $-1$ & $-\\fr52$  & $-2$ & $-2$ \\\\ \n     $12\\,\\Phi_{2,6}^{\\rm II}$ & $\\bold 0$ & $-1$ & $7$ & $-11$ & $-26$ & $-17$ & $-6$ & $-42$ & $-7$ & $10$ & $-59$  & $-76$ & $-25$ \\\\ \n     $\\Phi_{2,6}^{\\rm III}$ & $\\fr7{24}$ & $\\fr5{24}$ & $\\bold 0$ & $2$ & $6$ & $\\fr32$ & $-4$ & $\\fr7{24}$ & $\\fr{77}{24}$ & $-12$ & $12$  & $24$ & $-\\fr32$ \\\\ \n \\noalign{\\hrule height 1pt}\n    \\end{tabular}\\quad \\end{center}\n\\label{wt6tab}  \\caption{\\small{Jacobi and mock Jacobi forms of weight~2 and index~6}} \n\\end{table}\nWe mention in passing that the function $\\psi_{0,7}^{\\rm I} = 12^{7} A \\varphi_{2,6}^{\\rm I}$ corresponding to \\eqref{PhiA26} \nis precisely equal to $B$ times $\\psi^{\\rm opt}_{0,6}$ as given in Table~2. We do not know if this has any special significance. It may be connected \nwith the fact that $\\psi^{\\rm opt}_{0,6}$, which is equal to the quotient $\\vartheta_{1}(\\tau,4z)\/\\vartheta_{1}(\\tau,2z)$ \n(Watson quintuple product identity), is a $(-1)$--eigenvector of the involution~$W_{2} = W_{3}$ on~$\\widetilde J_{0,6}$.\n\n\nFor {\\bf m=10} the situation is similar. \nAgain, we can achieve $\\Delta \\ge -1$ but not uniquely because there is a weak Jacobi form ${\\cal K}_{10} \\in \\widetilde J_{2,10}$ \n(unique up to a normalizing factor, which we fix in the same way as before) satisfying the same\ndiscriminant bound and we can change $\\varphi_{2,10}$ an arbitrary multiple of ${\\cal K}_{10}$.  We again have three distinguished choices \n\\beta\n\\varphi_{2,10}^{\\rm II} \\= \\frac{\\psi_{0,11}^{\\rm II}}{12^{11}A} \\, , \\qquad \n\\varphi_{2,10}^{\\rm I} \\= \\varphi_{2,10}^{\\rm II}  \\+  \\frac1{12} \\,{\\cal K}_{10} \\, \\qquad \n\\varphi_{2,10}^{\\rm III} \\= \\varphi_{2,10}^{\\rm I} \\+ \\frac38 \\,{\\cal K}_{10} \\,, \n\\end{equation}\nwhere $\\psi_{0,11}^{\\rm II} \\in \\widetilde J_{0,11}$ is given (with $D=32(E_4^3-E_6^2)=2^{11}3^3\\Delta$ as in Table~2) by\n\\begin{eqnarray} \\Psi^{\\rm II}_{11} & = & B^{11} - 55 E_{4} A^{2} B^{9} + 330 E_{6} A^{3} B^{8} \n-990 E_{4}^{2} A^{4} B^{7} + 1848 E_{4} E_{6} A^{5} B^{6} -(2310 E_{4}^{3}-111 D) A^{6} B^{5} \\cr\n&& +1980 E_{4}^{2} E_{6} A^{7} B^{4} -(1155 E_{4}^{4} +120 E_{4} D) A^{8} B^{3} +(440 E_{4}^{3} E_{6} +20 E_{6} D) A^{9} B^{2} \\cr\n&& -(99 E_{4}^{5} -117 E_{4}^{2} D) A^{10} B + (10 E_{4}^{4} E_{6} -56 E_{4} E_{6} D) A^{11} \n\\end{eqnarray}\nand is the unique choice satisfying \\eqref{coeffCm}, $\\Phi_{2,10}^{\\rm III}$ is the unique \nchoice with no $\\Delta=0$ coefficients, and $\\Phi_{2,10}^{\\rm I}$ \nis the unique \nchoice whose invariant part under $W_{2}$ (or $W_{5}$) is strongly holomorphic. Explicitly, we have \n\\beta \\label{PHI2ten}  \\Phi_{2,10}^{\\rm I} \\=  -\\,2\\,{\\cal H}|\\mathcal V_{2,10}^{(1)} \\,-\\, \\frac1{12} \\,{\\cal F}_{10}\\,  \\end{equation}\nwhere ${\\cal F}_{10}$ is a mock Jacobi form of weight~2 and index~10 with Fourier coefficients given by\n  \\beta\\label{c10}  c({\\cal F}_{10};n,r) =  \\begin{cases} \\phantom{-\\,}C^{(10)}(40n-r^2) \\quad  \n                 & \\text{$r\\equiv\\pm1 \\; \\mod \\; {20}$ or $r\\equiv\\pm3 \\; \\mod \\; {20}$}  \\\\ \n  -\\, C^{(10)}(40n-r^2) \\quad  \n                 & \\text{$r\\equiv\\pm7 \\; \\mod \\; {20}$ or $r\\equiv\\pm9 \\; \\mod \\; {20}$}  \\end{cases}    \\end{equation}\nfor a function $C^{10}(\\Delta)$ whose first few values are\n  \\begin{center} \\begin{tabular}{|c|ccccccccccccc|ccc}  \n\\hline\n       $\\Delta$ & $-1$& $31$& $39$& $71$ & $79$ &$111 $& $119$& $151$ & $159$ & $191$ & $199$ & $231$ & $239$  \\\\  \\hline\n       $C^{(10)}(\\Delta)$ & $-1$ & 9 & 21 & 35 & 63 & 57 & $112$ & $126$ & $207$ & $154$ & $306$ & $315$ & $511$  \n       \\\\ \\hline\n\\end{tabular}\\;\\,, \\end{center}\nor equivalently, where ${\\cal F}_{10}$ has a theta expansion of the form\n\\beta\\label{F10exp} {\\cal F}_{10}(\\tau,z)\\=\\sum_{\\ell \\; \\mod \\; {20} \\atop (\\ell,10)=1} h_\\ell(\\tau)\\,\\vartheta_{10,\\ell}(\\tau,z) \\end{equation}\nwith mock modular forms $h_\\ell=h_\\ell(\\tau)$ having expansions beginning\n\\begin{eqnarray} &&h_1\\=-h_9\\=-h_{11}\\=h_{19}\\=q^{-1\/40}\\,\\bigl(-1\\+21q\\+63q^2\\+112q^3\\+\\cdots\\bigr)\\,, \\\\\n     &&h_3\\=-h_7\\=-h_{13}\\=h_{17}\\=q^{31\/40}\\,\\bigl(9\\+35q\\+57q^2\\+126q^3\\cdots\\bigr)\\,. \\end{eqnarray}\n\n\n\\subsection{Further data and formulation of main results for the weight 2 family \\label{Statements}}\n\nIn \\S\\ref{choosephi}, we found that the functions $\\Phi_{2,m}$ for $m \\le 11$ could all be obtained, using\nthe Hecke-like operators from \\S\\ref{hecke}, from just three functions ${\\cal H}$, ${\\cal F}_{6}$ and ${\\cal F}_{10}$. \nContinuing, we find that for ${\\bf m=12}$ the same statement holds: the $(+1,+1)$ eigencomponent of this form with respect\nto the involutions $W_4$ and $W_3$ can be chosen to be $-2{\\cal H}|{\\cal V }_{2,12}^{(1)}$ and the $(-1,-1)$ eigencomponent\ncan be chosen to be $\\frac1{12}{\\cal F}_6|{\\cal V }_{2,2}^{(6)}$, with ${\\cal V }_{k,t}^{(M)}$ as in~\\eqref{Vkt1} \n(so $V_{2,12}^{(1)} = \\frac{1}{4} (V_{12} - 2 V_{2} U_{3})$ and ${\\cal V }_{2,2}^{(6)} = \\frac12 V_{2}$). Thus \n\\beta\n\\Phi_{2,12} \\= - 2 {\\cal H}\\,|\\,{\\cal V }^{(1)}_{2,12} \\+ \\frac1{12}\\; {\\cal F}_6\\,|\\,{\\cal V }^{(6)}_{2,2}\n\\end{equation}\nis a possible choice for $\\Phi_{2,12}$. Similarly, for ${\\bf m=18}$ and ${\\bf m=24}$ we can take the functions\n\\beta\n\\Phi_{2,18} \\= - 2{\\cal H}\\,|\\,{\\cal V }^{(1)}_{2,18} \\+ \\frac1{24}\\; {\\cal F}_6\\,|\\,{\\cal V }^{(6)}_{2,3}\\,, \n\\qquad \\Phi_{2,24} \\= - 2{\\cal H}\\,|\\,{\\cal V }^{(1)}_{2,24} \\+ \\frac1{24}\\; {\\cal F}_6\\,|\\,{\\cal V }^{(6)}_{2,4}\n\\end{equation}\nas our representatives of the coset $\\Phi^{\\rm stand}_{2,m}+\\widetilde J_{2,m}$. \n\nOn the other hand, the indices ${\\bf m=14}$ and ${\\bf m=15}$ are similar to the indices $m=6$ and $m=10$ looked at previously: here\nwe find optimal mock Jacobi forms of the form\n\\beta\n\\Phi_{2,14}\\= -2{\\cal H}\\,|\\,{\\cal V }^{(1)}_{2,14} \\,-\\, \\frac18\\,{\\cal F}_{14}\\,, \n\\qquad \\Phi_{2,15}\\= -2{\\cal H}\\,|\\,{\\cal V }^{(1)}_{2,15} \\,-\\, \\frac16\\,{\\cal F}_{15}\\,, \n\\end{equation}\nwhere ${\\cal F}_{14}$ and ${\\cal F}_{15}$ are forms anti-invariant under $W_p$ ($p|m$) with Fourier expansions of the form \n  \\beta\\label{c14}  c({\\cal F}_{14};n,r) =  \\begin{cases} \\phantom{-\\,}C^{(14)}(56n-r^2) \\quad & \\text{for $r\\equiv\\pm1$, $\\pm3$ or $\\pm 5$ (mod~28),}  \\\\ \n    -\\, C^{(14)}(56n-r^2) \\quad  & \\text{for $r\\equiv\\pm9$, $\\pm11$, or $\\pm 13$ (mod~28),\\phantom{XXX}}  \\end{cases}    \\end{equation}\n  \\beta\\label{c15}  c({\\cal F}_{15};n,r) =  \\begin{cases} \\phantom{-\\,}C^{(15)}(60n-r^2) \\quad  \n                 & \\text{for $r\\equiv\\pm1$, $\\pm2$, $\\pm4$, or $\\pm 7$ (mod~30),}  \\\\ \n  -\\, C^{(15)}(60n-r^2) \\quad & \\text{for $r\\equiv\\pm8$, $\\pm11$, $\\pm13$, or $\\pm 14$ (mod~30),}  \\end{cases}  \\end{equation}\nwith coefficients $C^{14}(\\Delta)$ and $C^{15}(\\Delta)$ given for small $\\Delta$ by\n \\begin{center} \\begin{tabular}{|c|ccccccccccccc|}  \n \\hline\n       $\\Delta$ & $-1$& $31$& $47$& $55$ & $87$ &$103 $& $111$& $143$ & $159$ & $167$ & $199$ & $215$  & $223$ \\\\  \\hline\n       $C^{(14)}(\\Delta)$ & $-1$ & 3 & 10 & 15 & 15 & 30 & 42 & $20$ & $51$ & $65$ & $51$ & $85$ & $120$   \\\\\n       \\hline\n\\end{tabular} \\;, \\end{center}\n \\begin{center} \\begin{tabular}{|c|ccccccccccccc|}  \n \\hline\n       $\\Delta$ & $-1$& $11$& $44$& $56$ & $59$ &$71 $& $104$& $116$ & $119$ & $131$ & $164$ & $176$ & $179$  \\\\  \\hline\n       $C^{(15)}(\\Delta)$ & $-1$ & 2 & 10 & 4 & 14 & 9 & 28 & $24$ & $29$ & $26$ & $40$ & $28$ & $54$  \\\\\n       \\hline\n \\end{tabular} \\;. \\end{center}\n\n\nThis pattern continues: as we continue to higher values, we find that for each index~$M$ which is a product of an even number of distinct primes\n(i.e., for which $\\mu(M)=1$, where $\\mu(\\,\\cdot\\,)$ denotes the M\\\"obius function), we need to introduce a new function ${\\cal Q }_M$ that is anti-invariant \nunder~$W_{p}$ for all $p$ dividing $M$, but that these functions and their images under Hecke-like operators then suffice to give possible \nchoices for our special mock Jacobi forms in all indices, in the sense that the function\n  \\beta \\label{Qdecomp} \n  \\Phi_{2,m} \\= \\frac{1}{2^{\\omega(m)-1}} \\! \\sum_{M\\mid m \\atop \\mu(M)\\=+1} {\\cal Q }_{M} \\, | \\, {\\cal V }^{(M)}_{2,\\,m\/M} \\quad \\qquad \n  (\\, \\omega(m) := \\# \\{p\\mid \\text{$p$ prime, $p|M$}\\} \\, )\n  \\end{equation}\nbelongs to the coset $\\Phi^{\\rm stand}_{k,m}+\\widetilde J_{2,m}$ for all $m\\ge1$.\n(The factor~$2^{1-\\omega(m)}$ will be explained below.)\nIn fact more is true.  The functions ${\\cal Q }_M$ in~\\eqref{Qdecomp} are simply the $(-1,\\dots,-1)$ eigencomponents of $\\Phi_{2,M}$ for some choice of \nthese latter functions within their defining cosets, and the various terms in~\\eqref{Qdecomp} are the eigencomponents of $\\Phi_{2,m}$ for\nthe involutions $W_{m_1}$ ($m_1\\|m$).  In other words, we have the following theorem:\n  \\begin{thm} \\label{phimQm}  For any choice of the mock Jacobi forms $\\Phi_{2,m}\\in\\Phi^{\\rm stand}_{2,m}+\\widetilde J_{2,m}$ \n  $(m \\in \\mathbb{N})$, the eigencomponents $\\Phi_{2,m}^{\\varepsilon_1,\\dots,\\varepsilon_r}$  $(m=\\prod_{i} p_{i}^{\\nu_{i}}$, \n   $\\varepsilon_{i} = \\pm 1$, $\\prod \\varepsilon_{i} = +1)$\n  with respect to the decomposition\\footnote{or rather, its analogue for elliptic forms}~\\eqref{Jkmeigen}  are given by \n    \\beta \\label{eigenpieces}  \n  \\frac12  \\Phi_{2,m} \\, \\Bigl|\\; \\prod_{i=1}^{r} \\big(1+\\varepsilon_{i} W_{p_{i}^{\\nu_{i}}} \\big) \\= {\\cal Q }_{M} \\,\\bigl|\\,{\\cal V }_{2,m\/M}^{\\,(M)} \\quad\\bigl(\\text{mod }(J_{2,m}^{\\,!}\\bigr)^{\\varepsilon_1,\\dots,\\varepsilon_r}\\bigr)\\;, \\end{equation}\nwhere $M$ is the product of the $p_i$ with $\\varepsilon_i=-1$, ${\\cal Q }_{M}$ is defined by \n\\beta  \\label{defQmeven}\n{\\cal Q }_{M} \\= \\frac12 \\Phi_{2,M} \\, \\Bigl|\\; \\prod_{p|M} \\big(1- W_{p} \\big) \\= 2^{\\omega(M)-1} \\, \n\\Phi_{2,M}^{-,\\dots,-} \\, \\qquad (\\mu(M)\\=+1), \n\\end{equation}\nand ${\\cal V }_{2,m\/M}^{(M)}$ is the Hecke-like operator defined in~\\eqref{Vkt1}. \nIn particular, the decomposition~\\eqref{Qdecomp} is always true modulo $\\widetilde J_{2,m}$, \nand if we choose the functions~$\\Phi_{2,M}$ and consequently~${\\cal Q }_{M}$ arbitrarily for \nall~$M$ with~$\\mu(M)=1$, then the function defined by~\\eqref{Qdecomp} gives an admissible  \nchoice  of $\\Phi_{2,m}$ for all indices~$m\\ge 1$. \n\\end{thm}\n\n\\noindent{\\bf Examples.} \nThe functions ${\\cal Q }_{M}$ for the first few values of $M$ with $\\mu(M)=1$ are\n\\beta \\label{Qlist}\n{\\cal Q }_{1} = -  {\\cal H} \\,, \\; \\; {\\cal Q }_6 =-\\frac{{\\cal F}_6}{12}\\,, \\; \\;  {\\cal Q }_{10} =-\\frac{{\\cal F}_{10}}{6}\\,, \\; \\;  \n{\\cal Q }_{14}=-\\frac{{\\cal F}_{14}}{4} \\,, \\; \\; {\\cal Q }_{15}=-\\frac{{\\cal F}_{15}}{3} \\, , \\; \\; \\dots \\, , \n\\end{equation}\nwhere ${\\cal H}, {\\cal F}_{6}, \\dots , {\\cal F}_{15}$ are the functions introduced in \\S\\ref{MockJacobi}, \\S\\ref{choosephi},\nand this subsection. (The reasons for the various normalizations, and for having two symbols ${\\cal F}_{M}$\nand ${\\cal Q }_{M}$ to denote proportional functions, will be discussed later.)\n\n\nWe will prove Theorem \\ref{phimQm} towards the end of this section  \nby calculating the shadows of the mock Jacobi forms on the two sides \nof~\\eqref{eigenpieces}, since two mock Jacobi forms differ by a true \nJacobi form if and only if they have the same shadow. \n\n\\smallskip\n\n\\noindent{\\bf Remarks.} {\\bf 1.}\nWe wrote $\\bigl(J^{\\,!}_{2,m}\\bigr)^{\\varepsilon_1,\\dots,\\varepsilon_r}$ rather than $\\widetilde J_{2,m}^{\\,\\varepsilon_1,\\dots,\\varepsilon_r}$ in \nthe theorem because of the fact mentioned in~\\S\\ref{hecke} that the involutions $W_{m_1}$ do not preserve \nthe space of weak Jacobi forms (or weak elliptic forms), so that the left-hand side of eq.~\\eqref{eigenpieces} \nfor an arbitrary choice of $\\Phi_{2,m}$ may contain monomials $q^ny^r$ with $n<0$.  In fact, it will follow \nfrom the results given below that the functions $\\Phi_{2,M}$ can be chosen in such a way that the eigencomponents \n$\\Phi_{2,M}^{-,\\cdots,-}$ do not contain any negative powers of~$q$, in which case the expression on \nthe right-hand side of~\\eqref{Qdecomp} really does lie in $\\Phi^{\\rm stand}_{k,m}+\\widetilde J_{2,m}$ as asserted, and not merely \nin $\\Phi^{\\rm stand}_{k,m}+J^{\\,!}_{2,m}$.\\\\\n\\noindent{\\bf 2.} An exactly similar statement is true for the functions $\\Phi_{1,m}$, except that in that case we \nhave $\\prod_i\\varepsilon_i=-1$ (because in odd weight $W_m$ acts as multiplication by~$-1$), so that the $M$ that \noccur in the decomposition of $\\Phi_{1,m}$ are the divisors of~$m$ with~$\\mu(M)=-1$, and \nthe functions~${\\cal Q }_{M}$ that occur in the decomposition of $\\Phi_{1,M}$ are defined by\n\\beta \\label{defQmodd}\n{\\cal Q }_{M} \\= \\frac12 \\, \\Phi_{1,M} \\, \\Bigl|\\; \\prod_{p|M} \\big(1- W_{p} \\big) \\= 2^{\\omega(M)-1} \\; \n\\Phi_{2,M}^{-,\\cdots,-} \\, \\qquad (\\mu(M)\\=-1) \n\\end{equation}\ninstead of equation~\\eqref{defQmeven}.\nThe reason for the factor~$2^{\\omega(m)-1}$ in these two equations is now apparent:\nit is the minimal factor making~${\\cal Q }_{M}$ an integral linear combination of images of~$\\Phi_{k,m}$\nunder Atkin-Lehner involutions. (We could remove one factor~2 because for each decomposition~$M=M_{1}M_{2}$,\nthe two terms~$\\mu(M_{1}) \\Phi_{k,m}|W_{M_{1}}$ and~$\\mu(M_{2}) \\Phi_{k,m}|W_{M_{2}}$  \nare equal.)\n\n\\smallskip\n\n\n\nTheorem \\ref{phimQm} explains the first part of our observations, those connected with the eigencomponent \ndecomposition of the mock Jacobi forms $\\Phi_{2,m}$ and the fact that the collection of all of these \nfunctions can be expressed in terms of the special functions~\\eqref{Qlist} and their images under Hecke-like operators.\nBut it does not say anything about the orders of the poles of the functions in question, which was our original criterion for optimality.\nOn the other hand, from the tables of Fourier coefficients of the functions~${\\cal Q }_{1}, \\dots, {\\cal Q }_{15}$ that we have given \nwe see that, although none of these functions except ${\\cal Q }_{1}$ is strongly holomorphic, each of them \nsatisfies the only slightly weaker discriminant bound $\\Delta := 4mn-r^{2} \\ge-1$.\nThis property will turn out to play a prominent role. We will call Jacobi or elliptic forms \nsatisfying it {\\bf forms with optimal growth}\\footnote{An alternative name would be ``forms with \nonly simple poles'' because the forms $h_{\\ell}(\\tau) = \\sum_{\\Delta} C(\\Delta,\\ell) \\, q^{\\Delta\/4m}$ have no singularity \nworse than $q^{-1\/4m}$ as $q \\to 0$, i.e., they have at most simple poles with respect to the \nlocal parameter $q^{1\/4m}$ at $\\tau=i \\infty$. However, this terminology would be dangerous because \nit could also suggest simple poles with respect to~$z$. \nAnother reasonable name would be ``nearly holomorphic,'' but this terminology also carries a risk of \nconfusion since ``almost holomorphic'' is already a standard notation for modular forms which \nare polynomials in~$1\/\\tau_{2}$ with holomorphic coefficients.}, because their Fourier coefficients \ngrow more slowly than forms that do not obey this discriminant bound.\nWe observe that any Jacobi or elliptic form having optimal growth is automatically a weak \nform, since $4nm-r^2\\ge-1$ implies $n\\ge0$.\n\n\nAs we continued our computations to higher values of $M$, \nwe were surprised to find that this ``optimal growth'' property continued to hold for each \n$M$ of the form $pq$  ($p$ and $q$ distinct primes) up to 200.  It followed that the full function $\\Phi_{2,M}$ \ncould also be chosen to have optimal growth in these cases, since $ \\Phi_{2,M} \\= {\\cal Q }_1|{\\cal V }^{(1)}_{2,M}+{\\cal Q }_{M}$ \nand the image of the strongly holomorphic form ${\\cal Q }_1$ under ${\\cal V }^{(1)}_{2,M}$ is strongly holomorphic. \nThis made it particularly interesting to look at the case ${\\bf m=210}$, \nthe first product of four distinct primes, because in this case, even if the function ${\\cal Q }_{210} = \\Phi_{2,210}^{----}$\ncan be chosen to be have optimal growth, then the six middle terms in the decomposition~\\eqref{Qdecomp}, which here takes the form \n\\begin{eqnarray} 8 \\, \\Phi_{2,210} &\\= {\\cal Q }_{1}\\,|\\, {\\cal V }^{(1)}_{2,210} \\+ {\\cal Q }_{6}\\,|\\,{\\cal V }^{(6)}_{2,35} \\+ {\\cal Q }_{10}\\,|\\,{\\cal V }^{(10)}_{2,21} \n \\+ {\\cal Q }_{14}\\,|\\,{\\cal V }^{(14)}_{2,15} & \\cr\n  &\\qquad \\+ {\\cal Q }_{15}\\,|\\,{\\cal V }^{(15)}_{2,14}  \\+ {\\cal Q }_{21}\\,|\\,{\\cal V }^{(21)}_{2,10} \\+ {\\cal Q }_{35}\\,|\\,{\\cal V }^{(35)}_{2,6}\\+ {\\cal Q }_{210} \\; , & \\end{eqnarray}\nwill not be. (Even if ${\\cal Q }_{M}$ is chosen to have optimal growth, its image under the operator ${\\cal V }^{M}_{2,t}$ will in general satisfy\nonly  $\\Delta \\ge - t^{2}$. For $M$ of the form $pq$ we had no problems with this because ${\\cal Q }_1$, and hence also its image under ${\\cal V }^{(1)}_{2,M}$, is actually holomorphic.) The corresponding computation was somewhat forbidding since the space $\\widetilde J_{2,210}$ of forms \nby which we can change the initial choice $\\Phi^{\\rm stand}_{2,210}$ has dimension $3815$, \nand the conditions $C(\\Delta,r)=0$ for $\\Delta < -1$ give~3813 linear constraints, so that we have to find \nthe kernel of a roughly square matrix of size nearly~4000. Moreover, the coefficients of this matrix are \ngigantic, so that we have to work modulo some large prime. Nevertheless, this computation could \nbe carried out (using PARI\/GP), and it turned out that \nthe function ${\\cal Q }_{210}$ could again be chosen to have optimal growth! \nThis solution is given by ${\\cal Q }_{210} \\= -  {\\cal F}_{210}$, where ${\\cal F}_{210} = \\sum_{\\ell \\; \\mod \\; {420}} h_{\\ell} \\, \\vartheta_{210,\\ell}$\nwith the Fourier coefficients of the mock theta functions $h_{\\ell}$ given in Table~\\ref{fcq210}. \n\\begin{table}[h]\n{\\footnotesize{\n\\begin{center} \n\\begin{tabular}{!{\\vrule width 1pt}c|c!{\\vrule width 1pt}rcccccccccccc!{\\vrule width 1pt}ccc}  \n\\noalign{\\hrule height 1pt}\n       $\\ell \\text{ with } h_{\\ell} = h_{\\ell_{0}}$ &$ \\ell  \\text{ with } h_{\\ell} = -h_{\\ell_{0}}$ & \n     $n=0$ & 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12  \\\\  \n\\noalign{\\hrule height 1pt}\n       {\\bf 1}, 29, 41, 71 & 139, 169, 181, 209 &  $-2$& 13& 28& 34& 49& 46& 71& 59& 83& 77&102& 87&121 \\\\\n       {\\bf 11}, 31, 59, 101 & 109, 151, 179, 199  &  0&  9& 21& 27& 36& 41& 51& 44& 75& 62& 62& 82&104 \\\\\n       {\\bf 13}, 43, 83, 97 & 113, 127, 167, 197 &  0&  6& 17& 17& 35& 20& 49& 31& 57& 36& 77& 32& 94\\\\\n       {\\bf 17}, 53, 67, 73 & 137, 143, 157, 193 &  0&  4& 12& 16& 22& 19& 43& 17& 40& 50& 41& 27& 87 \\\\\n       {\\bf 19}, 61, 79, 89 & 121, 131, 149, 191 &  0&  3& 11& 12& 23& 14& 37& 17& 43& 28& 45& 30& 77  \\\\\n       {\\bf 23}, 37, 47, 103 & 107, 163, 173, 187 &  0&  1&  7&  4& 20& $-1$& 32&  3& 30& 10& 50&$-16$& 63 \\\\\n\\noalign{\\hrule height 1pt}\n\\end{tabular}\n\\end{center}\n}}\n\\caption{\\small Fourier coefficients of the function~${\\cal F}_{210}$}\n\\label{fcq210}\n\\end{table}\nIn this table, we have listed the values of $\\ell$ with $(\\ell,210)=1$ and $0 \\le \\ell < 210$ (since \n$h_{420n\\pm\\ell} = h_{\\ell}$ and $h_{\\ell}=0$ for $(\\ell,210)>1$) in groups of~8, with $h_{\\ell} = h_{\\ell_{0}}$\nfor the first four and  $h_{\\ell} = - h_{\\ell_{0}}$ for the second four, where $\\ell_{0}$ (shown in boldface)\nbelongs to $\\{ 1,11,13,17,19,23 \\}$ and $h_{\\ell_{0}}=\\sum_{n \\ge0} C(n) \\, q^{n-\\ell_{0}^{2}\/840}$. \nThus, the third line of the table means that \n$h_{\\pm 13} = h_{\\pm 43} = \\cdots  = -h_{\\pm 167} = -h_{\\pm 197} \n\\= q^{-{169}\/{840}} \\big(6q + 17 q^{2} + 17 q^{3} + \\dots  \\big)$.\nA striking property of these coefficients is that they are so small: for instance, \nthe largest value of~$c(n,\\ell)$ in the table is~$c(12,1)=121$, with corresponding discriminant \n$\\Delta=4\\cdot210\\cdot12-1^2=10079$, while for the Mathieu form $h^{(2)}$ the table following\neq.~(7.31) gave the much larger coefficient $132825$ for the much smaller discriminant\n$4\\cdot2\\cdot9-1^2=71$.  \nThis behavior is a direct consequence of the ``optimal growth''\nproperty together with the fact that the index is large (cf.~Theorem~\\ref{CardyProp} below and the \ndiscussion in~\\S\\ref{specialMTF}).\n\nIn fact, in this example even more happens: not only the function ${\\cal Q }_{210} = \\Phi_{2,210}^{- - - -}$, but the whole function $\\Phi_{2,M}$ \n(and hence each of its eigencomponents $\\Phi_{2,210}^{\\pm\\pm\\pm\\pm}$) can be chosen to have optimal growth.\nIt turns out that this statement holds for all square-free indices~$m$, whether they have an even or an odd\nnumber of prime factors. If~$m$ is not square-free, then this is not true in \ngeneral,\\footnote{It is true if and only if~$m$ is the product of a square-free number and a prime power.}\nbut it becomes true after a very small modification of the functions in question:\n\\begin{thm} \\label{QmOG}\nThe functions $\\{\\Phi_{2,m} \\}_{m \\ge 1}$ can be chosen to have the form \n\\beta \\label{Phi0mimp}\n  \\Phi_{2,m}(\\tau,z) \\= \\sum_{d^{2} \\mid m} \\Phi^{0}_{2,m\/d^{2}} (\\tau, \\, d z)\n\\end{equation}\nwhere $\\{\\Phi_{2,m}^{0} \\}_{m \\ge 1}$ are primitive mock Jacobi forms having optimal growth.  \n\\end{thm}\n\n\n\\begin{corollary} \\label{Qmcor}\nThe functions $\\{{\\cal Q }_{M} \\}_{M > 1, \\, \\mu(M)=1}$ can be chosen to have optimal growth. \n\\end{corollary}\n\\noindent (Here we have excluded the case $M=1$ because the function~${\\cal Q }_{1}=-{\\cal H}$ has already been\nchosen to have no poles at all.) Tables of Fourier coefficients of the functions~${\\cal Q }_{M}$ appearing \nin the corollary for~$M \\le 50$ are given in Appendix~\\ref{apptabs1}.\n\nIf we denote by ${\\cal E }_{m}^{\\rm OG}$ the space of elliptic forms of index~$m$ (cf.~\\S\\ref{hecke}) \nhaving optimal growth, then we have the sequence of inclusions \n$$ {\\cal E }_{m}^{0} \\; \\subset \\; {\\cal E }_{m} \\; \\subset \\; {\\cal E }_{m}^{\\rm OG} \\; \\subset \\; \\widetilde {\\cal E }_{m} \\; \\subset \\; {\\cal E }_{m}^{!}  \n\\qquad \\big( \\, \\varphi \\in {\\cal E }_{m}^{\\rm OG} \\;  \\underset{\\rm DEF}\\Longleftrightarrow  \\; \\text{$C_{\\varphi}(\\Delta, \\ell) = 0$ \nif $\\Delta < -1$} \\, \\big)\n$$\nof spaces of different types of elliptic forms (cusp, holomorphic, optimal growth, weak, weakly holomorphic)\nwith different orders of growth. Theorem~\\ref{QmOG} says that we can represent all of our functions \nin terms of new, ``primitive,'' forms $\\Phi_{m}^{0} \\in {\\cal E }_{m}^{\\rm OG}$ for all~$m$. \nThe function~$\\Phi_{m}^{0}$ is unique up to the addition of an element of the space \n$J_{2,m}^{\\rm OG}={\\cal E }_{m}^{\\rm OG} \\cap \\widetilde J_{2,m}$ of Jacobi forms of weight~2 and index~$m$ of optimal growth.  \nWe will study this latter space in the next subsection.\n\n\nTheorem~\\ref{QmOG} has an immediate consequence for the growth of the Fourier coefficients \nof the optimal choice of~$\\Phi_{2,m}$, since the Hardy-Ramanujan circle method implies that \nthe Fourier coefficients of a weakly holomorphic Jacobi (or mock Jacobi) form grow asymptotically \nlike~$e^{\\pi\\sqrt{\\Delta|\\Delta_{0}|}\/m}$, where~$\\Delta_{0}$ is the minimal value of the discriminants  \noccurring in the form. \nWe give a more precise version of this statement in the special case of \nforms of weight~2 having optimal growth. \n\n\n\\begin{thm}\\label{CardyProp}\nLet $\\varphi$ be a weak or mock Jacobi form of weight~$2$ and index~$m$ with optimal growth.\nThen the Fourier coefficients of $\\varphi$ satisfy the asymptotic formula\n  \\beta \\label{HRR} C_\\varphi(\\Delta,r) \\= \\kappa_r\\,e^{\\pi\\sqrt\\Delta\/m}\n   \\;+\\;\\text O\\bigl(e^{\\pi\\sqrt\\Delta\/2m}\\bigr)  \\end{equation}            \nas $\\Delta\\to\\infty$, where the coefficient $\\kappa_r=\\kappa(\\varphi;r)$ is given by \n  \\beta \\label{kappa} \\kappa_r \\= -\\sum_{\\ell\\; \\mod \\; {2m} \\atop \\ell^{2}\\,\\equiv1\\; \\mod \\; {4m}} \n    C_\\varphi(-1,\\ell)\\,\\cos\\Bigl(\\frac{\\pi\\ell r}m\\Bigr)\\;. \\end{equation}\n\\end{thm}\n\\begin{proof}\nWe only sketch the proof, since it is a standard application of the circle method. We write the \ncoefficient $c_\\varphi(\\Delta,r)$ as a Fourier integral $\\int_{-{\\frac12}+i\\varepsilon}^{{\\frac12}+i\\varepsilon}h_r(\\tau)\\,\\mathbf{e}(-\\Delta\\tau\/4m)\\,d\\tau\\,$,\nwith $h_\\ell$ defined by~\\eqref{defhltau} and~\\eqref{jacobi-theta}.  Its value is unchanged if we replace \n$h_r(\\tau)$ by its completion $\\widehat h_r(\\tau)$, since they differ only by terms with negative discriminant\nThe main contribution to the integral comes from a neighbourhood of $\\tau=0$, where we can estimate $\\widehat h_r(\\tau)$ to high accuracy as\n $$  \\widehat h_r(\\tau) \\= -\\frac{(i\/\\tau)^{3\/2}}{\\sqrt{2m}}\\;\n   \\sum_{\\ell\\; \\mod \\; {2m}} \\mathbf{e}\\Bigl(\\frac{r\\ell}{2m}\\Bigr)\\; \\widehat h_\\ell\\Bigl(-\\frac1\\tau\\Bigr)\n\\= -\\frac{\\kappa_r}{\\sqrt{2m}}\\;(i\/\\tau)^{3\/2}\\,\\mathbf{e}\\Bigl(\\frac1{4m\\tau}\\Bigr)\\+\\text O\\bigl(\\tau^{1\/2}\\bigr)\\,, $$\nthe first equation holding by the $S$-transformation equation (\\cite{Eichler:1985ja}, p.~59) and the second because \nthe ``optimal growth\" assumption implies that $\\widehat h_\\ell(-1\/\\tau)=C_\\varphi(-1,\\ell)\\,\\mathbf{e}(1\/4m\\tau)+\\text O(1)$ as $\\tau\\to0$\nand the contribution from the Eichler integral of the shadow of $\\widehat h_\\ell$ is at most~$\\text O(1\/\\tau)$. The rest\nfollows by a standard calculation, the only point worthy of note being that since the $h_\\ell$'s have weight~3\/2\nthe Bessel functions that typically arise in the circle method are simple exponentials here.\n \n \n \n \n \n \n \n \n \n \n\\end{proof}\nIt is interesting to compare the statement of Theorem~\\ref{CardyProp} with the tables given in the \nAppendix. This will be discussed in more detail in \\S\\ref{specialMTF}.\n\n\n\\subsection{Optimal choice of the function~$\\Phi_{2,m}$ \\label{optimal}}\n\nAs we already saw, Theorem~\\ref{QmOG} fixes the choice of the mock Jacobi form~$\\Phi_{2,m}^{0}$\nup to the addition of an element of the space \n$J_{2,m}^{\\rm OG}$ of Jacobi forms of weight~2 and index~$m$ having optimal growth.   \nThe structure of this latter space is described by the following theorem, proved in~\\S\\ref{structure}.  \n  \\begin{thm}\\label{chooseKm}   The quotient $J_{2,m}^{\\rm OG} \/ J_{2,m}$ has dimension $1$ for every~$m$.\n  An element ${\\cal K}_{m}\\in J_{2,m}^{\\rm OG}$ representing a non-trivial element of this quotient space can be chosen to satisfy\n    $$ {\\cal K}_{m}(\\tau,z) \\= y \\, - \\, 2 \\+ y^{-1} \\+ O(q)\\quad\\in\\;\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}[y,y^{-1}][[q]]\\,, $$\n   and then has polar coefficients given by \n  \\beta\n   C({\\cal K}_{m}; \\, \\Delta, \\, r) \\= \\begin{cases} \n   1 & \\text{if $\\Delta=-1$, $r^{2} \\equiv 1 \\; \\mod \\; {4m}$} \\, , \\\\  0 & \\text{if $\\Delta<-1$} \\, . \\end{cases} \n   \\end{equation}\n    The function ${\\cal K}_{m}$ can be chosen to be primitive and invariant under all the operators \n    $W_{m_1}$ $(m_1\\|m)$.\n  \\end{thm}\n\n\nThe choice of ${\\cal K}_m$ described in the theorem is unique up to the addition of a primitive holomorphic\nJacobi form invariant under all $W_{m}$ operators. This first occurs for $m=25$, so ${\\cal K}_m$ is\nunique for small values of~$m$. Because of Theorem~\\ref{chooseKm}, the function $\\Phi^{0}_{m}$ has only a very \nsmall amount of choice: we can change it without disturbing the ``optimal growth'' property by adding an arbitrary \nmultiple of ${\\cal K}_{m}$ and an arbitrary holomorphic Jacobi form, but this is the only freedom we have. In particular, we have \nthree special choices of ``optimal'' $\\Phi^{0}_{2,m}$, each unique for $m<25$, \ncorresponding to the three that we already encountered for $m=6$ and $m=10$ in \\S\\ref{choosephi}: \n\\begin{enumerate}\n\\item[(I)] The $(+,\\cdots,+)$-eigencomponent of  $\\Phi_{2,m}^{0, \\rm I}$ is strongly holomorphic.\n\\item[(II)] $\\Phi_{2,m}^{0, \\rm II}$ has $c(0,1)=0$. \n\\item[(III)] $\\Phi_{2,m}^{0, \\rm III}$ has $c(0,0)=0$. \n\\end{enumerate}\nEach of these three choices has certain advantages over the other ones. \nIn case~(I) we can choose~$\\Phi_{2,m}^{0, \\rm I}$  within its class modulo~$J_{2,m}^{0}$ \nso that its $(+,\\cdots,+)$-eigencomponent is precisely equal to $2^{1-\\omega(m)} {\\cal Q }_{1}|{\\cal V }^{(1)}_{2,m}$ \nand hence has Fourier coefficients expressible in terms of class numbers, like in the examples for \nsmall~$m$ that we already saw. \nAlso,~$\\Phi_{2,m}^{0,\\rm I}$ is strongly holomorphic when $m$ is a prime power, so that \nthis is certainly the best of the three choices in that special case. \nThe function occurring in~(II) has a Fourier expansion beginning $\\frac{1}{12}(m+1) + O(q)$, \nand the corresponding $\\psi^{0,\\rm II}_{m+1}$ satisfies~\\eqref{coeffCm} and belongs to the \nfamily of the weight zero, index~$m$ Jacobi forms discussed in~\\cite{Gritsenko:1999fk}.  \nThe choice~(III) looks at first sight like the least natural of the three, since it might seem  \nstrange to insist on killing a $\\Delta=0$ term when there are $\\Delta=-1$ terms that are more singular, \nbut in fact  it is the nicest in some ways, since it can be further improved by the addition of a holomorphic \nJacobi form to have {\\it all} of its $\\Delta=0$ coefficients equal to zero, and its \npolar coefficients are then given by an elegant formula, which we now state. \n\\begin{thm} \\label{PolarCoeffs}\nThe mock Jacobi form~$\\Phi_{2,m}^{0, \\rm III}$ in Theorem~\\ref{QmOG} can be chosen, \nuniquely up to the addition of a Jacobi cusp form of weight~$2$ and index~$m$, \nso that all its Fourier coefficients with~$\\Delta=0$ vanish. Its polar coefficients are \nthen given by \n\\beta \\label{Phi0IIIcoeffs}\nC(\\Phi_{2,m}^{0, \\rm III}; -1, r) \\= \\frac{1}{24} \\sum_{d^{2}|m} \\, \\mu(d) \\, \n\\Big(\\big(\\frac{r-1}{2}, \\frac{m}{d^{2}} \\big) \\+ \\big(\\frac{r+1}{2}, \\frac{m}{d^{2}} \\big)  \\Big) \n\\quad \\text{for $r^{2} \\equiv 1 \\; \\mod \\; {4m}$.} \n\\end{equation}\nMoreover, the function~$\\Phi_{2,m}^{0, \\rm III}$ can be chosen to be primitive without affecting these properties. \n\\end{thm}\n\\noindent This also gives a multiplicative formula for the polar coefficients of all eigencomponents \nof~$\\Phi_{2,m}^{0, \\rm III}\\,$: \n\\begin{corollary}\nWrite $m=\\prod_{i=1}^{s} P_{i}$, where $P_{i}=p_{i}^{\\nu_{i}}$ with distinct primes $p_{i}$\nand exponents $\\nu_{i} \\ge 1$. Then for any~$\\varepsilon_{i} \\in \\{\\pm 1\\}$ with~$\\varepsilon_{1}\\cdots \\varepsilon_{s}=1$, \nwe have  \n\\beta \\label{eigencoeffs}\nC\\Bigl( \\Phi_{2,m}^{0, \\rm III} \\, \\Bigl| \\, \\prod_{i=1}^{s} \\big( 1+ \\varepsilon_{i} W_{P_{i}} \\big); \\, -1 , \\, r \\Bigr) \\=\n\\frac{\\pm 1}{12} \\, \\prod_{i=1}^{s} \\; \\begin{cases} \\, (p_{i} + \\varepsilon_{i}) & \\text{if $\\nu_{i}=1$,}  \\\\\n\\, (p_{i}^{\\nu_{i}} - p_{i}^{\\nu_{i}-2}) & \\text{if $\\nu_{i}\\ge 2$,}  \n\\end{cases}\n\\end{equation}\nwhere the sign is the product of the $\\varepsilon_{i}$ for all $i$ for which $r \\equiv -1 \\; \\mod \\; {2 P_{i}}$. \nIn particular, for the functions~${\\cal Q }_{M}$ $(M>1$, $\\mu(M)=1)$, chosen by the Corollary\nto Theorem~\\ref{QmOG}\nto have optimal growth, the Fourier coefficients with non-positive discriminant are given by\n\\beta \\label{Qmpolar}\nC({\\cal Q }_{M}; \\Delta, r) \n\\= \\begin{cases}  \\qquad 0 \\,  & \\text{if $\\Delta = 0$} \\, , \\\\\n\\pm \\dfrac{\\varphi(M)}{24} \\,  &  \\text{if $\\Delta \\= -1$} \\, , \n\\end{cases} \n\\end{equation}\nwhere in the second line $r^{2} \\equiv 1 \\; \\mod \\; {4M}$ and the sign is determined \nby $\\mu((\\frac{r+1}{2}, M)) = \\pm 1$.\n\\end{corollary}\nThe final statement of the corollary can be compared with the examples in~\\eqref{Qlist}, \nin which each of the ${\\cal F}_{M}$'s occurring (for $M>1$)  was normalized with $C({\\cal F}_{M};-1,1)=-1$, \nand the numerical factor relating ${\\cal Q }_{M}$ and ${\\cal F}_{M}$ equals $-\\varphi(M)\/24$ in each case.\nTo give a feeling for the nature of these functions in general, \nand as an illustration of the formulas in Theorem~\\ref{PolarCoeffs} and its corollary, \nwe give in Table~\\ref{Kmphi0} below the first few Fourier coefficients $C(\\Phi; \\Delta,\\ell)$ for these \nand some related functions $\\Phi$ for $m=25$ (the first index where $J_{2,m}\\neq\\{0\\}$ and hence\n${\\cal K}_{m}$ is not unique), $m=37$~(the first index where $J_{2,m}^{0}\\neq\\{0\\}$ \nand hence $\\Phi_{2,m}^{0, \\rm III}$ is not unique),\n$m=50$~(the first case where \\eqref{eigencoeffs} is non-trivial), \nand $m=91$~(the first case where where $J_{2,m}^{0,--}\\neq\\{0\\}$ and hence ${\\cal Q }_m$ is not unique). \nThe free coefficients~``$a$'', ``$b$'' \nand ``$c$'' in the table arise because $\\dim J_{2,25} = \\dim J_{2,37}^{0} = \\dim J_{2,50}= \\dim J_{2,91}^0 =1$. \nIn the case of $m=91$ we show the two $\\ell$-values corresponding to each value of~$\\Delta$ in the format \n$\\,\\ell \\, \/ \\,\\ell^*\\,$ and give in the following line the Fourier coefficient $C({\\cal Q }_{91};\\Delta,\\ell)=-C({\\cal Q }_{91};\\Delta,\\ell^*)$.\nIn the cases of $m=25$ and $m=37$, we have given the (non-unique) holomorphic mock Jacobi \nform~$\\Phi^{0, \\rm I}_{2,2m}$ as well as~$\\Phi^{0, \\rm III}_{2,2m}$.\n\\begin{table}[h] \n{\\small\n\\begin{center}\n\\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}c|ccc|c|c|c|c|c|c!{\\vrule width 1pt}cc}  \n\\noalign{\\hrule height 1pt}\n       $\\Delta$ & $-1$&  \\multicolumn{3}{c|}{0}  & $4$& $11$ & $16$ &$19 $& $24$& $31$   \\\\  \n       $\\pm \\ell \\; \\mod \\; {50}$ & $  1$& $0$ & $  10$ & $  20$ & $  14$& $  17$ & $  22$ &$  9$& $  24$& $ 13$   \\\\  \\hline\n       $ {\\cal K}_{25}$ & $1$ & $-2$ & $a$ & $1-a$ & $1-a$ & $1+2a$ & $2+3a$ & $2-4a$ & $5-2a$ & $3+2a$   \\\\\n       $ \\Phi^{0, \\rm III}_{2,25} $ & $1$ & $0$ & $0$ & $0$ & $0$ & $1$ & $2$ & $-1$ & $2$ & $1$ \\\\\n       $ \\Phi^{0, \\rm I}_{2,25} $ & $0$ & $2$ & $c$ & $-1-c$ & $-1-c$ & $2c$ & $3c$ & $-3-4c$ & $-3-2c$ & $-2+2c$ \\\\\n\\noalign{\\hrule height 1pt}\n\\end{tabular} \n\\end{center}\n\n\\begin{center}\n\\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}c|c|c|c|c|c|c|c|c|c!{\\vrule width 1pt}ccc}  \n\\noalign{\\hrule height 1pt}\n       $\\Delta$ & $-1$& $0$ & 3 & 4 & $7$& $11$ & $12$ &$16 $& $27$& $28$   \\\\  \n       $\\pm \\ell \\; \\mod \\; {74}$ & $  1$& $0$ & $21$ & $12$ & $17$& $27$ & $32$ &$24$& $11$& $34$   \\\\  \\hline\n       $ {\\cal K}_{37} $ & $1$ & $-2$ & $a$ & $a$ & $1-a$ & $1+a$ & $2-a$ & $2-2a$ & $1-3a$ & $2+3a$ \\\\\n       $ \\Phi_{2,37}^{0, \\rm III} $ & $\\frac{19}{12}$ & $0$ & $b$ & $-\\frac16+b$ & $\\frac14-b$ & $\\frac{11}{12}+b$ \n       & $\\frac32-b$ & $1-2b$ & $-\\frac34-3b$ & $\\frac{13}{6}+3b$ \\\\\n       $ \\Phi_{2,37}^{0, \\rm I} $ & $0$ & $\\frac{19}{6}$ & $c$ & $-\\frac{1}{6}+c$ & $-\\frac{4}{3}-c$ & $-\\frac{2}{3}+c$ \n       & $-\\frac{5}{3}-c$ & $-\\frac{13}{6}-2c$ & $-\\frac{7}{3}-3c$ & $-1+3c$   \\\\\n\\noalign{\\hrule height 1pt}\n\\end{tabular} \n\\end{center}\n}\n\n{\\small \n\\begin{center}\n\\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}cc|ccc|c|c|c|cc|c|c!{\\vrule width 1pt}cc}  \n\\noalign{\\hrule height 1pt}\n       $\\Delta$ & \\multicolumn{2}{c|}{$-1$} &  \\multicolumn{3}{c|}{0} & $4$ & $16$ &$24 $& \\multicolumn{2}{c|}{31} & $36$ &39  \\\\  \n       $\\pm \\ell \\; \\mod \\; {100}$ & $1$& $49$ & $0$ & $20$ & $40$& $14$ & $28$ &$24$& $13$& $37$ &$42$ & $19$  \\\\  \\hline\n       $ {\\cal K}_{50} $ & $1$ & $1$ & $-2$ & $a$ & $1-a$ & $-a$ & $1+a$ & $1-2a$ & $1+2a$ & $1+2a$ &$2+3a$& $-2a$\\\\\n       $ \\Phi_{2,50}^{0, \\rm III} | (1+W_{2}) $ & $3$ & $3$ & $0$ & $0$ & $0$ & $-2$ & $2$ & $-2$ & $3$ & $3$ & $8$ &$-1$ \\\\\n       $ \\Phi_{2,50}^{0, \\rm III} | (1-W_{2}) $ & $1$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $1$ & $0$  &$-1$ \\\\\n\\noalign{\\hrule height 1pt}\n\\end{tabular} \n\\end{center}\n\n\n\n\\begin{center}\n\\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}c|c|c|c|c|c|c|c|c!{\\vrule width 1pt}c|c}  \n\\noalign{\\hrule height 1pt}\n       $\\Delta$ & $-1$& 3 & 12 & 27 & 40 & 48 & 55 & 68 & 75  \\\\  \n       $\\pm \\ell \\; \\mod \\; {182}$ & $1\\, \/ \\, 27$& $19\\, \/ \\,33$ & $38\\, \/ \\,66$ & $57\\, \/ \\,83$ & $18\\, \/ \\,60$& \n        $50\\, \/ \\,76$ & $71\\, \/ \\,85$ &$32\\, \/ \\,46$&  $17\\, \/ \\,87$   \\\\  \\hline\n       ${\\cal Q }_{91}$ & $3$ & $a$ & $-1+a$ & $-1-2a$ & $-1-3a$ & $-2+2a$ & $3a$ & $1+3a$ & $-5+2a$ \\\\ \n\\noalign{\\hrule height 1pt}\n\\end{tabular} \n\\end{center}\n\n} \n\\caption{\\small{Examples of non-unique Jacobi and mock Jacobi forms}} \n \\label{Kmphi0}\n\\end{table}\n\nIn summary, the ``optimal growth'' property of~$\\Phi_{2,m}^{0}$ and the near-uniqueness of the \nJacobi form~${\\cal K}_{m}$ have permitted us to pin down both of these functions in many cases, \nbut there still remains an additive ambiguity whenever there are Jacobi cusp forms.  \nWe do not know how to resolve this ambiguity in general, and state this as an open problem:\n\n\n\\smallskip\n\\noindent{\\it Question}: Is there a canonical choice of the forms ${\\cal K}_m$ and ${\\cal Q }_M$ for all\npositive integers~$m$ and all positive square-free integers~$M\\,$?\n\n\\smallskip\n{ \\noindent Should the answer to either of these questions be positive, the corresponding forms would be of\nconsiderable interest. In particular, one can speculate that the canonical choice of~${\\cal Q }_M$, if it exists,\nmight be canonically associated to the quaternion algebra of discriminant~$M$, since there\nis a natural bijection between positive square-free integers and quaternion algebras over~${\\mathbb Q}$,\nwith the latter being indefinite or indefinite according as $\\mu(M)=1$ (corresponding to our $k=2$\nforms and to mock theta functions of weight~3\/2) or $\\mu(M)=-1$ (corresponding to $k=1$ and \nto mock theta functions of weight~1\/2).\n}\n\nWe end this subsection by describing one possible approach to answering the above question\n  \nthat we tried.  This approach was not successful in the sense that, although it did produce\na specific choice for ${\\cal K}_m$ and $\\Phi_{2,m}^0$ (or $\\Phi_{1,m}^0$), these did not\nseem to have arithmetic coefficients and therefore had to be rejected, but the\nmethod of calculation involves a result that is of interest in its own right and\nmay be useful in other contexts.  The idea is to fix the choice of the Jacobi\nforms (or mock Jacobi forms) in question by demanding that they (or their completions)\nbe orthogonal to Jacobi cusp forms with respect to the Petersson scalar product.\n(As an analogy, imagine that one wanted to \ndetermine the position of~$E_{12}=1+\\cdots$ in $M_{12}=\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \\, E_{4}^{3} + \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \\, \\Delta$ \nbut did not know how to compute the Fourier expansion of $\\sum (m\\tau+n)^{-12} \\,$. Then one  \ncould simply write $E_{12}=E_{4}^{3}+a\\Delta$ and use the orthogonality of Eisenstein series \nand cusp forms to compute the number~$a=-432000\/691$ numerically\nas~$-(E_{4}^{3},\\Delta)\/(\\Delta,\\Delta)$, where $( \\, \\, , \\,)$ denotes the Petersson scalar product.)  \nTo apply this idea, we need a good way to compute the Petersson scalar product\nof Jacobi forms. This is the result of independent interest referred to above.\nFor simplicity, we state it only in the case~$k=2$.\n\\begin{thm}\\label{JCPSP}\nLet $\\varphi_1$ and $\\varphi_2$ be two Jacobi cusp forms of weight~$2$ and index~$m$, and set\n \\beta \\label{AHdef} A(\\Delta) \\,=\\, \\sum_{\\ell\\; \\mod \\; {4m}} C_{\\varphi_{1}}(\\Delta,\\ell)\\,\\overline{C_{\\varphi_{2}}(\\Delta,\\ell)}\\,,\\;\\quad \n H(t) \\,=\\, \\sum_{\\Delta=1}^\\infty\\,\\frac{A(\\Delta)\/\\sqrt\\Delta}{\\exp(2\\pi \\sqrt{t\\Delta\/m})\\,-\\,1}\\;.  \\end{equation}\nThen the function $\\dfrac{tH(t)-H(1\/t)}{t-1}$ $(t\\ne1)$ has a constant value $c_H$\nwhich is proportional\\footnote{The exact constant of proportionality plays no role for our\npurposes, since we will only be concerned with ratios of Petersson scalar products.\nIf the scalar product is normalized as in~\\cite{Eichler:1985ja}, its value is 1\/2.\n}\nto the Petersson scalar product $(\\varphi_1,\\varphi_2)$. \\end{thm}\n\\noindent{\\it Sketch of proof}: We refer to~\\cite{Eichler:1985ja} for the definition of \n$(\\varphi_1,\\varphi_2)$ (eq.~(13), p.~27) and for its expression as a simple multiple of \n$\\sum_\\ell(h_\\ell^{(1)},h_\\ell^{(2)})$, where $h_\\ell^{(1)}$ and~$h_\\ell^{(2)}$ are the coefficients \nin the theta expansions of $\\varphi_1$ and $\\varphi_2$ (Theorem 5.3, p.~61). \nCalculating these weight 3\/2 scalar products by the Rankin-Selberg method as explained in \\S\\ref{modularbasic},\nwe find that $(\\varphi_1,\\varphi_2)$ can be expressed as a simple multiple of the residue at $s=3\/2$\nof the Dirichlet series $L(s)=\\sum_{\\Delta>0}A(\\Delta)\\Delta^{-s}$.  The  Rankin-Selberg method also shows that \nthe function $\\widetilde L(s)=2\\,(2\\pi)^{-s}m^s\\,\\Gamma(2s)\\,\\zeta(2s)\\,L(s+\\frac12)$ has a meromorphic continuation \nto all~$s$, with poles only at $s=0$ and $s=1$, and is invariant under~$s\\to1-s$.  But $\\widetilde L(s)$ is\nsimply the Mellin transform of $H(t)$, so this implies the proposition by a standard argument. \n\\hfill $\\square$\n\nA rather surprising aspect of this theorem is that it is actually {\\it easier} to compute Petersson scalar \nproducts for Jacobi forms than it is for ordinary modular forms of integral weight. In that case the analogue \nof $H(t)$ would have an infinite sum of $K$-Bessel functions multiplying each coefficient~$A(\\Delta)$,\nwhereas here the Bessel functions are replaced by simple exponentials (because the modular forms $h_\\ell^{(i)}$\nhave half-integral weight, so that the usual gamma factor $\\Gamma(s)\\Gamma(k+s-1)$ in the \nRankin-Selberg method becomes a single gamma function $\\Gamma(2s)$)  and at the same time \nthe infinite sum becomes a geometric series that can be summed explicitly.\n\nTo apply the proposition to our situation in the case of ${\\cal K}_{37}$, we start with an initial choice\n(like the one given in Table~\\ref{Kmphi0}) and replace it by ${\\cal K}_{37}+a\\varphi_{37}$, where~$\\varphi_{37}$\nis the generator of~$J_{2,37}^{0}$ (normalized by~$C(\\varphi_{37}\\,; 3,21)=1$) and where $a\\in{\\mathbb R}$ is\nchosen to make this function orthogonal to $\\varphi_{37}$, i.e., $a=-({\\cal K}_{37},\\varphi_{37})\/(\\varphi_{37},\\varphi_{37})$,\nin the hope that this~$a$ will be an integer or simple rational number. \n(Of course ${\\cal K}_{37}$ is not a cusp form, but the integral defining the Petersson scalar product \nstill converges,\\footnote{The precise condition needed for the convergence of the Petersson scalar \nproduct of two weakly holomorphic Jacobi forms~$\\varphi_{1}$ and~$\\varphi_{2}$ is that $\\, \\text{ord}_{\\infty}(h_{\\ell}^{(1)})+\n\\text{ord}_{\\infty}(h_{\\ell}^{(2)})>0$ for all~$\\ell \\in {\\mathbb Z}\/2m{\\mathbb Z}$. This is amply satisfied in the \ncase~$\\varphi_1=\\varphi_{37}$, $\\varphi_2={\\cal K}_{37}$, since the only negative value of $\\, \\text{ord}_{\\infty}(h_{\\ell}^{(2)})$ \nis~$-1\/148$ for~$\\ell \\equiv \\pm 1 \\; \\mod \\; {74}$, and there~$\\, \\text{ord}_{\\infty}(h_{\\ell}^{(1)}) = 147\/148$.}\nthe sum defining $H(t)$ converges for $t>\\frac{1}{148}$ by Theorem~\\ref{CardyProp}, and Theorem~\\ref{JCPSP}\nstill holds.\\footnote{The proof given above breaks down, since the numbers~$A(\\Delta)$ grow exponentially\nin~$\\sqrt{\\Delta}$, so that the Dirichlet series~$\\sum_{\\Delta}A(\\Delta)\/\\Delta^{s}$ does not converge for any \nvalue of~$s$, but the function~$f(y)=y^{3\/2}\\sum_{\\Delta} A(\\Delta) e^{-\\pi\\Delta y\/m}$, which is the constant \nterm of an~$SL(2,{\\mathbb Z})$-invariant function that is small at infinity, still has a Mellin transform with a \nmeromorphic continuation to all~$s$, and if we define~$\\widetilde L(s)$ as the product of this Mellin transform\nwith~$\\pi^{-s}\\Gamma(s)\\zeta(2s)$, then~$\\widetilde L(s)$ has the same analytic properties as before and its inverse Mellin\ntransform~$H(t)$ is still given by the sum in~\\eqref{AHdef} for~$t$ sufficiently large and satisfies the \nsame functional equation.})\nThe numerical calculation for\nthe case $\\varphi_1=\\varphi_2=\\varphi_{37}$ gives the value $c_H=c_1=-0.95284748097797917403$ (a value that can be\nchecked independently, since up to a factor $-\\pi$ it must be equal to the even period of the elliptic\ncurve $y^2-y=x^3-x$, as calculated by GP-PARI as {\\tt ellinit}([0,0,1,-1,0])[15]). The corresponding\nnumber for $\\varphi_1=\\varphi_{37}$ and $\\varphi_2={\\cal K}_{37}$ (with ${\\cal K}_{37}$ chosen as in \nTable~\\ref{Kmphi0}, with~$a=0$) is $c_2= 0.26847600706319893417$, which is not a simple \nmultiple of $c_1$, but when we calculate again with $\\varphi_2=\\Phi^{0,{\\rm III}}_{2,37}$ (again as in Table~\\ref{Kmphi0},  \nwith~$b=0$) we find a value $c_3 = 0.107471184190738587618$ that is related to the two previous values\nby $12c_3-19c_2=4c_1$ numerically to high precision. Thus in this case we have indeed found\na form orthogonal to cusp forms and with rational coefficients.  This at first looks promising, but\nin fact has a simple and disappointing explanation: the linear combination \n$\\Phi^{0,{\\rm III}}_{2,37}-\\frac{19}{12}{\\cal K}_{37}-\\frac{1}{3}\\varphi_{37} =\\Phi^{0,{\\rm I}}_{2,37}-\\frac{1}{3}\\varphi_{37}$ \nis nothing other than ${\\cal Q }_1|V_{37}$ (as one can see in Table~\\ref{Kmphi0},\nwhere setting $c=-\\frac{1}{3}$ gives entries equal to $-H(\\Delta)$), and since the function ${\\cal Q }_1=-{\\cal H}$ is\nconstructed from an Eisenstein series, albeit a non-holomorphic one, it is automatically orthogonal\nto all cusp forms. And indeed, when we look at the case of~${\\cal Q }_{91}$ and~$\\varphi_{91} \\in J_{2,91}^{0,--}$, \nwhere there are no Eisenstein series\nin sight, then the calculation of the quotient $(\\widehat{\\cal Q }_{91},\\varphi_{91})\/(\\varphi_{91},\\varphi_{91})$ failed to \nyield a simple number. (The non-holomorphy of $\\widehat{\\cal Q }_{91}$ does not affect the application \nof Theorem~\\ref{JCPSP}, since its non-holomorphic part has Fourier coefficients only for~$\\Delta<0$, \nwhere the coefficients of $\\varphi_{91}$ vanish.)\nAs an incidental remark, we mention that the cusp form $\\varphi_{37}$, whose first few Fourier\ncoefficients were obtained only with difficulty in~\\cite{Eichler:1985ja} (pp.~118--120 and 145),\ncan now be obtained very easily using the ``theta blocks\" of~\\cite{GSZ} as\n$$  \\varphi_{37}(\\tau,z) \\= \\frac1{\\eta(\\tau)^6}\\,\\prod_{i=1}^{10}\\vartheta(\\tau,a_iz)\\,,\n   \\qquad (a_1,\\dots,a_{10})\\=(1,1,1,2,2,2,3,3,4,5)\\,, $$\n\nand similarly $\\varphi_{2,91}$ can be obtained easily as the difference of the two similarly defined theta blocks \nfor the 10-tuples $(a_1,\\dots,a_{10})=(1,1,2,3,4,4,5,5,6,7)$ and $(1,1,2,2,3,3,4,5,7,8)$.\n\n\n\\subsection{Observations on the weight one family, integrality, and positivity  \\label{specialMTF}}\n\nIn the last two subsections we discussed experimental properties of the family of weight~2 mock \nJacobi forms that were introduced in~\\S\\ref{simplest} and formulated a number of general results that will\nbe proved in \\S\\ref{structure}.  In this subsection we report on the experimental data for the \nweight~1 family and discuss various properties of both families that play a role in connection with\nother investigations in mathematics and physics (like ``Umbral Moonshine\") or that seem to be of\nindependent interest.  \n\n\nThe discussion of the weight~1 family is quite different in nature from that of the weight~2 case, because\nhere there do not seem to be general structural results as there were there, but only a small number of\nspecial functions for small values of the index having nice properties.  In particular, the analogue\nof Theorem~\\ref{QmOG} no longer holds, and we find instead that (at least up to $m=100$, and \npresumably for all~$m$) the appropriately defined primitive mock Jacobi forms $\\Phi_{1,m}^0$  \n \ncan be chosen to have optimal growth only for the indices  \n   \\beta\\label{mList}  m \\= 2,\\;3,\\;4,\\;5,\\;6,\\;7,\\;8,\\;9,\\;10,\\;12,\\;13,\\;16,\\;18,\\;25\\,. \\end{equation}\nSimilarly, although the analogue of Theorem~\\ref{phimQm} is still valid in the \nweight~1 case (now with the special mock Jacobi forms~${\\cal Q }_M$ labelled by square-free integers~$M$ having an\nodd rather than an even number of prime factors), the functions ${\\cal Q }_M$ can apparently be chosen to have \noptimal growth only for \n\\beta\\label{QMList}  M \\= 2,\\;3,\\;5,\\;7,\\;13,\\;30,\\;42,\\; 70,\\;78\\;. \\end{equation}\n(This list has a precise mathematical interpretation; see {\\it 3.} below.)\nThus here a small number of special examples is singled out. Interestingly enough, these\nturn out to include essentially all of the special mock theta functions that have been \nstudied in the literature in the past. \n\nFor convenience, we divide up our discussion into several topics.\n\n\\newpage\n\\bigskip\\noindent{\\it 1. Optimal choices for the weight one family  \\label{wt1optimal}}\n\\smallskip\n\nAs in the weight~2 case, we begin our discussion by looking at small values of~$m$.  We will \nconsider the original forms $\\Phi_{1,m}$ first, starting with the ``standard\" choices~\\eqref{defphi1m}\nand trying to modify them by the addition of weak Jacobi forms to attain the property of having optimal growth.  \nActually, it is better to look at the primitive mock Jacobi forms, now defined by\n  \\beta \\label{DefPhi01m}\n  \\Phi_{1,m}^0\\;\\equiv\\;\\sum_{d^2|m}\\frac{\\mu(d)}d\\,\\Phi_{1,m\/d^2}|U_d\\,, \\qquad\n  \\Phi_{1,m}\\;\\equiv\\;\\sum_{d^2|m}\\frac1d\\,\\Phi^0_{1,m\/d^2}|U_d\\qquad\\; \\mod \\; {\\widetilde J_{1,m}}\\,, \\end{equation}\ninstead of~\\eqref{Phi0mimp} and ~\\eqref{defPhi02m}, for the same reason for which we introduced the\nfunctions~$\\Phi_{2,m}^0$ in~\\S\\ref{Statements}: residue arguments similar to those that we will give \nin~\\S\\ref{structure} for the weight~2 case show here that $\\Phi_{1,m}$ can never have optimal growth unless\nit agrees with $\\Phi_{1,m}^0$ (which happens if and only if $m$ is square-free or the square\nof a prime number), so that the latter class is certain to be more productive.  Finally, we\nwill look at the forms~${\\cal Q }_M$ where~$M$ is a product of an odd number of distinct\nprime factors, since according to Theorem~\\ref{phimQm} all of the forms~$\\Phi_{1,m}$ can be constructed from these.\nUnlike the situation for~$k=2$, here we do not have to worry about any choices,\nbecause of the following theorem (to be compared with Lemma~2.3 of~\\cite{Cheng:2012tq} and to be proved, \nlike all its companions, in~\\S\\ref{structure}) and corollary.\nThe only issue is therefore whether the ``optimal growth\" condition can be fulfilled at all.\n\n\\begin{thm} \\label{Wt1OSP}\nThere are no weight one weak Jacobi forms of optimal growth.  \n\\end{thm}\n\n\\begin{corollary} \\label{UniqCor}\nIf any of the mock Jacobi forms $\\Phi_{1,m}$, $\\Phi_{1,m}^0$, or ${\\cal Q }_{M}$ $(\\mu(M)=-1)$ can \nbe chosen to have optimal growth, then that choice is unique.\n\\end{corollary}\n\nWe now look at some small individual indices.  For {\\bf m=1} there is nothing\nto discuss, since the function $\\Phi^{\\rm st}_{1,1}=(C\/2A)^{\\rm F}$ vanishes identically, as already\nexplained in~\\S\\ref{MJExamples} (Example~1).  (This, by the way, is the reason why $\\Phi_{1,m}=\\Phi_{1,m}^0$\nwhen~$m$ is the square of a prime.) The case of {\\bf m=2} has also already been treated \nin the same place (Example~2), where we saw that \n\\beta  \\label{PHI12} \n\\v^{\\text{opt}}_{1,2} \\= \\frac{B}{24\\,A}\\,C\\,, \\qquad \\Phi^{\\rm opt}_{1,2} \\= -\\frac1{24}\\,{\\cal F}_2\\,, \\end{equation}\nwith ${\\cal F}_2$ the mock modular form defined in~\\S\\ref{Mock} and related to the Mathieu moonshine story.\nFor~{\\bf m=3}, we find that the (unique) choice of $\\Phi_{1,m}$ having optimal growth is given by\n \\beta  \\label{PHI13}\n\\v^{\\text{opt}}_{1,3} \\= \\frac{B^2-E_4A^2}{288A}\\,C\\,, \\qquad \\Phi^{\\rm opt}_{1,3} \\= \\v^{\\text{opt}}_{1,3}\\,-\\,\\mathfrak{F}_P{1,3} \\= \\frac1{12}\\,{\\cal F}_3\\,, \\end{equation}\nwhere the normalizing factor 1\/12 has been chosen to give ${\\cal F}_3$ integral Fourier coefficients. These\ncoefficients are determined by virtue of their periodicity and oddness by the values $C^{(3)}(\\Delta)=C({\\cal F}_3;\\Delta,r)$ \nfor $r\\in\\{1,2\\}$, the first few values of $C^{(3)}(\\Delta)$ being given by\n$$ \n\\begin{tabular}{c|ccccccccccc}     $\\Delta$ & $-1$ & 8 & 11 & 20 & 23 & 32 & 35 & 44 & 47 & 56 & 59 \\\\ \n \\hline  $C^{(3)}(\\Delta)$ & $-1$ & 10 & 16 & 44 & 55 & 110 & 144 & 280 & 330 & 572 & 704  \\\\    \\end{tabular}  \\;\\,. \n$$\n(A more extensive table of these coefficients, listed separately for the two  values~$r=1$ and~$r=2$,\nis given in Appendix~\\ref{apptabs2}, where we also give the corresponding values for the other\nforms treated here, so that we no longer will give tables of coefficients in the text.)\nFor {\\bf m=4} and {\\bf m=5}, the functions $\\Phi_{1,m}$ can again be chosen (of course uniquely, \nby the corollary above) to have optimal growth, the corresponding meromorphic Jacobi forms~$\\v^{\\text{opt}}_{1,4}$\nand~$\\v^{\\text{opt}}_{1,5} $ being given by\n\\beta  \\label{4and5}\n \\frac{B^3-3E_4A^2B+2E_6A^3}{2\\cdot 12^2\\,A}\\,C \\quad \\text{and}\\quad\n\\frac{B^4-6E_4A^2B^2+8E_6A^3B-3E_4^2A^4}{2\\cdot 12^3\\,A}\\,C\\,, \n\\end{equation}\nrespectively. Note that in the case of~$m=4$, according to the decomposition result given in\nTheorem~\\ref{phimQm}, we could also make the choice ${\\cal Q }_2|{\\cal V }_{1,2}^{(2)}$ for the function~$\\Phi_{1,4}$,\nbut then it would have the worse discriminant bound $\\Delta_{\\rm min}=-4$. We also observe that, although~4\nis not square-free, we do not have to distinguish between~$\\Phi_{1,m}$ and~$\\Phi_{1,m}^0$ here, because\nthey differ by the function $\\Phi_{1,1}^0|U_2$ and the function $\\Phi_{1,1}$ vanishes identically.\n\nIt is now very striking that the numerators of the fractions in~\\eqref{PHI13} and~\\eqref{4and5}\nare the {\\it same} as the expressions that we already saw in eqs.~\\eqref{PHI21}, \\eqref{defX2}\nand \\eqref{PSIF4} in~\\S\\ref{choosephi}, meaning that for these values of~$m$ we have\n\\beta \\label{coincidence}\n\\v^{\\text{opt}}_{1,m} \\= \\frac CA\\,\\psi_{0,m-1}^{\\rm opt} \\= C\\,\\v^{\\text{opt}}_{2,m-2} \n\\end{equation}\nwith the same weight~0 weak Jacobi modular forms $\\psi_{0,m-1}^{\\rm opt}$ as were tabulated in Table~\\ref{Cmcoeffs}.\nThe same thing happens for several other small values of the index.  We do not know the deeper\nreason (if any) for this coincidence, but merely remark that the weight~0 weak Jacobi forms obtained\nfrom the weight~1 meromorphic Jacobi forms in our family by dividing by~$C$ also play a prominent role\nin the recent work of Cheng, Duncan and Harvey~\\cite{Cheng:2012tq} on ``umbral moonshine,\" and also that\nthe the forms $\\psi_{0,m}^{\\rm opt}$ occur in a somewhat different context in earlier work of \nGritsenko\\cite{Gritsenko:1999fk} related to Borcherds products, as was already mentioned in \\S\\ref{Statements} when we introduced the ``second standard choice\" $\\Phi_{2,m}^{0,\\rm II}$ of the primitive mock Jacobi form $\\Phi_{2,m}^0$.\n\nContinuing to larger values, we look first at primes, since for $m$ prime the three functions of interest\n$\\Phi_{1,m}$, $\\Phi_{1,m}^0$ and ${\\cal Q }_m$ all coincide.  For {\\bf m=7} we find that ${\\cal Q }_7=\\Phi^{\\rm opt}_{1,7}$ again has\noptimal growth and again satisfies~\\eqref{coincidence} with $\\v^{\\text{opt}}_{0,6}$ as given in Table~\\ref{Cmcoeffs}.  \nHowever, for {\\bf m=11} the function ${\\cal Q }_M$ {\\it cannot} be chosen to be of optimal growth; the best possible\nchoice, shown in the table~\\ref{apptabs3} in the appendix, has minimal discriminant~$-4$. For {\\bf m=13}\nthe function ${\\cal Q }_{13}$ again has optimal growth and is again given by~\\eqref{coincidence},\nbut for larger primes this never seems to happen again, the minimal value of~$\\Delta$ for the optimal \nchoices of~${\\cal Q }_M$ for the first few prime indices being given by\n$$ \n\\begin{tabular}{c|ccccccccccccccc}     $M$ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 \\\\ \n \\hline  $\\Delta_{\\rm min}$ & 1 & 1 & 1 & 1 & 4 & 1 & 4 & 4 & 8 & 5 & 8 & 4  \\\\    \\end{tabular}  \\;\\,. \n$$\n\nThe next simplest case is $m=p^2$, since here $\\Phi_{1,m}$ is still primitive.  For {\\bf m=9} \nand {\\bf m=25} this function can be chosen to have optimal growth,\nbut apparently not after that. The following case to look at is $m=p^\\nu$ with $\\nu\\ge2$, since here the \neigendecomposition~\\eqref{Qdecomp} of $\\Phi_{1,m}$ is trivial.  (The eigenvalue of $W_m$ must be~$-1$ \nbecause the weight is odd, so $\\Phi_{1,m}=\\Phi_{1,m}^-={\\cal Q }_p|{\\cal V }_{p^{\\nu-1}}^{(p)}$.)  Here we find optimal\ngrowth only for {\\bf m=8} and {\\bf m=16}.  Next, for $M$ square-free but not prime, where the decomposition~\\eqref{QmOG}\nis trivial, the only two cases giving optimal growth seem to be~{\\bf m=6} and {\\bf m=10}. And \nfinally, for $M$ neither square-free nor a power of a prime we also find two examples {\\bf m=12} and {\\bf m=18}. \nAltogether this leads to \nthe list~\\eqref{mList} given above.  The first few coefficients of the forms $\\Phi_{1,m}^0$ for all of these values\nof~$m$ are tabulated in Appendix~\\ref{apptabs2}.\n \nHowever, this is not the end of the story.  If $\\Phi_{1,m}$ has optimal growth for some~$m$, then of course\nall of its eigencomponents with respect to the Atkin-Lehner operators also do, but the converse is not true,\nso there still can be further square-free values of~$M$ for which the function \n${\\cal Q }_M=2^{1-\\omega}  \\def\\omega{\\omega(M)}\\,\\Phi_{1,M}^{-,\\dots,-}$ has optimal growth, and this indeed happens.  Since primes\n have already been treated and products of five or more primes are beyond the range of our ability to\ncompute, we should consider products of three primes. Here we find four examples, for the indices\n   $$  30\\=2\\cdot3\\cdot5\\,, \\qquad 42\\=2\\cdot3\\cdot7\\,, \\qquad \n     70\\=2\\cdot5\\cdot7\\,,\\qquad 78\\=2\\cdot3\\cdot13\\,,   $$ \nbut for no other values less than~200 (and probably for no other values at all), giving\nthe list~\\eqref{QMList}.  Again the\nFourier coefficients of the corresponding forms, as well as those for all other $M<150$ of the\nform $p_1p_2p_3$, have been listed in the appendix (part~\\ref{apptabs3}).  These examples are of \nparticular interest, for a reason to which we now turn.\n\n\n\n\\bigskip\\noindent{\\it 2. Relationship to classical  mock theta functions} \\smallskip\n\nRecall from \\S\\ref{basicdefs} that a mock modular form is called a mock theta function if its\nshadow is a unary theta series, and that this definition really does include all of Ramanujan's\noriginal mock theta functions that began the whole story.  The theta expansion coefficients $h_\\ell(\\tau)$\nof our special mock Jacobi forms $\\Phi_{1,m}$ and $\\Phi_{2,m}$ are always mock theta functions,\nsince their shadows are multiples of the unary theta series $\\varphi_{m,\\ell}^1$ and $\\varphi_{m,\\ell}^0$,\nrespectively.  In fact, it turns out that many of our examples actually coincide with the mock theta\nfunctions studied by Ramanujan and his successors.  This is the ``particular interest\" just mentioned.\n\n\nLet us consider first the case {\\bf m=30}, the first of our ${\\cal Q }_M$ examples with composite~$M$.\nBecause it is so highly composite, there are very few orbits of the group of Atkin-Lehner involutions\non the group of residue classes modulo~$2m$ and hence very few components in the theta decomposition\nof~${\\cal Q }_{30}$. In fact there are only two, so that up to sign we get only two distinct coefficients\n$h_1$ and $h_7$ and ${\\cal Q }_{30}$ has a decomposition\n\\begin{eqnarray*} -3\\,{\\cal Q }_{30}  \\= &h_1\\,\\bigl(\\vartheta_{30,1}+\\vartheta_{30,11}+\\vartheta_{30,19}+\\vartheta_{30,29}\n   -\\vartheta_{30,31}-\\vartheta_{30,41}-\\vartheta_{30,49}-\\vartheta_{30,59}\\bigr) \\cr\n + & \\,h_7\\,\\bigl(\\vartheta_{30,7}+\\vartheta_{30,13}+\\vartheta_{30,17}+\\vartheta_{30,23}\n -\\vartheta_{30,37}-\\vartheta_{30,43}-\\vartheta_{30,47}-\\vartheta_{30,53}\\bigr)\\,.\n\\end{eqnarray*}\n(The normalizing factor $-3$ is included for convenience: the function $-3{\\cal Q }_{30}$ is the function ${\\cal F}_{30}$\ndefined in~\\ref{DefOfcM} below.) The mock theta functions $h_1$ and $h_7$ \nhave Fourier expansions beginning\n\\begin{eqnarray*}  h_1(\\tau) &\\= q^{-1\/120}\\,(-1\\+q\\+q^2\\+2q^3\\+q^4\\+3q^5\\+\\cdots)\\,\n, \\cr\n   h_7(\\tau) &\\= q^{71\/120}\\,(1\\+2q\\+2q^2\\+3q^3\\+3q^4\\+4q^5\\+\\cdots)\\,. \n\\end{eqnarray*}\nThese expansions are easily recognized: up to trivial modifications they are two of the ``mock\ntheta functions of order~5\" in the original letter of Ramanujan to Hardy, namely\n\\begin{align}  \n  2\\,+\\,q^{1\/120}\\,h_1(\\tau) &\\= \\chi_1(q) \\= \\sum_{n=0}^\\infty\\frac{q^n}{(1-q^{n+1})\\cdots(1-q^{2n})}\\;, \\notag \\\\\n q^{-71\/120}\\,h_2(\\tau) &\\= \\chi_2(q) \\= \\sum_{n=0}^\\infty\\frac{q^n}{(1-q^{n+1})\\cdots(1-q^{2n+1})}\\;.\\notag\n\\end{align}\n(Actually, Ramanujan gave five pairs of mock theta functions of ``order~5\" in his letter, any two differing \nfrom one another by weakly holomorphic modular forms, but it was pointed out in~\\cite{Zagier:2007} that of the \nfive associated pairs of mock modular forms only $(q^{-1\/120}(2-\\chi_1(q)),\\,q^{71\/120}\\chi_2(q))$ has the \nproperty that its completion transforms as a vector-valued modular form for the full modular group.)  For {\\bf m=42},\nexactly the same thing happens, and the three distinct components $h_1$, $h_5$ and $h_{11}$ of ${\\cal Q }_{42}$,\nwhose initial coefficients are tabulated in Appendix~\\ref{apptabs3}, turn out to be, up to powers of~$q$ and signs, \nidentical with Ramanujan's three ``order~7\" mock theta functions ${\\cal F}_{7,i}$ ($i=1,\\,2,\\,3$) that were given as our\nfirst examples of mock modular forms in~\\S\\ref{basicdefs}.  What's more, this behavior continues: in~\\cite{Zagier:2007}\nit was mentioned that one could construct mock theta functions of arbitrary prime\\footnote{actually, only\n$(p,6)=1$ is needed} ``order\"~$p$ generalizing these two examples as quotients by $\\eta(\\tau)^3$ of certain\nweight~2 indefinite theta series, and an explicit example \n$$  M_{11,j}(\\tau) \\=\\frac1{\\eta(\\tau)^3}\\,\\sum_{m>2|n|\/11 \\atop n\\equiv j\\; \\mod \\; {11}} \n \\Bigl(\\frac{-4}m\\Bigr)\\,\\Bigl(\\frac{12}n\\Bigr)\\,\\bigl(m\\,\\text{sgn}(n)\\,-\\,\\frac n6\\bigr)\\,q^{m^2\/8-n^2\/264}\n  \\qquad(j\\in{\\mathbb Z}\/11{\\mathbb Z})$$\nwas given for~$p=11$.  Computing the beginnings of the Fourier expansions of these five mock theta functions\n(only five rather than eleven because $M_{11,j}=-M_{11,-j}$) and comparing with the table in Appendix~\\ref{apptabs3}, \nwe find perfect agreement with the coefficients in the theta expansion of~${\\cal Q }_{66}$, the first \nexample in that table of a form that is not of optimal growth.\n\nThis discussion thus clarifies the precise mathematical meaning of the number designated by Ramanujan as the \n``order\" of his mock theta functions, but which he never defined: it is indeed related to the {\\it level} of \nthe completions of the corresponding mock modular forms, as was clear from the results of Zwegers and other authors \nwho have studied these functions, but it is even more closely related to the {\\it index} of the mock \nJacobi forms that have these mock theta functions as the coefficients of its theta expansion.  \n\nOf course the above cases are not the only ones where there is a relationship between the special\nmock Jacobi forms~${\\cal Q }_M$ and classically treated mock theta functions, but merely the ones where the\nrelationship between the ``order\" and the index is clearest.  Without giving any further details, we\nsay only that Ramanujan's order~3 functions are related in a similar way\\footnote{but not ``on the nose,'' for instance, our function $h_{3,1}$ differs from Ramanujan's 3rd order mock theta function $-3\\, q^{1\/12} f(q^2)$ by an eta product.} to our~${\\cal Q }_3$ and that his\norder~10 functions (not given in his letter to Hardy, but contained in the so-called ``Lost Notebook\")\nare related to~${\\cal Q }_5$.  We also refer once again to the beautiful recent work of Cheng, Duncan and \nHarvey~\\cite{Cheng:2012tq} on ``umbral moonshine,\" in which many of these same functions are discussed\nin detail in connection with representations of special finite groups.\n\nFinally, we should mention that all of Ramanujan's functions were $q$-hypergeometric series but that \nin general no $q$-hypergeometric expression for the theta expansion coefficients of the functions\n$\\Phi_{k,m}$ or ${\\cal Q }_M$ is known.  This seems to be an interesting subject for further research.\nOne could look in particular for representations of this sort for the functions ${\\cal Q }_{13}$ and\n$\\Phi_{1,25}^0$ as given in Table~\\ref{apptabs2}, since these have particularly small and smoothly\ngrowing coefficients that suggest that such a representation, if it exists, might not be too complicated.\nIt would also be reasonable to look at ${\\cal Q }_{70}$ and ${\\cal Q }_{78}$, the only two other known weight~1\nexamples with optimal growth.\n\n\n\n\\bigskip\\noindent{\\it 3. Integrality of the Fourier coefficients} \\smallskip\n\n\nIn Subsections~\\ref{choosephi} and~\\ref{Statements} and above, we encountered several\nexamples of functions ${\\cal Q }_{M}$ of optimal growth that after multiplication by a suitable factor gave \na function ${\\cal F}_M$ having all or almost all of the following  nice arithmetic properties:\n\\begin{enumerate}[itemsep=0pt,topsep=3pt]\n  \\item [(A)] All Fourier coefficients $C({\\cal F}_M;\\Delta,r)$ are integral.\n  \\item [(B)] The polar Fourier coefficient $C({\\cal F}_M;-1,1)$ equals $-1$.\n  \\item [(C)] The non-polar Fourier coefficients $C({\\cal F}_M;\\Delta,r)$ have a sign depending only on~$r$.\n  \\item [(D)] $C({\\cal F}_M;\\Delta,r)$ is always positive for $r=r_{\\rm min}$, the smallest \n     positive number whose square equals $-\\Delta\\; \\mod \\; {4m}$. \\end{enumerate}\nIn particular, for $k=2$ this occurred for $M=6$, 10, 14 and 15 (compare eqs.~\\eqref{defC6}, \\eqref{c10}, \n\\eqref{c14} and~\\eqref{c15}, which describe the coefficients $C^{(M)}(\\Delta)=C({\\cal F}_M;\\Delta,r_{\\rm min})$\nfor these indices) and for $k=1$, for the values $M=2$, 3, 5, 7, 13 and 30 (with ``non-negative\" \ninstead of ``positive\" in~(C) in the case of~$M=13$), as well as for the functions $\\Phi_{1,m}^0$ for \nthe prime powers $M=4$, 8, 9, 16 and 25 (again with ``positive\" replaced by ``non-positive,\" and \nalso allowing half-integral values for the Fourier coefficients with~$\\Delta=0$).\nIt is natural to ask for what values of~$M$ these properties can be expected to hold. We will discuss\nproperties~(A) and~(B) here and properties~(C) and~(D) in point {\\it 4.} below.\n\n\n\nClearly either (A) or (B) can always be achieved by multiplying ${\\cal Q }_M$ by a suitable scalar factor.\nWe define a renormalized function~${\\cal F}_M$ in all cases by setting\n\\beta\\label{DefOfcM}  {\\cal F}_M(\\tau,z) \\= -c_M\\,{\\cal Q }_M(\\tau,z)\\,,  \\end{equation}\nwhere $c_M$ is the unique positive rational number such that ${\\cal F}_M$ has integer coefficients with no common\nfactor.  Then~(A) is always true and (B) and (C) will hold in the optimal cases.\nThis normalization is also convenient for the presentation of the coefficients in tables, since there are\nno denominators and very few minus signs.  Tables of the first few Fourier coefficients of all ${\\cal F}_M$ up to~$M=50$\nin the weight~2 case (i.e., for $M$ with $\\mu(M)=1$) and of all known~${\\cal F}_M$ having optimal growth in the weight~1 \ncase (i.e., for $M$ with $\\mu(M)=-1$) are given in~\\ref{apptabs1} and~\\ref{apptabs3} of the Appendix, respectively.\nLooking at these tables, we see\nthat the values of $M$ for which all three properties above hold are $M=6$, 10, 14, 15, 21 and 26 in the case $k=2$ \nand $M=2$, 3, 5, 7, 13, 30 and 42 in the case $k=2$, and that in the $k=2$ case we also get ${\\cal F}_M$ satisfying~(C)\nbut not~(B) for $M=22$ and ${\\cal F}_M$ satisfying~(B) but not~(C) for $M=35$ and 39.  This list is\nnot changed if we look at a larger range of~$M$ (we have calculated up to $M=210$ in the weight~2 case and $M=100$\nin the weight~1 case) or ~$\\Delta$ (in each case we have calculated about twice as many Fourier coefficients\nas those shown in the tables), so that it seems quite likely that it is complete.\n\nFor the weight~2 functions, the corollary of Theorem~\\ref{Phi0IIIcoeffs} tells us that $C({\\cal F}_M;-1,1)$ will\n always be negative with the normalization~\\eqref{DefOfcM} and that we have \n\\beta\\label{tConj}  c_M \\= t \\cdot {\\text{denom}\\big(\\frac{\\varphi(M)}{24}\\big)}\\,, \\qquad C({\\cal F}_M;-1,1) \n\\= -t\\cdot{\\text{numer}\\big(\\frac{\\varphi(M)}{24}\\big)}\\end{equation}\nfor some positive rational number~$t$. In all cases we have looked at, $t$ turned out to be an integer, \nand in fact usually had the value~1. For instance, the second statement is true for all values of~$M$ tabulated in \nAppendix~\\ref{apptabs}, and also for~$M=210$, where Table~\\ref{fcq210} in~\\S\\ref{Statements} shows that \n$c_{M}=1$ and hence~$t=1$, but the table below giving all values up to~$M=100$ shows that it fails for~$M=65$ and $M=85$.\n\n \\small  \n\\begin{center}\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}  \\hline\n       $M$ & 6 &10& 14& 15& 21 & 22 & 26 & 33 & 34 & 35 & 38 & 39 &46  \\\\  \\hline\n       $c_{M}$ &12 & 6 & 4 & 3 & 2 & 12 & 2 & 6 & 3 & 1 & 4& 1& 12 \\\\\n       $C({\\cal F}_{M};-1,1)$ &1&1&1&1&1&5&1&5&2&1&3&1&11 \\\\ \n       $t$&1&1&1&1&1&1&1&1&1&1&1&1&1 \\\\\n    \\hline \\end{tabular}\\quad \n\n\\vspace{0.4cm}\n\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}  \\hline\n  51 & 55 & 57 & 58 & 62 & 65 & 69 & 74 & 77 & 82 & 85 & 86 & 87 & 91 & 93 & 94 & 95 \\\\  \\hline\n    3 & 3 & 2 & 6 & 4 & 2 & 6 & 2 & 2 & 3 & 6 & 4 & 3 & 1 & 2 & 12 & 1 \\\\\n   4&5&3&7&5&4&11&3&5&5&16&7&7 &3 & 5 & 23 & 3\\\\ \n   1&1&1&1&1&2&1&1&1&1&2&1&1&1&1&1&1 \\\\\n    \\hline \\end{tabular}\\quad \n    \\end{center} \n   \n\\normalsize\n\nIf it is true that $t$ is integral, then equation~\\eqref{tConj} shows that the above list of~${\\cal F}_M$\nsatisfying property~(B) is complete, because we can then only have $|C({\\cal F}_M;-1,1)|=1$ if $\\varphi(M)$ divides~$24$,\nwhich in turn implies that~$M$ is at most~90 and hence within the range of our calculations. We do not know\nwhether the integrality is true in general, but in any case~$12^{M+1}t$ must be an integer, since \ncorrecting the initial choice~$\\Phi_{2,M}^{\\rm stand} = 12^{-M-1} B^{M+1}A^{-1} \\, - \\, \\mathfrak{F}_P{2,M}$ \nby weak Jacobi forms cannot make its denominator any worse.\n\nExactly similar comments apply in the weight~1 case.  Here a residue calculation similar to the ones\ndescribed in the next section (but with the residue~${\\cal R}$ as defined in~\\eqref{RasRes} below replaced by \nits weight~1 analogue as defined in~\\eqref{RasReswt1}) tells us that in the case of optimal growth\neq.~\\eqref{Qmpolar} still holds, so that we find that eq.~\\eqref{tConj} still holds for some postive rational \nnumber~$t$.  If this~$t$ turns out to be an integer, it again follows that property~(B) can only hold if\n$\\varphi(M)|24$, and the list of all integers~$M$ with $\\mu(M)=-1$ satisfying this latter condition coincides exactly with the \nlist~\\eqref{QMList}.  Thus, although we cannot yet prove that that list is complete, we can at least understand\nwhere it comes from.\n\n\n\\bigskip\\noindent{\\it 4.~Positivity of the Fourier coefficients} \\smallskip\n\nWe now turn to properties~(C) and (D). Here we can use the asymptotic formula given in Theorem~\\ref{CardyProp}\nto understand the question of the signs and asymptotics of the Fourier coefficients. \n(We will discuss only the weight~2 case in detail, but a similar theorem and similar discussion could \nequally well be given for the case~$k=1$.)  This theorem implies that~(C) ``almost holds\" in the sense\nthat for any fixed value of~$r$ the Fourier coefficients $C(\\Delta,r)$ eventually have a constant sign, so\nthat there are at most finitely many exceptions to the statement.  We therefore have to consider~(D)\nand also the question of which values of~$r$ are most likely to give rise to the exceptional signs.\n\nFor the function~${\\cal Q }_{M}$ (which we remind the reader differs from~${\\cal F}_{M}$ by a negative proportionality factor,\nso that the expected behavior~(D) would now say that~$C(\\Delta,r)$ is always negative when \n$r=r_{\\rm min}$ is the smallest positive square root of~$-\\Delta$), the value of the constant~$\\k_{r}$ \nappearing in that theorem as given according to~\\eqref{kappa} and~\\eqref{Qmpolar} by\n\\beta \\label{didnt}\n\\k({\\cal Q }_{M},r) \\= - \\frac{\\varphi(M)}{24} \\, \\cdot \\, 2 \\! \\sum_{0<\\ell < M \\atop \\ell^{2}\\,\\equiv1\\; \\mod \\; {4M}}\n\\mu\\Big( \\big(\\frac{\\ell+1}{2}, M \\big) \\Big) \\cos\\Bigl(\\frac{\\pi\\ell r}M\\Bigr) \\, .\n\\end{equation}\nA similar formula applies in the weight~1 case with ``cos\" resplaced by ``sin.\"  In the case when~$M$\nis a prime, this immediately gives the positivity of $\\k({\\cal Q }_M,r_{\\rm min})$, and in the case\nwhen~$M$ has three prime factors, the tables in~\\ref{apptabs3} show that~(D) usually fails.  We therefore \nconsider the weight~2 case, where $\\mu(M)=+1$.\n\nIf~$M$ (which we always assume to be larger than~1) is a product \nof two primes, and in particular for all~$M<330$ with~$\\mu(M)=1$ except~210, the sum on\nthe right-hand side has only two terms and reduces to simply~$\\cos(\\pi r\/M)-\\cos(\\pi r^{*}\/M)$, \nand since~$0<r_{\\rm min}<r_{\\rm min}^{*}<2M$ and~$\\cos(\\pi x)$ is decreasing on the \ninterval~$[0,\\pi]$, the value of~$-\\k({\\cal Q }_{M}, r_{\\rm min})$ is indeed always strictly negative.  \nIt then follows from Theorem~\\ref{CardyProp} that almost all~$C({\\cal F}_{M};\\Delta, r_{\\rm min})$ are positive,\nbut there can be, and in general are, finitely many exceptions. However, for small~$M$ these \nare fairly rare, e.g., the following table lists all triples~$(M,\\,r\\!=\\! r_{\\rm min},\\, n)$ within the \nrange of Appendix~\\ref{apptabs} for which~$c({\\cal F}_{M};n,r)$ is negative.\n\n\\begin{table}[h]\n\\begin{center}\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|cc|c|c|c|c|}  \\hline\n       $M$ & 6--22 &$33$& $34$& $35$& $38$ & $39$ & $46$ \\\\  \\hline\n       $r$ &--- & $8$ & $15$ & $2$ & $17$ & $10$ & $21$ \\\\\n       $n$ &--- & $1,\\,5,\\,7$ & $5,\\,11$ & $1,\\,5$ & $5,\\,7,\\,11$ & $5$ & $5,\\,7,\\,11,\\,13$  \\\\ \n    \\hline \\end{tabular}\\quad \n    \\end{center} \n   \n\\end{table}\n\\noindent (The corresponding $r$-values are marked with a star in the appendix, and a search \nup to a much larger~$n$-limit yields no further negative coefficients for these values of~$M$.)\nHere again the asymptotic formula~\\eqref{HRR} helps us to understand this behavior:\nthe negative values are much more likely to occur when~$\\k_{r}$ is small. For instance, \nfor~$M=33$ the table\n\n\\begin{table}[h]\n \\begin{center} \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}  \\hline\n       $r\\;\\,(\\,=\\,r_{\\rm min}\\,)$ & 1& 2& 4& 5 & 7 & 8 &10&13&16&19 \\\\  \\hline\n       $\\cos(\\frac{\\pi r}{66})\\vphantom{\\Bigl(}-\\cos(\\frac{\\pi r^{*}}{66})$ \n       & $1.58$ & $1.31$ & $1.71$ & $0.94$ & $1.71$ & $0.49$ & $1.58$ & $1.31$ & $0.94$ & $0.49$ \\\\\n    \\hline \\end{tabular}\\quad \\end{center}\n\\end{table}\n\n\\noindent shows that the residue classes~$r_{\\rm min} \\; \\mod \\; {2M}$ most likely to give some negative \nFourier coefficients are~$8$ and~19, for which the values of~$\\k_{r}$ (which are the same; this \nduplication always occurs for odd~$M$) are minimal, and indeed the class~$r=8$ did lead to three  \nnegative coefficients.  The class~$r=19$ did not contain any negative values of~$c(n,r)$, but instead \nachieved its small initial average by the initial zeros forced by the optimal growth \ncondition, which says that~$c(n,r)=0$ for~$0 \\le n\\le (r^2-2)\/4m$. \n\nFor values of~$M$ for more than two prime factors on the other hand, it does not have to be true \nthat~$\\k({\\cal F}_{M}, r_{\\rm min})>0$. It is true for the first such index, $M=210$, and even in a very strong form:\nnot only is~$\\k_{r}$ positive for each of the six possible values of~$r_{\\rm min}$ (which are nothing other\nthan the values~$\\ell_{0}$ printed in boldface in Table~\\ref{fcq210}), but of the seven ``partners'' of \neach~$r_{\\rm min}$, i.e., the seven other values of~$r\\in[0,210]$ with~$r^{2}\\equiv r_{\\rm min}^{2} \\; \\mod \\; {840}$, the three that have the same sign of~$C(\\Delta,r)$ as for~$C(\\Delta,r_{\\rm min})$ (i.e., which differ \nfrom~$r_{\\rm min} \\; \\mod \\; {2p}$ for exactly two of the four prime divisors~$p$ of~210) all belong to the \nfirst half of the interval~$[0,210]$, while the four that have the opposite sign (i.e., which differ \nfrom~$r_{\\rm min} \\; \\mod \\; {2p}$ for one or three of these~$p$) belong to the second half, \nso that all eight terms in the sum in~\\eqref{didnt} are positive. The value of~$\\k({\\cal F}_{M},r_{\\rm min})$ \nis again positive for all~$r_{\\rm min}$ for the next two values~$M=330$ and~$M=390$, but for \n$M=462=2\\cdot 3\\cdot 7\\cdot 11$ and $r\\; (=r_{\\rm min})\\, =\\, 31$ we have~$\\k({\\cal F}_{M},r)<0$,\nso that in this case all but finitely many coefficients $C({\\cal F}_{M};\\Delta,r)$ will be negative rather than positive. \nWhat does hold for any~$M$ and any~$r \\; \\mod \\; {2M}$ is that the values of~$C({\\cal F}_{M}; \\Delta,r)$ \neventually all have the same sign, because the coefficient~$\\k({\\cal F}_{M},r)$ in~\\eqref{HRR} can never \nvanish, since the primitive~$(2M)^{\\rm th}$ roots of unity are linearly independent over~${\\mathbb Q}$. \n  \n\n\\subsection{Higher weights} \\label{highweight}\n\nSo far we have studied two families of meromorphic Jacobi forms $\\varphi_{1,m}(\\tau,z)$ and $\\varphi_{2,m}(\\tau,z)$ with \nparticularly simple singularities, and the corresponding mock Jacobi forms.  We now discuss in much less detail the analogous families for \nhigher weights.  \nWe mention that other cases of the decomposition $\\varphi = \\varphi^{F} + \\varphi^{P}$ for meromorphic Jacobi forms with \nhigher order poles\\footnote{These were considered in~\\cite{BringFolsom} while answering a question \ndue to Kac concerning the modularity of certain characters of affine Lie superalgebras.} \n\\cite{BringFolsom, Olivetto}, and the asymptotics of their Fourier \ncoefficients~\\cite{BringmannFolsom2}, have been studied in the literature. \n\n\n\n\nThe starting point is a meromorphic Jacobi form $\\varphi(\\tau,z)=\\varphi_{k,m}(\\tau,z)$ of weight $k$ and arbitrary index $m\\in{\\mathbb N}$ having as \nits unique singularity (modulo the period lattice ${\\mathbb Z}\\tau+{\\mathbb Z}$) a pole of order $k$ at $z=0$, which we normalized by requiring the leading term of $\\varphi$ \nto be  $(k-1)!\/(2\\pi iz)^k$. By subtracting a linear combination of functions $\\varphi_{k',m}$ with $k'<k$ we can then fix the entire principal part of $\\varphi$ at $z=0$ to be\n $$  \\varphi(\\tau,z) \\= \\frac{(k-1)!}{(2\\pi iz)^k}\\,\\sum_{0\\le2j<k}\\Bigl(\\frac {m\\,E_2(\\tau)}{12}\\Bigr)^j \\,\\frac{(2\\pi iz)^{2j}}{j!} \\;+\\;\\text O(1)\\qquad\\text{as $z\\to 0$.} $$\nWe can make a standard choice, generalizing the previously defined forms \\eqref{defphi1m} and \\eqref{defphi2m}, by taking $\\varphi$ to be the unique polynomial \nin $A^{-1}$ and $B$ (resp.~$C$ times such a polynomial if $k$ is odd) with modular coefficients of the appropriate weight having this principal\npart, the next three of these being\n  $$  \\v^{\\text{stand\\vphantom{p}}}_{3,m} \\= \\frac{B^m\\,C}{12^mA^2}\\,, \\;\\quad  \\v^{\\text{stand\\vphantom{p}}}_{4,m} \\= \\frac{6\\,B^{m+2}}{12^{m+2}A^2}\\,,\\;\\quad\n   \\v^{\\text{stand\\vphantom{p}}}_{5,m} \\= \\frac{(B^2-mE_4A^2\/2)\\,B^{m-1}\\,C}{12^m\\,A^3}\\,.   $$\nAny other allowed choice of $\\varphi_{k,m}$ will differ from this standard one by a weak Jacobi form of weight~$k$ and index~$m$, and any two choices\nwill have the same polar part.  To write it down, we introduce Euler's functions\n$$ \\mathcal E_k(y) \\= \\text{Av}_\\IZ\\Bigl[\\frac{(k-1)!}{(2\\pi iz)^k}\\Bigr] \\= \\Bigl(-y\\,\\frac d{dy}\\Bigr)^{k-1}\\text{Av}_\\IZ\\Bigl[\\frac1{2\\pi iz}\\Bigr]\n   \\= \\bigl(-y\\,\\frac d{dy}\\bigr)^{k-1}\\biggl(\\frac12\\,\\frac{y+1}{y-1}\\biggr)\\;\\in\\;{\\mathbb Q}\\Bigl[\\frac1{y-1}\\Bigr] $$\n(which for $k=1$ and $k=2$ coincide with the functions $\\mathcal R_0$ and $\\mathcal R^{(2)}_0$ from \\S\\ref{MockfromJacobi}) and their index $m$ averages \n $$   \\mathfrak{F}_P{k,m}(\\tau,z) \\=  \\text{Av}^{(m)}_{\\IZ\\tau+\\IZ}\\Bigl[\\frac{(k-1)!}{(2\\pi iz)^k}\\Bigr] \\= \\Av m{\\mathcal E_k(y)} \\=\\sum_{s\\in{\\mathbb Z}} q^{ms^2}y^{2ms}\\mathcal E_k(q^s y)  $$\ngeneralizing \\eqref{AP1AP2}.\nThese functions all have wall-crossing, with Fourier expansions given by\n$$ \\mathcal E_k(y) \\= \\begin{cases}\\phantom-\\frac12\\,\\delta_{k,1}\\+\\sum_{t>0}t^{k-1}y^{-t} &\\text{if $|y|>1$} \\\\\n     -\\frac12\\,\\delta_{k,1}\\,-\\,\\sum_{t<0}t^{k-1}y^{-t} &\\text{if $|y|<1$} \\end{cases}\\;\\,=\\;\\,\\sum_{t\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(t)-\\mbox{sgn}(z_2)}2\\,t^{k-1}y^{-t}$$\nand\n\\beta \\label{APkm} \\mathfrak{F}_P{k,m}(\\tau,z) \\= \\sum_{s,\\,t\\in{\\mathbb Z}}\\frac{\\mbox{sgn}(t)\\,-\\,\\mbox{sgn}(s+z_2\/\\tau_2)}2\\,t^{k-1} q^{ms^2-st} y^{2ms-t}\\;.  \\end{equation}\n(Note that in the latter expression the coefficient of $y^r$ is a Laurent polynomial in~$q$ for every~$r$ and that the discriminant $4nm-r^2$ \nis the negative of a perfect square for each monomial $q^ny^r$ occurring.)  Then the polar part of $\\varphi_{k,m}$ is given by\n$$  \\varphi_{k,m}^P(\\tau,z) \\= \\sum_{0\\le2j<k}\\frac{(k-1)!}{(k-1-2j)!\\,j!}\\,\\Bigl(\\frac {m\\,E_2(\\tau)}{12}\\Bigr)^j \\,\\mathfrak{F}_P{k-2j,m}(\\tau,z)\\;. $$\n(To see this, observe that by~\\eqref{APkm} each of the functions $\\mathfrak{F}_P{k,m}$, and hence any linear combination of them with coefficients depending\nonly on~$\\tau$, is in the kernel of the operator $\\varphi\\mapsto h_\\ell$ defined by~\\eqref{Defofhl} for all~$\\ell\\in{\\mathbb Z}$.)\nIts finite part $\\Phi_{k,m}=\\varphi_{k,m}^F=\\varphi_{k,m}-\\varphi_{k,m}^P$ is a weakly holomoprhic elliptic form that depends on the particular choice of $\\varphi_{k,m}$.\nFrom the results of \\S\\ref{structure} (specifically, Theorem~\\ref{bigweight}) it follows that for $k\\ge3$ the function $\\varphi_{k,m}$ can always be chosen \nso that $\\Phi_{k,m}$ is strongly holomorphic, this choice then obviously being unique up to the addition of a strongly holomorphic Jacobi form.\n\nAs an example, consider $k=3$ and $m=2$. (This is the smallest possibility since index~1 cannot occur in odd weight.)  The finite part of our standard choice is\n$$    \\Phi^{\\rm stand}_{3,2} \\= \\frac{B^2C}{144A^2} \\,-\\,\\mathfrak{F}_P{3,2}\\,-\\,\\frac{E_2}3\\,\\mathfrak{F}_P{3,1} \\= h_{3,2}^{\\rm stand}\\,\\bigl(\\vartheta_{2,1}\\,-\\,\\vartheta_{2,3}\\bigr) $$\nwith\n$$  h_{3,2}^{\\rm stand} \\,=\\,\\frac1{144}\\, q^{-1\/8}\\,\\bigl(1+51q+2121q^2+14422q^3+60339q^4+201996q^5+588989q^6+\\cdots\\bigr)\\,.$$\nThis is weakly holomorphic and has large Fourier coefficients, but by subtracting a multiple of the unique weak Jacobi form $E_4C$ of\nthis weight and index we get a new choice\n$$  \\Phi^{\\rm opt}_{3,2} \\= \\Phi^{\\rm stand}_{3,2} \\,-\\,\\frac{E_4}{144}\\,C\\= h_{3,2}^{\\rm opt}\\,\\bigl(\\vartheta_{2,1}\\,-\\,\\vartheta_{2,3}\\bigr) $$\nthat is strongly holomorphic and has much smaller coefficients\n$$  h_{3,2}^{\\rm opt} \\=  h_{3,2}^{\\rm stand} \\,-\\,\\frac{E_4}{144\\,\\eta^3} \\= -\\frac43\\,q^{-1\/8}\\,\\bigl(q+4q^2+5q^3+11q^4+6q^5+21q^6+\\cdots\\bigr)\\,,$$\nin accordance with the theorem just cited.  \n\nHowever, this choice, although optimal from the point of view of the order of growth of its Fourier coefficients, is not the\nmost enlightening one.  By subtracting a different multiple of $E_4C$ from $\\Phi^{\\rm stand}_{3,2}$, we get a third function\n$$  \\Phi_{3,2} \\= \\Phi^{\\rm stand}_{3,2} \\+\\frac{7E_4}{144}\\,C \\= \\Phi^{\\rm opt}_{3,2} \\+ \\frac{E_4}{18}\\,C\\= h_{3,2}\\,\\bigl(\\vartheta_{2,1}\\,-\\,\\vartheta_{2,3}\\bigr) $$\nwith\n$$  h_{3,2} \\=  h_{3,2}^{\\rm stand} \\+\\frac{7E_4}{144\\,\\eta^3}  \\=  h_{3,2}^{\\rm opt} \\+\\frac{E_4}{18\\,\\eta^3} \n   \\= \\frac1{18}\\,q^{-1\/8}\\,(1+219q+2793q^2+15262q^3 +\\cdots)\\,.$$\nThis function can be identified: it is given by\n$$   h_{3,2} \\= \\frac13\\,\\dRS{1\/2}(h_2)\\,, $$\nwhere $h_2(\\tau)=q^{-1\/8}(-1+45q+231q^2+\\cdots)$ is the mock theta function defined in~\\eqref{defh2} (the one occurring in the Mathieu moonshine story) and\n$\\dRS{1\/2}$ is the Ramanujan-Serre derivative as defined in~\\eqref{RSderiv}. We can also write this at the level of mock Jacobi forms as\n$$   \\Phi_{3,2} \\=  \\frac1{24}\\,{\\cal L}_{1,2}\\bigl({\\cal F}_2\\bigr) \\,, $$\nwhere ${\\cal F}_2$ is the mock Jacobi form defined in~\\eqref{defCF2} and ${\\cal L}_{1,2}$ the modified heat operator defined in~\\eqref{LRS}.\n\nAn analogous statement holds in general, e.g., for weight 3 we have\n$$ \\Phi_{3,m} \\= -\\,{\\cal L}_{1,m}\\bigl(\\Phi_{1,m}\\bigr) \\quad\\pmod{\\widetilde J_{3,m}} $$\nfor any index $m$ and any choice of the functions $\\Phi_{3,m}$ and $\\Phi_{1,m}$ in their defining classes, and similarly for higher weight,\nthe specific formulas for $k\\in\\{3,4\\}$ and the standard choices being\n\\begin{eqnarray*} \\Phi^{\\rm stand}_{3,m}\\+{\\cal L}_{1,m}\\bigl(\\Phi^{\\rm stand}_{1,m}\\bigr)&\\=&-\\,\\frac{m-1}{12^m}\\,\\Bigl(4(m-2)E_6A\\+(2m+3)E_4B\\Bigr)\\,B^{m-3}C\\,, \\\\\n  \\Phi^{\\rm stand}_{4,m}\\+{\\cal L}_{2,m}\\bigl(\\Phi^{\\rm stand}_{2,m}\\bigr)&\\=&-\\,\\frac{m+1}{72\\cdot 12^m}\\,\\Bigl(4mE_6A\\+(2m-3)E_4B\\Bigr)\\,B^{m-1}\\,. \\end{eqnarray*}\nThus in one sense, the higher-weight families are better than the weight~1 and weight~2 families (because they can be made strongly holomorphic by \nsubtracting an appropriate weak Jacobi form), but in another, they are much less interesting (because they are always combinations of derivatives of \nlower-weight examples and ordinary Jacobi forms, so that they do not produce any essentially new functions).\n\n\n\n\\section{Structure theorems for Jacobi and mock Jacobi forms \\label{structure}} \nIn the last section we introduced two special families of meromorphic Jacobi forms and the\nassociated mock Jacobi forms and described various observations concerning them.  In this\nsection we try to give theoretical explanations for some of these. We begin by studying in\n\\S\\ref{relations} how the holomorphic Jacobi forms of weight~$k$ and index~$m$ lie in the\nspace of weak Jacobi forms of the same weight and index, or equivalently, what the relations\nare among the polar Fourier coefficients (i.e., the coefficients of $q^ny^r$ with $4nm < r^2$)\nof weak Jacobi forms.  The results described here are relatively easy consequence of results\nin the literature but are, as far as we know, new.  The remaining two subsections describe\nthe application of these results to mock Jacobi forms in general and to our two special \nfamilies in particular, providing fairly complete explanations of the experimentally\nfound properties of the weight two family and partial explanations in the weight one case.\n\n\n\\subsection{Relations among the polar coefficients of weak Jacobi forms \\label{relations}}\n\nThe definitions and main properties of holomorphic and weak Jacobi forms were\nreviewed in~\\S\\ref{Jacobi}, following the more detailed treatment in \\cite{Eichler:1985ja}.\nHere we investigate the relationship of these two spaces in more detail.\n\nBy definition, a weak Jacobi form has coefficients $c(n,r)$ which are non-zero only for $n\\ge0$,\nwhereas a holomorphic one has coefficients that vanish unless $4nm\\ge r^2$.  By virtue of\nthe periodicity property~\\eqref{cnrprop}, this stronger condition need only be checked for\n$r$ running over a set of representatives for ${\\mathbb Z}\/2m{\\mathbb Z}$, say $-m<r\\le r$, and since condition~\\eqref{modtransform}\nwith $\\sm abcd=\\sm{-1}00{-1}$ also implies that $c(n,-r)=(-1)^kc(n,r)$ and consequently $C(\\Delta,-r)=(-1)^kC(\\Delta,r)$,\nit in fact suffices to consider the coefficients with $0\\le r\\le m$ if $k$ is even and those with $0<r<m$ if $k$ is odd.\nIn other words, for all positive integers $k$ and $m$ we have an exact sequence\n  \\beta\\label{exseqJkm} 0 \\longrightarrow J_{k,m} \\longrightarrow \\widetilde J_{k,m}\n  \\stackrel{P}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{\\pm,m}} \\qquad(\\,(-1)^k=\\pm1\\,)\\,,  \\end{equation}\nwhere \n\\begin{eqnarray*}\n{\\cal N}_{+,m} & \\= & \\big\\{ (r,n) \\in {\\mathbb Z}^{2} \\, \\mid \\, 0 \\le r \\le m, \\; 0 \\le n < r^{2}\/4m  \\big\\} \\; , \\\\\n{\\cal N}_{-,m} & \\= & \\big\\{ (r,n) \\in {\\mathbb Z}^{2} \\, \\mid \\, 0 < r < m, \\; 0 \\le n < r^{2}\/4m  \\big\\} \\; , \n\\end{eqnarray*}\nand where the map $P$ (``polar coefficients'') associates to $\\varphi \\in \\widetilde J_{k,m}$ the collection of its\nFourier coefficients $c_{\\varphi}(n,r)$ for $(n,r) \\in {\\cal N}_{\\pm,m}$. In particular we have \n\\beta\\label{dimJkmineq}\n\\dim J_{k,m} \\;\\ge\\; j(k,m) \\; := \\; \\dim \\widetilde J_{k,m} \\, - \\, \\bigl|{\\cal N}_{\\pm,m}\\bigr| \\qquad (\\,(-1)^{k} \\= \\pm1\\,)\\,, \n\\end{equation}\nwith equality if and only if the map $P$ in \\eqref{exseqJkm} is surjective.  (Cf.~\\cite{Eichler:1985ja}, \np.~105.)\n\nUsing the description of $\\widetilde J_{*,*}$ given in \\S\\ref{Jacobi} (eqs.~\\eqref{jacovermod} and \\eqref{WJF}) and \nthe definition of ${\\cal N}_{\\pm,m}$, we can write this lower bound on the dimension explicitly as \n\\beta \\label{jkmformgen}\n  j(k,m)\\= \\begin{cases} \\sum_{0\\le j\\le m} \\big(\\dim M_{k+2j}-\\ceil{j^2\/4m}\\big) & \\text{if $k$ is even,} \\\\ \n  \\sum_{0 < j < m} \\big(\\dim M_{k+2j-1}-\\ceil{j^2\/4m}\\big) & \\text{if $k$ is odd,} \\end{cases} \n\\end{equation}\nwhere $\\ceil x$ denotes the ``ceiling\" of  a real number~$x$ (\\,= smallest integer $\\ge x$).\nThis formula is computable and elementary since $\\dim M_{k} = k\/12 + {\\cal O}(1)$ with the ${\\cal O}(1)$ term \ndepending only on $k\\; \\mod \\; {12}$, but is nevertheless not a good formula because the two \nterms $\\sum \\dim M_{k+2j}$ and $\\sum \\ceil{j^2\/4m}$ are each equal to $m^{2}\/12 + {\\cal O}(m)$, \nand $j(k,m)$ is only of the order of~$m$. In \\cite{Eichler:1985ja}, pp.~122--125 it was shown that, \nsomewhat surprisingly, the numbers $j(k,m)$ can be given by a completely different-looking formula involving \nclass numbers of imaginary quadratic fields.  This formula has the form \n\\beta \\label{jkmformula} j(k,m) \\= \\frac{2k-3}{24} \\, (m+1) \\+ \\frac{b}{4} \\, - \\, \\frac14 \\sum_{d|4m} h(-d) \\+ {\\cal O}(1) \\, ,\\end{equation}\nwhere $m=ab^{2}$ with $a$ square-free and where the ${\\cal O}(1)$ term depends only on $m$ and $k$ modulo~12.  \nIn particular, $j(k,m)=\\frac{2k-3}{24}\\,m \\+{\\cal O}(m^{1\/2+\\varepsilon})$ as $m\\to\\infty$ with $k$ fixed. \nThe formula \\eqref{jkmformula} has an\ninteresting interpretation as the dimension of a certain space of classical modular forms of level~$m$ and weight~$2k-2$,\ndiscussed in detail on pp.~125--131 of~\\cite{Eichler:1985ja}.\n\n\nIn \\cite{Eichler:1985ja}, it was shown that the inequality \\eqref{dimJkmineq} is an equality for $k\\ge m$  \nand it was conjectured that this holds for all $k \\ge 3$. This was later proved in \\cite{SkoruppaZagier} as a consequence\nof a difficult trace formula calculation. In fact, a slightly stronger statement was proved there:\nIt is clear that the exact sequence \\eqref{exseqJkm} remains valid if one replaces $J_{k,m}$ by $J_{k,m}^0$\nand the map $P:\\widetilde J_{k,m}\\to{\\mathbb C}^{{\\cal N}_{\\pm,m}}$ by the analogously defined map $P^0:\\widetilde J_{k,m}\\to{\\mathbb C}^{{\\cal N}^0_{\\pm,m}}$, \nwhere ${\\cal N}^0_{\\pm,m}$ is defined exactly like ${\\cal N}_{\\pm,m}$ but with the strict \ninequality $n<r^2\/4m$ replaced by $n \\le r^2\/4m$. \nThen the dimension formula proved in~\\cite{SkoruppaZagier} says that $\\dim J_{k,m}^0=\\dim\\widetilde J_{k,m}-|{\\cal N}^0_{\\pm,m}|$ and\nhence that the map $P^0$ is surjective, which is a stronger statement than the surjectivity of $P$ since\nthe set ${\\cal N}^0_{\\pm,m}$ is larger than ${\\cal N}_{\\pm,m}$. In summary, the results of \\cite{Eichler:1985ja} and \\cite{SkoruppaZagier} \ntogether give the following complete description of the ``polar coefficients\" maps for both holomorphic and cuspidal Jacobi forms when $k\\ge3$:\n\\begin{thm}\\label{bigweight} For $k>2$ and all $m\\ge1$ the map $P^0:\\widetilde J_{k,m}\\to{\\mathbb C}^{{\\cal N}^0_{\\pm,m}}$ is surjective and we have a\ncommutative diagram of short exact sequences \n\\begin{eqnarray} 0 \\longrightarrow J_{k,m}^0   \\longrightarrow  & \\widetilde J_{k,m} \n  & \\stackrel{P^0}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{\\pm,m}^0} \\longrightarrow{} 0 \\cr \\cap \\; \\; \\qquad & || & \\qquad  \n   \\ensuremath{\\rotatebox[origin=c]{-90}{$\\twoheadrightarrow$}} \\cr\n  0 \\longrightarrow J_{k,m} \\longrightarrow & \\widetilde J_{k,m} & \\stackrel{P}{\\longrightarrow}\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{\\pm,m}} \\longrightarrow 0 \n\\end{eqnarray} \\end{thm}\n\n\nWe now turn to the case $k=2$, the one of main interest for us.  Here formula \\eqref{jkmformula} takes the explicit form\n\\beta \\label{j2m} j(2,m) \\= \\frac12\\,\\Bigl(\\Bigl\\lfloor\\frac{m+1}3\\Bigr\\rfloor\\+\\Bigl\\lfloor\\frac{m+1}3\\Bigr\\rfloor\\Bigr)\n   \\,-\\, \\frac{m-b-1}{4} \\, - \\, \\frac14 \\sum_{d\\mid 4m} h(-d) \\,, \\end{equation}\nbut for this value of $k$ it was observed in \\cite{Eichler:1985ja} (p.~131) that equality could not always hold \nin \\eqref{dimJkmineq} (indeed, the right-hand side is often negative) and that, for reasons explained in detail there, one\nshould replace the number $j(k,m)$ by the modified number\n\\beta \\label{j2mform} j^{*}(2,m) \\;:=\\; j(2,m) \\+ \\delta(m)\\,, \\;\\quad {\\rm where} \\qquad \n  \\delta(m) \\;:=\\;\\sum_{d \\le \\sqrt{m} \\atop d|m, \\; d^{2} \\nmid m} 1 \\; . \\end{equation}\nWe note that $\\delta(m)$ is a linear combination of three multiplicative functions of $m$, namely\n  \\beta \\label{deltam} \\delta(m) \\= \\frac12 \\sigma_{0}(m) \\+ \\frac12 \\,  \\delta_{a,1} \\, - \\, \\sigma_{0}(b) \\end{equation}\n(with $m=ab^{2}$, $a$ square-free as before), or equivalently in terms of a generating Dirichlet series\n \\beta \\sum_{m=1}^{\\infty} \\frac{\\delta(m)}{m^{s}} \\= \\frac12\\,\\zeta(s)^2 \\+\\frac12\\,\\zeta(2s)\\,-\\,\\zeta(s) \\zeta(2s) \\;. \\end{equation}\nWe also note that $\\delta(m)$ depends only on the ``shape\" of~$m$ (\\,= the collection of exponents \n$\\nu_i$ if $m$ is written as $\\prod p_i^{\\nu_i}$), the first three cases being\n\\begin{eqnarray}\n\\delta(p^{\\nu})  & \\= & 0  \\; , \\label{deltapnu} \\\\\n\\delta(p_1^{\\nu_1}p_2^{\\nu_2}) &\\= & n_1n_2  \\; , \\\\\n\\delta(p_1^{\\nu_1}p_2^{\\nu_2}p_3^{\\nu_3}) & \\= & 3n_1n_2n_3 + \\varepsilon_1n_2n_3 + \\varepsilon_2n_1n_3 + \\varepsilon_3n_1n_2\\; , \n\\end{eqnarray}\nin which we have written each $\\nu_{i}$ as $2n_i+\\varepsilon_i-1$ with $n_i\\ge1$ and $\\varepsilon_i\\in\\{0,1\\}$. \n\nThe conjecture made in \\cite{Eichler:1985ja} that $\\dim J_{k,m}=j^*(2,m)$ for all $m\\ge1$ was proved \nin \\cite{SkoruppaZagier}, but here we want to give a sharper statement describing the cokernel of the map $P$ explicitly. \nTo do this, we follow \\cite{Eichler:1985ja}, pp.~132, where an elementary argument (containing a small error\nwhich will be corrected below) was given to show that $\\dim J_{2,m} \\ge j^{*}(2,m)$ for square-free $m$. \nActually, it is more convenient to work with cusp forms, since the corresponding statement here is\n  \\beta\\label{dimJ2m0} \\dim J^{0}_{2,m} \\= \\dim\\widetilde J_{2,m} \\, - \\, |{\\cal N}_{+,m}^0|  \\,+\\, \\delta^{0}(m) \\end{equation}\nwith \n  \\beta \\delta^0(m) \\= \\sum_{d|m,\\;d\\le\\sqrt m}1 \\= \\sum_{m=m_{1}\\cdot m_{2} \\atop {\\text{up to order}}} 1 \\; , \\end{equation}\nand $\\delta^0(m)$ is a slightly simpler function than $\\delta(m)$. (The third term in \\eqref{deltam} is missing.)\nWe want to explain formula~\\eqref{dimJ2m0} by finding an explicit map \n\\beta\\label{defDelta0m}\n\\beta \\, : \\; \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}^{0}_{+,m}} \\, = \\! \\bigoplus_{0\\le r \\le m \\atop 0 \\le n \\le r^{2}\/4m} \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \n\\; \\, {\\longrightarrow}  \\; \\, \\Delta^{0}(m) :=\n  \\bigoplus_{m=m_{1}\\cdot m_{2} \\atop {\\text{up to order}}} \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}\n\\end{equation}\nsuch that the sequence\n \\beta\\label{exseqJ2mcusp} 0 \\longrightarrow J_{2,m}^{0} \\longrightarrow \\widetilde J_{2,m}\n  \\stackrel{P^0}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}^{0}_{+,m}} \n  \\stackrel{\\beta}{\\longrightarrow}   \\Delta^{0}(m) \\longrightarrow 0 \\; .\n \\end{equation}\nis exact.\n\nTo give a natural definition of the map $\\beta$, we work, not with $\\widetilde J_{2,m}$ or even with $J^{\\,!}_{2,m}$, but with the larger \nvector spaces $\\widetilde\\CE_{+,m}$ and $\\CE^{\\,!}_{+,m}$   of even weak or weakly holomorphic elliptic forms of index~$m$, \nas defined at the end of \\S\\ref{hecke}.  We define a ``residue map'' ${\\cal R}:\\CE^{\\,!}_{+,m}\\to{\\mathbb C}$ by \n\\beta\\label{RasRes}\\varphi \\=\\sum_{n \\ge 0,\\,r \\in {\\mathbb Z}} c(n,r)\\, q^{n} \\, y^{r} \\quad\\mapsto\\quad \n  {\\cal R}[\\varphi]\\=\\text{Res}_{\\tau=\\infty}\\bigl(\\varphi(\\tau,0)\\,d\\tau\\bigr) \\= \\sum_{r\\in{\\mathbb Z}} c(0,r)\\;. \\end{equation}\nThen ${\\cal R}[\\varphi] =0$ if $\\varphi \\in J^{\\,!}_{2,m}$ because in that case $\\varphi(\\,\\cdot\\,,0)$ belongs to $M_{2}^{\\,!}$ and \nthe $q^{0}$ term of any weakly holomorphic modular form $f$ of weight two is zero by \nthe residue theorem applied to the $SL(2,{\\mathbb Z})$-invariant 1-form $f(\\tau) d\\tau$. (The map ${\\cal R}$ was already used in\n\\cite{Eichler:1985ja}, but it was erroneously stated there that for $\\varphi\\in\\widetilde J_{2,m}$ the function $\\varphi(\\tau,0)$\nitself, rather than just its residue at infinity, vanishes.) Now for each decomposition $m=m_1\\cdot m_2$ (up to order, \ni.e., without distinguishing the splittings $m_1\\cdot m_2$ and $m_2\\cdot m_1$) we define $R_{m_1,m_2}:\\,\\CE^{\\,!}_{+,m} \\to\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}$ \nby setting $R_{m_1,m_2}(\\varphi)={\\cal R}\\bigl[\\varphi\\bigl|W_{m_1}\\bigr]$ if $(m_1,m_2)=1$, where $W_{m_1}$ is the involution \ndefined in \\S\\ref{hecke}, while if $(m_{1},m_{2})=t >1$ then we define $R_{m_1,m_2}(\\varphi)$ as $R_{m_1\/t,m_2\/t}(\\varphi|u_t)$, \nwhere $u_t$ is the map from $\\CE^{\\,!}_{+,m}$ to $\\CE^{\\,!}_{+,m\/t^2}$ defined in~\\S\\ref{hecke}.\n(It was pointed out there that both $W_{m_1}$ for $(m_1,m\/m_1)=1$ and $u_t$ for $t^2|m$\nmake sense on all of $\\CE^{\\,!}_{+,m}$, and not just on the subspace of  holomorphic Jacobi forms, but that this\nis \\emph{not} true for weak elliptic or weak Jacobi forms, which is why we are now working with weakly holomorphic forms.) \nWe put all of these maps together to a single map $\\,R:\\,\\CE^{\\,!}_{+,m}\\to \\Delta^{0}(m)$ and then \ndefine the desired map~$\\beta$ by the commutative diagram \n\\begin{eqnarray}\\label{triangle} \n& \\qquad \\qquad \\widetilde \\CE_{+,m}  \\quad  \\stackrel{P^0}{\\longrightarrow} \\qquad \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^0} & \\cr\n& \\qquad \\cap \\qquad \\qquad \\qquad \\cap &\\cr\n& \\qquad \\qquad  \\CE^{\\,!}_{+,m}  \\quad {\\longrightarrow} \\qquad \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{!,0}} &\\\\\n\\nonumber & \\qquad R\\;\\searrow \\qquad  \\swarrow\\;\\beta  & \\cr\n &   \\quad \\qquad \\Delta^{0}(m)  & \\end{eqnarray}\nwith $ \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{!,0}}$ defined as in \\eqref{defN!} below. \n(To see that such a map $\\beta$ exists, observe that the coefficients $c(0,r)$ occurring in~\\eqref{RasRes} all\nhave discriminant $\\Delta=-r^2\\le0$, that the operators $W_{m_1}$ and $u_t$ can be defined purely in terms of\nthe coefficients $C_\\varphi(\\Delta,r)$ and preserve the condition $\\Delta\\le0$, and finally that the coefficients\n$C_\\varphi(\\Delta,r)$ for $\\varphi\\in \\widetilde \\CE_{+,m}$ can all be expressed in terms of the coefficients $c_\\varphi(n,r)$ with\n$(n,r)\\in{\\cal N}_{+,m}$ by virtue of the periodicity and evenness conditions defining $\\widetilde \\CE_{+,m}$.)\nThe discussion above shows that the composition $\\beta\\circ P^0$ in~\\eqref{exseqJ2mcusp} vanishes, \nsince ${\\cal R}$ vanishes on weakly holomorphic and hence in particular on weak Jacobi forms, and combining this\nstatement with the dimension formula~\\eqref{dimJ2m0} proved in~\\cite{SkoruppaZagier} we obtain \nthe main result describing strong and weak Jacobi forms of weight 2 and arbitrary index:\n\\begin{thm} \\label{exactseqallm}\nThe sequence \\eqref{exseqJ2mcusp} with $\\beta$ defined by \\eqref{triangle} is exact for all integers $m \\ge 1$. \n\\end{thm}\n\nThe two theorems above give a complete description of the space of Jacobi forms of all weights $k \\ge 2$.\nWe end this subsection by discussing the case of weight one.  Here the description of Jacobi forms is very\nsimple to state (though its proof is quite deep), since we have the following theorem of Skoruppa \n(\\cite{Skoruppa}, quoted as Theorem~5.7 of \\cite{Eichler:1985ja}):\n\\begin{thm} \\label{Skorthm}\nWe have $J_{1,m} = \\{0\\}$ for all $m\\ge 1$. \n\\end{thm}\nCombining this with the discussion above, we find that in this case we have an exact sequence \n \\beta\\label{exseqJ1m} 0 \\longrightarrow \\widetilde J_{1,m} \n \\longrightarrow  \\bigoplus_{0 < r < m \\atop 0 \\le n < r^{2}\/4m} \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \n \\longrightarrow  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{|j(1,m)|} \\longrightarrow 0 \\; .\n \\end{equation}\nwith \n$$ |j(1,m)| \\= -j(1,m) \\= \\frac m{24} \\+{\\cal O}\\bigl(m^{1\/2+\\varepsilon}\\bigr)\\;. $$\nWe have not been able to find a natural description of the last space, or explicit description of the last map,\nin \\eqref{exseqJ1m}, analogous to the results given for $k=2$. \n\n\n\n\n\n\\subsection{Choosing optimal versions of weak elliptic forms \\label{optimalforms}} \n\nIn this subsection, we will apply the theorem just explained to show how to make ``optimal versions''\nof elliptic forms by counting actual Jacobi forms to get poles of small order. \n\n\nLet $\\widetilde\\CE_{+,m}$ be as in the previous subsection and $\\widetilde\\CE_{-,m}$ be the similarly defined \nspace with $c(n,-r) = -c(n,r)$, and let $\\CE_{\\pm,m}$ and $\\CE_{\\pm,m}^0$ be the corresponding \nspaces with the condition \\eqref{weakjacobi} replaced by \\eqref{holjacobi} and \\eqref{cuspjacobi},\nrespectively. The discussion of \\S\\ref{relations} can be summarized as saying that we have a commutative \ndiagram of short exact sequences, \n\\begin{eqnarray} \\label{Jkmdiagram}\n0 \\longrightarrow \\CE_{\\pm,m}  \\longrightarrow  & \\widetilde\\CE_{\\pm,m}\n& \\stackrel{P}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{\\pm,m}} \\longrightarrow 0 \\cr\n  \\cup \\; \\; \\qquad & \\cup & \\qquad  || \\cr\n0 \\longrightarrow J_{k,m} \\longrightarrow & \\widetilde J_{k,m} & \\longrightarrow  \n\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{\\pm,m}} \\longrightarrow 0\n\\end{eqnarray}\nfor $k>2$, $(-1)^k=\\pm1$, and similarly with $J_{k,m}$ and $\\CE_{\\pm,m}$ replaced by\n$J_{k,m}^0$ and $\\CE_{\\pm,m}^0$, ${\\cal N}_{\\pm,m}$ by ${\\cal N}_{\\pm,m}^0$ and $P$ by $P^0$. \nFor $k=2$, we have instead: \n\\begin{eqnarray} \\label{J2mdiagram}\n0 \\longrightarrow \\CE_{+,m}^0  \\longrightarrow  & \\widetilde\\CE_{+,m} \n& \\stackrel{P^0}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^0} \\longrightarrow 0 \\cr\n  \\cup \\; \\; \\qquad & \\cup & \\qquad  || \\cr\n0 \\longrightarrow J^{0}_{2,m} \\longrightarrow & \\widetilde J_{2,m} & \\longrightarrow  \n\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^0} \\stackrel\\beta\\longrightarrow  \\Delta^{0}(m)\n \\longrightarrow 0 \n\\end{eqnarray}\nin the cuspidal case, where $\\Delta^{0}(m)$ was defined in \\eqref{defDelta0m} \nand~$\\beta$ is defined by~$\\beta \\circ P^{0}=R$. \nAs a consequence of these diagrams and of the vanishing of $\\delta(p^{\\nu})$ ({\\it cf.} \nequation \\eqref{deltapnu}), which implies that the diagram \\eqref{Jkmdiagram} also holds \nfor $k=2$ when $m$ is a prime power, we obtain:\n\\begin{thm} \\label{mpnuthm}\nLet $k\\ge2$, and $\\varphi \\in \\widetilde \\CE_{\\pm,m}$, where $(-1)^{k} = \\pm 1$. \nThen $\\varphi$ is the sum of an element of $\\widetilde J_{k,m}$ and an element of $ \\CE^{0}_{\\pm,m}$\nif and only if either \n\\null \\quad \\newline{\\rm(i)\\phantom{ii}} $\\quad k>2\\,,$ or \n\\newline{\\rm(ii)\\phantom{i}} $\\quad k=2$ and $R(\\varphi)\\;(=\\beta(P^0(\\varphi)))\\;=0\\,;$ \\\\\nIf $k=2$ and $m$ is a prime power, then $\\varphi$ is the sum of an element of \n$\\widetilde J_{k,m}$ and an element of~$ \\CE_{\\pm,m}$. \n\\end{thm}\nTheorem \\ref{mpnuthm} for $k=2$ makes a negative statement: if $\\varphi \\in \\widetilde \\CE_{+,m}$\nhas ${\\cal R}[\\varphi]\\neq 0$, and $m$ is not a prime power, then $\\varphi$ \\emph{cannot} in general be decomposed as the \nsum of a weak Jacobi form of weight 2 and a holomorphic elliptic form. \nHowever, it turns out that we can come very close to this, writing \n$\\varphi$ as a sum of images under Hecke-like operators of \nforms that ``have optimal growth'' in the sense of \\S\\ref{Statements}.\nWe now explain this in two steps. We first use the decomposition \\eqref{primdecomp}, \nand its variants for weakly holomorphic Jacobi or elliptic forms,  \nto reduce to the case of primitive forms. We then show that any primitive even elliptic form \ncan be corrected by the addition of a weakly holomorphic weight 2 Jacobi form \nto give a new elliptic form whose non-zero Fourier coefficients \n$c(n,r)$ all have discriminant $4mn-r^{2}\\ge -1$.  \n\n\nSince the decomposition \\eqref{primdecomp}, as was already observed in \\S 4.3 fails for weak Jacobi forms \nbut holds for weakly holomorphic ones, we must first extend the above results to \nweakly holomorphic forms. \n\\begin{proposition}\nAll statements in Theorem~\\ref{mpnuthm} remain true \\emph{mutatis mutandis} if \nall $\\widetilde \\CE$, $\\widetilde J$ are replaced by $\\CE^{!}$, $J^{!}$. \n\\end{proposition}\nThe proposition is equivalent to saying that \nthe analog of \\eqref{J2mdiagram} still holds \nwith weakly holomorphic rather than weak forms, i.e. we have a diagram \n\\begin{eqnarray} \\label{wklyholdiagram}\n0 \\longrightarrow \\CE_{+,m}^0  \\longrightarrow  & \\CE^{\\,!}_{+,m} \n& \\stackrel{P^0}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,!}} \\longrightarrow 0 \\cr\n  \\cup \\; \\; \\qquad & \\cup & \\qquad  || \\cr\n0 \\longrightarrow J^{0}_{2,m} \\longrightarrow & J^{\\,!}_{2,m} & \\longrightarrow  \n\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,!}} \\stackrel\\beta\\longrightarrow  \\Delta^{0}(m)\n \\longrightarrow 0 \n\\end{eqnarray}\nwhere \n\\beta\\label{defN!}\n{\\cal N}_{+,m}^{0,!} \\=  \\big\\{ (r,n) \\in {\\mathbb Z}^{2} \\, \\mid \\, 0 \\le r \\le m, \\; -\\infty < n \\le r^{2}\/4m  \\big\\} \\; .\n\\end{equation}\nWe again prove this by comparing dimensions, even though the middle \nspaces in each row in the diagram \\eqref{wklyholdiagram} are infinite-dimensional. \nSince any elliptic form $\\varphi$ has a minimal value of $n$ with $c_{\\varphi}(n,r) \\neq 0$\nfor some $r$, it suffices to prove the exactness of the rows in \\eqref{wklyholdiagram} \nwith $\\CE_{+,m}^{\\,!}$, $J^{\\,!}_{2,m}$, and $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,!}}$ replaced by the \nspaces $\\CE_{+,m}^{\\ge N}$, $J^{\\ge N}_{2,m}$, and $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,\\ge N}}$,  \nof forms with non-zero coefficients only for $n \\ge -N$ for some $N \\ge 0$. \nThe only non-trivial part is  the exactness at \n$\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,\\ge -N}}$, and since the composition of the maps \n$P^{0}$ and $\\beta$ vanishes, we \nonly need to check that the alternating sum of dimensions is zero. For this,\nwe observe that the dimensions of $J^{\\ge -N}_{2,m}$ and $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,\\ge -N}}$ \nare both larger by $(m+1)N$ than their values at $N=0$: \nfor $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,\\ge -N}}$, this is obvious since ${\\cal N}_{+,m}^{0,\\ge -N}$ \ndiffers from ${\\cal N}_{+,m}^{0}$ by the addition of $N$ rows of length $m+1$, and \nfor $J^{\\ge -N}_{2,m}$ it follows from the computation\n\\begin{eqnarray}\n\\dim J_{2,m}^{\\ge -N} & \\= & \\dim \\widetilde J_{12N+2,m} \\qquad  \n   \\; \\big(\\,\\text{because $J_{2,m}^{\\ge -N} = \\Delta(\\tau)^{-N} \\widetilde J_{12N+2,m}$}\\,\\big)\\cr \n& \\= & \\sum_{j=0}^{m} \\dim M_{k+12N+2j}  \\=  \\dim \\widetilde J_{2,m} \\+ (m+1)N \\, . \n\\end{eqnarray}\nThis completes the proof of the proposition. \\hfill $\\square$\n\n\n\n\n\nNow we can apply the decomposition \\eqref{primdecomp} to the whole diagram \\eqref{wklyholdiagram}\nto find that everything can be reduced to primitive forms (and their images under the $U_{t}$ operators). \nThe key point is that although $U_{t}$ and $u_{t}$ change the discriminant $\\Delta = 4 mn -r^{2}$\nby multiplying or dividing it by $t^{2}$, respectively, they do not change its sign, and therefore all \nof the spaces in our diagram can be decomposed as direct sums of primitive parts of lower index \nand their images under $U_{t}$. This reduces everything to case of primitive forms.\n\nFor primitive forms our diagram becomes\n\\begin{eqnarray} \\label{wklyholdiagramprim}\n0 \\longrightarrow \\CE_{+,m}^{0, \\rm prim}  \\longrightarrow  & \\CE^{!, \\rm prim}_{+,m} \n& \\stackrel{P^0}{\\longrightarrow}  \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,!, \\rm prim}} \\longrightarrow 0 \\cr\n  \\cup \\; \\; \\qquad & \\cup & \\qquad  || \\cr\n0 \\longrightarrow J^{0, \\rm prim}_{2,m} \\longrightarrow & J^{!, \\rm prim}_{2,m} & \\longrightarrow  \n\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0,!, \\rm prim}} \\stackrel\\beta\\longrightarrow  \\Delta^{0, \\rm prim}(m)\n \\longrightarrow 0 \n\\end{eqnarray}\nwhere \n\\beta\\label{delta0prim}\n\\Delta^{0, \\rm prim}(m) =  \n\\bigoplus_{{m=m_{1}\\cdot m_{2} \\atop (m_{1}, m_{2}) =1} \\atop {\\text{up to order}}} \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \\;.\n\\end{equation}\nThe map $\\beta$ in \\eqref{wklyholdiagramprim} is again defined by the requirement that $\\beta \\circ P^{0} = R$ on \n$\\CE_{+,m}^{!,\\rm prim}$, and lands in $\\Delta^{0,\\rm prim}(m)$ because $\\varphi|u_{t}=0$ for $\\varphi$ primitive and $t>1$.\nThe space $\\Delta^{0, \\rm prim}(m)$ has dimension $\\delta^{0, \\rm prim}(m) = 2^{s-1}$ if $m>1$, where \n$s$ is the number of primes dividing $m$. (The failure of this formula for $m=1$ corresponds to \nthe second term in \\eqref{deltam}.)\n\nWe also have the non-cusp-form analogs of the diagrams \\eqref{wklyholdiagram} and \n\\eqref{wklyholdiagramprim}, where all superscripts ``0'' have been dropped and\nwhere ${\\cal N}_{+,m}^{\\,!}$ is defined as in \\eqref{defN!} but  \nwith the condition $4 mn - r^{2}  \\le 0$ replaced by the strict inequality $4 mn - r^{2} <0$, \nwhile $\\Delta^{\\rm prim}(m)$ is a space of dimension \n$\\delta^{\\rm prim}(m)=\\delta^{0, \\rm prim}(m)-1 = 2^{s-1}-1$ (compare equation \\eqref{deltam}). \nThis space is the quotient of the space in \\eqref{delta0prim} by the one-dimensional \nsubspace of vectors with all components equal. This is essentially the idea of the proof of \nTheorem~\\ref{chooseKm} given below. \n\n\nHaving completed these preparations, we now proceed to the proofs of our main results.\nWe consider the case of square-free index first and then give a brief description of the \nmodifications needed to extend them to primitive forms of arbitrary index. \n\n\n\nFor $m>1$ square-free, we will show that any $\\varphi\\in\\widetilde\\CE_{+,m}$ can be corrected by a \nweak Jacobi form of weight 2 to get a another elliptic form whose non-zero Fourier coefficients \n$c(n,r)$ all have discriminant $4mn-r^{2}\\ge -1$, i.e., that we have a decomposition\n\\beta \\label{thJminone}\n\\widetilde\\CE_{+,m} \\= \\widetilde J_{2,m} \\+ \\CE^{\\rm OG}_{+,m} \\, \\qquad \\text{($m$ square-free),} \n\\end{equation} \nwith $\\CE^{\\rm OG}_{+,m}$ defined as in \\S\\ref{Statements}.\nActually, this is still not quite the statement we want, since it is slightly wasteful:\nthe codimension of $\\CE_{+,m}^0$ in $\\CE_{+,m}^{\\rm OG}$ is $2^{s-1}+1$,  where~$s$\nas before is the number of prime factors of~$m$ (there are $2^{s-1}$ pairs\n$(r,n)\\in{\\cal N}_{+,m}^0$ with $\\Delta:=4nm-r^2$ equal to $-1$, and one pair (0,0) with $\\Delta=0$), and there are \nonly $\\dim \\Delta^{0}(m)=2^{s-1}$ constraints $R_{m_1,m_2}(\\varphi)=0$ coming from \ndiagram~\\eqref{J2mdiagram}.  Correspondingly,\nthe intersection of the two spaces on the right-hand side of~\\eqref{thJminone} contains $J_{2,m}$\n($=J_{2,m}^{0}$ in this case)\nas a subspace of codimension~1.  To get a sharp statement, we should therefore replace~$\\CE_{+,m}^{\\rm OG}$\nby a suitable codimension one subspace.  There are three natural ways to do this, each of which\nleads to a sharpening of~\\eqref{thJminone}, corresponding exactly (in the square-free case) \nto the various choices of $\\Phi_{2,m}^{0}$ introduced in the discussion preceding Theorem~\\ref{PolarCoeffs}.\nWhat's more, all statements remain true verbatim for arbitrary~$m>1$ if we restrict to primitive forms. \nWe state all three variants in a single theorem.\n\\begin{thm} \\label{Emstar}\nSuppose that $m>1$ is square-free and let $\\CE^{\\rm OG,\\star}_m\\;\\,(\\star=\\rm I,\\,\\rm II,\\, \\rm III)$ denote \nthe codimension one subspace of $\\CE_{+,m}^{\\rm OG}$ defined by one of the three conditions \n\\newline\\null\\qquad {\\rm(I)} $\\quad \\sum_{0<r<m,\\;r^2\\equiv1\\pmod{4m}} C_\\varphi(-1;r)=0$,\n\\newline\\null\\qquad {\\rm(II)} $\\quad c_\\varphi(0,1)=0$,\n\\newline\\null\\qquad {\\rm(III)} $\\quad c_\\varphi(0,0)=0$,\n\\newline respectively.  Then we have an exact sequence\n\\beta\n0 \\longrightarrow J_{2,m} \\stackrel{\\text{\\rm diag}}\\longrightarrow \n\\widetilde J_{2,m} \\oplus \\CE_m^{\\rm OG, \\star} \\stackrel-\\longrightarrow \\widetilde\\CE_{+,m} \\longrightarrow 0 \\; ,\n\\end{equation}\ni.e., any even weak elliptic form of index~$m$ can be decomposed as the sum of a weak Jacobi form \nof weight~2 and index~$m$ and an element \nof $\\CE_m^{\\rm OG, \\star}$, and this decomposition is unique up to a holomorphic Jacobi form \n$($which can be added to one term and subtracted from the other$)$. \n\nFor~$m>1$ the same statement remains true if we replace~$\\widetilde \\CE_{+,m}$ by its \nsubspace~$\\widetilde \\CE_{+,m}^{\\rm res-prim}$ of forms~$\\varphi$ satisfying~$R(\\varphi|u_{t})=0$\nfor all~$t>1$ and~$\\CE_m^{\\rm OG, \\star}$ by its intersection with~$\\widetilde \\CE_{+,m}^{\\rm res-prim}$. \n\\end{thm}\n\n\n\n\\begin{proof}\nBy the exactness of the top line of diagram \\eqref{J2mdiagram}, we can find functions \n$\\psi_{m;(r,n)} \\in \\widetilde \\CE_{+,m}$, unique up to the addition of cusp forms, mapping to the canonical basis \n$\\delta_{(r,n)}$ of $\\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}^{{\\cal N}_{+,m}^{0}}$. \nClearly, the space $\\CE_{+,m}^{\\rm OG,\\rm III}\/\\CE_{+,m}^{0}$ is spanned by \n$\\psi_{m;(r,n)}$ with $0< r < m$ and $4mn-r^{2}=-1$, and has dimension $2^{s-1}$. \n(There are exactly two square roots of 1 modulo~$2P$ for each exact prime power divisor~$P$ of~$m$ and \nthese can be combined arbitrarily by the Chinese remainder theorem. This gives~$2^{s}$ \nresidue classes $r \\; \\mod \\; {2m}$ with $r^{2} \\equiv 1 \\; \\mod \\; {4m}$, and if we choose the \nrepresentatives of smallest absolute values then half of them are positive.) \nMoreover, the basis elements can be chosen to be a single orbit of the group \n$\\{W_{m_{1}} \\}_{m_{1} | m}$ \nintroduced in \\S\\ref{hecke}, whose cardinality is \nthe same number $2^{s-1}$. Namely, if $A_{m}$ is any choice of $\\psi_{m;(1,0)}$, i.e. \nany function in $\\widetilde \\CE_{+,m}$ whose unique non-zero coefficient \nin ${\\cal N}^{0}_{+.m}$ is $c(0,1)=1$, then the other functions $\\psi_{m;(r,n)}$\nwith $(r,n) \\in {\\cal N}^{0}_{+,m}$ and $4 mn-r^{2} = -1$, \ncan be chosen to be the images of $A_{m}$ under the Atkin-Lehner involutions $W_{m_{1}}$, \n$m=m_{1}m_{2}$. \nMore explicitly, with this choice $\\psi_{m;(r,n)}$ equals $A_{m}|W_{m_{1}}$ \nwith~$m_{1}=\\gcd\\big(\\frac{r+1}{2},m\\big)$, $m_{2}=\\gcd\\big(\\frac{r-1}{2},m\\big)$,\nand conversely~$A_{m}|W_{m_{1}}$ for any decomposition~$m=m_{1}m_{2}$ equals\n$\\psi_{m;(|r_{0}|,(r_{0}^{2}-1)\/4m)}$ \nwhere $r_{0}$ ($=1^{*}$ in the notation of \\S\\ref{hecke}) is the unique $r_{0}$ satisfying\n\\beta \\label{defr0}\nr_{0} \\equiv -1 \\; \\mod \\; {2m_{1}} \\; ,  \\qquad r_{0} \\equiv 1 \\; \\mod \\; {2m_{2}}  \\; , \\qquad  |r_{0}| < m\\; . \n\\end{equation}\n\nOn the other hand, for any $\\varphi \\in \\CE_{+,m}^{\\rm OG, III}$, we have \n\\beta\\label{Ronem}\nR_{1,m}(\\varphi) \\= {\\cal R}[\\varphi] \\=2 c_{\\varphi}(0,1) \\; \n\\end{equation}\n(because $c_{\\varphi}(0,-1) = c_{\\varphi}(0,1)$ and $c_{\\varphi}(0,r)=0$ for all $r \\neq \\pm 1$\nby the defining property of~$\\CE_{+,m}^{\\rm OG, III}$), and more generally, \n\\beta\nR_{m_{1}, m_{2}}(\\varphi) \\=2 \\, c_{\\varphi}(r_{0},(r_{0}^{2}-1)\/4m) \\; \n\\end{equation}\nfor each decomposition $m=m_{1}m_{2}$, and with $r_{0}$ as in \\eqref{defr0}. \nIt follows that, for any two decompositions $m=m_{1}m_{2}$ and $m=m_{1}'m_{2}'$,\nwe have \n\\beta\nR_{m_{1},m_{2}}\\big(A_{m} \\! \\mid \\! W_{m_{1}'} \\big)  \n\\= \\begin{cases} \n2 \\qquad \\text{if $m_{1}'=m_{1}$ or $m_{1}'=m_{2}$} \\; , \\\\\n0 \\qquad {\\rm otherwise} \\; .\n\\end{cases}\n\\end{equation}\nThis shows that the map  \n\\beta\n\\CE_{+,m}^{\\rm OG, III}\/\\CE_{+,m}^{0} \\stackrel{R}{\\longrightarrow} \n\\bigoplus_{m=m_1\\cdot m_2 \\atop {\\text{up to order}}} \\relax\\hbox{$\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$} \\, \n\\end{equation}\nis an isomorphism, and in combination with Theorem \\ref{mpnuthm}, completes the proof \nof the theorem in the case of square-free~$m$ and~$\\star=\\rm III$: any~$\\varphi \\in \\widetilde\\CE_{+,m}$ \ncan be decomposed, uniquely up to Jacobi cusp forms, as the sum of \n$\\frac12 \\sum_{m=m_{1}m_{2}} R_{m_{1},m_{2}}(\\varphi) A_{m}|W_{m_{1}} \\in \\CE_{+,m}^{\\rm OG, III}$ \nand an element of~$\\widetilde J_{2,m}$.\n\nThe proofs for $\\star=\\rm I$ and $\\star=\\rm II$ are almost identical,\nchanging the basis $\\{\\psi_{m;(r,n)} = A_{m}|W_{m_{1}}\\}$ of~$\\CE_{+,m}^{\\rm OG, III}\/\\CE_{+,m}^{0}$\nin both cases in such a way as to satisfy the new defining condition without changing the residues, i.e., \nin the case~$\\star=\\rm II$ by replacing the first basis element~$\\psi_{m;(1,0)}$ by~$2 \\psi_{m;(0,0)}$, and in \nthe case~$\\star=\\rm I$ by adding $2 \\psi_{m;(0,0)} - 2^{-s+1} \\sum_{r'} \\psi_{m;(r',n)}$ to each basis\nelement. \n\nThe proof of the statement for forms of arbitrary index~$m$ is very similar,  since there are still~$2^{s-1}$ polar \ncoefficients of a form in~$\\CE_{+,m}^{\\rm OG}$, in one-to-one correspondence with the decompositions of~$m$ \ninto two coprime factors. The details are left to the reader.\n\\end{proof}\nTheorem~\\ref{Emstar} says that the first line in the diagram \n\\begin{eqnarray} \\label{primdiag}\n0 \\longrightarrow  \\; \\; J_{2,m}^{\\rm prim}  \\quad \\stackrel{\\text{\\rm diag}}\\longrightarrow \n& \\widetilde J_{2,m}^{\\rm prim} \\oplus \\CE_m^{\\rm OG, \\rm prim, \\star} & \\stackrel-\\longrightarrow \n\\widetilde\\CE_{+,m}^{\\rm prim} \\longrightarrow 0  \\cr\n  \\cap  \\qquad \\qquad &  \\;  \\cap  \\qquad & \\qquad  || \\\\\n0 \\longrightarrow  J_{2,m}^{\\rm OG, prim}   \\stackrel{\\text{\\rm diag}}\\longrightarrow \n&\\widetilde J_{2,m}^{\\rm prim} \\oplus \\CE_m^{\\rm OG, \\rm prim} & \\stackrel-\\longrightarrow \n\\widetilde\\CE_{+,m}^{\\rm prim} \\longrightarrow 0  \\nonumber\n\\end{eqnarray}\nis exact. The exactness of the second line is a trivial consequence, and the fact \nthat~$\\CE_{+,m}^{\\rm OG, prim, \\star}$ has codimension exactly one in~$\\CE_{+,m}^{\\rm OG, prim}$ \nimplies that~$J_{2,m}^{\\rm prim}$ has codimension one in~$J_{2,m}^{\\rm OG, prim}$.\nMoreover, the proof of the theorem also shows that~$J_{2,m}^{\\rm OG, \\star} = J_{2,m}$. \n(If~$\\varphi\\in J_{2,m}^{\\rm OG}$, then the vanishing of the residues~$R_{m_{1}m_{2}}(\\varphi)$ for all decompositions \nwith~$(m_{1},m_{2})=1$ implies that the polar coefficients~$C_{\\varphi}(-1,r)$ are all equal to one another \nand to~$-{\\frac12} c_{\\varphi}(0,0)$, so that imposing any of the conditions~$(\\rm I)$, $(\\rm II)$, $(\\rm III)$ \nforces~$\\varphi$ to be holomorphic. Conversely, if~$\\varphi\\in J_{2,m}$, then the conditions $\\rm OG$, (I), (II) are \ntrivial, and (III) holds because ${\\cal R}[\\varphi]=0$.) It follows that the codimension of~$J_{2,m}$ \nin~$J_{2,m}^{\\rm OG}$ is also one.\n\n\n\n\\subsection{The residues of mock Jacobi forms} \\label{MJFappl}\n\nWe can apply the general theorems of the previous subsection \nto mock Jacobi forms. In particular, if we have any weak mock \nJacobi form of weight $k\\ge3$ and index $m$, it will be in $\\widetilde \\CE_{\\pm,m}$, and can be \ncorrected by a weak Jacobi form to get a holomorphic mock Jacobi form. \nIn the case $k=2$, there is an obstruction $R(\\varphi)$ to this being true, but, by the way we defined it, this\nobstruction can only depend on the shadow of~$\\varphi$. In this subsection, we show how to compute $R(\\varphi)$ \nexplicitly in terms of the shadow. Since $R(\\varphi)$ is the vector with components ${\\cal R}\\bigl[\\varphi|u_{(m_1,m_2)}|W_{m_1\/(m_1,m_2)}\\bigr]$,\nit suffices to compute $R[\\varphi]$ for a weak mock Jacobi form of weight~2 in terms of the shadow of~$\\varphi$.\n \nLet $\\varphi(\\tau,z)$ be a (strong or weak) mock Jacobi form of weight 2 and index $m$. Then, by definition, $\\varphi$ has a \ntheta expansion \\eqref{jacobi-theta} where each $h_{\\ell}$ is a (in general weakly holomorphic) mock modular form \nof weight~3\/2 and has a completion $\\widehat h_{\\ell} = h_{\\ell} + g_\\ell^*$ for some modular form $g_{\\ell}$ of weight~1\/2 \nsuch that the completion \n$\\widehat \\varphi = \\sum_{\\ell} \\, \\widehat h_{\\ell} \\, \\vartheta_{m,\\ell} = \\varphi + \\sum_{\\ell} \\, g_{\\ell}^{*} \\, \\vartheta_{m,\\ell}$\ntransforms like a Jacobi form of weight 2 and index $m$. \n\n\nAs usual, we denote by $\\vartheta_{m,\\ell}^0(\\tau)$ the Thetanullwert $\\vartheta_{m,\\ell}(\\tau,0)$. \n\\begin{thm} \\label{Rphishad}\nLet $\\varphi$ be as above. Then\n\\beta \\label{Rphivalue}\n{\\cal R}[\\varphi] \\= \\frac{\\sqrt{\\pi}}{6} \\, \\sum_{\\ell \\; \\mod \\; {2m}} \\bigl(\\vartheta_{m,\\,\\ell}^{0}, g_\\ell \\bigr)\\,, \n\\end{equation}\nwhere $(\\,\\cdot\\,,\\,\\cdot\\,)$ denotes the Petersson scalar product in weight 1\/2.\n\\end{thm}\n\\begin{proof}\nWith the normalization of the Petersson scalar product as given in \\eqref{Peterkabeta},\nwe have\n\\beta\n\\sum_{\\ell \\; \\mod \\; {2m}}  \\bigl(\\vartheta_{m,\\ell}^{0}, g_\\ell \\bigr) \\= \\frac{3}{\\pi} \\int_{{\\cal F}} \\, F(\\tau) \\, \nd \\mu(\\tau) \\; , \n\\end{equation}\nwhere\n\\beta \\label{defFtau}\nF(\\tau) \\=  \\tau_{2}^{1\/2}\\, \\sum_{\\ell \\; \\mod \\; {2m}}  \\overline{g_{\\ell}(\\tau)} \\,  \\vartheta_{m,\\ell}^{0}(\\tau) \\; , \n\\end{equation}\nand ${\\cal F}$ denotes a fundamental domain for $\\Gamma \\equiv SL_{2}({\\mathbb Z})$. This makes sense since the function \n$F(\\tau)$ is $\\Gamma$-invariant (the individual terms in \\eqref{defFtau} are not, which is why we had to \nbe a little careful in the normalization of the scalar product) and is convergent because the weight of the\nmodular forms $g_\\ell$ and $\\vartheta_{m,\\ell}^0$ is less than~1. From eq.~\\eqref{starinv} it follows that the\n$\\overline\\tau$-derivative of the completion\n \\beta\\label{phihatismock}\n  \\widehat\\varphi(\\tau,0) \\= \\varphi(\\tau,0) \\+ \\sum_{\\ell \\; \\mod \\; {2m}}  g_{\\ell}^{*}(\\tau) \\,  \\vartheta_{m,\\ell}^{0}(\\tau) \\, , \\end{equation}\n of $\\varphi(\\tau,0)$ satisfies\n\\beta \\label{Fisshadow} \\frac{\\pi}{\\pi \\overline\\tau} \\, \\widehat \\varphi (\\tau,0) \\= \\frac{1}{4i\\sqrt\\pi} \\frac{F(\\tau)}{\\tau_2^{2}} \\,. \\end{equation}\n\nThe fact that $\\widehat \\varphi(\\tau,0)$ transforms like a modular form of weight 2 on $\\Gamma$ \nimplies that the differential one-form $\\omega = \\widehat \\varphi(\\tau,0) \\, d\\tau$ is $\\Gamma$ invariant, and \nequation \\eqref{Fisshadow} implies \n\\beta  d\\omega}  \\def\\omega{\\omega \\= d\\big(\\widehat\\varphi(\\tau,0)\\,d\\tau\\big) \\= -\\frac{\\pi\\,\\widehat\\varphi(\\tau,0)}{\\pi \\bar{\\tau}} \\, d\\tau \\, d\\bar{\\tau} \n\\= \\frac{1}{2 \\sqrt\\pi} \\, F(\\tau) \\, d \\mu(\\tau) \\; .\n\\end{equation}\nTherefore, by Stokes's theorem, we have \n\\beta\n\\frac{\\sqrt\\pi}{6} \\, \\sum_{\\ell \\; \\mod \\; {2m}}  \\bigl(\\vartheta_{m,\\ell}^{0}, g_\\ell \\bigr) \\= \\int_{{\\cal F}} d \\omega}  \\def\\omega{\\omega \\= \\int_{\\pi {\\cal F}} \\omega \n\\= \\lim_{T \\to \\infty} \\int_{\\pi {\\cal F}(T)} \\omega\n\\end{equation}\nwhere ${\\cal F}(T)$ is the truncated fundamental domain \n\\beta\n\\{\\tau \\in {\\cal H} \\mid |\\tau|>1\\, , \\; |\\tau_{1}| < \\frac12 \\, , \\; \\tau_{2} < T \\} \\; .\n\\end{equation}\nOn the other hand, we have \n\\beta\n\\int_{\\pi {\\cal F}(T)} \\omega \\= \\int_{-\\frac12 + iT}^{\\frac12 + iT} \\omega\n\\= \\int_{-\\frac12 + iT}^{\\frac12 + iT} \\varphi(\\tau,0) \\, d\\tau  \\+ O\\big(\\frac{1}{\\sqrt{T}}\\big) \\; .\n\\end{equation}\nThe first equality holds because $\\omega$ is $\\Gamma$-invariant, and \nall the edges of ${\\cal F}(T)$ except the top edge $\\tau_{2}=T$\ncome in pairs which are $\\Gamma$-equivalent, but of opposite orientation. \nThe second equality follows from \\eqref{phihatismock}, and \nbecause $\\vartheta^{0}_{m,\\ell}$ is $O(1)$ and $g_{\\ell}^{*}(\\tau)$ is $O(T^{-1\/2})$ by \\eqref{defstar} \nwith $k=3\/2$. The theorem then follows.\n\\end{proof}\n\\noindent The theorem immediately implies a more general formula for the residue \n$R_{m_1,m_2}(\\varphi) = {\\cal R}\\bigl[\\varphi\\bigl|W_{m_1}\\bigr]$ when $m=m_{1}m_{2}$ with \n$(m_{1}, m_{2})=1$  in terms of Petersson scalar products, namely, \n\\beta \\label{Rm1m2ans}\nR_{m_1,m_2}(\\varphi)  \\= \\frac{\\sqrt{\\pi}}{6} \\, \\sum_{\\ell \\; \\mod \\; {2m}} \\bigl(\\vartheta_{m,\\,\\ell}^{0}, g_{\\ell^{*}} \\bigr)\\,, \n\\end{equation}\nwhere $\\ell \\mapsto \\ell^{*}$ is the involution on $({\\mathbb Z}\/2m{\\mathbb Z})^{*}$ appearing in the definition of~$W_{m_{1}}$. \n\n\n\n\n\\subsection{Remaining proofs of theorems from \\S\\protect\\ref{flyJac} \\label{proofs}} \n\n\nIn this subsection, we apply the results proved so far to the special family studied in~\\S\\ref{flyJac}, namely,\nthe meromorphic Jacobi forms~$\\varphi_{2,m}$ having a pole~$1\/(2 \\pi i z)^{2}$ at the origin\nand their associated mock Jacobi forms~$\\Phi_{2,m}$. Specifically, we shall prove \nTheorems \\ref{phimQm}, \\ref{QmOG}, \\ref{chooseKm}, \\ref{PolarCoeffs}, and \\ref{Wt1OSP}. \n\n\nWe begin with a preliminary remark.\nTwo of the results we need to prove are Theorem~\\ref{phimQm} and the statement that the \nfunctions~$\\Phi_{2,m}^{0}$ \ndefined implicitly by~\\eqref{Phi0mimp} are primitive modulo weak Jacobi forms. Both of these have the form \nthat a certain elliptic form is in fact a (weak) Jacobi form, i.e., that its shadow vanishes. Since the forms \nin question are defined using the Hecke-like operators defined in~\\S\\ref{hecke}, we first have to \nexplain how the action of these operators extends to non-holomorphic elliptic forms, since then \nwe can apply them to the completions of our mock Jacobi forms and verify the necessary identities \nbetween shadows. The definitions of the Hecke-like operators were given in terms of the Fourier coefficients~$c(n,r)$ \nor $C(\\Delta,\\ell)$ defined by~\\eqref{fourierjacobi} and \\eqref{cnrprop}.\nIn the case of the completions of mock modular forms~$\\Phi$, when we apply equation~\\eqref{defUl} or~\\eqref{defVl} \nfor the action of $U_{t}$ or $V_{t}$ with the~$\\varphi$ there replaced by~$\\Phi^{C}:=\\widehat \\Phi - \\Phi$ \n(the ``correction term'' discussed at the end of~\\S\\ref{MockDecomposition}), \nwe must interpret~$c(n,r)$ as the \ncoefficient of $\\beta(|n|\\tau_{2}) \\, q^{n} \\, y^{r}$ rather than simply $q^{n} y^{r}$ as in the holomorphic case,\nwhere $\\beta(t)=\\sqrt{\\pi\/4m} \\; \\text{erfc}(2\\sqrt{\\pi t})$ in the weight~2 case, \nand similarly for equation~\\eqref{Ucech2} and $C(\\Delta, \\, r\\! \\; \\mod \\; {2m})$. \nIn view of~\\eqref{phi2mhat} and~\\eqref{defstar}, these coefficients in the case of~$\\Phi_{2,m}$ are given by  \n\\beta \\label{phiCcfs}\nC(\\Phi^{C}_{2,m} ; \\Delta , \\, \\ell \\; \\mod \\; {2m}) \\= \\sum_{\\lambda^{2} = - \\Delta \\atop  \\lambda \\equiv \\ell \\; \\mod \\; {2m}} |\\lambda| \\; . \n\\end{equation}\n\n\n\n\n\\bigskip\n\n\n\\noindent {\\bf Proof of Theorem~\\ref{phimQm}}:\nWe have to show that for $m=P_{1} \\cdots P_{r}$ ($P_{i} = p_{i}^{\\nu_{i}}$, $p_{i}$ distinct primes) \nwe have \n\\beta\n\\Phi_{2,m} \\, \\big| \\, \\prod_{i=1}^{r} (1+\\varepsilon_{i} W_{P_{i}}) \\, \\equiv \\, \n\\Bigl(\\Phi_{2,M} \\, \\big| \\, \\prod_{p|M} (1-W_{p}) \\Bigr) \\,\\bigl|\\,{\\cal V }_{2,m\/M}^{\\,(M)} \\, , \n\\end{equation}\nwhere $M$ is the product of the primes~$p_{i}$ for which~$\\varepsilon_{i}=-1$ and ``$\\, \\equiv \\, $''\nmeans that the two sides of the equation differ by a weak Jacobi form. Because of the \nmultiplicativity of the Hecke-like operators $U_{d}$, $V_{t}$ and $W_{m_{1}}$, this formula \ncan be rewritten as \n\\beta\n\\Phi_{2,m} \\, \\big| \\, \\prod_{i=1}^{r} (1+\\varepsilon_{i} W_{P_{i}}) \\, \\equiv \\, \n\\Phi_{2,M} \\, \\big| \\,  \\prod_{\\varepsilon_{i}=-1} (1-W_{p_{i}}) {\\cal V }_{P_{i}\/p_{i}}^{(p_{i})} \\, \\big| \\, \n\\prod_{\\varepsilon_{i}=+1} {\\cal V }_{P_{i}}^{(1)} \\, , \n\\end{equation}\nso by induction on $r$, it suffices to show that if $P=p^{\\nu} || m$ then\n\\beta\\label{induction}\n\\Phi_{2,m} \\! \\mid \\! (1\\+W_{P}) \\, \\equiv \\, \\Phi_{2,m\/P}  \\! \\mid \\! {\\cal V }_{2,P}^{(1)} \\, , \\qquad \n\\Phi_{2,m} \\! \\mid \\! (1\\, - \\, W_{P}) \\, \\equiv \\, \\Phi_{2,mp\/P}  \\! \\mid \\! (1-W_{p}) \\, {\\cal V }_{2,P\/p}^{(p)} \\, .\n\\end{equation} \nFor this, it is enough to show that in each the difference of the left-hand side and the right-hand side \nhas shadow zero. \n\nFor the first equation in \\eqref{induction}, eq.~\\eqref{phiCcfs} implies that \nthe ``correction term'' for the left-hand side is given by \n\\begin{eqnarray}\\label{LHS1}\n&& C(\\Phi^{C}_{2,m} | (1+W_{P}) \\, ; \\Delta , \\ell \\; \\mod \\; {2m})  =  C(\\Phi^{C}_{2,m} \\, ; \\Delta , \\ell \\; \\mod \\; {2m}) + C(\\Phi^{C}_{2,m} \\, ; \\Delta , \\ell^{*} \\; \\mod \\; {2m}) \\cr\n&& \\; =  \\begin{cases} D (\\delta^{(2m)}_{D,\\ell} + \\delta^{(2m)}_{D,-\\ell} + \\delta^{(2m)}_{D,\\ell^{*}}\n+ \\delta^{(2m)}_{D,-\\ell^{*}}) & \\text{if $\\Delta=-D^{2}$, for some $D \\in \\mathbb{N}$} \\\\\n0 & \\text{if $-\\Delta$ is not a perfect square}\n\\end{cases} \n\\end{eqnarray}\nwhere $\\ell^{*} \\; \\mod \\; {2m}$ is defined by \n$\\ell^{*} \\equiv  -\\ell \\; \\mod \\; {2P}$, $\\ell^{*} \\equiv +\\ell \\; \\mod \\; {2m\/P}$, and the notation~$\\delta^{(n)}_{a,b}$\nmeans $\\delta_{a,b \\; \\mod \\; {n}}$.\nThe correction term for the right-hand side, on the other hand, is given \n(since ${\\cal V }_{2,P}^{(1)}=V_{2,P}-pV_{2,P\/p^{2}} U_{p}$, where the second term is omitted if $p^{2} \\nmid P$) \nby\n\\begin{eqnarray}\n&& \\sum_{d|(\\frac{\\Delta+\\ell^{2}}{4m},\\ell,P)} \\! d \\, C\\Big(\\Phi^{C}_{2,\\frac{m}{P}} \\, ; \n\\frac{\\Delta}{d^2}, \\frac{\\ell}{d}\\; \\mod \\; {2\\frac{m}{P}} \\Big) \\; \\quad -  p \\sum_{d|(\\frac{\\Delta+\\ell^{2}}{4m},\n\\ell,\\frac{P}{p^{2}})} \\! d \\, C\\Big(\\Phi^{C}_{2,\\frac{m}{P}} \\, ; \\frac{\\Delta }{p^{2} d^2}, \n\\frac{\\ell}{pd}\\; \\mod \\; {2\\frac{m}{P}} \\Big) \\cr\n&& \\quad \\= \\sum_{d|(\\frac{\\Delta+\\ell^{2}}{4m},\\ell,P)} \n\\sum_{\\lambda^{2} = - \\Delta  \\atop  \\lambda \\equiv \\ell \\; \\mod \\; {\\frac{2md}{P}}} |\\lambda| \\; \\quad - \n\\sum_{d|(\\frac{\\Delta+\\ell^{2}}{4m},\\ell,\\frac{P}{p^{2}})} \n\\sum_{\\lambda^{2} = - \\Delta  \\atop \\lambda \\equiv \\ell \\; \\mod \\; {\\frac{2md}{P}}} |\\lambda| \\; , \\cr\n&& \\quad \\= \\sum_{d \\in \\{P,P\/p\\} \\atop d|(\\frac{\\Delta+\\ell^{2}}{4m},\\ell,P)} \n\\sum_{\\lambda^{2} = - \\Delta  \\atop  \\lambda \\equiv \\ell \\; \\mod \\; {\\frac{2md}{P}}} |\\lambda|  \\; ,\n\\end{eqnarray}\nwhere the last line holds because the terms in the first sum in the second line \nwith~$d|Pp^{-2}$ cancel with the terms in the second sum.\nThe terms with~$d=P$ in the third line give the first two terms $\\sqrt{-D}=\\pm \\ell^{*}$ \nof the right hand side of~\\eqref{LHS1},\nwhile the terms with~$d=P$ in the third line give the second two terms $\\sqrt{-D}=\\pm \\ell^{*}$.\n\n\nThe correction term for the left-hand side of the second equation in \\eqref{induction} is given \nby the the right-hand side of~\\eqref{LHS1} with the signs of the last two terms changed. \nThis is now equal to zero unless~$(\\ell,m)=1$, in which case it is equal to~$\\sqrt{-\\Delta}$ (resp.~$-\\sqrt{-\\Delta}$) \nif $D \\equiv \\pm \\ell \\; \\mod \\; {2m}$ (resp.~$D \\equiv \\pm \\ell^{*} \\; \\mod \\; {2m}$).\nThe correction term for the right-hand side, on the other hand, is given by\n\\begin{eqnarray}\\label{RHS2}\n&& \\sum_{d|(\\ell,\\frac{P}{p},\\frac{\\Delta+\\ell^{2}}{4m})} \\! d \\, \\bigg( C\\Big(\\Phi^{C}_{2,mp\/P} \\, ; \n\\frac{\\Delta}{d^2} \\,, \\frac{\\ell}{d}\\; \\mod \\; {\\frac{2mp}{P}} \\Big) \\, - \\, C\\Big(\\Phi^{C}_{2,mp\/P} \\, ; \n\\frac{\\Delta}{d^2} \\,, \\frac{\\ell^{*}}{d}\\; \\mod \\; {\\frac{2mp}{P}} \\Big) \\bigg)\\cr\n&& \\qquad \\=  \\sum_{d|(\\ell,\\frac{P}{p},\\frac{\\Delta+\\ell^{2}}{4m})} \n\\Big(\\sum_{\\lambda^{2} = - \\Delta  \\atop  \\frac{\\lambda}{d} \\equiv \\frac{\\ell}{d} \\; \\mod \\; {\\frac{2mp}{P}}}\n\\, - \\, \\sum_{\\lambda^{2} = - \\Delta  \\atop  \\frac{\\lambda}{d} \\equiv \\frac{\\ell^{*}}{d} \\; \\mod \\; {\\frac{2mp}{P}}} \\Big) |\\lambda| \\; ,\n\\end{eqnarray}\nwhere $\\ell^{*} \\; \\mod \\; {2mp\/P}$ is now defined by \n$\\ell^{*} \\equiv  -\\ell \\; \\mod \\; {2p}$, $\\ell^{*} \\equiv +\\ell \\; \\mod \\; {2m\/P}$.\nIf~$p|\\ell$, then~$\\ell^{*}=\\ell$ and the expression vanishes, so that only~$d=1$\ncontributes in the first summation. The condition~$4m|(\\Delta+\\ell^{2})$ combined \nwith~$\\lambda \\equiv \\ell \\; \\mod \\; {2mp\/P}$ implies~$\\lambda \\equiv \\ell \\; \\mod \\; {2m}$ (and similarly \nfor $\\ell^{*}$ since $\\ell^{2} \\equiv \\ell^{*2} \\; \\mod \\; {2mp\/P}$), thus proving the second \nequation in~\\eqref{induction}.  \\hfill $\\square$\n\n\n\n\n\\bigskip\n\n\n\n\\noindent {\\bf Proof of Theorem~\\ref{QmOG}}:\nBy M\\\"obius inversion, the mock Jacobi forms~$\\Phi_{2,m}$  associated to any choice of minimal meromorphic \nJacobi forms~$\\varphi_{2,m}$ can be represented in the form~\\eqref{Phi0mimp}, where~$\\Phi_{2,m}^{0}$ \nis defined by \n  \\beta \\label{defPhi02m}\n  \\Phi^{0}_{2,m} \\= \\sum_{d^{2} \\mid m} \\mu(d) \\; \\Phi_{2,m\/d^{2}}\\,|\\,U_{d}  \\, .\n  \\end{equation}\nWe have to show that $\\Phi^{0}_{2,m}$ can be chosen to be primitive and to have optimal growth. \n\n\nTo prove the second of these statements, it is enough to show that $\\Phi^{0}_{2,m}| u_{p}$ has no shadow \nfor all primes $p$ with $p^{2} | m$. \nThen, by Theorem~\\ref{Emstar}, the weak elliptic form~$\\Phi^{0}_{2,m}$ can be chosen (by adding \na weak Jacobi form) to have optimal growth.\nWriting the square divisors of~$m$ as $d^{2} p^{2i}$ with $p \\nmid d$ and noting that $\\mu(d p^{i})=0$ for $i>1$ \nand that $U_{dp}=U_{d} U_{p}$, $U_{d} u_{p} = u_{p} U_{d}$, and $U_{p} u_{p} = p$, \nwe find from~\\eqref{defPhi02m} that  \n\\begin{eqnarray}\n\\Phi^{0}_{2,m} \\! \\mid \\! u_{p} &\\=& \\sum_{d^{2}\\mid m, \\, p \\nmid d} \\mu(d) \n\\big[\\Phi_{2,m\/d^{2}} \\! \\mid \\! U_{d} \\, - \\, \\Phi_{2,m\/d^{2} p^{2}} \\! \\mid \\! U_{d}  \\! \\mid \\! U_{p} \\big] \\mid u_{p} \\cr\n& \\= & \\sum_{d^{2}\\mid m, \\, p \\nmid d} \\mu(d) \n\\big[\\Phi_{2,m\/d^{2}} \\! \\mid \\! u_{p} \\, - \\, p \\, \\Phi_{2,m\/d^{2} p^{2}} \\big] \\mid U_{d} \\, , \\nonumber\n\\end{eqnarray}\nso it suffices  to show that $\\Phi_{2,m} | u_{p}$ and $p\\, \\Phi_{2,m\/p^{2}}$ \nhave the same shadow for any $m$ with~$p^{2}|m$. But the shadow of $\\Phi_{2,m} = \\varphi^{\\rm F}_{2,m}$ \nis determined by $\\varphi^{C}_{2,m} = \\widehat \\Phi_{2,m} - \\Phi_{2,m}$ (cf.~comments at the end of \\S\\ref{MockDecomposition}), \nand by \\eqref{phiCcfs}, we have: \n\\beta\nC(\\varphi^{C}_{2,m} \\, ; \\Delta p^{2}, \\, rp  \\; \\mod \\; {2m} ) \n \\=  \\begin{cases} p |\\lambda| & \\text{if $\\Delta = - \\lambda^{2}$, \\,  $\\lambda \\equiv r \\; \\mod \\; {2m\/p}$} \\, ,  \\\\ \n\\; 0 & \\text{otherwise} \\, .\n\\end{cases}\n\\end{equation}\nThen, from \\eqref{Ucech2}, we have:\n\\begin{eqnarray}\nC(\\varphi^{C}_{2,m}|u_{p} \\, ; \\Delta, \\, \\ell \\; \\mod \\; {2m\/p^{2}}) \n&\\= &\\sum_{r \\; \\mod \\; {2m\/p} \\atop r \\equiv \\ell \\; \\mod \\; {2m\/p^{2}}} \nC(\\varphi^{C}_{2,m} \\, ;  \\Delta p^{2} ,\\, rp  \\; \\mod \\; {2m} ) \\; , \\cr\n& \\= & \\begin{cases} p |\\lambda| & \\text{if $\\Delta= - \\lambda^{2}$, \\,  $\\lambda \\equiv \\ell \\; \\mod \\; {2m\/p^{2}}$} \\, ,  \\\\ \n\\; 0 & \\text{otherwise} \\, .\n\\end{cases} \\cr\n& \\= & p \\, C(\\varphi^{C}_{2,m\/p^{2}} \\, ; \\Delta,\\ell) \\, .\n\\end{eqnarray}\nThis completes the proof of the fact that $\\Phi^{0}_{2,m}$ can be chosen to have optimal growth. \nNow, by formula~\\eqref{Ucech2}, we have that~$\\Phi^{0}_{2,m} | u_{t}$, and hence \nalso~$\\Phi^{0}_{2,m} | u_{t}|U_{t}$, is holomorphic for $t >1$, so by replacing~$\\Phi^{0}_{2,m}$ \nby~$\\pi^{\\rm prim}(\\Phi^{0}_{2,m})$ as defined in~\\eqref{primproj} we can also assume \nthat~$\\Phi^{0}_{2,m}$ is primitive.  \\hfill $\\square$\n\n\n\\bigskip\n\n\\noindent {\\bf Proof of Theorem~\\ref{chooseKm}}:\nThe proof of Theorem~\\ref{Emstar} given in~\\S\\ref{optimalforms} the following discussion already showed \nthat $J_{2,m}^{\\rm OG}\/J_{2,m}$ is one-dimensional and that any representative ${\\cal K}_{m}$ of this \nquotient space, if we normalize it to have $c({\\cal K}_{m}\\,; 0,0)=2$, has polar coefficients as given in \nthe statement of Theorem~\\ref{chooseKm}. Since all the polar coefficients $C({\\cal K}_{m} \\, ; -1,r)$ are equal, \nwe have that~${\\cal K}_m|W_{m_{1}}-{\\cal K}_m \\in J_{2,m}$ for all $m_{1} || m$, so (by averaging) we can \nchoose~${\\cal K}_m$ to be invariant under all Atkin-Lehner operators. Finally, we can make~${\\cal K}_{m}$ \nprimitive by the argument used at the end of the proof of Theorem~\\ref{QmOG}. \n\\hfill{$\\square$}\n\\vspace{0.2cm}\n\n\\noindent {\\bf Remark}. For square-free $m$, since all holomorphic Jacobi forms are in fact cusp forms, ${\\cal K}_{m}$\nis unique up to Jacobi cusp forms, and its coefficients with $\\Delta=0$ are unique. In this case the only such \ncoefficient, up to translation, is~$c(0,0)$. If~$m$ is not square-free, there is more than one coefficient~$c(n,r)$ such \nthat~$\\Delta=0$. Notice that $r^{2}\\equiv 0 \\; \\mod \\; {4m} \\Leftrightarrow (r,2m)=2m\/t$ with~$t^{2}|m$.\nFor each~$t$ with $t^{2}|m$, we have \n\\beta\\label{ckm00}\n\\sum_{j \\; \\mod \\; {t}} C({\\cal K}_{m} \\, ; 0, \\frac{2mj}{t} \\; \\mod \\; {2m}) \\= \n\\begin{cases} -2   & \\text{if $t=1$} \\, , \\\\ \\; 0  & \\text{if $t>1$}\\, . \\end{cases} \n\\end{equation}\n(For~$t=1$, the sum on the left reduces to the single term~$c({\\cal K}_{m}\\,;0,0)=-2$, and for~$t>1$,\nit equals~$C({\\cal K}_{m}|u_{t}; 0,0)$, which vanishes because~${\\cal K}_m | u_{t} \\in J_{2,m\/t^{2}}$.)\nWe can choose the~$\\Delta=0$ coefficients arbitrarily subject to this one constraint, and once we have \nmade a choice, then~${\\cal K}_{m}$ is unique up to cusp forms even in the non-square-free case. \nFor instance, at the expense of introducing denominators, we could fix~${\\cal K}_{m}$ uniquely up to cusp forms by requiring \nthat~$C({\\cal K}_{m};0,r)$ depends only on the gcd of~$r$ and~$2m$. (This would correspond to choosing~$a={\\frac12}$\nin the first and third tables at the end of~\\S\\ref{Statements}.) Another choice, more similar to our ``$\\star = {\\rm II}$''\ncondition, would be to choose~$C({\\cal K}_{m};0,r)=0$ for all~$r$ with~$4m|r^{2}$ and~$r \\nmid 2m$, in which case\nthe remaining coefficients~$C({\\cal K}_{m};0,2m\/t)$ ($t^{2}|m$) are uniquely determined by~\\eqref{ckm00}. (This would \ncorrespond to choosing~$a=1$ in the first of the tables at the end of~\\S\\ref{Statements}.) The reader can easily\nwork out the complete formula for all of the coefficients~$C({\\cal K}_{m};0,r)$ for either of these two special choices. \n\n\nOur next task is to prove Theorem~\\ref{PolarCoeffs}. For this we need to calculate the residues of~$\\Phi_{2,m}$,\nusing the results of the previous subsection. We therefore have to apply the formula~\\eqref{Rm1m2ans} to the mock modular \nform~$\\Phi_{2,m}=\\varphi_{2,m}^{\\rm F}$. (Note that the residue of this form will be independent \nof the specific choice of~$\\varphi_{2,m}$, since any two choices differ by a Jacobi form\nand the residues of Jacobi forms vanish.) Since the completion of $\\Phi_{2,m}$ \ninvolves the weight~1\/2 unary theta series~$\\vartheta_{m,\\ell}^{0}\\,$, the first step is to \ncompute the scalar products of these theta series. \n\\begin{proposition} \\label{thetapeter}\nFor $m \\in \\mathbb{N}$, and $\\ell_{1} , \\ell_{2} \\in {\\mathbb Z}\/2m{\\mathbb Z}$, we have\n\\beta\n(\\vartheta_{m, \\, \\ell_{1}}^0 \\,, \\,\\vartheta_{m, \\, \\ell_{2}}^0) \\= \\frac{1}{2 \\sqrt{m}} \\big( \n\\delta_{\\ell_{1}, \\ell_{2} \\! \\; \\mod \\; {2m}}\n+ \\delta_{\\ell_{1}, -\\ell_{2} \\! \\; \\mod \\; {2m}} \\big)\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFormula~\\eqref{RS} gives \n\\begin{equation*}\n(\\vartheta_{m, \\, \\ell_{1}}^0 \\,, \\,\\vartheta_{m, \\, \\ell_{2}}^0) \\= \\frac{1}{2} \\, \\text{Res}_{s={\\frac12}} \n\\biggl( \\sum_{r>0} \\frac{(\\delta_{r, \\, \\ell_{1}\\;(2m)}+ \\delta_{r, \\, -\\ell_{1}\\;(2m)})\n(\\delta_{r, \\, \\ell_{2}\\;(2m)}+ \\delta_{r, \\, -\\ell_{2}\\;(2m)})}{(r^{2}\/4m)^{s}} \\biggr) \\; . \n\\end{equation*}\nThe propostition follows.\n\\end{proof}\n\n\n\n\n\n\n\\begin{proposition} \\label{RPhi2mval}\nLet $\\varphi_{2,m}$ be any meromorphic Jacobi form of weight 2 and index $m$ with pole\n$1\/(2 \\pi i z)^{2}$ and its translates, and let $\\Phi_{2,m}$ be its finite part $\\varphi^{\\rm F}_{2,m}$.\nThen\n\\beta \\label{Rm1m2}\nR_{m_{1},m_{2}}(\\Phi_{2,m}) \\= \\frac{m_{1}+m_{2}}{12}  \\, .\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy equation \\eqref{phi2mhat}, \nthe $\\ell^{\\rm th}$ component  $g_{\\ell}$ of the shadow for $\\Phi_{2,m}$ is $-\\sqrt{\\frac{m}{4 \\pi}} \\vartheta^{0}_{m,\\ell}$. \nWhen $(m_{1}, m_{2})=1$, $R_{m_{1},m_{2}}(\\Phi_{2,m}) = {\\cal R}[\\Phi_{2,m} \\! \\mid \\! W_{m_{1}}]$. \nTheorem~\\ref{Rphishad} then gives:\n\\begin{eqnarray} \n{\\cal R}[\\Phi_{2,m} \\! \\mid \\! W_{m_{1}}] &\\= & \\frac{\\sqrt{m}}{\\sqrt{4 \\pi}} \\frac{\\sqrt{\\pi}}{6}  \n\\sum_{\\ell  \\! \\; \\mod \\; {2m}} \\bigl( \\vartheta^{0}_{m,\\ell}, \\vartheta_{m,\\ell^{*}}^{0} \\bigr) \\cr\n& \\= & \\frac{1}{24} \\, \\sum_{\\ell  \\! \\; \\mod \\; {2m}}  \\big( \\delta_{\\ell, \\ell^{*} \\! \\; \\mod \\; {2m}}\n+ \\delta_{\\ell, -\\ell^{*} \\! \\; \\mod \\; {2m}} \\big) \\cr\n&\\=& \\frac{1}{12} \\, (m_{1}+m_{2}) \\, . \\nonumber \n\\end{eqnarray}\nIn this calculation, we used Proposition~\\ref{thetapeter} to get the second line. For \nthe third line, recall that $\\ell^{*}$ is defined by \n$(\\ell^{*} \\equiv -\\ell \\; \\mod \\; {2m_{1}}, \\ell^{*} \\equiv \\ell \\; \\mod \\; {2m_{2}})$, so that \n$\\ell^{*} \\equiv \\ell \\; \\mod \\; {2m} \\Leftrightarrow m_{1} \\! \\mid \\! \\ell$ (which is true for~$2 m_{2}$ values of~$\\ell$ in \n${\\mathbb Z}\/2m{\\mathbb Z}$), \nand similarly, $\\ell^{*} \\equiv -\\ell \\; \\mod \\; {2m} \\Leftrightarrow m_{2} \\! \\mid \\! \\ell$ (which is true for~$2m_{1}$ values of~$\\ell$).  \nWhen $(m_{1},m_{2})=t>1$, we have \n$R_{m_{1},m_{2}}(\\Phi_{2,m}) = R_{m_{1}\/t,m_{2}\/t}(\\Phi_{2,m} \\! \\mid \\! u_{t})$.\nAs we have shown in the proof of Theorem~\\ref{QmOG},  \nthe shadow of $\\Phi_{2,m} \\! \\mid \\! u_{t}$\nis the same as the shadow of $t \\Phi_{2,m\/t^{2}}$, so $R_{m_{1}\/t,m_{2}\/t}(\\Phi_{2,m} \\! \\mid \\! u_{t}) = \nt R_{m_{1}\/t,m_{2}\/t}(\\Phi_{2,m\/t^{2}}) = t (\\frac{m_{1}}{t} + \\frac{m_{2}}{t})\/12 = (m_{1}+m_{2})\/12$ by the \nspecial case already proved.\n\\end{proof}\n\nThe reader might find it helpful to compare the statement of this proposition with the table \nin \\S\\ref{choosephi} presenting the data for the case~$m=6$, \nin which the function $\\Phi=\\Phi_{2,6}$ (in any of its three versions I, II, or III) has \n\\begin{eqnarray*} \nR_{1,6}(\\Phi) &\\= 2 C(\\Phi ; -1, \\, \\pm 1\\! \\; \\mod \\; {12}) + C(\\Phi ; 0, \\, 0\\! \\; \\mod \\; {12}) \\= 7\/12 \\, , \\\\\nR_{2,3}(\\Phi) &\\= 2 C(\\Phi ; -1, \\, \\pm 5\\! \\; \\mod \\; {12}) + C(\\Phi ; 0, \\, 0\\! \\; \\mod \\; {12}) \\= 5\/12 \\, . \n\\end{eqnarray*}\n\n\\vspace{0.2cm}\n\n\\noindent {\\bf Proof of Theorem~\\ref{PolarCoeffs}}: We can now complete the proof of Theorem~\\ref{PolarCoeffs}}\nand its corollary. For $m$ square-free, any choice of~$\\Phi=\\Phi_{2,m}^{0}$ in ${\\cal E }_{m}^{\\rm OG}$, and \nany~$r$ with $r^{2} \\equiv 1 \\; \\mod \\; {4m}$, we have \n\\beta\n2 C(\\Phi; \\, -1, \\ell) + C(\\Phi; \\, 0,0) \\= {\\cal R}[\\Phi \\!\\mid \\! W_{m_{1}}] \\= \\frac{m_{1}+m_{2}}{12} \n\\end{equation}\nby Proposition~\\ref{RPhi2mval}, where $m=m_{1} m_{2}$ is the decomposition of~$m$ for which~$r^{*}=1$.\nSince $C(\\Phi^{\\rm III}; 0,0)=0$, this proves Theorem~\\ref{PolarCoeffs} and also gives the \ncorresponding formulas \n$$\nC(\\Phi^{\\rm I}; -1,r) \\= -\\frac{(m_{1}-1)(m_{2}-1)}{12} \\, , \\qquad  C(\\Phi^{\\rm II}; -1,r) \\= -\\frac{1}{12} \\, \n\\biggl( \\frac{m_{1}+m_{2}}{2} \\, - \\, \\prod_{p|m} \\frac{p+1}{2} \\biggr)  \n$$\nfor $\\Phi^{\\rm I}$ and $\\Phi^{\\rm II}$, as well as formula~\\eqref{Qmpolar} for the polar coefficients of~${\\cal Q }_{M}$\nwhen $\\mu(M)=1$. \nIf~$m$ is not square-free, then the solutions~$r \\; \\mod \\; {2m}$ of~$r^{2} \\equiv 1 \\; \\mod \\; {4m}$\nare still in~1:1 correspondence with the decompositions of~$m$ into coprime factors~$m_{1}$ and~$m_{2}$. \nTo compute the corresponding Fourier coefficient of~$\\Phi_{2,m}^{0}$, we look at each term \nin~\\eqref{defPhi02m} separately, writing each~$d$ as $d_{1}d_{2}$ with~$d_{i}^{2}|m_{i}$ and noting \nthat $\\frac{m_{1}}{d_{1}^{2}}=(\\frac{r-1}{2},\\frac{m}{d^{2}})$,  $\\frac{m_{2}}{d_{2}^{2}}=(\\frac{r+1}{2},\\frac{m}{d^{2}})$,\nafter which~\\eqref{Phi0IIIcoeffs} follows from~\\eqref{Rm1m2} and the fact that ${\\cal R}[\\varphi  \\! \\mid \\! U_{d}] \\= {\\cal R}[\\varphi]$. \n\\hfill $\\square$\n\n\n\\vspace{0.2cm}\n\n\\noindent {\\bf Proof of Theorem~\\ref{Wt1OSP}}: It is enough to show that any weak Jacobi form of optimal growth \nis actually holomorphic since, as we have seen in Theorem \\ref{Skorthm}, there are no holomorphic Jacobi forms \nof weight~1. To show this, we use the ``residue operator on weight~1 forms'' defined by\n\\beta\\label{RasReswt1}  {\\cal R}^{(1)}[\\varphi]\\=\\text{Res}_{\\tau=\\infty}\\bigl(\\varphi'(\\tau,0)\\,d\\tau\\bigr) \\= \\sum_{r\\in{\\mathbb Z}} r \\, c_{\\varphi}(0,r)\\,\\end{equation}\ninstead of~\\eqref{RasRes}, where, as usual,~$\\varphi'(t,z) = \\frac{1}{2 \\pi i} \\frac{d}{dz} \\,\\varphi(t,z)$. \nThis residue vanishes if~$\\varphi\\in \\widetilde J_{1,m}$, since then~$\\varphi'(\\cdot, 0) \\in M_{2}^{!}$. \nIf, further, $\\varphi$ has optimal growth, then $2  c_{\\varphi}(0,1)={\\cal R}^{(1)}[\\varphi]=0$, so~$C_{\\varphi}(-1,1)$ vanishes. \nBy the same argument with~$\\varphi$ replaced by $\\varphi$ acted upon by each of the Atkin-Lehner operators, \none has that~$C_{\\varphi}(-1,r)=0$ for all~$r$ with~$r^{2} \\equiv -1 \\; \\mod \\; {4m}$,  \nand thus~$\\varphi \\in J_{1,m}$. \\hfill $\\square$\n\n\n\\section{Quantum black holes and mock modular forms \\label{MockforDyons}}\n\n\nWe now have all the ingredients required to address the questions posed\nin the introduction. These answers follow from applying our results of\n\\S{\\ref{MockfromJacobi}}, in particular theorem~\\ref{Thmdouble}, to the quarter-BPS partition \nfunction of the the ${\\cal N}=4$ theory. It turns out that in this case, many of the mathematical objects \nin the theorem simplify. A reader who has not gone through  the details of \\S{\\ref{MockfromJacobi}} could still\nfruitfully read this section to obtain the relevant results for this example.\n\nIn particular,  we show in  \\S{\\ref{Degen}} that the immortal black hole degeneracies \n extracted from the asymptotic partition function (\\ref{inverse3})   are \nFourier coefficients of a (mixed) mock Jacobi form. The completion transforms as a true Jacobi form and obeys a first order partial differential equation~\\eqref{holanom}\nwhich can be understood as a  holomorphic anomaly equation. \nIn \\S{\\ref{Anomaly}} we discuss the physical origin of the meromorphy of the asymptotic counting function in this example and how it is related\nto the noncompactness of the microscopic brane dynamics. \nIn \\S{\\ref{Mlimit}} we analyze the contour corresponding to the M-theory limit to embed our analysis in the context of  $AdS_{3}\/CFT_{2}$ holography near the horizon of a black string. \nWe conclude in \\S{\\ref{generalization}} with  comments and open problems.\n\n\n\\subsection{Mock Jacobi forms, immortal black holes, and $AdS_{2}$ holography \\label{Degen}}\n\n\nWe first discuss the consequences of the  $S$-duality symmetry and its relation to the spectral flow symmetry for the moduli-independent immortal  degeneracies  $d^{*}(n,\\ell,m)$ defined by \\eqref{inverse3}.\nRecall that under the S-duality transformations\n\\beta\\label{11b1}\n\\begin{pmatrix} 1 & b  \\\\\n    0 & 1\\end{pmatrix} \\ \n\\end{equation} \nwith integer $b$, \nthe S-modulus \\eqref{Sdef} transforms as:\n\\beta\nS  \\to  S +  b \\, . \n\\end{equation}\nwhich means that the axion transforms as $a \\to a +b$. Since the axion arises from a dimensional reduction of  the 3-form field in the M-theory frame (discussed in \\S\\ref{Mlimit}), this subgroup \\eqref{11b1} of $S$-duality \nhas its origins  in large gauge transformations of the 3-form field. When the axion transforms as above, the effective theta angle changes and as a result the charge vector of the dyon  transforms as $(Q, P) \\rightarrow (Q + P b, P)$  due to the Witten effect \\cite{Witten:1979ey}.\nConsequently, the charge-invariants transform as\n\\beta\\label{nmltrans}\n n  \\to  n + m b^2 + b \\ell \\, , \\qquad  \\ell  \\to  \\ell + 2 m b \\, , \\qquad m \\rightarrow m .\n\\end{equation}\nwhich are nothing but the spectral flow transformations.  Note that $m$ is left fixed by this subgroup of S-duality whereas  other elements of the S-duality group would change $m$. Thus,  only this subgroup is expected to act on the sector with a fixed $m$. The covariance of the spectrum under these $S$-duality transformations implies\n\\beta\nd(n, \\ell, m) |_{S, \\mu} \\= d(n + m b^2 + b, \\ell + 2 m b, m) | _{S + b, \\mu} \\, .\n\\end{equation}\nSince the degeneracies of immortal black holes are independent of the moduli by definition, \nthey are actually invariant under~$S$-duality:\n\\beta\\label{spectral-immortal}\nd^*(n + m b^2 + b, \\ell + 2 m b, m) \\=  d^* (n, \\ell, m) \\, .\n\\end{equation} \n{}From the mathematical point of view, this is precisely the action of the elliptic part of the Jacobi transformations \\eqref{elliptic} on the Fourier coefficients of a Jacobi (or mock Jacobi) form. \n\nThe immortal degeneracies  $d^{*}(n,\\ell,m)$ are computed using the attractor contour~\\eqref{contourattract}, \nfor which the imaginary parts of the potentials are proportional to the charges and scale as \n\\begin{equation}\\label{Attractorcontour}\n\\Im (\\sigma) \\= 2n\/\\varepsilon \\, , \\qquad \n\\Im (\\tau) \\= 2m\/\\varepsilon \\, ,\\qquad \n\\Im (z) \\= -\\ell\/\\varepsilon \\, ,\n\\end{equation}\nwith $\\varepsilon$ very small and positive. In other words,  \n\\begin{eqnarray}\\label{attractor2}\n|p| = \\lambda^{2n}\\, , \\quad |q| = \\lambda^{2m} \\, , \\quad  |y| = \\lambda^{-\\ell} \\, , \\qquad \\text{with $\\lambda  = \\exp(-2\\pi\/\\epsilon) \\rightarrow 0$}\n\\end{eqnarray}\non the attractor contour.\nWe  assume that $n > m$ without loss of generality, otherwise one can \nsimply exchange  $\\sigma$ and $\\tau$. Moreover, using the spectral flow symmetry \\eqref{spectral-immortal} we can always bring $\\ell$ to the window $0 \\leq \\ell <2m$ for the immortal degeneracies. \n\nTo extract the Fourier coefficient from  the product representation \\eqref{final2} of the counting \nfunction~\\eqref{igusa} using \\eqref{attractor2}, we need to expand a typical term in the product of the form\n\\beta\n\\frac{1}{(1- p^{r}q^{s} y^{t})^{2C_0(4rs -t^{2})}} \\, .\n\\end{equation}\nwhere $2 C_0$ is the Fourier coefficient of the elliptic genus of a single copy \nof $K3$. Depending on whether~$|p^{r}q^{s}y^{t}|$ is less than one or greater than one, we expand \nthe denominator using \n\\beta\\label{xsmall}\n\\frac{1}{1-x} \\= \\begin{cases} 1+ x + x^{2} + \\ldots & \\text{for $|x| < 1$}\\,, \\\\\n -(x^{-1} + x^{-2} + x^{-3} + \\ldots)  & \\text{for $|x| > 1$}\\,.  \n  \\end{cases}\n\\end{equation}\nWe see that it is not \\textit{a priori} obvious  that expanding first in $p$  is the same as using the \nattractor contour because if  the powers of $t$ are sufficiently large and positive, the attractor contour may \ncorrespond to expanding some terms effectively in   $p^{-1}$ instead of in $p$. \n\nTo address this question  we now show that for the terms that appear in the product representation,  $|p^{r}q^{s}y^{t}|$ is always less than one using the fact that\n\\beta\nC_{0}(\\Delta) = 0 \\qquad \\text{for $\\Delta  <-1$} \\, .\n\\end{equation}\nTo begin with, the $p$-dependent terms with $\\Delta=0, -1$   arise only for $s=0$ and  are of the form\n\\beta\n\\frac{1}{(1-py )^{2}(1- py^{-1})^{2} (1- p^{r})^{20}} \\,\n\\end{equation}\nThese involve only single  powers $y$ or $y^{-1}$ and positive powers of $p$.  The absolute value of \n$py$ or $py^{-1}$ for the attractor contour goes as $\\lambda^{2m\\pm l}$  which is always much less than one for $0 \\leq |l|  < 2m$.   \nThe remaining  $p$-dependent terms with $s > 0$  have $\\Delta= 4rs -t^{2}>0 $ and $r >0$. The absolute value of $p^{r}q^{s} y^{t}$  goes as\n\\beta\n\\lambda^{2nr+ 2ms -\\ell t} \\, .\n\\end{equation}\nThe exponent is most negative when $\\ell$ takes its maximum positive value $\\ell_{max} = \\sqrt{2mn}$ and $t$ takes its  maximum positive value $t_{max}=\\sqrt{2rs}$. \nFor black hole configurations with a smooth  horizon,   $4mn- \\ell ^{2}>0 $ because the classical area of the horizon is the square-root of this quantity.  Hence the most negative value of the exponent  would be\n\\beta\n2nr+ 2ms - 2 \\sqrt{mnrs} \\= 2(\\sqrt{nr} -\\sqrt{ms})^{2} >  0 \\, , \n\\end{equation}\nand the absolute value of $p^{r}q^{s} y^{t}$ is always less than or equal to\n\\beta\n\\lambda^{ 2(\\sqrt{nr} -\\sqrt{ms})^{2}} \n\\end{equation}\nwhich is always much less than one. This implies that to extract the attractor Fourier coefficients, we expand all terms in the product in the small $p$ expansion using~\\eqref{xsmall} thus giving us the Jacobi form \n$\\psi_{m}$ of~\\eqref{reciproigusa}  as a  partition function in the remaining  two chemical potentials.\n\n\n\nThe degeneracies extracted from $\\psi_{m}(\\tau, z)$ still experience many wall-crossings, and we should now \nuse the rest of the attractor contour to analyze the immortal part of the degeneracies of \n$\\psi_{m}$.  We do this in two steps by first performing the inverse Fourier transform in $z$, \nand then in $\\tau$. \nFor the first step, we need to specify a contour for the imaginary part of $z$ relative to that of $\\tau$. \nThe conditions~\\eqref{Attractorcontour}  on the imaginary parts of $(\\tau,z)$ implies that \n$\\Im(z) = -\\frac{\\ell}{2m} \\Im(\\tau)$. \nSince by moving the contour in a purely horizontal direction we do not cross any poles, \nwe can add an arbitrary real part to $z$, which we can choose so that $z = -\\ell\\tau\/2m$. This \ngives the result of the first integral to be:\n\\beta\\label{fmlpsim}\nf_{m,\\ell}^*(\\tau)  \\=  e^{-\\pi i l^2 \\tau \/ 2m} \\int_{P}^{P+1} \\psi_m(\\tau,z) \\, e^{- 2 \\pi i \\ell z} \\, dz \\, , \n\\qquad P = - \\ell \\tau\/2m \\, . \n\\end{equation}\nWe then have to perform a~$\\tau$ integral with the contour specified by the attractor \nvalues~\\eqref{Attractorcontour}, but since the integrand~$f_{m,\\ell}^{*}(\\tau)$ is holomorphic \nin~$\\tau$, the answer is independent of the contour. We thus get the degeneracies $d^{*}(n,\\ell,m)$ to be:\n\\beta\\label{dstarfml}\nd^*(n,\\ell,m) \\= \\int e^{\\pi i \\ell^2 \\tau \/ 2m}\\, f_{m,\\ell}^*(\\tau) \\, d\\tau \\, , \n\\end{equation}\nwhere the integral is over an interval of length~$1$ for the fixed imaginary value~\\eqref{Attractorcontour} of $\\tau$. \n\n\nTo extend to other values of $\\ell$, we use the spectral flow invariance \\eqref{spectral-immortal} and sum over all values of~$\\ell$ to get a canonical partition function in the two variables~$(\\tau,z)$. \nBecause of the factor~$e^{\\pi i \\ell^2 \\tau \/ 2m}$ in \\eqref{dstarfml}, the sum over spectral-flow images yields the function~$\\vartheta_{m, l}(\\tau,z)$. \nPutting all this together, and comparing the above equations \\eqref{fmlpsim},~\\eqref{dstarfml} \nwith our definition \\eqref{phiF} of the finite part of a meromorphic Jacobi form, \nwe see that the single centered black hole degeneracies are Fourier coefficients of \nprecisely the {\\it finite} part of $\\psi_{m}$ as defined in~\\S\\ref{MockfromJacobi}:\n\\beta\n\\sum_{\\ell \\, {\\rm mod} \\, (2m)} f_{m,\\ell}^*(\\tau) \\, \\vartheta_{m,\\ell}(\\tau,z) \n\\= \\psi_{m}^{\\rm F}(\\tau,z) \\, .\n\\end{equation}\nWe see that the seemingly unconventional charge-dependent choice for the contour \nin~\\S\\ref{MockfromJacobi} for defining the finite part is completely natural from the point \nof view of the attractor mechanism (indeed, it was motivated there by these physical considerations).\nAccording to our analysis in~\\S\\ref{MockfromJacobi}, this finite part $\\psi_{m}^{\\rm F}$ is a {\\it mock} Jacobi form. \nThis is one of the main conclusions of our analysis, so we summarize the entire discussion by the \nfollowing concise statement:\n\\begin{quote}\n{\\it The degeneracies of single centered (immortal) black holes with magnetic charge \ninvariant $M^2\/2=m$ are Fourier coefficients of  a mock Jacobi form of index $m$.} \n\\end{quote}\n\nWe stress here that the converse is not true. \nThe single centered black hole solution exists only when the \ndiscriminant $4mn-\\ell^2$ is positive, and its degeneracy grows exponentially \nfor large values of the discriminant. On the other hand, as we saw in \\S\\ref{Jacobi}, \nany Jacobi form with exponential growth necessarily has negative discriminant states. \nThis means that the partition function for single centered black holes alone with fixed magnetic charge \nand varying electric charge cannot have good modular properties. Indeed the partition \nfunction $\\psi_{m}^{\\rm F}$ does contain negative discriminant states for example for $n<m$. These low-lying states \ndo not correspond to single centered black holes~\\cite{Dabholkar:2007vk,  Sen:2011mh}. \n\n \nWe now turn to the physical interpretation of the polar part $\\psi_{m}^{\\rm P}(\\tau,z)$ of the meromorphic Jacobi form defined in~\\eqref{phiPdoub}. As mentioned in the introduction,   the only multi-centered gravitational configurations that contribute \nto the supersymmetric index of quarter-BPS dyons in our~${\\cal N}=4$ string theory have exactly two centers \\cite{Dabholkar:2009dq} each of which is a half-BPS state.  At a wall, one of these two-centered configurations decays. A basic decay at a wall-crossing  is when  one center is purely electric and the other is \npurely magnetic. Other decays are related to this basic decay by duality \\cite{Sen:2007vb}.\nFor this   two-centered configuration, with one electric and one magnetic center, the indexed \npartition function after summing over the electric charges is:\n\\beta\\label{basicmulti}\np_{24}(m+1) \\cdot \\frac{1}{\\eta^{24}(\\tau)} \\, \\cdot \\sum_{\\ell > 0} \\, \\ell \\, y^{\\ell} \\, , \n\\end{equation}\nwhere the first factor is  the degeneracy of the half-BPS magnetic center, \nthe second factor is the partition function of the electric center ~\\cite{Dabholkar:1989jt},  \n and the third factor is the partition function counting the dimensions of the $SU(2)$  multiplets with  angular momentum $(\\ell -1)\/2$ contained in the electromagnetic field generated by the two centers~\\cite{Denef:2000nb} for which $N \\cdot M = \\ell$.\nThe third factor is an expansion of the function\n\\beta\\label{basicmulti2}\n \\frac{y}{(1-y)^{2}}\n \\end{equation}\nin the range  $|q| < |y| < 1$ (see equation~\\eqref{BBm}). \nHere we already see the basic wall-crossing of the theory because different \nways of expanding this meromorphic function for~$|y|<1$ and~$|y|>1$ will give different degeneracies. Other wall-crossings  are related to this basic wall-crossing by S-duality which as we have explained is nothing but spectral flow for a given fixed $m$. Thus  the full partition function that captures all wall-crossings  is obtained by `averaging'  over the spectral-flow images of ~\\eqref{basicmulti2}. \nThis averaging  is nothing but the  averaging operation defined in \\eqref{AP1AP2} and  \\eqref{AP1AP2again}\n\\beta\n \\AvB m{ \\frac{y}{(y-1)^2} } \\, = \\, \\frac{q^{ms^2 +s}y^{2ms+1}}{(1 -q^s y)^2}  \\,\n\\end{equation}\nIt is now easy to see that this averaging over  spectral-flow  images gives \nexactly the polar part~$\\psi_{m}^{\\rm P}$ of the meromorphic Jacobi form~$\\psi_{m}$:\n\\beta\\label{Tm} \n \\frac{p_{24}(m+1) }{\\eta^{24}(\\tau)} \\, \\sum_{s\\in{\\mathbb Z}} \\, \\frac{q^{ms^2 +s}y^{2ms+1}}{(1 -q^s y)^2} \n \\= \\psi_{m}^{\\rm P}(\\tau,z) \\, .\n\\end{equation}\nThis follows from the fact that the poles of~$\\psi_{m}(\\tau,z)$ are exactly at~$z \\in {\\mathbb Z} \\tau +{\\mathbb Z}$ and the residue at~$z=0$ is $$\\frac{p_{24}(m+1) }{\\eta^{24}(\\tau)} \\, .$$  as mentioned in  \\eqref{psifromphim}.\n\n\nWe thus see that  the decomposition theorem~(\\ref{Thmdouble}) \nas applied to our dyon partition function~$\\psi_{m}(\\tau,z)$ has a natural physical interpretation. It is simply the statement  that  the full partition function is a sum of \nits immortal and decaying constituents:\n\\beta \\label{singlepart} \n\\psi_{m}(\\tau,z) \\; = \\;  \\psi_{m}^{\\rm F}(\\tau,z) \\, + \\, \\psi^{\\rm P}_{m}(\\tau,z) \\, .\n\\end{equation}\nThe nontrivial part is of course  the implication of such a decomposition for modularity. By separating part of the function $\\psi_{m}$, we have evidently broken modularity, and $\\psi_{m}^{\\rm F}(\\tau,z)$  \nis not a Jacobi form any more. However,  the decomposition theorem~(\\ref{Thmdouble}) guarantees \nthat~$\\psi_{m}^{\\rm F}$ still has a very special modular behavior in that \nit is a mock Jacobi form. The mock behavior can be summarized by the following partial differential equation\n obeyed by its completion: \n\\beta \\label{holanom}\n \\tau_2^{3\/2} \\; \\frac{\\partial} {\\partial \\overline{\\tau}}   \\, \\widehat \\psi_{m}^{\\rm F}(\\tau,z) \\; = \\; \n\\sqrt{\\frac{m}{8 \\pi i}} \\; \\frac{ p_{24}(m+1)}{ \\Delta(\\tau)} \\,  \n \\sum_{\\ell \\, {\\rm mod} \\, (2m)}  {\\overline{\\vartheta_{m,\\ell}(\\tau,0)}} \\, \\vartheta_{m,\\ell} (\\tau,z) \\, .\n\\end{equation}\nNote that~$\\psi_{m}^{\\rm P}$ also admits a completion~$\\widehat{\\psi_{m}^{\\rm P}}$ which is modular.\nHowever, the function~$\\widehat{\\psi_{m}^{\\rm P}}$, unlike~$\\widehat{\\psi_{m}^{\\rm F}}$, is meromorphic,\nand therefore has wall-crossings. We refer the reader to the summary at the end of~\\S\\ref{MockDecomposition} \nfor a detailed discussion of the relationship between meromorphy (in~$z$), holomorphy (in~$\\tau$) and modularity. \n\n\nLet us now discuss the implications of these results for $AdS_{2}\/CFT_{1}$ holography. \nThe near horizon  geometry of a supersymmetric black hole is a two-dimensional anti de Sitter space $AdS_{2}$ and the dual theory is a one-dimensional conformal field theory  $CFT_{1}$. The partition function of this $CFT_{1}$ is nothing but the integer $d^*(n, \\ell, m)$ which gives the number of microstates of the black hole~\\cite{Sen:2008yk,Sen:2008vm}. Our results show that that these integers are the Fourier coefficients of a mock modular form and as a result there is a hidden modular symmetry. Such a modular symmetry would make it possible to use\npowerful techniques such as the  Hardy-Ramanujan-Rademacher expansion to process  the integers $d^*(n, \\ell, m)$ into a form that can be more readily identified with a bulk partition function of string theory in $AdS_{2}$. There has been recent progress in evaluating the bulk partition function using localization techniques which makes it possible to study the subleading nonperturbative corrections in the Rademacher expansion in a systematic way~\\cite{Dabholkar:2010uh,Dabholkar:2011ec,Gupta:2012cy,Murthy:2013xpa}. It would be interesting to see if the additional terms in the Rademacher expansion  arising from the `mock' modular nature can be given a physical bulk interpretation. \n\n\n\\subsection{Meromorphy, mock modularity,  and noncompactness \\label{Anomaly}}\n\n\nFrom the mathematical point of view, we have seen how the mock modularity of\n$\\psi_{m}^{F}$ is a consequence of meromorphy of the Jacobi form $\\psi_{m}$. In\nthe ${\\cal N}=4$ example above, we can understand the physical origin of this\nmeromorphy as coming from the non-compactness of the underlying brane dynamics. To see this\nmore clearly, we write the meromorphic Jacobi form $\\psi_m(\\tau, z)$ as\n  \\beta\\label{meroformtwo}  \\psi_m(\\tau, z) =  \\frac 1{\\varphi_{-2,1}(\\tau,z)} \\, \\frac{1}{\\Delta(\\tau)} \\,\n   \\, \\chi (\\tau,z; \\textrm{Sym}^{m+1}(K3)) \\, . \\end{equation}\nThe three factors in the above equations have the following physical interpretation~\\cite{David:2006yn}. \nThe function $\\psi_m(\\tau, z)$\ncan be viewed as an indexed partition function of the following $(0, 4)$ SCFT\n  \\beta\\label{scft}  \\sigma(\\textrm{TN}) \\times \\sigma_L(\\widetilde K\\text{-}n) \\times \\sigma(\\textrm{Sym}^{m+1}(K3)) \\,. \\end{equation}\nEach CFT factor has a natural interpretation from the microscopic derivation of\nthe dyon counting formula \\cite{David:2006yn} using the 4d-5d lift~\\cite{Gaiotto:2005gf}. \nThe $\\sigma_L(\\widetilde K$-$n)$ factor is the purely left-moving bosonic\nCFT associated with $\\widetilde K$-$n$ bound states of the Kaluza-Klein monopole\nwith momentum. The $ \\sigma(\\textrm{TN})$ factor is a $(4, 4)$ SCFT associated\nwith the motion of the center of mass of motion in the Kaluza-Klein background in\ngoing from five dimensions to four dimensions in the 4d-5d lift. The\n$\\sigma(\\textrm{Sym}^{m+1}(K3))$ is the $(4, 4)$ SCFT that is associated with the\nfive-dimensional Strominger-Vafa $Q1$-$Q5$-$n$ system. Note that the SCFT\n$\\sigma(\\textrm{TN})$ has a \\textit{noncompact} target space and the double pole\nof  $1\/{\\varphi_{-2,1} (\\tau,z)}$ can be traced back to this noncompactness.\n\nGoing over from the partition function to the Fourier coefficients corresponds to going over from the\ncanonical ensemble with fixed $z$ to the micro-canonical ensemble with fixed $\\ell$.\nThe Fourier coefficient  is supposed to count the number of\nright-moving ground-states in this fixed charge sector. Right-moving states with nonzero energy cancel in pairs in this\nindex. Now, counting ground states for a compact theory is a well-defined operation.\nHowever, when the target-space is noncompact, the wavefunctions are not properly\nnormalized without putting an infrared regulator in the target space. In the\npresent context, the essential physics of the noncompactness is captured by the point particle motion of the center of\nmass in the Taub-NUT geometry.  A simple regulator can be introduced by turning on the axion\nfield  which according to \\eqref{hetBrel} corresponds  in the Type-IIB frame  to  taking the modulus $U_{B} = U_{1B}+ iU_{2B}$ to have a small \nnonzero real part $U_{1B}$ \\cite{David:2006yn}. This introduces a nonzero potential on the\nTaub-NUT geometry proportional to $U_{1B}^2 U_{2B}$  that regulates the infrared behavior to\ngive a well defined counting problem \\cite{Gauntlett:1999vc,Pope:1978zx}.\nHowever, the index thus computed depends on the sign of $U_{1B}$ and jumps as $U_{1B}$ goes\nfrom being slightly negative to slightly positive.  Indeed the partition function for these modes is precisely the function\n  \\beta\\label{pole-part}  \\frac{y}{(1-y)^2}  \\end{equation}\nwhich can be expanded either in power of $y$ or in powers of $y^{-1}$ depending on whether $U_{1B}$ is positive or negative. \nFor $\\ell$ positive, there are $\\ell$ normalizable states  when $U_{1B} < 0$ and none when $U_{1B} >0$.  This is the essential source\nof the wall-crossing behavior in this context.  We should emphasize that even though turning on $U_{1B}$ appears to make the\nproblem effectively compact, this is true only up to some maximal $\\ell_{max}$ which scales as  \n$U_{1B}^2 U_{2B}$ \\cite{Gauntlett:1999vc,Pope:1978zx}.\nFor a given $\\ell $ positive, one can always make the problem well-defined with a compact target space by choosing the moduli\nso that $\\ell_{max}$ is larger than the $\\ell$ of interest. The partition function in that case is well-defined without any poles and goes as\n  \\beta   y + 2y^{2} + \\ldots +  \\ldots + \\ell_{max} y^{\\ell_{max}}  \\, . \\end{equation}\nHowever,  if one wishes to compute the canonical partition function that sums over all $\\ell$, one has to essentially take \n$U_{1B}^2 U_{2B}$ to go to infinity which is what leads to the pole at $y=1$ in (\\ref{pole-part}).\nThis in essence is the origin of the meromorphy of the canonical counting \nfunction of the asymptotic states, and therefore indirectly also of the mock modularity of the immortal near-horizon counting function.\n\nAs we have seen, this meromorphy of the asymptotic counting function $\\psi_{m}$ implies that the counting \nfunction of the near-horizon immortal states cannot be both modular and holomorphic\nat the same time. One can either have the holomorphic function $\\psi_{m}^{\\rm F}$ which is not quite \nmodular, or a modular function $\\widehat \\psi_{m}^{\\rm F}$ which is not quite holomorphic. \n{}From this point of view, the shadow can be regarded as a `modular anomaly' in $\\psi_{m}^{\\rm F}$ \nor as a `holomorphic anomaly' in the completion~$\\widehat \\psi_{m}^{\\rm F}$. \n\nA natural question that arises from this discussion is if the  mock Jacobi form and the holomorphic anomaly can be given a direct physical interpretation rather than arrive at it via a meromorphic Jacobi form as we have done. To address this question, it is useful to view the problem from the  perspective of $AdS_{3}$ holography in M-theory as we now discuss.\n\n\\subsection{The M-Theory limit  and $AdS_{3}$ holography \\label{Mlimit}}\n\n {}From the  perspective of $AdS_{2}\/CFT_{1}$ holography discussed above, there is no \\textit{a priori} reason why the black hole degeneracy should have  anything to do with modularity. It is only when we can  view the black hole as an excitation of a black string that we can have an \\textit{a priori} physical expectation of modularity. This is because the near horizon geometry of a black string is a three-dimensional anti de Sitter space $AdS_{3}$. The Euclidean thermal $AdS_{3}$ has a conformal torus as a boundary whose complex structure parameter is $\\tau$. The partition function of the dual boundary conformal field theory $CFT_{2}$ depends on on $\\tau$.  In this framework the  modular symmetry is identified with the $SL(2, \\mathbb{Z})$ mapping class group of the boundary \ntorus~\\cite{Maldacena:1998bw}, whereas the  elliptic symmetry is identified with large gauge transformations of the \n3-form field~\\cite{deBoer:2006vg} as we have already discussed in \\S\\ref{Degen}. \n  \nWe now describe how the quarter-BPS black holes considered  thus far can be regarded  as excitations of a black string in an appropriate M-theory frame. This  will enable us to make contact with $AdS_3\/CFT_2$ holography. The M-theory framework is \n natural also from the point of view of generalizing  these considerations to BPS black holes  in  ${\\cal N}=2$  compactifications on a\ngeneral Calabi-Yau manifold $X_6$.\n\nTo go to the M-theory frame, we consider the $T^2$ to be  a product\nof two circles $S^1 \\times {\\tilde S}^1$ and consider the charge configuration \\eqref{final charges} in the Type-IIB frame.   We first  $T$-dualize the $\\tilde S$ circle to the $\\hat S^{1}$ circle to go to the Type-IIA frame and then use  mirror symmetry of $K3$ to map the Type-IIB configuration \\eqref{final charges} to  a Type-IIA configuration of D4-F1-n-NS5 charges. We then  lift it to M-theory  to obtain a charge configuration consisting of M5-M2 bound states carrying momentum along $S^{1}$.  \nThis configuration now has  M5-branes \nwrapping  $D \\times S^{1}$ where $D$ is a divisor in $K3 \\times \\hat T^2$, an M2-brane wrapping the $\\hat T^2 = \\hat S^{1} \\times S^{1}_{m}$,  with some momentum  along  $S^{1}$. This  is a precisely a  configuration of the kind considered by Maldacena, Strominger, and Witten \\cite{Maldacena:1997de}. Since all M5-branes wrap $S^{1}$ and the momentum is along $S^{1}$, it is natural  flip $S^{1}_{m}$ and $S^1$  and regard the $S^{1}$ as the new M-circle.   In the limit when the radius $R$ of the  new M-circle $S^{1}$ is very large, the low-lying excitations of this M5-brane are described by the MSW string wrapping the M-circle. \nTo write the final configuration in the M-theory frame,  let $C^1$ to be the homology 2-cycle of the $\\hat T^2$, and let   $C^2$ and $C^3$ be two 2-cycles in $K3$ with intersection matrix \n  \\beta \\left(  \\begin{array}{cc}  0 & 1 \\\\  1 & 0 \\\\  \\end{array}\\right) \\, . \\end{equation}\nThe intersection number of these three 2-cycles is \n  \\beta \\int_{K3 \\times \\hat T^2} C^1 \\wedge C^2 \\wedge C^3 =1 \\,. \\end{equation}\nLet  $\\{ D_{a}\\}$ be the  4-cycles   dual to $ \\{ C^{a} \\}$,  $D_{a} \\cap C^{b} = \\delta_{a}^{b}$ .\nIn this basis, the M5-brane is wrapping the  4-cycle $$D =  \\sum_{a=1}^3p^{{a}} D_{a} $$ with $p^1=\\tilde K$, $p^2=Q_1$ and $p^3=Q_5$.\nThe M2-brane charges are $q_2=q_3=0$ and $q_1=\\tilde n$, and M-momentum or equivalently the D0-brane charge from the Type-IIA\nperspective is $q_0=n$.  Using this notation we see that\n\\begin{myitemize} \n\\item The electric charges in \\eqref{final charges} map to $n$ units of momentum along the M-circle\n$S^1$, and $\\tilde K$  M5-branes wrapping   $D_1 \\times S^1$.\n  \\item The magnetic charges in \\eqref{final charges} map to $Q_1$ M5-branes  wrapping $D_2 \\times S^1$, $Q_5$\nM5-branes  wrapping $D_3 \\times S^1$,  and $\\tilde n$ M2-branes wrapping $\\hat T^2$.\n\\end{myitemize}\nIn summary, the charge configuration of our interest is\n  \\beta\\label{M-charges}  \\Gamma = \\left[ \\begin{array}{c}  N \\\\ M \\end{array}\\right] \n     \\= \\left[ \\begin{array}{cccc} 0,& n ; & 0, & \\tilde K \\\\ Q_1 ,& \\tilde n; & Q_5, & 0\\\\ \\end{array} \\right]_{B}  \n     \\=  \\left[ \\begin{array}{cccc} 0,& q_0 ; & 0, &  p^1 \\\\ p^2 ,&  q_1; & p^3, & 0\\\\ \\end{array} \\right]_{M}\\, \\end{equation}\nReduction of this charge configurations along $S^{1}$ gives a configuration in Type-IIA frame consisting of only D4-D2-D0-branes.\n\n\nSince the effective MSW string wraps the M-circle, we have to take the radius $R$  of this circle to be large  keeping other scales fixed to obtain an $AdS_3$ near-horizon geometry of a long black string.\nThis implies that  in the original  heterotic frame in which we have labeled the charges \\eqref{chargevector},  the radius $R$ goes to infinity keeping other scales \nand moduli fixed. Since the 2-torus in the heterotic frame  is also a simple product of two circles $ S^1 \\times \\tilde S^{1}$,  the moduli in \\eqref{metric-B}   take the form\n  \\beta \\label{M-moduli} G_{ij} =  \\left( \\begin{array}{cc} \\tilde R^2 & 0\\\\  0 & R^2 \\\\  \\end{array} \\right) \\, , \\quad B_{ij} =0 \\end{equation}\nwhich implies that for the three moduli $S, T, U$ we have \n  \\beta S\\= a + i \\frac{R\\tilde R}{g_6^2}\\,, \\qquad T \\= i R \\tilde R\\,, \\qquad U = i\\,\\frac{R}{\\tilde R} \\,,  \\end{equation}\nwhere $g_6$ is the six-dimensional string coupling constant of heterotic compactified on $T^4$ and we have allowed a nonzero\naxion field $a$. This means in particular that\n  \\beta S_2 \\sim R, \\quad T_2 \\sim R, \\quad U_2 \\sim R \\end{equation}\nand for the charge configuration (\\ref{M-charges}),  the central charge matrix scales as\n  \\beta\\label{MSW-central-charge}  {\\cal Z} \\sim  \\left( \\begin{array}{cc} R &   0 \\\\ \n     0 & 0 \\\\ \\end{array} \\right) + \\frac 1{R}\\left( \\begin{array}{cc} 2n \\quad & \\quad   \\ell + {a g_6^2}\/{\\tilde R} \\\\ \n     \\ell + {a g_6^2}\/{\\tilde R} & \\quad 2m + (Q_1^2\/{\\tilde R}^2 + Q_5^{2 }{\\tilde R}^2 )\\\\ \\end{array} \\right) \\,  \\, . \\end{equation}\nFrom the contour prescription (\\ref{contour}), we conclude that \nthe imaginary parts  for the Fourier integral scale as \n  \\begin{eqnarray}\\label{Mcontour} \\Im(\\sigma) &\\sim & R \\\\ \\Im (\\tau) &\\sim & 1\/R \\\\ \\Im (z) & \\sim & 1\/R \\, . \\end{eqnarray}\nTherefore, in the region of the moduli space corresponding to the M-theory limit,  $p := \\exp{(2\\pi i \\sigma)}$ is becoming much\nsmaller compared to $q:= \\exp{(2\\pi i \\tau)}$ and $y:= \\exp{(2\\pi i z)}$. Hence one must first expand around $p=0$:\n  \\beta\\label{defpsin} \\frac 1{\\Phi_{10}(\\Omega)} \\= \\sum_{m=-1}^{\\infty} p^m \\psi_m (\\tau,z) \\ . \\end{equation}\nSince the function $\\Phi_{10}(\\Omega)$ has double zeros at $z=0$, the function $\\psi_m(\\tau,z)$ is meromorphic with double\n poles at $z=0$ and its translates.\n\nWe conclude that the  function $\\psi_{m}(\\tau,z)$  can  be interpreted as the asymptotic  partition function for counting the BPS excitation of this MSW M5-brane  for fixed value of the \nmagnetic charge $M^2\/2= m$ and is  a \\textit{meromorphic} Jacobi \nform of weight $-10$ and index $m$.\n This is the essential new ingredient compared to other situations encountered earlier such as the partition function of the  \nD1-D5 string \\cite{Strominger:1996sh} which is a \\textit{holomorphic} weak Jacobi form\\footnote{Here we mean holomorphic \nor meromorphic in the $z$ variable that is conjugate to $\\ell$. This  should not be confused with the nomenclature used  in \\S\\ref{Jacobi} while discussing  growth conditions \\eqref{holjacobi}.}.\n\n Note that $\\psi_{m}(\\tau,z)$ is the asymptotic counting function  in the canonical ensemble for a fixed chemical potential $z$ where one sums\nover all M2-brane charges $\\ell$ for a fixed chemical potential $z$. To obtain the micro-canonical partition function for a fixed M2-brane\ncharge, one has to perform the Fourier integral in $z$.  There is an infinity of double poles in the $z$ plane at $z = \\alpha\\tau $ for\n$\\alpha \\in {\\mathbb Z}$. The $z$ contour can be made to lie between the poles at $(\\alpha-1)\\tau$ and $\\alpha\\tau$ depending on how $\\Im(z)$ compares with\n $\\Im(\\tau)$ given  the $O(1\/R)$ terms in the central charge matrix (\\ref{MSW-central-charge}).   It is easy to see that by varying\nthe moduli, the contour can be made to cross all poles. Thus, there are an infinity of walls still accessible in the regime\nwhen the radius of the M-circle becomes large. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=6cm]{tikz-walls}\n\\end{center}\n\\caption{\nThe diagram on the left shows the moduli space projected onto the $\\Sigma$ upper-half plane divided into chambers separated \nby walls where  a quarter-BPS state decays into two half-BPS states. The M-theory limit, shown on the right, corresponds to taking\n$\\Im(\\Sigma)$ to be very large. In this limit several walls are no longer visible.  There are still an infinite number of\nwalls  which can be crossed by varying $\\Re(\\Sigma)$. Spectral flow transformation maps one chamber to another chamber.\n}\n\\label{walls}\n\\end{figure}\n \nTo display this graphically it is useful to define a complex scalar field $\\Sigma = \\Sigma_1 + i \\Sigma_2$ by\n \\beta \n {{\\cal Z } } \\sim   \\frac 1{\\Sigma_2}\\left(\\begin{array}{cc} |\\Sigma | ^2& \\Sigma_1 \\\\  \\Sigma_1& 1\\\\   \\end{array}\\right) \\, . \\end{equation}\nFor fixed charges, the MSW region which corresponds to taking $\\Sigma_2$ large with $\\Sigma_1$ fixed corresponding to \nvanishing axion. Varying $\\Sigma_1$  allows one to access the infinite number of chambers separated by walls \nas shown in Fig. \\ref{walls}. \nDegeneracies in these different chambers can be obtained simply by choosing the Fourier contour in $z$ for an appropriate\nvalue of $\\alpha$. Note that the field $\\Sigma_1$ depends both on the moduli  and    on the charges and thus varies as\nmoduli are varied for fixed charges or as the charges are varied for fixed moduli.   In particular, even for fixed values of\nmoduli, one can cross all walls by varying $\\ell$ appropriately.  \n\nWith this embedding into M-theory, we  now outline what we regard as a consistent physical picture to interpret $\\widehat \\psi_{m}^{F}$ even though we do not yet fully understand the details.   In the  limit of large radius $R$, we effectively have a five-dimensional theory obtained from compactifying M-theory on $K3 \\times \\hat T^{2}$. For a given a set of M5-brane charges $\\{p^{i}\\}$ at asymptotic infinity,  the five-dimensional supergravity   admits a single-centered black string solution carrying magnetic charges $\\{p^{i}\\}$. The near horizon geometry of this solution is an $AdS_{3}$ geometry where all vector multiplet moduli of the five-dimensional supergravity are fixed to their attractor values.   Following the usual reasoning of holography, we  expect that the superconformal field theory describing the low energy excitations of the M5-brane will be dual to this near horizon $AdS_{3}$. The indexed partition function of this SCFT is expected to describe the BPS gravitational configurations inside the $AdS_{3}$.  \n\nWhat is the indexed partition function of this SCFT? It  cannot possibly be the meromorphic Jacobi form $\\psi_{m}$ that exhibits many wall-crossings, because  all vector-multiplet moduli are fixed in the attractor $AdS_{3}$ geometry and no wall-crossings are possible.   It cannot possibly be $\\psi_{m}^{F}$, because it is not modular as would be expected from the symmetry under the $SL(2, \\mathbb{Z})$ mapping class group of the boundary torus. We propose that  the most natural  object to identify with   the indexed partition function of the $SCFT_{2}$ dual to the $AdS_{3}$ is the modular completion $\\widehat \\psi_{m}^{F}(\\tau, z)$.  It satisfies (in a rather nontrivial way) both requirements of being properly modular and  not having any wall-crossings. The non-holomorphy is not something that we would naively expect for the indexed partition function but this can be a reflection of noncompactness as we comment upon later. The quarter-BPS black holes discussed in this paper are  excitations inside this single $AdS_{3}$ represented as BTZ black holes and their degeneracies should then  be captured by the Fourier coefficients   of $\\psi_{m}^{F}(\\tau, z)$ which is indeed the case for $n>m$.\n\nThis \\textit{microscopic} picture is consistent with the \\textit{macroscopic}  analysis of\nmulti-centered black string solutions \\cite{deBoer:2008fk} of  five dimensional supergravity.  These solutions can be obtained by lifting  multi-centered black hole solutions of four-dimensional supergravity. According to this supergravity analysis, given a set of asymptotic magnetic charges $\\{p^{i}\\}$ of a wrapped M5-brane,  the single-centered $AdS_{3}$ throat described above is not the only allowed solution \\cite{deBoer:2008fk}.   There exist multi-centered solutions with multiple $AdS_{3}$ throats with the same total asymptotic charge.  Now,  as we have seen from the microscopic analysis, \n there are still  infinite chambers separated by walls visible in the M-theory limit as shown in Fig~\\ref{walls}. These chambers  can thus  correspond to chambers in which different   multi-string configurations exist as stable solutions.  The  wall-crossings can result  not from decays of multi-centered black holes inside a single $AdS_{3}$ throat but from the decays of these multi-string solutions. \nThus, it is reasonable to identify~$\\widehat\\psi_{m}^{F}$,  which has no wall-crossings, with the (indexed) partition function of the CFT dual to the single~$AdS_{3}$ throat.\n\n\\subsection{Open problems and comments\\label{generalization}}\n\nThe interpretation  of $\\widehat \\psi_{m}^{F}(\\tau, z)$ proposed above raises a number of interesting questions. The essentially new feature of our proposal  is the fact  that the modular partition function $\\widehat \\psi_{m}^{F}(\\tau, z)$  is nonholomorphic and correspondingly the holomorphic counting function $\\psi_{m}^{F}$ is mock modular. It would clearly be important to understand the physical origin of this nonholomorphy and mock modularity from the boundary and bulk perspectives. \n\n\n\\begin{itemize}\n\n\\item \n\\textit{Mock modularity from  the boundary perspective}:\n\\vspace{2mm}\n\n We have seen in \\S\\ref{Anomaly} that the mock modularity of $\\psi^{F}_{m}$  is tied  to the noncompactness of the asymptotic SCFT. A more interesting question is to understand this mock modularity directly from the point of view of the near horizon SCFT. It is natural to assume that the near horizon SCFT is also noncompact and this noncompactness is what is responsible for the mock modularity.  The connection between mock modularity\/nonholomorphy and noncompactness has been noted  earlier in physics. For example, the partition function\nof topological $SO(3)$  ${\\cal N} =4$ Yang-Mills theory on $\\mathbb{CP}^2$ is a mock modular form and it was suggested in \\cite{Vafa:1994tf} that it is  related to  the noncompactness of  flat directions in field space. The nonholomorphy of the elliptic genus has also been noted in the context of superconformal \nfield theories \\cite{Eguchi:2004yi,Troost:2010ud} with the noncompact $SL(2, \\mathbb{R})\/U(1)$ target \nspace. \n \n\nIn this context,  the nonholomorphy  can be understood in complete detail \\cite{Troost:2010ud, Ashok:2011cy} as a consequence of the continuum of spectrum of  the  noncompact SCFT. \nIt would be interesting to see if  the holomorphic anomaly \\eqref{holanom} of the completion of $\\widehat \\psi_{m}^{F}$ \ncan be understood from such a path integral perspective of the conformal field theory. \n\nIn the framework of  $AdS_{2}$ holography, the quantity of interest  is the integer $d^{*}(n, m, \\ell)$ which is identified with the partition function of the boundary $CFT$. Using the fact that the  completion $\\widehat \\psi_{m}^{F}(\\tau, z)$ is modular, it should be possible to develop a Rademacher-like expansion for the $d^{*}(n, m, \\ell)$ \nalong the lines of \\cite{BringOno, BringRicht, Bringmann:2010sd}. This method has been used to prove \nthat the black hole degeneracies are positive integers~\\cite{Bringmann:2012zr}. \n\n\n\n\\item \n\\textit{Mock modularity from the bulk perspective}:\n\nIf mock modularity is indeed a consequence of the noncompactness of the  boundary $CFT$, what are the implications of this noncompactness in the bulk $AdS_{3}$? In particular, if the spectrum of conformal dimensions is continuous, then the bulk theory must have a continuum of masses. \nPerhaps these are to be identified with some extended long-string like excitations. It would be interesting to see if the mock modularity of the elliptic genus can be understood  in terms of such bulk excitations. \nIndeed, recent investigations seem to indicate that the bulk string partition function in \nstring theories~\\cite{Harvey:2013mda} \n based on the non-compact $SL(2, \\mathbb{R})\/U(1)$ CFT~\\cite{Giveon:1999px, Murthy:2003es} \nshow a mock modular behavior. \n\nThe M-theory limit corresponds to the attractor contour  for $n>m$ and $0 \\leq \\ell <2m$ but the converse is not necessarily true, as explained in \\S\\ref{Degen}. For $n \\leq m$, we have the possibility that there will be multi-centered configurations. In particular, if  $n \\leq m$, then even if $\\ell <2m$, it is possible to have states with non-positive discriminant ($\\Delta = 4mn - \\ell^{2} \\leq 0)$.  Since, the discriminant must be positive for a black hole horizon, such states cannot possibly correspond to single-centered black holes and must be realized in supergravity  as multi-centered configurations if there are smooth solutions corresponding to them at all.  Hence, if  \n$\\widehat \\psi_{m}^{F}$ is indeed  the elliptic genus of a single-centered $AdS_{3}$, then there must be supergravity configurations corresponding to these  states for $n \\leq m$ that fit inside a single  $AdS_{3}$. \n\nThis microscopic prediction is  plausible  from the supergravity perspective. According to \\cite{deBoer:2008fk}, all two-centered black hole solutions where each center is a D4-D2 bound state,  are scaled out of  a single~$AdS_{3}$ throat. However,  the two-centered solutions that have one center carrying D6-brane charge and the other carrying anti-D6-brane charge can continue to exist as stable solutions inside a single $AdS_{3}$.  Such configurations can give the supergravity realization of the  negative discriminant states.  Further analysis  is required to verify these consequences of  our proposed interpretation of $\\widehat \\psi_{m}^{F}$. \n\n\\item\n\\textit{Generalization to ${\\cal N}=2$ compactifications}:\n\nOne motivation for the present investigation is to abstract some general lessons which\n could be applicable to the more\nchallenging case of dyons in ${\\cal N}=2$ compactifications on a general Calabi-Yau\nthree-fold. \nIn the ${\\cal N}=4$ case, the mock modularity of $\\psi_{m}^{F}$ was a consequence of meromorphy of $\\psi_{m}$. As we have seen in \\S\\ref{Anomaly}, this meromorphy is in turn a consequence of noncompactness of Taub-NUT factor $ \\sigma(\\textrm{TN})$ in   the asymptotic SCFT (\\ref{scft})\\footnote{Note that this is not to be confused with $\\mathbb{R}^{3}$ factor corresponding to the center of mass motion. It corresponds to the relative motion of the KK-monopole and the D1D5 system \\cite{David:2006yn} and thus  would  belong to  what is called the ``entropic\" factor\n\\cite{Minasian:1999qn} for  the MSW string in the M-theory frame.}.  This suggests that the SCFT dual to the near horizon $AdS_{3}$ is also noncompact. In the  ${\\cal N}=2$\ncontext also, there is no a priori reason to\nexpect that the MSW SCFT will be compact in general  everywhere in the moduli space.\nIf this SCFT is noncompact, then our results suggest that mock Jacobi forms would provide the right framework for making the  holographic\n$SL(2, {\\mathbb Z})$ symmetry manifest for the indexed partition function considered, for example, in \\cite{Dijkgraaf:2000fq,deBoer:2006vg,Manschot:2007ha}.  \nIn the ${\\cal N}=4$ case, the  mock modularity  is closely\ncorrelated with the wall-crossing phenomenon. The wall-crossing phenomenon in ${\\cal N}=2$ case corresponds to jumps in the\nDonaldson-Thomas invariants  and is known to be much\nmore complicated \\cite{Denef:2007vg, Konstevich:2008, Joyce:2008, Manschot:2010qz}. \nIt would be interesting though to define the  analog of the counting function for the immortal degeneracies in this case. Since these degeneracies  do not change under wall-crossing, such a counting function is likely to have an interesting mathematical interpretation. Moreover, it is expected to have nice (mock) modular properties from the considerations of holography discussed above.\n\n\n\\item\n\\textit{Mock modularity and indefinite theta series}:\n\nMock modular forms appear in Zwegers's work as members of  three quite different families of functions:\n\\begin{myenumerate}\n\\item Fourier coefficients of meromorphic Jacobi forms, \n\\item Appell-Lerch sums,  \n\\item Indefinite theta series. \n\\end{myenumerate}\nWe have seen that the first two families appear naturally in the context of black hole physics with \nnatural physical interpretations. This  suggests that the indefinite theta series could also play a role \nin this physical context. It  would be  interesting to develop this connection further. \nFor some earlier work in this direction, see~\\cite{Manschot:2009ia, Alexandrov:2012au}.\n\\end{itemize}\n\n     \n\n\\smallskip\n\n\n\\section*{Acknowledgments}\nWe would like to acknowledge useful conversations with Kathrin Bringmann, Miranda Cheng, Jan de\nBoer, Rajesh Gopakumar, Jeff Harvey, Jan Manschot, Shiraz Minwalla, Boris Pioline, Jan Troost, \nErik Verlinde, Sander Zwegers, and especially Ashoke Sen. We would also like to thank \nAlexander Weisse for technical help. It is a pleasure to thank the\norganizers of the 2007 workshop on `Black Holes and Modular Forms' in Paris and of the\n2008 workshop on `Number Theory and Physics at Crossroads' in Banff for hospitality\nwhere this research was initiated. The work of A.~D. was supported in part by the\nExcellence Chair of the Agence Nationale de la Recherche (ANR). The work of S.~M.\nwas supported by the ERC Advanced Grant no.~246974,\n{\\it ``Supersymmetry: a window to non-perturbative physics''}. \\\\\nv2: The work of A.~D.~was conducted within the framework of  the ILP LABEX (ANR-10-LABX-63)  supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02. S.~M.~would like to thank the SFTC for support from Consolidated grant number ST\/J002798\/1.\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn art investigations often multiple imaging systems, such as visual light photography (VIS), infrared reflectography (IRR), ultraviolet fluorescence photography (UV), and x-radiography (XR), are utilized. For instance, IR is used to reveal underdrawings, UV to visualize overpaintings and restorations, and XR to highlight white lead. \nSince the multi-modal images are acquired using different imaging systems, we have to take into account different image resolutions and varying viewpoints of the devices.\nThus, for a direct comparison on pixel level, image registration is crucial to align the multi-modal images. \n\n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/flowchart}\n\t\\caption{Our multi-modal registration and visualization tool CraquelureNetApp. Users can select the images and registration options in the GUI. The registration itself is performed fully automatically using the CraquelureNet, a CNN for multi-modal keypoint detection and description, which was ported from PyTorch to C++. Due to the high-resolution images of paintings, the CraquelureNet is applied patch-wise. The registration results are depicted inside the GUI and the users can interact with the visualizations, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot joint zooming of the views. \nImage sources: Workshop Lucas Cranach the Elder, Katharina of Bora (Detail), visual light and UV-fluorescence photographs, captured by Ulrike H\u00fcgle, Stiftung Deutsches Historisches Museum, Berlin, Cranach Digital Archive, KKL-No IV.M21b, all rights reserved\n}\n\\label{fig-01:overview}\n\\end{figure} \n\nImage registration methods for art imaging can mainly be split into intensity\\mbox{-,} control point- and feature-based approaches.\nThe intensity-based method of Cappellini~\\emph{et al}\\onedot~\\cite{CappelliniV2005} uses mutual information to iteratively register multispectral images.\nThe control point- and feature-based methods compute a geometric transform based on detected correspondences in both images.\nMurashov~\\cite{MurashovD2011} detects local grayscale maxima as control points in VIS-XR image pairs and applies coherent point drift~\\cite{MyronenkoA2010}.\nConover \\emph{et al}\\onedot~\\cite{ConoverDM2015} uses  Wavelet transform, phase correlation, and disparity filtering for control point detection in sub-images of VIS, IR, and XR.\nThe feature-based method of Zacharopoulos~\\emph{et al}\\onedot~\\cite{ZacharopoulosA2018} uses SIFT~\\cite{LoweDG2004} to align multispectral images of artworks.\nCraquelureNet~\\cite{SindelA2021} uses a convolutional neural network (CNN) to detect keypoints and descriptors based on branching points in the crack structure (craquelure) of the paint, as it is visible by all imaging systems, in contrast to the depicted image content that can be very different in the multi-modal images. \n\nMore and more museums or art projects provide the ability to inspect their artworks in interactive website viewers. Specifically for multi-modal images website viewer have been designed that allow a synchronized scrolling and zooming of the image views~\\cite{BoschProject,FransenB2020} or a curtain viewer~\\cite{BoschProject} that allows to interactively inspect multiple images in a single view. For these projects, the specific multi-modal images of artworks were pre-registered offline.\n\nFor the daily work, it would be practical for art technologists and art historians to have a tool, with which they can easily perform the registration themselves and also can interactively inspect the registered images.\nIn the field of medical 2D-2D or 3D-3D registration, there are open source tools such as MITK~\\cite{SteinD2010} that provides a graphical user interface (GUI) application for iterative rigid and affine registration and also a developer software framework. Since these software tools are very complex, domain knowledge is required to adapt the algorithms for multi-modal registration of paintings.\nThe ImageOverlayApp~\\cite{SindelA2020} is a small and easy to handle GUI application that allows the direct comparison of two registered artworks using image superimposition and blending techniques. However, here as well as in the online viewers, image registration is not provided.\n\nIn this paper, we propose a software tool for automatic multi-modal image registration and visualization of artworks. We designed a GUI to receive the user's registration settings and to provide an interactive display to show the registration results, such as superimposition and blending of the registered image pair, and both, individual and synchronized zooming and movement of the image views. As registration method, we integrate the keypoint detection and description network CraquelureNet into our application by porting it to C++. A quantitative evaluation and qualitative examples show the effective application of our software tool on our multi-modal paintings dataset and its transferability to our historical prints dataset.\n\n\\section{Methods}\nIn this section, the software tool for multi-modal registration and visualization is described.\n\n\\subsection{Overview of the Registration and Visualization Tool}\nIn Fig.~\\ref{fig-01:overview}, the main building blocks of the CraquelureNetApp are shown, consisting of the GUI, the preprocessing such as data loading and patch extraction, the actual registration method, which is split into the keypoint detection and description network CraquelureNet that was ported from PyTorch to LibTorch and into descriptor matching, homography estimation, and image warping, and the computation of visualizations of the registration results and its interactions with the user.\n \n\\subsection{Registration Method}\nCraquelureNet~\\cite{SindelA2021} is composed of a ResNet backbone and a keypoint detection and a keypoint description head. The network is trained on small $32 \\times 32 \\times 3$ sized image patches using a multi-task loss for both heads. The keypoint detection head is optimized using binary cross-entropy loss to classify the small patches into ``craquelure'' and ``background'', i.e. whether the center of the patch (a) contains a branching or sharp bend of a crack structure (``craquelure'') or (b) contains it only in the periphery or not at all (``background''). The keypoint description head is trained using bidirectional quadruplet loss~\\cite{SindelA2021} to learn cross-modal descriptors. Using online hard negative mining, for a positive keypoint pair the hardest non-matching descriptors are selected in both directions within the batch. Then, the positive distances (matching keypoint pairs) are minimized while the negative distances (hardest non-matching keypoint pairs) are maximized.\n\nFor inference, larger image input sizes can be fed to the fully convolutional network. Due to the architectural design, the prediction of the keypoint detection head and description head are of lower resolution than the input image and hence are upscaled by a factor of $4$ using bicubic interpolation for the keypoint heatmap and bilinear interpolation for the descriptors based on the extracted keypoint positions. Keypoints are extracted using non-maximum suppression and by selecting all keypoints with a confidence score higher than $\\tau_\\text{kp}$ based on the keypoint heatmap. \nSince the images can be very large, CraquelureNet is applied patch-based and the keypoints of all patches are merged and reduced to the $N_\\text{max}$ keypoints with the highest confidence score. Then, mutual nearest neighbor descriptor matching is applied and Random Sample Consensus (RANSAC)~\\cite{FischlerMA1981} is used for homography estimation~\\cite{SindelA2021}.\n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/gui}\n\t\\caption{The graphical user interface (GUI) of our CraquelureNetApp. The different elements of the GUI are marked in orange: the menu, the toolbar items for registration and visualization and the views.}\n\t\\label{fig-02:gui}\n\\end{figure} \n\nCraquelureNet~\\cite{SindelA2021} is completely implemented in Python using PyTorch and OpenCV. To transfer the trained PyTorch model to C++, we made use of the TorchScript conversion functionality and restructured the source code accordingly using a combination of TorchScript modules and Tracing. We have directly reimplemented the other parts of the registration pipeline in C++ using the same OpenCV functions as in Python.\n\nFor homography estimation, we provide the option to choose between different robust estimators of the OpenCV library: RANSAC~\\cite{FischlerMA1981} which is already used in~\\cite{SindelA2021,SindelA2022} is the default option. RANSAC is an iterative method to estimate a model, here homography, for data points that contain outliers. In each iteration, a model is estimated on a random subset of the data and is scored using all data points, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot the number of inliers for that model is computed. In the end, the best model is selected based on the largest inlier count and can optionally be refined.  \nUniversal RANSAC (USAC)~\\cite{RaguramR2013} includes different RANSAC variants into a single framework, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot different sampling strategies, or extend the verification step by also checking for degeneracy. We use the USAC default method of OpenCV 4.5.5 that applies iterative local optimization to the so far best models~\\cite{ChumO2003}.\nFurther, we enable two RANSAC variants which are also included in the OpenCV USAC framework:\nGraph-Cut RANSAC (GC-RANSAC)~\\cite{BarathD2018} uses the graph-cut algorithm for the local optimization and MAGSAC++~\\cite{BarathD2020} applies a scoring function that does not require a threshold and a novel marginalization method.\n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/menu_dialogs}\n\t\\caption{The menu and dialogs to configure registration and save options.}\n\t\\label{fig-03:menu_dialogs}\n\\end{figure} \n\n\\subsection{Graphical User Interface}\nThe GUI is implemented in C++ as a Qt5 desktop application. In Fig.~\\ref{fig-02:gui} the main GUI elements are marked in orange. The user can interact with the GUI using the menu or the toolbar that provides items specified for the registration and visualization task or can also directly interact with the views. The size of the views depends on the size of the application window.\n\n\\subsubsection{User interaction prior to registration:} To perform registration, two images have to be selected. They can either be chosen via the menu (Fig.~\\ref{fig-03:menu_dialogs}a) using a file opener or per drag and drop of the image file to the image view. The selected images are visualized in the two image views in the top row. \nPer click on the configure button in the menu (Fig.~\\ref{fig-03:menu_dialogs}b) or on the registration toolbar (``gear'' icon), a user dialog (Fig.~\\ref{fig-03:menu_dialogs}c) is opened to choose the registration options, such as patch size, number of maximum keypoints $N_\\text{max}$, image input size, method for homography estimation (RANSAC, USAC, GC-RANSAC, or MAGSAC++), whether to run on the GPU (using CUDA) or on the CPU and whether to visualize keypoint matches.\nCustom settings of patch size, input image size, and maximum number of keypoints are also possible when the predefined selections are not sufficient. The settings are saved in a configuration file to remember the user's choice. It is always possible to restore the default settings. The registration is started by clicking the run button in the menu (Fig.~\\ref{fig-03:menu_dialogs}b) or in the toolbar (``triangle'' icon).\n\n\\subsubsection{Visualizations and user interaction:} We include two different superimposition techniques to visually compare the registration results, a false-color image overlay (red-cyan) of the registered pair and a blended image overlay, similarly to~\\cite{SindelA2020}. After registration, the views in the GUI are updated with the transformed moving image (view 2) and the image fusions in the bottom row (view 3 and 4). The red-cyan overlay (view 4), stacks the reference image (view 1) into the red and the transformed moving image (view 2) into the green and blue channel. The blended image (view 3) is initially shown with alpha value 0.5. By moving the slider in the visualization toolbar, the user can interactively blend between both registered images. For the optional visualization of the keypoint matches, a separate window is opened that shows both original images with superimposed keypoints as small blue circles and matches as yellow lines using OpenCV.\n\nAdditionally to the image overlays, we have implemented a synchronization feature of the views that enriches the comparison. It can be activated by the ``connect views'' icon in the visualization toolbar. All interactions with one view are propagated to the other views. Using the mouse wheel the view is zoomed in or out with the focus at the current mouse position. Using the arrow keys the image view can be shifted in all directions. By activating the ``hand mouse drag'' item in the toolbar, the view can be shifted around arbitrarily. By pressing the ``maximize'' icon in the toolbar, the complete image is fitted into the view area preserving aspect ratio. This can also be useful after asynchronous movement of single views to reset them to a common basis. \n\nTo save the registration results to disk, the user can click the ``save'' button in the menu (Fig.~\\ref{fig-03:menu_dialogs}a) to choose the files to save in a dialog (Fig.~\\ref{fig-03:menu_dialogs}d) or the ``save all'' button to save them all. \n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/vis_irr}\n\t\\caption{VIS-IRR registration of paintings using CraquelureNetApp. In (a) the keypoint matches between the reference image (VIS) and moving image (IRR) are visualized as yellow lines. In (b) the registered image pair and the blended overlay and the false color overlay are depicted as complete images and in (c) they are synchronously zoomed in.\n\tImage sources: Meister des Marienlebens, Tempelgang Mari\\\"a, visual light photograph and infrared reflectogram, Germanisches Nationalmuseum, Nuremberg, on loan from Wittelsbacher Ausgleichsfonds\/Bayerische Staats\\-gem\\\"alde\\-samm\\-lungen, \\mbox{Gm 19}, all rights reserved}\n\t\\label{fig-04:vis_irr}\n\\end{figure} \n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/vis_uv}\n\t\\caption{VIS-UV registration of paintings using CraquelureNetApp. In (a) the keypoint matches of VIS and UV are densely concentrated at the  craquelure in the facial region. (b) and (c) show the registered pair and image fusions of the complete images and synchronized details.\t\n\tImage sources: Workshop Lucas Cranach the Elder or Circle, Martin Luther, visual light photograph, captured by Gunnar Heydenreich and UV-fluorescence photograph, captured by Wibke Ottweiler, Lutherhaus Wittenberg, Cranach Digital Archive, KKL-No I.6M3, all rights reserved} \n\t\\label{fig-05:vis_uv}\n\\end{figure} \n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/vis_xr}\n\t\\caption{VIS-XR registration of paintings using CraquelureNetApp. In (a) the keypoint matches of VIS and XR are mostly concentrated at the craquelure in the lighter image area such as the face, hands, and green background. (b) and (c) show the qualitative registrations results as overall view and as synchronized zoomed details.\n\tImage sources: Lucas Cranach the Elder, Martin Luther as ``Junker J\u00f6rg'', visual light photograph Klassik Stiftung Weimar, Museum, X-radiograph HfBK Dresden (Mohrmann, Ri\u00dfe), Cranach Digital Archive, KKL-No II.M2, all rights reserved}\n\t\\label{fig-06:vis_xr}\n\\end{figure} \n\n\\begin{table}[t]\n\\centering\n\\caption{Quantitative evaluation for multi-modal registration of paintings for the VIS-IRR, VIS-UV, and VIS-XR test datasets (each 13 image pairs) using CraquelureNet with different robust homography estimation methods. Registration results are evaluated with the success rates (SR) of mean Euclidean error (ME) and maximum Euclidean error (MAE) of the control points for different error thresholds $\\epsilon$. Best results are highlighted in bold.} \n\\label{tab-01}\n\\scriptsize\n\\begin{tabular*}{\\textwidth}{l @{\\extracolsep{\\fill}} ll rrr rrr}\n\\toprule\nMulti-modal & CraquelureNet & Homography  & \\multicolumn{3}{c}{SR of ME [\\%] $\\uparrow$} & \\multicolumn{3}{c}{SR of MAE [\\%] $\\uparrow$}\\\\\nDataset & Model & Method & $\\epsilon=3$ & $\\epsilon=5 $ & $\\epsilon=7$ & $\\epsilon=6$ & $\\epsilon=8$ & $\\epsilon=10$ \\\\ \n\\midrule\n & PyTorch (Python) & RANSAC & 84.6 & \\textbf{100.0} & \\textbf{100.0} & 38.5 & \\textbf{69.2} & \\textbf{84.6} \\\\ \n & LibTorch (C++) & RANSAC & \\textbf{92.3} & 92.3 & \\textbf{100.0} & 38.5 & \\textbf{69.2} & \\textbf{84.6} \\\\ \nVIS-IRR & LibTorch (C++) & USAC & \\textbf{92.3} & 92.3 & 92.3 & \\textbf{46.2} & \\textbf{69.2} & \\textbf{84.6} \\\\ \n & LibTorch (C++) & GC-RANSAC & \\textbf{92.3} & 92.3 & 92.3 & \\textbf{46.2} & \\textbf{69.2} & \\textbf{84.6} \\\\ \n & LibTorch (C++) & MAGSAC++ & \\textbf{92.3} & 92.3 & 92.3 & \\textbf{46.2} & \\textbf{69.2} & \\textbf{84.6} \\\\ \n\\midrule\n& PyTorch (Python) & RANSAC  & \\textbf{92.3} & \\textbf{100.0} & \\textbf{100.0} & 46.2 & 53.8 & 61.5 \\\\ \n & LibTorch (C++) & RANSAC & 84.6 & 92.3 & \\textbf{100.0} & \\textbf{53.8} & 53.8 & 53.8 \\\\ \nVIS-UV & LibTorch (C++) & USAC & \\textbf{92.3} & 92.3 & 92.3 & \\textbf{53.8} & \\textbf{69.2} & \\textbf{69.2} \\\\ \n & LibTorch (C++) & GC-RANSAC & \\textbf{92.3} & 92.3 & 92.3 & \\textbf{53.8} & \\textbf{69.2} & \\textbf{69.2} \\\\ \n & LibTorch (C++) & MAGSAC++ & 84.6 & 92.3 & 92.3 & 46.2 & 61.5 & \\textbf{69.2} \\\\ \n\\midrule\n & PyTorch (Python) & RANSAC  & 69.2 & \\textbf{84.6} & 84.6 & 23.1 & 38.5 & 61.5 \\\\ \n & LibTorch (C++) & RANSAC & 53.8 & \\textbf{84.6} & \\textbf{92.3} & \\textbf{30.8} & 46.2 & 61.5 \\\\ \nVIS-XR & LibTorch (C++) & USAC & 76.9 & \\textbf{84.6} & 84.6 & 23.1 & 38.5 & \\textbf{76.9} \\\\ \n & LibTorch (C++) & GC-RANSAC & 76.9 & \\textbf{84.6} & 84.6 & 23.1 & 38.5 & \\textbf{76.9} \\\\ \n & LibTorch (C++) & MAGSAC++ & \\textbf{84.6} & \\textbf{84.6} & 84.6 & 15.4 & \\textbf{53.8} & \\textbf{76.9} \\\\ \n\\bottomrule \n\\end{tabular*}\n\\end{table}\n\n\\begin{figure}[t]\t\t\t\t\t\t\t\t \n\\centering\t\n\\begin{tikzpicture}\n\t\\begin{customlegend}[\n\t\t\tlegend entries={SIFT,D2-Net,SuperPoint,CraquelureNet (C++)},\n\t\t\tlegend style={\n\t\t\t\t\/tikz\/every even column\/.append\tstyle={column sep=.75cm}},\n\t\t\tlegend columns=-1,\n\t\t\tlegend cell align=left]\n\t\t\\csname pgfplots@addlegendimage\\endcsname{oorange,mark=square*}\n\t\t\\csname pgfplots@addlegendimage\\endcsname{red,mark=square*}\t\t\n\t\t\\csname pgfplots@addlegendimage\\endcsname{blue,mark=square*}\n\t\t\\csname pgfplots@addlegendimage\\endcsname{cyan,mark=square*}\t\t\t\t\n\t\\end{customlegend}\n\\end{tikzpicture}\n\n\\subcaptionbox{VIS-IRR\\label{fig-07:paint_VIS-IRR_me}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n        ylabel = {SR of ME [\\%]},\n      \tymin=0,\n        ymax=100,\n        symbolic x coords= {$\\epsilon=3$,$\\epsilon=7$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-IRR_success_rates_me.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-IRR_success_rates_me.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-IRR_success_rates_me.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-IRR_success_rates_me.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\n\t}\n\\subcaptionbox{VIS-UV\\label{fig-07:paint_VIS-UV_me}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n      \tymin=0,\n        ymax=100,\n        symbolic x coords= {$\\epsilon=3$,$\\epsilon=7$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-UV_success_rates_me.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-UV_success_rates_me.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-UV_success_rates_me.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-UV_success_rates_me.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\n\t}\n\\subcaptionbox{VIS-XR\\label{fig-07:paint_VIS-XR_me}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n      \tymin=0,\n        ymax=100,\n        symbolic x coords= {$\\epsilon=3$,$\\epsilon=7$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-XR_success_rates_me.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-XR_success_rates_me.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-XR_success_rates_me.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-XR_success_rates_me.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\t\n\t}\t\n\t\n\\subcaptionbox{VIS-IRR\\label{fig-07:paint_VIS-IRR_mae}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n      \tymin=0,\n        ymax=100,\n        ylabel = {SR of MAE [\\%]},\n        symbolic x coords= {$\\epsilon=6$,$\\epsilon=10$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-IRR_success_rates_mae.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-IRR_success_rates_mae.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-IRR_success_rates_mae.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-IRR_success_rates_mae.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\n\t}\n\\subcaptionbox{VIS-UV\\label{fig-07:paint_VIS-UV_mae}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n      \tymin=0,\n        ymax=100,\n        symbolic x coords= {$\\epsilon=6$,$\\epsilon=10$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-UV_success_rates_mae.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-UV_success_rates_mae.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-UV_success_rates_mae.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-UV_success_rates_mae.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\n\t}\n\\subcaptionbox{VIS-XR\\label{fig-07:paint_VIS-XR_mae}}{\n\\begin{tikzpicture}\n    \\begin{axis}[\n        width  = 0.33\\textwidth,\n        height = 3cm,\n        major x tick style = transparent,\n        ybar,\n        bar width=0.15cm,\n    \t\tx=1.3cm, \n    \t\tenlarge x limits={abs=0.7cm},\n    \t\tenlarge y limits=false,\n        ymajorgrids = true,\n      \tymin=0,\n        ymax=100,\n        symbolic x coords= {$\\epsilon=6$,$\\epsilon=10$},\n        xtick = data,\n        nodes near coords={\\pgfmathprintnumber[fixed zerofill,precision=2]\\pgfplotspointmeta},\n        every node near coord\/.append style={font=\\tiny, xshift = 0 , rotate=90, anchor=west},\n        nodes near coords align = {center},\n        ]\n        \\addplot [style={oorange,fill=oorange,mark=none}]\n        table[x=Metric, y=SIFT] \n        {tables\/paintings_VIS-XR_success_rates_mae.dat};             \n        \\addplot [style={red,fill=red,mark=none}]\n        table[x=Metric, y=D2-Net] \n        {tables\/paintings_VIS-XR_success_rates_mae.dat};  \n        \\addplot [style={blue,fill=blue,mark=none}]\n        table[x=Metric, y=SuperPoint] \n        {tables\/paintings_VIS-XR_success_rates_mae.dat}; \n        \\addplot [style={cyan,fill=cyan,mark=none}]\n        table[x=Metric, y=CraquelureNetC++] \n        {tables\/paintings_VIS-XR_success_rates_mae.dat};                      \n\t\\end{axis}\n\\end{tikzpicture}\t\t\n\t}\n\\caption{Quantitative evaluation for our multi-modal paintings dataset using SR of ME with $\\epsilon=3,7$ and SR of MAE with $\\epsilon=6,10$.}\n\\label{fig-07:eval_paintings_sota}\n\\end{figure} \n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=\\textwidth]{graphics\/print}\n\t\\caption{VIS-VIS registration of prints using CraquelureNetApp. CraquelureNet detects a high number of good matches for the two prints in (a), although it did not see images of prints during training. (b) and (c) show the qualitative registration results as overall and detail views.\n\tImage sources: Monogramist IB (Georg Pencz), Philipp Melanchthon with beret and cloak, (left) Germanisches Nationalmuseum, Nuremberg, K 21148, captured by Thomas Klinke and (right) Klassik Stiftung Weimar, Museen, Cranach Digital Archive, DE{\\_}KSW{\\_}Gr-2008-1858, all rights reserved}\n\t\\label{fig-08:print}\n\\end{figure} \n\n\n\\section{Applications and Evaluation}\nIn this section, we apply the CraquelureNetApp to multi-modal images of paintings and to images of different prints of the same motif. We use the test parameters as defined in~\\cite{SindelA2021}: GPU, patch size of $1024$, $N_\\text{max}=8000$, $\\tau_\\text{kp}=0$, resize to same width, and RANSAC with a reprojection error threshold $\\tau_\\text{reproj}=5$.\n\nFor the quantitative evaluation, we measure the registration performance based on the success rate of successfully registered images. The success rate is computed by calculating the percentage of image pairs for which the error distance of manual labeled control points of the registered pair using the predicted homography is less or equal to an error threshold $\\epsilon$.\nAs metric for the error distance, we use the mean Euclidean error (ME) and maximum Euclidean error (MAE)~\\cite{SindelA2022}.\n\nAs comparison methods, we use the conventional keypoint and feature descriptor SIFT~\\cite{LoweDG2004} and the pretrained models of the two deep learning methods, SuperPoint~\\cite{DeToneD2018} and D2-Net~\\cite{DusmanuM2019}, which we apply patch-based to both the paintings and prints. SuperPoint is a CNN with a keypoint detection head and a keypoint description head and is trained in a self-supervised manner by using homographic warpings. D2-Net simultaneously learns keypoint detection and description with one feature extraction CNN, where keypoints and descriptors are extracted from the same set of feature maps.\nFor all methods, we use the same test settings as for our method (patch size, $N_\\text{max}$, RANSAC). \n\n\\subsection{Multi-modal Registration of Historical Paintings}\nThe pretrained CraquelureNet~\\cite{SindelA2021}, which we embedded into our registration tool, was trained using small patches extracted from multi-modal images of 16th century portraits by the workshop of Lucas Cranach the Elder and large German panel paintings from the 15th to 16th century. We use the images of the test split from these multi-modal datasets (13 pairs per domain: VIS-IRR, VIS-UV, VIS-XR) and the corresponding manually labeled control point pairs (40 point pairs per image pair)~\\cite{SindelA2021} to test the CraquelureNetApp. \n\nThe qualitative results of three examples (VIS-IRR, VIS-UV and VIS-XR) are shown in~\\cref{fig-04:vis_irr,fig-05:vis_uv,fig-06:vis_xr}. For each example, the keypoint matches, the complete images (Fit to view option in the GUI), and a zoomed detail view in the synchronization mode are depicted showing good visual registration performance for all three multi-modal pairs. \n\nSecondly, we compare in \\cref{tab-01} the success rates of ME and MAE of the registration for the C++ and PyTorch implementation of CraquelureNet using RANSAC for homography estimation or alternatively using USAC, GC-RANSAC, or MAGSAC++.  \nThe C++ implementation using RANSAC achieves comparable results as the PyTorch implementation (both $\\tau_\\text{reproj}=5$): For VIS-IRR and VIS-UV, all images are registered successfully using ME with an error threshold $\\epsilon=5$ (PyTorch) and $\\epsilon=7$ (C++). For VIS-XR, $12$ (C++) or $11$ (PyTorch) out of $13$ images were successfully registered using ME ($\\epsilon=7$). \nThe small deviations between the two models are due to slightly different implementations \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot of non-maximum suppression that were necessary for the conversion to C++.\nUsing the more recent USAC, GC-RANSAC, or MAGSAC++ is slightly less robust for VIS-IRR and VIS-UV, since not more than $12$ out of $13$ image pairs can be registered using ME ($\\epsilon=7$) as one image pair fails completely. VIS-XR registration is the most difficult part of the three domain pairs due to the visually highly different appearance of the VIS and XR images. Here, we can observe similar performance of ME, with a slightly higher percentage of USAC, GC-RANSAC, and especially MAGSAC++ for $\\epsilon=3$, but for $\\epsilon=7$ those are on par with the PyTorch model and are slightly inferior to the C++ model using RANSAC.\nRegarding the success rates of MAE, we observe a bit higher values for USAC, GC-RANSAC, and MAGSAC++ than for both RANSAC models for VIS-UV and VIS-XR ($\\epsilon=\\{8,10\\}$), while for VIS-IRR they are the same.\n\nIn \\cref{fig-07:eval_paintings_sota}, the multi-modal registration performance of our C++ CraquelureNet (RANSAC) is measured in comparison to SIFT~\\cite{LoweDG2004}, SuperPoint~\\cite{DeToneD2018}, and D2-Net~\\cite{DusmanuM2019}. CraquelureNet achieves the highest success rates of ME and MAE for all multi-modal pairs. D2-Net is relatively close to CraquelureNet for VIS-IRR with the same SR of ME but with a lower SR of MAE. For VIS-IRR and VIS-UV, all learning-based methods are clearly better than SIFT. For the challenging VIS-XR domain, the advantage of our cross-modal keypoint detector and descriptor is most distinct, since for $\\epsilon=7$, CraquelureNet still achieves high success rates of $92.3$\\,\\% for ME and $61.5$\\,\\% for MAE, whereas for $\\epsilon=7$, D2-Net only successfully registers $61.5$\\,\\% for ME and $15.3$\\,\\% for MAE, SuperPoint only $23$\\,\\% for ME and none for MAE, and lastly, SIFT does not register any VIS-XR image pair successfully.\nD2-Net was developed to find correspondences in difficult image conditions, such as day-to-night or depiction changes~\\cite{DusmanuM2019}, hence it also is able to detect to some extend matching keypoint pairs in VIS-XR images. SuperPoint's focus is more on images with challenging viewpoints~\\cite{DeToneD2018,DusmanuM2019}, thus the pretrained model results in a lower VIS-XR registration performance. In our prior work~\\cite{SindelA2021}, we fine-tuned SuperPoint using image patches extracted from our manually aligned multi-modal paintings dataset, which did not result in an overall improvement. On the other hand, CraquelureNet is robust for the registration of all multi-modal pairs and does not require an intensive training procedure, as it is trained in efficient time only using very small image patches. \n\nThe execution time of CraquelureNet (C++ and PyTorch) for the VIS-IRR registration (including loading of network and images, patch extraction of size $1024 \\times 1024 \\times 3$ pixels, network inference, homography estimation using RANSAC and image warping) was about 11\\,s per image pair on the GPU and about 2\\,min per image pair on the CPU. We used an Intel Xeon W-2125 CPU 4.00 GHz with 64 GB RAM and one NVIDIA Titan XP GPU for the measurements. \nThe comparable execution times of the C++ and PyTorch implementation of CraquelureNet (the restructured one for a fair comparison) is not surprising, as PyTorch and OpenCV are wrappers around C++ functions.\nThe relatively fast inference time for the registration of high-resolution images makes CraquelureNet suitable to be integrated into the registration GUI and to be used by art technologists and art historians for their daily work.\n\n\\subsection{Registration of Historical Prints}\nTo evaluate the registration performance for prints, we have created a test dataset of in total $52$ images of historical prints with manually labeled control point pairs. The dataset is composed of $13$ different motifs of 16th century prints with each four exemplars that may show wear or production-related differences in the print. For each motif a reference image was selected and the other three copies will be registered to the reference, resulting in $39$ registration pairs. For each image pair, $10$ control point pairs were manually annotated.\n\nWe test the CraquelureNet, that was solely trained on paintings, for the 16th century prints. One qualitative example is shown in~\\cref{fig-08:print}. \nCraquelureNet also finds a high number of good matches in this engraving pair, due to the multitude of tiny lines and branchings in the print as CraquelureNet is mainly focusing on these branching points.\n\nFor the quantitative evaluation, we compute the success rates of ME and MAE for the registration of the print dataset using our CraquelureNet C++ model in comparison to using SIFT~\\cite{LoweDG2004}, SuperPoint~\\cite{DeToneD2018}, and D2-Net~\\cite{DusmanuM2019}.\nFor the registration, the images are scaled to a fixed height of $2000$ pixel, as then the structures in the prints have a suitable size for the feature detectors. In the GUI of CraquelureNetApp this option is ``resize to custom height''. The results of the comparison are depicted in~\\cref{fig-09:prints_SR}. The CNN-based methods obtain clearly superior results to SIFT for both metrics. Overall, CraquelureNet and SuperPoint show the best results, as both achieve a success rate of ME of $100$\\,\\% at error threshold $\\epsilon=4$, where they are closely followed by D2-Net at $\\epsilon=5$ and all three methods achieve a success rate of MAE close to $100$\\,\\% at $\\epsilon=10$. For smaller error thresholds, CraquelureNet is slightly superior for MAE and SuperPoint for ME. As none of the methods was fine-tuned for the print dataset, this experiment shows the successful possibility of applying the models to this new dataset. \n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\t\\begin{customlegend}[\n\t\t\tlegend entries={SIFT,D2-Net,SuperPoint,CraquelureNet (C++)},\n\t\t\tlegend style={\n\t\t\t\t\/tikz\/every even column\/.append\tstyle={column sep=.75cm}},\n\t\t\tlegend columns=-1,\n\t\t\tlegend cell align=left]\n\t\t\\csname pgfplots@addlegendimage\\endcsname{oorange,mark=square*}\n\t\t\\csname pgfplots@addlegendimage\\endcsname{red,mark=*}\t\t\n\t\t\\csname pgfplots@addlegendimage\\endcsname{blue,mark=triangle*}\n\t\t\\csname pgfplots@addlegendimage\\endcsname{cyan,mark=star}\t\t\t\t\n\t\\end{customlegend}\n\\end{tikzpicture}\n\n\\subcaptionbox{~\\label{fig-09:prints_SR_ME}}{\n\\begin{tikzpicture}\n\\begin{axis} [\n\t\t\tscale = 0.6, \t\t\t\n\t\t\txmajorgrids,\n\t\t\tymajorgrids,\n\t\t\tscaled ticks=false,\n\t\t\tylabel = {SR of ME [\\%]},\n\t\t\txlabel = Error threshold,\n\t\t\txtick ={1,2,3,4,5,6},\n\t\t\tymax   = 100,\t\n\t\t\tymin   = 0\t\n\t\t\t]\n\t\\addplot [oorange,mark=square*]\n        table[x=Thresh, y=SIFT] \n        {tables\/prints_success_rates_me.dat}; \t\n\t\\addplot [red,mark=*]\n        table[x=Thresh, y=D2-Net] \n        {tables\/prints_success_rates_me.dat}; \n\t\\addplot [blue,mark=triangle*]\n        table[x=Thresh, y=SuperPoint] \n        {tables\/prints_success_rates_me.dat};\n\t\\addplot [cyan,mark=star]\n        table[x=Thresh, y=CraquelureNet] \n        {tables\/prints_success_rates_me.dat};        \n\\end{axis}\n\\end{tikzpicture}\n}\n\\hfill\n\\subcaptionbox{~\\label{fig-09:prints_SR_MAE}}{\n\\begin{tikzpicture}\n\\begin{axis} [\n\t\t\tscale = 0.6, \t\t\t\n\t\t\txmajorgrids,\n\t\t\tymajorgrids,\n\t\t\tscaled ticks=false,\n\t\t\tylabel = {SR of MAE [\\%]},\n\t\t\txlabel = Error threshold,\n\t\t\txtick ={5,6,7,8,9,10},\n\t\t\tymax   = 100,\t\n\t\t\tymin   = 0\t\n\t\t\t]\n\t\\addplot [oorange,mark=square*]\n        table[x=Thresh, y=SIFT] \n        {tables\/prints_success_rates_mae.dat}; \t\n\t\\addplot [red,mark=*]\n        table[x=Thresh, y=D2-Net] \n        {tables\/prints_success_rates_mae.dat}; \n\t\\addplot [blue,mark=triangle*]\n        table[x=Thresh, y=SuperPoint] \n        {tables\/prints_success_rates_mae.dat};\n\t\\addplot [cyan,mark=star]\n        table[x=Thresh, y=CraquelureNet] \n        {tables\/prints_success_rates_mae.dat};      \n\\end{axis}\n\\end{tikzpicture}\n}\n\\caption{Quantitative comparison of success rates for registration of prints ($39$ image pairs). For all methods RANSAC with $\\tau_\\text{reproj}=5$ was used. None of the methods was fine-tuned for the print dataset.\nIn (\\subref{fig-09:prints_SR_ME}) the success rate of mean Euclidean error (ME) for the error thresholds $\\epsilon=\\{1,2,...,6\\}$ and in (\\subref{fig-09:prints_SR_MAE}) the success rate of maximum Euclidean error (MAE) for the error thresholds $\\epsilon=\\{5,6,...,10\\}$ is plotted.} \n\\label{fig-09:prints_SR}\n\\end{figure}\n\n\\section{Conclusion}\nWe presented an interactive registration and visualization tool for multi-modal paintings and also applied it to historical prints. The registration is performed fully automatically using CraquelureNet. The user can choose the registration settings and can interact with the visualizations of the registration results. In the future, we could extend the application by including trained models of CraquelureNet on other datasets, such as the RetinaCraquelureNet~\\cite{SindelA2022} for multi-modal retinal registration. A further possible extension would be to add a batch processing functionality to the GUI to register a folder of image pairs.\n\n\\subsubsection{Acknowledgements} \nThanks to Daniel Hess, Oliver Mack, Daniel G\\\"orres, Wib\\-ke Ottweiler, Germanisches Nationalmuseum (GNM), and Gunnar Heydenreich, Cranach Digital Archive (CDA), and Thomas Klinke, \\mbox{TH K\\\"oln}, and Amalie H\\\"ansch, FAU Erlangen-N\\\"urnberg for providing image data, and to Leibniz Society for funding the research project ``Critical Catalogue of Luther portraits (1519 - 1530)'' with grant agreement No. SAW-2018-GNM-3-KKLB, to the European Union's Horizon 2020 research and innovation programme within the Odeuropa project under grant agreement No. 101004469 for funding this publication, and to NVIDIA for their GPU hardware donation. \n\n \\bibliographystyle{splncs04}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThis is the first part of a series of papers on  the spreading and\nvanishing dynamics of  diffusive equations with free boundaries of the form,\n\\begin{equation}\n\\label{main-eq}\n\\begin{cases}\nu_t=u_{xx}+uf(t,x,u),\\quad &t>0,\\,\\,  0<x<h(t)\\cr h^{'}(t)=-\\mu u_x(t,h(t)),\\quad &t>0,\\cr\nu_x(t,0)=u(t,h(t))=0, \\quad &t>0,\\cr h(0)=h_0, u(0,x)=u_0(x),\\quad &0\\leq x\\leq h_0,\n\\end{cases}\n\\end{equation}\nwhere $x=h(t;u_0,h_0)$ is the moving boundary to be determined, $\\mu$, $h_0$ are given\npositive constants, and the initial function $u_0(x)$ satisfies\n \\begin{equation}\n \\label{initial-value}\n u_0\\in C^2([0,h_0]),  \\  u^{'}_0(0)=u_0(h_0)=0, \\ {\\rm and} \\ u_0>0 \\ {\\rm in} \\ [0,h_0).\n \\end{equation}\n\nWe assume that\n $f(t,x,u)$ is a $C^1$ function in $t\\in\\RR$, $x\\in\\RR$, and\n$u\\in\\RR$; $f(t,x,u)<0$ for $u\\gg 1$; $f_u(t,x,u)<0$ for $u\\ge 0$;\nand $f(t,x,u)$ is almost periodic in $t$ uniformly with respect to\n$x\\in\\RR$ and $u\\in\\RR$ (see (H1), (H2) in subsection 2.1 for detail). Here is a typical example\nof such functions,\n$f(t,x,u)=a(t,x)-b(t,x)u$, where $a(t,x)$ and $b(t,x)$ are almost periodic in $t$ and periodic in $x\\in\\RR$, and\n$\\inf_{t\\in\\RR,x\\in\\RR}b(t,x)>0$.\n\n\n Observe  that for any given $u_0$ satisfying \\eqref{initial-value},\n \\eqref{main-eq} has a  unique (local) solution  $(u(t,x;u_0,h_0)$, $h(t;u_0,h_0))$ with\n  $u(0,x;u_0,h_0)=u_0(x)$ and $h(0;u_0,h_0)=h_0$ (see \\cite{DuGuPe}). Moreover,\nby comparison principle for parabolic equations, $(u(t,x;u_0,h_0),h(t;u_0,h_0))$ exists for all\n$t>0$ and $u_x(t,h(t))<0$. Hence $h(t;u_0,h_0)$ increases as $t$ increases.\n\nEquation \\eqref{main-eq}  with $f(t,x,u)=u(a-bu)$ and $a$ and $b$ being\ntwo positive constants was introduced by Du and Lin in \\cite{DuLi}\nto understand the spreading of species. A great deal of previous mathematical investigation on the spreading of species (in one space dimension case)\nhas been based on diffusive equations of the form\n\\begin{equation}\n\\label{kpp-general-eq}\nu_t=u_{xx}+u f(t,x,u),\\quad x\\in\\RR,\n\\end{equation}\nwhere $f(t,x,u)<0$ for $u\\gg 1$ and $f_u(t,x,u)<0$ for $u\\ge 0$. Thanks to the pioneering works of Fisher (\\cite{Fis}) and Kolmogorov, Petrowsky, Piscunov\n(\\cite{KPP}) on the following special case of \\eqref{kpp-general-eq}\n\\begin{equation}\n\\label{kpp-special-eq}\nu_t=u_{xx}+u(1-u),\\quad x\\in\\RR,\n\\end{equation}\n\\eqref{main-eq}, resp. \\eqref{kpp-general-eq}, is referred to as diffusive Fisher or KPP equation.\n\n\n\nOne of the central problems for both \\eqref{main-eq} and \\eqref{kpp-general-eq} is\nto understand their spreading dynamics. For \\eqref{kpp-general-eq}, this is closely related to\nspreading speeds and transition fronts of \\eqref{kpp-general-eq} and has been widely studied.\nWhen $f(t,x,u)$ is independent of $t$ and $x$ or is periodic in $t$ and $x$, the spreading dynamics\nfor \\eqref{kpp-general-eq} is quite well understood. For example, assume  that $f(t,x,u)$ is periodic in\n$t$ with period $T$ and periodic in $x$  with period $p$, and that\n$u\\equiv 0$ is a linearly unstable solution of \\eqref{kpp-general-eq} with respect to periodic perturbations. Then it is known that\n\\eqref{kpp-general-eq} has a unique positive periodic solution $u^*(t,x)$ ($u^*(t+T,x)=u^*(t,x+p)=u^*(t,x)$)  which is asymptotically stable with\nrespect to periodic perturbations   and\n it\nhas been proved\n that   there is a positive constant $c^*$ such that for every $c\\geq c^*$, there is a periodic\n traveling wave solution $u(t,x)$ connecting $u^*$ and $u\\equiv 0$\n with speed $c$ (i.e. $u(t,x)=\\phi(x-ct,t,x)$ for some $\\phi(\\cdot,\\cdot,\\cdot)$ satisfying that\n $\\phi(\\cdot,\\cdot+T,\\cdot)=\\phi(\\cdot,\\cdot,\\cdot+p)=\\phi(\\cdot,\\cdot,\\cdot)$ and $\\phi(-\\infty,\\cdot,\\cdot)=u^*(\\cdot,\\cdot)$ and\n $\\phi(\\infty,\\cdot,\\cdot)=0$), and there is no such traveling\n wave solution of slower speed (see \\cite{LiZh, Nad, NoXi, Wei}).  Moreover, the minimal wave speed $c^*$ is\n of the following spreading  property and is hence called the spreading speed of \\eqref{kpp-general-eq}:\n for any given $u_0\\in C_{\\rm unif}^b(\\RR,\\RR^+)$ with non-empty support,\n \\begin{equation}\n \\label{kpp-spreading-eq}\n \\begin{cases}\n \\lim_{|x| \\le c^{'}t,t\\to\\infty}[u(t,x;u_0)-u^*(t,x)]=0\\quad \\forall\\,\\, c^{'}<c^*\\cr\n \\lim_{|x| \\ge c^{''}t,t\\to\\infty} u(t,x;u_0)=0\\quad \\forall\\,\\, c^{''}> c^*,\n \\end{cases}\n \\end{equation}\n where $u(t,x;u_0)$ is the solution of \\eqref{kpp-general-eq} with $u(0,x;u_0)=u_0(x)$\n (see \\cite{LiZh, Wei}).\n\n\n\n\nThe spreading property \\eqref{kpp-spreading-eq} for \\eqref{kpp-general-eq} in the case that $f(t,x,u)$ is periodic in $t$ and $x$\n implies that spreading always happens for a solution of\n\\eqref{kpp-general-eq} with a positive initial function, no matter how small the positive initial function is.\nThe following strikingly different  spreading scenario has been proved for  \\eqref{main-eq} in the case that\n$f(t,x,u)\\equiv f(u)$ (see \\cite{DuGu, DuLi}):  it exhibits a spreading-vanishing dichotomy in the sense that for any given positive initial data\n$u_0$ satisfying \\eqref{initial-value} and $h_0$, either vanishing occurs (i.e. $\\lim_{t\\to\\infty}h(t;u_0,h_0)<\\infty$ and $\\lim_{t\\to\\infty}u(t,x;u_0,h_0)=0$) or\nspreading occurs (i.e. $\\lim_{t\\to\\infty}h(t;u_0,h_0)=\\infty$ and $\\lim_{t\\to\\infty}u(t,x;u_0,h_0)=u^*$ locally uniformly in $x\\in \\RR^+$, where\n$u^*$ is the unique positive solution of $f(u)=0$). The above spreading-vanishing dichotomy for \\eqref{main-eq} with\n$f(t,x,u)\\equiv f(u)$ has also been extended to the cases that $f(t,x,u)$ is periodic in $t$ or that $f(t,x,u)$ is independent of $t$ and periodic\nin $x$ (see \\cite{DuGuPe,DuLia}). The spreading-vanishing dichotomy proved for \\eqref{main-eq} in \\cite{DuGu, DuGuPe, DuLi, DuLia}\n is well supported by some empirical evidences, for example, the introduction of several\nbird species from Europe to North America in the 1900s was successful only after many initial\nattempts (see \\cite{LoHoMa, ShKa}).\n\n\n\n\nIn reality, many evolution systems in biology are subject to non-periodic time and\/or space variations. It is therefore\nof great importance to investigate the spreading dynamics for both \\eqref{main-eq} and \\eqref{kpp-general-eq} with general time and space\ndependent $f(t,x,u)$.  The spreading dynamics for \\eqref{kpp-general-eq} with non-periodic time and\/or space dependence has been\nstudied by many people recently\n(see \\cite{BeHa, BeHaNa, BeNa, HuSh, KoSh, NaRo, NoXi1, She1, She2, She3, TaZhZl, Zla}, etc.). However, there is little study\non the spreading dynamics for \\eqref{main-eq} with non-periodic time and space dependence.\n\n\nThe objective of the current series\nof papers is to investigate the spreading-vanishing dynamics of \\eqref{main-eq} in the case that $f(t,x,u)$ is almost periodic in $t$,\nthat is, to investigate  whether the population will\nsuccessfully establishes itself\nin the entire space (i.e. spreading occurs), or  it fails to establish and vanishes\neventually (i.e. vanishing occurs).\nRoughly speaking, for given $(u_0,h_0)$, if $h_\\infty=\\lim_{t\\to\\infty}h(t;u_0,h_0)=\\infty$ and for any $M>0$,\n$\\liminf_{t\\to\\infty}\\inf_{0\\le x\\le M}u(t,x;u_0,h_0)>0$,\nwe say {\\it spreading} occurs. If $h_\\infty<\\infty$ and $\\lim_{t\\to\\infty}u(t,x;u_0,h_0)=0$, we say {\\it vanishing} occurs (see Definition\n\\ref{spreading-vanishing-def} for detail). We say a positive number $c^*$ is a {\\it spreading speed} of \\eqref{main-eq} if for any $(u_0,h_0)$ such that the spreading occurs,\n$$\\lim_{t\\to\\infty}\\frac{h(t;u_0,h_0)}{t}=c^*$$\n and\n $$\n \\liminf_{0\\le x\\le c^{'}t,t\\to\\infty}u(t,x;u_0,h_0)>0\\quad \\forall \\,\\, c^{'}<c^*\n $$\n (see Definition\n\\ref{spreading-vanishing-def} for detail).\n\n\n\nIn this first part of the series of the papers, we focus on the study of spreading and vanishing dichotomy\nscenario for \\eqref{main-eq}.  Among others,\nwe  prove the following spreading and vanishing dichotomy:\n\n\\medskip\n\n\\noindent $\\bullet$  {\\it Assume (H1)-(H5) stated in subsection 2.1. For any given $u_0$ satisfying\n\\eqref{initial-value}, either spreading occurs or vanishing occurs.\nMoreover, there is $l^*>0$ such that for any given $u_0$ satisfying \\eqref{initial-value}, vanishing occurs if and only if\n$h_\\infty\\le l^*$} (see Theorem \\ref{main-thm2}).\n\n\\medskip\n\n\n\nTo characterize the detailed  spreading and vanishing dynamics of \\eqref{main-eq},\nwe also consider the following  fixed boundary problem on half line,\n\\begin{equation}\n\\label{aux-main-eq1}\n\\begin{cases}\nu_t=u_{xx}+uf(t,x,u),\\quad x\\in (0,\\infty)\\cr\nu_x(t,0)=0.\n\\end{cases}\n\\end{equation}\n\nObserve that if $u^*(t,x)$ is a solution of \\eqref{aux-main-eq1} and  $u_0(x)\\le u^*(0,x)$ for $x\\in [0,h_0]$, then\n$u(t,x;u_0,h_0)\\le u^*(t,x)$ for $0\\le x\\le h(t;u_0,h_0)$.  Among others, we prove that\n\n\\medskip\n\n\\noindent $\\bullet$ {\\it Assume (H1)-(H5) stated in subsection 2.1.\n \\eqref{aux-main-eq1} has a unique time almost periodic positive solution $u^*(t,x)$}  (see Theorem \\ref{main-thm1}) {\\it and\nfor any given $u_0$ satisfying \\eqref{initial-value}, if spreading occurs in \\eqref{main-eq}, then $u(t,x;u_0,h_0)-u^*(t,x)\\to 0$ as $t\\to\\infty$ locally uniformly in $x\\ge 0$} (see\nTheorem \\ref{main-thm2}).\n\n\\medskip\n\n\nWe note that the techniques for \\eqref{main-eq}  can be modified to study the\nfollowing double fronts free boundary problem:\n\\begin{equation}\n\\begin{cases}\n\\label{main-doub-eq}\nu_t=u_{xx}+uf(t,x,u), \\quad  &t>0,\\,\\,  g(t)<x<h(t), \\cr\nu(t,g(t))=0, g^{'}(t)=-\\mu u_x(t,g(t)),  \\quad &t>0,\\cr\nu(t,h(t))=0, h^{'}(t)=-\\mu u_x(t,h(t)), \\quad &t>0,\\cr\nu(0,x)=u_0(x),\\quad & h_0\\le x\\le g_0 \\cr\nh(0)=h_0,\\,\\, g(0)=g_0\n\\end{cases}\n\\end{equation}\nwhere both $x=g(t)$ and $x=h(t)$ are to be determined and $u_0$ satisfies\n\\begin{equation}\n\\label{initial-value-1}\n\\begin{cases}\nu_0\\in C^2([g_0,h_0]) \\cr\nu_0(g_0)=u_0(h_0)=0 \\ \\ and  \\ \\ u_0>0 \\ \\ in \\ \\ (g_0,h_0).\n\\end{cases}\n\\end{equation}\nUnder the assumptions (H1), (H2), $(H4)^*$, and (H5) (see section 6 for $(H4)^*$),  spreading-vanishing dichotomy for \\eqref{main-doub-eq} also holds.\nIn particular, we prove that\n\n\\medskip\n\\noindent $\\bullet$ {\\it Assume (H1), (H2), $(H4)^*$, and (H5). For given $h_0>0$ and  $u_0$ satisfying \\eqref{initial-value-1},\n either\n $h_\\infty-g_\\infty<\\infty$ and\n$\\lim_{t\\to+\\infty} u(t,x;u_0,h_0,g_0)=0$ uniformly in $x$,\nor   $h_{\\infty}=-g_{\\infty}=\\infty$ and\n$\\liminf_{t\\to\\infty}\\inf_{|x|\\le M}u(t,x;u_0,h_0,g_0)>0$\nfor any $M>0$}\n(see Proposition \\ref{spreading-vanishing-doub-prop}).\n\n\\medskip\n\n\nIn the second part of the series of the papers, we will study the existence of  spreading speeds\nfor \\eqref{main-eq} and the existence of time almost periodic semi-wave solutions of the\nfollowing free boundary problem associated to \\eqref{main-eq},\n\\begin{equation}\n\\label{aux-main-eq3}\n\\begin{cases}\nu_t=u_{xx}+uf(t,x,u),\\quad &t>0, \\,\\, -\\infty<x<h(t)\\cr\nu(t,h(t))=0,\\quad &t>0,\\cr\nh^{'}(t)=-\\mu u_x(t,h(t)), \\quad &t>0.\n\\end{cases}\n\\end{equation}\n If $(u(t,x),h(t))$ is an entire solution of \\eqref{aux-main-eq3},\nit is called a {\\it semi-wave solution} of \\eqref{aux-main-eq3}.\n\n\n The rest of this paper is organized as follows.  In Section 2, we introduce the definitions and standing assumptions and\n state the main results of the paper. We present preliminary materials in Section 3 for the use in later sections.\n Section 4 is devoted to the investigation of time almost periodic KPP equation \\eqref{aux-main-eq1}\n  on the half line and to the proof of Theorem \\ref{main-thm1}. In Section 5,\n we explore the spreading and vanishing dichotomy scenario of \\eqref{main-eq} and prove\n Theorem \\ref{main-thm2}.  The paper is ended with some remarks on spreading-vanishing dichotomy for \\eqref{main-doub-eq} in Section 6.\n\n\n\\section{Definitions, Assumptions, and Main Results}\n\nIn this section, we introduce the  definitions and standing assumptions, and state the main results.\n\n\\subsection{Definitions and assumptions}\n\nIn this subsection, we introduce the definitions and standing assumptions. We first recall the definition of almost periodic\nfunctions, next recall the definition of principal Lyapunov exponents for some linear parabolic equations, then state\nthe standing assumptions, and finally introduce the definition of spreading and vanishing for \\eqref{main-eq}.\n\n\\begin{definition}[Almost periodic function]\n\\label{almost-periodic-def}\n\\begin{itemize}\n\n\\item[(1)] A continuous function $g:\\RR\\to \\RR$ is called {\\rm almost periodic} if\nfor any $\\epsilon>0$, the set\n$$\nT(\\epsilon)=\\{\\tau\\in\\RR\\,|\\, |f(t+\\tau)-f(t)|<\\epsilon\\,\\, \\, \\text{for all}\\,\\, t\\in\\RR\\}\n$$\nis relatively dense in $\\RR$.\n\n\n\n\\item[(2)] Let $g(t,x,u)$ be a continuous function of $(t,x,u)\\in\\RR\\times\\RR^m\\times\\RR^n$. $g$ is said to be {\\rm almost periodic in $t$ uniformly with respect to $x\\in\\RR^m$ and\n$u$ in bounded sets} if\n$g$ is uniformly continuous in $t\\in\\RR$,  $x\\in\\RR^m$, and $u$ in bounded sets and for each $x\\in\\RR^m$ and $u\\in\\RR^n$, $g(t,x,u)$ is almost periodic in $t$.\n\n\\item[(3)] For a given almost periodic function $g(t,x,u)$, the hull $H(g)$ of $g$ is defined by\n\\begin{align*}\nH(g)=\\{\\tilde g(\\cdot,\\cdot,\\cdot)\\,|\\, & \\exists t_n\\to\\infty \\,\\,\\text{such that}\\,\\, g(t+t_n,x,u)\\to \\tilde g(t,x,u)\\,\\, \\text{uniformly in}\\,\\, t\\in\\RR,\\\\\n&\n\\,\\, (x,u)\\,\\, \\text{in bounded sets}\\}.\n\\end{align*}\n\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\n\\label{almost-periodic-rk}\n(1) Let $g(t,x,u)$ be a continuous function of $(t,x,u)\\in\\RR\\times\\RR^m\\times\\RR^n$. $g$ is almost periodic in $t$ uniformly with respect to\n$x\\in\\RR^m$ and $u$ in bounded sets  if and only if\n $g$ is uniformly continuous in $t\\in\\RR$,  $x\\in\\RR^m$, and $u$ in bounded sets and for any sequences $\\{\\alpha_n^{'}\\}$,\n$\\{\\beta_n^{'}\\}\\subset \\RR$, there are subsequences $\\{\\alpha_n\\}\\subset\\{\\alpha_n^{'}\\}$, $\\{\\beta_n\\}\\subset\\{\\beta_n^{'}\\}$\nsuch that\n$$\n\\lim_{n\\to\\infty}\\lim_{m\\to\\infty}g(t+\\alpha_n+\\beta_m,x,u)=\\lim_{n\\to\\infty}g(t+\\alpha_n+\\beta_n,x,u)\n$$\nfor each $(t,x,u)\\in\\RR\\times\\RR^m\\times\\RR^n$ (see \\cite[Theorems 1.17 and 2.10]{Fink}).\n\n(2) We may write $g(\\cdot+t,\\cdot,\\cdot)$ as $g\\cdot t(\\cdot,\\cdot,\\cdot)$.\n\\end{remark}\n\nFor a given positive constant $l>0$ and a given $C^1$ function $a(t,x)$ which is almost periodic in $t$ uniformly in $x$ in bounded sets, consider\n\\begin{equation}\n\\label{linearized-eq1}\n\\begin{cases}\nv_t=v_{xx}+a(t,x)v,\\quad 0<x<l\\cr v_x(t,0)=v(t,l)=0.\n\\end{cases}\n\\end{equation}\n\nLet\n$$\nY(l)=\\{u\\in C([0,l])\\,|\\, u(l)=0\\}\n$$\nwith the norm $\\|u\\|=\\max_{x\\in [0,l]}|u(x)|$ for $u\\in Y(l)$.\nLet $A=\\Delta$ acting on $Y(l)$ with $\\mathcal{D}(A)=\\{u\\in C^2([0,l])\\cap Y(l)\\,|\\, u_x(0)=0\\}$.\nNote that  $A$ is a sectorial operator. Let $0<\\alpha<1$ be such that $\\mathcal{D}(A^\\alpha)\\subset C^1([0,l])$. Fix such an $\\alpha$.\nLet\n\\begin{equation}\n\\label{bounded-domain-space-eq1}\nX(l)=\\mathcal{D}(A^\\alpha).\n\\end{equation}\nThen $X(l)$ is strongly ordered Banach spaces with positive cone\n$$\nX^+(l)=\\{u\\in X(l)\\,|\\, u(x)\\ge 0\\}.\n$$\n\nLet\n$$\nX^{++}(l)={\\rm Int}(X^+(l)).\n$$\nIf no confusion occurs, we may write $X(l)$ as $X$.\n\n\nBy semigroup theory (see \\cite{PA}), for any  $v_0\\in X(l)$, \\eqref{linearized-eq1}\nhas a unique solution $v(t,\\cdot;v_0,a)$ with $v(0,\\cdot;v_0,a)=v_0(\\cdot)$.\n\nFor a given positive constant $l>0$ and a given $C^1$ function\n $a(t,x)$ which is almost periodic function in $t$ uniformly in $x$ in bounded sets, consider\nalso\n\\begin{equation}\n\\label{aaux-linearized-eq1}\n\\begin{cases}\nv_t=v_{xx}+a(t,x)v,\\quad 0<x<l\\cr\nv(t,0)=v(t,l)=0.\n\\end{cases}\n\\end{equation}\nLet\n$$\n\\tilde Y(l)=\\{u\\in C([0,l])\\,|\\, u(0)=u(l)=0\\}.\n$$\nLet $A=\\Delta$ acting on $\\tilde Y(l)$ with $\\mathcal{D}(A)=\\{u\\in C^2([0,l])\\cap \\tilde Y(l)\\}$.\nNote that  $A$ is a sectorial operator. Let $0<\\alpha<1$ be such that $\\mathcal{D}(A^\\alpha)\\subset C^1([0,l])$. Fix such an $\\alpha$.\nLet\n\\begin{equation}\n\\label{bounded-domain-space-eq2}\n\\tilde X(l)=\\mathcal{D}(A^\\alpha).\n\\end{equation}\nThen, for any $v_0\\in \\tilde X(l)$, \\eqref{aaux-linearized-eq1} has a unique solution\n$\\tilde v(t,\\cdot;v_0,a)$ with $\\tilde v(0,\\cdot;v_0,a)=v_0(\\cdot)$.\n\n\n\\begin{definition}[Principal Lyapunov exponent]\n\\label{principal-lyapunov-exp}\n\\begin{itemize}\n\\item[(1)] Let $V(t,a)v_0=v(t,\\cdot;v_0,a)$ for $v_0\\in X(l)$ and\n$$\n\\lambda(a,l)=\\limsup_{t\\to\\infty}\\frac{\\ln \\|V(t,a)\\|_{X(l)}}{t}.\n$$\n$\\lambda(a,l)$ is called the {\\rm principal Lyapunov exponent} of \\eqref{linearized-eq1}.\n\n\\item[(2)] Let\n$$\n\\tilde\\lambda(a,l)=\\limsup_{t\\to\\infty}\\frac{\\ln\\|\\tilde V(t,a)\\|_{\\tilde{X(l)}}}{t}\n$$\nwhere $\\tilde V(t,a)v_0=\\tilde v(t,\\cdot;v_0,a)$ for $v_0\\in\\tilde X(l)$.\n$\\tilde\\lambda(a,l)$ is called the {\\rm principal Lyapunov exponent} of \\eqref{aaux-linearized-eq1}.\n\\end{itemize}\n\\end{definition}\n\n\n\nLet (H1)-(H5) be the following standing assumptions.\n\n\\medskip\n\n\n\\noindent {\\bf (H1)} {\\it $f(t,x,u)$ is $C^1$ in $(t,x,u)\\in\\RR^3$, $Df=(f_t,f_x,f_u)$ is bounded in $(t,x)\\in\\RR\\times\\RR$ and\nin $u$ in bounded sets, and $f$ is monostable in $u$ in the sense that\nthere are $M>0$ such that $$\\sup_{t\\in\\RR,x\\in\\RR,u\\ge M}f(t,x,u)<0$$ and\n$$\\sup_{t\\in\\RR,x\\in\\RR,u\\ge 0}f_u(t,x,u)<0.$$}\n\n\\medskip\n\n\\noindent{\\bf (H2)} {\\it $f(t,x,u)$ and $Df(t,x,u)=(f_t(t,x,u),f_x(t,x,u),f_u(t,x,u))$ are  almost periodic in $t$ uniformly with respect to $x\\in\\RR$ and\n$u$ in bounded sets.\n}\n\n\n\n\\medskip\n\\noindent {\\bf (H3)} {\\it There is $l^*>0$ such that\n$\\lambda(a(\\cdot,\\cdot),l)>0$ for $l>l^*$, where $a(t,x)=f(t,x,0)$.\n}\n\n\\medskip\n\n\n\\noindent{\\bf (H4)} {\\it There are $y^*\\ge 0$ and $L^*\\ge 0$ such that\n $\\tilde\\lambda(a(\\cdot,\\cdot+y),l)>0$ for $y\\ge y^*$ and $l\\ge L^*$.}\n\n \\medskip\n\n\\noindent {\\bf (H5)}\n {\\it For any given sequence $\\{y_n^{'}\\}\\subset \\RR$ and $\\{g_{n}^{'}\\}\\subset H(f)$, there are  subsequences\n$\\{y_n\\}\\subset\\{y_n^{'}\\}$ and $\\{g_n\\}\\subset\\{g_n^{'}\\}$ such that\n$\\lim_{n\\to\\infty} g_n(t,x+y_n,u)$ exists uniformly in $t\\in\\RR$ and $(x,u)$ in bounded sets.\n}\n\n\\medskip\n\nAssume (H1) and (H2).\nWe remark that, if $f(t,x,u)\\equiv f(t,u)$, then  (H3) (resp. (H4))  holds if and only if\n$\\lim_{t\\to\\infty}\\frac{1}{t}\\int_0^t f(s,0)ds>0$ (see Lemma \\ref{lyapunov-exponent-lm3} for the reasoning).\nIf $f(t,x,u)\\equiv f(t,u)$ or $f(t,x,u)$ is periodic in $x$, (H5) is automatically true.\n\n\nConsider \\eqref{main-eq}. Throughout this paper, we assume (H1) and (H2).\nFor any given $u_0$ satisfying \\eqref{initial-value},  \\eqref{main-eq} has a unique solution\n$(u(t,x;u_0,h_0),h(t;u_0,h_0))$ with $u(0,x;u_0,h_0)=u_0(x)$\nand $h(0;u_0,h_0)=h_0$ (see \\cite{DuGuPe}). By comparison principle for parabolic equations, $u(t,x;u_0,h_0)$ exists for all $t>0$ and\n$u_x(t,x;u_0,h_0)\\le 0$ for $t>0$.\nHence $h(t;u_0,h_0)$ is monotonically increasing, and therefore\nthere exists $h_{\\infty}\\in(0,+\\infty]$ such that\n$\\lim_{t\\to+\\infty}h(t;u_0,h_0)=h_{\\infty}$.\n\n\\begin{definition}[Spreading-vanishing and spreading speed]\n\\label{spreading-vanishing-def}\nConsider \\eqref{main-eq}.\n\\begin{itemize}\n\\item[(1)] For any given $u_0$ satisfying \\eqref{initial-value}, let $h_\\infty=\\lim_{t\\to\\infty}h(t;u_0,h_0)$.\n It is said that the {\\rm vanishing occurs} if $ h_\\infty<\\infty$ and\n$\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0,h_0)\\|_{C([0,h(t)])}=0$. It is said that the {\\rm spreading occurs}\n if $h_\\infty=\\infty$ and\n $\\liminf_{t\\to\\infty} u(t,x;u_0,h_0)>0$ locally uniformly in\n$x\\in [0,\\infty)$.\n\n\\item[(2)] A real number $c^*>0$ is called the {\\rm spreading speed} of \\eqref{main-eq} if for any $(u_0,h_0)$ such that\n\\eqref{initial-value} is  satisfied and the spreading occurs,\nthere holds\n$$\n\\lim_{t\\to\\infty}\\frac{h(t;u_0,h_0)}{t}=c^*\n$$\nand\n$$\n\\liminf_{0\\le x\\le c^{'}t,t\\to\\infty}u(t,x;u_0,h_0)>0,\\quad \\forall\\,\\, c^{'}<c^*.\n$$\n\\end{itemize}\n\\end{definition}\n\nBiologically, spreading means that the free boundary  $x=h(t;u_0,h_0)$ goes to infinity\nas $t\\to\\infty$ (i.e., $h_{\\infty}=\\infty)$, and population $u(t,x;u_0,h_0)$\nsuccessfully establishes itself\nin the entire space. On the other\nhand, vanishing means that the free boundary fails to move eventually, and\nthe population fails to establish and vanishes\neventually.\n\n\\subsection{Main results}\n\nIn this subsection, we state the main results of this paper. The first theorem is about the existence of time almost\nperiodic positive solution of \\eqref{aux-main-eq1}.\n\n\\begin{theorem}[Almost periodic solutions]\n\\label{main-thm1}\n Consider \\eqref{aux-main-eq1} and assume (H1)-(H5). Then there is a unique time almost periodic positive solution\n$u^*(t,x)$ of \\eqref{aux-main-eq1} and for any $u_0\\in C_{\\rm unif}^b([0,\\infty),\\RR^+)$ with $\\inf_{x\\in [0,\\infty)}u_0(x)>0$,\n$$\n\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0)-u^*(t,\\cdot)\\|_{C([0,\\infty))}=0,\n$$\nwhere $u(t,x;u_0)$ is the solution of \\eqref{aux-main-eq1} with $u(0,x;u_0)=u_0(x)$. If, in addition, $f(t,x,u)\\equiv f(t,u)$,\nthen $u^*(t,x)\\equiv V^*(t)$, where $V^*(t)$ is the unique time almost periodic positive solution of the following\nODE,\n\\begin{equation}\n\\label{ode-eq1}\n\\dot u=uf(t,u).\n\\end{equation}\n\\end{theorem}\n\n\nThe following theorem is about the spreading and vanishing dichotomy of \\eqref{main-eq}.\n\n\\begin{theorem}[Spreading-vanishing dichotomy]\n\\label{main-thm2}\nAssume (H1)-(H5). For any given $h_0>0$ and $u_0(\\cdot)$ satisfying  \\eqref{initial-value},\nthe following hold.\n\\begin{itemize}\n\\item[(1)]\nEither\n\n\n(i)\n$h_{\\infty}\\le l^*$\nand  $\\lim_{t\\to+\\infty} u(t,x;u_0,h_0)=0$\n\nor\n\n\n(ii)  $h_{\\infty}=\\infty$ and\n$\\lim_{t\\to\\infty}[u(t,x;u_0,h_0)-u^*(t,x)]=0$\nlocally uniformly for $x\\in[0,+\\infty)$, where $u^*(t,x)$ is as\nin Theorem \\ref{main-thm1}.\n\n\n\n\\item[(2)] If $h_0\\ge l^*$, then $h_\\infty=\\infty$.\n\n\\item[(3)]\nSuppose $h_{0}<l^*$. Then there exists $\\mu^{*}>0$\nsuch that spreading occurs if $\\mu>\\mu^{*}$ and vanishing occurs if $\\mu\\le \\mu^{*}$.\n\\end{itemize}\n\\end{theorem}\n\nWe remark that similar results as those in Theorems \\ref{main-thm1} and \\ref{main-thm2} hold for \\eqref{main-doub-eq}\n(see Propositions \\ref{positive-almost-periodic-solution-doub-prop} and \\ref{spreading-vanishing-doub-prop}).\n\n\\section{Preliminary}\n\nIn this section,\nwe present some preliminary results to be applied in later sections, including basic properties for principal Lyapunov exponents (see subsection 3.1),\nnon-increasing property of the so called part metric associated to diffusive KPP equations in both bounded and unbounded domains (see subsection 3.2.),\nthe asymptotic\ndynamics of   diffusive KPP equations  with time almost periodic dependence in fixed bounded environments (see subsection 3.3),  and comparison principles\nfor free boundary problems (see subsection 3.4).\n\n\n\n\n\\subsection{Principal Lyapunov exponents}\n\nConsider \\eqref{linearized-eq1}. Let $X=X(l)$, where $X(l)$ is as in \\eqref{bounded-domain-space-eq1}. We denote by\n$\\|\\cdot\\|$ either the norm in $X$ or in $\\mathcal{L}(X,X)$.\nRecall that for any $v_0\\in X$, \\eqref{linearized-eq1} has a unique solution $v(t,\\cdot;v_0,a)$ and\n$$\n\\lambda(a,l)=\\limsup_{t\\to\\infty}\\frac{\\ln\\|V(t,a)\\|_{X(l)}}{t},\n$$\nwhere $V(t,a)v_0=v(t,\\cdot;v_0,a)$.\nFor any $b\\in H(a)$, consider also\n\\begin{equation}\n\\label{linearized-eq1-1}\n\\begin{cases}\nv_t=v_{xx}+b(t,x)v,\\quad 0<x<l\\cr\nv_x(t,0)=v(t,l)=0.\n\\end{cases}\n\\end{equation}\nFor any $v_0\\in X$,\n\\eqref{linearized-eq1-1} has also a unique solution $v(t,\\cdot;v_0,b)$ with $v(0,\\cdot;v_0,b)=v_0(\\cdot)$.\n\n\\begin{lemma}\n\\label{lyapunov-exponent-lm0}\nThere is $\\phi^l:H(a)\\to X^{++}$   satisfying the following properties.\n\\begin{itemize}\n\\item[(i)] $\\|\\phi^l(b)\\|=1$ for any $b\\in H(a)$ and $\\phi^l:H(a)\\to X^{++}$ is continuous.\n\n\\item[(ii)] $v(t,\\cdot;\\phi^l(b),b)=\\|v(t,\\cdot;\\phi^l(b),b)\\|\\phi^l(b(\\cdot+t,\\cdot))$.\n\n\\item[(iii)] $\\lim_{t\\to\\infty}\\frac{\\ln \\|v(t,\\cdot;\\phi^l(b),b)\\|}{t}=\\lambda(a,l)$ uniformly in $b\\in H(a)$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nIt follows from \\cite{MiSh1} (see also \\cite{MiSh3, ShYi}).\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\\label{lyapunov-exponent-lm1}\n$\\lambda(a,l)$ is a monotone\nincreasing function of $a$ and $l$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nFor any fixed $a$, suppose\n$0<l_{1}\\leq l_{2}$. Note that $v(t,\\cdot,\\phi^{l_1}(a),a)$  and\n$v(t,\\cdot,\\phi^{l_2}(a),a)$ are solutions for the following problems, respectively,\n\\begin{equation*}\n\\begin{cases}\n\\label{linearized-eq3}\nv_t=v_{xx}+a(t,x)v, \\quad 0<x<l_{1}\\cr\nv_x(t,0)=v(t,l_{1})=0\n\\end{cases}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{cases}\n\\label{linearized-eq4}\nv_t=v_{xx}+a(t,x)v,\\quad 0<x<l_{2}\\cr\nv_x(t,0)=v(t,l_{2})=0.\n\\end{cases}\n\\end{equation*}\nChoose $0<\\epsilon \\ll 1$ such that $\\phi^{l_2}(a)>\\epsilon \\phi^{l_1}(a)$ on $[0,l_1]$. Then,  by\n comparison principle for parabolic equations, we have that\n$$v(t,x;\\epsilon\\phi^{l_1}(a),a)<v(t,x;\\phi^{l_2}(a),a)\\quad \\forall \\,\\, 0\\le x\\le l_1.$$\nBy Lemma \\ref{lyapunov-exponent-lm0} and a priori estimates for parabolic equations, we have that\n\\begin{equation*}\n\\begin{split}\n\\lambda(a,l_{2})=&\\lim_{t\\rightarrow\\infty}\\frac{\\ln\\|v(t,\\cdot,\\phi^{l_2}(a),a)\\|_{X(l_2)}}{t} \\\\\n\\geq&\\lim_{t\\rightarrow\\infty}\\frac{\\ln\\|v(t,\\cdot,\\epsilon\\phi^{l_1}(a),a)\\|_{X(l_1)}}{t} \\\\\n=&\\lim_{t\\rightarrow\\infty}\\frac{\\ln\\epsilon + \\ln\\|v(t,\\cdot,\\phi^{l_1}(a),a)\\|_{X(l_1)}}{t} \\\\\n=&\\lambda(a,l_{1})\n\\end{split}\n\\end{equation*}\nThus, $\\lambda(a,l)$ is a monotone increasing function of $l$.\n\nIf we fix $l$, we can use comparison principle and a priori estimates for parabolic\nequations again to get that $\\lambda(a,l)$ is a monotone increasing function of $a$.\n\\end{proof}\n\nIn the following, if no confusion occurs, we will write $\\phi^l(b)$ as $\\phi(b)$.\n\n\n\\begin{lemma}\n\\label{lyapunov-exponent-lm2} $\\lambda(a,l)$ is continuous in $a$.\n\\end{lemma}\n\n\\begin{proof}\nFor any $k\\ge 1$, Consider the following problem\n\\begin{equation}\n\\begin{cases}\n\\label{linearized-eq5}\nv_t=v_{xx}+v(a(t,x)\\pm \\frac{1}{k}),\\quad 0<x<l\\cr\nv_x(t,0)=v(t,l)=0.\n\\end{cases}\n\\end{equation}\nNote that\n$e^{\\pm \\frac{1}{k}t}v(t,x;\\phi(a),a)$ is a solution of\n\\eqref{linearized-eq5}. It follows from Lemma \\ref{lyapunov-exponent-lm0} that\n\\begin{equation*}\n\\begin{split}\n\\lambda(a\\pm \\frac{1}{k},l)=&\\lim_{t\\to\\infty}\n\\frac{\\ln \\|v(t,\\cdot;\\phi(a),a)\\|}{t}\\pm \\frac{1}{k} \\\\\n=&\\lambda(a,l)\\pm \\frac{1}{k}.\n\\end{split}\n\\end{equation*}\nLet $k\\to\\infty$, we can get that\n$\\lambda(a\\pm \\frac{1}{k},l)-\\lambda(a,l)\\to 0$.\nThis together with Lemma \\ref{lyapunov-exponent-lm1} implies that\n $\\lambda(a,l)$ is continuous in $a$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lyapunov-exponent-lm3} Suppose that $a(t,x)\\equiv a(t)$. Then\n$$\n\\lambda(a,l)=\\hat a+\\lambda_0(l),\n$$\nwhere $\\hat a=\\lim_{t\\to\\infty}\\frac{1}{t}\\int_0^t a(s)ds$ and $\\lambda_0(l)$ is the principal\neigenvalue of\n\\begin{equation}\n\\label{egv-eq}\n\\begin{cases}\nu_{xx}=\\lambda u,\\quad 0<x<l\\cr\nu_x(0)=u(l)=0.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\tilde v(t,x)=v(t,x)e^{-\\int_0^t a(s)ds}$.  Then \\eqref{linearized-eq1} becomes\n$$\n\\begin{cases}\n\\tilde v_t=\\tilde v_{xx},\\quad 0<x<l\\cr\n\\tilde v_x(t,0)=\\tilde v(t,l)=0.\n\\end{cases}\n$$\nIt then follows that $\\lambda(a,l)=\\hat a+\\lambda(0,l)$. It is clear that\n$\\lambda(0,l)=\\lambda_0(l)$. The lemma then follows.\n\\end{proof}\n\n\\begin{remark}\n\\label{Lyapunov-exp-rk}\n Let $a(t,x)$ be a given $C^1$ function which is  almost periodic function in $t$ uniformly in $x$ in bounded sets and $\\gamma\\in\\RR$.\nConsider\n\\begin{equation}\n\\label{aux-linearized-eq1}\n\\begin{cases}\nv_t=v_{xx}+\\gamma v_x+a(t,x)v,\\quad 0<x<l\\cr\nv(t,0)=v(t,l)=0.\n\\end{cases}\n\\end{equation}\nLet $\\tilde X(l)$ be as in \\eqref{bounded-domain-space-eq2}.\nThen, for any $v_0\\in \\tilde X(l)$, \\eqref{aux-linearized-eq1} has a unique solution\n$v(t,\\cdot;v_0,a)$ with $v(0,\\cdot;v_0,a)=v_0(\\cdot)$. Let\n$$\n\\tilde\\lambda(a,\\gamma,l)=\\limsup_{t\\to\\infty}\\frac{\\ln\\|V(t,a)\\|_{\\tilde X(l)}}{t}\n$$\nwhere $V(t,a)v_0=v(t,\\cdot;v_0,a)$.\n$\\tilde\\lambda(a,\\gamma,l)$ is called the {\\it principal Lyapunov exponent} of \\eqref{aux-linearized-eq1}.\nPrincipal Lyapunov exponent theory for \\eqref{linearized-eq1} also holds for \\eqref{aux-linearized-eq1}.\nIn particular, $\\tilde\\lambda(a,\\gamma,l)$ is continuous in $a$ and $\\gamma$.\n\n\\end{remark}\n\n\n\\subsection{Part metric associated to diffusive KPP equations}\n\nIn this subsection, we present the non-increasing property of the so called part metric associated to \\eqref{aux-main-eq1},\nand the following\ndiffusive KPP equations  with time almost periodic dependence in fixed bounded domain,\n\\begin{equation}\n\\label{fixed-boundary-main-eq1} \\begin{cases} u_t=u_{xx}+uf(t,x,u),\\quad\n0<x<l\\cr u_x(t,0)=u(t,l)=0.\n\\end{cases}\n\\end{equation}\nThroughout this subsection, we assume (H1) and (H2).\nLet\n$$\nH(f)={\\rm cl}\\{f(\\cdot+\\tau,\\cdot,\\cdot)\\,|\\, \\tau\\in\\RR\\},\n$$\nwhere the closure is taken in the open compact topology. Observe that for any $g\\in H(f)$, $g$ also satisfies (H1) and (H2).\n\nFirst, consider \\eqref{fixed-boundary-main-eq1}. For given $g\\in H(f)$, we also consider\n\\begin{equation}\n\\label{fixed-boundary-eq3} \\begin{cases} u_t=u_{xx}+ug(t,x,u),\\quad\n0<x<l\\cr u_x(t,0)=u(t,l)=0\n\\end{cases}\n\\end{equation}\n Let $X(l)$ be as in \\eqref{bounded-domain-space-eq1}.\nBy semigroup theory, for any $g\\in H(f)$ and $u_0\\in X(l)$, \\eqref{fixed-boundary-eq3}\nhas a unique (local) solution $u(t,x;u_0,g)$ with $u(0,x;u_0,g)=u_0(x)$.\nNote that $u(t,x;u_0,s):=u(t-s,x;u_0,f(\\cdot+s,\\cdot,\\cdot))$ is the solution of\n\\eqref{fixed-boundary-main-eq1} with $u(s,x;u_0,s)=u_0(x)$. By (H1) and comparison principle for\nparabolic equations, if $u_0\\in X^+(l)$, then $u(t,\\cdot;u_0,g)$ exists and $u(t,\\cdot;u_0,g)\\in X^+(l)$ for all $t>0$.\nMoreover, if $u_0\\in X^+(l)\\setminus\\{0\\}$, then $u(t,\\cdot;u_0,g)\\in X^{++}(l)$ for $t>0$.\n\n\nFor any $u_1,u_2\\in X^{++}(l)$, we can define the so called part metric, $\\rho(u_1,u_2)$, between $u_1$ and $u_2$, as follows,\n\\begin{equation}\n\\label{part-metric-eq1}\n\\rho(u_1,u_2)=\\inf\\{\\ln\\alpha\\,|\\, \\alpha\\ge 1,\\,\\, \\frac{1}{\\alpha}u_1(\\cdot)\\le u_2(\\cdot)\\le\\alpha u_1(\\cdot)\\}.\n\\end{equation}\nNote that if $u_1,u_2\\in X^{++}(l)$, then $u(t,\\cdot;u_i,g)\\in X^{++}(l)$ $(i=1,2$) for any $t>0$ and $g\\in H(f)$. Hence\n$\\rho(u(t,\\cdot;u_1,g),u(t,\\cdot;u_2,g))$ is also well defined.\n\n\nNext, consider  \\eqref{aux-main-eq1} and  consider also\n\\begin{equation}\n\\label{unbounded-eq3}\n\\begin{cases}\nu_t=u_{xx}+ug(t,x,u),\\quad 0<x<\\infty\\cr\nu_x(t,0)=0\n\\end{cases}\n\\end{equation}\nfor all $g\\in H(f)$.\n\n\nLet\n\\begin{equation}\n\\label{tidle-x-space}\n\\tilde X=\\{u\\in C([0,\\infty))\\,|\\, u\\quad \\text{is uniformly continuous and bounded on}\\,\\, \\, [0,\\infty)\\}\n\\end{equation}\nwith norm $\\|u\\|=\\sup_{x\\in[0,\\infty)}|u(x)|$ and\n$$\n\\tilde X^+=\\{u\\in \\tilde X \\,|\\, u(x)\\ge 0\\},\n$$\n$$\n\\tilde X^{++}=\\{u\\in \\tilde X \\,|\\, \\inf u(x)>0\\}.\n$$\nNote that $\\tilde X^{++}$ is not empty and is an open subset of $\\tilde X^+$.\nBy semigroup theory (see \\cite{PA}), for any $g\\in H(f)$ and $u_0\\in \\tilde X$, \\eqref{unbounded-eq3}\nhas a unique solution $u(t,x;u_0,g)$ with $u(0,x;u_0,g)=u_0(x)$.\nBy (H1) and comparison principle for parabolic equations, if $u_0\\in \\tilde X^+$, then\n$u(t,\\cdot;u_0,g)$ exists and $u(t,\\cdot;u_0,g)\\in \\tilde X^+$ for all $t>0$. Moreover, if $u_0\\in\\tilde X^{++}$, then\n$u(t,\\cdot;u_0,g)\\in \\tilde X^{++}$ for all $t>0$.\n\nFor given $u_1,u_2\\in \\tilde X^{++}$, we can also define the part metric, $\\rho(u_1,u_2)$, between $u_1$ and $u_2$ as follows,\n$$\n\\rho(u_1,u_2)=\\inf\\{\\ln\\alpha\\,|\\, \\alpha\\ge 1, \\,\\, \\frac{1}{\\alpha}u_1(\\cdot)\\le u_2(\\cdot)\\le \\alpha u_1(\\cdot)\\}.\n$$\nNote that if $u_1,u_2\\in\\tilde X^{++}$, then $\\rho(u(t,\\cdot;u_1,g),u(t,\\cdot;u_2,g))$ is well defined for $t>0$.\n\n\n\nWe now have the following proposition about the non-increasing of part metric.\n\n\\begin{proposition}\n\\label{part-metric-prop}\n\\begin{itemize}\n\n\\item[(1)] Consider \\eqref{fixed-boundary-eq3} and let $u(t,\\cdot;u_0,g)$ denote the solution of \\eqref{fixed-boundary-eq3}\nwith $u(0,\\cdot;u_0,g)=u_0(\\cdot)\\in X(l)$.\n For given $u_0,v_0\\in X^{++}(l)$ with $u_0\\not = v_0$, $\\rho(u(t,\\cdot;u_0,g),u(t,\\cdot;v_0,g))$ is strictly decreasing as $t$\nincreases.\n\n\\item[(2)] Consider \\eqref{unbounded-eq3} and let $u(t,\\cdot;u_0,g)$ denote the solution of \\eqref{unbounded-eq3}\nwith $u(0,\\cdot;u_0,g)\\in \\tilde X$.\n\\item[] (i) Given any $u_0,v_0\\in \\tilde X^{++}$ and $g\\in H(f)$, $\\rho(u(t,\\cdot;u_0,g),u(t,\\cdot;v_0,g))$\ndecreases as $t$ increases.\n\n\\item[] (ii)  For any $\\epsilon>0$, $\\sigma>0$, $M>0$,  and $\\tau>0$  with $\\epsilon<M$ and\n$\\sigma\\le \\ln \\frac{M}{\\epsilon}$, there is $\\delta>0$   such that\nfor any $g\\in H(f)$, $u_0,v_0\\in \\tilde X^{++}$ with $\\epsilon\\le u_0(x)\\le M$, $\\epsilon\\le v_0(x)\\le M$ for $x\\in\\RR^{+}$ and\n$\\rho(u_0,v_0)\\ge\\sigma$, there holds\n$$\n\\rho(u(\\tau,\\cdot;u_0,g),u(\\tau,\\cdot;v_0,g))\\le \\rho(u_0,v_0)- \\delta.\n$$\n\\end{itemize}\n\\end{proposition}\n\n\n\n\\begin{proof}   The proposition can be proved by the similar arguments as in \\cite[Proposition 3.4]{KoSh}.\nFor the completeness, we provide a proof in the following.\n\n (1)  For any $u_0,v_0\\in  X^{++}(l)$ with $u_0\\not = v_0$,  there is $\\alpha^*> 1$ such that  $\\rho(u_0,v_0)=\\ln \\alpha^{*}$\nand $\\frac{1}{\\alpha^{*}}u_0\\leq v_0\\leq \\alpha^{*} u_0$.\nBy comparison principle for parabolic equations,\n$$u(t, \\cdot;v_0,g)\\le u(t, \\cdot; \\alpha^{*}u_0,g)\\quad {\\rm for}\\quad t>0.\n$$\nLet\n $$v(t, x)=\\alpha^{*}u(t, x; u_0,g).$$\nWe then have\n\\begin{align*}\nv_{t}(t,x)&=v_{xx}(t,x)+v(t,x)g(t,x, u(t,x;u_0,g))\\\\\n& =\nv_{xx}(t,x)+v(t,x)g(t, x,v(t,x))+v(t,x)g(t, x,u(t,x;u_0,g))-v(t,x)g(t,x,v(t,x))\\\\\n&> v_{xx} (t,x)+ v(t,x)g(t, x,v(t,x))\\quad \\ for \\ all  \\ t>0,\\quad 0\\le x<l,\n\\end{align*}\nand\n$$\nv_x(t,0)=0,\\quad v(t,l)=0 \\  \\ {\\rm for} \\ {\\rm all} \\ t>0.\n$$\nBy strong comparison principle  for parabolic equations,\n$$\nu(t,x;\\alpha^* u_0,g)< \\alpha^* u(t,x;u_0,g)\\quad {\\rm for}\\quad 0\\le x < l.\n$$\nThen by Hopf lemma for parabolic equations, there is $\\tilde\\alpha^*<\\alpha^*$ such that\n$$\nu(t,x;\\alpha^* u_0,g)\\le \\tilde \\alpha^* u(t,x;u_0,g)\\quad {\\rm for}\\quad 0\\le x\\le l\n$$\nand hence\n$$\nu(t,\\cdot;v_0,g)\\leq \\tilde \\alpha^* u(t,\\cdot;u_0,g)\n$$\nfor $t>0$.\nSimilarly, we can prove that\n$$\n\\frac{1}{\\bar \\alpha^*}u(t,\\cdot;u_0,g)\\le u(t,\\cdot;v_0,g)\n$$\nfor some $\\bar \\alpha^*<\\alpha^*$ and  $t>0$. It then follows that\n$$\n\\rho(u(t,\\cdot;u_0,g),u(t,\\cdot;v_0,g))< \\rho(u_0,v_0)\\quad \\ for \\ all \\,\\, t\\ge 0\n$$\nand then\n$$\n\\rho(u(t_2,\\cdot;u_0,g),u(t_2,\\cdot;v_0,g))< \\rho(u(t_1,\\cdot;u_0,g),u(t_1,\\cdot;v_0,g))\n\\quad \\ for \\ all \\,\\, 0\\le t_1< t_2.\n$$\n\n(2) (i) It follows from the arguments in (1).\n\n(ii)\nLet $\\epsilon>0$, $\\sigma>0$, $M>0$, and $\\tau>0$ be given and $\\epsilon<M$, $\\sigma<\\ln \\frac{M}{\\epsilon}$. First,  we claim that\nthere are $\\epsilon_1>0$ and $M_1>0$ such that\nfor any $g\\in H(f)$ and $u_0\\in \\tilde{X}^{++}$ with $\\epsilon\\le u_0(x)\\le M$ for $x\\in\\RR^+$, there holds\n\\begin{equation*}\n\\label{part-metric-eq1}\n\\epsilon_1\\le u(t,x;u_0,g)\\le M_1\\quad \\ for \\ all \\,\\, t\\in[0,\\tau],\\,\\, x\\in \\RR^+.\n\\end{equation*}\nIn fact, let $\\tilde M>0$ be such that  $f(t,x,u)<0$ for $u\\ge \\tilde M$. Then for $0<\\tilde\\epsilon<\\max\\{\\epsilon,\\tilde M\\}$,\n$u(t,\\cdot;u_{\\tilde \\epsilon},g)\\le \\tilde{M}$ for all $t\\ge 0$ and $g\\in H(f)$, where $u_{\\tilde\\epsilon}(x)\\equiv \\tilde \\epsilon$.\nNote that $g(t,x,u)\\ge \\tilde \\alpha =\\inf_{t\\in\\RR,x\\in\\RR^+}f(t,x,\\tilde M)$ for $u\\le \\tilde M$.\nHence by comparison principal for parabolic equations,\n$$\n\\tilde M\\ge u(t,x;u_{\\tilde\\epsilon},g)\\ge e^{\\tilde{\\alpha} t} \\tilde \\epsilon\\quad \\ for \\ all \\,\\, t\\ge 0,\\quad x\\in \\RR^+.\n$$\nThe claim then follows.\n\nLet\n\\begin{equation*}\n\\label{part-metric-eq2}\n\\delta_1=\\epsilon_1^2 e^\\sigma (1-e^\\sigma)\\sup_{g\\in H(f),t\\in[0,\\tau],x\\in \\RR^+,u\\in[\\epsilon_1,M_1M\/\\epsilon]}g_u(t,x,u).\n\\end{equation*}\nThen $\\delta_1>0$ and there is $0<\\tau_1\\le\\tau$ such that\n\\begin{equation}\n\\label{part-metric-eq3-1}\n\\frac{\\delta_1}{2}\\tau_1<e^\\sigma\\epsilon_1\n\\end{equation}\nand\n\\begin{equation}\n\\label{part-metric-eq3-2}\n\\Big|\\frac{\\delta_1}{2}tv g_u(t,x,w)\\Big|+\\Big|\\frac{\\delta_1}{2}tg(t,x,v-\\frac{\\delta_1}{2}t)\\Big|\\le\\frac{\\delta_1}{2}\n\\end{equation}\nfor all  $g\\in H(f),\\,\\,  t\\in [0,\\tau_1],\\,\\, x\\in \\RR^+,\\,\\, v,w\\in[0,M_1M\/\\epsilon]$.\nLet\n\\begin{equation}\n\\label{part-metric-eq4}\n\\delta_2=\\frac{\\delta_1\\tau_1}{2 M_1}.\n\\end{equation}\nThen $\\delta_2<e^\\sigma$ and $0<\\frac{\\delta_2 \\epsilon}{M}<1$. Let\n\\begin{equation}\n\\label{part-metric-eq5}\n\\delta=-\\ln\\big(1-\\frac{\\delta_2\\epsilon}{M}\\big).\n\\end{equation}\nThen $\\delta>0$. We prove that  $\\delta$ defined in \\eqref{part-metric-eq5}  satisfies the property in the proposition.\n\nFor any $u_0,v_0\\in\\tilde{X}^{++}$ with $\\epsilon\\le u_0(x)\\le M$ and $\\epsilon\\le v_0(x)\\le M$ for $x\\in \\RR^+$ and\n$\\rho(u_0,v_0)\\ge\\sigma$,  there is $\\alpha^*> 1$ such that  $\\rho(u_0,v_0)=\\ln \\alpha^{*}$\nand $\\frac{1}{\\alpha^{*}}u_0\\leq v_0\\leq \\alpha^{*} u_0$.\n\n\nNote that $e^\\sigma\\le\\alpha^*\\le \\frac{M}{\\epsilon}$. Let\n $$v(t, x)=\\alpha^{*}u(t, x; u_0,g)$$\n\\begin{align*}\nv_{t}(t,x)&=v_{xx}(t,x)+v(t,x)g(t,x, u(t,x;u_0,g))\\\\\n& =\nv_{xx}(t,x)+v(t,x)g(t, x,v(t,x))+v(t,x)g(t, x,u(t,x;u_0,g))-v(t,x)g(t,x,v(t,x))\\\\\n&\\ge  v_{xx} (t,x)+ v(t,x)g(t, x,v(t,x))+\\delta_1\\quad \\forall 0<t\\le\\tau_1,\\,\\, x\\in \\RR^+.\n\\end{align*}  This together with \\eqref{part-metric-eq3-1}, \\eqref{part-metric-eq3-2}\nimplies that\n$$\n(v(t,x)-\\frac{\\delta_1}{2}t)_t\\ge \\big(v(t,x)-\\frac{\\delta_1}{2}t\\big)_{xx}+\\big(v(t,x)-\\frac{\\delta_1}{2}t\\big)g\\big(t,x,v(t,x)-\\frac{\\delta_1}{2}t\\big)\n$$\nfor $0<t\\le \\tau_1$.\nThen by comparison principle for parabolic equations,\n$$\nu(t,\\cdot;\\alpha^* u_0,g)\\leq \\alpha^* u(t,\\cdot; u_0,g)-\\frac{\\delta_1}{2}t\\quad {\\rm for}\\quad 0<t\\le\\tau_1.\n$$\nBy \\eqref{part-metric-eq4},\n$$\nu(\\tau_1,\\cdot;v_0,g)\\le (\\alpha^*-\\delta_2) u(\\tau_1,\\cdot;u_0,g).\n$$\nSimilarly, it can be proved that\n$$\n\\frac{1}{\\alpha^*-\\delta_2}u(\\tau_1,\\cdot;u_0,g)\\le u(\\tau_1,\\cdot;v_0,g).\n$$\nIt then follows that\n$$\n\\rho(u(\\tau_1,\\cdot;u_0,g),u(\\tau_1,\\cdot;v_0,g))\\le \\ln(\\alpha^*-\\delta_2) =\\ln\\alpha^*+\\ln (1-\\frac{\\delta_2}{\\alpha^*})\\le \\rho(u_0,v_0)-\\delta.\n$$\nand hence\n$$\n\\rho(u(\\tau,\\cdot;u_0,g),u(\\tau,\\cdot;v_0,g))\\le\\rho(u(\\tau_1,\\cdot;u_0,g),u(\\tau_1,\\cdot;v_0,g))\n\\le\\rho(u_0,v_0)-\\delta.\n$$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Asymptotic dynamics of diffusive KPP equations  with time almost periodic dependence on fixed bounded domain}\n\nIn this subsection, we consider the asymptotic dynamics of \\eqref{fixed-boundary-main-eq1}.\n Throughout this section, we assume that $f$ satisfies (H1) and (H2).\n\nLet $X(l)$ be as in \\eqref{bounded-domain-space-eq1} and $u(t,\\cdot;u_0,g)$ be the solution of\n\\eqref{fixed-boundary-eq3}\nwith $u(0,\\cdot;u_0,g)=u_0(\\cdot)$.\nObserve that \\eqref{fixed-boundary-eq3} generates a skew-product\nhemodynamical system,\n$$\\Pi_{t}: X^{+}(l)\\times H(f)\\to X^{+}(l)\\times H(f)$$\nof the following form:\n$$\\Pi_{t}(u_{0},g)=(u(t,\\cdot,u_{0},g), g_{t})\\quad \\forall(u_{0},g)\\in X^{+}(l)\\times H(f).$$\nThe system $\\Pi_{t}$ is strongly monotone in the sense that\n$u(t,\\cdot;u_{0},g)\\ll u(t,\\cdot;v_{0},g)$ for any $0\\leq u_{0}\\leq v_{0}$\nwith $u_{0}\\neq v_{0}$ and any $t>0$, where we write $u_{0}\\ll v_{0}$ if\n$v_{0}-u_{0}\\in X^{++}(l)$.\n\n\n\n\\begin{proposition}\n\\label{fixed-boundary-prop1}\nLet $a(t,x)=f(t,x,0)$.\n\\begin{itemize}\n\\item[(1)] If $\\lambda(a,l)< 0$, then for any $u_0\\in X^+(l)$,\n$\\|u(t,\\cdot;u_0,g)\\|\\to 0$ as $t\\to\\infty$ uniformly in $g\\in H(f)$. In particular,\n$\\|u(s+t,\\cdot;u_0,s)\\|\\to 0$ as $t\\to\\infty$ uniformly in\n$s\\in\\RR$.\n\n\\item[(2)] If $\\lambda(a,l)>0$, then\nthere exists $u^l:H(f)\\to X^{++}(l)$ such that $u^l(g)$ is continuous in $g\\in H(f)$,\n$u(t,\\cdot;u^l(g),g)=u^l(g\\cdot t)(\\cdot)$ for any $g\\in H(f)$, and for any $u_0\\in X^+(l)\\setminus\\{0\\}$,\n$$\n\\|u(t,\\cdot;u_0,g)-u(t,\\cdot;u^l(g),g)\\|\\to 0\n$$\nas $t\\to\\infty$ uniformly in $g\\in H(f)$. In particular,\n$u^{*,l}(t,x):=u(t,x;u^l(f),f)$ is almost periodic in $t\\in\\RR$ and for any $u_0\\in X^+(l)\\setminus\\{0\\}$,\n$$\n\\|u(s+t,\\cdot;u_0,s)-u^{*,l}(s+t,\\cdot)\\|\\to 0\n$$\nas $t\\to\\infty$ uniformly in $s\\in\\RR$, where $u(s+t,\\cdot;u_0,s)=u(t,\\cdot;u_0,f(\\cdot+s,\\cdot,\\cdot))$.\n\\end{itemize}\n\\end{proposition}\n\n\nThe proposition follows from \\cite[Theorem A]{MiSh2}. For completeness,\nwe provide a proof in the following.\n\n\n\\begin{proof}[Proof of Proposition \\ref{fixed-boundary-prop1}]\n(1) Let $b(t,x)=g(t,x,0)$ for any $g\\in H(f)$. Since $\\lambda(a,l)<0$,\nit is well known that $\\|v(t,x;\\phi^l(b),b)\\|\\to 0$ as $t\\to\\infty$, where $v(t,x;\\phi^l(b),b)$ is\nthe solution of \\eqref{linearized-eq1-1} with $v(0,\\cdot;\\phi^l(b),b)=\\phi^l(b)$ and\n$\\phi^l(b)$ is as in Lemma \\ref{lyapunov-exponent-lm0}.\nFor any $u_{0}\\in X^{+}(l)$, we can choose $M>0$ such that $u_{0}\\leq M\\phi^l(b)$\nfor $x\\in(0,l)$. It follows from comparison principle for parabolic equations\nthat\n$0\\leq u(t,x,u_{0},g)\\leq Mv(t,x,\\phi^l(b),b)$ for $x\\in[0,l]$.\nThis together with a priori estimates for parabolic equations implies that\n $\\|u(t,\\cdot,u_{0},g)\\|\\to 0$\nas $t\\to\\infty$.\n\n(2) Choose $\\xi>0$ such that $\\lambda(a,l)-\\xi>0$. Let\n$\\bar{a}(t,x)=f(t,x,0)-\\xi$\nand\n$\\phi^l: H(\\bar a)\\to X^{++}(l)$ be as in Lemma \\ref{lyapunov-exponent-lm0}.\nFor any $g\\in H(f)$, we choose $\\bar{b}\\in H(\\bar{a})$ such that\n$\\bar{b}(t,x)=g(t,x,0)-\\xi$. Let\n$v(t,x;\\phi^l(\\bar{b}),\\bar{b})$ be the solution of\n\\begin{equation*}\n\\begin{cases}\n\\label{linearized-eq7}\nv_t=v_{xx}+\\bar{b}(t,x)v,\\quad  0<x<l \\cr v_x(t,0)=v(t,l)=0\n\\end{cases}\n\\end{equation*}\nwith $v(0,x;\\phi^l(\\bar b),\\bar b)=\\phi^l(\\bar b)(x)$.\n\nSince $\\lambda(\\bar a,l)=\\lambda(a,l)-\\xi>0$, we can find $T>0$, such that\n$$\\|v(T,\\cdot;\\phi^l(\\bar{b}),\\bar{b})\\|\\geq1 \\, \\,\\,\\forall\\,\\, \\bar{b}\\in H(\\bar{a}).$$\nChoose $0<\\epsilon\\ll1$ such that for any $g\\in H(f)$,\n$$ug(t,x,u)\\geq(g(t,x,0)-\\xi)u \\quad {\\rm for}\\,\\,\n0\\leq u\\leq \\sup_{0\\leq t\\leq T}\\|v(t,\\cdot;\\epsilon\\phi^l(\\bar{b}),\\bar{b})\\|.$$\nUsing comparison principle for parabolic equations we obtain that\n$v(t,\\cdot;\\epsilon\\phi^l(\\bar{b}),\\bar{b})  (0\\leq t\\leq T)$ is a subsolution of\nthe problem \\eqref{fixed-boundary-eq3}. We then have\n$$\nu(T,\\cdot;\\epsilon \\phi^l(\\bar b),g)\\ge \\epsilon \\phi^l(\\bar b_{T})\n$$\nand then\n$$\nu(nT,\\cdot;\\epsilon \\phi^l(\\bar b),g)\\ge \\epsilon \\phi^l(\\bar b_{nT}).\n$$\nLet  $\\omega(\\epsilon\\phi^l(\\bar a),f)$ be the $\\omega$-limit set of $\\Pi_t(\\epsilon \\phi^l(\\bar a),f)$. We then have\n$\\omega(\\epsilon \\phi^l(\\bar a),f)\\subset X^{++}(l)\\times H(f)$.\n\nWe claim that for any $g\\in H(f)$,\nthere is unique $u^l(g)\\in X^{++}(l)$ such that $(u^l(g),g)\\in\\omega(\\epsilon \\phi^l(\\bar a),f)$.\nIn fact, if there is $g\\in H(f)$ such that there are $u_1,u_2\\in X^{++}(l)$ with\n$(u_i,g)\\in \\omega(\\epsilon \\phi^l(\\bar a),f)$ and $u_1\\not = u_2$, then $(u(t,\\cdot;u_i,g),g_t)\\in \\omega(\\epsilon \\phi^l(\\bar a),f)$ for all\n$t\\in\\RR$. By Proposition \\ref{part-metric-prop}(1), there is $\\rho_\\infty>0$ such that\n$\\rho(u(t,\\cdot;u_1,g),u(t,\\cdot;u_2,g))\\to \\rho_\\infty$ as $t\\to -\\infty$. For any $t_n\\to -\\infty$, without loss of\ngenerality, assume that\n$g_{t_n}\\to g^*$ and $u(t_n\\cdot;u_i,g)\\to u_i^*$. Then\n$$\nu(t,\\cdot;u_i^*,g^*)=\\lim_{n\\to\\infty}u(t+t_n,\\cdot;u_i,g)\n$$\nand\n$$\n\\rho(u(t,\\cdot;u_1^*,g^*),u(t,\\cdot;u_2^*,g^*))=\\lim_{n\\to\\infty}\\rho(u(t+t_n,\\cdot;u_1,g),u(t+t_n,\\cdot;u_2,g))=\\rho_\\infty\n$$\nfor all $t\\in\\RR$, which contradicts to Proposition \\ref{part-metric-prop}(1). Therefore,\nthe claim holds and $u^l:H(f)\\to X^{++}$ is continuous. In particular,\n$u^{*,l}(t,x)=u(t,x;u^l(f),f)$ is an almost periodic solution.\nMoreover, by the above arguments, for any $u_0\\in X^{++}$,\n$\\omega(u_0,f)=\\omega(\\epsilon \\phi^l(\\bar a),f)$ and then\n$$\n\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0,f)-u^{*,l}(t,\\cdot)\\|=0.\n$$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison principal for free boundary problems}\n\nIn order for later application, we need a comparison principle which can\nbe used to estimate both $u(t,x)$ and the free boundary $x=h(t)$.\n\\begin{proposition}\n\\label{comparison-principle} Suppose that $T\\in (0,\\infty)$, $\\bar{h}\n\\in C^{1}([0,T])$, $\\bar{u}\\in C(\\bar{D}^{*}_{T})\\cap C ^{1,2}(D^{*}_{T})$\nwith $D^{*}_{T}=\\{(t,x)\\in\\mathbb R^{2}:0<t\\leq T, 0<x<\\bar{h}(t)\\}$, and\n\\begin{equation*}\n\\begin{cases}\n\\bar{u}_t\\geq\\bar{u}_{xx}+\\bar{u}f(t,x,\\bar{u}),\\quad &t>0, 0<x<\\bar{h}(t)\n\\cr \\bar{h}^{'}(t)\\geq-\\mu\\bar{u}_x(t,\\bar{h}(t)),\\quad &t>0, \\cr\n\\bar{u}_x(t,0)\\leq 0,u(t,\\bar{h}(t))=0,\\quad  &t>0.\n\\end{cases}\n\\end{equation*}\nIf $h_{0}\\leq \\bar{h}(0)$ and $u_{0}(x)\\leq\\bar{u}(0,x)$ in $[0,h_{0}]$,\nthen the solution $(u,h)$ of the free boundary problem \\eqref{main-eq}\nsatisfies\n$$h(t)\\leq \\bar{h}(t)  \\ for\\ all\\ t\\in(0,T],\\ u(t,x)\\leq\\bar{u}(x,t) \\ for \\\nt\\in(0,T]\\ and\\ x\\in(0,h(t)).$$\n\n\\begin{proof}\nThe proof of this Proposition is similar to that of Lemma 3.5 in\n\\cite{DuLi} and Lemma 2.6 in \\cite{DuGu}.\n\\end{proof}\n\n\\begin{remark}\nThe pair $(\\bar{u},\\bar{h})$ in Proposition \\ref{comparison-principle}\nis called an upper solution of the free boundary problem. We can\ndefine a lower solution by reversing all the inequalities in the\nobvious places.\n\n\\end{remark}\n\n\\end{proposition}\n\n\\begin{proposition}\n\\label{comparison-principle1} Suppose that $T\\in (0,\\infty)$, $\\bar{g}, \\bar{h}\n\\in C^{1}([0,T])$, $\\bar{u}\\in C(\\bar{D}^{*}_{T})\\cap C ^{1,2}(D^{*}_{T})$\nwith $D^{*}_{T}=\\{(t,x)\\in\\mathbb R^{2}:0<t\\leq T, \\bar{g}(t)<x<\\bar{h}(t)\\}$, and\n\\begin{equation*}\n\\begin{cases}\n\\bar{u}_t\\geq\\bar{u}_{xx}+\\bar{u}f(t,x,\\bar{u}),\\quad &t>0, \\bar{g}(t)<x<\\bar{h}(t)\n\\cr \\bar{u}(t, \\bar{h}(t))=0, \\bar{h}^{'}(t)\\geq-\\mu\\bar{u}_x(t,\\bar{h}(t)),\\quad &t>0 \\cr\n\\bar{u}(t, \\bar{g}(t))=0, \\bar{g}^{'}(t)\\leq -\\mu\\bar{u}_x(t,\\bar{g}(t)),\\quad  &t>0.\n\\end{cases}\n\\end{equation*}\nIf $[g_0, h_0] \\subset [\\bar{g}(0), \\bar{h}(0)]$ and $u_{0}(x)\\leq\\bar{u}(0,x)$ in $[g_0,h_0]$,\nthen the solution $(u,g,h)$ of the free boundary problem \\eqref{main-doub-eq} satisfies\n$$g(t)\\ge \\bar{g}(t), h(t)\\leq \\bar{h}(t)  \\ for\\ all\\ t\\in(0,T],\\ u(t,x)\\leq\\bar{u}(x,t) \\ for \\\nt\\in(0,T]\\ and\\ x\\in(g(t),h(t)).$$\n\n\\begin{proof}\nThe proof of this Proposition only requires some obvious modifications as in\nProposition \\ref{comparison-principle}.\n\\end{proof}\n\n\\end{proposition}\n\n\n\\begin{proposition}\n\\label{global-existence} For any given $h_0>0$ and $u_0\\ge\n0$, $(u(t,x;u_0,h_0), h(t;u_0,h_0))$ exists for all $t\\ge 0$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof is similar to that of Theorem 4.3 in \\cite{DuGu}.\n\\end{proof}\n\n\\begin{remark}\nFrom the uniqueness of the solution to \\eqref{main-eq} and\nsome standard compactness argument, we can obtain that the\nunique solution $(u,h)$ depends continuously on $u_{0}$ and\nthe parameters appeared in \\eqref{main-eq}.\n\\end{remark}\n\n\n\n\\section{Asymptotic Dynamics of Diffusive KPP Equations on\n Fixed Unbounded Domain and Proof of Theorem \\ref{main-thm1}}\n\n\nIn this section, we consider the asymptotic dynamics of \\eqref{aux-main-eq1} and\nprove Theorem \\ref{main-thm1}. Throughout this section, we assume that  $f$ satisfies (H1)-(H5).\nWe let $\\tilde X$ be as in \\eqref{tidle-x-space} and\n$u(t,\\cdot;u_0,g)$ be the solution of \\eqref{unbounded-eq3} with $u(0,\\cdot;u_0,g)=u_0(\\cdot)\\in\\tilde X$.\nThe main results of this section are stated in the following proposition.\n\n\n\\begin{proposition}\n\\label{unbounded-prop1}\nAssume (H1)-(H5). There is $u^*:H(f)\\to \\tilde X^{++}$ satisfying the following properties.\n\\begin{itemize}\n\\item[(1)] (Almost periodicity in time)  $u^*(g)(x)$ is continuous in $g\\in H(f)$ in open compact topology with respect to $x$\n(that is, if $g_n\\to g$ in $H(f)$, then $u^*(g_n)(x)\\to u^*(g)(x)$ locally uniformly in $x$)\nand  $u(t,x;u^*(g),g)=u^*(g\\cdot t)(x)$ (hence $u^*(g\\cdot t)(x)$ is an almost periodic solution of \\eqref{unbounded-eq3}).\n\n\\item[(2)] (Stability) For any $u_0\\in\\tilde X^{++}$,\n$$\n\\|u(t,\\cdot;u_0,g)-u^*(g(\\cdot+t,\\cdot,\\cdot))(\\cdot)\\|_{\\tilde X}\\to 0\n$$\nas $t\\to\\infty$ uniformly in $g\\in H(f)$.\n\n\\item[(3)] (Uniqueness) For given $g\\in H(f)$, if $\\tilde u^{*}(t,x)$ is an entire positive solution of \\eqref{unbounded-eq3}, and $\\inf_{t\\in\\RR,x\\in\\RR^+}\\tilde u^{*}(t,x)>0$, then $\\tilde u^{*}(t,x)=u(t,x;u^*(g),g)$.\n\n \\item[(4)] (Spatial homogeneity) If $f(t,x,u)\\equiv f(t,u)$, then $u^*(g)(x)$ is independent of $x$ and\n $V^*(t;g)=u^*(g\\cdot t)$ is the unique time almost periodic solution of\n \\begin{equation}\n \\label{ode-eq2}\n u_t=ug(t,u).\n \\end{equation}\n\\end{itemize}\n\\end{proposition}\n\n\n\\begin{proof} [Proof of  Theorem \\ref{main-thm1}]\nLet $u^*(t,x)=u^*(f \\cdot t)(x)$, where $u^*(f \\cdot t)$ is as in\n Proposition \\ref{unbounded-prop1}. Theorem \\ref{main-thm1} then follows.\n \\end{proof}\n\n We  remark that\n the existence and uniqueness of positive solutions which are bounded away from $0$\nof KPP equations in heterogeneous unbounded domains have been studied in \\cite{BeHaNa} (see \\cite[Propositions 1.7, 1.8]{BeHaNa}).\nThe almost periodicity and stability results in Proposition \\ref{unbounded-prop1} are new.\n\n\n\n\nTo prove Proposition \\ref{unbounded-prop1}, we first prove two lemmas.\n\nFor any $L\\ge L^*$ and $y\\ge y^*$, consider\n\\begin{equation}\n\\label{unbounded-domain-eq1}\n\\begin{cases}\nu_t=u_{xx}+u g^y(t,x,u),\\quad 0<x<L\\cr\nu(t,0)=u(t,L)=0,\n\\end{cases}\n\\end{equation}\nwhere $ g^y(t,x,u)=g(t,x+y,u)$ for $0\\le x\\le L$.\nBy (H4),\n$\\tilde \\lambda(g^y(\\cdot,\\cdot,0),L)>0$ for $y\\ge y^*$. Then by the arguments of Proposition \\ref{fixed-boundary-prop1},\n\\eqref{unbounded-domain-eq1} has a unique time almost periodic\n positive solution $u^*(t,x;g,y,L)$.\nNote that\n$$\nu^*(t,x;g,y,L)=u^*(0,x;g\\cdot t,y,L).\n$$\n\n\n\\begin{lemma}\n\\label{unbounded-domain-lm2}\nAssume (H1)-(H5). Fix a $L\\ge L^*$. Then\n\\begin{equation}\n\\label{bounded-away-from-zero-eq1}\n\\inf_{y\\ge y^*, L\/4\\le x\\le 3L\/4, g\\in H(f)}u^*(0,x;g,y,L)>0.\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nAssume that \\eqref{bounded-away-from-zero-eq1} does not hold. Then there are $y_n\\ge y^*$, $g_n\\in H(f)$, and $x_n\\in [L\/4,3L\/4]$ such that\n$$\n\\lim_{n\\to\\infty} u^*(0,x_n;g_n,y_n,L)=0.\n$$\nBy (H5), without loss of generality, we may assume that\n$$\ng_n^{y_n}(t,x,u)\\to g^*(t,x,u)\n$$\nuniformly in $t\\in\\RR$ and $(x,u)$ in bounded sets.\n\n\nLet $a_n(t,x)=g_n^{y_n}(t,x,0)$ and $a^*(t,x)=g^*(t,x,0)$. By (H2), $g^*$ is almost periodic in $t$.\nThen\n$$\n\\tilde \\lambda(a_n,L)\\to\\tilde  \\lambda(a^*,L)\n$$\nand hence $\\tilde \\lambda(a^*,L)>0$. Note that  for any $\\epsilon>0$,\n$$\ng_n^{y_n}(t,x,u)\\ge g^*(t,x,u)-\\epsilon\\quad \\forall\\,\\, n\\gg 1,\\,\\, 0\\le x\\le L.\n$$\nand\n\\begin{equation*}\n\\label{unbounded-domain-eq2}\n\\begin{cases}\nu_t=u_{xx}+u\\big( g^*(t,x,u)-\\epsilon\\big),\\quad 0<x<L\\cr\nu(t,0)=u(t,L)=0\n\\end{cases}\n\\end{equation*}\nhas a unique positive almost periodic solution $\\tilde u^{L}(t,x)$ with\n$\\inf_{ L\/4\\le x\\le  3L\/4,t\\in\\RR} \\tilde u^{L}(t,x)>0$.  By comparison principle for parabolic equations,\nwe have\n$$\nu^*(t,x;g_n,y_n,L)\\ge \\tilde u^{L}(t,x)\\quad \\forall\\,\\, n\\gg 1.\n$$\nThis implies that\n$$\nu^*(0,x_n;g_n,y_n,L)\\not \\to 0\n$$\nas $n\\to\\infty$, which is a contradiction. Hence\n$$\\inf_{y\\ge y^*, L\/4\\le x\\le 3L\/4, g\\in H(f)}u^*(0,x;g,y,L)>0.$$\n\\end{proof}\n\n\\begin{lemma}\n\\label{unbounded-domain-lm3}\nAssume (H1)-(H5). Let $u_0\\equiv M(\\gg 1)$. Then\n$u(t,\\cdot;u_0,g\\cdot(-t))$ decreases as $t$ increases. Let $u^*(g)(x)=\\lim_{t\\to\\infty} u(t,x;u_0,g\\cdot(-t))$ for\n$x\\in  [0,\\infty)$.\nThen $u(t,\\cdot;u^*(g),g)=u^*(g\\cdot t)(\\cdot)$ and $\\inf_{x\\in\\RR^+,g\\in H(f)}u^*(g)(x)>0$.\n\\end{lemma}\n\n\n\\begin{proof}\nFirst of all, by comparison principle for parabolic equations, we have $u(t,\\cdot;u_0,g)\\le u_0$ for any $t>0$ and $g\\in H(f)$. Hence\n$$u(t+s,\\cdot;u_0,g\\cdot(-t-s))=u(t,\\cdot;u(s,\\cdot;u_0,g\\cdot(-t-s)),g\\cdot(-t))\\le u(t,\\cdot;u_0,g\\cdot(-t))$$\nfor any $t,s\\ge 0$. Therefore, $u(t,\\cdot;u_0,g\\cdot(-t))$ decreases as $t$ increases.\nLet\n$$\nu^*(g)(x)=\\lim_{t\\to \\infty}u(t,x;u_0,g\\cdot(-t))\\quad \\forall \\, x\\in [0,\\infty).\n$$\n\nNext,  for any $g\\in H(f)$ and $y\\ge y^*$,\n$$\nu(t,x+y;u_0,g\\cdot(-t))\\ge u^*(t,x;g\\cdot(-t),y,L)\\quad \\forall\\,\\, 0<x<L.\n$$\nIt then follows that $\\inf_{x> y^*+L\/4,g\\in H(f)}u^*(g)(x)>0$.\nChoose $l>y^*+L\/4$ and fix it.  By Proposition \\ref{fixed-boundary-prop1},\n$$\nu(t,x;u_0,g\\cdot (-t))\\ge u^l(g)(x)\\quad {\\rm for}\\quad 0\\le x\\le l.\n$$\nNote that\n$$\n\\inf_{g\\in H(f),0\\le x\\le y^*+L\/4}u^l(g)(x)>0.\n$$\nIt then follows that\n$$\n\\inf_{x\\ge 0,g\\in H(f)} u^*(g)(x)>0.\n$$\n\nNow, note that\n$$\nu(s,x;u_0,g\\cdot(-s))\\to u^*(g)(x) \\ as \\ s\\to\\infty\n$$\nuniformly in bounded sets. This implies that\n\\begin{align*}\nu(t,x;u^*(g),g)&=\\lim_{s\\to\\infty} u(t,x;u(s,\\cdot;u_0,g\\cdot(-s)),g)\\\\\n&=\\lim_{s\\to\\infty}u(t+s,x;u_0,g\\cdot(-s))\\\\\n&=\\lim_{s\\to\\infty} u(t+s,x;u_0,(g\\cdot t)\\cdot(-t-s))\\\\\n&=u^*(g\\cdot t)(x)\n\\end{align*}\nuniformly in bounded sets. The lemma is thus proved.\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{unbounded-prop1}]\n(1) Let $u^*(g)$ be as in Lemma \\ref{unbounded-domain-lm3} for $g\\in H(f)$. We prove that $g\\mapsto u^*(g)$ satisfies the conclusions in (1).\n\nFirst, assume that $g_n\\to g^*$ as $n\\to\\infty$. By regularity and\na priori estimates for parabolic equations, there is\n$n_k\\to \\infty$ such that\n$$\nu^*(g_{n_k})(x)\\to u^{**}(x)\n$$\nuniformly in bounded sets.\nWe prove that\n$u^{**}(x)=u^*(g^*)(x)$. Suppose that $u^{**}(x)\\not \\equiv u^*(g^*)(x)$. Note that\n$u(t,x;u^{**},g^*)$ and $u(t,x;u^*(g^*),g^*)$ exist globally (i.e. exist for all $t\\in\\RR$)\nand\n$$\\inf u(t,x;u^{**},g^*)>0,\\quad \\inf u(t,x;u^*(g^*),g^*)>0.\n$$\n Therefore,\n $$\n \\sup_{t\\in\\RR} \\rho(u(t,\\cdot;u^{**},g^*),u(t,\\cdot;u^*(g^*),g^*)<\\infty\n $$\n and there is $\\rho^*>0$ such that\n$$\n\\rho(u^{**}(\\cdot),u^*(g)(\\cdot))=\\rho^*.\n$$\nThen by Proposition \\ref{part-metric-prop}(2),\nfor any $\\tau>0$ there is $\\delta>0$ such that\n\\begin{align*}\n\\rho^*\\leq\\rho(u(-n\\tau,\\cdot;u^{**},g^*),u(-n\\tau,\\cdot;u^*(g^*),g^*))-n\\delta\n\\ for \\ n\\in\\NN.\n\\end{align*}\nLetting $n\\to\\infty$, we get a contradiction.\nHence $u^{**}(\\cdot)=u^*(g^*)(\\cdot)$ and\n$u^*(g)(x)$ is continuous in $g$ in open compact topology with respect to $x$.\n\nNext, by Lemma \\ref{unbounded-domain-lm3},\n we have that, for any $g\\in H(f)$, $u(t,\\cdot;u^*(g),g)=u^*(g\\cdot t)(\\cdot)$.\n\n\nWe prove  now that $u^*(g\\cdot t)(x)$ is almost periodic in $t$ uniformly in $x$ in bounded sets.\nNote  that for any given $\\{\\alpha_n^{'}\\}\\subset\\RR$ and $\\{\\beta_n^{'}\\}\\subset\\RR$,\nthere are $\\{\\alpha_n\\}\\subset\\{\\alpha_n^{'}\\}$ and $\\{\\beta_n\\}\\subset\\{\\beta_n^{'}\\}$  such that\n$\\lim_{n\\to\\infty}\\lim_{m\\to\\infty}g(t+\\alpha_n+\\beta_m,x,u)=\\lim_{n\\to\\infty}g(t+\\alpha_n+\\beta_n,x,u)$ for $(t,x,u)\\in\\RR^3$.\nAssume $\\lim_{m\\to\\infty}g(t+\\beta_m,x,u)=g^*(t,x,u)$ and $g^{**}(t,x,u)=\\lim_{n\\to\\infty}g(t+\\alpha_n+\\beta_n,x,u)$.\nIt then follows that\n$$\n\\lim_{m\\to\\infty} u(t+\\beta_m,x;u^*(g),g)=u^*(g^*\\cdot t)(x)\n$$\nuniformly in $x$ in bounded sets,\n\\begin{align*}\n\\lim_{n\\to\\infty}\\lim_{m\\to\\infty} u(t+\\alpha_n+\\beta_m,x;u^*(g),g)&=\\lim_{n\\to\\infty} u(\\alpha_n,x;u^*(g^*\\cdot t),g^*\\cdot t)\\\\\n&=\\lim_{n\\to\\infty}u(t,x;u^*(g^*\\cdot\\alpha_n),g^*\\cdot \\alpha_n)\\\\\n&=u^*(g^{**}\\cdot  t)(x)\n\\end{align*}\nuniformly in $x$ in bounded sets, and\n$$\n\\lim_{n\\to\\infty} u(t+\\alpha_n+\\beta_n,x;u^*(g),g)=u^*(g^{**}\\cdot t)(x)\n$$\nuniformly in $x$ in bounded set. Therefore\n$\\lim_{n\\to\\infty}\\lim_{m\\to\\infty} u(t+\\alpha_n+\\beta_m,x;u^*(g),g)=\\lim_{n\\to\\infty} u(t+\\alpha_n+\\beta_n,x;u^*(g),g)$.\nBy regularity and a priori estimates for parabolic equations,\n$u(t,x;u^*(g),g)$ is uniformly continuous in $t\\in\\RR$ and $x\\in\\RR^+$.\nHence, $u^*(g\\cdot t)(x)$ is almost periodic in $t$ uniformly in $x$ in bounded set.\n\n(2) For any $u_0\\in\\tilde{X}^{++}$ and $g\\in H(f)$.\nBy Proposition \\ref{part-metric-prop}(2),\n$\\rho(u(t,\\cdot;u_0,g),u^*(g\\cdot t)(\\cdot))$ decreases as $t$ increases.\nIt suffices to prove that\n$\\lim_{t\\to\\infty}\\rho(u(t,\\cdot;u_0,g),u^*(g\\cdot t)(\\cdot))=0$.\nSuppose that this is not true. Then there are $t_n\\to\\infty$, $g^*\\in H(f)$, $u^{**},\\tilde u^{**}\\in \\tilde X^{++}$\nwith $u^{**}\\neq\\tilde u^{**}$\n such that $g\\cdot t_n\\to g^*$,\n$u^*(g\\cdot t_n)(x)\\to u^{**}(x)$ and $u(t_n,\\cdot;u_0,g)\\to \\tilde u^{**}(x)$ locally uniformly in $x\\ge 0$.\nNote that $u(t,\\cdot;u^{**},g^*)$ and $u(t,\\cdot;\\tilde u^{**},g^*)$ exists for all $t\\in\\RR$,\n$$\n\\sup_{t\\in\\RR}\\rho(u(t,\\cdot;u^{**},g^*),u(t,\\cdot;\\tilde u^{**},g^*))<\\infty,\n$$\nand there is $\\rho^*>0$ such that\n$$\n\\rho(u^{**},\\tilde u^{**})=\\rho^*.\n$$\nBy the arguments in (1) and Proposition \\ref{part-metric-prop}(2), $u^{**}=\\tilde u^{**}$, a contradiction. Therefore\n$$\\lim_{t\\to\\infty}\\rho(u(t,\\cdot;u_0,g),u^*(g\\cdot t)(\\cdot))=0$$ and then\n$$\n\\lim_{t\\to\\infty} \\|u(t,x;u_0,g)-u^*(g)(x)\\|_{\\tilde X}=0\n$$\nuniformly in $g\\in H(f)$.\n\n(3) Suppose that $\\tilde u^{*}(t,x)$ is an entire positive solution of \\eqref{unbounded-eq3}, and $\\inf_{t\\in\\RR,x\\in\\RR^+}\\tilde u^{*}(t,x)>0$.\nAssume  $\\tilde u^*(0,x)\\not \\equiv u^*(g)(x)$. By the arguments in (1) and Proposition \\ref{part-metric-prop}(2), there is\n$\\delta_1>0$ such that\n$$\n\\rho(\\tilde u^*(0,\\cdot),u^*(g)(\\cdot))\\le \\rho(\\tilde u^*(-n,\\cdot),u^*(g\\cdot(-n))(\\cdot))-n\\delta_1\n$$\nfor $n\\ge 1$. Letting $n\\to\\infty$, we get a contradiction. Therefore $\\tilde u^*(0,x)\\equiv u^*(g)(x)$ and then $\\tilde u^*(t,x)\\equiv\nu^*(g\\cdot t)(x)$.\n\n(4) It follows from the fact that, if $f(t,x,u)\\equiv f(t,u)$, then  for $u_0\\equiv M$, $u(t,x;u_0,g)$ is independent of $x$.\n\\end{proof}\n\n\n\n\n\\section{Spreading-Vanishing Dichotomy in Diffusive KPP Equations with a Free Boundary and Proof of Theorem \\ref{main-thm2}}\n\nIn this section, we study the spreading\/vanishing scenario of \\eqref{main-eq} and prove Theorem \\ref{main-thm2}.\nThroughout this section, we assume (H1)-(H5).\n\nWe first prove a lemma. For any given $h_0>0$ and  $u_0$ satisfying \\eqref{initial-value},\nrecall that $(u(t,x;u_0,h_0)$, $h(t;u_0,h_0))$ is the solution of \\eqref{main-eq},\nand  $x=h(t;u_0,h_0)$ is increasing, and therefore\nthere exists $h_{\\infty}\\in(0,+\\infty]$ such that\n$\\lim_{t\\to+\\infty}h(t;u_0,h_0)=h_{\\infty}$.\n To stress the dependence of $h(t;u_0,h_0)$ on $\\mu$, we now\nwrite $h_{\\mu}(t;u_0,h_0)$ instead of $h(t;u_0,h_0)$ and $h_\\infty(\\mu)$\ninstead of $h_\\infty$ in the following. If no confusion occurs, we write $h_\\mu(t;u_0,h_0)$ as $h_\\mu(t)$.\n\n\n\\begin{lemma}\n\\label{monotone-increase}\nFor any $t\\in(0,+\\infty)$, $h_{\\mu}(t)$ is a strictly\n increasing function of $\\mu$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $0<\\mu_{1}<\\mu_{2}$. Let $(u_{1},h_{\\mu_{1}})$\nand $(u_{2},h_{\\mu_{2}})$ are the solutions of the following\nfree boundary problems\n\\begin{equation*}\n\\label{free-eq1}\n\\begin{cases}\nu_t=u_{xx}+uf(t,x,u),\\quad &t>0, 0<x<h_{\\mu_{1}}(t)\\cr\nh_{\\mu_{1}}^{'}(t)=-\\mu_{1} u_x(t,h_{\\mu_{1}}(t)),\\quad &t>0 \\cr\nu_x(t,0)=u(t,h_{\\mu_{1}}(t))=0,\\quad &t>0 \\cr\nh_{\\mu_{1}}(0)=h_{0},u(0,x)=u_{0}(x),\\quad &0<x\\leq h_{0}\n\\end{cases}\n\\end{equation*}\nand\n\\begin{equation*}\n\\label{free-eq2}\n\\begin{cases}\nu_t=u_{xx}+uf(t,x,u),\\quad &t>0, 0<x<h_{\\mu_{2}}(t)\\cr\nh_{\\mu_{2}}^{'}(t)=-\\mu_{2} u_x(t,h_{\\mu_{2}}(t)),\\quad &t>0 \\cr\nu_x(t,0)=u(t,h_{\\mu_{2}}(t))=0,\\quad &t>0 \\cr\nh_{\\mu_{2}}(0)=h_{0},u(0,x)=u_{0}(x),\\quad &0<x\\leq h_{0}.\n\\end{cases}\n\\end{equation*}\nSince $0<\\mu_{1}<\\mu_{2}$, then we have\n$$h_{\\mu_{1}}^{'}(t)=-\\mu_{1} u_x(t,h_{\\mu_{1}}(t))\n<-\\mu_{2} u_x(t,h_{\\mu_{1}}(t)).$$\nBy Proposition \\ref{comparison-principle},\nwe can obtain\n$$h_{\\mu_{1}}(t)\\leq h_{\\mu_{2}}(t)  \\ \\ for \\\nt\\in[0,+\\infty).$$\n\nNow we prove $h_{\\mu}(t)$ is strictly increasing.\nIf not, then we can find a first $t^{*}>0$ such\nthat $h_{\\mu_{1}}(t)<h_{\\mu_{2}}(t)$ for\n$t\\in(0,t^{*})$ and $h_{\\mu_{1}}(t^{*})=\nh_{\\mu_{2}}(t^{*})$. It follows that\n$$h'_{\\mu_{1}}(t^{*})\\geq h'_{\\mu_{2}}(t^{*}).$$\nCompare $u_{1}$ and $u_{2}$ over the region\n$$\\Omega_{t^{*}}:=\\{(t,x)\\in\\RR: 0<t\\leq t^{*},\n0\\leq x <h_{\\mu_{1}}(t)\\}.$$\nThe Strong maximum principle yields $u_{1}(t,x)<\nu_{2}(t,x)$ in $\\Omega_{t^{*}}$. Hence $w(t,x):=\nu_{2}(t,x)-u_{1}(t,x)>0$ in $\\Omega_{t^{*}}$ with\n$w(t^{*},h_{\\mu_{1}}(t^{*}))=0$. It follows that\n$w_{x}(t^{*},h_{\\mu_{1}}(t^{*}))< 0$, from which\nwe deduce, in view of\n$(u_{1})_{x}(t^{*},h_{\\mu_{1}}(t^{*}))<0$ and\n$\\mu_{1}<\\mu_{2}$, that\n$$-\\mu_{1}(u_{1})_{x}(t^{*},h_{\\mu_{1}}(t^{*}))<\n-\\mu_{2}(u_{2})_{x}(t^{*},h_{\\mu_{2}}(t^{*})).\n$$\nThus $h'_{\\mu_{1}}(t^{*})<h'_{\\mu_{2}}(t^{*})$.\nBut this is a contradiction, which proved our conclusion\nthat $h_{\\mu_{1}}(t)<h_{\\mu_{2}}(t)$ for all $t>0$.\n\\end{proof}\n\n\\begin{remark}\nIf we consider \\eqref{main-doub-eq}, for any $t\\in(0,+\\infty)$,\nby Proposition \\ref{comparison-principle1} and using the same\nargument as Lemma \\ref{monotone-increase} we have $g_\\mu(t)$\nis a strictly monotone decreasing function of $\\mu$.\n\\end{remark}\n\nWe now prove Theorem \\ref{main-thm2}.\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm2}]\n\n(1)(i) Suppose that $h_\\infty<\\infty$. First of all, we claim that\n $h^{'}(t;u_0,h_0)\\to 0$ as $t\\to\\infty$. Assume that the claim is not true. Then there is $t_n\\to \\infty$ ($t_n\\ge 1$) such that\n $\\lim_{n\\to\\infty}h^{'}(t_n;u_0,h_0)>0$. Let $h_n(t)=h(t+t_n;u_0,h_0)$ for $t\\ge -1$.\n  Note that $h_n(t)\\to h_\\infty$ as $n\\to\\infty$ uniformly in $t\\ge -1$.\n  By \\cite[Theorem 2.1]{DuLi},  $\\{h_n^{'}(t)\\}$ is uniformly bounded and equicontinuous on $[-1,\\infty)$.  We then may assume that\n  there is a continuous function $h^*(t)$ such that $h_n^{'}(t)\\to h^*(t)$ as $n\\to\\infty$ uniformly in $t$ in bounded sets of $[-1,\\infty)$.\n  It then follows that $h^*(t)=\\frac{d h_\\infty}{dt}\\equiv 0$ and then $\\lim_{n\\to\\infty}h^{'}(t_n;u_0,h_0)=0$, which is a contradiction.\n  Hence the claim holds.\n\n\n\n\n By regularity and a priori estimates for parabolic equations,\n for any sequence $t_n\\to\\infty$, there are $t_{n_k}\\to\\infty$ and\n$u^*\\in C^1(\\RR\\times [0,h_\\infty])$ and $g^*\\in H(f)$ such that\n$$\nf\\cdot t_{n_k}\\to g^*\n$$\nand\n$$\n\\|u(t+t_{n_k},\\cdot;u_0,h_0)-u^*(t,\\cdot)\\|_{C^1([0,h(t+t_{n_k})])}\\to 0\n$$\nas $t_{n_k}\\to\\infty$. Moreover, we have that $u^*(t,x)$ is an entire solution\nof\n\\begin{equation}\n\\label{limit-eq1}\n\\begin{cases}\nu_t=u_{xx}+u g^*(t,x,u),\\quad 0<x<h_\\infty\\cr\nu_x(t,0)=u(t,h_\\infty)=0.\n\\end{cases}\n\\end{equation}\n\n\n\nNext, we show that $h_{\\infty}<\\infty$\nimplies $h_{\\infty}\\leq l^*$.\nAssume that $h_{\\infty}\\in(l^*,\\infty)$. Then\n there exists $\\tilde{T}>0$ such that $h(t)>h_{\\infty}-\\epsilon>l^*$\nfor all $t\\geq\\tilde{T}$ and some small $\\epsilon>0$.\nConsider\n\\begin{equation}\n\\label{limit-eq2}\n\\begin{cases}\nv_t=v_{xx}+vf(t,x,v),\\quad  0< x< h_{\\infty}-\\epsilon \\cr\nv_{x}(t,0)=v(t,h_{\\infty}-\\epsilon)=0.\n\\end{cases}\n\\end{equation}\nBy comparison principle for parabolic equations,\n$$\nu(t+\\tilde T,\\cdot;u_0,h_0)\\ge v(t+\\tilde{T},\\cdot;u(\\tilde T,\\cdot;u_0,h_0),\\tilde T)\\quad {\\rm for}\\quad t\\ge 0,\n$$\nwhere $v(t+\\tilde{T},\\cdot;u(\\tilde T,\\cdot;u_0,h_0),\\tilde T)$ is the solution of \\eqref{limit-eq2}\nwith $u(\\tilde T,\\cdot;u(\\tilde T,\\cdot;u_0,h_0),\\tilde T)=u(\\tilde T,\\cdot;u_0,h_0)$.\nBy Proposition \\ref{fixed-boundary-prop1}, \\eqref{limit-eq2} has a unique time almost periodic positive solution $v_{h_{\\infty}-\\epsilon}(t,x)$.\nMoreover, for any $v_0\\ge 0$ and $v_0\\not\\equiv 0$,\n\\begin{equation}\n\\label{limit-eq3}\n\\|v(t+\\tilde T,\\cdot;v_0,\\tilde T)-v_{h_{\\infty-\\epsilon}}(t+\\tilde T,\\cdot)\\|\\to 0\n\\end{equation}\nas $t\\to\\infty$.  By \\eqref{limit-eq3} and\ncomparison principle for parabolic equations,\n$$\nu^*(t,x)>0\\quad \\forall\\,\\, t\\in\\RR,\\,\\, x\\in (0,h_\\infty).\n$$\nIt then follows that $u^*_x(t,h_\\infty)<0$. This implies that\n$$\n\\limsup_{t\\to\\infty}u_x(t,h(t);u_0,h_0)<0\n$$\nand then\n$$\n\\liminf h^{'}(t)=\\liminf_{t\\to\\infty} -\\mu u_x(t,h(t);u_0,h_0)>0,\n$$\nwhich is a contradiction. Therefore $h_\\infty\\le l^*$.\n\n\n\n\n\nWe  now  show that $\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0,h_0)\\|\n_{C([0,h(t)])}=0$. Let $\\bar{u}(t,x)$\ndenote the solution of the problem\n\\begin{equation*}\n\\label{fixed-boundary-eq14}\n\\begin{cases}\n\\bar{u}_t=\\bar{u}_{xx}+\\bar{u}f(t,x,\\bar{u}),\\quad &t>0, 0<x<h_{\\infty} \\cr\n\\bar{u}_{x}(t,0)=\\bar{u}(t,h_{\\infty})=0,\\quad &t>0 \\cr\n\\bar{u}(0,x)=\\tilde{u}_{0}(x), \\quad &0\\leq x\\leq h_{\\infty}\n\\end{cases}\n\\end{equation*}\nwhere\n$$\\tilde{u}_{0}(x)=\n\\begin{cases}\nu_{0}(x) & \\ for \\ 0\\leq x\\leq h_{0} \\\\\n0 & \\ for \\ x> h_{0}\n\\end{cases}\n$$\nThe comparison principle implies that\n$$0\\leq u(t,x;u_0,h_0)\\leq\\bar{u}(t,x) \\  \\ for \\ t>0 \\ and \\\nx\\in[0,h(t)]$$\n\nIf $h_{\\infty}<l^*$, then  $\\lambda(a,h_{\\infty})<0$\nand  by Proposition \\ref{fixed-boundary-prop1},\n $\\bar{u}\\to0$ uniformly for $x\\in[0,h_{\\infty}]$ as\n$t\\to\\infty$. Hence, $\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0,h_0)\\|\n_{C([0,h(t)])}=0$.\n\nIf $h_\\infty=l^*$, assume that  $\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0,h_0)\\|\n_{C([0,h(t)])}\\not =0$. Then there are $t_n\\to\\infty$ and $u^*\\not =0$, $g^*\\in H(f)$ such that\n$\\|u(t_n,\\cdot;u_0,h_0)-u^*(\\cdot)\\|_{C([0,h(t_n)])}\\to0$ and $f\\cdot t_n\\to g^*$ as $t_n\\to\\infty$. We have $u(t,\\cdot;u^*,g^*)$ is an entire solution of\n$$\n\\begin{cases}\nu_t=u_{xx}+ug^*(t,x,u),\\quad 0<x<l^*\\cr\nu_x(t,0)=u(t,l^*)=0.\n\\end{cases}\n$$\nBy Hopf lemma  for parabolic equations, we have  $u_x(t,l^*;u^*,g^*)<0$. This implies that\n$$\n\\lim_{n\\to\\infty} h^{'}(t_n)=-\\lim_{n\\to\\infty}\\mu u_x(t_n,h(t_n);u_0,h_0)>0,\n$$\nwhich is a contradiction again.\n\n\n\n(1)(ii)\nFirst note that for any fixed $x$, $u^l(g)(x)$ is increasing in $l$ and $u^l(g)(x)\\le u^*(g)(x)$.\nThen there is $\\tilde u^*(g)(x)$ such that\n$$\n\\lim_{l\\to\\infty}u^l(g)(x)=\\tilde u^*(g)(x)\\le u^*(g)(x)\n$$\nlocally uniformly in $x$.\n\nWe claim that\n$$\n\\tilde u^*(g)(x)\\equiv u^*(g)(x).\n$$\nIn fact, by Lemma \\ref{unbounded-domain-lm2},\n$$\n\\inf_{x\\ge 0,g\\in H(f)}\\tilde u^*(g)(x)>0.\n$$\nNote that $u(t,x;\\tilde u^*(g),g)=\\tilde u^*(g\\cdot t)(x)$. Then by Proposition \\ref{unbounded-prop1},\n  $u^*(g)(x)\\equiv \\tilde u^*(g)(x)$.\n\n\nNote that for any $T>0$ satisfying $h(T)>l^*$,\n$$\nu(t+T,x;u_0,h_0)\\ge u^l(t,x;u(T,\\cdot;u_0,h_0),f\\cdot T)\\quad \\forall\\,\\, t\\ge 0,\n$$\nwhere $u^l(t,x;u(T,\\cdot;u_0,h_0),f\\cdot T)$ is the solution of \\eqref{fixed-boundary-eq3} with\n$g=f\\cdot T$, $l=h(T;u_0,h_0)$, and $u^l(0,x;u(T,\\cdot;u_0,h_0),f\\cdot T)=u(T,x;u_0,h_0)$.\nNote also that\n$$\nu^l(t,x;u(T,\\cdot;u_0,h_0),f\\cdot T)-u^l(f\\cdot (t+T))(x)\\to 0\n$$\nas $t\\to\\infty$ uniformly in $x\\in [0,l]$ and\n$$\nu^l(f\\cdot (t+T))(x)-u^*(f\\cdot(t+T))(x)\\to 0\n$$\nas $l\\to\\infty$ locally uniformly in $x\\in [0,\\infty)$. It then follows that\n$$\nu(t,x;u_0,h_0)-u^*(f\\cdot t)(x)\\to 0\n$$\nas $t\\to\\infty$ locally uniformly in $x\\in [0,\\infty)$.\n\n\n\n(2) If $h_0\\ge l^*$, then $h_\\infty> h_0\\ge l^*$. (2) then follows from (1).\n\n\n(3) Assume that $h_{0}<l^*$.\nLet\n$$\n\\mu^{*}=\\sup\\{\\mu\\,|\\, h_\\infty(\\mu)<\\infty\\}.\n$$\n\nWe claim that $\\mu^{*}\\in\\{\\mu|h_\\infty(\\mu)<\\infty\\}$\nwhen $\\{\\mu\\,|\\,h_\\infty(\\mu)<\\infty\\}\\not =\\emptyset$. Otherwise\n$h_\\infty(\\mu^{*})=\\infty$. It means that we can find $T>0$\nsuch that $h_{\\mu^{*}}(T)>l^*$. By the continuous dependence of\n$h_\\mu$ on $\\mu$, there is $\\epsilon>0$ small such that\n$h_\\mu(T)>l^*$ for all $\\mu\\in[\\mu^{*}-\\epsilon,\\mu^{*}+\\epsilon]$.\nHence, for all such $\\mu$ we have\n$$h_\\infty(\\mu)=\\lim_{t\\to\\infty}h_\\mu(t)>h_\\mu(T)>l^*$$\nThus, $h_\\infty(\\mu)=\\infty$. This implies that\n$[\\mu^{*}-\\epsilon,\\mu^{*}+\\epsilon]\\cap\\{\\mu|h_\\infty(\\mu)<\\infty\\}=\\emptyset$,\nand it is a contradiction to the definition of $\\mu^{*}$. So we proved the claim\nthat $\\mu^{*}\\in\\{\\mu|h_\\infty(\\mu)<\\infty\\}$.\n\nFor $\\mu>\\mu^{*}$, we get $h_\\infty(\\mu)=\\infty$. If not,\nit must have $\\mu\\leq\\mu^{*}$, and it is a contradiction.\nThen spreading happens.\n\nFor $\\mu\\leq\\mu^{*}$, by the Lemma \\ref{monotone-increase}\nwe can obtain\n$$h_\\mu(t)\\leq h_{\\mu^{*}}(t) \\ for \\ all \\ t\\in(0,+\\infty)$$\nIt follow that $h_\\infty(\\mu)\\leq h_\\infty(\\mu^{*})<\\infty$,\nand vanishing happens.\n\\end{proof}\n\n\n\\section{Remarks}\nWe have examined the dynamical behavior of the population $u(t,x)$\nwith spreading front $x=h(t)$ determined by \\eqref{main-eq}, and\nproved that for this problem a spreading-vanishing dichotomy holds\n(see Theorem \\ref{main-thm2}). In this section, we discuss how  the techniques for \\eqref{main-eq} can be modified to study the\n double fronts free boundary \\eqref{main-doub-eq}.\n\nFirst, note that the existence and uniqueness results for solutions of \\eqref{main-eq} with given initial datum $(u_0,h_0)$\n can be\nextended to \\eqref{main-doub-eq} using the same arguments as in Section 5\n\\cite{DuLi}, except that we need to modify the transformation in the\nproof of Theorem 2.1 in \\cite{DuLi} such that both boundaries are\nstraightened. In particular, for given\n$g_0<h_0$ and $u_0$ satisfying \\eqref{initial-value-1},\nthe system \\eqref{main-doub-eq} has a unique global solution $(u(t,x;u_0,h_0,g_0),h(t;u_0,h_0,g_0),g(t;u_0,h_0,g_0))$ with\n$u(0,x;u_0,h_0,g_0)=u_0(x)$, $h(0;u_0,h_0,g_0)=h_0$, $g(0;u_0,h_0,g_0)=g_0$.\nMoreover, $g(t)$ decreases and $h(t)$ increases as $t$ increases. Let\n$g_\\infty=\\lim_{t\\to\\infty}g(t;u_0,h_0,g_0)$ and $h_\\infty=\\lim_{t\\to\\infty}h(t;u_0,h_0,g_0)$.\n\n\n\nWe next consider the spreading-vanishing dichotomy for \\eqref{main-doub-eq}. To this end,\nWe assume $(H4)^*$ instead of $(H4)$,\n\n\\medskip\n\\noindent {\\bf $(H4)^*$} {\\it There is $L^*\\ge 0$ such that\n $\\inf_{y\\in\\RR, \\, l\\ge L^*} \\tilde\\lambda(a(\\cdot,\\cdot+y),l)>0$.}\n\\medskip\n\n\n\\noindent  Consider the following  fixed boundary problem on $\\RR^1$,\n\\begin{equation}\n\\label{glob-eq}\nu_t=u_{xx}+uf(t,x,u) \\quad x\\in(-\\infty,\\infty).\n\\end{equation}\nFor given $u_0\\in C^b_{\\rm unif}(\\RR,\\RR^+)$, let $u(t,x;u_0)$ be the solution of\n\\eqref{glob-eq} with $u(0,x;u_0)=u_0(x)$.\nBy the similar arguments as those in Theorem \\ref{main-thm1}, we can prove\n\n\\begin{proposition}\n\\label{positive-almost-periodic-solution-doub-prop}\nAssume (H1), (H2), $ (H4)^*$, and (H5).\n \\eqref{glob-eq} has a unique time almost periodic positive solution\n$u^*(t,x)$ and for any $u_0\\in C^b_{\\rm unif}(\\RR,\\RR^+)$ with\n$\\inf_{x\\in(-\\infty,\\infty)}u_0(x)>0$,\n$\\lim_{t\\to\\infty}\\|u(t,\\cdot;u_0)-u^*(t,\\cdot)\\|_{C(\\RR)}=0$.\n\\end{proposition}\n\nWe now have the following spreading-vanishing dichotomy for \\eqref{main-doub-eq}.\n\n\\begin{proposition}\n\\label{spreading-vanishing-doub-prop}\nAssume (H1), (H2), $(H4)^*$, and (H5). For given $h_0>0$ and  $u_0$ satisfying \\eqref{initial-value-1},\nthe following hold.\n\\begin{itemize}\n\\item[(1)]\nEither\n\n\n(i) $h_\\infty-g_\\infty\\le L^*$ and\n$\\lim_{t\\to+\\infty} u(t,x;u_0,h_0,g_0)=0$ uniformly in $x$\n\nor\n\n\n(ii)  $h_{\\infty}=-g_{\\infty}=\\infty$ and\n$\\lim_{t\\to\\infty}[u(t,x;u_0,h_0,g_0)-u^*(t,x)]=0$\nlocally uniformly for $x\\in(-\\infty,\\infty)$, where $u^*(t,x)$ is the\nunique time almost periodic positive solution of \\eqref{glob-eq}.\n\n\n\n\\item[(2)] If $h_0-g_0\\ge L^*$, then $h_\\infty=-g_{\\infty}=\\infty$.\n\n\\item[(3)]\nSuppose $h_0-g_0<L^*$. Then there exists $\\mu^{*}>0$\nsuch that spreading occurs if $\\mu>\\mu^{*}$ and vanishing occurs if $\\mu\\le \\mu^{*}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\n(1) Observe that we have either $h_\\infty-g_\\infty<\\infty$ or $h_\\infty-g_\\infty=\\infty$.\n\nSuppose that $h_\\infty-g_\\infty<\\infty$. By $(H4)^*$ and the similar arguments as those in Theorem \\ref{main-thm2}(1)(i),\nwe must have $h_\\infty-g_\\infty\\le L^*$ and $u(t,x;u_0,h_0,g_0)\\to 0$ as $t\\to\\infty$.\n\nSuppose that $h_\\infty-g_\\infty=\\infty$. We first claim that $h_\\infty=-g_\\infty=\\infty$.\nIn fact, if the claim does not hold, without loss of generality, we may assume that $-\\infty<g_\\infty<h_\\infty=\\infty$.\nBy the similar arguments as those in Theorem \\ref{main-thm2}(1)(i), we have $g^{'}(t)\\to 0$ as $t\\to\\infty$.\nLet $T^*>0$ be such that $h(T^*)-g(T^*)>L^*$. Then by $(H4)^*$,\n$$\n\\inf_{t>T^*,x\\in [g(T^*),h(T^*)]}u(t,x;u_0,h_0,g_0)>0.\n$$\nLet $t_n\\to\\infty$ be such that $f(t+t_n,x,u)\\to g^*(t,x,u)$ and $u(t+t_n,x;u_0,h_0,g_0)\\to u^*(t,x)$.\nThen $u^*(t,x)$ is the solution of\n$$\n\\begin{cases}\nu_t=u_{xx}+u g^*(t,x,u),\\quad g_\\infty<x<\\infty\\cr\nu(t,g_\\infty)=0,\n\\end{cases}\n$$\nand\n$\\inf_{t\\in\\RR,x\\in [g(T^*),h(T^*)]}u^*(t,x)>0$. Then by Hopf Lemma for parabolic equations,\n$$\nu^*_x(t,g_\\infty)>0.\n$$\nThis implies that\n$$g^{'}(t+t_n;u_0,h_0,g_0)\\to -\\mu u^*_x(t,g_\\infty)<0,\n$$\nwhich contradicts to the fact that $g^{'}(t;u_0,h_0,g_0)\\to 0$ as $t\\to\\infty$. Hence $(g_\\infty,h_\\infty)=(-\\infty,\\infty)$.\nBy the similar arguments as those in Theorem \\ref{main-thm2}(1)(ii), we have\n$$\n\\lim_{t\\to\\infty}[u(t,x;u_0,h_0,g_0)-u^*(t,x)]=0\n$$\nlocally uniformly in $x$ in bounded sets.\n\n(2) and (3) follows from the similar arguments as those in Theorem \\ref{main-thm2} (2) and (3), respectively.\n\\end{proof}\n\n\\section*{Acknowledgements}\n\nFang Li would like to thank the China Scholarship Council for\nfinancial support during the two years of her overseas study and to\nexpress her gratitude to the  Department of Mathematics and\nStatistics, Auburn University  for its kind hospitality. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDue to the huge bandwidth, millimeter wave (mmWave) communications have been recognized as one of the key technologies to meet the demand for unprecedentedly high data rate transmission in the future mobile networks \\cite{A. L. Swindlehurst}. By equipping large-scale antenna arrays, massive multiple-input multiple-output (MIMO) can provide sufficiently large array gains for spatial multiplexing and beamforming \\cite{L. Lu}. MmWave massive MIMO communications can obtain the merits of both of them and thus have attracted significant interest \\cite{B. Wang}. However, the expensive and power-hungry hardwares used in mmWave bands become the main obstacle to equipping a dedicated radio frequency (RF) chain for each antenna. The mainstream solution for this problem is to use the two-stage hybrid architecture, where a large number of antennas are connected to much fewer RF chains via phase shifters \\cite{L. Liang}, \\cite{L. Pan}.\n\n\\subsection{Related Work}\n\nFor mmWave massive MIMO systems with the hybrid architecture, both the analog and digital processing should be carefully designed to achieve the comparable performance to the fully-digital systems. In \\cite{L. Liang}, a low-complexity hybrid precoding scheme at the base station (BS) has been proposed for the massive MIMO downlink with single-antenna users. The hybrid architecture has been further introduced to the user side in \\cite{W. Ni}, where hybrid block diagonalization (HBD) has been used for the analog and digital processing design. By exploiting the sparsity of mmWave channels, the hybrid precoding and combining at both the transmitter and receiver have been optimized in \\cite{O. E. Ayach}. The heuristic hybrid beamforming design in \\cite{F. Sohrabi} can approach the performance of the fully-digital architecture. The alternating minimization algorithms for both fully-connected and sub-connected hybrid architectures in \\cite{X. Yu} are with low complexity and limited performance loss. In \\cite{L. Zhao}, the hybrid processing along with channel estimation has been designed and analyzed for both the sparse and non-sparse channels. The uniform channel decomposition and nonlinear digital processing have been introduced in \\cite{Y. Lin} for hybrid beamforming design. In the existing works, the hybrid processing matrices at the transmitter and receiver are usually optimized separately due to the intractability of the joint optimization with non-convex constraints, which makes the further performance improvement possible with joint optimization.\n\nDeep learning (DL) has achieved great success in various fields, including computer vision \\cite{K. He_a}, speech signal processing \\cite{A. Graves}, natural language processing \\cite{R. Collobert}, and so on, due to its unique ability in extracting and learning inherent features. It has been recently introduced to wireless communications and shown quite powerful in the optimization of communication systems \\cite{T. O'hea}--\\cite{G. Gui} and resource allocation \\cite{L. Liang_b}--\\cite{Z. Yang}. In \\cite{H. Ye_a}, DL has been successfully applied in pilot-assisted signal detection for orthogonal frequency division multiplexing (OFDM) systems with non-ideal transceiver and channel conditions. For wideband mmWave massive MIMO systems in time-varying channels, channel correlation has been exploited by deep convolutional neural network (CNN) in \\cite{P. Dong_b} to improve the accuracy and accelerate the computation for the channel estimation. Deep neural network (DNN) has been utilized in \\cite{S. Gao} to model the mapping relationship among antennas for reliable channel estimation in massive MIMO systems with mixed-resolution ADCs. An autoencoder-like DNN has been developed in \\cite{C.-K. Wen} to reduce the overhead for channel state information (CSI) feedback in the frequency duplex division massive MIMO system. In \\cite{C. Lu_a}, CNN has been utilized in CSI compression and uncompression to significantly improve the recovery accuracy. By combining the residual network and CNN, an efficient channel quantization scheme has been proposed from the perspective of bit-level in \\cite{C. Lu_b}. The DL based end-to-end optimization has been developed in \\cite{S. Dorner} and \\cite{H. Ye_b} by breaking the block structures at the transceiver. DL has been recently used to design the hybrid processing matrices for massive MIMO systems with various transceiver architectures \\cite{H. Huang}--\\cite{X. Bao}. In \\cite{H. Huang}, the analog and digital precoder design has been modeled as the DNN mapping based on geometric mean decomposition. In \\cite{T. Lin}, DNN has been applied to design the analog precoder for massive multiple-input single-output (MISO) systems. Deep CNN has been applied to learn the phases of the analog precoder and combiner for mmWave massive MIMO systems in \\cite{A. M. Elbir}. For the same system, channel estimation and analog processing have been jointly optimized by DL with reduced pilot overhead in \\cite{X. Li}. In \\cite{X. Bao}, deep CNN along with an equivalent channel hybrid precoding algorithm have been proposed to design the hybrid processing matrices.\n\n\\subsection{Motivation and Contribution}\n\nThe research on the DL based hybrid processing for mmWave massive MIMO systems is still in the exploratory stage and has many open issues. The existing works have applied DL to design the analog precoder \\cite{T. Lin}, the analog combiner \\cite{X. Bao}, the analog precoder and combiner \\cite{A. M. Elbir}, \\cite{X. Li}, and the analog and digital precoders \\cite{H. Huang}. Currently, only partial hybrid processing is designed by DL for the mmWave transceiver. In addition, conventional hybrid processing schemes are usually used to generate label matrices for the DNN to approximate, which limits the performance of the DL based approaches. The problems in the existing works motivate us to propose a general DL based joint hybrid processing framework (DL-JHPF) with the following two unique features:\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] The framework jointly optimizes the analog and digital processing matrices at both the transmitter and receiver in an end-to-end manner without pre-designed label matrices. By doing this, it can be applied to various types of mmWave transceiver architectures and will have the potential to break through the performance of the existing schemes.\n\n\\item[2)] The framework enables end-to-end optimization but still preserves the block structures at the transceiver considering the hardware and power constraints in practical implementation for the hybrid architecture, which is quite different from the end-to-end optimization in \\cite{S. Dorner} and \\cite{H. Ye_b}.\n\\end{itemize}\nThe main contributions of this paper are summarized as follows.\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] We model the joint analog and digital processing design for the transceiver as a DL based framework, which consists of the NN based hybrid processing designer, signal flow simulator, and NN based signal demodulator. For the sake of practical implementation, it does not break the original block structures at the transceiver but still allows the back-propagation (BP) based end-to-end optimization by minimizing the cross-entropy loss between recovered and original bits. The trainability of DL-JHPF is proved theoretically.\n\n\\item[2)] We extend the proposed framework to OFDM systems by simply modifying the structure of the training data. The extension does not complicate the framework architecture and guarantees the relatively short training time even if the number of subcarriers is large.\n\n\\item[3)] We verify the effectiveness of the proposed framework by numerical results based on the 3rd Generation Partnership Project (3GPP) channel model that can well depict the real channel environment. The proposed DL-JHPF achieves remarkable improvement in bit-error rate (BER) performance even with mismatched CSI and channel scenarios. Thanks to the careful design, DL-JHPF reduces the runtime significantly by sufficiently exploiting the parallel computing and thus is more suitable for rapidly varying mmWave channels.\n   \n\\end{itemize}\n\nThe rest of the paper is organized as follows. Section II describes the channel model and signal transmission process for the considered mmWave massive MIMO system. The proposed DL-JHPF is elaborated in Section III. Simulation results are provided in Section IV to verify the effectiveness of the proposed framework and finally Section V gives concluding remarks.\n\n\\emph{Notations}: In this paper, we use upper and lower case boldface letters to denote matrices and vectors, respectively. $\\lVert\\cdot\\rVert_{F}$, $(\\cdot)^T$, $(\\cdot)^H$, and $\\mathbb{E}\\{\\cdot\\}$ represent the Frobenius norm, transpose, conjugate transpose, and expectation, respectively. $\\mathcal{CN}(\\mu,\\sigma^2)$ represents circular symmetric complex Gaussian distribution with mean $\\mu$ and variance $\\sigma^2$. $[\\mathbf{X}]_{i,j}$ and $[\\mathbf{x}]_{i}$ denote the $(i,j)$th element of matrix $\\mathbf{X}$ and the $i$th element of vector $\\mathbf{x}$, respectively. $|\\cdot|$ denotes the amplitude of a complex number.\n\n\\section{System Model}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{Fig1}\n\\caption{MmWave massive MIMO system model.}\\label{system_model}\n\\end{figure}\n\nAs shown in Fig.~\\ref{system_model}, we consider a point-to-point massive MIMO systems working at mmWave bands, where the transmitter and the receiver are with $N_{\\textrm{T}}$ and $N_{\\textrm{R}}$ antennas, respectively. To reduce the hardware cost and power consumption, $N_{\\textrm{T}}^{\\textrm{RF}}(< N_{\\textrm{T}})$ and $N_{\\textrm{R}}^{\\textrm{RF}}(< N_{\\textrm{R}})$ RF chains are used at the transmitter and the receiver, respectively, and are connected to the large-scale antennas via phase shifters.\n\n\\subsection{Channel Model}\n\nDue to the sparse scattering property, the Saleh-Valenzuela channel model has been used to well depict the mmWave propagation environment, where the scattering of multiple rays forms several clusters. According to \\cite{O. E. Ayach}, the $N_{\\textrm{R}}\\times N_{\\textrm{T}}$ channel matrix between the receiver and the transmitter can be represented as\n\\begin{eqnarray}\n\\label{eqn_H_tau}\n\\mathbf{H}=\\sqrt{\\frac{N_{\\textrm{T}}N_{\\textrm{R}}}{N_{\\textrm{cl}}N_{\\textrm{ray}}}}\\sum_{n=1}^{N_{\\textrm{cl}}}\\sum_{m=1}^{N_{\\textrm{ray}}} \\alpha_{n,m}\\mathbf{a}_{\\textrm{R}}(\\varphi_{n,m})\\mathbf{a}_{\\textrm{T}}^{H}(\\phi_{n,m}),\n\\end{eqnarray}\nwhere $N_{\\textrm{cl}}$ and $N_{\\textrm{ray}}$ denote the number of scattering clusters and the number of rays in each cluster, respectively, $\\alpha_{n,m}\\sim \\mathcal{CN}(0, \\sigma_{\\alpha}^2)$ is the propagation gain of the $m$th path in the $n$th cluster with $\\sigma_{\\alpha}^2$ being the average power gain, $\\varphi_{n,m}$ and $\\phi_{n,m}\\in[0,2\\pi]$ are the azimuth angles of arrival and departure (AoA\/AoD) at the receiver and the transmitter, respectively, of the $m$th path in the $n$th cluster.\\footnote{The path gain $\\alpha_{n,m}$ is the fast fading and varies in the time scale of channel coherence interval. Other parameters, $N_{\\textrm{cl}}$, $N_{\\textrm{ray}}$, $\\varphi_{n,m}$, $\\phi_{n,m}$, are slow fading and may be unchanged in a large time scale compared to $\\alpha_{n,m}$. The Doppler spread determines how often these channel parameters change.} For a uniform linear array with $N$ antenna elements and an azimuth angle of $\\theta$, the response vector can be expressed as\n\\begin{eqnarray}\n\\label{eqn_au}\n\\mathbf{a}(\\theta)=\\frac{1}{\\sqrt{N}} \\left[1,e^{-j2\\pi\\frac{d}{\\lambda}\\sin(\\theta)},\\ldots,e^{-j2\\pi\\frac{d}{\\lambda}(N-1)\\sin(\\theta)}\\right]^{T},\n\\end{eqnarray}\nwhere $d$ and $\\lambda$ denote the distance between the adjacent antennas and carrier wavelength, respectively.\n\nIn the above channel model, we assume the transmitted signal is with narrowband and therefore, channel matrix is independent of frequency. For wideband transmission, OFDM is used to convert a frequency-selective channel into multiple flat fading channels and the corresponding channel matrices will be different at different subcarriers. Accordingly, the design of DL-JHPF in Section III will start at the narrowband systems and is then extended to the wideband OFDM systems.\n\n\\subsection{Signal Transmission}\n\nThe transmitter sends $N_{\\textrm{s}}$ parallel data streams to the receiver through the wireless channel. The bits of each data stream are first mapped to the symbol by the $M$-ary modulation. The symbol vector intended for the receiver, $\\mathbf{x}\\in\\mathbb{C}^{N_{\\textrm{s}}\\times 1}$ with $\\mathbb{E}\\left\\{\\mathbf{x}\\mathbf{x}^{H}\\right\\}=\\frac{1}{N_{\\textrm{s}}}\\mathbf{I}_{N_{\\textrm{s}}}$, is successively processed by the digital precoder, $\\mathbf{F}_{\\textrm{BB}}\\in\\mathbb{C}^{N_{\\textrm{T}}^{\\textrm{RF}}\\times N_{\\textrm{s}}}$, at the baseband and the analog precoder, $\\mathbf{F}_{\\textrm{RF}}\\in\\mathbb{C}^{N_{\\textrm{T}}\\times N_{\\textrm{T}}^{\\textrm{RF}}}$, through the phase shifters, yielding the transmitted signal\n\\begin{eqnarray}\n\\label{eqn_s}\n\\mathbf{s}=\\sqrt{P}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x},\n\\end{eqnarray}\nwhere $P$ denotes the transmit power. $\\mathbf{F}_{\\textrm{RF}}$ represents the phase-only modulation by the phase shifters and thus has the constraint of $\\left|[\\mathbf{F}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}$, $\\forall \\,i, \\,j$. $\\mathbf{F}_{\\textrm{BB}}$ is normalized as $\\|\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\|_{F}^2=N_{\\textrm{s}}$ to satisfy the total power constraint at the transmitter. Then the received signal at the receiver is given by\n\\begin{eqnarray}\n\\label{eqn_y}\n\\mathbf{y}=\\sqrt{P}\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x}+\\mathbf{n},\n\\end{eqnarray}\nwhere $\\mathbf{n}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times 1}$ is additive white Gaussian noise (AWGN) with $\\mathcal{CN}(0,1)$ elements.\n\nThe received signal $\\mathbf{y}$ is then processed by the hybrid architecture at the receiver as\n\\setlength{\\arraycolsep}{0.05em}\n\\begin{eqnarray}\n\\label{eqn_r}\n\\mathbf{r}&&=\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{y} =\\sqrt{P}\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x} +\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{n},\n\\end{eqnarray}\nwhere $\\mathbf{W}_{\\textrm{RF}}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times N_{\\textrm{R}}^{\\textrm{RF}}}$ and $\\mathbf{W}_{\\textrm{BB}}\\in\\mathbb{C}^{N_{\\textrm{R}}^{\\textrm{RF}}\\times N_{\\textrm{s}}}$ represent the analog combiner and digital combiner, respectively. A hardware constraint is imposed on $\\mathbf{W}_{\\textrm{RF}}$ such that $\\left|[\\mathbf{W}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}$, $\\forall \\,i, \\,j$ similar to $\\mathbf{F}_{\\textrm{RF}}$. Then the detected signal vector, $\\mathbf{r}$, is demodulated to recover the original bits of $N_{\\textrm{s}}$ data streams.\n\nSince the performance of the digital communication system is ultimately determined by BER, we aim to jointly design $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ to minimize the BER between the original and demodulated bits, that is\n\\begin{eqnarray}\n&&\\min\\limits_{\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}}\\quad P_{\\textrm{e}}\\left(\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}\\right),\\label{eqn_optim_problem}\\\\\n&&\\quad\\quad\\:\\:\\textrm{s.t.}\\qquad\\qquad\\!\\! \\left|[\\mathbf{F}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}, \\forall \\,i, \\,j,\\label{const1}\\\\\n&&\\qquad\\qquad\\qquad\\quad\\;\\; \\left|[\\mathbf{W}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}, \\forall \\,i, \\,j,\\label{const2}\\\\\n&&\\qquad\\qquad\\qquad\\quad\\;\\; \\|\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\|_{F}^2=N_{\\textrm{s}}\\label{const3}.\n\\end{eqnarray}\nThe BER in (\\ref{eqn_optim_problem}) is a complicated nonlinear function of $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ without closed-form expression and the constraints in (\\ref{const1}) and (\\ref{const2}) are non-convex, which make this optimization problem intractable to be solved by the traditional approaches. DL is a potential solution by using the BP algorithm and thus we develop DL-JHPF to address this problem.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=6.6in]{Fig2}\n\\caption{Proposed DL-JHPF.}\\label{DL_framework}\n\\end{figure*}\n\n\\section{Proposed DL-JHPF}\n\nIn this section, we first briefly review the existing work on the DNN based end-to-end communications. Then we propose DL-JHPF, where the framework is first described, followed by the details of training, deployment, and testing along with the corresponding complexity analysis. Finally, we extend the framework to OFDM systems over wideband mmWave channels.\n\n\\subsection{DNN based End-to-End Communications}\n\nPrior works have shown that DNN based end-to-end optimization is an efficient tool to minimize BER. The BP algorithm makes the DNN based end-to-end communications over the air possible so long as the optimized performance metric is differentiable \\cite{T. O'hea}, \\cite{S. Dorner}, \\cite{H. Ye_b}. For the DNN based end-to-end communication system, the modules at the transmitter and the receiver are replaced by two DNNs, respectively. Specifically, the DNN at the transmitter encodes the original symbols into the transmitted signal and the one at the receiver recovers the original symbols from the output of the wireless channel. In the training stage, the error between the original and recovered symbols is computed and the weights of the two DNNs are adjusted iteratively based on the error gradient propagated from the output layer of the DNN at the receiver to optimize the recovery accuracy.\n\nIn this paper, we focus on the DL based joint analog and digital processing design for the transceiver in mmWave massive MIMO systems. Then, the existing DNN based end-to-end communication is not suitable for this task since it integrates the modules of the transceiver into two DNNs and thus cannot meet the hardware and power constraints in practical implementation. To address this challenge, we design DL-JHPF in the following.\n\n\\subsection{Framework Description}\n\n\n\nAs shown in Fig.~\\ref{DL_framework}, the proposed DL-JHPF consists of three parts: hybrid processing designer, signal flow simulator, and NN demodulator, which are elaborated as follows.\n\n\\emph{Hybrid processing designer:} It plays the role of outputting the hybrid processing matrices for the transceiver by using NNs based on the channel matrix. It includes six fully-connected NNs and is used to generate the analog and digital processing matrices for the transmitter and the receiver based on the channel matrix, $\\mathbf{H}$. Specifically, $\\mathbf{H}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times N_{\\textrm{T}}}$ is first converted to a $2N_{\\textrm{T}}N_{\\textrm{R}}\\times 1$ real-valued vector.\\footnote{In Fig.~\\ref{DL_framework}, only the main process of the framework is shown while the matrix and vector reshaping process is omitted.} Then it is input into two NNs, called precoder phase NN (PP-NN) and combiner phase NN (CP-NN), to generate the corresponding phases, $\\boldsymbol{\\phi}_{\\textrm{P}}\\in\\mathbb{R}^{N_{\\textrm{T}}N_{\\textrm{T}}^{\\textrm{RF}}\\times 1}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}\\in\\mathbb{R}^{N_{\\textrm{R}}N_{\\textrm{R}}^{\\textrm{RF}}\\times 1}$, respectively, for phase shifters. With $\\boldsymbol{\\phi}_{\\textrm{P}}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}$, two complex-valued vectors with constant amplitude elements are generated as\n\\begin{eqnarray}\n\\label{eqn_fRF}\n\\bar{\\mathbf{f}}_{\\textrm{RF}}=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}e^{j\\boldsymbol{\\phi}_{\\textrm{P}}},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_wRF}\n\\bar{\\mathbf{w}}_{\\textrm{RF}}=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}e^{j\\boldsymbol{\\phi}_{\\textrm{C}}},\n\\end{eqnarray}\nbased on which, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ are given by\n\\begin{eqnarray}\n\\label{eqn_FRF}\n\\mathbf{F}_{\\textrm{RF}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{f}}_{\\textrm{RF}}),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_WRF}\n\\mathbf{W}_{\\textrm{RF}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{w}}_{\\textrm{RF}}),\n\\end{eqnarray}\nwhere $\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\cdot)$ denotes the operation reshaping a vector to a matrix.\nThen, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with $\\mathbf{H}$ are used to generate a low-dimensional equivalent channel, i.e.,\n\\begin{eqnarray}\n\\label{eqn_Heq}\n\\mathbf{H}_{\\textrm{eq}}=\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}.\n\\end{eqnarray}\n$\\mathbf{H}_{\\textrm{eq}}\\in\\mathbb{C}^{N_{\\textrm{R}}^{\\textrm{RF}}\\times N_{\\textrm{T}}^{\\textrm{RF}}}$ is converted to a $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}\\times 1$ real-valued vector before it is input into four parallel NNs. The first two NNs, corresponding to the real part digital combiner NN (ReDC-NN) and the imaginary part digital combiner NN (ImDC-NN), output two $N_{\\textrm{s}}N_{\\textrm{R}}^{\\textrm{RF}}\\times1$ vectors, $\\bar{\\mathbf{w}}_{\\textrm{BB,re}}$, $\\bar{\\mathbf{w}}_{\\textrm{BB,im}}$, respectively. Then $\\mathbf{W}_{\\textrm{BB}}$ can be obtained as\n\\begin{eqnarray}\n\\label{eqn_WBB}\n\\mathbf{W}_{\\textrm{BB}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{w}}_{\\textrm{BB,re}}+j\\bar{\\mathbf{w}}_{\\textrm{BB,im}}).\n\\end{eqnarray}\nAnother two NNs, corresponding to the real part digital precoder NN (ReDP-NN) and the imaginary part digital precoder NN (ImDP-NN), output two $N_{\\textrm{s}}N_{\\textrm{T}}^{\\textrm{RF}}\\times1$ vectors, $\\bar{\\mathbf{f}}_{\\textrm{BB,re}}$, $\\bar{\\mathbf{f}}_{\\textrm{BB,im}}$, respectively. Then the unnormalized digital precoder $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ is given by\n\\begin{eqnarray}\n\\label{eqn_unnorm_FBB}\n\\bar{\\mathbf{F}}_{\\textrm{BB}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{f}}_{\\textrm{BB,re}}+j\\bar{\\mathbf{f}}_{\\textrm{BB,im}}).\n\\end{eqnarray}\nThe following normalization utilizes $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ and $\\mathbf{F}_{\\textrm{RF}}$ in (\\ref{eqn_FRF}) to output the final digital precoder as\n\\begin{eqnarray}\n\\label{eqn_DP_norm}\n\\mathbf{F}_{\\textrm{BB}}=\\frac{\\sqrt{N_{\\textrm{s}}}}{\\|\\mathbf{F}_{\\textrm{RF}}\\bar{\\mathbf{F}}_{\\textrm{BB}}\\|_{F}}\\bar{\\mathbf{F}}_{\\textrm{BB}}.\n\\end{eqnarray}\n\n\\emph{Signal flow simulator:} In the training stage, it simulates the process from the original bits, $\\mathbf{X}_{\\textrm{b}}$, to the detected signal, $\\mathbf{r}$, over the channel, $\\mathbf{H}$, with AWGN, $\\mathbf{n}$, where $\\mathbf{X}_{\\textrm{b}}$ with the size of $N_{\\textrm{s}}\\times\\log_2 M$, $\\mathbf{H}$, and $\\mathbf{n}$ are generated in the simulation environment. It bridges the back propagation of the error gradient from NN demodulator to hybrid processing designer as we will elaborate in Section III.C. In the deployment and testing stage, the signal flow simulator is replaced by the actual transceiver and the actual wireless fading channel. In these two stages, the analog and digital processing matrices at the transceiver are provided by the hybrid processing designer based on the simulated or actual $\\mathbf{H}$.\n\n\\emph{NN demodulator:} It is a fully-connected NN, which receives the detected signal, $\\mathbf{r}$, from the signal flow simulator (in the training stage) or the actual receiver (in the testing stage) and outputs recovered bits $\\hat{\\mathbf{x}}_{\\textrm{b}}\\in \\mathbb{R}^{N_{\\textrm{s}}\\log_2 M\\times1}$ with each element lies in the interval $[0,1]$. $\\hat{\\mathbf{x}}_{\\textrm{b}}$ is then reshaped to $\\hat{\\mathbf{X}}_{\\textrm{b}}$ with the same size as $\\mathbf{X}_{\\textrm{b}}$.\n\n\\textbf{Remark 1.} \\emph{The learning of hybrid processing matrices, $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{W}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, and $\\mathbf{W}_{\\textrm{BB}}$, in DL-JHPF is embedded into the signal transmission and demodulation process instead of approximating pre-designed label matrices. All NNs are optimized jointly sharing the mapping principle from $\\mathbf{X}_{\\textrm{b}}$ at the transmitter to $\\mathbf{X}_{\\textrm{b}}$ at the receiver that resembles an\nautoencoder. By minimizing the error between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$, each NN in hybrid processing designer can learn to output the appropriate vectors with specific meaning implicitly, i.e., phases of phase shifters and real and imaginary parts of the digital precoder and combiner. By doing this, DL-JHPF will have the potential to break through the performance of the existing schemes.}\n\n\\subsection{Framework Training}\n\nThe goal of offline training is to determine the weights of the NNs in hybrid processing designer and NN demodulator based on the training samples with the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ and the label $\\mathbf{X}_{\\textrm{b}}$, where $\\mathbf{H}$ is generated by certain channel model and $\\mathbf{n}$ is generated according to the $\\mathcal{CN}(0,1)$ distribution. By minimizing the end-to-end error between the original bits, $\\mathbf{X}_{\\textrm{b}}$, and the recovered bits, $\\hat{\\mathbf{X}}_{\\textrm{b}}$, the weights of each NN in DL-JHPF are adjusted iteratively and the training procedure is elaborated as follows.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.0in]{Fig3}\n\\caption{Training model for proposed DL-JHPF.}\\label{visual_model}\n\\end{figure}\n\nThe proposed DL-JHPF is actually an integrated DNN consisting of neuron layers and custom layers. The training model in Fig.~\\ref{visual_model} demonstrates the detailed training process of the framework. For each training sample, $\\mathbf{H}$ is converted into a real-valued vector by matrix-to-vector reshaping and real and imaginary parts stacking, which is input into PP-NN and CP-NN consisting of dense and batch normalization (BN) layers to generate the corresponding phases, $\\boldsymbol{\\phi}_{\\textrm{P}}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}$, respectively. Then (\\ref{eqn_fRF}) and (\\ref{eqn_wRF}) are executed by the same custom layer. Afterwards, the output vectors are reshaped according to (\\ref{eqn_FRF}) and (\\ref{eqn_WRF}) to generate $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, respectively. Next, (\\ref{eqn_Heq}) is executed by a custom layer to generate $\\mathbf{H}_{\\textrm{eq}}$, followed by matrix-to-vector reshaping and real and imaginary parts stacking. This  vector is input into four NNs consisting of dense and BN layers, i.e., ReDC-NN, ImDC-NN, ReDP-NN, and ImDP-NN, respectively. The output vectors of the former two NNs are used to generate $\\mathbf{W}_{\\textrm{BB}}$ through real and imaginary parts combining and vector-to-matrix reshaping as (\\ref{eqn_WBB}). Using the same operation, the output vectors of the latter two NNs are used to generate $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ as (\\ref{eqn_unnorm_FBB}). After obtaining $\\bar{\\mathbf{F}}_{\\textrm{BB}}$, a custom layer is added to perform the normalization in (\\ref{eqn_DP_norm}) to generate $\\mathbf{F}_{\\textrm{BB}}$. Then (\\ref{eqn_r}) is executed through a custom layer by using the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ and the generated $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ to yield the detected signal, $\\mathbf{r}$. After real and imaginary stacking, $\\mathbf{r}$ is converted to a real-valued vector and input into the NN demodulator consisting of dense and BN layers to output the recovered bits, $\\hat{\\mathbf{x}}_{\\textrm{b}}$, which is then reshaped to $\\hat{\\mathbf{X}}_{\\textrm{b}}$. The binary cross-entropy (BCE) loss between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$ is calculated as\n\\begin{eqnarray}\n\\label{eqn_loss}\n\\mathcal{L}&&=-\\frac{1}{N_{\\textrm{tr}}}\\sum_{n=1}^{N_{\\textrm{tr}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}\\biggl( [\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}\\ln([\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\nonumber\\\\\n&&\\quad+(1-[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j})\\ln(1-[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\biggr),\n\\end{eqnarray}\nwhere $N_{\\textrm{tr}}$ denotes the number of training samples, superscript $n$ is added to indicate the index of the training sample, and $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}$ is expressed as the function of the parameter set of all NNs in DL-JHPF, i.e., $\\boldsymbol\\Theta$.\n\nRecall the optimization problem in (\\ref{eqn_optim_problem}), the BER over the training set can be written as\n\\begin{eqnarray}\n\\label{eqn_BER}\n&&P_{\\textrm{e,tr}}\\left(\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}\\right) =P_{\\textrm{e,tr}}\\left(\\boldsymbol\\Theta\\right) \\nonumber\\\\\n&&\\quad=\\frac{\\sum_{n=1}^{N_{\\textrm{tr}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}|[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}-[\\hat{\\mathbf{X}}_{\\textrm{b,bin}}^{n}(\\boldsymbol\\Theta)]_{i,j}|}{N_{\\textrm{tr}}N_{\\textrm{s}}\\log_2 M},\\;\\;\\;\\;\n\\end{eqnarray}\nwhere $\\hat{\\mathbf{X}}_{\\textrm{b,bin}}^{n}(\\boldsymbol\\Theta)$ is the binary demodulated bit matrix with $[\\hat{\\mathbf{X}}_{\\textrm{b,1hot}}^{n}(\\boldsymbol\\Theta)]_{i,j}=0$ for $[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j}<0.5$ and $1$ otherwise. With $\\mathbf{X}_{\\textrm{b}}^{n}$ fixed, minimizing $\\mathcal{L}$ in (\\ref{eqn_loss}) with respect to $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)$ yields $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)=\\mathbf{X}_{\\textrm{b}}^{n}$, which also minimizes $P_{\\textrm{e,tr}}\\left(\\boldsymbol\\Theta\\right)$ in (\\ref{eqn_BER}). Therefore, DL-JHPF can directly minimize the BER over the training set by minimizing the BCE loss and the feasibility is guaranteed by the following theorem.\n\n\\textbf{Theorem 1.} \\emph{The proposed DL-JHPF is trainable and can minimize the BCE loss through BP algorithm.}\n\\begin{IEEEproof}\nConsidering the mini-batch training, the BCE loss over a batch is written as\n\\begin{eqnarray}\n\\label{eqn_loss_batch}\n\\mathcal{L}_{\\textrm{bat}}&&=-\\frac{1}{N_{\\textrm{bat}}}\\sum_{n=1}^{N_{\\textrm{bat}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}\\biggl( [\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}\\ln([\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\nonumber\\\\\n&&\\quad+(1-[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j})\\ln(1-[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\biggr),\n\\end{eqnarray}\nwhere $N_{\\textrm{bat}}$ denotes the batch size. Then $\\boldsymbol\\Theta$ will be updated $\\lceil\\frac{N_{\\textrm{tr}}}{N_{\\textrm{bat}}}\\rceil$ times in each epoch.\n\nTo prove Theorem 1, we need to show that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to each parameter in $\\boldsymbol\\Theta$. According to \\cite{Y. LeCun}, the outputs are differentiable with respect to the corresponding weights and inputs for each NN in DL-JHPF. Since DL-JHPF can be viewed as an integrated DNN consisting of neuron layers and custom layers, the proof can be further simplified to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to the outputs of each NN due to chain rule. In the following, we prove the differentiability of $\\mathcal{L}_{\\textrm{bat}}$ with respect to the outputs of each NN by incorporating the custom layers.\n\n\\emph{NN demodulator:} From (\\ref{eqn_loss_batch}), $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j}, \\forall i, j$.\n\n\\emph{Re\/ImDC-NN:} As mentioned in Section III.B, $\\bar{\\mathbf{w}}_{\\textrm{BB,re}}$ and $\\bar{\\mathbf{w}}_{\\textrm{BB,im}}$ are the outputs of ReDC-NN and ImDC-NN, respectively. Without loss of generality, we will prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$. According to (\\ref{eqn_r}) and (\\ref{eqn_WBB}), $[\\mathbf{r}]_{1}$ is the function of $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$, that is\n\\begin{eqnarray}\n\\label{eqn_r1}\n\\!\\!\\!\\!\\![\\mathbf{r}]_{1}&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}-j[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1})[\\mathbf{z}]_{1}+C_1\\nonumber\\\\\n&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}-j[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1})([\\mathbf{z}]_{1,\\textrm{re}}+j[\\mathbf{z}]_{1,\\textrm{im}})+C_1\\nonumber\\\\\n&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}[\\mathbf{z}]_{1,\\textrm{re}}+[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}[\\mathbf{z}]_{1,\\textrm{im}}+C_{1,\\textrm{re}})\\nonumber\\\\\n&&\\quad+j([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}[\\mathbf{z}]_{1,\\textrm{im}}-[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}[\\mathbf{z}]_{1,\\textrm{re}}+C_{1,\\textrm{im}})\\nonumber\\\\\n&&=[\\mathbf{r}]_{1,\\textrm{re}}+j[\\mathbf{r}]_{1,\\textrm{im}},\n\\end{eqnarray}\nwhere $\\mathbf{z}=\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{y}$ and $C_1$ denotes the component of $[\\mathbf{r}]_{1}$ independent of $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$ with the subscripts `re' and `im' indicating the real and imaginary parts, respectively. Since $[\\mathbf{r}]_{1,\\textrm{re}}$ and $[\\mathbf{r}]_{1,\\textrm{im}}$ are a part of inputs of NN demodulator, $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\mathbf{r}]_{1,\\textrm{re}}$ and $[\\mathbf{r}]_{1,\\textrm{im}}$. Then we have\n\\begin{eqnarray}\n\\label{eqn_derL_wre_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}}\\nonumber\\\\\n&&=[\\mathbf{z}]_{1,\\textrm{re}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} +[\\mathbf{z}]_{1,\\textrm{im}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_wim_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}}\\nonumber\\\\\n&&=[\\mathbf{z}]_{1,\\textrm{im}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} -[\\mathbf{z}]_{1,\\textrm{re}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}.\n\\end{eqnarray}\n\n\\emph{Re\/ImDP-NN:} Since $\\bar{\\mathbf{f}}_{\\textrm{BB,re}}$ and $\\bar{\\mathbf{f}}_{\\textrm{BB,im}}$ are the outputs of ReDP-NN and ImDP-NN, respectively, we also aim to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$. Considering the normalization in (\\ref{eqn_DP_norm}), we first calculate the derivatives of $\\mathcal{L}_{\\textrm{bat}}$ with respect to the real and imaginary parts of $[\\mathbf{F}_{\\textrm{BB}}]_{1,1}$, i.e., $\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}$ and $\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}$, which can be obtained similarly to (\\ref{eqn_derL_wre_1}) and (\\ref{eqn_derL_wim_1}). According to (\\ref{eqn_DP_norm}), we have\n\\begin{eqnarray}\n\\label{eqn_FBB11re}\n[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}=\\frac{[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}}{f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_FBB11im}\n[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}=\\frac{[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}}{f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})},\n\\end{eqnarray}\nwhere $f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})\\!=\\![([\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{re}}[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1} \\!-\\![\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{im}}[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}+C_{2,\\textrm{re}})^2+([\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{re}}[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1} \\!+\\![\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{im}}[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}+C_{2,\\textrm{im}})^2+C_3]^{\\frac{1}{2}}$ with $C_2$ and $C_3$ independent of $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$. Then we can find that $[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}$ and $[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}$ are differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$, which leads to\n\\begin{eqnarray}\n\\label{eqn_derL_fBBre_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}} &&\\!=\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}},\\nonumber\\\\\n&&\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_fBBim_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}} &&\\!=\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}}.\\nonumber\\\\\n&&\n\\end{eqnarray}\n\n\\emph{PP-NN:} We still aim to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}$ that is one of the output of PP-NN and generates $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$ as\n\\begin{eqnarray}\n\\label{eqn_fRF_reim}\n\\frac{1}{\\sqrt{N_{\\textrm{T}}}}e^{j[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}}&&=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}(\\cos[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1} +j\\sin[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1})\\nonumber\\\\\n&&=[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}+j[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}.\n\\end{eqnarray}\nFrom (\\ref{eqn_r}) and (\\ref{eqn_Heq}), $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$ influence the values of $[\\mathbf{r}]_{i}, i=1,\\ldots,N_{\\textrm{s}}$ and $[\\mathbf{H}_{\\textrm{eq}}]_{j,1}, j=1,\\ldots,N_{\\textrm{R}}^{\\textrm{RF}}$. According to the previous proof, $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to the real and imaginary parts of each element in $\\mathbf{r}$ and $\\mathbf{H}_{\\textrm{eq}}$. $[\\mathbf{r}]_{i,\\textrm{re}}$, $[\\mathbf{r}]_{i,\\textrm{im}}$, $[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}$, and $[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}$, $i=1,\\ldots,N_{\\textrm{s}}, j=1,\\ldots,N_{\\textrm{R}}^{\\textrm{RF}}$ are also differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$. Resorting to chain rule, we have\n\\begin{eqnarray}\n\\label{eqn_derL_fRF_1re}\n&&\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} =\\sum_{i=1}^{N_{\\textrm{s}}}\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}\\right)\\nonumber\\\\\n&&+\\!\\sum_{j=1}^{N_{\\textrm{R}}^{\\textrm{RF}}}\\!\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}\\right)\\!,\\nonumber\\\\\n&&\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_fRF_1im}\n&&\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} =\\sum_{i=1}^{N_{\\textrm{s}}}\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}\\right)\\nonumber\\\\\n&&+\\!\\sum_{j=1}^{N_{\\textrm{R}}^{\\textrm{RF}}}\\!\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}\\right)\\!.\\nonumber\\\\\n&&\n\\end{eqnarray}\nBy considering (\\ref{eqn_fRF_reim})$-$(\\ref{eqn_derL_fRF_1im}), we arrive at\n\\begin{eqnarray}\n\\label{eqn_derL_pha_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} \\frac{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} \\frac{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}}\\nonumber\\\\\n&&=-\\sin[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} +\\cos[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}},\n\\end{eqnarray}\n\n\\emph{CP-NN:} The proof is similar to that of PP-NN and thus is omitted for simplicity.\n\nNow we have shown $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to each parameter in $\\boldsymbol\\Theta$, which completes the proof.\n\\end{IEEEproof}\n\n\nIt can be seen that the proposed DL-JHPF is abstracted into an integrated DNN, where the hybrid processing matrices, $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$, are essentially the trainable weights therein. From the proof of Theorem 1, each weight of this integrated DNN can be optimized iteratively through BP algorithm by minimizing the BCE loss. Therefore, the optimal precoding and combining matrices on training set are obtained.\n\nFor the NNs in Fig.~\\ref{visual_model}, each dense layer is with rectified linear unit (ReLU) activation function and followed by a BN layer to avoid gradient diffusion and overfitting. The number of dense layers and the number of neurons in each dense layer need to be adjusted according to the input and output dimensions. Since the outputs of the NNs will be used for hybrid processing at the transmitter and the reciever, the activation functions of the output layers should be carefully designed and are elaborated as follows.\n\n\\emph{PP-NN and CP-NN:} The two NNs generate the phases for $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, respectively. Since (\\ref{eqn_fRF}) and (\\ref{eqn_wRF}) are periodic functions, ReLU activation function is used in the output layer to provide the unbiased output for all possible phases. We may also use Sigmoid or hyperbolic tangent as the activation function, after which the outputs are multiplied by $2\\pi$ or $\\pi$ to obtain the final phases with the range of $[0,2\\pi]$ or $[-\\pi,\\pi]$. According to the simulation trails, ReLU and hyperbolic tangent achieve almost the same performance while Sigmoid performs worse. Therefore, ReLU is preferable since it is simple and free of the operation of exponential functions.\n\n\\emph{Re\/ImDP-NN and Re\/ImDC-NN:} The four NNs generate the real and imaginary parts for $\\mathbf{F}_{\\textrm{BB}}$ and $\\mathbf{W}_{\\textrm{BB}}$, respectively. Since $\\mathbf{F}_{\\textrm{BB}}$ can be normalized by (\\ref{eqn_DP_norm}) while $\\mathbf{W}_{\\textrm{BB}}$ has no constraint, the output layers do not apply any activation function to impose constraints and directly output the values that are input into the neurons.\n\n\\emph{NN demodulator:} This NN approximates the original bits, $\\mathbf{X}_{\\textrm{b}}$, based on $\\mathbf{r}$. The approximation for each element in $\\mathbf{X}_{\\textrm{b}}$ is a binary classification and thus the Sigmoid activation function is used for the output layer of the NN demodulator.\n\n\\vspace{-0.2cm}\n\\subsection{Deployment and Testing}\n\nIn this subsection, we elaborate the deployment and testing of the trained DL-JHPF for practical implementation, where $\\mathbf{H}$ is assumed to be available at both the transmitter and the receiver.\\footnote{Although only the estimated channel is available in practical implementation, it has been shown by the simulation results that the relatively accurate channel estimate hardly causes performance loss and is almost equivalent to $\\mathbf{H}$.}\n\nThe practical deployment of DL-JHPF includes the following three parts:\n\n\\emph{Deployment of hybrid processing designer:} PP-NN and CP-NN will be deployed together at \\emph{both the transmitter and the receiver} to output the analog processing matrices, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, based on which the equivalent channel, $\\mathbf{H}_{\\textrm{eq}}$, can be generated via (\\ref{eqn_Heq}). ReDP-NN and ImDP-NN are equipped at the \\emph{transmitter} to generate the digital precoder, $\\mathbf{F}_{\\textrm{BB}}$, while ReDC-NN and ImDC-NN are equipped at the \\emph{receiver} to generate the digital combiner, $\\mathbf{W}_{\\textrm{BB}}$, both based on $\\mathbf{H}_{\\textrm{eq}}$.\n\n\\emph{Deployment of signal flow simulator:} It is only used for the training stage and will be replaced by the actual transceiver and wireless fading channel in the deployment and testing stage.\n\n\\emph{Deployment of NN demodulator:} It will be deployed at the \\emph{receiver} to output the recovered bits, $\\hat{\\mathbf{X}}_{\\textrm{b}}$, based on the detected signal, $\\mathbf{r}$, after compensating the impact of the fading channel.\n\nWhen testing the trained DL-JHPF in real world, the channel may change rapidly due to the relative motion of the transceiver and scatterers, in which case DL-JHPF will be faced new propagation scenarios with different channel statistics from the training stage. This channel scenario discrepancy poses a high requirement on the robustness of DL-JHPF. Fortunately, the offline trained framework in Section III.C is quite robust to the new channel scenarios that are not observed before as shown from our simulation results (Figs.~\\ref{HBD_BS_proDL_UMaipCSI} and \\ref{HBD_BS_proDL_ofdm_UMaipCSI}). The further online fine-tuning may only provide marginal performance improvement but requires a relatively large overhead and needs to be performed frequently in the rapidly changed channel scenario. In addition, only the NNs at the receiver can be fine-tuned and thus the performance after fine-tuning will still have an intrinsic loss compared to the end-to-end training in Section III.C. To sum up, the proposed framework can cope with the mismatch of the channel scenario without relying on the fine-tuning in most cases.\n\n\\subsection{Complexity Analysis}\n\nIn this subsection, we analyze the computational complexity of the proposed DL-JHPF in testing stage by using the metric of required number of floating point operations (FLOPs). According to Fig.~\\ref{visual_model}, the total required FLOPs of all neural layers in DL-JHPF is given by\n\\begin{eqnarray}\n\\label{eqn_Complexity_NN}\n\\mathcal{C}_{\\textrm{NN}}\\sim\\mathcal{O}\\left(\\sum_{\\Delta\\in\\mathcal{N}}\\sum_{i=1}^{L^{\\Delta}-1}N^{\\Delta}_{i}N^{\\Delta}_{i+1}\\right),\n\\end{eqnarray}\nwhere $\\mathcal{N}$ denotes the set including all NNs in DL-JHPF, $L^{\\Delta}$ and $N^{\\Delta}_{i}$ represent the number of neural layers and the number of neurons of the $i$th neural layer of the NN $\\Delta$.\n\nIn addition, the complexity of matrix multiplications in the framework is given by\n\\begin{eqnarray}\n\\label{eqn_Complexity_matrix}\n\\mathcal{C}_{\\textrm{Mat}}\\sim\\mathcal{O}\\bigl(N_{\\textrm{R}}^{\\textrm{RF}}N_{\\textrm{T}}N_{\\textrm{R}}).\n\\end{eqnarray}\n\nThen, the total complexity of the proposed DL-JHPF can be expressed as\n\\begin{eqnarray}\n\\label{eqn_Complexity_DL_JHPF}\n\\mathcal{C}_{\\textrm{DL-JHPF}}\\sim\\mathcal{C}_{\\textrm{NN}}+\\mathcal{C}_{\\textrm{Mat}}.\n\\end{eqnarray}\nIt is noted that the NNs can be run efficiently via parallel computing on the graphic processing unit (GPU) and the simple matrix multiplications only cause negligible computational load for the central processing unit (CPU) compared with the existing schemes. Therefore, the proposed DL-JHPF is with low complexity and consumes the very limited runtime.\n\n\\subsection{Extension to OFDM Systems}\n\nIn this subsection, we extend the proposed DL-JHPF to the wideband OFDM systems. Two key issues need to be considered for the extension:\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] In the OFDM systems, the digital precoder and combiner can be designed independently for different subcarriers while the analog precoder and combiner must be shared by all subcarriers. It is critical to design the unified analog precoder and combiner performing well for all subcarriers.\n\n\\item[2)] It is important to maintain the relatively small size, i.e., the number of hidden layers and the number of neurons in each layer in the NNs, and short training time for DL-JHPF when the number of subcarriers is large.\n\\end{itemize}\n\nIn the following, we study how to address the two issues when extending DL-JHPF to the OFDM systems.\n\nAccording to \\cite{P. Dong_b}, the $N_{\\textrm{R}}\\times N_{\\textrm{T}}$ channel matrix between the receiver and the transmitter of the $k$th subcarrier can be expressed as\n\\begin{eqnarray}\n\\label{eqn_Hk}\n\\mathbf{H}[k]&&=\\beta\\sum_{n=1}^{N_{\\textrm{cl}}}\\sum_{m=1}^{N_{\\textrm{ray}}} \\alpha_{n,m}e^{-j2\\pi\\tau_n f_s\\frac{k}{K}}\\mathbf{a}_{\\textrm{R}}(\\varphi_{n,m})\\mathbf{a}_{\\textrm{T}}^{H}(\\phi_{n,m}),\n\\end{eqnarray}\nwhere $\\beta=\\sqrt{\\frac{N_{\\textrm{T}}N_{\\textrm{R}}}{N_{\\textrm{cl}}N_{\\textrm{ray}}}}$, $\\tau_n$, $f_s$, and $K$ denote the delay of the $n$th cluster, the sampling rate, and the number of OFDM subcarriers, respectively.\nThe signal transmission model in (\\ref{eqn_r}) becomes subcarrier dependent and the detected signal of the $k$th subcarrier is given by\\footnote{Although $\\mathbf{x}$ and $\\mathbf{n}$ are also different for different subcarriers, they are independent of the channel and thus the index $k$ in them is omitted.}\n\\setlength{\\arraycolsep}{0.05em}\n\\begin{eqnarray}\n\\label{eqn_rk}\n\\mathbf{r}[k]&&=\\sqrt{P}\\mathbf{W}_{\\textrm{BB}}^H[k]\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}[k]\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}[k]\\mathbf{x}\\nonumber\\\\\n&&\\quad+\\mathbf{W}_{\\textrm{BB}}^H[k]\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{n}.\n\\end{eqnarray}\n\nIn the following, we propose a simple method to design the structure of training data so that the DL-JHPF in Section III.C can be flexibly extended to OFDM systems without changing the framework architecture. That is, both the framework size and training time will not be increased. The process of training and testing is detailed as follows.\n\n\\emph{Training:} Compared to the training sample with the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ in Section III.C, we modify the input tuple as $\\langle\\bar{\\mathbf{H}}, \\mathbf{H}[i], \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$, where $\\bar{\\mathbf{H}}$ is the channel matrix of a given subcarrier, e.g., the $q$th subcarrier, same for all training samples while $\\mathbf{H}[i]$ is the channel matrix of an uncertain subcarrier with $i$ randomly generated from the set $\\{1,2,\\ldots,K\\}$ for each training sample. As shown in Fig.~\\ref{DL_framework_ofdm}, when inputting each training sample into the framework, $\\bar{\\mathbf{H}}$ will be used to generate $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ via PP-NN and CP-NN. Then $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with $\\mathbf{H}[i]$ are used to generate the equivalent channel of the $i$th subcarrier, $\\mathbf{H}_{\\textrm{eq}}[i]$, based on which, $\\mathbf{F}_{\\textrm{BB}}[i]$ and $\\mathbf{W}_{\\textrm{BB}}[i]$ can be obtained through Re\/ImDP-NN and Re\/ImDC-NN. On the other hand, $\\mathbf{H}[i]$ is also input into the signal flow simulator to act as the fading channel since this training sample is used to simulate the transmission of the $i$th subcarrier. Then the end-to-end training can be performed by minimizing the BCE loss between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$. Through training, we can obtain the unified analog precoder and combiner that match the channel of each subcarrier well without complicating the architecture of DL-JHPF.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{Fig4}\n\\caption{Extension of the proposed DL-JHPF to OFDM systems.}\\label{DL_framework_ofdm}\n\\end{figure}\n\n\\emph{Testing:} With $\\mathbf{H}[k], k=1,2,\\ldots,K$, available at the transceiver, choose the channel matrix of the $q$th subcarrier as $\\bar{\\mathbf{H}}$. Input $\\bar{\\mathbf{H}}$ into PP-NN and CP-NN to generate the unified $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ for all subcarriers. The unified $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with the channel of each subcarrier, $\\mathbf{H}[k], k=1,2,\\ldots,K$, are used to generate the corresponding equivalent channel, which will be input into Re\/ImDP-NN and Re\/ImDC-NN to generate $\\mathbf{F}_{\\textrm{BB}}[k]$ and $\\mathbf{W}_{\\textrm{BB}}[k]$ for channel equalization in each subcarrier. The NN demodulator will be used to recover the original bits for each subcarrier based on the detected signal, $\\mathbf{r}[k]$.\n\n\\section{Simulation Results}\n\nIn this section, the effectiveness of the proposed DL-JHPF is verified in several cases. Six hybrid processing schemes and the fully-digital transceiver architecture are used as the baseline schemes for comparison: 1) HBD scheme in \\cite{W. Ni}; 2) Beam sweeping (BeS) scheme in \\cite{O. E. Ayach}; 3) Discrete Fourier transform (DFT) codebook based joint digital beamforming (DCJDB) scheme, where the analog precoder and combiner are searched from the DFT codebook by the method in \\cite{A. M. Elbir} while the digital precoder and combiner are jointly optimized according to \\cite{D. P. Palomar}; 4) Joint digital beamforming with alternating minimization (JDB-AltMin), where the optimal precoding and combining matrices are first designed according to \\cite{D. P. Palomar}, based on which the hybrid precoding and combining matrices are constructed according to the PE-AltMin algorithm in \\cite{X. Yu}; 5) Hybrid beamforming via deep learning (HBDL) scheme in \\cite{A. M. Elbir}; 6) Deep learning for direct hybrid precoding (DLDHP) scheme in \\cite{X. Li}; 7) Fully-digital transceiver architecture.\n\n\\subsection{Simulation Settings}\n\n\\begin{table}[!t]\n\\centering\n\\caption{Architectures of DNNs in Proposed DL-HPF}\n\\label{table_1}\n\\begin{tabular}{p{1.4cm}<{\\centering}|c|c|c}\n\\hline\n~ & Layer type & \\makecell{Number of \\\\neurons} & \\makecell{Activation \\\\function}\\\\\n\\hline\n\\multirow{5}{*}{PP-NN} & Input & $2N_{\\textrm{T}}N_{\\textrm{R}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 512 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 256 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 128 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{T}}N_{\\textrm{T}}^{\\textrm{RF}}$ & ReLU\\\\\n\\hline\n\\multirow{6}{*}{CP-NN} & Input & $2N_{\\textrm{T}}N_{\\textrm{R}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 512 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 256 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 128 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 64 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{R}}N_{\\textrm{R}}^{\\textrm{RF}}$ & ReLU\\\\\n\\hline\n\\multirow{5}{*}{\\makecell{Re\/ImDP-\\\\NN}} & Input & $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 40 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{s}}$ & - \\\\\n\\hline\n\\multirow{5}{*}{\\makecell{Re\/ImDC-\\\\NN}} & Input & $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 40 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{R}}^{\\textrm{RF}}N_{\\textrm{s}}$ & - \\\\\n\\hline\n\\multirow{5}{*}{\\makecell{NN\\\\demodulator}} & Input & $2N_{\\textrm{s}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 50 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{s}}\\log_{2} M$ & Sigmoid \\\\\n\\hline\n\\end{tabular}\n\\vspace{-0.3cm}\n\\end{table}\n\n\\emph{1) System Settings:} We set $N_{\\textrm{T}}=32$ and $N_{\\textrm{T}}^{\\textrm{RF}}=3$ for the transmitter and $N_{\\textrm{R}}=16$ and $N_{\\textrm{R}}^{\\textrm{RF}}=3$ for the receiver. The number of data streams is set as $N_{\\textrm{s}}=3$. The channel data are generated according to the 3GPP TR 38.901 Release 15 channel model \\cite{3GPP}. Specifically, we use the clustered delay line models with $N_{\\textrm{cl}}=3$ clusters and $N_{\\textrm{ray}}=20$ rays in each cluster. The carrier frequency is $f_c=28$ GHz. For OFDM systems, the sampling rate is $f_s=100$ MHz and the number of subcarriers is $K=64$. Two channel scenarios, urban micro (UMi) street non-line of sight (NLOS) scenario and urban macro (UMa) NLOS scenario, are considered.\\footnote{According to the parameters for UMi NLOS scenario and UMa NLOS scenario defined by \\cite{3GPP}, we use the system object, nr5gCDLChannel, embedded in 5G Library for LTE System Toolbox in MATLAB to generate the corresponding channel data.} Quadrature phase shift keying (QPSK) is used as the modulation method.\n\n\\emph{2) Proposed DL-JHPF Settings:} The training set, validation set, and testing set contain $261$,$000$, $29$,$000$, and $10$,$000$ samples, respectively. The training set and validation set are generated in UMi NLOS scenario while the testing set is generated in both UMi NLOS and UMa NLOS scenarios. Adam is used as the optimizer. The number of epochs in the training stage is set as $800$ while the corresponding learning rates are $10^{-3}$ for the first $500$ epochs and $10^{-4}$ for the rest $300$ epochs, respectively. The batch size is $256$. The architecture of each NN in DL-JHPF is listed in Table~\\ref{table_1}, where the BN layer is added after each dense layer and thus is not listed in the table for simplicity.\n\n\n\\subsection{Performance Evaluation}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig5}\n\\caption{BER performance of the proposed DL-JHPF and the existing hybrid processing schemes.}\\label{HBD_BS_proDL}\n\\end{figure}\n\nIn Figs.~\\ref{HBD_BS_proDL}$-$\\ref{HBD_BS_proDL_UMaipCSI}, the proposed DL-JHPF is first evaluated in narrowband systems while the performance in wideband OFDM systems is presented in Figs.~\\ref{HBD_BS_proDL_ofdm} and \\ref{HBD_BS_proDL_ofdm_UMaipCSI}.\n\nFig.~\\ref{HBD_BS_proDL} shows the BER performance of HBD, BeS, DCJDB, JDB-AltMin, HBDL, DLDHP, the proposed DL-JHPF, and the fully-digital architecture versus signal-to-noise ratio (SNR) in UMi NLOS scenario with perfect CSI. From the figure, DL-JHPF has a larger slope for the BER curve and outperforms the other six hybrid processing schemes after $\\textrm{SNR}=0$ dB although it performs not very well in the low SNR regime. When $\\textrm{BER}=10^{-2}$, the proposed DL-JHPF achieves about $0.2$ dB, $1$ dB, $1.2$ dB, $2$ dB, $6$ dB, and $8$ dB gains compared to JDB-AltMin, DLDHP, DCJDB, BeS, HBDL, and HBD, respectively. The advantage of DL-JHPF becomes more obvious as SNR increases and the BER is smaller than $10^{-4}$ when $\\textrm{SNR}=10$ dB while the performance of other four schemes is larger than $10^{-3}$. With the significantly increased number of RF chains, the fully-digital beamforming obtains substantial diversity gains, which directly leads to the better BER performance than all the hybrid processing schemes. The performance gap between the proposed DL-JHPF and the fully-digital beamforming is about 4dB.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig6}\n\\caption{Robustness of the proposed DL-JHPF with mismatched CSI.}\\label{HBD_BS_proDL_ipCSI}\n\\end{figure}\n\nPerfect CSI is used in framework training while only estimated CSI is available in the practical transmitter and receiver, which leads to the CSI mismatch. In Fig.~\\ref{HBD_BS_proDL_ipCSI}, we investigate the robustness of the proposed DL-JHPF with mismatched CSI, where the BER curve tested with perfect CSI in Fig.~\\ref{HBD_BS_proDL} is also plotted as the lower bound. We use the approach in \\cite{P. Dong_b} to estimate channels at $\\textrm{SNR}=10$ dB and $20$ dB, respectively, for hybrid processing design. From Fig.~\\ref{HBD_BS_proDL_ipCSI}, when tested with the CSI estimated at $20$ dB, DL-JHPF achieves almost the same BER performance as the perfect CSI case and outperforms the other six hybrid processing schemes after $\\textrm{SNR}=0$ dB, indicating that DL-JHPF is hardly impacted by the mismatched CSI estimated at $20$ dB. When tested with the CSI estimated at $10$ dB, performance loss occurs at an acceptable level for DL-JHPF. The loss is less than $1$ dB when $\\textrm{BER}=10^{-2}$ and DL-JHPF still has the clear performance superiority after $\\textrm{SNR}=2.5$ dB even compared to other hybrid processing schemes with the CSI estimated at $20$ dB.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig7}\n\\caption{Robustness of the proposed DL-JHPF with mismatched channel scenario and CSI.}\\label{HBD_BS_proDL_UMaipCSI}\n\\end{figure}\n\nAs mentioned in Section III.D, it is very likely to face with different channel scenarios in the practical testing for DL-JHPF. In Fig.~\\ref{HBD_BS_proDL_UMaipCSI}, we further consider this channel scenario mismatch and test the robustness of DL-JHPF to the aggregate impact caused by channel scenario and CSI mismatch. For DL-JHPF, the BER performance is tested in UMi NLOS scenario with perfect CSI, in UMa NLOS scenario with perfect CSI (mismatched channel scenario), and in UMa NLOS scenario with the CSI estimated at $20$ dB (mismatched channel scenario and CSI), respectively. The performance curves of the baseline schemes evaluated in UMa NLOS scenario with the CSI estimated at $20$ dB are also plotted for comparison. From Fig.~\\ref{HBD_BS_proDL_UMaipCSI}, the channel scenario mismatch causes only less than $0.5$ dB performance loss for DL-JHPF. The total loss caused by the aggregate impact of channel scenario and CSI mismatch is only less than $1$ dB. The proposed DL-JHPF has learned the inherent structure of the mmWave channels and thus is able to maintain its advantage even with mismatched channel scenarios and CSI.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig8}\n\\caption{BER performance of the proposed DL-JHPF and the existing hybrid processing schemes in OFDM systems.}\\label{HBD_BS_proDL_ofdm}\n\\end{figure}\n\nFig.~\\ref{HBD_BS_proDL_ofdm} shows the BER performance of HBD, BeS, DCJDB, JDB-AltMin, HBDL, DLDHP, the proposed DL-JHPF, and the fully-digital architecture in OFDM systems with UMi NLOS scenario and perfect CSI, which is similar to that in Fig.~\\ref{HBD_BS_proDL}. In addition, we plot the BER performance of an ideal case with matched analog processing (AP) for DL-JHPF, where different analog processing matrices are designed for different subcarriers to match the corresponding channels. This is impossible to be implemented in practical systems and we just use it to quantify the performance loss caused by using the unified analog processing matrices for all subcarriers. From Fig.~\\ref{HBD_BS_proDL_ofdm}, only about $1$ dB loss is incurred, which proves the effectiveness of DL-JHPF in OFDM systems by simply modifying the structure of training data without changing the framework architecture and increasing the training time.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig9}\n\\caption{Robustness of the proposed DL-JHPF with mismatched channel scenario and CSI in OFDM systems.}\\label{HBD_BS_proDL_ofdm_UMaipCSI}\n\\end{figure}\n\nIn Fig.~\\ref{HBD_BS_proDL_ofdm_UMaipCSI}, we further test the robustness of DL-JHPF in OFDM systems with the mismatched channel scenario and CSI. The aggregate impact of channel scenario and CSI mismatch is still limited, and DL-JHPF tested in UMa NLOS scenario with the CSI estimated at $20$ dB (mismatched channel scenario and CSI) even outperforms that tested in UMi NLOS scenario with perfect CSI after $\\textrm{SNR}=8$ dB, which verifies the effectiveness and robustness of the proposed DL-JHPF in OFDM systems. In addition, the performance gap between the proposed DL-JHPF and the fully-digital beamforming maintains at about 4dB.\n\n\\subsection{Computational Complexity Comparison}\n\n\\begin{table}\n  \\centering\n  \\caption{Runtime of Hybrid Processing Schemes}\n  \\label{runtime}\n  \\begin{tabular}{c|c}\n  \\hline\n  ~ & Runtime (in ms)\\\\\n  \\hline\n  HBD & $6.98$\\\\\n  \\hline\n  BeS & $10.61$\\\\\n  \\hline\n  DCJDB & $338.73$\\\\\n  \\hline\n  JDB-AltMin & $1.56$\\\\\n  \\hline\n  HBDL & $0.46$\\\\\n  \\hline\n  DLDHP & $0.16$\\\\\n  \\hline\n  Proposed DL-JHPF & $0.06$\\\\\n  \\hline\n  \\end{tabular}\n\\end{table}\n\nFor mmWave mobile communications, the length of coherence time becomes smaller compared to that in sub-6 GHz and thus the runtime of a hybrid processing scheme is vital. Based on the simulation settings mentioned above, we compare the runtime of the proposed DL-JHPF in the testing stage with the baseline schemes in Table~\\ref{runtime}. The HBD, BeS, DCJDB, and JDB-AltMin schemes are run on the Intel(R) Core(TM) i7-3770 CPU while the proposed DL-JHPF are run on the NVIDIA GeForce GTX 2080 Ti GPU. For HBDL and DLDHP, the predictions of $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ are implemented via DNN on the GPU while the following design of $\\mathbf{F}_{\\textrm{BB}}$ and $\\mathbf{W}_{\\textrm{BB}}$ are executed on the CPU. By moving the time-consuming design of analog processing to the GPU that enables the efficient parallel computing, the DL based schemes reduce the runtime significantly compared to the conventional schemes. Through carefully design, the proposed DL-JHPF is fully GPU-driven when generating hybrid processing matrices and thus consumes the minimum time among the three DL based schemes. Therefore, the proposed DL-JHPF is more suitable for mmWave communications, especially for the high-mobility scenario.\n\n\\section{Conclusion}\n\nIn this paper, DL is applied for joint hybrid processing design at the transceiver in mmWave massive MIMO systems. A novel DL-JHPF is developed to learn the optimal analog and digital processing matrices by minimizing the end-to-end BCE loss between the original and recovered bits. The elaborate architecture of the proposed DL-JHPF guarantees the BP-enabled training of each NN therein. By simply modifying the structure of training data, DL-JHPF can be flexibly extended to OFDM systems without changing the framework architecture and increasing the training time. Simulation results show the superiority and robustness of DL-JHPF in various non-ideal conditions with the significantly reduced runtime.\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\\section{Introduction}\n\nDue to the huge bandwidth, millimeter wave (mmWave) communications have been recognized as one of the key technologies to meet the demand for unprecedentedly high data rate transmission in the future mobile networks \\cite{A. L. Swindlehurst}. By equipping large-scale antenna arrays, massive multiple-input multiple-output (MIMO) can provide sufficiently large array gains for spatial multiplexing and beamforming \\cite{L. Lu}. MmWave massive MIMO communications can obtain the merits of both of them and thus have attracted significant interest \\cite{B. Wang}. However, the expensive and power-hungry hardwares used in mmWave bands become the main obstacle to equipping a dedicated radio frequency (RF) chain for each antenna. The mainstream solution for this problem is to use the two-stage hybrid architecture, where a large number of antennas are connected to much fewer RF chains via phase shifters \\cite{L. Liang}, \\cite{L. Pan}.\n\n\\subsection{Related Work}\n\nFor mmWave massive MIMO systems with the hybrid architecture, both the analog and digital processing should be carefully designed to achieve the comparable performance to the fully-digital systems. In \\cite{L. Liang}, a low-complexity hybrid precoding scheme at the base station (BS) has been proposed for the massive MIMO downlink with single-antenna users. The hybrid architecture has been further introduced to the user side in \\cite{W. Ni}, where hybrid block diagonalization (HBD) has been used for the analog and digital processing design. By exploiting the sparsity of mmWave channels, the hybrid precoding and combining at both the transmitter and receiver have been optimized in \\cite{O. E. Ayach}. The heuristic hybrid beamforming design in \\cite{F. Sohrabi} can approach the performance of the fully-digital architecture. The alternating minimization algorithms for both fully-connected and sub-connected hybrid architectures in \\cite{X. Yu} are with low complexity and limited performance loss. In \\cite{L. Zhao}, the hybrid processing along with channel estimation has been designed and analyzed for both the sparse and non-sparse channels. The uniform channel decomposition and nonlinear digital processing have been introduced in \\cite{Y. Lin} for hybrid beamforming design. In the existing works, the hybrid processing matrices at the transmitter and receiver are usually optimized separately due to the intractability of the joint optimization with non-convex constraints, which makes the further performance improvement possible with joint optimization.\n\nDeep learning (DL) has achieved great success in various fields, including computer vision \\cite{K. He_a}, speech signal processing \\cite{A. Graves}, natural language processing \\cite{R. Collobert}, and so on, due to its unique ability in extracting and learning inherent features. It has been recently introduced to wireless communications and shown quite powerful in the optimization of communication systems \\cite{T. O'hea}--\\cite{G. Gui} and resource allocation \\cite{L. Liang_b}--\\cite{Z. Yang}. In \\cite{H. Ye_a}, DL has been successfully applied in pilot-assisted signal detection for orthogonal frequency division multiplexing (OFDM) systems with non-ideal transceiver and channel conditions. For wideband mmWave massive MIMO systems in time-varying channels, channel correlation has been exploited by deep convolutional neural network (CNN) in \\cite{P. Dong_b} to improve the accuracy and accelerate the computation for the channel estimation. Deep neural network (DNN) has been utilized in \\cite{S. Gao} to model the mapping relationship among antennas for reliable channel estimation in massive MIMO systems with mixed-resolution ADCs. An autoencoder-like DNN has been developed in \\cite{C.-K. Wen} to reduce the overhead for channel state information (CSI) feedback in the frequency duplex division massive MIMO system. In \\cite{C. Lu_a}, CNN has been utilized in CSI compression and uncompression to significantly improve the recovery accuracy. By combining the residual network and CNN, an efficient channel quantization scheme has been proposed from the perspective of bit-level in \\cite{C. Lu_b}. The DL based end-to-end optimization has been developed in \\cite{S. Dorner} and \\cite{H. Ye_b} by breaking the block structures at the transceiver. DL has been recently used to design the hybrid processing matrices for massive MIMO systems with various transceiver architectures \\cite{H. Huang}--\\cite{X. Bao}. In \\cite{H. Huang}, the analog and digital precoder design has been modeled as the DNN mapping based on geometric mean decomposition. In \\cite{T. Lin}, DNN has been applied to design the analog precoder for massive multiple-input single-output (MISO) systems. Deep CNN has been applied to learn the phases of the analog precoder and combiner for mmWave massive MIMO systems in \\cite{A. M. Elbir}. For the same system, channel estimation and analog processing have been jointly optimized by DL with reduced pilot overhead in \\cite{X. Li}. In \\cite{X. Bao}, deep CNN along with an equivalent channel hybrid precoding algorithm have been proposed to design the hybrid processing matrices.\n\n\\subsection{Motivation and Contribution}\n\nThe research on the DL based hybrid processing for mmWave massive MIMO systems is still in the exploratory stage and has many open issues. The existing works have applied DL to design the analog precoder \\cite{T. Lin}, the analog combiner \\cite{X. Bao}, the analog precoder and combiner \\cite{A. M. Elbir}, \\cite{X. Li}, and the analog and digital precoders \\cite{H. Huang}. Currently, only partial hybrid processing is designed by DL for the mmWave transceiver. In addition, conventional hybrid processing schemes are usually used to generate label matrices for the DNN to approximate, which limits the performance of the DL based approaches. The problems in the existing works motivate us to propose a general DL based joint hybrid processing framework (DL-JHPF) with the following two unique features:\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] The framework jointly optimizes the analog and digital processing matrices at both the transmitter and receiver in an end-to-end manner without pre-designed label matrices. By doing this, it can be applied to various types of mmWave transceiver architectures and will have the potential to break through the performance of the existing schemes.\n\n\\item[2)] The framework enables end-to-end optimization but still preserves the block structures at the transceiver considering the hardware and power constraints in practical implementation for the hybrid architecture, which is quite different from the end-to-end optimization in \\cite{S. Dorner} and \\cite{H. Ye_b}.\n\\end{itemize}\nThe main contributions of this paper are summarized as follows.\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] We model the joint analog and digital processing design for the transceiver as a DL based framework, which consists of the NN based hybrid processing designer, signal flow simulator, and NN based signal demodulator. For the sake of practical implementation, it does not break the original block structures at the transceiver but still allows the back-propagation (BP) based end-to-end optimization by minimizing the cross-entropy loss between recovered and original bits. The trainability of DL-JHPF is proved theoretically.\n\n\\item[2)] We extend the proposed framework to OFDM systems by simply modifying the structure of the training data. The extension does not complicate the framework architecture and guarantees the relatively short training time even if the number of subcarriers is large.\n\n\\item[3)] We verify the effectiveness of the proposed framework by numerical results based on the 3rd Generation Partnership Project (3GPP) channel model that can well depict the real channel environment. The proposed DL-JHPF achieves remarkable improvement in bit-error rate (BER) performance even with mismatched CSI and channel scenarios. Thanks to the careful design, DL-JHPF reduces the runtime significantly by sufficiently exploiting the parallel computing and thus is more suitable for rapidly varying mmWave channels.\n   \n\\end{itemize}\n\nThe rest of the paper is organized as follows. Section II describes the channel model and signal transmission process for the considered mmWave massive MIMO system. The proposed DL-JHPF is elaborated in Section III. Simulation results are provided in Section IV to verify the effectiveness of the proposed framework and finally Section V gives concluding remarks.\n\n\\emph{Notations}: In this paper, we use upper and lower case boldface letters to denote matrices and vectors, respectively. $\\lVert\\cdot\\rVert_{F}$, $(\\cdot)^T$, $(\\cdot)^H$, and $\\mathbb{E}\\{\\cdot\\}$ represent the Frobenius norm, transpose, conjugate transpose, and expectation, respectively. $\\mathcal{CN}(\\mu,\\sigma^2)$ represents circular symmetric complex Gaussian distribution with mean $\\mu$ and variance $\\sigma^2$. $[\\mathbf{X}]_{i,j}$ and $[\\mathbf{x}]_{i}$ denote the $(i,j)$th element of matrix $\\mathbf{X}$ and the $i$th element of vector $\\mathbf{x}$, respectively. $|\\cdot|$ denotes the amplitude of a complex number.\n\n\\section{System Model}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{Fig1}\n\\caption{MmWave massive MIMO system model.}\\label{system_model}\n\\end{figure}\n\nAs shown in Fig.~\\ref{system_model}, we consider a point-to-point massive MIMO systems working at mmWave bands, where the transmitter and the receiver are with $N_{\\textrm{T}}$ and $N_{\\textrm{R}}$ antennas, respectively. To reduce the hardware cost and power consumption, $N_{\\textrm{T}}^{\\textrm{RF}}(< N_{\\textrm{T}})$ and $N_{\\textrm{R}}^{\\textrm{RF}}(< N_{\\textrm{R}})$ RF chains are used at the transmitter and the receiver, respectively, and are connected to the large-scale antennas via phase shifters.\n\n\\subsection{Channel Model}\n\nDue to the sparse scattering property, the Saleh-Valenzuela channel model has been used to well depict the mmWave propagation environment, where the scattering of multiple rays forms several clusters. According to \\cite{O. E. Ayach}, the $N_{\\textrm{R}}\\times N_{\\textrm{T}}$ channel matrix between the receiver and the transmitter can be represented as\n\\begin{eqnarray}\n\\label{eqn_H_tau}\n\\mathbf{H}=\\sqrt{\\frac{N_{\\textrm{T}}N_{\\textrm{R}}}{N_{\\textrm{cl}}N_{\\textrm{ray}}}}\\sum_{n=1}^{N_{\\textrm{cl}}}\\sum_{m=1}^{N_{\\textrm{ray}}} \\alpha_{n,m}\\mathbf{a}_{\\textrm{R}}(\\varphi_{n,m})\\mathbf{a}_{\\textrm{T}}^{H}(\\phi_{n,m}),\n\\end{eqnarray}\nwhere $N_{\\textrm{cl}}$ and $N_{\\textrm{ray}}$ denote the number of scattering clusters and the number of rays in each cluster, respectively, $\\alpha_{n,m}\\sim \\mathcal{CN}(0, \\sigma_{\\alpha}^2)$ is the propagation gain of the $m$th path in the $n$th cluster with $\\sigma_{\\alpha}^2$ being the average power gain, $\\varphi_{n,m}$ and $\\phi_{n,m}\\in[0,2\\pi]$ are the azimuth angles of arrival and departure (AoA\/AoD) at the receiver and the transmitter, respectively, of the $m$th path in the $n$th cluster.\\footnote{The path gain $\\alpha_{n,m}$ is the fast fading and varies in the time scale of channel coherence interval. Other parameters, $N_{\\textrm{cl}}$, $N_{\\textrm{ray}}$, $\\varphi_{n,m}$, $\\phi_{n,m}$, are slow fading and may be unchanged in a large time scale compared to $\\alpha_{n,m}$. The Doppler spread determines how often these channel parameters change.} For a uniform linear array with $N$ antenna elements and an azimuth angle of $\\theta$, the response vector can be expressed as\n\\begin{eqnarray}\n\\label{eqn_au}\n\\mathbf{a}(\\theta)=\\frac{1}{\\sqrt{N}} \\left[1,e^{-j2\\pi\\frac{d}{\\lambda}\\sin(\\theta)},\\ldots,e^{-j2\\pi\\frac{d}{\\lambda}(N-1)\\sin(\\theta)}\\right]^{T},\n\\end{eqnarray}\nwhere $d$ and $\\lambda$ denote the distance between the adjacent antennas and carrier wavelength, respectively.\n\nIn the above channel model, we assume the transmitted signal is with narrowband and therefore, channel matrix is independent of frequency. For wideband transmission, OFDM is used to convert a frequency-selective channel into multiple flat fading channels and the corresponding channel matrices will be different at different subcarriers. Accordingly, the design of DL-JHPF in Section III will start at the narrowband systems and is then extended to the wideband OFDM systems.\n\n\\subsection{Signal Transmission}\n\nThe transmitter sends $N_{\\textrm{s}}$ parallel data streams to the receiver through the wireless channel. The bits of each data stream are first mapped to the symbol by the $M$-ary modulation. The symbol vector intended for the receiver, $\\mathbf{x}\\in\\mathbb{C}^{N_{\\textrm{s}}\\times 1}$ with $\\mathbb{E}\\left\\{\\mathbf{x}\\mathbf{x}^{H}\\right\\}=\\frac{1}{N_{\\textrm{s}}}\\mathbf{I}_{N_{\\textrm{s}}}$, is successively processed by the digital precoder, $\\mathbf{F}_{\\textrm{BB}}\\in\\mathbb{C}^{N_{\\textrm{T}}^{\\textrm{RF}}\\times N_{\\textrm{s}}}$, at the baseband and the analog precoder, $\\mathbf{F}_{\\textrm{RF}}\\in\\mathbb{C}^{N_{\\textrm{T}}\\times N_{\\textrm{T}}^{\\textrm{RF}}}$, through the phase shifters, yielding the transmitted signal\n\\begin{eqnarray}\n\\label{eqn_s}\n\\mathbf{s}=\\sqrt{P}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x},\n\\end{eqnarray}\nwhere $P$ denotes the transmit power. $\\mathbf{F}_{\\textrm{RF}}$ represents the phase-only modulation by the phase shifters and thus has the constraint of $\\left|[\\mathbf{F}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}$, $\\forall \\,i, \\,j$. $\\mathbf{F}_{\\textrm{BB}}$ is normalized as $\\|\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\|_{F}^2=N_{\\textrm{s}}$ to satisfy the total power constraint at the transmitter. Then the received signal at the receiver is given by\n\\begin{eqnarray}\n\\label{eqn_y}\n\\mathbf{y}=\\sqrt{P}\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x}+\\mathbf{n},\n\\end{eqnarray}\nwhere $\\mathbf{n}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times 1}$ is additive white Gaussian noise (AWGN) with $\\mathcal{CN}(0,1)$ elements.\n\nThe received signal $\\mathbf{y}$ is then processed by the hybrid architecture at the receiver as\n\\setlength{\\arraycolsep}{0.05em}\n\\begin{eqnarray}\n\\label{eqn_r}\n\\mathbf{r}&&=\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{y} =\\sqrt{P}\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\mathbf{x} +\\mathbf{W}_{\\textrm{BB}}^H\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{n},\n\\end{eqnarray}\nwhere $\\mathbf{W}_{\\textrm{RF}}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times N_{\\textrm{R}}^{\\textrm{RF}}}$ and $\\mathbf{W}_{\\textrm{BB}}\\in\\mathbb{C}^{N_{\\textrm{R}}^{\\textrm{RF}}\\times N_{\\textrm{s}}}$ represent the analog combiner and digital combiner, respectively. A hardware constraint is imposed on $\\mathbf{W}_{\\textrm{RF}}$ such that $\\left|[\\mathbf{W}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}$, $\\forall \\,i, \\,j$ similar to $\\mathbf{F}_{\\textrm{RF}}$. Then the detected signal vector, $\\mathbf{r}$, is demodulated to recover the original bits of $N_{\\textrm{s}}$ data streams.\n\nSince the performance of the digital communication system is ultimately determined by BER, we aim to jointly design $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ to minimize the BER between the original and demodulated bits, that is\n\\begin{eqnarray}\n&&\\min\\limits_{\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}}\\quad P_{\\textrm{e}}\\left(\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}\\right),\\label{eqn_optim_problem}\\\\\n&&\\quad\\quad\\:\\:\\textrm{s.t.}\\qquad\\qquad\\!\\! \\left|[\\mathbf{F}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}, \\forall \\,i, \\,j,\\label{const1}\\\\\n&&\\qquad\\qquad\\qquad\\quad\\;\\; \\left|[\\mathbf{W}_{\\textrm{RF}}]_{i,j}\\right|=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}, \\forall \\,i, \\,j,\\label{const2}\\\\\n&&\\qquad\\qquad\\qquad\\quad\\;\\; \\|\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}\\|_{F}^2=N_{\\textrm{s}}\\label{const3}.\n\\end{eqnarray}\nThe BER in (\\ref{eqn_optim_problem}) is a complicated nonlinear function of $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ without closed-form expression and the constraints in (\\ref{const1}) and (\\ref{const2}) are non-convex, which make this optimization problem intractable to be solved by the traditional approaches. DL is a potential solution by using the BP algorithm and thus we develop DL-JHPF to address this problem.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=6.6in]{Fig2}\n\\caption{Proposed DL-JHPF.}\\label{DL_framework}\n\\end{figure*}\n\n\\section{Proposed DL-JHPF}\n\nIn this section, we first briefly review the existing work on the DNN based end-to-end communications. Then we propose DL-JHPF, where the framework is first described, followed by the details of training, deployment, and testing along with the corresponding complexity analysis. Finally, we extend the framework to OFDM systems over wideband mmWave channels.\n\n\\subsection{DNN based End-to-End Communications}\n\nPrior works have shown that DNN based end-to-end optimization is an efficient tool to minimize BER. The BP algorithm makes the DNN based end-to-end communications over the air possible so long as the optimized performance metric is differentiable \\cite{T. O'hea}, \\cite{S. Dorner}, \\cite{H. Ye_b}. For the DNN based end-to-end communication system, the modules at the transmitter and the receiver are replaced by two DNNs, respectively. Specifically, the DNN at the transmitter encodes the original symbols into the transmitted signal and the one at the receiver recovers the original symbols from the output of the wireless channel. In the training stage, the error between the original and recovered symbols is computed and the weights of the two DNNs are adjusted iteratively based on the error gradient propagated from the output layer of the DNN at the receiver to optimize the recovery accuracy.\n\nIn this paper, we focus on the DL based joint analog and digital processing design for the transceiver in mmWave massive MIMO systems. Then, the existing DNN based end-to-end communication is not suitable for this task since it integrates the modules of the transceiver into two DNNs and thus cannot meet the hardware and power constraints in practical implementation. To address this challenge, we design DL-JHPF in the following.\n\n\\subsection{Framework Description}\n\n\n\nAs shown in Fig.~\\ref{DL_framework}, the proposed DL-JHPF consists of three parts: hybrid processing designer, signal flow simulator, and NN demodulator, which are elaborated as follows.\n\n\\emph{Hybrid processing designer:} It plays the role of outputting the hybrid processing matrices for the transceiver by using NNs based on the channel matrix. It includes six fully-connected NNs and is used to generate the analog and digital processing matrices for the transmitter and the receiver based on the channel matrix, $\\mathbf{H}$. Specifically, $\\mathbf{H}\\in\\mathbb{C}^{N_{\\textrm{R}}\\times N_{\\textrm{T}}}$ is first converted to a $2N_{\\textrm{T}}N_{\\textrm{R}}\\times 1$ real-valued vector.\\footnote{In Fig.~\\ref{DL_framework}, only the main process of the framework is shown while the matrix and vector reshaping process is omitted.} Then it is input into two NNs, called precoder phase NN (PP-NN) and combiner phase NN (CP-NN), to generate the corresponding phases, $\\boldsymbol{\\phi}_{\\textrm{P}}\\in\\mathbb{R}^{N_{\\textrm{T}}N_{\\textrm{T}}^{\\textrm{RF}}\\times 1}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}\\in\\mathbb{R}^{N_{\\textrm{R}}N_{\\textrm{R}}^{\\textrm{RF}}\\times 1}$, respectively, for phase shifters. With $\\boldsymbol{\\phi}_{\\textrm{P}}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}$, two complex-valued vectors with constant amplitude elements are generated as\n\\begin{eqnarray}\n\\label{eqn_fRF}\n\\bar{\\mathbf{f}}_{\\textrm{RF}}=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}e^{j\\boldsymbol{\\phi}_{\\textrm{P}}},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_wRF}\n\\bar{\\mathbf{w}}_{\\textrm{RF}}=\\frac{1}{\\sqrt{N_{\\textrm{R}}}}e^{j\\boldsymbol{\\phi}_{\\textrm{C}}},\n\\end{eqnarray}\nbased on which, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ are given by\n\\begin{eqnarray}\n\\label{eqn_FRF}\n\\mathbf{F}_{\\textrm{RF}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{f}}_{\\textrm{RF}}),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_WRF}\n\\mathbf{W}_{\\textrm{RF}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{w}}_{\\textrm{RF}}),\n\\end{eqnarray}\nwhere $\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\cdot)$ denotes the operation reshaping a vector to a matrix.\nThen, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with $\\mathbf{H}$ are used to generate a low-dimensional equivalent channel, i.e.,\n\\begin{eqnarray}\n\\label{eqn_Heq}\n\\mathbf{H}_{\\textrm{eq}}=\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}\\mathbf{F}_{\\textrm{RF}}.\n\\end{eqnarray}\n$\\mathbf{H}_{\\textrm{eq}}\\in\\mathbb{C}^{N_{\\textrm{R}}^{\\textrm{RF}}\\times N_{\\textrm{T}}^{\\textrm{RF}}}$ is converted to a $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}\\times 1$ real-valued vector before it is input into four parallel NNs. The first two NNs, corresponding to the real part digital combiner NN (ReDC-NN) and the imaginary part digital combiner NN (ImDC-NN), output two $N_{\\textrm{s}}N_{\\textrm{R}}^{\\textrm{RF}}\\times1$ vectors, $\\bar{\\mathbf{w}}_{\\textrm{BB,re}}$, $\\bar{\\mathbf{w}}_{\\textrm{BB,im}}$, respectively. Then $\\mathbf{W}_{\\textrm{BB}}$ can be obtained as\n\\begin{eqnarray}\n\\label{eqn_WBB}\n\\mathbf{W}_{\\textrm{BB}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{w}}_{\\textrm{BB,re}}+j\\bar{\\mathbf{w}}_{\\textrm{BB,im}}).\n\\end{eqnarray}\nAnother two NNs, corresponding to the real part digital precoder NN (ReDP-NN) and the imaginary part digital precoder NN (ImDP-NN), output two $N_{\\textrm{s}}N_{\\textrm{T}}^{\\textrm{RF}}\\times1$ vectors, $\\bar{\\mathbf{f}}_{\\textrm{BB,re}}$, $\\bar{\\mathbf{f}}_{\\textrm{BB,im}}$, respectively. Then the unnormalized digital precoder $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ is given by\n\\begin{eqnarray}\n\\label{eqn_unnorm_FBB}\n\\bar{\\mathbf{F}}_{\\textrm{BB}}=\\mathcal{T}_{\\textrm{v}\\rightarrow \\textrm{m}}(\\bar{\\mathbf{f}}_{\\textrm{BB,re}}+j\\bar{\\mathbf{f}}_{\\textrm{BB,im}}).\n\\end{eqnarray}\nThe following normalization utilizes $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ and $\\mathbf{F}_{\\textrm{RF}}$ in (\\ref{eqn_FRF}) to output the final digital precoder as\n\\begin{eqnarray}\n\\label{eqn_DP_norm}\n\\mathbf{F}_{\\textrm{BB}}=\\frac{\\sqrt{N_{\\textrm{s}}}}{\\|\\mathbf{F}_{\\textrm{RF}}\\bar{\\mathbf{F}}_{\\textrm{BB}}\\|_{F}}\\bar{\\mathbf{F}}_{\\textrm{BB}}.\n\\end{eqnarray}\n\n\\emph{Signal flow simulator:} In the training stage, it simulates the process from the original bits, $\\mathbf{X}_{\\textrm{b}}$, to the detected signal, $\\mathbf{r}$, over the channel, $\\mathbf{H}$, with AWGN, $\\mathbf{n}$, where $\\mathbf{X}_{\\textrm{b}}$ with the size of $N_{\\textrm{s}}\\times\\log_2 M$, $\\mathbf{H}$, and $\\mathbf{n}$ are generated in the simulation environment. It bridges the back propagation of the error gradient from NN demodulator to hybrid processing designer as we will elaborate in Section III.C. In the deployment and testing stage, the signal flow simulator is replaced by the actual transceiver and the actual wireless fading channel. In these two stages, the analog and digital processing matrices at the transceiver are provided by the hybrid processing designer based on the simulated or actual $\\mathbf{H}$.\n\n\\emph{NN demodulator:} It is a fully-connected NN, which receives the detected signal, $\\mathbf{r}$, from the signal flow simulator (in the training stage) or the actual receiver (in the testing stage) and outputs recovered bits $\\hat{\\mathbf{x}}_{\\textrm{b}}\\in \\mathbb{R}^{N_{\\textrm{s}}\\log_2 M\\times1}$ with each element lies in the interval $[0,1]$. $\\hat{\\mathbf{x}}_{\\textrm{b}}$ is then reshaped to $\\hat{\\mathbf{X}}_{\\textrm{b}}$ with the same size as $\\mathbf{X}_{\\textrm{b}}$.\n\n\\textbf{Remark 1.} \\emph{The learning of hybrid processing matrices, $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{W}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, and $\\mathbf{W}_{\\textrm{BB}}$, in DL-JHPF is embedded into the signal transmission and demodulation process instead of approximating pre-designed label matrices. All NNs are optimized jointly sharing the mapping principle from $\\mathbf{X}_{\\textrm{b}}$ at the transmitter to $\\mathbf{X}_{\\textrm{b}}$ at the receiver that resembles an\nautoencoder. By minimizing the error between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$, each NN in hybrid processing designer can learn to output the appropriate vectors with specific meaning implicitly, i.e., phases of phase shifters and real and imaginary parts of the digital precoder and combiner. By doing this, DL-JHPF will have the potential to break through the performance of the existing schemes.}\n\n\\subsection{Framework Training}\n\nThe goal of offline training is to determine the weights of the NNs in hybrid processing designer and NN demodulator based on the training samples with the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ and the label $\\mathbf{X}_{\\textrm{b}}$, where $\\mathbf{H}$ is generated by certain channel model and $\\mathbf{n}$ is generated according to the $\\mathcal{CN}(0,1)$ distribution. By minimizing the end-to-end error between the original bits, $\\mathbf{X}_{\\textrm{b}}$, and the recovered bits, $\\hat{\\mathbf{X}}_{\\textrm{b}}$, the weights of each NN in DL-JHPF are adjusted iteratively and the training procedure is elaborated as follows.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.0in]{Fig3}\n\\caption{Training model for proposed DL-JHPF.}\\label{visual_model}\n\\end{figure}\n\nThe proposed DL-JHPF is actually an integrated DNN consisting of neuron layers and custom layers. The training model in Fig.~\\ref{visual_model} demonstrates the detailed training process of the framework. For each training sample, $\\mathbf{H}$ is converted into a real-valued vector by matrix-to-vector reshaping and real and imaginary parts stacking, which is input into PP-NN and CP-NN consisting of dense and batch normalization (BN) layers to generate the corresponding phases, $\\boldsymbol{\\phi}_{\\textrm{P}}$ and $\\boldsymbol{\\phi}_{\\textrm{C}}$, respectively. Then (\\ref{eqn_fRF}) and (\\ref{eqn_wRF}) are executed by the same custom layer. Afterwards, the output vectors are reshaped according to (\\ref{eqn_FRF}) and (\\ref{eqn_WRF}) to generate $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, respectively. Next, (\\ref{eqn_Heq}) is executed by a custom layer to generate $\\mathbf{H}_{\\textrm{eq}}$, followed by matrix-to-vector reshaping and real and imaginary parts stacking. This  vector is input into four NNs consisting of dense and BN layers, i.e., ReDC-NN, ImDC-NN, ReDP-NN, and ImDP-NN, respectively. The output vectors of the former two NNs are used to generate $\\mathbf{W}_{\\textrm{BB}}$ through real and imaginary parts combining and vector-to-matrix reshaping as (\\ref{eqn_WBB}). Using the same operation, the output vectors of the latter two NNs are used to generate $\\bar{\\mathbf{F}}_{\\textrm{BB}}$ as (\\ref{eqn_unnorm_FBB}). After obtaining $\\bar{\\mathbf{F}}_{\\textrm{BB}}$, a custom layer is added to perform the normalization in (\\ref{eqn_DP_norm}) to generate $\\mathbf{F}_{\\textrm{BB}}$. Then (\\ref{eqn_r}) is executed through a custom layer by using the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ and the generated $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$ to yield the detected signal, $\\mathbf{r}$. After real and imaginary stacking, $\\mathbf{r}$ is converted to a real-valued vector and input into the NN demodulator consisting of dense and BN layers to output the recovered bits, $\\hat{\\mathbf{x}}_{\\textrm{b}}$, which is then reshaped to $\\hat{\\mathbf{X}}_{\\textrm{b}}$. The binary cross-entropy (BCE) loss between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$ is calculated as\n\\begin{eqnarray}\n\\label{eqn_loss}\n\\mathcal{L}&&=-\\frac{1}{N_{\\textrm{tr}}}\\sum_{n=1}^{N_{\\textrm{tr}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}\\biggl( [\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}\\ln([\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\nonumber\\\\\n&&\\quad+(1-[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j})\\ln(1-[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\biggr),\n\\end{eqnarray}\nwhere $N_{\\textrm{tr}}$ denotes the number of training samples, superscript $n$ is added to indicate the index of the training sample, and $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}$ is expressed as the function of the parameter set of all NNs in DL-JHPF, i.e., $\\boldsymbol\\Theta$.\n\nRecall the optimization problem in (\\ref{eqn_optim_problem}), the BER over the training set can be written as\n\\begin{eqnarray}\n\\label{eqn_BER}\n&&P_{\\textrm{e,tr}}\\left(\\mathbf{F}_{\\textrm{RF}},\\mathbf{F}_{\\textrm{BB}},\\mathbf{W}_{\\textrm{RF}},\\mathbf{W}_{\\textrm{BB}}\\right) =P_{\\textrm{e,tr}}\\left(\\boldsymbol\\Theta\\right) \\nonumber\\\\\n&&\\quad=\\frac{\\sum_{n=1}^{N_{\\textrm{tr}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}|[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}-[\\hat{\\mathbf{X}}_{\\textrm{b,bin}}^{n}(\\boldsymbol\\Theta)]_{i,j}|}{N_{\\textrm{tr}}N_{\\textrm{s}}\\log_2 M},\\;\\;\\;\\;\n\\end{eqnarray}\nwhere $\\hat{\\mathbf{X}}_{\\textrm{b,bin}}^{n}(\\boldsymbol\\Theta)$ is the binary demodulated bit matrix with $[\\hat{\\mathbf{X}}_{\\textrm{b,1hot}}^{n}(\\boldsymbol\\Theta)]_{i,j}=0$ for $[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j}<0.5$ and $1$ otherwise. With $\\mathbf{X}_{\\textrm{b}}^{n}$ fixed, minimizing $\\mathcal{L}$ in (\\ref{eqn_loss}) with respect to $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)$ yields $\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)=\\mathbf{X}_{\\textrm{b}}^{n}$, which also minimizes $P_{\\textrm{e,tr}}\\left(\\boldsymbol\\Theta\\right)$ in (\\ref{eqn_BER}). Therefore, DL-JHPF can directly minimize the BER over the training set by minimizing the BCE loss and the feasibility is guaranteed by the following theorem.\n\n\\textbf{Theorem 1.} \\emph{The proposed DL-JHPF is trainable and can minimize the BCE loss through BP algorithm.}\n\\begin{IEEEproof}\nConsidering the mini-batch training, the BCE loss over a batch is written as\n\\begin{eqnarray}\n\\label{eqn_loss_batch}\n\\mathcal{L}_{\\textrm{bat}}&&=-\\frac{1}{N_{\\textrm{bat}}}\\sum_{n=1}^{N_{\\textrm{bat}}}\\sum_{i=1}^{N_{\\textrm{s}}}\\sum_{j=1}^{\\log_2 M}\\biggl( [\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j}\\ln([\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\nonumber\\\\\n&&\\quad+(1-[\\mathbf{X}_{\\textrm{b}}^{n}]_{i,j})\\ln(1-[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j})\\biggr),\n\\end{eqnarray}\nwhere $N_{\\textrm{bat}}$ denotes the batch size. Then $\\boldsymbol\\Theta$ will be updated $\\lceil\\frac{N_{\\textrm{tr}}}{N_{\\textrm{bat}}}\\rceil$ times in each epoch.\n\nTo prove Theorem 1, we need to show that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to each parameter in $\\boldsymbol\\Theta$. According to \\cite{Y. LeCun}, the outputs are differentiable with respect to the corresponding weights and inputs for each NN in DL-JHPF. Since DL-JHPF can be viewed as an integrated DNN consisting of neuron layers and custom layers, the proof can be further simplified to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to the outputs of each NN due to chain rule. In the following, we prove the differentiability of $\\mathcal{L}_{\\textrm{bat}}$ with respect to the outputs of each NN by incorporating the custom layers.\n\n\\emph{NN demodulator:} From (\\ref{eqn_loss_batch}), $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\hat{\\mathbf{X}}_{\\textrm{b}}^{n}(\\boldsymbol\\Theta)]_{i,j}, \\forall i, j$.\n\n\\emph{Re\/ImDC-NN:} As mentioned in Section III.B, $\\bar{\\mathbf{w}}_{\\textrm{BB,re}}$ and $\\bar{\\mathbf{w}}_{\\textrm{BB,im}}$ are the outputs of ReDC-NN and ImDC-NN, respectively. Without loss of generality, we will prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$. According to (\\ref{eqn_r}) and (\\ref{eqn_WBB}), $[\\mathbf{r}]_{1}$ is the function of $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$, that is\n\\begin{eqnarray}\n\\label{eqn_r1}\n\\!\\!\\!\\!\\![\\mathbf{r}]_{1}&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}-j[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1})[\\mathbf{z}]_{1}+C_1\\nonumber\\\\\n&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}-j[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1})([\\mathbf{z}]_{1,\\textrm{re}}+j[\\mathbf{z}]_{1,\\textrm{im}})+C_1\\nonumber\\\\\n&&=([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}[\\mathbf{z}]_{1,\\textrm{re}}+[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}[\\mathbf{z}]_{1,\\textrm{im}}+C_{1,\\textrm{re}})\\nonumber\\\\\n&&\\quad+j([\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}[\\mathbf{z}]_{1,\\textrm{im}}-[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}[\\mathbf{z}]_{1,\\textrm{re}}+C_{1,\\textrm{im}})\\nonumber\\\\\n&&=[\\mathbf{r}]_{1,\\textrm{re}}+j[\\mathbf{r}]_{1,\\textrm{im}},\n\\end{eqnarray}\nwhere $\\mathbf{z}=\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{y}$ and $C_1$ denotes the component of $[\\mathbf{r}]_{1}$ independent of $[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}$ with the subscripts `re' and `im' indicating the real and imaginary parts, respectively. Since $[\\mathbf{r}]_{1,\\textrm{re}}$ and $[\\mathbf{r}]_{1,\\textrm{im}}$ are a part of inputs of NN demodulator, $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\mathbf{r}]_{1,\\textrm{re}}$ and $[\\mathbf{r}]_{1,\\textrm{im}}$. Then we have\n\\begin{eqnarray}\n\\label{eqn_derL_wre_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,re}}]_{1}}\\nonumber\\\\\n&&=[\\mathbf{z}]_{1,\\textrm{re}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} +[\\mathbf{z}]_{1,\\textrm{im}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_wim_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{w}}_{\\textrm{BB,im}}]_{1}}\\nonumber\\\\\n&&=[\\mathbf{z}]_{1,\\textrm{im}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{re}}} -[\\mathbf{z}]_{1,\\textrm{re}}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{1,\\textrm{im}}}.\n\\end{eqnarray}\n\n\\emph{Re\/ImDP-NN:} Since $\\bar{\\mathbf{f}}_{\\textrm{BB,re}}$ and $\\bar{\\mathbf{f}}_{\\textrm{BB,im}}$ are the outputs of ReDP-NN and ImDP-NN, respectively, we also aim to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$. Considering the normalization in (\\ref{eqn_DP_norm}), we first calculate the derivatives of $\\mathcal{L}_{\\textrm{bat}}$ with respect to the real and imaginary parts of $[\\mathbf{F}_{\\textrm{BB}}]_{1,1}$, i.e., $\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}$ and $\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}$, which can be obtained similarly to (\\ref{eqn_derL_wre_1}) and (\\ref{eqn_derL_wim_1}). According to (\\ref{eqn_DP_norm}), we have\n\\begin{eqnarray}\n\\label{eqn_FBB11re}\n[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}=\\frac{[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}}{f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_FBB11im}\n[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}=\\frac{[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}}{f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})},\n\\end{eqnarray}\nwhere $f([\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}, [\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1})\\!=\\![([\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{re}}[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1} \\!-\\![\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{im}}[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}+C_{2,\\textrm{re}})^2+([\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{re}}[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1} \\!+\\![\\mathbf{F}_{\\textrm{RF}}]_{1,1,\\textrm{im}}[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}+C_{2,\\textrm{im}})^2+C_3]^{\\frac{1}{2}}$ with $C_2$ and $C_3$ independent of $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$. Then we can find that $[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}$ and $[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}$ are differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}$ and $[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}$, which leads to\n\\begin{eqnarray}\n\\label{eqn_derL_fBBre_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}} &&\\!=\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,re}}]_{1}},\\nonumber\\\\\n&&\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_fBBim_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}} &&\\!=\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{F}_{\\textrm{BB}}]_{1,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{BB,im}}]_{1}}.\\nonumber\\\\\n&&\n\\end{eqnarray}\n\n\\emph{PP-NN:} We still aim to prove that $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to $[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}$ that is one of the output of PP-NN and generates $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$ as\n\\begin{eqnarray}\n\\label{eqn_fRF_reim}\n\\frac{1}{\\sqrt{N_{\\textrm{T}}}}e^{j[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}}&&=\\frac{1}{\\sqrt{N_{\\textrm{T}}}}(\\cos[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1} +j\\sin[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1})\\nonumber\\\\\n&&=[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}+j[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}.\n\\end{eqnarray}\nFrom (\\ref{eqn_r}) and (\\ref{eqn_Heq}), $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$ influence the values of $[\\mathbf{r}]_{i}, i=1,\\ldots,N_{\\textrm{s}}$ and $[\\mathbf{H}_{\\textrm{eq}}]_{j,1}, j=1,\\ldots,N_{\\textrm{R}}^{\\textrm{RF}}$. According to the previous proof, $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to the real and imaginary parts of each element in $\\mathbf{r}$ and $\\mathbf{H}_{\\textrm{eq}}$. $[\\mathbf{r}]_{i,\\textrm{re}}$, $[\\mathbf{r}]_{i,\\textrm{im}}$, $[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}$, and $[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}$, $i=1,\\ldots,N_{\\textrm{s}}, j=1,\\ldots,N_{\\textrm{R}}^{\\textrm{RF}}$ are also differentiable with respect to $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}$ and $[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}$. Resorting to chain rule, we have\n\\begin{eqnarray}\n\\label{eqn_derL_fRF_1re}\n&&\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} =\\sum_{i=1}^{N_{\\textrm{s}}}\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}\\right)\\nonumber\\\\\n&&+\\!\\sum_{j=1}^{N_{\\textrm{R}}^{\\textrm{RF}}}\\!\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}\\right)\\!,\\nonumber\\\\\n&&\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{eqn_derL_fRF_1im}\n&&\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} =\\sum_{i=1}^{N_{\\textrm{s}}}\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{re}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{r}]_{i,\\textrm{im}}} \\frac{\\partial[\\mathbf{r}]_{i,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}\\right)\\nonumber\\\\\n&&+\\!\\sum_{j=1}^{N_{\\textrm{R}}^{\\textrm{RF}}}\\!\\left(\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{re}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} \\!+\\!\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}} \\frac{\\partial[\\mathbf{H}_{\\textrm{eq}}]_{j,1,\\textrm{im}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}\\right)\\!.\\nonumber\\\\\n&&\n\\end{eqnarray}\nBy considering (\\ref{eqn_fRF_reim})$-$(\\ref{eqn_derL_fRF_1im}), we arrive at\n\\begin{eqnarray}\n\\label{eqn_derL_pha_1}\n\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}} &&=\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} \\frac{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}} +\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}} \\frac{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}}{\\partial[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}}\\nonumber\\\\\n&&=-\\sin[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{re}}} +\\cos[\\boldsymbol{\\phi}_{\\textrm{P}}]_{1}\\frac{\\partial\\mathcal{L}_{\\textrm{bat}}}{\\partial[\\bar{\\mathbf{f}}_{\\textrm{RF}}]_{1,\\textrm{im}}},\n\\end{eqnarray}\n\n\\emph{CP-NN:} The proof is similar to that of PP-NN and thus is omitted for simplicity.\n\nNow we have shown $\\mathcal{L}_{\\textrm{bat}}$ is differentiable with respect to each parameter in $\\boldsymbol\\Theta$, which completes the proof.\n\\end{IEEEproof}\n\n\nIt can be seen that the proposed DL-JHPF is abstracted into an integrated DNN, where the hybrid processing matrices, $\\mathbf{F}_{\\textrm{RF}}$, $\\mathbf{F}_{\\textrm{BB}}$, $\\mathbf{W}_{\\textrm{RF}}$, and $\\mathbf{W}_{\\textrm{BB}}$, are essentially the trainable weights therein. From the proof of Theorem 1, each weight of this integrated DNN can be optimized iteratively through BP algorithm by minimizing the BCE loss. Therefore, the optimal precoding and combining matrices on training set are obtained.\n\nFor the NNs in Fig.~\\ref{visual_model}, each dense layer is with rectified linear unit (ReLU) activation function and followed by a BN layer to avoid gradient diffusion and overfitting. The number of dense layers and the number of neurons in each dense layer need to be adjusted according to the input and output dimensions. Since the outputs of the NNs will be used for hybrid processing at the transmitter and the reciever, the activation functions of the output layers should be carefully designed and are elaborated as follows.\n\n\\emph{PP-NN and CP-NN:} The two NNs generate the phases for $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, respectively. Since (\\ref{eqn_fRF}) and (\\ref{eqn_wRF}) are periodic functions, ReLU activation function is used in the output layer to provide the unbiased output for all possible phases. We may also use Sigmoid or hyperbolic tangent as the activation function, after which the outputs are multiplied by $2\\pi$ or $\\pi$ to obtain the final phases with the range of $[0,2\\pi]$ or $[-\\pi,\\pi]$. According to the simulation trails, ReLU and hyperbolic tangent achieve almost the same performance while Sigmoid performs worse. Therefore, ReLU is preferable since it is simple and free of the operation of exponential functions.\n\n\\emph{Re\/ImDP-NN and Re\/ImDC-NN:} The four NNs generate the real and imaginary parts for $\\mathbf{F}_{\\textrm{BB}}$ and $\\mathbf{W}_{\\textrm{BB}}$, respectively. Since $\\mathbf{F}_{\\textrm{BB}}$ can be normalized by (\\ref{eqn_DP_norm}) while $\\mathbf{W}_{\\textrm{BB}}$ has no constraint, the output layers do not apply any activation function to impose constraints and directly output the values that are input into the neurons.\n\n\\emph{NN demodulator:} This NN approximates the original bits, $\\mathbf{X}_{\\textrm{b}}$, based on $\\mathbf{r}$. The approximation for each element in $\\mathbf{X}_{\\textrm{b}}$ is a binary classification and thus the Sigmoid activation function is used for the output layer of the NN demodulator.\n\n\\vspace{-0.2cm}\n\\subsection{Deployment and Testing}\n\nIn this subsection, we elaborate the deployment and testing of the trained DL-JHPF for practical implementation, where $\\mathbf{H}$ is assumed to be available at both the transmitter and the receiver.\\footnote{Although only the estimated channel is available in practical implementation, it has been shown by the simulation results that the relatively accurate channel estimate hardly causes performance loss and is almost equivalent to $\\mathbf{H}$.}\n\nThe practical deployment of DL-JHPF includes the following three parts:\n\n\\emph{Deployment of hybrid processing designer:} PP-NN and CP-NN will be deployed together at \\emph{both the transmitter and the receiver} to output the analog processing matrices, $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$, based on which the equivalent channel, $\\mathbf{H}_{\\textrm{eq}}$, can be generated via (\\ref{eqn_Heq}). ReDP-NN and ImDP-NN are equipped at the \\emph{transmitter} to generate the digital precoder, $\\mathbf{F}_{\\textrm{BB}}$, while ReDC-NN and ImDC-NN are equipped at the \\emph{receiver} to generate the digital combiner, $\\mathbf{W}_{\\textrm{BB}}$, both based on $\\mathbf{H}_{\\textrm{eq}}$.\n\n\\emph{Deployment of signal flow simulator:} It is only used for the training stage and will be replaced by the actual transceiver and wireless fading channel in the deployment and testing stage.\n\n\\emph{Deployment of NN demodulator:} It will be deployed at the \\emph{receiver} to output the recovered bits, $\\hat{\\mathbf{X}}_{\\textrm{b}}$, based on the detected signal, $\\mathbf{r}$, after compensating the impact of the fading channel.\n\nWhen testing the trained DL-JHPF in real world, the channel may change rapidly due to the relative motion of the transceiver and scatterers, in which case DL-JHPF will be faced new propagation scenarios with different channel statistics from the training stage. This channel scenario discrepancy poses a high requirement on the robustness of DL-JHPF. Fortunately, the offline trained framework in Section III.C is quite robust to the new channel scenarios that are not observed before as shown from our simulation results (Figs.~\\ref{HBD_BS_proDL_UMaipCSI} and \\ref{HBD_BS_proDL_ofdm_UMaipCSI}). The further online fine-tuning may only provide marginal performance improvement but requires a relatively large overhead and needs to be performed frequently in the rapidly changed channel scenario. In addition, only the NNs at the receiver can be fine-tuned and thus the performance after fine-tuning will still have an intrinsic loss compared to the end-to-end training in Section III.C. To sum up, the proposed framework can cope with the mismatch of the channel scenario without relying on the fine-tuning in most cases.\n\n\\subsection{Complexity Analysis}\n\nIn this subsection, we analyze the computational complexity of the proposed DL-JHPF in testing stage by using the metric of required number of floating point operations (FLOPs). According to Fig.~\\ref{visual_model}, the total required FLOPs of all neural layers in DL-JHPF is given by\n\\begin{eqnarray}\n\\label{eqn_Complexity_NN}\n\\mathcal{C}_{\\textrm{NN}}\\sim\\mathcal{O}\\left(\\sum_{\\Delta\\in\\mathcal{N}}\\sum_{i=1}^{L^{\\Delta}-1}N^{\\Delta}_{i}N^{\\Delta}_{i+1}\\right),\n\\end{eqnarray}\nwhere $\\mathcal{N}$ denotes the set including all NNs in DL-JHPF, $L^{\\Delta}$ and $N^{\\Delta}_{i}$ represent the number of neural layers and the number of neurons of the $i$th neural layer of the NN $\\Delta$.\n\nIn addition, the complexity of matrix multiplications in the framework is given by\n\\begin{eqnarray}\n\\label{eqn_Complexity_matrix}\n\\mathcal{C}_{\\textrm{Mat}}\\sim\\mathcal{O}\\bigl(N_{\\textrm{R}}^{\\textrm{RF}}N_{\\textrm{T}}N_{\\textrm{R}}).\n\\end{eqnarray}\n\nThen, the total complexity of the proposed DL-JHPF can be expressed as\n\\begin{eqnarray}\n\\label{eqn_Complexity_DL_JHPF}\n\\mathcal{C}_{\\textrm{DL-JHPF}}\\sim\\mathcal{C}_{\\textrm{NN}}+\\mathcal{C}_{\\textrm{Mat}}.\n\\end{eqnarray}\nIt is noted that the NNs can be run efficiently via parallel computing on the graphic processing unit (GPU) and the simple matrix multiplications only cause negligible computational load for the central processing unit (CPU) compared with the existing schemes. Therefore, the proposed DL-JHPF is with low complexity and consumes the very limited runtime.\n\n\\subsection{Extension to OFDM Systems}\n\nIn this subsection, we extend the proposed DL-JHPF to the wideband OFDM systems. Two key issues need to be considered for the extension:\n\\begin{itemize}[\\IEEEsetlabelwidth{Z}]\n\\item[1)] In the OFDM systems, the digital precoder and combiner can be designed independently for different subcarriers while the analog precoder and combiner must be shared by all subcarriers. It is critical to design the unified analog precoder and combiner performing well for all subcarriers.\n\n\\item[2)] It is important to maintain the relatively small size, i.e., the number of hidden layers and the number of neurons in each layer in the NNs, and short training time for DL-JHPF when the number of subcarriers is large.\n\\end{itemize}\n\nIn the following, we study how to address the two issues when extending DL-JHPF to the OFDM systems.\n\nAccording to \\cite{P. Dong_b}, the $N_{\\textrm{R}}\\times N_{\\textrm{T}}$ channel matrix between the receiver and the transmitter of the $k$th subcarrier can be expressed as\n\\begin{eqnarray}\n\\label{eqn_Hk}\n\\mathbf{H}[k]&&=\\beta\\sum_{n=1}^{N_{\\textrm{cl}}}\\sum_{m=1}^{N_{\\textrm{ray}}} \\alpha_{n,m}e^{-j2\\pi\\tau_n f_s\\frac{k}{K}}\\mathbf{a}_{\\textrm{R}}(\\varphi_{n,m})\\mathbf{a}_{\\textrm{T}}^{H}(\\phi_{n,m}),\n\\end{eqnarray}\nwhere $\\beta=\\sqrt{\\frac{N_{\\textrm{T}}N_{\\textrm{R}}}{N_{\\textrm{cl}}N_{\\textrm{ray}}}}$, $\\tau_n$, $f_s$, and $K$ denote the delay of the $n$th cluster, the sampling rate, and the number of OFDM subcarriers, respectively.\nThe signal transmission model in (\\ref{eqn_r}) becomes subcarrier dependent and the detected signal of the $k$th subcarrier is given by\\footnote{Although $\\mathbf{x}$ and $\\mathbf{n}$ are also different for different subcarriers, they are independent of the channel and thus the index $k$ in them is omitted.}\n\\setlength{\\arraycolsep}{0.05em}\n\\begin{eqnarray}\n\\label{eqn_rk}\n\\mathbf{r}[k]&&=\\sqrt{P}\\mathbf{W}_{\\textrm{BB}}^H[k]\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{H}[k]\\mathbf{F}_{\\textrm{RF}}\\mathbf{F}_{\\textrm{BB}}[k]\\mathbf{x}\\nonumber\\\\\n&&\\quad+\\mathbf{W}_{\\textrm{BB}}^H[k]\\mathbf{W}_{\\textrm{RF}}^H\\mathbf{n}.\n\\end{eqnarray}\n\nIn the following, we propose a simple method to design the structure of training data so that the DL-JHPF in Section III.C can be flexibly extended to OFDM systems without changing the framework architecture. That is, both the framework size and training time will not be increased. The process of training and testing is detailed as follows.\n\n\\emph{Training:} Compared to the training sample with the input tuple $\\langle\\mathbf{H}, \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$ in Section III.C, we modify the input tuple as $\\langle\\bar{\\mathbf{H}}, \\mathbf{H}[i], \\mathbf{X}_{\\textrm{b}}, \\mathbf{n}\\rangle$, where $\\bar{\\mathbf{H}}$ is the channel matrix of a given subcarrier, e.g., the $q$th subcarrier, same for all training samples while $\\mathbf{H}[i]$ is the channel matrix of an uncertain subcarrier with $i$ randomly generated from the set $\\{1,2,\\ldots,K\\}$ for each training sample. As shown in Fig.~\\ref{DL_framework_ofdm}, when inputting each training sample into the framework, $\\bar{\\mathbf{H}}$ will be used to generate $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ via PP-NN and CP-NN. Then $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with $\\mathbf{H}[i]$ are used to generate the equivalent channel of the $i$th subcarrier, $\\mathbf{H}_{\\textrm{eq}}[i]$, based on which, $\\mathbf{F}_{\\textrm{BB}}[i]$ and $\\mathbf{W}_{\\textrm{BB}}[i]$ can be obtained through Re\/ImDP-NN and Re\/ImDC-NN. On the other hand, $\\mathbf{H}[i]$ is also input into the signal flow simulator to act as the fading channel since this training sample is used to simulate the transmission of the $i$th subcarrier. Then the end-to-end training can be performed by minimizing the BCE loss between $\\mathbf{X}_{\\textrm{b}}$ and $\\hat{\\mathbf{X}}_{\\textrm{b}}$. Through training, we can obtain the unified analog precoder and combiner that match the channel of each subcarrier well without complicating the architecture of DL-JHPF.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{Fig4}\n\\caption{Extension of the proposed DL-JHPF to OFDM systems.}\\label{DL_framework_ofdm}\n\\end{figure}\n\n\\emph{Testing:} With $\\mathbf{H}[k], k=1,2,\\ldots,K$, available at the transceiver, choose the channel matrix of the $q$th subcarrier as $\\bar{\\mathbf{H}}$. Input $\\bar{\\mathbf{H}}$ into PP-NN and CP-NN to generate the unified $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ for all subcarriers. The unified $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ along with the channel of each subcarrier, $\\mathbf{H}[k], k=1,2,\\ldots,K$, are used to generate the corresponding equivalent channel, which will be input into Re\/ImDP-NN and Re\/ImDC-NN to generate $\\mathbf{F}_{\\textrm{BB}}[k]$ and $\\mathbf{W}_{\\textrm{BB}}[k]$ for channel equalization in each subcarrier. The NN demodulator will be used to recover the original bits for each subcarrier based on the detected signal, $\\mathbf{r}[k]$.\n\n\\section{Simulation Results}\n\nIn this section, the effectiveness of the proposed DL-JHPF is verified in several cases. Six hybrid processing schemes and the fully-digital transceiver architecture are used as the baseline schemes for comparison: 1) HBD scheme in \\cite{W. Ni}; 2) Beam sweeping (BeS) scheme in \\cite{O. E. Ayach}; 3) Discrete Fourier transform (DFT) codebook based joint digital beamforming (DCJDB) scheme, where the analog precoder and combiner are searched from the DFT codebook by the method in \\cite{A. M. Elbir} while the digital precoder and combiner are jointly optimized according to \\cite{D. P. Palomar}; 4) Joint digital beamforming with alternating minimization (JDB-AltMin), where the optimal precoding and combining matrices are first designed according to \\cite{D. P. Palomar}, based on which the hybrid precoding and combining matrices are constructed according to the PE-AltMin algorithm in \\cite{X. Yu}; 5) Hybrid beamforming via deep learning (HBDL) scheme in \\cite{A. M. Elbir}; 6) Deep learning for direct hybrid precoding (DLDHP) scheme in \\cite{X. Li}; 7) Fully-digital transceiver architecture.\n\n\\subsection{Simulation Settings}\n\n\\begin{table}[!t]\n\\centering\n\\caption{Architectures of DNNs in Proposed DL-HPF}\n\\label{table_1}\n\\begin{tabular}{p{1.4cm}<{\\centering}|c|c|c}\n\\hline\n~ & Layer type & \\makecell{Number of \\\\neurons} & \\makecell{Activation \\\\function}\\\\\n\\hline\n\\multirow{5}{*}{PP-NN} & Input & $2N_{\\textrm{T}}N_{\\textrm{R}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 512 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 256 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 128 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{T}}N_{\\textrm{T}}^{\\textrm{RF}}$ & ReLU\\\\\n\\hline\n\\multirow{6}{*}{CP-NN} & Input & $2N_{\\textrm{T}}N_{\\textrm{R}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 512 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 256 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 128 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 64 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{R}}N_{\\textrm{R}}^{\\textrm{RF}}$ & ReLU\\\\\n\\hline\n\\multirow{5}{*}{\\makecell{Re\/ImDP-\\\\NN}} & Input & $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 40 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{s}}$ & - \\\\\n\\hline\n\\multirow{5}{*}{\\makecell{Re\/ImDC-\\\\NN}} & Input & $2N_{\\textrm{T}}^{\\textrm{RF}}N_{\\textrm{R}}^{\\textrm{RF}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 40 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{R}}^{\\textrm{RF}}N_{\\textrm{s}}$ & - \\\\\n\\hline\n\\multirow{5}{*}{\\makecell{NN\\\\demodulator}} & Input & $2N_{\\textrm{s}}$ & - \\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 50 & ReLU\\\\\n\\cline{2-4}\n~ & Dense & 20 & ReLU\\\\\n\\cline{2-4}\n~ & Output & $N_{\\textrm{s}}\\log_{2} M$ & Sigmoid \\\\\n\\hline\n\\end{tabular}\n\\vspace{-0.3cm}\n\\end{table}\n\n\\emph{1) System Settings:} We set $N_{\\textrm{T}}=32$ and $N_{\\textrm{T}}^{\\textrm{RF}}=3$ for the transmitter and $N_{\\textrm{R}}=16$ and $N_{\\textrm{R}}^{\\textrm{RF}}=3$ for the receiver. The number of data streams is set as $N_{\\textrm{s}}=3$. The channel data are generated according to the 3GPP TR 38.901 Release 15 channel model \\cite{3GPP}. Specifically, we use the clustered delay line models with $N_{\\textrm{cl}}=3$ clusters and $N_{\\textrm{ray}}=20$ rays in each cluster. The carrier frequency is $f_c=28$ GHz. For OFDM systems, the sampling rate is $f_s=100$ MHz and the number of subcarriers is $K=64$. Two channel scenarios, urban micro (UMi) street non-line of sight (NLOS) scenario and urban macro (UMa) NLOS scenario, are considered.\\footnote{According to the parameters for UMi NLOS scenario and UMa NLOS scenario defined by \\cite{3GPP}, we use the system object, nr5gCDLChannel, embedded in 5G Library for LTE System Toolbox in MATLAB to generate the corresponding channel data.} Quadrature phase shift keying (QPSK) is used as the modulation method.\n\n\\emph{2) Proposed DL-JHPF Settings:} The training set, validation set, and testing set contain $261$,$000$, $29$,$000$, and $10$,$000$ samples, respectively. The training set and validation set are generated in UMi NLOS scenario while the testing set is generated in both UMi NLOS and UMa NLOS scenarios. Adam is used as the optimizer. The number of epochs in the training stage is set as $800$ while the corresponding learning rates are $10^{-3}$ for the first $500$ epochs and $10^{-4}$ for the rest $300$ epochs, respectively. The batch size is $256$. The architecture of each NN in DL-JHPF is listed in Table~\\ref{table_1}, where the BN layer is added after each dense layer and thus is not listed in the table for simplicity.\n\n\n\\subsection{Performance Evaluation}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig5}\n\\caption{BER performance of the proposed DL-JHPF and the existing hybrid processing schemes.}\\label{HBD_BS_proDL}\n\\end{figure}\n\nIn Figs.~\\ref{HBD_BS_proDL}$-$\\ref{HBD_BS_proDL_UMaipCSI}, the proposed DL-JHPF is first evaluated in narrowband systems while the performance in wideband OFDM systems is presented in Figs.~\\ref{HBD_BS_proDL_ofdm} and \\ref{HBD_BS_proDL_ofdm_UMaipCSI}.\n\nFig.~\\ref{HBD_BS_proDL} shows the BER performance of HBD, BeS, DCJDB, JDB-AltMin, HBDL, DLDHP, the proposed DL-JHPF, and the fully-digital architecture versus signal-to-noise ratio (SNR) in UMi NLOS scenario with perfect CSI. From the figure, DL-JHPF has a larger slope for the BER curve and outperforms the other six hybrid processing schemes after $\\textrm{SNR}=0$ dB although it performs not very well in the low SNR regime. When $\\textrm{BER}=10^{-2}$, the proposed DL-JHPF achieves about $0.2$ dB, $1$ dB, $1.2$ dB, $2$ dB, $6$ dB, and $8$ dB gains compared to JDB-AltMin, DLDHP, DCJDB, BeS, HBDL, and HBD, respectively. The advantage of DL-JHPF becomes more obvious as SNR increases and the BER is smaller than $10^{-4}$ when $\\textrm{SNR}=10$ dB while the performance of other four schemes is larger than $10^{-3}$. With the significantly increased number of RF chains, the fully-digital beamforming obtains substantial diversity gains, which directly leads to the better BER performance than all the hybrid processing schemes. The performance gap between the proposed DL-JHPF and the fully-digital beamforming is about 4dB.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig6}\n\\caption{Robustness of the proposed DL-JHPF with mismatched CSI.}\\label{HBD_BS_proDL_ipCSI}\n\\end{figure}\n\nPerfect CSI is used in framework training while only estimated CSI is available in the practical transmitter and receiver, which leads to the CSI mismatch. In Fig.~\\ref{HBD_BS_proDL_ipCSI}, we investigate the robustness of the proposed DL-JHPF with mismatched CSI, where the BER curve tested with perfect CSI in Fig.~\\ref{HBD_BS_proDL} is also plotted as the lower bound. We use the approach in \\cite{P. Dong_b} to estimate channels at $\\textrm{SNR}=10$ dB and $20$ dB, respectively, for hybrid processing design. From Fig.~\\ref{HBD_BS_proDL_ipCSI}, when tested with the CSI estimated at $20$ dB, DL-JHPF achieves almost the same BER performance as the perfect CSI case and outperforms the other six hybrid processing schemes after $\\textrm{SNR}=0$ dB, indicating that DL-JHPF is hardly impacted by the mismatched CSI estimated at $20$ dB. When tested with the CSI estimated at $10$ dB, performance loss occurs at an acceptable level for DL-JHPF. The loss is less than $1$ dB when $\\textrm{BER}=10^{-2}$ and DL-JHPF still has the clear performance superiority after $\\textrm{SNR}=2.5$ dB even compared to other hybrid processing schemes with the CSI estimated at $20$ dB.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig7}\n\\caption{Robustness of the proposed DL-JHPF with mismatched channel scenario and CSI.}\\label{HBD_BS_proDL_UMaipCSI}\n\\end{figure}\n\nAs mentioned in Section III.D, it is very likely to face with different channel scenarios in the practical testing for DL-JHPF. In Fig.~\\ref{HBD_BS_proDL_UMaipCSI}, we further consider this channel scenario mismatch and test the robustness of DL-JHPF to the aggregate impact caused by channel scenario and CSI mismatch. For DL-JHPF, the BER performance is tested in UMi NLOS scenario with perfect CSI, in UMa NLOS scenario with perfect CSI (mismatched channel scenario), and in UMa NLOS scenario with the CSI estimated at $20$ dB (mismatched channel scenario and CSI), respectively. The performance curves of the baseline schemes evaluated in UMa NLOS scenario with the CSI estimated at $20$ dB are also plotted for comparison. From Fig.~\\ref{HBD_BS_proDL_UMaipCSI}, the channel scenario mismatch causes only less than $0.5$ dB performance loss for DL-JHPF. The total loss caused by the aggregate impact of channel scenario and CSI mismatch is only less than $1$ dB. The proposed DL-JHPF has learned the inherent structure of the mmWave channels and thus is able to maintain its advantage even with mismatched channel scenarios and CSI.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig8}\n\\caption{BER performance of the proposed DL-JHPF and the existing hybrid processing schemes in OFDM systems.}\\label{HBD_BS_proDL_ofdm}\n\\end{figure}\n\nFig.~\\ref{HBD_BS_proDL_ofdm} shows the BER performance of HBD, BeS, DCJDB, JDB-AltMin, HBDL, DLDHP, the proposed DL-JHPF, and the fully-digital architecture in OFDM systems with UMi NLOS scenario and perfect CSI, which is similar to that in Fig.~\\ref{HBD_BS_proDL}. In addition, we plot the BER performance of an ideal case with matched analog processing (AP) for DL-JHPF, where different analog processing matrices are designed for different subcarriers to match the corresponding channels. This is impossible to be implemented in practical systems and we just use it to quantify the performance loss caused by using the unified analog processing matrices for all subcarriers. From Fig.~\\ref{HBD_BS_proDL_ofdm}, only about $1$ dB loss is incurred, which proves the effectiveness of DL-JHPF in OFDM systems by simply modifying the structure of training data without changing the framework architecture and increasing the training time.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[trim=0 0 0 0, width=3.6in]{Fig9}\n\\caption{Robustness of the proposed DL-JHPF with mismatched channel scenario and CSI in OFDM systems.}\\label{HBD_BS_proDL_ofdm_UMaipCSI}\n\\end{figure}\n\nIn Fig.~\\ref{HBD_BS_proDL_ofdm_UMaipCSI}, we further test the robustness of DL-JHPF in OFDM systems with the mismatched channel scenario and CSI. The aggregate impact of channel scenario and CSI mismatch is still limited, and DL-JHPF tested in UMa NLOS scenario with the CSI estimated at $20$ dB (mismatched channel scenario and CSI) even outperforms that tested in UMi NLOS scenario with perfect CSI after $\\textrm{SNR}=8$ dB, which verifies the effectiveness and robustness of the proposed DL-JHPF in OFDM systems. In addition, the performance gap between the proposed DL-JHPF and the fully-digital beamforming maintains at about 4dB.\n\n\\subsection{Computational Complexity Comparison}\n\n\\begin{table}\n  \\centering\n  \\caption{Runtime of Hybrid Processing Schemes}\n  \\label{runtime}\n  \\begin{tabular}{c|c}\n  \\hline\n  ~ & Runtime (in ms)\\\\\n  \\hline\n  HBD & $6.98$\\\\\n  \\hline\n  BeS & $10.61$\\\\\n  \\hline\n  DCJDB & $338.73$\\\\\n  \\hline\n  JDB-AltMin & $1.56$\\\\\n  \\hline\n  HBDL & $0.46$\\\\\n  \\hline\n  DLDHP & $0.16$\\\\\n  \\hline\n  Proposed DL-JHPF & $0.06$\\\\\n  \\hline\n  \\end{tabular}\n\\end{table}\n\nFor mmWave mobile communications, the length of coherence time becomes smaller compared to that in sub-6 GHz and thus the runtime of a hybrid processing scheme is vital. Based on the simulation settings mentioned above, we compare the runtime of the proposed DL-JHPF in the testing stage with the baseline schemes in Table~\\ref{runtime}. The HBD, BeS, DCJDB, and JDB-AltMin schemes are run on the Intel(R) Core(TM) i7-3770 CPU while the proposed DL-JHPF are run on the NVIDIA GeForce GTX 2080 Ti GPU. For HBDL and DLDHP, the predictions of $\\mathbf{F}_{\\textrm{RF}}$ and $\\mathbf{W}_{\\textrm{RF}}$ are implemented via DNN on the GPU while the following design of $\\mathbf{F}_{\\textrm{BB}}$ and $\\mathbf{W}_{\\textrm{BB}}$ are executed on the CPU. By moving the time-consuming design of analog processing to the GPU that enables the efficient parallel computing, the DL based schemes reduce the runtime significantly compared to the conventional schemes. Through carefully design, the proposed DL-JHPF is fully GPU-driven when generating hybrid processing matrices and thus consumes the minimum time among the three DL based schemes. Therefore, the proposed DL-JHPF is more suitable for mmWave communications, especially for the high-mobility scenario.\n\n\\section{Conclusion}\n\nIn this paper, DL is applied for joint hybrid processing design at the transceiver in mmWave massive MIMO systems. A novel DL-JHPF is developed to learn the optimal analog and digital processing matrices by minimizing the end-to-end BCE loss between the original and recovered bits. The elaborate architecture of the proposed DL-JHPF guarantees the BP-enabled training of each NN therein. By simply modifying the structure of training data, DL-JHPF can be flexibly extended to OFDM systems without changing the framework architecture and increasing the training time. Simulation results show the superiority and robustness of DL-JHPF in various non-ideal conditions with the significantly reduced runtime.\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{ACKNOWLEDGMENTS}\nThe authors thank Matthew G. Knepley (Rice University) \nfor his invaluable advice. The authors also thank\nthe Los Alamos National Laboratory (LANL)\nInstitutional Computing program. \nJC and KBN acknowledge the financial support from the \nHouston Endowment Fund and from the Department of Energy \nthrough Subsurface Biogeochemical Research Program. \nSK thanks the LANL LDRD program and the LANL Environmental\nPrograms Directorate for their support.\nThe opinions expressed in this paper are those of \nthe authors and do not necessarily reflect that \nof the sponsors.\n\n\\bibliographystyle{plain}\n\n\\section{GOVERNING EQUATIONS AND ASSOCIATED NON-NEGATIVE NUMERICAL METHODOLOGIES}\n\\label{Sec:Governing_Equations}\nLet $\\Omega \\subset \\mathbb{R}^{nd}$ be a bounded open \ndomain, where ``$nd$'' is the number of spatial dimensions. \nThe boundary of the domain is denoted by $\\partial \\Omega \n= \\overline{\\Omega} - \\Omega$, which is assumed to be \npiecewise smooth. A spatial point is denoted by $\\mathbf{x} \n\\in \\overline{\\Omega}$. The gradient and divergence operators \nwith respect to $\\mathbf{x}$ are, respectively, denoted as \n$\\mathrm{grad}[\\cdot]$ and $\\mathrm{div}[\\cdot]$. As usual, \nthe boundary is divided into two parts: $\\Gamma^{\\mathrm{D}}$ \nand $\\Gamma^{\\mathrm{N}}$. $\\Gamma^{\\mathrm{D}}$ is that part \nof the boundary on which Dirichlet boundary conditions \nare  prescribed, and $\\Gamma^{\\mathrm{N}}$ is the part of \nthe boundary on which Neumann boundary conditions are \nprescribed. For mathematical well-posedness, we assume \n$\\Gamma^{\\mathrm{D}} \\cup \\Gamma^{\\mathrm{N}} = \\partial \n\\Omega$ and $\\Gamma^{\\mathrm{D}} \\cap \\Gamma^{\\mathrm{N}} \n= \\emptyset$. \nThe unit outward normal to boundary is denoted as \n$\\widehat{\\mathbf{n}}(\\mathbf{x})$. The diffusivity \ntensor is denoted by $\\mathbf{D}(\\mathbf{x})$, which \nis assumed to symmetric, bounded above and uniformly \nelliptic. That is,\n\\begin{align}\n  \\mathbf{D}(\\mathbf{x}) = \\mathbf{D}^{\\mathrm{T}}\n  (\\mathbf{x}) \\quad \\forall \\mathbf{x} \\in \\Omega\n\\end{align}\nand there exists two constants $\\mathrm{0 < \n  \\xi_1 \\leq \\xi_2 < +\\infty}$ such that \n\\begin{align}\n  \\label{Eqn:Helmholtz_positive_definiteness_D}\n  \\xi_1 \\mathbf{y}^{\\mathrm{T}} \\mathbf{y}\\leq \\mathbf{y}^{\\mathrm{T}} \n  \\mathbf{D}(\\mathbf{x}) \\mathbf{y} \\leq \\xi_2 \\mathbf{y}^{\\mathrm{T}} \n  \\mathbf{y} \\quad \\forall \\mathbf{x} \\in \\Omega \\; \\mathrm{and} \\; \n  \\forall \\mathbf{y} \\in \\mathbb{R}^{nd}\n\\end{align} \n\n\\subsection{Governing equations for steady-state response}\nWe shall denote the steady-state concentration field \nby $c(\\mathbf{x})$. The governing equations can be \nwritten as follows: \n\\begin{subequations}\n  \\label{Eqn:LargeScale_BVP}\n  \\begin{alignat}{2}\n  \\label{Eqn:LargeScale_GE}\n  -&\\mathrm{div}[\\mathbf{D}(\\mathbf{x}) \n    \\mathrm{grad}[c]] = f(\\mathbf{x}) & \\qquad & \n  \\mathrm{in} \\; \\Omega  \\\\\n  %\n  \\label{Eqn:LargeScale_Dirichlet}\n  &c(\\mathbf{x}) = c^{\\mathrm{p}} (\\mathbf{x}) & \\qquad &\n  \\mathrm{on} \\; \\Gamma^{\\mathrm{D}} \\\\\n  %\n  \\label{Eqn:LargeScale_Neumann}\n  -&\\mathbf{\\widehat{n}}(\\mathbf{x}) \\cdot \\mathbf{D}\n  (\\mathbf{x}) \\mathrm{grad}[c] =\n  q^{\\mathrm{p}}(\\mathbf{x}) & \\quad & \\mathrm{on} \\;\n  \\Gamma^{\\mathrm{N}}\n  \\end{alignat}\n\\end{subequations}\nwhere $f(\\mathbf{x})$ is the volumetric \nsource\/sink, $c^{\\mathrm{p}}(\\mathbf{x})$ is the \nprescribed concentration, and $q^{\\mathrm{p}}\n(\\mathbf{x})$ is the prescribed flux. For \nuniqueness, we assume $\\Gamma^{\\mathrm{D}} \n\\neq \\emptyset$. \n\n\\subsubsection{Maximum principle and the non-negative constraint}\nThe above boundary value problem is a self-adjoint \nsecond-order elliptic partial differential equation \n(PDE). It is well-known that such PDEs possess an \nimportant mathematical property -- the classical \nmaximum principle \\cite{Evans_PDE}. The mathematical \nstatement of the classical maximum principle can be \nwritten as follows: \nIf $c(\\mathbf{x}) \\in C^{2}(\\Omega) \\cap C^{0}(\\overline{\\Omega})$, \n$\\partial \\Omega = \\Gamma^{\\mathrm{D}}$, and $f(\\mathbf{x}) \\leq 0$  \nin $\\Omega$ then \n\\begin{align}\n  \\max_{\\mathbf{x} \\in \\overline{\\Omega}} c(\\mathbf{x}) \n  = \\max_{\\mathbf{x} \\in \\partial \\Omega} c^{\\mathrm{p}}(\\mathbf{x})\n\\end{align}\nSimilarly, if $f(\\mathbf{x}) \\geq 0$ in $\\Omega$ then\n\\begin{align}\n  \\label{Eqn:LargeScale_MP}\n  \\min_{\\mathbf{x} \\in \\overline{\\Omega}} c(\\mathbf{x}) \n  = \\min_{\\mathbf{x} \\in \\partial \\Omega} c^{\\mathrm{p}}(\\mathbf{x})\n\\end{align}\nTo make our presentation on maximum principles \nsimple, we have assumed stronger regularity on \nthe solution (i.e., $c(\\mathbf{x}) \\in C^{2} \n\\cap C^{0}(\\overline{\\Omega})$), and assumed that \nDirichlet boundary conditions are prescribed on \nthe entire boundary. However, maximum principles \nrequiring milder regularity conditions on the \nsolution, even for the case when Neumann boundary \nconditions are prescribed on the boundary, can be \nfound in  literature (see \n\\cite{Mudunuru_Nakshatrala_arXiv_2015,\nMudunuru_Nakshatrala_ADR_arXiv_2015}). \n\nIf $f(\\mathbf{x}) \\geq 0$ in $\\Omega$ and \n$c^{\\mathrm{p}}(\\mathbf{x}) \\geq 0$ on the \nentire $\\partial \\Omega$ then the maximum \nprinciple implies that $c(\\mathbf{x}) \n\\geq 0$ in the entire domain, which is \nthe non-negativity of the concentration \nfield.  \n\n\\subsubsection{Single-field Galerkin weak formulation}\nThe following function spaces will be \nused in the rest of this paper: \n\\begin{align}\n  \\mathcal{U} &:= \\left\\{c(\\mathbf{x}) \\in H^{1}(\\Omega) \n  \\; \\big| \\; c(\\mathbf{x}) = c^{\\mathrm{p}}(\\mathbf{x}) \\; \n  \\mathrm{on} \\; \\Gamma^{\\mathrm{D}}\\right\\} \\\\\n  %\n  \\mathcal{W} &:= \\left\\{w(\\mathbf{x}) \\in H^{1}(\\Omega) \n  \\; \\big| \\; w(\\mathbf{x}) = 0 \\; \\mathrm{on} \\; \n  \\Gamma^{\\mathrm{D}}\\right\\} \n\\end{align}\nwhere $H^{1}(\\Omega)$ is a standard Sobolev space \n\\cite{adams2003sobolev}. The single-field Galerkin \nweak formulation corresponding to equations \n\\eqref{Eqn:LargeScale_GE}--\\eqref{Eqn:LargeScale_Neumann} \nreads: Find $c(\\mathbf{x}) \\in \\mathcal{U}$ \nsuch that we have \n\\begin{align}\n  \\label{Eqn:Single_field_formulation}\n  \\mathcal{B}(w;c) = L(w) \\quad \\forall \n  w(\\mathbf{x}) \\in \\mathcal{W}\n\\end{align}\nwhere the bilinear form and linear functional \nare, respectively, defined as \n\\begin{subequations}\n  \\label{Eqn:functionals_B_L}\n  \\begin{align}\n    \\mathcal{B}(w;c) &:= \\int_{\\Omega} \\mathrm{grad}[w(\\mathbf{x})] \n    \\cdot \\mathbf{D}(\\mathbf{x}) \\mathrm{grad}[c(\\mathbf{x})] \n    \\; \\mathrm{d} \\Omega \n    \\\\\n    L(w) &:= \\int_{\\Omega} w(\\mathbf{x}) f(\\mathbf{x}) \n    \\; \\mathrm{d} \\Omega \n    + \\int_{\\Gamma^{\\mathrm{N}}} w(\\mathbf{x}) \n    q^{\\mathrm{p}}(\\mathbf{x}) \\; \\mathrm{d} \\Gamma \n  \\end{align}\n\\end{subequations}\nSince $\\mathbf{D}(\\mathbf{x})$ is symmetric, \nby Vainberg's theorem \\cite{Hjelmstad}, the \nsingle-field Galerkin weak formulation given \nby equation \\eqref{Eqn:Single_field_formulation} \nis equivalent to the following variational \nproblem: \n\\begin{align}\n  \\label{Eqn:LargeScale_var_stat}\n  \\mathop{\\mathrm{minimize}}_{c(\\mathbf{x}) \\in \\mathcal{U}} \n  \\quad \\frac{1}{2} \\mathcal{B}(c;c) - L(c)\n\\end{align}   \n\n\\subsubsection{A methodology to enforce the maximum principle \n  for steady-state problems}\nOur methodology is based on the finite element method. \nWe decompose the domain into ``$Nele$'' non-overlapping \nopen element sub-domains such that \n\\begin{align}\n  \\overline{\\Omega} = \\bigcup_{e = 1}^{Nele} \\overline{\\Omega}^{e}\n\\end{align}\n(Recall that a superposed bar denotes the set closure.) The \nboundary of $\\Omega^e$ is denoted by $\\partial \\Omega^{e} \n:= \\overline{\\Omega}^{e} - \\Omega^{e}$. We shall define the \nfollowing finite dimensional vector spaces of $\\mathcal{U}$ \nand $\\mathcal{W}$:\n\\begin{subequations}\n  \\begin{align}\n    \\mathcal{U}^{h} &:= \\left\\{c^{h}(\\mathbf{x}) \\in \\mathcal{U} \n    \\; \\big| \\; c^{h}(\\mathbf{x}) \\in C^{0}(\\overline{\\Omega}), \n    c^{h}(\\mathbf{x}) \\big|_{\\Omega^e} \\in \\mathbb{P}^{k}(\\Omega^{e}), \n      e = 1, \\cdots, Nele \\right\\} \\\\\n   \n    \\mathcal{W}^{h} &:= \\left\\{w^{h}(\\mathbf{x}) \\in \\mathcal{W} \n    \\; \\big| \\; w^{h}(\\mathbf{x}) \\in C^{0}(\\overline{\\Omega}), \n    w^{h}(\\mathbf{x}) \\big|_{\\Omega^e} \\in \\mathbb{P}^{k}(\\Omega^{e}), \n    e = 1, \\cdots, Nele \\right\\}       \n  \\end{align}\n\\end{subequations}\nwhere $k$ is a non-negative integer, and \n$\\mathbb{P}^{k}(\\Omega^{e})$ denotes the linear \nvector space spanned by polynomials up to $k$-th \norder defined on the sub-domain $\\Omega^{e}$. The \nfinite element formulation for equation \n\\eqref{Eqn:Single_field_formulation} can be \nwritten as: Find $c^{h}(\\mathbf{x}) \\in \\mathcal{P}^{h}$ \nsuch that we have \n\\begin{align}\n  \\label{Eqn:FE_formulation}\n  \\mathcal{B}(q^{h};c^{h}) = L(q^{h}) \\quad \n  \\forall q^{h}(\\mathbf{x}) \\in \\mathcal{Q}^{h}\n\\end{align}\nIt has been documented in the literature that the \nabove finite element formulation violates the \nmaximum principle and the non-negative constraint \\cite{liska2008enforcing,\nNakshatrala_JCP_2009_v228_p6726,Nagarajan_IJNMF_2011_v67_p820}.\n \nWe now outline an optimization-based methodology that \nsatisfies the maximum principle and the non-negative \nconstraint on general computational grids. \nTo this end, we shall use the symbols $\\preceq$ \nand $\\succeq$ to denote component-wise inequalities \nfor vectors. That is, for given any two vectors \n$\\boldsymbol{a}$ and $\\boldsymbol{b}$ \n\\begin{align}\n  \\boldsymbol{a} \\preceq \\boldsymbol{b} \\quad \n  \\mbox{means that } \\quad a_i \\leq b_i \\; \\forall i\n\\end{align}\nThe symbol $\\succeq$ can be similarly defined as well.\nLet $<\\cdot;\\cdot>$ denote the standard inner-product \nin Euclidean space. After finite element discretization, \nthe discrete equations corresponding to equation \n\\eqref{Eqn:FE_formulation} take the form\n\\begin{align}\n  \\label{Eqn:Helmholtz_discrete}\n  \\boldsymbol{K} \\boldsymbol{c} = \\boldsymbol{f}\n\\end{align}\nwhere $\\boldsymbol{K}$ is a symmetric positive definite matrix, $\\boldsymbol{c}$ is the \nvector containing nodal concentrations, and $\\boldsymbol{f}$ is the force vector. \nEquation~\\eqref{Eqn:Helmholtz_discrete} is equivalent to the following minimization problem \n\\begin{align}\n  \\label{Eqn:Helmholtz_minimization}\n  \\mathop{\\mbox{minimize}}_{\\boldsymbol{c} \\in \\mathbb{R}^{ndofs}} \\quad  \\frac{1}{2} \n  \\langle\\boldsymbol{c}; \\boldsymbol{K}  \\boldsymbol{c}\\rangle \n  - \\langle\\boldsymbol{c}; \\boldsymbol{f}\\rangle\n\\end{align}\nwhere ``$ndofs$'' denotes the number of degrees of freedom \nfor concentration. Equation~\\eqref{Eqn:Helmholtz_discrete} \ncan lead to unphysical negative solutions.\n\nFollowing \\cite{Nagarajan_IJNMF_2011_v67_p820,\nliska2008enforcing}, a methodology corresponding to equation \n\\eqref{Eqn:Helmholtz_minimization} that satisfies the \nnon-negative constraint can be written as follows: \n\\begin{subequations}\n  \\label{Eqn:non-negative}\n  \\begin{align}\n    &\\mathop{\\mbox{minimize}}_{\\boldsymbol{c} \\in \\mathbb{R}^{ndofs}} \\quad  \n    \\frac{1}{2} <\\boldsymbol{c}; \\boldsymbol{K}  \\boldsymbol{c}>  - \n    <\\boldsymbol{c}; \\boldsymbol{f}> \\\\\n   \n    \\label{Eqn:non-negative_constraint}\n    &\\mbox{subject to} \\quad \\boldsymbol{0} \\preceq \\boldsymbol{c}  \n  \\end{align}\n\\end{subequations}\nwhere $\\boldsymbol{0}$ is a vector of size $ndofs$ \ncontaining zeros. Since $\\boldsymbol{K}$ is positive \ndefinite, equation \\eqref{Eqn:non-negative} has a \nunique global minimum \\cite{Boyd_convex_optimization}. \nSeveral robust numerical methods can be used to solve \nequation \\eqref{Eqn:non-negative}, which include active \nset strategy, interior point methods \n\\cite{Boyd_convex_optimization}. In this paper, to \nsolve the resulting optimization problems, we shall \nuse the parallel optimization toolkit TAO \n\\cite{tao-user-ref}, which has the active-set Newton trust \nregion (TRON) and quasi-Newton-based bounded limited memory\nvariable metric (BLMVM) algorithms.\n\n\\subsection{Governing equations for transient response}\nWe shall denote the time by $t \\in [0,\\mathcal{I}]$, where \n$\\mathcal{I}$ denotes the length of the time interval of \ninterest. We shall denote the time-dependent concentration \nby $c(\\mathbf{x},t)$. The initial boundary value problem \ncan be written as follows:\n\\begin{subequations}\n  \\begin{alignat}{2}\n    \\label{Eqn:LargeScale_GE_unsteady}\n    &\\frac{\\partial c}{\\partial t} \n    = \\mathrm{div}[\\mathbf{D}(\\mathbf{x}) \\mathrm{grad}[c]] \n    + f(\\mathbf{x},t) \n    && \\quad \\mathrm{in} \\; \\Omega \\times (0,\\mathcal{I}) \\\\\n    %\n    &c(\\mathbf{x},t) = c^{\\mathrm{p}}(\\mathbf{x},t)  \n    && \\quad \\mathrm{on} \\; \\Gamma^{\\mathrm{D}} \n    \\times (0,\\mathcal{I}) \\\\\n    %\n    -&\\widehat{\\mathbf{n}}(\\mathbf{x}) \\cdot \n    \\mathbf{D}(\\mathbf{x}) \\mathrm{grad}[c] \n    = q^{\\mathrm{p}}(\\mathbf{x},t) \n    && \\quad \\mathrm{on} \\;  \\Gamma^{\\mathrm{N}} \n    \\times (0,\\mathcal{I}) \\\\ \n    %\n    \\label{Eqn:LargeScale_IC}\n    &c(\\mathbf{x},0) = c_{0}(\\mathbf{x}) \n    && \\quad \\mathrm{in} \\; \\Omega \n    %\n  \\end{alignat}\n\\end{subequations}\nwhere $c_0(\\mathbf{x})$ is the prescribed initial \nconcentration, $f(\\mathbf{x},t)$ is the time-dependent \nvolumetric source\/sink, $c^{\\mathrm{p}}(\\mathbf{x},t)$ is \nthe time-dependent prescribed concentration on the \nboundary, and $q^{\\mathrm{p}}(\\mathbf{x},t)$ is the \nprescribed time-dependent flux on the boundary. \n\n\\subsubsection{Maximum principle and the non-negative constraint}\nThe maximum principle of a transient diffusion equation \nasserts that the maximum can occur only on the boundary \nof the domain or in the initial condition if $f(\\mathbf{x},\nt) \\leq 0$ and $\\Gamma^{\\mathrm{D}} = \\partial \\Omega$. \nMathematically, a solution to equations \n\\eqref{Eqn:LargeScale_GE_unsteady}--\\eqref{Eqn:LargeScale_GE_unsteady} \nwill satisfy:\n\\begin{align}\n  c(\\mathbf{x},t) \\leq \\max \\left[\\max_{\\mathbf{x} \\in \\Omega} \n    c_0(\\mathbf{x}), \\max_{\\mathbf{x} \\in \\partial \\Omega} \n    c^{\\mathrm{p}}(\\mathbf{x},t)\\right] \\quad \\forall t\n\\end{align}\nprovided $f(\\mathbf{x},t) \\leq 0$. Similarly, the \nminimum will occur either on the boundary or in the \ninitial condition if $f(\\mathbf{x},t) \\geq 0$. That \nis, if $f(\\mathbf{x},t) \\geq 0$ then a solution to \nequations \\eqref{Eqn:LargeScale_GE_unsteady}--\\eqref{Eqn:LargeScale_GE_unsteady} \nsatisfies:\n\\begin{align}\n  c(\\mathbf{x},t) \\geq \\min \\left[\\min_{\\mathbf{x} \\in \\Omega} \n    c_0(\\mathbf{x}), \\min_{\\mathbf{x} \\in \\partial \\Omega} \n    c^{\\mathrm{p}}(\\mathbf{x},t)\\right] \\quad \\forall t\n\\end{align}\n\nIf $f(\\mathbf{x},t) \\geq 0$ in $\\Omega$, $c^{\\mathrm{p}}\n(\\mathbf{x},t) \\geq 0$ on the entire $\\partial \\Omega$, \nand $c_0(\\mathbf{x}) \\geq 0$ in $\\Omega$ then the maximum \nprinciple implies that $c(\\mathbf{x},t) \\geq 0$ in the \nentire domain  at all times, which is the non-negative \nconstraint for the concentration field for transient \nproblems. \n\n\\subsubsection{A methodology to enforce the \n  maximum principle for transient problems}\nWe divide the time interval of interest into \n$\\mathcal{N}$ sub-intervals. That is, \n\\begin{align}\n  [0,\\mathcal{I}] := \\bigcup_{n = 0}^{\\mathcal{N}} [t_n,t_{n+1}]\n\\end{align}\nwhere $t_n$ denotes the $n$-th time-level. We \nassume that the time-step is uniform, which \ncan be written as:\n\\begin{align}\n  \\Delta t = t_{n+1} - t_{n}\n\\end{align}\nFollowing the recommendation provided in  \n\\cite{Nakshatrala_Nagarajan_Shabouei_Arxiv_2013} to \nmeet maximum principles, we employ the backward Euler \nmethod for temporal discretization. We shall denote \nthe nodal concentrations at the $n$-th time-level \nby $\\boldsymbol{c}^{(n)}$. We shall denote the minimum \nand maximum values for the concentration by $c_{\\mathrm{min}}$ \nand $c_{\\mathrm{max}}$, which will be provided by the maximum \nprinciple and the non-negative constraint. At each time-level, \none has to solve the following convex quadratic program:\n\\begin{subequations}\n\\label{Eq:non-negative}\n  \\begin{align}\n    &\\mathop{\\mathrm{minimize}}_{\\boldsymbol{c}^{(n+1)}} \\quad \n    \\frac{1}{2} \\langle {\\boldsymbol{c}^{(n+1)}}; \n    \\widetilde{\\boldsymbol{K}} \\boldsymbol{c}^{(n+1)} \\rangle \n    - \\langle {\\boldsymbol{c}^{(n+1)}}; \n    \\widetilde{\\boldsymbol{f}}^{(n+1)} \\rangle \\\\\n    &\\mbox{subject to} \\quad c_{\\mathrm{min}} \\mathbf{1}  \\preceq \n    \\boldsymbol{c}^{(n+1)} \\preceq c_{\\mathrm{max}} \\mathbf{1}\n  \\end{align} \n\\end{subequations}\nwhere\n\\begin{align}\n  &\\widetilde{\\boldsymbol{K}} := \\frac{1}{\\Delta t} \\boldsymbol{M} + \n  \\boldsymbol{K} \\\\\n  &\\widetilde{\\boldsymbol{f}}^{(n+1)} := \\boldsymbol{f}^{(n+1)} + \n  \\frac{1}{\\Delta t} \\boldsymbol{M} \\boldsymbol{c}^{(n+1)}\n\\end{align}\nIn the above equation, $\\boldsymbol{M}$ is the capacity \nmatrix \\cite{Nakshatrala_Nagarajan_Shabouei_Arxiv_2013}. \n\n\\section{CONCLUDING REMARKS}\n\\label{Sec:Concluding_Remarks}\nWe presented a parallel non-negative computational framework \nsuitable for solving large-scale steady-state and transient \nanisotropic diffusion equations. The main contribution is \nthat the proposed parallel computational framework satisfies \nthe discrete maximum principles for large-scale diffusion-type \nequations even on general computational grids. \nThe parallel framework is built upon PETSc's DMPlex\ndata structure, which can handle unstructured meshes, \nand TAO for solving the resulting optimization problems \nfrom the discretization formulation. We have conducted \nsystematic performance modeling and strong-scaling \nstudies to demonstrate the efficiency, both in the parallel and\nhardware sense of the computational framework. The robustness of \nthe proposed framework has been illustrated by solving a \nlarge-scale realistic problem involving the transport of \nchromium in the subsurface at Los Alamos, New Mexico.\nFuture areas of research include: \n(a) extending the proposed parallel framework to \nadvective-diffusive and advective-diffusive-reactive \nsystems, and (b) posing the discrete problem as a \nvariational inequality, which will be valid even \nfor non-self-adjoint operators, and use other PETSc\ncapabilities to solve such variational \ninequalities. \n\n\n\n\n\n\\section{REPRESENTATIVE NUMERICAL RESULTS}\n\\label{Sec:Numerical_Results}\nIn this section, we compare the performance of our \nnon-negative methodology using the TAO solver to \nthat of the Galerkin formulation using the Krylov \nSubspace (KSP) solver. We examine the performance \nusing two problems: \n\\begin{enumerate}[(i)]\n\\item a unit cube with a hole under steady-state, and \n\\item a transient Chromium transport problem.\n\\end{enumerate}\nThe diffusivity tensor is assumed to depend \non the flow velocity through \n\\begin{align}\n  \\mathbf{D}(\\mathbf{x}) = \\left(\\alpha_T \\|\\mathbf{v}\\|+D_{M}\\right) \\mathbf{I} + \n(\\alpha_L - \\alpha_T) \\frac{\\mathbf{v} \\otimes \\mathbf{v}}{\\|\\mathbf{v}\\|} \n\\label{Eqn:S5_dispersion_tensor}\n\\end{align}\nwhere $\\alpha_L$, $\\alpha_T$, and $D_M$ denote the longitudinal dispersivity,\ntransverse dispersivity and molecular diffusivity, respectively. We employ the \nconjugate gradient method and the block Jacobi\/ILU(0) preconditioner for solving \nthe linear system from the Galerkin formulation and employ TAO's TRON and BLMVM \nmethods for the non-negative methodology. The relative convergence tolerances for \nboth KSP and TAO solvers are set to $10^{-6}$, and $\\Delta t$ for the transient \nresponse in the Chromium problem is initially set to 0.2 days. For strong-scaling \nstudies shown here, we used OpenMPI v1.6.5 for message passing and bound processes to cores \nwhile mapping by sockets. ParaView\\cite{paraview} and VisIt\\cite{VisIt} were \nused to generate all contour and mesh plots.\n\n\\begin{remark}\nThroughout the paper, the non-negative methodology that we refer to, is in fact a \ndiscrete maximum principle preserving methodology, in that, along with the \nnon-negative constraint we also enforce that the concentrations are less than \nor equal to 1.\n\\end{remark}\n\n\\subsection{Anisotropic diffusion in a unit cube with a cubic hole}\nLet the computational domain be a unit cube with a cubic hole of size \n$[4\/9, 5\/9] \\times [4\/9, 5\/9] \\times [4\/9, 5\/9]$. The concentration on the \nouter boundary is taken to be zero and the concentration on the interior \nboundary is taken to be unity. The volumetric source is taken as zero \n(i.e., $f(\\mathbf{x}) = 0$). The velocity vector field for this problem is \nchosen to be\n\\begin{align}\n\\mathbf{v}(\\mathbf{x}) = \\mathbf{e}_x + \\mathbf{e}_y + \\mathbf{e}_z\n\\end{align}\n\\begin{table}[t]\n  \\centering\n  \\caption{Cube with a hole: list of various mesh type and refinement \n    level combinations used \\label{Tab:S5_cube_information}}\n  \\begin{tabular}{lcccc}\n    \\hline\n    Case & Mesh type & Refinement level & Tetrahedrons & Vertices \\\\\n    \\hline\n    A1 & A & 1 & 199,296 & 36,378 \\\\\n    B1 & B & 1 & 409,848 & 75,427 \\\\\n    C1 & C & 1 & 793,824 & 140,190 \\\\\n    A2 & A & 2 & 1,594,368 & 278,194 \\\\\n    B2 & B & 2 & 3,278,784 & 574,524 \\\\\n    C2 & C & 2 & 6,350,592 & 1,089,562 \\\\\n    A3 & A & 3 & 12,754,994 & 2,175,330 \\\\\n    B3 & B & 3 & 26,230,272 & 4,483,126 \\\\\n    C3 & C & 3 & 50,804,736 & 9,172,044 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[t]\n  \\centering\n  \\caption{Cube with a hole: minimum and maximum concentrations \n    for each case \\label{Tab:S5_cube_dmp}}\n  \\begin{tabular}{lccc}\n    \\hline\n    Case & Min. concentration & Max. concentration & \\% nodes violated \\\\\n    \\hline\n    A1 & -0.0224825 & 1.00000 & 9,518\/36,378 $\\rightarrow$ 26.2\\% \\\\\n    B1 & -0.0139559 & 1.00000 & 32,247\/43,180 $\\rightarrow$ 42.8\\% \\\\\n    C1 & -0.0125979 & 1.00000 & 57,272\/140,190 $\\rightarrow$ 40.9\\% \\\\\n    A2 & -0.0311518 & 1.00103 & 82,983\/278,194 $\\rightarrow$ 29.2\\% \\\\\n    B2 & -0.0143857 & 1.00000 & 255,640\/574,524 $\\rightarrow$ 44.9\\% \\\\\n    C2 & -0.0119539 & 1.00972 & 453,766\/1,089,562 $\\rightarrow$ 41.6\\% \\\\\n    A3 & -0.0258559 & 1.00646 & 643,083\/2,175,330 $\\rightarrow$ 29.6\\% \\\\\n    B3 & -0.0115908 & 1.00192 & 2,073,934\/4,483126 $\\rightarrow$ 46.3\\%  \\\\\n    C3 & -0.0096186 & 1.00545 & 4,932,551\/9,172,044 $\\rightarrow$ 53.8\\%  \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[b]\n  \\centering\n  \\caption{Cube with a hole: wall-clock times (seconds) on Mustang for each solver \\label{Tab:S5_cube_times_mustang}}\n  \\begin{tabular}{lccccc}\n    \\hline\n    Case & Galerkin & TRON1 & TRON2 & TRON3 & BLMVM \\\\\n    \\hline\n    A1 & 0.337 & 0.933 & 0.981 & 1.14 & 2.62 \\\\\n    B1 & 0.790 & 1.72 & 2.06 & 2.71 & 5.04 \\\\\n    C1 & 2.24 & 4.34 & 5.80  & 7.74 & 13.5 \\\\\n    A2 & 7.21 & 15.2 & 21.7 & 32.5 & 72.0 \\\\\n    B2 & 15.4 & 30.0 & 43.7 & 57.5 & 109 \\\\\n    C2 & 40.4 & 67.8 & 113 & 118 & 286 \\\\\n    A3 & 121 & 225 & 414 & 599 & 1167 \\\\\n    B3 & 315 & 498 & 1061 & 1344 & 2524 \\\\\n    C3 & 997 & 1539 & 2490 & 4365 & 9679 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[b]\n  \\centering\n  \\caption{Cube with a hole: wall-clock times (seconds) on Wolf for each solver \\label{Tab:S5_cube_times_wolf}}\n  \\begin{tabular}{lccccc}\n    \\hline\n    Case & Galerkin & TRON1 & TRON2 & TRON3 & BLMVM \\\\\n    \\hline\n    A1 & 0.126 & 0.388 & 0.396 & 0.449 & 1.01 \\\\\n    B1 & 0.314 & 0.720 & 0.853 & 1.07 & 2.03 \\\\\n    C1 & 0.888 & 1.91 & 2.47 & 3.31 & 5.71 \\\\\n    A2 & 2.58 & 6.34 & 8.74 & 12.8 & 26.2 \\\\\n    B2 & 5.90 & 12.9 & 17.8 & 22.8 & 46 \\\\\n    C2 & 16.2 & 30.1 & 47.3 & 48.9 & 133 \\\\\n    A3 & 48.0 & 98.4 & 129 & 247 & 609 \\\\\n    B3 & 107 & 171 & 342 & 435 & 1060 \\\\\n    C3 & 281 & 467 & 870 & 1245 & 3131 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\nThe diffusion parameters are set as: $\\alpha_L = 1$, $\\alpha_T = 0.001$, and \n$D_M = 0$. The pictorial description of the computational domain and the three mesh\ntypes composed of 4-node tetrahedrons are shown in Figure \n\\ref{Fig:S5_cube_description}. We consider three unstructured mesh types \nwith three levels of element-wise mesh refinement, giving us nine total case \nstudies of increasing problem size as shown in Table \\ref{Tab:S5_cube_information}. \nWe ran a total of five different simulations for this study: \n\\begin{itemize}\n\\item Galerkin with CG\/block Jacobi\n\\item TRON1: with KSP tolerance of $10^{-1}$\n\\item TRON2: with KSP tolerance of $10^{-2}$\n\\item TRON3: with KSP tolerance of $10^{-3}$\n\\item BLMVM\n\\end{itemize} \nThe TRON solvers also use the CG and block Jacobi preconditioner but with \ndifferent KSP tolerances. Numerical results for both the Galerkin formulation and \nthe non-negative methodologies for some of the mesh cases are shown in Figure \n\\ref{Fig:S5_cube_results}.  The top row of figures arise from the Galerkin \nformulation where the white regions denote negative concentrations, and the \nbottom row arise from either TRON or BLMVM. Details concerning the \nviolation of the DMP for each case study can be found in Table \n\\ref{Tab:S5_cube_dmp}. Concentrations both negative\nand greater than one arise for all case studies.  Moreover, \nsimply refining the mesh does not resolve these issues; in fact, \nrefinment worsens the violation. These numerical results indicate that our \ncomputational framework can successfully enforce the DMP for diffusion problems \nwith highly anisotropic diffusivity.\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_solver_iterations.eps}}\n\\caption{Cube with a hole: solver iterations needed for Galerkin and BLMVM.}\n\\label{Fig:S5_cube_iterations_galerkin}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_solver_iterations_ksp.eps}}\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_solver_iterations_tao.eps}}\n\\caption{Cube with a hole: KSP (left) and TAO (right) solver iterations needed for TRON.}\n\\label{Fig:S5_cube_iterations_tron}\n\\end{figure}\n\n\\subsubsection{Performance modeling}\nWe first consider the wall-clock time spent in the solvers on a single core. \nTable \\ref{Tab:S5_cube_times_mustang} and \\ref{Tab:S5_cube_times_wolf} depict \nthe solver time for each mesh, and we first note that Mustang system requires \nsignificantly more wall-clock time to obtain a solution than Wolf; \nthis behavior is expected due to the difference in HPC hardware specifications \nlisted in Table \\ref{Tab:S3_HPC}, specifically, Mustang has the lower clock rate \nand lower bandwidth (as determined through STREAMS Triad). It can also be seen that \nthe various non-negative solvers consume varying amounts of wall-clock time.\nBLMVM can require as much as ten times the amount of wall-clock time as the standard\nGalerkin method. TRON on the other hand, does not consume nearly as much time but \ntightening the KSP tolerances will gradually increase the amount of time. \nWe are interested in determine why these optimization solvers consume more \nwall-clock time, whether it be mostly due to additional workload \nassociated with optimization-based techniques or due to the presence of \nrelatively more complicated and expensive data structures compared to the \nstandard solvers used for the Galerkin formulation. The first step is noting \nthe total KSP and TAO iterations needed and how they vary with respect to problem size. \nFigure \\ref{Fig:S5_cube_iterations_galerkin} depicts the KSP and TAO iterations for \nthe Galerkin and BLMVM methods respectively. It is well-known that block \nJacobi (also known as ILU(0)) requires more iterates as the size of the \nproblem increases. In other words, the solver may exhibit poor scaling for \nextremely large problems, but we see that the BLMVM algorithm has an even poorer \nscaling rate of the solver iterates. For the TRON solvers, we document both \nthe KSP and TAO iterates as shown in Figure \\ref{Fig:S5_cube_iterations_tron}. \nWe see that tightening the KSP tolerance increases the number of KSP iterates \nbut requires slightly fewer TAO iterates. This behavior indicates that the more \naccurate the computed gradient projection is, the fewer \noptimization loops the solver has to perform.\n\n\\begin{figure}[tp]\n\\centering\n\\subfloat[Mustang]{\\includegraphics[scale=0.55]\n{Figures\/cube_flopss_mustang.eps}}\n\\label{Fig:S5_cube_flopss_mustang}\n\\subfloat[Wolf]{\\includegraphics[scale=0.55]\n{Figures\/cube_flopss_wolf.eps}}\n\\label{Fig:S5_cube_flopss_wolf}\\\\\n\\caption{Cube with a hole: measured floating-point rate (FLOPS\/s) on a single core.}\n\\label{Fig:S5_cube_flopss}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_ai.eps}}\n\\caption{Cube with a hole: arithmetic intensity for all solvers and all cases  on a single processor.}\n\\label{Fig:S5_cube_ai}\n\\end{figure}\n\nWe also examine the measured floating-point rate provided by the PETSc performance logs,\nas shown in Figure \\ref{Fig:S5_cube_flopss}, of all five solvers across their respective \nmachines, and the floating point performance decreases as the problem size grows. One \ncould compare these numbers to the TPP and see that the hardware efficiencies are no \ngreater than $5\\%$, but it is difficult to draw any other conclusions with regard to the\ncomputational performance. The calculated AI, based on our proposed performance model, \nis shown in Figure \\ref{Fig:S5_cube_ai}. It is \ninteresting to note that the AI remains largely invariant with problem size \nunlike the wall-clock time, solver iterations, and floating point rates. \nAccording to the perfect cache model, the Galerkin formulation's AI is greater \nthan any of the non-negative methodologies. The optimization-based \nalgorithm based on TAO's BLMVM solver has significantly more streaming\/vector \noperations which explains the relatively lower AI. \nUsing these metrics in equation \\eqref{Eqn:S3_efficiency} as well as the \nSTREAMS Triad bandwidth of one core as shown in Figure \\ref{Fig:S3_streams}, \nthe estimated roofline-based efficiencies are shown in Figure \\ref{Fig:S5_cube_roofline}. \nAlthough the raw floating-point rate of BLMVM is lower than the Galerkin method, the roofline\nmodel suggests that BLMVM is actually more efficient in the hardware sense. The TRON methods\nhave much lower floating-point rates, but these metrics can be improved or ``gamed\" by \ntightening the KSP tolerances. This behavior leads us to believe that there is some \nlatency associated with setting up the data structures needed to compute gradient \ndescent projections.\n\n\\begin{figure}[tp]\n\\centering\n\\subfloat[Mustang]{\\includegraphics[scale=0.55]\n{Figures\/cube_efficiency_mustang.eps}}\n\\label{Fig:S5_cube_flopss_mustang}\n\\subfloat[Wolf]{\\includegraphics[scale=0.55]\n{Figures\/cube_efficiency_wolf.eps}}\n\\label{Fig:S5_cube_flopss_wolf}\\\\\n\\caption{Cube with a hole: estimated floating-point efficiency with respect to the \narithmetic intensity and measured memory bandwidth from STREAMS.}\n\\label{Fig:S5_cube_roofline}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_speedup_mustang.eps}}\n\\caption{Cube with a hole: speedup for all 9 mesh cases up to 64 processors on the Mustang system (16 cores per node).}\n\\label{Fig:S5_cube_speedup_mustang}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/cube_speedup_wolf.eps}}\n\\caption{Cube with a hole: speedup for all 9 mesh cases up to 64 cores on the Wolf system (8 cores per node).}\n\\label{Fig:S5_cube_speedup_wolf}\n\\end{figure}\n\\subsubsection{Strong-scaling}\nThe  metric of most interest to many computational scientists is the strong-scaling \npotential of any numerical framework. We conduct strong-scaling \nstudies to measure the speedup of all nine case studies over 64 cores. \nFour Mustang nodes with 16 cores each and 8 Wolf nodes with 8 cores \neach are allocated for this study. We do not fully saturate the compute\nnodes because the STREAMS benchmark indicates that there is little or no \ngain in memory performance when using a full node. Figure \\ref{Fig:S5_cube_speedup_mustang} \ndepicts the speedup on the Mustang system, and Figure \\ref{Fig:S5_cube_speedup_wolf}\ndepicts the speedup on the Wolf system. First, we note that the parallel efficiency \n(actual speedup over ideal speedup) increases with problem size due to \nAmdahl's Law. We also note that Wolf exhibits better strong-scaling due to the\nfaster speedups for the same test studies. For all problems and machines, \nthe TRON simulations are slightly less efficient in the parallel sense but can be \nimproved by tightening the KSP tolerances. Interestingly, the BLMVM algorithm not \nonly has the best roofline efficiency but also the best parallel speedup. \nWe can infer from these results that although BLMVM is the more efficient \noptimization in the hardware sense, TRON is more efficient in the algorithmic sense \ndue to its lesser time-to-solution. Our study has shown that one can draw \ncorrelations between the performance models conducted on a single-core and the \nactual speedup across multiple distributed memory nodes. As future solvers and \nalgorithms are implemented within PETSc, we can use this performance model to \nassess how efficient they are in both the hardware and algorithmic sense and how \nefficiently they will scale in a parallel setting.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.45]{Figures\/chromium_perspective.png}}\n\\caption{Chromium plume migration in the subsurface: Permeability field (m$^2$) and the locations of the pumping well (R28) and contaminant source (R42).}\n\\label{Fig:S5_chromium_description}\n\\end{figure}\n\\subsection{Transport of chromium in subsurface}\n\\begin{table}[b]\n  \\centering\n  \\caption{Chromium plume migration in the subsurface: parameters \\label{Tab:S5_chrom_properties}}\n  \\begin{tabular}{ll}\n    \\hline\n    Parameter & Value \\\\\n    \\hline\n    $\\alpha_L$ & 100 m \\\\\n    $\\alpha_T$ & 0.1 m \\\\\n    Contaminant source (R42) & $1\\times 10^{-4}$ kg\/m$^2$s$^2$ \\\\\n    $\\Delta t$ & 0.2 days \\\\\n    Domain size & 7000 km$\\times$6000 km$\\times$100 m \\\\\n    $D_M$ & $1\\times 10^{-9}$ m$^{2}$\/s \\\\\n    Permeability & Varies \\\\\n    Pumping well (R28) & -0.01 kg\/m$^2$s$^2$\\\\\n    Total hexahedrons & 1,984,512 \\\\\n    Total vertices & 2,487,765 \\\\\n    $\\mathbf{v}$ & Varies with position  \\\\\n    Viscosity & 3.95$\\times 10^{-5}$ Pa s \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{figure}[htp]\n\\centering\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_001.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_005.png}}\\\\\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_024.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_076.png}}\\\\\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_300.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_orig_900.png}}\n\\caption{Chromium plume migration in the subsurface: concentrations at select times using the Galerkin formulation.}\n\\label{Fig:S5_chromium_orig}\n\\end{figure}\n\\begin{figure}[htp]\n\\centering\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_001.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_005.png}}\\\\\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_024.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_076.png}}\\\\\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_300.png}}\n\\subfloat{\\includegraphics[scale=0.36]{Figures\/chrom_nonneg_900.png}}\n\\caption{Chromium plume migration in the subsurface: concentrations at select times using the non-negative methodology.}\n\\label{Fig:S5_chromium_nonneg}\n\\end{figure}\nSubsurface clean-up due to anthropogenic contamination \nis a big challenge \\cite{us2004cleaning}.  \nRemediation studies \\cite{hammond2010field,harp2013contaminant} need accurate predictions \nof transport of the involved chemical species, which are obtained \nusing limited data at monitoring wells and through \nnumerical simulations.\nTo accurately predict the fate of the contaminant,\na transport solver that: a) is robust, in that it will not \ngive unphysical solutions, and b) can handle field-scale \nscenarios, is needed. The computational framework that is \nproposed in this paper is an ideal candidate \nfor such problems. We now consider a realistic \nlarge-scale problem to predict the fate of \nchromium in the Los Alamos, New Mexico area. \nThe chromium was released into the Sandia canyon in the 50s up to \nearly 70s. Back then chromium was used as an anti-corrosion\nagent for the cooling towers at a power plant at the \nLos Alamos National Laboratory (see \\cite{heikoop2014isotopic} \nand references therein for details).   \n\nHere we study the effectiveness of our proposed framework\nto this real-world scenario of predicting the extent of \nchromium plume. The following is a conceptual model domain\nthat is considered:  A domain of size\n $[496,503]\\mathrm{km} \\times [536,542]\n\\mathrm{km} \\times [0,100]\\mathrm{m}$ with the permeability field (m$^2$) as shown \nin Figure \\ref{Fig:S5_chromium_description}. R42 in \nFigure \\ref{Fig:S5_chromium_description} is estimated to be the contaminant source\nlocation and a pumping well is located at R28. The parameters used for this \nproblem are shown in Table \\ref{Tab:S5_chrom_properties}, and we employ the \nfollowing boundary conditions:\n\\begin{align}\n \\label{Eqn:S5_chrom_bc_conc}\n &c^{\\mathrm{p}}(x=496\\mathrm{km},y,z) = c^{\\mathrm{p}}(x=503\\mathrm{km},y,z) = \nc^{\\mathrm{p}}(x,y=536\\mathrm{km},z) =  c^{\\mathrm{p}}(x,y=542\\mathrm{km},z) = 0\n\\end{align}\nFor this highly heterogeneous problem, we \nemploy PETSc's algebraic multi-grid preconditioner (GAMG) and\ncouple this with the TRON algorithm for the non-negative solver. Our goal is \nto examine its strong-scaling potential across 1024 cores. \nWe first solve the steady flow equation (based on mass balance and\nDarcy's model to relate pressure and mass flux) with the pumping well \nlocated at R28. Cell-wise velocity is obtained from the resulting pressure field and \nused to calculated element-wise dispersion tensor.  We then solve \nthe transient diffusion problem (with tensorial dispersion) with a \nconstant contaminant source located at R40 for up to 180 days. \nThe concentrations at select time levels for Galerkin formulation and \nnon-negative methodology are shown in Figures \\ref{Fig:S5_chromium_orig} \nand \\ref{Fig:S5_chromium_nonneg}, respectively. Negative concentrations arise with \nthe Galerkin formulation even as the solution approaches steady-state.\n\nFigure \\ref{Fig:S5_chromium_strongscale_first} depicts the amount of wall-clock \ntime with respect to the number of cores at the first time level. We see here \nthat the demonstrates good parallel performance across 1024 cores with up to \n35 percent parallel efficiency. Unlike the previous benchmark problem, we consider \na case where we completely saturate a Wolf node by using all 16 cores and notice\nthat the performance is slightly worse than using a partially saturated node \n(8 cores). This change of behavior can be attributed to the lack of memory \nperformance improvement one achieves when using all 16 cores as shown in \nFigure \\ref{Fig:S3_streams}. Interprocess communication becomes a major component of\nthe Wolf simulations so the parallel scalability gets worse the more efficiently TRON\nconducts its work. Nonetheless, the higher quality computing resources \nof a Wolf node results in faster solve times than the solve time on Mustang \neven with lesser parallel efficiency. \n\nAnother metric of interest is the number of solver iterations required for\nconvergence across various number of MPI processes. Figure \n\\ref{Fig:S5_chromium_iterations_first} depicts the number of KSP solver iterations\nand TAO solver iterations across 1024 cores, and we notice that there are \nsignificant fluctuations. This trend is largely attributed to the accumulation \nof numerical round-offs from the TRON algorithm. One can reduce\nthese fluctuations by tightening the solver tolerances, but the strong-scaling \nremains largely unaffected even for the results shown. This study suggests that \nthe proposed non-negative methodology using TRON with GAMG preconditioning \nis suitable for large-scale transient heterogeneious and anisotropic diffusion \nequations.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/chrom_strong_first.eps}}\n\\caption{Chromium plume migration in the subsurface: wall-clock time of the TRON \noptimization solver with multi-grid preconditioner (GAMG) versus number \nof processors after the first time level. Two Wolf cases are considered, where \nwe fully saturate a compute node (16 cores) and where we partially saturate a \ncompute node (8 cores per node). The parallel efficiency with respect to 16 \ncores is shown on the right hand side.}\n\\label{Fig:S5_chromium_strongscale_first}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/chrom_solver_iters.eps}}\n\\caption{Chromium plume migration in the subsurface: number of KSP (left hand side) \nand TAO (right hand side) solver iterations for the TRON optimization solver versus \nnumber of cores after the first time level.}\n\\label{Fig:S5_chromium_iterations_first}\n\\end{figure}\n\n\\section{INTRODUCTION}\n\\label{Sec:Introduction}\nThe modeling of flow and transport in subsurface is vital \nfor energy, climate and environmental applications. Examples \ninclude CO$_2$ migration in carbon-dioxide sequestration, \nenhanced geothermal systems, oil and gas production, \nradio-nuclide transport in a nuclear waste repository, \ngroundwater contamination, and  thermo-hydrology in the \nArctic permafrost due to the recent climate change \n\\cite{karra2014three,lichtner2014modeling,\nhammond2010field,kelkar2014simulator}. \nSeveral numerical codes (e.g., FEHM \\cite{zyvoloski2007fehm}, \nTOUGH \\cite{pruess2004tough}, PFLOTRAN \\cite{lichtner2015pflotran}) \nhave been developed to model flow and transport in subsurface \nat reservoir-scale. These codes typically solve unsteady Darcy \nequations for flow and advection-diffusion equation for \ntransport. The predictive capability of a numerical simulator \ndepends on the robustness of the underlying numerical methods. \nA necessary and essential requirement is to satisfy important \nmathematical principles and physical constraints. \nOne such property in transport and reactive-transport problems \nis that the concentration of a chemical species cannot be negative. \nMathematically, this translates to the satisfaction of the \ndiscrete maximum principle (DMP) for diffusion-type equations. \nSubsurface flow and transport applications typically encounter \ngeological media that are highly heterogeneous and anisotropic \nin nature, and it is well-known that the classical finite \nelement (or finite volume and finite difference, for that \nmatter) formulations do not produce non-negative solutions \non arbitrary meshes for such porous media \n\\cite{Ciarlet_Raviart_CMAME_1973_v2_p17,\nliska2008enforcing,\nNakshatrala_JCP_2009_v228_p6726,\nNagarajan_IJNMF_2011_v67_p820}.  \n\nSeveral studies over the years have focused on the \ndevelopment of methodologies that enforce the DMP \nand ensure non-negative solutions \\cite{Nakshatrala_JCP_2009_v228_p6726,\nNagarajan_IJNMF_2011_v67_p820,Payette_IJNME_2012_v91_p742,\nNakshatrala_JCP_2013_v253_p278}. However, these studies \ndid not address how these methods can be used for \nrealistic large-scale subsurface problems that have millions of grid nodes. \nFurthermore, complex coupling between different physical \nprocesses as well as the presence of multiple species amplify the \ndegrees-of-freedom (i.e., the number of unknowns).\nThe aim of this paper is to develop a parallel \ncomputational framework that solves anisotropic diffusion \nequations on general meshes, ensures non-negative solutions, \nand can be employed to solve large-scale realistic problems.\n\nLarge-scale problems can be tackled by using recent \nadvancements in high-performance computing (HPC) \nmethods and toolkits that can be used on the \nstate-of-the-art supercomputing architecture. \nOne such toolkit is PETSc \\cite{petsc-user-ref}, which \nprovides data structures and subroutines for setting up \nstructured and unstructured grids, parallel communication, \nlinear and non-linear solvers, and parallel I\/O. These \nhigh-level data structures and subroutines help in faster \ndevelopment of parallel application codes and minimize \nthe need to program low-level message passing, so that \nthe domain scientists can focus more on the application. \nTo this end, we develop a non-negative parallel framework\nby leveraging the existing capabilities within PETSc. Our \nframework ensures the DMP for anisotropic diffusion by using \nlower-order finite elements and the optimization-based \napproach in \\cite{Nakshatrala_JCP_2009_v228_p6726,\nliska2008enforcing,Nagarajan_IJNMF_2011_v67_p820}. The TAO \ntoolkit \\cite{tao-user-ref}, which is built on top of PETSc, \nis used for solving the resulting optimization problems. The \nrobustness of the proposed framework will be demonstrated by \nsolving realistic large-scale problems. \n\nThe rest of this paper is organized as follows. In Section \n\\ref{Sec:Governing_Equations}, we present the governing \nequations and the classical single-field Galerkin finite \nelement formulation for steady-state and transient diffusion \nequations. The optimization-based method to ensure non-negative \nconcentrations is also outlined in this section. \nIn Section \\ref{Sec:Parallel_Implementation}, the parallel \nimplementation procedure using PETSc and TAO is presented. \nWe also highlight the relevant data structures used in \nthis study and present a pseudo algorithm describing our parallel framework. \nIn Section \\ref{Sec:Performance_Modeling}, a performance model \nloosely based on the roofline model is outlined. This model is used \nto estimated the efficiency with respect to computing hardware \nof currently available solvers within PETSc and TAO.\nIn Section \\ref{Sec:Numerical_Results}, we first verify our \nimplementation using a 3D benchmark problem from the literature\nand present a detailed performance study using the proposed model. \nThen, we study a large-scale three-dimensional realistic \nproblem involving the transport of chromium in the subsurface\nand document the numerical results of the non-negative \nmethodology with the classical single-field Galerkin formulation. \nConclusions are drawn in Section \\ref{Sec:Concluding_Remarks}.\n\n\\section{PERFORMANCE MODELING}\n\\label{Sec:Performance_Modeling}\nPETSc is a constantly evolving open-source library that\nbrings out new features and algorithms almost every day.\nIt has capabilities to interface with a large number of\nother open-source software and linear algebra packages. \nHowever, it is not always known which of these algorithms \nwill have the best performance across multiple distributed \nmemory HPC systems, especially if these packages have little \ndocumentation and have to be used as black-box solvers. \nComputational scientists would like to know which solvers \nor algorithms to use for their specific need before running \njobs on the state-of-the-art HPC systems. The first and the \ntrivial metric to look for in answering this question is the \ntime-to-solution for a given solver or optimization method. \nHowever, additional information is needed in order to quantify \nthe hardware and algorithmic efficiency as well as the potential \nscalability across multiple cores in the strong sense.\n\\begin{table}[htp]\n  \\centering\n    \\caption{List of HPC systems used in this study \\label{Tab:S3_HPC}}\n  \\begin{tabular}{lcc}\n    \\hline\n    & Mustang (MU) & Wolf (WF) \\\\\n    \\hline\n    Processor & AMD Opteron 6176 & Intel Xeon E5-2670 \\\\\n    Clock rate& 2.3 GHz & 2.6 GHz\\\\\n    FLOPs\/cycle & 4 & 8\\\\\n    Sockets per compute node & 2 & 2\\\\\n    NUMA nodes per socket & 2 & 1\\\\\n    Cores per socket & 12 & 8\\\\\n    Total cores (compute nodes) & 38400 (1600) & 9856 (616)\\\\\n    Memory per compute node & 64 GB & 64 GB\\\\\n    L1 cache per core & 128 KB &32 KB\\\\\n    L2 cache per core & 512 KB & 256 KB\\\\\n    L3 cache per socket & 12 MB & 20 MB\\\\\n    Interconnect & 40 Gb\/s & 40 Gb\/s \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\nHardware specifications of HPC systems significantly impact the performance of any \nnumerical algorithm. Ideally we want our simulations to consume as little wall-clock\ntime as possible as the number of processing cores increases (i.e., achieving good \nspeedup), but several other factors including compiler vectorization, cache locality, \nmemory bandwidth, and code implementation may drastically affect the performance. \nTable \\ref{Tab:S3_HPC} lists the hardware specifications \nof the two HPC systems (Mustang and Wolf) that are used \nin our numerical experiments. The Mustang HPC system consists of\nrelatively older generation of processors so  \nit is expected to not perform as well. One could \nsimply measure wall-clock time across multiple compute \nnodes on the respective HPC machines and determine the \nparallel efficiency of a certain algorithm, but we are \ninterested in quantifying how different algorithms behave \nsequentially and what kind of parallel performance to expect \nbefore running numerical simulations on supercomputers. The wall-clock \ntime of any simulation can generally be summed up as a function of three things: \nthe workload, transfer of data between the memory and CPU register, \nand interprocess communication. Hardware efficiency in this \ncontext is defined as the amount of time spent performing work \nover waiting on memory fetching and cache registers to free up. \n\nThe limiting factor of performance for numerical methods on modern computing \narchitectures is upper-bounded by the memory bandwidth. That is, the floating point \nperformance given by FLOPS\/s will never reach the theoretical peak performance \n(TPP). This limitation is particularly important for iterative solvers and \noptimization methods that rely on numerous sparse matrix-vector (SpMV) \nmultiplications (see \\cite{may_ptatin} and the references within). The \nfrequent use of SpMV allows for little cache reuse and will result in a \nlarge number of very expensive cache misses. Such behavior is important to \ndocument when determining how efficient a scientific code is, so performance models \nsuch as the Roofline Model \\cite{williams,Lo_roofline}, which measures memory transfers, \nhave been used to better quantify the efficiency with respect to the hardware. \nPerformance models in general can help application developers \nidentify bottlenecks and indicate which areas of the code can be further optimized. \nIn other words, the code can be designed so that it maximizes the full \nbenefits of the available computing resources. In the next \nsection, we will demonstrate that such models can also be used to predict the \nparallel efficiency of various optimization solvers on the two very different LANL \nHPC systems. The key parameter for these performance models is the Arithmetic \nIntensity (AI) which is defined as:\n\\begin{align}\n   &\\mbox{AI} = \\frac{\\mbox{Total FLOPS}}\n   {\\mbox{Total Bytes Transferred}}\n  \\label{Eqn:S3_efficiency}\n\\end{align}\nwhere the Total Bytes Transferred (TBT) metric denotes\nthe amount of bandwidth needed for a given floating\npoint operation. The AI serves as a multiplier to the\nactual memory bandwidth and creates a ``roofline\" for\nthe estimation of ideal peak performance. A cache model \nis needed in order to properly define the TBT. \n\n\\begin{table}[tp]\n  \\centering\n    \\caption{Commonly used PETSc operations and their \n      respective Total Bytes Transferred. Here we note \n      $X,Y,Z$ as vectors with $i = 1,\\cdots,N$ entries, \n    $a$ is a scalar value, and $nz$ denotes the total \n      number of non-zeros. We assume that the sizes of \n      integers and doubles are 4 and 8 bytes respectively. \n    \\label{Tab:S3_TBT}}\n  \\begin{tabular}{lccc}\n    \\hline\n    PETSc function & Operation & Total Bytes Transferred \\\\\n    \\hline\n    VecNorm() & $a = \\sqrt{\\sum^{N}_{i} X(i)^2} $ & $8(N + 1)$ \\\\\n    VecDot() & $a = \\sum^{N}_{i}X(i)*Y(i)$ & $8(2N + 1)$ \\\\\n    VecCopy() & $Y \\leftarrow X$ & $8(2N)$ \\\\\n    VecSet() & $Y(i) = a$ & $8(2N)$ \\\\\n    VecScale() & $Y = a*Y$ & $8(2N)$ \\\\\n    VecAXPY() & $Y = a*X+Y$ & $8(3N)$ \\\\\n    VecAYPX() & $Y = X+a*Y$ & $8(3N)$ \\\\\n    VecPointwiseMult() & $Z(i) = X(i)*Y(i)$ & $8(3N)$ \\\\\n    MatMult() & SpMV & $4(N + nz) + 8(2N + nz)$ \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\nTo this end, we propose a roofline-like performance \nmodel where the TBT assumes a ``perfect cache\" -- each \nbyte of the data needs to be fetched from DRAM only \nonce. This assumption enables us to predict a slightly \nmore realistic upper bound of the peak performance than \nby simply comparing to the TPP. Table \\ref{Tab:S3_TBT} \nlists the key PETSc functions used for the solvers and their \nrespective estimates of TBT based on the perfect cache assumption. \nThe formula for SpMV follows the procedure outlined in \n\\cite{Kaushik99towardrealistic}. We assume that the TBT formula \nfor operations also involving a sparse matrix and vector \nlike the incomplete lower-upper (ILU) factorization to be \nthe same as MatMult(). Estimating the TBT for other \nimportant operations like the sparse matrix-matrix and \ntriple matrix products (which are important for multi-grid methods) \nis an area of future work. In short, our AI formulation \nrelies on the following four key assumptions:\n\\begin{enumerate}[(i)]\n\\item All floating-point operations (add, multiply, square roots, etc.) are treated \nequally and equate to one FLOP count. \n\\item There are no conflict misses. That is, each matrix and vector element is \nloaded into cache only once.\n\\item Processor never waits on a memory reference. That is, any number of loads and \nstores are satisfied in a single cycle.\n\\item Compilers are capable of storing scalar multipliers in the register only for \npure streaming computations.\n\\end{enumerate}\nTherefore, the efficiency based on this new roofline-like performance model as:\n\\begin{figure}[htp]\n\\centering\n\\subfloat{\\includegraphics[scale=0.55]{Figures\/streams_benchmark.eps}}\n\\caption{Estimated memory bandwidth of a single Mustang and Wolf compute node based \non the STREAMS Triad Benchmark}\n\\label{Fig:S3_streams}\n\\end{figure}\n\\begin{align}\n   &\\mbox{Efficiency (\\%)} = \\frac{\\mbox{Measured FLOPS\/s}}\n   {\\mathrm{min}\\left\\{\\begin{array}{l}\n    \\mbox{TPP} \\\\\n    \\mathrm{AI}\\times\\mbox{STREAMS}\n  \\end{array}\n  \\right.}\\times100\n  \\label{Eqn:S3_efficiency}\n\\end{align}\nwhere the numerator is reported by the PETSc program\nand the denominator is the ideal performance upper-bounded \nby both the TPP and the product of AI and STREAMS bandwidth. \nSTREAMS Triad \\cite{streams} is one of the most \npopular benchmarks for determining the achievable memory \nperformance of a given machine. Figure \\ref{Fig:S3_streams} denotes the \nestimated  memorybandwidth as a function of number of cores on a single \nMustang and Wolf node. It is interesting to note that although\nthe Wolf node has a greater bandwidth, there is\nno performance gain past eight cores. This means that an optimal use of a\nWolf compute node for memory-bandwidth bound algorithms would be \neight cores, whereas one would still see some performance gains when\nusing all 24 cores on a Mustang node.\n\nThe performance model that uses equation \\eqref{Eqn:S3_efficiency} \nis a serial model so the STREAMS metric for the Mustang and Wolf \nsystems are 5.65 GB\/s and 15.5 GB\/s respectively. It should be \nnoted that this performance model does account for cache effects. \nThat is, it does not quantify the useful bandwidth sustained for \nsome level of cache. The true hardware and algorithmic efficiency \nis not be reflected by this model, so our aim is to show relative \nperformance between select PETSc and TAO solvers. Comparing the \nAI and the measured FLOPS\/s with the STREAMS bandwidth will give \nus a better understanding of how high-performing the PETSc and TAO \nsolvers are for select problems.\n\n\\section{PARALLEL IMPLEMENTATION}\n\\label{Sec:Parallel_Implementation}\n\\subsection{PETSc and TAO}\n\\begin{figure}[htp]\n\\centering\n\\subfloat[Optimal 2D and 3D elements]{\\includegraphics[scale=0.75]{Figures\/DMPlex_simple.eps}}\n\\label{Fig:S3_dmplex_simple}\\\\\n\\subfloat[Interpolated 2D and 3D elements]{\\includegraphics[scale=0.75]{Figures\/DMPlex_interpolated.eps}}\n\\label{Fig:S3_dmplex_interpolated}\n\\caption{Representation of mesh points within the DMPlex data structure and their associated directed acyclic graphs}\n\\label{Fig:S3_dmplex}\n\\end{figure}\nWe leverage on existing scientific libraries such as PETSc and TAO to formulate our \nlarge-scale computational framework. PETSc is a suite of data structures and \nroutines for the parallel solution of scientific applications. It also provides \ninterfaces to several other libraries such as Metis\/ParMETIS\\cite{METIS} and HDF5 \n\\cite{hdf5} for mesh partitioning and binary data format handling respectively. The \nData Management (DM) data structure is used to manage all information including \nvectors and sparse matrices and compatible with binary data formats. To handle \nunstructured grids in parallel, a subset of the DM structure called DMPlex \n(see \\cite{Lange_easc,Knepley_scicomp,petsc-user-ref}), as shown in Figure \\ref{Fig:S3_dmplex},\nuses the direct acyclic graph to organize  all mesh information. This topology \nenables the freedom to mix and match various non vertex-based discretization such as\nthe two-point flux finite volume method and the classical mixed formulations\nbased on the lowest-order Raviart Thomas finite element space.\n\nAnother important feature within PETSc is TAO. The TAO library has a suite of data \nstructures and routines that enable the solution of large-scale optimization \nproblems. It can support any data structure or solver within PETSc. Our non-negative\nmethodology will use both the Newton-Trust Region (TRON) and Bounded Limited-Memory \nVariable-Metric (BLMVM) solvers available within TAO. BLMVM is a quasi-Newton method \nthat uses projected gradients to approximate the Hessian, which is useful for problems where the\nhessian is too complicated or expensive to compute. Other optimization algorithms \nsuch as TRON and the Gradient Projected Conjugate Gradient (GPCG) typically require \nHessian information and more memory, but they are expected to converge more rapidly\nthan BLMVM. Further details regarding the implementation of these various methods \nmay be found in \\cite{tao-user-ref} and the references within.\n \n\\subsection{Finite element implementation}\nPETSc abstractions for finite elements, quadrature rules, and function spaces have \nalso been recently introduced and are suitable for the mesh topology within DMPlex. \nThey are built upon the same framework as the Finite element Automatic Tabulator \n(FIAT) found within the FEniCS Project \\cite{Kirby2012a,Alogg,LoggMardalEtAl2012a}. \nThe finite element discretizations simply need the equations, auxiliary coefficients\n(e.g., permeability, diffusivity, etc.), and boundary conditions specified as \npoint-wise functions. We express all discretizations in nonlinear form so let \n$\\boldsymbol{r}$ and $\\boldsymbol{J}$ denote the residual and Jacobian respectively.\n\nFollowing the FEM model outlined in \\cite{2013arXiv1309.1204K}, we consider the weak\nform that depends on fields and gradients. The residual evaluation can be expressed\nas: \n\\begin{align}\n&\\boldsymbol{w}^{\\mathrm{T}}\\boldsymbol{r}(\\boldsymbol{c}) \\sim \\int_{\\Omega^{e}} \\left[w\\cdot \n\\mathcal{F}_{0}\\left(c,\\nabla c\\right) + \n\\nabla w \\cdot \\boldsymbol{\\mathcal{F}}_1(c,\\nabla c) \n\\right]\\mathrm{d} \\Omega = 0\n\\end{align}\nwhere $\\mathcal{F}_0(c,\\nabla c)$ and $\\boldsymbol{\\mathcal{F}}_1(c,\\nabla c)$ are \nuser-defined point-wise functions that capture the problem physics. This framework \ndecouples the problem specification from the mesh and degree of freedom traversal. \nThat is, the scientist need only focus on providing point function evaluations while\nletting the finite element library take care of meshing, quadrature points, basis \nfunction evaluation, and mixed forms if any. The discretization of the residual is \nwritten as: \n\\begin{align}\n&\\boldsymbol{r}(\\boldsymbol{c}) = \\mathop{\\huge \\boldsymbol{\\mathsf{A}}}_{e=1}^{Nele} \n\\left[\\begin{array}{lr}\\boldsymbol{N}^{\\mathrm{T}} & \\boldsymbol{B}^{\\mathrm{T}}\t\n\\end{array}\\right]\n\\boldsymbol{W}\\left[\\begin{array}{l}\t\\mathcal{F}_0(c_q,\\nabla c_q) \\\\ \n\\boldsymbol{\\mathcal{F}}_1(c_q,\\nabla c_q)\\end{array}\\right]\n\\label{Eqn:S3_residual}\n\\end{align}\nwhere $\\boldsymbol{\\mathsf{A}}$ represents the standard assembly operator, \n$\\boldsymbol{N}$ and $\\boldsymbol{B}$ are matrix forms of basis functions that \nreduce over quadrature points, $\\boldsymbol{W}$ is a diagonal matrix of quadrature \nweights (including the geometric Jacobian determinant of the element), and $c_q$ is \nthe field value at quadrature point $q$. The Jacobian of \\eqref{Eqn:S3_residual} \nneeds only the derivatives of the point-wise functions:\n\\begin{align}\n&\\boldsymbol{J}(\\boldsymbol{c}) = \\mathop{\\huge \\boldsymbol{\\mathsf{A}}}_{e=1}^{Nele} \n\\left[\\begin{array}{lr}\\boldsymbol{N}^{\\mathrm{T}} & \\boldsymbol{B}^{\\mathrm{T}}\t\\end{array}\\right]\n\\boldsymbol{W}\\left[\\begin{array}{ll} \\mathcal{F}_{0,0} & \\mathcal{F}_{0,1} \\\\ \n\\boldsymbol{\\mathcal{F}}_{1,0}& \\boldsymbol{\\mathcal{F}}_{1,1}\\end{array}\\right]\n\\left[\\begin{array}{c} \\boldsymbol{N} \\\\ \\boldsymbol{B}\n\\end{array}\\right], \\quad \\left[\\mathcal{F}_{i,j}\\right] = \n\\left[\\begin{array}{ll}\n\\frac{\\partial\\mathcal{F}_0}{\\partial c} & \\frac{\\partial\\mathcal{F}_0}{\\partial\\nabla c} \\\\\n\\frac{\\partial\\boldsymbol{\\mathcal{F}}_1}{\\partial c} & \\frac{\\partial\\boldsymbol{\\mathcal{F}}_1}{\\partial\\nabla c}\n\\end{array}\\right](c_q,\\nabla c_q)\n\\label{Eqn:S3_jacobian}\n\\end{align}\nThe point-wise functions corresponding to the weak form in \n\\eqref{Eqn:functionals_B_L} would be:\n\\begin{algorithm}[b]\n  \\begin{algorithmic}\n    \\State Create\/input DAG on rank 0\n    \\State Create\/input cell-wise velocity on rank 0\n    \\If {size $>$ 1}\n    \\State Partition mesh among all processors\n    \\EndIf\n    \\State Refine distributed mesh if necessary\n    \\State Create PetscSection and FE discretization\n    \\State Set $n=0$ and $\\boldsymbol{c}^{(0)}=10^{-8} $\n    \\State Insert Dirichlet BC constraints into  $\\boldsymbol{c}^{(0)} $\n    \\State Compute Jacobian $\\boldsymbol{J}$\n    \\While {true} \\Comment{Begin time-stepping scheme}\n\t    \\State Compute Residual $\\boldsymbol{r}^{(n)}$ \n\t    \\If {Classical Galerkin}\n\t    \\Comment{Solve without non-negative methodology}\n\t    \\State $\\boldsymbol{c}^{(n+1)} = \\boldsymbol{c}^{(n)} - \\boldsymbol{J}\\backslash\\boldsymbol{r}^{(n)}$\n\t    \\Else\\Comment{Solve with non-negative methodology}\n\t    \\State TaoSolve() for $\\boldsymbol{c}^{(n+1)}$ based on equations \\eqref{Eqn:S3_tao_objective} and \\eqref{Eqn:S3_tao_gradient}\n\t    \\EndIf\n\t    \\If {steady-state or $(n) == $ total number of time steps}\n\t    \\State break\n\t    \\Else\n\t    \\State $n += 1$\n\t    \\EndIf\n    \\EndWhile\n    \n  \\end{algorithmic}\n  \\caption{Pseudocode for the large-scale transport solver \\label{Algo:S3_pseudocode}}\n\\end{algorithm}\n\\begin{subequations}\n\\begin{align}\n&\\mathcal{F}_0 = -f(\\mathbf{x}),\\quad \\boldsymbol{\\mathcal{F}}_1 = \\mathbf{D}(\\mathbf{x})\n\\nabla c_q \\\\\n&\\mathcal{F}_{0,0} = 0, \\quad \\mathcal{F}_{0,1} = 0, \\quad \\boldsymbol{\\mathcal{F}}_{1,0} = \\mathbf{0}, \\quad \\boldsymbol{\\mathcal{F}}_{1,1} = \\mathbf{D}(\\mathbf{x})\n\\end{align}\n\\end{subequations}\nSimilarly, the point-wise functions for the transient response are:\n\\begin{subequations}\n\\begin{align}\n&\\mathcal{F}_0 = \\dot{c}_q-f(\\mathbf{x},t),\\quad \\boldsymbol{\\mathcal{F}}_1 = \\mathbf{D}(\\mathbf{x})\n\\nabla c_q \\\\\n&\\mathcal{F}_{0,0} = \\frac{1}{\\Delta t}, \\quad \\mathcal{F}_{0,1} = 0, \\quad \\boldsymbol{\\mathcal{F}}_{1,0} = \\mathbf{0}, \\quad \\boldsymbol{\\mathcal{F}}_{1,1} = \\mathbf{D}(\\mathbf{x})\n\\end{align}\n\\end{subequations}\nwhere $\\dot{c}_{q}$ denotes the time derivative. A similar discretization is used \nto project the Neumann boundary conditions into the residual vector. \nAssuming a fixed time-step, $[\\mathcal{F}_{i,j}]$ and the Jacobian in equation \n\\eqref{Eqn:S3_jacobian} do not change with time and have to be computed only once. \nIf $n$ denotes the time level ($n$ = 0 denotes initial condition) then \nthe residual and Jacobian can be defined as:\n\\begin{align}\n&\\boldsymbol{r}^{(n)} \\equiv \\boldsymbol{r}(\\boldsymbol{c}^{(n)})\\\\\n&\\boldsymbol{J} \\equiv \\boldsymbol{J}(\\boldsymbol{c}^{(0)})\n\\end{align}\nTo enforce the non-negative methodology, the following objective function $b$ \nand gradient function $\\boldsymbol{g}$ is provided: \n\n  \\begin{align}\n  \\label{Eqn:S3_tao_objective}\n    &b = \\frac{1}{2}\\boldsymbol{c}^{(n+1)}\\cdot\\boldsymbol{J}\\boldsymbol{c}^{(n+1)} + \\boldsymbol{c}^{(n+1)}\\cdot\\left[\\boldsymbol{r}^{(n)}-\\boldsymbol{J}\\boldsymbol{c}^{(n)}\\right]\\\\\n    \\label{Eqn:S3_tao_gradient}\n    &\\boldsymbol{g} = \\boldsymbol{J}\\left[\\boldsymbol{c}^{(n+1)}-\\boldsymbol{c}^{(n)}\\right] + \\boldsymbol{r}^{(n)}\n  \\end{align}\nBLMVM relies only on the above two equations, whereas TRON needs the Hessian which \nis equivalent to $\\boldsymbol{J}$. Algorithm \\ref{Algo:S3_pseudocode} outlines \nthe steps taken in our computational framework.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\textbf{INTRODUCTION}}\n\n\nCircumstellar disks play a fundamental role in the formation of \nstars and planets. \nA significant fraction of the mass of a star is thought to be \nbuilt up by accretion through the disk.\nThe gas and dust in the inner disk ($r<$10\\,AU) \nalso constitute the likely material from which planets form.  \nAs a result, observations of the gaseous \ncomponent of inner disks have the potential to provide critical clues to \nthe physical processes governing star and planet formation.\n\nFrom the planet formation perspective, probing the structure,\ngas content, and dynamics of inner disks is of interest, since\nthey all play important\nroles in establishing the architectures of planetary systems\n(i.e., planetary masses, orbital radii, and eccentricities).\nFor example, the lifetime of gas in the inner disk\n(limited by accretion onto the star, photoevaporation, \nand other processes)\nplaces an upper limit on the timescale for giant planet formation\n(e.g., {\\em Zuckerman et al.}, 1995).\n\nThe evolution of gaseous inner disks may also bear \non the efficiency of \norbital migration and the eccentricity evolution of giant and terrestrial\nplanets.  Significant inward orbital migration, induced\nby the interaction of planets with a gaseous disk, is \nimplied by the small orbital radii of extrasolar\ngiant planets compared to their likely formation distances\n(e.g., {\\em Ida and Lin}, 2004).  \nThe spread in the orbital radii of the planets (0.05--5\\,AU)\nhas been further taken to indicate that the timing of the dissipation \nof the inner disk sets the final orbital radius of the planet\n({\\em Trilling et al.}, 2002).\nThus, understanding \nhow inner disks dissipate may impact \nour understanding of the origin of planetary orbital radii.\nSimilarly, residual gas in the terrestrial planet region may play \na role in defining the final masses and eccentricities of \nterrestrial planets.  Such issues have a strong connection\nto the question of the likelihood of solar systems like our own.\n\nAn important issue from the perspective of both star and planet\nformation is the nature of the physical mechanism that is \nresponsible for disk accretion.  \nAmong the proposed mechanisms, \nperhaps the foremost is the magnetorotational instability \n({\\em Balbus and Hawley}, 1991) although other possibilities exist.   \nDespite the significant theoretical progress that has been made \nin identifying plausible accretion mechanisms \n(e.g., {\\em Stone et al.}, 2000), there is little observational \nevidence that any of these processes are active in disks. \nStudies of the gas in inner disks offer opportunities \nto probe the nature of the accretion process. \n\nFor these reasons, it is of interest to probe \nthe dynamical state, physical and chemical structure, and the \nevolution of the gas content of inner disks.\nWe begin this Chapter with a brief review of the \ndevelopment of this field and an overview of \nhow high resolution spectroscopy can be used to study the \nproperties of inner disks (Section 1). \nPrevious reviews provide additional background on these topics \n(e.g., {\\em Najita et al.}, 2000).  \nIn Sections 2 and 3, we review recent observational and theoretical \ndevelopments in this field, first describing \nobservational work to date on the gas in \ninner disks, and then describing theoretical models \nfor the surface and interior regions of disks. \nIn Section 4, we look to the future, highlighting several topics \nthat can be explored using the tools discussed in Sections 2 and 3. \n\n\n\n\\bigskip\n\\noindent\n\\textbf{1.1 Historical Perspective} \n\\bigskip\n\nOne of the earliest studies of gaseous inner disks was \nthe work by Kenyon and Hartmann on FU Orionis objects. \nThey showed that many of the peculiarities of these systems \ncould be explained in terms of \nan accretion outburst in a disk surrounding \na low-mass young stellar object (YSO; cf.\\ {\\em Hartmann and Kenyon},\\ 1996).\nIn particular, the varying spectral type of FU Ori objects in \noptical to near-infrared spectra, evidence for double-peaked \nabsorption line profiles, and the decreasing widths of absorption \nlines from the optical to the near-infrared argued for an origin \nin an optically thick gaseous atmosphere in the inner region of \na rotating disk.\nAround the same time, \nobservations of CO vibrational overtone emission, first in the BN object\n({\\em Scoville et al.},\\ 1983) and later in other high and low mass objects\n({\\em Thompson}, 1985; {\\em Geballe and Persson}, 1987; \n{\\em Carr}, 1989), revealed the \nexistence of hot, dense molecular gas plausibly located in a disk.\nOne of the first models for the CO overtone emission ({\\em Carr}, 1989) placed\nthe emitting gas in an optically-thin inner region of an accretion\ndisk. \nHowever, only the observations of the BN object had sufficient \nspectral resolution to constrain the kinematics of the emitting gas. \n\nThe circumstances under which a disk would produce emission or \nabsorption lines of this kind \nwere explored in early models\nof the atmospheres of gaseous accretion disks under the influence\nof external irradiation (e.g., {\\em Calvet et al.},\\ 1991).  The models\ninterpreted the FU Ori absorption features \nas a consequence of midplane accretion rates high enough to \noverwhelm external irradiation in establishing a temperature\nprofile that decreases with disk height.  At lower accretion rates,\nthe external irradiation of the disk was expected to induce a\ntemperature inversion in the disk atmosphere, producing emission\nrather than absorption features from the disk atmosphere.  \nThus the models potentially provided an explanation for the \nFU Ori absorption features and CO emission lines that had been \ndetected.\n\nBy PPIV ({\\em Najita et al.},\\ 2000), high-resolution spectroscopy \nhad demonstrated that CO overtone emission shows the dynamical signature of a\nrotating disk ({\\em Carr et al.},\\ 1993; {\\em Chandler et al.},\\ 1993), thus\nconfirming theoretical expectations and opening the door\nto the detailed study of gaseous inner disks in a larger number of\nYSOs.\nThe detection of CO fundamental emission (Section 2.3) and emission\nlines of hot H$_2$O (Section 2.2) had also added new probes of the \ninner disk gas.\n\nSeven years later, at PPV, we find both a growing\nnumber of diagnostics available to probe gaseous inner disks as\nwell as increasingly detailed information that can be gleaned from\nthese diagnostics.  Disk diagnostics familiar from PPIV \nhave been used to infer the intrinsic line broadening of disk\ngas, possibly indicating evidence for turbulence in disks (Section 2.1).  \nThey also demonstrate the differential rotation of disks,  \nprovide evidence for non-equilibrium molecular abundances (Section 2.2),  \nprobe the inner radii of gaseous disks (Section 2.3), \nand are being used to probe the gas dissipation timescale \nin the terrestrial planet region (Section 4.1).\nAlong with these developments, new spectral line diagnostics have \nbeen used as probes of the gas in inner disks.  These \ninclude transitions of molecular hydrogen at UV, near-infrared, and  \nmid-infrared wavelengths (Sections 2.4, 2.5) and the fundamental ro-vibrational \ntransitions of the OH molecule (Section 2.2).  Additional potential \ndiagnostics are discussed in Section 2.6.  \n\n\n\n\\bigskip\n\\noindent\n\\textbf{1.2 High Resolution Spectroscopy of Inner Disks}\n\\bigskip\n\nThe growing suite of diagnostics can be used to probe \ninner disks using standard high resolution spectroscopic \ntechniques. \nAlthough inner disks are typically too small to resolve \nspatially at the distance of the nearest star forming regions,\nwe can utilize the likely differential rotation \nof the disk along with high spectral resolution to separate \ndisk radii in velocity.  \nAt the warm temperatures ($\\sim$100\\,K--5000\\,K) and high densities of inner\ndisks, molecules are expected to be abundant in the gas phase and\nsufficiently excited to produce rovibrational features in the\ninfrared.  Complementary atomic transitions are likely to be good \nprobes of the hot inner disk and the photodissociated surface \nlayers at larger radii. \nBy measuring multiple transitions of different species, \nwe should therefore be able\nto probe the temperatures, column densities, and abundances\nof gaseous disks as a function of radius.\n\nWith high spectral \nresolution we can resolve individual lines, which facilitates \nthe detection of weak spectral features. \nWe can also work around  \ntelluric absorption features, using the radial velocity of the \nsource to shift its spectral features out of telluric \nabsorption cores. \nThis approach makes it \npossible to study a variety of atomic and molecular \nspecies, including those present in the Earth's atmosphere.\n\n\nGaseous spectral features are expected in a variety of situations.  \nAs already mentioned, significant vertical variation in the temperature \nof the disk atmosphere will produce emission (absorption) features if the \ntemperature increases (decreases) with height ({\\em Calvet et al.}, 1991; \n{\\em Malbet and Bertout}, 1991).  \nIn the general case, when the disk is optically thick, observed spectral\nfeatures measure only the atmosphere of the disk and are unable to \nprobe directly the entire disk column density, a situation familiar \nfrom the study of stellar atmospheres.  \n\n\nGaseous emission features are also expected \nfrom regions of the disk that are optically thin in the continuum. \nSuch regions might arise as a result of dust sublimation \n(e.g., {\\em Carr}, 1989) or as a consequence of grain growth \nand planetesimal formation. \nIn these scenarios, the disk would have a low continuum opacity \ndespite a potentially large gas column density. \nOptically thin regions can also be produced by a significant \nreduction in the total column density of the disk.  This situation \nmight occur as a consequence of giant planet formation, in which \nthe orbiting giant planet carves out a ``gap'' in the disk.  \nLow column densities would also be characteristic of a dissipating disk. \nThus, we should be able to use gaseous emission lines to probe \nthe properties of inner disks in a variety of interesting \nevolutionary phases.  \n\n\n\n\\section{\\textbf{OBSERVATIONS OF GASEOUS INNER DISKS}}\n\n\\bigskip\n\\noindent\n\\textbf{2.1 CO Overtone Emission}\n\\bigskip\n\nThe CO molecule is expected to be \nabundant in the gas phase over a wide range of \ntemperatures, from the temperature at which it condenses on grains \n($\\sim$20\\,K) up to its thermal dissociation temperature \n($\\sim$4000\\,K at the densities of inner disks). \nAs a result, CO transitions are expected to probe \ndisks from their cool outer reaches ($>$100\\,AU) \nto their innermost radii.\nAmong these, the overtone transitions of CO ($\\Delta v$=2, \n$\\lambda$=2.3$\\mu$m) were the emission line diagnostics first \nrecognized to probe the gaseous inner disk.\n\nCO overtone emission is detected in both low and high mass young\nstellar objects, but only in a small fraction of the objects observed.\nIt appears more commonly among higher luminosity objects. Among \nthe lower luminosity stars, it is detected from embedded protostars \nor sources with energetic outflows\n({\\em Geballe and Persson}, 1987; {\\em Carr}, 1989; \n{\\em Greene and Lada}, 1996; {\\em Hanson et al.}, 1997;\n{\\em Luhman et al.}, 1998; {\\em Ishii et al.}, 2001; \n{\\em Figueredo et al.}, 2002; {\\em Doppmann et al.}, 2005).\nThe conditions required to excite the overtone emission,  \nwarm temperatures ($\\gtrsim 2000$ K) and high densities \n($>$$10^{10}\\rm \\,cm^{-3}$), may be met in disks\n({\\em Scoville et al.}, 1983; {\\em Carr}, 1989; \n{\\em Calvet et al.}, 1991), inner winds ({\\em Carr}, 1989), \nor funnel flows ({\\em Martin}, 1997). \n\nHigh resolution spectroscopy can be used to distinguish among these \npossibilities.  The observations typically find strong \nevidence for the disk interpretation.  The emission line profiles \nof the $v$=2--0 bandhead in most cases \nshow the characteristic signature of bandhead emission from\nsymmetric, double-peaked line profiles originating in \na rotating disk (e.g., {\\em Carr et al.}, 1993; {\\em Chandler et al.}, 1993; \n{\\em Najita et al.}, 1996; {\\em Blum et al.}, 2004).  The symmetry of the observed \nline profiles argues against the likelihood that the emission arises \nin a wind or funnel flow, since inflowing or outflowing gas is \nexpected to produce line profiles with  red- or blue-shifted \nabsorption components (alternatively line asymmetries) of the \nkind that are seen in the hydrogen Balmer lines of T Tauri stars \n(TTS).\nThus high resolution spectra provide strong evidence for \nrotating inner disks. \n\nThe velocity profiles of the CO overtone emission are normally\nvery broad ($>$100\\,${\\rm km}\\,{\\rm s}^{-1}$).  In lower mass stars \n($\\sim$$1 M_\\odot$), the emission profiles show that the emission \nextends from very close to the star,\n$\\sim$0.05\\,AU, out to $\\sim$0.3\\,AU\n(e.g., {\\em Chandler et al.}, 1993; {\\em Najita et al.}, 2000).\nThe small radii are consistent with the high excitation\ntemperatures measured for the emission ($\\sim$1500--4000\\,K).\nVelocity resolved spectra have also been modeled in a number\nof high mass stars ({\\em Blum et al.}, 2004; {\\em Bik and Thi}, 2004), \nwhere the CO emission is found to arise at radii $\\sim 3$\\,AU.\n\nThe large near-infrared excesses of the sources in which CO \novertone emission is detected imply that the warm emitting gas \nis located in a vertical temperature inversion region in the disk atmosphere. \nPossible heating sources for the temperature inversion include: \nexternal irradiation by the star at optical through UV wavelengths \n(e.g., {\\em Calvet et al.}, 1991; {\\em D'Alessio et al.}, 1998) \nor by stellar X-rays ({\\em Glassgold et al.}, 2004; henceforth GNI04); \nturbulent heating in the disk atmosphere generated by a \nstellar wind flowing over the disk surface ({\\em Carr et al.}, 1993);  \nor the dissipation of turbulence generated by disk accretion \n(GNI04). \nDetailed predictions of how these mechanisms heat the gaseous \natmosphere are needed in order to \nuse the observed bandhead emission strengths and profiles to \ninvestigate the origin of the temperature inversion. \n\nThe overtone emission provides an additional clue \nthat suggests a role for turbulent dissipation \nin heating disk atmospheres.  \nSince the CO overtone bandhead is made up of closely spaced lines \nwith varying inter-line spacing and optical depth, the emission \nis sensitive to the intrinsic line broadening of the emitting \ngas (as long as the gas is not optically thin).  \nIt is therefore possible to distinguish intrinsic line \nbroadening from macroscopic motions such as rotation.  \nIn this way, \none can deduce from spectral synthesis modeling \nthat the lines are suprathermally broadened, with line widths \napproximately Mach 2 ({\\em Carr et al.}, 2004; {\\em Najita et al.}, 1996). \n{\\em Hartmann et al.}\\ (2004) find further evidence for \nturbulent motions in disks based on high resolution \nspectroscopy of CO overtone absorption in FU Ori objects.\n\nThus disk atmospheres appear to be turbulent.  The turbulence \nmay arise as a consequence of turbulent angular momentum \ntransport in disks, as in the magnetorotational instability \n(MRI; {\\em Balbus and Hawley}, 1991) or the global baroclinic instability \n({\\em Klahr and Bodenheimer}, 2003).  Turbulence in the upper disk \natmosphere may also be generated by a wind blowing over the \ndisk surface. \n\n\n\\bigskip\n\\noindent\n\\textbf{2.2 Hot Water and OH Fundamental Emission}\n\\bigskip\n\nWater molecules are also expected to be abundant in disks over \na range of disk radii, from the temperature at which water \ncondenses on grains ($\\sim$150 K) up to its thermal dissociation \ntemperature ($\\sim$2500 K).  Like the CO overtone \ntransitions, the rovibrational transitions of water are also \nexpected to probe the high density conditions in disks. \nWhile the strong telluric absorption produced by water vapor in the\nEarth's atmosphere will restrict the study of cool water to\nspace or airborne platforms, it is possible to observe from the\nground water that is much hotter than the Earth's atmosphere.\nVery strong emission from hot water can be detected in the near-infrared \neven at low spectral resolution (e.g., SVS-13; {\\em Carr et al.}, 2004).  \nMore typically, high resolution spectroscopy of\nindividual lines is required to detect much weaker emission lines.\n\nFor example, emission from individual lines of water in the\n$K$- and $L$-bands have been detected in a few stars (both low \nand high mass) that also show CO overtone emission \n({\\em Carr et al.}, 2004; {\\em Najita et al.}, 2000; {\\em Thi and Bik}, 2005).\nVelocity resolved spectra show that the widths of the\nwater lines are consistently narrower than those of the \nCO emission lines.  Spectral synthesis modeling further shows \nthat the excitation temperature of the water emission\n(typically $\\sim$1500\\,K),\nis less than that of the CO emission.  These results\nare consistent with both the water and CO originating in a\ndifferentially rotating\ndisk with an outwardly decreasing temperature profile.  That is,\ngiven the lower dissociation temperature of water ($\\sim$2500\\,K)\ncompared to CO ($\\sim$4000\\,K),\nCO is expected to extend inward to smaller radii than \nwater, i.e., to higher velocities and temperatures. \n\nThe $\\Delta v$=1 OH fundamental transitions \nat 3.6$\\mu$m have also been detected in the spectra of two actively \naccreting sources, SVS-13 and V1331~Cyg, that also show CO \novertone and hot water emission  (Carr et al., in preparation).  \nAs shown in Fig.~1, these \nfeatures arise in a region that is crowded with spectral \nlines of water and perhaps other species.  Determining the strengths of \nthe OH lines will, therefore, require making corrections for \nspectral features that overlap closely in wavelength. \n\nSpectral synthesis modeling of the detected CO, H$_2$O and OH features \nreveals relative abundances \nthat depart significantly from chemical equilibrium \n(cf.\\ {\\em Prinn}, 1993), with the \nrelative abundances of H$_2$O and OH a factor of 2--10 below that of CO \nin the region of the disk probed by both diagnostics \n({\\em Carr et al.}, 2004; Carr et al., in preparation; \nsee also {\\em Thi and Bik}, 2005).\nThese abundance ratios may arise from strong vertical abundance \ngradients produced by the external irradiation of the disk \n(see Section 3.4). \n\n\\begin{figure}[b!]\n\\plotfiddle{najita_fig1.eps}{0.05in}{270}{180}{250}{-8}{0}\n\\vspace{-.7in}\n\\caption{\\small\\baselineskip 10pt\nOH fundamental ro-vibrational emission from SVS-13  \non a relative flux scale.}\n\\end{figure}\n\n\n\\bigskip\n\\noindent\n\\textbf{2.3 CO Fundamental Emission}\n\\bigskip\n\nThe fundamental ($\\Delta v$=1) transitions of CO at 4.6$\\mu$m \nare an important probe of inner disk gas in part because of \ntheir broader applicability compared, e.g., to the CO overtone lines. \nAs a result of their comparatively small A-values, \nthe CO overtone transitions require large column densities of warm gas \n(typically in a disk temperature inversion region) in order to produce \ndetectable emission.  \nSuch large column densities of warm gas may be rare \nexcept in sources with the largest accretion rates, i.e., \nthose best able to tap a large accretion energy budget and \nheat a large column density of the disk atmosphere. \nIn contrast, the CO fundamental transitions, with their much larger \nA-values, should be detectable \nin systems with more modest column densities of warm gas, \ni.e., in a broader range of sources.   \nThis is borne out in high resolution spectroscopic surveys for CO \nfundamental emission from TTS ({\\em Najita et al.}, 2003) \nand Herbig AeBe stars ({\\em Blake and Boogert}, 2004) \nwhich detect emission from \nessentially all sources with accretion rates typical of these \nclasses of objects.  \n\n\n\\begin{figure}[t!]\n\\vspace{-1.6in}\n\\plotfiddle{najita_fig2.eps}{1.50in}{0.}{240}{240}{5}{150}\n\\vspace{-0.3in}\n\\caption{\\small\\baselineskip 10pt\nGaseous inner disk radii for TTS from CO fundamental emission \n(filled squares) compared with corotation radii for the same sources. \nAlso shown are dust inner radii from near-infrared interferometry \n(filled circles; {\\em Akeson et al.}, 2005a,b) or spectral energy \ndistributions \n(open circles; {\\em Muzerolle et al.}, 2003).  \nThe solid and dashed lines indicate an inner radius equal to, \ntwice, and 1\/2 the corotation radius.\nThe points for the three stars with measured inner radii\nfor both the gas and dust are connected by dotted lines.\nGas is observed to extend inward of the dust inner radius \nand typically inward of the corotation radius.\n} \n\\end{figure}\n\nIn addition, the lower temperatures\nrequired to excite the CO $v$=1--0 transitions make these transitions \nsensitive to cooler gas at larger disk radii, beyond the \nregion probed by the \nCO overtone lines.  Indeed, the measured line profiles for the \nCO fundamental emission are broad (typically 50--100\\,${\\rm km}\\,{\\rm s}^{-1}$ FWHM) and \ncentrally peaked, in contrast to the CO overtone lines which are \ntypically double-peaked.  These velocity profiles suggest \nthat the CO fundamental emission arises from a wide range of radii, \nfrom $\\lesssim$0.1\\,AU out to 1--2\\,AU in disks \naround low mass stars, i.e., the terrestrial planet region of \nthe disk ({\\em Najita et al.}, 2003). \n\nCO fundamental emission spectra typically show symmetric emission \nlines from multiple vibrational states (e.g., $v$=1--0, 2--1, 3--2); \nlines of $^{13}$CO can also be detected when the emission is strong \nand optically thick. \nThe ability to study multiple vibrational states as well as \nisotopic species within a limited spectral range makes the CO \nfundamental lines an appealing choice to probe gas in the inner disk \nover a range of temperatures and column densities.  \nThe relative strengths of the lines \nalso provide insight into the excitation mechanism for the emission.\n\nIn one source, the Herbig AeBe star HD141569, the excitation\ntemperature of the rotational levels ($\\sim$200\\,K) is much lower\nthan the excitation temperature of the vibrational levels ($v$=6\nis populated), which is suggestive of UV pumping of cold gas \n({\\em Brittain et al.}, 2003).  \nThe emission lines from the source are narrow, indicating\nan origin at $\\gtrsim$17\\,AU.  The lack of fluorescent emission\nfrom smaller radii strongly suggests that the region within 17\\,AU \nis depleted of gaseous CO.  \nThus detailed models of the fluorescence process can be used\nto constrain the gas content in the inner disk region \n(S. Brittain, personal communication).\n\nThus far HD141569 appears to be an unusual case.  For the majority \nof sources from which CO fundamental is detected, the relative line \nstrengths are consistent with emission from thermally excited\ngas.  They indicate typical excitation temperatures of \n1000--1500\\,K and CO column densities of $\\sim$$10^{18}\\rm \\,cm^{-2}$  \nfor low mass stars. \nThese temperatures are much warmer than the dust temperatures at\nthe same radii implied by spectral energy distributions (SEDs) \nand the expectations of some disk atmosphere models\n(e.g., {\\em D'Alessio et al.}, 1998).  \nThe temperature difference can be \naccounted for by   \ndisk atmosphere models \nthat allow for the thermal decoupling of the\ngas and dust (Section 3.2).\n\n\nFor CTTS systems in which the inclination is known, we can convert \na measured HWZI velocity for the emission to an inner radius.  The CO inner \nradii, thus derived, are typically $\\sim$0.04\\,AU for TTS \n({\\em Najita et al.}, 2003; {\\em Carr et al.}, in preparation), \nsmaller than the inner radii that are measured for the \ndust component either through interferometry \n(e.g., {\\em Eisner et al.}, 2005; {\\em Akeson et al.}, 2005a; \n{\\em Colavita et al.}, 2003; see chapter by {\\em Millan-Gabet et al.})\\ or \nthrough the interpretation \nof SEDs (e.g., {\\em Muzerolle et al.}, 2003).\nThis shows that gaseous disks extend inward \nto smaller radii than dust disks, a result that is not \nsurprising given the relatively low sublimation temperature \nof dust grains ($\\sim$1500--2000 K) compared to the \nCO dissociation temperature ($\\sim$4000 K). \nThese results are consistent with the suggestion that \nthe inner radius of the dust disk is defined by \ndust sublimation rather than by physical truncation\n({\\em Muzerolle et al.}, 2003; {\\em Eisner et al.}, 2005).\n\nPerhaps more interestingly, the inner radius of the CO emission \nappears to extend up to and usually within the corotation \nradius (i.e., the radius at which the disk rotates at the same \nangular velocity as the star; Fig.~2).  In the current paradigm for \nTTS, a strong stellar magnetic field \ntruncates the disk near the corotation radius. \nThe coupling between the stellar magnetic field and the gaseous inner disk \nregulates the rotation of the star, bringing \nthe star into corotation with the disk at the coupling radius. \nFrom this region emerge both energetic (X-)winds and magnetospheric \naccretion flows (funnel flows; {\\em Shu et al.}, 1994).  \nThe velocity extent of the CO fundamental emission shows that gaseous\ncircumstellar disks indeed extend inward beyond the dust destruction\nradius to the corotation radius (and beyond), \nproviding the material that feeds both X-winds and funnel flows.\nSuch small coupling radii are consistent with the rotational rates \nof young stars.  \n\n\n\\begin{figure}[t!]\n\\vspace{-0.1in}\n\\plotfiddle{najita_fig3.eps}{0.05in}{0.}{180}{180}{20}{0}\n\\vspace{-0.1in}\n\\caption{\\small\\baselineskip 10pt\nThe distribution of gaseous inner radii, measured with \nthe CO fundamental transitions, compared to the \ndistribution of orbital radii of short-period\nextrasolar planets.\nA minimum planetary orbital radius of $\\sim$0.04\\,AU \nis similar to the minimum gaseous inner radius\ninferred from the CO emission line profiles.\n} \n\\end{figure}\n\n\n\nIt is also interesting to compare the distribution of inner\nradii for the CO emission with the orbital radii of\nthe ``close-in'' extrasolar giant planets (Fig.\\ 3).\nExtra-solar planets discovered by radial velocity surveys\nare known to pile-up near a minimum radius of 0.04 AU.\nThe similarity between these distributions is roughly consistent \nwith the idea that the truncation of the inner disk \ncan halt the inward orbital migration of a giant \nplanet ({\\em Lin et al.}, 1996).  In detail, however, the planet \nis expected to migrate slightly inward \nof the truncation radius, to the 2:1 resonance, \nan effect that is not seen in the present \ndata.  A possible caveat is that the wings of the CO lines may \nnot trace Keplerian motion or that the innermost gas is not \ndynamically significant.  It would be interesting to explore \nthis issue further since the results impact our understanding \nof planet formation and the origin of planetary architectures.  \nIn particular, the existence of a stopping mechanism implies \na lower efficiency for giant planet formation, e.g., compared \nto a scenario in which multiple generations of planets form and \nonly the last generation survives  \n(e.g., {\\em Trilling et al.}, 2002). \n\n\n\n\\bigskip\n\\noindent\n\\textbf{2.4 UV Transitions of Molecular Hydrogen}\n\\bigskip\n\nAmong the diagnostics of inner disk gas \ndeveloped since PPIV, perhaps the most interesting \nare those of \nH$_2$.\nH$_2$\\ is presumably the dominant gaseous species in disks, due\nto high elemental abundance, low depletion onto grains, and robustness\nagainst dissociation. Despite its expected ubiquity, H$_2$\\ is difficult\nto detect because permitted electronic transitions are in the far\nultraviolet (FUV) and accessible only from space. Optical and \nrovibrational IR\ntransitions have radiative rates that are 14 orders of magnitude\nsmaller.\n\n\n\\begin{figure}[t!]\n\\plotfiddle{najita_fig4.eps}{0.05in}{0.}{230}{230}{2}{0}\n\\vspace{-0.1in}\n\\caption{\\small\\baselineskip 10pt\nLy\\,$\\alpha$\\ emission from TW Hya, an accreting T Tauri star,\nand a reconstruction of the Ly\\,$\\alpha$\\ profile seen by the circumstellar H$_2$.\nEach observed H$_2$\\ progression (with a common excited state) yields a\nsingle point in the reconstructed Ly\\,$\\alpha$\\ profile. The wavelength of\neach point in the reconstructed Ly\\,$\\alpha$\\ profile corresponds to the\nwavelength of the upward transition that pumps the progression.\nThe required excitation energies for the H$_2$\\ \\textit{before} the\npumping is indicated in the inset energy level diagram. There are\nno low excitation states of H$_2$\\ with strong transitions that overlap\nLy\\,$\\alpha$. Thus, the H$_2$\\ must be very warm to be preconditioned for\npumping and subsequent fluorescence.} \n\\end{figure}\n\n\n\nConsidering only radiative transitions with spontaneous rates above\n10$^7$ s$^{-1}$, H$_2$\\ has about 9000 possible Lyman-band (B-X)\ntransitions from 850-1650 \\AA\\ and about 5000 possible Werner-band\n(C-X) transitions from 850-1300 \\AA\\ ({\\em Abgrall et al.}, 1993a,b). However,\nonly about 200 FUV transitions have actually been detected in spectra\nof accreting TTS.\nDetected H$_2$\\ emission lines in the FUV all originate from about two\ndozen radiatively pumped states, each more than 11 eV above ground.\nThese pumped states of H$_2$\\ are the only ones connected to the ground\nelectronic configuration by strong radiative transitions that overlap\nthe broad Ly\\,$\\alpha$\\ emission that is characteristic of accreting TTS (see\nFig.\\ 4). Evidently, absorption of broad Ly\\,$\\alpha$\\ emission pumps the H$_2$\\\nfluorescence.\nThe two dozen strong H$_2$\\ transitions that happen to overlap the broad\nLy\\,$\\alpha$\\ emission are all pumped out of high $v$ and\/or high $J$ states at\nleast 1 eV above ground (see inset in Fig.~4). This means some\nmechanism must excite H$_2$\\ in the ground electronic configuration,\nbefore Ly\\,$\\alpha$\\ pumping can be effective. If the excitation mechanism is\nthermal, then the gas must be roughly 10$^3$ K to obtain a significant\nH$_2$\\ population in excited states. \n\n\nH$_2$\\ emission is a ubiquitous feature of accreting TTS. Fluoresced H$_2$\\\nis detected in the spectra of 22 out of 24 accreting TTS observed in the\nFUV by HST\/STIS ({\\em Herczeg et al.}, 2002; {\\em Walter et al.}, 2003; \n{\\em Calvet et al.}, 2004; {\\em Bergin et al.}, 2004; \n{\\em Gizis et al.}, 2005; {\\em Herczeg et al.}, 2005; \nunpublished archival data). Similarly, H$_2$\\ is detected in\nall 8 accreting TTS observed by HST\/GHRS ({\\em Ardila et al.}, 2002) and all\n4 published FUSE spectra ({\\em Wilkinson et al.}, 2002; \n{\\em Herczeg et al.}, 2002; 2004; 2005). \nFluoresced H$_2$\\ was even detected in 13 out of 39\naccreting TTS observed by IUE, despite poor sensitivity \n({\\em Brown et al.}, 1981; {\\em Valenti et al.,} 2000).\nFluoresced H$_2$\\ has not been detected in FUV spectra of \nnon-accreting TTS, despite observations of 14 stars with STIS \n({\\em Calvet et al.}, 2004; unpublished archival data), \n1 star with GHRS ({\\em Ardila et al.}, 2002), and 19 stars with IUE \n({\\em Valenti et al.}, 2000). However,\nthe existing observations are not sensitive enough to prove that the\ncircumstellar H$_2$\\ column decreases contemporaneously with the\ndust continuum of the inner disk.\nWhen accretion onto the stellar surface stops, fluorescent pumping\nbecomes less efficient because the strength and breadth of Ly\\,$\\alpha$\\\ndecreases significantly and the H$_2$\\ excitation needed to prime the\npumping mechanism may become less efficient. COS, if installed on\nHST, will have the sensitivity to set interesting limits on H$_2$\\\naround non-accreting TTS in the TW Hya association.\n\n\nThe intrinsic Ly\\,$\\alpha$\\ profile of a TTS is not observable at Earth, except\npossibly in the far wings, due to absorption by neutral hydrogen along\nthe line of sight. However, observations of H$_2$\\ line fluxes constrain\nthe Ly\\,$\\alpha$\\ profile seen by the fluoresced H$_2$. The rate at which a\nparticular H$_2$\\ upward transition absorbs Ly\\,$\\alpha$\\ photons is equal to the\ntotal rate of observed downward transitions out of the pumped state,\ncorrected for missing lines, dissociation losses, and propagation\nlosses. If the total number of excited H$_2$\\ molecules before pumping is\nknown (e.g., by assuming a temperature), then the inferred\npumping rate yields a Ly\\,$\\alpha$\\ flux point at the wavelength of each\npumping transition (Fig.\\ 4).\n\n\n{\\em Herczeg et al.}\\ (2004) applied this type of analysis to TW Hya,\ntreating the circumstellar H$_2$\\ as an isothermal, self-absorbing slab.\nFig.~4 shows reconstructed Ly\\,$\\alpha$\\ flux points for the upward pumping\ntransitions, assuming the fluoresced H$_2$\\ is at 2500 K. The smoothness\nof the reconstructed Ly\\,$\\alpha$\\ flux points implies that the H$_2$\\ level\npopulations are consistent with thermal excitation. Assuming an H$_2$\\\ntemperature warmer or cooler by a few hundred degrees leads to\nunrealistic discontinuities in the reconstructed Ly\\,$\\alpha$\\ flux points.\nThe reconstructed Ly\\,$\\alpha$\\ profile has a narrow absorption component\nthat is blueshifted by $-90~ {\\rm km}\\,{\\rm s}^{-1}$, presumably due to an intervening\nflow.\n\n\nThe spatial morphology of fluoresced H$_2$\\ around TTS is diverse.\n{\\em Herczeg et al.} (2002) used STIS to observe TW Hya with 50 mas\nangular resolution, corresponding to a spatial resolution of 2.8 AU at\na distance of 56 pc, finding no evidence that the fluoresced H$_2$\\ is\nextended. At the other extreme, {\\em Walter et al.} (2003) detected\nfluoresced H$_2$\\ up to 9 arcsec from T Tau N, but only in progressions\npumped by H$_2$\\ transitions near the core of Ly\\,$\\alpha$.\nFluoresced H$_2$\\ lines have a main velocity component at or near the\nstellar radial velocity and perhaps a weaker component that is\nblueshifted by tens of ${\\rm km}\\,{\\rm s}^{-1}$ ({\\em Herczeg et al.}, 2006). These two\ncomponents are attributed to the disk and the outflow, respectively.\nTW Hya has H$_2$\\ lines with no net velocity shift, consistent with\nformation in the face-on disk ({\\em Herczeg et al.}, 2002). On the other\nhand, RU Lup has H$_2$\\ lines that are blueshifted by $12~ {\\rm km}\\,{\\rm s}^{-1}$, suggesting\nformation in an outflow. In both of these stars, absorption in\nthe blue wing of the \\ion{C}{2} 1335 \\AA\\ wind feature strongly\nattenuates H$_2$\\ lines that happen to overlap in wavelength, so in\neither case H$_2$\\ forms inside the warm atomic wind \n({\\em Herczeg et al.}, 2002; 2005).\n\n\nThe velocity widths of fluoresced H$_2$\\ lines (after removing\ninstrumental broadening) range from $18~ {\\rm km}\\,{\\rm s}^{-1}$ to $28~ {\\rm km}\\,{\\rm s}^{-1}$ for the 7\naccreting TTS observed at high spectral resolution with STIS \n({\\em Herczeg et al.}, 2006). \nLine width does not correlate well with inclination.\nFor example, TW Hya (nearly face-on disk) and DF Tau (nearly edge-on\ndisk) both have line widths of $18~ {\\rm km}\\,{\\rm s}^{-1}$. Thermal broadening is\nnegligible, even at 2000 K. Keplerian motion, enforced corotation, and\noutflow may all contribute to H$_2$\\ line width in different systems.\nMore data are needed to understand how velocity widths (and shifts)\ndepend on disk inclination, accretion rate, and other factors.\n\n\n\n\\bigskip\n\\noindent\n\\textbf{2.5 Infrared Transitions of Molecular Hydrogen} \n\\bigskip\n\nTransitions of molecular hydrogen have also been studied at longer\nwavelengths, in the near- and mid-infrared.  \nThe $v$=1--0 S(1) transition of H$_2$ (at $2\\mu$m) has been detected in\nemission in a small sample of classical T Tauri stars (CTTS) and one weak\nT Tauri star (WTTS; {\\em Bary et al.}, 2003 and references therein). \nThe narrow emission lines ($\\lesssim$$10{\\rm km}\\,{\\rm s}^{-1}$), if arising in a disk, \nindicate an origin at\nlarge radii, probably beyond 10\\,AU.  The high temperatures\nrequired to excite these transitions thermally (1000s\\,K), in contrast\nto the low temperatures expected for the outer disk, suggest that\nthe emission is non-thermally excited, possibly by X-rays ({\\em Bary et\nal.}, 2003).  \nThe measurement of other rovibrational transitions of H$_2$ is needed\nto confirm this.\n\nThe gas mass detectable by this technique depends\non the depth to which the exciting radiation can penetrate the disk.\nThus, the emission strength may be limited\neither by the strength of the radiation field, if the gas column\ndensity is high, or by the mass of gas present, if the gas column\ndensity is low.\nWhile it is therefore difficult to measure total gas masses with \nthis approach, clearly non-thermal processes can light \nup cold gas, making it easier to detect.  \n\nEmission from a WTTS is surprising\nsince WTTS are thought to be largely devoid of circumstellar \ndust and gas, given \nthe lack of infrared excesses and the low accretion rates for these\nsystems.  The Bary et al.\\ results call this\nassumption into question and suggest that longer lived gaseous\nreservoirs may be present in systems with low accretion rates.  We\nreturn to this issue in Section 4.1.\n\nAt longer wavelengths, the pure rotational transitions of H$_2$ \nare of considerable interest because molecular hydrogen\ncarries most of the mass of the disk, and these mid-infrared\ntransitions are capable of probing the $\\sim$100 K temperatures\nthat are expected for the giant planet region of the disk.\nThese transitions present both advantages\nand challenges as probes of gaseous disks.\nOn the one hand, their small A-values make them sensitive, in\nprinciple, to very large gas masses (i.e., the transitions do not\nbecome optically thick until large gas column densities \n$N_H$=$10^{23}-10^{24}\\rm \\,cm^{-2}$ \nare reached).  On the other hand, the small A-values also \nimply \nsmall critical densities, which allows the\npossibility of contaminating emission from gas at lower densities\nnot associated with the disk,\nincluding shocks in outflows and UV excitation of ambient gas.\n\nIn considering the detectability of H$_2$ emission from gaseous \ndisks mixed with dust, one issue is that the dust \ncontinuum can become optically thick over column densities \n$N_H \\ll 10^{23}-10^{24}\\rm \\,cm^{-2}$.  \nTherefore, in a disk that is optically thick in the continuum \n(i.e., in CTTS), H$_2$ emission may probe smaller \ncolumn densities. \nIn this case, the line-to-continuum contrast may be low unless there  \nis a strong temperature inversion in the disk atmosphere, and \nhigh signal-to-noise observations may be required to detect the \nemission. \nIn comparison, in disk systems that are optically thin \nin the continuum (e.g., WTTS), H$_2$ could \nbe a powerful probe as long as there are sufficient heating \nmechanisms (e.g., beyond gas-grain coupling) to heat the H$_2$.\n\nA thought-provoking result from ISO was the report of approximately\nJupiter-masses of warm gas residing in $\\sim$20\\,Myr old debris\ndisk systems ({\\em Thi et al.}, 2001) based on the detection of the \n28~$\\mu$m and 17~$\\mu$m lines of H$_2$.  This result was surprising \nbecause of the advanced age of the sources in which the emission \nwas detected; gaseous reservoirs are expected to dissipate on \nmuch shorter timescales (Section 4.1).  \nThis intriguing result is, thus far, unconfirmed by either \nground-based studies ({\\em Richter et al.}, 2002; \n{\\em Sheret et al.}, 2003; \n{\\em Sako et al.}, 2005) or studies with Spitzer \n(e.g., {\\em Chen et al.}, 2004).  \n\n\nNevertheless, ground-based studies have detected pure rotational \nH$_2$ emission from some sources.  Detections to date include AB Aur \n({\\em Richter et al.}, 2002).  \nThe narrow width of the emission in AB Aur ($\\sim$$10~{\\rm km}\\,{\\rm s}^{-1}$ FWHM), \nif arising in a disk, locates the emission \nbeyond the giant planet region. \nThus, an important future direction for these studies is to search \nfor H$_2$ emission in a larger number of sources and \nat higher velocities, in the giant planet region of the disk.  \nHigh resolution mid-IR spectrographs on $>$3-m telescopes \nwill provide the greater sensitivity needed for such studies.\n\n\n\\bigskip\n\\noindent\n\\textbf{2.6 Potential Disk Diagnostics} \n\\bigskip\n\nIn a recent development, {\\em Acke et al.}\\ (2005) have reported\nhigh resolution spectroscopy of the [OI]\\,6300\\,\\AA\\ line in \nHerbig AeBe stars.  The majority of the sources show a\nnarrow ($<$$50~{\\rm km}\\,{\\rm s}^{-1}$ FWHM), fairly symmetric emission component\ncentered at the radial velocity of the star.  In some cases,\ndouble-peaked lines are detected.  These features are interpreted\nas arising in a surface layer of the disk that is irradiated by the star. \nUV photons incident on the disk surface are thought to photodissociate \nOH and H$_2$O, producing a non-thermal population of excited \nneutral oxygen that decays radiatively, producing the observed emission lines. \nFractional OH abundances of $\\sim$$10^{-7}-10^{-6}$ are needed \nto account for the observed line luminosities.\n\nAnother recent development is the report of strong absorption in the \nrovibrational bands of C$_2$H$_2$, HCN, and CO$_2$ in the 13--15~$\\mu$m \nspectrum of a low-mass class I source in Ophiuchus, IRS\\,46  \n({\\em Lahuis et al.}, 2006).  The high excitation temperature of \nthe absorbing gas (400-900\\,K) suggests an origin close\nto the star, an interpretation that is consistent with millimeter \nobservations of HCN which indicate a source size $\\ll$100\\,AU.\nSurprisingly, high dispersion observations of rovibrational CO\n(4.7~$\\mu$m) and HCN (3.0~$\\mu$m) show that the molecular absorption \nis {\\it blueshifted} relative to the molecular cloud.  If IRS\\,46\nis similarly blueshifted relative to the cloud, the absorption may\narise in the atmosphere of a nearly edge-on disk.  A disk origin \nfor the absorption is consistent with the observed relative abundances \nof C$_2$H$_2$, HCN, and CO$_2$ ($10^{-6}$--$10^{-5}$), which are\nclose to those predicted by {\\em Markwick et al.}\\ (2002) for\nthe inner region of gaseous disks ($\\lesssim$2\\,AU; see Section 3).\nAlternatively, if IRS\\,46 has approximately the same velocity as\nthe cloud, then the absorbing gas is blueshifted with respect to the star \nand the absorption may arise in an outflowing wind.  \nWinds launched from the disk, at AU distances, may have \nmolecular abundances similar to those observed if the chemical \nproperties of the wind change slowly as the wind is launched.  \nDetailed calculations of the chemistry of disk winds are needed to \nexplore this possibility.  The molecular abundances in the \ninner disk midplane (Section 3.3) provide the initial conditions for such \nstudies. \n\n\n\n\n\\section{\\textbf{THERMAL-CHEMICAL MODELING}}\n\n\\bigskip\n\\noindent\n\\textbf{3.1 General Discussion}\n\\bigskip\n\nThe results discussed in the previous section illustrate the growing\npotential for observations to probe gaseous inner disks.  While,\nas already indicated, some conclusions can be drawn directly from\nthe data coupled with simple spectral synthesis modeling, harnessing\nthe full diagnostic potential of the observations will likely rely\non detailed models of the thermal-chemical structure \n(and dynamics) of disks.  Fortunately, the development of \nsuch \nmodels has been an active area of recent research.\nAlthough much of the effort has been devoted to understanding the\nouter regions of disks ($\\sim$100\\,AU; e.g., {\\em Langer et al.}, 2000;\nchapters by {\\em Bergin et al.}\\ and {\\em Dullemond et al.}), recent \nwork has begun to focus on the region within 10\\,AU.\n\nBecause disks are intrinsically complex structures, the models\ninclude a wide array of processes.  These encompass heating sources\nsuch as stellar irradiation (including far UV and X-rays) and viscous\naccretion; chemical processes such as photochemistry and grain\nsurface reactions; and mass transport via magnetocentrifugal winds,\nsurface evaporative flows, turbulent mixing, and accretion onto the\nstar.  The basic goal of the models is to calculate the density,\ntemperature, and chemical abundance structures that result from\nthese processes.  Ideally, the calculation would be fully\nself-consistent, although approximations are made to simplify the\nproblem.\n\nA common simplification is to adopt a specified density distribution \nand then solve the rate equations that define the chemical model.  \nThis is justifiable where the thermal and chemical\ntimescales are short compared to the dynamical timescale.\nA popular choice is the $\\alpha$-disk model ({\\em Shakura and Sunyaev}, 1973; \n{\\em Lynden-Bell and Pringle}, 1974) in which a \nphenomenological parameter $\\alpha$ characterizes the efficiency of \nangular momentum transport; its vertically averaged \nvalue is estimated to be $\\sim$$10^{-2}$ for \nT Tauri disks on the basis of\nmeasured accretion rates ({\\em Hartmann et al.}, 1998). \nBoth vertically isothermal $\\alpha$-disk models  \nand the Hayashi minimum mass solar nebula \n(e.g., {\\em Aikawa et al.}, 1999) \nwere adopted in early studies. \n\nAn improved method removes the assumption of vertical \nisothermality and calculates the vertical thermal structure \nof the disk \nincluding viscous accretion heating at the midplane \n(specified by $\\alpha$) \nand stellar radiative heating\nunder the assumption that the gas and dust temperatures are the\nsame ({\\em Calvet et al.}, 1991; {\\em D'Alessio et al.}, 1999).\nSeveral chemical models have been built using the D'Alessio density\ndistribution (e.g., {\\em Aikawa and Herbst}, 1999; GNI04; \n{\\em Jonkheid et al.}, 2004).\n\nStarting about\n2001, theoretical models showed that the gas temperature can become\nmuch larger than the dust temperature in the \natmospheres of outer \n({\\em Kamp and van Zadelhoff}, 2001) \nand inner ({\\em Glassgold and Najita}, 2001) disks.\nThis suggested the need to treat the gas and dust as two independent\nbut coupled thermodynamic systems. \nAs an example of this approach, {\\em Gorti and Hollenbach} (2004) \nhave iteratively solved a system of chemical rate equations \nalong with the equations of hydrostatic equilibrium\nand thermal balance for both the gas and the dust. \n\nThe chemical models developed so far \nare characterized by diversity as well as uncertainty.  \nThere is diversity in \nthe adopted density distribution and external radiation field \n(UV, X-rays, and cosmic rays; the relative importance of these \ndepends on the evolutionary stage) and in \nthe thermal and chemical processes considered.\nThe relevant heating processes are less well understood than \nline cooling. \nOne issue is how UV, X-ray, and cosmic rays heat the gas.\nAnother is the role of mechanical heating associated\nwith various flows in the disk, especially accretion \n(GNI04). \nThe chemical processes are also less certain. \nOur understanding of astrochemistry is based mainly\non the interstellar medium, where densities and temperatures\nare low compared to those of inner disks,\nexcept perhaps in shocks and photon-dominated regions. \nNew reaction pathways or \nprocesses may be important at the higher densities \n($> 10^{7}\\rm \\,cm^{-3}$)\nand higher radiation fields of inner disks.  \nA basic challenge is to understand the thermal-chemical \nrole of dust grains and PAHs.  Indeed, perhaps the most significant \ndifference between models is the treatment of grain chemistry. \nThe more sophisticated models include adsorption of gas onto \ngrains in cold regions and desorption in warm  regions. \nYet another level of complexity is introduced \nby transport processes which can affect the\nchemistry through vertical or radial mixing. \n\nAn important practical issue in thermal-chemical modeling is that \nself-consistent calculations become increasingly\ndifficult as the density, temperature, and the number of species\nincrease. Almost all models employ truncated chemistries with \nwith somewhere from 25 to 215 species, compared with 396 in the UMIST data\nbase ({\\em Le Teuff et al.}, 2000). The truncation\nprocess is arbitrary, determined largely by the goals of the\ncalculations. {\\em Wiebe et al.}, (2003) have an\nobjective method for selecting the most important reactions from\nlarge data bases.  \nAdditional insights \ninto disk chemistry are offered in the chapter by {\\em Bergin et al.}\n\n\\bigskip\n\\noindent\n\\textbf{3.2 The Disk Atmosphere}\n\\bigskip\n\nAs noted above, {\\em Kamp and van Zadelhoff} (2001) concluded \nin their model of debris disks that \nthe gas and dust temperature can differ, \nas did {\\em Glassgold and Najita} (2001) for T Tauri disks. The\nformer authors developed a comprehensive thermal-chemical model where\nthe heating is primarily from the dissipation of the drift velocity\nof the dust through the gas. For T Tauri disks, stellar X-rays,\nknown to be a universal property of low-mass YSOs,\nheat the gas to temperatures \nthousands of degrees  \nhotter than the dust temperature.\n\n\n\\begin{figure}[t!]\n\\vspace{-0.2in}\n\\plotfiddle{najita_fig5.ps}{0.05in}{0.}{220}{220}{10}{0}\n\\vspace{-0.3in}\n\\caption{\\small\\baselineskip 10pt\nTemperature profiles from GNI04 for a protoplanetary disk atmosphere. \nThe lower solid line shows the dust \ntemperature of {\\em D'Alessio et al.} (1999) \nat a radius of 1\\,AU and a mass accretion rate of \n$10^{-8}M_\\odot$ yr$^{-1}$.  \nThe upper curves show the corresponding gas temperature as a function of the \nphenomenological mechanical heating parameter defined by\nEquation 1, $\\alpha_h$ = 1 (solid line), 0.1 (dotted line), and\n0.01 (dashed line).  The $\\alpha_h = 0.01$ curve closely follows\nthe limiting case of pure X-ray heating.  The lower vertical lines\nindicate the major chemical transitions, specifically CO forming\nat $\\sim 10^{21} {\\rm cm}^{-2}$, H$_2$ forming\nat $\\sim 6\\times 10^{21} {\\rm cm}^{-2}$, and\nwater forming at higher column densities.} \n\\end{figure}\n\nFig.~5 shows the vertical temperature profile obtained by {\\em\nGlassgold et al.}\\ (2004) with a thermal-chemical model\nbased on the dust model of {\\em D'Alessio et al.}\\ (1999) for a generic\nT Tauri disk. Near the midplane, the densities are high enough\nto strongly couple the dust and gas.  \nAt higher altitudes,\nthe disk becomes optically thin to the stellar optical\nand infrared radiation, and the temperature of the (small) grains\nrises, as does the still closely-coupled gas temperature. However,\nat still higher altitudes, the gas responds strongly to the \nless attenuated X-ray flux, and its temperature\nrises much above the dust temperature. \nThe presence of a hot X-ray heated layer above a cold midplane\nlayer was obtained independently by {\\em Alexander et al.} (2004).\n\nGNI04 also considered \nthe possibility that the surface layers of protoplanetary disks \nare heated by the dissipation of mechanical energy. \nThis might arise through \nthe interaction of a wind with the upper layers of the disk \nor through disk angular momentum transport. \nSince the theoretical understanding of such processes is incomplete, \na phenomenological\ntreatment is required. \nIn the case of angular momentum transport, \nthe most widely accepted mechanism\nis the MRI ({\\em Balbus and Hawley}, 1991; \n{\\em Stone et al.}, 2000), which leads to the local\nheating formula,\n\\begin{equation}\n\\label{accheat}\n\\Gamma_{\\rm acc} = \\frac{9}{4} \\alpha_h \\rho c^2 \\Omega, \n\\end{equation} \nwhere $\\rho$ is the mass density, $c$ is the isothermal sound speed,\n$\\Omega$ is the angular rotation speed, and $\\alpha_h$ is a\nphenomenological parameter \nthat depends on how the turbulence dissipates.\nOne can argue, on the basis of simulations by {\\em\nMiller and Stone} (2000), that midplane turbulence generates Alfv\\'en\nwaves which, on reaching the diffuse surface regions, produce shocks\nand heating. Wind-disk heating can be represented by a similar\nexpression on the basis of dimensional arguments.\nEquation 1 is essentially an adaptation of the expression for \nvolumetric heating in an $\\alpha$-disk model, where $\\alpha$ can in \ngeneral depend on position. GNI04 used the notation $\\alpha_h$ \nto distinguish its value in the disk atmosphere from the usual \nmidplane value. \n\n\nIn the top layers fully exposed to X-rays, the gas temperature\nat 1\\,AU is $\\sim$5000\\,K. Further down, there is a \nwarm transition region (500--2000\\,K) composed mainly of atomic \nhydrogen but with carbon fully associated into CO. \nThe conversion from atomic H to H$_2$ is reached at a column \ndensity of $\\sim$$6\\times 10^{21}\\rm \\,cm^{-2}$, with more complex \nmolecules such as water forming deeper in the disk. \nThe location and thickness of the warm molecular region \ndepends on the strength of the surface heating. The curves \nin Fig.~5 illustrate this dependence for a T Tauri disk at\n$r=1$\\,AU.  With $\\alpha_h = 0.01$, X-ray heating dominates this\nregion, whereas with $\\alpha_h > 0.1$, mechanical heating dominates.\n\nGas temperature inversions can also be produced by UV radiation\noperating on small dust grains and PAHs, as demonstrated by\nthe thermal-chemical models of \n{\\em Jonkheid et al.} (2004) \nand {\\em Kamp and Dullemond} (2004). {\\em\nJonkheid et al.}\\ use the {\\em D'Alessio et al.}\\ (1999) model and\nfocus on the disk beyond 50\\,AU. At this radius, the gas temperature\ncan rise to 800\\,K or 200\\,K, depending on whether small grains\nare well mixed or settled. For a thin disk and a high stellar UV\nflux, {\\em Kamp and Dullemond} obtain temperatures that asymptote\nto several 1000\\,K inside 50\\,AU. \nOf course these results are\nsubject to various assumptions that have been made about the stellar\nUV, the abundance of PAHs, and the growth and settling of dust\ngrains.\n\n\nMany of the earlier chemical models, oriented towards outer\ndisks (e.g., {\\em Willacy and Langer}, 2000; {\\em Aikawa\nand Herbst}, 1999; 2001; {\\em Markwick et al.}, 2002), adopt a value\nfor the {\\it stellar} UV radiation field that is $10^4$ times larger\nthan Galactic \nat a distance of 100\\,AU. This choice can be traced\nback to early IUE measurements of the stellar UV beyond 1400\\,\\AA\\, \nfor several TTS ({\\em Herbig and Goodrich}, 1986).  Although\nthe UV flux from TTS covers a range of values and is\nundoubtedly time-variable, detailed studies with IUE\n(e.g., {\\em Valenti et al.}, 2000; {\\em Johns-Krull et al.}, 2000)\nand FUSE (e.g., {\\em Wilkinson et al.}, 2002; {\\em Bergin et al.}, \n2003) indicate that it decreases into the FUV domain with\na typical value $\\sim$$10^{-15} {\\rm erg} \\, {\\rm cm}^{-2} {\\rm s}^{-1}$\\,\\AA $^{-1}$,\nmuch smaller than earlier estimates.\nA flux of $\\sim$$10^{-15}{\\rm erg} \\, {\\rm cm}^{-2} {\\rm s}^{-1}$\\,\\AA $^{-1}$ at Earth\ntranslates into a value at 100\\,AU of $\\sim$100 times the traditional \nHabing value for the interstellar medium.\nThe data in the FUV range are sparse, unfortunately, as a function \nof age or the evolutionary state of the system.  More measurements of \nthis kind are needed since \nit is obviously important to use realistic fluxes in the crucial FUV band\nbetween 912 and 1100\\,\\AA\\ where atomic C can be photoionized and\nH$_2$ and CO photodissociated \n({\\em Bergin et al.}, 2003 and the chapter by {\\em Bergin et al.}). \n\nWhether stellar FUV or X-ray radiation dominates the ionization, \nchemistry, and heating of protoplanetary disks is important because of \nthe vast difference in photon energy. \nThe most direct physical consequence\nis that FUV photons cannot ionize H, and thus the abundance of\ncarbon provides an upper limit to the ionization level produced by\nthe photoionization of heavy atoms, \n$x_e\\sim$$10^{-4}$--$10^{-3}$. Next,\nFUV photons are absorbed much more readily than X-rays, although\nthis depends on the size and spatial distribution of the dust grains,\ni.e, on grain growth and sedimentation. Using realistic numbers for\nthe FUV and X-ray luminosities of TTS, we estimate that\n$L_{\\rm FUV} \\sim L_{\\rm X}$. The rates used in many  early chemical\nmodels correspond to $L_{\\rm X} \\ll L_{\\rm FUV} $.\nThis suggests that future chemical modeling of protoplanetary disks \nshould consider both X-rays and FUV in their treatment of ionization,\nheating, and chemistry.\n\n\\bigskip\n\\noindent\n\\textbf{3.3 The Midplane Region}\n\\bigskip\n\nUnlike the warm upper atmosphere of the disk, which is accessible\nto observation,  \nthe optically thick midplane is much more difficult to\nstudy. Nonetheless, it is extremely important for understanding the dynamics\nof the basic flows in star formation such as accretion and outflow. \nThe important role of the\nionization level for disk accretion via the MRI was pointed out by\n{\\em Gammie} (1996). The physical reason is that collisional coupling\nbetween electrons and neutrals is required to transfer the turbulence\nin the magnetic field to the neutral material of the disk. Gammie\nfound that Galactic cosmic rays cannot penetrate beyond a \nsurface layer of the disk. He suggested that accretion only occurs\nin the surface of the inner disk (the ``active region'') and not in\nthe much thicker midplane region (the ``dead zone'') where the\nionization level is too small to mediate the MRI.  \n\n{\\em Glassgold et al.}\\ (1997) argued that the Galactic cosmic rays never\nreach the inner disk because they are blown away by the stellar\nwind, much as the solar wind excludes Galactic cosmic rays.  They\nshowed that YSO X-rays do almost as good a job as cosmic rays in\nionizing surface regions, thus preserving the layered accretion\nmodel of the MRI for YSOs. {\\em Igea and Glassgold} (1999) supported\nthis conclusion with a Monte Carlo calculation of X-ray transport\nthrough disks, demonstrating that scattering plays an important\nrole in the MRI by extending the active surface layer to column\ndensities greater than $10^{25} \\, {\\rm cm}^{-2}$, approaching the\nGalactic cosmic ray range used by {\\em Gammie} (1996).\nThis early work showed that the theory of disk ionization and\nchemistry is crucial for understanding the role of the MRI for YSO\ndisk accretion and possibly for planet formation. Indeed, {\\em Glassgold,\nNajita, and Igea} suggested that Gammie's dead zone might provide\na good environment for the formation of planets. \n\nThese challenges\nhave been taken up by several groups (e.g., {\\em Sano et al.}, 2000;\n{\\em Fromang et al.}, 2002; {\\em Semenov et al.}, \n2004; {\\em Kunz and Balbus}, 2004; {\\em Desch}, 2004; {\\em\nMatsumura and Pudritz}, 2003, 2005; and {\\em Ilgner and Nelson}, \n2006a,b). {\\em Fromang et al.}\\ discussed many of the issues that\naffect the size of the dead zone: differences in the disk model,\nsuch as a Hayashi disk or a standard $\\alpha$-disk; temporal evolution\nof the disk; the role of a small abundance of heavy atoms that\nrecombine mainly radiatively; and the value of the magnetic Reynolds\nnumber. {\\em Sano et al.} (2000) explored the role played by small dust\ngrains in reducing the electron fraction when it becomes as small\nas the abundance of dust grains. They showed that the dead zone \ndecreases and eventually vanishes as the grain size increases or\nas sedimentation towards the midplane proceeds.\nMore recently, {\\em Inutsuka and Sano} (2005) have suggested that\na small fraction of the energy dissipated by the MRI leads to the\nproduction of fast electrons with energies sufficient to ionize H$_2$. \nWhen coupled with vertical mixing of highly ionized surface regions,\nInutsuka and Sano argue that the MRI can self generate the ionization \nit needs to be operative throughout the entire disk.\n\n\nRecent chemical modeling ({\\em Semenov et al.}, 2004;\n{\\em Ilgner and Nelson}, 2006a,b) confirms that the level of ionization\nin the midplane is affected by many microphysical\nprocesses. These include the abundances of radiatively-recombining atomic\nions, molecular ions, small grains, and PAHs. The proper treatment of the\nions represents a great challenge for disk chemistry, one made\nparticularly difficult by the lack of observations of the dense gas at the\nmidplane of the inner disk. Thus the uncertainties in inner disk\nchemistry preclude definitive quantitative conclusions about the\nmidplane ionization of  protoplanetary disks. Perhaps the biggest\nwild card is the issue of grain growth, emphasized anew by {\\em\nSemenov et al.}, (2004). If the disk grain size\ndistribution were close to interstellar, then the small grains would\nbe effective in reducing the electron fraction and producing dead\nzones. But significant grain growth is expected {\\em and} observed\nin the disks of YSOs, limiting the extent of dead zones (e.g.,\n{\\em Sano et al.}, 2002).\n\n\nThe broader chemical properties of the {\\it inner} midplane region are\nalso of great interest since most of the gas in the disk is within\none or two scale heights.  \nThe chemical composition\nof the inner midplane gas is important because it provides the\ninitial conditions for outflows and for the formation of planets\nand other small bodies; it also determines whether the MRI operates. \nRelatively little work has been done on the midplane\nchemistry of the inner disk. For example, GNI04 excluded N and S\nspecies and restricted the carbon chemistry to species closely\nrelated to CO. However, {\\em Willacy et al.}\\ (1998), {\\em Markwick\net al.}\\ (2002), and {\\em Ilgner et al.}\\ (2004) have carried out\ninteresting calculations that shed light on a possible rich organic\nchemistry in the inner disk.\n\nUsing essentially the same chemical model, these authors follow\nmass elements in time as they travel in a steady accretion\nflow towards the star. At large distances, the gas is subject to\nadsorption, and at small distances to thermal desorption. In between\nit reacts on the surface of the dust grains; on being liberated\nfrom the dust, it is processed by gas phase chemical reactions. The\ngas and dust are assumed to have the same temperature, and all\neffects of stellar radiation are ignored. The ionizing sources are\ncosmic rays and $^{26}$Al. Since the collisional ionization of low \nionization potential atoms is ignored, a very low ionization level \nresults. {\\em Markwick et al.}\\ improve on {\\em Willacy et al.}\\ by\ncalculating the vertical variation of the temperature, and {\\em\nIlgner et al.}\\ consider the effects of mixing. Near 1\\,AU, H$_2$O\nand CO are very abundant, as predicted by simpler models, but {\\em\nMarkwick et al.}\\ find that CH$_4$ and CO have roughly equal abundances.\nNitrogen-bearing molecules, such as NH$_3$, HCN, and HNC are also predicted\nto be abundant, as are a variety of hydrocarbons such as CH$_4$, C$_2$H$_2$,\nC$_2$H$_3$, C$_2$H$_4$, etc. {\\em Markwick et al.}\\ also simulate\nthe presence of penetrating X-rays and find increased column densities\nof CN and HCO$^+$. \nDespite many uncertainties, these predictions \nare of interest for our future understanding of the midplane region.\n\nInfrared spectroscopic searches for hydrocarbons in disks may be \nable to test these predictions.  \nFor example, {\\em Gibb et al.}\\ (2004) searched for CH$_4$ in absorption \ntoward HL Tau.  The upper limit on the abundance of \nCH$_4$ relative to CO ($<$1\\%) in the absorbing gas \nmay contradict the predictions of {\\em Markwick et al.}\\ (2002) \nif the absorption arises in the disk atmosphere. \nHowever, some support for the {\\em Markwick et al.}\\ (2002) model comes \nfrom a recent report by {\\em Lahuis et al.}\\ (2006)\nof a rare detection by {\\it Spitzer} of C$_2$H$_2$ and HCN in\nabsorption towards a YSO, with ratios close to\nthose predicted for the inner disk (Section 2.6). \n\n\\bigskip\n\\noindent\n\\textbf{3.4 Modeling Implications}\n\\bigskip\n\nAn interesting implication of the irradiated disk atmosphere models\ndiscussed above is that the region\nof the atmosphere over which the gas and dust temperatures differ \nincludes the region that is accessible\nto observational study.  Indeed, the models have interesting \nimplications for some of the observations presented in Section 2. \nThey can account roughly \nfor the unexpectedly warm gas temperatures that have been found for\nthe inner disk region based on the CO fundamental (Section 2.3) and UV\nfluorescent H$_2$ transitions (Section 2.4).  \nIn essence, the warm gas temperatures arise from the direct heating \nof the gaseous component and the poor thermal coupling between the \ngas and dust components at the low densities characteristic \nof upper disk atmospheres.  \nThe role of X-rays in heating disk atmospheres has some support from  \nthe results of {\\em Bergin et al.}\\ (2004); they suggested that some of \nthe UV H$_2$ emission from TTS arises \nfrom excitation by fast electrons produced by X-rays. \n\nIn the models, CO is found to form at a column density \n$N_H$$\\simeq$$10^{21}\\rm \\,cm^{-2}$ \nand temperature $\\sim$1000\\,K in the radial range 0.5--2\\,AU \n(GNI04; Fig.~5), \nconditions similar to those deduced for the emitting gas \nfrom the CO fundamental lines ({\\em Najita et al.}, 2003). \nMoreover, CO is abundant in a region \nof the disk that is predominantly atomic hydrogen, a situation \nthat is favorable for exciting the rovibrational transitions \nbecause of the large collisional excitation cross section \nfor H + CO inelastic scattering.\nInterestingly, X-ray irradiation alone is probably \ninsufficient to explain the strength of the CO emission \nobserved in actively-accreting TTS.  \nThis suggests that \nother processes may be important in heating disk atmospheres. \nGNI04 have explored the role of mechanical \nheating.  Other possible heating processes are FUV irradiation  \nof grains and or PAHs. \n\nMolecular hydrogen column densities comparable to the UV \nfluorescent column of $\\sim$$5\\times 10^{18}{\\rm cm}^{-2}$ observed \nfrom TW Hya are reached at 1\\,AU \nat a total vertical hydrogen column density\nof $\\sim$$5\\times 10^{21}{\\rm cm}^{-2}$, where the fractional abundance \nof H$_2$ is $\\sim$$10^{-3}$ (GNI04; Fig.~5). \nSince Ly$\\alpha$ photons must traverse the entire \n$\\sim$$5\\times 10^{21}{\\rm cm}^{-2}$ in order to excite the emission, \nthe line-of-sight dust opacity through this column must be relatively low.  \nObservations of this kind, when combined with atmosphere models, \nmay be able to constrain the gas-to-dust ratio in disk atmospheres, \nwith consequent implications for grain growth and settling. \n\nWork in this direction has been carried out by {\\em Nomura and Millar} \n(2005).  They have made a detailed thermal model of a\ndisk that includes the formation of H$_2$ on grains, destruction via FUV\nlines, and excitation by Ly$\\alpha$ photons. The gas at the surface is\nheated primarily by the photoelectric effect on dust grains and PAHs,\nwith a dust model \nappropriate for interstellar clouds, i.e., \none that reflects little grain growth. \nBoth interstellar and stellar UV radiation\nare included, the latter based on observations of TW Hya.  The gas\ntemperature at the surface of their flaring disk model reaches 1500\\,K\nat 1\\,AU. They are partially successful in accounting for the\nmeasurements of {\\em Herczeg et al.}\\ (2002), but their model\nfluxes fall short by a factor of five or so.  A likely defect in\ntheir model is that the calculated temperature of the disk surface is\ntoo low, a problem that might be remedied by reducing the UV attenuation \nby dust and by including X-ray or other surface heating processes.\n\nThe relative molecular abundances that are predicted by \nthese non-turbulent, layered  \nmodel atmospheres are also of interest.  At a distance of 1\\,AU, the\ncalculations of GNI04 indicate that the relative abundance of \nH$_2$O to CO is $\\sim$$10^{-2}$ in the disk atmosphere for\ncolumn densities $<$$10^{22}\\rm \\,cm^{-2}$; \nonly at column densities $>$$10^{23}\\rm \\,cm^{-2}$ are H$_2$O and CO \ncomparably abundant.  \nThe abundance ratio in the atmosphere is significantly lower than \nthe few relative abundances measurements to date (0.1--0.5) \nat $<$0.3\\,AU \n({\\em Carr et al.}, 2004; Section 2.2). \nPerhaps layered\nmodel atmospheres, when extended to these \nsmall radii, will be able to account for the abundant water that is detected. \nIf not, the large water abundance may be evidence of strong vertical \n(turbulent) mixing that carries abundant water from deeper in the \ndisk up to the surface. \nThus, it would be of great interest to develop the modeling for the \nsources and regions where water is observed in the context of \nboth layered models and those with vertical mixing. \nWork in this direction has the potential to place unique constraints \non the dynamical state of the disk. \n\n\n\n\\section{\\textbf{CURRENT AND FUTURE DIRECTIONS}}\nAs described in the previous sections, significant progress has \nbeen made in developing both observational probes of gaseous \ninner disks as well as the theoretical models that are needed to \ninterpret the observations.  In this section, we describe some  \nareas of current interest as well as future directions \nfor studies of gaseous inner disks. \n\n\n\\bigskip\n\\noindent\n\\textbf{4.1 Gas Dissipation Timescale} \n\\bigskip\n\nThe lifetime of gas in the inner disk is of interest in the context \nof both giant and terrestrial planet formation. \nSince significant gas must be present in the disk in order for a \ngas giant to form, the gas dissipation timescale in the giant \nplanet region of the disk can help to identify dominant pathways \nfor the formation of giant planets.  A short dissipation time scale \nfavors processes such as gravitational instabilities which can \nform giant planets on short time scales ($< 1000$ yr; {\\em Boss}, 1997; \n{\\em Mayer et al.}, 2002). \nA longer dissipation time scale accommodates the more leisurely \nformation of planets in the core accretion scenario (few--10\\,Myr; \n{\\em Bodenheimer and Lin}, 2002). \n\n\nSimilarly, \nthe outcome of terrestrial \nplanet formation (the masses and eccentricities of the planets and \ntheir consequent habitability) may depend sensitively on   \nthe residual gas in the terrestrial planet region of the disk \nat ages of a few Myr. \nFor example, in the picture of terrestrial planet formation \ndescribed by {\\em Kominami and Ida} (2002), \nif the gas column density in this region is $\\gg$$1 \\rm \\,g\\,cm^{-2}$  \nat the epoch when protoplanets assemble to form terrestrial \nplanets, \ngravitational gas drag is strong enough to circularize the orbits \nof the protoplanets, making it difficult for them to collide and \nbuild Earth-mass planets.  In contrast, if the gas column density is \n$\\ll$$1 \\rm \\,g\\,cm^{-2}$, Earth-mass planets can be produced, but gravitational \ngas drag is too weak to recircularize their orbits. \nAs a result, only a narrow range of gas column densities around \n$\\sim$$1 \\rm \\,g\\,cm^{-2}$ is expected to lead to planets with the Earth-like \nmasses and \nlow eccentricities that we associate with habitability on Earth. \n\n\n\nFrom an observational perspective,  \nrelatively little is known about the evolution of the \ngaseous component.  Disk lifetmes are typically inferred from \ninfrared excesses that probe the dust component of the disk, \nalthough processes such as grain growth, planetesimal formation, \nand rapid grain inspiraling produced by gas drag ({\\em Takeuchi and Lin}, 2005) \ncan compromise dust as a tracer of the gas. \nOur understanding of disk lifetimes can be improved \nby directly probing the gas content of disks and using \nindirect probes of disk gas content such as stellar accretion rates \n(see {\\em Najita,} 2006 for a review of this topic).  \n\nSeveral of the diagnostics decribed in Section 2 may \nbe suitable as direct probes of disk gas content. \nFor example, transitions of H$_2$ and other molecules and atoms \nat mid- through far-infrared wavelengths are thought to be \npromising probes of the giant planet region of the disk \n({\\em Gorti and Hollenbach},\\ 2004). \nThis is a important area of investigation currently for the \nSpitzer Space Telescope and, in the future, for Herschel  \nand 8- to 30-m ground-based telescopes. \n\n\nStudies of the lifetime of gas in the terrestrial planet region are\nalso in progress.  The CO transitions are well suited for this purpose\nbecause the transitions of CO and its isotopes probe gas column\ndensities in the range of interest ($10^{-4}-1 \\rm \\,g\\,cm^{-2}$).  \nA current study by Najita, Carr, and Mathieu, \nwhich explores the residual gas content of optically thin disks \n({\\em Najita}, 2004),  \nillustrates some of the challenges \nin probing the residual gas content of disks.  \nFirstly,\ngiven the well-known correlation between IR excess and accretion\nrate in young stars (e.g., {\\em Kenyon and Hartmann}, 1995), CO emission\nfrom sources with optically thin inner disks may be intrinsically\nweak if accretion contributes significantly to heating disk\natmospheres.  \nThus, high signal-to-noise spectra may be \nneeded to detect this emission. \nSecondly, since the line emission may be intrinsically weak, \nstructure in the stellar photosphere may complicate the \nidentification of emission features. \nFig.~6 shows an example in which CO absorption in the stellar \nphotosphere of TW Hya likely veils weak emission from the \ndisk.  Correcting for the stellar photosphere would not only amplify \nthe strong $v$=1--0 emission that is clearly present \n(cf. {\\em Rettig et al.}, 2004), it would also uncover \nweak emission in the higher vibrational lines, confirming the \npresence of the warmer gas probed by the UV fluorescent lines \nof H$_2$ ({\\em Herczeg et al.}, 2002). \n\n\\begin{figure}[t!]\n\\vspace{-1.2in}\n\\plotfiddle{najita_fig6a.eps}{0.05in}{0.}{300}{300}{-30}{0}\n\\vspace{-2.6in}\n\\plotfiddle{najita_fig6b.ps}{0.05in}{0.}{300}{300}{-30}{0}\n\\vspace{-1.5in}\n\\caption{\\small\\baselineskip 10pt\n(Top) Spectrum of the transitional disk system TW Hya at 4.6~$\\hbox{$\\mu$m}$ \n(histogram). \nThe strong emission in the $v$=1--0 CO fundamental lines extend \nabove the plotted region. \nAlthough the model stellar photospheric spectrum (light solid line) fits \nthe weaker features in the TW Hya spectrum,   \nit predicts stronger absorption in the low vibrational CO \ntransitions (indicated by the lower vertical lines) than is \nobserved.  This suggests that the stellar \nphotosphere is veiled by CO emission from warm disk gas.  \n(Bottom) \nCO fundamental emission from the transitional disk system \nV836 Tau.  Vertical lines mark the approximate \nline centers at the velocity of the star.  The velocity widths \nof the lines locate the emission within a few AU of the star, \nand the relative strengths of the lines suggest optically thick \nemission.  Thus, a large reservoir of gas may be present in the \ninner disk despite the weak infrared excess from this portion \nof the disk. \n} \n\\end{figure}\n\n\n\n\nStellar accretion rates \nprovide a complementary probe of the gas content of inner disks. \nIn a steady accretion disk, the column density $\\Sigma$ is related to \nthe disk accretion rate $\\dot M$ \nby a relation of the form $\\Sigma\\propto \\dot M\/\\alpha T$, where \n$T$ is the disk temperature. \nA relation of this form allows us to infer $\\Sigma$ from $\\dot M$ \ngiven a value for the viscosity parameter $\\alpha$. \nAlternatively, the relation could be calibrated empirically \nusing measured disk column densities.  \n\nAccretion rates are available for many sources in the age \nrange 0.5--10\\,Myr (e.g., {\\em Gullbring et al.}, 1998; \n{\\em Hartmann et al.}, 1998; \n{\\em Muzerolle et al.}, 1998, 2000).  \nA typical value of $10^{-8}M_{\\odot}\\,{\\rm yr}^{-1}$ for TTS \ncorresponds to a(n active) disk column density of \n$\\sim$$100 \\rm \\,g\\,cm^{-2}$ at 1\\,AU for \n$\\alpha$=0.01 ({\\em D'Alessio et al.}, 1998). \nThe accretion rates show an overall decline with time \nwith a large dispersion at any given age.  \nThe existence of 10\\,Myr old sources with accretion rates as large \nas $10^{-8}\\,M_{\\odot}\\,{\\rm yr}^{-1}$ ({\\em Sicilia-Aguilar et al.}, 2005) suggests \nthat gaseous disks may be long lived in some systems.\n\nEven the lowest measured accretion rates may be dynamically significant.  \nFor a system like V836 Tau (Fig.\\ 6), a $\\sim$3\\,Myr old \n({\\em Siess et al.}, 1999) system with an optically thin inner disk, \nthe stellar accretion rate of $4\\times 10^{-10}M_{\\odot}\\,{\\rm yr}^{-1}$ \n({\\em Hartigan et al.}, 1995; {\\em Gullbring et al.}, 1998) would correspond\nto $\\sim$$4 \\rm \\,g\\,cm^{-2}$ at 1\\,AU.  Although the accretion rate \nis irrelevant for the buildup of the stellar mass, it corresponds \nto a column density that would \nfavorably impact terrestrial planet formation.  \nMore interesting perhaps\nis St34, a TTS with a Li depletion age of 25\\,Myr; \nits stellar accretion rate of $2\\times 10^{-10}M_{\\odot}\\,{\\rm yr}^{-1}$ \n({\\em White and Hillenbrand}, 2005) \nsuggests a dynamically significant reservoir of gas\nin the inner disk region.  These examples suggest that dynamically\nsignificant reservoirs of gas may persist even after inner disks \nbecome optically thin and over the timescales \nneeded to influence the outcome of terrestrial planet formation. \n\nThe possibility of long lived gaseous reservoirs can be confirmed \nby using the diagnostics described in Section 2 to measure total disk \ncolumn densities.  Equally important, a measured the disk \ncolumn density, combined with the stellar accretion rate, would allow \nus to infer a value for viscosity parameter $\\alpha$ for the system.  \nThis would be another way of constraining the disk accretion mechanism. \n\n\n\n\\bigskip\n\\noindent\n\\textbf{4.2 Nature of Transitional Disk Systems} \n\\bigskip\n\nMeasurements of the gas content and distribution in inner disks \ncan help us to identify systems in various states of planet \nformation.  Among the most interesting objects to study in this \ncontext are the transitional disk systems, which possess \noptically thin inner and optically thick outer disks.   \nExamples of this class of objects include TW Hya, GM Aur, DM Tau, \nand CoKu Tau\/4 ({\\em Calvet et al.}, 2002; {\\em Rice et al.}, 2003; \n{\\em Bergin et al.}, 2004; {\\em D'Alessio et al.}, 2005; \n{\\em Calvet et al.}, 2005).  \nIt was suggested early on that optically thin inner disks might \nbe produced by the dynamical sculpting of the disk by orbiting giant \nplanets ({\\em Skrutskie et al.}, 1990; see also {\\em Marsh and Mahoney}, 1992). \n\nIndeed, optically thin disks may arise in multiple phases of disk \nevolution.\nFor example, as a first step in planet formation (via core accretion), \ngrains are expected to grow into planetesimals and eventually \nrocky planetary cores, producing a region of the disk that has reduced \ncontinuum opacity but is gas-rich.  \nThese regions of the disk may therefore show strong line emission.\nDetermining the fraction of \nsources in this phase of evolution may help to establish the \nrelative time scales for planetary core formation and the accretion of  \ngaseous envelope. \n\nIf a planetary core accretes enough gas to produce a low mass \ngiant planet ($\\sim$1$M_J$), it is expected to carve out a gap in its \nvicinity (e.g., {\\em Takeuchi et al.}, 1996).  Gap crossing streams \ncan replenish an inner disk and allow further accretion onto \nboth the star and planet ({\\em Lubow et al.}, 1999). \nThe small solid angle subtended by the accretion streams would \nproduce a deficit in the emission from both gas and dust in the \nvicinity of the planet's orbit. \nWe would also expect to detect the presence of an inner disk. \nPossible examples of systems in this phase of evolution include \nGM Aur and TW Hya in which hot gas is detected close to the star as is \naccretion onto the star ({\\em Bergin et al.}, 2004; {\\em Herczeg et al.}, 2002; \n{\\em Muzerolle et al.}, 2000).  The absence of gas in the vicinity of \nthe planet's orbit would help to confirm this interpretation. \n\nOnce the planet accretes enough mass via the accretion streams to \nreach a mass $\\sim$5--10$M_J$, it is expected to cut off further accretion \n(e.g., {\\em Lubow et al.}, 1999).  The inner disk will accrete onto the \nstar, leaving a large inner hole and no trace of stellar accretion. \nCoKu Tau\/4 is a possible example of a system in this phase of \nevolution (cf.\\ {\\em Quillen et al.}, 2004) since it appears to have a large \ninner hole and a low to negligible accretion rate \n($<$few\\,$\\times 10^{-10}M_{\\odot}\\,{\\rm yr}^{-1}$). \nThis interpretation predicts little gas anywhere within the orbit \nof the planet. \n\nAt late times, \nwhen the disk column density around 10\\,AU has decreased sufficiently \nthat the outer disk is being photoevaporated away faster than \nit can resupply material to the inner disk via accretion, \nthe outer disk will decouple from the inner disk, which will accrete \nonto the star, leaving an inner hole that is devoid of gas and dust \n(the ``UV Switch'' model; {\\em Clarke et al.}, 2001).   \nMeasurements of the disk gas column density and the stellar \naccretion rate can be used to test this possibility. \nAs an example, TW Hya is in the age range ($\\sim$10\\,Myr) where \nphotoevaporation is likely to be significant. \nHowever, the accretion rate onto star, gas content of the inner disk \n(Sections 2 and 4), \nas well as the column density inferred for the outer disk \n($32 \\rm \\,g\\,cm^{-2}$ at 20\\,AU based on the dust SED; \n{\\em Calvet et al.}, 2002) \nare all much larger than is expected in the UV switch model. \nAlthough this mechanism is, therefore, unlikely to explain \nthe SED for TW Hya, it may explain the \npresence of inner holes in less massive disk systems of comparable age. \n\n\n\n\\bigskip\n\\noindent\n\\textbf{4.3 Turbulence in Disks} \n\\bigskip\n\nFuture studies of gaseous inner disks may also help to \nclarify the nature of the disk accretion process.  As indicated \nin Section 2.1, evidence for suprathermal line broadening \nin disks supports the idea of a turbulent accretion process. \nA turbulent inner disk may have important \nconsequences for the survival of terrestrial planets and the \ncores of giant planets.  An intriguing puzzle \nis how these objects avoid Type-I \nmigration, which is expected to cause the object to lose angular \nmomentum and spiral into the star on short timescales (e.g., {\\em Ward}, 1997).  \nA recent suggestion is that if disk accretion is turbulent, \nterrestral planets will scatter off \nturbulent fluctuations, executing a ``random walk'' which greatly \nincreases the migration time as well as the chances of \nsurvival ({\\em Nelson et al.}, 2000; see chapter by {\\em Nelson et al.}). \n\nIt would be interesting to explore this possible connection further by extending \nthe approach used for the CO overtone lines to a wider \nrange of diagnostics to probe the intrinsic line width as a \nfunction of radius and disk height.  By comparing the results \nto the detailed predictions of theoretical models, it may be \npossible to distinguish between the turbulent signature, produced \ne.g., by the MRI instability, from the turbulence that might \nbe produced by, e.g., a wind blowing over the disk. \n\nA complementary probe of turbulence may come from exploring the\nrelative molecular abundances in disks.  As noted in Section 3.4,\nif relative abundances cannot be explained by model\npredictions for non-turbulent, layered accretion flows, a \nsignificant role for strong vertical mixing produced by turbulence\nmay be implied.  Although model-dependent, this approach toward \ndiagnosing turbulent accretion appears to be less sensitive \nto confusion from wind-induced turbulence, especially if one can identify \ndiagnostics that require vertical mixing from deep down in the disk. \nAnother complementary approach toward probing the accretion process, \ndiscussed in Section 4.1, is to measure total gas column densities \nin low column density, dissipating disks in order to infer values \nfor the viscosity parameter $\\alpha$. \n\n\n\n\n\n\\section{\\textbf{SUMMARY AND CONCLUSIONS}} \n\nRecent work has lent new insights on the \nstructure, dynamics, and gas content of inner disks \nsurrounding young stars.  \nGaseous atmospheres appear to be hotter than the dust in \ninner disks.  This is a consequence of irradiative (and \npossibly mechanical) heating of the gas as well as \nthe poor thermal coupling between the gas and dust at the low \ndensities of disk atmospheres.\nIn accreting systems, the gaseous disk appears to be turbulent \nand extends inward beyond the dust sublimation radius\nto the vicinity of the corotation radius.\nThere is also evidence that dynamically significant reservoirs \nof gas can persist even \nafter the inner disk becomes optically thin in the continuum. \nThese results bear on important star and planet formation issues such as the\norigin of winds, funnel flows, and the rotation rates of young\nstars; the mechanism(s) responsible for disk accretion; and the role\nof gas in the determining the architectures of terrestrial and\ngiant planets.\nAlthough significant future work is needed to reach any conclusions \non these issues, the future for such studies is bright.  \nIncreasingly detailed studies of the inner disk region should be \npossible with the advent of \npowerful spectrographs and interferometers (infrared and submillimeter) \nas well as sophisticated models that describe the coupled \nthermal, chemical, and dynamical state of the disk. \n\n\n\n\n\n\\textbf{ Acknowledgments.} We thank Stephen Strom who contributed \nsignificantly to the discussion on the nature of transitional disk systems. \nWe also thank Fred Lahuis and Matt Richter for sharing \nmanuscripts of their work in advance of publication. \nAEG acknowledges support from the NASA Origins and NSF Astronomy \nprograms. \nJSC and JRN also thank the NASA Origins program for its support. \n\n\\bigskip\n\n\\centerline\\textbf{ REFERENCES }\n\\bigskip\n\\parskip=0pt\n{\\small\n\\baselineskip=11pt\n\n\n\\par\\noindent\\hangindent=1pc\\hangafter=1 Abgrall H., Roueff E., Launay F., Roncin J. 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(1998)\n{\\em \\aap, 338}, 995-1005. \n\n\\par\\noindent\\hangindent=1pc\\hangafter=1 Zuckerman B., Forveille T., and Kastner J.~H. (1995) \n{\\em Nature, 373}, 494-496\n\n\\end{document}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nCrowdsourcing websites aggregate judgments in order to\nhelp users discover high quality content. These systems typically combine choices of many people to algorithmically \\textit{rank} content so that better items---product reviews~\\cite{lim2015evaluating}, news stories~\\cite{PopularDynam,SocialInfluenceBias}, or answers on question-answering (Q\\&A) platforms~\\cite{BestAnswerYahoo,HighQualityQA}---are easier to find.\n\nDespite a long history of crowdsourcing~\\cite{JuryThm,Galton1908}, some of its limitations have only recently become apparent. Salganik et al. (\\citeyear{Salganik2006}) found that aggregating the votes of many people to rank songs increases the inequality and instability of song popularity. Even when starting from the same initial conditions, the same songs could end up with vastly different rankings. Other studies have shown that algorithmic ranking can amplify the inequality of popularity~\\cite{lerman14as,Burghardt2018} and bias collective outcomes in crowdsourcing applications~\\cite{MyopiaCrowd,dev2019quantifying}. In addition, information about the choices of other users affects decisions in complex ways~\\cite{SocialInfluenceBias,Hogg2015hcomp,talton2019people}. Unfortunately for crowdsourcing system designers, it is still not clear how these finding could help improve collective outcomes, in large part due to difficulty of quantifying the quality of options (e.g., the best answer to a question) and its impact on individual decisions. \n\nTo better understand and improve collective outcomes that emerge from individual decisions, we break down the crowdsourcing task into its basic elements: item quality, item ranking, and social influence. We create a controlled experiment to study how these elements jointly affect individual decisions and collective outcomes. We use experimental data to construct and validate a mathematical model of human judgements, and then use it to explore algorithmic ranking and identify strategies to improve crowdsourcing performance. Our study addresses the following research questions:\n\\begin{description}\n    \\item[RQ1] How does quality and presentation of options jointly impact individual decisions?\n    \\item[RQ2] When is algorithmic ranking unstable and does not reliably identify the best option? \n    \\item[RQ3] How can we stabilize algorithmic ranking such that the best option is typically ranked first?\n\\end{description}\n\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{figures\/Fig1.pdf}\n\\caption{\\label{fig:ExperimentSchematics} Schematic of the experiment conditions. Left: guess condition, where subjects write a guess for the correct value. Center: control condition, where subjects choose the best of two answers. Right: social influence condition, where a randomly chosen answer is called the ``most popular'' and is ranked first.\n}\n\\end{figure}\n\n\nOur experiment asks subjects to choose the best answer to questions with objectively correct answers, such as the number of dots in a picture. Figure~\\ref{fig:ExperimentSchematics} illustrates one such question, where we ask users to find the area ratio between the largest and smallest shapes. (Other questions used in the experiment are shown in Appendix Fig.~\\ref{fig:Questions}.) While these simple questions abstract away some of the complexity of the often-subjective decision making people do in crowdsourcing systems, they allow quality to be objectively measured and its effects on decisions better understood.\n\n\n\nThe experiment has three conditions shown in Fig.~\\ref{fig:ExperimentSchematics}. In the first condition, we let subjects write their answers to understand how their subjective guesses deviate from the correct answers. In the remaining conditions, we ask subjects to choose the best of two randomly generated answers that are randomly ordered. In the control condition, subjects are not told how they are ordered, while in the social influence condition, the first answer is labeled ``more popular''. These simple conditions allow us to disentangle the elements of crowdsourcing systems and begin quantifying how individual decisions affect collective outcomes.\n\nTo begin answering RQ1, we construct a mathematical model of the probability to choose an option as a function of its position and quality. The model requires only two parameters to measure \\textit{cognitive heuristics} (mental shortcuts people use to make quick and efficient judgements) and is in excellent agreement with experimental data. The first parameter reflects a user's preference to pick the first answer (known as ``position bias''~\\cite{lerman14as,Burghardt2018,MusicLabModel,PopularDynam}) and the other parameter measures the rate at which answers are guessed at random. Subjects otherwise pick an answer closest to their initial (unobserved) guess. We call this model the Biased Initial Guess (BIG) model. The BIG model demonstrates that the ``social influence'' experiment condition enhances position bias~\\cite{Burghardt2018}, \ntherefore cognitive heuristics that produce position bias and social influence can be quantified with a single parameter, a substantial simplification over previous work \\cite{MusicLabModel,PopularDynam}. Moreover, it helps explain why users often choose the worst answer when answer quality differences are small.\n\nThe BIG model not only improves our understanding of how answers are chosen, but it allows us to test different ranking policies in simulations to answer RQ2 and RQ3. Importantly, these simulations demonstrate that cognitive heuristics can make popularity-based ranking highly unstable.  When the quality difference between the options is small, initially minor differences in popularity can create a cumulative advantage \\cite{Rijt2014}, meaning the better answer does not always become the most popular.  However, when the quality difference passes a critical point, popularity-ranking is stable, and the better answer eventually becomes the most popular. These results may help explain an under-appreciated finding of Salganik et al. (\\citeyear{Salganik2006}) that the best and worst songs tended to be correctly ranked when songs were ordered by popularity, but intermediate-quality songs landed anywhere in between. \n\nFinally, to answer RQ3, we propose an algorithm that rectifies this instability by ordering answers based on their inferred quality, which we call RAICR: Rectifying Algorithmic Instabilities in Crowdsourced Ranking. This method is found to be stable and consistently order the better answer first, thus making best answers easier to find, even when they are only slightly better. RAICR ranks answers as well as, or better than, common baselines such as ordering by popularity or by recency (ranking by the last answer picked).\n\nOur work shows that individual decisions within crowdsourcing systems are strongly affected by cognitive heuristics, which collectively create instability and poor crowd wisdom. Designers of crowdsource systems need to account for these biases in order make good content easier to find. Algorithms such as RAICR, however, can correct for these biases, thereby improving the wisdom of crowds. \n\n\\section{Related Literature}\n\n\\subsection{Crowdsourcing}\nCrowdsourcing has a two-century long history demonstrating how a collective can outperform individual experts \\cite{JuryThm,Galton1908,surowiecki2005wisdom,Kaniovski,Simoiu2019}, thus creating the moniker ``wisdom of crowds''. Crowds have been shown to beat sport markets \\cite{Brown2019,Peeters2018}, corporate earnings forecasts \\cite{Zhi2019}, and improve visual searches \\cite{Juni2017}. One reason crowd wisdom works is due to the law of large numbers: assuming unbiased and independent guesses, the average guess should converge to the true value. \n\nIndividual decisions, including in online settings, are biased by cognitive heuristics, such as anchoring \\cite{shokouhi2015anchoring}, primacy \\cite{Primacy1}, prior beliefs \\cite{white2013beliefs} \nand position bias \\cite{Burghardt2018}. These biases are not necessarily canceled out with large samples \\cite{Prelec2017,Kao2018}. As a result, aggregating guesses of a crowd does not necessarily converge to the correct answer.\n\nGuesses are usually not independent, which can sometimes improve the wisdom of crowds. Social influence models, such as the DeGroot model \\cite{Degroot1974}, have been shown to push simulated agents to an optimal decision \\cite{Golub2010,Mossel2015,Bala1998,Acemoglu2011}. These results have been backed up experimentally \\cite{Becker2017,Becker2019,Ungar2012,Tetlock2017}, even when opinion polarization is included \\cite{Becker2019}. One reason social influence can be beneficial is that it encourages people who are way off the mark to improve their guess \\cite{Mavrodiev2013,Becker2017,Abeliuk2017www}. \n\n\nOften, however, social influence can reduce crowd wisdom. Corporate earnings predictions \\cite{Zhi2019}, jury decisions  \\cite{Kaniovski,BurghardtJury}, and other guesses can degrade with influence \\cite{Lorenz2011,Lorenz2015,Simoiu2019}, and malevolent individuals can manipulate people to make particular collective decisions \\cite{SocialInfluenceBias,Asch1951}. Too much influence by a single individual can also reduce the wisdom of collective decisions \\cite{Becker2017,Acemoglu2011,Golub2010}, and deferring to friends can sometimes make unpopular (and potentially low-quality) ideas appear popular \\cite{Lerman2016}. \n\nRecent work has also demonstrated how other cognitive heuristics can affect crowd wisdom. After the landmark study by Salganik et al. (\\citeyear{Salganik2006}), some researchers found that social influence has no effect on decisions \\cite{Antenore2018}, or that position bias, i.e., the preference to choose options listed first, largely explain biases in crowdsourcing \\cite{lerman14as,MusicLabModel}. Social influence instead enhances the position bias \\cite{Burghardt2018,MusicLabModel}. Burghardt et al. (\\citeyear{Burghardt2018}) have begun to tease apart these effects, showing that while social influence enhances position bias, it has no marginal effect when we control for position. This is consistent with our approach, which models and rectifies both biases with a single parameter. \n\n\\subsection{Algorithmic Ranking}\n\nThe goal of a ranking algorithm is to make good content easier to find. \nMany papers have begun to address this goal \\cite{Page1999,Jarvelin2002,Bendersky2011}, which has recently been applied to crowdsourced ranking. Because of human biases, however, algorithms that na{\\\"i}vely use human feedback to suggest content will end up forming echo chambers \\cite{Hilbert2018,Bozdag2013,Hajian2016}, or only recommend already-popular items \\cite{Abdollahpouri2017,Abdollahpouri2019}. This can also give some content a cumulative advantage \\cite{Rijt2014}, even when it is of similar quality to content that remains unpopular. \n\nTo correct for algorithmic bias in this paper, we create a ranking method that follows the strategy of Watts, who says, ``...we can instead measure directly how they respond to a whole range of possibilities and react accordingly'' \\cite{Watts2012}. In the present context, this strategy implies we can create better algorithms for option ranking by observing, and addressing, how people respond to social influence and position biases. We show this strategy applied in RAICR improves upon simple algorithms used in the past, which include ordering results by popularity \\cite{Burghardt2018} or recency \\cite{lerman14as}. While some crowdsourced ranking strategies use a two-tier platform model, in which researchers rank options based on whether content is downloaded and rated \\cite{Salganik2006,Antenore2018,Abeliuk2017www}, RAICR is based on a common simpler model in which we only observe if content is chosen \\cite{Burghardt2018,PopularDynam}. This subtle difference implies many previous ranking schemes are not applicable. The present paper also compliments previous work that uses features, such as answer position, to predict Q\\&A website quality \\cite{Shah2010,MyopiaCrowd}.\n\n\\section{Experiment}\n\nThe experiment asked subjects, hired through Amazon Mechanical Turk between August 2018 and September 2019, to answer a series of questions shown in the Appendix. The order of questions was randomized for each subject, and questions were not time-limited. Questions include specifying the ratio of the areas of two shapes or the lengths of two lines, or the number of dots in an image. We designed the experiment around quotidian tasks that do not require specific expertise but are difficult for most people. Despite their difficulty, the questions have objectively correct answers. We quantify the quality of an answer by the difference from the mean of all guesses, which better matches a log-normal distribution, as discussed later. In the Appendix, we define quality of an answer by the difference from the correct value and find results are qualitatively very similar. \n\nSubjects were assigned to one of three conditions. In the \\textit{guess condition}, shown in the left panel of Fig.~\\ref{fig:ExperimentSchematics}, the subjects could freely type their guesses. In the \\textit{control condition}, shown in the center panel of Fig.~\\ref{fig:ExperimentSchematics}, subjects were told to choose the best among two options, and were not told how the two answers were ordered. Finally, in the \\textit{social influence condition}, shown in the right panel of Fig.~\\ref{fig:ExperimentSchematics}, they were told the first among two answers was the most popular, but the layout was otherwise identical to the previous condition. We only showed ten questions to each subject to reduce the effects of performance depletion in Q\\&A systems~\\cite{Ferrara17}. Further, to reduce bias due to other phenomena, such as the decoy effect \\cite{Huber1982}, we showed subjects only two choices. Subjects in the same condition but assigned on different days have statistically similar behavior.\n\nApproximately 1800 subjects are evenly split between the conditions (596, 586, and 587 for the guess, control, and social influence condition, respectively). For guess values, we remove extreme outliers (guesses smaller than 1 or greater than $10^6$). All answers\nare supposed to be greater than 1, while values greater than $10^6$ may affect mean values and appear to represent throwaway answers. The number of valid participants for each question is shown in Table\\ref{tab:guesssummary}.\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Number of subjects for each guess question (after cleaning)}\n\\label{tab:guesssummary}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nQ1&Q2&Q3&Q4&Q5&Q6&Q7&Q8&Q9&Q10\\\\ \\hline\n595&594&593&595&592&593&589&591&584&590\\\\ \\hline\n\\end{tabular}\n\\begin{flushleft}\n\\end{flushleft}\n\\end{table}\n\n\nMechanical Turk workers were hired if they had an approval rate of over $95\\%$, completed more than 1000 Human Intelligence Tasks (HITs), and never participated in any of the experiment conditions before. Each worker was paid $\\$1.00$ for the guessing condition and $\\$0.50$ for the other two conditions. The assignment took $6$ minutes on average for the guess condition and $2$ minutes on average for the other conditions, equivalent to an hourly wage of $\\$10-\\$15$. The human experiment was approved by the appropriate IRB board.\n\n\\section{Results}\n\\subsection{A Mathematical Model of Decisions}\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{figures\/Fig2.pdf}\n\\caption{\\label{fig:CDFGuess} The CDF of guesses for each question. (a) Before normalization, and (b) after normalization.\n}\n\\end{figure}\n\nWe use data gathered from the experiment to answer \\textbf{RQ1}: \\textit{How does quality and presentation of options jointly impact individual decisions?}\n\n\\subsubsection{Open-Ended Experiment Condition} \nTo derive the model of how people make these decisions, we start by constructing the distribution of guesses for each question (the \\textit{guess} condition in the experiment). \nThe guesses, plotted in Appendix Fig.~\\ref{fig:GuessExperimentResults} and CDF shown in Fig.~\\ref{fig:CDFGuess}a, are highly variable (by as many as six orders of magnitude), while the correct answers vary by three orders of magnitude. The median guess may differ from the true answer, in agreement with previous work \\cite{Kao2018}, but values are typically the correct order of magnitude.\n\nWe normalize guesses by defining a new variable $A$:\n\\begin{equation*}\n    A = \\frac{\\text{ln}(X)-\\langle \\text{ln}(X)\\rangle}{\\sigma_{\\text{ln}(X)}},\n\\end{equation*}\nwhere $X$ is a guess value, and $\\langle \\text{ln}(X)\\rangle$ is the mean of the logarithm of guesses. Figure~\\ref{fig:CDFGuess}b shows that this simple normalization scheme effectively collapses answer guesses to a single distribution. We show in the Appendix that guesses are not normally distributed, but instead are better approximated as log-normal, in agreement with previous work on a different set of questions \\cite{Kao2018}. The normalized guesses $A$ can be thought of as the $z$-scores in log-normal distributions. Alternative ways to center data, shown in the Appendix, produce similar results. We use these normalized guesses and distributions in the remaining two experiment conditions. Intuitively, if the mean of all guesses converges to the correct answer, $A=0$ can be thought of as the best answer.\n\n\\subsubsection{Two-Choice Experiment Conditions}\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{figures\/Fig3.pdf}\n\\caption{\\label{fig:ChooseVsPos} Probability to choose an answer versus its position for the social influence and control condition.\n}\n\\end{figure}\n\nThe latter two experiment conditions require subjects to pick the best among two answers to the question. For both the control and social influence conditions, answers are ordered vertically, with one answer above the other. There is a significant position effect when answers are ordered this way, as shown in Fig.~\\ref{fig:ChooseVsPos}. In the control condition, there is a slightly greater probability (52\\%) to choose the first (top) answer over the last one (p-value $<0.001$). In the social influence condition, meanwhile, the probability to choose the first answer is substantially larger (59\\%) and statistically significantly different than the control condition (p-value $<0.001$). This is in agreement with previous work showing that social influence amplifies the position effect~\\cite{Burghardt2018}.\n\n\n\\subsubsection{The Biased Initial Guess Model}\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{figures\/Fig4.pdf}\n\\caption{\\label{fig:ModelSchematics} Schematic to calculate $s(A_1,A_2)$. Guesses follow an approximately log-normal distribution unique to each question, but the normalized guesses, $A$ are approximately standard normal distributed. In an experiment, assume two candidate answers are provided, $A_1$ is listed first, and $A_2$ is listed last. The variable $s(A_1,A_2)$ in Eq.~\\ref{eq:initselect} is the probability an unknown initial guess is closer to $A_1$ than $A_2$.\n}\n\\end{figure}\n\nWe now have the necessary ingredients to model how decisions to choose an answer are affected by its quality, position, and social influence. \nWe present the Biased Initial Guess (BIG) decision model and show it is consistent with the data.\n\nWe first discuss the simplest case where a user has to choose the better of two answers $A_1$, listed first, and $A_2$, listed last, in the absence of cognitive biases. Figure~\\ref{fig:ModelSchematics} shows probability of the answer, with the normalized values of the choices $A_1$ and $A_2$, as well as the user's initial guess $A_I$ about the true answer, which we do not observe. All things equal, the user will choose the first answer $A_1$ if it is closer to the initial guess, i.e., if $|A_1-A_I|<|A_2-A_I|$, and will otherwise choose $A_2$. The probability to choose $A_1$ is then:\n\\begin{equation}\ns(A_1,A_2) = \n\\begin{cases}\n\\text{Pr}(A_I > \\frac{A_1+A_2}{2}) & A_1 > A_2 \\\\\n1\/2 & A_1 = A_2\\\\\n\\text{Pr}(A_I < \\frac{A_1+A_2}{2}) & A_1 < A_2\n\\end{cases}\n\\end{equation}\n Assuming $A_1$ and $A_2$ follow a normal distribution quantified in the guess experiment condition,\n\\begin{equation}\n\\label{eq:initselect}\ns(A_1,A_2)= \n\\begin{cases}\n\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2 & A_1 \\ge A_2 \\\\\n\\left[1 + \\text{erf}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\\right]\/2, & A_1 < A_2\n\\end{cases}\n\\end{equation}\nwhere $\\text{erf}(.)$ is the error function, $\\text{erfc}(.) = 1-\\text{erf}(.)$ is the complimentary error function. When $s(A_1,A_2)>0.5$, $A_1$ is not only more likely to be closer to the initial guess ($A_I$) than $A_2$, but is also closer to zero than $A_2$, and therefore the objectively better answer. On the other hand, when $s(A_1,A_2)<0.5$, $A_1$ is further from most guesses compared to $A_2$ and the objectively worse answer.\n\nTo better model decision-making, we have to account for biases, due to cognitive heuristics and algorithmic ranking, to explain why people do not always choose the best option. As shown in Fig.~\\ref{fig:ChooseVsPos}, sometimes they choose the first answer even if it is not the best answer. We quantify this position bias by assuming that with probability $p$ participants choose the first answer regardless of its quality. This parameter should presumably be small in the control condition and large in the social influence condition. Subjects may also choose an answer regardless of its position or quality because there is no monetary incentive to choose good answers. We model this by allowing subjects to choose an answer at random with a probability $r$. Taking these two heuristics into account, we arrive at the BIG Model: \n\n\\begin{equation}\n\\label{eq:model_full}\n\\begin{split}\n\\text{Pr}(\\text{Choose}~A_{1}) &= r\/2 + (1 - r) [p + (1 - p) s(A_1,A_2)]\\\\\n&= \\begin{cases}\nr\/2 + (1 - r) \\left[p + (1 - p)\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right] & A_1 \\ge A_2 \\\\[7pt]\nr\/2 + (1 - r) \\left[p + (1 - p)\\left[1 + \\text{erf}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\\right]\/2\\right], & A_1 < A_2\n\\end{cases}\n\\end{split}\n\\end{equation}\nThe probability of choosing $A_2$ is simply the compliment of this probability. Because $A=0$ is expected to be the best answer, with some simple manipulation we can infer the probability the best answer is chosen.\n\n\\begin{equation}\n\\label{eq:model_best}\n\\text{Pr}(\\text{Choose}~A_{1}|A_{1}~\\text{Best}) = \\begin{cases}\nr\/2 + (1 - r) \\left[p + (1 - p) \\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right]  & \\frac{A_1+A_2}{2}<0\\\\[7pt]\nr\/2 + (1 - r) \\left[p + (1 - p) \\left(1-\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right)\\right] & \\frac{A_1+A_2}{2}\\ge0\\\\\n\\end{cases}\n\\end{equation}\nA similar equation can model $\\text{Pr}(\\text{Choose}~A_{2}|A_{2}~\\text{Best})$.\n\nAgreement between the model and data is shown in Fig.~\\ref{fig:ModelFit}. In the \\textit{control condition} (Fig.~\\ref{fig:ModelFit}a), the best parameters are $r=0.28\\pm0.02$ and $p=0.05\\pm0.02$. We find that the log-likelihood of the model, $\\ell=-3002.49$, is not statistically different from log-likelihood if the data came from the model itself: p-value $=0.40$. See Methods for how p-values and error bars are calculated. We also check if we need both parameters, $r$ and $p$, using the likelihood ratio test and Wilks' Theorem~\\cite{Wilks1938}. We compare the likelihood ratio of the two-parameter model to simpler models with $p$ or $r$ (or both) set to zero. The probability a simpler model could fit the data as well or better is $\\le 0.002$. We conclude that our model describes the control condition very well. \n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/Fig5.pdf}\n\\caption{\\label{fig:ModelFit} Decision model agreement with experiment data. Plots show the probability the best answer is chosen for the model (lines) and experiment (symbols) under (a) the control condition and (b) the social influence condition. Purple diamonds are probabilities when the best answer is the first answer, $A_1$ and green squares for when the best answer is $A_2$.\n}\n\\end{figure}\n\nThe agreement between data and model is similarly close in the \\textit{social influence condition} (Fig.~\\ref{fig:ModelFit}b). The position bias parameter $p=0.21\\pm 0.01$ is larger than in the control condition, in agreement with expectations. We also find $r=0.08\\pm 0.02$, thus social influence reduces the frequency of random guesses. In both experiment conditions, surprisingly, $\\approx 20\\%$ of users choose answers for reasons besides ``quality'' ($r\/2+(1-r)p = 18\\%$ and $23\\%$ for the control and social influence conditions, respectively). Similar to the control condition, we find that the model is consistent with the data. The log-likelihood of the empirical data ($\\ell=-3190.13$) is not statistically different from log-likelihood values if the data came from the model: p-value $0.47$. The probability a simpler model ($r$ or $p$ set to zero) could fit the data as well or better is $<10^{-5}$. In conclusion, we find the BIG model is consistent with both experiment conditions and its parameters are interpretable and meaningful. In the Appendix, we show that all these results are consistent when we look at a subset of experiment questions or center the data differently. \n\n\\subsection{Algorithmic Ranking Instability}\nCrowdsourcing websites automatically highlight what they consider the best choices to help their users more quickly discover them. For example, Stack Exchange (like other Q\\&A platforms) usually ranks answers to questions by the number of votes they receive. Despite problems with popularity-based ranking identified in previous studies~\\cite{Salganik2006,lerman14as}, it is widely used for ranking content in crowdsourcing websites. In this section we identify an instability in popularity-based ranking: the first few votes a worse answer receives can lock it in the top position, where it acquires cumulative advantage \\cite{Rijt2014}. This allows us to answer \\textit{RQ2: When is algorithmic ranking unstable and does not reliably identify the best option?} \n\n\n\\begin{figure*}[tbh!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/Fig6.pdf}\n\\caption{Comparison of ranking policies via simulations. Plots show the probability the best answer is ranked first after (a) 50, (b) 500, and (c) 20,000 votes, when answers are ranked by quality (black line), recency (orange dashed line), and popularity. Also shown in (c) is the critical value of $A_{\\text{worst}}$ based on Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect}. In these simulations, subjects choose answers following the BIG Model with $p=0.2$ and $r=0.09$. ``Popularity 0'' (green dashed line), ``Popularity -10'' (cyan dashed line), and ``Popularity -200'' (red dashed line) means that the worst answer starts with a 0, 10, or 200 vote advantage, respectively. Shaded areas are 95\\% confidence intervals. \n\\label{fig:InitialGuessModelRankData} \n}\n\\end{figure*}\n\nTo demonstrate the instability, we simulate a group of agents who choose answers according to the BIG model with $p=0.2$ and $r=0.09$. In the simulations, one answer is objectively best, e.g., exactly equal to the correct answer ($A_{\\text{best}}=0$), while the worst answer is larger than $A_{\\text{best}}$ (results are symmetric if $A_{\\text{worst}}<A_{\\text{best}}$). At each timestep, a new user arrives and independently chooses the first or last answer according to Eq.~\\ref{eq:model_full}. Answers reorder depending on the ranking algorithm. Figure~\\ref{fig:InitialGuessModelRankData} shows our results. If the worst answer, $A_{\\text{worst}}$, is not too large and has a few extra votes (a common occurrence when the worst answer is posted before the better answer), we find that popularity ordering completely breaks down---the worst answer usually becomes more popular and is ranked first. Even when both answers start with the same number of votes, the worse answer is often ranked first. Results remain stable even after 20K votes; the effect does not appear transient.\n\nWe can explain this result using our model. Using the compliment of Eq.~\\ref{eq:model_full}, we can define the probability the best answer is chosen, conditional on it being ranked last:\n\\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2) = r\/2 + (1 - r) (1 - p) s(A_1,A_2)\n\\end{equation}\nWe find that\n\\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2) = \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_1)-p(1-r)\n\\end{equation}\nwhere $\\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_1)$ is the probability of choosing the best answer when it is ranked first. If \\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2)>\\text{Pr}(\\text{Choose}~ A_{\\text{worst}}=A_1)\n\\end{equation} \nthen subjects are more likely to choose the better answer regardless of whether its ranked first or second, therefore answer order is stable. When the above inequality is not true, however, then subjects are more likely to pick the first answer regardless of its quality, and the popularity ranking is unstable. The critical value of $A_{\\text{worst}}$ between the two regimes is when \n\\begin{equation}\\label{eq:scrit}\n    s_{\\text{crit}}=\\frac{1}{2 (1-p)}\n\\end{equation}\nand is independent of $r$. Intuitively, if the answer is exceptionally bad, it will always be less popular. This is alike to the results in Salganik's MusicLab study \\cite{Salganik2006}, where particularly good and bad songs were ranked correctly. However, if the first answer is likely to be chosen regardless of quality, the worst answer will continue accumulating votes and remain in the top position. Given $A_{\\text{best}}=0$, we can use Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect} to numerically solve for the critical value of $A_{\\text{worst}}$. We plot the critical point as a function of $p$ and $A_{\\text{worst}}$ in phase diagram in Fig.~\\ref{fig:SimPhaseSpace}. We see that there is always a large part of the phase space where popularity-based ranking will be unpredictable if answer quality is close together. Based on Eq.~\\ref{eq:scrit}, if $p\\ge 0.5$, there will be no case where popularity-based ranking is guaranteed to correctly rank answers. While this is an extreme case, it still points to substantial limitations of popularity-based ranking.\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.44\\textwidth]{figures\/Fig7.pdf}\n\\caption{\\label{fig:SimPhaseSpace} Phase diagram for popularity-based ranking based on Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect}. We show the boundary between regimes where best answers are guaranteed to be ranked first with enough votes (white area) and where ranking is unstable (gray area) as a function of position bias ($p$) and the value of the worse answer ($A_{\\text{worst}}$). In this plot, larger values of $A_{\\text{worst}}>0$ correspond to worse answers and results are symmetric for $A_{\\text{worst}}<0$. \n}\n\\end{figure}\n\\subsection{Stabilizing Algorithmic Ranking}\n\nGiven the problems with popularity ranking, it is critical to create a more consistent ranking algorithm. Moreover, while we have so-far explored questions with numeric answers, we want a method that works for all types of answers. Using the BIG model, we can answer RQ3: \\textit{How can we stabilize algorithmic ranking such that the best option is typically ranked first?}\n\nAssume we can approximate $p$ and $r$, then we can invert Eq.~\\ref{eq:model_full} and use votes to infer the only unknown variable $s(A_{\\text{best}},A_{\\text{worst}})$. When $s(A_{\\text{best}},A_{\\text{worst}})>0.5$, $A_{\\text{best}}$ is the best answer, but if we incorrectly rank $A_{\\text{worst}}$ first, $s(A_{\\text{worst}},A_{\\text{best}})<0.5$. We can therefore rank the answer in which $s(A_{\\text{best}},A_{\\text{worst}})>0.5$ as the best answer. This is the backbone of the RAICR algorithm. The algorithm uses maximum likelihood estimation to solve for $s(A_{\\text{best}},A_{\\text{worst}})$, as shown in the Appendix. Therefore, if the data matches the BIG model with the correct $r$ and $p$ parameters, our method \\emph{optimally infers the correct raking} by having minimal variance and no bias \\cite{Newey1994}. Moreover, \\emph{this method only depends on the votes an answer receives rather than the type of answer}, such as a numerical or textual answer. We compare quality ranking to popularity-based ranking, and recency-based ranking (ranking by the last answer picked), as shown in Fig.~\\ref{fig:InitialGuessModelRankData}. The probability recency ranks the best answer first is calculated as the self-consistent equation: \n\\begin{dmath}\n    \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) = \\text{Pr}(\\text{Choose}~ A_{\\text{best}}|A_{\\text{best}}~\\text{First}) \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency})\\\\  +  \\text{Pr}(\\text{Choose}~ A_{\\text{best}}|A_{\\text{best}}~\\text{Last})(1 - \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) ).\n\\end{dmath}\nThis represents the limit answers acquire many votes. The solution to this equation is:\n\\begin{dmath}\n    \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) = \\frac{2 (1-p) (1-r) s(A_{\\text{best}},A_{\\text{worst}}\n    +r}{2-2 p (1-r)\n\\end{dmath}\nWe find that the RAICR algorithm performs at least as well as popularity-based ranking by ranking the better answer first, and often better after 20-50 votes (Fig.~\\ref{fig:InitialGuessModelRankData}a and Appendix Fig.~\\ref{fig:SimPRobust}). Moreover, RAICR always outperforms the recency-based algorithm. The benefit of the RAICR algorithm only improves as we collect more votes. \nFor example, Fig.~\\ref{fig:InitialGuessModelRankData}b shows after after 500 votes the advantage of RAICR is larger, and Fig.~\\ref{fig:InitialGuessModelRankData}c shows that after 20K votes the method is nearly optimal. Popularity-based ranking performs badly for $A_{worse}<0.6$, and recency performs worst if $A_{worse}>0.6$. \n\nWhile we show these results hold when we have 20 to 20,0000 votes, many platforms are underprovisioned, with  a large fraction of webpages \nreceiving little attention and votes~\\cite{baeza2018bias,Gilbert2013}. It is the webpages that receive many votes, however, which may be the most important. Correctly ranking options in these popular pages is therefore especially critical to crowdsourcing websites. Moreover, a moderate number of votes is generally needed to make a reasonable estimate of quality, so very few methods will accurately rank unpopular pages.\n\nOne caveat of the RAICR algorithm is that we need an approximate value for two parameters: $r$ and $p$. What happens if either of these parameters are far off? For example, we could assume $r=0$ when $r=0.09$. Results in the Appendix, however, show that the findings are quantitatively very similar, and therefore our model is robust to assumptions about $r$. On the other hand, what if we incorrectly estimate both $r$ and $p$? We show in the Appendix that even in this worst-case scenario, our method performs slightly worse, but is comparable or substantially better than popularity-based ranking. \n\n\\section{Future Work and Design Implications\n\nIn the experiment, we decomposed the crowdsourcing task to its basic components to reduce the complexity and variation inherent in real world tasks. While this helps to disentangle effects of option quality and position without confounding factors muddying the relationships, future work is needed to verify ecological validity of our results \\cite{ObservationalVsExp}. It is encouraging that the probability to choose an answer (Fig.~\\ref{fig:ChooseVsPos}b) is quantitatively similar to empirical data gathered from Stack Exchange \\cite{Burghardt2018}, despite Mechanical Turk workers not being representative of the general population \\cite{Munger2019}. \n\nFor simplicity, we only explored two-option questions; future work should aim to understand multi-option decision-making given options of variable quality. A generalization of RAICR should also address more complicated biases, such as preference for round numbers \\cite{Fitz1986}, anchoring \\cite{shokouhi2015anchoring,Furnham2011}, and biases that appear in multi-option decisions, such as the decoy effect \\cite{Huber1982}. Finally, while RAICR is found to be robust to moderate changes in its parameters, this algorithm and its extensions may fail to rank options properly if its parameters are far off, or if the BIG model is wrong. In the experiment, the model is backed by data, but future work needs to address whether other tasks or questions follow this model. \n\nOur results offer implications for crowdsourcing platforms. First, designers must recognize the limits of crowdsourcing due to biases implicit in their platform. In our experiment people often upvoted options at random (up to 20\\% of all votes), and chose an inferior option simply because it was shown first. This creates a ranking instability when options are of similar quality. Our controlled experiment and mathematical model point to ways we can counteract this instability. Designers should similarly create platform-tailored mathematical models and controlled experiments to rigorously test how crowds can better infer the best options. \n\nA key property of our RAICR algorithm is that it relies on accurate modeling of user decisions to counteract cognitive biases. In effect, each vote is weighted depending on the ranks of answers at the time the vote is cast. The idea is similar to one described by Abeliuk et al.~\\citeyear{Abeliuk2017www} that ranks items by their inferred quality in order to more robustly identify blockbuster items. Similar weighting schemes could be applied to future debiased algorithms to address the unique goals of each crowdsourcing platform. \n\nThere are also simple methods that platforms like  Reddit, Facebook, and Stack Exchange can try that may greatly outperform the baselines we mention in our paper. For example, items that have not yet acquired many votes can be ranked randomly to reduce initial ranking biases. Alternatively, new posts and links could be ranked appropriately but their popularity could be hidden until they gather enough votes. Our results suggest this could reduce social influence-based position bias up until the true option quality is more obvious. \n\n\\section{Conclusion}\nIn this paper, we introduce an experiment designed to inform how cognitive biases and option quality interact to affect crowdsourced ranking. Results from this experiment help us create a novel mathematical decision model, the BIG model, that greatly improves our understanding of how people find the best answer to a question as a function of answer quality, rank, and social influence. This model is then applied to the RAICR algorithm to better rank answers. The BIG model also helped us uncover instability in popularity-based ranking. The instability depends on the quality of options: when there are large differences between option qualities, popularity converges optimally and predictably. However, when the difference between the quality of options is small, the better option may not always become the most popular. These results can help us better understand the foundational empirical results of Salganik et al. (\\citeyear{Salganik2006}), who found that popularity-based ranking correctly ranked high and low quality songs, while the ranking of intermediate quality songs was highly unstable. Although our experimental setup is undeniably simpler than real crowdsourcing websites, our results suggest that accurate models of user behavior together with mathematically principled inference can improve the efficiency of crowdsourcing. \n\n\n\n\\begin{acks}\nOur work is supported by the US Army Research Office MURI Award No. W911NF-13-1-0340 and the DARPA Award No. W911NF-17-1-0077. Data as well as code to create experiments, create simulations, and analyze data is available at https:\/\/github.com\/KeithBurghardt\/QualityRankCodeAndData.\n\\end{acks}\n\n\n\n\n\n\\section{Introduction}\nCrowdsourcing websites aggregate judgments in order to\nhelp users discover high quality content. These systems typically combine choices of many people to algorithmically \\textit{rank} content so that better items---product reviews~\\cite{lim2015evaluating}, news stories~\\cite{PopularDynam,SocialInfluenceBias}, or answers on question-answering (Q\\&A) platforms~\\cite{BestAnswerYahoo,HighQualityQA}---are easier to find.\n\nDespite a long history of crowdsourcing~\\cite{JuryThm,Galton1908}, some of its limitations have only recently become apparent. Salganik et al. (\\citeyear{Salganik2006}) found that aggregating the votes of many people to rank songs increases the inequality and instability of song popularity. Even when starting from the same initial conditions, the same songs could end up with vastly different rankings. Other studies have shown that algorithmic ranking can amplify the inequality of popularity~\\cite{lerman14as,Burghardt2018} and bias collective outcomes in crowdsourcing applications~\\cite{MyopiaCrowd,dev2019quantifying}. In addition, information about the choices of other users affects decisions in complex ways~\\cite{SocialInfluenceBias,Hogg2015hcomp,talton2019people}. Unfortunately for crowdsourcing system designers, it is still not clear how these finding could help improve collective outcomes, in large part due to difficulty of quantifying the quality of options (e.g., the best answer to a question) and its impact on individual decisions. \n\nTo better understand and improve collective outcomes that emerge from individual decisions, we break down the crowdsourcing task into its basic elements: item quality, item ranking, and social influence. We create a controlled experiment to study how these elements jointly affect individual decisions and collective outcomes. We use experimental data to construct and validate a mathematical model of human judgements, and then use it to explore algorithmic ranking and identify strategies to improve crowdsourcing performance. Our study addresses the following research questions:\n\\begin{description}\n    \\item[RQ1] How does quality and presentation of options jointly impact individual decisions?\n    \\item[RQ2] When is algorithmic ranking unstable and does not reliably identify the best option? \n    \\item[RQ3] How can we stabilize algorithmic ranking such that the best option is typically ranked first?\n\\end{description}\n\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{figures\/Fig1.pdf}\n\\caption{\\label{fig:ExperimentSchematics} Schematic of the experiment conditions. Left: guess condition, where subjects write a guess for the correct value. Center: control condition, where subjects choose the best of two answers. Right: social influence condition, where a randomly chosen answer is called the ``most popular'' and is ranked first.\n}\n\\end{figure}\n\n\nOur experiment asks subjects to choose the best answer to questions with objectively correct answers, such as the number of dots in a picture. Figure~\\ref{fig:ExperimentSchematics} illustrates one such question, where we ask users to find the area ratio between the largest and smallest shapes. (Other questions used in the experiment are shown in Appendix Fig.~\\ref{fig:Questions}.) While these simple questions abstract away some of the complexity of the often-subjective decision making people do in crowdsourcing systems, they allow quality to be objectively measured and its effects on decisions better understood.\n\n\n\nThe experiment has three conditions shown in Fig.~\\ref{fig:ExperimentSchematics}. In the first condition, we let subjects write their answers to understand how their subjective guesses deviate from the correct answers. In the remaining conditions, we ask subjects to choose the best of two randomly generated answers that are randomly ordered. In the control condition, subjects are not told how they are ordered, while in the social influence condition, the first answer is labeled ``more popular''. These simple conditions allow us to disentangle the elements of crowdsourcing systems and begin quantifying how individual decisions affect collective outcomes.\n\nTo begin answering RQ1, we construct a mathematical model of the probability to choose an option as a function of its position and quality. The model requires only two parameters to measure \\textit{cognitive heuristics} (mental shortcuts people use to make quick and efficient judgements) and is in excellent agreement with experimental data. The first parameter reflects a user's preference to pick the first answer (known as ``position bias''~\\cite{lerman14as,Burghardt2018,MusicLabModel,PopularDynam}) and the other parameter measures the rate at which answers are guessed at random. Subjects otherwise pick an answer closest to their initial (unobserved) guess. We call this model the Biased Initial Guess (BIG) model. The BIG model demonstrates that the ``social influence'' experiment condition enhances position bias~\\cite{Burghardt2018}, \ntherefore cognitive heuristics that produce position bias and social influence can be quantified with a single parameter, a substantial simplification over previous work \\cite{MusicLabModel,PopularDynam}. Moreover, it helps explain why users often choose the worst answer when answer quality differences are small.\n\nThe BIG model not only improves our understanding of how answers are chosen, but it allows us to test different ranking policies in simulations to answer RQ2 and RQ3. Importantly, these simulations demonstrate that cognitive heuristics can make popularity-based ranking highly unstable.  When the quality difference between the options is small, initially minor differences in popularity can create a cumulative advantage \\cite{Rijt2014}, meaning the better answer does not always become the most popular.  However, when the quality difference passes a critical point, popularity-ranking is stable, and the better answer eventually becomes the most popular. These results may help explain an under-appreciated finding of Salganik et al. (\\citeyear{Salganik2006}) that the best and worst songs tended to be correctly ranked when songs were ordered by popularity, but intermediate-quality songs landed anywhere in between. \n\nFinally, to answer RQ3, we propose an algorithm that rectifies this instability by ordering answers based on their inferred quality, which we call RAICR: Rectifying Algorithmic Instabilities in Crowdsourced Ranking. This method is found to be stable and consistently order the better answer first, thus making best answers easier to find, even when they are only slightly better. RAICR ranks answers as well as, or better than, common baselines such as ordering by popularity or by recency (ranking by the last answer picked).\n\nOur work shows that individual decisions within crowdsourcing systems are strongly affected by cognitive heuristics, which collectively create instability and poor crowd wisdom. Designers of crowdsource systems need to account for these biases in order make good content easier to find. Algorithms such as RAICR, however, can correct for these biases, thereby improving the wisdom of crowds. \n\n\\section{Related Literature}\n\n\\subsection{Crowdsourcing}\nCrowdsourcing has a two-century long history demonstrating how a collective can outperform individual experts \\cite{JuryThm,Galton1908,surowiecki2005wisdom,Kaniovski,Simoiu2019}, thus creating the moniker ``wisdom of crowds''. Crowds have been shown to beat sport markets \\cite{Brown2019,Peeters2018}, corporate earnings forecasts \\cite{Zhi2019}, and improve visual searches \\cite{Juni2017}. One reason crowd wisdom works is due to the law of large numbers: assuming unbiased and independent guesses, the average guess should converge to the true value. \n\nIndividual decisions, including in online settings, are biased by cognitive heuristics, such as anchoring \\cite{shokouhi2015anchoring}, primacy \\cite{Primacy1}, prior beliefs \\cite{white2013beliefs} \nand position bias \\cite{Burghardt2018}. These biases are not necessarily canceled out with large samples \\cite{Prelec2017,Kao2018}. As a result, aggregating guesses of a crowd does not necessarily converge to the correct answer.\n\nGuesses are usually not independent, which can sometimes improve the wisdom of crowds. Social influence models, such as the DeGroot model \\cite{Degroot1974}, have been shown to push simulated agents to an optimal decision \\cite{Golub2010,Mossel2015,Bala1998,Acemoglu2011}. These results have been backed up experimentally \\cite{Becker2017,Becker2019,Ungar2012,Tetlock2017}, even when opinion polarization is included \\cite{Becker2019}. One reason social influence can be beneficial is that it encourages people who are way off the mark to improve their guess \\cite{Mavrodiev2013,Becker2017,Abeliuk2017www}. \n\n\nOften, however, social influence can reduce crowd wisdom. Corporate earnings predictions \\cite{Zhi2019}, jury decisions  \\cite{Kaniovski,BurghardtJury}, and other guesses can degrade with influence \\cite{Lorenz2011,Lorenz2015,Simoiu2019}, and malevolent individuals can manipulate people to make particular collective decisions \\cite{SocialInfluenceBias,Asch1951}. Too much influence by a single individual can also reduce the wisdom of collective decisions \\cite{Becker2017,Acemoglu2011,Golub2010}, and deferring to friends can sometimes make unpopular (and potentially low-quality) ideas appear popular \\cite{Lerman2016}. \n\nRecent work has also demonstrated how other cognitive heuristics can affect crowd wisdom. After the landmark study by Salganik et al. (\\citeyear{Salganik2006}), some researchers found that social influence has no effect on decisions \\cite{Antenore2018}, or that position bias, i.e., the preference to choose options listed first, largely explain biases in crowdsourcing \\cite{lerman14as,MusicLabModel}. Social influence instead enhances the position bias \\cite{Burghardt2018,MusicLabModel}. Burghardt et al. (\\citeyear{Burghardt2018}) have begun to tease apart these effects, showing that while social influence enhances position bias, it has no marginal effect when we control for position. This is consistent with our approach, which models and rectifies both biases with a single parameter. \n\n\\subsection{Algorithmic Ranking}\n\nThe goal of a ranking algorithm is to make good content easier to find. \nMany papers have begun to address this goal \\cite{Page1999,Jarvelin2002,Bendersky2011}, which has recently been applied to crowdsourced ranking. Because of human biases, however, algorithms that na{\\\"i}vely use human feedback to suggest content will end up forming echo chambers \\cite{Hilbert2018,Bozdag2013,Hajian2016}, or only recommend already-popular items \\cite{Abdollahpouri2017,Abdollahpouri2019}. This can also give some content a cumulative advantage \\cite{Rijt2014}, even when it is of similar quality to content that remains unpopular. \n\nTo correct for algorithmic bias in this paper, we create a ranking method that follows the strategy of Watts, who says, ``...we can instead measure directly how they respond to a whole range of possibilities and react accordingly'' \\cite{Watts2012}. In the present context, this strategy implies we can create better algorithms for option ranking by observing, and addressing, how people respond to social influence and position biases. We show this strategy applied in RAICR improves upon simple algorithms used in the past, which include ordering results by popularity \\cite{Burghardt2018} or recency \\cite{lerman14as}. While some crowdsourced ranking strategies use a two-tier platform model, in which researchers rank options based on whether content is downloaded and rated \\cite{Salganik2006,Antenore2018,Abeliuk2017www}, RAICR is based on a common simpler model in which we only observe if content is chosen \\cite{Burghardt2018,PopularDynam}. This subtle difference implies many previous ranking schemes are not applicable. The present paper also compliments previous work that uses features, such as answer position, to predict Q\\&A website quality \\cite{Shah2010,MyopiaCrowd}.\n\n\\section{Experiment}\n\nThe experiment asked subjects, hired through Amazon Mechanical Turk between August 2018 and September 2019, to answer a series of questions shown in the Appendix. The order of questions was randomized for each subject, and questions were not time-limited. Questions include specifying the ratio of the areas of two shapes or the lengths of two lines, or the number of dots in an image. We designed the experiment around quotidian tasks that do not require specific expertise but are difficult for most people. Despite their difficulty, the questions have objectively correct answers. We quantify the quality of an answer by the difference from the mean of all guesses, which better matches a log-normal distribution, as discussed later. In the Appendix, we define quality of an answer by the difference from the correct value and find results are qualitatively very similar. \n\nSubjects were assigned to one of three conditions. In the \\textit{guess condition}, shown in the left panel of Fig.~\\ref{fig:ExperimentSchematics}, the subjects could freely type their guesses. In the \\textit{control condition}, shown in the center panel of Fig.~\\ref{fig:ExperimentSchematics}, subjects were told to choose the best among two options, and were not told how the two answers were ordered. Finally, in the \\textit{social influence condition}, shown in the right panel of Fig.~\\ref{fig:ExperimentSchematics}, they were told the first among two answers was the most popular, but the layout was otherwise identical to the previous condition. We only showed ten questions to each subject to reduce the effects of performance depletion in Q\\&A systems~\\cite{Ferrara17}. Further, to reduce bias due to other phenomena, such as the decoy effect \\cite{Huber1982}, we showed subjects only two choices. Subjects in the same condition but assigned on different days have statistically similar behavior.\n\nApproximately 1800 subjects are evenly split between the conditions (596, 586, and 587 for the guess, control, and social influence condition, respectively). For guess values, we remove extreme outliers (guesses smaller than 1 or greater than $10^6$). All answers\nare supposed to be greater than 1, while values greater than $10^6$ may affect mean values and appear to represent throwaway answers. The number of valid participants for each question is shown in Table\\ref{tab:guesssummary}.\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Number of subjects for each guess question (after cleaning)}\n\\label{tab:guesssummary}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nQ1&Q2&Q3&Q4&Q5&Q6&Q7&Q8&Q9&Q10\\\\ \\hline\n595&594&593&595&592&593&589&591&584&590\\\\ \\hline\n\\end{tabular}\n\\begin{flushleft}\n\\end{flushleft}\n\\end{table}\n\n\nMechanical Turk workers were hired if they had an approval rate of over $95\\%$, completed more than 1000 Human Intelligence Tasks (HITs), and never participated in any of the experiment conditions before. Each worker was paid $\\$1.00$ for the guessing condition and $\\$0.50$ for the other two conditions. The assignment took $6$ minutes on average for the guess condition and $2$ minutes on average for the other conditions, equivalent to an hourly wage of $\\$10-\\$15$. The human experiment was approved by the appropriate IRB board.\n\n\\section{Results}\n\\subsection{A Mathematical Model of Decisions}\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{figures\/Fig2.pdf}\n\\caption{\\label{fig:CDFGuess} The CDF of guesses for each question. (a) Before normalization, and (b) after normalization.\n}\n\\end{figure}\n\nWe use data gathered from the experiment to answer \\textbf{RQ1}: \\textit{How does quality and presentation of options jointly impact individual decisions?}\n\n\\subsubsection{Open-Ended Experiment Condition} \nTo derive the model of how people make these decisions, we start by constructing the distribution of guesses for each question (the \\textit{guess} condition in the experiment). \nThe guesses, plotted in Appendix Fig.~\\ref{fig:GuessExperimentResults} and CDF shown in Fig.~\\ref{fig:CDFGuess}a, are highly variable (by as many as six orders of magnitude), while the correct answers vary by three orders of magnitude. The median guess may differ from the true answer, in agreement with previous work \\cite{Kao2018}, but values are typically the correct order of magnitude.\n\nWe normalize guesses by defining a new variable $A$:\n\\begin{equation*}\n    A = \\frac{\\text{ln}(X)-\\langle \\text{ln}(X)\\rangle}{\\sigma_{\\text{ln}(X)}},\n\\end{equation*}\nwhere $X$ is a guess value, and $\\langle \\text{ln}(X)\\rangle$ is the mean of the logarithm of guesses. Figure~\\ref{fig:CDFGuess}b shows that this simple normalization scheme effectively collapses answer guesses to a single distribution. We show in the Appendix that guesses are not normally distributed, but instead are better approximated as log-normal, in agreement with previous work on a different set of questions \\cite{Kao2018}. The normalized guesses $A$ can be thought of as the $z$-scores in log-normal distributions. Alternative ways to center data, shown in the Appendix, produce similar results. We use these normalized guesses and distributions in the remaining two experiment conditions. Intuitively, if the mean of all guesses converges to the correct answer, $A=0$ can be thought of as the best answer.\n\n\\subsubsection{Two-Choice Experiment Conditions}\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{figures\/Fig3.pdf}\n\\caption{\\label{fig:ChooseVsPos} Probability to choose an answer versus its position for the social influence and control condition.\n}\n\\end{figure}\n\nThe latter two experiment conditions require subjects to pick the best among two answers to the question. For both the control and social influence conditions, answers are ordered vertically, with one answer above the other. There is a significant position effect when answers are ordered this way, as shown in Fig.~\\ref{fig:ChooseVsPos}. In the control condition, there is a slightly greater probability (52\\%) to choose the first (top) answer over the last one (p-value $<0.001$). In the social influence condition, meanwhile, the probability to choose the first answer is substantially larger (59\\%) and statistically significantly different than the control condition (p-value $<0.001$). This is in agreement with previous work showing that social influence amplifies the position effect~\\cite{Burghardt2018}.\n\n\n\\subsubsection{The Biased Initial Guess Model}\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{figures\/Fig4.pdf}\n\\caption{\\label{fig:ModelSchematics} Schematic to calculate $s(A_1,A_2)$. Guesses follow an approximately log-normal distribution unique to each question, but the normalized guesses, $A$ are approximately standard normal distributed. In an experiment, assume two candidate answers are provided, $A_1$ is listed first, and $A_2$ is listed last. The variable $s(A_1,A_2)$ in Eq.~\\ref{eq:initselect} is the probability an unknown initial guess is closer to $A_1$ than $A_2$.\n}\n\\end{figure}\n\nWe now have the necessary ingredients to model how decisions to choose an answer are affected by its quality, position, and social influence. \nWe present the Biased Initial Guess (BIG) decision model and show it is consistent with the data.\n\nWe first discuss the simplest case where a user has to choose the better of two answers $A_1$, listed first, and $A_2$, listed last, in the absence of cognitive biases. Figure~\\ref{fig:ModelSchematics} shows probability of the answer, with the normalized values of the choices $A_1$ and $A_2$, as well as the user's initial guess $A_I$ about the true answer, which we do not observe. All things equal, the user will choose the first answer $A_1$ if it is closer to the initial guess, i.e., if $|A_1-A_I|<|A_2-A_I|$, and will otherwise choose $A_2$. The probability to choose $A_1$ is then:\n\\begin{equation}\ns(A_1,A_2) = \n\\begin{cases}\n\\text{Pr}(A_I > \\frac{A_1+A_2}{2}) & A_1 > A_2 \\\\\n1\/2 & A_1 = A_2\\\\\n\\text{Pr}(A_I < \\frac{A_1+A_2}{2}) & A_1 < A_2\n\\end{cases}\n\\end{equation}\n Assuming $A_1$ and $A_2$ follow a normal distribution quantified in the guess experiment condition,\n\\begin{equation}\n\\label{eq:initselect}\ns(A_1,A_2)= \n\\begin{cases}\n\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2 & A_1 \\ge A_2 \\\\\n\\left[1 + \\text{erf}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\\right]\/2, & A_1 < A_2\n\\end{cases}\n\\end{equation}\nwhere $\\text{erf}(.)$ is the error function, $\\text{erfc}(.) = 1-\\text{erf}(.)$ is the complimentary error function. When $s(A_1,A_2)>0.5$, $A_1$ is not only more likely to be closer to the initial guess ($A_I$) than $A_2$, but is also closer to zero than $A_2$, and therefore the objectively better answer. On the other hand, when $s(A_1,A_2)<0.5$, $A_1$ is further from most guesses compared to $A_2$ and the objectively worse answer.\n\nTo better model decision-making, we have to account for biases, due to cognitive heuristics and algorithmic ranking, to explain why people do not always choose the best option. As shown in Fig.~\\ref{fig:ChooseVsPos}, sometimes they choose the first answer even if it is not the best answer. We quantify this position bias by assuming that with probability $p$ participants choose the first answer regardless of its quality. This parameter should presumably be small in the control condition and large in the social influence condition. Subjects may also choose an answer regardless of its position or quality because there is no monetary incentive to choose good answers. We model this by allowing subjects to choose an answer at random with a probability $r$. Taking these two heuristics into account, we arrive at the BIG Model: \n\n\\begin{equation}\n\\label{eq:model_full}\n\\begin{split}\n\\text{Pr}(\\text{Choose}~A_{1}) &= r\/2 + (1 - r) [p + (1 - p) s(A_1,A_2)]\\\\\n&= \\begin{cases}\nr\/2 + (1 - r) \\left[p + (1 - p)\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right] & A_1 \\ge A_2 \\\\[7pt]\nr\/2 + (1 - r) \\left[p + (1 - p)\\left[1 + \\text{erf}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\\right]\/2\\right], & A_1 < A_2\n\\end{cases}\n\\end{split}\n\\end{equation}\nThe probability of choosing $A_2$ is simply the compliment of this probability. Because $A=0$ is expected to be the best answer, with some simple manipulation we can infer the probability the best answer is chosen.\n\n\\begin{equation}\n\\label{eq:model_best}\n\\text{Pr}(\\text{Choose}~A_{1}|A_{1}~\\text{Best}) = \\begin{cases}\nr\/2 + (1 - r) \\left[p + (1 - p) \\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right]  & \\frac{A_1+A_2}{2}<0\\\\[7pt]\nr\/2 + (1 - r) \\left[p + (1 - p) \\left(1-\\text{erfc}\\left(\\frac{1}{\\sqrt 2}\\frac{A_1+A_2}{2}\\right)\/2\\right)\\right] & \\frac{A_1+A_2}{2}\\ge0\\\\\n\\end{cases}\n\\end{equation}\nA similar equation can model $\\text{Pr}(\\text{Choose}~A_{2}|A_{2}~\\text{Best})$.\n\nAgreement between the model and data is shown in Fig.~\\ref{fig:ModelFit}. In the \\textit{control condition} (Fig.~\\ref{fig:ModelFit}a), the best parameters are $r=0.28\\pm0.02$ and $p=0.05\\pm0.02$. We find that the log-likelihood of the model, $\\ell=-3002.49$, is not statistically different from log-likelihood if the data came from the model itself: p-value $=0.40$. See Methods for how p-values and error bars are calculated. We also check if we need both parameters, $r$ and $p$, using the likelihood ratio test and Wilks' Theorem~\\cite{Wilks1938}. We compare the likelihood ratio of the two-parameter model to simpler models with $p$ or $r$ (or both) set to zero. The probability a simpler model could fit the data as well or better is $\\le 0.002$. We conclude that our model describes the control condition very well. \n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/Fig5.pdf}\n\\caption{\\label{fig:ModelFit} Decision model agreement with experiment data. Plots show the probability the best answer is chosen for the model (lines) and experiment (symbols) under (a) the control condition and (b) the social influence condition. Purple diamonds are probabilities when the best answer is the first answer, $A_1$ and green squares for when the best answer is $A_2$.\n}\n\\end{figure}\n\nThe agreement between data and model is similarly close in the \\textit{social influence condition} (Fig.~\\ref{fig:ModelFit}b). The position bias parameter $p=0.21\\pm 0.01$ is larger than in the control condition, in agreement with expectations. We also find $r=0.08\\pm 0.02$, thus social influence reduces the frequency of random guesses. In both experiment conditions, surprisingly, $\\approx 20\\%$ of users choose answers for reasons besides ``quality'' ($r\/2+(1-r)p = 18\\%$ and $23\\%$ for the control and social influence conditions, respectively). Similar to the control condition, we find that the model is consistent with the data. The log-likelihood of the empirical data ($\\ell=-3190.13$) is not statistically different from log-likelihood values if the data came from the model: p-value $0.47$. The probability a simpler model ($r$ or $p$ set to zero) could fit the data as well or better is $<10^{-5}$. In conclusion, we find the BIG model is consistent with both experiment conditions and its parameters are interpretable and meaningful. In the Appendix, we show that all these results are consistent when we look at a subset of experiment questions or center the data differently. \n\n\\subsection{Algorithmic Ranking Instability}\nCrowdsourcing websites automatically highlight what they consider the best choices to help their users more quickly discover them. For example, Stack Exchange (like other Q\\&A platforms) usually ranks answers to questions by the number of votes they receive. Despite problems with popularity-based ranking identified in previous studies~\\cite{Salganik2006,lerman14as}, it is widely used for ranking content in crowdsourcing websites. In this section we identify an instability in popularity-based ranking: the first few votes a worse answer receives can lock it in the top position, where it acquires cumulative advantage \\cite{Rijt2014}. This allows us to answer \\textit{RQ2: When is algorithmic ranking unstable and does not reliably identify the best option?} \n\n\n\\begin{figure*}[tbh!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/Fig6.pdf}\n\\caption{Comparison of ranking policies via simulations. Plots show the probability the best answer is ranked first after (a) 50, (b) 500, and (c) 20,000 votes, when answers are ranked by quality (black line), recency (orange dashed line), and popularity. Also shown in (c) is the critical value of $A_{\\text{worst}}$ based on Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect}. In these simulations, subjects choose answers following the BIG Model with $p=0.2$ and $r=0.09$. ``Popularity 0'' (green dashed line), ``Popularity -10'' (cyan dashed line), and ``Popularity -200'' (red dashed line) means that the worst answer starts with a 0, 10, or 200 vote advantage, respectively. Shaded areas are 95\\% confidence intervals. \n\\label{fig:InitialGuessModelRankData} \n}\n\\end{figure*}\n\nTo demonstrate the instability, we simulate a group of agents who choose answers according to the BIG model with $p=0.2$ and $r=0.09$. In the simulations, one answer is objectively best, e.g., exactly equal to the correct answer ($A_{\\text{best}}=0$), while the worst answer is larger than $A_{\\text{best}}$ (results are symmetric if $A_{\\text{worst}}<A_{\\text{best}}$). At each timestep, a new user arrives and independently chooses the first or last answer according to Eq.~\\ref{eq:model_full}. Answers reorder depending on the ranking algorithm. Figure~\\ref{fig:InitialGuessModelRankData} shows our results. If the worst answer, $A_{\\text{worst}}$, is not too large and has a few extra votes (a common occurrence when the worst answer is posted before the better answer), we find that popularity ordering completely breaks down---the worst answer usually becomes more popular and is ranked first. Even when both answers start with the same number of votes, the worse answer is often ranked first. Results remain stable even after 20K votes; the effect does not appear transient.\n\nWe can explain this result using our model. Using the compliment of Eq.~\\ref{eq:model_full}, we can define the probability the best answer is chosen, conditional on it being ranked last:\n\\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2) = r\/2 + (1 - r) (1 - p) s(A_1,A_2)\n\\end{equation}\nWe find that\n\\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2) = \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_1)-p(1-r)\n\\end{equation}\nwhere $\\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_1)$ is the probability of choosing the best answer when it is ranked first. If \\begin{equation}\n    \\text{Pr}(\\text{Choose}~ A_{\\text{best}}=A_2)>\\text{Pr}(\\text{Choose}~ A_{\\text{worst}}=A_1)\n\\end{equation} \nthen subjects are more likely to choose the better answer regardless of whether its ranked first or second, therefore answer order is stable. When the above inequality is not true, however, then subjects are more likely to pick the first answer regardless of its quality, and the popularity ranking is unstable. The critical value of $A_{\\text{worst}}$ between the two regimes is when \n\\begin{equation}\\label{eq:scrit}\n    s_{\\text{crit}}=\\frac{1}{2 (1-p)}\n\\end{equation}\nand is independent of $r$. Intuitively, if the answer is exceptionally bad, it will always be less popular. This is alike to the results in Salganik's MusicLab study \\cite{Salganik2006}, where particularly good and bad songs were ranked correctly. However, if the first answer is likely to be chosen regardless of quality, the worst answer will continue accumulating votes and remain in the top position. Given $A_{\\text{best}}=0$, we can use Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect} to numerically solve for the critical value of $A_{\\text{worst}}$. We plot the critical point as a function of $p$ and $A_{\\text{worst}}$ in phase diagram in Fig.~\\ref{fig:SimPhaseSpace}. We see that there is always a large part of the phase space where popularity-based ranking will be unpredictable if answer quality is close together. Based on Eq.~\\ref{eq:scrit}, if $p\\ge 0.5$, there will be no case where popularity-based ranking is guaranteed to correctly rank answers. While this is an extreme case, it still points to substantial limitations of popularity-based ranking.\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=0.44\\textwidth]{figures\/Fig7.pdf}\n\\caption{\\label{fig:SimPhaseSpace} Phase diagram for popularity-based ranking based on Eqs.~\\ref{eq:scrit} and~\\ref{eq:initselect}. We show the boundary between regimes where best answers are guaranteed to be ranked first with enough votes (white area) and where ranking is unstable (gray area) as a function of position bias ($p$) and the value of the worse answer ($A_{\\text{worst}}$). In this plot, larger values of $A_{\\text{worst}}>0$ correspond to worse answers and results are symmetric for $A_{\\text{worst}}<0$. \n}\n\\end{figure}\n\\subsection{Stabilizing Algorithmic Ranking}\n\nGiven the problems with popularity ranking, it is critical to create a more consistent ranking algorithm. Moreover, while we have so-far explored questions with numeric answers, we want a method that works for all types of answers. Using the BIG model, we can answer RQ3: \\textit{How can we stabilize algorithmic ranking such that the best option is typically ranked first?}\n\nAssume we can approximate $p$ and $r$, then we can invert Eq.~\\ref{eq:model_full} and use votes to infer the only unknown variable $s(A_{\\text{best}},A_{\\text{worst}})$. When $s(A_{\\text{best}},A_{\\text{worst}})>0.5$, $A_{\\text{best}}$ is the best answer, but if we incorrectly rank $A_{\\text{worst}}$ first, $s(A_{\\text{worst}},A_{\\text{best}})<0.5$. We can therefore rank the answer in which $s(A_{\\text{best}},A_{\\text{worst}})>0.5$ as the best answer. This is the backbone of the RAICR algorithm. The algorithm uses maximum likelihood estimation to solve for $s(A_{\\text{best}},A_{\\text{worst}})$, as shown in the Appendix. Therefore, if the data matches the BIG model with the correct $r$ and $p$ parameters, our method \\emph{optimally infers the correct raking} by having minimal variance and no bias \\cite{Newey1994}. Moreover, \\emph{this method only depends on the votes an answer receives rather than the type of answer}, such as a numerical or textual answer. We compare quality ranking to popularity-based ranking, and recency-based ranking (ranking by the last answer picked), as shown in Fig.~\\ref{fig:InitialGuessModelRankData}. The probability recency ranks the best answer first is calculated as the self-consistent equation: \n\\begin{dmath}\n    \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) = \\text{Pr}(\\text{Choose}~ A_{\\text{best}}|A_{\\text{best}}~\\text{First}) \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency})\\\\  +  \\text{Pr}(\\text{Choose}~ A_{\\text{best}}|A_{\\text{best}}~\\text{Last})(1 - \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) ).\n\\end{dmath}\nThis represents the limit answers acquire many votes. The solution to this equation is:\n\\begin{dmath}\n    \\text{Pr}(\\text{Rank $A_{\\text{best}}$ First}|\\text{Recency}) = \\frac{2 (1-p) (1-r) s(A_{\\text{best}},A_{\\text{worst}}\n    +r}{2-2 p (1-r)\n\\end{dmath}\nWe find that the RAICR algorithm performs at least as well as popularity-based ranking by ranking the better answer first, and often better after 20-50 votes (Fig.~\\ref{fig:InitialGuessModelRankData}a and Appendix Fig.~\\ref{fig:SimPRobust}). Moreover, RAICR always outperforms the recency-based algorithm. The benefit of the RAICR algorithm only improves as we collect more votes. \nFor example, Fig.~\\ref{fig:InitialGuessModelRankData}b shows after after 500 votes the advantage of RAICR is larger, and Fig.~\\ref{fig:InitialGuessModelRankData}c shows that after 20K votes the method is nearly optimal. Popularity-based ranking performs badly for $A_{worse}<0.6$, and recency performs worst if $A_{worse}>0.6$. \n\nWhile we show these results hold when we have 20 to 20,0000 votes, many platforms are underprovisioned, with  a large fraction of webpages \nreceiving little attention and votes~\\cite{baeza2018bias,Gilbert2013}. It is the webpages that receive many votes, however, which may be the most important. Correctly ranking options in these popular pages is therefore especially critical to crowdsourcing websites. Moreover, a moderate number of votes is generally needed to make a reasonable estimate of quality, so very few methods will accurately rank unpopular pages.\n\nOne caveat of the RAICR algorithm is that we need an approximate value for two parameters: $r$ and $p$. What happens if either of these parameters are far off? For example, we could assume $r=0$ when $r=0.09$. Results in the Appendix, however, show that the findings are quantitatively very similar, and therefore our model is robust to assumptions about $r$. On the other hand, what if we incorrectly estimate both $r$ and $p$? We show in the Appendix that even in this worst-case scenario, our method performs slightly worse, but is comparable or substantially better than popularity-based ranking. \n\n\\section{Future Work and Design Implications\n\nIn the experiment, we decomposed the crowdsourcing task to its basic components to reduce the complexity and variation inherent in real world tasks. While this helps to disentangle effects of option quality and position without confounding factors muddying the relationships, future work is needed to verify ecological validity of our results \\cite{ObservationalVsExp}. It is encouraging that the probability to choose an answer (Fig.~\\ref{fig:ChooseVsPos}b) is quantitatively similar to empirical data gathered from Stack Exchange \\cite{Burghardt2018}, despite Mechanical Turk workers not being representative of the general population \\cite{Munger2019}. \n\nFor simplicity, we only explored two-option questions; future work should aim to understand multi-option decision-making given options of variable quality. A generalization of RAICR should also address more complicated biases, such as preference for round numbers \\cite{Fitz1986}, anchoring \\cite{shokouhi2015anchoring,Furnham2011}, and biases that appear in multi-option decisions, such as the decoy effect \\cite{Huber1982}. Finally, while RAICR is found to be robust to moderate changes in its parameters, this algorithm and its extensions may fail to rank options properly if its parameters are far off, or if the BIG model is wrong. In the experiment, the model is backed by data, but future work needs to address whether other tasks or questions follow this model. \n\nOur results offer implications for crowdsourcing platforms. First, designers must recognize the limits of crowdsourcing due to biases implicit in their platform. In our experiment people often upvoted options at random (up to 20\\% of all votes), and chose an inferior option simply because it was shown first. This creates a ranking instability when options are of similar quality. Our controlled experiment and mathematical model point to ways we can counteract this instability. Designers should similarly create platform-tailored mathematical models and controlled experiments to rigorously test how crowds can better infer the best options. \n\nA key property of our RAICR algorithm is that it relies on accurate modeling of user decisions to counteract cognitive biases. In effect, each vote is weighted depending on the ranks of answers at the time the vote is cast. The idea is similar to one described by Abeliuk et al.~\\citeyear{Abeliuk2017www} that ranks items by their inferred quality in order to more robustly identify blockbuster items. Similar weighting schemes could be applied to future debiased algorithms to address the unique goals of each crowdsourcing platform. \n\nThere are also simple methods that platforms like  Reddit, Facebook, and Stack Exchange can try that may greatly outperform the baselines we mention in our paper. For example, items that have not yet acquired many votes can be ranked randomly to reduce initial ranking biases. Alternatively, new posts and links could be ranked appropriately but their popularity could be hidden until they gather enough votes. Our results suggest this could reduce social influence-based position bias up until the true option quality is more obvious. \n\n\\section{Conclusion}\nIn this paper, we introduce an experiment designed to inform how cognitive biases and option quality interact to affect crowdsourced ranking. Results from this experiment help us create a novel mathematical decision model, the BIG model, that greatly improves our understanding of how people find the best answer to a question as a function of answer quality, rank, and social influence. This model is then applied to the RAICR algorithm to better rank answers. The BIG model also helped us uncover instability in popularity-based ranking. The instability depends on the quality of options: when there are large differences between option qualities, popularity converges optimally and predictably. However, when the difference between the quality of options is small, the better option may not always become the most popular. These results can help us better understand the foundational empirical results of Salganik et al. (\\citeyear{Salganik2006}), who found that popularity-based ranking correctly ranked high and low quality songs, while the ranking of intermediate quality songs was highly unstable. Although our experimental setup is undeniably simpler than real crowdsourcing websites, our results suggest that accurate models of user behavior together with mathematically principled inference can improve the efficiency of crowdsourcing. \n\n\n\n\\begin{acks}\nOur work is supported by the US Army Research Office MURI Award No. W911NF-13-1-0340 and the DARPA Award No. W911NF-17-1-0077. Data as well as code to create experiments, create simulations, and analyze data is available at https:\/\/github.com\/KeithBurghardt\/QualityRankCodeAndData.\n\\end{acks}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nTrigonometric moment problems are ubiquitous in systems and control, such as spectral estimation, signal processing, system identification, image processing and remote sensing \\cite{bose2003multidimensional,ekstrom1984digital,stoica1997introduction}. In the  (truncated) multidimensional trigonometric moment problem we seek a nonnegative measure $\\derivd \\mu$ on $\\mathbb{T}^d$ satisfying  the moment equation \n\\begin{equation} \\label{eq:Cov}\nc_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}) \\quad\\text{for all ${\\boldsymbol k}\\in\\Lambda$},\n\\end{equation}\nwhere $\\mathbb{T}:=(-\\pi,\\pi]$, ${\\boldsymbol \\theta}:=(\\theta_1,\\ldots, \\theta_d)\\in\\mathbb{T}^d$, and $({\\boldsymbol k},{\\boldsymbol \\theta}):=\\sum_{j=1}^d k_j\\theta_j$ is the scalar product in $\\mathbb{R}^d$. Here $\\Lambda \\subset \\mathbb{Z}^d$ is a finite index set satisfying $0 \\in \\Lambda$ and $- \\Lambda = \\Lambda$. A necessary condition for \\eqref{eq:Cov} to have a solution is that the sequence \n\\begin{equation}\n\\label{eq:covariances}\nc:= [ c_{\\boldsymbol k} \\mid {\\boldsymbol k}:=(k_1,\\ldots, k_d) \\in \\Lambda ]\n\\end{equation}\nsatisfy the symmetry condition $c_{-{\\boldsymbol k}}=\\bar{c}_{\\boldsymbol k}$. The space of sequences \\eqref{eq:covariances} with this symmerty  will be denoted $\\mathfrak{C}$ and will be represented by vectors  $c$, formed by ordering the coefficient in some prescribed manner, e.g., lexiographical. Note that $\\mathfrak{C}$ is isomorphic to $\\mathbb{R}^{|\\Lambda|}$, where $|\\Lambda|$ is the cardinality of $\\Lambda$. \nHowever, as we shall see below, not all $c\\in\\mathfrak{C}$ are {\\em bona fide\\\/} moments for nonnegative measures $\\derivd \\mu$. \n\nIn many  of the applications mentioned above there is a natural complexity constraint prescribed by design specifications. In the context of finite-dimensional systems these constraints often arise in the requirement that transfer functions be rational. This leads to the {\\em rational covariance extension problem}, which  has been  studied in various degrees of generality in \\cite{georgiou2005solution,georgiou2006relative,karlsson2015themultidimensional,ringh2015themultidimensional,ringh2015multidimensional} and can be posed as follows. \n\nDefine $e^{i{\\boldsymbol \\theta}}:=(e^{i\\theta_1},\\ldots, e^{i\\theta_d})$ and let\n\\begin{subequations}\\label{eq:ratinaldmu}\n\\begin{equation}\\label{eq:decomp}\n\\derivd \\mu({\\boldsymbol \\theta}) = \\Phi(e^{i{\\boldsymbol \\theta}})\\derivd m({\\boldsymbol \\theta}) + \\derivd \\nu({\\boldsymbol \\theta}),\n\\end{equation}\nbe the (unique) Lebesgue decomposition of $d\\mu$  (see, e.g., \\cite[p. 121]{rudin1987real}), \nwhere $$dm ({\\boldsymbol \\theta}):=(1\/2\\pi)^d\\prod_{j=1}^d d\\theta_j$$ is the (normalized)\nLebesgue measure and $\\derivd \\nu$ is a singular measure. Then given a $c\\in\\mathfrak{C}$, we are interested in parameterizing solutions to \\eqref{eq:Cov} such that the absolutely continuous part of the measure \\eqref{eq:decomp}  takes the form\n\\begin{equation}\\label{eq:rat}\n\\Phi(e^{i{\\boldsymbol \\theta}})=\\frac{P(e^{i{\\boldsymbol \\theta}})}{Q(e^{i{\\boldsymbol \\theta}})},\\quad p, q \\in \\bar{\\mathfrak{P}}_+ \\backslash \\{0\\},\n\\end{equation}\n\\end{subequations}\nwhere $\\bar{\\mathfrak{P}}_+$ is the closure of the convex cone $\\mathfrak{P}_+$ of the coefficients $p\\in\\mathfrak{C}$ corresponding to trigonometric polynomials\n\\begin{equation}\\label{Ptrigpol}\nP(e^{i{\\boldsymbol \\theta}})=\n\\sum_{{\\boldsymbol k}\\in \\Lambda}p_{\\boldsymbol k} e^{-i({\\boldsymbol k},{\\boldsymbol \\theta})}, \\quad p_{-{\\boldsymbol k}}=\\bar{p}_{\\boldsymbol k}\n\\end{equation}\nthat are positive for all ${\\boldsymbol \\theta}\\in \\mathbb{T}^d$. \n\nThe reason for referring to this problem as a rational covariance extension problem is that the numbers \\eqref{eq:covariances} correspond to covariances $c_{\\boldsymbol k}:= \\ExpOp \\{y({\\bf t}+{\\boldsymbol k})\\overline{y({\\bf t})}\\}$ of a  discrete-time, zero-mean, and homogeneous\\footnote{Homogeneity  generalizes stationarity in the case $d=1$. }  stochastic process  $\\{y({\\bf t});\\,{\\bf t}\\in\\mathbb{Z}^d\\}$. The corresponding power spectrum, representing the energy distribution across frequencies, is defined as the nonnegative measure $d\\mu$ on $\\mathbb{T}^d$ whose Fourier coefficients are the covariances \\eqref{eq:covariances}.\nA scalar version of this problem ($d=1$) was first posed by Kalman \\cite{kalman1981realization} and has been extensively studied and solved in the literature \\cite{georgiou1983partial, byrnes1995acomplete, byrnes2002identifyability, enqvist2004aconvex, nurdin2006new, CarliGeorgiou2011, lindquist2013thecirculant, byrnes2000anewapproach, zorzi2014rational, musicus1985maximum}.\nIt has been generalized to more general scalar moment problems \\cite{byrnes2001ageneralized, georgio2003kullback, byrnes2006generalizedinterpolation} and to the multidimensional setting \\cite{georgiou2006relative, georgiou2005solution, ringh2015multidimensional, ringh2015themultidimensional, karlsson2015themultidimensional}. Also worth mentioning here is work by Lang and McClellan \\cite{lang1982multidimensional, lang1983spectral, mcclellan1982multi-dimensional, mcclellan1983duality, lang1982theextension, lang1981spectral} considering the multidimensional maximum entropy problem, which hence has certain overlap with the above literature.\n\nThe multidimensional rational covariance extension problem posed above has a solution if and only if $c\\in\\mathfrak{C}_+$, where $\\mathfrak{C}_+$ is the open convex cone \n\\begin{equation*}\n\\mathfrak{C}_+ := \\left\\{ c \\mid \\langle c, p \\rangle > 0, \\quad \\text{for all $p \\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$} \\right\\},\n\\end{equation*}\nwhere $\\langle c, p \\rangle := \\sum_{{\\boldsymbol k} \\in \\Lambda} c_{\\boldsymbol k} \\bar{p}_{\\boldsymbol k}$ is the inner product in $\\mathfrak{C}$ (Theorem~\\ref{thm:rational}).  However, the covariances $[ c_{\\boldsymbol k} \\mid {\\boldsymbol k}:=(k_1,\\ldots, k_d) \\in \\Lambda]$ are generally determined from statistical  data. Therefore the condition $c\\in\\mathfrak{C}_+$ may not be satisfied, and testing this condition is difficult in the multidimensional case. \nTherefore we may want to find a positive measure $\\derivd \\mu$ and  a corresponding $r\\in\\mathfrak{C}_+$, namely\n\\begin{equation}\n\\label{eq:r}\nr_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}), \\quad {\\boldsymbol k}\\in\\Lambda,\n\\end{equation}\nso that $r$ is close  to $c$ in some norm, e.g.,  the Euclidian norm $\\|\\centerdot\\|_2$. This is an ill-posed inverse problem which in general has an infinite number of  solutions $\\derivd \\mu$. As we already mentioned, we are interested in rational solutions  \\eqref{eq:ratinaldmu}, and to obtain such solutions we use  regularization as in \\cite{ringh2015multidimensional}. Hence, we seek a $d\\mu$  that minimizes\n\\begin{displaymath}\n\\lambda  \\mathbb D (P\\derivd m, \\derivd \\mu)+\\frac12\\|r-c\\|_2^2\n\\end{displaymath}\nsubject to \\eqref{eq:r},\nwhere $\\lambda\\in\\mathbb{R}$ is a regularization parameter and\n\\begin{equation}\\label{eq:normalizedKullbackLeibler}\n \\mathbb D (P\\derivd m, \\derivd \\mu) :=  \\int_{\\mathbb{T}^d}  \\left(  P \\log\\frac{P}{\\Phi} \\derivd m +d\\mu -Pdm \\right)\n\\end{equation}\nis the nomalized Kullback-Leibler divergence \\cite[ch. 4]{gzyl1995method} \\cite{csiszar1991why, zorzi2014rational}. As will be explained in Section~\\ref{sec:prel}, $ \\mathbb D (P\\derivd m, \\derivd \\mu)$ is always nonnegative and has the property $\\mathbb D (P\\derivd m,P\\derivd m)=0$. \n\nIn this paper we shall consider a more general problem in the spirit of \\cite{enqvist2007approximative}. To this end, for any Hermitian, positive definite matrix $M$,  we define the weighted vector norm $\\|x\\|_M:=(x^*Mx)^{1\/2}$ and consider the problem\n\\begin{align}\\label{eq:primal_relax}\n\\min_{\\derivd \\mu \\geq 0, \\, r}  \\qquad & \\mathbb D (P\\derivd m, \\derivd \\mu) + \\frac{1}{2}\\|r - c\\|_{W^{-1}}^2 \\\\\n\\st  \\quad &r_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}), \\quad {\\boldsymbol k}\\in\\Lambda, \\nonumber\n\\end{align}\nwhich is the same as the problem above with $W=\\lambda I$. We shall refer to $W$ as the {\\em weight matrix}.\n\nUsing the same principle as in \\cite{schott1984maximum}, we shall also consider the problem to minimize $\\mathbb D (P\\derivd m, \\derivd \\mu)$ subject to \\eqref{eq:r} and the hard constraint \n\\begin{equation}\n\\label{eq:hardconstraint}\n\\|r - c \\|^2 \\le \\lambda \\,.\n\\end{equation}\nSince \\eqref{eq:r} are {\\em bona fide\\\/} moments and hence $r\\in\\mathfrak{C}_+$, while $c\\not\\in\\mathfrak{C}_+$ in general, this problem will not have a solution if the distance from $c$ to $\\mathfrak{C}_+$ is greater than $\\sqrt{\\lambda}$. Hence the choice of $\\lambda$ must be made with some care. Analogously with the {\\em rational covariance extension with soft  constraints\\\/} in \\eqref{eq:primal_relax}, we shall consider the more general problem \n\\begin{align}\\label{eq:primal_relax_new}\n\\min_{\\derivd \\mu \\geq 0, \\, r}  \\qquad &\\mathbb D (P\\derivd m, \\derivd \\mu)  \\\\\n\\st  \\quad &r_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}), \\quad {\\boldsymbol k}\\in\\Lambda, \\nonumber \\\\\n     \\quad &\\|r - c \\|_{W^{-1}}^2 \\le 1, \\nonumber\n\\end{align}\nto which we shall refer as the {\\em rational covariance extension problem with hard constraints}. Again this problem reduces to the simpler problem by setting $W=\\lambda I$.\n\nAs we shall see, the soft-constrained problem \\eqref{eq:primal_relax} always has a solution, while the hard-constrained problem  \\eqref{eq:primal_relax_new} may fail to have a solution for some weight matrices $W$. However, in Section~\\ref{sec:homeomorphism} we show that the two problems are in fact equivalent in the sense that whenever \\eqref{eq:primal_relax_new} has a solution there is a corresponding $W$ in \\eqref{eq:primal_relax} that gives the same solution, and any solution of \\eqref{eq:primal_relax} can also be obtained from \\eqref{eq:primal_relax_new} by a suitable choice of $W$. The reason for considering both formulations is that  one formulation might be more suitable than the other for the particular application at hand. For example, an absolute error estimate for the covariances is more naturally incorporated in the formulation with hard constraints. A possible choice of the weight matrix $W$ in either formulation would be the covariance matrix of the estimated moments, as suggested in \\cite{enqvist2007approximative}. This corresponds to the Mahalanobis distance  and could be a natural way to incorporate uncertainty of the covariance estimates in the spectral estimation procedure.\n\nPrevious work in this direction can be found in \\cite{shankwitz1990onthe, enqvist2007approximative, byrnes2003theuncertain, schott1984maximum, karlsson2016confidence}, \nwhere \\cite{shankwitz1990onthe,karlsson2016confidence,byrnes2003theuncertain} consider the problem of selecting an appropriate covariances sequence to match in a given confidence region. The two approximation problems considered here are similar to the ones considered in \\cite{schott1984maximum} and \\cite{enqvist2007approximative}. (For more details, also see \\cite[Ch. B]{avventi2011spectral}.)\n\n\nWe begin in Section~\\ref{sec:prel}  by reviewing the regular multidimensional rational covariance extension problem for exact covariance matching  in a broader perspective. In Section~\\ref{sec:approxsoft} we present our main results on approximate rational covariance extension with soft  constraints, and in Section~\\ref{sec:wellposed} we show that the dual solution is well-posed. In Section~\\ref{sec:no_singular_part} we investigate conditions under which there are solutions without a singular part. The approximate rational covariance extension with hard  constraints is considered in \nSection~\\ref{sec:approxhard}, and in Section~\\ref{sec:homeomorphism} we establish a homeomorhism between the weight matrices in the two problems, showing that the problems are actually equivalent when solutions exist. We also show that under certain conditions the homeomorphism can be extended to hold between all sets of parameters, allowing us to carry over results from the soft-constrained setting to the hard-constrained one.  In Section~\\ref{sec:covest} we discuss the properties of various covariance estimators, in Section~\\ref{sec:example} we give a 2D example from spectral estimation, and in Section~\\ref{sec:ex_texture}  we apply our theory to system identification and texture reconstruction. \nSome of the results of this paper were announced in \\cite{ringh2016multidimensional} without proofs.\n\n\n\n\n\\section{Rational covariance extension with exact matching}\\label{sec:prel}\n\nThe trigonometric moment problem to determine a positive measure $\\derivd \\mu$ satisfying \\eqref{eq:Cov} is an inverse problem that has a solution if and only if $c\\in\\bar{\\mathfrak{C}}_+$ \\cite[Theorem 2.3]{karlsson2015themultidimensional}, where $\\bar{\\mathfrak{C}}_+$ is the closure of $\\mathfrak{C}_+$, and then in general it has infinitely many solutions. However, the nature of possible rational solutions \\eqref{eq:ratinaldmu} will depend on the location of $c$ in $\\bar{\\mathfrak{C}}_+$. To clarify this point we need the following lemma.\n\n\\begin{lemma}\\label{lem:PsubsetC}\n$\\bar{\\mathfrak{P}}_+ \\setminus \\{0\\} \\subset \\mathfrak{C}_+$.\n\\end{lemma}\n\n\\begin{proof}\nObviously the inner product $\\langle q, p \\rangle := \\sum_{{\\boldsymbol k} \\in \\Lambda} q_{\\boldsymbol k} \\bar{p}_{\\boldsymbol k}$ can be expressed in the integral form\n\\begin{equation}\n\\label{innerprod}\n\\langle q, p \\rangle = \\int_{\\mathbb{T}^d} Q(e^{i{\\boldsymbol \\theta}})\\overline{P(e^{i{\\boldsymbol \\theta}})}\\derivd m({\\boldsymbol \\theta}),\n\\end{equation}\nand therefore $\\langle q,p\\rangle >0$ for all $q,p\\in\\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$, as $P$ and $Q$ can have zeros only on sets of measure zero. Hence the statement of the lemma follows.\n\\end{proof}\n\nTherefore, under certain particular conditions, the multidimensional rational covariance extension problem has a very simple solution with a polynomial spectral density, namely\n\\begin{equation}\n\\label{eq:dmu=Pdm}\n\\derivd \\mu= P(e^{i{\\boldsymbol \\theta}})dm({\\boldsymbol \\theta}), \\quad p \\in \\bar{\\mathfrak{P}}_+ \\backslash \\{0\\}. \n\\end{equation}\n\n\\begin{proposition}\\label{prop:polsolution}\nThe multidimensional rational covariance extension problem has a unique polynomial solution \\eqref{eq:dmu=Pdm} if and only if $c\\in\\bar{\\mathfrak{P}}_+ \\backslash \\{0\\}$, namely $P=C$, where \n\\begin{displaymath}\nC(e^{i{\\boldsymbol \\theta}}):=\\sum_{{\\boldsymbol k}\\in \\Lambda}c_{\\boldsymbol k} e^{-i({\\boldsymbol k},{\\boldsymbol \\theta})}.\n\\end{displaymath}\n\\end{proposition}\n\nThe proof of Proposition~\\ref{prop:polsolution} is immediate by noting that any such $C$ is a {\\em bona fide\\\/} spectral density and noting that  $c_{\\boldsymbol k}=\\int_{\\mathbb{T}^d}e^{i({\\boldsymbol k},{\\boldsymbol \\theta})}C(e^{i{\\boldsymbol \\theta}})dm({\\boldsymbol \\theta})$. \n\nAs seen from the following result presented in \\cite[Section 6]{karlsson2015themultidimensional}, the other extreme occurs for $c\\in\\partial\\mathfrak{C}_+:=\\bar{\\mathfrak{C}}_+\\setminus\\mathfrak{C}_+$, when only singular solutions exist.\n\n\\begin{proposition}\\label{prop:singular}\nFor any $c\\in\\partial\\mathfrak{C}_+$ there is a solution $\\derivd \\mu$ of \\eqref{eq:Cov} with support in at most $|\\Lambda|-1$ points. There is no solution with a absolutely continuous part $\\Phi\\derivd m$. \n\\end{proposition}\n\nHowever, for any $c\\in\\mathfrak{C}_+$, there is a rational solution \\eqref{eq:ratinaldmu} parametrized by $p\\in\\bar{\\mathfrak{P}}_+\\backslash\\{0\\}$, as demonstrated in \\cite{ringh2015multidimensional} by considering a primal-dual pair of convex optimization problems.  In that paper the primal problem is a weighted maximum entropy problem, but as also noted in \\cite[Sec. 3.2]{ringh2015multidimensional}, it is equivalent to\n\\begin{align}\\label{eq:primal}\n\\min_{\\derivd \\mu \\geq 0} & \\quad \\int_{\\mathbb{T}^d} P \\log \\frac{P}{\\Phi} \\derivd m({\\boldsymbol \\theta}) \\\\\n\\st & \\quad c_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}), \\quad {\\boldsymbol k} \\in \\Lambda, \\nonumber\n\\end{align}\nwhere $\\Phi dm$ is the absolutely continuous part of $d\\mu$.\nThis amounts to minimizing the (regular) Kullback-Leibler divergence between $P\\derivd m$ and $\\derivd \\mu$, subject to $\\derivd \\mu$ matching the given data \\cite{georgio2003kullback,ringh2015multidimensional}. In the present case of exact covariance matching, this problem is equivalent to minimizing \\eqref{eq:normalizedKullbackLeibler} subject to \\eqref{eq:Cov}, since $P$ is fixed and the total mass of $\\derivd \\mu$ is determined by the $0$:th moment  $c_{\\boldsymbol 0} = \\int_{\\mathbb{T}^d} \\derivd \\mu$. \nHence both $\\int_{\\mathbb{T}^d} \\derivd \\mu$ and $\\int_{\\mathbb{T}^d} P\\derivd m$ are constants in this case. Hence problem \\eqref{eq:primal_relax} is the natural extension of \\eqref{eq:primal} for the case where the covariance sequence is not known exactly. \n\nThe primal problem \\eqref{eq:primal} is a problem in infinite dimensions, but with a finite number of constraints. The dual to this problem will then have a finite number of variables but an infinite number of constraints and is given by\n\\begin{align}\\label{eq:dual}\n\\min_{q\\in\\bar{\\mathfrak{P}}_+} & \\quad \\langle c, q \\rangle - \\int_{\\mathbb{T}^d} P \\log Q \\,\\derivd m({\\boldsymbol \\theta}). \n\\end{align}\nIn particular, Theorem  2.1 in \\cite{ringh2015multidimensional}, based on corresponding analysis in \\cite{karlsson2015themultidimensional}, reads as follows.\n\n\\begin{theorem}\\label{thm:rational}\nProblem \\eqref{eq:primal} has a solution if and only if $c \\in \\mathfrak{C}_+$. For every $c \\in \\mathfrak{C}_+$ and $p \\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$ the functional in \\eqref{eq:dual} is strictly convex and has a unique minimizer $\\hat{q} \\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$. Moreover, there exists a unique $\\hat{c} \\in \\partial \\mathfrak{C}_+$ and a (not necessarily unique) nonnegative singular measure $d\\hat\\nu$ with support \n\\begin{equation}\n\\label{supp(dnu)}\n\\supp(d\\hat\\nu) \\subseteq \\{ {\\boldsymbol \\theta} \\in \\mathbb{T}^d  \\mid \\hat{Q}(e^{i{\\boldsymbol \\theta}}) = 0 \\}\n\\end{equation}\nsuch that\n\\begin{subequations}\n\\begin{align}\n& c_{{\\boldsymbol k}} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\left( \\frac{P}{\\hat Q}  \\derivd m  + d\\hat\\nu \\right),\\quad {\\boldsymbol k} \\in \\Lambda, \\\\\n& \\hat{c}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} d\\hat\\nu, \\quad{\\boldsymbol k} \\in \\Lambda.\n\\end{align}\n\\end{subequations}\nFor any such $d\\hat\\nu$, the measure \n\\begin{equation}\n\\label{ }\nd\\hat\\mu({\\boldsymbol \\theta}) = \\frac{P(e^{i{\\boldsymbol \\theta}})}{\\hat{Q}(e^{i{\\boldsymbol \\theta}})}\\derivd m({\\boldsymbol \\theta}) + d\\hat\\nu({\\boldsymbol \\theta})\n\\end{equation}\nis an optimal solution to the problem  \\eqref{eq:primal}. Moreover, $d\\hat\\nu$ can be chosen with support in at most $|\\Lambda|-1$ points, where $|\\Lambda|$ is the cardinality of the index set $\\Lambda$.\n\\end{theorem}\n\nIf $c\\in \\partial \\mathfrak{C}_+$, only a singular measure with finite support would match the moment condition (Proposition~\\ref{prop:singular}). In this case, the problem \\eqref{eq:primal} makes no sense, since any feasible solution has infinite objective value.  \n\nIn \\cite{karlsson2015themultidimensional} we also derived the KKT conditions\n\\begin{subequations}\\label{eq:opt_cond_exact}\n\\begin{align}\n& \\hat{q} \\in \\bar{\\mathfrak{P}}_+, \\quad \\hat c \\in \\partial \\mathfrak{C}_+, \\quad \\langle \\hat{c}, \\hat{q} \\rangle = 0 \\label{eq:opt_cond_exact_slack} \\\\\n& c_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\frac{P}{\\hat Q}\\derivd m + \\hat{c}_{\\boldsymbol k},\\quad {\\boldsymbol k} \\in \\Lambda  ,\\label{eq:opt_cond_exact_KKT}\n\\end{align}\n\\end{subequations}\nwhich are necessary and sufficient for optimality of the primal and dual problems. \n\nSince  \\eqref{eq:primal}  is an inverse problem, we are interested in how the solution depends on the parameters of the problem. From Propositions 7.3 and 7.4 in \\cite{karlsson2015themultidimensional} we have the following result.\n\n\\begin{proposition}\\label{prop:cp2qhat}\nLet $c$,  $p$ and $\\hat{q}$  be as in Theorem~\\ref{thm:rational}. Then the map $(c,p)\\mapsto \\hat{q}$ is continuous.\n\\end{proposition}\n\nTo get a full description of well-posedness of the solution we would like to extend this continuity result to the map $(c,p)\\mapsto (\\hat{q},\\hat{c})$. However, such a generalization is only possible under certain conditions. The following result is a consequence of Proposition~\\ref{prop:cp2qhat} and \\cite[Corollary 2.3]{ringh2015multidimensional}.\n\n\\begin{proposition}\\label{prop:c2qhatchat2}\nLet $c$, $p$, $\\hat{q}$ and $\\hat{c}$ be as in Theorem~\\ref{thm:rational}. Then,  for  $d\\leq 2$ and all $(c,p)\\in\\mathfrak{C}_+\\times\\mathfrak{P}_+$, the mapping $(c,p)\\to(\\hat q, \\hat c)$  is continuous.  \n\\end{proposition}\n\nCorollary 2.3 in \\cite{ringh2015multidimensional} actually ensures that $\\hat{c}=0$ for $d\\leq 2$ and $p\\in\\mathfrak{P}_+$. However, in Section~\\ref{ssec:qhat2chat} we present a generalization of Proposition~\\ref{prop:c2qhatchat2} to cases with $d\\geq3$, where then $\\hat{c}$ may be nonzero. (The proof of this generalization can be found in \\cite{ringh2017further}.)\nHere we shall also consider an example where continuity fails when $p$ belongs to the boundary $\\partial\\mathfrak{P}_+:= \\bar{\\mathfrak{P}}_+ \\setminus \\mathfrak{P}_+$, i.e.,  the corresponding nonnegative trigonometric polynomial $P(e^{i{\\boldsymbol \\theta}})$ is zero in at least one point. \n\n\\section{Approximate covariance extension with soft constraints}\\label{sec:approxsoft}\n\nTo handle the case with noisy covariance data, when $c$ may not even belong to $\\mathfrak{C}_+$, we  relax the exact covariance matching constraint \\eqref{eq:Cov} in the primal problem \\eqref{eq:primal} to obtain the problem \\eqref{eq:primal_relax}. In this case  it is natural to reformulate the objective function in \\eqref{eq:primal} to include a term that also accounts for changes in the total mass of $\\derivd \\mu$. Consequently, we have exchanged the objective function in \\eqref{eq:primal} by the normalized Kullback-Leibler divergence \\eqref{eq:normalizedKullbackLeibler} plus a term that ensures approximate data matching.\n\nUsing the normalized Kullback-Leibler divergence, as proposed in  \\cite[ch. 4]{gzyl1995method} \\cite{csiszar1991why, zorzi2014rational}, is an advantage in the approximate covariance matching problem since this divergence is always nonnegative, precisely as is the case for probability densities. To see this, observe that, in view of the basic inequality  $x - 1 \\geq \\log x$,\n\\begin{align*}\n  \\mathbb D(P\\derivd m, \\derivd \\mu) &= \\int_{\\mathbb{T}^d}  \\left(  P \\left( - \\log\\frac{\\Phi}{P} \\right) \\derivd m +d\\mu -Pdm \\right) \\\\\n & \\geq \\int_{\\mathbb{T}^d}  \\left(  P (1 - \\frac{\\Phi}{P})\\derivd m + \\Phi\\derivd m -Pdm \\right) + \\int_{\\mathbb{T}^d} \\derivd \\nu \\geq 0,\n \\end{align*}\n since $\\derivd \\nu$ is a nonnegative measure. \nMoreover, $ \\mathbb D(P\\derivd m, P\\derivd m)=0$, as can be seen by taking $\\derivd \\mu=Pdm$ in \\eqref{eq:normalizedKullbackLeibler}.\n\nThe problem under consideration is to find a nonnegative measure $\\derivd \\mu=\\Phi dm +d\\nu$ minimizing $$\\mathbb D(P\\derivd m, \\derivd \\mu)+\\frac{1}{2}\\|r - c\\|_{W^{-1}}^2$$ subject to \\eqref{eq:r}. To derive the dual of this problem we consider the corresponding maximization problem and form the Lagrangian \n\\begin{align*}\n\\mathcal{L}(\\Phi,d\\nu, r, q) = & \\; -\\mathbb D(P\\derivd m, \\derivd \\mu) - \\frac{1}{2}\\|r - c\\|_{W^{-1}}^2 + \\sum_{{\\boldsymbol k} \\in \\Lambda} q_{\\boldsymbol k}^* \\Big( r_{\\boldsymbol k} - \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}) \\Big)\\\\\n=&\\;-\\mathbb D(P\\derivd m, \\derivd \\mu) - \\frac{1}{2}\\|r - c\\|_{W^{-1}}^2  +\\langle r,q\\rangle - \\int_{\\mathbb{T}^d}Q\\derivd \\mu \\, ,\n\\end{align*}\nwhere $q := [ q_{\\boldsymbol k} \\mid {\\boldsymbol k}:=(k_1,\\ldots, k_d) \\in \\Lambda]$ are Lagrange multitipliers and $Q$ is the corresponding trigonometric polynomial \\eqref{Ptrigpol}. However, \n\\begin{equation}\n\\label{D(Pdm,dmu)}\n\\mathbb D(P\\derivd m, \\derivd \\mu)=\\int_{\\mathbb{T}^d}P(\\log P -1)dm - \\int_{\\mathbb{T}^d}P\\log\\Phi dm + r_{\\boldsymbol 0}\\, ,\n\\end{equation}\nand therefore \n\\begin{align}\\label{eq:Lagrangian}\n\\mathcal{L}(\\Phi,d\\nu, r, q) = &\\; \\int_{\\mathbb{T}^d}P\\log\\Phi dm -  \\int_{\\mathbb{T}^d}Q\\Phi dm -  \\int_{\\mathbb{T}^d} Qd\\nu  -\\int_{\\mathbb{T}^d}P(\\log P -1)dm\\notag \\\\\n&\\; +\\langle r,q-e\\rangle - \\frac{1}{2}\\|r - c\\|_{W^{-1}}^2 \\, , \n\\end{align}\nwhere  $e := [e_{\\boldsymbol k}]_{{\\boldsymbol k} \\in \\Lambda}$, $e_{\\boldsymbol 0} = 1$ and $e_{{\\boldsymbol k}} = 0$ for ${\\boldsymbol k} \\in \\Lambda \\setminus \\{ \\boldsymbol 0 \\}$, and hence $r_{\\boldsymbol 0}=\\langle r,e\\rangle$.\n\nIn deriving the dual functional $$\\varphi(q)=\\sup_{\\Phi\\geq 0,d\\nu\\geq 0,r}\\mathcal{L}(\\Phi,d\\nu, r, q),$$ to be minimized, we only need to consider $q\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}$, as $\\varphi$ will take infinite values for $q\\not\\in\\bar{\\mathfrak{P}}_+$. In fact, following along the lines of \\cite[p. 1957]{ringh2015multidimensional}, we note that, if $Q(e^{i{\\boldsymbol \\theta}_0})<0$, \\eqref{eq:Lagrangian} will tend to infinity when $\\nu({\\boldsymbol \\theta}_0)\\to\\infty$. Moreover, since $p\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}$, there is a neighborhood where $P(e^{i{\\boldsymbol \\theta}})>0$, letting $\\Phi$ tend to infinity in this neighborhood, \\eqref{eq:Lagrangian} will tend to infinity if $Q\\equiv 0$. We also note that the nonnegative function $\\Phi$ can only be zero on a set of measure zero; otherwise the first term in \\eqref{eq:Lagrangian} will be $-\\infty$.\n\nThe directional derivative\\footnote{Formally, the Gateaux differential \\cite{luenberger1969optimization}.} of the Lagrangian \\eqref{eq:Lagrangian} in any feasible direction $\\delta\\Phi$, i.e., any direction $\\delta\\Phi$ such that $\\Phi +\\varepsilon \\delta\\Phi \\geq 0$ for sufficiencly small $\\varepsilon >0$,  is easily seen to be\n\\begin{displaymath}\n\\delta\\mathcal{L}(\\Phi,d\\nu, r, q;\\delta\\Phi)= \\int_{\\mathbb{T}^d}\\left(\\frac{P}{\\Phi}-Q\\right)\\delta\\Phi\\derivd m.\n\\end{displaymath} \nIn particular, the direction $\\delta\\Phi:=\\Phi\\,\\text{sign}(P-Q\\Phi)$ is feasible since $(1\\pm\\varepsilon)\\Phi\\geq 0$ for $0<\\varepsilon < 1$. Therefore, any maximizing $\\Phi$ must satisfy $\\int_{\\mathbb{T}^d}|P-Q\\Phi|\\derivd m\\leq 0$ and hence \n\\eqref{eq:rat}. Moreover, a maximizing choice of $d\\nu$ will require that \n\\begin{equation}\n\\label{eq:intQdnu=0}\n\\int_{\\mathbb{T}^d} Qd\\,\\nu=0, \n\\end{equation}\nas this nonnegative term can be made zero by the simple choice $d\\nu\\equiv 0$, and consequently \\eqref{supp(dnu)} must hold.  Finally, the directional derivative \n\\begin{displaymath}\n\\delta\\mathcal{L}(\\Phi,d\\nu, r, q;\\delta r)= \\langle \\delta r, q-e + W^{-1}(r-c)\\rangle\n\\end{displaymath}\nis zero for all $\\delta r\\in \\mathfrak{C}$ if\n\\begin{equation}\n\\label{eq:q2r}\nr=c+W(q-e).\n\\end{equation} \nInserting this together with \\eqref{eq:rat} and \\eqref{eq:intQdnu=0} into  \\eqref{eq:Lagrangian} then yields the dual functional \n\\begin{displaymath}\n\\varphi(q)=\\langle c, q\\rangle - \\int_{\\mathbb{T}^d} P\\log Q\\,\\derivd m +\\frac12\\|q-e\\|_W^2 +c_0.\n\\end{displaymath}\nConsequently the dual of the (primal) optimization problem \\eqref{eq:primal_relax} is equivalent to \n\\begin{align}\\label{eq:dual_relax}\n\\min_{q\\in\\bar{\\mathfrak{P}}_+} & \\quad \\langle c, q \\rangle - \\int_{\\mathbb{T}^d} P \\log Q\\, \\derivd m+ \\frac{1}{2}\\|q - e\\|_{W}^2 .\n\\end{align}\n\n\\begin{theorem}\\label{thm:softcontraints}\nFor every $p \\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$ the functional in \\eqref{eq:dual_relax} is strictly convex and has a unique minimizer $\\hat{q}\\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$. Moreover, there exists a unique $\\hat{r}\\in\\mathfrak{C}_+$, a unique $\\hat{c} \\in \\partial \\mathfrak{C}_+$ and a (not necessarily unique) nonnegative singular measure $d\\hat\\nu$ with support \n\\begin{equation}\n\\label{supp(dnu)soft}\n\\supp(d\\hat\\nu) \\subseteq \\{ {\\boldsymbol \\theta} \\in \\mathbb{T}^d  \\mid \\hat{Q}(e^{i{\\boldsymbol \\theta}}) = 0 \\}\n\\end{equation}\nsuch that\n\\begin{subequations}\\label{rhat+chat}\n\\begin{align}\n& \\hat{r}_{{\\boldsymbol k}} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\left( \\frac{P}{\\hat Q}  \\derivd m  + d\\hat\\nu \\right)\\, \\text{ for all } {\\boldsymbol k} \\in \\Lambda, \\label{rhat}\\\\\n& \\hat{c}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} d\\hat\\nu, \\text{ for all } {\\boldsymbol k} \\in \\Lambda \\, ,\\ \\label{chat}\n\\end{align}\n\\end{subequations}\nand the measure \n\\begin{equation}\n\\label{eq:optdmu(soft)}\nd\\hat\\mu({\\boldsymbol \\theta}) = \\frac{P(e^{i{\\boldsymbol \\theta}})}{\\hat{Q}(e^{i{\\boldsymbol \\theta}})}\\derivd m({\\boldsymbol \\theta}) + d\\hat\\nu({\\boldsymbol \\theta})\n\\end{equation}\nis an optimal solution to the primal problem \\eqref{eq:primal_relax}. Moreover, $d\\hat\\nu$ can be chosen with support in at most $|\\Lambda|-1$ points.\n\\end{theorem}\n\n\\begin{proof}\nThe objective functional $\\mathbb{J}$ of the dual problem \\eqref{eq:dual_relax} can be written as the sum of two terms, namely \n\\begin{displaymath}\n\\mathbb{J}_1(q) =  \\langle \\tilde{c}, q \\rangle - \\int_{\\mathbb{T}^d} P \\log(Q) \\derivd m\\quad\\text{and}\\quad \\mathbb{J}_2(q)=\\langle c-\\tilde{c}, q \\rangle +\\frac{1}{2}\\|q - e\\|_{W}^2\\, ,\n\\end{displaymath}\nwhere $\\tilde{c}\\in\\mathfrak{C}_+$.\nThe functional $\\mathbb{J}_1$ is strictly convex (Theorem~\\ref{thm:rational}), and trivially the same holds for $\\mathbb{J}_2$ since it is a positive definite quadratic form. Consequently, $\\mathbb{J}=\\mathbb{J}_1+\\mathbb{J}_2$ is strictly convex, as claimed. Moreover, $\\mathbb{J}_1$ is lower semicontinuous  \\cite[Lemma 3.1]{ringh2015multidimensional} with compact sublevel sets $\\mathbb{J}_1^{-1}(-\\infty,\\rho]$ \\cite[Lemma 3.2]{ringh2015multidimensional}. Likewise,  $\\mathbb{J}_2$ is continuous with compact sublevel sets. Therefore $\\mathbb{J}$ is lower semicontinuous with compact sublevel sets and therefore has a minimum $\\hat{q}$, which must be unique by strict convexity. \n\nIn view of \\eqref{eq:q2r}, the optimal value of $r$ is given by\n\\begin{equation}\n\\label{eq:rhat}\n\\hat{r}=c +W(\\hat{q}-e)\n\\end{equation}\nand is hence unique. Since therefore the linear term $c+W(q-e)$  in the gradient of $\\mathbb{J}$ takes the value $\\hat{r}$ at the optimal point, the analysis in \\cite[sect. 3.1.5]{ringh2015multidimensional} applies with obvious modifications, showing that there is a $\\hat{c}\\in \\bar{\\mathfrak{C}}_+$, which then must be unique, such that \n\\begin{displaymath}\n \\hat{r}_{{\\boldsymbol k}} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)}\\frac{P}{\\hat Q}  \\derivd m +\\hat{c}_{{\\boldsymbol k}}.\n\\end{displaymath}\nMoreover, there is a discrete measure $d\\hat\\nu$ with support in at most $|\\Lambda|-1$ points such that \\eqref{chat} holds; see, e.g., \\cite[Proposition 2.4]{karlsson2015themultidimensional}. Then \\eqref{rhat} holds as well. In view of \\eqref{eq:intQdnu=0}, \n\\begin{equation}\n\\label{eq:compslacknes}\n\\langle\\hat{c},\\hat{q}\\rangle =\\int_{\\mathbb{T}^d} \\hat{Q}d\\hat\\nu =0,\n\\end{equation}\nand consequently $\\hat{c}\\in\\partial\\mathfrak{C}_+$, and the support of $d\\hat\\nu$ must satisfy \\eqref{supp(dnu)soft}.\n\n\nFinally, let $r$ be given in terms of $\\derivd \\mu$ by \\eqref{eq:r}, and let $\\mathbb{I}(\\derivd \\mu)$ be the corresponding primal functional in \\eqref{eq:primal_relax}.  Then, for any such $\\derivd \\mu$, \n\\begin{displaymath}\n\\mathbb{I}(\\derivd \\mu)= \\mathcal{L}(\\Phi,d\\nu, r, \\hat{q})\\leq \\mathcal{L}(\\hat{\\Phi},d\\hat\\nu, \\hat{r}, \\hat{q}) =\\mathbb{I}(d\\hat\\mu),\n\\end{displaymath}\nand hence $\\derivd \\hat{\\mu}$ is an optimal solution to the primal problem \\eqref{eq:primal_relax}, as claimed. \n\\end{proof}\n\nWe collect the KKT conditions in the following corollary. \n\n\\begin{corollary}\\label{cor:KKTsoft}\nThe conditions\n\\begin{subequations}\\label{eq:opt_cond_relax}\n\\begin{align}\n& \\hat{q} \\in \\bar{\\mathfrak{P}}_+, \\quad \\hat c \\in \\partial \\mathfrak{C}_+, \\quad \\langle \\hat{c}, \\hat{q} \\rangle = 0 \\label{eq:opt_cond_relax_slack} \\\\\n& \\hat{r}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\frac{P}{\\hat Q}\\derivd m + \\hat{c}_{\\boldsymbol k},\\quad {\\boldsymbol k}\\in\\Lambda \\label{eq:opt_cond_relax_KKT} \\\\\n&\\hat{r}- c = W(\\hat q - e). \\label{eq:opt_cond_relax_matchError}\n\\end{align}\n\\end{subequations}\nare necessary and sufficient conditions for optimality of the dual pair  \\eqref{eq:primal_relax} and  \\eqref{eq:dual_relax} of optimization problems. \n\\end{corollary}\n\n\n\\section{On the well-posedness of the soft-constrained problem}\\label{sec:wellposed}\n\nIn the previous sections we have shown that the primal and dual optimization problems are well-defined. Next we investigate the well-posedness of the primal problem  as an inverse problem. Thus, we first  establish continuity of the solutions $\\hat{q} $ in terms of the parameters $W$, $c$ and $p$.\n\n\n\n\n\\newcommand{c}{c}\n\\newcommand{c^{(0)}}{c^{(0)}}\n\\newcommand{c^{(1)}}{c^{(1)}}\n\\newcommand{c^{(2)}}{c^{(2)}}\n\\newcommand{c^{(k)}}{c^{(k)}}\n\\newcommand{p}{p}\n\\newcommand{p^{(0)}}{p^{(0)}}\n\\newcommand{p^{(1)}}{p^{(1)}}\n\\newcommand{p^{(2)}}{p^{(2)}}\n\\newcommand{p^{(k)}}{p^{(k)}}\n\\newcommand{P}{P}\n\\newcommand{P^{(0)}}{P^{(0)}}\n\\newcommand{P^{(1)}}{P^{(1)}}\n\\newcommand{P^{(2)}}{P^{(2)}}\n\\newcommand{P^{(k)}}{P^{(k)}}\n\n\\newcommand{W}{W}\n\\newcommand{W^{(0)}}{W^{(0)}}\n\\newcommand{W^{(1)}}{W^{(1)}}\n\\newcommand{W^{(2)}}{W^{(2)}}\n\\newcommand{W^{(k)}}{W^{(k)}}\n\n\n\\newcommand{{\\ca, \\pa, \\Wa}}{{c, p, W}}\n\\newcommand{{\\cb, \\pb, \\Wb}}{{c^{(0)}, p^{(0)}, W^{(0)}}}\n\\newcommand{{\\cc, \\pc, \\Wc}}{{c^{(1)}, p^{(1)}, W^{(1)}}}\n\\newcommand{{\\cd, \\pd, \\Wd}}{{c^{(2)}, p^{(2)}, W^{(2)}}}\n\\newcommand{{\\ck, \\pk, \\Wk}}{{c^{(k)}, p^{(k)}, W^{(k)}}}\n\n\\subsection{Continuity of $\\hat{q}$ with respect to $c$, $p$ and $W$} \n\nWe start considering the continuity of the optimal solution with respect to the parameters. The parameter set of interest is\n\\begin{equation}\n\\label{eq:Pcal}\n\\mathcal{P}=\\{({\\ca, \\pa, \\Wa})\\mid c\\in \\mathfrak{C}, p\\in \\bar{\\mathfrak{P}}_+\\setminus\\{0\\}, W>0\\}.\n\\end{equation}\n\n\\begin{theorem}\\label{thm:W2qcont} \nLet \n\\begin{equation}\n\\label{eq:Jsoft}\n\\mathbb{J}_{{\\ca, \\pa, \\Wa}}(q)=\\langle c, q \\rangle - \\int_{\\mathbb{T}^d} P \\log Q\\, \\derivd m+ \\frac{1}{2}\\|q - e\\|_{W}^2.\n\\end{equation}\nThen the map $({\\ca, \\pa, \\Wa})\\mapsto \\hat{q}:=\\argmin_{q\\in\\bar{\\mathfrak{P}}_+} \\mathbb{J}_{{\\ca, \\pa, \\Wa}}(q)$ is continuous on  $\\mathcal{P}$.\n\\end{theorem}\n\n\n\\begin{proof}\nFollowing the procedure in  \\cite[Proposition  7.3]{karlsson2015themultidimensional} we use the continuity of the optimal value (Lemma~\\ref{lem:W2Jcont}) to show continuity of the optimal solution. To this end, let $({\\ck, \\pk, \\Wk})$ be a sequence of parameters in $\\mathcal{P}$ converging to $({\\ca, \\pa, \\Wa})\\in\\mathcal{P}$ as $k\\to\\infty$. Moreover, defining $\\mathbb{J}_k(q):=\\mathbb{J}_{\\ck, \\pk, \\Wk} (q)$ and $\\mathbb{J}(q):=\\mathbb{J}_{\\ca, \\pa, \\Wa} (q)$  for simplicity of notation, let $\\hat{q}_k = \\argmin_{q \\in \\bar{\\mathfrak{P}}_+} \\mathbb{J}_{k}(q)$ and $\\hat{q} = \\argmin_{q \\in \\bar{\\mathfrak{P}}_+} \\mathbb{J}(q)$. By Lemma~\\ref{lem:W2Jcont}, $(\\hat{q}_k)$ is bounded, and hence there is a subsequence, which for simplicity we also call $(\\hat{q}_k)$, converging to a limit  $q_\\infty$. If we can show that $q_\\infty = \\hat{q}$, then the theorem follows.  To this end,  choosing a $q_0 \\in \\mathfrak{P}_+$, we have\n\\begin{align*}\n\\mathbb{J}_{k}(\\hat{q}_k) &= \\mathbb{J}_{k} (\\hat{q}_k + \\varepsilon q_0) - \\langle c^{(k)}, \\varepsilon q_0 \\rangle + \\int_{\\mathbb{T}^d} P^{(k)} \\log \\left( \\frac{\\hat{Q}_k + \\varepsilon Q_0}{\\hat{Q}_k} \\right)\\derivd m \\\\ \n&\\quad+ \\frac12\\|\\hat{q}_k-e\\|_{W^{(k)}}^2 -\\frac12 \\|\\hat{q}_k+\\varepsilon q_0-e\\|_{W^{(k)}}^2 \\\\\n&\\geq \\mathbb{J}_{k} (\\hat{q}_k + \\varepsilon q_0) - \\langle c^{(k)}, \\varepsilon q_0 \\rangle + \\frac12\\|\\hat{q}_k-e\\|_{W^{(k)}}^2-\\frac12 \\|\\hat{q}_k+\\varepsilon q_0-e\\|_{W^{(k)}}^2.\n\\end{align*}\nConsequently, by Lemma~\\ref{lem:W2Jcont}, \n\\begin{equation*}\n\\mathbb{J}(\\hat{q}) = \\lim_{k \\to \\infty} \\mathbb{J}_{k}(\\hat{q}_k) \\geq \\lim_{k \\to \\infty} \\mathbb{J}_{k}(\\hat{q}_k + \\varepsilon q_0) - \\varepsilon \\langle c^{(k)}, q_0 \\rangle + \\frac12\\|\\hat{q}_k-e\\|_{W^{(k)}}^2 -\\frac12 \\|\\hat{q}_k+\\varepsilon q_0-e\\|_{W^{(k)}}^2.\n\\end{equation*}\nHowever $\\hat{q}_k + \\varepsilon q_0 \\in \\mathfrak{P}_+$, and, since $({\\ca, \\pa, \\Wa},q)\\mapsto\\mathbb{J}_{{\\ca, \\pa, \\Wa}}(q)$ is continuous in $\\mathcal{P}\\times\\mathfrak{P}_+$, we obtain\n\\begin{align}\n\\mathbb{J}(\\hat{q}) &\\geq \\lim_{k \\to \\infty} \\left(\\mathbb{J}_{k}(\\hat{q}_k + \\varepsilon q_0) - \\varepsilon \\langle c^{(k)}, q_0 \\rangle + \\frac12\\|\\hat{q}_k-e\\|_{W^{(k)}}^2-\\frac12 \\|\\hat{q}_k+\\varepsilon q_0-e\\|_{W^{(k)}}^2\\right)\\notag\\\\ \n&= \\mathbb{J}(q_\\infty + \\varepsilon q_0) - \\varepsilon \\langle c, q_0 \\rangle+ \\frac12\\|q_\\infty-e\\|_{W}^2 -\\frac12 \\|q_\\infty+\\varepsilon q_0-e\\|_{W}^2 .\\label{Jpestimate}\n\\end{align}\nLetting $\\varepsilon\\to 0$ in \\eqref{Jpestimate}, we obtain the inequality $\\mathbb{J}(\\hat{q})\\geq \\mathbb{J}(q_\\infty)$. By strict convexity of $\\mathbb{J}$ the optimal solution is unique, and hence $\\hat q=q_\\infty$. \n\\end{proof}\n\n\\subsection{Continuity of $\\hat{c}$ with respect to $\\hat{q}$} \\label{ssec:qhat2chat}\n\nWe have now established continuity from $({\\ca, \\pa, \\Wa})$ to $\\hat{q}$. In the same way as in Proposition~\\ref{prop:c2qhatchat2} we are also interested in continuity of the map $({\\ca, \\pa, \\Wa})\\mapsto (\\hat{q},\\hat{c})$. This would follow if we could show that the map from $\\hat{q}$ to $\\hat{c}$ is continuous. From the KKT condition \\eqref{eq:opt_cond_relax_matchError}, it is seen that $\\hat{r}$ is continuous in $c$, $W$ and $\\hat{q}$. In view of \\eqref{eq:opt_cond_relax_KKT}, i.e.,  \n\\begin{displaymath}\n\\hat{r}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\frac{P}{\\hat Q}\\derivd m + \\hat{c}_{\\boldsymbol k},\\quad {\\boldsymbol k}\\in\\Lambda\n\\end{displaymath}\ncontinuity of $\\hat c$ would  follow if $\\int_{\\mathbb{T}^d} P\\hat{Q}^{-1}\\derivd m$ is continuous in $(p,\\hat{q})$ whenever it is finite. If $p\\in\\mathfrak{P}_+$, this follows from the continuity the map $\\hat{q}\\mapsto \\hat{Q}^{-1}$ in $L_1(\\mathbb{T}^d)$.\nFor the case $d\\leq 2$, this is trivial since if  $\\int_{\\mathbb{T}^d} \\hat{Q}^{-1}\\derivd m$ is finite, then $\\hat q\\in \\mathfrak{P}_+$ and $\\hat Q$ is bounded away from zero (cf., Proposition~\\ref{prop:c2qhatchat2}).  However, for the case  $d>2$ the optimal $\\hat q$ may belong to the boundary $\\partial \\mathfrak{P}_+$, i.e., $\\hat Q$ is zero in some point.  The following proposition shows $L_1$ continuity of $\\hat{q}\\mapsto \\hat{Q}^{-1}$  for certain cases.\n\n\\begin{proposition} \\label{prp:qQinvLt}\nFor $d\\geq3$, let $\\hat q\\in\\bar{\\mathfrak{P}}_+$ and suppose that the Hessian $ \\nabla_{{\\boldsymbol \\theta}\\thetab} \\, \\hat Q$ is positive definite in each point  where $\\hat Q$ is zero. Then $\\hat Q^{-1}\\in L_1(\\mathbb{T}^{d})$ and the mapping from the coefficient vector $q\\in \\bar{\\mathfrak{P}}_+$ to $Q^{-1}$ is $L_1$ continuous in the point $\\hat q$.\n\\end{proposition}\n\nThe proof of this proposition is given in \\cite{ringh2017further}.  From Propositions~\\ref{prp:qQinvLt} and \\ref{prop:c2qhatchat2} the following continuity result follows directly.\n\n\\begin{corollary}\nFor all $c\\in \\mathfrak{C}, p\\in\\mathfrak{P}_+, W>0$, the mapping $(c,p,W)\\to(\\hat q, \\hat c)$ is continuous in any point $({\\ca, \\pa, \\Wa})$ for which the Hessian $ \\nabla_{{\\boldsymbol \\theta}\\thetab} \\, \\hat Q$ is positive definite in each point where $\\hat Q$ is zero.\n\\end{corollary}\n\nThe condition  $p \\in \\mathfrak{P}_+$ is needed, since we may have pole-zero cancelations in $P\/\\hat{Q}$ when $p\\in\\partial\\mathfrak{P}_+$, and then $\\int_{\\mathbb{T}^d} P\/\\hat{Q}\\derivd m$ may be finite even if $\\hat{Q}^{-1}\\not\\in L_1$. The following example shows that this may lead to discontinuities in the map $p\\mapsto \\hat{c}$ (cf. Example 3.8 in \\cite{karlsson2015themultidimensional}).\n\n\\begin{example}\nLet\n\\begin{displaymath}\nc = \\begin{bmatrix}1 \\\\ 3 \\\\ 1\\end{bmatrix} = \n\\begin{bmatrix}0 \\\\ 2 \\\\ 0\\end{bmatrix} +\\begin{bmatrix}1 \\\\ 1 \\\\ 1\\end{bmatrix}= \n\\int_{-\\pi}^\\pi\\begin{bmatrix}e^{-i\\theta} \\\\ 1 \\\\ e^{i\\theta}\\end{bmatrix} \\left( 2 \\derivd m +d\\nu_0\\right),\n\\end{displaymath}\nwhere $\\derivd m=d\\theta\/2\\pi$ and $d\\nu_0$ is the singular measure $\\delta_{0}(\\theta)d\\theta$ with support in $\\theta=0$. Since $\\derivd \\mu:=2 \\derivd m +d\\nu_0$ is positive, $c \\in \\bar{\\mathfrak{C}}_+$. Moreover, since\n\\[\nT_{c} = \\begin{bmatrix}\n3 & 1\\\\\n1 & 3\n\\end{bmatrix}\n> 0\n\\]\nwe have that $c \\in \\mathfrak{C}_+$ (see, e.g., \\cite[p. 2853]{lindquist2013thecirculant}). Thus we know \\cite[Corollary 2.3]{ringh2015multidimensional} that for each $p \\in \\mathfrak{P}_+$ we have a unique $\\hat{q} \\in \\mathfrak{P}_+$ such that $P\/ \\hat{Q}$ matches $c$, and hence $\\hat{c} = 0$.  However, for $p = 2 (-1, 2, -1)'$ we have that $\\hat{q} = (-1, 2, -1)'$ and $\\hat{c} = (1, 1, 1)'$ (Theorem~\\ref{thm:rational}). Then, for the sequence $(p_k)$, where $p_k=2 (-1, 2 + 1\/k, -1) \\in \\mathfrak{P}_+$, we have $\\hat{c}_k = 0$, so \n\\[\n\\lim_{k \\to \\infty} \\hat{c}_k = \\lim_{k \\to \\infty}\n\\begin{bmatrix}\n0 \\\\ 0 \\\\ 0\n\\end{bmatrix} \\neq \n\\begin{bmatrix}\n1 \\\\ 1 \\\\ 1\n\\end{bmatrix},\n\\]\nwhich shows that the mapping $p \\to \\hat{c}$ is not continuous.\n\\end{example}\n\n\\section{Tuning to avoid a singular part}\\label{sec:no_singular_part}\n\nIn many situations we prefer solutions where there is no singular measure $d\\nu$ in the optimal solution.  An interesting question is therefore for what prior $P$ and  weight $W$ we obtain $d\\hat\\nu=0$. The following result provides a sufficient condition. \n\n\\begin{proposition}\\label{boundlprop}\nLet $c\\in\\mathfrak{C}$  and let $p$ be the Fourier coefficients of the prior $P$. If the weight satisfies\\footnote{Here $\\|A\\|_{2,1}=\\max_{c\\neq 0}\\|Ac\\|_1\/\\|c\\|_2$ denotes the subordinate (induced) matrix norm.} \n\\begin{equation}\\label{eq:supnormbound}\n\\|W^{-1\/2}\\|_{2,1}<\\|c-p\\|_{W^{-1}}^{-1},\n\\end{equation}\n then the optimal solution of \\eqref{eq:primal_relax} is on the form  $$d\\hat{\\mu}=(P\/\\hat{Q}) dm,$$ i.e., the singular part $d\\hat{\\nu}$ vanishes.\n\\end{proposition}\n\n\\begin{remark}\nNote that for a scalar weight, $W=\\lambda I$ the bound \\eqref{eq:supnormbound} simplifies to \n\\begin{equation}\n\\label{eq:scalarbound}\n\\lambda> |\\Lambda|^{1\/2}\\|c-p\\|_2,\n\\end{equation}\n where $|\\Lambda|$ is the cardinality of index set $\\Lambda$.\n\\end{remark}\n\nFor the proof of Proposition~\\ref{boundlprop} we need the following lemma.\n\n\\begin{lemma}\\label{boundlemma}\nCondition \\eqref{eq:supnormbound} implies \n\\begin{equation}\\label{eq:newnormbound}\n\\|W^{-1}(\\hat{r}-c)\\|_1<1,\n\\end{equation}\nwhere $\\hat{r}$ is the optimal value of $r$ in problem \\eqref{eq:primal_relax}.\n\\end{lemma}\n\n\\begin{proof}\nLet \n\\begin{equation}\n\\label{eq:primalcost}\n\\mathbb{I}(\\derivd \\mu,r):=\\mathbb D (P\\derivd m, \\derivd \\mu) + \\frac{1}{2}\\|r - c\\|_{W^{-1}}^2 \n\\end{equation}\nbe the cost function of problem \\eqref{eq:primal_relax}, and let $(d\\hat{\\mu},\\hat{r})$ be the optimal solution. Clearly, $\\mathbb{I}(Pdm,p)\\geq \\mathbb{I}(d\\hat{\\mu},\\hat{r})$, and consequently \n\\begin{displaymath}\n\\|\\hat{r}-c\\|_{W^{-1}} \\leq \\|p-c\\|_{W^{-1}},\n\\end{displaymath}\nsince $\\mathbb D (P\\derivd m, d\\hat{\\mu})\\geq 0$ and $\\mathbb D (P\\derivd m,P\\derivd m)=0$. Therefore,\n\\begin{align*}\n\\|W^{-1}(\\hat{r} - c)\\|_1\n&\\le \\|W^{-1\/2}\\|_{2,1} \\|W^{-1\/2}(\\hat{r} - c)\\|_2\\\\\n&= \\|W^{-1\/2}\\|_{2,1}\\|\\hat{r} - c\\|_{W^{-1}}\\\\\n&\\le \\|W^{-1\/2}\\|_{2,1}  \\|p - c\\|_{W^{-1}},\n\\end{align*}\n which is less than one by \\eqref{eq:supnormbound}. Hence  \\eqref{eq:supnormbound} implies \\eqref{eq:newnormbound}.\n\\end{proof}\n\n\n\\begin{proof}[{Proof of Proposition~\\ref{boundlprop}}]\nSuppose the optimal solution has a nonzero singular part $d\\hat{\\nu}$, and form the  directional derivative of \\eqref{eq:primalcost} at $(d\\hat{\\mu},\\hat{r})$ in the direction $-d\\hat{\\nu}$. Then $\\Phi$ in \\eqref{eq:decomp} does not vary, and \n\\begin{displaymath}\n\\delta\\mathbb{I}(d\\hat{\\mu},\\hat{r};-d\\hat{\\nu},\\delta r)=-\\int_{\\mathbb{T}^d}d\\hat{\\nu} +\\delta r^*W^{-1}(\\hat{r} - c),\n\\end{displaymath}\nwhere\n\\begin{displaymath}\n\\delta r_{\\boldsymbol k} = -\\int_{\\mathbb{T}^d} \\expfunkd\\hat{\\nu}.\n\\end{displaymath}\nThen $|\\delta r_{\\boldsymbol k}|\\leq \\int d\\hat{\\nu}$ for all ${\\boldsymbol k}\\in\\Lambda$, and hence\n\\begin{displaymath}\n|\\delta r^*W^{-1}(\\hat{r} - c)|\\leq \\|W^{-1}(\\hat{r}-c)\\|_1 \\int_{\\mathbb{T}^d} d\\hat{\\nu} <\\int_{\\mathbb{T}^d} d\\hat{\\nu},\n\\end{displaymath}\nby \\eqref{eq:newnormbound} (Lemma~\\ref{boundlemma}). Consequently,\n\\begin{displaymath}\n\\delta\\mathbb{I}(d\\hat{\\mu},\\hat{r};-d\\hat{\\nu},\\delta r)<0\n\\end{displaymath}\nwhenever $d\\hat{\\nu}\\ne 0$, which contradicts optimality. Hence $d\\hat{\\nu}$ must be zero. \n\\end{proof}\n\nThe condition of Proposition~\\ref{boundlprop} is just sufficient and is in general conservative. To illustrate this, we consider a simple one-dimensional example ($d=1$). \n\n\\begin{example}\\label{ex:singular}\nConsider a covariance sequence $(1,c_1)$, where $c_1\\ne 0$, and a prior $P(e^{i\\theta})=1-\\cos\\theta$, and set  $W =\\lambda I$. Then, since \n\\begin{displaymath}\nc=\\begin{pmatrix} c_1\\\\ 1 \\\\c_1 \\end{pmatrix} \\quad \\text{and} \\quad p=\\begin{pmatrix} -1\/2\\\\ 1 \\\\-1\/2 \\end{pmatrix},\n\\end{displaymath}\nthe sufficient condition \\eqref{eq:scalarbound} for an absolutely continuous solution is \n\\begin{equation}\n\\label{bound}\n\\lambda > \\sqrt{\\tfrac32}\\, |1+2c_1|.\n\\end{equation}\nWe want to investigate how restrictive this condition is. \n\nClearly we will have a singular part  if and only if $\\hat{Q} = q_0 P$, in which case we have \n\\begin{displaymath}\n\\hat{q}=q_0\\begin{pmatrix} -1\/2\\\\ 1 \\\\-1\/2 \\end{pmatrix}\\quad \\text{and} \\quad\\hat c=\\beta\\begin{pmatrix} 1\\\\ 1 \\\\1\n\\end{pmatrix}\n\\end{displaymath}\nfor some $\\beta >0$. In fact, it follows from $\\langle \\hat{c}, \\hat{q} \\rangle = 0$ in \\eqref{eq:opt_cond_relax_slack} that $\\hat{c}_1=\\hat{c}_0$. Moreover, \\eqref{eq:opt_cond_relax_KKT} and  \\eqref{eq:opt_cond_relax_matchError} yield\n\\begin{align*}\n\\hat{r} =&\\int \\frac{P}{\\hat{Q}}\\begin{pmatrix} e^{i\\theta}\\\\ 1 \\\\e^{-i\\theta}\n\\end{pmatrix} dm +\\hat c=\n\\begin{pmatrix} \\beta\\\\ \\beta+1\/q_0 \\\\\\beta\n\\end{pmatrix}\\\\\nc = &\\,\\hat{r}-\\lambda(q-e)=\\begin{pmatrix} \\beta+\\lambda q_0\/2,\\\\ \\beta+1\/q_0-\\lambda  q_0+\\lambda  \\\\\\beta+\\lambda  q_0\/2\n\\end{pmatrix}.\n\\end{align*}\nBy eliminating $\\beta$, we get \n\\[\nc_1=1-\\frac{1}{q_0}-\\frac{3}{2}q_0 \\lambda  +\\lambda,\n\\]\nand solving for $q_0$ yields\n\\[\nq_0=\\frac{\\lambda  +c_1-1+ (6\\lambda  + (\\lambda  +c_1-1)^2 )^{1\/2}}{3\\lambda }\n\\]\n(note that $\\lambda >0$ and $q_0>0$). Again, using \\eqref{eq:opt_cond_relax_matchError} we have\n\\begin{align*}\n\\beta&=c_1-\\lambda  q_0\/2\\\\\n&=c_1-\\frac{1}{6}\\left( \\lambda  +c_1-1+ (6\\lambda  + (\\lambda  +c_1-1)^2)^{1\/2}\\right).\n\\end{align*}\nWe are interested in $\\lambda$ for which $\\beta >0$, i.e.,\n\\begin{equation}\n\\label{eq:inex}\n6c_1-(\\lambda  +c_1-1)>(6\\lambda  + (\\lambda  +c_1-1)^2)^{1\/2},\n\\end{equation}\nwhich is equivalent to the two conditions\n\\begin{subequations}\\label{twoconditions}\n\\begin{equation}\n1+5c_1>\\lambda \n\\end{equation}\n\\begin{equation}\n2c_1(1+2c_1)>\\lambda  (1+2c_1),\n\\end{equation}\n\\end{subequations}\nwhich could be seen by noting that the left member of \\eqref{eq:inex} must be positive and then squaring both sides.\nTo find out whether this has a solution we consider three cases, namely $c_1<-1\/2$, $-1\/2<c_1<0$, and $c_1>0$. For $c_1<-1\/2$, condition \\eqref{twoconditions} becomes \n$2c_1<\\lambda <1+5c_1$,\nwhich is impossible since $1+5c_1<2c_1$. Condition \\eqref{twoconditions} cannot be satisfied when $-1\/2<c_1<0$, because then $\\lambda$ would be negative which contradicts $\\lambda >0$. When $c_1>0$,  Condition \\eqref{twoconditions} is satisfied if and only if $\\lambda <2c_1$.\n\nConsequently, there is no singular part if either $c_1$ is negative or $$\\lambda\\geq 2c_1.$$This shows that the condition \\eqref{bound} is not tight.\n\\end{example}\n\n\n\n\n\n\\section{Covariance extension with hard constraints}\\label{sec:approxhard}\nThe alternative optimization problem  \\eqref{eq:primal_relax_new} amounts to minimizing $\\mathbb D (P\\derivd m, \\derivd \\mu)$ subject to the hard constraint  $\\|r - c \\|_{W^{-1}}^2 \\le 1$, where $r_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu$. Hard constraints of this type were used in \\cite{schott1984maximum} in the context of entropy maximization. In general the data $c\\not\\in\\bar{\\mathfrak{C}}_+$, whereas, by definition, $r\\in\\bar{\\mathfrak{C}}_+$. Consequently, a necessary condition for the existence of a solution is that $\\bar{\\mathfrak{C}}_+$ and the strictly convex set\n\\begin{equation}\n\\label{eq:SW}\n\\mathfrak{S}_W=\\{ r\\mid \\|r - c \\|_{W^{-1}}^2 \\le 1\\}\n\\end{equation}\nhave a nonempty intersection. In the case that $\\mathfrak{S}_W\\cap\\bar{\\mathfrak{C}}_+\\subset \\partial\\mathfrak{C}_+$, this intersection only contains one point \\cite[Section 3.12]{luenberger1969optimization}. In this case, any solution to the moment problem contains only a singular part (Proposition~\\ref{prop:singular}), and then the primal problem \\eqref{eq:primal_relax_new} has a unique feasible point $r$, but the objective function is infinite. Moreover, $\\mathbb D (P\\derivd m, \\derivd \\mu)\\geq 0$ is strictly convex with $\\mathbb D (P\\derivd m, P\\derivd m)= 0$, so if $p\\in\\mathfrak{S}_W$ then \\eqref{eq:primal_relax_new} has the trivial unique optimal solution $\\derivd \\hat{\\mu} =P\\derivd m$, and $\\hat{r}=p$. The remaining case, $p\\not\\in\\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset$ needs further analysis. \n\nTo this end, setting $\\derivd \\mu=\\Phi\\derivd m+d\\nu$, we consider the Lagrangian \n\\begin{align*}\n\\mathcal{L}(\\Phi,d\\nu, r, q, \\gamma) = & \\; -\\mathbb D (P\\derivd m, \\derivd \\mu)\n+ \\sum_{{\\boldsymbol k} \\in \\Lambda} q_{\\boldsymbol k}^* \\left( r_{\\boldsymbol k} - \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\derivd \\mu({\\boldsymbol \\theta}) \\right) \\\\\n& \\; +\\gamma \\left(1- \\|r - c \\|_{W^{-1}}^2 \\right)\\\\\n = & \\; -\\mathbb D (P\\derivd m, \\derivd \\mu) + \\langle r,q\\rangle - \\int_{\\mathbb{T}^d} Q \\derivd \\mu +\\gamma \\left(1- \\|r - c \\|_{W^{-1}}^2 \\right), \n\\end{align*}\nwhere $\\gamma\\ge 0$. Therefore, in view of \\eqref{D(Pdm,dmu)}, \n\\begin{align}\\label{eq:Lagrangian2}\n\\mathcal{L}(\\Phi,d\\nu, r, q,\\gamma) = &\\; \\int_{\\mathbb{T}^d}P\\log\\Phi dm -  \\int_{\\mathbb{T}^d}Q\\Phi dm -  \\int_{\\mathbb{T}^d} Qd\\nu -\\int_{\\mathbb{T}^d}P(\\log P -1)dm \\notag \\\\\n&\\; +\\langle r,q-e\\rangle +\\gamma \\left(1-\\|r - c \\|_{W^{-1}}^2 \\right) , \n\\end{align}\nwhere, as before,  $e := [e_{\\boldsymbol k}]_{{\\boldsymbol k} \\in \\Lambda}$, $e_{\\boldsymbol 0} = 1$ and $e_{{\\boldsymbol k}} = 0$ for ${\\boldsymbol k} \\in \\Lambda \\setminus \\{ \\boldsymbol 0 \\}$, and hence $r_{\\boldsymbol 0}=\\langle r,e\\rangle$. This Lagrangian differs from that in \\eqref{eq:Lagrangian} only in the last term that does not depend on $\\Phi$. Therefore, in deriving the dual functional $$\\varphi(q,\\gamma)=\\sup_{\\Phi\\geq 0,d\\nu\\geq 0,r}\\mathcal{L}(\\Phi,d\\nu, r, q,\\gamma),$$  we only need to consider $q\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}$, and a first variation in $\\Phi$ yields \\eqref{eq:rat} and  \\eqref{eq:intQdnu=0}. The directional derivative \n\\begin{displaymath}\n\\delta\\mathcal{L}(\\Phi,d\\nu, r, q,\\gamma;\\delta r)= q-e + 2\\gamma W^{-1}(r-c)\n\\end{displaymath}\nis zero for \n\\begin{equation}\n\\label{eq:q2rhard}\nr=c+\\frac{1}{2\\gamma}W(q-e).\n\\end{equation} \nThus inserting \\eqref{eq:rat} and \\eqref{eq:intQdnu=0} and \\eqref{eq:q2rhard} into  \\eqref{eq:Lagrangian2} yields the dual functional\n\\begin{equation}\n\\label{eq:harddual}\n\\varphi(q,\\gamma)=\\langle c, q\\rangle - \\int_{\\mathbb{T}^d} P\\log Q\\,\\derivd m +\\frac{1}{4\\gamma}\\|q-e\\|_W^2 +\\gamma -c_{\\boldsymbol 0}\n\\end{equation}\nto be minimized over all $q\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}$ and $\\gamma\\geq 0$. Since $\\frac{d\\varphi}{d\\gamma}= -\\frac{1}{4\\gamma^2}\\|q-e\\|_W^2 +1$, there is a stationary point\n\\begin{equation}\n\\label{eq:gammaopt}\n\\gamma =\\frac12\\|q-e\\|_W\n\\end{equation}\nthat is nonnegative as required. \n\nFor $\\gamma=0$ we must have $q=e$, and consequently $\\varphi(q,\\gamma)$ tends to zero as $\\gamma\\to 0$. By weak duality zero is therefore a lower bound for the minimization problem \\eqref{eq:primal_relax_new}, and $\\mathbb D (P\\derivd m, \\derivd \\hat{\\mu})=0$, which corresponds to the trivial unique solution $\\derivd \\hat{\\mu}=P\\derivd m$ and $\\hat{r}=p$ mentioned above. This solution is only feasible if $p\\in\\mathfrak{S}_W$. \nTherefore we can restrict our attention to the case $\\gamma>0$. Inserting \\eqref{eq:gammaopt} into \\eqref{eq:harddual} and removing the constant term $c_{\\boldsymbol 0}$, we obtain the modified dual functional\n\\begin{equation}\n\\label{eq:moddual}\n\\mathbb{J}(q)=\\langle c, q\\rangle - \\int_{\\mathbb{T}^d} P\\log Q\\,\\derivd m + \\|q-e\\|_W.\n\\end{equation}\nMoreover, combining \\eqref{eq:q2rhard} and \\eqref{eq:gammaopt}, we obtain\n\\begin{equation}\n\\label{eq:boundaryr}\n\\|r-c\\|_{W^{-1}}=1\\, ,\n\\end{equation}\nwhich also follows from complementary slackness since $\\gamma>0$ and restricts $r$ to the boundary of $\\mathfrak{S}_W$. \n\n\\begin{theorem}\\label{thm:hardcontraints}\nSuppose that $p\\in\\bar{\\mathfrak{P}}_+ \\setminus\\{0\\}$, $p\\not\\in\\mathfrak{S}_W$ and $\\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset$. Then the modified dual problem \n\\begin{equation}\n\\label{eq:moddualproblem}\n\\min_{q\\in\\bar{\\mathfrak{P}}_+}\\mathbb{J}(q)\n\\end{equation} \nhas a unique solution $\\hat{q}\\in\\bar{\\mathfrak{P}}_+ \\setminus\\{0\\}$. Moreover, there exists a unique $\\hat{r}\\in\\mathfrak{C}_+$, a unique $\\hat{c} \\in \\partial \\mathfrak{C}_+$ and a (not necessarily unique) nonnegative singular measure $d\\hat\\nu$ with support \n\\begin{equation}\n\\label{supp(dnu)hard}\n\\supp(d\\hat\\nu) \\subseteq \\{ {\\boldsymbol \\theta} \\in \\mathbb{T}^d  \\mid \\hat{Q}(e^{i{\\boldsymbol \\theta}}) = 0 \\}\n\\end{equation}\nsuch that\n\\begin{subequations}\\label{rhat2+chat2}\n\\begin{align}\n& \\hat{r}_{{\\boldsymbol k}} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\left( \\frac{P}{\\hat Q}  \\derivd m  + d\\hat\\nu \\right)\\, \\text{ for all } {\\boldsymbol k} \\in \\Lambda, \\label{rhat2}\\\\\n& \\hat{c}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} d\\hat\\nu, \\text{ for all } {\\boldsymbol k} \\in \\Lambda \\, ,\\ \\label{chat2}\n\\end{align}\n\\end{subequations}\nand the measure \n\\begin{equation}\n\\label{eq:optdmu(hard)}\nd\\hat\\mu({\\boldsymbol \\theta}) = \\frac{P(e^{i{\\boldsymbol \\theta}})}{\\hat{Q}(e^{i{\\boldsymbol \\theta}})}\\derivd m({\\boldsymbol \\theta}) + d\\hat\\nu({\\boldsymbol \\theta})\n\\end{equation}\nis an optimal solution to the primal problem \\eqref{eq:primal_relax_new}. Moreover, \n\\begin{equation}\n\\label{eq:rhat&c}\n\\|\\hat{r}-c\\|_{W^{-1}}=1\\, ,\n\\end{equation}\nand $d\\hat\\nu$ can be chosen with support in at most $|\\Lambda|-1$ points. \n\nIf $p\\in\\mathfrak{S}_W$, the unique optimal solution is $\\derivd \\hat{\\mu}=P\\derivd m$, and then $\\hat{r}=p$. If $\\mathfrak{S}_W\\cap\\bar{\\mathfrak{C}}_+\\subset \\partial\\mathfrak{C}_+$, any solution to the moment problem will have only a singular part.  Finally, if $\\mathfrak{S}_W\\cap\\bar{\\mathfrak{C}}_+=\\emptyset$, then the problem \\eqref{eq:primal_relax_new} will have no solution. \n\\end{theorem}\n\n\\begin{proof}\nWe begin by showing that the functional $\\mathbb{J}$ has a minimum under the stated conditions. To this end,  \nwe first establish that the functional $\\mathbb{J}$ has compact sublevel sets $\\mathbb{J}^{-1}(-\\infty,\\rho]$, i.e., $\\|q\\|_\\infty$ is bounded for all $q$ such  that  $\\mathbb{J}(q)\\leq\\rho$, where $\\rho$ is sufficiently large for the sublevel set to be nonempty. The functional \\eqref{eq:moddual} can be decomposed in a linear and a logarithmic term as\n\\begin{displaymath}\n\\mathbb{J}(q)=h(q) - \\int_{\\mathbb{T}^d} P\\log Q\\,\\derivd m +c_{\\boldsymbol 0}\\, ,\n\\end{displaymath}\nwhere $h(q):= \\langle c, q-e\\rangle +\\|q-e\\|_W$. The integral term will tend to $-\\infty$ as $\\|q\\|_\\infty\\to \\infty$. \nTherefore we need to have the linear term to tend to $+\\infty$ as $\\|q\\|_\\infty\\to \\infty$, in which case we can appeal to the fact that linear growth is faster than logarithmic growth.  However, if $c\\not\\in\\bar{\\mathfrak{C}}_+$ as is generally assumed, there is a $q\\in\\bar{\\mathfrak{P}}_+$ such that  $\\langle c, q\\rangle< 0$, so we need to ensure that the positive term $\\|q-e\\|_W$ dominates. \n\nLet $\\tilde{r}\\in\\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset$. Then,  by Theorem~\\ref{thm:rational},  there is a positive measure $d\\tilde{\\mu}=\\tilde{\\Phi}dm+ d\\tilde{\\nu}$ with a nonzero $\\tilde{\\Phi}$ such that \n\\begin{displaymath}\n\\tilde{r}=\\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} d\\tilde\\mu \\, ,\n\\end{displaymath} \nand $\\tilde{r}$ satisfies the constraints in the primal problem  \\eqref{eq:primal_relax_new}. Consequently,   \n\\begin{displaymath}\n\\varphi(q,\\gamma)\\geq\\mathcal{L}(\\tilde\\Phi,d\\tilde\\nu, \\tilde{r}, q,\\gamma)\\geq -\\mathbb{D}(P\\derivd m,d\\tilde{\\mu})\n\\end{displaymath}\n for all $q\\in\\bar{\\mathfrak{P}}_+$ and $\\gamma \\geq 0$, which in particular implies that \n\\begin{equation}\n\\label{eq:lowerbound}\n\\mathbb{J}(q)\\geq -\\mathbb{D}(P\\derivd m,d\\tilde{\\mu}) \\quad \\text{ for all $q\\in\\bar{\\mathfrak{P}}_+$}.\n\\end{equation}\nNow, if there is a $q\\in\\bar{\\mathfrak{P}}_+$ such that $h(q)\\leq 0$, then $\\mathbb{J}(\\lambda q)\\to -\\infty$ as $\\lambda\\to\\infty$, which contradicts \\eqref{eq:lowerbound}. Therefore, $h(q)> 0$ for all $q\\in\\bar{\\mathfrak{P}}_+$. Then, since $h$ is continuous, it has  a minimum $\\varepsilon$ on the compact set $K:=\\{q\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}\\mid \\|q-e\\|_\\infty =1\\}$. As $e\\not\\in K$,  $\\epsilon >0$. Therefore,\n\\begin{displaymath}\nh(q) \\geq  \\varepsilon  \\|q-e\\|_\\infty \\geq  \\varepsilon\\|q\\|_\\infty - \\varepsilon\\|e\\|_\\infty \\geq \\frac{\\varepsilon}{|\\Lambda |} \\|Q\\|_\\infty  - \\varepsilon\\|e\\|_\\infty \\, ,\n\\end{displaymath}\nsince $\\|Q\\|_\\infty \\leq |\\Lambda|\\|q\\|_\\infty$ \\cite[Lem. A.1]{ringh2015multidimensional}.  Likewise, \n\\begin{align*}\n\\int_{\\mathbb{T}^d} \\! \\! P \\log Q \\derivd m &=\\int_{\\mathbb{T}^d} \\! \\!  P \\log \\left[ \\frac{Q}{\\|Q\\|_\\infty} \\right] \\derivd m\\!  +\\! \\int_{\\mathbb{T}^d}\\! \\!  P  \\log \\|Q\\|_\\infty \\derivd m\\\\ & \\leq \\int_{\\mathbb{T}^d}\\! \\!  P \\log \\|Q\\|_\\infty \\derivd m\\, ,\n\\end{align*}\nsince $Q\/\\|Q\\|_\\infty \\leq 1$. Hence\n\\begin{equation}\\label{eq:Jhard_ineq}\n\\rho \\geq \\mathbb{J}(q) \\geq  \\frac{\\varepsilon}{|\\Lambda |} \\|Q\\|_\\infty -\\int_{\\mathbb{T}^d}\\! \\!  P \\log \\|Q\\|_\\infty \\derivd m  - \\varepsilon\\|e\\|_\\infty \\, .\n\\end{equation}\nComparing linear and logarithmic growth  we see that the sublevel set is bounded from above and below.  Moreover, a trivial modification of \\cite[Lemma 3.1]{ringh2015multidimensional} shows that $\\mathbb{J}$ is lower semi-continuous, and hence  $\\mathbb{J}^{-1}(-\\infty,\\rho]$ is compact. Consequently, the problem \\eqref{eq:moddualproblem} has an optimal solution $\\hat{q}$.\n\n\nNext we show that $\\hat{q}$ is unique. For this we return to the original dual problem to find a minimum of \\eqref{eq:harddual}. The solution $\\hat{q}$ is a minimizer of $\\varphi(q,\\hat{\\gamma})$, where \n\\begin{displaymath}\n\\hat{\\gamma}=\\frac12\\|\\hat{q}-e\\|_W\\, ,\n\\end{displaymath}\nand $\\mathbb{J}(\\hat{q})=\\varphi(\\hat{q},\\hat{\\gamma})+c_{\\boldsymbol 0}$.\nTo show that $\\varphi$ is strictly convex, we form the Hessian\n\\begin{displaymath}\nH=\\begin{bmatrix} \\int_{\\mathbb{T}^d}P\/Q^2\\derivd m &0\\\\0&0 \\end{bmatrix}\n+\\frac{1}{2\\gamma^3}\\begin{bmatrix} \\gamma^2 W& -\\gamma (q-e)^*W\\\\-\\gamma W(q-e)&(q-e)^*W(q-e)\\end{bmatrix}\\\n\\end{displaymath}\nand the quadratic form\n\\begin{displaymath}\n\\begin{bmatrix} x\\\\ \\xi\\end{bmatrix}^*H\\begin{bmatrix} x\\\\ \\xi\\end{bmatrix}=\nx^* \\left(\\int_{\\mathbb{T}^d}P\/Q^2\\derivd m\\right) x + \\frac{1}{2\\gamma^3} [\\gamma x - \\xi (q-e)]^*W[\\gamma x - \\xi (q-e)],\n\\end{displaymath}\nwhich is positive for all nonzero $(x,\\xi)$, since $(q-e)\\ne 0$ and $\\gamma>0$. Consequently, $\\varphi$ has a unique minimizer $(\\hat{q},\\hat{\\gamma})$, where $\\hat{q}$ is the unique minimizer of $\\mathbb{J}$.\n\nIt follows from \\eqref{eq:q2rhard} and \\eqref{eq:gammaopt} that \n\\begin{equation}\n\\label{eq:rhathard}\n\\hat{r}=c + \\frac{W(\\hat{q}-e)}{\\|\\hat{q}-e\\|_W}\\, ,\n\\end{equation}\nwhich consequently is unique. Moreover, $h(\\hat{q})=\\langle\\hat{r},\\hat{q}\\rangle -\\hat{r}_{\\boldsymbol 0}$, and hence we can follow the same line of proof as in Theorem~\\ref{thm:softcontraints} to show that there is a unique $\\hat{c}\\in\\partial\\mathfrak{C}_+$ such that $\\langle\\hat{c},\\hat{q}\\rangle=0$ and a positive discrete measure $d\\hat\\nu$ with support in $|\\Lambda| -1$ points so that \\eqref{supp(dnu)hard} and \\eqref{rhat2+chat2} hold. Next, let $\\mathbb{I}(\\derivd \\mu)=-\\mathbb{D}(Pdm,\\derivd \\mu)$ be the primal functional in \\eqref{eq:primal_relax_new}, where $\\derivd \\mu$ is restricted to the set of positive measures $\\derivd \\mu:=\\Phi\\derivd m + d\\nu$  such that $r$, given by \\eqref{eq:r}, satisfies the constraint $\\|r-c\\|_W\\leq 1$. \nIn view of \\eqref{eq:rhat&c}, \n\\begin{displaymath}\n\\mathbb{I}(\\derivd \\mu)= \\mathcal{L}(\\Phi,d\\nu, r, \\hat{q},\\hat{\\gamma})\\leq \\mathcal{L}(\\hat{\\Phi},d\\hat\\nu, \\hat{r}, \\hat{q},\\hat{\\gamma}) =\\mathbb{I}(d\\hat\\mu)\n\\end{displaymath}\nfor any such $\\derivd \\mu$, and hence $\\derivd \\hat{\\mu}$ is an optimal solution to the primal problem \\eqref{eq:primal_relax_new}. \nFinally, the cases $p\\in\\mathfrak{S}_W$, $\\mathfrak{S}_W\\cap\\bar{\\mathfrak{C}}_+\\subset \\partial\\mathfrak{C}_+$, and $\\mathfrak{S}_W\\cap\\bar{\\mathfrak{C}}_+=\\emptyset$ have already been discussed above.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:KKThard}\nSuppose that $p\\in\\bar{\\mathfrak{P}}_+ \\setminus\\{0\\}$ and $\\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset$. The KKT conditions\n\\begin{subequations}\\label{eq:opt_cond_relax_new}\n\\begin{align}\n& \\hat{q} \\in \\bar{\\mathfrak{P}}_+, \\quad \\hat c \\in \\partial \\mathfrak{C}_+, \\quad \\langle \\hat{c}, \\hat{q} \\rangle = 0 \\label{eq:opt_cond_relax_slack_new} \\\\\n& \\hat{r}_{\\boldsymbol k} = \\int_{\\mathbb{T}^d} e^{i(\\kb,\\thetab)} \\frac{P}{\\hat Q}\\derivd m + \\hat{c}_{\\boldsymbol k},\\quad {\\boldsymbol k}\\in\\Lambda \\label{eq:opt_cond_relax_KKT_new} \\\\\n&(\\hat{r}-c)\\|\\hat{q}-e\\|_W = W(\\hat{q}-e), \\quad \\hat{r}\\in\\mathfrak{S}_W \\label{eq:opt_cond_relax_matchError_new}\n\\end{align}\n\\end{subequations}\nare necessary and sufficient conditions for optimality of the dual pair  \\eqref{eq:primal_relax_new} and  \\eqref{eq:moddualproblem} of optimization problems. \n\\end{corollary}\n\nThe corollary follows by noting that, if $p\\in\\mathfrak{S}_W$, then we obtain the trivial solution $\\hat{q}=e$, which corresponds to the primal optimal solution $\\derivd \\hat{\\mu} =P\\derivd m$. \n\n\\begin{proposition}\\label{prop:conditionW}\nThe condition\n\\begin{equation}\n\\label{eq:W>cc*}\nW > cc^*\n\\end{equation}\nis sufficient for the pair \\eqref{eq:primal_relax_new} and \\eqref{eq:moddualproblem} of dual problems to have optimal solutions. \n\\end{proposition}\n\n\\begin{proof}\nIf $W>cc^*$, then $(q-e)^*W(q-e)\\geq \\langle c, q-e\\rangle ^2$ with equality only for $q=e$. Hence, if $q\\ne e$, $\\|q-e\\|_W > |\\langle c, q-e\\rangle|$, i.e., $h(q) > 0$ for all $q\\in\\bar{\\mathfrak{P}}_+\\setminus\\{0\\}$ except $q=e$. Then we proceed as in the proof of Theorem~\\ref{thm:hardcontraints}.\n\\end{proof}\n\n\\begin{remark}\nCondition \\eqref{eq:W>cc*} guarantees that $0 \\in {\\rm int}(\\mathfrak{S}_W)$ and hence in particular that $\\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset$ as required in Theorem~\\ref{thm:hardcontraints}. To see this, note that $0\\in\\bar{\\mathfrak{C}}_+$ and that $r=0$ satisfies the hard constraint in \\eqref{eq:primal_relax_new} if $c^*W^{-1}c\\leq 0$. However, since $W>cc^*$, there is a $W_0 > 0$ such that $W = W_0 + cc^*$. Then the well-known Matrix Inversion Lemma (see, e.g., \\cite[p. 746]{lindquist2015linear}) yields\n\\[\n(W_0 + cc^*)^{-1} = W_0^{-1} - W_0^{-1} c (1 + c^*W_0^{-1}c)^{-1} c^* W_0^{-1},\n\\]\nand therefore\n\\[\nc^*W^{-1}c = c^*W_0^{-1}c - c^*W_0^{-1} c (1 + c^*W_0^{-1}c)^{-1} c^* W_0^{-1}c = \\frac{c^*W_0^{-1}c}{1 + c^*W_0^{-1}c} < 1,\n\\]\nwhich establishes that $0 \\in {\\rm int}(\\mathfrak{S}_W)$. However, for $\\mathfrak{S}_W\\cap\\mathfrak{C}_+$ to be nonempty, $r=0$ need not be contained in this set. Hence, condition \\eqref{eq:W>cc*} is not necessary, although it is easily testable. In fact, this provides an alternative proof of Proposition~\\ref{prop:conditionW}.\n\\end{remark}\n\n\\section{On the equivalence between the two problems}\\label{sec:homeomorphism} \n\nClearly $\\mathfrak{S}_W\\cap\\mathfrak{C}_+$ is always nonempty if $c\\in\\mathfrak{C}_+$. Then both the problem \\eqref{eq:primal_relax} with soft constraints and the problem \\eqref{eq:primal_relax_new} with hard constraints  have a solution for any choice of W. On the other hand, if $c \\not \\in \\mathfrak{C}_+$, the problem with soft constraints will always have a solution, while the problem with hard constraints may fail to have one for certain choices of $W$. However, if the weight matrix in the hard-constrained problem -- let us denote it $W_{\\rm hard}$ -- is chosen in the set $\\mathcal{W}:=\\{W > 0 \\mid \\mathfrak{S}_W\\cap\\mathfrak{C}_+\\ne\\emptyset, p\\not\\in\\mathfrak{S}_W\\}$, then it can be seen  from Corollaries \\ref{cor:KKTsoft} and \\ref{cor:KKThard} that we obtain exactly the same solution $\\hat{q}$ in the soft-constrained problem by choosing \n\\begin{equation}\n\\label{eq:Whard2Wsoft}\nW_{\\rm soft}=W_{\\rm hard}\/\\|\\hat{q}-e\\|_{W_{\\rm hard}}. \n\\end{equation}\nWe note that  \\eqref{eq:Whard2Wsoft} can be written $W_{\\rm hard}=\\alpha W_{\\rm soft}$, where $\\alpha :=\\|\\hat{q}-e\\|_{W_{\\rm hard}}$. Therefore, substituting $W_{\\rm hard}$ in \\eqref{eq:Whard2Wsoft}, we obtain \n \\begin{displaymath}\nW_{\\rm soft}=\\frac{\\alpha W_{\\rm soft}}{\\|\\hat{q}-e\\|_{\\alpha W_{\\rm soft}}}=\\alpha^{1\/2}\\frac{W_{\\rm soft}}{\\|\\hat{q}-e\\|_{W_{\\rm soft}}},\n\\end{displaymath}\nwhich yields  $\\alpha=\\|\\hat{q}-e\\|_{W_{\\rm soft}}^2$. Hence the inverse of \\eqref{eq:Whard2Wsoft} is given by \n\\begin{equation}\n\\label{eq:Wsoft2Whard}\nW_{\\rm hard}=W_{\\rm soft}\\|\\hat{q}-e\\|_{W_{\\rm soft}}^2. \n\\end{equation}\nBy Theorem~\\ref{thm:W2qcont} $\\hat{q}$ is continuous in $W_{\\rm soft}$, and hence, by \\eqref{eq:Wsoft2Whard}, the corresponding $W_{\\rm hard}$ varies continuously with $W_{\\rm soft}$. In fact, this can be strengthened to a homeomorphism between the two weight matrices. \n\n\\begin{theorem}\\label{thm:W2Whom}\nThe map \\eqref{eq:Whard2Wsoft} is a homeomorphism between $\\mathcal{W}$ and the space of all (Hermitian positive definite) weight matrices, and the inverse is given by \\eqref{eq:Wsoft2Whard}.\n\\end{theorem}\n\n\\begin{proof}\n By \\cite[Lemma 2.3]{byrnes2007interior}, a continuous map  between two spaces of the same dimension is a homeomorphism if and only if it is injective and proper, i.e., the preimage  of any compact set is compact. To see that $\\mathcal{W}$ is open, we observe that $\\mathfrak{S}_W$ is continuous in $W$ and that $\\mathfrak{C}_+$ is an open set. As noted above, the map \\eqref{eq:Wsoft2Whard} -- let us call it $f$ -- is continuous  and  also injective, as it can be inverted. Hence it only remains to show that $f$ is proper. To this end, we take a compact set $K\\subset\\mathcal{W}$ and  show that $f^{-1}(K)$ is also compact. There are two ways this could fail. First, the preimage could contain a singular semidefinite matrix. However this is impossible by \\eqref{eq:Wsoft2Whard}, since $\\|\\hat{q}\\|_\\infty$ is bounded for $W_{\\text{hard}}\\in K$ (Lemma~\\ref{lem:qhatbounded}) and  a nonzero scaling of a singular matrix cannot be nonsingular. Secondly, $\\|W_{\\rm soft}\\|_F$ could tend to infinity. However, this is also impossible. To see this, we first show that there is a $\\kappa >0$ such that $\\|p-r\\|_{W_{\\rm hard}^{-1}}\\geq \\kappa$ for all $r\\in\\mathfrak{S}_{W_{\\rm hard}}$ and all $W_{\\rm hard}\\in K$. To this end, we observe that the minimum of $\\|p-r\\|_{W^{-1}}$ over all $W\\in K$ and $r$ satisfying the constraint $\\|r-c\\|_{W^{-1}}\\leq 1$ is bounded by \n \\begin{displaymath}\n\\kappa:= \\min_{W\\in K} \\|p-c\\|_{W^{-1}} -1\n\\end{displaymath}\n by the triangle inequality $\\|p-r\\|_{W^{-1}}\\geq\\|p-c\\|_{W^{-1}}-\\|c-r\\|_{W^{-1}}\\geq\\|p-c\\|_{W^{-1}}-1$.\n The minimum is attained, since $K$ is compact, and positive, since  $p\\not\\in\\bigcup_{W\\in K}\\mathfrak{S}_{W}$.\nNow, from Corollary~\\ref{cor:KKThard} we see that $\\hat{q}=e$ if and only if $\\hat{r}=p$. The map from $\\hat{q}\\mapsto\\hat{r}$ is continuous in $q=e$. In fact, $\\hat Q$ is uniformly positive in a neighborhood of $e$ and hence the corresponding $\\hat{c}=0$. \nDue to this continuity, if $\\hat{q}\\to e$, then $\\hat{r}\\to p$, which cannot happen since $\\|p-r\\|_{W^{-1}}\\geq \\kappa$ for all $W\\in K$. Thus, since $\\|\\hat{q}-e\\|_{W}$ is bounded away from zero, the preimage $f^{-1}(K)$ of $K$ is bounded. Finally, consider a convergent sequence $(W_k)$  in $f^{-1}(K)$ converging to a limit $W_\\infty$. Since the sequence is bounded and cannot converge to a singular matrix, we must have $W_\\infty >0$, i.e.,  $W_\\infty\\in f^{-1}(\\mathcal{W})$. By continuity, $f(W_k)$ tends to the limit $f(W_\\infty)$, which must belong to $K$ since it is compact.  Hence the preimage $W_\\infty$ must belong to $f^{-1}(K)$.  Therefore, $f^{-1}(K)$ is compact as claimed. \n\\end{proof}\n\nIt is illustrative to consider the simple case when $W=\\lambda I$. Then the two maps \\eqref{eq:Whard2Wsoft} and \\eqref{eq:Wsoft2Whard} become\n\\begin{equation}\n\\label{eq:Lambda2lambda}\n\\begin{split}\n\\lambda_{\\rm soft}&=\\frac{\\sqrt{\\lambda_{\\rm hard}}}{\\|\\hat{q}-e\\|_2}\\\\\n\\lambda_{\\rm hard}&=\\lambda_{\\rm soft}^2\\|\\hat{q}-e\\|_2^2\n\\end{split}\n\\end{equation}\nWhereas the range of $\\lambda_{\\rm soft}$ is the semi-infinite interval $(0,\\infty)$, for the homeomorphism to hold  $\\lambda_{\\rm hard}$ is confined to \n\\begin{displaymath}\n\\lambda_{\\rm min} <\\lambda< \\lambda_{\\rm max},\n\\end{displaymath}\nwhere $\\lambda_{\\rm min}$ is the distance from $c$ to the cone $\\bar{\\mathfrak{C}}_+$ and $\\lambda_{\\rm max}=\\|c-p\\|$. When $\\lambda_{\\rm soft}\\to\\infty$, $\\lambda_{\\rm hard}\\to\\lambda_{\\rm max}$ and $\\hat{q}\\to e$. If $\\lambda_{\\rm hard}\\geq\\lambda_{\\rm max}$, then the coresponding problem has the trivial unique solution $\\hat{q}= e$, corresponding to the primal solution $\\derivd \\hat{\\mu}=P\\derivd m$.\n\n\n\n\n\nNote that Theorem~\\ref{thm:W2Whom} implies that some continuity results in one of the problems can be automatically transferred to the other problem. In particular, we have the following result.\n\n\n\\begin{theorem}\\label{thm:W2qcont_hard}  \nLet \n\\begin{equation}\n\\label{eq:Jhard}\n\\mathbb{J}_W(q)=\\langle c, q \\rangle - \\int_{\\mathbb{T}^d} P \\log Q\\, \\derivd m+ \\|q - e\\|_{W}.\n\\end{equation}\nThen the map $W\\mapsto \\hat{q}:=\\argmin_{q\\in\\bar{\\mathfrak{P}}_+} \\mathbb{J}_W(q)$ is continuous.\n\\end{theorem}\n\n\\begin{proof}\nThe theorem follows by noting that $W\\mapsto \\hat{q}:=\\argmin_{q\\in\\bar{\\mathfrak{P}}_+} \\mathbb{J}_W(q)$ can be seen as a composition of two continuous maps, namely the one in Theorem~\\ref{thm:W2qcont}  and the one in Theorem~\\ref{thm:W2Whom}.\n\\end{proof}\n\nNext we shall vary also $c$ and $p$, and to this end we introduce a more explicit notation for $\\mathfrak{S}_W$ and $\\mathcal{W}$, namely $\\mathfrak{S}_{c,W}=\\mathfrak{S}_W$ in \\eqref{eq:SW} and \n\\[\n\\mathcal{W}_{c,p}:=\\{W>0\\mid \\mathfrak{S}_{c,W}\\cap\\mathfrak{C}_+\\ne\\emptyset, p\\not\\in\\mathfrak{S}_{c,W}\\}.\n\\]\nThen the corresponding set of parameters \\eqref{eq:Pcal} for the problem with hard constraints is given by\n\\begin{equation}\n\\label{eq:Pcalhard}\n\\mathcal{P}_{\\rm hard}=\\{({\\ca, \\pa, \\Wa})\\mid c\\in \\mathfrak{C}, p\\in \\bar{\\mathfrak{P}}_+\\setminus\\{0\\}, W\\in \\mathcal{W}_{c,p}\\},\n\\end{equation}\nthe interior of which is\n\\begin{equation*}\n{\\rm int}(\\mathcal{P}_{\\rm hard})=\\{({\\ca, \\pa, \\Wa})\\mid c\\in \\mathfrak{C}, p\\in \\mathfrak{P}_+, W\\in \\mathcal{W}_{c,p}\\}.\n\\end{equation*}\nTheorem~\\ref{thm:W2Whom} can now be modified accordingly to yield the following theorem, the proof of which is deferred to the appendix.\n\\begin{theorem}\\label{thm:W2Whom2}\nLet the map $(c,p,W_{\\rm hard})\\mapsto W_{\\rm soft}$ be given by \\eqref{eq:Whard2Wsoft} and the map $(c,p,W_{\\rm soft})\\mapsto W_{\\rm hard}$ by \\eqref{eq:Wsoft2Whard}. Then the map that sends $(c,p,W_{\\rm hard})\\in {\\rm int}(\\mathcal{P}_{\\rm hard})$ to $(c,p,W_{\\rm soft})\\in {\\rm int}(\\mathcal{P})$ is a homeomorphism\n\\end{theorem}\nNote that this theorem is not a strict amplification of Theorem~\\ref{thm:W2Whom} as we have given up the possibility for $p$ to be on the boundary $\\partial \\mathfrak{P}_+$. The same is true for the following modification of Theorem~\\ref{thm:W2qcont_hard}.\n\\begin{theorem}\\label{thm:par2qcont_hard} \nLet $\\mathbb{J}_{{\\ca, \\pa, \\Wa}}(q)$ be as in \\eqref{eq:Jhard}.\nThen the map $({\\ca, \\pa, \\Wa})\\mapsto \\hat{q}:=\\argmin_{q\\in\\bar{\\mathfrak{P}}_+} \\mathbb{J}_{{\\ca, \\pa, \\Wa}}(q)$ is continuous on  ${\\rm int}(\\mathcal{P}_{\\rm hard})$.\n\\end{theorem}\n\\begin{proof}\nThe theorem follows immediately by noting that $({\\ca, \\pa, \\Wa}_\\text{hard})\\mapsto \\hat{q}$ can be seen as a composition of two continuous maps, namely $({\\ca, \\pa, \\Wa}_\\text{hard})\\mapsto({\\ca, \\pa, \\Wa}_\\text{soft})$ of Theorem~\\ref{thm:W2Whom2} and $({\\ca, \\pa, \\Wa}_\\text{soft})\\mapsto\\hat{q}$ of Theorem~\\ref{thm:W2qcont}.\n\\end{proof}\nTheorem~\\ref{thm:par2qcont_hard}  is a counterpart of Theorem~\\ref{thm:W2qcont} for the problem with hard constraints, except that $p$ is restricted to the interior $\\mathfrak{P}_+$. It should be possible to extend the result to hold for all $p\\in\\bar{\\mathfrak{P}}_+\\setminus \\{0\\}$ via a direct proof along the lines of the proof of Theorem~\\ref{thm:W2qcont}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Estimating covariances from data}\\label{sec:covest}\n\nFor a scalar stationary stochastic process $\\{y(t);\\,t\\in\\mathbb{Z}\\}$, it is well-known that the biased covariance estimate \n\\[\nc_k = \\frac{1}{N} \\sum_{t = 0}^{N - k -1}  y_t\\bar{y}_{t+k} ,\n\\]\nbased on an observation record $\\{ y_t \\}_{t = 0}^{N-1}$, yields a positive definite Toeplitz matrix, which is equivalent to $c \\in \\mathfrak{C}_+$ \\cite[pp. 13-14]{ahiezer1962some} In fact, these estimates correspond to the ones obtained from the periodogram estimate of the spectrum (see, e.g., \\cite[Sec. 2.2]{stoica1997introduction}). On the other hand, the Toeplitz matrix of the unbiased estimate \n\\[\nc_k = \\frac{1}{N-k} \\sum_{t = 0}^{N -k-1}  y_t \\bar{y}_{t+k}\n\\]\nis in general not positive definite.\n\nThe same holds in higher dimensions ($d>1$) where the observation record is $\\{y_{\\boldsymbol t}\\}_{{\\boldsymbol t}\\in \\mathbb{Z}^d_N}$ with $$\\mathbb{Z}^d_N=\\{(\\ell_1,\\ldots, \\ell_d)\\,|\\, 0\\le \\ell_j\\le N_j-1, j=1,\\ldots, d\\}.$$ \nThe unbiased estimate is then given by\n\\begin{equation}\\label{eq:unbiased_est}\nc_{\\boldsymbol k} = \\frac{1}{\\prod_{j=1}^d (N_j-|k_j|)} \\sum_{{\\boldsymbol t} \\in \\mathbb{Z}^d_{N}} y_{\\boldsymbol t} \\bar{y}_{{\\boldsymbol t}+{\\boldsymbol k}},\n\\end{equation}\nand the biased estimate by\n\\begin{equation}\\label{eq:biased_est}\nc_{\\boldsymbol k} = \\frac{1}{\\prod_{j=1}^d N_j} \\sum_{{\\boldsymbol t} \\in \\mathbb{Z}^d_{N}} y_{\\boldsymbol t} \\bar{y}_{{\\boldsymbol t}+{\\boldsymbol k}},\n\\end{equation}\nwhere we define $y_{\\boldsymbol t}=0$ for ${\\boldsymbol t}\\notin \\mathbb{Z}^d_{N}$. The sequence of unbiased covariance estimates does not in general belong to $\\mathfrak{C}_+$, but  the biased covariance estimates yields $c\\in\\mathfrak{C}_+$ also in the multidimensional setting. In fact, this can be seen by noting that the biased estimate corresponds to the Fourier coefficients of the periodogram \\cite[Sec. 6.5.1]{dudgeon1984multidimensional}, i.e.,  if the estimates $c_{\\boldsymbol k}$ are given by \\eqref{eq:biased_est}, then \n\\begin{equation}\\label{eq:periodogram}\n\\Phi_{\\rm periodogram}({\\boldsymbol \\theta}) := \\frac{1}{\\prod_{j=1}^d N_j} \\Big| \\sum_{{\\boldsymbol t} \\in \\mathbb{Z}^d_N} y_{\\boldsymbol t} e^{i({\\boldsymbol t},\\thetab)} \\Big|^2=\\sum_{{\\boldsymbol k} \\in \\mathbb{Z}^d_N-\\mathbb{Z}^d_N} c_{{\\boldsymbol k}} e^{-i(\\kb,\\thetab)} ,\n\\end{equation}\nwhere $\\mathbb{Z}^d_N-\\mathbb{Z}^d_N$ denotes the Minkowski set difference. This leads to the following lemma.\n\n\\begin{lemma}\\label{lem:biased}\nGiven the observed data $\\{ y_{{\\boldsymbol t}} \\}_{{\\boldsymbol t} \\in \\mathbb{Z}^d_N}$, let $\\{ c_{\\boldsymbol k} \\}_{{\\boldsymbol k} \\in \\Lambda}$ be given by \\eqref{eq:biased_est}.  Then $c \\in \\mathfrak{C}_+$.\n\\end{lemma}\n\n\\begin{proof}\nGiven $\\{ y_{{\\boldsymbol t}} \\}_{{\\boldsymbol t} \\in \\mathbb{Z}^d_N}$, let $c = \\{ c_{\\boldsymbol k} \\}_{{\\boldsymbol k} \\in \\mathbb{Z}^d_N}$, where $c_{\\boldsymbol k}$ be given by \\eqref{eq:biased_est}. \nIn view of  \\eqref{innerprod} and \\eqref{eq:periodogram} we have\n\\[\n\\langle c, p \\rangle = \\int_{\\mathbb{T}^d} \\frac{1}{\\prod_{j=1}^d N_j} \\Big| \\sum_{{\\boldsymbol t} \\in \\mathbb{Z}^d_N} y_{\\boldsymbol t} e^{i({\\boldsymbol t},\\thetab)} \\Big|^2 P(e^{i{\\boldsymbol \\theta}}) \\derivd m({\\boldsymbol \\theta}),\n\\]\nwhich is positive for all $p \\in \\bar{\\mathfrak{P}}_+ \\setminus \\{0\\}$. Consequently $c \\in \\mathfrak{C}_+$.\n\\end{proof}\n\nAn advantage of the approximate procedures to the rational covariance extension problem is that they can also be used for cases where the biased estimate is not available, e.g., where the covariance is estimated from snapshots.\n\n\n\\section{Application to spectral estimation}\\label{sec:example}\n\nAs long as we use the biased estimate \\eqref{eq:biased_est}, we may apply exact covariance matching as outlined in Section~\\ref{sec:prel}, whereas in general approximate covariance matching will be required for biased covariance estimates. However, as will be seen in the following example, approximate covariance matching may sometimes be better even if $c\\in\\mathfrak{C}_+$.\n\nIn this application it is easy to determine a bound on the acceptable error in the covariance matching, so we use the procedure with hard constraints. Given data generated from a two-dimensional stochastic system, we test three different procedures, namely (i) using the biased estimate and exact matching, (ii) using the biased estimate and the approximate matching \\eqref{eq:primal_relax_new}, and (iii)  using the unbiased estimate and the approximate matching \\eqref{eq:primal_relax_new}. The procedures are then evaluated by checking the size of the error  between the matched covariances and the true ones from the dynamical system.\n\n\\subsection{An example} \n\nLet $y_{(t_1, t_2)}$ be the steady-state output of a two-dimensional recursive filter driven by a white noise input $u_{(t_1, t_2)}$. Let the transfer function of the recursive filter be\n\\[\n\\frac{b(e^{i\\theta_1}, e^{i\\theta_2})}{a(e^{i\\theta_1}, e^{i\\theta_2})} = \\frac{ \\sum_{{\\boldsymbol k} \\in \\Lambda_+} b_{{\\boldsymbol k}} e^{-i({\\boldsymbol k},{\\boldsymbol \\theta})}}{\\sum_{{\\boldsymbol k} \\in \\Lambda_+} a_{{\\boldsymbol k}} e^{-i({\\boldsymbol k},{\\boldsymbol \\theta})}},\n\\]\nwhere $\\Lambda_+=\\{(k_1,k_2)\\in \\mathbb{Z}^2\\mid 0\\le k_1\\le 2, 0\\le k_2\\le 2 \\}$ and the coefficients are given by\n$b_{(k_1,k_2)}=B_{k_1+1, k_2+1} $ and $a_{(k_1,k_2)}=A_{k_1+1, k_2+1}$, where\n\\begin{align*}\nB = \\!\n{\\scriptscriptstyle\n\\begin{bmatrix}\n    \\,0.9 &  -0.2\\phantom{0} &   0.05 \\\\\n    \\,0.2 &  \\phantom{-} 0.3\\phantom{0} &   0.05 \\\\\n   -0.05 &   -0.05 &   0.1\\phantom{0}\n\\end{bmatrix}}, \\;\nA = \\!\n{\\scriptscriptstyle\n\\begin{bmatrix}\n    \\phantom{-}1\\phantom{.0} &   \\phantom{-}0.1 &   \\phantom{-}0.1 \\\\\n   -0.2 &   \\phantom{-}0.2 &  -0.1 \\\\\n    \\phantom{-}0.4 &  -0.1 &  -0.2\n\\end{bmatrix}\n}.\n\\end{align*}\nThe spectral density $\\Phi$ of $y_{(t_1, t_2)}$, which is shown in Fig.~\\ref{fig:original} and is similar to the one considered in \\cite{ringh2015multidimensional}, is given by\n\\[\n\\Phi(e^{i\\theta_1}, e^{i\\theta_2}) = \\frac{P(e^{i\\theta_1}, e^{i\\theta_2})}{Q(e^{i\\theta_1}, e^{i\\theta_2})} = \\left|\\frac{b(e^{i\\theta_1}, e^{i\\theta_2})}{a(e^{i\\theta_1}, e^{i\\theta_2})}\\right|^2,\n\\]\nand hence the index set $\\Lambda$ of the coefficients of the trigonometric polynomials $P$ and $Q$ is given by $\\Lambda=\\Lambda_+-\\Lambda_+=\\{(k_1,k_2)\\in \\mathbb{Z}^2\\,|\\; |k_1|\\le 2, |k_2|\\le 2 \\}$. Using this example, we perform two different simulation studies.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{figures\/ \\spect_est_fig figure3.png}\n\\caption{Log-plot of the original spectrum.}\n\\label{fig:original}\n\\end{figure}\n\n\n\\subsection{First simulation study}\nThe system was simulated for $500$ time steps along each dimension, starting from $y_{(t_1, t_2)} = u_{(t_1, t_2)} = 0$ whenever either $t_1 < 0$ or $t_2 < 0$. Then covariances were estimated from the $9 \\times 9$ last samples, using both the biased and the unbiased estimator. With this covariance data we investigate the  three procedures (i), (ii) and (iii)  described above. In each case, both the maximum entropy (ME) solutions and solutions with the true numerator are computed.\\footnote{Maximum entropy: $P\\equiv 1$. True numerator: $P=P_{\\text{true}}$.} The weighting matrix is taken to be $W = \\lambda I$, where $\\lambda$ is $\\lambda_{\\text{biased}} := \\| c_{\\text{true}} - c_{\\text{biased}} \\|_2^2$ in procedure (ii) and $\\lambda_{\\text{unbiased}} := \\| c_{\\text{true}} - c_{\\text{unbiased}} \\|_2^2$ in procedure (iii)\n\\footnote{Note that this is the smallest $\\lambda$ for which the true covariance sequence belongs to the uncertainty set $\\{r\\,|\\, \\|r-c\\|_2^2\\le \\lambda\\}$.}\nThe norm of the error%\n\\footnote{Here we use the norm of the covariance estimation error as measure of fit. However, note that this is not the only way to compare accuracy of the different methods. The reason for this choice is that comparing the  accuracy of the spectral estimates is not straightforward since it depends on the selected metric or distortion measure.}      \n between the matched covariances and the true ones, $\\| \\hat{r} - c_{\\text{true}} \\|_2$, is shown in Table.~\\ref{tab:mean_std_2}. The means and standard deviations are computed over the $100$ runs.\n\nThe biased covariance estimates belong to the cone $\\mathfrak{C}_+$ (Lemma~\\ref{lem:biased}), and therefore procedure (i) can be used. The corresponding error in Table~\\ref{tab:mean_std_2} is the statistical error in estimating the covariance. This error is quite large because of a  short data record. Using approximate covariance matching in this case seems to give a worse match.\nHowever, approximate matching of the unbiased covariances gives as good a fit as exact matching of the biased ones. \n\n\\begin{table}[bt]\n\\caption{Norm differences $\\| \\hat{r} - c_{\\rm {true}} \\|_2$ for different solutions in the first simulation setup.}\n\\label{tab:mean_std_2}\n\\begin{center}\n\\begin{tabular}{l l l}\n\\toprule\n                         & Mean     & Std. \\\\\n\\midrule\nBiased, exact matching   & $3.2374$ & $1.7944$ \\\\\nBiased, approximate matching, ME-solution    & $3.7886$ & $1.3274$  \\\\\nBiased, approximate matching, using true $P$ & $3.8152$ & $1.6509$  \\\\\nUnbiased, approximate matching, ME-solution    & $3.2575$ & $1.4721$  \\\\\nUnbiased, approximate matching, using true $P$ & $3.2811$ & $1.7787$  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Second simulation study}\nIn this simulation the setup is the same as the previous one, except that the simulation data has been discarded if the {\\em unbiased\\\/} estimate belongs to $\\bar{\\mathfrak{C}}_+$. To obtain $100$ such data sets, $414$ simulations of the system were needed. (As a comparison, in the previous experiment $23$ out of the $100$ runs resulted in an unbiased estimate outside $\\bar{\\mathfrak{C}}_+$.)\nAgain, the norm of the error between matched covariances and the true ones is shown in Table~\\ref{tab:mean_std}, and the means and standard deviations are computed over the $100$ runs. \n\nAs before, the biased covariance estimates belong to the cone $\\mathfrak{C}_+$, and therefore procedure (i) can be used.\nComparing this with the results from procedure (ii)  suggests that there may be an advantage not to enforce exact matching, although we know that the data belongs to the cone.\nRegarding procedure (iii), we know that the unbiased covariance estimates do not belong to the cone $\\bar{\\mathfrak{C}}_+$, hence we need to use approximate covariance matching. In this example, this procedure turns out to give the smallest estimation error.\n\n\n\n\\begin{table}[bt]\n\\caption{\nNorm differences $\\| \\hat{r} - c_{\\rm {true}} \\|_2$ for different solutions in the second setup, where all unbiased estimate are outside $\\bar{\\mathfrak{C}}_+$.}\n\\label{tab:mean_std}\n\\begin{center}\n\\begin{tabular}{l l l}\n\\toprule\n                                     & Mean     & Std. \\\\\n\\midrule\nBiased, exact matching   & $2.9245$ & $2.2528$ \\\\\nBiased, approximate matching, ME-solution    & $1.9087$ & $1.1324$ \\\\\nBiased, approximate matching, using true $P$ & $1.8532$ & $1.1904$  \\\\\nUnbiased, approximate matching, ME-solution    & $1.5018$ & $0.6601$ \\\\\nUnbiased, approximate matching, using true $P$ & $1.4451$ & $0.7296$  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n \n\n\n\n\n\n\\section{Application to system identification and texture reconstruction}\\label{sec:ex_texture}\n\nNext we apply the theory of this paper to texture generation via Wiener system identification.\nWiener systems form a class of nonlinear dynamical systems consisting of a linear dynamic part composed with a static nonlinearity as illustrated in Figure \\ref{fig:blockdiagramRep}. This is a subclass of so called block-oriented systems \\cite{billings1980identification}, and Wiener system identification is a well-researched area (see, e.g., \\cite{greblicki1992nonparametric} and references therein) that is still very active \\cite{lindsten2013bayesian, wahlberg2015identification, abdalmoaty2016simulated}. Here, we use  Wiener systems to model and generate textures.\n\n\\begin{figure}[t\n\\centering\n\\input{figures\/blockdiagram.tex}\n\\caption{A Wiener system with thresholding as static nonlinearity.}\n\\label{fig:blockdiagramRep}\n\\end{figure} \n\nUsing dynamical systems for modeling of images and textures is not new and has been considered in, e.g., \\cite{Chiuso-F-P-05, picci2008modelling}. The setup presented here is motivated by \\cite{barman2015gaussian}, where thresholded Gaussian random fields are used to model porous materials for design of surface structures in pharmaceutical film coatings. Hence we let the static nonlinearity, call it $f$, be a thresholding with unknown thresholding parameter $\\tau$. In our previous work \\cite{ringh2017further} we applied exact covariance matching to such a problem. However, in general there is no guarantee that the estimated covariance sequence $c$ belongs to the cone $\\mathfrak{C}_+$. Consequently, here we shall use approximate covariance matching instead.\n\nThe Wiener system identification can be separated into two parts. We start by identifying  the nonlinear part. Using the notations of Figure~\\ref{fig:blockdiagramRep}, let $\\{u_{\\boldsymbol t}; \\, {\\boldsymbol t} \\in \\mathbb{Z}^d\\}$ be a zero-mean Gaussian white noise input, and let $\\{x_{\\boldsymbol t}; \\, {\\boldsymbol t} \\in \\mathbb{Z}^d\\}$ be the stationary output of the linear system, which we assume to be normalized so that $c_{\\mathbf 0}:= \\ExpOp [x_{\\boldsymbol t}^2]=1$. Moreover, let $y_{\\boldsymbol t}=f(x_{\\boldsymbol t})$ where $f$ is the static nonlinearity\n\\begin{equation}\\label{eq:staticnonlin}\nf(x)=\\begin{cases}1 & x>\\tau \\\\0 & \\mbox{otherwise}\\end{cases}\n\\end{equation}\nwith unknown thresholding parameter $\\tau$. Since $ \\ExpOp [y_{\\boldsymbol t}] = 1-\\phi(\\tau)$, where $\\phi(\\tau)$ is the Gaussian cumulative distribution function, an estimate of $\\tau$ is given by $\\tau_{\\rm est}= \\phi^{-1}(1- \\ExpOp [y_{\\boldsymbol t}])$.\n\nNow, let $c_{\\boldsymbol k}^x :=  \\ExpOp [x_{{\\boldsymbol t}+{\\boldsymbol k}} x_{\\boldsymbol t}]$ be the covariances of $x_{\\boldsymbol t}$, and let $c_{\\boldsymbol k}^y:=  \\ExpOp [y_{{\\boldsymbol t}+{\\boldsymbol k}} y_{\\boldsymbol t}] -  \\ExpOp [y_{{\\boldsymbol t} + {\\boldsymbol k}}] \\ExpOp [y_{\\boldsymbol t}]$ be the covariances of $y_{\\boldsymbol t}$. As was explained in \\cite{ringh2017further}, by using results from \\cite{price1958useful} one can obtain a relation between $c_{\\boldsymbol k}^y$ and $c_{\\boldsymbol k}^x$, given by \n\\begin{equation}\n\\label{eq:randcrelation}\n\\begin{split}\nc_{\\boldsymbol k}^y = \\int_0^{c_{{\\boldsymbol k}}^x} \\frac{1}{2\\pi \\sqrt{1-s^2}}\\exp\\left(-\\frac{\\tau^2}{1+s}\\right)ds .\n\\end{split}\n\\end{equation}\nThis is an invertible map, which we compute numerically, and given $\\tau_{\\rm est}$ we can thus get estimates of the covariances $c_{\\boldsymbol k}^x$ from estimates of the covariances $c_{\\boldsymbol k}^y$.   However, even if $c^y$ is is a biased estimate so that $c^y \\in \\mathfrak{C}_+$, $c^x$ may not be a {\\em bona fide} covariance sequence.\n\n\n\n\n\\subsection{Identifying the linear system}\\label{subsec:linear_syst}\n\nSolving \\eqref{eq:primal_relax} or \\eqref{eq:primal_relax_new} for a given sequence of covariance estimates $c$, we obtain an estimate of the absolutely continuous part of the power spectrum $\\Phi$ of that process. In the case $d=1$, $\\Phi =P\/Q$ can be factorized as \n\\[\n\\Phi(e^{i\\theta}) = \\frac{P(e^{i\\theta})}{Q(e^{i\\theta})} = \\frac{|b(e^{i\\theta})|^2}{|a(e^{i\\theta})|^2},\n\\]\nwhich provides a transfer function of a corresponding linear system, which fed by a  white noise input  will produce an autoregressive-moving-average (ARMA) process with an output signal with precisely the  power distribution  $\\Phi$ in steady state. For $d\\geq 2$, a spectral factorization of this kind is not possible in general  \\cite{dumitrescu2007positive}, but instead there is always a factorization as a sum-of-several-squares \\cite{dritschel2004factorization, geronimo2006factorization},\n\\[\n\\Phi(e^{i{\\boldsymbol \\theta}}) = \\frac{P(e^{i{\\boldsymbol \\theta}})}{Q(e^{i{\\boldsymbol \\theta}})} = \\frac{\\sum_{k = 1}^{\\ell} |b_k(e^{i{\\boldsymbol \\theta}})|^2}{\\sum_{k = 1}^m |a_k(e^{i{\\boldsymbol \\theta}})|^2},\n\\]\nthe interpretation of which in terms of a dynamical system is unclear when $m > 1$. Therefore we resort to a heuristic and apply the factorization procedure in \\cite[Theorem 1.1.1]{geronimo2004positive} although some of the conditions \nrequired to ensure the existence of a spectral factor may not be met.\n(See \\cite[Section 7]{ringh2015multidimensional} for a more detailed discussion.)\n\n\\subsection{Simulation results}\nThe method, which is summarized in Algorithm~\\ref{alg:WienerEst}, is tested on some textures from the Outex database  \\cite{ojala2002outex} (available online at \\url{http:\/\/www.outex.oulu.fi\/}). These textures are color images and have thus been converted to binary textures by first converting them to black-and-white and then thresholding them.%\n\\footnote{The algorithm has been implemented and tested in Matlab, version R2015b. The textures have been normalized to account for light inhomogenities using a reference image available in the database. The conversion from color images to black-and-white images was done with the built-in function \\texttt{rgb2gray}, and the threshold level was set to the mean value of the maximum and minimum pixel value in the black-and-white image.}\nThree such textures are shown in Figure~\\ref{subfig:texture_one} through \\ref{subfig:texture_three}.\n\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\algsetup{indent=12pt}\n\n\\begin{algorithm}\n\\caption{\n\\label{alg:WienerEst}\n\\begin{algorithmic}[1]\n\\REQUIRE $(y_{\\boldsymbol t})$\n\\STATE Estimate threshold parameter: $\\tau_{\\rm est}= \\phi^{-1}(1-E[y_{\\boldsymbol t}])$\n\\STATE Estimate covariances: $c_{\\boldsymbol k}^y := E[y_{{\\boldsymbol t}+{\\boldsymbol k}} y_{\\boldsymbol t}] - E[y_{{\\boldsymbol t} + {\\boldsymbol k}}]E[y_{\\boldsymbol t}]$\n\\STATE Compute covariances $c_{\\boldsymbol k}^x := E[x_{{\\boldsymbol t}+{\\boldsymbol k}} x_{\\boldsymbol t}]$  by using \\eqref{eq:randcrelation}\n\\STATE Estimate a rational spectrum using Theorem~\\ref{thm:softcontraints} or \\ref{thm:hardcontraints}\n\\STATE Apply the factorization procedure in \\cite[Theorem 1.1.1]{geronimo2004positive}\n\\ENSURE $\\tau_{\\rm est}$, coefficients for the linear dynamical system\n\\end{algorithmic}\n\\end{algorithm}\nIn this example there is no natural bound on the error, so we use the problem with soft constraints, for which we choose the weight $W = \\lambda I$ with $\\lambda=0.01$ for all data sets. Moreover, we do maximum-entropy reconstructions, i.e., we set the prior to $P \\equiv 1$. The optimization problems are then solved by first discretizing the grid $\\mathbb{T}^2$, in this case in $50 \\times 50$ points (cf. \\cite[Theorem 2.6]{ringh2015multidimensional}), and solving the corresponding problems using the CVX toolbox \\cite{cvx, grant2008graph}. The reconstructions are shown in Figures~\\ref{subfig:recon_texture_one}~-~\\ref{subfig:recon_texture_three}. Each reconstruction seems to provide a reasonable visual representation of the structure of the corresponding original. This is especially the case for the second texture.\n\n\n\\begin{figure*}%\n  \\centering\n  \\subfloat[First texture.\\label{subfig:texture_one}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/granite001-inca-100dpi-00_reg_0point01\/\" figure7_new.png}}%\n  \\hfil\n  \\subfloat[Second texture.\\label{subfig:texture_two}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/paper010-inca-100dpi-00_reg_0point01\/\" figure7_new.png}}%\n  \\hfil\n  \\subfloat[Third texture.\\label{subfig:texture_three}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/plastic008-inca-100dpi-00_reg_0point01\/\" figure7_new.png}}%\n  \\hfil \n  \\subfloat[Reconstruction of \\ref{subfig:texture_one}.\\label{subfig:recon_texture_one}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/granite001-inca-100dpi-00_reg_0point01\/\" figure37_new.png}}%\n  \\hfil \n  \\subfloat[Reconstruction of \\ref{subfig:texture_two}.\\label{subfig:recon_texture_two}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/paper010-inca-100dpi-00_reg_0point01\/\" figure37_new.png}}%\n  \\hfil\n  \\subfloat[Reconstruction of \\ref{subfig:texture_three}.\\label{subfig:recon_texture_three}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/plastic008-inca-100dpi-00_reg_0point01\/\" figure37_new.png}}%\n  \\hfil\n  \\subfloat[Close-up of \\ref{subfig:texture_one}.\\label{subfig:texture_one_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/granite001-inca-100dpi-00_reg_0point01\/\" figure6.png}}%\n  \\hfil\n  \\subfloat[Close-up of \\ref{subfig:texture_two}.\\label{subfig:texture_two_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/paper010-inca-100dpi-00_reg_0point01\/\" figure6.png}}%\n  \\hfil\n  \\subfloat[Close-up of \\ref{subfig:texture_three}.\\label{subfig:texture_three_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/plastic008-inca-100dpi-00_reg_0point01\/\" figure6.png}}%\n  \\hfil \n  \\subfloat[Close-up of \\ref{subfig:recon_texture_one}.\\label{subfig:recon_texture_one_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/granite001-inca-100dpi-00_reg_0point01\/\" figure36.png}}%\n  \\hfil \n  \\subfloat[Close-up of \\ref{subfig:recon_texture_two}.\\label{subfig:recon_texture_two_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/paper010-inca-100dpi-00_reg_0point01\/\" figure36.png}}%\n  \\hfil\n  \\subfloat[Close-up of \\ref{subfig:recon_texture_three}.\\label{subfig:recon_texture_three_closeup}]{\\includegraphics[width=0.3\\textwidth]{figures\/ \"text_gen\/plastic008-inca-100dpi-00_reg_0point01\/\" figure36.png}}%\n    \\caption{In Figures~\\ref{subfig:texture_one}~-~\\ref{subfig:texture_three} three different binary textures, of size $1200 \\times 900$ pixels, are shown. These are obtained from the textures granite001-inca-100dpi-00, paper010-inca-100dpi-00, and plastic008-inca-100dpi-00 in the Outex database, respectively. The textures in Figures~\\ref{subfig:texture_one}~-~\\ref{subfig:texture_three} are used as input $(y_{\\boldsymbol t})$ to Algorithm~\\ref{alg:WienerEst} and in Figures~\\ref{subfig:recon_texture_one}~-~\\ref{subfig:recon_texture_three} the corresponding reconstructed textures of size $500\\times 500$ are shown. In Figures~\\ref{subfig:texture_one_closeup}~-~\\ref{subfig:recon_texture_three_closeup}  close-ups of size $100\\times 100$ are shown of the original and reconstructed textures (areas marked in Figures~\\ref{subfig:texture_one}~-~\\ref{subfig:recon_texture_three}).\n  }%\n  \\label{fig:texture_generation}%\n\\end{figure*}\n\n\n\n\n\n\n\\section{Conclusions}\nIn this work we extend the results of our previous paper \\cite{ringh2015multidimensional}  on the multidimensional rational covariance extension problem to allow for approximate covariance matching. We have provided two formulations to this problem, and we have shown that they are connected via a homeomorphism. In both formulations we have used  weighted 2-norms to quantify the missmatch of the estimated covariances. However,  we expect that by suitable modifications of the proofs similar results can be derived for other norms, since all norms have directional derivatives in each point \\cite[p. 49]{deimling1985nonlinear}. \n\nThese results  provide a procedure for multidimensional spectral estimation, but in order to obtain a complete theory for multidimensional system identification and realization theory  there are still some open problems, such as spectral factorization and interpretations in terms of multidimensional stochastic systems, as briefly discussed in Section~\\ref{subsec:linear_syst}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Preliminaries\\label{sec:rathomotopy}}\nLet $M$ be a multiply connected Riemannian manifold. Then the conjugacy classes in the fundamental group $\\pi_1(M)$ may be mapped bijectively onto the free homotopy classes of closed loops on $M$. Closed geodesics are generated by undergoing an energy minimizing procedure. Thus, every closed multiply connected Riemannian manifold $M$ carries a closed geodesic. Likewise, every closed simply connected Riemannian manifold $M$ carries a closed geodesic according to Lyusternik and Fet \\cite{lyusternik-fet}. \nThe existence of at least one non-trivial, non-constant, closed geodesic on a closed multiply connected Riemannian manifold follows from properties of the gradient flow of the energy functional. \n\\begin{theorem}\\label{Cartan} Assume $M$ is closed and $\\pi_1(M)$ is non-trivial. There is a closed geodesic in each non-trivial free homotopy class or in each non-trivial conjugacy class of the fundamental group $\\pi_1(M)$.\n\\end{theorem}\nThe aforementioned theorem is only valid for $\\pi_1(M)$ non-trivial, i.e. when $M$ is multiply connected. The theorem of Lyusternik--Fet covers the case for $M$ simply connected.\n\\begin{theorem}[Lyusternik--Fet \\cite{lyusternik-fet}]\nAny closed simply connected Riemannian manifold carries at least one closed geodesic.\n\\end{theorem}\n\n\\begin{definition}\nA non-constant closed geodesic is said to be prime if it traces out its image exactly once and if it is not the iterate of another closed geodesic. Similarly, two geodesics are geometrically distinct if their images differ as subsets of $M$.\n\\end{definition} \nTwo geodesics have the same image if and only if their parameterizations differ modulo an affine transformation of $\\mathbb{R}$. Note, in general, prime closed geodesics are not simple; that is, they admit self-intersections. We henceforth address the following question.\n\\begin{question}\nDoes every closed Riemannian manifold $M$ admit infinitely many geometrically distinct, non-constant, prime closed geodesics?\n\\end{question}\nThe case for $S^2$ has been proven in the affirmative by Hingston \\cite{hingston}. However, the solution is not known in general. In the case of multiply connected manifolds, we can enumerate a class for which the fundamental group $\\pi_1(M)$ has infinitely many closed geodesics, whereby the minimization procedure produces infinitely many geometrically distinct geodesics, one for each homotopy class. As such, the essence of the problem is reduced to the simply connected case, which was addressed by Gromoll and Meyer \\cite{gromoll-meyer}. \n\n\\begin{notation}\nWe will use the following conventional notations.\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Omega M$ denotes the loop space of a manifold $M$.\n\\item $\\mathcal{L} M$ denotes the smooth free loop space on a manifold $M$.\n\\item $\\Lambda M$ denotes the free loop space of Sobolev class $H^1=W^{1,2}$ on a manifold $M$.\n\\item $\\mathbb{F}_p$ denotes the field with $p$ elements for $p\\ge 2$ prime.\n\\item $b_k(\\Lambda M;\\mathbb{F}_p):=\\text{rank }H_k(\\Lambda M;\\mathbb{F}_p)$ is the $k$-th Betti number of $\\Lambda M$ in $\\mathbb{F}_p$ coefficients, which is equivalent to the $k$-th Betti number of $\\mathcal{L} M$ in $\\mathbb{F}_p$ coefficients.\n\\end{enumerate}\n\\end{notation}\n\n\\begin{definition}\nClosed geodesics $\\gamma:S^1=\\mathbb{R}\/\\mathbb{Z}\\to M$ are critical points of the energy functional \n\\begin{equation}\\label{energyfunctional}\nE[\\gamma]=\\int_{S^1}\\|\\dot\\gamma(t)\\|^2dt,\n\\end{equation} which is defined on the free loop space $\\Lambda M=H^1(S^1,M)$ of Sobolev class $H^1$.\n\\end{definition}\nThus, the critical points of $E$ are determined by the topology of the Hilbert manifold $\\Lambda M$. Let the bilinear map $\\nabla:\\Gamma(TM)\\times\\Gamma(TM)\\to\\Gamma(TM)$ on sections of the tangent bundle $TM$ be the Levi-Civita connection on $M$. For a smooth curve $\\gamma: I\\subset\\mathbb{R}\\to M$ parameterized by $t$, the covariant derivative $\\nabla_{\\dot\\gamma}\\dot\\gamma :=\\nabla_{t}\\dot\\gamma$ defines a vector field along $\\gamma$. The curve $\\gamma$ is said to be a geodesic if it satisfies the autoparallel transport equation \n\\begin{equation}\\label{autoparallel}\n\\nabla_t\\dot\\gamma=0.\n\\end{equation} \nA geodesic has constant speed $\\|\\dot\\gamma(t)\\|$ because $\\frac{1}{2}\\frac{d}{dt}\\|\\dot\\gamma\\|^2=\\langle \\nabla_t\\dot\\gamma,\\dot\\gamma\\rangle$. Equation (\\ref{autoparallel}) is a second-order, non-linear, ordinary differential equation with smooth coefficients. \n\nLet $p,q\\in M$ be distinct points, and consider the space of smooth paths defined on $[0,1]$ from $p$ to $q$:\n\\begin{equation}\n\\mathcal{P}(p,q)=\\{\\gamma\\in C^{\\infty}([0,1],M):\\gamma(0)=p,\\gamma(1)=q\\}.\n\\end{equation}\nThe space of such paths $\\mathcal{P}(p,q)$ is a Fr\\'echet manifold and the tangent space at a path $\\gamma$ is defined as $T_{\\gamma}P(p,q)=\\{\\xi\\in\\Gamma(\\gamma^*TM):\\xi(0)=0,\\xi(1)=0\\}$. That is, an element $\\xi\\in T_{\\gamma}\\mathcal{P}(p,q)$ uniquely determines a curve \n\\begin{equation}\n\\alpha_{\\xi}:\\mathcal{V}(0)\\subset\\mathbb{R}\\to\\mathcal{P}(p,q),\\quad \\alpha_{\\xi}(s)(t):=exp_{\\gamma(t)}(s\\xi(t)),\n\\end{equation} where $\\alpha_{\\xi}(0)=\\gamma$ and $\\evalat[\\big]{\\frac{d}{ds} }{s=0}\\alpha_{\\xi}(s)(t)=\\xi(t),t\\in[0,1]$. The energy functional $E:\\mathcal{P}(p,q)\\to\\mathbb{R}$ for this Fr\\'echet manifold is defined as \n\\begin{equation}\\label{energyfunctional}\nE[\\gamma]:=\\int_0^1 \\|\\dot\\gamma(t)\\|^2dt.\n\\end{equation} Similarly, the differential of $E$ at $\\gamma$ is a linear map $dE[\\gamma]:T_\\gamma\\mathcal{P}(p,q)\\xrightarrow{\\sim}\\mathbb{R}$ given by the first variation formula via its action on a smooth section of the tangent bundle $\\xi\\in T_{\\gamma}\\mathcal{P}(p,q)$, \n\\begin{equation}\ndE[\\gamma]\\xi=2\\int_0^1\\langle\\nabla_t\\xi,\\dot\\gamma\\rangle=-2\\int_0^1\\langle\\xi,\\nabla_t\\dot\\gamma\\rangle.\n\\end{equation}\n\\begin{remark}\nThe curve $\\gamma\\in C^{\\infty}([0,1],M)$ is a critical point of $E$ if and only if $\\nabla_t\\dot\\gamma=0$; that is, $\\gamma$ is a geodesic from $p$ to $q$.\n\\end{remark} \nFor our purposes, we examine Banach and, more specifically, Hilbert manifolds. Consider the following space of paths of Sobolev class $H^1$ defined on $[0,1]$ from $p$ to $q$ as the domain for the energy functional:\n\\begin{equation}\n\\Omega(p,q)=\\{\\gamma\\in H^1([0,1],M):\\gamma(0)=p,\\gamma(1)=q\\}.\n\\end{equation}\nSuch paths are continuous so the endpoint conditions are well-defined. The space $\\Omega(p,q)$ is, in fact, a smooth Hilbert manifold with tangent space at $\\gamma\\in \\Omega(p,q)$ given by the vector space of vector fields along $\\gamma$ belonging to the Sobolev class $H^1$ and vanishing at endpoints:\n\\begin{equation}\nT_{\\gamma}\\Omega(p,q)=\\{\\xi\\in H^1(\\gamma^*TM):\\xi(0)=0,\\xi(1)=0\\}.\n\\end{equation}\nMoreover, the energy functional $E:\\Omega(p,q)\\to\\mathbb{R}$ is $C^2$. The critical points are the $H^1$-solutions of the elliptic autoparallel transport equation $\\nabla_t\\dot\\gamma=0$. As such, the geodesic solutions are necessarily smooth. \n\nConsider the length functional \n\\begin{equation}\nL:\\mathcal{P}(p,q)\\to\\mathbb{R}, \\quad L[\\gamma]:=\\int_0^1\\|\\dot\\gamma(t)\\|^2dt.\n\\end{equation} \nThe functional $L$ is invariant under the action of the infinite-dimensional group of diffeomorphisms on the interval [0,1], i.e. $L[\\gamma(gt)]=L[\\gamma(t)]$ for all $g\\in G:=\\text{Diffeo}([0,1])$. Thus, the critical points of $L$ form infinite-dimensional families of solutions. The degeneracy is rectified by observing that some path $\\gamma\\in\\mathcal{P}(p,q)$ admits positive reparameterizations on $[0,1]$ with constant speed \\cite{alexandru}. A path $\\gamma\\in\\mathcal{P}(p,q)$ is a geodesic if and only if it is a critical point of the length functional $L[\\gamma]$ and has constant speed.\n\nConsider the quotient space $S^1=\\mathbb{R}\/\\mathbb{Z}$ and the space of smooth loops in $M$: \n\\begin{equation}\n\\mathcal{L}M:=C^{\\infty}(S^1,M),\n\\end{equation} which is a Fr\\'echet manifold on which the $2$-dimensional orthogonal group $O(2)=SO(2)\\rtimes\\{\\pm 1\\}$ acts by $(e^{i\\theta}\\cdot\\gamma)(t):=\\gamma(t+\\theta)$ and $(-1\\cdot\\gamma)(t):=\\gamma(-t)$. For economy, we consider the smooth Banach manifold\n\\begin{equation}\n\\Lambda M:=H^1(S^1,M),\n\\end{equation} which is the so-called $\\textit{free loop space}$ of $M$. The inclusion map $\\mathcal{L}M\\hookrightarrow \\Lambda M$ is a homotopy equivalence, and the $O(2)$ action descends naturally to $\\Lambda M$. Equation (\\ref{energyfunctional}) defines a functional $E:\\Lambda M\\to\\mathbb{R}$, whose critical points are smooth periodic or closed geodesics. However, under present considerations,  $E$ is $O(2)$-invariant which induces critical point degeneracy. Closed geodesics may be classified crudely by bifurcating isotropy groups. In particular, a constant geodesic corresponds to a point in $M$ with isotropy group $O(2)$, whereas the isotropy group of a non-constant geodesic is $\\mathbb{Z}\/k\\mathbb{Z}$ with $1\/k$ its minimal period for $k\\in\\mathbb{Z}^+$. \n\nThe tangent space at a point $\\gamma\\in\\Lambda M$, identified with a path, is $T_{\\gamma}\\Lambda M=H^1(\\gamma^*TM)$, which is the space of sections of $\\gamma^*TM$ of Sobolev class $H^1$ for $\\gamma:S^1\\to M$ and $\\gamma^*:T^*M\\to T^*S^1$ its pullback. The free loop space $\\Lambda M$ of Sobolev class $H^1$ is a Hilbert manifold with respect to a $H^1$-inner product \n\\begin{equation}\n\\langle\\xi,\\eta\\rangle_1:=\\int_{S^1}\\langle\\xi,\\eta\\rangle+\\int_{S^1}\\langle\\nabla_t\\xi,\\nabla_t\\eta\\rangle=\\langle\\xi,\\eta\\rangle_0+\\langle\\nabla_t\\xi,\\nabla_t\\eta\\rangle_0.\n\\end{equation} \nThe Arzel\\'a-Ascoli theorem implies a compact inclusion $\\Lambda M=H^1(S^1,M)\\hookrightarrow C^0(S^1,M)$. This inclusion may be used to prove the following result.\n\\begin{proposition}[Klingenberg \\cite{klingenberg}]\nThe Riemannian metric on $\\Lambda M$ given by the $H^1$-inner product is complete.\n\\end{proposition}\n\n\\section{Morse Theory\\label{sec:morsetheory}}\n\nThe gradient $\\nabla E$ is a vector field on $\\Lambda M$ defined by the following inner product.\n\\begin{equation}\n\\langle\\nabla E[\\gamma],\\xi\\rangle_1=dE[\\gamma]\\xi=\\langle\\dot\\gamma,\\nabla_t\\xi\\rangle_0, \\quad \\xi\\in H^1(\\gamma^*TM)\n\\end{equation} \nFurthermore, if $\\gamma$ is smooth then $\\langle\\dot\\gamma,\\nabla_t\\xi\\rangle_0=-\\langle\\nabla_t\\dot\\gamma,\\xi\\rangle_0$ such that $\\nabla E[\\gamma]\\in\\Gamma(\\gamma^*TM)$ is the unique periodic solution of the differential equation: \n\\begin{equation}\n\\nabla_t^2\\eta(t)-\\eta(t)=\\nabla_t\\dot\\gamma(t).\n\\end{equation} \nThe following condition of Palais and Smale is necessary to extend Morse theory to the infinite-dimensional setting of Hilbert manifolds $\\Lambda M$.\n\\begin{theorem}[Palais-Smale \\cite{palais-smale}]\\label{Palais-Smale} The energy functional $E:\\Lambda M\\to\\mathbb{R}$ satisfies condition (C) of Palais and Smale:\n\n(C) Let $(\\gamma_m)\\in\\Lambda M$ be a sequence such that $E[\\gamma_m]$ is bounded and $\\|\\nabla E[\\gamma_m]\\|_1\\to 0$. Then $(\\gamma_m)$ has limit points and every limit point is a critical point of $E$.\n\\end{theorem}\n\nLet $\\text{Crit}($E$):=\\{\\gamma\\in\\Lambda M:dE[\\gamma]=0\\}$ denote the set of critical points of the energy functional, i.e., the set of closed geodesics on $M$. For $a\\ge 0$, we denote\n\\begin{align*}\n   \\Lambda^{\\le a}:&=\\{\\gamma\\in\\Lambda M:E[\\gamma]\\le a\\}, \\\\\n   \\Lambda^{<a}:&=\\{\\gamma\\in\\Lambda M:E[\\gamma]<a\\},\\\\\n\\intertext{and}\n\\Lambda^a:&=\\{\\gamma\\in\\Lambda M:E[\\gamma]=a\\}.\n\\end{align*} \n The following properties of the negative gradient flow $\\frac{d}{ds}\\phi_s=-\\nabla E(\\phi_s)$ of the energy functional are due to condition (C) of Palais Smale. \n\n\\begin{enumerate}[label=(\\roman*)]\n\\item Crit($E$)$\\cap\\Lambda^{\\le a}$ is compact for all $a\\ge 0$.\n\\item the flow $\\phi_s$ is defined for all $s\\ge 0$.\n\\item given an interval $[a,b]$ of regular values, there exists a real number $s_0\\ge 0$ such that $\\phi_s(\\Lambda^{\\le b})\\subset\\Lambda^{\\le a}$ for all $s\\ge s_0$.\n\\item $\\Lambda^0\\equiv M\\subset\\Lambda^{\\le\\varepsilon}$ is a strong deformation retract via $\\phi_s$ for $\\varepsilon>0$ sufficiently small.\n\\end{enumerate}\n\nSuppose $\\gamma\\in \\text{Crit(}E\\text{)}$ is a critical point. The $O(2)$-invariance of the energy functional $E$ means that the orbit $O(2)\\cdot \\gamma $ is contained entirely in Crit($E$).\n\\begin{definition} The $\\textit{index}$ $\\lambda(\\gamma)$ of $\\gamma$ is the dimension of the negative eigenspace of the Hessian $d^2E[\\gamma]$.\n\\end{definition}\n\\begin{definition}\nThe nullity $\\nu(\\gamma)$ of $\\gamma$ is the dimension of the kernel of $d^2E[\\gamma]$, i.e. $\\nu(\\gamma):=\\dim\\ker d^2E[\\gamma]$.\n\\end{definition} \nNote, $\\langle\\dot\\gamma\\rangle\\in \\ker d^2E[\\gamma]$ so $\\nu(\\gamma)\\ge 1$ if $\\gamma$ is non-constant because Crit($E$) is invariant under the $S^1$ reparameterization action. The Hessian of $E$ at $\\gamma$ is given by the second variation formula:\n\\begin{equation}\nd^2E[\\gamma](\\xi,\\eta)=-\\int\\langle\\xi,\\nabla_t^2\\eta+R(\\dot\\gamma,\\eta)\\dot\\gamma\\rangle.\n\\end{equation}\n\\begin{definition}\\label{morsefunction}\nA $C^2$-function $f$ defined on a Hilbert manifold is a Morse function if all of its critical points are non-degenerate, which means that the Hessian has a $0$-dimensional kernel at each critical point $c$ so $\\dim\\ker d^2f(c)=0$.\n\\end{definition}\nThe energy functional $E$ defined on $\\Lambda M$ will never satisfy the conditions of Definition \\ref{morsefunction} because the critical points at level $0$ form a closed $\\dim M$-dimensional manifold, which means that it is never non-degenerate. In a similar vein, the energy functional $E$ is $S^1$-invariant which means that the kernel of the Hessian of $E$ at a non-constant geodesic $\\gamma$ is always at least $1$-dimensional, $\\dim\\ker d^2E[\\gamma] \\ge 1$, because it contains the infinitesimal generator $\\dot\\gamma$ of the $S^1$-action.\n\\begin{definition}[Oancea \\cite{alexandru}] \nA $C^2$-function defined on a Hilbert manifold is said to be a Morse-Bott function if its critical set is a disjoint union of closed connected submanifolds $\\bigsqcup_i\\Sigma_i$ and, for each critical point $c_i$, the kernel of the Hessian at that critical point coincides with the tangent space to the respective connected component of the critical locus, $\\ker d^2f(c_i)\\cong T\\Sigma_i$. As such, the critical set is said to be non-degenerate. \n\\end{definition}\n\\begin{definition}\nThe index $\\lambda(p)$ of a critical point $p$ is defined as $\\iota(p)=\\dim\\{V_{\\text{max}}\\subset T_pM: d^2f(p)<0\\}$, the dimension of the maximal subspace of the tangent space at $p$ on which the Hessian is negative definite.\n\\end{definition}\n\n\\begin{definition} \nThe nullity $\\nu(p)$ of a critical point $p$ is defined as $\\nu(p)=\\dim\\ker d^2f(p)$, the dimension of the kernel of the Hessian at $p$.\n\\end{definition} \n\\begin{theorem}[Palais \\cite{palais-hilbertmanifolds}]\\label{criticalmorse}\nAssume that the Riemannian metric on $M$ is chosen in such a way that the energy functional $E$ on $\\Lambda M$ is Morse-Bott.\n\\begin{enumerate}[label=(\\roman*)]\n\\item The critical values of $E$ are isolated and there are only a finite number of connected components of Crit($E$) on each critical set.\n\\item The index and nullity of each connected component of Crit($E$) are finite.\n\\item If there are no critical values of $E$ in $[a,b]$ then $\\Lambda^{\\le b}$ retracts onto and is diffeomorphic to $\\Lambda^{\\le a}$.\n\\item Let $a<c<b$ and assume $c$ is the only critical value of $E$ in the interval [a,b]. Denote by $\\Sigma_1,\\cdots,\\Sigma_r$ the connected components of Crit($E$) at level $c$, and denote by $\\lambda_1,\\cdots,\\lambda_r$ their respective indices. \\begin{itemize}\n  \\item Each manifold $\\Sigma_i$ carries a well-defined vector bundle $\\nu^{-}\\Sigma_i$ of rank $\\lambda_i$ consisting of negative directions for $d^2E|_{\\Sigma_i}$.\n  \\item The sublevel set $\\Lambda^{\\le b}$ retracts onto a space homeomorphic to $\\Lambda^{\\le a}$ with the disk bundles $D\\nu^{-}\\Sigma_i$ disjointly attached to $\\Lambda^{\\le a}$ along their boundaries.\n\\end{itemize}\n\\end{enumerate}\n\\end{theorem}\nWe denote the disk bundles by $D\\nu^-\\Sigma_i$, which are associated to a fixed scalar product on the fibers $\\nu^-\\Sigma_i$, a sub-bundle of the normal bundle $T\\Sigma_i^{\\perp}$ with inner product given by the Hilbert structure of the manifold $\\Lambda M$. According to \\cite{alexandru}, there exist smooth embeddings $\\phi_i:\\partial D\\nu^-\\Sigma_i\\to\\partial\\Lambda^{\\le a}$ with disjoint images from which we can construct a quotient space $\\Lambda^{\\le a}\\cup\\bigcup_iD\\nu^{-}\\Sigma_i\/\\sim$ such that a point lying in the boundary $\\partial D\\nu^{-}\\Sigma_i$ can be identified with the image under $\\phi_i$ in $\\partial\\Lambda^{\\le a}$.\n\\begin{remark}\nWhen one passes the critical level $c$, the sublevel $c+\\varepsilon$, for $\\varepsilon>0$ sufficiently small, deformation retracts onto the the union of the sublevel $c-\\varepsilon$ and a $\\lambda(c)$-cell. The $\\lambda$-cell is the negative vector bundle over the critical manifold consisting of a single point. Such a retraction is forced by a modification of the negative gradient flow $\\frac{d}{ds}\\phi_s=-\\nabla f(\\phi_s)$ of $f$.\n\\end{remark} \n\\begin{corollary}\nAdopting the the aforementioned a posteriori hypotheses, the sublevel set $\\Lambda^{\\le b}$ is diffeomorphic to and deformation retracts onto $\\Lambda^{\\le c}$. Similarly, the sublevel set $\\Lambda^{<c}$ is diffeomorphic to and deformation retracts onto $\\Lambda^{\\le a}$. In particular, let $c\\in[a,b]$ be a non-degenerate critical point of $E$ of index $\\lambda(c)$ such that $E[c]=r$. Furthermore, suppose $E^{-1}[r-\\varepsilon,r+\\varepsilon]$ is compact and contains no critical points other than $c$. Then it follows that $\\Lambda^{\\le r+\\varepsilon}$ is homotopy equivalent to the union of $\\Lambda^{\\le r-\\varepsilon}$ and a $\\lambda(c)$-cell.\n\\end{corollary}\n\\begin{corollary}\nSuppose $E:\\Lambda M\\to\\mathbb{R}$ is a real-valued smooth functional on $\\Lambda M$, $a<b$, $E^{-1}[a,b]$ is compact, and there are no critical values between $a$ and $b$. Then the sublevel set $\\Lambda^{\\le b}$ is diffeomorphic to and deformation retracts onto $\\Lambda^{\\le a}$.\n\\end{corollary}\nThe above provides a useful heuristic for determining the homology groups of $\\Lambda M$ by way of the non-decreasing filtration $\\Lambda^{\\le a}$, whereby \\begin{equation} \\Lambda M=\\varinjlim_{a\\to\\infty}\\Lambda^{\\le a}. \\end{equation} Thus, the homology of $\\Lambda M$ is $H_{\\bullet}(\\Lambda M)=\\varinjlim H_{\\bullet}(\\Lambda^{\\le a})$. Suppose $E$ is non-degenerate such that the sequence $0=c_0<c_1<c_2<\\cdots$ of its critical values is diverging. Then the following holds \\cite{alexandru}.\n\\begin{enumerate}[label=(\\roman*)]\n\\item If $c_i<a<c_{i+1}$, an inclusion $\\Lambda^{\\le c_i}\\hookrightarrow\\Lambda^{\\le a}$ induces an isomorphism on homology $H_{\\bullet}(\\Lambda^{\\le c_i})\\cong H_{\\bullet}(\\Lambda^{\\le a})$ which means the homology groups of $\\Lambda^{\\le a}$ change when passing a critical value.\n\\item The change in the homology upon crossing a critical level $c$ of $E$ is entirely captured by the long exact sequence of the pair $(\\Lambda^{\\le c},\\Lambda^{<c})$:\n\\begin{equation*}\n\\cdots\\longrightarrow H_{\\bullet}(\\Lambda^c)\\longrightarrow H_{\\bullet}(\\Lambda^{\\le c})\\longrightarrow H_{\\bullet}(\\Lambda^{\\le c},\\Lambda^{<c})\\stackrel{[-1]}{\\longrightarrow}\\cdots\n\\end{equation*}\n\\item If there are no critical values of $E$ in $[a,b]$ then $\\Lambda^{\\le b}$ retracts onto and is diffeomorphic to $\\Lambda^{\\le a}$. \nLet $\\Sigma^c:=\\bigsqcup_{i=1}^r \\Sigma_i$ be the critical set of $E$ at level $c$. Therefore, the following isomorphisms hold. \n\\begin{equation*}\nH_{\\bullet}(\\Lambda^{\\le c},\\Lambda^{<c})\\cong H_{\\bullet}\\left(\\bigsqcup_iD\\nu^-\\Sigma_i,\\bigsqcup_i\\partial D\\nu^-\\Sigma_i\\right)\\cong\\bigoplus_iH_{\\bullet-\\lambda_i}(\\Sigma_i;o_{\\nu^-\\Sigma_i})\n\\end{equation*} Note, $H_{\\bullet-\\lambda_i}(\\Sigma_i;o_{\\nu^-\\Sigma_i})$ denotes the homology groups of $\\Sigma_i$ with coefficients given by the local system of orientations of the bundle $\\nu^-\\Sigma_i$. The first isomorphism follows from excision and (iv) of Theorem \\ref{criticalmorse}. Likewise, the second isomorphism is induced by the Thom isomorphism for finite rank vector bundles. \n\\end{enumerate}\nWe present the useful and well-known theorem of Gromoll and Meyer.\n\\begin{theorem}[Gromoll-Meyer \\cite{gromoll-meyer}]\\label{Gromoll\/Meyer} Let $M$ be a simply connected closed manifold. If the sequence $\\{b_k(\\Lambda M;\\mathbb{F}_p)\\}_{k\\ge 0}$ is unbounded for some prime $p\\ge 2$, then any Riemannian metric on $M$ admits infinitely many prime closed geodescis.\n\\end{theorem}\n\nIt was shown by Vigu\\'e-Poirrier and Sullivan \\cite{poirrier-sullivan} that for a simply connected closed manifold $M$, the sequence of rational Betti numbers $b_k(\\Lambda M;\\mathbb{Q})=\\text{rank }H^k(\\Lambda M;\\mathbb{Q})$ is unbounded if and only if the rational cohomology ring $H^*(M;\\mathbb{Q})=\\bigoplus_{k\\in\\mathbb{N}}H^k(M;\\mathbb{Q})$ has at least two generators. That is, $M$ does not have rational cohomology that is the quotient of a polynomial ring. There are a few simply connected compact globally symmetric spaces of rank $>1$ whose rational cohomology ring $H^*(M;\\mathbb{Q})$ is not a truncated polynomial ring. In fact, it was shown by Ziller \\cite{ziller} that compact globally symmetric spaces of rank $>1$ are such that the sequence of Betti numbers for their free loop space is unbounded with coefficients in $\\mathbb{F}_2$. On the other hand, compact globally symmetric spaces of rank $1$ have rational cohomology generated by a single element.\n\nWe must address two complications that arise in view of Gromoll and Meyer \\ref{Gromoll\/Meyer}, the first being that iterates $\\gamma_m:S^1\\to M$ with $\\gamma_m(t)=\\gamma(mt), m\\in\\mathbb{Z}$ of a closed geodesic $\\gamma$ are themselves closed geodesics (under the action of infinite-dimensional reparameterization), and thereby constitute a  part of the homology of free loop space $\\Lambda M$. Second, we would like to handle degeneracy for closed geodesics. It was shown by Gromoll and Meyer \\cite{gromoll-meyer} that a single degenerate closed geodesic adds to homology by at most $\\nu(\\gamma)$ for degrees contained in $[\\lambda(\\gamma),\\lambda(\\gamma)+\\nu(\\gamma)]$. This result follows from the lemmata which dictate the behavior of index and nullity subject to iteration.\n\\begin{lemma}[Gromoll-Meyer Index Iteration \\cite{gromoll-meyer}]\\label{Index Iteration}\nThe index is either $\\lambda(\\gamma_m)=0$ for all $m\\ge 1$ or there is an $\\varepsilon>0$ and $C>0$ such that for all $m,s\\ge 1$:\n\\begin{equation}\n\\lambda(\\gamma_{m+s})-\\lambda(\\gamma_m)\\ge\\varepsilon s-C.\n\\end{equation}\n\\end{lemma}\n\\begin{lemma}[Gromoll-Meyer Nullity Iteration \\cite{gromoll-meyer}]\\label{Nullity iteration} The nullity is either $\\nu(\\gamma_m)=1$ for all $m\\ge 1$ or there are finitely many iterations $\\gamma_{m_1},\\cdots,\\gamma_{m_s}$ of $\\gamma$ such that for all $m\\ge 1$ there is an $i$ and $j$ for which $m=jm_i$ and $\\nu(\\gamma_m)=\\nu(\\gamma_{m_i})$. That is, the sequence $\\{\\nu(\\gamma_m)\\}_{m\\ge 1}$ only takes on finitely many values. \n\\end{lemma}\n\nBy virtue of a combinatorial argument and the Gromoll-Meyer index and nullity iteration lemmata, if the number of simple closed geodesics is bounded, there is a bound on the Betti numbers of $\\Lambda M$.\n\nLet\n\\begin{equation}\nA_{\\gamma}:H^1(\\gamma^*TM)\\to L^2(\\gamma^*TM),\\quad \\xi\\mapsto -\\xi''-R(\\dot\\gamma,\\xi)\\dot\\gamma\n\\end{equation} be the self-adjoint $\\textit{asymptotic elliptic operator}$ at the closed geodesic $\\gamma$ with compact resolvent such that $d^2E[\\gamma](\\xi,\\eta)=\\langle A_{\\gamma}\\xi,\\eta\\rangle_{L^2}$. More precisely, \n\\begin{equation}\n\\begin{split}\n \t\\langle A_{\\gamma}\\xi,\\eta\\rangle_{L^2}&=\\langle A_{\\gamma}\\eta,\\xi\\rangle_{L^2}\\\\ \n\t\t&=\\int\\langle-\\eta''-R(\\dot\\gamma,\\eta)\\dot\\gamma,\\eta\\rangle \\\\\n           &= -\\int\\langle\\xi,\\nabla_t^2\\eta+R(\\dot\\gamma,\\eta)\\dot\\gamma\\rangle.\n\\end{split}\n\\end{equation} Its spectrum is discrete with no accumulation point besides $\\pm\\infty$ \\cite{kato}. The eigenvectors of $A_{\\gamma}$ are smooth by elliptic regularity. The eigenvalue problem $-\\xi''-R(\\dot\\gamma,\\xi)\\dot\\gamma=-\\lambda\\xi$ has no non-trivial periodic solutions for $\\lambda\\gg 0$. Moreover, the spectrum of the elliptic operator $A_{\\gamma}$ is bounded from below which implies the finiteness of the index of $\\gamma$, given by \n\\begin{equation}\n\\lambda(\\gamma)=\\sum_{\\text{Spec}(A_{\\gamma})\\ni\\lambda<0}\\text{mult}(\\lambda)<+\\infty\n\\end{equation} for $\\text{mult}(\\lambda)$ the multiplicity of the eigenvalue $\\lambda$. The nullity of $\\gamma$ is \n\\begin{equation}\n\\nu(\\gamma)=\\dim\\ker A_{\\gamma},\n\\end{equation} for which we have the bounds $1\\le\\nu(\\gamma)\\le 2n-1$. The first inequality is due to $\\langle\\dot\\gamma\\rangle\\in\\ker A_{\\gamma}$, and the second inquality follows because an element in $\\ker A_{\\gamma}$ solves a second-order ordinary differential equation.\n\nIt was shown by Takens and Klingenberg that for a $C^4$-generic metric on a closed manifold $M$, there exist closed geodesics of $\\textit{twist type}$ for which the eigenvalues of the Poincar\\'e return map have modulus unity, or there exist closed geodesics of $\\textit{hyperbolic type}$ for which the eigenvalues of the Poincar\\'e return map lie outside or inside the unit circle. The Birkhoff-Lewis fixed point theorem implies that for the former case there are infinitely many prime closed geodesics in a neighborhood of the twist geodesic. It is then left to show that this is true for hyperbolic geodesics. It should be noted that if all geodesic orbits are hyperbolic, then $\\lambda(\\gamma_m)=m\\lambda(\\gamma)$.\n\\begin{theorem}[Rademacher \\cite{rademacher}]\\label{Rademacher} \nLet $M$ be a simply connected closed Riemannian manifold with rational cohomology a truncated polynomial ring in one even variable and all geodesics hyperbolic. Then there are infinitely many geometrically distinct hyperbolic geodesics.\n\\end{theorem}\n\\begin{theorem} \nLet $M$ be a simply connected closed Riemannian manifold with rational homotopy type that of a rank $1$ symmetric space (i.e. $S^n,\\mathbb{CP}^n$, $\\mathbb{HP}^n$, $\\mathbb{P}^2(\\mathbb{O})$) and suppose all closed geodesics on $M$ are hyperbolic. Then the number $\\pi(x)$ of prime closed geodesics of length less than or equal to $x$ is on the order of \n\\begin{equation}\n\\pi(x)=|\\{\\gamma\\in \\Lambda M: L[\\gamma]\\le x\\}| \\sim\\frac{x}{logx}.\n\\end{equation}\n\\end{theorem} Recall that the isotropy group at a non-constant closed geodesic is $\\mathbb{Z}_m$ with $1\/m$ the minimal period. Since $\\mathbb{Z}_m,m\\ge 2$ are $\\mathbb{Q}$-acyclic, we have the following isomorphism on equivariant cohomology:\n\\begin{equation*} H_{\\mathbb{Z}_m}^*(\\Lambda M,M;\\mathbb{Q})\\cong H^*(\\Lambda M\/{\\mathbb{Z}_m}, M;\\mathbb{Q})\n\\end{equation*} where $H_{\\mathbb{Z}_m}^*(\\bullet):=H^*({\\bullet}_{\\mathbb{Z}_m})$ is $\\mathbb{Z}_m$-equivariant cohomology.\n\nThe most significant consequence of $O(2)$-invariance for $E$ is that the $m$-fold iterate $\\gamma_m$ of a closed geodesic adds to $H_{\\bullet}^{O(2)}(\\Lambda M)$ by $H_{\\bullet}(B\\mathbb{Z}_m)$. Thus \\cite{brown}, \n\\begin{align*}\n       H_{\\bullet}(B\\mathbb{Z}_m;\\mathbb{Z}) &=  \\begin{cases} \n      \\mathbb{Z}, & \\text{if }\\bullet=0, \\\\\n      \\mathbb{Z}_m, & \\text{if} \\hspace{.05 cm}\\bullet\\text{odd}, \\\\\n      0, & \\text{if }\\bullet>0\\text{ even}. \n   \\end{cases}\n\\end{align*} By the universal coefficient theorem, we deduce that \n\\begin{align*}\nH_{\\bullet}(B\\mathbb{Z}_m;\\mathbb{Z}_{\\ell})&= \\begin{cases} \n      \\mathbb{Z}_{\\ell}, & \\text{if }\\bullet=0, \\\\\n      \\mathbb{Z}_d, & \\text{if }\\bullet>0,\n   \\end{cases}\n\\end{align*} where $d=\\text{gcd}(m,\\ell)$, and \n\\begin{align*}\nH_{\\bullet}(B\\mathbb{Z}_m;\\mathbb{Q})=  \n\\begin{cases} \n      \\mathbb{Q}, & \\text{if }\\bullet=0, \\\\\n      0, & \\text{if }\\bullet>0. \n   \\end{cases}\n\\end{align*}\n\n\\section{Rational Homotopy Theory\\label{sec: homotopytheory}}\n\nAn important consequence of rational homotopy theory is that every simply connected closed Riemannian manifold $M$ with rational cohomology not isomorphic to a truncated polynomial has infinitely many geometrically distinct prime closed geodesics.\n\\begin{definition}\nSuppose $M,N$ are simply connected topological spaces. A continuous map $f:M\\to N$ is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with $\\mathbb{Q}$.\n\\end{definition}\n\\begin{definition}\nA rational space is a simply connected CW complex whose homotopy groups are vector spaces over $\\mathbb{Q}$.\n\\end{definition}\n\\begin{definition}\nThe rationalization of a simply connected CW complex $M$ is the rational space $M_{\\mathbb{Q}}$ for which the map from $M$ to $M_{\\mathbb{Q}}$ descends to an isomorphism on rational homology, i.e. $\\pi_k(M_{\\mathbb{Q}})\\cong\\pi_k(M)\\otimes\\mathbb{Q}$ and, by the universal coefficient theorem, $H_k(M_{\\mathbb{Q}},\\mathbb{Z})\\cong H_k(M,\\mathbb{Z})\\otimes\\mathbb{Q}\\cong H_k(M,\\mathbb{Q})$ for all $k>0$.\n\\end{definition}\nThe rational cohomology ring $H^*(M,\\mathbb{Q})$ is a graded commutative algebra and $\\pi_*(M)\\otimes\\mathbb{Q}$ is a graded homotopy Lie algebra. Let $\\Omega M$ be the $\\textit{loop space}$ of $M$, then $\\pi_*(\\Omega M)\\otimes\\mathbb{Q}$ is also a graded Lie agebra by the Whitehead product structure. \n\\begin{theorem}\nThe homology of loop space $H_{\\bullet}(\\Omega M,\\mathbb{Q})$ is the (graded) universal enveloping algebra of the (graded) Lie algebra $\\pi_{\\bullet}(\\Omega M,\\mathbb{Q})\\cong\\pi_{\\bullet-1}(M,\\mathbb{Q})$.\n\\end{theorem}\n\\begin{definition}\nLet $M$ be a simply connected (CW complex) space for $H^*(M,\\mathbb{Q})$ finite-dimensional over $\\mathbb{Q}$. A simply connected space $M$ is said to be rationally elliptic if the homotopy Lie algebra $\\pi_*(M)\\otimes\\mathbb{Q}$ is finite-dimensional over $\\mathbb{Q}$, or else it is rationally hyperbolic.\n\\end{definition}\n\nAccording to F\\'elix and Halperin \\cite{felix-halperin}, if $M$ is rationally hyperbolic, there exists an $\\alpha>1$ and an integer $N$ such that for all $n\\ge N$,\n\\begin{equation}\n\\sum_{i=1}^n\\dim_{\\mathbb{Q}}\\pi_i(M)\\otimes\\mathbb{Q}\\ge \\alpha^n.\n\\end{equation}\n\\begin{remark}\nLet $M$ be a simply connected closed Riemannian manifold. Then the rational cohomology ring $H^*(M;\\mathbb{Q})$ requires more than one generator if and only if $\\pi_{\\text{odd}}(M)\\otimes\\mathbb{Q}$ is more than $1$-dimensional, i.e. $\\dim\\pi_{\\text{odd}}(M)\\otimes\\mathbb{Q}>1$.\n\\end{remark}\n\\begin{proof}\nA basis for rational homotopy $\\pi_k(M)\\otimes\\mathbb{Q}$ corresponds to a set of algebra generators of the minimal model. Then the claim follows from Proposition 1 of Vigu\\'e-Poirrier and Sullivan \\cite{poirrier-sullivan}.\n\\end{proof}\n\\begin{corollary}\nThe rational space $M_{\\mathbb{Q}}$ cannot be a $K(G,n)$ Eilenberg--MacLane space for $G$ singly generated if $\\dim\\pi_{\\text{odd}}(M)\\otimes\\mathbb{Q}>1$.\n\\end{corollary}\n\\begin{proof}\nSuppose the rational space $M_{\\mathbb{Q}}$ is a $K(G,n)$ Eilenberg--MacLane space. Then for $n$ a positive integer, $\\pi_n(K(G,n))\\cong G$, and $\\pi_i(M_{\\mathbb{Q}})=0$ for all $i\\ne n$. Thus, $\\dim\\pi_i(M_{\\mathbb{Q}})=\\dim\\pi_i(M)\\otimes\\mathbb{Q}=\\delta_{in}$, which implies that $\\dim\\pi_{\\text{odd}}(M)\\otimes\\mathbb{Q}\\le 1$. It follows that $H^*(M;\\mathbb{Q})$ is singly generated so $M_{\\mathbb{Q}}\\ne K(G,n)$, a contradiction. \n\\end{proof} \n\\begin{lemma}\nIf the rational cohomology ring $H^*(M;\\mathbb{Q})$ is not singly generated, $M_{\\mathbb{Q}}=M\\times E\\mathbb{Q}\\ne K(G,n)$ for any positive integer $n$ or group $G$ acting freely on $M$.\n\\end{lemma}\n\\begin{theorem}[Hurewicz Theorem]\\label{hurewicz}\nIf $X$ is an $(k-1)$-connected topological space, then for a positive integer $n$, the group homomorphism\n\\begin{equation*}\nf_*:\\pi_n(X)\\to H_n(X)\n\\end{equation*} is an isomorphism for all $n\\le k$ and an abelianization for $n=1$.\n\\end{theorem}\n\\begin{definition}\nLet $G$ be an abelian group. The group $G \\otimes_{\\mathbb{Z}} \\mathbb{Q}$ is the divisible hull of $G$. The dimension of  $G \\otimes_{\\mathbb{Z}} \\mathbb{Q}$ over $\\mathbb{Q}$ is the rational rank of $G$. We denote by $\\text{rank }_{\\mathbb{Q}}(G)$ the rational rank of $G$. If $G$ is torsion free then the map $G\\to G\\otimes\\mathbb{Q}$ is injective and the rank of $G$ is the minimum dimension of the $\\mathbb{Q}$-vector space containing $G$ as an abelian subgroup.\n\\end{definition} \n\\begin{lemma}\\label{rathyperbolic}\nIf $M$ is a rationally hyperbolic manifold then $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}\\ge 2$ and $\\dim_{\\mathbb{Q}}\\pi_k(M)\\otimes\\mathbb{Q}>\\alpha^{k-1}$ for an integer $k>1$ and a real number $\\alpha>1$. Moreover, $\\dim_{\\mathbb{Q}}\\pi_{\\text{odd}}(M)\\otimes\\mathbb{Q}>1$ so the rational cohomology ring $H^*(M;\\mathbb{Q})$ has at least two generators.\n\\end{lemma}\n\\begin{proof}\nIf $M$ is rationally hyperbolic, there exists a real number $\\alpha>1$ and an integer $N$ such that \n\\begin{equation*}\n\\sum_{i=1}^n\\dim_{\\mathbb{Q}}\\pi_i(M)\\otimes\\mathbb{Q}\\ge \\alpha^n\n\\end{equation*} for all $n\\ge N$. If $n=1$ then $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}\\ge \\alpha$ for $\\alpha>1$ so $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}>1$. If $n=2$ then $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}+\\dim_{\\mathbb{Q}}\\pi_2(M)\\otimes\\mathbb{Q}\\ge \\alpha^2$. Assume $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}= \\alpha$ so that $\\dim_{\\mathbb{Q}}\\pi_2(M)\\otimes\\mathbb{Q}\\ge \\alpha^2-\\alpha$. If $n=3$ then $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}+\\dim_{\\mathbb{Q}}\\pi_2(M)\\otimes\\mathbb{Q}+\\dim_{\\mathbb{Q}}\\pi_3(M)\\otimes\\mathbb{Q}\\ge \\alpha^3$. Let $\\dim_{\\mathbb{Q}}\\pi_2(M)\\otimes\\mathbb{Q}=\\alpha^2-\\alpha$ so $\\alpha+(\\alpha^2-\\alpha)+\\dim_{\\mathbb{Q}}\\pi_3(M)\\otimes\\mathbb{Q}\\ge \\alpha^3$ or $\\dim_{\\mathbb{Q}}\\pi_3(M)\\otimes\\mathbb{Q}\\ge \\alpha^3-\\alpha^2$. It follows by induction that $\\dim_{\\mathbb{Q}}\\pi_k(M)\\otimes\\mathbb{Q}\\ge \\alpha^k-\\alpha^{k-1}$ for $\\alpha>1$. More generally, $\\dim_{\\mathbb{Q}}\\pi_k(M)\\otimes\\mathbb{Q}\\ge \\alpha^{k-1}(\\alpha-1)>\\alpha^{k-1}$ or $\\dim_{\\mathbb{Q}}\\pi_k(M)\\otimes\\mathbb{Q}> \\alpha^{k-1}$ for $\\alpha>1$. As per the above, for $k=1$, it is already known that $\\dim_{\\mathbb{Q}}\\pi_1(M)\\otimes\\mathbb{Q}>1$. \n\\end{proof}\n\nAnother program for determining the dimension of $\\pi_i(M)\\otimes\\mathbb{Q}$ is by way of minimal models. If $M$ is a closed, simply connected smooth manifold, then the rank of the rational homotopy groups $\\pi_i(M)\\otimes\\mathbb{Q}$ equals the number of degree $i$ generators introduced in the construction of the minimal model for $M$. For manifolds that are $\\textit{formal}$, e.g. if $M$ is K\\\"{a}hler, their cohomology ring with trivial differential is quasi-isomorphic as a differential graded algebra to the minimal model. Thus, we can use the cohomology ring to determine the rank of rational homotopy groups.\n\n\\begin{theorem}\\label{betti-inequality}\nLet $E:\\Lambda M\\to\\mathbb{R}$ be a Morse function on the free loop space of Sobolev class $H^1$. Let $C_{\\lambda}$ denote the number of critical points of the energy functional $E$ of index $\\lambda$. Then we have the following constraints:\n\\begin{equation*}\n\\begin{split}\n& \\text{rank } H_{\\lambda}(\\Lambda M)=b_{\\lambda}\\le C_{\\lambda} \\text{ for each } \\lambda,\\\\ &\n\\chi(\\Lambda M)=\\sum_{\\lambda}(-1)^{\\lambda}C_{\\lambda}.\n\\end{split}\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nLet $\\Lambda_1$ be obtained from $\\Lambda_0$ by attaching a handle of index $\\lambda$ for $M_0\\subset M_1$, then $H_k(\\Lambda_1,\\Lambda_0)=H_k(D^{\\lambda},S^{\\lambda-1})$. Thus, $H_k(\\Lambda_1,\\Lambda_0)=\\mathbb{Z}$ if $k=\\lambda$ and $0$ otherwise. Denote by $c_1<c_2<\\dots<c_n$ the critical values of $E$. Consider the sequence of real numbers $\\{a_i\\}$ such that $c_i<a_i<c_{i+1}$ for $0\\le i\\le n-1$, and let $\\Lambda_i=\\Lambda^{\\le a_i}$. Thus, we obtain the inclusions of an increasing sequence of manifolds with boundary, $D^{\\dim \\Lambda M}=\\Lambda_0\\subset \\Lambda_1\\subset\\Lambda_2\\subset\\dots\\subset\\Lambda_n=\\Lambda M$, where $\\Lambda_{i+1}$ is $\\Lambda_i$ with finitely many cells attached for each critical point in the set $\\Lambda^{[a_i,a_{i+1}]}$. Since homology groups $H_k$ are sub-additive, it follows that \n\\begin{equation*}\n\\text{rank } H_k(\\Lambda M;\\mathbb{Z})\\le\\sum_i\\text{rank } H_k(\\Lambda_{i+1},\\Lambda_i)\n\\end{equation*} so the rank of $H_k(\\Lambda M;\\mathbb{Z})$ is less than or equal to the number of critical points of index $k$.\n\\end{proof}\n\nSuppose $\\Lambda M$ is elliptic with top non-zero rational homology group in degree $n$. Then the Euler characteristic of the infinite-dimensional CW complex $\\Lambda M$ is $\\chi(\\Lambda M)=\\sum_{\\lambda=0}^{\\infty}(-1)^{\\lambda}C_{\\lambda}$ and $b_{\\lambda}\\le C_{\\lambda}$. Furthermore, $b_{\\lambda}\\le{n\\choose \\lambda}$ so $C_{\\lambda}={n\\choose \\lambda}$ as demonstrated in the following lemma.\n\\begin{lemma}\nIf $\\Lambda M$ is an elliptic space with top non-zero rational cohomology in degree $n$, then the rational cohomology ring $H^*(\\Lambda M;\\mathbb{Q})$ is the complete intersection ring and does not have at least two generators because the sequence of Betti numbers $\\{b_{\\lambda}(\\Lambda M;\\mathbb{Q})\\}_{\\lambda\\ge 0}$ is bounded by  $\\{{n\\choose\\lambda}\\}_{\\lambda\\ge 0}$.\n\\end{lemma}\nLet $M$ be rationally elliptic so that $b_{2k+1}(M)=0$, for $k\\in\\mathbb{N}$, if $\\chi(M)>0$ and let $\\pi_i(M)\\otimes\\mathbb{Q}=0$ for $i\\le\\dim M-1$. Let $n:=\\dim M$ such that \n\\begin{equation}\n\\begin{split}\n\\chi(M)&=\\sum_{\\lambda=0}^{\\infty}(-1)^{\\lambda}\\text{rank }H_{\\lambda}(M;\\mathbb{Q})\\\\\n&=\\sum_{\\lambda=0}^{\\infty}(-1)^{\\lambda}\\dim\\pi_{\\lambda}(M)\\otimes\\mathbb{Q}\\\\&\n=(-1)^n\\dim\\pi_n(M)\\otimes\\mathbb{Q}>0.\n\\end{split}\n\\end{equation} Recall the isomorphism $\\pi_n(\\Lambda M;\\mathbb{Q})\\cong\\pi_{n+1}(M;\\mathbb{Q})$, which implies $\\pi_n(\\Lambda M_{\\mathbb{Q}})=\\pi_n(\\Lambda M)\\otimes\\mathbb{Q}\\cong\\pi_{n+1}(M)\\otimes\\mathbb{Q}\\cong H_{n+1}(M)\\otimes\\mathbb{Q}$ where it is assumed that $\\pi_i(M)\\otimes\\mathbb{Q}=0$ for $i\\le n-1.$ Then \n\\begin{equation}\n\\begin{split}\nH_{n-1}(\\Lambda M_{\\mathbb{Q}};\\mathbb{Z})\\cong H_{n-1}(\\Lambda M;\\mathbb{Q})&\\cong\\pi_{n}(M;\\mathbb{Q})\\cong\\pi_n(M)\\otimes\\mathbb{Q}\\\\&\\cong H_n(M)\\otimes\\mathbb{Q}\\cong H_n(M;\\mathbb{Q}),\n\\end{split}\n\\end{equation} where the first isomorphism follows from the rational Hurewicz theorem and the last by $H_i(M_{\\mathbb{Q}};\\mathbb{Z})\\cong H_i(M;\\mathbb{Z})\\otimes\\mathbb{Q}\\cong H_i(M;\\mathbb{Q})$. We a priori assume the odd Betti numbers $b_{2i+1}$ vanish. Then we obtain the following bounds:\n\\begin{equation}\n\\begin{split}\n0<\\chi(M)&=\\sum_{i=0}^{\\dim M}(-1)^ib_i(M;\\mathbb{Q})\\\\\n&=\\sum_{i=1}^{\\dim M}(-1)^{i-1}b_{i-1}(\\Lambda M;\\mathbb{Q})\\\\\n&\\le\\sum_{i=0}^{\\infty}(-1)^i\\text{rank }H_i(\\Lambda M;\\mathbb{Q})=\\chi(\\Lambda M)\n\\end{split}\n\\end{equation} so $0<\\chi(M)\\le\\chi(\\Lambda M)$ or $\\chi(\\Lambda M)>0$. \n\\begin{theorem}\nLet $M$ be a closed $(\\dim M-1)$-connected Riemannian manifold. If the Euler characteristic $\\chi(\\Lambda M)$ of free loop space of Sobolev type $H^1=W^{1,2}$ is positive, then $M$ is rationally elliptic. \n\\end{theorem}\nBott's conjecture posits that any simply connected closed Riemannian manifold with nonnegative sectional curvature should be rationally elliptic. Assuming Bott's conjecture holds, if the sectional curvature of $M$ is nonnegative, i.e. $K\\ge 0$, then $M$ is rationally elliptic. As before, let $M$ be a closed orientable even $n$-dimensional Riemannian manifold without boundary, and suppose $\\Omega$ is the curvature form of the Levi-Civita connection $^{\\text{L.C.}}\\nabla$ associated with $M$ of $\\dim M:=n=2k$, $k\\in\\mathbb{N}$. Then $\\Omega$ is an $\\frak{so}(n)$-valued $2$-form on $M$, so it is a skew-symmetric $n$-by-$n$ matrix of $2$-forms over $\\wedge^{\\text{even}}T^*M$. Let $Pf(\\Omega)$ denote the $n$-form Pfaffian. Since the sectional curvature is nonnegative, by the generalized Gauss-Bonnet theorem,\n\\begin{equation}\n\\begin{split}\n\\chi(M)&=\\frac{1}{\\sqrt{(2\\pi)^n}}\\int_M Pf(\\Omega)\\\\&=\\frac{1}{\\sqrt{(2\\pi)^n}}\\int_Mi^{\\frac{n^2}{4}}\\exp\\left(\\frac{1}{2}\\text{Tr}\\log((\\sigma_y\\otimes I_{n\/2})^T\\cdot\\Omega) \\right)\\ge 0.\n\\end{split}\n\\end{equation}\n\nThe prime geodesic theorem describes the asymptotic distribution of prime geodesics on an $n$-dimensional hyperbolic manifold $M=\\mathbb{H}^n\/\\Gamma$ with $\\Gamma$ a discrete subgroup of $SO_{(1,n)}^{+}\\mathbb {R}$.\n\\begin{theorem} [Prime Geodesic Theorem \\cite{sarnak}]\nLet $M$ be a hyperbolic manifold of dimension $\\dim M=m+1$. Furthermore, let $\\Gamma=\\pi_1(M)$ be its fundamental group. For any element $\\gamma\\in\\Gamma$ there exists a closed geodesic representative $\\gamma$ in $M$. Let $\\ell(\\gamma)$ denote the length of the geodesic $\\gamma$, $N(\\gamma):=e^{\\ell(\\gamma)}$ its norm, and $\\pi_{\\Gamma}(x)$ the number of primitive elements $\\gamma\\in\\Gamma$ such that $N(\\gamma)\\le x$. Moreover, let $\\{z_1,\\dots,z_N\\}$ denote the zeros of the Selberg zeta function and $\\text{li}(x)=\\int_0^x\\frac{dt}{lnt}$ the logarithmic integral. Then $\\pi_{\\Gamma}(x)$ satisfies the following equality.\n\\begin{equation}\n\\pi_{\\Gamma}(x)=\\text{li}(x^m)+\\sum_{n=0}^Nli(x^{z_n})+(\\text{error term})\n\\end{equation}\n\\end{theorem}\nAs before, let $C_{\\lambda}$ denote the number of critical points of $E:\\Lambda M\\to\\mathbb{R}$ of index $\\lambda$ where the index of a critical point $\\gamma\\in\\Lambda M$ is $\\lambda(\\gamma):=\\dim\\{V_{\\text{max}}\\subset T_{\\gamma}\\Lambda M:d^2E<0\\}<+\\infty$. Consider the energy functional $E:\\Lambda M\\to\\mathbb{R}$ for $\\Lambda M$ an infinite-dimensional Hilbert manifold. For $a,b\\in\\mathbb{R}$ regular values of $E$, let $C_{\\lambda}(a,b)$ denote the number of critical points of index $\\lambda$ in $E^{-1}[a,b]$ with homology rank $r_{\\lambda}(a,b)=\\text{rank }(H_{\\lambda}({\\Lambda M}^b,{\\Lambda M}^a))$ and torsion rank $t_{\\lambda}(a,b)=t(H_{\\lambda}({\\Lambda M}^b,{\\Lambda M}^a))$ for ${\\Lambda M}^c=E^{-1}(-\\infty,c]$. That is,\n\\begin{equation}\n\\begin{split}\n& r_{\\lambda}(a,b)+t_{\\lambda}(a,b)+t_{\\lambda -1}(a,b)\\le C_{\\lambda}, \\\\\n& \\sum_{i=0}^{\\lambda}(-1)^{\\lambda-i}r_i(a,b)\\le\\sum_{i=0}^{\\lambda}(-1)^{\\lambda-i}C_i,\n\\end{split}\n\\end{equation} for $\\lambda=0,1,\\dots$. For sufficiently large $\\lambda$, the last inequality becomes an equality. Thus, for $\\chi(\\Lambda M)=\\sum_{\\lambda=0}^{\\infty}(-1)^{\\lambda}b_{\\lambda}(\n\\Lambda M; \\mathbb{Q})$ and $b_{\\lambda}+t_{\\lambda}+t_{\\lambda-1}\\le C_{\\lambda}$, we have $b_{\\lambda}\\le C_{\\lambda}$. Therefore the total number of primitive elements $\\gamma\\in\\Gamma$ is\n\\begin{equation*}\n\\lim_{t\\to\\infty^-}\\int_0^t\\pi_{\\Gamma}(x)dx=\\int_0^{\\infty}\\pi_{\\Gamma}(x)dx.\n\\end{equation*} Recall, the total number of critical points is given by the sum over all indices of the number of critical points of a specific index, i.e., $\\sum_{\\lambda}C_{\\lambda}$. As such,\n\\begin{equation}\n\\sum_{\\lambda}C_{\\lambda}=\\sum_{\\lambda}b_{\\lambda}(\\Lambda M;\\mathbb{Q})=\\int_0^{\\infty}\\pi_{\\Gamma}(x)dx.\n\\end{equation} That is, for $\\Lambda M=\\mathbb{H}^{\\infty}\/\\Gamma$ hyperbolic, \n\\begin{equation*}\n\\begin{split}\n\\sum_{\\lambda=0}^{\\dim\\Lambda M}b_{\\lambda}(\\Lambda M;\\mathbb{Q})&=\\int_0^{\\infty}\\text{li}(x^{\\dim \\Lambda M-1})dx+\\sum_{n=0}^N\\text{li}(x^{z_n})dx+(\\text{error term})dx \\\\\n&=\\lim_{k\\to\\infty}\\int_0^{\\infty}\\text{li}(x^{k-1})dx+\\sum_{n=0}^N\\int_0^{\\infty}\\text{li}(x^{z_n})dx+\\int_0^{\\infty}(\\text{error term})dx.\n\\end{split}\n\\end{equation*} Moreover, for $\\Lambda M=\\mathbb{H}^{\\infty}\/\\Gamma$ hyperbolic with $\\Gamma\\subset SO^+_{(1,\\infty)}\\mathbb{R}$, the \\textit{Poincar\\'e polynomial} of $\\Lambda M$ is the generating function of Betti numbers of $\\Lambda M$, $P_{\\Lambda M}(z)=b_0(\\Lambda M)+b_1(\\Lambda M)z+b_2(\\Lambda M)z^2+\\cdots$, so $P_{\\Lambda M}(1)=\\sum_{\\lambda=0}^{\\infty}b_{\\lambda}(\\Lambda M)=\\int_0^{\\infty}\\pi_{\\Gamma}(x)dx$.\n\\begin{lemma} By the main result \\ref{main result}, \n\\begin{equation*}\n\\int_0^{\\infty}\\pi_{\\Gamma}(x)dx=P_{\\Lambda M}(1)=+\\infty.\n\\end{equation*}\nThus, there are infinitely many such prime closed geodesics.\n\\end{lemma}\nWe remark that we have only shown this to be true for $\\Lambda M$ hyperbolic. Thus, we resort to a holonomic classification of simply connected manifolds due to Berger. \n\n\\section{Holonomy, The Berger Classification, and The Classification of Symmetric Spaces \\label{sec: holonomyclassification}}\nWe begin with preliminary definitions of reducibility and irreducibility of holonomy representations.\n\\begin{definition}\nThe Riemannian holonomy of a Riemannian manifold $(M,g)$ is the holonomy of the Levi-Civita connection on the tangent bundle of the manifold.\n\\end{definition}\n\\begin{definition}\nA Riemannian manifold $(M,g)$ is said to be (resp. locally) reducible if it is (resp. locally) isometric to a Riemannian product.\n\\end{definition}\n\\begin{theorem}[de Rham \\cite{deRham}]\\label{de Rham} If a Riemannian manifold is complete, simply connected, and if its holonomy representation is reducible, then $(M,g)$ is a Riemannian product. \n\\end{theorem}\n\nBerger provided a complete classification of holonomy for simply connected Riemannian manifolds which are irreducible and non-symmetric (see Table \\ref{table}).\n\n\\begin{table}[H]\n\\centering\n    \\begin{tabular}{ | c | c | c | c | }\n    \\hline\n    $\\text{Hol}(g)$ & $\\dim_{\\mathbb{R}}(M)$ & $G$-structure & Description \\\\ \\hline\n    $SO(n)$ & n & Orientable manifold & - \\\\ \\hline\n    $U(n)$ & $2n$ & K\\\"{a}hler & K\\\"{a}hler \\\\ \\hline\n    $SU(n)$ & $2n$ & Calabi-Yau Manifold & Ricci-flat, K\\\"{a}hler\\\\ \\hline\n    $Sp(n)\\cdot Sp(1)$ & $4n$ & Quaternion-K\\\"{a}hler manifold & Einstein\\\\ \\hline\n    $G_2$ & $7$ & $G_2$ manifold & Ricci-flat\\\\ \\hline\n    $Spin(7)$ & $8$ & $Spin(7)$ manifold & Ricci-flat\\\\ \\hline\n    \\end{tabular}\n    \\caption{The Berger classification of holonomy groups for irreducible and non-symmetric simply connected Riemannian manifolds.}\n    \\label{table}\n\\end{table}\n\nThere is a sequence of inclusions $Sp(n)\\subset SU(2n)\\subset U(2n)\\subset SO(4n)$ so every hyperk\\\"{a}hler manifold is Calabi-Yau, every Calabi-Yau manifold is K\\\"{a}hler, and every K\\\"{a}hler manifold is also orientable. \n\\begin{theorem}\\label{holonomytheorem}\nIf $M$ is simply connected and has reducible holonomy, then the de Rham decomposition theorem implies that $M$ is a product, and hence does not have rational cohomology generated by one element.\n\\end{theorem}\n\\begin{proof}\nLet $M$ be a simply connected $n$-dimensional manifold such that there is a complete decomposable reduction of the tangent bundle $TM=T^{(0)}M\\oplus T^{(1)}M\\oplus\\cdots\\oplus T^{(k)}M$ under the action of holonomy and the tangent bundles $T^{(i)}M$ parameterizing Frobenius-integrable distributions. If $\\text{Hol}(M)$ is reducible then $M=V_0\\times V_1\\times\\cdots\\times V_k$ for each $V_i$ an integral manifold for the respective tangent bundle $T^{(i)}M$ where $V_0$ is an open subset of $\\mathbb{R}^n$. Moreover, the holonomy group $\\text{Hol}(M)$ splits as a direct product of the holonomy groups of each $M_i$. Thus, if $M=M_0\\times M_1\\times\\cdots\\times M_k$ then $\\text{Hol}(M)=\\text{Hol}(M_0\\times M_1\\times\\cdots\\times M_k)=\\text{Hol}(M_0)\\times \\text{Hol}(M_1)\\times\\cdots\\times \\text{Hol}(M_k)$. So $M=M_0\\times M_1\\times\\cdots\\times M_k$ means $H^*(M;\\mathbb{Q})$ cannot be singly generated.\n\\end{proof}\nCoupling the Berger classification with the de Rham decomposition, one obtains a classification of reducible holonomy groups by requiring that each factor is one of the examples from Berger's list, realized by compact simply connected manifolds. One can find metrics on product manifolds with irreducible holonomy so the converse of Theorem \\ref{holonomytheorem} is false. More precisely, a generic manifold, with $O(n)$ holonomy or $SO(n)$ if it is orientable, will have irreducible holonomy and have homology that is not generated by a single element.\n\\begin{corollary}\nIf $M$ is oriented then $\\text{Hol}(M)=SO(n)$ so there are at least two non-trivial classes in $H^*(M;\\mathbb{Q})$. Thus, if $\\text{Hol}(M)=\\prod_{i=1}^k \\text{Hol}(M_i)$ then $H^*(M;\\mathbb{Q})$ must have at least two generators.\n\\end{corollary}\n\nSince Ziller \\cite{ziller} showed $b_k(\\Lambda M;\\mathbb{F}_2)$ to be unbounded for $M$ a compact globally symmetric space of rank $>1$, we only consider simply connected Riemannian manifolds which are irreducible and non-symmetric, as well as non-compact symmetric spaces of rank $1$. We obtain the following classification of reducible holonomy groups from Berger's classification and de Rham's decomposition theorem. We require that each factor in the decomposition is one from Berger's list or the holonomy of an irreducible non-compact symmetric space of rank $1$, which may be interchanged with a compact symmetric space of rank $1$ under bijection in the decomposition.\nLet $N$ be a simply connected Riemannian symmetric space. Then it may decomposed as $N=N_0\\times N_1\\times\\cdots\\times N_r$ for $N_0$ Euclidean and the other $N_i$ irreducible. As such, the classification of symmetric spaces is reduced to the the irreducible case. By duality de Rham (see \\cite{helgason}, P. 199),\ncompact and noncompact simply connected irreducible symmetric spaces may be interchanged, which therefore simplifies the classification to that of compact irreducible symmetric spaces. \n\nLet $M$ be an $n$-dimensional simply connected Riemannian manifold that is not a compact symmetric space of rank $>1$. The holonomy is then a direct product decomposition of: \n\\begin{equation*}\n\\begin{split}\n\\text{Hol}(M)=&\\prod_{i_1=1}^{N_1}SO(n_1)\\prod_{i_2=1}^{N_2} U(n_2)\\prod_{i_1=3}^{N_3} SU(n_3)\\prod_{i_4=1}^{N_4} (Sp(n_4)\\cdot Sp(1))\\prod_{i_5=1}^{N_5} Sp(n_5)\\prod_{i_6=1}^{N_6} G_2\\prod_{i_7=1}^{N_7} Spin(7)\\\\ &\\times \\text{Hol}(\\text{compact symmetric spaces of rank}= 1)\n\\end{split}\n\\end{equation*}\nwhere it is required that a given decomposition has a least two irreducible factors and $1\\le N_k<\\infty$ for every $k$ in the expansion. Adopting the Cartan labeling convention, we consider the label AIII, BDI, CII, FII symmetric spaces $G\/K$.\n\\begin{remark}\nFor label AII, $G\/K=SU(p+q)\/S(U(p)\\times U(q))$ has $\\text{rank}(G\/K)=\\min(p,q)$ and $\\dim(G\/K)=2pq$. So we let $p=1$, such that $K=S(U(1)\\times U(q))$. If we let $\\dim(G\/K)=n=2q$ then $q=\\frac{n}{2}$ or $K=S\\left(U(1)\\times U\\big(\\frac{n}{2}\\big)\\right)$.\n\\end{remark}\n\\begin{remark}\nFor label BDI, $G\/K=SO(p+q)\/SO(p)\\times SO(q)$ has $\\text{rank}(G\/K)=\\min(p,q)$ and  $\\dim(G\/K)=pq$. So if we let $p=1$, $SO(1+q)\/SO(1)\\times SO(q)$ has dimension $q$. Thus, $K=SO(1)\\times SO(n)$ for $\\dim(G\/K)=n$.\n\\end{remark}\n\\begin{remark}\nFor label CII, $G\/K=Sp(p+q)\/Sp(p)\\times Sp(q)$ has $\\text{rank}(G\/K)=\\min(p,q)$ and $\\dim (G\/K)=4pq$. We let $p=1$ so $F_4\/Spin(9)$ has dimension $n=4q$ or $K=Sp(1)\\times Sp\\left(\\frac{n}{4}\\right)$.\n\\end{remark}\n\\begin{remark}\nFor label FII, $G\/K=F_4\/Spin(9)$ has $\\text{rank}(G\/K)=1$ and $\\dim(G\/K)=16$.\n\\end{remark}\nNote, if $M=\\mathbb{R}^n$ with the flat Euclidean metric then parallel translation is simply a translation in $\\mathbb{R}^n$. In particular, if $P_{\\gamma} : T_{\\gamma(0)}M\\to T_{\\gamma(1)}M$ for $\\gamma:[0,1]\\to M$, then $P_{\\gamma}$ is the identity for each $\\gamma$ so $\\text{Hol}(\\mathbb{R}^n)$ is the trivial group, and it may be ignored in the decomposition. Thus, the decomposition of compact symmetric spaces for the problem under consideration is $\\text{Hol(compact simply connected symmetric spaces)}=S\\left(U(1)\\times U\\big(\\frac{n}{2}\\big)\\right)\\times (SO(1)\\times SO(n))\\times \\left(Sp(1)\\times Sp\\big(\\frac{n}{4}\\big)\\right)\\times Spin(9).$\nTo be more precise, the holonomy of a reducible simply connected Riemannian manifold $M$ (excluding compact symmetric spaces of rank $>1$) is given by the following.\n\\begin{theorem} (Classification of Reducible Holonomy.) \nIf $M$ is an $n$-dimensional simply connected Riemannian manifold with decomposition $M=M_1\\times\\cdots\\times M_j$ of irreducible $M_k$ ($k=1,\\dots, j)$, then it has holonomy decomposition given by the direct product:\n\\begin{equation*}\n\\begin{split}\n\\text{Hol}(M)=&\\prod_{i_1=1}^{N_1}SO(n_1)\\prod_{i_2=1}^{N_2} U(n_2)\\prod_{i_3=1}^{N_3} SU(n_3)\\prod_{i_4=1}^{N_4}(Sp(n_4)\\cdot Sp(1))\\prod_{i_5=1}^{N_5} Sp(n_5)\\prod_{i_6=1}^{N_6} G_2\\prod_{i_7=1}^{N_7} Spin(7)\\\\ &\\prod_{i_8=1}^{N_8} S(U(1)\\times U(n_6)) \\prod_{i_9=1}^{N_9} (SO(1)\\times SO(n_7))\\prod_{i_{10}=1}^{N_{10}} (Sp(1)\\times Sp(n_8))\\prod_{i_{11}=1}^{N_{11}} Spin(9)\n\\end{split}\n\\end{equation*} where it is required that there are at least two factors in a given decomposition and $\\sum_i\\dim_{\\mathbb{R}}(M_i)=n$.\n\\end{theorem}\n\\begin{corollary}\nIf $M$ is a simply connected Riemannian manifold with reducible holonomy given by the previous theorem, the rational cohomology ring $H^*(M;\\mathbb{Q})$ has at least two generators.\n\\end{corollary}\nSince Riemannian spaces that are locally isometric to the homogeneous spaces $G\/H$ have local holonomy isomorphic to $H$, the holonomy decomposition is\n\\begin{equation*}\n\\begin{split}\n\\text{Hol}(M)=&\\prod_{i_1=1}^{N_1}SO(n_1)\\prod_{i_2=1}^{N_2} U(n_2)\\prod_{i_3=1}^{N_3} SU(n_3)\\prod_{i_4=1}^{N_4} (Sp(n_4)\\cdot Sp(1))\\prod_{i_5=1}^{N_5} Sp(n_5)\\prod_{i_6=1}^{N_6} G_2\\prod_{i_7=1}^{N_7} Spin(7)\\\\ & \\prod_{i_8=1}^{N_8} \\text{Hol}\\left(\\frac{SU(n_6+1)}{S(U(1)\\times U(n_6))}\\right) \\prod_{i_9=1}^{N_9} \\text{Hol}\\left(\\frac{SO(n_7+1)}{SO(1)\\times SO(n_7)}\\right)\\prod_{i_{10}=1}^{N_{10}} \\text{Hol}\\left(\\frac{Sp(n_8+1)}{Sp(1)\\times Sp(n_8)}\\right)\\\\ & \\prod_{i_{11}=1}^{N_{11}} \\text{Hol}\\left(\\frac{F_4}{Spin(9)}\\right).\n\\end{split}\n\\end{equation*}\n\nLet $M$ be any space whose rational cohomology ring is a free graded-commutative algebra. Then the rationalization $M_{\\mathbb{Q}}$ is a product of Eilenberg-MacLane spaces. This hypothesis on cohomology also applies to compact Lie groups. Consider the decomposition of $M$, i.e. $M=M_1\\times\\cdots\\times M_k$, then the rationalization of holonomy is $\\text{Hol}(M)_{\\mathbb{Q}}=\\text{Hol}(M)\\times_{\\mathbb{Q}}E\\mathbb{Q}$ and $\\text{Hol}(M_{\\mathbb{Q}})=\\text{Hol}(M\\times_{\\mathbb{Q}}E\\mathbb{Q})=\\text{Hol}(M)\\times \\text{Hol}(E\\mathbb{Q})$. The total space of the universal bundle over $B\\mathbb{Q}$ is $E\\mathbb{Q}=B\\mathbb{Q}\\rtimes\\mathbb{Q}$ since $B\\mathbb{Q}=E\\mathbb{Q}\/\\mathbb{Q}$. Note, $B\\mathbb{Q}$ is the Eilenberg-MacLane space $K(\\mathbb{Q},1)$ so $E\\mathbb{Q}=K(\\mathbb{Q},1)\\rtimes\\mathbb{Q}$.\n\nWe can obtain the space $K(\\mathbb{Q},1)$ by the following procedure. Take the circle $S^1$, consider the sequence of maps $f_n:S^1\\to S^1$ of degree $n$, and form an infinite mapping telescope of $f_n$, a special case of a homotopy colimit. Observe, $\\mathbb{Q}$ is the filtered colimit $\\mathbb{Z}\\to\\mathbb{Z}\\to\\cdots$ where successive maps are multiplication by $1,2,\\dots$. Note, since $\\Omega\\mathbb{Q}\\cong\\ast$ regardless of its topology, the resulting classifying space $B\\mathbb{Q}$ will be a $K(\\mathbb{Q},1)$ space. It follows that $\\text{Hol}(M)_{\\mathbb{Q}}=\\text{Hol}(M)\\times_{\\mathbb{Q}}E\\mathbb{Q}$ and $\\text{Hol}(M_{\\mathbb{Q}})=\\text{Hol}(M)\\times \\text{Hol}(E\\mathbb{Q})$, so $\\text{Hol}(M)_{\\mathbb{Q}}=\\frac{\\text{Hol}(M_{\\mathbb{Q}})}{\\text{Hol}(E\\mathbb{Q})}\\times_{\\mathbb{Q}}E\\mathbb{Q}$. By first computing the rationalization space $M_{\\mathbb{Q}}$, we can determine the equivariant cohomology ring $H^*_{\\mathbb{Q}}(M;\\mathbb{Q})=H^*(E\\mathbb{Q}\\times_{\\mathbb{Q}}M;\\mathbb{Q})$, noting that the rationalization $M_{\\mathbb{Q}}$ is the product of Eilenberg-MacLane spaces whose rational cohomology rings are easier to compute. Since the decomposition of $M$ is given by:\n\\begin{equation*}\n\\begin{split}\nM=&\\prod_{i_1=1}^{N_1}M_{\\text{orientable, }SO(n_1)}\\prod_{i_2=1}^{N_2} M_{\\text{K\\\"{a}hler, } U(n_2)}\\prod_{i_3=1}^{N_3} M_{\\text{Calabi-Yau, }SU(n_3)}\\prod_{i_4=1}^{N_4} M_{\\text{quaternion-K\\\"{a}hler, }Sp(n_4)\\cdot Sp(1)}\\\\\n& \\prod_{i_5=1}^{N_5} M_{\\text{hyperk\\\"{a}hler, }Sp(n_5)}\\prod_{i_6=1}^{N_6} M_{G_2}\\prod_{i_7=1}^{N_7} M_{Spin(7)}\\prod_{i_8=1}^{N_8}\\frac{SU(n_6+1)}{S(U(1)\\times U(n_6))}\\prod_{i_9=1}^{N_9}\\frac{SO(n_7+1)}{SO(1)\\times SO(n_7)}\\\\\n&\\prod_{i_{10}=1}^{N_{10}} \\frac{Sp(n_8+1)}{Sp(1)\\times Sp(n_8)}\\prod_{i_{11}=1}^{N_{11}} \\frac{F_4}{Spin(9)},\n\\end{split}\n\\end{equation*} we can compute the rational cohomology ring $H^*(M;\\mathbb{Q})$ by the Kunneth formula.\n\\begin{equation*}\n\\begin{split}\nH^*(M;\\mathbb{Q})\\cong & \\bigotimes_{i_1=1}^{N_1}H^*(M_{\\text{orientable, }SO(n_1)};\\mathbb{Q})\\bigotimes_{i_2=1}^{N_2} H^*(M_{\\text{K\\\"{a}hler, } U(n_2)};\\mathbb{Q}) \\bigotimes_{i_3=1}^{N_1} H^*(M_{\\text{Calabi-Yau, }SU(n_3)};\\mathbb{Q})\\\\ &\\otimes\\cdots\\otimes\\bigotimes_{i_{11}=1}^{N_{11}} H^*(F_4\/Spin(9);\\mathbb{Q})\n\\end{split}\n\\end{equation*}\n\\section{Palais-Smale Local Approximation for Free Loop Space}\nLet $\\phi$ be a chart on the Hilbert manifold $\\Lambda M$ defined by \n\\begin{equation}\n\\begin{split}\n\\phi:\\quad& \\mathcal{U}\\to \\mathcal{V}\\subset\\mathbb{R}^n \\\\ \n& \\gamma\\mapsto  \\left(\\int_{S^1}\\|\\dot\\gamma(t)\\|^2dt,\\int_{S^1}\\|\\dot\\gamma(t)\\|^2dt,\\dots \\right),\n\\end{split}\n\\end{equation} satisfying the following commutative diagram:\n\\newline\n\\[\n\\begin{tikzcd}[column sep=large, row sep=large]\n    \\mathcal{U}\n     \\arrow[swap]{d}{E}\n \\arrow{r}{\\phi}\n& \\mathcal{V}\\subset\\mathbb{R}^n \\\\\n  \\mathbb{R} \\arrow[swap]{ru}{\\phi\\circ E^{-1}}\n&\n\\end{tikzcd}\n\\]\nThe Morse Lemma can be generalized to the infinite-dimensional setting. On separable Hilbert spaces, it takes the following form.\n\\begin{theorem}\\label{generalizedmorse}\nLet $H$ be a separable Hilbert space and $f:H\\to\\mathbb{R}$ a $C^k$ function $f$ with $k\\ge 3$ in the sense of Fr\\'echet differentiability, for which $c$ is a non-degenerate critical point. Then there exist convex neighborhoods $\\mathcal{U}$ and $\\mathcal{V}$ of $c$, and a diffeomorphism $\\varphi:\\mathcal{U}\\to\\mathcal{V}$ of class $C^{k-2}$ with $\\varphi(c)=c$ and a bounded orthogonal projection $P: H\\to H$ such that \n\\begin{equation}\nf(x)=f(c)-\\|P(\\varphi(x))\\|_H^2+\\|\\varphi(x)-P(\\varphi(x))\\|_H^2,\n\\end{equation} where $\\dim\\text{im }P$ is the Morse index of the critical point $c$.\n\\end{theorem}\n\\begin{theorem} \\label{result}\nIf the energy functional $E:\\Lambda M\\to\\mathbb{R}$, defined by $E[\\gamma]=\\int_{S^1}\\|\\dot\\gamma(t)\\|^2dt$ for $\\gamma:S^1\\to M$, is a $C^3$-function, then $M$ is a union of infinitely many Baire spaces $S^{\\infty}$. \n\\end{theorem}\n\\begin{proof}\nAssume that the Riemannian metric on $M$ is chosen such that the energy functional $E:\\Lambda M\\to\\mathbb{R}$ is Morse-Bott, i.e., $E$ is a $C^2$-function. Assume that $E$ is also a $C^3$-function by choice. Let $H_{\\alpha}\\subset\\Lambda M$ and let $\\Lambda M=\\bigcup_{\\alpha\\in A}H_{\\alpha}\\oplus H_{\\alpha}^{\\perp}$ for $A$ the set of critical points of $E$. Further, let ${\\mathcal{U}}_{\\alpha}\\subset\\Lambda M$ be a local neighborhood of the critical point $\\beta_{\\alpha}$. Then the projection map $P:\\mathcal{U}_{\\alpha}\\to H_{\\alpha}$ has $P(x)=m$ for $x=m+m',m\\in H_{\\alpha},m'\\in H_{\\alpha}^{\\perp}$. Recall that $\\dim\\text{im }P=\\lambda(\\beta_{\\alpha})$. Let $\\{\\gamma_{\\alpha k}\\}_{k=1}^{\\lambda(\\beta_{\\alpha})}$ be an orthonormal basis for $H_{\\alpha}$. That is, the orthonormal basis satisfies the following criteria.\n\\begin{enumerate}[label=(\\roman*)]\n\\item Orthogonality: Every two different basis elements are orthogonal, i.e. $\\langle\\gamma_k,\\gamma_j\\rangle=0$ for all $k,j=1,\\dots,\\lambda(\\beta_{\\alpha})$ with $k\\ne j$.\n\\item Normalization: Every element of this basis has norm $1$, i.e. $\\|\\gamma_{\\alpha k}\\|=1$ for all $k=1,\\dots,\\lambda(\\beta_{\\alpha})$.\n\\item Completeness: The linear span of the family $\\gamma_{\\alpha k}$, for $k=1,\\dots,\\lambda(\\beta_{\\alpha})$, is dense in $H_{\\alpha}$.\n\\end{enumerate} \nSimilarly, let $\\{\\gamma_{\\alpha k}^{\\perp}\\}_{k\\in B}$, for $B:=(\\lambda(\\beta_{\\alpha})+1,\\lambda(\\beta_{\\alpha})+2,\\dots)$ a countable but infinite set, be an orthonormal basis for $H_{\\alpha}^{\\perp}$, satisfying the above properties, of $\\dim H_{\\alpha}^{\\perp}=\\text{codim }H_{\\alpha}=\\infty$ for $\\dim H_{\\alpha}=\\lambda(\\beta_{\\alpha})$. Then $\\langle\\gamma_{\\alpha k},\\gamma_{\\alpha j}^{\\perp}\\rangle=0$ for all $k=1,\\dots,\\lambda(\\beta_{\\alpha}),j=\\lambda(\\beta_{\\alpha})+1,\\dots$. We may construct $\\Lambda M$ as a patching over neighborhoods for sufficiently many critical points, which are homeomorphic to $H_{\\alpha}\\oplus H_{\\alpha}^{\\perp}$ for $\\alpha\\in A$. Thus, for $\\mathcal{U}_{\\alpha}$ an open neighborhood of $\\beta_{\\alpha}\\in\\Lambda M$, $\\mathcal{U}_{\\alpha}=H_{\\alpha}\\oplus H_{\\alpha}^{\\perp}$ such that for $\\gamma_{\\alpha}\\in\\mathcal{U}_{\\alpha}$, there exists a unique $\\theta_{\\alpha}\\in H_{\\alpha}$ and $\\varepsilon_{\\alpha}\\in H_{\\alpha}^{\\perp}$ such that $\\gamma_{\\alpha}=\\theta_{\\alpha}+\\varepsilon_{\\alpha}$ whereby $\\theta_{\\alpha}=\\sum_{k=1}^{\\lambda(\\beta_{\\alpha})}c_{\\alpha k}\\gamma_{\\alpha k}$ and $\\varepsilon_{\\alpha}=\\sum_{j=\\lambda(\\beta_{\\alpha})+1}^{\\infty}\\widehat{c_{\\alpha j}}\\gamma_{\\alpha j}^{\\perp}$. Thus,\n\\begin{equation}\n\\gamma_{\\alpha}=\\sum_{k=1}^{\\lambda(\\beta_{\\alpha})}c_{\\alpha k}\\gamma_{\\alpha k}+\\sum_{j=\\lambda(\\beta_{\\alpha})+1}^{\\infty}\\widehat{c_{\\alpha j}}\\gamma_{\\alpha j}^{\\perp}\n\\end{equation} where $\\langle\\gamma_{\\alpha k},\\gamma_{\\alpha j}^{\\perp}\\rangle=0$. We invoke Theorem \\ref{approximation} to conclude that $E:\\Lambda M\\to\\mathbb{R}$ may be approximated by a smooth functional $E_{\\varepsilon}:\\Lambda M'\\to\\mathbb{R}$ with infinitely many critical points so that $A$ in $\\Lambda M=\\bigcup_{\\alpha\\in A}H_{\\alpha}\\oplus H_{\\alpha}^{\\perp}$ is uncountable or countably infinite. Note, this follows because $\\mathcal{U}_{\\alpha}$ is dense. \n\nFurthermore, let $\\{\\gamma_{\\alpha k}\\}_{k=1}^{\\infty}$ be an orthonormal basis for $\\mathcal{U}_{\\alpha}=H_{\\alpha}\\oplus H_{\\alpha}^{\\perp}$ such that if $\\gamma_{\\alpha}\\in\\mathcal{U}_{\\alpha}$ then $\\gamma_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\gamma_{\\alpha k}$ and $P:\\mathcal{U}_{\\alpha}\\to H_{\\alpha}$ means $P(\\gamma_{\\alpha})=P\\left(\\sum_{k=1}^{\\infty}c_{\\alpha k}\\gamma_{\\alpha k}\\right)$. Since $P$ is a linear operator, $P(\\gamma_{\\alpha})=\\theta_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}P(\\gamma_{\\alpha k})$. Let $\\{\\theta_{\\alpha k}\\}_{k=1}^{\\infty}$ be an orthonormal basis for $H_{\\alpha}$ where $\\theta_{\\alpha k}=0$ for $k>\\lambda(\\beta_{\\alpha})$. That is, $\\{\\theta_{\\alpha k}\\}_{k=1}^{\\lambda(\\beta_{\\alpha})}$ is an orthonormal basis for $H_{\\alpha}$. Then $P(\\gamma_{\\alpha k})=\\theta_{\\alpha k}$ and $P(\\gamma_{\\alpha k})=\\theta_{\\alpha k}=0$ for $k>\\lambda(\\beta_{\\alpha})$. Let $\\{\\theta_{\\alpha k}^{\\perp}\\}_{k=\\lambda(\\beta_{\\alpha})+1}^{\\infty}$ be an orthonormal basis for $H_{\\alpha}^{\\perp}$. Then for $P_{\\perp}:\\mathcal{U}_{\\alpha}\\to H_{\\alpha}^{\\perp}$ and $P_{\\perp}(\\gamma_{\\alpha k})=\\theta_{\\alpha k}^{\\perp}$, we have $P_{\\perp}(\\gamma_{\\alpha})=\\varepsilon_{\\alpha}=\\sum_{k=1}^{\\infty}\\widehat{c_{\\alpha k}}P_{\\perp}(\\gamma_{\\alpha k})$ such that $P_{\\perp}(\\gamma_{\\alpha k})=\\theta_{\\alpha k}^{\\perp}=0$ for $k<\\lambda(\\beta_{\\alpha})+1$. Therefore $\\gamma_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\gamma_{\\alpha k}$ and \n\\begin{equation*}\n\\begin{split}\nP(\\gamma_{\\alpha})=\\theta_{\\alpha}&=\\sum_{k=1}^{\\infty}c_{\\alpha k}P(\\gamma_{\\alpha k})\\\\&=\\sum_{k=1}^{\\lambda(\\beta_{\\alpha})}c_{\\alpha k}P(\\gamma_{\\alpha k}) \\text{ with }\nP(\\gamma_{\\alpha k})=\\begin{cases} \n      \\theta_{\\alpha k}, & \\text{if }k\\le\\lambda(\\beta_{\\alpha}), \\\\\n      0, & \\text{if }k>\\lambda(\\beta_{\\alpha}), \n         \\end{cases} \n  \\end{split}\n  \\end{equation*}\n\\begin{equation*}\n\\begin{split}\nP_{\\perp}(\\gamma_{\\alpha})=\\varepsilon_{\\alpha}&=\\sum_{k=1}^{\\infty}\\widehat{c_{\\alpha k}}P_{\\perp}(\\gamma_{\\alpha k})\\\\&=\\sum_{k=\\lambda(\\beta_{\\alpha})+1}^{\\infty}\\widehat{c_{\\alpha k}}P_{\\perp}(\\gamma_{\\alpha k})\\text{ with } P_{\\perp}(\\gamma_{\\alpha k})=\\begin{cases} \n      \\theta_{\\alpha k}^{\\perp}, & \\text{if }k\\ge\\lambda(\\beta_{\\alpha})+1, \\\\\n      0, & \\text{if }k<\\lambda(\\beta_{\\alpha})+1, \n         \\end{cases} \n\\end{split}\n\\end{equation*} where $\\langle\\theta_{\\alpha k},\\theta_{\\alpha j}^{\\perp}\\rangle=0$ for all $k=1,\\dots,\\lambda(\\beta_{\\alpha}), j=\\lambda(\\beta_{\\alpha})+1,\\dots.$ Thus, by the Morse-Palais Lemma \\ref{generalizedmorse}, $E[\\gamma_{\\alpha}]=E[\\beta_{\\alpha}]-\\|P(\\varphi(\\gamma_{\\alpha}))\\|^2_{\\Lambda M}+\\|\\varphi(\\gamma_{\\alpha})-P(\\varphi(\\gamma_{\\alpha}))\\|^2_{\\Lambda M}$ where $\\varphi:\\mathcal{U}\\to\\mathcal{V}$ is a diffeomorphism such that for $\\gamma_{\\alpha}\\in\\mathcal{U}_{\\alpha}$, $\\varphi(\\gamma_{\\alpha})=\\gamma_{\\alpha}$, i.e. $\\varphi=\\text{id}_{\\mathcal{U}_{\\alpha}}$ by identification. It follows that \n\\begin{equation}\n\\begin{split}\n&E[\\gamma_{\\alpha}]=E[\\beta_{\\alpha}]-\\|P(\\varphi(\\gamma_{\\alpha}))\\|^2_{\\Lambda M}+\\|\\varphi(\\gamma_{\\alpha})-P(\\varphi(\\gamma_{\\alpha}))\\|^2_{\\Lambda M} \\\\\n&=E[\\beta_{\\alpha}]-\\|P(\\varphi(\\gamma_{\\alpha}))\\|^2_{\\Lambda M}+\\|\\gamma_{\\alpha}\\|^2_{\\Lambda M}-2Re\\langle\\gamma_{\\alpha},P(\\gamma_{\\alpha})\\rangle+\\|P(\\gamma_{\\alpha})\\|^2_{\\Lambda M} \\\\ &=E[\\beta_{\\alpha}]+\\|\\gamma_{\\alpha}\\|^2_{\\Lambda M}-2Re\\langle\\gamma_{\\alpha},P(\\gamma_{\\alpha})\\rangle.\n\\end{split}\n\\end{equation} \nLet \\begin{equation*}\n\\begin{split}\n\\varphi_{\\alpha}:\\quad &\\mathcal{U}_{\\alpha}\\cap H_{\\alpha}\\to\\mathbb{R}^{\\lambda(\\beta_{\\alpha})} \\\\\n&\\gamma_{\\alpha}\\mapsto \\left(\\gamma_{\\alpha 1},\\dots,\\gamma_{\\alpha \\lambda(\\beta_{\\alpha})}\\right)\n\\end{split}\n\\end{equation*} since $H_{\\alpha}$ is a vector space, so the tangent space $T_{\\gamma_{\\alpha}}H_{\\alpha}$ at any point $\\gamma_{\\alpha}\\in H_{\\alpha}$ is canonically isomorphic to $H_{\\alpha}$ itself. Similarly, let\n\\begin{equation*}\n\\begin{split}\n\\widehat{\\phi_{\\alpha}}:\\quad &\\mathcal{U}_{\\alpha}\\cap H_{\\alpha}^{\\perp}\\to\\mathbb{R}^{\\infty}\\\\\n&\\gamma_{\\alpha}\\mapsto(\\gamma_{\\alpha 1},\\gamma_{\\alpha 2},\\dots)\n\\end{split}\n\\end{equation*} with $H_{\\alpha}\\cap H_{\\alpha}^{\\perp}=\\{\\beta_{\\alpha}\\}$, affinely translated $\\{0\\}$. Then let \n\\begin{equation*}\n\\begin{split}\n\\phi_{\\alpha}:\\quad&\\mathcal{U}_{\\alpha}\\to\\mathbb{R}^{\\infty} \\\\\n& \\gamma_{\\alpha}\\mapsto (\\gamma_{\\alpha 1},\\gamma_{\\alpha 2},\\dots)\n\\end{split}\n\\end{equation*} be a chart on $\\mathcal{U}_{\\alpha}$. Thus, \n\\begin{equation*}\n\\begin{split}\n\\phi_{\\alpha}^i:\\quad&\\mathcal{U}_{\\alpha}\\to\\mathbb{R}\\\\\n&\\gamma_{\\alpha}\\mapsto \\gamma_{\\alpha i}\n\\end{split}\n\\end{equation*}\nare local coordinate charts such that $\\gamma_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\gamma_{\\alpha k}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\phi_{\\alpha}^k$ so $\\|\\gamma_{\\alpha}\\|^2_{\\Lambda M}=\\langle \\gamma_{\\alpha},\\gamma_{\\alpha}\\rangle_{\\Lambda M}=\\left\\|\\sum_{k=1}^{\\infty}c_{\\alpha k}\\phi_{\\alpha}^k\\right\\|^2_{\\Lambda M}=\\sum_{k=1}^{\\infty}\\left\\|c_{\\alpha k}\\phi_{\\alpha}^k\\right\\|^2_{\\Lambda M}$ since the $\\phi_{\\alpha}^k$'s are orthogonal by choice, and $\\|\\gamma_{\\alpha}\\|^2_{\\Lambda M}=\\sum_{k=1}^{\\infty}c_{\\alpha k}^2\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}$ where $\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=\\|\\phi_{\\alpha}^k\\|^2_0+\\|\\nabla_t\\phi_{\\alpha}^k\\|^2_0$. It follows that \n\\begin{equation*}\n\\|\\gamma_{\\alpha}\\|^2_{\\Lambda M}=\\sum_{k=1}^{\\infty}c_{\\alpha k}^2\\left(\\|\\phi_{\\alpha}^k\\|^2_0+\\|\\nabla_t\\phi_{\\alpha}^k\\|^2_0\\right).\n\\end{equation*}\nLikewise, $\\theta_{\\alpha}=\\sum_{k=1}^{\\lambda(\\beta_{\\alpha})}\\widehat{c_{\\alpha k}}P(\\phi_{\\alpha}^k)=\\sum_{k=1}^{\\lambda(\\beta_{\\alpha})}\\widehat{c_{\\alpha k}}\\phi_{\\alpha}^k$ since $P(\\phi_{\\alpha}^k)=\\phi_{\\alpha}^k$ for $k\\le\\lambda(\\beta_{\\alpha})$. Recall $\\gamma_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\gamma_{\\alpha k}$. Therefore, $\\langle \\gamma_{\\alpha}, P(\\gamma_{\\alpha})\\rangle_{\\Lambda M}=\\left\\langle\\sum_{k=1}^{\\infty}c_{\\alpha k}\\phi_{\\alpha}^k, \\sum_{j=1}^{\\lambda(\\beta_{\\alpha})}\\widehat{c_{\\alpha j}}\\phi_{\\alpha}^j\\right\\rangle_{\\Lambda M}=\\sum_{(k,j)\\in K\\times J}c_{\\alpha k}\\widehat{c_{\\alpha j}}\\langle\\phi_{\\alpha}^k, \\phi_{\\alpha}^j\\rangle=c_{\\alpha k}\\widehat{c_{\\alpha k}}:=C_{\\alpha}$. As such, $E[\\gamma_{\\alpha}]=E[\\beta_{\\alpha}]+\\sum_{k=1}^{\\infty}c_{\\alpha k}^2\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}-2C_{\\alpha}$ or, for $\\widehat{C_{\\alpha}}=E[\\gamma_{\\alpha}]-E[\\beta_{\\alpha}]+2C_{\\alpha}$, we have $\\sum_{k=1}^{\\infty}c_{\\alpha k}^2\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=\\widehat{C_{\\alpha}}$ where $\\gamma_{\\alpha}=\\sum_{k=1}^{\\infty}c_{\\alpha k}\\phi_{\\alpha}^k=c_{\\alpha 1}\\phi_{\\alpha}^1+c_{\\alpha 2}\\phi_{\\alpha}^2+\\cdots$, and $\\phi_{\\alpha}^k$ is defined on $\\mathcal{U}_{\\alpha}$. Then $c_{\\alpha k}=\\int_{\\mathcal{U}_{\\alpha}}\\phi_{\\alpha}^k(\\gamma_{\\alpha})d\\gamma_{\\alpha}$. Thus, $\\sum_{k=1}^{\\infty}\\frac{C_{\\alpha k}^2}{\\widehat{C}_{\\alpha}}\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=1$ so letting $a_{\\alpha k}=C^2_{\\alpha k}\/\\widehat{C_{\\alpha}}$ we have\n\\begin{equation}\n\\sum_{k=1}^{\\infty}a_{\\alpha k}\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=1.\n\\end{equation} The locus of all points $\\{\\phi_{\\alpha}^1,\\phi_{\\alpha}^2,\\dots\\}$, i.e. the coordinates on the patch $\\mathcal{U}_{\\alpha}$, corresponds to an infinite-dimensional ellipsoid with $a_{\\alpha i}\\ne 0$ for $k\\in\\mathbb{Z}^+$ because, otherwise, $a_{\\alpha i}= 0$ would obstruct the uniqueness and existence of the orthonormal basis. Thus, $\\sum_{k=1}^{\\infty}a_{\\alpha k}\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=1$ for each $\\mathcal{U}_{\\alpha}$. It follows that\n\\begin{equation}\n\\Lambda M=\\bigcup_{\\alpha\\in A}\\mathcal{U}_{\\alpha}=\\bigcup_{\\alpha\\in A}\\left\\{(\\phi_{\\alpha}^1,\\phi_{\\alpha}^2,\\dots)\\in\\mathbb{R}^{\\infty}\\Bigg | \\sum_{k=1}^{\\infty}a_{\\alpha k}\\|\\phi_{\\alpha}^k\\|^2_{\\Lambda M}=1\\right\\}\\cong \\bigcup_{\\alpha\\in A}S^{\\infty}.\n\\end{equation} Suppose there exists a suitable $C^0$-perturbation $E_{\\varepsilon}$ of $E$ such that there is an atlas with $\\Lambda M=\\bigcup_{\\alpha\\in A}\\mathcal{U}_{\\alpha}$ where $\\mathcal{U}_{\\alpha}$ is locally a Hilbert space for all $\\alpha\\in A$. Then given such a distribution of critical points on $\\Lambda M$, we can compute the cohomology groups via a nerve of a good cover and, thus, the sequence of Betti numbers. Note, such a construction of an atlas for $\\Lambda M$ can always be made because $E_{\\varepsilon}$ is an arbitrary $C^0$-perturbation of $E$. Then the rational cohomology of the Baire space $H^k(S^{\\infty};\\mathbb{Q})$ is the inverse limit $\\varprojlim_nH^k(S^{n};\\mathbb{Q})$.\n\\end{proof}\n\\begin{lemma}\nLet $\\Lambda M\\subset\\mathbb{R}^{\\infty}$ be a paracompact connected Hilbert manifold. Moreover, let $E:\\Lambda M\\to\\mathbb{R}$ be a $C^2$-function. Then the simplicial complex $\\Delta:=\\{\\gamma\\in \\Lambda M:dE[\\gamma]=0\\}$, for vertices identified with the intersections of a refined open cover $\\{U_{\\alpha}\\}_{\\alpha\\in A}$ the non-degenerate critical points of $E$, is homotopy equivalent to $\\Lambda M$.\n\\end{lemma}\n\\begin{proof}\nA $C^0$-perturbation of the energy functional $E:\\Lambda M\\to\\mathbb{R}$ produces infinitely many critical points. Suppose we form a good cover $\\mathcal{U}=\\{U_{\\alpha_k}\\}_{\\alpha_k\\in A}$ such that each $U_{\\alpha_k}$ is a neighborhood of the non-degenerate critical point ${\\gamma}_k$. Then, in general, this should be a cover of $\\Lambda M$ and the map from the geometric realization of the simplicial space, i.e. the topological nerve of $\\{U_{a_k}\\}$, to $\\Lambda M$ is a homotopy equivalence. Likewise, the geometric realization of the topological nerve $R(\\text{Top-Nrv}(\\mathcal{U}))$ is homotopy equivalent to the geometric realization of the simplicial set for which the $|A|$-simplices are $|A|$-tuples $(\\alpha_0,\\dots,\\alpha_{|A|})$, whereby the intersection $\\bigcap_{k\\in\\mathbb{Z}}U_{\\alpha_k}$ is nonempty and $|A|\\to\\infty$ under $C^0$-perturbation of $E$.\n\\end{proof}\n\nTreating $\\Lambda M$ as a CW complex, by Theorem \\ref{result}, it is given by the union of Baire spaces, namely $M=\\bigcup_{\\alpha\\in A}S^{\\infty}$ without information on intersections. The only requirement is that the union of such Baire spaces covers $\\Lambda M$. Let $\\Sigma=\\{S^{\\infty},\\dots,S^{\\infty}\\}$ be a finite collection of sets, with cardinality $|A|$. The nerve consists of all sub-collections whose sets have a non-empty common intersection, $\\text{Nrv}(\\Sigma)=\\left\\{X\\subseteq\\Sigma:\\bigcap X\\ne\\emptyset\\right\\}$, which is an abstract simplicial complex. That is, suppose we form a good cover $\\mathcal{U}=\\{U_{\\alpha_k}\\}_{\\alpha_k\\in A}$ such that each $U_{\\alpha_k}$ is a neighborhood of the non-degenerate critical point $\\gamma_k$. Since all Hilbert manifolds are paracompact, we may compute homology of the CW model of $\\Lambda M$ via \\v{C}ech homology:\n\\begin{equation*} \\check{H}_k(\\mathcal{U},\\mathcal{F})=\\varinjlim_{\\mathcal{U}\\in A}H_k(\\mathcal{U},\\mathcal{F})\n\\end{equation*} for $\\mathcal{F}$ a presheaf of abelian groups on $\\Lambda M$, and $A$ a directed set consisting of all open covers of $\\Lambda M$, which is directed by  $\\mathcal{U}<\\mathcal{V}$ for $\\mathcal{V}$ a refinement of $\\mathcal{U}$. \\v{C}ech homology is similarly given by $\\check{H}_k(\\Lambda M,\\mathcal{F})=\\varinjlim_{\\mathcal{U}\\in A}H_k(\\text{Nrv}(\\mathcal{U}),\\mathcal{F})$.\n\n\\section{Integral Cellular Homology and Smith Normal Form for Free Loop Space\\label{sec: SNF}}\n\\begin{lemma}\nEvery continuous map $f:M\\to\\mathbb{R}^n$ from a Hilbert manifold can be arbitrarily approximated by a smooth map $g:M\\to\\mathbb{R}^n$ which has no critical points.\n\\end{lemma}\n\\begin{theorem}\\label{approximation}\nA continuous map $f:M\\to\\mathbb{R}^n$ from a Hilbert manifold may be approximated to create infinitely many local minima and maxima by a small $C^0$-perturbation. This is impossible, in general, for a $C^1$-perturbation.\n\\end{theorem}\n\\begin{proof}\nWorking in local charts, let $\\sigma(t):=\\max (0,\\min(2t,1))$ for $t\\in\\mathbb{R}$. For $\\varepsilon>0$, define $f_{\\varepsilon}(x)=f(\\sigma(\\|x\\|\/\\varepsilon)x)$. Then $f_{\\varepsilon}$ is constant in the ball of radius $\\varepsilon\/2$, it coincides with $f$ outside the ball of radius $\\varepsilon$, and $\\|f-f_{\\varepsilon}\\|_{\\infty}=o(1)$ for $\\varepsilon\\to 0$.\n\\end{proof}\n\n\\begin{lemma}[Homotopy Invariance]\\label{invariance}\nThe respective homotopy types of a closed Riemannian manifold $M$ and its (free) loop space $\\Lambda M$ are invariant under a $C^0$-perturbation of the energy functional $E:\\Lambda M\\to\\mathbb{R}$ in an open neighborhood of a critical point $\\gamma\\in\\Lambda M$ of index $\\lambda$. For $\\varepsilon>0$, such a local $C^0$-perturbation of $E$ is given by $E_{\\varepsilon}[\\gamma]=E[\\sigma(\\|\\gamma\\|\/\\varepsilon)\\gamma]$ where $\\sigma(t):=\\max(0,\\min(2t,1))$ for $t\\in\\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nThe canonical fibration $\\Omega M\\hookrightarrow\\Lambda M\\overset{\\text{ev}}\\longrightarrow M$ admits the section $M\\hookrightarrow\\Lambda M$ given by the inclusion of constant loops, which implies that the associated long exact sequence of homotopy reduces to the split short exact sequences $0\\to\\pi_k(\\Omega M)\\to\\pi_k(\\Lambda M)\\to\\pi_k(M)\\to 0$. Thus, for $k\\ge 1$ we have canonical isomorphisms\n\\begin{equation*}\n\\pi_k(\\Lambda M)\\cong\\pi_k(\\Omega M)\\rtimes\\pi_k(M)\\cong\\pi_{k+1}(M)\\rtimes\\pi_k(M).\n\\end{equation*} Then for $\\Lambda M=E^{-1}(-\\infty,\\infty)$, define by $\\Lambda M':=E^{-1}_{\\varepsilon}(-\\infty,\\infty)$ the free loop space corresponding to the perturbed $M$, say $M'$, after a $C^0$-perturbation $E_{\\varepsilon}$ of the energy functional. For a $C^0$-perturbation, $E_{\\varepsilon}$ is constant in the ball of radius $\\varepsilon\/2$ and it coincides with $E$ outside the ball of radius $\\varepsilon$. The point $\\gamma\\in\\Lambda M$ has index $\\lambda$. Then let $\\alpha$ be the number of critical points, excluding $\\gamma$, of index $\\lambda$. All of the infinitely many local minima or maxima in the $\\varepsilon$-neighborhood created by the $C^0$-perturbation are also of index $\\lambda$. Therefore, the number of critical points of index $\\lambda$ changes from $1+\\alpha$ to $+\\infty$ since infinitely many critical points of index $\\lambda$ are added in addition to the already existing $(1+\\alpha)$-many. It follows that $\\Lambda M\\cong \\Lambda M'$ under homotopy. Likewise, $\\pi_k(\\Omega M)\\cong \\pi_k(\\Omega M')$.\n\nWe will now use this homotopy equivalence $\\pi_k(\\Lambda M)\\cong\\pi_k(\\Lambda M')$ to show that the homotopy type of $M$ is fixed. From the canonical isomorphisms, $\\pi_k(\\Lambda M)\\cong \\pi_k(\\Omega M)\\rtimes\\pi_k(M)\\cong \\pi_k(\\Lambda M')\\cong \\pi_k(\\Omega M')\\rtimes\\pi_k(M')$. By $\\pi_k(\\Omega M)\\cong \\pi_k(\\Omega M')$, it follows that $\\pi_k(M)\\cong\\pi_k(M')$. Let $f:M\\to M'$, then, because $\\pi_k(M,x_0)\\cong\\pi_k(M',f(x_0))$ for $x_0\\in M$, it follows that $f$ is a homotopy equivalence so $M$ and $M'$ are of the same homotopy type. This is similarly seen through the isomorphism $\\pi_k(\\Omega M)\\cong\\pi_{k-1}(M)$.\n\\end{proof} \n\nLet $Y_k=Y_{k-1}\\bigcup_{\\alpha_k\\in A_k}^{\\theta_k} H^{\\lambda_k}$ be a manifold $Y_{k-1}$ with $|A_k|$-many $\\lambda_k$-handles $H^{\\lambda_k}=D^{\\dim Y_{k-1}-\\lambda_k}\\times D^{\\lambda_k}$ attached along the embedding map $\\theta_k:S^{\\lambda_k-1}\\times D^{\\dim Y_{k-1}-\\gamma_k}\\to\\partial Y_{k-1}$ for $1\\le k\\le N$ and $1\\le \\alpha_k\\le |A_k|$ with $Y_0:=D^{\\infty}$ a Baire space and $Y:=Y_N$. Thus, free loop space has a handle decomposition given by $\\Lambda M:=Y=D^{\\infty}\\bigcup_{k=1}^N\\left[\\bigcup_{\\alpha_k\\in{A_k}}^{\\theta_k}H^{\\lambda_k}\\right]$ where $\\lambda_k$ is the index of $|A_k|$-many critical points of the Morse functional $E:\\Lambda M\\to\\mathbb{R}.$\n\nThere is a deformation retraction of $Y_k=Y_{k-1}\\bigcup_{\\alpha_k\\in A_k}^{\\theta_k} H^{\\lambda_k}$ onto the core $Y_{k-1}\\bigcup_{\\alpha_k\\in A_k}^{\\theta_k}(D^{\\lambda_k}\\times \\{0\\})$ so homotopically attaching a $\\lambda_k$-handle is the same as attaching a $\\lambda_k$-cell. Thus, $Y_k\\approx Y_{k-1}\\bigcup_{\\alpha_k\\in A_k}^{\\theta_k} (D^{\\lambda_k}\\times \\{0\\})\\cong Y_{k-1}\\bigcup_{\\alpha_k\\in A_k}^{\\theta_k}e^{\\lambda_k}$ for $e^{\\lambda_k}$ a $\\lambda_k$-cell. In the handle decomposition, we collapse each handle $D^m\\times D^{n-m}$ to obtain a homotopy equivalent CW complex $X\\approx_G\\Lambda M$ under homotopy $G:X\\to Y$. Thus, let $n_{\\lambda_i}$ denote the number of $\\lambda_i$-handles of $\\Lambda M$. Then $\\Lambda M$ has an infinite-dimensional CW structure with $n_{\\lambda_i}$-many $\\lambda_i$-cells for each $1\\le i\\le N$.\n\nIn particular, we construct the CW complex for $\\Lambda M$ as follows:\n\n\\begin{enumerate}[label=(\\roman*)]\n\\item Set $X_0:=D^{\\infty}$, which is regarded as an $\\infty$-cell $D^\\infty\\times \\{0\\}$ modulo a point $\\{0\\}$.\n\\item Form the $k$-skeleton $X_k$ from $X_{k-1}$ by attaching $\\lambda_k$-cells $e_{\\alpha}^{\\lambda_k}$ via attaching maps $\\phi_{\\alpha}^k:S^{k-1}\\to X_{k-1}$ for $e_{\\alpha}^{\\lambda_k}$. Here, we abbreviate $X_{\\lambda_k}$ by $X_k$.\n\\item We stop the inductive process with $\\Lambda M\\approx X=X_N$ for $N< \\infty$.\n\\end{enumerate} \n\\begin{definition}\nEvery cell $e_{\\alpha}^k$ in the cell complex $X$ has characteristic map $\\Phi_{\\alpha}^k: D_{\\alpha}^k\\to X$ extending the attaching map $\\phi_{\\alpha}^k$. It is a homeomorphism from the interior of $D_{\\alpha}^k$ onto $e_{\\alpha}^k$. Namely, $\\Phi_{\\alpha}^k$ is the composition $D_{\\alpha}^k\\hookrightarrow X_{k-1}\\sqcup_{\\alpha} D_{\\alpha}^k\\to X_k\\hookrightarrow X$ where the middle map is a quotient map defining $X_k$. \n\\end{definition}\nThe CW complex $X$ is infinite-dimensional because the $0$-th skeleton $X^0=D^{\\infty}$ is infinite-dimensional. We topologize $X$ by saying a set $A\\subset X$ is open (resp. closed) if and only if $A\\cap X_k$ is open (resp. closed) in $X_k$ for each $k$. That is, a set $A\\subset X$ is open (resp. closed) if and only if $\\Phi_{\\alpha}^{-1}(A)$ is open (resp. closed) in $D_{\\alpha}^k$ for each characteristic map $\\Phi_{\\alpha}$. The first direction follows from continuity of the characteristic maps $\\Phi_{\\alpha}$. In the other direction, suppose $\\Phi_{\\alpha}^{-1}(A)$ is open in $D_{\\alpha}^k$ for each $\\Phi_{\\alpha}$. Further suppose by induction on $k$ that $A\\cap X_{k-1}$ is open in $X_{k-1}$. Since $\\Phi_{\\alpha}^{-1}(A)$ is open in $D_{\\alpha}^k$ for all $\\alpha$, $A\\cap X_k$ is open in $X_k$ by definition of the quotient topology on $X_k$. Thus $A$ is open in $X$, from which it follows that $X$ is the quotient space of $\\bigsqcup_{k,\\alpha}D_{\\alpha}^k$. The $(k+1)$-th skeleton is obtain from the $k$-skeleton by attaching $(k+1)$-cells; that is, there is a pushout diagram:\n\\begin{equation*}\n\\begin{tikzcd}[column sep=large, row sep=large]\n\\bigsqcup S^k \\arrow{d} \\arrow{r}\n& X_k \\arrow{d} \\\\\n\\bigsqcup D^{k+1} \\arrow{r}\n& X_{k+1}\n\\end{tikzcd}\n\\end{equation*}\nwhere the disjoint unions are over all $(k+1)$-cells of $X$.\n\n\\begin{lemma}\\label{CWcomplex}\nSince $X$ is a CW complex, it follows that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $H_k(X_n,X_{n-1})=\\begin{cases}\n\\mathbb{Z}^{\\#\\lambda_n-\\text{cells}}, & \\text{if } k =n\\\\\n0, & \\text{if } k\\ne n.\n\\end{cases}$ \\\\ That is, $H_k(X_n,X_{n-1})$ is free abelian for $k=n$ with a basis in one-to-one correspondence with the $n$-cells of $X$.\n\\item $H_k(X_n)=0$ if $k>n$. In particular, if $X$ is finite-dimensional, then $H_k(X)=0$ if $k>\\dim (X)$. Note, we exclude the latter condition because $X$ under present consideration is, indeed, infinite-dimensional.\n\\item The map $H_k(X_n)\\to H_k(X)$ induced by the inclusion $i:X_n\\hookrightarrow X$ is an isomorphism for $k<n$ and a surjection for $k=n$.\n\\end{enumerate} \n\\end{lemma}\\label{cellularhomology}\n\\begin{proof}\n(i) Consider the CW pair $(X_n, X_{n-1})$. The skeleton $X_n$ is obtained from $X_{n-1}$ by attaching the $\\lambda_n$-cells $(e_{\\alpha}^{\\lambda_n})_{\\alpha}$. Choose a point $x_{\\alpha}$ at the center of each $\\lambda_n$-cell $e_{\\alpha}^{\\lambda_n}$. Furthermore, let $A=X_n-\\{x_{\\alpha}\\}_{\\alpha}$. Thus, $A$ deformation retracts to $X_{n-1}$ so that $H_k(X_n,X_{n-1})\\cong H_k(X_n,A)$. It follows that $H_k(X_n,A)\\cong \\bigoplus_{\\alpha} H_k(D_{\\alpha}^n, D_{\\alpha}^n-\\{x_{\\alpha}\\})$ by excision of $X_{n-1}$. From the long exact sequence for the pair $(D_{\\alpha}^n,D_{\\alpha}^n-\\{x_{\\alpha}\\})$, we obtain\n\\begin{equation*}\nH_k(D_{\\alpha}^n, D_{\\alpha}^n-\\{x_{\\alpha}\\})\\cong \\tilde{H}_{k-1}(S_{\\alpha}^{n-1})\\cong\\begin{cases}\n\\mathbb{Z}, & \\text{if }k=n\\\\\n0, & \\text{if }k\\ne n.\n\\end{cases}\n\\end{equation*} The first assertion follows at once.\\\\\n(ii) Consider the long exact sequence for the CW pair $(X_n, X_{n-1})$:\n\\begin{equation*}\n\\cdots\\to H_{k+1} (X_n, X_{n-1})\\to H_k(X_{n-1})\\to H_k(X_n)\\to H_k(X_n,X_{n-1})\\to\\cdots\n\\end{equation*} Then if $k+1\\ne n$ and $k \\ne n$, from (i), it follows that $H_{k+1}(X_n,X_{n-1})=0$ and $H_k(X_n,X_{n-1})=0$. Therefore, $H_k(X_{n-1})\\cong H_k(X_n)$. If $k>n$, i.e. for $n\\ne k +1$ and $n\\ne k$, we have that \n\\begin{equation*}\nH_k(X_n)\\cong H_k(X_{n-1})\\cong H_k(X_{n-2})\\cong\\dots\\cong H_k(X_0).\n\\end{equation*} In the case under consideration, $X_0$ is a Baire space $D^{\\infty}$ so $H_k(D^{\\infty})=0$ if $k\\ne 0$. Then for $k>n$, $H_k(X_n)=0$ as was to be shown. \\\\\n(iii) We prove the statement for an infinite-dimensional CW complex $X$. Let $k<n$ and consider the long exact sequence for the CW pair $(X_{n+1},X_n)$ of sub-complexes:\n\\begin{equation*}\n\\cdots\\to H_{k+1}(X_{n+1},X_n)\\to H_k(X_n)\\to H_k(X_{n+1})\\to H_k(X_{n+1}, X_n)\\to\\cdots\n\\end{equation*} Recall that $k<n$ so $k+1\\ne n+1$ and $k\\ne n+1$ or, by (i), $H_{k+1}(X_{n+1},X_n)=0$ and $H_k(X_{n+1},X_n)=0$. Then from exactness of the above sequence, $H_k(X_n)\\cong H_k(X_{n+1})$ and, by iteration, we obtain:\n\\begin{equation*}\nH_k(X_n)\\cong H_k(X_{n+1})\\cong H_k(X_{n+2})\\cong\\dots\\cong H_k(X_{n+l})=H_k(X),\n\\end{equation*} where $l$ is chosen such that $X_{n+l}=X$ which is admissible since the indices of the handles of $X$ are finite by definition and $X:=X_N=X_{N-1}\\cup_{\\theta} H^{\\lambda_N}$ where $\\lambda_i<\\infty$ for $1\\le i\\le N$.\n\\end{proof}\n\n\\begin{definition}\nThe cellular homology $H_*^{CW}(X)$ of the CW complex $X$ under consideration. is defined as the homology of the cellular chain complex $(C_*(X),d_*)$ indexed by the cells of $X$. That is, the $n$-th abelian group in the chain complex is $C_n(X):=H_n(X_n,X_{n-1})=\\mathbb{Z}^{\\#\\lambda_n-\\text{cells}}$ with boundary operators $d_n:C_n(X)\\to C_{n-1}(X)$ defined by the diagram:\n\n\\fontsize{9}{12}\\selectfont{\n\\begin{tikzcd}\n & & & H_n(X_{n+1},X_n) = 0 & \\\\\nH_n(X_{n-1}) = 0 \\ar[rd] & & H_n(X_{n+1})\\cong H_n(X) \\ar[ru] & & \\\\\n & H_n(X_n) \\ar[ru,\"i_n\"] \\ar[rd,\"j_n\"] & & & \\\\\nH_n(X_{n+1},X_n) = 0 \\ar[ru,\"\\partial_{n+1}\"] \\ar[rr,\"d_{n+1}\"] & & H_n(X_n,X_{n-1}) \\ar[dr,\"\\partial_n\"'] \\ar[rr,\"d_n\"] & & H_{n-1}(X_{n-1},X_{n-2}) \\\\\n & & & H_{n-1}(X_{n-1}) \\ar[ru,\"j_{n-1}\"'] & \\\\\n & & H_{n-1}(X_{n-2}) = 0 \\ar[ru] & & \\\\\n\\end{tikzcd}}\n\\end{definition}\n\nThe diagonal arrows are due to the long exact sequences of relative homology, and we invoke Lemma \\ref{cellularhomology} to conclude $H_n(X_{n-1})=0$, $H_{n-1}(X_{n-2})=0$ and $H_n(X_{n+1})\\cong H_n(X)$ in the above diagram. Then we define \n\\begin{equation*} \nd_n=j_{n-1}\\circ\\partial_n:C_n(X)\\to C_{n-1}(X)\n\\end{equation*} and observe that $d_n\\circ d_{n+1}=0$. Moreover, $d_n\\circ d_{n+1}=j_{n-1}\\circ \\partial_n \\circ j_n\\circ \\partial_{n+1}=0$ since the composition of two consecutive maps in a long exact sequence is the zero map $\\partial_n\\circ j_n=0$.\n\nWe henceforth use the notations of Section \\ref{sec:rathomotopy}. For two real numbers $a,b$ where $b>a$, define the sets $\\Lambda^{[a,b]}$ by $\\Lambda^{[a,b]}=\\{p\\in\\Lambda M: a\\le E(p)\\le b\\}$. Recall, if $\\Lambda^{[a,b]}$ is compact and contains no critical point of $E:\\Lambda M\\to\\mathbb{R}$ then $\\Lambda^a$ is diffeomorphic to $\\Lambda^b$. On the other hand, suppose that $\\Lambda^{[a,b]}$ is compact and contains exactly one non-degenerate critical point of $E$. If the index of the critical point is $\\lambda$ then $\\Lambda^b$ is obtained from $\\Lambda^a$ by successively attaching a handle of index $\\lambda$ and a collar.\n\\begin{proposition}\nEvery smooth Hilbert manifold admits a Morse function $E:\\Lambda M\\to\\mathbb{R}$ such that $\\Lambda^a$ is compact for every $a\\in\\mathbb{R}$.\n\\end{proposition}\n\\begin{theorem}\\label{handledecomposition}\nLet $E:\\Lambda M\\to\\mathbb{R}$ be a Morse function on a paracompact manifold $\\Lambda M$ such that each $\\Lambda^a$ is paracompact. Let $p_1,p_2,\\dots,p_k,\\dots$ be the critical points of $E$ and let the index of $p_i$ be $\\lambda_i$. Then $\\Lambda M$ has a handle decomposition which admits a handle of index $\\lambda_i$ for each $i$. \n\\end{theorem}\n\\begin{proof}\nConsider the Morse function $E:\\Lambda M\\to\\mathbb{R}$. Let $c_1<c_2<\\dots<c_k<\\dots$ be the $m$ critical values of $E$. Then consider the sequence of real numbers $\\{a_i\\}$ for which $c_k<a_k<c_{k+1}$. For real $a,b$ with $b>a$, we let $\\Lambda^{[a,b]}=\\{p\\in\\Lambda M:a\\le E[p]\\le b\\}$. Consequently, $\\Lambda^{a_{k+1}}=\\Lambda^{a_k}\\cup\\Lambda^{[a_k,a_{k+1}]}$ and $\\Lambda M=\\bigcup_{k=0}^{m-1}\\Lambda^{[a_k,a_{k+1}]}$. The set $\\Lambda^{[a_k,a_{k+1}]}$ is paracompact so it only contains finitely many critical points of $E$. If $\\lambda_{k+1}$ is the index of $c_{k+1}$, it has the homotopy type of a handle of index $\\lambda_{k+1}$.\n\\end{proof}\n\\begin{corollary}\nEvery smooth manifold has the homotopy type of a CW complex.\n\\end{corollary}\n\\begin{proof}\nThe set $\\Lambda^b$ is homotopy equivalent to $\\Lambda^a$ with a $1$-cell attached.\n\\end{proof}\n\n\\begin{definition}\\label{Smithnormal}\nLet $A\\in\\text{Mat}_{m,n}(\\mathbb{Z})$ be a matrix over the principal ideal domain $\\mathbb{Z}$. Then the Smith normal form is a factorization $A=UDV$ for which:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $D\\in\\text{Mat}_{m,n}(\\mathbb{Z})$ is diagonal where $d_{i,j}=0$ if $i\\ne j$.\n\\item Every diagonal entry of $D$ divides the next, i.e. $d_{i,i}|d_{i+1,i+1}$. Such diagonal entries are called the elementary divisors of $A$.\n\\item $U\\in\\text{Mat}_{m,m}(\\mathbb{Z})$ and $V\\in\\text{Mat}_{n,n}(\\mathbb{Z})$ are invertible over $\\mathbb{Z}$. That is, $\\det U,\\det V=\\pm 1$ for $U,V$ in $SL$(---,$\\mathbb{Z})$.\n\\end{enumerate} \n\\end{definition}\nSimilarly, for $A\\in\\text{Mat}_{m,n}(\\mathbb{Z})$ an integer matrix, the Smith normal form $A=UDV$ may be written as $D=U^{-1}AV^{-1}$, which is permissible by the invertibility of $U$ and $V$. Here, the product $U^{-1}AV^{-1}$ is\n\\begin{equation}\nSNF(A)=\\begin{bmatrix}\\alpha_1 & 0 & 0 & & \\cdots & & 0\\\\0& \\alpha_2 & 0 & & \\cdots & & 0\\\\ 0 & 0 & \\ddots & & & & 0\\\\ \\vdots& & & \\alpha_r & & & \\vdots \\\\ & & & & 0 & & & \\\\ & & & & & \\ddots & \\\\ 0 & & & \\cdots & & & 0 \\end{bmatrix}\n\\end{equation} where the diagonal elements $\\alpha_i$, unique up to multiplication by a unit of the principal ideal domain, satisfy $\\alpha_i|\\alpha_{i+1}$ for all $1\\le i <r$. The elements $\\alpha_i$ are the elementary divisors of $A$, determined by\n\\begin{equation}\n\\alpha_i=\\frac{d_i(A)}{d_{i-1}(A)}\n\\end{equation} where $d_i(A)$, known as the $i$-\\textit{th determinant divisor}, is the greatest common divisor of all $i\\times i$ matrix of minors of $A$, and $d_0(A):=1$.\n\\begin{remark}\nThe Smith normal form exists for matrices over arbitrary principal ideal domains $R$ whereby $U$ and $V$ must be invertible over $R$ so that their determinants are any units in $R$. The Smith form is unique up to multiplication by units in $R$.\n\\end{remark}\n\\begin{lemma}[Omar Camarena]\\label{cokernel}\nLet the non-zero elementary divisors of an $m\\times n$ integer matrix $A$ be $\\alpha_1,\\dots,\\alpha_r$ where $r:=\\text{rank }(A)$. Then $\\text{coker}(A):=\\mathbb{Z}^m\/\\text{im}(A)\\cong\\bigoplus_{i=1}^r\\mathbb{Z}\/a_i\\oplus\\mathbb{Z}^{m-r}$.\n\\end{lemma}\n\\begin{proof}\nConsider the Smith normal form $UDV$ of $A$. Then $\\text{coker}(A)\\cong\\text{coker}(D)$ under the map $x+\\text{im}(A)\\mapsto U^{-1}x+\\text{im}(D)$. To compute $\\text{coker}(D)$, if $e_1,\\dots,e_m$ is the standard basis of $\\mathbb{Z}^m$, then the image of $D$ is spanned by $a_1e_1,\\dots,a_re_r$.\n\\end{proof}\n\\begin{theorem}[Omar Camarena]\\label{Smithhomology}\nThe homology of a complex $\\mathbb{Z}^n\\overset{A}\\longrightarrow\\mathbb{Z}^m\\overset{B}\\longrightarrow\\mathbb{Z}^k$ (with $BA=0$) is $\\ker(B)\/\\text{im}(A)\\cong\\bigoplus_{i=1}^r\\mathbb{Z}\/{\\alpha}_i\\oplus\\mathbb{Z}^{m-r-s}$ where $r:=\\text{rank(B)}$ and $\\alpha_1,\\dots,\\alpha_r$ are the non-zero elementary divisors of $A$.\n\\end{theorem}\n\\begin{proof}\nThe boundary map $B$ is zero on $im(A)$ so it descends to a homomorphism $\\bar{B}:\\mathbb{Z}^m\/\\text{im}(A)\\to\\mathbb{Z}^k$. We will now show that $\\ker(B)\/\\text{im}(A)$ is isomorphic to $\\ker(\\bar{B})$. From Lemma \\ref{cokernel}, $\\mathbb{Z}^m\/\\text{im}(A)\\cong\\bigoplus_{i=1}^r\\mathbb{Z}\/\\alpha_i\\oplus\\mathbb{Z}^{m-r}$. The target $\\mathbb{Z}^k$ of $B$ is free so that the kernel contains all of the torsion of its domain group. It follows that\n\\begin{equation*}\n\\ker(\\bar{B})=\\bigoplus_{i=1}^r\\mathbb{Z}\/\\alpha_i\\oplus\\ker\\left(\\bar{B}|_{\\mathbb{Z}^{m-r}}\\right).\n\\end{equation*} The kernel of the restriction of $\\bar{B}$ to its free part is also free and, thus, the rank is \n\\begin{equation}\n\\text{rank }\\ker(\\bar{B})=m-r-s\n\\end{equation} for $s$ the rank of $\\text{im}(B)=\\text{im}(\\bar{B})=\\text{im}\\left(\\bar{B}|_{\\mathbb{Z}^{m-r}}\\right)$.\n\\end{proof}\n\n\\begin{theorem} [Main Result]\\label{main result}\nEvery closed Riemannian manifold $M$ admits infinitely many geometrically distinct, non-constant, prime closed geodesics.\n\\end{theorem}\n\\begin{proof}\nLet $X\\approx_{G} \\Lambda M$ be the homotopy equivalence defined in the construction preceding Lemma \\ref{CWcomplex}. Consider a section of the cellular chain complex for the CW pair $(X_k,X_{k-1})$:\n\\begin{equation}\\label{eq:complex}\n\\cdots\\longrightarrow H_{k+1}(X_{k+1}, X_k)\\overset{M_{k+1}}\\longrightarrow H_k(X_k, X_{k-1})\\overset{M_k}\\longrightarrow H_{k-1}(X_{k-1},X_{k-2})\\longrightarrow\\cdots \\cdots\n\\end{equation}\nor equivalently, by the definition of $C_k(X):=H_k(X_k,X_{k-1})=\\mathbb{Z}^{\\#\\lambda_k-\\text{cells}}$,\n\\begin{equation*}\n\\cdots\\longrightarrow\\mathbb{Z}^{\\#\\lambda_{k+1}-\\text{cells}}\\overset{M_{k+1}}\\longrightarrow \\mathbb{Z}^{\\#\\lambda_k-\\text{cells}}\\overset{M_k}\\longrightarrow \\mathbb{Z}^{\\#\\lambda_{k-1}-\\text{cells}}\\longrightarrow \\cdots\n\\end{equation*} where\n\\begin{equation}\nM_k:  H_k(X_k, X_{k-1})\\to H_{k-1}(X_{k-1}, X_{k-2})\n\\end{equation}\nis a homomorphism from one free abelian group, indexed by $\\lambda_k$-cells attached to go from $X_{k-1}$ to $X_k$, to another. Any such homomorphism is given by an integer-valued matrix $M_k$. In what follows, we determine the $(i,j)$-entries of the matrix\n\\begin{equation*}\nM_k:  H_k(X_k, X_{k-1})\\to H_{k-1}(X_{k-1}, X_{k-2})\n\\end{equation*} which is given by a particular CW presentation of the CW complex $X$. For each entry of this matrix, take the $i$-th $\\lambda_k$-cell represented by a continuous map $f_i: S^{k-1}\\to X_{k-1}$, and take the $j$-th $\\lambda_{k-1}$-cell represented by a map $g_j: (D^{k-1}, S^{k-2})\\to (X_{k-1}, X_{k-2})$. Then choose any point $y$ in the interior of $D^{k-1}$, seen as a point $g_j(y)$ in $X_{k-1}$, and look at the points $x_1,\\dots, x_r$ in the inverse image $f^{-1}_i(y)$. Then we observe whether the restriction of $f_i$ to a small neighborhood of $x_t$ (for $1\\le t\\le r$), mapping to a small neighborhood of $y$, preserves or reverses the orientation. We put $+1$ if it preserves orientation and $-1$ if it reverses orientation; the sum of these signs is the corresponding entry of the matrix.\n\nMore precisely, we should take an \\textit{interior} point $y$ in $D^{k-1}$; a small open neighborhood $V$ of $y$ maps homeomorphically onto its image in $X_{k-1}$ under the map $g_j: (D^{k-1}, S^{k-2})\\to (X_{k-1}, X_{k-2})$  determined by the particular attaching map $S^{k-2}\\to X_{k-2}$. This is a homeomorphism so the restriction of the map $f_i$, which takes $U = f_i^{-1}(g_j(V))$, an open neighborhood of $S^{k-1}$, to $g_j(V)$, is a map $U\\to V$ between open subsets of manifolds. This applies in the case of a general CW complex, which might not itself be a manifold. For any continuous map $f: U\\to V$ between manifolds, there is a \\textit{continuously differentiable} map within the same homotopy class. Thus, we exclude the possibility that the attaching maps are pathological in nature.\n\nAssuming $f_i$ is continuously differentiable, almost all points $y$ in $D^{k-1}$ are regular values of $f_i$ in the technical sense of measure theory. This follows by Sard's theorem which says that the set of critical values has measure zero. For a point $y$ to be \\textit{regular} means that for each point $x$ in $f^{-1}_i(y)$, working in chosen coordinate patches, the derivative $Df_i$ at $x$, represented by a Jacobian matrix whose entries are partial derivatives at $x$ as a linear map from the tangent space at $x$ to the tangent space at $y$, has maximal rank. In the case where $U$ and $V$ have the same dimension, maximal rank means that $Df_i$ is invertible at $x$. Then, by the implicit function theorem, in the case where $y$ is regular, this inverse image $f^{-1}_i(y)$ in $S^{k-1}$ will be a compact $0$-dimensional manifold, and will hence consist of a finite number of points $x_1, \\dots, x_r.$ \n\nThe manifolds $S^{k-1}$ and $D^{k-1}$ are orientable. We may select standard Euclidean coordinates $(y_1, \\dots, y_{k-1})$ for $y$ in $D^{k-1}$, which are the same coordinates for the open neighborhood $V$. As for each point $x$ in the inverse image of $y$, we can take any coordinate patch consistent with the standard orientation of $S^{k-1}$. Then, computing the derivative $Df_i$ at $x$ in terms of the chosen coordinatizations at $x$ and $y$ as a Jacobian matrix $\\text{Jac}_x(f_i)$, the sign of the determinant of $\\text{Jac}_x(f_i)$ is well-determined. The sign is either $+1$ or $-1$ since $Df_i$ is invertible. The \\textit{degree} of $f_i$ is then computed as \n\\begin{equation*}\n\\sum_{x \\in f^{-1}_i(y)}  \\text{sgn } \\det \\text{Jac}_x(f_i).\n\\end{equation*}\nIt is well-defined and independent of which regular value $y$ we pick. It is also invariant with respect to the homotopy class of the attaching map.\n\nTo write down the $(i,j)$-entry of the matrix $M_k$, we order the attaching maps that we used to get from $X_{k-1}$ to $X_k$:\n\\begin{equation*}\nf_1: S^{k-1} \\to X_{k-1},  f_2: S^{k-1}\\to X_{k-1},\\dots, f_{m_k}: S^{k-1} \\to X_{k-1}\n\\end{equation*} where $m_k:=n_{\\lambda_{k-1}}$ is the number of $\\lambda_{k-1}$-cells of $X$. We assume there are only finitely many attaching maps for each skeletal dimension, although we can also drop the finiteness assumption. Likewise, we order the attaching maps one dimension down:\n\\begin{equation*}\ng_1: S^{k-2}\\to X_{k-2}, g_2: S^{k-2}\\to X_{k-2},\\dots, g_{n_k}: S^{k-2}\\to X_{k-2}\n\\end{equation*} where $n_k:=n_{\\lambda_k}$ is the number of $\\lambda_k$-cells of $X$.\nWe think of the latter attaching maps in terms of giving maps of \\textit{pairs} of spaces  $(D^{k-1}, S^{k-2})\\to (X_{k-1}, X_{k-2})$, where the first map component  $D^{k-1}\\to X_{k-1}$  identifies interior points of $D^{k-1}$ with points of $X_{k-1}$, and the second component is the actual attaching map $g_j: S^{k-2}\\to X_{k-2}$. Then the matrix entry for $M_k\\in\\text{Mat}_{m_k,n_k}(\\mathbb{Z})$ is \n\\begin{equation*}\n(M_k)_{ij}:=\\deg(f_i)_j=\\sum_{f_i(x) = y}  \\text{sgn } \\det \\text{Jac}_x(f_i)\n\\end{equation*} for any choice of regular value $y$ of $f_i$ in the interior of the $j$-th cell attachment  $(D^{k-1}, S^{k-2})\\to (X_{k-1}, X_{k-2})$ and for any choice of coordinatizations of small coordinate patches for the points $y$ and $x$ in the inverse image $f_i^{-1}(y)$ chosen compatibly with the orientations.  For $V$ an open neighborhood of $y$ in $D^{k-1}$, $f_i$ is a map between open neighborhoods of manifolds:\n\\begin{equation*}\n\\begin{split}\nf_i:&S^{k-1}\\longrightarrow X_{k-1},\\\\\n& U=f_i^{-1}(g_j(V))\\mapsto g_j(V)\n\\end{split}\n\\end{equation*} from which we may write the $(i,j)$-entry of $M_k$ as\n\\begin{equation}\n(M_k)_{ij}:=\\sum_{x\\in f_i^{-1}(g_j(V))}\\text{sgn }\\det \\text{Jac}_x(f_i).\n\\end{equation} Thus, \n\\begin{equation}\nM_k=\\begin{bmatrix} \\deg(f_1)_1 & \\deg(f_1)_2 & \\dots & \\deg(f_1)_{n_{\\lambda_k}}\\\\ \\deg(f_2)_1& \\deg(f_2)_2 & \\dots & \\deg(f_2)_{n_{\\lambda_k}}\\\\ \\vdots &\\vdots & \\ddots & \\vdots \\\\ \\deg\\left(f_{n_{\\lambda_{k-1}}}\\right)_1 & \\deg\\left(f_{n_{\\lambda_{k-1}}}\\right)_2 & \\dots & \\deg\\left(f_{n_{\\lambda_{k-1}}}\\right)_{n_{\\lambda_k}}\\end{bmatrix}.\n\\end{equation}\nThe homology of the cellular complex does not depend on the order chosen for the attaching maps. Since we have a cellular chain complex, the composite of any two matrices $M_{k+1}$ and $M_k$ is 0, and one wants to compute $\\ker M_k \/ \\text{im }M_{k+1}$  to get the $k$-th homology group of the CW complex $\\Lambda M\\approx_G X$.\n\nWe henceforth compute cellular homology for $X$ using the standard basis of $\\mathbb{Z}^k$ for the free modules $C_k(X)$. To do so, we invoke Smith normal form of the matrix representation for the boundary homomorphisms. It should be remarked that if we want to determine homology with a basis over a field $K$, we would simply use Gaussian elimination. Note, $\\mathbb{Z}$-modules are direct sums $\\mathbb{Z}\/\\alpha_1\\oplus\\mathbb{Z}\/\\alpha_2\\oplus\\cdots\\oplus\\mathbb{Z}\/\\alpha_{l}$ for integers $\\alpha_i$ with $\\alpha_i | \\alpha_{i+1}$. We reduce an integer matrix $M_k$ to $SNF(M_k)$ using row and column operations as in standard Gaussian elimination; but, the caveat is that the entries must remain integers at all stages. Recall $M_k$ is $m_k\\times n_k$ where $m_k=n_{\\lambda_{k-1}}$ and $n_k=n_{\\lambda_k}$. The row and column operations correspond to a change of basis for $M_k$ and $M_{k+1}$, respectively. Thus, the Smith normal form gives a factorization $M_k=U_kD_kV_k$ with\n\\begin{equation}\n\\begin{split}\nD_k=\\begin{bmatrix} B_k & \\bm{0} \\\\ \\bm{0} &\\bm{0} \\end{bmatrix}, \\text{ where }\nB_k=\\begin{bmatrix} \n\\alpha_1 & & & & 0 \\\\ \n & & \\ddots & &  \\\\\n & & & & \\\\\n 0 & & & & \\alpha_{l_k}\n\\end{bmatrix}\n\\end{split}\n\\end{equation} for $B_k$ a matrix with $l_k$ non-zero diagonal entries which satisfy $\\alpha_i\\ge 1$. The SNF matrices for $M_{k+1}$ and $M_k$ completely characterize the $k$-th homology group $H_k$. The rank of the boundary group $\\text{im }M_{k+1}$ is the number of non-zero rows of $D_{k+1}$, i.e. $l_{k+1}=\\text{rank }\\text{im }M_{k+1}=\\min(n_{\\lambda_k},n_{\\lambda_{k+1}})$. The rank of the cycle group $\\ker M_k$ is the number of zero columns of $D_k$, i.e. $n_k-l_k$. The torsion coefficients of $H_k$ are the diagonal entries $\\alpha_i$ of $M_{k+1}$ that are greater than one. The $k$-th Betti number of the CW complex $\\Lambda M$ is thus:\n\\begin{equation}\nb_k(\\Lambda M;\\mathbb{Z})=\\text{rank }\\ker M_k-\\text{rank }\\text{im } M_{k+1}=n_k-l_k-l_{k+1}.\n\\end{equation}\n\nSimilarly, the elementary divisors are given by $\\alpha_i=d_i(M_k)\/d_{i-1}(M_k)$ where $d_i(M_k)$ is the $i$-determinant divisor for $1\\le i \\le l_k$. In particular, $d_i(M_k)$ is the greatest common divisor of all $i\\times i$ minors of $M_k$. Denote by $\\Sigma_{i,k}$ the $i\\times i$ sub-matrix of $M_k$. Then an $i\\times i$ minor of $M_k$, i.e. the minor determinant of order $i$, is the determinant of an $i\\times i$ matrix obtained from $M_k$ by deleting $m_k-i$ rows and $n_k-i$ columns. The minor of order zero is defined to be $+1$. For the matrix $M_k$, there are ${m_k\\choose i}{n_k\\choose i}$ minors of size $i\\times i$. One such minor $|\\Sigma_{i,k}|$ is the determinant of the $i$-th sub-matrix $\\Sigma_{i,k}$:\n\\begin{equation} |\\Sigma_{i,k}|=\n\\begin{vmatrix}\n(M_k)_{\\sigma,\\rho} & \\dots & & (M_k)_{\\sigma,\\rho+i} & \\\\ \n\\hspace*{4mm} (M_k)_{\\sigma+1,\\rho} & \\dots & & \\hspace*{3.5mm}(M_k)_{\\sigma+1,\\rho+i} & \\\\\n\\vdots & \\ddots & & \\vdots \\\\\n\\hspace*{2mm}(M_k)_{\\sigma+i,\\rho} & \\dots & & \\hspace*{3.4mm} (M_k)_{\\sigma+i,\\rho+i} &\n\\end{vmatrix}.\n\\end{equation} for $1\\le\\sigma\\le m_k$ and $1\\le\\rho\\le n_k$. There are $i!$ orderings of the columns $(\\rho,\\rho+1,\\dots,\\rho+i)$ that go in each possible order $(\\alpha,\\beta,\\dots,\\omega)$. We choose $(M_k)_{\\sigma,\\alpha}$ from row $1$ of $\\Sigma_{i,k}$ (row $\\sigma$ of $M_k$), $(M_k)_{\\sigma+1,\\beta}$ from row $2$ of $\\Sigma_{i,k}$ (row $\\sigma+1$ of $M_k$), and eventually $(M_k)_{\\sigma+i,\\omega}$ from row $i$ of $\\Sigma_{i,k}$ (row $\\sigma+i$ of $M_k$). Then the determinant contains the product $(M_k)_{\\rho,\\alpha}(M_k)_{\\rho+1,\\beta}\\dots (M_k)_{\\rho+i,\\omega}$ times $+1$ or $-1$. Let $P=(\\alpha,\\beta,\\dots,\\omega)$ be the permutation matrices corresponding to a certain ordering of the columns of $\\Sigma_{i,k}$. Then the determinant of $\\Sigma_{i,k}$ is the sum of the $i!$ simple determinants (up to sign), which are given by choosing one entry from every row and column of $\\Sigma_{i,k}$. Therefore,\n\\begin{equation}\n|\\Sigma_{i,k}|=\\sum_{P=(\\alpha,\\beta,\\dots,\\omega)}(\\det P)(M_k)_{\\sigma,\\alpha}(M_k)_{\\sigma+1,\\beta}\\dots (M_k)_{\\sigma+i,\\omega}\n\\end{equation} for $\\rho\\le \\alpha,\\beta,\\dots,\\omega\\le \\rho+i$. Then the $i$-th determinant divisor of $M_k$ is the greatest common divisor of all $i\\times i$ minors $|\\Sigma_{i,k}|$. Let $|\\Sigma_{i,k}^r|$ be the $r$-th possible $i\\times i$ minor of order $i$ of $M_k$ for $1\\le r\\le {m_k\\choose i}{n_k\\choose i}$. Then the $i$-th determinant divisor is\n\\begin{equation}\nd_i(M_k)=\\text{gcd}\\left(\\big |\\Sigma_{i,k}^1\\big |,\\dots,\\biggl |\\Sigma_{i,k}^{{m_k\\choose i} {n_k\\choose i}}\\biggr|\\right).\n\\end{equation} Recall that for $a,b\\in\\mathbb{Z}$, $\\text{gcd}(a,b)=\\sum_{\\substack{q|a, \\\\ q|b}}\\varphi(q)$ where $\\varphi(\\cdot)$ is the Euler totient function. It follows that if we let $\\Phi_j:=\\text{gcd}(\\Phi_{j-1},|\\Sigma_{i,k}^j|)$ for $2\\le j\\le {m_k\\choose i}{n_k\\choose i}$ and $\\Phi_1:=|\\Sigma_{i,k}^1|$, then\n\\begin{equation}\n\\Phi_j=\\sum_{\\substack{q_{j-1}|\\Phi_{j-1} \\\\ q_{j-1}||\\Sigma_{i,k}^j|}}\\varphi(q_{j-1})=\\sum_{\\substack{q_{j-1}|\\Phi_{j-1} \\\\ q_{j-1}||\\Sigma_{i,k}^j|}}q_{j-1}\\prod_{p|q_{j-1}}\\left(1-\\frac{1}{p}\\right)\n\\end{equation} for $p$ prime and $q_j|\\Phi_j$ for all $j$. Moreover, we let $\\tau(i,k):={m_k\\choose i} {n_k\\choose i}$ to obtain $d_i(M_k)=\\Phi_{\\tau(i,k)}$:\n\\begin{equation}\n\\begin{split}\nd_i(M_k)&=\\text{gcd}\\left(|\\Sigma_{i,k}^1|,\\dots,|\\Sigma_{i,k}^{\\tau(i,k)}|\\right)\\\\&=\\sum_{\\substack{q_{\\tau(i,k)-1}|\\Phi_{\\tau(i,k)-1} \\\\ q_{\\tau(i,k)-1} \\big | \\left|\\Sigma_{i,k}^{\\tau(i,k)}\\right |}}q_{\\tau(i,k)-1}\\prod_{p|q_{\\tau(i,k)-1}}\\left(1-\\frac{1}{p}\\right).\n\\end{split}\n\\end{equation}\n\nRecall that $X$ is the CW complex homotopy equivalent to $\\Lambda M$. Applying Theorem \\ref{Smithhomology} to the complex\n\\begin{equation}\n\\mathbb{Z}^{n_{\\lambda_{k+1}}}\\overset{M_{k+1}}\\longrightarrow\\mathbb{Z}^{n_{\\lambda_k}}\\overset{M_k}\\longrightarrow\\mathbb{Z}^{n_{\\lambda_{k-1}}}\n\\end{equation} defined in equation (\\ref{eq:complex}), we conclude that the $k$-th homology group of free loop space $\\Lambda M$ is \n\\begin{equation}\n\\begin{split}\nH_k(\\Lambda M;\\mathbb{Z})&=\\ker M_k\/\\text{im }M_{k+1}=\\bigoplus_{i=1}^{l_{k+1}}\\mathbb{Z}\/\\alpha_i\\oplus\\mathbb{Z}^{n_k-l_{k+1}-l_k}\\\\ &=\\left. \\bigoplus_{i=1}^{l_{k+1}}\\mathbb{Z} \\right\/ \\left(\\frac{d_i(M_k)}{d_{i-1}(M_k)}\\right)\\oplus\\mathbb{Z}^{n_k-l_{k+1}-l_k} \\\\ \n&=\\left. \\bigoplus_{i=1}^{l_{k+1}}\\mathbb{Z} \\right\/ \\left(\\frac{\\text{gcd}\\biggl(\\big |\\Sigma_{i,k}^1\\big |,\\dots,\\biggl |\\Sigma_{i,k}^{{m_k\\choose i} {n_k\\choose i}}\\biggr|\\biggr)}{\\text{gcd}\\biggl(\\big |\\Sigma_{i-1,k}^1\\big |,\\dots,\\biggl |\\Sigma_{i-1,k}^{{m_k\\choose i-1} {n_k\\choose i-1}}\\biggr|\\biggr)}\\right)\\oplus\\mathbb{Z}^{n_k-l_{k+1}-l_k}\n\\end{split}\n\\end{equation} where $l_{k+1}=\\text{rank }\\text{im }M_{k+1}=\\min(n_{\\lambda_k},n_{\\lambda_{k+1}})$. Therefore, the $k$-th Betti number is\n\\begin{equation}\nb_k(\\Lambda M;\\mathbb{Z})=n_k-l_k-l_{k+1}=n_{\\lambda_k}-\\min\\left(n_{\\lambda_{k-1}},n_{\\lambda_k}\\right)-\\min\\left(n_{\\lambda_k},n_{\\lambda_{k+1}}\\right). \n\\end{equation}\n\nWe use a $C^0$-perturbation of $E:\\Lambda M\\to\\mathbb{R}$. That is, let $\\sigma(t):=\\max(0,\\min(2t-1,1))$ for $t\\in\\mathbb{R}$. Then for $\\varepsilon>0$, define \n\\begin{equation}\nE_{\\varepsilon}(x):=E(\\sigma(\\|x\\|\/\\varepsilon)x)=\\int_{S^1}\\|\\dot{\\sigma}(\\|\\gamma\\|\/\\varepsilon)\\gamma(t)+\\sigma(\\|\\gamma\\|\/\\varepsilon)\\dot{\\gamma}(t)\\|^2dt.\n\\end{equation}\nTherefore, $E_{\\varepsilon}$ is constant in the ball of radius $\\varepsilon\/2$, it coincides with $E$ outside the ball of radius $\\varepsilon$, and $\\|E-E_{\\varepsilon}\\|_{\\infty}=o(1)$ for $\\varepsilon\\to 0$. The homotopy types of $\\Lambda M$ and $M$ are fixed under such a perturbation by Lemma \\ref{invariance}, i.e. for $f:M\\to M'$ and $x_0\\in M$, $\\pi_k(M,x_0)\\cong\\pi_k(M',f(x_0))$ for all $k$. Assume that we perturb $\\Lambda M$ via $E_{\\varepsilon}$ such that we create infinitely many minima or maxima in a local neighborhood of a critical point $\\gamma\\in X_k\\subset \\Lambda M$ of index $\\lambda_k$. Then, by Lemma \\ref{invariance}, the homotopy type of $M$ is fixed and the number of critical points of index $\\lambda_k$ tends to infinity, i.e. $n_{\\lambda_k}\\to\\infty$. Then the $k$-th Betti number of $H_k(\\Lambda M;\\mathbb{Z})$ becomes \n\\begin{equation*}\n\\lim_{n_{\\lambda_k}\\to\\infty}n_{\\lambda_k}-\\min\\left(n_{\\lambda_{k-1}},n_{\\lambda_k}\\right)-\\min\\left(n_{\\lambda_k},n_{\\lambda_{k+1}}\\right)=+\\infty\n\\end{equation*} where it is assumed that $n_{\\lambda_i}$, $1\\le i\\le N$, is finite for all $i\\ne k$. It follows that the sequence of Betti numbers $\\{b_k(\\Lambda M;\\mathbb{Z})\\}_{k\\ge 0}$ is unbounded. Since the homotopy type of $M$ is invariant under the $C^0$-perturbation of the energy functional, this can be done for an arbitrary closed Riemannian manifold. Thus, every closed Riemannian manifold $M$ carries infinitely many geometrically distinct, non-constant, prime closed geodesics. \n\nIf we instead compute cellular homology over a field $K$, say $\\mathbb{Q}$, then we may simply use Gaussian elimination to reach the factorization $M_k=U_kD_kV_k$ for $U_k\\in\\text{Mat}_{m_k,m_k}(\\mathbb{Q})$ and $V_k\\in\\text{Mat}_{n_k,n_k}(\\mathbb{Q})$ where $U_kD_k$ is the inverse of the Gaussian elimination matrix $E_k$ which transforms $M_k$ to $V_k$. Note, each diagonal entry of $U_k$ is $+1$ since $D_k$ is the diagonal matrix that contains all the pivots of $M_k$. Then the rank of $\\text{im }M_{k+1}$ is the number of non-zero rows of $D_{k+1}$, i.e. $l_{k+1}$, and the rank of $\\ker M_k$ is the number of zero columns of $D_k$, i.e. $n_k-l_k$. Then the $k$-th homology group of $\\Lambda M$ is\n\\begin{equation}\nH_k(\\Lambda M;\\mathbb{Q})=\\ker M_k\/\\text{im }M_{k+1}=\\mathbb{Q}^{n_k-l_{k+1}-l_k}\n\\end{equation} and the $k$-th Betti number is given by $b_k(\\Lambda M;\\mathbb{Q})=n_k-l_k-l_{k+1}$. That is, all torsion for homology vanishes.\n\\end{proof} \n\\begin{remark}\nFor a finite CW complex $X$, the Euler characteristic $\\chi(X)$ is defined to be the alternating sum $\\sum_k(-1)^kC_k$ where $C_k$ is the number of $k$-cells of $X$. The Euler characteristic of $X$ is independent of the choice of CW structure on $X$. We similarly define the Euler characteristic of  an infinite-dimensional CW complex $\\Lambda M$ to be the alternating sum $\\sum_k(-1)^kC_k$ where $C_k$ is the number of $k$-cells of $\\Lambda M$ if and only if all homology groups have finite rank and all but finitely many are zero. In this case, $\\Lambda M\\approx_GX_N$ is such that $H_k(X_N)=0$ if $k>N$ by Lemma \\ref{CWcomplex} so that all but finitely many homology groups are zero because the index of any critical point is finite. \n\\end{remark}\n\\begin{definition}\nThe Poincar\\'e polynomial of $\\Lambda M$ is given by \n\\begin{equation*}\nP_{\\Lambda M}(t)=\\sum_{k=0}^{\\infty}(\\text{rank }H_k(\\Lambda M;\\mathbb{Z}))t^k.\n\\end{equation*} If the Euler characteristic is well-defined, i.e. if $\\text{rank }H_k(\\Lambda M;\\mathbb{Z})=0$ for $k\\ge N$ and a fixed $N$, then $\\chi(\\Lambda M)=P_{\\Lambda M}(-1)$.\n\\end{definition}\nSince $H_k(\\Lambda M;\\mathbb{Z})\\cong H_k(X_N;\\mathbb{Z})=0$ for $k>N$, we have:\n\\begin{equation} \n\\begin{split}\n\\chi(\\Lambda M)& =\\sum_{k=0}^{\\infty}(-1)^k\\Big(n_{\\lambda_k}-\\min\\left(n_{\\lambda_{k-1}},n_{\\lambda_k}\\right)-\\min\\left(n_{\\lambda_k},n_{\\lambda_{k+1}}\\right)\\Big) \\\\ \n&=\\sum_{k=0}^N(-1)^k\\Big(n_{\\lambda_k}-\\min\\left(n_{\\lambda_{k-1}},n_{\\lambda_k}\\right)-\\min\\left(n_{\\lambda_k},n_{\\lambda_{k+1}}\\right)\\Big).\n\\end{split}\n\\end{equation} Homology $H_k$ is sub-additive due to the increasing sequence of inclusions $D^{\\infty}=X_0\\subset X_1\\subset X_1\\subset X_2\\subset\\dots\\subset X_N\\approx\\Lambda M$ so we find\n\\begin{equation}\n\\text{rank }H_k(\\Lambda M;\\mathbb{Z})\\le \\sum_i\\text{rank }H_k(X_i,X_{i-1};\\mathbb{Z})=\\text{rank }H_k(X_k,X_{k-1};\\mathbb{Z})=n_{\\lambda_k}\n\\end{equation} from \n\\begin{equation*}\nH_k(X_i,X_{i-1};\\mathbb{Z})=\\begin{cases}\n\\mathbb{Z}^{\\#\\lambda_i-\\text{cells}}, &\\text{if }k=i\\\\\n0, & \\text{if }k\\ne i.\n\\end{cases}\n\\end{equation*} That is, the rank of $H_k(\\Lambda M;\\mathbb{Z})$ is less than or equal to the number of critical points of index $\\lambda_k$. Equivalently, $\\text{rank }H_k(\\Lambda M;\\mathbb{Z})=n_{\\lambda_k}-l_k-l_{k+1}\\le n_{\\lambda_k}$. Since the $k$-th Betti number of the free loop space $\\Lambda M$ is given by $b_k(\\Lambda M;\\mathbb{Z})=n_{\\lambda_k}-\\min\\left(n_{\\lambda_{k-1}},n_{\\lambda_k}\\right)-\\min\\left(n_{\\lambda_k},n_{\\lambda_{k+1}}\\right)$, the least upper bound of the sequence of Betti numbers is $\\sup_{k\\in\\mathbb{N}}b_k(\\Lambda M;\\mathbb{Z})=n_{\\lambda_k}$. Thus, the supremum of the $k$-th Betti number is controlled by the number of handles of index $\\lambda_k$.\n\n\\section{Acknowledgements}\nI would like to sincerely thank Dr. John Pardon for his guidance and suggestions regarding the content of this paper. I also thank Dr. Todd Trimble for his many useful conversations on techniques employed in this analysis.\n\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Pump and Stokes Rabi frequency calibration}\n\nWe calibrate the pump Rabi frequency by measuring the Feshbach (FB) molecules loss rate as a function of laser power when the laser is resonant with the $\\ket{J'=1,M'_J=1}$ state. In the regime where $\\Omega_P \\ll \\Gamma$, the depletion rate is $1\/\\tau \\approx \\Omega_P^2\/\\Gamma$. We fit the inverse FB molecule lifetime versus pump power as shown in Fig.~\\ref{fig:pumpRabi}, and extract a Rabi frequency of $\\Omega_P\/2\\pi = 6.2(8)~{\\rm MHz} \\times \\sqrt{\\frac{P_P}{1~{\\rm mW}}}$.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[]{smfig1.pdf}\n    \\caption{Depletion rate of Feshbach molecules as a function of pump laser power, for calibration of pump laser Rabi frequency.}\n    \\label{fig:pumpRabi}\n\\end{figure}\n\nWe calibrate the Stokes laser Rabi frequency using Autler-Townes spectroscopy. After locating the rovibrational ground state, we tune the Stokes laser onto resonance with the $\\ket{J'=1,M'=1}$ state, and measure the depletion of Feshbach molecules as a function of pump laser frequency. We observe an Autler-Townes doublet feature as shown in Fig.~\\ref{fig:ATdoublet}, and fit it to a model function for the 2-body survival probability $P_{\\rm Na+Cs}$ after a pulse duration $t$ \\cite{Fleischhauer2005}\n\\begin{equation}\n    P_{\\rm Na+Cs}(t) = P_{\\rm atom} + P_{\\rm mol} \\exp{\\left( - \\Omega_P^2 t \\frac{4 \\Gamma (\\Delta_P-\\Delta_S)}{|\\Omega_S^2 + 2 i (\\Delta_P-\\Delta_S) (\\Gamma + 2 i \\Delta_P)|^2} \\right)},\n\\end{equation}\nwhere $P_{\\rm atom}$ is an atomic background population, $P_{\\rm mol}$ is the Feshbach molecule creation fidelity, and $\\Delta_{P,S}$ are the one-photon detunings of pump and Stokes lasers, respectively. We obtain a value of $\\Omega_S\/2\\pi=283(9)~{\\rm MHz}$ at $P_S=3.2~{\\rm mW}$, or $\\Omega_S\/2\\pi=158(8)~{\\rm MHz} \\times \\sqrt{P_S\/(1~{\\rm mW})}$ for the Stokes laser Rabi frequency. \n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[]{smfig2.pdf}\n    \\caption{Autler-Townes splitting of $\\ket{c^3\\Sigma_1,v'=26,J'=1,M'_J=1}$ used for calibration of Stokes laser Rabi frequency.}\n    \\label{fig:ATdoublet}\n\\end{figure}\n\n\n\\section{Motional ground state fraction}\nIn Ref.~\\cite{Zhang2020}, we estimated that the center of mass (COM) ground state fraction of Feshbach molecules was 77(5)\\% using Raman sideband thermometry. In that work, we also observed a Feshbach molecule creation fidelity of 47(1)\\%, consistent with a relative motional ground state population of 58(4)\\%. Presently, we observe a slightly reduced Feshbach molecule creation fidelity of 38(3)\\%, while the atomic ground state cooling conditions have not changed from Ref.~\\cite{Zhang2020}. The reduced fidelity of Feshbach molecule creation arises from an axial misalignment of the dual species optical tweezers, which primarily leads to heating of the Na atom during the merge process. Because Na is much lighter than Cs, heating of the Na atom primarily contributes to excitation of the relative motional degree of freedom, as opposed to the COM. Using the Feshbach molecule creation fidelity as a proxy for thermometry, we estimate that the COM ground state fraction of Feshbach molecules under present conditions is 75(5)\\%. In the following, we neglect state changes of the $\\sim 25\\%$ initially motionally excited molecules.\n\nTwo main effects cause COM motional excitation of the molecule during Raman transfer. The first arises from the differential wavenumber of the pump and Stokes lasers, $\\Delta k$. In the Raman transfer process, motional sidebands of the COM degree of freedom are not resolved: The Raman Rabi frequency is of order $\\Omega_R\\approx 2\\pi \\times 200~{\\rm kHz}$, while the axial trap frequency during transfer is $\\omega_z\\approx 2 \\pi \\times 3~{\\rm kHz}$. Coherent absorption of a pump photon and emission of a Stokes photon during Raman transfer thus creates a coherent state of the molecule's COM motion with momentum $\\hbar \\Delta k$. The mean occupation number resulting from this kick is $\\bar{n} \\approx 0.085$, giving a 92\\% probability to remain in the ground motional state.\n\nA second source of motional excitation during Raman transfer is the mismatch of initial and final state trap frequencies, due to a difference in polarizability of each state to the trapping laser. The Feshbach state polarizability is estimated as ${\\rm Re}[\\alpha_{\\rm Na+Cs}(1064~{\\rm nm})] = 1397~a_0^3$ from the sum of atomic polarizabilities, while the $X^1\\Sigma$ polarizability is ${\\rm Re}[\\alpha_{X^1\\Sigma}(1064~{\\rm nm})] = 936~a_0^3$ from \\cite{Vexiau2017}. The ratio of trap depths is then $1397\/936 \\approx 1.5$, so that ratio of trap frequencies is $\\sqrt{1.5}$ which we call $f$ below, and the ratio of harmonic oscillator lengths is $(1.5)^{1\/4}$. The probability of remaining in the motional ground state is\n\\begin{equation}\n    P_{0 \\leftarrow 0} = |\\bra{n'_x=0,n'_y=0,n'_z=0} e^{i \\Delta k z} \\ket{n_x=0,n_y=0,n_z=0}|^2.\n\\end{equation}\nThe initial Feshbach molecule COM spatial wavefunction is essentially projected onto the rovibrational ground state COM wavefunction. Because the Raman transfer lasers propagate along the $z$ axis, we can separate the integrals and evaluate them straightforwardly, finding\n\\begin{equation}\n    P_{0\\leftarrow 0} = \\frac{8 f^{3\/2}}{(f+1)^3} \\exp{\\left(- \\frac{f \\, \\eta^2}{2f + 2}\\right)},\n\\end{equation}\nwhere $\\eta = \\Delta k \\, z_{\\rm ho} \\approx 0.413$ is the Lamb-Dicke parameter. We find $P_{0\\leftarrow 0} \\approx 0.94$. \n\nIn the worst case, in which we neglect any cancellation of these two motional excitation effects, we would expect a reduction of the motional ground state population by a factor 0.862. We estimate the overall COM ground state fraction of the rovibrational ground state to be 75(5)\\% $\\times$ 0.862 = 65(5)\\%. Both excitation effects could be reduced by increasing the trap frequency in order to perform Raman transfer in a sideband-resolved regime, however this will be limited by scattering of both trap and Raman lasers. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Entropy: In the Eye of the Beholder}\nInformation is a central concept in our daily life. We rely on information in order to make sense of the world: to make ``informed\" decisions. We use information technology in our daily interactions with people and machines. Even though most people are perfectly comfortable with their day-to-day understanding of information, the precise definition of information, along with its properties and consequences, is not always as well understood. I want to argue in this comment that a precise understanding of the concept of information is crucial to a number of scientific disciplines. Conversely, a vague understanding of the concept can lead to profound misunderstandings, within daily life and within the technical scientific literature.  My purpose is to introduce the concept of information--mathematically defined--to a broader audience, with the express intent of eliminating a number of common misconceptions that have plagued the progress of information science in different fields.\n\nWhat is information? Simply put, information is that which allows you (who is in possession of that information) to make predictions with accuracy better than chance. Even though the former sentence appears glib, it captures the concept of information fairly succinctly. But the concepts introduced in this sentence need to be clarified. What do I mean with prediction? What is ``accuracy better than chance\"? Predictions of what? \n\nWe all understand that information is useful. When is the last time that you have found information to be counterproductive? Perhaps it was the last time you watched the News. I will argue that, when you thought that the information you were given was not useful, then what you were exposed to was most likely not information. That stuff, instead, was mostly entropy (with a little bit of information thrown in here or there). Entropy, in case you have not yet come across the term,  is just a word we use to quantify how much isn't known.\n\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``But, isn't entropy the same as information?\"}\n\\end{center}\nOne of the objectives of this comment is to make the distinction between the two as clear as possible. Information and entropy are two very different objects. They may have been used synonymously (even by Claude Shannon--the father of information theory--thus being responsible in part for a persistent myth) but they are fundamentally different. If the only thing you will take away from this article is your appreciation of the difference between entropy and information, then I will have succeeded.\n\nBut let us go back to our colloquial description of what information is, in terms of predictions. ``Predictions of what\"? you should ask. Well, in general, when we make predictions, they are about a system that we don't already know. In other words, an {\\em other} system. This other system can be anything: the stock market, a book, the behavior of another person. But I've told you that we will make the concept of information mathematically precise. In that case, I have to specify this ``other system\" as precisely as I can. I have to specify, in particular, which states the system can take on. This is, in most cases, not particularly difficult. If I'm interested in quantifying how much I don't know about a phone book, say, I just need to tell you the number of phone numbers in it. Or, let's take a more familiar example (as phone books may appeal, conceptually, only to the older crowd among us), such as the six-sided fair die. What I don't know about this system is which side is going to be up when I throw it next. What you do know is that it has six sides. How much don't you know about this die? The answer is not six. This is because information (or the lack thereof) is not defined in terms of the number of unknown states. Rather, it is given by the {\\em logarithm} of the number of unknown states. \n\\vskip 0.5cm\n\\begin{center}\n{\\em  ``Why on Earth introduce that complication?\"}, you ask.\n\\end{center}\nWell, think of it this way. Let's quantify your uncertainty (that is, how much you don't know) about a system (System One) by the number of states it can be in. Say this is $N_1$. Imagine that there is another system (System Two), and that one can be in $N_2$ different states. How many states can the joint system (System One And Two Combined) be in? Well, for each state of System One, there can be $N_2$ number of states. So the total number of states of the joint system must be $N_1\\times N_2$. But our uncertainty about the joint system is not $N_1\\times N_2$. Our uncertainty adds, it does not multiply. And fortunately the logarithm is that one function where the log of a product of elements is the sum of the logs of the elements. So, the uncertainty (let's call it $H$) about the system $N_1\\times N_2$ is the logarithm of the number of states\n$$H(N_1N_2)=\\log(N_1N_2)=\\log(N_1) + \\log(N_2).$$\nLet's return to the six-sided die. You know, the type you've known most of your life. What you don't know about the state of this die (your uncertainty) {\\em before} throwing the die is $\\log 6$. When you peek at the number that came up, you have reduced your uncertainty (about the outcome of this throw) to zero. This is because you made a perfect measurement. (In an imperfect measurement, you only got a glimpse of the surface that rules out a ``1\" and a ``2\", say.) \n\nWhat if the die wasn't fair? Well that complicates things. Let us for the sake of the argument assume that the die is so unfair that one of the six sides (say, the ``six\") can never be up. You might argue that the {\\em a priori} uncertainty of the die (the uncertainty before measurement) should now be $\\log 5$, because only five of the states can be the outcome of the measurement. But how are you supposed to know this? You were not told that the die is unfair in this manner, so as far as you are concerned, your uncertainty is still $\\log 6$. \n\nAbsurd, you say? You say that the entropy of the die is whatever it is, and does not depend on the state of the observer? Well I'm here to say that if you think that, then you are mistaken. Physical objects do not have an intrinsic uncertainty. I can easily convince you of that. You say the fair die has an entropy of $\\log 6$? Let's look at an even more simple object: the fair coin. Its entropy is $\\log 2$, right? What if I told you that I'm playing a somewhat different game, one where I'm not just counting whether the coin comes up heads or tails, but am also counting the angle that the face has made with a line that points towards True North. And in my game, I allow four different quadrants, like in Fig.~\\ref{fig:coin} below.\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=4in]{Die.pdf} \n   \\caption{A fair coin with entropy $\\log_2 8=3$ bits. In the figure on the left, the outcome is ``Tails, quadrant II\", while the coin on the right landed as ``Heads, quadrant I\".  }\n   \\label{fig:coin}\n\\end{figure}\n\nSuddenly, the coin has $2\\times4$ possible states, just because I told you that in my game the angle that the face makes with respect to a circle divided into 4 quadrants is interesting to me. It's the same coin, but I decided to measure something that is actually measurable (because the coin's faces can be in different orientation, as opposed to, say, a coin with a plain face but two differently colored sides). And you immediately realize that I could have divided the circle into as many quadrants as I can possibly resolve by eye. \n\nAll right, fine, you say, so the entropy is $\\log(2\\times N)$ where $N$ is the number of resolvable angles. But you know, what is resolvable really depends on the measurement device you are going to use. If you use a microscope instead of your eyes, you could probably resolve many more states. Actually, let's follow this train of thought. Let's imagine I have a very sensitive thermometer that can sense the temperature of the coin. When throwing it high (the coin, not the thermometer) the energy the coin absorbs when hitting the surface will raise the temperature of the coin slightly, compared to a coin that was tossed gently. If I so choose, I could include this temperature as another characteristic, and now the entropy is $\\log(2\\times N\\times M)$, where $M$ is the number of different temperatures that can be reliably measured by the device. And now you realize that I can drive this to the absurd, by deciding to consider the excitation states of the molecules that compose the coin, or of the atoms composing the molecules, or nuclei, the nucleons, or even the quarks and gluons. \n\nThe entropy of a physical object, it dawns on you, is not defined unless {\\em you} tell me which degrees of freedom are important to {\\em you}. In other words, it is defined by the number of states that can be resolved by the measurement that you are going to be using to determine the state of the physical object. If it is heads or tails that counts for you, then $\\log 2$ is your uncertainty. If you play the ``4-quadrant\" game, the entropy of the coin is $\\log 8$, and so on. Which brings us back to six-sided die that has been mysteriously manipulated to never land on ``six\". You (who do not know about this mischievous machination) expect six possible states, so this dictates your uncertainty. Incidentally, how do you even know the die has six sides it can land on? You know this from experience with dice, and having looked at the die you are about to throw. This knowledge allowed you to quantify your a priori uncertainty in the first place. (I'll discuss prior knowledge in much more detail in the next section)\n\nNow, you start throwing this weighted die, and after about twenty throws or so without a ``six\" turning up, you start to become suspicious. You write down the results of a longer set of trials, and still note this curious pattern of ``six\" {\\em never} showing up, but you find that the other five outcomes occur with roughly equal frequency. What happens now is that you adjust your expectation. You now hypothesize (a posteriori) that it is a weighted die with five equally likely outcome, and one outcome that never actually occurs. Now your {\\em expected} uncertainty is $\\log 5$. (Of course, you can't be 100\\% sure because you only took a finite number of trials.)\n\nBut you did learn something through all these measurements. You gained information. How much? Easy! It's the difference between your uncertainty before you started to be suspicious, minus the uncertainty after it dawned on you. The information you gained is just $\\log 6-\\log5$. How much is that? Well, you can calculate it yourself. You didn't give me the base of the logarithm you say? \nWell, that's true. Without specifying the logarithm's base, the information gained is not specified. It does not matter which base you choose: each base just gives units to your information gain. It's kind of like asking how much you weigh. Well, my weight is one thing. The number I give you depends on whether you want to know it in kilograms, or pounds. Or stones, for all it matters.\n\nIf you choose the base of the logarithm to be 2, well then your units will be called ``bits\" (which is what we all use in information theory land). But you may choose the Eulerian $e$ as your base. That makes your logarithms \"natural\", but your units of information (or entropy, for that matter) will be called \"nats\".  You can define other units, but we'll keep it at that for the moment. \nSo, if you choose base 2 (bits), your information gain is $\\log_2(6\/5)\\approx 0.263$ bits. That may not sound like much, but in a Vegas-type setting this gain of information might be worth, well, a lot. Information that you have (and others do not) can be moderately valuable (for example, in a stock market setting), or it could mean the difference between life and death (in a predator\/prey setting). In any case, we should value information.  \n\nAs an aside, this little example where we used a series of experiments to ``inform\" us that one of the six sides of the die will not, in all likelihood, ever show up, should have convinced you that we can never know the {\\em actual} uncertainty that we have about any physical object, unless the statistics of the possible measurement outcomes of that physical object are for some reason known with infinite precision (which you cannot attain in a finite lifetime). It is for that reason that I suggest to the reader to give up thinking about the uncertainty of any physical object, and be only concerned with differences between uncertainties (before and after a measurement, for example). The uncertainties themselves we call entropy. Differences between entropies (for example before and after a measurement) are called information. Information, you see, is real. Entropy on the other hand, is in the eye of the beholder.\n\n\\section{The Things We Know}\nIn the first section I have written mostly about entropy. How the entropy of a physical system (such as a die, a coin, or a book) depends on the measurement device that you will use for querying that system. That, come to think of it, the uncertainty (or entropy) of any physical object really is infinite, and made finite only by the finiteness of our measurement devices. \nOf course the things you could possibly know about any physical object is infinite! Think about it! Look at any object near to you. OK, the screen in front of you. Just imagine a microscope zooming in on the area framing the screen, revealing the intricate details of the material. The variations that the manufacturing process left behind, making each and every computer screen (or iPad or iPhone), essentially unique.\nIf this was a more formal article (as opposed to a comment), I would now launch into a discussion of how there is a precise parallel (really!) to renormalization in quantum field theory... but it isn't. So, let's instead delve head-first into the matter at hand, to prepare ourselves for a discussion of the concept of information.\n\nWhat does it even mean to have information? Yes, of course, it means that you know something. About something. Let's make this more precise. I'll conjure up the old ``urn\". The urn has things in it. You have to tell me what they are. So, now imagine that.....\n\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``Hold on, hold on. Who told you that the urn has things in it? Isn't that information already? Who told you that?\"}\n\\end{center}\n\nOK, fine, good point. But you know, the urn is really just a stand-in for what we call \"random variables\" in probability theory. A random variable is a ``thing\" that can take on different states. Kind of like the urn, that you draw something from? When I draw a blue ball, say, then the ``state of the urn\" is blue. If I draw a red ball, then the ``state of the urn\" is red. So, ``urn=random variable\". OK?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``OK, fine, but you haven't answered my question. Who told you that there are blue and red balls in it? Who?\"}\n\\end{center}\nLet me think about that. Here's the thing. When a mathematician defines a random variable, they tell you which state it can take on, and with what probability. Like: ``A fair coin is a random variable with two states. Each state can be taken on with equal probability one-half.\" When they give you an urn, they also tell you how likely it is to get a blue or a red ball from it. They just don't tell you what you will {\\em actually} get when you pull one out. \n\\vskip 0.5cm\n\\begin{center}\n{\\em ``But is this how real systems are? That you know the alternatives before asking questions?\"}\n\\end{center}\nAll right, all right. I'm trying to teach you information theory, the way it is taught in any school you would set your foot in. I concede, when I define a random variable, then I tell you how many states it can take on, and what the probability is that you will see each of these states, when you ``reach into the random variable\". Let's say that this info is magically conferred upon you. Happy now?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``Not really.\"}\n\\end{center}\nOK, let's just imagine that you spend a long time with this urn, and after a while of messing with it, you realize that:\n\n\\noindent A) This urn has balls in it.\\\\\n\\noindent B) From what you can tell, they are blue and red.\\\\\n\\noindent C) Reds occur more frequently than blues, but you're still working on what the ratio is.\\\\\nIs this enough?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``At least now we're talking. Do you know that you assume a lot when you say \"random variable?\"}\n\\end{center}\nAll right, you're making this more difficult than I intended it to be. According to standard lore, it appears that you're allowed to assume that you know something about the things you know nothing about. Let's just call these things ``common sense\". Like, that a coin has two sides. Or an urn that has red and blue balls in it.  They could be any pair of colors, you do realize. And the things you {\\em don't know} about the random variable are the things that go beyond common sense. The things that, unless you had performed dedicated experiments to ascertain the state of the variables, you wouldn't already know. \n\nHow much don't you know about it?  Easily answered using our good buddy Shannon's insight. How much you don't know is quantified by the ``entropy\" of the urn. That's calculated from the fraction of blue balls known to be in the urn, and  the fraction of red balls in the urn. You know, these fractions that are common knowledge. So, let's say that fraction of blue is $p$. The fraction of red then is of course (you do the math) $1-p$. And the entropy of the urn is\n\\begin{eqnarray}\n                                H(X)=-p\\log p-(1-p)\\log(1-p)\\;.          \\label{ent}\n\\end{eqnarray}\n\\vskip 0.5cm\n\\begin{center}\n{\\em In the first section you wrote that the entropy is $\\log N$, where $N$ is the number of states of the system. Are you changing definitions on me?}\n\\end{center}\nI'm not, actually. I just used a special case of the entropy to get across the point that the uncertainty\/entropy is additive. It was the special case where each possible state occurs with equal likelihood. In that case, the probability $p$ is equal to $1\/N$, and the above formula (\\ref{ent})  turns into the first one. But let's get back to our urn. I mean random variable. And let's try to answer the question: \n\\vskip 0.5cm\n\\begin{center}\n{\"How much is there to know (about it)?\"}\n\\end{center}\nAssuming that we know the common knowledge stuff that the urn only has read and blue balls in it, then what we don't know is the identity of the next ball that we will draw. This drawing of balls is our experiment. We would love to be able to predict the outcome of this experiment exactly, but in order to pull off this feat, we would have to have some information about the urn. I mean, the contents of the urn. \n\nIf we know nothing else about this urn, then the uncertainty is equal to the log of the number of possible states, as I wrote before. Because there are only red and blue balls, that would be log 2. And if the base of the log is two, then the result is $\\log_2 2=1$ bit.  So, if there are red and blue balls only in an urn, then I can predict the outcome of an experiment (pulling a ball from the urn) just as well as I can predict whether a fair coin lands on heads or tails. If I correctly predict the outcome (I will be able to do this about half the time, on average) I am correct purely by chance. Information is that which allows you to make a correct prediction with accuracy better than chance, which in this case means, more than half of the time. \n\\vskip 0.5cm\n\\begin{center}\n{\\em ``How can you do this for the case of the fair coin, or the urn with equal numbers of red and blue balls?\"}\n\\end{center}\nWell, you can't unless you cheat. I should say, the case of the urn and of the fair coin are somewhat different. For the fair coin, I could use the knowledge of the state of the coin before flipping, and the forces acting on it during the flip, to calculate how it is going to land, at least approximately. This is a sophisticated way to use extra information to make predictions (the information here is the initial condition of the coin) but something akin to that has been used by a bunch of physics grad students to predict the outcome of casino roulette in the late 70s (you can read about it in~\\cite{Bass1985}.) \n\nThe coin is different from the urn because for the urn you won't be able to get any ``extraneous\" information. But suppose the urn has blue and red balls in {\\em unequal} proportions. If you knew what these proportions were [the $p$ and $1-p$ in Eq. (\\ref{ent}) above] then you could reduce the uncertainty of 1 bit to $H(X)$. A priori (that is, before performing any measurements on the probability distribution of blue and read balls), the distribution is of course given by $p=1\/2$, which is what you have to assume in the absence of information. That means your uncertainty is 1 bit. But keep in mind (from section 1)  that it is only one bit because you have decided that the color of the ball (blue or red) is what you are interested in predicting.\n\nIf you start drawing balls from the urn (and then replacing them, and noting down the result, of course) you would be able to estimate $p$ from the frequencies of blue and red balls. So, for example, if you end up seeing 9 times as many red balls as blue balls, you should adjust your prediction strategy to ``The next one will be red\". And you would likely be right about 90\\% of the time, quite a bit better than the 50\/50 prior.\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``So what you are telling me, is that the entropy formula (\\ref{ent}) assumes a whole lot of things, such as that you already know to expect a bunch of things, namely what the possible alternatives of the measurement are, and even what the probability distribution is, which you can really only know if you have divine inspiration, or else made a ton of measurements!\"}\n\\end{center}\nYes, dear reader, that's what I'm telling you. Well, actually, instead of divine inspiration you might want to use a theory. Theories can sometimes constrain probability distributions in such a way that you know certain things about them {\\em before} making any measurements, that is, theory can shape your {\\em prior} expectation. For example, thermodynamics tells you something about the probability distribution of molecules in a container filled with gas, say, and knowing things like the temperature of that gas allows you to make predictions better than chance. But even in the absence of that, you already come equipped with {\\em some} information about the system you are interested in (built into your common sense) and if you can predict with accuracy better than chance (because for example somebody told you the $p$ of the probability distribution and it is not one half), then you have some extra information. And yes, most people won't tell you that. But if you want to know about information, you first need to know.... what it is that you already know.\n\n\\section{Everything is conditional}\nLet us take a few steps back for a second to contemplate the purpose of this article. \nI believe that Shannon's theory of information is a profound addition to the canon of theoretical physics. Yes, I said theoretical physics. I can't get into the details of why I think this here (but if you are wondering about this you can find my musings here~\\cite{Adami2011}). But if this theory is so fundamental (as I claim) then we should make an effort to understand the basic concepts in walks of life that are not strictly theoretical physics. I tried this for molecular biology~\\cite{Adami2004}, and evolutionary biology~\\cite{Adami2012a}.  \n\nBut even though the theory of information is so fundamental to several areas of science, I find that it is also one of the most misunderstood theories. It seems, almost, that {\\em because} ``everybody knows what information is\", a significant number of people (including professional scientists) use the word, but do not bother to learn the concepts behind it.  But you really have to. You end up making terrible mistakes if you don't. The theory of information, in the end, teaches you to think about knowledge, and prediction. I'm trying to give you the entry ticket to all that. \n\nHere's the quick synopsis of what we have learned in the first two sections:\n\n1.) It makes no sense to ask what the entropy of any physical system is. Because technically, it is infinite. It is only when you specify {\\em what} questions you will be asking (by specifying the measurement device that you will use in order to determine the state of the random variable in question) that entropy (a.k.a. uncertainty) is finite, and defined.\n\n2.) When you are asked to calculate the entropy of a mathematical (as opposed to physical) random variable, you are usually handed a bunch of information you didn't realize you have. Like, what's the number of possible states to expect, what those states are, and possibly even what the likelihood is of experiencing those states. But given those, your prime directive is to predict the state of the random variable as accurately as you can. And the more information you have, the better your prediction is going to be.\n\nNow that we've got these preliminaries out of the way, it seems like high time that we get to the concept of information in earnest. I mean, how long can you dwell on the concept of entropy, really? Actually, just a bit longer as it turns out. \n\nI think I confused you a bit in the first two sections. One time, I write that the entropy is just $\\log N$, the logarithm of the number of states the system can take on, and later I write Shannon's formula for the entropy of random variable $X$ that can take on states $x_i$ with probability $p_i$ as    \n\\begin{eqnarray}\nH(X)=-\\sum_{i=1}^N p_i\\log p_i \\label{enter}\\;. \\label{ent2}\n\\end{eqnarray} \nActually, to be perfectly honest, I didn't even write that formula. I wrote one where there are only two states, that is, $N=2$ in Eq.~(\\ref{ent2}).\nAnd then I went on to tell you that the expression $\\log N$ was ``just a special case\" of Equation~(\\ref{ent}). But I think I need to clear up what happened here.\n\nIn section 2, I talked about the fact that you really are given some information when a mathematician defines a random variable. Like, for example, in Eq. (\\ref{ent2}) above. If you know nothing about the random variable, you don't know the $p_i$. You may not even know the range of $i$. If that's the case, we are really up the creek, with paddle {\\it in absentia}. Because you wouldn't even have any idea about how much you don't know. So in the following, let's assume that you know at least how many states to expect, that is, you know $N$.\nIf you don't know anything else about a probability distribution, then you have to assume that each state appears with equal probability. Actually, this isn't a law or anything. I just don't know how you would assign probabilities to states if you have zero information. Nada. You just have to assume that your uncertainty is maximal in that case. And this happens to be a celebrated principle: the ``maximum entropy principle\". The uncertainty (\\ref{ent2}) is maximized if $p_i=1\/N$ for all $i$.  And if you plug in $p_i=1\/N$ in (\\ref{ent2}), you get\n\\begin{eqnarray}\n                                                 H_{\\rm max}=\\log N\\;.   \\label{ent3}\n\\end{eqnarray}\nIt's that simple. So let me recapitulate. If you don't know the probability distribution, the entropy is (\\ref{ent3}). If you do know it, it is (\\ref{ent2}). The difference between the entropies is knowledge\\footnote{The statement that the most objective prior is the maximum-entropy one (implying  that it is un-informative) should be a little bit more technical than appears here, because if the probability distribution of a random variable is known to be different than the uniform distribution, then the un-informative prior is not the uniform distribution, but one that maximizes the entropy under the constraint of the likelihood of the data given the probability distribution. This is discussed in Jaynes's book~\\cite{Jaynes2003},  and cases where a seemingly un-informative (uniform) prior clashes with the underlying probability distribution to make it actually informative, are discussed in~\\cite{ZhuLu2004,Neumann2007}.}\nThe uncertainty (\\ref{ent3}) does not depend on knowledge, but the entropy (\\ref{ent2}) does. On a more technical note,  Eq.~(\\ref{ent3}) is really just like the entropy in statistical physics when using the microcanonical ensemble, while Eq. (\\ref{ent2}) is the Boltzmann-Gibbs entropy in the canonical ensemble, where the $p_i$ are given by the Boltzmann distribution.\nIf you have noticed that I've been using the words ``entropy\" and ``uncertainty\" interchangeably, I did this on purpose because they are one and the same thing here. You should use one or the other interchangeably too. But you should never say ``information\" when you don't know what you can predict with the entropy at hand.\n\nSo, getting back to the narrative, one of the entropies is conditional on knowledge, while another is not. But, you think while scratching your head, wasn't there something in Shannon's work about ``conditional entropies\"? Indeed, and those are the subject of this section. The section title kind of gave it away, I'm afraid. To introduce conditional entropies more formally, and then connect to (\\ref{ent2}), we first have to talk about conditional probabilities. What's a conditional probability? I know, some of you groan ``I've known what a conditional probability is since I was seven!\" But even you may learn something. After all, you learned something reading this article even though you're somewhat of an expert? Right? Why else would you still be reading? \n\\vskip 0.5cm\n\\begin{center}\n                             {\\em ``Infinite patience\"}, you say? Moving on. \n \\end{center}\nA conditional probability characterizes the likelihood of an event, when another event has happened at the same time. So, for example, there is a (generally small) probability that you will crash your car. The probability that you will crash your car while you are texting at the same time is considerably higher. On the other hand, the probability that you will crash your car while it is Tuesday at the same time is probably unchanged, that is, unconditional on the ``Tuesday\" variable. (Unless Tuesday is your texting day, that is.)\n\nSo, the probability of events depends on what else is going on at the same time. ``Duh\", you say. But while this is obvious, understanding how to quantify this dependence is key to understanding information.  In order to quantify the dependence between ``two things that happen at the same time\", we just need to look at two random variables. In the case I just discussed, one variable is the likelihood that you will crash your car, and the other is the likelihood that you will be texting. The two are not always independent, you see. The problems occur when the two occur simultaneously. You know, if this was another article (like, the type where I veer off to discuss topics relevant only to theoretical physicists) I would now begin to remind you that the concept of simultaneity is totally relative, so that the concept of a conditional probability cannot even be unambiguously defined in relativistic physics (but concepts such as ``before\" and ``after\" can, so that helps a lot). But this is not that article, so I will just let it go.    \n\nOK, here we go: $X$ is one random variable (think: $p_i$ is the likelihood that you crash your car while you conduct maneuver $X=x_i$, where each $x_i$ is a particular maneuver or action).  The other random variable is $Y$. That variable has only two states: either you are texting ($Y=1$), or you are not ($Y=0$) And those two states have probabilities $q_1$ (texting)  and $q_0$ (not texting)  associated with them. \nI can then write down the formula for the uncertainty of crashing your car while texting, using the probability distribution\n\\begin{eqnarray}\n                                                    P(X=x_i|Y=1)\\; .\n\\end{eqnarray}\nThis you can read as ``the probability that random variable $X$ is in state $x_i$ given that, at the same time, random variable $Y$ is in state $Y=1$.\"  This vertical bar ``$|$'' is always read as ``given\".\n\nSo, let me write  $P(X=x_i|Y=1)$ as $p(i|1)$. It's much simpler that way. I can also define $P(X=x_i|Y=0)=p(i|0)$. $p(i|1)$ and $p(i|0)$ are two probability distributions that may be different (but they don't have to be if my driving is unaffected by texting). Fat chance for the latter, by the way. \n\nI can then write the entropy while texting as\n\\begin{eqnarray}\n                               H(X|{\\rm texting})=-\\sum_{i=1}^N p(x_i|1)\\log p(x_i|1)\\;.  \\label{cond1}\n\\end{eqnarray}\nOn the other hand, the entropy of the driving variable while {\\em not} texting is \n\\begin{eqnarray}\n                           H(X|{\\rm not\\  texting})=-\\sum_{i=1}^N p(x_i|0)\\log p(x_i|0)\\;.  \\label{cond2}\n\\end{eqnarray}\nNow, compare Eqs. (\\ref{cond1}) and (\\ref{cond2}) to Eq. (\\ref{ent2}). The latter two are conditional entropies, conditional in this case on the co-occurrence of another event, here texting. They look just like the Shannon formula for entropy, which I told you was the one where ``you already knew something\", like the probability distribution. In the case of (\\ref{cond1}) and (\\ref{cond2}), you know exactly what it is that you know, namely whether random variable $X$ is texting while driving, or not. \n\nSo here's the gestalt idea that I want to get across. Probability distributions are born being uniform. In that case, you know nothing about the variable, except perhaps the number of states it can take on. Because if you didn't know {\\em that}, then you wouldn't even know how much you don't know. That would be the ``unknown unknowns\", that a certain political figure once injected into the national discourse. \n\nThese probability distributions become non-uniform (that is, some states are more likely than others) once you acquire information about the states. This information is manifested by conditional probabilities. You really only know that a state is more or less likely than the random expectation if you at the same time know something else (like in the case discussed, whether the driver is texting or not). Put in another way, what I'm trying to tell you here is that any probability distribution that is not uniform (same probability for all states) is necessarily conditional. When someone hands you such a probability distribution, you may not know what it is conditional about. But I assure you that it is conditional. I'll state it as a theorem:\n\\vskip 0.5cm\n\\begin{center}\n{\\bf All  probability distributions that are not uniform are in fact conditional probability distributions.}\n\\end{center}\nThis is not what your standard textbook will tell you, but it is the only interpretation of ``what do we know\" that makes sense to me. ``Everything is conditional\" thus, as the title of this section promised.\n\nWe can also write down what the {\\em average} uncertainty for crashing your car is, given your texting status. It is simply the average of the uncertainty while texting and the uncertainty while not texting, weighted by the probability that you engage in any of the two behaviors.  Thus, the conditional entropy $H(X|Y)$, that is the uncertainty of crashing your car given your texting status, is\n\\begin{eqnarray}\n                           H(X|Y)=q_0H(X|Y=0)+q_1H(X|Y=1)\\;.   \\label{condav}\n\\end{eqnarray}\nThat's obvious, right? $q_0$ being the probability that you are texting while executing any maneuver $i$, and $q_1$ the probability that you are not (while executing any maneuver).\nWith this definition of the entropy of one random variable given another, we can now finally tackle information.\n\n\\section{Information}\nBefore going on, let me quickly summarize the take-home points of sections 1-3.\n\n1.) Entropy, also known as ``uncertainty\", is something that is mathematically defined for a ``random variable\". But physical objects aren't mathematical. They are messy complicated things. They become mathematical when observed through the looking glass of a measurement device that has a finite resolution. We then understand that a physical object does not ``have an entropy\". Rather, its entropy is defined by the measurement device I choose to examine it with.  Information theory therefore is a theory of the relative state of measurement devices. \n\n2.) Entropy, also known as uncertainty, quantifies how much you don't know about something (a random variable). But in order to quantify how much you don't know, you have to know {\\em something} about the thing you don't know. These are the hidden assumptions in probability theory and information theory. These are the things you didn't know you knew.\n\n3.) Shannon's entropy is written in terms of ``$p \\log p$\", but these ``$p$\" are really conditional probabilities if you know that they are {\\em not} uniform, that is, all $p$ the same for all states. They are not uniform given what else you know.\n\nI previously defined the unconditional entropy, which is the one where we know nothing about the system that the random variable describes. We call that $H_{\\rm max}$, because an unconditional entropy must be maximal: it tells us how much we don't know if we don't know anything except how many states my measurement device has. Then there is the conditional entropy $H=-\\sum_i p_i\\log p_i$, where the $p_i$ are conditional probabilities. They are conditional on some knowledge. Thus, $H$ tells you what remains to be known. So finally, I give you:\n\\vskip 0.5cm\n\\begin{center}\nInformation is: ``What you don't know minus what remains to be known given what you know\". \n\\end{center}\nThere it is. Clear?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``Hold on, hold on. Hold on for just a minute.\"}\n\\end{center}\nWhat?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``This is not what I've been reading in textbooks.\"}\n\\end{center}\nSo tell me what it is that you read.\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``It says there that the mutual information is the difference between the entropy of random variable $X$,  $H(X)$, and the conditional entropy $H(X|Y)$, which is the conditional entropy of variable $X$ given you know the state of variable $Y$. Come to think of it, you yourself defined that conditional entropy at the end of section 3. I think it is Equation (\\ref{condav}) there!\" And there is this Venn diagram on Wikipedia. It looks like Figure 2!\"}\n\\end{center}\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=4in]{entropy.pdf} \n   \\caption{Entropy Venn diagram, showing conditional and mutual entropies of two variables. Source: Wikimedia}\n   \\label{fig:venn}\n\\end{figure}\n\nAh, yes. That's a good diagram. Two variables $X$ and $Y$. The red circle represents the entropy of $X$, the blue circle the entropy of $Y$. The purple thing in the middle is the shared entropy $I(X:Y)$, which is what $X$ knows about $Y$. Also what $Y$ knows about $X$. They are the same thing.\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``You wrote $I(X:Y)$ but Wiki says $I(X;Y)$. Is your semicolon key broken?\"}\n\\end{center}\nActually, there are two notations for the shared entropy (a.k.a information) in the literature. One uses the colon, the other the semicolon. Thanks for bringing this up. It confuses people. In fact, I wanted to bring up this other thing....\n\\vskip 0.5cm\n\\begin{center} {\\em ``Hold on again. You also keep on saying ``shared entropy\" when Wiki says ``shared information\". You really ought to pay more attention.\"}\n\\end{center}\nWell, you. That's a bit of a pet-peeve of mine. Just look at the diagram above. The thing in the middle, the purple area, it's a shared entropy. Information is shared entropy. ``Shared information\" would be, like, shared shared entropy. I think that's a bit ridiculous, don't you think?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``Well, if you put it like this. I see your point. But why do I read `shared information' everywhere?\"}\n\\end{center}\nThat is, dear reader, because people are confused about what to call entropy, and what to call information. A sizable fraction of the literature calls what we have been calling ``entropy\" (or uncertainty) ``information\". You can see this even in the book by Shannon and Weaver~\\cite{ShannonWeaver1949} (which, come to think of it, was edited by Weaver, not Shannon). When you do this, then what is shared by the ``informations\" is ``shared information\". But that does not make any sense, right?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``I don't understand. Why would anybody call entropy `information'? Entropy is what you don't know, information is what you know. How could you possibly confuse the two?\"}\n\\end{center}\nI'm with you there. Entropy is ``potential information\". It quantifies ``how much you could {\\em possibly} know\". But it is not what you {\\em actually} know. I think, between you and me, that it was just sloppy writing at first, which then ballooned into a massive misunderstanding. Both entropy and information are measured in bits, and so people would just flippantly say: ``a coin has two bits of information\", when they mean to say ``two bits of entropy\". And it's all downhill from there. \n\nI think I've made my point here, I hope. Being precise about entropy and information really matters. Colon vs. semicolon does not. Information is ``unconditional entropy minus conditional entropy\". When cast as a relationship between two random variables $X$ and $Y$, we can write it as\n\\begin{eqnarray}\nI(X:Y)=H(X)-H(X|Y)\\;.\n\\end{eqnarray}\nAnd because information is symmetric in the one who measures and the one who is being measured (remember: ``a theory of the relative state of measurement devices\") this can also be written as\n\\begin{eqnarray}\nI(X:Y)=H(Y)-H(Y|X)\\;.\n\\end{eqnarray}\nAnd both formulas can be verified by looking at the Venn diagram above.\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``OK, this is cool.\"}\n\\end{center}\n\\begin{center} {\\em \"Hold on, hold on!\"}\n\\end{center}\nWhat is it again?\n\\vskip 0.5cm\n\\begin{center}\n{\\em ``I just remembered. This was all a discussion that came after I brought up that information was $I(X:Y)=H(X)-H(X|Y)$, while you said it was $H_{\\rm max}-H$, where the $H$ was clearly an entropy that you write as $H=-\\sum_i p_i\\log p_i$. All you have to do is look back a few pages, I'm not dreaming this!\"}\n\\end{center}\nSo you are saying that textbooks say\n\\begin{eqnarray}\nI=H(X)-H(X|Y) \\label{info1}\n\\end{eqnarray}\nwhile I write instead\n\\begin{eqnarray}\nI=H_{\\rm max}-H(X)\\;,   \\label{info2}\n\\end{eqnarray}\nwhere $H(X)=-\\sum_i p_i\\log p_i$. Is that what you're objecting to?\n\\vskip 0.5cm\n\\begin{center}\n{\\em \n``Yes. Yes it is.\"}\n\\end{center}\nWell, here it is in a nutshell. In (\\ref{info1}), information is defined as the difference between the actual observed entropy of $X$, minus the actual observed entropy of $X$ given that I know the state of $Y$ (whatever that state may be). \nIn (\\ref{info2}), information is defined as what I don't know about $X$ (without knowing any of the things that we may implicitly know already), and the actual uncertainty of $X$, given a particular probability distribution that is non-uniform. The latter entropy does not mention a system $Y$. It quantifies my knowledge of $X$ without stressing {\\em what it is} that I know about $X$. If the probability distribution with which I describe $X$ is not uniform, then I do know something about $X$. My $I$ in Eq. (\\ref{info2}) quantifies that. Eq. (\\ref{info1}) quantifies what I know about $X$ above and beyond what I already know via Eq. (\\ref{info2}), namely using my knowledge of $Y$. It quantifies specifically the information that $Y$ conveys about $X$. So you could say that the total information that I have about $X$, given that I also know the state of $Y$, would be\n\\begin{eqnarray}\nI_{\\rm total}=H_{\\rm max}-H(X) + H(X)-H(X|Y)=H_{\\rm max}-H(X|Y)\\;.\n\\end{eqnarray}\nSo the difference between what I would write and what textbooks write is really only in the unconditional term: it should be maximal. But in the end, Eqs. (\\ref{info1}) and (\\ref{info2}) simply refer to different informations. Eq.~(\\ref{info2}) is information, but I may not be aware how I got into possession of that information. Eq.~(\\ref{info1}) tells me exactly the source of my information: the variable $Y$. Is it clear now?\n\\begin{center}\n{\\em ``I'll have to get back to you on that. I'm still reading. I think I have to read it again. It sort of takes some getting used to.\"}\n\\end{center}\nI know what you mean. It took me a while to get to that place. But, as I hinted at in the introduction, it pays off big time to have your perspective adjusted, so that you know what you are talking about when you say ``information\". I have been (and will be) writing a good number of articles that reference ``information\", and many of those are a consequence of research that was only possible when you understand the concept precisely. I wrote a series of articles on information in black holes already~\\cite{AdamiVerSteeg2014,AdamiVerSteeg2015,BradlerAdami2014,BradlerAdami2015}. That's just the beginning. There are others, for example on how to measure how much information is stored in DNA~\\cite{AdamiCerf2000} or proteins~\\cite{GuptaAdami2015}, and the relationship between information and cooperation~\\cite{Iliopoulosetal2010,AdamiHintze2013,HintzeAdami2015} (I mean, how you can fruitfully engage in the latter only if you have the former), and information processing in the brain~\\cite{Edlundetal2011,Marstalleretal2013}. And there are more to come, on information in DNA binding sites~\\cite{CliffordAdami2015} or even how to use information theory to estimate the likelihood of a spontaneous origin of life~\\cite{Adami2015}. I know it sounds more like a threat than a promise. I really mean it to be a promise. \n\n\\section{Epilogue}A kind reviewer brought to my attention an elegant and insightful article by E.T. Jaynes~\\cite{Jaynes1965}, in which Jaynes not only accurately characterizes the difference between Boltzmann and Gibbs thermodynamic entropies and derives thermodynamics' second law, but also makes essentially the same statement about entropy that I have made here, by writing: \n\n``From this we see that entropy is an anthropomorphic concept, not only in the well-known statistical sense that it measures the extent of human ignorance as to the microstate. {\\em Even at the purely phenomenological level, entropy is an anthropomorphic concept.} For it is a property, not of the physical system, but of the particular experiments you or I choose to perform on it.\"\n\nWhile this statement pertained to thermodynamics entropy, it applies just as well to Shannon entropy as the two are intimately related. Jaynes dissects this relationship cogently in his article and I see no need to repeat it here as my focus is on information, not entropy. I have written about the relationship between thermodynamic and Shannon entropy~\\cite{Adami2011} but wish I would have known Ref.~\\cite{Jaynes1965} then. Perhaps this link is best summarized by noting that thermodynamics is a special case of information theory where ``all the fast things have happened, but the slow things have not\" (as Richard Feynman described thermodynamical equilibrium~\\cite{Feynman1972}). In other words,  in equilibrium information about the fast things has disappeared, but there may still be information about the slow things: that 's the things we don't always know we know.  \n\n\\noindent{\\bf Acknowledgememnts}\nI would like to thank Julyan Cartwright for suggesting to turn my blog series on information into this comment, Eugene Koonin for lively discussions about information, evolution, and genomics, as well as three referees for insightful comments.\nThis work was supported in part by NSF's BEACON Center for the Study of Evolution in Action, under Contract No. DBI-0939454.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe study of uniform embeddings of metric spaces into Hilbert space was\ninitiated by Gromov.\n\n\\begin{defn}\\label{def1}\nLet $(\\Gamma,d)$ be a metric space. Let $\\mathcal{H}$ be a separable\nHilbert space. A map $f\\colon\\Gamma\\to\\mathcal{H}$ is said to be a\n{\\em uniform embedding\\\/}~{\\rm\\cite{Gr}} if there exist non-decreasing\nfunctions $\\rho_1$, $\\rho_2$ from $[0,\\infty)$ to itself such that\n\\begin{itemize}\n\\item[(1)] $\\rho_1(d(x,y))\\le \\|f(x)-f(y)\\|_{\\mathcal{H}}\\le \\rho_2(d(x,y))$\nfor all $x,y\\in\\Gamma$;\n\\item[(2)] $\\lim_{r\\to+\\infty}\\rho_i(r)=+\\infty$ for $i=1,2$.\n\\end{itemize}\n\\end{defn}\n\nGromov suggested that a finitely generated group $G$ (viewed as a metric space with a\nword length metric) uniformly embeddable into Hilbert space should satisfy the Novikov\nConjecture~\\cite[page~67]{Gr1}. That was proved in \\cite{Yu}, see also~\\cite{skandalis,Hig}.\n\n\n\nGuoliang Yu introduced a F\\o lner-type condition on finitely generated groups $G$,\ncalled property A, which is a weak form of amenability and which guaranties the\nexistence of a uniform embedding of the metric space into Hilbert space~\\cite{Yu}.\nThat property is interesting in itself because it turned out to be equivalent to\nthe exactness of $G$, and to the existence of an amenable action of $G$ on a compact\nspace (see \\cite{Tu} for a detailed discussion of property A).\n\nAmong ``classical\"  groups for which property A has not been proved so far, is the\nR.\\,Thompson group $F$, which is the group of all piecewise linear orientation\npreserving self-homeomorphisms of the unit interval with finitely many dyadic\nsingularities and all slopes integer powers of $2$.\n\nIt is not known whether $F$ is amenable or not, so the question about a weak\namenability is interesting.\n\nIt is worth noticing that, by a result of Niblo and Reeves, a group acting properly\nand cellularly by isometries on a CAT(0) cubical complex has the Haagerup\nproperty~\\cite{Niblo}. In particular, by a result of Farley \\cite{Farley}, this holds\nfor diagram groups over finite semigroup presentations. Moreover, this holds for all\ncountable diagram groups because they can be embedded into a certain ``universal\"\ndiagram group~\\cite{GS3} . It follows immediately from the definition of the Haagerup\nproperty that such a group can be embedded uniformly into Hilbert space. However,\nthe interaction between the Haagerup property and Guoliang Yu's property A is rather\ncomplicated~\\cite[Ch.~1.3]{cherix},~\\cite{GuKa,skandalis}. Notice that amenable groups\nsatisfy both the Haagerup property and property A.\n\n\\small\n$$\n\\begin{array}{ccccc}\nG\\ \\hbox{has property A}\\hspace{-1cm} & &\\Longrightarrow & &G\\ \\hbox{{\\small is\nuniformly }}\\\\\n                         &  &      &        & \\strut \\ \\hbox{{\\small  embedded\ninto }}\\\\\n\\Big\\Uparrow\\vcenter{\\rlap{{\\bf ?}}} &  &      &        &\\strut  \\ \\hbox{{\\small\nHilbert space}}\\\\\n& & &    & \\Big\\Uparrow\\\\\n& & &    & \\strut \\\\\nG\\ \\hbox{is a countable} & \\Longrightarrow& G\\ \\hbox{{\\small acts properly and}} &\n\\Longrightarrow & G \\ \\hbox{has Haagerup}\\\\\n\\ \\hbox{diagram group}& &\\strut  \\ \\hbox{cellularly by isometries }& &\\strut\\\n\\hbox{property}\\\\\n & &\\strut \\ \\hbox{on a CAT(0) }  & &\\\\\n & &\\strut \\ \\hbox{cubical complex}  & &\n\\end{array}\n$$\n\\normalsize\n\\bigskip\n\nGuentner and Kaminker \\cite{GuKa} introduced a natural quasi-isometry invariant of a\ngroup that shows how close to a quasi-isometry can a uniform embedding of a group\ninto a Hilbert space be.\n\n\\begin{defn}{\\rm (cf.\\cite[Def.~2.2]{GuKa})}\nThe Hilbert space compression of a finitely generated discrete group $G$ is the number\n$R(G)$, which is the supremum of all $\\alpha\\ge0$ for which there exists a uniform\nembedding of $G$ into a Hilbert space with $\\rho_1(n)=Cn^\\alpha$ with a constant $C>0$\nand linear $\\rho_2$ (see Definition \\ref{def1}).\n\\end{defn}\n\nBy ~\\cite{GuKa}, Hilbert space compression strictly greater than 1\/2 implies property A.\nAny group that is not uniformly embeddable into a Hilbert space (such groups exist\nby \\cite{random}) has Hilbert space compression 0. It is proved in \\cite{GuKa} that\nfree groups have Hilbert space compression 1 (although there are no quasi-isometric\nembeddings of a free group of rank $>1$ into a Hilbert\nspace~\\cite{bourg,linial}\\footnote{We thank D.~Sonkin for bringing these papers to our\nattention.}). More generally, by Campbell and Niblo \\cite{nibloS}, any discrete group\nacting properly, co-compactly on a finite dimensional CAT(0) cubical complex has Hilbert\nspace compression 1.\n\nNote that till now there were no examples of groups with compression\nstrictly between 0 and 1.\n\nOne of our main results is the following.\n\n\\begin{thm}\\label{Fhalf}\nThe Hilbert space compression of R.\\,Thompson's group $F$ equals $1\/2$.\n\\end{thm}\n\nIn particular, this shows that the  result by Guentner and Kaminker\non groups with compression strictly greater than $1\/2$ cannot be\napplied to $F$. However, one can extract a stronger fact\nfrom~\\cite{GuKa}. Suppose that a finitely generated group $G$ has a\nuniform embedding into Hilbert space with linear $\\rho_2(n)$ and\n$\\rho_1\\gg\\sqrt{n}$ (i.e.\n$\\lim_{n\\to\\infty}\\rho_1(n)\/\\sqrt{n}=\\infty$). The proof\nof~\\cite[Theorem~3.2]{GuKa} shows that property A holds for $G$ in\nthis case. For R.\\,Thompson's group $F$ we actually show that\n$\\rho_1\\le C\\sqrt{n} \\log n$ for some constant $C$.\n\n\n\\begin{quest}\n\\label{sqd}\nIs it true that there exists a uniform embedding of $F$ into Hilbert space\nwith linear $\\rho_2$ and $\\rho_1\\gg\\sqrt{n}$?\n\\end{quest}\n\nIf the answer is positive, then $F$ has property A.\n\\vspace{1ex}\n\n\nIt is known \\cite{GS1,GS5,Farley} that geometric properties of $F$\nare better understood when $F$ is considered as a diagram group.\nRecall that a diagram group is the fundamental group of the space of\npositive paths on a directed 2-complex~\\cite{GS3} (an equivalent\ndefinition in terms of semigroup presentations from~\\cite{GS1} is\ngiven below). For example, $F$ is the diagram group of the dunce hat\nconsidered as a directed $2$-complex with one vertex, one edge $x$ and\none directed cell $x=x^2$.\n\nOur interest in uniform embeddings for diagram groups was stimulated\nin part by a connection with certain metric properties of diagram\ngroups which have been studied independently before. Recall that\nelements of a diagram group $G$ can be represented by diagrams, which are\nplane cellular complexes subdivided into a number of regions (cells).\nThis allows one to introduce a canonical {\\em diagram metric\\\/} ${\\mathrm{dist}}_d$ on\n$G$ such that the ${\\mathrm{dist}}_d(g_1,g_2)$ is the number of cells in the diagram\nwith minimal number of cells representing the element $g_1^{-1}g_2$. This\nmetric is left invariant. It is proper provided the corresponding directed\n2-complex (semigroup presentation) is finite or when the diagram group is\nfinitely generated. It is known (this is proved in Burillo~\\cite{burillo} using\na different terminology) that for the R.\\,Thompson group $F$ considered as a\ndiagram group of the dunce hat, the diagram metric is bi-Lipschitz equivalent\nto the word metric. We say that a finitely generated diagram group satisfies\n{\\em property B\\\/} if its diagram metric is bi-Lipschitz equivalent to the word\nmetric.\n\n\\begin{thm}\\label{dghalf}]\nThe Hilbert space compression of a finitely generated diagram group with property B\nis at least $1\/2$.\n\\end{thm}\n\n\\begin{quest} \\label{prb} Do all finitely generated diagram groups satisfy property B?\n\\end{quest}\n\nWe think that is a very interesting question.\n\\vspace{2ex}\n\nIn \\cite{GS1,GS3}, it is proved that there are (finitely presented) diagram groups\ncontaining all countable diagram groups as subgroups. Such diagram groups are called\n{\\em universal\\\/}. One of these diagram groups (in this paper it is denoted by $U$)\ncorresponds to the semigroup presentation $\\langle\\, x,a\\mid  x^2=x^3,ax=a\\,\\rangle$.\n\n\n\\begin{thm}\\label{BurU}\nThe universal diagram group $U$ satisfies property B.\n\\end{thm}\n\n\n\nNotice that Theorem \\ref{BurU} does not imply (at least directly) a\npositive answer to Question \\ref{prb}. Indeed, $U$ contains the direct\nproduct $G={\\mathbb F}_2\\times{\\mathbb F}_2$ of two free groups of rank 2\n(which is a countable diagram group). By a result of Mikhailova \\cite{Mikh},\nthe group $G$ contains a non-recursively distorted subgroup $H$. Hence the word\nmetric on a diagram group may not be equivalent to the metric induced by the\nembedding of $H$ into $U$. Certainly Theorem \\ref{BurU} implies that all\nundistorted subgroups of $U$ satisfy property B. Clearly the diagram metric\non a finitely generated diagram group does not exceed a constant times the word\nmetric. An easy argument shows that for any finitely generated diagram group,\nthere exists a recursive function $f(n)$ such that the word metric does not\nexceed $f({\\mathrm{dist}}_d)$.\n\nBesides the ~R.\\,Thompson group $F$, the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ is another typical representative of the class of diagram groups.\nIt corresponds to the presentation\n$\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2, b_2=b\\,\\rangle$ \\cite{GS5}.\nIt also satisfies property B. Hence Theorem \\ref{dghalf} can be applied. The\ngroup ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ is amenable, so it satisfies property A, but the\nnext theorem shows that the compression of that group is not $1$.\n\n\\begin{thm} \\label{wr}\nThe Hilbert space compression of the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ belongs to $[1\/2,3\/4]$.\n\\end{thm}\n\n\\begin{quest} What is the Hilbert space compression of ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$?\n\\end{quest}\n\nFinally we prove the following general result.\n\n\\begin{thm} \\label{wr1} Let $H$ be a finitely generated group with\na su\\-per-po\\-ly\\-no\\-mial growth function. Then the Hilbert space compression of\n${\\mathbb{Z}}\\wr H$ is at most $1\/2$.\n\\end{thm}\n\nThis theorem immediately implies\n\n\\begin{cy} The Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$\nis at most $1\/2$.\n\\end{cy}\n\nNote that the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$ is amenable and so it satisfies property A.\nThus property A does not imply that Hilbert space compression is bigger than $1\/2$.\nIt is not known whether ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$ is a diagram group (the most probable answer\nseems to be negative). Thus Theorem \\ref{dghalf} does not apply and we do not know if the\nHilbert space compression of this group is exactly $1\/2$.\n\n\\begin{quest} What is the Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$?\nAre there amenable groups with Hilbert space compression 0 or arbitrary close to 0?\n\\end{quest}\n\n\nFinally, we would like to mention that our proof of Theorem~\\ref{dghalf} yields actually a\nstronger result. Namely, a lower bound on the the {\\it equivariant\\\/} Hilbert space\ncompression $R_G(G)$ which is the Hilbert space compression defined by restricting\nto $G$-equivariant uniform embeddings of $G$ into Hilbert spaces equiped with actions\nof $G$ by {\\it affine isometries\\\/}~\\cite[Section 5]{GuKa}.\n\n\\begin{thm}\\label{B-equiv}\nThe equivariant Hilbert space compression of a finitely generated diagram group\nwith property B is at least $1\/2$.\n\\end{thm}\n\nThe estimate $1\/2$ cannot be improved since $R_G(G)>1\/2$ implies amenability~\\cite{GuKa}\nand there are non-amenable diagram groups with property B~\\footnote{We thank Y. de Cornulier\nfor this remark.}. Moreover, one can see from the proof that the diagram metric of a\nfinitely generated diagram group is a conditionally negative definite function on the group.\nRecall that a function $\\psi : G\\to \\mathbb{R}$, satisfying $\\psi(g)=\\psi(g^{-1})$\nis said to be conditionally negative definite if $\\sum\\psi(g_ig_j^{-1})c_ic_j\\le 0$\nfor all $n\\in\\mathbb{N}$ and $g_1,\\dots ,g_n\\in G, c_1, \\dots , c_n\\in \\mathbb{R}$ such that\n$\\sum c_i=0$. The existence of a continious conditionally negative definite function $\\psi : G\\to [0,\\infty)$ with\n$\\lim_{g\\to\\infty}\\psi(g)=\\infty$ is equivalent to the a-T-menability ~\\cite{ch-b-v}, hence\nto the Haagerup approximation property~\\cite{cherix}. For more applications see~\\cite{GuK} and references therein.\n\n\n\n\\section{Preliminary information about diagram groups}\n\n\\subsection{Diagram groups}\\label{diagram}\n\nLet us briefly recall the concept of a diagram group and some\nterminology from this area. Details can be found in~\\cite{GS1}. For\nthe reasons of the present paper, it is enough to use the definition\nin terms of semigroup presentations given in \\cite{GS1} rather than\nan equivalent definition from \\cite{GS3}.\n\nLet $X$ be an alphabet. We denote by $X^*$  the set of all words over $X$\nand by $X^+$ the set of all nonempty words. The length of a word $w$\nis denoted by $|w|$.\n\nLet ${\\cal P}=\\langle\\, X\\mid{\\cal R}\\,\\rangle$ be a semigroup presentation. Here ${\\cal R}$ consists of\nordered pairs of nonempty words over $X$. We always assume that if $(u,v)\\in{\\cal R}$,\nthen $(v,u)\\notin{\\cal R}$.\n\nA triple of the form $(p,u=v,q)$ is called an {\\em atomic $2$-path\\\/}, where\n$p,q\\in X^*$, $(u,v)\\in{\\cal R}\\cup{\\cal R}^{-1}$. The following labelled plane graph is\ncalled an {\\em atomic diagram\\\/} (over ${\\cal P}$):\n\n\\begin{center}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(110.66,26.33)\n\\put(0.00,13.33){\\circle*{1.33}}\n\\put(40.00,13.33){\\circle*{1.33}}\n\\put(70.00,13.33){\\circle*{1.33}}\n\\put(110.00,13.33){\\circle*{1.33}}\n\\bezier{200}(40.00,13.33)(55.00,33.33)(70.00,13.33)\n\\bezier{200}(40.00,13.33)(55.00,-6.67)(70.00,13.33)\n\\put(20.00,17.33){\\makebox(0,0)[cc]{$p$}}\n\\put(55.00,25.33){\\makebox(0,0)[cc]{$u$}}\n\\put(90.00,17.33){\\makebox(0,0)[cc]{$q$}}\n\\put(55.00,0.00){\\makebox(0,0)[cc]{$v$}}\n\\put(0.33,13.33){\\line(1,0){39.67}}\n\\put(70.00,13.33){\\line(1,0){40.00}}\n\\end{picture}\n\\end{center}\n\n\\noindent Here each segment labelled by a word $w$ is subdivided into $|w|$ edges\nlabelled by the letters of $w$. (Segments have a left-to-right orientation.)\n\nNotice that each atomic diagram has the {\\em top path\\\/} (in our case it is\nlabelled by $puq$) and the {\\em bottom path\\\/} (labelled by $pvq$). This atomic\ndiagram is defined uniquely by a $2$-path $(p,u=v,q)$ so we will often use\nthis notation for the atomic diagram itself.\n\nSuppose that we have a sequence $\\Delta_1$, \\dots, $\\Delta_k$ of atomic\ndiagrams. Let us consider the case when the bottom label of each $\\Delta_i$\n($1\\le i<k$) coincides with the top label of $\\Delta_{i+1}$. We can thus\nidentify the bottom path of $\\Delta_i$ with the top path of $\\Delta_{i+1}$\n($1\\le i<k$) to obtain the new labelled plane graph denoted by\n$\\Delta:=\\Delta_1\\circ\\cdots\\circ\\Delta_k$ (the {\\em concatenation of\\\/}\nthe above {\\em atomic diagrams\\\/}). One can naturally define the top path\n(which will be the top path of $\\Delta_1$) and the bottom path (the bottom\npath of $\\Delta_k$) of $\\Delta$.\n\nFor every word $w\\in X^+$, we define the {\\em trivial diagram\\\/} denoted by\n$\\varepsilon(w)$, which is the interval labelled by $w$. Its top and bottom paths\ncoincide with itself.\n\n\\begin{defn}\nA {\\em diagram $\\Delta$ over a semigroup presentation ${\\cal P}$\\\/} is either a trivial\ndiagram, or a concatenation of atomic diagrams over ${\\cal P}$. The diagrams are\nconsidered identical whenever they are isotopic as labelled plane graphs.\n\\end{defn}\n\nFor a diagram $\\Delta$, its top and bottom paths will be denoted by\n$\\topp{\\Delta}$ and $\\bott{\\Delta}$, respectively. They start at the\nsame vertex denoted by $\\iota(\\Delta)$ and end at the same vertex\ndenoted by $\\tau(\\Delta)$. The diagram itself is situated ``between\"\nits top and bottom paths. If $w'$ is the label of the top path of\n$\\Delta$ and $w''$ is the label of its bottom path, then we say that\n$\\Delta$ is a $(w',w'')$-diagram.\n\nGiven two diagrams $\\Delta'$ and $\\Delta''$, one can define their\n{\\em concatenation\\\/} denoted by $\\Delta'\\circ\\Delta''$, provided the\nlabels of $\\bott{\\Delta'}$ and $\\topp{\\Delta''}$ coincide.\n\nFor every word $w\\in X^+$, the set of all $(w,w)$-diagrams over ${\\cal P}$\nforms a monoid with $\\varepsilon(w)$ as a unit.\n\nFor any diagram $\\Delta$, by $\\Delta^{-1}$ we denote the mirror image of\n$\\Delta$ with respect to the axis connecting $\\iota(\\Delta)$ and $\\tau(\\Delta)$).\nClearly, the top (bottom) path of $\\Delta$ becomes the bottom (top) path of\n$\\Delta^{-1}$.\n\\vspace{1ex}\n\nHere is an example of two diagrams over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$:\n\n\\begin{center}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(126.33,25.67)\n\\put(80.33,12.00){\\circle*{1.33}}\n\\put(95.33,12.00){\\circle*{1.33}}\n\\put(110.33,12.00){\\circle*{1.33}}\n\\put(125.33,12.00){\\circle*{1.33}}\n\\bezier{200}(80.33,12.00)(95.33,32.00)(110.33,12.00)\n\\bezier{212}(95.33,12.00)(110.33,-10.00)(125.33,12.00)\n\\put(88.33,20.00){\\circle*{1.33}}\n\\put(102.33,20.00){\\circle*{1.33}}\n\\put(102.33,4.00){\\circle*{1.33}}\n\\put(118.33,4.00){\\circle*{1.33}}\n\\put(4.33,1.67){\\circle*{1.33}}\n\\put(19.33,1.67){\\circle*{1.33}}\n\\put(34.33,1.67){\\circle*{1.33}}\n\\put(49.33,1.67){\\circle*{1.33}}\n\\put(64.33,1.67){\\circle*{1.33}}\n\\bezier{180}(19.33,1.67)(34.33,18.67)(49.33,1.67)\n\\put(27.33,8.00){\\circle*{1.33}}\n\\put(41.33,8.67){\\circle*{1.33}}\n\\bezier{152}(27.33,8.00)(10.33,18.67)(4.33,1.67)\n\\bezier{152}(41.33,8.67)(60.33,17.67)(64.33,1.67)\n\\put(7.33,7.67){\\circle*{1.33}}\n\\put(62.33,6.67){\\circle*{1.33}}\n\\put(18.33,11.67){\\circle*{1.33}}\n\\put(51.33,11.67){\\circle*{1.33}}\n\\bezier{228}(7.33,7.67)(16.33,34.67)(27.33,8.67)\n\\bezier{240}(41.33,8.67)(54.33,35.67)(62.33,6.67)\n\\put(11.50,17.00){\\circle*{1.33}}\n\\put(22.33,17.67){\\circle*{1.33}}\n\\put(46.33,17.67){\\circle*{1.33}}\n\\put(58.33,17.67){\\circle*{1.33}}\n\\put(34.33,17.67){\\circle*{1.33}}\n\\put(4.33,1.67){\\line(1,0){60.33}}\n\\put(22.33,17.67){\\line(1,0){24.00}}\n\\put(80.33,12.00){\\line(1,0){45.00}}\n\\end{picture}\n\\end{center}\n\nThe label of each edge of these diagrams is $x$.\n\nNotice that each cell in a diagram can be considered as a diagram\nitself. If the cell corresponds to a relation $u=v$, we shall denote\nthis diagram by $(u=v)$.\n\nLet $\\Delta$ be a diagram over ${\\cal P}$. Suppose that it contains a pair\nof cells $\\pi'$, $\\pi''$ such that the bottom path of $\\pi'$ coincides\nwith the top path of $\\pi''$ and the top label of $\\pi'$ equals the\nbottom label of $\\pi''$. In this case we say that the cells $\\pi'$\nand $\\pi''$ form a {\\em dipole\\\/} in $\\Delta$. One can define the\noperation of removing the dipole. Namely, given a dipole formed by\n$\\pi'$, $\\pi''$, we first remove the common boundary of these cells\nand then glue $\\topp{\\pi'}$ with $\\bott{\\pi''}$ (the paths we glue\nhave the same label). We get a new diagram over ${\\cal P}$ that has fewer\ncells. Proceeding in such a way, we get to a diagram that has no dipoles,\ncalled a {\\em reduced\\\/} diagram.\n\nKilibarda \\cite{KilDiss,Kil} proved that the process of cancelling\ndipoles in diagrams is confluent, that is, the result does not\ndepend on the order in which we remove the dipoles. Two diagrams are\ncalled {\\em equivalent} if the corresponding reduced diagrams are\nthe same.\n\nFor every word $w\\in X^+$, the set of all reduced $(w,w)$-diagrams\nwith operation ``concatenation followed by removing all dipoles\" is\na group ~\\cite[Lemma~5.2]{GS1}, denoted by ${\\cal D}({\\cal P},w)$.\n\n\\begin{defn}\nThe group ${\\cal D}({\\cal P},w)$ is called a {\\em diagram group\\\/} over a semigroup\npresentation ${\\cal P}$ with {\\em base\\\/} $w$.\n\\end{defn}\n\n\\begin{example}{\\rm \\cite[Ex.~6.4]{GS1}}\nThe diagram group ${\\cal D}({\\cal P},x)$ over the semigroup presentation\n${\\cal P}=\\langle\\, x\\mid x^2=x\\,\\rangle$ with base $x$ is R.\\,Thompson's group $F$\nthat has the following group presentation:\n$$\n\\langle\\, x_0,x_1,x_2,\\ldots\\mid x_jx_i=x_ix_{j+1}\\ (j>i)\\,\\rangle.\n$$\n\\end{example}\n\nThere is one more important binary operation on diagrams. Given two diagrams\n$\\Delta_1$, $\\Delta_2$ over ${\\cal P}$, one can identify $\\tau(\\Delta_1)$ and\n$\\iota(\\Delta_2)$. This gives a new diagram over ${\\cal P}$ denoted by\n$\\Delta_1+\\Delta_2$. This operation is associative but not commutative.\n\n\n\\subsection{Universal diagram groups}\\label{universal}\n\n\nIn \\cite{GS3}, Guba and Sapir found fi\\-ni\\-tely presented diagram\ngroups each of which contains every countable diagram group as a\nsubgroup. Such diagram groups are called {\\em universal\\\/}. For\ninstance, the diagram group $U={\\cal D}({\\cal P},a)$, where ${\\cal P}=\\langle\\, x,a\\mid\nx^3=x^2,ax=a\\,\\rangle$ is universal. It has the following Thompson-like\ngroup presentation:\n$$\n\\langle\\, x_0,x_1,x_2,\\ldots\\mid x_jx_i=x_ix_{j+1}\\ (j-i>1)\\,\\rangle.\n$$\nIt is easy to see that $U$ can be generated by $x_0$, $x_1$, $x_2$.\nIndeed, $x_n=x_2^{x_0^{n-2}}$ for all $n\\ge3$ ($a^b=b^{-1}ab$ by\ndefinition). (In fact this group is finitely presented \\cite{GS3}.)\n\nWe are going to describe explicitly the procedure of expressing an\nelement of $U$ as a word in generators $x_0$, $x_1$, $x_2$, \\dots\\,.\nLet $g\\in U$ be represented by an $(a,a)$-diagram $\\Delta$ over ${\\cal P}$.\nOne can decompose $\\Delta$ into a product of atomic diagrams. It is easy\nto see that the atomic diagrams that can occur have the form\n$(1,a=ax,x^m)^{\\epsilon}$, where $m\\ge0$, $\\epsilon=\\pm1$ or the form\n$(ax^k,x^2=x^3,x^m)^{\\epsilon}$, where $k,m\\ge0$, $\\epsilon=\\pm1$. In the first case\nwe assign the identity to the atomic diagram and in the second case we assign\nto it the element $x_m^{\\epsilon}$. Multiplying these elements, we get an expression\nfor the element $g\\in U$, see~\\cite[Section~6]{GS3}.\n\nGiven a diagram $\\Xi$ over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$, we can canonically\nassign a diagram $\\Delta$ over $\\langle\\, x,a\\mid x^3=x^2,ax=a\\,\\rangle$ as follows.\nFirst we take the sum $\\varepsilon(a)+\\Xi$ and then add a number of cells of\nthe form $a=ax$ on the top to make the top label equal $a$. Then we add\na number of cells of the form $ax=a$ on the bottom in order to make the\nbottom label also equal $a$. The result is an $(a,a)$-diagram that\nrepresents an element of the group $U$.\n\nOn the above picture, where two diagrams over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$ are\ndrawn, the described operation leads to the following elements of $U$,\nrespectively: $x_3x_1^{-1}x_0^{-1}x_3^{-2}x_1^{-1}$ and $x_1^{-1}x_0$.\nNotice that the way to decompose a diagram into the product of atomic\nfactors is not unique in general. For the first case, we always chose the\nrightmost cell that can be included into the next atomic factor. This\nprocedure will be described later in details.\n\n\n\\section{Main Results}\n\n\\begin{defn}\nLet $M$ be a set and $p,q\\colon M\\to[0,\\infty)$ be arbitrary functions. Then\n$p\\preceq q$ if there exists a constant $c>0$ such that $p(m)\\le cq(m)$ for every\n$m\\in M$. These functions are equivalent, $p\\sim q$, if $p\\preceq q$ and $q\\preceq p$.\n\\end{defn}\n\nLet $G$ be a group generated by a set $\\Sigma$. For any $g\\in G$, we denote by\n$\\ell_\\Sigma(g)$ the length of the shortest word over $\\Sigma^{\\pm1}$ that\nrepresents $g$. Notice that if $G$ is finitely generated, then all functions of the\nform $\\ell_\\Sigma$ are equivalent for all finite sets of generators. Therefore, the\nfunction $g\\mapsto\\ell(g)$ is unique up to equivalence. The function $\\ell$ on $G$\ndefines the {\\em word length metric\\\/} on $G$ as follows: the (word) distance\nbetween $g_1,g_2\\in G$ is $\\ell(g_1^{-1}g_2)$.\n\nWe will also omit the subscript $\\Sigma$ on $\\ell$ if the generating set of $G$ is clear.\n\nLet $G={\\cal D}({\\cal P},w)$ be a diagram group over a semigroup presentation ${\\cal P}$ with base $w$.\nLet $\\psi\\colon H\\hookrightarrow G$ be an embedding of a finitely generated group $H$\ninto $G$.\n\n\\begin{defn}\nWe say that $\\psi\\colon H\\hookrightarrow G$ is a $B$-{\\em embedding\\\/}, whenever\n$\\#(h)\\sim\\ell(h)$ for all $h\\in H$. Here $\\#(h)$ is the number of cells in the\nreduced $(w,w)$-diagram representing $\\psi(h)$.\n\\end{defn}\n\nAn obvious argument implies that $\\#(g)\\preceq\\ell(g)$ in all cases. Indeed, let $C$\nbe the maximum number of cells that represent generators of $H$. Then any element\nthat has length $n$ in these generators can be represented by a diagram over ${\\cal P}$\nwith at most $Cn$ cells.\n\nHowever, not every embedding of the above form is a $B$-embedding.\n\n\\begin{example}\nLet us consider the direct product $G={\\mathbb F}_2\\times{\\mathbb F}_2$ of\ntwo free groups of rank $2$. It is known that this is a diagram group, see,\nfor example, \\cite[Section~8]{GS1}. By a well-known result of Mikhailova \\cite{Mikh},\nthere is a finitely generated subgroup $H$ in $G$ with undecidable membership problem.\nThis implies that no recursive function $f(n)$ can have the property\n$\\ell(h)\\le f(\\#(h))$ for all $h\\in H$.\n\\end{example}\n\nThe most important case is when $G={\\cal D}({\\cal P},w)$ is a finitely generated diagram\ngroup and $H=G$ with the identical embedding $\\psi$.\n\n\\begin{defn}\nA finitely generated diagram group $G={\\cal D}({\\cal P},w)$ over a semigroup\npresentation ${\\cal P}$ with base $w$ {\\em has property\\\/} B whenever the\nidentical embedding $G\\hookrightarrow G$ is a $B$-embedding.\n\\end{defn}\n\nIn other words, for all $g\\in G$, one has $\\#(g)\\sim\\ell(g)$, where $\\#(g)$\nis the number of cells in the reduced diagram representing $g$.\n\nNotice that we cannot simply say that $G$ has property B itself.\nThis concept depends on the presentation ${\\cal P}$ and base $w$.\n\nR.\\,Thompson's group $F$ has property B as a diagram group over $\\langle\\, x\\mid\nx^2=x\\,\\rangle$ with base $x$. This immediately follows from a result of\nBurillo \\cite{burillo}.\n\n\\begin{proof}[Proof of Theorem \\ref{dghalf}] Let $G$ be a finitely generated\ngroup that can be B-embedded into a diagram group. We need to show\nthat the Hilbert space compression of $G$ is at least $1\/2$.\n\nGiven a diagram group ${\\cal D}({\\cal P},w)$, let us build the following geometric object\nassociated with this group.\nLet us take all reduced diagrams over ${\\cal P}$ that have $w$ as a top label. We\nidentify all top paths of these diagrams. This gives a 1-path $p$\nlabelled by $w$.\n\nSuppose that there are two reduced $(w,\\cdot)$-diagrams $\\Delta_1$,\n$\\Delta_2$ with decompositions of the form\n$\\Delta_i\\equiv\\Delta_i'\\circ\\Delta_i''$, where $i=1,2$ and\n$\\Delta_1'$, $\\Delta_2'$ are isotopic. Then we identify $\\Delta_1'$\nwith $\\Delta_2'$ via this isotopy. We do that for all pairs of\ndiagrams and all decompositions of them with the above property. The\nobject we get as a result is a directed 2-complex $\\cal T$, which turns\nout to be a rooted 2-tree in the sense of \\cite{GS3}. This directed\n$2$-complex can be viewed as a semigroup presentation if we assign\ndifferent labels to different edges and consider pairs of words\nwritten on the boundaries of the cells as relations. It is proved in\n\\cite{GS3} that for any path $q$ in $\\cal T$ with the same endpoints as\n$p$ there exists a unique reduced $(p,q)$-diagram over $\\cal T$.\n\nLet $\\mathbf F$ be the set of all geometric 2-cells of $\\cal T$. By\n${\\mathbb R}^{\\mathbf F}$ we denote the vector space over ${\\mathbb R}$\nwith ${\\mathbf F}$ as a basis. Clearly, this vector space is a\nsubset in Hilbert space ${\\cal H}=\\ell_2(\\mathbf F)$.\n\nEvery element $g$ of the diagram group ${\\cal D}({\\cal P},w)$ can be uniquely\nrepresented by a reduced $(w,w)$-diagram $\\Delta$. This diagram can\nbe naturally embedded into $\\cal T$. (The top of the diagram is identified\nwith the path $p$ under this embedding.) Let us assign to $g$ a function\n$\\phi_g$ from $\\mathbf F$ to $\\mathbb R$, where $\\phi_g(f)=1$ if\n$f\\in\\mathbf F$ is contained in the image of $\\Delta$ under the\nabove embedding and $\\phi_g(f)=0$ otherwise.\n\nSo we have a mapping $\\phi\\colon{\\cal D}({\\cal P},w)\\to{\\cal H}$ from the diagram group\n${\\cal D}({\\cal P},w)$ to ${\\cal H}$ defined by the rule $g\\mapsto\\phi_g$.\n\nAs we have already mentioned in the introduction, one can define a\ncanonical {\\em diagram metric\\\/} on ${\\cal D}({\\cal P},w)$ as follows: given\ntwo elements $g_1,g_2\\in{\\cal D}({\\cal P},w)$, one can define the {\\em diagram\ndistance\\\/}, denoted by ${\\mathrm{dist}}_d(g_1,g_2)$ between these elements as\nthe number of cells in the reduced diagram over ${\\cal P}$ representing\n$g_1^{-1}g_2$.\n\nSuppose that the diagram distance between two diagrams $\\Delta_1$,\n$\\Delta_2$ from ${\\cal D}({\\cal P},w)$ equals $n$. Let us consider the images\nof the diagrams $\\Delta_i$ ($i=1,2$) in $\\cal T$. They can be\ndecomposed as $\\Delta_i\\equiv\\Psi\\circ\\bar\\Delta_i$ ($i=1,2$), where\n$\\Psi$ is the ``greatest common divisor\" of $\\Delta_1$ and\n$\\Delta_2$. In this case we see that $\\bar\\Delta_1$, $\\bar\\Delta_2$\ndo not have common cells in $\\cal T$. The total number of cells in\n$\\bar\\Delta_1$ and $\\bar\\Delta_2$ equals $n$. Since\n$\\phi_{g_1}-\\phi_{g_2}$, as an element of ${\\mathbb R}^{\\mathbf F}$\nis a vector whose coordinates are $0, 1$ or $-1$, and the number of\nnon-zero coordinates is $n$, we conclude that the norm of\n$\\phi_{g_1}-\\phi_{g_2}$ is $\\sqrt{n}$.\n\nNow given a finitely generated group $G$, which is B-embedded into a\ndiagram group ${\\cal D}({\\cal P},w)$, we see that the word length metric in\n$G$ is equivalent to the diagram metric induced on $G$ as a subset\nin ${\\cal D}({\\cal P},w)$. Therefore, for some positive constants $C_1$, $C_2$\none has inequalities\n$$C_1\\sqrt{{\\mathrm{dist}}(g_1,g_2)}\\le\\|\\phi_{g_1}-\\phi_{g_2}\\|\\le\nC_2\\sqrt{{\\mathrm{dist}}(g_1,g_2)}\\le C_2{\\mathrm{dist}}(g_1,g_2).$$ Hence the Hilbert\nspace compression of $G$ is at least $\\frac12$. \n\\end{proof}\n\n We will use\nthe following result which is a generalization of the well-known\nparallelogram theorem to higher dimensions \\cite{DeMa}. Namely, let\n$E^n\\subset {\\mathbb R}^n$ be an $n$-dimensional hypercube. Suppose\nthat we have a mapping of the set of vertices of $E^n$ into a metric\nspace $M$. In this case we will say that we have a {\\em skew cube\\ }\nin $M$. For every edge of $E^n$ (there are exactly $2^{n-1}n$ of\nthem), by an edge of the skew cube we will mean the distance in $M$\nbetween the images of the endpoints of the edge. Similarly, for each\n(long) diagonal of $E^n$ (which connects opposite vertices of $E^n$)\nwe consider the corresponding {\\em diagonal\\\/} of the skew cube.\n\n\\begin{lm}{\\rm (Skew Cube Inequality \\cite{DeMa})}\\label{cube}\nFor every skew cube in a Hilbert space, the sum of squares of its\ndiagonals does not exceed the sum of squares of its edges.\n\\end{lm}\n\n\\begin{proof}[Proof of Theorem~\\ref{Fhalf}] We need to prove that the\nHilbert space compression of R.\\,Thompson's group $F$ equals $1\/2$.\n\nThe fact that the compression is at least $1\/2$, follows from\nTheorem \\ref{dghalf}.\n\nFor any $n\\ge0$, let us define $2^n$ elements of $F$ that commute\npairwise. All these elements will be reduced $(x,x)$-diagrams over\n${\\cal P}=\\langle\\, x\\mid x^2=x\\,\\rangle$. For $n=0$, let $\\Delta$ be the diagram\nthat corresponds to the generator $x_0$. Namely, if $\\pi$ is\nthe diagram that consists of one cell of the form $x=x^2$, then\n$\\Delta$ is $\\pi\\circ(\\varepsilon(x)+\\pi)\\circ(\\pi+\\varepsilon(x))^{-1}\\circ\\pi^{-1}$ by\ndefinition. It has 4 cells.\n\nSuppose that $n\\ge1$ and we have already constructed diagrams $\\Delta_i$\n($1\\le i\\le2^{n-1}$) that commute pairwise. For every $i$ we consider two\n$(x^2,x^2)$-diagrams: $\\varepsilon(x)+\\Delta_i$ and $\\Delta_i+\\varepsilon(x)$. We get $2^n$\nspherical diagrams with base $x^2$ that obviously commute pairwise. It remains\nto conjugate them to obtain $2^n$ spherical diagrams with base $x$ having the\nsame property. Namely, we take $\\pi\\circ(\\varepsilon(x)+\\Delta _i)\\circ {\\pi}^{-1}$ and\n$\\pi\\circ(\\Delta_i+\\varepsilon(x))\\circ\\pi ^{-1}$.\n\nLet us denote the elements of $F$ obtained in this way by $g_i$ ($1\\le i\\le2^n$).\nThese elements define a $2^n$-dimensional skew cube in $F$. It follows easily from\nthe construction that each $g_i$ has exactly $2n+4$ cells as a diagram. So the word\nlength of each $g_i$ is $O(n)$. Now for any $\\epsilon_i=\\pm1$ ($1\\le i\\le2^n$) we consider\nthe product of the form $g=g_1^{\\epsilon_1}\\cdots g_{2^n}^{\\epsilon_{2^n}}$. It is easy to see\nfrom definitions that the diagram that represents $g$ has the form\n$\\Gamma _n\\circ(\\Delta^{\\epsilon _1}+\\cdots+\\Delta^{\\epsilon_{2^n}})\\circ\\Gamma _n^{-1}$,\nwhere $\\Gamma_n$ is defined by induction in the following way: $\\Gamma _0=\\pi$,\n$\\Gamma_{k+1}=\\pi\\circ(\\Gamma_k+\\Gamma_k)$ ($k\\ge0$). In particular,\nthe number of cells in $\\Gamma_n$ equals $2^n-1$ and so $g$ is represented by a\ndiagram with exactly $2(2^n-1)+4\\cdot2^n=3\\cdot2^{n+1}-2$ cells. Since $F$ satisfies\nproperty $B$, the word length of an element $g=g_1^{\\epsilon_1}\\cdots g_{2^n}^{\\epsilon_{2^n}}$\nwill be at least $C2^n$, where $C>0$ is a constant that does not depend on the\n$\\epsilon_i$'s.\n\nNow consider a uniform embedding of $F$ into a Hilbert space ${\\cal\nH}$ with linear $\\rho_2$. In the image of our skew cube in $F$\nformed by $g_1$, \\dots, $g_{2^n}$, each edge will be equal to\n$O(n)$. The Skew Cube Inequality implies that there exists a\ndiagonal of the corresponding skew cube in ${\\cal H}$ that does not\nexceed $O(n)\\cdot\\sqrt{2^n}=O(n2^{\\frac{n}{2}})$. This means that\nsome points in $F$ that were at distance $d\\ge C2^n$ from each other\nwill be mapped to points in ${\\cal H}$ at distance\n$O(n2^{\\frac{n}{2}})=O(\\sqrt{d}\\log_2 d)$. Therefore, the\ncompression cannot exceed $1\/2$. \\end{proof}\n\n\n\nIn order to prove Theorem \\ref{BurU}, we need the  following simple\nconstruction, called the {\\em rightmost decomposition\\\/} of a\ndiagram. The idea of such a decomposition applied to the\npresentation $\\langle\\, x\\mid x^2=x\\,\\rangle$ was used in \\cite{GS2} to get a\nnew normal form for the elements of R.\\,Thompson's group $F$.\n\n\\begin{defn}{\\rm (Rightmost decomposition of a diagram)}\nLet $\\Delta$ be a diagram over a semigroup presentation ${\\cal P}$. A\n{\\em rightmost decomposition\\\/} of the diagram $\\Delta$ into a\nproduct of atomic factors is defined as follows. We proceed by\ninduction on the number of cells in $\\Delta$. If $\\Delta$ has no\ncells, then the product has no factors. Let $\\Delta$ have cells.\nThen it has at least one top cell (a cell whose top path is a\nsubpath in the top path of $\\Delta$). Let us consider the rightmost\ntop cell $\\pi$ of $\\Delta$. Let $\\topp{\\Delta}=p\\topp{\\pi}q$. Then\n$\\Delta$ is a concatenation of an atomic diagram\n$\\Pi=\\varepsilon(p)+\\pi+\\varepsilon(q)$ and some diagram $\\Delta'$ that has fewer\ncells. (Here $\\Delta'$ is obtained from $\\Delta$ by deleting the\npath $\\topp{\\pi}$.) Taking the product of $\\Pi$ and the rightmost\ndecomposition of $\\Delta'$, we obtain the rightmost decomposition of\n$\\Delta$.\n\\end{defn}\n\n\\begin{proof}[Proof of Theorem~\\ref{BurU}] Let ${\\cal P}=\\langle\\, x,a\\mid\nx^3=x^2,ax=a\\,\\rangle$. We need to prove that the universal diagram group\n$U={\\cal D}({\\cal P},a)$ has property B.\n\nLet $\\Delta$ be an $(a,a)$-diagram over ${\\cal P}$ with $N$ cells. It\nsuffices to prove that the element $g\\in U$ represented by $\\Delta$\nhas length at most $KN$ in generators $x_0$, $x_1$, $x_2$, where\n$K>0$ is a constant independent on $g$.\n\nLet $e_0$ be the top edge of $\\Delta$. Suppose that there is an\n$(a,ax)$-cell whose top edge coincides with $e_0$. Then we denote by\n$e_1$ the edge labelled by $a$ on the bottom path of this cell. If\n$e_1$ is the top path of an $(a,ax)$-cell, then $e_2$ denotes the\nedge on the bottom path of this cell labelled by $a$. Proceeding in\nsuch a way, we finally obtain a sequence of edges $e_0$, \\dots,\n$e_k$ ($k\\ge0$).\n\nAnalogously, changing in the previous paragraph top by bottom, we define the sequence\nof edges $f_0$, \\dots, $f_m$  ($m\\ge0$), where $f_0$ is the bottom edge of $\\Delta$.\nA very easy geometric observation is that $e_k$ must coincide with $f_m$. It is also\neasy to see that $\\Delta$ is a concatenation of the form\n$\\Delta=\\Delta_1\\circ(\\varepsilon(a)+\\Delta')\\circ\\Delta_2$, where $\\Delta_1$ consists of\n$(a,ax)$-cells only, $\\Delta_2$ consists of $(ax,a)$-cells only and $\\Delta'$ is an\n$(x^k,x^m)$-diagram over ${\\cal P}'=\\langle\\, x\\mid x^3=x^2\\,\\rangle$.\n\nLet us apply the rightmost decomposition to $\\varepsilon(a)+\\Delta'$. Each factor is an\natomic diagram of the form $(ax^s,x^2=x^3,x^t)^\\epsilon$, where $s,t\\ge0$, $\\epsilon=\\pm1$.\nAccording to the description of $U$ given in subsection~\\ref{universal}, this atomic\ndiagram corresponds to $x_t^\\epsilon$. (Notice also that no generators correspond to atomic\ndiagram with $(a,ax)$-cells and\/or their inverses.)\n\nTherefore, the rightmost decomposition of $\\Delta'$ allows us to decompose $g$ as a\nproduct (in $U$) of the form\n\\be{epss}\ng=x_{j_1}^{\\epsilon_1}\\cdots x_{j_r}^{\\epsilon_r}\\hbox{ with }\\epsilon_i=\\pm 1,\n\\end{equation}\nwhere $r=N-k-m$ is the number of cells in $\\Delta'$. We need to establish some easy\nproperties of the subscripts and the superscripts in (\\ref{epss}).\n\nSuppose that (\\ref{epss}) contains a subword of the form $x_i^{\\epsilon}x_j$. Then we claim\nthat $i\\le j+1$. Indeed, otherwise the cell that corresponds to $x_j$ is located to\nthe right of the cell corresponding to $x_i^{\\epsilon}$. (One can also see that $i>j+1$\nwould imply that $x_i^{\\epsilon}x_j$ equals $x_jx_{i+1}^{\\epsilon}$ in $U$.)\n\nNow suppose that (\\ref{epss}) contains a subword of the form\n$x_i^{\\epsilon}x_j^{-1}$. In this case we claim that $i\\le j+2$. (Now\n$i>j+2$ would also imply that the cell that corresponds to $x_j^{-1}$ is located to\nthe right of the one corresponding to $x_i^{\\epsilon}$. The element $x_i^{\\epsilon}x_j^{-1}$\nwould be also equal to $x_j^{-1}x_{i-1}^{\\epsilon}$ in $U$.)\n\nFor every $j\\ge0$ we replace each letter $x_j$ in (\\ref{epss}) by\nthe word $u_j^{-1}v_ju_j$, where $u_j=1$, $v_j=x_j$ if $j=0,1$ and\n$u_j=x_0^{j-2}$, $v_j=x_2$ if $j\\ge2$. We get the following word\n$W$ in generators $x_0$, $x_1$, $x_2$:\n$$\nW=u_{j_1}^{-1}v_{j_1}^{\\epsilon_1}u_{j_1}u_{j_2}^{-1}v_{j_2}^{\\epsilon_2}\\cdots\nv_{j_r}^{\\epsilon_r}u_{j_r}.\n$$\n\nSince $x_j=u_j^{-1}v_ju_j$ in $U$, the word $W$ represent the same\nelement $g\\in U$. Now let $W_i$ denote the freely irreducible form\nof the word $u_{j_i}u_{j_{i+1}}^{-1}$ for all $1\\le i<r$. It is\nalso convenient to define the words $W_0=u_{j_1}^{-1}$ and\n$W_r=u_{j_r}$. We are going to estimate the length of the word\n$$\n\\overline{W}=W_0v_{j_1}^{\\epsilon_1}W_1\\cdots v_{j_r}^{\\epsilon_r}W_r.\n$$\n\nWe know that $|W_0|=j_1\\le k-2$ (the first cell chosen in\n$\\Delta'$ is at a distance $j_1$ from the rightmost point of\n$\\Delta$ and the length of its top path is at least $2$).\nAnalogously, $|W_r|=j_r\\le m-2$. Also $|v_{j_1}|+\\cdots+|v_{j_r}|=r$. So\nwe estimate some part of the length of $\\overline{W}$ as follows:\n\\be{wv}\n|W_0|+|W_r|+|v_{j_1}|+\\cdots+|v_{j_r}|\\le k+m+r-4<N.\n\\end{equation}\n\nIt remains to estimate the sum $|W_1|+\\cdots+|W_{r-1}|$. It is\neasy to see from the definitions that $|W_i|\\le|j_{i+1}-j_i|$ for\nall $1\\le i<r$. Let $I$ be the set of all $1\\le i<r$ such that\n$j_i>j_{i+1}$. Then $|W_1|+\\cdots+|W_{r-1}|\\le\\sum_{1\\le i<r}|j_{i+1}-j_i|=S_1+S_2$,\nwhere $S_1=\\sum_{i\\in I}(j_i-j_{i+1})$ and $S_2=\\sum_{i\\notin I}(j_{i+1}-j_i)$.\n\nFor every $i\\in I$, we know that $j_i-j_{i+1}\\le2$. Therefore,\n$S_1\\le2|I|\\le2(r-1)<2r$. On the other hand,\n$$\nS_2-S_1=\\sum_{1\\le i<r}(j_{i+1}-j_i)=j_r-j_1\\le j_r\\le m-2<k+m=N-r.\n$$\nThis gives $S_2=S_1+(S_2-S_1)<N+r$ and so $S_1+S_2<N+3r\\le4N$.\nUsing (\\ref{wv}), we finally have\n$$\n|\\overline{W}|=(|W_0|+|W_r|+|v_{j_1}|+\\cdots+|v_{j_r}|)+(|W_1|+\\cdots+|W_{r-1}|)<5N.\n$$\nSo our statement is true for $K=5$.\n\nNotice that the constant here is not optimal. \\end{proof}\n\nLet us recall the definition of the (restricted) wreath product of groups. Let $G$, $H$\nbe two groups. If $\\phi$ is a function from $H$ to $G$, then the {\\em support\\\/} of $\\phi$\nis the set $\\mathop{\\mathbf{supp}}(\\phi)=\\{\\,b\\in H\\mid \\phi(b)\\ne1\\,\\}$. The restricted wreath product\n$G\\wr H$ of the above groups will consist of all pairs of the form $(b,\\phi)$, where $b\\in H$\nand $\\phi$ is a function from $H$ to $G$ with finite support. The group operation is defined\nin the standard way, that is, $(b_1,\\phi_1)(b_2,\\phi_2)=(b_1b_2,\\phi_1^{b_2}\\phi_2)$, where\n$\\phi^\\beta(b)=\\phi(b\\beta)$ by definition ($\\beta\\in H)$. For each of the groups $G$, $H$ we\nhave a canonical embedding into $G\\wr H$. Namely, $a\\mapsto(1,\\phi_a)$, where $\\phi_a(1)=a$,\n$\\phi_a(b)=1$ whenever $b\\ne1$. Also $b\\mapsto(b,\\mathbf{1})$, where $\\mathbf{1}$ is the\nfunction from $H$ to $G$ that takes all elements to 1.\n\nWe will deal with restricted wreath products of finitely generated groups. If we fix some\ngenerating sets for $G$ and for $H$, then the union of these sets (via canonical embeddings)\nwill generate $G\\wr H$. Hence we have three length functions: on $G$, on $H$, and on $G\\wr H$.\nFor simplicity, we will always freely speak about lengths in each of these three groups\nprovided the generating sets for $G$ and for $H$ are clear. The same will concern Cayley\ngraphs of these groups. The same symbol $\\ell$ will be used to denote the length function\nfor each of these groups. This will not lead to a confusion.\n\nWorking with lengths, we need the following fact proved in \\cite{Parry} (see Theorem 1.2).\n\n\\begin{lm}\n\\label{Parr}\nLet $G$, $H$ be finitely generated groups with fixed generating sets. Then the length\nof an element $(b,\\phi)\\in G\\wr H$ is equal to the sum $\\sum\\ell(\\phi(\\beta))+M$, where\n$\\ell(a)$ is the length of $a\\in G$ and $M$ is the length of the shortest path in the\nCayley graph of $H$ starting at the identity, ending at $b$ and visiting each element\nof $\\mathop{\\mathbf{supp}}(\\phi)$ at least once. The sum is taken over all $\\beta\\in\\mathop{\\mathbf{supp}}(\\phi)$.\n\\end{lm}\n\nLet $G={\\cal D}({\\cal P},w)$ be a diagram group. We say that the base $w$ is {\\em protected\\\/}\nif $w$ is a letter such that $w=uwv$ modulo ${\\cal P}$ implies that both words $u$, $v$ are\nempty (for any $u$, $v$). For any diagram group $G={\\cal D}({\\cal P},w)$, one can construct a new\nsemigroup presentation ${\\cal P}'$ adding 3 new letters $a$, $b$, $c$ to the alphabet and one\nnew defining relation $a=bwc$. It is not hard to see that $G$ will be isomorphic to the\ndiagram group ${\\cal D}({\\cal P}',a)$, where the base $a$ is now protected.\n\nIt is shown in \\cite[Theorem 11]{GS5} that, given a diagram group $G={\\cal D}({\\cal P},y)$ with\nprotected base $y$, one can construct a new semigroup presentation $\\bar{\\cal P}$ adding two\nnew letters $x$, $z$ to the alphabet and two new defining relations $x=xy$, $yz=z$\n(see also Example 10 before the theorem we refer to). In this situation, the diagram\ngroup ${\\cal D}(\\bar{\\cal P},xz)$ will be isomorphic to the restricted wreath product $G\\wr{\\mathbb{Z}}$.\n(The concept of a protected base we gave slightly differs from the original one although\nthe proof goes without any essential changes.) Now we are going to prove that if $G$ has\nproperty B (as a diagram group over ${\\cal P}$) then $G\\wr{\\mathbb{Z}}$ also has property B (as a diagram\ngroup over $\\bar{\\cal P}$).\n\n\\begin{lm}\n\\label{GwrZB}\nIn the above notation, if $G={\\cal D}({\\cal P},y)$ has property B, then $G\\wr{\\mathbb{Z}}={\\cal D}(\\bar{\\cal P},xz)$\nalso has property B.\n\\end{lm}\n\n\\proof\nIt suffices to prove that any diagram over $\\bar{\\cal P}$ with $N$ cells has length $O(N)$ in\nthe group $G\\wr{\\mathbb{Z}}$. Any reduced $(xz,xz)$-diagram over $\\bar{\\cal P}$ can be obtained as\nfollows. Let us take finitely many $(y,y)$-diagrams over ${\\cal P}$ and let $\\Xi$ be their\nsum. Now let us add a few cells of the form $x=xy$ and a few cells of the form $z=yz$ to\nthe diagram $\\varepsilon(x)+\\Xi+\\varepsilon(z)$ on the top in order to make the top label equal $xz$. Then\nwe add a few cells of the form $xy=x$ and $yz=z$ on the bottom to make the bottom label also equal $xz$.\nThe result is an $(xz,xz)$-diagram $\\Delta$. Let us denote by\n$o$ the vertex on the bottom path of $\\Delta$ that splits this path into the product of two\nedges. Clearly, the vertex $o$ is contained in $\\Xi$ and splits it into a sum. Let\n$\\Xi=\\Xi_k+\\cdots+\\Xi_0+\\Xi_{-1}+\\cdots+\\Xi_{-m}$, where each\n$\\Xi_j$ ($-m\\le j\\le k$) is a $(y,y)$-diagram and the vertex $o$ subdivides $\\Xi$\ninto the sum of the two summands. All the $\\Xi_j$ that form the first summand satisfy\n$j\\ge0$, the ones that form the second summand satisfy $j<0$. (Notice that each of the\nsummands can be empty.)\n\nLet $g_j\\in G$ be the element of ${\\cal D}({\\cal P},y)$ represented by $\\Xi_j$ ($-m\\le j\\le k$).\nWe also define $g_j=1$ for any integer $j$ not in $[-m,k]$. Then, according to\n\\cite[Example 10]{GS5}, the diagram $\\Delta$ represents the element $h$ of  the group\n$G\\wr\\langle\\, t\\,\\rangle$ of the following form: $(t^d,\\phi)$, where $d$, $\\phi$ are defined as\nfollows. The exponent $d$ on $t$  is the difference between the number of $(z,yz)$-cells\nand the number of $(yz,z)$-cells in $\\Delta$. The function $\\phi$ from ${\\mathbb{Z}}=\\langle\\, t\\,\\rangle$ is\ndefined by the rule $\\phi(t^j)=g_j$ for any integer $j$.\n\nIt remains to estimate the length of $h$. We will use Lemma \\ref{Parr}. We see that\nthe length of $h$ equals\n\\be{sumss}\n\\sum\\limits_{j=-m}^k\\ell(g_j)+M,\n\\end{equation}\nwhere $M$ is the length of the shortest path in the Cayley graph of ${\\mathbb{Z}}=\\langle\\, t\\,\\rangle$ that\nstarts at the identity, ends at $t^d$ and visits all those vertices $t^j$ for which\n$g_j\\ne1$. It is clear that $M\\le2(k+m+1)+|d|$. Obviously, $|d|$ does not exceed the\nnumber of cells in $\\Delta$. It is also clear that $k+m+1$ is the number of $(x,xy)$-cells\nplus the number of $(z,yz)$-cells in $\\Delta$. This implies $M=O(N)$.\n\nSince $G={\\cal D}({\\cal P},y)$ has property B, there exists a constant $C>0$ such that the\nlength of each $g\\in G$ does not exceed $C\\cdot \\#(g)$, where $\\#(g)$ denotes the\nnumber of cells in the reduced diagram over ${\\cal P}$ representing $g$. Therefore, the\nfirst summand in~(\\ref{sumss}) does not exceed\n$C(\\#(g_{-m})+\\cdots+\\#(g_0)+\\cdots+\\#(g_k))\\le CN$. Thus we finally have that the\nlength of $h$ in $G\\wr{\\mathbb{Z}}$ is $O(N)$.\n\\endproof\n\n\\begin{cy}\n\\label{zwrzb}\nLet ${\\cal P}=\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2,b_2=b\\,\\rangle$. Then\nthe restricted wreath product ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ has property B as a diagram group\n${\\cal D}({\\cal P},ac)$.\n\\end{cy}\n\nIndeed, ${\\mathbb{Z}}$ is the diagram group over $\\langle\\, b,b_1,b_2\\mid b=b_1,b_1=b_2,b_2=b\\,\\rangle$\nwith protected base $b$.\n\\vspace{2ex}\n\nLet $H$ be a finitely generated group with fixed generating set. By its\n{\\em growth function\\\/} $\\gamma(n)$ (with respect to this set of generators)\nwe mean the number of elements in the ball of radius $n$ around the identity\nin the Cayley graph of $H$.\n\nTheorems \\ref{wr} and \\ref{wr1} will follow from the next result.\n\n\\begin{thm}\\label{wr2}\nLet $H$ be a finitely generated group with fixed generating set. Suppose\nthat its growth function satisfies the condition $\\gamma(n)\\succeq n^k$ for\nsome $k>0$. Then the Hilbert space compression $\\alpha$ of the group ${\\mathbb{Z}}\\wr H$\nsatisfies the following inequality:\n$$\n\\alpha\\le\\frac{1+k\/2}{1+k}.\n$$\n\\end{thm}\n\n\\proof Let $x$ be a generator of ${\\mathbb{Z}}$ in the wreath product ${\\mathbb{Z}}\\wr H$. Every\nelement in ${\\mathbb{Z}}\\wr H$ is a pair $(b,\\phi)$ where $b\\in H$, $\\phi$ is a function\n$g\\mapsto x^{m_g}$ from $H$ to ${\\mathbb{Z}}$ with finite support.\n\nLet $B_n$ be the ball of radius $n$ around $1$ in the Cayley graph of $H$.\nConsider the set\n$$\nX_n=\\{\\,w_b=b^{-1}x^{2n+1-|b|}b\\mid b\\in B_n\\,\\},\n$$\nwhere $\\ell(b)$ is the length of $b$. Obviously, $w_b$ has the form $(1,\\phi_b)$ in\n${\\mathbb{Z}}\\wr H$, where the support of $\\phi_b$ is $\\{\\,b\\,\\}$ and $\\phi_b(b)=x^{2n+1-\\ell(b)}$.\nNotice that $\\ell(x^r)=|r|$ in $\\langle\\, x\\,\\rangle$ for all $r\\in{\\mathbb{Z}}$. Hence by Lemma~\\ref{Parr},\nthe length of $w_b$ in ${\\mathbb{Z}}\\wr H$ equals $2n+1-\\ell(b)+M$, where $M=2\\ell(b)$.\nTherefore, $\\ell(w_b)=2n+1+\\ell(g)$ is always between $2n+1$ and $3n+1$. Clearly, all\nelements of $X_n$ pairwise commute. The number of them is exactly $\\gamma(n)$.\n\nIt is easy to see that for every choice of $\\epsilon_b\\in\\{\\,1,-1\\,\\},b\\in B_n$, the\nlength of the element $w=\\prod_{b\\in B_n}w_b^{\\epsilon_b}$ is greater than $n\\gamma(n)$.\nIndeed, $w$ has the form $(1,\\phi)$ in ${\\mathbb{Z}}\\wr H$, where $\\mathop{\\mathbf{supp}}(\\phi)=B_n$ and\n$\\phi(b)=x^{2n+1-\\ell(b)}$ for all $b\\in B_n$. Hence the length of $w$ is at least\n$$\n\\sum_{b\\in B_n}\\ell(\\phi(b))=\\sum_{b\\in B_n}(2n+1-\\ell(b))>n\\gamma(n)\n$$\nby Lemma~\\ref{Parr}.\n\nConsider the skew cube spanned by $X_n$ in ${\\mathbb{Z}}\\wr H$ and its image\nin ${\\cal H}$ under the uniform embedding $f$. The sides of the skew\ncube in $\\cal H$ do not exceed $Cn$ for some constant $n$ since our\ngroup is finitely generated. So the sum of squares of sides does not\nexceed $C^22^{\\gamma(n)-1}\\gamma(n)n^2$. On the other hand, the number of\ndiagonals is $2^{\\gamma(n)-1}$. Hence by the Skew Cube Inequality,\nthere exists a diagonal of the skew cube in $\\cal H$ which does not\nexceed $Cn\\sqrt{\\gamma(n)}$. Thus there exist two points in the\nCayley graph of $G$ at distance $d\\ge n\\gamma(n)$ whose images under\n$f$ are at distance at most $Cn\\sqrt{\\gamma(n)}$. If $\\rho_1(d)=Kd^\\alpha$\nfor some constant $K>0$ in the definition of the uniform embedding, then\n$Cn\\sqrt{\\gamma(n)}\\ge K(n\\gamma(n))^\\alpha$. Hence\n$$\nn^{1-\\alpha}\\succeq(\\gamma(n))^{\\alpha-1\/2}\\succeq n^{k(\\alpha-1\/2)}.\n$$\nThis implies $1-\\alpha\\ge k(\\alpha-1\/2)$ and so $\\alpha\\le(1+k\/2)\/(1+k)$.\n\\endproof\n\n\\begin{thm}\n\\label{wrprod} The Hilbert space compression of the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ belongs to $[1\/2;3\/4]$.\n\\end{thm}\n\n\\proof The lower bound follows from Theorem \\ref{dghalf} and the fact that\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ satisfies property B as a diagram group ${\\cal D}({\\cal P},ac)$ over\n${\\cal P}=\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2,b_2=b\\,\\rangle$\n(Corollary~\\ref{zwrzb}).\n\nThe upper bound follows from Theorem \\ref{wr2} with $G={\\mathbb{Z}}$. Indeed, the growth\nfunction of ${\\mathbb{Z}}$ is linear, that is, $\\gamma(n)\\succeq n^k$ for $k=1$. Hence\n$\\alpha\\le(1+k\/2)\/(1+k)=3\/4$.\n\\endproof\n\nRecall that a finitely generated group has a {\\em su\\-per-po\\-ly\\-no\\-mial\\\/}\ngrowth whenever its growth function $\\gamma(n)$ exceeds $n^k$ for all $k\\ge1$.\n\n\\begin{thm}\nLet $H$ be a finitely generated group with su\\-per-po\\-ly\\-no\\-mial growth. Then the\nHilbert  space compression of ${\\mathbb{Z}}\\wr H$ is at most $1\/2$.\n\\end{thm}\n\n\\proof Indeed, $\\gamma(n)\\succeq n^k$ implies $\\alpha\\le(1+k\/2)\/(1+k)$. The right-hand\nside of this inequality approaches $1\/2$ as $k\\to\\infty$.\n\\endproof\n\n\\begin{cy} The Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$\nis at most $1\/2$.\n\\end{cy}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Intro} Introduction}\n\nIt is common to determine the oscillation frequencies of atoms in a magnetic trap by exciting their motion. A sudden displacement of the trap excites a dipole (centre-of-mass) oscillation of the cloud at the trap frequency, while a sudden compression or de-compression induces quadrupole (width) oscillation of the cloud. In many experiments, the trapped cloud is long and thin, with transverse frequencies of 1\\,kHz or above. The oscillation amplitude is then typically below the resolution of the imaging system and therefore difficult to observe directly. In that case, the oscillation may be driven, and the excitation of the cloud may be detected through the increase in its length, which results from heating. This exhibits a resonant behaviour, with a maximum temperature at a specific driving frequency, related to the natural transverse oscillation frequency.\n\nThis paper presents temperature resonances of magnetically trapped atoms, induced by dipole oscillation, with six different trap frequencies. These traps are formed on an atom chip using the field of a permanently magnetised videotape. A full characterisation of the videotape traps is carried out with particular emphasis on the effects of anharmonicity. Our experimental results are compared with a numerical model that provides valuable insight into the dynamics of the cold atoms moving in the real confining potential of the trap under the important influences of collisions and atom loss.\n\nSeveral references already consider the dipolar excitation of trapped particles by shaking the trap centre \\cite{st1,st2,st10,ions1,st13}. The first three are theory papers \\cite{st1,st2,st10} that describe heating due to laser noise in harmonic optical traps far from resonance.  Ref. \\cite{st2} calculates the evolving energy distribution, accounting for atom loss, in a truncated harmonic trap. Ref. \\cite{ions1} describes measurements of the resonant frequencies and laser cooling rates for ions in a shaken Penning trap. In Ref.\\,\\cite{st13} M. Kumakura et al.  investigate the excitation of neutral atoms in a shaken cloverleaf magnetic trap. They measure resonances in atom loss and temperature and discuss how different ratios of atomic temperature to effective trap depth result in either heating or cooling after shaking. They use a classical 1D equation of motion, without collisions, to illustrate some features of their experiment.\n\nThere have also been discussions of parametric excitation, i.e. modulation of the trap frequency, in various contexts. These include heating and cooling of neutral atoms in optical dipole traps \\cite{st1,st2,st10,st3,st6,st5}, one-dimensional optical lattices \\cite{st4,st6,st10} or magnetic traps \\cite{st17}, and measurement of oscillation frequencies for electrons in a Penning trap \\cite{st11,st12}, ions in a quadrupole trap \\cite{st16} or neutral atoms in a MOT \\cite{st14,st15}. We do not study parametric resonance here because there is no straightforward way to modulate the frequency of our videotape trap without also shaking its position. The same is true of any permanent-magnet atom trap.\n\n\nThe experiments described here determine both the position and the shape of the dipolar temperature resonances. We also investigate these resonances theoretically with the aid of a three-dimensional numerical model that accounts simultaneously for anharmonicity of the trap, atom loss due to finite trap depth, interatomic collisions and evolution of the collision rate during the excitation. All these aspects of our model go beyond what has been done before. Our model yields good quantitative agreement between experiment and theory, even though we use a rather simplified collision model. We anticipate that our simplifying assumption could be exported to a number of ensemble dynamic problems in order to achieve faster simulations.\n\n\\section{\\label{sec:experim} The experiment}\n\n\\begin{figure*}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1x.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1y.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1z.eps}\n\\end{minipage}\n\\\\\n\\vspace{0.3cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2x.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2y.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2z.eps}\n\\end{minipage}\n\\caption{Total field strength of magnetic trap versus position. The transverse bias field is $B_b=5.9\\,\\mathrm{G}$ along $\\hat{x}$. Upper row: overview along each Cartesian direction. Lower row: zoom into the centre shows the extrema of the modulation induced by the shaking field $b=105\\,$mG.}\n\\label{fig:howPotChanges}\n\\end{figure*}\n\nOur experiment uses the field gradient of a magnetized videotape to trap the atoms. The tape, lying in the $xz$ plane, has magnetisation $M_1\\cos(kx)\\hat{x}$, where $k=(2\\pi\/110)\\,\\mu\\mbox{m}^{-1}$. Above the surface, this magnetisation  produces a transverse (i.e. in the $xy$ plane) magnetic field of $\\{B_{x},B_{y}\\}=B_{1}e^{-k y}\\{-\\cos(kx),\\sin(ky)\\}$ \\cite{HindsHughesReview} with $B_1\\simeq 100\\,$G.  The addition of a transverse bias field, $B_{b}\\{\\cos(\\theta),\\sin(\\theta)\\}$, produces an array of zeros in the total transverse field located at a height $y=\\ln(B_{1}\/B_{b})\/k$. On Taylor expanding the field around any of these zeros one finds a quadrupole field in the $xy$ plane whose orientation depends on the angle $\\theta$ of the bias field. The magnitude of the field grows linearly with the cylindrical radius $r$ according to $|B|=k B_{b} r$, providing a cylindrically symmetrical, linear confining potential $\\mu_{B}g_{F}m_{F}|B|$ for weak-field seeking atoms with magnetic quantum number $m_{F}$ and $g$-factor $g_{F}$ ($\\mu_B$ is the Bohr magneton).  With the addition of a further bias field $B_{z}\\hat{z}$, the trapping potential for transverse displacements becomes approximately harmonic in the region close to the axis, where $k B_{b} r\\ll B_{z}$. In that region, atoms oscillate in the trap with a transverse frequency of\n\\begin{equation}\\label{eqn:radialVideofreq}\nf_{r}\\approx \\frac{k B_b}{2\\pi}\\sqrt{\\frac{\\mu_B g_F m_F}{m\nB_{z}}}\\, ,\n\\end{equation}\nwhere $m$ is the mass of the atom. We use $^{87}$Rb atoms in the $(F=2, m_{F}=2)$ ground state, for which $g_{F}m_{F}=1$. The bias field $B_{b}$ is tuneable over the range $6-36$\\,G, which varies the distance from the trap to the videotape between $50$ and $18\\,\\mu$m. Over this range of positions the axial bias $B_{z}$ drops from $2.58$ to $2.51\\,$G. These parameters give transverse harmonic oscillation frequencies in the range $3-20\\, \\mathrm{kHz}$. The axial bias field has a minimum value near the centre of the videotape and increases roughly quadratically with $z$ to form a weak axial trap with a frequency of $15\\,$Hz. In summary, this combination of videotape fields and bias fields produces an array of cigar-shaped 3D traps with strong transverse confinement and weaker trapping along $\\hat{z}$. For more details of videotape traps, see \\cite{HindsHughesReview,myVideoPaper}.\n\nThe upper graphs in Figure\\,\\ref{fig:howPotChanges} show how the total magnetic field strength $|B|$ varies along the three Cartesian axes through the centre of the trap. These are calculated using a full numerical model of the apparatus, with $B_b=5.9\\,$G, directed along $\\hat{x}$. The trap along $x$ repeats every $110\\,\\mu$m because of the periodic magnetisation of the videotape, but only one of these is shown as only one trap is used in the experiments. Along $y$, we see an asymmetric trap, with a strong repulsive wall as the videotape is approached and an asymptote far from the tape that is equal to the total bias field strength. Along $z$, there is the weak trapping due to the inhomogeneous axial bias field.\n\n\n\n\nWe load one such trap using an experimental sequence similar to that described in \\cite{myVideoPaper}. The temperature of the atom cloud depends on the bias field and ranges from $13\\,\\mu$K at $B_b=5.9\\,$G up to $130\\,\\mu$K at $B_b=35.6\\,$G. We then add a modulation field $b \\cos(2\\pi f t)$ along $\\hat{x}+\\hat{y}$, to displace the trap by a small distance $-\\tfrac{b}{k B_b}\\cos(\\omega t)$ along $\\hat{x}+\\hat{y}$. We choose $b=105\\,$mG, which gives a shaking amplitude of $50-300\\,$nm over the range of transverse bias fields used. Zooming into the centre of the trap, the three lower graphs in Fig.\\,\\ref{fig:howPotChanges} show the extrema of this modulation. The first two show the (equal) movements along $x$ and $y$ resulting from our particular arrangement of fields. The third shows the absence of movement along $z$ and also illustrates the modulation of the minimum field, due to the variation of $B_z$ with transverse displacement of the trap. Since $B_z$ affects the transverse trap frequency, this could cause parametric heating, but the effect is considered in \\cite{MyThesis} and found to be negligible.\n\n\\begin{figure}[b]\n\\includegraphics[width=0.4\\textwidth]{resonances.eps}\n\\caption{Temperature of cloud after shaking for $5\\,$s, plotted as a function of shaking frequency $f$. (a) Experimental results. The points are measurements taken at six transverse bias fields, $B_b$. Lines are Lorentzian fits to the data.  (b) Simulation. Points are calculated and lines are Lorentzian fits.}\n\\label{fig:resonances}\n\\end{figure}\n\n\nWith $B_b$ initially set at $5.9\\,$G, the atoms are shaken for $5\\,\\mathrm{s}$ and held for $1\\,$s, after which the temperature of the trapped cloud is determined by measuring its density distribution along the $z$-direction. This is done using a CCD camera to record the absorption of resonant laser light \\cite{myVideoPaper}, taking into account the inhomogeneous Zeeman shift of the trapped atoms. The experiment is repeated for a range of shaking frequencies to map out a resonance curve, shown in the leftmost peak of Fig. \\ref{fig:resonances}(a). The next two curves are obtained in the same way with $B_b=8.9\\,$G and $B_b=14.8\\,$G. At still higher transverse bias fields ($B_b=21.9, 29.6$ and $35.6\\,$G), the atoms are too close to the videotape to yield clean absorption images. In these three cases the trap is moved away from the surface by lowering the bias to $3\\,$G over $3\\,$s before taking the image. This weakens the trap, thereby cooling the cloud, and it is the lower temperature in this final trap that we plot in Fig. \\ref{fig:resonances}(a).\n\nThese temperature resonances show that the atoms absorb energy most efficiently near a particular frequency. In the case of a one-dimensional harmonic trap, the physics would be that of a driven, weakly damped harmonic oscillator, whose Lorentzian resonance would be centred on the oscillator frequency with a width given by the collision rate. The actual widths are much greater than the collision rate, but nevertheless, motivated by this thought, we fit a Lorentzian to each resonance curve and plot the centre frequencies (the red squares) as a function of $B_b$ in Fig.\\,\\ref{fig:FreqVsBias}. For comparison, the line in Fig.\\,\\ref{fig:FreqVsBias} shows the frequency of small transverse oscillations, given by Eq.\\,\\ref{eqn:radialVideofreq}, with $B_z$ evaluated at the centre of the trap for each value of transverse bias.  The data points lie below this line because (i) the most energetic atoms move out of the region of small ${x,y}$ where the harmonic approximation of Eq.\\,(\\ref{eqn:radialVideofreq}) is valid and (ii) the atoms are also displaced from the centre along $z$, where the increased value of $B_z$ reduces the radial frequency. These effects also produce inhomogeneous broadening of the temperature resonances, making them wider for the hotter clouds at higher bias fields, as seen in Fig.\\,\\ref{fig:resonances}(a).\n\n\\begin{figure}[b]\n  \\includegraphics[width=0.9\\columnwidth]{FreqVsBias.eps}\n  \\caption{Frequency of temperature resonance versus bias magnetic field. Solid line: calculated harmonic frequency of transverse oscillations at trap centre. Red squares: measured centre frequencies of the temperature resonances in Fig.\\,\\ref{fig:resonances}(a). Blue circles: centre frequencies of the simulated temperature resonances plotted in Fig.\\,\\ref{fig:resonances}(b).}\n  \\label{fig:FreqVsBias}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{FreqVsAmpliBias2.eps}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{FreqvsAmpliBias7.eps}\n\\end{minipage}\n\\caption{Frequency of oscillations along $\\hat{x}$ through centre of trap as a function of amplitude. Upper: $B_b=5.9\\,\\mathrm{G}$. Lower: $B_b=35.6\\,\\mathrm{G}$. Solid red line: frequency calculated from numerical solution to full equation of motion. Dotted green line: Harmonic approximation. Dashed black line: linear potential approximation. Circles: rms cloud radius at initial temperature.  Squares: rms cloud radius after heating at resonance.}\n\\label{fig:FreqsVsAmpliBias2and7}\n\\end{figure}\n\nNumerically integrating the equations of motion in the full trap potential, we have calculated the period of oscillation along the $x$ direction through the centre of the trap. The inverse of this is plotted as a function of amplitude $A_x$  by the solid lines in Fig.\\,\\ref{fig:FreqsVsAmpliBias2and7}, the upper(lower) panel being for the case of $B_b=5.9(35.6)\\,$G. At small amplitude, the frequency coincides with the harmonic approximation indicated by the dotted line. At large amplitude, the potential approaches that of  linear trap, and the frequency tends correspondingly to $\\tfrac{1}{4}\\sqrt{\\mu_B k B_b\/(2 m A_x)}$, indicated by the dashed line. The filled circles(squares) mark the rms radius of the cloud in this direction before(after) resonant heating. These show that the atoms explore the anharmonic region of the transverse trap even before the cloud is heated, and move further into this region after heating. These frequency shifts are larger when the bias field is larger. Figure\\,\\ref{fig:RadialFreqVsZ} shows how \\emph{axial} displacement from the centre of the trap lowers the harmonic frequency for small transverse oscillations. The essence of this effect is already captured in Eq.\\,(\\ref{eqn:radialVideofreq}) through the dependence of $f_r$ on $B_z$, but here we show the result of the full numerical model of our experiment. Again, the circles(squares) represent the rms size, this time along $z$ before(after) heating. Both of these mechanisms contribute appreciably to the inhomogeneous broadening seen in Fig.\\,\\ref{fig:resonances}(a) and the lowering of the trap frequency seen in Fig.\\,\\ref{fig:FreqVsBias}.\n\nFor a more quantitative understanding of the resonances it is necessary to build a dynamical model that allows the atoms to collide. In the next section we develop such a model and use it to simulate the data presented in Fig.\\,\\ref{fig:resonances}(a).\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.36\\textwidth]{freqradialVsZ.eps}\n  \\caption{Frequency of small (harmonic) transverse oscillations as a function of displacement $z$ from the trap centre. \\textbf{Circles}: rms cloud radius (half-length) at initial temperature.  \\textbf{Squares}: rms cloud radius (half-length) after heating at resonance.}\n  \\label{fig:RadialFreqVsZ}\n\\end{figure}\n\n\n\\section{\\label{sec:simul} Simulation}\n\\subsection{\\label{sec:simul:a} The role of collisions}\nOur goal here is to find a numerical model that is simplified as far as possible, while still reproducing the data of Fig.\\,\\ref{fig:resonances}(a). We use the known currents and videotape magnetisation to determine an accurate 3D  potential,  $U(x,y,z,t)$, for the shaken magnetic trap. We note that the inclusion of gravity has no significant effect because these traps are so strong vertically. The cloud of approximately $10^5$ atoms is represented by an ensemble of 500-5000 point particles, this being sufficient to represent the average properties of the ensemble. The three initial velocity components for each particle are chosen at random from the Maxwell-Boltzmann distribution corresponding to initial temperature $T_i$, which is the baseline temperature in Fig.\\,\\ref{fig:resonances}(a) for that particular trap. Similarly, the initial positions are distributed with a probability density proportional to $\\exp [-U(x,y,z,0)\/(k_{B}T_{i})]$, where $k_{B}$ is the Boltzmann constant. The classical equations of motion are then integrated numerically to follow the movement of the particles.\n\nThe two curves in Fig.\\,\\ref{fig:peak5}(a) show how the energy of the cloud increases with time when the trap having $B_b = 21.9\\,$G  is shaken at 7.2kHz. These parameters correspond to the peak of the pale blue resonance curve at $7\\,$kHz in Fig.\\,\\ref{fig:resonances}(a). A simple model without collisions produces the green (lower) curve in Fig.\\,\\ref{fig:peak5}(a). Particles are placed in the trap at time $t=0$, which remains static for the first $0.1\\,$s. Then the shaking is switched on and the energy rises rapidly, increasing by $20\\%$ over the next $0.1\\,$s. This is  due to the excitation of particles whose transverse oscillation frequency is close to the drive frequency.  Once they are sufficiently excited, the anharmonicity moves these particles out of resonance. Because the resonant group has been depleted, there is almost no subsequent energy increase even though the shaking continues until $t=5.1\\,$s. The shaking is then switched off leaving the cloud to evolve freely over the last second of the simulation. This behaviour disagrees with the experiment. In reality, the temperature doubles (though not in Fig.\\,\\ref{fig:resonances}(a) because there the trap was relaxed before recording the temperature). The discrepancy is removed when we allow the simulation to redistribute the momentum through collisions.\n\n\\begin{figure}[t]\n\\begin{minipage}{0.65\\linewidth}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{Peak5-1.eps}\n\\end{minipage}\\\\\n\\begin{minipage}{0.65\\linewidth}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{Peak5-2.eps}\n\\end{minipage}\\\\\n\\caption{Simulation of $10^5$ atoms in a trap having $B_b=21.9\\,\\mathrm{G}$.  The trap is shaken by a modulation field of  $105\\,$mG amplitude and  $7.2\\,\\mathrm{kHz}$ frequency over the time interval  $t=0.1-5.1\\,$s. The initial temperature is $71\\,\\mu \\mathrm{K}$. These parameters correspond to the resonant peak of the pale blue curve in Fig.\\,\\ref{fig:resonances}(a). This simulation uses 500 particles. Green curves: no collisions. Blue curves: momentum is redistributed through collisions. (a) Increase of total energy with time. (b) Number of atoms remaining within the volume of the trap, with the initial atom number being $N_i=10^5$.}\n\\label{fig:peak5}\n\\end{figure}\n\n\nThe following very simple model of momentum redistribution is sufficient for our purpose. The thermally averaged atom-atom scattering cross-section is $\\sigma=8 \\pi a^{2}\/(1+2\\pi a^2 m k_{B} T \/\\hbar^2)$, where $a=5.53\\times10^{-9}\\,\\mathrm{m}$ is the s-wave scattering length \\cite{scatteringLength} and $T$ is the temperature of the cloud. The average relative velocity between particles is $v = \\sqrt{16 k_{B} T\/(\\pi m)}$ \\cite{PhysChem}. Knowing the mean density of trapped atoms, $ n $, we obtain a mean collision rate per atom,  $ n \\sigma v$. At appropriate time intervals (short compared with the inverse collision rate), the numerical integration is paused and a random number is generated for each particle to determine whether it has had a collision. Ideally, this should account for the local variations of density \\cite{Bird94}, which we ignore here in order to keep the model simple. If there is no collision, the atom continues unperturbed. Otherwise, the momentum of the particle is redirected by the collision, according to some angular distribution. For the data presented here, we used that of elastic scattering from an infinitely heavy sphere. However, we find that the results are quite insensitive to the chosen distribution and there is nothing special about this particular one. Given our experimental conditions, the average time between collisions is $30-100\\, \\mathrm{ms}$ at the start of the excitation, and this becomes longer as the cloud heats up.\n\nWith the momentum redistribution thus incorporated,  we obtain the dark blue (higher) curve in Fig.\\,\\ref{fig:peak5}(a).  Now, the ensemble is able to continue absorbing energy after the initial absorption because the depleted velocity group is steadily replenished through collisions. The refilling rate slows down as the atoms become more energetic, and this is responsible for the saturation of heating, seen in Fig.\\,\\ref{fig:peak5}(a). In this case, the energy doubles over the $5\\,$s of shaking and our model approximates well  the measured heating of the atom cloud.\n\nFigure \\,\\ref{fig:peak5}(b) plots the number of atoms in the trap as a function of time, with collisions (blue, lower) and without (green, higher). In the collision-free case, few atoms leave the trap because the heating is weak and because the phase space is not efficiently sampled in the absence of collisions.  By contrast, when collisions are included, the cloud heats much more strongly and the energetic atoms are more easily able to find an exit route from the trap. There is competition between the heating rate due to resonant excitation and the cooling rate due to evaporation from the trap. In this example, the heating is dominant because the trap is deep in comparison with the mean energy absorbed by each atom.\n\n\\begin{figure}\n \\includegraphics[width=0.4\\textwidth]{energySpectra.eps}\n \\caption{Simulated energy spectra of $10^5$ atoms, initially at $71\\,\\mu \\mathrm{K}$ in a trap having $B_b=21.9\\,\\mathrm{G}$. They are shaken over the time interval  $t=0.1-5.1\\,$s by a $7.2\\,\\mathrm{kHz}$  field of  amplitude $b=105\\,$mG. These are the same parameters used in Fig.\\,\\ref{fig:peak5} and at the resonant peak of the pale blue curve in Fig.\\,\\ref{fig:resonances}(a). This simulation uses 5000 particles. (a) Without collisions. (b) With collisions. Red curves: Initial thermal distribution just before shaking. Green curves: After $0.1\\,$s of shaking. Blue curves: After $5\\,$s of shaking.}\n \\label{fig:energySpectra}\n\\end{figure}\n\n\nFigure \\ref{fig:energySpectra}(a) shows calculated energy spectra in the absence of collisions. The ordinate is the probability density for a given energy, normalised to unity, while the abscissa shows that energy,  normalised to the trap depth $U_0=1.2\\,$mK. The red curve ($t=0.1\\,$s), showing the initial distribution just before the trap starts to shake, has a single peak just below $E=0.2 U_0$. After only $100\\,$ms of shaking (green curve, $t=0.2\\,$s) a deep notch appears in the distribution close to the energy of the initial peak. This shows that atoms close to that energy, having an oscillation period close to the period of the drive, are the ones excited by the shaking. Their excitation causes a second peak in the distribution at approximately $E\/U_{0}=0.3$. There is no further significant change in the distribution, even after $5\\,$s of shaking, as shown by the blue curve. This behaviour is to be compared with Figure \\ref{fig:energySpectra}(b), which shows energy distributions for the same simulated experiment when collisions are included. The initial distribution is the same, as is the notch appearing at $0.2\\,$s, but in this case continued shaking does produce additional heating because the collisions refill the velocity group that absorbs energy from the drive. This is clearly seen in the growing probability on the high-energy end of the spectrum.\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.4\\textwidth]{simulatedResonAtomLoss.eps}\\\\\n  \\caption{Simulated ratio of final atom number $N$ to initial number $N_i$ as a function of the excitation frequency. The colour code corresponds to that of Fig. \\ref{fig:resonances}, with coloured arrows marking the temperature resonance frequencies. For the lowest three bias fields, the lowest barrier to escape is along \\emph{y} ($360\\, \\mu \\mathrm{K}$, $520\\, \\mu \\mathrm{K}$ and $880\\, \\mu \\mathrm{K}$ in order of increasing bias field), while for the highest three, it is along \\emph{z} ($\\sim 1200\\, \\mu \\mathrm{K}$). For comparison, the measured initial cloud temperatures before the excitation take values of $13.4\\, \\mu \\mathrm{K}$, $13.3\\, \\mu \\mathrm{K}$, $16.7\\, \\mu \\mathrm{K}$, $71\\, \\mu \\mathrm{K}$, $98\\, \\mu \\mathrm{K}$ and $132\\, \\mu \\mathrm{K}$, in order of increasing bias field.}\n  \\label{fig:simulatedResonAtomLoss}\n\\end{figure}\n\n\\subsection{\\label{sec:simul:b} Simulated Resonances}\n\nFigure \\ref{fig:resonances}(b) shows our simulated temperature resonances at each of the six different values of the bias field. After the shaking stops in the experiment, we allow the atoms to thermalise for one second before measuring the temperature. In the simulation, we simply take the final energy as a measure of the temperature that would be reached in equilibrium. The central frequencies of these simulated resonances agree very closely with the experiment, as indicated by the blue circles in Fig.\\,\\ref{fig:FreqVsBias}.\n\nThe peak temperature rises are also remarkably well reproduced by the simulations, given the simplicity of the model. In particular, these model collisions redistribute momentum, but do not permit re-thermalisation of the energy distribution because the energy of a given atom is conserved in the collision. Despite that, the energy increase in the model reproduces all the measured temperature rises to well within a factor of two and that remains the case for a variety of model angular distributions.\n\nThe widths of the simulated resonant peaks are a little too large - by a factor of $1.5 - 2$. In searching for an explanation we simulated the yellow ($8.9\\,$G) resonance with the initial temperature reduced from $13.3\\,\\mu$K to $7\\,\\mu$K. This only reduced the width by $10\\%$, so we do not think a temperature calibration error can explain the discrepancy between the measured and calculated widths. It could well be that our very simple collision model causes the resonances to be too broad, although we do not see a clear reason why that should be so.\n\nFigure \\ref{fig:simulatedResonAtomLoss} shows the simulated resonances in atom loss for the same six bias fields used in Fig.\\,\\ref{fig:resonances}. We see that the dips in atom number are on the low-frequency side of the temperature resonances indicated by arrows. This is because the energetic atoms most likely to be driven out of the trap are also those most able to explore the anharmonic regions and hence to oscillate at  lower frequencies. The same behaviour has been reported by several groups for both parametric shaking \\cite{st4,st5,st9,st17,st19}, and for shaking of the trap centre \\cite{st13}, in agreement with the results of our simulations.\\\\\n\n\\section{\\label{sec:Conclusions} Summary and Conclusions}\nWe have measured the temperature rise in a cigar-shaped cloud of cold atoms after shaking it sinusoidally in the transverse direction. Unlike most previous measurements, we have modulated the position of the trap, not its curvature. We have recorded temperature resonances as a function of frequency for several values of the bias field that controls the curvature of the transverse trapping potential.  Essential to the interpretation of our measurements is the understanding that,  at the temperatures involved, atoms oscillate with a wide range of transverse frequencies in the trap and hence that the central frequencies of the observed resonances lie below the calculated harmonic frequencies for small oscillations.\n\nWe have developed a simple numerical model that has provided a clear quantitative understanding of the driven dynamics. When compared with other models and simulations in the literature, this provides one of the most detailed and complete attempts to reproduce the observed resonances. No other models include both the collisions between particles and the anharmonicity of the trapping potential, yet both of these are shown to be essential for a full understanding of the behaviour.\n\n\\begin{acknowledgments}\nThe authors thank Robert Nyman and Michael Trupke for useful discussions. We are indebted to the FastNet and AtomChips European networks and to the UK EPSRC and Royal Society for their funding support.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material: Controlling spin relaxation in hexagonal BN-encapsulated graphene with a transverse electric field}\n\n\n                            \n                             \n\\maketitle\n\n\\subsection{Fabrication of the stacks}\nThe hBN-graphene-hBN stacks were fabricated using a pick-up method \\cite{DeanScience2013} described elsewhere \\cite{WeesAppliedPhysicsLetters2014} in more details.\nThe graphene and hBN flakes were obtained by mechanical cleavage of HOPG (SPI Supplies) and hBN powder (HQ Graphene).\nThe preparation of the stacks starts with the exfoliation and optical microscopy selection of hBN on a glass mask covered by a thin layer of polycarbonate (HQ Graphene).\nThe single layer graphene and bottom hBN flakes are exfoliated on Si\/SiO$_{2}$ substrates and selected by optical microscopy.\nThe mask containing the top hBN flake is then aligned and pressed against the substrate containing the graphene flake and heated to $\\approx$75 $^{\\circ}$C.\nWhen the mask is retracted from the substrate the graphene flake adhere strongly to the hBN flake and releases from the substrate sticking to the mask.\nThe mask containing the hBN-graphene stack is then aligned to the bottom hBN flake, brought in contact to the substrate and heated to temperatures up to $\\approx$150 $^{\\circ}$C.\nAt these temperatures the polycarbonate film melts on the substrate and releases from the mask together with the stack.\nThe sample is then left in chloroform for at least 15 hours in order to completely dissolve the polycarbonate film and obtain a clean graphene surface ensuring good graphene-contact interfaces.\n\n\\subsection{Contact deposition}\nThe contacts and top gate electrodes in this work were all fabricated at the same step to avoid further contamination of the non-encapsulated graphene regions.\nThe electrodes were patterned using standard e-beam lithography techniques using a PMMA 950K (270 nm thick) resist.\nThe markers for the e-beam lithography were patterned prior to the electrodes in the same PMMA resist and developed leaving openings which were used for the alignment of the electrodes.\nWe then used an e-beam evaporator with a base pressure below 10$^{-6}$ Torr to evaporate a 0.4 nm thick layer of Ti which was then naturally oxidized by inserting pure oxygen gas in the vacuum chamber to achieve pressures above 1 Torr.\nAfter 15 minutes the chamber was pumped down once again to the initial base pressure and the procedure was repeated to deposit and oxidize another 0.4 nm thick Ti layer.\nThis procedure is then followed by the deposition of a 67 nm thick Co layer with a capping layer of Al (5 nm) to protect the Co layer from oxidizing.\nThe large thickness of the Co layer is necessary to both overcome the height of the bottom hBN flake and also to facilitate the alignment of the electrodes by the perpendicular magnetic field.\nAfter the resist lift-off and wire bonding, the samples were loaded in a continuous He flow cryostat which stands in a room temperature electro-magnet.\nThe contact resistances were in the range 2 - 100 k$\\Omega$.\n\n\\subsection{Transport measurements}\nThe charge transport measurements were performed by standard low-frequency lock-in techniques with currents up to 100 nA.\nAll spin transport measurements presented here were done using the non-local geometry in which the charge contribution to the measured signal is minimized by separating the charge current path from the voltage detection circuit.\nWe used standard low-frequency lock-in techniques with applied currents of 1 $\\mu$A in order to preserve the high resistive contact interface barriers and avoid device heating. \n\n\\subsection{Magnetoresistance effects on the nonlocal signal}\nTo be sure that the magneto-resistance contribution is minimal in our nonlocal signals we characterize the graphene sheet resistance as a function of the applied magnetic field.\nIn Fig. \\ref{fig:SI-magres} we show a typical curve of the $R_{sq}$ versus $V_{tg}$ for B = 0 and 1 T measured in a 4-probe geometry.\nIt can be seen that the change in $R_{sq}$ with $B$ is maximum around the charge neutrality point and small at high values of $n$.\nThis observation also reflects on our nonlocal measurements in which we observe the monotonic increase in the nonlocal resistance as a function of $B$ characteristic to magneto-resistance effects only for low $n$.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{SI-magres.pdf}\n\t\\caption{Graphene square resistance as a function of $V_{tg}$ for $V_{bg}=-16.65$ V for sample 1 at 4.2 K and B = 0 (dashed blue line) and 1 T (red solid line). \\textit{Inset:} Ratio between the two curves in the main graph showing the magneto-resistance in the sample.}\n\t\\label{fig:SI-magres}\n\\end{figure}\n\n\\subsection{Simulation of Hanle precession data for non-homogeneous systems}\n\nIn order to describe the different regions in our system (encapsulated and non-encapsulated) we use a model for spin transport in an inhomogeneous system described elsewhere \\cite{GuimaraesNanoLetters2012}.\nWe solve the Bloch equations for spin diffusion in one dimension for a system with three distinct regions connected to one another.\nEach one of these regions can have independent values for the relevant parameters.\nWe assume the outer regions identical, with square resistance $R_{o}$, spin diffusion coefficient $D_{o}$ and spin relaxation time $\\tau_{o}$.\nThe inner region has distinct parameters $R_{i}$, $D_{i}$ and $\\tau_{i}$.\nWe assume that the spins are injected at the left boundary of the inner region and detected on the right, which results in a nonlocal spin signal $R_{sim}$.\nFor this work we get $R_{o}$, $D_{o}$, $\\tau_{o}$, $R_{i}$ and $D_{i}$ by charge and spin transport experiments as described in the main text.\nWe then fit the simulated curves for $R_{sim}$ using the solution for the Bloch equation for homogeneous systems (exactly in the same way we analyze our experimental results).\nFrom this fit we extract an effective spin diffusion coefficient $D_{fit}$ and relaxation time $\\tau_{fit}$.\nAn example of such simulated curve and fitting is shown in Fig. \\ref{fig:SI-simhanle}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.39\\textwidth]{SI-simhanle.pdf}\n\t\\caption{Hanle precession curve calculated using the model for a non-homonegeous system (black dots) with a fit using the expression for the solution of the Bloch equation in a homogeneous system.}\n\t\\label{fig:SI-simhanle}\n\\end{figure}\n\nFor a fixed $V_{bg}$, the effect of $V_{tg}$ is to change the spin resistance ratio between the inner and outer regions, $\\frac{R_{i}\\lambda_{i}}{R_{o}\\lambda_{o}}$, where $\\lambda_{i(o)}$ is the spin relaxation length for the inner (outer) regions.\nThe change of this ratio affects strongly the extracted spin relaxation times $\\tau_{fit}$ but leaves $D_{fit}$ mostly unaffected \\cite{GuimaraesNanoLetters2012}.\nIn order to get a good estimation of $\\tau_{i}$, we take the experimental values for $V_{bg}$=-52.5 V and simulate a few sets of points with only changing $R_{i}$.\nAs it can be seen in Fig. 3b in the main text, we find a close match between our simulated values of $\\tau_{fit}$ to our experimentally obtained values of $\\tau_{s}$ for $\\tau_{i}$ ranging from 3 to 5 ns.\nMeaning that the spin relaxation time for the encapsulated region is higher than, but still within the same order of magnitude of, the experimentally obtained values using the solution for the Bloch equations in a homogeneous system.\n\nIt is known that $R_{c}$ can affect the measurement of $\\tau_{s}$, especially in the case of very long values of $\\lambda_{s}$.\nHowever, even though we have a wide range of contact resistances $R_{c}$ 2 - 100 k$\\Omega$ with one sample having all contact resistances above 50 k$\\Omega$, we did not see any clear correlation between $R_{c}$ and the measured $\\tau_{s}$.\nWe would expect that, in the case of contact induced spin relaxation or poor graphene quality in the regions underneath the contacts, $\\tau_{s}$ would be mostly affected in the non-encapsulated regions.\nTherefore, the measured spin relaxation time for the non-encapsulated will be an effective spin relaxation time that includes all these effects, and consequently is taken into account in our model by setting $\\tau_{o}$ as the measured spin relaxation time for these regions.\n\n\\subsection{Experimental Hanle precession data}\n\nFor most of our devices we observe a small asymmetry in our Hanle precession curves between negative and positive perpendicular magnetic fields \\ref{fig:SI-hanle} which might arise due to some anisotropy in the magnetization of the electrodes.\nAlthough we cannot be certain about the reason for this asymmetry, the parameters extracted by fitting just one or the other side of the curve do not change significantly.\nComparing the parameters extracted from each side, $\\tau_{s}$ deviates by a factor of $\\approx$ 1.06 and $D_{s}$ by a maximum of a factor 2.\nSince the values for $D_{s}$ extracted by fitting the whole Hanle curve match the values for $D_{c}$ extracted by charge transport measurements, we believe that our procedure is justified.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[width=0.39\\textwidth]{SI-hanle-alt.pdf}\n\t\\caption{Top: Experimental data for $R_{s}=(R_{nl}^{P}-R_{nl}^{AP})\/2$ at $V_{bg}$ = -52.5V and $V_{tg}$ = +3.744 V, where $R_{nl}^{P(AP)}$ is the non-local resistance obtained for the electrodes in a (anti)parallel magnetization (dots) as a function of an applied perpendicular magnetic field. The fit (red line) used to extract $D_{s}$ and $\\tau_{s}$ is also shown. In the inset it is shown the data for $R_{nl}^{P(AP)}$ in black (red). Bottom: Hanle precession curve for $V_{bg}$ = $V_{tg}$ = 0 V with the respective non-local spin-valve shown in the inset.}\n\t\\label{fig:SI-hanle}\n\\end{figure}\n\nThe measured spin polarization of our electrodes were in the range $P$ = 1 - 6 $\\%$.\nThese values for $P$ are in the lower range of the values reported in literature \\cite{MihaiPhys.Rev.B2009,KawakamiPhys.Rev.Lett.2011}, however we believe that this value can be improved by increasing the quality of the high resistive barriers \\cite{KawakamiPhys.Rev.Lett.2010}.\n\n\\subsection{Results for spin and charge transport at 4.2 K and room temperature}\n\nOur results presented in the main text were all performed in sample 1 at 4.2 K.\nFor comparison we also include here the square resistance as a function of $V_{tg}$ and $V_{bg}$ and the extracted spin transport parameters for the same sample at room temperature (Fig. \\ref{fig:samples}a) and another sample (sample 2) at room (Fig. \\ref{fig:samples}b) and liquid Helium (Fig. \\ref{fig:samples}c) temperatures.\nThe samples dimensions and their electronic mobilities at room temperature and 4.2 K are given in table \\ref{fig:samples}.\nSample 3 (not shown here) showed similar results.\n\n\\begin{figure*}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{samples-diftemp.pdf}\n\t\\caption{Charge and spin transport measurements for (a) sample 1 at room temperature and sample 2 at (b) room temperature and (c) 4.2 K.}\n\t\\label{fig:samples}\n\\end{figure*}\n\n\\begin{table*}[h]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n & $\\ell_{tg}$ & $\\ell_{enc}$ & $L$ & $W$ & $\\mu_{RT}$ & $\\mu_{4K}$ \\\\\n\\hline\nSample 1 & 10.35 $\\mu$m & 12 $\\mu$m & 13.8 $\\mu$m & 1.3 $\\mu$m & 1.5 m$^{2}$\/Vs & 2.3 m$^{2}$\/Vs \\\\\n\\hline\nSample 2 & 4.5 $\\mu$m & 7.6 $\\mu$m & 8.72 $\\mu$m & 2.8 $\\mu$m & 2.4 m$^{2}$\/Vs & 3.3 m$^{2}$\/Vs \\\\\n\\hline\n\\end{tabular}\n\\label{tab:samples}\n\\caption{Length of the top gated region ($\\ell_{tg}$), encapsulated region ($\\ell_{enc}$), inner electrodes spacing ($L$), graphene flake width ($W$) and electronic mobilities at room temperature ($\\mu_{RT}$) and 4.2 K ($\\mu_{4K}$) for samples 1 and 2.}\n\\end{table*}\n\n\\subsection{Estimation of $\\Delta_{R}$}\n\nIn order to give a rough estimation for the spin orbit coupling strength $\\Delta_{R}$ we assume that the spin relaxation is composed of the sum of an electric field independent and electric field dependent terms: $1 \/ \\tau_{\\bot} = 1 \/ \\tau_{ind} + 1 \/ \\tau_{E}$.\nThe spin relaxation time $\\tau_{||}$ is taken equal to the spin relaxation experimentally ($\\tau_{s}$) and $\\tau_{ind}$ is $\\tau_{s}$ multiplied by the ratio $r = \\tau_{\\bot} \/ \\tau_{s}$ at \\linebreak $\\bar{E}$ = 0 V\/nm: $\\tau_{ind} = 0.75 \\tau_{s}$.\nIf we assume a D'Yakonov-Perel spin relaxation mechanism for $1 \/ \\tau_{E} = \\frac{4 \\Delta_{R}^2}{\\hbar^{2}} \\tau_{p}$, we have: $\\Delta_{R}=\\zeta \\bar{E}=(\\hbar \/ 2) (\\tau_{E} \\tau_{p})^{-1\/2}$ \\cite{FabianPhys.Rev.B2009a}, where $\\tau_{p}=2 D_{c} \/ v_{F}$ is the momentum relaxation time, $D_{c}$ the charge diffusion constant and $v_{F}\\approx 10^{6}$ m\/s the Fermi velocity.\nWe further take $D_{c} \\approx D_{s}$.\nSolving it for $\\tau_{E}$ we have:\n\n\\begin{equation}\n\\tau_{E} = \\tau_{s} \\left( \\frac{1}{r} - \\frac{1}{0.75} \\right)^{-1},\n\\end{equation}\n\nand\n\n\\begin{equation}\n\\zeta = \\frac{\\Delta_{R}}{\\bar{E}} = \\frac{\\hbar}{2 \\bar{E}} \\frac{1}{\\sqrt{\\tau_{s} \\tau_{p}}} \\left( \\frac{1}{r} - \\frac{1}{0.75} \\right)^{-1\/2}\n\\end{equation}\n\n\\section{D'Yakonov-Perel \\textit{versus} Elliott-Yafet mechanisms for spin relaxation}\nTwo mechanisms are often considered in order to try to explain spin relaxation in graphene: they are the D'Yakonov-Perel (DP) and Elliott-Yafet (EY) mechanisms for spin relaxation.\nThese two mechanisms show different behavior of the spin relaxation time as a function of the momentum relaxation time ($\\tau_{p}$): $\\tau_{EY} \\propto \\tau_{p}$ and $\\tau_{DP} \\propto 1 \/ \\tau_{p}$, with $\\tau_{EY(DP)}$ been the spin relaxation time due to the EY (DP) mechanism.\n\nIt was theoretically demonstrated that $\\tau_{EY}$ for the specific case of graphene for many different type of scatterers is given by \\cite{GuineaPhys.Rev.Lett.2012}: \n\n\\begin{equation}\n\\tau_{EY} \\approx \\frac{E_{F}^{2}}{\\Delta_{SO}^2} \\tau_{p},\n\\end{equation}\n\n\\noindent where $E_{F}$ is the Fermi energy and $\\Delta_{SO}$ the spin orbit coupling.\nThe DP mechanism obeys:\n\n\\begin{equation}\n\\tau_{DP} \\approx \\frac{\\hbar^{2}}{\\Delta_{SO}^2} \\frac{1}{\\tau_{p}}.\n\\end{equation}\n\nFollowing the reasoning of Huertas-Hernando et al. \\cite{BrataasPhys.Rev.Lett.2009}, to obtain the relative importance of these mechanisms we take the ratio:\n\n\\begin{equation}\n\\frac{\\tau_{EY}}{\\tau_{DP}} \\approx \\left( \\frac{E_{F} \\tau_{p}}{\\hbar} \\right)^{2}.\n\\end{equation}\n\nIf $\\tau_{EY} \/ \\tau_{DP} >$ 1, the spin relaxation time due to the DP mechanism is smaller and it dominates over the EY mechanism.\nUsing typical values for our samples of $E_{F} \\approx$ 30 meV (equivalent to a charge carrier density $n \\approx$ 10$^{12}$ cm$^{-2}$) and $\\tau_{p} \\approx$ 10$^{-13}$ s, we obtain $\\tau_{EY} \/ \\tau_{DP} \\approx$ 20, indicating a higher importance of the DP mechanism for the spin relaxation in our samples.\n\nAlthough this is not a definite proof that the DP is the dominant mechanism for spin relaxation in graphene, we believe that it is the most appropriate way to obtain the spin orbit coupling strength due to the electric field in our devices.\n\n\n\n\\section{Introduction}\nThe generation, manipulation and detection of spin information has been the target of several studies due to the implications for novel spintronic devices \\cite{WolfScience2001,ZuticActaPhysicaSlovaca2007}.\nIn the recent years graphene has attracted a lot of attention in spintronics due to its theoretically large intrinsic spin relaxation time and length of the order of $\\tau_{s}\\approx$ 100 ns and $\\lambda_{s}\\approx$ 100 $\\mu$m respectively \\cite{BrataasPhys.Rev.Lett.2009,BarnafmmodePhys.Rev.B2011}.\nAlthough experimental results still fall short of these expectations \\cite{MihaiPhys.Rev.B2009,KawakamiPhys.Rev.Lett.2011,OezyilmazPhys.Rev.Lett.2011,GuimaraesNanoLetters2012}, graphene has already achieved the longest measured nonlocal spin relaxation length \\cite{KawakamiPhys.Rev.Lett.2011,WojtaszekPhys.Rev.B2013} and furthest transport of spin information at room temperature \\cite{ZomerPhys.Rev.B2012}.\nHowever, the mechanisms for spin relaxation in graphene are still under heavy debate with various theoretical models proposed \\cite{BrataasPhys.Rev.Lett.2009,BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.Lett.2014,WuNewJournalofPhysics2012,RocheJPD2014,GuineaPhys.Rev.Lett.2012}.\n\nTo take advantage of the long spin relaxation times in graphene, e.g. for spin logic devices, one requires easy control of the spin information, for example by an applied electric field.\nSingle layer graphene is an ideal system for this purpose, not only because of its high mobilities and low intrinsic spin-orbit fields (SOF), but also due to the simple relation between the carriers' wavevector, the applied perpendicular electric field and the induced Rashba SOF \\cite{BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.B2009,BrataasPhys.Rev.Lett.2009,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,MacDonaldPhys.Rev.B2006,RashbaPhys.Rev.B2009,FabianPhys.Rev.B2009a}.\nIn bilayer graphene a more complicated behavior is expected when spin-orbit coupling is considered \\cite{FabianPhys.Rev.B2012}.\n\nHere we report nonlocal spin transport measurements on single layer graphene in which we address both topics specified above.\nOur devices consist of a single layer graphene flake on hexagonal Boron Nitride (hBN) of which a central region is encapsulated with another hBN flake and hence protected from the environment.\nThe presence of a top and bottom gate give rise to two independent electric fields that are experienced by the graphene: $E_{tg}= - \\epsilon_{tg}(V_{tg}-V_{tg}^{0})\/d_{tg}$ and $E_{bg}=\\epsilon_{bg}(V_{bg}-V_{bg}^{0})\/d_{bg}$, respectively \\cite{WangNature2009}, where $\\epsilon_{tg(bg)}\\approx$ 3.9 is the dielectric constant, $d_{tg(bg)}$ is the dielectric thickness and $V_{tg(bg)}^{0}$ the position of the charge neutrality point for the top (bottom) gate.\nTheir difference controls the carrier density in the graphene ($n = (E_{bg} - E_{tg}) \\epsilon_{0} \/ e$) and their average gives the effective electric field experienced by the graphene ($\\bar{E}=(E_{tg}+E_{bg})\/2$), which breaks the inversion symmetry in the encapsulated region, where $\\epsilon_{0}$ is the electric constant and $e$ the electric charge.\nOur devices show enhanced spin relaxation times of at least 2 ns and also, due to the higher electronic mobility, spin relaxation lengths above 12 $\\mu$m at room temperature (RT) and 4.2 K.\nBy a simple model we show that the measured spin relaxation times are a lower bound due to the influence of the non-encapsulated regions.\n\nBy comparing the spin relaxation time for spins out-of-plane ($\\tau_{\\bot}$) to spins in-plane ($\\tau_{||}$) as a function of the electric field we get insight on the nature of the SOF that cause spin relaxation in graphene.\nFor SOF pointing preferentially in the graphene plane, e.g. for adatoms and impurities, we expect: $\\tau_{\\bot}\\approx 0.5\\tau_{||}$ \\cite{ZuticActaPhysicaSlovaca2007,BrataasPhys.Rev.Lett.2009,TombrosPhys.Rev.Lett.2008,FabianPhys.Rev.B2009a}.\nIf the SOF point out-of-plane, as for ripples\\cite{BrataasPhys.Rev.Lett.2009}, we have: $\\tau_{\\bot} \\gg \\tau_{||}$.\nHowever, if the main relaxation mechanism is through random magnetic impurities or defects, no preferential direction for the spins is expected: $\\tau_{||} \\approx \\tau_{\\bot}$.\nHere we obtain $\\tau_{\\bot} \/ \\tau_{||}\\approx$ 0.75 at $\\bar{E}$ = 0 V\/nm$^{-1}$.\nThis ratio decreases with increasing $\\bar{E}$, in agreement with an electric field induced Rashba SOF pointing in the graphene plane.\n\nDevice number 1 is illustrated in Fig. \\ref{fig:figure-1}a and b.\nThe hBN-graphene-hBN stack sits on a 300 nm thick SiO$_{2}$ layer on a heavily doped Si substrate which is used as a back-gate.\nThe sample preparation is described in detail in the supplementary information and follows Ref. \\cite{DeanScience2013,WeesAppliedPhysicsLetters2014}.\nWe use Co electrodes with a thin TiO$_{2}$ interface barrier to perform spin transport measurements.\nThree devices were studied, all showing similar results.\nHere we show the results for the device with the longest encapsulated region ($\\approx$ 12 $\\mu$m) and spacing between the inner contacts (13.8 $\\mu$m).\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-1.pdf}\n\t\\caption{(a) Side-view schematics and (b) Top-view optical microscope image of a hBN encapsulated graphene spin-valve. The numbers show the contact electrodes and the top (bottom) gates electrodes are indicated as TG (BG). The graphene is outlined by the dashed line. (c) Square resistance (R$_{sq}$) as a function of $V_{tg}$ and $V_{bg}$.}\n\t\\label{fig:figure-1}\n\\end{figure}\n\nThe charge transport properties of the encapsulated region are measured by applying a current between electrodes 1 and 5, Fig. \\ref{fig:figure-1}a and b, and scanning the top and bottom gate voltages ($V_{tg}$ and $V_{bg}$, respectively) while recording the voltage between electrodes 2 and 3.\nFig. \\ref{fig:figure-1}c shows the square resistance ($R_{sq}$) as a function of $V_{tg}$ and $V_{bg}$.\nThe charge neutrality point depends on both $V_{tg}$ and $V_{bg}$ in a linear fashion.\nThe slope of the line gives the ratio between the bottom and top gate capacitances: $\\alpha_{bg}\/\\alpha_{tg}\\approx$ 0.036.\nA small top gate independent resistance peak around $V_{bg}$ = -16.6 V (not visible in Fig. \\ref{fig:figure-1}c) arises from the non-top gated regions between the two inner contacts.\nThe electronic mobility for this device is $\\mu\\approx$ 1.5 m$^{2}$\/Vs at RT and $\\mu\\approx$ 2.3 m$^{2}$\/Vs at 4.2 K.\nAlthough the mobilities of our devices are above the best devices based on SiO$_{2}$ they are still one order of magnitude lower than the best devices on hBN \\cite{DeanScience2013} which can be attributed to small bubbles or contamination visible on the graphene\/hBN stack.\n\nSpin dependent measurements are performed using a standard nonlocal geometry in which the current path is separated from the voltage detection circuit \\cite{MihaiPhys.Rev.B2009}.\nThe current is driven between electrodes 1 and 2 and the voltage measured between electrodes 3 and 5, which are on the other side of the encapsulated region (Fig. \\ref{fig:figure-1}a and b).\nTo obtain the spin relaxation time ($\\tau_{s}$) and the spin diffusion coefficient ($D_{s}$) we perform Hanle precession measurements where the nonlocal signal is measured as a function of a perpendicular magnetic field $B$.\nWe then fit the data with the solution to the Bloch equations \\cite{MihaiPhys.Rev.B2009}.\n\nThe results for $D_{s}$, $\\tau_{s}$ and the spin relaxation length ($\\lambda_{s}=\\sqrt{D_{s}\\tau_{s}}$) as a function of the $V_{tg}$ for three values of $V_{bg}$ at 4.2 K are shown in Fig. \\ref{fig:figure-2}.\nA similar set of measurements was performed at RT and for other samples where only a small difference was observed (see supplementary information \\cite{supinfo}).\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-2.pdf}\n\t\\caption{(Spin) diffusion coefficient ($D_{s}$), spin relaxation time $\\tau_{s}$ and spin relaxation length $\\lambda_{s}$ as a function of $V_{tg}$ for three different values of $V_{bg}$. The error bars are smaller than the dot size. The dashed red (blue) lines show the charge diffusion coefficient $D_{c}$ for $V_{bg}$ = +52.5 V (-52.5 V).}\n\t\\label{fig:figure-2}\n\\end{figure}\n\nDue to our device mobility, $D_{s}$ is higher than for regular graphene devices on SiO$_{2}$ ($D_{s}$ $\\approx$ 0.02 m$^{2}$\/s) \\cite{KawakamiPhys.Rev.Lett.2011,MihaiPhys.Rev.B2009} and comparable to suspended \\cite{GuimaraesNanoLetters2012} and non-encapsulated hBN supported devices \\cite{ZomerPhys.Rev.B2012} ($D_{s}$ $\\approx$ 0.05 m$^{2}$\/s).\nAs an extra confirmation, we check that $D_{s}$ agrees with the charge diffusion coefficient $D_{c}=[R_{sq}e^{2}\\nu(E_{F})]^{-1}$\\footnote{A resistance of 3.2 k$\\Omega$ was subtracted in the calculation of $D_{c}$ for $V_{bg}=-52.5$V to account for the non-top gated regions.}, where $e$ is the electron charge and $\\nu(E_{F})$ the density of states at the Fermi energy $E_{F}$.\nNext, we observe that the obtained spin relaxation times are higher than those on regular SiO$_{2}$ substrates ($\\tau_{s}\\approx$ 0.1 - 1 ns)\\cite{KawakamiPhys.Rev.Lett.2011,MihaiPhys.Rev.B2009} and in non-encapsulated hBN supported devices ($\\tau_{s}\\approx$ 0.1 - 0.5 ns)  \\cite{ZomerPhys.Rev.B2012}, reaching up to $\\tau_{s}$=(1.9$\\pm$0.2) ns at 4.2 K and $\\tau_{s}$=(2.4$\\pm$0.4) ns for RT.\nThese values surpass all previous nonlocal measurements of $\\tau_{s}$ in single layer graphene both at room and low temperatures \\footnote{2-terminal local measurements on epitaxial graphene estimated $\\tau_{s}\\approx$ 100 ns at 4.2 K \\cite{FertNatPhys2012}.} \\cite{FertNatPhys2012}.\nWe obtain a maximum of $\\lambda_{s}$ = 12.3 $\\mu$m at 4.2 K and $\\lambda_{s}$ = 12.1 $\\mu$m at RT.\n\n\nComparing $\\tau_{s}$ obtained in our devices with non-encapsulated hBN based devices ($\\tau_{s}\\approx$ 0.2 ns) \\cite{ZomerPhys.Rev.B2012}, we can conclude that the encapsulation of graphene on hBN significantly increases the spin relaxation times.\nNote that the non-encapsulated devices had comparable electronic mobilities which indicates that $\\tau_{s}$ is not linked to the momentum relaxation time in a trivial manner \\cite{K.NanoLetters2012}.\nBy measuring a region about 5 $\\mu$m away from the encapsulated part, we find $D_{s}\\approx$ 0.03 m$^{2}$\/s, $\\tau_{s}\\approx$ 0.3 ns and $\\lambda_{s}\\approx$ 3 $\\mu$m, in agreement with the previously reported results.\nThe increase in $\\tau_{s}$ for the encapsulated region can be due to several factors.\nThis region is protected from polymer remains or other contamination which can increase spin scattering.\nIn addition to that, the inversion asymmetry, which can generate an extra term for the spin-orbit coupling, is also reduced and controlled by tuning $V_{tg}$ and $V_{bg}$ separately as explained earlier.\n\nAs can be seen in Fig. \\ref{fig:figure-2}, $\\tau_{s}$ is modulated by $V_{tg}$, showing a dip close to the charge neutrality point in the encapsulated region (e.g. $V_{tg}$=1.1 V and $V_{bg}$=-52.5 V).\nFor larger charge carrier densities in the non-encapsulated regions, the modulation in $\\tau_{s}$ by $V_{tg}$ is smaller, although still present.\nFurthermore, the average value of $\\tau_{s}$ is maximum at low carrier densities in the non-encapsulated regions (large negative $V_{bg}$) and decreases with increasing $V_{bg}$.\nThis difference on the measured $\\tau_{s}$ for different carrier densities on the outer regions can be explained by simulations that treat the full device \\cite{GuimaraesNanoLetters2012}.\nSince the transport is diffusive, the spins can explore both the encapsulated and the non-encapsulated regions before been detected.\nWe take this into account by describing our sample as two outer regions connected by a central region \\cite{supinfo}.\nThe relevant parameters ($R_{sq}$, $D_{s}$ and $\\tau_{s}$) are set for each region individually.\nThe values for the outer regions and $R_{sq}$ and $D_{s}$ for the inner region can be extracted from our charge and spin transport measurements.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-3-alt.pdf}\n\t\\caption{Effective spin relaxation times extracted for different values for $\\tau_{o}$ and $\\tau_{i}$ (lines) compared to our experimental data for $V_{bg}$ = -52.5 V (dots). The inset shows a schematics of the simulated system.}\n\t\\label{fig:figure-3}\n\\end{figure}\n\nChanging the values of the spin relaxation time for the outer regions, $\\tau_{o}$, around the experimentally obtained values and changing the values for the spin relaxation time in the inner region, $\\tau_{i}$, we simulate Hanle precession curves that are fitted in the same way done for our experiments to obtain an effective value for the spin relaxation time $\\tau_{fit}$.\nWe get a reasonable quantitative agreement between our simulations and experiment at $V_{bg}$ = -52.5V for $\\tau_{i}$ = 3 ns.\nThis means that the spin relaxation time for the encapsulated region is higher ($\\tau_{s}\\approx$ 3 ns), but still within the same order of magnitude as the values obtained by analyzing the data using a homogeneous system (Fig. \\ref{fig:figure-2}).\nThe trend in $V_{tg}$ is also reproduced, which indicates that it is given by the ratio of the resistivities of the inner and outer regions.\n\nEven though the experimentally obtained value for $\\tau_{s}$ depends on the gate voltages in a non-trivial way due to the influence of the non-encapsulated regions, we can still study how the electric field affects the ratio between the spin relaxation times for out-of-plane to in-plane spins: $r=\\tau_{\\bot}\/\\tau_{||}$.\nAs explained in the introduction, this way we can get insight about the SOF in our system.\n\nTo compare the spin relaxation for spins parallel and perpendicular to the graphene plane we perform the Hanle precession measurements as described before, but increase the perpendicular magnetic field to higher values, $B>$ 1 T.\nAt such high magnetic fields the magnetization of the electrodes rotates out-of-plane and the injected spins do not precess anymore.\nThis is seen as a saturation of the nonlocal signal at high B, Fig. \\ref{fig:figure-4}a.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-4-altb.pdf}\n\t\\caption{(a) $R_{nl}$ as a function of $B$ showing Hanle precession at low fields and a saturation of the signal at high fields when the magnetization of the electrodes point out-of-plane. Inset: A cartoon showing the magnetization of the contacts: in-plane at low fields and out-of-plane at high fields. (b) The ratio $\\tau_{\\bot} \/ \\tau_{||}$ as a function of electric field $\\bar{E}$ for different values of carrier density $n$. \\textit{Inset}: The same data points for $\\tau_{\\bot} \/ \\tau_{||}$ used in the main graph, but plotted as a function of $n$ where different colors and symbols represent the different values of $\\bar{E}$.}\n\t\\label{fig:figure-4}\n\\end{figure}\n\nWe observe that different combinations of $V_{tg}$ and $V_{bg}$ result in a saturation of $R_{nl}$ at high magnetic fields at values always smaller than $R_{nl}$ at $B$=0 T, with the saturation occuring at 43-57$\\%$ of the initial value.\nGiven that the nonlocal spin signal is given by $\\Delta R_{nl}=\\frac{P^{2}R_{sq}\\lambda_{s}}{W}e^{-L\/\\lambda_{s}}$, where $P$ is the polarization of the electrodes, $L$ the distance between electrodes and $W$ the channel width, we can estimate the anisotropy in the spin relaxation times assuming that $P$ and $R_{sq}$ do not change significantly with field.\nDue to a large magneto-resistance at low carrier densities, our analysis is done only for points at large enough carrier densities for both the inner and outer regions \\footnote{At low $n$, $R_{sq}$ of our devices scales with B$^{2}$ leading to a background in our signal that overcomes the nonlocal spin signal at large B.}.\nWe can relate the ratio of the nonlocal spin signal and the ratio of the spin relaxation times by:\n$R_{nl}^{\\bot} \/ R_{nl}^{||} = \\sqrt{r}e^{\\frac{L}{\\lambda_{||}}\\left( \\frac{\\sqrt{r}-1}{\\sqrt{r}} \\right)}$, where $\\lambda_{||}=\\sqrt{D_{s}\\tau_{||}}$ is obtained via our Hanle precession measurements.\nIn Fig. \\ref{fig:figure-4}b we plot the ratio $\\tau_{\\bot} \/ \\tau_{||}$ as a function of $\\bar{E}$ for different values of $n$ where we see a clear decrease of this ratio, from 0.75 at $\\bar{E}\\approx$ 0 V\/nm to about 0.65 at $\\bar{E}\\approx$  -0.7 V\/nm.\nThe inset on Fig. \\ref{fig:figure-4}b shows the dependence of $r$ as a function of $n$ for different values of $\\bar{E}$ where no clear trend can be seen.\nThe value for $\\tau_{\\bot} \/ \\tau_{||}$ for zero electric field is similar to the values found previously on SiO$_{2}$ based devices ($\\tau_{\\bot} \\approx$ 0.8 $\\tau_{||}$) \\cite{TombrosPhys.Rev.Lett.2008}.\nIn the case of a inversion symmetric graphene layer with no extrinsic sources for SOF we would not expect any particular preference for direction of the spins, meaning that $\\tau_{\\bot} \\approx \\tau_{||}$ \\cite{ZuticActaPhysicaSlovaca2007}.\nThe fact that even at $\\bar{E}$ = 0 V\/nm the ratio between $\\tau_{\\bot}$ and $\\tau_{||}$ is below 1 means that even without an externally applied electric field there are probably remanent SOF pointing preferentially in the graphene plane.\n\nThe decrease of $r$ with increasing $\\bar{E}$ is in agreement with theories that dictate an increase in in-plane Rashba SOF with the increase of electric field \\cite{BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.B2009,RashbaPhys.Rev.B2009,MacDonaldPhys.Rev.B2006,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,FabianPhys.Rev.B2009a}.\nThe Rashba-type spin-orbit Hamiltonian for graphene is given by: $\\mathcal{H}_{SO}=\\frac{\\Delta_{R}}{2} \\left( \\mathbf{\\sigma} \\times \\mathbf{s} \\right)_{z}$, where $\\mathbf{\\sigma}$ and $\\mathbf{s}$ are the pseudospin and real spin Pauli matrices respectively.\nFor single layer graphene the spin-orbit constant is known to depend on the $\\bar{E}$ in a linear way $\\Delta_{R}=\\zeta \\bar{E}$ \\cite{FabianPhys.Rev.B2009,RashbaPhys.Rev.B2009}, where $\\zeta$ is the coupling constant.\nTheoretical values range from $\\zeta$ = 0.3 - 66 $\\mu$eV\/Vnm$^{-1}$ \\cite{MacDonaldPhys.Rev.B2006,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,FabianPhys.Rev.B2009}.\nWe can roughly estimate $\\zeta$ by assuming a D'Yakonov-Perel mechanism for spin relaxation \\cite{DP1971,FabianPhys.Rev.B2009a} with different values for the spin orbit coupling for in- and out-of-plane spins.\nOur analysis \\cite{supinfo} results in $\\zeta \\approx$ (40$\\pm$20) $\\mu$eV\/Vnm$^{-1}$, within the range of the theoretical predictions.\n\nIn conclusion, we measured the spin transport characteristics of a single layer graphene device encapsulated with hBN.\nWe measured spin relaxation times up to $\\tau_{s}\\approx$ 2 ns and spin relaxation lengths above $\\lambda_{s}$= 12 $\\mu$m.\nBy taking into consideration that the non-encapsulated regions of our devices play a role in our measurements of $\\tau_{s}$, we estimate the actual spin relaxation time in the encapsulated region to be $\\tau_{s}\\approx$ 3 ns.\nFurthermore, we showed that the ratio between out-of-plane and in-plane spin relaxation times changes from $\\tau_{\\bot} \/ \\tau_{||}\\approx$ 0.75 to 0.65 with increasing the applied out-of-plane electric field.\nThis observation is in agreement with an electric field induced Rashba-type spin orbit.\nOur results show not only that $\\tau_{s}$ in graphene can be improved by improving the quality of the devices, but also that electrical control of spin information in graphene is possible, paving the way to new graphene spintronic devices.\n\n\\textit{Note added:} During the preparation of this manuscript we became aware of a work in which $\\lambda_{s}$ up to 10 $\\mu$m for single and few-layer graphene were achieved for a single gated structure which did not allow the study of $\\tau_{s}$ as a function of $\\bar{E}$ \\cite{arxiv-aachen}.\n\nWe would like to acknowledge A. Kamerbeek and E. Sherman for insightful discussions and J. G. Holstein, H. M. de Roosz and H. Adema for the technical support.\n\nThe research leading to these results has received funding from the Dutch Foundation for Fundamental Research on Matter (FOM), the European Union Seventh Framework Programme under grant agreement n$^{\\circ}$604391 Graphene Flagship, the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7\/2007-2013\/ under REA grant agreement n$^{\\circ}$607904-13 Spinograph, NWO, NanoNed, the Zernike Institute for Advanced Materials and CNPq, Brazil.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTurbulence is considered to be the last major unsolved problem of classical physics~\\cite{Frisch:1995tu,FalkovichSreenivasan}. It is usually studied in the context of non-relativistic incompressible fluid flows, which are described by the Navier-Stokes (NS) equations\n\\begin{equation}\n\\partial_t v_i + v_j \\partial_j v_i = -\\partial_i P + \\nu \\partial_{jj} v_i + f_i\n\\qquad\n\\partial_i v_i = 0,\n\\end{equation}\nwhere $v_i$ is the fluid velocity, $P$ is the pressure (divided by the constant density), $\\nu$ is the fluid viscosity and $f_i$ some external force. Due to the strong non-linearity of the NS equations, the study of turbulent flows has focused on the statistical properties of the solutions. There are strong indications that these properties exhibit some degree of universality but a full analytical understanding is still missing.\n\nSolutions to the NS equations are characterized by the Reynolds number, defined by:\n\\begin{equation}\nRe = \\frac{L v}{\\nu},\n\\end{equation}\nwhere $L$ is the characteristic length scale of velocity difference and $v$ is the typical velocity difference. Flows with $ Re \\ll 1$ are dominated by the linear terms and are laminar while flows with $Re \\gg 100$ are dominated by the non-linear term and become turbulent. One defined also the viscous scale $l\\sim \\left(\\nu^3\/\\epsilon\\right)^{1\/4}$.\n\nA statistical turbulent steady-state can be maintained by some stationary random external force constantly being applied at the scale $L$. One usually studies the longitudinal structure functions of order $n$:\n\\begin{equation}\\label{tur:structurefuncs}\nS_n (r) \\equiv \\left\\langle \\left( \\delta \\vec{v}(\\vec{r}) \\cdot \\frac{\\vec{r}}{r} \\right)^n \\right\\rangle .\n\\end{equation}\nwhere $ \\delta \\vec{v} (\\vec{r})$ is the velocity difference on displacement $\\vec{r}$. A very well known, and one of few exact results for the statistics of turbulence, is Kolmogorov's four-fifths law \\cite{Kol:1941a}\n\\begin{equation}\\label{tur:Kol45law}\nS_3 (r) = - \\frac{12}{d\\left(d+2\\right)} \\epsilon r,\n\\end{equation}  \nwhere $d$ is the number of space dimensions, $\\epsilon$ is the mean rate of energy dissipation per unit volume and $l\\ll r\\ll L$ ($Re \\rightarrow \\infty$).\n \nKolmogorov \\cite{Kol:1941b} assumed that in this limit the velocity statistics in the inertial range are scale invariant. This assumption and dimensional analysis then lead to the conclusion that:\n\\begin{equation} \\label{eq:univeral_scaling}\nS_n (r) \\propto (\\epsilon r )^{n\/3},\n\\end{equation}\nwhich is known as \\emph{K41 scaling}. Experimental and numerical evidence suggests that turbulent flows in the inertial range do exhibit universal behaviour, such that the structure functions scale as power laws:\n\\begin{equation}\\label{eq:anomscaling}\nS_n(r) \\propto r^{\\xi_n} \\ .\n\\end{equation}\nHowever, in $d\\ge3$ space dimensions, $\\xi_n$ is not linear in $n$, and deviates from the prediction of $\\xi_n = n\/3 $ obtained from the self-similarity (or scale invariance) assumption (except for $n=3$, for which \\eqref{tur:Kol45law} holds). The calculation of the anomalous exponents $\\xi_n$ remains an open problem.\n\nThe Euler equation, which is the Navier Stokes equation with zero viscosity, is a \\emph{dual scale} invariant theory -- it's scale invariant independently in both the space dimensions and the time dimension. The NS equations are of-course not dual scale invariant, and in the limit of zero viscosity the theory is still anomalous and the scale symmetry is broken to $z=2\/3$, as can be seen from \\eqref{tur:Kol45law} (which should be valid at $\\nu\\rightarrow0$ but not  vanishing). This breaking can be thought to be spontaneous \\cite{Oz:2017ihc} and in that sense, the study of turbulence has strong connections with the study of spontaneous symmetry breaking and anomalies in non relativistic field theories (see \\cite{Arav:2017plg} for a related discussion of non-relativistic scale symmetry breaking and anomalies).\n\nEling and Oz proposed \\cite{Eling:2015mxa} an analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence which is a Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation \\cite{Knizhnik:1988ak,Rhodes:2013iua} and is claimed to be valid in any number of space dimensions. In this formula the anomalous exponents are the K41 exponents \\eqref{eq:univeral_scaling} dressed via a coupling to a lognormal random geometry:\n \\begin{equation} \\label{eq:kpz_anomalous_exponents}\n  \\xi_n - \\frac{n}{3} = \\gamma^2\\left(d\\right)\\xi_n\\left(1-\\xi_n\\right) \\ .\n \\end{equation}\nThis formula has one real parameter $\\gamma$ that depends on the number of space dimensions. The use of the KPZ formula is inspired by the fluid\/gravity correspondence \\cite{Bhattacharyya:2008kq, Eling:2009pb, Eling:2010vr}, the fluid is thought to be coupled to random geometry. A derivation of \\eqref{eq:kpz_anomalous_exponents} was suggested in \\cite{Oz:2017ihc}, relying on the effective action for the spontaneous symmetry breaking pattern we will discuss in this paper.\n\nAs mentioned above, the turbulence problem exhibits a two-dimensional scale symmetry which is thought to be spontaneously broken to a regular one dimensional Lifshitz scaling. The above proposal and this symmetry breaking mechanism suggest we should study the spontaneous symmetry breaking of this dual scale, or ``arbitrary $z$'' scale symmetry and the Nambu-Goldstone mode that should exist in such theory. A particle with a logarithmic correlation function may serve as the mediator for the random geometry process needed to establish the KPZ relation, therefore, a logarithmic-correlated NG mode will be of a great interest.\n\n\n\\section{Dual Scale Invariant Effective Action}\nWe are looking for an effective action which would be scale invariant under an \\emph{arbitrary} scale symmetry, i.e., the whole set of transformations\n\\begin{equation}\n \\left({ \\vec{x},t} \\right) \\rightarrow \\left( { \\lambda \\vec{x}, \\lambda^z t } \\right)\n\\end{equation}\nfor arbitrary $z$.\nWe can also parametrize these tranformations differently by\n\\begin{equation}\n \\left({ \\vec{x},t} \\right) \\rightarrow \\left( { \\lambda_1 \\vec{x}, \\lambda_2 t } \\right) \\ .\n\\end{equation}\nWe are interested in the dilaton \\emph{steady-state effective action} which we expect to be a space(-only) integration of a lagrange density $L$.\nA natural guess would be\n\\begin{equation} \\label{random_geometry_lagrangian}\n L = \\tau \\left(\\nabla^2\\right)^{d\/2}\\tau\n\\end{equation}\nwhich induces an effective action invariant under a general tranformation $\\vec{x} \\rightarrow \\lambda \\vec{x} $, regardless of $z$ (or $\\lambda_2$). It would be nice if this was the only option since a particle with such action in $d$ space dimensions has a logarithmic correlation function, the problem is that by declaring the tranformation rule $\\tau \\rightarrow \\tau + \\sigma$ for $\\lambda = e^\\sigma$, we get that we can also have\n\\begin{equation}\n L = e^{-\\alpha\\tau} \\left(\\nabla^2\\right)^{\\beta\/2}\\tau\n\\end{equation}\nas long as $\\alpha+\\beta=d$\n \nNote that the original symmetry group had 2 degrees of freedom -- $\\lambda$ and $z$, or $\\lambda_1 $ and $\\lambda_2$, but the symmetry we required the dilaton lagrangian to obey was parametrized by only one of them $\\lambda$ (or $\\lambda_1$). This seems like a simplification of the requirement and we will see that although we don't introduce time dependence or time integration (by the steady-state assumption), there is a way to require both $\\lambda$ and $z$ (or $\\lambda_1$ and $\\lambda_2$), and therefore getting stricter requirements.\n \nNote also that usually when one deals with Lifshitz scaling with fixed dynamical exponent $z$, the dilaton transformation is defined to be\n\\begin{equation}\n \\tau\\left(\\vec{x},t\\right) \\rightarrow \\tau \\left( {e^{-\\sigma} \\vec{x} , e^{-z\\sigma}t } \\right) + \\sigma \n\\end{equation}\nsuch that the dilaton is able to directly compensate for space variations, by the $\\sigma$ contribution added to $\\tau$ which corresponds to the space tranformation $\\vec{x} \\rightarrow e^{-\\sigma}\\vec{x}$. However, we can suggest a different representation:\n\\begin{equation}\n \\tilde\\tau\\left(\\vec{x},t\\right) \\rightarrow \\tilde\\tau \\left( {e^{-\\sigma\/z} \\vec{x} , e^{-\\sigma}t } \\right) + \\sigma \n\\end{equation}\nsuch that the tilded dilaton $\\tilde\\tau$ is able to directly compensate for time variations since in this representation the added $\\sigma$ contribution corresponds to the time transformation $t \\rightarrow e^{-\\sigma}t$.\n\nBy moving to this representation, it is justified to require the space part of the dilaton lagrangian to be invariant under the two parameters tranformation:\n\\begin{equation} \\label{new_dilaton_transformation}\n \\tilde\\tau \\rightarrow \\tilde\\tau + \\sigma,  \\vec{x} \\rightarrow \\lambda\\vec{x}\n\\end{equation}\nwith no connection between $\\sigma$ and $\\lambda$, which are in fact related by the hidden (and arbitrary) $z$. In this representation, $\\sigma$ defines the time scaling, and $\\lambda$ defines the space scaling, so that it is clear that the lagrangian should be invariant under the dual scaling -- space and time independently.\n \nSuppose we are left with an unbroken scale symmetry with some specific $z$, so that $\\sigma_{1,2} = z_{1,2}\\sigma$, i.e. $z = z_2\/z_1$ is unbrokem while other $z$'s are broken. We should expect $\\tau$ to be invariant under the unbroken symmetry, thus, up to overall normalization factor,\n\\begin{equation}\n \\tau \\rightarrow \\tau + \\alpha\\sigma_1 + \\beta\\sigma_2\n\\end{equation}\nwith \n\\begin{equation}\n \\alpha z_1 + \\beta z_2 = 0 \\ .\n\\end{equation}\nIn our case $z=2\/3$ so we have $\\tau \\rightarrow \\tau - 2\\sigma_1 + 3\\sigma_2$.\n\nIn general, for when we have $\\tau \\rightarrow \\tau + \\alpha\\sigma_1 + \\beta\\sigma_2$ with nonzero $\\beta$, the possible leading order space only lagrangian is only $\\tau \\left( \\nabla^2 \\right) ^ {d\/2}\\tau$ because any $\\tau$ exponent would allow an arbitrary time scaling to ruin the symmetry since the exponent term would scale while all space derivatives wouldn't.\n\nWe can have higher order terms. Every $\\tau$ should be accompanied by a space derivative, and the total dimension (number of derivatives) should be $d$. So for example $(\\partial\\tau)^d$ is a possibility. Rotation invariance requires of course all derivative indices to be contracted in pairs, which is a constraint. This constraint seems to forbid all local terms in the odd space dimensionality case, we may keep $\\tau \\left( \\nabla^2 \\right) ^ {d\/2}\\tau$ but recognize the fact that it is non-local. For even dimensionality we have more terms the higher the dimensionality is. For example, in 2d we have only $\\tau \\Delta \\tau$ (which is equivalent to $\\partial\\tau \\partial\\tau$), in 4d we have $\\tau \\Delta^2 \\tau$, $(\\partial\\tau \\partial\\tau) \\Delta \\tau$, $(\\partial\\tau \\partial\\tau)^2$.\n\n\\section*{Acknowledgments}\nWe would like to thank Igal Arav and Yaron Oz for valuable discussions, and specifically Yaron Oz for introducing this problem to me. This work is supported in part by the I-CORE program of Planning and Budgeting Committee (grant number 1937\/12), the US-Israel Binational Science Foundation, GIF and the ISF Center of Excellence.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nAutomated methods proposed by the natural language processing and information retrieval communities often form the basic building blocks in an application. However, in this paper, we argue that such automated tools, even though they have achieved some level of maturity, are not enough for the needs of the end-users especially for\ndomains that require higher-level information assimilation and cognition like foraging and sensemaking over spatial information. For example,~\\citet{lai2022nlp} have recently used natural language processing (NLP) to understand context in the extraction and geocoding of historical floods, storms, and adaptation measures. They extend the state-of-the-art for low-level information extraction, e.g., named entity extraction and geocoding, but do not provide a holistic understanding of the story underlying these events.\n\nWe posit that our research community needs to ``see the forest for the trees.\" Sensemaking is an integral part of\ninformation processing, and tighter coupling between the lower levels (information extraction) and the higher levels (information understanding and sensemaking) can improve the state-of-the-art. Specifically, we call for the community to look more at ``higher-level tools and systems\" that enable end-users to complete tasks. Towards this goal, we study the case of extracting geospatial information from text using visual analytics (VA)~\\cite{andrienko2020va,yuan2021va} to perform tasks over the extracted data.\n\nSince text is unstructured data and the information within the text is often messy, the output from computational techniques includes associated errors and is not sufficient to explore mentions of movement in text without human expertise. VA can address this issue through human-in-the-loop strategies that enable analysts to work iteratively with computational methods that extract knowledge from messy data, cope with uncertainties in computational results, and improve those results over time~\\citep{endert2014viz,robinson2017geoviz}. VA is especially suitable for big, diverse, messy data that can be interpreted differently~\\citep{tapia2021spatio,angelini2018visual,ninkov2019vaccine,snyder2020,MacEachren2011a}.\n\nOur research objective was to determine if computational techniques and geovisual analytics can leverage \\textbf{large volumes of movement statements to enable an end-user to understand the movement described} quickly. If successful, research can then take advantage of the wealth of movement data found in written descriptions about people, wildlife, goods, and other things moving throughout our world. Text statements about movement can be used to understand what is moving, when it is moving, why it is moving, and how it is moving.\n\nFor our research, ``geographic movement\" refers to the movement of people, animals, objects, goods, information, natural physical processes, and similar things through spaces ranging in size from multiple buildings to the whole earth. We applied computational methods to identify and extract movement statements,\nand present them in GeoMovement, a human-in-the-loop\nweb-based geovisual analytics system for  identification, processing, and exploration of descriptions of movement. \nGeoMovement involves computational 1) cleaning of the messy text, 2) predicting the statements that describe movement using a machine learning (ML) model, 3) applying Geographic Information Retrieval (GIR) techniques to identify places mentioned, and 4) predicting statements that describe restricted movement or desired movement that is not possible (hereafter referred to as ``impaired movement\"). While there is substantial research on some of these subtasks, \\textbf{integrating these techniques with VA} and demonstrating its success in our chosen domain is the main contribution of this paper.\n\nWhile some progress has happened in processing descriptions of movement in text, there is very little work on detecting and understanding descriptions of impaired movement. For example, the COVID-19 pandemic prompted us to focus on impaired movement due to the importance of disruptions and restrictions in global movement patterns of people and, perhaps equally important, the movement of goods like food and medicines. Therefore, another important contribution of this work is that we show an adaptation of an existing approach developed to detect negation in picture descriptions~\\citep{van2016pragmatic} can successfully uncover impaired geographic movement in text documents.\n\nSpecifically, we integrated (and adapted or extended) many existing computational and VA methods to produce a system that supports information foraging related to geographic movement as reflected in text statements. GeoMovement is unique in identifying movement statements and filtering them by place and time.\n\nFigure~\\ref{fig:fig1} shows GeoMovement's user interface. Users can search and filter based on search terms, the statements' dates, and impaired movement status. The statements originate from three sources ingested into GeoMovement to demonstrate its capabilities and utility for investigating movement:\n\n\\begin{itemize}\n\t\\item 398 thousand News articles from August 2019 to May 2020,\n\t\\item 328 million Twitter tweets from February 2020 to May 2020,\n\t\\item 15.6 thousand Scientific articles from August 2019 to November 2020.\n\\end{itemize}\n\n\\noindent\nOver 520 million total statements contain diverse movement patterns, things moving, geographic coverage, and temporal differences.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure1.eps}\n\t\\caption{The GeoMovement user interface: A) Main mapping interface with hexagons selected for place aggregation B) Connection lines between co-occurring place mentions from an area near New York City selected C) Free text key term search D) Text data source buttons E) Normal movement vs. impaired movement buttons F) Geo-binning buttons G) Classification method for bins and connection lines colors and the number of classes chooser H) Statements that match the search I) Most common bi-grams that match the search J) Two-sided Temporal Bar Chart that shows the number of statements, by date, with a divided bar depicting normal movement on the right (green) and negated movement on the left (purple) K) Time range slider.}\n\t\\label{fig:fig1}\n\\end{figure*}\n\nExisting geographic movement research has improved analysis methods~\\citep{dodge2012similarity,dodge2016future,dodge2016movement,dodge2016traj,graser2021movement,graser2020massive,graser2020trajectory,soares2017,huang2017} and shown how these methods can derive valuable information about human movement and wildlife movement~\\citep{wang2020trajectories,dodge2014vultures,miller2019movement,zhu2021travel,li2021trans}. However, partly due to the challenges and because computational techniques to address them are relatively new, this prior research focused on geographic movement in precise movement trajectories from sensors such as GPS and mostly ignored movement described in text documents. The limited existing research to analyze movement described in text has only focused on narrow tasks like mapping route descriptions.\nThis work fills a gap in the research to support the study of geographic scale movement by developing and demonstrating methods for analyzing movement described in text documents at a broad scale. \nWe anticipate that our effort will spur more research to leverage this under-utilized geospatial movement information in text documents.\n\nWe present two case studies (and a third case study in a supplementary video demonstration) related to the global pandemic that demonstrate the potential of our approach and shows, despite the messy data and imprecise computational predictions, a human can derive meaningful and essential information about geographic movement described in text with GeoMovement. The case studies also demonstrate that GeoMovement can support multi-scale information foraging through vast volumes of messy text statements about geographic movement. Furthermore, the space-time concept\/attribute filtering methods implemented effectively narrow in on information relevant to an analyst's objectives.\n\n\\section{Related Work}\\label{sec:sec2}\n\nFirst, we review related efforts to analyze various geographic movement types described in text documents. Existing research into studying descriptions of geographic movement has primarily focused on reconstructing route descriptions. These routes (driving directions~\\citep{jaiswal2012geocam,drymonas2010routes}, hiking and general route descriptions~\\citep{moncla2014hiking,moncla2014itinerary,moncla2016itinerary,Piotrowski2010a}, and historical exploration routes~\\citep{bekele2016historic,blank2015historic}) form a constrained subset of movement statements that simplify and thus do not address many\nof the\nchallenges with a broader set of movement statements. Other research took the opposite approach of textualizing (convert routes to description) to take advantage of the benefits of text~\\citep{chu2014taxi,aldohuki2017taxi}. \nAdditionally,~\\citet{huang2020nlp} showed how geovisual analytics could improve the retrieval of trajectories in a search. But, their focus on text analytics was only on users' queries of the data (which was precise sensor-based trajectories). Furthermore, in addition to addressing narrow domains, most of this research has restricted the data to a small amount of text.\n\nThe GeoCAM project~\\citep{jaiswal2012geocam} created an application that identified, extracted, and generated maps of route directions found on webpages providing textual (often formatted) directions to reach a location. While the GeoCAM project complements the research we present here, their problem is\nsimpler since route directions \nare just a small subset of movement descriptions. Furthermore, route directions typically follow a semi-structured pattern that allows for simpler ML models and rules-based approaches. In a precursor to the GeoCAM project, \\citet{drymonas2010routes} used similar techniques to map route directions. These projects encouraged future work like ours to go beyond route directions to general movement descriptions~\\citep{klippel2008linguists}.\n\nSecond, we describe related efforts that use geovisual analytics on geographic movement described in text documents. The complicated nature of analyzing place in text documents, especially the need to represent spatial relationships best rendered visually on a map, has prompted other researchers to take a geovisual analytics approach. SensePlace was a system to analyze place mentions in text and pull information from other sources to aid analysis~\\citep{Tomaszewski2011}.\nSensePlace2 developed and applied geovisual analytics to methods that focus on the extent to which Twitter users' tweet location compared to the places they discuss in their tweets~\\citep{MacEachren2011a}. \\citet{Robinson2013b} performed a user study with experts that showed both the advantages, usefulness, and difficulties of such a system for crisis management. SensePlace3 extended this effort by advancing the geovisual analytics techniques and scaling the system to work on millions of tweets per month, thereby improving analysis~\\citep{Pezanowski2017}.\n\nThe SMART system complements the SensePlace versions' focus by providing a visual interface enabling human analysts to explore text's spatial, temporal, and topical components~\\citep{snyder2020,karimzadeh2019geovisual}. SMART implemented advanced geovisual analytics techniques, including a tweet classifier to filter semantically and a cluster lens to visualize keywords at a large scale. However, like SensePlace3, their system did not focus on analyzing movement. Finally, the NewsStand system~\\citep{teitler2008,samet2020} mapped places where news articles are written compared to the places they discuss but focuses less on geovisual analytics for analysis and more on correctly mapping the text.\n\nA few other efforts use geovisual analytics and mapping systems to analyze places mentioned in tweets~\\citep{Thom2012,Bosch2013,Bosch2011,felmlee2020women} and show the potential to take advantage of this geographic data source, albeit mainly focusing on tweets with a geocoded location that makes the challenge different. Mapping and geovisualizations have also been combined with topic analysis and network graph analysis to show similarities between cities~\\citep{hu2017extracting}.\n\n\\citet{jamonnak2020videos} combined location information associated with videos and the narrations of those videos to show their locations on the map and the topics and sentiment discussed at those locations.\n\\citet{xu2018reviews} created a system to explore Yelp business reviews in areas and their change over time.\n\\citet{ma2020local} showed that geovisual analytics is vital to understanding\ncritical local places that need immediate help in a disaster from 911 call transcripts and clusters of specific crime types from police reports. Although these systems successfully demonstrated geovisual analytics on text, they focused on particular topics and did not consider movement. Therefore, they could not be applied to our goal of analyzing wide-ranging types of movement described in text. Since impaired movement detection is not the primary focus of this research, we describe work related to it in Section~\\ref{sec:impaired}.\n\n\\section{Text Computational Processing Predictions}\\label{sec3}\n\nWe first acquired three different sets of documents consisting of 398 thousand news articles, 328 million tweets, and 15.6 thousand scientific articles. Our document sources and the keyword and time parameters used to obtain them are described in detail in Appendix~\\ref{sec:textsources}. We cleaned the documents using typical text pre-processing methods and applied computational techniques to identify places in the text, predict the statements that describe movement, and predict statements that describe impaired movement.\n\n\\subsection{Predicting Movement}\\label{sec:predmovement}\n\n\nIn \\citet{pezanowski2020geomovement}'s work, humans label the statements with a binary class as either describing geographic movement or not. Since they took this initial step and created a corpus to train a model, we can use this model to predict statements that describe movement. The prediction of this ML model is a probability value between zero (no movement) and one (movement). We set an arbitrary cut-off for GeoMovement to only show statements with a probability greater than 0.6 that the statements are about movement. This relatively loose threshold was arrived at by trial and error and is acceptable for all three sources.\n\n\n\\subsection{Predicting Geographic Location}\n\nWe used the GeoTxt system to perform GIR on our text sources~\\citep{karimzadeh2013geotxt,karimzadeh2019geotxt}. We chose this because GeoTxt performs comparably or better than other state-of-the-art geoparsers and performs best without case sensitivity, which is common in Twitter data (one of our data sources)~\\citep{gritta2018geoparsing}. \nA small evaluation corpus had an F1-score of 0.78 for geoparsing place names and an accuracy of 0.91 in resolving those place names correctly. Moreover, \nit performed even better on higher-order administrative places such as countries and states, which are common in our statements. For enterprise projects, paid commercial sources also exist from software companies such as Esri, Google, and Microsoft. We chose not to use these products because they do not allow for customization compared to GeoTxt, which is open-source software and\nallows adjustments to the software in the future.\n\n\\subsection{Predicting Impaired Movement}\\label{sec:impaired}\n\nThe global pandemic of 2020-2021 brought attention to global movement and how it spreads, and how the pandemic disrupted or prevented regular global movement. Because the pandemic highlighted the importance of analyzing disruptions to movement, we investigated potential strategies for detecting statements about impaired movement.\n\nTo detect impaired movement in our statements, we looked to adapt existing methods of negation detection in text. Negation detection strategies can potentially uncover statements about formal restrictions on movement (of the sort imposed by governments), decisions not to move taken by individuals for their safety, and impediments to movement created by limited public\/commercial transport such as canceled flights due to the lack of passengers or ill crew.\n\nIn this section, first, we review existing related research on negation detection and its everyday use cases. \nSecond, we describe how we adapted an existing approach from the literature that detects negation in picture descriptions~\\citep{van2016pragmatic} to our challenge of detecting impaired geographic movement described in text. Third, we show how we improved upon our initial attempt to detect impaired movement using our geovisual analytics methods to analyze initial mistakes in predictions and then modify the rules specifically to detect impaired movement.\n\n\\subsubsection{Existing Approaches to Negation Detection}\n\nAddressing negation in text has been identified as a problem in several existing works. For example, \\citet{Fialho} had remarked that ``when a negation was involved in a sentence, the classifiers found more difficult to return the appropriate label\" in the context of negation in sentences as identified as part of discourse representation structures. Negation detection in text is vital in challenges like automated summarization of medical reports~\\citep{Vincze2008,slater2021negation}, summarizing picture descriptions~\\citep{van2016pragmatic}, and as a hint in identifying sarcasm~\\citep{reyes2014sarcasm}. \\citet{hiremath2021sarcasm} show that sarcasm detection depends upon detecting negative sentences in positive situations and positive sentences in negative situations.\n\nMuch of the current state-of-the-art research on negation detection was influenced by a Workshop titled \\textit{Resolving the Scope and Focus of Negation}~\\citep{Morante2012}, which also produced labeled datasets that continue to be used in training and evaluating the success of new methods. Supervised ML-based solutions such as the LSM Network~\\citep{zhao2021sentiment} have learned negative terms while performing sentiment mining automatically from large-scale training data. The current state-of-the-art method, NegBert, is based on ML~\\citep{khandelwal-sawant-2020-negbert,khandelwal2020multitask}. Although this ML approach is the current state-of-the-art, the challenge in using ML approaches is the need for time-consuming labeling of large amounts of training data. NegBert was trained and tested on datasets designed explicitly for negation detection evaluation and therefore could not be used for our tangential challenge of detecting impaired movement. In the absence of training data, we show that rules-based approaches can still be used.\n\n\\citet{van2016pragmatic} and \\citet{van2016building} have used rules to detect negation. They had humans annotate Flickr picture descriptions for negations and define categories of negations. This annotation exercise produced simple clues for negation, thereby allowing their rules-based approach to be effective on picture descriptions. In general, rules-based approaches have been proven to work in negation detection when the domain is relatively narrow and,\nlike most rule-based systems typically provides high precision but low recall. Rule-based methods can work fine in our application, where a sample of the negative sentences suffices,\nbut having false positives can result in incorrect conclusions.\n\n\\subsubsection{Applying Negation Detection to Descriptions of Geographic Movement}\n\nDetecting impaired movement is similar to previous negation detection using specific key terms. However, what constitutes a negated word is ill-defined and varies\nfrom domain to domain and problem to problem \nsince terms, like \\textit{canceled} or \\textit{diverted}, would not always be considered negated. But, when applied to movement, they are. This ambiguity complicates the task. We investigated if we could adapt the current methods that are focused on the negation of words (ex. She does \\textit{not} have cancer. The alarm clock did \\textit{not} have the feature I wanted.) to detect impaired geographic movement (ex. Our flight to England was \\textit{canceled}. The fruit was \\textit{stuck} in Brazil because of initial fears early in the pandemic.). \n\nWe used~\\citet{van2016pragmatic}'s rules-based approach for negation detection. As discussed above, we adapted this rules-based approach, as opposed to an ML approach, to 1) avoid costly labeling of training data, 2) achieve transparency in how the results are obtained, and 3) because our needs in detecting impaired movement are relatively narrow, which the literature suggests~\\citep{van2016pragmatic,van2016building,slater2021negation} is a good fit for a rules-based approach.\n\n\\citet{van2016pragmatic}'s method tags part-of-speech in text (such as verbs, nouns, adjectives, etc.) and then searches for a list of negation keywords, prefixes, and suffixes to detect negations. Some negation rules examples are a) exact negated words like ``no\" and ``not,\" b) verbs that start with ``de,\" ``mis,\" ``dis,\" or c) adjectives that end with ``less.\" Overall, all rules are relatively simple and are easily understandable and reproducible. If the input sentence matches a rule, it containing negation.\n\nWe first applied \\citet{van2016pragmatic}'s exact rules for negation to our statements previously predicted to describe movement (as described in Section~\\ref{sec:predmovement}) to predict impaired movement. After 800,000 movement statements were predicted, we stopped and obtained a summary count. The model predicted about 28\\% of these movement statements as describing impaired movement.\n\nWe selected a stratified random sample of 50 predicted impaired movement statements and 50 predicted normal movement statements. The initial model did not do very well to predict impaired movement correctly. There were 23 true positives where the statement was correctly predicted as impaired movement compared to 27 false positives where the statement was predicted to be impaired movement, but it was normal movement. It did slightly better in correctly predicting normal movement with 42 true negatives as opposed to eight false negatives. Table~\\ref{tab:matrix} shows the confusion matrix for these predictions. These values equate to a precision of 0.46, a recall of 0.74, an F1-score of 0.57, and an accuracy of 0.65 on the stratified sample.\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption{The confusion matrix for prediction of negated movement with unmodified rules from general negation.}\\label{tab:matrix}%\n\\begin{tabular}{@{}|c|c|c|@{}}\n\\hline\n & Actual impaired  & Actual normal \\\\\n\\hline\nPredicted impaired   & 23 (TP)   & 27 (FP)  \\\\ \\hline\nPredicted normal    & 8 (FN)   & 42 (TN)  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Improving the Detection of Impaired Movement}\n\nTo improve our detection of impaired movement, we either added new rules or removed existing rules. Since there were more false positives than false negatives, this hints that rules should mostly be removed so that fewer statements are predicted to have impaired movement.\n\nNot only did we analyze the errors in statement predictions, but we also consulted a list online of verbs common to movement see Appendix~\\ref{sec:A} and \\url{https:\/\/archiewahwah.wordpress.com\/2019\/04\/16\/movement-verbs-list\/}). First, we removed rules specific to verbs that had the prefixes ``de,\" ``mis,\" and ``dis.\" These prefixes could rarely negate the verbs we found related to movement. Second, based on an online list of adjectives related to movement (see Appendix~\\ref{sec:A} and \\url{https:\/\/dspace.ut.ee\/bitstream\/handle\/10062\/18053\/adjectives\\_and\\_adverbs\\_of\\_movement.html}), we removed the adjectives' rules beginning with ``a\" and ``dis\" because they are much too broad and did not make sense with those commonly related to movement. Third, we added the specific lemmas ``cancel,\" ``postpone,\" ``prevent,\" and ``avoid\" because they relate to impaired movement. Lemmas allow us to match many different synonyms of these words that all mean the same.\n\nThese rules modifications improved predictions on the existing stratified random sample with a precision of 0.74, a recall of 0.76, an F1-score of 0.75, and an accuracy of 0.79. Therefore, the F1-score improved from 0.57 to 0.75.\n\nSince we used the first set of 100 sampled statements to determine some of our rules, it would not be fair to rely on these improved metrics alone to conclude we improved our method. The rules may over-fit the sample. Therefore, we selected a second unseen test set of 100 statements using the same random stratified technique. On this unseen set, the original rules' prediction F1-score was 0.59, while our new rules were again much better with a 0.65 F1-score. Overall, our minor modifications to the rules produced substantial improvement. \n\nFinally, since we selected the two sets of 100 statements from new articles, as a final assessment, we sampled 100 statements from both the tweets and scientific articles using the same stratified random sampling technique. \nPredictions using our modified rules for impaired movement on the tweets resulted in an F1-score of 0.61 and an F1-score of 0.60 on scientific articles.\nTherefore, our predictions for impaired movement performed slightly less accurately for these two sources than news articles, but still respectable. This lower accuracy is likely because both sources contain language more common for that audience (i.e., slang and other informal languages in tweets and technical language in scientific articles). Based on our experience modifying the rules for news articles, we estimate that additional rule changes would also improve predictions for tweets and scientific articles. However, based on our experience and the literature, we surmise that given resources to label a large amount of training data, using an ML approach as in~\\citet{khandelwal-sawant-2020-negbert} would likely produce more accurate predictions. In summary, we are using a rules-based approach as a proof-of-concept prediction that is important to show the potential benefits of detecting impaired movement statements and use this as an attribute to analyze geographic movement described in text.\n\n\\section{Geovisual Analytics to Find Meaning in Descriptions of Movement}\n\nGeoMovement is a web-based geovisual analytics system that allows users to explore descriptions of geographic movement. GeoMovement serves as an interface between the human and the data described above and is summarized in Figure~\\ref{fig:fig1}. The statements' content, place mentions, date of creation, and impaired movement prediction are all searchable. The map visualizes place mentions in multiple levels of aggregate geo-bins. A geo-bin is many smaller places (e.g., cities) aggregated and displayed as one larger place that they all reside within (e.g., country). Geo-bins allow for a clear summary of location-based data.  By choosing particular places of interest on the map, the user can visualize the co-occurrences of places. The individual statements view completes the overview-first + detail approach. This workflow matches the information-seeking mantra of overview-first to gain an awareness of the information and details of interest on-demand~\\citep{shneiderman1996eyes}. This section is divided into three subsections that follow the information seeking mantra that starts with the overviews, then options for the user to search to filter to statements of interest, then detailed views of the filtered statements.\n\nFor detailed information on GeoMovement, Appendix~\\ref{sec:textsources} describes the sources and nature of the data sources and statements. In addition, Appendix~\\ref{sec:B4} includes technical details of the application development that allow for fast user queries on large volumes of text, thereby enabling efficient sensemaking. An important technical component of our approach is our use of Elasticsearch (\\url{https:\/\/www.elastic.co\/})) as the primary information storage and retrieval software for GeoMovement. Elasticsearch is a search engine that accepts many search parameters like free text and time, and returns matching results ordered by most relevant to the user's search. In addition, it can group results by attributes like place mentions in the text. Most impressively, Elasticsearch does all of this and returns results very quickly, most often in the matter of milliseconds.\n\n\\subsection{Overviews}\n\nThere are two primary overview means to explore the movement descriptions. The first overview is the map (Figure~\\ref{fig:fig1} at point A) that displays the number of statements spatially aggregated by their place mentions. Since we have a large number of statements from multiple sources, GeoTxt extracted over 98,000 unique place mentions from them. Displaying this large number of places on a map using points would likely be very confusing for users. Many places would overlap and quickly seeing overall patterns along and comparing quantities between places would be difficult. This is the primary reason for aggregating the statements place mentions. Coloring the polygon bins that contain places mentions a lot in the statements darker than those that are not mentioned frequently in the statements. Users get a clear overview of place mentions in the statements and a way to visually compare places.\n\nWe chose five shapes to spatially aggregate the number of statements by their place mentions. The geo-bins scales include continent, country, administrative 1 (the worldwide equivalent of a state in the United States), and two sizes of a hexagonal pattern. Each scale allows users to explore different types of movement data, like long-range bird migrations by continent, traded goods by country or administrative level, and detailed movement through the hexagons that do not adhere to political boundaries like the spread of disease. We chose hexagons as an aggregation shape since they tessellate and will distort values less than squares~\\citep{birch2007hexagons,esri2021hexagons}. The user interface provides an option shown in Figure~\\ref{fig:fig1} at point F where the user can choose the aggregate level.\n\nThe bin counts are divided into classes to ease visual comparisons between bins. The user has control of the bin count classification technique and number of classes, as shown in Figure~\\ref{fig:fig1} at point G. The user has five choices on how to classify the aggregate counts. Jenks Natural Breaks, Equal Interval, Standard Deviation, Arithmetic Progression, and Quantile are options, and they can enter the number of classes between two to seven. These classification techniques are well accepted statistical methods to make the data more understandable.\n\nWe used the geostats JavaScript library (\\url{https:\/\/github.com\/simogeo\/geostats}) to calculate the class breakpoints for the chosen users' classification method. Each classification method is valuable depending on the user search and resulting data. Darker colored bins represent areas that have a larger number of place mentions within the bin. The color scheme is a sequential color scheme chosen from ColorBrewer (\\url{https:\/\/colorbrewer2.org\/})~\\citep{brewer2003color} to ensure the classes of statement counts are easily distinguishable.\n\nThe second overview in GeoMovement is a Two-sided Temporal Bar Chart. Research has shown that there are differences in how people understand geographic movement such as the those who think more spatially than others~\\citep{liben1993space,ishikawa2016gis}. Based on this research, GeoMovement provides the user multiple views of the data. The Two-sided Temporal Bar Chart shows aggregate counts of statements grouped by the month they were published (Figure~\\ref{fig:fig1} at point J). The oldest month is at the top and the youngest month is at the bottom. The Two-sided Temporal Bar Chart allows for visualization of the amount and changes over time, both in impaired movement on the left side and normal movement on the right side. When hovering the mouse over the chart, the number of statements that match the current user search are shown in bars in the foreground while the total number of statements in GeoMovement are shown in bars in the background. Figure~\\ref{fig:fig6} shows an example of the number of statements that match the user search in the foreground once the user has hovered the cursor over the chart.\n\n\\subsection{Search}\n\nAfter the overviews give the user an understanding of the data, they can begin filtering it through search options. The five key ways to search the data are 1) free-text search (Figure~\\ref{fig:fig1} at point J), 2) buttons to select from the three sources for statements --- news articles, tweets, and scientific articles (Figure~\\ref{fig:fig1} at point J), 3) buttons to select impaired movement statements or normal movement statements (Figure~\\ref{fig:fig1} at point J), 4) a time-range slider (Figure~\\ref{fig:fig1} at point J), 5) and clicking location(s) on the map to filter by location. Multiple features can be chosen by holding the Ctrl-key on their keyboard and selecting the next feature with a mouse-click. After any of the first four searches are performed, the geo-bins and Two-sided Temporal Bar Chart update to show counts of statements that match the searches. After the fifth search, the detailed views appear.\n\n\\subsection{Detailed Views}\n\nThe detailed views of statements and their attributes include a) connecting great circles drawn\non the map between place mentions in the statements, and co-occurring place mentions in the same statement, b) the five most common bi-grams for the set of statements matching the search, c) and a table showing the actual statements that match the search. All connections between places are aggregated to show places commonly used together. The aggregate count classes adhere to the users' currently chosen map classification method and total class number in a sequential green color scheme that is color-blind friendly, also selected from ColorBrewer~\\citep{brewer2003color}.\n\nIt is important to note that the connection lines are completely accurate in showing movement between the places since they are drawn solely by the places' co-occurrence in a statement. For example, the statement below has three place mentions and therefore three possible movement pairs: Sydney\u2013New York, Sydney\u2013London, New York\u2013London. The first two are correct concerning movement references in the statement, but the third is not correct because there is no direct movement between New York and London. One could argue that the movement could be from New York to Sydney and then next to London from Sydney, but this is not probable. However, these connection lines to give the user overall patterns of interest that they can confirm through inspection of the statements.\n\n\\begin{quote}\nBut this will be the first time a commercial flight is flown from Sydney to New York, and just the second time from Sydney to London, Qantas said~\\citep{Garber2019}.\n\\end{quote}\n\nSecond, the ten most common bi-grams are shown for the set of statements that match the search and have place mentions in the chosen location bin. This provides details about the actual statements behind the overviews by showing the most common words and topics in the selected statements. Since many of the statements contain place mentions and the search location is an essential parameter for the matching statements, two-word place names are often in this bi-grams list. These place mentions may be valuable, but we also found that it was often more important to see bi-grams about topics and not necessarily places when exploring the data. Therefore, we allow users to double-click a bi-gram to remove the bi-gram from the list, and the next most common bi-gram appears, up to the 20th most common bi-gram. To re-populate the bi-grams list, the user can re-run the search. The user can also select multiple bins by holding the Ctrl-key and clicking another map bin. Users can then compare the most common bi-grams for both map bins to see differences between statements with place mentions in each bin.\n\nTo complete the overview first + detail approach to analysis, once the user found sets of statements of interest in the overviews and chose a map bin, the user can see the actual statements matching the search in a paged list. The displayed statements match all search parameters (when selecting multiple bins, statements can have place mentions in either bin). The user can scroll through the pages of statements. Each statement's published date is also shown. If there is a particular statement of interest, holding the Ctrl-key while clicking on the statement will open the original document on the Web in a new browser window.\n\nGeoMovement's tight use of modern search engine technologies and Information Retrieval (IR) allows for extremely fast human-in-the-loop sensemaking for the most relevant information on movement. Again, Appendix~\\ref{sec:B} provides further details, and our supplemental video showcases a real-time live demonstration of efficient knowledge discovery using GeoMovement.\n\n\\section{GeoMovement Assessment}\\label{sec:assess}\n\nFirst, we\nshow the challenge of interpreting movement described in statements without GeoMovement by discussing summary statistics of the data. Second, we present two case studies that show how our approach can retrieve information about movement from vast quantities of statements. A third case study is included in a supplemental video. It is recorded in real-time to show that a user can quickly extract meaningful information about movement despite the challenges \nposed by the large quantity of messy text, ambiguity in text descriptions, and imprecise computational predictions. \nThird, we show how these case studies also generated future GeoMovement needs. \nFinally, in Appendix~\\ref{sec:C}, we discuss the skill level and hardware and software requirements for GeoMovement users.\n\n\\subsection{Illustrative Data Summary Statistics}\n\nTo show the value of GeoMovement, we created summary statistics of the data to clearly illustrate how it is unreasonable to think a human can analyze and understand large quantities of movement statements without such a system. We chose three keywords related to our case studies and three prominent place names: one being a country, one a state, and one a city. The number of our statements that match these keywords and place names is shown in Table~\\ref{tab:counts}. To relate these statistics to our first case study in Section~\\ref{sec:cases}, we filter GeoMovement's 520 million statements (36 thousand of those contain the term \\textit{smuggling}, 275 thousand of them include the term \\textit{gold} and 201 thousand mention \\textit{India}) and efficiently identify important gold smuggling patterns around and in India. To produce these statistics, we used our GIR extracted place names to determine the number of statements that contain each selected place name. And, to find the number of statements containing each of the keywords, we searched the statements in a Postgresql database using a full-text search (\\url{https:\/\/www.postgresql.org\/docs\/14\/textsearch.html}) so that different variations of the same word will be matched (e.g. smuggling, smuggled, smuggle).\n\n\\begin{table*}[htb]\n\t\\begin{center}\n\t\t\\caption{Summary statistics of the number of statements that contain selected key terms and place mentions.}\\label{tab:counts}%\n\t\t\\begin{tabular}{@{}|c|c|c|c|c|c|c|@{}}\n\t\t\t\\hline\n\t\t\tsmuggling & gold & sports & London & California & India & statements \\\\\n\t\t\t\\hline\n\t\t\t36 K & 275 K & 336 K & 589 K & 165 K & 201 K & 520 M  \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\nThis exercise to generate summary statistics is meant to show that without such a geovisual analytics system like ours, it would be extremely difficult, if not unfeasible, to perform geographic, temporal, and attribute sensemaking of the described movement. Although GeoMovement's 520 million statements are substantial, it is still a fraction of the accessible text available that could be included and analyzed in GeoMovement, given more development and computational resources.\n\n\\subsection{Case Studies}\\label{sec:cases}\n\nTo confirm our claim that the geovisual analytics interface helps users understand and make sense from the statements and multiple computational predictions, we provide two detailed case studies below from different types of (prototypical, fictitious) potential users. The case studies presented provide evidence of usability. A third case study, given only in the video supplement (due to space limitations in the text), adds additional evidence about the flexibility and utility of GeoMovement to explore the mix of text data sources from different perspectives.\n\n\\subsubsection{Understanding International Crime Affecting India}\n\nJennifer Lang is a college student who wants to write a class report about different types of international crime affecting India. She opens her web browser to GeoMovement and types \\textit{smuggling} to begin her search. \nShe notices that many statements involve smuggling are in October 2019, despite that month having fewer statements overall. She adjusts the time range slider to filter statements to that month and sees a hotspot of activity in England. She clicks the hexagon bin in England and is reminded of a significant human smuggling event in that month where many people lost their lives after being trapped in a truck that was smuggling them (Figure~\\ref{fig:fig2}). Although this is a significant smuggling event from a British perspective and also highlighted in the U.S. news, she decides to look for other ways to focus on India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure2.eps}\n\t\\caption{By filtering the search results for \\textit{smuggling} to only show statements from October 2019, she sees many statements from a significant human trafficking event in England where many people lost their lives.}\n\t\\label{fig:fig2}\n\\end{figure*}\n\nAs her next step, she chooses to aggregate place mentions by the country level and select India. Figure~\\ref{fig:fig3} shows the results, and she sees that most results related to smuggling are affecting India from the neighboring countries of Pakistan and Bangladesh, and a few more countries. In the bigrams list, she sees many references to drugs and gold being smuggled into India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure3.eps}\n\t\\caption{A search for \\textit{smuggling} shows important location sources for India and a spike of activity in October 2009.}\n\t\\label{fig:fig3}\n\\end{figure*}\n\nNext, after reading some individual statements, she finds out more about multiple cases of gold smuggling from both outside and within India (Figure~\\ref{fig:fig4}). To get a more detailed analysis of the movement, she types \\textit{gold} in the search box and chooses to aggregate place mentions by the state level. By clicking the neighboring state of Sindh, Pakistan, which has many place mentions in the statements, connections with many states in India are highlighted, including a state in southeast India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure4.eps}\n\t\\caption{A search for \\textit{gold} produces many statements about gold smuggling and possible routes.}\n\t\\label{fig:fig4}\n\\end{figure*}\n\nShe clicks this state in India and discovers that the state is Tamil Nadu, where the large city of Chennai is located. From reading a few statements, she knows that gold smuggling in Chennai is arriving at the airport and through the Chennai Express train from Mumbai. The connection with Mumbai through the train is confirmed on the map by the strong connection with Mumbai's state, Maharashtra, on the west coast closer to Pakistan (Figure~\\ref{fig:fig5}). She can now clearly visualize and report on some of the prominent drugs and gold smuggling sources into and throughout India and read more detailed descriptions about individual incidents.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure5.eps}\n\t\\caption{While looking for more detail about gold smuggling, statements suggest a smuggling route from Mumbai to Chennai along the Chennai Express train.}\n\t\\label{fig:fig5}\n\\end{figure*}\n\n\\subsubsection{Examining the Impact of the Pandemic on Travel for Tourism}\n\nArti Reddy is a travel agent in India. She uses GeoMovement to understand the pandemic's impact on global travel and travel related to India. As of May 2020, like other countries in the World, India was dealing with a global pandemic. Since Arti had previously planned to advertise to potential customers traveling for sporting events, she chose this topic to investigate. She loads GeoMovement and enters the term \\textit{sports} in the text search box. With the geospatial hexagon geo-bins as a layer, she quickly sees a hotspot of discussion in Japan. She selects to filter statements for impaired movement. While mousing over the Two-sided Temporal Bar Chart, she sees that the impaired movement statements are more prevalent in recent months (except May, where there is less data), as seen in the Two-sided Temporal Bar Chart in Figure~\\ref{fig:fig6}.\n\n\\begin{figure}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure6.eps}\n\t\\caption{Mousing over the Two-sided Temporal Bar Chart shows the number of filtered statements for impaired movement (purple bars to the left of the axis) involving sports has been increasing since the pandemic began (bars are aggregate counts of statements by month, with the oldest month at the top). Filtered statements by the search are shown in the foreground, while all statements are shown in the background on the mouse hover.}\n\t\\label{fig:fig6}\n\\end{figure}\n\nArti switches to the country geo-binning, chooses to view both types of movement again, and moves the timeline from July 2019 to December 2019, and Japan is still a popular location. After selecting Japan on the map, she sees in the bi-grams list that the upcoming Olympics in Japan are prominent, and on the map, she sees that there are connections between Japan with many other places in the World (Figure~\\ref{fig:fig7}). There is much discussion about the potential impact of the pandemic on the Olympics. She is intrigued to see that there are connections between Japan and her home country that implies her business will be affected.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure7.eps}\n\t\\caption{Sports have been disrupted during the pandemic, including the Olympics, where athletes travel from around the World. Mousing over the Two-sided Temporal Bar Chart shows that the filtered statements for impaired movement have increased since the pandemic began.}\n\t\\label{fig:fig7}\n\\end{figure*}\n\nSince she is most familiar with India's geography, she changes the geo-binning to the state level for more detail and selects two states in India with much discussion (Figure~\\ref{fig:fig8}). A quick look through the statements shows that the Olympics are in jeopardy, and a closer-to-home event of the under-17 women's soccer World Cup scheduled to be in India was unfortunately postponed. Teams would have come from around the World for this event. This sad news prompts her to follow up on the story by clicking the statements to view the original articles in her web browser to see if it will be rescheduled and thus if there will be a future need for travel. From reading the statements, she sees that an auto show, the AP World Indoor Sporting Championships, and the Australian Grand Prix auto racing's China leg are three events postponed in China. Cricket in Australia was also severely affected, with teams planning to come from India, England, and many other countries to compete. Major sporting leagues in the U.S., like the NBA basketball league, were also interrupted. The disruptions to nearby events like cricket are particularly concerning given the sport's popularity with Indians and their potential spectator travel related to her business.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure8.eps}\n\t\\caption{Sports in India have also been affected, including competitions where local teams compete against other teams from around the World.}\n\t\\label{fig:fig8}\n\\end{figure*}\n\nLastly, to see how widespread the pandemic's impact is on global sporting events where Indians may travel, Arti explores other countries and sees that the pandemic has significantly impacted professional soccer in Spain and other European countries. Madrid, Milan, and cities in Germany where prominent soccer clubs play all show up clearly on the map as having their games affected (Figure~\\ref{fig:fig9}).\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure9.eps}\n\t\\caption{Europe is another example where the pandemic postponed many of their beloved soccer matches.}\n\t\\label{fig:fig9}\n\\end{figure*}\n\nThese use cases highlight the potential for information foraging and acquisition from statements describing movement. Geovisual analytics that combines multiple views of the data and overview + detail capabilities allow users to quickly identify important locations, connections between locations, time periods, and topics of interest. We showed how the map view could show critical hotspots, and the Two-sided Temporal Bar Chart can show essential time periods, prompting the user to investigate details.\n\nIn this section, our use cases show GeoMovement's current sensemaking capabilities. Moreover, the examples of computational prediction errors and suggested solutions highlight how geovisual analytics is well suited for allowing user interactions to either explicitly or implicitly improve GeoMovement's computational predictions. Improved prediction accuracy can make the sensemaking process more efficient by reducing the human effort to sift through incorrect intermediate results.\n\n\\subsection{Case Studies Needs Assessment}\n\nIn addition to our case studies demonstrating the effectiveness of GeoMovement in extracting information about movement, we used them as a needs assessment to propose additional functionality. In the first case study, Jennifer identified statements from an event that she was not interested in. Although she quickly found what she was looking for, other problems may require much further analysis, and future results should not include statements deemed not relevant. Therefore, a future addition to GeoMovement can consist of a mechanism to either mark statements as completely ``not relevant\" and not show them or as ``less important.\" If user accounts were added to GeoMovement, these preferences in statements could be stored. If certain types of statements will never be relevant to that user, their marked statements can be used to affect their future search results.\nSuch user input can be used as feedback to the system and in the future, statements that are very similar to those that are marked not of interest will also be filtured out; i.e., the system learns from the feedback.\nElasticsearch allows for on-the-fly criteria provided in searches that promote or demote statements or exclude them, allowing for these criteria to be personalized if user accounts were added and changed for each each search.\n\nIn the second case study, Arti found that impaired movement became more prevalent in recent months. It would be a fair assumption that this is because of travel restrictions from the pandemic. However, Arti may ask for more detail about ``How is impaired movement changing over time?\" \nTo do this, she should easily access statements by month. Currently, she would have to change the time slider filter to each month and select the impaired movement button. A straightforward way to answer this question would be to choose any bar in the Two-sided Temporal Bar Chart to update the time range and impaired movement filters. For example, clicking on the impaired movement purple bar for January 2020 would update all views to show these statements. Similarly, when Jennifer found a spike in smuggling activity in India in October 2009, she would likely ask herself why this is and want a quick way to filter to those statements.\n\nFinally, both Jennifer and Arti were interested in India. A question they both may have is: ``What other locations have similar problems as India?\" Currently, GeoMovement allows selection on the map of multiple locations to show bi-grams lists for both locations' statements next to each other. However, there are automated ways to give users hints on locations with similar statements. For example, Elasticsearch provides a ``percolate\" query where, after the user already selects a location, they could choose a user interface control and click a second location. The statements from the first location can be used in the percolate query. The result would be statements like the first set, and therefore other places like that place will show.\n\nOur evaluation of the sensemaking capabilities of GeoMovement includes a statistical summary illustrating the challenge in extracting meaningful information without it, case study demonstrations, a needs assessment, and a list of user requirements. We show that despite the large volume of messy data that is at times ambiguous, GeoMovement can quickly extract meaningful information about geographic movement.\n\n\\section{Results and Implications}\n\nAnalyzing a large volume of text describing geographic movement requires imprecise computational processing and predictions on already messy data that different people can perceive differently, resulting in some errors. We present the data in a geovisual analytics web application that follows the overview-first + detail mantra~\\citep{shneiderman1996eyes}. Users can find exciting patterns in the overviews and areas that need further investigation. They can also search the data and eventually drill down to the actual statements and articles. Results that appear to be erroneous or merely uninteresting can be hidden so that it is easier for the analyst to focus on more helpful information.\n\nOur approach tightly integrates geovisual analytics with behind-the-scenes computational predictions for statements describing movement from most text that does not, GIR methods to extract geography for mapping and analysis and a novel prediction for impaired movement. We show the value of GeoMovement through case studies where humans quickly learn about and visualize geographic movements. Furthermore, we demonstrate that a multi-scale system that applies a space-time-concept\/attribute filtering process can effectively make sense from large volumes of messy movement statements.\n\nBecause of the challenges of utilizing messy big text data for understanding geographic movement, previous work has either focused only on movement data from precise sensors or addressed narrow domains of movement described in text. However, the need for improved methods to utilize big text data about movement is illustrated in work described in Section~\\ref{sec:sec2}, and \\citet{hultquist2020crowd}, \\citet{hultquist2019nuclear}, and \\citet{Janowicz2019semantic}, where geography in text complements sensor data or stands alone to solve real-world problems. \\citet{maceachren2017bigdata} also encouraged such advances in his positional paper. GeoMovement 1) tightly integrates computational predictions and geovisual analytics, and 2) is built with modern web computing technologies that return results from user queries in milliseconds. These features allow users to make sense of and uncover movement patterns quickly. We show how this efficient and broad movement exploration is unfeasible without computational and visual means to assist a human. Moreover, this completes our research objective to show that modern advances in NLP and IR can leverage large volumes of movement statements to understand the movement described quickly.\n\nAlthough we stopped collecting text for GeoMovement, our techniques can be adapted in a straightforward technical way to accommodate a continually updating source given modest computational and storage requirements. In addition, all of our code is available with an open-source license to prompt future advances \\citep{pez2022geomovement}.\n\nWe anticipate GeoMovement or a production version to be beneficial for scientists, journalists, or even the general public to find information about the movement of humans, wildlife, goods, disease, and much more. Advances to analyzing movement statements can provide similar knowledge gains as movement recorded by sensors with precise geospatial and temporal data.\n\n\\section{Conclusions and Future Work}\n\nOur research shows that modern computational techniques can be combined with a human-in-the-loop geovisual analytics system to overcome significant challenges and identify, process, and explore large volumes of movement statements to quickly obtain an overview of movement patterns and forage for detailed information of interest. Future research should take advantage of the wealth of context information found alongside geographic movement in text documents about why, when, and how the movement is occurring that is not often present in precise GPS data. Our research methods can likely be adapted to analyze statements involving other attributes like time and more complex spatial analysis like correlation. In addition, future research should explore the integration of precise geospatial movement data with movement in text. One initial way is to link them spatially and temporally. For example, the following steps can involve connecting entities in the text to other information using the Semantic Web \\citep{bernerslee2001semantic} and linked geographic data \\citep{Stadler2012linked,janowicz2012linked}. Future research needs to identify what makes movement statements different from other statements.  \\citep{pezanowski2022movement}, took initial steps towards this goal by identifying vital characteristics of movement statements that humans use to differentiate the movement described.\n\nAlso, since we chose to illustrate the effectiveness of GeoMovement through the case studies and other assessments in Section~\\ref{sec:assess}, future work should include a more thorough evaluation of user needs, especially focusing on the additions suggested in this Section.\n\nMoreover, prior research shows that a geovisual analytics system combined with ML predictions can continually improve the accuracy of the ML predictions by having users correct machine errors~\\citep{snyder2020,andrienko2022va}. An extension of GeoMovement can make it an intermediary between computational predictions and humans. As more humans use GeoMovement, humans can iteratively correct any errors they encounter, thereby improving the predictions. As an example, in Section~\\ref{sec:impaired}, we discuss how an ML model would likely be superior to a rules-based approach detecting impaired movement but requires a large amount of training data. GeoMovement can show initial predictions of normal movement and impaired movement from an ML model that used a small amount of training data. A simple tool can allow users to correct errors in the predictions. Once many users are using GeoMovement, the training data set can proliferate. This technique has also shown success in commercial production mapping systems like Google Maps, where users reach their destination, and Maps asks about driving directions' accuracy.\n\nTwo overall ways that GeoMovement can improve the computational predictions are, first, explicitly asking for feedback on any incorrect predictions and second, through implicit user actions. For example, in GeoMovement's statements view, a button can be added to explicitly mark any statements that do not describe movement. Once a sufficient number of users mark a statement to conclude that it is incorrect, the corrected statement can be added to the corpus of statements used to train the model. As an example of implicit actions, the statements' list is currently returned in order by the search engine software that roughly corresponds to how closely they match the search terms entered and how frequently those search terms appear in the statement. Skipped statements can be recorded when users page through results to find what they are looking for. If many users skip certain statements, they can be deemed less valuable and given a weighting that lets them be listed lower in the order they are returned, thereby promoting more critical statements. There are many other possibilities to obtain either explicit feedback (users correct place mentions that were assigned to the incorrect location using the GIR techniques~\\citep{karimzadeh2019geoannotator}) or implicit feedback from users (identifying important locations to highlight based on where previous users navigated to on the map) to improve GeoMovement's sensemaking ability.\n\nAdditionally, none of the existing GIR systems have ideal performance metrics. There is often a trade-off between different geoparsers' false positives and false negatives and other ways to rank geocoding results on large datasets. Future work can allow user controls in GeoMovement to choose different geoparsing back-ends. Also, the open-source GIR GeoTxt can be modified to allow GeoMovement users to set a tolerance to allow more false-positives in situations where missing relevant statements are most critical or allow fewer false-positives in situations where the key is a quick overview and having every relevant statement does not matter. Our use of geovisual analytics for GeoMovement adds substantial potential for future improved analysis of movement statements.\n\nFinally, it is important to revisit that GeoMovement only supports English. Therefore, a valuable extension to our research would be the addition of other languages. The higher-level software, programming languages, and APIs (discussed in Section~\\ref{sec:B}) we used in our research to harvest, process, store, retrieve, and visualize data are all capable of handling many languages and character sets. However, incorporating documents in other languages into GeoMovement would require significant work on the three main computational predictions we used. First, identifying statements that describe movement would require a large amount of labeled training data in the added language. Also, for higher accuracy in predictions, language models like ELMo would need to be re-trained for that language like the research of \\citet{che2018lang} and \\citet{fares2017lang}. Second, our geoparsing methods to extract place mentions and geocode them to their correct location on Earth would need to be adapted \\citep{mandl2008lang}. Third, our negation detection techniques would also require adaption to support other languages \\citep{morante2021lang}. These changes are possible but need significant work as existing research is less robust than the equivalent in English.\n\nThe World is a dynamic place, and understanding geographic-scale movements of things is essential in many domains like business, public health, and environmental science. Leveraging information about movement found in text can complement precise sensor-based movement data. Also, movement described in text is valuable since precise movement data is often unavailable because of high costs or impracticality to deploy sensors. GeoMovement and the combination of computational and visual methods it integrates are steps toward that objective.\n\n\\backmatter\n\n\\bmhead{Supplementary information}\n\nWe include a supplementary video of GeoMovement to show how geovisual analytics helps derive valuable patterns about geographic movement from text documents.\n\n\\bmhead{Acknowledgments}\n\nWe would like to thank the Information Technology group in the College of Information Sciences and Technology at The Pennsylvania State University for providing computing resources to host GeoMovement. Specifically, Adam McMillen, a Systems Administrator in the group, provided his expertise in establishing a virtual server and deploying GeoMovement to it.\n\n\\section*{Statements and Declarations}\n\nNo funds, grants, or other support was received.\\\\\n\n\\noindent\nThe authors have no competing interests to declare that are relevant to the content of this article.\\\\\n\n\\noindent\nAuthor Contributions: Conceptualization: Scott Pezanowski, Prasenjit Mitra; Methodology: Scott Pezanowski, Prasenjit Mitra, Alan M. MacEachren; Software: Scott Pezanowski; Data curation: Scott Pezanowski; Formal analysis: Scott Pezanowski; Validation: Scott Pezanowski, Prasenjit Mitra; Resources: Prasenjit Mitra; Visualization Design: Scott Pezanowski, Alan M. MacEachren; Visualization Implementation: Scott Pezanowski; Project administration: Scott Pezanowski; Writing - original draft: Scott Pezanowski; Writing - review \\& editing: Scott Pezanowski, Prasenjit Mitra, Alan M. MacEachren; Supervision: Prasenjit Mitra, Alan M. MacEachren \\\\\n\n\n\\begin{appendices}\n\n\\section{Terms Related to Movement}\\label{sec:A}\n\nTable~\\ref{tab:verbs} shows the list of verbs related to movement used in our negation detection techniques as retrieved from \\url{https:\/\/archiewahwah.wordpress.com\/2019\/04\/16\/movement-verbs-list\/} on January 21, 2022.\n\n\\begin{table*}[!htb]\n\\caption{List of verbs related to movement used in our detection of impaired movement.}\\label{tab:verbs}%\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{ll}\n\\hline\nAdvance - move forward.                                    & Pootle {[}informal{]} - proceed casually.                 \\\\ \\hline\nAim (for) - go in the direction of.                        & Pop (in, to, over etc) - quickly visit\/pass etc.          \\\\\nAmble - walk casually.                                     & Potter - move in an unhurried way.                        \\\\\nAngle (towards etc) - turn one - s steps towards.          & Pound - proceed with fast, heavy steps.                   \\\\\nBack (out, away etc) - move in reverse.                    & Prance - move flamboyantly, with effected grace.          \\\\\nBarrel - move in a forceful, uncontrolled way.             & Progress - advance.                                       \\\\\nBeetle - hurry like an insect.                             & Proceed - go forward.                                     \\\\\nBelt - move swiftly.                                       & Promenade - take a leisurely walk.                        \\\\\nBez {[}informal, dialect{]} - zip around.                  & Prowl - move in a shifty or predatory manner.             \\\\\nBluster - move forcefully yet ungraciously.                & Race - move quickly, in competition.                      \\\\\nBolt - move swiftly.                                       & Ramble - walk far and wide.                               \\\\\nBounce - move with elastic motions.                        & Regress - go back.                                        \\\\\nBound - move quickly with large steps.                     & Return - go back.                                         \\\\\nBumble - proceed in a clumsy fashion.                      & Roam - proceed with no direction in mind.                 \\\\\nCanter - move fairly, like a horse. &\n  Roll - {[}literal{]} proceed in turning motions like a wheel \/ {[}figurative{]} move steadily. \\\\\nCareen - speed forward uncontrollably                      & Rove - wander far and wide.                               \\\\\nCareer - speed forward with little control.                & Run - proceed quickly, both feet leaving the floor.       \\\\\nCharge - move aggressively towards something.              & Rush - move with haste.                                   \\\\\nCrawl - {[}literal{]} go on all fours \/ {[}figurative{]} proceed slowly. &\n  Sashay - move in a confident and flamboyant way. \\\\\nCreep - move sneakily or slowly.                           & Saunter - walk arrogantly, confidently.                   \\\\\nDance - move rhythmically.                                 & Scamper - run like an agitated animal.                    \\\\\nDart - go swiftly.                                         & Scarper - run away.                                       \\\\\nDash - run quickly.                                        & Scoot - proceed at a fair pace \/ shuffle to one side.     \\\\\nDawdle - proceed slowly and reluctantly.                   & Scud - move quickly as if blown by the wind.              \\\\\nDive - descend quickly.                                    & Scuff - walk in a careless, friction-producing way.       \\\\\nDodder - move unsteadily, as if elderly.                   & Scurry - hurry like a small animal.                       \\\\\nDogtrot - move at a brisk, comfortable pace, like a dog.   & Scuttle - hurry like an insect.                           \\\\\nEmerge (from) - come out of.                               & Seethe - proceed like oozing liquid.                      \\\\\nEscape - move out of danger\/confinement.                   & Shuffle - walk slowly, without lifting one - s feet.      \\\\\nFile (in) - {[}of multiple people{]} go one-by-one.        & Skedaddle - depart in haste.                              \\\\\nFlee - run away from.                                      & Skip - proceed bouncing from one foot to the other.       \\\\\nFlounce - move in a flamboyant way.                        & Skitter - move hurriedly.                                 \\\\\nFlop - move loosely.                                       & Slide - move frictionlessly.                              \\\\\nFly - {[}literal{]} move through the air \/ {[}figurative{]} proceed swiftly. &\n  Slink - go smoothy\/sensuously. \\\\\nFootslog - march a long distance.                          & Slip - move frictionlessly \/ make an accidental movement. \\\\\nForge (on, ahead) - proceed strongly and steadily.         & Slither - slide forward like a snake.                     \\\\\nGallop - move quickly, like a horse.                       & Slope - as sneak.                                         \\\\\nGambol - proceed in a playful, energetic manner.           & Sneak - Proceed surreptitiously.                          \\\\\nGlide - move frictionlessly.                               & Speed - move very fast.                                   \\\\\nGo - basic movement verb \/ depart.                         & Split - depart.                                           \\\\\nHare - proceed extremely quickly, like the animal.         & Sprint - run at top speed.                                \\\\\nHasten - move with haste.                                  & Stagger - move unbalanced, unsteadily.                    \\\\\nHead (towards, for, to etc) - proceed in the direction of. & Stalk - move as though hunting.                           \\\\\nHie {[}archaic{]} - go quickly. &\n  Stampede {[}multiple people{]} - progess chaotically \/in agitation. \\\\\nHightail {[}informal{]} - move quickly.                    & Steam - power forward.                                    \\\\\nHike - go a long distance.                                 & Step - move with the feet.                                \\\\\nHop - {[}literal{]} proceed on one foot \/ {[}figurative{]} make a short journey &\n  Streak - move quickly, as if leaving a line of light behind you. \\\\\nHurtle - move quickly, violently and recklessly.           & Stride - walk purposefully.                               \\\\\nIssue (from) - come out of.                                & Stroll - walk in a brisk, leisurely manner.               \\\\\nJog - move at a medium pace\/half-run.                      & Strut - walk stiffly \/ arrogantly.                        \\\\\nJump - propel oneself through the air.                     & Swagger - move arrogantly.                                \\\\\nJaunt - go on a short trip.                                & Sweep - proceed swiftly.                                  \\\\\nJourney - travel a distance.                               & Tank - progress swiftly and forcefully.                   \\\\\nLabour - move with difficulty, requiring force.            & Tiptoe - proceed lightly, silently on the toes.           \\\\\nLeap - jump far.                                           & Traipse - walk a distance.                                \\\\\nLeg (it) - run (away).                                     & Tramp - walk a distance.                                  \\\\\nLimp - proceed unevenly \/ with an injured leg.             & Trample - walk without precision or care.                 \\\\\nLollop - proceed in ungainly bounds.                       & Travel - move a distance.                                 \\\\\nLope - move in large strides.                              & Tread - move using the feet.                              \\\\\nLunge - jump forward to attack.                            & Trek - travel a long time \/ distance.                     \\\\\nMarch - move steadily\/forcefully\/with purpose.             & Trip - proceed lightly, gaily.                            \\\\\nMeander - proceed in an indirect way.                      & Tromp - walk heavy-footed.                                \\\\\nMooch {[}informal{]} - go around in a skulking manner.     & Troop - march with effort.                                \\\\\nMosey - walk in a leisurely manner.                        & Trot - move briskly like a horse.                         \\\\\nMove - basic verb of movement.                             & Trundle - move arduously like a cart.                     \\\\\nNip (into, across, over etc) - quickly go.                 & Tumble - fall \/ spiral forward                            \\\\\nPace - walk steadily.                                      & Undulate - proceed in wavy motions                        \\\\\nPad - walk casually\/softly\/steadily like an animal. &\n  Waddle - Walk in ungainly fashion, from side to side. \\\\\nParade - proceed in an extrovert manner.                   & Walk - go by foot.                                        \\\\\nPatrol - walk around in order to guard.                    & Wander - travel without a direction in mind.              \\\\\nPatter - go with a light tripping sound.                   & Wend - travel by a circuitous route.                      \\\\\nPass - move beyond.                                        & Whizz - go speedily.                                      \\\\\nPelt - move quickly, like a hurled stone.                  & Wobble - move unsteadily.                                 \\\\\nPerambulate {[}formal, rare{]} - walk.                     & Zip - move swiftly.                                       \\\\\nPlod - move with heavy, laborious motions.                 &                                                           \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\n\nTable~\\ref{tab:adjs} shows the list of adjectives related to movement used in our negation detection techniques as retrieved from \\url{https:\/\/dspace.ut.ee\/bitstream\/handle\/10062\/18053\/adjectives\\_and\\_adverbs\\_of\\_movement.html} on January 21, 2022.\n\n\\begin{table*}[!htb]\n\\caption{List of adjectives related to movement used in our detection of impaired movement.}\\label{tab:adjs}%\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l}\n\\hline\nIn the text, some nouns are qualified with adjectives (sharp, sudden) and some verbs with adverbs (dramatically, significantly).\\\\\nAdjectives used to describe movement include: slow, slight, moderate, gradual, steady, quick, rapid, significant, sharp, substantial, dramatic. \\\\\nUsed to show a small change: slight \\\\\nUsed to show a regular movement: gradual, steady \\\\\nUsed to show considerable, striking or unexpected change: significant, substantial, dramatic (both positive and negative change), sharp, sudden \\\\\nAdverbs are formed by adding -ly to the adjective, and sometimes one or two other letters change as well. \\\\\nDegree of change: \\\\\ndramatically, considerably, significantly, substantially, sharply, moderately, slightly \\\\\nNote that ``dramatically\" can refer to both good and bad changes. \\\\\nSpeed of change: \\\\\nrapidly, quickly, suddenly, gradually, steadily, slowly\n          \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\\section{Detailed Description of GeoMovement}\\label{sec:B}\n\n\\subsection{Text Data Sources}\\label{sec:textsources}\n\nFirst, we purchased news articles from Webhose.io (\\url{https:\/\/webhose.io\/}). Webhose.io collects text documents from various Web sources and distributes them in a clean semi-structured XML format. To save costs and reduce the number of irrelevant statements, we used their web archive tool to obtain text documents that match the key term \\textit{travel}, that is in the \\textit{English} language, and that was published on a website categorized as \\textit{news}. We downloaded all articles in their archive between August 1, 2019, and May 10, 2020, that fit these filters (397,919 articles).\n\nSecond, we acquired and incorporated tweets using the GeoCov19 dataset spanning February 1, 2020, to May 2, 2020~\\citep{qazi2020}. This dataset contains hundreds of millions of tweets related to the Covid-19 pandemic. We harvested only the tweets in English, which resulted in 328 million tweets.\n\nThe third source for statements is the Covid-19 Open Research Dataset (CORD-19)~\\citep{wang2020cord}. This dataset contains scientific articles about the Covid-19 pandemic that were harvested and placed in a clean semi-structured XML format. The collection has over 50 thousand articles retrieved from PubMed Central (\\url{https:\/\/www.ncbi.nlm.nih.gov\/pmc\/}), bioRxiv (\\url{https:\/\/www.biorxiv.org\/}), and medRxiv (\\url{https:\/\/www.medrxiv.org\/}) that match search terms related to all Coronaviruses and is updated daily. We downloaded the dataset on November 14, 2020, and selected documents from that date back until August 1, 2019, which totaled about 15,600 documents.\n\n\\subsection{Application Development for Efficient Sensemaking}\\label{sec:B4}\n\nGeoMovement is an entirely web-based application. We built the web client using the React JavaScript Library (\\url{https:\/\/reactjs.org\/}) as its framework. We also used Material UI (\\url{https:\/\/material-ui.com\/})  React Components to make development efficient and provide a friendly and familiar user experience. We used Mapbox GL JS (\\url{https:\/\/docs.mapbox.com\/mapbox-gl-js\/api\/}) in the map portion of GeoMovement and Deck GL Libraries (\\url{https:\/\/deck.gl\/})~\\citep{Wang2017deck} to provide advanced mapping and geovisualization capabilities along with state-of-the-art rendering speeds for large amounts of data. The web client application is a visual interface for humans to search the server's data in Elasticsearch. Other software used to pre-process the data includes Postgresql (\\url{https:\/\/www.postgresql.org\/}) for data storage and PostGIS (\\url{http:\/\/postgis.net\/}) for geospatial computations. The result of this application design is a web application accessible in any major modern web browser, both desktop, and mobile that produces rapid results to user searches of hundreds of millions of descriptions of geographic movement. Almost instantaneous responses to searches enable users to more easily forage through data in the sensemaking process and identify important information. Figure~\\ref{fig:fig1} shows the user interface of GeoMovement.\n\n\\section{GeoMovement User Requirements}\\label{sec:C}\n\nIn Section~\\ref{sec:cases}, we address the utility of GeoMovement through two case study scenarios that demonstrate the ability of GeoMovement to support information foraging through the large volumes of text coming from multiple kinds of sources. The scenarios make specific assumptions about the required capabilities, skills, and prior knowledge of users.\n\n\\begin{center}\n\tUser Capabilities\n\\end{center}\n\nCurrently, GeoMovement works only with text in English; thus, users are expected to be English speakers. Given the focus on interactive maps and graphical displays, users need manual dexterity sufficient to handle interaction with the map and various controls and adequate visual acuity to read the maps and graphs.\nAlthough we did not test GeoMovement on many computer configurations, based on our internal testing, web development expertise, and established guidelines and recommendations for the development technologies, we can estimate that the user needs modest computing capabilities. The user should have a computer with a minimum monitor screen resolution of $1024 \\times 768$ pixels. A modern JavaScript-enabled browser (Chrome, Firefox, Safari, and Microsoft Edge) is required, and minimum computer hardware specifications to support these browsers. GeoMovement was tested and worked on a modern smartphone with a mobile Chrome browser; however, GeoMovement's sensemaking capabilities are restricted because of the small touch screen. Although GeoMovement's design and architecture provide swift search responses from the server software, performance depends on the users' Internet connection speed because it is web-based. Therefore, GeoMovement will function, but user Internet connection speeds under the typical broadband minimum of 25 Mbps will likely produce a lag in user search results that inhibits the analysis.\n\n\\begin{center}\n\tUser Skills\n\\end{center}\n\nGeoMovement expects familiarity with web browsers, graphs, and maps. We assume some experience with thematic maps (maps that depict data geographically), but no particular expertise is required. Thus, users familiar with the kinds of maps routinely found on news sites like those of the New York Times of the Guardian are sufficient. An ability to understand simple statistical graphs (bar charts) is assumed.\n\n\\begin{center}\n\tUser prior knowledge\n\\end{center}\n\nSince GeoMovement is designed for user-led information foraging, it does not provide suggestions to users about what to explore or specifically what to look for. The assumption made is that each user has some knowledge of the topic they are interested in, that they can apply, to leverage the visual interface and computational methods included in GeoMovement. That is, in fact, the point. This is a human-in-the-loop system intended for users who have domain expertise that is complemented by the ability of GeoMovement to process very large volumes of text and potential for interactive filtering tools to support quick narrowing in on items of interest through the use of that domain knowledge to decide what is interesting and what is not interesting. Beyond domain knowledge, however, users also need some understanding of the limitations of the computational methods that underlie GeoMovement. Specifically, they should understand that any system like this will generate some false positives and some false negatives; thus, it will show them some irrelevant information (which their domain expertise is likely to allow them to ignore) and miss some relevant information. Therefore, they need to understand that absence of evidence from GeoMovement does not necessarily mean the absence of the phenomena they are interested in.\n\n\n\n\n\\end{appendices}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOur purpose was to build a high performance supercomputer from PC elements.\nWe use PCs for two reasons. They have excellent cost\/performance ratios \n\\cite{pricewatch} and\n can easily be upgraded when faster motherboards and CPUs will be available.\nThe PMS project\nstarted in 1998, and the machine is now ready for physical calculations.\nOur first PMS machine (PMS1) consists of 32 PCs arranged in a three-dimensional \n$2 \\times 4 \\times4$ mesh. Each node has two special\ncommunication cards providing fast communication through flat cables \nto the six neighbours.\nThis gives a much better performance than simple Ethernet link.\n\nSince the machine is built from PCs, the latest versions of all programming\nlanguages (such as C and Fortran) can be used for coding. Writing applications is\nstraightforward. One only has to keep in mind the 3-dimensional mesh \nstructure of the\nmachine; no further deep understanding of how the communication works is\nrequired. There\nare some routines written in C that make communication of data between the\nnodes easy. The machine works in Single Instruction Multiple Data (SIMD) mode: all\nprocessors execute the same program, while the data they work on may differ.\n\nNowadays double precision floating point arithmetic is necessary for accurate\nresults. PMS offers this precision since the  processors have double\nprecision Floating Point Units (FPUs). In cases when single precision is\nenough, the special MMX instruction set of AMD K6-2 processors can be used. \nThis provides a much higher performance, in principle 8 times higher, \nthan the standard double precision mode.\n\nThe following sections describe the hardware and software architectures\nof PMS. We first give a short overview of the machine and then describe \nthe hardware and the software in more detail. Some performance results are \nalso presented, and an outlook is given.\n\n\n\n\\section{Overview}\nThe nodes in PMS are based on PC components. In our first PMS machine (PMS1)\nthe rack containes four trays (see Fig. \\ref{pms_full}).\nThese trays  hold eight nodes in two rows.\nEach node is powered by its own standard PC power supply located at the\nbottom of the rack. \nAppropriate air cooling by ventilation is provided for.\nNotice, that rack mounting is not\nrequired for the present communication speed of PMS1.\n Even PCs within their own cases can be used.\n\n Each node in PMS is an almost complete PC.\nIn PMS1 the configuration of a node consists of a \n100MHz motherboard (SOYO SY-5EHM), \na single 450MHz  AMD K6-II processor,\n128Mbyte (7nsec) SDRAM, \n10Mbit Ethernet card, \nand a hard disk of caoacity 2.1 Gbyte.\n\n\\begin{figure}\n  \\centerline{\n    \\epsfig{file=trays.eps,height=7cm}\n  }\n  \\caption[hl]{The PMS1 rack and the trays.\n  \\label{pms_full}\n  }\n\\end{figure}\n\nThe nominal speed of each node is 225 Mflop, since floating point\noperations of the AMD processor require two clock cycles. The price of one\nnode was approximately \\$350 (for the price of such a computer see\n\\cite{pricewatch}). The additional cost for the communication cards (see\nlater) is \\$40 for each node. So the price\/performance ratio is reasonably\nlow.\n\n\nPMS uses special hardware  for communication (PMS CH)\nto make high speed parallel\ncalculations possible.   \nThe basic idea behind the hardware is that the PMS CH  provides direct connection \nof each node to its nearest neighbours (NN). \nThe PMS CH handles both polled port (PP) IO operations and direct memory access (DMA)\nbetween two selected nodes.\n\nThe PP IO mode is quite  straightforward; \nthe sender node sends a word and a flag indicating that\na word is being sent. The sender holds the data until the receiver node sends an \nacknowledgement signal after reading the data. \nIn case of the DMA the problem is  synchronization\nbetween the sender and the receiver. Typical usage of DMA assumes data transfer between a PC\nand an another device. This device is usually either a slave or a bus master with \nappropriate circuits. \nIf DMA is used to transfer data between two PCs then one has to deal with\nsynchronization between two DMA chips. Both problems are solved by using \nFirst In First Out (FIFO) data buffers for read(receive) and write(send) \noperations and appropriate hardware hand-shaking circuits.\n\nPrior to data transfer the direction of the data flow must be decided.\nAlso, using DMA transfer receiving and sending procedures must be \ninitiated in a synchronized fashion since long delays in DMA\nmay lead to system crash. The PC architecture limits this synchronization\nto 2-30 bus clocks since both DMA and the interrupt have an \ninvoking latency (which is not defined by the bus protocol and may depend on the\nchip-set applied on the motherboard).\n\nProgramming  the PMS CH is a fairly simple task. It can be done under \n Linux and 32-bit (extended) DOS operating systems. \nAll low-level device drivers are written\nin C and the programmer may use all kinds of commercial, share-ware or free-ware\ncompilers. Using the communication drivers requires the knowledge of only a \nfew functions.\nA simple programming example for Linux users can be found in the Appendix.\n\nThe PMS architecture does not define the bus used for the PMS CH.\nIn PMS1 we have used the simplest ISA bus (see Fig. \\ref{pms_ISA}).\nImplementation of  the PMS CH for other, faster buses is under development.\n\n\n\n\nOur first implementation of the PMS CH includes two plug-in ISA cards, \nthe PMS CPU card and the PMS Relay card. \n\\begin{figure}\n  \\centerline{\n    \\epsfig{file=ISA.eps,height=7cm}\n     \\epsfig{file=c2.eps,height=7cm}\n  }\n  \\caption[a]{a, The PMS CPU and Relay cards. b, Nodes connected by flat cables in one direction.\n  \\label{pms_ISA} \n  }\n\\end{figure}\nThe nodes are arranged in a $2 \\times 4 \\times4$  mesh as shown in Fig. \n\\ref{pms_grid}.\nIn each node both the  PMS CPU and the PMS Relay cards are installed\nproviding a fast communication to the six NNs.\nAt the boundaries periodic boundary conditions are realized as indicated in \nFig. \\ref{pms_grid} where the links at the boundaries correspond to the ones on the other sides.\nThis determines the hardware architecture of the machine, which is\nsimilar to that of the APE machines \\cite{APE}.\n\n\n\n\\begin{figure}\n  \\centerline{\\epsfig{file=pmscluster.eps,width=7cm}}\n  \\caption[h]{The PMS cluster \\label{pms_grid}}\n\\end{figure}\n\n Debian Linux 2.1 is installed on each node. After turning the power on\neach node boots from its own hard-disk. All nodes can be accessed through the\nEthernet network. There is a main computer that controls the whole cluster.\nSome simple Linux scripts have been written to copy the executable\nprogram code\nand the appropriate data to and from the nodes and to execute the programs.\nCollecting the results is also managed by a simple Linux script.\nIn principle, Ethernet connection could be used for data transfer between the nodes during simulations. However, \nthis turns out to be too slow in most cases. One major reason for this is\nthat any data transfer between two machines makes the whole network busy.\nSince building up the Ethernet connection is quite slow even for two\ncomputers, the Ethernet connection is not satisfactory. The  \ntheoretical maximum \nof the Ethernet connection speed is 1 Mbyte\/sec.\nThis bottleneck could be avoided by using switch boxes, however efficient \nswitch boxes are far too expensive and would dominate  the price of the\nentire machine (cf. Fig. \\ref{compare} and the corresponding discussion \nin the text below.)\n\nThe special communication cards --described in more detail in the next\nsection-- provide faster communication between adjacent nodes. However, this\nmakes the machine applicable to local problems only, where\ncommunication only between the neighbouring nodes is necessary. In PMS1 \nthe speed of \ncommunication through these cards --limited essentially by the ISA bus speed-- \nis 2 Mbyte\/sec, which is greater than\nthe speed of our Ethernet by a factor of two. Furthermore to build up our \ncommunication practically no time is needed,  while the build-up of the \nEthernet connection is rather slow.\nEven more 16 pairs of machines can\nsimultaneously communicate. Altogether we get two orders of magnitude \nbetter performance. Note that the system is scalable.\nOne can build machines with larger number of nodes. The\ntotal inter-node communication performance is proportional to the\nnumber of nodes.\n\nThe whole computer is rack mounted. This way the flat cables needed for \ncommunication are relatively short. At current communication speeds\nthis is not necessary. However, for future developments when communication\ngets much faster this can be crucial.\n\n\\section{Description of  PMS Communication Hardware (CH) }\n\nThere are two communication cards in each machine. The CPU card contains\nthe main\ncircuits needed for transmitting data, while the Relay card contains the\nconnectors for the flat cables connecting the adjacent nodes and some\nadditional circuits.\n\\begin{figure}\n  \\centerline{\\epsfig{file=pmscard.eps,width=7cm}}\n  \\caption[h]{The PMS communication card \\label{card_scem}}\n\\end{figure}\nThe block diagram of the cards is shown in Fig. \\ref{card_scem}. \nThe circuits of both\ncards are included. There are two 16-bit buffers,\nthe output buffer and the input buffer, which accept the data coming from the\ncomputer and from one of the adjacent nodes, respectively. The Control\nRegister\nis used --among other things-- to clear the buffers and set the node to\neither\nsender or receiver state.\nThe Cable Select\ncircuit selects which direction the data is sent to or received from. The six\ndirections are labelled as left (L), right (R), up (U), down (D), front (F),\nback (B).\nIf the same physical cable is selected by two adjacent nodes, one of them set\nas sender\nand the other as receiver, a physical connection is established and the\ncontent of the output register of the sender is\nimmediately transferred to the receiver's input register.\nThe Local State Indicator (LSI) and the Remote State Indicator (RSI) are two\nregisters\nto indicate the states of the nodes. There are 12 LSI and 12 RSI lines. They\ncorrespond\nto sending to and receiving from the six directions.\nEach node can indicate its request for sending or receiving through the LSI\nlines.\nThe RSI lines are identical to the six neighbours' LSI lines.\nIf there is a match between RSI and LSI signals (i.e. a send and a\ncorresponding receive request coincide) then the Match Any (MA) bit is set\nand an IRQ\nis generated on the PC bus. The interrupt is generated on both nodes at the\nsame time, so\nthe interrupt handlers on both machines can safely start transferring data\nwithout any extra syncronization. The data can be transferred either \nby DMA or by consecutive I\/O operations.\n\nThe Cable Select circuit, the LSI, RSI and MA registers together with the\nflat\ncable connectors are located on the relay card, while all other circuits are\nlocated on the CPU card.\n\n\n\\section{Software}\n\nAs  mentioned above, the whole cluster is connected via an Ethernet\nnetwork. Each node has a special hostname that corresponds\nto its location in the cluster. \nThe hostnames\nare s000,s001,s002,s003,s010 $\\dots$ s133. The numbers in the \nhostname correspond to the coordinates of the node in the\nthree dimensional mesh.\nWhen it is necessary for a node to identify itself (e.g. write\/read priorities\nduring inter-node communication, see below) the\nfile '\/etc\/hostname' is used. The Ethernet network is used only \nfor job management, i.e. to distribute\nand collect data to and from the nodes. It is not used for \ninter-node communication during simulations. This is achieved \nby the special hardware described in the previous section.\n\nIn order to take advantage of the fast communication from applications\n(e.g. high level C, C++ or Fortran code),\na low-level Linux kernel driver has been developed to access all the registers\nof the communication card. From the user level cards can simply be reached by\nreading or writing the device files '\/dev\/pms0, \/dev\/pms1 ... \/dev\/pms5'.\nThe six device files correspond to the six directions. A write operation\nto one of these device files will         \ntransmit data to the corresponding direction, and reading from these device \nfiles reads\nout previously transferred data. All the necessary input\/output operations for\ntransferring data blocks are performed by the device driver. Notice the\nimportant feature that this can be reached from any high level \nC, C++, Fortran code for which compilers are available.\n\nThe main structure of the device driver is similar to that of the card.\nThere are six read buffers, one for each direction in the main memory of the\nmachine and there is one write buffer. The data are always written to the write\nbuffer and read out from one of the read buffers.\n\nThe driver has two main parts. The first part is accessed from applications when\nthe user writes or reads any of the device files '\/dev\/pms*'. The other part is\nthe interrupt handler where the real data transfer takes place.\n\nWhenever data are written to one of the device files, all the driver does is to\ncopy the written data to the write buffer and set the corresponding LSI send\nsignal to\nindicate that a data send is requested. If the buffer is already full,\nan error byte is returned to the application. Reading from the device files is\nsimilar: if there are data in the corresponding read buffer, they are sent to the\napplication and the LSI receive line is set, since the node is ready to receive\nnew data. If the read buffer is empty, an 'End Of File' byte is returned to the\napplication.\n\nWhen there is a coincidence between corresponding LSI and RSI signals,\nan interrupt is generated by the card, which invokes the driver's\ninterrupt handler. It is the task of the handler to transfer data from the\nsender to the receiver.\nThe interrupt handlers on the two communicating nodes start almost at\nthe same time. The difference may only be a few clock cycles. There is,\nhowever, a need for synchronization. If the machines are ready to send\nor receive the first byte they indicate it with their LSI lines. Notice \nthat this will not cause an extra interrupt since interrupts are\ndisabled within the interrupt handler.\nThe sender first transmits the size of the package that will\nfollow in 16-bit words.\nIn the present version this is a 16-bit value, so the maximum size of a\npackage that can be transferred is 128 kbytes. Then the given\nnumber of 16-bit values follow. Each word from the sender's write buffer\nis copied to the receiver's read buffer. Finally, a 32-bit checksum is sent.\nThe receiver computes its own checksum and if it does not match the\nreceived checksum, it is indicated to the sender and the whole transfer\nis repeated. The final step in the interrupt handler is to clear the \nLSI lines of both nodes. On the one hand this indicates for the sender that the data\nhave been transferred and\nthe write buffer is empty again. On the other hand this tells the\nreceiver that the corresponding read buffer is full, so no new data can\narrive unless the buffer is emptied.\n\nThe buffer sizes are set in the driver to constant values. From the\nprevious paragraph it is clear that the maximum reasonable buffer size\nis 128 kbytes. In order to save memory, while allowing large packages\nat the same time the buffer size is set to only 64 kbytes at present.\n\nThe driver makes application programming quite easy. Communication can\nbe achieved by accessing the above mentioned device files. \nHowever, some C functions\nhave also been written to make writing applications even\nsimpler. These functions are the following:\n\n{\\it pms\\_open} is used to initialize the card. It clears all\nbuffers, sets the LSI receive lines, clears the LSI send lines and\nenables interrupts. The node thus becomes ready to receive data from any of\nthe neighbours.\n\n{\\it pms\\_close} is used to close the card. All LSI lines\nare cleared and interrupts are disabled. No further\ncommunication may take place after this function call.\n\n{\\it pms\\_send, pms\\_recv}  are used to send and\nreceive data. Their parameters are the direction, the number of bytes\nto send, and a pointer to the beginning of the data. On success they\nreturn a positive value, otherwise a negative one. If there is no data\nin the read buffer, {\\it pms\\_recv} returns 0.\n\nThe driver does not take care of any priority problems. It is possible\nto write applications that will not work since all nodes are waiting\nfor data while none of them is sending anything. This is often the case\nwhen the same code is running on all the nodes without any priority\ncheck. There is a simple solution to these kind of problems. The\nparity of the node is simply the\nparity of the sum of the three digits in its hostname. Each time when\ncommunication is performed, even nodes send data first and receive\nafterwards, while odd nodes receive first and send their data afterwards.\nThis simple method works in most cases. Note that the code for it \nis located in the application (C, C++ or Fortran) and not in the driver.\n\n\\section{Performance}\n\nWe carried out lattice gauge theory simulations on our parallel\ncomputer PMS. We used double precision variables.  \nTwo types of theories were studied.\\\\\na. We analyzed the pure SU(3) gauge theory with the \nsimplest Wilson action. Overrelaxation and heatbath \nupdating algorithms were used for the link variables.\nThe most CPU time consuming part, manipulation with\n$3\\times 3$ matrices, was written in assembly language.\nThis increased the speed of the code by more than a factor \nof two.\\\\\nb. We studied the bosonic sector of the minimal\nsupersymmetric extension of the standard model (MSSM). \nSU(2) and SU(3) gauge fields were included with\ntwo complex Higgs doublets and with left and\nright handed coloured scalar quarks. Again, \noverrelaxation and heatbath  updating algorithms were \nused for both gauge link variables and scalar site variables.\nThe microcanonical overrelaxation update for scalar quarks \nis quite complicated and new, its details  will be\ndiscussed elsewhere \\cite{mssm_ewpt_prd}.\nManipulation with $3\\times 3$ and $2 \\times 2$ matrices \nwas written in assembly language, which again increased \nthe speed of the code essentially.\n\nHaving carried out lattice simulations for these two theories we \nobtained similar results for the speed of the\ncode and for the communication between nodes. \nThe MSSM results for communication are actually somewhat better. The reason\nfor that is quite simple. The number of variables in the MSSM\nis  larger by a factor of two than in pure SU(3) gauge theory; \nhowever, the number of floating point operations \nneeded for a full update is more than an order of magnitude larger.\nThus, for the same lattice size the time needed to\ntransfer the surface variables --done by the\ncommunication cards-- compared to the update time\nis smaller for MSSM than for the pure SU(3) theory. \nWe expect that a  similar though less pronounced effect \nshould be observed for fermionic systems e.g. \nwith multiboson \\cite{multiboson}\nalgorithms. \nSince it is much more straightforward to compare our SU(3) results \nthan those of the MSSM simulations with \nthe results of the literature, \nin the following paragraphs we discuss the SU(3) case, too. \n\nFor small lattice sizes the most economical way to use our\n32 PC cluster is to put independent lattices on the different\nnodes. The maximum lattice size in the SU(3) theory \nfor 128 Mbyte memory is $\\sim 20^4$,\nor for finite temperature systems $6 \\cdot 32^3$.\nOne thermalizes such a system on a single node, then distributes\nthe configuration to the other nodes and continues the\nupdating on all 32 nodes.\nWe measured the sustained performance of the cluster in this\ncase, which gave  $32 \\times 152$Mflop=4.9Gflop. This\n152Mflop\/node performance means that one double precision operation is\ncarried out practically for every\nthird clock cycle of the 450 MHz,\nwhereas the nominal maximum of the \nprocessor is one operation for\nevery second clock cycle. As it was mentioned above, without assembly \nprogramming an approximate reduction factor of two  \nin the performance was observed. \n\nIncreasing the volume of the simulated system one can divide\nthe lattice between 2 nodes (the $2\\times 4\\times 4$ topology\nhas 2 nodes in one of the directions). For even larger lattices\none can use 4 nodes (4 in one direction), 8 nodes\n($2\\times 4$ in two directions) 16 nodes ($4\\times 4$ in\ntwo directions) or 32 nodes ($2 \\times 4 \\times 4$ in three\ndirections). Again, the most economical way to perform the simulations\nis to prepare one thermalized configuration and put it on\nother nodes (this method obviously can not be used for the\n$2\\times 4\\times 4$ topology, because in this case the whole \nmachine with 32 nodes is just one lattice). \n\n\\begin{figure}\n\\centerline{\\epsfig{file=perform.eps,width=8cm}}\n\\caption[a]{Sustained performance of  PMS  as a function\nof the lattice volume for pure SU(3) gauge theory and for MSSM. \nThe endpoints of the lines correspond to the largest\nvolumes which can be simulated on a 32 PC cluster.\n\\label{perform}}\n\\end{figure}\n\nBased on our measurements we determined the sustained performance\nof a 32-node PMS cluster as a function of the lattice volume. The\nresult for a set of lattice volumes for finite temperature systems \nwith temporal extension,\n$L_t=6$ for SU(3) and $L_t=4$ for MSSM can be seen on Fig. \n\\ref{perform}. Clearly, the largest volume one can reach is approximately\ntwice as large for the SU(3) gauge theory than for the MSSM. \nFor both cases there are regions where the performance increases\nwith the volume. This can be easily explained by the fact that larger \nvolume means better surface\/volume ratio, thus better performance.\nThere are three drops in the performance for both SU(3) and MSSM. \nThey correspond to\nlattice volumes for which new communication directions were opened\n(or, in other words, the dimension of the mesh of the nodes on which the lattice\nwas divided, increased by one) in order to fit the lattice into the \navailable RAMs. As it can be seen the performance for the MSSM \nis still very high even at the largest volume with three-dimensional\ncommunication: it is just 10\\% smaller than the performance without\ncommunication. This plot gives us the optimum architecture of\nsuch a parallel computer. The number of nodes and the\nnumber of communication directions used for a given lattice \nshould be as small as possible simultaneously. \nThis means  a $2\\times 4\\times 4$\ntopology for 32 nodes and a $2\\times 4\\times 8$ topology 64 nodes. \n\nDespite the fact that the speed of the communication between two nodes\nis not that high (2 Mbit\/sec) the performance of the cluster is quite good. The\nreason for this is the high speed of the individual nodes (450 MHz) and the\nlarge RAM on each node. This sort of design does not need a division of the\nlattice to hundreds of sub-lattices, thus it does not need a very fast\ncommunication. Clearly, the use of PCI communication instead of ISA\ngives an order of magnitude faster communication, which increases the\npotential of such a machine.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=compare.eps,width=10cm,height=10cm}}\n\\vspace{-10cm}\n\\centerline{\\epsfig{file=compare2.eps,width=10cm}}\n\\caption[a]{Sustained performance of different machines\nused in lattice gauge theory as a function of price\/performance.\nOur PMS machine is presented as a 32 PC based system.\n\\label{compare}}\n\\end{figure}\n\nIn order to compare our machine with other existing (and proposed) \nparallel systems\nused in lattice gauge theory, we use the plot of N. Christ \\cite{christ}.\nAs it can be seen in Fig. \\ref{compare},\n PMS1 is perhaps the best existing machine  \nas far as price\/sustained performance is concerned. \nIt is particularly important to emphasize \n that the presented performance 3\\$\/Mflop with $\\approx$4Gflop total \n performance of  PMS1  is calculated\nfor double precision operations (while some of the performances of the \nmachines on the plot are given for single precision operations). \nFor single precision simulations one can use the MMX instruction set which is\n8 times faster than the double precision operations (4 operations for each\nclock cycle). We estimated the single precision performance by assuming that\nMMX programming results in 20\\% decrease in performance. The total performance\nof PMS1 is $\\approx$27Gflop with 0.45\\$\/Mflop price-to-sustained performance\nratio. (Notice that our measured performance is based on double precision\ncode, whereas single precision numbers are estimated ones.) \nOur PMS1 machine, similarly to other workstation farms has a moderate \nmaximum sustained\nperformance as compared to Teraflop-scale machines (CP-PACS \\cite{CP-PACS} or QCDSP \\cite{QCDSP}). \nHowever, PMS1 has a  much better price\/performance ratio than  other workstation\nfarms. The reason for this is threefold.\\\\\na. At the day of purchase our choice for the PC components was optimized \nfor price\/performance.\\\\\nb. Machines with 100Mbit fast Ethernet (e.g. Indiana) are inexpensive, but\ncommunication is slow, resulting in a high price\/performance ratio. \\\\\nc. Machines with much faster networks using Myrinet (e.g. Alice \\cite{alpha} \nand Altacluster)\ngive higher performance. However, this is balanced in the price\/performance\nratio by the higher cost of the network.\\\\\nThe essential feature of  PMS architecture is that the speed of the \ncommunication (depending on the size of the machine) is comparable to that of\nMyrinet. However, the price for such a communication is as low as \n$\\approx$40\\$\/node, which is more than an order of magnitude less than the\ncost of Myrinet.\n\n\n\n\\section{Conclusion}\nWe have presented a description of the status of  PMS project. Details of \nthe communication hardware and  software have been explained. The existing \n32-node machine (PMS1) is able to perform medium size LGT simulations in case \nof nearest neighbour interactions, distributing the lattice on several nodes, \nif necessary. This is done at a very good price\/sustained performance ratio.\nProblems requiring more number crunching and less communication give a  \nbetter performance. \n\nWe plan to increase the number of connected nodes and implement the PMS CH \nfor faster buses in order to increase the maximum size of programs and enhance\ncommunication speed.\n\nFinally, we would like to emphasize that the design of our communication hardware and the\ndeveloped Linux drivers are freely available for non-profit organizations.\nThose who intend to build a PMS architecture machine have simply to buy PCs \nproduce the communication cards and connect the nodes by flat cables.\nFurthermore they have to install Linux and our Linux driver.\nQuestions and requests for information should be sent to pms@labor.elte.hu.\n\n\n\\section{Acknowledgement}\nThis work was supported by Hungarian Science Foundation Grants under Contract\nNos. OTKA-T22929-T29803-F17310-M28413\/FKFP-0128\/1997.\n\n\n\\section{Appendix}\n\nThe following C code is a simple example to illustrate the usage of PMS\ncommunication cards in a given application. This example transfers random numbers\nin each direction. It can be used to test all channels. We performed\nthis test on all cards by sending 100 Gbytes of data on\neach channel. We observed no errors.\nThe header file 'pms.h' that contains the function prototypes\nis also included.\\\\\n\n\n\\baselineskip=-0.4truecm\n\\noindent\n\n{\\small\n\\begin{verbatim}\n#include<stdio.h>\n#include<stdlib.h>\n#include<string.h>\n#include\"pms.h\"               \/* PMS function prototypes *\/\n#define MUL 1579              \/* Random number generator multiplier *\/\n#define N 1000                \/* Number of packages to transfer *\/\n#define K 16384               \/* Size of each package in words *\/\nunsigned short j_r=1, j_s=1;  \/* initial values for random number generators *\/\n\nint send(int dir) {\nstatic unsigned short buf[K];\nint i,err;\n   \n  for (i=0; i<K; i++) {       \/* generate a series of random numbers *\/\n    j_s*=MUL;\n    j_s++;\n    buf[i]=j_s;\n  }\n  err=pms_send(buf,K,dir);    \/* send the random numbers to direction 'dir' *\/\n  return err;\n} \/* end of send() *\/\n\/*********************************************\/\n\nint recv(int dir) {\n  static unsigned short buf[K];\n  int i,err;\n\n  while((err=pms_recv(buf,K,dir))==0); \/* Wait for data: read until data comes *\/\n  if (err>0) {\n    err=0;\n    for (i=0; i<K; i++) {     \/* Generate the same random numbers *\/\n      j_r*=MUL;               \/* and check for errors *\/\n      j_r++;\n      if (buf[i]!=j_r) {\n        err++;\n        printf(\"Error \n        printf(\"Sent: 0\n      }\n    }\n  }\n  return err;\n}\/* end of recv() *\/\n\/*********************************************\/\n\nvoid main(int argc, char * argv[]) {\nint i;\nFILE * hostname;\nchar host[10];\nint pos[3],par;\n\n  hostname=fopen(\"\/etc\/hostname\",\"r\"); \/* Get the hostname of the current node *\/\n  fscanf(hostname,\n  pos[0]=host[1]-48;\n  pos[1]=host[2]-48;\n  pos[2]=host[3]-48;\n  par=(pos[0]+pos[1]+pos[2])\\&1; \/* Get the parity of the node *\/\n  printf(\"position:\n  printf(\"parity:\n  if (pms_open()<0) {         \/* Initialize the card *\/\n    printf(\"PMS card can not be opened\\n\");\n    exit(-1);\n  }\n  for (i=0; i<N; i++) {\n    printf(\"Testing 1D+ -> 1D-\\n\"); \/* Test the first direction *\/\n    if (!par) {  \/* Priority check: even nodes send first, odd nodes *\/\n                 \/* receive first *\/\n      if (send(0) <=0) printf(\"Send error\\n\");\n      if (recv(1) !=0) printf(\"Recv error\\n\");\n    } else {\n      if (recv(1) !=0) printf(\"Recv error\\n\");\n      if (send(0) <=0) printf(\"Send error\\n\");\n    }\n\n    printf(\"Testing 1D- -> 1D+\\n\"); \/* Test the second direction *\/\n    if (!par) {\n      if (send(1) <=0) printf(\"Send error\\n\");\n      if (recv(0) !=0) printf(\"Recv error\\n\");\n    } else {\n    if (recv(0) !=0) printf(\"Recv error\\n\");\n    if (send(1) <=0) printf(\"Send error\\n\");\n  }\n  printf(\"Testing 2D+ -> 2D-\\n\"); \/* Test the first direction *\/\n  if (!par) {\n    if (send(2) <=0) printf(\"Send error\\n\");\n    if (recv(3) !=0) printf(\"Recv error\\n\");\n  } else {\n    if (recv(3) !=0) printf(\"Recv error\\n\");\n    if (send(2) <=0) printf(\"Send error\\n\");\n  }\n\n  printf(\"Testing 2D- -> 2D+\\n\"); \/* Test the second direction *\/\n  if (!par) {\n    if (send(3) <=0) printf(\"Send error\\n\");\n    if (recv(2) !=0) printf(\"Recv error\\n\");\n  } else {\n    if (recv(2) !=0) printf(\"Recv error\\n\");\n    if (send(3) <=0) printf(\"Send error\\n\");\n  }\n  printf(\"Testing 3D+ -> 3D-\\n\"); \/* Test the first direction *\/\n  if (!par) {\n    if (send(4) <=0) printf(\"Send error\\n\");\n    if (recv(5) !=0) printf(\"Recv error\\n\");\n  } else {\n    if (recv(5) !=0) printf(\"Recv error\\n\");\n    if (send(4) <=0) printf(\"Send error\\n\");\n  }\n\n  printf(\"Testing 3D- -> 3D+\\n\"); \/* Test the second direction *\/\n  if (!par) {\n    if (send(5) <=0) printf(\"Send error\\n\");\n    if (recv(4) !=0) printf(\"Recv error\\n\");\n  } else {\n    if (recv(4) !=0) printf(\"Recv error\\n\");\n    if (send(5) <=0) printf(\"Send error\\n\");\n  }\n  pms_close(); \/* Close the card *\/\n}\/* end of main() *\/\n\/*********************************************\/\n\n(file: pms.h)\n\nextern int mynode;\nextern int node_par;\n\nint pms_open(void);\nvoid pms_close(void);\nint pms_send(void * buf, int count, int dir);\nint pms_recv(void * buf, int count, int dir);\n\n\n\\end{verbatim}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA quantum walk (QW) is the quantum version of a classical random walk~\\cite{Aharonov1993PhysRevA,Kempe2003introductory}.\nDue to the principle of superposition in quantum mechanics, \na QW gives rise to impressive applications in quantum information science, from quantum computing~\\cite{LCE+10,Childs2009PhysRevLett,Childs2013Science,Childs2004PhysRevA} to quantum simulation~\\cite{Schreiber2012Science2DQW}, and from implementing quantum measurement~\\cite{Kurzy2013PhysRevLettGeneralizedMeasuring,Bian2015PhysRevLettPOVM,Zhao2015PhysRevA} \nto exploring topological phases~\\cite{Kitagawa2010PhysRevAExploring,2012PhysRevB,Kitagawa2012Observation,xiao2017observation,Wang2018PhysRevA,Wang2019PhysRevLett,Xu2019PhysRevResearch,WZS19}.\nFor the discrete-time QW~(DTQW), entanglement can be generated between coin~(c) and walker~(w) position degrees of freedom of the walker~\\cite{Carneiro2005Entanglement}, which is a key resource for quantum information processing~\\cite{Horodecki2009RevModPhys}. Particularly, qubit-qudit entanglement generated in DTQW provides richer contents beyond qubit-qubit entanglement, such as nonlocality~\\cite{Lee2007PhysRevA,Li2015SR}, entanglement detection~\\cite{Zhao2011PhysRevA_Endetection,Chen2012PhysRevA,Shen2020PhysRevA}, entanglement of formation~\\cite{Gerjuoy2003PhysRevA, Chen2005PhysRevLett,Lastra2012PhysRevA}, survival of entanglement~\\cite{Dajka2008PhysRevA,Giordani2021NJP}, separability~\\cite{Johnston2013PhysRevA,Ha2005PhysRevA},  steering~\\cite{Chen2018PhysRevA},  concurrence~\\cite{Chen2005PhysRevLett,Zhao2011PhysRevA,Mintert2004PhysRevLett} and discord~\\cite{Sai2012,Girolami2012PhysRevLett,Ma2015SR}.\nThe DTQW is particularly valuable for maximal entanglement generation \n(MEG),\nand our goal is guaranteed MEG after just a few steps. \n\nIn a dimensional (1D) DTQW with static coin operations (unchanging coin operation during evolution), \nentanglement generation depends on the initial coin state and cannot reach a maximal value~\\cite{Abal2006PhysRevA}.\nCounterintuitively, by introducing disorder into the DTQW~\\cite{Chandrashekar2013ArXiv}, e.g., randomly choosing SU(2) coin operation\n\\begin{equation*}\n\\hat{C}(\\xi,\\gamma,\\zeta)\n=\\begin{pmatrix}\\text{e}^{\\text{i}\\xi}\\cos\\gamma&\\text{e}^{\\text{i}\\zeta}\\sin\\gamma\\\\\\text{e}^{-i\\zeta}\\sin\\gamma&-\\text{e}^{-i\\xi}\\cos\\gamma\\end{pmatrix},\n4\\gamma,\\xi,\\zeta\\in[0, 2\\pi],\n\\end{equation*}\nat each step,\ngenerated entanglement is significantly enhanced and achieves MEG asymptotically\nindependent of initial conditions~\\cite{Vieira2013PhysRevLett,Vieira2014PhysRevA}. \nMotivated by robust entanglement generation under experimental conditions with imperfections and disorder,\nrandom-coin DTQWs have been studied for various disorder configurations~\\cite{Zeng2017,Orthey2019,Pires2021,Laneve2021PhysRevResearch}\nand been observed experimentally in a 20-step optical network~\\cite{Wang2018Optica}.\n\nThe main obstacle to utilizing a disordered DTQW for MEG is that many steps are required to reach asymptotic maximal entanglement~\\cite{Vieira2013PhysRevLett}, which is problematic for current experimental technologies.\nAlternative strategies in optimising\ncoin-operation sequence during the evolution have been proposed aiming at MEG for few steps. Universal and optimal coin sequences are proposed to generated highly entangled states for fewer than $10$ steps~\\cite{Gratsea_2020_Universal_optimal_coinSequences,Gratsea_2020_Generation_of_hybrid}. However, the universal sequence works for an odd number of steps and for the states with vanishing relative phase,\nwhereas the optimal sequence depends on the initial coin states.\nParrondo sequences have been proposed to generate maximal entanglement at steps $T=3$ and $T=5$, \nwith the expectation that many steps are needed to for MEG~\\cite{govind2020generating}. \nRecently, MEG has been realized with a position-inhomogeneous quantum walk~\\cite{Zhang2022PRA}.\nIdeal MEG via a DTQW  should work for any step number \\emph{and} independent of initial conditions,\nbut previous experiments only achieve either one or the other, not both;\nwe achieve both simultaneously here\nfor all steps beyond the second\nby solving an optimisation problem.\nExperimentally, we demonstrate the DTQW with requisite coin sequences up to 10 steps with linear optics\nand observe significant enhancement of entanglement generation compared to other entanglement-generation schemes.\n\\begin{figure*}[ht!\n \t\\includegraphics[width=\\linewidth]{Fig1_V5} \n \t\\caption{(a) The geometric representation of the optimal coin sequence $\\mathrm{C}_T$ that can generate maximal entanglement irrelevant of initial state $\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}$. (b) Results of maximal $\\mathcal F_{\\mathrm{C}_T}$ with coin set $\\{\\hat{C}(\\gamma_1), \\hat{C}(\\gamma_2)\\}$ at $T=5$, $T=10$ and $T=20$. (c) The maximal $\\mathcal F_{\\mathrm{C}_T}$ in QW with five different coin sets from $T\\in[20]$.\n \t(d) 50 optimized~$\\bm b$ at $T=20$ with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ where 0 represents $\\hat{H}$ and 1 represents $\\hat{\\mathds1}\\}$.} \n \t\\label{Fig:conceptandoptimazation}\n \\end{figure*}\n \nThe 1D DTQW coin-walker Hilbert space is $\\mathscr{H}\n    =\\mathscr{H}_\\text{c}\\otimes\\mathscr{H}_\\text{w}$\nwith\n\\begin{equation}\n\\label{eq:Hcw}\n    \\mathscr{H}_\\text{c}=\\text{span}\\{\\ket0_\\text{c},\\ket1_\\text{c}\\},\\,\n    \\mathscr{H}_\\text{w}=\\text{span}\\{\\ket{x};\nx \\in \\mathbb{Z}\\}\n\\end{equation}\nThe walker is initially localized position,\n$\\ket0_\\text{w}$,\nwith arbitrary initial coin \n$\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}=\\cos(\\nicefrac\\theta2)\\ket0_\\text{c}+\\text{e}^{\\text{i}\\phi}\\sin(\\nicefrac\\theta2)\\ket1_\\text{c}$,\nwhere $2\\theta,\\phi \\in [0,2\\pi]$. At step~$t$, coin operator $\\hat{C}_t$ is applied.\nThen the walker moves left or right entangled with the coin\nrepresented by the conditional shift operator\n\\begin{equation}\n\\label{eq:shiftoperator}\n\\hat{S}=\\ket0_\\text{c}\\bra{0}\\otimes\\sum_{x}\\ket{x+1}_\\text{w}\\bra{x}+\\ket1_\\text{c}\\bra{1}\\otimes\\sum_x\\ket{x-1}_\\text{w}\\bra{x},\n\\end{equation}\nwhich is independent of~$t$.\n\nFor $t\\in[N]:=\\{1,\\ldots,N\\}$,\nthe evolution is\n\\begin{equation}\n\\label{eq:evolution}\n\\ket{\\theta,\\phi}_{\\text{f}}\n=\\prod_{t\\in[T]}\\hat{U}_t\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}\\otimes\\ket0_\\text{w},\\,\n\\hat{U}_t=\\hat{S}(\\hat{C}_t\\otimes\\hat{\\mathds1}_\\text{w}),\n\\end{equation}\nfor $\\hat{\\mathds1}_\\text{w}=\\sum_{x}\\ket{x}\\bra{x}$\nthe identity operator on $\\mathscr{H}_\\text{w}$,\nand~f short for `final'.\nThe sequence\n$\\bm{C}_T:=\\left(\\hat{C}_t\\right)_{t\\in[T]}$\ndescribes coin operations applied to the walker.\nAs varying coin parameters~$\\xi$ and~$\\zeta$ is equivalent to varying~$\\phi$ of initial state\n$\\ket{\\theta,\\phi}_\\text{in}$~\\cite{Bach2004}, \nwe replace\n$\\hat{C}(\\gamma)\\gets\\hat{C}(0,\\gamma,0)$.\nFurthermore, we restrict construction of~$\\bm{C}_T$\nby allowing only two coin operations:\n$\\gamma_{0,1}$\nlabelled by one bit with values~0 and~1.\nThe sequence~$\\bm{C}_T$ is then expressed as a bit string $\\bm b\\in\\{0,1\\}^T$ of length~$T$ with~$0$ labelling~$C(\\gamma_0)$ and~$1$ labelling~$C(\\gamma_1)$.\n\nAchieving walker-coin MEG at step~$T$ regardless of $\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}$ corresponds to designing $\\bm{C}_T$ that maps any $\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}\\otimes\\ket0_\\text{w}$ to maximally entangled walker-coin state $\\ket{\\theta,\\phi}_{\\text{f}}$.\nEntanglement of $\\ket{\\theta,\\phi}_{\\text{f}}$ \nis quantified by entropy\n\\begin{equation}\n\\mathcal S_{\\text{E}}(\\ket{\\theta,\\phi}_\\text{f})\n=-{\\rm tr}\\left(\\rho_\\text{c}^{\\text{f}} \\log_2 \\rho_\\text{c}^{\\text{f}}\\right)\n=-\\sum_{\\varepsilon\\in\\pm}\n\\lambda_\n\\varepsilon\\log_2\\lambda_\\varepsilon,\n\\label{Eq:vonneumannentropy}\n\\end{equation} \nof the reduced coin state~\\cite{Abal2006PhysRevA,Bennett1996PhysRevA}\n$\\rho_\\text{c}^{\\text{f}}:={\\rm tr}_\\text{w}(\\ket{\\theta,\\phi}_\\text{f}\\bra{\\theta,\\phi})$ and $\\lambda_{\\pm}$ are the eigenvalues of $\\rho_\\text{c}^{\\text{f}}$.\nNote that $0\\leq \\mathcal S_{\\text{E}}\\leq 1$, and $\\mathcal S_{\\text{E}}\\equiv0$ for separable states and~1 for maximally entangled states.\n\nThus, MEG evolution~(\\ref{eq:evolution}) yields maximally entangled $\\ket{\\theta,\\phi}_{\\text{f}}$,\nwhich is equivalent to  $\\mathcal E_{\\bm{C}_T}(\\rho_\\text{c}^{\\text{in}}=\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}\\bra{\\theta,\\phi})=\\nicefrac{\\mathds1}{2}$ in~$\\mathscr{H}_\\text{c}$, where $\\mathcal E_{\\bm{C}_T}$ is a completely-positive linear map determined by $\\bm{C}_T$.\nA geometric illustration of~$\\mathcal E_{\\bm{C}_T}$ for MEG\nis in~\\cref{Fig:conceptandoptimazation}(a), which is the depolarizing channel $\\mathcal E^{\\text{DP}}(\\rho_\\text{c}^{\\text{in}})=(1-\\eta)\\rho_\\text{c}^{\\text{in}}+\\eta\\nicefrac{\\mathds1}{2}$ with $\\eta=1$~\\cite{Nielsen2011Quantum}.\nProcess fidelity\n$\\mathcal F_{\\bm{C}_T}\n:={\\rm tr}\\left(\\sqrt{\\chi_{\\bm{C}_T}}\\chi_{\\text{DP}}\\sqrt{\\chi_{\\bm{C}_T}}\\right)^2$,\nis our figure of merit to design $\\bm{C}_T$,\nwhere $\\chi_{\\bm{C}_T}$ is the Pauli-matrix representation of quantum channel $\\mathcal E_{\\bm{C}_T}$.\nNote that $\\mathcal F_{\\bm{C}_T}=1$ indicates MEG at step~$T$ regardless of $\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}$, and we refer to the corresponding coin sequence $\\bm{C}_T$ as the \\emph{optimal} coin sequence.\n\n\\begin{figure*}[htb\n \t\\includegraphics[width=\\linewidth]{Fig2_V4} \n \t\\caption{(a) Detailed sketch of the setup to realize the 10-step DTQW. (b) Coin operations realized in experiment. A half-wave plate~(HWP) set at 22.5$^\\circ$ corresponds to operation $\\hat{H}$ and a quarter-wave plate~(QWP) set at 45$^\\circ$ corresponds to operation $\\hat{F}$. No waveplate needs to be arranged if the operation is $\\hat{\\mathds1}$. (c) Symbols used in (a) and (b): periodically poled potassium titanyl phosphate (PPKTP), polarization beam splitter (PBS), half-wave plate (HWP), quarter-wave plate (QWP), beam displacer (BD), single photon detector (SPD).} \n \t\\label{Fig:Setup}\n \\end{figure*} \n\n\nThree parameters determine~$\\bm{C}_T$: $\\gamma_{0,1}$ and corresponding~$\\bm b$.\nWe address designing optimal $\\bm{C}_T$ as an optimisation problem.\nFor fixed~$T$, optimizing~$\\bm b$\nand~$\\gamma_{0,1}$\nis achieved by minimising the cost function $1-\\mathcal F_{\\bm{C}_T}$,\nin our case using an annealing algorithm.\nResults of optimisation of $\\gamma_{0,1}$\nat $T\\in\\{5,10,20\\}$\nare shown in~\\cref{Fig:conceptandoptimazation}(b) ($\\bm b$ is not shown here).\nEvidently,\nthe minimal $1-\\mathcal F_{\\bm{C}_T}$\nis obtained for two coin sets:\n$\\{\\hat{C}(0), \\hat{C}(\\nicefrac{\\pi}4)\\}$ and $\\{\\hat{C}(\\nicefrac{\\pi}2), \\hat{C}(\\nicefrac{\\pi}4)\\}$. Note that $\\hat{C}(0)=\\hat{\\sigma}_z$, $\\hat{C}(\\nicefrac{\\pi}4)=\\hat{H}$ and $\\hat{C}(\\nicefrac{\\pi}2)=\\hat{\\sigma}_x$.\nThe evolution unitary operator with coin operator $\\hat{\\sigma}_z$, i.e.,  $\\hat{U}=\\hat{S}\\hat{\\sigma}_z$,\nmakes the components~$\\ket0_\\text{c}$ and~$\\ket1_\\text{c}$ propagate in the opposite direction without interference, which has the similar effect of $\\hat{U}=\\hat{S}\\hat{\\mathds1}$ with identity operator $\\hat{\\mathds1}$ yielding a phase $\\text{e}^{\\text{i}\\pi}$ on component~$\\ket1_\\text{c}$.\n\nWe conjecture that the coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ with optimized~$\\bm b$ leads to MGE as well.\nAlong this spirit, we fix $T$ and coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$\nand optimize~$\\bm b$ with cost function $1-\\mathcal F_{\\bm{C}_T}$. The results of $\\mathcal F_{\\bm{C}_T}$ with~$T$ up to~20 are shown with blue dots in~\\cref{Fig:conceptandoptimazation}(c), in which we observe that $\\mathcal F_{\\bm{C}_T}=1$ since step $T=3$.\nTo give a comparison, we also show the maximum $\\mathcal F_{\\bm{C}_T}$ over $2^T$ possible~$\\bm b$ with coin sets\n$\\{\\hat{H}, \\hat{\\sigma}_z\\}$, $\\{\\hat{H}, \\hat{\\sigma}_x\\}$, $\\{\\hat{H}, \\hat{F}\\}$ and $\\{\\hat{H}\\}$,\nwhere $\\hat{F}=[1,\\text{i};\\text{i}, 1]\/\\sqrt{2}$. The results with coin set $\\{\\hat{H}, \\hat{\\sigma}_z\\}$ is as same as that with $\\{\\hat{H}, \\hat{\\mathds1}\\}$. For the coin set $\\{\\hat{H}, \\hat{\\sigma}_x\\}$, $\\mathcal F_{\\bm{C}_T}=1$ is achieved at step $T=5$ and $T\\geq 7$.\nAsymptomatic behavior is observed with coin set $\\{\\hat{H}, \\hat{F}\\}$ and oscillating behavior is observed in Hadamard walk. \n\n\n\nWe note that optimal~$\\bm b$ at step~$T$ is not unique.\nFor instance, we obtain 1104 optimal~$\\bm b$ at $T=20$,\nand we list 50 among them in~\\cref{Fig:conceptandoptimazation}(d).\nThere are no obvious features of regularities and generalities of these optimal~$\\bm b$.\nAn optimal~$\\bm b$ containing~0 ($\\hat{H}$) as less as possible is preferred in experiment, so we choose the optimal~$\\bm b$ containing one~0 or two~0s in our realization ( The explicit form of $\\bm b$ and its corresponding proof are in~\\cref{App:optimalsequence} ).\n\\begin{figure*}[ht!bp\n \\includegraphics[width=\\linewidth]{Fig3_V3}\n\\caption{%\n(a) Experimental results of $\\mathcal F_{\\bm{C}_T}$\n with coin sets $\\{\\hat{H}, \\hat{\\mathds1}\\}$, $\\{\\hat{H},\\hat{F}\\}$ and $\\{\\hat{H}\\}$ at $T=2$ to $T=10$.\n(b) Average entanglement $\\langle\\mathcal{S}_\\text{E}\\rangle$\nover 296 initial coin states with the reconstructed\n$\\chi^\\text{exp}_{\\bm{C}_T}$.\n(c) Results of Geometric representation of reconstructed $\\chi^{\\text{exp}}_{\\bm{C}_T}$ with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ at $T=4,6,8$ and $10$.\n(d) Geometric representation of reconstructed $\\chi^{\\text{exp}}_{\\bm{C}_T}$\nwith coin set\n$\\{\\hat{H},\\hat{F}\\}$\nat $T=4,6,8$ and $10$.}\n\\label{Fig:Geometricplot}\n\\end{figure*}\n\nThe experimental setup to implement 1D DTQW is shown in~\\cref{Fig:Setup}(a). The coin state is encoded in the photon's polarization degree of freedom by $\\ket{H(V)}=\\ket{0(1)}$, where $\\ket{H(V)}$ denotes the horizontal (vertical) polarization.\nThe position state is encoded in the photon's spatial degree of freedom, i.e., the longitudinal spatial modes.\nTwo photons in state $\\ket{H}\\ket{V}$ with central wavelength at 810~nm are generated from a periodically poled potassium titanyl phosphate (PPKTP) crystal pumped by an ultraviolet CW laser diode with the central wavelength at 405~nm. The two photons are then separated by a polarizing beam splitter (PBS), which transmits the horizontal polarization and reflects vertical polarization.\nThe reflected photon is detected by single-photon detector (SPD) to serve as a trigger. The transmitted photon is sent into the photonic network consisting of waveplates and birefringent calcite beam displacers (BDs), in which the longitudinal spatial mode of injected photon is denoted as the start position of the walker $\\ket0_\\text{w}$. The coin operations $\\hat{C}_t$ are realized by waveplates which rotates the polarization of photon, and the BD transmits the vertical polarization while deviating from the horizontal polarization so that BD acts as shift operation $\\hat{S}$. \n\n\n\nTo reconstruct the process matrix $\\chi^{\\text{exp}}_{\\bm{C}_T}$,\nwe prepare four states as initial coin states $\\ket{\\theta,\\phi}_\\text{c}^{\\text{in}}$, i.e., $\\ket{H}$, $\\ket{V}$, $\\ket{+}$ and $\\ket{L}$. For each step~$t$, we set the the optimal coin sequence $\\bm{C}_T$ accordingly, and reconstruct $\\rho_\\text{c}^{\\text{f}}$ using quantum state tomographic technology~\\cite{Nielsen2011Quantum}. The experimental results of $\\mathcal F_{\\bm{C}_T}$ with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ are shown with blue circles in~\\cref{Fig:Geometricplot}(a).\nWe observe that the average $\\mathcal F_{\\bm{C}_T}$ from $T=3$ to $T=10$ is $0.9954\\pm0.0008$, which is much better than the results with coin set $\\{\\hat{H}, \\hat{F}\\}$ as shown with red squares. \nFor the Hadamard QW, $\\mathcal F_{\\bm{C}_T}<0.8$\nand oscillates as~$T$ increases (shown with yellow triangles). We calculate the average entanglement $\\langle\\mathcal{S}_\\text{E}\\rangle$ over 296 initial coin states with the reconstructed $\\chi^\\text{exp}_{\\bm{C}_T}$, and the results are shown in~\\cref{Fig:Geometricplot}(b). The error bar indicates initial-state-independence, and we observe a stronger initial-state-independence with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ than the other two. This is also reflected by the geometric interpretations of $\\mathcal F_{\\bm{C}_T}$ as shown in~\\cref{Fig:Geometricplot}(c) (coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$) and~\\cref{Fig:Geometricplot}(d) (coin set $\\{\\hat{H}, \\hat{F}\\}$) at $T=4, 6, 8$ and $10$, respectively. It is obviously that the results with $\\{\\hat{H}, \\hat{\\mathds1}\\}$ is much more dense than that with $\\{\\hat{H}, \\hat{F}\\}$, which indicates the entanglement generation with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ has stronger independence of initial coin states than that with the other two. More details are shown in~\\cref{App:processodd}. \n\nBesides $\\mathcal F_{\\bm{C}_T}$, we investigate spreading over the walker Hilbert space $\\mathscr{H}_\\text{w}$. \nSpread of the probability distribution $\\mathcal P(x, T)$ can be characterized by the normalized Shannon entropy \n\\begin{equation}\n\\mathcal S_\\text{S}(T)=\\frac{-\\sum_x \\mathcal P(x, T) \\ln\\mathcal P(x, T)}{\\ln(T+1)}, \n\\end{equation}  \nwith $\\frac1{\\ln(T+1)}$ the normalisation parameter.\nThe walker is able to occupy $T+1$ positions after~$t$ steps so that the maximal value of $-\\sum_x \\mathcal P(x, T) \\ln\\mathcal P(x, T)$ is $\\ln (T+1)$, which corresponds to the uniform distribution over $T+1$ positions~\\cite{2020_martin-vazquez_Phys.Rev.A_Optimizing}.\nLarger $\\mathcal S_\\text{S}(T)$ implies $\\mathcal P(x, T)$ is more uniform. For a $T$-step QW associated with corresponding optimal $\\bm{C}_T$, we measure the probability distribution $\\mathcal P(x,T)$ at step~$T$, according to which we calculate the normalized Shannon's entropy $\\mathcal S_\\text{S}(T)$. The results of $\\mathcal P(x,T)$ with initial coin state $\\ket{H}$ and $\\ket{L}$ are shown in~\\cref{Fig:spread}(a,c), and the corresponding $\\mathcal S_\\text{S}(T)$ are shown in~\\cref{Fig:spread}(b,d) respectively. Compared with the other two cases, the uniformity of QW with coin set $\\{\\hat{H}, \\hat{\\mathds1}\\}$ shows enhancement at $T=3$ and $T=7$.  We also analyze the trend of probability distributions with different coin sets. More experimental results and analysis can be found in~\\cref{App:transport}. \n\n\\begin{figure*}[htbp\n\\includegraphics[width=\\linewidth]{Fig4_V5} \n\\caption{%\nMeasured $\\mathcal P(x,T)$ with initial coin state (a) $\\ket{H}$ and (c) $\\ket{L}$, according to which the calculated $\\mathcal S_\\text{S}(T)$ are shown in (b) and (d), respectively. } \n\\label{Fig:spread}\n\\end{figure*}\n\nIn conclusion, we design coin sequences that can rigorously generate maximal entanglement between coin and her walker in 1D DTQW with three key features, that are: to be available at any $T\\geq3$, to be independent of initial coin state and to be the simplest for experimental implementation. A comparison of our coin sequence~$\\bm{C}_T$ with the other coin sequences is shown in~\\cref{App:comparison}, and MEG with our coin sequence significantly outperforms all other proposed coin sequences in three features mentioned above.\n\nExperimentally, we realize a 10-step 1D DTQW with an optical network, and observe the MEG with with our proposed coin sequence as well as other coin sequences. The results show a significant enhancement of MEG with our coin sequences, which benefits the intermediate quantum information processing that requires maximal qubit-qudit entanglement. \nMoreover, for few steps, the spread of probability distributions with our coin sequence is more uniform, which is favorable and useful in various quantum algorithms and quantum simulation of biological processes~\\cite{2003_kendon_Phys.Rev.A_Decoherence,2007_maloyer_NewJ.Phys._Decoherence,2020_martin-vazquez_Phys.Rev.A_Optimizing}. Our protocol can also be generalized to $p$-diluted disorder QW~\\cite{Geraldi2019PhysRevLett,Geraldi2021PhysRevResearch}, in which transport behavior can be engineered by controlling probability of coin operations.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the advent of Internet of Things (IoT), multi-sensor systems have become increasingly ubiquitous. They are used to make control decisions in systems ranging from smart homes to vehicular networks. Sensors in these systems measure time varying quantities and communicate with a central monitoring station. It is crucial that the monitoring station receives timely updates from the sensors to make critical decisions. \n\nIn this work, we consider \\textit{n} sensors that measure real time updates and update a monitoring station through a shared unreliable communication channel. We divide time into slots and model the channels as ON-OFF channels that have independent realizations across sensors and time slots. At most one sensor is allowed to be scheduled in each time slot under the constraints of the scheduling problem. The goal is to design a scheduling policy that optimizes a freshness objective and hence reduces the staleness in information.\n\n\\textit{Age of Information} (AoI) has emerged as a popular metric to measure freshness of information at a destination and is defined as the time elapsed since the last update from a source. It grows linearly with time and resets to zero when an update is received from a source. AoI minimization has been studied extensively for scheduling applications in \\cite{7492912}-\\cite{8514816}. \nIn a multi-sensor setting with a shared channel, the AoI of a sensor increases due to two reasons. The first is when an attempted update by a sensor fails due to poor channel conditions and the second is when the shared channel is used by some other sensor. The original AoI metric does not distinguish between these two events. It is evident that the first event is fundamental and is caused by the physics of the problem and the second event is a resource allocation issue and therefore, controllable by the algorithm designer. Given this, we propose a variant of the AoI which distinguishes between these two events by imposing a penalty by increasing the ``age\" of the information of a sensor at the monitoring station only in the latter case. We refer to this metric as the \\textit{Channel-Aware Age of Information} (CA-AoI) and by definition, the CA-AoI of a sensor is a measure of the number of missed opportunities to send an update since its recent most successful update.\n\n The goal of our work is to design scheduling policies to minimize the time-average of the weighted sum of CA-AoIs of the sensors. Another important issue with AoI is that it grows to infinity as the channel conditions of a sensor become poor irrespective of the scheduling policy employed. The following example highlights the difference in schedules designed to minimize the time-average of the sum of the AoIs vs. the CA-AoIs. Consider a two sensor system with equal weights such that the first sensor has near reliable channel conditions and the second sensor has an extremely poor channel connection. On average, the AoI of the second sensor will be higher than the AoI of the first sensor irrespective of the fraction of times it is scheduled for transmission. As a result, a scheduling algorithm designed to optimize for AoI will attempt to schedule the second sensor at a higher rate in spite of the system conditions limiting its communication capacity. Similarly, the throughput achieved in a collection of sensors that aim at capturing the same physical process albeit with some diversity in measurements, will be low due to the disproportionate amount of resources allocated to the sensors with poor channel conditions. On the other hand, CA-AoI shrinks to zero as the channel conditions deteriorate and the resource allocation will thus be fairer across the sensors. However, using a scheduling policy that blindly maximizes the throughput will be unable to meet the timeliness requirements of the system. As we discuss ahead in Section \\ref{sec:discuss}, a scheduling algorithm designed to optimize for CA-AoI will be better suited to balance throughput and freshness compared to AoI. \n\n\n\nOptimizing this objective has several challenging aspects. For most sources the channel state information (CSI) is unknown. Hence the scheduler needs to guess when the channel state is ON so that the source can push an update. On the other hand, for some sources CSI is known before every time-slot. But even with CSI, designing a scheduling policy is highly nontrivial because the scheduler is allowed to schedule only a single source in every time slot. Computing the optimal policy using dynamic programming is difficult for such problems as the state-space grows exponentially with the number of sources. Hence low complexity policies are required that can make this constrained scheduling problem computationally efficient. \n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[scale = 0.40]{Figures\/PS.pdf}\n    \\caption{A multi-source system consisting of $n$ sensors communicating with a central monitoring station via unreliable communication channels. $p_{i}$ denotes the channel ON probabilities. }\n    \\label{fig:PS}\n\\end{figure}\n\n\\subsection{Our Contributions}\n \nWe consider three cases, namely, systems without CSI, systems with CSI, and systems with partial CSI. \n \\begin{itemize}\n     \\item[--] We propose a new AoI-like freshness metric called Channel Aware-Age of Information (CA-AoI). Unlike vanilla AoI, CA-AoI is not agnostic of channel reliability. \n     \\item[--] We derive universal lower bounds on the average weighted sum of CA-AoI for different CSI models.  \n      \\item[--] We view the scheduling problem through the lens of a restless multi-armed bandit framework (RMAB) and prove that all three problem instances are indexable. We derive structural properties for the optimal policy. Therefore we calculate closed form expressions for the Whittle index and propose low complexity scheduling algorithms. \n     \\item[--] We propose a suite of stationary randomized scheduling algorithms to optimize the objective for all three problem instances. We find closed form expressions for the cost under these policies and thus optimal parameters for the algorithms. \n     \\item[--] We compare the performance of the Whittle index and randomized policies against the greedy policy via simulations. We show the efficacy of our scheduling algorithms and conclude the superiority of the Whittle index policy in diverse settings.\n     \\item[--] We finally demonstrate the advantage of using the sum of CA-AoIs over the sum of AoIs as the freshness metric in handling the diversity-throughput trade off. \n \\end{itemize}\n\n\\subsection{Related Work}\n\nAs mentioned above, there exists a rich body of work that uses AoI minimization in scheduling applications \\cite{7492912}-\\cite{8514816}. Several existing works model the scheduling problem as restless bandits and use Whittle index policies to solve them. The work in \\cite{9027444} derives the Whittle index for AoI minimization in a system having iid\/Markovian channels and multiple information settings (without CSI\/ with CSI\/ delayed CSI).  In \\cite{article}, the authors use Whittle's methodology for the setting where realization of the stochastic arrival process for a given time-slot is known apriori. In \\cite{8486307}, the focus is on designing Whittle index policies for AoI minimization when the system has throughput constraints. In \\cite{8919842}, the authors consider Whittle index policies to minimize non-decreasing functions of AoI and prove optimality of the index policy for a system with 2 sources and reliable channels. Moreover in \\cite{8514816}, a lower bound for the sum of AoIs metric is derived and various low complexity algorithms with theoretical guarantees are designed. Very recently in \\cite{NIPS2019_8348}, the authors focus on scheduling a web engine under bandwidth constraints to track content changes in sources with different observabilities  and propose randomized algorithms to optimize a harmonic freshness objective. \n\n\\section{Setting}\n We consider a system of \\textit{n} sensors, each communicating with a central monitoring station via a shared unreliable communication channel. Each sensor measures a time varying quantity and can push real time updates to the central monitoring station. \n \n \\subsection{Communication Model}\n Time is divided into slots. In a time slot, a maximum of a single sensor can push updates to the monitoring station. Here the sensors are assumed to be active sources and can thus generate updates in every time slot. \n\\subsection{Channel Model}\nWe model the channels as ON-OFF channels with channel ON probability for channel $i$ denoted as $p_{i}$. Here we assume that channel realizations are independent across sensors, and independent and\nidentically distributed (i.i.d.) over time-slots. \n\n\\subsection{Channel State Information (CSI) Models}\nWe consider the following three cases:\n\\begin{enumerate}\n    \\item[--] \\emph{Systems without CSI}: In this case, the state of the channels for the various sensors (ON\/OFF) in a time-slot is not known to the scheduler before making the scheduling decision for that time-slot.  \n    \\item[--] \\emph{Systems with CSI}: In this case, the state of the channels for the various sensors (ON\/OFF) in a time-slot is known to the scheduler before making the scheduling decision for that time-slot.\n    \\item[--] \\emph{Systems with partial CSI}: In this case, the state of the channels for some (a fixed subset) of the sensors is known to the scheduler before making the scheduling decision for that time-slot.\n\\end{enumerate}\nIn all three cases, channel statistics are known to the scheduler.\n\\subsection{Our Metric: Channel-Aware Age of Information (CA-AoI)}\n\nLet $X_{i}(t)$ denote the CA-AoI of sensor $i$ at time $t$. We denote the channel state at time $t$ with respect to sensor $i$ as $\\Lambda_{i}(t)$, where $\\Lambda_{i}(t)$ = 1 means that the channel is ON and $\\Lambda_{i}(t)$ = 0 implies that the channel is OFF. We indicate the action taken for sensor $i$ with $a_{i}(t)$, where $a_{i}(t)$ = 1 means that sensor $i$ is scheduled as time $t$ and $a_{i}(t)$ = 0 otherwise. Given the above definitions, CA-AoI for a sensor evolves as follows:\n\\begin{align*}\nX_{i}(t+1) = \\begin{cases}\n0, & \\text{if } a_{i}(t) = 1 \\text{ and } \\Lambda_{i}(t) = 1,\\\\\nX_{i}(t)+1, & \\text{if } a_{i}(t) = 0 \\text{ and } \\Lambda_{i}(t) = 1, \\\\\nX_{i}(t), & \\text{otherwise}.\n\\end{cases}\n\\end{align*}\nNote that as opposed to the vanilla AoI metric, CA-AoI doesn't penalize the age of a sensor if the channel is OFF at time $t$. In other words the age of a sensor is incremented only when the channel is in the ON-state and the sensor isn't scheduled.\n\n\\subsection{Objective}\nThe objective of the scheduling policy $\\pi$ is to minimize the time average of the weighted sum of CA-AoI:\n\\begin{gather}\n    \\text{minimize }J^{\\pi} = \\limsup\\limits_{T\\rightarrow \\infty} \\frac{1}{T} \\mathbb{E}_{\\pi} \\left[\\sum_{t=1}^{T}\\sum_{i=1}^{n} w_{i}X_{i}(t) \\right], \\label{eq:1}\n    \\\\\n    \\text{subject to } a_{i}(t) \\in \\{0,1\\} \\ \\forall i, \\sum_{i=1}^{n} a_{i}(t) \\leq 1. \\nonumber\n\\end{gather}\nIn \\eqref{eq:1}, $w_{i}$ denotes the weight assigned to sensor $i$ and $\\mathbb{E}_{\\pi}$ denotes the expected weighted sum of CA-AoIs when a policy $\\pi$ is used.\n\n\n\n\\section{Main results and discussion}\n\nWe discuss our key results here. All proofs are provided in the Appendix. We first focus on deriving universal lower bounds for the freshness objective $J^{\\pi}$ . \n\n\\subsection{Universal Lower Bounds}\n\n\\begin{definition}(Admissible Policies $\\Pi$)\nIn an infinite horizon setting $T\\rightarrow \\infty$, the class of admissible policies $\\Pi$ omits all those policies that stop scheduling a particular sensor after a time slot $T' < \\infty$ with a positive probability.\n\\end{definition}\nOur first result characterizes a lower bound on the solution of $J^{\\pi}$ for a system without CSI among the class of admissible policies $\\Pi$. We denote the set of sensors in this system by $n^{-}$.\n\\begin{theorem}\n\\label{theorem:lower_bound_ucsi}\nFor a system without CSI,  $J^{\\pi^{-}} \\geq \\mathcal{L}^{-}\\quad \\forall \\pi^{-} \\in \\Pi$  where\n\n\\begin{equation}\n    \\mathcal{L}^{-} = \\frac{(\\Sigma_{i \\in n^{-}} \\:\\sqrt{w_{i}p_{i}})^{2} - \\Sigma_{i \\in n^{-}} w_{i}p_{i} }{2}  \\nonumber\n\\end{equation}\n\\end{theorem}\n\nThe proof of Theorem \\ref{theorem:lower_bound_ucsi} can be found in Appendix \\ref{appe:lower_bound_ucsi}. The proof follows a sample path argument and uses Fatou's lemma and is derived from the technique in \\cite{8514816} used to find a lower bound for the AoI objective. \n\nIn the next result we derive the lower bound on  $J^{\\pi}$ for a system with CSI having a set of sensors $n^{+}$, among the class of admissible policies $\\Pi$. \n\n\\begin{theorem}\n\\label{theorem:lower_bound_csi}\nFor a system with CSI, $J^{\\pi^{+}} \\geq \\max\\left(0,\\mathcal{L}^{+}\\right) \\quad \\forall \\pi^{+} \\in \\Pi$  where\n\n\\begin{equation}\n    \\mathcal{L}^{+} = \\frac{(\\Sigma_{i \\in n^{+}} \\:\\sqrt{w_{i}}p_{i})^{2} - \\Sigma_{i \\in n^{+}} w_{i}p_{i} }{2}  \\nonumber\n\\end{equation}\n\\end{theorem}\nThe proof of Theorem \\ref{theorem:lower_bound_csi} can be found in Appendix \\ref{appe:lower_bound_csi}. \\footnote{For a system with partial CSI  the lower bound can be found by summing the lower bounds derived on sensors with and without CSI.\nNote that it may be possible to further tighten these lower bounds.}\n\n\\subsection{Whittle Index Scheduling Policies}\nThe scheduling problem can be viewed as a Restless Multi Armed Bandit (RMAB) with the sensors as arms whose age evolve even when they are not scheduled for transmission. Solving the RMAB problem is P-SPACE hard. However relaxing the constraints on the number of arms that can be played in every time slot and instead placing a constraint on its expectation, makes this problem tractable. The relaxed problem helps decouple the problem into $n-$subproblems, one for each sensor and the solution of this problem is called the Whittle index policy. Hence each subproblem consists of a single sensor, a single channel and a playing charge c (Lagrange multiplier) for the arm. Let $c(s(t),a(t))$ denote the cost incurred by action $a(t)$ in state $s(t)$.   A policy $\\pi = \\{a(1),a(2) \\cdots\\}$ decides whether to idle the arm or play the arm at each time step. A policy is cost-optimal if it minimizes the average cost, defined as follows:\n\\[\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\sum_{t=1}^{T}\\mathbb{E}_{\\pi}\\left[c(s(t),a(t))\\right|s(0)=0].\\]\n\\begin{definition} (Indexability)\nGiven a playing charge c, let S(c) be the set of states for which the optimal policy is to idle. Then the subproblem is called \\textit{indexable} if S(c) monotonically increases from the empty state to the entire state space as c increase from $-\\infty \\text{ to } \\infty$. \n\\end{definition}\n\\begin{definition}(Whittle index)\nThe Whittle index of a state is the playing charge that makes it equally lucrative to play or idle the arm.\n\\end{definition}\nWe now focus on three cases, namely, systems without CSI, systems with CSI, and systems with partial CSI, separately.\n\\subsubsection{System without CSI} Here the state of the channel is not known a priori and can be determined only after scheduling an information packet via the channel. We first formulate the decoupled subproblem. We drop the sensor index $i$ for further analysis as each sub-problem deals with a single sensor.\\\\\n\\textit{Decoupled subproblem}:\\\\\n\\textit{States}: The CA-AoI ($X(t)$) of the source defines its state $s(t)$\\\\\n\\textit{Actions}: The actions $\\in\\{0,1\\}$ are denoted by $a(t)$, where $a(t)=1$ indicates that sensor is scheduled to update the central station and $a(t)=0$ signifies that the sensor is kept idle.\\\\\n\\textit{Transition probabilities}:\nThe transition probability to state $s'$ after taking action a from state $s=x$ is:\n\\begin{align*}\n&\\mathbb{P}(s' = x | a=1,s=x) = 1-p\\\\\n&\\mathbb{P}(s' = 0 | a=1,s=x) = p\\\\\n&\\mathbb{P}(s' = x | a=0,s=x) = 1-p\\\\\n&\\mathbb{P}(s' = x+1 | a=0,s=x) = p\n\\end{align*}\n\\textit{Cost}: Let $\\Lambda(t) \\in \\{0,1\\}$ denote the channel state. Then the cost incurred by choosing action a at state $s = x$ is:\n\\[c(s,a) = w[x(1-a\\Lambda)+\\Lambda(1-a)]+ca.\\]\nThe above cost function can be simply deduced as the sum of the post-action age and the charge incurred when playing an arm.\n\nThe Markov Decision Process (MDP) for this subproblem has been shown in Fig. \\ref{fig:mdp} (a). Our next result shows that systems without CSI are indexable and characterizes the Whittle index in closed form.\n\n\\begin{theorem}\n\\label{theorem:UCSI_whittle}\n The scheduling problem for a system without CSI is indexable and the Whittle index of a sensor in state x is given by:\n \\[\\mathcal{W}(x,p) = \\frac{w(x+1)(x+2)}{2(2-p)}\\]\n\\end{theorem}\n\nWe use this result to design a Whittle index based scheduling policy which is to schedule the sensor with the highest Whittle Index in each time-slot. See Appendix \\ref{appe:Th1} for the proof.\\footnote{Note that it is not possible to know the channel states of the unscheduled sensors, under the system without CSI. To update the CA-AoI of an unscheduled sensor $i$ we set it's channel state $\\Lambda$ to be the outcome of a Bernoulli random variable with mean $p_i$.}\n\n\n\\subsubsection{System with CSI}\nIn this case the scheduler knows the states of the channel before scheduling. We begin by describing the subproblem for this case:\\\\\n\\textit{Decoupled subproblem}:\\\\\n\\textit{States}: The state $s(t)$ of the arm is defined by the tuple $(X(t),\\Lambda(t))$\\\\\n\\textit{Actions}: $a(t)=1$ indicates that sensor is scheduled to update the central station and $a(t)=0$ signifies that the sensor is kept idle.\\\\\n\\textit{Transition probabilities}: The transition probability to state $s'$ after taking action a from state $s=x$ is:\n\\begin{align*}\n&\\mathbb{P}(s' = (x,0) | a=0,s=(x,0)) = 1-p\\\\\n&\\mathbb{P}(s' = (x,1) | a=0,s=(x,0)) = p\\\\\n&\\mathbb{P}(s' = (x,0) | a=1,s=(x,0)) = 1-p\\\\\n&\\mathbb{P}(s' = (x,1) | a=1,s=(x,0)) = p\\\\\n&\\mathbb{P}(s' = (x+1,0) | a=0,s=(x,1)) = 1-p\\\\\n&\\mathbb{P}(s' = (x+1,1) | a=0,s=(x,1)) = p\\\\\n&\\mathbb{P}(s' = (0,0) | a=1,s=(x,1)) = 1-p\\\\\n&\\mathbb{P}(s' = (0,1) | a=1,s=(x,1)) = p.\n\\end{align*}\n\n\\begin{figure}[t]\n    \\centering\n    \\subfloat[System without CSI]{{\\includegraphics[width=0.5\\linewidth]{Figures\/UCSI.pdf} }}%\n    \\qquad\n    \\subfloat[System with CSI]{{\\includegraphics[width=.4\\linewidth]{Figures\/CSI.pdf} }}%\n    \\caption{MDP for decoupled subproblems}%\n    \\label{fig:mdp}\n\\end{figure}\n\\textit{Cost}:  The cost incurred by choosing action a at state $s = x$ is:\n\\[c(s,a) = w[x(1-a\\Lambda)+\\Lambda(1-a)]+ca.\\]\nThe MDP for this subproblem has been represented in Fig. \\ref{fig:mdp} (b). Our next result shows that systems with CSI are indexable and characterizes the Whittle index in closed form.\n\n\\begin{theorem}\n\\label{theorem:CSI_whittle}\n The scheduling problem for a system with CSI is indexable and the Whittle index of a sensor in state (x,$\\Lambda$) is given by:\n \\begin{align*}\n \\mathcal{W}(x,p) = \\begin{cases}\n \\dfrac{w(x+1)(x+2)}{2} & \\text{if } \\Lambda = 1,\\\\\n 0 & \\text{otherwise.}\n \\end{cases}\n \\end{align*}\n\\end{theorem}\nWe use this result to design a Whittle index based scheduling policy which is to schedule the sensor with the highest Whittle Index in each time-slot. Note that the Whittle index gives us an online algorithm because it is independent of the channel statistics.  See Appendix \\ref{appe:Th2} for the proof. \n\n\\subsubsection{System with partial CSI}\nWe now consider the case for which CSI is available only for a subset of the sensors. We refer to this setting as systems with partial CSI. As in the previous two cases, the scheduling problem reduces to $n$ subproblems and the previous results hold true for this case as well. Let $n^{-}$ and $n^{+}$ be the set of sensors without CSI and with CSI respectively.  \nOur next result shows that systems with partial CSI are indexable and characterizes the Whittle index in closed form. This result follows from Theorems \\ref{theorem:UCSI_whittle} and \\ref{theorem:CSI_whittle}.\n\\begin{theorem}\n\\label{theorem:mixed_whittle}\n The scheduling problem for a system having partial CSI is indexable and the Whittle index of a sensor in state x is given by:\n \\begin{align*}\n  \\mathcal{W}(x_{i},p_{i}) = \\begin{dcases}\n \\frac{w_{i}(x_{i}+1)(x_{i}+2)}{2(2-p_{i})} & \\text{if } i \\in n^{-},\\\\ \n \\frac{w_{i}(x_{i}+1)(x_{i}+2)}{2} & \\text{if } \\Lambda_{i} = 1, i\\in n^{+},\\\\\n 0 & \\text{otherwise.}\n \\end{dcases}\n \\end{align*}\n\\end{theorem}\n\n\n\n\n\\subsection{Randomized Scheduling Policies}\nWe refer to a policy as a randomized scheduling policy if it specifies a probability\ndistribution on the set of scheduling decisions at every time slot. To characterize the optimal\\footnote{Optimal among policies belonging to the given class of randomized scheduling policies.} randomized scheduling policy $\\pi^{*}$ for each of the three cases, we first find closed form expressions for the objective under a class of randomized policies and give algorithms to find policy parameters that optimize the objective.\n\n\\subsubsection{System without CSI}\n The randomized policy $\\pi^{-}$ in this case is to schedule source $i$ according to a Bernoulli process with parameter $\\Delta_{i}$. Let $n^{-}$ denote the set of all sensors without CSI and $\\vec{\\Delta}$ the vector of parameters $\\Delta_{i}$. Then the policy is:\n\\begin{gather*}\n    \\pi^{-} = \\{\\text{Sched}(i) \\sim \\text{Bernoulli}(\\Delta_{i}) \\quad \\forall  i \\in n^{-} | 1 \\geq \\Delta_{i} > 0 \\}.\n\\end{gather*}\n\nTo optimize the parameters $\\Vec{\\Delta}$ of the policy we relax the constraints on $\\pi^{-}$ from $\\sum_{i=1}^{n} a_{i}(t) = 1$ to $\\mathbb{E}\\left[\\sum_{i=1}^{n} a_{i}(t)\\right] = 1.$\nOur next result characterizes the performance of a randomized policy as a function of the parameters $\\vec{\\Delta}$.\n\\begin{theorem}\n\\label{theorem:UCSI_random}\nWithout loss of generality assume that $\\vec{p}>0$. Then under the policy $\\pi^{-}$, the freshness objective $J^{\\pi^{-}}$ as defined in Eq. \\ref{eq:1} is \n\\begin{equation}\n   J^{\\pi^{-}} = \\sum_{i \\in n^{-}}\\frac{w_{i}(1-\\Delta_{i})}{\\Delta_{i}}.\n\\end{equation}\n\\end{theorem}\n\nCheck Appendix \\ref{appe:Th4} for the proof. Next, we use this result to design an optimization problem for finding the optimal parameters $\\vec{\\Delta}$.\n\n\\textit{Problem 1: Finding optimal $\\pi^{*} \\in \\pi^{-}$} -- Finding the best policy reduces to the following convex optimization problem.\n\\begin{gather*}\n    \\text{minimize }J^{\\pi^{-}} = \\sum_{i\\in n^{-}}\\frac{ w_{i}(1-\\Delta_{i})}{\\Delta_{i}},\n    \\\\\n    \\text{subject to } \\sum_{i\\in n^{-}} \\Delta_{i} = 1 , 1\\geq \\Delta_{i}> 0 \\quad \\forall i \\in n^{-}. \\nonumber\n\\end{gather*}\n\nNote that the first constraint comes from the relaxation of the scheduling problem constraints. Using the Lagrange multiplier method, finding the optimal policy reduces to solving the following system of equations:\n\\begin{gather}\n    \\Delta_{i} = \\sqrt{\\frac{w_{i}}{\\lambda}} \\quad \\forall i \\in n^{-},\\label{eq_deltas}\\\\\n    \\sum_{i\\in n^{-}} \\Delta_{i} = 1. \\label{eq_lamda}\n\\end{gather}\n\n\nNote that an exact solution to $\\vec{\\Delta}$ can be obtained from the above system of equations. The optimization procedure is given in Algorithm \\ref{algo_1}.\n\n\n\\IncMargin{1em}\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\\Input{$p_{i}$ - Channel ON probability $\\forall i \\in n^{-}$ \\\\ $w_{i}$ - Weight of sensor $i$ $\\forall i \\in n^{-}$ }\n\\Output{$\\vec{\\Delta}^{*}$ }\n\\BlankLine\n$\\sqrt{\\lambda^{*}}$ $\\longleftarrow \\: \\sum_{i \\in n^{-}} \\sqrt{w_{i}}$ \\;\n \\For{i $\\in n^{-}$}{\n$\\Delta_{i}^{*}\\longleftarrow \\frac{\\sqrt{w_{i}}}{\\sqrt{\\lambda^{*}}}$\n}\n\\Return{$\\vec{\\Delta}^{*}$} \n\\caption{Finding $\\pi^{*} \\in \\pi^{-}$ (Problem 1)}\\label{algo_1}\n\\end{algorithm}\\DecMargin{1em}\n\n See Appendix \\ref{appe:alg1} for more details. Hence we schedule sensor $i$ with a probability $\\Delta^*_i$.\n\n\\subsubsection{System with CSI} For systems with CSI, the scheduling problem is again non-trivial since only one sensor can be scheduled in every time slot. Hence we schedule each sensor for only a fraction of the times for which the channel is in the ON-state. Let $n^{+}$ denote the set of sensors with CSI and $\\vec{\\alpha}$ denote the vector of parameters $\\alpha_{i}$. The policy $\\pi^{+}$ is:\n\\begin{gather*}\n    \\pi^{+} = \\{\\text{Schedule sensor }i\\text{ with probability }\\alpha_{i}\\text{ if channel state = ON } \\: \\forall i \\in n^{+} | 1 \\geq \\alpha_{i} > 0. \\}\n\\end{gather*}\n Like the previous case we find an optimal policy for relaxed constraints. Our next result characterizes the performance of a randomized policy as a function of the parameters $\\vec{\\alpha}$.\n \\begin{theorem}\n \\label{theorem:CSI_random}\nWithout loss of generality assume that $\\vec{p}>0$. Then under the policy $\\pi^{+}$, the freshness objective $J^{\\pi^{+}}$ as defined in Eq. \\ref{eq:1} is \n\\begin{equation}\n   J^{\\pi^{+}} = \\sum_{i\\in n^{+}}\\frac{ w_{i}(1-\\alpha_{i})}{\\alpha_{i}}.   \n\\end{equation}\n \\end{theorem}\n Check Appendix \\ref{appe:Th5} for the proof. Note that sensor $i$ has channel ON probability $p_{i}$ and is scheduled with probability $\\alpha_{i}$ when channel state is ON. Thus the average rate at which it is scheduled is $p_{i}\\alpha_{i}$. This problem can also be setup as an optimization problem. \n\n\n\\textit{Problem 2: Finding optimal $\\pi^{*} \\in \\pi^{+}$} -- Finding the optimal policy can be reduced to the following convex optimization problem.\n\\begin{gather*}\n    \\text{minimize }J^{\\pi^{+}} = \\sum_{i\\in n^{+}}\\frac{ w_{i}(1-\\alpha_{i})}{\\alpha_{i}}\n    \\\\\n    \\text{Subject to } \\sum_{i\\in n^{+}} p_{i}\\alpha_{i} = 1 ,\\:1 \\geq \\alpha_{i}> 0 \\quad \\forall i \\in n^{+}.\n\\end{gather*} \n\nThese constraints are derived from the relaxation of the constraints of the original scheduling problem. We ignore the $\\alpha_{i}\\leq 1$ constraint and optimize this using the Lagrange multiplier method. We describe how we account for this omitted constraint ahead. The optimal solution is obtained by solving the following system of equations:\n\\begin{gather}\n    \\alpha_{i} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda}} \\quad \\forall i \\in n^{+}, \\label{csi_opt1}\\\\\n    \\sum_{i\\in n^{+}} p_{i}\\alpha_{i} = 1. \\label{eq_csi}\n\\end{gather}\n\nAn exact solution of the above set of equations can be found. Algorithm \\ref{algo_2} describes the optimization procedure.\\\\\n\\IncMargin{1em}\n\\begin{algorithm}\n\\SetKwData{R}{R}\\SetKwData{ActiveSet}{ActiveSet}\\SetKwData{violation}{violation}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\\Input{$p_{i}$ - Channel ON probability $\\forall i \\in n^{+}$ \\\\ $w_{i}$ - Weight of sensor $i$ $\\forall i \\in n^{+}$}\n\\Output{$\\vec{\\alpha}^{*}$ }\n\\BlankLine\n\\R$\\longleftarrow 1$\\;\n\\ActiveSet$\\longleftarrow n^{+}$\\;\n\\BlankLine\n\\While{\\ActiveSet $\\neq \\phi$ }{\n\\violation $\\longleftarrow$ False\\;\n\\BlankLine\n$\\sqrt{\\lambda^{*}}$ $\\longleftarrow \\sum_{i \\in {ActiveSet}}\\frac{\\sqrt{w_{i}p_{i}}}{R}$ \\;\n\n\\For{i $\\in$ \\ActiveSet}{\n$\\alpha_{i}^{*} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda^{*}}}$\\;\n\\uIf{$\\alpha_{i}^{*} \\geq$ 1}{\n$\\alpha_{i}^{*} = 1$\\;\n\\violation = True\\;\n\\R= \\R $-\\:p_{i}$\\;\n\\ActiveSet = \\ActiveSet $\\backslash \\{i\\}$\n}\n}\n\\uIf{\\violation == False}{Break}\n}\n\n\\Return{$\\vec{\\alpha}^{*}$} \n\\caption{Finding $\\pi^{*} \\in \\pi^{+}$ (Problem 2)}\\label{algo_2}\n\\end{algorithm}\\DecMargin{1em}\nIn Algorithm \\ref{algo_2}, $\\sqrt{\\lambda^{*}}$ is obtained by solving the following set of equations:\\\\\n\\begin{gather}\n    \\alpha_{i} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda^{*}}} \\quad \\forall i \\in ActiveSet\\\\\n    \\sum_{i\\in ActiveSet} \\alpha_{i} p_{i} = R.\n\\end{gather}\nIn Algorithm \\ref{algo_2}, we initialize the active set with $n^{+}$. If any policy parameter exceeds the inequality constraints i.e. $\\alpha_{i}^{*}\\geq 1$ then we set $\\alpha_{i}^{*}= 1$, remove it from the active set and modify the constraints. This loop is run until the active set is empty or all policy parameters satisfy the inequality constraints. Since the policy found through Algorithm \\ref{algo_2} is optimal for the relaxed constraint scheduling problem  ($\\mathbb{E}\\left[\\sum_{i=1}^{n} a_{i}(t)\\right] = 1$) we need to make some implementational modification for the original problem ($\\sum_{i=1}^{n} a_{i}(t) = 1$). When more than one sensors are to be scheduled according to the policy, we schedule them greedily i.e. schedule the sensor having the highest weighted age. See Appendix \\ref{appe:alg2} for more details. \n\n\\subsubsection{System with partial CSI} We combine the policies derived for systems with and without CSI  dealt with in the previous sections to design a scheduling policy for the system having partial CSI. Let $n^{-}$ and $n^{+}$ denote the set of channels without CSI and with CSI respectively. Like the previous cases we relax the constraints and make implementational modifications to account for them. The policy in this case is:\n\\begin{align*}\n\\pi= \\begin{cases}\n\\{\\text{Schedule(i)} \\sim \\text{Bernoulli}(\\Delta_{i}) \\quad \\forall  i \\in n^{-} | 1 \\geq \\Delta_{i} > 0 \\},\\\\\n\\{\\text{Schedule(i) with probability }\\alpha_{i}\\\\\\text{ if channel state = ON } \\: \\forall i \\in n^{+} | 1 \\geq \\alpha_{i} > 0 \\} .\n\\end{cases}\n\\end{align*}\n\nOur next result characterizes the performance of a randomized policy as a function of the parameters $\\vec{\\Delta}$ and $\\vec{\\alpha}$.\n\\begin{theorem}\n\\label{theorem:mixed_random}\n Without loss of generality assume that $\\vec{p}>0$. Then under the policy $\\pi$, the freshness objective $J^{\\pi}$ as defined in Eq. \\ref{eq:1} is \n\\begin{equation}\n   J^{\\pi} = \\sum_{i\\in n^{-}}\\frac{ w_{i}(1-\\Delta_{i})}{\\Delta_{i}} + \\sum_{i\\in n^{+}}\\frac{ w_{i}(1-\\alpha_{i})}{\\alpha_{i}}.   \n\\end{equation}\n\\end{theorem}\nThis result is a straightforward extension of Theorems \\ref{theorem:UCSI_random} and \\ref{theorem:CSI_random}. Hence the scheduling problem can be setup as an optimization problem as follows:\n\n\\textit{Problem 3: Finding optimal $\\pi^{*} \\in \\pi$} -- Finding the optimal policy can be reduced to the following convex optimization problem.\n\\begin{gather*}\n    \\text{minimize } J^{\\pi} = \\sum_{i\\in n^{-}}\\frac{ w_{i}(1-\\Delta_{i})}{\\Delta_{i}} + \\sum_{i\\in n^{+}}\\frac{ w_{i}(1-\\alpha_{i})}{\\alpha_{i}},   \n    \\\\\n    \\text{subject to } \\sum_{i\\in n^{-}}\\Delta_{i} + \\sum_{i\\in n^{+}} p_{i}\\alpha_{i} = 1,\\\\\n    1 \\geq \\alpha_{i}> 0 \\quad \\forall i \\in n^{+}\\:,  1 \\geq \\Delta_{i}> 0 \\quad \\forall i \\in n^{-}.\n\\end{gather*} \nWe ignore the inequality constraints for $\\alpha_{i}$ and use the Lagrange multiplier method to reduce the optimization problem to solving the following set of equations:\n\\begin{gather}\n    \\Delta_{i} = \\sqrt{\\frac{w_{i}}{\\lambda}} \\quad \\forall i \\in n^{-},\\\\\n     \\alpha_{i} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda}} \\quad \\forall i \\in n^{+},\\\\\n     \\sum_{i\\in n^{-}}\\Delta_{i} + \\sum_{i\\in n^{+}} p_{i}\\alpha_{i} = 1. \\label{eq_3}\n\\end{gather}\n\nAn exact solution to the above system of equations can be found and the optimization procedure is described in Algorithm \\ref{algo_3}. \n\n\\IncMargin{1em}\n\\begin{algorithm}\n\\SetKwData{R}{R}\\SetKwData{ActiveSet}{ActiveSet}\\SetKwData{violation}{violation}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\\Input{$p_{i}$ - Channel ON probability $\\forall i \\in n$ \\\\ $w_{i}$ - Weight of sensor i $\\forall i \\in n$}\n\\Output{$\\vec{\\Delta}^{*},\\vec{\\alpha}^{*}$ }\n\\BlankLine\n\\R$\\longleftarrow 1$\\;\n\\ActiveSet$\\longleftarrow n^{+}$\\;\n\\BlankLine\n\\While{\\ActiveSet $\\neq \\phi$ }{\n\\violation $\\longleftarrow$ False\\;\n$\\sqrt{\\lambda^{*}}$ $\\longleftarrow \\sum_{i \\in n^{-}}\\frac{\\sqrt{w_{i}}}{R}+ \\sum_{i \\in {ActiveSet}}\\frac{\\sqrt{w_{i}p_{i}}}{R}$ \\;\n\\BlankLine\n\\For{i $\\in$ \\ActiveSet}{\n$\\alpha_{i}^{*} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda^{*}}}$\\;\n\\uIf{$\\alpha_{i}^{*} \\geq$ 1}{\n$\\alpha_{i}^{*} = 1$\\;\n\\violation = True\\;\n\\R= \\R $-\\:p_{i}$\\;\n\\ActiveSet = \\ActiveSet $\\backslash \\{i\\}$\n}\n}\n\\uIf{\\violation == False}{Break}\n}\n\\For{i $\\in n^{-}$}{\n$\\Delta_{i}^{*}\\longleftarrow \\sqrt{\\frac{w_{i}}{\\lambda^{*}}}$\n}\n\\Return{$\\vec{\\alpha}^{*},\\vec{\\Delta}^{*}$} \n\\caption{Finding $\\pi^{*} \\in \\pi$ (Problem 3)}\\label{algo_3}\n\\end{algorithm}\\DecMargin{1em}\n\nIn Algorithm \\ref{algo_3} $\\sqrt{\\lambda^{*}}$ is obtained by solving the following set of equations:\n\\begin{gather}\n    \\Delta_{i} = \\sqrt{\\frac{w_{i}}{\\lambda^{*}}} \\quad \\forall i \\in n^{-},\\\\\n     \\alpha_{i} = \\sqrt{\\frac{w_{i}}{p_{i}\\lambda^{*}}} \\quad \\forall i \\in \\text{ActiveSet},\\\\\n     \\sum_{i\\in n^{-}}\\Delta_{i} + \\sum_{i\\in \\text{ActiveSet}} p_{i}\\alpha_{i} = R. \\label{eq_4}\n\\end{gather}\nIt is straightforward to check that Algorithm \\ref{algo_3} utilizes the optimization procedure formulated in Algorithms \\ref{algo_1} and \\ref{algo_2}. When more than one sensors are to be scheduled according to the policy, we schedule the sensors in a greedy manner.\n\n\n\n\n\n\\section{Simulation Results}\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n    \\includegraphics[width=1\\linewidth]{Figures\/ucsi_3.png}\n    \\caption{Average sum of weighted age on changing channel ON probability of sensor 3 ($p_{3}$) in a system without CSI }\n    \\label{fig:UCSI3}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}{.45\\textwidth}\n \\centering\n    \\includegraphics[width=1\\linewidth]{Figures\/ucsi_2.png}\n    \\caption{Average sum of weighted age on changing channel ON probability of sensor 2 ($p\n    _{2}$) in a system without CSI }\n    \\label{fig:UCSI2}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n    \\includegraphics[width = 1\\linewidth]{Figures\/n_UCSI.png}\n    \\caption{Average sum of weighted age on varying number of sensors in a system without CSI}\n    \\label{fig:UCSIn}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}{.45\\textwidth}\n \n\\centering\n    \\includegraphics[width = 1\\linewidth]{Figures\/csi_3.png}\n    \\caption{Average sum of weighted age on changing channel ON probability of sensor 3 ($p_{3}$) in a system with CSI }\n    \\label{fig:CSI3}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\includegraphics[width = 1\\linewidth]{Figures\/csi_2.png}\n    \\caption{Average sum of weighted age on changing channel ON probability of sensor 2 ($P_{2}$) in a system with CSI }\n    \\label{fig:CSI2}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}{.45\\textwidth}\n \n\\centering\n    \\includegraphics[width = 1\\linewidth]{Figures\/n_CSI.png}\n    \\caption{Average sum of weighted age on varying number of sensors in a system with CSI}\n    \\label{fig:CSIn}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\includegraphics[width=1\\linewidth]{Figures\/mixed_csi.png}\n    \\caption{Average sum of weighted age on varying channel ON probability ($p_{2}^{+}$) of sensor 2 with CSI in a partial CSI system}\n    \\label{fig:mixed_o}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}{.45\\textwidth}\n \n\\includegraphics[width=1\\linewidth]{Figures\/mixed_ucsi.png}\n    \\caption{Average sum of weighted age on varying channel ON probability ($p_{2}^{-}$) of sensor 2 without CSI in a partial CSI system}\n    \\label{fig:mixed_uo}\n\\end{minipage}\n\\end{figure}\n\n\n\nIn this section we compare the performance of our policies against the greedy policy via simulations.\nWhile evaluating the objective we normalize the weights so that they sum to unity. The greedy policy for the system without CSI is to schedule the sensor having the highest weighted age times the channel ON probability in that time slot, i.e., $\\argmax_i\\left(w_{i}X_{i}(t)p_{i}\\right)$. In Fig. \\ref{fig:UCSI3} we simulate a system without CSI consisting of three sensors. We set $p_{1}=0.1,\\: p_{2} = 0.9$ and vary the channel ON probability of sensor 3 ($p_{3}$) from 0 to 1. The corresponding weights are set as $w_{1}=1,\\:w_{2}=1,\\:w_{3}=100$. We give sensor 3 a considerably larger weight to clearly observe the effect of changing it's channel ON probability. The Whittle's index policy outperforms the other two policies in this system. In Fig. \\ref{fig:UCSI2} we test our policies on a system without CSI consisting of two sensors where an important sensor has a poor connection. We fix $p_{1}=0.1$ and vary $p_{2}$ between 0 to 1. The weights are set as $w_{1} =1000,\\: w_{2} = 1$. The Whittle index policy and randomized policy outperform the greedy policy by a considerable margin in this system. In Fig. \\ref{fig:UCSIn} we investigate the effect of the size of the system on the performance of the policies. We vary the number of sensors from 10 to 40. In each system the channel statistics are chosen uniformly from 0 to 1. Similarly the weights are chosen uniformly at random from 1 to 100. The Whittle index policy outperforms both the policies. \n\n\nWe run similar experiments for a that system with CSI. The greedy policy for a system having CSI is to schedule the sensor having the highest weighted age i.e. $\\argmax_i\\left(w_{i}X_{i}(t)\\right)$. Fig. \\ref{fig:CSI3} shows the simulation of a system with CSI having 3 sensors. The parameters of the system remain the same. Once again the Whittle index policy performs the best and the randomized policy attains similar performance at higher channel ON probabilities. In Fig. \\ref{fig:CSI2} we simulate our policies in a system with CSI having important sensors transmitting through poor channels. The parameters remain the same as before. The Whittle index policy and the randomized policy perform substantially better than the greedy policy. We simulate the effect of varying system size on the performance of the policies in Fig. \\ref{fig:CSIn}. The greedy policy and the Whittle index policy yield good performance.\n\n\nIn  Fig. \\ref{fig:mixed_o} we simulate a system having partial CSI. The system has 2 sensors with CSI and 2 sensors without CSI. The channel statistics of the sensors without CSI are $p_{1}^{-}=0.1,\\;p_{2}^{-}=0.9$. The channel ON probability of one of the sensors with CSI is set to $p_{1}^{+} = 0.1$ and the probability of the other sensor is varied from 0 to 1. The weights are setup as $w_{1}^{-}=w_{2}^{-} = w_{1}^{+}=1, \\;w_{2}^{+}=100$ . The Whittle index policy performs considerably better than the other policies. In Fig. \\ref{fig:mixed_uo} we simulate a similar system albeit wth parameters between sensors with and without CSI exchanged. The channel ON probabilities are $p_{1}^{-}=p_{1}^{+}=0.1,\\;p_{2}^{+}=0.9$ and $p_{2}^{-}$ is varied from 0 to 1. The weights are $w_{1}^{+}=w_{2}^{+} = w_{1}^{+}=1, \\;w_{2}^{-}=100$ Once again the Whittle index policy outperforms the others. \n\nFrom these results, and others which we omit due to lack of space, we conclude that the randomized scheduling policies work well when sensors have poor channel communication. However the Whittle index policy is effective across all types of system settings and achieves minimum cost. It outperforms the randomized policy by a substantial margin in larger systems and systems that have better connections.\n\n\\section{Discussion: AoI vs CA-AoI}\n\\label{sec:discuss}\n\n\\label{sec:discuss}\nIn this section, we discuss why the CA-AoI metric is more suitable than the original AoI metric for some multi-sensor systems. \n\nAs discussed before, the AoI of a sensor increases due to two reasons. The first is when an attempted update by a sensor fails due to poor channel conditions and the second is when the shared channel is used by some other sensor. The CA-AoI distinguishes between these two events by imposing a penalty by increasing the \"age\" of the information of a sensor at the monitoring station only in the latter case.\n\n\nMany systems deploy multiple sensors to measure the same quantity, for example  temperature, close to each other to increase robustness to sensor failures as well as diversity. Consider a system with two sensors deployed in the same area. However, due to their placement, one sensor has significantly better channel conditions than the other. This can happen for instance if there is an obstruction between only one of the two sensors and the destination. If the sum AoI metric is used, under any ``good\" policy, a larger fraction of resources will be allotted to the sensor with poor channel conditions than if the sum CA-AoI metric is used. As a result, the throughput, i.e., the total number of measurements received by the destination over time will be lower in the first case. Since the quantity being measured by both sensors is roughly the same, CA-AoI might be the more appropriate metric for this case. Sum of CA-AoIs strikes a balance between rewarding the diversity obtained from getting measurements of the same physical process from two independent sources and ensuring that the resource allocation is not heavily tilted towards the sensor with poor channel conditions. On the other hand, using a greedy policy that blindly maximizes the throughput will be unable to meet the freshness requirements of the application. Whittle index policies for either choice of the metric have been demonstrated to work well in the unknown CSI setting. In Fig. \\ref{fig:throughput} we show the throughput achieved by Whittle index policies that optimize for AoI and CA-AoI  in an equally weighted two sensor system, where the channel ON probability of the sensor with poor channel conditions is fixed to 0.1 and that of the other sensor is varied. We also simulate the ``greedy'' policy where the sensor with better channel conditions is scheduled in every time slot. We simulate the same setting in Fig. \\ref{fig:resources} and measure the fraction of resources allotted to the poor sensor. It can be clearly observed that the throughput obtained by using the CA-AoI metric is higher and the performance gap widens as the channel ON probability of the sensor with better channel conditions increases. In Fig. \\ref{fig:age}, we compare the freshness performance obtained from the AoI and the CA-AoI policies via the vanilla AoI metric. As expected, the greedy policy doesn't preserve any timeliness and the Whittle AoI policy outperforms the CA-AoI minimizing policy. While the AoI metric forces a higher scheduling rate of the sensor with poor channel conditions as the probability increases to meet the vanilla freshness requirement, the CA-AoI metric causes much fewer resources to be allocated to the poor sensor. However like the greedy policy, the Whittle CA-AoI policy doesn't completely discount the sensor with poor channel conditions and yields a more fairer allocation of resources and thus freshness of information at the cost of maximum throughput. \n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}[b]{.45\\textwidth}\n  \\centering\n    \\includegraphics[width=1\\linewidth]{Figures\/throughput_3.png}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}[b]{.45\\textwidth}\n\\centering\n    \\includegraphics[width=1\\linewidth]{Figures\/resources_poor_3.png}\n\\end{minipage}\n\\par\n\\begin{minipage}[t]{.45\\textwidth}\n\\centering\n    \\caption{Throughput achieved by minimizing CA-AoI vs AoI}\n    \\label{fig:throughput}\n\\end{minipage}\n\\qquad\n\\begin{minipage}[t]{.45\\textwidth}\n\\centering\n   \\caption{Fraction of resources allocated to poor sensor while minimizing CA-AoI vs AoI}\n    \\label{fig:resources}\n\\end{minipage}\n\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}[b]{.45\\textwidth}\n  \\centering\n    \\includegraphics[width=1\\linewidth]{Figures\/age_plot_3.png}\n\\end{minipage}%\n\\qquad\n\\begin{minipage}[b]{.45\\textwidth}\n\\centering\n    \\includegraphics[width=1.1\\linewidth]{Figures\/aoi_thput.png}\n\\end{minipage}\n\\par\n\\begin{minipage}[t]{1\\textwidth}\n\\centering\n    \\caption{Average sum of weighted AoIs while  minimizing CA-AoI vs AoI. Left plot shows all three policies and the right plot compares AoI and CA-AoI Whittle policies}\n    \\label{fig:age}\n\\end{minipage}\n\\end{figure}\n\\section{Conclusions}\nWe propose a variant of the AoI metric called Channel-Aware Age of Information (CA-AoI). The CA-AoI of a source at the intended destination is defined as the number of time-slots elapsed since the recent most update from the source was received in which the channel conditions were good enough for the source to send an update. The CA-AoI of a sensor is therefore, a measure of the number of missed opportunities to send an update since its recent most successful update. \nWe focus on a multi-source system that updates a monitoring station through a shared unreliable channel under different CSI models and characterize lower bounds on the metric. We model our scheduling problem as a Restless Multi Armed Bandit (RMAB) problem and prove indexability under all the CSI models.  We show the threshold type structure of the optimal policy and derive the Whittle index for the problems. Besides this we also design stationary randomized scheduling policies and show that finding the optimal parameters reduces to simple convex optimization procedures. Via extensive simulations we show that the proposed policies outperform the greedy policies in several settings with the Whittle index policy being the most effective across all system settings. Finally, we discuss the effectiveness of using the sum of CA-AoI metric in a fair resource allocation across sensors and thus in achieving a higher throughput. \n\n\n\n\n\n\n\\bibliographystyle{.\/bibliography\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe formation of Sun-like stars is set in motion by the collapse of a cold,\ndense cloud. Many different physical processes take place in the protostellar stage --  first few $10^5$ yrs that are critical to the subsequent evolution of the star and its planetary system \\citep{Lada1987}. The mass of the star and that of its\ncircumstellar disk are determined during this embedded phase\n\\citep{Hueso2005} and the first steps of planet formation must take\nplace then \\citep{Greaves2010, Williams2012, ALMA2015,Manara2018,Harsono2018,Tobin2020,Tychoniec2018a,Tychoniec2020,SeguraCox2020}. At the\nsame time, on larger scales, the collapsing envelope is dispersed by\nthe energetic action of bipolar jets and winds emanating from the star-disk\nsystem which create outflows of entrained gas and dust \\citep{Arce2006,Offner2014}\n\nRotational transitions of molecules are a powerful tool to probe other\ncomponents of the system and can be used to infer densities,\ntemperatures, UV fields, chemical abundances and kinematics\n\\citep{vanDishoeck1998,Evans1999}. Until recently, studies of low-mass protostars have suffered from insufficient spatial resolution to disentangle these different physical\ncomponents. The advent of submillimeter interferometry opened the possibility to study protostellar systems at much smaller scales than with single-dish observations \\citep[e.g.,][]{Chandler1996,Hogerheijde1999,Schilke1992, Wilner2000, Jorgensen2005b, Tobin2011}.\n\nWith the Atacama Large Millimeter\/submillimeter Array (ALMA), it is possible to\nimage many molecular lines on the relevant physical scales with achievable observing times at sub-arcsecond resolution. Impressive ALMA studies of individual low-mass protostars\nhave been presented, focusing both on simple species (< 6 atoms) and complex molecules (> 6 atoms)\n\\citep[e.g.,][]{Sakai2014b, Jorgensen2016, LopezSepulcre2017,Lee2019,  Codella2018, LeeJE2019, Manigand2020, vanGelder2020, Bianchi2020}. The first ALMA surveys of complex molecules on a larger sample are taking place \\citep{Yang2021} and of Class I disks \\citep{Podio2020, Garufi2020}. The CALYPSO survey with NOEMA studied a larger sample of protostars but at more limited resolution and sensitivity \\citep{Belloche2020, Maret2020, Podio2021}.\n\nHere we present ALMA data of 16 protostellar sources covering rotational transitions of various molecules; we use these data to build a complete picture of what types of molecules trace which physical structures in protostars. This sample constitutes one of the largest combinations of high-resolution ALMA observations of Class 0\/I protostars to date in ALMA Band 3, 5, and 6. Covering a broad range of protostellar properties\nwithin the low-mass regime, the aim is to identify and describe key molecular tracers of future Sun-like stars and what physical components of star-forming sources they correspond to. Parts of the ALMA datasets presented here have been already published, focusing on different aspects: complex organic molecules \\citep{vanGelder2020,Nazari2021}; Class I disks temperature structure \\citep{vantHoff2020b}; outflows and high-velocity jets in Serpens \\citep{Hull2016, Tychoniec2019}, molecular emission associated with magnetic fields \\citep{Hull2017, LeGouellec2019}. In this work we make a comprehensive overview of these different datasets, making full use of those observations, with uniform analysis methods. This allows to reveal and systematise the molecular tracers of Class 0\/I protostars. Out of 16 presented sources 11 are included in upcoming {\\it James Webb Space Telescope} (JWST) observations with the Mid-Infrared Instrument (MIRI;  \\citealt{Wright2015})  (\\citeyear{JWST1290,JWST1257}) and Near-Infrared Spectrograph (NIRSpec) ( \\citeyear{JWST1960,JWST1186,JWST1798,JWST2104})\n\n\n\\subsection{Physical components of a protostellar system}\n\nClass 0 sources are defined by their strong excess of submillimeter\nluminosity and very low bolometric temperatures \\textless 70~K\n\\citep{Andre1993, Chen1995}. These sources are associated with\npowerful outflows, and the envelope mass dominates the mass of the\nentire system.  Class I sources are defined by having an infrared\nspectral index indicating strong reddening \\citep{Lada1987} with\nbolometric temperatures of 70--650 K \\citep{Chen1995}. Those systems\nhave already converted most of their envelope mass into disk and\nprotostar \\citep{Crapsi2008, vanKempen2009a, Maury2011}. For the\ntypical envelope masses of sources presented here and average disk\nmasses found by \\citet{Tychoniec2020}, the $M_{\\rm disk}\/M_{\\rm\n  env}\\simeq$ 1\\% for Class 0 and $\\simeq$ 20\\% for Class I, with\nvalues up to 75--98\\% in cases of rotationally supported disks \\citep{Jorgensen2009}.\n\n\\begin{table*}\n\\caption{Targeted protostellar systems}            \n\\label{table:targets}     \n\\centering                         \n\\begin{tabular}{l l c c c c c  c c}       \n\\hline\\hline                \nSource name & R.A. & Decl. & $d$ & Class & $L_{\\rm bol}$ &   $T_{\\rm bol}$ & $M_{\\rm env}$ &  Ref. \\\\\n & (J2000) & (J2000) & (pc) & & (L$_{\\odot}$) &   (K) & (M$_{\\odot}$) &   \\\\\n\n\\hline                                  \nSerpens SMM1 &  18:29:49.8 & +01:15:20.5 & 439 & 0 & 109 &   39  & 58 &  (1)\\\\\n\nSerpens S68N & 18:29:48.1 & +01:16:43.3 & 439 & 0 & 6 &   58  & 10 &  (2) \\\\\n\nSer-emb 8 (N) & 18:29:48.7 & +01:16:55.5 & 439 & 0 & --- &  ---  & --- &  ---\\\\\nSerpens SMM3 & 18:29:59.2   & +01:14:00.3  & 439 & 0 & 28 & 38 &  13  & (1) \\\\\n\n\nBHR 71 & 12:01:36.3 &  -65:08:53.0 & 200 & 0 & 15 &  44  & 2.7  &  (1)\\\\\n\nIRAS 4B & 03:29:12.0  &  +31 13 08.1 & 293 & 0 & 7 &  28  & 4.7 &  (1)\\\\\n\nPer-emb-25 & 03:26:37.5   &  +30:15:27.8 & 293 & 0\/I & 1.9 &  61  & 2.0 &  (2) \\\\\n\nB1-c &  03:33:17.9   & +31:09:31.8 & 293 & 0 & 5 &  48  & 15 &  (2) \\\\\n\nHH211-mm  & 03:43:56.8   & +32:00:50.2 & 293  & 0 & 2.8 &  27  & 19 &  (2) \\\\\nL1448-mm & 03:25:38.9   & +30:44:05.3 & 293 & 0 & 13 & 47 &  15  & (2) \\\\\n\n\nL1527 IRS & 04:39:53.9   &  +26:03:09.5   & 140 & 0\/I & 1.6 & 79 & 0.12 & (3) \\\\\n\nB5-IRS1 & 03:47:41.6 & +32:51:43.7 & 293 & I & 7 &  181  & 3.5 &  (2) \\\\\nTMC1 & 04:41:12.7   & +25:46:34.8  &140 & I  & 0.9 &  101  & 0.14 &  (1) \\\\\nIRAS 04302 & 04:33:16.5   & +22:53:20.4 &140 & I & 0.7 &  300  & 0.05 & (1) \\\\\nL1489 IRS & 04:04:43.0   & +26:18:57.0 &140 & I  & 3.8 &  200  & 0.2 &  (1)\\\\\nTMC1A & 04:39:34.9  &  +25:41:45.0 &  140 & I & 2.7  &  118 & 0.2 & (1)\\\\\n\n\n\n\n\\hline                                  \n\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\item{(1) \\citealt{Kristensen2012}, (2) \\citealt{Enoch2009}, (3) \\citealt{Green2013} }\n\n\\end{tablenotes}\n\\end{table*}\n\nThe different components of protostellar systems vary significantly in\ntheir physical conditions, such as density and temperature, molecular\nenrichment, and dynamics.  Our current knowledge about them is described briefly below to set the scene for the interpretation of our data.\n\n{\\it Envelope.} The envelope surrounding a protostar is the material\nthat fuels the accretion process onto the star and disk. The\nphysical conditions in the outer envelope on scales of a few 1000 au\nare reminiscent of those of starless cores with heavy freeze-out, and their chemical\ncomposition is directly inherited from the cloud out of which the star\nis being born \\citep{Caselli2012}. Systematic motions such as infall\nor expansion can occur but otherwise they are characterized by low turbulence and\nnarrow (FWHM < 0.5 - 1 km s$^{-1}$) line profiles indicative of quiescent gas \\citep{Jorgensen2002}. \n\n{\\it Warm inner envelope}. In the innermost part of the envelope on scales of the\ndisk, temperatures rise above 100 K, so any water and complex organic\nmolecules (COMs) contained in ices are released from the grains back\ninto the gas where they are readily observed at submillimeter\nwavelengths. This region with its unique chemical richness is called\nthe hot core, or to distinguish it from its high-mass counterpart, hot\ncorino \\citep{Herbst2009}.\n\n{\\it Jets and outflows}. As the material is accreting from the envelope onto the disk, excess angular momentum has to be transported via a still elusive process to allow material to accrete on the growing protostar. Jets and outflows constitute compelling  candidates to extract angular momentum via magnetic fields. In the earliest stages when the mass loss is at its peak, the densities are high enough to form molecules in the internal shocks in the jet \\citep{Bachiller1992, Tafalla2010}. Much slower (\\textless 20 km s$^{-1}$) and less collimated gas moving away from the protostar is called an outflow. Their origin remains debated. Large scale outflows reveal bow-shock shells and cavities possibly driven by the fast intermittent jet \\citep{Gueth1996, Gueth1999, Tychoniec2019}. Temperatures in shocked regions are much higher than in the surrounding envelope, up to a few thousand K, and sputtering of grain cores and ice mantles can further result in unique chemical signatures \\citep{Arce2008, Flower2013}. \n\n{\\it Outflow cavity walls.} These are the narrow zones in between the\ncold dense quiescent envelope material and the lower-density warm cone\nwhere outflows are propagating at large velocities. Cavity walls are\nexposed to UV radiation from the accreting star-disk boundary layer,\nwhich can escape through the outflow cavity without being extincted\n\\citep{Spaans1995}. This creates conditions similar to those found in\nPhoton Dominated Regions (PDRs), which occur throughout the\ninterstellar medium near sources of intense UV radiation\n\\citep{Hollenbach1997}.  In units of the interstellar radiation field\n(ISRF, \\citealt{Draine1978}), typical values of 10$^2$--10$^3$ are found\non scales of $\\sim 1000$ au \\citep{vanKempen2009c, Yildiz2012,Benz2016,\n  Karska2018}.\n  \n\n\n{\\it Young disk.} In the inner envelope, a protoplanetary disk starts\nto form as the natural outcome of a rotating collapsing core\n\\citep{Ulrich1976, Cassen1981,Terebey1984}. \nA young disk should be rotating in Keplerian motion. At early stages it is difficult to identify whether the so-called embedded disk is\nrotationally supported, since any molecular emission from the disk is entangled with that from the envelope. In recent years several\nembedded disks have been identified to have Keplerian rotational\nstructure on scales of $\\sim$100 au \\citep{Tobin2012,Murillo2013,Ohashi2014, Yen2017}. Molecular tracers in young disks, apart from\nproviding the kinematic information, can probe their temperature\nstructure as well \\citep{vantHoff2018a}. \n\n\nThis work is organized as follows. In Section \\ref{section:observations}, the observations used in this work are presented, Section \\ref{section:results} presents the results of this work including detections and morphology of the targeted molecules. In Section \\ref{section:discussion} the results are discussed. with special focus on which molecular tracers are corresponding to each of the physical components. The focus is on a qualitative description, rather than quantitative analyses for which source specific models and more rotational transitions of a\ngiven molecule would be needed. We summarize our\nwork in Section \\ref{section:conclusions}.\n\n\\section{Observations}\n\\label{section:observations}\n\n\\subsection{Datasets}\n\nSix different ALMA 12m datasets at Band 3, 5, and 6 are used in this work to cover 14 out of 16 sources. The spatial resolution of all ALMA 12m datasets is comparable (0\\farcs3--0\\farcs6), except for 2017.1.01174.S, where Band 3 observations are obtained at 3\\arcsec. Additionally for 6 out of 16 sources in Band 6, ACA observations with 7m antennas were obtained at 6\\arcsec\\  resolution. \n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=1\\linewidth,trim={0cm 0cm 0cm 0cm}]\n \n   {plots\/fig1.jpeg}\n  \\caption{ Continuum emission at 1.3 mm of four example protostellar systems obtained with ALMA at 0\\farcs5 resolution. Symbols of stars point to confirmed protostellar sources, circles show condensations of continuum emission, without confirmed protostellar nature, dotted lines show outflow cavity walls, and dashed lines show streams of envelope material. Arrows indicate outflow directions.}\n \\label{fig:continuum_showcase_v1}\n\\end{figure*}\n\n\nThe targeted protostars are well-known objects located in different\nstar-forming regions. They span a range of properties within the\nlow-mass regime; the probed range of $L_{\\rm bol}$, $T_{\\rm bol}$ and\n$M_{\\rm env}$ is shown in Table  \\ref{table:targets}, and are provided by a suite of observations across the infrared and submillimeter spectrum  \\citep{Enoch2009,Kristensen2012,Green2013}.\n\nThe details of the observations  are summarized in Table \\ref{table:observations}.  The spatial resolution allows to observe protostellar systems at solar-system scales; Band 5 and 6 observations provide a resolution of $\\sim$0\\farcs5, which corresponds to a 70--220 au diameter for sources in our sample. Thus, regions down to 35-110 au radius in the inner envelope are probed. The Band 3 data achieve moderate resolution of $\\sim$3\\arcsec\\   which provides information on intermediate envelope scales of 500--1500 au. The ACA observations of six sources at 6\\arcsec\\  resolution probe envelope scales of 800--2000 au. \n\n\\subsection{Spectral setup of the observations}\n\nA collection of different datasets using different ALMA bands implies\nvarying spectral and spatial resolution as well as spectral coverage\nacross the analysis. This is the reason that throughout this paper the\nsources shown in the figures differ when presenting detections and maps of different\nmolecules. In all cases, when the molecule is discussed, only those\nsources where the given transition has been targeted are\ndiscussed. All non-detections are explicitly\nstated. Table \\ref{table:table_mols} provides a list of targeted\nmolecular transitions, with sources that have a particular line covered and\ndetected or not detected.\n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={2cm 1cm 2cm 0cm}]\n  {plots\/fig2.jpeg}\n  \n  \\caption{Maps of key envelope tracers toward B1-c (Class 0, top) and TMC1 (Class I, bottom) obtained with ACA. Contours represent continuum emission at 1.3 mm observed with ACA. Note different distances to B1-c and TMC1 resulting in different spatial resolutions of the maps. {\\it Left:} C$^{18}$O 2 -- 1 {\\it Middle:} N$_2$D$^+$ 3 -- 2. {\\it Right:} DCO$^+$ 3 -- 2. All moment 0 maps are integrated from -2.5 to 2.5 km s$^{-1}$ w.r.t $\\varv_{\\rm sys} $.}\n \\label{fig:envelope_plot_no1_v1}\n\\end{figure*}\n\n\nThe ALMA observations presented here target different spectral setups across Band 3, 5, and 6. This allows to probe the strongly varying physical scales and conditions. In particular, our Band 3 data grants access to lines at very low excitation levels that enable tracing more extended material. Dust is less optically thick in Band 3 compared with Band 6 which potentially allows to peek inside the densest inner regions. Band 6 offers a usual set of tracers of outflows: CO, SiO and SO. \n\nCold outer envelopes with temperatures \\textless~20~K are probed\nwith low $E_{\\rm up}$ transitions. Additionally, non-thermal processes\nsuch as sputtering of material from the grains in the outflow, will\nalso be seen in low $E_{\\rm up}$ due to their lower critical\ndensities. On the other hand, thermal desorption from grains in\nthe innermost regions are best probed with lines with high $E_{\\rm up}$. With the large span of frequencies of the observations, different transitions of the same\nmolecule can be detected and used to trace different components of the system (e.g., a HNCO line at $E_{\\rm up}=$  15 K is available in Band 3 and lines at 70 and 125 K are covered in Band 6). \n\n\\section{Results}\n\\label{section:results}\n\n\\subsection{Continuum emission from protostars }\n\nFigure \\ref{fig:continuum_showcase_v1} presents continuum emission maps toward four example protostars obtained with ALMA at $\\sim$ 0\\farcs5. The examples illustrate most characteristic features observed in the continuum maps. The continuum images for all sources are presented in  Fig. \\ref{fig:continuum_no1_v1}.\n\nThe continuum emission observed at millimeter wavelengths (1.3 - 3 mm in our observations) traces thermal dust emission from the inner envelope and the embedded disk. In the Class 0 protostars, the central continuum source generally appears compact $R$ \\textless 100 au (e.g., B1-c, Fig. \\ref{fig:continuum_showcase_v1}). The example of SMM3 (Fig. \\ref{fig:continuum_no1_v1}) shows a large resolved dust structure perpendicular to the outflow, but its classification as a disk is not certain. The fact that we observe primarily compact continuum emission towards Class 0 sources is consistent with observations of confirmed rotationally-supported disks \\citep[e.g.,][]{Tobin2018, Maury2019, Tobin2020} and predictions of models \\citep{Visser2010,Harsono2015b, Machida2016}. While we assume that this compact emission belongs to the young, embedded disk, we do not have spatial and kinematic resolution to confirm the presence of Class 0 Keplerian disks in our sample.\n\nThe extended continuum emission in Class 0 sources is consistent with a significant amount of envelope material surrounding the protostar.\nIn the case of the SMM1 system, presented in Fig. \\ref{fig:continuum_showcase_v1} (left), the continuum clumps outside the central emission are  components of multiple protostellar systems, marked with stars, confirmed by the presence of individual molecular outflows  \\citep{Hull2016, Hull2017}. Binary components are also seen in TMC1, BHR71, and IRAS 4B (Fig. \\ref{fig:continuum_no1_v1}). In the case of S68N presented in Fig. \\ref{fig:continuum_showcase_v1}, two emission peaks that stand out from the diffuse envelope emission are marked with circles, but their protostellar nature is not confirmed.\n\n\n\n\\subsection{Protostellar envelope}\n\nWe present molecules that trace the bulk of the protostellar envelope in Fig. \\ref{fig:envelope_plot_no1_v1}. The protostellar envelope has a typical radius on the order of a few 1000 au  \\citep{Jorgensen2002,Kristensen2012}.  Thus, the sub-arcsecond ALMA 12m array observations tend to resolve-out the envelope emission. For instance, the maximum recoverable scale (MRS) of ALMA Band 6 observations at 0\\farcs4  presented here is 5\\arcsec, which is between 600--2000 au diameter depending on distance to the source. For that reason, we discuss in this section mainly the ALMA-ACA observations obtained at lower spatial resolution (6\\arcsec; 750--2500 au) for six sources in our datasets; Class 0 sources: B1-c, BHR71, Per-emb-25, SMM3, and IRAS 4B, and Class I source TMC1. The ACA can zoom-in on what was previously contained in a single-dish beam of 15--20\\arcsec, while the MRS of ACA (30\\arcsec) enables us to preserve sensitivity to large-scale emission.  The MRS of all observations presented here are reported in Table \\ref{table:observations}. \n\n\n\nIn Fig. \\ref{fig:envelope_plot_no1_v1} typical envelope tracers C$^{18}$O 2--1 ($E_{\\rm up} = 16$ K),  DCO$^+$ 3--2 ($E_{\\rm up} = 21$ K), and N$_2$D$^+$ 3--2 ($E_{\\rm up} = 22$ K), observed at 6\\arcsec\\    resolution with the ACA, are presented toward example Class 0 and Class I sources -- B1-c and TMC1, respectively. The emission from the presented molecules exhibits similar behaviour for all Class 0 sources, therefore B1-c serves as a representative case; TMC1 is the only Class I source in the sample with 7m observations available. The maps for all sources for which these molecules have been targeted can be found in Appendix \\ref{AppendixD}. All envelope tracers presented here are characterized by narrow line profiles with FWHM $\\sim 1$ km s$^{-1}$.\n\nThe C$^{18}$O emission peak coincides with the continuum peak for our six sources and appears to be compact, less than 1000 au diameter for B1-c and TMC1. For B1-c and all Class 0 sources (Fig. {\\ref{fig:envelope_co18_7m}), low-level extended C$^{18}$O emission is seen along the outflow direction. For the only Class I source targeted with the ACA, emission is marginally resolved in the direction perpendicular to the outflow.\n\n\nIn our observations N$_2$D$^+$ is seen extended in the direction perpendicular to the core major axis toward B1-c (Fig. \\ref{fig:envelope_plot_no1_v1}) and other Class 0 sources except IRAS 4B (Fig.{\\ref{fig:envelope_n2d+_7m}). In the case of IRAS 4B the emission from this molecule appears dominated by large-scale emission from the filament detected toward this source, connecting it with IRAS 4A \\citep{Sakai2012}. The peak of the N$_2$D$^+$ emission is significantly shifted from the continuum peak in all cases, with a significant decrease in the inner regions in some cases (see BHR 71 in Fig. {\\ref{fig:envelope_n2d+_7m}). Similar extended N$_2$D$^+$ emission in other Class 0 sources was seen by \\cite{Tobin2013} based on lower resolution SMA and IRAM-30m data. For TMC1 the N$_2$D$^+$  molecule is not detected.  \n\nThe DCO$^+$ emission is seen extended  in a similar fashion to what is observed for N$_2$D$^+$. However, contrary to N$_2$D$^+$, DCO$^+$ is brightest on the continuum peak for all sources except TMC1 and Per-emb-25. For these two sources, the emission peak is offset by 1000--2000 au from the continuum source in the direction perpendicular to the outflow. In the Class I source TMC1, DCO$^{+}$ is present on much smaller scales ($< 2000$ au radius) than in Class 0 sources.\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm}]\n  {plots\/fig3.jpeg}\n\\caption{Maps of the EHV jets observed in SiO. Moment 0 maps are presented in color scale with continuum emission at 1.3 mm presented in black contours, both obtained with 12m observations. SiO 4--3 map of HH211 integrated from -20 to -10 and from 10 to 20 km s$^{-1}$ w.r.t $\\varv_{\\rm sys}$; B1-c ntegrated from -70 to -40 and from 40 to 70 km s$^{-1}$ w.r.t $\\varv_{\\rm sys} $;  SMM3 integrated from -60 to -40 and from 20 to 35 km s$^{-1}$ w.r.t $\\varv_{\\rm sys}$. }\n \\label{fig:ehv_no1}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={2cm 0cm 2cm 2cm}]\n  {plots\/fig4.jpeg}\n\n  \\caption{Zoom-in on molecular bullets from the SMM3 jet (see  Fig. \\ref{fig:ehv_no1}). CO, SiO, SO, and H$_2$CO molecular transitions are presented. {\\it Top:}  Norther\/blueshifted bullet. Moment 0 maps are integrated from -60 to -40 km s$^{-1}$ w.r.t $\\varv_{\\rm sys}$. The map center is offset from the SMM3 continuum center by (--3\\farcs7, +10\\farcs3). {\\it Bottom:}  Southern\/redshifted bullet. Moment 0 maps are integrated  from 20 to 40 km s$^{-1}$.  The map center is offset from the SMM3 continuum center by (+2\\farcs7, --7\\farcs).}\n \\label{fig:ehv_no2}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm}]\n  {plots\/fig5.jpeg}\n\n  \\caption{Low-velocity outflow in CO, SO, and SiO. Moment 0 maps of CO toward TMC1, SO and SiO map toward B1c are integrated from -10 to 10 km s$^{-1}$ w.r.t $\\varv_{\\rm sys}$.}\n \\label{fig:lowvel}\n\\end{figure*}\n\n\n\\subsection{Outflows and jets}\n\\label{section:results_outflows}\n\nFigure \\ref{fig:ehv_no1} presents the extremely high-velocity (EHV)\nmolecular jet component for HH211, B1c, and SMM3 observed in\nSiO with the ALMA 12m array; the L1448-mm EHV is shown in Fig.   \\ref{fig:outflow_showcase_no1}. The latter and HH211 are well-known EHV sources\n\\citep{Guilloteau1992, Lee2007} while SMM3 and B1-c are new detections\nof the jet component. HH211 shows SiO emission at low-velocities because the outflow is almost in the plane of the sky, but the high velocities are\nevident from the large proper motion movements of the bullets\n($\\sim$115 km s$^{-1}$; \\citealt{LeeCF2015}). CO, H$_2$CO, and SiO in\nthe EHV jets of Emb8N and SMM1 are presented in detail in\n\\cite{Hull2016} and \\cite{Tychoniec2019}.\n\nThese data are particularly interesting as still only few molecules\ntracing EHV jets have been identified to date (see \\citealt{Lee2020} for\nreview). Apart from those presented here -- CO, SiO, SO, and H$_2$CO-- \nmolecules such as HCO$^+$ and H$_2$O have been seen in this\nhigh-velocity component \\citep{Kristensen2012,LeeCF2014}. What is\nespecially important to highlight is the third detection of a H$_2$CO\nbullet in SMM3 after IRAS 04166 \\citep{Tafalla2010} and Emb8N\n\\citep{Tychoniec2019}. These detections mean that either a significant fraction of ice-coated dust is released with the jet, or that the H$_2$CO is efficiently produced in the jet through gas-phase chemistry.\n\n\nIn the case of L1448-mm and HH211, there are\nseveral molecular bullets along the jet axis with velocities up to 100 km~s$^{-1}$, while both B1-c and SMM3\nshow a much simpler structure with two bullets detected on one side in\nthe former and a single pair of symmetrically placed bullets observed\nin the latter case.  B1-c actually has a pair of bullets $\\sim$~200 au from the continuum peak, with the other bullet at 2500 au only seen in the redshifted part of the jet. The emission in SiO and SO appears very similar (Fig. \\ref{fig:sio_outflows_no2}, Fig. \\ref{fig:so_outflow}). There are no other molecules tracing the high-velocity component toward this source. H$_2$CO and $^{12}$CO are not targeted with our ALMA 12m datasets toward B1-c. The SMM3 jet has two distinct high-velocity bullets at $\\sim$~3200 au from the source which appear similar in CO, SiO and SO (see zoom-in  on Fig \\ref{fig:ehv_no2}). Additionally, the redshifted bullet shows faint, but significant emission from H$_2$CO. No traces of H$_2$CO are found in the blueshifted outflow.\n\n\n\nFigure \\ref{fig:lowvel} presents low-velocity outflow tracers CO 2--1  ($E_{\\rm up}=16$ K) for TMC1, SO 5$_6$--4$_5$  ($E_{\\rm up}=35$ K), and  SiO 4--3 ($E_{\\rm up}=21$ K) for B1-c. In Fig. \\ref{fig:co_outflows} we present an overview of CO 2--1 emission for five sources obtained with the ALMA 12 m array at  0\\farcs4 resolution. The sources show a variety of emission structures in the low velocity gas   (\\textless 20 km s$^{-1}$). In all cases, we do not capture the entirety of the outflows as they extend beyond the primary beam of observations ($\\sim$ 30\\arcsec). SMM3 and Emb8N have very narrow outflow opening angles ($<$ 20 degrees), while SMM1, S68N and TMC1 present larger opening angles.\n\nCO emission is especially prominent in the cavity walls, which can be related to both the limb brightening effect as well as higher (column) density of the material in the outflow cavity walls. This is especially highlighted in the Class I source, TMC1, where CO emission is almost exclusively seen in the outflow cavity walls. Even though CO is piling up in this region, it is observed at velocities up to 15 km s$^{-1}$ so it is clearly tracing the entrained material and not the envelope. The lower envelope density in Class I results in less material to be entrained in the outflow. In SMM1, three CO outflows from SMM1-a, SMM1-b, and SMM1-d  are overlapping \\citep{Hull2016,Tychoniec2019}.\n\n\n\nIn Fig. \\ref{fig:sio_outflows_no2}, SiO maps are presented: Band 5 SiO 4--3 ($E_{\\rm up}=21$ K) and Band 6 SiO 5--4 ($E_{\\rm up}=31$ K) data are shown in velocity ranges corresponding to the low-velocity outflow. In contrast to CO emission, the low-velocity SiO is mostly observed in clumps of emission instead of tracing the entirety of the outflowing gas. Several such clumps can be seen in the S68N source. In some cases, the clumps are relatively symmetric (Emb8N, B1-c), while monopolar emission is seen in other examples (L1448-mm, SMM1-d). In the case of SMM1-a and SMM1-b, very weak SiO emission at low velocities is observed. SiO emission in outflows is exclusively present in the Class 0 sources, while absent in the Class I sources, TMC1 and B5-IRS1, covered in these data sets.\n\nFig. \\ref{fig:so_outflow} presents SO 5$_6$--4$_5$  ($E_{\\rm up}=35$ K)  and SO 6$_7$--5$_6$ ($E_{\\rm up}=47$ K)  observations in Band 6. The emitting regions of SO are comparable with those of SiO for the Class 0 sources. The cases of S68N and B1-c show that SO emission also peaks at the source position while SiO is absent there. Thus, SO and SiO do not always follow each other and some SO might be associated with hot core emission \\citep{Drozdovskaya2018}. Important differences are observed for TMC1, where SO seems to be associated with the remainder of the envelope or the disk, while the SiO is not detected toward this source, as mentioned above.\n\n\nHCN 1--0 ($E_{\\rm up}=4$ K) and H$^{13}$CN 2--1 ($E_{\\rm up}=12$ K) maps are presented in Fig. \\ref{fig:hcn_outflow} and \\ref{fig:h13cn}, respectively. HCN is clearly seen in outflowing material enhanced in similar regions as low-velocity SiO. For Emb8N the HCN emission has been associated with intermediate velocity shock \\citep{Tychoniec2019}. In B1-c and L1448-mm weak extended emission along the outflow direction is detected but H$^{13}$CN strongly peaks on source. In the case of HH211, H$^{13}$CN is seen only in the outflow, with a geometry consistent with the outflow cavity walls, but with velocity profiles that are consistent with the outflowing material.\n\n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={2cm 1cm 2cm 0cm}]\n  {plots\/fig6.jpeg}\n  \\caption{Maps of the ice mantle tracers toward the S68N outflow, with the CO low-velocity outflow map for reference. {\\it Top:} CO, H$_2$CO and CH$_3$OH moment 0 maps obtained in Band 6 at 0\\farcs5 resolution. Circles show regions from which spectra were obtained for analysis in Section 7.1.  {\\it Bottom:} CH$_3$CHO, HNCO, and CH$_3$CN moment 0 maps obtained in Band 3 at 2\\farcs5 resolution. The emission is integrated from -10 to -1 km s$^{-1}$ and from 1 to 10 km s$^{-1}$ w.r.t  $\\varv_{\\rm sys} $.} \n \\label{fig:icemantle_no1}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={2cm 0cm 2cm 0cm}]\n  {plots\/fig7.jpeg}\n\\caption{Maps of the outflow cavity wall tracers toward SMM3 and B1c, with low-velocity outflow map for reference. {\\it Top:} Moment 0 maps toward SMM3 of CO 2--1, c-C$_3$H$_2$ 6$_{1,6}$--5$_{0,5}$ , and $^{13}$CS 5--4  obtained in Band 6 at 0\\farcs5 resolution and CN 1--0 in Band 3 at 3\\arcsec  with continuum emission at the same band and resolution in black contours. {\\it Bottom:} Moment 0 maps toward B1c of SO 6$_{7}$--5$_{6}$, C$_2$H 3$_{2.5,3}$--2$_{1.5,1}$  and H$^{13}$CO$^+$ 3--2  obtained at 0\\farcs5 and CN 1--0  at 3\\arcsec with continuum emission at the same band and resolution in black contours. The emission is integrated from -5 to -1 km s$^{-1}$ and from 1 to 5 km s$^{-1}$ w.r.t  $\\varv_{lsr} $. Outflow directions and delineated cavity walls are shown in C$_2$H and C$_3$H$_2$ maps.} \n \\label{fig:hydrocarbons_no1}\n\\end{figure*}\n\n\n \nIce-mantle tracers are a different class of molecules detected in low-velocity protostellar outflows. They are produced and entrained through interactions between the jet and the envelope. Here we present ALMA 12m array observations in Band 6 at 0\\farcs5 resolution for CO 2 -- 1  ($E_{\\rm up}=17$ K), CH$_3$OH 2$_{1,0}$ -- 1$_{0,0}$ ($E_{\\rm up}=28$ K), and  H$_2$CO 3$_{0,3}$ -- 2$_{0,2}$ ($E_{\\rm up}=21$ K) , and in Band 3 at 3\\arcsec\\ for CH$_3$CN  6$_1$ -- 5$_1$  ($E_{\\rm up}=26$ K), CH$_3$CHO 6$_{1,6,0}$ -- 5$_{1,5,0}$ ($E_{\\rm up}=21$ K), and HNCO 5$_{0,5}$ -- 5$_{0,4}$ ($E_{\\rm up}=16$ K). Fig. \\ref{fig:icemantle_no1} compares maps of integrated emission from those molecules with those of CO for S68N. All ice-mantle tracers detected in the outflow are observed in their low-energy transitions. \nAdditional maps for S68N are presented in the Appendix (Fig. \\ref{fig:fig9_extra}): Band 6  0\\farcs5  resolution image of CH$_3$CHO 14$_{0,14}$--13$_{0,13}$ ($E_{\\rm up}=96$ K) and H$_2$CCO 13$_{1,13}$--12$_{1,12}$  ($E_{\\rm up}=101$ K), which is overlapping with NH$_2$CHO 12$_{2,10}$--12$_{2,10}$  ($E_{\\rm up}=92$ K), and the Band 3 image at 3\\arcsec\\   resolution of CH$_3$OCHO 10$_{0,10}$--9$_{0,9}$ ($E_{\\rm up}=30$ K).  \n\nAround the frequency of H$_2$CCO 3$_{1,13}$--12$_{1,12}$  ($E_{\\rm up}=101$ K) line an extended emission can be seen in the outflow, this is however coincident with the NH$_2$CHO which is only 4 km s$^{-1}$ apart (Fig. \\ref{fig:fig9_extra}).  NH$_2$CHO is more commonly observed in the shocked regions than H$_2$CCO \\citep{Ceccarelli2017, Codella2017}; it is possible that those lines are blended. More low-energy transitions would be required to confirm the identification of these lines.\n\n\nThe velocities observed for ice-mantle tracers in the outflow are \\textless 15 km s$^{-1}$ with respect to the systemic velocity. This is slower than the CO and SiO outflow line wings which have velocities up to 20--30 km s$^{-1}$. On the other hand, the lines are clearly broader than those of molecular tracers of UV-irradiated regions that trace passively heated gas (see Section \\ref{section:outflow_cavity_walls}). \n\nIce-mantle tracers are also detected in SMM3 and B1-c, two Class 0 protostars (Fig. \\ref{fig:161} and \\ref{fig:151}). For SMM3 only a lower $A_{ij}$ transition of CH$_3$OH with $E_{\\rm up}=$ 61 K was targeted and not detected towards this source. B1-c has its methanol emission confused with the high-velocity SO emission, whereas emission from other COMs in the outflow is weak and appears only on the redshifted part of the outflow. Thus, S68N is the best case to study the composition of shock-released ice mantles.\n\n\\subsection{Outflow cavity walls}\n\nIn this section, we highlight key molecules detected in the outflow cavity walls. It is challenging to precisely distinguish cavity walls from the outflowing material. The velocity of the gas in the cavity walls should be lower than in the outflow, as the cavity wall contains envelope material at rest but which is  passively heated by UV radiation. We first discuss maps of species associated with cavity walls and then their line profiles.\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm},draft=False]\n  {plots\/fig8.jpeg}\n\n  \\caption{Spectra obtained at the cavity wall positions for hydrocarbons (red) and ice-mantle tracers (blue). {\\it Left:} C$_2$H 3$_{2.5, 2}$--2$_{1.5, 1}$ ($E_{\\rm up}=$ 25 K) and CH$_3$OH 2$_{1,0}$--1$_{0,1}$ ($E_{\\rm up}=$ 28 K) spectra for S68N, {\\it Middle:}   C$_2$H 3$_{2.5, 2}$--2$_{1.5, 1}$ ($E_{\\rm up}=$ 25 K ) and SO 6$_7$--5$_6$ ($E_{\\rm\n  up}=48$ K)  spectra for B1-c, {\\it Right:}  c-C$_3$H$_2$ $4_{4,1}$--$3_{3,0}$ ($E_{\\rm\n  up}=32$ K), H$_2$CO 3$_{2,1}$--2$_{2,0}$ ($E_{\\rm\n  up}=68$ K), and $^{13}$CS 5--4 ($E_{\\rm\n  up}=33$ K) spectra for SMM3.}\n \\label{fig:spectra_coms_outflow_no1}\n\\end{figure*}\n\n\n\\begin{figure}[h]\n\\centering\n    \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm}]\n  {plots\/fig9.jpeg}\n  \n  \\caption{Compact emission for B1-c and SMM3 for various molecules tracing the warm inner envelope (hot core). Moment 0 maps shown in colorscale integrated from -3 to 3  km s$^{-1}$ w.r.t $\\varv_{\\rm sys}$. 1.3 mm continuum presented in contours.} \n \\label{fig:hotcore_v1}\n\\end{figure}\n\nFigure \\ref{fig:hydrocarbons_no1} presents integrated emission maps of key tracers discussed in this section for two examples: SMM3 and B1-c. Plots for the remaining sources (S68N, Emb8N, and TMC1) are presented in the Appendix in Fig. \\ref{fig:hydrocarbons_no2}. Key tracers of the outflow cavity walls are simple unsaturated\nhydrocarbon molecules: c-C$_3$H$_2$ $4_{4,1}$--$3_{3,0}$ ($E_{\\rm\n  up}=32$ K) for SMM3 and TMC1, and C$_2$H $3_{2.5,3}$--$2_{1.5,2}$\n($E_{\\rm up}=25$ K) for S68N, B1-c, and Emb8N, as seen in maps\nobtained with ALMA Band 6 at 0\\farcs5 resolution.  \n\n\nIn Emb8N and SMM3 the emission from C$_2$H and c-C$_3$H$_2$, respectively, is symmetric; it appears similar in extent and shape on both sides of the continuum source. Comparison with CO emission, which traces the bulk of the outflowing gas, indicates that the hydrocarbons are located in the outflow cavity walls close to the source. A higher energy transition of c-C$_3$H$_2$ 7$_{2,6}$--7$_{1,7}$ $E_{\\rm up}$=61 K is seen towards SMM3 closer to the protostar compared with the lower-energy transition. This reflects the increase of temperature of the cavity walls closer to the source.\nFor B1-c, the emission from C$_2$H is U-shaped suggestive of a cavity wall, stronger on the blueshifted side of the outflow, which could be either a projection effect or an asymmetry in the envelope structure. For B1-c no ALMA CO observations exist to compare with the bulk of the outflow at comparable resolution; however, the other outflow tracer, SO, confirms the outflow direction and rough extent of the outflow cavity walls. Moreover, the shape of the cavity walls is consistent with the appearance of the CO 3--2 outflow observed with SMA toward this source at 4\\arcsec\\   resolution \\citep{Stephens2018}.\n\n\nS68N presents a chaotic structure (Fig. \\ref{fig:hydrocarbons_no2}, top), but C$_2$H is found elongated in the outflow direction. While it is difficult to identify the cavity wall, the C$_2$H emission surrounds the CO outflow emission. The C$_2$H  emission toward this source is asymmetric, with stronger emission in the blueshifted part of the outflow. Emission of c-C$_3$H$_2$ toward the Class I source TMC1  (Fig. \\ref{fig:hydrocarbons_no2}, bottom) is not directly related to the cavity walls, but is extended perpendicular to the outflow, which suggests that c-C$_3$H$_2$ traces the envelope or extended disk material. \n\nFigure \\ref{fig:hydrocarbons_no1} also shows CN 1--0 ($E_{\\rm up}$ = 5 K) observed at 3\\arcsec\\  resolution in Band 3 for SMM3 and B1-c. Compared with C$_2$H, CN is tracing similar regions. In S68N CN has a similar extent as C$_2$H but not over the full extent of the outflow traced by CO. In B1-c, the CN emission has a similar shape of the cavity wall cone as seen in C$_2$H, but also a significant contribution from larger scales is detected. In all cases the CN emission avoids the central region, which likely results from on-source absorption by the foreground CN molecules. \n\nIn some cases, like SMM3, the extent of the CN is broader than hydrocarbons, more comparable with CO, but the narrow linewidths suggest that this emission is still associated with passively irradiated envelope rather than with the entrained outflow.\n\nTMC1 presents a high-resolution example of CN emission (Fig. \\ref{fig:hydrocarbons_no2}). The offset between CO and CN reveals a physical\nstructure of the inner regions of the protostellar system: the\nentrained outflow traced with CO appears closer to the jet axis, while\nCN highlights the border between the outflow cavity wall and quiescent\nenvelope. CN is sensitive to UV radiation, as it can be produced with atomic C and N, whose abundances are enhanced in PDRs, with UV photodissociation of HCN contributing as well\n\\citep{Fuente1993,Jansen1995a,Walsh2010,Visser2018}.\n\nH$^{13}$CO$^{+}$ emission is presented for B1c (Fig. \\ref{fig:hydrocarbons_no1}), and S68N and Emb8N (Fig. \\ref{fig:hydrocarbons_no2}) observed in Band 6 at 0\\farcs4 resolution. The bulk of the emission from this molecule appears to be related to the cold envelope, however streams of material can be seen in B1-c and S68N. The streams of gas observed in  H$^{13}$CO$^+$ are coincident with the cavity wall observed in C$_2$H. As H$^{13}$CO$^+$ is expected to probe the dense envelope, the similarity of the morphology of the traced material between H$^{13}$CO$^+$ and C$_2$H and CN shows that the envelope material is UV-irradiated. $^{13}$CS observed in Band 6 with the 12m array is detected for SMM3 (Fig. \\ref{fig:hydrocarbons_no1}). The morphology of  $^{13}$CS emission is very similar to that of c-C$_3$H$_2$. \n\n\nIn Fig. \\ref{fig:spectra_coms_outflow_no1} spectra of ice mantle\ntracers, CH$_3$OH and H$_2$CO, and the hydrocarbon molecules, C$_2$H and\nC$_3$H$_2$ are presented. All spectra are shifted by their source\nvelocity to zero km s$^{-1}$. In case of S68N, it is seen that C$_2$H\nand CH$_3$OH have very similar line profiles indicating that they\ntrace similar material. The width of $\\sim 10$ km$^{-1}$ suggests\nthat this material is entrained with the outflow. A narrow component\nappears to be superposed at systemic velocities. Note that fine splitting of C$_2$H blends\nthe spectra although the other transition at +2 km s$^{-1}$ does\nnot affect the blueshifted velocity component. In contrast, B1c shows\nonly remarkably narrow C$_2$H line profiles with a FWHM of $\\sim$ 2 km\ns$^{-1}$, and SMM3 has similarly narrow c-C$_3$H$_2$ and $^{13}$CS lines compared with broader H$_2$CO emission (Fig. \\ref{fig:spectra_coms_outflow_no1}, right). \n\n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={3cm 1cm 3cm 0cm}]\n  {plots\/fig10a.jpeg}\n     \\includegraphics[width=0.95\\linewidth,trim={3cm 1cm 3cm 0cm}]\n    {plots\/fig10b.jpeg}\n  \\caption{Images of Class I disks observed with ALMA 12m in Band 6 \\citep{vantHoff2020b}. {\\it Top:} Moment 0 maps of C$^{17}$O at 0\\farcs4 resolution integrated from  -10 to 10 km s$^{-1}$ w.r.t  $\\varv_{\\rm sys} $.  {\\it Bottom:} Moment 0 maps of CN integrated from -2 to 2 km s$^{-1}$ w.r.t  $\\varv_{\\rm sys} $. Continuum contours in black.}  \n \\label{fig:disks_no1_v1}\n\\end{figure*}\n\n\\subsection{Inner envelope}\n\\label{section:results_inner_env}\n\nFig. \\ref{fig:hotcore_v1} shows emission from H$_2$CCO, HNCO, t-HCOOH,\nSO, H$_2$CS, and H$^{13}$CN for the Class 0 protostar B1-c, and H$_2$S\nand OCS lines for SMM3; all observed with the ALMA\n12m array at 0\\farcs5. Several molecules tracing the inner hot envelope are detected in 7m data and are presented in Fig. \\ref{fig:102}. CCS is detected for SMM3, BHR71, and IRAS 4B, while non-detected for B1-c, TMC1, and Per-emb-25. OCS and H$_2$S (Fig. \\ref{fig:102}) for all targeted in 7m Class 0 sources while non-detected for Class I sources.\n\n\n\n\nThe SO 6$_7$--5$_6$ $E_{\\rm up}=48$ K line is seen to\npeak on the central source for B1-c and S68N (Fig. \\ref{fig:so_outflow}).  SO has already been discussed in the outflow (Section \\ref{section:low-velocity_outflow}), but it is also\nprominent in the inner envelope. While the spatial resolution does not\nallow to disentangle the hot core emission from the small-scale\noutflow on a few hundred au scale, there is a difference between these\ntwo sources and SMM3 and Emb8N. The latter two sources show a\nsubstantial decrease in SO intensity towards the continuum peak, i.e.,\nthey have prominent SO emission in the outflow, but not from the hot\ncore. This suggests that sources like B1-c and S68N, which are\nbright in SO toward the continuum emission peak, have an additional\ncomponent responsible for SO emission. This is highlighted by the\nnarrower lines of SO toward the continuum peak compared with the\noutflow in S68N (Fig. \\ref{fig:so_spectra_v1}). The narrow component, visible\nin spectra taken on-source, has a width of $\\sim$ 5 km s$^{-1}$. The main component of the spectrum taken in the blueshifted outflow has a similar width but has a more prominent line wing up to 20 km s$^{-1}$.\n\n\nTMC1 clearly shows SO emission toward both components of the\nbinary system, slightly offset from their peak positions\n(Fig. \\ref{fig:so_outflow}). There is also a molecular ridge present in SO close to the disk-envelope interface.\n\nHNCO and HN$^{13}$CO are detected toward  B1-c and S68N peaking on source in higher $E_{\\rm up}$ transitions \\citep{Nazari2021}.  For lines with $E_{\\rm up} < $90 K an extended component is also detected in the outflow. SMM3 and Emb8N have no detections of HNCO on source, but for SMM3 this molecule appears in the outflow. For all Class I sources where the relatively strong HNCO 11$_{0,11}$--10$_{0,10}$ line ($A_{ij}=2\\times10^{-4}$ s$^{-1}$, $E_{\\rm up}$=70 K)  was targeted, it was not detected.\n\nOCS is detected toward SMM3 peaking in the center; S68N and SMM1 show centrally peaked O$^{13}$CS detection, a minor isotopologue signalling a high abundance of OCS (Fig. \\ref{fig:o13cs_appendix}). In all cases the emission is moderately resolved; of size $\\sim$ 200 au in case of SMM3 and detected up to 500 au away from source for S68N and SMM1.\n\nH$_2$S shows strong emission toward SMM3 and is also weakly present in TMC1. For SMM3 the emission is resolved along the outflow direction and perpendicular to the expected disk axis. Those are the only two\nsources for which H$_2$S 12m array data were taken. Additionally, the 7m data\npresented in the Appendix (Fig. \\ref{fig:102}) show prominent, centrally peaked H$_2$S\nemission for four more Class 0 sources. \n\nH$_2$CS is detected for B1-c, S68N and L1448-mm through a line with\n$E_{\\rm up}=$ 38 K. Another transition with $E_{\\rm up}=46 $ K is found\nin Class I disks: IRAS-04302, L1489, and TMC1A. In B1-c, the emission is marginally\nresolved, while in IRAS-04302 the molecule is clearly seen across the\nmidplane, indicating sublimation from icy grains at temperatures of at least 20 K\n\\citep{vantHoff2020b,Podio2020}. \n\n\nH$_2$CCO is detected for B1-c and S68N. For the lower $E_{\\rm up}=100$~K transition, the molecule is also detected in the outflow.  For Class I sources, the transition at comparable energy is not detected. HCOOH is detected for B1-c and S68N with low-energy transitions that are seen both on source and in the outflow, while the higher energy line ($E_{\\rm up}=83$~K) is seen only on source. H$^{13}$CN is detected in B1-c and L1448-mm on source, additionally to the outflow component.\n\n\n\nDisks are commonly observed in Class I sources \\citep{Harsono2014,Yen2017}, as the envelope clears out.  Fig. \\ref{fig:disks_no1_v1} presents maps of\nthe C$^{17}$O 2--1 ($E_{\\rm up}=$ 16 K) and CN 2--1 ($E_{\\rm up}=$16\nK) lines toward Class I disks and L1527-IRS, which is identified as a Class 0\/I object, observed with ALMA 12m at 0\\farcs3\nresolution. The C$^{17}$O and H$_2$CO emission for disks in Taurus\nusing these data are analyzed in detail by \\cite{vantHoff2020b}. Here\n  we discuss CN in comparison with C$^{17}$O. C$^{17}$O is observed concentrated towards the continuum emission for all disks, and is a much cleaner tracer of the disk than any other more\nabundant CO isotopologues, although even C$^{17}$O still shows some trace\nemission from the surrounding envelope. \n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm},draft=False]\n\n{plots\/fig11.jpeg}\n  \\caption{Summary cartoon presenting key molecular tracers of different components, limited to molecules presented in this work except for NH$_3$ and N$_2$H$^{+}$ which are important large scale envelope tracers. CN molecule is shown on top of the disk as it traces the disk atmosphere and not the midplane. Protostar indicated in the center in yellow.} \n \\label{fig:cartoon_summary}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{section:discussion}\n\n\nFigure \\ref{fig:cartoon_summary} summarizes current understanding about which molecule traces which component in Class 0\/I protostars. Table \\ref{table:table_components_mols}\n lists all molecules discussed in this work with indication of the physical component that they trace. Below, we will address several implications of the results. We note that the discussion is confined to the molecules targeted and detected in our datasets, but is by no means complete. Other relevant tracers not discussed here are NH$_3$ and N$_2$H$^{+}$, which are tracing the envelope well \\citep[e.g.,][]{Tobin2011, Chen2007}. PO and PN have been recently suggested as a cavity walls or outflow tracers, but have very low abundances \\citep{Bergner2019}. SO$_2$ is an important tracer of the warm gas in the disk-envelope interface \\citep{ArturV2019}. Finally, this work focuses on low-energy transitions observed with ALMA $E_{\\rm up} <$100 K with the exception for hot core tracers.\n \n\\input{table_components.tex}\n\n\n\\subsection{Cold envelope}\n\n\n\\subsubsection{C$^{18}$O}\n\nC$^{18}$O is a good tracer of high column density material because of its low critical density. However, it becomes less abundant as soon as the dust temperature drops below the CO freeze-out temperature ($\\sim$ 20--25 K). There is also a density threshold: CO freeze-out only occurs at densities above $\\sim 10^4$--$10^5$ cm$^{-3}$, because at lower densities the timescales for freeze-out are longer than the lifetime of the core \\citep{Caselli1999,Jorgensen2005a}. \n\n\nFor 4 out of 6 sources presented here at 6\\arcsec\\  resolution, \\cite{Kristensen2012} performed modeling of the SED and sub-mm spatial extent using the DUSTY code \\citep{Ivezic1997}. The results provide, among other properties, a temperature structure throughout the envelope and the radius at which the temperature drops below 10 K, which is considered as the border between envelope and the parent cloud. \\cite{Kristensen2012} obtained radii of 3800, 5000, 6700, and 9900 au for IRAS4B, TMC1, SMM3, and BHR71, respectively. The compact C$^{18}$O emission observed on-source and its\nnon-detection over the full expected extent of the protostellar\nenvelope can be explained by CO freeze-out occurring already within\nthe inner 1800 -- 2500 au radius, which is the spatial resolution\nof our observations for Class 0 sources. This upper limit on the CO\nsnowline is consistent with CO snowlines typically observed and\nmodeled toward other Class 0 protostars \\citep{Jorgensen2004d,\n  Anderl2016, Hsieh2019}. \n\nEquation 1 from \\cite{Frimann2017} allows to calculate the expected CO snow\nline for the current luminosity of the example sources B1-c and TMC1\npresented in Fig. \\ref{fig:envelope_plot_no1_v1} in the absence of an\noutburst. For B1-c the snowline is expected to be at\n200--400 au radius depending on the assumptions of the sublimation\ntemperature (larger radii for 21 K and smaller for 28 K), while the\nTMC1 CO snowline is expected to be at 100--200 au. For the most luminous \nsource with C$^{18}$O 7m observations available, the expected radius\nis at 400--750 au. Therefore clearly in all cases the expected\nCO snowline is well within the 7m beam.  In high-resolution studies, the CO emission is often\nseen at greater distances than expected from the current luminosities\nof those protostars. This is attributed to accretion bursts of\nmaterial which increase their luminosities resulting in a shift of the\nobserved CO emission radius up to a few times its expected\nvalue (but usually still within a 1000 au radius)\n\\citep{Jorgensen2015, Frimann2017,Hsieh2019}. \n\n\\subsubsection{N$_2$D$^+$ and DCO$^{+}$}\n\nBoth DCO$^{+}$ and N$_2$D$^+$ are considered cold gas\ntracers \\citep[e.g.,][]{Qi2015}. N$_2$D$^{+}$ is efficiently destroyed by CO in the gas-phase,\ntherefore freeze-out of CO results in N$_2$D$^+$ being retained in the\ngas-phase at larger radii of the envelope, where temperatures are\nlower. This behaviour has been demonstrated in several other protostellar sources by \\cite{Tobin2011,Tobin2013,Tobin2019}.\n\n\n\nBoth DCO$^+$ and N$_2$D$^+$ are produced through reactions with H$_2$D$^ +$.  At cold\ntemperatures the H$_2$D$^+$ abundance is enhanced through the\nH$_3^+$ + HD $\\rightarrow$ H$_2$D$^+$ + H$_2$ reaction, which is exothermic by\n230 K. As the reverse reaction is endothermic, low temperatures\nincrease H$_2$D$^+$. Additionally, both H$_3^+$ and\nH$_2$D$^+$ are enhanced in gas where CO has been depleted. However, the CO\nmolecule is still needed for the production of DCO$^+$ through the\nH$_2$D$^+$ + CO reaction. Therefore, DCO$^{+}$ is expected to be most\nabundant around the CO snowline \\citep{Jorgensen2004d,Mathews2013}.\nWarmer production routes through CH$_2$D$^+$ + CO are also possible\n\\citep{Wootten1987, Favre2015, Carney2018}.\n\nThe difference between DCO$^{+}$ and N$_2$D$^{+}$ chemistry is reflected in morphology of both molecules in the dense regions close to the continuum peak. As DCO$^+$  requires gas-phase CO for its formation, it peaks close to the CO snowline, which is within the resolution of our observations ($\\sim$1800--2500 au radius), while N$_2$D$^+$ is only located where CO is not present in the gas phase. Therefore we observe a significant decrease of N$_2$D$^+$ in the inner envelope. If the warm production of DCO$^+$ is triggered in the inner regions, this will additionally produce DCO$^+$ within the beam of our observations, hence DCO$^+$ does not decrease in the inner envelope. The extent of the DCO$^+$ and N$_2$D$^+$ emission in each source is comparable, ranging from $\\sim$ 5000 au in B1-c and BHR71 to 1500 au in TMC1, suggesting that their outside radii trace the region where CO becomes present again in the gas phase due to the low density.\n \nThe morphology of the emission from cold gas tracers such as DCO$^{+}$ and N$_2$D$^+$  is sensitive to the density and temperature profile of the system, which can be affected by system geometry (i.e., outflow opening angle, disk flaring angle, flattening of the envelope). DCO$^{+}$ has been shown to increase its abundance in the cold shadows of a large embedded disk \\citep{Murillo2015}.  Emission from  DCO$^{+}$ and N$_2$D$+$ is consistent with a picture of a dissipating envelope in Class I sources, resulting in less dense, warmer gas surrounding the protostar. TMC1 has an order of magnitude lower envelope mass compared to the Class 0 sources (Table \\ref{table:targets}; \\citealt{Kristensen2012}). This causes the  extent of the cold and dense region to shrink, preventing  N$_2$D$^+$ from being detected, and limiting the extent of  DCO$^+$ emission. The dense gas toward TMC1 is clearly present only in the flattened structure surrounding the binary system, likely forming a young, embedded disk. In fact this source is suggested to have a rotationally-supported circumbinary disk \\citep{Harsono2014, vantHoff2020b}. The geometry of the disk can create favourable conditions for the  DCO$^+$ enhancement in the cold shadows of the disk.\n\n\nOther relevant molecules that trace the quiescent envelope material but are not presented here are HCO$^+$ and  H$^{13}$CO$^+$ (\\citealt{Hogerheijde1997, Jorgensen2007, Hsieh2019}, van 't Hoff et al. in prep.). These molecules have been shown to probe the material outside of the water snowline  \\citep{Jorgensen2013,vantHoff2018}. As water sublimates at temperatures $\\sim 100 $ K, much higher than CO, HCO$^+$ can be seen throughout the envelope, except for the warmest inner regions. N$_2$H$^+$ is tracing the envelope material and CO snowline and has been shown to peak closer to the central protostar than N$_2$D$^+$ \\citep{Tobin2013}. Their ratio can potentially be used as an evolutionary tracer of protostars \\citep{Emprechtinger2009}. NH$_3$ is also observed to map similar components as N$_2$H$^+$ \\citep{Tobin2019}. \n\nIn summary,  the quiescent envelope material is traced by dense and\/or cold gas tracers. Chemical interactions result in N$_2$D$^+$ tracing the outer envelope where CO is frozen-out, whereas DCO$^{+}$ is seen both in the outer envelope as well as in the inner regions, tracing the unresolved CO snowline. C$^{18}$O is a good tracer of dense ($n$ $> 10^5$ cm$^{-3}$) and warm  ( $T$ $>$ 30 K) regions in the inner 2000 au radius of the protostellar systems. The evolution of the physical conditions from Class 0 to Class I is evident as the envelope becomes less dense and the protostellar luminosity can heat up dust and gas more easily. \n\n\n\\subsection{Outflows and jets}\n\nOutflowing material from protostellar systems is best analyzed with kinematic information. In the following section, we will discuss three different componfents of protostellar outflows observed with the ALMA 12m array:  1) the high-velocity jet  (\\textgreater 30 km s$^{-1}$), 2) the low-velocity outflow (\\textless 30 km s$^{-1}$), 3) the gas that results from the interaction with the outflow -- ice sputtering products at velocities close to that of ambient material, but with linewidths significantly broader (up to 15 km s$^{-1}$) than the quiescent envelope tracers. The three components are presented in Fig. \\ref{fig:outflow_showcase_no1} with examples of S68N, a representative case of a prominent outflow and ice sputtering, and L1448-mm, a source with a prototypical high-velocity jet.\n \n\\subsubsection{Extremely high-velocity (EHV) jet}\n\\label{section:extremely_high}\n\nThere are 6 out of 7 Class 0 sources targeted at high-resolution strengthen the conclusion that EHV jets are more common\nthan previously thought in Class 0 sources \\citep[][]{Podio2021}. It is argued that the molecular jet tracers have a very different physical origin than the protostellar outflows; contrary to the low-velocity outflow, which consists mostly of entrained envelope material, the EHV jet is expected to be directly launched from the innermost region of the system \\citep{Tafalla2010, Lee2020}.  The high-velocity jet is comprised of atomic material which readily forms molecules in the high-density clumps \\citep[internal working surfaces;][]{Raga1990, Santiago-Garcia2009} that are resulting from shocks in the jet, which are produced by the velocity variations of the ejection. This in turn means that by observing the high-velocity bullets, one gains insight on the variability of the accretion process \\citep{Raga1990,Stone1993}. The new EHV sources B1c and SMM3 show bullet spacings of 1200 au in B1c and 3200 au in SMM3, which can be converted using the terminal velocity of the jets (not corrected for inclination) to the dynamical ages of 80 and 250 years, respectively. If the central mass of the protostar can be estimated this can be used to provide the orbital period of the component causing the variability \\citep{Lee2020}.\n\n\nThe fact that the EHV jet tracers are dominated by O-bearing molecules has been associated with a low C\/O ratio in the jet material \\citep{Tafalla2010}. For high mass-loss rates, molecules are produced efficiently in the jet from the launched atomic material \\citep{Glassgold1991, Raga2005, Tabone2020a}. Additionally the ratio of SiO-to-CO can indicate the presence of dust in the launched material, which can in turn inform about the jet launching radius, i.e., whether it is inside or outside the dust sublimation radius \\citep{Tabone2020a}. The new detections of high-velocity jets suggest that this process may be occurring in every young Class 0 object. Studying large samples of objects with ALMA and combining with multi-transition observations can unveil the atomic abundances of the inner regions, which are difficult to measure otherwise directly \\citep{McClure2019}.  \n\nThe presence of H$_2$CO in the jet could result from gas-phase production through the reaction of CH$_3$ with O. Alternatively, if the icy grains were launched with the jet, they could be sputtered in the internal working surfaces at high velocities \\citep{Tychoniec2019}.\n\nIn summary: O-bearing species such as CO, SiO, SO, and H$_2$CO observed at high velocities are  excellent tracers of the chemistry within the protostellar jet. Those molecules most likely formed in the internal working surfaces from the material carried away from the launching region of the jet.\n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.95\\linewidth,trim={0cm 0cm 0cm 0cm}]\n  {plots\/fig12.jpeg}\n  \\caption{Maps of three different components of the outflow. Moment 0 maps are presented in color scale with continuum emission at 1.3 mm in black contours, both obtained with ALMA 12m observations. \n   {\\it Left:} An extremely high-velocity (EHV) molecular jet illustrated with the SiO (4--3) map for L1448-mm integrated from -70 to -50 and from 50 to 70 km s$^{-1}$ w.r.t $\\varv_{\\rm sys} $.\n  {\\it Middle:} Low-velocity outflow illustrated with the CO 2--1 map for S68N integrated from -15 to -3 and from 3 to 15 km s$^{-1}$ w.r.t $\\varv_{\\rm sys} $. {\\it Right:} Ice mantle content released with shock sputtering presented with the CH$_3$OH (2$_{1,0}$ -- 1$_{0,0}$) map for S68N integrated from -8 to -1 and from 1 to 8 km s$^{-1}$ w.r.t $\\varv_{\\rm sys} $. The CO outflow from Ser-emb-8N is present at the edge of the map.}\n \\label{fig:outflow_showcase_no1}\n\\end{figure*}\n\n\\subsubsection{Low-velocity outflow}\n\\label{section:low-velocity_outflow}\n\nThe CO molecule traces the bulk of the gas as it is the most abundant molecule detectable in the sub-mm regime. It  serves as an indicator of the outflow extent and gas morphology, as it is not affected by chemical processing in shocks. It can also be used to quantify the total mass-loss rates, using the dense cloud abundance ratio of CO\/H$_2$ $\\sim$ 10$^{-4}$. A well-known correlation between molecular outflows and protostellar luminosities indicates a strong link between the accretion and ejection processes \\citep{Cabrit1992, Bontemps1996, Mottram2017}. The decrease in accretion and total envelope mass with evolution of the system also results in fainter, less powerful outflows. The weak emission from the Class I source TMC1, which is an order of magnitude lower in intensity with respect to outflows from Class 0 protostars, is consistent with this trend.\n\n\n\nSiO is a molecule that is enhanced by several orders of magnitude in shocks compared with gas in cold and dense clouds where most of the Si is locked in the grains \\citep[e.g.,][]{Guilloteau1992, Dutrey1997}. Shocks release Si atoms from the grains by means of sputtering and grain destruction, leading to subsequent reactions with OH, another product of shocks, forming SiO  \\citep{Caselli1997,Schilke1997,Gusdorf2008,Gusdorf2008a}. Thus, SiO is much more prominent in high-velocity gas, where grains are more efficiently destroyed (see Section \\ref{section:extremely_high}), than in low-velocity shocks.\n\nSO is enhanced in shocks through reactions of atomic S released from the grains with OH, as well as through H$_2$S converted to SO with atomic oxygen and OH \\citep{Hartquist1980,Millar1993}. Shocks could also explain the emission toward TMC1, where weak accretion shocks onto the disk could enhance the SO abundances \\citep{Sakai2014a,Yen2014, Podio2015}.\nOverall, there is a clear decrease of both SiO and SO low-velocity\nemission from Class 0 to Class I. Either the less powerful jet cannot\ndestroy the grains and create conditions for the production of SiO and\nSO and\/or the much less dense envelope and outflow cavity walls do not\nprovide enough dust grains for creating large column densities of those\nmolecules; additionally, the excitation conditions might change significantly\n with evolution of the protostellar system, hampering the\ndetection of even low-$J$ SO and SiO transitions with critical\ndensities in the $10^5-10^6$ cm$^{-3}$ range.\n\nHCN is associated with the most energetic outflows \\citep{Jorgensen2004, Walker-Smith2014} and is enhanced at high temperatures in shocks. H$^{13}$CN is likely associated with both the hot inner regions (not detected in HCN due to optical thickness of the line) and the low-velocity outflow \\citep{Tychoniec2019, Yang2020}. The geometry of H$^{13}$CN emission seen in HH211 resembles a cavity wall: it could be a result of CN produced by UV-photodissociation and subsequent production of HCN via the H$_2$ + CN reaction, which requires high temperatures \\citep{Bruderer2009,Visser2018}. The fact that H$^{13}$CN is seen at outflow velocities shows that shocks are required to produce HCN, which is likely released at the cavity walls and then dragged with the outflow.\n\n\n\n\n\n\n\n\\subsubsection{Shock sputtering products}\n\\label{section:shock_sputtering}\n\nIce-mantle tracers are detected in outflows of SMM3, B1-c and S68N protostars. As indicated in Section \\ref{section:results_outflows} S68N is the best test-case to study the composition of shock-released ice mantles. The morphology of the emission of ice-mantle tracers in S68N is somewhat uniform. CH$_3$CN and HNCO are clearly brighter on the redshifted part of the outflow (south-east) and CH$_3$CHO are brighter in the blueshifted (north-west) side. CH$_3$OH and H$_2$CO show an even distribution between the two lobes. The peak intensity for all species occurs at significant distances from the source ($\\sim$ 5000 au) and in some cases the emission drops below the detection limit closer to the source. This is contrary to the CO emission, which can be traced all the way back to the central source. In all tracers the emission is also detected at the continuum position, however, this emission has a narrow profile and results from thermal sublimation of ices in the hot core of S68N \\citep{vanGelder2020}.\n\nThe velocities observed for CH$_3$OH and other ice-mantle tracers are high enough for these molecules to be material near the outflow cavity walls, where ice mantles could be sputtered. These molecules therefore most likely trace low-velocity entrained material with a considerable population of ice-coated grains that are sputtered in the shock \\citep{Tielens1994,Buckle2002, Arce2008, JimenezSerra2008, Burkhardt2016}. Thermal desorption of molecules from grains is not likely as dust temperatures  at distances of few times 10$^3$ au from the protostar are well below the desorption temperatures for most COMs. The fact that there is no enhancement of these tracers closer to the source also argues against emission being related to high temperature. \n\n\n\\subsection{Outflow cavity walls}\n\\label{section:outflow_cavity_walls}\n\n\nEmission from hydrocarbons and CN in the outflow cavity walls is directly related to the exposure of those regions to the UV radiation from the accreting protostar. Both c-C$_3$H$_2$ and C$_2$H have been prominently observed in PDRs such as the Horsehead Nebula and the Orion Bar tracing the layers of the cloud where UV-radiation photodissociates molecules, which helps to maintain high atomic carbon abundance in the gas-phase that is needed to build these molecules. C$_2$H is enhanced in the presence of UV radiation at cloud densities \\citep{Fuente1993, Hogerheijde1995} and c-C$_3$H$_2$ usually shows a good correlation with C$_2$H \\citep{Teyssier2004}. Both molecules have efficient formation routes involving C and C$^+$, although  models with only  PDR chemistry tend to underpredict their abundances, especially for c-C$_3$H$_2$. A  proposed additional mechanism is the top-down destruction of PAHs \\citep{Teyssier2004, Pety2005, Pety2012, Guzman2015}; the spatial coincidence of PAH emission bands with hydrocarbons in PDRs is consistent with that interpretation \\citep{vanderWiel2009}. \n\nIt is instructive to compare the conditions between classic PDRs and outflow cavity walls around low-mass protostars.The $G_0$ value for Orion Bar is estimated at 2.6 $\\times$ 10$^4$ \\citep {Marconi1998}, while the Horsehead Nebula has a much more moderate radiation field of $~ 10^2$ \\citep{Abergel2003}. The UV radiation field around low-mass protostars measured by various tracers is 10$^2$--10$^3$ at $\\sim$ 1000 au from the protostar \\citep{Benz2016, Yildiz2015, Karska2018}, therefore the PDR origin of small hydrocarbons is plausible. The top-down production of hydrocarbons due to PAHs destruction does not appear to be an efficient route here, as PAHs are not commonly observed in low-mass protostellar systems \\citep{Geers2009}, and the UV-fields required for this process are above 10$^3$ \\citep{Abergel2003}.\n\nThe difference in morphology of hydrocarbons between Class 0 and Class I systems -- outflow cavity walls in Class 0 versus rotating disk-like structure in Class I -- is most likely related to the evolution of the protostellar systems. Class 0 sources have a dense envelope and the UV radiation can only penetrate the exposed outflow cavity walls, while for Class I it is likely much easier for both the UV radiation from the accreting protostar and the interstellar radiation to reach deeper into the envelope or disk. This is consistent with emission from well-defined cavity walls seen toward those sources.\n\n\n\nThe case of moderate $\\sim 10$ km s$^{-1}$ velocity material observed\ntoward S68N indicates that the C$_2$H line does not in all sources\ntrace exclusively the quiescent cavity walls. The profile is consistent with the\nobserved morphology of the line (see Fig. \\ref{fig:hydrocarbons_no2}) -- emission is seen up to a few thousand au from the source and its shape does not resemble a\ncavity wall as clearly as in other sources. The narrow component centered at systemic velocity seen\nin Fig. \\ref{fig:spectra_coms_outflow_no1} indicates that while the broad\ncomponent might be dominating the emission, the UV-irradiated cavity\nwall also contributes to the emission observed for S68N. \n\nS68N could be a very young source, as the chaotic structure of its\noutflow and envelope indicates \\citep{LeGouellec2019}. The high\nabundances of freshly released ice-mantle components described in\nSection \\ref{section:shock_sputtering} are consistent with this interpretation. It is also\npossible that UV radiation produced locally in shocks is causing the\nenhancement of C$_2$H emission at higher velocities.\n\nThe $^{13}$CS molecule, as a high-density tracer likely traces the material piling up on the cavity walls pushed by the outflow. This emission is usually slow ($\\pm$ 2  km s$^{-1}$ ) indicating that this is not outflowing gas but rather envelope material on the outflowing cavity walls. The non-detection of $^{13}$CS in 12m data (Fig. \\ref{fig:hydrocarbons_no2}) towards the Class I source TMC1 is consistent with the dissipating envelope as the source evolves, hence no high-density material is seen in the remnant cavity walls, even though they are still highlighted by the CO emission. In the remaining Class 0 sources with the $^{13}$CS detections -- Emb8N, SMM1 and S68N (Fig. \\ref{fig:fig_E2}), -- emission also follows other cavity wall tracers. SMM1 is the only source that shows higher velocity structure (> 7 km s$^{-1}$) of the  $^{13}$CS emission on one side of the cavity wall. As this is spatially coincident with the high-velocity jet observed in CO \\citep{Hull2016}, this emission might be related to the material released with the jet from the envelope.  \n\n\nTo summarize, we observe the hydrocarbons C$_2$H and c-C$_3$H$_2$, as\nwell as CN, H$^{13}$CO${^+}$ and $^{13}$CS in the outflow cavity walls\nof Class 0 protostars. C$_2$H, as the most abundant of the hydrocarbons presented here, is also seen prominently across the envelope at\nvelocities comparable to the low-velocity outflow whereas c-C$_3$H$_2$ appears as a clean tracer of the quiescent, UV-irradiated gas in the cavity walls in the Class 0 sources.\n\n\n\\subsection{Warm inner envelope}\n\n\\subsubsection{Compact emission}\n\\label{section:compact_emission}\n\n\n\nThe inner regions of young protostellar systems are characterized by\nhigh temperatures which result in a rich chemistry as molecules that\nform efficiently in ices on grains in cold clouds sublimate into the\ngas phase.  Complex organic molecules (COMs) detected in these\ndatasets are discussed quantitatively in detail elsewhere, for both\nO-bearing by \\cite{vanGelder2020} and for N-bearing species by \\citep{Nazari2021} In this section we focus on smaller molecules that also\ntrace the innermost hot core regions and therefore are likely abundant in ices. This includes several small S-bearing molecules.\n\n\n\n\nBoth B1-c and S68N are sources characterized as hot cores\n\\citep{Bergner2017, vanGelder2020}, which means that the conditions in\ntheir inner regions are favourable for release of molecules from the\nice mantles. For the well-studied case of IRAS16293-2422,\n\\cite{Drozdovskaya2018} have also identified a hot core component of\nSO based on isotopologue data, in addition to an SO component in the large scale outflow. Overall, SO observed in protostars appears to be related mostly to evaporation of grain mantle.\n\nThe SO appearance close to the central protostar could be related to accretion shocks onto the disk, which are weaker than shocks that cause the SO emission seen in the outflows (Section 4); in accretion shocks,  SO can be released from the icy mantles with the infalling material \\citep{Sakai2014a}.  However, narrow linewidths of SO toward TMC1 seem to rule out the accretion shock scenario, and points to emission along the cavity walls \\citep{Harsono2020}.\n\nHNCO emission has been modeled by \\cite{HernandezGomez2018} and suggested to be a superposition of both warm inner regions of the envelope as well as the colder, outer envelope. Its similar behaviour to sulphur-bearing species, also observed in our work, is proposed to be related to the fact that O$_2$ and OH are involved in formation of  species like SO and HNCO \\citep{RodriguezFernandez2010}. \n\n\nH$_2$S is expected to be the dominant sulphur carrier in\nices \\citep{Taquet2020}. However, it has not yet been detected in ice absorption spectra\nto date \\citep{Boogert2015}. The weak emission from H$_2$S in dark\nclouds has been modeled as a result of the photodesorption of ices at\nthe outside of the cloud (like in the case of H$_2$O,\n\\citealt{Caselli2012b}), while chemical desorption is important for\ngrains deeper inside the cloud but outside the water snowline\n\\citep{NavarroAlmaida2020}. These models are consistent with \nH$_2$S ice containing most of the  sulphur. \nMultiple lines of H$_2$CS are a\npowerful tool to probe the warm > 100 K, innermost regions of the\nprotostellar systems \\citep{vantHoff2020a}. \n\n\n\n\nWhile B1-c and S68N are characterized as hot cores with many COM lines detected \\citep[][]{vanGelder2020}, and L1448-mm has warm water in the inner regions \\citep{Codella2010}, SMM3 does not appear to have significant emission from COMs. Moreover, simple molecules associated with the hot cores for B1-c, such as SO and H$_2$CO, are only seen outside of the SMM3 central source. While it is  possible that the optically thick continuum prevents a detection of COMs in its inner envelope \\citep{DeSimone2020}, differences in chemistry or physical structure (e.g., a large cold disk, see Section \\ref{section:results_inner_env}) between the SMM3 and the hot core sources are also possible.\nThe fact that emission from OCS and H$_2$S is centrally peaked (Fig. \\ref{fig:hotcore_v1}, bottom row) suggests that continuum optical depth is not an issue, although both those species could be a result of grain destruction or ices sputtering, therefore not necessarily coming from the midplane but rather from the surface of a disk-like structure. The additional detection of CCS in 7 data towards SMM3, which is on the other hand not detected in B1-c could hint at different chemical composition of the two protostellar systems. \n\n\n\n\n\n\n\n \n\n\\subsubsection{Embedded disks}\n\nIn the case of large edge-on\ndisks like IRAS-04302 and L1527, the vertical structure of the emission can be\nprobed, as well as the radius where CO freeze-out occurs. Overall,\nClass I disks are warmer than their Class II counterparts with CO freeze-out taking place only in the\noutermost regions \\citep{Harsono2015,vantHoff2018,vantHoff2020b,Zhang2020}.\n\n\n\nWe detect CN in all observed Class I disks, although it does not trace the midplane of the disk. In the near edge-on example of  IRAS-04302, the CN emission originates from the upper layers of the disks, in the same direction as the outflow, which is perpendicular to the disk in this source. This opens a possibility that the emission is also related to the irradiated residual cavity walls in those sources. TMC1A is a clear example where CN is tracing the same material as probed by \\cite{Bjerkeli2016} in CO, which is attributed to a disk wind. TMC1 and L1527 show CN oriented in the same direction as the disk; in TMC1 there is also a clear filament structure on larger scales irradiated by the UV from the protostar, seen also in other tracers.\n\nIn comparison with the C$^{17}$O emission, which traces the midplane disk, CN thus appears in the upper layers and in the outflow, therefore in most cases the two molecules are mutually exclusive. This picture is consistent with the bulk density traced by CO and the irradiated layers of the disk and envelope exposed to UV traced by CN. Recent observations of a sample of Class I sources including IRAS-04302 by \\cite{Garufi2020} is consistent with CN not tracing the disk midplane.\n\nIn the younger Class 0 or borderline Class 0\/I sources characterization of the disk is much more difficult because of the strong envelope emission. Nevertheless, several Keplerian disks have been identified with observations of CO\nisotopologues like C$^{18}$O and $^{13}$CO\n\\citep{Tobin2012,Murillo2013}.\nOur data allow us to investigate this for the case of SMM3. Fig. \\ref{fig:smm3_disk} shows the red and blue-shifted emission from C$^{18}$O toward SMM3. There is a clear rotational signature in the direction perpendicular to the outflow on scales of a few hundred au. However, to unambiguously identify the disk and its radius, higher spatial and spectral resolution data are necessary. \n\n\n\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=0.99\\linewidth,trim={1cm 0cm 0cm 0cm}]\n \n {plots\/fig13.jpeg}\n  \\caption{Ratio of complex organic molecules with respect to CH$_3$OH in S68N. The ratios are shown for a 3\\arcsec\\   region in both redshifted and blueshifted part of the outflow. The abundance ratios in the hot cores are taken from \\cite{vanGelder2020} and \\cite{Nazari2021}. The errorbars mark the lower and upper limit on the ratios, as calculated from the optically thick CH$_3$OH emission (lower limit) and $^{13}$CH$_3$OH (upper limit). All outflow abundances obtained from the Band 6 data at 0\\farcs45 resolution except for  CH$_3$OCHO and CH$_3$CN which were derived from Band 3 data at 3\\arcsec. } \n \\label{fig:barplot}\n\\end{figure*}\n\n\n\n\n\\subsection{COMs in outflow versus hot core}\n\nThe complex organic molecules are already detected in the prestellar stage of star formation \\citep{Bacmann2012, Scibelli2020}, where they are efficiently produced on the surfaces of icy grains \\citep[e.g.,][]{Watanabe2004,Oberg2009}. COMs can be subsequently released back into the gas through various non-thermal desorption processes and\/or be re-formed by gas-phase reactions. In the inner envelope, temperatures are high enough that thermal desorption of ices is enabled. In the outflow cavity walls, sputtering by shocks can release ices from the grains. In the hot core, COMs are detected through high excitation lines, while in the outflow, they are observed primarily through transitions with low $E_{\\rm up}$. Therefore, comparing the observations of molecular complexity in outflows with hot cores can unveil if there is any warm temperature processing of the ices and whether some molecules also have a gas-phase production route. \n\n\n\n\nTherefore, we compare abundance ratios between different ice\nmantle species and methanol at three positions for S68N: one on\nthe source, obtained from \\cite{vanGelder2020} and \\cite{Nazari2021}, and\none each in the blueshifted and redshifted part of the outflow, which are calculated from the fluxes obtained in this work.  We\nmeasure the abundance of the species in a 3\\arcsec\\   region centered on\nCH$_3$OH peak on the blueshifted and redshifted regions. The size of the\nregion is based on the spatial resolution of the Band 3 data. The regions are\nindicated in Fig. \\ref{fig:icemantle_no1}. From these regions, we extracted spectra and calculated column densities using the spectral analysis tool CASSIS  \\footnote{http:\/\/cassis.irap.omp.eu\/}.\n\nFor the CASSIS model, we assume $T_{\\rm ex}$ = 20 K, typical of subthermally excited molecules with large dipole moments in outflow gas. The FWHM of the lines was fixed at 3 km s$^{-1}$, which is the width of the CH$_3$OH line. With those parameters, an LTE calculation provides column densities for the observed line intensities under the assumption of optically thin emission (see \\cite{Tychoniec2019} for a discussion on uncertainties in this method.)\n\n\nOnly a single CH$_3$OH line   2$_{1,0}$--1$_{0,1}$ ($E_{\\rm up}$ = 28 K) is detected in the outflow of S68N. Escape probability calculations with RADEX \\citep{Tak2007} show that even at the lower densities in outflows this CH$_3$OH line is likely optically thick. No optically thin CH$_3$OH isotopologue is detected in the outflow; therefore, we provide an upper limit for the $^{13}$CH$_3$OH 2$_{1,1}$-1$_{0,1}$ ($E_{\\rm up}=$28 K). The upper limit on this $^{13}$CH$_3$OH line of $1\\times 10^{14}$ cm$^{-2}$ translates to an upper limit on the CH$_3$OH column density of $7\\times 10^{15}$ cm$^{-2}$ assuming $^{12}$C \/ $^{13}$C = 70 \\citep{Milam2005}. \n\n\nThe column density calculated from the flux measurement of the CH$_3$OH line gives  of $2\\times 10^{15}$ cm$^{-2}$; thus, the methanol abundance can be underestimated by a factor of 4. With the information available, we assume that the CH$_3$OH column density is between $2\\times 10^{15}$ cm$^{-2}$ and $7\\times 10^{15}$ cm$^{-2}$ and we compare abundances of other molecules with respect to CH$_3$OH using this range of values. The hot core methanol column density of S68N in \\cite{vanGelder2020} is corrected for optical thickness with CH$_3^{18}$OH. \n\n\n\n\n\n\nFigure \\ref{fig:barplot} compares the abundance ratios of various species with respect to methanol for S68N. First, it shows that the relative abundances are remarkably similar on both sides of the outflow, well within the uncertainties. Second, most abundances relative to CH$_3$OH are found to be comparable to the S68N hot core within our uncertainties and those of\n\\cite{vanGelder2020} and \\cite{Nazari2021}, noting that due to lack of optically thin  CH$_3$OH line detected in the outflow, our uncertainties are larger. The greatest difference is seen for CH$_3$CHO and H$_2$CCO, which could be attributed to the additional gas-phase formation in shocks. While the difference is well beyond the conservative uncertainties assumed here, this result should be confirmed with a larger sample of sources with different properties. The jump in CH$_3$CHO abundance with respect to methanol in the outflow  has been also reported for two other sources - L1157 and IRAS4A \\citep{DeSimone2020b}.\n\nFor the well-studied L1157 outflow, \\cite{Codella2020} find\nCH$_3$CHO\/CH$_3$OH is 0.5\\%. which they find in agreement with ratios\nin hot cores toward different protostars, but not including L1157\nitself. In the case of S68N we uniquely show a comparison between the\nemission from molecules in the outflows and hot core for the same\nsource.  While modeling by \\cite{Codella2020} shows that the similar\nspatial origin of CH$_3$OH and CH$_3$CHO does not imply that they are\nboth solely grain sputtering products, an agreement between hot cores\nand outflows could mean that similar processes are responsible for\nemission in both regions. \n\n\nIt is interesting that the SMM3 outflow shows ice mantle tracers in\nthe outflow but not on the source. Thus, the lack of hot core\nemission is likely due to the physical conditions in the inner regions such as a large disk (Section \\ref{section:results_inner_env}) or\ncontinuum optical depth \\citep{DeSimone2020} and not to the lack of\ncomplex molecules on the grains. Since methanol is not detected, we cannot\nprovide abundance ratios for this outflow. B1-c - another source\nwith an outflow containing ice mantle products, has methanol lines\noverlapping with the high-velocity SO outflow, therefore precise\nabundance measurements are not possible.\n\n\n{\\it JWST} will be able to provide information on the ice content due\nto rich absorption spectra in the mid-IR. The abundances of COMs in\nthe gas-phase provided by ALMA \\citep{vanGelder2020,Nazari2021} can then be directly compared with the ice\ncontent. So far, the ice content observed on cloud scales with {\\it\n  Spitzer} does not show a correlation with the gas content but those\ndata do not probe thermally desorbed ices \\citep{Perotti2020}.\n\n\n\\subsection{Carbon-chains and other hydrocarbons vs COMs}\n\nEarly observations at low spatial resolution suggested that large\ncarbon-chains such as C$_5$H, HC$_7$N, and HC$_9$N and complex organic\nmolecules are mutually exclusive, and therefore the two were proposed\nto be tracers of two different categories of protostellar sources\ndriven by different chemistry, namely warm carbon-chain chemistry\n(WCCC) and hot core chemistry \\citep{Sakai2013}. Warm carbon-chain\nchemistry is thought to occur above the sublimation temperature of CH$_4$ $\\sim$ at\n30 K.  The proposed difference then lies in the WCCC sources\ncollapsing more rapidly than the hot core sources, which prevents CO\nto accumulate on the ices, leaving a higher CH$_4$ ice content. Thus,\nin WCCC sources, there would be less CO to form more complex organic\nmolecules such as CH$_3$OH, and an underabundance of COMs is indeed\nobserved. This scenario has been supported by the lower deuteration observed in\nWCCC sources \\citep{Sakai2009a}. With ALMA observations at\nhigher resolution and sensitivity, sources harboring both COMs and\nsmall hydrocarbons, c-C$_3$H$_2$, and C$_2$H, have now been observed\n\\citep{Imai2016, Oya2017}.\n\n\n \n\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[width=0.92\\linewidth,trim={0cm 0cm 0cm 0cm}]\n  {plots\/fig14.jpeg}\n  \\caption{Emission maps of an example COM and hydrocarbon toward B1-c. {\\it Left:} CH$_3$CHO moment 0 map, {\\it Right:} C$_2$H moment 0 map; both integrated from -2 to 2 km s$^{-1}$ w.r.t  $\\varv_{\\rm sys} $.} \n \\label{fig:coms_vs_hydrocarbons}\n\\end{figure}\n\nOur data also show detection of small hydrocarbons and COMs in the\nsame sources (Fig. \\ref{fig:coms_vs_hydrocarbons}).  However, our high resolution maps now show those\nhydrocarbons to originate in the UV irradiated outflow cavity walls\n(Section 6). Both c-C$_3$H$_2$ and C$_2$H can be solely explained by\nPDR chemistry \\citep{Guzman2015, Cuadrado2015} even in moderate UV\nfields of low-mass protostars and therefore WCCC chemistry does not\nneed to be invoked for the sources presented in this work. The PDR\norigin is consistent with our observed geometry\nat the edges of the cavities, where lower extinction allows the UV\nradiation from the protostellar accretion to easily reach this\nregion. It also explains the confinement of this emission to the inner\n2000 au of the envelope as the UV radiation decreases with distance\nfrom the source. A deep search for larger hydrocarbons like HC$_5$N in\ncavity walls could help to understand their origin.\n\n\n\n\n\\subsection{Class 0 vs Class I in different components}\n\n\nMany processes take place within the protostellar\nlifetime of a few $10^5$ yr that alter the chemical composition and\nphysical conditions. First, the envelope is dissipating, and the radius\nwhere key molecules are in the gas-phase is becoming smaller\n\\citep{Jorgensen2005a}. With high-resolution studies, it is possible\nto probe different snowlines in unprecedented detail now\n\\citep{vantHoff2018,Hsieh2019}. There is a clear evolutionary trend observed in temperature -- Class 0 disks are generally warmer than Class II disks, with Class I sources in between, which has major impact on the richness of their spectra \\citep{vantHoff2020b,vantHoff2020a}. This is also observed in our sample where all Class 0 show either complex organic molecules or small molecules related to grain evaporation but no evidence of those molecules is found in Class I sources (Section \\ref{section:compact_emission}). Table \\ref{table:evolution_summary} summarizes the evolution of the molecular composition of protostellar systems.\n\nComplex organic molecules are a key tracer of chemistry at the early stages. We see them in the outflows only in Class 0 sources, likely due to the decrease of the outflow force and mass and envelope density. In the inner regions, physical conditions such as temperature and density also favour the early stages for the abundances of COMs. In rare cases of rich chemistry in Class I disks, it is usually attributed to accretion outbursts heating up the disk \\citep{vantHoff2018c,Lee2019}. \n\nMolecular EHV jets are only present in the Class 0 phase \\citep{Nisini2015}. In our sample 6 out of 7 sources have high-velocity component and none is detected in Class I (Section \\ref{section:extremely_high}), whereas the lower velocity outflows in these stages are rich in ice mantle tracers (Section \\ref{section:shock_sputtering}). The Class I jets are invisible in molecular tracers, and their outflows show CO primarily, with much less contribution from shock tracers such as SO and SiO (Section \\ref{section:low-velocity_outflow}).\n\n\nCavity walls are seen prominently in the Class 0 sources in molecules\nwhose abundances are sensitive to UV radiation. In Class I sources, a\nflared disk and weak outflow remain. While there is less envelope to\nattenuate the UV radiation, there is also less material to irradiate and accretion luminosity of older sources decrease,\nwhich results in less prominent emission from UV tracers on large\nscales in Class I (Section \\ref{section:outflow_cavity_walls}). Another example of a transition of molecular appearance between Class 0 and I is SO: this molecule traces the outflow in Class 0 and the disk-envelope interface in Class I primarily. \n\n\n\n\n\n\\begin{table*}\n\\caption{Summary of evolution of chemical tracers}            \n\\label{table:evolution_summary}     \n\\centering                         \n\\begin{tabular}{p{2cm} p{7cm} p{7cm}}       \n\\hline\\hline                \nComponent & Class 0 & Class I \\\\\n\n\\hline                                  \nEnvelope & Cold, dense envelope results in cold tracers (DCO$^+$, N$_2$H$^+$, N$_2$D$^+$) & Envelope dissipates, extent of cold tracers is much smaller \\\\\n\\hline\nJet & O-bearing molecules (CO, SiO, SO, H$_2$CO) are present in high-velocity bullets & Molecular jet disappears, seen only in atomic and ionised gas \\\\\n\\hline\nOutflow & Ice sputtering (CH$_3$OH, HNCO) and grain destruction (SO, SiO) tracers are present & Decreased outflow mass and less dense envelope result in no tracers of sputtering and grain destruction, only faint CO remains\\\\\n\\hline\nCavity walls & Prominent signs of UV-irradiated cavity walls (CN, C$_2$H, c-C$_3$H$_2$) & No prominent signs of hydrocarbons in cavity walls, CN still present \\\\\n\\hline\nHot core & COMs and simple tracers of ice sublimation and high-temperature chemistry  (H$_2$S, OCS) & Small extent plus disk shadowing results in less complexity, except for outbursting sources\\\\\n\\hline\nDisk & Warm disk with COMs, dust obscuration & Colder disk molecules, Keplerian rotation seen in H$_2$CO, C$^{18}$O, CN in the disk surface.\\\\\n\n\\hline                                  \n\n\\end{tabular}\n\\end{table*}\n\n\n\n \n\n\n\\section{Conclusions}\n\\label{section:conclusions}\n\nIn this work, we have presented an overview of the molecular tracers of the physical components in protostellar Class 0 and Class I systems. Table \\ref{table:table_components_mols} and Fig. \\ref{fig:cartoon_summary} present an overview of the results, which are summarized below. \n\n\\begin{itemize}\n\\item Protostellar envelopes are primarily traced with the dense gas tracers such as C$^{18}$O and CO, and snowline tracers such as DCO$^{+}$, N$_2$D$^+$, H$^{13}$CO$^+$. For this component, it is essential to use observations that are sensitive to large scales of $\\sim$ 5000 au.\n\n\\item Protostellar outflows are separated into different components. High-velocity molecular jets are traced by O-bearing molecules CO, SiO, SO, and H$_2$CO. Entrained outflow material can be probed by low-velocity CO, SiO, SO, H$_2$CO, HCN as well as molecules released from ices by sputtering, such as CH$_3$OH,  CH$_3$CHO, HNCO, CH$_3$CN, and CH$_3$OCHO.\n\n\n\\item Outflow cavity walls are pronounced in UV irradiation tracers: C$_2$H, c-C$_3$H$_2$, and CN, and high-density tracers such as $^{13}$CS. \n\n\\item Hot cores, the inner warm envelopes, are probed predominantly by COMs and simple molecules released from ice mantles and produced by high-temperature chemistry such as SO, H$_2$S, H$_2$CS, OCS, H$^{13}$CN, HCOOH, and HNCO.\n\n\\item Embedded disks, which are most clearly seen in Class I sources, can be seen in C$^{17}$O and H$_2$CO while CN traces their upper layers. CO isotopologues can be used to probe disks in Class 0 sources. However, the confusion with the envelope emission requires a detailed analysis of the kinematics to confirm that a Keplerian disk is present. \n\n\\end{itemize}\n\nHydrocarbons and COMs are found to  co-exist routinely in protostellar\nsources: the former are present in the UV-irradiated cavity walls and\nthe latter in the hot core and in the outflows. It is plausible that\nPDR chemistry for the formation of hydrocarbons is sufficient to\nexplain the presence of molecules such as C$_2$H and c-C$_3$H$_2$, and\nthat no warm carbon-chain chemistry is required, which relies on\nabundant CH$_4$ in ices. Observations of more complex carbon chains in\ncavity walls and envelopes are necessary to understand whether the WCCC\nplays a role there. {\\it JWST} will be able to probe the CH$_4$ ice\ncontent at  spatial scales comparable to ALMA, which will show if there are any differences in the ice\ncomposition between sources.\n\nThroughout this study, the availability of both low-lying molecular\nlines to probe the cold extended envelope and outflow, as well as\nhigh-excitation lines to trace compact hot cores has been a key for the\nanalysis. The occurrence of the same molecules in different physical\ncomponents provides an opportunity to study the processes\ninvolved in their formation and excitation. For example, ice mantle\ntracers sputtered from the grains in outflows give useful insight on\nthe ice composition, whereas comparison with abundances of thermally\nreleased COMs in the inner region is a powerful tool to probe whether\ngas-phase formation routes contribute to complex molecules formation.\n\n{\\it JWST}-MIRI will provide unprecedented resolution and sensitivity\nin mid-IR that will allow probing the emission from hot gas in shocks\nas well as ice absorption features. {\\it JWST} observations can then be combined with our\nunderstanding of the kinematics and spatial origin of molecules\nrevealed by ALMA.\n\n{\\it Acknowledgements. }The authors are grateful to the anonymous referee for comments that improved the quality of the paper.\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA\\# 2017.1.01174.S, ADS\/JAO.ALMA\\# 2017.1.01350.S, ADS\/JAO.ALMA\\# 2017.1.01371.S, ADS\/JAO.ALMA\\# 2017.1.01413.S,  ADS\/JAO.ALMA\\# 2013.1.00726.S, ADS\/JAO.ALMA\\# 2016.1.00710.S.  ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in co-operation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ. Astrochemistry in Leiden is supported by the Netherlands Research School for Astronomy (NOVA). \u0141.T.  acknowledges the ESO Fellowship Programme. M.L.vG. acknowledges support from the Dutch Research Council (NWO) with project number NWO TOP-1 614.001.751. C.L.H.H. acknowledges the support of the NAOJ Fellowship and JSPS KAKENHI grants 18K13586 and 20K14527. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThis research has made use of NASA's Astrophysics Data System Bibliographic Services. This research made use of NumPy \\citep{harris2020array}; Astroquery \\citep{2019AJ....157...98G}; Astropy, a community-developed core Python package for Astronomy \\citep{2018AJ....156..123A, 2013A&A...558A..33A}; Pandas \\citep{McKinney_2010, McKinney_2011}; and Matplotlib \\citep{Hunter2007}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe existence of intergalactic magnetic fields has been speculated for nearly half a century \nlargely motivated by the observation of micro Gauss fields in galaxies and clusters of \ngalaxies~\\citep{1992ApJ...387..528K,2001ApJ...547L.111C,2008Natur.454..302B,2010A&A...513A..30B}.\nA primordial magnetic field can be a crucial ingredient for the formation \nof stars and galaxies~\\citep{Rees:1987} and its properties can open a window to\nthe early universe in the domain of very high energy particle \ninteractions of matter~\\citep{Vachaspati:1991nm,Vachaspati:2001nb,Copi:2008he,Chu:2011tx,Long:2013tha}. \nIt is not surprising then that substantial theoretical effort has been devoted to discover ways to\ngenerate primordial magnetic fields~(for recent reviews, see \\citealt{2011PhR...505....1K, 2012SSRv..166...37W, 2013A&ARv..21...62D}), understand their implications for the present\nstate of the universe~(e.g.,~\\citealt{aharonian, kim}),\nand to observe them using an array of\ntools~(e.g., see \\citealt{2012PhR...517..141Y} and\nreferences therein).\n\nUntil relatively recently, various cosmological observables yielded an upper bound on the\nmagnetic field strength at the nano Gauss level~(e.g. \\citealt{2013arXiv1303.5076P}), and blazar observations \nhave placed lower bounds at the $\\sim 10^{-16}~{\\rm G}$ level~\\citep{Neronov:1900zz,Tavecchio:2010mk,Dolag:2010ni,Essey:2011,Chen:2014}. An important\ndevelopment in the last few years is an appreciation of the importance of magnetic\nhelicity in the generation and evolution of cosmological magnetic fields. This has led\nto the exciting possibility to use the CP odd nature of the helicity to discover intergalactic\nmagnetic fields~\\citep{2004PhRvD..69f3006C,2005PhRvD..71j3006K,\nKahniashvili:2005yp,Tashiro:2013bxa,Tashiro:2014}. \nFurther, the helicity of the intergalactic magnetic field provides \nan unmistakable handle by which one can study detailed stochastic properties. \n\nWe have implemented these ideas to observe and measure \nthe strength of the intergalactic magnetic field and its correlation functions using diffuse\ngamma ray data from the Fermi-LAT satellite\\footnote{http:\/\/fermi.gsfc.nasa.gov}~\\citep{Tashiro:2013ita}. \nOur analysis and (updated) results are reviewed in Sec.~\\ref{sec:review} where we also test and confirm \npredictions made in~\\citet{Tashiro:2013ita}. \n\nThe successful predictions bolster confidence in the intergalactic magnetic field hypothesis.\nYet there are some other tests that are also essential to rule out other more mundane\nexplanations related to the observational techniques. Foremost among these is that\nFermi-LAT observations are not performed uniformly on the sky; instead different parts\nof the sky are sampled with (slightly) different time exposures. We now take the time\nexposure map into account in our Monte Carlo simulations which we use to calculate\nstatistical error bars. This results in larger error bars but the signal found in~\\citet{Tashiro:2013ita}\npersists with high significance. Another test is that Fermi-LAT provides \ntwo data sets for the diffuse gamma ray sky, namely the CLEAN and ULTRACLEAN\ndata sets, the former being recommended for analyses of the diffuse background but the latter \nis the most conservative.\nThe analysis in~\\citet{Tashiro:2013ita} used the CLEAN data set but,\nthroughout this paper, we use ULTRACLEAN, with essentially no change in results. A third\ntest is that a cosmological signal should be present over the whole sky, in particular\nit should be  seen in both the northern and southern hemispheres. So we analyze the\nnorth and south data sets separately. The results show a much stronger signal in the\nnorth and a weaker signal in the south. At present we do not have a good explanation for the \ndifference in the strengths of the signal.\n\nIn Sec.~\\ref{sec:QRMH} we turn our attention to connecting the observed correlator of gamma\nrays described in Sec.~\\ref{sec:review} to the correlation function of magnetic fields. The analytical\nframework has already been set up in \\citet{Tashiro:2014}. We now use that framework to give a \n``proof of principles'' reconstruction of the intergalactic magnetic field helical correlator. \n\nWe conclude in Sec.~\\ref{conclusions}.\n\n\n\\section{Parity odd correlators: test of predictions}\n\\label{sec:review}\n\nConsider the location vectors, ${\\bm n}(E)$, of gamma rays of energy $E$ on the galactic \nsky. As motivated in~\\citet{Tashiro:2013ita}, we consider the triple-product correlator at energies \n$E_3 > E_2 > E_1$,\n\\begin{equation}\nQ \\left( R ; E_1,E_2,E_3 \\right)  = \\frac{1}{N_1N_2N_3} \n\t  \\sum_{i=1}^{N_1}\\sum_{j=1}^{N_2}\\sum_{k=1}^{N_3} \n                    W_R({\\bm n}_i (E_1) \\cdot {\\bm n}_k (E_3))\\, \n\t\t    W_R({\\bm n}_j (E_2) \\cdot {\\bm n}_k (E_3))\\,\n\t\t    {\\bm n}_i (E_1) \\times  {\\bm n}_j (E_2) \\cdot {\\bm n}_k (E_3),\n\\label{Qdef}\n\\end{equation}\nwhere the indices $i$, $j$, and $k$ refer to different photons at energy $E_1$, $E_2$ and $E_3$ respectively. \nThe top-hat window function $W_R$ is given by\n\\begin{numcases}\n{W_R(\\cos\\alpha) =}\n1, \\nonumber & \\qquad for $\\alpha \\leq R$\n\\\\\n0,  & \\qquad otherwise.\n\\end{numcases}\nThe statistic can also be written as\n\\begin{equation}\nQ(E_1,E_2,E_3,R)  = \\frac{1}{N_3} \\sum_{k=1}^{N_3} \n                                   {\\bm \\eta}_1\\times {\\bm \\eta}_2 \\cdot {\\bm n}_k(E_3)\n\\label{Qdef2}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bm \\eta}_a = \\frac{1}{N_a} \\sum_{i \\in D(n_k(E_3),R)} {\\bm n}_i (E_a), \\ \\ \\ a=1,2\n\\end{equation}\nand $D(n_k(E_3),R)$ is the ``disk'' or ``patch'' with center at the location of \n${\\bm n}_k(E_3)$ and with angular radius $R$. \n\nTo calculate $Q$ from data, we bin the data in the energy ranges (10,20),\n(20,30), (30,40), (40,50) and (50,60)~GeV. (The energies $E$ will refer to the lower\nend of the bin.) We also realize that the data will be \ncontaminated by gamma rays from the Milky Way and from other identified sources~\\citep{Ackermann:2014usa}. \nThese are avoided by only considering $E_3$ photons at very high galactic latitudes,\n$|b| > 80^\\circ$,\nand by excising a patch of angular radius $1.5^\\circ$ centered on sources identified in the First LAT High-Energy Catalog~\\citep{TheFermi-LAT:2013xza}.\n\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{52ultra_isotropic}\n  \\end{center}\n\\caption{$Q(R)\\times 10^6$ versus $R$ for the ULTRACLEAN data set for weeks 9-328 for\n$R \\leq 30^\\circ$. The data points are shown with standard-error error bars. The Monte Carlo %\nerror bars (magenta) are generated under the isotropic assumption. Data points that deviate \nby more than 2$\\sigma$ are colored in red.\n}\n\\label{upto30degrees}\n\\end{figure}\n\nIn contrast to the analysis in~\\citet{Tashiro:2013ita} which used the Fermi-LAT CLEAN data set, in this \npaper we will use the more conservative ULTRACLEAN data set for Fermi-LAT observation\nweeks 9-328. With the latest release of the Fermi Science Tools, {\\tt v9r33p0}, \nwe follow the Fermi team recommendations to select good quality data\nwith a zenith angle cut of $100^\\circ$.\nHowever, the results are very similar with the use of either data set. In Fig.~\\ref{upto30degrees} \nwe show plots\nof $Q(R)$ for various energy combinations $(E_1,E_2)$; $E_3$ is always taken to be\n$50~{\\rm GeV}$. In these plots we also show the spreads obtained from Monte Carlo\nsimulations assuming that diffuse gamma ray background is isotropically distributed.\nIn Sec.~\\ref{sec:exposure} we will improve on the Monte Carlo simulations by using the Fermi-LAT \ntime exposure map.\n\n\nAnother feature in Fig.~\\ref{upto30degrees} is that we have extended our analysis to $R=30^\\circ$.\nThis is to test the prediction in~\\cite{Tashiro:2014} that $|Q(R)|$ should have a\npeak at\n\\begin{equation}\nR_{\\rm peak}(E_2) \\approx R_{\\rm peak, 0}  \\left ( \\frac{E_2^{(0)}}{E_2} \\right )^{3\/2}\n\\label{Rpeak}\n\\end{equation}\nwhere $R_{\\rm peak, 0}$ is the location of the peak when $E_2 = E_2^{(0)}$.\nThe plot with $E_1=10~{\\rm GeV}$ and $E_2=40~{\\rm GeV}$ has a peak at $\\approx 14^\\circ$.\nThis implies that all plots with $E_2=40~{\\rm GeV}$ should also have peaks at\n$\\approx 14^\\circ$, a feature that is confirmed in Fig.~\\ref{upto30degrees}. Then \\eref{Rpeak} can\nbe used with $E_2=30~{\\rm GeV}$, $E_2^{(0)}=40~{\\rm GeV}$ to predict peaks at $\\approx 21^\\circ$ \nin the $(10,30)$ and $(20,30)$ plots. This feature is clearly seen in the $(10,30)$ plot in \nFig.~\\ref{upto30degrees}, but a clear peak is not present in the $(20,30)$ plot.\nThere should also be an inverted peak for $E_2=20~{\\rm GeV}$ at $R \\approx 40^\\circ$. \nHowever, as we will see in Sec.~\\ref{constructnuE},\nat such large angles, the Milky Way contribution becomes very strong and we do not see \nany indication of the cosmological signal. We also note that all the identified peaks in \nFig.~\\ref{upto30degrees} are inverted, as we would expect from magnetic fields with \nleft-handed helicity provided the spectrum is not too steep~\\citep{Tashiro:2014}.\n\nWe summarize the peak data in Table~\\ref{peaktable} as we will use it in later sections when\nwe model the magnetic field.\n\n\n\\begin{table*}\n  \\begin{tabular}{|c||c|c|c|c|c|c| } \\hline\n   $(E_1,E_2)$ & (10,20) & (10,30) &  (10,40)  & (20,30) & (20,40)  & (30,40) \\\\ \\hline\n   $(R_{\\rm peak})_{\\rm data}$ & ? & $19^\\circ$ &  $13^\\circ$ & ? &\n\t\t       $13^\\circ$  &$11^\\circ$ \\\\ \\hline\n$(Q_{\\rm peak})_{\\rm data}\\times 10^6$ &? &  $-92$ &$-242$  &?  & $-204$& $-177$ \\\\ \\hline\n  \\end{tabular}\n \\caption{\n The peak locations and amplitudes. There is\n no well-identified peak in the (10,20) and (20,30) energy combinations.\n }\n \\label{peaktable}\n\\end{table*}\n\n\n\\section{Time-Exposure and Resampling analyses}\n\\label{sec:exposure}\n\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{timeexposurewithlats}\n\\end{center}\n\\caption{Fermi-LAT time exposure maps in the five energy bins. The $|b|=60^\\circ, 80^\\circ$ galactic\nlatitudes are shown as dashed white curves. \n }\n\\label{timeexposure}\n\\end{figure}\n\nFermi-LAT observations do not cover the sky uniformly. Using the latest release\nof the Fermi Science Tools, {\\tt v9r33p0}, we construct the time exposure map \nby first creating a livetime cube with {\\tt gtltcube} and then using\n\t{\\tt gtexpcube2} to obtain full sky exposure maps corresponding to the {\\em ultraclean} response function for each \nenergy bin.\nA plot of the time exposure over weeks 9-328 is shown in \nFig.~\\ref{timeexposure}.\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{52ultra_fermiexp}\n  \\end{center}\n\\caption{The data overlaid on Monte Carlo generated error bars (magenta) by using the\ntime-exposure maps shown in Fig.~\\ref{timeexposure}.\n }\n\\label{Qwithexposure}\n\\end{figure}\n\n\nWith the time exposure maps shown in Fig.~\\ref{timeexposure} we run Monte Carlo simulations.\nThe error bars in Fig.~\\ref{Qwithexposure} are the statistical spread in the Monte Carlo $Q(R)$. In \nFig.~\\ref{Qwithexposure} we also overlay the data points on the Monte Carlo error bars. \n\nAn alternate way to account for the time exposure is to ``resample'' the data points. \nFor this we take the Fermi data and count the number of events in each energy bin. We then strip the data \nof all energy information. Then we randomly resample the observed number counts for each energy\nbin from this data set with replacement, choosing the 50~GeV photons first and ensuring that they are \nabove $80^\\circ$\nlatitude. $Q(R)$ is then calculated from this resampled data set and the whole procedure is repeated\nwith 10000 resamples. The result is shown in Fig.~\\ref{Qwithresampling} together with the observed\ndata points. As expected the Monte Carlo error bars are wider than those in Fig.~\\ref{upto30degrees} \nand are close to those of the time-exposure results in Fig.~\\ref{Qwithexposure}. A disadvantage\nin the resampling analysis compared to the time-exposure analysis is that resampling loses any \nenergy-dependent factors present in the observations. However, it is an independent check of\nthe signal.\n\nA cursory look at either Fig.~\\ref{Qwithexposure} or Fig.~\\ref{Qwithresampling} shows that the \nindividual points for $R=13^\\circ - 15^\\circ$ occurring in the $(10,40)~{\\rm GeV}$ energy panel are \nstill significant at more than $2\\sigma$; while 13 consecutive points, $R=7^\\circ-19^\\circ$, deviate at \nmore than $1\\sigma$. However the values of $Q(R)$ are correlated among various values of $R$ and \nalso among panels of different energies. In the absence of independent random variables,  the relevant \nquantity to calculate is the probability of the overall patterns of deviations in all the different energy panels.\n\nWe have performed a simple evaluation of the statistical significance of the signal by counting\nMonte Carlo runs that deviate in the (10,40)~GeV panel by more than $+1\\sigma$ at 13 consecutive\n$R$ values. This gives a probability of $\\simeq 1\\%$. We could also include data in the other \nenergy panels but have resisted doing so because of the danger that there may be a ``look elsewhere'' \neffect. \n\n\nWhile the above test gives an estimate of how likely the data is to occur in the Monte-Carlo runs, \nthe criteria relies on the data itself and so is not completely satisfactory. Alternately, as\ndescribed in the following sections, our model predicts peaks and also that the peaks are roughly in \nthe same locations when $E_2$ is the same as is the case for panels (10,40)~GeV and (20,40)~GeV.  \nAs another estimate of significance, we sample the Monte Carlo for panels (10,40)~GeV and (20,40)~GeV \nat values of $R=5^\\circ,10^\\circ,15^\\circ,20^\\circ$ and ask which of them have peaks at $10^\\circ$ or \n$15^\\circ$ in both panels. We define a peak by the maximum value being larger than the standard \ndeviation at the corresponding value of $R$.  This procedure gives a probability of $\\simeq 3\\%$.\n\nThe correlation between $Q(R)$ for different values of $R$ and also across different panels makes\nit difficult to propose a clean test for significance. We have given the result for two such tests. \nHowever, we expect the issue to become clearer when more data is available and we also plan to \ndevelop Monte Carlo methods for alternate hypotheses which will allow us to perform Bayesian inference.\n\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{mc-resampled_galcut80nummc10000}\n  \\end{center}\n\\caption{\nThe data overlaid on Monte Carlo generated error bars (magenta) by using the\nresampling method as described in the text.\n}\n\\label{Qwithresampling}\n\\end{figure}\n\n\\section{North-South analysis}\n\\label{sec:northsouth}\n\nIf the signal we see in Fig.~\\ref{upto30degrees} is cosmological, we expect  it to exist both\nin the northern and the southern hemisphere. So we have split the $E_3=50~{\\rm GeV}$\nFermi-LAT data according to whether the galactic latitude is larger than $80^\\circ$\n(northern hemisphere) or less than $-80^\\circ$ (southern hemisphere). The resulting\nvalues of $Q(R)$ are shown in Fig.~\\ref{northsouthfig}, where we also plot the full sky results\nof Fig.~\\ref{upto30degrees} for comparison. \nFor clarity, we do not show the Monte Carlo error bars, which will be larger\nthan those for full sky shown in Fig.~\\ref{upto30degrees} by a factor $\\approx \\sqrt{2}$.\n\nThe peak values of $Q(R)$ obtained in the northern polar region have larger amplitude \nthan those obtained in the southern polar region. Yet the north-south plots have similarities, \nmost striking in the (10,30) and (30,40) panels. \nThere can be several reasons for a stronger signal in one polar region than the other.\nThe existence of the signal depends on the number of TeV blazars in the polar regions. \nThe Fermi-LAT time exposure is also slightly different in the north and south, yielding fewer \nphotons in almost all bins in the south (see Table~\\ref{photonnumber}).\n\n\n\\begin{table*}\n  \\begin{tabular}{|l||c|c|c|c|c|} \\hline\n   &10-20 GeV &  20-30 GeV &   30-40 GeV &  40-50 GeV &   50-60 GeV \\\\ \\hline\n\t  North($>50^\\circ$) & 10329& 2598& 1187 & 596 & 350 \\\\\n\t  South($>50^\\circ$) & 8058 & 2002& 869 & 386 & 267\\\\ \\hline\n\t  Total ($>50^\\circ$) & 18387 &  4600 & 2056 & 982 & 617 \\\\ \\hline \\hline\n         North($>80^\\circ$) &  798 & 208& 119 & 58 & 32 \\\\\n    South($>80^\\circ$) & 500 & 124 & 37 & 28 & 23 \\\\ \\hline\n    Total ($>80^\\circ$) &  1298 &  332 & 156 & 86 & 55 \\\\ \\hline\n  \\end{tabular}\n \\caption{\nNumber of ULTRACLEAN photons for each energy bin collected in weeks 9-328 and\nthe North-South distribution for $|b| > 50^\\circ$ and $|b| > 80^\\circ$ without any source\nremoval.\n}\n \\label{photonnumber}\n\\end{table*}\n\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{North-South-Combined_80}\n  \\end{center}\n\\caption{$Q(R)$ vs. $R$ for northern (red), southern (green), and north+south (light blue) data with the\npatch center absolute galactic latitudes $> 80^\\circ$. We have\nnot shown error bars for clarity but they are comparable to $\\sqrt{2}$ times those shown in \nFig.~\\ref{upto30degrees} with the isotropic assumption and Fig.~\\ref{Qwithexposure} with time \nexposure included.\n }\n\\label{northsouthfig}\n\\end{figure}\n\n\\section{From $Q(R)$ to $M_H$}\n\\label{sec:QRMH}\n\nIn this section, we will describe how to reconstruct the intergalactic magnetic helicity power spectrum,\n$M_H$, from the gamma ray correlators, $Q(R)$. As shown in~\\citet{Tashiro:2014}, the plot of $Q(R)$ is\nexpected to have a peak if there is an intergalactic helical magnetic field. The structure arises\nbecause $Q(R)$ must vanish at small $R$ for mathematical reasons (explained below), and it must also vanish\nat large $R$ because non-cascade photons, assumed to be isotropically distributed, dilute the signal \ncoming from the cascade photons\\footnote{In the realistic case, the Milky Way starts contributing\nat very large values of $R$ and can give a non-zero signal. This is the reason why \\citet{Tashiro:2013ita}\nhad restricted attention to $R < 20^\\circ$; in Fig.~\\ref{upto30degrees} we have gone up to $R=30^\\circ$ to\ntest the prediction of a peak at $\\approx 21^\\circ$ in the (10,30) and (20,30) panels.}. Therefore,\nin between, $Q(R)$ has to have a peak.\n\nIn the following analysis, we will adopt the model for $Q(R)$ described in~\\citet{Tashiro:2014}.\nThe first step is to define the bending angle for a gamma ray of observed energy $E_\\gamma$~\\citep{Tashiro:2013bxa}\n\\begin{equation}\n \\Theta (E_\\gamma) \\approx\n  {e D_{\\rm TeV}  D_e \\over E_e D_s}  ~{v}_{L} {B}\n \\approx\n 7.3 \\times 10^{-5} ~ \\left( { B_0  \\over 10^{-16} ~{\\rm G}}\\right) \\left( {E_\\gamma\n  \\over 100 ~{\\rm GeV}}\\right)^{-3\/2} \\left( {D_s\n  \\over 1 ~{\\rm Gpc}}\\right)^{-1} \\left({1+z_s}\\right)^{-4},\n\\label{Theta}\n\\end{equation}\nwhere $e$ is the electron charge, $v_L \\sim 1$ is the speed, and $z_s$ the source\nredshift. Second we model the ratio of the number of cascade photons to the total number\nof photons in a patch of radius $R$\n\\begin{equation}\n\\frac{N_c}{N_t} = \n\\left ( 1 + \\frac{0.63~ \\nu(E) {\\cal A}(R, E)}{1- \\exp(-{\\cal A}(R,E))} \\right )^{-1},\n\\label{NcbyNt}\n\\end{equation}\nwhere ${\\cal A}(R,E) = A(R)\/A(\\Theta (E))$ and $A(R) = 2 \\pi (1-\\cos R)$ is the area of a patch\nof radius $R$. \\eref{NcbyNt} encapsulates all the information about the noise and signal photons\nin the function $\\nu(E)$ which can be viewed as the ratio of the number of non-cascade to\ncascade photons within a patch of radius $\\Theta(E)$. We will estimate this function below.\n\nThe model of~\\citet{Tashiro:2014} now gives\n\\begin{equation}\nQ(R) = \n\\left ( 1 + \\frac{0.63~ \\nu_1 {\\cal A}(R, E_1)}{1- \\exp(-{\\cal A}(R,E_1))} \\right )^{-1}\n\\left ( 1 + \\frac{0.63~ \\nu_2 {\\cal A}(R, E_2)}{1- \\exp(-{\\cal A}(R,E_2))} \\right )^{-1}\n\\frac{1}{1+\\nu_3}   (1-e^{-R\/\\Theta_1})(1-e^{-R\/\\Theta_2}) Q_\\infty ,\n\\label{QRQc}\n\\end{equation}\nwhere we are using the short-hand notation: $\\nu_a = \\nu(E_a)$, $\\Theta_a = \\Theta (E_a)$\n($a=1,2,3$).\nThe last factor, $Q_\\infty$, is related to the magnetic field correlator, \n\\begin{align}\nQ_\\infty =\n- \\frac{10^{27}}{(1+z_s)^{12} { \\rm G^2}}  \\left [\n \\frac{{q_{12}} |d_{12}| M_H (|d_{12}|)}{{\\cal E}_1^{3\/2} {\\cal E}_2^{3\/2}}\n+\\frac{{q_{23}} |d_{23}| M_H (|d_{23}|)}{{\\cal E}_2^{3\/2} {\\cal E}_3^{3\/2}}\n-\\frac{{q_{13}} |d_{13}| M_H (|d_{13}|)}{{\\cal E}_3^{3\/2} {\\cal E}_1^{3\/2}} \n\\right ] \n\\left ( \\frac{1~{\\rm Gpc}}{D_s} \\right )^2,\n\\label{Qinfty}\n\\end{align}\nwhere $q_{12}=\\pm 1$ refers to a ``charge ambiguity'' (see Sec.~\\ref{reconstructMH}), \n${\\cal E}_a = E_a \/(10~{\\rm GeV})$, and $d_{ab}$ is the distance at\nwhich $Q(R)$ probes the magnetic field\n\\begin{equation}\nd_{ab} \\approx \\left ( \\frac{\\delta_{\\rm TeV}}{E_b} \\right )^{1\/2} - \\left ( \\frac{\\delta_{\\rm TeV}}{E_a} \\right )^{1\/2}\n\\label{dab}\n\\end{equation} \nwhere $\\delta_{\\rm TeV} \\approx 5.6 \\times 10^5\/(1+z_s)^{4}~{\\rm GeV\\text{-}Mpc^2} \n\\approx 8.8 \\times 10^{40}\/(1+z_s)^{4}~{\\rm Gpc}$. \n\n\nGiven the observation energies $E_1$, $E_2$ and $E_3$, assuming a magnetic field\nstrength -- more correctly a value for the combination $B_0 (1~{\\rm Gpc}\/D_s) (1+z_s)^{-4}$\n-- we can find $\\Theta_1$ and $\\Theta_2$\nfrom \\eref{Theta}. Note that $\\nu_a$ and $Q_\\infty$ do not depend on $R$. So we can\n(in principle) find the extremum of $Q(R)$ by differentiating \\eref{QRQc}. This will tell\nus $R_{\\rm peak}(E_1,E_2,E_3)$ which we can insert into \\eref{QRQc} to get the\nratio $Q_{\\rm peak}(E_1,E_2,E_3)\/Q_\\infty$. If we use data to fix $Q_{\\rm peak}$, we can\ninfer $Q_\\infty$ for every energy combination, and that can be related to the magnetic\nhelicity power spectrum via \\eref{Qinfty}. Thus we can reconstruct $M_H$.\n\nA simplified partial analysis of this program yielded the interesting results that $R_{\\rm peak}$\nis approximately independent of $E_1$ ($E_3$ is fixed for the entire analysis). Hence the\npeak position of $Q(R)$ primarily depends on $E_2$ as given in \\eref{Rpeak}.\nFor magnetic power spectra that are not too steep, and if $\\nu (E)$ is also constant,\n$Q_{\\rm peak}$ also only depends on $E_2$~\\citep{Tashiro:2014}.\n\n\n\n\n\\subsection{Noise to signal ratio}\n\\label{constructnuE}\n\n\nThe function $\\nu (E)$ is a crucial ingredient in the reconstruction of $M_H$. It is defined as\nthe number of noise photons divided by the number of cascade photons in a patch whose\nangular radius is $\\Theta (E)$,\n\\begin{equation}\n\\nu (E) \\equiv \\frac{N_n(\\Theta(E))}{N_c(\\Theta(E))}.\n\\label{nuE}\n\\end{equation}\nThe noise photons are assumed to be isotropically distributed -- for the time being we ignore\nthe Milky Way contributions -- and so \n\\begin{equation}\nN_n \\propto A(R) = 2 \\pi (1-\\cos R)\n\\end{equation}\nwhile we expect the cascade photons to be clustered at small $R$. Hence if we plot the total\nnumber of photons in a patch of radius $R$, $N_t(R)$, we expect to see a peak at small $R$, \nand areal growth at large $R$. \n\nThis expectation is, however, confounded by two factors: first, at some large $R$, we expect \ngamma rays from the Milky Way to start out-numbering the cosmological photons, and second,\nwe have removed Fermi identified sources in our computation of $Q(R)$ to reduce the number \nof non-cascade photons. We will first discuss the Milky Way contamination and find that this\ncontribution is minimal for $R \\lesssim 20^\\circ{\\rm -}40^\\circ$ depending on the energy. The \nsource cuts remove both noise and possibly some signal (cascade photons). This fact has\nprevented us from deriving $\\nu_a$ from the source-cut data. Hence we will leave $\\nu_a$\nas free parameters  with the additional mild assumption that $\\nu_a > 1$, as suggested by\nestimates of $\\nu_a$ with the data that includes sources.\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{NtvsR}\n  \\end{center}\n  \\caption{The average number of photons with energies $10, 20, 30, 40~{\\rm GeV}$ in \nhigh-latitude ($|b|>80^\\circ$) patches of size $R$ in the full data {\\it i.e.} without source cuts. \nUp to intermediate values of $R$, the data is fit well by the functions shown in the figures and\ndrawn in red. At larger $R$ the data deviates from the fit because those large patches extend to\nlower galactic latitudes where the Milky Way starts contributing to the photon numbers.\n }\n\\label{NtvsR}\n\\end{figure}\n\nAt some large $R$, we start sampling lower galactic latitudes, and so we expect gamma rays \nfrom the Milky Way to start out-numbering the cosmological photons. We do not want to extend \nour analysis to such large $R$. To determine these values of $R$, we plot the average number\nof photons in a patch of radius $R$ versus $R$, in the data without source cuts. The results\nare shown in Fig.~\\ref{NtvsR} and matches the clustering at low $R$, a flat\npart representing constant areal density, and then a rising part which is due to the Milky\nWay contamination. Thus we see that for $E=10,~20~{\\rm GeV}$, Milky Way contamination\nis minimal for $R \\lesssim 20^\\circ$, for $E=30~{\\rm GeV}$ we have $R \\lesssim 30^\\circ$,\nand for $E=40~{\\rm GeV}$ we have $R \\lesssim 40^\\circ$.\nSince we are not interested in gamma rays from the Milky Way, we only consider the curves up \nto the point where they start  rising. Then, as in Fig.~\\ref{NtvsR}, we fit the plots to\n\\begin{equation}\n\\left [ \\frac{N_t(R)}{A(R)} \\right ]_{\\rm fit; ~all} = c_1(E) + \\frac{c_2(E)}{R^{3\/2}}.\n\\end{equation}\nwhere the subscript ``all'' means that sources are included in these fits.\nThe coefficient $c_1(E)$ is the areal density of noise photons (including sources), while $c_2(E)$\nis related to the number of cascade photons. Thus we get\n\\begin{equation}\n\\nu_{\\rm all} (E) = \\frac{c_1(E)\\Theta(E)^{3\/2}}{c_2(E)}\n\\end{equation}\nNote that $\\nu_{\\rm all}(E)$ depends on the magnetic field through $\\Theta(E)$ as given in\n\\eref{Theta}.\n\nThe scheme described above does not\nwork for $E=50~{\\rm GeV}$ photons because our patches are centered on these photons\nand there is precisely one $50~{\\rm GeV}$ photon per patch. Then to estimate $\\nu_{\\rm all}(50)$, we\nfit a power law to the values of $\\nu_{\\rm all} (E)$ determined at the lower energies with the\nresult $\\nu_{\\rm all} (E) \\propto E^{-2.6}$,\nand then extrapolate to $E=50~{\\rm GeV}$. The resulting numbers for \n$\\nu_{\\rm all} (E)$ for $B = 5.5 \\times 10^{-14}~{\\rm G}$\nare shown in Table~\\ref{nuEtable}. The values for other $B$ are easily determined by\nnoting that $\\nu_{\\rm all} (E) \\propto \\Theta^{3\/2} \\propto B^{3\/2}$ and we can also give an\nexplicit fitting function\n\\begin{equation}\n\\nu_{\\rm fit; all} (E) = 166 \\left ( \\frac{10~{\\rm GeV}}{E} \\right )^{2.5}\n                             \\left ( \\frac{B}{5.5 \\times 10^{-14}~{\\rm G}} \\right )^{3\/2}\n                             \\left [ \\left (\\frac{D_s}{1~\\rm Gpc}\\right ) (1+z_s)^4  \\right ]^{-3\/2}.\n\\label{eq:nuE_50Gev}\n\\end{equation}\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=150mm]{NtnosourcevsR}\n  \\end{center}\n\\caption{\n\tSame as in Fig.~\\ref{NtvsR}, but now with source cuts.\n}\n\\label{NtnosourcevsR}\n\\end{figure}\n\nNext we consider the effect of source cuts and plot the average number of photons\nin a patch versus patch radius in Fig.~\\ref{NtnosourcevsR}. Note that the area of a patch\nof radius $R$ is no longer given by $A(R)=2\\pi (1-\\cos R)$ since the patches have holes\nin them. For Fig.~\\ref{NtnosourcevsR}, we have used the correct average area of a\npatch of radius $R$ that we evaluated by Monte Carlo methods. However, the plots in\nFig.~\\ref{NtnosourcevsR} do not have a clear excess at small $R$ that we can identify\nwith cascade photons, nor a clear flat region that we can identify with a constant areal\ndensity of non-cascade photons. For these reasons, we are unable to extract values\nof $\\nu_a$ with confidence and will leave these as free parameters for the most part.\nBelow, when we do need to insert numerical values of $\\nu_a$, we will take \n$\\nu(E) = \\nu_{\\rm all}(E)$ as given in Table~\\ref{nuEtable}. We hope that the values\nof $\\nu_a$ will be determined more satisfactorily in the future.\n\n\n\\begin{table*}\n  \\begin{tabular}{|c||c|c|c|c|c|c| } \\hline\n   Energy bin & 10-20 &  20-30  & 30-40 & 40-50  &50-60 \\\\ \\hline\n$\\Theta (E)$ &  $73^\\circ$ & $26^\\circ$     &$14^\\circ$ &$9^\\circ$ &\n\t\t       $7^\\circ$ \\\\ \\hline\n$\\nu_{\\rm all} (E)$ &  156 & 30     & 8    &4 & 2 \\\\ \\hline\n\\end{tabular}\n \\caption{\n Values of the bending angles from \\eref{Theta} for $B = 5.5 \\times 10^{-14}~{\\rm G}$ and \n $\\nu_{\\rm all}(E)$ deduced from the fits in Fig.~\\ref{NtvsR} that counts ``all'' photons,\n including those from sources. Note that $\\nu_{\\rm all}(50)$ is \nestimated by using a power law fit to the data for $\\nu_{\\rm all}(E)$ at other energies.\n }\n \\label{nuEtable}\n\\end{table*}\n\n\n\n\\subsection{$|{\\bf B}|$ from peak position}\n\\label{Bpeakposition}\n\nTo obtain the location of the peaks of $Q(R)$, note that all the $R$ dependent factors in \\eref{QRQc} can be\nseparated out as\n\\begin{equation}\nq(R) \\equiv \\frac{(1+\\nu_3) Q(R)}{Q_\\infty} =\n\\left ( 1 + \\frac{0.63~ \\nu_1 {\\cal A}(R, E_1)}{1- \\exp(-{\\cal A}(R,E_1))} \\right )^{-1}\n\\left ( 1 + \\frac{0.63~ \\nu_2 {\\cal A}(R, E_2)}{1- \\exp(-{\\cal A}(R,E_2))} \\right )^{-1}\n(1-e^{-R\/\\Theta_1})(1-e^{-R\/\\Theta_2}).\n\\end{equation}\nNext we define $x = R\/\\Theta_1$, $\\beta = \\Theta_1 \/\\Theta_2$. We also note that\nwe are only interested in relatively small $R$ and so $A(R) \\approx \\pi R^2$. Further,\nwith the assumption $\\nu_i \\gtrsim 1$, we can simplify the first two factors around\n${\\cal A} \\approx 1$ and then\n\\begin{equation}\nq(x) \\approx \n\\left ( \\frac{0.63~ \\nu_1 x^2}{1- \\exp(-x^2)} \\right )^{-1}\n\\left ( \\frac{0.63~ \\nu_2 \\beta^2 x^2}{1- \\exp(-\\beta^2 x^2)} \\right )^{-1}\n(1-e^{-x})(1-e^{-\\beta x}).\n\\end{equation}\nThe location of the extremum of this function does not depend on the $\\nu_a$ since\nthose are just multiplicative factors. So $q(x)$ will have a peak at $x = x_{\\rm peak}(\\beta )$.\nHowever, from \\eref{Theta}, $\\beta = \\Theta_1\/\\Theta_2$ does not depend on $B$.\nSo the position of the peak in $x$ is independent of the magnetic field, and the peak position\nin $R$ depends linearly on $B$ {\\it i.e.} $R_{\\rm peak} \\propto B$. \n\nFig.~\\ref{RpeakvsB} shows the peak position in the model $Q(R)$ as a function of $B$ with the \nchoice  $\\nu_a =\\nu_{\\rm all}(E_a)$. Also, the observed  locations of the peaks in Table~\\ref{peaktable} \nare shown in Fig.~\\ref{RpeakvsB}. We find that all four peak positions line up and can be consistently \nexplained with\n\\begin{equation}\nB \\approx 5.5\\times 10^{-14} \\left [\\left (\\frac{D_s}{1~\\rm Gpc}\\right ) (1+z_s)^4  \\right ] ~{\\rm G} .\n\\end{equation}\nEven though we have adopted $\\nu_a =\\nu_{\\rm all}(E_a)$ for drawing purposes in Fig.~\\ref{RpeakvsB}, \nas argued above, the estimate of $B$ is independent of this choice provided $\\nu_a \\gtrsim 1$.\n\n\\begin{figure}\n  \\begin{center}\n \\includegraphics[width=80mm]{RpeakvsB}\n  \\end{center}\n\\caption{\nPeak position versus $B$. The observed values, with an error of $\\pm 1^\\circ$,\nare shown as horizontal bands. The observations are consistent for\n$B \\approx 5.5\\times 10^{-14}({D_s}\/{1~\\rm Gpc} ) (1+z_s)^4 ~{\\rm G}$. \nThe plots for $(E_1,E_2) = (10,20), (20,30)~{\\rm GeV}$\n(top line and blue line)\nare drawn though the data does not show any well-defined peaks for these energy\ncombinations.}\n\\label{RpeakvsB}\n\\end{figure}\n\n\n\\subsection{Reconstruction of $M_H$}\n\\label{reconstructMH}\n\nWe now take the values of $\\Theta (E)$ from Table~\\ref{nuEtable} and the \npeak locations, $(R_{\\rm peak})_{\\rm data}$, and amplitudes, $(Q_{\\rm peak})_{\\rm data}$, \nshown in Table~\\ref{peaktable}, insert them\nin \\eref{QRQc} to find $Q_\\infty$ for the energy combinations in which a definite peak is\nobserved. In addition we assume that $\\nu_a \\gtrsim 1$. Then the values of $Q_\\infty$ \nare shown in Table~\\ref{Qinftytable}. \nAdditionally, we can find the distances $d_{ab}$ for each energy combination\nfrom \\eref{dab} with the numerical values shown in Table~\\ref{dabtable}.\n\n\\begin{table*}\n  \\begin{tabular}{|c||c|c|c|c|c|c| } \\hline\n   $(E_1,E_2,E_3=50)$ [GeV] & (10,20) & (10,30) & (10,40) & (20,30)  & (20,40) & (30,40) \\\\ \\hline\n$Q_\\infty$ &  ? & -2.2 & -4.1 & ? & -0.3 & -0.1 \\\\ \\hline\n\\end{tabular}\n \\caption{\n $Q_\\infty$ for different energy combinations with the assumption $\\nu(E) = \\nu_{\\rm all}(E)$,\nrounded to one decimal place. \n}\n \\label{Qinftytable}\n\\end{table*}\n\n\\begin{table*}\n  \\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c| } \\hline\n   $(E_1,E_2)$ [GeV]& (10,20) & (10,30) & (10,40) & (10,50) & (20,30)  & (20,40) & (20,50)& (30,40)& (30,50)&(40,50) \\\\ \\hline\n$d_{ab}$ [Mpc] &  69 & 100  & 118 & 131 & 31 & 49 & 61 & 18 & 31 &12 \\\\ \\hline\n\\end{tabular}\n \\caption{$d_{ab}$ for different energy combinations.\n }\n \\label{dabtable}\n\\end{table*}\n\nThe magnetic helicity spectrum is now given by \\eref{Qinfty} which we re-write as\n\\begin{align}\n \\frac{m_{ab}}{{\\cal E}_a^{3\/2} {\\cal E}_b^{3\/2}}\n+\\frac{m_{bc}}{{\\cal E}_b^{3\/2} {\\cal E}_c^{3\/2}}\n-\\frac{m_{ac}}{{\\cal E}_c^{3\/2} {\\cal E}_a^{3\/2}} \n= -  Q_\\infty (E_a,E_b,E_c)\n\\label{mabeqns}\n\\end{align}\nwhere ${\\cal E}_a = E_a\/(10~{\\rm GeV})$, $E_a < E_b < E_c $ and\n\\begin{equation}\nm_{ab} \\equiv \\frac{ q_{ab} |d_{ab}| M_H( |d_{ab}|) }\n     {\\left [ 3\\times 10^{- 14} (1+z_s)^6  \\frac{D_s}{1~{\\rm Gpc}} \\right ]^2  {\\rm G}^2 }\n     \\label{mabdef}\n\\end{equation}\nIn our case, we have observed four peaks, and so we have four such equations,\n\\begin{eqnarray}\n\\hskip 6 cm\n \\frac{m_{13}}{3^{3\/2}}\n +\\frac{m_{35}}{15^{3\/2}}\n-\\frac{m_{15}}{5^{3\/2}} \n&=& 2.2\\\\\n\\hskip 6 cm\n \\frac{m_{14}}{4^{3\/2}}\n+\\frac{m_{45}}{20^{3\/2}}\n-\\frac{m_{15}}{5^{3\/2}} \n&=& 4.1 \\\\\n\\hskip 6 cm\n \\frac{m_{24}}{8^{3\/2}}\n+\\frac{m_{45}}{20^{3\/2}}\n-\\frac{m_{25}}{10^{3\/2}} \n&=& 0.3\\\\\n\\hskip 6.2 cm\n \\frac{m_{34}}{12^{3\/2}}\n+\\frac{m_{45}}{20^{3\/2}}\n-\\frac{m_{35}}{15^{3\/2}} \n&=& 0.1\n\\label{simultaneous}\n\\end{eqnarray}\nThese 4 simultaneous linear equations involve 8 unknowns: \n$m_{13}, ~m_{14}, ~m_{15}, ~m_{24}, ~m_{25}, ~m_{34}, ~m_{35}, ~m_{45}$. \nThus the system does not have a unique solution and additional physical input is required.\nFor the present analysis, we will consider three cases: ultra-blue spectrum in which the \nspectrum vanishes at large distances, ultra-red spectrum in which the spectrum\nvanishes at small distances, and a power law spectrum.\n\nIn the case of the ultra-red spectrum, we assume that the spectrum on the smallest length scales\nvanish: $m_{24}=0=m_{34}=m_{35}=m_{45}$ and solve for $m_{13}, ~m_{14}, ~m_{15}, ~m_{25}$.\nThis does not work, however, because the assumption is inconsistent with \\eref{simultaneous},\nas the left-hand side vanishes while the right-hand side is non-vanishing. Thus, at least with\nthese simple assumptions, the data does not fit an ultra-red spectrum.\n\nIn the ultra-blue spectrum case, we set\n$m_{ab} = 0$ for the larger values of $d_{ab}$ in Table~\\ref{dabtable}. Then \n$m_{13}=0=m_{14}=m_{15}=m_{25}$ and the only non-zero unknowns to solve for are\n$m_{24}, ~m_{34}, ~m_{35}, ~m_{45}$. The solution is\n\\begin{equation}\n(d, m)_{45} = (12,367), \\ \n(d, m)_{34} = (18,-75), \\  \n(d, m)_{35} = (31,128),  \\ \n(d, m)_{24} = (49,-93), \n \\ \\ \\ ({\\rm ultra-blue\\ assumption})\n \\label{bluesoln}\n\\end{equation}\nwhere we also show the distance scale of the correlation (in Mpc).\n\nNext we would like to go from $m_{ab}$ to $|d_{ab}|M_H(|d_{ab}|)$ by using\n\\eref{mabdef}. Here we need to resolve the discrete ``charge ambiguity'' factors $q_{ab}$.\nTo understand how these factors arise, recall that pair production by TeV gamma rays\nresults in electrons and positrons. These carry opposite electric charges and are bent\nin opposite ways in a magnetic field. The cascade gamma rays we eventually observe\ncould have arisen due to inverse Compton scattering of an electron or a positron. This\nambiguity is encapsulated in $q_{ab} = q_a q_b$ where $q_a = \\pm 1$ represents the sign of \nthe charge of the particle that resulted in the observed gamma ray of energy $E_a$.\n\nTo determine $q_{ab}$ we work with the assumption that the magnetic helicity power\nspectrum, $M_H(d)$, has the same sign over the distance scales of interest. Then\nthere are two cases to consider: $M_H > 0$ or $M_H < 0$. If we assume $M_H > 0$,\nthen with the signs of $m_{ab}$ in \\eref{bluesoln},\nwe get $q_4 q_5 =+1$, $q_3 q_4 =-1$, $q_3 q_5 =+1$ and $q_2 q_4 =-1$. \nMultiplying the first three relations gives $q_3^2 q_4^2 q_5^2 =-1$, which has no real\nsolutions. Hence there is no consistent solution with $M_H > 0$. On the other hand, if\nwe assume $M_H < 0$, we need $q_4 q_5 = -1$, $q_3 q_4 =+1$, $q_3 q_5 =-1$ and\n$q_2 q_4 =+1$. Then $q_2 = q_3 = q_4 = - q_5$ is a solution (with either choice of \n$q_2=\\pm 1$). With this solution, we find\n\\begin{equation}\n(d_{ab}, |d_{ab}| M_H(|d_{ab}|) =\n(12,-3.7), \\ \n(18,-0.7), \\  \n (31,-1.3),  \\ \n(49,-0.9), \n \\ \\ \\ ({\\rm ultra-blue\\ assumption})\n\\end{equation}\nwhere $d_{ab}$ is in Mpc and the correlator is in units of \n$[3\\times 10^{-13} (1+z_s)^6 (D_s\/1~{\\rm Gpc}) ~{\\rm G}]^2$. \nHowever, this estimate of the \nhelical power spectrum with the ultra-blue assumption is in tension with the estimate of the \nmagnetic field strength\nbased on the peak location, $\\sim 5\\times 10^{-14}~{\\rm G}$, since the magnetic helicity\ndensity is bounded by the energy density of the magnetic field via the so-called\n``realizability condition'' \\citep{Moffat:1978}. \n\n\nAnother approach is to assume that the helicity power spectrum has a power law dependence on\ndistance,\n\\begin{equation}\nr M_H(r) = A \\left ( \\frac{r}{\\rm Mpc} \\right )^p \\left [ 3\\times 10^{- 14} (1+z_s)^6  \\frac{D_s}{1~{\\rm Gpc}} \\right ]^2  {\\rm G}^2 \n\\end{equation}\nand then determine the best fit values of $A$ and $p$ by minimizing the error function\n\\begin{equation}\nE(A,p) = \\sum_{i=1}^4 \\left ( \\frac{\\rm lhs}{\\rm rhs} -1 \\right)_i^2\n\\end{equation}\nwhere $i=1,2,3,4$ labels the four simultaneous equations in \\eref{simultaneous},\n``lhs'' and ``rhs'' refer to the left-hand side and right-hand side of those equations, and we\nset $q_{ab}=+1$. Note that the construction of $E(A,p)$ gives equal weight to the four equations;\nfor example, if we define the error via $\\sum ({\\rm lhs}-{\\rm rhs})_i^2$, this would give greater\nweight to equations with larger ${\\rm rhs}$. Then we obtain the best fit values\n\\begin{equation}\nA = 2.08, \\ \\ \\ p = +0.56\n\\label{MHbestfit}\n\\end{equation}\nwhich suggests a mildly red spectrum.\n\n\nWe would like to caution the reader that the results of this section depend on the assumption \n$\\nu(E) = \\nu_{\\rm all}(E)$ as given in Table~\\ref{nuEtable} which is probably not accurate.\nThe intent of the above analysis is to show that a reconstruction of the helical power spectrum\nmay be possible once a definitive model for $Q(R)$ is established.\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\n\nWe have re-examined the finding in \\cite{Tashiro:2013ita} that the parity-odd correlator, $Q(R)$, of \nFermi-LAT observed gamma rays does not vanish. We have used the most recent data (weeks 9-328)\nand find that the signal originally found in data up to September 2013 persists and is, in fact, a\nlittle stronger (see~\\url{https:\/\/sites.physics.wustl.edu\/magneticfields\/wiki\/index.php\/Search_for_CP_violation_in_the_gamma-ray_sky} for a time sequence of results).\n We have also successfully tested the prediction \nof a peak in the (10,30) data around $R \\approx 21^\\circ$ \\citep{Tashiro:2014} and the locations\nof the peaks in the other panels as seen in Fig.~\\ref{upto30degrees}. Building on the model of\n\\cite{Tashiro:2014}, we find that all the peak locations in Fig.~\\ref{upto30degrees} can be explained\nby a single value of the magnetic field strength: $B \\sim 5.5\\times 10^{-14}~{\\rm G}$ \n(see Fig.~\\ref{RpeakvsB}). The plots of $Q(R)$ in the northern and southern hemispheres\nseparately show similar signals in some energy panels but not in others\n(Fig.~\\ref{northsouthfig}), for which we do not have an explanation.\nThe peak amplitudes have then been used to reconstruct the\nmagnetic helicity spectrum under the assumptions of either an ultra-red spectrum or an \nultra-blue spectrum. The assumption of an ultra-red spectrum does not \nyield a solution; the \nassumption of an ultra-blue spectrum yields the helicity power spectrum at four distance scales,\nhowever the field amplitude is in tension with the realizability condition. \nThe sign of the helicity depends on the\nsign of the charged particles that are responsible for the inverse Compton scattering of the\nCMB photons. We find that this sign ambiguity has a unique resolution if we assume that the\nspectrum is either everywhere positive or everywhere negative {\\it i.e.} $M_H > 0$ or\n$M_H < 0$ at all distance scales, in which case the helicity is  \nnegative (left-handed). We have also worked with the assumption\nthat the helicity spectrum is a power law to find the best fit amplitude and exponent as given in\n\\eref{MHbestfit}. The reconstruction of the power spectrum should be considered a ``proof of principle'' \nand not a definitive claim since it assumes a noise to signal ratio, $\\nu(E)$, and also certain other\nassumptions, {\\it e.g.} that the helical spectrum does not change sign over the length scales of\ninterest.\n\nWe have made several improvements on the\nanalysis to check the robustness of the results. The change from the Fermi-LAT CLEAN to\nULTRACLEAN data set makes no difference to the signal; the error bars obtained from Monte Carlo\nsimulations that include the Fermi-LAT time exposure are larger but the signal is still statistically\nsignificant at the $\\lesssim 1\\%$ level. \n\n\nAnother outcome of our analysis in Sec.~\\ref{constructnuE} is that the Milky Way starts \ncontributing to the gamma ray data set at $R \\gtrsim 20^\\circ$ for the 10~GeV bin and at yet \nlarger $R$ for higher energies (Fig.~\\ref{NtvsR}). Even though our non-trivial signals occur\nat $R \\lesssim 20^\\circ$, this raises the question if the Milky Way is somehow responsible for the \nsignal we are detecting. This seems unlikely to us for several reasons:\n(i) we observe a signal even in the (30,40)~GeV data set where contamination is seen to be minimal, \n(ii) the signal has a peak structure whereas Milky Way contamination would presumably lead to a\nmonotonically increasing signal at large $R$, and \n(iii) that the whole pattern of peak locations fits the helical magnetic field hypothesis very well. \nTo further confirm the signal, as more data is accumulated, we could restrict attention to only the \nhigher energies but reduce the bin size, {\\it e.g.} in 5~GeV bin widths instead of the current \n10~GeV bins. The higher energies would limit the Milky Way contamination and the several (smaller) \nbins would still give us the magnetic helicity spectrum over a range of distance scales.\n\n\n\\section*{acknowledgements}\nWe are grateful to a large number of colleagues who have made suggestions for further tests\nof our results in~\\citet{Tashiro:2013ita}. We would especially like to thank \nJim Buckley, Dieter Horns, \nOwen Littlejohns, Andrew Long, Guenter Sigl, David Spergel, and Neal Weiner for their remarks.\nThis work was supported by MEXT's Program for Leading Graduate Schools ``PhD professional: \nGateway to Success in Frontier Asia,'' the Japan Society for Promotion of Science (JSPS) Grant-in-Aid \nfor Scientific Research (No.~25287057) and the DOE at ASU and at WU.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{introduction}\nLet $\\chi$ be a character $\\bmod{\\,q}$ for some integer $q > 1$. Throughout the paper we will use the term character for Dirichlet character. Let $S_{\\chi}(M,N)$ be defined as follows\n\\begin{equation*}\nS_{\\chi}(M,N) = \\sum_{M<n\\le M+N} \\chi(n).\n\\end{equation*}\nHistorically, studying this sum has proven fruitful in analytic number theory to bound the least $k$-th power non-residue, to bound class numbers, to bound the least inert prime in a number field, and to bound the least primitive root, among other applications. The first non-trivial bound for this sum was proven independently by P\\'olya and Vinogradov in 1918; namely, they showed that there exists an absolute constant $c>0$ such that $|S_{\\chi}(M,N)|\\le c\\sqrt{q}\\log{q}$. The P\\'olya--Vinogradov inequality is very useful when $N$ is big compared to $\\sqrt{q}$, but not very useful otherwise (since trivially $|S_{\\chi}(M,N)| \\le N$). What we want is an inequality that gives a nontrivial result even when $N$ is small compared to $\\sqrt{q}$. The best theorem for short character sums is known as the Burgess inequality (\\cite{Bur1957},\\cite{Bur1962}, \\cite{Bur1963}, \\cite{Bur1986}) and allows us to take $N$ as small as $q^{\\frac{1}{4}-o(1)}$.\\footnote{To get $-o(1)$ instead of $+o(1)$, one needs a clever use of the large sieve as done by Hildebrand in \\cite{Hil1986}.} We state the theorem below:\n\\begin{theorema}[Burgess]\nLet $\\chi$ be a primitive character $\\bmod{\\,q}$ with $q > 1$, and let $M$ and $N$ be non-negative reals with $N\\ge 1$. Then $$|S_{\\chi}(M,N)| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2} + \\varepsilon}$$ for $r = 2,3$ and for any $r \\geq 1$ if $q$ is cubefree, the implied constant depending only on $\\varepsilon$ and $r$.\n\\end{theorema}\n\nTo prove the Burgess inequality, one of the keys is the following inequality which relies on a deep theorem of Weil \\cite{Weil}:\n\\begin{theore}\\label{main inequality Burgess}\nFor $p$ a prime number, $r$ a positive integer and $B$ a positive real number satisfying $r \\le 9B$, let $\\chi$ be a non-principal character to the modulus $p$. Then\n$$\\sum_{x \\bmod{\\,p}}\\left|\\sum_{1\\le b\\le B} \\chi(x + b)\\right|^{2r} \\le (2r-1)!! B^{r} p + (2r-1)B^{2r}\\sqrt{p}.,$$\nwhere $(2r-1)!! = (2r-1)(2r-3)\\ldots(1).$\n\\end{theore}\nThe above theorem was proven with weaker constants by Erd{\\\"o}s and Davenport in \\cite{DE1952}, and Burgess improved it to better constants and used it to get the Burgess inequality. In \\cite{Boo2006}, Booker proved it with these constants for quadratic characters. In \\cite{ETk}, the author extended it to all characters. The reliance on the Weil estimate makes it difficult to improve the Burgess inequality asymptotically.\n\nRecently, some problems have required getting explicit estimates on the Burgess inequality. Booker in \\cite{Boo2006} needed an explicit form of the inequality to compute a 32-digit discriminant. McGown in \\cite{McG3} used an explicit form of the inequality to show that there are no norm-Euclidean Galois cubic fields of discriminant greater than $10^{140}$. The goal of this paper is to improve their explicit estimates in the ranges they work in and give an explicit estimate that works regardless of the range of $N$. We apply these estimates to a question about $k$-th power non-residues $\\bmod{\\,p}$.\n\nThe work of Booker and McGown relies on the exposition of the Burgess inequality in \\cite{IK2004}. In that book, Iwaniec and Kowalski sketch the proof of the following:\n\\begin{theoremab}\\label{theorem Iwaniec}\nLet $p$ be a large enough prime. Let $\\chi$ be a non-principal character $\\bmod{\\,p}$. Let $r$ be a positive integer, and let $M$ and $N$ be non-negative integers with $N\\ge 1$. Then\n$$|S_{\\chi}(M,N)| \\le 30 N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}.$$\n\\end{theoremab}\n\nIn Section \\ref{burgess section 1} we improve Theorem B to\n\\begin{theorem} \\label{burgess kiks 1}\nLet $p$ be a prime. Let $\\chi$ be a non-principal character $\\bmod{\\,p}$. Let $M$ and $N$ be non-negative integers with $N\\ge 1$, let $2\\le r\\le 10$ be a positive integer, and let $p_0$ be a positive real number. Then for $p \\ge p_0$, there exists $c_1(r)$, a constant depending on $r$ and $p_0$ such that\n$$|S_{\\chi}(M,N)| \\le c_1(r) N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nwhere $c_1(r)$ is given by Table \\ref{super table burgess}.\n\\begin{table}[h]\n\\begin{center}\n\t\\begin{tabular}{|c| c | c| c|}\n\\hline\n\t$r$ & $p_0  = 10^7$ & $p_0= 10^{10}$ & $p_0= 10^{20}$ \\\\ \\hline\n2 & 2.7381 & 2.5173 & 2.3549 \\\\  \\hline\n\n3 & 2.0197 & 1.7385 & 1.3695 \\\\  \\hline\n\n4 & 1.7308 & 1.5151 & 1.3104 \\\\  \\hline\n\n5 & 1.6107 & 1.4572 & 1.2987 \\\\  \\hline\n\n6 & 1.5482 & 1.4274 & 1.2901 \\\\  \\hline\n\n7 & 1.5052 & 1.4042 & 1.2813 \\\\  \\hline\n\n8 & 1.4703 & 1.3846 & 1.2729 \\\\  \\hline\n\n9 & 1.4411 & 1.3662 & 1.2641 \\\\  \\hline\n\n10 & 1.4160 & 1.3495 & 1.2562 \\\\  \\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Values for the constant $c_1(r)$ in the Burgess inequality.}\\label{super table burgess}\n\\end{table}\n\\end{theorem}\n\nIn the spirit of Theorem B, where we have no restriction on $r$, we also prove the following corollary:\n\\begin{corollary}\\label{burgess corollary 1}\nLet $p$ be a prime such that $p \\ge 10^7$. Let $\\chi$ be a non-principal character $\\bmod{\\,p}$. Let $r$ be a positive integer, and let $M$ and $N$ be non-negative integers with $N\\ge 1$. Then\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}.$$\n\\end{corollary}\n\nRestricting $N$ to be less than $4 p^{\\frac{1}{2} + \\frac{1}{4r}}$, McGown in \\cite{McG3} proved an explicit version of Burgess with worse constants but with a better exponent in $\\log{p}$. Indeed, he proved:\n\\begin{theoremac}\\label{McGown's theorem burgess}\nLet $p\\ge 2\\cdot 10^{4}$ be a prime number. Let $M$ and $N$ be non-negative integers with $1\\le N\\le 4 p^{\\frac{1}{2} + \\frac{1}{4r}}$. Suppose $\\chi$ is a non-principal character $\\bmod{\\,p}$. Then there exists a computable constant $C(r)$ such that\n\\begin{equation*}\n|S_{\\chi}(M,N)| < C(r) N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}},\n\\end{equation*}\nwhere $C(r)$ is given by Table \\ref{table mcgown burgess}.\n\\begin{table}[h]\n\\begin{center}\n\t\\begin{tabular}{|c| c | c| c|}\n\\hline\n\t$r$ & $C(r)$ & $r$ & $C(r)$ \\\\ \\hline\n\t2 & 10.0366 & 9 & 2.1467 \\\\ \\hline\n\t3 & 4.9539 & 10 & 2.0492 \\\\ \\hline\n\t4 & 3.6493 & 11 & 1.9712 \\\\ \\hline\n\t5 & 3.0356 & 12 & 1.9073 \\\\ \\hline\n\t6 & 2.6765 & 13 & 1.8540 \\\\ \\hline\n\t7 & 2.4400 & 14 & 1.8088 \\\\ \\hline\n\t8 & 2.2721 & 15 & 1.7700\\\\ \\hline\n\t\n\t\\end{tabular}\n\\end{center}\n\\caption{Values for the constant $C(r)$ in the Burgess inequality.}\\label{table mcgown burgess}\n\\end{table}\n\\end{theoremac}\n\nThe restriction that $N \\le 4p^{\\frac{1}{2} + \\frac{1}{4r}}$ is used to get the exponent $\\frac{1}{2r}$ in the $\\log{p}$ term of the inequality. In Section \\ref{burgess section 2}, we improve McGown's Theorem to have better constants in a similar range.\n\\begin{theorem}\\label{burgess kiks 2}\nLet $p$ be a prime. Let $\\chi$ be a non-principal character $\\bmod{\\,p}$. Let $M$ and $N$ be non-negative integers with $1\\le N\\le 2 p^{\\frac{1}{2} + \\frac{1}{4r}}$, let $r\\le 10$ be a positive integer, and let $p_0$ be a positive real number. Then for $p \\ge p_0$, there exists $c_2(r)$, a constant depending on $r$ and $p_0$ such that\n\\begin{equation*}\n|S_{\\chi}(M,N)| < c_2(r) N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}},\n\\end{equation*}\nwhere $c_2(r)$ is given by Table \\ref{table kiks burgess}.\n\n\\begin{table}\n\\begin{center}\n\t\\begin{tabular}{|c| c | c| c|}\n\\hline\n\t$r$ & $p_0 = 10^{10}$ & $p_0 = 10^{15}$ & $p_0 = 10^{20}$ \\\\ \\hline\n2 & 3.6529 & 3.5851 & 3.5751 \\\\  \\hline\n\n3 & 2.5888 & 2.5144 & 2.4945 \\\\  \\hline\n\n4 & 2.1914 & 2.1258 & 2.1078 \\\\  \\hline\n\n5 & 1.9841 & 1.9231 & 1.9043 \\\\  \\hline\n\n6 & 1.8508 & 1.7959 & 1.7757 \\\\  \\hline\n\n7 & 1.7586 & 1.7066 & 1.6854 \\\\  \\hline\n\n8 & 1.6869 & 1.6384 & 1.6187 \\\\  \\hline\n\n9 & 1.6283 & 1.5857 & 1.5654 \\\\  \\hline\n\n10 & 1.5794 & 1.5410 & 1.5216 \\\\  \\hline\n\\end{tabular}\n\\end{center}\\caption{Values for the constant $c_2(r)$ in the Burgess inequality.}\\label{table kiks burgess}\n\\end{table}\n\n\\end{theorem}\nUsing an idea from \\cite{MV2007}, we can get rid of the restriction on $N$ for $r\\ge 3$.\n\n\\begin{corollary}\\label{burgess kiks corollary 2}\nLet $p\\ge 10^{10}$ be a prime number. Let $M$ and $N$ be non-negative integers with $N\\ge 1$. Suppose $\\chi$ is a non-principal character $\\bmod{\\,p}$ and that $p \\ge p_0$ for some positive real $p_0$. Then for $r\\ge 3$, there exists a computable constant $c_2(r)$ depending on $r$ and $p_0$, such that\n\\begin{equation*}\n|S_{\\chi}(M,N)| < c_2(r) N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}},\n\\end{equation*}\nwhere $c_2(r)$ is the same as that of Table \\ref{table kiks burgess} whenever $r\\ge 3$.\n\\end{corollary}\n\nPutting an extra restriction on $N$ (namely, $N \\le 2\\sqrt{p})$, Booker in \\cite{Boo2006} gave better bounds in the special case of quadratic characters.\n\\begin{comment}\n\\begin{theorema} \\label{Booker's thm}\nLet $p > 10^{20}$ be a prime number with $p \\equiv 1 \\pmod{4}$. Let $r\\in \\{2,3,4,\\ldots,15\\}$. Let $M$ and $N$ be real numbers such that $0 < M , N \\le 2\\sqrt{p}$. Let $\\chi$ be a non-principal quadratic character $\\bmod{\\,p}$.\nThen\n$$|S_{\\chi}(M,N)| \\le \\alpha(r) N^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}\\left(\\log{p} + \\beta(r)\\right)^{\\frac{1}{2r}},$$\nwhere $\\alpha(r)$ and $\\beta(r)$ are given by Table \\ref{table booker}.\n\\begin{table}\n\\begin{center}\n\t\\begin{tabular}{c c c | c c c}\n\t$r$ & $\\alpha(r)$ & $\\beta(r)$ & $r$ & $\\alpha(r)$ & $\\beta(r)$ \\\\ \\hline\n\t2 & 1.8221 & 8.9077 & 9 & 1.4548 & 0.0085 \\\\\n\t3 & 1.8000 & 5.3948 & 10 & 1.4231 & -0.4106\\\\\n\t4 & 1.7263 & 3.6658 & 11 & 1.3958 & -0.7848\\\\\n\t5 & 1.6526 & 2.5405 & 12 & 1.3721 & -1.1232\\\\\n\t6 & 1.5892 & 1.7059 & 13 & 1.3512 & -1.4323\\\\\n\t7 & 1.5363 & 1.0405 & 14 & 1.3328 & -1.7169\\\\\n\t8 & 1.4921 & 0.4856 & 15 & 1.3164 & -1.9808\n\t\n\t\\end{tabular}\n\\end{center}\n\\caption{Explicit constants on the Burgess inequality for quadratic characters.}\\label{table booker}\n\\end{table}\n\\end{theorema}\n\\end{comment}\n\n\\begin{remark}\nUsing Theorem A one could extend Booker's theorem to all orders of $\\chi$ (with slightly worse constants). The reason we would get slightly worse constants is that in the quadratic case, the inequality in Theorem A can be improved slightly to $(2r-1)!! B^{r} p + (2r-2)B^{2r}\\sqrt{p}$. Every other part of Booker's proof extends naturally, but this part of the inequality fails when looking at higher orders.\n\\end{remark}\n\nIn Section \\ref{section p^1\/6}, we apply these estimates to a question about $k$-th power non-residues $\\bmod{\\,p}$. Indeed, let $p$ be a prime and let $k$ be an integer with $k \\mid p-1$ and $k > 1$. Let $g(p,k)$ be the least $k$-th power non-residue $\\bmod{\\,p}$. The case $k = 2$, i.e., the question of how big the least quadratic non-residue is, has been studied extensively. A probabilistic heuristic using that a prime $q$ is a quadratic non-residue $\\bmod{\\,p}$ half of the time, suggests that $g(p,2) \\ll \\log{p}\\log{\\log{p}}$ and that $g(p,2) \\gg \\log{p}\\log{\\log{p}}$ for infinitely many $p$. Assuming the Generalized Riemann Hypothesis for Dirichlet L-functions (GRH), Ankeny \\cite{Ank1952} showed that $g(p,k) \\ll (\\log{p})^2$ and Bach \\cite{Bach1990} made this explicit by proving (under GRH) that $g(p,k) \\le 2(\\log{p})^2.$ The best unconditional results (for $g(p,k)$) are due to Burgess \\cite{Bur1957}, who, building on work by Vinogradov \\cite{Vinogradov}, showed that $$g(p,k) \\ll_{\\varepsilon} p^{\\frac{1}{4\\sqrt{e}}+\\varepsilon}.$$\n\nFor $k \\ge 3$, better estimates which depend upon $k$ have been proven by Wang Yuan(\\cite{Yuan}) building on work of Vinogradov (\\cite{Vinogradov}) and Buh{\\v{s}}tab (\\cite{Bu1949}).\n\nAll of the work described so far has been of asymptotic nature. In terms of getting explicit bounds, Karl Norton \\cite{Nor1971}, building on a technique of Burgess \\cite{Bur19632}, was able to show that $g(p,k) \\leq 3.9 p^{1\/4}\\log p$ unless $k =2$ and $p\\equiv 3 \\pmod 4$ for which he showed  $g(p,k) \\leq 4.7 p^{1\/4}\\log p$. In \\cite{ETk}, the author improved Norton's bounds to $0.9 p^{1\/4}\\log{p}$ and $1.1 p^{1\/4}\\log{p}$, respectively. These bounds are far from the asymptotic bound of $p^{\\frac{1}{4\\sqrt{e}}+\\varepsilon}$. In this paper, as an application of the explicit Burgess inequality, we find an upper bound on how large $p$ has to be to ensure that there is a $k$-th power non-residue less than $p^{1\/6}$.\n\n\\begin{theorem}\\label{theorem p^1.6}\nLet $p$ be a prime number and $k >1$ be a positive divisor of $p-1$. Then for $p \\ge 10^{4732}$, the least $k$-th power non-residue $\\bmod{\\,p}$ is less than or equal to $p^{1\/6}$.\n\\end{theorem}\n\n\\begin{remark}\nThe techniques involved in the proof can be used to answer this question for $p^{\\alpha}$ whenever $\\alpha > \\frac{1}{4\\sqrt{e}}$.\n\\end{remark}\n\n\\section{Preliminary lemmas}\nLet $A$ and $N$ be positive integers. Let $v(x)$ be the number of representations of $x$ as $\\bar{a}n \\pmod{p}$, where $\\bar{a}$ is the inverse of $a \\pmod{p}$,  $1\\le a\\le A$ and $M < n \\le M+N$, that is,\n\\begin{equation}\\label{burgess v(x)}\nv(x) = \\# \\left\\{(a,n)\\in\\mathbb{N}^2\\mid 1\\le a\\le A, \\mbox{ } M < n\\le M+N \\mbox{ and } n\\equiv ax \\bmod{\\,p}\\right\\}.\n\\end{equation}\n\nThe main lemma in this section is the following:\n\\begin{lemma}\\label{burgess V2 lemma}\nLet $p$ be a prime and let $N < p$ be a positive integer. Let $A\\ge 28$ be an integer satisfying $A< \\frac{N}{12}$, then\n\\begin{equation}\\label{burgess V2 claim}\nV_2 = \\sum_{x\\kern-3pt\\mod{p}}v^2(x) \\leq 2AN\\left(\\frac{AN}{p} + \\log(1.85 A)\\right).\n\\end{equation}\n\\end{lemma}\n\nTo prove the lemma regarding $V_2$ we will need a couple of estimates involving the $\\phi$ function (Lemmas \\ref{burgess claim 2} and \\ref{burgess claim 1}), an estimate on a sum of logarithms (Lemma \\ref{burgess claim 3}) and a non-trivial combinatorial count (Lemma \\ref{burgess counting lemma}).\n\n\\begin{lemma}\\label{burgess claim 2}\nFor $x\\ge 1$ a real number we have:\n\\begin{equation}\\label{burgess claim 2 eq}\n\\sum_{n\\leq x}\\frac{\\phi(n)}{n} \\leq \\frac{6}{\\pi^2} x + \\log{x} + 1.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} For $1\\le x < 2$, the left hand side of \\eqref{burgess claim 2 eq} is $1$, while the right hand side is at least $1$. We can manually check that for all integers $x$ satisfying $2\\le x \\le 41$ we have\n$$\\sum_{n\\le x} \\frac{\\phi(n)}{n} \\le \\frac{6}{\\pi^2}(x-1) + \\log{(x-1)} + 1,$$\nimplying that \\eqref{burgess claim 2 eq} is true for $x< 41$. Therefore, we may assume that $x \\ge 41$.\n\nLet's work with the sum:\n\\begin{equation}\\label{kakaro}\n\\sum_{n\\leq x}\\frac{\\phi(n)}{n} = \\sum_{n \\leq x}\\frac{1}{n}\\sum_{d | n}\\mu(d)\\frac{n}{d} = \\sum_{d\\leq x}\\sum_{n\\leq \\frac{x}{d}}\\frac{\\mu(d)}{d} = \\sum_{d\\leq x}\\left\\lfloor\\frac{x}{d}\\right\\rfloor\\frac{\\mu(d)}{d}.\n\\end{equation}\n\nFrom \\cite[Theorem $422$]{Hardy} it follows that for $x \\ge 1$\n\\begin{equation}\\label{Hardylin}\n\\sum_{d\\le x}\\frac{1}{d} < \\log{x} + \\gamma + \\frac{1}{x}.\n\\end{equation}\n\nUsing \\eqref{Hardylin} in \\eqref{kakaro} yields\n\\begin{equation}\\label{almost there 1}\n\\sum_{n\\le x}\\frac{\\phi(n)}{n} \\le x\\sum_{d\\ge 1} \\frac{\\mu(d)}{d^2} -x\\sum_{d>x}\\frac{\\mu(d)}{d^2}+ \\sum_{d\\leq x} \\frac{1}{d} \\leq\\frac{6}{\\pi^2}x + \\log{x} + \\gamma+\\frac{1}{x} - x\\sum_{d>x}\\frac{\\mu(d)}{d^2}.\n\\end{equation}\n\nMoser and Macleod \\cite{MM1966} gave a simple proof that for $x\\ge 2$ we have\n\\begin{equation}\\label{moser}\n\\left|\\sum_{d>x}\\frac{\\mu(d)}{d^2}\\right|\\le \\frac{1}{3x} + \\frac{8}{3x^2}.\n\\end{equation}\n\nCombining \\eqref{moser} with \\eqref{almost there 1} yields for $x\\ge 41$ that\n$$\\sum_{n\\le x}\\frac{\\phi(n)}{n} \\le \\frac{6}{\\pi^2}x + \\log{x} + \\gamma + \\frac{1}{x}+\\frac{1}{3} + \\frac{8}{3x} \\le \\frac{6}{\\pi^2}x + \\log{x} + 1.$$\n\\end{proof}\n\n\\begin{lemma}\\label{burgess claim 1}\nFor $x\\ge 1$ a real number we have:\n\\begin{equation}\\label{burgess claim 1 eq}\n\\sum_{n \\leq x} n\\phi(n) \\leq \\frac{2}{\\pi^2}x^3 + \\frac{1}{2}x^2\\log{x} + x^2.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor $1\\le x<2$, the left hand side of \\eqref{burgess claim 1 eq} is $1$, while the right hand side is at least $x^2 \\ge 1$. Therefore it is true for $1\\le x<2$. Now for $2\\le x<3$, the left hand side is $3$, while the right hand side is at least $x^2 \\ge 4$. Therefore \\eqref{burgess claim 1 eq} is true for $1\\le x < 3$. In the rest of the proof we will assume that $x\\ge 3$.\n Let's work with the sum:\n\\begin{multline*}\n\\sum_{n\\leq x} \\phi(n)n = \\sum_{n \\leq x}\\sum_{d | n} \\frac{\\mu(d)n^2}{d} = \\sum_{d \\leq x} \\mu(d)d \\sum_{dm \\leq x}m^2\\\\ = \\sum_{d\\leq x}\\frac{\\mu(d)d}{6}\\left\\lfloor\\frac{x}{d}\\right\\rfloor\\left(\\left\\lfloor\\frac{x}{d}\\right\\rfloor + 1\\right)\\left(2\\left\\lfloor\\frac{x}{d}\\right\\rfloor + 1\\right).\n\\end{multline*}\n\nNow, let $\\theta_d =\\frac{x}{d}-\\left\\lfloor\\frac{x}{d}\\right\\rfloor.$ Then we have  \\begin{multline}\\label{choco2}\n\\sum_{n\\le x} \\phi(n) n = \\frac{x^3}{3}\\sum_{d\\leq x} \\frac{\\mu(d)}{d^2}  + \\frac{x^2}{6}\\sum_{d\\le x}\\frac{(3-6\\theta_d )\\mu(d)}{d} \\\\+ \\frac{x}{6}\\sum_{d\\le x}\\left(6\\theta_d^2 - 6\\theta_d + 1\\right)\\mu(d) - \\frac{1}{6}\\sum_{d\\le x} \\theta_d(1-\\theta_d )(1-2\\theta_d )\\mu(d)d .\n\\end{multline}\n\nFrom \\eqref{Hardylin} it follows that for $x\\ge 3$\n\\begin{equation}\\label{hardy no mames}\n\\sum_{d\\le x} \\frac{1}{d} < \\log{x} + \\gamma + \\frac{1}{x} < \\log{x} + 1-\\frac{1}{60}-\\frac{1}{60x}.\n\\end{equation}\n\nUsing that $0\\le\\theta_d \\leq 1$ we have that $|3-6\\theta_d|\\le 3$, that $|6\\theta_d^2 - 6\\theta_d+1| \\le 1$ and $|(1-\\theta_d)(1-2\\theta_d)(-\\theta_d)| \\le \\frac{1}{10}$. Therefore, using \\eqref{choco2}, \\eqref{hardy no mames}, that $\\displaystyle\\sum_{d\\ge 1}\\frac{\\mu(d)}{d^2} = \\frac{6}{\\pi^2}$, and that $|\\mu(d)| \\leq 1$,  we get\n\\begin{multline}\\label{burgess lemma 2 eq}\n\\sum_{n\\le x} \\phi(n) n \\leq \\frac{x^3}{3}\\sum_{d\\leq x} \\frac{\\mu(d)}{d^2}  + \\frac{x^2}{2}\\sum_{d\\leq x}\\frac{1}{d} + \\frac{x}{6}\\sum_{d\\leq x} 1 +  \\frac{1}{60}\\sum_{d \\leq x} d \\\\\n\\leq \\frac{2}{\\pi^2}x^3 - \\frac{x^3}{3}\\sum_{d>x}\\frac{\\mu(d)}{d^2} + \\frac{1}{2}x^2\\log{x} + \\frac{x^2}{2} -\\frac{x^2}{120} -\\frac{x}{120}+\\frac{x^2}{6}+ \\frac{1}{60}\\left(\\frac{x(x+1)}{2}\\right).\n\\end{multline}\n\nFrom \\eqref{moser} we have (for $x \\ge 2$)\n\\begin{equation*}\\label{burgess key to lemma 2}\n\\sum_{d> x}\\frac{\\mu(d)}{d^2} \\ge \\frac{1}{3x} - \\frac{8}{3x^2} \\ge -\\frac{1}{x}.\n\\end{equation*}\nCombining this with \\eqref{burgess lemma 2 eq} yields the lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{burgess claim 3}\n For $x \\geq 1$ we have:\n$$\\sum_{d\\leq x} \\log{\\left(\\frac{x}{d}\\right)} \\leq x -1$$\n\\end{lemma}\n\\begin{proof} For $1\\leq x <2$ we have $\\displaystyle\\sum_{d\\leq x} \\log{\\left(\\frac{x}{d}\\right)} = \\log{x} \\leq x -1$. Therefore, we may assume $x \\geq 2$. Now,\n\\begin{equation}\\label{logsum}\n\\sum_{d\\leq x} \\log{\\left(\\frac{x}{d}\\right)} = \\left\\lfloor x\\right\\rfloor \\log{x} - \\sum_{d\\leq x} \\log{d} \\leq \\left\\lfloor x\\right\\rfloor \\log{x} - \\left\\lfloor x\\right\\rfloor \\log{\\lfloor x\\rfloor}+\\lfloor x\\rfloor -1.\n\\end{equation}\nTo get the second inequality we used that $$\\sum_{d\\leq x} \\log{d} = \\sum_{2\\le d\\le x}\\log{d} \\geq \\int_{1}^{\\lfloor x\\rfloor}\\log{t}\\,dt = \\lfloor x \\rfloor\\log{\\lfloor x\\rfloor} -\\lfloor x\\rfloor + 1.$$\n\nNow, notice that $x = \\lfloor x \\rfloor + \\{x\\}$ and $\\log{(1+y)}\\leq y$. Therefore we have\n\\begin{equation}\\label{logito}\n\\lfloor x\\rfloor \\log{x} = \\lfloor x\\rfloor\\log{\\lfloor x\\rfloor} + \\lfloor x\\rfloor\\log{(x\/\\lfloor x\\rfloor)}\\leq \\lfloor x\\rfloor\\log{\\lfloor x\\rfloor} + \\{x\\}.\n\\end{equation}\n\nCombining equations (\\ref{logsum}) and (\\ref{logito}) yields\n\n\\begin{equation*}\n\\sum_{d\\leq x} \\log{\\left(\\frac{x}{d}\\right)}\\leq \\{x\\} + \\lfloor x\\rfloor - 1 = x-1.\n\\end{equation*}\n\\end{proof}\n\n\\begin{lemma}\\label{burgess counting lemma}\nLet $A \\ge 2$, $N \\ge 1$, $a_1$, $a_2$ and $M$ be integers. Let $p > N$ be a prime number. Suppose, $1\\le a_1,a_2 \\le A$ with $a_1 \\neq a_2$. Then the number of pairs of integers $(n_1,n_2)$ satisfying $M < n_1,n_2 \\le N+M$ and $a_1 n_2 - a_2 n_1 = kp$ is bounded above by\n$$N \\frac{\\gcd{(a_1,a_2)}}{\\max\\{a_1,a_2\\}} + 1.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $d = \\gcd{(a_1,a_2)}$. Since $a_1n_2 -a_2n_1 = kp$, we have that $d | k$. Let $a_1=a_1^{'}d, a_2 = a_2^{'}d$ and $k = k^{'} d$. Now, we also have\n\\begin{equation}\\label{goku}\nn_2 = \\frac{k p+a_2n_1}{a_1} = \\frac{k^{'} p + a_2^{'}n_1}{a_1^{'}}.\n\\end{equation}\nThe right hand side of \\eqref{goku} must be an integer. Therefore $k^{'}p + a_2^{'}n_1 \\equiv 0 \\bmod{a_1^{'}}$. Since this is a linear equation in terms of $n_1$, there is at most one solution $\\bmod{\\,a_1^{'}}$. Therefore, in the interval $(M, M+N]$ there are at most $\\displaystyle\\frac{N}{a_1^{'}} + 1$ choices of $n_1$. Since $n_2$ is uniquely determined from $n_1$, the number of pairs $(n_1, n_2)$ satisfying the conditions of the lemma is bounded by\n$$\\frac{N}{a_1^{'}} + 1 = N\\frac{\\gcd{(a_1,a_2)}}{a_1} + 1.$$\nAnalogously, the number of pairs is bounded by $N\\displaystyle\\frac{\\gcd{(a_1,a_2)}}{a_2} + 1$. The statement of the lemma is now an easy consequence.\n\\end{proof}\n\nNow we are ready to prove Lemma \\ref{burgess V2 lemma}.\n\n\\begin{proof}[Proof of Lemma \\ref{burgess V2 lemma}]\nWe'll begin by noting that $V_2$ is the number of quadruples $(a_1,a_2,n_1,n_2)$ with $1\\leq a_1,a_2 \\leq A$ and $M < n_1,n_2 \\leq M + N$ such that $a_1n_2 \\equiv a_2n_1\\pmod p$. If $a_1 = a_2$, since $N < p$, we have that $n_1 = n_2$ because $n_1 \\equiv n_2 \\pmod p$ while $|n_1 - n_2| \\leq N < p$. Therefore, the number of quadruples in this case is $AN$. Fix $a_1$ and $a_2$ with $a_1 \\neq a_2$. Let $k$ be an integer satisfying\n\\begin{equation}\\label{burgess the equation}\na_1n_2 - a_2n_1 = kp,\n\\end{equation}\nfor some $n_1$ and $n_2$. We can put a bound on possible values for $k$. First of all, $k$ must be a multiple of $\\gcd{(a_1,a_2)}$. Now, if we write $n_1 = n_1^{'} + M$ and $n_2 = n_2^{'} + M$, we have, using $kp - (a_1-a_2)M = a_1n_2^{'} - a_2n_1^{'}$, that\n$$-a_2 N \\le - a_2n_1^{'} < kp - (a_1-a_2)M < a_1n_2^{'} \\le a_1 N  .$$\nTherefore $k$ lies in an interval of length at most $\\frac{(a_1 + a_2) N}{p}$. Since $k$ is a multiple of $\\gcd{(a_1,a_2)}$ and $k$ lies in such an interval, then there are at most\n$$\\frac{(a_1 + a_2) N}{\\gcd{(a_1,a_2)} p} + 1,$$\nchoices for $k$.\n\nGiven $a_1,a_2$ and $k$ we can count the number of pairs $(n_1,n_2)$ which would satisfy (\\ref{burgess the equation}). By Lemma \\ref{burgess counting lemma}, the number of pairs is bounded by $N \\frac{\\gcd{(a_1,a_2)}}{\\max\\{a_1,a_2\\}} + 1$. Therefore we get\n\\begin{multline}\\label{bubu}\nV_2 \\leq AN + 2\\sum_{a_1 < a_2} \\Big( \\frac{(a_1 + a_2) N}{\\gcd{(a_1,a_2)}p} + 1   \\Big) \\Big( \\frac{\\gcd{(a_1,a_2)} N}{\\max\\{a_1,a_2\\}} + 1  \\Big) \\\\ = AN + \\frac{2N^2}{p} S_1 + \\frac{2N}{p} S_2  + 2N S_3 + A^2 - A,\n\\end{multline}\nwhere\n\\begin{equation*}\\label{S_1}\nS_1 = \\sum_{a_1 < a_2} \\frac{a_1 + a_2}{\\max\\{a_1,a_2\\}} \\mbox{ , }\n\\end{equation*}\n\\begin{equation*}\\label{S_2}\nS_2 = \\sum_{a_1 < a_2} \\frac{a_1 + a_2}{\\gcd{(a_1,a_2)}} \\mbox{ , }\n\\end{equation*}\nand\n\\begin{equation}\\label{S_3}\nS_3 = \\sum_{a_1 < a_2} \\frac{\\gcd{(a_1,a_2)}}{\\max\\{a_1,a_2\\}}.\n\\end{equation}\n\nDealing with $S_1$ is straightforward, in fact $S_1$ is\n\\begin{equation}\\label{burgess S1}\n\\sum_{a_2 \\leq A}\\sum_{a_1 < a_2} \\frac{a_1 + a_2}{a_2} = \\sum_{a_2 \\leq A} \\left(a_2 -1 + \\frac{a_2(a_2-1)}{2a_2}\\right) = \\frac{3}{2}\\frac{A(A-1)}{2} = \\frac{3}{4}A^2- \\frac{3}{4}A.\n\\end{equation}\n\nNow, let's estimate $S_2$:\n\\begin{multline*}\nS_2 = \\sum_{a_1< a_2 \\leq A} \\frac{a_1+a_2}{\\gcd{(a_1,a_2)}} = \\sum_{d \\leq A}\\sum_{b_2 \\leq \\frac{A}{d}}\\sum_{b_1 < b_2, (b_1,b_2) = 1} \\left(b_1 + b_2\\right)\\\\ = \\sum_{d\\leq A}\\sum_{2\\leq b_2 \\leq \\frac{A}{d}} \\left(\\frac{\\phi(b_2)}{2}b_2+ \\phi(b_2)b_2 \\right) = \\frac{3}{2}\\sum_{d\\leq A}\\sum_{2\\leq b_2 \\leq \\frac{A}{d}} \\phi(b_2)b_2.\n\\end{multline*}\nUsing Lemma \\ref{burgess claim 1}, we get\n\\begin{equation*}\nS_2 \\leq \\frac{3}{\\pi^2}\\sum_{d\\leq A}\\big(\\frac{A}{d}\\big)^3 + \\frac{3}{4}\\sum_{d\\leq A} \\big(\\frac{A}{d}\\big)^2\\log{(\\frac{A}{d})} + \\frac{3}{2}\\sum_{d\\leq A} \\big(\\frac{A}{d}\\big)^2.\n\\end{equation*}\nUsing that $\\log{(\\frac{A}{d})} = \\log{A}-\\log{d}$,  and that $\\displaystyle\\sum_{d\\ge 1} \\frac{1}{d^s} = \\zeta(s)$, yields\n\\begin{equation*}\nS_2 \\le \\frac{3\\zeta(3)}{\\pi^2}A^3 + \\frac{3\\zeta(2)}{4}A^2\\log{A} - \\frac{3}{4}A^2\\sum_{d\\le A}\\frac{\\log{d}}{d^2} + \\frac{3}{2}A^2\\zeta(2).\n\\end{equation*}\nUsing that for $A\\ge 11$ we have $\\frac{3\\zeta(2)}{2} - \\frac{3}{4}\\sum_{d\\le A}\\frac{\\log{d}}{d^2} < 2$ yields\n\\begin{equation}\\label{burgess S2}\nS_2 \\leq \\frac{3\\zeta(3)}{\\pi^2}A^3 + \\frac{3\\zeta(2)}{4}A^2\\log{(A)} + 2 A^2.\n\\end{equation}\n\nLet's estimate $S_3$. We have\n$$S_3 = \\sum_{a_1< a_2 \\leq A} \\frac{\\gcd{(a_1,a_2)}}{\\max(a_1,a_2)} = \\sum_{d \\leq A}\\sum_{b_2 \\leq \\frac{A}{d}}\\sum_{b_1 < b_2, (b_1,b_2) = 1} \\frac{1}{b_2} = \\sum_{d \\leq A} \\sum_{2\\leq b_2 \\leq \\frac{A}{d}} \\frac{\\phi(b_2)}{b_2}.$$\nUsing Lemma \\ref{burgess claim 2} yields\n$$S_3 \\leq \\sum_{d\\leq A} \\left(\\frac{A}{d}\\frac{1}{\\zeta(2)} + \\log{(\\frac{A}{d})}\\right) = \\frac{6}{\\pi^2} A\\sum_{d\\le A}\\frac{1}{d} + \\sum_{d\\le A} \\log{(\\frac{A}{d})}. $$\n\nFrom \\eqref{Hardylin} it follows that for $A\\ge 27$\n\\begin{equation*}\n\\sum_{d\\le A} \\frac{1}{d} < \\log{A} + \\gamma + \\frac{1}{A} < \\log{(1.85 A)}.\n\\end{equation*}\nUsing this and Lemma \\ref{burgess claim 3} yields\n\\begin{equation}\\label{burgess S3}\nS_3 \\le \\frac{6}{\\pi^2} A\\log{(1.85A)} + A -1.\n\\end{equation}\n\n Using (\\ref{burgess S1}), (\\ref{burgess S2}) and (\\ref{burgess S3}) in (\\ref{bubu}) yields the following upper bound for $V_2$:\n\\begin{equation}\\label{*}\n2AN\\Big(\\frac{3}{2} + \\frac{A-1}{2N} + \\frac{3AN}{4p} - \\frac{3N}{4p} + \\frac{3\\zeta(3)A^2}{\\pi^2 p} + \\frac{3\\zeta(2)A\\log{A}}{4p} + \\frac{6}{\\pi^2}\\log{(1.85A)} -\\frac{1}{A} + \\frac{2A}{p}\\Big)\n\\end{equation}\n\nFor $A\\ge 4$, we have\n\\begin{equation}\\label{burgess *1}\n\\frac{3\\zeta(3)A^2}{\\pi^2 p} + \\frac{3\\zeta(2)A\\log{A}}{4p} < \\frac{3}{4}\\frac{A^2}{p}.\n\\end{equation}\nSince $N > 3A$ we have the following two inequalities:\n\\begin{equation}\\label{1}\\frac{AN}{4p} > \\frac{3}{4}\\frac{A^2}{p} \\mbox{ and } \\frac{3N}{4p} > \\frac{2A}{p}.\n\\end{equation}\n\nCombining \\eqref{burgess *1} and \\eqref{1} yields\n\\begin{equation}\\label{burgess *2}\n\\frac{3AN}{4p} +\\left(\\frac{3\\zeta(3)A^2}{\\pi^2 p} + \\frac{3\\zeta(2) A\\log{A}}{4p}\\right)+\\left(\\frac{2A}{p} - \\frac{3N}{4p}\\right) < \\frac{AN}{p}.\n\\end{equation}\nFinally, using that $A\\ge 28$ and that $N > 12A$, yields\n\\begin{equation}\\label{3}\\Big(1-\\frac{6}{\\pi^2}\\Big)\\log{(1.85 A)} \\geq \\Big(1-\\frac{6}{\\pi^2}\\Big)\\log{(51.8)} \\geq 1.54766 > \\frac{3}{2} + \\frac{1}{24}  \\geq \\frac{3}{2} + \\frac{A}{2N} .\\end{equation}\n\nCombining \\eqref{burgess *2} and \\eqref{3} in \\eqref{*} yields \\eqref{burgess V2 claim}.\n\\end{proof}\n\n\\begin{remark} The main term will come from the $\\log{(1.85A)}$ term and the $1.85$ can be changed to a smaller number (the limit being $e^{\\gamma}$), forcing $A$ to be slightly larger to make the inequalities work. Also, the coefficient on $\\log{(1.85A)}$ can be changed to be as close to $\\frac{6}{\\pi^2}$ as we want as long as $A$ is big enough. It is important to note that big $A$'s will mean forcing $p$ to be much bigger in the estimates for the Burgess inequality.\n\\end{remark}\n\n\\begin{remark}\nThe constraint $A\\ge 28$ is used to get the main term to be $\\log{(1.85 A)}$; however, we can relax the condition on $A$ and get a slightly worse main term. We chose our values this way to get the constants in Tables \\ref{super table burgess} and \\ref{table kiks burgess} as low as possible for small values of $r$. Relaxing the $A\\ge 28$ condition would make these constants worse, but improve the constants for larger values of $r$. Since the small values of $r$ seem to be the most useful in applications, we decided to focus on minimizing these cases.\n\\end{remark}\n\n\n\n\\section{Explicit Burgess inequality}\\label{burgess section 1}\n\\begin{proof}[Proof of Theorem \\ref{burgess kiks 1}]\nLet $M$ and $N\\ge 1$ be non-negative integers. Let $r \\ge 2$ be a positive integer. Fix a constant $c_1(r) \\ge 1$ (which we will name later). We will prove the Theorem by induction. Assume that for all positive integers $h< N$, we have\n$$|S_{\\chi}(M,h)|\\le c_1(r) h^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}.$$\nThe idea is to estimate $S_{\\chi}(M,N)$ by shifting by $h$ ($n \\mapsto n + h$) and getting an error that we can deal with by induction.\n\nNote that, for all $h < N$,\n\\begin{equation*}\nS_{\\chi}(M,N) = \\sum_{M < n \\leq M+N} \\chi(n + h) + \\sum_{M<n\\leq M+h} \\chi(n) - \\sum_{M+N < n \\leq M+N + h} \\chi(n).\n\\end{equation*}\nTherefore\n\\begin{equation*}\nS_{\\chi}(M,N)  = \\sum_{M < n \\leq N+M} \\chi(n + h)+ 2\\theta E(h),\n\\end{equation*}\nwhere $|\\theta|\\le 1$ which depends upon $h$, and $E(h) = \\displaystyle\\max_{K}|S_{\\chi}(K,h)|$.\n\nLet $A$ and $B$ be positive reals and let $H = \\lfloor A\\rfloor\\lfloor B\\rfloor$. We will use shifts of length $h = ab$ where $a$ and $b$ are positive integers satisfying $a \\leq A$ and $b \\leq B$. After averaging over all the pairs $(a,b)$ we get\n\\begin{equation}\\label{bur eq 1}\nS_{\\chi}(M,N) = \\frac{1}{H}\\sum_{a,b} \\sum_{M < n \\leq M + N} \\chi(n + ab) + \\frac{1}{H}\\sum_{a,b}2\\theta E(ab).\n\\end{equation}\nLet $v(x)$ be defined as in (\\ref{burgess v(x)}), then\n\\begin{equation}\\label{bur eq 2}\n\\left|\\sum_{a,b}\\sum_{M < n \\leq M + N} \\chi(n + ab)\\right| \\le \\sum_{x\\kern-3pt\\mod{p}} v(x)\\left|\\sum_{b\\le B} \\chi(x+b)\\right|.\n\\end{equation}\nLet $$V:= \\sum_{x\\kern-3pt\\mod{p}} v(x)\\left|\\sum_{b\\le B}\\chi(x+b)\\right|.$$\nThen, combining (\\ref{bur eq 1}) with (\\ref{bur eq 2}), we get\n\\begin{equation}\\label{burgess ineq V\/H + 2\/H}\n|S_{\\chi}(M,N)| \\leq \\frac{V}{H} + \\frac{2}{H}\\sum_{a,b} E(ab).\n\\end{equation}\n\nWe can now focus on estimating $V$. Now define $V_1 := \\displaystyle \\sum_{x \\pmod p} v(x)$,\\\\ $V_2 := \\displaystyle \\sum_{x \\pmod p} v^2(x)$ and $W := \\displaystyle \\sum_{x \\pmod p} \\Big|\\displaystyle \\sum_{1 \\leq b \\leq B} \\chi(x + b)\\Big|^{2r}$. Using H\\\"older's Inequality we get\n\\begin{equation}\\label{burgess V}\nV \\leq V_1^{1- \\frac{1}{r}}V_2^{\\frac{1}{2r}}W^{\\frac{1}{2r}}.\n\\end{equation}\n\nFirst note that\n\\begin{equation*}\\label{burgess V1}\nV_1 = \\lfloor A\\rfloor N\\le AN.\n\\end{equation*}\nFrom Lemma \\ref{burgess V2 lemma}, for $\\lfloor A\\rfloor\\ge 28$ and $\\lfloor A\\rfloor<\\frac{N}{12}$, we have\n\\begin{equation}\\label{burgess V2}\nV_2  \\leq 2AN\\left(\\frac{AN}{p} + \\log(1.85 A)\\right).\n\\end{equation}\nWe can also bound $W$, since by Theorem A, we have (for $r\\le 9B$):\n\\begin{equation}\\label{burgess W}\nW \\le \\frac{(2r)!}{2^r r!}B^r p + (2r-1)B^{2r}\\sqrt{p} = (2r-1)!! B^r p + (2r-1) B^{2r}\\sqrt{p}.\n\\end{equation}\n\nLet's head back to proving the Burgess bound. We will let $AB = kN$ for $k$ a real number to be chosen later. Using the inequalities of $V_1,V_2$ and $W$ together with \\eqref{burgess V} yields the following bound upper bound for $\\frac{V}{H}$:\n\\begin{multline}\\label{Opti}\n\\frac{V}{H} \\le \\frac{1}{\\lfloor A\\rfloor \\lfloor B\\rfloor}V_1^{1-\\frac{1}{r}}V_2^{\\frac{1}{2r}}W^{\\frac{1}{2r}} \\leq \\frac{\\frac{AB}{\\lfloor A\\rfloor\\lfloor B\\rfloor} }{(AB)^{\\frac{1}{2r}}}\\cdot\\frac{(2WB)^{\\frac{1}{2r}}}{B}\\left(\\frac{AN}{p} + \\log{(1.85A)}\\right)^{\\frac{1}{2r}}N^{1-\\frac{1}{2r}}\n\\\\\\le \\frac{A}{A-1}\\cdot\\frac{B}{B-1}\\cdot\\frac{1}{k^{\\frac{1}{2r}}}\\cdot\\frac{(2WB)^{\\frac{1}{2r}}}{B}\\left(\\frac{AN}{p} + \\log{(1.85A)}\\right)^{\\frac{1}{2r}}N^{1-\\frac{1}{r}}.\n\\end{multline}\n\n\nBecause of \\eqref{Opti} we can see that a good choice for $B$ is the one that minimizes $\\frac{WB}{B^{2r}}$. Using \\eqref{burgess W}, we seek to minimize the expression  $(2r-1)!!\\frac{p}{B^{r-1}} + (2r-1)Bp^{\\frac{1}{2}}$. We take the derivative with respect to $B$ and equal it to zero. After this process we get that a good $B$ is\n\\begin{equation}\\label{burgess B}\nB = \\big((2r-3)!!(r-1)\\big)^{\\frac{1}{r}}p^{\\frac{1}{2r}}.\n\\end{equation}\n\nUsing this value of $B$ we get\n\\begin{equation}\\label{WB}\n\\frac{(2WB)^{\\frac{1}{2r}}}{B} \\leq \\left(\\frac{2r(2r-1)}{r-1}\\right)^{\\frac{1}{2r}}(r-1)^{\\frac{1}{2r^2}}\\big((2r-3)!!\\big)^{\\frac{1}{2r^2}}p^{\\frac{r+1}{4r^2}}.\n\\end{equation}\n\n\nNow we must try to bound $\\frac{AN}{p} + \\log{(1.85A)}$. To do this, we can use the P\\'olya--Vinogradov inequality to give an upper bound for $N$, since for $N$ large, the P\\'olya--Vinogradov inequality would be a better bound than the Burgess inequality. Indeed, if\n\\begin{equation}\\label{range}\nN \\geq p^{\\frac{1}{2} + \\frac{1}{4r}}\\log{p},\n\\end{equation}\nthen, since $c_1(r)\\ge 1$, we have $$c_1(r)N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}} \\ge \\sqrt{p}\\log{p}.$$\nTherefore, from the P\\'olya--Vinogradov inequality (see Section $9.4$ in \\cite{MV2007}) we can conclude that $|S_{\\chi}(M,N)|\\le c_1(r)N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}\\left(\\log{p}\\right)^{\\frac{1}{r}},$ whenever we have \\eqref{range}.\n\nIf we have $r\\ge 3$, then we can use the Burgess inequality with $r-1$ instead of the P\\'olya--Vinogradov inequality, to get a better upper bound on $N$. Indeed, if we let $s$ be a real number that satisfies\n\\begin{equation}\\label{s condition}\nc_1(r-1) \\le s^{\\frac{1}{r(r-1)}} c_1(r),\n\\end{equation}\nthen if\n$$N\\ge s\\, p^{\\frac{1}{4} + \\frac{1}{2r} + \\frac{1}{4r(r-1)}}\\log{p},$$\nthen\n$$c_1(r)N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}\\ge c_1(r-1)N^{1-\\frac{1}{r-1}}p^{\\frac{r}{4(r-1)^2}}(\\log{p})^{\\frac{1}{r-1}}.$$\n\nSimilarly, we can put a lower bound on $N$, by noting that $|S_{\\chi}(M,N)| \\le N$. Indeed,\n$$c_1(r)N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}} \\ge N,$$\nwhenever\n\\begin{equation*}\\label{range low}\nN\\le c_1(r)^r p^{\\frac{1}{4} + \\frac{1}{4r}}\\log{p}.\n\\end{equation*}\n\nTherefore, we may assume that\n\\begin{equation}\\label{range for N}\nc_1(2)^2 p^{\\frac{3}{8} }\\log{p} < N < p^{\\frac{5}{8}}\\log{p},\n\\end{equation}\nfor $r=2$, and that\n\\begin{equation}\\label{range2 for N}\nc_1(r)^r p^{\\frac{1}{4} + \\frac{1}{4r}}\\log{p} < N < s\\, p^{\\frac{1}{4} + \\frac{1}{2r} + \\frac{1}{4r(r-1)}}\\log{p},\n\\end{equation}\nfor $r\\ge 3$.\n\nUsing that $A = \\frac{kN}{B}$, the upper bound for $N$ in \\eqref{range for N}, and \\eqref{burgess B}, we get \\begin{equation}\\label{log}\n\\frac{AN}{p} = \\frac{kN^2}{pB} \\le \\frac{k p^{\\frac{5}{4}}\\log^2{p}}{p B} \\le k\\log^2{p} ,\n\\end{equation}\nfor $r=2$, and for $r\\ge 3$, we get\n\\begin{equation}\\label{log2}\n\\frac{AN}{p} = \\frac{kN^2}{pB} \\le \\frac{s^2 k p^{\\frac{1}{2} +\\frac{1}{r}+\\frac{1}{2r(r-1)}}\\log^2{p}}{p B} = \\frac{s^2 k}{\\left((2r-3)!!(r-1)\\right)^{\\frac{1}{r}}p^{\\frac{1}{2}-\\frac{1}{2r}-\\frac{1}{2r(r-1)}}}\\log^2{p} ,\n\\end{equation}\n\nNow we consider what happens to $\\log{(1.85A)}$.\n\\begin{equation}\\label{la otra vez}\n \\log{(1.85A)} = \\log{\\left(\\frac{1.85kN}{B}\\right)} \\leq \\log{\\left(1.85k\\log{p}\\right)}+\\frac{3\\log{p}}{8},\n \\end{equation}\nfor $r=2$, and for $r\\ge 3$, we get\n\\begin{equation}\\label{la otra vez2}\n \\log{(1.85A)} = \\log{\\left(\\frac{1.85kN}{B}\\right)} \\leq \\log{\\left(\\frac{1.85 s\\,k\\log{p}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}+\\frac{\\log p}{4} + \\frac{\\log{p}}{4r(r-1)} .\n \\end{equation}\n\n\nNow, let's bound the error term, the part we have labeled as $E(h)$.\n\nFor any $a,b$ such that $ab = h < N$, we have by induction hypothesis $E(h) \\leq c_1(r)(ab)^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log p)^{\\frac{1}{r}}$. Therefore,\n\\begin{multline}\\label{error}\n\\frac{1}{c_1(r)p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}}\\cdot\\frac{2}{H}\\sum_{a,b} E(ab) \\le\\frac{2}{\\lfloor A\\rfloor \\lfloor B\\rfloor}\\sum_{1\\leq a\\leq A}\\sum_{1 \\leq b \\leq B} (ab)^{1-\\frac{1}{r}}\\\\\n\\le 2\\frac{1}{AB}\\left(\\int_1^{A+1} t^{1-\\frac{1}{r}}\\,dt\\right)\\left(\\int_1^{B+1} t^{1-\\frac{1}{r}}\\,dt\\right)\\frac{AB}{(A-1)(B-1)}\\\\\\le\n(AB)^{1-\\frac{1}{r}}\\frac{2}{(2-\\frac{1}{r})^2} \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}\n\\frac{AB}{(A-1)(B-1)}\n\\\\ = \\frac{2r^2}{(2r-1)^2}k^{1-\\frac{1}{r}} N^{1-\\frac{1}{r}}\\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}\n\\frac{AB}{(A-1)(B-1)}.\n \\end{multline}\n\n\nCombining equations \\eqref{Opti}, \\eqref{WB}, \\eqref{log}, \\eqref{la otra vez} and \\eqref{error} with \\eqref{burgess ineq V\/H + 2\/H} yields (for $r=2$)\n\\begin{multline}\\label{burgess super messy}\n\\frac{|S_{\\chi}(M,N)|}{N^{\\frac{1}{2}}p^{\\frac{3}{16}}(\\log{p})^{\\frac{1}{2}}}\\leq \\frac{AB}{(A-1)(B-1)}\\left(12\\right)^{\\frac{1}{4}}\\left(1 + \\frac{3}{8k\\log{p}}  + \\frac{\\log{\\left(1.85k\\log{p}\\right)}}{k \\log^2{p}}\\right)^{\\frac{1}{4}} \\\\\n+ \\frac{8}{9}k^{\\frac{1}{2}}c_1(2) \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{\\frac{3}{2}}\\frac{AB}{(A-1)(B-1)}. \\end{multline}\n\nSimilarly, for $r\\ge 3$, combining equations \\eqref{Opti}, \\eqref{WB}, \\eqref{log2}, \\eqref{la otra vez2} and \\eqref{error} with \\eqref{burgess ineq V\/H + 2\/H} yields\n\\begin{multline}\\label{burgess super messy2}\n\\frac{|S_{\\chi}(M,N)|}{N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}}\\leq \\left(\\frac{2r(2r-1)}{r-1}\\right)^{\\frac{1}{2r}}\\big((2r-3)!!(r-1)\\big)^{\\frac{1}{2r^2}}\\frac{AB}{(A-1)(B-1)}\n\\\\ \\left(\\frac{s^2}{\\left((2r-3)!!(r-1)\\right)^{\\frac{1}{r}}p^{\\frac{r-2}{2(r-1)}}} + \\frac{1}{4k\\log{p}} + \\frac{1}{4r(r-1) k \\log{p}} + \\frac{\\log{\\left(\\frac{1.85s\\,k\\log{p}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}}{k \\log^2{p}}\\right)^{\\frac{1}{2r}} \\\\\n+ \\frac{2r^2}{(2r-1)^2}k^{1-\\frac{1}{r}}c_1(r) \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}\\frac{AB}{(A-1)(B-1)}. \\end{multline}\n\nNow, if we let $c_1(r)$ be defined as follows\n\\begin{equation}\\label{c}\nc_1(2) = \\frac{AB}{(A-1)(B-1)}\\left(12\\right)^{\\frac{1}{4}}\\frac{\\left(1 + \\frac{3}{8k\\log p}  + \\frac{\\log{\\left(1.85k\\log{p}\\right)}}{k \\log^2{p}}\\right)^{\\frac{1}{4}}}{1 - \\frac{8}{9}k^{\\frac{1}{2}} \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{\\frac{3}{2}}\\left(\\frac{AB}{(A-1)(B-1)}\\right)},\n\\end{equation}\nfor $r =2$, and\n\\begin{multline}\\label{c2}\nc_1(r) = \\frac{AB}{(A-1)(B-1)}\\left(\\frac{2r(2r-1)\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{r-1}\\right)^{\\frac{1}{2r}}\\\\\n\\cdot\\frac{\\left(\\frac{s^2}{\\left((2r-3)!!(r-1)\\right)^{\\frac{1}{r}}p^{\\frac{1}{2}-\\frac{1}{2r}-\\frac{1}{2r(r-1)}}} + \\frac{1}{4k\\log{p}} + \\frac{1}{4r(r-1) k \\log{p}} + \\frac{\\log{\\left(\\frac{1.85s\\,k\\log{p}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}}{k \\log^2{p}}\\right)^{\\frac{1}{2r}}}{1 - \\frac{2r^2}{(2r-1)^2}k^{1-\\frac{1}{r}}\\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}\\left(\\frac{AB}{(A-1)(B-1)}\\right)},\n\\end{multline}\nfor $r\\ge 3$. Therefore from \\eqref{burgess super messy} and \\eqref{burgess super messy2}, we get that\n$$|S_{\\chi}(M,N)| \\le c_1(r) N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}}.$$\n\nAll we have to do is pick $k$ to minimize $c_1(r)$ in such a way that $\\lfloor A\\rfloor\\ge 28$, and that $N\\ge 12 A$. First, we'll start by showing that\n\\begin{equation*}\\label{burgess B>12}\nB \\ge 15.\n\\end{equation*}\n\nSince $B = ((2r-3)!!(r-1))^{\\frac{1}{r}}p^{\\frac{1}{2r}}$, we can just manually check for $ 2\\le r\\le 20$  that the inequality is satisfied. To show that it works for $r\\ge 21$, we can show that\n\\begin{equation}\\label{machin32}\n((2r-3)!!(r-1))^{\\frac{1}{r}} \\ge 15,\n\\end{equation}\nby noticing that it works for $r=21$ and that the left hand side of \\eqref{machin32} is increasing. Indeed, the left hand side is increasing; by noticing that $(2r-3)(r-1) < (2r-1)(r+1)$, we get\n$$(2r-3)!!(r-1) < \\frac{(2r-1)^{r-1}(r+1)^{r-1}}{(r-1)^{r-2}} < \\frac{(2r-1)^{r}(r+1)^{r}}{(r-1)^{r}},$$\nimplying that\n$$\\frac{1}{r}\\log{((2r-3)!!(r-1))} < \\log{((2r-1)(r+1))} - \\log{(r-1)},$$\nwhich implies\n$$\\frac{r+1}{r}\\log{((2r-3)!!(r-1))} < \\log{((2r-1)!!)} + \\log{(r+1)},$$\nand hence\n$$\\log{\\left(((2r-3)!!(r-1))^{\\frac{1}{r}}\\right)} < \\log{\\left(((2r-1)!!)(r+1))^{\\frac{1}{r+1}}\\right)}.$$\n\\begin{comment}We also have to make sure that $\\log{(1.85A)} < \\frac{\\log{p}}{2} -\\frac{\\log{p}}{4r} + \\log{(1.9)}.$ Then, we have to pick such a $k$ in terms of $r$ to make $c$ as small as possible.\n\\end{comment}\n\nUsing that $B \\ge 15$, since $A = \\frac{kN}{B}$, then\n\\begin{equation*}\\label{burgess A}\nA = \\frac{kN}{B} < \\frac{kN}{12} < \\frac{N}{12},\n\\end{equation*}\nwhenever $k < 1$.\n\\begin{comment}\nby remembering that $k = \\frac{AB}{N}$. Let's assume that $p \\ge 10^7$. It is not hard to check that $N\\ge 12A$, since $\\log{(10^7)} \\ge 16$ and hence for $N \\ge 48$, we have $A \\le \\frac{N}{16}+1 \\le \\frac{N}{12}$. Now for $p \\ge 10^4$, we have $N \\ge 48$ because $N \\ge c_1(r)^r p^{\\frac{1}{4} + \\frac{1}{4r}}\\log{p} \\ge p^{\\frac{1}{4}}\\log{p} > 48$.\n\\end{comment}\n\nLet $k \\ge \\frac{1}{30}$. To check that $\\lfloor A\\rfloor \\ge 28$, we use \\eqref{range2 for N} and \\eqref{burgess B} and we note that\n\\begin{equation*}\n\\lfloor A\\rfloor \\ge A-1 \\ge\\frac{N}{30B}-1 \\ge \\frac{c_1(r)^r p^{\\frac{1}{4}-\\frac{1}{4r}}\\log{p}}{30((2r-3)!!(r-1))^{\\frac{1}{r}}}-1.\n\\end{equation*}\nTable \\ref{burgess table lower bounds for c} shows the lower bound $c_1(r)$ must satisfy to have $\\lfloor A\\rfloor\\ge 28$ in different situations.\n\n\\begin{table}\n\\begin{center}\n\t\\begin{tabular}{|c| c | c| c|}\n\\hline\n\t$r$ & $p\\ge 10^7$ & $p\\ge 10^{10}$ & $p\\ge 10^{20}$ \\\\ \\hline\n$2$ & $2.68289$ & $1.45765$ & $0.24442$ \\\\  \\hline\n\n$3$ & $1.88354$ & $1.13939$ & $0.251637$ \\\\  \\hline\n\n$4$ & $1.6153$ & $1.06881$ & $0.305418$ \\\\  \\hline\n\n$5$ & $1.48379$ & $1.04807$ & $0.363232$ \\\\  \\hline\n\n$6$ & $1.40512$ & $1.04167$ & $0.417191$ \\\\  \\hline\n\n$7$ & $1.35216$ & $1.04007$ & $0.465518$ \\\\  \\hline\n\n$8$ & $1.31369$ & $1.04016$ & $0.508197$ \\\\  \\hline\n\n$9$ & $1.28422$ & $1.04077$ & $0.545749$ \\\\  \\hline\n\n$10$ & $1.26077$ & $1.04147$ & $0.578819$ \\\\  \\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Lower bounds for the constant $c_1(r)$ in the Burgess inequality to satisfy $\\lfloor A\\rfloor\\ge 28$.}\\label{burgess table lower bounds for c}\n\\end{table}\n\n\\begin{comment}The last thing to check is that $\\log{(1.85A)} < \\frac{\\log{p}}{2} - \\frac{\\log{p}}{4r} + \\log{(1.9)}$. Using \\eqref{range for N} and \\eqref{burgess A} we get\n\\begin{multline*}\n\\log{(1.85A)} \\le \\log{\\left(\\frac{1.85N}{p^{\\frac{1}{2r}}\\log{p}} + 1.85\\right)} \\le \\log{\\left(1.85 p^{\\frac{1}{2}-\\frac{1}{4r}} + 1.85\\right)} \\\\= \\left(\\frac{1}{2} - \\frac{1}{4r}\\right)\\log{p} + \\log{(1.85)} + \\log{\\left(1+\\frac{1}{p^{\\frac{1}{2}-\\frac{1}{4r}}}\\right)} < \\frac{\\log{p}}{2} - \\frac{\\log{p}}{4r} + \\log{(1.9)},\n\\end{multline*}\nthe last inequality being true for $p\\ge 10^5$ and $r\\ge 2$.\n\nNow, using \\eqref{burgess B}, \\eqref{burgess A} and \\eqref{range for N}, we can place the following bounds on $k$:\n\\begin{equation*}\\label{burgess range of k}\nk\\ge \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{\\log{p}} - \\frac{1}{p^{\\frac{1}{2r}}\\log{p}} ,\n\\end{equation*}\nand\n\\begin{equation*}\nk \\le \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{\\log{p}}  + \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{p^{\\frac{1}{4}-\\frac{1}{4r}}\\log{p}}.\n\\end{equation*}\n\\end{comment}\n\nNow, $B$ is defined in terms of $r$ and $p$. By fixing an $r$ and a $p_0$ (a fixed lower bound for $p$), we can calculate $B$ in terms of $r$ and $p_0$. Let $c_1'(r)$ be a parameter satisfying $c_1'(r) < c_1(r)$. $A$ is written in terms of $k$ and $N$ and from \\eqref{range for N} and \\eqref{range2 for N} we have a range for $N$ in terms of $c_1(r)$, $p$, $r$, $k$ and $s$. From this we can find a lower bound for $A$ in terms of $c_1'(r)$, $k$, $r$, $s$ and $p$. The parameter $s$ is optimal when it is as small as possible so we fix $s$ (in terms of $r$, $c_1'(r)$ and $c_1(r-1)$) to be the smallest real satisfying \\eqref{s condition}. After plugging in $A$, $B$, $r$, $s$, $k$, $p_0$ and $c_1'(r)$ to the equations \\eqref{c} and \\eqref{c2}, we can find a good value of $k \\in [\\frac{1}{30},1)$, and a good value of $c_1'(r)$ for each $r$ and $p_0$ to find the smallest $c_1(r)$ we can. After making the choices of $k$ and $c_1'(r)$ described in Table \\ref{last table 1}, we conclude that $c_1(r)$ has the values listed in Table \\ref{super table burgess} as upper bounds.\n\n\\begin{table}[h]\n\\begin{center}\n\t\\begin{tabular}{c| c | c| c| c | c | c |}\n\\cline{2-7} & \\multicolumn{2}{|c|}{$p_0 = 10^7$} & \\multicolumn{2}{|c|}{$p_0 = 10^{10}$} & \\multicolumn{2}{|c|}{$p_0 = 10^{20}$ }\\\\ \\hline\n\n\\multicolumn{1}{|c|}{$r$} & $k$ & $c_1'(r)$ & $k$ & $c_1'(r)$ & $k$ & $c_1'(r)$ \\\\ \\hline\n\n\\multicolumn{1}{|c|}{$2$} & 2\/45 & 2.738 & 1\/30 & 2.517 & 1\/30 & 2.354\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$3$} & 1\/16 & 2.019 & 11\/150 & 1.737 & 2\/15 & 1.369\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$4$} & 1\/12 & 1.729 & 31\/300 & 1.515 & 37\/300 & 1.310\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$5$} & 1\/12 & 1.610 & 7\/75 & 1.456 & 31\/300 & 1.298\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$6$} & 1\/12 & 1.548 & 1\/12 & 1.426 & 7\/75 & 1.289\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$7$} & 11\/150 & 1.504 &11\/150 & 1.404 & 1\/12 & 1.281\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$8$} & 19\/300 & 1.470 & 19\/300 & 1.383 & 1\/12 & 1.272\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$9$} & 19\/300 & 1.441 & 19\/300 & 1.366 & 11\/150 & 1.264\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$10$} & 4\/75 & 1.415 & 4\/75 & 1.349 & $11\/150$ & 1.256\\\\  \\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Values chosen for $k$ and $c_1'(r)$ to build Table \\ref{super table burgess}.}\\label{last table 1}\n\\end{table}\n\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary \\ref{burgess corollary 1}]\nWe begin by pointing out that Theorem \\ref{burgess kiks 1} proves this for $2\\le r\\le 10$ and $p \\ge 10^{7}$. We also know that it is true for the $r=1$ case by the P\\'olya--Vinogradov inequality (Vinogradov proved it with the constant 1 in \\cite{Vinogradov}).\n\nFollowing the proof of Theorem \\ref{burgess kiks 1}, we also have that $B \\ge 15$ for all $r$ and hence, for any $k < 1$, we have $A < \\frac{N}{12}$. It is also worth pointing out that we can use $s = 1$, since now the constant $2.74$ is fixed as the constant in our upper bound, instead of a constant depending on $r$.\n\nWe need to show that you can pick a $k$ such that $\\lfloor A\\rfloor \\ge 28$. First, let's prove that $2.74^r \\ge ((2r-3)!!(r-1))^{\\frac{1}{r}}$. Indeed, for all $r \\ge 1$ we have\n$$2.74^r > 2r \\ge ((2r -3)!!(r-1))^{\\frac{1}{r}}.$$\n\nNow we have\n$$A = \\frac{kN}{B} \\ge \\frac{k (2.74)^r p^{\\frac{1}{4}-\\frac{1}{4r}}\\log{p}}{((2r-3)!!(r-1))^{\\frac{1}{r}}} \\ge k p^{\\frac{1}{4}-\\frac{1}{4r}}\\log{p} > 29,$$\nwhenever $k >  \\frac{29}{p^{\\frac{1}{4}-\\frac{1}{4r}}\\log{p}}$.\n\nWe replace $B$ by 15 in \\eqref{c2} (since $B \\ge 15$), and we can see that the only factors that don't decrease with $r$ are the $k^{1-\\frac{1}{r}}$ term which appears in the denominator, and the $\\left(2-\\frac{1}{r}\\right)$ exponent in the denominator. With this in mind, let $c(r)$ be defined as follows for $r \\ge 4$:\n\\begin{multline}\\label{cmatad}\nc(r) = \\frac{15A}{14(A-1)}\\left(\\frac{2r(2r-1)\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{r-1}\\right)^{\\frac{1}{2r}}\\\\\n\\cdot\\frac{\\left(\\frac{1}{\\left((2r-3)!!(r-1)\\right)^{\\frac{1}{r}}p^{\\frac{1}{2}-\\frac{1}{2r}-\\frac{1}{2r(r-1)}}} + \\frac{1}{4k\\log{p}} + \\frac{1}{4r(r-1) k \\log{p}} + \\frac{\\log{\\left(\\frac{1.85\\,k\\log{p}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}}{k \\log^2{p}}\\right)^{\\frac{1}{2r}}}{1 - \\frac{2r^2}{(2r-1)^2}k\\left(\\frac{16(A+1)}{15A}\\right)^2\\left(\\frac{15A}{14(A-1)}\\right)}.\n\\end{multline}\n\nLetting $k = \\frac{11}{64}$, $A \\ge kp^{\\frac{1}{4}-\\frac{1}{4r}}$ and $p \\ge 10^7 $ we confirm that $c(r) \\le 2.74$ whenever $r\\ge 3$. Since it is also true for $r\\le3$, we conclude our corollary.\n\\end{proof}\n\n\\section{Improving McGown's theorem}\\label{burgess section 2}\nThe main obstacle in improving the $(\\log{p})^{\\frac{1}{r}}$ factor in the Burgess inequality is the bound on $V_2$. However, if we put a bound on $N$, we can make the proof cleaner while also improving the exponent in $\\log{p}$ to $\\frac{1}{2r}$. First we prove a lemma regarding $V_2$ and then we will be able to prove Theorem \\ref{burgess kiks 2}.\n\n\\begin{lemma}\\label{burgess V2 improved}\nLet $p$ be a prime, and $N$ be a positive integer. Let $A \\ge 30$ be an integer such that $N>7A$ and $2AN< p$. Let $v(x)$ be defined as in \\eqref{burgess v(x)}. Then\n\\begin{equation*}\\label{burgess V2 eq}\nV_2 = \\sum_{x\\kern-3pt\\mod{p}}v^2(x) \\leq 2AN \\log(1.85 A).\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is essentially the same as that of Lemma \\ref{burgess V2 lemma}. Recall that $V_2$ is the number of quadruples $(a_1,a_2,n_1,n_2)$ with $1\\leq a_1,a_2 \\leq A$ and $M < n_1,n_2 \\leq M + N$ such that $a_1n_2 \\equiv a_2n_1\\pmod p$. If $a_1 = a_2$, since $N < p$, we have that $n_1 = n_2$ because $n_1 \\equiv n_2 \\pmod p$ while $|n_1 - n_2| \\leq N < p$. Therefore, the number of quadruples in this case is $AN$. Fixing $a_1\\ne a_2$ and writing\n$$a_1n_2-a_2n_1 = kp,$$\nwe can put a bound on possible values for $k$.  As shown in the proof of Lemma \\ref{burgess V2 lemma}, there are at most $\\frac{(a_1+a_2)N}{\\gcd{(a_1,a_2)}p}  + 1$ values of $k$. Since $2AN < p$, then we have that $k$ is uniquely determined.\n\nIn the proof of Lemma \\ref{burgess V2 lemma}, we showed that given $a_1,a_2$ and $k$, the number of pairs $(n_1,n_2)$ is bounded by $N\\frac{\\gcd{(a_1,a_2)}}{\\max\\{a_1,a_2\\}} + 1$.\n\nNow, for $A\\ge 30$ and $N > 7A$ we have\n\\begin{equation}\\label{orales}\n\\left(1-\\frac{6}{\\pi^2}\\right)\\log{(1.85 A)} \\ge \\left(1-\\frac{6}{\\pi^2}\\right)\\log{(55.5)} = 1.57471 > \\frac{3}{2} + \\frac{1}{14} > \\frac{3}{2} + \\frac{A}{2N}.\n\\end{equation}\n\nUsing the definition of $S_3$ as in \\eqref{S_3}, using the inequalities \\eqref{burgess S3} and \\eqref{orales}, for $A\\ge 30$ and $N>7A$, we have\n\\begin{multline*}\nV_2 \\le AN + 2\\sum_{a_1 < a_2}\\left(\\frac{\\gcd{(a_1,a_2)}N}{\\max\\{a_1,a_2\\}} + 1\\right)\\\\\n= AN + 2NS_3 + A^2-A \\le 2AN\\log{(1.85A)}.\n\\end{multline*}\n\\end{proof}\nNow we are ready to prove Theorem \\ref{burgess kiks 2}.\n\n\\begin{proof}[Proof of Theorem \\ref{burgess kiks 2}]\nThe proof is very similar to the proof of Theorem \\ref{burgess kiks 1}. We proceed by induction, assuming that for all $h < N$ we have $|S_{\\chi}(M,h)|\\le c_2(r) p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}}.$\n\nMost of the work in the proof of Theorem $\\ref{burgess kiks 1}$ can be replicated. So I'll just point out the things that change.\n\nThe first change is that by employing Lemma \\ref{burgess V2 improved}, \\eqref{burgess V2} becomes\n\\begin{equation*}\\label{burgess V2 2}\nV_2 \\le 2AN\\log{(1.85A)}.\n\\end{equation*}\nThis change affects \\eqref{Opti}, by deleting $\\frac{AN}{p}$ inside the parenthesis. Now it looks as follows:\n\\begin{equation}\\label{burgess V\/H3}\n\\frac{V}{H} \\le \\frac{AB}{(A-1)(B-1)}\\frac{1}{k^{\\frac{1}{2r}}}\\frac{(2WB)^{\\frac{1}{2r}}}{B}(\\log{(1.85A)})^{\\frac{1}{2r}}\nN^{1-\\frac{1}{r}}.\n\\end{equation}\n\nThe next change is the range for $N$, which we deduced by using the P\\'olya--Vinogradov inequality, the trivial bound, and the case for $r-1$. Instead of \\eqref{range for N}, using our hypothesis and the trivial bound, we now have\n\\begin{equation}\\label{new range for N}\nc_2(2)^r p^{\\frac{3}{8}} \\sqrt{\\log{p}} < N < 2 p^{\\frac{5}{8}},\n\\end{equation}\nfor $r=2$. Assuming $c_2(r-1) \\le s^{\\frac{1}{r(r-1)}}c_2(r)$ for a real number $s$, and using the Burgess inequality for $r-1$ we have, for $r\\ge 3$, the following range for $N$\n\\begin{equation}\\label{new range for N2}\nc_2(r)^r p^{\\frac{1}{4} + \\frac{1}{4r}} \\sqrt{\\log{p}} < N < \\min\\{2 p^{\\frac{1}{2} + \\frac{1}{4r}},s\\,p^{\\frac{1}{4}+\\frac{1}{2r}+\\frac{1}{4r(r-1)}}\\sqrt{\\log{p}}\\}.\n\\end{equation}\n\nUsing that $A=\\frac{kN}{B}$ and \\eqref{new range for N}, we get\n\\begin{equation}\\label{la otra vez 2}\n\\log{(1.85A)}=\\log{\\left(\\frac{1.85kN}{B}\\right)}\\le \\log{\\left(3.7k\\right)}+\\frac{3\\log{p}}{8} \\le \\frac{3\\log{p}}{8},\n\\end{equation}\nfor $r=2$ (we're assuming $k < 1\/4$, which implies $\\log{(3.7k)} < 0$). Using \\eqref{new range for N2}, yields\n\\begin{equation}\\label{la otra vez 22}\n\\log{(1.85A)}=\\log{\\left(\\frac{1.85kN}{B}\\right)}\\le \\log{\\left(\\frac{1.85s\\,k\\sqrt{\\log{p}}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}+\\frac{\\log{p}}{4} + \\frac{\\log{p}}{4r(r-1)},\n\\end{equation}\nfor $r\\ge 3$.\n\nThe bound for $E(h)$ is almost the same as in \\eqref{error}, the only difference being the exponent of $\\log{p}$, which is now $\\frac{1}{2r}$ instead of $\\frac{1}{r}$. Making this change and using both \\eqref{WB} and \\eqref{la otra vez 2} with \\eqref{burgess V\/H3} yields (for $r=2$)\n\\begin{multline}\\label{burgess super messy 2}\n\\frac{|S_{\\chi}(M,N)|}{N^{\\frac{1}{2}}p^{\\frac{3}{16}}(\\log{p})^{\\frac{1}{4}} }\\le \\frac{AB}{(A-1)(B-1)}\\left(12\\right)^{\\frac{1}{4}} \\left(\\frac{3}{8k} \\right)^{\\frac{1}{4}} \\\\\n+ \\frac{AB}{(A-1)(B-1)}\\frac{8}{9}k^{\\frac{1}{2}}c_2(2) \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{\\frac{3}{2}}. \\end{multline}\n\nFor $r\\ge 3$, using \\eqref{WB} and \\eqref{la otra vez 22} with \\eqref{burgess V\/H3} yields\n\\begin{multline}\\label{burgess super messy 22}\n\\frac{|S_{\\chi}(M,N)|}{N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}} }\\le \\frac{AB}{(A-1)(B-1)}\\left(\\frac{2r(2r-1)\\left((2r-3)!!(r-1)\\right)^{\\frac{1}{r}}}{r-1}\\right)^{\\frac{1}{2r}}\n\\\\\n\\cdot \\left(\\frac{\\log{\\left(\\frac{1.85s\\,k\\sqrt{\\log{p}}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}}{k\\log{p}}+\\frac{1}{4k} +\\frac{1}{4r (r-1)k}\\right)^{\\frac{1}{2r}} \\\\\n+ \\frac{AB}{(A-1)(B-1)}\\frac{2r^2}{(2r-1)^2}k^{1-\\frac{1}{r}}c_2(r) \\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}. \\end{multline}\n\nNow, if we let $c_2(r)$ be defined as follows\n\\begin{equation}\\label{c21}\nc_2(2) = \\frac{A}{A-1}\\frac{B}{B-1}\\frac{\\left(12\\right)^{\\frac{1}{4}}\\left(\\frac{3}{8k}\\right)^{\\frac{1}{4}}}{1 - \\frac{8}{9}k^{\\frac{1}{2}}\\left(\\frac{(A+1)(B+1)}{AB}\\right)^{\\frac{3}{2}}\\left(\\frac{AB}{(A-1)(B-1)}\\right)},\n\\end{equation}\nand, for $r\\ge 3$,\n\\begin{equation}\\label{c22}\nc_2(r) = \\frac{A}{A-1}\\frac{B}{B-1}\\frac{\\left(\\frac{2r(2r-1)\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{r-1}\\left(\\frac{\\log{\\left(\\frac{1.85s\\,k\\sqrt{\\log{p}}}{((2r-3)!!(r-1))^{\\frac{1}{r}}}\\right)}}{k\\log{p}}+\\frac{1}{4k} +\\frac{1}{4r (r-1)k}\\right)\\right)^{\\frac{1}{2r}}}{1 - \\frac{2r^2}{(2r-1)^2}k^{1-\\frac{1}{r}}\\left(\\frac{(A+1)(B+1)}{AB}\\right)^{2-\\frac{1}{r}}\\left(\\frac{AB}{(A-1)(B-1)}\\right)},\n\\end{equation}\nfor $r\\ge 3$. Then, from \\eqref{burgess super messy 2} and \\eqref{burgess super messy 22}, we get that\n$$|S_{\\chi}(M,N)| \\le c_2(r) N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}}.$$\n\nAll we have to do is pick $k$ to minimize $c_2(r)$ in such a way that $\\lfloor A\\rfloor\\ge 30$, that $N> 7 A$ and $2AN < p$.\n\\begin{comment}All we have to do is pick $k$ in such a way that $A$ is an integer, i.e., $\\frac{kN}{B}$ is an integer. Make sure that $A\\ge 30$, that $N > 7 A$ and that $2AN < p$. We also have to make sure that $\\log{(1.85A)} < \\frac{\\log{p}}{2} -\\frac{\\log{p}}{4r}.$ Then, we have to pick such a $k$ in terms of $r$ to make $c$ as small as possible.\n\\end{comment}\n\n\\begin{comment}\nwhen\n\\begin{equation}\\label{burgess A2}\nA = \\left\\lfloor\\frac{N}{9 p^{\\frac{1}{2r}}}\\right\\rfloor + 1,\n\\end{equation}\n\\end{comment}\nUsing that $B \\ge 15$, it is not hard to check that $N\\ge 7A$. Indeed, since $A = \\frac{kN}{B}$, we have $A \\le \\frac{kN}{15} < \\frac{N}{7}$.\n\nTo check that $\\lfloor A\\rfloor \\ge 30$ for $k\\ge \\frac{3}{64}$, we do the following:\n\\begin{equation*}\n\\lfloor A\\rfloor\\ge A-1 \\ge \\frac{3N}{64B} -1 \\ge \\frac{3c_2(r)^r p^{\\frac{1}{4}-\\frac{1}{4r}}\\sqrt{\\log{p}}}{64((2r-3)!!(r-1))^{\\frac{1}{r}}}-1.\n\\end{equation*}\nTable \\ref{burgess table lower bounds for c 2} shows the lower bound $c$ must satisfy to have $\\lfloor A\\rfloor\\ge 30$ in different situations.\n\n\\begin{table}[h]\n\\begin{center}\n\t\\begin{tabular}{|c| c | c| c|}\n\\hline\n\t$r$ & $p\\ge 10^{10}$ & $p\\ge 10^{15}$ & $p\\ge 10^{20}$ \\\\ \\hline\n$2$ & $2.78392$ & $1.22500$ & $0.55514$ \\\\  \\hline\n\n$3$ & $1.75393$ & $0.86474$ & $0.43480$ \\\\  \\hline\n\n$4$ & $1.47708$ & $0.81850$ & $0.46029$ \\\\  \\hline\n\n$5$ & $1.35767$ & $0.82260$ & $0.50431$ \\\\  \\hline\n\n$6$ & $1.29240$ & $0.83775$ & $0.54839$ \\\\  \\hline\n\n$7$ & $1.25127$ & $0.85450$ & $0.58848$ \\\\  \\hline\n\n$8$ & $1.22279$ & $0.87022$ & $0.62388$ \\\\  \\hline\n\n$9$ & $1.20171$ & $0.88422$ & $0.65489$ \\\\  \\hline\n\n$10$ & $1.18536$ & $0.89649$ & $0.68202$ \\\\  \\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Lower bounds for the constant $c_2(r)$ in the Burgess inequality to satisfy $\\lfloor A\\rfloor\\ge 30$.}\\label{burgess table lower bounds for c 2}\n\\end{table}\n\nLet's now verify that $2AN < p$. Indeed, from the fact that $A=\\frac{kN}{B}$ and from \\eqref{new range for N}, we have\n\\begin{equation*}\n2AN = \\frac{2kN^2}{B} \\le \\frac{8kp}{((2r-3)!!(r-1))^{\\frac{1}{r}}} < p,\n\\end{equation*}\nwhenever $k < \\min\\left\\{\\frac{((2r-3)!!(r-1))^{\\frac{1}{r}}}{8},1\\right\\}.$\n\\begin{comment}\nthe last inequality being true for $p \\ge 15000$ and $r\\ge 2$. The last thing to check is that $\\log{(1.85A)} < \\frac{\\log{p}}{2} - \\frac{\\log{p}}{4r}$. Using \\eqref{new range for N} and \\eqref{burgess A2} we get\n\\begin{multline*}\n\\log{(1.85A)} \\le \\log{\\left(\\frac{1.85N}{9 p^{\\frac{1}{2r}}} + 1.85\\right)} \\le \\log{\\left(\\frac{3.7 p^{\\frac{1}{2}-\\frac{1}{4r}}}{9} + 1.85\\right)} \\\\= \\left(\\frac{1}{2} - \\frac{1}{4r}\\right)\\log{p} + \\log{\\left(\\frac{3.7}{9}+\\frac{1.85}{p^{\\frac{1}{2}-\\frac{1}{4r}}}\\right)} < \\frac{\\log{p}}{2} - \\frac{\\log{p}}{4r},\n\\end{multline*}\nthe last inequality being true for $p\\ge 23$ and $r\\ge 2$.\n\nNow, using \\eqref{burgess B}, \\eqref{burgess A2} and \\eqref{new range for N}, we can place the following bounds on $k$:\n\\begin{equation*}\nk \\ge \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{9} - \\frac{1}{9 p^{\\frac{1}{2r}}},\n\\end{equation*}\nand\n\\begin{equation*}\nk \\le \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{9} +  \\frac{\\left((2r-3)!! (r-1)\\right)^{\\frac{1}{r}}}{p^{\\frac{1}{4}-\\frac{1}{4r}}\\sqrt{\\log{p}}} .\n\\end{equation*}\n\\end{comment}\n\nAs in the proof of Theorem 1, we define can find bounds for $c_2(r)$ by controlling the parameters $p_0$, $c_2'(r)$ and $k$. We find a good value of $k \\in [\\frac{3}{64},\\frac{((2r-3)!!(r-1))^{\\frac{1}{r}}}{8})$ and a good value of $c_2'(r)$ for each $r$ and $p_0$, and plug in the values of $B$, $k$, and a lower bound bound for $A$ on \\eqref{c21} to find $c_2(2)$ and on \\eqref{c22} to find $c_2(r)$ for $r\\ge 3$ in Table \\ref{table kiks burgess} and conclude the theorem. The values of $k$ and $c_2'(r)$ we chose can be found on Table \\ref{last table 2}.\n\n\\begin{table}[h]\n\\begin{center}\n\t\\begin{tabular}{c| c | c| c| c | c | c |}\n\\cline{2-7} & \\multicolumn{2}{|c|}{$p_0 = 10^{10}$} & \\multicolumn{2}{|c|}{$p_0 = 10^{15}$} & \\multicolumn{2}{|c|}{$p_0 = 10^{20}$ }\\\\ \\hline\n\n\\multicolumn{1}{|c|}{$r$} & $k$ & $c_2'(r)$ & $k$ & $c_2'(r)$ & $k$ & $c_2'(r)$ \\\\ \\hline\n\n\\multicolumn{1}{|c|}{$2$} & 0.124 & 3.65 & 0.124 & 3.58 & 0.124 & 3.57\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$3$} & 0.126 & 2.58 & 0.131 & 2.51 & 0.135 & 2.49\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$4$} & 0.106 & 2.19 & 0.116 & 2.12 & 0.120 & 2.10\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$5$} & 0.091 & 1.98 & 0.101 & 1.92 & 0.107 & 1.90\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$6$} & 0.080 & 1.85 & 0.090 & 1.79 & 0.095 & 1.77\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$7$} & 0.072 & 1.75 & 0.079 & 1.70 & 0.084 & 1.68\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$8$} & 0.064 & 1.68 & 0.071 & 1.635 & 0.077 & 1.61\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$9$} & 0.058 & 1.625 & 0.065 & 1.58 & 0.070 & 1.56\\\\  \\hline\n\n\\multicolumn{1}{|c|}{$10$} & 0.054 & 1.579 & 0.060 & 1.54 & 0.064 & 1.52\\\\  \\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Values chosen for $k$ and $c_2'(r)$ to build Table \\ref{table kiks burgess}.}\\label{last table 2}\n\\end{table}\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary \\ref{burgess kiks corollary 2}]\nBy Theorem \\ref{burgess kiks 2}, we have our desired result whenever $N < 2 p^{\\frac{1}{2} + \\frac{1}{4r}}$. Therefore, the only thing we need to prove is that for $p\\ge 10^{10}$ and $r\\ge 3$, $N < 2 p^{\\frac{1}{2}+\\frac{1}{4r}}$. Since the induction in the proof of Theorem \\ref{burgess kiks 2} relied on the upper bound for $N$, we can't use the Burgess inequalities in Theorem \\ref{burgess kiks 2} to give an upper bound for $N$ in this corollary. However, we can use the Burgess inequalities from Theorem \\ref{burgess kiks 1} to improve the upper bound for $N$. Indeed, for $p\\ge 10^{10}$, we have\n$$|S_{\\chi}(M,N)| \\le 2.6 N^{1-\\frac{1}{2}}p^{\\frac{3}{16}}(\\log{p})^{\\frac{1}{2}}.$$\nIf\n$$N \\ge (2.6)^{\\frac{2r}{r-1}}p^{\\frac{3}{8}-\\frac{1}{8r} - \\frac{3}{8r(r-1)}}\\sqrt{\\log{p}},$$\nthen\n$$N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{2r}} \\ge 2.6 N^{1-\\frac{1}{2}}p^{\\frac{3}{16}}\\sqrt{\\log{p}} \\ge |S_{\\chi}(M,N)|.$$\n\nTherefore, we may assume that\n\\begin{equation}\\label{burgess corollary 2 eq}\nN \\le (2.6)^{\\frac{2r}{r-1}}p^{\\frac{3}{8}-\\frac{1}{8r} - \\frac{3}{8r(r-1)}}\\sqrt{\\log{p}}.\n\\end{equation}\n\nNow, all we need to conclude is to show that the right hand side of \\eqref{burgess corollary 2 eq} is less than $2p^{\\frac{1}{2} +\\frac{1}{4r}}$. Using that $p \\ge 10^{10}$, we can verify this manually for $r\\in\\{3,4,\\ldots,21\\}$. Now, for $r\\ge 22$ we have\n$$N \\le (2.6)^{\\frac{2r}{r-1}}p^{\\frac{3}{8} -\\frac{1}{8r} - \\frac{3}{8r(r-1)}}\\sqrt{\\log{p}} \\le (2.6)^{\\frac{44}{21}}p^{\\frac{3}{8}}\\sqrt{\\log{p}} < 2p^{\\frac{1}{2}}.$$\nThe last inequality is true whenever $p\\ge 10^{10}$.\n\\end{proof}\n\n\\begin{remark}\nBooker and McGown in their proofs have $A$ range through only prime numbers. This idea makes the constants converge quicker. For large enough $p$, it doesn't improve the numbers, but it does for smaller $p$. To save space, we ommited using that technique here, instead focusing on other techniques that made an impact on the \"asymptotic\" constant. One of the nice ideas not used by McGown or Booker is the idea of using Burgess for smaller $r$ to help out with the larger $r$. This allows the theorems to extend to the whole range when $r \\ge 3$.\n\\end{remark}\n\n\\begin{remark}\nTheorem A is a little stronger in \\cite{ETk} when the order of the character is bigger. Therefore, one could use that theorem to get better constants for cubic characters, quartic characters and so on.\n\\end{remark}\n\n\\section{Least $k$-th power non-residue}\\label{section p^1\/6}\nTo prove our results on the least $k$-th power non-residues, we will need the following estimates from \\cite{RS1962}:\n\n\\begin{lemma}\\label{Rosser and Scho}\nLet $B = \\displaystyle\\lim_{m\\rightarrow\\infty} \\sum_{p\\le m}\\frac{1}{p} - \\log{\\log{x}}$, and let $\\pi(x)$ be the number of primes up to $x$. Then the following estimates are true:\n\\begin{align*}\n\\log{\\log{x}} + B - \\frac{1}{2\\log^2{x}} < \\sum_{p\\le x} \\frac{1}{p} & \\mbox{ for } x > 1,\\\\\n\\sum_{p\\le x} \\frac{1}{p} < \\log{\\log{x}} + B + \\frac{1}{2\\log^2{x}} & \\mbox{ for } x \\ge 286,\\\\\n\\pi(x) < \\frac{x}{\\log{x}}\\left(1 + \\frac{3}{2\\log{x}}\\right) & \\mbox{ for } x > 1.\n\\end{align*}\n\\end{lemma}\nFrom it we derive the following immediate corollary:\n\\begin{corollary}\\label{Vino corollary}\nFor real numbers $x,y$ satisfying $x > y > 1$ and $x \\ge 286$, the following estimate is true:\n\\begin{equation*}\n\\sum_{y < p\\le x} \\frac{1}{p} < \\log{\\log{x}}-\\log{\\log{y}} + \\frac{1}{2\\log^2{x}} + \\frac{1}{2\\log^2{y}}.\n\\end{equation*}\n\\end{corollary}\n\nNow we are ready to prove the key lemma (a lower bound on a character sum), which is the essence of Vinogradov's trick.\n\\begin{lemma}\\label{lower bound non-residue}\nLet $x\\ge 286$ be a real number, and let $ y = x^{\\frac{1}{\\sqrt{e}} + \\delta}$ for some $\\delta > 0$. Let $\\chi$ be a non-principal character $\\bmod{\\,p}$ for some prime $p$. If $\\chi(n) = 1$ for all $n\\le y$, then\n\\begin{equation*}\n\\left|\\sum_{n\\le x} \\chi(n)\\right| \\ge x\\left(2\\log{(\\delta\\sqrt{e}+1)} - \\frac{1}{\\log^2{x}} - \\frac{1}{\\log^2{y}} - \\frac{1}{x}\\right).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $\\chi(n)$ is totally multiplicative, $\\chi(n) = 1$ for all $n \\le y$, and $x < p$, then\n\\begin{align*}\\label{vinogradov trick step 1}\n\\left|\\sum_{n\\le x} \\chi(n)\\right|\n&= \\left|\\sum_{\\substack{n\\le x\\\\ \\chi(n)=1}}\\chi(n) + \\sum_{\\substack{n\\le x\\\\ \\chi(n)\\ne 1}}\\chi(n)\\right|\\\\\n&= \\left|\\sum_{n\\le x} 1 -\\sum_{\\substack{n\\le x\\\\ \\chi(n)\\ne 1}}1 + \\sum_{\\substack{n\\le x\\\\ \\chi(n)\\ne 1}}\\chi(n)\\right|\\\\\n&\\ge \\sum_{n\\le x} 1 - 2\\sum_{\\substack{n \\le x\\\\ \\chi(n) \\ne 1}}\\chi(n)\\\\\n&\\ge \\sum_{n\\le x} 1 - 2\\sum_{\\substack{y < q \\le x\\\\ \\chi(q) \\ne 1}}\\sum_{n\\le \\frac{x}{q}}1,\n\\end{align*}\nwhere the sum ranges over $q$ prime. Therefore we have\n\\begin{equation*}\n\\left|\\sum_{n\\le x} \\chi(n)\\right| \\ge \\left\\lfloor x\\right\\rfloor - 2\\sum_{y < q \\le x} \\left\\lfloor\\frac{x}{q}\\right\\rfloor \\ge x - 1 - 2x\\sum_{y < q \\le x} \\frac{1}{q} .\n\\end{equation*}\nUsing Corollary \\ref{Vino corollary} to estimate the sum of the reciprocals of primes we get the desired inequality.\n\\end{proof}\n\nWe can now prove Theorem 3. We will use the explicit Burgess inequality proved as Corollary \\ref{burgess corollary 1} because it works for all $r$.\n\\begin{proof}[Proof of Theorem \\ref{theorem p^1.6}]\nLet $\\chi$ be a character $\\bmod{\\,p}$. Then if $n < p$ and $\\chi(n) \\neq 1$, $n$ is a $k$-th power non-residue. Let $r$ be an integer. Let $x\\ge 286$ be a real number and let $y = x^{\\frac{1}{\\sqrt{e}}+\\delta} = p^{1\/6}$ for some $\\delta>0$. Assume that $\\chi(n) = 1$ for all $n\\le y$. Now by Corollary \\ref{burgess corollary 1} and Lemma \\ref{lower bound non-residue} we have\n\\begin{equation*}\n2.74x^{1-\\frac{1}{r}} p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}} \\ge x\\left(2\\log{(\\delta\\sqrt{e}+1)} - \\frac{1}{\\log^2{x}} - \\frac{1}{\\log^2{y}} - \\frac{1}{x}\\right).\n\\end{equation*}\nNow, letting $x = p^{\\frac{1}{4} + \\frac{1}{2r}}$ we get\n\\begin{equation}\\label{p^1\/6}\n2.74 p^{\\frac{\\log{\\log{p}}}{r \\log{p}} - \\frac{1}{4r^2}} \\ge 2\\log{(\\delta\\sqrt{e}+1)} - \\frac{1}{\\log^2{x}} - \\frac{1}{\\log^2{y}} - \\frac{1}{x}.\n\\end{equation}\nPicking $r = 22$, one finds that $\\delta = 0.00458\\ldots$. For $p \\ge 10^{4732}$, the right hand side of \\eqref{p^1\/6} is bigger than the left hand side, showing that $\\chi(n)$ is not always 1 for $n \\le y = p^{1\/6}$, and hence the theorem is true.\n\\end{proof}\n\n\\begin{remark}\nTo be able to use Theorem \\ref{burgess kiks 2} to improve Theorem \\ref{theorem p^1.6}, we would need to calculate what happens for $r > 20$ since the restriction $y = p^{1\/6}$ implies $r > 20$. Since we know that $p$ will be large, we can also pick a large $p_0$ and then find a good constant for the Burgess inequality when $p$ is very large and $r > 20$. After doing all of this work, one could show that Theorem \\ref{theorem p^1.6} works for $p \\ge 10^{3850}$.\n\\end{remark}\n\\nocite{Fri1987}\n\\nocite{PPP}\n\n\\section*{Acknowledgements} I would like to dedicate this paper to Paul Bateman and Heini Halberstam. I would like to thank Carl Pomerance for suggesting the problem, his useful comments and his encouragement. I would like to express my gratitude to Andrew Granville, Hugh Montgomery and Robert Vaughan for allowing me to look at unpublished manuscripts that helped me on my work on the Burgess inequality. I would also like to thank Paul Pollack who directed me to McGown's work. Finally, I'd like to thank an anonymous referee that read the paper carefully and suggested many improvements.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\subsection{Experimental Setup}\n\nMotivated by the incidence levels defined by the Brazilian Ministry of Health, we evaluate each model according to the area under the receiver operating characteristic curve (AUC) considering incidence levels as labels. The receiver operating characteristic curve is created by plotting the true positive rate against the false positive rate at multiple threshold values, used for defining whether the predicted class is positive or negative. In this sense, it indicates the trade-off between sensitivity and specificity. An area close to 1 indicates that one can increase the model's sensitivity by changing the threshold incurring in few false positive errors, while smaller areas indicate more false positive errors when increasing sensitivity. For each week and city under study, we verify the attained and predicted incidence levels. We calculate the AUC for each level on an one-against-all strategy and then compute the average over the three levels.\n\nPredictions were issued always with $\\beta = 4$ weeks of antecedence. Choosing the antecedence value implies in dealing with a trade-off: shorter antecedence leads to more data and potentially more accurate predictions, but less time for health authorities to act in advance to mitigate the impact of future outbreaks. We chose $\\beta = 4$ weeks as a balance between utility and accuracy, allowing some time for health authorities to act while still managing to obtain accurate predictions.\n\nWe also defined that the first two years (weeks 1 to 104) would be used for training only. Predictions are issued from the third to the sixth year, always with 4 weeks in advance, as indicated previously. This means that our available training set is continuously growing with time, as we collect more data.\n\nFinally, models are always compared using a paired Wilcoxon test with 95\\% confidence level. Bonferroni corrections were applied whenever multiple statistical tests were required to ensure statistical significance.\n\n\\subsection{Evaluation of Proposed Framework}\n\nIn order to use the proposed framework, we must define the correlation threshold that defines whether estimated DIR values are going to be integrated into the training set or not. For that, we instantiated four versions of the proposed framework, each with a distinct threshold value. We compared the difference in accuracy assuming four values of delay in the epidemiological data for each threshold used when compared to simply extending predictive antecedence, as shown in Figure \\ref{fig:difference}. Positive values indicate that the proposed framework achieved better results, while negative values indicate the opposite scenario. Note that, although it is difficult to identify the best threshold value, the proposed framework obtained more accurate predictions for all delays and threshold values used in the experiment, obtaining statistically superior results for delays of at least 4 weeks. Besides that, the difference seems to grow with larger delays, indicating that the proposed framework is indeed more robust to delays in epidemiological data.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/delayDifference.png}\n  \\caption{Mean difference in accuracy when comparing the proposed framework with extending predictive antecedence to deal with epidemiological data delay. Whiskers indicate 95\\% confidence intervals.}\n  \\label{fig:difference}\n\\end{figure}\n\nHaving verified that the proposed framework leads to more accurate predictions in general, we would like to better understand which cities are improved by the proposed methodology and which cities are not. For this analysis, we considered the threshold value of 0.5 and fixed the epidemiological delay as 6 weeks to investigate the role of the amount of Twitter data available on the accuracy improvement obtained. Figure \\ref{fig:differenceNumberTweets} shows that cities that obtained higher improvements in accuracy tend to have more Twitter data available. This result corroborates with the intuitive notion that more Twitter data eases the identification of stronger relationships with epidemiological dengue data.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/gainByNumberTweets.png}\n  \\caption{Impact of the amount of Twitter data in the observed difference of accuracy when considering predictions provided by the proposed methodology and provided by the original model with extended predictive antecedence. Each symbol denotes a city, with its shape and color indicating the highest incidence level it achieved in the period under study.}\n  \\label{fig:differenceNumberTweets}\n\\end{figure}\n\nWe now evaluate the spatial distribution of accuracy improvements. Figure \\ref{fig:differenceSpatialDist} shows the spatial distribution of accuracy improvements obtained when applying the proposed framework. Note that improvements are mostly concentrated within the Southeast region of Brazil. Since the Southeast region of Brazil is the most developed region of the country, this distribution reflects the availability of Internet access in Brazil. Having more people capable of posting on Twitter, this region was the most benefited by the proposed framework. We also highlight that, for only 17 cities out of 213, applying the proposed framework was detrimental, indicating that the proposed framework is safe to use regardless of where cities are located.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{images\/spatialDistributionDifferences.png}\n  \\caption{Spatial distribution of the observed difference of accuracy when considering predictions provided by the proposed methodology and provided by the original model with extended predictive antecedence. Each symbol denotes a city, with its shape and color indicating the gain\/loss in accuracy provided by the proposed methodology.}\n  \\label{fig:differenceSpatialDist}\n\\end{figure}\n\nAlthough bigger improvements in accuracy where concentrated in the Southeast region of Brazil, we did not observe significant spatial autocorrelation between improvements, as shown in Figure \\ref{fig:differenceSpatialCor}.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/spatialCorrelationDifferences.png}\n  \\caption{Spatial autocorrelation of the observed difference of accuracy when considering predictions provided by the proposed methodology and provided by the original model with extended predictive antecedence.}\n  \\label{fig:differenceSpatialCor}\n\\end{figure}\n\n\\subsection{Analysis of Predictions}\n\nIn the last section, we evaluated whether the proposed methodology led to more accurate predictions than the increased antecedence approach. We now evaluate the appropriateness of the proposed methodology and analyze the final predictions issued by it.\n\nWe begin by evaluating two assumptions of the proposed model: homoscedasticity and normality of residuals. In other words, the GP-based model assumes that residuals are Gaussian, independent and identically distributed. Figure \\ref{fig:qqplot} shows the quantile-quantile plot of residuals. Note that observed quantiles are associated with theoretical quantiles in a approximately linear fashion, indicating that residuals are approximately Gaussian, independent and identically distributed.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/qqplotResiduals.png}\n  \\caption{Quantile-quantile normal plot of residuals.}\n  \\label{fig:qqplot}\n\\end{figure}\n\nFigure \\ref{fig:cdfAUC} shows empirical cumulative distribution function of AUC. Almost 77\\% of the cities obtained AUC higher than 0.6, while about 20\\% of the cities obtained AUC higher than 0.8.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/cdfAUC.png}\n  \\caption{Empirical cumulative distribution function of the observed accuracy when considering predictions provided by the proposed methodology.}\n  \\label{fig:cdfAUC}\n\\end{figure}\n\nFigure \\ref{fig:aucNumberTweets}, on the other hand, shows the impact of the volume of Twitter data on the accuracy of predictions. Note that the amount of Twitter data is not associated with final predictions. We believe this result is mainly due to the module deciding whether to use estimated DIR values or not, which makes the whole methodology safe to use even when Twitter data is scarce.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/aucByNumberTweets.png}\n  \\caption{Impact of the amount of Twitter data in the observed accuracy when considering predictions provided by the proposed methodology.}\n  \\label{fig:aucNumberTweets}\n\\end{figure}\n\nThe spatial distribution of final accuracies is shown in Figure \\ref{fig:aucSpatialDist}. Differently from Figure \\ref{fig:differenceSpatialDist}, high final accuracies are not strongly concentrated only within the Southeast region. In fact, a significant number of cities from North and Northeast regions obtained high AUC values. These results suggest that, although Twitter was not strongly helpful for cities within these regions, the GP-based EEWS is still capable of issuing accurate predictions with increased predictive antecedence. In this direction, one could argue that the proposed framework is capable of improving predictions exactly where delays in epidemiological data are more detrimental to EEWSs.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{images\/spatialDistributionAUC.png}\n  \\caption{Spatial distribution of the observed accuracy when considering predictions provided by the proposed methodology.}\n  \\label{fig:aucSpatialDist}\n\\end{figure}\n\nFigure \\ref{fig:predictions} shows predictions issued by the proposed framework for all Brazilian capital cities that reached at least medium incidence from 2013 to 2016. Cities are sorted by the number of dengue-related tweets collected, where S\\~{a}o Paulo was the city the with highest number of tweets and Palmas was the city with the lowest number of tweets. Note that, even for cities with low amount of Twitter data, the proposed framework typically detects major dengue fever outbreaks, reaching the correct incidence level (indicated by the dashed lines). The city of Belo Horizonte, although being the capital city with third highest number of dengue-related tweets, demonstrated to be particularly challenging to forecast, since it exhibited very low incidence during all period under study, except from the beginning of 2013 and 2016, where very drastic outbreaks were observed. However, we note that our proposed methodology, although failing to capture the outbreak in 2013, was able to capture the increase in incidence associated with the outbreak in 2016, reaching the correct incidence level.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{images\/predictions.png}\n  \\caption{Predictions issued for all Brazilian capital cities under study. The solid black line denotes the observed DIR, while the cyan line indicates the predicted DIR. The gray shaded area exhibits the 95\\% confidence interval. Dashed lines indicate the boundaries between incidence levels.}\n  \\label{fig:predictions}\n\\end{figure*}\n\n\n\\section{Introduction} \\label{sec:introduction}\n\n\\input{introduction}\n\n\\section{Related Work} \\label{sec:related}\n\n\\input{related}\n\n\\section{Data Collection}\n\n\\input{data}\n\n\\section{Incorporating online data into epidemiological models}\n\n\\input{methodology}\n\n\\section{Experimental Analysis} \\label{sec:experiments}\n\n\\input{experiments}\n\n\\section{Conclusions and Future Work}\n\n\\input{conclusions}\n\n\\subsection*{Competing Interests}\nThe authors have declared that no competing interests exist.\n\n\\subsection*{Acknowledgements} \nWe would like also to thank the authors from \\citet{souza2015latent} for sharing their dataset of geolocalized dengue-related tweets, enabling this study.\n\n\nThis work was partially supported by CNPq, FAPEMIG, CAPES, InWeb, MASWeb, BIGSEA, INCT-Cyber and INCT-Dengue.\n\n\\balance\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\subsection{EEWS for Dengue Fever Incidence Forecasting}\n\nWe begin by defining the final component of the proposed framework: an epidemiological model to forecast future incidence based on the available training set, possibly extended by integrating estimated delayed DIR values.\n\nHere, we define a model based on \\citep{albinati2016accurate}, where the authors proposed to model log-transformed DIR values of Brazilian cities using a GP model equipped with a covariance function capable of exploiting temporal local dependences typical of infectious diseases, as well as the annual seasonality commonly observed in dengue data \\citep{gharbi2011time,martinez2011predicting,promprou2006forecasting}. In summary, we propose to use the following two-part covariance function\n\\begin{eqnarray}\n  k_s(t, t') & = & k_{loc}(t-t') + k_{qp}(t-t') \\label{eq:cov} \\\\\n  k_{loc}(\\tau = t-t') & = & \\sigma_{loc}^2 \\left( 1 + \\frac{\\sqrt{5}\\tau}{\\ell_{loc}} + \\frac{5\\tau^2}{\\ell_{loc}^2} \\right) exp \\left( - \\frac{\\sqrt{5}\\tau}{\\ell_{loc}} \\right) \\nonumber \\\\\n  k_{qp}(\\tau = t-t') & = & \\sigma_{qp}^2 \\left( 1 + \\frac{\\sqrt{5}\\tau}{\\ell_{qp}} + \\frac{5\\tau^2}{\\ell_{qp}^2} \\right) exp \\left( - \\frac{\\sqrt{5}\\tau}{\\ell_{qp}} \\right) \\nonumber \\\\\n  & & exp \\left( -2 * \\frac{sin^2(\\pi |\\tau| \/ p)}{\\ell_{per}} \\right) \\nonumber\n\\end{eqnarray}\nwhere $k_s(t,t)$ denotes the covariance between incidence values at city $s$ during weeks $t$ and $t'$ and $\\sigma_{loc}$, $\\ell_{loc}$, $\\sigma_{qp}$, $\\ell_{qp}$, $\\ell_{per}$ and $p$ are hyperparameters learned from likelihood maximization.\n\nThe intuition of the covariance function in Equation \\ref{eq:cov} is the following. The function is formed by two temporal components, which are used to exploit two temporal patterns, namely local dependences and seasonality. Local dependences are exploited by $k_{loc}$, which correlates weeks nearby and is expressed in terms of a Mat\\'ern covariance function, a covariance function commonly used in the GP literature to enforce smoothness by indicating that nearby data points are highly correlated, while distant data points are loosely correlated \\citep{rasmussen2006gaussian}. Seasonality is exploited using a quasi-periodic function formulated as the product between a Mat\\'ern and a periodic covariance function. By using this formulation, the model is capable of using information from incidence of past years, but is forced to give more relevance to more recent years than very distant ones. Figure \\ref{fig:tempCorrelation} shows the covariance function obtained after hyperparameter optimization (Table \\ref{tab:hypers}). Note that covariance is high for $\\tau$ close to 0, with smaller peaks for $\\tau$ around 1 year (52 weeks).\n\n\\begin{table}[h!]\n  \\centering\n  \\caption{Hyperparameters obtained after likelihood maximization}\n  \\label{tab:hypers}\n  \\begin{tabular}{cccc}\n    \\hline\n    & 1st Quartile & Median & 3rd Quartile \\\\\n    \\hline\n    $\\sigma_{loc}^2$ & 0.039 & 0.101 & 0.186 \\\\\n    $\\sigma_{qp}^2$ & 0.484 & 1.427 & 2.311 \\\\\n    $\\ell_{loc}$ & 1.000 & 2.000 & 7.000 \\\\\n    $\\ell_{qp}$ & 41.000 & 58.000 & 99.000 \\\\\n    $\\ell_{per}$ & 1.000 & 1.000 & 2.000 \\\\\n    $p$ & 54.000 & 59.000 & 73.000 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{images\/temporalCorrelation.png}\n  \\caption{Covariance function considering temporal components in Equation \\ref{eq:cov} with optimized hyperparameters according to Table \\ref{tab:hypers}.}\n  \\label{fig:tempCorrelation}\n\\end{figure}\n\nHaving defined the covariance function, the model used in this work can be expressed as follows:\n\\begin{eqnarray}\n  DIR_{s,t} & = & exp\\left( y_{s,t} + \\overline{y_s} \\right) - 1 \\label{eq:eews} \\\\\n  y_{s,\\cdot} & \\sim & \\mathcal{GP} \\left( 0, k_s(t,t') \\right) \\nonumber\n\\end{eqnarray}\nwhere $DIR_{s,t}$ denotes the DIR value at city $s$ during week $t$, $\\overline{y_s}$ indicates the mean of log-transformed DIR values at city $s$ and $k_s(t,t')$ is the covariance function indicated in Equation \\ref{eq:cov}.\n\n\\subsection{Estimating Delayed Epidemiological Data}\n\nThe first step on the proposed framework is to exploit a known relationship between online and epidemiological data. Based on the study provided by \\citet{gomide2011dengue}, which identified a linear association between the number of dengue-related tweets and DIR, and on the results obtained by the Gaussian process-based model proposed by \\citet{albinati2016accurate}, we opted for a Gaussian process model equipped with the covariance function indicated in Equation \\ref{eq:cov}, with an additional component designed to exploit the linear association between Twitter and epidemiological data.\n\nFollowing \\citet{albinati2016accurate}, we first applied a logarithmic transformation of DIR values to avoid using Gaussian distributions to model count data. We then centered the resulting values and estimated the zero-mean Gaussian process model. In summary, this component is implemented by the following model:\n\\begin{eqnarray}\n  DIR_{s,t} & = & exp(y_{s,t} + \\overline{y_s}) - 1 \\label{eq:comp1} \\\\\n  y_{s,\\cdot} & \\sim & \\mathcal{GP} \\left( 0, k_s(t,t') + \\frac{x_{s,t} * x_{s,t'}}{\\ell_{tw}} \\right) \\nonumber\n\\end{eqnarray}\nwhere $x_{s,t}$ indicates the log-transformed number of dengue-related tweets posted during week $t$ at city $s$ and $k_s(t,t')$ is given by Equation \\ref{eq:cov}. Note that $\\ell_{tw}$ is an additional hyperparameter, which is also learned from likelihood maximization using available training data.\n\n\\subsection{Deciding Whether to Use Estimates}\n\nIn order to avoid using poorly estimated DIR values, which could introduce noise into epidemiological data, we defined an extra module responsible for deciding whether estimated epidemiological data is reliable or not. Since we are assuming a linear relationship between Twitter and dengue data, we opted for using a threshold based on the correlation between epidemiological and Twitter data. Whenever the correlation exceeds the threshold, estimated dengue data is considered. Otherwise, only actual epidemiological dengue data is used. Details on how to define this threshold are given on Section \\ref{sec:experiments}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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#1 $}}}\n\t\\newcommand{\\zentrieren}[1]{\\vcenter{\\hbox{$ \\overset{\\textcolor{white}{a}}{#1} $}}}\n\n\\maketitle\n\n\n\\begin{center}\n\t\\begin{minipage}{0.875\\textwidth}\n\\begin{center}\n\t\t\\textbf{Abstract}\\\\\n\\end{center}\nThe segmentation of data into stationary stretches also known as multiple change point problem is important for many applications in time series analysis as well as signal processing. Based on strong invariance principles, \nwe analyse data segmentation methodology using moving sum (MOSUM) statistics for a class of regime-switching multivariate processes where each switch results in a change in the drift. In particular, this framework includes the data segmentation of multivariate partial sum, integrated diffusion and renewal processes even if the distance between change points is sublinear. \nWe study the asymptotic behaviour of the corresponding change point estimators, show consistency and derive the corresponding localisation rates which are minimax optimal in a variety of situations including an unbounded number of changes in Wiener processes with drift. Furthermore, we derive the limit distribution of the change point estimators for local changes --  a result that can in principle be used to derive confidence intervals for the change points.  \\end{minipage}\n\\end{center}\n\\noindent\\textbf{Keywords:} Data segmentation, Change point analysis, moving sum statistics, multivariate processes, invariance principle, regime-switching processes\\\\\\\\\n\\textbf{MSC2020 classification:} 62M99; 62G20, 62H12\n\t\n\\section{Introduction}\\label{introduction}\nThe detection and localisation of structural breaks has a long tradition in statistics, dating back to \\cite{Page}. Nevertheless, there is still a large maybe even increasing interest in this topic surely also because \nchange point analysis  is broadly applicable in a number of fields such as neurophysiology (see \\cite{Messer14}), genomics (compare \\cite{Olshen}, \\cite{NiuZhang}, \\cite{LiMunkSieling}, \\cite{ChanChen}), finance (\\cite{Aggarwal}, \\cite{ChoFryzlewicz}), astrophysics (see \\cite{FischEckleyFearnhead}) or oceanographics (\\cite{Killick10}). \n\nA large amount of research deals with the detection of changes in univariate time series in particular changes in the mean (compare \\cite{CsorgoHorvath97} for an overview) where recently also applications to continuous time stochastic processes,  functional or high-dimensional panel data (see e.g.\\ \\cite{horvath2014extensions}). However, extensions to the multiple change point problem that aims at segmenting the data into stationary stretches beyond changes in the mean in time series data are much more scarce.\n\nGenerally, data segmentation methods can roughly be split up in two approaches: The first approach first introduced by \\cite{Yao} in the context of i.i.d.\\ normally distributed data using the Schwarz' criterion aims at optimizing suitable objective functions.\n\\cite{Kuehn} extended this approach to processes in a setting closely related to the one in this paper albeit only allowing for univariate processes and a finite number of change points. Further approaches include e.~g.\\ least-squares (\\cite{YaoAu}) or  the quasi-likelihood-function (\\cite{Braun}). Generally, such approaches are computationally expensive, such that there is another body of work proposing fast algorithms e.g.\\ using dynamic programming (\\cite{Killick12}, \\cite{Maidstone}).\n\nA second approach is based on hypothesis testing, where e.g.\\ binary segmentation introduced by \\cite{Vostrikova} recursively uses tests constructed for the at-most-one-change situation. This arises several problems including the observation that detection power can be poor if the set of change points is unfavourable, such that several extensions have been proposed in the literature such as circular binary segmentation (\\cite{Olshen}) or wild binary segmentation (\\cite{Fryzlewicz}).\n\n\n\\subsection*{Connection to existing work}\nAnother class of test-based methods uses moving sum (MOSUM) statistics which were first introduced by \\cite{BauerHackl}. They are particularly useful in the context of localising multiple change points and recently have broadly been used for the detection and  the estimation of change points, see e.g.\\ \\cite{yau2016inference} for changes in autoregressive time series,\n\\cite{EichingerKirch} in a hidden Markov framework and \\cite{ChoKirchMeier} as well as\n\\cite{ChoKirch} who proposed a two-stage data segmentation procedure based on multiscale MOSUM\\  statistics. This work extends the results of \\cite{EichingerKirch} to a more general setting including multivariate mean changes, changes in diffusion as well as in renewal processes. Our results also lays the foundations for the analysis of a two-step procedure as in \\cite{ChoKirch}.\n\t \n\t\n\t\n\n\t\n\t \\cite{Messer14} propose a bottom-up-approach combining several moving-sum statistics to obtain change point estimators in univariate renewal processes. \n\t Our work extends these results in several ways: First, \\cite{Messer14} do not show consistency of the change point estimators neither do they derive localisation rates, which is one of the main results of this work. Furthermore, in addition to results for MOSUM\\  procedures with linear bandwidth (in the sample size) as in \\cite{Messer14} we obtain results for sublinear bandwidths allowing in particular to obtain consistent estimators in situations where the distance between change points is sublinear. Additionally, we go beyond the univariate case including some multivariate point processes based on renewal processes in our analysis. \n\t Sequential change point methodology for renewal processes has been proposed by~\\cite{GutSteinebach02} and \\cite{GutSteinebach09}, for diffusion processes by \\cite{Mihalache}.\n\t\n\t We analyse a more general model of regime-switching multivariate processes including multivariate partial sum, renewal as well as diffusion processes.  We require the processes to fulfill a multivariate invariance principle, where processes switch (possibly with a number increasing to infinity with increasing sample size)  between finitely many regimes with each switch resulting in a change in the drift.\n\t A univariate version of that model with  at-most-one change point has  been considered by \\cite{HorvathSteinebach} and \\cite{KuehnSteinebach}. A univariate version for finitely many change points has been considered by \\cite{Kuehn} where consistency for the number of change points has been shown. Those results are now extended to include MOSUM\\  methodology for the estimation of a multiple (possibly unbounded) number of change points in a multivariate setting, where we achieve a minimax optimal separation rate in addition to a minimax optimal localisation rate (for the change point estimators) in case of a bounded number of change points as well as for Wiener processes with drift (see Remark~\\ref{Remark_minimax} below). \n\n\n\t  \\subsection*{Organization of the material}\n\t  In Subsection \\ref{subsection_model}, we introduce the multiple change point model we consider followed by some examples of processes fulfilling the model in Subsection \\ref{subsection_examples}. In Section \\ref{section_procedures}, we describe how to  estimate change points based on MOSUM\\  statistics: First, we introduce the MOSUM\\  statistics in  \\ref{subsection_mosum_statistics}, before  presenting the estimators for the structural breaks in  \\ref{subsection_cp_estimators}. In \\ref{subsection_threshold} we derive some asymptotic results for the MOSUM\\  statistics that are required for threshold selection and can also be used in a testing context. In Section \\ref{section_consistency} we show that the corresponding data segmentation procedure is consistent. Finally,  we derive the localisation rates in addition to the corresponding asymptotic distribution of the change point estimators for local changes. In Section \\ref{subsection_simulation}, we present some results from a small simulation study. The proofs can be found in Appendix \\ref{appendix}.\n\n\\section{Multiple change point problem}\\label{section_seg_procedure}\nIn this section we introduce the  general multiple change model for which we derive the theoretic results.  In particular, this model includes changes in multivariate renewal processes as a special case which was the original motivation for this work.\n\\subsection{Model} \\label{subsection_model}\nConsider $P<\\infty$ time-continuous $p$-dimensional stochastic processes $\\{\\mathbf{R}_{t,T}^{(j)}:0\\le t\\le T\\}$ with (unknown) drift $(\\boldsymbol{\\mu}_T^{(j)}\\cdot t)$ and (unknown) covariance $(\\boldsymbol{\\Sigma}_{j,T}\\cdot t)$ fulfilling the following joint invariance principle.\n\nThe observed process is then assumed to switch between these $P$ processes (states).\n\n\\begin{Assumption}\\label{invariance_principle}\\ \\\\\n\tDenote the joint process by $\\mathbf{R}_{t,T}=\\left({\\mathbf{R}_{t,T}^{(1)}}^{\\prime},\\ldots,{\\mathbf{R}_{t,T}^{(P)}}^{\\prime}\\right)^{\\prime}$ as well the joint drift by $\\boldsymbol{\\mu}_T=\\left({\\boldsymbol{\\mu}_T^{(1)}}^{\\prime},\\ldots,{\\boldsymbol{\\mu}_T^{(P)}}^{\\prime}\\right)^{\\prime}$, where $^{\\prime}$ indicates the matrix transpose.\n\t For every $ T>0 $ there exist $(p\\cdot P)$-dimensional Wiener processes $\\mathbf{W}_{t,T}$ with covariance matrix $\\boldsymbol{\\Sigma}_T$ and\t\\begin{align*}\n\t\t\\boldsymbol{\\Sigma}_{T}^{(i)}=(\\boldsymbol{\\Sigma}_T(l,k))_{l,k=p\\,(i-1)+1,\\ldots,p\\,i}\n\t\\end{align*}\n\twith\n\t\\begin{align*}\n\t\t\\|{\\boldsymbol{\\Sigma}_{T}^{(i)}}\\|=O(1),\\qquad \\|{\\boldsymbol{\\Sigma}_{T}^{(i)}}^{-1}\\|=O(1),\n\\end{align*}\n\tsuch that, possibly after a change of probability space,  it holds that for some sequence $\\nu_T\\to 0$\n\t\\begin{align*}\n\t\\sup_{0\\le t\\le T}\\|\\widetilde{\\mathbf{R}}_{t,T}-\\mathbf{W}_{t,T}\\|=\n\t\\sup_{0\\le t\\le T}\\|\\left(\\mathbf{R}_{t,T}-\\boldsymbol{\\mu}_T\\,t\\right)-\\mathbf{W}_{t,T}\\|=\t\tO_P\\left(T^\\frac{1}{2}\\,\\nu_T\\right),\n\t\\end{align*}\n\twhere $\\widetilde{\\mathbf{R}}_{t,T}=\\mathbf{R}_{t,T}-\\boldsymbol{\\mu}_T\\,t$ denotes the centered process.\n\\end{Assumption}\nIf these $P$ processes  are independent, which is a reasonable assumption in a switching context, the joint invariance principle reduces to the validity of an invariance principle for each process.\n\nThe assumption on the norm of the covariance matrices is equivalent to the smallest eigenvalue of $\\boldsymbol{\\Sigma}_T^{(i)}$ being bounded in addition to being bounded away from zero (both uniformly in $T$). In many situations, the covariance matrices will not depend on $T$, in which case this assumption is automatically fulfilled under positive definiteness. The convergence rate $\\nu_T$  in the invariance principle typically depends on the number of moments that exist. Roughly speaking, the more moments the original process has, the faster $\\nu_T$ converges.\n\nWe now observe a process $\\mathbf{Z}_{t,T}$ with increments switching between the above processes  at some unknown change points $ 0=c_{0}<c_{1}<\\ldots<c_{q_T}<c_{q_T+1}=T$, where $q_T$ can be bounded or unbounded.\nMore precisely, we observe for $c_{\\ell}<t\\le c_{\\ell+1}$\n\\begin{align}\n\t\\mathbf{Z}_{t,T}=\\left(\\mathbf{R}_{t,T}^{(c_{\\ell}+1)}-\\mathbf{R}_{c_{\\ell},T}^{(c_{\\ell}+1)}\\right)+\\sum_{j=1}^{\\ell\n}\\left(\\mathbf{R}_{c_j,T}^{(c_j)}-\\mathbf{R}_{c_{j-1},T}^{(c_j)}\\right).\n\t\\label{eq_model}\n\\end{align}\nThe upper index $^{(c_j)}$ at the process $\\mathbf{R}_{\\cdot,T}$ indicates (with a slight abuse of notation) the active regime between the $(j-1)$-st and the $j$-th change point, from which the increments come in that stretch.\nBecause we concentrate on the detection of changes in the drift, we need to assume that the drift changes between two neighboring regimes,\n i.e.\\\n\\begin{align*}\n\t\\mathbf{d}_{i,T}:=\\boldsymbol{\\mu}_T^{(c_i+1)}-\\boldsymbol{\\mu}_T^{(c_i)} \\neq 0 \\quad \\text{for all }i=1,\\ldots,q_T,\n\\end{align*}\nwhere $\\mathbf{d}_{i,T}$ is bounded but we allow for $\\mathbf{d}_{i,T}\\to 0$ as long as the convergence is slow enough (see Assumption \\ref{bandwidth}).\nFor ease of notation we drop the dependency on $T$ for all above quantities except $q_T$ in the following except in situations where it helps clarify the argument. The aim of this paper is to estimate the number and location of the change points and prove consistency of the estimator for the number of change points in addition to deriving localisation rates for the change point estimators.\n\nThe corresponding univariate model with at most one change was first considered by \\cite{HorvathSteinebach} and extended to a gradual change setting by \\cite{steinebach2000some}.  \\cite{KirchSteinebach} \nprove validity for  corresponding permutation tests. Furthermore, \\cite{GutSteinebach02} \ndevelop sequential change point tests and analyse the corresponding stopping time (\\cite{GutSteinebach09}).\n\nA related univariate multiple change situation\n with a bounded number of change points has been considered by  \\cite{KuehnSteinebach}, who propose to use a Schwarz information criterion for change point estimation. However, this methodology is computationally expensive with quadratic computational complexity, which is one of the reasons why we propose an alternative methodology  based on a single-bandwidth moving sum (MOSUM) statistic in order to estimate the change points. We will show that the rescaled change point estimators are consistent and derive the corresponding localisation rates.\n\n\n\\subsection{Examples}\\label{subsection_examples}\nIn this section, we give three important examples fulfilling the above model assumptions, namely partial sum-processes, renewal processes as well as integrals of diffusion processes including Ornstein-Uhlenbeck and Wiener processes with drift. A detailed analysis of MOSUM\\  procedures  for detecting changes in (univariate) renewal processes extending the work by ~\\cite{Messer14} was the original motivation for this work and is covered by this much broader framework.\n\n\\subsubsection{Partial-Sum-Processes}\\label{sec_partial_sums}\nThis first example extends the classical multiple changes in the mean model:\n\nLet $\\mathbf{X}^{(i)}_1,\\mathbf{X}^{(i)}_2,\\ldots$ be a time series with $ \\EW{\\mathbf{X}^{(i)}_1}=0$ and $\\Covmulti{\\mathbf{X}^{(i)}_1}=I_p$  and all $i=1,\\ldots,P$.\nLet \n\\begin{align*}\n\t\\mathbf{R}_{t}^{(i)}=\\sum_{j=1}^{\\lfloor t\\rfloor}\\left(\\boldsymbol{\\mu}^{(i)}+ {\\boldsymbol{\\Sigma}_{T}^{(i)}}^{1\/2}\\mathbf{X}^{(i)}_j\\right).\n\\end{align*} \nThe corresponding process fulfills Assumption \\ref{invariance_principle} in a wide range of situations. For example, \\cite{Einmahl} \nshows the validity in the case that  $\\mathbf{X}_1, \\mathbf{X}_2,\\ldots$ with $\\mathbf{X}_j=\\left(\\mathbf{X}^{(1)}_j,\\ldots,\\mathbf{X}^{(P)}_j  \\right)^{\\prime}$ are i.i.d.~with $ \\EW{\\|\\mathbf{X}_1\\|^{2+\\delta}}<\\infty $ for some $\\delta>0$.\nAdditionally,  \\cite{KuelbsPhilipp} state an invariance  principle for mixing random vectors in Theorem 4, additionally there are many corresponding univariate results under many different weak-dependency formulations.\n\nFor $\\mathbf{X}^{(i)}=\\mathbf{X}^{(1)}$ (and $\\boldsymbol{\\Sigma}^{(i)}=\\boldsymbol{\\Sigma}^{(1)}$)  for all $i$, then we are back to the classical multiple mean change problem that has been considered  in many papers in particular for the univariate situation, see e.g.~the recent survey papers by \\cite{fearnhead2020relating} or \\cite{cho2020data}.\n\n\n\n\n\\subsubsection{Renewal and some related point processes}\\label{sec_ex_renewal}\nThe second example aims at finding structural breaks in the rates of renewal and some related point processes:\n\nWe consider $P$ independent sequences of $p$-dimensional point processes that are related to renewal processes in the following way: For each $i=1,\\ldots,P$ we start with $\\tilde{p}\\ge p$ independent renewal processes $\\widetilde{R}_{t,j}^{(i)}$, $j=1,\\ldots,\\tilde{p}$, from which we derive a $p$-dimensional point process $\\mathbf{R}_t^{(i)}=\\mathbf{B}^{(i)} \\left(\\widetilde R_{t,1}^{(i)},\\ldots,\\widetilde R_{t,\\tilde p}^{(i)}\\right)^{\\prime}$, where $\\mathbf{B}^{(i)}$ is a $(p\\times\\tilde{p})$ - matrix with non-negative integer-valued entries.\nBy  Lemma 4.2 in \\cite{SteinebachEastwood}  Assumption \\ref{invariance_principle} is fulfilled for a block-diagonal $\\boldsymbol{\\Sigma}_T$ with\n\\begin{align*}\n\t&\\boldsymbol{\\Sigma}_T^{(i)}=\\mathbf{B}^{(i)}\\mathbf{D}\\left(\\frac{\\boldsymbol{\\sigma}^2(i)}{\\boldsymbol{\\mu}^{3}(i)}\\right){\\mathbf{B}^{(i)}}^{\\prime},\\\\\n\t&\\text{with }\\mathbf{D}\\left(\\frac{\\boldsymbol{\\sigma}^2(i)}{\\boldsymbol{\\mu}^{3}(i)}\\right)=\\operatorname{diag}\\left(\\frac{\\sigma_1^2(i)}{\\mu_1^{3}(i)},\\ldots,\\frac{\\sigma_{\\tilde{p}}^2(i)}{\\mu_{\\tilde{p}}^{3}(i)}\\right),\n\\end{align*}\n where $\\mu_j(i)$ and $\\sigma_j^2(i)$ are the mean and variance of the corresponding inter-event times. \\cite{SteinebachEastwood} and \\cite{Csenki} consider $\\tilde{p}=p$ but use inter-event times that are dependent for $j=1,\\ldots,p$. In such a situation, the invariance principle in  Assumption \\ref{invariance_principle} still holds if the intensities are the same across components with  $\\Sigma_T^{(i)}=\\Sigma_{\\IET}^{(i)}\/\\mu_1^3(i)$, where $\\Sigma_{\\IET}^{(i)}$ is the covariance of the vector of inter-event times -- a setting that we adopt in the simulation study. If the intensities differ, then by \\cite{SteinebachEastwood} an invariance principle towards a Gaussian process can still be obtained, but this is no longer a multivariate Wiener process. While each component is a Wiener process, the increments from one component may depend on the past of another. Many of the below results can still be derived in such a situation, however, such a model does not seem to be very realistic for most applications as the stochastic behavior of the increments of one component depends on the lagged behavior of the other components, where the lag increases with time. While a lagged dependence is realistic in many situations, in most situations one would expect this lagged-dependence to be constant across time.\n\n\n\t\t\\cite{Messer14} \n\t\tconsider this model for univariate renewal processes with varying variance. They propose a multiscale procedure based on MOSUM\\  statistics related to those we will discuss in the next section using linear bandwidths. In \\cite{Messer17}, they extend the procedure to processes with weak dependencies. They show convergence in distribution of the MOSUM\\   statistics to functionals of Wiener processes similar to the results that we obtain and analyze the behaviour of the signal term in \\cite{Messer172}. However, they have not derived any consistency results for the change point estimators. In this paper, we extend their results to sublinear bandwidths and prove the consistency of the corresponding estimators as well as their localisation rates.\n\\subsubsection{Diffusion processes}\nClearly, switching between independent (or components of a multivariate) Brownian motion with drift is included in this framework. Additionally,\n\\cite{Heunis} and  \\cite{Mihalache} derive invariance principles in the context of diffusion processes including Ornstein-Uhlenbeck processes among others. Let $ \\left(\\mathbf{X}_t\\right)_{t\\ge0} $ be a stochastic process in $ \\R^N $ satisfying a stochastic differential equation \n\\[ \nd\\mathbf{X}_t=\\boldsymbol{\\mu}\\left(\\mathbf{X}_t\\right)dt+ \\boldsymbol{\\Sigma}\\left(\\mathbf{X}_t\\right)d\\mathbf{B}_t\n \\]\n with respect to an $ n $-dimensional standard Wiener process $ \\left(\\mathbf{B}_t\\right)_{t\\ge0} $ and let $ \\boldsymbol{\\mu},\\boldsymbol{\\Sigma} $ be globally Lipschitz-continuous. \n Under some conditions on $ f:\\R^N\\to\\R^p $, as given by \\cite{Heunis},   relating to $ \\boldsymbol{\\mu},\\boldsymbol{\\Sigma} $, which in particular guarantee that the function $f$ applied to the (invariant) diffusion results in a centered process, \n there exists a $ p $-dimensional Wiener process $ \\left(\\mathbf{W}_t\\right)_{t\\ge0} $ and some $ \\eta>0 $ such that\n \\[ \n \\left\\|\\int_0^T f(\\mathbf{X}_s)\\ ds-\\mathbf{W}_T\\right\\|=O\\left(T^{1\/2-\\eta}\\right),\n  \\]\n  where $ \\left(\\mathbf{X}_t\\right)_{t\\ge0} $ either is a solution to the SDE with fixed starting value $ \\mathbf{X}_0=y_0 $ or a strictly stationary solution  with respect to an invariant distribution.\\\\\n  Furthermore, in the case of a one-dimensional stochastic diffusion process, \\cite{Mihalache} showed for some $ L^2 $-functions fulfilling constraints depending on $ \\boldsymbol{\\mu}, \\boldsymbol{\\Sigma} $ that there exists a strong invariance principle for the integrals of diffusion processes with a rate of $ O(\\left(T\\log_2 T\\right)^{1\/4}\\sqrt{\\log T}) $. \n\n\n\n\\section{Data segmentation procedure} \\label{section_procedures}\nNow, we are ready to introduce a MOSUM-based data segmentation procedure for stochastic processes following the above model:\n\n\n\\subsection{Moving sum statistics} \\label{subsection_mosum_statistics}\nAt every change point, the drift of the process that is active to the left of that change point is different from the drift of the  process that is active to the right of the change point. \nConsequently, the increment of the process $\\mathbf{R}$ to the left will systematically differ from the one to the right. On the other hand, in a stationary stretch away from any change point both increments will be approximately the same as they are estimating the same drift. It is this observation that gives rise to the following moving sum (MOSUM) statistic that is based on the moving difference of increments with bandwidth $h=h_T$\n\\begin{align}\\label{eq_mosum}\n\t\\mathbf{M}_t=\\mathbf{M}_{t,T,h_T}(\\mathbf{Z})&=\\frac{1}{\\sqrt{2h}}\\,\\left[\\left(\\mathbf{Z}_{t+h}-\\mathbf{Z}_t\\right)-\\left( \\mathbf{Z}_t-\\mathbf{Z}_{t-h} \\right)\\right]\\notag\\\\\n\t&=\\frac{1}{\\sqrt{2h}}\\,\\left(\\mathbf{Z}_{t+h}-2\\mathbf{Z}_t+\\mathbf{Z}_{t-h}\\right).\n\\end{align}\nIf there is no change, then this difference will fluctuate around 0.  On the other hand close to a change point, this difference will be different from 0. \nIdeally, the bandwidth should be chosen to be as large as possible (to get a better estimate obtained from a larger 'effective sample size' of the order $h$). On the other hand, the increments should not be contaminated by a second change as this can lead to situations where the change point can no longer be reliably localised by the signal.\nIndeed, we need the following assumptions on the bandwidth for a change to be detectable:\n\\begin{Assumption}\\ \\label{bandwidth}\nFor $\\nu_T$ as in Assumption~\\eqref{invariance_principle}\tthe bandwidth $h<T\/2$ fulfills\n\t\t\\begin{align*}\n&\t\\frac{\\nu_T^2\\,T\\,\\log T}{h}\\to 0.\n\\end{align*}\t\nFurthermore, it isolates the $i$-th change point in the sense of\n\\begin{align}\n\t\t&\t\th\\le \\frac 1 2 \\, \\Delta_i,\\qquad \\text{where }\\Delta_i=\\min(c_{i+1}-c_i,c_i-c_{i-1}). \\label{condition_bandwidth}\n\\end{align}\t\nAdditionally, the signal needs to be large enough to be detectable by this bandwidth,\ni.e.\\\n\t\\begin{align}\n\t\t\\frac\n\t\\|\\mathbf{d}_i\\|^2\\,h}{\\log\\left( \\frac{T}{h}\n\t\t\n\t\\right)}\\to\\infty.\\label{condition_mean_change}\n\t\t\\end{align}\n\\end{Assumption}\nCombining \\eqref{condition_bandwidth} and \\eqref{condition_mean_change} shows that -- with an appropriate bandwidth $h$ -- changes are detectable as soon as\n\\begin{align}\\label{eq_separation}\n\t\t\\frac\n\t\\|\\mathbf{d}_i\\|^2\\,\\Delta_i}{\\log\\left( \\frac{T}{\\Delta_i}\\right)\n\t\t\n\t}\\to\\infty.\n\t\t\\end{align}\n\t In case of the classical mean change model as in Subsection~\\ref{sec_partial_sums} this is known to be the minimax-optimal separation rate that cannot be improved (see Proposition 1 of \\cite{AriasCastro}).\n\n\t The assumption on the distance of the first and last change point to the boundary of the process in \\eqref{condition_bandwidth} can be relaxed as no boundary effects can occur there.\n\n\n\n\t\t\\subsection{Change point estimators} \\label{subsection_cp_estimators}\n\t\tThe MOSUM\\  statistic $\\mathbf{M}_t= \\mathbf{m}_{t}+ \\boldsymbol{\\Lambda}}%{\\mathbf{N}_t$ as in \\eqref{eq_mosum}  decomposes into a piecewise linear signal term $ {\\mathbf{m}_t=\\mathbf{m}_{t,h,T}} $ and a centered noise term $\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t=\\boldsymbol{\\Lambda}}%{\\mathbf{N}_{t,h,T}$ with\n\t\t\\begin{align}\n\t\t&{\\sqrt{2h}}\t\\,\\mathbf{m}_t=\\begin{cases}\n\t\t\t\t(h-t+c_i)\\,\\mathbf{d}_i,&\\ \\text{for }c_i<t\\le c_i+h,\\\\\n\t\t\t\t0,&\\ \\text{for }c_i+h<t\\le c_{i+1}-h,\\\\\n\t\t\t\t(h+t-c_{i+1})\\,\\mathbf{d}_{i+1},&\\ \\text{for }c_{i+1}-h<t\\le c_{i+1},\n\t\t\t\\end{cases} \\label{signal}\\\\[3mm]\n\t\t\t&\t\\sqrt{2h}\\;\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t= \\sqrt{2h}\\;\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t \\label{noise}(\\widetilde{\\mathbf{R}}) \t\\\\*\n\t\t&\\quad =\\begin{cases}\n\t\t\t\t\\widetilde{\\mathbf{R}}_{t+h}^{(c_{i+1})}-2\\widetilde{\\mathbf{R}}_{t}^{(c_{i+1})}+\\widetilde{\\mathbf{R}}_{c_i}^{(c_{i+1})}\n\t\t\t\t-\\widetilde{\\mathbf{R}}_{c_i}^{(c_{i})}+\\widetilde{\\mathbf{R}}_{t-h}^{(c_{i})},&\\ \\text{for }c_i<t\\le c_i+h,\\\\\n\t\t\t\t\\widetilde{\\mathbf{R}}_{t+h}^{(c_{i+1})}-2\\widetilde{\\mathbf{R}}_{t}^{(c_{i+1})}+\\widetilde{\\mathbf{R}}_{t-h}^{(c_{i+1})},\n\t\t\t\t&\\ \\text{for }c_i+h<t\\le c_{i+1}-h,\\\\\n\t\t\t\t\\widetilde{\\mathbf{R}}_{t+h}^{(c_{i+2})}-\\widetilde{\\mathbf{R}}_{c_{i+1}}^{(c_{i+2})}+\\widetilde{\\mathbf{R}}_{c_{i+1}}^{(c_{i+1})}-2\\widetilde{\\mathbf{R}}_{t}^{(c_{i+1})}+\\widetilde{\\mathbf{R}}_{t-h}^{(c_{i+1})},\n\t\t\t\t&\\ \\text{for }c_{i+1}-h<t\\le c_{i+1},\t\n\t\t\t\\end{cases} \\notag\n\\end{align}\nwhere $ \\widetilde{\\mathbf{R}}_t:=\\mathbf{R}_t-t\\boldsymbol{\\mu} $ for $ i=0,\\ldots,q_T $  and the upper index $c_{j}$ denotes the active regime between the $(j-1)$st and $j$-th change point (with a slight abuse of notation).\\\\\nThe\nsignal term is a\npiecewise linear function that takes its extrema at the change points and is 0 outside $h$-intervals around the change points. Additionally,  the noise term is asymptotically negligible compared to the signal term (see Theorem \\ref{theorem_threshold} for the corresponding theoretic statement and Figure \\ref{figure_signal_noise} for an illustrative example).\n\\begin{figure}\n\t\\includegraphics*[scale=0.44]{Signal_noise2.pdf}\n\t\\caption{Univariate MOSUM\\  statistic with $T=100,\\, 1\\,000,\\, 10\\,000$ (from left to right), where the noise term (fluctuating around the signal)  becomes smaller and smaller relative to the signal term.}\\label{figure_signal_noise}\n\\end{figure} \n\\begin{figure}[t]\n\t\t\t\\caption{In the upper panel, the observed event times of a univariate renewal process with 3 change points (i.e.\\ 4 stationary segments) are displayed (where the plot needs to be read like a text: It starts in the upper row on the left, then continues in the first row and jumps to the second row and so on). The grey and white regions mark the estimated segmentation of the data while the red intervals mark the true segmentation.\\\\\n\t\tIn the lower panel, the corresponding MOSUM\\  statistic with (relative) bandwidth $h\/T=0.07$ is displayed. The grey areas are the regions where the threshold ($\\alpha=0.05$ as in Remark~\\ref{rem_choice_threshold})  is exceeded (in absolute value). The blue solid lines  indicate the change point estimates obtained as local extrema that fall within the grey area (making them \\emph{significant}). The true change points are indicated by the red dashed lines. }\\label{figure_rp_example}\n\t\\includegraphics*[width=\\textwidth]{RP_example13.pdf}\n\n\t\\end{figure} \n\t\n\n\nThese observations motivate the following data segmentation procedure, that considers local extrema that are big enough (in absolute value) as change point estimators: \n\nTo this end,  a suitable threshold $\\beta=\\beta_{h,T}$ is needed to define \\emph{significant} time points, where a point $t^*$ is \\emph{significant} if\n\\begin{align}\\label{eq_mosum_signif}\n\t\\mathbf{M}_{t^*}^{\\prime}\\,\\widehat{\\mathbf{A}}^{-1}_{t^*}\\,\\mathbf{M}_{t^*}\\ge\\beta\n\\end{align}\nwith $\\widehat{\\mathbf{A}}_{t^*}$ is some symmetric positive definite matrix that may depend on the data fulfilling\n\\begin{Assumption}\n\\label{variance_estimator}\n\\begin{align*}\n\t&\\sup_{h\\le t\\le T-h}\\left\\|\\widehat{\\mathbf{A}}_{t,T}^{-1}\\right\\|=O_P\\left(1\\right),\\quad\n\t\\sup_{i=1,\\ldots,q_T}\\sup_{|t-c_i|\\le h}\\left\\|\\widehat{\\mathbf{A}}_{t,T}\\right\\|=O_P(1).\n\\end{align*}\n\\end{Assumption}\nA good (non data-driven) choice  fulfilling this assumption is given by\n\\begin{align}\\label{eq_sigma_t}\n\t\\boldsymbol{\\Sigma}_{t}=\\boldsymbol{\\Sigma}_{t,T}=\\boldsymbol{\\Sigma}_T^{(c_{i})}\n\\end{align}\nfor $c_{i-1}<t\\le c_{i}$, which guarantees scale-invariance of the procedure and allows for nicely interpretable thresholds (see Section~\\ref{subsection_threshold}). The latter remains true for estimators as long as they fulfill\n\\begin{align}\n\t\\sup_{i=1,\\ldots,q_T}\\sup_{|t-c_i|> h}\\left\\|\\widehat{\\boldsymbol{\\Sigma}}_{t,T}^{-1\/2}-\\boldsymbol{\\Sigma}_t^{-1\/2}\\right\\|=o_P\\left(\\left(\\log \\frac{T}{h}\\right)^{-1}\\right) \\label{variance_estimator2}\n\\end{align}\nin addition to the above boundedness assumptions. In particular, this permits local estimators that are consistent only away from change points but contaminated by the change in a local environment thereof. The latter is typically the case for covariance estimators, think e.g.\\ of the sample variance contaminated by a change point. In order to not reduce detection power in small samples, it is beneficial if the estimator is additionally consistent directly at the change point, which is also achievable (see e.g.\\ \\cite{EichingerKirch}).\n\nThe choice of the threshold $\\beta$ will be discussed in the next section, where we can make use of asymptotic considerations in the no-change situation. \n\nTypically, there are intervals of significant points (due to the continuity of the signal) such that only local extrema of such intervals actually indicate a change point. To define what a local extremum is, we require  a tuning parameter $0<\\eta <1 $. This parameter defines the locality requirement on the extremum, where a  point $t^*$ is a local extremum if it maximizes the absolute  MOSUM\\  statistic within its $\\eta h$-environment, i.e.\\ if\n\\begin{align}\n\tt^*=\\min\\left\\{ \\argmaxx{t^*-\\eta h\\le t\\le t^*+\\eta h}\\|\\mathbf{M}_t\\|.\n\t\\right\\} \\label{cp_estimator}\n\\end{align}\n\nThe threshold $\\beta$  distinguishes between \\emph{significant}  and \\emph{spurious} local extrema that are purely associated with the noise term.\nThe set of all significant local extrema is the set of change point estimators with its cardinality an estimator for the number of the change points.\n \n\n\tFigure~\\ref{figure_rp_example} shows an example illustrating these ideas: Away from the change points the MOSUM\\  statistic fluctuates  around 0 (within the white area that is beneath the threshold in absolute value) while it falls within the grey area close to the change points -- making corresponding local extrema significant.\n\tFurthermore, the statistic does not need to return to the white area in order to have all changes estimated, as can be seen between the first and second change point. This is one of the major advantages of the $\\eta$-criterion based on \\emph{significant} local maxima as described here (in comparison to the $\\epsilon$-criterion originally investigated by \\cite{EichingerKirch} in the context of mean changes, see also the discussion in \\cite{ChoKirchMeier}). Nevertheless, results for the $\\epsilon$-criterion can be obtained along the lines of our proofs below.\n\t\n\t\n\t\n\n\n\n \t\\subsection{Threshold selection}\\label{subsection_threshold}\n\tAs pointed out above we need to choose a threshold $\\beta=\\beta_{h,T}$ that can distinguish between significant and spurious local extrema. The following theorem gives the magnitudes of signal as well as noise terms:\n\t\\begin{Theorem}\\label{theorem_threshold}\\\n\t\tLet the Assumptions~\\ref{invariance_principle}, \\ref{bandwidth} and \\ref{variance_estimator} hold.\n\n\t\n\t\\begin{enumerate}[(a)]\n\t\t\\item \tFor the signal $\\mathbf{m}_t$ with $c_i-h<t<c_i+h$, it holds\n\t\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\t\\mathbf{m}_t^{\\prime}\\widehat{\\mathbf{A}}_t^{-1}\\mathbf{m}_t \\ge \\frac{1}{2\\|\\widehat{\\mathbf{A}}_t\\|}\\frac{(h-\\abs{t-c_i})^2}{h}\\|\\mathbf{d}_i\\|^2.\n\t\t\t\\end{align*}\n\t\t\tAt other time points the noise term is equal to zero.\n\t\t\t\\item For the noise term it holds for $q_T=0$, i.e.\\ in the no-change situation\t\t\t\\begin{enumerate}[(i)]\n\t\t\t\t\\item for a linear bandwidth $h=\\gamma T$ with $0<\\gamma<1\/2$ \t\t\t\\begin{align*}\n\t\t\t\t\t\t&\\sup_{\\gamma T \\le t\\le T-\\gamma t} \n\t\\boldsymbol{\\Lambda}}%{\\mathbf{N}_{t}^{\\prime}\\boldsymbol{\\Sigma}_{T}^{-1}\\boldsymbol{\\Lambda}}%{\\mathbf{N}_{t}\\\\\n\t&\\qquad\\overset{\\mathcal{D}}{\\longrightarrow} \\sup_{\\gamma\\le s\\le 1-\\gamma}\\frac{1}{2\\gamma}\\left({\\mathbf{B}}_{s+\\gamma}-2{\\mathbf{B}}_s+{\\mathbf{B}}_{s-\\gamma}\\right)^{\\prime}\\left({\\mathbf{B}}_{s+\\gamma}-2{\\mathbf{B}}_s+{\\mathbf{B}}_{s-\\gamma}\\right), \n\t\t\t\\end{align*}\n\t\twhere ${\\mathbf{B}} $ denotes a multivariate standard Wiener process.\\\\ \n\t\t\tIn particular, the squared noise term is of order $O_P(1)$ in this case.\n\t\\item for a sublinear bandwidth $h\/T\\to 0$ but Assumption \\ref{bandwidth} fulfilled, it holds that\n\t\\begin{align*}\n\t\ta\\left(\\frac{T}{h}\\right)\\supp{h\\le t\\le T-h}\\sqrt{\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t^{\\prime}\\boldsymbol{\\Sigma}_T^{-1}\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t }-b\\left(\\frac{T}{h}\\right)\\overset{\\mathcal{D}}{\\longrightarrow} E,\n\t\\end{align*}\n\twhere $ E $ follows a Gumbel distribution with $ \\Prob{E\\le x}=e^{-2e^{-x}} $ and \n\t\\begin{align*}\n\t\ta(x)=&\\ \\sqrt{2\\log x}\\\\\n\t\tb(x)=&\\ 2\\log x+\\frac{p}{2}\\log\\log x+\\log\\frac{3}{2}-\\log\\Gamma\\left(\\frac{p}{2}\\right).\n\t\\end{align*}\n\t\tIn particular, the above squared noise term is of order $O_P\\left( \\log(T\/h)  \\right)$ in this case.\n\\end{enumerate}\nThe assertions remain true if an estimator for the covariance is used fulfilling \\eqref{variance_estimator2} uniformly over all $h\\le t\\le T-h$.\n\\item In the situation of multiple change points, it holds that\n\\begin{align*}\n\t\\sup_{h \\le t\\le T-h} \\|\\boldsymbol{\\Lambda}}%{\\mathbf{N}_t\\|=&\\ O_P(\\sqrt{\\log(T\/h)}).\n\n\\end{align*}\n\\end{enumerate}\n\\end{Theorem}\n\n In order to obtain consistent estimators, on the one hand,\n the threshold needs to be asymptotically negligible compared to the squared signal term as in Theorem~\\ref{theorem_threshold} (a). This guarantees that every change is detected with asymptotic probability 1. On the other hand, the threshold needs to grow faster than the squared noise term in Theorem~\\ref{theorem_threshold} (c) so that  false positives occur with asymptotic probability 0. Hence, both conditions are fulfilled under the following assumption:\n\n\\begin{Assumption}\\label{threshold} The threshold fulfills:\n\\begin{align*}\n\t\\frac{\\beta_{h,T}}{h_T\\ \\min\\limits_{i=1,\\ldots,q_T}\\|\\mathbf{d}_i\\|^2}&\\ \\to 0,\\qquad\n\t\\frac{\\log\\frac{T}{h_T}}{\\beta_{h,T}}\\ \\to 0 \\qquad (T\\to \\infty).\n\\end{align*}\n\\end{Assumption}\n\nThe following remark introduces a threshold that has a nice interpretation in connection with change point testing:\n\\begin{Remark}\\label{rem_choice_threshold}\n\tA common choice for the threshold is obtained as the asymptotic $\\alpha_T$-quantile obtained from Theorem~\\ref{theorem_threshold} (b) for some sequence $\\alpha_T\\to 0$. In the simulation study in Section~\\ref{subsection_simulation} we use this  threshold with $\\alpha_T=0.05$.\nUsing the asymptotic $\\alpha$-quantile (from the no-change situation) guarantees that all change point estimators are significant at a global level $\\alpha$, where significant is meant in the usual testing sense. This gives this choice a nice interpretability.  In fact, Theorem~\\ref{theorem_threshold}\nshows that such a threshold with a constant sequence $\\alpha$ yields an asymptotic test at level $\\alpha$ which has asymptotic power one by Theorem~\\ref{theorem_consistency}. Nevertheless, often, tests designed for the at-most-one-change as in \\cite{HuskovaSteinebach00}, \\cite{HuskovaSteinebach02} have a better power, but are not as good at localising change points (see Figure~1 in \\cite{cho2020data} for an illustration).\n\\end{Remark}\n\n\n\n\n\n\n\\section{Consistency of the segmentation procedure}\\label{section_consistency}\nIn this section, we will show consistency of the above segmentation procedure for both the estimators of the number and locations of the change points. \nFurthermore, we derive localisation rates for the estimators of the locations of the change points for some special cases showing that they cannot be improved in general. This is complemented by the observations that these localisation rates are indeed minimax-optimal if the number of change points is bounded in addition to observing Wiener processes with drift. Otherwise the generic rates that are obtained based solely on the invariance principle will not be tight in the sense that the proposed procedure can provide better rates than suggested by the invariance principle.\n\n\nThe following theorem shows that the change point estimators defined in \\eqref{cp_estimator} are consistent for the number and locations of the change points.\n\\begin{Theorem}\\label{theorem_consistency}\nLet Assumptions \\ref{invariance_principle}, \\ref{bandwidth} -- \\ref{threshold} hold.  Let $ 0<\\hat{c}_1<\\ldots<\\hat{c}_{\\hat{q}_T} $ be the change point estimators of type \\eqref{cp_estimator}. Then for any $\\tau>0$ it holds\n\\begin{align*}\n\t\\lim_{T\\to\\infty}\\Prob{\\max_{i=1,\\ldots,\\min(\\hat{q}_T,q_T)}\\left|\\hat{c}_i-c_i\\right|\\le \\tau h,\n\t\\hat{q}_T=q_T}=\\ 1.  \n\\end{align*}\n\\end{Theorem}\n\n\nThe theorem shows in particular that the number of change points is estimated consistently. For the linear bandwidth we additionally get consistency of the change point locations in rescaled time, while for the sublinear bandwidths we already get a convergence rate of $h\/T$ towards the rescaled change points.\n\n\n\nUnder the following stronger assumptions, the localisation rates can be further improved:\n\n\\begin{Assumption}\\label{Hajek_Renyi}\\begin{enumerate}[(a)]\n\t\t\t\\item It holds  for any of the centered processes $\\widetilde{\\mathbf{R}}^{(j)}$ as in \\eqref{noise} and any value $\\theta_i=\\theta_{i,T}$ (which will be $c_i$ or $c_i\\pm h$ when the assumption is applied) for any sequence $D_T\\ge 1$ (bounded or unbounded)\n\t\\begin{align*}\n\t\t\\sup_{\\frac{D_T}{\\|\\mathbf{d}_i\\|^2}\\le s \\le h} \\frac{\\sqrt{D_T} \\left\\|\\widetilde{\\mathbf{R}}_{\\theta_i}^{(j)}-\\widetilde{\\mathbf{R}}_{\\theta_i\\pm s}^{(j)}\\right\\|}{s\\left\\|\\mathbf{d}_i\\right\\|}=O_P(\\omega_T).\n\t\\end{align*}\n\\item Let now the upper index $\\theta_i$ denote the active stretch in the stationary segment $(\\theta_i,\\theta_i+s)$ respectively $(\\theta_i-s,\\theta_i)$.   Then, it holds for any sequence $D_T>0$\n\t\\begin{align*}\n\t\t\\max_{i=1,\\ldots,q_T}\t\\sup_{\\frac{D_T}{\\|\\mathbf{d}_i\\|^2}\\le s \\le h} \\frac{\\sqrt{D_T} \\left\\|\\widetilde{\\mathbf{R}}_{\\theta_i}^{(\\theta_i)}-\\widetilde{\\mathbf{R}}_{\\theta_i\\pm s}^{(\\theta_i)}\\right\\|}{s\\left\\|\\mathbf{d}_i\\right\\|}=O_P(\\tilde{\\omega}_T).\n\t\\end{align*}\n\t\\end{enumerate}\n\\end{Assumption}\nThe localisation rates of the MOSUM\\  procedure are determined by the  rates $\\omega_n,\\tilde{\\omega}_n$ which need to be derived for each example separately (at least for the tight ones). In the context of partial sum processes these results are well known. For example, the suprema in (a) are stochastically bounded by the H\\'aj\\'ek-R\\'enyi inequality which has been shown for partial sum processes even with weakly dependent errors. In that context, the assertion in (b) is fulfilled with a polynomial rate in $q_T$ (see \\cite{ChoKirch}, Proposition 2.1 (c)(ii)).\n\n\n\n\\begin{Remark}\\label{rem_rates}\n\t\\begin{enumerate}[(a)]\n\t\t\\item  For Wiener processes with drift $\\omega_T=1$ and $\\tilde{\\omega}_T=\\sqrt{\\log(q_T)}$ (see Proposition~\\ref{prop_WP_rates} below).\t\n\t\t\\item \tBy the invariance principle in Assumption~\\ref{invariance_principle}, all rates are clearly dominated by $T^{1\/2}\\nu_T$.\t\t\tHowever, this is often far too liberal a bound (see Proposition 2.1 in \\cite{ChoKirch} for some tight bounds in case of partial sum processes). \n\t\t\\item Often, for each regime there exist forward and backwards invariance principles from some arbitrary starting value $\\theta_i$. This is the case for partial sum processes and for (backward and forward) Markov processes due to the Markov property. For renewal processes, this can be shown along the lines of the original proof for the invariance principle (\\cite{CsorgoHorvathSteinebach}) because the time to the next (previous) event is asymptotically negligible; see also Example 1.2 in \\cite{KuehnSteinebach}).\n\t\t\tIn this case, the H\\'aj\\'ek-R\\'enyi results for Wiener processes carry over (see Proposition~\\ref{prop_WP_rates}) to the different processes underlying each regime, resulting in $\\omega_T=1$. For the situation with a bounded number of change points this carries over to $\\tilde{\\omega}_T$.\n\t\\end{enumerate}\n\\end{Remark}\n\n\n\n\\begin{Theorem}\\label{theorem_convergence_rate}\\ \\\\\n\tLet Assumptions \\ref{invariance_principle}, \\ref{bandwidth} -- \\ref{threshold} in addition to \\ref{Hajek_Renyi} hold.\n\tFor $\\hat{q}_T<q_T$ define $\\hat{c}_i=T$ for $i=\\hat{q}_T+1,\\ldots,q_T$.\n\\begin{enumerate}[(a)]\n\t\\item\n\tFor a single change point estimator the following localisation rate holds \t\t\\[ \n\t\t\\left\\|\\mathbf{d}_i\\right\\|^2\\left|\\hat{c}_{i}-c_i\\right|\t=O_P\\left(\\omega_T^2\\right).\n\t\\]\n\\item The following uniform rate holds true:\n\t\\[ \n\t\t\\max_{i=1,\\ldots,q_T}\\left\\|\\mathbf{d}_i\\right\\|^2\\left|\\hat{c}_{i}-c_i\\right|\t=O_P\\left(\\tilde{\\omega}_T^2\\right).\n\t\\]\n\\end{enumerate}\n\n\\end{Theorem}\n\n\\begin{Remark}[Minimax optimality]\\label{Remark_minimax}\n\tWe have already mentioned beneath \\eqref{eq_separation} that the separation rate given there is minimax optimal (see Proposition 1 of \\cite{AriasCastro}).\n\n\tMinimax optimal localisation rates (derived in the context of changes in the mean of univariate time series, which is  covered by the partial sum processes in our framework) are known for a few special cases: First, the minimax optimal localisation rate for a single change point and in extension also for a bounded number of change points is given by $\\omega_T=1$ in the above notation (see e.g.\\ Lemma 2  in \\cite{wang2020univariate}).  In particular this shows that our procedures achieves the minimax optimality in case of a bounded number of change points under weak assumptions (as pointed out in Remark~\\ref{rem_rates} (c)). \nSecondly, the optimal localisation rate for unbounded change points under sub-Gaussianity (attained for partial sum process of i.i.d.\\ errors) is given by $\\tilde{\\omega_T}=\\sqrt{\\log T}$ (see Proposition 6 in \\cite{Verzelenetal} and Proposition 2.3 in \\cite{ChoKirch}). Indeed, we match this rate for Wiener processes with drift.\n\\end{Remark}\n\n\nThe following theorem derives the limit distribution of the change point estimators for local changes which shows in particular that the rates are tight. In principle, this result can be used to obtain asymptotically valid confidence intervals for the change point locations. In case of fixed changes, the limit distribution depends on the underlying distribution of the original process (see \\cite{AntochHuskova_fixed}\n for the case of partial sum processes), where the proof can be done along the same lines.\nWe need the following assumption:\n\n\\begin{Assumption}\\label{ass_cp_distr}\n\tLet $\\mathbf{d}_i=\\mathbf{d}_{i,T}=\\|\\mathbf{d}_i\\| \\mathbf{u}_i + o(\\|\\mathbf{d}_i\\|)$ with $\\|\\mathbf{u}_i\\|=1$ and $\\|\\mathbf{d}_{i,T}\\|\\to 0$.\nAssume that\t $\\mathbf{Y}^{(j)}_s=\\mathbf{Y}^{(j)}_s(c_i,D)$ with\n\t\\begin{align*}\n\t\t&\t\t\\mathbf{Y}^{(1)}_s=\\widetilde{\\mathbf{R}}^{(c_i)}_{c_i-h+\\frac{s-D}{\\|d_i\\|^2}}-\\widetilde{\\mathbf{R}}^{(c_i)}_{c_i-h-\\frac{D}{\\|d_i\\|^2}},\\\\\n&\t\t\\mathbf{Y}^{(21)}_s=\\widetilde{\\mathbf{R}}^{(c_i)}_{c_i+\\frac{s-D}{\\|d_i\\|^2}}-\\widetilde{\\mathbf{R}}^{(c_i)}_{c_i-\\frac{D}{\\|d_i\\|^2}},\\qquad\t\\mathbf{Y}^{(22)}_s=\\widetilde{\\mathbf{R}}^{(c_{i+1})}_{c_i+\\frac{s-D}{\\|d_i\\|^2}}-\\widetilde{\\mathbf{R}}^{(c_{i+1})}_{c_i-\\frac{D}{\\|d_i\\|^2}},\\\\\n&\t\t\\mathbf{Y}^{(3)}_s=\\widetilde{\\mathbf{R}}^{(c_{i+1})}_{c_i+h+\\frac{s-D}{\\|d_i\\|^2}}-\\widetilde{\\mathbf{R}}^{(c_{i+1})}_{c_i+h-\\frac{D}{\\|d_i\\|^2}}\n\t\\end{align*}\n\tfulfill the following multivariate functional central limit theorem for any constant $D>0$ in an appropriate space equipped with the supremum norm\n\t\\begin{align*}\n\t\t\\left\\{\\|\\mathbf{d}_i\\|\\,(\\mathbf{Y}^{(1)}_s,\\mathbf{Y}^{(21)}_s,\\mathbf{Y}^{(22)}_s,\\mathbf{Y}^{(3)}_s)^{\\prime}:0\\le s\\le 2D\\right\\}\\overset{w}{\\longrightarrow}\\left\\{\\widetilde{\\mathbf{W}}_s:0\\le s\\le 2D\\right\\},\n\t\\end{align*}\n\twhere $\\widetilde{\\mathbf{W}}$ is a Wiener process with covariance matrix $\\boldsymbol{\\Xi}$ (not depending on $D$). For $-D\\le t\\le D$ denote $\\mathbf{W}_t=(\\mathbf{W}^{(1)}_t,\\mathbf{W}^{(21)}_t,\\mathbf{W}^{(22)}_t,\\mathbf{W}^{(3)}_t)^{\\prime}=\\widetilde{\\mathbf{W}}_{D+t}-\\widetilde{\\mathbf{W}}_D$.\n\\end{Assumption}\n\nBy Assumption~\\ref{bandwidth} it holds $h\\|\\mathbf{d}_i\\|^2\\to \\infty$, such that the distance $h-\\frac{2D}{\\|d_i\\|^2}$ between $\\mathbf{Y}^{(1)}$ and $\\mathbf{Y}^{(2j)}$ (resp.\\ between $\\mathbf{Y}^{(2j)}$ and $\\mathbf{Y}^{(3)}$) diverges to infinity. \nAs such for processes with independent increments the processes $\\mathbf{Y}^{(1)}$, $(\\mathbf{Y}^{(21)},\\mathbf{Y}^{(22)})^{\\prime}$, $\\mathbf{Y}^{(3)}$  are independent for $T$ large enough. \nAdditionally, under weak assumptions such as mixing conditions this independence still holds asymptotically in the sense that $\\mathbf{W}^{(1)}$, $(\\mathbf{W}^{(21)},\\mathbf{W}^{(22)})^{\\prime}$, $\\mathbf{W}^{(3)}$  are independent. \n\nFunctional central limit theorems for these processes follow from invariance principles as in Assumption~\\ref{invariance_principle} with $\\boldsymbol{\\Sigma}_T\\to \\boldsymbol{\\Sigma}$  as long as such invariance principles still hold with an arbitrary (moving) starting value, which is typically the case (see also Remark~\\ref{rem_rates} (c)). As such, it typically holds that $\\boldsymbol{\\Xi}^{(1)}=\\boldsymbol{\\Xi}^{(21)}=\\boldsymbol{\\Sigma}^{(c_i)}$ and $\\boldsymbol{\\Xi}^{(3)}=\\boldsymbol{\\Xi}^{(22)}=\\boldsymbol{\\Sigma}^{(c_{i+1})}$\nwhere $\\boldsymbol{\\Xi}^{j}=\\operatorname{Cov}(\\mathbf{W}^{(j)}_1)$ and $\\boldsymbol{\\Sigma}^{(c_i)}$ is the covariance matrix associated with the regime between the $(i-1)$st and $i$th change point.\n\nThe following theorem gives the asymptotic distribution for the change point estimators in case of local change points.\n\\begin{Theorem}\\label{limit_distribution}\\ \\\\\n\tLet Assumptions \\ref{invariance_principle}, \\ref{bandwidth} -- \\ref{threshold}, \\ref{Hajek_Renyi} (a) with $\\omega_T=1$  and \\ref{ass_cp_distr} hold. For $\\hat{q}_T<q_T$ define $\\hat{c}_i=T$ for $i=\\hat{q}_T+1,\\ldots,q_T$.\n\tLet \t\n\t\\begin{align*}\n\t\\Psi_t^{(i)}:=-\\abs{t}+\\begin{cases}\n\t\t\\mathbf{u}_i^{\\prime}\\mathbf{W}_{t}^{(1)}-2\\,\\mathbf{u}_i^{\\prime}\\mathbf{W}_{t}^{(21)}+ \\mathbf{u}_i^{\\prime}\\mathbf{W}_t^{(3)},&\\ t<0\\\\\t\n\t\t\\mathbf{u}_i^{\\prime} \\mathbf{W}_{t}^{(1)}-2\\,\\mathbf{u}_i^{\\prime}\\mathbf{W}_{t}^{(22)}+\\mathbf{u}_i^{\\prime}\\mathbf{W}_t^{(3)},&\\ t\\ge 0.\n\t\\end{cases}\n\t\\end{align*}\n\tThen, for all $ i=1,\\ldots,q_T, $ it holds that for $T\\to\\infty$\n\t\t\\[ \n\\left\\|\\mathbf{d}_i\\right\\|^2\\left(\\hat{c}_{i}-c_i\\right)\\overset{\\mathcal{D}}{\\longrightarrow} \\argmax\\geschweift{\\Psi_t^{(i)}\\Big| t\\in\\R}\n\t\\]\n\n\tIf there is a fixed number of changes $q_T=q$ with $q$ fixed and a functional central limit theorem as in Assumption~\\ref{ass_cp_distr} holds jointly for all $q$ change points, then the result also holds jointly.\n\\end{Theorem}\n\nDue to the Markov property of Wiener processes, $\\{\\Psi_t^{(i)}:t\\ge 0\\}$ is independent of $\\{\\Psi_t^{(i)}:t\\le 0\\}$.\n\n\n\n\\begin{Remark}\n\t\\begin{enumerate}[(a)]\n\t\t\\item If $\\mathbf{W}^{(1)}$, $(\\mathbf{W}^{(21)},\\mathbf{W}^{(22)})^{\\prime}$, $\\mathbf{W}^{(3)}$  are independent which is typically the case (see discussion beneath Assumption~\\ref{ass_cp_distr}), then \t$\\Psi_t^{(i)}$ simplifies to\n\\begin{align*}\n\t\\Psi_t^{(i)}:=-\\abs{t}+\\begin{cases}\n\t\t\\sqrt{\\sigma^2_{(1)}+4\\,\\sigma^2_{(21)}+ \\sigma^2_{(3)}}\\,B_t,&\\ t<0\\\\\t\n\t\t\\sqrt{\\sigma^2_{(1)}+4\\, \\sigma^2_{(22)}+\\sigma^2_{(3)}}\\,B_t,&\\ t\\ge 0,\n\t\\end{cases}\n\t\\end{align*}\n\twhere $B$ is a (univariate)  standard Wiener process and  $\\sigma_{(j)}^2=\\mathbf{u}_i^{\\prime}\\boldsymbol{\\Xi}^{(j)}\\mathbf{u}_i$. Usually (see discussion beneath Assumption~\\ref{ass_cp_distr})  $\\sigma_{(21)}=\\sigma_{(1)}$ and $\\sigma_{(22)}=\\sigma_{(3)}$ further simplifying the expression.\tFor some examples such as partial sum processes it holds $\\boldsymbol{\\Sigma}_t=\\boldsymbol{\\Sigma}$ for all $t$, such that all $\\sigma_{(j)}$ coincide. In this case this further simplifies to \n\\begin{align*}\n\t\\Psi_t^{(i)}:=-\\abs{t}+\\sqrt{6}\\, \\sigma_{(1)}\\,B_{t}.\n\\end{align*}\nFor univariate partial sum processes this result has already been obtained in Theorem 3.3 of \\cite{EichingerKirch}. However, the assumption of $\\boldsymbol{\\Sigma}_t=\\boldsymbol{\\Sigma}$ is typically not fulfilled for renewal processes because the covariance depends on the changing intensity of the process.\n\t\\item If  $\\mathbf{W}^{(1)}$, $(\\mathbf{W}^{(21)},\\mathbf{W}^{(22)})^{\\prime}$, $\\mathbf{W}^{(3)}$  are independent and $\\mathbf{M}_t$  in \\eqref{cp_estimator} is replaced by $\\boldsymbol{\\Sigma}_t^{-1\/2}\\mathbf{M}_t$, then the Wiener processes $\\mathbf{W}^{(j)}$ are standard Wiener processes, such that $\\Psi_t^{(i)}$ simplifies to\n\t\t\\begin{align*}\n\t\t\t\t\\Psi_t^{(i)}:=-\\abs{t}+\\sqrt{6}\\, B_{t}.\n\t\t\\end{align*}\n\t\tThis shows that in this case the limit distribution of $\\hat{c}_{i}-c_i$ does only depend on the magnitude of the change $\\mathbf{d}_i$ but not on its direction $\\mathbf{u}_i$.\n\t\t\t\n\t\t\t\tStatistically, however, this is difficult to achieve as it requires a uniformly (in $t$) consistent estimator for the usually unknown covariance matrices $\\boldsymbol{\\Sigma}_t$.\n\n\\end{enumerate}\n\\end{Remark}\n\n\\section{Simulation study}\\label{subsection_simulation}\nIn this section, we illustrate the performance of our procedure for multivariate renewal processes by means of a simulation study. Related simulations in addition to a variety of  data examples for partial sum processes have been conducted by \\cite{EichingerKirch,ChoKirchMeier} and for univariate renewal processes by \\cite{Messer14,Messer17}.\n\nMore precisely, we analyse three-dimensional renewal processes with $T=1600$, where the increments of the\ninter-event times for each component are \n$ \\Gamma- $distributed with intensity changes at 250, 500, 900 and 1150, where the expected time  $\\mu$ between events is given by $1.3$, $0.9$, $0.6$, $0.8$ and $1.3$.  We use a bandwidth of $ h=120$ and the parameter $\\eta=0.75$. Smaller values of $\\eta$ as suggested by \\cite{ChoKirchMeier} for partial sum processes tend to produce duplicate change point estimators by having two or more significant local maxima for each change point if the variance is too large. For a single-bandwidth MOSUM\\ procedure as suggested here, this should be avoided but can be relaxed if a post-processing procedure is applied as e.g.\\ by \\cite{ChoKirch} for partial sum processes.\n\nIn contrast to partial sum processes, it is natural for renewal processes that the variances change with the intensity, therefore \nwe consider the following three scenarios:  (i) standard deviations of constant value 0.7 (referred to as {\\texttt{constvar}}), (ii)  standard deviations being $ 5\/6  \\mu $ (referred to as {\\texttt{smallvar}}) and (iii) multivariate Poisson processes (referred to as {\\texttt{Poisson}}).\n\nWe consider both the case of independence and dependence between the three components.\nIn the latter case, we generate for each regime $i$ an independent (in time) sequence of $ \\Gamma- $distributed inter-event-times $Y_{j}=Y_{j}^{(i)}$, $j=1,2,3$,  with a correlation of $ 0.2 $ (for all pairs) as $ Y_{j}=X_{j}+X_{4}$, where $X_j\\sim\\Gamma(s,\\lambda) $ for $j=1,2,3$ and $X_{4}\\sim\\Gamma(s\/4,\\lambda) $ for appropriate values of $s$ and $\\lambda$ (resulting in the above intensities and standard deviations for each regime).\n\n\\begin{table}[t]\n\t\\begin{subtable}[h]{\\textwidth}\n\t\t\\caption{{\\texttt{constvar}}: Constant standard deviation of $0.7$.}\n\t\t\\begin{tabular}{c||c|c|c|c|c|c}\n\t\t\tChange point at& 250 & 500 & 900 & 1150& spurious & duplicate \\\\ \\hline\\hline\n\t\t\tindependent, estimator (A)  & 0.9992  & 0.9985 & 0.9253 & 0.9998 & 0.0243 & 0.0035\\\\ \n\t\t\tindependent, estimator (B)  & 0.9962  & 0.9727 & 0.6149 & 0.9998 & 0.0033 & 0.0003\\\\ \\hline\n\t\t\tdependent, estimator (A) & 0.9966  & 0.9959 & 0.9052 & 1 & 0.0326 & 0.0030\\\\\n\t\t\tdependent, estimator (B) & 0.9879  & 0.9551 & 0.6245 &  0.9993 & 0.0072 & 0.0004\\\\\n\t\t\tdependent, estimator (C)\t& 0.9439 & 0.8360 & 0.3534 & 0.9975 & 0.0049 & 0.0004\\\\\n\t\t\\end{tabular}\t\n\t\t\\vspace{3mm}\n\t\t\\label{Table_constvar}\n\t\\end{subtable}\n\t\n\t\\begin{subtable}[h]{\\textwidth}\n\t\t\\caption{{\\texttt{smallvar}}: Standard deviation of $ 5\/6 $ the expected time between events.}\n\t\t\\begin{tabular}{c||c|c|c|c|c|c}\n\t\t\tChange point at & 250 & 500 & 900 & 1150& spurious & duplicate\\\\ \\hline\\hline\n\t\t\tindependent, estimator (A) & 0.9790  & 1 & 0.9707 &  1 & 0.0273 & 0.0023\\\\ \n\t\t\tindependent, estimator (B) & 0.9354  & 1 & 0.9302 &  0.9999 & 0.0038 & 0.0002\\\\ \\hline\n\t\t\tdependent, estimator (A) & 0.9657  & 0.9999 & 0.9527 &  0.9996 & 0.0347 & 0.0026\\\\\n\t\t\tdependent, estimator (B) & 0.9197  & 0.9982 & 0.9089 & 0.9986 & 0.0071 & 0.0013\\\\\n\t\t\tdependent, estimator (C)\t&  0.7421  & 0.9882 & 0.7137 &  0.9896 & 0.0043 & 0.0003\\\\\n\t\t\\end{tabular}\n\t\t\\vspace{3mm}\n\t\t\\label{Table_smallvar}\n\t\\end{subtable}\n\t\\begin{subtable}[h]{\\textwidth}\n\t\t\\caption{ {\\texttt{Poisson}}-distributed inter-event times.}\n\t\t\\begin{tabular}{c||c|c|c|c|c|c}\n\t\t\tChange point at& 250 & 500 & 900 & 1150& spurious & duplicate\\\\ \\hline\\hline\n\t\t\tindependent, estimator (A) & 0.8913  & 0.9963 & 0.8615 & 0.9976 &  0.0390 & 0.0042 \\\\ \n\t\t\tindependent, estimator (B) & 0.7338  & 0.9844 & 0.7174 &  0.9876 & 0.0029 & 0.0009\\\\ \\hline\n\t\t\tdependent, estimator (A) & 0.8654 & 0.9910 & 0.8331 & 0.9923 & 0.0457 & 0.0047\\\\\n\t\t\tdependent, estimator (B) & 0.7138 & 0.9756 & 0.6961 & 0.9749 & 0.0056 & 0.0012\\\\\n\t\t\tdependent, estimator (C)\t& 0.4525 & 0.8883 & 0.4217 & 0.8963 & 0.0042 & 0.0001\\\\\n\t\t\\end{tabular}\n\t\t\\vspace{2mm}\n\t\t\\label{Table_poisson}\n\t\\end{subtable}\n\t\\caption{Detection rates for each change point as well as the average number of spurious and duplicate estimators for different distributions of the inter-event times.}\n\t\\label{Table_simstudy}\n\\end{table}\n\n\n\n\n\nIn the simulations, we use a threshold as in Remark~\\ref{rem_choice_threshold} with $\\alpha_T=0.05$. By Section~\\ref{sec_ex_renewal} and \\eqref{eq_sigma_t} it holds that $\\boldsymbol{\\Sigma}_t=\\Covmulti{(Y_1,Y_2,Y_3)^{\\prime}}\/\\EW{Y_1}^3$\nwhile we use the following choices for the matrix $\\widehat{\\mathbf{A}}_t$ as in \\eqref{eq_mosum_signif}: (A) Diagonal matrix with locally estimated variances $\\widehat{\\boldsymbol{\\Sigma}}_t(j,j)$ on the diagonal, $j=1,2,3$,  (B) with the true variances $\\boldsymbol{\\Sigma}_t(j,j)$ on the diagonal and (C) in case of dependent components (non-diagonal) true covariance matrix $\\boldsymbol{\\Sigma}_t$. While only (A) is of relevance in applications, this allows us to understand the influence of estimating the variance on the procedure. For dependent data, the distinction between (B) and (C) is important for applications, because a good enough estimator (resulting in a reasonable estimator for the inverse) is often not available for the full covariance matrix as in (C) for moderately high or high dimensions, while it is much less problematic to estimate (B).\nIn (A) the variances at location $t$ are estimated as \n\n\\begin{align}\n\t\\widehat{\\boldsymbol{\\Sigma}}_t(j,j)=\\min\\geschweift{\\frac{ \\widehat{\\sigma}_{j,-}^2(t)}{\\widehat{\\mu}_{j,-}^3(t)},\\frac{ \\widehat{\\sigma}_{j,+}^2(t)}{\\widehat{\\mu}_{j,+}^3(t)}}, \\label{eq_var_est}\n\\end{align}\nwhere  $\\widehat{\\sigma}_{j,\\pm}^2(t)$ and $\\widehat{\\mu}_{j,\\pm}(t)$ are the sample variance and sample mean respectively based on the inter-event times of the $j$th-component within the windows $(t-h,t]$ for $'-'$ respectively $(t,t+h]$ for $'+'$. The first and last inter-event times that have been censored by the window are not included. Using the minimum of the left and right local estimators takes into account that the variance can (and typically will) change with the intensity which has already been discussed by \\cite{ChoKirchMeier} in the context of partial sum processes.\n\nThe results of the simulation study can be found in Table \\ref{Table_simstudy}, where  we consider a change point to be detected if there was an estimator in the interval $ [c_i-h,c_i+h] $.  Additional significant local maxima in such an interval are called \\emph{duplicate} change point estimators, while additional significant local maxima outside any of these intervals are called \\emph{spurious}. The procedure performs well throughout all simulations with high detection rate, few spurious and very few duplicate estimators. The results improve further for smaller variance, in which case the signal-to-noise ratio is better.\n\n\n\n\n\nWhen the diagonal matrix with the estimated variance is being used, the detection power is larger in all cases than when the true variance is being used. In case of the changes at location 900 this is a substantial improvement, such that the use of this local variance estimator can help boost the signal significantly. This comes at the cost of having an increased but still reasonable amount of spurious and duplicate change point estimators. \n\\begin{figure}[tb]\n\t\\includegraphics*[width=\\textwidth]{Mix.pdf}\n\t\\caption{MOSUM\\ statistics with bandwidths of $ h=30,60,90,120 $ (top to bottom) for a three-dimensional renewal process with \\emph{multiscale} change\n\t\twith increasing distance between change points in combination with decreasing magnitude of the changes in intensity. The dashed vertical lines indicate the location of the true changes, while the solid lines indicate the change point estimators. \\\\In this multiscale situation no single bandwidth can detect all changes: The changes to the left are well estimated by smaller bandwidth, the ones in the middle by medium-sized bandwidths and the one to the right by the largest bandwidth. } \\label{figure_mix}\n\\end{figure}\nThis effect stems from using the minimum in \\eqref{eq_var_est}, which was introduced to gain detection power if the variance changes with the intensity. \nAdditionally, the use of the true (asymptotic) covariance matrix leads to worse results than only using the corresponding diagonal matrix possibly because the asymptotic covariances do not reflect the small sample (with respect to $h$) covariances well enough. \nFrom a statistical perspective this is advantageous because the local estimation of the inverse of a covariance matrix in moderately large or large dimensions is a very hard problem leading to a loss in precision, while the diagonal elements are far less difficult to estimate consistently.\n\n\n\nIn the above situation the changes are \\emph{homogeneous} in the sense that the smallest change in intensity is still large enough compared to the smallest distance to neighboring change points (for a detailed definition we refer to \\cite{ChoKirch}, Definition 2.1, or \\cite{cho2020data}, Definition 2.1).\nIn particular, this guarantees that all changes can be detected with a single bandwidth only.\n\nIn some applications with \\emph{multiscale} signals, where frequent large changes as well as small isolated changes are present, this is no longer the case as Figure \\ref{figure_mix} shows. In such cases, several bandwidths need to be used and the obtained candidates pruned down in a second step (see \\cite{ChoKirch} for an information criterion based approach for partial sum processes as well as \\cite{Messer14} for a bottom-up-approach for renewal processes). Similarly, if the distance to the neighboring change points is unbalanced MOSUM\\ procedures with asymmetric bandwidths as suggested by \\cite{ChoKirchMeier} may be necessary.\n\\section{Discussion and Outlook}\nIn this paper, we considered a class of multivariate processes that, possibly after a change of probability space fulfill a uniform strong invariance principle. We assumed that the process switches possibly infinitely many times between finitely many regimes, with each switch inducing a change in the drift. This setup includes several important examples, including multivariate partial sum processes, diffusion processes as well as renewal processes.\nIn order to localise these changes, we extended the work of \\cite{EichingerKirch} and \\cite{Messer14} and proposed a single-bandwidth procedure using MOSUM\\  statistics in order to estimate changes, allowing for local changes. We were able to show consistency for the estimators. Further, we were able to derive (uniform) localisation rates in the form of exact convergence rates, which are indeed minimax-optimal.\\\\\nOne drawback of the procedure is the use of a single bandwidth. In practice, the identification of the optimal bandwidth turns out to be rather difficult as pointed out e.~g.~by \\cite{ChoKirch} and \\cite{Messer14}: On the one hand, one wants to choose a large bandwidth in order to have maximum power, while on the other hand, choosing a too large bandwidth may lead to misspecification or nonidentification of changes. Furthermore, as can be seen in the simulation study, in a multiscale change point situation (see Definition~2.1 of \\cite{ChoKirch}) no single bandwidth can detect all change points. Therefore, one future topic of interest would be to extend the proposed procedure to a true multiscale setup as  in \\cite{ChoKirch}.\n\\section*{Acknowledgements}\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.\n\n\\printbibliography\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nCollective migration occurs when a group of individual cells exhibit coordinated movements with similar speed and directionality \\cite{Rorth2009}. This phenomenon is intriguing in cancer biology as a demonstration of how heterogeneous phenotypes cooperate to enhance solid tumor progression within a dysregulated microenvironment \\cite{Tabassum2015}. Indeed, patient histopathology sections often include packs of tightly connected cancer cells organized as rounded clusters or linearly extended strands within an aberrant tissue architecture \\cite{Friedl2012ncb}. Such behaviors are reminiscent of leader-follower interactions and epithelial-mesenchymal transitions (EMT) in development and wound healing \\cite{VilchezMercedes2021}. Collective migration is also intriguing from a physical science perspective as an emergent phenomenon driven by interactions between many individuals \\cite{Trepat2018}. These behaviors are further influenced by abnormal solid and fluid stresses from their surroundings, which impact metastatic dissemination and drug delivery \\cite{Jain2020}. Thus, a cross-disciplinary approach is required to understand how heterogeneous groups of invasive cells interact with the surrounding matrix via biological and physical mechanisms.\n\nRecent advances in assay development have enabled new mechanistic insights into collective migration via exquisitely sensitive measurements as well as biomimetic culture conditions. For example, single cell molecular profiling technologies have revealed the interplay of myriad signaling pathways, which previously involved dissociation of tissues into single cells \\cite{Satija2019}, but now can retain information about tissue architecture via spatial transcriptomics \\cite{rao2021exploring}. New light microscopy techniques based on structured illumination permit deeper imaging with improved temporal resolution and reduced phototoxicity \\cite{Lemon2020}. Fluorescent protein reporters can be used for longitudinal monitoring of transcriptional state or force transduction \\cite{Specht2017}. In parallel, cells cultured within engineered 3D biomaterials experience controlled biochemical and physical cues, which can be dynamically modulated \\cite{Leggett2017,Beri2018}. The mechanical properties of single cells and local matrix architecture can be further probed at subcellular resolution \\cite{Leggett2021, LI20211863}. These early stage technologies are now being translated with primary mouse or human organoids in order to predict disease progression and therapeutic response \\cite{LeSavage2021}.  \n\nIn this chapter, we highlight recent developments in collective tumor invasion from a biological and physical perspective. We first review the essential concepts of collective cell migration and leader-follower interactions. We then examine how patterns of collective motion can be analyzed using physical concepts such as order parameters and phase transitions, as well as the interplay of metabolism with phenotypic heterogeneity. Next, we review biomaterials and the ECM, along with precision measurement 3D matrix mechanics. Finally, we explore EMT as a representation of phenotypic heterogeneity, including recent results on partial or intermediate states.  We close with our critical perspective on the field and discussion of future directions.\n\n\\section{Collective Cell Migration and Leader-Follower Interactions}\n\\label{sec:leader}\n\nEarly work on eukaryotic cell migration investigated how individual cells (e.g. fibroblasts, leukocytes) cyclically adhere and propel themselves along planar (2D) substrates, revealing physically conserved mechanisms across various cell types \\cite{lauffenburger1996cell}. First, cells polarize with a leading and trailing edge, often in response to asymmetric stimuli of soluble or immobilized biochemical factors, substrate stiffness, or local topography \\cite{sengupta2021principles}. Next, cells extend protrusions from the leading edge driven by actin polymerization, which form integrin-mediated focal adhesions with ligands on the substrate \\cite{ridley2003cell}. Finally, actomyosin contractility via stress fibers pulls the cell body forward and retract the trailing edge \\cite{levayer2012biomechanical}. Subsequent work using compliant substrates revealed that cell adhesion and migration were modulated by the relative strength of cell-substrate interactions and intracellular contractility \\cite{Janmey2020}.\n\nIn comparison, 3D matrix is a more mechanically confined environment that can impede cell migration relative to 2D substrates \\cite{yamada2019mechanisms}. Migratory cells can traverse a narrow region by local remodeling of the matrix into a wider path, as well as by deforming their intracellular components (e.g. nucleus, cytoskeleton) to squeeze through. Mesenchymal migration phenotypes are associated with elongated, spindle-like morphologies, which extend narrow actin-driven protrusions (via RAC1 and CDC42) that apply strong cell-matrix adhesions and matrix metalloproteinases to degrade and remodel the ECM \\cite{yilmaz2009emt}. Mesenchymal cells express the intermediate filament vimentin \\cite{Leggett2021}, which is highly stretchable (without breaking) and can protect cells undergoing large deformations \\cite{Hu2019, Patteson2019}. Alternatively, amoeboid migration phenotypes exhibit compact, rounded morphologies, with propel themselves forward via actomyosin contractility (via RHOA) and do not require strong adhesions or matrix metalloproteinases for effective migration \\cite{paluch2016focal}. Weak ECM adhesion and limited remodeling allow cells utilizing amoeboid migration (e.g. immune cells) to migrate more rapidly than mesenchymal cells (e.g. fibroblasts) which tend to migrate more slowly \\cite{Liu2015, Ruprecht2015}. Migratory cells are further capable of switching between mesenchymal and amoeboid migration modes, depending on local ECM adhesivity and matrix remodeling capabilities \\cite{yamada2019mechanisms}.  Empirically, cancer cell lines cannot traverse confined spaces smaller than 7 $\\mu$m$^2$ without matrix remodeling \\cite{wolf2013physical}, which is gauged by nuclear deformation as an intracellular ``ruler'' \\cite{Lomakin2020}. Further, groups of mesenchymal or amoeboid cells may migrate as multicellular ``streams'' without strong cell-cell adhesions in response to some asymmetric external stimulus, where contact guidance by matrix architecture is permissive for similar migration direction and speed \\cite{roussos2011chemotaxis}. In the absence of these asymmetric stimuli, cells typically exhibit more random and less directed motility \\cite{sengupta2021principles}.\n\nCarcinomas arise from healthy epithelial tissues, which consist of epithelial cells with strong cell-cell adhesion and apicobasal polarity organized into tightly packed layers (Fig. \\ref{fig:leader_cells}A) \\cite{rodriguez1989morphogenesis}. During tumor progression, tissues become progressively disorganized due to uncontrolled proliferation, then transition to malignant invasion via degradation of the basement membrane \\cite{Kelley2014a}.  Groups of epithelial cells exhibit collective migration when they remain mechanically connected, sustaining cell polarization for directed motion, as well as coordinating longer ranged responses to environmental stimuli (Fig. \\ref{fig:leader_cells}B,C) \\cite{Rorth2009}. Within solid tumors, the spatial organization of these groups can vary widely, from relatively compact clusters to elongated single or multi-file strands to large masses \\cite{Friedl2012ncb}. These collectives may be categorized broadly into distinct functional groups based on parameters of multicellular morphology, the degree of cell-cell adhesion, and supracellular coupling of intercellular signaling  \\cite{mayor2016front}. Furthermore, collective migration patterns are shaped (in part) by the surrounding ECM architecture and presence of various stromal cells. \n\nDespite differences in spatial organization, 3D collective invasion is most often characterized by the presence of highly motile cells at the front of invasive groups, termed ``leader cells'' (Fig. \\ref{fig:leader_cells}C), \\cite{VilchezMercedes2021}. Leader cells \noften exhibit actin-rich protrusions with strong cell-matrix adhesions and actomyosin contractility to locally remodel the ECM (Fig. \\ref{fig:leader_cells}C), analogous to the mesenchymal motility phenotype described previously \\cite{mayor2016front}. Leader cells adhere to collagen I or fibronectin mediated primarily by integrin $\\beta_1$ and $\\beta_3$. They further generate traction forces to deform the ECM via contractility of intracellular stress fibers, which can associate with vimentin intermediate filaments in order to enhance tractions \\cite{Leggett2021} (Fig. \\ref{fig:leader_cells},i), which can occur in combination with ECM degradation via matrix metalloproteinases (Fig. \\ref{fig:leader_cells},ii). Thus, through repeated protrusion-contraction-deformation events, leader cells pull on intrinsically ``wavy'' collagen fibers to stretch and bundle fibers into regions of aligned, stiffened ECM (Fig. \\ref{fig:leader_cells},iii). Furthermore, collagen fibers can be cross-linked enzymatically by lysyl oxidase to locally enhance ECM stiffness and promote tumor invasion though focal adhesion kinase (FAK) signaling \\cite{Levental2009}. Finally, leader cells can deposit new ECM (Fig. \\ref{fig:leader_cells},iv), such as fibronectin, to reinforce adhesive migratory cues \\cite{Winkler2020}. Altogether, these leader cell-driven ECM remodeling activities result in the generation of ECM \"microtracks\" (Fig. \\ref{fig:leader_cells},v) which are permissive for cell migration and contain aligned, bundled ECM proteins that present contact guidance cues. For the remainder of this review, we primarily focus on leader cells originating from epithelial tumors, or carcinomas, but note that leader cells may also arise from stromal cells (e.g. fibroblasts) (Fig. \\ref{fig:leader_cells}D) or immune cells (e.g. macrophages) \\cite{VilchezMercedes2021}. Regardless of cell type, leader cell activity typically results in ECM architecture that is more permissive for cell migration.\n\n\n\n\n\\begin{figure}[h!]\n \\centering\n\\includegraphics[width=5in]{leader_cells}\n\\caption{Schematic of collective cancer cell invasion. A. Epithelial cells, B. Follower cells, C. Cancer Leader Cell, D. Cancer-Associated Fibroblast. ECM remodeling via force-mediated matrix deformations (i), MMP degradation (ii), alignment \/ crosslinking (iii), ECM deposition (iv), and formation of ECM microtracks (v).}\n\\label{fig:leader_cells}      \n\\end{figure}\n\nConcomitant with ECM remodeling activities, leader cells may guide the migration of adjacent cells termed \"followers\"  (Fig. \\ref{fig:leader_cells}B) along ECM microtracks. Follower cells are defined primarily by a rearward spatial position relative to leaders but may also possess migratory and ECM remodeling capabilities. Indeed, follower cells traversing wider spaces generated by leader cells encounter less mechanical resistance and can expend less energy to remodel ECM or undergo large deformations \\cite{VilchezMercedes2021}. Leader cells coordinate the activity of followers through a combination of cell-cell mechanical adhesions and biochemical signaling. \n\nFirst, cell-cell junctions are formed when two cells connect adhesion proteins to form a stable mechanical junction (e.g. adherens junctions and desmosomes), typically utilizing proteins in the cadherin superfamily (e.g. E, N, P-cadherins or desmosomal cadherins) \\cite{cavallaro2004cell}, which are intracellularly coupled to actin microfilaments or keratin intermediate filaments, respectively \\cite{jacob2018types}. As a stationary sheet-like structure, epithelial cells remain tightly-connected to neighboring cells via desmosome adhesions and adhere to the basement membrane via hemidesmosomes containing $\\alpha_6\\beta_4$ integrins (Fig. \\ref{fig:leader_cells}A). Loss of apical-basal polarity is a common step in cancer progression, which often accompanies loss of cell-cell adhesions and acquisition of cell-ECM adhesions, which establish front-back polarity and induce cell motility (Fig. \\ref{fig:leader_cells}B). Motile follower cells may lose desmosomes and keratin expression and gain expression of vimentin IF's, which are associated with EMT. However, hybrid phenotypes expressing keratin and vimentin have been observed clinically \\cite{Thomas1999}. Mesenchymal leader and followers may remain adherent through N-cadherin junctions, or maintain transient adhesions that result in weak cell-cell coordination. Nonetheless, epithelial followers often retain E-cadherin to tightly coordinate migration. Finally, leader cells are connected to followers via adherens junctions. For example, cancer leader cells in partial EMT states may connect to followers via homotypic E-cadherin junctions (Fig. \\ref{fig:leader_cells}C). Leaders in later stages of EMT may express N-cadherin and adhere to followers via heterotypic E\/N-cadherin junctions, which have also been documented for fibroblast leader cells with epithelial follower cells (Fig. \\ref{fig:leader_cells}D) \\cite{Labernadie2017}. Overall, cell-cell adhesions maintain coherent collective motion by mechanical coupling via the cell cytoskeleton, as well as reinforcement of multicellular front-rear polarity. From a biochemical standpoint, cells in close proximity to one another may also coordinate motility by juxtacrine signaling (e.g. NOTCH1-DLL4 signaling) \\cite{Riahi2015}. Moreover, contact-independent autocrine signaling across gap junctions may be utilized for intercellular coordination of contractility and size \\cite{Aasen2019}, via connexins \\cite{Han2020a, khalil2020collective}.\n\nIt should be noted that leader cells may not always guide collective migration, as exemplified by the diversity of collective migration behaviors in development, such as the attraction-repulsion dynamics observed in neural crest-placode migration \\cite{Theveneau2013a}. Furthermore, followers may or may not have the potential to become leader cells. Interestingly, followers can ``switch'' with leaders during collective migration, which may occur if leader cells undergo cell division, and could also redistribute metabolic costs of migration. Nevertheless, the functional role and molecular definition of leader cells remains unresolved, and may depend on biological context \\cite{theveneau2017leaders}. In this chapter, we consider leader cell behaviour and leader-follower dynamics in the context of epithelial cancer invasion. As discussed previously, leader cells often utilize cell-cell adhesions through proteins such as E-cadherin, which is an epithelial biomarker \\cite{Leggett2019}. Thus, these leader cells may represent a hybrid or partial EMT phenotype, which we consider in detail in a subsequent section. \n\n\\section{Analyzing Collective Migration using Phase Transitions and Order Parameters}\n\\label{sec:order}\n\nHealthy epithelial tissues are comprised of mechanically connected cells with limited motility, as well as uniform shape and regular geometric packing \\cite{Guillot2013}. This mechanically arrested state can transition to more motile states during tumor progression, development, and wound healing \\cite{Oswald2017}. For example, cells can disperse individually (e.g. gas-like), can migrate in closer proximity (e.g. fluid-like), or move in coordinated groups (e.g. active nematics with long-ranged orientational order) (Fig. \\ref{fig:jamming}A) \\cite{Porta2020,fredberg2021origin}. In these latter contexts, cells may exhibit varying cell-cell adhesion strength along with larger fluctuations in cell shape and packing. Analogous phase transitions that emerge from many interacting particles can be captured using statistical physics, assuming that the particles are relatively similar, move with constant velocity and can alter direction, interact over some distance, and are subject to ``noise'' of varying strength \\cite{Vicsek2012}. These phase transitions can be quantified based on changes in some order parameter that represents a symmetry property of the system \\cite{chaikin1995principles}. For instance, the jamming transition describes how soft materials exhibit a liquid to solid-like response when varying control parameters of interparticle adhesion, density, or shear \\cite{Liu2010}. Fredberg proposed that a biological jamming transition could occur similarly with respect to cell-cell adhesion, cell density, or motility in 2D epithelial monolayers \\cite{Sadati2013}.\n\n\\begin{figure}[h]\n \\centering\n\\includegraphics[width=4.5in]{jamming.png}\n\\caption{A. ``Jamming'' phase diagram of individual and collective cell migration governed by cell-cell adhesion and cell density. Redrawn from \\cite{Porta2020}. B. Representative single cell migration tracks (blue lines) from multicellular spheroids in low density collagen matrix, revealing coordinated and persistent motion for vehicle control (shNT), which is lost after knockdown of E-cadherin cell-cell junctions (sdCDH1). Reproduced from \\cite{Ilina2020}. C. Triangulation-based configurational fingerprints compare deformed and (relaxed) reference states. D. Late (invasive) spheroids exhibit increased volumetric order parameter (per cell) relative to early spheroids, but comparable shear order parameters (per cell). Reproduced from \\cite{Yang2021}. E. Topological data analysis represents discrete cell positions based on the ``persistence'' of topological loops at varying length scales around empty areas, which can then be mapped to computational simulations with varying adhesion and propulsion. Reproduced from \\cite{Bhaskar2021}.}\n\\label{fig:jamming}       \n\\end{figure}\n\n\nAn intuitive order parameter for collective migration is cell velocity, and particle image velocimetry (PIV) can be used to analyze spatial and temporal correlations \\cite{Adrian2005}. Early work used PIV to visualize cooperative motions within 2D epithelial monolayers. For instance, Silberzan and coworkers utilized PIV for collective migration of epithelial monolayers into unoccupied regions \\cite{Poujade2007, Petitjean2010}.  Angelini et al. used PIV to measure how compliant substrates were deformed by epithelial monolayers, revealing characteristic length and time scales over which cells coordinate their migration, analogous to dynamic heterogeneity in a physical glass transition \\cite{Angelini2010, Angelini2011}. More recently, Scita and coworkers applied PIV to reveal the onset of angular rotation and local cellular rearrangements in 3D spheroids \\cite{Palamidessi2019,Cerbino2021}. \n\nO. Ilina et al. utilized PIV to establish a physical picture of 3D collective migration based the crosstalk of cell-cell adhesion with ECM confinement \\cite{Ilina2020}. Multicellular spheroids were embedded within collagen matrix in varying assay geometries. When ECM porosity was sufficiently large to permit cell migration (without ECM degradation), cells with downregulated E-cadherin typically migrated individually in a fluid-like state. As ECM porosity decreased, the increasing spatial confinement resulted in cells moving together in close proximity, despite their relatively weak cell-cell junctions. Nevertheless, higher resolution tracking of cell velocity and spatial correlations revealed that neighboring cells within a group were locally uncorrelated when cell-cell junctions were weak (Fig. \\ref{fig:jamming}B). Thus, these results highlight the importance of mature cell-cell adhesions (e.g. E-cadherin) to coordinate effective group migration during tumor invasion and metastasis.\n\n\nGeometric order parameters can also be defined based on cell shape or local connectivity. Historically, honeycomb-like hexagon-dominated cellular structures were observed empirically in animal and plant tissues \\cite{thompson_1968}. Interestingly, proliferating cells within epithelial tissues result in a wider distribution of polygons, particularly pentagons and heptagons \\cite{Gibson2006}. Bi and Manning captured these transitions based on a cell shape index (i.e. ratio of cell perimeter to area), which are indicative of shear modulus of the cell monolayer \\cite{Bi2015}, effective diffusivity of single cells \\cite{Bi2016} and cellular stresses \\cite{Yang2017a}. Fredberg and coworkers subsequently validated this parameter in asthmatic airway epithelium \\cite{Park2015}, and revealed that cell shape index and variability were also strongly correlated with unjamming \\cite{Atia2018, Mitchel2020}. These concepts were also extended to 3D collective migration by analogy to phase transitions from solid to liquid to gas \\cite{Wong2014,Kang2021, Grosser2021}.\n \nH. Yang et al. proposed new order parameters based on the deformation of a triangular lattice relative to an idealized equilibrium (Fig. \\ref{fig:jamming}C) \\cite{Yang2021}. For some spatial configuration of cells within a tissue, the cell positions (e.g. nuclei) can be connected by fictitious lines to partition the surrounding space into triangles (e.g. Delaunay triangulation). The deformation gradient can then be expressed using invariants $vol_n = \\det{(\\mathbf{F})}$, representing how the area of each triangle deviates from the average, along with $shear_n = \\text{tr}\\,(\\mathbf{F}^T\\mathbf{F})\/\\det{\\mathbf{(F)}}-2$, corresponding to how a given triangle is distorted relative to an equilateral triangle. Corresponding order parameters are defined by ensemble averaging the natural logarithm of these invariants over all triangles, i.e. $\\Phi_{vol} = \\langle [\\log{(vol_n)}]^2\\rangle$ and $\\Phi_{shear} = \\langle \\log{(shear_n)}\\rangle$. This analysis was first validated based on ventral furrow formation of the \\emph{Drosophila} germband epithelium, which transitions from an arrested, jammed-like state towards a more motile, unjammed-like state. This macroscopic phenomenon is associated with characteristic shearing and shrinking behavior within the tissue, resulting in a sharp increase in the $\\Phi_{shear}$ and $\\Phi_{vol}$ at a characteristic timescale where this phase transitions occurs. Conversely, the proliferation of kidney epithelial cells (e.g. MDCK) first results in a suppression of density fluctuations in $\\Phi_{vol}$, analogous to a transition from gas to liquid, followed by a further suppression of packing disorder in $\\Phi_{shear}$, analogous to a transition from liquid to solid. Lastly, these concepts were used to analyze the collective invasion of mammary epithelial spheroids (e.g. MCF-10A). Remarkably, spheroids at earlier times with minimal invasion (e.g. acini) exhibited similar $\\Phi_{shear}$ but smaller $\\Phi_{vol}$ relative to spheroids at later time with more invasive strands (Fig. \\ref{fig:jamming}D). For these spheroids at later times, cells within these peripheral strands exhibited increased $\\Phi_{shear}$ relative to the interior. Although this framework does not require the reference triangles to be the same, and any experimental frames can be used as the reference, special caution should still be taken when choosing appropriate reference frames, especially when dealing with naturally heterogeneous biological systems. Indeed, Y. Han et al. observed that cells within these multicellular spheroids dynamically modulate their size and stiffness as they shuttle between the interior and peripheral strands \\cite{Han2020a}. Further investigations will help to establish the validity of this approach across different biological systems.\n\nD. Bhaskar et al. utilized topological data analysis to visualize the ``shape'' of a tissue based on discrete cell positions \\cite{Bhaskar2021}. Different spatial configurations of cells can be compared by computing the ``cost'' of rearranging topological features (i.e. persistence homology). Briefly, the spatial connectivity between cell positions is now sampled across varying length scales, generating a topological signature (``barcode'') that emphasizes features that are present across multiple length scales. For pairwise connected components (i.e. dimension 0 homology), this analysis will be skewed by population sizes, making it challenging to compare biological specimens with different numbers of cells. Instead, Bhaskar et al. considered how nuclei can be linked as connected loops around an empty area (i.e. dimension 1 homology), which is less sensitive to population size and also samples larger scale spatial structure. This unsupervised machine learning approach correctly classified experimental data of cell positions in individual, compact clusters, or branching phases (Fig. \\ref{fig:jamming}E). Further, this approach could classify distinct phases and identify phase transitions for simulated self-propelled particles with varying motility and adhesion strength. The success of this approach can be explained by mapping discrete data points to a continuous shape (i.e. manifold), which remains highly effective even with relatively sparse sampling, as well as variability in the position of data points. Indeed, different phases could be successfully distinguished even when 80\\% of the cells were randomly removed. Ongoing research is extending this approach to multiple cell types.\n\nOrder parameters represent a powerful approach to understand non-living physical systems, particularly in soft matter \\cite{chaikin1995principles}. From a reductionist perspective, it is appealing to think that these approaches could also describe collective migration \\cite{Oswald2017,fredberg2021origin}, although these are living systems that are far from thermodynamic equilibrium. Indeed, groups of cancer cells are likely to be quite heterogeneous and in smaller numbers (than in soft matter systems), and the nature of biological ``noise'' remains poorly understood. For instance, a given population of cancer cells may be comprised of distinct subpopulations with ``epithelial'' or ``mesenchymal'' states, associated with distinct motility and adhesion phenotypes \\cite{GamboaCastro2016}. Thus, some care is warranted when using order parameters that average biological behaviors across populations and time. An intriguing possibility is that unsupervised machine learning could be used to classify spatiotemporal patterns of cell migration and infer new order parameters \\cite{Cichos2020}. Although so-called deep learning is computationally expensive and typically requires extensive training data, constraining convolutional neural networks with some physical principles (e.g. physics-informed machine learning \\cite{Karniadakis2021}) could be highly effective for smaller experimental datasets.\n\n\\section{Metabolic Heterogeneity and Mechanobiological Phenotype}\n\\label{sec:metabolism}\nTumor cells reprogram their metabolic activity in order to sustain proliferation and invasion in dysregulated microenvironments, an emerging hallmark of cancer \\cite{pavlova2016emerging}. Indeed, it has been hypothesized that cancer cells may optimize their metabolic activity to ``go or grow,'' depending on local microenvironmental conditions \\cite{DeDonatis2010}. Ordinarily, when ample oxygen is available, normal cells utilize mitochondrial oxidative phosphorylation and glycolysis in the cytoplasm to produce chemical energy in the form of ATP (Fig. \\ref{fig:metabolism_background}A). Oxidative phosphorylation converts glucose to glycolysis intermediate products to pyruvate, which is oxidized in the mitochondria under aerobic conditions to produce 36 ATP (per glucose molecule) (Fig. \\ref{fig:metabolism_background}A, i). In comparison, glycolysis occurs under anaerobic conditions to produce lactate and 2 ATP (per glucose molecule) (Fig. \\ref{fig:metabolism_background}A, ii). There exists a metabolic trade-off whereby oxidative phosphorylation results in higher yield of ATP per glucose molecule with slower kinetics, while glycolysis results in lower yield of ATP per glucose molecule with considerably faster kinetics. Cancer cells often encounter hypoxic conditions, and may activate glycolysis via hypoxia-inducible factor 1$\\alpha$ (HIF-1$\\alpha$). This induces downstream transcription of glycolytic enzymes like hexokinase and lactate dehydrogenase \\cite{semenza2013hif}, as well as enhanced expression of GLUT glucose transporters (Fig. \\ref{fig:metabolism_background}A). Indeed, cancer cells benefit from using glycolysis to fulfil their metabolic requirements even when abundant oxygen is available. In particular, abundant ATP is advantageous for the nearly continuous cytoskeletal reorganization associated with directed cell migration \\cite{dewane2021fueling}. Furthermore, enhanced lactate excretion via glycolysis by cancer cells acidifies the extracellular space (Fig. \\ref{fig:metabolism_background}A, ii), which has been linked to the production of matrix metalloproteinases that facilitate migration through the ECM \\cite{romero2016lactate}. Moreover, uncontrolled proliferation also requires glycolytic intermediates necessary for synthesis of nucleic acids, proteins, and lipids, as well as lactate which crucially maintains the NAD$^+$\/NADH redox balance \\cite{lunt2011aerobic}.  Cell metabolism, morphology, and migration are intimately linked, and thus targeting cell metabolism may present a novel strategy to inhibit the cellular energy production fueling collective cancer invasion. \n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=3.5in]{metabolism_background}\n\\caption{A. Cell metabolism converts glucose to ATP via oxidative phosphorylation (i) or glycolysis (ii), depending on oxygen availability. B. Mechanical cues alter cell metabolism via substrate stiffness and TRIM 21 (i), weak cell-matrix adhesion (ii), strong cell-matrix adhesion with actomyosin contractility (iii), or force-mediated E-cadherin stabilization (iv).}\n\\label{fig:metabolism_background}      \n\\end{figure}\n\n\nJ.S. Park et al investigated how the metabolism of individual human bronchial epithelial cells was modulated by soft or stiff planar substrates \\cite{Park2020}. Morphologically, cells on stiff substrates exhibited larger footprints with extensive actomyosin stress fibers, while cells on soft substrates had smaller footprints with weak contractility. Consequently, cells on soft substrates showed reduced expression of all phosphofructokinase (PFK) enzymes, which catalyze a rate-limiting step of glycolysis by phosphorylating fructose 6-phosphate, and thereby determine the overall glycolytic rate (Fig. \\ref{fig:metabolism_background}B,i). Loss of the PFK isoform PFKP on soft substrates could be attributed to degradation by the proteosome, specifically by the E3 ubiquitin ligase TRIM21. Indeed, TRIM21 associates with stress fibers and remains inactive when bound, thus maintaining PFKP activity. Remarkably, PFKP expression is elevated in patients with lung cancer, which could explain why tumors exhibit abnormally high levels of glycolysis. Genetic manipulation of these bronchial epithelial cells with a KRAS mutation resulted in consistently high PFK levels, which became insensitive to substrate stiffness. It should be noted that metabolic activity in breast cancer cell lines exhibit varying sensitivity when cultured over planar collagen hydrogels of varying density, thus modulating cytoskeletal activity (Fig. \\ref{fig:metabolism_background}B, ii-iii) \\cite{Mah2018}. Shear forces acting on cell-cell junctions can also increase glucose uptake, driving increased ATP production and glycolysis through AMPK activity (Fig. \\ref{fig:metabolism_background}B, iv) \\cite{bays2017linking}.\n\n\nS.J. Decamp et al. showed that epithelial monolayers of canine kidney cells (e.g. MDCK II) exhibit spatially varying metabolic activity at collective fronts moving to occupy empty areas (Fig. \\ref{fig:metabolism}A) \\cite{DeCamp2020}. Indeed, collectively migrating cells at the front exhibited higher traction forces, elongated morphologies, and faster migration compared to cells towards the rear with small aspect ratios and low tractions. Moreover, cells at the collective front also display decreased NAD$^+$\/NADH ratio, reduced NADH lifetime, and increased glucose uptake, which suggests a shift towards glycolysis. Interestingly, cells far to the rear also displayed a decrease in NADH lifetime and enhanced glucose uptake, indicating a shift towards glycolysis. However, these rearward cells showed an elevated cytoplasmic NAD$^+$\/NADH ratio, which differs from cells at the collective front. This long distance signaling could be mediated through mechanical deformation of the compliant substrate, priming rearward cells for increased proliferation and migration. It should be noted that cell migration and proliferation into empty regions on top of planar substrates is relatively unimpeded and could differ significantly from metabolic behaviors in 3D.\n\n\\begin{figure}[h!]\n\\includegraphics[width=3.5in]{metabolism}\n \\centering\n\\caption{A. Collective unjamming of epithelial monolayers reveals increase of traction forces and metabolic activity near invasion front. Reproduced from \\cite{DeCamp2020}. B. Breast cancer leader cells exhibit increased glycolysis, but switch places regularly with followers. Reproduced from \\cite{Zhang2019b}. C. Lung cancer leader cells exhibit increased oxidative phosphorylation, and invasion can be suppressed by targeting both pyruvate dehydrogenase and glucose uptake. Reproduced from \\cite{Commander2020}.}\n\\label{fig:metabolism}      \n\\end{figure}\n\n\nReinhart-King's group utilized a ratiometric fluorescent reporter of ATP: ADP ratio (PercivalHR) to elucidate how cellular energetics are regulated during individual and collective migration in 3D collagen matrix. First, highly metastatic breast adenocarcinoma cells (MDA-MB-231) expressing the PercivalHR construct invaded individually through narrow bifurcating channels in collagen I \\cite{Zanotelli2019}. Cells exhibited greater glucose uptake and ATP:ADP ratio as they migrated through narrower channels, indicating that increased cell deformation was associated with higher energetic cost. Indeed, pharmacological treatment to perturb cytoskeletal stiffness and cell deformability was sufficient to modulate cell energetic state. Further, increasing matrix stiffness also increased energy costs, since cells required enhanced contractility to deform the matrix and enter confined spaces. When cells were presented with a choice of bifurcating into a wide or narrow channel, their decision-making was shown to be biased by the relative energetic cost of these two choices. Thus, cells were more likely to choose the narrower channel when the cells themselves or the matrix were more compliant.\n\nIn a complementary study, this lab also investigated how these same cells invaded collectively from multicellular spheroids, revealing that leader and follower cells invading in 3D matrix can dynamically switch their roles (Fig. \\ref{fig:metabolism}B) \\cite{Zhang2019b} \\ref{fig:metabolism}B i).  Collective invasion was also impeded by denser 3D matrix with greater mechanical resistance, in qualitative agreement with the previous study using individual cells \\cite{Zanotelli2019}. Nevertheless, collective invasion was associated with cooperative behaviors where leaders exhibited higher glucose uptake and energy consumption relative to followers. Once a leader cell exhausted its available ATP, it would switch with a follower, and this switching frequency increased in denser matrix. Moreover, expression of a fluorescent cell cycle reporter (e.g. CycleTrak) revealed that leader and follower cells exhibited behaviors consistent with the ``grow-or-go'' hypothesis \\cite{DeDonatis2010}, whereby leader cells were less proliferative than follower cells, which was also observed using breast tumor organoids (e.g. MMTV-PyMT mice). \n\nIn contrast, Commander et al. also observed metabolic heterogeneity in leader and follower cells invading from a different spheroid model, but concluded that followers exhibit higher glucose uptake than leaders (Fig. \\ref{fig:metabolism}C ii) \\cite{Commander2020}. Briefly, a non-small cell lung cancer line (H1299) was sorted for leaders and followers by photoswitching of a fluorescent label, followed by FACS \\cite{Konen2017}. Interestingly, followers maintained distinct morphological and invasive phenotypes for several passages before reverting to the parental phenotype, while leader cells retained their phenotype indefinitely. Metabolic analysis based on extracellular flux (e.g. Seahorse Bioscience) revealed that follower cells typically exhibited increased glucose uptake, glycolysis and decreased oxidative phosphorylation \\ref{fig:metabolism}C i). Instead, leader cells exhibited mitochondrial respiration and a pyruvate dehydrogenase dependency, which could be therapeutically targeted with alexidine dihydrocholoride \\ref{fig:metabolism}C iii). A combination drug treatment against both pyruvate dehydrogenase and glucose uptake was effective at suppressing leader cell invasion and follower cell proliferation, respectively. It should be noted that this study utilized a different cell line (H1299) than Reinhart-King's investigations (MDA-MB-231), with varying degrees of phenotypic plasticity. Moreover, subtle differences in spheroid formation and matrix embedding procedures could result in very different phenotypic outcomes \\cite{Peirsman2021}. \n\nAn advantage of using engineered biomaterials is that cancer cells encounter highly consistent and homogeneous mechanical cues, so that the effect of stiffness or other matrix properties on metabolic activity can be systematically elucidated. However, it should be noted that cancer cells in vivo may exhibit very different metabolic states as they migrate through varying mechanical and biochemical microenvironments. Further, these cancer cells may interact dynamically with other stromal and immune cells, which could further act to enhance or suppress tumor progression. Thus, additional work is needed to understand how phenotypic plasticity of metabolic and migratory phenotypes is regulated in the context of mechanical cues from the surrounding microenvironment.\n\n\\section{The Extracellular Matrix and Engineered Biomaterials\n}\n\\label{sec:ecm}\n\nThe extracellular matrix (ECM) is a complex structural network of proteins, glycoproteins, proteoglycans and polysaccharides that mechanically supports tissues while presenting instructive and permissive cues \\cite{Cox2021}. ECM is continually being remodeled by a combination of protein deposition, post-translational chemical modifications, proteolytic degradation, as well as mechanically-driven reorganization \\cite{Winkler2020}. These myriad processes are tightly regulated in normal tissues to maintain homeostasis, but become increasingly dysregulated during aging, fibrosis, and cancer progression \\cite{Piersma2020}. Moreover, alterations in ECM structure, mechanics, and composition may regulate dormancy or metastatic colonization at distant tissue sites (i.e. the premetastatic niche) \\cite{Peinado2017}. We briefly review relevant features of the ECM in the context of collective migration, and refer interested readers to more comprehensive reviews elsewhere \\cite{Winkler2020,Cox2021,Piersma2020}. \n\nThe basement membrane is a sheet-like structure that encloses epithelial tissues, comprised of laminin and collagen IV, linked through bridging proteins such as nidogen and perlecan (Fig. \\ref{fig:ecmcartoon}A,i) \\cite{Pozzi2017}. Indeed, laminin is more prominently displayed on the inner (epithelial-facing) side and instructs epithelial phenotype, while collagen IV may be more localized on the outer (stromal-facing) side to provide more structural support \\cite{plodinec2015}. Altogether, these ECM proteins are assembled with a net-like architecture that is relatively thin (hundreds of nanometers) with small pores (tens to hundreds of nanometers) (Fig. \\ref{fig:ecmcartoon}A,ii). Since the basement membrane is chemically crosslinked, high density, with very low porosity (Fig. \\ref{fig:ecmcartoon}A,iii-v), it is not permissive for epithelial cell migration \\cite{Kelley2014a}. Thus, enzymatic or mechanical remodeling of the basement membrane is crucial for carcinoma cells to invade into the surrounding stroma \\cite{Chang2019a}.  Historically, reconstituted basement membrane (e.g. Matrigel) has been widely used as a compliant biomaterial that mimics the protein composition of native basement membrane \\cite{Benton2014}. However, it should be noted that Matrigel is not of human origin, and is derived from mouse sarcoma cells (e.g. Engelbreth-Holm-Swarm) which exhibits appreciable batch-to-batch variability \\cite{Aisenbrey2020}.\n\n\\begin{figure}[h!]\n\\includegraphics[width=4.5in]{ecmcartoon.png}\n \\centering\n\\caption{Comparison of basement membrane (A), interstitial matrix (B) and tumor \/ fibrotic matrix (C) based on composition (i), topography (ii), crosslinking (iii), density (iv), porosity (v), stiffness (vi), and viscosity (vii). Abbreviations: HA, hyaluronic acid, PG, proteoglycans.}\n\\label{fig:ecmcartoon}      \n\\end{figure}\n\nThe interstitial matrix is a three-dimensional fibrous network comprised of fibrillar collagens (I, III, V), fibronectin, elastin, proteoglycans, glycosaminoglycans, and others (Fig. \\ref{fig:ecmcartoon}B,i) \\cite{Gritsenko2012}. In particular, collagen I is organized as randomly oriented fibers that are physically entangled at moderate density and porosity (Fig. \\ref{fig:ecmcartoon}B,ii-v) \\cite{Wolf2009}. Further, Keely and coworkers observed that these fibers exhibited a wavy (``crimped'') conformation \\cite{Provenzano2006}, which has recently been reported to impede cell polarization \\cite{Fischer2021} and enhance dormancy \\cite{di2021tumor}. The interstitial matrix is occupied by stromal cells (e.g. fibroblasts), which play a crucial role in depositing and organizing ECM proteins \\cite{Plikus2021}. It should be noted that the interstitial matrix can be spatially and mechanically non-uniform due to the presence of cell-sized spaces surrounded by denser matrix. Nevertheless, the typical pore structure of healthy interstitial matrix impedes cell migration without significant remodeling \\cite{Wolf2009}. A biomimetic model for these fibrous architectures is to reconstitute collagen I from rat tail or bovine skin, which can be tuned based on collagen concentration, polymerization temperature and pH (see reviews in \\cite{Hapach2015,Sapudom2018}). \n\nPathological states such as fibrosis and tumor progression are associated with aberrant deposition of collagens, fibronectin, elastin, laminin, hyaluronic acid, proteoglycans (e.g. versican, syndecan, glypican, etc.), and other matrisome proteins (e.g. tenascin C, periostin, osteopontin, SPARC, thrombospondin) (Fig. \\ref{fig:ecmcartoon}C,i) \\cite{Piersma2020}. Structurally, tumor-associated collagen I is increasingly dense, straight and aligned, with cell-sized tracks leading into the stroma (Fig. \\ref{fig:ecmcartoon}C,ii-v) \\cite{Provenzano2006,Provenzano2008b,Conklin2011}. Indeed, collagen I can be further strengthened by chemical crosslinking by lysyl oxidase and transglutaminase (Fig. \\ref{fig:ecmcartoon}C,iii) \\cite{Levental2009}, which has been recently linked to the inflammatory activity of tumor-associated macrophages \\cite{Maller2021}.\n\nThe stiffness of tumor associated ECM is dramatically higher than healthy ECM, an indicator of aberrant ECM composition and architecture (Fig. \\ref{fig:ecmcartoon}B,vi;C,vi) \\cite{Piersma2020}. For any solid material, these mechanical properties can be quantified based on a reversible deformation in response to an applied stress (i.e. elasticity), which is typically linear for small deformations (i.e. strains) \\cite{fung_tong_2001}. Interestingly, fibrous materials often exhibit a nonlinear elasticity, so that they can stiffen at increasing strain, attributed to the increasing difficulty of reorienting fibers, or the cost of straightening and stretching fibers \\cite{Broedersz2014}. For sufficiently large deformations, this structural reorientation may be permanent (i.e. plastic) and will not be reversible after the stress is removed. Finally, it should be noted that biomaterials incorporate significant (viscous) fluid, resulting in a time-dependent mechanical response as some energy is dissipated via viscoelastic or poroelastic mechanisms \\cite{Chaudhuri2020}. An unresolved question is the stiffness and viscosity of the basement membrane, which varies across tissues and also is challenging to probe due to its small thickness and tight integration with surrounding tissue (Fig. \\ref{fig:ecmcartoon}A,vi-vii). Interestingly, H. Li et al. recently reported that cell-deposited basement membranes exhibit nonlinear strain stiffening, which has not been observed using other mechanical measurements that operate in the linear regime \\cite{Li2021}.\n\nSynthetic hydrogels can be prepared with comparable stiffness as ECM using flexible polymers with tunable crosslinks and ligand density, resulting in spatially uniform materials with nanoscale pores \\cite{Leggett2017,Beri2018}. For example, polyethylene glycol and polyacrylamide have been widely explored to elucidate how cancer cells respond in vitro to linear elastic material properties (see review in \\cite{Janmey2020}). In particular, polyethylene glycol permits systematic control of ligand density, crosslinking and degradability to investigate cancer cell invasion \\cite{gill2012synthetic, beck2013independent, singh2015synthetic,pradhan2019tunable}. Alternatively, alginate is a seaweed-derived polysaccharide where viscoelasticity (e.g. stress relaxation) can be tuned using reversible ionic crosslinks or by molecular weight \\cite{Chaudhuri2017}. Furthermore, synthetic hydrogels with spatially anisotropic architectures are able to mimic ECM cues, such as the aligned topographical features of collagen fibers \\cite{Prince2019}.  Such ``designer biomaterials'' may permit improved control over distinct mechanical and biochemical features of the cell microenvironment. For example, aligned ECM cues can be patterned within a homogeneous matrix using magnetic nanoparticle self-assembly, allowing for hydrogels with modulated alignment cues while maintaining comparable elastic moduli and ligand densities \\cite{kim2016independent,paul2019probing}. \n\n\n\\section{Precision Measurement of ECM Mechanics\n}\n\\label{sec:mechanics}\n\nInvading cancer cells both sense and respond to the structure and composition of the ECM, a bi-directional interaction known as dynamic reciprocity \\cite{Xu2009e}. Fluorescence imaging of cell morphology is widely used to elucidate how cells are shaped by the local ECM properties \\cite{Driscoll2015a}. However, visualizing how cells act on the surrounding ECM remains challenging, particularly in 3D geometries \\cite{Polacheck2016}. Seminal work by Legant et al. visualized how fibroblasts applied mechanical tractions within a synthetic 3D hydrogel (e.g. polyethylene glycol diacrylate, PEGDA) by tracking the motion of microscale tracer particles relative to an undeformed reference state (e.g. traction force microscopy) \\cite{Legant2010}. It should be noted that PEGDA is a purely elastic material, with a well-established constitutive equation that permits the cell-generated stresses to be quantitatively extracted from the matrix strain field. Nevertheless, naturally-derived biomaterials exhibit a more complex rheological response that is nonlinear and viscoelastic, and can be irreversibly remodeled by cell secreted factors \\cite{Hall2013}.\n\nBoth J. Steinwachs et al. and M. Hall et al. measured tractions exerted by single breast cancer cells invading within reconstituted collagen I matrix labeled with fluorescent microparticles \\cite{Steinwachs2016,Hall2016}. These breast cancer cells (e.g. MDA-MB-231) mechanically interact with discrete collagen fibers, which become aligned along the cell axis to locally stiffen the matrix. This nonlinear strain-stiffening response suggests that cells within these fibrillar materials can mechanically interact over longer distances than in more homogeneous materials that exhibit a linear elastic response. In order to compute cell-generated stresses, both groups formulated constitutive equations optimized for random fiber networks, validated using bulk rheology of cell-free collagen I matrix. Interestingly, recent work by van Oosten et al. shows that these strain-stiffening fiber networks switch to strain-\\emph{softening} behavior in the presence of cell-sized inclusions, reminiscent of the architecture and mechanics of living tissues \\cite{vanOosten2019}. Thus, non-uniformity of the ECM at multiple length scales is likely to impact cellular mechanobiology.\n\nY. Han et al. sought to directly measure matrix mechanical properties in the neighborhood of single cells using optical tweezers \\cite{Han2018} (Fig. \\ref{fig:ecm}A,i,ii). In these experiments, microparticles entangled within a collagen I matrix were perturbed under a controlled displacement $x$ and the resulting force $F$ was optically measured, which is a readout of the local matrix stiffness (Fig. \\ref{fig:ecm}A,iii). This approach mapped the spatial variation in stiffness to a nonlinear stress profile Fig. \\ref{fig:ecm}A,iv). Notably, matrix stiffness was strongly elevated near cancer cells (e.g. MDA-MB-231) as the fibrillar matrix was plastically deformed into more aligned architectures, which was also observed for reconstituted basement membrane and fibrin. Although this method is a very sensitive and localized measurement of matrix rheology, it is relatively time and labor intensive since microparticles are probed one at a time.\n\n\\begin{figure}[h]\n\\includegraphics[width=4in]{ecmmechanics.png}\n \\centering\n\\caption{A. Direct measurement of ECM stress stiffening using optical tweezers. (i) MDA-MB-231 cell (blue) in a 3D collagen network (green). (Scale bar, 10 \u00b5m.) (ii-iv) Schematics of the force\u2013displacement measurement using optical tweezers, and the relation between matrix stiffening (blue potential wells) and the cell-generated stress field along the cell axis. Reproduced from \\cite{Han2018}. B. High-throughput confocal imaging on a 96 well plate (i), visualizing multicellular clusters (green) and tracer particles (red) embedded in 3D silk collagen matrix (ii). Representative displacement field exhibits spatially non-uniform regions of local outward (yellow) and inward (purple) displacement (iii). Reproduced from \\cite{Leggett2020}. C. Dynamic light scattering based on cells encapsulated within 3D collagen \/ Matrigel, read out via fluctuations in scattering intensity over varying lag times. Reproduced from \\cite{Krajina2021}. D. Optical coherence tomography of multicellular spheroids in 3D collagen matrix for label-free readout of matrix deformation, revealing enhanced invasion for mammary epithelial cells (MCF-10A) co-cultured with obese adipose stem cells (ob\/ob ASC). Reproduced from \\cite{Ling2020}.}\n\\label{fig:ecm}       \n\\end{figure}\n\nMulticellular clusters in 3D matrix may coordinate their tractions to facilitate collective migration, which adds additional complexity to be explored. S. Leggett et al. investigated how mammary epithelial acini (e.g. MCF-10A) transitioned towards invasion after controlled induction of EMT via the master regulator Snail \\cite{Leggett2020}. Cells were embedded within composite silk-collagen matrix \\cite{Khoo2019} in a 96-well plate, enabling increased experimental throughput to evaluate drug response (Fig. \\ref{fig:ecm}B). Using optimized topology-based particle tracking, localized patterns of protrusive and contractile matrix displacement could be resolved with submicron resolution \\cite{Patel2018}. Notably, epithelial clusters were usually compact and exhibited several regions of local protrusion or contraction. Induction of a transitory EMT state results in clusters with extended protrusions with more regions of local contraction, indicative of outgrowth and invasion. Finally, fully mesenchymal states were associated with spindle-like morphologies with only a few localized contractile regions and minimal protrusions. These results show that collective migration and EMT are associated with spatially non-uniform patterns of matrix displacement, which can be used as a mechanophenotypic ``signature'' of cell state. This platform is scalable for a broader range of cell types and biomaterials, which is particularly relevant or clinical translation using patient-derived cells and drug screening.\n\nIn order to longitudinally probe matrix viscoelasticity over a wide frequency range, B. Krajina et al. utilized dynamic light scattering to measure fluctuations of dilute microparticles within a biomaterial (Fig. \\ref{fig:ecm}C) \\cite{Krajina2021}. Based on correlations in scattering intensity, an ensemble averaged mean square displacement can be extracted and converted to a frequency-dependent shear modulus using the generalized Stokes-Einstein relation \\cite{Squires2010}. In principle, this approach could resolve both thermally driven motion as well as actively driven motion due to cell-generated forces. For instance, mammary epithelial spheroids (e.g. MCF-10A) embedded in composites of collagen I and Matrigel drove considerable changes in shear modulus over 6 days. With the addition of TGF-$\\beta$ to drive invasion and EMT, there was an effective increase in matrix stiffness at high frequencies and matrix fluidization at low frequencies, which was explained based on elevated matrix degradation as well as cellular contractility. This dynamic light-scattering approach promises to provide new insights into how biomaterials are altered over extended durations in response to cellular activity, although it will likely require complementary measurements with conventional fluorescence microscopy to map back to cell morphology and migratory behaviors. \n\nFinally, Adie and coworkers utilized optical coherence tomography for label-free imaging of cells and collagen fibers \\cite{Mulligan2019}. As a case study, mammary epithelial spheroids (e.g. MCF-10A) were characterized with or without co-culture with adipose stem cells \\cite{Ling2020}. Notably, mammary epithelial spheroids only remained relatively localized with limited matrix deformation (Fig. \\ref{fig:ecm}D). In comparison, the addition of adipose stem cells from obese mice resulted in extensive collective invasion as well as matrix remodeling, which could be rescued by pharmacological inhibition of matrix degradation or Rho kinase activity. This technique has great potential since it can be scaled to larger imaging volumes in highly scattering media, directly imaging biomaterial deformations with minimal phototoxicity.\n\nEmerging technologies to visualize bi-directional interactions between cancer cells and 3D matrix will enable new insights into collective invasion and drug response, particularly when combined with patient-derived organoids, as well as stromal and immune cells. Thus far, these early proof-of-concepts have investigated cell-generated tractions in relatively simplified biomaterials such as collagen I or reconstituted basement membrane (e.g. Matrigel). As the composition and architecture of biomaterials increases in complexity to mimic the in vivo environment, an improved fundamental understanding of these material properties will be needed to understand how cancer cells are affected. Lastly, these biophysical measurements all require specialized equipment and training for optimal results, and further work is needed to translate these techniques into conventional biomedical settings.\n\n\n\n\\section{Epithelial-Mesenchymal Plasticity and Collective Cell Migration\n}\n\\label{sec:EMT}\nClassically, EMT has been understood as a phenotypic switch in which adherent cells lose epithelial biomarkers (e.g. E-cadherin) and gain mesenchymal biomarkers (e.g. vimentin) in order to disseminate individually for embryonic development and wound healing \\cite{Nieto2016}. EMT is often observed at tumor invasion fronts, mediated by tumor-stromal interactions and the interstitial matrix \\cite{christofori2006new}. More recently, epithelial-mesenchymal plasticity (EMP) has been proposed to represent a broader spectrum of intermediate or hybrid cell states that may exhibit some mixture of epithelial and mesenchymal features \\cite{Yang2020b}. Functionally, EMP is associated with a decrease in differentiated features (e.g. apico-basal polarity, strong cell-cell adhesions), resulting in reorganization of the cytoskeleton and cell-matrix adhesions towards front-back polarity and invasion across the basement membrane, which may or may not be captured by certain biomarkers or classical EMT transcription factors (e.g. Snail, Twist, ZEB, etc.) \\cite{Yang2020b}. These morphological phenotypes often exhibit appreciable variability at the single cell level, even for relatively controlled induction of transcription factors \\cite{leggett2016morphological}. Nevertheless, the mechanistic role of EMT in human tumor metastasis remains unresolved, partially due to the difficulty of measuring a rare and dynamic process at the single cell level (within patients) \\cite{Diepenbruck2016}. For instance, if EMT occurs transiently as tumor cells disseminate from the primary site and they revert back to an epithelial state (MET) at a secondary metastatic site, measurements of the primary and metastatic site will most likely yield epithelial cells \\cite{Brabletz2012}. Several recent papers have combined live cell imaging with deep molecular profiling in order to better elucidate the potential role of EMP.\n\n\\begin{figure}[h]\n\\includegraphics[width=3in]{emt}\n \\centering\n\\caption{A. Immunofluorescence staining of keratin 14 (K14) and vimentin (Vim) (red) expression in tumor cells (GFP) across epithelial, early\/late hybrid and mesenchymal states (i), corresponding to changes in EpCAM, CD106\/Vcam1, CD51\/Itgav, and CD61\/Itgb3 (ii). Reproduced from \\cite{Pastushenko2018,Pastushenko2019}. B. Leader cells in MMTV-PyMT breast tumor organoids exhibit a basal epithelial phenotype with high keratin 14 expression. Reproduced from \\cite{Cheung2013}. C. Lineage tracing of partial EMT via Tenascin C gene expression reveals enhanced migration and plasticity relative to late EMT. Reproduced from \\cite{Luond2021}}.\n\\label{fig:emt}       \n\\end{figure}\n\nI. Pastushenko et al. utilized fluorescence activated cell sorting (FACS) and single cell RNA-seq to map an EMT landscape in genetically engineered mouse models of skin squamous cell carcinoma (KRas$^{LSL-G12D}$p53$^{fl\/fl}$) and breast cancer (MMTV-PyMT) that overexpressed fluorescent proteins \\cite{Pastushenko2018, Pastushenko2019}. They defined an epithelial state based on high expression of Keratin 14 (K14) with no vimentin expression, which exhibited a compact cell morphology and (high E-cadherin expression) (Fig. \\ref{fig:emt}A,i). Further, hybrid states were observed which coexpressed K14 and vimentin with partially elongated morphology (and low E-cadherin expression), associated with collective migration. Finally, mesenchymal states expressed vimentin with no K14 (or E-cadherin), and exhibited a fully elongated morphology. The epithelial state also expressed high EpCAM, which was absent in hybrid and mesenchymal states (Fig. \\ref{fig:emt}A,ii). Instead, an ``early hybrid'' state was defined based on the absence of CD106 (Vcam1), (Itgav), and CD61 (Itgb3), whereas a ``hybrid state'' was associated with CD106 expression only. Next, a ``late hybrid'' state was associated with expression of CD51 and\/or CD106. A mesenchymal state was associated with co-expression of CD51 and CD61 or CD51, CD106, and CD61. Functionally, epithelial cells proliferated faster than cells in hybrid or mesenchymal states. However, hybrid or mesenchymal cells exhibited much higher tumor initiating potential than epithelial cells when implanted at successively smaller dilutions and displayed increased invasion. Nevertheless, hybrid cells exhibited the most plasticity to transition to other states, as well as the greatest metastatic potential. These hybrid or mesenchymal states were also promoted by the local microenvironment, including blood and lymphatic vessels as well as macrophages.\n\nInstead, K.J. Cheung et al demonstrated that epithelial cells expressing K14 from genetically engineered mouse models of breast cancer (MMTV-PyMT) act as leader cells for collective migration (Fig. \\ref{fig:emt}B) \\cite{Cheung2013}. Mouse tumors were isolated and digested into fragments, which were then embedded in 3D collagen I matrix. Leader cells were associated with basal markers (e.g. p63, P-cadherin, and Keratin 5), but did not express common EMT biomarkers (e.g. vimentin, Snail, Twist). Further, these leader cells maintained E-cadherin expression and were mechanically connected to their followers, which were K14 negative. Indeed, K14 positive leader cells were preferentially localized at the tumor periphery and in lung metastases in mouse models, as well as in organoids and histology slides from human patients. Interestingly, K14 positive cells were present but did not function as leaders when these organoids were cultured in Matrigel. Re-transplantation of these organoids from Matrigel to collagen I rescued the K14 positive leader cell phenotype and aggressive invasion. \n\nF. L{\\\"u}{\\\"o}nd et al. developed a dual recombinase fluorescence reporter construct to lineage trace breast tumor cells that had undergone an EMT (reversible or irreversible) \\cite{Luond2021}. Briefly, a first reporter labeled epithelial cells with mCherry, but irreversibly switched to GFP expression when early EMT (e.g. Tenascin C, Tnc) was induced with Tamoxifen treatment. A second reporter similarly labeled epithelial cells, but irreversibly switched to GFP expression during late EMT (e.g. N-cadherin, Cdh2), also under Tamoxifen treatment. Live cell imaging of primary tumor slices revealed that some early (partial) EMT cells exhibited an elongated morphology as individuals with directional migration (Fig. \\ref{fig:emt}C,1,2). Other early (partial) EMT cells associated with multicellular ``colonies,'' which maintained cell-cell contacts, although cells at the periphery of these colonies were elongated and motile (Fig. \\ref{fig:emt}C,3). In comparison, late (full) EMT cells were also elongated but remained localized in perivascular regions near capillaries. Moreover, early (partial) EMT cells invaded collectively and aggressively as leader cells when seeded on collagen I matrix, while late (full) EMT cells invaded individually. Further, early (partial) EMT cells were enriched in lung metastases, while late (full) EMT cells were enriched in response to chemotherapy. Other recent reports corroborate that partial EMT is associated with collective invasion and plasticity \\cite{Aiello2018}, as well as enhanced metastatic potential \\cite{Simeonov2021}. \n\n\n\\section{Perspective and Future Directions\n}\n\\label{sec:perspective}\nThe statistician G.E.P. Box famously commented that ``\\emph{all models are wrong; the practical question is how wrong do they have to be to not be useful}.'' \\cite{boxmodel}\nThis aphorism is relevant in the context of cancer biology, where modeling refers to vitro or in vivo experimental systems that represent key features of patient tumors \\cite{Ben-David2019}, as well as the physical sciences, where modeling refers to theoretical and computational approaches \\cite{Altrock2015}. In both instances, an unresolved question is how a group of cancer cells with significant (intratumoral) heterogeneity are able to coordinate robust behaviors \\cite{Tabassum2015}. As presented here, physical science approaches represent a promising approach to elucidate generalized principles of collective invasion, given that cancer has common set of requirements for metastasis despite high intratumoral heterogeneity. We argue that such collective processes are crucial for invasion and metastasis as well as therapy resistance, which are ultimately responsible for most cancer-related fatalities in human patients \\cite{Weiss2022}. Thus, in this chapter, we have considered new biological and engineering approaches that have the potential for useful insights, given the ethical and practical limits on direct experimental access to human patient tumors. \n\nModern single cell -omics enable unprecedented insights into molecular signaling pathways that regulate cellular behavior \\cite{Satija2019}. However, such deep phenotyping typically requires the removal and destruction of cells at an endpoint, making it difficult for dynamic measurements. In comparison, live cell imaging and fluorescent reporters can visualize how cells migrate and proliferate with unprecedented spatiotemporal resolution \\cite{Lemon2020,Specht2017}. However, these imaging techniques are limited to a few fluorescence channels, so that only a few signaling pathways or cellular features can be monitored. This mismatch makes it challenging to elucidate how myriad molecular signaling pathways are integrated into mechanobiological phenotypes such as cell migration and proliferation. The Connectivity Map approach has proven quite powerful to computationally link gene expression profiles to drug response \\cite{lambcmap}. An intriguing possibility is to further couple such molecular analyses with cellular-scale computational models that account for the mechanobiology of the cytoskeleton, cell-matrix adhesions, etc. \\cite{Buttenschon2020} This could make testable predictions about therapeutic regimens that could inhibit invasion or proliferation. Given interpatient heterogeneity and the challenges of drug development, even predicting which treatments may \\emph{not} work would be useful to inform clinical practice. \n\nAberrant metabolism represents a promising target to inhibit both collective migration as well as uncontrolled proliferation. Leader cells represent an intriguing target, either via inhibition of glycolysis (e.g. 2-Deoxyglucose) or oxidative phosphorylation (e.g. metformin). Given the observed phenotypic plasticity and heterogeneity of leader and follower cells, it may be necessary to target both glycolysis and oxidative phosphorylation. It should also be noted that various FDA-approved drugs that inhibit metabolism can in turn mitigate aggressive EMT features of tumors \\cite{ramesh2020targeting}. For example, when activated by excess glucose, the polyol metabolic pathway can induce EMT through autocrine TGF-$\\beta$ stimulation \\cite{Schwab2018}, which is known to potentiate tumor aggressiveness. The polyol pathway inhibitor epalrestat, used in the treatment of diabetic neuropathy, is under a Phase II clinical trial to evaluate its potential to inhibit triple-negative breast cancer \\cite{Wu2017a}. Thus, an improved fundamental understanding of how metabolism and mechanobiology are mechanistically coupled may inform the design of targeted therapies, likely acting in combination on different hallmarks of cancer.\n\n\nAlthough visualizing cell morphology and migration with fluorescence microscopy is relatively straightforward, inferring cell-cell interactions remains challenging, and is often indirectly inferred from cell morphology and migration. For instance, physics-inspired analyses based on PIV or order parameters are sufficient to reveal when groups of cells migrate with similar speed and direction \\cite{Ilina2020}, but provide limited quantitative information on how these cells interact. One promising approach for analyzing multicellular configurations is to analyze the mechanical cost of rearrangements relative to some idealized reference state \\cite{Yang2021}. Alternatively, the topological differences between different multicellular architectures can be compared using persistence homology \\cite{Bhaskar2021}. It would be intriguing to develop a hybrid machine learning approach that accounts for the mechanical cost of cell reorganization but does not assume a reference state. However, it should be noted that tumor cells are intrinsically heterogeneous, and approximating them as identical agents with exactly the same cell-cell and cell-matrix adhesions may not be valid. Establishing meaningful readouts of heterogeneity and cooperative behavior may nevertheless be useful as a therapeutic target (beyond cell viability) that can be perturbed. Moreover, understanding when groups of cells exhibit more homogeneous behaviors may provide insight into how ``noise'' is regulated by homeostasis in normal epithelial tissues.\n\nSimilarly, elucidating reciprocal mechanical interactions between cells and ECM is experimentally difficult \\cite{Polacheck2016}. Early groundbreaking work utilized planar 2D substrates with controlled biochemical ligand density and stiffness, which were non-cell-degradable and exhibited a well-defined linear elastic response \\cite{Janmey2020}. Subsequent experiments using 3D biomaterials are based on a similar premise - that cells should exhibit relatively similar behaviors when cultured within (relatively) reproducible and spatially homogeneous mechanical environments \\cite{LeSavage2021}. Nevertheless, cells also exhibit the capability to substantially remodel biomaterials, so that the local architecture and rheology may vary considerably at later times. Thus, recent technology development seeks to mechanically probe alterations of ECM \\cite{Han2018,Mulligan2019,Krajina2021}, although there remain trade-offs in spatial and temporal resolution. Alternatively, spatially heterogeneous patterns of forces may be sufficient to infer distinct cell states \\cite{Leggett2020}. Thus, we envision that dynamic, stimuli-responsive biomaterials will be increasingly used to reveal dynamic cell responses \\cite{Uto2017}. Moreover, it should be noted that ECM in vivo is spatially \\emph{in}homogeneous in composition and structure, particularly during fibrosis or tumor progression. Highly tunable biomaterials capable of recapitulating desmoplastic ECM architectures in the tumor microenvironment \\cite{Prince2019} will be needed to understand why current therapies, such as CAR-T immunotherapy, often fail to penetrate immunosuppressive solid tumors \\cite{Pires2021}. High resolution mapping of local ECM architecture in vivo, combined with a physical understanding of how cells migrate collectively in different homogeneous biomaterials in vitro, could also predict how cell might invade inhomogeneous microenvironments in vivo \\cite{Madsen2015}. Indeed, robust biomaterial-based models serve as an important first-step in elucidating collective invasion mechanisms in a controlled environment relative to in vivo models, where mechanisms can be further corroborated. \n\nFinally, tumor cells not only exhibit phenotypic heterogeneity but also phenotypic plasticity, which is often a relatively rare event. Thus, it remains challenging to test the role of epithelial-mesenchymal plasticity in human patients without complete single cell information at high temporal resolution \\cite{Yang2020b}. Indeed, it is conceivable that tumor cells could adapt their phenotype as they encounter different microregions of the primary tumor, colonize the secondary metastatic site, or in response to therapeutic treatments \\cite{gupta2006cancer}. Recent advances in biofabrication and microfluidics enable compartmentalized engineered tissues, where soluble signaling conditions can be rapidly switched \\cite{Ayuso2021a}. This capability to rapidly and sharply switch microenvironmental cues is likely to be quite powerful to directly observe how cells respond. Further, improvements in epigenetic profiling will be useful to understand when cells mechanically adapt or retain a ``memory'' of prior phenotypes  \\cite{Nemec2021}. An ongoing translational challenge may be to understand how patient tumor cells individually and collectively adapt to changing microenvironmental conditions via leader cells or  EMP, and to limit heterogeneity in an evolutionary context \\cite{Gatenby2020}.\n\nIn conclusion, we have considered how the mechanobiology of cell-cell and cell-matrix interactions mediate how tumor cells collectively invade into 3D matrix. We highlight how these coordinated behaviors with leader-follower interactions and epithelial-mesenchymal plasticity represent a caricature of embryonic development, as tumor cells repurpose the associated molecular signaling and biophysical machinery. We also highlight emerging technologies in ex vivo measurements of cell migration based on 3D biomaterials with live cell imaging. In combination, these technologies can quantify how groups of cells cooperate mechanically or metabolically with each other and to remodel the surrounding ECM. These exquisitely sensitive measurements of single cell dynamics will enable new physical models that could be integrated with molecular profiling to predict how patient cells respond to targeted treatments, revealing new fundamental insights and accelerating clinical translation. Ultimately, we envision that the integration of new bioengineering assays with physical modeling represent a highly ``useful'' representation of human tumors to address unresolved questions of intratumoral and interpatient heterogeneity in cancer progression.\n\n\n\\section*{Acknowledgements}\nWe apologize to authors whose work could not be included due to space constraints. This work was supported by the National Institute of General Medical Sciences (R01GM140108, P20GM109035).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Brinkman problem governs the flow of a viscous fluid in an inhomogeneous material where fractures, bubbles, or channels are present within a porous matrix.\nMathematically, this problem translates into a system of partial differential equations with saddle-point structure which can be regarded as a superposition of the Stokes and Darcy systems.\nAs pointed out in \\cite{Mardal.Tai.ea:02}, the construction of finite element approximations that are uniformly well-behaved across the entire range of (Stokes- or Darcy-dominated) regimes is not straightforward; a representative, but by far not exhaustive, list of references is \\cite{Burman.Hansbo:05,Burman.Hansbo:07,Juntunen.Stenberg:10,Konno.Stenberg:11,Anaya.Gatica.ea:15,Evans.Hughes:13,Caceres.Gatica.ea:17,Vacca:18,Araya.Harder.ea:17}.\nIn \\cite{Botti.Di-Pietro.ea:18}, we introduced a numerical method for the Brinkman problem on matching simplicial meshes and derived what appears to be the first error estimate accounting for the local regime through a dimensionless number which can be interpreted as a friction coefficient.\nThanks to the presence of cutoff factors, this error estimate holds in all situations, including the Stokes problem as well as the singular limit corresponding to the pure Darcy problem.\n\nIn this work, we provide a positive answer to an open question left in the above reference, namely whether similar robustness features and error estimates can be obtained on general polytopal meshes.\nAs for the original method of \\cite{Botti.Di-Pietro.ea:18}, the discretisation of the Stokes term is inspired by Hybrid High-Order (HHO) methods \\cite{Di-Pietro.Ern.ea:14,Di-Pietro.Ern:15,Di-Pietro.Droniou:20} while, for the Darcy and forcing terms, a novel construction inspired by discrete de Rham methods \\cite{Di-Pietro.Droniou.ea:20,Di-Pietro.Droniou:21*1} (see also \\cite{Di-Pietro.Ern:17} for an antecedent) replaces the one based on the Raviart--Thomas--N\\'ed\\'elec space \\cite{Raviart.Thomas:77,Nedelec:80}.\nThe first central element in this construction is a discrete vector potential that changes in nature depending on the value of the local friction coefficient.\nThe other key ingredient are regime-dependent stabilisation terms.\nThanks to these novel tools, we are able to derive a robust estimate of the adjoint error for the discrete divergence, which is the pivot result for the extension of the techniques of \\cite{Botti.Di-Pietro.ea:18} to polytopal meshes.\nThe resulting error estimate, stated in Theorem~\\ref{thm:error.estimate} below, is valid on the entire range of values for the local friction coefficient, from $0$ (pure Stokes) to $+\\infty$ (pure Darcy).\n\nThe rest of the work is organised as follows.\nIn Section~\\ref{sec:setting} we briefly recall the continuous and discrete settings.\nIn Section~\\ref{sec:scheme} we formulate the numerical scheme and state the main stability and convergence results.\nExtensive numerical validation of these results on a variety of meshes and regimes for analytical solutions is provided in Section~\\ref{sec:numerical.tests}, where a more physical three-dimensional test case is also considered.\nFinally, the proofs of the main results are collected in Section~\\ref{sec:analysis}.\n\n\n\n\\section{Setting}\\label{sec:setting}\n\n\\subsection{Continuous problem}\n\nLet $\\Omega\\subset\\mathbb{R}^d$, $d\\in\\{2,3\\}$, denote a bounded connected open polytopal (i.e., polygonal if $d=2$ and polyhedral if $d=3$) domain with boundary $\\partial\\Omega$.\nFor the sake of simplicity, and without loss of generality, we assume that $\\Omega$ has unit diameter.\nLet two functions $\\mu:\\Omega\\to\\mathbb{R}$ and $\\nu:\\Omega\\to\\mathbb{R}$ be given.\nIn what follows, we assume that there exist real numbers $\\underline{\\mu},\\overline{\\mu}$, and $\\overline{\\nu}$ such that, almost everywhere in $\\Omega$,\n\\begin{equation}\\label{eq:mu.nu:bounds}\n  0<\\underline{\\mu}\\le\\mu\\le\\overline{\\mu},\\qquad\n  0\\le\\nu\\le\\overline{\\nu}.\n\\end{equation}\nLet $\\bvec{f}:\\Omega\\to\\mathbb{R}^d$ and $g:\\Omega\\to\\mathbb{R}$ denote volumetric source terms.\nThe Brinkman problem reads:\nFind the velocity $\\bvec{u}:\\Omega\\to\\mathbb{R}^d$ and the pressure $p:\\Omega\\to\\mathbb{R}$ such that\n\\begin{subequations}\\label{eq:strong}\n  \\begin{alignat}{2}\n    \\label{eq:strong:momentum}\n    -\\boldsymbol{\\nabla}{\\cdot}(\\mu\\boldsymbol{\\nabla}\\bvec{u}) + \\nu\\bvec{u} + \\boldsymbol{\\nabla} p &= \\bvec{f} &\\qquad&\\text{in $\\Omega$},\n    \\\\ \\label{eq:strong:mass}\n    \\boldsymbol{\\nabla}{\\cdot}\\bvec{u} &= g &\\qquad&\\text{in $\\Omega$},\n    \\\\ \\label{eq:bc}\n    \\bvec{u} &= \\bvec{0} &\\qquad&\\text{on $\\partial\\Omega$},\n    \\\\ \\label{eq:zero.mean.pressure}\n    \\int_\\Omega p &= 0.\n  \\end{alignat}\n\\end{subequations}\nA few simplifications are made to make the exposition more compact while retaining all the difficulties related to the robustness across the entire range of values for $\\mu$ and $\\nu$.\nFirst of all, in \\eqref{eq:strong:momentum} we have considered a viscous term expressed in terms of the full gradient instead of its symmetric part $\\GRAD_{\\rm s}$.\nThe changes to replace $\\boldsymbol{\\nabla}$ with $\\GRAD_{\\rm s}$ are standard in the HHO literature; see, e.g., \\cite{Di-Pietro.Ern:15,Botti.Di-Pietro.ea:19*3} and \\cite[Chapter~7]{Di-Pietro.Droniou:20}.\nSecond, we assume henceforth that both $\\mu$ and $\\nu$ are piecewise constant on a polytopal partition $P_\\Omega$ of the domain.\nThe extension to coefficients that vary smoothly inside each element, and are possibly full tensors, is also standard; see, in particular, \\cite[Section~4.2]{Di-Pietro.Droniou:20}.\n\n\\subsection{Discrete setting}\n\n\\subsubsection{Mesh and notation for inequalities up to a constant}\n\nWe consider polytopal meshes $\\mathcal{M}_h\\coloneq\\elements{h}\\cup\\faces{h}$ matching the geometrical requirements detailed in \\cite[Definition 1.4]{Di-Pietro.Droniou:20}, with $\\elements{h}$ set of elements and $\\faces{h}$ set of faces.\nTo avoid dealing with jumps of the problem coefficients $\\mu$ and $\\nu$ inside mesh elements, we additionally assume that $\\elements{h}$ is compatible with $P_\\Omega$, meaning that, for each $T\\in\\elements{h}$, there exists $\\omega\\in P_\\Omega$ such that $T\\subset\\omega$.\nWe then set $\\mu_T\\coloneq \\mu_{|T}$ and $\\nu_T\\coloneq \\nu_{|T}$ for all $T\\in\\elements{h}$, noticing that these constant values are uniquely defined in each element.\nFor any $Y\\in\\mathcal{M}_h$, we denote by $h_Y$ its diameter, so that $h=\\max_{T\\in\\elements{h}}h_T>0$.\nFor every mesh element $T\\in\\elements{h}$, we denote by $\\faces{T}$ the subset of $\\faces{h}$ containing the faces that lie on the boundary $\\partial T$ of $T$.\nFor any mesh face $F\\in\\faces{h}$, we fix once and for all a unit normal vector $\\bvec{n}_F$ and, for any mesh element $T\\in\\elements{h}$ such that $F\\in\\faces{T}$, we let $\\omega_{TF}\\in\\{-1,+1\\}$ denote the orientation of $F$ relative to $T$, selected so that $\\omega_{TF}\\bvec{n}_F$ points out of $T$.\nBoundary faces lying on $\\partial\\Omega$  are collected in the set $\\faces{h}^\\boundary$.\n\nOur focus being on the $h$-convergence analysis, we assume that $\\mathcal{M}_h$ belongs to a sequence of refined polygonal or polyhedral meshes that is regular in the sense of \\cite[Definition 1.9]{Di-Pietro.Droniou:20}.\nThis implies, in particular, that the number of faces of each mesh element is bounded from above by an integer independent of $h$; see \\cite[Lemma 1.12]{Di-Pietro.Droniou:20}.\n\nFrom this point on, $a \\lesssim b$ (resp. $a \\gtrsim b$) means $a\\le Cb$ (resp. $a\\ge Cb$) with $C$ only depending on $\\Omega$, the mesh regularity parameter, and the polynomial degree $k$ of the scheme defined in Section~\\ref{sec:scheme}.\nWe stress that this means, in particular, that $C$ is independent of the problem parameters $\\mu$ and $\\nu$.\nWe also write $a\\simeq b$ as a shorthand for ``$a\\lesssim b$ and $b\\lesssim a$''.\n\n\\subsubsection{Polynomial spaces}\n\nGiven $Y\\in\\elements{h}\\cup\\faces{h}$ and an integer $m\\ge 0$, we denote by $\\Poly{m}(Y)$ the space spanned by the restriction to $Y$ of $d$-variate polynomials of total degree $\\le m$.\nThe symbols $\\vPoly{m}(Y;\\mathbb{R}^d)$ and $\\tPoly{m}(Y;\\mathbb{R}^{d\\times d})$ respectively denote the sets of vector- and tensor-valued functions over $Y$ whose components are in $\\Poly{m}(Y)$.\nFor $T\\in\\elements{h}$, we will need the following direct decomposition of $\\vPoly{m}(T;\\mathbb{R}^d)$ (see, e.g., \\cite[Corollary 7.4]{Arnold:18}):\n\\\n  \\vPoly{m}(T;\\mathbb{R}^d)\n  = \\Goly{m}(T) \\oplus \\cGoly{m}(T),\n  \\\n  with\n  \\begin{equation}\\label{eq:Goly.cGoly}\n    \\text{%\n      $\\Goly{m}(T)\\coloneq\\boldsymbol{\\nabla}\\Poly{m+1}(T)$\\quad and\\quad\n      $\\cGoly{m}(T)\\coloneq\\begin{cases}\n      (\\bvec{x} - \\bvec{x}_T)^\\bot\\Poly{m-1}(T) & \\text{if $d=2$},\n      \\\\\n      (\\bvec{x} - \\bvec{x}_T)\\times\\vPoly{m-1}(T;\\mathbb{R}^3) & \\text{if $d=3$},\n      \\end{cases}$\n    }\n  \\end{equation}\nwhere $\\bvec{x}_T$ is a point such that $T$ is star-shaped with respect to a ball of radius $\\gtrsim h_T$ and, in the case $d=2$, for any $\\bvec{v}\\in\\mathbb{R}^2$ we denote by $\\bvec{v}^\\bot$ the vector obtained rotating $\\bvec{v}$ by $-\\frac\\pi2$ radians.\nGiven a polynomial (sub)space $\\mathcal{X}^m(Y)$ on $Y\\in\\elements{h}\\cup\\faces{h}$, the corresponding $L^2$-orthogonal projector is denoted by $\\pi_{\\mathcal{X},Y}^m$.\nBoldface fonts will be used when the elements of $\\mathcal{X}^m(Y)$ are vector-valued.\nThe set of broken polynomials of total degree $\\le m$ on the mesh is denoted by $\\Poly{m}(\\elements{h})$, and the corresponding $L^2$-orthogonal projector by $\\lproj{m}{h}$.\n\n\\subsubsection{Local friction coefficient}\n\nThe regime inside each mesh element $T\\in\\elements{h}$ is identified by the following dimensionless number, which can be interpreted as a friction coefficient:\n\\begin{equation}\\label{eq:CfT}\n  \\Cf{T}\\coloneq\\frac{\\nu_T h_T^2}{\\mu_T}.\n\\end{equation}\nElements for which $\\Cf{T}<1$ are in the Stokes-dominated regime, while elements for which $\\Cf{T}\\ge1$ are in the Darcy-dominated regime.\nThe values $\\Cf{T} = 0$ and $\\Cf{T} = +\\infty$ correspond to pure Stokes and pure Darcy, respectively.\nNotice that $\\Cf{T} = +\\infty$ is a singular limit which, despite requiring to modify the continuous formulation \\eqref{eq:strong}, can be handled seamlessly by the method developed in the next section; see Remark~\\ref{rem:pure.darcy} below.\n\n\n\\section{A robust numerical scheme for the Brinkman problem}\\label{sec:scheme}\n\n\\subsection{Spaces}\n\nLet an integer $k\\ge 0$ be fixed.\nWe define the following HHO space:\n\\begin{multline*}\n  \\uvec{U}_h^k\\coloneq\\Big\\{\n  \\uvec{v}_h = \\big(\n  (\\bvec{v}_T)_{T\\in\\elements{h}}, (\\bvec{v}_F)_{F\\in\\faces{h}}\n  \\big):\n  \\\\\n  \\text{%\n    $\\bvec{v}_T\\in\\vPoly{k}(T;\\mathbb{R}^d)$ for all $T\\in\\elements{h}$\n    and $\\bvec{v}_F\\in\\vPoly{k}(F;\\mathbb{R}^d)$ for all $F\\in\\faces{h}$%\n  }\n  \\Big\\}.\n\\end{multline*}\nThe meaning of the polynomial components in $\\uvec{U}_h^k$ is provided by the interpolator $\\uvec{I}_h^k:\\bvec{H}^1(\\Omega;\\mathbb{R}^d)\\to\\uvec{U}_h^k$ such that, for all $\\bvec{v}\\in\\bvec{H}^1(\\Omega;\\mathbb{R}^d)$,\n\\[\n  \\uvec{I}_h^k\\bvec{v}\\coloneq\\big(\n  (\\vlproj{k}{T}\\bvec{v})_{T\\in\\elements{h}},\n  (\\vlproj{k}{F}\\bvec{v})_{F\\in\\elements{h}},\n  \\big)\\in\\uvec{U}_h^k,\n\\]\nwhere it is understood that $L^2$-orthogonal projectors are applied to restrictions or traces as needed.\nThe restrictions of $\\uvec{U}_h^k$, $\\uvec{v}_h\\in\\uvec{U}_h^k$, and $\\uvec{I}_h^k$ to a mesh element $T$, respectively denoted by $\\uvec{U}_T^k$, $\\uvec{v}_T\\in\\uvec{U}_T^k$, and $\\uvec{I}_T^k$, are obtained collecting the components attached to $T$ and its faces.\n\nIn what follows, given a logical proposition $P$, we denote by $\\tv{P}$ its truth value such that\n\\begin{equation}\\label{eq:tv}\n  \\tv{P} \\coloneq \\begin{cases}\n    0 & \\text{if $P$ is false,}\n    \\\\\n    1 & \\text{if $P$ is true.}\n  \\end{cases}\n\\end{equation}\nWe define the following $L^2$-like product in $\\uvec{U}_h^k$:\nFor all $(\\uvec{w}_h,\\uvec{v}_h)\\in\\uvec{U}_h^k\\times\\uvec{U}_h^k$,\n\\begin{equation}\\label{eq:prod.U}\n  \\begin{gathered}\n    (\\uvec{w}_h, \\uvec{v}_h)_{\\bvec{U},h}\n    \\coloneq\\sum_{T\\in\\elements{h}}(\\uvec{w}_T, \\uvec{v}_T)_{\\bvec{U},T}%\n    \\quad\\text{with}\\quad%\n    \\\\\n    (\\uvec{w}_T, \\uvec{v}_T)_{\\bvec{U},T}\n    \\coloneq\\lambda_T\\int_T\\bvec{w}_T\\cdot\\bvec{v}_T\n    + h_T\\sum_{F\\in\\faces{T}}\\tv{\\text{$\\Cf{T}<1$ or $F\\not\\in\\faces{h}^\\boundary$}}\\int_F\\bvec{w}_F\\cdot\\bvec{v}_F,\n  \\end{gathered}\n\\end{equation}\nwhere $\\lambda_T\\simeq 1$ is a factor, based on the regularity of the element $T$, chosen to balance out the element and face contributions to $(\\cdot,\\cdot)_{\\bvec{U},T}$ (see Section \\ref{sec:numerical.tests}).\nThe corresponding local and global seminorms are obtained setting, for $\\bullet\\in\\elements{h}\\cup\\{h\\}$,\n\\begin{equation}\\label{eq:norm.U}\n  \\norm{\\bvec{U},\\bullet}{\\uvec{v}_\\bullet}\\coloneq (\\uvec{v}_\\bullet, \\uvec{v}_\\bullet)_{\\bvec{U},\\bullet}^{\\nicefrac12}.\n\\end{equation}\nThe following boundedness property of the interpolator in the $\\norm{\\bvec{U},h}{{\\cdot}}$-norm follows from the definition of this norm along with the uniform boundedness of the $L^2$-orthogonal projectors $\\vlproj{k}{Y}$, $Y\\in\\mathcal{M}_h$, and continuous trace inequalities (cf.\\ \\cite[Lemma~1.31]{Di-Pietro.Droniou:20}):\nFor all $T\\in\\elements{h}$ and all $\\bvec{v}\\in\\bvec{H}^1(T;\\mathbb{R}^d)$,\n\\begin{equation}\\label{eq:boundedness:IT:norm.U}\n  \\norm{\\bvec{U},T}{\\uvec{I}_T^k\\bvec{v}}\\lesssim\n  \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}} + h_T\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}.\n\\end{equation}\n\nThe velocity and pressure spaces, respectively incorporating the boundary and zero-average conditions, are\n\\[\n\\uvec{U}_{h,0}^k\\coloneq\\left\\{\n\\uvec{v}_h\\in\\uvec{U}_h^k\\,:\\,\\text{%\n  $\\bvec{v}_F = \\bvec{0}$ for all $F\\in\\faces{h}^\\boundary$\n}\n\\right\\},\\qquad\nP_h^k\\coloneq\\Poly{k}(\\elements{h})\\cap L^2_0(\\Omega),\n\\]\nwhere, as usual, $L^2_0(\\Omega)=\\left\\{q\\in L^2(\\Omega)\\,:\\,\\int_\\Omega q=0\\right\\}$.\n\n\\begin{remark}[Boundary degrees of freedom]\n  Note that the degrees of freedom on the boundary faces of a vector in $\\uvec{U}_h^k$ may not be controlled by the seminorms $\\norm{\\bvec{U},\\bullet}{{\\cdot}}$. This is, however, not an issue as the final problem will be set on $\\uvec{U}_{h,0}^k$ (see also Remark \\ref{rem:pure.darcy} for the handling of boundary values in the limiting case of the pure Darcy problem).\n\\end{remark}\n\n\\subsection{Viscous term}\n\nLet $T\\in\\elements{h}$ be fixed.\nFor the discretisation of the viscous term, we define the \\emph{discrete gradient} $\\btens{G}_T^k:\\uvec{U}_T^k\\to\\tPoly{k}(T;\\mathbb{R}^{d\\times d})$ and the \\emph{Stokes potential} $\\bvec{P}_{\\stokes,T}^{k+1}:\\uvec{U}_T^k\\to\\vPoly{k}(T;\\mathbb{R}^d)$ such that, for all $\\uvec{v}_T\\in\\uvec{U}_T^k$,\n\\begin{equation}\\label{eq:GT}\n  \\int_T\\btens{G}_T^k\\uvec{v}_T:\\btens{\\tau}\n  = -\\int_T\\bvec{v}_T\\cdot\\boldsymbol{\\nabla}{\\cdot}\\btens{\\tau}\n  + \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_F\\bvec{v}_F\\cdot\\btens{\\tau}\\bvec{n}_F\n  \\qquad\n  \\forall\\btens{\\tau}\\in\\tPoly{k}(T;\\mathbb{R}^{d\\times d}),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:PST}\n  \\boldsymbol{\\nabla}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T = \\Gproj{k}{T}\\btens{G}_T^k\\uvec{v}_T,\\qquad\n  \\int_T\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T = \\int_T\\bvec{v}_T,\n\\end{equation}\nwith $\\Gproj{k}{T}$ applied to tensor-valued fields also acting row-wise.\nLikewise, in the formulas above, $\\boldsymbol{\\nabla}{\\cdot}$ and $\\boldsymbol{\\nabla}$ are understood to act row-wise.\n\nThe Stokes term in \\eqref{eq:strong:momentum} is discretised through the bilinear form $a_{\\mu,h}:\\uvec{U}_h^k\\times\\uvec{U}_h^k\\to\\mathbb{R}$ such that, for all $(\\uvec{w}_h, \\uvec{v}_h)\\in\\uvec{U}_h^k\\times\\uvec{U}_h^k$,\n\\begin{equation}\\label{eq:a.mu.h}\n  a_{\\mu,h}(\\uvec{w}_h, \\uvec{v}_h)\\coloneq\\sum_{T\\in\\elements{h}}\\mu_T a_{\\mathrm{S},T}(\\uvec{w}_T, \\uvec{v}_T),\n\\end{equation}\nwhere, for all $T\\in\\elements{h}$,\n\\begin{equation}\\label{eq:a.stokes.T}\n  a_{\\mathrm{S},T}(\\uvec{w}_T, \\uvec{v}_T)\n  \\coloneq\\int_T\\btens{G}_T^k\\uvec{w}_T:\\btens{G}_T^k\\uvec{v}_T\n  + \\frac{\\min(1,\\Cf{T}^{-1})}{h_T^2}(\n  \\uvec{w}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{w}_T,\n  \\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T\n  )_{\\bvec{U},T}.\n\\end{equation}\nWe define the following induced seminorms: For all $\\uvec{v}_h\\in\\uvec{U}_h^k$,\n\\begin{equation}\\label{eq:norm.mu.stokes}\n  \\norm{\\mu,h}{\\uvec{v}_h}\\coloneq a_{\\mu,h}(\\uvec{v}_h,\\uvec{v}_h)^{\\nicefrac12}\n  \\quad\\text{and}\\quad\n  \\text{\n    $\\norm{\\mathrm{S},T}{\\uvec{v}_T}\\coloneq a_{\\mathrm{S},T}(\\uvec{v}_T,\\uvec{v}_T)^{\\nicefrac12}$ for all $T\\in\\elements{h}$.\n  }\n\\end{equation}\n\n\\begin{lemma}[Norm equivalence]\\label{lem:hho.stabilisation}\n  Let $T\\in\\elements{h}$ and $\\uvec{v}_T\\in\\uvec{U}_T^k$. Then, it holds\n  \\begin{equation}\\label{eq:seminorm.1.T:upper.bound}\n    \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{v}_T}^2\n    + \\frac{1}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{v}_F}^2\n    \\gtrsim \\norm{\\mathrm{S},T}{\\uvec{v}_T}^2.\n  \\end{equation}\n  Assuming, moreover, $\\Cf{T}< 1$, we also have\n  \\begin{equation}\\label{eq:seminorm.1.T:lower.bound}\n    \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{v}_T}^2\n    + \\frac{1}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{v}_F}^2\n   \\lesssim \\norm{\\mathrm{S},T}{\\uvec{v}_T}^2.\n  \\end{equation}\n    \n\\end{lemma}\n\n\\begin{proof}\n  For the sake of brevity, we only prove \\eqref{eq:seminorm.1.T:lower.bound}.\n  The proof of \\eqref{eq:seminorm.1.T:upper.bound} hinges on similar arguments, together with the fact that $\\min(1,\\Cf{T}^{-1})\\le 1$, and is left to the reader.\n  Taking $\\btens{\\tau} = \\boldsymbol{\\nabla}\\bvec{v}_T$ in \\eqref{eq:GT}, integrating by parts the first term in the right-hand side, and using Cauchy--Schwarz and discrete trace inequalities (see \\cite[Lemma~1.32]{Di-Pietro.Droniou:20}) as in the proof of \\cite[Eq.~(2.25)]{Di-Pietro.Droniou:20}, we get, after simplifying and raising to the square,\n  \\begin{equation}\\label{eq:est.norm.grad.vT}\n    \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{v}_T}^2\n    \\lesssim\\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\btens{G}_T^k\\uvec{v}_T}^2\n    + h_T^{-1}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{v}_F}^2.\n  \\end{equation}\n  To estimate the second term, for any $F\\in\\faces{T}$, we insert $\\pm(\\vlproj{k}{T}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T - \\vlproj{k}{F}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T)$ and use triangle inequalities to get\n  \\begin{align}\n      h_T^{-1}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{v}_F}^2\n      &\\lesssim\n      h_T^{-1}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\vlproj{k}{T}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      + h_T^{-1}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\vlproj{k}{F}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      \\nonumber\\\\\n      &\\quad\n      + h_T^{-1}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\vlproj{k}{F}(\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T - \\vlproj{k}{T}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T)}^2\n      \\nonumber\\\\\n      &\\lesssim\n      h_T^{-2}\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\vlproj{k}{T}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      + h_T^{-1}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\vlproj{k}{F}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      \\nonumber\\\\\n      &\\quad\n      + h_T^{-2}\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T - \\vlproj{k}{T}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      \\nonumber\\\\\n      &\\lesssim\n      h_T^{-2}\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      + \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      \\nonumber\\\\\n      &\\lesssim\n      \\frac{\\min(1,\\Cf{T}^{-1})}{h_T^2}\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}^2\n      + \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\btens{G}_T^k\\uvec{v}_T}^2\n      = \\norm{\\mathrm{S},T}{\\uvec{v}_T}^2,\n  \\label{eq:est.vT-vF}\n  \\end{align}\n  where we have used the $L^2$-boundedness of $\\vlproj{k}{F}$ along with discrete trace inequalities in the second passage,\n  the definition \\eqref{eq:norm.U} of $\\norm{\\bvec{U},T}{{\\cdot}}$ along with $\\Cf{T}<1$ for the first two terms and the approximation properties of $\\vlproj{k}{T}$ for the last term in the third passage,\n  and concluded noticing that $1 = \\min(1,\\Cf{T}^{-1})$ and that $\\boldsymbol{\\nabla}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T$ is by definition the $L^2$-orthogonal projection of $\\btens{G}_T^k\\uvec{v}_T$ on $\\Goly{k}(T)^d$ (see \\eqref{eq:PST}), so that $\\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}\\le\\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\btens{G}_T^k\\uvec{v}_T}$.\n  Plugging \\eqref{eq:est.vT-vF} into \\eqref{eq:est.norm.grad.vT} and using the fact that $\\card(\\faces{T})\\lesssim 1$ by mesh regularity, we get\n  $\\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{v}_T}^2\\lesssim\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2$,\n  which is the sought estimate for the first term in the left-hand side of \\eqref{eq:seminorm.1.T:lower.bound}.\n  The fact second term is $\\lesssim\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2$ is an immediate consequence of \\eqref{eq:est.vT-vF} along with $\\card(\\faces{T})\\lesssim 1$.\n\\end{proof}\n\n\\begin{remark}[HHO stabilisation\n  It is not difficult to check that the bilinear form $\\uvec{U}_T^k\\times\\uvec{U}_T^k\\ni(\\uvec{w}_T,\\uvec{v}_T)\\mapsto(\\uvec{w}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{w}_T, \\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T)_{\\bvec{U},T}$ matches \\cite[Assumption~8.10]{Di-Pietro.Droniou:20}.\n  As a matter of fact, this bilinear form is clearly positive-semidefinite, it satisfies the requested seminorm equivalence by \\eqref{eq:seminorm.1.T:upper.bound} and \\eqref{eq:seminorm.1.T:lower.bound}, and is polynomially consistent since it only depends on its arguments through the difference operators defined by \\cite[Eq.~(8.30)]{Di-Pietro.Droniou:20}.\n\\end{remark}\n\n\\subsection{Darcy term}\n\nLet again $T\\in\\elements{h}$.\nThe discretisation of the Darcy and coupling terms hinges on the \\emph{discrete divergence} $D_T^k:\\uvec{U}_T^k\\to\\Poly{k}(T)$ such that\n\\begin{equation}\\label{eq:DT}\n  D_T^k\\uvec{v}_T \\coloneq \\tr(\\btens{G}_T^k\\uvec{v}_T)\\qquad\\forall\\uvec{v}_T\\in\\uvec{U}_T^k.\n\\end{equation}\nBased on this operator, we define the \\emph{Darcy potential} $\\bvec{P}_{\\darcy,T}^k:\\uvec{U}_T^k\\to\\vPoly{k}(T;\\mathbb{R}^d)$ such that, for all $\\uvec{v}_T\\in\\uvec{U}_T^k$ and all $(q,\\bvec{w})\\in\\Poly{k+1}(T)\\times\\cGoly{k}(T)$,\n\\begin{equation}\\label{eq:PDT}\n  \\int_T\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T\\cdot(\\boldsymbol{\\nabla} q + \\bvec{w})\n  = -\\int_TD_T^k\\uvec{v}_T~q\n  + \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_F(\\bvec{v}_F\\cdot\\bvec{n}_F)~q\n  + \\int_T\\bvec{v}_T\\cdot\\bvec{w}.\n\\end{equation}\nThis Darcy potential will play a key role in the discretisation of the source term, to ensure that the scheme is fully robust in the whole range of friction coefficients; see Remark \\ref{rem:disc.source}.\n\n\\begin{remark}[Link with DDR]\\label{rem:link.Xdivh}\n  Recall the Discrete De Rham $\\Hdiv{\\Omega}$-like space\n  \\[\n  \\begin{aligned}\n  \\Xdiv{h}\\coloneq\\Big\\{\n  &\\uvec{v}_h = \\big(\n  (\\bvec{v}_{\\cvec{G},T}, \\bvec{v}_{\\cvec{G},T}^{\\rm c})_{T\\in\\elements{h}},\n  (v_F)_{F\\in\\faces{h}}\n  \\big)\\,:\\,\n  \\\\\n  &\\qquad\\text{%\n    $\\bvec{v}_{\\cvec{G},T}\\in\\Goly{k-1}(T)$ and\n    $\\bvec{v}_{\\cvec{G},T}^{\\rm c}\\in\\cGoly{k}(T)$\n    for all $T\\in\\elements{h}$,\n  }\n  \\\\\n  &\\qquad\\text{%\n    $v_F\\in\\Poly{k}(F)$ for all $F\\in\\faces{h}$\n  }\n  \\Big\\}.\n  \\end{aligned}\n  \\]\n  Noticing that $\\Goly{k-1}(T)\\subset \\Goly{k}(T)$ (cf.\\ \\eqref{eq:Goly.cGoly}), this space injects into $\\uvec{U}_h^k$ through the mapping\n  \\[\n  \\Xdiv{h}\\ni\\uvec{v}_h\\mapsto\\big((\\mathfrak{R}_{\\cvec{G},T}^k(\\bvec{v}_{\\cvec{G},T},\\bvec{v}_{\\cvec{G},T}^{\\rm c}))_{T\\in\\elements{h}}, (v_F\\bvec{n}_F)_{F\\in\\faces{h}}\\big)\\in\\uvec{U}_h^k,\n  \\]\n  where $\\mathfrak{R}_{\\cvec{G},T}^k:\\Goly{k}(T)\\times\\cGoly{k}(T)\\to\\vPoly{k}(T;\\mathbb{R}^d)$ denotes the recovery operator \\cite[Eq.~(2.17)]{Di-Pietro.Droniou:21*1}, which satisfies $\\Gproj{k}{T}\\mathfrak{R}_{\\cvec{G},T}^k(\\bvec{v}_{\\cvec{G},T},\\bvec{v}_{\\cvec{G},T}^{\\rm c})=\\bvec{v}_{\\cvec{G},T}$ and $\\cGproj{k}{T}\\mathfrak{R}_{\\cvec{G},T}^k(\\bvec{v}_{\\cvec{G},T},\\bvec{v}_{\\cvec{G},T}^{\\rm c})=\\bvec{v}_{\\cvec{G},T}^{\\rm c}$ (where $\\cGproj{k}{T}$ is the $L^2$-orthogonal projector on $\\cGoly{k}(T)$.\n  It can be checked that the discrete divergence \\eqref{eq:DT} and the Darcy potential \\eqref{eq:PDT} only depend on the polynomial components shared by $\\uvec{U}_T^k$ and $\\Xdiv{T}$, and that they coincide with the corresponding DDR operators respectively defined by \\cite[Eqs.~(3.32) and~(4.9)--(4.10)]{Di-Pietro.Droniou:21*1}.\n\\end{remark}\n\nAccounting for the previous remark and recalling \\cite[Eq.~(4.12) and (4.13)]{Di-Pietro.Droniou:21*1}, it holds\n\\begin{alignat}{4}\\label{eq:lproj.k-1.PDT}\n  \\vlproj{k-1}{T}\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T &= \\vlproj{k-1}{T}\\bvec{v}_T\n  &\\qquad&\\forall\\uvec{v}_T\\in\\uvec{U}_T^k,\n  \\\\\n  \\label{eq:polynomial.consistency:PDT}\n  \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v} &= \\bvec{v}\n  &\\qquad&\\forall\\bvec{v}\\in\\vPoly{k}(T;\\mathbb{R}^d).\n\\end{alignat}\nThe approximation properties of $\\bvec{P}_{\\darcy,T}^k$ in the $L^2$-norm have been studied in \\cite[Theorem~6]{Di-Pietro.Droniou:21*1}.\nThe following proposition extends the above results to general Hilbert seminorms.\n\n\\begin{proposition}[Approximation properties of the Darcy potential]\n  Let an integer $r\\in\\{0,\\ldots,k\\}$ be given.\n  Then, for all $T\\in\\elements{h}$, all $\\bvec{v}\\in\\bvec{H}^{r+1}(T;\\mathbb{R}^d)$, and all $m\\in\\{0,\\ldots,r+1\\}$,\n  \\begin{equation}\\label{eq:approximation:PDT}\n    \\seminorm{\\bvec{H}^m(T;\\mathbb{R}^d)}{\\bvec{v} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}\n    \\lesssim h_T^{r+1-m}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{v}}.\n  \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n  By \\cite[Proposition~1.35]{Di-Pietro.Droniou:20}, $\\bvec{P}_{\\darcy,T}^k\\circ\\uvec{I}_T^k:\\bvec{H}^1(T;\\mathbb{R}^d)\\to\\vPoly{k}(T;\\mathbb{R}^d)$ is a projector owing to \\eqref{eq:polynomial.consistency:PDT}.\n  By \\cite[Lemma~1.43]{Di-Pietro.Droniou:20}, it then suffices to prove that, for all $\\bvec{v}\\in\\bvec{H}^1(T;\\mathbb{R}^d)$,\n  \\begin{alignat}{4}\\label{eq:approximation:PDT:condition.m=0}\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}\n    &\\lesssim\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}}\n    + h_T\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}\n    &\\qquad&\\text{if $m=0$},\n    \\\\\\label{eq:approximation:PDT:condition.m>=1}\n    \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}\n    &\\lesssim\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}\n    &\\qquad&\\text{if $m\\ge 1$}.\n  \\end{alignat}\n  To prove \\eqref{eq:approximation:PDT:condition.m=0}, it suffices to recall Remark~\\ref{rem:link.Xdivh} and use \\cite[Eqs.~(4.24) and~(4.28)]{Di-Pietro.Droniou:21}.\n  To prove \\eqref{eq:approximation:PDT:condition.m>=1}, we write\n  \\[\n  \\begin{alignedat}{4}\n    \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}\n    &= \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k(\\bvec{v} - \\vlproj{0}{T}\\bvec{v})}\n    \\\\\n    &\\lesssim\n    h_T^{-1}\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k(\\bvec{v} - \\vlproj{0}{T}\\bvec{v})}\n    &\\qquad&\n    \\text{\\cite[Eq.~(1.46)]{Di-Pietro.Droniou:20}}\n    \\\\\n    &\\lesssim\n    h_T^{-1}\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v} - \\vlproj{0}{T}\\bvec{v}}\n    + \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v} - \\vlproj{0}{T}\\bvec{v}}\n    &\\qquad&\n    \\text{Eq.~\\eqref{eq:approximation:PDT:condition.m=0}}\n    \\\\\n    &\\lesssim\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}},\n  \\end{alignedat}\n  \\]\n  where the first line follows using the polynomial consistency \\eqref{eq:polynomial.consistency:PDT} of $\\bvec{P}_{\\darcy,T}^k$ to write $0 = \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\vlproj{0}{T}\\bvec{v}} = \\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\vlproj{0}{T}\\bvec{v}}$,\n  while the conclusion follows from a Poincar\\'e--Wirtinger inequality on the zero-average function $\\bvec{v} - \\vlproj{0}{T}\\bvec{v}$.\n\\end{proof}\n\nLet $\\widetilde{\\bvec{P}}_{\\darcy,T}^k:\\uvec{U}_T^k\\to\\Poly{k}(T;\\mathbb{R}^d)$ be such that\n\\begin{equation}\\label{eq:tPDT}\n  \\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n  \\coloneq\\tv{\\Cf{T}<1}\\bvec{v}_T + \\tv{\\Cf{T}\\ge 1}\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T\n  \\qquad\\forall\\uvec{v}_T\\in\\uvec{U}_T^k.\n\\end{equation}\nThe Darcy term in \\eqref{eq:strong:momentum} is discretised by means of the bilinear form $a_{\\nu,h}:\\uvec{U}_h^k\\times\\uvec{U}_h^k\\to\\mathbb{R}$ such that, for all $(\\uvec{w}_h,\\uvec{v}_h)\\in\\uvec{U}_h^k\\times\\uvec{U}_h^k$,\n\\begin{equation}\\label{eq:a.nu}\n  a_{\\nu,h}(\\uvec{w}_h,\\uvec{v}_h)\n  \\coloneq\\sum_{T\\in\\elements{h}}\\nu_Ta_{\\mathrm{D},T}(\\uvec{w}_T,\\uvec{v}_T)\n\\end{equation}\nwith, for all $T\\in\\elements{h}$,\n\\begin{equation}\\label{eq:a.darcy.T}\n  a_{\\mathrm{D},T}(\\uvec{w}_T,\\uvec{v}_T)\\coloneq\n  \\int_T\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{w}_T\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T + \\min(1,\\Cf{T})(\n  \\uvec{w}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{w}_T,\n  \\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T,\n  )_{\\bvec{U},T}.\n\\end{equation}\n\nWe define the following induced norms: For all $\\uvec{v}_h\\in\\uvec{U}_h^k$,\n\\begin{equation}\\label{eq:norm.nu.darcy}\n  \\norm{\\nu,h}{\\uvec{v}_h}\\coloneq a_{\\nu,h}(\\uvec{v}_h,\\uvec{v}_h)^{\\nicefrac12}\n  \\quad\\text{and\\quad $\\norm{\\mathrm{D},T}{\\uvec{v}_T}\\coloneq a_{\\mathrm{D},T}(\\uvec{v}_T,\\uvec{v}_T)^{\\nicefrac12}$ for all $T\\in\\elements{h}$.\n  }\n\\end{equation}\n\n\\subsection{Coupling}\n\nThe coupling terms in \\eqref{eq:strong:momentum} and \\eqref{eq:strong:mass} are discretised by the bilinear form $b_h:\\uvec{U}_h^k\\times\\Poly{k}(\\elements{h})\\to\\mathbb{R}$ such that, for all $(\\uvec{v}_h,q_h)\\in\\uvec{U}_h^k\\times\\Poly{k}(\\elements{h})$,\n\\begin{equation}\\label{eq:bh}\n  b_h(\\uvec{v}_h,q_h)\n  \\coloneq-\\sum_{T\\in\\elements{h}}\\int_TD_T^k\\uvec{v}_T~q_T,\n\\end{equation}\nwhere $q_T$ denotes the restriction of $q_h$ to $T$.\nRecalling \\cite[Eq.~(8.36)]{Di-Pietro.Droniou:20}, it holds:\nFor all $\\bvec{v}\\in\\bvec{H}^1(\\Omega;\\mathbb{R}^d)$,\n\\begin{equation}\\label{eq:consistency:bh}\n  b_h(\\uvec{I}_h^k\\bvec{v},q_h) = -\\int_\\Omega\\boldsymbol{\\nabla}{\\cdot}\\bvec{v}~q_h\n  \\qquad\\forall q_h\\in\\Poly{k}(\\elements{h}).\n\\end{equation}\n\n\\subsection{Discrete problem and main results}\n\nThe discrete problem reads:\nFind $(\\uvec{u}_h,p_h)\\in\\uvec{U}_{h,0}^k\\timesP_h^k$ such that\n\\begin{equation}\\label{eq:discrete}\n  \\begin{alignedat}{4}\n    a_{\\mu,h}(\\uvec{u}_h,\\uvec{v}_h)\n    + a_{\\nu,h}(\\uvec{u}_h,\\uvec{v}_h)\n    + b_h(\\uvec{v}_h,p_h)\n    &= \\sum_{T\\in\\elements{h}}\\int_T\\bvec{f}\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n    &\\qquad&\\forall\\uvec{v}_h\\in\\uvec{U}_{h,0}^k,\n    \\\\\n    -b_h(\\uvec{u}_h,q_h)\n    &= \\int_\\Omega gq_h\n    &\\qquad&\\forall q_h\\inP_h^k.\n  \\end{alignedat}\n\\end{equation}\n\nThe equivalent variational formulation is:\nFind $(\\uvec{u}_h,p_h)\\in\\uvec{U}_{h,0}^k\\timesP_h^k$ such that\n\\begin{equation}\\label{eq:discrete:variational}\n  \\mathcal{A}_h((\\uvec{u}_h,p_h),(\\uvec{v}_h,q_h))\n  = \\sum_{T\\in\\elements{h}}\\int_T\\bvec{f}\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n  + \\int_\\Omega gq_h,\n\\end{equation}\nwith $\\mathcal{A}_h:\\big(\\uvec{U}_h^k\\timesP_h^k\\big)^2\\to\\mathbb{R}$ such that, for all $(\\uvec{w}_h,r_h)$ and all $(\\uvec{v}_h,q_h)$ in $\\uvec{U}_h^k\\timesP_h^k$,\n\\begin{equation}\\label{eq:Ah}\n  \\mathcal{A}_h((\\uvec{w}_h,r_h),(\\uvec{v}_h,q_h))\n  \\coloneq\n  a_{\\mu,h}(\\uvec{w}_h,\\uvec{v}_h)\n  + a_{\\nu,h}(\\uvec{w}_h,\\uvec{v}_h)\n  + b_h(\\uvec{v}_h,r_h)\n  -b_h(\\uvec{w}_h,q_h).\n\\end{equation}\n\nRecalling \\eqref{eq:norm.mu.stokes} and \\eqref{eq:norm.nu.darcy}, we equip the space $\\uvec{U}_{h,0}^k$ with the following natural energy norm: For all $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k$,\n\\begin{equation}\\label{eq:norm.mu.nu.h}\n  \\norm{\\mu,\\nu,h}{\\uvec{v}_h}\\coloneq\\left(\n  \\norm{\\mu,h}{\\uvec{v}_h}^2 + \\norm{\\nu,h}{\\uvec{v}_h}^2\n  \\right)^{\\nicefrac12}\n\\end{equation}\nand, given a linear form $\\ell_h:\\uvec{U}_{h,0}^k\\to\\mathbb{R}$, we denote its dual norm by\n\\[\n\\norm{\\mu,\\nu,h,*}{\\ell_h}\\coloneq\\sup_{\\uvec{v}_h\\in\\uvec{U}_{h,0}^k\\setminus\\{\\uvec{0}\\}}\\frac{\\ell_h(\\uvec{v}_h)}{\\norm{\\mu,\\nu,h}{\\uvec{v}_h}}.\n\\]\nThe bilinear form $a_{\\mu,h} + a_{\\nu,h}$ is $\\norm{\\mu,\\nu,h}{{\\cdot}}$-coercive with unit coercivity constant.\nThe well-posedness of \\eqref{eq:discrete} then classically follows from the theory of mixed methods (see, e.g., \\cite[Lemma~A.11]{Di-Pietro.Droniou:20}) thanks to the inf-sup condition on $b_h$ stated in the following lemma.\n\\begin{lemma}[Inf-sup condition on $b_h$]\\label{lem:inf-sup}\n  Letting $\\beta\\coloneq\\left(\\overline{\\mu} + \\overline{\\nu}\\right)^{-\\nicefrac12}$, it holds, for all $q_h\\inP_h^k$,\n  \\[\n  \\beta\\norm{L^2(\\Omega)}{q_h}\n  \\lesssim\n  \\sup_{\\uvec{v}_h\\in\\uvec{U}_{h,0}^k\\setminus\\{\\uvec{0}\\}}\\frac{b_h(\\uvec{v}_h,q_h)}{\\norm{\\mu,\\nu,h}{\\uvec{v}_h}}.\n  \\]\n\\end{lemma}\n\n\\begin{proof}\n  See Section \\ref{sec:analysis:stability}.\n\\end{proof}\n\nThanks to the presence of cut-off factors, the following error estimate is robust across the entire range of (local) regimes.\n\n\\begin{theorem}[Error estimate]\\label{thm:error.estimate}\n  Denote by $(\\bvec{u},p)\\in\\bvec{H}^1_0(\\Omega;\\mathbb{R}^d)\\times L^2_0(\\Omega)$ the unique solution to the standard weak formulation of \\eqref{eq:strong} and by $(\\uvec{u}_h,p_h)\\in\\uvec{U}_{h,0}^k\\timesP_h^k$ the unique solution of the numerical scheme \\eqref{eq:discrete} (or, equivalently, \\eqref{eq:discrete:variational}).\n  Then, recalling the notation \\eqref{eq:tv} for the truth value of a logical proposition and assuming, for some $r\\in\\{0,\\ldots,k\\}$, $\\bvec{u}\\in\\bvec{H}^{r+2}(\\elements{h};\\mathbb{R}^d)$, $p\\in H^1(\\Omega)$, and, for all $T\\in\\elements{h}$, $p\\in H^{r+1+\\tv{\\Cf{T}\\ge 1}}(T)$, it holds,\n  \\begin{equation}\\label{eq:error.estimate}\n    \\begin{aligned}\n      &\n      \\norm{\\mu,\\nu,h}{\\uvec{u}_h - \\uvec{I}_h^k\\bvec{u}}^2+\\norm{L^2(\\Omega)}{p_h - \\lproj{k}{h} p}^2\n      \\\\\n      &\\quad\n      \\lesssim\\frac{1}{\\gamma^2}\\Bigg[\n        \\begin{aligned}[t]\n          &\\sum_{T\\in\\elements{h}}\\mu_T\\min(1,\\Cf{T}^{-1})h_T^{2(r+1)}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{u}}^2\n          + \\sum_{T\\in\\elements{h}}\\nu_T\\min(1,\\Cf{T})h_T^{2(r+1)}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{u}}^2\n          \\\\\n          &+ \\sum_{T\\in\\elements{h}}\\left(\n          \\mu_T^{-1}\\tv{\\Cf{T}<1}h_T^{2(r+1)}\\seminorm{H^{r+1}(T)}{p}^2\n          + \\nu_T^{-1}\\tv{\\Cf{T}\\ge 1}h_T^{2(r+1)}\\seminorm{H^{r+2}(T)}{p}^2    \n          \\right)\n          \\Bigg],\n        \\end{aligned}\n    \\end{aligned}\n  \\end{equation}\n  where $\\gamma^{-2}\\coloneq 4\\beta^{-4} + 8\\beta^{-2} + 1$ with $\\beta$ as in Lemma~\\ref{lem:inf-sup}, while, for all $T\\in\\elements{h}$, $\\nu_T^{-1}\\tv{\\Cf{T}\\ge 1}\\coloneq 0$ if $\\nu_T = 0$.\n\\end{theorem}\n\n\\begin{proof}\n  See Section \\ref{sec:analysis:convergence}.\n\\end{proof}\n\\begin{remark}[Robustness of the error estimate and application to the Darcy problem]\\label{rem:pure.darcy}\n  In the spirit of \\cite[Remark~13]{Botti.Di-Pietro.ea:18}, the presence of the cutoff factors $\\min(1,\\Cf{T}^{-1})$, $\\min(1,\\Cf{T})$, $\\mu_T^{-1}\\tv{\\Cf{T}<1}$, and $\\nu_T^{-1}\\tv{\\Cf{T}\\ge 1}$ makes the above estimate robust across the entire range $\\Cf{T}\\in\\lbrack 0,+\\infty)$.\n  \n  The case $\\Cf{T} = +\\infty$ corresponds to the pure Darcy problem, which is the singular limit obtained assuming $\\min_\\Omega\\nu>0$ and $\\Cf{T} = +\\infty$ for all $T\\in\\elements{h}$.\n  In this case, a more in-depth discussion is in order.\n  Denoting by $\\gamma_{\\bvec{n}}$ the normal trace operator on $\\partial\\Omega$, the space for the velocity becomes $\\HZdiv{\\Omega}\\coloneq\\{\\bvec{v}\\in\\Hdiv{\\Omega}\\,:\\,\\text{$\\gamma_{\\bvec{n}}(\\bvec{v}) = 0$ on $\\partial\\Omega$}\\}$, and the weak formulation of \\eqref{eq:strong} yields the Darcy problem in mixed form.\n  The error estimate \\eqref{eq:error.estimate} remains valid under the regularity assumption $\\bvec{u}\\in\\bvec{H}^{r+1}(\\elements{h};\\mathbb{R}^d)$, and provided the following conventions are adopted:\n  $\\mu_T^{-1}\\tv{\\Cf{T}<1}\\coloneq 0$ and, for any $\\bvec{v}\\in\\HZdiv{\\Omega}\\cap\\bvec{H}^1(\\elements{h};\\mathbb{R}^d)$, all the components of the boundary values of $\\uvec{I}_h^k\\bvec{v}$ are forced to zero, i.e., $(\\uvec{I}_h^k\\bvec{v})_F \\coloneq \\bvec{0}$ for all $F\\in\\faces{h}^\\boundary$.\n  Notice that the tangential components of the velocity on boundary faces do not appear in the formulation of the method when $\\mu = 0$.\n    To check this fact:\n  \\begin{itemize}\n  \\item Concerning the Darcy contribution $a_{\\mathrm{D},T}$ (cf.\\ \\eqref{eq:a.darcy.T}), recall Remark~\\ref{rem:link.Xdivh} for the consistent term while, for the stabilisation term, notice that, by \\eqref{eq:prod.U}, boundary faces are not present in $(\\cdot,\\cdot)_{\\bvec{U},T}$ since $\\Cf{T}\\ge 1$ for all $T\\in\\elements{h}$ ;\n  \\item Concerning the coupling term $b_h$ (cf.\\ \\eqref{eq:bh}), notice that the following equivalent formulation results applying the definition \\eqref{eq:GT} of $\\btens{G}_T^k$ with $\\btens{\\tau} = q_T\\btens{I}_d\\coloneq (q_h)_{|T}\\btens{I}_d$ for all $T\\in\\elements{h}$:\n    \\[\n    b_h(\\uvec{v}_h,q_h)\n    = \\sum_{T\\in\\elements{h}}\\left(\n      \\int_T\\bvec{v}_T\\cdot\\boldsymbol{\\nabla} q_T\n      - \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_F(\\bvec{v}_F\\cdot\\bvec{n}_F)~q_T\n      \\right),\n      \\]\n      clearly showing that $b_h$ is independent of the tangential component of $\\bvec{v}_F$ for all $F\\in\\faces{h}$.\n  \\end{itemize}\n  \n  The method obtained for the pure Darcy problem has more unknowns than, say, the mixed method of \\cite{Di-Pietro.Ern:17} or a similar one that could be obtained starting from the space $\\Xdiv{h}$ of \\cite{Di-Pietro.Droniou:21*1}.\n  In particular, the tangential components of interface unknowns are not present in the consistency term of $a_{\\mathrm{D},T}$ (see again Remark~\\ref{rem:link.Xdivh}), but are controlled by the stabilisation term.\n  Despite this difference in the discrete space for the flux, the estimate for the error on $\\bvec{u}$ resulting from \\eqref{eq:error.estimate} in the pure Darcy case is analogous to the one given in \\cite[Theorem~6]{Di-Pietro.Ern:17} (where the highest regularity case corresponding to $r = k$ is considered).\n\\end{remark}\n\n\n\\section{Numerical tests}\\label{sec:numerical.tests}\n\n  In this section we numerically assess the convergence properties of the scheme \\eqref{eq:discrete} for different values of the friction coefficient (including the limit cases) and on both standard and genuinely polyhedral meshes.\n\nThe code used for the numerical tests is part of the open source C++ \\texttt{HArDCore3D} library; see \\url{https:\/\/github.com\/jdroniou\/HArDCore}.\nIn order to reduce the size of the global linear systems, static condensation was applied the scheme \\eqref{eq:discrete} in accordance with the principles outlined in \\cite[Appendix B]{Di-Pietro.Droniou:20}; see \\cite[Section 6]{Di-Pietro.Ern.ea:16*1} for a discussion specific to the Stokes equations and \\cite{Botti.Di-Pietro:22} for a study of the effect of static condensation on $p$-multilevel preconditioners for the Stokes problem.\nWe have chosen to locally eliminate all element degrees of freedom except for the average value of the pressure inside each element. The linear systems were solved using the \\texttt{Intel MKL PARDISO} library (see \\url{https:\/\/software.intel.com\/en-us\/mkl}). \n\nThe parameter $\\lambda_T$ in \\eqref{eq:prod.U} was chosen as $\\frac{h_T^3}{|T|}\\card(\\faces{T})$, to give a larger weight to the element contribution in \\eqref{eq:norm.U} when $T$ is elongated or has many faces: this compensates the relatively larger contribution, in these circumstances, of the boundary terms in this local norm. We have also applied scalings to the stabilisation terms in \\eqref{eq:a.stokes.T} and \\eqref{eq:a.darcy.T}: 3 for the Stokes stabilisation, 0.3 for the Darcy stabilisation.\nIntroducing scalings in the stabilisation terms is not strictly necessary to observe the convergence of the scheme at the expected rates, but we noticed that they improve the magnitudes of the relative errors. Understanding the optimal scaling of stabilisations involved in polytopal methods is an ongoing subject of investigation; here, these numbers were found by quick trial and error on unexpensive tests (low degree $k$, coarse meshes), before being used in all the tests below.\n\n\\subsection{Convergence in various regimes}\n\nFollowing \\cite{Botti.Di-Pietro.ea:18}, we consider a constant viscosity $\\mu$ and inverse permeability $\\nu$, and we evaluate the relative velocity--pressure error\n\\[\nE_{\\bvec{u},p}=\\frac{\\left(\\norm{\\mu,\\nu,h}{\\uvec{u}_h-\\uvec{I}_h^k\\bvec{u}}^2+\\norm{L^2(\\Omega)}{p_h-\\lproj{k}{h}p}^2\\right)^{1\/2}}{\\left(\\norm{\\mu,\\nu,h}{\\uvec{I}_h^k\\bvec{u}}^2+\\norm{L^2(\\Omega)}{\\lproj{k}{h}p}^2\\right)^{1\/2}},\n\\]\nwhen the nature of the exact solution $(\\bvec{u},p)$ is determined by the global friction coefficient $\\Cf{\\Omega}=\\nu\/\\mu$, with the convention $\\Cf{\\Omega}=+\\infty$ if $\\mu=0$. Specifically, we consider the domain $\\Omega=(0,1)^3$ and, setting $\\chi_\\mathrm{S}(\\Cf{\\Omega}) \\coloneq \\exp(-\\Cf{\\Omega})$, the pressure and velocity are chosen as\n\\begin{align*}\np(x,y,z)={}&\\sin(2\\pi x)\\sin(2\\pi y)\\sin(2\\pi z)\\quad\\forall (x,y,z)\\in\\Omega,\\\\\n\\bvec{u}={}&\\chi_\\mathrm{S}(\\Cf{\\Omega})\\bvec{u}_\\mathrm{S}+(1-\\chi_\\mathrm{S}(\\Cf{\\Omega}))\\bvec{u}_\\mathrm{D},\n\\end{align*}\nwhere $\\bvec{u}_\\mathrm{S}$ and $\\bvec{u}_\\mathrm{D}$ are the velocity obtained in the Stokes ($\\Cf{\\Omega}=0$) and Darcy ($\\Cf{\\Omega}=+\\infty$) limits, and are given by\n\\begin{align*}\n  \\bvec{u}_\\mathrm{S}(x,y,z)={}& \\frac12\\left[\\begin{array}{c}\n                \\sin(2\\pi x)\\cos(2\\pi y)\\cos(2\\pi z)\\\\\n                 \\cos(2\\pi x)\\sin(2\\pi y)\\cos(2\\pi z)\\\\\n                -2\\cos(2\\pi x)\\cos(2\\pi y)\\sin(2\\pi z)\n          \\end{array}\\right]\\quad\\forall (x,y,z)\\in\\Omega,\\\\\n  \\bvec{u}_\\mathrm{D}={}&\\left\\{\\begin{array}{ll}-\\nu^{-1}\\boldsymbol{\\nabla} p&\\mbox{ if $\\nu>0$},\\\\\n      \\bvec{0}&\\mbox{ otherwise}.\n      \\end{array}\\right.\n\\end{align*}\nWe notice that $\\boldsymbol{\\nabla}{\\cdot} \\bvec{u}_\\mathrm{S}=0$ and that $\\nu\\bvec{u}_\\mathrm{D}+\\boldsymbol{\\nabla} p=0$; these are expected relations, respectively, for a solution of an incompressible Stokes equation, and for a solution of a Darcy equation in mixed form (when gravity is neglected).\n  The meshes used for the test correspond to the families of Voronoi meshes ``Voro-small-0'', of tetrahedral meshes ``Tetgen-Cube-0'' and of random hexahedral meshes ``Random-Hexahedra'' available on the \\texttt{HArDCore3D} repository.\nThe errors as a function of $h$ are presented in Figures \\ref{fig:conv.voro}, \\ref{fig:conv.tets} and \\ref{fig:conv.hexa}, showing that the predicted convergence is observed in practice for all the considered mesh families and polynomial degrees, and that both orders of convergence and magnitudes of errors are robust in all regimes.\n\\begin{figure}[h!]\\centering\n  \\ref{br.conv.voro}\n  \\vspace{0.50cm}\\\\\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} [legend columns=4, legend to name=br.conv.voro]  \n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k0_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=0$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k1_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=1$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k2_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=2$}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k3_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=3$}\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=\\nu=1$}\n  \\end{minipage}\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} \n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k0_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k1_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k2_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k3_sv1_sp0\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=1$, $\\nu=0$}\n  \\end{minipage}\n  \\\\[0.5em]\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} \n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k0_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k1_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k2_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Voro-small-0_k3_sv0_sp1\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=0$, $\\nu=1$}\n  \\end{minipage}\n  \\caption{Voronoi meshes: errors $E_{u,p}$ with respect to $h$}\n  \\label{fig:conv.voro}\n\\end{figure}\n\n\\begin{figure}[!h]\\centering\n  \\ref{br.conv.tets}\n  \\vspace{0.50cm}\\\\\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} [legend columns=4, legend to name=br.conv.tets]  \n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k0_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=0$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k1_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=1$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k2_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=2$}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k3_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=3$}\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=\\nu=1$}\n  \\end{minipage}\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k0_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k1_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k2_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k3_sv1_sp0\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=1$, $\\nu=0$}\n  \\end{minipage}\n  \\\\[0.5em]\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k0_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k1_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k2_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Tetgen-Cube-0_k3_sv0_sp1\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=0$, $\\nu=1$}\n  \\end{minipage}\n  \\caption{Tetrahedral meshes: errors $E_{u,p}$ with respect to $h$}\n  \\label{fig:conv.tets}\n\\end{figure}\n\n\\begin{figure}[!h]\\centering\n  \\ref{br.conv.hexa}\n  \\vspace{0.50cm}\\\\\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} [legend columns=4, legend to name=br.conv.hexa]  \n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k0_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=0$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k1_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=1$;}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k2_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=2$}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k3_sv1_sp1\/data_rates.dat};\n        \\addlegendentry{$k=3$}\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=\\nu=1$}\n  \\end{minipage}\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k0_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k1_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k2_sv1_sp0\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k3_sv1_sp0\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=1$, $\\nu=0$}\n  \\end{minipage}\n  \\\\[0.5em]\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis}\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{1}{black};\n        \\addplot [mark=star, red] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k0_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{2}{black};\n        \\addplot [mark=*, blue] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k1_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{3}{black};\n        \\addplot [mark=o, olive] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k2_sv0_sp1\/data_rates.dat};\n        \\logLogSlopeTriangle{0.90}{0.4}{0.1}{4}{black};\n        \\addplot [mark=o, black] table[x=MeshSize,y=EnergyError] {dat\/convergence\/Random-Hexahedra_k3_sv0_sp1\/data_rates.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{$\\mu=0$, $\\nu=1$}\n  \\end{minipage}\n  \\caption{Random hexahedral meshes: errors $E_{u,p}$ with respect to $h$}\n  \\label{fig:conv.hexa}\n\\end{figure}\n\n\\subsection{Lid-driven cavity in porous medium}\\label{sec:lid.cavity}\n\nThe tests in this section are inspired by situations described in \\cite{Bernardi.Hecht.ea:10,Alvarez.Gatica.ea:16}. In these references, a V-crack is realised at the top of a homogeneous porous medium, and plays the role of a lid-driven cavity (with a Stokes-dominated model in this cavity, while the rest of the medium is modelled using pure Darcy flow), and low-order mixed finite elements on triangles\/tetrahedra are used to simulate the flow. \n\nWe consider here a cavity, where a pure Stokes flow occurs, sitting in a porous medium, with pure Darcy flow; the porous medium is heterogeneous, with permeability equal to $10^{-7}$ in the surrounding ``box'' and $10^{-2}$ in a ``wedge'' at the outset of the cavity; see Figure \\ref{fig:cavity-domain}, left. The domain is $\\Omega=(-1,2)\\times (-1,2)\\times(-2,0)$, with the cavity being $(0,1)^3$ and the wedge $\\{(x,y,z)\\in\\mathbb{R}^3\\,:\\,1<x<2\\,,\\;0<y<1\\,,\\;-0.75(x-1)+0.25<z<0\\}$. The domain has been meshed using \\texttt{gmsh} (\\href{https:\/\/gmsh.info\/}{https:\/\/gmsh.info\/}), with cubic elements in the cavity, and mostly tetrahedral elements in the porous medium (together with a few pyramidal elements at the junctions cavity--porous medium); see Figure \\ref{fig:cavity-domain}, right, for an example of mesh, and Table \\ref{tab:meshes} for the characteristic of all meshes. The files describing the geometry are available in the \\texttt{HArDCore} repository.\n\n\\begin{figure}\\centering\n  \\begin{minipage}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{cavity.png}\n  \\end{minipage}\n  \\hspace{0.25cm}\n  \\begin{minipage}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{cavity-mesh.png}\n  \\end{minipage}\n  \\caption{Left: geometry of the cavity (green) inside the porous medium, comprising a wedge (green) and the surrounding box (shadow). Right: example of mesh used in the simulations.}\n  \\label{fig:cavity-domain}\n\\end{figure}\n\n\\begin{table}\\centering\n  \\begin{tabular}{c|c|c|c|c|c}\n    \\toprule\n    Mesh index & 1 & 2 & 3 & 4 & 5 \\\\\n    \\midrule\n    Mesh size & 0.95 & 0.61 & 0.54 & 0.22 & 0.17\\\\\n    Num. of elements & 1,326 & 5,935 & 7,963 & 99,748 & 201,653\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Characteristics of the mesh family for the tests in Section \\ref{sec:lid.cavity}.\\label{tab:meshes}}\n\\end{table}\n\nThe forcing term $\\bvec{f}=(0,0,-0.98)$ represents the gravity, while we fix $g=0$. The boundary conditions on the velocity are $\\bvec{u}(x,y,z)=(x(1-x),0,0)$ on top of the cavity, and $\\bvec{u}=\\bvec{0}$ elsewhere. Figure \\ref{fig:cavity-stream} presents the streamlines obtained on the third mesh in the family with $k=2$. These streamlines show the usual form of circulation inside the cavity for a pure Stokes lid-driven cavity, which drives some (slower) motion inside the wedge section of the porous medium; given the very low permeability of the rest of the medium, little material is transferred into this medium, in which the velocity remains almost zero; in the region $z<-1$ below the cavity, for example, the maximum of the vertex values (obtained by averaging the potential reconstructions in each element surrounding the vertices) of the velocity is below $6\\times 10^{-5}$.\n\n\\begin{figure}\\centering\n    \\includegraphics[width=.8\\textwidth]{cavity-stream.png}\n  \\caption{Streamlines for the test case of Section \\ref{sec:lid.cavity} (cavity and wedge displayed in shadow).}\n  \\label{fig:cavity-stream}\n\\end{figure}\n\nTo qualitatively assess the impact of increasing the degree of approximation $k$ of the method, we evaluate for various meshes and degrees the flux across the interface $\\Gamma=\\{0\\}\\times (0,1)\\times (-0.75,1)$ between the cavity and the wedge. All the meshes $\\mathcal{M}_h$ we consider are compatible with this interface, that is, setting $\\Gamma_h=\\{F\\in\\faces{h}\\,:\\,F\\subset \\Gamma\\}$ we have $\\Gamma=\\cup_{F\\in\\Gamma_h}F$.\nWe then consider the numerical convergence of the numerical flux defined by\n\\[\n\\sum_{F\\in\\Gamma_h}\\int_F \\bvec{u}_F\\cdot\\bvec{n}_{\\Gamma},\n\\]\nwhere $\\bvec{n}_\\Gamma=(1,0,0)$ is the unit normal to $\\Gamma$ pointing inside the wedge.\nThe values of this flux for different degrees of approximations $k$ are provided in Figure \\ref{fig:conv.flux} (left: w.r.t.\\ the mesh size; right: w.r.t.\\ the total wall time, including assembly and solution time -- notice that the \\texttt{HArDCore} library uses multi-threading processes). \nThese results show that the lowest order of approximation struggles to provide what seems to be a correct value of the flux, and that the mesh must be extremely fine to get close to this value; on the contrary, for $k\\ge 1$, all results, even on coarse meshes and with a low computational cost, seem to  be very close to a given value, indicating that convergence has already occurred in these cases. These results corroborate a conclusion already highlighted in \\cite{Anderson.Droniou:18}: even on a problem where the solution is not expected to be very regular, slightly increasing the order of approximation of the scheme (here, going from $k=0$ to $k=1$) can lead to a vastly improved accuracy of the numerical outputs at a very low computational cost.\n\n\\begin{figure}[!h]\\centering\n  \\ref{br.flux.conv}\n  \\vspace{0.50cm}\\\\\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{axis} [legend columns=4, legend to name=br.flux.conv]  \n        \\addplot [mark=star, red] table[x=MeshSize,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k0\/flux_file.dat};\n        \\addlegendentry{$k=0$;}\n        \\addplot [mark=*, blue] table[x=MeshSize,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k1\/flux_file.dat};\n        \\addlegendentry{$k=1$;}\n        \\addplot [mark=o, olive] table[x=MeshSize,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k2\/flux_file.dat};\n        \\addlegendentry{$k=2$}\n        \\addplot [mark=o, black] table[x=MeshSize,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k3\/flux_file.dat};\n        \\addlegendentry{$k=3$}\n      \\end{axis}            \n    \\end{tikzpicture}\n    \\subcaption{w.r.t.~mesh size}\n  \\end{minipage}\n  \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}[scale=0.85]\n      \\begin{loglogaxis} \n        \\addplot [mark=star, red] table[x=TwallTotal,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k0\/flux_file.dat};\n        \\addplot [mark=*, blue] table[x=TwallTotal,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k1\/flux_file.dat};\n        \\addplot [mark=o, olive] table[x=TwallTotal,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k2\/flux_file.dat};\n        \\addplot [mark=o, black] table[x=TwallTotal,y=Flux] {dat\/cavity\/CubeCavityWedge-tets-hexas_k3\/flux_file.dat};\n      \\end{loglogaxis}            \n    \\end{tikzpicture}\n    \\subcaption{w.r.t.~wall time (seconds)}\n  \\end{minipage}\n  \\caption{Convergence of flux values from the cavity to the wedge.}\n  \\label{fig:conv.flux}\n\\end{figure}\n\n\n\n\\section{Analysis}\\label{sec:analysis}\n\n\\subsection{Stability}\\label{sec:analysis:stability}\n\n\\begin{proposition}[$\\norm{\\mu,\\nu,h}{{\\cdot}}$-boundedness of the interpolator]\n  With $\\beta$ as in Lemma \\ref{lem:inf-sup}, it holds, for all $\\bvec{v}\\in\\bvec{H}^1(\\Omega;\\mathbb{R}^d)$,\n  \\begin{equation}\\label{eq:boundedness:IT:norm.mu.nu}\n    \\beta\\norm{\\mu,\\nu,h}{\\uvec{I}_h^k\\bvec{v}}\\lesssim\\norm{\\bvec{H}^1(\\Omega;\\mathbb{R}^d)}{\\bvec{v}}.\n  \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n  It holds, by definition,\n  $\n  \\norm{\\mu,\\nu,h}{\\uvec{I}_h^k\\bvec{v}}^2\n  = \\sum_{T\\in\\elements{h}}\\left[\n  \\mu_T\\mathfrak{T}_1(T) + \\nu_T\\mathfrak{T}_2(T) + \\nu_T\\mathfrak{T}_3(T)\n  \\right]\n  $\n  with\n  \\[\n  \\begin{gathered}\n    \\mathfrak{T}_1(T) \\coloneq\n    \\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\btens{G}_T^k\\uvec{I}_T^k\\bvec{v}}^2\n    + \\frac{\\min(1,\\Cf{T}^{-1})}{h_T^2}\\norm{\\bvec{U},T}{\\uvec{I}_T^k(\\bvec{v} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\bvec{v})}^2,\n    \\\\\n    \\mathfrak{T}_2(T) \\coloneq\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}^2,\\qquad\n    \\mathfrak{T}_3(T) \\coloneq\n    \\min(1,\\Cf{T})\\norm{\\bvec{U},T}{\\uvec{I}_T^k(\\bvec{v} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v})}^2.\n  \\end{gathered}\n  \\]\n  For the first term, combining \\eqref{eq:seminorm.1.T:upper.bound} and \\cite[Eqs.~(8.25)]{Di-Pietro.Droniou:20}, we obtain $\\mathfrak{T}_1(T) \\lesssim\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}^2$.\n  For the second term, if $\\Cf{T}<1$, we can write $\\mathfrak{T}_2(T)= \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\vlproj{k}{T}\\bvec{v}}^2\\le\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}}^2$ using the boundedness of $\\vlproj{k}{T}$, while, if $\\Cf{T}\\ge 1$,\n  \\eqref{eq:approximation:PDT:condition.m=0} gives $\\mathfrak{T}_2(T)\\lesssim\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}}^2 + h_T^2\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}^2\\le\\norm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}^2$, where the conclusion follows observing that $h_T\\le 1$ since $\\Omega$ has unit diameter by assumption.\n  Finally, for the third term, the boundedness \\eqref{eq:boundedness:IT:norm.U} of the interpolator in the $\\norm{\\bvec{U},T}{{\\cdot}}$-norm followed by the approximation properties \\eqref{eq:approximation:PDT} of $\\bvec{P}_{\\darcy,T}^k\\circ\\uvec{I}_T^k$ with $(r,m) = (0,0)$ and $(r,m)=(0,1)$ yield\n  \\[\n  \\mathfrak{T}_3(T)\n  \\lesssim\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}^2\n  + h_T^2\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{v}}^2\n  \\lesssim\n  h_T^2\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{v}}^2.\n  \\]\n  Gathering the above estimates and recalling the bounds \\eqref{eq:mu.nu:bounds} on $\\mu$ and $\\nu$, the result follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:inf-sup}]\n  Classical consequence of the continuous inf-sup condition for the divergence $\\boldsymbol{\\nabla}{\\cdot}:\\bvec{H}^1_0(\\Omega;\\mathbb{R}^d)\\to L^2_0(\\Omega)$ (see, e.g., \\cite{Girault.Raviart:86,Bogovskii:80,Solonnikov:01,Duran.Muschietti:01}) along with the Fortin properties for the interpolator corresponding to \\eqref{eq:consistency:bh} and \\eqref{eq:boundedness:IT:norm.mu.nu};\n  see, e.g., \\cite[Section~5.4.3]{Boffi.Brezzi.ea:13} for further details.\n\\end{proof}\n\n\\subsection{Convergence}\\label{sec:analysis:convergence}\n\nThe purpose of this section is to prove Theorem \\ref{thm:error.estimate}.\nThe proof rests on consistency results for the Stokes, Darcy, and coupling bilinear forms as well as the forcing  term linear form which make the object of the following subsections.\n\n\\subsubsection{Consistency of the Stokes bilinear form}\n\n\\begin{lemma}[Consistency of the Stokes bilinear form]\n  Given $\\bvec{w}\\in\\bvec{H}_0^1(\\Omega;\\mathbb{R}^d)$ such that ${\\boldsymbol{\\nabla}{\\cdot}(\\mu\\boldsymbol{\\nabla}\\bvec{w})}\\in\\bvec{L}^2(\\Omega;\\mathbb{R}^d)$, let the Stokes consistency error linear form $\\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\cdot):\\uvec{U}_{h,0}^k\\to\\mathbb{R}$ be such that, for all $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k$,\n  \\begin{equation}\\label{eq:Errsto}\n    \\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\uvec{v}_h)\n    \\coloneq\n    -\\sum_{T\\in\\elements{h}}\\int_T\\boldsymbol{\\nabla}{\\cdot}(\\mu_T\\boldsymbol{\\nabla}\\bvec{w})\\cdot\\bvec{v}_T\n    - a_{\\mu,h}(\\uvec{I}_h^k\\bvec{w},\\uvec{v}_h).\n  \\end{equation}\n  Then, further assuming $\\bvec{w}\\in\\bvec{H}^{r+2}(\\elements{h};\\mathbb{R}^d)$ for some $r\\in\\{0,\\ldots,k\\}$, it holds\n  \\begin{equation}\\label{eq:estimate:Errsto}\n    \\norm{\\mu,\\nu,h,*}{\\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\cdot)}\n    \\lesssim\\left(\n    \\sum_{T\\in\\elements{h}}\\mu_T\\min(1,\\Cf{T}^{-1})h_T^{2(r+1)}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}^2\n    \\right)^{\\nicefrac12}.\n  \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  Let $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k\\setminus\\{\\uvec{0}\\}$.\n  Proceeding as in \\cite[Point~(ii) in Lemma~2.18]{Di-Pietro.Droniou:20} using an integration by parts for the first term in the definition of $\\mathcal{E}_{\\stokes,h}^k$ along with the definitions \\eqref{eq:a.mu.h} of $a_{\\mu,h}$ and \\eqref{eq:GT} of $\\btens{G}_T^k$ for the second term, we get the following reformulation of the error:\n  \\[\n  \\begin{aligned}\n    \\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\uvec{v}_h)\n    &= \\sum_{T\\in\\elements{h}}\\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_F\\mu_T(\\boldsymbol{\\nabla}\\bvec{w} - \\btens{G}_T^k\\uvec{I}_T^k\\bvec{w})\\bvec{n}_F\\cdot(\\bvec{v}_F - \\bvec{v}_T)\n    \\\\\n    &\\quad\n    - \\sum_{T\\in\\elements{h}}\\frac{\\mu_T\\min(1,\\Cf{T}^{-1})}{h_T^2}\n    (\\uvec{I}_T^k(\\bvec{w} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\bvec{w}), \\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T)_{\\bvec{U},T}.\n  \\end{aligned}\n  \\]\n  Using Cauchy--Schwarz and H\\\"older inequalities along with $\\norm{\\bvec{L}^\\infty(F;\\mathbb{R}^d)}{\\bvec{n}_F}\\le 1$ for all $F\\in\\faces{h}$, we can write\n  \\begin{equation}\\label{eq:consistency.stokes:basic}\n    \\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\uvec{v}_h)\\lesssim\\sum_{T\\in\\elements{h}}\\left[\n      \\mathfrak{T}_1(T) + \\mathfrak{T}_2(T)\n      \\right]\n  \\end{equation}\n  with\n  \\[\n  \\begin{aligned}\n    \\mathfrak{T}_1(T)&\\coloneq\n    \\mu_T^{\\nicefrac12} h_T^{\\nicefrac12}\\norm{\\btens{L}^2(\\partial T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{w} - \\btens{G}_T^k\\uvec{I}_T^k\\bvec{w}}\n    ~\\left(\n    \\frac{\\mu_T}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{v}_T}^2\n    \\right)^{\\nicefrac12},\n    \\\\\n    \\mathfrak{T}_2(T)&\\coloneq\n    \\frac{\\mu_T\\min(1,\\Cf{T}^{-1})}{h_T^2}\n    \\norm{\\bvec{U},T}{\\uvec{I}_T^k(\\bvec{w} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\bvec{w})}\n    \\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\stokes,T}^{k+1}\\uvec{v}_T}.\n  \\end{aligned}\n  \\]\n  \n  Let us estimate $\\mathfrak{T}_1(T)$.\n  Recalling that $\\btens{G}_T^k\\circ\\uvec{I}_T^k = \\tlproj{k}{T}$ and using the approximation properties of this projector (cf.~\\cite{Di-Pietro.Droniou:17} and \\cite[Chapter~1]{Di-Pietro.Droniou:20} concerning the extension to non-star-shaped elements), it is readily inferred for the first factor\n  \\[\n  \\mu_T^{\\nicefrac12} h_T^{\\nicefrac12}\\norm{\\btens{L}^2(\\partial T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{w} - \\btens{G}_T^k\\uvec{I}_T^k\\bvec{w}}\n  \\lesssim\n  \\mu_T^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}.\n  \\]\n  The estimate of the second factor depends on the regime.\n  If $\\Cf{T}<1$, using \\eqref{eq:seminorm.1.T:lower.bound} we write\n  \\begin{equation}\\label{eq:est.seminorm.1.pT:CfT<1}\n    \\frac{\\mu_T}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{v}_T}^2\n    \\lesssim\\mu_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2\n    = \\mu_T\\min(1,\\Cf{T}^{-1})\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2,\n  \\end{equation}\n  where the conclusion follows observing that $1 = \\min(1,\\Cf{T}^{-1})$.\n  If, on the other hand, $\\Cf{T}\\ge 1$ (which implies, in particular, $\\nu_T>0$), we insert $\\pm\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T$ into the norm and use triangle and discrete trace inequalities to write\n    \\begin{align*}\n      &\\frac{\\mu_T}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{v}_T}^2\n      \\\\\n      &\\qquad\\lesssim\n      \\nu_T\\Cf{T}^{-1}\\left(\n      h_T\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2\n      + \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2\n      \\right)\\\\\n      &\\qquad\\lesssim\n      \\nu_T\\Cf{T}^{-1}\\Bigg(\n      h_T\\sum_{F\\in\\faces{T}\\cap \\faces{h}^\\boundary}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2\n      +\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2\n      \\Bigg)\n    \\end{align*}\n    where we have additionally used the definition \\eqref{eq:CfT} of $\\Cf{T}$ in the first inequality,\n    and continued invoking the definition of $\\norm{\\bvec{U},T}{{\\cdot}}$ (see \\eqref{eq:prod.U}--\\eqref{eq:norm.U}) to bound the element term and the non-boundary face terms in the second line by $\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2$.\n    For all $F\\in\\faces{T}\\cap\\faces{h}^\\boundary$, we have $\\bvec{v}_F = \\bvec{0}$ by definition of $\\uvec{U}_{h,0}^k$ and, using discrete trace inequalities and the mesh regularity to write $\\card(\\faces{T})\\lesssim 1$, we infer that\n    \\[\n    \\begin{aligned}\n      \\frac{\\mu_T}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{v}_T}^2\n      &\\lesssim\\nu_T\\Cf{T}^{-1}\\left(\n      \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2+\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}^2\n      \\right)\n      \\\\\n      &\\lesssim\\nu_T\\Cf{T}^{-1}\\norm{\\mathrm{D},T}{\\uvec{v}_T}^2,\n    \\end{aligned}\n    \\]\n    where the last passage follows recalling the definitions \\eqref{eq:norm.nu.darcy} of the $\\norm{\\mathrm{D},T}{{\\cdot}}$-norm, \\eqref{eq:tPDT} of $\\widetilde{\\bvec{P}}_{\\darcy,T}^k$ (which is equal to $\\bvec{P}_{\\darcy,T}^k$ since $\\Cf{T}\\ge 1$), and observing that $1 = \\min(1,\\Cf{T})$.\n    Hence, further observing that $\\Cf{T}^{-1} = \\min(1,\\Cf{T}^{-1})$, we can go on writing\n    \\begin{equation}\\label{eq:est.seminorm.1.pT:CfT>=1}\n      \\frac{\\mu_T}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_F - \\bvec{v}_T}^2\n      \\lesssim\\nu_T\\min(1,\\Cf{T}^{-1})\\norm{\\mathrm{D},T}{\\uvec{v}_T}^2.\n    \\end{equation}\n  Gathering \\eqref{eq:est.seminorm.1.pT:CfT<1} and \\eqref{eq:est.seminorm.1.pT:CfT>=1}, we arrive at\n  \\begin{equation}\\label{eq:consistency.stokes:T1}\n    \\mathfrak{T}_1(T)\n    \\lesssim\\mu_T^{\\nicefrac12}\\min(1,\\Cf{T}^{-1})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}\\left(\n    \\mu_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2\n    + \\nu_T\\norm{\\mathrm{D},T}{\\uvec{v}_T}^2\n    \\right)^{\\nicefrac12}.\n  \\end{equation}\n\n  Moving to $\\mathfrak{T}_2(T)$, using the $\\norm{\\bvec{U},T}{{\\cdot}}$-boundedness \\eqref{eq:boundedness:IT:norm.U} of $\\uvec{I}_T^k$ followed by the approximation properties of $\\bvec{P}_{\\stokes,T}^{k+1}\\circ\\uvec{I}_T^k$ (consequence, for each of its components, of \\cite[Eq.~(2.14) and Theorem~1.48]{Di-Pietro.Droniou:20}), we have\n  \\[\n  \\norm{\\bvec{U},T}{\\uvec{I}_T^k(\\bvec{w} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\uvec{w})}  \n  \\lesssim\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{w} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\uvec{w}}\n  + h_T\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{w} - \\bvec{P}_{\\stokes,T}^{k+1}\\uvec{I}_T^k\\uvec{w}}\n  \\lesssim h_T^{r+2}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}.\n  \\]\n  Plugging this estimate into the definition of $\\mathfrak{T}_2(T)$ and recalling the definition \\eqref{eq:norm.mu.stokes} of $\\norm{\\mathrm{S},T}{{\\cdot}}$, we get\n  \\begin{equation}\\label{eq:consistency.stokes:T2}\n    \\mathfrak{T}_2(T)\n    \\lesssim \\mu_T^{\\nicefrac12}\\min(1,\\Cf{T}^{-1})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}~\n    \\mu_T^{\\nicefrac12}\\norm{\\mathrm{S},T}{\\uvec{v}_T}.\n  \\end{equation}\n\n  Using \\eqref{eq:consistency.stokes:T1} and \\eqref{eq:consistency.stokes:T2} to estimate the right-hand side of \\eqref{eq:consistency.stokes:basic}, we obtain\n  \\[\n  \\begin{aligned}\n    \\mathcal{E}_{\\stokes,h}^k(\\bvec{w};\\uvec{v}_h)\n    &\\lesssim\\sum_{T\\in\\elements{h}}\\mu_T^{\\nicefrac12}\\min(1,\\Cf{T}^{-1})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}\\left(\n    \\mu_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2\n    + \\nu_T\\norm{\\mathrm{D},T}{\\uvec{v}_T}^2\n    \\right)^{\\nicefrac12}\n    \\\\\n    &\\le\n    \\left(\n    \\sum_{T\\in\\elements{h}}\\mu_T\\min(1,\\Cf{T}^{-1})h_T^{2(r+1)}\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{w}}^2\n    \\right)^{\\nicefrac12}\\norm{\\mu,\\nu,h}{\\uvec{v}_h},\n  \\end{aligned}\n  \\]\n  where the conclusion follows using a discrete Cauchy--Schwarz inequality on the sum over $T\\in\\elements{h}$ along with the definition \\eqref{eq:norm.mu.nu.h} of $\\norm{\\mu,\\nu,h}{{\\cdot}}$.\n  Dividing by $\\norm{\\mu,\\nu,h}{\\uvec{v}_h}$ and passing to the supremum concludes the proof of \\eqref{eq:estimate:Errsto}.\n\\end{proof}\n\n\\subsubsection{Consistency of the Darcy bilinear form}\n\n\\begin{lemma}[Consistency of the Darcy bilinear form]\n  Given $\\bvec{w}\\in\\bvec{H}^1(\\Omega;\\mathbb{R}^d)$, let the Darcy consistency error linear form $\\mathcal{E}_{\\darcy,h}^k(\\bvec{w};\\cdot):\\uvec{U}_h^k\\to\\mathbb{R}$ be such that, for all $\\uvec{v}_h\\in\\uvec{U}_h^k$,\n  \\begin{equation}\\label{eq:Errdar}\n    \\mathcal{E}_{\\darcy,h}^k(\\bvec{w};\\uvec{v}_h)\n    \\coloneq\n    \\sum_{T\\in\\elements{h}}\\int_T\\nu_T\\bvec{w}\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n    - a_{\\nu,h}(\\uvec{I}_h^k\\bvec{w},\\uvec{v}_h).\n  \\end{equation}\n  Then, further assuming $\\bvec{w}\\in\\bvec{H}^{r+1}(\\elements{h};\\mathbb{R}^d)$ for some $r\\in\\{0,\\ldots,k\\}$, it holds\n  \\begin{equation}\\label{eq:estimate:Errdar}\n    \\norm{\\mu,\\nu,h,*}{\\mathcal{E}_{\\darcy,h}^k(\\bvec{w};\\cdot)}\n    \\lesssim\\left(\n    \\sum_{T\\in\\elements{h}}\\nu_T\\min(1,\\Cf{T}) h_T^{2(r+1)}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}^2\n    \\right)^{\\nicefrac12}.\n  \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  Let $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k\\setminus\\{\\uvec{0}\\}$.\n  Expanding $a_{\\nu,h}$ according to its definition \\eqref{eq:a.nu}, we get\n  \\begin{equation}\\label{eq:estimate:Errdar:basic}\n    \\mathcal{E}_{\\darcy,h}^k(\\bvec{w};\\uvec{v}_h)\n    = \\sum_{T\\in\\elements{h}}\\left[\n    \\mathfrak{T}_1(T) + \\mathfrak{T}_2(T)\n    \\right],\n  \\end{equation}\n  with\n  \\[\n  \\begin{aligned}\n    \\mathfrak{T}_1(T) &\\coloneq\n    \\int_T\\nu_T(\\bvec{w} - \\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w})\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T,\n    \\\\\n    \\mathfrak{T}_2(T) &\\coloneq\n    -\\nu_T\\min(1,\\Cf{T})(\\uvec{I}_T^k(\\bvec{w} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w}), \\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T)_{\\bvec{U},T}.\n  \\end{aligned}\n  \\]\n\n  The estimate of $\\mathfrak{T}_1(T)$ depends on the regime.\n  Let us start with the case $\\Cf{T}\\ge 1$.\n  Recalling \\eqref{eq:tPDT} to replace $\\widetilde{\\bvec{P}}_{\\darcy,T}^k$ with $\\bvec{P}_{\\darcy,T}^k$ and applying a Cauchy--Schwarz inequality, we get\n  \\[\n  \\begin{aligned}\n    |\\mathfrak{T}_1(T)|\n    &\\lesssim\\nu_T\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{w} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w}}\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}\n    \\\\\n    &\\lesssim\\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}~\\nu_T^{\\nicefrac12}\\norm{\\mathrm{D},T}{\\uvec{v}_T},\n  \\end{aligned}\n  \\]\n  where, to pass to the second line, we have used the approximation properties \\eqref{eq:approximation:PDT} of $\\bvec{P}_{\\darcy,T}^k\\circ\\uvec{I}_T^k$ with $m=0$,\n  the definition \\eqref{eq:norm.nu.darcy} of the $\\norm{\\mathrm{D},T}{{\\cdot}}$-norm,\n  and observed that $1 = \\min(1,\\Cf{T})$.\n  \n  Let us now consider the case $\\Cf{T}<1$.\n  Recalling that $\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w} = \\vlproj{k}{T}\\bvec{w}$ in this case, we can write $\\mathfrak{T}_1(T) = \\int_T\\nu_T(\\bvec{w} - \\vlproj{k}{T}\\bvec{w})\\cdot(\\bvec{v}_T - \\vlproj{0}{T}\\bvec{v}_T)$ and, using Cauchy--Schwarz inequalities, continue with\n  \\begin{equation}\\label{eq:T1.basic}\n    |\\mathfrak{T}_1(T)|\n    \\le\\nu_T\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{w} - \\vlproj{k}{T}\\bvec{w}}\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\vlproj{0}{T}\\bvec{v}_T}.\n  \\end{equation}\n  Using the approximation properties of $\\vlproj{k}{T}$, it is readily inferred that the first factor is $\\lesssim h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}$.\n  To estimate the last factor, we use a Poincar\\'e--Wirtinger inequality to write $\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\vlproj{0}{T}\\bvec{v}_T}\\lesssim h_T\\norm{\\btens{L}^2(T;\\mathbb{R}^{d\\times d})}{\\boldsymbol{\\nabla}\\bvec{v}_T}\\lesssim h_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}$, where the conclusion follows from \\eqref{eq:seminorm.1.T:lower.bound}.\n  Plugging the above estimates into \\eqref{eq:T1.basic}, we can go on writing\n  \\[\n    \\begin{aligned}\n    |\\mathfrak{T}_1(T)|\n    &\\lesssim\\nu_T^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}\n    ~\\nu_T^{\\nicefrac12}h_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}\n    \\\\\n    &=\n    \\nu_T^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}\n    ~\\mu_T^{\\nicefrac12}\\Cf{T}^{\\nicefrac12}\\norm{\\mathrm{S},T}{\\uvec{v}_T},\n    \\\\\n    &=\n    \\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}\n    ~\\mu_T^{\\nicefrac12}\\norm{\\mathrm{S},T}{\\uvec{v}_T},\n    \\end{aligned}\n  \\]\n  where we have used the definition \\eqref{eq:CfT} of $\\Cf{T}$ to pass to the second line and, after rearranging the factors,\n  the fact that $\\Cf{T} = \\min(1,\\Cf{T})$ to conclude.\n  Gathering the above estimates, we thus have\n  \\begin{equation}\\label{eq:estimate:Errdar:T1}\n    |\\mathfrak{T}_1(T)|\n    \\lesssim\n    \\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}\n    ~\\left(\n    \\mu_T\\norm{\\mathrm{S},T}{\\uvec{v}_T}^2\n    + \\nu_T\\norm{\\mathrm{D},T}{\\uvec{v}_T}^2\n    \\right)^{\\nicefrac12}.\n  \\end{equation}\n  \n  To estimate $\\mathfrak{T}_2(T)$, we use a Cauchy--Schwarz inequality to write\n  \\begin{equation}\\label{eq:estimate:Errdar:T2}\n    \\begin{aligned}\n      &|\\mathfrak{T}_2(T)|\n      \\\\\n      &\\quad\n      \\le\\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}\\norm{\\bvec{U},T}{\\uvec{I}_T^k(\\bvec{w} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w})}\n      ~\\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}\\norm{\\bvec{U},T}{\\uvec{v}_T - \\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}\n      \\\\\n      &\\quad\n      \\lesssim\n      \\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}\\left(\n      \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{w} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w}}\n      + h_T\\seminorm{\\bvec{H}^1(T;\\mathbb{R}^d)}{\\bvec{w} - \\bvec{P}_{\\darcy,T}^k\\uvec{I}_T^k\\bvec{w}}\n      \\right)\n      ~\\nu_T^{\\nicefrac12}\\norm{\\mathrm{D},T}{\\uvec{v}_T}\n      \\\\\n      &\\quad\n      \\lesssim\n      \\nu_T^{\\nicefrac12}\\min(1,\\Cf{T})^{\\nicefrac12}h_T^{r+1}\\seminorm{\\bvec{H}^{r+1}(T;\\mathbb{R}^d)}{\\bvec{w}}\n      ~\\nu_T^{\\nicefrac12}\\norm{\\mathrm{D},T}{\\uvec{v}_T},\n      \\end{aligned}\n  \\end{equation}\n  where we have used the $\\norm{\\bvec{U},T}{{\\cdot}}$-boundedness \\eqref{eq:boundedness:IT:norm.U} of $\\uvec{I}_T^k$ along with the definition \\eqref{eq:norm.nu.darcy} of the $\\norm{\\mathrm{D},T}{{\\cdot}}$-norm in the second inequality and\n  the approximation properties \\eqref{eq:approximation:PDT} of $\\bvec{P}_{\\darcy,T}^k\\circ\\uvec{I}_T^k$ with $m = 0$ and $m = 1$ to conclude.\n  Plugging \\eqref{eq:estimate:Errdar:T1} and \\eqref{eq:estimate:Errdar:T2} into \\eqref{eq:estimate:Errdar:basic}, using discrete Cauchy--Schwarz inequalities, dividing by $\\norm{\\mu,\\nu,h}{\\uvec{v}_h}$, and passing to the supremum, the conclusion follows.\n\\end{proof}\n\n\\subsubsection{Consistency of the coupling bilinear form}\n\nThe quantity estimated in the following lemma can be interpreted as an adjoint consistency error for the discrete divergence.\n\n\\begin{lemma}[Consistency of the coupling bilinear form]\n  Given $q\\in H^1(\\Omega)$, let the coupling consistency error linear form $\\mathcal{E}_{\\mathrm{c},h}^k(q;\\cdot):\\uvec{U}_{h,0}^k\\to\\mathbb{R}$ be such that, for all $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k$,\n  \\begin{equation}\\label{eq:Errc}\n    \\mathcal{E}_{\\mathrm{c},h}^k(q;\\uvec{v}_h)\n    \\coloneq\n    \\sum_{T\\in\\elements{h}}\\int_T\\boldsymbol{\\nabla} q\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T - b_h(\\uvec{v}_h,\\lproj{k}{h}q).\n  \\end{equation}\n  Then, further assuming, for some $r\\in\\{0,\\ldots,k\\}$, $q\\in H^{r+1+\\tv{\\Cf{T}\\ge 1}}(T)$ for all $T\\in\\elements{h}$, it holds\n  \\begin{multline}\\label{eq:estimate:Errc}\n    \\norm{\\mu,\\nu,h,*}{\\mathcal{E}_{\\mathrm{c},h}^k(q;\\cdot)}\n    \\\\\n    \\lesssim\\left[\n      \\sum_{T\\in\\elements{h}}\\left(\n      \\mu_T^{-1}\\tv{\\Cf{T}<1} h_T^{2(r+1)}\\seminorm{H^{r+1}(T)}{q}^2\n      + \\nu_T^{-1}\\tv{\\Cf{T}\\ge 1} h_T^{2(r+1)}\\seminorm{H^{r+2}(T)}{q}^2\n      \\right)\n      \\right]^{\\nicefrac12},\n  \\end{multline}\n  where $\\nu_T^{-1}\\tv{\\Cf{T}\\ge 1}\\coloneq 0$ if $\\nu_T = 0$ as in Theorem~\\ref{thm:error.estimate}.\n\\end{lemma}\n\n\\begin{proof}\n  Let $\\uvec{v}_h\\in\\uvec{U}_{h,0}^k\\setminus\\{\\uvec{0}\\}$.\n  We start by noticing that, expanding the bilinear form $b_h$ according to its definition \\eqref{eq:bh},\n  \\begin{equation}\\label{eq:Errc:estimate:basic}\n    \\mathcal{E}_{\\mathrm{c},h}^k(q;\\uvec{v}_h)\n    = \\sum_{T\\in\\elements{h}}\n    \\left(\n    \\int_T\\boldsymbol{\\nabla} q\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n    + \\int_T\\lproj{k}{T}q~D_T^k\\uvec{v}_T\n    - \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_Fq~(\\bvec{v}_F\\cdot\\bvec{n}_F)\n    \\right),\n  \\end{equation}\n  where the insertion of the last term in parenthesis is made possible by the single-valuedness of $q~(\\bvec{v}_F\\cdot\\bvec{n}_F)$ at interfaces along with the fact that $\\bvec{v}_F\\cdot\\bvec{n}_F = 0$ for all $F\\in\\faces{h}^\\boundary$.\n  Denote by $\\mathfrak{T}(T)$ the argument of the summation in \\eqref{eq:Errc:estimate:basic}.\n  To estimate this quantity, we distinguish two cases based on the value of $\\Cf{T}$.\n  \n  If $\\Cf{T}<1$, $\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T = \\bvec{v}_T$ by \\eqref{eq:tPDT}, so that\n  \\[\n  \\mathfrak{T}(T)\n  = \\int_T\\boldsymbol{\\nabla} q\\cdot\\bvec{v}_T\n  + \\int_T\\lproj{k}{T}q~D_T^k\\uvec{v}_T\n  - \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_Fq~(\\bvec{v}_F\\cdot\\bvec{n}_F).\n  \\]\n  Thus, proceeding as in the derivation of \\cite[Eq.~(8.41)]{Di-Pietro.Droniou:20}, we get\n   \\begin{equation}\\label{eq:termT.estimate:CfT<1}\n    \\begin{aligned}\n      \\mathfrak{T}(T)\n      &\\lesssim\n      h_T^{r+1}\\seminorm{H^{r+1}(T)}{q}\\left(\n      \\frac{1}{h_T}\\sum_{F\\in\\faces{T}}\\norm{\\bvec{L}^2(F;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{v}_F}^2\n      \\right)^{\\nicefrac12}\n      \\\\\n      &\\lesssim\n      \\mu_T^{-\\nicefrac12}\\tv{\\Cf{T}<1}^{\\nicefrac12}\n      h_T^{r+1}\\seminorm{H^{r+1}(T)}{q}~\\mu_T^{\\nicefrac12}\\norm{\\mathrm{S},T}{\\uvec{v}_T},\n    \\end{aligned}\n   \\end{equation}\n   where we have used \\eqref{eq:seminorm.1.T:lower.bound} to conclude.\n\n   If $\\Cf{T}\\ge 1$, on the other hand, we have $\\widetilde{\\bvec{P}}_{\\darcy,T}^k = \\bvec{P}_{\\darcy,T}^k$ (cf.\\ \\eqref{eq:tPDT}), so that\n   \\[\n   \\mathfrak{T}(T)\n   =\n   \\left(\n   \\int_T\\boldsymbol{\\nabla} q\\cdot\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T\n   + \\int_T\\lproj{k}{T}q~D_T^k\\uvec{v}_T\n   - \\sum_{F\\in\\faces{T}}\\omega_{TF}\\int_Fq~(\\bvec{v}_F\\cdot\\bvec{n}_F)\n   \\right).\n   \\]\n   Using the definition \\eqref{eq:PDT} of $\\bvec{P}_{\\darcy,T}^k$ to proceed as in \\cite[Theorem~11]{Di-Pietro.Droniou:21*1}, we get\n   \\begin{equation}\\label{eq:termT.estimate:CfT>=1}\n     \\mathfrak{T}(T)\n     \\lesssim h_T^{r+1}\\seminorm{H^{r+2}(T)}{q}~\\norm{\\mathrm{D},T}{\\uvec{v}_T}\n     \\le \\nu_T^{-\\nicefrac12}\\tv{\\Cf{T}\\ge 1}^{\\nicefrac12}\n     h_T^{r+1}\\seminorm{H^{r+2}(T)}{q}~\\nu_T^{\\nicefrac12}\\norm{\\mathrm{D},T}{\\uvec{v}_T},\n   \\end{equation}\n   where we have additionally noticed that $\\Cf{T}\\ge 1$ implies $\\nu_T>0$.\n\n   To conclude, we plug \\eqref{eq:termT.estimate:CfT<1} and \\eqref{eq:termT.estimate:CfT>=1} into \\eqref{eq:Errc:estimate:basic},\n   use a Cauchy--Schwarz inequality on the sum over $T\\in\\elements{h}$,\n   recall the definition \\eqref{eq:norm.mu.nu.h} of the $\\norm{\\mu,\\nu,h}{{\\cdot}}$-norm,\n  and pass to the supremum after dividing by $\\norm{\\mu,\\nu,h}{\\uvec{v}_h}$.\n\\end{proof}\n\n\\begin{remark}[Discretisation of the source term]\\label{rem:disc.source}\n  The use of $\\bvec{P}_{\\darcy,T}^k$ in the discretisation of the source term when $\\Cf{T}\\ge 1$ (see \\eqref{eq:discrete:variational} and \\eqref{eq:tPDT}) is crucial to ensure that, in this case, the consistency error of the coupling bilinear form can be bounded above using the Darcy norm instead of the Stokes norm; compare \\eqref{eq:termT.estimate:CfT>=1} and \\eqref{eq:termT.estimate:CfT<1}. This bound is  key to establishing an error estimate in $h^{r+1}$ that remains robust in the Darcy limit. \n\\end{remark}\n\n\n\\subsubsection{Consistency of the forcing term linear form}\n\nThe following lemma estimates the difference between the standard HHO right-hand side linear form and the one obtained, as in \\eqref{eq:strong:momentum}, using $\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T$ instead of $\\bvec{v}_T$ as a test function.\n\n\\begin{lemma}[Consistency of the forcing term]\n  For any $\\bvec{\\varphi}\\in\\bvec{L}^2(\\Omega;\\mathbb{R}^d)$, define the right-hand side consistency error linear form $\\mathcal{E}_{\\mathrm{rhs},h}^k(\\bvec{\\varphi};\\cdot):\\uvec{U}_h^k\\to\\mathbb{R}$ such that, for all $\\uvec{v}_h\\in\\uvec{U}_h^k$,\n  \\begin{equation}\\label{eq:Errrhs}\n    \\mathcal{E}_{\\mathrm{rhs},h}^k(\\bvec{\\varphi};\\uvec{v}_h)\\coloneq\n    \\sum_{T\\in\\elements{h}}\\int_T\\bvec{\\varphi}\\cdot(\\bvec{v}_T - \\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T).\n  \\end{equation}\n  Further assuming $\\bvec{\\varphi}\\in\\bvec{H}^r(\\elements{h};\\mathbb{R}^d)$ for some $r\\in\\{0,\\ldots,k\\}$, it holds\n  \\begin{equation}\\label{eq:estimate:Errrhs}\n    \\norm{\\mu,\\nu,h,*}{\\mathcal{E}_{\\mathrm{rhs},h}^k(\\bvec{\\varphi};\\cdot)}\n    \\lesssim\\left(\n    \\sum_{T\\in\\elements{h}}\\mu_T^{-1}\\min(1,\\Cf{T}^{-1}) h_T^{2(r+1)}\\seminorm{\\bvec{H}^r(T;\\mathbb{R}^d)}{\\bvec{\\varphi}}^2\n    \\right)^{\\nicefrac12}.\n  \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  Denote by $\\mathfrak{T}(T)$ the argument of the summation in \\eqref{eq:Errrhs}.\n  If $\\Cf{T}<1$, the definition \\eqref{eq:tPDT} of $\\widetilde{\\bvec{P}}_{\\darcy,T}^k$ yields $\\mathfrak{T}(T) = 0$.\n  Consider now the case $\\Cf{T}\\ge 1$ (which implies, in particular, $\\nu_T>0$). We first notice that, letting $\\vlproj{k-1}{T}\\bvec{\\varphi} \\coloneq \\bvec{0}$ if $k=0$,\n  \\begin{equation}\\label{eq:approx.pik}\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{\\varphi}-\\vlproj{k-1}{T}\\bvec{\\varphi}}\\lesssim h_T^r\\seminorm{\\bvec{H}^r(T;\\mathbb{R}^d)}{\\bvec{\\varphi}},\n  \\end{equation}\n  where the result is trivial if $k=0$ (which imposes $r=0$) and otherwise follows from the approximation properties of $\\vlproj{k-1}{T}$, see \\cite[Theorem 1.45]{Di-Pietro.Droniou:20}. Recalling that, for $\\Cf{T}\\ge 1$, we have $\\widetilde{\\bvec{P}}_{\\darcy,T}^k = \\bvec{P}_{\\darcy,T}^k$ by \\eqref{eq:tPDT} and invoking \\eqref{eq:lproj.k-1.PDT} (which trivially holds also for $k=0$), we then write\n  \\[\n  \\begin{aligned}\n    \\mathfrak{T}(T)\n    &= \\int_T(\\bvec{\\varphi} - \\vlproj{k-1}{T}\\bvec{\\varphi})\\cdot(\\bvec{v}_T - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T)\n    \\\\\n    &\\le\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{\\varphi} - \\vlproj{k-1}{T}\\bvec{\\varphi}}\n    \\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}\n    \\\\\n    &\\lesssim\n    \\mu_T^{-\\nicefrac12} h_T^r \\seminorm{\\bvec{H}^r(T;\\mathbb{R}^d)}{\\bvec{\\varphi}}\n    ~h_T\\mu_T^{\\nicefrac12}h_T^{-1}\\norm{\\mathrm{D},T}{\\uvec{v}_T}\n    \\\\\n    &=\n    \\mu_T^{-\\nicefrac12} h_T^{r+1} \\seminorm{\\bvec{H}^r(T;\\mathbb{R}^d)}{\\bvec{\\varphi}}\n    ~\\nu_T^{\\nicefrac12}\\min(1,\\Cf{T}^{-1})^{\\nicefrac12}\\norm{\\mathrm{D},T}{\\uvec{v}_T},\n  \\end{aligned}\n  \\]\n  where we have used Cauchy--Schwarz inequalities in the first passage,\n  the approximation properties \\eqref{eq:approx.pik} of the $L^2$-orthogonal projector for the first factor together with the definitions \\eqref{eq:norm.U} and \\eqref{eq:norm.nu.darcy} of $\\norm{\\bvec{U},T}{{\\cdot}}$ and $\\norm{\\mathrm{D},T}{{\\cdot}}$ to write $\\norm{\\bvec{L}^2(T;\\mathbb{R}^d)}{\\bvec{v}_T - \\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}\\le\\norm{\\bvec{U},T}{\\uvec{v}_T-\\uvec{I}_T^k\\bvec{P}_{\\darcy,T}^k\\uvec{v}_T}\\le\\norm{\\mathrm{D},T}{\\uvec{v}_T}$ in the second passage,\n  while the conclusion follows from the definition \\eqref{eq:CfT} of $\\Cf{T}$ along with $\\Cf{T}^{-1} = \\min(1,\\Cf{T}^{-1})$.\n  Using the above estimate in \\eqref{eq:Errrhs}, applying a Cauchy--Schwarz inequality on the sum over $T\\in\\elements{h}$, and recalling the definition \\eqref{eq:norm.mu.nu.h} of $\\norm{\\mu,\\nu,h}{{\\cdot}}$, \\eqref{eq:estimate:Errrhs} follows.\n\\end{proof}\n\n\\subsubsection{Proof of Theorem \\ref{thm:error.estimate}}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:error.estimate}]\n  Since $a_{\\mu,h}+a_{\\nu,h}$ is $1$-coercive and has norm $1$ for the $\\norm{\\mu,\\nu,h}{{\\cdot}}$ norm, Lemma~\\ref{lem:inf-sup} and \\cite[Lemma A.11]{Di-Pietro.Droniou:20} show that $\\mathcal{A}_h$ is $\\gamma$-inf-sup stable for the norm in the left-hand side of \\eqref{eq:error.estimate}. Hence, in the spirit of the third Strang lemma \\cite{Di-Pietro.Droniou:18}, this error estimate follows if we bound the consistency error by the bracketed term in the right-hand side.\n  The consistency error for the scheme \\eqref{eq:discrete:variational} is\n  \\begin{align}\n    \\mathcal{E}_h^k(\\bvec{u},p;\\uvec{v}_h)\n    &\\coloneq\n    \\sum_{T\\in\\elements{h}}\\int_T\\bvec{f}\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T + \\int_\\Omega gq_h\n    -\\mathcal{A}_h((\\uvec{I}_h^k\\bvec{u},\\lproj{h}{k}p),(\\uvec{v}_h,q_h))    \n    \\nonumber\\\\\n    &=\n    \\sum_{T\\in\\elements{h}}\\int_T\\boldsymbol{\\nabla}{\\cdot}(\\mu_T\\boldsymbol{\\nabla}\\bvec{u})\\cdot(\\bvec{v}_T - \\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T)\n    -\n    \\sum_{T\\in\\elements{h}}\\int_T\\boldsymbol{\\nabla}{\\cdot}(\\mu_T\\boldsymbol{\\nabla}\\bvec{u})\\cdot\\bvec{v}_T - a_{\\mu,h}(\\uvec{I}_h^k\\bvec{u},\\uvec{v}_h)\n    \\nonumber\\\\\n    &\\quad\n    + \\sum_{T\\in\\elements{h}}\\int_T\\nu_T\\bvec{u}\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n    - a_{\\nu,h}(\\uvec{I}_h^k\\bvec{u},\\uvec{v}_h)\n    + \\sum_{T\\in\\elements{h}}\\int_T\\boldsymbol{\\nabla} p\\cdot\\widetilde{\\bvec{P}}_{\\darcy,T}^k\\uvec{v}_T\n    - b_h(\\uvec{v}_h,\\lproj{k}{h}p)\n    \\nonumber\\\\\n    &\\quad\n    + \\cancel{%\n      \\int_\\Omega g q_h\n      + b_h(\\uvec{I}_h^k\\bvec{u}, q_h)%\n    }\n    \\nonumber\\\\\n    &=\n    \\mathcal{E}_{\\mathrm{rhs},h}^k(\\boldsymbol{\\nabla}{\\cdot}(\\mu\\boldsymbol{\\nabla}\\bvec{u});\\uvec{v}_h)\n    + \\mathcal{E}_{\\stokes,h}^k(\\bvec{u};\\uvec{v}_h)\n    + \\mathcal{E}_{\\darcy,h}^k(\\bvec{u};\\uvec{v}_h)\n    + \\mathcal{E}_{\\mathrm{c},h}^k(p;\\uvec{v}_h),\n  \\label{eq:error.estimate:basic}\n  \\end{align}\n  where we have we have replaced $\\bvec{f}$ with the left-hand side of \\eqref{eq:strong:momentum}, expanded $\\mathcal{A}_h$ according to its definition \\eqref{eq:Ah}, and used \\eqref{eq:consistency:bh} along with \\eqref{eq:strong:mass} to cancel the last term in the first passage,\n  and used the definitions of the consistency errors \\eqref{eq:Errrhs} with $\\bvec{\\varphi} = \\boldsymbol{\\nabla}{\\cdot}(\\mu\\boldsymbol{\\nabla}\\bvec{u})$, \\eqref{eq:Errsto} and \\eqref{eq:Errdar} with $\\bvec{w} = \\bvec{u}$, and \\eqref{eq:Errc} with $q = p$ to conclude.\n\n  Using, respectively, \\eqref{eq:estimate:Errrhs}\n  (further noticing that $\\seminorm{\\bvec{H}^r(T;\\mathbb{R}^d)}{\\boldsymbol{\\nabla}{\\cdot}(\\mu_T\\boldsymbol{\\nabla}\\bvec{u})}\\lesssim \\mu_T\\seminorm{\\bvec{H}^{r+2}(T;\\mathbb{R}^d)}{\\bvec{u}}$ for all $T\\in\\elements{h}$),\n  \\eqref{eq:estimate:Errsto},\n  \\eqref{eq:estimate:Errdar},\n  and \\eqref{eq:estimate:Errc} to estimate the terms in the right-hand side of \\eqref{eq:error.estimate:basic}, the result follows.\n\\end{proof}\n\n\\section*{Acknowledgements}\n\nThis research received support from the ANR ``NEMESIS'' (ANR-20-MRS2-0004) and the Australian Research Council's Discovery Projects funding scheme (DP210103092). The authors would also like to thank Ricardo Ruiz-Baier for sharing Gmsh geometry files at the source of the tests in Section \\ref{sec:lid.cavity}.\n\n\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nJEAN-PATRICK MANCHETTE (1942\u20131995) was a genre-redefining French crime novelist, screenwriter, critic, and translator. Born in Marseille to a family of relatively modest means, Manchette grew up in a southwestern suburb of Paris, where he wrote from an early age. While a student of English literature at the Sorbonne, he contributed articles to the newspaper _La Voie communiste_ and became active in the national students' union. In 1961 he married, and with his wife M\u00e9lissa began translating American crime fiction\u2014he would go on to translate the works of such writers as Donald Westlake, Ross Thomas, and Margaret Millar, often for Gallimard's S\u00e9rie noire. Throughout the 1960s Manchette supported himself with various jobs writing television scripts, screenplays, young-adult books, and film novelizations. In 1971 he published his first novel, a collaboration with Jean-Pierre Bastid, and embarked on his literary career in earnest, producing ten subsequent works over the course of the next two decades and establishing a new genre of French novel, the _n\u00e9o-polar_ (distinguished from traditional detective novel, or _polar,_ by its political engagement and social radicalism). During the 1980s, Manchette published celebrated translations of Alan Moore's Watchmen graphic novels for a _bande-dessin\u00e9e_ publishing house co-founded by his son, Doug Headline. In addition to _Fatale_ (also available as an NYRB Classic), Manchette's novels _Three to Kill_ and _The Prone Gunman,_ as well as Jacques Tardi's graphic-novel adaptations of them (titled _West Coast Blues_ and _Like a Sniper Lining Up His Shot,_ respectively), are available in English.\n\nDONALD NICHOLSON-SMITH's translations of noir fiction include Manchette's _Three to Kill;_ Thierry Jonquet's _Mygale_ (a.k.a. _Tarantula);_ and (with Alyson Waters) Yasmina Khadra's _Cousin K._ He has also translated works by Paco Ignacio Taibo II, Henri Lefebvre, Raoul Vaneigem, Antonin Artaud, Jean Laplanche, Guillaume Apollinaire, and Guy Debord. For NYRB Classics he has translated Manchette's _Fatale_ and is presently working on Jean-Paul Cl\u00e9bert's _Paris Insolite._ Born in Manchester, England, he is a longtime resident of New York City.\n\nJAMES SALLIS's recent and forthcoming books include the novel _Others of My Kind,_ a reissue of his novel _Death Will Have Your Eyes,_ and _Black Night's Gonna Catch Me Here: Selected Poems 1968\u20132012._ He is also the author of _Drive_ and of _Chester Himes: A Life,_ and the translator of Raymond Queneau's novel _Saint Glinglin._\nTHE MAD AND THE BAD\n\nJEAN-PATRICK MANCHETTE\n\nTranslated from the French by\n\nDONALD NICHOLSON-SMITH\n\nIntroduction by\n\nJAMES SALLIS\n\nNEW YORK REVIEW BOOKS\n\nNew York\nTHIS IS A NEW YORK REVIEW BOOK\n\nPUBLISHED BY THE NEW YORK REVIEW OF BOOKS\n\n435 Hudson Street, New York, NY 10014\n\nwww.nyrb.com\n\nCopyright \u00a9 1972 by \u00c9ditions Gallimard, Paris\n\nTranslation copyright \u00a9 2014 by Donald Nicholson-Smith\n\nIntroduction copyright \u00a9 2014 by James Sallis\n\nAll rights reserved.\n\nFirst published in French as _\u00d4 Dingos, \u00d4 Ch\u00e2teaux_ by \u00c9ditions Gallimard.\n\n_Cet ouvrage a b\u00e9n\u00e9fici\u00e9 du soutien des Programmes d'aide \u00e0 la publication de l'Institut fran\u00e7ais._\n\nThis work, published as part of a program of aid for publication, received support from the Institut Fran\u00e7ais.\n\nCover image: Miwa Yanagi, The White Casket (detail), 1994; courtesy of the artist and Yoshiko Isshiki Office  \nCover design: Katy Homans\n\nThe Library of Congress has cataloged the earlier printing as follows:\n\nManchette, Jean-Patrick, 1942\u20131995, author.\n\n[O dingos, \u00f4 ch\u00e2teaux. English]\n\nThe mad and the bad \/ Jean-Patrick Manchette ; translated by Donald Nicholson-Smith.\n\npages cm. \u2014 (New York Review Books Classics)\n\nTranslated from the French.\n\nISBN 978-1-59017-720-4 (paperback)\n\nI. Nicholson-Smith, Donald, translator. II. Title.\n\nPQ2673.A452O313 2014\n\n843'.914\u2014dc23\n\n2013050861\n\nISBN 978-1-59017-740-2  \nv1.0\n\nFor a complete list of books in the NYRB Classics series, visit www.nyrb.com or write to: Catalog Requests, NYRB, 435 Hudson Street, New York, NY 10014.\nBiographical Notes\n\nTitle page\n\nCopyright and More Information\n\nTranslator's Note\n\nIntroduction\n\n_I should like to thank Doug Headline for his moral support, editorial assistance, and patience. Thank you too to Alyson Waters for her great advice on translation issues. I am grateful to everyone concerned with this project at New York Review Books. And, as so many times in the past, I must express my hopeless indebtedness to Mia Rublowska for all her help._\n\n_Donald Nicholson-Smith_\n\n# INTRODUCTION\n\nIN AMERICA it was Hammett and Chandler: Hammett who took murder out of the manor houses and gave it back to the people who actually commit it; Chandler who fashioned of bus stations, diners, and cheap hotel rooms, at the frontier's last raw edge, a mythology specifically American. In France the new maps were drawn by Jean-Patrick Manchette.\n\nWhen Manchette began to write his novels in the mid-1970s, the French _polar_ had become a still pool of police procedurals and tales of Pigalle lowlife. Manchette wanted to throw in rocks, disturb the calm surface, bring up all the muck beneath\u2014to demonstrate that the crime novel could be (as he said again and again) \"the great moral literature of our time.\"\n\nFor Manchette and the generation of writers who succeeded him, then, these novels became far more than simple entertainment; they became a means of facing society's failures head on. One after another the curtains will be torn back. Pretense. Deceit. Manipulation. Till there in the small, choked room behind it all we witness society's true engines\u2014greed and violence\u2014grinding away.\n\n\"He was like an electroshock to the chloroformed country of literature and the French thriller,\" Jean-Francois Gerault noted.\n\nManchette published ten novels with Gallimard from 1971 to 1982. Before and after, he worked as an editor, reviewed movies, wrote scripts for film and TV and numerous essays on thrillers and crime fiction. He also published translations of Ross Thomas, Donald Westlake, Alan Moore, and others\u2014at least thirty books. By 1989, treatment of and complications from a pancreatic tumor made work difficult. He died in 1995 in Paris of lung cancer, aged fifty-three, before completing a new novel, _La Princesse du sang_ (Princess of the Blood), intended to be the first of a five-book cycle covering five decades from the postwar period to the present.\n\n_The Mad and the Bad,_ original title _\u00d4 dingos, \u00d4 ch\u00e2teaux!,_ came early in the game, in 1972, following close upon the prior year's collaborative _Laissez bronzer les cadavres_ (Corpses in the Sun, with Jean-Pierre Bastid) and solo _L'Affaire N'Gustro_ (The N'Gustro Affair), winning that year's grand prize for crime fiction.\n\nThe novel's tale of a young woman and a boy set upon by deadly forces beyond their understanding shows the coopting of classic noir plots that we see in all Manchette's novels. In _Three to Kill_ a businessman witnesses a murder and, pursued by the killers, steps away from his ordinary life to turn the killing back on them. In _The Prone Gunman_ a hired killer yearning to give it all up returns catastrophically to his hometown. The pleasure lies in the many ways Manchette twists and turns his story on the spit of plot, how he transforms the expected, how much weight he manages to pack into scenes that remain lean and muscular. Things move fast, almost at a blur\u2014then excruciatingly slow. Sentences are clipped, headlong. Charged language everywhere, sometimes to the point of the incantatory.\n\nHere also are Manchette's trademark disavowed individuals, ill-fitting stones in societal walls that will crumble at the first wayward blow.\n\n\"The nursemaid before you. Completely off her rocker. Fifty if she was a day. And an idiot. What about you? What's your thing?\"\n\n\"I don't understand at all,\" said Julie. \"My thing? What do you mean?\"\n\n\"The thing that's screwy with you.\"\n\n\"I'm cured,\" Julie stated.\n\n\"The hell you are!\" exclaimed the driver. \"The boss's way of doing good is over the top. He only hires retards. He sets up factories for cripples to work in, can you figure that?\"\n\n\"Not really.\"\n\n\"Those guys who go around in little motorized wheelchairs? He's got them working on a production line! In this house it's the same baloney. The cook is epileptic. The gardener has only one arm, pretty handy for using the shears. His private secretary is blind. His valet suffers from locomotor ataxia\u2014no wonder his meals arrive cold! The snotty brat's old nanny\u2014well, I told you about her. As for you, you must know yourself.\"\n\n\"What about you?\" asked Julie.\n\nShe had taken out a pack of Gauloises and a Criquet lighter. She lit a cigarette and, throwing her head back, blew smoke through her nostrils.\n\n\"What about you?\" she repeated.\n\nA parade of grotesquerie, dialogue rich in subtext, and a parody of labor in capitalist society\u2014all in less than a page.\n\nThere's much that's quintessentially French about Manchette: his political stance, the stylish hard surface of his prose, his adoption of a \"low\" or demotic art form to embody abstract ideas. Like any great illusionist, he directs our attention one way as the miraculous happens in another. He tells us a simple story. This occurred. That. But there's bone, there's gristle. Floors give way, and wind heaves its shoulder against the door. His stories of cornered individuals become an indictment of capitalism's excesses, its unchallenged power, its reliance on distraction and spectacle.\n\nFor Manchette the world is a giant marketplace in which gangs of thugs\u2014be they leftist, reactionary, terrorist, police, or politicians\u2014compete relentlessly; one in which tiny groups of individuals, \"torn to pieces by the enemy and sodomized by [their] own leaders,\" stay afloat by clinging to the flotsam. In his work he alludes to and parodies literary writers such as Baudelaire and Stendhal, juxtaposes the vulgar and the precious, enjambs depictions of quotidian life against scenes of such extreme and often implicit violence as to call into question all the myriad fictions of bourgeois, accepted existence. Like Hammett, he affirms that everyone lies; like Rimbaud, that everything we are taught is false.\n\nManchette revered Chandler and Hammett as founders of the form in which he worked, and in Chandler's lyrical description from \"The Simple Art of Murder\" found a world he well recognized.\n\nThe realist in murder writes of a world in which gangsters can rule nations and almost rule cities, in which hotels and apartment houses and celebrated restaurants are owned by men who made their money out of brothels, in which a screen star can be the fingerman for a mob, and the nice man down the hall is a boss of the numbers racket; a world where a judge with a cellar full of bootleg liquor can send a man to jail for having a pint in his pocket, where the mayor of your town may have condoned murder as an instrument of moneymaking, where no man can walk down a dark street in safety because law and order are things we talk about but refrain from practicing. . . .\n\nIt is not a very fragrant world, but it is the world you live in, and certain writers with tough minds and a cool spirit of detachment can make very interesting and even amusing patterns out of it. It is not funny that a man should be killed, but it is sometimes funny that he should be killed for so little, and that his death should be the coin of what we call civilization.\n\nThough dredged from the same dark sense of purloined promise as Chandler's, Manchette's profoundly leftist, distinctly European stance may be something of a problem for American readers. Like many of his generation, Manchette was influenced by the Situationist Guy Debord, whose theories, elaborated in _The Society of the Spectacle,_ were everywhere during France's 1968 insurrections. Situationists held that capitalism's overweening successes came only at the expense of increased alienation, social dysfunction, and a general degradation of daily life; that the acquisition, exchange, and consumption of commodities had forcefully supplanted direct experience, creating a kind of life by proxy; and that liberation might be found in fashioning moments that reawakened authentic desires, a sense of adventure, a ransom from dailiness.\n\nAgain and again one finds similar ideas in Manchette, here as a loose scaffolding holding story parts together, there like bones poking through broken skin. Manchette's stories clip along at breakneck speed, breath be damned, skimming over polarized societies and forfeited lives, momentum never flagging. And in that disjunction, lightness of surface supporting the heaviness beneath, Manchette found his voice.\n\nBack in the hills of the rural South where I grew up, squirrel hunters often nailed their game to trees and, with a knife and brute strength, tore the body from the skin in a single hard pull. As a method it was clean, quick, and efficient. The skins stayed behind on the trees, dozens of them, all around cabins and favorite hunting sites, constant reminders.\n\nBooks like Manchette's are those skins.\n\n\u2014JAMES SALLIS\n\n# THE MAD AND THE BAD\n# 0\n\nTHE MAN whom Thompson was supposed to kill\u2014a pederast guilty of seducing the son of a businessman\u2014entered his bedroom. As he closed the door behind him, he had time to recoil at the sight of Thompson standing against the wall beside the hinges. Then Thompson stabbed him in the heart with a rigid hacksaw blade mounted on a large cylindrical hilt with a circular sheet-metal guard. While the guard prevented the blood from spurting, Thompson pumped the cylindrical hilt vigorously, and the homosexual's heart was sliced into two or more pieces. The victim opened his mouth and a single spasm shook him. His rump struck the door and he slumped forward dead. Thompson stepped aside. For the last half hour his stomach cramps had grown almost intolerable. He left the bedroom. No one had seen him enter; no one saw him leave. It was two o'clock in the morning. Thompson had an appointment in Paris at eleven. He made his way on foot to the Perrache railroad station. The cramps had him almost doubled over. The killer resolved to give up his trade. Soon. Every time it was worse. For the last ten hours he had been unable to eat or drink anything. Now that he had killed, hunger gnawed at him in the most repellent way. Eventually he reached the station buffet. He ordered a choucroute and devoured it. He ordered another, which he savored. His stomach had calmed down. His mind likewise: Thompson had just earned a tidy sum of money. It was three in the morning. The killer paid his bill, returned to his gray Rover, which was parked at a meter, and headed for the autoroute A6.\n\nLater on, somewhere between Lyon and Paris, he pulled off into a rest area and snoozed until daybreak.\n\nAt eleven in the morning he was prompt for his appointment. His new client wore dark glasses and Thompson smiled at this childishness. Seated in a booth, the two men drank Scottish beer. The new client placed a photograph facedown on the table.\n\n\"It's going to be a bit tricky,\" he said. \"It will have to look as though . . . well, I'll explain. What's the matter? Aren't you well?\"\n\nThompson was massaging his belly.\n\n\"I'm okay, I'm okay,\" he replied.\n\nHe turned the photograph over. It was a color snapshot. A half-length portrait of a redheaded boy with a sullen expression.\n\n\"Does this bother you?\"\n\n\"Not at all,\" said Thompson.\n\nWhat bothered him was his stomach. It was starting again. The pain was back.\n\n# 1\n\nIT WAS a black Lincoln Continental. Tinted side windows made it impossible to discern the occupants. The car was having some difficulty making the tight turns of the narrow road. All around was forest: a profusion of beeches and a carpet of rotting dead leaves that encroached on the road.\n\nFrom the middle of a partial clearing in the trees some fifty meters long, a driveway led off to the right. It was flanked by wide grassy shoulders and punctuated by white marker stones linked by a decorative chain. To make the turn into the drive, the Lincoln had first to veer in the opposite direction so that it briefly filled the left side of the roadway before turning into the drive with its white graveled surface. Flintstones and dust sprayed up under the mudguards.\n\nThe driveway led directly to a Louis XIII manor house. The place had three corner towers. One stood in water, and water lilies floated beneath its windows. The Lincoln slowed down. The manor was getting closer.\n\nIt was surrounded by a vast expanse of grass. Here and there paths plunged off into the woods. Groups of strollers were to be seen, clad in long lab coats, pink, blue, or pistachio green. The big car passed a hunched man with long hair and glasses who had unbuttoned his blue coat to urinate on a molehill. He had knelt to improve the accuracy of the stream, carefully directing it into the hole made by the animal. He seemed serious and malevolent. He paid no attention to the imposing automobile.\n\nThe Lincoln proceeded past other strange figures. There were men in blue and women in pink. Those in pistachio green, men and women, had an air of efficiency. They were obviously staff.\n\nThe car came to the end of the drive and pulled up on the forecourt in front of the manor's front entrance\u2014a low white double perron. Cutting the ignition, the driver got out. He was a man of about thirty-five, thickset, round-bodied, round-faced, and short-legged. He wore a blue chauffeur's uniform, a white shirt, a red tie, and a cap. He took the cap off, revealing his hair\u2014what looked like a prison cut. The man opened the back door of the car. An individual of the same age emerged. He wore bell-bottoms and a battered silver corduroy safari jacket. His hair, short, light red, almost tow, was very fine. His long, intelligent, mobile, haughty face would have brought Franchot Tone to the mind of anyone who knew who Franchot Tone was. His pink skin was covered by a mass of freckles barely distinguishable from his overall complexion. His eyes were a watery green. He looked like a mutant in a television series.\n\nA shower of gravel landed with a crackle on the rear of the Lincoln. The driver and the redhead turned toward its source, an unshaven forty-year-old in a blue lab coat. A young woman in pistachio green hurried over.\n\n\"Are you throwing gravel on the car, Guillaume?\"\n\n\"Uh-huh.\"\n\n\"Why are you doing that? Are you trying to scratch it up?\"\n\nThe forty-year-old shrugged, turned on his heel, and stalked off angrily. The pistachio-green young woman addressed the new arrivals affably.\n\n\"Gentlemen?\"\n\n\"Hartog,\" said the redhead. \"I'm expected.\"\n\n\"For an admission?\"\n\n\"No, a release. Do I look crazy to you?\"\n\nThe young woman laughed. \"No more than the next person. You shouldn't use that word here. You might shock people.\"\n\n\"I like to shock.\"\n\n\"You might cause the residents distress.\"\n\n\"I'm not sure I don't want to cause them distress.\"\n\n\"What's that?\" asked the young woman, leaning forward, nonplussed by his syntax.\n\n\"Enough of this,\" said the redhead. \"I'm expected. At least I should be. I'm here to pick someone up.\"\n\n\"Take the steps,\" said the young woman, suddenly practical. \"Go on in. There's a receptionist in the hall. Would you excuse me please?\"\n\n\"Wait a second.\"\n\nThe redhead inspected the rear of the Lincoln, then straightened up.\n\n\"No damage. Why don't you stop them from throwing stones?\"\n\n\"Self-discipline. You wouldn't understand.\"\n\n\"You pathetic bitch.\"\n\nThe young woman flushed, and smiled.\n\n\"That'll do,\" added the redhead. \"Go away.\"\n\nShe went away, still flushed. The smile was gone.\n\n\"Stay in the car,\" the redhead told the driver. \"Keep them from throwing gravel. Clobber them if need be.\"\n\nThe driver sat sideways on his seat, letting his legs rest outside the car, as his boss went up the white steps and entered the manor. It was chilly in the lobby. The redhead shivered. The floor was tiled with marble. Trompe l'oeil glass doors lined the whole reception area. A dark man with Latin features sat behind a mahogany table reading the satirical paper _Charlie-Hebdo._\n\n\"Michel Hartog,\" said the redhead. \"I'm expected. Appointment with Doctor Rosenfeld.\"\n\n\"Ah yes. Please follow me.\"\n\nThe dark man rose, put his paper down on the table, and walked ahead of Hartog. He opened a genuine glass door and went to the end of a narrow padded hallway. He rang a bell by a cushioned white leather door.\n\n\"Come in,\" said the intercom.\n\nThe dark guy opened the door. \"Monsieur Hartog,\" he announced.\n\nHe stood back to let the redhead go in, closed the door behind him, and left.\n\nDoctor Rosenfeld came up to Hartog, his hand outstretched. The two were roughly the same height. Rosenfeld was beginning to go bald. He had a lively expression and wore a plaid tie and no vest.\n\n\"I'm delighted to see you,\" he said.\n\n\"Is the girl ready?\"\n\n\"Mademoiselle Ballanger is packing. She'll be here in a moment. I'll let her know right away.\"\n\nHe went back behind his desk and fiddled with an intercom. They were in the corner tower that overlooked the water. A breath of damp air wafted through the open window. Hartog went over to the casement and looked out.\n\n\"Monsieur Hartog is here,\" said Rosenfeld. \"If mademoiselle could come right away with her suitcase\u2014\"\n\nThe intercom buzzed. Rosenfeld nodded and broke the connection. He arched back in his office chair and contemplated Hartog, who was looking down at the water with an irritated expression. The doctor hunted in drawers, extracted a straight-stemmed pipe and some Jean Bart tobacco, and began loading the pipe. A half smile hovered on his lips. Hartog turned round briskly and faced him.\n\n\"I'll write you a check.\"\n\nThe doctor raised his eyebrows.\n\n\"A gift,\" said the redhead emphatically. \"A gift for your institution.\"\n\n\"Very well, if you wish. But it is not necessary.\"\n\n\"What you do is interesting.\"\n\n\"Anti-psychiatry, you mean?\"\n\n\"I don't know,\" said Hartog. \"I mean minding crazy people.\"\n\nRosenfeld grimaced, almost spoke, but changed his mind and lit his pipe. Hartog leant on the corner of the desk and made out a check for ten thousand francs. When he had finished writing, he held the check out to the doctor.\n\n\"This is a lot of money,\" said Rosenfeld.\n\n\"It's nothing to me,\" answered Hartog.\n\n# 2\n\nIN THE manor's community room, an audience of inmates sat on benches. Taking advantage of the staff's lack of attentiveness, the patients were passing around a bottle of Kiravi red wine and sipping it through a straw. On a stage, a dozen people were dancing about, singing and playing music enthusiastically. They had drums, an upright piano, a cornet, and a tenor saxophone.\n\n\"Oh, the thrill, the fearsome thrill of the first embrace!\" went the singers.\n\nSeveral spectators were clapping incessantly.\n\nIn Julie's bedroom the sounds of the concert were distantly audible, but the words were unintelligible.\n\nThe room was roughly cubiform, with pale green walls, a table, a chair, blinds, and a reproduction of a van Gogh painting of a wheat field. Julie was standing next to her luggage, a cardboard suitcase and a skai bag. She was a thin, tall girl with hollow cheeks and luxuriant black\u2014 very black\u2014hair. She had a porcelain complexion and brightly colored lips. She was beautiful, but in a startling way: she might have been taken for a transvestite. Her tweed suit was too warm for the season. Large tan hands protruded from her sleeves like bunches of beans hung up to dry.\n\nA nurse with a benevolent horsey face came in.\n\n\"He is here,\" she announced.\n\n\"Already?\"\n\n\"Why, aren't you happy about it?\"\n\n\"I'm scared.\"\n\n\"Come on, sweetheart. Don't you worry. He's supposed to be a nice man.\"\n\n\"Hmmm,\" said Julie.\n\nThe nurse picked up the suitcase and left the room. Julie picked up the skai bag and followed. The two women went outside and made for the corner tower. It was a fine day. It was spring. As they passed in front of the manor Julie looked at the Lincoln. The driver was reading the sports paper _L'\u00c9quipe._ He had put on tinted glasses. He returned Julie's gaze.\n\nThe two women entered the tower by a small side door. A few meters down a hallway they came to the white leather padding, the bell, and the intercom.\n\n\"Come in, come in.\"\n\nOnce in the office, Julie contemplated the redhead and was taken aback by his youthful, modish appearance. Rosenfeld had risen to his feet, his pipe clamped in his mouth, less jovial than usual.\n\n\"Michel Hartog, Julie Ballanger,\" he said by way of introduction.\n\nHartog scrutinized Julie.\n\n\"I suggest we talk as we drive. Let's go.\"\n\n\"What? Already?\"\n\n\"You might like to take a little stroll around the grounds,\" Rosenfeld put in. \"A chance to get acquainted. Julie is naturally overwhelmed by the idea of leaving us forever after living here for five years. In her shoes, I am sure you would be a little panicked.\"\n\n\"Sure, it would be perfectly awful for me,\" said Hartog. He turned to Julie. \"Come on, let's go.\"\n\n\"I could have refreshments brought out for you by the pond,\" Rosenfeld persisted, his voice trailing off.\n\nHartog did not deign to respond. Grabbing hold of Julie's suitcase, he held his hand out to the doctor. The doctor shook it weakly.\n\n\"Julie,\" began Rosenfeld, \"I hardly need tell you\u2014 \"\n\n\"Quite so,\" said Hartog rudely.\n\nHe grasped the young woman by the elbow and towed her towards the door.\n\n# 3\n\nAS THE Lincoln started off, Julie, in the back seat, turned round. Through the rear window she saw her analyst and the nurse, Madame C\u00e9cile, holding a Kleenex, both waving goodbye. The two figures grew smaller. The tires sang on the gravel. Then the Lincoln was back on the asphalt of the road, the manor lurched and disappeared, the car picked up speed, and the trees and rotting leaves raced by.\n\nJulie admired the automobile. It was like being in a boat. Genuine leather and mahogany adorned the interior. There were all kinds of recessed features behind the front seats. Julie ran her hand over the controls set into the leather.\n\n\"Look at this,\" said Hartog.\n\nHe opened the alcoves one by one, showing Julie a bar, a radio-telephone, a tiny television, and a miniature steno keyboard.\n\n\"It's not a magic car,\" said Hartog. \"It was made by people, you know.\"\n\n\"It's impressive all the same. I'm just a poor girl, aren't I?\"\n\nJulie tinkered with the electric latches. Her manipulation opened another compartment. Within it was a revolver, which she took to be a Colt. In fact it was a short-barreled Arminius, a German weapon. The sides of the grip were plastic, and it looked like a toy. Julie quickly closed the compartment. Hartog laughed.\n\n\"It's just for protection. Sorry, but I'm not a crime lord!\"\n\n\"No, you're a soap lord!\"\n\nThey both laughed.\n\n\"You didn't picture me like this, did you?\"\n\n\"I certainly didn't. I imagined a polite old gentleman.\"\n\n\"That's because of my reputation for doing good. Everyone thinks I'm gaga. Would you like a drink?\"\n\n\"I'm not allowed it.\"\n\n\"My ass you're not!\" cried Hartog, and Julie's eyebrows danced up and down.\n\nThe young man opened the minibar and fixed two Ballantine's on the rocks. He put one of them in Julie's hand.\n\n\"D\u00e9d\u00e9!\" he called. \"How about a drink?\"\n\n\"Sure thing,\" replied the driver.\n\nHartog passed him a whiskey, which the thickset man emptied as he drove. The Lincoln got onto the Autoroute de l'Ouest and immediately picked up speed. It hit 140 kph and stayed there, eating up the fast lane. The passengers were as snug as if they were in a railroad sleeping car.\n\n\"What do you think of me?\" Hartog asked. \"What do you know about me? Do you get the feeling you are in a fairy tale?\"\n\n\"I don't believe in fairy tales.\"\n\n\"Okay. But what then?\"\n\n\"You are a soap, oil, and detergent magnate. You are rich and you are a philanthropist.\"\n\n\"Let's not exaggerate.\"\n\n\"You do Good. You are probably trying to compensate for the feeling of being a usurper. Because your wealth is not the fruit of your own labor. Only the death of your brother and his wife made you the owner of it. You must have developed a strong sense of guilt, even if you had no wish for them to die. Anyway, one always wishes for the death of one's brother at some level.\"\n\n\"Congratulations!\" said Hartog in a toneless voice. \"Is that what they teach at the asylum?\"\n\n\"It's not an asylum. It's an open establishment. I could have left any time I wanted.\"\n\n\"But you stayed there for five years. Why?\"\n\n\"You've seen my records. You know why.\"\n\n# 4\n\nTHE LINCOLN followed the Seine. Portuguese workers in plastic hard hats were operating jackhammers. Hartog got out a pack of Gitanes and offered one to Julie, who accepted.\n\n\"You won't recognize much. In five years there has been a great deal of construction.\"\n\n\"So I've heard.\"\n\n\"Are you interested in city planning?\"\n\n\"Not particularly. What about you?\"\n\n\"A bit more than that.\"\n\nJulie smiled, blowing smoke through her nose.\n\n\"I know,\" she said, \"that you designed the Hartog Foundation building yourself.\"\n\n\"The Hans-Peter and Marguerite Hartog Foundation.\"\n\n\"Yes, but people call it the Hartog Foundation.\"\n\n\"But it's the Hans-Peter and Marguerite Hartog Foundation.\"\n\n\"Your brother and his wife.\"\n\nThe redhead nodded. The corners of his mouth were tense. Shreds of tobacco adhered to his lower lip. His cigarette was damp with saliva.\n\n\"You might say,\" he added, \"that I have made up for my wish for my brother's death, made up for it by constructing the Foundation.\"\n\n\"Are you interested in psychoanalysis?\"\n\n\"No more than you are in city planning.\"\n\nThe Lincoln crossed the Seine and made its way through Neuilly. On Rue de Longchamp it turned at a building adorned with a plastic fresco. Sliding doors opened automatically as the car approached. It nosed down a cement ramp into an underground garage where two Volkswagen minibuses, a Citro\u00ebn DS 21, and a Porsche were parked, and pulled into a demarcated slot.\n\n\"We're home,\" said Hartog. \"This whole building belongs to me.\"\n\nThe driver got out and went to open Hartog's door. Julie let herself out on the other side. She saw a broad-shouldered figure emerge from between the DS 21 and a minibus. It was a man in a white raincoat with epaulettes who came up to them with long rapid strides. He held a rolled-up magazine in his left hand.\n\n\"You shit! You piece of shit!\" he shouted.\n\nThe driver of the Lincoln turned briskly and flung himself at the man with fists raised. The other man struck him in the middle of the chest. Stunned, Julie heard the thud. Fists still raised, the driver staggered backward and fell onto the cement, his skull resounding as it hit the ground.\n\nHartog was getting back into the car. White Raincoat came over and slammed the door against his torso. Hartog let out a cry of pain. The man grabbed him by the collar of his safari jacket, jerked him out of the Lincoln, and threw him to the ground.\n\n\"Stop! Stop!\" shouted Julie.\n\nThe brute paid her no mind, took a run-up, and kicked Hartog in the ribs. Hartog groaned. The blood drained from his face. Julie got back in the Lincoln and opened the compartment with the revolver. She trained the weapon on White Raincoat through the car's open door.\n\n\"Stop or I'll kill you!\" she cried.\n\nThe guy looked at her. He had a flat face, a pug nose, and large, very pale gray eyes. On the top of his head the hair was beginning to thin. Yellowish strands lay there limply.\n\n\"The safety isn't even off,\" he said.\n\nWith a burst of laughter he whacked the revolver with his rolled-up magazine. The Arminius went skittering across the cement. The guy shrugged and made off with his long strides for the exit ramp. Julie quickly retrieved the revolver. She did not know whether to stop the brute or attend to Hartog, who was moaning on the ground. Before she could make up her mind, White Raincoat was past the doors, which closed automatically behind him.\n\n\"Help! Somebody help!\" called Julie.\n\nThe driver struggled uncertainly to his feet, clutching at his stomach.\n\n\"Call a doc,\" said Hartog. \"I have broken ribs, I can feel them. Don't touch me.\"\n\n\"The police,\" stammered Julie.\n\nShe was grinding her teeth from nervousness.\n\n\"No. Just my doctor. Quickly!\"\n\nBent double, wracked by nausea, the driver called the doctor on the car's radio-telephone. Julie, distraught, stayed by Hartog.\n\n\"I have to thank you, kid,\" said the injured man in a flat voice. \"Put the gun away, there's a good girl.\"\n\n\"This is dreadful.\"\n\n\"You haven't seen anything yet.\"\n\n# 5\n\nTHE BUILDING had five floors. While the doctor was seeing to Hartog, the driver shepherded the young woman through the place. The ground floor and the second floor housed empty offices and conference rooms with thick carpeting and teak furniture. Mondrians on the walls.\n\nThe elevator stopped at the third floor and its doors opened, but the driver pressed the button to close them again.\n\n\"That's the boss's quarters,\" he said. \"No going in there unless you're invited.\"\n\nThe elevator car went up another floor.\n\n\"This is where you'll be staying. We get out here. The top floor is for the snotty brat.\"\n\n\"Little Peter?\"\n\n\"Yes. The snotty brat.\"\n\n\"The whole floor is his?\"\n\nThe driver nodded.\n\n\"The boss will show you. Come and see your room.\"\n\nJulie followed the man down a windowless hallway. A white door opened into a room about ten by four meters. Blue carpeting, white walls. A very big window at the far end, overlooking Rue de Longchamp. A Pollock on the left wall, storage units along the right. Cupboards and shelves were covered by a fine-grained white plastic laminate. Likewise the bed, which had a red blanket. There was no bedspread. A white table and a white chair in the middle of the room completed the decor. Which left a good deal of empty space. Julie shivered and crossed her arms. The driver was now in the middle of the room with her suitcase. He turned round.\n\n\"My name is Andr\u00e9. What's yours?\"\n\n\"Julie. Julie Ballanger.\"\n\n\"Not related to that Communist politician, are you?\"\n\n\"No, I'm not related to anyone.\"\n\n\"I'm no longer married myself,\" said the driver. \"Would you care for a drink?\"\n\nHe put the suitcase down and went to open a low cabinet. Julie noted an assortment of bottles as well as a record player. The driver noticed her glance.\n\n\"For this kind of thing, you can't say the boss doesn't do it right. Quite a guy. What are you drinking? Another scotch?\"\n\nJulie nodded. The driver handed her the drink. He had poured himself a Ricard. He drank half of it and wiped his mouth with his sleeve.\n\n\"Physically, you are far better built than Old Polio.\"\n\n\"Old Polio?\"\n\n\"The nursemaid before you. Completely off her rocker. Fifty if she was a day. And an idiot. What about you? What's your thing?\"\n\n\"I don't understand at all,\" said Julie. \"My thing? What do you mean?\"\n\n\"The thing that's screwy with you.\"\n\n\"I'm cured,\" Julie stated.\n\n\"The hell you are!\" exclaimed the driver. \"The boss's way of doing good is over the top. He only hires retards. He sets up factories for cripples to work in, can you figure that?\"\n\n\"Not really.\"\n\n\"Those guys who go around in little motorized wheelchairs? He's got them working on a production line! In this house it's the same baloney. The cook is epileptic. The gardener has only one arm, pretty handy for using the shears. His private secretary is blind. His valet suffers from locomotor ataxia\u2014no wonder his meals arrive cold! The snotty brat's old nanny\u2014well, I told you about her. As for you, you must know yourself.\"\n\n\"What about you?\" asked Julie.\n\nShe had taken out a pack of Gauloises and a Criquet lighter. She lit a cigarette and, throwing her head back, blew smoke through her nostrils.\n\n\"What about you?\" she repeated.\n\nThe driver shrugged. \"I was a paratrooper. I landed on a mine. I can't walk properly anymore.\"\n\n\"What about the brute from before?\"\n\n\"He's something different,\" said the driver. \"A friend of the boss's.\"\n\n\"Some friend!\"\n\n\"Ex-friend then. At the beginning Hartog was a complete zero. As broke as the next man. He was in construction, you know, an architect, with the other idiot, Fuent\u00e8s, the joker who attacked us in the garage.\"\n\n\"What do you mean, 'with the other idiot'?\"\n\nThe driver was pouring himself a third drink. He went and sat down on the white chair at the white table.\n\n\"Nothing sexual. They worked together. They had an architectural partnership, as they call it. It wasn't working out, and then kaboom! Hartog's brother flew his plane into a palm tree. It was the brother and his wife who had all the dough. In a flash Hartog found himself guardian of the kid and owner of all that loot. He dumped the other idiot, Fuent\u00e8s, and Fuent\u00e8s has never forgiven him. He comes round every now and again to beat him up.\"\n\n\"Every now and again? What fun!\"\n\n\"If only we could knock him off.\" The driver sighed. \"It's the valet and I who make the nightly rounds, and we have a Colt. Twice, maybe three times, we had a chance to blow his head off, that Fuent\u00e8s. But Hartog wants none of it.\"\n\nJulie downed the rest of her scotch and shuddered. The driver gave her a friendly smile.\n\n\"Nervous?\" he inquired.\n\n# 6\n\nONCE THE driver had left, Julie settled in. The closets were far too large for the girl's few possessions, but when she opened them she found that they were already half full. There were clothes on hangers, others folded on shelves. Julie took a rapid inventory. The things were new and in her size. Looking around for a mirror, she discovered the bathroom, which was through an almost invisible door near the Pollock. Like the bedroom it was fully supplied: there was soap, a horsehair glove, toothpaste (three kinds), bath salts, etc. Even a leg-shaving kit. Julie clenched her teeth in exasperation.\n\nThere was a large mirror on the bathroom wall and the girl looked at herself in it. She tried on various outfits that appealed to her. She also looked at herself naked and did not like what she saw. She found herself boyish, built like a horse, her breasts too flat, her shoulders too muscular, her hips too narrow, and her waist not narrow enough. Her very dark hair, shoulder length, carefully done, and artificially curled at the ends, looked to her like a wig. In short, she thought she looked like a post-op transsexual.\n\nShe did not dare wear any of the new clothes, even though some of them were to her liking despite herself. She went and opened her suitcase and slipped into an out-of-fashion little black dress. Then she put all her own things away. Some of the clothes were five years old. She had no jewelry, not even costume jewelry. A toiletry bag went onto a glass shelf over the sink. A medicine bag was next: She looked in her striped Kelton and decided it was time for a Tofranil. She washed one down with a little scotch. Last, she put away her books\u2014a few crime novels, some cheap editions of works by Freud, and a set of small English guides to common birds, wildflowers, and the like.\n\nWhile arranging her effects, Julie found that next to the built-in record player were a good many LPs. Mozart, Bartok, New Orleans jazz. She turned the system's radio on. She went over to the window, had difficulty figuring out how to operate the pivot, but eventually succeeded and leant her elbows on the sill and surveyed Rue de Long-champ. The sidewalks exuded peace and affluence. Behind her the radio chattered. Claude Francois emoted vociferously. Then, to a violin accompaniment, a female voice sang the praises, in succession, of a vermifugal drug, rain tires, and a men's magazine. Julie went and turned the radio off and the record player on and played \"King Porter Stomp.\"\n\nReturning to the window, she got a shock. The evening now beginning was languid and hot. In the half darkness she thought she glimpsed a massive figure wearing a white raincoat with epaulettes. She leant out farther for a better view. \"King Porter Stomp\" had brought tears to her eyes. By the time she wiped them away, the figure had disappeared. Or else had been nothing more than an illusion, a wraith.\n\nA knock came at her door and Julie flinched. She went and opened the door. Hartog came in. His smile resembled the coin slot of a parking meter. He was wearing a white crewneck sweater beneath which the slight bulge of a bandage could be seen.\n\n\"You're up and about!\" exclaimed Julie.\n\n\"Why not?\"\n\n\"What about your ribs?\"\n\n\"Two are cracked. I'm bandaged tightly. Don't you worry. D\u00e9d\u00e9 has got you settled, I see. I came to make sure that all was well. And to tell you that we'll be eating dinner in a quarter of an hour. Ordinarily, you'll be eating separately with Peter, but on this first evening I wanted to chat with you a little. Come on down and have something to drink.\"\n\nThe living room on the third floor where they were soon ensconced was dotted with enormous brown leather armchairs. Julie accepted another glass.\n\n\"You must be concerned, I take it, to know why that savage attacked me earlier?\"\n\n\"No. Andr\u00e9 explained things to me.\"\n\n\"I see. Fuent\u00e8s is a bitter man\u2014a failure. But I hesitate to set the police on him. Are your teeth chattering?\"\n\n\"I am sort of allergic to the word p . . . p . . .\" Julie leant forward. \"Excuse me,\" she went on. \"I'm allergic to the word 'police.' \"\n\n\"How come? A trauma?\"\n\n\"I don't know. When I was six, the farmer's wife I'd been placed with had me locked up for an hour in the main police station to give me a healthy fear of authority. That's the only thing I have in common with Alfred Hitchcock. Afterwards I went into convulsions.\"\n\n\"I know that, of course. I've read your records.\"\n\nSilence.\n\n\"All right then,\" said Hartog. \"To get back to Fuent\u00e8s, I am reluctant to take action. After all, he's a childhood friend. We go back to the hardscrabble days. That makes for a bond. And, I have to say, he fascinates me.\"\n\nThe man gave a little laugh.\n\n\"That's only human, isn't it?\" said Julie. \"He is what you almost became. A failed architect.\"\n\nHartog's left shoulder jumped nervously. The redhead snickered again.\n\n\"Yes, okay, but he's not an architect anymore. He despises architecture. He works as a foreman, as a manual laborer.... We no longer know where he lives.\"\n\n\"A few minutes ago I thought I saw him down in the street.\"\n\n\"Quite possibly. He's on the prowl.\"\n\n\"How reassuring!\"\n\nHartog laughed\u2014a different laugh this time, more relaxed\u2014and offered Julie a Gitane, which he lit for her with a large jade desktop lighter. A nude woman in gold was inlaid in the stone and the tips of her breasts were two tiny rubies. The redhead got up.\n\n\"Let me show you Peter. The cook has given him his dinner. You can try and put him to bed.\"\n\nJulie put her glass down and followed him. The elevator took them to the top of the building.\n\n\"Andr\u00e9 explained to me how you employ only handicapped people,\" said Julie. \"So I understand better now.\"\n\n\"What do you understand better?\"\n\n\"Why you hired me.\"\n\n\"You\u2014you are different.\"\n\n\"How so?\"\n\n\"You need love,\" said Hartog. \"Like Peter.\"\n\nThe elevator stopped. The pair walked down a thickly carpeted, gloomy passage. A grayish light emanated from an open door. Peter was watching television.\n\nThe Hartog heir must have been six or seven years old. He was redheaded and freckled like his uncle, his body plump and soft. He was transfixed by the television set, which was broadcasting a report on famine in Asia.\n\n\"Go right ahead,\" said Hartog. \"He knows that you have arrived, that you are replacing Marcelle.\"\n\nJulie entered the room.\n\n\"I am Julie.\"\n\nPeter looked her over then turned back to the screen.\n\n\"Put him to bed,\" said Hartog.\n\nJulie moved farther in.\n\n\"Come on now,\" she said in a tone that came out wrong, \"it's time to go beddy-byes.\"\n\nPeter was in red flannel pajamas. Julie took his hand. He withdrew it violently. Julie took it once more.\n\n\"Come on, come on.\"\n\nPeter went limp.\n\n\"Come on now, get up!\"\n\nPeter stayed limp. Pulling him by the arm, she dragged the boy along. He let himself be dragged. With his other hand, which was trailing along the floor, he grabbed a wooden dog.\n\n\"Come on, Peter,\" said Julie. \"Stand up!\"\n\nThe boy came to his feet. His free arm described a quadrant in the air and brought the wooden dog down on the bridge of Julie's nose. The blow made a dull sound. The girl's eyes filled with tears. She let go of the little boy and staggered. She held both hands to her nose. Tears streamed down her cheeks.\n\nPeter, suddenly panic-stricken, threw his arms around Julie's waist and began kissing her side noisily, desperately. He seized her hand and kissed that. He said not a word.\n\n\"It's okay,\" said Julie.\n\nHer voice was a snuffle; her nostrils were clogged with blood. She contemplated Peter. In this house of defectives, she was almost surprised that he did not have webbed feet.\n\n\"You hurt me,\" she said. \"Really hurt me. But let's start over. I want to be your friend. Tomorrow we'll get to know each other. Right now it's time for little boys to go beddy-bye. Agreed?\"\n\n\"What about my TV?\"\n\n\"Enough with the TV! It's time to sleep. No more TV!\"\n\nPeter grabbed the wooden dog and hurled it into the television. The cathode-ray tube imploded. Shards of hot glass, valves, transistors, and scraps of metal and plastic sprayed noisily in all directions. The tuner ricocheted off the wall.\n\n\"Kaya!\" yelled Peter at the top of his lungs.\n\nJulie let him have a stiff-armed, resounding slap across the face. The boy was flung against the wall. He rebounded, regained his balance, and stood, as it were, to attention, fists clenched and eyes closed. His eyelids trembled. Julie was horrified at having struck him so hard. She shot a sideways glance at Hartog, who was crossing the room to unplug the demolished television set. The redhead was unperturbed.\n\n\"Okay,\" said Julie to Peter. \"I hit you. You hit me first. We'll start from scratch again tomorrow. Agreed?\"\n\n\"Agreed, agreed, agreed, agreed!\" cried Peter. \"Stop asking me if I agree!\"\n\nHe climbed into his bed and buried himself under the covers. Hartog put a hand on Julie's shoulder.\n\n\"Come and have dinner.\"\n\n# 7\n\nTHE TELEPHONE woke Julie. As she picked up the receiver, she glanced at her watch. Six thirty-five. Her head ached; her mouth was dry.\n\n\"Am I waking you up?\" came Hartog's voice.\n\n\"Yes.\"\n\n\"Would you be so good as to come down to my office?\"\n\n\"Where is it?\"\n\n\"Ground floor, door K. I'll expect you. There's coffee.\"\n\n\"Fine.\"\n\n\"You have ten minutes,\" said the phone.\n\nThey hung up. Julie disentangled herself from the sheets, almost fell, and sat on the edge of the bed. She had a terrible hangover. She rubbed her eyes with her fists.\n\nIn the bathroom, the fluorescent tube over the sink shed a nasty oyster-hued light on her. Julie brushed her teeth, untangled her hair, and swallowed two Tofranils. No time to take a shower. She made herself up cursorily and went back into her room. Her little Hermes Baby typewriter was open on the table with a sheet of paper inserted in it. Julie leant over and read:\n\nNEUILLY, 5 JUNE\n\nDoctor Y. Rosenfeld\n\nCh\u00e2teau des Bauges\n\n78\u2014Gouzy\n\nDear Doctor,\n\nI realize that I tried to put off leaving because . . .\n\n\"Oh my God!\" Julie muttered. \"You must have been completely plastered!\"\n\nShe tore the paper from the machine, rolled it into a ball, and tossed it into the cylindrical aluminum wastepaper basket. She opened the closet and slipped into black pants and a yellow blouse.\n\n\"You are going to get yourself canned,\" she remarked.\n\nShe went and took the elevator. On the ground floor she had no difficulty finding door K. (The K, in gilded metal, stood out in relief in the middle.) Julie knocked. She heard Hartog's voice within.\n\n\"Come in!\"\n\nThe girl complied, closing the door behind her. The office was square, all white, with a white table cluttered with papers and files, a white chair, and two white leather armchairs and a matching reclining couch. Hartog was seated on the edge of the couch, talking on a telephone handset with a built-in dial and a long coiled cord. The redhead was unshaven. He wore a white nylon dressing gown over black-and-blue pajamas. He was smoking. A standing metal ashtray was beside him.\n\n\"I don't give a damn about Goujon's project,\" he shouted into the phone. \"I've told you what I want. I've sent you a sketch. That's not enough? Shit!\"\n\nJulie hovered in the middle of the pale gray carpet. Hartog gestured for her to sit down. He dropped ash on the floor.\n\n\"He can stick it up his ass!\" he yelled into the phone. \"Walkways between the workers' housing units, freeway up to the site. Costs? What costs? The costs are my business!\"\n\nJulie heard not _co\u00fbts_ (costs) but _coups_ (hits), and she understood strictly nothing.\n\n\"Good, that's better. Call me the day after tomorrow in Munich.\"\n\nHe rang off without saying goodbye and turned towards Julie. The redhead's face was glistening. Sweat pearled at the roots of the seriously thinning hair on his liver-spotted cranium. He lit a Gitane from the butt of his last one.\n\n\"I asked for coffee! Where in the hell is my coffee?\"\n\nA knock at the door.\n\n\"Come in!\"\n\nThe valet came in with coffee on a white tray.\n\n\"It's high time, Georges,\" grumbled the redhead.\n\n\"I had to make it myself,\" said Georges in a rebellious tone of voice.\n\n\"How come? I pay a cook, don't I?\"\n\n\"Madame Boudiou is not at all well,\" answered Georges. \"She's had a seizure. Almost swallowed her tongue again.\"\n\n\"Put that down here and get out!\"\n\n\"Yes, sir.\"\n\nGeorges left the room. Hartog got to his feet and wiped his face with a black handkerchief. He went over to a side door and disappeared, though he remained within hearing distance.\n\n\"You were drunk last night,\" he shouted.\n\n\"Possibly,\" said Julie in a sullen tone. \"I can't remember anything.\"\n\n\"Alcohol and tranquilizers, huh?\" said the invisible millionaire in an oddly cheerful tone. \"Better not make a habit of it. Not while you're working.\"\n\n\"I wasn't working.\"\n\n\"Okay. Pour the coffee, there's a good girl.\"\n\nJulie served the coffee. Hartog reappeared wearing pants, his torso and feet bare. He was bandaged from waist to pectorals. He was holding a battery-powered electric shaver. He swatted the telephone.\n\n\"What a bunch of assholes!\"\n\nHe rummaged through the papers piled on the table, extracted a large watercolor drawing, and spread it out noisily in front of Julie.\n\n\"Look. I was the one who made this plan. Can you read a city planner's drawing?\"\n\n\"No.\"\n\nThe redhead looked crestfallen.\n\n\"Well, screw it.\" He sighed. \"I know it's good.\"\n\n\"Did you get me up at half past six in the morning just to show me your little drawings?\"\n\nHartog took a swallow of coffee. He looked at Julie curiously.\n\n\"Quite the little rebel,\" he observed. \"I know all about you. Pickpocket. Arsonist. Congratulations.\"\n\n\"Of course you do,\" replied Julie. \"It's all in my file.\"\n\n\"You, all you poor people, are just too stupid. You go about things in the dumbest way.\"\n\n\"Everyone can't inherit money.\"\n\nHartog shrugged.\n\n\"For my part I do something with my inheritance. You people wouldn't know what to do with one. You, Fuent\u00e8s and company, people like you. What I do is create a work.\"\n\n\"It's all about money,\" said Julie. \"Money and little drawings.\"\n\n\"Little drawings, little drawings,\" Hartog repeated vaguely.\n\nThe lower part of his face was rigid. Pulling himself together, he took a three-part folder from his desk and shook it, causing a stream of 21 cm x 27 cm photographs to cascade out.\n\n\"A work\u2014goddammit, a work!\"\n\nHe was swearing but seemed calm. He struck his sternum with the flat of his hand. Perspiration trickled abnormally, obscenely down his hairless white body. The redhead stood up and turned his shaver on. Julie examined the photos distractedly. Houses. Buildings. On the back of each were noted place and date. Julie finished her coffee. Hartog switched his shaver off.\n\n\"Don't you admire my work?\"\n\n\"With lots of cash you can do whatever you want.\"\n\n\"I create beauty,\" said Hartog.\n\nJulie did not bother to reply. The redhead put his shaver down without wiping it.\n\n\"I have some jobs for you,\" he said in a suddenly altered tone of voice. \"I'm taking a plane at eight o'clock. I'll be gone for three days. You'll have to manage without me. Mademoiselle Boyd, my secretary, will be here. You can call on her if you need funds.\"\n\nJulie nodded absently. She was looking at a photograph.\n\n\"This one I like,\" she murmured. \"Definitely.\"\n\nHartog took the photo from her hands and scrutinized it. It showed an almost chaotic group of low buildings spread across a mountain crest. It looked as if ruins from various places had been assembled, linked up by more recent masonry, and added to over the years by fresh disordered elements. Between drystone walls and slate platforms were slung rickety catwalks, and here and there were sugarloaf structures never more than three or four meters tall. Vegetation flourished in patios, crannies, and roof corners.\n\nA violent flush darkened Hartog's features and spread visibly down his narrow torso as far as the bandage. The redhead scratched his scalp in agitation.\n\n\"You do have your nutty side,\" observed Julie benevolently. \"All this stuff here is your work, isn't it?\"\n\nHartog nodded. \"My folly.\"\n\n\"Do people like it?\"\n\n\"My folly,\" the redhead said again. \"Folly\u2014meaning, as you well know, a place of pleasure and fantasy. An aerie . . .\"\n\nHe seemed to regain his self-confidence as he talked.\n\n\"This is where I gave full rein to my imagination,\" he said with satisfaction. \"Perhaps it was even a bit childish. But every man needs a place to be alone, to get away from himself. The place is more restful for me than a Trappist monastery.\"\n\nA drop of perspiration fell from Hartog's round forehead onto the glossy print. He tossed the picture towards the table and turned away. The photo fluttered, hovered, and fell into Julie's lap.\n\n\"Would you like a drink?\" asked Hartog, his back still turned.\n\n\"At this hour?\"\n\n\"I'm having one. Feel free to help yourself.\"\n\nLeaning over a built-in panel, he was opening a minibar. The house was full of bars. A drinker's paradise.\n\n\"Goodbye,\" said Hartog without turning around.\n\nStiff-backed, glass in hand, he left through the side door, and Julie hesitated for a moment before pouring herself a brandy, which she downed, standing, in a single gulp, reminded of a time when, freezing cold at dawn, she would stand at a bar and wash down black coffee with four shots of calvados at the start of a day of wandering, tears, fatigue, and despair.\n\n# 8\n\nAFTER Hartog left, Julie went to the kitchen to get Peter's breakfast. There she found the valet Georges eating a sloppy omelet at one corner of the immense table. The fellow was in suspenders. His eyes were bloodshot. He got to his feet as Julie approached.\n\n\"Madame Boudiou is doing better,\" he offered, his mouth full. \"I'll get you something for the snotty brat.\"\n\nEgg yolk dribbled down his chin.\n\n\"Don't go to the trouble,\" said Julie. \"I'll manage.\"\n\nShe piled little boxes of cereal, which looked like little boxes of laundry detergent, on a tray. She could hear the valet chewing behind her.\n\n\"While you're at it, would you open me a bottle of Guinness?\"\n\nJulie complied silently, placing the beer and a glass in front of Georges. The valet drank greedily and it seemed to make him feel better. Color returned to his cheeks.\n\n\"Is it true what they say, that you were in the nuthouse?\"\n\n\"It's true.\"\n\nGeorges was embarrassed. \"Before that, you were a domestic?\"\n\n\"Before that, I was a juvenile delinquent.\"\n\nGeorges was even more embarrassed.\n\n\"Would you excuse me please,\" said Julie.\n\nShe left the room with her tray, taking the service elevator. Clamped between her elbow and her side was the photograph of Hartog's folly. She was not conscious of having kept it.\n\nThe girl entered Peter's room. The little boy was awake, playing with toy cars. He consulted an electric watch.\n\n\"You're late.\"\n\n\"Good morning,\" said Julie.\n\n\"Where is Marcelle? Where is Old Polio?\"\n\n\"She has left. I'm her replacement.\"\n\nPeter shrugged. \"You're late,\" he repeated.\n\n\"Your uncle is going away. I had to say goodbye to him.\"\n\n\"He never comes to see me.\"\n\nJulie put the tray down on the table and poured hot milk over instant cocoa and cold milk into a bowl.\n\n\"That's because he is very busy.\"\n\n\"No, it's because he doesn't love me. Nobody loves me, except for Marcelle. She told me so herself.\"\n\n\"She was wrong,\" replied Julie flatly.\n\nPeter made no reply and went and sat at the table. He poured all kinds of cereal into the bowl of milk and began eating greedily. A knock came at the door. Julie opened the door. In the hallway was a very tall baby-faced man, very blond, wearing a blue suit.\n\n\"\u2014oiselle,\" he was saying, \"I have the television set.\" He gestured towards a large cardboard box beside him on the floor.\n\n\"Who are you?\" asked Julie, taken aback.\n\n\"I've brought the television. Isn't this the place?\" He consulted a slip of paper. \"Downstairs they told me it was here.\"\n\n\"A TV! A TV!\" cried Peter, jumping up and down.\n\n\"Yes, it's definitely here,\" said Julie. \"But I wasn't expecting you.\"\n\nThe blond giant brought the large box into the room. Immediately nervous, Julie went over to open the window, raising the blind.\n\n\"Where should I set it up?\"\n\nJulie, stock-still, was gazing down at the street, at a white raincoat with epaulettes. A bus moved forward, blocking her view of it.\n\n\"Mademoiselle?\"\n\nThe bus passed on. No more white raincoat. Julie turned.\n\n\"Mademoiselle. Where should I set it up?\"\n\n\"Anywhere you like. On the floor. Over there. Near the socket for the antenna.\"\n\nJulie was jittery. The blond giant was opening the cardboard box. He was in no hurry. Julie stroked Peter's fine\u2014 too fine\u2014hair.\n\n\"You see, he does love you, your uncle,\" she said distractedly. \"This new television is his way of saying goodbye.\"\n\n\"Anything I break,\" said Peter, \"they give me a new one.\"\n\n# 9\n\nAFTER breakfast Julie decided to take Peter to the Jardin du Luxembourg.\n\n\"I don't want to go,\" said the boy.\n\n\"Do as I say! Get dressed.\"\n\nJulie stamped her foot. Peter shrugged.\n\n\"Is it true, what Marcelle says, that you were in the nuthouse?\"\n\nJulie's white face got even whiter. Her violet eyes flashed. She took a step toward Peter. The little boy jumped back. He looked at her in alarm. Julie turned on her heel and left the room. A bright flush crept up her throat. In the elevator she pounded the sides with her fist. She went down the hallway and into her room. She crossed it, zigzagging and stumbling and beginning to cry. She was seeing red. She had a photograph in her hand. She tossed it onto the table. She punched the table. She tore off her yellow blouse and her pants. In panties and bra, she circled the room, staying close to the wall and rubbing her head against it. Her tears fell onto her toes. She flung open cabinets. Getting dressed again did nothing to calm her.\n\nShe slipped into smoky-gray tights, pistachio-green shorts, and a sort of T-shirt, long and orange. She strode up and down the room, the muscles of her legs rippling superbly. She went to look at herself in the bathroom mirror for consolation.\n\n\"Fuck the whole lot of you!\" she declared.\n\nMockingly, the bathroom tiles resounded with hate. Julie took four Tofranils and washed them down with fifteen centiliters of scotch. She shivered.\n\n\"Brrr!\"\n\nShe went for her yellow suede handbag, which she had thrown onto the table. Beneath it was the photograph. Julie looked at it. What a beautiful place! A labyrinth. A house to get away from oneself in. Julie turned the photo over. On the back, written with a nylon-tip marker: \"Moorish Tower. Canton of Olliergues. Massif Central, 1967.\" The girl stuffed the photo into her bag. Her teeth were chattering. The inevitable adjustment crisis. Relax. Drop a line to Doctor Rosenfeld. Julie looked around for her Hermes Baby. Nowhere to be seen. Impossible to find it. Impossible to remember where the hell she stuck it. Damn it then! Julie went to the door. Go for a walk.\n\n# 10\n\nIT WAS ten in the morning when Julie and Peter arrived by taxi at the Jardin du Luxembourg. The girl's anger was gently congealing in her mind.\n\n\"I've never been here before,\" said Peter.\n\n\"You must have been.\"\n\n\"No, never. Marcelle used to take me to the Bois de Boulogne.\"\n\nThe boy walked along hunched forward, hands in the pockets of his velvet jeans, eyes glued to the pebbles on the path, which he kept kicking. He looked like a furious little old man.\n\n\"What are we going to do?\"\n\n\"Walk. Talk.\"\n\n\"Rats!\"\n\nJulie sighed. \"You don't like me much, do you?\"\n\n\"I liked Marcelle better.\"\n\n\"There's some kid stuff over that way,\" said the girl. \"I'll take you.\"\n\nShe made for the western edge of the park, with its tennis courts, boule pitches, children's rides, merry-go-round, and Punch-and-Judy show. Along the way Julie and Peter passed students, mothers, old folks. Overwhelmed by the atmosphere, the colors, the sounds, the girl opened her eyes and nostrils wide.\n\n\"I'll take you to lots of places,\" she said in a conciliatory tone.\n\nPeter could not have cared less. He climbed unenthusiastically onto the merry-go-round and allowed himself to have a strap fastened around him. He sat astride a large wooden lion. Distractedly, he fingered the beast's fangs. The ride started up. Peter gazed off into space.\n\n\"Rats it is then!\" muttered Julie.\n\nShe sat on a nearby bench. Before getting the cab, she had bought a copy of _Vogue._ She thumbed through the magazine. Sumptuous long-limbed women floating in fabulous fabrics. What expressions they had! What hair! What jawlines! What legs! What ecstasy! If only I could be a model, thought Julie. The pages of _Vogue_ dripped with luxury. The girl wiped her nose with a Kleenex.\n\n\"Hey!\" came a voice from close by.\n\nJulie started, uncrossing her legs. She cast a haughty look at the young man with a long nose, a smile, blue eyes, and a thick mop of brown hair who was leaning towards her holding a copy of _Le Monde._ He looked like a permanent student with his navy-blue pea jacket, jeans, and espadrilles. Grasping the back of the bench, he bent even closer to the girl.\n\n\"Be careful, don't scream\u2014it'll be better for you. See this?\"\n\nHe unfolded his newspaper. In his hand was an MAB Model C automatic.\n\n\"That's a toy!\" exclaimed Julie.\n\n\"Don't you believe it. And don't mess up. We are here to take the kid. There are two other armed guys around you. If you mess up, you're dead. And the tiny tots over there will catch the stray rounds. So you see, it's not worth it.\"\n\n\"You seem quite serious.\"\n\n\"You bet. This is a kidnapping. You're going to call Peter Hartog over here just as soon as the merry-go-round stops. And do it fast.\"\n\nStraining her neck, Julie looked all around. She saw nothing but seemingly harmless passersby.\n\n\"Don't try anything,\" said the brown-haired young man patiently. \"What's in that for you? We won't hurt you. We're going to let you go later, with a letter. Hartog will pay up. We'll return the kid. Everybody will be happy. Hey, the merry-go-round is slowing down. Call him now! Do it! Don't think about it.\"\n\nSweat greased the young man's upper lip. A tic animated the corner of an eyelid.\n\n\"Take it easy,\" said Julie. \"I'll do what you say.\"\n\nThe merry-go-round was stopping.\n\n\"Peter!\" called the girl. \"Peter! Come here!\"\n\nPeter shook his head and clung to his lion.\n\n\"He's not going to listen to me. Let's go and get him.\"\n\n\"Okay.\"\n\nShe got to her feet. The young man took her arm. The two of them entered the fenced-off enclosure around the merry-go-round.\n\n\"I want to stay!\" cried Peter.\n\n\"No arguments.\"\n\nJulie undid his strap. The boy got off grudgingly.\n\n\"We're all going for a walk,\" the brown-haired young man told him. \"My name is Bibi\u2014like Bibi Fricotin, it's easy to remember.\"\n\n\"Bibi Fricotin? Who's that?\"\n\n\"Come on, let's walk, and I'll tell you.\"\n\nBibi took Peter's hand and pushed Julie ahead. They headed straight for the park's Rue Vavin exit. No sooner were they past the railings than a blue Renault 16 pulled up at the curbside. Bibi opened the rear door and got in first, pulling Peter behind him. Julie hesitated for a split second then felt herself pushed forward and into the car. Someone got in after her and slammed the door. It was the blond giant who had delivered the television that morning.\n\n\"You? . . . You!\" stammered Julie.\n\nShe was half sitting on the floor. The blond giant took her arm and hoisted her onto the seat. His large blue eyes had barely any lashes. The R 16 sped down the side of the Jardin du Luxembourg, turned left, snaked through a few small streets, emerged onto Boulevard Raspail, and headed for Denfert-Rochereau. Julie was terrified, and her stomach growled. Cramps doubled her over. Bibi and the blond giant lit cigarettes.\n\n\"Let me have one please,\" Julie mumbled.\n\nBibi passed her the one he had just lit, a Craven Filtre. A caramel smell invaded the car. The blond giant opened his window.\n\n\"Close that!\" yelled the driver without turning round.\n\nThe giant obeyed. Of the driver Julie could see no more than the shaven, reddish nape of his neck below a green, vaguely Tyrolean hat.\n\n\"You won't get away with this,\" said the girl. \"It's not too late to let us out.\"\n\n\"Shut up.\"\n\nPeter was looking at Julie anxiously. He dared not speak. His eyes were moist with tears held back, tears of sheer fright. The girl stroked his hair.\n\nThe R 16 passed the Lion de Belfort and headed down towards the Porte d'Orl\u00e9ans. Avenue du G\u00e9n\u00e9ral Leclerc was clogged. The car crawled along, stopping for half a dozen red lights. They saw traffic cops. Bibi and the blond giant were continually looking sharply in every direction. Bibi kept his pistol in his lap and his hand on the butt. The driver turned the car radio on. From it came first jazz, then Indian flute music, and then a Viennese waltz. The R 16 exited at Porte d'Orl\u00e9ans, got onto the Autoroute du Sud, threaded its way through the tunnel, and reemerged into the open air. Its speed increased to 120 kph.\n\n\"Take it slow, N\u00e9nesse,\" warned the blond giant.\n\n\"I know my job, okay?\"\n\n\"You can't know when some asshole might slam into you.\"\n\n\"Let 'em try! Right now, you just shut up and let me do the driving.\"\n\n\"At the tollbooth,\" Bibi said quietly to Julie, \"you'd better behave, or the kid will get it.\"\n\n\"Okay, okay,\" Julie replied. \"Relax.\"\n\nShe was breathing through her mouth. She was calm for now. She could not quite believe the evidence of her senses.\n\n# 11\n\nTHE HOUSE was not much of a house. It was a sort of chalet standing on the sandy bottom of a shallow valley among tall pines. The exterior walls were constructed of varnished logs. Inside, however, the walls of the sole room were smooth: plasterboard covered with enamel paint. There was a small cooking area in one corner. In another, a partition, its base ten centimeters from the floor and its top ten centimeters from the ceiling, enclosed a narrow cubicle containing a shower and a chemical toilet.\n\nThe valley's flanks were covered with heather, with patches of bare sandstone and a profusion of pine and birch. The slope was steep. Though less than a hundred kilometers from Paris, the place was remote and hard to reach.\n\nAt its deepest, the valley was some fifty meters wide and two hundred meters long. To the south it came to a dead end; to the north was a kind of notch, a sandy gulley strewn with outcrops of sandstone. Thompson was sitting on the chalet's steps eyeing the mouth of the gulley.\n\nHe was a man of around fifty with a British look about him. His dark face was shaped like a Vienna sausage. His hair looked like pieces of straw crudely stuck to his cranium, and his little mustache was likewise snaggy. His eyes were blue. In Thompson's lap was a Sauer-made Weatherby rifle with a nine-lug bolt. He was wearing a taupe sports suit and a beige turtleneck. He listened to the sound of the car's motor as it came nearer.\n\nThe R 16 powered into the gulley, slithered briefly in the sand, then picked up speed on the valley floor carpeted with pine needles. It drew up in a clump of trees in front of the chalet. The doors opened and the occupants got out.\n\n\"Where is Fuent\u00e8s?\"\n\n\"Who is Fuent\u00e8s?\"\n\n\"She's got this idea in her head,\" said Bibi. \"She absolutely wants us to have been sent by some guy called Fuent\u00e8s.\"\n\n\"Not so, I assure you,\" said Thompson flatly, getting to his feet.\n\nHe was very tall and loose-limbed. Julie took in the surroundings. The R 16 had left the autoroute at Nemours then taken a host of detours via tracks and little roads through the woods. The girl was completely disoriented.\n\n\"Do come in,\" said Thompson. \"Bring the boy.\"\n\n\"You're gangsters!\" cried Peter.\n\n\"I won't deny that.\"\n\nThey went into the chalet. There were four windows and two pairs of bunks, upper and lower. In the middle of the room were a table and folding stools.\n\n\"Sit down,\" Thompson ordered. \"I've made coffee. The kid must be thirsty too. He can have a glass of water.\"\n\n\"I was supposed to be let go with a letter,\" said Julie.\n\nThompson smiled. \"That was a simplification. You'll have to wait.\"\n\n\"For how long?\"\n\n\"You'll see.\"\n\n\"What's your gun?\" asked Peter. \"A Winchester?\"\n\nThompson made no reply and went and put his weapon down on a bunk. Then he got the coffeepot and four enameled tin mugs from the cooking area. They sat down at the table. The blond giant remained standing near the door, arms crossed. He looked like the bouncer in a bar. His eyes were porcine and imbecilic. The driver of the R 16 resembled him closely. The same pink piggy face; the same little eyes embedded deeply in the flesh. He sat down at the table and kept his hat on.\n\n\"So you're going to keep me prisoner here?\" asked Julie.\n\nNobody answered. Thompson poured coffee for everyone. He went for the sugar and brought Peter a glass of water.\n\n\"You dirty bastards!\" said Julie.\n\n\"Can it!\" shouted the driver.\n\n\"Let's not get excited,\" said Thompson. \"This young woman is not wrong. But who can pride themselves on not being dirty bastards? Nonentities? Not even.\"\n\n\"I know the story,\" said Julie. \"In a world of wolves, etc., etc.\"\n\nThompson sat down and laid his forearms flat on the table.\n\n\"But it's so true,\" he said sanctimoniously. \"In a world of wolves\u2014\"\n\nJulie flung her hot coffee in his face and raced for the door. Peter dashed after her. The blond giant caught the girl with a left to the chin and she fell. Peter began to yell and went for the rifle. As the boy passed, Thompson, without getting up from the table, grabbed him by the hair.\n\n\"Don't move. Don't move or I'll hurt you.\"\n\nHe had a good handful of red hair and he twisted it. He was dripping with coffee. With his left hand he got out a white handkerchief and dabbed at his face.\n\nMeanwhile Bibi and the driver had pinned Julie's arms. They dragged her to the wall and backed her up against it. The blond giant grunted and punched the girl in the stomach. She howled.\n\n\"Stop!\" yelled Thompson. \"No marks on her\u2014she must not be marked.\"\n\n\"But she hasn't got it yet,\" raged the giant.\n\n\"She'll get it,\" said Thompson. \"Look at this, mademoiselle.\"\n\nSpread-eagled between the two men holding her, Julie was convulsed with pain and trying vainly to bring her knees up against her belly. She was producing little staccato hoarse moans. Thompson, still holding Peter by the hair, lifted the boy off his feet.\n\n\"Look over here, mademoiselle, be reasonable.\"\n\nThe child wriggled and shouted. He began to cry as hard as he could.\n\n\"Stop it!\" Julie pleaded.\n\nIt was hard for her to keep her head straight. Hair was falling into her eyes.\n\n\"Do you get it?\" \"Stop it, for God's sake!\"\n\n\"Do you get it?\"\n\n\"I get it!\"\n\nThompson put Peter down without letting go of his hair.\n\n\"I can do worse,\" he observed. \"You had better both behave from now on.\"\n\n\"We'll behave, you son of a bitch.\"\n\nJulie was released. She rushed to Peter and took him in her arms. Thompson surrendered the little boy to her. He was crying his eyes out and drooling, and his face was scarlet. The man wiped his sweaty hands with his handkerchief.\n\n\"I loathe this sort of thing,\" he said.\n\n# 12\n\nN\u00c9NESSE, the driver, took charge of cooking the day's two meals. For lunch, steaks grilled with fines herbes followed by cheese. For dinner, fried sardines. The gangsters had brought ten liters of Corbi\u00e8res with them. They drank in moderation. Thompson stuck to water. Before each meal he swallowed two black capsules.\n\nDuring lunch, Julie kept asking for more wine, and after a while Thompson stopped giving her the chance to ask. Instead he refilled her glass generously as soon as it seemed to be getting empty.\n\nFrom the floor came the drone of a little transistor radio. In the news headlines there was no mention of either Peter or Julie.\n\nThe girl's head began to droop. Thompson cleared the table, throwing the cardboard plates into a plastic bag. Julie lay down on a bunk. Her head hurt and so did her belly. She tried vaguely to distract the boy with a game of Chinese portraits. Peter's voice trembled as he played, and his eyes were red. He stretched out alongside Julie.\n\n\"When you threw your coffee,\" he asked, \"were you going to get away with me or on your own?\"\n\nJulie shook her head. \"I don't know.\"\n\nPeter snuggled closer to her.\n\n\"I like you better than Marcelle,\" he whispered.\n\nThe blond giant had left the chalet in a leisurely way. Julie surmised that he was going the rounds or mounting a watch on some hill. At the top of the valley's left flank a jumble of rocks rose above the trees and could easily offer cover to a lookout while affording him a long view.\n\nN\u00e9nesse and Bibi were throwing dice on the front steps of the house. They could be heard keeping score in low, unexcited tones.\n\nThompson was lying on an upper bunk, the rifle alongside him between his body and the wall. He was gazing at the ceiling. His hands were crossed on his stomach. He was experiencing quiet little burps.\n\nAn ulcer?\n\n\"What are we waiting for?\" Julie asked at length.\n\nThompson sat on the edge of his bed, letting his long thin legs dangle. He had to lower his head so as not to bang it on the ceiling. His eyes were red. Coffee stains were visible on his jacket and the front of his sweater.\n\n\"Oh yes, I have a letter for you to sign.\"\n\n\"The letter?\"\n\nThompson lowered himself clumsily to the floor. He bent down and reached under the lower bunk for a worn brown leather briefcase. He rooted in it and withdrew a typewritten letter, which he laid out on the table.\n\n\"Come and sign.\"\n\n\"Let me read it at least!\"\n\n\"As you wish.\"\n\nJulie sat at the table. The letter was addressed to Hartog and crudely typed. The girl read:\n\nMonsieur Hartog, I am writing to tell you that I have little Peter with me. I took him away on the spur of the moment, on an impulse. Now that I have calmed down, it would be easy enough to come back with the boy. Too easy, in fact. I can now see things clearly at last. I have had it up to here with all the humiliation that you cause me, you and others like you. Death to the pigs!!! Death to the rich!!! I have just as much right as you to get enjoyment out of life. I want money. If you call the police, I will hang Peter with a rope. Or else I'll cut his little body up every which way with my knife. So you had better follow my orders, keep your mouth shut, and wait for my next letter for me to tell you how to deliver the money to me. Get a million new francs together in small bills. No need to worry about Peter so long as you are sensible.\n\nJulie looked at Thompson. She dug her fingernails into the wood of the table. A nervous twitch pulled at the edges of her mouth and her teeth were bared in a ghastly grin.\n\n\"You might as well laugh it off, mademoiselle, I assure you.\" Thompson retreated towards the bunk where he had left his weapon. \"You're bound to be found innocent in the end.\"\n\n\"But this letter!\" Julie managed to say. \"This letter!\"\n\n\"It's the letter of a madwoman, granted. We know your history. But I'm sure you understand that Hartog must be reduced to a state of terror. Do you understand, or are you really balmy?\"\n\n\"I'd like another large glass of wine.\"\n\nThompson nodded. With his rifle under his arm, he went over to the kitchen area, poured wine into a tin mug, and held it out to Julie. She drank it down.\n\n\"You really expect me to sign that?\"\n\nThompson opened Julie's bag, which was on the bunk near Peter. He foraged in it for a moment and found a ballpoint pen.\n\n\"You have no choice,\" he told her. \"You don't want me to start pulling the lad's hair again, do you? Or tearing off his ear? Or breaking one of his fingers?\"\n\n\"Can I think it over?\"\n\n\"It'll do you no good. I'm sorry. Sign.\"\n\nJulie took the pen that the man was holding out to her.\n\n\"You wrote this on a typewriter,\" she observed. \"You wrote it with my Hermes, which that other fat fuck stole this morning.\"\n\n\"Yes,\" said Thompson. \"The perfect touch.\"\n\nJulie signed.\n\n# 13\n\nOVER DINNER the gangsters listened to Europe-Soir radio. Still no news. After the meal N\u00e9nesse left in the R 16 with the letter. Julie and Peter were told to go to bed. Thompson lay down when they did, at the other end of the room, without getting undressed. Night had just fallen. Bibi turned off the hurricane lamp hanging from the ceiling.\n\n\"I'll take the first watch.\"\n\nThe blond giant grunted assent.\n\n\"I'll come outside with you for a smoke or two.\"\n\n\"Don't go to bed too late, Coco,\" came Thompson's voice. \"You'll have to get up at one for the second watch.\"\n\nThe giant grunted again and followed Bibi out. Through the unshuttered windows Julie followed the pair's movements. Their silhouettes leant towards each other. A struck match flared, then went out.\n\nThe girl tried to relax. She was full of wine but strung out. The lack of tranquilizers was making itself felt. She laid her arms alongside her body, palms up.\n\n\"Julie!\" whispered Peter from the lower bunk. \"Are you asleep?\"\n\n\"No.\"\n\n\"Nor am I.\"\n\n\"Try. Think of something nice. Think of flowers.\"\n\n\"I can't.\"\n\n\"Quiet, please!\" cried Thompson.\n\nJulie and Peter were quiet. In a moment or two, overcome by fatigue and emotion, the boy fell asleep. But not Julie. She was wide awake when the blond giant came back into the chalet, stubbed out his cigarette on the sole of his shoe, and groped his way to the bunk below Thompson's. And she was still not sleeping when headlights briefly lit up the valley, silhouetting the trunks of pines, then went out. Its way lit by sidelights alone, the R 16 pulled up beneath the trees. In the darkness Julie could not tell the time by her watch. Outside, Bibi and N\u00e9nesse exchanged a few words in low voices and laughed. They stayed together for a few moments, smoking, then the interior lights of the R 16 went on. Julie discerned N\u00e9nesse arranging a plaid traveling blanket inside the car, lying down on the banquette and switching the lights off.\n\nJulie did not sleep a wink all night. She heard the watch changing; she heard Peter moaning in his sleep; and she heard Thompson getting up to go to the chemical toilet, where he remained for a long time groaning quietly.\n\nThompson did not go on watch. At dawn the blond giant woke N\u00e9nesse from his sleep in the R 16. It was five o'clock. There was just enough light for Julie to see the dial of her watch. The landscape was truly beautiful. Through the windows Julie could see one flank of the valley contoured against a blood-red sky. The sandstone outcrops and the silhouettes of the tall trees were like petrified monsters. The depths of the valley turned blue, then yellow. Julie heard the two men talking outside.\n\n\"I'm afraid I'll never get back to sleep at this hour.\"\n\n\"You want to go make coffee?\"\n\n\"I'm liable to wake Thompson.\"\n\n\"What the fuck do we care?\"\n\n\"He wouldn't like that.\"\n\n\"So what? What the fuck is it to us? He's not the one out here freezing his tail off.\"\n\n\"You go then.\"\n\nSilence.\n\n\"Oh, the hell with it. I don't want coffee anyway. Wait, I have something in the crate that will pick us up.\"\n\nBustling sounds. Julie watched the two figures moving in the dawn's yellow glare. A car door squealed, then slammed.\n\n\"Brrr! That feels good!\"\n\n\"Warms you up.\"\n\n\"Pass it back.\"\n\nJulie craned her neck to see the other end of the room, still wreathed in grayish, grainy shadows. She caught Thompson's eye. He was still immobile, arms by his sides, rifle near to hand. The girl felt sure that the man had not slept a wink either.\n\nPeter was awoken by the daylight at a quarter past six, and he got up immediately to make sure Julie was still there. The young woman drew him to her and held him tight.\n\n\"I like you better than Marcelle,\" said Peter once more.\n\nThompson left his bed and went to make coffee. The noise woke Bibi, who sat up grumpily. The brother heavies came back into the chalet. A fuzzy conversation in sleep-filled voices was gradually struck up. The second day had begun.\n\nThompson served coffee, and flageolet beans in paper bowls. Stomachs heaved at first, but the beans were warming. Peter refused to eat, refused coffee. Thompson gave him a mug of water.\n\n\"I'm sick of this,\" said the boy. \"I want to go home. I have to go home. Why are you doing this to me?\"\n\nHe began to stagger about in tears. He cried desperately for a long time. Eventually he ran out of tears, but he went on moaning dry-eyed: \"Why are you doing this to me?\"\n\n\"Shut up, for Christ's sake,\" shouted N\u00e9nesse.\n\n\"Yes,\" said Thompson to Julie. \"Have him be quiet, mademoiselle. He's starting to get on our nerves.\"\n\n# 14\n\nSHORTLY after one in the afternoon, with everyone seated at the table around a pork roast, the radio mentioned Peter and Julie. \"This may be the beginning of another kidnapping,\" said the lead newscaster in grave tones, between two commercials. \"Yes indeed, Jacques Paoli,\" intoned his second fiddle. \"To recap, a seven-year-old boy, little Peter Hartog, nephew of G\u00e9rard Hartog, who had gone out with his nanny on Wednesday morning, never returned to the businessman's home. Nor did the nanny, and the police are not excluding the possibility of a kidnapping. What is more, the nanny was discharged only recently from a mental institution.\" Etc., etc. Jacques Paoli responded by saying that it was too early to draw any conclusions, but they would be following developments closely. Meanwhile they would move on to other things, right after messages from their sponsors. Thompson turned the radio off.\n\n\"Let's finish our meal,\" he said.\n\nHe poured wine for Julie and gave her a thin smile. Suddenly he retched. With his hand to his mouth, he rushed to the toilet. The sound of violent vomiting was heard. The girl looked at each of the diners in turn.\n\n\"The police are on to this,\" she said. \"It's not too late to let us go. Kidnapping never works and you know it.\"\n\n\"Shut up!\" shouted N\u00e9nesse.\n\nThompson reappeared, his face dripping with sweat and gray as papier-m\u00e2ch\u00e9. Withdrawing a tin from his brown leather briefcase, he took spoonfuls of black granules from it, which he started chewing, seated on the edge of a bunk.\n\nJulie emptied her glass of wine to help her calm down. She realized straightaway that it was spiked.\n\n\"Have you put something in this?\" she demanded.\n\nThere was a bitter taste on her tongue.\n\n\"Just a sedative,\" said Thompson. \"My guys and I have certain things to talk over without you present, and I can't leave you alone in the house in full possession of your faculties, even if we go only a few meters away. But don't be afraid. You'll be deeply asleep, that's all.\"\n\nHe got up. The granules had blackened his lips.\n\n\"While you're still lively, you're going to write a little note.\"\n\nJulie was drooping. You should ask him what it is, the drug he's given you, she thought sluggishly. It's wild how it relaxes you. She yawned. Her fingers tickled. Thompson placed a blank sheet of paper in front of her and put the ballpoint pen in her hand.\n\nHe guided her hand.\n\n\"Write 'I warned you . . .' \"\n\nJulie wrote. Out of the corner of her eye she noticed that Peter was slumped over the table.\n\n\"Peter! What have you done to him?\"\n\n\"He's sleeping. Cross out what you just wrote.\"\n\n\"What? Why? Cross it out? Why?\"\n\n\"Quickly! Cross it out! Write underneath: 'I've had it. I can't take it anymore.' \"\n\nJulie wrote: \"Had it. Can't anymore.\"\n\n\"Good.\"\n\nThompson's voice was far away. Julie had water in her ears. She clung to the table and saw a streak of saliva settle on her hand.\n\n\"It's snail slime,\" she observed. They were lifting her up. Carrying her outside.\n\n\"Peter . . .\"\n\n\"Keep still. Relax.\"\n\n\"Yes, Doctor.\"\n\nThe four men laid Julie and Peter in the R 16's trunk, which they closed and locked. They then went into the chalet and removed all traces of their stay. They closed the shutters. N\u00e9nesse carried the bag of trash to the car and stowed it under the front passenger seat. Thompson joined him carrying his rifle and Julie's handbag. He disassembled the weapon. He arranged the various parts in a tan leather case with purple velvet lining. He paused frequently, doubled over by cramps. Eventually he settled in to the right of N\u00e9nesse, who had taken the wheel. Bibi and the blond giant, after locking up the chalet, soon joined them and got into the back. The four men were all sweating profusely.\n\n\"A few more minutes and it will be done,\" said Thompson.\n\n\"This is the most dangerous part,\" said Bibi.\n\nThompson closed his door.\n\n\"We know that.\"\n\n\"Are you sure they're unconscious?\"\n\n\"A bull would be unconscious in their place.\"\n\n\"The girl used to pop a load of pills. She could be immune. I wouldn't like it if she opened her eyes and looked at us in the middle of it.\"\n\n\"She's asleep, I guarantee you that.\"\n\nLaid out on her back in the trunk, incapable of any act of will, Julie was staring upward and listening to the men talk.\n\nN\u00e9nesse had finished carefully warming up the R 16, which shuddered into motion, passed through the gulley, and left the valley behind. Following a rough back road, it traveled several kilometers through sandy barrens covered with heather and dotted with pine and birch. At length it reached a straight, deserted departmental road and picked up speed.\n\n\"Take it easy, N\u00e9nesse.\"\n\n\"Don't tell me my job.\"\n\n\"Drop me here,\" said Thompson suddenly.\n\nN\u00e9nesse braked.\n\n\"Here? You'll have a long trek to Nemours.\"\n\n\"I'll have six kilometers. The fresh air will do me good.\"\n\n\"Are you sure you don't want to finish the trip with us?\"\n\n\"I'm sticking to the plan,\" said Thompson. \"I don't know anything about this jaunt.\"\n\n\"You're pushing it, Monsieur Thompson,\" said the blond giant unexpectedly.\n\nThompson opened his door and got out.\n\n\"I'll find you in the late afternoon at the Blason du Roi. I'll be at the bar from four o'clock on.\"\n\nHe picked up the tan case that held his rifle and slammed the door.\n\n\"Goodbye, gentlemen,\" he said through the open window.\n\nHe walked away at a brisk pace. The car started off again. \"He's pushing it,\" said the giant.\n\n\"Thompson is a pro,\" said N\u00e9nesse. \"There's no one else like him. Not in France anyway.\"\n\n\"He doesn't seem well,\" growled the giant.\n\nThe R 16 slowed down again. It jolted. Branches lashed its sides. They had left the road and were making their way through closely packed birch trees on a narrow track invaded by ground cover. In the trunk Julie was being tossed about. She was aware of Peter breathing deeply beside her. Her back hurt; she was lying on coiled rope.\n\nThe car pulled up.\n\n\"Here we are,\" said N\u00e9nesse. \"This is it.\"\n\nSilence.\n\n\"Who is going to do it?\" asked Bibi in a faltering tone.\n\n\"The three of us\u2014what a question!\"\n\n\"It makes me feel sick. A kid . . .\"\n\n\"Me too,\" said the blond giant. \"It makes me feel sick too. N\u00e9nesse, if it doesn't make you feel sick, you do it.\"\n\n\"I tell you what we'll do,\" said the driver. \"We'll throw dice for it. It'll be quick. The first ace rolled, okay?\"\n\n\"Fair enough.\"\n\nJulie heard the car doors. N\u00e9nesse was getting dice from his pocket. With his hand he swept the dust from the track. The dice bounced on the dry earth.\n\n\"A five. Whose turn next?\"\n\n\"Give it to me,\" said Bibi.\n\n\"Ace!\"\n\n\"It's not fair,\" said Bibi. \"Coco hasn't rolled.\"\n\n\"It's settled. It's settled,\" said N\u00e9nesse. \"My brother and me will go and have a smoke over there.\"\n\nThe two turned away and went off into the trees and undergrowth. Bibi went to open the car's trunk. Julie had closed her eyes. Even through her eyelids, the sunshine hurt.\n\nBibi manhandled her like a sack to get at the ropes. There were two of them. The young man made slip knots at the end of each, then went over to a large round rock with a birch growing right next to it. The place had been chosen beforehand. Bibi knew exactly what he was supposed to do. He scrambled up the rock and, leaning over, tied the ropes around a fork in the tree. They hung down the trunk, the nooses about two meters above the ground. The young man wiped away the sweat that was trickling into his eyes. He got down off the rock and returned to the car. He looked in every direction. He no longer saw any sign of the two brothers among the trees, which were close-set and festooned with creepers.\n\n\"Hey!\" shouted Bibi into the woods. \"At least give me a hand getting the girl up onto the rock.\"\n\n\"Up yours!\" came N\u00e9nesse's voice from a good thirty meters away.\n\n\"Bastards!\" replied Bibi without much conviction.\n\nHe leant down and clumsily got his arms around the girl. She was heavy and limp. He bounced her to make sure he had her tight and lugged her over to the round rock. Pushing her against the limestone, he gripped her by the hips and shoved her up ahead of him. It was a devil, but he managed it. The rock sloped slightly. A rather practical place for a hanging.\n\nBibi finished heaving Julie up and hauled himself up over her. The noose had to go around the girl's neck. Just push her off into space and let her swing. She would never be able to get her footing on the rock. Easy. Must not forget to throw the scribbled note on the ground that said \"Had it. Can't anymore.\"\n\nBibi leant in to shove Julie up farther. She was breathing little short warm breaths on his neck. Nervous cramps assailed the young man's leg muscles. The victim's body rubbed against his. He gulped anxiously as he felt an erection coming on.\n\nAt this juncture, Julie moved her right hand. She groped briefly at Bibi's chest. She grasped the MAB in the inside pocket of his pea jacket and fired a shot through the young man's body.\n\n# 15\n\nTHE 9MM bullet entered below Bibi's ribs, burst his liver, and exited through a buttock. The little gangster fell backwards screaming. He tumbled from the rock and landed on his back in the heather. Julie jumped after him and ended up on all fours on top of him. Bibi was shaking and wailing. The girl ran stumbling to the car. She could not see clearly, and she was dizzy. She was amazed when she realized that Peter was fast asleep. She swept the boy into her arms and carried him away, still sleeping.\n\nMeanwhile, once their initial shock had passed, the two brothers charged through the brush like bulls. They emerged not far from the car and saw Bibi lying on the ground and Julie disappearing into the greenery.\n\n\"She shot him!\" cried Coco.\n\n\"Stay with him.\"\n\nN\u00e9nesse scrambled in pursuit of Julie. He could no longer see the young woman. Branches slashed at his eyes. He could hear her up ahead battling her way forward. She was not more than twenty-five meters away. As he ran, N\u00e9nesse drew his revolver, a Spanish Ruby. This was a reflex: He knew he must not shoot the girl. What mattered was to capture both her and the boy.\n\nJust then he spotted them. Julie was running, the boy in her arms. She was tripping and bumping into trees. Where the woods were thick her silhouette was unclear. Then it disappeared. N\u00e9nesse upped his pace. It reappeared. She had stopped. She could not escape him now. N\u00e9nesse sneered, then he got the feeling that a metal rod was being thrust into his thorax, heard the report of a pistol shot, fell onto all fours, and lost hold of his pistol.\n\n\"This is stupid!\" he exclaimed.\n\nN\u00e9nesse was in shock. He felt himself. There was a hole in his flank. He was lying on his side. He heard his brother's voice.\n\n\"There you are, for Christ's sake! What the fuck are you doing? I've been calling you for ten minutes! Where's the girl?\"\n\n\"Help me up,\" said N\u00e9nesse. \"That little bitch put a slug in my chest!\"\n\n# 16\n\nJULIE was far away. She was out of breath. She was not running now, save over brief stretches of open terrain. Soon she found herself in completely wild country. The woods thinned out and gave way to rolling sandy ground interrupted here and there by jumbled rocks surmounted by pines. There were hollows with great masses of bracken. There was no trace of human settlement.\n\nGradually, as she walked, the girl's mind became clearer. Her teeth chattered. She had to resist the urge to lie down and sleep. Peter was still unconscious in her arms. Her shoulders hurt, her calves hurt, her feet hurt. Her shoes were not made for walking. They were full of sand and bits of heather that got between her toes. Her smoky-gray tights were in shreds. It was three in the afternoon. It was fine and sunny, but clouds were gathering in the west. Julie navigated by the sun. It was her guiding star.\n\nThough she was holding the little boy, the automatic pistol was still in her hand. It was hard for her to believe that she had shot two men. The idea made her burst out laughing. The girl was exhausted. She halted in the shade of a stand of pines, her back to a rock, on a mound offering a long view. She got her breath. She tended to Peter. Laid out on the ground, the little boy was sleeping with his mouth open. Julie shook him but got no reaction. She pushed his eyelids open. His eyes were slightly rolled up. He shook his head in his sleep. He was not exactly comatose but he was in a very deep, involuntary slumber, and there was nothing she could do but wait. With a dead branch she drew a large heart in the sand in front of her, and inside it she wrote: HERE LIVED JULIE THE RABID BITCH.\n\nThe girl was thirsty. After a while, carrying Peter, she set off once more. She forgot the MAB in the sand. She crossed somewhat less arid heathland. The birches were more numerous again. Eventually Julie pushed on through a veritable palisade of trees. Once through them, she found herself at the top of a rather bare slope. A valley stretched before her. At the far end of it the gray houses of a settlement could be seen. A road ran through the village. Julie set off down the hillside. Along the way, realizing what she must look like, she stood behind a rock to shed her ripped tights and untangle her hair. The sky, meanwhile, was clouding over. As she got closer to the village, from the open windows of the houses she heard live radio commentary on a football match, part of the European Champion Clubs' Cup. The girl reached the edge of little vegetable gardens behind the houses. She clambered over an ironwork fence, followed a stony path, and emerged onto the main street, which was also the main road through the village.\n\nJulie waited for a moment, expecting anyone and everyone to rush up to her and greet her or ask her questions. But the two or three passersby she saw paid her not the slightest attention. As for some youths seated at a refreshment stand, they were mainly interested in her legs. One of them whistled. Stiff-backed, Julie walked on.\n\nShe stopped at the local purveyor (according to the enameled frontage) of \"Newspapers, Tobacco, and Novelties.\" That day's _France-Soir_ was displayed on a plywood rack, and Julie's photo was on the front page. A bad picture, taken several years earlier. The words \"Little Peter's nanny had been in psychotherapy\" appeared in medium-size type below the picture, and underneath that, in smaller characters: \"She has disappeared with the child (page 3).\"\n\nJulie did not go in and buy the paper. She hurried on, then froze in her tracks. On the other side of the road, at the end of the village and some fifty meters from Julie, the tricolor flag of the National Gendarmerie was clearly visible. At this point it began to rain. It would be so easy to run to the building, shelter there from the rain, and throw herself into the solid lap of the police. Julie turned on her heel on the narrow sidewalk. A car was approaching under the downpour. The girl stuck her arm out, thumb upward. Oh, she thought, I've lost my pistol. The automobile, a pale blue Peugeot 204, pulled over in a spray of dirty water. The door opened. The driver was a red-faced man in his forties.\n\n\"Get in. Pithiviers?\"\n\n\"Yes,\" said Julie. \"Yes, I'm going to Pithiviers.\"\n\n# 17\n\nWALKING cross-country at a good pace, filling his lungs with fresh air but still tormented by stomach cramps, Thompson had covered the six kilometers separating him from Nemours with great difficulty. The first thing he did was go and fetch his gray Rover from the garage where he had left it. He parked it on the town's main square, took a small suitcase from the trunk, and rented a room for an hour at an inn. At four in the afternoon, freshly shaven and wearing a white turtleneck and an oak-brown sports suit, the man settled himself at the bar of the Blason du Roi. His face was pale. His cramps were incessant. Cold sweat coated his brow. He consulted his watch. By this time his victims should have been hanged. He pictured the scene: the two bodies jerking at the end of ropes, never having regained consciousness, tongues protruding, swollen, black, obscene. His pain lessened slightly. The barman leant towards Thompson looking worried. The killer waved his hands vaguely, fighting off the temptation to crush the man's cervical vertebrae. He had him serve a Campari, which he drowned in soda water. At the first sip he was overcome by frightful nausea. He ran to the men's room and threw up a remarkable amount of stomach acid. He returned to the bar. The server looked over.\n\n\"Are you ill, monsieur?\"\n\nTight-lipped, Thompson shook his head. The barman let it drop and went about his business, but from time to time he cast a concerned glance at the motionless killer. Thompson did not touch his glass again.\n\nCoco came into the bar. Now, thought Thompson, I am about to feel better. He is going to tell me how they died and I won't feel ill anymore and I'll eat. He scrutinized the blond giant and realized immediately that something had gone wrong.\n\n\"Come quickly,\" said Coco.\n\nThompson got off his barstool and left a five-franc piece on the counter. He did not wait for his change.\n\nOutside it was pouring\u2014a sudden storm. Coco and Thompson ran to the R 16. As he ran, Thompson flung his head back to catch raindrops in his mouth. The two men got into the front of the car. N\u00e9nesse was sitting in the back with a Celtique in his maw and a raincoat over his shoulder that fell askew, toga-like, across his chest.\n\n\"Are you hurt?\" asked Thompson promptly. \"Where is Bibi?\"\n\n\"Bibi is dead,\" said Coco. \"N\u00e9nesse got a slug in the side but it went right through. It's not serious. But the girl and the kid got away.\"\n\nNo wonder I'm in pain!\n\n\"What in God's name happened?\" roared Thompson.\n\n\"It was the girl. We were in the middle of hanging her and she suddenly came to. Bibi was holding her, he was taken by surprise, she grabbed his gun and killed him.\"\n\n\"And then?\"\n\n\"The girl and the kid got away. We were going after them, N\u00e9nesse and me. But he told me to take care of Bibi, and a second later I heard a shot. You couldn't see a thing on account of the bushes. It took me minutes to find my brother. She had winged him like I told you. I hadn't the faintest idea where she was, the girl I mean. I looked, I swear to God, but she had a head start.\"\n\n\"Did the kid wake up too?\"\n\nCoco looked embarrassed. He licked his fat lips. His tongue was covered with foam.\n\n\"Half awake. Not completely. The girl dragged him away, you might say.\"\n\n\"I can't believe this,\" said Thompson. \"How can it be? What did you do with Bibi?\"\n\n\"He didn't go right away. We waited a while. We tried to see if we could do anything, but his liver was shot through. There was no point in hanging about. N\u00e9nesse told me to put him out of his misery, and then we buried him.\"\n\n\"Right there?\"\n\n\"Well, yes.\"\n\n\"Idiots!\" said Thompson. \"The girl will lead the cops to the place in no time. They'll believe her story all the sooner. Idiots!\"\n\n\"We have to get out of here,\" N\u00e9nesse put in. \"We came to tell you. We're pulling out. Right now.\"\n\nThompson's eyes flashed. His mustache quivered.\n\n\"You'll pull out when I say!\" he said in a violent undertone. \"You screwed up the job. You have accounts to settle.\"\n\n\"The money we received,\" said Coco, \"we are keeping. We are very sorry Monsieur Thompson, but we took risks. If things went wrong, it's because the girl was not completely knocked out.\" He looked Thompson straight in the eye. \"That was your fault,\" he concluded.\n\nIn the back of the car, N\u00e9nesse's arm moved vaguely under the raincoat and Thompson divined the revolver aimed at him beneath the fabric. The rain drummed frantically on the car's metal roof.\n\n\"What are you thinking?\" asked Thompson contemptuously. \"Are you going to start firing here? I am deeply disappointed in you. Here is what we are going to do.\"\n\n\"Monsieur Thompson, it's no use\u2014\"\n\n\"Shut up! You are going to make your way home. Your brother can take care of his wound. I'll contact my client. I'll come back and tell you whether or not the money already paid out must be reimbursed. Don't hope for too much. The fault is yours.\"\n\n\"Don't you hope for too much either, Thompson,\" said N\u00e9nesse through gritted teeth. \"Reimbursement-wise, I mean.\"\n\n\"No sense in discussing that yet,\" said Thompson. \"Goodbye, gentlemen. I'll get back in touch with you at your place, tomorrow night at the latest.\"\n\nThe man got out of the Renault. He stood motionless for a moment, his shoulders hunched against the rain, which was soaking his suit. Then he walked briskly to his Rover and drove off.\n\n# 18\n\nAFTER the storm the sun came out, shining more brightly than ever between cloud banks scudding eastward. The road surfaces shone brightly as well. The red-faced motorist hummed as he drove.\n\n\"He's a great sleeper, that kid of yours, I must say. My goodness, what a sleepyhead! Ha! Ha! Is he your son?\"\n\n\"No, he's the younger son of my boss,\" said Julie, putting on an accent.\n\n\"Are you French?\"\n\n\"No, I'm English.\"\n\n\"I guessed it from your complexion. You know, lily-and-rose.\"\n\n\"What's a lily like?\"\n\n\"It's a white flower that symbolizes purity and beauty.\"\n\n\"Oh.\"\n\n\"A lily-and-rose complexion is a poetic way of referring to a fine English complexion.\"\n\n\"Oh, I see.\"\n\n\"I suppose the men in France flirt with you a bit?\"\n\n\"So do the men in England.\"\n\nJulie was having a grand old time with the vocabulary. She pictured men flirting with her\u2014and her shooting them point-blank. I must be in a manic phase, she told herself.\n\n\"Yes, but Frenchmen,\" said the red-faced motorist, \"what do you think of them, the way they flirt?\"\n\n\"I don't know. Some of them are crude.\"\n\n\"Crude? You mean brutal?\"\n\n\"No, crude. They talk dirty to me!\"\n\nThis gave the motorist pause.\n\n\"Well, you know how it is, a girl in shorts, it's inevitable. Are you from London? A student?\"\n\n\"At Oxford,\" declared Julie. \"I study economics.\"\n\n\"Well, that's amazing.\" the man exclaimed enthusiastically. \"I'm a salesman myself. I could tell you a thing or two about economics! Aren't you going beyond Pithiviers?\"\n\nJulie stretched in her seat. Her thigh muscles rippled.\n\n\"Are you going farther yourself?\"\n\n\"I'm stopping for five minutes, just to see a customer, then going on. Where are you headed?\"\n\n\"South.\"\n\n\"That's perfect. I go to Sully, then Bourges. That will get you along.\"\n\nJulie contemplated the man. He was wearing a blue pinstriped suit. His face was square and ruddy, and his brown hair fell in curls over his forehead. He had little eyes behind rectangular glasses. He was piglike.\n\n\"You are a nice man,\" she said.\n\nWith her right hand she gave the motorist a friendly little tap on the shoulder, then pressed her palm against his chest and drew her nails raspingly across the material of his jacket. The man turned beet red. An idiotic smile tugged at his lips. Julie withdrew her hand. Flushed and perspiring, he kept on driving, darting frequent sideways glances at the young woman. He was wondering whether she was the genuine article. The sweat gathered like drool on his glistening curls.\n\n\"Couldn't we stop for a moment?\" asked Julie.\n\n\"Stop? What, pull over? Yes, sure. Why?\"\n\n\"There!\" cried Julie. \"A dirt road!\"\n\nShe was pointing. The 204 braked sharply, turned, and bumped onto the dirt road.\n\n\"Stop here.\"\n\nThe car pulled up. The driver put the handbrake on. He looked back furtively at Peter asleep on the back seat. Julie opened her door.\n\nThe young woman got out. Through the windshield the motorist, smiling like an imbecile, watched her indecisively. He saw the girl vanish behind a hedge. Was she going to piss, the man asked himself, or insert her diaphragm? He was trembling with apprehension. Suddenly Julie reappeared. She was waving an arm in an odd fashion.\n\n\"Bring your starting handle over here!\" she shouted.\n\nThe motorist opened the door on his side and reached into the back of the car.\n\n\"What's happening?\"\n\n\"Quick! Your starting handle! Bring your starting handle!\"\n\n\"But what for? Oh, okay, screw it!\" said the man.\n\nWith the implement in his hand, he ran over to Julie. He had short legs and his pants ballooned over his fat backside. Julie was bent double, gazing at something within the hedge. The motorist contemplated her spread legs.\n\n\"Give it here! Quick! It's still in there!\"\n\nThe motorist felt Julie wrench the starting handle from his grasp. The girl was frantically pointing to the bottom of the hedge.\n\n\"There! There!\"\n\nDiscombobulated, he leant over. Julie brought the handle down on his skull. I thought so, he told himself as he fell onto all fours.\n\n\"You pig! You swine! You're disgusting!\" said Julie.\n\nHe tried to get up. Julie struck him on the forehead. His scalp split. Blood streamed down the good citizen's face.\n\n\"Stop!\" he pleaded.\n\nJulie hit him twice more. He subsided onto the dusty track. He was moaning. Almost unconscious. He tried to grab Julie's ankle, meaning to bring it to his lips. Or perhaps to make the girl fall\u2014he no longer knew which. A final blow from the starting handle finished him. He stopped moving. Julie searched him. He had an opened pack of Gitanes filters on him, along with a packet of condoms, a receipt book with counterfoils, a silver ballpoint pen, and pocket change. In his wallet Julie found various papers in the name of \u00c9mile Ventr\u00e9e and a five-hundred-franc note. She slipped the bill into her shorts. Then she took \u00c9mile Ventr\u00e9e's shoes off and tossed them well away. She tugged off his pants and undershorts and systematically tore them up. She returned to the car. The key was still in the ignition. Peter was sleeping deeply. Julie started the motor, got back onto the main road, and drove away quickly. Less than an hour later the 204 took the Autoroute du Sud at Courtenay and headed in the direction of the Mediterranean.\n\n# 19\n\nHIS SUIT still damp from the rain, Thompson had parked his Rover in a lot at Orly and taken an air taxi. The next morning another air taxi brought him back. He took a room at the Hilton for an hour or so. He had his two suits cleaned and pressed. He sat waiting in his room wearing a blue-and-brown-striped flannel robe, drank some Vittel, and promptly went and threw up in the washbowl. His face had taken on a ghastly pallor, his eyes were bloodshot, and his attacks of nausea kept degenerating into uncontrollable coughing fits, quite awful. He was shivering. His nose was stuffed up and very dry. His skin was burning hot. He took a shower and his teeth began to chatter.\n\nAs soon as his clean clothes were delivered Thompson began to hurry. He put the taupe suit back on over an iron-gray turtleneck. He settled his bill, retrieved the Rover, and sped towards Paris. Waves of nausea made it very hard for him to drive. Leaving the Paris ring road at the Porte Brancion, he drove into Malakoff. Not far from the railroad, on a grimy street with grass pushing up between the paving stones, Thompson parked in front of a crumbling villa flanked by a yard and a rusting shed. The killer got out and went to ring the bell at an ironwork gate. The potholed sidewalk was strewn with trash, textile tailings, and metal junk. Stray cats went slinking by. An old, half-illegible graffito read RIDGEWAY GET OUT! Thompson's expression was tense.\n\nThe gate opened and Thompson did not relax on seeing Coco in front of him clad in mechanic's overalls.\n\n\"I have to bring my car in.\"\n\nCoco checked the street with a mean look.\n\n\"The money? What's been decided?\"\n\n\"We'll talk later.\"\n\nThompson got back into the Rover. Coco opened the double gates and the car entered the yard, which was cluttered with wheel hubs without tires, gutted laundry boilers, a Dodge truck cabin, and a Buick Roadmaster on its rims. Coco closed the gates behind the Rover. He stood still, legs apart, arms akimbo, as Thompson climbed out of the car, coughing.\n\n\"How is your brother doing?\"\n\n\"Not so bad. You want to see him?\"\n\n\"Yes.\"\n\n\"Ha! Ha! Always careful, aren't we, Mr. Thompson?\"\n\n\"Don't be silly. We need to talk.\"\n\nCoco pouted skeptically, then led Thompson up a short steep stairway sheltered by a glass awning. The glass was cracked in several places. The boot scraper at the head of the stairs was buried under a thick layer of dried mud.\n\nN\u00e9nesse greeted them at the entrance to the villa. It was obvious that he had not shaved or washed since the day before. Skintight blue jeans emphasized his considerable genital apparatus. Beneath a tank top the slight bulge of a bandage was discernible. He smelled bad, he smelled of salami, and leveled in front of him he held a sawed-off Tarzan shotgun. Thompson closed the door behind him.\n\n\"I'm here for a friendly talk,\" he said. \"This sort of thing I cannot accept.\"\n\nN\u00e9nesse hesitated, then placed his gun, butt downward, in an umbrella stand.\n\n\"But I'm making no promises,\" he noted. \"You want a little drink?\"\n\nThompson shook his head. The three men repaired to a petty-bourgeois living room with dumpy furniture and a waxed parquet floor. The window, with its cretonne curtains, overlooked railroad tracks. They sat around a table covered with an oilcloth. Coco produced a bottle of pear brandy and three tiny glasses from a hideous sideboard. He poured for all. Thompson made no objection.\n\n\"What about your wound?\"\n\n\"It's nothing. It's clean. And it's not the first time.\"\n\n\"I'm happy for you,\" said Thompson. \"Now, something is happening. I need a driver. I'm carrying on with the job on a new basis. I've seen my client. He is very unhappy. In fact he's in a blind fury. We were off to a bad start in our talk, then some new developments changed things. Have you been listening to the news?\"\n\n\"Yes,\" said Coco. \"But they don't tell you anything on the radio.\"\n\n\"Quite so. The fact is, the girl didn't go to the cops. She took off. And it's up to me,\" added Thompson, \"to kill her.\"\n\nHe clutched his stomach with both hands.\n\n\"She took off?\" echoed Coco.\n\n\"Yesterday afternoon\u2014but we only learnt this overnight\u2014she assaulted a motorist who picked her up with the kid. She beat the guy to death with a starting handle. Stole his wallet and his car. And the car was found empty, out of gas, in a parking lot down the A6 forty-five kilometers north of Lyon.\"\n\n\"She's whacko!\" said Coco.\n\n\"Well, yes.\"\n\nCoco shook his head.\n\n\"Where d'you get your information?\" asked N\u00e9nesse.\n\n\"My client,\" said Thompson. \"He gets it straight from the police, where he has good friends.\"\n\n\"And what does he want us to do?\"\n\n\"He wants me to find her before the police do, and kill her. And I need a driver. I can't drive anymore. I'm ill.\"\n\n\"Your client,\" said Coco, \"what he wants, it's not doable.\"\n\nThompson winced. \"No, I must kill her. The kid too. I have to.\"\n\nThe two brothers exchanged glances. Thompson was off his trolley\u2014that was obvious. On the other hand, there was dough in this.\n\n\"This client of yours, does he pay two big ones a head?\"\n\nThompson nodded.\n\n\"We can give it a shot,\" offered N\u00e9nesse.\n\n\"I need only one driver. Only Coco. You are injured.\"\n\n\"N\u00e9nesse and me both,\" said Coco. \"We work together or it's not on.\"\n\n\"Very well, very well,\" said Thompson. He rubbed his red eyes. He sighed. \"An air taxi is waiting for us. I'll explain in the car, give you all the details.\"\n\nThompson got up abruptly. His chair fell over behind him. He noticed a woman's handbag on the floor by the wall. He stooped and picked it up.\n\n\"Idiots!\" he said softly, almost in a whisper. \"Idiots! The girl's bag. It must be destroyed. I'll take it.\"\n\nHe headed for the door. The brothers emptied their glasses and got to their feet.\n\n# 20\n\nFOR THE second night in a row Julie had not slept at all.\n\nPeter woke up at six in the morning after sleeping for seventeen hours. From about midnight his sleep had gradually become more normal. When he awoke he cried out. Julie leapt from her bed and hurried over to the little boy. He was sitting up and gazing uncomprehendingly into the gray half-light of the room.\n\n\"I'm here. Don't be afraid.\"\n\nPeter flung his arms around the girl's neck, squeezing with all his might and almost strangling her.\n\n\"Where are we, Julie? Where are the bad guys?\"\n\n\"Shh! We're in a hotel. We got away.\"\n\n\"Are they chasing us?\"\n\n\"No.\"\n\n\"Did the police catch them? Did you tell the police?\"\n\nJulie disentangled herself, shivering. The boy looked around the room. It was a very large room, with white roughcast walls, an old-fashioned rustic wooden washstand with a built-in bowl and a pitcher, and a large oval mirror mounted lengthwise between two wooden uprights.\n\n\"The police!\" Peter repeated. \"What about them?\"\n\n\"I didn't go to the police.\"\n\n\"Why?\"\n\nJulie shook her head in exasperation and her dark hair swirled about her white neck.\n\n\"You're all naked,\" noted Peter with interest.\n\n\"I'm getting dressed. You get dressed too. We're going to have breakfast.\"\n\n\"What about the police?\"\n\nOh, the little devil! The rotten little devil! thought Julie. But then she thought: No, it's me that's screwy.\n\n\"I didn't go to the police because I'm afraid of the police. I hate the police. Police! Police! Police! That's why! Now you know!\" _Sotto voce,_ Julie was ranting.\n\n\"Why?\"\n\n\"Oh Christ!\" the girl exclaimed, and she sat down on the edge of the bed.\n\nShe did not know whether to cry or burst out laughing. She was still in doubt as she slipped on her shorts and sweat-soaked T-shirt. After abandoning the Peugeot 204, she had walked for kilometers and kilometers with Peter in her arms, following lanes and back roads and cutting across fields. Ten, twenty kilometers\u2014she was not sure. Her head weighed heavy and all her joints ached.\n\n\"I used to be a criminal,\" she blurted, and she looked at Peter to see what effect her words had on him.\n\nHe gulped. \"You're lying.\"\n\n\"No, I've been in prison.\"\n\n\"What for? Murder?\"\n\n\"Get dressed.\"\n\nBeyond the window shutters it was daylight. Ever since Peter had uttered the word \"police,\" Julie had been feeling stifled in the dimness.\n\n\"I want you to tell me!\" cried the boy. \"When did I go to sleep?\"\n\nThe girl took his head between her hands.\n\n\"Do you really want to know? Really see things the way they are? Listen, I'm an escaped prisoner. Do you believe me?\"\n\n\"Sure I do. You mean like in _Branded._ And you have to prove your innocence?\"\n\n\"That's right.\" Julie sighed.\n\n\"What are you supposed to have done?\"\n\n\"Murder. Okay?\"\n\n\"Well, I trust you,\" Peter said firmly.\n\n\"Get dressed then.\"\n\nThe girl helped the little boy get dressed.\n\n\"How are you going to prove you are innocent?\"\n\n\"We'll go and find Uncle Hartog,\" said Julie, \"and I'll explain everything to him.\"\n\nShe stood slack-jawed for a moment or two.\n\n\"No, what am I talking about? I'm such a fool. I'm horribly confused, it's stupid. We have to go to . . . the police. I don't know what to do.\"\n\n\"Tie my laces,\" suggested Peter.\n\nFrom outside, audible through the slots in the shutters, came the discreet sound of a car slowing down and coming to a halt in the sandy hotel parking lot. Julie ran to the window. Through the gaps she could see the roof of a black Peugeot 403. Figures in light blue raincoats were getting out. One of them looked up to inspect the hotel's facade. He was a young man with a crew cut. He had his hands in his pockets. He was smoking a cigarette.\n\n\"Everyone's sleeping,\" the man said.\n\n\"Shit,\" said another raincoat. \"The sun is up. I'm up. Let's go.\"\n\nThe young man lowered his head. The two raincoats left Julie's field of vision. A moment later the girl heard the front-door bell downstairs ringing for a long time.\n\n\"The cops,\" said Peter.\n\n\"Let's get the hell out of here,\" said Julie.\n\nShe took the boy by the hand. They went out into the hallway. The ringing was louder now. Footsteps resounded from the ground floor.\n\n\"Okay, okay,\" said a voice, \"we're coming.\"\n\nJulie was bathed in sweat. She raced down the hallway past closed doors. At the end was a window giving onto a corrugated metal roof. In the distance bulky green mountains billowed up, densely wooded and pied by patches of mist. Julie jumped down onto the tin roof. The boy was laughing. The two of them slid down. The lean-to was not very high. Julie leapt off holding Peter and almost sprained her ankle as they landed in a farmyard bordered by cowsheds. A rooster crowed in a horrible way. The fugitives slithered through the mud, followed a path between low stone walls, and debouched onto a slope covered with broom. Closely mown grass, slick from dew, was greasy underfoot. Julie slipped and fell, rolling down through the broom with Peter. They got up, passed through a line of trees, and found themselves at a loop in the road just below the hotel. A big Chausson motor coach appeared from round the bend. Julie signaled. The bus braked and stopped. Julie and Peter climbed in.\n\n\"Lucky for you you're so cute,\" remarked the driver, who was wearing a white work smock.\n\nThe door closed with a pneumatic din and the bus went on down the hill. The inside smelled like wet dog. Country folk snoozed on the tatty seats with hats on their heads and baskets on their knees. Julie took a seat. She felt dizzy. There was a metallic taste in her mouth. The bus was juddering through an endless succession of hairpin bends . . .\n\n. . . The driver was tapping Julie on the shoulder.\n\n\"Are you all right?\"\n\nThe girl cast a stunned glance at the tree-lined square where the bus was parked.\n\n\"I fell asleep,\" she said in a daze.\n\n\"Last stop. Five francs fifty.\"\n\nJulie paid. They got off, she and Peter. They walked away, left the square, spotted a deserted caf\u00e9, and took a table outside. Julie ordered a large caf\u00e9 cr\u00e8me and a large hot chocolate. The place had no croissants.\n\n\"Wait here a minute,\" said Julie to Peter.\n\nThere was a bakery nearby and a tobacconist almost next door to it. The girl soon returned to their table with croissants, cigarettes, and newspapers. As Peter ate, Julie frantically lit a Gauloise and began paging through the papers. She still merited a small headline but the accompanying story was brief. Clearly the journalists had no fresh information. There should be a headline such as THE NET TIGHTENS, but, of course, thought Julie, when it really is tightening, they don't say so. Her lip tensed slightly. The back of her neck hurt.\n\nThe newspaper people embroidered a good deal on Hartog: \"young financier,\" they wrote, and \"meteoric rise,\" and \"not a few envious rivals.\"\n\n\"At present it is impossible to reach the industrialist,\" Julie read. \"According to some sources, M. Hartog, who returned hastily from Munich upon learning of the disappearance of his nephew, has left Paris once more so as to avoid publicity as the search goes on.\"\n\n\"It's obvious,\" murmured Julie. \"He has to be at his Moorish Tower.\"\n\nShe felt more confident now, more sure of her plan. She could not be very far from Hartog's fabulous house, his mountain labyrinth. She would get there. Throw herself at his feet. She could see the scene now, and she was excited. A scene worthy of Delacroix. The rich man would help her to her feet, understand her, forgive her, congratulate her, and seat her at his right hand on a cloud-wreathed mountaintop overlooking the Massif Central\u2014far from the world of men, as the saying went. The girl stood and pulled Peter up, then turned back to pay for the breakfast. She was trembling. She asked the way to the station. She came to a grade crossing with a view of the platforms, bordered by dusty privet in cement planters. BOEN, said a kind of sign above the platform, blue letters against a white background\u2014a hollow, boxlike sign supposed to light up at night like the signs outside police stations. The name BOEN meant nothing to Julie. She turned back. Peter trotted along beside her, finishing off a croissant. He observed his surroundings with curiosity; he was calm. They went into some shops, and Julie bought a rather hideous gray raincoat that came down to her knees and some Michelin maps. As they came out of the bookstore, she noticed an immense white poster on the other side of the street bearing a text in scarlet letters:\n\nLAW AND ORDER  \nCAN WE LIVE WITHOUT POLICE?  \nWITHOUT GENDARMES?\n\nTHE BATTLE OF THAT GREAT DAY  \nOF GOD ALMIGHTY (REV. 16:14)  \nWHICH MEANS THE DISSOLUTION  \nOF VALUES?  \nIN THE LAST DAYS PERILOUS TIMES  \nSHALL COME (2 TIM. 3:1)\n\nPAUL ZWICKAU\n\nWith her Michelin maps under her arm, holding on to Peter with the opposite hand, Julie crossed the street and went through a doorway just beneath the poster. In a little anteroom with a tiled floor country people wearing hats were talking in hushed tones. Beyond them was a conference hall. A sparse audience was seated on benches. Old people and children were in the majority. A guy on a stage was talking in front of a trompe l'oeil rendering of red-and-gold curtains. As Julie went to sit down, the guy gestured towards the wings.\n\n\"Paul Zwickau!\" he announced in a high-pitched voice.\n\nA young man came running from the wings. He wore a black suit and a tan polo-neck shirt. He had glasses and a part in his hair.\n\n\"Would you like to live,\" he cried, \"under a system that freed humanity from crime, violence, injustice, and poverty?\"\n\nThe audience did not seem opposed. Zwickau began striding up and down the stage. He frequently tossed his head back in a violent manner. It was not hard to fear for his neck.\n\n\"Law and order reign in the Universe!\" he exclaimed. \"When we contemplate the vault of the heavens, so much is obvious. Even though the deepest meaning of the movements of the heavenly bodies may escape us, we must respect the views of the greatest scholars, astronomers, and nuclear physicists, who tell us how, night after night, season after season, year after year, and century after century the stars hew fast to their course in space and follow their orbits in so regular a fashion that it is possible to predict eclipses centuries in advance!\"\n\nThe orator cleared his throat.\n\n\"My brothers, my sisters, listen! Our Eternal God has established a harmonious order in the Universe. Do you really believe that He is incapable of imposing that order and harmony on our planet, on this wretched little lump of dirt whirling through interstitial space?\"\n\n\"I have seen His name!\" shouted a fat woman in black.\n\nZwickau ignored the interruption.\n\n\"What do we see?\" he cried.\n\n\"I have seen His name!\" said the fat woman again.\n\nA muscular steward grabbed her arm and whispered something in her ear. She immediately took on a guilty air.\n\n\"What do we see?\" Zwickau insisted with a derisive smile. \"Has man himself been able to institute law and order, an end to violence, injustice, and poverty? No! Would you like to know what happens in a big city when there are no more police? That is what occurred in Montreal on October 7, 1969. The police were on strike. Did citizens respect the law once they knew the police were no longer there to make arrests? Not at all! Right away Montreal became the scene of rioting, arson, looting, and fighting among taxi drivers. The rioters armed themselves with clubs and rocks and engaged in an orgy of senseless destruction. They smashed the windows of the Queen Elizabeth Hotel and stole merchandise. They vandalized the fine IBM Building. They plundered the Windsor and Mount Royal hotels. Without police, respect for law and order completely vanished. According to government spokesmen, the city was 'on the verge of anarchy'!\"\n\nZwickau stood for a moment on tiptoe, arm raised, to underscore the horror of it all.\n\n\"What do these events mean?\" he went on. \"Why this riotousness? And above all, what is to be done about it? Man has striven mightily and in all sincerity. He has tried every kind of regime. But he has never turned towards his Creator! His Creator who knows man's problems better than anyone, because He has been observing everything that happens since the beginning of time.\"\n\n\"Yes!\" cried many voices.\n\n\"Hallelujah!\"\n\n\"Hallelujah! Schmallelujah!\" cried Julie.\n\n\"Be quiet, my sister,\" said the muscular steward.\n\n\"I'm not your sister,\" retorted Julie. But she was quiet.\n\n\"He is the Creator!\" Zwickau continued. \"It is our God who will bring order here below by instituting the state that we need. It is written that 'the God of heaven shall set up a kingdom, which shall never be destroyed; and the kingdom shall not be left to other people, but it shall break in pieces and consume all those kingdoms, and it shall stand for ever.' Daniel two, forty-four.\"\n\nJulie stood up.\n\n\"Pig! Disgusting pig!\" she shouted.\n\nZwickau leapt down from the stage.\n\n\"Listen to me!\"\n\nJulie grabbed Peter and hoisted him onto her shoulders. The preacher pursued them out into the street. The girl raced towards the station. Zwickau gave up the chase. Julie was in the station. She had to get rid of all these bastards who were out to destroy her. This was no time to lose her head. She would have loved to open fire with a machine gun and create a bloodbath.\n\n\"Answer me, for heaven's sake!\" complained Peter. \"What are we doing? Where are we going?\"\n\n\"We're running away. We're going to Hartog's place. His magnificent house over the horizon.\"\n\nJulie consulted the departures board, and with help from her road maps she eventually figured out where she was: sixty-odd kilometers west of Lyon. It was another sixty kilometers or so to the canton of Olliergues and the magnificent house. A local train drew in. Julie bought two tickets for Saint-\u00c9tienne. The train bore the fugitives away. It was hot. Julie was sweating under her formless raincoat. Peter was silent, remarkably well-behaved, gazing at Julie with eyes wide, green, and suspicious. Julie began looking at her maps again.\n\n\"We're going to be traveling in the mountains,\" she told Peter. \"In the mountains no one can catch us. We'll go into the mountains and we'll find the magnificent house.\"\n\n\"You already told me that.\"\n\nThey got off at Montbrison. It was half past one in the afternoon. This surprised Julie. It should have been earlier. She and Peter crossed the sweltering esplanade fronting the station and had lunch in a sort of brasserie.\n\n\"When are we leaving?\" asked Peter.\n\n\"We're not in a hurry.\"\n\n\"Yes, we are. We are hunted animals,\" observed the boy.\n\n\"Do you wish we were there already?\"\n\n\"Yes.\"\n\n\"Aren't you happy just to be with me?\"\n\n\"Oh yes, I am.\"\n\n\"Would you like me to buy you some toys?\"\n\n\"I don't know. If you like. What kind of toys?\"\n\n\"I don't know. Whatever you like.\"\n\n\"I don't mind.\"\n\n\"Listen, Peter,\" said Julie, \"we could stop in the mountains and live as a mother and her son. No one would ever find us.\"\n\n\"The police always find criminals.\"\n\nJulie grimaced. \"Finish your dessert.\"\n\n\"Not hungry anymore.\"\n\nJulie paid the bill. She counted the money she had left. Less than four thousand francs. How fast it went!\n\n\"Come on,\" she said.\n\n\"Where to?\"\n\n\"Let's go.\"\n\nThey wandered through the town. It was market day. The center of the little place was completely clogged with crowds of people and multicolored stalls. Julie bought the boy an ice cream. She was vaguely looking for a bus station. Was there even a bus station? At last, on a sort of circular avenue, near an empty caf\u00e9, she came upon a blue post bearing the words COACH STOP. No timetable was in evidence. Map in hand, taking her bearings by the sun, the girl tried to work out which way buses stopping there would be headed. At that moment a black Simca 1500 passed by, and through its open back window Julie saw Coco looking straight at her.\n\n# 21\n\nTHEY HAD left the Rover at the Orly West parking lot.\n\n\"Wait for me,\" said Thompson to the two brothers once they were in the terminal. \"Wait for me. I'm going to see if there's not a message for me.\"\n\nHe disappeared briefly and returned hunched over, carefully tearing up a slip of paper. Coco and N\u00e9nesse were looking around the terminal with curiosity, concentrating chiefly on the legs of stewardesses. They both wore cheap slate-gray suits and checked shirts. Each carried a small suitcase containing a change of underwear and a gun.\n\n\"The girl,\" said Thompson, \"has been almost nabbed twice.\"\n\n\"That a message from your client?\"\n\nThe Englishman nodded. His eyes had horrible dark circles under them and the edges of his mouth were white.\n\n\"The police just missed her at a hotel where she spent the night. Then seemingly she caused a scandal at a hot gospel meeting a hundred kilometers away later in the morning. She's been reported seen in other regions, in Rouen, in the Alps\u2014but that's perfectly impossible.\"\n\n\"It's always like that with regular citizens,\" sneered Coco. \"They see evil everywhere.\"\n\n\"Your client,\" said N\u00e9nesse, \"he has his ear to the ground.\"\n\n\"He stays informed,\" said Thompson with a sigh. \"Come on.\"\n\nThe three men went on foot to the far end of the airfield. A few corporate and charter aircraft stood there, near a gray temporary building. Some young men in short-sleeved shirts were playing boule on the grass. Thompson hailed one of them.\n\n\"Finish up without me, fellows,\" said the man to the other players.\n\n\"Where are you going?\"\n\n\"Lyon.\"\n\nThompson waited until they were seated in the plane then leant over to the pilot.\n\n\"Actually, I'm not certain that we are going to Lyon,\" he said. \"I need to get as close as possible to Bo\u00ebn, between Roanne and Saint-\u00c9tienne.\"\n\nThe pilot scratched his head. He was dark, with brown eyes, a crew cut, and a lively, healthful mien.\n\n\"There's Villeneuve, near Feurs,\" he said. \"That's the closest. Otherwise, farther south, you have the Saint-\u00c9tienne airfield, which is actually at Andr\u00e9zieux.\"\n\n\"None of that means much to me. Let's take off anyway. I'll take a look at the map.\"\n\nThe pilot put on his earphones and sunglasses with nylon frames and exchanged cabalistic signals with a mechanic. The engines sputtered into life. The aircraft had two of them. It was a yellow-and-red machine, quite graceful though a little garish, with scarlet nacelles on either wing that probably housed reserve fuel tanks. A Cessna 421. The cabin had room for six passengers, comfortable seats complete with armrests and ashtrays. The grass outside lay down flat in the prop wash. The twin-engine plane went into motion, maneuvering on the tarmac. The pilot chattered into his radio. The craft took up its holding position, brakes on.\n\n\"At Orly,\" the pilot confided to Thompson, \"it's always a bitch on account of the commercial traffic.\"\n\nHe gabbled into his microphone. The brakes were off. The aircraft raced over the concrete for the longest time, then tipped up and took off. Thompson rejoined the brothers aft. His color was greenish, his eyes half closed.\n\n\"I've never been up in a plane before,\" said Coco.\n\nThompson was consulting his maps, marking places with a gold mechanical pencil. He stepped away for a moment, went into the lavatory, and vomited in an almost absentminded way; he was getting used to his condition.\n\nMeanwhile the brothers were in ecstasies at seeing the earth from the air.\n\nWhen Thompson emerged from the lavatory, his mouth dry, he went and shouted into the pilot's ear, \"At Villeneuve, could I easily hire a car?\"\n\n\"You mean a taxi?\"\n\n\"No. A hire car without a driver.\"\n\n\"That, no way, old pal.\"\n\n\"I'm not your old pal,\" said Thompson.\n\nThe pilot blanched. \"Excuse me, sir.\"\n\nThompson smiled. \"Take us to that airfield you mentioned before, near Saint-\u00c9tienne. I fancy we'll find a car there.\"\n\nThey found one right away, a black Simca 1500, rather the worse for wear but the best thing on offer.\n\nN\u00e9nesse was grumpy. \"I'm quite fit to drive,\" he insisted. He took the wheel. The others did not argue.\n\nThompson had rented the car under the name of Andre Proust, producing all the paperwork needed.\n\n\"Make for Montbrison,\" Thompson told N\u00e9nesse. \"Then to Boen. We'll make inquiries at the train station or the bus company office.\"\n\n\"We'll never corner her before the cops,\" said N\u00e9nesse. \"It's hopeless.\"\n\n\"Yes, we will,\" countered Thompson. \"We have to.\"\n\nThe 1500 was going at top speed. Coco fidgeted in the back seat.\n\n\"Take it easy!\"\n\n\"Shut up!\"\n\nThompson sighed and contemplated his knees. He had fastened his seat belt. The road was very straight. The Simca was eating it up at roughly 100 kph. It was three in the afternoon when it entered Montbrison.\n\n\"Slow down,\" Thompson ordered. \"Look for signs to Boen or Roanne.\"\n\n\"Christ alive!\" shouted Coco. \"There! Look there! Stop! Over there! The girl! She's here!\"\n\nN\u00e9nesse slammed on the brakes. The Simca pulled to the left as it slowed. Eyeing his rearview mirror, N\u00e9nesse spun the wheel frenziedly. He felt a violent force applied to the side of his body. Skidding, the Simca performed a U-turn on the spot, blocking a Renault 4CV coming the other way. Coco and Thompson were half thrown from their seats. Their car had gone some fifty meters past Julie and Peter. The girl stood motionless, stricken, at the edge of the sidewalk, which was lined with plane trees.\n\n\"We kill her immediately and leave via the National 496,\" declared Thompson.\n\nHe reached inside his jacket.\n\n\"We're not hanging her anymore?\" asked Coco in bewilderment.\n\n\"We kill her. That's all that matters.\"\n\nThe Simca hurtled towards Julie with its engine roaring. The girl seemed to come alive. She took Peter by the arm and ran between cars parked on the sidewalk.\n\nThompson's hand emerged from his jacket armed with a bizarre-looking SIG automatic of the kind used for target practice. It could have been mistaken for a rather unrealistic toy gun. With his other hand he rolled down the window of his car door as quickly as he could.\n\nTen meters from his goal N\u00e9nesse downshifted. Braked by its transmission, the Simca slowed rapidly and leant forward on its tired suspension. Thompson heard Coco's shot detonate right by his ear. Julie dived headlong into the dust, but Thompson spotted the bullet's impact point, too high, on the roof of a parked Renault 4. Julie was crawling as fast as she could around the car. Thompson's innards were in the grip of an iron hand. He saw Peter's pale face lined up with his sighting mark and squeezed the trigger of the SIG just as N\u00e9nesse gave the steering wheel a violent twist. The round passed just beneath the little boy's ear.\n\n\"I'm going to crush them,\" said N\u00e9nesse.\n\nThe Simca, turning so sharply that it almost flipped over, mounted the sidewalk, lost contact with the ground, and swiveled head-to-tail.\n\n\"Fucking shit!\" cried N\u00e9nesse.\n\nJulie, still holding Peter by the hand, set off in the opposite direction, zigzagging among the parked cars. Coco fired just under Thompson's nose for the second time, his hot powder scorching the Britisher's face. The Simca, still skidding, collided with the R 4 and tore off a fender. Julie dashed between two vehicles.\n\n\"Pull out of here, N\u00e9nesse!\" yelled Coco. \"We're fucked.\"\n\nCoco emptied his revolver at random. Rounds ricocheted wildly off bodywork. Triplex glass rained down. Its motor still roaring, the Simca jounced back onto the roadway, leaving Julie behind.\n\n\"Stop, you idiot. I order you to stop,\" said Thompson in a steely voice.\n\nN\u00e9nesse was not listening. His lips were blue. Thompson struck his fingers with the barrel of his automatic weapon. N\u00e9nesse braked hard.\n\n\"What do you . . . you want?\" he stammered. \"You want . . . want us to be picked up on the spot?\"\n\n\"The kid and the girl. They must be killed.\"\n\n\"In three minutes the cops will be here.\"\n\n\"In three minutes I'll have killed them. U-turn!\"\n\nN\u00e9nesse did nothing.\n\n\"U-turn or I kill you,\" said Thompson, digging the barrel of the SIG into the man's ribs.\n\nN\u00e9nesse blinked and started up.\n\n\"Three minutes,\" he said between gritted teeth.\n\nA hundred meters behind them a crowd was gathering. Farther away, Thompson saw Julie and Peter disappearing down a side street. People were running. The Simca turned round and headed for the throng.\n\n\"There they are,\" cried the locals. \"It's them!\"\n\n\"Drive right through the lot of them,\" said Thompson. \"Then take the first left.\"\n\nThe Simca drove right through the lot of them. The locals scattered, shouting. N\u00e9nesse clung tightly to the steering wheel. The tires sang as the car turned sharply into a little cobbled street. At the far end, Julie and Peter were running. They were crossing a bridge. The Simca bounded after them. The roadway was crowded with people. They were forced to press themselves against the walls and the shopwindows. Cries of protest filled the air.\n\nOn the far side of the bridge milling hordes completely blocked the street. Peter and Julie melted into the mass. N\u00e9nesse braked vigorously. Once again the car pulled to the left as it came to a halt.\n\n\"That didn't get us very far,\" said the driver.\n\nJust round a sharp turn thousands of people were besieging stalls set up in the middle of the street. Vehicular traffic was out of the question. Julie's gray-and-brown silhouette could be glimpsed in the crowd. Thompson punched violently at the inside of the car door.\n\n\"Coco and I will continue on foot. N\u00e9nesse, you go and park this car on the boulevard and then steal another one.\"\n\n\"You're cracked,\" N\u00e9nesse hissed.\n\n\"We'll meet in the caf\u00e9-tobacconist you just saw on the boulevard. It's called Les Fleurs. In a quarter of an hour. Don't forget my case.\"\n\n\"A quarter of an hour!\" groaned N\u00e9nesse.\n\n\"Bye-bye,\" said Thompson.\n\nHe got out of the Simca and shouldered his way roughly into the crowd. Coco did not move.\n\n\"Let's get the fuck out of here,\" he said.\n\n\"No,\" sighed his brother. \"He's the boss. And he's a pro. Just do what he says.\"\n\n# 22\n\nJULIE was dazed. Her head, her eyes were in a fog. When a gap in the crowd opened up, she bounded through it, tugging Peter along behind her. The little boy was stupefied with terror. Over the hatted heads a clear space could be seen. Julie turned to look back. Amid the crush she spotted Thompson getting nearer with great long strides, a tall silhouette, thin and dried-up and gray-haired. If he got a clear shot he would be able to pot the girl like a target at a fairground shooting gallery. From a hundred meters Julie could make out the man's teeth gleaming in his lined face. She raced straight towards a Prisunic fronting the street and entered through the glass doors.\n\nShe charged down the aisles. The store occupied the ground-floor level of an entire block. Beyond the vast accumulation of commodities more glass doors opened onto another street and an esplanade black with people. Julie charged in that direction. She must get out ahead of Thompson. Melt into the crowd. She jostled housewives as she passed.\n\nThe girl was no more than a few meters from the exit doors when Coco materialized on the other side of the glass. Blinking, he looked at Julie, who had pulled up short. He seemed hesitant, almost fearful.\n\nJulie made an about-turn, twisting Peter's arm. The boy began to cry.\n\n\"Oh, shut up! Shut up!\" cried Julie. \"It's over.\"\n\nShe rushed up to a salesgirl.\n\n\"Mademoiselle, call the police right away.\"\n\n\"What?\"\n\n\"The police! Call the police!\"\n\n\"But what's going on?\" demanded the salesgirl, taking a step back.\n\nShe scrutinized Julie with a suspicious smile tugging at her lips; twenty meters away, Coco came in through the glass doors. Suddenly he dashed forward. Julie whirled round. Tableware was on display close by, and she swept a pile of unbreakable plates onto the floor. They did not break.\n\n\"You're crazy!\" exclaimed the salesgirl, taking another leap backwards.\n\n\"Murderer!\" yelled Julie with all her might.\n\nPirouetting once more, she slapped the salesgirl violently across the face and set off at a run. She never let go of Peter, who lost his balance and fell forward still firmly in Julie's grasp. She did not release him, hauling him along at top speed, his feet dragging on the tile floor. He was bawling at the top of his lungs. At the other end of the Prisunic, Thompson had entered the store and stood motionless, his pistol dangling at the end of his arm, barrel pointing towards the floor.\n\n\"Murderer!\" yelled Julie again, heightening the skepticism of the housewives.\n\nShe kept running, zigzagging among the shelves. As she proceeded she grabbed products and threw them on the floor. A store employee with a badge on his white cashier's smock suddenly posted himself in her way, legs and arms spread like a goalie.\n\n\"Stop right there!\" he commanded in a measured tone.\n\nJulie delivered a head blow to his face. The girl's hard skull struck the man's chin, snapping his head back and causing him to collapse into a heap on the tiled floor. Julie leapt over him. He grabbed Peter and held on. Julie grabbed a stainless steel paring knife from a display and stabbed at the air above the head of the department manager. He immediately let go of Peter and curled into a ball, using his elbows to protect his eyes and his knees to protect his genitals.\n\n\"Police!\" he cried in a falsetto voice.\n\n\"About time too!\" said Julie, and a bullet passed through her right arm.\n\n# 23\n\nTHOMPSON had decided that he could wait no longer. With stunning speed the store was transformed into a madhouse. More and more people began to run. A wake of detritus marked Julie's trajectory through the aisles. Women were screaming. Several shopgirls had started ringing the little handbells they usually rang to attract a supervisor when they needed change or to check the identity of a customer wishing to pay by check. Above the hullabaloo, by way of background, floated the sweet yet cannonading tones of an old Joan Baez hit, piped in through speakers. The place was a bear garden.\n\nThompson stretched out his arm to fire, his stomach hurting so badly he felt as though he was being torn apart, and the figure of Julie wobbled in the SIG's sights. The girl dropped to the ground. Thompson immediately revised his aim for a second shot through the delirious crowd. The round (a 9mm Parabellum) exploded the head of a frantic customer. The man was running. He put his hands out in front of him like a diver and executed a noisy belly flop on the floor of the store. Thompson shuddered. His stomach was a raging furnace. His nostrils dilated at the scent of gunpowder. He had paid no mind to the detonations, but all around him the crowd was going utterly crazy. People went on running and screaming in the line of fire between Julie sitting on the ground and the automatic still searching for her. Thompson could no longer see either Julie or Peter. He bounded down an aisle, knocking over an old woman who began to wail in terror. He trotted by Boys Apparel, his mouth full of bile. He heard a deafening explosion and assumed that Coco had decided to open up. Fragments of plastic flew from a display. The store was filled by an immense tumult. This is exciting, I am enjoying this, Thompson told himself as he spat gastric juice onto the ground. More and more people were getting down on the floor, huddling against the bottom shelves. Mothers were shielding their children by covering them with their bodies. The whole mass was shrieking. Thompson was doubled over with laughter.\n\n# 24\n\nCOCO WATCHED fragments of plastic toys spraying into the air along the path of his bullet. He was trembling. In his hand was an old Colt revolver, solid, crude, and with a tendency to shoot to the right. For a split second he caught sight of Julie and Peter down an aisle and he fired again, winging a carton of laundry detergent.\n\nCoco was expecting people to tackle him. Instead, they got away from him as fast as they could, squawking and falling over one another. A granny lay three meters from him, her arms cradling her head and her legs beating a hysterical tattoo. She had varicose veins. Coco turned away and looked behind him towards the glass doors that gave on to the esplanade. People were going out, some of them on all fours. Others were running in. Coco fired twice into the doors and large segments of glass tumbled down and shattered. The crowd drew back and dispersed in an uproar.\n\n\"Thompson!\" cried the thug. \"Let's blow.\"\n\nNo reply could be heard above the hubbub. The Muzak stopped abruptly with a horrible screech. Someone had seriously jolted the turntable. Then a tremulous voice resounded from one end of the store to the other.\n\n\"Nobody move! Everyone lay on the floor! We are being robbed!\"\n\nCoco looked up in irritation and saw the rattled announcer, wearing a white overshirt, perched in a glassed-in compartment above the food section. He directed a round to the top of the man's cubicle. The hero of the hour, cursing, dropped swiftly from his chair and disappeared from view.\n\n\"Moron,\" observed Coco.\n\nIt is amazing how much can happen in twenty-five seconds. Coco heard Thompson firing, once, twice, three times. The screaming continued. The blond giant, now quite furious, circumvented Sports Gear and slipped down through Wines and Spirits.\n\n\"Thompson!\" he called.\n\nJulie popped into his line of sight, just by Cleaning Products. Her right arm was covered in blood. It looked as if she was wearing a high glove. With her left hand she lobbed a flaming bottle of denatured alcohol at Coco. The thug pressed the trigger of his big Colt and the round buried itself in the ceiling. Coco fell over backwards, uttering a shocked exclamation when his skull struck the floor. Close by him there was a sound like a lightbulb giving up the ghost and he found himself in the middle of a slick of blazing liquid. Little blue flames licked at his pants. Coco felt the hairs on his legs igniting. He put his Colt down on the floor and swatted his thighs in desperation. Julie had vanished. Bottles came whirling over from behind the liquor shelves, shattering around Coco and unleashing a veritable sea of fire. The hired man wrenched off his pants, jumped about frenziedly in the blaze, and burned himself as he retrieved his Colt. In his underwear he ran towards the exit. His legs were frying and he caught the smell of bacon emanating from his burnt skin.\n\n\"Run ahead! Run ahead!\" cried Julie to Peter.\n\nShe was holding another liter bottle of alcohol, its neck flaming like a soldering iron. As she followed Peter, she doused a display of sweaters with fire. Eventually she turned and tossed the half-empty bottle randomly, as far away from her as she could, into the swirling smoke. It exploded. Screaming redoubled. Pushing Peter ahead of her, the young woman leapt over a patch of blue flames. A housewife crossed their path. Her skirt was on fire, as was the grocery cart she was pushing. The woman rammed into a stand with books and fell down along with it. The books caught fire. Weeping, the woman rolled into a ball.\n\nOn every side the cries grew hoarser. People were coughing. Indistinct figures stumbled over the wreckage or bent double amid the swirling smoke. Far behind her, Julie heard the doors through which she had entered shatter under the pressure of people in flight. A great gust of wind blew through the store from one end to the other. The flames shot up to the ceiling. The girl passed a squad of heroic department managers manning fire extinguishers. Another, ax in hand, was desperately hewing at a display rack.\n\nAll of a sudden Julie and Peter were outside, treading on broken glass. A crowd was gathering at a respectful distance. Escapees were streaming out of the store on either side of the girl. Women were breaking down. Some were being carried out in men's arms, screaming and wriggling. The sidewalk was strewn with shoes and commodities. Julie and Peter dashed into the crowd, only to be greeted by helping hands.\n\n\"Are you hurt? And the lad?\"\n\n\"It's okay, we're all right, thank you . . .\"\n\nJulie strove to extricate them. Fortunately other victims came tumbling out, bemoaning their singed perms, and the pair pushed their way even deeper into the human maelstrom. They were out of the spotlight now. Concealing her bleeding arm as best she could, the girl threaded their way forward. The firehouse alarm wailed in the distance.\n\n\"Make way! Clear the road!\"\n\n\"It's arsonists!\"\n\n\"They're Maoists! They fired machine guns at cars over on the avenue.\"\n\nSirens blaring, red fire trucks swept past Julie as she emerged from the crowd. Firefighters were already leaping to the ground.\n\nThe girl and the little boy followed the street for fifty meters or so until they came back to the circular avenue. Julie turned, staggering slightly, took her raincoat off, and wrapped it round her bloody arm. Then she took Peter's hand once more. Over the heads of the people she saw a thick cloud of white smoke emerging from the store and cloaking the base of the building. On the upper floors the windows were open and heads, open mouths, and waving arms could be seen.\n\nAt the turnoff to the Prisunic cars on the circular road were slowing down and the drivers were craning their necks to see what was happening. Two firemen came up at a trot and, turning their backs to the fire, took it upon themselves to keep the traffic moving.\n\nOther cars had already pulled up any old how. Some drivers were standing beside their vehicles, staring; others had run towards the disaster. Julie opened the door of a Renault 2CV whose engine was still running. Nobody took any notice.\n\n\"Get in,\" she told Peter.\n\n\"In the car?\"\n\n\"Come on, get in.\"\n\n\"But it's not ours.\"\n\n\"Get in, for Christ's sake!\"\n\nShe grabbed Peter, shoved him in, slammed the door behind him, walked round, and got behind the wheel. Her teeth were chattering again.\n\n\"But you're stealing the car!\" exclaimed the little boy.\n\nJulie shifted into first and grimaced. She felt as though the prong of a pitchfork had been driven into her arm.\n\n\"We'll go to jail,\" said Peter.\n\nOne of the firemen signaled to Julie to hurry up. The girl accelerated down the circular road. At the first intersection she turned the 2CV down a narrow winding road. Above the last houses of the town, the horizon was filled by rounded green mountains. Good, thought Julie. Good . . .\n\n# 25\n\nEYELASHES singed, hands streaked with soot and slobber, and his pistol in his pocket, Thompson crossed the little bridge against the flow of the surging crowd.\n\nPeople were hastening towards the cataclysm. Customers had come out of the newsagent's onto the sidewalk to see. Their jostling overturned the newspaper rack. Kids bounded up, full of the excitement that grips children at moments of catastrophe.\n\nAbove the rooftops rose an ever thicker and darker plume of smoke. The firehouse alarm blared continuously. A jet of water appeared amid the smoke, gleaming in the sunlight.\n\nBatteries of water cannon had been positioned to either side of the blazing store. The front of the building was being hosed down as the ground floor flooded. Another siren with a different sound signaled the arrival of an Estafette that immediately disgorged policemen. They began to push the press of people back. The tactics of the firefighters and police were hindered by the throngs of country people down from the mountains to shop. Stalls cluttered the main street. In the pushing and shoving many of them had been knocked over. Fruit rolled about on the ground. People trampled on it, squashed it, slipped on it. Some fell. A three-note klaxon sounded insistently: an ambulance was struggling through to evacuate burn victims.\n\nThe floors of the building's second story had caught fire. Occupants were leaping from windows.\n\nThompson walked at a swift pace, hunched forward. He did not look back. Around him the crowd thinned out. He reached the circular road at a point about fifty meters from the caf\u00e9 called Les Fleurs.\n\nN\u00e9nesse was sitting on the terrace with a beer in front of him and his face covered by tiny beads of sweat. He rose as Thompson approached and left some coins on the round table. At the curbside in front of the caf\u00e9 stood a Ford Capri, empty, its engine running. The uproar that had broken out in the town center had made things easier for N\u00e9nesse. He'd had no problem discreetly dumping the Simca and appropriating another car. The town had gone mad.\n\n\"Where is Coco?\" asked N\u00e9nesse.\n\nThompson shook his head. His lips were white with dried saliva. N\u00e9nesse's eyes narrowed dangerously.\n\n\"Did you start the fire?\"\n\n\"That little bitch did it,\" panted Thompson. \"Set the damn department store on fire. I got her, though. Ha!\" He clutched his stomach. \"But what about Coco?\"\n\n\"He was on the other side. I saw him in flames. Cops. Firemen. Out of here. Gotta get out of here.\"\n\nThompson turned away and opened the Capri's passenger door. His whole back was quivering. The hairs at the nape of his neck stood on end. The blood throbbed in his ears, which were scarlet. N\u00e9nesse grabbed his shoulder, whirled him around roughly, and shook him.\n\n\"You're going to take off, Thompson, and leave my brother behind?\"\n\n\"Where's my gun?\" asked Thompson in a zombielike voice. \"You didn't forget my gun, did you?\"\n\nHe struggled distractedly in N\u00e9nesse's grip, surveying the inside of the Ford. He saw the case on the back seat and sighed with relief. N\u00e9nesse shook him more and more violently.\n\n\"You're figuring to leave Coco!\"\n\nA police Estafette passed them, its siren wailing, and continued down the circular road. Thompson wrenched his shoulders away and grimaced.\n\n\"Have to get out of here! Get your hands off me!\"\n\n\"No way!\" cried N\u00e9nesse.\n\nMustering all his strength, Thompson kicked the man in the crotch. N\u00e9nesse staggered back groaning and piled into the railing of the caf\u00e9. Doubled up in pain, he reached inside his cheap jacket. Thompson dealt him a double forearm smash to either side of the neck. N\u00e9nesse groaned again and sat at the foot of the railing. His eyes were bulging. Slowly he tried to draw his weapon. Thompson grabbed his forearm and broke the man's wrist over his knee. N\u00e9nesse passed out briefly.\n\nCoco, meanwhile, in his underpants, brandishing his Colt, was running every which way.\n\n\"Stop him! Stop him!\" came shouts in his wake.\n\nThe blond giant reached the circular road and hesitated.\n\n\"Arsonist!\" yelled his pursuers, who were more and more numerous. \"He's armed! Watch out!\"\n\nCoco's burnt legs were hurting. He took off once more down the sidewalk towards the caf\u00e9 called Les Fleurs, which he had spotted. A garnet-red Ford Capri had just pulled out from in front of the caf\u00e9 and was racing away. Coco recognized his brother sitting on the ground at the edge of the caf\u00e9's terrace. N\u00e9nesse was swaying. He is drunk! thought the giant.\n\n\"Halt! You are ordered to halt!\" came a new voice, distinct, military.\n\nCoco ran faster.\n\n\"Halt!\"\n\nCoco heard the report of a service pistol, very likely from a shot in the air. The giant turned and saw four gendarmes running towards him like a line of rugby players mounting a perfect attack. Behind them a horde of spectators were cheering as if in a stadium. Coco emptied his revolver at random. He still had three rounds. He noted with satisfaction that one of the gendarmes had fallen on all fours. The other three stopped dead, stood with legs apart, took aim with arms outstretched, and opened fire more or less simultaneously.\n\n\"They're killing him,\" muttered N\u00e9nesse.\n\nHis whole body was wracked with pain. He watched Coco being cut down like a blade of wheat. Fragments of knee flew into the air and the blond giant twirled screaming into the dirt, one leg bent and the other oddly straight. N\u00e9nesse had struggled to his feet and managed to draw his weapon with his left hand. He fired over his brother but hit nobody. The gendarmes returned a disorderly fire. N\u00e9nesse saw rounds exiting his brother's back in a shower of flesh and tissue. Coco rolled back onto his disarticulated leg without dropping his empty Colt, whose trigger he continued to press. N\u00e9nesse sighed, and two large tears sprang from his little eyes. He tossed his weapon aside and waited to be arrested. At that moment the caf\u00e9's owner crossed the terrace in three strides and emptied both barrels of a shotgun into N\u00e9nesse's ear.\n\n# 26\n\nTHE FORD Capri was doing 130 kph on National 496. Thompson felt as if he had swallowed boiling oil. He clung to the steering wheel, which his chin collided with every time convulsive retching overtook him.\n\nVery sharp bends obliged him to slow down. The car jolted over the uneven surface of the roadway. Subjected to erratic forces, the tires screeched. Thompson zigzagged.\n\nSome fifteen kilometers along, he braked violently and swung the Ford onto a grassy track, which met the road at a right angle, and followed it between clusters of pine trees. The dirt road was deeply rutted. The car danced grotesquely. A brutal impact threw Thompson forward. The man's forehead rammed into the windshield. He recovered himself, reversed, and promptly speeded up. The motor howled. The automobile's underbelly was scraping the center of the track and crashing into granite boulders three-quarters concealed in the dirt. From time to time the wheels slithered on the slick grass and spun wildly. The overheated tires were wreathed in smoke.\n\nBrush and trees closed in quickly on either side. Branches lashed the sides of the Ford. The way got steeper and the car's hood loomed over Thompson like the prow of a boat. The killer ground his teeth. He was now climbing into a real forest.\n\nEventually he felt the rear axle strike an obstacle. The shock ran through the whole vehicle, which seemed to slump. A continuous whine rose from the chassis. Thompson looked over to the left, towards a narrow opening. The Ford flattened a bush before passing between two trees and bouncing down into a tiny hollow. The front bumper buried itself in moss and earth, the engine groaned and began racing, the transmission was shot, the car had lost all propulsion, and the machine sank onto its ruined shock absorbers.\n\nThompson turned off the ignition.\n\nHe remained in his seat for a moment, motionless. He was not touching the back of it, for he was bent forward, his torso pressed against the wheel and his tense bony buttocks perched on the edge of the leather. His spasms died down. He could hear birds singing in the woods.\n\nIn the glove compartment were an open pack of Camels and a lady's Flaminaire lighter. Thompson stuck a cigarette between his lips, lit it, and stepped outside coughing. First he went back towards the dirt road. His tires had left impressions in the muddy ruts. The grassy center of the track was all broken up as though by a crude plow. The killer could do nothing about that. He grabbed handfuls of moss from the side of the hollow and used this to hide his trail, the place where the Ford had suddenly turned off and lunged into the undergrowth. He righted the flattened bush.\n\nThe vegetation was thick and sturdy. The car was virtually invisible from the track. Thompson went back and walked once all around it. It was still making little groaning sounds. His cigarette remained tight between his lips, the ash falling regularly and spreading over his jacket.\n\nThe Ford was useless. The clutch was shot, the suspension destroyed, and a viscous liquid was dripping from the back axle and soaking into the moss. Thompson opened the door and inventoried the vehicle's contents. No radio\u2014 which was a real pity. No luggage, except for the rifle case and the brothers' two suitcases. At the foot of the rear seat Thompson came upon Julie's handbag. It took him a few moments to identify it. Then he was shaken by a wave of nausea. That bitch! He knew that he had not killed her. His body was telling him that. He was trembling convulsively. He opened the bag and furiously emptied the contents onto the grass. Using his heel, he stamped on the few items that had fallen out\u2014a wallet, a handkerchief, a photograph, a nail file . . .\n\nThe killer collected himself. Kneeling on the moss, he gathered up the crushed objects and put them back in the bag. Then, bag in hand, he moved off through the brush and under the old-growth trees to reconnoiter.\n\nIt was fortunate in a way that the Ford had given up the ghost where it had. A hundred meters farther on, Thompson reached the fringe of the woods. First the pines thinned, leaving large clearings, then the trees came completely to an end. Crouched in half darkness beneath a clump of ill-favored pines, Thompson surveyed the upslope before him. It rose evenly, clear, sunlit, and flecked with yellow and pink dots, which were flowers. The late-afternoon light made for long shadows and put everything into sharp relief. Up here the garnet-red Ford would have stood out like a fly on a baby's head.\n\nBlinking, Thompson looked up at a pale blue sky where barely perceptible tendrils of mist were floating. By now the region's roads would be blocked; the helicopters of the gendarmeries must surely be hovering over the valleys. The killer shrugged and went back into the undergrowth.\n\nConcerned not to get lost, he left a trail of discreet markers behind him\u2014sprigs of brushwood, stones, or tiny grazes on the soft bark of the pines.\n\nZigzagging down the hill, he eventually found the spot he needed, a muddy gulley where a little brook bubbled, half covered by greenery. With his bare hands Thompson explored the blackish mud of the brook's sides. The water had worn a channel through the earth. Thompson excavated the vertical side of the channel. When he had a hole big enough he buried Julie's bag in it, tamping down the soil with his heel and covering all with mud. He worked with a fury. Once finished, he wondered whether he should repeat the operation for the brothers' suitcases. But making them disappear meant a great deal of work and it would scarcely slow the thought processes of the police once the Ford was found. Thompson gave up on the idea.\n\nHe went back to the Ford. He felt curiously relaxed. He had not eaten for two days. He had become used to the pain from his stomach.\n\nThe light was fading in the clearing. Thompson opened the case where the parts of his rifle nestled so cozily. He oiled them and polished them with love. Then he went and sat in the car and waited for nightfall.\n\n# 27\n\nJULIE was driving in second the whole time. Her right arm was no longer able to work the gearshift. It hung down by her side lacquered with dried blood.\n\n\"Does it hurt?\" asked Peter.\n\n\"No. I mean yes. I don't know.\"\n\n\"You must know if it hurts, though!\"\n\nJulie shook her head. The road was very steep and winding. At bends the girl struggled to steer properly. Her left wrist was now hurting more than her injured arm.\n\n\"Did the bullet come out?\" Peter inquired.\n\n\"I don't know.\"\n\n\"If it didn't come out by itself, it has to be pulled out,\" observed the little boy.\n\n\"Be quiet,\" said Julie. \"No, talk to me.\"\n\n\"Will we be there soon?\"\n\n\"I don't know. I don't have my maps anymore.\"\n\n\"Are we lost?\"\n\n\"No. I can remember the map. More or less, anyway. We have to go west. That's why the sun is in our eyes.\"\n\nThe engine coughed, spluttered, and died abruptly. A little red light on the dashboard came on. The 2CV was freewheeling, suddenly silent, the wind whistling against the windshield. Julie let go of the wheel to pull on the starter and the motor turned over and caught. The gear lever vibrated. Then the motor quit again. The little car was approaching another steep hill. Julie steered it onto the shoulder, where it came to a crooked halt. The girl applied the handbrake, shifted to neutral, and pulled on the starter again. The 2CV revved mightily but the motor would not turn over. Julie looked at the instruments. The needle of the gas gauge had fallen well below zero. The girl sighed a high-pitched sigh that almost broke into a sob.\n\n\"Isn't it working?\" asked Peter.\n\nJulie got out of the car. The air was cooler now and she shivered. She took her bloodstained raincoat from the front seat and draped it clumsily over her shoulders. Then she fell suddenly into a sitting position on the asphalt. Peter leapt from the car, took Julie by the shoulder, and tried frantically to get her to her feet.\n\n\"I'm all right,\" the girl murmured. \"I'll be okay.\"\n\nClinging to the car, she got to her feet. She staggered a little.\n\n\"We'll continue on foot,\" she said.\n\n\"Oh drat!\" said Peter. \"I'm fed up with walking.\"\n\n\"If you'd rather,\" said Julie evenly, \"I'll leave you here on the edge of these deep woods haunted by big gray owls.\"\n\n\"Shut up! Shut up!\" exclaimed the little boy. \"Okay, okay.\"\n\nJulie started off, unsteady on her feet. Peter trotted behind her. Darkness began creeping into the hollows, but beyond the crest of the hill ahead the sun was a pool of copper. The road, narrow and potholed, was deserted. Since the 2CV had left Montbrison the two travelers had encountered very few vehicles and only the odd pedestrian in the close vicinity of the town. For the last hour or more they had seen no one, and no sign of habitation.\n\nJulie and Peter climbed towards the sunlit heights. A track crossed the little road. An improvised sign dangled from a post at the intersection. It bore the name of a hamlet that meant nothing to Julie.\n\nMaybe, she thought, I'll get to the top of this hill, then I'll lie down and die and nothing will have mattered at all.\n\nHer head drooped. She walked on like an automaton.\n\n\"Carry me,\" Peter asked.\n\nShe made no reply. She kept on walking. The sky was misting over. Behind Julie, high above her, the clouds were taking on a blue-gray hue.\n\nWhen the girl reached the top of the hill she almost failed to notice. But since the ground was no longer rising, her chin bumped against her chest and she raised her head and stopped short, swaying, before a glorious setting sun that whacked her full in the face with its light.\n\nBelow her feet lay a somewhat nondescript valley full of trees. Beyond it, almost immediately, the ground rose once more, precipitously, forming a mountainside even higher than the one the fugitives had just climbed. At Julie's eye level, the trees vanished from its darkening slopes, and the summit sprang up above like a greenish skull. And there, far off, silhouetted against the sunset, and pointed out so to speak by the sun, a low, chaotic, grotesque structure adhered to that skull close to the summit.\n\n\"Look at the castle!\" exclaimed Peter.\n\n\"That's Hartog's Moorish Tower,\" Julie stammered.\n\n\"It's all squished! It's not a tower. It's ugly.\"\n\n\"No, no,\" said Julie. \"Come on. Let's go down.\"\n\nThe road looped into the valley, then followed the valley floor, leading away from the distant structure. Julie set off again, abandoning the dilapidated road, cutting straight down across fields and skirting copses. On the steep slope, helped by her own weight, she broke into a run. As she tried to slow down, her heels slammed into the ground and the shock carried to her injured arm. Peter frolicked around her, his fatigue forgotten.\n\nThe sun disappeared behind the Moorish Tower. Julie and Peter reached the bottom of the valley, where a fast-moving stream tumbled along. Anxiously, Julie contemplated its white eddies swirling in the shadows. Peter pulled her by the sleeve.\n\n\"There's a bridge.\"\n\nHe was pointing to a crude conglomeration of branches, planks, and logs a short way upstream. Julie followed him towards it.\n\nThe makeshift bridge, like something built by drunken boy scouts, was precariously balanced on large round rocks. The ropes that had once moored it securely to nearby trees were frayed, and so rotted that they almost crumbled at Julie's touch. The plank walkway had holes in it at several points. Below, the water seethed black and gray.\n\nBy now night had fallen in the valley. Julie started onto the walkway and felt the breath of the torrent below upon her face. The girl clung to the rudimentary handrail that ran alongside the planking. The bridge trembled beneath her and she trembled along with it. She had the impression of being enormously heavy. The planks bowed beneath her feet and squeaked. Julie stopped halfway across and watched in horrible fascination as a rusty nail emerged slowly, like a penis, from the wood of a plank. The plank split, the twisted nail flew into the air and fell into the foam below, and Julie's foot slipped through the walkway. The girl bruised herself as she grabbed onto the planking. Everything was shaking, around her and inside her head. She found herself on all fours, crawling along above the boiling race, desperately looking around for Peter. She heard the little boy call her, then saw him, already on the other side, hopping up and down with impatience.\n\n\"Hey, can't you hurry up?\"\n\nThe bridge gave way, and Julie flung herself forward. She ended up kneeling in the mud of the bank. Behind her, the walkway overturned. One end crashed into the creek. The whole contraption swung round into alignment with the current and plunged swiftly into the foaming waters, bouncing off rocks, coming apart, disintegrating, and disappearing with startling speed. The darkness swallowed it up.\n\nSmeared with mud, Julie clambered up the bank. She was burning up with fever. She did not know where Peter was. She was driven on by a single idea: a hundred meters, fifty meters from where she was now, once past the foot of the mountain, there would be nothing ahead but a vast carpet of grass and flowers, and upon it would be the Moorish Tower, where Hartog was waiting for her.\n\n# 28\n\nHARTOG, his features drawn and his lower lip covered with cold sores and fading bite marks, was slumped in a chaise lounge on a tiled deck. An enormous glass with a sprig of mint trailing in it sat on the ground beside him alongside an overflowing ashtray. A stained cigarette dangled from the redhead's mouth. Dark glasses hid his eyes. He was wearing white pants and a mesh undershirt.\n\nD\u00e9d\u00e9, the driver, wearing a dark suit, tiptoed up to him from behind.\n\n\"Any news?\" asked the redhead between gritted teeth.\n\n\"The second goon died on the way to the hospital. The roadblocks have come up with nothing yet. The police are combing the region by helicopter.\"\n\n\"I don't understand what happened,\" muttered Hartog. \"Where has she gone, that loony bitch?\"\n\nThe driver shrugged, stuck his hands in his pockets, and let his gaze wander over the Mediterranean, which was slapping at the pebbles below the deck.\n\n# 29\n\nCLOSE UP, the Moorish Tower appeared even more peculiar than in photographs. The place was obviously built upon several preexisting structures\u2014the mountain cattle sheds that the Auvergnats call _jasseries._ But these had been literally buried by masonry, by low domes, barely level terraces, and formless piles of stones. There was nothing towerlike about the Moorish Tower. It had spread across the surface of the mountain without growing upward. It was like a Siamese temple flattened by power hammers.\n\nPeter had halted near an outgrowth of the building with a dark opening like the entrance to a tunnel. Julie joined him, panting. She was shivering. The sky was violet, shadows cloaked the mountainside, and a pitted, yellowish moon floated at the horizon.\n\n\"There's nobody here,\" said Peter.\n\nThe labyrinthine stonework was indistinct. Julie approached the dark opening, stumbling amid pebbles and wild grass. Something rolled and clattered beneath her feet; she thought it was a can of food. A vague light twinkled in the tunnel. Julie went farther in and her forehead struck a bead curtain which began to jiggle and clink. Between its glass and wooden beads a room was visible. With a brusque movement of her left arm, Julie pushed the curtain aside and went on in.\n\nShe found herself at the side of a vaulted chamber furnished in nondescript fashion. A kitchen table, chairs, an ugly tiled floor.\n\n\"Hey!\" she shouted. \"Anyone here?\"\n\nA post office calendar was pinned to the enamel-painted wall. The picture was of cats in a basket. It was hideous. Julie contemplated it, swaying. She blinked. The calendar was immense. Was it a hallucination? The girl hobbled over to the wall and touched her hand to the color print. It was fifty centimeters long, at least. Julie gave a strangled cry, backed away, bumped into a chair, and felt her hair stand on end. The seat came up to the middle of her body and the table was almost on a level with her chin.\n\n\"It's the Giant's Castle,\" cried Peter.\n\nEyes wide with terror, Julie wheeled around and saw a man emerge from the darkness at the far end of the crazy room. He wore blue work overalls. Yellowish strands of hair fell down over his broad brow. It was Fuent\u00e8s. Julie tried to step back and felt herself falling.\n\n# 30\n\nTHOMPSON awoke sweltering. The sun was shining on his face through a gap in the foliage and the killer was bathed in sweat. Hastily he got into a crouch, his head swiveling like a weathervane, but the surrounding woods were calm. The singing of the birds and the sighing of the wind were the only sounds. Thompson consulted his watch. It had stopped. He looked up. Judging by the sun, the morning was well advanced. The man clicked his tongue with irritation. For years he had never slept so late.\n\nHe got to his feet. He felt ill. He had difficulty walking as far as the upper fringe of the wood. Prone in the grass, he scanned the round-topped mountains. Here and there, kilometers away and shimmering, were little brownish groupings of what were long-horned cattle. No human beings were to be seen.\n\nThompson scratched his cheek. His growth of beard itched. The killer was exasperated. He had to nourish himself, yet the very idea turned his stomach. If he could just manage to slip through the roadblocks, reach a town, and get in to see a doctor, perhaps he could be set to rights by a blood transfusion.\n\nBut no. The doctor would ask questions. And ask himself questions. The whole region must be in an uproar by now. \"Massacre in Montbrison,\" the headlines probably read. Thompson got up. His leg muscles were jittery beneath him. He made his way, under cover, to the stream. Once there, he lay on his stomach to drink.\n\nAs he lowered his lips to the running water something splashed downstream, then reached him in a flash of gray. Thompson extended his arm. He rolled over sideways, ending on his back in the grass with his fingers clasped about the gills of a wriggling trout. Thompson observed the fish's struggles, how it opened and closed its mouth convulsively. It was interesting. The killer placed his thumbs beneath the throat and forced the trout's head back. It fought even more frantically. Thompson increased the pressure and felt the neck snap in his hands. The killer felt a rush of happiness. He immediately gutted the fish with his fingers. It was no longer wriggling. He proceeded to devour its sides. The flesh was insipid and hard like that of raw mollusks. Thompson gulped it down feverishly. Bones scratched his throat but he forced himself to carry on, to swallow as much of the meat as he could while the creature's death still filled him with exaltation.\n\nHis nausea did not return until twenty minutes later, once digestion had begun. So all was not lost from the nutritional point of view. Thompson waited for the spasms to end before going back to his vantage point on the bank of the stream to look out for another trout. As he waited he began to think. There was not just trout. There were little animals too. He had never thought of that. Never in his life had Thompson hunted. But the idea of catching an animal and\u2014yes\u2014breaking it was perhaps a temporary solution, a way of countering his lack of appetite, and of surviving, for a little while, yes.\n\n# 31\n\nWHEN JULIE came round for the first time, she was alone and naked in a low bed in the middle of a vaulted room. Anxiety gnawed at her insides. Where was Peter? She called out.\n\n\"Peter!\"\n\nShe could hardly hear her own voice; she thought she must be deaf. She tried to sit up and managed only to get onto her elbows. Her head fell back onto a bolster without a pillowcase. With difficulty the girl rolled onto her side. Again she tried to call out.\n\n\"Uhh\" was the only sound that issued from her dry throat.\n\nHad that sound traveled sixty centimeters, it would have been like reaching the ends of the earth. Julie saw a flagged floor, a red perforated-plastic chair of the kind you see at the outdoor tables of provincial caf\u00e9s, and a large bare stone wall with a French window. Outside the light was blindingly bright. Julie could see nothing beyond the glass except for milky shadows overwhelmed by whiteness.\n\nA very short time later, the girl came to for a second time. The light had turned orange now. Julie pushed herself to the edge of the bed and let herself drop to the floor. There was a vague pain in her arm. She touched the spot. A crepe bandage encircled her biceps and she could not bend her right elbow. Fuent\u00e8s appeared in the orange opening.\n\n\"What the hell are you doing on the floor?\" he demanded.\n\n\"You murderer!\" Julie responded feebly, looking about in vain for a weapon.\n\nFuent\u00e8s leant down and took hold of the girl. He was bare-chested, wearing white linen pants, and their skin touched as he lifted Julie back into the bed.\n\n\"Where is Peter? Where is Hartog? What do you want? Are you going to rape me?\"\n\nIt was exhausting Julie to talk. She sensed that the brute was tucking her in.\n\n\"Sleep. You have nothing to fear.\"\n\nThe girl wanted to ask again where Peter was, but succeeded only in producing a bubble of saliva, like a newborn baby. After that she kept experiencing brief moments of relative lucidity. It was sometimes day outside, sometimes night. Fuent\u00e8s had her drink broth through a straw. She would choke on it and spill most of it down her front, which was no longer naked, she noticed, because she was wearing a man's shirt. Once she thought she saw Peter, but since he seemed to be wearing a wig she had to acknowledge that this was a dream.\n\n\"I'm able to talk now,\" said the girl at last.\n\nShe opened her eyes, astonished to have said it. Fuent\u00e8s was sitting next to the bed dressed in khaki shorts and an apple-green shirt, a gypsy shirt. He had a thick beard. His eyes had dark circles under them.\n\n\"Your fever has passed, at any rate.\"\n\n\"I'm completely whacked,\" said Julie.\n\nShe pressed her wound. The site was covered by a large square adhesive bandage. It did not hurt very much.\n\n\"Do you realize that you were fired at with an expanding bullet?\" asked Fuent\u00e8s. \"Part of it was still in you. I got it out. The rest had come out by itself. You're lucky you still have your arm. Who was the bastard that did it?\"\n\n\"Thompson,\" said Julie. \"As if you didn't know!\"\n\n\"Oh yes, right,\" he answered. \"Peter told me all about it. How I was the chief killer and all that.\"\n\n\"Peter? What have you done with him?\"\n\nFuent\u00e8s scratched his beard. Noisily. \"He's playing on the hillside.\"\n\n\"I don't believe you.\"\n\n\"Try a bit harder.\"\n\nJulie laughed nervously. Fuent\u00e8s began to laugh too. He took a pack of Gitanes from the pocket of his apple-green shirt and lit up.\n\n\"I'm not offering you one. It would only make you cough. You're not in any state.\"\n\n\"Have I been here long?\"\n\n\"Soon be eight days.\"\n\nJulie made an unintelligible exclamation. Fuent\u00e8s shrugged.\n\n\"I haven't killed you. Looked after you, more like.\"\n\nHis tone was even. He was checking things off. He might as well have been counting on his stubby fingers.\n\n\"The trouble with all this is that it wasn't me. Not you, either, according to Peter. But I couldn't make head or tail of his story. You know how kids are. He told me no end of crazy stuff. Gangsters, buildings on fire . . . You tell me your version. Maybe I'll believe you.\"\n\nJulie told him. A large sun shone in through the doorway. Dust danced in its beams. Fuent\u00e8s dragged on his cigarette, the paper of which was burning unevenly. As Julie told her story, the man seemed to nod off.\n\n\"Your tale,\" the man said at last, \"has absolutely everything. Even the calvary\u2014the ascent to the castle\u2014as the finale. You must have had a shock when you saw me.\"\n\nJulie nodded. In her mind she felt disarmed.\n\n\"This, where we are,\" he said, \"is my place. It never belonged to Hartog, and he didn't build it.\"\n\n\"He told me just the opposite.\"\n\n\"Yes. The rat. The dirty rat.\"\n\n\"Be quiet!\" cried Julie.\n\nFuent\u00e8s pursed his lips. He seemed very tired.\n\n\"Now it's my turn to tell a story,\" he said. \"Once upon a time there were two young men . . . Listen, once there were two young men . . .\"\n\n# 32\n\nONCE UPON a time there were two young men. Hartog. Fuent\u00e8s. Architects, working together. Penniless, more or less. They got along with each other pretty well. It was Fuent\u00e8s who had most of the ideas, if I do say so myself. Hartog merely followed suit. Seemingly in agreement. It was thanks to him that the partnership stayed afloat. His family, who were rich, would commission projects that were rarely realized but brought in fat fees.\n\nAll of a sudden Hans-Peter and Marguerite Hartog were killed in an airplane crash. Young Hartog became rich. His architectural projects could be realized. And it became apparent that they did not really agree, Fuent\u00e8s and he. Fuent\u00e8s did not want to build factories, or workers' housing. In fact it wasn't obvious what Fuent\u00e8s wanted. What is more, he was finding it harder and harder to work at all.\n\nHartog split up with him. Hartog busied himself with Hartog's money. On the side, he built. A museum, factories of his design. Fuent\u00e8s was furious. Should I carry on? The whole story had already been made into a novel, and a film by King Vidor with Gary Cooper, inspired by the life of Frank Lloyd Wright.\n\nFuent\u00e8s gave up practicing as an architect. Just like Gary Cooper, he took odd construction jobs as a laborer, a stone breaker, occasionally a foreman. Then he bought some sheepfolds in the Massif Central and started building on them in his leisure time, whenever he wasn't drunk (for it must be said that by now he was a serious lush). What he built was this idiotic kind of labyrinth.\n\nNow and then, when he tied one on in Paris, he would pay Hartog a little visit. He would insult him. Accuse him of stealing his ideas. Occasionally knock him around a bit.\n\nJulie was listening open-mouthed. Fuent\u00e8s got up, went out, and came back almost right away with a bottle of beer. He drank straight from the bottle and did not sit down again. He strode up and down the room, punctuating his narrative with grand gestures and scathing laughter.\n\nThe funniest thing about it is that Hartog is jealous of Fuent\u00e8s. I am quite sure of it. He is jealous\u2014it's as plain as day. He has taken pictures of this stupid labyrinth. Put them in his files. And the day you told him that the place was beautiful he pretended that it was his own private hermitage. The moron! The imbecile!\n\nFuent\u00e8s burst out laughing, then began to cough. One got the impression that he would not be able to stop.\n\n# 33\n\nJULIE was wearing her bloodstained shorts and the man's shirt, whose sleeves she had rolled up. She was beginning to be able to walk again, leaning on Fuent\u00e8s's arm. It was now nine days since she had left the burning store. She was inspecting the room with the outsize furniture where she had passed out on the night of her arrival.\n\n\"A fantasy,\" said Fuent\u00e8s. \"We tend to forget what it's like to be a kid. In here you feel as if you were only a meter twenty tall all over again. You must have had a terrible shock.\"\n\n\"Well, by that point . . .\"\n\nJulie perched herself on a gigantic chair. She giggled like a little girl. Fuent\u00e8s, beer bottle in hand, watched her sardonically. Through the window, narrow as a castle loophole, the girl could see Peter some fifty meters from the labyrinth. The boy was shooting arrows from a crude bow that Fuent\u00e8s had made for him. Julie could not get used to the new figure Peter made. In a single week he had acquired a tan and slimmed down; his hair had grown thicker and his attitude had changed. He was running now among the flowers. It was like a commercial for some kind of return to nature.\n\n\"I'm already fit enough to leave here,\" said Julie, who was no fan of nature.\n\nShe got down from the giant chair, went through the bead curtain, and picked her way through the debris beyond. The sun, still glorious, now shone full in her face. She heard Fuent\u00e8s coming up behind her.\n\n\"You have no place to go,\" the man told her.\n\n\"Straight to the police.\"\n\n\"That's a long walk.\"\n\n\"You don't have a car?\"\n\nJulie had turned round. Fuent\u00e8s shook his head.\n\n\"I had an old Jeep. The right thing for hereabouts. But I drove it into a ravine one night when I was drunk. Now I'll have to start saving up for another one. It pisses me off.\"\n\nHe put his empty bottle down carefully beside the wall.\n\n\"Come and see my kiln,\" he said.\n\nThey walked the perimeter of the labyrinth. Julie noticed a strictly horizontal stairway along a drystone wall and an inaccessible garden on top of a squat circular tower three meters high.\n\n\"You could stay here for weeks, even months,\" remarked Fuent\u00e8s as though the idea appealed to him.\n\n\"In my handbag is that photo,\" Julie reminded him. \"Sooner or later they'll figure it out, and turn up here.\"\n\n\"I have all I need to give them a warm welcome,\" said Fuent\u00e8s.\n\nThey were approaching his kiln. It was a stone-and-mortar excrescence on the outer wall of the labyrinth. Smoke issued from its base.\n\n\"What do you mean?\" asked Julie.\n\n\"I have a gun.\"\n\n\"You don't know Thompson. He's a frightening man. A gun won't stop him.\"\n\nFuent\u00e8s laughed derisively. \"A gun stops anybody.\"\n\n\"You don't get it. He'll be at us before you know it, and he'll kill us.\"\n\n\"You've seen too many horror films,\" said Fuent\u00e8s.\n\nHe opened one of the kiln's two doors, which were of thick rusted metal.\n\n\"I nicked these doors from a Nazi blockhouse in Normandy. Look, this is where I put the wood in.\"\n\nFuent\u00e8s piled logs onto the embers deep inside the opening. Julie touched the stone wall of the kiln. The stone was hot.\n\n\"My pots are above. I'm not opening up there, because they're being fired. They're all duds though. I mess up everything I do.\"\n\nHe slammed the lower door.\n\n\"As for firewood,\" he noted, \"there's no shortage. I simply have to go down the hill.\"\n\nJulie pictured him bare-chested, splitting logs. The picture was pleasant.\n\n\"I'll go down tomorrow,\" she said. \"Down to the cops, I mean.\"\n\n\"I'll go with you,\" said Fuent\u00e8s. \"You'll be in a really tight spot trying to explain everything and getting them to believe you. Do you really think they'll believe you? Anyway, you'll need someone. I'll find you a lawyer.\"\n\n\"I'm not afraid of going to the cops anymore,\" said the girl.\n\nA hundred meters away, Peter was playing in the flowers and shouting for joy.\n\n# 34\n\nHARTOG was spending more and more time in his chaise lounge on the Mediterranean coast. The redhead's lips were badly swollen from continual biting. His fair eyelashes were blinking faster than usual. He let his gaze roam over the sea. In the distance sailboats slid sluggishly along\u2014the wind was light. Nearer to the strand a small, potbellied rowboat was riding the swell.\n\nD\u00e9d\u00e9 brought the mail.\n\n\"Open it,\" said Hartog. \"Read it to me.\"\n\nD\u00e9d\u00e9 hesitated, then produced a nail file from his breast pocket and opened the envelopes. He dealt with each piece in turn before tossing it onto a low table near the redhead.\n\n\"Bill from a local supplier . . . Subscription offer for a series of books entitled Martyrology of Eros . . . A report from Mademoiselle Boyd.\"\n\n\"Read that.\"\n\n\"Hmm. 'Dear Sir, I must bring it to your attention that\u2014' \"\n\n\"Not aloud,\" Hartog interrupted. \"Read it, then tell me if there's anything important in it.\"\n\nD\u00e9d\u00e9 went quiet and, standing, perused the two sheets of paper.\n\n\"She says,\" he said at last, \"that your absence is more and more awkward. She lists the jobs requiring attention. She assures you that she quite understands, but must insist that you either return to Paris or delegate your decision-making authority.\"\n\n\"What business is it of hers?\" snapped Hartog.\n\nD\u00e9d\u00e9 made no reply and, after replacing it in its envelope, put the letter down with the others on the low table.\n\n\"All right, that's enough, that'll do,\" said the redhead. \"I don't want to be disturbed.\"\n\n\"Yes, sir.\"\n\nD\u00e9d\u00e9 left the deck on tiptoe. He had lost virtually all respect for his boss. The man had let himself be destroyed by this business\u2014you could hardly believe it. A kid he hadn't given a shit about before! True, there was less and less hope of ever seeing them again, the kid and the nutty girl. But so what? The kid was a pest. The one he, D\u00e9d\u00e9, felt bad about was the crazy girl. She was pretty, and she was fun in some ways.\n\nHe sat down in the hall of the villa and opened a copy of _Playboy._ From this post his job was to keep anyone from reaching Hartog.\n\nThe redhead remained stretched out on his chaise lounge, alone now, facing the sea, his eyes closed and his body rigid. He started when he heard something rattling the pebbles, and sat up straight. The squat little boat was significantly closer now, indeed it had just run aground four meters from Hartog. Thompson was stepping from it onto dry land. He was gaunt and his worn clothing was washed out; he had a large bag over his shoulder and a thick growth of beard had invaded his face. Hartog, however, had no difficulty recognizing his hired killer.\n\nHe rose from his beach chair, striking his shin painfully on the low table but paying it no mind.\n\n\"You are mad,\" he said. \"What do you mean by coming here?\"\n\nThompson came up onto the deck. He had no shoes on. His soles left wet imprints on the tiles. Hartog looked around nervously but there was no one to see them.\n\n\"Are they dead?\" he asked in an undertone.\n\nHis swollen lips quivered with excitement. Thompson shook his head. Hartog imitated him mechanically and his mouth gaped. In his despair he was like a little boy. He grabbed Thompson by the collar. The man struck him briskly on the wrists to make him let go. Hartog backed away, looking at the killer. This, he thought\u2014this guy was a professional? Why, it was sickening! To think how carefully he had planned his moves, burdening himself month after month with cripples, nut jobs, and freaks until it seemed quite normal, the crazy girl making off with the kid, hanging him, hanging herself. Such a beautiful scheme loused up by this incompetent, this underling, this wreck. But then the redhead took stock of Thompson's appearance. The man was clearly exhausted. His cheeks, beneath the beard, were hollow, his eyes sunken and red-rimmed.\n\n\"Can we talk calmly?\" Thompson asked. \"Is there somewhere we can talk things over for a few hours without fear of prying eyes?\"\n\n\"I don't want to talk things over with you for a few hours,\" said Hartog. \" I want you to leave. You were supposed to kill them and . . . and\u2014\" The redhead stamped his foot on the tiles.\n\n\" I am going to kill them,\" said Thompson in a very weary voice. \"I want to kill them if it's the last thing I do. That's why I need to talk with you. I need information.\"\n\n\"I gave you all the information you needed,\" said Hartog in a shrill murmur. \"Just leave. Get out of here. I have nothing to say to you.\"\n\n\"It's an architectural question, Monsieur Hartog. It concerns a . . . a structure. A sort of labyrinth. In the Massif Central.\"\n\nHartog looked at Thompson as if the man was out of his mind.\n\n# 35\n\nIT WAS getting to be dusk. D\u00e9d\u00e9 put his _Playboy_ down and went into the villa's kitchen. Hartog was giving no sign of wanting to go out. Once again the driver would have to make dinner. Muttering, he opened the white cupboards and the gigantic refrigerator. Their stores were diminishing. For ten days now Hartog had barely left his beach chair save to go and sit at a corner of the table to eat the eggs or grilled meat that D\u00e9d\u00e9 cooked inexpertly for him.\n\nOn this occasion D\u00e9d\u00e9 chose a can of cassoulet. He opened a drawer in search of a can opener. At that moment he caught the sound of voices on the other side of the wall. The house was solidly built, largely soundproof. D\u00e9d\u00e9 could not make out any of the conversation. But it could not be the television\u2014the stationary set was at the other end of the villa and the portable one was on the blink.\n\nThe driver picked up the home intercom and called the living room by pressing a button with his thumb. At the other end of the line there was no immediate answer.\n\n\"What is it?\" came Hartog's voice at last.\n\n\"I heard voices, sir. I was wondering. . .\"\n\nD\u00e9d\u00e9 hesitated. At the other end Hartog hesitated too. The driver could hear his whistling breath.\n\n\"I'm with someone,\" said Hartog. \"We're talking business. He came via the beach. I don't want to be disturbed. Be discreet, Andre my boy. Discreet.\"\n\n\"Very well, sir.\"\n\n\"Hold on a second, Andre. Would you run into town and buy a chicken or a rabbit. Alive, please. Call me on the intercom when you get back.\"\n\n\"Very well, sir.\"\n\n\"That'll be all. But hurry up.\"\n\n\"Very well, sir.\"\n\nD\u00e9d\u00e9 hung up. Hartog must be off his head. A live rabbit! The driver went back to the hall, took the keys to the Fiat from their hook, and went out. He locked the front door behind him. Discreet, discreet . . .\n\nIn the living room, Hartog had hung up a moment before his man. He was on his feet, shoulders propped against the wall by the intercom. Thompson was sitting opposite him in an armchair, hunched forward, hands dangling between his legs.\n\n\"And just when did this thing about the photograph come back to you?\"\n\n\"The day before yesterday,\" said the killer. \"You see, I never stopped going over things in my mind. I was looking for some detail that would get me back on the trail. I have no idea why I didn't think of the photo earlier. But it was not like an address . . . Not something that you would remember automatically. And I only saw it once. And when I did see it, I was angry.\"\n\n\"Then what?\"\n\n\"On the back of the photo were the words 'Massif Central.' Rather a large area. I could scarcely start searching it. And I was very weak. It occurred to me that you, as an architect, might know. One would not easily forget the structure shown.\"\n\n\"No,\" said Hartog. \"True enough.\"\n\nThe redhead had regained his self-confidence.\n\n\"I feel you're going to tell me where it is, no?\" said the killer.\n\n\"I'm not sure. You don't seem to be good for much.\"\n\nThompson stood up. His fingers with their long, dirty nails scratched nervously at his tangled beard.\n\n\"Killing them, that girl and that kid, is all that I am good for. If I don't kill them, it'll kill me.\"\n\n\"What you need is a good psychoanalysis. You have a psychosomatic ulcer. Listen, I'll give you money to disappear. Go wherever you want, so long as it's very far away.\"\n\nThompson shook his head. Hartog pushed away from the wall and strode up and down on the blue-gray carpet, his hands behind his back, his chin sunk onto his chest, as though deep in thought.\n\n\"I do have you over a barrel,\" said Thompson. \"I'm not out to profit from that. I just want to finish my job. And to do so I need information from you.\"\n\n\"The girl may never have gone up there at all,\" said Hartog in a smarmy tone. \"And even if she did, she may very well no longer be there.\"\n\nHe went behind a kind of desk, quickly opened a drawer, and just as he was taking out the Arminius in order to gun Thompson down, was struck in the pit of the stomach by a glass ashtray whose impact was like the kick of a mule. Hartog doubled over and fell. He clutched at the drawer, which came out and overturned. The Arminius ended up on the carpet thirty centimeters from his hand. Curled up in pain, the redhead rolled over and reached for the little revolver, but Thompson kicked the weapon and it slid away along the wall for three or four meters.\n\n\"Be reasonable, Hartog,\" said the killer, showing no trace of anger.\n\nHe leant over the redhead, helped him to sit up and relax, massaging his stomach in a practiced way. Hartog's pain ceased to be excruciating. He sighed.\n\n\"You still have your reflexes,\" he observed.\n\n\"You see, you can still count on me,\" said the killer.\n\nHartog got gingerly to his feet. He looked at the floor, smiling distractedly.\n\n\"All right,\" he murmured. \"But I'm going with you. There is a man up there, a man who . . . Listen, don't interrupt me.\"\n\nThompson leant against the desk.\n\n\"That bastard,\" Hartog went on. \"I know what he's been telling that girl. I know what he thinks. I'm a murderer, am I? I have little children killed. For money, nothing but money. But it's not true, Thompson. I create beauty. I have already told you that.\" The redhead was shaking. \"He's up there with the girl and the little boy, and they aren't going to the police to denounce me because they are waiting for me. They've got it in for me, Thompson. I know it.\"\n\n\"You know it?\"\n\nHartog nodded.\n\n\"They'll have to be killed, all three of them,\" he said. \"The kid, the girl, the man.\"\n\n\"That'll be twenty thousand francs extra,\" noted Thompson.\n\n\"Yes, yes,\" said Hartog. \"That's fine with me.\"\n\n# 36\n\nONCE HARTOG and Thompson saw eye to eye, the killer washed, shaved, combed his hair, and donned clean clothes. In his closets Hartog kept regular-size garments not his own just in case the odd guest needed helping out. Thompson was provided with white pants, an operetta sailor's jersey, and a blue blazer. He had kept a healthy mustache and a short beard and looked like a Royal Navy reservist\u2014albeit an insomniac one on account of the bags under his eyes, the cavernous cheeks, and the dry skin.\n\nD\u00e9d\u00e9 the driver returned at nightfall and called Hartog on the intercom. The redhead emerged from his sequestration to get the red hen that his man had procured and carried it into the living room. Thompson took the fowl and shut himself up in the bathroom. Hartog made no attempt to monitor him. The killer had told him about his problems. Hartog had no wish to see this.\n\nWhen Thompson reappeared, his face and clothing bore not the slightest trace of blood. As for the bird's feathers and carcass, they had been gobbled up by the waste disposal unit.\n\nAs for Hartog, just imagining the business made him lose his appetite for quite a while. Along with Thompson he studied maps of southwest France. The killer and the redhead were at present not far from the Spanish border. They decided on the route they would follow to the Moorish Tower and reckoned the time it would take. They were agreed that they should move in at dawn. Hartog informed D\u00e9d\u00e9 that he was taking the Fiat and would be gone until the next evening. Around ten o'clock the redhead and the killer went straight to the garage and set off.\n\nDuring this time, in the labyrinth, Julie was sleeping peacefully.\n\n# 37\n\nSHE SAT up straight. A noise had woken her, but perhaps she had dreamt it.\n\nShe saw a white light through the curtains. Her watch said it was five thirty. Dawn. Julie had no desire to get up at dawn. She laid her dark head once more on the pillow without a pillowcase. But again there came a slight noise, from the hallway, like the sound of a large mouse. Julie got out of bed and slipped into her shorts. She went out of her room. In the blue half-light of the hallway, Peter, with his bow under his arm, was picking up arrows that he had dropped on the floor.\n\n\"Go back to bed this minute!\"\n\n\"I'm not sleepy anymore.\"\n\n\"It's far too early to get up.\"\n\n\"This is our last day,\" said the boy. \"We're going to go home, and then where will I play with my bow and arrows?\"\n\nJulie untangled her hair with her fingers.\n\n\"Oh to heck with it!\" She sighed. \"I might as well make coffee.\"\n\n\"While you're doing it,\" cried Peter, dashing down the hallway, \"I'll play with my bow and arrows. I'll come back in when the coffee is ready.\"\n\nHe raced off.\n\n\"Peter!\"\n\nHe was gone.\n\n\"Goddammit!\" said Julie.\n\nShe went into her room and sat down on the bed, but she no longer felt like going back to sleep. She drew the curtains. The sky was the color of milk, the vegetation dripping with dew. Peter was going to catch his death of cold. Drat! The girl left her room again, went down the hallway and out onto an interior courtyard with a stream running through it. She brushed her teeth, freshened up, then went into a hexagonal kitchen and put a coffeepot on the propane stove. The panes of the kitchen's two windows turned yellow as the coffee began to brew.\n\nWhen the coffee had finished brewing, Julie opened one of the windows and looked for Peter, meaning to call him, but he was not to be seen on the broad terrace outside. The boy had gone beyond the edge. Down below, in a wooded hollow, was a muddy track. The Fiat was stuck there, and Thompson and Hartog were trying vainly to free it up. The two were infuriated. They were way behind their planned arrival time. They had spent the last two hours losing their way down primitive, barely passable roads that never led where they wanted to go.\n\nPeter could hardly make out the car and the two men through the pine branches. Persuaded that what he was looking at was a stagecoach in difficulty, he tried to creep up soundlessly and remain out of sight until he could shoot an arrow at the palefaces.\n\nJulie poured herself a bowl of coffee, touched it to her lips and burnt herself. She put the vessel down and left the kitchen. She almost lost her way in the web of corridors and rooms. Then she stepped into Fuent\u00e8s's room. The failed architect was lying on his back in bed, wearing khaki shorts. Empty beer bottles were strewn across a good half of the room. He had dried beer on his chest. His thoracic hair was sticky with it. He was snoring. Julie contemplated him with commiseration and chagrin. She regretted the fact that he was not a handsome young man and that he had not tried to possess her. She would have struggled, scratched his face no doubt, and in any case men did nothing for her, but, all the same, she regretted it.\n\nShe was still regretting it when she heard the first gunshot.\n\n\"To blazes with it!\" Thompson had just said. \"To blazes with the car. We're only a few kilometers at most from that labyrinth affair. Let's walk it.\"\n\n\"We'll have to haul ass,\" replied Hartog, \"once the job is done. We'll need the car.\"\n\nWith great nervous energy he began breaking off pine branches and sliding them under the Fiat's wheels. Thompson shrugged, taking his sailor's canvas bag from the car and extracting the case containing his weapon. Opening it on the back seat, he assembled the gun with great care. The magazine was full. Rifle in hand, he turned round smiling. At that instant, through the branches, at ground level some fifty meters away, he spied Peter's face. A hemorrhage made itself felt in his stomach. In a single motion he shouldered the gun and pressed the trigger. The rising sun was briskly evaporating the dew, causing a slight distortion of the light. Instead of hitting the boy exactly in the middle of the face, the round tore off an ear.\n\n# 38\n\nJULIE jumped and spun three hundred and sixty degrees. Less than a second after the shot she was already unsure whether it was gunfire. She rushed to the window and scanned the broad terrace, now bathed in sunshine, with its pink and yellow sprinkling of high-stemmed flowers. Peter was nowhere to be seen.\n\n\"Wake up!\" shouted the girl, without turning round, to the sleeping architect.\n\nPeter was coming back up the hill as fast as he could run, holding his bow and arrows tight in a loving hug. The idea of crying out did not even occur to him, for he was shocked and terrified to the highest degree. Blood streamed from his torn ear. Thompson emptied his magazine firing at the boy, and the bullets ricocheted wildly among the branches, spraying pine needles, fragments of wood, and drops of dew and sap in every direction. None hit Peter. Doubled over in pain as he was, the killer was shooting like a clod. He retched and vomited a jet of frothy blood mixed with bile onto his rifle.\n\n\"Fucking Christ!\" hollered Hartog.\n\nThompson did not even hear him. The killer forged ahead through the branches. Pine needles lashed at him, scratching his face. His viscous pink spittle was left dangling behind him in repellent long strands. The mucus shimmered as the sun rose higher. Thompson climbed the hill like a hare. He slowed down when his rifle's pin struck nothing. Grimacing, he searched his pockets without halting and stuck a handful of cartridges between his teeth. He detached the magazine from the rifle and reloaded it nimbly, still climbing. There was bile and blood on the cartridges he loaded into the magazine.\n\nJulie was shaking Fuent\u00e8s violently.\n\n\"Oh, leave me the fuck alone!\" grumbled Fuent\u00e8s.\n\n\"Wake up, for Christ's sake!\"\n\nFuent\u00e8s sat up straight. \"Z'appening?\"\n\n\"It's Peter! They're attacking!\"\n\nFuent\u00e8s covered his eyes with his hand and sighed. Julie shook him again, near her breaking point.\n\n\"Where's your gun?\"\n\nBy this time Hartog had sized things up: Thompson racing ahead and the car stuck in the mud. He swore between gritted teeth, leant inside the Fiat, grabbed the Arminius, and stuffed a box of S&W .32 cartridges into the pocket of his wool jacket. Then he rushed off in the wake of his hired killer.\n\nAs Thompson was reloading, Peter covered thirty meters of open ground and reached the edge of the terrace. He raced towards the Moorish Tower, still about a hundred meters away, and began shouting at the top of his lungs. At last his eyes filled with tears.\n\nFuent\u00e8s threw open a cabinet. In the corner, leaning straight up against the side, was his gun. He knelt down and scrabbled frantically through the jumble at the bottom. There were cans of food, old decorating and architecture magazines, and all kinds of cardboard boxes.\n\n\"Where in God's name did I put my ammo?\"\n\nThompson was clambering up the steep slope on all fours, shaken by fits of coughing and nausea, his rifle under his arm and his index finger in the muzzle to keep water out. Hartog caught up with him and overtook him, skidding on the wet grass.\n\nPeter was getting close to the Moorish Tower, still yelling.\n\nFuent\u00e8s found the box he was looking for and opened it frenziedly, spilling the contents all over the floor: Manufrance 4 x 60 C buckshot cartridges, roughly cylindrical with a yellow head and a greenish cardboard case with an arboreal design. The man picked up his gun and began to load it. It was a four-shot model.\n\n\"You call that thing a gun!\" cried Julie as she left the room and ran towards the kitchen.\n\n\"It is a gun,\" noted Fuent\u00e8s commonsensically.\n\nPeter was still running towards the labyrinth. The right side of his face was covered with blood. When she saw him through the window, Julie moaned. She grabbed a carving knife.\n\n\"Come on!\" she shouted to Fuent\u00e8s.\n\nThe man came into the hexagonal room. He was still in shorts, barefoot, bare-chested, and locks of hair fell over his forehead to just above his eyes. He was holding his gun rather uncertainly. He came over to Julie, to the window, and saw Peter running up crying. Hartog and Thompson appeared on the far side of the terrace, about sixty meters behind the boy.\n\n\"It's Thompson,\" said Julie.\n\n\"And it's Hartog.\"\n\n\"Yes . . .\"\n\nFeet firmly planted, Thompson brought his weapon up to his shoulder. Fuent\u00e8s struck the window with the barrel of his gun. The pane collapsed. He fired. Hartog threw himself flat on his stomach. Thompson raised the barrel of his rifle and set off running on a diagonal.\n\n\"Watch out!\" said Julie. \"He's getting out of your sight.\"\n\nFuent\u00e8s made no reply. He fired a second time, and a third. After each shot he operated the forearm of the semiautomatic shotgun. The ejected head and singed cardboard case of the old cartridge would then fly up and bounce on the kitchen floor. The grass and flowers trembled twice over two or three square meters in close proximity to Hartog. The man moved in an odd fashion, crouched down and frantic, like a little spider.\n\n\"Watch out for Thompson!\" cried Julie again, tugging at Fuent\u00e8s's arm.\n\nThompson was running doubled up and leaping from side to side. As for Peter, he was headed straight for the kitchen.\n\n\"Get that kid out of the way!\" cried Fuent\u00e8s.\n\nPeter slipped and fell. Fuent\u00e8s breathed a sigh of relief and pressed the trigger of the shotgun. Hartog's silhouette flew up into the air. Like an acrobat. For a moment his feet were higher than his head. He landed in the thick grass and rolled into a ball. The corner of the Moorish Tower now hid Thompson from view.\n\n\"Ha!\" went Fuent\u00e8s with satisfaction.\n\nJulie dashed out of the kitchen. Fuent\u00e8s ejected his spent shell casing. Hartog squirmed in the grass. Fuent\u00e8s glanced about impatiently.\n\n\"Where are my cartridges?\" he demanded.\n\nJulie was no longer there to answer. She had emerged into the open air and run to get Peter, who was weeping and squirming. He was still clutching his bow and arrows. Just as they got back to the threshold of the Moorish Tower, something caused Julie's right foot to slip. She fell violently onto her back and heard a first shot echoing across the terrace. She got up right away and rushed with Peter through the open door. Something buried itself in the wall of the entrance with the sound of a branch snapping and then another shot rang out. Julie noticed absently that the sole of her right shoe had been completely torn off by the first. The sole of her foot was burning hot. Thompson had fired at her from the far extremity of the labyrinth. The girl had heard him breaking a window when he reached the corner of the structure. She examined Peter's ripped ear. His skull had not been touched.\n\n\"Hurry! Go and hide! Quick!\" she said, pushing him forward.\n\nHe took three steps, then turned back to Julie. He was in tears. He held his arms out to her.\n\n\"Go and hide!\" screamed Julie.\n\nShe succeeded in frightening him. He scampered off down the hallway. The girl got to her feet and started running back to the kitchen.\n\nHartog's shoulder was shattered, but he managed to stand. He held the Arminius in his left hand. Walking very unsteadily, he followed Thompson's example and made for cover.\n\nThe killer had disappeared into the Moorish Tower after breaking a window. He dashed down a shadowy passageway, shouldered a door open, and came out into an interior courtyard with a stream running through it. Hugging the walls, he whirled at great speed from one side to the next. Continually pirouetting, he was like a self-parody. All his reflexes were twice as alert as usual. He crossed the open-air space like a spinning top, went through a doorway, and found himself in an empty bedroom. He gnashed his teeth and rolled his eyes. His little beard was gummy. He was drooling.\n\nJulie entered the kitchen. Fuent\u00e8s was no longer there. He had returned to the cabinet. He was reloading. He stood up straight, gun at the ready, just as Thompson materialized at the open door. Against all logic, Fuent\u00e8s was taken aback to see a strange face. His finger hesitated on the trigger. Thompson dropped to one knee, below the shotgun's line of fire, and put a round into Fuent\u00e8s's shinbone, shattering it. The former architect fell to the earthen floor and screamed in pain.\n\n\"Drop your popgun,\" Thompson ordered. \"Tell me where the girl and the kid are.\"\n\nFuent\u00e8s shook his head. Without taking aim, shooting from the hip, Thompson shot him in the right elbow. The joint disintegrated. Fuent\u00e8s howled. His gun fell next to him on the soft earth.\n\n\"Don't be a dolt,\" said Thompson.\n\nWith his left thumb, Fuent\u00e8s pressed the shotgun's trigger. The buckshot traveled at ground level, throwing up a shower of soil and gravel and demolishing Thompson's foot. The killer almost let go of his weapon but caught it with one hand. With the other he clung to the door casing. Miraculously, he did not fall. Flabbergasted, swaying, he contemplated his completely crushed and lacerated foot in disbelief. Bone and flesh were mangled\u2014much was missing\u2014and blood was pouring out like water from a tap.\n\n\"I wouldn't have harmed you,\" he said in a sad tone. \"It's just the boy, and the girl too. The girl I must kill\u2014you wouldn't understand.\"\n\nConvulsed with pain, Fuent\u00e8s was trying with his left hand to work the slide action of his shotgun. Leaning his shoulder against the doorjamb, Thompson fired a third shot, hitting the wounded man in the stomach. At that moment Julie appeared behind the killer and plunged her carving knife into his lower back.\n\n# 39\n\nHARTOG was staggering on through the labyrinth. His shoulder was numb. He was also losing all sensation in his head. The redhead's temples were throbbing with fever. Otherwise, he was almost happy. He liked the surroundings. He crossed a room with giant furniture, went down a damp, dark, and probably underground passageway, then climbed steps that brought him back out into the open air amid a profusion of flowers in an elevated garden. From this practical vantage point he viewed the maze of roofs, roof terraces, and little courtyards that made up the Moorish Tower. He felt envy for Fuent\u00e8s, which reminded him that he had to kill the man. The Arminius was in his left hand. Hartog crouched among the flowers and kept watch. From not far away, behind the walls, came the sound of gunfire. He counted four reports. He waited.\n\n# 40\n\nTHOMPSON whirled round so quickly that the handle of the knife escaped Julie's grasp. The steel blade remained buried in the back of the killer, who was hovering in the middle of the room on his one good foot. The girl stood stricken in the doorway, white with terror, lips open but teeth clenched. Fuent\u00e8s for his part lay on his back giving no further sign of life. He was covered with blood.\n\nThompson grimaced oddly and tried to level his rifle, but he lost his balance and had to use his weapon like a crutch for support. He remained like this for a moment, doubled over, with the stock in his armpit and the handle of the carving knife quivering in his back.\n\nJulie fled.\n\nShe ran as fast as she could down the corridor, her mouth open wide in a virtual cry.\n\n\"I'm going to kill you, you bitch!\" declared Thompson, and then a string of obscenities in English poured from his mouth. He drew himself up. All the muscles of his face contracted and relaxed in chaotic fashion. He forgot his destroyed foot. Brandishing his rifle, now plugged up with earth, he set off, even walking on his stump, in pursuit of Julie.\n\nThe girl had gone racing up a staircase. Thompson glimpsed her heels just as they vanished. Julie came out into the open air on a roof garden. She gave an anguished cry when she saw that there was no way off. Then she saw Hartog pop up like a jack-in-the-box amid the flowers on another roof. She saw him raise his left hand, and the flash of a shot. She felt a violent impact\u2014here goes, she thought, this time it's the other arm\u2014and then she twirled and fell back down the staircase. She went bouncing from one step to the next, and she cried out.\n\n\"Uncle Hartog!\" exclaimed Peter, on the other roof garden.\n\nHartog turned and saw the kid ten meters from him. The redhead was shaking uncontrollably. Quickly, without first weighing the import of the death of people one kills oneself, he raised the Arminius once more.\n\nBelow, Thompson saw the girl who had given him such grief land at the foot of the stairs. He put his gun to his shoulder, aimed for the heart and fired. The muzzle of his rifle being plugged with dirt, the weapon exploded, and the explosion ripped off both hands of the killer as well as his jawbone. He fell flat on his face, dead.\n\n\"You bastard,\" said Peter meanwhile, shooting an arrow into his uncle's face.\n\nThe projectile, lacking feathers and poorly balanced, struck its target sideways, whipping across Hartog's eyes. Taken by surprise, the redhead gave a nervous yelp and took a little jump back. The ground crumbled beneath his feet. The man toppled backwards and crashed headfirst onto the potter's kiln three meters below. Under the impact, the vault, built of large stone blocks poorly bound by crumbling mortar, gave way. Hartog and the large blocks of stone toppled pell-mell onto the glazed pottery being fired below. The floor of the ware chamber gave way in its turn and the whole mass collapsed into the burning embers. Hartog's red hair caught fire, as did his clothing. His bodily fluids bubbled and evaporated. For a few short moments the heap of rubble shifted slightly, like a molehill. Then all movement ceased.\n\n# 41\n\nSUBSEQUENTLY, Fuent\u00e8s would undergo several operations and survive, and Julie would be detained for a week while her confused explanations were disentangled. Before that she'd had to muster enough strength to walk several kilometers before she ran into a shepherd who came to her aid. Later on she would spend more time in a sanitarium before vanishing into the great wide world. And she never saw either Peter or Fuent\u00e8s again.\n\nBut, before any of this, the boy, from the roof of the labyrinth, contemplated the smoking mass that contained the incinerated body of his uncle. He did not clearly comprehend how his uncle had ended up on the side of the bad guys, but since he had always detested the redhead the question did not strike him as of any great import.\n\nLeaving his vantage point, he made for the staircase down which Julie had disappeared. On the way, the boy performed pointless acrobatic tricks. But eventually he was at the bottom of the stairs, where he found Julie sobbing.\n\nJulie looked at him and pulled him close to her, still weeping convulsively and incapable of uttering a word, terrified as she was by the idea that Hartog might reappear. Peter wriggled out of her embrace, which made him uncomfortable, proceeded down the hallway, and examined the hideous remains of Thompson with curiosity.\n\n\"You're dead,\" he declared.\n\nTo make sure of it, he prodded the corpse with the tip of his bow. Then he went on, searching for Fuent\u00e8s, but he almost got lost and failed to find the gravely injured man. Eventually he came out into the open air on the side of the Moorish Tower opposite from the potter's kiln. The air was pure. It was a super day. Peter was delighted by the delay that the shooting had caused to his departure. His ear hurt when he touched it, but he didn't want alcohol put on it. He went off to play cowboys and Indians.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nLEGEND OF A MUSICAL CITY\n\nMAX GRAF\n\nPHILOSOPHICAL LIBRARY\n\nNEW YORK\n\nAll copyrighted material within is Attributor Protected.\nTO POLLY\nTable of Contents\n\nTitle Page  \nDedication  \n1 \\- The Musical Centre of the World\n\nThe March into Vienna  \nHouses of Music  \nLandscape and People  \nThe Viennese  \nFame and Traditions\n\n2 \\- The Last Chapter of Great Music in Vienna\n\nOn The Ringstrasse  \nThe New Vienna  \nEmperor Franz Josef\n\n3 \\- Meeting Great Composers\n\nRecollections of Johannes Brahms  \nThe Composer of the \"Queen of Sheba \"  \nHours with Hugo Wolf  \nStudies with Anton Bruckner\n\n4 \\- In the Opera House and Concert Halls\u20131891-1914\n\nThe Musical Capital of Nations  \nIn the Opera House  \nNew Concert Halls and Old Traditions  \nModern Music in Vienna  \nPopular music-Johann Strauss\n\n5 \\- The Last Hours 1918-1938\n\nRichard Strauss in Vienna\u2014The festivals in Salsburg\u2014Toscanini in Salsburg\n\n6 \\- Past and Future\n\nAn Outlook on History and New Life\n\nCopyright Page  \nNotes\n\n1\n\nThe Musical Centre of the World\nThe March into Vienna\n\nOn March 13, 1938, the soldiers of the German Reich, in full panoply of war, marched into peaceful Vienna. Through its ancient streets pounded the tread of infantry and the heavy clanking of cannon, tanks and Panzer trucks. The eyes of the young soldiers, beneath their steel helmet brims, stared straight ahead. They turned neither right nor left; neither toward the palaces nor the old baroque churches, nor upward toward the golden spire of St. Stephen's Cathedral. Gazing into the distance, the eyes of these marching men were cold, unemotional and earnest, and in their depths lay war.\n\nAeroplanes cruised over houses, towers and church domes like grey birds of prey. Few Viennese were in the streets. Inside the houses, families were just sitting down to supper. Through closed windows, the unified sound of marching feet and clanging noise penetrated the rooms, chilling the hearts of all who heard with gloom and apprehension. Hours passed before the march of the German soldiers finally ceased, and the drone of aeroplanes persisted far into the night.\n\nIn the streets, no rejoicing greeted the victorious army. Vienna, which usually took such great delight in festive celebrations, was silent. There were no smiling strollers to be seen, no lovers embracing in shadowy doorways or sitting on the benches of the parks and promenades. This was no time for love. And from the taverns came no revelers, faces flushed with wine, singing songs of Vienna and its lovely women. The night seemed not only darker, but more oppressive.\n\nIn that hour, when a brutal hand grasped the city by the scruff of its neck, there was more involved than war. All European culture was threatened. In Vienna, which found itself transformed overnight into a typical German \"big town,\" all the independent spirit, the joy of living, the freedom, the natural intelligence, were harshly forced under the yoke; while a national fanaticism, imported from Germany, murderously stormed through this city which had never been fanatical, but always humane, gay and friendly. What could Vienna mean to the world without its famous joy of living, which had survived so many troublous times, without its culture, nourished from so many different sources, without its sensuality\u2014Vienna, the \"Falstaff of German cities,\" as it was once termed by one of its ironic poets? A legendary city of pleasure, it was quickly being transformed into a prison like those erected throughout Europe in the wake of soldiers in field grey uniforms. What could it mean for the world which loved this city as one loves a beautiful, smiling woman?\n\nFor the world at large, Vienna was, above all, a city of music, or better, the city of music; in fact, the only city in the world which would have been unthinkable without music. One could no more imagine Vienna without it than Rome without St. Peter's and the Vatican, Paris without its boulevards, or New York without Wall Street and skyscrapers. Vienna was the city where great composers had lived. It was the city of Haydn, Mozart, Beethoven and Schubert; of Brahms and Bruckner, Gustav Mahler and Arnold Schoenberg. From it had emanated the Strauss waltzes which flowed around the world, everywhere preaching the gospel of life's enjoyment in three-quarter time. From Vienna, too, came the operettas of Lehar, Oscar Strauss and Fall. The city was as full of music as a vineyard with grapes. Not only concert halls and theatres vibrated with sound, but the air itself. As Paris was the city of the mind, Rome the center of the Catholic world, London the capital of the greatest empire of modern times, so was Vienna the acknowledged music capital of the world. Musicians from all corners of the globe came there to study. Virtuosi came, too, because success in Vienna was a prerequisite of being \"world-famous.\" The Vienna State Opera, the Vienna Philharmonic Orchestra, the Vienna Conservatory were internationally recognized. Every tourist visited the houses where the great classics had lived just as, in Rome, he would visit St. Peter's and Michael Angelo's statue of Moses. Every musician who came to Vienna visited the graves of the classic musicians in the same spirit as a pious Catholic makes a pilgrimage to the tombs of the saints in Rome. Everything of greatness and of value which had been accomplished in Vienna\u2014the famous medical school, the Art Academy, the new living quarters for workers\u2014was dwarfed beside its place in music.\n\nIn the course of centuries, Vienna had become a kind of fairy-tale city. And, as in all fairy-tales, life there was easier, more brilliant and more exciting than anywhere else. Everywhere there was song and sound. The fairy-tale omits to say, of course, that there, as in all modern metropolises, poverty and misery stalked through the streets, in its outskirts, and that, despite the fabled gaiety, hard and earnest work was done. The tale does not speak of the political battles which took place there, nor of the colossal rise of the workingman to political rights, to education, culture, and finally, to political might. According to the saga of Vienna, there was only loving, dancing, singing, drinking and music-making. This legend has been propagated primarily by the motion picture which, for the masses of our time, is the biggest story-teller of them all. Everyone has seen films whose scenes were laid in Vienna. In such pictures, certain connoisseurs of mass taste depicted the fairy-tale Vienna. There were always poor but decent girls who fell in love with dashing officers, nobles and grand dukes. Lovers sat in Prater inns under chestnut trees, and everywhere there was music. Orchestras played waltzes, peasants sang Viennese songs. There were kisses and embraces during the music. Although this type of Viennese film belongs to the most mendacious products of fabrication, it is none the less characteristic that in every case Vienna and music are inseparable.\n\nOn that sad March day which was the beginning of a great world catastrophe, the city of music was struck dumb. As in Haydn's \"Farewell\" Symphony, the musicians packed up their music and their instruments, and the candles burned down in the music stands. A great epoch of music came to an end on this day.\n\nTen generations had labored at the development of Vienna into a great music capital. The long process had begun at the end of the 17th century, when Vienna had achieved the rank of a European city noted for music. From that time on, the graph of its development rose higher and higher in an unbroken ascent. Its peak was reached at the time when the classic musicians lived in Vienna, and on this marvelous construction, the powerful towers of the Haydn symphony, the Mozart Opera, the Beethoven Music of Humanity, and the Schubert song stood in all their glory. Then the graph descends to points which are, nevertheless, still high and strong\u2014Brahms and Bruckner. Finally came the great demolisher of the architecture of classic music, Arnold Schoenberg. There ended a unique and great development such as the world had seen only twice before: first, when Greek art rose to the brilliance of the Periclean Age, to Phidias, to the tragic poets, to Plato, Aristotle, to the divine laughter of Aristophanes, and to the mighty marble pillars which looked down from the Acropolis to the blue sea. And a second time, when the painting of the Renaissance had risen to Raphael, Michael Angelo and Titian. Vienna's development was a similar one, showing a steady growth of artistic fantasy, an increasingly large creative scope, the work of generations, a continuously richer unfolding of the powers of intellect. It was like a symphony, which flowed on to more and more powerful crescendos, to magnificent climaxes, and then, in 1938, died away with a dreary and mournful sound.\n\nFor almost three centuries there was great music in Vienna. While Venice, Rome, Naples, Paris, Dresden, Munich, Leipzig and Berlin existed simultaneously as musical centers, they could offer no comparison in the extent and length of their development. In their cases, the superb picture of a clear-cut epoch of history to be seen in Vienna, was lacking, as well as the close bond between music and life which characterized the city. Music had traversed its length and breadth; the Emperor's castle, the nobles' palaces, the townspeople's houses, the city proper and the suburbs, the squares and gardens, the churches, and the winehouses on the fringe of the Vienna woods. Each new generation which sprang up in Vienna wished to outdo the old one which had just been lowered into the grave, in the cultivation of music. Children were brought up as musicians and music lovers. Thus it was possible that even the rather foolish and superficial aristocrats of the 18th century comprised a most sensitive musical public who understood and loved Haydn, Mozart, and even Beethoven. One of the most stupid rulers of Austria, Francis I, used to sit, on summer evenings, in his castle on the Danube, playing violin in a string quartet. In the period of the most narrow-minded reaction after the Napoleonic Wars, during which the intellectual life of Vienna was suppressed and the greatest Austrian poet, Grillparzer, locked his finest works in his desk because he did not wish them to undergo police censoring, the elderly Beethoven, Schubert, Lanner and Strauss the Father, uninhibited by censor or police, kept on writing, and the city was full of music. Here music was not taken as occasional entertainment. It belonged to the people, like the homes in which they lived and died, like the clothes they wore, like the people whom they loved, and like the happy and unhappy hours they experienced.\n\nUpon the German occupation, all this came to a dead end, just as did the chatter at the caf\u00e9s on the Parisian boulevards, the gossip in the Brussels restaurants, the pilgrimages to Rome, the sensuous gypsy music in Budapest, the elegant frivolity in Warsaw, a short time later. Europe was becoming a battleground, a police barracks and a dreary prison. Disturbances raged everywhere. Death rained out of the air. A new age, a new social order, a new world history were being born amid a thousand pains, cries of anguish and streams of blood. The artistic construction of Viennese musical life was shattered like the other priceless treasures which belonged to the greatness of European culture. An old and noble epoch of Vienna's music became a pile of debris.\n\nAs a world music capital, Vienna was an artistic masterpiece. How many elements had to be assembled before such a city stood, and had grown into music! The people who lived in Vienna, the landscape surrounding it, the history which unrolled there, the social order of its society\u2014all these factors combined to make it what it was. The city's geographical position and the temperament of its population had their part in this development, from the baroque period of pompous clothes, wavy perruques and luxurious display until the modern times of electric street-lighting and the automobile, which brought this combination of nature and history to a completion. The inherent force which guided Vienna to its unique musical destiny was that which we call talent or genius. And during three hundred years this talent was equally original, equally strong and equally creative. When one considers the comparative brevity of the Elizabethan Age in England, one comprehends the extent of Vienna's creative talent which produced great music of all kinds for such a long time without becoming exhausted.\n\nIt required three centuries to make Vienna a musical city. One single day sufficed to destroy this historical edifice.\nHouses of Music\n\nAs from the pages of an old chronicle, from its buildings one can read the history which shaped Vienna into a great musical city. Although it was since the beginning of the second half of the 19th century that Vienna developed into a modern city, there are still parts of it which either have not changed at all, or changed only unessentially. Right next to fashionable streets with modern houses and the noisy traffic of the new era, there are silent old streets which are so narrow that they are dark and shadowy. Many of these streets, where houses dating from the middle ages still stand, are very crooked. The old houses have low doors, and small, lightless courtyards. There are palaces whose pillars support balconies and mythological figures carved out of sandstone, over the heavy cornices. Through these same dim alleys went Haydn, Mozart and Beethoven, and at night when all is still, one imagines he can still hear their echoing steps.\n\nThus the past is vividly alive in Vienna. Its musical history is bound to the present and does not lie dead in books. Walking through these streets, one realizes that here dwell contemporary musicians who are grandsons of the classical musicians of old.\n\nStarting a musical stroll at St. Stephen's Cathedral, from whose open \"Giant Door\" comes the sound of the old organ, one remembers, perhaps that in its choir Haydn and Schubert sang as boys; and that on a stormy winter's day, at the south door, Mozart's shabby coffin was blessed. I saw Anton Bruckner seated at the organ there, his short legs treading the pedal. On the manual, his bony fingers improvised variations on the Austrian national hymn, and his clear-cut, patriarchal face shone when the sound of the organ streamed forth, growing in strength, finally rising to a brilliant Gloria. Ever since the 16th century, when the famous Paul Hoffheymer played his Te Deum on the organ while Emperor Maximilian knelt praying, great musicians were at home in St. Stephen's.\n\nA few steps from the cathedral, at the head of a narrow, dark street stands a house from the time of Charles VI, where Mozart composed the music for \"The Marriage of Figaro,\" and not far away is the \"Deutsches Haus\" where he lived when he came to Vienna as a member of the suite of the Archbishop of Salzburg. In this massive building, Mozart sat at the servants' table with the Archbishop's lackeys, cook and baker. Here he came for orders from his strict master as to in what noble's palace he was to play piano. Here, too, the Archbishop's chamberlain, Count Arco, showed him the door with a kick, when madly in love and furious, he requested his release from service. Through the narrow Wollzeile (\"Wool Row\"), which runs eastward from St. Stephen's, Mozart's coffin was carried to St. Mark's cemetery. Neither his ailing wife nor his friends followed his coffin which was hurriedly borne to the cemetery and, during a snowstorm, lowered into a mass grave, where it disappeared. Also, the house where, with the pallid hand of a dying man, Mozart wrote the score of his \"Requiem\" and died, still stands in a street which time has scarcely changed. A \"Haydn House\" is in the neighborhood. Here Haydn, already old, composed the pious prayer of the Austrian National Hymn.\n\nNot far from the cathedral is a little square where the bustle of the city traffic never penetrates. Here all is as still and peaceful as on an isolated island. Here is old Vienna, the Vienna of the 18th century, preserved as if through enchantment. Three buildings enclose the square. One is a baroque church from the time of the Thirty Years' War, in magnificent Jesuit style. From the roof, colored frescoes gleam. High red porphyry pillars frame the chapels, and marble statues of the saints adorn the altars. The second house is a low, broad one which, at one time, was a Jesuit cloister. The silent corridors are vaulted, and one's step echoes on the stair. In this building, Franz Schubert spent four years as a young student in the City School. Here the boy with the round Viennese face and blond hair played violin in the student orchestra. Here, too, he wrote down his first compositions. At that time, the clear laughter of boys rang out in the dusky halls, and on the stairs, Schubert and his friends used to play hide-and-seek. The third house was built during the reign of Empress Maria Theresia as a university. In it is a state-room with frescoes by an Italian painter, illuminated by concealed lights, and there, on the 27th of March, 1808, occurred the performance of Haydn's \"Creation\" at which the old composer made his last public appearance. The carriage of Prince Esterhazy brought him to the building. The Rector of the University and several musicians, Beethoven among them, assisted the old gentleman from the carriage. Trumpets and trombones sounded a fanfare as Haydn was escorted to the concert hall, and the ladies of the aristocracy wrapped him in their shawls so that he would not feel cold. After the first half of the concert, the composer, who was deeply moved by his work and the premonition of his death, was again taken to the carriage. Only a few years after, in the same hall, a no less famous Beethoven concert took place, when his Seventh Symphony and his musical battle picture, \"Wellington's Victory,\" were played for the first time. The most prominent musicians in Vienna played in the orchestra. Court Conductor Salieri beat the time for the cannonade, Hummel played the timpani, young Meyerbeer, Spohr, Mayseder, Dragonetti, Moscheles, Romberg sat at the music stands. At the piano was the deaf composer, shouting without being able to hear, springing up at the fortissimi, bending low during the soft passages, and shaking his leonine grey head. That was on the 8th and 12th of December, 1813. Thirty-five years later in the same room, fiery students made freedom speeches, harbingers of the Vienna Revolution of 1848. Across the University Square stormed the Viennese youth, wearing black, red and gold bands across their chest, crying \"Hoch!\" for freedom. Then for another hundred years all was quiet in the old square where time seemed to stand still.\n\nIf one returns again to the noisy, traffic-cluttered neighborhood of St. Stephen's, and follows the stream of promenaders who chatter, laugh and flirt along the Graben, one passes the house where Mozart composed \"The Abduction from the Seraglio\"\u2014music of springtime and a first great love. Nearby is a quiet street of nobles' palaces. The largest of these, massive and broad, is that of the Esterhazy princes. When Haydn came from Eisenstadt as conductor for this wealthy noble family, he lived here. At that time he wore the uniform of a house-servant of the Prince\u2014a light blue frock-coat with silver lace and buttons, white neck-band, blue vest, white stockings, dagger, wig and buckled shoes. At the palace door Prince Esterhazy's soldiers kept watch, and Haydn used to doff his three-cornered hat when he passed an officer. Then from his pocket he would pull the silver snuff box the Prince had given him, take a pinch, and think of the theme for a new string quartet which had just occurred to him.\n\nPalaces of other nobles are nearby, because this is the vicinity of the Imperial Castle, and the aristocracy built their own dwellings around it, with heavy doors, broad steps, decorative pieces and frescoes, with forged iron lanterns hanging in entrances broad enough to admit great, gold-ornamented carriages. The Palace of Prince Lobkowitz, the rich Bohemian nobleman, contained a magnificent music salon. High pilasters of grey marble supported a ceiling of colored paintings. Here, in 1807, Beethoven gave two concerts, the first presenting the Fourth Symphony, the G Major Piano Concerto and the Coriolanus Overture; the second including his first three symphonies, and arias from his opera, \"Fidelio.\" The aristocratic audience sat on richly embroidered chairs, while lackeys stood against the marble walls holding candles.\n\nBefore arriving at the square in front of the Imperial Palace, one passes a stately town house which attracts the eye because of its air of quiet distinction. Next to the grey house is a church, on whose roof stands a statue of St. Michael with drawn sword. Here the courtiers of the Emperor lie buried. In an attic of the town house Haydn lived, above the apartment of the court-poet, Metastasio. Metastasio permitted the young musician to instruct his ten-year-old proteg\u00e9e, Marianne Martinez, in piano. In return for this privilege, Haydn was allowed to clean the poet's boots and clothes.\n\nFrom his attic room, Haydn could see the Imperial Palace and the little theatre which snuggled against it like a baby chicken at the breast of a mother hen. Into this theatre, Gluck used to go, when he was conductor there\u2014very dignified, wearing embroidered silk clothes, sword at his side, and a costly walking stick in his hand. Here his \"Orpheus\" was performed for the first time, as well as his \"Alceste,\" which latter work the Viennese found a sad and boring \"Requiem.\" Mozart's \"Abduction from the Seraglio,\" the \"Marriage of Figaro\" and \"Cosi fan Tutte\" also had their premieres in this theatre.\n\nOn fast days and days of mourning decreed by the Catholic Church, the pious Hapsburgs permitted no operas to be given in the theatre. On such days the emperors prayed in the court chapel, and on Maundy Thursdays, they washed the feet of twelve poor men; while on Good Fridays they fasted. The only theatrical events allowed on such occasions were performances in the court theatre, but in the pauses between two parts of an oratorio, virtuosi were heard. In 1798, Haydn conducted his new military symphony in this theatre. Time and time again Mozart sat there at the piano, and he gave most of his subscription concerts there. Between the two sections of a Dittersdorf oratorio, he played a new piano concerto (1783) with cadenzas between themes in which his whole heart sang out. Emperor Josef II often came to his box on the left side of the stage and, as connoisseur, observed the fingers of the slight, pale Mozart tenderly flying over the keys. Beethoven, too, played here. During the intermission of an Italian oratorio, he played his B Major Piano Concerto in 1795. He played it once more in the same year when Mozart's widow sponsored a performance of her husband's opera \"Titus\"; and in the intermission, he played Mozart's D minor Piano Concerto, whose elegiac tenderness he loved especially. Where in the world was there a theatre like this, where Gluck, Haydn, Mozart and Beethoven passed regularly in and out?\n\nWhen I went to high school in Vienna, this theatre, replete with so much glory, was still standing. At that time, it was the most famous theatre in the world, with which only the Com\u00e9die Fran\u00e7aise in Paris could compare in splendor and tradition. Every evening, the greatest German actors played before the nobility and the wealthy citizens of Vienna in the small hall. Everyone present knew everyone else, and Emperor Franz Josef, who sat in the same box where Empress Maria Theresia and Josef II had sat before him, looked down over the society of his residence-city. At this theatre, Franz Josef used to call for his friend of long years' standing\u2014the actress Katherina Schratt, whose Viennese laughter rippled through the comedies like silver bells. Here, too, it was that Crown Prince Rudolph met the dark-eyed Maria Vecera, with whom he went to his death.\n\nWe young people used to stand in the gallery, kept in order by an ancient Sergeant, and clap till our hands were sore when Sonnenthal recited the Ring parable with the pathos of a rabbi, when Wolter emitted her famous tragic scream, or Baumeister, strong and simple, played his Judge of Zalamea and the entire audience wept. For us, it was a deep personal grief when the little theatre was demolished and the magnificent new Burgtheater was built on the Ring.\n\nIn the Imperial Palace itself was a second theatre, the \"Redoutensaaltheater,\" which was built in 1706 for Italian operas, but which was also used for the Austrian court balls. Mozart liked very much to attend the public masked balls which were held here. Dressed in the colored costume of a harlequin, he danced and joked, and enjoyed enormously squeezing the respective waists of his masked partners. Haydn, Mozart and Beethoven composed dance music for the Redoutensaal balls. Concert performances, too, belonged to the history of this theatre, and the names of all three classicists are linked to it. Haydn presented a student recital here, in 1795, which is especially noteworthy because his young pupil, Beethoven, played a piano concerto. As a famous composer, Beethoven returned to the Redoutensaal in 1814. Then he conducted his great battle picture, \"The Battle of Victoria,\" and sections of \"The Ruins of Athens.\" A short time later, his Eighth Symphony had its premiere here.\n\nIn Vienna and its environs, there are no less than twenty-eight houses in which Beethoven lived. Most of them are quite unchanged. Among the most imposing of these is the beautiful Pasqualatti House, its door decorated with coats-of-arms, which stands on a remnant of the old city fortifications. From his window, Beethoven could see the Vienna Woods. The house in the D\u00f6blinger Hauptstrasse, where the tempests of the \"Eroica\" first thundered, was, in his time, still surrounded by vineyards and stood in a rural neighborhood. In no Beethoven house is one so near the spirit of the immortal as in the small, low house on the Probusgasse in Heiligenstadt. It looks today as it did when Beethoven left it. Through the house door, one enters a small court where a flight of stairs leads to the modest rooms where Beethoven spent a summer. The windows of these rooms open upon a garden. Here, when the leaves of the garden trees were already beginning to fall from the branches, in 1802, he wrote the stirring \"Heiligenstadt Testament\" when he gave up hope of being cured of his deafness. The exalted, sorrowful voice of his great Adagios was in this room and in his soul when he wrote those lines. Nearby are three other houses where Beethoven spent summers. All stand in rural surroundings, far away from the city traffic. From all of them, it is not far to the vineyards, fields and the border of the woods where the composer used to rest and write notes in his sketch book. In Baden bei Wien and in Moedling are the old houses with ample courtyards and thick walls where the gigantic labors of the Ninth Symphony and the Missa Solemnis were undertaken.\n\nThus one can almost revive Beethoven's entire life in these walks through Vienna. But there is also an entire Schubert quarter, an idyll in the great city. Schubert was at home in the suburbs, with simple modest people who lived in small houses. The house where Schubert was born still stands\u2014an insignificant looking building, five windows broad. In its small courtyard, he played as a boy, and in the little garden adjoining, he first saw the flowers blooming and the trees adorned with green leaves. From here, one can see the baroque tower of the church where he was baptized, and where he often sang in the choir or played the organ. Here, too, he first fell in love with the beautiful daughter of a Viennese manufacturer, for she was also a member of the choir. Nearby is the school house where Schubert taught the suburban children reading and writing while his first master songs, \"Gretchen am Spinnrad\" and the \"Erlk\u00f6nig\" ran through his brain. Between the house where he was born and a large house in the Kettenbrueckengasse which belonged to his brother and in which Schubert died, his entire life was lived. The houses where he visited friends, the taverns where he drank wine, are all still standing in Vienna, and should Schubert return today, he could find his way about without any difficulty whatsoever. Vienna, large, modern city that it is, is still a Schubert town. It goes without saying that the people christened that most beautiful of all graceful Biedermeier houses still standing in Vienna: the \"Dreimaederlhaus.\" Schubert continues to inspire the imagination of the Viennese folk. He himself belongs to these simple people, and he has not become a historic figure but remained a living man\u2014the stocky, good-natured school-master who spoke the dialect of the folk.\n\nWith the exception of most of the Schubert houses and those where Beethoven summered, all the houses of musical memory in question, are in the narrow confines of the old fortress-city, the \"City of Vienna.\" But the suburbs, too, hold memories of musicians of bygone days. There is the little house, for instance, which belonged to Haydn as an old man, and where his aged, trembling fingers played the piano. He died in this house. French soldiers kept watch at the door, and presented arms when his oaken coffin was carried out to the nearby cemetery. In a town house on the main thoroughfare of Vienna, an inscription recalls that Gluck once had owned it, and also died there. One may wander as far as one will, even to the mountains which enclose Vienna, but one encounters reminders of the musical great at every turn.\n\nIn the little rococo theatre of the Castle of Schoenbrunn Gluck conducted. In one of its halls, the child-prodigy Mozart first played before the Empress and the Princesses, and then sat himself down on one Princess's lap. In the Schoenbrunn garden, Haydn romped widly as a boy, much to the annoyance of the Empress, who saw to it that he was properly spanked.\n\nHowever, it is not only classical music history's mementos that the music lover encounters in his Vienna rambles. In Hietzing, on the bank of the River Wien, stands a yellow country house with a tower, where Richard Wagner worked on the composition of his \"Meistersinger von N\u00fcrnberg.\" In the shadow of Salmannsdorf Mountain, on a steep, rocky path, is an ivy-covered cottage where Johann Strauss, as a boy, wrote his first wonderful waltzes. The \"Beautiful Blue Danube\" was composed in the garden of a house in the Praterstrasse which is still standing today; and the liveliest of all operettas, \"Die Fledermaus,\" in a Hietzing villa. There too, is the house where Bruckner wrote most of his symphonies, as well as the little gardener's house in the Belvedere, where he died. The house where Brahms lived and died stands in a shady street near the green cupola of the Karlskirche.\n\nOnly recently the old \"Freihaus\" was torn down. This had been a city within a city, a large edifice which was erected by Count Starhemberg in the 17th century. In one of its courts was a garden which, in the course of time, became very dilapidated. Here stood the little wooden pavilion where Mozart composed \"The Magic Flute,\" a bottle of champagne lying with his notes on a table nearby. (This pavilion can now be seen in Salzburg.) In another court was the small theatre where \"The Magic Flute\" had its premiere. This theatre was ideally suited to the suburbs, a rude barn with two galleries. One paid seventeen kreuzer for parterre seats, seven kreuzer for those in the gallery. On both sides of the stage were painted figures: a knight with a sword and a lady with a mask. In such humble surroundings one of the greatest masterpieces of all music had its first performance. Mozart, already ill and pallid, sat at the conductor's desk and laughed at the jokes on the stage.\n\nThe Saint Stephens Cathedral\n\nGuesthouse of the Monastery Heiligenkreuz in the City of Vienna\n\nI often passed through the three courtyards of the \"Freihaus,\" near the spot where the theatre had stood, and where later there was a garage (the \"Mozart\" Garage, of course). At night, if one's head were agreeably glowing with wine, one could detect a ghostly ringing in the air in this neighborhood. The tender sound would become more distinct and form itself into a jolly, folk-like melody with the words: \"Ein Maedchen oder Weibchen wuenscht Papageno sich\" (\"A sweetheart or a wife is Papageno's plea\"). The Freihaus was the real Mozart section of Vienna.\n\nA short time after the premiere of \"The Magic Flute,\" Mozart died in poverty. However, Schickaneder, the impressario who had produced the opera, was a rich man, and partly from the profits of this production, built a large new theatre on the River Wien. This Theater-an-der-Wien holds many cherished Beethoven memories. For a long time the composer lived in this theatre, working on his opera \"Fidelio.\" In November, 1805, it was performed for the first time. French officers sat in the parterre seats, because Vienna was occupied at that time by Napoleon's army. The court and the high society had fled Vienna. The atmosphere in the theatre that evening was bad, and Beethoven's opera was a fiasco.\n\nTwo years previously, in this lovely theatre hall, Beethoven had presented his first and second symphonies, and the oratorio, \"Christ on the Mount of Olives.\" The \"Eroica,\" too, was performed here, in 1805, for the first time publicly, and a shoemaker's apprentice shouted from the gallery: \"I'd give a kreuzer if it'd only stop!\" Three years later Beethoven presented his fifth and sixth symphonies, and parts of his Mass in C and his Choral Fantasy in the same hall. At this performance he himself played his Piano Concerto in G, and improvised. His Violin Concerto also had its premiere here.\n\nAnother theatre which, like the Theater-an-der-Wien, still stands in Vienna, was festively opened with Beethoven's music. The overture, \"Zur Weihe des Hauses\" (For the Dedication of the House) was composed for the opening of the Josefstaedter Theater.\n\nVienna's churches, too, contain brilliant chapters of musical history. As mentioned before, the St. Stephen's Cathedral is the Haydn Church, the Lichtenthaler Church, the Schubert Church. For the Waisenhaus Church in the Third District, young Mozart composed a mass and conducted its performance there. The Court Chapel, where Anton Bruckner was organist, is the real Bruckner Church. This small Gothic chapel is directly in the Imperial Castle. Here I often saw Franz Josef, Sundays, praying in his box, right above the altar. In the two galleries, sat the members of the Austrian nobility, raised above the people who stood in the chapel nave. In the third gallery was the organ, presided over by Bruckner, and the orchestra and chorus which played under the baton of the broad-shouldered and dignified Hans Richter. Before Mass began, the musicians assembled in the court of the chapel. Here came the \"Saengerknaben\" (Boys' Choir) in their embroidered uniforms, small swords at their waists; and the opera singers who were soloists.\n\nOne ends the musical walk through Vienna at the cemetery where the great musicians are buried. Here Haydn, Mozart, Beethoven, Schubert, Johann Strauss, Brahms, Hugo Wolf and many others have their homes below the earth. These homes, too, belong to Vienna, just as those others where they dwelled in their lifetimes.\n\nWhile no one accompanied Mozart's coffin to the cemetery, all Vienna was present when Beethoven's was borne to the Waehringer Cemetery. The magnificent funeral of a great musician was a great spectacle for the Viennese, and many a composer who, during his lifetime, was fought, obstructed and ridiculed by the people, would be carried to his grave amid the most extravagant pomp. I was present when Bruckner's coffin was blessed at the Karlskirche, before, according to his last wish, it was taken to St. Florian to be interred beneath the great organ. From the choir sounded the mournful music of Bruckner's Seventh Symphony. Beside the flower-decked coffin the dark-robed priests were praying, while the university students, whom Bruckner so loved, stood at attention with drawn swords. Near the coffin stood a short peasant who kept looking anxiously about him. This was Bruckner's brother who may well have been amazed at the enormous crowd gathered for the funeral. He resembled Bruckner, but his face was merely that of a peasant while the composer's had radiated a kind of holiness.\n\nA sad burial was that of Gustav Mahler in the Grinzing Cemetery, because only a few loyal members of his circle of friends were convinced of his greatness as a composer; and these followed his coffin like the adherents of a new religion who had lost their Messiah.\n\nIn any case, all Vienna was present at Johann Strauss's funeral, and the cemetery was crowded with people, of whom even the most humble knew the Strauss melodies by heart. The Mayor of Vienna made the eulogy. For, after all, Strauss's music was the true folk-music of Vienna, and he was the musical potentate of Austria whose kingdom and reign lasted longer than Emperor Franz Josef's.\n\nThus music in Vienna was not an isolated province where only musicologists and historians went to dig for treasure. It was no dead memory, but an ever active power, belonging to the intellect, to the way of life and the atmosphere of the city. And if one should destroy every building in Vienna, the earth would still be there over which so many great musicians passed; and the air of which they breathed.\nLandscape and People\n\nLike Vienna's streets, so its surrounding landscape is rich in memories of its great musicians. This same vivid landscape which holds incomparable charm for visitors today was often the source of inspiration for the glorious works of the past. At the Schreiber brook, which flows down from the Kahlenberg through the vineyards, Beethoven discovered his Pastoral Symphony. The brook ripples and rustles on today just as it did in Beethoven's time, and the birds sing in the trees along its banks.\n\nThe real poet of the Viennese landscape was Schubert. There is no music which is so close to nature as his. It sounds like nature, with its fragrance of blooming lilacs. Natural, too, is the line of his melody. His great C Major Symphony was already understood and described by Robert Schumann as a hymn to the Viennese landscape. It was Schumann, too, who rediscovered in this music the cathedral of St. Stephen, the Danube, and the Vienna woods.\n\nNot far from the brook where Beethoven heard his pastoral music is the charming village of Grinzing. Here the Viennese sit every fine evening in the gardens and courtyards of the many wine-houses, drinking the wine which is so famous and \"Maulfreundlich\" (friendly to the mouth), as the Viennese say, when they first smell the earthy fragrance which rises from the glass and disappears on the tongue. Here Schubert often sat with his friends, played waltzes on out-of-tune pianos, or thought out such music as the last movement of the G Major string quartet, which has all the exciting flavor of Grinzing wine. In an inn garden here, Schubert wrote his song \"Hark, hark the lark,\" on the back of a menu.\n\nThe moonlight passages of Arnold Schoenberg's \"Verklaerte Nacht\" grew out of the Viennese landscape. And Richard Wagner, in the surroundings of the Vienna woods, composed the poetic end of the second act of \"Meistersinger,\" with the moon rising in the silent night, casting its silver threads over the housetops.\n\nFrom the heights of the Kahlenberg one has the finest view of the great city, encircling the St. Stephen's Cathedral as a wheel its axle. The broad stream of the Danube winds like a silver ribbon through the Viennese plains, which extend as far as the Hungarian mountains, dimly visible in a misty haze. To the south, the mighty Alps appear on the horizon; and in the west lie forests. All and north along the forests' borders, grapes flourish on the clay-like soil which once the Roman soldiers transplanted from Italy when they established their camp on the Danube.\n\nEvery one of the classical musicians has looked down on Vienna from the Kahlenberg. Mozart, inspired by the Kahlenberg, wrote the merry \"Magic Flute\" duet between Papageno and Papagena, happy and guileless as the tender cooing of love birds. Beethoven stood here, serious, moved, and, as always, religiously impressed by nature, and here he wrote down the wonderful words: \"O God, what magnificence in such forest places. On the heights there, is peace\u2014peace in which to serve Him.\"\n\nFrom time immemorial, on these wide plains, many races have mingled. Here the roads of Europe meet. To the north, there are Slavic people whose melancholy and tender music is that of a people who have long lived under oppression, full of plaintive desire. From this Slavic north, from Silesia, Franz Schubert's grandfather emigrated to Vienna. In Schubert's music one often hears an echo of Slavic minor harmony, as well as in that of Dvorak and Smetana.\n\nWhere the Danube flows to the east and the plain disappears in mist, is Hungary, and farther on, Turkey and the Orient. From the Hungarians comes the gypsy music\u2014sobbing clarinets, passionate violins, crashing cymbals, melancholy preludes and twirling dances with the rattle of spurs, for the Hungarians are a nation of horsemen. In the music of all the classicists, one hears the sound of this intoxicating music. In his musical library, Haydn had a large collection of gypsy music; and in his string quartet, Opus 20, Number 4 (Allegretto alla Zingarese), in his G Major Piano Trio (Rondo alla Ongarese), and elsewhere he has either used the original or imitated Hungarian music.\n\nBeethoven wrote Hungarian music in his \"King Stephen Overture\"; and Schubert captured the sound which blew over the Viennese plains from the east in his \"Divertissement \u00e0 la Hongrois.\" For years, including the period of Brahms, who wrote his famous Hungarian dances and made the clarinet sob like an instrument in a gypsy band in his Clarinet Quintet, Hungarian music has been an important influence on much of the music Vienna produced.\n\nThe Orient, too, lent its colorful sounds to Viennese music. Turkish merchants, wearing turbans and silk robes, were no unusual sights in Vienna even during Mozart's time. When he wrote his opera, \"The Abduction from the Seraglio\" where, right at the beginning, Turkish cymbals, drums and triangles conjure up the fairy-tale atmosphere of the Orient, one hundred years had already passed since Sultan Soliman's Turkish army had pitched its tents around Vienna. As the army withdrew, the first coffee house was opened in Vienna, and coffee was served in the Turkish way, brewed heavy and fragrant. Since that time Vienna remained an oriental coffee-house city, where one dreamed, planned and did business while having coffee. Tobacco, too, came from Turkey, and still in the time of Franz Josef, one saw in all stores where tobacco was sold, the picture of a Turk with a long pipe in his hand, puffing forth clouds of smoke. Vienna was near the Orient, and Count Metternich liked to have it said that the Orient began right outside the eastern city limits. Therefore it is no wonder that Mozart and Beethoven wrote Turkish Marches, and that Goldmark became one of the masters of modern oriental painting in music.\n\nIn the south, the Alpine roads lead from the Vienna plain to Italy; from the beeches, oaks and firs of Vienna to cypresses and pines. Along these roads for centuries came Italian musicians, bringing their music to Vienna. The Viennese enjoyed the beauty and sensuality of the Italian music. They loved the music of the blue sea, the sounds and harmonies of the fishermen's barks, the melodies which the Venetian gondoliers sang when they rowed their black boats through the canals and under the marble bridges of Venice. Until the year 1859, Milan and Lombardy belonged to Austria. In 1866, only, it lost Venice. On the large piazzas of Milan, Verona and Venice, the Austrian officers used to promenade in the evening before attending performances at the opera. The Milan Scala was, until 1859, an Austrian opera house. As a matter of fact, for a certain time, the State Opera of Vienna, the Scala of Milan, and the San Carlo Theatre of Naples were all under the management of one director, the clever, stocky Barbaja, who organized opera theatres, gambling houses, coffee houses with the same enthusiasm and business acumen. Viennese soldiers returned from service in Italy, whistling Italian opera melodies. From Italy, too, Viennese officers imported the greeting \"Ciao.\" Though it sounded Chinese, it was a contraction of the Italian word \"schiavo\" (\"servant\"). From Italy the soldiers brought the Virginia cigar which has become the \"people's cigar\" of Austria. Austrian military bands brought back Italian music to all the towns of Austria. When Richard Wagner, after the love catastrophe with Mathilde Wesendonck, went to the then Austrian Venice, and walked, evenings, on St. Mark's Square he heard a military band there whose playing he praised highly. Thus, Italian music, even in comparatively recent times, crossed over the Alps to Vienna.\n\nThe characteristic music of all these other nations mingled with Viennese music and left its traces. The softness, the melancholy, the folk-song-like human quality of Slavic music took away the German hardness from Viennese music. The voluptuous richness of Hungarian music lent it sparkle; and the Italian melodies augmented the quality of its sound. The basis of Viennese music, however, was German, for even in ancient times, in Vienna, German blood had mingled with that of the Celts and Romanic peoples.\n\nThe Bavarian peasants who settled in Austria and changed dense woods into thriving farmland, must surely have had musical talent. Mediaeval folk-songs from Austria, as they were sung in village inns and on the \"dance meadows,\" have a decided power. With the Babenberger Dukes, who built their castles in Vienna, the agile, clever Franks entered the country and soon after started to fiddle and sing on the Babenberger estates. The greatest German poet of the middle ages, Walther von der Vogelweide, told all Germany that in Austria, he had learned \"singen und sagen\" (\"singing and saying\"). But its great period as a land of music did not commence for Austria, until, in the 17th century, the Italian musicians came to Vienna; when the Emperors and high society of Vienna spoke and sang Italian, and the city was like an Italian town. Vienna did not become the musical capital of the world as a national city, but as a supernational one. Supernational, too, were the Catholic ideas which waved their battle-flags in Vienna during the time of the Turkish Wars and the Counter-Reformation. Supernational, also, was the idea of the Austrian Empire which comprised peoples of different blood and language; supernational the broadening of the intellectual horizon of the time. A Frenchman of Italian ancestry, Prince Eugene, led the Austrian armies, and Austrian viceroys resided in Brussels and Naples. In the 17th century it was Catholicism which had made Vienna Europe's capital, and had given its music the rich background formed from the unification of French, German and Italian elements. During the 18th century, the great humanistic idea stepped into the place of the religious one. Mozart's \"Magic Flute\" and Beethoven's \"Fidelio\" and Ninth Symphony are perfect expressions of this humanism which originated in the musical city Vienna. Thus arose the music capital of the world which has been invaded by German nationalism.\n\nWhen one stands on the Kahlenberg, one comprehends all this at a glance. One sees the plain with its roads leading into the distance, the great river which is a broad bridge on which the cultures of Asia and Europe meet. One sees St. Stephen's with its golden cross shining over a land which, for centuries, has been accustomed to kneel in prayer under its high, Gothic arches. One visualizes the Roman soldiers and merchants coming from Italy, on the south; the armies of Prince Eugene marching in the east; from the west, merchants from Ulm and Augsburg carrying goods from the Orient on their carts; from the north, Czech businessmen and Slovakian toy-merchants, and on the Danube, ships carrying Turkish carpets and tobacco being pulled against the current by strong horses. One can picture the musicians who came from all sides: Italian opera singers, Hungarian cymbal-players, gypsies with contra-basses on their backs and violins in hand, Czech clarinet players and other orchestral musicians. All this colorful, bright, exotic and sensual band emigrated to Vienna, the capital of a great empire in whose army four languages were spoken, and their music mingled there with German music in the way spices mingle with the taste of a roast. It was this process, which continued through centuries, which brought Viennese music to its unique and unparalleled glory.\n\nSuch mixtures and minglings occurred elsewhere in Europe besides in Vienna. In the 17th century, in Munich, Dresden and Hannover, Italian and German music came together. Everywhere there were Italian court composers, court poets, singers and instrumentalists. In Berlin, later the capital of German nationalism, Frederick the Great, in the 18th century, sat in the new opera house, listened to Italian operas which were written by German composers, and gave his orders in French. A German composer, Gluck, and an Italian composer, Piccini, wrote operas for Paris. In London, during the time of Haydn, musicians from all over Europe streamed together during the \"season\" just as they did, a century later, in New York. From many sides Europe aimed at one goal\u2014a European music which should summarize the best in the music of all nations\u2014a music transcending the national, the music of a unified culture and education. What so many European towns had attempted, however, succeeded only in Vienna. Perhaps this success was due to the fact that, geographically, Vienna lay in the center of the map of Europe. Moreover, in the 17th century, Vienna became the political capital of Europe, for the all-powerful Hapsburgs resided there.\n\nDuring the time of Emperor Franz Josef, Vienna was still the center of all European nations. The aristocracy came there from their castles in Bohemia, Poland, Hungary, Croatia and Italy. The Emperor's officials came from all parts of Germany and Austria. The Minister-Premier of long years' standing, Count Taaffe, was a descendant of an Irish family, while his aides were mainly Hungarians and Poles.\n\nEven today in humble suburban homes, where the past is preserved like the scent of lavender in old chests, Viennese mothers sing their babies to sleep with a song which begins with the enigmatic words: \"Heidi Puppeja.\" They are nothing else than the Greek words \"Aeide Bubaion\" (\"Sleep, my little boy\"). But how did Greek words travel to Vienna and fit themselves into a lullaby? History gives the answer. In the early middle ages, Austrian rulers married princesses of the Greek Imperial Court in Constantinople, and with these princesses, Greek priests, scholars and courtiers came to the court of the Viennese Babenbergers. Greek nurses sang the children of the Dukes to sleep. And from the ducal castle, the lullabies stole into the more humble Viennese houses where small children lay waiting for sleep. Thus a Greek song came to cling to Viennese soil like gossamer, whose threads are borne from a distance by the wind and remain clinging to the branches of a tree or shrub in the harvest-field.\n\nMarvelous was Vienna's talent to change all men and music from the four corners of the globe into its own possession. Immigrants, after a short time, became genuine Viennese, just as if they had drunk some sort of magic potion to effect the transformation. Their temperament, their thoughts, feelings, lives and loves, all become Viennese. They spoke the Viennese dialect and made Viennese music.\n\nNo one has even been a greater glorifier of \"Viennadom\" than Johann Strauss. What the world knows of Vienna, it knows mainly from his waltzes. They contained all of Vienna\u2014the gaiety, the laughter, the love, the beautiful women, wine and song, the landscape, woods and streams. And did not the music of \"Fledermaus\" exactly depict the chatter of the Viennese salon, the elegance of the men, the amiability of its ladies, the hilarity of champagne's intoxication? In the Strauss polkas lay the babble, the merry, mocking wit of Vienna. In the Strauss marches, one saw the typical Viennese soldier, his hat cocked over one eye, Virginia cigar in the corner of his mouth. All this was Vienna in its Sunday best\u2014not the everyday Vienna, but Vienna in a festive mood. This was the ideal picture of Vienna, the dream, the fairy-tale. And yet this musician who invented the fairy-tale which the whole world believed, was not a complete Viennese. His grandmother had been born in Spain, and had the surprising maiden-name of Roger. Family history recounts that she was the daughter of a Spanish aristocrat who had had to flee his homeland because he had killed a grandee in a duel. She played the Spanish guitar in Vienna, and sang romances to its accompaniment. All the Strausses are a foreign, non-Viennese, southern type. Strauss's brother, Edward, had the jet-black hair of a foreign race, with an ivory complexion. His brother Josef had an olive complexion, dark eyes, and looked like a romantic gypsy. Johann Strauss, too, was dark, and the writer Heinrich Laube, when he saw Strauss for the first time, playing violin at the head of his orchestra, said that he was \"black as a Moor, with curly hair, the typical King of the Blackamoors.\"\n\nThere were other glorifiers of Vienna who themselves were foreigners, or who had had foreign ancestors. The greatest Viennese popular actor of Franz Josef's time, who made the people laugh and cry like no one else, was Girardi. As his name shows, he was of Italian descent. The writer who pictured the humble, modest class of Viennese with greatest success was named Vincenz Chiavacci. No other Viennese writer described so vividly the earthy market-women, the bourgeois men, with golden watch-chains across their stomachs, sitting, talking politics at inn tables. No other writer painted this scene so true to life as did this Viennese who, also, had his origin in Italy. The greatest actor of the Burgtheater, whom Emperor Franz Josef elevated to the nobility, and whom Vienna considered its most representative portrayer of men, was Sonnenthal, who came from Hungary.\n\nThe most popular song of Franz Josef's time was the \"Fiakerlied\" (\"Song of the Fiacre Driver\"). Old and young, rich and poor sang it, but only a Viennese could sing it in the correct dialect. Crown Prince Rudolph had his favorite fiacre-driver, Bratfisch, whistle the song to him before he went, drunken, to his death in Mayerling. And who composed this song, the most Viennese of all songs, in which, even in this age of the automobile, the fiacre-driver lives on, cracks his whip and makes his black horses trot smartly? A Jew from Hungary who had immigrated to Vienna.\n\nThe Viennese could never become a nationalist, since in his town, people of many other countries, and of many other languages had gathered. All these people were Viennese like himself. The neighbor who talked with a Czech or Hungarian accent was just as much a Viennese as he who was born there. One did not laugh at his pronunciation and his peculiarities as at a foreigner, but one saw in him a man like oneself. Thus sprang up the spirit of friendliness, companionship and popular humanism which brought Viennese together, caused them to gather in the smoky coffee houses and inns and made them become friends. This spirit found expression, in music of a higher style, in the hilarity of Haydn's symphonies, the humanity of Mozart's melodies, the popular attraction of Schubert's music; and soared to the heights\u2014the truly great heights where one can see the open Heavens, the throne of God and the legions of angels\u2014in the great humanitarian choruses of Beethoven's Ninth Symphony.\n\nEven Johannes Brahms, earnest Protestant and severe musician, who came from North Germany to Vienna, thawed in the sociable, human atmosphere and the serene landscape of this city of mingled races. Here he wrote his most beautiful melodies, and almost became a Viennese in his \"Liebeslieder\" Waltzes. Like the great masters whose heir he was, Brahms, too, loved the Viennese countryside. Every Sunday he made excursions to the Vienna woods with his friends, wearing the grey-green coat and knickerbockers of the Alpine hunter. I frequently met him at the Hoeldrichs Mill, a rustic restaurant, which stands among beech and pine groves. In front of the building, which had once been a mill, is an old linden tree; and the notes of Schubert's song, \"Der Lindenbaum\" which are inscribed on the wall of the house, remind one that Schubert is said to have written this song here. Brahms loved to sit in the garden of the inn, under the shadowy trees, drinking coffee. He would beam with delight and contentedness, so pleased was he with his surroundings. The lovely graciousness of the Viennese landscape was in direct contrast to his own severe nature. He had come here from a land of fog where the sea raged and the brown heath stretched for miles on end, and like all northerners, he longed for sunshine and gaiety.\n\nHugo Wolf, in many of his songs, was inspired by the atmosphere of this landscape. The wonderful M\u00f6rike songs were composed in a summer house at the edge of the P\u00f6tzleinsdorfer Heath, near Vienna.\n\nThe Viennese landscape has something unique which cannot be found in any other town of the world. The surrounding woods are not merely a background but belong to the city and are merged with it. They descend from the hills, swing into the town and finally spread themselves out into the innumerable suburban gardens. The suburbs themselves were once country villages outside the walls and portals of the old fortress town. The magnificent Vienna parks, too, are parts of the Vienna woods which the art of the gardener has ennobled. The \"Augarten,\" which Emperor Josef II, in his humanitarian way, opened to the public, and in which Mozart conducted his morning concerts, is old woodland. The Park of Schoenbrunn, with its clipped hedges, artistic fountains, its graceful Gloriette with its sandstone statues, and the rococo pavilion in the menagerie where Empress Maria Theresia drank tea with the aristocracy, had formerly been the Emperor's hunting grounds. Only the Prater, Vienna's amusement park, was permitted to preserve its original forest character, and the lovers of Vienna are grateful for it.\n\nVienna's suburbs even today lie in the green. In the inn gardens where the contemporary bourgeois sit over their beer, Strauss and Lanner played their waltzes. The chestnut trees were blooming and scented buds fell from the lindens. It is almost self-understood that the Blue Danube Waltz was composed in just such a garden, at 54 Praterstrasse. The greatest of all Viennese waltzes came into existence in the Viennese landscape which it glorified, under trees, near the Danube.\nThe Viennese\n\nVienna is unthinkable without its typical, middle-class citizens. It was they who built the city and lived in its houses. They battled on the walls against the Turks. In the year 1848 they stood at the barricades, workers, artists and students, and fought for a new era. They sat in the workshops and slaved in the factories, frequented the inns and took their wives and children to the Vienna Woods on Sundays. As young men, they wore the uniform of the Emperor's soldiers, and went to Bohemia, to the Balkans, to Italy, where their sovereign was always waging wars. These anonymous Viennese gave the city on the Danube its own characteristic personality, which is recognizable in all the music which was written there.\n\nDuring the many centuries which passed from the time of Vienna's founding, and its development from a Roman military camp to a dark mediaeval city, and, finally, to a great European capital, the inhabitants showed the same characteristic personality. The \"Viennese\" was a strong and definite conception. The Viennese, by a greater adaptability, was differentiated from his South German relatives, the Bavarian and the Suabian. He was less heavy-handed, more amiable, more lively. He was not so clumsy in love-affairs, not so ill-mannered in eating. He had good taste and many talents, among which was that of acting. The Viennese played with life a little. He did not take it entirely seriously. He kept a sharp eye out for the weaknesses of his fellow man, whom he delighted in mocking or mimicking. In all Germany, there was no such original Folk Theatre as the 18th century Viennese Folk Theatre. In the small wooden shacks on the Graben, Viennese folk-spirit found its laughing outlet. Here Viennese dialect flourished in all its rudeness, its freshness, and lack of inhibition. He who wished to captivate the people had to speak their dialect. During the time of the Turkish Wars, the great Viennese preacher, Abraham a Santa Clara, although a Suabian by birth, always used Viennese dialect when he spoke from the pulpit of St. Stephen's. Sometimes he spoke as if he had been born in a peasant farm-house near Vienna, or sometimes, even in a pig-sty. When he thundered against the Turks or against the sins of the world, he talked like a native Viennese, sitting at an inn-table after several glasses of potent wine. He made his listeners laugh when he wished to reform them. Paris had its great preachers\u2014its Bossuet, its P\u00e8re Lacordaire\u2014but they were classical, master orators who spoke the literary language of the universities and salons. For that a Viennese would not stand, for he was essentially a skeptic who considered elevated words \"swollen.\" He wanted to hear his own language which, though not always refined, was gay, natural and witty\u2014the rough language of human common sense.\n\nWhen, during the 17th century, everywhere in Europe and, of course, in Austria, too, the absolute power of the reigning sovereign developed, and a feudalistic society gathered at the royal courts, the joyous temperament of the Viennese took possession of even this society. The aristocrats who circulated in the courts of the Austrian baroque emperors like planets around the sun, had come from all over Europe, but they learned Viennese dialect. They spoke and lived as Viennese\u2014that is, they took life as a game which was meant as entertainment. Empress Maria Theresia spoke the dialect like a Viennese bourgeoise, and the Emperor Franz Josef had a slight dialect accent. During his reign, a special kind of Viennese dialect developed among the aristocrats and officers\u2014an elegant, somewhat fatigued and refined Viennese.\n\nThus a common bond existed between the court and aristocracy on one hand, and the people on the other; a bond created by a common dialect and a common philosophy of life. Externally, the dividing line between \"high\" and \"low\" society was not so outspoken in Vienna as it was elsewhere in Europe. In the wine houses and gardens of the \"Heurigen,\" fine society and common people sat side by side, enjoying themselves, singing folk songs like: \"Menschen, Menschen san mir alle, Fehler hat ein jeder gnua; S'koennen ja nicht alle gleich sein, S'is schon so von der Natur\" (Human, human, we're only human; everyone has faults enough; we can't all be alike\u2014that's how Nature made us.\") Archdukes, princes, rich burghers, shoemakers and tailors alike sang such songs. In Vienna, even if nowhere else, in the 19th century there existed a kind of democracy in the taverns which prevailed as long as the effect of the wine. The language of this democracy was the Viennese folk dialect. If Haydn, therefore, in his symphonies, mixed the higher forms of music with popular dance tunes, he was only exemplifying the Viennese spirit.\n\nIn Vienna, the joy of music was a part of the general joy of life. Even in mediaeval times, the Viennese wanted to hear music. The town hummed with fiddlers, harpists and singers who made music in fifteenth century rooms and taverns. In the 17th century, a musician became a sort of legendary figure in Vienna, and by rights, he should be included in the city's coat-of-arms. This was Augustin, a bag-piper and folk-singer, who fared from inn to inn, playing, singing. Even to-day the Viennese sing a song which was his favorite: \"O du lieber Augustin, alles ist hin.\" (O my dear Augustin, everything is gone.\") And in new Vienna, a monument has been erected in his memory. There he is depicted, blowing his bagpipes, a little shaky on his legs, for the legendary singer loved wine. When, in 1679, the plague broke out in Vienna, he staggered toward home one night and fell in a pit where the corpses of the plague victims were lying. There he slept off his drunkenness, and when he awoke, he climbed out of the pit and gaily and healthily went on living and drinking wine. In the course of time Augustin became a symbol of \"Viennesedom,\" which remained indestructible in the face of any misfortune. This same Augustinian spirit emerges in the song of another Viennese folk-singer of the 19th century: \"Allweil lustig, fesch und munter, denn der Wiener geht nicht unter.\" (\"Always jolly, gay and merry, for no Viennese ever goes under.\") The Viennese considered himself indestructible even in hard times. He was the born optimist. This quality, along with his sensitiveness and his appreciation of the beauty of Nature, has made him a musician. His unworried, carefree attitude toward life he expressed by singing and playing and dancing, over wine at the inns, at festivals on the village green, on mountain-climbing excursions. One sang or listened to music because one drank, and one drank because one wanted to sing. All this may have been primitive music, even animalistic music, but through the centuries it developed the Viennese into a musician and Vienna into a city of music.\n\nIn the 17th and 18th centuries, in no other European city could one experience what one could in Vienna on summer evenings. There is a poem, written by Abraham a Santa Clara toward the end of the 17th century, which describes the \"night-music\" or serenades in the Vienna streets. It tells how musicians with flutes, kettle-drums, trumpets and lutes serenade in front of the houses of beautiful Viennese women, and how they run away when the police patrol appears.\n\nDuring the 17th century this serenading became more and more popular, and this led to the composition of masses of \"Serenades,\" \"Cassations,\" \"Divertimenti\" and \"Nocturnes.\" In the \"Vienna Newspaper\" on July 16, 1873, the publisher Toricella advertised his \"very special night-pieces,\" in order \"to provide the many lovers of night-music with new material for enjoyment.\" Early in his career, Haydn was connected with this serenade music. Since he had been discharged from his post as choir-boy at St. Stephen's, he needed to earn money, and did so by participating in the serenades, on which occasions he played timpani. This experience aided him in acquiring the folk-like tone of the themes of his symphonies, and the early string quartets he composed are still entitled \"Quadri,\" \"Cassatio,\" \"Divertimenti\" and \"Notturno\" from the terminology of the serenade.\n\nMozart, too, composed such serenade music, among which is the marvelous \"Haffner\" Serenade. And in Vienna, he himself was honored by a serenade. In a letter of November 3, 1781, he wrote his father: \"At eleven at night I was serenaded by two clarinets, two horns and two bassoons. The six gentlemen who executed it are poor devils, but who nevertheless blow nicely together.\" Beethoven's \"Serenade\" and the romantic Schubert Octet are the classic fulfillments of Viennese night-music, and the serenades of Brahms and of Richard Strauss are its last fading notes.\n\nIn Emperor Franz Josef's time, in the streets the Viennese could hear music which also was a part of his folk-music. This was military music. Every regiment of the Austrian army had its band, and the people used to swarm in the streets when the first sounds of the deep wind-instruments, then the bright clarinets, could be heard. Most of the crowd marched in front of the band, in step with the beat of the great drum, which was carried on a small wagon, drawn by a pony. The drum-major, who according to an unwritten tradition, always had a long, martial beard, raised and lowered his commander's baton. The drummer artistically tossed his sticks in the air, and just as artistically caught them.\n\nOn the stroke of twelve, at the old Imperial Palace, the guard was changed, under the Emperor's window. Then, throughout the city, all life in the streets seemed to come to a standstill, for the people would hear the music of the military band on its way to the palace. Hundreds of Viennese, first boys, then grown-ups, would crowd into the very courtyard of the castle, where music-stands were set up. The concert began while the officers changed watch and the soldiers marched to their posts in the different castle courts. Often one would see the old monarch, from the window of his office, look down at the crowd while the band played for hundreds of his subjects.\n\nThere was another man whom I used to see, listening as if electrified, when a military band paraded the streets of Vienna. He was the composer, Gustav Mahler. I used frequently to sit with Mahler, of evenings, on the terrace of a Ringstrasse caf\u00e9. When, in the distance, the \"boom-boom\" of a military band's drum became audible, he would interrupt the most serious conversation about music and the world, and hurry out to the curb. There, with the wind blowing through his hair, he would stand as if spell-bound, not returning until the last boom of the drum died away. The great role military music played in Mahler's compositions is known to everyone familiar with his symphonies, especially his third. He himself told me that, as a child, he had lived opposite a barracks, and that the military music and signals he heard at that time made a deep impression upon him.\n\nSome military marches belonged to the music of the folk; above all, the famous \"Radetzky\" March by Johann Strauss's father. To its accompaniment, Viennese soldiers went to Italy in 1848. Marshal Radetzky, with drooping white mustaches, sat, bent forward on his horse, and the soldiers marched to the bloody battle-fields of Novara and Custozza.\n\nThe trio of the Radetzky March is a yodel-song of the Austrian mountains which the recruits from Upper Austria used to sing when they were marching to Vienna in 1848. So the mountains, too, made their contribution to Viennese music. The greatest composers like Haydn, Beethoven and Anton Bruckner, did not consider it beneath their dignity to compose military music.\n\nWhen Goethe, in 1823, was taking the cure at Karlsbad, he heard the band of an Austrian hunter regiment play. He was, as he wrote, \"touched.\" And when Rossini heard Austrian regimental music, he wrote: \"There is no artistic enjoyment which may be called democratic to such a high degree as the playing of regimental bands. Everyone may take part without entrance fees and without evening dress.\" Military bands achieved an inestimable contribution toward the music education of the Viennese. These bands, at their concerts, played in addition to their military marches and waltzes, opera and symphonic music also, in the public parks; and every fine evening, hundreds of Viennese surrounded the concert podium and listened to the music which formed one part of the city's musical life which was accessible to everybody.\n\nThe church, too, played an important part in the musical life of the Viennese. On Sundays and holidays, it was their real concert-hall. There was good music in all Vienna's churches. Well-trained choirs sang and orchestras played. So it was since the middle ages, when students of the University of Vienna sang in St. Stephen's Cathedral choir, up to the period of baroque when the Catholic Church employed the most brilliant musical means to win souls for Catholicism. It must be admitted that the church-music of this fanatical battle-period of Catholicism was what determined the musical taste of the Viennese until present times. From that time on, the Viennese have loved brightness and splendor in music; the sensual enchantment of sounds streaming powerfully through 17th century churches, making the Mass a brilliant concert.\n\nThe role of Catholicism in the development of Vienna into a music capital has not, as yet, been sufficiently acknowledged. The Viennese has been a devout Catholic ever since the time the holy Severin preached this faith in the vineyards of Sievering. Under the leadership of the Archbishop of Vienna, Count Kollonitz, the Viennese defended his city against the Turks in order to save the Catholic faith for all Europe. He loved the pomp of the Church, the brightness and color of its processions, and, during pilgrimages, walked ecstatically behind the cross which was borne by the priest.\n\nThis Viennese Catholicism is different from Spanish or Italian Catholicism. It was not sombre like that of Spain, nor was it theatrical or operatic like that of Italy. It was a friendly, comfortable Catholicism which gave free play to the Viennese joy of living and its hilarious spirit of life. On the many Catholic holidays which the Viennese celebrated, the festival meal was the most important. On the feast-day of St. Martin, when one said the prayer: \"You shall multiply the geese and also the cool wine,\" the first geese were slaughtered. On this day, too, emissaries came from Pressburg, bringing geese to the emperor. This was an old Austrian court-custom. The cloisters of Vienna, Klosterneuberg, Melk and Gottweih, provided wine for the Viennese. Each of these ancient monasteries had its wine-cellar in Vienna proper where the people went to drink \"cloister wine.\" Here, the behavior was not on a very pious level, and the laughter of Viennese girls rang out in the low-ceilinged rooms. On St. Leopold's Day, hundreds of Viennese went with their wives and children to Klosterneuburg to drink wine in honor of the good saint, who is buried there in the famous altar of Verdun. The monks provided good food and sweet wine, and they did not hesitate to sit down with the Viennese and join the celebration. This was Viennese Catholicism. All the pilgrimages ended up at the inns, after the religious ceremonies. The same held true for funerals. Near every Catholic cemetery, even to-day, in Vienna, there is an \"Inn of the Eye of God\" where God can look down on the Viennese, who, after a funeral, enjoy life and their drinking. Most of the Viennese went to church on Sundays, and at any hour of the day, one could see hundreds of people from all walks of life kneeling in prayer before the image of the Blessed Virgin Mary in St. Stephen's after lighting their votive candles. Understanding Father-confessors in Vienna's churches forgave the people's sins, and from the pulpits, priests spoke in dialect about God, Heaven and salvation in the faith. One of the great cardinals of Vienna in Franz Josef's time said: \"In Vienna, one can appoint anyone and use him as cardinal, even a complete idiot. Just no fanatic.\" Therefore Catholicism in Vienna wore popular dress.\n\nOne has only to listen to the Masses of the classic masters in order to feel that Viennese Catholicism was human and merry. Haydn's masses are full of folk-like melodies, and when he was reproached for composing church music which was too gay, he answered: \"Why shouldn't I be gay when I think of dear God?\" When Goethe related this story, tears rolled down his cheeks, so moved was he by Haydn's child-like soul. Child-like, pure and melodious, too, are Mozart's masses. The most beautiful piece of church music he ever wrote was the \"Ave Verum,\" composed at Baden-bei-Wien. It sounds like a piece sung by angels who had been Vienna choir boys. Beethoven, in his two masses, wrote great Catholic festival music containing the pathos of the baroque. Bruckner's masses, too, were baroque services.\n\nThe masses of Schubert all have their roots in Vienna. In this music, one feels one is looking from the open church door out over the vineyards and woods. The melodies are like folk songs sung by happy young girls. This music has all the splendor of Catholicism, but also Viennese humanness.\n\nEvery Sunday, the masses of Haydn and Mozart were performed in the churches. Viennese girls sang in the choirs, and in the pews, the citizens in their Sunday best sat enjoying the sound of fresh young voices. Only in the court chapel and at St. Stephen's was there a boys' choir, for the Viennese preferred the sound of women's voices in church. Here, too, he loved sensuality in the music; and had little understanding for the spiritual charm of Palestrina or the mediaeval tartness of the Netherland mass music. In church, he wanted to hear the sound of instruments, beautiful voices, the roar of the organ, trumpet fanfares praising God above. The Catholic Church, therefore, which so richly maintained the 17th century tradition in Vienna, has been one of the main influences in preserving the musical taste of the baroque among the people.\n\nWhen one speaks of influences which made a musician out of the Viennese, and a musical city out of Vienna, one must not omit mentioning the most important. Beside the Viennese man stands the Viennese woman, and though music history, which was written by dry pedants, makes no reference to this fact, it is surely because no ancient manuscript reveals specific instances of her influence on Vienna's music and musical taste. And yet this influence was strong. For ages Vienna has been a town of beautiful women and girls. They are at home not only in the palaces of the aristocrats, but in the small suburban houses, where they appear at their best and most original.\n\nThe Viennese woman has given sensual color to the city's life. Her gay, natural laugh\u2014different from the spirituel laugh of the Frenchwoman\u2014could be heard in Vienna through the centuries. It was her chatter, kisses and embraces which inspired great music and small music, folk songs and dances. Though there exist no statistics by which one can measure exactly what Viennese music owes to women, there is one instance where a definite relationship between the two can be established.\n\nIf there exists a form of music which is a direct expression of sensuality, it is the Viennese waltz. It was the dance of the new romantic period after the Napoleonic Wars, and the contemporaries of the first waltzes were highly shocked at the eroticism of this dance in which a lady clung to her partner, closed her eyes as in a happy dream, and glided off as if the world had disappeared. The new waltz melodies overflowed with longing, desire and tenderness. No composer gave the waltz more passionate warmth than Johann Strauss. When he was composing his seduction waltzes, Strauss was continually surrounded by women, and the titles of many of his waltzes show by which of them he was inspired. There are \"Irene\" waltzes, \"Siren,\" \"Lady Bug\" and \"La Viennoise\" waltzes. Then there are \"Wilhelmina\" quadrilles, \"Martha,\" \"Maria,\" \"Anika,\" \"Sophie\" and \"Alexandra\" quadrilles. On goes the bacchantic train of the waltz from \"My Life is Love and Desire\" to \"Wine, Women and Song.\"\n\nThe great erotic whose love of life and sensuality flamed up so overwhelmingly in the three-quarter time of the waltz, had three wives. The first one, Henriette Treffz, a prima donna at the K\u00e4rntnerthor Theatre, he married in 1862. This woman came from low origins. She was the daughter of a Czech laborer, Challupetsky, and all Vienna knew her as the mistress of Baron Tedesko, by whom she had several children. For Strauss, she was apparently the primitive erotic element which gave him his artistic energy. During this marriage, in addition to other music which became world-famous, \"Die Fledermaus\" was written. After the death of \"Jetty,\" Strauss married Angelica Dietrich who called herself \"Lilly.\" At that time he was fifty-three years old. His wife, who was young, seductive and full of temperament, eloped with a friend of Strauss's, and disappeared from his horizon. Apart from his marriages, not very much is known of the more personal aspects of the composer's life. Like the Pied Piper, the swarthy Strauss, bowing his violin, would lead his orchestra, and each waltz would plead his suit.\n\nIn the year 1859, Strauss fell in love with a beautiful lady in St. Petersburg. This was a romantic passion, with Werther tears, melancholia and soul-suffering, which streamed into some of the waltzes Strauss wrote around this time.\n\nIn 1883, fifty-eight years old, Strauss married Adele Deutsch, a young Viennese widow. In his soul, feelings of love were still storming ardently. When Johann, or \"Jean\" as he called himself in this marriage, composed at night, he would interrupt his work frequently to write messages and greetings on slips of paper to his beloved wife, who was in the next room. \"How you have made your Jean quite beside himself! There you are! How excited he becomes! There you are! He would like to laugh, leap, even dance... Let's be merry, Adele. On ne vit qu'une fois!\" The music of the Cagliostro Waltz, which had just then occurred to Strauss, confirms these happy lines. Another time he wrote her: \"It's merry in my soul, gay melodies hum in my head, and my heart overflowing with joy and happiness, is gaily beating the measure.\" And again: \"Sleep sweet, you blue-black-eyed Adele and only wife on earth.\" Or, on another occasion, \"I am wishing you a good-night, peaceful sleep, and a great deal of humor when you awaken. Humor is half the joy of living. To wander merrily through life is my desire.\" For sixteen years, Johann Strauss lived most happily with Adele. She was the last woman who brought the joy of love to the life-loving composer. On one of the slips of paper he had written her, he swears that he has been \"faithful, yes, faithful. I am astonished at myself.\" Strauss dedicated the \"Adele\" Waltz to \"my beloved wife.\" After so many Viennese women who loved, are loved, laugh and shine with happiness through his waltzes, he had found the woman. Each of the three wives, and the other anonymous women who had, from time to time, enflamed the heart of Johann Strauss\u2014small fires which quickly died and great conflagrations\u2014are all, somehow, still living in Strauss's waltzes.\n\nViennese women had refined the waltz rhythm, which was originally that of a peasant dance, even by the time of the Schubert waltzes. With Lanner and Strauss, the father, the women lent romantic grace to the waltz; and they gave this rhythm the elan of great passion to be found in the music of Johann Strauss, the son.\n\nBut as a woman stood beside Strauss, so did a woman stand beside another composer who wrote in Vienna. In Mozart's \"Abduction from the Seraglio,\" at the beginning of the opera, how tenderly sounded the love-call \"Constanza!\" This meant a great deal more to Mozart's dark-eyed, frivolous and childlike Constanza, than to the opera-character Constanza. But Constanza was there, too, when he wrote the music for his coquettish Zerlina in \"Don Giovanni\" and the melodies for his Susanna in \"The Marriage of Figaro.\" She is there, completely, in his Papagena. When Mozart wrote the tender melodies of his Despinetta in \"Cosi fan Tutte,\" he probably was thinking about the lovely Viennese chamber-maids whom the artist, Liotard, so liked to paint, with the little white cap on their heads, and the chocolate-cup in their hands. Mozart loved to put his arms around the waists of such Viennese girls. Laughingly, he used to speak of his \"chamber-maiding.\"\n\nWhen Beethoven improvised on the piano, he was surrounded by lovely Viennese women who stared, with admiration, at this musician who, according to the testimony of his most intimate friend, \"was always in the clutches of love, and that to the highest degree.\"\n\nWhen Schubert made Sunday excursions with his friends, the wagon was full of laughing girls. And so it was with all the composers until Brahms, the clumsy, North German, in whom a passion burned which he hid by force behind his silence. How many of his love songs may have been inspired by Viennese women! Of one, however, we are certain\u2014the lovely melody of the \"Liebeslieder\" Waltzes: \"Upon the Danube bank there stands my house; from it my dearest darling looks out.\" Of all the melodies which Brahms composed in Vienna, this is the most tender and most beautiful, and it contains all the grace of the Viennese.\n\nBeethoven's House in Heiligenstadt\n\nParc of the Imperial Castle Schonbrunn\n\nThe men and women of Vienna contributed in building the music city. Not only the aristocracy and the court helped, but the bourgeoisie, and the \"folk\" of Vienna. This folk, the nameless, little people, the many hundreds of thousands who lived a humble life in narrow rooms, prepared the ground which later bore such rich fruits. Without their musical talent and love of music, the music of higher society would have presented a lifeless, artificial picture such as music presented at the court of Frederick the Great in Berlin.\n\nMusic filled every corner of Viennese life. From the music of the people came the themes of Haydn symphonies; from their folk-dances came the melodies of the scherzos; from the folk-songs came the Schubert \"Lied.\" Only so long as the music of the Viennese people possessed this vital strength, derived from the folk, was there great Viennese music, and a great musical city, Vienna.\nFame and Traditions\n\nIn the year 1540, a man from Suabia came to Vienna. On a wooden ship, he sailed up the Danube which, from Ulm, flows past peasant houses, fields and the mountains of Upper Austria. In all the gardens, the trees were laden with apples and cherries, and off to the south, ranged the mountains of the Salzkammergut. From a promontory, the Benedictine Monastery of Melk overlooked the river. In Wachau, vineyards began to accompany the river on its way, and on one of the mountains, now higher than in the previous country, stood the Goettweih Monastery. Then the hills along the Danube banks grew lower and the land flatter. Only small trees lined the shores, and in their branches the water-birds nested. At last Vienna with its churches and St. Stephen's Cathedral at its centre, came into view. The ship which carried this much-traveled, forty-year old man to Vienna anchored at the foot of a hill where the gate to the city was located.\n\nThe way by which this man, Wolfgang Schmelzl, journeyed to Vienna was the ancient one along which, in bygone times, the Nibelungs were drawn to Hungary. Later, on the same highway, Mozart and Beethoven, Bruckner and Brahms, merchants, journalists, artists, came to Vienna, and last of all, Hitler and his armies. This was the road from South Germany to Vienna.\n\nWolfgang Schmelzl, as a qualified musician, found a place in the Schotten Monastery. He became a schoolmaster and a priest, and wrote plays which were performed in the monastery and in the Town Hall. He assembled a song book of Viennese music of those times, for the most part \"Quodlibets,\" in which beloved melodies were shaken and tossed about together like dice in a cup. Like so many others who came there, Schmelzl quickly became a Viennese, and in gratitude to the hospitable city, wrote a poem entitled: \"Eulogy to the Praiseworthy, World-famous, Royal City, Vienna\" in the year 1548. In it, one may read the oldest praise of Vienna, the musical city: \"Ich lob diss Ort fuer alle Land, Hier seid viel Singer, Saytenspiel, Allerley gesellschaft Freuden viel, Mehr Musikos und Instrument, Findt man gwisslich an keinem End.\"\n\nThus Vienna rises out of the darkness of history as a city of music, and begins to sound.\n\nSince the time when Wolfgang Schmelzl wrote his Eulogy, the musical fame of Vienna kept growing greater and greater, spreading out over the entire world. Its praises were sung by countless musicians and music-lovers who visited it through the ages. Reichardt, the Berlin Court-Conductor, calls the Vienna of 1793 \"the first city for practical music.\" The composer, Spohr, in 1812, greeted Vienna as \"the uncontested capital of the music world.\" The composer Zelter, a keen and rude thinker, very beautifully described the Vienna of 1819 and the Viennese musical public in a letter to Goethe: \"Here people know something about music, and that in contrast to Italy which believes itself to be the beatific church. Here people are really deeply educated. Though they are pleased by everything, nevertheless the best sticks permanently. They enjoy hearing a mediocre opera which is well cast. An excellent work, even though it is not so well cast, they retain.\"\n\nOf the opinions of more modern musicians, I shall mention, because it is not so well-known, the one Wagner had of Vienna when he came there in the revolutionary year 1848 with reformatory ideas for the Viennese opera: \"Vienna, which on a beautiful clear Sunday I saw again, has, I must confess, completely enchanted me. I have re-discovered Paris, only a more beautiful, more lively and German one.\" In a second letter, Wagner wrote to his wife: \"This Vienna is a glorious city; here everything with which I come in contact is so warm and so pleasant! The atmosphere is home-like, yet wonderful. What surroundings! I had forgotten everything, and now, for the first time, I am really enchanted.\" Still more enthusiastic is Wagner's opinion of Vienna in the section of his autobiography where he tells of the Vienna \"Lohengrin\" performances in 1861, and of the \"uninterrupted, hot-blooded ovations which I have experienced nowhere except with the Viennese public.\" Wagner also speaks here of \"the great gifts of the Viennese public in respect to music,\"\u2014high praise, indeed, from the greatest musician of the 19th century.\n\nIn a bad humor, Hans von B\u00fclow sprays poison over Vienna which, as unwilling praise, appears especially valuable. In a letter to Raff, he wrote, in 1860, on Vienna: \"Moreover, the Berliners are musically more intelligent than the Viennese, although all the means here, especially the official, are of the highest rank in Germany, in any case. Here they have the best orchestras, the best soloists.\" And in another letter of the same year, written to Draesecke, it is again a case of mixing gall and praise: \"Orchestra and opera in Vienna magnificent\u2014otherwise pure filth, stupidity and colossal meanness.\"\n\nThere was some sort of enchantment in the musical ground where Vienna stood. Great musicians trod this ground and were held fast by it. Thus one day Beethoven came from the Rhineland to Vienna and never went away again. Brahms came there from the cold north, and never deserted Vienna. For a time it looked as if Richard Strauss, too, would be one of those composers who came only to remain permanently; for in 1919 he became Director of the Vienna Opera, and built a house in Vienna. This house stands in the Botanical Garden, the Director of which was once the famous botanist Jacquin in whose home Mozart composed so much wonderful music; and near the Belvedere Garden where Bruckner spent the last years before his death. But what vicinity in Vienna was not a musician's neighborhood?\n\nIn his Vienna home, Richard Strauss worked in his high studio, which was adorned with a costly wooden ceiling he had brought from Italy; among Greek terra cottas and old pictures, writing his notes in a most elegant script. Often I have sat there waiting for him after opera performances which he had conducted. When he came home, stimulated and red in the face, after the opera, I have often heard him exclaim enthusiastically: \"One can do such things only in Vienna!\"\n\nThere were but a few musicians in recent times who either could not find a footing in Vienna or were pushed aside by it. Among these was Robert Schumann, who came to Vienna during the Revolution of 1848 and went away again, because he found there no interest for his music which, at that time, was considered revolutionary. Lortzing, too, one of the great masters of German Comic Opera, was not held fast by Viennese soil.\n\nLortzing was active only three years in Vienna as Conductor of the Theatre-an-der-Wien. He had rejoiced when he received this offer. \"Anyone who doesn't go to Vienna is a nobody,\" he wrote in May, 1845, when Director Pokorny made him the offer to come to Vienna. \"How beautiful Vienna is!\" he cried out in July, 1846, when he was honored there as conductor. But soon the tone of his letters changed. As soon as March, 1847, it was: \"Life in Vienna is quite nice; but theatre conditions are less so\u2014no repertory, no system!\" Then in May, 1847, Lortzing wrote that Vienna was \"no place\" for him, and that \"here not a soul speaks to me.\" In November, 1847, he complains: \"I have been here now for fifteen months, but nevertheless, I still cannot feel much at home in the so-called beautiful Vienna . . . Its musical taste is the most spoiled one can find.\" In 1848, he shouts: \"The musical life displeases me. The taste is horrible.\" Great artists, too, are only human. Human nature colors opinion, and Lortzing was influenced in his because his operas, as he himself communicates, \"have not been in the repertoire for half a year.\" Italian taste predominated in Vienna, or, as Lortzing complained, \"Just tootling and more tootling and trilling.\"\n\nBut such phenomena and manifestations were exceptions. In general, through the centuries, Vienna showed an astounding force in attracting and holding musicians fast. Their various nationalities and races joined to create the great tradition which made Vienna's music rich and colorful.\n\nThus the sound of the famous Viennese violin is unthinkable without the influence of the Hungarian. The great violin teacher Joseph Boehm, who kept educating generation after generation of violinists at the Vienna Conservatory, came to Vienna from Budapest in 1816. He had the whispering, voluptuous sound of gypsy violins in his blood. One of his most famous pupils was Joseph Joachim, who studied under him at the Conservatory, along with H. W. Ernst, Hellmesberger, Gruen and others. Everyone who heard Joachim play the great Beethoven quartets knew that this classical violinist carried the dark sound of Hungarian violins over into his classical greatness. It was this Hungarian-colored violin tone which was passed on by Boehm, Hellmesberger, and Gruen, who later became violin teachers themselves at the Vienna Conservatory. Almost all the violinists of the Vienna Royal Opera Orchestra under Hans Richter and Gustav Mahler were pupils either of Gruen or of Hellmesberger. Fritz Kreisler was the last great pupil of Gruen.\n\nThe Viennese violin-tone is like the fragrance exhaled by famous cross-bred roses. The same holds true for all the other forms of Viennese music. They are the result of cross-breeding, like Vienna itself.\n\nWhen I mentioned this drop of Hungarian perfume the Vienna violin tone gives forth, I have not by far exhausted the history of this sound which the Vienna Philharmonic, the Rose Quartet, Joachim and Fritz Kreisler made world-famous.\n\nJoseph Boehm was not only a violin teacher, but also the leader of a string quartet, about which the Leipzig Musikzeitung of 1821 delightedly exclaimed: \"So must one hear Beethoven's and Mozart's quartets played!\" Also members of this quartet were Weiss and Linke, who had played in the famous Rasumoffsky Quartet which Count Rasumoffsky had engaged, and who first played in his great palace Beethoven's last quartets. Weiss and Linke had often seen Beethoven at rehearsals, and had heard his thunderous voice cry out: \"Does he believe that I give a thought to his miserable violin when the spirit overcomes me?\"\n\nWhen members of the Schuppanzig Quartet, in 1816, gave a farewell concert which consisted entirely of Beethoven's works, the composer was present, and the newspapers announced: \"Beethoven attended and seemed very much pleased.\" The Schuppanzig Quartet was the authentic Beethoven quartet.\n\nWhen Weiss and Linke later played with Boehm, the concerts took place under the trees of the Prater, in a coffee house, at eight o'clock in the morning, so early did the Viennese arise to hear music. And the spirit of Beethoven was among them. Thus, the sound of these violins became Beethoven's sound, which was always preserved in Vienna wherever violins played.\n\nThe sensual Viennese violin sound which streamed out of the orchestra had still another effect. In the center of the Viennese Opera Orchestra, great conductors\u2014Hans Richter, Arthur Nikisch, Franz Schalk, and last, Arthur Bodansky\u2014sat as musicians, and all of them with the exception of Hans Richter, who played horn, were violinists who had retained in their ears the sound of this warm, streaming violin-tone. Later, when they stood at the conductor's desk, they carried something of this sound over into their orchestral visions. The full-bodied sound of Hans Richter's orchestra, the romantic, luxuriant tone of Nikisch's, was inspired by that of the Viennese violins.\n\nThus, in the course of centuries, a great musical city arose in the center of the Danube lands, a city where old traditions were piled up like the ancient crowns and swords, the jewel-adorned glasses and ornate gold plate, the silken mantles and tapestries in the Treasure Chamber of the Viennese Court. These traditions worked so long as living forces in Vienna that they were no more memory and history, but the present. There was no musician in Vienna who could escape them. Great musicians had created this tradition and each new generation of musicians carried it on. Even revolutionaries like Arnold Schoenberg and Alban Berg recognized this tradition which they sought to disturb, since their hate was only reversed love. In this Viennese music tradition all the spiritual streams of north, east, south and west met. Haydn's symphonies bound together all the characteristics of northern and southern German music\u2014Austrian folk-music and the strong logic of the developments of Philipp Emmanuel Bach, the forms of the Mannheim Symphony and the rhythms of the folk dances of the Alpine landscape. In his Military Symphony, the Adagio is built upon a French song. His Folk Hymn originates from a Gregorian melody. Hungarian and even Croatian music left their traces upon his music. And everywhere Italian melody is mingled. Along this road Viennese music was fused into world-music. All Europe made music in Vienna. And that made Vienna a great musical city.\n2\n\nThe Last Chapter of Great Music in Vienna\nOn The Ringstrasse\n\nI myself lived through the last chapter of the great musical city Vienna. We who were born in Vienna, and grew up there, had no idea, during the city's brilliant period before the first world war that this epoch was to be the end of the greatest development of history of music known to the world; and still less did we suspect that the Hapsburg Monarchy, of which Vienna was the capital, was destined to decline.\n\nWe grew up in a beautiful, large city which had adorned itself with new palaces, new gardens and new avenues. A new Royal Palace, too, rose out of the earth, next to the low, old one which had seen so much history, and before which, as children, we had played among blooming elder trees and proud equestrian statues. From the great, open, pillared hall of the new palace, one could see the marble columns of the new Parliament, which was a Greek temple, stylistically; the iron man on the Gothic Town Hall's tower; the garden of the Town Hall and the fortress; and from there on out to the Kahlenberg. We enjoyed the splendid city which was so elegantly beautiful, and never thought that the light which shone over it could ever be that of a colorful sunset.\n\nWe saw the traffic of modern times stream through the streets and watched, amazed, when the first electric lights went on in a shop window, and the first small electric streetcar started its way to the M\u00f6dling woods. The new time of technical wonders was apparent everywhere in the streets. Big-city life had lively aspects. Well-dressed men and women, chatting and laughing, walked slowly through the new streets. Elegant riders sat on full-blooded horses. Fiacres drove through the streets and when the driver cracked his whip, the horses raised their feet as if in a graceful waltz-step. There were Austrian army officers in colored uniforms and white kid gloves; the Emperor's guards, marching past, in red-braided uniforms, white leather breeches and highly-polished boots, halberds in hand, on their way to the old palace. There were mounted guards, too, with leopard skins and long white helmet-plumes, riding Arabian horses. The great city seemed festive, lively, full of color.\n\nEven the modest citizen had an easy existence. At ten o'clock in the morning, after two hours' work, he would go to his favorite inn and fortify himself for further accomplishments with a glass of fresh beer and a portion of goulash. Evenings, with wife and children, he would saunter in the Prater where in good weather the inns were crowded, where games and entertainments were continuously patronized, and where music sounded from all sides\u2014shrill barrel organs, blaring, brass bands, or waltz. violins. Or he might sit in a corner of the tavern he regularly frequented, and talk politics with his cronies. The waiters kept carrying glass after glass of foaming beer, while smoke rose to the roof from numerous meerschaum pipes or long Virginia cigars.\n\nThe greatest German poet called the Viennese \"Phaecians on whose hearth one always kept turning the skewer.\" Before 1918 this statement certainly was a true one. The Viennese knew it, and sang: \"There is only one Imperial city, there is only one Vienna!\"\n\nThis beautiful, jovial, well-to-do city which today has become a legendary one, was for us, too, a musical city. We became musicians without knowing why or how. Everywhere we went we encountered music. We sang and fiddled. As a student in high school, I took my violin every evening and went to other homes and played classical string-quartets with minor official, teachers or business people, just as if that were self-understood. On Sundays, I played Haydn's or Mozart's Masses in church choirs. On excursions we all sang choruses and canons. Or we stood, evenings, in front of restaurant gardens or in the parks and listened to band concerts. Thus we prepared outselves for the holiday when we went for the first time to the Opera palace on the Ring. From noontime on we stood, boys and girls, in front of the small entrance to the Opera House, which still remained locked, with piano scores in our hands, heatedly debating about music. When the door at last was opened, we would storm excitedly to the box-office, and then up four flights of stairs like competing foot-racers\u2014up the still dimly-lighted passageway to the fourth gallery from where we could look down on the stage. Slowly, after some time, the musicians came into the comfortable orchestra pit. The violinists tuned their instruments and little runs rose on high like butterflies. Wood-winds tried passages, and the low brasses tossed growling, antedeluvian tones into the bustle of the orchestral voices. The sounds kept swelling louder and louder, like the buzzing of a teakettle about to boil. Then the lights in the hall were extinguished, the great chandelier went slowly out and one could see only the ghostly glow of the lights over the musicians' desks. When all had become silent and one's heart pounded excitedly, the conductor came, sat at his desk, opened the sorcerer's book and raised his magic wand. This was a festive moment. For then, for the first time, were we taken in to the great Viennese music community which assembled every evening in the new opera house. A new generation of musicians and of music had joined itself to the old. We had become full-fledged citizens of the old music city.\n\nLater on, as critic, I sat in one of the comfortable red-plush orchestra seats. But I never entered the opera hall without throwing a glance toward the gallery. It always was a desirous, longing glance, for nowhere else was the enthusiasm for music and the delight in it more genuine than there where young people crowded and sat in corners, looking down at the stage or at their piano scores. It was also a glance of gratitude, for here I was brought up and educated to the position of opera critic. Like almost all Viennese music-lovers and musicians, I began my musical life in the fourth gallery of the Vienna Opera. The Opera House belonged to my existence. I was young there and became old there; and it was one of the most beautiful moments in my career as critic when, on one of my birthdays, the young people of the fourth gallery presented me with an address which lauded my critical activity with flattering and highly exuberant words. In the Opera of Vienna we were a great musical community. The house was a sort of musical Town Hall of Vienna to which each new generation of Viennese sent its representatives. Thus the Vienna Opera House meant something different, something more significant, in the life of the city than did that of the splendid opera houses elsewhere. It stood in the center of Viennese life, and not on its outside.\n\nWhen we were growing up, Vienna was still the musical center of the world. To make this clear, I shall invite the reader to accompany me along the Ringstrasse, Vienna's most festive thoroughfare, on a sunny day in 1890. At this period, between twelve and one every day, Vienna's fashionable society took its promenade on the broad street between the City Park and the new Opera House. Everyone whom one might call \"all Vienna\" crowded on that small section of the Ringstrasse. Here everyone seemed to know everyone else, greeted, chattered and sauntered along; with the bright laughter of the women as the upper voice over the babble and flirting, over the malicious remarks and anecdotes of the metropolis. Here one met politicians and industrialists, young and old aristocrats who spoke the nasal dialect of the castles, officers among whom the dragoons with golden helmets, and Hungarian hussars with braided coats and tightly fitted breeches were the most elegant; well-known artists, women of whom Vienna spoke, the \"wits\" who were invited to all the parties and whose latest bits of gossip flew swiftly along the Ring. What was the subject of all this conversation? The last theatre premiere, the latest excess of the young Archduke, the newest divorce, an interesting guest, a new affair, sometimes politics\u2014which were joked about\u2014always love, which was likewise not taken seriously. One laughed over the greatest vicissitudes of life and was always just as witty, charming, frivolous and superficial as people of every old, overcultivated society are bound to be.\n\nCourtyard of an old Viennese Commoners House\n\nStaircase in the Winter Palace of Prince Eugene\n\nI can still see the distinguished Count Berchtold on a summer's day in 1914, standing in the doorway of a Ringstrasse Hotel. He had just signed the declaration of war on Serbia. Now he stood here, slender, laughing ironically, a gold-tipped cigarette in his well-manicured fingers, watching the crowds and conversing with the passersby. Thus cultivated Ringstrasse society entered the world-war which disrupted it. It had lived laughing and joking, and laughing and joking it died.\n\nOn the Ringstrasse bridle-path, one saw distinguished gentlemen and ladies in long black fitted riding habits returning from their morning ride in the Prater. Fiacres passed along the road, or smart English carriages which were driven by the aristocrats themselves. Military music sounded in the distance, and sometimes the old Emperor passed by in an open, low carriage, with his adjutant seated beside him, and his old valet on the coach-seat with the driver. Then the officers would interrupt their jokes, stand straight and stiff as poles on the rim of the Ringstrasse, and salute, while men in civilian clothes removed their hats. Slowly, then, the promenaders started into motion again under the fragrant acacia trees, and went on with their gossip.\n\nThis was the Vienna of the 1890 period. In the so-called \"Ringstrasse Period,\" a slightly different picture would have been presented to the reader during a promenade. At first glance, he could recognize Johannes Brahms as one of the strollers. Almost daily, Brahms would come from his nearby dwelling, his bowler hat pushed up over his forehead and his hands crossed behind his back. He was, at that time, rather corpulent, a bourgeois-looking man with a flowing beard whom one might have taken for a professor, had not his blue eyes been genuine artist's eyes, and had not his rocking gait been that of a dreamer.\n\nThere was still another famous musician one often met on the Ringstrasse who looked like a professor. Going unhurriedly on his way to the opera, he was big and broad-shouldered, and impressive. This was the famous Wagnerian conductor, Hans Richter, first conductor of the Vienna Opera. When Wagner made him his conductor, Richter was young and had a blond beard like the youthful Wotan. Now his beard was already greying, but the eyes behind their spectacles still preserved their steel-blue glitter.\n\nBrahms and Richter lifted their hats to a small man with heavy, bushy eyebrows, white goatee, and hook-nose which looked like the beak of an old hawk. This was the famous music critic and Wagner-slayer, Eduard Hanslick, who wrote witty and intellectual music criticisms for fashionable Ringstrasse society. Brahms, Richter and Hanslick, though, could not help smiling when, in the distance, they saw a little man approaching who looked strange indeed in these surroundings. He did not wear a smartly-cut town suit, but instead, a wide jacket of heavy material which he had had sent from Upper Austria, where his home was. His trousers fell in countless wrinkles to his small feet, and their bagginess gave his legs an elephantine appearance. His face was that of an old peasant, weathered by air, sun and rain; but it was a peasant face with Roman features and the profile of the Roman Emperor Claudius. The singular appearance of the man became more marked when he lifted the broad-brimmed artist's hat from his bald, round head, and, with a deep bow, bent almost to the ground. This strange man was Anton Bruckner, who, at that time, was still ridiculed and mocked. Now a great bust of him, delineating all the wrinkles of his old face, stands in Vienna's City Park. He is raising the thumb of his right hand as if he wished to explain something, or as if he was just hearing a melody, and wanted to say to the twittering birds who perch on his shoulder: \"Be still!\"\n\nA younger man was there, also. He had a gloomy fanatical glance, wore a short brown beard, and dressed most of the time in a brown velvet coat. He looked admiringly at Bruckner. This was Hugo Wolf. Nearby was a very short man, almost lost under his artist's hat. His face had a yellow tinge and showed unmistakable Hungarian features, his eyes brown and kindly, a white mustache falling over the sensuous old mouth. It was the famous composer of the opera, \"The Queen of Sheba,\" Carl Goldmark.\n\nThere was only one musician who did not go on foot, but drove along the Ringstrasse, comfortably leaning back in his luxurious fiacre. His beard was cut like the Emperor's and dyed black, because this man with the large burning eyes did not want to grow old. He had always been the singer of the Viennese joy of life, of the countryside, of \"Wine, Woman and Song,\" and he wanted always to be the most famous Viennese: Johann Strauss.\n\nHad the guest whom I accompanied along the Ringstrasse come ten years later, still during the reign of Franz Josef, he would have met an unusual man. This man always carried his soft hat in his hand, and walked with a strange, stamping gait, limping with his right leg from time to time. His dark face, framed by long hair, had a sharp profile and eyes which shot out dark looks through his glasses. It was the ascetic face of a mediaeval monk. In this man the nerves pulled tensely and from him a spiritual strength streamed forth. It could be either a good or a bad spirit flashing from the high forehead and the eyes. This was the new Opera Director of the Emperor, Gustav Mahler. Still later, after the first World War, the greatest German musician, Richard Strauss, came to Vienna as Director of the Opera. One could see him on his way there, every day\u2014tall, slender and with a smile that Claude Debussy compared to the smile of a victorious conqueror striding among the huts of an African negro town. It was a new era in which, also, young, audacious musicians were to be seen on the Ringstrasse, much to the horror of the public who saw the palace of classical music practically in flames. Among these were Arnold Schoenberg and Alban Berg.\n\nThus musical history promenaded on the Ringstrasse between noon and one o'clock. The greatest musicians of the new era were at home there. They had not yet become monuments, but enjoyed, like everyone who lived in Vienna, the beautiful city in which they worked and struggled, and from where their works went forth into the wide world. One encountered the musical great at all kinds of social gatherings, at the opera and concerts, on excursions, in the restaurants and taverns of Vienna. At the \"Gause\" or \"The Red Porcupine\" \u2014taverns in the inner city\u2014one could often see Brahms with his friends, a glass of light beer in front of him, and at another table, Anton Bruckner with his pupils, like Franz Schalk or Ferdinand Loewe, both of whom later became famous conductors. Bruckner would eat his beloved roast pork with a mountain of cabbage, and drink his Pilsener beer, just like other mortals, and would look anxiously, from time to time, toward Brahms' table. When Bruckner rose to go, he made the same kind of deep bow which, as organist of the St. Florian Monastery, he had made to the Archbishop Rudigier of Linz, to Brahms, who laughed mockingly. As one of the Directors of the \"Society of Friends of Music,\" Brahms was Bruckner's superior. The Conservatory where Bruckner was a poorly-paid teacher had been founded by the Society, and was its own school. Thus we see that Brahms and Bruckner were also only human, and had not yet become historical figures.\n\nGreat musical history distinguished Franz Josef's entire lifetime in Vienna. When the Emperor was born, Beethoven was still alive and shook his grey forest of hair when he threw his last string quartets at the manuscript paper like a Vulcan tossing forth lava and fire. Only three years, then, had passed since Schubert's death. But when Franz Josef died and, in his coffin, was borne along the Ringstrasse for the last time, throughout the world was a new music which rose from the chaos and the war which was to destroy the Empire of the Hapsburgs. Arnold Sch\u00f6nberg was born in Franz Josef's reign. The whole development of classic to romantic, on to modern music took place under Franz Josef, and the great musicians who promenaded on the Ringstrasse were the leading characters in this development.\nThe New Vienna\n\nThe new Vienna, in which the last chapter of great musical history was set, arose during the reign of Emperor Franz Josef.\n\nThis Emperor it was who gave the order to tear down the fortress walls of the city of Vienna. In 1858 the first brick was broken from them. The old city gate where police officials used suspiciously to examine the papers of arriving visitors was demolished. The old Viennese, who had always known their city as a narrow fortress, stood about and watched how the laborers knocked down the thick walls while the dust flew.\n\nUntil this time, Vienna, for centuries, had been a small city of high town houses clustered together in dark streets around churches and palaces, with three or four small squares where market booths stood. From the tower of St. Stephen's cathedral in the center of Vienna, the watchman kept his vigil. The city was protected by the old walls and trenches, which the Turks had stormed and broken in.\n\nEvenings the Viennese went walking on the ramparts to take their breath of air. On the \"Glacis\"\u2014a green meadow beyond the city walls\u2014they watched the soldiers exercising, and looked off in the distance at the outlying districts. Of these, only a few had a citified character, such as \"Mariahilf,\" which is now Vienna's 6th district and the site of numerous silk factories. Here lived the wealthy factory owners, so the people called the suburb the \"Diamond Ground.\" Most of the suburbs were such villages. Here the Viennese noblemen, in the 17th and 18th centuries, had built their summer palaces, with magnificent parks in French style, with ponds where goldfish swam, and sandstone statues lining both sides of the paths. In these gardens they gave their spendthrift summer parties, to which the guests drove in gold adorned carriages, with runners preceding, and lackeys behind the seat.\n\nWhen the Viennese promenaders looked southward from the ramparts, they saw, surrounded by an enormous garden, the ornate palace of Count Fondi, which later belonged to Prince Schwarzenberg; and beside it the great park with lakes and Egyptian sphinxes, on the brow of a hill where stands Prince Eugene's summer palace. Nearer to the city there was a whole group of palaces\u2014that of Duke Trautson, Master of the Court, that of Marquis Rofrano (whom one knows from \"Der Rosenkavalier\" as the youthful lover of Maria Theresia), and, farther on, the palace and garden of the princely bishop, Count Schoenborn. The Emperor, too, had a summer palace here\u2014the \"Favorita\"\u2014summer castle of Emperor Leopold I, Josef I and Charles VI. There was a summer palace in the Augarten, and, most important of all, the castle of Schoenbrunn, in whose large garden Emperor Franz Josef died.\n\nFrom the woods of the Kahlenberg, a fresh wind blew in the evening. Viennese in high buttoned black coats, with broad ties and wide collars sat with ladies in crinolines and beribboned straw hats in the coffee booths which stood against the rampart walls, and sipped lemonade.\n\nIn the 19th century, the fortress city became too small for the many people who came there from all over Europe, and room had to be made for them in a new and larger Vienna. In the classical period, the city had two hundred and fifty thousand inhabitants. When Emperor Franz Josef came to the throne in 1848, it had four hundred and eight thousand. In 1888, the population had risen to seven hundred and seventy-five thousand when room had to be created for it to expand. And when Franz Josef died, the population stood at two million.\n\nIn 1865, there were the walls and gates, the trenches and the \"Glacis\" had been, the new Ringstrasse was laid out\u2014the festival street of new Vienna, fifty-seven meters wide with one path for horsemen and another for pedestrians. Along the Ringstrasse the great new public buildings were erected: the Parliament, where the free word of the new era was spoken by orators of all nations, with Greek temple pillars, marble friezes and bronze statues; the City Hall, with the loggia of a Venetian palace, the two great museums, the new Burgtheater, the ornate Gothic Votive Church, the University with its Renaissance halls, the splendid Stock Exchange, the new Opera House. Archduke Ludwig Victor built his interesting and beautiful Renaissance palace on a corner of the Ringstrasse. Archduke Eugene built a similar town residence nearby. The princes of the new Stock Exchange wealth also built homes on the Ringstrasse, such as the show-place of the banker Tedesco, which was decorated with frescoes by Rahl. There were colored frescoes, too, on the great \"Heinrichshof,\" which the Danish architect, Hansen, had built nearby. Among these new buildings were large and small parks, and when spring came to new Vienna, the fragrance of lilacs floated along the Ringstrasse.\n\nWhen, in 1876, the Ringstrasse was completed, there was a festival procession the like of which no one had seen since the time of the Renaissance. The greatest Viennese painter had conceived the procession's plan. Austria's trades and arts, drawn in carriages, passed in review before the Emperor's pavilion, just as Hans Makart had sketched it. The small, pale artist was the painter of the new and color-happy Vienna, just as Johann Strauss was its musician. In his palace on the \"Wieden,\" Makart stood in old-fashioned German attire before his easel and let the glowing colors run over the canvas like streams of fire. The most beautiful women of Viennese aristocracy posed nude for him, and one can still see them to-day in his great historical pictures where, naked, they surround Emperor Maximilian during his march into Bruges; or when one of their number, the lovely Princess Liechtenstein, lay on a luxurious couch as Cleopatra. Now the painter rode at the head of the procession, and swept off his feathered hat before the Emperor, who put his hand to his military cap in salute.\n\nIn a short time a new metropolis had grown up out of the old fortress town. In 1868, the Danube was brought closer to the city and on its new banks rose warehouses, dock yards and cranes. In 1870, the new water supply system was built, which brought spring water from the Alps directly to all the city's houses. Vienna was the bustling capital of a new time, with free trade, speculation growing more and more. Minds, too, became freer and more active. At the University of Vienna, famous professors taught; students from all nations came there to study. In the School of Applied Arts, Viennese artisans became artists who produced costly glass vessels and leather articles. Writers and artists from many lands came there. In the great Viennese newspapers, one read editorials which resounded throughout Europe, and critiques which were considered authoritative everywhere\u2014all in a clear, intellectual style which, in the feuilletons, also had the light conversational tone of the Viennese salon.\n\nIn the new city, a new society had assembled. From the Austrian provinces, which were multi-lingual, men came to Vienna, especially from Bohemia and Moravia, but also from Poland and Hungary. Vienna was still a city of mixed nationalities. From all sections of Austria, urged by the liberal ideas of the time, came Jews who provided a progressive element in business and trades as well as in the intellectual field, and whose women ornamented the new society with an eastern charm. This new society bound itself to the old court aristocracy which still set the tone.\n\nUp to 1914, Vienna was basically an aristocratic city, and behind its democratic forms, the old feudal order of the baroque period was but badly hidden. The court certainly made its compromises with the modern national and social movements, generally at the last moment, when it could not do otherwise\u2014and too late. But Emperor Franz Josef nevertheless felt himself a follower of the old Hapsburgs, who had been a proud race and who knew only ruler and subjects. The social life of Vienna always, as in the 17th century, revolved around the court, like a wheel around its axis; and the new industrial and Stock Exchange men, streaming with wealth, crowded themselves into the court society to receive orders and titles from the Emperor. When they gave parties in their new houses, they invited aristocrats as window-dressing. The young sons of Viennese manufacturers dressed themselves like aristocrats and aped their accent and manner. A part of the new society was elevated to the rank of nobility. Between the courtly feudal society of Vienna and the \"folk,\" comprised of modest bourgeois and workers, there was still a dividing line which had sprung up in the 18th century, at which time the aristocrats surrounded their palaces with iron chains so as to hold the people at a safe distance away. One can see even to-day, in Vienna, such iron chains hanging on the walls, as decoration, on many of these noble palaces.\n\nThe courtly society and newly-rich citizenry which surrounded the Hapsburgs, took possession of new Vienna in which the old order, and that of industry, trade and the stock market mingled. These minglings brought forth complicated mental activities and many break-ups. Here was old and new, history and the present. Here was old baroque Vienna with its court which still lived according to the dictates of the Spanish Ceremonial. Here were the aristocrats who held high posts in military and official circles, the officers who felt themselves personal officers of the Emperor, and here was the Church which sent its confessors to the court and the palaces, and before whose Cardinal of Vienna the Emperor himself knelt. But here were also the modern forms of life of an industrial and a technical age. All these opposite elements which finally destroyed the Hapsburg Empire mingled before 1914 in a general joy of living as in a boiling kettle.\n\nVienna's new society sat in fiacres which drove out, decked with flowers, on the first of May, along the main all\u00e9e of the Prater. It occupied the boxes and orchestra of the theatres. It drove out to the fashionable restaurants in Vienna's suburbs\u2014to the \"Stelzer\" on the border of the Rodaun woods, and to the \"Red Barn\" where it ordered \"milk-cream strudel.\" It went to horse races at Freudenau and then on to the fashionable Prater inns where the fresh roasted young geese were served, or cray-fish, or the fragrant \"May bowl.\" This society, which we already encountered on the Ringstrasse, was elegant, sensuous and gay. Its members loved Vienna's life, the wealth, and the women, among whom the girls of the Opera Ballet were especially sought after. They bought villas in the Salzkammergut to which they moved after the big races, and where they could be near the court of the Emperor which was installed in his summer villa at Ischl. When, after the autumn hunts, the court returned to Vienna, the new season could begin with balls and receptions, theatre premieres and Prater excursions.\n\nAs in every feudal society, in new Vienna the officers played an important role. The old Emperor wore the military uniform, and we used to see him in a marshal's uniform when, bent slightly, the right foot forward and his hand on his sword-hilt, he greeted a deputation or inaugurated a ball. As the Emperor stood, so stood the officers on the Ringstrasse. They copied the Emperor's bearing, the way he held his sword, and his manner of speaking the dialect. The young aristocrats served in the noble Cavalry regiments which accepted no commoners. They built their own caste. Commoner-women had to content themselves with infantry officers. For the Viennese woman, however, an officer counted as the only man. Officers were the best dancers and paid the most ardent court. All spoke the easy Viennese dialect, at court and in the salons. To the theatres and to the streets, to balls and to society, the colored uniforms of the officers lent bright spots which made the whole picture of Viennese life rich and brilliant.\n\nThat was the society of new Vienna which, according to old traditions, was a musical public. Opera performances and great concerts belonged to the sensuous, intellectual, happy life in new Vienna, and the music had to correspond to this atmosphere. At the time when the Ringstrasse was opened, Italian opera, the brilliance of Meyerbeer's works, the wit of the French Opera Comique, waltzes and operettas of Johann Strauss was what the public demanded. Classical music belonged to Vienna and was a heritage of the city, but even in Franz Josef's time, Vienna, in its musical inclinations, was a half-Italian city which, above all, loved melody, lively rhythm and the sensuality of sound. The Viennese\u2014the new Viennese\u2014was not of a profound and earnest temperament, but elegant, gay and formal. He had something of the lightness of a dancer and the motion of a comedian. To this was added the witty, superficial mind of a salon habitu\u00e9, and the sensuousness of a lover. \"My life course is love and desire,\" announces the title of one of the loveliest Strauss waltzes. This, too, was the prevailing atmosphere of the old city which, also in new times, was an aristocratic one; even when the manifestation of new forces and new social movements increased, working under the brilliant surface, and shaking the ground upon which the Vienna of Emperor Franz Josef stood.\nEmperor Franz Josef\n\nAmid all the music which flowed through the palaces, homes, gardens and streets of new Vienna, there was one single man who remained entirely unmoved. This was Emperor Franz Josef who gave his name to the last chapter of Viennese musical history.\n\nAt six in the morning, he sat at his desk in the palace, or at Schoenbrunn and signed documents. On the desk stood a box of Virginia cigars. Nearby were the ministers in frock-coats and black cravats, who reported the order of the day.\n\nWhen Franz Josef entered the Royal Palace as Emperor, he was a youth of eighteen. Now he was an old man with a white beard, and the oldest Viennese could never remember him except as an old man, who traveled through all the provinces of his empire dressed in the uniform of a Field Marshal, or who reviewed military parades, sitting on a brown horse. Wherever he went, mayors, city officials, Catholic priests and rabbis stood, with scrolls in hand, to welcome him. Children in national costumes timidly recited poems, and the old man's friendly blue eyes shone on the speaker. Only in the summer did the Emperor exchange his uniform for the suit of a hunter, when he lived at his country residence at Ischl. On his head he wore a green hunting cap, with a \"chamois beard.\" He wore the grey hunter's coat and short leather trousers which leave the knees bare, when he went hunting in the mountains for woodcock and chamois. At night, he put on an old military coat as night-dress and lay down on an iron army-cot.\n\nHe was a lonely old man. During the sixty-eight years that he ruled Austria, there were more unhappy hours for him than joyous ones. His Italian possessions were lost. In the War of 1866, Bismarck's German armies defeated the Austrians. Hungary freed itself from the remaining possessions of the Hapsburgs. The nations of the empire fought for their independence. Emperor Franz Josef had the feeling that he was sitting in an old, decaying house where one dared not move for fear of having the whole thing fall apart. Only unwillingly and hesitatingly did he give the command when the first World War broke out. He had a premonition that the war meant the end of his monarchy.\n\nIn the Emperor's own house misfortune pursued him. The Empress traveled about in foreign countries and her romantic spirit was also foreign to this correct and sober man. The Crown Prince was shot on a night of drunkenness. The imperial home was lonely. That music brings solace and that tones can speak with the voice of spirits, the Emperor did not know. He had no connections with music. The singing and organ playing in the court chapel, a court concert on festival occasions, and the daily military concert in the palace court-yard provided all the music which reached the Emperor's ears.\n\nHistorians have never ceased debating since the old monarchy was destroyed in 1918, whether the Emperor was a great or a mediocre man. Several wished to see in this Hapsburg a more than average ruler, because of his long political experience, his courtly dignity, his cleverness in forestalling the end of the monarchy. Some saw in him a dry official nature, a zealous worker, a mediocre man who derived a certain routine from long experience. If there is one sure proof that Emperor Franz Josef had an average nature without vision, it lies in the fact that he remained in a city which was full of music without displaying musical interest.\n\nThe Summer Palace of Prince Eugene (The \"Belvedere\")\n\nThe Horse-Ballet \"La Contesa dell'Aria e dell'Acqua\" (January 24, 1667)\n\nThe industrious old official who had virtues but no imagination had no connection with the arts. In his younger years he had taken an interest in the Burgtheater where his dear friend, Katherina Schratt, had played. This simple, sensible Viennese woman was for years the special friend of the Emperor, but he was not very romantic in this respect. When the Emperor left his castle of Sch\u00f6nbrunn and sought his beloved in a nearby villa at Hietzing, she had to receive him at six in the morning in formal salon attire. It was the Hapsburg who called upon her and who remained the Hapsburg even in the drawing-room. The Empress, slender, tall, with a dark crown of hair, had a drop of diseased blood in her veins which drove her to foreign places. She galloped on horseback across the Hungarian plains or climbed in the mountains or voyaged in the south, where, in the hills of Corfu, she had built a marble palace which was like a theatrical setting. Here she walked among laurels and cypresses, past marble statues, gazed nostalgically on the blue sea and had Homer read aloud to her by a small, hunchbacked Greek. The bright laughter of Katherina Schratt, meanwhile, in the Emperor's loneliness, resounded over an abyss which was dry and cold.\n\nWhen Emperor Franz Josef was older, he seldom went to the Burgtheater. His greatest artistic interest was in paintings. He showed preference for diligently executed still-lifes such as the artist Schoedel painted, with iridescent silk cloths and ivory and glasses which reflected the sunlight. In painting, too, he shunned the imaginative, preferring clever handwork. Each year Franz Josef opened the Art Exhibition, viewed the pictures with interest or politeness and took his leave with words which have become proverbial: \"It was very beautiful. I enjoyed it very much.\" These words were impersonal, because a Hapsburg did not reveal his inner feelings except to the confessor of the Catholic Church, who, during successive Hapsburg reigns, had come to the Imperial Palace, in black monks' robes, prudent and taciturn.\n\nAt the Vienna Opera, I seldom saw the Emperor present. He went, naturally, when the Spanish Court Ceremonial prescribed his appearance. It is self-understood that he was present when the new opera house was opened. He went, too, when foreign potentates came to Vienna, and in whose honor festival performances were given. The last time I saw Emperor Franz Josef at the Opera was when he celebrated his seventieth birthday, and the German Emperor Wilhelm, with all the princes of the German Confederation came to Vienna. At that time, Gustav Mahler was the Opera Director, and he conducted the first act of Goldmark's opera, \"The Queen of Sheba,\" which had been newly mounted with great splendor for the occasion. In addition, ballets were presented. The Opera was filled with noble court society. Everywhere were uniforms with many orders and decorations, and white-bosomed shirts. In the boxes glittered the jewels of the ladies, and bosoms and arms gleamed for exhibition. The Emperor sat with his guests in the great Royal Box, from which was suspended the rug embroidered with his initials. Beside him sat Emperor Wilhelm with his turned-up mustache. As the ballet girls made their entrance, Wilhelm slapped his thigh with the palm of his hand. This gesture he repeated at every new scenic effect. It would be interesting to know what the two rulers thought of the little man at the conductor's desk who now directed the opera. Probably they did not spend any thought upon him.\n\nBehind the Royal Box was a room, decorated with frescoes, where the court society ate a buffet supper during intermissions, because the Royal Opera was not only a palace of music, but also a society salon.\n\nIf Franz Josef, on less festive occasions, visited the Opera, it was generally at ballet performances. At such times, he would sit in his private box at the left of the stage. His large opera glass lay on the parapet of the box, and from time to time, he raised it to his eyes. He was always alone in his box and gave one the impression that he was an old man who wished to rest.\n\nThe Viennese Opera was supported by the Emperor. A high official of the aristocracy\u2014the Lord High Steward\u2014took care of its administrative business, and it was his duty to suggest, when the post was vacant, a new Opera Director to the Emperor. It belonged to the most distinguished tradition of Franz Josef's period that when this position was open, the greatest musician available was sought to fill it. During the forty years I was opera critic in Vienna, only one single time was a so-called \"business manager\" named Opera Director\u2014Hans Gregor, and, despite his genuine diligence, one found him a foreign element in Vienna. The Royal Opera should belong to musicians, not to routine business. Wilhelm Jahn and Hans Richter, Gustav Mahler, Felix Weingartner, Franz Schalk, Richard Strauss and Clemens Krauss were the names of the Vienna Opera directors of my time\u2014all of them important musicians who created musical traditions or kept good tradition alive.\n\nWhen the Royal Opera Director was named, he was completely free to carry out his artistic plans. It is understandable that many interested persons who wished favors from the Director sought the assistance of influential court circles, or of Katherina Schratt. The Emperor himself was inaccessible to such intervention. Since Gustav Mahler's unbending character did not fulfil the many wishes of high-standing persons, the way to the Emperor was zealously and frequently sought. But Franz Josef would always say, in such cases, \"He is the Opera Director. He must understand it better.\" Once Mahler decided not to re-engage a very much \"protected\" singer. A lady of the highest court society, a princess, intervened with the Emperor. Mahler, who had already begun his summer vacation, was ordered to the imperial villa at Ischl. There was nothing for Mahler to do but comply, so from the Woerthersee, where he was composing his Ninth Symphony, he traveled to Ischl and appeared, as officially ordered, on the next forenoon, in frock-coat. The Emperor required him to give a detailed account of why the Swedish singer, Karin, had not been re-engaged. When Mahler had finished, Franz Josef pulled at his collar, embarrassed, and said, \"Well, you understand it better.\"\n\nMahler delighted in telling the story of how, one day, an opera composed by one of the young archdukes was brought to him, along with the information that it was the Emperor s wish that this work be performed. To this Mahler replied, as he laughingly related: \"I am sorry. I cannot fulfil his Majesty's wishes\u2014only his commands. If the Emperor commanded me to perform this opera, then I would do so. I would print on the theatre-posters: 'Upon the command of His Majesty, for the first time, the opera so-and-so of Archduke Peter Ferdinand.' \" The opera of the archduke remains unperformed up to now.\n\nFranz Josef, like many official-minded men, was extremely correct. He permitted his Opera Directors, whose maintenance he paid, greatest liberty in all things. There was only one thing which he disliked: that an Opera Director give up his position of his own free will. This held true for Ministers as well as Opera Directors. \"I will not permit you to throw your chair before my feet,\" shouted the Emperor at his Minister, Schaeffle, when the latter wished to resign his post. When Gustav Mahler resigned as Opera Director, the Lord High Steward, Prince Montenuovo, did his utmost to dissuade Mahler from his decision, since he realized that the Emperor would take the news very much amiss. Also, when Felix Weingartner, who was Mahler's successor, resigned as Opera Director after two years, on grounds of private nature, the Lord High Steward scarcely dared to inform the Emperor. As Weingartner told me, Mahler's resignation had already angered Franz Josef considerably, and a second resignation following so shortly after the first, was bound to make him bitter.\n\nNot only did I see the Emperor seldom at the opera, but also at the concerts. Once he came to the Grossen Musikvereins Saal on the occasion of a festival concert in honor of the fiftieth anniversary of the Philharmonic Concerts (1910). It was then that he heard Beethoven's Ninth Symphony for the first time in his life. It remains unknown what impression the work made on the then eighty-year-old monarch. The last time I saw the Emperor in a concert hall was three years later (1913) when the new \"Konzerthaus\" was opened. For its dedication, Richard Strauss composed his \"Festival Prelude.\" A gay legion of young girls welcomed the venerable sovereign on the steps of the new building, and he thanked them with a friendly smile of his large blue eyes, which had seen so many different eras.\n\nIn the service of this Emperor who was so little touched by his city's music, were three of the best musicians of his time\u2014Johann Strauss, Anton Bruckner, and his Opera Director, Gustav Mahler.\n\nJohann Strauss was Franz Josef's Court Ball Music Director, beginning in 1864, in which year Strauss had composed \"The Blue Danube.\" At the great court balls, Strauss, in a red frock-coat, conducted his orchestra. First he would rest his violin on his hip while he turned it, and then beat the time for the introduction with his bow. When the aristocratic society had set itself in motion\u2014old and young, archdukes and princesses\u2014he would begin to play his magic violin and everyone responded to the beat and the melody of the dark-eyed man on the podium. Many Strauss waltzes were composed for festive court occasions. For the Emperor's wedding, Strauss wrote his \"Myrtle Sounds\"; for the wedding of the Crown Prince, his \"Myrtle Blossoms\"; and on the occasion of \"the Happy Return of the Emperor,\" his \"Salvo of Joy.\" For the old Emperor he wrote his \"Emperor's Jubilee Waltz\" and his \"Emperor Waltz.\" In the \"Myrtle Sounds,\" even the Austrian folk-hymn danced to waltz time.\n\nThe Spanish Court Ceremonial put an end to the activity of the world-famous musician, since it proved stronger than the Emperor, even if not stronger than Strauss's music. In order to marry for the third time, Strauss became a Protestant, but Protestants were not permitted to hold positions at the Catholic Austrian Court which had fought Protestantism for so long. Therefore, Strauss had to give up conducting waltzes at court festivals, and hung his beautiful red frock-coat in the closet. At that time, he was fifty-eight years old and the most famous musician in the world. But the Spanish Ceremonial was three hundred years old.\n\nFranz Josef played an especially beautiful r\u00f4le in the life of Anton Bruckner, who was his court organist. The Emperor provided funds for the publication of the Third and Eighth Symphonies of Bruckner. Since the composer was growing old, Franz Josef gave him a house in the Belvedere Garden. In gratitude for this, Bruckner dedicated his Eighth Symphony \"to Emperor Franz Josef.\"\n\nIt must have been a singular scene when the seventy-nine year old Bruckner went for an audience with the old emperor, to thank him for the kindness he had shown his court organist. Bruckner had to put on his frock-coat which scarcely fitted his broad, short figure, when he went to the Imperial Palace. Franz Josef held such audiences one day each week. In the great anteroom of the palace were assembled the high ranking officers who had received honors, priests who had to introduce themselves to the Emperor, or to thank him for a donation, industrial magnates who had been raised to the peerage or who had received a court title, and male and female peasants from the provinces, dressed in colored national costumes, who had some favor to beg of the Emperor. (Baron Prilesky, the court official who arranged these audiences, told me that Franz Josef was highly irritated if too few people sought an audience.) In the baroque hall stood the Emperor's guards in red coats, embroidered with gold thread, flowing white plumes in their helmets, swords in hand.\n\nBruckner made a deep, clumsy bow before each one of these guards. Finally the door opened to the room where Franz Josef stood at his desk on which the papers lay with the names of the people who had come for an audience. The Emperor, with cordiality, came toward Bruckner, who stammered a few words of thanks. The Emperor asked Bruckner if he had any further wish, and the composer replied: \"Please, Your Majesty, can you not put in a little word with Councilor Hanslick so that he will criticize me better?\" Bruckner himself told us this story. But I do not know whether the Emperor was able to fulfil this wish. The critic was never so kindly disposed to Bruckner as was the Emperor.\n\nWhat Franz Josef did for music and musicians, and to that belongs the conferring of high orders on Brahms and Bruckner\u2014he did, not out of an active relationship with music but as a correct fulfilment of his regal duty which was regulated by the Spanish Court Ceremonial, his Catholic piety, and by the documents on his writing desk.\n\nThe Empress had more feeling for music than the Emperor, for sick souls seek music. The Empress, who was a romantic woman, was a great enthusiast for Wagner's music. She also sent Wagner one thousand gulden when he was weighed down by debts in 1863. The Archdukes of the Austrian court interested themselves primarily in ballet performances, with the exception of Archduke Eugene, who was a good pianist. The relations between the Austrian court and aristocracy, on one hand, and the Viennese corps de ballet on the other, were quite lively for awhile. Folk-singers and art-whistlers catered to the musical needs of Crown Prince Rudolf. His fiacre-driver, Bratfisch, had to whistle Viennese songs to him. With this fiacre-driver, the Crown Prince and Baroness Vecera drove to the hunting castle, Mayerling; and Bratfisch's songs of beautiful Vienna, of St. Stephen's Cathedral, of lovely Viennese women, of the fiacre and of the blue Danube were interrupted by the death-knell of the Austrian monarchy.\n\nEmperor Franz Josef lived music-less and died music-less. The monotonous litanies of the priests at his death-bed in Sch\u00f6nbrunn were the only sounds which recalled music when the Emperor of the greatest musical capital of the world died.\n3\n\nMeeting Great Composers\nRecollections of Johannes Brahms\n\nI\n\nBy the time I first met Brahms on the Ringstrasse he was the greatest musical authority in Vienna. By degrees he had overcome the resistance of the public, with which he met when he had come to Vienna, and was recognized as one of the most serious and solid composers of the time. There he was regarded as a sort of \"Burgomaster\" of the musical life of the city. He was on the boards of governors of the most distinguished musical societies, and his decisions were final in the choice of professors for the Conservatory and awards of state scholarships to musicians. A close friend of Hanslick, Brahms' opinions were certainly duly reflected in the Hanslick criticisms, as well as in many of the other criticisms, since most of the critics were friends of Hanslick and bowed before the verdict of their powerful colleague. In short, Brahms became head of an influential conservative clique and whether he wanted to be thus identified or not, he was the musical pope of the same ultra-conservative society who had hissed even the first performance of his 4th Symphony by the Vienna Philharmonic. This group now gathered around Brahms to wall out modern composers like Wagner, Liszt, Bruckner and Hugo Wolf. Brahms, the Conservative, had come a long way from Brahms the Romantic, the enthusiast who had written passionate music full of love, and heartbreak, who had laid bare his soul, crowded with love for his friend Schumann's wife and the tragic passion of his feeling. Brahms as a composer had learned to quell his vibrant enthusiasm and barricade it behind classical form. He used his craftsmanship to cover his romanticism with a classical facade. He adapted himself to the bourgeois and middle-class Viennese society and in their houses became an honored guest, thought of as an ingenious and conservative composer belonging to the middle class.\n\nBrahms Walking on the \"Ringstrasse\"\n\nMy earliest recollection of Brahms is as remarkable as it is surprising. There are few people who have heard Brahms play dance music. I am one. When I was a student at the University of Vienna, I met my friends each night to discuss the happenings of the day. For reasons of strict economy, since we had very little money, our choice of meeting places was narrowed down to a reasonably priced eating place located in one of the oldest streets of Vienna. Here for little or nothing we were able to have a slice of sausage and a glass of beer in one of the two rooms which the establishment boasted for catering to a clientele made up for the most part of hucksters, tradeswomen waiting for the markets to open, and a few others like ourselves. The low-ceilinged rooms were dense with smoke and acrid with the smell of beer. The walls were exceptionally thick and like those of all the other houses in the street, had been standing since the 15th century. In a place like this\u2014and only the Americanism \"dive\" really describes it\u2014one would never have expected to find a man like Brahms.\n\nOne bitter cold night in the middle of winter the door opened and in walked an elegant company of fashionably dressed ladies and gentlemen apparently returning from a late party. One of the group was Brahms and we were all surprised to see the famous composer in our midst. He was also in evening clothes and with his friends sat at a table and ordered beer.\n\nA short time later the door opened again, admitting a rush of snow and three more patrons. In contrast to the other newcomers, these last were eminently more suited to the environment, the feminine member of the trio being a well-known street-walker and inhabitant of this old and dark section of the city. Her escorts were gigolos and all were very drunk. They seated themselves at a table next to Brahms and his friends and ordered drinks\u2014as if they had not already drunk enough\u2014all of them. Suddenly above the talking, the \"lady\" called over to Brahms: \"Professor, play us some dance music. We want to dance.\" I had the impression that she knew him, which is not unlikely, for Brahms, like Beethoven, was somewhat divided in his patronage of idealistic companionship and the more downright variety found in dark and unfrequented streets. At any rate, he rose from the table and with slow and deliberate steps went toward the old untuned piano which leaned against the dirty wall and began to play\u2014waltzes, quadrilles\u2014tunes which were for the most part pass\u00e9 and dated back a good many years. The lady who had succeeded so remarkably in getting Brahms to play, danced with her friends and others joined in. Brahms played uninterruptedly for an hour and then returned to his table, paid his bill and left. I have often thought the proprietor of this establishment should have put up a tablet in memory of this evening: \"Brahms played dance music in this pub.\"\n\nThe following day I went to my coffee house and met an elderly gentleman who had been in Brahms' party the night before. This was Mr. Bela Haas, a well-to-do socialite famous for his wit. I said to him: \"Please tell me what ever induced Brahms to play dance music for such a gathering,\" and he answered, \"Well, I too was surprised and asked Brahms the same question. He told me: 'When I was a boy in Hamburg I used to play in just such a place the whole night. I played dance music for drunken sailors and their girls. The place reminded me of that time and the pieces I played were those I used to play every night in Hamburg.' \"\n\nShortly after this first acquaintance with Brahms I met him again under no less interesting circumstances.\n\nIn the year 1892 a great theatre and music exposition was inaugurated in Vienna. The center of this international exposition was \"die Rotunde,\" a famous building in the Prater, in which rare and valuable treasures of musical and theatrical life gathered from all over the world were on display. Next to the Rotunde in the park, a vast theatre was built, and it was here that Smetana's \"Bartered Bride\" went forth, after a sensational Viennese success, to acclaim throughout the world. However, the main attraction for the general public was \"Old Vienna,\" a skillfully constructed replica of an 18th century Vienna marketplace, with house faithfully reproduced, a stage for comedians who performed old Viennese plays, and a large restaurant where the famous Viennese \"Schrammeln\" played popular music in the garden. This group of musicians was known for its perfect renditions of Viennese music and was made up of two violins, one guitar and one accordion supplemented by whistlers and yodelers, usually from the audience. Vienna of Franz Josef's time was very enthusiastic about these musicians. They were invited to the palaces of the aristocrats and Crown Prince Rudolf himself liked to frequent the places where the Schrammeln played and it was not unusual for him to join in the singing of Viennese tunes.\n\nIt was at this restaurant that I encountered Brahms for the second time. He was in the company of Alice Barbi, a marvelous singer who had come to Vienna for the first time in the year 1888, as an unknown artist and rose overnight to fame, acclaimed as the greatest singer of classical songs. I had been present at her first concert, the audience limited to fifty who had been given complimentary tickets to this preview, so to speak, by the concert management. The second concert was sold out as were all which followed. Unfortunately for us she early terminated her career, marrying an Italian aristocrat and going into retirement. She was one of the most striking figures we ever had\u2014a dark Italian beauty with dreamy black eyes, luxurious dark hair which she wore in a low knot on the nape of her neck. The expression in her rich contralto voice filled the soul and she could make an audience joyful or sad with her song. Her program regularly consisted of songs of Schubert, Schumann and Brahms. She never sang music of other composers and I have never heard Brahms' pieces\u2014the simpler composition like \"Vergebliches Staendchen\" nor the more difficult like \"Wie bist du meine Koenigin\"\u2014sung more lovingly than by Barbi. The combination of Italian sense of melody, southern beauty, true musical form and deep expression flowing from the soul was never more perfect than in this singer.\n\nBrahms had come with Alice Barbi to the \"Schrammeln\" to show her the popular Viennese musicians and typical Viennese popular playing. It was not difficult to see that Brahms was not merely being a polite host to Barbi; he was captivated by her. Usually a somber man, he was in good spirits, his face wreathed in smiles and completely engrossed in this great artist and beautiful woman. He doted upon her and one would have had to be blind not to have seen his devotion and pleasure in her company.\n\nThe Schrammeln after playing several Viennese selections struck up an American song popular at that time\u2014\"TA-RA-RA-BOOM-DE-AY.\" As the \"BOOM\" was played, it was customary to strike the table with a walking stick or thump it with a beer glass, and to this day I can see Brahms exuberantly rapping his umbrella on the table\u2014a little boy with a gray beard. There was love, kindled and sparkling, in Brahms that day beneath the old trees of the Prater which overhung dining tables as the musicians played an American song. Perhaps Barbi reminded him of another great singer with whom he had been in love in his youth and who, like Barbi, was also a contralto\u2014Hermine Spiese.\n\nAt the last of the concerts which Barbi gave, an all-Brahms concert was announced. Barbi, wanting to show Brahms her devotion, dedicated a whole evening of songs to him.\n\nWe young admirers of Alice Barbi would not have missed the Brahms concert for anything in the world and stood waiting for her appearance. Finally the door to the stage opened and Barbi came out dressed in one of the simply-cut white gowns which she always wore, her black hair knotted softly, and behind her to our surprise, and the surprise of the whole audience, in the place of the usual accompanist, came Brahms. He was somewhat at a loss and came on the stage with the awkwardness of a great Newfoundland dog following its mistress. Later we were told that Brahms appeared unannounced at the singer's dressing room and astounded her by saying that he wished to accompany her at the piano. That night the audience not only heard Brahms' most beautiful songs sung by Barbi but played by the composer himself.\n\nBrahms never accompanied in the manner to which we are accustomed to-day. Accompanists of great singers perform in the same manner as lackeys laying a carpet at the feet of their mistresses. They are in the background, obsequious and bending to the whims of the artist and never step forth to attract attention. But not Brahms. He was still a great musician as an accompanist. At this concert there were two musicians performing with an equally distributed effort and attention. The singer was not the main feature, the song was important, and both singer and pianist worked toward the same goal in the service of the composer.\n\nBrahms' accompaniment had a strong foundation of basses, even in sweet songs like the Wiegenlied, the accompaniment of which is usually sublimated and pampering. Brahms himself always used firmness in the basses. His hand was somewhat heavy and his playing was devoid of the complicated shading and nuances of colors which characterizes players of the Liszt school. He was simple and strong. There was spiritual and musical potency in his playing\u2014no nervous over-sensitiveness running amuck in hundreds of little color patches. When Brahms played, the design was important and not the colors themselves.\n\nThere was another trait in Brahms' accompaniment which is no longer found. He built up his accompaniment like a symphonic composition. There was unity, gradation and development. The music reached its climax from within by virtue of the musical forces. When he played his songs he was the creator of great symphonic forms, used in the development of a vast musical composite of several ideas, which grew like petals to form a single beautiful flower.\n\nBarbi's concert was the first time I had heard Brahms interpret his own songs, but later I was to have other opportunities of attending concerts at which he played his own compositions. He enjoyed accompanying the Viennese tenor, Gustav Walter. Walter was the first in Vienna to give a concert as a soloist in a concert at which no other artist was programmed. He had a mellifluous tenor voice which he used with fine taste and was accomplished in developing discriminating half-tones and falsettos. He is identified with the history of Brahms' music since it was he who sang the solo at the first performance of Brahms' cantata \"Rinaldo.\" He was especially well known for his charming renditions of Brahms' semi-classical songs like \"St\u00e4ndchen\" and \"Wiegenlied.\" Brahms, whose hands were heavy, experienced some difficulty in adapting his accompaniment to Walter's hovering pianos.\n\nThe last time I saw Brahms on the concert stage was in 1891 when he appeared at the first performance of his Piano trio in B major in the Kleiner Musikvereinssaal of Vienna.\n\nIt is well known that while in Vienna Brahms revised his first chamber music composition which he had written in 1854. Previously the music was the excited product of a romantic composer\u2014wild, fanciful and copious, but when revised it became a clear logical, classic pattern. Brahms put into the transformation of this music all the many endowments of an experienced and great composer, rounding it out toward greater coherence and unity. When the revised music was first performed, Arnold Rose played the violin and Brahms himself was at the piano. He played in the powerful and tightly woven classic style which was so characteristic of him and as usual introduced thundering basses. Upon this ponderous structure a magnificent and uniform building was erected. In the great climaxes of the composition ran the undertone of subterranean rumbling like the echo of a remote earthquake which served to remind listeners that beneath the heavy boulders of classic form the romanticism of Brahms' youth was buried.\n\nBrahms attended all the big concerts in Vienna. In the Society of Friends' Concert hall where he had conducted \"Gesellschafts\" concerts for three years, and where Philharmonic and choral concerts were given, Brahms always sat in the Director's box. It was here that he heard his last concert on the occasion of the performance of his 4th symphony, March 7th, 1897. Certainly he must have recalled the first inauspicious playing of his composition\u2014by the same orchestra and conductor (Hans Richter), which was roundly hissed by an earlier Viennese audience. On this night I mention, we were deeply shocked by his appearance. He had the sallow, gray pallor of a dying man, and it was only a month later that he passed away in his Karlsgasse (now Brahmsgasse) apartment where he had lived and worked for a quarter of a century.\n\nII\n\nIt was at this apartment that I spoke with Brahms. Then I was a student at the University of Vienna at the time and ambitious to become a musician\u2014or better still, a composer. Without benefit of a great deal of training I had been composing day and night\u2014songs, violin pieces, chamber music\u2014and like many young people I was obsessed with the one thought of giving utterance to my musical feeling. My mother was a friend of Celestine Truxa, the widow of an Austrian naval officer, from whom Brahms rented his apartment and she asked Mrs. Truxa if it would be possible to have Brahms look at some of my work and pass judgment on my talent. Also she wanted his advice as to what she should do with a son so ambitious to become a composer, that he gave himself over to writing musical notes instead of law notes, as a legal career had long been planned for me.\n\nBrahms bade me come and I went to his house carrying under my arm the precious bundle which held my wonderful compositions. I remember the apartment had three rooms furnished in the old Viennese period style. I entered the room that was his \"library,\" with a writing table, and an easy chair with a cushion. The next room was his \"workroom\" where his new compositions lay on a large old-fashioned table illuminated by the soft glow of the oil lamp. There was also a rocking chair which one found in all modest homes and which seemed to symbolize the quiet-loving German mentality of that time and went with a hearth, slippers, cap and dressing gown. The only adornment the workroom boasted was a picture of Raphael's Sistine Madonna; his piano was placed near the windows and held, besides notes, a large box of cigars. One door of this room opened into the third\u2014Brahms' bedroom, and over the bed hung Bach's picture. White lace curtains hung at all the windows and the apartment was kept meticulously clean by Mrs. Truxa. There was an air of lower middle-class simplicity about these rooms. There was nothing artistic about them and they were certainly a direct contrast to the luxurious suite in which Wagner worked and very different from Richard Strauss' surroundings. Probably Brahms' attachment to his old-fashioned accommodations grew out of their likeness to his father's home in Hamburg.\n\nWhen I approached Brahms in his workroom, he received me with a few mumbled words which he growled into his beard\u2014as a dozing dog would if disturbed\u2014and took my compositions from my hand. At once he plunged two fingers between the first and third staffs of the score. I was somewhat surprised to see him cover up my middle voices of which I was inordinately proud. At this time the songs of Richard Strauss were new, and songs like \"Traum durch die Daemmerung\" and \"Heimliche Aufforderung\" had impressed me deeply, so I had more or less adopted the style, writing the piano part with many voices. But the very nicest of all my notes Brahms hid\u2014the many semiquavers with which I had been so lavish. Seeing my astonishment he growled, \"When I look over a new song I always cover the middle voices. I only want to see the melody and the bass. If these two are right, everything is right.\" Finally he said, \"You must learn counterpoint. Go to a good counterpoint teacher for two years and then come back and we shall see what is left of your talent.\" I asked him if he would recommend someone and with the same gruff tone he said, \"In the country there are some old organists who know counterpoint. Perhaps you can find one who will teach you.\"\n\nWith this interview ringing in my ears I left, bent on following his suggestion and I did find an \"old organist\" and went to him as a pupil. My teacher, however, was not a country organist, or more correctly, was no longer a country organist. He was Anton Bruckner.\n\nBrahms was the Great God of musical life in Vienna, an authority of the first order and a torch-bearer of the classical heritage of a musical city. The whole conservative society paid him homage. But Bruckner lived on the thin edge of Vienna's musical life. He was surrounded and admired by his own circle of enthusiastic friends and students, but the public had no taste for his music. At a concert performance of one of his symphonies the hall emptied after the first movement except, of course, for our own ranks who made up the standees, and we stayed and cheered him until we were hoarse. The critics dubbed his music \"drunken Wagnerian.\" Brahms said of him to a friend of mine, \"The whole Bruckner movement is nothing but a swindle. In 30 years no one will know who Bruckner was,\" and this was Brahms' opinion of Bruckner, Brahms who was surrounded by mediocre musicians who wrote symphonies, devoid of ideas, according to theory-book prescription and were soon forgotten. We students were anxious to have Bruckner's opinion of Brahms and went to a Philharmonic concert at which one of Brahms' symphonies was performed. Brahms was enthroned as usual in the director's box while Bruckner stood among his students quite far back in the standing-room section. When the concert was over we asked Bruckner how he liked Brahms' music and he answered, \"Doff your caps, gentlemen. He is Dr. Brahms\u2014I like my music better.\"\n\nFridays Brahms always attended the meetings of the Tonkuenstlerverein, an association of Viennese musicians. The new compositions of the members were usually performed at these gatherings, and Brahms' presence was the b\u00eate noir of every composer, for he was a sharp and scurrilous censurer, sparing not even his best friends. His well-sharpened tongue lashed out with poisonous criticism. Looking back, I think Brahms' sadistic pleasure in giving offense, was a front to cover up vulnerable spots in his own soul\u2014that the poison was the defense mechanism of his softness. A wit of Vienna, who was invited to all parties, invented the story that Brahms leaving a supper said to the hostess, \"I apologize if there were any here whom I neglected to offend.\"\n\nThe inventor of this anecdote was Mr. Bela Haas, a wealthy man and friend of Hanslick and Brahms. Many of Brahms' new compositions were rehearsed in his apartment. Brahms liked Haas' witticisms and since they were always barbed and aggressive thrusts, it is understandable that he appreciated a talent so like his own. Bela Haas did not hesitate to nick even Brahms with his stinging foils. He told me that he and Brahms were walking in the Stadtpark where the beautiful memorial statue of Schubert stands. Brahms stood looking at the statue\u2014an artist with a notebook on his knee and pen poised as if to write down some heaven-sent melody which had come winging into his heart. After some moments of reverent study, Bela Haas said, \"Look, Dr. Brahms\u2014in thirty years there will be another musician's monument erected here and people will pause to read the name of JOHANNES BRAHMS in high golden letters.\" Brahms was quietly basking in speculated glory when Haas continued, \"and reading the name Johannes Brahms, they will turn to one another and say, 'Brahms? Brahms? Whoever on earth was he?' \"\n\nIII\n\nBrahms' character was certainly not simple. Everyone familiar with his music knew him as a man of delicate, warm and beautiful sentiment. There are tunes of unusual tenderness and grace in his works, but his shyness forced him to hide the free melodious expression behind a barrier of complicated rhythm and intricate syncopation and it comes to the surface only as light through chinks in a wall. It is often true that strong personalities are ashamed of softness and prefer to seem hard and high handed. One of the most interesting and revealing stories about Brahms was told me by Bela Haas and shows clearly how he used rudeness in fear of betraying his sensitiveness.\n\nWhen Brahms celebrated his sixtieth birthday, he was awarded the \"Leopoldsorder,\" the highest order of Austria which Emperor Franz Josef had never bestowed upon a musician, minor orders usually being reserved for them, if given at all. In honor of the occasion a party was given for Brahms and he asked Bela Haas to go with him. Together they walked through the Ringstrasse to the Franz Josef Quai where the festivities were to take place in the home of one of Brahms' friends. It was dark and about midway to the house Brahms suddenly pushed Bela Haas into one of the public lavatories near the pavement, saying, \"Come on! Come on!\" Bela Haas was dumbfounded, but Brahms finally succeeded in getting him into the washroom and extracted from his pocket a small box saying, \"Here, help me put this thing on.\" The \"Thing\" was the medal of the Leopoldsorder and standing in the men's room, Haas helped him to adjust the ribbon about his neck.\n\nBeyond any question of doubt Brahms was delighted and honored to receive this decoration, but was deeply ashamed of his joy and selected the most ignominious place he could think of to put it on in order to conceal any vanity or pleasure which might escape him.\n\nThere were many conflicts in Brahms' soul, many inhibitions. His outer roughness covered inward tenderness and his almost ascetic simplicity in living covered his romanticism. His attacks on his friends were only one of the symptoms which belied the complicated organization of his personality and deep tension in his soul.\n\nLike all great musicians living in Vienna, Brahms loved the landscape around the city. The country to the south of the city was specially dear to him as it was full of memories of Beethoven where on the gates of houses and gardens are marked tablets reminding the passerby that Beethoven had walked there.\n\nOne of Brahms' friends, the theorist and musicologist Mandyczewsky, told me a story of Brahms' strolling in the Vienna woods. After many hours of wandering he and his friends had come to an inn and asked for black coffee. The coffee was made with chicory\u2014an economy exercised by many cooks\u2014and Brahms did not like chicory in his coffee. He called the proprietress to his table and said, \"My dear old lady, have you some chicory?\" When she said she had, he continued in an even more gracious tone, \"It's not possible! May I see it?\" The old woman retreated to the kitchen and returned with two packages of chicory which she handed to Brahms. He looked them over solemnly and inquired, \"Is that all you have?\" When she said yes, he pocketed both boxes and said, \"Well, now you can go back and make us some black coffee.\"\n\nIV\n\nBrahms generally spent his summers at Ischl, the site of Emperor Franz Josef's summer palace. Johann Strauss had his summer house there too, and Brahms used to visit him and play \"Tarock,\" a favorite card game, the Austrian pinochle. It was in Ischl that Brahms composed his last work, the melancholy \"Vier Ernste Gesaenge\"\u2014an astounding presentiment of death.\n\nAs all great composers were\u2014Wagner, Berlioz and Liszt among them\u2014Brahms was fascinated by the melodies of Johann Strauss. The Strauss waltzes were the inspiration for his \"Liebesliederwalzer,\" the most beautiful tribute Brahms could have paid Vienna. Once at a ball in Vienna a lady asked Brahms to write something on her fan. He wrote the opening bars of the Blue Danube and the words \"Unfortunately not by Johannes Brahms.\"\n\nI attended the first performances of the String Quartet in G Major, the revised Piano Trio in B, the Quinetet in B Minor and the two sonatas for clarinet. Along with these the \"Volkslieder\" and several compositions for piano were played for the first time. Brahms was nearly always present at the first performances of his new compositions and would come to the stage to take his bows. His personality and bearing were arresting wherever one saw him; of medium height he gave the impression of being a great boulder of a man and would walk heavily to the stage, his flushed face always belying his sensitiveness and pleasure in being recognized and applauded.\n\nFrequently I saw Brahms in the Vienna Opera House. Of Wagner's work he specially liked \"Die Meistersinger von Nuernberg\" for its festive, brilliant music, and it seemed to arouse his patriotism. For all his many years in Vienna, he was a German patriot\u2014ardently enthusiastic over German victories in France and his political inclinations were those of a middle class man in the Bismarck epoch. When Hanslick was about to publish his vitriolic criticism of Meistersinger in a book, Brahms asked him to temper it a little, which he did, but it is still damningly harsh. As composers went, Brahms loved Mozart best, especially for his Nozze di Figaro. Professor Mandyczewsky told me that Brahms thought the finale of the second act the finest music ever written for the theatre, and the music the most perfect from a musicodramatic standpoint. It has movement, theatrical action, lifeblood, and is a wonderfully artistic characterization of contrasting personalities. Brahms preferred Mozart to Beethoven. While Beethoven's music had a divine quality, was compelling and often violent, Mozart coped with the reality of human struggling. He often said, \"There are spots in Beethoven's music which indicate that under circumstance Beethoven could have become a criminal.\"\n\nNext to Mozart the greatest musician, in Brahm's opinion, was Bach. He once said to Mandyczewsky, \"When the new Handel edition comes out and is sent me, I put it in my library and say, 'As soon as I have time I will look it over.' But when a new Bach edition appears, I let everything else go, for there is nothing more important to me than studying this volume. I know that I shall learn something new.\"\n\nBrahms had a special predilection for Bizet's Carmen and kept the score in his library. He admired this music for its spirited clarity and jewel-like, sparkling orchestration. His own orchestration was compact and heavy and had no brilliancy or color, but he could admire in Bizet the sense of fitness which made him leave a free space between sounds and his aptitude for noting in music true Mediterranean zest and color.\n\nBrahms could well appreciate Bizet, for his music was so different from his own. But he was ever a man of contrasts. Coming from the foggy north, he loved the South. He needed its sunlight and warmth, for there was much that was dark and turbulent in his soul; and he who was so stern and grave took his heart to the gaiety and cheer of Vienna.\nThe Composer of the \"Queen of Sheba \"\n\nIn the epoch of Emperor Franz Josef, Carl Goldmark was the oldest of the many great musicians who contributed to Vienna's glory as a center of musical life. The lines and wrinkles on Goldmark's face spoke of the many years of hard struggling and working, of despairing and praying he had undergone. He had the face of a kind, good and calm man. He always reminded me of a wise old Rabbi, whose mind was above this life, and who lived in the thought of God. He was eighty-three years old when he died. There were few men in Vienna who grew so old. Franz Josef also lived to the same age as Goldmark, who was a boy when the eighteen-year old Emperor ascended the throne in the midst of the storms of the Revolution of 1848. Goldmark lived through the entire epoch of Franz Josef, experienced all the misfortunes and catastrophes of the crumbling monarchy, and, like the Emperor, finally died during the great war which destroyed it.\n\nIt was music which supplied the greatest splendor to the Austrian monarchy during Franz Josef's reign. The waltzes of Johann Strauss, the symphonies of Anton Bruckner, the songs of Hugo Wolf, the brilliant Vienna Opera directed by Gustav Mahler, the famous Vienna Philharmonic Orchestra, the glory of the Vienna Conservatory which supplied the whole world with composers, conductors, singers and instrumentalists\u2014all this comprised the greatest treasure of the Austrian monarchy. During Franz Josef's reign, Austria may have lost wars. It may have lost the Italian provinces and the supremacy over Germany. The everlasting struggles of the nationalities may have destroyed the empire of the Hapsburgs. But the colorful display of music, the wealth of musical life, the sounds which filled the air, and the great composers who were born in Austria, or later came there, gave the time of Franz Josef imperishable fame and glory. To that abundance of sounds, colors and melodies, the music of Goldmark was a very original and peculiar contribution. Little music existed which could compare with his in brightness, exuberance and voluptuousness. An admixture of oriental luxury increases still more the general impression of sensual enjoyment which cannot be separated from Goldmark's music.\n\nGoldmark was born in Hungary, which, of all European countries, is nearest to the Orient. The music of the Hungarian gypsies is oriental music, and the cymbal, which accompanies all gypsy music, an oriental instrument. The primitive colors which the Hungarian peasants use in painting their jugs and plates are oriental colors. These impressions of his youth must have accompanied Goldmark throughout his life. But its oriental color is only one of the elements which join in creating the peculiar value of Goldmark's music. Goldmark is one of the artistic representatives of Franz Josef's era which succeeded for the last time in the history of Austria in gathering the spiritual and artistic forces of German, Slavic and Hungarian peoples in the valley of the Danube, and shaping them into a higher general culture. Goldmark represents the spirit of Austrian culture under Franz Josef, the artistic wealth, the refinement of taste, the predilection for sensual enjoyment, the inclination for decorative luxury which were the gifts of an old European capital. In this sense Goldmark belongs to Austria and Vienna, and the exuberant colors of his music remind us that at the time of sunset, all colors are brighter and more radiant.\n\nCarl Goldmark came to Vienna when he was fourteen years old. He was a poor Jewish boy from Hungary. His father was a cantor and notary of Polish descent. There were twelve children in the family, all of whom grew up with little more than the traditional religious education. In the synagogue, little Goldmark heard the old tunes of the Hebrew service which accompanied the Jewish people from their native oriental country to the distress and persecutions of Europe. Saturdays and feast days he heard the book of the Torah and the prophets sung with the ancient cantillation which adorned the holy words. He did not begin to learn reading and writing until he was twelve years old. He was one of many poor Jewish boys who came to the Austrian capital to study. He lived with his brother Josef, a medical student whose boots and clothes he had to clean. Every day, like other poor Jewish boys, he had his lunch at the home of a well-to-do, charitable family. In Jewish life, it was a tradition that the rich feed the poor. Such meals were called \"Taege-essen\" (\"day's eating\").\n\nGoldmark studied violin in the Conservatory with Josef Boehm. Then he was engaged as violinist at the Carltheater where operettas and popular plays used to be performed. Sitting in the orchestra, he listened attentively to the orchestral sounds. He told me often how he acquired experience in how to write for orchestra in the small pit where twenty to thirty musicians sat. He tried studiously to become acquainted with the compass and the technique of producing tones from the various instruments, and questioned his colleagues indefatigably about the peculiarities of their instruments. He studied sounds and the mingling of sounds. While the comedians sang their ditties, he studied orchestration. Goldmark completed these studies of his in a small theatre in Budapest. There were only two first violinists and when the concertmaster fell ill, Goldmark played alone at the first desk of the first violins. He told me how once, as the only violinist, he played a performance of Meyerbeer's \"The Huguenots,\" and remarked in his mild and wise way: \"I learned orchestration when I was playing in that orchestra. You cannot learn orchestration in a good orchestra. Music played by a good orchestra always sounds well.\"\n\nGoldmark was the noblest example of an autodidact I have ever known. He loved to study, even when he was a very old man. When he was more than seventy, I once met him in Gmunden, walking along the beautiful all\u00e9e which borders the lake. I asked him where he was going and, noticing a small red book in his hand, what he was reading. He replied that every day he made a round-trip on one of the lake steamers, and that, sitting on board, he studied an English dictionary. And he explained further: \"Just as one has to exercise in order to keep the body fit, so does one have to exercise one's mind. The best exercise for mental fitness is studying the vocabulary of a foreign language.\"\n\nIn 1848, the storms of revolution were sweeping through the narrow streets of Vienna. Goldmark's brother Josef was one of its leaders, and in the hall of the old university, he shouted for freedom and the brotherhood of man; and finally he had to leave Vienna when the army of Duke Windischgraetz suppressed the glorious call. He went to the United States, where his daughters still live, wonderful old ladies who preserve the noble features of the Goldmark family. The American composer, Rubin Goldmark, was his nephew, and Justice Brandeis his brother-in-law.\n\nCarl Goldmark, a reserved and shy man, kept away from the bloody struggle and returned to Hungary for three years. When, once more, he went back to Vienna, the new city had been built.\n\nIn 1869, the new Opera House was opened in Vienna, and it was for this theatre that Carl Goldmark's greatest work, the opera \"The Queen of Sheba\" was written. \"The Queen of Sheba\" was more than a grand opera with a splendid display of voices, many-part choruses, richly colored orchestra, impressive scenery, and imposing processions of masses. Like Verdi's Egyptian opera, \"Aida,\" Goldmark's first great opera which was finished the same year that Verdi's opera was performed in Cairo for the first time (1872), was a masterwork of modern painting of landscape with the lights, sun and the colors of the Orient. Both were romantic operas. A French romantic poet, Gerard de Nerval, who traveled to Cairo, Syria and Constantinople, living outside reality and dreaming fantastic dreams, was the first to use the tale of the Queen of Sheba for an opera book written for Meyerbeer. But the composer of \"Faust,\" Gounod, composed the opera, \"The Queen of Sheba,\" which was performed in Paris ten years before Goldmark's work. Goldmark's \"Queen of Sheba\" surely connected with the French romantic tradition. Like \"Aida,\" it is an oriental opera in the style of the French romantic opera school. In both works, the style of the operas of Rossini, Meyerbeer and Hal\u00e9vy, reaches its climax.\n\nBut despite all superficial resemblance between Goldmark's \"Queen of Sheba\" and the great French opera, there is something quite new in the former. The operas of Meyerbeer and other composers of the French romantic opera school were products or fancy, technique and calculation. Goldmark's \"Queen of Sheba\" is a product of fancy, technique and unconscious remembrances, impressions, feelings and dreams of a Jew who, as a boy, saw the lights in the synagogue and on the festive Sabbath table, heard the recitation of the Torah; and who read the description of the Tabernacle which a Jew in Babylonian exile had imagined as a fairy tale\u2014a wonderful dream of precious material, jewels, gold and silver.\n\nIt is strange that Goldmark himself did not want to admit the connection between his work and his Jewish soul. I once asked him if he never had a mind to compose an opera \"Esther.\" I reminded him that Grillparzer, the greatest Austrian poet, had left a wonderful \"Esther\" fragment and that it certainly would be an artistic deed to unite Grillparzer's verses with his music. He became furious. \"It seems that the Jews consider my \"Queen of Sheba\" a national opera. But in composing this work, I had no other interest than to write music to an interesting and efficient text. I was captivated by the oriental colors. I was interested in this type of subject purely on account of artistic motives. After having composed this oriental opera, I had exhausted this subject. I turned to other subjects.\"\n\nIt was true. After the \"Queen of Sheba,\" Goldmark composed the opera, \"Merlin,\" the text of which the poet and philosopher, Siegfried Lipiner, took from a Celtic legend; another opera, \"The Captive\" with Greek heroes as characters and the Briseis episode from the \"Iliad\" as subject; \"The Cricket on the Hearth,\" after Dickens' \"Christmas Carol\"; and the old German \"Goetz von Berlichingen\" based on Goethe's play\u2014an opera of mediaeval times in which the composer made his farthest departure from the artistic world and mood of the \"Queen of Sheba.\"\n\nAnother time I had an experience which again illustrated how Goldmark misunderstood the very sources of his creative fancy and his unique artistic force. I had been in Prague. I loved this old and somewhat stern city where I had studied at the High School, and where, as a boy, I had often walked to the old houses of the Ghetto, through the dusky streets; and where I had passed the weather-worn tombstones of the old Jewish cemetery, and the Jewish City Hall with the Hebrew ciphers on the clock whose hands moved from right to left, arriving finally at the Gothic \"Altneuschule.\" This synagogue always made a very strong impression on me. The mediaeval room with the pointed arches of its pillars and the traceries of its small windows was always dark, and in the gloom, old Jews prayed ecstatically, their prayer-desks shaking, now murmuring, now singing. When visiting Prague, I never missed going to this synagogue, in which the dark ages seemed still to be alive. After returning, from a trip to Prague, I happened to meet Goldmark, and I told him of this synagogue and how I had been impressed by some phrases of the prayers I heard there, which reminded me of certain passages in his \"Queen of Sheba.\" I asked him whether he had used those Jewish prayer tunes consciously or unconsciously. He was very angry, and absolutely refused to admit that he used synagogal music in his opera. The resemblance of some phrases in the \"Queen of Sheba\" with the turns and phrases of the Jewish prayers, he insisted, was purely coincidental. Goldmark seemed to have forgotten that he was the son of a cantor, and that the first music he had heard as a child was that of the synagogue. These melodies must have been buried in the deepest layers of his soul, for such recollections of youth are indestructible. Composing his opera, he could not have helped being shaken in the unconscious regions where, as with every great artist, childhood impressions kept living and streaming upward through the roots of his fancy into the images and forms of his works.\n\nWhen I spoke to the famous chief Rabbi of Vienna, Professor Chajes, about Goldmark, I expressed my bewilderment that this great and wise composer who represented, in his personality, the best Jewish traditions, wanted to escape his Judaism. He did not want to see the fact that his greatest opera was a perfect masterpiece because its biblical subject expressed all the forces of his Jewish soul, his Jewish environment and the beauty of his Jewish religion. As a Jew, he was uneasy and timid, although his greatness was part of the greatness of the people to whom he belonged not only by birth but by the best of his feeling. His wisdom was rabbinical wisdom. His ambition for learning and his impulse to become more and more proficient were an inheritance from the people of the \"Book.\" Excepting Professor Chajes, Professor Freud who honored me by discussing with me some aspects of Jewish music, and Professor Einstein, he was the most perfect representative of the Jewish personality I ever met. But this he did not wish to see.\n\nI asked Professor Chajes if he did not think that the \"Queen of Sheba\" was a wonderful picture of the Palestinian landscape in sounds, a colorful image of oriental light; and how it could happen that Goldmark had accomplished this without having been in the Orient. The great scholar answered: \"Goldmark did not have to go to the Orient. He had the Orient in his soul,\" and he repeated this profound sentence at the dedication of the honorary tomb of Goldmark.\n\nEven if Goldmark did not want to recognize the Jewish heritage in his creative faculty, he certainly was intimidated by the anti-semitic students who stood in the gallery at the Opera House and hissed at a performance of the \"Queen of Sheba.\" The music critics of anti-semitic Viennese newspapers did not fail to depreciate Goldmark's music as that of a Jew.\n\nI always was sorry that so great a composer as Hugo Wolf howled with the wolves in this respect. When Wolf set Goethe's poem, \"Gutmann und Gutweib\" to music, he accompanied the words \"Und da kamen Juden\" (\"and there Jews were coming\") with the contrapuntal combination of two themes. The first of these is one of the themes of Goldmark's \"Queen of Sheba,\" the other a theme of Adalbert von Goldschmidt, one of the most interesting composers in Vienna, whose trilogy \"Gaea\" is a profound and philosophical work. Hugo Wolf was the last person who should have indulged in anti-semitic allusions. One of his best friends, at whose expense his first songs were published and in whose apartment he lived for a long time, was my friend, Friedrich Eckstein, to whom I am indebted for having introduced me to Hugo Wolf.\n\nGoldmark could expect no other methods of dealing from his enemies than with abuse and hatred, and the greater and the more successful his work, the greater the abuse and hatred had to be. It was more difficult for him to bear the reluctant praise of friends. He suffered keenly when he heard the criticisms of Eduard Hanslick, who had the greatest esteem for Goldmark's pure and modest personality, but who confessed, in his review of the first performance of the \"Queen of Sheba\" that he could \"put up with this kind of music only for a short time.\" I quote: \"With the obstinacy of a spirited man, Goldmark nestled down into the preference for oriental music with its lamenting, whining tunes, with its augmented fourths and diminished sixths, its disagreeable fluctuating between major and minor, its heavily growling basses above which thousands of dissonant and small tones are crossing. Only a few colleagues of Goldmark have such a lasting pleasure in sharp dissonances as he. I have not always succeeded in becoming friends with Goldmark's music even though my efforts were facilitated by my personal esteem and sympathy for the composer. I never missed spirit and independence in Goldmark's compositions, but I did miss clarity, natural feeling and sense of beauty.\" In Hanslick's criticisms on the \"Queen of Sheba,\" allusions to the Jewish color of Goldmark's music are not lacking. The song of Astharoth which lures Assad to the Queen is, for instance, a tune \"with which one may push pious Jews into a synagogue, but no lover to a rendezvous.\" The melody of this criticism was the same as that of Goldmark's enemies, only the orchestration was finer and more refined.\n\nThe most bitter experience Goldmark had to endure was the harsh, even rude treatment he received from Brahms. Brahms did not like the music of the \"Queen of Sheba.\" He felt about it as did Hugo Wolf, and Goldmark relates in his memoirs some very unpleasant stories which demonstrate how unkindly Brahms acted toward him. One of these stories was told me by Ignatz Bruell, Goldmark's most intimate friend, whose own opera, \"The Golden Cross\" was one of the most successful and popular works of the time. Bruell told it somewhat differently than did Goldmark himself. The latter tried to extenuate Brahms' very ugly behavior. According to Bruell, he gave a party in honor of Goldmark after the beautiful chorus, \"Wer sich der Musik erkieset\" (\"Who chooses music for himself\") was performed for the first time in concert by the Society of Friends of Music. The words of this chorus were by Luther, who loved music. Goldmark's new composition was a great success and the audience took the opportunity of showing him their admiration for his artistic personality, venerable in accomplishment as well as in actual age. All the composer's friends were invited to the party afterwards, and they awaited Brahms' arrival with excitement. Finally Brahms arrived, gloomy, and he did not speak one word. After some taciturn brooding, he finally grumbled: \"Wonderful text. Sorry that a Jew composed the music to it!\" Perhaps Brahms, when he made this malicious remark, was irritated because he himself had not set this poetical stanza of Luther's to music of his own serious, manly and ingenious composition. Instead of such music, we have inherited the much less agreeable memory of a cruel offense to one of the noblest of his colleagues.\n\nGoldmark's \"Queen of Sheba\" rises high over the lowland of human weakness, envy and meanness. Since its premiere, it has been recognized in all the great opera houses of the world as one of the most impressive operas ever written, and as the production of an artist who possessed a fancy of his own and a new kind of imagination nourished upon the images of the Bible, the sublime solemnity of the Psalms, and the intensive and pathetic feeling of the songs of the Synagogue. The \"Queen of Sheba\" is an opera written for audiences who like beautiful melodies, ingenious climaxes of choruses and ensembles, resounding orchestra, dramatic passions and conflicts. No opinion on Goldmark was more unjust and false than the sentence of Friedrich Nietzsche calling him \"the skilful ape of Wagner.\" There is more in the music of the \"Queen of Sheba\" than the utilization of Wagner's orchestral devices, which are self-evident in the work of a modern composer. But there is color, the conjuration of a vanished world, glorification of the past, a great epoch of the history of a people illuminated by the lights of legend. The Queen of Sheba's visit with Solomon had often been depicted in painting, frescoes and sculptures by the greatest artists of the dark ages and of the Renaissance. No one of them displayed the imagination, the vividness and the exuberance of Goldmark.\n\nI made the acquaintance of Carl Goldmark in the year 1899, shortly after the premiere of his opera, \"The Captive\" (\"Die Kriegsgefangene\") in the Vienna Opera House.\n\nAt this time, I had begun writing music criticism in a small Viennese music paper which had to tolerate my first attempts at reviewing opera and concert. My criticisms on the new tragic and sombre Greek opera was animated by my high personal esteem for Goldmark, and especially by my admiration for his opera \"Merlin,\" a great and fantastic work, which was the most Wagnerian of all his operas, and in which he wished to surpass even the \"Queen of Sheba.\" Only the largest and best staffed and equipped opera houses could perform \"Merlin,\" which requires great voices, a great display of settings, and all the resources that a theatre possesses. Even Goldmark's friends were surprised when, ten years after the first performance of \"Merlin\" (1896), he wrote an opera in the emotional style of a simple musical comedy. This was his melodious \"Cricket on the Hearth,\" in which, after having been an over-pathetic composer, he transformed himself into a modest, good-natured, typical citizen.\n\nI was sitting at my desk, perpetrating some new critical essay, when my bell rang. I went to answer it, looked through the peep-hole of the door, but I could see nothing but the top of a felt hat. I opened the door and there was Goldmark, a man of such small stature, that he did not reach the level of the peep-hole. I recognized him, of course. He was about sixty-seven at this time, and I had often seen him at concerts, so I showed him, with the natural respect of a young man, to my room. Goldmark told me that he wanted to thank me for my criticism of his new opera, and he made me feel ashamed by his goodness and his tranquillity.\n\nSince that day, I often spoke to Goldmark. I always felt better after having been in his company. It was a kind of moral efficacy which emanated from him that so impressed me. In my lifetime, I have met many great men, artists as well as scholars. No one else, however, had the superior detachment of mind Goldmark possessed. When I read, later on, the sentence of the philosopher Leibnitz that \"justice is the most perfect union of wisdom and kindness,\" I immediately thought of Goldmark. He was a wise and good man.\n\nShortly after this first meeting, I was appointed lecturer at the Conservatory of the Friends of Music. My first lecture on the Aesthetics of Music was published in a musical journal. A few days later I received a kind letter from Goldmark, who had chanced to read the article, in which he said friendly words praising the content and ideas as well as the form of my lecture. That was why I went to thank Goldmark. I found him in the coffee-house where he used to drink coffee after his lunch and read the important European newspapers. He was an avid reader, and wanted to know everything that was going on in the world. The coffee-house was on the Ringstrasse, close to the Burgtheater (Caf\u00e9 Landmann), and it had a magnificent view of the new City Hall and the park surrounding it. Goldmark was sitting there at a table on the outside terrace with his old friend, Ignatz Bruell. On one side of the table, Goldmark was reading his papers; on the other, Bruell was hidden behind some great daily. Neither one of them spoke a word. In this way, they understood each other perfectly. One was silent, the other listened. I did not want to disturb the thoughtful silence. So I ordered a cup of coffee, took up a newspaper and read, thereby becoming the third silent man.\n\nAs in the life of every great artist, there were in Goldmark's, too, happy days and days of sadness and distress. Throughout all those fluctuations of life, Goldmark was always restful, kind and good. Seldom did he complain of the injustices of destiny. And never have I heard him make one of those malicious, ugly and envious remarks which I so frequently encountered among even the greatest artists when they discussed their colleagues or rivals.\n\nAll Goldmark's operas had their premieres in the Vienna Opera House. The \"Queen of Sheba\" from here started its triumphal march to Germany, Italy and other European countries, and so did Goldmark's other operas. But when he wrote his \"Goetz von Berlichingen,\" Gustav Mahler was convinced that the opera was a weak work, and refused to perform it. Goldmark was confounded, but did not lose the balance of his spirit. He recalled how, years ago, another Director of the Vienna Opera House had at first refused to perform the \"Queen of Sheba,\" and that, when it had been decided to perform it, the conductor practically cut the work to pieces after the dress rehearsal; and, finally, that notwithstanding all the prophesies of coming failure, the premiere resulted in a glorious success for composer and artists. So, instead of the Vienna Opera House, the Opera House of Budapest was given the honor of presenting \"Goetz von Berlichingen\" for the first time. Goldmark asked me to come to Budapest for the occasion. Attending a rehearsal in the beautiful and luxurious Opera House in Budapest, I was astonished at the energy and command the usually reserved and bashful Goldmark displayed. He was not an easy-going artist. His will and intentions were definitely determined. The little man knew exactly how to impose his orders, and the conductor, singers and stage-director had to obey his instructions, even though he expressed them in a soft, polite tone.\n\nThe first night of \"Goetz von Berlichingen\" was a great success. The Hungarians are an enthusiastic people. Since Goldmark had been born in Hungary, the audience loudly cheered the composer; and after the performance, Goldmark's victory was celebrated with inspiring speeches, Hungarian wine and gypsy music at a hotel near the Danube where Hungarian society congregated to celebrate festive events, and where during the reign of Franz Josef, the Hungarian joy of life overflowed every night.\n\nThe success of Goldmark's opera in Budapest was so great that the Vienna Opera House had to follow with a performance there. Despite an excellent Vienna presentation, the opera soon disappeared from the repertory. Goldmark's artistic imagination could not easily adapt itself to the musical description of the ancient Germany of rude and simple customs, of knights of the dark ages, of strong castles with thick walls and high watch-towers and of soldiers in armor. His fancy leaned to the burning sun and the brilliant colors of the south. His was (and I quote a word of Nietzsche) a \"Mediterranean\" fancy. The first composition with which Goldmark had attracted the attention of the world had been the overture \"Sacuntala\" which was played in all the concert halls, and became famous because of its gleaming Oriental colors. All Goldmark's great music had the splendor and magnificence of Oriental carpets. Even in his simple and idyllic musical comedy, \"The Cricket on the Hearth,\" the orchestra is like a trunk full of precious jewels. In his popular orchestral suite, the \"Rural Wedding,\" the orchestra is anything but rural. It is artistically embroidered, heavily loaded with gorgeousness, refined and ingenious. His imagination was related to the one which created the tales of \"A Thousand and One Nights.\" How true were Professor Chaje's words: \"He had the Orient in his soul!\"\n\nWhen I spent my summer vacations in Gmunden, one of the most charming resorts in the Austrian Alps, I met Goldmark almost every day. He liked the place, and had bought a low farmhouse on the outskirts of the town. On the wall of that house there is now a commemorative tablet with his portrait.\n\nWhile we walked together, he would tell me of his youth, the memoirs he intended to write (he began writing them when he was eighty) and especially of his meeting with Richard Wagner. He told me how well he loved the Alps and how he spent every vacation in the mountains. Until his seventieth birthday, he used to climb high mountains and swim in the icy, green waters of the Alpine lakes. Goldmark was very angry at the physicians who, when he was seventy, prohibited him from continuing such youthful activities.\n\nSometimes I accompanied Goldmark to other sections of the Salzkammergut when he wanted to visit friends who summered in that wonderful country of snowy peaks and green lakes. One might say that the Salzkammergut was divided into several musical districts. In Gmunden Carl Goldmark composed. Not far from there was the charming village of Traunkirchen where Hugo Wolf was busy writing songs. At Ischl, two famous Johanns\u2014Strauss and Brahms\u2014spent their vacations. On the shores of the Attersee, Gustav Mahler had rented a small house which had once been used as a laundry, and there, on a meadow under flowering fruit trees, he composed his Second Symphony. A few miles farther on, Ignatz Bruell and his family owned a wonderful house, and it was here that Goldmark liked best to visit.\n\nIgnatz Bruell was not only a successful composer, but a man of most fascinating personality. Nothing could be more impressive than visiting this fine and cultivated man who lived with his beautiful wife and his two just as beautiful daughters in perfect harmony. Bruell's great blue eyes were like those of a good child. He was a friendly, naive man who did not understand that there could be wickedness and ugliness in human beings. Bruell had the reputation of being one of the most musical pianists in Vienna, and Brahms liked hearing his own new piano compositions played by Bruell. Sometimes I had the honor of being invited by Bruell to play classical symphonies with him. It was a pleasure to play with this friendly and indulgent man. Still greater was the pleasure when Bruell, lured to the piano by his flattering wife, played compositions of Schumann. His whole soul was filled with romanticism, and I have seldom heard Schumann's works played with such poetic feeling. While he played, Goldmark used to sit in a corner, listening attentively.\n\nI spoke to Goldmark for the last time in Vienna, at his apartment in the Neubaugasse. He was sitting at his table writing notes. Curious, I asked the old man who at that time was over eighty, what new composition he was planning. He said: \"I am writing exercises for my grandchild to whom I am teaching piano.\" As a young musician in Vienna, Goldmark had made his living by teaching piano. He taught for fifteen years. His favorite pupil had been a seven-year-old girl to whom he gave lessons, receiving a free meal at her home as payment. That little girl later became a famous soprano, Caroline Bettelheim, and she sang the role of the Queen of Sheba in the first performance of the opera. So when, in his old age, Goldmark wrote exercises for his grandchild, he remembered Caroline Bettelheim and his days as a poor young musician in Vienna. The circle of his life was reaching completion.\n\nSince during the first World War, I was an officer in the Austrian army and served far from Vienna, I could not be present at the funeral of Goldmark. But when, after the war, a commemorative tablet was affixed to the house in the Schuettelstrasse, from where, sixty years before, Goldmark had walked with his violin on his way to the Carltheater, and where he had written his first successful compositions, I was asked to make the speech of the day. Representatives of the Ministry of Education were present, with those of Vienna's musical life. I was given the opportunity of recalling the figure of the little man who was a great composer, and whose imagination was one of the most personal and unique in the history of music.\nHours with Hugo Wolf\n\nIt was in the standing-room section at the Philharmonic Concerts that I met Hugo Wolf for the first time. He stood there during one of the rare performances of Anton Bruckner's Fourth Symphony, leaning with his back to the great mirror which ornamented the rear wall of the great concert hall. He was a slender man in his thirties, dressed in a brown velvet jacket, a wide black artist's tie around his neck, extremely pale in the face and with eyes which burned like live coals\u2014the eyes of a fanatic.\n\nAround him stood young people, enthusiastic followers of Wagner, Liszt and Berlioz, the stormtroopers of the new musical generation. Many of them had long, streaming hair and wild looks. All excitedly awaited the appearance of Hans Richter, the conductor whose baton was the magic wand that exorcized the ghosts of great music. Everyone discussed music with enthusiasm. Some held scores which they studied like mystical books.\n\nHugo Wolf was one of the most passionate enthusiasts. Although his wonderful \"Moericke Lieder,\" which now belong to the repertoire of all concert singers, had been published at that time, Wolf still had to live in the outskirts of Vienna as if in exile. A ferocious revolutionary, he was expelled by Vienna's fashionable society which detected in his songs strong traces of the damned Wagnerian poison. In writing the modern chromatic harmonies and expressive declamatory lines of a new musical epoch, he had committed a crime which was punished with the death sentence by the conservative music critics of Vienna and by conservative musical society. Besides writing modern music, he dared, as music critic of a weekly, to attack Eduard Hanslick and Brahms. Wolf was enthusiastic about the music of Wagner and Liszt, and excited about that of Berlioz. Bruckner was his saint, before whom he knelt. He was furious about the superficial taste of high society, and stormed like a crusader against the walls of the old music fortress Vienna, swinging the banner of Wagner. He was fanatical, enthusiastic, moody, like all true artists. He was unjust, of course, subjective, but courageous and brave. The conservative music critics of Vienna who pretended to defend classical music when they were really defending their own narrow-mindedness, never forgave Wolf the enthusiastic fanaticism of his criticisms. The name Hugo Wolf disappeared from the columns of the Viennese daily papers where musical conservatism had entrenched itself. He was shunned like a leper and had to live outside the pale of a musical society which did not even know his name. He had to be satisfied with the love and admiration of a small group of friends who recognized his greatness and his genius, and who could feel the delicacy of his soul behind the pricks of his personality.\n\nWe young musicians, devoted to Wagner, fanatical believers in modern music, who hated the sensual conservatism of Vienna's wealthy society, believed in Hugo Wolf. His songs were new music. The words were filled with a new kind of emotion. A modern, nervous tempo swung through his harmonies. There was a new and fascinating lyrical feeling in his melodies, and more air and atmosphere in them than in other songs. There were new shades and colors which corresponded to those to be found in the impressionistic paintings of those days. Hugo Wolf belonged to us and we belonged to him. There is a secret understanding among young men of the same generation. So we stared at the pale man who stood in the standing-room section just like ourselves, while Brahms sat in a box like God sitting on clouds. Hugo Wolf was as excited about Bruckner's music as we were; and when he shouted before the beginning of the fourth movement of the symphony, menacingly and imperiously raising his fist: \"Attention! Now the sublimest part is starting!\" it sounded like a battle cry.\n\nShortly after seeing Wolf for the first time, I was given the welcome opportunity of making his acquaintance.\n\nIn the \"Prater\" of Vienna where people, by the hundreds, sat in the many restaurants, or rode on the merry-go-rounds; where young couples, in love, walked under old chestnut trees, and where on lovely summer evenings the air was filled with music, songs and kisses, in 1902 a new amusement park had been opened. It was called \"Venice in Vienna.\" A sort of make-believe city had been built of canvas and wood. On water-filled canals, gondolas were manned by gondoliers imported from Venice. They shouted their melodious \"Sta-li\" and beat the dirty water with their long, black oars, standing on the high stern of the gondola and rocking to and fro. Vienna's best society sat in the restaurants and coffee-houses of the amusement section, or walked among the cheap settings of Gothic palaces, laughing, flirting and throwing \"confetti.\" Italian music floated seductively from the great open-air theatre, and in all the coffee-houses and restaurants, small groups of Italian singers shook their tambourines and entertained with folk-songs of Venice and Naples.\n\nI had been standing in front of one of the stages, listening to Neapolitan songs, canzonettas and barcarolles, which reminded me of moonlit nights in Naples, of the white-sailed fishing boats heading slowly for Capri, of the wide Gulf of Naples and of Vesuvius. I was so absorbed in reminiscing, that I was not aware of two men who stood beside me and listened quite as attentively as myself. One of them was my friend Fritz Eckstein, a manufacturer and world-traveler, a great scholar in philosophy and calculus, a mystic and music-lover, who was serving as a volunteer secretary to Anton Bruckner, and who had the symphonies of Bruckner and the songs of Hugo Wolf printed at his own expense. (This great man died in Vienna, not long ago, after the Nazis had expelled him from his home where, among books on philosophy and mathematics, piled from floor to ceiling, Hugo Wolf had lived as his guest for a long time.) The other listener was Hugo Wolf. He was extremely pale, and in his eyes a disastrous fire was burning. The approaching catastrophe was already painted in the pallor of that suffering face and in the restlessness of the eyes.\n\nFritz Eckstein introduced me to Wolf, but a conversation did not start for some time. Wolf remained silent, as if submerged in his thoughts. His glance was stinging, and I was careful not to disturb him, for, to all appearances, he wanted to be left alone with the dark powers in his soul.\n\nLike many great artists, Wolf went from deep depression to excitement and fits of elation. He struggled between melancholy and agitation, and like Beethoven, the greatest musician of this kind, he knew hell and heaven, the deepest abysses and the highest peaks. I was greatly moved when, after Wolf's death, the director of the lunatic asylum where he died, showed me the chart where the physicians of that institution had recorded day by day the sufferings and tortures of his sick mind. This diary of a diseased soul, as it were, proved that even in the changing moods of mental illness there was the same regular cycle of depression and excitement, of shining day and black night, restlessly revolving until his death, as there had been in the happier days of the creative artist.\n\nFritz Eckstein, Hugo Wolf and I, after some fruitless attempts to start a conversation with the silent composer, sat down at a table in one of the coffee-houses and ordered coffee, the smell of which accompanies all phases of life in Vienna. The Italian singers there were singing the world-famous canzonetta, \"Funiculi-Funicula\"\u2014the rattling rhythms of which had been sung for the first time when Cook opened the cog-railway carrying tourists to the summit of Vesuvius. It seemed as if this melody awakened Wolf out of a deep dream. He became excited, gave the singers money, and had them repeat the tune, which was new to him, again and again. He shouted enthusiastically, \"I shall write a symphony about that melody!\" He was not, of course, accorded the time to write any such symphony, since he soon became lost in his sickness as in a dark swamp. But somewhat later, the young Richard Strauss used the merry tune in the last movement of his symphonic poem, \"Aus Italien,\" mingling the vivacious rhythms of the \"funiculi\" with the noises of the streets of Naples.\n\nShortly after that meeting, Wolf sent me, at the instigation of Fritz Eckstein who knew of my admiration for the composer's songs, the two volumes of his new \"Spanish Song Book\" (\"Spanisches Liederbuch\"). Southern countries had been for Hugo Wolf what they had been for the philosopher Friedrich Nietzsche, who was always longing for the splendor and the sun of the south\u2014a kind of spiritual native home. The \"Italian\" and \"Spanish\" songs of Hugo Wolf, his exciting \"Spanish Serenade,\" his operas \"Corregidor\" and \"Manuel Venegas\" testify to this longing. In his blood was the ardent desire for the colors of the Mediterranean, for the pines and laurel bushes. He was born in southern Styria whose alpine mountains descend to friendly hills where red-wine grapes grow and fan-blowers merrily tinkle in the vineyards. In that country, where on the summit of every hill a little white church is standing, the sun is already as warm as the southern sun. One feels as if one were in a friendly ante-room to Italy.\n\nThe ports of Trieste, Aquileja, Dalmatia and Venice are not far away from that country where Slavic races mingle with races of German or Roman descent. Hugo Wolf's grandmother had a Slavic name, Stanko, and her family originated in Malborgetto, an Italian village. The unrest in Wolf's blood proves the heritage of different races in the Austrian mixture.\n\nWhen I opened the second volume of the \"Spanish Song Book\" which contains the secular songs, I felt as if I were wandering on the streets and squares of Spanish towns under green balconies. I heard the tinkling of mandolins and the rattling of castagnettes. I saw dark eyes glancing beneath black mantillas and great combs on the top of the heads of the Spanish girls. I heard the clatter of shoes dancing the same dances which, even in Roman times, had brought a bright flush to the faces of the dancers. In those songs was Spain\u2014 not a masked-ball Spain, but the real soul of the country, the colors of the Spanish day and of the Spanish night, the smell of the dust on the streets, the passions and the loves of that country where the blood of Romans, Berbers of Carthage, Visigoths, Arabs and Moors had mingled, heated by the sun of the south like mulled wine by fire.\n\nBut still more of the real Spain I found in the first volume of the \"Spanish Song Book,\" in the sacred songs. The striking Catholic devotion of that music is not the Austrian devotion, and Hugo Wolf's sacred songs do not belong to the specific Austrian piety as do the masses of Haydn, Mozart, Beethoven, Schubert and Bruckner. Wolf's is the music of a visionary who evokes in his soul, like the Spanish Jesuits, the apparition of Jesus and Mary by prayer and penance, crying and beating his breast. Whoever knows anything about the devotional exercises of St. Ignatius will recognize that kind of devotion in the songs of Hugo Wolf. Spanish Catholic devotion had ruled for many years in Vienna under Leopold the First. Friars of Spain had built houses and monasteries in Vienna, and in one of those houses Beethoven had died. It was a strange occurrence that a modern composer like Wolf renewed the Spanish spirit of the 17th century. He had been a student in the convent of St. Paul in Carinthia. The convent had been built, on the summit of a hill, like a fortress with thick walls and mighty vaults where thirteen Hapsburg princes lie buried. It was as if the Spanish devotion of these dead Hapsburgs had transferred itself to the composer. The past is great in Austria. Grillparzer, the most famous Austrian poet, wrote plays in Spanish verses, and a modern poet, Hugo von Hoffmannsthal, was inspired by Calderon. Vienna was a city on whose spiritual life many cultures left their traces.\n\nHugo Wolf was, in his imagination, living in sunny Spain when his spirit dipped into darkness. While I was studying his new songs, he was walking under the arcades of the Vienna Opera House, shouting: \"I am the new Director of the Opera House!\" Friends tried to calm him, and told him, at last, that he had to drive to the Imperial Castle in order to present himself, as new Director of the Opera, to the Emperor. He consented, and his friends took him to a lunatic asylum.\n\nI shall never forget the day, February 19, 1903, when we brought the mortal frame of Hugo Wolf to the Votivkirche in Vienna. When the priest walked solemnly, with the crucifix and incense, to the door of the church where the coffin had been deposited, intoning the very old Catholic prayers for the dead, from behind the altar a choir began one of the sacred choruses of Hugo Wolf, \"Ergebung\" (\"Submission\"). This chorus of the dead composer sounded mysteriously through the Gothic aisles of the church and proclaimed the greatness of the man who, unruly and unrestrained in life, struggling with passionate dark powers in his soul, prayed: \"Be a pitiful judge to us sinners.\" No other composer found more vibrant melodies and harmonies for all the moods of desperation and struggle, no other was more moving and more human in the confession of sin. He was a suffering mortal who prayed, in his sacred songs, for pardon. Heaven rewarded him with fame and immortality.\nStudies with Anton Bruckner\n\nIn 1890 I entered the University of Vienna, where Anton Bruckner was teaching harmony and composition. That great composer who, with Brahms, had brought the tradition of the classical symphony to a close, was at that time still unrecognized.\n\nBruckner at the Organ\n\nThe most influential music critic of Vienna, and the great adversary of Wagner, Eduard Hanslick, tenaciously opposed any music in which he scented the Wagneresque. He failed to understand Anton Bruckner's music, which proceeds, essentially, from other sources than the Wagnerian, and is imbued, in a Catholic spirit, with the sacredness of the Mass.\n\nNor did the musical public of that time care for Bruckner's music. When his symphonies were performed at Philharmonic concerts in Vienna, half of the audience left the hall before his number had begun and the other half followed after the close of the first or second movement. By the fourth movement, there was but a handful of young enthusiasts among the standees, who cheered and bravoed until they were hoarse while Anton Bruckner stepped onto the platform and bowed, humble and childlike, to Hans Richter (the great Wagnerian conductor), and threw kisses to the members of the orchestra. The fashionable audiences of the Philharmonic concerts would decide that their behavior had been quite justified when they read the next day in the newspapers that Bruckner composed \"like a drunkard,\" or that he was a composer who had become \"confused by Wagner's music.\"\n\nThis attitude on the part of the Viennese public must be viewed in the atmosphere and in the attitude of that specific era. A city which laughed, sang, loved and drank deeply of all worldly happiness could only with reluctance seek or find access to the music of Anton Bruckner, for his music sprang from deeper sources than the superficial life of such a town and had in it nothing akin to the materialism of modern existence. Bruckner's was a profoundly religious nature.\n\nIn Vienna his personality was often a matter of mirth. The elegant society could not understand Bruckner, and he was like a stranger in the sensuous, new city\u2014a man who hailed from another planet. Up to his forty-fifth year, Anton Bruckner had lived in rural districts among people who spoke the peasant dialect of Upper Austria. He had spent many years as a teacher and organist among the priests of the Monastery of St. Florian, whose lives were passed between the cloister cells and the chapel. The Abbot of St. Florian and the Archbishop were for Bruckner the representatives of God. He could not understand the great city, with its beautifully-dressed people who chaffed, laughed and flirted so easily, as if there were no Heaven and no God. He dedicated his Ninth Symphony to \"Our beloved Father Almighty.\" Like the mediaeval musicians, he felt that he was a \"Musician of God.\" But in a great cosmopolitan capital, proud of its own enlightenment, a Musician of God seemed absurd. This international city believed in cement and iron, in electricity and machines, in the Stock Exchange and in business speculations.\n\nSuch was Anton Bruckner's status in the life of Vienna when I entered the University there.\n\n\"Do come to Bruckner's lecture to-day\u2014there's always lots of fun!\" said one of my fellow-students, and I promptly agreed to come. It was the first time I had heard the name of Anton Bruckner. But the prospect of having \"lots of fun\" was irresistible for a freshman aged eighteen. I arrived in good time at the lecture-room of the Philosophical faculty where this musician, of whom I knew nothing up to that time, gave his lectures on counterpoint. My colleague had described him as a \"queer fool\" and I expected to be amused.\n\nI knew the room in which the entertainment was supposed to take place. It was the same room in which Professor Hanslick gave his lectures on musicology, for which I had registered. How well I remember the quaint, old-fashioned piano next to the reading-desk. How well, too, I recall the expectant tension with which I had been looking forward to the famous critic's first lecture\u2014and how distinctly I remember my disappointment! The little old man who had dared fight Richard Wagner, the mightiest genius of modern music, with the graceful periods of a drawing-room hero, mounted the podium without raising his eyes, almost timidly. He drew a manuscript from his pocket and, in a high-pitched voice, read us a lecture which dealt with the life of Beethoven in a most superficial way, bare of scientific foundation. Now and then he interrupted his monotonous report, tripped to the piano, played some passages from one of Beethoven's compositions with a very out-of-date fingering, and thereafter resumed the dull ripple of his lecture.\n\nPrevious to this I had attended courses of several important men of the University of Vienna, and I was well aware how much inspiration and stimulation the fascinating courses of brilliant teachers could convey to the students. I had become enthusiastic over the elegantly-phrased and witty lectures of Adolf Exner, who, through the description of a Roman journey, had introduced us to the field of Roman Law; and I had been captivated by the stagy declamation of the idealistic philosopher Brentano. Bernatzik's lectures on anarchism, Reich's lectures on Ibsen and Menger's on questions of social politics had introduced me to the spiritual movements of the time. I was most eager to learn and went about listening to all sorts of courses without, as yet, knowing where I would finally settle down. Eduard Hanslick, however, had neither the slightest gift of speech nor the ability freely to communicate with his students. He therefore did not appeal to me and I never attended another of his lectures.\n\nAnton Bruckner had become a lecturer on theory of music at the University of Vienna contrary to Eduard Hanslick's wishes. \"That's what he'll never forgive me!\" Bruckner exclaimed again and again. Hanslick, extremely touchy like all vain little men, actually never forgave Bruckner in spite of all the bowing and scraping the latter indulged in whenever he encountered the dreaded critic.\n\nThe small lecture-room where I sat with my colleague awaiting the merry performance soon was crammed, and after the academic quarter had elapsed there entered a crowd of people: first Anton Bruckner, bowing again and again, and behind him, like a rustling train, old and young men, and a very pretty girl, too. All of us greeted the queer musician by solemn trampling which my friend and I prolonged for fun, much to Bruckner's joy. We were highly amused at his bows and appearance. Of course he wore his Upper-Austrian loose jacket, comfortably cut out of home-woven fabric. His trousers were loose and baggy. The big head was close shaven, and innumerable wrinkles furrowed his face. When he began telling us a lengthy story in Upper-Austrian dialect, he reminded me of an old peasant, battered by wind and weather.\n\nThis was the year 1891 and Bruckner had just arrived from Berlin where Siegfried Ochs had made his \"Te Deum\" a triumphal success\u2014the work of a \"foolish\" composer who had been laughed and sneered at in Vienna. Bruckner could not stop talking about the \"marvelous director,\" or the \"highly ingenious\" performance, or the \"most charming\" choir. And he did not omit the fact that he had kissed the girls of the choir, one after the other, amid the applause of the audience. With obvious emotion, he reported that there had not been a single Berlin paper which had disapproved. He glanced about furtively when he told us: \"Just imagine, gentlemen, one of the honorable critics wrote that I was a second Beethoven! Good Lord, how can anyone dare say such a thing!\" and he quickly made the sign of the cross on his forehead as if to avert the sin.\n\nThe \"fun\" I had expected during that first lecture was only partly realized, for suddenly something happened that I can never forget. From a nearby church the Angelus sounded and when that little bell rang, Anton Bruckner interrupted his lecture, knelt down and began to pray: \"Ave Maria . . .\"\n\nIn the churches of many cities I have watched the devout at their prayers. In Vienna, in the great and glorious old St. Stephen's Cathedral one could daily see people of all sorts and of all conditions in life worship before the statue of the Blessed Virgin. In the Cathedral of St. Anthony of Padua, in St. Peter's in Rome, in churches in Barcelona and Valencia I have seen hundreds in ecstatic prayer, but I have never seen anyone pray as Anton Bruckner did. He seemed to be transfigured, illuminated from within. His old peasant face with the countless wrinkles that covered it like furrows in a field, had become the face of a priest. His expression may best be compared with that of the Apostles in the paintings of Giotto. He looked like an aged saint, his countenance transfigured as one imagines that of St. Francis was when he knelt before the Crib and prayed.\n\nI stopped smiling when I saw his face, humble and blissful, refulgent with celestial light, radiant as a church window through which the sun is shining. I saw this face before me when I listened to his first Symphony, and I have always seen it whenever I have heard his symphonies well-played.\n\nHaving finished his prayer, Bruckner stepped up to the music black-board on the wall, and began his lecture.\n\nBruckner had been a pupil of Simon Sechter, that mighty fortress of counterpoint to whose home he made a pilgrimage every week from Linz, where Bruckner was cathedral organist at the time. Sechter's entire life was dedicated to counterpoint. Every day, after he got up, he sat at his writing desk, working out a new fugue. Not before having finished the last organ-point, over which the four voices streamed to the final chord, did he have his breakfast.\n\nSechter's doctrine, which was delivered to us by Bruckner like a holy heritage, was built upon two strong pillars. The one which inspired Bruckner with greatest respect was the theory of the \"Fundamental Basses,\" a world of spirits in the bass, which accompanied the harmonies like shadows in the depths; and the theory of \"natural harmonies\" which form the laws of all beauty of harmonic progression. Everywhere there was law and order, even holiness. The fundamental steps of the bass which Bruckner invariably noted in his scores under the last line of the staff, had cosmic importance. Thus we understood the greatness and sometimes, the rigidity and solemnity of Bruckner's harmonies. Bruckner, the pupil of Sechter who was a kind of architect of harmonies, pondered over chords and chord associations as a mediaeval architect contemplates the original forms of a Gothic cathedral. They were his path to the Kingdom of God.\n\nThe conductor Ferdinand Loewe, one of Bruckner's pupils, showed me several folio volumes with exercises written by Bruckner when he studied with Sechter. He must have worked on them day and night. Sometimes when Bruckner came to Vienna he visited the opera house. Here he attended performances of Wagner's operas. He was deeply moved when he heard \"Tannh\u00e4user.\" He stood in the standing-room section when he heard Winkelman sing the story of his pilgrimage to the Pope in Rome. When Tannh\u00e4user related how he was cursed by the Pope, Bruckner began to cry and shouted: \"Why did he not pardon him! Why did he not pardon him!\"\n\nBruckner explained Sechter's principles to us by using many comparisons, taken from everyday life, in the dialect of Upper Austria. He sounded like a peasant telling humorous stories to the children in a farmhouse. For instance, he compared an unexpected dissonance in music with a \"dear auntie\" who unexpectedly came on a visit, spreading terror and confusion in the household until she is cajoled out of the place again. \"Where is dear auntie, where is dear auntie?\" he shouted, bending over the piano-keys. He struck a chord again and again until the necessary missing note was sounded. Then Bruckner would beam and bow and exclaim: \"Oh, there she is, our dear auntie . . .!\" The interval of a second was a \"poor fellow\" who had \"to wait in a corner till he was called\"; and the seventh, a \"damned wretch whom you may not trust.\"\n\nSuch metaphors were intermingled with memories from Bruckner's life or complaints about Hanslick. One might find such intermezzi scurrilous or naive, but they were the stammerings of a child-like soul in an artificial and complicated world which this pure and saint-like man could not comprehend. It was difficult not to become aware of the radiance in Bruckner's face. There was something in the mobile features of this old man which was far greater than the erudition, knowledge and profundity which we found in other professors at the University.\n\nOne time, with infinite care, Bruckner wrote some chords on the black-board and pointed out a certain dissonant sequence which he said was \"quite wrong, perfectly incorrect\" and \"strictly forbidden.\" Remarking that this very dissonance which he had just criticized so harshly had been used only twice, once by Beethoven, once by himself, he sat down at the piano and began playing a passage from the Adagio of his Seventh Symphony, where this \"perfectly incorrect\" and \"strictly forbidden\" dissonance occurred. It was the first of Bruckner's music I ever heard.\n\nThe impression was so glorious that from that day on I had not a moment's respite until I heard the living orchestral interpretation of one of Bruckner's symphonies. I still recall vividly Bruckner's serene expression while playing his magnificent music to the audience. I still see the picture of the old man's pathetic face, radiant with divine genius.\n\nDuring my four years of university studies I did not miss a single one of Bruckner's lectures. I had been led to him, first, by the prospect of \"having fun.\" I had been disappointed in this expectation. The new \"Brother Gaudeamus\" (that was Bruckner's name for his students), had forgotten how to laugh in this first lecture, but he had been taught how to admire ingenious creative power.\n\nSometimes Bruckner honored me by allowing me to accompany him, after his lecture, to a nearby coffee-house. We used to sit at the round table which stood in the middle of the room. Since this particular coffee-house was situated near the Stock Exchange on the Ringstrasse, the guests, for the most part, were stock-jobbers who gazed curiously at the strange composer who ordered coffee and read the newspapers. We young students would sit and wait patiently until Bruckner finished reading. Once he read about the great pogrom in Kishinew where hundreds of Jews had been killed. He looked distressed and timid when he laid the newspaper down. Anxiously he looked at the guests who were sitting in the cafe, discussing rates of exchange and market quotations, and said to us, with a half-shy, half-respectful glance at the Jewish guests: \"Jesus and Mary! Now they have killed, in Russia, so many (\u2014with lowered voice) Jewish gentlemen!\"\n\nI often heard Bruckner play the organ. In the Court Chapel, in St. Stephen's Cathedral, in the church at Klosterneuberg, I admired his art of improvising. The sound of his organ-playing is still preserved in his symphonies. Their great climaxes are the kind Bruckner built up at the end of his improvisations, pulling all the registers of his instrument, unchaining all the glory of magnificent sounds. Sitting at the organ, treading the pedal and loosing all the power of shining chords, he saw the Heavens open, with God and the saints in the clouds. The concept of God descending from Heaven is the final vision in all Bruckner's symphonies; and that was exactly what he wished to express on the organ when, before the close, his music soared in celestial splendor of sound, soul and song. Often he would end his improvisation with a great fugue, which was crowned by a chorale. The colossal last movement of his Fifth Symphony, when, over the massive orchestral fugue, brass choruses unfold, rising majestically above the climax like the voice of eternity, is essentially the same as one of his organ improvisations. Had Bruckner ever written an opera, it would have had the same character. He often said to us that if he were to write an opera, he would need a libretto whose theme would be the descent of the Holy Spirit.\n\nAnton Bruckner was the last greatest musician of the nineteenth century whose music ascended from Earth into Heaven. His pathos and splendor came in large part from the Catholic Church, which, during the Counter-Reformation, had developed the style which inspired him. Bruckner may be called the last great musician of the Baroque, which was still alive in new Vienna, in his music. He had nothing in common with modern realism and materialism, which reproduced the visage of the world masterfully and vividly. The metropolis in which he dwelt was foreign to him. He was a man who lived in God and whose music is a cathedral with high pillars, bold arches, brilliant windows, divine services and a glowing altar at which a lonely worshipper kneels and prays.\n\nMeeting this humble genius in Vienna was the greatest experience of my life. I let elegant Viennese society laugh about Bruckner. But I quietly sat down at my writing-desk and wrote an essay, which was published in one of the great German magazines, and gave the first thorough account of the greatness of Bruckner's music. Thus I entered the musical life of Vienna as music critic. I pushed open the door, shouting: \"Hats off! A genius!\"\n4\n\nIn the Opera House and Concert Halls\u20131891-1914\nThe Musical Capital of Nations\n\nThe presence of so many great musicians gave meaning to musical life in the period of Emperor Franz Josef. Each year new works by Brahms were performed. Hugo Wolf had his songs published. Bruckner's symphonies were new works. All this music had not, as yet, been catalogued by the music historians. It was the music of the day, and came fresh from the pots of old Vienna's music-kitchen. The music-critics took their testing ladles, dipped them in the steaming pots, tasted, and found the fare good or bad.\n\nLike any educated, cultivated society, Viennese music circles enjoyed well-written, witty or malicious newspaper criticisms. The daily papers which appeared in Vienna were a great power. In 1867, the \"New Free Press\" was founded which, along with the London Times and the Paris \"Figaro,\" quickly became the most widely-read newspaper in Europe. For it, the best pens wrote on politics and art. The jewel-piece of a feuilleton was zealously cultivated after Parisian models. But here it was a Vienna feuilleton: tasteful, cultured, charming, instructive in a gracious form. The strongest German stylist in the world of journalism, the Suabian Ludwig Speidel, wrote on the theatre, and all Vienna waited for four or five days after a premiere for his feuilleton; since, after first-nights, for a few days he went walking under the Prater trees to assemble his thoughts and bring them into clear form. When, distinguished by his majestic scholar's head, he appeared in the Burgtheater and lowered himself heavily into his aisle-seat, his hands resting on the silver knob of his cane, everyone would turn respectfully toward this great critic who would let the Viennese public know how the new play had pleased him. After the performance Speidel would go to the \"Winter Beer House,\" where the best Pilsener was served, and sit there until the early morning hours with a few learned friends.\n\nHugo Wittman, the second great Viennese feuilletonist, was also a Suabian, but he wrote in a light conversational tone, like a Parisian. The third of the \"New Free Press\" feuilletonists was the music critic, Eduard Hanslick, and what he wrote about music was read by all Europe.\n\nEvery evening Hanslick went to the concert hall or to the opera, and when the little old man made his appearance, a movement would go through the audience. Seldom did a musical society and a critic understand each other as well as the Viennese public and Hanslick. Viennese society loved classical music, Italian opera arias and French music, and so did Hanslick. For a long time Viennese society wanted to hear nothing of Wagner, and Hanslick wanted this still less and still longer. Viennese society was elegant. So was Hanslick. Society loved wit, and Hanslick was witty. Society enjoyed life, and was superficial, and so was Hanslick, even though he possessed a keen mind and great culture. Hanslick said in his criticisms what the most mentally limited member of Viennese society thought, but he wrote it down in most ingenious form.\n\nHanslick and Bruckner  \n(O. Bohler)\n\nWhen I began to write music criticisms, I had to introduce myself to my famous old colleague, and I learned to know him as an amiable man. On this occasion, he removed a silver-snuff-box from the back pocket of his black Biedermeyer coat and offered me some of the black powder. Then he himself took an ample pinch, and, taking out a colored handkerchief, sneezed with quite some thoroughness. Then he began to recount the latest anecdotes. He loved the lively things in life. He liked to go to parties, enjoyed good food, was gallant to the ladies, told jokes, and played Strauss waltzes.\n\nHanslick came to Vienna from Prague, and his name is undoubtedly Czech. Indeed, most of the critics and writers of Franz Josef's time came from outside Vienna: Hanslick from Bohemia, Speidel and Wittmann from Suabia, the great art-critic, Luwig Hevesi, and the brilliant feuilletonist Theodore Herzl, who was the founder of Zionism, from Hungary. All of them became Viennese, and leaders of Viennese taste. What one calls the Viennese spirit sprang from such mixtures and minglings. The great architects who had built the palaces on the Ringstrasse were a North German, Schmidt, a Dane, Hansen; and only one of them, Ferstel, was a native Viennese. The foremost professors of the University of Vienna came from many different countries. Without this racial combining and fusing, Vienna would not have become a European capital; nor would the city's musical life have preserved its richness; its universality and its color. Music in Vienna was brought to life through the musical talent of many nations.\n\nOf these contributing nations, the Italian was especially important up to the beginning of the nineteenth century. Even in Emperor Franz Josef's time there was a large Italian percentage in Vienna's population. Austria, until 1859, possessed Milan and Lombardy, and only in 1866 did it lose Venice. From this time on there were many families of Italian officials in Vienna. For ages, the trade in Italian fruits and oils had been in the hands of Italian merchants who resided in Vienna. Italians, too, were the peddlers who sold salami and cheese in all the Prater inns and whom the Viennese nicknamed \"Salamucci.\" The Viennese folk-dialect still to-day contains many Italian words.\n\nEven in more modern times Italian music had a second home in Vienna. From 1815 to 1859, the Milan La Scala was, of course, an Austrian theatre and the Austrian officers and soldiers who were garrisoned there went there to hear the operas of Verdi and Donizetti. The greatest Italian singers appeared in Vienna, and Italian composers were quite at home in the Vienna Opera. In 1821 Rossini conducted there. Donizetti, who also was an Austrian court-composer, wrote his operas \"Linda di Chamounix\" and \"Maria Rohan\" for the Vienna Opera. In 1871, Verdi conducted his \"Requiem\" and \"Aida\" in this theatre.\n\nWhen Emperor Franz Josef began his reign in the revolutionary year 1848, enthusiastic nationalists ripped the Italian placards from the Vienna Royal Opera. But Vienna musical society refused to be deprived of Italian singers and operas. Each spring, when the Prater chestnut-trees flamed into bloom, an Italian opera season moved into the Vienna Opera House, and the Viennese went wild over the brilliance of the Italian voices.\n\nWith each new Italian opera composer, Vienna's enthusiasm for Italian voices and operatic works was renewed. Just such a wave of excitement resulted from Mascagni's \"Cavalleria Rusticana\" when the naturalistic opera was given in Vienna in 1891. And enthusiasm came to an even higher pitch the following year when Mascagni himself, famous overnight, appeared in Vienna.\n\nWhen I met Mascagni at that time, he was a slendet young man with jet-black hair who looked like a Roman Proconsul. The handsome Italian, who had a modest and charming manner, had completely devastated Viennese women of all circles of society and of all ages. Every morning the postman brought hundreds of love-letters to the Hotel Imperial where the composer was living. Frequently Mascagni conducted his successful opera himself at the Vienna Opera House, and on such occasions was showered with applause. Of his later operas, \"L'Amico Fritz\" and \"Die Rantzau\" were performed in Vienna, but despite Mascagni's personal popularity, they met with no great success.\n\nIn 1892 Leoncavallo's \"Pagliacci\" was given for the first time by an Italian opera troupe which presented guest performances frequently at the theatre of the Music Exhibition in Vienna. Leoncavallo himself came to conduct this performance. He was a stocky man with a bushy moustache who looked more like a butcher than a musician. In his conversation, he had difficulty disguising his envy of Mascagni's triumph. Leoncavallo visited the editors of all the Viennese papers and declared himself a Jew, which was quite untrue. When I happened to mention this to Mascagni, he was furious. He said that Leoncavallo was a swindler if he pretended he was Jewish. It seems that Leoncavallo had heard that many managers and owners of Viennese papers were Jewish, and he felt he could obtain their good graces by posing as one of them. He also thought that they would beat their drums louder for a Jewish composer than for a Christian. Nevertheless, Jew or non-Jew, the composer and his \"Pagliacci\" had a great success. The role of Nedda was sung by a young Italian soprano with burning dark eyes, who completely captivated her audience by the nervous intensity of her expression. She represented a new type of tragic opera singer. She was no pompous woman with a voluptuous bosom, as the case had always been up to that time on Italian opera stages. She was cat-like, slender and haggard. This was Bellincioni. When she appeared on the stage in one of the new naturalistic operas, she was like a woman of the people, an Italian from the south; and in roles of this sort she had no peer. She was to the operatic stage what Duse, with still stronger spiritual force, was to the dramatic\u2014the artist with the soul and temperament of the new era.\n\nIn 1892, to her Santuzza and her Nedda, she added a third great role\u2014the main feminine one in the opera \"Mala Vita\" by Giordano, who with this naturalistic work, introduced himself to the Viennese public. The heroine is a daughter of joy, the hero a consumptive, who makes a vow to marry a fallen woman. The opera closes with the prostitute, after an unhappy marriage, returning to the house of her profession. I can still see Bellincioni before me\u2014pale, pulling a shawl over her dark head, walking back with weary steps to the house of commercial love.\n\nThe following year, in the Theater an der Wien, there was another Italian opera season presented. Once more, only Italian realistic operas comprised the repertory. Blood flowed, on the stage, in streams. One had the choice of seeing the leading man stabbed with dagger or knife, throw himself under a moving railroad train, or end his operatic life in a hemmorrhage. The favorite means of death in these Italian operas was, of course, knife-blows, which, in \"Cavalleria Rusticana\" and \"Pagliacci\" had world-wide success.\n\nDuring this season, an opera, \"The Willis,\" by the then unknown composer Puccini, was presented to Viennese audiences for the first time. During the general rehearsal, Puccini, slender and pale, with a well-tended moustache, stood calmly and quietly in the theatre, gazing dreamily at the stage. Quite unlike so many excited composers at dress-rehearsals of their works, he did not say a single word. To my astonishment, I noticed that Mascagni had borrowed from the score of this opera some orchestral effects for his famous \"Intermezzo\" in \"Cavalleria.\" Even in this opera of his youth, Puccini was already an artist of interesting orchestral sound. His orchestra was far superior to Mascagni's, and much more artistic. He was a more delicate and more subtle artist. Nevertheless, at the time, the performance of his opera did not attract much attention. The great Puccini triumphs only began with \"La Boh\u00eame\" which was not performed at the Court Opera but at the Theater an der Wien, for the first time, because Gustav Mahler, then Director of the Court Opera, considered the \"Boh\u00eame\" by Leoncavallo better than Puccini's.\n\nPuccini regularly came to Vienna for the premieres of his works; and the Vienna Opera, where in Jeritza and Lotte Lehmann he found famous interpreters of his feminine roles, was one of his favorite opera houses. His \"La Rondine\" was composed especially for a Viennese operetta-theater, the Carltheater. Puccini was the last great Italian opera composer who delighted the Viennese public in the same way as, in earlier centuries, Rossini, Donizetti and Verdi had done. Puccini's was a tender, soft, melancholy love melody which breathed out the perfume of violets when the violins began to sing. I heard this new love melody of Puccini's for the first time at a concert of the marine band in Pola, which played a potpourri of the then new \"Boh\u00eame.\" A young conductor stood on the podium, wearing a white linen uniform, and a marine cap on his head. It was Franz Lehar, who, in his own operettas, made the Puccini love melody even more popular.\n\nThrough more than three hundred years, Italian music furnished one of the most important and influential components of Viennese music. The feeling for melody and for sound, which distinguished Vienna, was stimulated by Italian music. The most popular Viennese musician, Johann Strauss, was primarily a melody-inventor, like the Italian opera composers. Just how much Italian music flowed in the blood of Viennese composers is demonstrated, in modern times, by Hugo Wolf, in his \"Italian Serenade\" and in his \"Italian Song Book.\"\n\nThe relationship between Vienna and French music was established through the French education of the nobility and of the upper classes of Vienna. The Viennese aristocrats all spoke French, which was a kind of international language, and the members of high society read the latest French novels. In Vienna, the French mind and the bright and polished elegance of French thought were very much admired. Something of this French clarity was apparent in the French grand operas of Boieldieu and Auber. It appeared, in a new form, in Massenet.\n\nWhen Massenet's \"Manon\" was first performed, November 9, 1890, at the Vienna Court Opera, it achieved a sensational success which far surpassed its reception in Paris. At this performance, I was standing in the highest gallery of the opera house, and in the St. Sulpice scene, where Manon persuades her abb\u00e9 to follow her, I clapped, along with the rest of the audience, until my hands were sore. The leading artists, Renard as Manon, and Van Dyck as Des Grieux, took curtain call after curtain call. This was a genuine, roaring Viennese success.\n\nEveryone looked forward with special excitement to Massenet's new work, \"Werther,\" which was to be produced at the Court Opera the following year. The dress-rehearsal was depressing and it was feared that the new opera would be a fiasco, because the gloomy melancholy of the last act took the listener by surprise. As each act ended, faces became longer. However, to the general amazement, \"Werther's\" premiere was a huge success, and Massenet was so delighted that he could not say enough flattering things about Vienna and the Viennese public. Massenet again came to Vienna for the hundredth performance of his \"Manon\" and personally conducted. The admirable man was no conductor, and he directed the performance with broad, stiff arm-movements. Nevertheless, the jubilation was great.\n\nOf the famous French composers after Massenet, Debussy and Ravel came to Vienna.\n\nClaude Debussy conducted a concert of his own works in Vienna in 1910. I described him at that time as \"a broad-shouldered man, with coal-black hair and pointed beard, slightly indolent and ponderous in his movements, with a somewhat sullen and weary glance.\"\n\nAt that time I had already known Debussy personally. Twenty years before, I had spent a year in Paris. That had been a period of artistic movement. There were veritable streams of new ideas in poetry, where the mystic Mallarm\u00e9 was the last word, and in music, where French musicians were busy attacking Wagner's omnipotence. Up on the hill of the Montmartre, which the young artists had proudly named \"The Brains of Paris,\" the young poets, painters and musicians discussed new formulas of art, and daily made new experiments. In the Latin Quarter I encountered Strindberg, brooding and sunk in mystic thought, sitting outside a caf\u00e9, with an absinthe glass before him. Surrounded by a group of friends was Oscar Wilde, already fat, wearing an unusual green ring on his right hand which held a lighted cigar. At the studio of L\u00e9andre, the excellent caricaturist, I met Debussy, shortly after I had seen several songs of his, steeped in a mist of color, published in the progressive \"R\u00e9vue Blanche.\"\n\nDebussy rarely came out of his shell. He liked to muse and meditate, and only displayed outward enthusiasm when he spoke about the masses of Palestrina, which were sung by the famous choir in the Church of St. Gervais. These he considered the perfect works of religious ecstasy.\n\nWhen Debussy came to Vienna for the first time, he was forty-eight years old. He had already become famous in Paris through the performance of \"Pell\u00e9as and Melisande\" (1902). Debussy's music was heard in Vienna for the first time at a concert, conducted by the composer. The programme included the \"Petite Suite,\" \"L'Apres midi d'un Faun\" and \"Iberia.\" This music, transparent, gleaming, shimmering, was a music of hovering impressions, of melting, indefinite contours; a \"music of nerves\" belonging to a new age.\n\nRavel visited Vienna very frequently. The first reception accorded him there was not very friendly. The composer, with the violinist Szigeti, performed his Violin Sonata. Its scherzo, in blues rhythm, seemed, to the conservative Viennese public, to violate classic traditions. While one danced to \"Blues\" in Vienna as one did to all the members of the Fox-trot family, nevertheless the public did not deem this modern rhythm worthy enough to be employed in chamber-music; although Haydn, Mozart and Beethoven not only had composed dance music, but had incorporated contemporary popular dance music in their symphonies and chamber-music. After the Ravel \"Blues\" Scherzo, one of those famous Viennese concert riots occurred with which the musical public accompanied the performance of all modern music up to Schoenberg and Stravinsky. But the artistic qualities of Ravel's ingenious and finely polished music quickly conquered opposition, and when, in 1929, he first conducted a concert of his compositions, he was warmly acclaimed. After that, Ravel frequently and gladly returned to Vienna.\n\nAfter the performance of his beautiful ballet, \"L'Enfant et les Sortil\u00e9ges,\" at the Vienna Opera, I had the pleasure of meeting Ravel at the home of one of his friends who had planned an evening of Viennese wine and music in his honor. I learned to admire and esteem the intellectual refinement of this musician who looked like a jockey; however, one could easily discern the signs of the malignant disease which eventually destroyed Ravel in the lost gaze in his eyes, and in his absolute immersion in his thoughts.\n\nVienna was bound to two great European nations through its enjoyment and appreciation of Italian melody and the French intellect. An especially interesting color was added to Viennese life by the proximity of Hungary.\n\nSince 1867, Austria and Hungary had been bound in a double monarchy, and Emperor Franz Josef had received the crown of St. Stephen in Budapest. Most of the Hungarian aristocrats spent the winter in Vienna, and the two sister cities of the Danube were joined by close social ties. Hungarian music was played in the restaurants of the inner city and in the Prater where in the \"Czarda\" (so the country inns were called in Hungary), the Hungarian soldiers and the Viennese girls met. The fiery rhythms of Hungarian music were popular throughout all Vienna. The frenzy of a \"Czardas\" melody was known to old and young; and Hungarian \"Goulash\" had become a kind of national dish in Vienna.\n\nHungary, the land of fiery women, fiery horses and fiery wines added still more sensuousness to Viennese life. A Hungarian woman set Vienna's whole life of pleasure in motion: this was Princess Metternich or, as the Viennese intimately called her, \"Princess Pauline.\" The little, ugly woman with the thick, protruding lips, scintillated intellect, and had the temperament of a Hungarian horse of the steppes. Her father, Count Sandor, was a daring horseman. When he rode in the Prater, he jumped his horse over moving carriages. Princess Metternich had inherited her father's temperament, and first demonstrated it in Paris, where, as the wife of the Austrian ambassador, she led the gaiety of the social life at Napoleon III's court, and stormed the gates of the Paris Opera in order to secure the performance of Wagner's \"Tannh\u00e4user.\" She was also popular in Vienna for her brilliant energy. Every year she organized balls and concerts, theatricals and folk-festivals, for charitable purposes, and Viennese society spent money freely on them. No P. T. Barnum knew his business better than she, nor devised cleverer ways of attracting huge audiences.\n\nJohann Strauss, who understood Viennese life and atmosphere better than anyone else, allotted considerable place in his works to music in Hungarian style. One of his most popular operettas, \"The Gypsy Baron\" owes its success in no little degree to its enchanting Hungarian melodies. The only \"grand\" opera that the Waltz King composed, \"Ritter Pazmann,\" contains some of the most tempestuous Hungarian ballet music imaginable. While the Emperor ruled over Austria and Hungary, Johann Strauss ruled over Austrian and Hungarian melodies.\n\nFrom Hungary came the composer Carl Goldmark, the conductors Hans Richter and Arthur Nikisch; Franz Lehar, the most successful operetta composer in the world. From Hungary, too, came Kalmann, who in his operetta, \"The Czardas Princess,\" had made the passionate Hungarian tone popular. Lehar's and Kalmann's operetta successes went forth from Vienna, which was a sounding-board for music of every description and of every nationality. Hungary was inexhaustible in talent, which sprang to its full bloom in Vienna's air.\n\nIf the Hungarians gave additional color to the life and music of Vienna, the Slavs gave it softness. They contributed more than other influences toward banishing the German stiffness and rudeness from Viennese culture. From Vienna, Anton Dvorak's music started its course through the world. It was Brahms who discovered this primitive musician and who procured him a publisher, Simrock, in Leipzig.\n\nAlso in Vienna Bedrich Smetana's music was discovered for the world at large. I was present at the performance of \"The Bartered Bride\" in Vienna, on June 1st, 1892, which secured for Smetana's masterpiece its first real international recognition. For up to that time the opera was known only to the Czech audiences in Prague, where it was very popular.\n\nI first heard the opera in a little Czech village, during my boyhood, when I was spending a vacation there. It was a typical Czech village which lay in a valley surrounded by woods. At the entrance of the village was the castle, where a toll-gate barred the road. Then came a long row of farmhouses with their accompanying stables and barns. In the center of the village, on the promontory, stood the church, and nearby, the inn. On Sundays the peasant boys and girls, who all week had worked in the fields, went to dance at the inn. The girls wore many petticoats, and tied colored kerchiefs around their heads. The young men kept their pipes in their mouths if they wanted to dance; walked up to one of the girls, removed the pipe, and spat the tobacco juice on the wooden floor. This was the invitation to the dance. The musicians sat at a table, their music propped up against beer glasses. Thus they played Smetana's music, while the dancers revolved in the smoky room, and we children stood at the door and listened and watched. And that was how I heard Smetana's music in the landscape from whence it sprang.\n\nIn Vienna, it was played again in different surroundings. At the Vienna Theatre and Music Exhibition of 1892 there was a theatre where troupes of different nationalities presented their performances. Here Judic played French comedies. Italian singers sang Italian operas. Reinhardt came from Berlin and gave Shakespeare's \"Winter Tale.\" The Czech National Theatre gave Smetana's \"Bartered Bride.\" Right from the end of the overture the surprised and delighted audience started applauding. The duet, \"You know a girl\" had to be sung four times, and the beautiful quintet in the last act was also encored. This was the greatest operatic success that Vienna had experienced. Hesch, the bass who sang the role of Kezal, was engaged for the Vienna Opera by Gustav Mahler, and became one of Vienna's most beloved singers.\n\nVienna again demonstrated its faculty of understanding and evaluating foreign culture when it uncovered Janacek's opera, \"Jenufa,\" to the eyes of the world. Old Janacek came to Vienna for the performance of his work. With his bushy moustache, he looked like a colonel, but his shining blue eyes betrayed the artist. Seldom have I met a more charming old gentleman. He had radical and original views about music. In fact, his opinions might well have belonged to a very young man. He attended a festival of modern music, one time, at Salzburg. After a concert, where the most modern music had been played, I spent an entire night with him, walking through the ancient streets of the town, discussing the problems of modern music until the sun rose over the mountains. At that time Janacek was seventy years old, but he was younger than the youngest of us all.\n\nJanacek's opera, \"Jenufa,\" kneaded out of the strong Slavic earth, was at first refused by the Czech public and music critics. Its keen realism, related to the realism of Moussorgsky, was not understood, and the great Slavic compassion which, at the end of the opera, transfigures the Czech village tragedy, struck few responsive chords in the hearts of that first audience. Vienna had no prejudices in its regard, and the Viennese success of this genial work opened to it the doors of the opera houses of the world. In Vienna, the title role was sung by Jeritza, who came of Slavic ancestry. Leo Slezak, the tenor, came from Moravia. Both artists still had their Czech accent. Young Jeritza, with her Slavic nose, her blonde hair, her expansive, sensuous smile, always looked like one of the young peasant-girls who worked and laughed in the Moravian fields. In her was the unbroken strength of nature, and in her radiant voice sang the female sex.\n\nIn the period of Franz Josef, the Italian, the Hungarian and the Slavic still mingled in Vienna's music. A great river has many tributaries.\nIn the Opera House\n\nNew Vienna's best society congregated every evening in the boxes and parquet of the new Opera House which was opened by Emperor Franz Josef on May 25, 1869.\n\nUp to this time the \"K\u00e4rntnerthor Theatre,\" which had stood in the old fortress-city near the southern town portal, had been Vienna's opera theatre. It had been built during Empress Maria Theresia's reign and operas had been performed there since 1810. From 1819 on it had been named the \"Royal Court Opera Theatre.\" An old Viennese music lover told me that in the old theatre the acoustics were so wonderful that one not only heard the tones, but the very breath of air which surrounded them.\n\nFor the Viennese this theatre had been the scene of great enthusiasm for sixty-nine years. Here the ovations of the public had roared about Rossini and Donizetti. Here the delight had been great over the great and brilliant operas of Meyerbeer. History was at home in this theatre, as in all Vienna's theatres. Weber had composed his opera \"Euryanthe\" expressly for the K\u00e4rntnerthor Theatre. Here Beethoven's \"Fidelio\" was produced in 1814 for the first time in the version we know it today. Here in 1824, the deaf Beethoven performed his Ninth Symphony for the first time. The house was filled to capacity. Beethoven, wearing a black frock-coat, with black satin knee-breeches, silk stockings and buckled shoes, stood at the conductor's desk. He no longer could hear anything at all, and when, at the end, the audience shouted and applauded, one of the singers had to turn the deaf master around so that he could see the members of the audience waving their handkerchiefs.\n\nDistinguished history and living present dwelled in this theatre. Nothing could bring more fame to it than the letter which Richard Wagner wrote when, in 1861, he heard his \"Lohengrin\" here for the first time: \"That is a heavenly Opera! A mass of wonderful voices, one more beautiful than the other; chorus, orchestra\u2014enchanting!\"\n\nFor new Vienna, which had found the old fortress walls too confining, the old opera house was likewise too small. People wanted a new opera house, beautiful and large, like the new city; elegant like the city's society. They wanted a palace, not an old building with dark nooks and corners and low-ceilinged corridors. A prize-competition was instituted for the building of a new opera house, in which Van der Nuell and Siccardsburg were the winners. Both architects were Viennese, and they proudly erected the new theatre on the Ringstrasse, which was the Via Triumphalis of Vienna, at the entrance to the city. Past this new music palace the life of the new city flowed into the old from dawn to dusk. The majestic building faced the Ringstrasse, and its bright loggia was decorated by Schwind, Vienna's greatest painter, with illustrations from legends. This was a festival theatre in early French Renaissance style, with marble entrance-halls and a broad, splendid staircase, which the beautiful women in fur wraps and jewels slowly ascended on the way to their boxes. The great auditorium itself, with its rows of boxes laid out on the Italian model, was lavishly decorated with gold and extravagantly lighted. The ceiling was bordered by an enormous wreath of golden fruit, and in the center of glorious frescoes, hung a brilliant chandelier. The orchestra pit was comfortable and the stage was broad and deep. It was a magnificent theatre, worthy of a rich and magnificent city.\n\nStrangely enough, the Viennese at first were not very well satisfied with their new opera. They were accustomed to the more intimate acoustics of the old opera house and found that they did not hear well in the new one. The comic journals made fun of the bronze winged horses which stood atop the building, saying that they were too thin, and went so far as to hang feed-bags around their necks. Finally, the problematic horses were sold, in 1877, to the city of San Francisco. One of the architects, Siccardsburg, committed suicide by shooting himself because of the continuous attacks and criticisms of the building. Only when the Viennese had gotten used to the house did they recognize its beauty.\n\nAmong the important opera houses of Europe, the Vienna Opera had its own special style. The Paris Opera was more elaborate, and, as the theatre of the financial and stock-exchange period, was overloaded with costly porphyry pillars, marble and gold. The Vienna Opera is festive, but middle-class. It is not obtrusively elegant. It is, to be sure, a large theatre, but when one enters the auditorium, one does not have the feeling of being in a large theatre. The society people, who sit in the boxes or in the orchestra on comfortable red plush chairs, form, like the society of a salon, a complete unit.\n\nIn this opera house, Vienna's society was like a painting within a golden frame. In the boxes at the right and left of the stage, and in the large festival box, sat the members of the Imperial family. In the orchestra were the officers, decorated with many orders, and the brilliantly dressed men and women of the wealthy official families. Toward the rear stood the young officers, who, as guests of the Emperor, were admitted to the opera for a very small sum, amounting to about ten cents. In the boxes were the families of the old noble houses, and the new plutocrats of Vienna. In the galleries were the young music enthusiasts, many holding piano scores\u2014girls and young men, sometimes in a close embrace for double enjoyment.\n\nIn the Vienna Opera, one encountered more color, more vitality and more musical elan than in other German opera houses. It lay on the map between Germany and Italy and bound the operatic art of two great nations.\n\nIn the first third of the 19th century, during Rossini's time, the most famous Italian singers sang in the Vienna Opera. Every year one heard Lablache, Rubini, Donzelli, Ambrogi, the mesdames Lalande, Dardanelli and other great international favorite. The uproar over Rossini, which almost made the Viennese forget that Beethoven was living among them, is quite understandable when one considers that such amazing artists performed his works. Even Weber, when he came to Vienna for the performance of his \"Euryanthe,\" was caught up in the general delirium, and wrote: \"I have never encountered such artists before\" and that they \"represent the highest, purest perfection.\" After the period of the Rossini opera came (after 1830), the Meyerbeer period, with new singers like Adolphe Nourrit who, in Auber's \"La Muette de Portici\" and in Meyerbeer's operas, from \"Robert le Diable\" to \"Les Huguenots,\" first sang the leading tenor roles, and brought an elegant, French style of tenor singing to the operatic stage. Another exponent of this energetic, yet charming French style was Roger, who was the first Johann van Leyden in Meyerbeer's \"Le Proph\u00e8te.\" After Meyerbeer, it was Verdi who, in Italy, evolved a new, vigorous tenor type\u2014the stretta singer with the stormy high \"C.\" The Vienna Opera, during two generations, reveled in such singers and such vocal artistry. This created a tradition which remained on its stage and with its public. For this reason it was that the Viennese derived more joy from beautiful voices than opera audiences in other German cities. In Vienna, the charm of sound held sway as it did in Italian opera houses, and this delight in sheer tonal beauty reigned also in the chorus and orchestra.\n\nWhen the Vienna Opera was opened, the Wagnerian period had already dawned. The older Viennese wanted nothing to do with Wagner, but the women and the youth displayed a colossal enthusiasm for the composer's colorful romanticism. Since the younger generation belonged to him, the world belonged to Wagner. \"Die Meistersinger von N\u00fcrnberg,\" \"Der Ring des Nibelungen\" and \"Tristan und Isolde\" were set in the golden frame of the Vienna Opera, and Franz von Jauner, who became Opera Director in 1872, engaged Hans Richter, the famous Wagnerian conductor, to perform them.\n\nHans Richter's life was closely bound with Wagner's. He had, in Wagner's villa, Triebschen, copied the score of \"Meistersinger.\" He had been present when Wagner laid the cornerstone of his Bayreuth Theatre; and had played the tympani in the performance of Beethoven's Ninth Symphony which consecrated the festival theatre. Again at Triebschen, Richter had played the trumpet at the performance of the \"Siegfried Idyll\" with which Wagner greeted his new-born son. Richter prepared the chorus for the first \"Meistersinger\" performance. At Bayreuth, he conducted the first performance of the \"Ring.\" Finally, he died in Bayreuth, where he had spent the last years of his life, like Kurvenal at the feet of his master. In Hans Richter, the Vienna Opera encountered Wagner's great representative and the most famous general in the Wagnerian wars.\n\nI have often seen Hans Richter conduct at the Vienna Opera, and I dare say that never again, as many great conductors as I have seen standing in his place there and elsewhere, have I seen a conductor so full of naturalness and simple strength. Nikisch, the pale salon gypsy with the romantic curls, was interesting; Gustav Mahler a bundle of nerves with the highest kind of intelligence; Weingartner, elegant and formally impressive; Toscanini, a firebrand and a volcano. None of these was so imposing as Hans Richter, of whom Debussy wrote, after hearing him conduct in London: \"He conducts as dear God would conduct if He had learned conducting from Hans Richter.\" I was deeply moved when I read this sentence of Debussy's, since I had often asked myself whether my great admiration for Hans Richter had not been influenced by the fact that he was the great conductor of my youth, and that I had heard all the masterworks of music first under his powerful baton. But here I saw that Debussy felt exactly the same as I on the subject.\n\nIn the Vienna Opera, Hans Richter sat on a comfortable wicker chair in the centre of his musicians. He was a massive man, broad, heavy, and his large blue eyes behind professor-like spectacles looked calmly out as far as the most removed members of his orchestra. His beard flowed down over his breast. Holding his baton powerfully in his hand, his beat was clear and visible from the distance. At the climaxes he swung out with a broad movement. Strength and brilliance emanated from Richter's conducting, and all details merged into a great symphonic line. Every kind of pathos, whether it was the festive pathos of the \"Meistersinger,\" the tragic pathos of Beethoven's Ninth Symphony or the religious pathos of the Bach Mass, Richter achieved without exertion. He did not have to raise himself on his toes or strain his nerves when he wished to draw the fullest degree of sound from his orchestra. As complete master of his craft, he was natural in every moment; he possessed an instinctive feeling for perfect truth which was undimmed by the slightest shadow of a mannerism. I have never heard anything so powerful as Hans Richter's conducting of the third act of \"Walkyrie,\" which was unified and rounded from the Ride of the Walkyries of the beginning\u2014which really sounded like a ride through clouds and tempest\u2014to the great climaxes of the end, where the orchestra roared up from the depths. Nor have I ever heard anything so solemn and awe-inspiring as his prelude to the third act of \"Meistersinger,\" which rose from the depths to the heights, all feeling and thought, solitude and pomp.\n\nHans Richter conducted everything at the Vienna Opera, not only Wagner, but Mozart, Rossini and Bizet. For Mozart, Richter, whose hand was heavy and mighty, lacked a certain lightness. Even Mozart's \"Don Giovanni,\" conducted by him, took on great symphonic lines, as if Beethoven had composed it. But whenever Mozart's music became really monumental, as in the Overture to the \"Magic Flute,\" Richter was unsurpassable. This music really ascended to the light and had a sunny brilliance. Richter conducted Rossini, Verdi and Bizet with more routine than enthusiasm, and most of the time with his baton in his left hand, which he had schooled to the same extent as his right. But in Wagner's works, he sat commandingly and filled with his mission as Wagner's apostle, in the middle of the orchestra, with all the strings grouped at the left and the winds at the right, like two armies which he led to victory.\n\nFor the Wagner performances, the Vienna Opera had at its disposal singers who, like Herman Winkelmann, Amalia Materna and Theodore Reichmann, had been trained by Wagner himself for his new style. Winkelmann sang the role of Parsifal at the first \"Parsifal\" performance in Bayreuth. Materna had been the first Br\u00fcnnhilde, and Kundry, Reichmann the first Amfortas. Winkelmann's manly heroic voice, in which there also seemed to flow a tear, his heroic gestures, his straightforward glance, made him the born singer of Tannh\u00e4user and Tristan. He was the bold knight in \"Meistersinger\" who is a poet and a lover. As Lohengrin, too, he was manly and knightly, when he left his swan-boat and stood there, holding a sword over his breast, his glance directed upward as in prayer. His high \"a\" sparkled like a golden shield on which the sun is falling. In Materna's voice, there was heroic jubilation; and Reichmann's elegiac baritone was shrouded as if by a veil. All these singers had been favorites of Wagner, especially Materna, whom Wagner, in his letters, addressed with particular tenderness: \"My dear, good child\u2014my true, faithful Wagner-daughter.\" Once he closed a letter to her: \"God, when I think of the last Kundry evenings! Adieu, my dearest, good and best one!\" Such artists made Wagner's performances into impressive events which had far-reaching influence. In the Vienna performance of \"Meistersinger,\" a young baritone sang the role of the Town Watchman. Later, this singer, Angelo Neumann, traveled through the world with a Wagnerian touring company. One of the most magnificent conductors of our times, Arthur Nikisch, was a violinist in the Wagnerian orchestra of the Vienna Opera.\n\nBut the Vienna Opera was not alone a great Wagner theatre. Since 1891, when Mascagni's \"Cavalleria Rusticana\" brought filled houses, it was also a Mascagni theatre. After the premiere of Massenet's \"Manon\" in 1890, it became a Massenet theatre; and it was a Goldmark and Verdi theatre as well. A French writer on art, Ritter, in one of his books, called the Vienna Opera \"the gayest theatre in the world. There they perform earliest what there is of note in Europe; here singers and dancers from the four corners of the globe congregate, and costly productions and ballet performances play an eminent part.\"\n\nThe Vienna Opera would not have been a typical Viennese theatre had not ballet performances played a significant role. Vienna was the city of beautiful women, and, moreover, the city of the dance. Although, according to an unwritten law, the prima ballerina had to be an Italian, the other ballet members came from Vienna homes, had the Viennese laugh and the Viennese love of life. For them, new ballets were written which, like Baier's famous \"Puppenfee\" (\"Doll Fairy\"), were graceful, playful and charming. Ballet performances attracted all Vienna's best society. Young aristocrats, after the performance, took the pretty ballerinas to supper at the Hotel Sacher nearby. Wealthy old men sat in the first row and gazed through large opera-glasses at the girls on the stage, as if this was their principal occupation in life. The Viennese called these old gentlemen \"ballet uncles,\" and folk-singers delighted in singing mocking songs about them.\n\nThe opera was Vienna's focal point. The great musicians who were its artistic leaders came from many foreign countries. This had been the case also at the old opera house, the K\u00e4rntnerthor Theatre, where Otto Nicolai (the founder of the Vienna Philharmonic concerts) had come from K\u00f6nigsberg, Heinrich Esser from Mannheim and Otto Dessoff, the greatest conductor of earlier days, from Leipzig. In the new opera house, there were, among the leading musicians, only three native Viennese: Herbeck, Franz Schalk, Clemens Krauss. Hans Richter came from Hungary; Wilhelm Jahn, the excellent Opera Director from 1880-1887, from Moravia; Gustav Mahler also from the Bohemian-Moravian borderland; Felix von Weingartner from Dalmatia; Richard Strauss from Bavaria; Bruno Walter from Berlin. All these men created the tradition which covered the new opera house with its patina. Their achievements there became history, and the ghosts, which cast their spell in the beautiful halls of the Vienna Opera, bound the present with the past and future.\nNew Concert Halls and Old Traditions\n\nOne year after the new Opera House, the new Concert House\u2014the Music Society Building\u2014was dedicated. Here, ever since 1870, the large concerts have been given. This building, too, is near the Ringstrasse. Emperor Franz Josef had donated the ground, and among the sponsors of the new house were the Empress, the Archdukes, Princes, Counts like the Liechtensteins, Schoenborns, Schwarzenbergs, Lobkowitzes \u2014all members of the eighty aristocratic families which, according to the words of Napoleon, ruled Austria. Other sponsors included bankers and industrialists. In compliance with the taste of new Vienna's society, the building was decorated in that pompous style which characterizes all the buildings of the Ringstrasse epoch: paintings of the Nine Muses on the ceiling, golden Caryatides whose hands were clasped beneath their bosoms, framing the boxes, and with gas-lit chandeliers.\n\nTwo years later, in the Herrengasse where the Liechtenstein, Gallas and Kinsky palaces stood near the Imperial Palace, the intimate \"Konzertsall\" (B\u00f6sendorfersaal) was opened. Here Rubinstein, von B\u00fclow and Liszt played the piano, the Joachim quartet performed, and Barbi sang. The piano manufacturer, Boesendorfer, built it. On this site had formerly been the riding school of the Duke of Liechtenstein. Because of this fact, the Viennese pianists, when it came time for the building's dedication, refused to play there. Hans von B\u00fclow was courageous enough to perform at its inauguration, on November 19, 1872. Since the lighting had not been completed in time, stable lanterns were brought for illumination. The Emperor's Chief Stable-Master, Count Gruenne, regretfully remarked: \"It's a shame such a beautiful riding-school is gone!\"\n\nThe most important concerts were those of the Vienna Philharmonic. The members of the orchestra of the Opera managed these performances and themselves chose their conductors. This fact explains the enthusiasm which poured forth from the orchestra under great conductors, the freedom of execution and the brightness of its sound. The personnel comprised the best musicians of the city who played in these concerts for their own enjoyment. They would not tolerate much pressure from the side of the conductor, any more than a spirited, thoroughbred horse will stand hard pressure from the spurs. Hans Richter told me: \"One must allow this orchestra its free-play and lead it only to the point where it can develop its forces freely.\" All of the great conductors observed this rule when they directed the Philharmonic, best of all Hans Richter during the sixteen years of his leadership, and Felix von Weingartner who was at its helm for eighteen years. Two other conductors could boast of having aroused the orchestra's greatest enthusiasm and admiration\u2014Arthur Nikisch and Arturo Toscanini. Men like they, and Muck, Mottl, Furtwaengler, Richard Strauss and Mahler knew how to unleash all its powers of brightness and virtuosity. Mediocre conductors were lost with this orchestra.\n\nHans Richter leaves the stand after a  \nBruckner performance  \n(O. B\u00f6hler)\n\nThe most eminent members of the orchestra were, at the same time, professors at the Conservatory which was housed in the same building. They educated pupils who adopted their way of playing, and who later became teachers themselves of the following generation of musicians. Thus there developed a special style of Viennese orchestral playing which was transmitted from one epoch to another like a precious heirloom. No concert orchestra in Europe had a similar tradition.\n\nThe Viennese public was musical, sensitive and capable of enthusiasm, but was, to a high degree, conservative, because it was surrounded by ghosts of old traditions which had become memories and habits. Each new musical movement had to battle these ghosts. Vienna had little interest in the progressive development of music. Every new musician encountered resistance in Vienna, even Robert Schumann. Up to the year 1854, the Society of the Friends of Music had not played a single one of his works. When finally, in this year, his C Major symphony was performed, it was coolly received.\n\nThe first orchestral work of Franz Liszt, \"Les Preludes,\" was performed by Hellmesberger. It was hissed by a part of the audience. In 1858, Liszt himself came to Vienna. Twice his \"Graner Festival Mass\" was performed, both times with dubious success, although Liszt at that time was one of the most famous composers in Europe. In 1861, Carl Taussig gave three orchestral concerts of Liszt's compositions. Attendance was very poor.\n\nUntil 1850, nobody in Vienna had even thought of performing an opera by Richard Wagner, although, through the successes of his \"Rienzi\" and \"Tannh\u00e4user,\" he was Germany's most famous musician and the man of the hour. When finally the Director of the Vienna Opera wanted to perform \"Tannh\u00e4user,\" the censor forbade it because of its \"immoral\" libretto. At that time, Wagner's music was known in Vienna only through the concerts given by the Waltz King, Johann Strauss. Even the music of \"Tristan and Isolde,\" which had been declared \"unplayable,\" was first played by Johann Strauss at his concerts in the Volksgarten. The first performance of \"Tannh\u00e4user\" in Austria did not take place in Vienna, but in Graz (where Richard Strauss's \"Salome\" was also performed before its Vienna presentation). When finally \"Tannh\u00e4user\" was given in Vienna, it was at the Thalia Theatre in Ottakring\u2014a theatre far out in the suburbs which was nothing more than a wooden barn. The performance, according to all reports, must have been terrible. Tannh\u00e4user was hoarse or voiceless, Wolfram shouted, Venus sang off-pitch. Hanslick called the Singer's Contest \"the Battle of the Invalids.\" Wagner considered the performance only as a means for making money, and wrote to the Theatre Director: \"The devil take Vienna if it doesn't bring me in some money. Otherwise the cozy nest means damned little to me.\"\n\nThis suburban performance of \"Tannh\u00e4user\" at least had the merit of having paved the way toward the Wagnerian performances at the Vienna Court Opera. In 1858, \"Lohengrin\" was performed for the first time, at the old K\u00e4rntnerthor Theatre. \"Tannh\u00e4user\" followed in 1859. Fourteen years had passed since this opera's Dresden premiere, and it had been produced on all the German stages, as well as in Br\u00fcnn, Lemberg and Linz before its performance in Vienna. The criticisms in the Vienna newspapers were so malicious that the Leipzig \"New Magazine for Music\" reproached the authors for their narrow-mindedness and lack of restraint.\n\nIn 1861, Richard Wagner came to Vienna from exile. Here he heard his \"Lohengrin\" for the first time; and the audience gave an ovation to the pale, small man who sat in a second-tier box. Wagner was so enthusiastic about the performance and the public that he immediately wanted to bring \"Tristan and Isolde\" to Vienna for its premiere, and wished to compose \"Meistersinger\" expressly for Vienna. At this point, Vienna had the wonderful opportunity of becoming Wagner's music city\u2014perhaps his Bayreuth. The spiteful attacks on the composer in the newspapers destroyed this plan. I have read a letter of Wagner's which is still unpublished, where he exclaims, up in arms, that he will never return to a city where he could not even go to the men's-room without being scolded. His \"Tristan and Isolde\" was removed from the repertoire after seventy-seven rehearsals as \"voice-murdering.\" And when, in 1879, in the new Vienna Opera, the \"Meistersinger\" was given for the first time, the \"Pr\u00fcgel\" (\"Thrashing\") Scene at the end of Act II was continued in the orchestra. One section of the audience hissed, and the other half attacked the hissers.\n\nIn 1872, the Vienna Philharmonic rejected the second symphony of Anton Bruckner as \"unplayable.\" When Bruckner's symphonies were finally performed, the public was so antagonistic and the criticisms so spiteful that Bruckner writes in a letter: \"They scared me to death in Vienna.\" In a letter of 1875, he complains desperately that he had to go begging in Vienna. In 1875, he asked the Philharmonic not to perform his Seventh Symphony because of the opposition of Brahms and Hanslick. When Bruckner was seventy years old and, in 1892, his Eighth Symphony, dedicated to Emperor Franz Josef, was performed, the Viennese hailed the shy old man, who stood helplessly on the concert platform, with great enthusiasm. The Viennese public made life hard for composers, but if a musician were great enough to outlast a period of being misunderstood, or if by any chance he had meanwhile become famous somewhere else, nowhere could more enthusiasm be demonstrated than by this same public.\n\nFor all great composers of modern times, this period finally arrived. When \"Thus Spake Zarathustra\" was performed under Hans Richter, the audience laughed at the cock'scrow of the trumpet and hissed at the end. The same thing occurred with \"Ein Heldenleben\" (\"A Hero's Life\"). A few years later, no more violent applause could be heard in Vienna than after Strauss performances. First performances of works by Bartok, Ravel and Hindemith were duly hissed. At the first performance of Stravinsky's \"Sacre du Printemps,\" the whole hall changed into a real inferno. People laughed, jeered, mocked, hissed and it looked as if this genial work could not be heard to the end. A man shouted from the gallery into the tumult to the conductor (it was Franz Schalk): \"Play Bruckner!\" Thirty years before, the same man would probably have shouted: \"Play Mozart!\" In Vienna, the town of the musical classics, it was a favorite sport, if a problematic work had been played and a classical one followed, to greet the latter with demonstrative applause. When Liszt's \"Prometheus\" was played for the first time, the G Minor Symphony of Mozart was given a riotous ovation, although Mozart certainly needed the applause far less than Liszt.\n\nThe biggest concert scandals in Vienna occurred at the Sch\u00f6nberg concerts. Even as a very young musician, Sch\u00f6nberg had a talent for infuriating the public. When, in 1900, two songs of his were sung in a concert hall, the public laughed as if it had just heard the most enormous joke. At the end came violent hissing. Somebody kept whistling on his house-key (another favorite Viennese concert-sport) and Gustav Mahler turned upon the indefatigable whistler and shouted: \"Sir, won't you stop?\" Whereupon the whistler calmly retorted: \"Why? I hiss your symphonies, too.\"\n\nEven in 1912, when Sch\u00f6nberg arranged a performance of a song by his later famous pupil, Alban Berg, the public had just as good a time as if it had been composed by Sch\u00f6nberg himself. A Sch\u00f6nberg adherent slapped a Sch\u00f6nberg adversary in the face, and the concert had to be stopped. When, twenty years later, Berg's \"Wozzeck\" was performed at the Vienna Opera, it had a spectacular success. Berg was, by that time, already a famous composer abroad, and there is nothing which so enthuses the Viennese public as great fame to which it has contributed nothing.\n\nThe fame of most of the great Viennese musicians originated abroad. The successes of Bruckner began with the performance of his Seventh Symphony, under Nikisch, in Leipzig in 1884, and the performance of the \"Romantic\" Symphony in Munich the same year under Hermann Levi. The acknowledgment of Hugo Wolf as song composer originated in Mannheim where Wolf had made a large circle of friends. The success of \"Wozzeck\" by Alban Berg started with the performances in Berlin under Kleiber, and in Philadelphia under Stokowski. All the Viennese musicians who, in official musical circles, established themselves slowly, had a group of friends who supported and extolled them. The unconditional admiration of these friends provided these musicians with what the general public lacked in warmth and interest.\n\nThe fault for the public's lack of understanding was certainly, to a degree, on the side of the Viennese music critics, but not entirely. After all, those critics who are most read by the public stand dependent, to a certain degree, upon the taste of that public. \"Que voulez-vous, je suis leur chef. Il faut que je les suivre,\" said a witty French party-leader. The power that the influential and widely-read critics wielded over the public consisted to a degree in the fact that they wrote what their public enjoyed reading. More important than the critics, however, was the spirit of the town, which was unfavorable to new and great art in other fields as well as in music. The city of Vienna was, at the time when the new industrial age roared through its streets, full of the enjoyment of life. From the arts, too, it demanded entertainment, not mental exertion. Music should be melodious, lively, clever. For such Viennese music, Brahms was too serious, Bruckner too full of Catholic piety and Mass solemnity, Mahler too cabalistic and too mystic, Sch\u00f6nberg too intensely occupied with problems. Vienna, so long accustomed to sensuous enjoyment of its music, was weak in that intellectual power which was more prevalent in northern Germany. Intellectual problems in music, which the works of every great new composer presented, were uncomfortable in Vienna.\n\nAnother important instance which demonstrated that while Vienna was a musical city, it was not one which showed understanding for new musicians, was the conservative mind of the monarchy whose capital was Vienna. The Austrian empire was an old monarchy, and the old order penetrated new forms of life. In new Vienna, a time of technical wonders had arrived, but the spirit of the city was old. The construction of society was the same as that of the old feudal court-society of the seventeenth century. And the modern political innovations, like the parliament, and political movements such as those of the workers, changed nothing of this. Despite its Ringstrasse and new palaces, Vienna was a baroque city, just as Rome will always be.\n\nThe Spanish Ceremonial, which had ruled the life at the Court of Charles V still ruled at Emperor Franz Josef's. As in the Madrid of the seventeenth century, the Emperor's guards passed with slow steps through the streets where now modern traffic hurried. The horses of the imperial riding school lifted their feet to the beat of a bygone age. On Good Friday, all Vienna was changed into old Spain. On the road from St. Stephen's Cathedral to the Imperial Palace, four altars, decorated with fresh green, were erected where the four evangels were read. From the giant portal of St. Stephen's came a great procession. After the cross came armies of priests and monks, and under a baldachin walked the Cardinal Archbishop of Vienna, carrying a chalice which blazed with gold and precious stones. Behind the baldachin walked the old Emperor, bareheaded, holding a lighted candle and the Archdukes, the ministers and generals, the high officials and courtiers. All the spectators fell to their knees, and the bells rang. One felt himself back in seventeenth century Madrid, where an ancient Hapsburg, in the Escorial monastery, had himself laid, alive, in his coffin, and prayed the litany of the dead which the monks were chanting in the church. An hour later, automobiles passed through the same streets.\n\nThe Viennese lived in modern houses, but the German they spoke was full of old forms dating from the middle-ages. In no other dialect of the German language can one find so many expressions extant which dated back to the time that the Song of the Nibelungs was composed. Besides, the whole history of Vienna is preserved in the folk speech: French words from the time when Napoleon lived in the Castle of Schoenbrunn, Spanish expressions from the time of Leopold I, who had married a Spanish woman, and Italian words. The customs, too, are filled with the past. Viennese men kissed the hands of the ladies whom they met the way the Spaniards had done in the 17th century. From the same period dated the practice of addressing people of higher social standing by the noble title \"Herr von\"\u2014a custom specially beloved by waiters, who flattered every well-dressed guest by thus addressing him. If the waiter expected an especially good tip, he called the customer \"Baron.\" If he did not get it, the guest had to be satisfied with the more simple \"Herr von.\" One lived in an ancient noble city where the world began with a baron.\n\nEvery second Viennese had a long-winded title of some sort. The longest and most pompous, of course, belonged to the Emperor. Upon them lay the dust of ancient history. So even the title \"Emperor of Jerusalem\" was not missing, as if Vienna still dwelled in the time of the Crusades. The Court was old, the Church of Austria was old, and the Emperor himself was old. An empire of this sort is not easy to set in motion, even though it is through modern, technical, political and social movements. For a long time during Emperor Franz Josef's reign, the militia was called out at every worker's strike, and fired upon the strikers. The demands of the nationalities and social classes were only slowly satisfied. This conservative spirit naturally worked upon the mentality of the Viennese. In an empire where all was regulated, music could not progress rapidly. In an old police state where every free word was considered dangerous and where the police watched any casual movement with suspicion, strong personalities were apt to be thought dangerous. The appreciation of strong artistic personalities who had destroyed a bit of the old spirit, could not readily occur. So it was with musical life. Every new artist was a threat and a danger.\n\nClassical music was, in such a state and in a city so bound by the old order, of especial importance. It was itself an old order which had been given over in flesh and blood. Even Beethoven's greatest and most daring innovations were not taken into the old order without opposition. It surely took time until the works of Beethoven's last period, even the Ninth Symphony, were given recognition and appreciation. The greatest Austrian poet, Grillparzer, heard in this music only revolution and chaos, and Hanslick only recognized this music against his will. But finally these works, too, became old music, and, as such, in Vienna, classical music. Classical music was the same for the Viennese public as the Constitution was for political life. It was the cornerstone of Viennese musical art and the temple to which one fled in times of need.\n\nNot even in the time of the classicists was the great musical city great because it had understood new music better or more rapidly than other European cities. As early as 1738, Reichardt, the Prussian Court Conductor, complained: \"The works of foreign masters have a hard time succeeding here.\" In music, Vienna was an aristocratic city, proud of its past, rich in tradition. It was the great musical city because here, more than anywhere else, music was a part of its life and the lives of its inhabitants. The city had a musical soul. There were other cities in Europe which thought more intellectually or conversed more brilliantly, as Paris, for instance. Others had stronger political power, like London. Vienna was the musician.\n\nFranz Schalk at the desk.\nModern Music in Vienna\n\nRichard Strauss, Gustav Mahler, Arnold Sch\u00f6nberg\n\nAround the year 1900, the old conservative music city was set into motion. Along with the national and social movements came a general unrest upon artistic fronts. A new epoch was announcing itself. Old orders were shaken. Revolution was in the air, and the life-enjoying, amusement-seeking public of Vienna was frightened.\n\nIn literature, painting, theatre and music, the naturalistic tendency had the upper hand. Every year there would be a new social drama by Henrik Ibsen who had turned away from the romantic, and now analyzed the morals of bourgeois society, the modern marriage and the stricken conscience. Strindberg's plays portrayed the battle of the sexes. Zola, Maupassant and Flaubert delighted in depicting everyday life. Tolstoi and Dostoievsky, in powerful novels, described people who lived in hate and love, passion and crime. From France came new impressionistic pictures which released the landscape in a play of colored spots and disintegrated the sunlight. Claude Monet, Sisley and Pissaro painted momentary shadings and color vibrations. Liebermann painted factories. In Germany, a new poet, Gerhardt Hauptmann, appeared on the scene, and with his \"Die Weber\" (\"The Weavers\") and \"Hannele's Himmelfahrt\" (Hannele's Ascension to Heaven\") awoke the social conscience of the time.\n\nIn the colored, make-believe world of the theatre, too, changed perception and feeling of a new time were reflected in a new and naturalistic style of dramatic portrayal. Each year Duse came and transformed the stage into reality with her pale, Mater Dolorosa face, her sad eyes, her tired hands, and the voice which was pure soul. In Vienna's Burgtheater, a fortress of the classic dramatic style which Goethe had founded in Weimar, modern actors like Mitterwurzer became established, and when this genial and unpredictable actor died in 1897, Joseph Kainz, the greatest of the new \"nerve\" actors was a famous and worthy successor. In Berlin, Brahm assembled actors for a naturalistic Ibsen ensemble. He was followed by Max Reinhardt, who in \"A Mid-Summer Night's Dream\" had the woods fantastically staged with shimmering birch boughs, and the gloomy Cyclops wall in \"Elektra\" built up high. In Paris, Antoine played, in his \"L'Oeuvre\" Theatre, the dramas of Ibsen, Strindberg, Hauptmann and Bjoernson.\n\nAll at once in Vienna, there was a new literature which bound modern naturalism with Viennese dramatic art and Viennese tenderness. Arthur Schnitzler brought the ladies of Vienna society and the girls of the suburbs to a stage become realistic. Hugo von Hofmannsthal, even when he was still a boy, wrote marble-smooth verse, in which the Viennese baroque lived once more. A great modern critic, Hermann Bahr, traveled throughout Europe and visited all the battle-fields of modern art between Paris and St. Petersburg. All at once, Vienna, too, had a famous modern architect, Otto Wagner, who no longer, like the designers of the Ringstrasse buildings, thought in terms of Greek pillars or Renaissance arches, but in those of cement and iron. In 1903, he erected a new Postal Savings Bank Building in the most modern style on the Ringstrasse, and Vienna conservatism cried and screamed. The same thing had happened in 1900, when the painter Klimt exhibited his ceiling paintings which he had done for the university.\n\nThe Viennese public was terrified by so much new art, and saw the end of the world coming when Ibsen and Gerhardt Hauptmann, even, managed to penetrate the classic portals of the Burgtheater. I was present when in the \"German Folk Theatre,\" Ibsen's \"The Wild Duck\" was presented for the first time. Ibsen himself had come for this performance. He sat in the box nearest the stage, with his ruffled white hair and white beard. He watched, without changing his expression, while the audience laughed at every phrase in the last act; while it applauded everything which was meant ironically, and ridiculed everything which was meant seriously, with jeers and cat-calls. When the curtain fell amid the monstrous clamor of aroused theatre-patrons, Ibsen rose in his box and laughing heartily, tossed his high top-hat to the parquet below. Similar uproars followed every performance of a modern play. When Maeterlinck's \"L'Intruse\" was given at the Josef-stadt Theatre, every breathing pause called forth laughter. No comedy could have made the audience more hilarious.\n\nMusic was carried along with the intellectual movement of the era, and it became naturalistic and impressionistic. The classic harmony began to dissolve. Classic forms, like all the constructions of the period, began to shake and tremble. In France, Claude Debussy, influenced by the impressionistic painters, and by the poetry of Mallarm\u00e9 and Verlaine, began to compose oscillating tone colors. His faun began to play the flute in the sunlight of an antique landscape. His Spain was a canvas full of color daubs and dust. In Germany, Richard Strauss became the leader of the modern music movement, after he had turned away from the Parsifal-like solemnity of \"Guntram,\" the opera of his youth. He boldly applied himself to every-day life, became the greatest naturalistic champion of the orchestra, and painted the noise of Naples' streets. When his Neapolitan descriptions were hissed in Munich, he wrote von B\u00fclow: \"I had a wonderful time. The first step toward independence!\" Daring, ironic, full of zest for life, for pranks and jokes, rich in voluptuous melody, young Richard Strauss appeared in a musical world which was tired of Wagner's pompous romanticism. Once I had dinner with Strauss (who was a young man at the time) and Zemlinsky in Vienna. Strauss scratched his fork on his plate and it gave out a squeaking noise. \"Could you reproduce that in the orchestra?\" he asked Zemlinsky. And as Zemlinsky was shaking his head, Strauss said confidently: \"I can do it!\" What sound in the world was Strauss unable to copy with his orchestra?\n\nI became acquainted with Strauss in Munich when he was a young court-conductor. The \"beer-friendly\" city on the Isar had its epoch of genius, and about 1900 Strauss introduced Frank Wedekind to me. Wedekind had already written his bold drama of adolescence, \"The Awakening of Spring,\" which had completely horrified the whole city. Also through Strauss, in Munich, I met the artist Thoma, whose sharp, stinging drawings appeared every week in \"Simplicissimus.\" This paper was the most courageous opposition paper that appeared in Germany. On every street corner one saw its placard\u2014a red pug-dog, showing his teeth. For this paper, Thoma drew his cartoons of Prussian Junkers, German university students, fat industrialists on whose laps young girls were sitting, and conceited officers. It was fresh opposition air in Munich. Berlin, the German Kaiser Wilhelm, the Junkers were the enemy. Richard Strauss lived in this air. He composed with preference to revolutionary modern poets, Dehmel, Liliencron, Bierbaum, Falke, Christian Morgenstern, Hart; and his song, \"Der Arbeitsmann\" (\"The Workman\"), on a text by Dehmel, which waves the red flag, will outlive many other songs he has written. At that time, I already knew the songs of Richard Strauss, the naturalistic declamation of which, combined with their lyric tenderness, was something new. Especially the \"Traum durch die Dammerung (\"Dream through the Twilight\") had made an impression upon me with its fragrance and atmosphere. A girl of my acquaintance sang the beautiful melody, I accompanied at the piano, and at the end, we kissed each other. This was certainly not the worst way of hearing modern music.\n\nOscar Fried, who later made himself a name as conductor and composer, spoke to me in Munich about Strauss, saying that he was a genius, and I was anxious to meet him. With great interest, I had heard him conduct a performance of \"Tannh\u00e4user\" at the Munich Court Opera. At that time, he still had his bush of red hair, and his slender hands, of which he was a trifle vain, and which also, later on, he kept carefully manicured. He held the baton gracefully, almost floating in his hand, as opposed to the conductors of the old school who used to grasp their batons firmly, like a commander's staff. Most striking was the supple, delicate sense of sound, the nervous shading, and above all, the many accelerations and retardings of tempi, the modern accentuation. These shiftings of tempi were still more obvious to me when Strauss conducted Mozart's \"Don Giovanni\" in the small Residence Theatre. His was a non-academic Mozart which stressed the orchestra and the drama on the stage\u2014Mozart, 1900.\n\nWhen Oscar Fried introduced me to Strauss, who was his teacher, I saw a tall, slender man with fine blue eyes, who was simple and natural. His nature was sharply critical. One had the impression that he knew exactly what he wanted and clearly spoke his thoughts. I well remember our conversation that day. We spoke of Mozart, and Strauss was very polemical against the conception that Mozart was a naive musician. He insisted that Mozart had always chosen sensational subjects for his operas. \"The Marriage of Figaro\" was the revolutionary play of Mozart's day, and it had been banned in Paris. \"Don Giovanni\" had an erotic subject, and Mozart was accused of immorality when he composed it. \"The Magic Flute,\" too, with its employment of Masonic rituals, had a sensational subject for its day and age. His love for Mozart, Strauss had inherited from his father. Later on, when I visited Strauss at his country house in Garmisch, I would often see him, after he returned from an automobile excursion, lie down on a sofa and study the score of a Mozart string quartet which, for that purpose, always lay ready on a music stand nearby. This was his refreshment and inspiration too. Afterwards when Strauss had become Director of the Vienna Opera and gave parties at his home there, he always had a Mozart string quartet played at the beginning of the festivities. On one such occasion, I observed that Strauss, who was sitting in a corner, was hardly able to restrain his tears. I asked him later what had moved him so much in this quartet (it was the C Major). He answered that his father, in his will, had ordered the quartet played in church at his funeral. This had been done, and Strauss recalled it. With a young modern musician like Strauss, this admiration for Mozart was striking. We young musicians were first Wagnerians, then Mozartians. Strauss was first a Mozartian.\n\nOtherwise, Strauss was respectless, not at all pathetic, and Munich was shocked when he wrote his opera \"Feuersnot\" (\"Fire Famine\"), on a text by Wolzogen, a cabaret director. In this opera, Strauss ridiculed the people of Munich and glorified the first love-night of a young girl in a naturalistic symphony. The bold opera failed in Munich and Strauss went to Berlin. Here he was in a modern atmosphere. He always needed spiritual inspiration. He found it, mainly, in the new theatre of Max Reinhardt where, for the first time, he saw Oscar Wilde's \"Salome\" and Hugo von Hofmannstahl's \"Elektra.\" He found it, also, in \"The Weavers\" of Gerhardt Hauptmann, and wanted to compose an opera based on the drama. Berlin, during this period, was rich in intellectual forces of all kinds. There were excellent modern magazines, brilliant journalists and critics (like Maximilian Harden, who wrote articles against the Emperor, and on the theatre in a lush, colored style), the best German actors of the younger generation like Bassermann, Sorma and Lehmann. Nikisch conducted the Philharmonic, Weingartner at the Royal Opera. For Richard Strauss, a musician with modern temperament and elan, this Berlin was the right place. Here he became the musician who collected all modern intellectual and artistic movements in his compositions as in a burning-lens.\n\nRichard Strauss first appeared personally in Vienna as a composer of songs in 1902. He gave a \"Liederabend\" (Song Recital) at the Boesendorfer Hall, and his wife, Pauline de Ahna, sang his songs. Strauss sat modestly at the piano, like someone who did not wish to draw attention to himself. He was a fine, delicate pianist who shaded the tones with good taste. The Vienna public, who so often had gathered in the same hall to hear the first performance of songs by Brahms and Hugo Wolf, found the Strauss songs beautiful and full of warmth of feeling, but not exactly original. Songs like \"Wiegenlied\" (\"Cradle Song\"), with its easily grasped melody and the bells in the accompaniment appeared quite banal and based on a cheap effect. Also, the impressive melody of the \"Heimliche Aufforderung\" did not impress the audience as being something novel. People had been expecting, in Richard Strauss, a wild innovator, and found, instead, a tasteful lyricist. Hanslick, Vienna's chief critic, called Pauline, the singer of the evening, \"the better half of the composer.\"\n\nStrauss's music had already been heard at the Philharmonic concerts. Hans Richter had conducted \"Don Juan\" in 1892, and \"Tod und Verkl\u00e4rung\" (\"Death and Transfiguration\") in 1893. But even in these performances, no one felt that he had heard the works of a new musical genius. It all sounded too beautiful and too virtuoso. Only with the performance of \"Also Sprach Zarathustra\" (\"Thus Spake Zarathustra\") did the uproar and tumult begin like the ones reserved for a genius.\n\nStrauss first became known as a conductor in Vienna the year he came there as guest, directing the Munich Chamber Orchestra. With the long steps of a stork he stormed on to the podium, and began to conduct Wagner's Prelude to \"Meistersinger\" with the stabbing arm movements which resembled the sallies of a fencer, and the jerking knee-bends which, not much later, made him a notable conductor in the concerts he directed. Vienna was accustomed to the all-powerful, prevailing calm of Hans Richter, and to the Richter breadth of baton technique. Now there stood a young musician of a new generation who was not mighty and broad, but slender and nervous on the podium. His red hair gleamed like a torch borne at the head of a revolutionary mob. In the course of years, Strauss became more concise and more calm as a conductor, and later conducted almost entirely from the wrist with quite brief, short motions.\n\nOnce I went to congratulate Strauss, after he had conducted the first act of \"Tristan and Isolde\" in a most thrilling way, using only short, precise movements. \"Feel me here!\" Strauss said to me, and placed my hand on his armpit. \"Absolutely dry! I can't stand conductors who perspire!\" But at that time, in concert, he must have perspired since he often, in conducting the \"Meistersinger\" prelude, gave the impression of being ecstatic.\n\nIn his conducting of this Prelude, also, Strauss showed every inclination to special accent and to the dramatic quickening and retarding of tempi which had already impressed me in Munich as the nervous characteristics of a new time. Even as an old man, conducting this prelude, he still used many of the self-willed tempo variations. He loved to conduct it with the Vienna Philharmonic, to improvise with all the accents, and enjoyed it immensely when the orchestra followed him perfectly through the sudden changes of tempi. Once, after conducting the Prelude of the \"Meistersinger,\" he rushed up to me, red and excited, and cried out, radiant with joy: \"Everything improvised! All without rehearsal!\"\n\nLater on, Strauss disliked anything unusual in conducting. He used to like to tell me how his father, who had played horn in the Munich Orchestra, had come home after Strauss conducted his first opera performance, and said: \"You young conductors imagine God-knows-what when you sit at the conductor's desk. We old musicians already know when you walk to the podium what you are. You don't even have to hold your baton in your hand before we know all.\" And another time, he remarked: \"How many good conductors did I have to see\u2014B\u00fclow, Hermann Levi, Hans Richter, Fischer, Muck, Mottl, Schuch\u2014and how much did I have to conduct myself, until I could do what I now can do.\"\n\nBut this was the wisdom of old age. As a young musician, Strauss conducted as young musicians have always done, and will always do\u2014with enthusiasm, with tense nerves, with new nuances which correspond to the transformed mode of feeling of a new epoch; and never in a calm and clarified way, for these are the qualities either of the highest maturity or of stagnant routine.\n\nAs opera composer, Strauss first appeared in Vienna with his exuberant work of youth, \"Die Feuersnot.\" At the dress rehearsal, Strauss, interrupting the conductor, Gustav Mahler, suddenly rushed forward and shouted: \"Please, dear Mahler, start again at the 'Nachtlager in Granada' ('A Night's Camp in Granada') spot.\" Actually, in \"Feuersnot,\" there is a quotation from the Biedermeyer opera \"Das Nachtlager von Granada,\" by Kreutzer, just as in \"Salome\" there is a quotation from \"The Barber of Seville,\" concerning the use of which in the over-ripe beauty of the \"Salome\" music, Strauss himself has made the best of jokes. In the \"Feuersnot\" music, again Strauss did not impress the public as revolutionary. Some people found the libretto frivolous in many respects, and others were shocked about the way Strauss glorified himself in it, but the music, with its folk-like choruses and its zestful lyrics, appeared neither offensive nor particularly bold, though it was harmonious and brisk. Public opinion changed when the horrifying \"Salome\" was performed for the first time at a private theatre, the \"Deutsches Volkstheater.\" At that performance there was already an actively hissing opposition.\n\nThus Richard Strauss entered Viennese music life when the great Brahms and Bruckner epoch had come to an end Bruckner had died 1896, Brahms had died in 1897, and with these two symphonists, classical music itself terminated. One was now in a new world of orchestral sound, of virtuoso description, and of crass realism, and Richard Strauss appeared more and more as the eminent master of the period.\n\nFor Vienna's musical life, it was of greatest importance that, at that time, Gustav Mahler took over the direction of the Vienna Royal Opera. Mahler conducted a performance of \"Lohengrin\" on May 11, 1897, and was appointed director on October 8th. Emperor Franz Josef had, in Mahler, appointed a man who, next to Richard Strauss, was the strongest musical personality of the time. Franz Josef was sixty-seven years old, Mahler thirty-seven. Old and new eras stood opposite each other in those two men, the aged monarch and the fanatical musician. With Gustav Mahler at its helm, the Vienna Royal Opera of Franz Josef moved itself right into the center of the European intellectual world, and became, like the theatre of Max Reinhardt in Berlin, and L'Oeuvre Theatre of Antoine in Paris, one of the leading theatres of the intellectual movements of the time.\n\nFrom the personality of Gustav Mahler, who was a demonic man, streams of nervous energy emanated, and pervaded stage, orchestra and audience at the Vienna Opera. Long before Mahler appeared in the orchestra pit, the audience became excited. When the house grew dark, the small man with sharply-chiseled features, pale and ascetic-looking, literally rushed to the conductor's desk. His conducting was striking enough in his first years of activity in Vienna. He would let his baton shoot forward suddenly, like the tongue of a poisonous serpent. With his right hand, he seemed to pull the music out of the orchestra as out of the bottom of a chest of drawers. He would let his stinging glance loose upon a musician who was seated far away from him, and the man would quail. Giving a cue, he would look in one direction, at the same time pointing his baton in another. He would stare at the stage and make imploring gestures at the singers. He would leap from his conductor's chair as if he had been stung. Mahler was always in full movement like a blazing flame. Later he became calmer. Evidently he controlled himself which only augmented his inner tension.\n\nMahler was conscious of the extreme tension which emanated from him into the theatre. He once told me, during one of our first talks, \"Believe me, people only realize what I am when I am gone. Then it is as if a storm had broken over the theatre.\" The Viennese public, directly after Mahler's arrival, realized the additional artistic energy which had been bestowed on it in such a personality. In the first year of his direction, the operas of Mozart were performed ten times more often than in other seasons. Wagner had twenty evenings more than in the previous year. Even smaller-scale, lighter operas like Lortzing's \"Czar und Zimmermann\" played to sold-out houses. Mahler worked according to a broad programme, on the renovation of the ensemble and the repertory. He told me shortly after his arrival that within the next ten years he wanted to revive and re-stage \"Fidelio\" and the works of Wagner, Mozart, Weber, Lortzing, Gluck; and to put these works together in the form of a great cycle. Then, this accomplished, he wanted to leave the opera.\n\nRarely has any theatre operated according to a larger-scale plan, and in the first half of Mahler's directorship, what was accomplished corresponded exactly with the ideal picture set forth at the beginning. The intensity with which Mahler attacked the re-studying of a classic work was unusual. Of every work which he prepared and performed, Mahler said: \"This is the greatest opera which has ever been written!\" He was filled with solemn enthusiasm after he had brought out a new work. Stage and orchestra, decor and singing had to form a complete artistic unity according to his ideas. Each opera was a dramatic work of art. The Wagnerian conception of a comprehensive art work was applied by Mahler to every opera.\n\nFor this purpose, Mahler developed a new ensemble. The great Viennese heroic singers of the Wagner epoch had grown old. Winkelmann, Reichmann and Materna had to be replaced by younger forces, and Mahler found the noble tenor Schmedes and young Slezak. In Mildenburg, he had a modern Wagner tragedienne, in Weidemann and Deumuth great baritones. He had Gutheil-Schoder as an interesting singer for lighter roles. Hers was a realistic talent of the first rank for tragic as well as comic parts. In Kurz he had found a new coloratura. All these singers blindly followed the dictates of the genial musician who had formed and trained them. Mahler clung to this ensemble with great love. I can remember how disturbed he was, when he had Gutheil-Schoder sing the Eva in \"Meistersinger,\" and read the unfavorable criticisms about her. He sat sunken down behind the mountain of newspapers, like a wicked dwarf, and nervously chewed his finger-nails\u2014something which he was accustomed to do in moments of stress.\n\nOne time, to Mahler's horror, his tenor Schmedes walked into a smoke-filled coffee-house where Mahler was sitting, holding a fat cigar which emitted powerful blasts of smoke. Mahler used all his wiles to get his Siegfried out of the place. He asked him to go, he begged him, and when nothing helped, he flattered him and told Schmedes how the audience only wanted to hear him, and described how horribly disappointed people would be if he had to cancel and someone else replaced him. Schmedes hung on Mahler's every word while so much flattery was being poured over him, and finally left the smoky place, swollen with pride like a freshly inflated balloon. Surely he must have sung especially well that night. However, Mahler could also be very hard and unjust toward singers when he felt any opposition from them.\n\nMahler was a modern musician with the temperament of his period. As such, he introduced a new Mozart style to the Vienna Opera which was taken up by young musicians and carried to all the German opera houses. Mahler's Mozart performances had spirit and dramatic liveliness, grace and wit. The orchestra had rich nuances, polished like Venetian glass, and worked out in detail like fine lace. Thus, through the artistic mind of an artist of the Ibsen period, Mozart's music achieved a new popularity. \"The Marriage of Figaro\" and \"Cosi fan Tutte\" shone like jewels. \"The Magic Flute,\" which became a great box-office attraction, gleamed with intellectual sparkle. As in all people of high intelligence, there was in Mahler a kind of child-like merryness which came as a release after work. When in \"The Magic Flute,\" the animals appeared\u2014lion, hippopotamus, rabbit and monkeys\u2014attracted by Tamino's flute-playing, none of the opera audience laughed more than Mahler at his conductor's desk. He fairly shook with laughter. All comic opera music, whether it was Lortzing, or French music like Boieldieu's \"La Dame Blanche,\" found in Mahler a magnificent interpreter, for no one understood liveliness better than this man who also knew demons in his breast.\n\nWith Mahler, the tragic reigned in his interpretation of \"Tristan.\" There are several possible interpretations of this Wagnerian work. Hans Richter conducted it as a symphony with tremendous climaxes and a unified line. Felix Mottl, whom Toscanini specially praised to me, produced the most nature-like sound when working toward his climaxes, and the richest fulness in the music. Strauss conducted the music of the second act\u2014Strauss-like\u2014as an ecstatic hymn, as he did the love music of \"Heros Life.\" Mahler conducted the same music with modern temperament, making vibrant, intense, hysterical climaxes of the period of Charcot and Freud.\n\nAt the performance of \"Tristan\" in 1903, which Mahler arranged on the anniversary of Wagner's death, for the first time stage decor designed by Professor Roller was used. Roller, that artist who translated Mahler's visions into painting, was one of the principal leaders of the group of modern Viennese painters who, in 1897, banded together in the so-called \"Secession.\" According to the pattern of Claude Monet, Roller painted the same landscape in the changing lights of the different months of the year, and studied the influence of light on color to an almost scientific degree. His new \"Tristan\" sets, too, were studies in light and color, the first act a study in orange, the second in purple, the third in grey. In 1905, with Roller, Mahler began work on the revival and new staging of the \"Ring of the Nibelung,\" with the new \"Rheingold.\" The same year saw the new Roller mounting of Mozart's \"Don Giovanni,\" with four grey towers as frame for the stage, which facilitated changes of scenery. When Roller designed the new \"Don Giovanni\" sets, he was strongly influenced by the ideas of Gordon Craig. Modern ideas of staging and painting found a most forceful expression in Roller's stage.\n\nMahler at the desk\n\nThrough the combined efforts of Mahler and Roller, the Vienna Opera had suddenly become the leading opera house of Europe. One may safely say that during the first years of Mahler's direction, the highest level was achieved which can be attained in an opera house where performances are given daily, in the perfection of these performances, in artistic spirit, and in the unity of musical and scenic faculties.\n\nThe great successes of Mahler led to Hans Richter's departure from the theatre. Mahler and Richter were direct opposites. Physically, Richter was tall and broad-shouldered, Mahler small and slender. They belonged to different generations, Richter the pathetic and powerful, Mahler the intellectual and nervous. Richter had a great nature, Mahler a great intelligence. Richter was a blond German with Wotan's eye, Mahler a dark Jewish type. Mahler did everything in his power not to hurt Richter, but he could not do away with their differences. The enthusiasm which swept about Mahler after each new revival gave Richter the feeling that his work in Vienna was at an end. He had brought Wagner's works of art to a whole generation of Viennese opera-goers. For this he had fought, for a large part of the public had wanted nothing to do with Wagner's music dramas. When I started to go to the Vienna opera, the house was half empty for \"Tristan und Isolde,\" in spite of Winkelmann and Materna. \"Siegfried,\" too, at that time had not as yet any public, and \"Goetterdaemmerung\" could only be given with cuts. At performances of \"Meistersinger,\" half of the audience in the orchestra rushed away after the third act quintet. Richter had been Wagner's apostle. That fact gave his conducting of Wagner the inner greatness, the elan and morality it possessed. To Richter, measured by his own achievement, Mahler's Wagner must have seemed less great than it was, for Mahler no longer had to fight; while Richter and the other enthusiasts who had surrounded Wagner, had.\n\nI observed the difference between the two conductors when Mahler conducted \"Walkyrie.\" After the performance, Mahler rushed to me and asked: \"Well, how was it?\" I answered: \"A marvelous performance, but the tempi at the end seemed too fast to me.\" Mahler frowned, and I quickly noted that I had committed a faux-pas. Artists, like cooks, when they come from the kitchen, want to hear praise and not criticism. Therefore I added, in order to put him in better humor, that probably I was used to the broad tempi of Hans Richter. Mahler became furious and said: \"Stop that! Richter has no idea about tempi!\" Now it was my turn to be angry, and I reported that Richter had conducted the first \"Walkyrie\" in Bayreuth under Wagner, and therefore must know the Wagnerian tempi. But Mahler replied curtly: \"Maybe he knew the right tempi then. Since then he has forgotten them.\" I realized it was no use talking to Mahler at this moment, and went on my way. Walking home, I saw Hans Richter ahead of me. Joining him, I asked him whether he had heard the \"Walkyrie\" performance, and when he replied that he had, I asked him whether he had not thought the tempi at the end had been too fast. He answered: \"What of the music, at the end, is \"Fire Spell\"\u2014that was brought out very well by Mahler. But that this end is a transfiguration (and he put his hand on his heart) and a painful resignation\u2014of that Mahler has no idea!\"\n\nWhen the news of Hans Richter's resignation from the Vienna Opera became known, there were demonstrations by the Wagnerian party at the opera whose sound took on an anti-Semitic character. Just during those years when Mahler was opera director, the anti-Semitic party which Dr. Karl Lueger had founded became successful in the elections of the Municipal Council. Dr. Lueger, a Viennese with the wit of a folk-singer, became mayor of Vienna in 1897 after Emperor Franz Josef had twice rejected him because he hated demagogues. Gustav Mahler, as a Jew\u2014even though he had been baptized, as required by the Spanish Court Ceremonial of all holders of court positions\u2014had been the object of anti-Semitic attacks. At such moments, Mahler felt as a Jew. Once he told me that an artist who is a Jew had to achieve twice as much as another who is not, just as a swimmer who has shorter arms has to make double efforts in order to gain his goal.\n\nA large part of the opera orchestra was connected with the anti-Semitic party, so that Mahler, after a short time, gave up the direction of the Philharmonic concerts. In the newspapers, too, hostilities began. Disgruntled opera singers sought and found newspaper connections. Incidents which easily occur in any theatrical organization were published and widely spread abroad, by reporters. Mahler once said to me: \"I am not afraid of the opera critics. I'm only afraid of opera reporters.\" However all this unpleasantness for a long time had no influence on Mahler's position. The high court officials who were his superiors lent their support and confidence to the successful and interesting Opera Director. The opera critics of the important newspapers with whom Mahler had social relations and with whom, at tea, he developed and elaborated his plans, also stood by him. The opera public celebrated him.\n\nNot Gustav Mahler, the Opera Director, but Gustav Mahler, the composer, was the one who, in a fit of anger, gave up his Vienna position. The great symphonies which Mahler composed during his Vienna period, the fifth to the seventh, show that the inner excitement in his soul had increased during this time, and that he sensed as tragic the collision of the ego with the world. External events could increase this excitement, but it lay in the blood of this artist who, like Strindberg, belonged to the tragic modern type. A brother of Mahler's, who also was a musician, died a suicide. Mahler freed himself from his inner tensions, from time to time, through his creative work. Then material for conflict again piled up in his soul. Occasionally there were explosions. Such an explosion was his offering his resignation in 1907.\n\nGustav Mahler was conscious of his importance as a composer. \"In forty or fifty years, they will play my symphonies at orchestral concerts as they now play Beethoven's,\" Mahler once told me. These symphonies he composed in the summer, during opera vacations, mostly at the Woerthersee. There near his summer house, he had a hut in the forest for composing, where he could be alone and undisturbed. Once I asked him why, apart from songs which are pre-studies to his symphonies, he wrote only symphonies and not, like many other composers, chamber-music and independent choral music as well. Mahler answered: \"I have time to compose only in summer. During this short holiday, I have to write large works if I want to go down into posterity.\" But that is not the only explanation for Mahler's urge to symphony, and to colossal creations at that. There is no doubt that through his philosophic inclinations, Mahler was driven to symphony. In his music, he wanted to solve the riddles of life and death, of nature and humanity, of destiny and the individual. He had the nature of a Faust who wanted to encompass the world's spirit. From that came the great conception of his symphonies, from that the unredeemed, Ahasueristic, deep excitement of his personality.\n\nWith the second symphony, which philosophized about death and resurrection, Mahler in 1899 gave proof to the Viennese public for the first time that he was endeavoring to solve the riddle of the cosmos. In 1901 he performed his \"Klagende Lied\" (\"Plaintive Song\"), a work of his youth; in 1902, the Fourth Symphony with the heavenly music of the angels, in 1904 the Third Symphony, in 1905 the Fifth. The success of the Second Symphony was unquestionable. Later on, an opposition appeared, as it usually did with new works in Vienna. The holy saints of the classic arts were invoked to aid against the modern devils. The Viennese humorists made their jokes about the modern musicians as well as about the painters and architects. And a composer of bad operettas wrote daily in a widely read Vienna newspaper that Gustav Mahler had no talent and no musical invention.\n\nNevertheless Mahler continued his magnificent work during the uproar of the attacks. Each revival was still a festival, as was the last, \"Iphigenia in Aulis\" with its sublime Greek style. But in his heart, Mahler, who fought for his symphonic work, was stirred up, and in his excitement, he handed in his resignation. It was certainly a sudden, impulsive action which Mahler soon regretted and would have liked to take back. But Mahler's superiors, who for some time had tried to change his mind, had become embittered, by this time, and there was no longer any turning back.\n\nWhat Mahler left behind him in the Vienna Opera was a repertory, prepared in the style of the new era, a new ensemble, Roller as principal scenic designer, and two conductors whom he had engaged and trained as his helpers\u2014Franz Schalk and Bruno Walter. Moreover, he left the memory of the Vienna Opera's most splendid artistic period which became the yardstick against which everything which was achieved later was measured.\n\nMahler's successor was Felix von Weingartner, who as an elegant, amiable, smiling, worldly musician was a kind of antithesis of the demonic Mahler. When after three years Weingartner resigned, the operatic businessman, Hans Gregor, succeeded him. Weingartner left no lasting traces behind him upon the opera, and Gregor was merely a businessman. This time was far removed from that when Mahler elevated the opera to a masterpiece of his high, striving spirit. The greatest artist during this period who came to the Vienna Opera was Richard Strauss, who always came back as guest when one of his new works was performed. Strauss conducted \"Elektra\" especially often, and with great pleasure. One time, years later, after he had conducted \"Elektra\" with particular elan, he came home and called to me: \"I am still the only one who can conduct Eyektra.\" He conducted this work with very brief movements, and appeared, in the first scene, unintentionally to collect the brooks of music until they flowed together into a river. Everything melodious in the following scenes was broadened out in a noble manner. The breaking-out of the sound in the Elektra-Orestes scene, when after the recognition, the keys rush together, was held and welded by a strong hand, and at the end, the dance rhythms were mightily formed into one bow. At the climaxes, the usually self-controlled conductor sprang excitedly from his chair. The orchestra was always kept subordinate to the stage and the singing, no matter how strongly its naturalistic painting and vivid coloring flared up. One time when I visited Strauss in Garmisch, he was telling about the Italian performance of his opera which he recently had seen in Milan, and went on: \"In Milan, I realized for the first time that my 'Elektra' is an Italian opera. The singers can stand on the ramp quite calmly and all they have to do is sing\u2014then there is everything there that the work requires.\" I heard \"Elektra\" at least twenty times under Strauss's direction. There was always the most perfect equality between the singing and the orchestra.\n\nAt this time, Strauss had advanced to the point where he was the recognized leader and greatest exponent of modern music. Despite this fact, his most Viennese work, which brings to the stage the old Vienna of the period of Empress Maria Theresia, \"Der Rosenkavalier,\" had only a very moderate success at its Vienna premiere. After the second act, there was scarcely any applause. Strauss had not expected that. At the dress-rehearsal he sat in the first row and enjoyed the antics of the excellent Ochs von Lerchenau, to whom he often called \"bravo\"; and he beamed when the horns rang out their waltz coloraturas in a brilliant virtuoso manner. And now the work was presented before a cold audience who considered the many waltzes in this score unworthy of being called opera music. Strauss was greatly distressed, and finally said to me, sadly: \"If only the people knew how hard it is to compose such waltzes!\"\n\nIn \"Rosenkavalier,\" to my astonishment, I discovered a place which recalled to my mind a little experience I had had with Strauss in Vienna. During the time when he was working on \"Rosenkavalier,\" I went walking with him in the vicinity of Hietzing. From an inn came the sound of dance music, and Strauss stood there for a long time, listening with interest. Finally he said, laughing: \"Do you hear the false basses?\" I said yes, since the false basses, some of which were being drawn from their instruments by the contrabass players, were obvious enough. Imagine my surprise when in the inn-music of \"Rosenkavalier's\" third act, I heard the jolly false bass notes! They are a small, characteristic trait in the Strauss picture of Vienna, and they intensify my delight at every performance of the work as reminiscences of my walk with Strauss.\n\nThe modern music of the Richard Strauss period was not rejected by the Vienna Opera, no matter what difficulties the Viennese public made every time a new work was performed. Still under Mahler's direction, Hans Pfitzner's \"Rose vom Liebesgarten,\" an opera with pure, romantic fairy-tale atmosphere, mystic solemnity with gloriously-colored nature pictures, was given for the first time. Under Gregor's direction, Debussy's \"Pelleas and Melisande\" was performed, with its dreamy scenes\u2014a modern, sad ballad in fragrant, floating, vibrating tones. Under the same director, Schreker's \"Das Spielwerk und die Prinzessin\" (\"The Toy and the Princess\"). a symbolic opera and a glowing color fantasy, had its premiere. The music of the new era had gotten a foot-hold at the Vienna Opera.\n\nDuring the time when Gustav Mahler was Opera Director, Richard Strauss was writing his finest operatic and symphonic works. Debussy set people's nerves tingling with his impressionistic tone poems which had the artistic delicacy, the tenderness and refinement of Japanese wood-cuts; Hans Pfitzner bound the atmosphere of \"Tristan\" and \"Parsifal\" with that of the German fairy-tale and saga; a new generation of musicians was growing up in Vienna.\n\nThe most mature of the Viennese musicians was Alexander von Zemlinsky who, while still a pupil at the Vienna Conservatory, had attracted Brahms' attention by composing a colorful symphony, and whose opera, \"Once Upon a Time\"\u2014a tender, spiritual work\u2014had been performed by Mahler at the Vienna Opera. Zemlinsky belonged to my circle of friends who, of evenings, used to meet over black coffee at a coffee-house near the old Burgtheater. This coffee-house\u2014the Caf\u00e9 Griensteidel\u2014was the headquarters of literature in Vienna. In its three low rooms could be seen all the men who, as poets or as critics, played their part in the city's intellectual life. The coffee-houses of Vienna still had their special types of patron, like their special kind of coffee. At the Caf\u00e9 Fenstergucker, for instance, one found the fashionable Viennese society. At Caf\u00e9 Imperial, the owners of stables and race-horses held forth; at Caf\u00e9 Pucher, the ministers and high state officials from the ministries situated nearby; at Caf\u00e9 Central, the journalists, the chess-players and the intellectuals, among them for a few years a fugitive from Russia, a Dr. Bornstein, who later, as Trotsky, made world history.\n\nAt the Caf\u00e9 Griensteidel, Hermann Bahr, the leader of modern criticism, gave his fiery talks every evening, with a Virginia cigar in his hand and a brown lock on his forehead. Arthur Schnitzler came there, elegant and melancholy. Hugo von Hofmannsthal was there, too\u2014still almost a boy and already famous, drawling in the tired aristocratic Viennese dialect. A strange man who wore large horn-rimmed glasses and had the drooping mustaches of a walrus\u2014Peter Altenberg\u2014had just published his first sketches which glorified women's soft, tender legs and innocent children's eyes. A small hunchback, Karl Kraus, was the greatest satirist of the time.\n\nSchoenberg\n\nWe young musicians sat in a corner. Alexander von Zemlinsky, although he was only twenty-eight at the time, in the maturity of his judgment, had a visible superiority over the rest of the musicians. Another young musician of our circle was Arthur Bodansky, who was then a violinist in the opera orchestra, and shortly after, became a conductor at an operetta theatre. Zemlinsky brought with him a young man who was a pupil of his. He had the face of a Pierrot, and tossed pointed and stinging paradoxes about. One of them was the assertion that a C Major triad is an effect which can only be applied rarely, and then after most careful preparation. In this sentence, all classic harmony was turned upside down. The young man was Arnold Sch\u00f6nberg. A short time later, one of his first works was performed at a concert\u2014a string quartet in D Minor in classic style influenced by Brahms and Schumann, with moonlight sounds in the slow movement which were quite novel. I had, meanwhile, been promoted to the post of music critic of a weekly newspaper, and wrote an article praising that quartet; and I closed with the prophetic words: \"One should remember his name. It is Arnold Sch\u00f6nberg.\" Of course, Sch\u00f6nberg himself took good care that his name should be remembered. Soon after the performance of the quartet, he brought me a sextet, \"Verklaerte Nacht\" (\"Transfigured Night\"). The music sounded new, the harmonies were unusual. Since I did not trust my judgment I gave the score to Gustav Mahler, at that time Opera Director. Mahler wavered in his opinion, as I had done, and asked Arnold Rose to play the sextet with his musicians, in Mahler's office. He invited me to come to this private performance, and we both were enthusiastic. Mahler said to Rose: \"You must play that!\" and Rose played it at the next of his chamber-music evenings, to the great displeasure of the Viennese public who hissed loudly. This hissing increased in volume in proportion to Sch\u00f6nberg's boldness, in Vienna and in other cities.\n\nThe age of classic harmony which had been founded in Vienna came to an end there. The world had changed. Political, economic and social problems rose, like dark storm-clouds, on the sky. Vienna, the gay, elegant town, saw shadows falling over the Ringstrasse which seemed to belong exclusively to the rich, when in 1905, thousands of workers paraded over it and demanded equal voting rights. The people of Vienna\u2014the \"common men\"\u2014were not satisfied any longer to stop at the curb of the street and cry \"Heil\" when a court carriage passed. They demanded their social and political rights. The Viennese small citizenry united in a big party, with anti-Semitic slogans, which took possession of the Town Hall in 1895. Dr. Lueger, its leader, raged against capitalists, Jews and the Stock Exchange. He succeeded in frightening the court, the high clergy and wealthy society. The Social Democratic workers strengthened their organizations, too. They had a great leader in Dr. Victor Adler who, since he stuttered, gained power by purely intellectual means. Reason, idealism, calm superiority were his weapons, and even the Emperor esteemed his wisdom which was created by passion. Thus in Vienna, the old town of the court and the aristocracy, the masses pressed restlessly upwards.\nPopular music-Johann Strauss\n\nThe political, social and intellectual changes which became apparent in Vienna around 1900 were also of the greatest importance to musical life.\n\nUp to that time, the small Viennese citizen contented himself with simple folk-like music. He stood, so to speak, on the outside of the festive music life of the great city. His life took place in the low houses of the suburbs where the women washed their laundry in the small courtyards and the children played in the streets. In the suburban inns, popular musicians played, mainly harpists and violinists. Such folk-musicians had played back in mediaeval Vienna, and only in 1889, the last of the harpists, Magdalena Hagenauer, died.\n\nWhat these folk-musicians played was music of the Austrian Alps. The Austrian mountain dances were called \"Laendler\" (country-music), and turned slowly in three-quarter time. Lanner, the waltz-composer, still wrote \"Laendlers.\" Johann Strauss, the son, still wrote waltzes in \"Laendler style\"\u2014his opus 247. His waltz \"Grillenf\u00e4nger\" (Whim Catcher) bears the sub-title \"In the style of Laendler.\" Suburban inn music was accompanied by whistling and hand-clapping. There were yodelers, too. Old Vienna knew famous lady-yodelers, such as \"Tanzerl Leni\" (Little dancing Leni) who died in 1870, or \"Juden Lisl\" (Jewish Lisa) who sang at the inn \"Zum Mondschein\" in Matzleinsdorf, one of the southern suburbs of Vienna. In gay Vienna of the '70's, the aristocrats amused themselves with that kind of suburban folk-music. They had their favorite yodelers and whistlers. One of these latter, whose nickname was \"Baron Jean,\" whistled at all the fashionable parties of Vienna around 1880. I knew him myself.\n\nMost of this folk-music came on the ships which traveled from Linz to Vienna. There would be Linz musicians aboard, either two fiddlers, a guitarist and a double-bass player, or one fiddler, one clarinettist and a double-bass. During the trip, they played their country dance music, and its sound hovered over the Danube and the landscape. Arriving in Vienna, they played in the inns along the Danube's banks\u2014\"The White Lamb,\" \"The White Cock,\" \"The Golden Bear\" or \"The Blue Star.\" In one of these Danube inns, Johann Strauss's father was born. There he heard the Laendler of Upper Austria, as a boy, and later on he ennobled it to the rank of waltz.\n\nSome of these Linz musicians would wander to the Spittelberg, where the inns and taverns of Vienna's high life were situated, and where there was music day and night. Not far from these inns is the house where Lanner, Vienna's first waltz composer, was born.\n\nMusic and song were not absent, either, from the small wine gardens of the Vienna environs. In the beginning of the nineteenth century, quartets were the principal performers here. The most famous of these Viennese \"wine-quartets\" of Emperor Franz Josef's time were the \"Schrammeln.\" Viennese painters often used such quartets as subjects for their paintings. The motion pictures, too, introduced them for local color in their sentimental Viennese films. Members of the quartet used to sit on a small platform in the winegarden. On the table stood wine glasses on which they propped their music. Their Viennese faces would become redder and redder in the course of the evening and their music warmer. The violins would sound sweet and soft. A grateful Vienna has not forgotten these musicians. In Ottakring, one of the suburbs, stands a fountain. At each of its four corners is the figure of a musician, round bowler hat with a flat brim on his head, with violins, harmonica and guitar in hand. When older Viennese pass this fountain, they smile appreciatively and understandingly.\n\nNumerous popular songs were written and composed for the so-called \"folk-singers.\" When the oldest of these, Johann Baptist Moser, sang in 1828, the fiacres of the Viennese aristocrats drew up in long lines in front of the inn where he was performing. The same thing happened with Johann F\u00fcrst, whom Princess Metternich introduced into the palaces of the nobility. There were also famous women folk-singers like Antonia Mansfeld, who was tall, slender, and piquant when she sang her songs. She was revered by society as well as by the people, and when the Emperor Napoleon III came to Vienna in 1873, she was called to sing for him. Another songstress, \"Fiacre Milly\" (the name of this 100 per cent Viennese woman was Czech\u2014Turrecek), was embodied in an opera of Richard Strauss's\u2014\"Arabella\"\u2014and in the second act she sings her yodel song.\n\nMany of the songs which were composed for the folk-singers became, in turn, folk-music. Vienna had no \"Tin Pan Alley.\" Simple musicians wrote these songs for the folk-singers, who paid them a few gulden as recompense. The most popular of them were written by Karl Lorens (\"Ja, da fahr'n wir halt nach Nussdorf 'raus\"; \"So lang der alte Steffel\"2; \"Menschen, Menschen san mir Alle\"3; \"Die Linzer Buben\"4); by Alexander Krakauer, \"Du guter Himmelsvater\" 5; by Johann Sioli, \"Der alte Draher\"6; by Siczinsky, \"Wien, Wien nur du allein\"7; and Ludwig Gruber (\"Es wird ein Wein sein\"8; \"Mei Mutter ist a Wienerin\"9). Old and young, noble and commoner sang these songs. The Viennese was touched when he sang the praises of his town, the tower of St. Stephen's, Viennese women and wine. As long as there is a Vienna, those songs will be sung.\n\nOf all the songs of Emperor Franz Josef's time, none became so popular as the Fiacre Song. This song is as characteristic of the period as the fiacre itself. On every corner of the city stood the fiacres\u2014distinguished carriages with two horses whose owner was a great, great man, proud of his art of holding the reins and whip lightly in his hand. On his head, he wore the bowler hat of the aristocrat when he sat on the coachbox. Black jacket and striped trousers completed his costume. He kept a meerschaum pipe in his mouth. When a stroller passed, he would tip his hat gently and ask casually: \"Shall we go, your Worship?\" If one said yes, then the \"waterer\" who had watered the horses, and the \"coach-door opener,\" both of whom belonged to the entourage, would appear and receive their respective tips. These preliminaries completed, the driver would click his tongue and the horses began trotting. As a man of high standing, the fiacre-driver was discreet. If one drove out to the surrounding districts of Vienna with a lady, one only had to say the word \"Porzellan\" (\"China ware\"), and the driver knew that he must drive carefully in order not to hurt the fragile cart-load which was resting happily in the arms of his passenger.\n\nThis was the fiacre driver which a genuine Viennese, Gustav Pick, glorified in a song which became a kind of Viennese folk-hymn. Pick, who like the old-fashioned poets, composed both text and music of his song, like so many other \"genuine\" Viennese came to Vienna from Hungary. He enjoyed the gay life in Vienna, and loved to go driving in a fiacre. Where he was to find the money for such excursions did not worry him very much, so long as he had friends. From the rhythm of trotting fiacre-horses he evolved the song in which a fiacre-driver describes his art; and the trotting rhythm changes into a waltz when the fiacre-driver sings: \"Anyone can become a coachman, but only a Viennese knows how to drive a fiacre.\" The Tirtaeus of the Viennese fiacre-drivers whistled the melody to a musician, since he himself knew nothing about notes; and the musician wrote down the easy song, which became immortal.\n\nIn 1885, the new song was sung in Vienna for the first time. This was at a charity festival in the great Rotunda of the Prater. Girardi, the most popular actor in all Vienna, appeared, seated on the coach-box of a fiacre, and drove right into the Rotunda. When he reached the center of the gigantic enclosure, he brought his horses to a halt with a jerk of the bridle, stood up on the seat, flourished his hat in greeting and sang the new song which swept through the city like wildfire and is popular even to-day. Not once, even when the fiacre-driver was replaced by the automobile chauffeur, did the song become old. After the war, it took on a new meaning. It became the ballad of happy Old Vienna, where life was as easy and carefree as in a fairy-tale; and the fiacre-driver, not the old Emperor, became the symbolic figure of this golden age.\n\nOnly two other songs achieved a popularity similar to that of the Fiacre Song. One was the \"Deutschmeister Lied\" (\"Teutonic Order Song\"). The \"Deutschmeister Regiment\" (Regiment of the Teutonic Order) was the real Vienna regiment in which all native Viennese served, and whose blue lapels belonged to the color-picture of Vienna. This regiment was considered especially plucky and dashing. Viennese wit and frivolity were at home in its barracks. A Viennese by the name of Tureck who was an official of the state printing-works, belonged to this regiment. He was an under-officer and sat in his office very bored. His duty was to stamp the passes which permitted the soldiers to enter the barracks after the retreat. This was not a very entertaining business, and to distract himself, Tureck began to hum as he stamped the seal of the regiment again and again on the pass where one could read the regiment's full title: \"K. und K. Infanterieregiment Hoch und Deutschmeister Numero 4\" (\"Imperial and Royal Infantry Regiment of the Teutonic Order Number 4\"). From the rhythm of these words and the methodical stamping a melody began to form itself\u2014the melody of a march song which Tureck sang to his comrades when he marched with the other soldiers to the drill grounds. Soon the whole town was singing it, because wherever the soldiers went, they sang it.\n\nDuring the 17th century, when Austrian soldiers marched to Serbia under Prince Eugene, the famous \"Prince Eugene Song\" developed in similar fashion: one of the soldiers sang it, and the others joined in the dashing melody. In the same manner, all true folk-songs came into being. Verses flew from somewhere into the mind of a man of the people. These verses began to sound. A poem, a melody arose and took wings.\n\nThe last Viennese folk-song dates from the year 1918\u2014the \"Grinzing Song\" of Ralph Benatzky. An excursion to the wine gardens in Grinzing was responsible for the song, and there it was sung first. (\"Ich moecht' wieder einmal in Grinzing sein, beim Wein\"\u2014\"I would love to be in Grinzing again, at wine). From here it traveled all through the city, for the desire to be in Grinzing drinking wine was a lively one in the soul of every Viennese. In gratitude for the song, a memorial plaque, with Benatzky's portrait, was put up near the Grinzing wine garden which the song glorified.\n\nAll this music belonged to the people of Vienna. \"Art\" music for the common man was available only in restaurant gardens where military bands or civilian orchestras gave concerts. These circumstances began to change about 1900. Vienna had grown enormously. In 1892 the suburbs were incorporated into the city, and the small townspeople of the suburbs became \"big-city\" Viennese. They also developed into a political power and wanted their say in Vienna's musical life. They demanded more and more good music for low prices. The fruitful work of the \"Volksbildungverein\" (\"Society for the Education of the People\") and the cultural organizations of the workers contributed to the fostering of musical activity. Musical Vienna, like the town itself, was growing.\n\nIn 1898, the inhabitants of the Vienna suburbs decided to build a second national opera house in the outskirts of the city, which should make the masterworks of the opera accessible to people of modest means at modest prices. Rainer Simons, the Director of this theatre who named it the \"Volksoper\" (the People's Opera), was a very clever theatre expert. From small beginnings, he brought the theatre to great heights. Simons had a special feeling for young talent which gave promise of a great future. A long line of singers became famous in this theatre, while being badly-paid but well-directed artistically. None of these singers attained such fame as Jeritza, whom Simons discovered in a small operetta theatre at Ischl. I remember her first performances. She was a glowing, laughing blonde with a Czech snub-nose, long hands and legs which gave her quite some difficulty on the stage. But her voice was sensuous and brilliant, and Simons knew exactly how to make talent of this sort free and uninhibited.\n\nWhen I asked Simons what he had done to make Jeritza lose her clumsiness on the stage, he told me that he had had the impression that the young singer was being dominated entirely by her husband. Her inhibitions were apparent when he gave Jeritza her first great role\u2014the feminine lead in Giordano's \"Siberia.\" During rehearsals, in the beginning, she had no idea of what to do during the main scene, in which a girl who has been deserted by her lover, a Russian officer, meets him again by chance, and whips him. Director Simons during this scene stood in the wings, and when Jeritza raised her whip, called to her: \"Imagine that it's your husband!\" Jeritza flew into a fury and beat the unfortunate tenor so passionately that the scene became one of her first great realistic interpretations. It was a tremendous success for Jeritza. Soon after, she was engaged at the Court Opera where she became a sensational singer of naturalistic roles, and from where she journeyed through the world with her wonderful \"Tosca.\"\n\nWhen, in 1913, Wagner's works became public domain, the Volksoper made them available to the public in good performances. Richard Strauss conducted \"Feuersnot\" and \"Salome\" in this theatre; Battistini sang there as a guest. Zemlinsky was the first conductor. Later Felix von Weingartner became Director. Even beside the Opera of Gustav Mahler, the Volksoper could be esteemed and considered a very serious opera theatre.\n\nIn 1900, two new symphony orchestras were founded\u2014the \"Konzertvereinsorchester\" (Concert Society Orchestra) and the \"Tonk\u00fcnstler Orchester,\" both of which attracted a people's public, by virtue of their low admission prices. At these concerts, also, famous conductors of the present day conducted\u2014Weingartner, Strauss, Bruno Walter, Furtw\u00e4ngler and others. Bruckner's Ninth Symphony, Richard Strauss's \"Sinfonia Domestica,\" and Mahler's \"Das Lied von der Erde\" (Song of the Earth) were given their first performances in Vienna by these orchestras. The Vienna Social Democratic workers also had their own symphony concerts\u2014\"the Workers' Symphony Concerts.\"\n\nThus the tree of musical Vienna put forth new shoots. Enjoyment and understanding of serious music did not remain confined to the wealthy and the aristocratic. Great new layers of the population had entered the cultural circle of music. When Emperor Franz Josef began his reign, there was no such thing as a large public concert in Vienna. There were neither regular orchestral concerts nor choral concerts, and just a very few virtuoso recitals to be heard in Vienna. But after the Emperor died there were in Vienna, in addition to two opera houses, three large orchestras, two large choral societies, two large concert houses with five halls where concerts took place every day. The old musical city had broadened out, and the radio, which conveyed serious and popular music to the smallest house in Vienna, increased the democratizing of musical enjoyment to an extent which previously had been unimaginable. Music came into the houses like water and electric light. Opera performances, symphony concerts and chamber-music were accessible to the poorest people, like the public parks in Vienna which belonged to everyone, and whose lilacs and roses had an equal fragrance for all.\n\nThus around 1900, in Vienna, there arose a broad popular society around the ancient courtly and feudal musical society. They were separate groups. With the courtly-feudal society, which was destroyed in the war years 1914-1918, ended the three-hundred year development period of Vienna's past. With popular society, a new development began. Their paths ran parallel for a time, then crossed and were borne on by the political movements of the present. And there was a man in Vienna who united both societies. What the old Emperor had not been successful in accomplishing\u2014the harmonization of the social antithesis\u2014a musician who was a genius succeeded in doing. This musician was Johann Strauss. His music was just as much music for the high-born as for the man in the street. It was perfect art-music, and at the same time, perfect folk-music. His waltzes were highly esteemed by Richard Wagner, Hector Berlioz, Franz Liszt and Johannes Brahms, and loved by all the people.\n\nThe two great forerunners of Strauss, the son: Lanner and Father Strauss, also grew with Vienna. Then it was small, with many dreamy, quiet corners, with ladies in crinolines, and clean-shaven men in buttoned black coats and high hats. This little Vienna, under the reign of a silly emperor, led a comfortable existence. A pound of beef cost only twelve kreuzer and a bottle of wine just as little. People called this period of Vienna's history the \"Backhendlzeitalter\" (\"The Age of the Breaded Chicken\"), because even the poorest could eat the cheap chickens in his small restaurant. Dance halls were in every nook and corner. Dangerous modern intellectual ideas were not allowed to penetrate the city. The police and censor saw to that. Pleasure and sensuality seemed less dangerous than thinking, and the new Viennese waltz developed from sensuality. When Lanner began to play his romantic waltzes, he played with two violins and a guitar. Later his orchestra grew to five musicians, and afterwards it become larger. Strauss, the father, played at \"Sperl's\" with a small orchestra. All that changed when Johann Strauss, the son, wrote his waltzes. Hanslick was still complaining about the Wagnerian noise and the Wagnerian dissonances of the Strauss waltzes, and the old Viennese sentimentally cherished the quiet, feeling old waltz music of the Lanner period. Johann Strauss, though, wrote waltzes for a new epoch in which Vienna was a metropolis. He was fiery and brilliant, powerful and elaborate. He raised his violin and played like a teller of fairy tales\u2014stories of new Vienna, the Vienna woods and the Blue Danube. He grew up with Vienna, but his Vienna belonged to the whole world. His waltzes were no longer like those of Lanner and of his father still, romantic dreams\u2014but a hymn to Vienna which sounded throughout the world.\n\nWhen I first met Strauss, he was leading the life of a well-to-do Viennese aristocrat in his small palace on the \"Wieden.\" He was fashionably dressed in old Viennese style, in dark coat and striped trousers. Strauss was not a musician who gave much reflection to his art. I can quote no sentence of his which would cast a new light on music, any more than I can quote any wise speeches of Schubert on the subject, because he was full of music which streamed forth from his inner soul without constraint, and without reflection. Strauss was the same sort of musician as Schubert. His music sang out when, at night, he sat at his desk, knowing that the woman he loved lay sleeping in the next room, and an erotic atmosphere hovered about him. This erotic atmosphere he needed, for he wrote music for dancing couples who held each other embraced. That these melodies were secretly bound to the city of Vienna and its landscape, no one was less aware than Strauss. By day he was like the other Viennese, speaking the dialect the way they did, and playing cards like them. In the night, all the voices of Vienna became loud in him, and he wrote down what he heard without thinking much about it.\n\nIn no work does the voice of Vienna sound more enchanting than in the famous \"Blue Danube Waltz,\" which still to-day symbolizes Vienna for the world. This waltz of waltzes was first sung in 1867, at a concert of the Vienna Men's Choral Society, since it was originally a waltz for male voices with orchestra. Strauss meant it to console the Viennese over the defeat in the War of 1866, and the chorus begins: \"Viennese, be glad!\" Not only the Viennese, but the whole world was glad that Strauss composed this waltz. The horn call at the beginning sounds like Oberon's, calling the fairies, elves and pixies together. The atmosphere of Shakespeare's \"Mid-Summer Night's Dream\" is there\u2014moonlight, woodland charm, dances and games of the fairies. It is a far cry from the customary introductions to waltzes. An enchanted light is cast even over the old Danube when Strauss makes music. The Danube looked nothing but a dirty grey to a Viennese. But with Johann Strauss it became blue, and since that time, the whole world believes it is blue.\n\nIn his other great waltzes, like \"Tales from the Vienna Woods,\" \"Wine, Women and Song\" and \"Vienna Blood,\" Strauss also told such fairy tales. He is the greatest poet of Vienna, and gave it the colors of a beautiful dream, like a city in Paradise. Every Viennese saw his city in this Strauss light.\n\nStrauss waltzes were the last fairy-tales told in Vienna. Reality was earnest, care-worn and gloomy. Social tensions kept becoming greater, the labor pains of a new period more severe. A crisis arose, from which feudal society could find no exit except the outbreak of the first World War. To the sound of merry march-music, the Viennese regiments of 1914 passed through the city and on to the battle-fields of Russia and Italy. In this war, the old feudal order, which, in many guises, had reigned in Vienna from the end of the sixteenth century on, finally perished. The social foundations upon which Vienna's musical life had been erected, were drowned in blood and destroyed.\n5\n\nThe Last Hours 1918-1938\nRichard Strauss in Vienna\u2014The festivals in Salsburg\u2014Toscanini in Salsburg\n\nOn a dull November day in 1916, in the midst of the first World War, Emperor Franz Josef died. During the last two years of his lifetime, the frail old man arose daily from his army cot, put on his old military coat, sat at his desk and received his generals who reported on the battles in Russia, France and Italy. Then he went into the chapel to pray.\n\nI, an Austrian officer, stood with my fellow-officers on the Ringstrasse when the enormous hearse passed, drawn by six heavy horses. On the same street, I had often seen the Emperor ride. Now it was a sad procession which passed the Court Opera and progressed to the old Capuchin monastery where, in a dark cellar room, the coffins of the Hapsburgs rested, guarded by the brown-garbed monks. When the procession arrived, the monastery gate was closed. A dignitary of the court, according to the prescription of the Spanish Court Ceremonial, knocked at the door. A Capuchin monk opened, and asked \"Who is there?\" The dignitary answered, \"Franz Josef.\" Then the Emperor's coffin was carried through the low passages and down the staircase to the crypt.\n\nTwo years later the Austrian monarchy fell apart. What was left of old Austria was a small, mountainous country of six million inhabitants, a third of whom lived in the capital, Vienna. Most of the great industries of Austria belonged to the new state of Czechoslovakia, the oil wells to the new Poland, the harbors of the Adriatic Sea to Italy. Austria was small and poor, cut off from world-traffic, and Vienna seemed too large for such a small state. Could Vienna preserve itself as a music city? That was the question which all of us who had contributed to its musical life asked ourselves. Vienna's position as capital of the Austro-Hungarian double-monarchy had been destroyed. The political affairs of half Europe no longer were concentrated there. The various national states had become independent, and took, in part, a hostile or distrustful attitude toward Vienna. What could Vienna mean as capital of a small alpine country, from which the neighboring states closed themselves off? Nothing proves more positively the vital powers which existed in Vienna even after the world-shattering chaos of the first World War than the general conviction that Vienna had to be preserved as the center of European music.\n\nEach of the ruling political parties was convinced of the greatness of Vienna as a musical city and of the importance of the Vienna Opera. Every opportunity to set forth the weight of small Austria as a music center was utilized. The Social Democratic Party, which governed Vienna, immediately after the end of the war, arranged a great music festival in Vienna which described the whole heritage of Austrian music from Haydn and Schubert up to Sch\u00f6nberg. The Beethoven Centenary in 1927 was festively dedicated as a celebration for the world, and from all parts of Europe representatives came from the great nations as they did to the League of Nations. From Paris came Herriot, Minister of Education. He compared Vienna, as the music city of the world, to the Acropolis as a symbol of the cultural consciousness of humanity. One year later, according to plans laid out by me, Vienna had a large Schubert Festival. The modern Viennese composers, right after the war, invited the musicians of the world to a music festival at Salzburg. Composers came, bringing the scores which they had written during the war. They all understood each other, and in music spoke a common language of the new era which turned away from romanticism and no longer wanted to dream, but wished to be matter-of-fact. The \"International Society for Modern Music\" was founded on this occasion, at the suggestion of the young Austrian musicians.\n\nAbove all, the Vienna Opera had to be preserved in a grand style. After the break-down of the monarchy, it was no longer a court theatre, but a state theatre. As such, it was to proclaim the cultural will of Austria to the world. Two new directors were appointed. One was the oldest conductor of the Vienna Opera, Franz Schalk, a man of great musical culture and sharp intellect. The other was the most famous musician of the present, Richard Strauss. \"The goal will be set high and great, pure throughout and without concessions,\" Franz Schalk wrote me after taking office on November 26, 1918. \"Once more we want to try whether it is possible to direct a theatre without a glance forward or backward, up or down, but with a higher artistic intention alone.\" Richard Strauss came to Vienna in 1919. In a rather long conversation which I had with him, he assured me he wanted to devote himself especially to the direction of the classical operatic works. As if to underscore these words, he began his Viennese activities by conducting Beethoven's \"Fidelio\" and Mozart's \"Magic Flute.\"\n\nIn the past years, Richard Strauss had become more closely bound to Vienna. Every year he appeared there, conducting his own works. In Rodaun, near Vienna, lived the poet and librettist of Strauss's operas, Hugo von Hofmannsthal, who drew him deeper and deeper into the magic circle of the Viennese baroque. The collaboration of Strauss and Hofmannsthal was a perfect one. Hofmannsthal had an inspiring mind, and Strauss needed such inspiration. In his letters and in his conversation, Hofmannsthal met the need of the musician whose bright spirit liked to enflame itself with the modern spirit of artists. Hofmannsthal, who was at home in the baroque theatre of Goldoni, in the Venetian comedy and opera, in the Viennese miracle plays, led Strauss in this field. With a fine mind, he guided Strauss away from Wagner to a lighter, more gracious style of musical creation. For Strauss, he created a whole series of female characters, from the Marschallin in \"Rosenkavalier\" to the Viennese \"Arabella.\" He led Strauss from comic plays to meaningful fairy-plays. When Hofmannsthan sent the text of \"Rosenkavalier\" to the composer, Strauss was so filled with music that he could not get the continuation of the libretto quickly enough. This Viennese comedy of Hofmannsthal's, with its tender renunciation and the elegy of age, brought all Strauss's most inner music to light, and \"Rosenkavalier\" became his most successful opera. When, in 1924, I went to Garmisch, Strauss showed me Hofmannsthal's libretto for \"Egyptian Helen,\" and I saw on the margins of the manuscript, already pencilled in, the most important motifs which had resounded in Strauss's mind immediately upon his first reading of the text. Hofmannsthal possessed the gift of awakening music in Strauss. Sometimes, however, things came to a standstill. Such an occasion occurred when Strauss was supposed to compose the second act of \"Egyptian Helen.\" For a long time he had no inspiration. To overcome this hindrance, he made a trip to Spain and Greece. On February 15, 1925, Strauss wrote me from Paris that he hoped that at the walls of Granada, something nice for the Desert King, Altair (one of the characters in the second act of the opera), would occur to him. This hope was fulfilled. On May 20, 1926, he sent me \"best regards from the native country of Helen and Elektra,\" from Athens.\n\nRichard Strauss  \n(portrait of Mopp)\n\nStrauss was not an uncritical musician who accepted the opera texts of Hofmannsthal without having looked at them. The second act of \"Rosenkavalier\" was exactly indicated by him, since the original sketch of the poet was not to his liking. The beautiful \"Elektra\" aria, also, after the recognition of Elektra and Orestes (\"Orest, Orest\"), was written by Hofmannsthal according to the wishes of the composer. To me, the most perfect understanding between Strauss and Hofmannsthal seems to be in \"Ariadne on Naxos,\" in the beautiful prelude to this opera which Strauss composed for the Vienna Opera. This prelude is an example of a winged and soulful modern comic opera style, in its spirit of tenderness and in its vivacity.\n\nVienna attracted Strauss more and more and finally captured him completely. He felt at home there. At first he lived in one of the Mozart neighborhoods, not far from the square where the \"Magic Flute\" theatre had been. From the balcony of his apartment, one could see the fountain erected in honor of the \"Magic Flute\" where Tamino, playing the flute, walked with Pamina. At that time Strauss loved to compare himself with Mozart. When I once told him that \"Rosenkavalier\" was his \"Marriage of Figaro\" and the \"Frau ohne Schatten\" his \"Magic Flute,\" he said jokingly: \"Then there is nothing left for me to compose but my Requiem.\" In 1926 Strauss moved to the beautiful new house near the Belvedere Park, which he had had built on grounds which the Austrian state had given him as fee for his conducting at the Vienna State Opera. This house, too, stood on Mozart ground, and Strauss enjoyed my telling him this repeatedly. The street where it stood was named the Jaquingasse, so called after Mozart's friend, Gottfried von Jaquin. Young Mozart had composed his opera, \"Bastien and Bastienne\" for the well-known hypnotist, Dr. Mesmer, who also possessed a house with a theatre not far from there.\n\nSince \"Rosenkavalier,\" Strauss felt more and more like the Mozart of his time. One day he surprised me with the information that he had composed a symphony in four movements in classic style which was not programme music\u2014a symphony like the F Minor symphony of his youth. I had my doubts that this conservatism of the mature Strauss could be productive. His real field was virtuoso tone-painting which depicted modern realism in glowing colors. Here was his greatness and his progress. Strauss himself must have had similar doubts, for he locked the manuscript in his desk and did not even show it to his friends. Apparently it was easier to be a Mozart in the eighteenth century than in the twentieth.\n\nStrauss's Vienna house was built by the Munich architect, Seidel, who had also built his home in Garmisch. It was a beautiful house. Strauss was especially proud of some of his art treasures, and of a beautiful Greco which hung in his studio. He loved to rummage in antique shops for old pictures and other objects of art. In Partenkirchen, on his walks, I often accompanied him into such shops. He had no piano in his studio because he did not use one when composing. In the foyer of the studio, however, stood an upright, above which hung a portrait of Alexander Ritter, the Munich composer who had guided him to the symphonic poems of Liszt and thus had had a decisive influence on young Strauss.\n\nStrauss's practical wife had seen to the arrangement of the house. Strauss loved his original and energetic Pauline whose acquaintance he had made when she was a young singer in Weimar. Strauss told me how he became engaged. Pauline de Ahna was standing on the stage with the other singers at a rehearsal of Strauss's early opera, \"Guntram.\" At the conductor's desk was Strauss, who, as is usual at such rehearsals, had stopped some of the singers and made various corrections. Suddenly Pauline began to cry, threw the vocal score at the conductor's desk and sobbed: \"You criticize everybody except me.\" The consequences of this scene were that Strauss and Pauline became engaged. The art of making scenes, Pauline also exercised as wife of the master. Once I arrived at Strauss's home in Vienna at an inopportune moment when Frau Pauline was just making a very lively scene with her husband. She was excited, screamed, cried, and finally rushed out of the door which she slammed loudly behind her. Strauss looked after her, smiling and said to me: \"You see, such a wife I needed!\"\n\nThere is no doubt that Strauss loved his wife's originality and temperament. Repeatedly he told me that as a young man, he had had a strong inclination to dissipate and make nothing of himself. Then his wife, through her resolute nature, really cured him. Strauss would never tolerate anyone hurting her through a joke or otherwise. When I once permitted myself a little jest about Frau Pauline's love of neatness, Strauss looked at me seriously and immediately said: \"Don't make fun of my wife's love of cleanliness!\" The regulating of his life was determined by his wife. She was the earthy element in which he found the complement to his genial nervous nature. He laughed when she declared to him that when she had married him\u2014a mere musician\u2014she had made a m\u00e9salliance. After all, she was the daughter of a Bavarian general, and according to Munich's standards, a general had a higher rank than a musician. Strauss also laughed when she insisted that he must have Jewish blood in his veins. When he wrote the poem for his opera \"Intermezzo,\" in which he portrayed his marriage and his home-life on the stage, Strauss worked some of these expressions of his wife's into his text with a great deal of love and humor. When Strauss was in New York for the first time, he conducted a performance of Beethoven's Ninth Symphony at Wanamaker's. At that time, I attacked him sharply since performing one of the greatest and most solemn of all musical works in a department store seemed to me a profanation of art. Strauss replied to me: \"It is no shame to earn money for wife and child!\"\n\nIn Strauss, in addition to his genius, there was also a bourgeois streak. It probably was inherited from his father. Strauss felt comfortable in his family circle, with his wife and child, and led a family life without complication. He subordinated himself to his wife. Once I asked him why he never conducted \"Meistersinger\" in Vienna. I had heard him conduct it in Berlin, and his had been a beautiful animated and festive interpretation. He answered: \"There you'll have to talk to my wife. She doesn't let me conduct long operas.\" Frau Pauline was tenderly solicitous of his health. She saw that his meals were light, as the doctors recommended, worried about his digestion, and had but one concern\u2014maintaining her husband's good health. And she spread her wings protectively over her son like a hen over a chick.\n\nThe bourgeois element was a necessary part of the Strauss family life. As high as the flight of the composer's fantasy was, as bold as he was an artist, just so firmly did this genial man keep his feet on the ground in life. When he returned from his world of imagination into everyday life, his existence exactly resembled that of a middle class citizen who was a good husband and father, happy at his family table. Once I made an automobile trip with Strauss from Garmisch to a place in the Bavarian highlands where some relatives of his lived. He invited these simple, modest ladies into the village inn, and ordered Bavarian white sausages, which are a national dish of the Bavarians. He ate the sausages with great gusto, and as a connoisseur, and talked with his relatives in great detail about the composition, flavor and seasoning as if he had no other interests except Bavarian white sausages.\n\nTo his bourgeois side, too, belonged the pleasure which Strauss took in card games. He played Skat, the favorite game of the German middle class. He was very proud of his prowess. When I once asked why he enjoyed cards so much, he answered: \"There, at least, one doesn't think of silly music.\" In Garmisch, Strauss had his regular card \"party,\" which consisted of the head forester, a retired officer and similar citizens. But Strauss also liked to play regularly in Vienna. At the bankers' houses where he was an honored guest, a card table was set up after dinner, and Strauss would play for several hours. The slip on which he noted his gains and losses in his fine script, he would leave to his host as an autograph, after meticulously writing the date over his signature. Generally he won at these games. The circles in which Strauss moved in Vienna were mainly the financial ones of the wealthy bourgeoisie who had acquired tremendous fortunes during the years of the inflation. When I once asked Strauss if he enjoyed moving in such circles, he said: \"There, at least, I see that something has become of me. In all those houses, people are anxious to fulfil my every wish.\" The new barons of finance were, of course, very much flattered to have Strauss in their homes. However, his relationship with the bankers of the inflation had its disadvantages for Strauss. When, in 1929, the house of Morgan led its campaign against speculation in francs, Strauss lost his fortune which had been invested with these bankers.\n\nPauline Strauss laid special store upon their relations with aristocrats, and she herself came of a noble family. At the large parties which Strauss gave in Vienna, the leaders of the political world, ministers, high officials, the Mayor of Vienna and the most important Social Democrats were invited. I had the impression that Strauss sympathized politically with the Social Democrats. By no means did he love the clerical party. He was a free thinker in religious questions and therefore did not let his son Franz have any religious instruction. That created difficulties when Franz Strauss was to take his entrance exam for the university, since he was also required to pass an examination in religion. Strauss sent his son to a priest to whom it was no secret that he was a freethinker. At the examination, as Strauss related gaily, the priest asked his son: \"Do you think your father is convinced in what he believes?\" When Franz naturally said \"yes,\" the clever priest replied, \"You see, then he too, is a true Catholic,\" and dismissed the son of this somewhat skeptical believer who thus passed his examination.\n\nLike all Bavarians, Strauss disliked Prussians, even more after the war than before. I remember that after the war, Strauss made a long political discourse to me during a walk at Garmisch. He declared that the best political solution of the problem would be the uniting of Bavaria, Austria and Hungary in a league from which Prussia would be excluded. After the recent upheaval in Germany, as one knows, Strauss put himself, for a period of time, at the disposition of National Socialism. He wanted thus to gain influence on musical development in Germany; and he told me when we met in Salzburg that he had sent three memorandums to the government asking that its attitude on the race question be made more mild in the music field. Doubtless, Richard Strauss made a false step when he allied himself with National Socialism. He placed himself in a difficult position; because not only were his best friends Jews and his librettist, Hugo von Hofmannsthal, a half-Jew, but his daughter-in-law was Jewish and his grandchild, therefore, a half-Jew. A free, European mind, like Strauss, who was bound to all the intellectual movements of Europe and who thought so much in European terms that he had his ballet, \"Joseph's Legend\" performed first in the Paris Opera by the Russian ballet, is greater than the culture-inimical National Socialism.\n\nI never knew Strauss as anything but amiable, fine, cultivated and thoroughly good-hearted. He could be stubborn, however, and when his face reddened in anger, it was better not to argue with him. But in general, in his great superiority over the circumstances of life, Strauss made an impressive appearance, and lost nothing of his greatness in simpler and more natural forms. I have seen Strauss in the most various situations, and he was always self-controlled and superior. He was a man who always had himself securely in hand and always remained true to character. Of all the genial men I have known in my life, Strauss was the most natural, the most clever and the best behaved. In his eminent self-control, Goethe was his model.\n\nI often visited Strauss in Garmisch where his house overlooked the Krammer. The house, whose windows gave on the powerful mass of the Zugspitze, was protected from the noise and turmoil of the street by a great park whose trees hid it from the eyes of the passersby. Strauss spent his summers there, composing. \"In winter I need light occupation,\" said Strauss to me, \"therefore I conduct. You cannot imagine what it means for me when I conduct \"Tristan\" once in winter, and how it stimulates me in my work.\" At Garmisch, his mornings were dedicated mainly to composing. Strauss wrote his works either in the beautiful studio, or in the garden where a work table stood under an awning. The afternoons belonged to automobile excursions, to his beloved card games, and to letter-writing which took up a great part of his time, since Strauss was a great correspondent. The number of letters which Strauss wrote in his lifetime must run up to the thousands. They are all in his pointed, nervous yet nonetheless definite and stubborn handwriting which is just as personal as his musical manuscripts.\n\nThe first sketches of Strauss's compositions were made in regular notebooks with pencil. These sketch books he could carry comfortably in his pocket. The care with which he wrote, even in these sketch books, is great. There is nothing confused by excitement, as it so often was in the case of Beethoven. Strauss's writing is always clear, as if engraved. Neatness and clarity were essential to Strauss in his work, also. There was none of the disorder which usually surrounds genius in his studio. After he had finished his work for the day, he would collect his sketches and notes from his desk and put them away.\n\nStrauss thought very keenly about the theoretical problems of his art. This was especially the case at the time when the atonal composers appeared upon the musical scene. Though Strauss had stretched classic and romantic harmony to their utmost boundaries in his \"Elektra,\" nevertheless he had built upon a firm foundation of classic harmony. With pleasure he told me, after returning from the modern music festival at Donaueschingen, that he himself had asked Hindemith after the performance of the latter's string quartet: \"Why do you write atonally? After all, you have talent!\" In spite of this, he showed me, with definite satisfaction, a spot in the score of his \"Alpine Symphony\" where he had written several atonal bars. It is the rising progress of the contrabasses at the beginning of the storm, which, as an atonal spot of color, is quite impressionistically inserted in the picture of the storm. In \"Elektra,\" too, there are similar harmonically free coloristic passages. In general, Strauss thought that modern composers knew how to do too little. \"They can't do it! They can't do it!\" he exclaimed again and again. He meant that modern musicians bring up new problems, but that they are incapable of solving them. How well and how sharply Strauss theorized one sees best from the preface to his opera \"Intermezzo\" where he analyzes his comic opera style. When he wrote this preface, he had me lecture to him in detail on the history of the opera recitative, and he used the musico-historic information which I could give him in its preface.\n\nThe Vienna Opera, the direction of which Strauss had taken over in difficult times, together with Franz Schalk, he loved particularly. He admired the orchestra especially. When, in 1923, he made a concert tour of South America with the opera orchestra, he wrote me on August 7th, from La Plata: \"Up to now I have given twenty-one concerts with the marvelous Philharmonic in Rio de Janeiro, San Paolo and Montevideo with greatest success. The first concerts were partly splendidly improvised, the later ones, after polishing rehearsals, a fantastic rhythmic precision in wonderful delivery and paradisiac sound. The orchestra, too, apparently takes great pleasure in playing with me.\" On August 11th, he followed with a postal card from Buenos Aires in which he said: \"The Philharmonic yesterday played Schubert's C Major and the 'Domestica' as if in a fairy-tale.\"\n\nRepeatedly, in my hearing, Strauss called the Vienna Opera his \"Bayreuth.\" He had already left the Vienna Opera when on June 6, 1931, he wrote from Garmisch: \"The precious possession of the Vienna Opera must be preserved at any price. It cannot be that the whole German culture will go to pieces with these damned swine-politics.\"\n\nIn those years when the soil of Austria was shaken, it is immeasurable what Strauss meant for the Vienna Opera and Austrian musical life. Not only had he been the artistic authority which protected the opera against the dilettante meddling of officials and politicians, but he also filled the beautiful house with enthusiasm. He was at the peak of his successes as composer and his work had become established.\n\nStrauss liked especially well to conduct \"Tristan and Isolde,\" to which he gave passionate expression, as if it were a relative of \"Elektra,\" and \"Lohengrin,\" with wonderful solemnity in the Prelude. Strauss's Wagner style was the Wagner style of a new era. He preferred faster tempi, great climaxes, and sometimes\u2014as at the end of \"Walkyrie\"\u2014real Strauss strettas. Strauss also conducted Mozart's works often, and paid no heed when the Viennese music critics said his tempi were too fast. \"I understand that better,\" he said curtly, and asserted his right as greatest living musician to understand Mozart best. Despite his justified self-reliance, Strauss also knew how to evaluate and honor the opinions of others. He wrote me on May 26, 1922, from Karlsbad, where he was taking the cure: \"Your criticism of 'Meistersinger' was a masterpiece of productive criticism\u2014matter-of-fact, instructive in the best sense. It gave me such enjoyment that I especially want to thank you for it.\"\n\nUnder the direction of Strauss and Schalk, a second intimate theatre opened in the Redoutens\u00e4le (Masquerade Halls) of the Imperial Palace. It was a true jewel-box\u2014a small, aristocratic splendid hall with large chandeliers of Venetian glass and Gobelins of the Netherland school on the walls. The stage was a mere platform, and from its rear, a double staircase led to a side hall in which the court society had drunk tea in the eighteenth century. The time of Emperor Josef I was still alive in this delightful hall. It was built during his reign, in 1706. One of the halls was dedicated to Italian opera which was presented each year on the empress's birthday. The other was destined for court festivals. In 1744, the last opera was performed, \"Hypermnestra\" by Hasse. After this, these Halls were used for grand balls. Haydn and Mozart composed dance music for them, and Beethoven conducted there. Now, the greatest composer of modern times sat at the conductor's desk while on the stage there was song and dance amid stylized decorations. There was modern rococo atmosphere in the lovely theatre. Strauss conducted \"Cosi fan Tutte\" there, and surprised us once with a revival of the old French opera, \"Jean de Paris,\" which had been dear to his heart ever since his youth. For this theatre, where the opera ballet had danced ballets by Ravel and Stravinsky, Strauss composed his \"Couperin Ballet.\" His \"Le Bourgeois Gentilhomme,\" with its clever music, was also presented there.\n\nIn the great Opera on the Ringstrasse, Strauss regularly conducted his own operas. The Vienna State Opera, with its exemplary performance of his works, for several years was really his \"Bayreuth.\" Here his great \"Elektra\" was enthusiastically celebrated by the youth; his \"Salome\" again and again presented. The warm fairy-tale of the \"Frau ohne Schatten\" gripped the listener, the Marschallin said good-bye to her youth, and the \"Rosenkavalier\" waltzes played and joked. \"Ariadne auf Naxos\" delighted audiences with its intellect and its melody. \"Egyptian Helen\" presented a modern comedy in Greek frame and a modern problem of marriage in legendary form. \"Joseph's Legend\" unfolded, in a modern ballet, the beauty of a baroque festival. With \"Intermezzo\" the present day and Strauss' family life came to the stage. Color, brilliance, modern thought were spread out in the Vienna Opera, and the great composer who had created these works sat there, impressive and powerful, ruling stage and orchestra, surrounded by the reflected light of the orchestra lamps.\n\nWith Richard Strauss as leading musician, the Vienna State Opera once again stood in the center of universal worth and meaning.\n\nThe opera and the music of Vienna had been saved by the collapse of the Austrian nation. It was more than chance that Hugo von Hofmannsthal, when he wrote opera poems for Strauss, had bound himself to Viennese baroque traditions. For Hofmannsthal, the Austria and the Vienna of the eighteenth century, the Vienna of Charles VI and Empress Maria Theresia, of Prince Eugene and Metastasio, the Vienna of battling Catholicism and of the Roman Empire, was the true one and the one which outlasted all changes of time. This Vienna he revived in his poems for \"Rosenkavalier\" and \"Arabella.\" The poems of the \"Frau ohne Schatten\" and \"Ariadne auf Naxos,\" too, were related to the theatre of this Vienna. For the baroque fantasy of Richard Strauss which bathed in splendor and brightness of sound, this world was the right opera world. Strauss loved Vienna and Austria, and one of the most beautiful compositions of his Vienna period was his setting of the \"Austrian Song\" by Anton Wildgans, a Viennese poet whose ideas were similar to those of Hofmannsthal. In one of the most difficult hours of Vienna's history, Richard Strauss came there as the last of the great musicians who was at home within its confines.\n\nIn 1924, differences between Strauss and his co-director, Franz Schalk, arose, in which the wives of the two artists were not without a part. At this time, I received a letter from the Austrian Minister of Education, Dr. Schneider, who invited me to have a talk with him. He told me that one day Richard Strauss came to him and handed in his resignation, and the next day Franz Schalk came and did likewise. I immediately took the standpoint that the departure of the greatest musician of the present would be an unbearable loss for the musical life of Vienna. The Minister seemed to agree with me, but failed to do anything when Strauss handed in his resignation on November 1, 1924. In a letter of April 13, 1925, Strauss wrote me from Garmisch that to that day he had received no official confirmation from the Minister: \"Nothing, absolutely nothing! Not even my theatre dress coat did they send me.\"\n\nThe thought that Richard Strauss would be allowed to leave Vienna without anyone's having made an attempt to hold him seemed to me, at that time where every effort had to be made to maintain the Vienna Opera at its high level, so senseless that I went to the Austrian Chancellor, Dr. Ramek, to present the facts to him. The Chancellor listened to me calmly, and promised me that an attempt would be made to change Strauss's mind. A representative of the Ministry of Education was sent to Garmisch to see Strauss, but the gentleman was neither very apt nor did he endeavor very earnestly to mend the torn threads of the situation. In a letter which Strauss wrote me after this visit, dated January 17, 1925, he talked of a \"polite beating-around-the-bush about something which one wants to glue together only superficially,\" and he adds: \"If Vienna wants something from me, it should put its cards upon the table, and not only ask what my wishes are, but fulfil them, too. They are completely un-egotistical and refer only to the well-being of the State Opera, about which I still know more than most other people.\"\n\nStrauss took an active part in the development of the Vienna State Opera's affairs, even from the distance. He saw it, since his departure, in \"uncheckable decay\" (letter of March 30, 1926) and stormed about the \"whole nest of lies and impotence\" which had formed there, \"from the administrative to the artistic direction\" (letter of July 4, 1926). Again and again he made suggestions as to who should be put in the place of Director Schalk, and he collaborated in the nomination of Schneiderhahn as Intendent and Clemens Krauss as Director in 1926. After the appointment of the new Director, he returned for a time as conductor, again, but he would no longer have any part in the management. He had crossed the border of the sixties and looked at life from a higher level. In a letter, he writes: \"Complete indifference is already a very strong expression for that which still moves me.\" (Strauss meant, in regard to the Vienna Opera). He knew that the opera was in good hands with Clemens Krauss. That sufficed him. The Strauss chapter of the Vienna State Opera\u2014a great light in a dark time\u2014was closed for good.\n\nIn 1933, commissioned with the direction of a music festival for the city of Vienna, I invited Richard Strauss to conduct \"Tristan and Isolde\" on the anniversary of the fiftieth death-day of Wagner. Strauss declined and wrote me: \"The day before yesterday I conducted, in Dresden, for the last time in this life, \"Tristan,\" in honor of the death of the great master, and with it I definitely finished my activity as conductor of the Wagnerian wonder works. I shall be sixty-nine in June, and I have to husband my strength if I want to complete the compository tasks which still lie ahead of me, with passable freshness. Wagner's dramas are strenuous enough for younger men . . . so it really is not possible, even with the best of will. Such a renunciation is painful, but sometime one has to say good-bye.\"\n\nOne year later I met Strauss, in the company of Opera Director Clemens Krauss, in Salzburg, for the last time. Shortly before, Strauss had celebrated his seventieth birthday, and he looked in the bloom of health. The conversation did not touch upon the Vienna Opera, but on the best methods of catching trout. Strauss held forth in an expert discourse on the subject. One could see he was far distant from the Vienna Opera and its worries.\n\nThe fame of the Vienna Opera had become greater and greater. Guest appearances abroad were furthered by the Austrian government. In 1923, the Vienna Opera was sent to Geneva, where it solicited the interest of the League of Nations with its Mozart performances. In 1934, the Vienna Opera presented Strauss's \"Frau ohne Schatten\" and Mozart's \"Cosi fan Tutte\" in Venice. Nine years later it traveled to Rome with \"Ariadne auf Naxos,\" which Mussolini wanted to hear. The Opera Orchestra, the Vienna Philharmonic, toured half of Europe and gave its concerts, enthusiastically acclaimed. In 1935, the Philharmonic and the Opera Chorus went to Florence to perform the Beethoven Ninth Symphony under Felix von Weingartner's direction. The music city Vienna seemed to send out rays of its old glory into a new world. Austria had become small but was still large on the map of music. It no longer had any political might, but its musical power seemed unshaken.\n\nNew musical fame came to Austria through the festivals which were held every summer in Salzburg, and which, since 1927, became more and more the music festivals of the whole world. In the old Bishop's town on the Salzach, the last great chapter of Vienna's musical history took place.\n\nIt can be regarded as symbolic that during the first World War, at a time when the unhappy ending could already be foreseen, in 1917, the \"Salzburg Festival Committee\" was founded in Vienna. It issued a proclamation for the holding of festivals in the Mozart town. From the very beginning, the programme for the planned festivals was conceived on a grand scale. It was mapped out by Hugo von Hofmannsthal, and approved by the Art Council of the Festivals, which consisted of Franz Schalk, Max Reinhardt, Richard Strauss, Professor Roller and Hofmannsthal. Mozart, of course, was meant to be the center of the festivals, but from this center, one proceeded in the most various directions. The \"Idomeneo\" of Mozart seemed to lead to the Greek operas of Gluck and to the antique drama. From Mozart's \"Don Giovanni,\" one arrived, according to the ideas of Hofmannsthal, at Moli\u00e8re, Calderon, and at the spiritual dramas of the middle ages. From the \"Magic Flute,\" a double path opened, one branch leading to Goethe's \"Faust,\" the other to Weber, Raimund, the Vienna Folk Theatre, to Gozzi and Goldoni and to Johann Strauss's \"Fledermaus.\" Thus from the outset, the frame of the festivals was widely spanned.\n\nThis festival programme was fitted to the spirit of the town of Salzburg, or better, conceived from its spirit by Hofmannsthal. For in Salzburg, the baroque time was not a thing of the past, but something which was still alive in the panorama of the town, in the churches, palaces, yes, even in the simple houses of the people. Lying on the banks of a green alpine stream, and encircled by high mountains, Salzburg is still dominated by its baroque buildings. With the Archbishop Wolf Dietrich (1587-1612) who, as a relative of Pope Pius IV and of Cardinal Borromeo, had Italian blood in his veins, the great Italian epoch of Salzburg began. Under this prince of the church, the cornerstone of the new cathedral was laid, the first plan of which was drawn up by Vincenzo Scamozzi.\n\nItalian art and Italian artists began to overflow Salzburg under this bishop. Tiburtius Massainus of Verona became the first conductor, and Italian falsetto singers were heard in the cathedral. Dozens of bourgeois houses were torn down by Wolf Dietrich to make room for the new bishop's residence, which was to look like a Roman palazzo with a large court where later the Triton fountain of Antonio Darios let fall its splashing cascades.\n\nThe Castle of Mirabell was begun under Wolf Dietrich. Under the melancholy Marcus Sitticus (1612-1619), who, like his predecessor, had been educated in Rome, the new cathedral by Santino Solari rose high in marble splendor akin to St. Peter's in Rome. In the middle of a woodland park, the Castle of Hellbrunn rose up, with water art and grottos, with fish ponds and a stone theatre, with game parks and statues. Under Marcus Sitticus, who sought distraction in festivals and music, Salzburg was filled with theatrical performances of all kinds. In the Hellbrunn stone theatre, worldly as well as religious operas were performed, and Emperor Ferdinand was greeted with the performance of an opera, \"Orfeo\" on the occasion of his visit in 1618. A Venetian masque-procession was held in the town that same year\u2014the Town Councillors of Salzburg in Venetian costume sat on a galley, pulled by four white horses and driven by Neptune with his three-pronged fork.\n\nThe \"Gymnasium\" (high school) founded by Bishop Marcus Sitticus in 1617, became the fostering-place of theatre-art in Salzburg. In the school plays which were performed there, all the gods of Greek legend appeared normally, or by means of \"flying\" machines. Choirs sang, rockets flared. The Bishop's praises were sung in choruses and arias. \"Labor\" entered on a triumphal cart amid the blare of trumpets and Mercury distributed prizes to the best pupils while a large chorus sang hymns. The Court Chapel was completely Italianized under Marcus Sitticus. The court conductor, Peter Gutfreund, had to change his name to \"Pietro Bonamico,\" and his successor was Francesco Turco. In 1628, under Archbishop Paris Lodron, the new marble cathedral was dedicated. Stefano Bernardi, who had become Cathedral Conductor in 1627, had composed a twelve voiced Te Deum for the occasion. From the four pillars of the cupola, the fifty-three part festival mass, which the young Roman, Oratio Benevoli, had composed, resounded solemnly. With its sixteen vocal parts, thirty-four instrumental parts, two organ parts and basso continuo, this music filled the cathedral's huge interior. Before Mass, the relics of the cathedral had been ordered to their new resting place, in a festival procession on triumphal floats where Heaven, the Blessed Virgin, the Saints and Christ on the Cross were depicted. In the afternoon, there was a festival play in the Gymnasium, in which Neptune, Diana and mermaids greeted the festival guests. The finale showed the new cathedral on the stage, above which were angels on a rainbow. The chorus sang and there were comic interludes. At Hellbrunn, in the stone theatre, an ecclesiastical opera, dealing with the conversion of Mary Magdalen, was performed in Italian.\n\nThe alpine landscape which surrounds Salzburg and the buildings in Roman style grew together in a unique way. In the seventeenth century, Salzburg was a town of eight thousand inhabitants at the most. Nevertheless, it could boast of churches and palaces which even in Rome, were considered out of the ordinary. The wealth of the old monasteries, of which the Benedictine Order's St. Peter's is the oldest on German soil, was surpassed by only very few places in Europe. All these buildings stand thronged together in the small space between the Capuchin and the Monk's Mountains, between burghers' houses with narrow stairs and flat roofs. German middle-ages as well as Roman baroque splendor meet in Salzburg. Here is alpine country as well as Italy.\n\nThis is the town which became the town of Mozart, in whose music German and Italian elements, alpine landscape and the Mediterranean are united. Mozart was born in one of the old merchant-houses in the narrow, crooked Getreidegasse (Grain Street). His first masses were played in the cathedral and in St. Peter's Church. He was engaged at the court of the Archbishop, whose musicians were still, for the most part, Italians. In the theatre of the University he saw his first theatrical performances. All the characteristics and atmosphere of his beautiful native town made their impressions upon the sensitive, receptive soul of the boy Mozart. And it was this native town of Mozart's which laid the germ of the phenomenal development of his music which became the most perfect synthesis of all European spiritual movements. For Salzburg was a world by itself, and although at that time, it was situated in Germany, it was more closely connected with Rome and Italy.\n\nIn Salzburg, just as in Vienna, the earth was drenched with music. From Salzburg the song went forth which became the Christmas song of the whole world, \"Silent Night, Holy Night.\" A school teacher had written the music, a priest the poem. Both lived in a small village near Salzburg, both spoke the Salzburg dialect, both were men of the people.\n\nIt was Christmas Eve\u2014December 24th, 1818. In the old church of the small village, Oberndorf, near Salzburg, the priest was celebrating Midnight Mass. Candles burned on the altar, and rosy-cheeked peasant boys, in white surplices, were kneeling behind the priest, swinging the censer. On the side of the altar, as every year, the crib, with freshly painted wooden figures, had been erected. Peasants, in thick winter cloaks, sat praying on the church benches.\n\nThe priest had just sung the \"Benedicamus Domino\" in his shaking, aged voice, when from the choir, two male voices, accompanied by a guitar, began the song. The tenor part was sung by Joseph Mohr, the priest who had written the text, the bass by Franz Gruber, the school-master who composed the melody. At the end of every stanza of \"Silent Night, Holy Night,\" the choir of blond schoolboys, who till this moment had been looking down on the altar, the crib and their parents, joined in with their clear voices.\n\nNone of those present, least of all the poet and the composer, had the slightest idea that the song would spread over the whole world. The authors never once thought of having it published. It had been written for Christmas, 1818, and belonged to the divine service, just as the candles flickering on the altar, the crib with the ox and the ass, and the silver censer from which great clouds of incense floated up to the roof of the church.\n\nThe melody of this song composed by a peasant's son had a folk-like character. The school-teacher began his \"Silent Night, Holy Night\" as one sings to a child who is to be sung to sleep. It sounds like a lullaby in which there is something shy and withheld, as it must have been in the looks of the shepherds when they were led by the angels to the Christ Child, and stood there, awed and pious, their caps in their rough hands. Unwittingly the composer gave the melody the rhythm of the songs which are sung in Italy by the Venetian gondoliers and by the fishermen in the boats on the Gulf of Naples. The tune's Italian character is derived from the Salzburg landscape, through which the road leads from Italy to Germany. The houses in Salzburg have the flat roofs of Italian houses, and the sun of the south falls on the Salzburg countryside. Thus the Italian color of the melody belongs to the landscape to which Schoolmaster Gruber was bound. The song is a piece of nature.\n\nNowhere in the world, during the nineteenth century, was a similar song composed which traveled the whole world. It belonged to the Christmas celebration and brought a powerful message to all humanity.\n\nWith such a past, Salzburg was the ideal place for the world's festivals. In 1920, on August 22nd, Max Reinhardt produced the religious play, \"Jedermann\" (\"Everyman\"), adapted by Hugo von Hofmannsthal from the English mystery, on the square in front of the Salzburg Cathedral. A simple wooden stage had been erected. From the portal of the church organ music sounded, and at the end of the drama, the bells tolled while the sun set behind the mountains and dyed the marble of the cathedral red. The spectators sat on rude benches and watched the play of the life and death of the wealthy Everyman. At the end, the shroud was placed upon him, and Faith and Good Works led him into his grave.\n\nThe theatre had returned to its mediaeval form and had become once more a primitive mystery play which was equally understandable, plastic and touching for the educated as well as for the folk. Since that day, \"Jedermann\" was presented every Sunday and holiday during the festival, there, in front of the Cathedral, and peasants came from even the most distant mountain villages to see it.\n\nIn 1922, Max Reinhardt produced the \"Salzburg Great World Theatre,\" from Hofmannsthal's adaptation of a play by Calderon, in the Kollegien Church. This time Reinhardt had erected his stage in mediaeval form on the high altar, and the baroque church of Fischer von Erlach took part in the drama.\n\nBruno Walter  \n(Sketched by Dolbin)\n\nThe new \"Festspielhaus\" which Clemens Holzmeister had designed as a simple building which stood between the summer and winter riding schools of the old Bishops, was opened in 1925. A Salzburg painter, Faistauer, decorated its foyer with frescoes. Originally, only plays were given in this theatre, while the performances of the Vienna Opera were held in the small Stadtheater. But in 1927, the Beethoven Centennial year, Franz Schalk had performed \"Fidelio\" in the Festspielhaus, and from that time on it became the opera theatre of the Salzburg festivals. There, operas were performed, mounted by the great scenic-designers and stage-directors of the present day. Schalk performed \"Fidelio\" with Roller and the \"Magic Flute\" with Strnad. For Gluck's \"Iphigenia in Tauris,\" again Professor Roller was engaged. New ideas and inspiration were brought to Salzburg by Bruno Walter. He conducted the much admired \"Orfeo.\" Weber's \"Oberon\" displayed its oriental magic in the scenic designs by Strnad. Then came the \"Abduction from the Seraglio.\" Later followed the \"Marriage of the Figaro\" done again with Italian singers. Weber's \"Euryanthe\" and Hugo Wolf's \"Corregidor\" were among the revivals conducted by Bruno Walter. Walter commissioned the leading stage-directors for his performances \u2014Karl Heinz Martin, Ludwig Hoert, Herbert Graf and Lothar Wallerstein. Walter's performances, completely thought out, with brilliant intellect, and the greatest intensity within complete unity of style, belong to the art of Salzburg, and were artistic events for the whole world. Clemens Krauss conducted \"Cosi fan Tutte,\" with the charming stage d\u00e9cors by Siebert. The great operatic works of Richard Strauss were introduced, one by one, by the composer, supported in the same spirit by Lothar Wallerstein as stage-director. Thus, the Salzburg opera repertory spread in various directions with Mozart as the central point, in performances which might be considered the most perfect in modern stage technique.\n\nAround the opera performances, concerts by the Vienna Philharmonic were grouped, under the most important conductors; concerts in the Salzburg Cathedral, Mozart Serenades in the courtyards of the Bishops' Residence. In 1934, Max Reinhardt realized his old plan of performing Goethe's \"Faust\" in Salzburg. He had a theatre built in the courtyard of the Bishops' summer riding school, and Clemens Holzmeister erected a whole town for the drama. Goethe's \"Faust\" became a Salzburg \"Faust.\" It was not an educational play\u2014nothing literary\u2014but all theatre, all spectacle and \"show.\" It was as if Goethe's \"Faust\" had returned to the popular play from which it had emerged. The angels stood high upon the mountain. God's voice thundered from the heights at Mephistopheles below. In one of the openings in the mountain wall, Dr. Faust sat at his work table illuminated by the nocturnal lamp, Mephistopheles behind him. In front of Marguerite's house, a real garden, with green grass and daisies, had been planted. With their girls, the men danced around a real linden tree, and from the church, behind which one saw Salzburg's house-tops, came the sound of choir and organ. Theatre and reality were bound together.\n\nThe highest point in the development of the Salzburg Festivals was reached in 1935, when Arturo Toscanini stepped into the circle of great artists who contributed to Salzburg's fame. With this most famous conductor of the time, a man of moral independence and most noble artistic conscience had been won. He was not only a representative of music, but of those ideas of all humanity which had made eighteenth century Vienna the capital of classical music. When in Haydn's \"Creation,\" the creation of man is heralded in the sublime aria, \"With dignity and highness arrayed,\" when in Mozart's \"Magic Flute\" the godliness of man is glorified in solemn choruses, when in Beethoven's Ninth Symphony men are hailed as brothers\u2014all that was for Toscanini not only music, but faith. No one should approach classical music who is not imbued with the humanistic conception of great music and great artists. Toscanini rejected any compromise such as could be found even in musicians like Furtwaengler and Strauss after the victory of the Hitler movement in Germany, in 1933. One cannot perform great music and at the same time accept the enslaving and brutalizing of humanity. Classical music means music of the noblest humanity. Whoever did not serve this ideal or wished to make compromises had no business in the temple of music.\n\nThe three years in which Toscanini was active in Salzburg were three years of perfect artistic achievement. Music lovers from all parts of the world came to Salzburg to enjoy music which belonged to everybody. For Europe, which was full of nationalistic slogans, of hatred and fear, of force and turmoil, the Salzburg Festival with Toscanini was the last spot left on earth where music was not only beauty, but also faith.\n\nToscanini\n\nToscanini began with Beethoven's \"Fidelio.\" Then followed the Italian performance of Verdi's \"Falstaff,\" the wonder-work of the eighty-year old composer. In 1936, the \"Meistersinger von N\u00fcrnberg\" was performed. In the intimate hall of the Festspielhaus, it had the effect of a true comedy, every word understandable, so that every jest, and the art speeches in the shoemaker's room, had wonderful vitality. It was a festival play with joyousness and folklike qualities, with tenderness and resignation. It had a second act, built up high by the stage-director (Dr. Herbert Graf), with many convenient nooks for conversation, for jolly pranks, for confessions, for beatings and for silver moonlight. In 1937, Toscanini conducted the \"Magic Flute.\" Once again one heard a work of humanity with symbolic solemnity, with folk-wit, mystic ceremonies and lively jokes. Toscanini, who was nearly seventy, applied a burning energy to the work. Here was a master-conductor who was only satisfied with the nearest possible approach to perfection, who demanded the finest efforts from artists and colleagues in the production, and who himself undertook the highest and most grandiose.\n\nIn addition to his opera performances, Toscanini conducted orchestral concerts and large choral works with classical programmes; and from far away, the whole world came to hear his interpretations. Salzburg could scarcely accommodate the guests who flooded the city on the Salzach on performance days.\n\nViennese music and Austrian music, during these years, experienced their highest level. For the last time they demonstrated their power of holding the world in their grip. For the last time, also, they confirmed that great way of thinking through which they had become great. It is said that those who are dying often find their moments of greatest well-being right before their passing. All which is life force in dying assembles in a last mounting of strength. From every part, the forces stream together, and a feeling of happiness, health and of a new birth of life takes possession of the dying man, who is deluded into feeling strong and happy. Such a moment in the musical life of Vienna was provided by the Salzburg Festivals. It seemed as if all the powerful forces of Viennese and of Austrian music assembled themselves again at this time; and as if Vienna and Austria were again capable of sending forth the ideas which had made Vienna the music city of the world\u2014a city of mingling and uniting. There is great meaning in the fact that Vienna (for Vienna's musical spirit reigned in Salzburg) returned, in the Salzburg Festivals, to the foundations of that time in which it became great. The rise of Vienna to the status of a European music capital began with the Italian composers and conductors of the Viennese court\u2014with Cesti, Draghi and Caldara\u2014and its glory ends with an Italian conductor\u2014Toscanini. The circle, the first lines of which were drawn in sixteenth century Vienna, was closed in 1937, in Salzburg.\n6\n\nPast and Future\nAn Outlook on History and New Life\n\nIt was the end of a development such as had never taken place in a more unified and complete way in the history of music. The city which, through so many centuries, had made music for the world, was destroyed. The war of 1914 ruined the old Austria, the invasion of the year 1938, the European music city, Vienna. Between the spirit which had created Vienna and nationalism, there could be no relationship. They belong to entirely different worlds.\n\nShortly before the destruction of Vienna as a musical city, the devastation of Europe began. The two events were connected. All over Europe, an old epoch went up in flames, to the accompaniment of murder and hideous cruelty. Both great wars, that of 1914 and of 1938, represent, perhaps, only horrible episodes of a great revolution which destroyed an old order, and which will lead, through streams of blood, to a new one. The social and national tensions in Europe were already great around 1900, society shaken, and its balance destroyed. All these tensions exploded, like a great tempest, in the two wars. As it has been in any historical crisis, this too, is not only the bloody disruption of an old epoch which is no longer capable of controlling its intellectual, social and political problems, and which resorted to force to prevent the forward surge of a new power. It is, at the same time, the painful birth of a new era. From the crisis a new Europe will arise. The way may be long, and lead through want, devastation and misery; new crises may mount up before the ground is prepared for new harvests in a peaceful world, and a new Europe of work, community of nations and harmony of all forces be created. Only the past will be dead, and a fresh chapter of world history will begin. To-day, too, one can say what Goethe said, in the midst of the storms of the French Revolution, when the cannons of the Battle of Valmy were thundering: \"From now and from here on begins a new chapter of world history, and you can say that you have been there!\"\n\nA new era and a new society, new intellectual values and new forms of life also mean new art and a new relationship of life and society to art. In this connection, the question of Vienna's future as a music city becomes important. Vienna can never rise again in its old form. Here, too, a new development begins, and I should not like to close this book without having answered the question as to under what circumstances and conditions Vienna could become a great musical city once more.\n\nWhen I conducted the reader on a trip through new Vienna, along the Ringstrasse, the past kept peeping through into the present. The new city of trolleys and busses again and again changed itself back to the old fortress city of the seventeenth and eighteenth centuries. The society which reigned in Vienna until 1918 was only the old feudal society of the baroque period in modern dress. Vienna was old and new at the same time. The past, in this vivid and beautiful city, was everywhere alive, and belonged to its life and existence.\n\nVienna had scarcely changed since the seventeenth century. imperial Palace, nobles' palaces, churches and town houses ire almost as they were. To be sure, modern banks and a few modern apartment houses have been built, but they did not change the essential picture of the city. Should Mozart return there to-day, he would find the neighborhood where he wrote his \"Marriage of Figaro\" quite unchanged. Schubert would find his old inns in quiet streets, Beethoven his stately Pasqualatti House. In modern Vienna there are still many narrow, winding streets through which vehicles have difficulty passing.\n\nIn a city of this type, the past is a strong power. It was so in the political, social and musical life. Old musical periods did not die when those musicians who lived and worked in them passed away. In the sound of instruments and voices, there were ghostly voices from bygone days. Thus past and present were united. Tradition bound to-day to yesterday. One cannot speak of Vienna's future as a music city without understanding its past. One and the same law ruled in the whole development from the seventeenth to the twentieth century.\n\nIt is this law of life of Vienna that we wish to understand that, unchanged, set in motion all the changing characteristics of Viennese music and which will continue to work into the future. This law, as long as Vienna was a great musical city, possessed living strength. Costumes had changed. External forms of life had changed. Generation followed generation, new buildings were erected. Only the personality of the music city remained unchanged during the change of times and customs. Should Vienna begin a new development as a musical city in a new epoch of world history, it would only be possible if it continues to observe this law of its existence. That does not mean that it should copy its past. Nothing is further from my mind than to attempt to preach a dusty conservatism. A great, modern city is no museum. A new time is here with its demands, with new people, new aims and new work. The past is thoroughly dead\u2014a cemetery and a pile of debris. A return to the old forms of life is impossible, for from dead life only a dead art could emerge. Music can only be vital as the expression of a new time, and only from living forces can new music grow. The only thing which remains unchanged is the law which formed Vienna into a great musical city.\n\nThe question which is to be answered is what, exactly, made Vienna into a world music city? When and how did Vienna become this music city? What spiritual forces were active which produced in Vienna music which had meaning for the whole world? What is the law which ruled these spiritual forces?\n\nOnly history can answer these questions; and we must ask it, if we wish to receive a reply about the future which would be more than mere idle thought. Therefore, after I have sketched the last chapter of the music city, Vienna, from my own point of view and from my personal experience, we must call up the vision of the time when Vienna raised itself to the position of first musical city of the world.\n\nI\n\nThere had always been music in Vienna ever since the Roman soldiers built the fortress on the Danube. We know of the music and of the musical instruments of this period. But Vienna, in those days, was nothing but a fortress with a market-place, and not a music city. In the following eras, twice in the city's history, the musical life became richer and music began to sound more loudly through the Danube lands. One of these times fell in the days when the Babenberger dukes built their castle on the Danube and the Minnesingers, in the thirteenth century, sang their songs of spring, of women, and of love. The greatest of all German poets before Goethe, Walter von der Vogelweide, composed and sang, accompanying himself on the viola. But Minnesingers also sang in other castles, and in the chateaux of southern France there was considerably more music than in thirteenth century Vienna. In the fifteenth century, under Maximilian I, there was again a great deal of music in Vienna. The court chapel was founded where boys from the Netherlands sang the Masses. Great musicians of that period, like Paul Hoffheimer, Ludwig Senfl and Heinrich Isaak were in the service of music-loving rulers. This fact demonstrated, however, more the love for music of the princes than of Vienna's love of music. There were, at that time, many devotees of music among the aristocrats\u2014the Estes in Ferrara, the Gonzagas in Mantua, the Sforzas in Milan, the Kings of France and England, the Dukes of Burgundy. The fifteenth century was a great musical era. The cities of Holland and Paris, not Vienna, were the great music centers of Europe.\n\nIt was otherwise in the sixteenth century. At this time, the Austrian landscape\u2014Graz, Innsbruck, Vienna\u2014began to sound. From 1567 on, there were Italian musicians at the court of Graz. At Innsbruck, half of the court chapel orchestra were Italians, directed by a musician from Milan. In Vienna, in 1619, Archduke Ferdinand of Steiermark (Styria), mounted the throne as Emperor Ferdinand II, and brought his orchestra from Graz. He had sixty musicians, mainly Italians, who were then joined to the Vienna Court Orchestra under its conductor, Priuli. Vienna, which even under Rudolf II had had an Italian Court Conductor (Camillo Zanotti, 1586, Orologioo, 1603) now became an Italian music city and grew more so when Ferdinand II married a princess from Mantua. With the new empress came a whole suite of Italians\u2014secretaries, poets, musicians. It was the same under Ferdinand III whose wife, likewise, was an Italian.\n\nDuring the same time, Vienna developed into a capital where the destiny of European politics was decided. As early as 1438, Vienna had been the capital of the German Empire, and its rulers, who resided in the Imperial Palace of Vienna, were crowned with the crown of the German Emperors. In 1526, the Austrian Empire was formed, in which Bohemia and Hungary were joined to Austria. Ferdinand of Austria was also crowned King of Bohemia and Hungary, and the Slavic, Hungarian and German peoples were united. The great historical hour of Vienna had struck. The Emperors, who lived in Vienna, could consider themselves the center of the world. They were the German kings, the rulers of the Danube territories. They were the champions of the Catholic religion which, in Rome, assembled all its forces to fight against Protestantism. They felt themselves more than Austrian and German rulers. They had become monarchs of the European world. In the sixteenth century, Vienna became a supernational city which remained so until 1918.\n\nFor all Europe, these faithful rulers of the new monarchy waged the wars against the Turks who first besieged Vienna in 1527. All Europe waited for the news from Vienna when, in 1683, the Turks pitched their tents on the slopes of the Kahlenberg. The bells were rung throughout Germany and the Te Deum Laudamus was sung when the citizens of Vienna defeated the Turks and the great bells of St. Stephen's announced the freeing of the city. Again it was a European event when the Austrian armies of Prince Eugene drove the Turks out of Hungary and moved victoriously to the Balkans. As principal power of Europe, Vienna fought these battles\u2014and as leader of Christian thought in Europe.\n\nII\n\nDuring the sixteenth and seventeenth centuries, European history was formed in Vienna. Another important European event was the battle of the Austrian princes for the reestablishment of the Catholic religion. In Austria, in Bohemia, and in Hungary, Protestantism had swept through the nobility and the bourgeoisie. Vienna itself had maintained a Protestant administration. In Upper Austria and in Styria, almost the whole country had become Protestant. In 1554, in Hungary, a Lutheran was elected imperial chief minister. With fire and sword, Rudolf II and the gloomy, Jesuit-educated Ferdinand II led the war on Protestantism. The Protestants were expelled from their positions and from their homes, and were tried before clerical courts. Two hundred Protestant nobles of Bohemia were beheaded on the market-place of Prague after the disastrous battle on the White Mountain. Catholic priests journeyed through the land. Education was handed over to the Jesuits. Jesuits taught at the universities of Prague and Vienna, and in 1585, the University of Graz was founded expressly for them. There were church processions in the streets. Emperor Rudolf himself, carrying a lighted candle, used to march in such processions, even in the cold of winter. In 1578, when after a long period, the first Mass was read at St. Stephen's, Emperor Ferdinand walked behind the tabernacle in the long train of priests and brothers.\n\nIn the smallest Austrian town, one can still see to-day the power of this fanatical faith movement. In almost every market-place stands a \"Blessed Trinity Pillar,\" on which God is depicted wearing a long beard and sitting in the clouds, with Jesus, Holy Ghost and chubby-cheeked angels, as a memorial to the time of the Counter-Reformation. Naturally, the largest of these columns stands in Vienna. Emperor Leopold I had it erected on the Graben, not far from the Imperial Palace. It was designed by an Italian theatre architect, Burnacini. On it, too, the Trinity is enthroned in the clouds, and from the heights plunges an angel with shield and sword, and chases away the devil, who has the form of a shriveled old woman. The devil, of course, symbolizes Protestant Unbelief, and Emperor Leopold I, in full regalia, observes this edifying play, kneeling, his hands folded in prayer.\n\nAt this time, all Austria decked itself with new churches and in every town, even in the most remote alpine valley, a baroque church tower rose toward the sky. The great Benedictine and Augustine orders transformed their old monasteries into powerful baroque strongholds which, from their mountain heights, looked down over the land. In Vienna, between 1622 and 1627, almost every year a new monastery arose, and in some years, two.\n\nLike the Emperor and the Archdukes, the Viennese nobility also exhibited the new Catholic piety of the Counter-Reformation period. Many princes, like the Esterhazys, were reconverted to Catholicism from Protestantism, and let themselves be bathed in the sun of the Emperor's favor. Prince Piccolomini, the imperial commander in the Thirty Years' War, had the Serviten Church, one of the largest in Vienna, built at his expense. Another nobleman, Prince Paul Esterhazy, in 1692, with the inhabitants of all his lands, in all, eleven thousand two hundred people, undertook a pilgrimage to Mariazell, to which point Emperor Franz Josef, and the nobility of his time later made pilgrimages. Countesses and princesses, citizens, musicians, priests and three hundred thousand eight hundred and sixty boys walked in the procession, singing. The coat-of-arms of the Esterhazy family was borne at the head of the procession. Thus they traveled six days on foot through the Styrian mountains where the famous church stood, at the edge of a forest. Five hundred and ten men stretched out their arms in the form of a cross. Prince Paul Esterhazy, the wealthiest and most powerful man in Hungary who, at coronations, placed the crown on the monarch's head, walked behind his musicians in the festive train. That was the pathos of the seventeenth century and the fanatical faith of the time.\n\nThus Vienna was great in the sixteenth and seventeenth centuries, and the leader in political and religious thought. Here one never felt in national terms. The idea of the empire was a universal one, as was the idea of Catholicism. The ruler in Vienna who only in 1805 took the title \"Emperor of Austria,\" was \"Roman Emperor of the German Nation,\" and as such, the successor of the Roman Emperors and the crowned head of a universal monarchy, not a national state. As Roman Emperor, he also wielded the sword of the Catholic Church, which stood above all nations.\n\nAt this time, Italian music was not foreign music, but a universal language which was just as understandable in London and Paris, Madrid and St. Petersburg, as in Vienna. All modern forms of music were created in the sixteenth and seventeenth century Italy\u2014opera, cantata, concert and symphony, chamber-music and oratorio. Italian music was simply music for all Europe. In this sense it was taken up in Vienna at that period. Italian music was international just as the Latin language was. To Mozart, Italian music meant just that. Musical beauty, musical brilliance in its greatest perfection was, in Mozart, Italian music.\n\nIn this spirit, Vienna became the musical city of the world in the seventeenth century. It was the representative of general European ideas and was so considered by everyone who assembled there at the time\u2014by nobles and burghers, immigrants of all nationalities, the officials, the clergy, by Metastasio, the court poet, by Leibnitz, the philosopher, by the musicians, by the adventurers like Casanova and Da Ponte. In Vienna, the politics and business of all nations met.\n\nEven in the age of the musical classicists, this state of thought was prevalent. Beethoven often said sharp words about the Vienna court and the Austrian state. Once he spoke of the \"complete moral corruption of the Austrian state.\" Nevertheless, Beethoven did not designate himself as a German musician but as an \"Austrian musician\" (Letter of January 8, 1812). And in 1809, one reads in Beethoven that he would never \"cease counting himself among the Austrian artists.\" At this time, the Austrian Emperors were no longer German Emperors. But Austrian soil was always being considered universal by anyone who made his residence there.\n\nThis conception of Vienna and Austria rose, in the sixteenth and seventeenth centuries, to brilliant heights, and with it, the conception of a musical city whose task it was to cultivate music in the largest and most universal sense.\n\nIII\n\nThe highpoint of this development was achieved between 1640 and 1740. Music appeared in all phases of Viennese life.\n\nBetween 1640 and 1740, four Austrian Emperors were themselves composers, Ferdinand III (1637-1657), Leopold I (1658-1705), Joseph I (1705-1711) and Charles VI (1711-1740). The Thirty Years War, the Turkish Wars, the War of the Spanish Succession, did not hinder them from covering manuscript paper with notes. Leopold I left behind him sixty-seven church sonatas, one hundred and fifty-five opera arias, nine Theatrical Festivals, and seventeen collections of ballet music. When a member of the imperial house died, he would compose funeral music. Archdukes and Archduchesses were required to learn singing and dancing. Leopold's sisters, Marianne and Leonore, took part in the court cantata performances. In the city of the Music Emperor, musicians from everywhere had congregated. Browne, a traveler who journeyed through seventeenth century Europe, wrote in his account of Vienna that \"there are so many musicians in Vienna\" that \"it would be difficult to meet more of them anywhere else.\"\n\nThe majority of these musicians were Italians. They were the court composers and conductors. The long line of these prominent court musicians begins with Giovanni Valentini (1629 or 1630), and progresses to Bertali (1605-1669), Johann F. Sances (1600-1697), Marcantonio Cesti (1666-1669), Antonio Draghi (1635-1700), P. A. Ziani (1706-1711), Antonio Caldara (1670-1736) and Francesco Conti (1682-1732). In addition, came Italians as court poets, among them the greatest Italian poet of his time, Pietro Metastasio; and the Burnacini family as theatre painters and festival designers. One understands how a biographer of Emperor Leopold I can say: \"The German language is, in Austria, almost in a foreign country.\" Vienna where, under Leopold I, Spanish was spoken, was precisely more and something else than just a German city.\n\nThe whole glory of the court unfolded at opera performances. The number of opera and oratorio performances under Leopold I in the time between 1658 and 1705 was over four hundred. Twelve to fourteen new operas were performed between 1701 and 1711 under Joseph I. The scenery of these operas amazed the world. When Lady Montague visited Vienna, she wrote, on the opera, that never in all Europe had she seen anything which could compare with it in splendor and beauty. Opera performances were festival music at court christenings, engagements, weddings and birthdays. With their mixture of Greek mythology, arias, choruses and ballets, with scenic wonders, \"flying machines\" and numerous scene changes, with their homage to the Emperor, and their processions, they surely were colorful show-pieces, like the great ceiling paintings in the new Viennese palaces, where the heavens opened and all Olympus was visible.\n\nThe performance of Cesti's \"Il Pomo Oro\" (The Golden Apple) in 1666 was the wonder of all Europe. For this performance, a special theatre was built with three galleries, and with the portal of a Greek temple in front of the stage. On the canopy above the stage was the double-eagle of the Emperor with the sceptre and the imperial globe in his talons. In the orchestra sat Emperor Leopold, surrounded by halberd bearers, with his newly-married bride, the Spanish Infanta Margarita Teresa, and the entire court. In the opera's prelude, the might and brilliance of the court were celebrated. In the center of the stage stood an equestrian statue of the Emperor. On both sides, in pillared halls, were portraits of earlier Hapsburg rulers. In the clouds, \"Austrian Fame\" rode on Pegasus, accompanied by Amor and Hymen. On the stage, Austria's possessions\u2014Austria, Hungary, Bohemia, Italy, Sardinia, Spain and America\u2014sang. There was a festive atmosphere in the hall and on the stage. The opera formed a part of life, and was the jewel of a festive hour.\n\nAfter the prelude, Burnacini's colored scenes unfolded: Earth, Heaven and the Underworld, temples, harbors, caves, woods and villas. In the sixty-seven scenes of the opera, there were always new wonders to be seen\u2014lightning, rain, hail, the Furies fleeing in the Underworld, Neptune and Venus in their chariots, accompanied by Tritons and Nereids, the ship of Paris battling the sea-waves. At the end, all the members of the imperial house, past and present, appeared in the clouds and the chorus announced the great future of the children which would stem from this new marriage.\n\nMusic was entwined with all the events of court life. Ballets were introduced by opera scenes. Thus in 1681, on the Emperor's birthday, a \"Baletto di Fate\" (Ballet of Elves) was performed, and its introduction was a scene from Draghi's opera, \"L'Albero del Ramo Oro\" (\"The Tree with the Golden Bough\"). A year later, again on the Emperor's birthday, a ballet was performed, preceded by an opera scene, from \"Li Tributi\" (The Tributes) by Draghi. In 1685, there was still another birthday ballet for the Emperor. Vulcan, Peace and War sang their arias. There were similar operatic prologues and intermezzi in the Vienna court theatrical productions, and many of the opera scenes were sung by archdukes and archduchesses. When, in 1682, the ballet \"Il Sogno delle Grazie\" (The Dream of the Graces) was produced at the Castle of Laxenburg. Archduchess Marie Antonia and her ladies-in-waiting sang the arias and ensembles of the operatic introduction. Emperor Charles VI himself conducted opera performances at his court. A pompous man, he sat at the piano and directed the performance of Caldara's \"Euristeo,\" in which archdukes danced in the ballet and nobles sang. The court poet, Zeno, paid the Emperor the compliment that he played like a professor, which pleased the Emperor far more than a victory on the battle-field.\n\nThe imperial summer palaces, too\u2014the \"Favorita\" and Laxenburg\u2014were the scenes of many opera, ballet and cantata performances. The last opera at \"Favorita\" was given in 1740. On that occasion, the opera \"Zenobia\" by the vice-conductor, Predieri, was performed. In the same year, Charles VI died, surrounded by music, in his summer palace. Wherever he traveled, his opera singers and orchestra accompanied him. In Linz, Prague, Wiener-Neustadt and in Oedenburg, there were court performances. When, in 1723, the imperial court journeyed to Prague for the Emperor's coronation, and, on the Empress's birthday, arrived at Znaim, a stage was erected on a wagon and Caldara's festival opera, \"La Concordia dei Planeti\" (The Concord of the Planets) was performed.\n\nThe imperial parks resounded with arias. In 1679, a Greek temple was built in the park of Sch\u00f6nbrunn, where Draghi's opera, \"Il Tempio di Diana\" (the Temple of Diana) was given for the Empress's birthday. Iphigenia sang her arias on the steps of the temple under the trees. In the same park, an opera by Draghi and Minato, \"Il Trionfator di Centauri\" was sung in 1674, and pixies, nymphs and centaurs danced under the trees. The sandstone statues which were erected later in the Park of Sch\u00f6nbrunn were all mythological opera figures. Similar statues, costumed as opera singers of the baroque period, stand on the walls of the Vienna nobles' palaces which were built during this period, as a reminder of the epoch where the atmosphere of every festive moment became music.\n\nThe great equestrian ballets did not lack music, arias and choruses. The most famous performance of this kind was \"La Contesa di Aria e dell'Aqua\" (The Contest of Air and Water) of 1667. The best theatre architect of Italy, Carlo Pasetti, was called to Vienna to stage the ballet. On the fortress square, he built a temple of Diana, made of marble, bronze and lapis lazuli, in which the chorus, hidden, sang. Court-conductor Bertali composed the music. The text of the operatic introduction was written by the court-poet, Francesco Sbarra. The performance began with an aria sung by \"Fame,\" who, holding a golden trumpet, arrived on the festival square in the ship of the Argonauts, heralded by trumpet fanfares. Then came riders who performed jousts, and finally the Emperor himself as Roman Imperator, wearing a golden crown, and gold breast-plates adorned with a rose formed of diamonds and topazes, carrying his gold sceptre. He rode a black horse whose bridle and saddle shimmered with precious stones. He was preceded by trumpeters, and behind him, the members of the Austrian nobility, whose robes were studded with jewels and whose horses were caparisoned with gold. They were followed by the sixty men of the Emperor's mounted body-guard, then a carriage drawn by eight white horses, and finally by four \"Saltatori\" (Dancers) whose horses walked in Spanish gait. The actual equestrian ballet commenced to the measures of the ballet-music which Heinrich Schmelzer had composed.\n\nA few months later, a similar equestrian ballet was performed in the park of \"Favorita.\" In this ballet, which had been composed by Cesti, the Emperor, with the Duke of Lorraine, also took part. Again the horses gleamed with gold and the riders with gems. On one of the carriages, \"Germania\" sat enthroned, the sceptre in her right hand, the globe in her left. On both sides of the goddess stood the Austrian provinces, with shield and sword. Germania sang an aria praising peace. Then followed the ballet of the Emperor, the Duke of Lorraine and the knights, and at the end, the Emperor rode up to the Empress and paid her homage.\n\nAll these festivals and musical events were not entertainment or some special province parallel to existence, but life itself in its most festive hour. The Vienna Court Orchestra was the most famous in Europe. In the middle of the sixteenth century, it had thirty-six members; in 1705, one hundred and five; in 1723, one hundred and thirty-four. Emperor Leopold I allotted seven thousand more gulden to the orchestra than he did to the Imperial Privy Council\u2014a total sum of thirty-six thousand gulden. Each day the orchestra played sonatas during meals in the palace, and on New Year's Day, when the court ate from golden dishes, special festive music was performed.\n\nThe last of the great musical emperors in Vienna was Joseph II. He had studied music with Gassmann and had Salieri as his adviser. Every day, according to schedule, he devoted an hour to music, and three times a week, opera excerpts were performed at which the Emperor accompanied on the piano and sang arias. Occasionally, he had music played in the Park of Sch\u00f6nbrunn where the people could listen. This was the Emperor who sat in the box of the National Theatre when Mozart played the piano there, and performed his \"Marriage of Figaro\" for the first time. Emperor Joseph's brother was the Archduke Maximilian who in 1785 was Elector of Cologne, and as such, was the Lord of young Beethoven. As member of this ruler's court orchestra, Beethoven wore its uniform\u2014green frock coat, green breeches, white silk vest, shoes with black laces, and a sword. Like Haydn and Mozart, Beethoven, too, grew up in the baroque society which came to power in Vienna during the seventeenth century, and which transformed the imperial city into a musical city.\n\nIV\n\nThere was a great deal of music, also, in the palaces of the nobility. After 1683, when the Turks had been expelled from Vienna, the aristocrats built, in the city and its environs, their large new palaces with Caryatides framing wide portals, and mythological gods on the roofs. Prince Eugene had three palaces built, Prince Esterhazy three, the Princes Liechtenstein, two. The Lobkowitzes, Morzins, Waldsteins and Kinskys built in Prague. In 1683, the Esterhazys began a great castle in Eisenstadt, before whose gates their own soldiers kept watch; and in 1766, they built another castle on the Neusiedler Lake which could only be compared with the Palace of Versailles. Its magnificent park had a theatre and a marionette theatre. Many of these palaces had their places in musical history. Gluck and Mozart conducted performances in the Auersperg Palace in Vienna. Haydn conducted in the Esterhazy Palace, and was resident conductor at Count Morzin's summer palace. As a young musician, Gluck was at home at the Lobkowitz Palace in Prague.\n\nEven in Leopold I's time, there were aristocrats in Vienna who composed. Baron Storzenau composed ballet music for one of Draghi's operas (1699). Heinrich Gottfried von Kielmannseg wrote an overture for a Serenade which was held in 1681 at the \"Favorita.\" Nobles and emperors were musicians. Where else in the world had there ever been anything like this?\n\nIn Moravia, Count Johann Adam von Questenberg built a castle in Jarschmeritz where he engaged instrumentalists for performing operas, whom he had ordered from Vienna, Venice, Rome and Parma. His conductor was his valet, Mitscha, who composed operas, congratulatory cantatas, ballets and church music. In 1728, the music-loving count ordered an opera by Court-Conductor Caldara for which the honorarium was two hundred and fifty gulden. Count Rottal of Holleschau, Moravia, also maintained an orchestra during Charles VI's reign. Holzbauer, the symphonist, was a member of this orchestra before he went to Mannheim and there became a predecessor of Haydn. Another Mannheim composer, Franz Xavier Richter, like Holzbauer, played in the Holleschau Orchestra before establishing himself in Mannheim. Each of these castles was a center of culture from which music poured forth.\n\nMusic spilled forth, too, over all the Austrian provinces, from the churches, monasteries and the ecclesiastically-directed cloister schools. The importance of the ecclesiastical school in the development of Vienna as a music city is still scarcely appreciated. One must only consider that in the seventeenth century in Austria, the teaching in high schools and universities lay entirely in the hands of the clergy, and that music there was a very important subject, to realize the meaning of these schools. All the aristocrats whom we have met as enthusiastic friends of music were educated in such schools.\n\nIn Vienna and Austria of the seventeenth century, ecclesiastical society developed side by side with the worldly. Like the worldly, ecclesiastical society was powerful, brilliant and wealthy. The old monastic buildings of the Danube valley and of the woodland valleys of the Austrian Alps were always rebuilt in the magnificent baroque style which, in Rome, had become the expression of a new and passionate Catholicism. At the beginning of the seventeenth century, new monasteries were built in Vienna. Ferdinand II, the fanatical general of the Counter-Reformation in Austria, endowed the building of the Capuchin Monastery where the Hapsburgs were laid to rest (1632) and the Carmelite Cloister in Vienna (1632). He had a great church and school building erected for the Jesuits (1628-1631), and a church and monastery for the Netherland Paulist monks. His successor, Ferdinand III, built the cloister of St. Anna, the Augustine monastery of St. Rochus and the new Dominican Church (1631). Leopold I had the Carmelite monastery of St. Joseph built. He had the old gothic church of the \"Nine Angel Choirs\" in Vienna rebuilt in baroque style, like the professed house of the Jesuits nearby. At that time, the pompous facade with its angel figures, which is one of the great monuments of the Counter-Reformation, was constructed. The same emperor built the monastery of the Black Spanish monks, the house where Beethoven died during a thunderstorm; the Barnabite monastery of the White Spanish monks, the monastery of the Spanish Piarists. The widow of Ferdinanl III had the Ursuline Convent built.\n\nIn 1739, the rebuilding of the Melk Monastery began. When one travels along the Danube toward Vienna, one can see this mighty monastery from far off, where the Danube bends toward the woods and vineyards of the Wachau. The great baroque terrace overlooks the Danube valley, and between the summer vestry, which had been decorated with light frescoes by the theatrical engineer of the court, Beduzzi, and the library, stood the Diezenhofer church. This monastery is like a fortress of baroque Catholicism, splendid and festive, like theatrical decoration.\n\nNot far from this most beautiful of all monasteries is the monastery of St. Florian, which had been begun in 1686 by Carlone of Milan, with a wonderful stairway, marble pillars, Gobelins and frescoes, and with the beautiful church, and the organ under which, as he had wished, Anton Bruckner's coffin was laid to rest. From this his church stems the colored gleam of the Bruckner symphony since the Austrian baroque spirit had lasted into Franz Josef's time.\n\nV\n\nThere was, of course, music in all these churches. It was the festive church music of the seventeenth century, with many choirs, brilliant orchestra and arias full of feeling. Music was zealously cultivated. In the Kremsmuenster Monastery, Italian operas were performed by students in the monastery theatre. From this period, 1651-1780, more than one hundred text-books of such opera performances have come down to us. Young monks were sent by the monastery to Italy for musical training. The same practice held true in the monasteries of Lambach and St. Florian. Leopold I attended an opera performance at St. Florian in 1680. In Salzburg, the Benedictine University was opened, and in 1636, the Archbishop presented it with a great hall and stage where ecclesiastical plays with music were produced. Another small theatre was opened at the university in 1657, a third large theater with twelve back-drops in 1662. Forty poet monks who wrote plays for this theatre are known to us, and thirty composers. All these theatrical pieces were operatic in character. Figures from Greek history and Roman sagas were the main personages involved. Genii came flying through the air and sang arias of homage to the Archbishop, who sat in the orchestra with the clergy. In one of these plays, which took place in the period of the Turkish Wars\u2014\"Corwin,\" produced in 1688\u2014there are thirty-eight vocal roles which were sung by students and singers of the court. Mozart had seen such plays at the Salzburg University Theatre. When he later composed the \"Magic Flute,\" he remembered this well, for his charming, fairy-tale opera originates in this type of ecclesiastical opera of the baroque time.\n\nThe most important clerical order of the seventeenth century was the Jesuit. Almost all Austrian teaching lay in the hands of this order which had placed all the modern means of the theatre and of opera in the service of educating the young. The most famous of the Jesuit schools in Rome\u2014the \"Collegium Germanicum,\" had its musical tradition. Carissimi had been active there as conductor. Ecclesiastical operas were splendidly produced there, and similar performances were given in all Jesuit schools, in order to win souls for Catholicism, or to reconvert them. In Vienna, the Jesuit performances comprised the second Court Opera Theatre. The Emperor was regularly present, and with him, the entire court. In 1620, the Jesuits built their own theatre in Vienna, and in 1650, a small practice-theatre for the students. Young Gluck received his first impressions of opera at a Jesuit school in Komotau, Bohemia.\n\nDuring Leopold I's reign, in the Jesuit theatre in Vienna, the work, \"Pietas Victrix\" of the South Tyrolean Jesuit poet, Avancini, was given. This ecclesiastical festival opera dealt with the victory of Christianity over the Roman Emperor Constantine. It was a festival play, with opera music, dance and stage mechanics. One saw land and sea battles, flying angels, and heard the arias and choruses of the students. A prologue began the piece, and an allegoric intermezzo glorified the verity of Catholicism. The division of the place of action into a heavenly and earthly world, as was customary in Jesuit pieces, had its effect upon the Vienna folk theatre. One can still find this division in the fairy-plays of Raimund. Goethe, too, employs the device in \"Faust.\" Thus lasting effects upon theatrical history have been produced by these Jesuit pieces.\n\nThe spiritual plays which were performed during Lent in the Vienna churches, and especially in the Court Chapel, also had operatic forms. In every Viennese church, a holy sepulchre was put up during Holy Week, before which the devout prayed. Behind the sepulchre a stage was erected, on which there was a cross; and beneath it Mary, the saints and allegoric characters sang of penance and redemption in opera arias. All the Italian court poets of Vienna, like Minato, Stampiglia and Pariati, made texts for these spiritual operas, and the court-composers wrote the music. Emperor Ferdinand III's widow, Eleanora, specially furthered these spiritual operas. She wrapped herself in a veil of gloomy piety and spent her last years in the cloister of the Ursulines. Holy services and pious meditation filled her life. The court-composer Draghi wrote ecclesiastical operas for her which were performed in the chapel during Easter Week. Even when she traveled, she never missed an opportunity of hearing religious operas. When she journeyed to Prague in 1680, her orchestra had to perform opera by Draghi and Minato on Holy Thursday. It was called \"La Sacra Lancia\" (The Holy Spear). The Catholic Church militant, Austria, Bohemia and allegorical figures like \"Regret\" and \"Devotion\" sang in this ecclesiastical opera, which was in costume. The scene must have looked like one of Guido Reni's pictures, with its saintly figures, come to life upon the opera stage.\n\nSometimes members of the nobility sang leading roles in these operas. In 1689, \"Le Cinque Virgine\" was performed at the monastery of St. Joseph in Vienna. The main roles were sung by the Countesses Czernin, Rappach, Fuersteinberg, Kinigl, and Opperdorf. In 1723, on the birthday of Charles VI, the \"Melodrama di Sancto Wenzeslao\" was given at the Jesuit monastery in Prague. The music was written by the Czech composer, J. D. Zelenka in which the Bohemian St. Wenceslaus sang Italian arias.\n\nVI\n\nThe \"people\" of Vienna were separated from this courtly-aristocratic society. Only the music of the churches was equally accessible to both strata of Viennese society since, as in Italy, the church was the concert hall of the people.\n\nEverywhere on Austrian soil during the seventeenth century, we encounter the music of the folk\u2014or better, the music of the nations who were united in Austria. The ballet conductor of Emperor Leopold I, Heinrich Schmelzer, made use of much of this folk-music in his works. One time one finds an \"Aria Viennese,\" another time \"Hannakische Themen\" (in the Kremsiere Suites); another time an \"Aria Styriaca.\" In Hungarian manuscripts of the seventeenth century, one finds Hungarian dances, Slovakian songs and dances, and gypsy music; a Wallachian dance and several Polish dances. Naturally, the greater part of this folk-music was not written down. It was played at village inns, fairs and weddings by folk-musicians upon whom the art-musicians looked down. In 1839, Lanner, the originator of the Viennese waltz, was rejected by the Vienna Union of Musicians\u2014the \"Tonkunstlersocietat,\" because \"he's in dance music.\" Only in Haydn does this folk-music flow out into symphonic form, and in Bruckner, who had played at peasant weddings in his youth\u2014in the scherzi of his symphonies.\n\nThe Viennese folk sang the songs which the \"Hanswurst\" (Harlequin) sang in the folk theatres. In 1686, the first \"Song Book\" was printed in Vienna. In 1683, the melody of the \"Prince Eugene Song\" was known all over Vienna, and even in the pious pieces of the Jesuit theatre, songs in Viennese dialect were interpolated; and in 1685, Viennese market-scenes were performed with songs as \"Intermezzi.\"\n\nThus Viennese music, even in the seventeenth century, had its foundation of folk-like quality. It had strong roots in the earth, in the fields and woods, and grew from here, a century later, into the music of Haydn, Mozart, Beethoven and Schubert. Only thus did music become a great whole. Only in this way did Vienna become a music city. Only thus did Haydn's symphonies, Mozart's \"Entf\u00fchrung aus dem Serail\" (\"Abduction from the Seraglio\" ) and \"Magic Flute,\" Beethoven's dance themes, Schubert's \"Heidenr\u00f6slein,\" the waltzes of Lanner and the two Strausses come into being. The great, festive, brilliant style of the baroque which loses none of its strength in Bruckner, and the liveliness of the folk-music together left their marks on Vienna's music and music-making.\n\nTo the wealth and the originality of their folk-music, Vienna and Austria owes the fact that here alone, of all the other cities and lands of Europe, the great Italian period of the seventeenth century did not remain a passing episode. Classical music drew everything together on Viennese soil\u2014great baroque music and folk-music. In works in a grand manner, Viennese classical music is always baroque, because all forms of baroque are employed in it. Though the religious atmosphere of baroque music changed into a philosophical, humanistic atmosphere, in the greatest works by classic masters, in the adagios of Haydn string quartets and symphonies, in Mozart's \"Requiem\" and \"Magic Flute,\" in Beethoven's Ninth Symphony, in Hugo Wolf's \"Ecclesiastical Songs\" and \"Michel Angelo's Songs,\" the full religious baroque atmosphere is still there. Baroque brilliance and feeling flows through all of Mozart's music and gives it its sunny brightness. Bruckner unconsciously returns to baroque style which had been imparted to him at the festival masses at Linz and St. Florian by the swelling organ sound and by his own strong Catholic faith. Vienna has always remained a baroque city. When one analyzes art, whether music\u2014as it is here the case\u2014or theatre, painting or architecture, one keeps encountering its baroque sources, just as one finds in the features of grandchildren the traits of their ancestors. Just as Viennese life and social order maintained their seventeenth century forms until 1918, these forms were preserved in music until the first World War, the social and political battles, the revolution of the world, and the great revolutionist, the Viennese Arnold Sch\u00f6nberg, destroyed the old order by force.\n\nVII\n\nWe wished to recognize the law of life of the music city, and trusted ourselves to the direction of history. Now the time has come to draw conclusions. Before us as an entity lies the musical life of the great city. From the end of the sixteenth century on to the beginning of the twentieth, it is these same musical forces which were active in Vienna. Generations come and go. The law remains.\n\nWe call this law, too, the personality of a city. Cities are living beings\u2014\"Stamped from that living develops,\" as Goethe put it. We say the names: Rome, Naples, Florence, Athens, Paris, London\u2014and the cities rise up in our minds. They have their distinguishing, outstanding features like impressive people who, upon entering a room, immediately attract attention to themselves. Such a city is musical Vienna. Even in the seventeenth century, its definite individuality could be noted when all Europe began to recognize it as a city which was destined to make music which belonged to the world.\n\nThere were three things which caused Vienna to appeathen, as the great music city. First, it was not a national city and not the center of life of a single nation, but an international city. It was the cross-roads of Europe to which not only peoples and goods journeyed, but also ideas and artworks.\n\nSecond, Vienna connected music closely with life and drenched it with music. High society as well as common people adorned their whole lives with music. Vienna was unique in this connection of every-day life with music. The whole city was like a sounding-board and vibrated with music like an Italian violin.\n\nThird, only in Vienna was the music of the folk the foundation of the great musical life. In Vienna, folk-music nourished opera, symphony, chamber-music and art songs. There is no city in Europe where the case has been similar.\n\nFrom this, the following conclusions for the future of Vienna's music life may be drawn:\n\nOf all the foundations on which musical life in Vienna has been developed for more than three centuries, one has been preserved unchanged: the folk-earth from which Vienna's music has grown. Vienna's old society has been destroyed. There is no Emperor, no court, no nobility, no feudal society. The baroque Vienna is only a memory, a closed chapter of history, of which only the old buildings, the palaces and churches remind us. After the upheavals of the great wars, in Vienna the society of the people will come to power and complete the rising which began about 1900. The character of the Viennese man of the people has not changed in the course of time. From the middle ages to the nineteenth century, it has kept its essential qualities and will also survive the great social revolutions of our time.\n\nThere are two more conditions which are demonstrated by Vienna's musical history. Vienna did not become a great musical town on a national basis, and it can only become a great musical town again on a super-national basis. Now, as before, the destiny of Vienna is pre-designed by its geographical situation on Europe's largest river which flows from west to east, and on whose banks various nations make their homes. Only as a European music town which absorbs and forwards all movements of European intellectuality can Vienna again become a music city which means something to everybody. Since the world in this age of telegraph, radio and airplanes, has become much smaller, this task is an easier one than it was in the past. Europe has become small and the countries on the other side of the ocean have moved much closer. This will enrich the spiritual life of Vienna. It will be able to remain a world town in the field of intellect if it preserves its talent of absorbing what new and great things are created in the world. And if, after the comparatively short period during which German Nationalism has oppressed Vienna, it again finds its unprejudiced sense for the characteristics of other nations, it will again discover its peaceful and human way of thinking.\n\nThe third condition for the re-emergence of Vienna as a music town is its capability of combining music with folk-society and daily life. Vienna had already proved after World War I that it possessed the will and ability to foster music as a public task. It was conscious of its mission and one can expect that the old city of world music will clearly recognize that it will have to assemble all its powers to make music for the world on its new social basis: the powers of the state, of the city and of the representatives of the people.\n\nThat is what history teaches. Out of war and devastation, need and social struggles, a new time is dawning, a new music and a new relationship between music and society. The old music city has become a legend which, like all legends, enchants us with its color and splendor. And if I told that legend, it was in order to show where the path to the future is leading. It would be bold to try to predict this future. We can only wish, hope and work. Vienna still stands as the old city, the same friendly landscape still surrounds it, the same people still live there, and the same river flows by. The same music is in the earth and in the air, in rustling leaves and rushing wind, in Beethoven's forests and in Schubert's vineyards, in Hugo Wolf's gardens and Anton Bruckner's churches. Great composers may hear this music and listen to the melodies. It is they who will add a new chapter to the glorious old book: \"The Legend of a Musical City,\" the leaves of which the roaring storm of our time is turning up.\nCopyright 1945\n\nBy PHILOSOPHICAL LIBRARY, INC.   \n15 East 40th Street, New York, N. Y.\n\nBy F. HUBNER & Co., INC., Printers   \nNew York, N. Y.\n\nISBN: 978-0-8065-3717-7\nNotes\n\n\"Ah, lovely Phyllis hear,  \nAh, hear our music sighing.   \nCome let us pass a night   \nTogether sweetly lying.\"\n\n(Thus closes the poem of the saintly folk-preacher.)\n\n\"I praise this place above all lands,  \nHere are many singers, players of strings,   \nEverybody enjoys life.   \nMore music and instruments surely can't be found elsewhere.\"\n\nGoldmark's Memoirs were translated into English by his niece, Alice Brandeis, and published in New York, 1927. (\"Notes of a Viennese Composer.\")\n\n\"What would you, I am their leader. I have to follow them.\"\n\n\"Then we will drive to Nussdorf.\"\n\n2 \"As long as the old Stephen.\"\n\n3 \"Human, human are we all.\"\n\n4 \"The boys from Linz.\"\n\n5 \"Good Father in Heaven.\"\n\n6 \"The old stroller.\"\n\n7 \"Vienna, Vienna, you alone.\"\n\n8 \"There will be a wine.\"\n\n9 \"My mother is a Viennese.\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTranscribed from the 1825 George Eyre and Andrew Strahan edition by David\nPrice, email ccx074@pglaf.org.  Many thanks to Royal Borough of\nKensington and Chelsea, Local Studies and Archives, for allowing their\ncopy to be used for this transcription.\n\n                   [Picture: Public domain book cover]\n\n\n\n\n\n                                    AN\n                                   ACT\n                            TO PROVIDE FOR THE\n                      PAVING, GRAVELLING, LIGHTING,\n                               AND WATCHING\n                    Certain Footways and Carriageways\n                               IN AND NEAR\n                             BROMPTON SQUARE\n                             IN THE PARISH OF\n                      SAINT MARY ABBOTTS KENSINGTON,\n                       _IN THE COUNTY OF MIDDLESEX_\n\n\n                          AND TO PROVIDE FOR THE\n\n                  Maintenance of a Garden and Shrubbery\n\n                           IN THE SAID SQUARE.\n\n                          [Picture: Royal Arms]\n\n                                _LONDON_:\n                PRINTED BY GEORGE EYRE AND ANDREW STRAHAN,\n              PRINTERS TO THE KING'S MOST EXCELLENT MAJESTY.\n\n                                  1825.\n\n\n\n\nINDEX.\n\nA.\nACCOUNTS, Book of, to be open to the gratuitous                      8\nInspection of the Trustees and Persons rated.  _See_\nTrustees.\n\u2014 Copies and Extracts may be gratuitously taken by duly          ibid.\nqualified Persons.\nACT, Limits of, defined.                                             2\n\u2014 Persons for putting the Act in Execution defined,               ibid\n\u2014 Expense of obtaining, &c. to be paid out of the first             43\nMonies raised by virtue of the Act.\n\u2014 to be deemed a public Act, and judicially taken Notice            44\nof as such, without being specially pleaded.\nACTION.  In every Action, the Designation of Defendant as           30\nset forth by the Act, being indebted to the Plaintiff in\nthe Sum, &c. shall be sufficient.  _See_ Clerk,\nTreasurer.\nACTION or SUIT, proper Names of, to be given.  _See_                41\nNotice.\n\u2014 not to be commenced after a Tender of sufficient Amends        ibid.\nto the Parties aggrieved, or (on their Behalf) to their\nAttorney or Agent.\n\u2014 not to be commenced after the Expiration of Six Months            42\nnext after the Fact committed.\n\u2014 the Venue to be laid in the County where the Cause of          ibid.\nAction shall have arisen, and not elsewhere.\nACTS.  _See_ Proceedings.\nADJOURNMENTS.  Neglect of, &c.                                      14\nAGGRIEVED.  Persons aggrieved by any Rate or Assessment             38\nafter Payment thereof, may appeal to the Trustees at\ntheir next Meeting.\n\u2014 Persons aggrieved by any Decision of the Trustees or              38\nJustice of the Peace in the Execution of this Act, may\nappeal to the General Quarter Sessions to be held next\nafter the Expiration of One Calendar Month from the Time\nwhen the Cause of Complaint shall have arisen, but not\nafterwards.\nAMBASSADOR, Houses occupied by, the Landlord or                     26\nProprietor of such Houses to be rated as the Occupier.\nAMENDMENT of any Rate, Assessment, or Matter of Form in         38, 39\nany Conviction or Proceeding connected with the Execution\nof this Act, may be made by the Justices at the Quarter\nSessions.\nANNOYANCES, &c.  Penalty for.                                       35\nANNUITY, Form of the Grant of.  _See_ Money.                        30\n\u2014 Money raised by way of Annuity, to be deemed Personal          ibid.\nEstate.\n\u2014 Restrictions upon the granting of\u2014not to be granted for           31\nthe Lives of more than Two Persons under the Age of\nThirty-five.\nAPPEAL may be made to the Trustees by Persons aggrieved             38\nby any Rate made in pursuance of the Act, at the next\nMeeting of the Trustees, after Payment of such Rate or\nAssessment.\n\u2014 may be made against Decision of the Trustees or any            ibid.\nJustice of Peace, at the next Quarter Sessions.\n\u2014 to the Quarter Sessions against Decision of Trustees or        ibid.\nJustice, not to be made later than One Month after the\nCause of Complaint arising.  _See_ Aggrieved.\n\u2014 to the next Quarter Sessions against Decision of              38, 39\nTrustees or Justice\u2014the Appellant to give Fourteen Days\nwritten Notice of the Appeal and of the Matter thereof to\nthe Clerk of the Trustees, and within Five Days after, to\nenter into a Recognisance, with Two sufficient Sureties,\nbefore a Justice of Peace.  _See_ Sureties.\n\u2014 to the Quarter Sessions, &c. if against a Conviction,             39\nthe Sureties of the Appellant to be conditioned to pay\nthe Penalty, if the Conviction be affirmed.  _See_\nSureties.\n\u2014 Matters of, to be determined in a summary Way by the           ibid.\nJustices at the General Quarter Sessions.  _See_\nJustices.\nAREA, enclosed, of Brompton Square, first Expense of                43\nenclosing, &c. not to be defrayed by the Trustees.  _See_\nTrustees.\n\u2014 Provision in favour of the Occupiers of Houses having             22\nInterest in the enclosed Area, previously to the passing\nof the Act.\nARMS and Clothing for Watchmen to be provided.                      17\n\u2014 for Watchmen, &c. vested in the Trustees.                         23\nASHES, DUST, &c.  _See_ Mud.\nASSESSMENT.  _See_ Rate.\nB.\nBASKET.  _See_ Stall.\nBOG-HOUSE, or NIGHT-SOIL.  Penalty \u00a35 for emptying it,              33\nexcept between the Hours of Twelve at Night and Five in\nthe Morning.\nBONFIRE making, &c. in Brompton Square.                             32\nBOOK.  _See_ Names.\nBREAKING in. &c.  _See_ Horse.\nBROMPTON SQUARE, STREETS, &c. on the Land claimed to be              1\nseised by Wm. Farlar, Esq. partly laid out prior to the\npassing of the Act.\nBUILDING MATERIALS.  Penalty not exceeding \u00a35, for                  33\nplacing them within the Jurisdiction of this Act.  _See_\nProvision.\nBUSINESS, Room or Building may be hired for Transaction             41\nof.  _See_ Trustees.\nC.\nCARRIAGEWAYS, &c.  Trustees to contract for cleansing               21\nthem.  _See_ Footways.\nCARPETS.  Beating or dusting them, &c.  Penalty \u00a35 for.             32\nCHAIRMAN, how to be appointed.                                       4\n\u2014 to have a casting Vote.                                            5\nCHAISE.  _See_ Coach.\nCHELSEA WATER-WORKS.  _See_ Water-Works.\nCINDERS.  _See_ Dust.\nCLERK to be appointed, and how.                                  ibid.\n\u2014 &c. not to be Treasurer.  _See_ Trustees.                          7\n\u2014 to keep a Book of Accounts.  _See_ Trustees.  _See_                8\nInspection of Accounts.\n\u2014 to the Trustees, his Death or Removal, or that of the             40\nTreasurer, no Ground for the Discontinuance of an Action.\n\u2014 to be deemed Plaintiff or Defendant in an Action, but             41\nnot personally answerable for Damages.  _See_ Damages.\n\u2014 competent Witness in an Action, though nominally               ibid.\nPlaintiff or Defendant.\n\u2014 to enter Contracts in a proper Book.                              23\nCLOATHING.  _See_ Arms.\nCOACH, Chaise, Waggon, Cart, Truck, Wheelbarrow, or other           34\nCarnage whatever, driven upon any Footway within the\nJurisdiction of the Act, Penalty for, not exceeding \u00a35.\nCOACH-HOUSES.  _See_ Stables.\nCOLLECTOR of RATES to be appointed, and how.                         5\nCOMPLAINTS, Costs incident to, how to be levied, if not         36, 37\npaid on Conviction.  _See_ Penalties.\nCONTRACT for doing WORKS, &c. connected with the                    23\nExecution of the Act, may be yearly, or otherwise.\n\u2014 &c. not to continue in force for more than Three Years         ibid.\nfrom the Time of contracting.\n\u2014 all Contracts for doing Works, &c. connected with the          ibid.\nExecution of the Act, to be entered in a proper Book.\nCOMMISSIONERS of SEWERS.  _See_ Sewers.\nCOMPENSATION for special Damage done by Parties'                    39\nirregularly distraining.  _See_ Irregularity.\nCONVICTION of OFFENDERS, &c.  _See_ Justice.\n\u2014 Form of.                                                       ibid.\n\u2014 Appeal against.  _See_ Appeal.                                    41\nCOSTS and EXPENSES respecting the Contamination of Water            16\nby any Escape of Gas, to be determined (if requisite) by\na Justice of Peace.  _See_ Water.\n\u2014 and Expenses occasioned by Distress, &c. to be                    29\ndetermined by Justice of the Peace.  _See_ Justice.\n\u2014 and Expenses occasioned by Distress, &c. may be sued           ibid.\nfor, &c. with full Costs of Suits, &c. by the Trustees.\n_See_ Action.\nCOSTS, &c. incident to a Complaint, hour to be levied, if           36\nnot paid on Conviction.  _See_ Penalties.\n\u2014 awarded by the Trustees at the Quarter Sessions,                  39\nSureties for an Appellant must be conditioned to pay\nsuch.  _See_ Sureties.\n\u2014 if not paid on Decision of the Justices at the Quarter         ibid.\nSessions, the Delinquent to be committed to the Common\nGaol or House of Correction.  _See_ Justices.\n\u2014 _See_ Damages.\n\u2014 when Plaintiff is to pay Treble, and how recoverable.             42\n_See_ Verdict.\n\u2014 of Performance of Conditions for renting Ground for a             22\nPleasure Garden, to be paid out of Rates made by virtue\nof the Act.\nCost of every Lease for renting Ground for a Pleasure            ibid.\nGarden, to be paid out of the Rates made by virtue of the\nAct.\n\u2014 Costs and Expenses of obtaining and passing the Act to            43\nbe paid out of the first Monies raised by virtue of it.\nD.\nDAMAGE done by fixing of Lamps.  _See_ Lamps.\n\u2014 Special, done by Parties' irregularly distraining,                38\nCompensation for, how recoverable.  _See_ Irregularity.\n\u2014 Damages or Costs recovered against the Clerk or                   41\nTreasurer, not to be paid by them, but by the Trustees,\nout of the Money raised under the Act.\n\u2014 wilfully done to the Pavements, Lamps, Fences, Trees,             24\nWalks, &c. &c. in Brompton Square, Penalty for.\n\u2014 done to the Pavements, Lamps, &c.  Satisfaction to be          ibid.\nmade for.\n\u2014 &c.  Satisfaction fur, how to be obtained.                     ibid.\n\u2014 and Injury.  To be made good to the Trustees by the               16\nWater-works Company, all Damage which may have been\ncaused by an Examination of the Pipes respecting the\nContamination of Water by Gas, if the Gas be not the\nOccasion of the Nuisance.\n\u2014 the Amount of Loss, Injury, Damage, &c. done to                ibid.\nFootways, &c. to be determined, if requisite, by a\nJustice of the Peace.  _See_ Justice.\nDEFENDANT in an Action may plead specially or the general           42\nIssue, and at the Trial thereof give this Act and the\nspecial Matter in Evidence.\n\u2014 Verdict to be found for, if, in an Action, the Thing           ibid.\ncomplained of appear to have been done under Authority of\nthe Act; or if the Action be brought contrary to the\nRestrictions, &c. which the Act has laid down.  _See_\nVerdict.\nDEMURRER.  If, on Demurrer, Judgment be given against            ibid.\nPlaintiff, Defendant to have Treble Costs.  _See_\nVerdict.\nDISCHARGE.  _See_ Payment.\nDISTRESS and SALE by, of the Goods, &c. of any Officer,              6\nhaving Money for which he has not accounted, and having\nrefused to give Account, &c.  _See_ Notice.\nDISTRESS, Warrant of.  _See_ Warrant.\n\u2014 not illegal from Informality.                                     38\n\u2014 Persons making, not Trespassers from any Informality in        ibid.\nthe Proceedings.\n\u2014 Persons making, not Trespassers _ab initio_, on account        ibid.\nof any subsequent Irregularity.\nDUNG.  _See_ Mud.\nDUST, Trustees to contract for taking away.                         21\nDUST, ASHES, &c.  _See_ Mud.\nE.\nEXAMINATION respecting Gas; upon Examination, should it             15\nappear that Contamination has not arisen from an Escape\nof Gas, the Expenses and Costs of Digging to be borne by\nthe Water Company, &c.\nEXECUTION of the Act.  _See_ Notices, Form, Proceedings.\nEXEMPTION from General Rates.  _See_ Rates.\nEXPENSE of removing Gas Pipes from private Dwelling\nHouses, &c.  _See_ Pipes.\nEXPENSES incident to inclosed Area of Brompton Square, to           19\nbe wholly borne by the Tenants and Occupiers of Houses.\n_See_ Costs.\nF.\nFARLAR, WILLIAM, Esq.                                                1\nFILTH.  _See_ Mud.\nFIREWORKS, discharging them in Brompton Square.                     32\nFOOTBALL, or any other Game, playing at, in Brompton             ibid.\nSquare.\nFOOTWAYS and CARRIAGEWAYS, Trustees shall cause to be                9\nproperly formed, repaired, cleansed, watered, drained,\nprotected with Posts, &c. or altered, as they think\nnecessary.  _See_ Penalty.\n\u2014 &c. if broken up for the laying of Pipes or any other             10\nPurpose, the Work to be expeditiously completed, and the\nFootways, &c. put in as good a State as before, at the\nExpense of the Company or Persons so breaking them up.\n\u2014 &c. if not repaired by the Parties causing them to be             11\nbroken up, for 24 Hours after the Completion of the Work\nfor which they were broken up, the Trustees to have them\nproperly repaired, and may compel the Parties who caused\nthem to be broken up, to defray the Cost of repairing.\n\u2014 &c.  Expense of repairing, how to be recovered by the          ibid.\nTrustees of Persons failing to reinstate such Footways,\n&c. after causing them to be broken up.\n\u2014 &c. first Expense of, not to be defrayed by Trustees.\n_See_ Trustees.\nFORFEITURES, how recoverable.  _See_ Penalties.\nFORM, Matter of, in any Conviction or Proceeding under              38\nthe Act, may be amended by the Trustees at the Quarter\nSessions.\n\u2014 Want of, no Ground for quashing or vacating any                ibid.\nProceedings connected with the Execution of this Act.\n_See_ Proceedings.\nG.\nGAME, i.e. Play, &c.  _See_ Football.\nGAOL, or House of Correction, any Officer having received         6, 7\nMoney for which he has not accounted, and having refused\nto give Accounts, may be committed to.  _See_ Notice.\nSatisfaction.\nGARDEN.  _See_ Pleasure Garden.\nGARDEN-POT or TUB, Penalty for placing or putting out,              33\nupon or before any Building next or towards any Footway\nwithin the Jurisdiction, &c. unless properly secured from\nfalling.\nGAS found to escape, speedy Means to be taken to prevent            16\nit.  _See_ Notice.  Penalty.\nGAS PIPES.  _See_ Pipes.\nGAS WASHINGS.  _See_ Penalty.\nGATES to be erected at the Entrance of Brompton Square.             21\n\u2014 to be kept up and preserved by the Trustees.                   ibid.\n\u2014 to be kept closed during such Hours as the Trustees            ibid.\nshall direct.\n\u2014 a Key or Keys of the Gates to be left in the Charge of         ibid.\nthe Watchmen, during the Hours appointed by the Trustees\nfor the Gates to be closed.\nGOODS.  _See_ Stall.\nGOODS and CHATTELS liable to Distress, in case of the           35, 36\nNon-payment of Penalties.  _See_ Penalties.\nGRAND JUNCTION WATER-WORKS.  _See_ Water-works.\nGRAVELLING. &c.  _See_ Owners.\nGROUND for Pleasure Garden, to be subject to                    21, 22\nStipulations, agreed upon between the Trustees and Wm.\nFarlar, his Heirs, &c.\n\u2014 Lessees of, their Estate and Interest, to vest in the          ibid.\nTrustees for the Time being.\nH.\nHORSE, BEAST, &c. wilfully suffering them to go upon any            35\nFootway, &c.\n\u2014 &c. fastening any Horse, &c. to any House, Wall, Fence,        ibid.\nPost.  Rail, &c. or across any Footway, &c.\n\u2014 &c. riding or leading a Horse, &c. on any Footway              ibid.\nwithin the Jurisdiction of the Act.\n\u2014 &c. driven by any Person in any Part of Brompton Square        ibid.\nfor the Purpose of breaking him in, offering him for\nSale, &c., Penalty not exceeding \u00a35.\nHOUSE of CORRECTION.  _See_ Gaol.\nHOUSES.  Trustees to order them to be numbered.  _See_\nPenalty.\nI.\nINFORMALITY of Proceedings in making Distress, _See_\nDistress.  Irregularity.\nINFORMATION in Writing, not necessary to the Conviction             36\nof an Offender.\nINHABITANTS of any House, &c. within the Jurisdiction of            40\nthis Act, may be a competent Witness in any Action, &c.\nrelating to its Execution, or to any Rate made in\npursuance thereof.\nINSPECTION of ACCOUNTS, Clerk refusing to permit.  _See_\nPenalty incurred, &c.\nIRREGULARITY, subsequent, of Parties' distraining, does             38\nnot render them Trespassers _ab initio_.  _See_ Distress.\n\u2014 of Parties' distraining.  Persons aggrieved by, may            ibid.\nrecover Compensation for special Damage by Action,\nsubject to certain Limitations.\nISSUE.  _See_ Defendant.\nJ.\nJAIL.  _See_ Gaol.\nJUDGMENT, when, on Demurrer, Judgment is given against              42\nPlaintiff, Defendant to have Treble Costs.  _See_\nVerdict.\nJURISDICTION of the Trustees defined.                                9\nJUSTICES of Peace for Middlesex, Penalties recoverable              36\nbefore.\n\u2014 to summon Offenders liable to Penalty under the Act.           ibid.\n\u2014 to determine Complaints and convict Offenders on Proof,        ibid.\neither by their own Confession or the Oath of credible\nWitness or Witnesses, even without Information in\nWriting.\n\u2014 to grant Warrants for Distress on Goods and Chattels,          ibid.\nin case of the Nonpayment of Penalties.\n\u2014 Penalties recoverable before.                                  ibid.\n\u2014 may order Offenders to be kept in Custody till Return          ibid.\ncan be made to Warrant of Distress, unless satisfactory\nSecurity for their Appearance be given.\n\u2014 in what Form he may cause the Conviction of Offenders             37\nto be drawn up.\n\u2014 may take Security for the Appearance of Offenders                 36\nconvicted under this Act, by way of Recognisance or\notherwise.\n\u2014 at the General Quarter Sessions, on Proof of due Notice           37\ngiven and Recognisance entered into, to determine Matters\nof Appeal, award Costs, levy Costs by Distress, and\ncommit the Delinquent to the Common Gaol for Six Months,\nor until the Costs awarded be paid.\n\u2014 at Quarter Sessions, may amend any Rate, Assessment, or           39\nMatter of Form in any Conviction or Proceeding under this\nAct.\n\u2014 of Peace for Middlesex may act as such in the Execution           40\nof the Act, though also a Trustee.\n\u2014 of Peace, his Decision may be appealed against at the         38, 39\nnext Quarter Sessions.  _See_ Appeal.  Aggrieved.\nK.\nKEY or KEYS of the Gates, to be left in the Charge of the           21\nWatchmen during the Hours appointed by the Trustees for\nthe Gates to be closed.\nKEYS of the Pleasure Garden, to be granted only to the              22\nOccupiers of Houses in the Square, except with the\nwritten Consent of the Majority of the Trustees.\n\u2014 to the enclosed Area of Brompton Square, not to be made        ibid.\nwithout written Consent of Trustees.\n\u2014 false, to the enclosed Area of Brompton Square, Penalty        ibid.\nfor making, selling, or knowingly purchasing.\n\u2014 false, &c., may be seised, &c. by Authority of                    25\nTrustees.\nL.\nLAMPS, &c. to be set up, in and about the Footways, &c.             10\nwithin Jurisdiction of the Act, on or against Walls or\nPalisadoes belonging to any contiguous House, &c.\n\u2014 to be lit with Oil or Gas, during such Hours, &c. as           ibid.\nthe Trustees judge proper.\n\u2014 if unlawfully extinguished, the Offender to pay a              ibid.\nPenalty not exceeding \u00a35.\n\u2014 Damage done to any Building, &c. by fixing of, to be           ibid.\nrepaired by the Trustees out of the Money raised under\nthis Act.\n\u2014 Lamp Irons, &c. vested in the Trustees.                           23\nLAND, Description of, at Brompton Square, claimed to be              1\nseised by Wm. Farlar, Esq.\nLANDLORD, claiming or receiving the Rent of Lodgers, &c.,           26\nto be rated as the Occupiers, &c.\nLEASE of Ground for an enclosed Area in Brompton Square,            22\nmay be contracted for in the Names of Five Trustees.\n\u2014 of Ground for Pleasure Garden, to be taken and accepted        ibid.\nby the Trustees from Wm. Farlar, his Heirs, &c.\n\u2014 Cost of, to be paid out of the Rates made by virtue of            22\nthe Act.\n\u2014 of Room or Building for Transaction of Business                   41\nrelating to the Act, may be taken and accepted by\nTrustees to themselves, any of them, or any Person in\nTrust for them, for any Number of Years, at such Rent as\nthey shall from Time to Time think fit.  _See_ Trustees.\n\u2014 of enclosed Area in Brompton Square, when obtained, how           25\naffecting Rates.  _See_ Rates.\nLESSEES of Ground for Pleasure Garden; their Estate and             24\nInterest to vest in the Trustees for the Time being.\nLIME, slacking or sifting, &c. in Brompton Square.                  32\nPenalty for.  _See_ Provision.\nM.\nMATERIALS for the Footways, &c. vested in the Trustees.             23\nMEETINGS, Expenses of, by whom to be defrayed.                    4, 5\n\u2014 Five a sufficient Number to form a Meeting.                    ibid.\n\u2014 Five, or any Number under Five, may adjourn.                   ibid.\n\u2014 First Meeting to be held on the Fourth Monday, &c. at          ibid.\nthe Grapes Tavern at Brompton.  _See_ Chairman.\nMEETINGS, adjourned, where and when.                             ibid.\n\u2014 Notice respecting adjourned Meetings.                          ibid.\n\u2014 Special.                                                       ibid.\n\u2014 Neglect or Omission to adjourn.                                ibid.\nMERCHANDISE.  _See_ Stall.\nMIDDLESEX, West, Water-works.  _See_ Water-works.\nMONIES received by an Officer, or collected by him by             5, 6\nvirtue of the Act, and upon Inspection of his Accounts,\nbefore refused to be given, not accounted for, on\nNonpayment thereof, any Justice, by Warrant under his\nHand, may cause such Monies to be levied by Distress,\n&c., or, if Goods and Chattels be not sufficient, &c. &c.\nto be committed to the Common Gaol or House of\nCorrection, &c. for Six Calendar Months.\n\u2014 may be raised on the Credit of the Rates by way of Loan           30\non Annuity.\n\u2014 charged in consequence of Alteration or Amendment of              28\nRates, may be recovered and levied, &c. as an original\nInsertion.\n\u2014 arising from Penalties, to whom to be paid, and how to            37\nbe disposed of.  _See_ Penalties.\nMONTH.  A Month's Notice to be given of intended Action.            41\nMORTAR, making it in Brompton Square.  _See_ Provision.             32\nMUD, Dust, Sand, Rubbish, Filth, Dung, Ashes,\u2014Penalty not           33\nexceeding \u00a35 for placing, within the Jurisdiction of the\nAct.\nN.\nNAMES of Streets, Numbers of Houses, Trustees to order              20\nthem to be set up.  _See_ Penalties.\nNAMES, Orders, and Proceedings of the Trustees at                   32\nMeetings to be entered into a Book kept for the Purpose.\n_See_ Trustees.  Proceedings.\nNIGHT-SOIL.  _See_ Bog-house.\nNOTICE respecting adjourned Meetings.                                4\n\u2014 signed by Three or more Trustees, to be given to                   6\nOfficers who shall refuse to give true and perfect\nAccounts of the Monies, &c. before any other legal Step\ncan be taken.\n\u2014 by Parol or in Writing, to be given to the Trustees or            17\nGas Light Company respecting the Escape of Gas.\n\u2014 to be given in case Gas Washings, &c. shall be conveyed        ibid.\ninto any River, Stream. &c.\n\u2014 to be given to Persons neglecting to put up Fences,               34\nLights, &c. during Repairs, and also to remove them in\nproper Time.  _See_ Penalty.  Repairs.\n\u2014 in Writing properly signed. &c. respecting the                    14\nContamination of Water by Gas, to be given to the\nTrustees or Gas Company, before the Penalty of \u00a310 can be\nexacted, or any legal Step can be taken.  _See_ Pipes.\n\u2014 written, of intended Action, stating fully the Cause of           41\nAction, Name, and Place of Abode of intended Plaintiff,\nand of his Attorney or Agent, to be left at the usual\nResidence of the intended Defendant, a Month previously\nto the Commencement of the Action.\n\u2014 of Appeal to the Quarter Sessions against Decision of             39\nTrustees or Justice of Peace, to be given to the Clerk of\nthe Trustees.  _See_ Appeal.\nNOTICES or SUMMONSES, requisite for carrying this Act               40\ninto Execution, may be written or printed, or partly\nboth.\n\u2014 may be served by delivering them to the Persons to whom        ibid.\nthey are directed, or leaving them at the last known\nResidence of such Persons, or at, or upon, any House, &c.\nwhereunto they respectively relate.\nNUMBERS.  _See_ Names of Streets.\nNUISANCES and ANNOYANCES, Penalty for.                              34\nO.\nOATH, to be taken and subscribed to, by every Owner or               3\nOccupier of a House, &c. before he can act as Trustee.\n_See_ Unqualified Persons.\nOCCUPIER or LODGER, to be liable, under a Modification,             27\nto be distrained for Rates, &c. due at the Time of making\nthe Distress.\n\u2014 having been distrained, &c. may deduct the Amount, &c.         ibid.\nout of the Rent, payable by him to the Person or Landlord\nrated.\nOFFENDERS, may be apprehended without any other warrant             35\nor Authority than the Act.\n\u2014 liable to Penalty, to be summoned before any Justice of\nthe Peace for Middlesex.  _See_ Justice.\n\u2014 not having Goods and Chattels whereon sufficient                  37\nDistress can be taken, to be committed to the Common\nGaol.  _See_ Warrant.\n\u2014 Form of Conviction of.                                         ibid.\nOFFICERS, how to be appointed.                                       5\n\u2014 if requisite, how to be removed.                               ibid.\n\u2014 to give true and perfect Accounts in Writing of all             5, 6\nReceipts and Payments of Monies to the Trustees, or to\nsuch Persons as they shall nominate.  _See_ Monies.\n\u2014 neglecting to give up Papers, &c. within Seven Days             6, 7\nafter being required by a Notice in Writing, any Justice\nof the Peace, upon Complaint of the Trustees, may issue a\nWarrant.  _See_ Warrant.  Justice.\nOFFICERS or PERSONS engaged in the Execution of this Act,           35\nPenalty for obstructing them.\nORDERS and Determinations of Trustees to be made at                  4\nMeetings to be held pursuant to the Act, and not\notherwise.  _See_ Names.\n\u2014 none to be made at any Meeting, without the Concurrence            5\nof the Majority of the Trustees present.  _See_ Chairman.\nNames.\n\u2014 Revocation of, and Exception.                                  ibid.\n\u2014 written Notice respecting the Revocation of Orders, to         ibid.\nbe given by Three or more Trustees under their Hands, at\na previous Meeting.\nOWNERS and OCCUPIERS of HOUSES, &c. beneficial to them,              1\nand also of public Advantage, to provide by Law for the\npaving, gravelling, &c. and for the Maintenance of a\nPleasure Garden in the Square.  _See_ Owners.\nP.\nPAVING, gravelling, forming, &c.  Footways or                       43\nCarriageways, first Expence of, not to be defrayed by\nTrustees.  _See_ Trustees.  Owners.\nPAYMENTS of RATES after a Distress upon Occupier, Lodger,           27\n&c. to be a sufficient Discharge for so much Rent as\nshall be due to the Landlord.\nPENALTY for unqualified Persons acting as Trustees, Fifty            3\nPounds.\n\u2014 to whom to be paid.                                            ibid.\n\u2014 how and where to be recovered.                                 ibid.\n\u2014 for the Offices of Clerk and Treasurer being accepted              8\nby the same Person or disqualified Parties, One hundred\nPounds.  _See_ Trustees.\n\u2014 for the Offices of Clerk and Treasurer, being accepted         ibid.\nby the same Person or disqualified Parties, or for the\nTreasurers' accepting another Place of Profit or Trust\nunder the Trustees, to be recovered with full Costs of\nSuit in any of His Majesty's Courts of Record at\nWestminster, by Action of Debt or on the Case, or by\nBill, Suit or Information, without Allowance of Essoign,\nProtection or Wager of Law, or of more than one\nImparlance.\n\u2014 incurred by Clerk for refusing to permit Inspection of             9\nthe Accounts, not to exceed \u00a310.\n\u2014 for taking up or breaking Footways, &c. without Consent           10\nof Trustees, not to exceed \u00a310.\n\u2014 for unlawfully extinguishing Lamps.  _See_ Lamps.\n\u2014 for not removing Contamination of Water by Gas, of \u00a310            15\n_per Diem_ (proper Notice having been previously given,)\nto be paid by the Trustees or Gas Company to the\nSecretary or chief Clerk of the Water Works Company, or\nelse legally recovered with Costs.  _See_ Pipes.\n\u2014 of \u00a35 _per Diem_, for neglecting to use Measures for              17\nstopping the Escape of Gas, within Twenty-four Hours\nafter written Notice shall have been given.\n\u2014 of \u00a3200 for conveying, or causing to be conveyed, Gas             18\nWashings, &c. into any River, Stream, Sewer, &c.\n\u2014 to be paid to any Person or Persons who shall sue for          ibid.\nthe same in any of His Majesty's Courts, &c.\n\u2014 no Essoign, Protection of Law, and not more than one           ibid.\nImparlance allowed.\n\u2014 not recoverable, unless sued for within Twelve Calendar        ibid.\nMonths next after an Annoyance.\n\u2014 of \u00a320 in Addition to the \u00a3200 for each and every Day,            19\nafter a written Notice shall have been given that Gas\nWashings &c. are conveyed into any River, &c.\n\u2014 to be paid to the Informer, &c. or to the Person who           ibid.\nshall have sustained any Annoyance.\n\u2014 not exceeding 10_l._ for unlawfully removing Dust.                21\n\u2014 for certain Nuisances and Annoyances.                             24\n\u2014 for placing Garden-pot or Tub, &c.  _See_ Garden-pot.\n\u2014 for not erecting or placing proper Enclosures to\nprevent Accidents, during the Repairs of Houses.  _See_\nRepairs.\n\u2014 for keeping Swine.  _See_ Swine.\n\u2014 not exceeding \u00a310 for obstructing Trustees in the                 35\nExecution of the Act.\n\u2014 not exceeding \u00a310 on Person or Persons summoned to give        ibid.\nEvidence before a Justice of Peace, respecting an Offence\npunishable under the Act, and failing to attend, &c.\n\u2014 not exceeding \u00a35 for assaulting a Watchman.                       20\n\u2014 on every Victualler who shall harbour or entertain a           ibid.\nWatchman during the Time appointed for his being on Duty.\n\u2014 not exceeding \u00a32 on a Watchman for Misconduct.                 ibid.\n\u2014 not exceeding \u00a35 for defacing Names of Streets or              ibid.\nNumbers of Houses, unless, within Seven Days after a\nwritten Notice shall have been given, the same be\nreinstated.\n\u2014 for wilfully damaging the Pavements, Lamps, Fences,               24\nTrees, Walks, &c. &c. in Brompton Square.\n\u2014 for making, selling, or knowingly purchasing false Keys           25\nto the enclosed Area of Brompton Square.\n\u2014 for not securing Gas Pipes.  _See_ Pipes.                         13\nPENALTIES.  All those not before particularly directed,             36\nrecoverable before any Justice of Peace for Middlesex, on\nComplaint made within Six Months after the Commission of\nthe Offence.\n\u2014 if not paid on Conviction, to be levied by Distress and           36\nSale of Goods and Chattels of the Offender, by Warrant\nunder the Hand and Seal of the Justice of the Peace for\nMiddlesex.\n\u2014 Money arising from, unless where otherwise directed, to           37\nbe paid to the Treasurer of the Trustees; and (except\nthat Part which the Trustees are empowered to dispose of\nin Rewards,) to be applied for the Purposes of this Act.\nPIGS.  _See_ Swine.\nPIPES, Gas, not to be affixed to a House, &c. without the           12\nOccupier's Permission.\n\u2014 Regulations for the laying down of.                            ibid.\n\u2014 Penalty for not properly securing, so as to prevent the           13\nEscape of Gas, \u00a35\n\u2014 to be removed from Dwelling Houses, &c. within Seven           ibid.\nDays next after Service of a written Notice has been\ngiven to the Trustees or Gas Company by the Owners or\nOccupiers of such House, &c.\n\u2014 Expenses of removing Gas Pipes from Dwelling House, to         ibid.\nbe paid by Trustees or Gas Company, and legally recovered\nbefore a Justice of Peace.\n\u2014 if Water be contaminated by, the Trustees or Gas                  14\nCompany shall, within 24 Hours after receiving proper\nNotice from the Water Works Company, take Measures to\nremedy such Contamination, under Penalty of 10_l._ for\nevery Day during which the Water remains contaminated.\nPLEAD, Defendant in an Action may plead specially, or               42\nGeneral Issue.\nPLEASURE GARDEN in the Square.  _See_ Owners.\n\u2014 to be maintained and preserved, subject to Stipulations           21\nof Lease.\n\u2014 Lease of Ground for the laying out of, lawful to be               21\ncontracted for, by the Trustees in the Names of Five.\n\u2014 Ground for, to be subject to Stipulations agreed upon          ibid.\nbetween the Trustees and William Farlar, his Heirs, &c.\n\u2014 Lessees of Ground for, their Estate and Interest to            ibid.\nvest in the Trustees for the Time being.\n\u2014 to be kept for the exclusive Use of Persons, occupying         ibid.\nHouses in the Square.\n\u2014 to be fenced in with substantial Iron Railing, and             ibid.\nsecured with proper Entrance Gates and Locks.\n\u2014 Access to, to be confined to the Occupiers of Houses in        ibid.\nthe Square.\n\u2014 Railing, Gales, Locks, Plants, Materials for Gravel               23\nWalks of, &c., to be vested in the Trustees during the\nContinuance of any Lease of the enclosed Area granted to\nthem.\nPOST, Rails, &c. vested in the Trustees.                         ibid.\nPROCEEDINGS of unqualified Persons acting as Trustees in             3\nthe Execution of the Act, previous to Conviction, to be\ngood and valid.\n\u2014 no Proceeding, connected with the Execution of this               39\nAct, to be quashed for Want of Form, or removed by any\nWrit or Process into any of His Majesty's Courts of\nRecord.\n\u2014 an Entry of, to be signed by the Chairman at the next             32\nMeeting.  _See_ Names.\nPRODUCE of Annuity not to exceed more than One-fifth of         31, 32\nthe Income which a Sum equal to the Purchase Money would\nproduce, &c. according to the Rate prescribed by Two Acts\npassed in the Reign of His late Majesty, intituled \"An\nAct to enable, &c.\" and \"An Act for amending, &c.\"\nPROMISSORY NOTE, &c.  Form of, for Money raised on the              30\nCredit of Rates.\n\u2014 or Form of Security, to be transferrable by                       31\nIndorsement, &c.\nPROVISION, that no Person shall be subject to any Penalty           34\nfor Rubbish before Houses when repairing.\nPUBLIC, this Act to be judicially noticed as such.  _See_\nAct.\nQ.\nQUARTER SESSIONS.  Appeal to, against Decision of                   39\nTrustees or Justice.  _See_ Appeal.  Aggrieved.\nJustices.\n\u2014 General, Determination of Justices at, to be final and         ibid.\nconclusive.\n\u2014 \u2014 Justices at, may amend any Rate, Assessment, or              ibid.\nMatter of Form, in any Conviction or Proceeding under\nthis Act.\nR.\nRAILS.  _See_ Posts.\nRATE, First, by virtue of the Act, to be for Two                    28\nQuarters, ending at Christmas, after passing the Act.\nRATES, all, to be just and equal.                                ibid.\n\u2014 may be amended by the Trustees at any Meeting pursuant         ibid.\nto the Act.\n\u2014 every Rate subsequent to the first, shall be for One           ibid.\nwhole Year, commencing, &c.\n\u2014 to become due immediately upon the making thereof; but         ibid.\nmay be collected in one or several Payments, &c.  _See_\nTrustees.\n\u2014 and Monies due in pursuance of the Act, to be vested in        ibid.\nthe Trustees.  _See_ Trustees.\n\u2014 to be paid to such Person or Persons as the Trustees           ibid.\nshall from Time to Time appoint to receive the same.\n_See_ Trustees.  Money.  Loan.  Annuity.\n\u2014 to be applied, &c. by the Trustees in defraying the            ibid.\nExpenses of, and incident to, the Execution of this Act,\nand otherwise as in, or by, this Act is directed.  _See_\nTrustees.\n\u2014 to be levied, upon refusing or neglecting to pay them.            28\n\u2014 upon Refusal or Neglect to pay any Part, &c. of Rates,         ibid.\n&c. any Justice for the County is empowered (after due\nNotice, &c. issued,) to grant a Warrant.  _See_ Warrant.\n\u2014 Arrears &c. to be levied with incidental Costs, &c. by         ibid.\nDistress of the Goods, &c. of Persons neglecting or\nrefusing to pay.\n\u2014 &c. if not paid within Five Days after Distress shall          ibid.\nbe taken,\u2014Goods and Chattels sufficient to pay the said\nRates, &c. to be appraised and sold.\n\u2014 Overplus, if any, to be returned to the Owner.                 ibid.\n\u2014 &c.  In Default of a Distress, &c. the Person or               ibid.\nPersons to be committed to the Common Gaol, &c. there to\nremain without Bail for any Time not exceeding Six\nCalendar Months, unless, &c.\n\u2014 for Houses let out furnished, or in Lodgings, &c. how             26\nto be charged.\n\u2014 to be made by the Trustees on Tenants and Occupiers of            25\nall Lands, Houses, Shops, Coach Houses, Stables,\nErections, Buildings, and Tenements within the\nJurisdiction of the Act.\n\u2014 to be just and equal Pound Rates, according to the             ibid.\nannual Rent or Value of the Premises assessed.\n\u2014 to appear distinctly in each, what Part shall be raised        ibid.\nin respect of Costs and Expenses.\n\u2014 not to exceed, in any one Year, 2_s._ in the Pound of             26\nthe annual Rent or Value of the Property rated, except\nafter a Lease of the enclosed Area shall have been\nobtained.\n\u2014 in respect of Houses in the Square, after a Lease of           ibid.\nthe enclosed Area shall have been obtained, not in any\none Year to exceed 2_s._ 6_d._ in the Pound of the Rent\nor annual Value.\n\u2014 in making, Stables and other Buildings belonging to               26\nHouses in the Square, not to be rated as separate\nBuildings.\n\u2014 to facilitate the making of, Trustees, or those                ibid.\nappointed by them, may examine Books of Assessment\naffecting Houses, &c. within the Jurisdiction of the Act.\n\u2014 Persons exempted from the Payment of; how the Rates of         ibid.\nHouses occupied by such Persons are to be charged.  _See_\nAmbassador.\n\u2014 Persons rated under this Act not liable, in respect of            42\nthe same Premises, to any general Rate for paving,\nlighting, or watching any Part of Kensington, or any\nparticular Street, Road, or Place.\n\u2014 Persons aggrieved by, may appeal after Payment.  _See_            38\nAppeal.\n\u2014 Payment of, to be made before it is appealed against.          ibid.\n\u2014 Assessment, or any Matter of Form in any Conviction or            40\nProceeding, connected with the Execution of the Act, may\nbe amended by the Justices at the Quarter Sessions.\nRECOGNISANCE, Justice empowered to take Security for                36\nAppearance of Offenders by.  _See_ Justice.\n\u2014 to be entered into before a Justice of Peace by                ibid.\nAppellants against Decision of Trustees.  _See_ Appeal.\nRECOVERY of Costs.  _See_ Costs.\nRENT, Rates.  _See_ Landlord.  Occupier.  Payment of\nRates.\n\u2014 of Lease for Pleasure Garden, not in the whole, at any            22\none Time, to exceed \u00a330 per Annum.\n\u2014 of Ground for Pleasure Garden to be paid out of Rates          ibid.\nmade by virtue of the Act.\n\u2014 of Room or Building engaged by Trustees for transacting           41\nBusiness, to be paid out of the Money raised by virtue of\nthe Act.\nREPAIRS, Houses or Buildings under Repair to be properly            34\nenclosed, and convenient Space left for Carriages, &c. to\npass, &c. with Ease and Safety.\n\u2014 to prevent Accidents from, a Light to be kept up from          ibid.\nSun-set to Sun-rise.\n\u2014 when finished, all Rubbish, Building Materials, &c. to         ibid.\nbe removed within a reasonable Time.\n\u2014 Houses, &c. under Repair, if not properly enclosed and            35\nsufficient Light kept up from Sun-set to Sun-rise, to\nprevent Accidents, Notice having first been given by the\nClerk of the Trustees to the Party offending,\u2014Penalty not\nexceeding \u00a310\nRIDING, &c.  _See_ Horse.\nROOM or Building may be hired by Trustees for Transaction           41\nof Business relating to the Act.  _See_ Trustees.\nS.\nSATISFACTION being made to the Trustees by an Officer,               6\nhaving before refused to give or deliver up to the\nTrustees proper Accounts or Vouchers respecting Monies,\n&c. and having been committed to Prison,\u2014to be released.\nSECURITY, Form of, for Money borrowed on the Credit of              30\nRates.\n\u2014 sufficient, to be given to the Justice of Peace for the           36\nAppearance of Offenders convicted under this Act.  _See_\nJustice.\nSERVING of Notices, how to be executed.  _See_ Justices.        41, 42\nSEWERS, Commissioners of, for Westminster and Part of               43\nMiddlesex, their Rights and Authorities not to be\nprejudiced by this Act.\nSPECIAL MEETINGS.  _See_ Meetings.\nSPECIAL PLEADING of this Act, not necessary to its being            44\njudicially noticed as a Public Act.\nSTABLES, Coach Houses, &c. not to be rated as separate              26\nBuildings.  _See_ Rates.\nSTALL, Stall Board, Basket, Goods, Wares, Merchandize,              33\nPenalty not exceeding \u00a35 for placing them within the\nJurisdiction of the Act.\nSUIT.  _See_ Action.\nSUMMONED, the Person or Persons to give Evidence before             35\nany Justice of Peace respecting an Offence punishable\nunder the Act failing to attend, &c. and Qualification\nrespecting Excuse.\u2014Penalty not exceeding \u00a310.\nSUMMONS.  _See_ Notice.\nSURETIES, Two necessary for an Appellant against Decision           39\nof Trustees or Justice; they must be conditioned to try\nsuch Appeal at, abide the Order of, pay the Costs\nawarded, and pay the Penalty of a Conviction, if\naffirmed.  _See_ Appeal.\nSWINE, Penalty not exceeding \u00a35 for keeping or breeding             34\nwithin the Jurisdiction of the Act.\nT.\nTENDER of sufficient Amends to Parties aggrieved, to                42\nprevent the Commencement of an Action.  _See_ Action.\nTRANSACTION of Business, Room or Building may be hired              41\nfor.  _See_ Trustees.\nTREASURER to be appointed, and how.                                  5\n\u2014 &c. not to be Clerk.  _See_ Trustees.                              7\n\u2014 his Death or Removal, no Ground for the Discontinuance            40\nof an Action.\n\u2014 to be deemed Plaintiff or Defendant in an Action, but           ibid\nnot personally answerable for Damages.  _See_ Damages.\n\u2014 competent Witness in an Action, though nominal                 ibid.\nPlaintiff or Defendant.\n\u2014 to accept no Place of Profit or Trust under the                    8\nTrustees, under Penalty of One hundred Pounds.  _See_\nPenalty.\nTRUSTEES, how to be described in Actions.                           24\n\u2014 Qualification of.                                                  2\n\u2014 may appoint or remove Officers, and regulate their                 5\nSalaries.\n\u2014 present at Meetings, empowered to act for themselves,          ibid.\nand for those who are absent.\n\u2014 to cause to be entered in a Book, kept for the Purpose,           32\nthe Names and Proceedings at Meetings.\n\u2014 may not appoint their Clerk or his Partner, or anyone              7\nemployed by him, to be their Treasurer; nor their\nTreasurer, his Partner, or any one employed by him, to be\ntheir Clerk.\n\u2014 to cause their Clerk to keep a Book of Accounts of the             8\nMoney received and expended on account of the Act; which\nBook or Books shall be open to the gratuitous Inspection\nof Trustees and Persons rated under the Act.\n\u2014 their Jurisdiction defined.                                        9\n\u2014 may compel Parties breaking up the Footways, &c. to pay           10\nthe Expense of their Reparation.  _See_ Footways.\n\u2014 to cause Lamp Posts, &c. to be set up.  _See_ Lamps.              11\n\u2014 to repair Damage done by fixing of Lamps.  _See_ Lamps.        ibid.\n\u2014 to cause Nuisance from Gas to be removed.  _See_ Pipes.           14\n\u2014 Gas Light Company, &c. to bear the Costs of searching             16\nand digging, when Water is contaminated by an Escape of\nGas.  _See_ Notice, Costs, Expenses, Examination, Damage,\nPenalty, and Justice.\n\u2014 to appoint Watchmen, and to determine respecting their            19\nWages and Hours of watching.\n\u2014 to provide Arms, Clothing, Watch Boxes.                        ibid.\n\u2014 to provide a Watch House.                                      ibid.\n\u2014 to appoint when the Gates of the Square are to be                 21\nclosed, and when to be opened.\n\u2014 to contract with Persons to cleanse the Carriageways,          ibid.\nand to take away Dust, Cinders, &c.\nW.\nWATCH BOXES to be provided and set up where the Trustees            18\nshall appoint.\n\u2014 Arms, Clothing for Watchmen, &c. vested in the                    23\nTrustees.\nWATCH HOUSE, for the Custody of Persons apprehended by              18\nthe Watchman while on Duty.\nWATCHMEN to be appointed to patrol, &c.  Places within           ibid.\nthe Jurisdiction of the Act.\n\u2014 to have the Charge of the Keys of the Gates at the                21\nEntrance of the Square.  _See_ Gates.\nWATER, if contaminated by any Escape of Gas, the Costs of           15\nsearching and digging, &c. to be borne by the Trustees,\nGas Light Company, &c.\nWATER WORKS COMPANY, lawful for them to dig to and about,           14\n&c. all the Pipes, &c. to remove the Contamination of the\nWater by Gas.  _See_ Pipes.\n\u2014 of Chelsea, West Middlesex, and Grand Junction, their             43\nrespective Rights and Powers, &c. not to be prejudiced by\nthis Act.\nWHEELBARROW, &c.  _See_ Coach.\nWITNESS.\u2014Trustee or Occupier of House within Jurisdiction           40\nof this Act may be a competent Witness in any Action\nrelating to the Execution of it, &c.  _See_ Trustees.\nInhabitant.\n\u2014 Clerk or Treasurer may be a competent one, though                 41\nnominally Plaintiff or Defendant.  _See_ Clerk,\nTreasurer.\n\n\n\n\nANNO QUINTO\nGEORGII IV. Regis.\n\n\n_Cap._ cviii.\n\n\nAn Act to provide for the paving, gravelling, lighting, and watching\ncertain Footways and Carriageways in and near _Brompton Square_ in the\nParish of _Saint Mary Abbotts Kensington_ in the County of _Middlesex_;\nand to provide for the Maintenance of a Garden and Shrubbery in the said\nSquare.  [4th _June_ 1824.]\n\nWHEREAS _William Farlar_ of _Brompton_ in the County of _Middlesex_,\nEsquire, claims to be seised of a Piece of Freehold Land at _Brompton_,\nin the Parish of _Saint Mary Abbotts Kensington_, in the County of\n_Middlesex_, which abuts South on the Turnpike Road from _Knightsbridge_\nto _Fulham_, West on Land belonging to the Trustees of _Saint George's_\nHospital, North on Land now or lately belonging to the Heirs of the late\nDuke of _Kingston_, and East on Ground now or lately belonging to _Elisha\nBiscoe_, and a Road leading to _Brompton_ Chapel: And whereas a Square\ncalled _Brompton Square_, and also divers Streets and public Places, have\nin part been laid out and formed, and are intended to be speedily\ncompleted on the said Piece of Land; and it would be beneficial to the\nOwners and Occupiers of Houses in the said Square, Streets, and public\nPlaces, and also be of public Advantage, to provide by Law for the\npaving, gravelling, watering, lighting, and watching of the said Square,\nStreets, and public Places, and for the Maintenance of a Pleasure Garden\nin the said Square: May it therefore please Your Majesty that it may be\nenacted; and be it enacted by the King's most Excellent Majesty, by and\nwith the Advice and Consent of the Lords Spiritual and Temporal, and\nCommons, in this present Parliament assembled, and by the Authority of\nthe same, That the Limits of this Act shall for all the Purposes thereof\nbe deemed and taken to comprize all the Land of which the said _William\nFarlar_ claims to be seised as aforesaid, and all Messuages, Erections,\nand Buildings thereon.\n\nII.  And be it further enacted, That every Person who now is or for the\nTime being shall be in his own Right or in the Right of his Wife in the\nactual Possession, or in the Receipt of the Rents and Profits of a\nDwelling House or other Tenement within the Limits of this Act, of the\nclear yearly Value of Forty Pounds above Reprisals, and also every Person\nwho now is, or, for the Time being, shall be, actually and _bon\u00e2 fide_ an\nOccupier of a Dwelling House rated or liable to be rated by virtue or for\nthe Purposes of this Act, shall be and they respectively are hereby\nconstituted Trustees for putting this Act into Execution, and the said\nTrustees shall be called and known as \"The Trustees for regulating and\nimproving _Brompton Square_, and certain adjacent Places.\"\n\nIII.  Provided always, and be it further enacted, That no Person shall be\ncapable of acting as a Trustee in the Execution of this Act, except in\nadministering the Oath herein-after mentioned (which Oath any One of the\nsaid Trustees is hereby authorized to administer at any Meeting which\nshall be held pursuant to this Act), until such Person shall have taken\nand subscribed an Oath to the Effect following: that is to say,\n\n    'I \u2014 \u2014 do swear, That I truly and bon\u00e2 fide am in my own Right [or in\n    the Right of my Wife] in the actual Possession [or in the Receipt of\n    the Rents and Profits] of a Dwelling House or other Tenement of the\n    clear yearly Value of Forty Pounds above Reprises [or that I am an\n    Occupier of a Dwelling House rated or liable to be rated by virtue\n    and for the Purposes and] within the Limits of an Act of the Fifth\n    Year of the Reign of King _George_ the Fourth, intituled [_set forth\n    the Title of this Act_.]\n\n                                                         'So help me GOD.'\n\nAnd if any Person not being qualified as aforesaid, or not having taken\nand subscribed the Oath herein-before mentioned, shall presume to act as\na Trustee in the Execution of this Act, every such Person for every such\nOffence shall forfeit and pay the Sum of Fifty Pounds to any Person or\nPersons who shall sue for the same, to be recovered with full Costs of\nSuit in any of His Majesty's Courts of Record at _Westminster_, by Action\nof Debt or on the Case, wherein no Essoign, Protection, or Wager of Law,\nand not more than One Imparlance, shall be allowed; and every Person\nagainst whom any such Action shall be brought shall prove that he was at\nthe Time of acting qualified as aforesaid, or shall pay the said Penalty,\nwithout any other Proof on the Part of the Plaintiff than that such\nPerson acted as a Trustee in the Execution of this Act: Provided always\nnevertheless, that all Acts and Proceedings of any Person acting as a\nTrustee in the Execution of this Act, though not qualified as aforesaid,\nprevious to his being convicted, shall notwithstanding such Conviction be\nas good, valid, and effectual as if such Person had been duly qualified.\n\nIV.  And be it further enacted, That the said Trustees shall hold their\nFirst Meeting for executing this Act on the Fourth _Monday_ next after\nthe passing of this Act, at the _Grapes Tavern_ at _Brompton_ aforesaid,\nif then open, and if not, then at some other House or Place in the said\nParish of _Saint Mary Abbotts Kensington_, between the Hours of Nine and\nEleven in the Forenoon, and may then and from Time to Time afterwards\nadjourn to such Time or Times, and to such Place or Places within the\nsaid Parish, as the Majority of the Trustees present at any Meeting shall\nfrom Time to Time appoint; and that Notice of every such adjourned\nMeeting shall be delivered or left at each Dwelling House within the\nLimits of this Act, Two Days at the least previously to every such\nMeeting; and if at any Time appointed for a Meeting Five or more Trustees\nshall not appear within One Hour after the Time appointed for first\nassembling, any One or more of the said Trustees then present may adjourn\nthe Meeting to another Day; and in every Case of Neglect or Omission to\nadjourn (either by reason of the Non-attendance of any Trustee or\notherwise), and also whenever any Three or more Trustees shall deem it\nnecessary to have a Special Meeting before the Day of Adjournment, any\nThree or more of the said Trustees, or their Clerk, may call a Meeting at\nthe Place where the last Meeting was held, by a Notice similar to the\nNotice directed to be given of adjourned Meetings; and that the said\nTrustees shall at all their Meetings defray their own Expences, except\nany Sum not exceeding Ten Shillings a Day for the Use of the Room wherein\nthey shall meet; and at every Meeting of the said Trustees the Majority\npresent at the Hour appointed for first assembling shall appoint One of\nthe Trustees to act as a Chairman; and all Orders and Determinations of\nthe Trustees shall be made at Meetings to be held pursuant to this Act\nand not otherwise; and no Order or Determination shall be made at any\nsuch Meeting, unless a Majority of the Trustees present shall concur\ntherein, except in Cases where there shall be an Equality of Votes, when\nthe Chairman (in addition to his own Vote as a Trustee) shall have a\ncasting Vote; and all Acts, Orders, and Proceedings relating to this Act,\nwhich are hereby authorized or directed to be made, done, or executed by\nor before the said Trustees, and all the Powers and Authorities vested in\nthem generally, may be made, done, and executed by the Majority of the\nTrustees present at the respective Meetings to be held pursuant to this\nAct (the whole Number present not being less than Five), and all such\nActs, Orders, and Proceedings shall have the same Force and Effect as if\nthe same were made, done, or executed by or before all the Trustees; and\nno Order or Determination made, agreed upon, or entered into at any\nMeeting of the said Trustees shall be revoked or altered at any\nsubsequent Meeting, unless Notice of the Intention to propose such\nRevocation or Alteration shall have been given by Three or more Trustees,\nby Writing under their Hands, to the Trustees at a previous Meeting, nor\nunless such Revocation or Alteration shall be agreed to by a greater\nNumber of Trustees than concurred in the making of the Order or\nDetermination which it shall be proposed to revoke or alter.\n\nV.  And be it further enacted, That the said Trustees shall from Time to\nTime, by Writing under their Hands, appoint a Treasurer and Clerk, and\nalso a Collector of the Rates to be made as herein-after mentioned, and\nmay also from Time to Time appoint or employ such other Officers and\nPersons as they shall think necessary in the Execution of this Act; and\nthe said Trustees may from Time to Time remove such Officers and Persons\nrespectively, or any of them, and upon the Death, Resignation, or Removal\nof such Officers and Persons respectively, or any of them, may appoint\nothers in their Stead, and may out of any Money to be raised or collected\nby virtue of this Act make such Allowances as they shall think proper\nunto such Officers and Persons respectively; and all such Officers and\nPersons respectively shall, at such Times and in such Manner as shall be\ndirected by the said Trustees, deliver to the said Trustees, or to such\nPerson or Persons as they shall nominate, true and perfect Accounts in\nWriting of all their Receipts and Payments by virtue of this Act, with\nproper Vouchers for the Payments, and shall pay all such Monies as shall\nappear to be in their Hands or owing by them respectively unto the said\nTrustees, or to such Person or Persons as they shall nominate, within\nsuch Time as the said Trustees shall appoint; and if any such Officer or\nPerson shall refuse or neglect to deliver to the said Trustees, or to\nsuch Person or Persons as they shall nominate, such an Account as\naforesaid, or to deliver up the Vouchers relating to the same, or to make\nPayment as aforesaid, or if any such Officer or Person shall not deliver\nup to the said Trustees, or to such Person or Persons as they shall\nnominate, all Books, Papers, and Writings in his Custody or Power\nrelating to the Execution of this Act, within Seven Days after being\nthereto required by a Notice in Writing signed by Three or more of the\nsaid Trustees, and left at his last or usual Place of Abode, then and in\nevery such Case, upon Complaint being made thereof by or on behalf of the\nsaid Trustees to any Justice of the Peace, such Justice is hereby\nrequired to issue a Summons or Warrant for bringing such Officer or\nPerson before him, and upon his appearing or not being to be found to\nhear and determine the Matter of such Complaint in a summary Way, and to\nexamine and adjust the said Account or Accounts if produced; and if upon\nthe Confession of the Officer or Person against whom the Complaint shall\nbe made, or by the Oath of any credible Witness or Witnesses, or upon\nInspection of the said Accounts if produced, it shall appear to such\nJustice that any Money which shall have been collected or received by\nvirtue of this Act, shall be in the Hands of or be owing from such\nOfficer or Person, such Justice is hereby authorized and empowered\nforthwith, on Nonpayment thereof, by a Warrant under his Hand and Seal to\ncause such Money to be levied by Distress and Sale of the Goods and\nChattels of such Officer or Person respectively; and if no Goods and\nChattels shall be found sufficient to answer and satisfy the said Money,\nand the Charges of making and selling such Distress, or if it shall\nappear to such Justice that any Officer or Person shall have refused or\nneglected to render such Account as aforesaid, or to deliver up the\nVouchers relating thereto, or that any Books, Papers, and Writings\nrelating to the Execution of this Act shall be in the Custody or Power of\nsuch Officer or Person, then such Justice shall forthwith commit such\nOfficer or Person to the Common Gaol or House of Correction, there to\nremain without Bail or Mainprize, in case he shall be committed for\nNonpayment of any Money, until he shall have accounted for and paid the\nfull Amount thereof, or compounded with the said Trustees in respect\nthereof, or in case he shall be committed for not delivering any Books,\nPapers, Writings, or Vouchers as aforesaid, until he shall have delivered\nup such Books, Papers, Writings, and Vouchers respectively, or made\nSatisfaction in respect thereof to the said Trustees; but no Person who\nshall be committed for want of sufficient Distress, shall be detained in\nPrison by virtue of this Act for any longer Term than Six Calendar\nMonths.\n\nVI.  Provided always, and be it further enacted, That it shall not be\nlawful for the said Trustees to appoint the same Person who may be\nappointed their Clerk in the Execution of this Act, or the Partner or any\nPerson in the Service or Employ of any such Clerk, to be the Treasurer\nfor the Purposes of this Act, or to appoint the Person who may be\nappointed Treasurer, or the Partner or any Person in the Service or\nEmploy of any such Treasurer, to be the Clerk to the said Trustees; and\nif any Person shall accept both the Offices of Clerk and Treasurer for\nthe Purposes of this Act, or if any Person, being the Partner or in the\nService or Employ of any such Clerk, shall accept the Office of\nTreasurer, or being the Partner or in the Service or Employ of any such\nTreasurer, shall accept the Office of Clerk in the Execution of this Act,\nor if any such Treasurer shall accept any Place or Office of Profit or\nTrust under the said Trustees, other than that of Treasurer, every such\nPerson so offending shall for every such Offence forfeit and pay the Sum\nof One hundred Pounds to any Person or Persons who shall sue for the\nsame, to be recovered, with full Costs of Suit, in any of His Majesty's\nCourts of Record at _Westminster_, by Action of Debt or on the Case, or\nby Bill, Suit, or Information, wherein no Essoign, Protection, or Wager\nof Law, and not more than One Imparlance, shall be allowed.\n\nVII.  And be it further enacted, That the said Trustees shall and they\nare hereby required to order and direct a Book or Books to be provided\nand kept by their Clerk for the Time being, in which Book or Books such\nClerk shall enter or cause to be entered true and regular Accounts of all\nSums of Money received, paid, laid out, and expended for or on account of\nthis Act, and of the several Articles, Matters, and Things for which such\nSums of Money shall have been disbursed, laid out, and paid, which Book\nor Books shall at all seasonable Times be open to the Inspection of the\nsaid Trustees or any Person or Persons rated or assessed for the Purposes\nof this Act, without Fee or Reward; and the said Trustees, or Persons\nassessed as aforesaid, or any of them, may take Copies of or Extracts\nfrom the said Book or Books, or any Part or Parts thereof, without paying\nany thing for the same, and in case the said Clerk shall refuse to permit\nor shall not permit the said Trustees or either of them, or any Person\nrated or assessed as aforesaid, to inspect such Book or Books, or to take\nsuch Copies of or Extracts from the same as aforesaid, when required,\nsuch Clerk shall for every such Offence forfeit and pay a Sum not\nexceeding Ten Pounds.\n\nVIII.  And be it further enacted, That all the Footways and Carriageways\nnow formed and which hereafter shall be formed in the said Square, and\nall other Footways and Carriageways which now are or hereafter may be\nlaid out and formed in or upon the said Piece of Land belonging to the\nsaid _William Farlar_, and also the Footway next the Turnpike Road, such\nFootway extending from the Entrance of the Mews behind _Brompton Row_,\nand behind the East Side of the said Square, Westward along the South End\nof the said Square and in front of _Brunswick Place_, to the Entrance of\n_Cottage Place_, and the Mews behind the West Side of the said Square,\nand also all the Houses and Buildings now erected and which hereafter\nshall be erected upon the said Piece of Land belonging to the said\n_William Farlar_, shall be deemed to be within the Jurisdiction of this\nAct; and that it shall be lawful for the said Trustees to cause all the\nsaid Footways and Carriageways to be properly formed, paved, or\ngravelled, amended and kept in repair, upon such Levels and in such\nManner and with such Materials as they shall judge necessary, and to\ncause the said Carriageways to be cleansed and watered, and Drains,\nSinks, and Watercourses to be made for conveying the Water off and from\nthe said Footways and Carriageways, and Posts and Rails or Chains to be\nset up for the Protection and Preservation of the Foot Pavements or any\nPart thereof, and to cause to be done or made all such Works,\nAlterations, and Improvements in and about the said Footways and\nCarriageways as the said Trustees shall from Time to Time deem necessary\nand proper for the making, maintaining, and supporting the same; and if\nany Person or Persons shall dig, break, or take up the said Footways or\nCarriageways, or any Part thereof, without the Consent of the said\nTrustees, or of some Person empowered by them to give such Consent, every\nsuch Person shall for every such Offence forfeit and pay a Sum not\nexceeding Ten Pounds.\n\nIX.  And be it further enacted, That if and as often as any of the\nFootways or Carriageways within the Jurisdiction of this Act, or any Part\nor Parts thereof, shall be dug up, opened, or disturbed for the Purpose\nof laying down, altering, repairing, renewing, or removing any Pipe or\nPipes, or for any other Purpose whatsoever, then and in every such Case\nall and every the Works, Business, and Matters for or in respect whereof\nsuch Footways or Carriageways respectively, or any Part or Parts thereof,\nshall be so dug up, opened, or disturbed, shall be executed and completed\nwith all possible Expedition, and in every such Case (whether such\nFootways or Carriageways shall be so dug up, opened, or disturbed with or\nwithout the Consent of the said Trustees) the Company or Companies,\nPerson or Persons respectively, who from Time to Time shall cause or\ndirect the said Footways or Carriageways, or any Part or Parts thereof\nrespectively, to be so dug up, opened, or disturbed, shall at their own\nCosts and Charges cause the same to be forthwith well and substantially\nfilled up, rammed down, repaired, and gravelled, and put into as good,\nsound, and perfect a State and Condition in all respects as the said\nFootways and Carriageways respectively were in immediately before the\nsame being so dug up, opened, or disturbed; and in case any such Company\nor Companies, Person or Persons, shall neglect or refuse so to do for the\nSpace of Twenty-four Hours next after the Completion of the Works,\nBusiness, or Matters in respect whereof the same shall be so dug up,\nopened, or disturbed, it shall be lawful for the said Trustees to cause\nsuch Footways and Carriageways respectively to be forthwith well and\nsubstantially filled up, rammed down, repaved, gravelled, and put into\nsuch good, sound, and perfect Condition as aforesaid; and all the Costs,\nCharges, and Expences thereof shall be repaid to the said Trustees by the\nCompany or Companies, Person or Persons respectively, who shall have\ncaused or directed the same Footways or Carriageways respectively to be\nso dug up, opened, or disturbed; and unless all such Costs, Charges, and\nExpences shall be so repaid to the said Trustees (on Demand made thereof\nby the Clerk or Collector for the Time being appointed or employed by the\nsaid Trustees), then all such Costs, Charges, and Expences may be\nrecovered from the Company or Companies, Person or Persons, hereby made\nliable to repay the same as aforesaid, in the same Manner as any Penalty\nor Forfeiture incurred under the Provisions contained in this Act may be\nlevied or recovered before any Justice of the Peace.\n\nX.  And be it further enacted, That it shall be lawful for the said\nTrustees from Time to Time to cause such Lamp Posts, Lamp Irons, and\nLamps to be set up in or about the Footways and Carriageways within the\nJurisdiction of this Act, or upon or against the Walls or Palisadoes of\nor belonging to any adjoining or contiguous House or Building, as they\nshall deem necessary for lighting the said Footways and Carriageways, and\nto cause such Lamps to be lighted with Oil or Gas during such Hours and\nSeasons as they shall from Time to Time deem necessary or proper; and if\nany Person or Persons shall extinguish the Light in any such Lamp whilst\nthe same ought to be kept burning, every such Person shall for every such\nOffence forfeit and pay a Sum not exceeding Five Pounds: Provided always,\nthat if any Damage shall be occasioned to any Building or Wall, or to any\nRails or Palisadoes, by the affixing, setting up, taking down, altering,\nor repairing any such Lamp or Lamp Iron, or the Fastenings thereof, the\nsaid Trustees shall immediately cause such Injury or Damage to be well\nand sufficiently repaired, and the Expence thereof shall be paid out of\nany Money which shall be raised under the Authority of this Act: Provided\nalso, that nothing herein contained shall authorize the said Trustees to\ncause any Pipe or Pipes for the Conveyance of Gas to be affixed or\ncontinued upon or against any House or Building, except it be with the\nConsent of the Occupier of such House or Building.\n\nXI.  And be it further enacted, That all Pipes and other Conduits to be\nlaid or used for the Conveyance of Gas in, under, through, along, across,\nor round the said Square called _Brompton Square_, or any other Place\nwithin the Jurisdiction of this Act, shall be laid at the greatest\npracticable Distance from any Water Pipe now or for the Time being laid\ndown for the Conveyance of Water under, through, along, across, or round\nthe said Square, or any such other Place as aforesaid, except in Cases\nwhere it shall be unavoidably necessary to lay the Gas Pipes across any\nof the Water Pipes, in which Case the Gas Pipes shall be laid over and\nabove the Water Pipes at the greatest practicable Distance therefrom, and\nshall form therewith a Right Angle, and in that Case the Gas Pipes so\ncrossing the Water Pipes shall be at the least Six Feet in Length, so\nthat no Joint of any such Gas Pipe shall be nearer to any Part of any\nsuch Water Pipes than Three Feet at the least; and in laying down the Gas\nPipes, the said Trustees, or the Company, Contractor, or other Person or\nPersons supplying Gas or laying down Pipes for the Conveyance thereof,\nshall in no Case join Two or more Pipes together previously to their\nbeing laid in the Trench, but shall lay each Pipe as nearly as may be in\nits Place in the Trench, and shall in such Trench properly form the\nJointing with the other Pipes to be added thereto with proper and\nsufficient Material, and shall also make and keep all and every such\nTrench and all Pipes connected or communicating therewith and all the\nScrews Joints, Inlets, and Apertures therein respectively, air-tight, and\nin every respect prevent the Gas from escaping therefrom, upon pain of\nforfeiting for every Offence the Sum of Five Pounds.\n\nXII.  And be it further enacted, That in case any Pipe, Cock, or Branch\nshall in pursuance of the Powers herein contained be carried or laid from\nany Main or Pipe belonging to the said Trustees, or to any Gas Light\nCompany, or other Person or Persons contracting to light or lighting with\nGas the said Square or any other Place within the Limits or Jurisdiction\nof this Act, into, through, or against any private Dwelling House,\nBuilding, Yard, or Ground, for any Purpose whatsoever, and the Owner or\nOccupier of any such private Dwelling House, Building, Yard, or Ground\nshall be desirous that the same Pipe, Cock or Branch, shall be altered or\nremoved and taken away, and of such Desire shall give Notice in Writing\nto the said Trustees, or to such Gas Light Company, or other Person or\nPersons so contracting to light or lighting with Gas as aforesaid, then\nand in every such Case the said Trustees, or such Gas Light Company, or\nother Person or Persons so contracting to light or lighting with Gas as\naforesaid, shall within Seven Days next after the Service of such Notice\nat the usual Office or Place of transacting Business of the said\nTrustees, Gas Light Company, or other such Person or Persons as aforesaid\nrespectively, alter or remove and take away, or cause to be altered or\nremoved and taken away, all or any such Pipes, Cocks, and Branches\nrespectively, and in default thereof it shall be lawful for such Owner or\nOccupier, or any Person or Persons acting under the Authority of such\nOwner or Occupier, to cause such Pipes and Branches respectively to be\naltered or removed and taken away, as the Case may require; and all the\nCosts and Expences of and incident to every such Alteration or Removal\nshall be paid by the said Trustees, or by such Gas Light Company, or\nother Person or Persons contracting to light or lighting with Gas as\naforesaid, and may be recovered in the same Manner as any Penalty or\nForfeiture which shall be incurred under the Provisions contained in this\nAct may be levied or recovered, before a Justice of the Peace.\n\nXIII.  And be it further enacted, That whenever the Water which shall be\nsupplied by any Water Works Company, or Company established for the\nSupply of Water, shall be contaminated or affected in any way whatsoever\nby the Gas of the said Trustees, or of any Gas Light Company, or other\nPerson or Persons contracting to light or lighting with Gas the said\nSquare called _Brompton Square_, or any other Place or Places within the\nJurisdiction of this Act, then and in every such Case the said Trustees,\nGas Light Company, or other Person or Persons so contracting to light or\nlighting with Gas as aforesaid, shall, within Twenty-four Hours next\nafter Notice thereof in Writing, signed by any One of the Directors, or\nby the Secretary or Chief Clerk for the Time being of any such Water\nWorks Company, and left at the usual Office or Place of transacting\nBusiness of the said Trustees, Gas Light Company, or other Person or\nPersons contracting to light or lighting with Gas as aforesaid, cause\nMeasures to be taken effectually to stop and prevent such Gas from\ncontaminating or affecting the Water of any such Water Works Company; and\nin case the said Trustees, Gas Light Company, or other Person or Persons\ncontracting to light or lighting with Gas as aforesaid, shall not, within\nTwenty-four Hours next after each and every such Notice so left as\naforesaid, effectually remove the Cause of every such Complaint, and\nprevent all and every such Contamination whereof Notice shall be given as\naforesaid, then and in every such Case the said Trustees, or such Gas\nLight Company, or other Person or Persons contracting to light or\nlighting with Gas as aforesaid, shall on each and every Complaint whereof\nNotice shall be given as aforesaid forfeit and pay to the Secretary or\nChief Clerk of such Water Works Company, for the Use and Benefit of such\nCompany, the Sum of Ten Pounds for each and every Day during which the\nWater supplied by any such Water Works Company shall be and remain\ncontaminated or affected by the Gas of the said Trustees, Gas Light\nCompany, or other Person or Persons contracting to light or lighting with\nGas as aforesaid; and every such Penalty or Forfeiture shall and may be\nlevied and recovered for the Use of such Water Works Company, in the same\nManner as any other Penalty or Forfeiture which shall be incurred under\nthe Provisions contained in this Act may be levied and recovered before a\nJustice of the Peace, with Costs, to be assessed by such Justice.\n\nXIV.  And whereas it may be or become a Question upon such Complaint as\naforesaid whether the said Water be contaminated or affected by the Gas\nof the Trustees, Gas Light Company, or other Person or Persons so\ncontracting to light or lighting with Gas the said Square, or such other\nPlace or Places as aforesaid; be it therefore enacted, That in every such\nCase it shall be lawful for any such Water Works Company at aforesaid to\ndig to and about, and to search and examine all or any of the Mains,\nPipes, Conduits, and Apparatus of the said Trustees, Gas Light Company,\nor other Person or Persons contracting to light or lighting with Gas as\naforesaid, for the Purpose of ascertaining whether such Contamination\nproceed or be occasioned by the Gas of the said Trustees, or of each Gas\nLight Company, or other Person or Persons; and if it shall appear that\nthe said Water has been contaminated or affected by any Escape of Gas,\nthen and in every such Case the Costs and Expences of the said Digging,\nSearch, and Examination, and of the Repair of the Pavement of the Street\nor Place which shall be taken up or disturbed, shall be borne and paid by\nthe said Trustees, Gas Light Company, or other Person or Persons by whose\nGas the said Water shall be so contaminated as aforesaid, which Costs and\nExpences shall be ascertained and determined (if necessary) by any\nJustice of the Peace, and may be levied and recovered in the same Manner\nas any Penalty or Forfeiture which shall be incurred under the Provisions\ncontained in this Act may be recovered before a Justice of the Peace:\nProvided always, that if upon such Examination it shall appear that such\nContamination has not arisen from any such Escape of Gas from any of the\nMains, Pipes, or Conduits of the said Trustees, Gas Light Company, or\nother Person or Persons contracting to light or lighting with Gas as\naforesaid, then and in every such Case the Water Works Company by whom or\non whose Behalf such Examination and Search shall be made shall bear and\npay all the Costs, Charges, and Expences of and incident to such\nExamination and Search, and shall also make good to the said Trustees,\nGas Light Company, or other Person or Persons so contracting to light or\nlighting with Gas as aforesaid, any Loss, Injury, or Damage which may be\noccasioned to the Mains, Pipes, Conduits, or Apparatus of the said\nTrustees, Gas Light Company or other Person or Persons as aforesaid, in\nand by such Search and Examination, and also any Injury or Damage which\nmay be done in or about any of the Footways or Carriageways which shall\nbe broken up or disturbed in such Search or Examination; the Amount of\nsuch Injury, Loss, or Damage to be ascertained and determined, if\nnecessary, by any such Justice of the Peace as aforesaid.\n\nXV.  And be it further enacted, That whenever any Gas shall be found to\nescape from any Pipe or Pipes which shall be laid down, set up, or used\nin or about the lighting _Brompton Square_, or any other Place or Places\nwithin the Jurisdiction of this Act, the said Trustees, or the Gas Light\nCompany, or other Person or Persons contracting to light or lighting with\nGas the said Square, or such other Place or Places as aforesaid, shall,\nimmediately after Notice given to them or him respectively, by Parol or\nin Writing, of any such Escape of Gas, from any Person or Persons\nwhomsoever, cause the most speedy and effectual Measures to be taken to\nstop and prevent such Gas from escaping; and in case the said Trustees,\nGas Light Company, or other Person or Persons so contracting to light or\nlighting with Gas the said Square or other Place or Places respectively,\nshall not, within Twenty-four Hours next after such Notice given,\neffectually stop and prevent any future Escape of Gas, and wholly and\nsatisfactorily remove the Cause of Complaint, then and in every such Case\nthe said Trustees, or the Gas Light Company, or other Person or Persons\nso contracting to light or lighting with Gas as aforesaid, shall for\nevery such Offence forfeit and pay the Sum of Five Pounds for each Day\nafter the Expiration of Twenty-four Hours from the Time of giving any\nsuch Notice, during which the Gas shall be suffered to escape as\naforesaid; which Penalty or Penalties shall from Time to Time be\nrecoverable in a summary Way, in the same Manner as any other Penalty or\nForfeiture which shall be incurred under the Provisions contained in this\nAct may be recovered.\n\nXVI.  And be it further enacted, That if the said Trustees, or any Gas\nLight Company, or other Person or Persons whomsoever contracting to light\nor lighting with Gas the said Square called _Brompton Square_, or any\nother Place or Places within the Jurisdiction of this Act, shall at any\nTime drain, carry, or convey, or cause or suffer to be drained, carried,\nor conveyed, or to run or flow, any Washings or other Liquid, or any\nother Matter or Thing whatsoever which shall arise or be made in or about\nthe Prosecution of their Gas Works or the Manufacture of Gas, into the\nRiver _Thames_, or any other River, or into any Brook, Stream, Reservoir,\nCanal, Aqueduct, Waterway, Feeder, Pond, or Spring Head, or into any\nDrain, Sewer, or Ditch communicating with any of them, or shall do any\nAnnoyance, Act, or Thing to the Water contained in any of them whereby\nsuch Water or any Part thereof shall or may be soiled, fouled, or\ncorrupted, then and in every such Case the said Trustees or such Gas\nLight Company, or other Person or Persons so contracting to light or\nlighting with Gas the said Square, or such other Place or Places as\naforesaid, shall for every such Offence forfeit and pay the Sum of Two\nhundred Pounds to any Person or Persons who shall sue for the same in any\nof His Majesty's Courts of Record at _Westminster_, by Action of Debt or\non the Case, wherein no Essoign, Protection, or Wager of Law, and not\nmore than One Imparlance, shall be allowed; Provided always, that no such\nPenalty or Forfeiture shall be recoverable, unless the same be sued for\nwithin Twelve Calendar Months next after such Annoyance, Injury, Damage,\nAct, or Thing shall have ceased: Provided also, that in addition to the\nsaid Penalty of Two hundred Pounds (and whether such Penalty shall or\nshall not be sued for or recovered), in case any Washings or other\nLiquid, or any other Matter or Thing whatsoever which shall arise or be\nmade in or about the Prosecution of such Gas Works, or the Manufacture of\nGas as aforesaid, shall be drained, conducted, or conveyed, or caused or\nsuffered to run or flow into the said River _Thames_, Or any other River,\nor into any Brook, Stream, Reservoir, Canal, Aqueduct, Waterway, Feeder,\nPond, or Spring Head, or into any Drain, Sewer, or Ditch, communicating\nwith any of them, or if any such Annoyance, Act, or Thing shall be done\nor caused to be done as aforesaid, and Notice thereof in Writing shall be\ngiven by any Person or Persons whomsoever to the said Trustees, or to the\nGas Light Company, or other Person or Persons so contracting to light or\nlighting with Gas the said Square, or such other Place or Places as\naforesaid, and if the said Trustees, or such Gas Light Company, or other\nPerson or Persons as last aforesaid, shall not, within Twenty-four Hours\nnext after every such Notice as aforesaid, stop and prevent all such\nWashings or other Liquids, and all such other Substances and Things as\naforesaid from being drained, conducted, or conveyed, or from running or\nflowing in manner aforesaid, and also cause any such other Annoyance,\nAct, or Thing as aforesaid to be discontinued, then and in every such\nCase the said Trustees, Gas Light Company, or other Person or Persons so\noffending shall forfeit and pay the Sum of Twenty Pounds for each and\nevery Day such Washings or other Liquids, or other Matters or Things,\nshall be so drained, conducted, or conveyed, or caused or suffered to run\nor flow in manner aforesaid, or such other Annoyance, Act, or Thing shall\nbe so done or caused to be done as aforesaid; and such last-mentioned\nPenalty shall and may be levied and recovered in the same Manner as any\nother Penalty or Forfeiture which shall be incurred under the Provisions\ncontained in this Act may be levied or recovered, and shall be paid to\nthe Informer or Informers, or to the Person or Persons who in the\nJudgment of the Justice or Justices before whom the Conviction shall take\nplace shall have sustained any Annoyance, Injury, or Damage by the Act,\nMatter, or Thing in respect whereof such Penalty or Forfeiture shall be\ninflicted.\n\nXVII.  And be it further enacted, That it shall be lawful for the said\nTrustees from Time to Time to appoint so many able-boded Men to act as\nWatchmen, and to patrol, watch, and guard the said Square, and all other\nPlaces within the Jurisdiction of this Act, during such Hours and at such\nWages as the said Trustees shall think proper, and to provide suitable\nArms and Clothing for such Watchmen, and also to provide and set up\nWatchboxes for their Reception at such Places as the said Trustees shall\nthink necessary; and also to provide a proper Watchhouse or Place for the\nsafe Custody of such Persons as may be apprehended by such Watchmen while\non Duty; and that such Watchmen shall act as Constables during the Hours\nappointed for their watching, and shall have, exercise, and enjoy all\nsuch and the like Powers and Authorities, Privileges and Immunities, as\nany Constable may have and enjoy by Law; and every Person who shall\nassault or resist, or promote or encourage the assaulting or resisting,\nany such Watchman in the Execution of his Duty, and also every Victualler\nwho shall harbour or entertain any Watchman during any Part of the Time\nappointed for his being on Duty, shall for every such Offence forfeit and\npay a Sum not exceeding Five Pounds; and in case and as often as any such\nWatchman shall misbehave himself or neglect his Duty, he shall forfeit\nand pay a Sum not exceeding Two Pounds.\n\nXVIII.  And be it further enacted, That it shall be lawful for the said\nTrustees from Time to Time to cause the Name of the said Square, and of\nany other Street or Place within the Jurisdiction of this Act, to be\npainted or otherwise described in a conspicuous Part of any House or\nBuilding within such Square, Street, or Place respectively; and also from\nTime to Time to cause all the Houses, Buildings, and Tenements within the\nJurisdiction of this Act to be distinguished by Numbers, to be painted or\naffixed upon the Entrance Doors thereof respectively; and if any Person\nor Persons shall alter or wilfully deface or remove any such Name or\nNumber, or any Part thereof, or if any Person or Persons for the Time\nbeing chargeable with the Rates which shall be made in pursuance of this\nAct in respect of any House, Building, or Tenement of which the Number\nshall be so altered, defaced, or removed, shall not perfectly reinstate\nsuch Number within Seven Days next after such Person or Persons\nrespectively shall be required so to do by a Notice from the said\nTrustees or their Clerk, every such Person shall for every such Offence\nforfeit and pay a Sum not exceeding Five Pounds.\n\nXIX.  And be it further enacted, That proper Gates shall be erected and\nat all Times kept up and preserved by the said Trustees, across the\nFootways and Carriageways at the Entrance to _Brompton Square_ next the\nTurn-pike Road, so as to inclose and keep the same Footways and\nCarriageways (such footways and Carriageways not being now public\nHighways) private, as far as may be consistent with the general\nConvenience of the Occupiers of Houses in the said Square; and which said\nGates shall be kept closed during such Hours between Ten of the Clock at\nNight and Seven of the Clock in the Morning, as the said Trustees shall\ndirect, and during those Hours a Key or Keys thereof shall be left in the\nCharge of the Watchman or Watchmen on Duty in the said Square.\n\nXX.  And be it further enacted, That the said Trustees shall contract\nwith or employ some Person or Persons to cleanse the Carriageways, and\ntake away the Dust, Cinders, Ashes, and Rubbish (not being Rubbish\narising from Building or Repairs) from the Houses of the Inhabitants\nwithin the Jurisdiction of this Act; and if any Person or Persons (not\nbeing duly authorized by the said Trustees, and not acting under or on\nbehalf of a Person so authorized) shall go about to collect or gather, or\nshall ask for, receive, or carry away any Dust, Cinders, or Ashes from\nany House, Building, or Place within the Jurisdiction of this Act, every\nPerson so offending shall for every such Offence forfeit and pay a Sum\nnot exceeding Ten Pounds.\n\nXXI.  And, to the Intent that a Pleasure Garden or Shrubbery may be\nmaintained and preserved within the inclosed Area of the said square, for\nthe Embellishment thereof, and for the Use of the Inhabitants of the\nHouses in the said Square; be it further enacted, That it shall be lawful\nfor the said Trustees from Time to Time to contract for, and in the Names\nof any Five or more of the said Trustees, to take and accept from the\nsaid _William Farlar_, his Heirs or Assigns, a Lease or Leases of all\nGround now laid out and which hereafter shall be laid out as or for an\ninclosed Area in the said Square, at such Rent or Rents, and subject to\nsuch Stipulations and Conditions as shall from Time to Time be settled\nand agreed upon between the said Trustees and the said _William Farlar_,\nhis Heirs or Assigns, so that such Rent or Rents shall not in the whole\nat any one Time exceed the yearly Sum of Thirty Pounds, and the Estate\nand Interest of the Lessees in every such Lease shall by virtue of this\nAct absolutely vest in the Trustees for the Time being; and such Rent or\nRents, and the Costs and Charges of and incident to the Performance of\nsuch Stipulations and Conditions, and also the Costs and Charges of every\nsuch Lease, shall be paid out of the Rates to be made by virtue of this\nAct; and during the Continuance of every such Lease the said inclosed\nArea (subject to the Stipulations and Conditions which shall be contained\nin every such Lease respectively) shall be kept and maintained as and for\na private Pleasure Garden and Shrubbery, for the exclusive Use and\nEnjoyment of the Persons for the Time being occupying Houses in the said\nSquare (subject to the Regulations and Provisions contained in this Act),\nand shall be kept securely fenced in and inclosed with good and\nsubstantial Iron Railing, and proper Entrance Gates, and with good and\nsecure Locks on all such Gates; and no Person or Persons whomsoever other\nthan the Occupiers of Houses in the said Square (not being menial\nServants) shall be entitled to have or use any Key or Keys of the said\ninclosed Area, except it be with the Consent in Writing of the Majority\nof the said Trustees, who for the Time being shall be Occupiers of Houses\nin the said Square: Provided always, that the Owners and Occupiers of\nHouses in the said Square called _Brompton Square_ built previously to\nthe passing of this Act, shall, notwithstanding any thing herein-before\ncontained, have and enjoy all Right and Interest, if any, which such\nOwners or Occupiers respectively may have, or be entitled to, in the\ninclosed Area of the said Square under or by virtue of any Covenant or\nAgreement now in existence.\n\nXXII.  And be it further enacted, That it shall be lawful for the said\nTrustees to contract, by the Year or otherwise, with any Person or\nPersons for the doing of any Works or Business whatsoever relating to or\nconnected with the Execution of this Act, so that no such Contract shall\ncontinue in force for more than Three Years from the Time of contracting,\nand the Clerk to the said Trustees shall enter all such Contracts in a\nBook or Books to be provided and kept for that Purpose.\n\nXXIII.  And be it further enacted, That all the Paving Stones, Gravel,\nand other Materials of which the Footways and Carriageways within the\nJurisdiction of this Act shall for the Time being consist, and the Posts\nand Rails which shall be set up for the Protection of the said Footways,\nand also all the Lamp Posts, Lamp Irons, Lamps, and other Utensils which\nthe said Trustees shall from Time to Time cause to be set up or used for\nlighting the said Footways and Carriageways, and the Watchboxes, Arms,\nand Clothing which shall be provided by the said Trustees for the Use of\nthe Watchmen who shall be employed in pursuance of this Act, and also\n(during the Continuance of any Lease of the inclosed Area of the said\nSquare which shall be granted to the said Trustees as aforesaid) the Iron\nRailing, Fences, Gates, and Locks of and belonging to the said inclosed\nArea, and the Fences, Plants, Shrubs, and Herbage in the said inclosed\nArea for the Time being, and also the Materials of which the Gravel Walks\ntherein for the Time being shall be formed and consist, shall for all\nPurposes be deemed and considered as vested in the said Trustees, and\nshall in all Actions, Suits, Indictments, Complaints, Convictions, and\nother Proceedings whatsoever relating thereto, be described generally as\nthe Property of the Trustees for regulating and improving _Brompton\nSquare_ and certain adjacent Places, without naming the said Trustees or\nany of them.\n\nXXIV.  And be it further enacted, That if any Person or Persons shall\nwilfully or maliciously destroy or damage any of the Footways or\nCarriageways within the Jurisdiction of this Act, or any Part thereof, or\nany Post, Rail, Lamp Post, Lamp Iron, Lamp, Watchbox, Arms, Clothing, or\nother Utensil or Thing which shall be provided by or in pursuance of the\nDirections of the said Trustees, or any of the Iron Railing, Fences,\nGates, Locks, Trees, Plants, Shrubs Herbage, or Gravel Walks of or\nbelonging to the inclosed Area of the said Square, or any of them, or any\nPart thereof respectively, every such Person shall for every such Offence\nforfeit and pay a Sum not exceeding Five Pounds; and every such Person,\nand also every Person by whom any of the said Footways and Carriageways,\nLamp Posts, Lamp Irons, Lamps, Watchboxes, Arms, Clothing, Iron Railing,\nFences, Gates, Locks, Trees, Plants, Shrubs, Herbage, Gravel Walks,\nUtensils and Things, or any of them or any Part thereof respectively,\nshall be carelessly or accidentally destroyed or damaged, shall make full\nSatisfaction for the Damage or Injury done or committed by him, her, or\nthem respectively; and unless such full Satisfaction shall be forthwith\nmade, any Justice of the Peace, upon Application to him for the Purpose\nby or on behalf of the said Trustees, may award such Sum of Money to be\npaid to the said Trustees by way of Satisfaction for such Damage as such\nJustice shall think reasonable, and the same shall be recovered in the\nManner herein-after provided for the Recovery of any Fine, Penalty, or\nForfeiture to be inflicted or imposed under the Provisions of this Act.\n\nXXV.  And be it further enacted, That if at any Time or Times during the\nContinuance of any Lease or Leases of the inclosed Area of the said\nSquare, any Person or Persons shall, without the Consent in Writing of\nthe said Trustees, make or cause to be made, or shall sell or cause to be\nsold, or shall offer for Sale any Key or Keys for unlocking or opening\nthe Gates of the inclosed Area of the said Square, or any of them, or if\nany Person or Persons shall knowingly purchase or use any Key or Keys so\nunlawfully made or sold, every such Person shall for every such Offence\nforfeit and pay a Sum not exceeding Ten Pounds, and it shall be lawful\nfor the said Trustees or any of them, or any Person or Persons in their\nService or Employ, or acting under their Authority, to seize and to\ndetain and keep or to destroy every such Key.\n\nXXVI.  And be it further enacted, That to enable the said Trustees to\ncarry this Act into Execution it shall be lawful for them from Time to\nTime to make such Rates as they shall deem needful upon the Tenants and\nOccupiers of all Lands, Houses, Shops, Coach-houses, Stables, Erections,\nBuildings, and other Tenements whatsoever within the Jurisdiction of this\nAct, in such Sums of Money as the said Trustees shall think proper, so\nthat such Rates shall be just and equal Pound Rates, according to the\nannual Rent or Value of the Premises thereby assessed, and so that the\nCosts and Expences of and incident to the enclosed Area of the said\nSquare shall be wholly borne by the Tenants and Occupiers of Houses in\nthe said Square, and so that in every such Rate it shall distinctly\nappear what Part thereof shall be raised in respect of the said\nlast-mentioned Costs and Expences, and so that the said Rates shall not\nin any One Year exceed the Sum of Two Shillings in the Pound of the\nannual Rent or Value of the Property rated, except as to Houses in the\nsaid Square, after such Lease of the enclosed Area of the said Square\nshall be obtained as aforesaid, and during the Continuance of every such\nLease, and so that in the last-mentioned Cases the Rates in respect of\nthe Houses in the said Square shall not in any One Year exceed the Sum of\nTwo Shillings and Sixpence in the Pound of the Rent or annual Value\nthereof; and in making the said Rates, all Stables, Coach-houses, and\nother Buildings which shall belong to or be held and occupied with any of\nthe Houses in the said Square, and within the Jurisdiction of this Act,\nshall be respectively rated with such Houses, and not as separate\nBuildings; and to facilitate the making of the said Rates, it shall be\nlawful for the said Trustees, or any Person or Persons appointed by them,\nat all seasonable Times to examine the Books of Assessment to any Public,\nParliamentary.  Parochial, or Local Tax charged upon or affecting the\nHouses, Buildings, or Tenements within the Jurisdiction of this Act, or\nany of them.\n\nXXVII.  And be it further enacted, That in every Rate to be made as\naforesaid, the Landlord or Proprietor of every House or Tenement within\nthe Jurisdiction of this Act, which shall be occupied by any Ambassador,\nForeign Minister, or other Person not compellable by Law to pay such\nRate, shall be deemed to be and shall be rated as the Occupier thereof\nrespectively.\n\nXXVIII.  And be it further enacted, That whenever any House within the\nJurisdiction of this Act shall be let out ready furnished, or shall be\nlet out, furnished or unfurnished, in separate Tenements, or wholly or in\npart to a Lodger or Lodgers, the Landlord or Owner, being the Person or\nPersons claiming or receiving the Rent payable by the Occupier or\nOccupiers of every such furnished House, Tenement, or furnished Lodging\nrespectively, shall be deemed to be and shall be rated as the Occupier\nthereof: Provided always, that the Goods and Chattels of every Occupier\nof any such ready-furnished House or separate Tenement or Lodging, shall\n(without Prejudice to the Liability of the Person or Persons so rated) be\nliable to be distrained for raising such Rates, and all Arrears thereof,\nso that in no Case any greater Sum shall be levied by Distress of the\nGoods and Chattels of any such Occupier than shall at the Time of making\nsuch Distress be actually due from such Occupier for the Rent of the\nPremises in which such Distress shall be made: Provided also, that every\nsuch Occupier who shall pay any such Rate or Rates, or upon whose Goods\nand Chattels the same shall be levied, shall and may deduct the Amount of\nthe Sum or Sums which shall be so paid or levied out of any Rent payable\nby such Occupier to the Person or Persons rated; and such Payment shall\nbe a sufficient Discharge to every such Occupier for so much of such Rent\naccordingly.\n\nXXIX.  And be it further enacted, That every Rate which shall be made for\nthe Purposes of this Act may from Time to Time be altered and amended by\nthe said Trustees at any Meeting pursuant to this Act, either by\ninserting the Name or Names of any Person or Persons who ought to have\nbeen rated or who shall have come into the Possession of any rated or\nrateable House, Building, or Tenement during the Time for which any Rate\nmay be payable, or by striking out the Name or Names of any Person or\nPersons who ought not to have been rated, or by raising or lowering the\nSum or Sums charged upon any Person or Persons, or by apportioning\nbetween any outgoing and incoming Tenant any Sum or Sums which shall be\ncharged upon either in such Rate or otherwise as the said Trustees shall\nthink necessary for making such Rate a just and equal Rate; and the\nMonies charged to any Person or Persons in consequence of any such\nAlteration or Amendment may be recovered and levied in the same Manner as\nif such Alteration or Amendment had been an original Insertion.\n\nXXX.  And be it further enacted, That the first Rate which shall be made\nby virtue of this Act shall be for Two Quarters ending at _Christmas Day_\nnext after the passing of this Act, and that every subsequent Rate shall\nbe for One whole Year, commencing from the Expiration of the Term for\nwhich the preceding Rate shall have been made; and that all the said\nRates shall become due and be payable immediately upon the making\nthereof, but may be collected in one or several Payments, or yearly,\nhalf-yearly, or quarterly, as the said Trustees shall from Time to Time\ndirect; and that such Rates and all other Monies which shall become due\nor be payable to the said Trustees in pursuance of the Provisions in this\nAct contained shall be and are hereby vested in the said Trustees, and\nshall be paid to such Person or Persons as the said Trustees shall from\nTime to Time appoint to collect and receive the same, and shall be\napplied and disposed of by the said Trustees in defraying the Expences of\nand incident to the Execution of this Act, and otherwise as in or by this\nAct is directed or authorized.\n\nXXXI.  And be it further enacted, That if any Person or Persons liable to\npay any Rate or Rates which shall be made in pursuance of this Act, or\nany Part or Parts of any such Rate or Rates, shall refuse or neglect to\npay the same, it shall be lawful for any Justice of the Peace for the\nCounty of _Middlesex_, and every such Justice is hereby authorized and\nempowered (such Defaulter having been first duly summoned to appear\nbefore such Justice or Justices to shew Cause for such Refusal or\nNeglect, and not having appeared upon such Summons, or having appeared\nand not shewn good and sufficient Cause to such Justice that he, she, or\nthey, is or are not chargeable with such Rate or Rates) to grant a\nWarrant under his Hand and Seal, authorizing the Person appointed to\ncollect the same Rates as aforesaid, or any other Person or Persons, to\nlevy such Rate or Rates, and all Arrears thereof, together with all\nincidental Coats and Expences, by Distress of the Goods and Chattels of\nthe Person or Persons so neglecting or refusing; and if the said Rate or\nRates, and all Arrears thereof, together with all such Costs as\naforesaid, and also all Costs of and incident to the taking and detaining\nthe Distress, shall not be paid within Five Days next after the Distress\nshall be taken, the Person appointed to collect the said Rates, or other\nPerson authorized by the said Warrant, shall cause the Goods and Chattels\ndistrained to be appraised and sold, or such Part thereof as shall be\nsufficient to pay the said Rate or Rates, and all Arrears thereof, and\nall such Costs and Expences as aforesaid, and the Costs and Expences of\nkeeping, appraising, and selling the Distress, and after full Payment of\nthe same, shall return the Overplus (if any) to the Owner or Owners of\nsuch Goods and Chattels respectively, on Demand made thereof by such\nOwner or Owners; and in default of such Distress it shall be lawful for\nany such Justice to commit the Person or Persons so neglecting or\nrefusing to pay such Rate or Rates, and all Arrears thereof, and all\nattendant and incidental Costs and Expences as aforesaid, to the Common\nGaol or House of Correction, there to remain without Bail or Mainprize\nfor any Time not exceeding Six Calendar Months, unless Payment shall be\nsooner made of such Rate or Rates, and all Arrears thereof as aforesaid,\ntogether with all Costs and Expences attendant upon and incident to the\nRecovery thereof (such Costs and Expences to be ascertained and\ndetermined by such Justice); and subject and without Prejudice to the\nPowers and Remedies herein before provided for the Recovery of the said\nRates, the said Trustees may sue for and recover the same, with full\nCosts of Suit, in any Court of Request, or by Action of Debt in any of\nHis Majesty's Courts of Record; and in every such Action it shall be\nsufficient to declare that the Defendant is indebted to the Plaintiff in\nthe Sum sought to be recovered by virtue of a Rate (or of several Rates,\nas the Case may require) made in pursuance of this Act.\n\nXXXII.  And be it further enacted, That it shall be lawful for the said\nTrustees to borrow at Interest, or to raise by the Sale of One or more\nAnnuity or Annuities for a Life or Lives (subject to the Restrictions\nherein-after contained), any Sum or Sums of Money which the said Trustees\nshall from Time to Time require, in or about the Execution of this Act;\nand the said Trustees shall make and deliver to the Person or Persons\nfrom whom any Money shall be so borrowed One or more Promissory Note or\nNotes in the Form or to the Effect following, to the Amount of the Money\nso borrowed; namely,\n\n    'THE Trustees for regulating and improving _Brompton Square_ and\n    certain adjacent Places, acting in the Execution of an Act of the\n    Fifth Year of King _George_ the Fourth, intituled [_here insert the\n    Title of this Act_], hereby promise to pay to \u2014 \u2014, \u2014 \u2014 Executors or\n    Administrators, or \u2014 \u2014 or their Order, by Indorsement hereon, out of\n    the Rates to be made in pursuance of the said Act, the Sum of \u2014\n    Pounds on the \u2014 Day of \u2014 [_here name a Day for the Payment of the\n    Money_, _not being less than Five Years nor more than Fifteen Years\n    from the Date of the Note_], and also Interest for the same from the\n    Date hereof, at the Rate of \u2014 _per Centum per Annum_, by half-yearly\n    Payments, on every \u2014 Day of \u2014 and \u2014 Day \u2014 in the meantime.  In\n    witness whereof Three of the said Trustees have hereunto subscribed\n    their Names, the \u2014 Day of \u2014.'\n\nAnd the said Trustees shall also make and deliver to the Purchaser or\nPurchasers of every such Annuity, a Grant thereof in the Form or to the\nEffect following:\n\n    'THE Trustees for regulating and improving _Brompton Square_ and\n    certain adjacent Places, acting in the Execution of an Act of the\n    Fifth Year of King _George_ the Fourth, intituled [_here insert the\n    Title of this Act_] in consideration of the Sum of \u2014 paid to them by\n    \u2014 of \u2014 do hereby grant to \u2014 an Annuity of \u2014 a Year during \u2014, and to\n    be paid half-yearly on the \u2014 Day of \u2014 and the \u2014 Day of \u2014 out of the\n    Rates to be made in pursuance of the said Act.  In witness whereof\n    Three of the said Trustees have hereunto set and affixed their Hands\n    and Seals, the Day \u2014 of \u2014.'\n\nAnd the Money expressed to be made payable in every such Promissory Note\nas aforesaid, and the Interest thereof, and also every Annuity which\nshall be granted as aforesaid, shall be and are hereby respectively\ncharged upon and made payable out of the Rates to be made by virtue of\nthis Act, and shall be paid accordingly; and every such Promissory Note\nshall be transferable by Indorsement and Delivery, in the same Manner as\nany other Promissory Note is by Law transferable; and every such Annuity\nshall be and be deemed Personal Estate; but no such Annuity shall be\ngranted for the Lives of more than Two Persons, or be made to depend on\nthe Life or Lives of any Person on Persons who at the Time of granting\nsuch Annuity shall be under the Age of Thirty-five Years, nor shall any\nsuch Annuity be granted so as to produce to the Purchaser an Income\nexceeding by more than One-fifth the Income which a Sum equal in Amount\nto the Purchase Money for the same would produce if then invested in the\nPurchase of a similar Annuity, according to the Rate prescribed by Two\nseveral Acts, one made in the Forty-eighth Year of the Reign of His late\nMajesty King _George_ the Third, intituled _An Act to enable the\nCommissioners for the Reduction of the National Debt to grant Life\nAnnuities_, and the other made in the Fifty-second Year of the Reign of\nHis said late Majesty, intituled _An Act for amending Two Acts passed in\nthe Forty-eighth and Forty-ninth Years of His present Majesty_, _for\nenabling the Commissioners for the Reduction of the National Debt to\ngrant Life Annuities_; and none of the said Trustees shall be personally\nliable for the Payment of any Promissory Note or Annuity to be issued or\ngranted in pursuance of this Act.\n\nXXXIII.  And be it further enacted, That the Names of the Trustees\npresent at each Meeting in pursuance of this Act, and also all their\nOrders and Proceedings at each such Meeting, shall be entered in a Book\nor Books to be kept for that Purpose; and the Entry of the Proceedings at\neach Meeting shall be signed by the Chairman at the next Meeting, and\nsuch Entries being so signed shall be deemed Originals, and may be read\nin evidence in all Courts whatsoever.\n\nXXXIV.  And be it further enacted, That if any Person or Persons shall\nbeat or dust, or cause to be beat or dusted, any Carpet or Carpets, or\nshall slack, sift, or screen, or cause to be slacked, sifted, or\nscreened, any Lime or Rubbish, or shall mix any Mortar, or shall make or\nassist in making any Bonfire, or shall discharge any Fireworks, or play\nat Football, or any other Game in the said Square, or any Part thereof;\nor if any Person or Persons shall set or place or cause to be set or\nplaced any Timber, Bricks, Stone, Lime, Mortar, or other Building\nMaterials, or any Stall, Stallboard, or Basket, or any Goods, Wares, or\nMerchandize whatsoever, or any Mud, Dirt, Sand, Rubbish, Filth, Dung,\nAshes, or Dust, upon any Footway or Carriageway within the Jurisdiction\nof this Act; or if any Person or Persons shall run, drive, or place, or\ncause to be run, driven, or placed, any Coach, Chaise, Waggon, Cart,\nTruck, Wheelbarrow, or other Carriage whatsoever, upon any Footway within\nthe Jurisdiction of this Act; or if any Person or Persons shall cause or\nwilfully permit or suffer any Horse or other Beast or Cattle, which such\nPerson or Persons may be riding, driving, or leading, to go upon any such\nFootway, or shall tie or fasten any Horse or other Beast or Cattle to any\nHouse, Wall, Fence, Post, Rail, or other Thing whatsoever, across any\nFootway or Carriageway within the Jurisdiction of this Act; or if any\nPerson or Persons shall place or put out, or cause or permit to be placed\nor put out upon or before any House or Building next or towards any\nFootway within the Jurisdiction of this Act, any Garden or other Pot or\nTub (except such Pot or Tub shall by Iron Rails or Bars be perfectly\nsecured from falling); or if any Person or Persons shall lead or drive\nany Horse or other Beast in any Part of the said Square, either with or\nwithout any Carriage, for the Purpose of breaking, exercising, trying, or\noffering for Sale any such Horse or other Beast; or if any Person or\nPersons shall empty or begin to empty any Boghouse, or to take away any\nNight Soil from any House or Place within the Jurisdiction of this Act,\nor shall come with Carts or Carriages for that Purpose (except between\nthe Hours of Twelve of the Clock at Night and Five of the Clock in the\nMorning); or if any Person or Persons shall occasion any other Kind of\nObstruction or Annoyance whatsoever, in or upon any of the Footways or\nCarriageways within the Jurisdiction of this Act, or shall obstruct or\nincommode, hinder or prevent, the free Passage of any such Footway or\nCarriageway, or prejudice or annoy in any Manner whatsoever any Person or\nPersons travelling, passing, or going thereon; or if any Person or\nPersons shall breed, feed, or keep any Kind or Species of Swine in any\nHouse or Building within the Jurisdiction of this Act, or within any\nYard, Garden, or other Tenement held with any such House or Building;\nevery such Person shall forfeit and pay a Sum not exceeding Five Pounds\nfor every such Offence.\n\nXXXV.  Provided always, and be it further enacted, That no Person shall\nbe subject to any Penalty on account of any Rubbish, Building Materials,\nor Scaffolding, being necessarily deposited or fixed in or upon any\nFootway or Carriageway, before any House or Building which shall be\npulling down, rebuilding, or under Repair, so that the same be properly\nenclosed, and convenient Space left for Carriages to pass and repass with\nease and Safety, and so that a sufficient Light be kept up and maintained\nthere from Sun-set to Sun-rise, to prevent Accidents, and so that all\nsuch Rubbish, Building Materials, Enclosures, and Scaffolding\nrespectively, be removed within a reasonable Time after such pulling\ndown, rebuilding, or repairing shall be finished, or upon Notice signed\nby the Clerk to the said Trustees, and given to the Owner or Occupier of\nthe House or Building which shall be so pulling down, rebuilding, or\nunder Repair, or affixed upon some conspicuous Part of such House,\nBuilding, Enclosure, or Scaffolding: Provided also, that in case any\nPerson or Persons so depositing or affixing or causing to be deposited or\naffixed any such Rubbish, Building Materials, or Scaffolding as\naforesaid, in or upon any of the said Footways or Carriageways as\naforesaid, shall not properly enclose the same, and leave convenient\nSpace for Carriages to pass and repass with Ease and Safety as aforesaid,\nand also keep up a good and sufficient Light there from Sun-set to\nSun-rise, so as to prevent Accidents, or shall not remove any such\nRubbish, Building Materials, or Scaffolding upon such Notice as\naforesaid, every Person making default in any of the Cases aforesaid\nshall forfeit and pay a Sum not exceeding Ten Pounds.\n\nXXXVI.  And be it further enacted, That it shall be lawful for any\nOfficer or Person employed in the Execution of this Act, without any\nother Warrantor Authority than this Act, to seize and detain, and for any\nother Person or Persons to assist in seizing and detaining any unknown\nPerson or Persons who shall commit any Offence punishable under this Act,\nand to take such Offender or Offenders before a Justice of the Peace, in\norder to his or her Conviction for such Offence.\n\nXXXVII.  And be it further enacted, that if any Person or Persons shall\nobstruct, hinder, or molest the said Trustees or any of them, or any\nOfficer or other Person whomsoever, who shall be employed or acting in\nthe Execution of this Act, in the Performance of any Act, Matter, or\nThing to be done by virtue of or under the Authority of this Act, or if\nany Person or Persons who shall be summoned to give Evidence before any\nJustice of the Peace touching or concerning any Offence or Matter\npunishable under this Act, shall not attend pursuant to such Summons, nor\nproduce a sufficient Excuse to the Satisfaction of such Justice for his\nNon-attendance, every Person or Persons so offending shall for every such\nOffence forfeit and pay a Sum not exceeding Ten Pounds.\n\nXXXVIII.  And be it further enacted, That all Penalties and Forfeitures\nwhich shall be incurred under the Provisions herein-before contained, the\nManner of levying or recovering whereof is not herein-before particularly\ndirected, shall be recoverable before any Justice of the Peace for the\nCounty of _Middlesex_, upon Complaint made to any such Justice within Six\nCalendar Months next after the Offences respectively shall be committed;\nand it shall be lawful for every such Justice to whom such Complaint\nshall be made, to summon the Party or Parties complained against before\nhim, and on such Summons to hear and determine the Matter of such\nComplaint, and on Proof of the Offence or Offences respectively, either\nby the Confession of the Offender or Offenders, or by the Oath of any\ncredible Witness or Witnesses, to convict such Offender or Offenders, and\nto adjudge him or them to pay the Penalty or Forfeiture incurred,\nalthough no Information in Writing shall be exhibited before such\nJustice; and in every Case where any such Penalty or Forfeiture, together\nwith all Costs of and incident to the Complaint, Summons, Hearing, and\nConviction, shall not be forthwith paid by the Offender or Offenders at\nthe Time of Conviction, such Penalty and also such Costs as aforesaid\nshall be levied by Distress and Sale of the Goods and Chattels of the\nOffender or Offenders, by Warrant under the Hand and Seal of such\nJustice, (and which Warrant such Justice is hereby empowered and required\nto grant); and in every such Case it shall be lawful for such Justice to\norder the Offender or Offenders so convicted to be detained and kept in\nsafe Custody until Return can conveniently be made to such Warrant of\nDistress, unless the Offender or Offenders shall give sufficient\nSecurity, to the Satisfaction of such Justice, for his or their\nAppearance before such Justice on such Day or Days as shall be appointed\nfor the Return of such Warrant of Distress (not being more than Seven\nDays from the Time of taking such Security), and which Security the said\nJustice is hereby empowered to take by way of Recognizance or otherwise;\nand if upon the Return of such such Warrant it shall appear that no\nsufficient Distress can be had thereupon, or in case it shall appear to\nthe Satisfaction of such Justice, either by the Confession of the\nOffender or Offenders or otherwise, that such Offender or Offenders hath\nor have not sufficient Goods and Chattels whereon such Penalties or\nForfeitures with all Costs as aforesaid can be levied, if a Warrant of\nDistress were issued (in which Case it shall not be necessary to issue\nany Warrant of Distress), then and in every such Case it shall be lawful\nfor such Justice to cause such Offender or Offenders to be committed to\nthe Common Gaol or House of Correction, there to remain without Bail or\nMainprize for any Term not exceeding Six Calendar Months, unless such\nPenalties and Forfeitures, and all attendant and incidental Costs, shall\nbe sooner paid; and the Monies arising by such Penalties and Forfeitures\nrespectively shall from Time to Time, in every Case where the\nAppropriation thereof is not herein-before otherwise directed, be paid to\nthe Treasurer to the said Trustees, and (subject to the Power hereby\ngiven to the said Trustees to dispose of any Part or Parts thereof, not\nexceeding One Half, in rewarding the Informer or Informers, or any other\nPerson or Persons who shall have aided in detecting or convicting the\nOffender or Offenders respectively) shall be applied and disposed of for\nthe Purposes of this Act.\n\nXXXIX.  And be it further enacted, That every Justice of the Peace before\nwhom any Person shall be convicted of any Offence against this Act, shall\nand may cause the Conviction to be drawn up in the following Form of\nWords, or in any other Form to the like Effect:\n\n    'Middlesex to wit. } BE it remembered, That on the \u2014 Day of \u2014 in the\n    Year of our Lord \u2014 _A. B._ is convicted before \u2014 One of His Majesty's\n    Justices of the Peace for the County of _Middlesex_, of [_here\n    shortly state the Offence_, _and when and where committed_] contrary\n    to the Form of a Statute made in the Fifth Year of the Reign of His\n    Majesty King _George_ the Fourth, intituled [_here set forth the\n    Title of this Act_], and I do adjudge that the said _A.B._ hath\n    therefore forfeited the Sum of \u2014.  Given under my Hand and Seal, the\n    Day and Year first above written.'\n\nXL.  And be it further enacted, That where any Distress shall be made for\nany Rate or Penalty to be levied in pursuance of the Provisions contained\nin this Act, the Distress shall not be deemed unlawful, nor the Party or\nParties making the same a Trespasser or Trespassers, on account of any\nDefect or Want of Form in any Proceeding relating thereto, nor shall the\nParty or Parties distraining be deemed a Trespasser or Trespassers _ab\ninitio_, on account of any subsequent Irregularity, but the Person or\nPersons aggrieved by any such Irregularity may recover full Compensation\nfor any special Damage by an Action on the Case, subject to the\nLimitations herein-after made.\n\nXLI.  And be it further enacted, That if any Person shall think himself\nor herself aggrieved by any Rate to be made in pursuance of this Act, he\nor she, having first paid such Rate or Assessment, may appeal to the said\nTrustees at their next Meeting after the Payment thereof, and the said\nTrustees, if they shall think such Person aggrieved, shall give such\nRelief in the Premises as they shall deem reasonable; and if any such\nPerson shall be dissatisfied with the Determination of the said Trustees,\nor if any Person shall think himself or herself aggrieved by any thing\nwhich shall be done by any Justice of the Peace in the Execution of this\nAct, such Person may appeal to the Justices of the Peace for the County\nwherein the Cause of Complaint shall arise, at their General Quarter\nSessions to be held next after the Expiration of One Calendar Month from\nthe Time when the Cause of Complaint shall arise, but not afterwards,\nsuch Appellant first giving Fourteen Days Notice at the least in Writing\nof his or her Intention to make such Appeal, and of the Matter thereof,\nto the Clerk to the said Trustees, and within Five Days next after such\nNotice entering into a Recognizance before some Justice of the Peace for\nsuch County, with Two sufficient Sureties conditioned to try such Appeal\nat and abide the Order of and pay such Costs as shall be awarded by the\nJustices at such Sessions, and also (when the Appeal shall be against a\nConviction) to pay the Penalty or Forfeiture, if the Conviction shall be\naffirmed; and the said Justices at such Sessions, upon Proof of such\nNotice having been given as aforesaid, and of such Recognizance as\naforesaid having been entered into, shall hear and finally determine the\nMatter of such Appeal in a summary Way, and award such Costs to the\nAppellant or Respondent as the said Justices shall think proper, and may\nby their Order or Warrant levy such Costs by Distress and Sale of the\nGoods and Chattels of the Person or Persons ordered to pay the same, and\nfor want of sufficient Distress may commit such Person or Persons to the\nCommon Gaol or House of Correction for any Term not exceeding Six\nCalendar Months, or until Payment of such Costs, and the Determination of\nthe said General Quarter Sessions shall be final and conclusive to all\nIntents and Purposes; and no such Rate or Assessment as aforesaid, nor\nany Proceeding touching the Conviction of any Offender or Offenders, or\nany other Matter or Thing done or transacted in or about the Execution of\nthis Act, shall be quashed or vacated for Want of Form, or removed by\nCertiorari or any other Writ or Process whatsoever, into any of His\nMajesty's Courts of Record at _Westminster_; but any such Rate or\nAssessment, or any Matter of Form in any Conviction or Proceeding as\naforesaid, may be amended as the Justices at any General Quarter Sessions\nshall direct.\n\nXLII.  And be it further enacted, That any Justice of the Peace for the\nCounty of _Middlesex_ may act as such in the Execution of this Act,\nnotwithstanding he may at the same Time be a Trustee under this Act; and\nnone of the said Trustees, nor any rated Inhabitant or Occupier of any\nHouse, Building, or Tenement within the Jurisdiction of this Act, shall\nby reason thereof be deemed an incompetent Witness in any Action,\nProsecution, Information, Complaint, Appeal, or Proceeding whatsoever\nrelating to the Execution of this Act, or to any Rate made in pursuance\nthereof; and all Notices and Summonses which are required to be given by\nthis Act, or which may be necessary for carrying this Act into Execution,\nmay be either written or printed, or partly written and partly printed,\nand may (in all Cases in which the Manner of serving the same is not\nparticularly directed by this Act) be served either by delivering the\nsame to the Person or Persons to whom such Notices or Summonses\nrespectively are directed, or by leaving the same at the usual or last\nknown Place of Abode of such Person or Persons, or at or upon any House,\nBuilding, or Tenement whereunto such Notices or Summonses respectively\nshall relate.\n\nXLIII.  And be it further enacted, That the said Trustees may sue and be\nsued in the Name of their Clerk or Treasurer for the Time being, and not\notherwise; and no such Action or Suit shall abate or be discontinued by\nthe Death or Removal of such Clerk or Treasurer, but the Clerk or\nTreasurer for the Time being to the said Trustees shall always be deemed\nPlaintiff or Defendant, as the Case may be, but such Clerk or Treasurer\nshall not be personally answerable for any Damages or Costs which may be\nrecovered against him by reason of his being so made Plaintiff or\nDefendant, but such Damages and Costs respectively shall be paid by the\nsaid Trustees out of the Money to be raised under this Act; and such\nClerk or Treasurer shall be a competent Witness in such Action,\nnotwithstanding he may be the nominal Plaintiff or Defendant.\n\nXLIV.  And be it further enacted, That it shall be lawful for the said\nTrustees from Time to Time to hire any Room or Building, as they shall\nthink fit, in any convenient Place within the Limits of this Act, to be\nused as and for an Office or Offices for keeping their Accounts, holding\ntheir Meetings, and transacting the Business relating to this Act, or for\nother the Purposes of this Act, and to accept and take a Lease of such\nRoom or Building to themselves, or any of them, or to any Person or\nPersons in Trust for them, for any Term or Number of Years, at and under\nsuch Rent or Rents as they shall from Time to Time think fit, and to pay\nsuch Rent or Rents out of the Money to arise by virtue of this Act.\n\nXLV.  And be it further enacted, That no Action or Suit shall be\ncommenced against any Person or Persons for any thing done by such Person\nor Persons in or about the Execution of this Act, until the Expiration of\nOne Calendar Month next after a Notice in Writing, fully and explicitly\nstating the Cause or Causes of Action, and the Name or Names, and Place\nor Places of Abode of the intended Plaintiff or Plaintiffs, and of his or\ntheir Attorney or Agent, shall have been delivered to or left at the\nusual Place of Abode of the intended Defendant or Defendants; and no such\nAction or Suit shall be commenced after a Tender of sufficient Amends\nshall have been made, either to the Party aggrieved or (on his or their\nBehalf) to the Attorney or Agent named in such Notice, nor after the\nExpiration of Six Calendar Months next after the Fact committed, and the\nVenue in every such Action or Suit shall be laid in the County where the\nCause of Action shall have arisen and not elsewhere; and the Defendant or\nDefendants may at his or their Election plead specially or the General\nIssue, and at the Trial thereof give this Act and the special Matter in\nEvidence; and if the Matter or Thing complained of shall appear to have\nbeen done under the Authority of this Act, or if any such Action or Suit\nshall be brought contrary to the foregoing Restrictions and Limitations,\nor any of them, then the Jury shall find for the Defendant or Defendants;\nand upon such Verdict, or if the Plaintiff or Plaintiffs shall be\nnonsuited, or discontinue his, her, or their Action, after the Defendant\nor Defendants shall have appeared, or if upon Demurrer Judgment shall be\ngiven against the Plaintiff or Plaintiffs, then the Defendant or\nDefendants shall have Treble Costs, and have such Remedy for the Recovery\nthereof as any Defendant may have for the Recovery of Costs in any Case\nby Law.\n\nXLVI.  And be it further enacted, That no Person who shall be an Occupier\nof any House, Building, or Tenement within the Jurisdiction of this Act,\nand shall be charged in respect thereof towards any of the Rates to be\nmade by virtue of this Act, shall be liable or charged in respect of the\nsame Premises to or with any general Rate or Rates for paving, lighting,\nor watching the Town or Parish of _Kensington_ or any Part thereof, or to\nor with any Rate for or towards paving, lighting, or watching any\nparticular Street, Road, or Place; any Law or Usage to the contrary\nnotwithstanding.\n\nXLVII.  Provided always, and be it further enacted.  That nothing herein\ncontained shall extend to authorize the said Trustees to expend any Money\nwhich shall be raised under the Authority of this Act in or about the\nfirst laying out, forming, paving, or gravelling any of the Footways or\nCarriageways on the East, West, or North Sides of _Brompton Square_, or\nin or about the first enclosing, forming, or planting any Part of the\nArea of the said Square.\n\nXLVIII.  Provided always and be it further enacted, That nothing in this\nAct contained shall extend or be deemed or construed to extend to\nprejudice, diminish, alter, or take away any of the Rights, Powers, or\nAuthorities vested in the Commissioners of Sewers for the City and\nLiberty of _Westminster_ and Part of the County of _Middlesex_; but all\nthe Rights, Powers, and Authorities vested in them shall be as good,\nvalid, and effectual as if this Act had not been made.\n\nXLIX.  Provided also, and be it further enacted, That nothing in this Act\ncontained shall operate or be construed to take away any Right, Power,\nInterest, Privilege, Advantage, or Authority, which is by Law now vested\nin the Governor and Company of _Chelsea_ Water Works, or in the Company\nof Proprietors of the West _Middlesex_ Water Works, or in the Grand\nJunction Water Works Company respectively; but that all and every the\nsaid Rights, Powers, Interests, Privileges, Advantages, and Authorities\nrespectively, may be exercised and enjoyed in as full and ample a Manner\nto all Intents and Purposes as the same were exercised and enjoyed\nimmediately before the passing of this Act.\n\nL.  And be it further enacted, That all Costs and Expences of and\nincident to the obtaining and passing this Act shall be paid out of the\nfirst Monies which shall be raised by virtue of this Act.\n\nLI.  And be it further enacted, That this Act shall be deemed and taken\nto be a Public Act, and shall be judicially taken notice of as such by\nall Judges, Justices, and others, without being specially pleaded.\n\n                                * * * * *\n\n            LONDON: Printed by GEORGE EYRE and ANDREW STRAHAN,\n           Printers to the King's most Excellent Majesty. 1825.\n\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\u00a9 1960 by Propertius Co., Ltd.\n\nPublished in 1983 by\n\nAcademy Chicago, Publishers\n\n425 N. Michigan Ave\n\nChicago, IL 60611\n\nAll rights reserved. No part of this book may be reproduced in any form, mechanical or electronic, without the advance written permission of the publisher.\n\nPrinted and bound in the U.S.A.\n\nSecond printing, 1987\n\nLibrary of Congress Cataloging in Publication Data\n\nBruce, Leo. 1903-1980.\n\nFurious old women.\n\nI. Title.\n\nPR6005.R673F8 1983b 823'.912 83-15683\n\nISBN 0-89733-084-6 (pbk.)\n\n# 1\n\n\"ANGRY young men?\" said Mrs Bobbin. \"Ridiculous. Whining about their maladjustment or disillusionment or whatnot. I'm seventy-one years old, I lost my husband in the first world war, my only son in the second, and my income has been whittled down to a pittance. Now my sister has been battered to death by hooligans. If I were to start talking the self-pitying rubbish of these people there might be some reason for it. But I don't and if I hear or read another word about 'the beat generation' I shall scream. Quite literally, wherever I happen to be. Now give me a cigarette and tell me how you're going to find out who murdered Millicent.\"\n\nCarolus Deene looked at the old lady appreciatively. Seventy-one, was she? She looked twelve years younger and when she snorted as she did now she made one think of a war-horse.\n\n\"Well, I'm going to start by asking you some questions,\"\n\n\"That won't get you far. I was in London at the time as I've told you. But ask what you like. If it's only to satisfy my curiosity I must know who killed Milly.\"\n\n\"She was the eldest of you?\"\n\n\"Yes. I came second, my brother Arthur third\u2014he died in 1940\u2014and Flora was the youngest. She was a late child and is only fifty-four.\"\n\n\"You three sisters lived together?\"\n\n\"We had done since the war, when I gave up my London flat and came down here.\"\n\n\"This village was your family home?\"\n\n\"Yes. We were all born here. Nothing feudal but we have lived in this house, Crossways, for three generations. Millicent and Flora never really left it. They were always wrapped up in parish affairs. I married in 1910 and didn't return till my son was killed in 1940. I couldn't afford to keep a home of my own and joined my sisters.\"\n\n\"You liked living here?\"\n\n\"Bored stiff,\" said Mrs Bobbin. \"Church affairs meant nothing to me and my sisters were immersed in them. Local residents pretentious and unintelligent. I was almost reduced to television. As I say, I have a right to be annoyed with life.\"\n\n\"But you had no financial worries?\"\n\n\"Hadn't we? You didn't know Millicent. She had most of the money, the largest share from my father and a big sum from my brother's estate. Flora and I had meagre little incomes, yet Millicent insisted that we should all pay the same share. She had played the stock market for twenty years and had made herself an extremely rich woman. But you'd never have known it. Every single household expense was divided into three equal parts and we all shelled out the same. I haven't had a penny to play with, _really_ play with, since the very fair income left me by my husband has dwindled to nothing. I may not have been in want or hunger perhaps but there are worse things than those. There is perpetual niggling, calculating and balancing which goes on for years and never leaves one any room for a small extravagance. We had that here at Crossways.\"\n\n\"Will your sister's death relieve you of it?\"\n\n\"I don't know. The will is enormous and very complicated and most of her wealth seems to be going to help Milly's pet church charities. But this money is so tied up with conditions on how the church is to be run that it will produce problems of its own. There's some for Flora and me but I don't know yet how much. Certainly not enough to compensate us for the annoyance of the murder.\"\n\n\"Annoyance, Mrs Bobbin?\"\n\n\"I'm extremely angry about it. Milly may have been mean and sanctimonious but she was my sister. To club her to death on her way to church was quite damnable.\"\n\n\"You said 'on her way to church'. What makes you think that?\"\n\n\"Her body was found in an open grave in the churchyard. Dug for a Mr Chilling, I believe, a retired shopkeeper who lived in the village.\"\n\n\"Yes, but it might have been taken there, surely?\"\n\n\"I don't think so. You will find as you come to look into this that the whole thing's bound up with the church, just as my sister was.\"\n\n\"Even so, what makes you say 'on her way' to church? She may have been coming away from it.\"\n\nMrs Bobbin thought for a minute, then said, indifferently, \"I suppose so.\"\n\n\"It happened about a week ago?\"\n\n\"Today's Friday, so it was eight days ago. Last Thursday, February 18th. It was a dry cold day which grew cloudy in the afternoon so that the night was dark. I went up to London by the ten o'clock train in the morning and got back at eight-fifteen. I walked from the station and let myself in to find the house completely dark and empty. I at once telephoned the police.\"\n\n\"No one normally slept in the house but the three of you?\"\n\n\"No one. Our only domestic help is a young woman called Naomi who comes in during the day and leaves after lunch. I suppose we are lucky to have her.\"\n\n\"Local girl?\"\n\n\"Yes. Naomi Chester. I often wondered why she stayed with us. Millicent was unbearable with her.\"\n\n\"In what way?\"\n\n\"Inquisitive. My sister, Mr Deene, had rather a nasty mind and a habit of prying into other people's lives. Naomi's was a sad case. She is in love with a young man called Grey who married about three years ago a girl who has since been certified insane. There is a small child by the marriage and it appears that Naomi is going to have a baby by Grey. My sister's curiosity about this embarrassed us all, particularly Naomi. But I suppose the girl continued to work for us to remain in the village near Grey. My sister Flora and I were quite fond of her.\"\n\n\"Where was your sister Flora on the day of the murder?\"\n\n\"She left the house during the afternoon and went by bus to Burley, the nearest town. She did some shopping, had tea at a cafe then went to the pictures. She got in soon after I did.\"\n\n\"When did you first hear of your sister's death?\"\n\n\"Not till next day. We had a most worrying night, of course. It appears that next morning the sexton, a man called Rumble, who works for us in his spare time as a gardener, noticed that the grave he had dug for Chilling had been partly filled in and began to investigate. He found my sister's body loosely covered with earth.\"\n\nCarolus Deene nodded.\n\n\"That was ingenious,\" he said. \"If it hadn't been noticed and the coffin lowered the chances are the body would never have been found. It would have been just another mysterious disappearance.\"\n\n\"But not quite ingenious enough. Why didn't the murderers dig deeper? Then they _would_ have got away with it.\"\n\n\"Who was the last person to see your sister alive?\"\n\n\"So far as we know, Naomi Chester. She usually leaves the house about two-thirty but was later that day. She says that there was nothing unusual when she went. My sister Flora had already caught the 2.40 bus for Burley so that when Naomi went Millicent was left alone in the house. But that was not unusual. Any one of the three of us was quite accustomed to being alone. It's always been a very quiet and orderly village.\"\n\n\"So nothing is known of your sister's movements after Naomi Chester left her here?\"\n\n\"Unless the police have discovered anything. But it's fairly easy to guess. My sister decided to go round to the church.\"\n\n\"There was no service?\"\n\n\"I think not. Mr Waddell, the vicar, did once try to introduce what he called daily evensong but the attendance was so small he had to give it up. No, I think Millicent went there for one of her chores\u2014cleaning the brass or something.\"\n\n\"Flowers, perhaps?\"\n\n\"No. She disapproved of flowers in church. Millicent was what is called Low Church. Flowers were only a shade worse than candles which were anathema, while as for incense... However, you'll hear all that in good time. She must have gone to polish her brass\u2014she took a pride in an idiotic-looking eagle that served as a lectern\u2014and on her way was clubbed and robbed. As simple as that.\"\n\n\"But was she known to carry anything worth robbing?\"\n\n\"She had a rather ostentatious taste in jewellery. There were usually enough diamonds on her to make a fair-sized tiara. And she nearly always carried a fairly large sum of money.\"\n\n\"Both had been stolen, I take it, when the corpse was found?\"\n\n\"Oh yes. I've told you she was done to death by hooligans. Damnable.\"\n\n\"How did she carry the money?\"\n\n\"It was my sister's habit to carry a most capacious bag. More like a shopping basket, I often told her. In this was her purse. The bag was found with her body but the purse was missing.\"\n\n\"Nothing has been recovered?\"\n\n\"Certainly not. The police say they are investigating but beyond asking me a number of questions even more moronic than yours they have done nothing to my knowledge. That is why when Mrs Kensington said she knew you I told her to ring you up. I want someone hanged for this. Yes, Mr Deene, hanged by the neck till he's thoroughly well dead. I have never been more infuriated in my life. Have a drink?\"\n\n\"Thank you.\"\n\n\"Whisky?\"\n\n\"Yes, please.\"\n\n\"I'll get it for you. Flora's still more or less prostrate with shock and Naomi left hours ago.\"\n\n\"Naomi Chester is still working for you, then?\"\n\n\"Why not? She inherited nothing from Millicent and she's got her coming child to think of.\"\n\nSipping his whisky and looking about him Carolus saw no sign of the 'meagre little incomes' 'whittled down to a pittance' and 'dwindling to nothing'. They sat in the central room of the house from which a fine stone staircase curved upward to the first floor far above them. The furniture was of the eighteenth century, sherry-dark mahogany beautifully polished, and there were several good portraits.\n\n\"It is only since my sister's death that I am able to offer you a drink in this house. Millicent was quite rabid on the subject. Not a drop of alcohol must cross the door. I often wondered whether she had some secret temptation to drink. Such violent opinions often mean that, I'm told. I drink very little myself so it was no sacrifice to me, and Millicent carried Flora with her in this as in most other things. But I like to be able to offer ordinary hospitality.\"\n\nCarolus said he appreciated it.\n\n\"Does the case interest you?\" asked Mrs Bobbin sharply.\n\n\"Yes. But there are one or two things I should tell you before we go any farther. First, I do not necessarily accept your view that your sister was clubbed to death by ruffians for the sake of immediate robbery.\"\n\n\"Why not?\"\n\n\"It doesn't quite ring true. That kind of murder could happen here I suppose, as it could happen anywhere in the world, but all my instincts are against it. And I don't think thugs of that kind would have waited to cover the body at the bottom of an open grave. It's not impossible but I don't _think_ that's the way it happened.\"\n\n\"Go on,\" commanded Mrs Bobbin.\n\nCarolus looked at her strong resolute face and beautiful white hair and tried not to make his words sound opinionated and peremptory.\n\n\"You see, I think it could have happened almost anywhere. It was, as you say, a dark night and there would have been plenty of time to take your sister's body to the churchyard and cover it, for persons who knew that the grave was already dug. Moreover the actual killing could have been the work of anyone\u2014even a woman.\"\n\n\"You seem to wish to widen the possibilities instead of narrowing them.\"\n\n\"Well, they are pretty wide. I take it there isn't much known even about the time of your sister's death?\"\n\n\"I believe not. She had been under the earth for some hours.\"\n\n\"So we do not know when, why, where or by whom.\" \"If you put it like that....\"\n\n\"Then another thing I must tell you. I can't give much time at present to investigation. The Spring term is a month old and as you know I'm a schoolmaster. I think the police will get at the truth long before I shall.\"\n\n\"They may. But I'm prepared to chance that. I have one thing in common with your angry young men, Mr Deene\u2014I do not like policemen. They are paid to protect the public and their failure to do so in the last twenty years has turned the country into a paradise for criminals while decent citizens who overstep some silly little law are harassed and persecuted. I believe the police are often corrupt and unscrupulous and I'm almost as angry when I think of their failure in this affair as when I think of the murderer's success. If the police should hit on a solution before you it will either be a miracle or a mistake. So go ahead even if it's a part-time job with you.\"\n\n\"Your sister will wish that, too?\"\n\n\"My sister is just as angry as I am over this. Perhaps angrier, in an Old Testament sort of way. She's calling for vengeance like someone on Mount Sinai. She will be delighted if you help to discover the murderer. So far as we're concerned you have a free hand. The case is wide open.\"\n\n\"It certainly is. I have never approached anything with such a range of possibilities. If one doesn't accept the theory of robbery one can't even see a motive.\"\n\n\"I wouldn't say that. A number of people will benefit financially by my sister's death. I myself, my sister Flora, our nephew Dundas Griggs, the vicar Bonar Waddell and a former chauffeur of my father's who keeps one of the local inns, a man named George Larkin. He has a son called Bill who is also, rather unaccountably, in the will.\"\n\n\"Quite a collection.\"\n\n\"But as I've told you I'm convinced that there was nothing like a motive beyond the immediate one of robbery.\"\n\n\"Had your sister any enemies?\"\n\nMrs Bobbin considered this for a moment.\n\n\"I'm tempted to say 'No' because the enemies she had are so obviously unconnected with the case. But there are two women in the village who hated Millicent and believe me, Mr Deene, in a little community like this hate can be fierce. One was the vicar's wife, Agatha Waddell. I suppose it was some kind of jealousy because her husband gave a good deal of time and attention to Millicent. Agatha is a lean and hungry-looking woman who seems thoroughly dissatisfied with life.\"\n\n\"And the second?\"\n\n\"The second is Grazia Vaillant, the vicar's _other_ best parishioner. I told you my sister's life turned on the parish.\"\n\n\"Tell me a little about this lady.\"\n\n\"Grazia Vaillant, like Millicent, is rich and immersed in church affairs. Each of these women wanted her own way. But there the similarity ends. Where Millicent was downright Grazia is gushing. Millicent was Low Church, Grazia, High, which left the vicar torn between them trying to look both ways at once. Grazia is somewhat eccentric, Millicent was outwardly rather a common-place woman. Needless to say they detested one another.\"\n\n\"Have you seen Miss Vaillant since your sister's death?\"\n\n\"Yes. Rather subdued, oddly enough. I think she misses Millicent's opposition. Flora will be no match for her, so she'll get it all her own way and the church will become so High it will stink of incense.\"\n\n\"You speak as though your vicar had nothing to say in the matter?\"\n\n\"What could he do between two women like Millicent and Grazia? His great ambition is to be all things to all men, a good mixer, popular, tolerant\u2014you know the type. Grazia will bowl him over like a ninepin. As for the curate, he's considered 'good with boys'. Scouts and that.\"\n\n\"Who else is on the parochial scene?\"\n\n\"Only one Churchwarden who counts\u2014Commander Fyfe. He's not what you would imagine a retired Naval Officer to be like. Then there's Rumble who is both verger and sexton and who found the body. You'll meet him for yourself. I won't spoil your first impressions.\"\n\n\"Who else am I likely to meet?\"\n\n\"The publican of the Black Horse, George Larkin. Perhaps you had better hear straight away what someone in the village is bound to tell you sooner or later, that is that when he was chauffeur to my father many years ago there was a scandal of some sort. I was at boarding school at the time and have never from that day to this heard the story in full but I believe he ran away with Millicent. My father found them a week later down at Brighton and everything was hushed up. Larkin had a pub bought for him up in Westmorland or somewhere on condition that he did not return here. He waited till after my father's death then returned, since this is after all his native place. His own pub had flourished and he bought the Black Horse. He is a widower now and in his seventies though his son is barely twenty.\"\n\n\"Anyone else?\"\n\n\"Oh, the village has the usual local characters. Rather irritating, most of them, I find. There's a poacher called Mugger who is rather notorious. The village policeman, Slatt, I consider to be of sub-normal intelligence but doubtless I'm prejudiced. You'll come on others unless you produce a solution almost at once.\"\n\n\"I don't think that's likely. It's all too vague.\"\n\n\"But tempting?\"\n\n\"Yes.\"\n\n\"Then do your damnedest. Because, as I say, I'm furious. This is the _end._ I've been pretty annoyed for years with this silliness called modern life. It irritates me to make no progress at all except towards old age. I could strangle the self-pitying young people who moan at the mess around them and do nothing about it. But I've never yet been angry on a personal score, I mean at something which happened to me. When I hear that my elder sister has been battered to death and thrown in an open grave, I find it too much. I'm livid. So don't take too long in solving the thing. I understand you're clever. For goodness' sake show it. And help yourself to another drink\u2014don't sit looking at an empty glass.\"\n\n\"Thank you.\"\n\n\"When will you start your enquiries?\"\n\n\"Tomorrow,\" said Carolus.\n\n# 2\n\nCAROLUS drove the forty miles from the village of Gladhurst to the town of Newminster where, in the Queen's School, he was Senior History Master.\n\nThe Queen's School, Newminster, is, as its pupils find themselves under the necessity rather often of explaining, a public school. A minor, a small, a lesser-known one, they concede, but still in the required category. It has a hundred and sixty day-boys, ninety boarders and a staff of eighteen. Its buildings are old, picturesque and very unhygienic and one of its class-rooms is a show-piece untouched from the Elizabethan Age in which the school was founded.\n\nAt the school Carolus occupied an equivocal position. He was an extraordinarily good teacher but as a member of the staff he gave the headmaster, Mr Gorringer, many unquiet hours. His excessively large private income and Bentley Continental motor-car, his habit of dressing rather too well and keeping up an extravagant establishment for himself did not endear him to the staff, who were sparsely paid, conscientious men. His known interest in criminology and the fact that he had solved a number of much-publicized murder mysteries made the headmaster 'tremble as he put it, for the good name of the school.\n\nCarolus, moreover, had an unfortunately casual manner which Mr Gorringer could not but feel only narrowly missed the disrespectful. Mr Gorringer took his head-mastership seriously and was highly averse to the flippancy of Carolus. He was a large man with a pale solemn face and a magnificent pair of pendulous, hairy ears. His verbose and earnest manner of speech became him.\n\nCarolus was aware that in school the boys were apt to take advantage of his known interest in crime both ancient and modern. A master with a hobby-horse is easily led away from the tiresome lesson in hand into the realms of his fancy. He may or may not realize this as the end of the school period comes and he finds that he has talked for three-quarters of an hour on his favourite subject and forgotten what he was supposed to be teaching.\n\nCarolus Deene was very well aware of his weakness but he regarded his twin interests of crime and history as almost indistinguishable. The history of men is the history of their crimes, he said. Crippen and Richard III, Nero and the latest murderer to be given headlines in newspapers were all one to him, as his pupils delightedly discovered. However Carolus enjoyed his job of teaching history and was conscientious over his work.\n\nAs he reached Newminster he realized that it was nearly nine o'clock and expected no mercy from his housekeeper Mrs Stick. The dinner she had cooked for him would be spoilt and her temper with it. If in addition she suspected that his lateness was caused by anything connected with criminal investigation she would be nearly as angry as Mrs Bobbin.\n\n\"Fortunately it was chops,\" she said breathing hard when Carolus was seated. She was a small, thin severe woman who cooked superbly and with her husband made Carolus's home the most comfortable in Newminster.\n\n\"That's all right then,\" said Carolus brightly.\n\n\"It's a good thing I didn't put them on,\" said Mrs Stick. \"Nor the soufflay either. Else they'd have been ruined. It's nine o'clock.\"\n\n\"Is it really? Time goes by when you're interested, doesn't it?\"\n\nThis earned him a long suspicious stare from Mrs Stick.\n\n\"I don't know what you've got to be interested in, I'm sure,\" she said. \"It wasn't anything to do with the school because Stick heard tonight from the school porter that you were Missed at the house match this afternoon. The headmaster was enquiring for you afterwards.\"\n\n\"Oh dear, was there a house match?\"\n\n\"Yes, Sir, there was, and you wasn't at it so when you say you've been interested you can't blame me if I think the worst.\"\n\nCarolus said nothing while Mrs Stick was bringing in his grilled chops.\n\n\"All my nice saltay potatoes have gone dead and greasy, of course. Well, you can't expect anything else. I was only saying to Stick that I don't mind as long as it doesn't mean you're thinking of starting anything of _that_ kind again. We can't have murderers and such all over the house.\"\n\n\"Surely we never have?\"\n\n\"Well, coming here at all hours and us not knowing whether you'd ever come back alive each time you go out. Look at that time they tried to poison you! And how our holiday was upset that year when they kept finding bodies.\"\n\n\"The chops are excellent, Mrs Stick.\"\n\n\"So they may be. But I was saying to Stick today I shan't be cooking for Mr Deene much longer if he gets up to any more of these larks with murderers. It will never do, I said. People will begin to think there's something funny about _us_ next.\"\n\n\"And what did Stick say?\" asked Carolus curiously, for in all the years he had been hearing of Mrs Stick's remarks to her husband he had never once been supplied with the reply.\n\n\"Stick says what _I_ say,\" said Mrs Stick vehemently, and Carolus did not doubt it.\n\n\"I don't think there's much fear of your being involved this time, Mrs Stick.\"\n\n\"Then you _are_ up to something,\" said the little woman triumphantly. \"I knew it as soon as you came in late like that. Not that poor lady battered to death over at Gladhurst, is it?\"\n\n\"You wouldn't want to see an elderly woman murdered and the murderer go unpunished, would you?\"\n\n\"There's police to see to that and no call for you to start upsetting us all. It's no good, Sir. Stick and I will have to leave and go somewhere where we don't have the horrors every day of our lives. Put into an open grave, wasn't she? Look at that and tell me how I'm expected to sleep sound at night. Besides, there's what my sister will say. Last time I saw her she was on about my working where there are all these murders. I don't know. I really don't.\"\n\nBut Carolus had an even more difficult interview next morning when Mr Gorringer the headmaster sent for him during the Break.\n\nCarolus was speeding towards the common-room in order to seize _The Times_ before Hollingbourne could despoil the virginity of its crossword puzzle with the two clues he usually managed to solve, when he was waylaid by Muggeridge, the school porter.\n\n\"He wants you,\" said Muggeridge sulkily, not needing to enlarge on this.\n\n\"Damn,\" said Carolus.\n\n\"I know. That's what I said when he rang that blasted bell of his.\" Muggeridge found his life a wearisome one and resented the uniform, including a gold-braided top-hat, on which Mr Gorringer insisted. \"He could just as well have come and spoken to you himself. But no, he has to sit at his blasted desk and ring his blasted bell and send me chasing all over the blasted place looking for you while my tea's getting cold. He's got something up his sleeve, too.\"\n\n\"Thank you, Muggeridge.\"\n\n\"Oh, I don't blame _you,\"_ conceded Muggeridge. \"You can't help it any more than what I can if he wants to get on his high horse. I shall tell him, one of these days.\"\n\nCarolus found Mr Gorringer if not on his high horse at least very upright and pontifical as he sat at his enormous study table.\n\n\"Ah, Deene,\" he greeted Carolus in a manner neither over-friendly nor hostile. It was hard to guess what direction the interview would take. \"I wanted a word with you.\"\n\nCarolus nodded.\n\n\"Pray take a seat,\" said Mr Gorringer. \"We were disappointed not to see you at the house match yesterday. A very creditable show on the part of Plantagenet.\" The houses at the Queen's School, Newminster, apart from School House, were known as Plantagenet and Stuart. \"But doubtless you had more pressing matters to attend to?\"\n\n\"Yes,\" said Carolus.\n\n\"It is in some way connected with this that I wished to see you. I cannot help feeling, my dear Deene, that, while you carry out your duties as Senior History Master with conspicuous care and success, your interest in what we might call the life of the school is apt to wander into regions remote from ours. I recognize your right to occupy your own time as you wish but it has come to my ears that the other men are a trifle hurt by your indifference to our extra-academic activities.\"\n\nCarolus waited.\n\n\"It happens, though, that a situation has arisen which will solve this little difficulty in a way most satisfactory to us all. _Breadman has decided to retire.\"_\n\nFor a moment Carolus did not catch the awful implications of these last five words.\n\n\"As you know, he has been at the school as boy and master since the turn of the century and his going will leave a gap difficult indeed to fill. It will also mean that Plantagenet House will require a housemaster. You, Deene, are Breadman's successor by right of seniority and I have decided to appoint you.\"\n\nCarolus gasped.\n\n\"You will have the care of twenty-five boys together with their boarding fees which after due allowance for generous catering leave a very satisfactory balance to your benefit. You will thus, I hope, be drawn into the integral life of the school. Allow me to be the first to congratulate you on this promotion.\"\n\n\"But headmaster, you can't possibly be serious,\" said Carolus. \"As a widower...\"\n\n\"I have considered that. While it is admittedly desirable that our housemasters should be married men, I have decided not to allow this to bar your way. A good housekeeper can surely be found.\"\n\n\"But it's quite outside my scope. I know nothing about catering. I couldn't think of accepting the responsibility.\"\n\n\"Come, come, Deene. You are too modest. You will find that once ensconced in Plantagenet you will be absorbed by the small problems it will produce.\"\n\n\"No!\" said Carolus desperately. \"Let me make it clear at once, headmaster. I appreciate your confidence in me but I decline absolutely.\"\n\nMr Gorringer eyed him with severity.\n\n\"Surely I am not to understand, Deene, that your personal, perhaps I may be permitted to say selfish, interests stand in the way? Your unfortunate hobby, for instance? If I thought _that_...\"\n\nCarolus had a sudden inspiration.\n\n\"What about Hollingbourne?\" he asked. \"He's a family man. Catering for twenty-five more beyond his own children would be child's play to him. He'd _like_ it.\"\n\n\"Hollingbourne is your junior on the staff,\" said Mr Gorringer dubiously.\n\nCarolus followed up his advantage.\n\n\"Does that really matter? It's right up his street. I couldn't possibly deprive him of it.\"\n\n\"I should like to say that is generous of you, Deene. But I fear me that it is far otherwise. However, Hollingbourne certainly would be gratified by the opportunity. I shall have to consider it.\"\n\nCarolus found perspiration on his forehead when he left the headmaster's study.\n\nBut it was a half holiday and though a bitterly cold day the air was crisp and clean. Carolus decided to drive over to Gladhurst immediately after lunch.\n\nHe found the case as put to him by the irate old lady Mrs Bobbin was an unusually intriguing one. There had been a marked inconsistency in her account of it to Carolus yesterday, which he had noted at the time and now reconsidered. While insisting more than once that her sister had met her death at the hands of hooligans whose motive was robbery, she had said twice that he would find it a 'parish' affair and had spoken of the various persons connected with the parish whom he would meet. It was difficult to know what the old lady really believed.\n\nHe had read the newspaper account of the inquest and found that this, like everything else connected with the thing, was inconclusive. The injuries to the skull from which Millicent Griggs had died could have been caused by her being struck from behind with something like a club or could be the result of her head striking something immovable with sufficient force. There was almost no indication of the time of death but the body had been covered for at least ten hours when it was found by Rumble in the morning. No weapon had been found and nothing to indicate where Millicent Griggs had met her death. Nobody strange to the village had been observed that afternoon. Nothing had been traced of the jewellery Millicent Griggs had worn nor the money she was carrying.\n\nSo the horizon was clear. Not only was there no hint of the murderer's identity\u2014it could not even be guessed whether there had been one or two or even more concerned in the crime. The time of it was unknown, so was the place. The way in which the woman was killed, the weapon, all the information which an investigator usually had at the outset was totally lacking. The only thing on which there seemed to be agreement was motive, but here Carolus found himself highly sceptical of the general conclusion, that Millicent Griggs had been murdered for the jewels and money she carried. It could be but it seemed a trite and unlikely solution.\n\nYet this did not mean that the case would prove to be extraordinarily baffling. It might even be better to start in such uncertainty. Facts would come with time.\n\nHe stopped at a point about a mile from Gladhurst from which the little village could be seen nestling\u2014there was no other word\u2014in its wooded valley. The clear winter afternoon gave its outlines sharpness, the square-towered church, the red roofs, the smoke curling upward. It looked compact as though its houses were huddling together to keep warm. It looked friendly, too, a Christmas card village in which one would have supposed there lived only jovial farmers and their amiable workers, kindly cosy people with no enmity or cruelty among them. Well, he had known scenes like that before and seen the outwardly cheery people he met turn into malicious, jealous and violent brutes before his eyes.\n\nFor one moment he hesitated. He had a strange superstitious feeling that ahead of him lay great evil and perhaps danger, that the death of Millicent Griggs, horrible though it was, would prove to be only a part of something more menacing and beastly, and that he would find the worst qualities in those he had to meet. Why should he once again walk into the slime when there was so much goodness and kindliness to be found?\n\nBut of course he drove on. Curiosity, if nothing more, was sufficient temptation. Why had Millicent Griggs died? Why above all had she been buried in that open grave when the murderer could have escaped at once? Where was she murdered? These were questions enough for the moment without even considering the foremost and final one\u2014who had killed her?\n\n# 3\n\nTHE first person he interviewed in Gladhurst confirmed his feeling of vague repugnance at the case, for he realized almost at once that Naomi Chester was a very frightened young woman. Carolus hated to see fear in any human being, it was to him as embarrassing as nudity. Naomi did her best to conceal what she felt but unsuccessfully.\n\nShe was a thin tall girl, a trifle sallow and anaemic-looking yet rather pretty in a thoughtful way. She looked as though she could get and keep anything she wanted by merely clinging to it. She sat opposite Carolus and moved her thin arms and long fingers as though she were about to play cats' cradles with them. Her brown eyes were watchfully, indeed fearfully, on Carolus.\n\n\"How did you get on with the late Miss Griggs?\" asked Carolus in a friendly reassuring voice.\n\n\"Oh, all right,\" said Naomi at once.\n\nLie One, thought Carolus, remembering what Mrs Bobbin had told him.\n\n\"You liked her?\"\n\n\"Well, I wasn't much gone on all that church business,\" said Naomi.\n\n\"Did she talk to you much about that?\"\n\n\"No. Not about that, she didn't.\"\n\n\"Other things, perhaps?\"\n\n\"I never had much to do with her,\" said Naomi sulkily.\n\nThe conviction was growing on Carolus that Naomi was not only frightened but had something to conceal.\n\n\"She didn't want to know a lot about your private life?\"\n\nNaomi gave a small bitter smile.\n\n\"You haven't any private life in a place like Glad-hurst,\" she said. \"Everyone knows your business.\"\n\n\"But not everyone discusses it in front of you. Did Miss Griggs?\"\n\n\"No!\" said Naomi loudly and twined her fingers.\n\n\"Coming to the Thursday of her death. What time did you get here in the morning?\"\n\n\"Nine. As usual.\"\n\n\"And leave?\"\n\n\"I've gone over all this with the police. About half-past three.\"\n\n\"Was that your usual time?\"\n\n\"I hadn't got one, really.\"\n\n\"I understand you normally left at about two-thirty.\"\n\nNaomi was growing perceptibly more uncomfortable.\n\n\"When I can I do. It's often later.\"\n\n\"And that day it was later?\"\n\n\"A bit, yes.\"\n\n\"Why?\"\n\n\"I don't know.\"\n\n\"Why?\"\n\n\"No reason. I just hadn't finished my work.\"\n\n\"Surely you'd washed up the lunch things?\"\n\n\"Yes. Before Miss Flora left.\"\n\n\"Then what kept you?\"\n\n\"Nothing. I was just running about finishing.\"\n\n\"Were you talking to Miss Griggs?\"\n\n\"No. She was resting.\"\n\n\"Where?\"\n\n\"Where she usually rested. In the drawing-room.\"\n\n\"You didn't go in there?\"\n\n\"No.\"\n\n\"I should like to clear this up, you know. I know enough about housework to know that you'd have got all your rooms done in the morning so that after you'd washed up and put things away and just done the last oddments you could get off. You must have finished the washing-up at half-past two because Miss Flora caught the 2.40 bus. What did you do for another hour?\"\n\n\"I don't know now. I was busy.\"\n\n\"Where?\"\n\n\"Nowhere special. There were things to be done.\"\n\nCarolus decided to use one of his diversions which so often brought out the truth.\n\n\"The drawing-room where Miss Griggs was resting is on the ground floor, isn't it? It has French windows?\"\n\n\"Yes.\"\n\nNaomi was staring at him now with a fixed melancholy.\n\n\"Were you near enough to hear if anyone had entered and talked to her there?\"\n\n\"Oh yes. No one did that.\"\n\n\"Then where were you?\"\n\n\"As a matter of fact I found I'd forgotten to do my hall.\"\n\n\"That's the central room of the house?\"\n\n\"Yes. They don't like it being called the lounge. It's the hall.\"\n\n\"Wouldn't that be the first room you'd do?\"\n\n\"No. The last. After I'd done those round it. Yes, I remember what it was kept me now. I found I'd forgotten it in the morning.\"\n\n\"Does it take an hour?\"\n\n\"Oh yes. With all the staircase and that.\"\n\n\"I see. And all this time you heard and saw nothing of Miss Griggs?\"\n\n\"No. When I looked in before leaving she was asleep.\"\n\n\"How was she lying?\"\n\n\"On her back on the big settee with a lot of cushions under her head.\"\n\n\"Was there much light?\"\n\n\"No. Not much. She always pulled the blinds before going to sleep.\"\n\n\"But you could see her face?\"\n\n\"Only just.\"\n\n\"Where did you go from there?\"\n\nIt was quite evident to Carolus that this was a new one. Whatever the police had asked Naomi they had stopped at the point of her leaving the house.\n\n\"Home,\" she said.\n\n'You live with your mother?\"\n\n\"Yes.\"\n\n\"Just the two of you?\"\n\n\"S'right.\"\n\n\"Was she in when you got home?\"\n\n\"No. She was at work. She's the head cook at Highcliff House.\"\n\n\"Who lives there?\"\n\n\"It's a private nursing home. Just up the road. Mum's got a good job there.\"\n\n\"Didn't you ever think of working with her there?\"\n\n\"I did only it upset me. All those poor things. So I went to work for these three. There wasn't the strain.\"\n\n\"You still haven't told me where you went that afternoon.\"\n\n\"Why? What's it to do with it? You're trying to find out who killed Miss Griggs, not worry about what I do in my own time.\"\n\n\"What time did you meet Grey?\"\n\n\"Who said I met him that day?\"\n\n\"Really, you are being rather silly, you know. You seem to want to make me think you had something to do with the death of Miss Griggs. I'll tell you frankly that I believe you know something you're afraid to tell. I can't see why you should be so evasive otherwise. Now don't start crying.\"\n\n\"I'm not crying,\" said Naomi unconvincingly. \"Only you keep on at me with questions which have nothing to do with it.\"\n\n\"If they have nothing to do with it, why not answer them?\"\n\n\"What is it you want to know?\"\n\n\"What time you met Grey.\"\n\n\"He got off early that afternoon. He came home about four.\"\n\n\"To your home?\"\n\n\"Yes, for a minute. Then he went off to see Estelle. That's his daughter. She's only two. Then he got changed and came back here to pick me up. Then we went off to the pictures.\"\n\n\"By bus?\"\n\n\"No. Laddie, that's Laddie Grey, everyone calls him that, has got a motor-bike and side-car. We went to Burley in that.\"\n\n'\" What is Grey's job?\"\n\n\"He's on the building. Only there's not much doing this month.\"\n\n\"That's why he was able to get off early that afternoon?\"\n\n\"I suppose so.\"\n\n\"Where was he working?\"\n\n\"At Commander Fyfe's, I believe. Painting a room.\"\n\n\"How do you think Miss Griggs was killed?\"\n\n\"Me? Oh I think the same as everyone else. Knocked on the head and robbed.\"\n\nOn her way to the church?\"\n\n\"I suppose so.\"\n\n\"You don't see anything difficult to believe about a woman leaving her afternoon snooze by a warm fire and walking through a dark afternoon to clean brass in an empty church?\"\n\n\"Not with her I don't. She was church mad.\"\n\n\"You don't think she could have been killed anywhere but in the churchyard?\"\n\n\"Well, it would be a bit risky in the road. Where else could she have been killed?\"\n\n\"Anywhere. Here for instance.\"\n\nNaomi goggled.\n\n\"In this house?\"\n\n\"Why not?\"\n\n\"I should have heard it.\"\n\n\"But after you had gone?\"\n\n\"Of course I don't know what may have happened then.\"\n\n\"No. Did you come back that day?\"\n\n\"Me? Come back? Whatever for?\"\n\n\"You might have forgotten something.\"\n\n\"No. I never came back.\"\n\n\"You have your key?\"\n\n\"Of the back door, yes.\"\n\n\"When did you hear about Miss Griggs's death?\"\n\n\"Next morning. While I was at work. Slatt, the village policeman, came to see Mrs Bobbin and told her.\"\n\n\"You knew she was missing?\"\n\n\"Only when I got to work. Miss Flora told me.\"\n\n\"When you left the house that afternoon did you walk home?\"\n\n\"Yes. Of course.\"\n\n\"I wondered whether you had a bicycle,\" said Carolus mildly.\n\n\"No. I always walk.\"\n\n\"How far did you have to go?\"\n\n\"About half a mile. Beyond the church.\"\n\n\"Whom did you pass on the way home?\"\n\n\"No one. Why?\"\n\n\"I quite understand that Gladhurst is a quiet little place, but surely between half past three and four on a fine weekday afternoon you'd have met _someone_?\"\n\n\"Not that I remember I didn't.\"\n\n\"Please try to remember. It may be quite important.\"\n\n\"I think I saw the Reverend Slipper. That's the curate. He was just nipping into Jevonses the grocers. I don't know whether he saw me.\"\n\n\"No one else?\"\n\n\"Not to my recollection.\"\n\n\"You didn't call anywhere?\"\n\n\"Don't think so.\"\n\nCarolus handed Naomi his cigarette-case and lit both cigarettes.\n\n\"Look here,\" he said, \"won't you take my advice? I don't know _why_ you're holding something back or who has persuaded you to, but whatever it is it's fatal not to be open in a case like this.\"\n\nThis time Naomi began to cry in earnest.\n\n\"I daresay you've been through a lot,\" said Carolus sympathetically, \"but I'm afraid you'll have a good deal more unless you tell the truth. Why don't you?\"\n\n\"There's nothing,\" blurted out Naomi between her sobs. \"I've told you all I know.\"\n\n\"All right,\" said Carolus. \"Have it your own way. Only if you decide to be sensible get hold of me or the police at once. Don't tell anyone else you're going to tell the truth. Really. You can phone me at the number I'm writing down or go to the police. But for heaven's sake don't hesitate.\"\n\nCarolus saw her staring fixedly at him through tearful eyes. But she asked no question and in a moment went still sobbing from the room.\n\n\"I _knew_ it would be a beastly case,\" he reflected.\n\nA few minutes later Mrs Bobbin came into the room.\n\n\"You've made the girl cry,\" she said.\n\n\"I'm afraid shell cry a good deal more before the case is over unless she decides to speak the truth. She knows something which she is determined not to reveal.\"\n\n\"But why?\"\n\n\"I don't know. I shall in time. And she lies.\"\n\n\"What about?\"\n\n\"I wanted to know what she did between 2.30 when your sister Flora left for Burley and 3.30 when, she says, she started out for home.\"\n\n\"What does she say she did?\"\n\n\"Forgot to clean the hall in the morning. Remembered it and did it. Your sister Millicent meanwhile was sleeping in the drawing-room.\"\n\n\"What's wrong with that?\"\n\n\"A woman doesn't _forget_ to do the principal room of a house. And if by some freak she did she wouldn't spend an hour on it when it was time for her to go home.\"\n\n\"But the hall was done that day, very thoroughly. I remember noticing the smell of furniture polish when I came in. I congratulated Naomi on it next morning before we knew anything about Millicent. She said 'Yes, I did it last thing before going home'. So it doesn't sound as though she was lying at all.\"\n\n\"Then she told me she got on quite well with your sister Millicent though she never had much to do with her. After what you told me yesterday I knew that wasn't true.\"\n\n\"Well, she certainly did not get on with Milly as well as with Flora and me but then few people did. She probably did see less of her than of us.\"\n\n\"These are not perhaps very important. But I know that girl is hiding something and if you have any influence with her I hope you'll use it to persuade her to tell the truth.\"\n\n\"You're very sure of yourself, Mr Deene.\"\n\n\"When do you think I could meet your sister, Miss Flora?\" asked Carolus.\n\n\"Tomorrow or the next time you come. She's not up to it today. It has been a severe shock to her.\"\n\n\"I quite understand. I have plenty of work to do. Tell me, how was your sister dressed when she was found?\"\n\nMrs Bobbin looked at Carolus sharply.\n\n\"Exactly the clothes she was wearing when Flora left except for her fur coat.\"\n\n\"What about shoes?\"\n\n\"Oh, she had changed those, of course. She had on a strong pair of walking shoes.\"\n\n\"She did not wear galoshes, of any sort? You must forgive my being old-fashioned enough to call them galoshes.\"\n\n\"She did, yes.\"\n\n\"Always, in winter?\"\n\n\"Nearly always.\"\n\n\"And she wasn't wearing them?\"\n\n\"No. Not that day. It was dry.\"\n\n\"Are they still in her room?\"\n\n\"I suppose so.\"\n\n\"May I see them?\"\n\n\"If Naomi hasn't gone.\" She went to the door. \"Naomi!\" she called rather stridently.\n\nWhen the girl appeared Carolus saw that she had repaired her face and was looking more self-confident.\n\n\"Naomi,\" said Mrs Bobbin, \"where are Miss Millicent's rubber shoes?\"\n\n\"Her overshoes, you mean?\" asked Naomi watchfully. \"Wasn't she wearing them?\"\n\n\"I believe not.\"\n\n\"I'll see,\" said Naomi.\n\nWhen she returned she said simply that they were nowhere to be found.\n\n\"Perhaps if she had not got them on when she was found she left them in the church?\" suggested Naomi.\n\n\"Then surely Mrs Rumble would have come on them?\" said Mrs Bobbin.\n\n\"Well, you know what _she_ is,\" said Naomi meaningly.\n\n\"Yes,\" admitted Mrs Bobbin.\n\n\"You mean,\" said Carolus, \"that your sister probably wore her galoshes and took them off while she was in the church?\"\n\n\"It seems possible.\"\n\n\"Then either she forgot them when she left or else something happened to her while she was still in the building?\"\n\n\"If Mrs Rumble found them there, yes.\"\n\n\"I shall have to see her tomorrow, then. Would your sister have taken anything with her from here if she went to clean brass in the church?\"\n\n\"No. The things she used were kept in a cupboard in the vestry.\"\n\n\"Mrs Rumble will know if they had been disturbed?\"\n\n\"Most unlikely. She had nothing to do with the brass.\"\n\n\"She lives near the church?\"\n\n\"Just opposite.\"\n\n\"Well, thank you, Mrs Bobbin. I shan't have to trouble you again, I hope, though I would like to see Miss Flora as soon as possible.\"\n\n\"Have a cup of tea before you go? You run along, Naomi. I'll get it myself.\"\n\nHalf an hour later Carolus left the house and walked towards the church. It would be too late, he decided, to see much of the building but it was at about this time, according to.Naomi's and Mrs Bobbin's conjectures, that Millicent Griggs would have taken this way nine days ago.\n\nHe could make out the tall shape of the square tower which stood apart from the village. Beside it was a fair-sized house which he took to be the vicarage and a small cottage which was probably Rumble's.\n\nAs he stood there a woman's figure approached him from the direction of the church and cottage.\n\n\"Good-night, Naomi,\" he said as the tall girl passed.\n\n\"Oh! Oh, good-night.\"\n\n\"Been to see Mrs Rumble?\" asked Carolus.\n\n\"No! What? I was just... Why do you... Oh, leave me alone. I've told you everything. Why do you keep on?\"\n\n\"I just said good-night,\" said Carolus. \"Don't forget when you do want to tell the truth....\"\n\nBut Naomi was gone.\n\n# 4\n\nTHAT evening Carolus paid his first visit to the Black Horse. He found it fairly crowded and after a while wondered what recommended it. Most pubs have something in their favour, something with which to enter the more or less bitter competition with their fellows. Where all have grown so stereotyped, each has to find a way faintly to distinguish it from the rest. So one will have a waggish, or a generous or a popular landlord, another a personable landlord's wife or daughter; in another there will always be a bright warm fire while yet another has beer from a brewery whose advertising has been successful in making men believe its beer is different from and better than the rest. In one there will be good darts, yet another is supposed to give a larger than ordinary measure of spirits. Yet all, being brewery-owned, licensing-hours-observing, well-regulated, standardized swilling-houses for standardized products at standardized prices, have lost all character, and in a few years their customers will be as standardized as they are. English pubs have ceased to differ one from another except superficially, their signs, their furniture, the way in which their antiquity has been restored may vary slightly, but there it ends.\n\nWhat made the gathering in the Black Horse choose it from among the three in the village? Not the landlord, surely, for a more dull and surly-looking man than George Larkin it would be hard to find. He stood back from his counter and said \"Eh?\" whenever anyone shouted for a drink. His most effusive welcome was a sudden small nod. It may have been the beer which attracted customers, but as there were several complaints about it that evening it seemed scarcely likely.\n\nThen Carolus decided not to be carping. Perhaps the beer was better than it seemed. Perhaps George Larkin had a heart of gold.\n\n_... Folks of a surly tapster tell_ ,\n\n_And daub his visage with the smoke of Hell;_\n\n_... He's a Good Fellow, and 'twill all be well_.\n\nWell, he might be, but Carolus, quietly sipping a Scotch and soda from a not too transparent glass still wondered at the inn's prosperity.\n\nAs the evening wore on he thought he found an explanation for it. Her name was Flo. She was a laughing buxom woman in her thirties who seemed to be everyone's friend. It was a tribute to her good nature that she was referred to as \"old Flo\". Of her it was said\u2014\"Flo doesn't mind\", and that, in all its implications, was the secret of her popularity.\n\nShe was evidently a nightly customer and her jolly laugh, which doubled her plump body and brought tears to her eyes, could be heard at the end of each smutty story which Flo, quite evidently, \"didn't mind\". Was she, Carolus wondered in good old English phraseology, the village whore? No. She just didn't mind.\n\nCarolus found himself being addressed by a small, mild, informative man.\n\n\"I come from what they call Hellfire Corner,\" he said surprisingly. \"Well, it's a name they give. It's only like a building estate really. There are one or two rough lots up there, though. Old Slatt, that's the copper, doesn't come up there more than he has to. I don't blame him. The last time he was nosing round someone dropped his bicycle down an old well there. He never did find out who did it and of course it made him look very silly with the Inspector. But there you are. Those that will keep hanging round at closing time what do you expect?\"\n\nCarolus recognized the question as rhetorical, bought two more drinks and settled down to listen.\n\nHis friend indicated a small stocky man with a flat face and a ready grin.\n\n\"That's Rumble,\" he said. \"Sexton and verger!\"\n\n\"He looks very cheerful for a sexton and verger.\"\n\n\"He is very cheerful. Why shouldn't he be? He enjoys his work, does Rumble. Very thorough he is. Well, you have to be with his job. His wife's just the opposite. Funny, isn't it, how you get that? She's as sour and nasty-tempered as he's easy-going. You don't often see her in here. She works for Miss Vaillant at the Old Vicarage.\"\n\n\"Is that the big house opposite the church?\"\n\n\"Yes. The vicar's had to let it. He lives in a smaller place. Miss Vaillant's a character.\"\n\n\"In what way?\"\n\n\"Queer old stick, she is. Well, not so old. Plenty of money, mind. Ah, here's Mrs Chester. I thought she'd be in. She's in charge of the kitchens up at Highcliff House. That's a big place for invalids. Sort of nursing home. You know, rich people who can afford it. Mrs Chester's worked there for years. She's a good sort. You ought to hear her and Flo if they get together. Laugh? Well, you _would_ laugh. The things they come out with. Flo doesn't mind.\"\n\n\"That's a good thing, anyway.\"\n\n\"Old George Larkin takes no notice. I often wonder if he hears half the time. His son's a bit better than what he is but they're both pretty quiet.\"\n\n\"So I've noticed.\"\n\nThe younger Larkin had in fact joined his father behind the bar and seemed just as remote and discontented as the landlord.\n\nCarolus found himself vigorously nudged by his informative new acquaintance.\n\n\"This is a real character, just come in,\" he said. \"Mugger, his name is. Lives a few doors from me.\"\n\n\"What does he do for a living?\"\n\n\"That's just it. Anyone would be hard put to it to say. Odd jobs here and there. Could be a wonderful gardener if he took the trouble. Now'n again he does a bit. Gives them a hand round at the Swan when they're busy. Goes in for keeping rabbits and pigeons and I don't know what. Breeds ferrets. Chickens, turkeys, anything like that. You ought to see his place. You wouldn't think it to look at him, would you?\"\n\nWithout knowing quite what 'it' was, Carolus wouldn't. Mugger was a tall ginger-haired man, rather stern and serious in expression.\n\n\"Proper old fox, he is,\" went on the little man. \"He doesn't half give old Slatt a time. Does it to pull his leg. Last winter Slatt caught him with a sack full of rabbits in a wood belonging to Commander Fyfe. Thought he'd got him at last, he did, and took him down to the police station to send him over to Burley. Old Mugger never said a word and when Slatt got on the phone to Commander Fyfe he found he'd given him permission to be there to keep the rabbits down.\"\n\n\"It seems to be a lively village,\" said Carolus.\n\n\"Of course this murder's upset things a bit. Scared a lot of people to think that such a thing could happen. Well, it's not very nice, is it? An old lady like that. They say the back of her head was smashed in something cruel. Makes you think.\"\n\n\"It does.\"\n\n\"She was a great churchgoer, Miss Griggs was. And her sister. Well, it's a place for churchgoing. I'm a bell ringer myself. Only a lot have fallen off lately because of the larks this Miss Vaillant persuades the vicar to get up to. Wants him to start incense and she's got him dressed up like I don't know what with lace all over him. It doesn't do in a place like this. Miss Griggs was very much against it. I don't understand it myself but my wife says all that bobbing up and down and bowing and scraping is nothing less than roaming Catholic. There's talk of them starting idols next so I don't know what we are coming to!\"\n\n\"Idols?\"\n\n\"Well, images, then. It seems this Miss Vaillant brought one home from Italy and wanted it put in the church. Miss Griggs said if it went in she'd never come again and Miss Vaillant said if it didn't she wouldn't, so there you are. The poor vicar didn't know which way to turn. But they made it up somehow, because about a week before Miss Griggs died she started going to tea with Miss Vaillant. From what I hear she went twice in a week. It seems they came to terms. I never did hear what they arranged but Rumble heard from his wife that they got quite friendly all of a sudden. You don't know where you are with people like that, do you?\"\n\n\"I don't.\"\n\n\"That's what poor old Chilling used to say....\"\n\n\"Chilling?\"\n\n\"Bert Chilling. Just Passed On. It was in his grave as you remember that they'd put the body. He was a very nice chap, old Chilling. Used to come in here of an evening and very often ask you to have one. Can't say fairer than that, can I? Well, he was in the choir. Had been for years. And when this Miss Vaillant started these larks with the vicar and most of them were against it, old Chilling surprised them all. 'You give 'em plenty of it, Mr Waddell!' he used to say. 'Livens things up,' he told them. 'I like to see it', he said. 'Candles and vestments\u2014the lot. I don't know why you don't put in confession boxes and be done with it.' Of course this came to Miss Griggs's ears and she nearly passed out on the spot. But Miss Vaillant was ever so pleased. So there he was till suddenly he heard, on his deathbed as it turned out, that they'd made friends again. I shouldn't be surprised but what it Hastened his End. The shock of it, I mean. But you never know what to think, do you?\"\n\n\"Not always,\" admitted Carolus and refilled their glasses.\n\n\"Lovibond's my name,\" admitted the small man rather coyly. \"Fred Lovibond. I've got a little electrical business here. I was going to tell you how I came to have that....\"\n\n\"No, you weren't,\" said Carolus gently, \"you were going to tell me the name of that young chap in the corner.\"\n\n\"That? That's Laddie Grey. Used to help Rumble in the churchyard in the old days. His wife's been Shut Up, poor chap. About this business of mine....\"\n\n\"I thought you were going to tell me about the murder of Miss Griggs.\"\n\n\"Oh well, that. That was a funny business. Right from the start I said it was a funny business. I mean, where was she going at that time of the night?\"\n\n\"What time of the night?\" asked Carolus.\n\n\"When she was done for. It can't have been in plain daylight can it? I mean, it would call attention.\"\n\n\"But it's dark by four.\"\n\n\"All right. But where was she going after four? It's all very well but I don't see anyone going into a cold church at that time. And where else could it have been? No one's seen her, don't forget.\"\n\n\"Then what do you think happened, Mr Lovibond?\"\n\n\"Oh, I couldn't say. It doesn't bear thinking about really, does it? I mean it could have been anyone, almost. It came out at the inquest that it might have been a woman just as well as a man. I think myself it must have been a stranger. I can't see anyone in Gladhurst doing it. I suppose we shall know in good time. I hear they've got all the biggest policemen on it. Can't let anything like that go past, can they? Old Slatt's been going round like a dog with two tails. ' _We_ shall probably be making an arrest shortly,' he says. Makes you laugh, doesn't it?\"\n\n\"I don't know Slatt,\" said Carolus.\n\n\"You'll find him out in the square now. Always out there in the evening, waiting for ten o'clock. He likes to have a chat while he's standing there looking at his watch for closing time. He talks quick enough. It's a job to stop him talking.\"\n\nCarolus found him without difficulty, a long-faced, solemn man who looked suspiciously at the world as though he feared it was going to burst into an explosion of laughter at the sight of him.\n\nCarolus addressed him cheerfully.\n\n\"You're the Gladhurst policeman, aren't you?\"\n\n\"Police _Officer_ ,\" said Slatt in a gloomy voice. \"P'raps you wasn't aware that the status has been changed? Police Officer, please.\"\n\n\"Good evening, Police Officer,\" said Carolus obligingly. \"Nice evening for murders.\"\n\n\"I take it you're a stranger here,\" said Slatt. \"Otherwise you wouldn't make jokes of that sort.\"\n\nA motor-bike sped by and immediately the whole bearing of Slatt changed. He ceased to be a lethargic individual leaning on a bicycle and became a crouching lion.\n\n\"See the way they go by?\" he said. \"It's disgraceful, really it is. You can't do anything about it either. We've got half a dozen or more in the village itself and they come through all the time.\"\n\n\"What's wrong with them?\"\n\n\"Wrong with them? They're dangerous, smelly, noisy and you never know what the young people who ride on them are up to.\"\n\n\"Does it matter?\"\n\n\"Of course it matters. You can't have people doing what they like. These motor-bikes bring trouble all round. We're always getting complaints. When we're busy with something like this murder we don't want trouble with motor-bikes, do we? If I had my way....\"\n\n\"You will, I expect, before long. Everything points to it anyway. But how do motor-bikes add to the confusion of the murder?\"\n\n\"Because part of the investigation I carried out concerned their movements on that day. It was thought necessary to know which of them were where at the time of the crime. How was I to discover that with them flying round like wasps? There's young Bill Larkin the son of this House opposite got one and the Reverend Slipper the curate just bought one for running about on. Then Laddie Grey, he's had one for a long time and Albert Chilling whose father died the other day and two or three more. What they've got up at Hellfire Corner no one knows\u2014half of them not insured I daresay. But I can't be everywhere at once, can I?\"\n\n\"I suppose not.\"\n\n\"As soon as I've stopped one and given him a warning for speeding there's another I find with his licence out of date. What am I to do? It's the same with the pubs. If I'm here at ten o'clock when they close I can't be outside the Swan as well and I heard it was nearly five past before they were all out of there the other night. That's only the half of it. There's swedes been missing from a pile beside the road to Burley and all the poaching to think of. I have to have eyes at the back of my head.\"\n\n\"Poaching?\" prodded Carolus.\n\n\"We've one or two really bad characters round here,\" said Slatt darkly.\n\n\"Oh? ' A policeman's lot is not a happy one ', I gather.\"\n\n\"A police _officer_.\"\n\n\"I beg your pardon. 'A police officer's lot is not a happy one', I should say.\"\n\n\"No. It's very difficult sometimes. Do you live here?\"\n\n\"No.\"\n\n\"Staying in the village?\"\n\n\"No.\"\n\n\"Is that your car over there?\"\n\n\"Yes,\"\n\n\"Isn't that what you call a Bentley Continental?\"\n\n\"It is.\"\n\n\"You seem interested in what goes on here.\"\n\n\"I am. Particularly murder.\"\n\n\"That's a matter I can't discuss. It would never do for a police officer.\"\n\n\"No. Of course not. Besides, I'm a stranger.\"\n\n\"Yes,\" said Slatt suspiciously. \"And we don't get many strangers in Gladhurst.\"\n\n\"None on the day of the murder, I understand?\"\n\n\"Not so far as we're aware. And we've covered it pretty thoroughly. Bus, train, roads, shops, private houses. No one saw anyone they didn't know.\"\n\n\"It must be very difficult for you. A dark afternoon, motor-bicycles and no strangers. You don't even know the time it happened, I'm told.\"\n\n\"Not to the minute, we don't.\"\n\n\"You should know that. They say 'if you want to know the time ask a policeman'.\"\n\n\"Police officer,\" corrected Slatt.\n\n\"Of course. 'If you want to know the time, ask a police officer'.\"\n\n\"That's better,\" said Slatt. \"No, we don't know the exact time but if you had the experience of these cases that I have you'd know that time is not the all-important factor. That's an idea put about by these detective writers.\" He paused to look at his watch. \"Still four minutes to go,\" he said.\n\nCarolus smiled.\n\n\"So time has its importance in matters of the licensing laws?\"\n\n\"That's another matter. It would never do to let people hang about in pubs half the night, would it? What would happen to their work next day, do you think?\"\n\n\"I should have thought that the time he goes to bed might have been a man's own affair.\"\n\n\"His own affair?\" said Slatt, horrified. \"Whatever do you think would happen to the country if we let people do what they liked? I'm surprised at a man like you suggesting such a thing.\"\n\n\"Anarchic, you think?\"\n\n\"Worse than that. Why very soon we'd have them up at all hours dancing and singing and I don't know what. How would you expect me to keep law and order then, I should like to know.\"\n\n\"Do you sleep well?\" asked Carolus.\n\n\"No, I don't,\" said Slatt. \"That's my trouble.\"\n\n\"I thought it might be. Makes me think of Kipling\u2014\n\nOver the edge of the purple down,\n\nWhere the single lamplight gleams,\n\nKnow ye the road to the Merciful Town\n\nThat is hard by the Sea of Dreams\u2014\n\nWhere the poor may lay their wrongs away,\n\nAnd the sick may forget to weep?\n\nBut we\u2014pity us! Oh, pity us!\n\nWe wakeful; ah, pity us!\u2014\n\nWe must go back with Policeman Day\u2014\n\nBack from the City of Sleep!\"\n\n\"Police _officer_ \" Slatt reminded him.\n\n\"Of course\u2014\n\nWe must go back with Police Officer Day\n\nBack from the City of Sleep\"\n\n\"Time they was out,\" said Slatt and strode across to the Black Horse.\n\n# 5\n\nWHEN Carolus next came to Gladhurst he decided to act on Mrs Bobbin's assurance that he would find it a parish matter. He asked for the vicarage and was directed to a small new semi-detached villa. The door was opened by a young woman with bright professional cheerfulness, like that of a hospital nurse.\n\n\"Yes, my father's at home,\" she beamed. \"Come in if you can squeeze your way along this passage. Mrs Bobbin told us to expect you. Father's in here.\"\n\nMr Waddell was as bright and beaming as his daughter, a bald, round, obliging sort of man.\n\n\"De-lighted,\" he said. \"De-lighted, my dear fellow. Anything I can tell you is at your disposal. I shall be only too happy.\" Suddenly he looked serious. \"Unless of course it conflicts with my duty to the police.\"\n\nCarolus, not yet used to the vicar's facility for discovering the horns of a dilemma and being tossed from one to the other, said\u2014\" I don't think so.\"\n\n\"I've always understood that private detectives and CID men were at loggerheads,\" the vicar continued anxiously. \"I shouldn't wish to find myself....\"\n\n\"No, no,\" said Carolus briskly. \"What did you do that afternoon, Mr Waddell?\"\n\nThe vicar recovered his jollity and gift for being all things to all men.\n\n\"Ah hum!\" he beamed. \"You're certainly not afraid of convention, Mr Deene. That, if I may use the expression, is an old one. What's more I think I can answer it pretty accurately. I lunched at home. My wife and daughter. Cottage pie, I remember. A little argument about whether it should be called Cottage or Shepherd's. My wife became quite impassioned about it. I found myself weighing the balance between them. I dislike disagreement. We can't all hold the same opinions about everything but we needn't be dogmatic. A little sweet reasonableness here, a small compromise there and the world....\"\n\n\"You were telling me what you did that afternoon.\"\n\n\"Yes. Yes. I get carried away. I went to see my Churchwarden. Commander Fyfe.\"\n\n\"Is he as keen as you are on agreement among people?\"\n\n\"He has recently joined me in a little conspiracy which, until this sad event in the parish, we thought successful. Two good and sincere women\u2014sincere each in her own manner\u2014had been long in conflict. Miss Griggs might be described as evangelical in her views, Miss Vaillant held opposite opinions. Miss Vaillant wished me to introduce a form of ritual which Miss Griggs considered out of keeping... inappropriate... not in accordance with her conception of the Church of England. In a word, Miss Vaillant was High Church, Miss Griggs, Low.\"\n\n\"And you?\"\n\n\"Oh, Broad. Broad. I could not subscribe to some of the more exuberant notions of Miss Vaillant and felt her use of the word 'Mass' was open to misinterpretation. On the other hand I felt that Miss Griggs, sincere though she was, might wish to deprive our fine old church of some of its grandeur.\"\n\n\"So how did you manage the situation?\"\n\n\"Well, I had liturgical colours, you know, and we turned to the East for the Creed. I had to draw the line at holy water but I allowed those of the choir who wished it to make the sign of the cross. I had six candlesticks on the altar but kept a plain cross and felt bound to refuse the large crucifix presented by Miss Vaillant. We used the term Sung Eucharist but I wouldn't allow Matins to be ousted from the hour of eleven o'clock. As for Confessions, I expressed myself willing to hear them but chiefly in cases of illness. I agreed rather reluctantly to the choir wearing the lace cottas which Miss Vaillant presented after their surplices were worn out but I would not go so far as scarlet cassocks. Recently, after much careful consideration of the subject, I took the important step of wearing vestments at Sung Eucharist though not at the Early Morning Communion Service which the Misses Griggs attended. I would not allow incense but we were considering a side chapel.\"\n\n\"It all sounds very sensible,\" said Carolus.\n\n\"But of course the parties at both extremes were dissatisfied. It is so often the way. I had distressing scenes with Miss Vaillant when she wished to prevail on me to wear a biretta and Miss Griggs refused to attend the church on one occasion until I had removed a Paschal Light. But recently, after years of moving among these discordances, I seemed to perceive a ray of light. With the aid of my Churchwarden, splendid chap, we at least induced the two ladies to be on speaking terms. It appears that an agreement was reached that when together they should discuss everything except religion, the church, our services. It seemed to work so well that Miss Griggs actually went to Miss Vaillant's home on two occasions recently. Then came this dreadful murder to throw a shadow over all.\"\n\n\"Thank you for those interesting details,\" said Carolus. \"Now may I bring you back to the afternoon of the tragedy.\"\n\n\"I was saying I walked round to Commander Fyfe's pleasant home, The Fairway. Perhaps you know Commander Fyfe?\"\n\n\"Not yet.\"\n\n\"An excellent fellow. A really grand chap. But apt\u2014how shall I put it?\u2014to see mystery, even intrigue, where none is noticeable to others. He reads newspapers which are inclined to stress the more sensational aspects of national life. But the best of men. We discussed the hopeful situation that had risen between the two ladies and one or two other parish matters. An unfortunate rift has arisen between our organist-choirmaster, Mr Waygooze and my enthusiastic curate Peter Slipper. Slipper's splendid with boys. Scouts, Boys' Club, Camping, splendid. Waygooze naturally wants them for the choir and the two enthusiasms clash. I see Waygooze's point of view, of course, and I support Slipper in his endeavours. I find myself...\"\n\n\"Just so. That afternoon?\"\n\n\"I took a cup of tea with Commander and Mrs Fyfe, then decided to call at the church on my way home.\"\n\n\"May I be impertinent enough to ask why?\"\n\n\"Certainly, my dear chap. Certainly. No impertinence at all. It is a custom of mine. From time to time I spend a while there in contemplation. We do not Reserve, of course, but the church is something more than a mere meeting-place. My verger Rumble locks it at night. Most conscientious, Rumble. A loyal employee. He has a very happy disposition in spite of his domestic trouble. His wife, you know. They don't Get On. Most unfortunate. How many times have I tried to pour oil on those troubled waters? Mrs Rumble works for Miss Vaillant and is apt to take her view of the little controversy which has disturbed my flock. Rumble does a certain amount of gardening at Crossways when he has time. Perhaps because of that he takes the view so ardently held by the Misses Griggs. Frankly I do not know which, if either, was to blame. On the one hand Rumble seems to have a happy disposition, on the other it is not always the more bitter ones who are responsible for quarrels.\"\n\n\"You were saying that you went to the church?\"\n\n\"Yes. At about five o'clock.\"\n\n\"That would mean that its lights were on and could be seen from outside?\"\n\n\"Yes. When I arrived the church was in darkness. I switched on the main light over the chancel.\"\n\n\"Was there anything unusual or out-of-place?\"\n\n\"Not that I observed. All seemed quite as it should be. Mrs Rumble keeps the church in order. I saw nothing to criticize though I was not that moment looking for anything.\"\n\n\"Was there any sign of the brass having been recently cleaned?\"\n\n\"Now there you have me,\" said the vicar. \"I did not notice the brass.\"\n\n\"Not even the lectern?\"\n\n\"No. The lectern is in the form of a magnificent brass eagle with outspread wings. It was a gift from the father of the Misses Griggs and Mrs Bobbin before I had the living. You may know the kind of thing?\"\n\n\"I do indeed.\"\n\n\"I certainly did not notice that afternoon if it had been recently polished.\"\n\n\"Pity. Did you happen to see a pair of galoshes near the door?\"\n\n\"Galoshes?\" repeated the vicar.\n\n\"Rubber overshoes.\"\n\n\"Near the door?\"\n\n\"Yes, Mr Waddell. Near the door.\"\n\n\"No. That would be most unusual.\"\n\n\"But it was dark when you arrived? You could have passed them?\"\n\n\"I think not. The switches for all the lights are at the West end of the church.\"\n\n\"Still, you can't be certain?\"\n\n\"No. I can't be certain.\"\n\n\"You see, Mr Waddell, there is a suggestion that Millicent Griggs went to the church that afternoon to clean her brass. She may have been murdered on the way to the church, or coming away from it, or even _in_ it, for all we know.\"\n\n\"Terrible. Terrible.\"\n\n\"So anything you could tell me which would suggest that she had been there, or had not been there....\"\n\n\"I hardly know what to say. On the one hand all seemed so peaceful yet on the other I suppose that shortly before or after my visit.... Terrible. My wife would like you to take a cup of tea with us.\"\n\n\"I shall be delighted. You saw nothing?\"\n\n\"Nothing unusual, no.\"\n\n\"And heard nothing?\"\n\n\"Not in the church. Not at that time.\"\n\n\"Then?\"\n\n\"After returning here I set out to call on Miss Vaillant. Miss Vaillant, as you may know, occupies the Old Vicarage just opposite the church. I couldn't possibly afford to keep it up and got permission to let it while we have moved into these more modest quarters. It was past six when I approached Miss Vaillant's house. The church was, of course, in darkness. The village seemed quiet. A dark peaceful night. Then, just before I rang the bell, I heard the sound of a motor-bike engine starting up.\"\n\n\"Where?\"\n\n\"It _sounded_ as though it came from the lane behind the church.\"\n\n\"Was that very unusual?\"\n\n\"Well, no. That lane is used, I gather, by the young people of the village. Harmless, I'm sure... I suppose... I hope. Our village policeman seems perturbed about it. He explained to me that once the young folk imagine they can do what they want....\"\n\n\"Yes, he gave me his views on public morality. So why did you particularly notice that motor-cycle being started?\"\n\n\"I hardly know. But I did notice it.\"\n\nThe cheerful daughter came to the door.\n\n\"Tea's ready,\" she said, and Carolus found himself following her to a small drawing-room where Mrs Waddell was already seated behind a silver tray. \"A lean and hungry-looking woman who seems thoroughly dissatisfied with life\" was Mrs Bobbin's description of her and it fitted. She did not look too pleased at the inclusion of Carolus in the party. She gave him a toothy and unwilling smile and after a minute or two spoke to him.\n\n\"I may as well tell you, Mr Deene, that I was against my husband giving you any information in this matter at all. It seems to me wholly a police affair\"\n\n\"You think the police will discover the murderer, Mrs Waddell?\"\n\n\"I daresay. I'm not much interested. But I see no reason to assist a private detective. Milk and sugar?\"\n\n\"Please. I hope the police are successful,\" said Carolus and added mischievously\u2014\" It would be disgraceful if a kind, generous and devout old lady could...\"\n\nThe thin face of Agatha Waddell had turned scarlet.\n\n\"Is that the impression you have been given about the late Millicent Griggs?\" she asked.\n\n\"Agatha, my dear...\" protested the vicar.\n\n\"Now, mother,\" said his daughter.\n\n\"I don't know who can have told you that,\" said Agatha Waddell. \"Millicent Griggs was kind to no one but herself, she was generous only to gain her own ends and as for her devoutness she was the most hypocritical, narrow-minded....\"\n\n\"Agatha, my dear, what _will_ Mr Deene think of us?\"\n\n\"I don't care what he thinks. If he's fool enough to believe that Millicent Griggs was kind....\"\n\n\"You must forgive my wife's strong feelings, Mr Deene. She had her difficulties with the late Miss Griggs, as we all had.\"\n\n\"Perhaps you were of the other faction, Mrs Waddell?\" suggested Carolus.\n\n\"Grazia Vaillant's? Certainly not. A gushing insincere woman.\"\n\n\"My dear, we must be charitable in our judgments.\"\n\n\"There's a limit to charity. I've seen my husband's life here made a purgatory by these two self-centred, bigoted women.\"\n\n\"Aren't you somewhat overstating the case, dear? They have been most generous.\"\n\n\"When it suited them. More tea, Mr Deene? I wonder how you would like being a vicar's wife in a small country parish.\"\n\n\"I'm sure I should find it a most difficult transformation,\" said Carolus, pacifically.\n\n\"Intrigue, suspicion, back-biting, jealousy the whole time.\"\n\n\"You are not suggesting that Millicent Griggs was murdered from sheer malice, are you?\"\n\n\"I'm not suggesting anything. But there's enough malice in Gladhurst to murder fifty people.\"\n\n\"You alarm me.\"\n\n\"Millicent Griggs herself was capable of it.\"\n\n\"Really?\"\n\n\"I know for a fact that she wrote to the Bishop.\"\n\n\"What about?\"\n\n\"Prejudice again. Fortunately the Bishop knows my husband. He has been here to lunch. Otherwise it might have meant all sorts of difficulties for us. She said that my husband intended to ride into church on an ass on Palm Sunday.\"\n\n\"And didn't he?\" asked Carolus who unfortunately knew very little about ritual.\n\n\"Certainly not. She even implied that there was something between my husband and Grazia Vaillant.\"\n\n\"And...\" Carolus stopped himself in time. \"And that was an obvious lie,\" he said.\n\n\"Of course. There was no limit to what that woman would say.\"\n\nThere was a long pause.\n\n\"I wonder whether just for the sake of form, Mrs Waddell, you will tell me how you spent that afternoon? I have to ask everyone that.\"\n\nTo Carolus's surprise she acceded without protest.\n\n\"I had my Mothers from four to six.\"\n\n\"Where?\" said Carolus not revealing his ignorance of her meaning.\n\n\"At the Institute.\"\n\n\"Oh, I see. And afterwards?\"\n\n\"I came home.\"\n\n\"And you, Miss Waddell?\"\n\n\"Who? Me?\" laughed Rosa Waddell. \"I was out on my bike. Went over to Burley, as a matter of fact.\"\n\n\"For anything particular?\"\n\n\"Not really. More for the run than anything.\"\n\n\"See anyone you knew?\"\n\n\"No. Can't say I did. Why? Don't say I'm a suspect? I often felt like bashing Millicent Griggs but you surely don't think I actually did it?\"\n\n\"I haven't got to the point of having suspects. I'm only just beginning to get a notion _where_ it happened. I certainly don't pretend to guess when or through whom. Were you back before your mother?\"\n\n\"Yes. I got my own tea. I was hungry, too. Then I settled down to a book. Thirkell. You know, goes on and on but you feel you have to find out what happens to the dreary people. Daddy popped in for a few minutes and rushed out again. Then mother arrived.\"\n\n\"You didn't notice any times?\"\n\n\"I suppose daddy came around six and mother a good bit later.\"\n\n\"What do you call a good bit?\"\n\n\"Well, it was about a quarter to seven when mummy turned up.\"\n\n\"Thank you, Miss Waddell. I think you said, Mr Waddell, that you called on Miss Vaillant after coming home?\"\n\n\"I said I went to her house. I did so. In fact I had arranged to do on the previous morning.\"\n\n\"Did you stay long?\"\n\nThe vicar beamed good-humouredly. \"I didn't stay at all. I didn't enter. Miss Vaillant was out. I rang several times without result.\"\n\n\"That's interesting,\" said Carolus seriously.\n\n\"Yes,\" smiled the vicar, who seemed to think he had made a good remark, \"Miss Vaillant was out.\"\n\n# 6\n\nMUCH though Carolus wanted to meet Grazia Vaillant he felt it more important to see the Rumbles, husband and wife separately. In piecing together his picture of events that afternoon, in fixing times and forming an idea of where the various inhabitants of Gladhurst were likely to have been, he needed to hear what the Rumbles would tell him.\n\nAnother fine clear day found Carolus over at Gladhurst, making for the churchyard, since he understood that at this time the sexton would be working there. As he approached he heard a cheerful voice singing Rockin' _Along in the Breeze_.\n\nHe found Rumble, who stopped singing and looked up with a grin.\n\n\"You're the one who is going to find out who did for her, aren't you?\" he said.\n\n\"I hope so,\" returned Carolus briefly. Coming straight to the point, he said: \"You found the body, I understand?\"\n\nIt seemed that Rumble did not like this short cut across his reminiscences.\n\n\"I found the body,\" he said, \"but _how_ did I find it? How was it I came to disturb the grave waiting for Mr Chilling? How is it Miss Griggs isn't down there now and no one the wiser?\"\n\n\"Ah!\" said Carolus who from many cross-examinations had learnt the value of this long-drawn monosyllable in such contexts.\n\nRumble grinned.\n\n\"There's a lot to know about burying,\" he said. \"And from what I hear we're not a patch on the ancient Egyptians. What do we do, when all's said and done? Make a box for 'em with brass fittings and drop it under six foot of soil. Well, not six foot really because the ground doesn't have to be dug more than six foot down. I always do mine seven. Seven to the inch. That's how I came to know Chilling's had been tinkered with. I dropped my tape down the evening before and said to old Mugger who was helping me, there you are, I said, that's seven foot to the inch so Chilling'll have the best part of six foot on top of him when he's in tomorrow. Next morning when I measured it, it was scarcely more than five. So I said to myself this is funny, I said, something peculiar's been going on here and I jumped in and started digging down again. It wasn't two minutes before I came on Miss Griggs.\"\n\nRumble smiled broadly.\n\n\"I suppose that was rather unnerving for you?\"\n\n\"Well, you get used to anything like that in this job.\"\n\n\"Do you really?\"\n\n\"I've never found what you might call a recent one before but I've come on old 'uns. Been there hundreds of years and not much left. It's not so bad as it was but my dad and grandad had the job here before me and I've heard it was a proper sardine tin before they enlarged the churchyard. There was vaults under the church cram-packed with coffins. You could smell it. Well, I think you can still, very often. Sort of musty smell. Outside in the old graveyard, which only went as far as that yew tree, they used to pop 'em in one on top of the other till they were only just under the surface. So it's not surprising that now'n again when I'm digging a new grave I come on something, is it?\"\n\n\"I suppose not.\"\n\n\"Though of course not like this. She was lying on her back and the earth had been just shoved in over the top. It looked as though it might have been done in the dark. It was lucky I looked; otherwise she'd have been under old Chilling till Judgment Day and that would never have done, would it?\"\n\n\"Never,\" agreed Carolus sternly. \"What did you do when you found the body?\"\n\n\"Nipped straight over to see the Reverend Waddell. Course, he was very upset. He got on to the police right away and I will say they weren't long in coming out. Then\u2014you should have seen them. You've never seen such a pantomime in all your life. Every bit of ground was gone over inch by inch. They must have taken away a couple of hundredweight of soil. They was taking photographs and measuring up and examining the ground. But after all that, they said it was all right. We could go ahead burying Chilling that afternoon as arranged. That was a relief to everyone.\"\n\n\"Why?\"\n\n\"Well, the expense. He'd been taken over to Worsley's the undertakers at Burley and was Lying there. It all costs money. And though I believe he's left his widow comfortable no one likes to see it wasted, do they? Besides it was all arranged for that afternoon. Reverend Waddell was going to bury him himself. It would have meant time wasted. But the police seemed to understand that. They said it was all right to go ahead. So we did and Chilling's down there on his own. Just the one bell, he had, and the organist without the choir. All very nice but nothing extravagant. You should have seen it when the Miss Griggs's father was buried. That was a funeral. You'd have thought it was royalty. Of course he'd been very good to the parish. Fiver for me every Christmas and fifty quid in the Easter Collection. Not to mention other things.\"\n\n\"Were you digging the grave on that Thursday afternoon?\"\n\n\"That's right. Me'n Mugger.\"\n\n\"What time did you finish?\"\n\n\"Must have been well before four. It was beginning to get dark and you know what time that happens. I daresay we'd finished and put our tools away by half-past three.\"\n\n\"Where did you put your tools?\"\n\n\"In the furnace room.\"\n\n\"Is that locked?\"\n\n\"No. No need in a place like this.\"\n\n\"So anyone can have used those tools to throw the earth over Miss Griggs's body?\"\n\n\"I don't suppose anyone knew they were there. The police examined 'em all for finger prints and there was only mine on them.\"\n\n\"I see. What did you and Mugger do when you'd finished?\"\n\n\"I went home. He lives over the other side of the village. He left me near my cottage and went on.\"\n\n\"While you were digging did anyone come to the churchvard?\"\n\n\"Not while we were digging, they didn't.\"\n\n\"But afterwards? Did you meet anyone as you came away?\"\n\nRumble looked somewhat perplexed.\n\n\"It doesn't seem worth mentioning,\" he said, \"but as me'n Mugger left the churchyard we did happen to meet George Larkin and his son Bill.\"\n\n\"Where were they going?\"\n\n\"They went into the churchyard.\"\n\n\"To the church?\"\n\n\"I don't know. It was getting a bit dark and we wanted to get home.\"\n\n\"Were you surprised to see the Larkins?\"\n\n\"I was a bit. George isn't much of a churchgoing man though he does come sometimes on Sunday mornings.\"\n\n\"Did you speak to them?\"\n\n\"Just passed the time of day.\"\n\n\"They didn't say where they were going?\"\n\n\"No. They never say much, as you'd know if you went to the Black Horse. Anyhow we didn't stop and ask questions.\"\n\n\"Have you ever seen them there before?\"\n\n\"Can't say I have. But that's not to say they've never been. No. I said good-night to George and Bill and went over to my home.\"\n\n\"Was there anyone at your home?\"\n\n\"Do you mean my old woman? There was no one else to be at home. No. She hadn't come back from Vaillant's then. She came in about five. I remember that because she said there was a light on in the church. Not often she'll bother to speak. She's a funny-tempered woman, my wife. Still, she did say that afternoon, there's a light on over in the church. So I went out to look and she was right.\"\n\n\"What did you do about it?\"\n\n\"I was just going over when I saw old Flo coming across.\" Rumble paused to laugh reminiscently. \"You know Flo?\"\n\n\"By sight.\"\n\n\"She's all right, is Flo. She's a good sport! Liked all round except by some of the wives. They hate her, some of them. Think their husbands are too friendly with her. Well, so they may be. Flo doesn't mind. Anyway, there she was. 'It's only Mr Waddell,' she said, 'I just saw him go in.' I asked her where she was going and she said wouldn't I like to know, then I heard my old woman coming so I shut the door to save trouble and argument.\"\n\n\"Quite.\"\n\n\"I didn't bother any more about the church till after I'd had my tea, then when I looked out the lights were off. So I walked over and locked up.\"\n\n\"Did you look round before doing so?\"\n\n\"No. I didn't go in at all. There was nothing to go in for.\"\n\n\"You didn't meet anyone when you went across?\"\n\n\"Not a soul.\"\n\n\"Nor hear anything?\"\n\n\"Nothing unusual.\"\n\n\"Thanks, Rumble. You've been very helpful.\"\n\nRumble grinned.\n\n\"Going to try the old woman?\"\n\n\"There are one or two things I want to ask her.\"\n\nRumble seemed amused at that.\n\n\"I wish you luck,\" he said darkly.\n\n\"I gather you don't approve of Miss Vaillant's religious views?\"\n\n\"It's not that. It's the work it makes. Processions and that. She wants to start burning incense all over the place. It was Commander Fyfe who stopped that by asking what the Insurance people would say about it. I've got my job as verger without dressing up in lace and carrying a banner on a pole. Besides people don't like it. The collections have fallen off since she started persuading the Reverend Waddell. We don't get the weddings we did, either. Baptisms are down. And Confirmations are nowhere. It all means less for me and it's little enough as it is. There was a lot of trouble last Ash Wednesday when she wanted the vicar to start smudging everyone's faces with charcoal. Have you ever heard anything like it? Miss Griggs, the old lady that's been done for, asked if she wanted to turn us all into chimney sweeps. The Reverend Waddell didn't know what to do.\"\n\n\"What _did_ he do?\"\n\n\"Well, he did her and no one else. That was his way out of it. She walked about looking as though she'd been cleaning out the grate and forgotten to wash. She couldn't complain because she'd got what she wanted and Miss Griggs couldn't say much either because no one else had it. But they were like two tiger cats, those two. Never knew two women hate one another as they did and the vicar's wife hated the pair of them. Nice state of affairs.\"\n\n\"Very.\"\n\n\"You go across and see the old woman. She's at home now. But don't blame me if she won't let you in. She's very funny about things.\"\n\n\"I can only try,\" said Carolus.\n\n\"I hope you _do_ find out who done it,\" chuckled Rumble. \"Though with no disrespect I can't help smiling when I think how cross the old lady would have been if she'd known she was going to be banged on the head. She was one to fly into a fit of temper anytime. Well, all three of them were\u2014Mrs Bobbin perhaps most of the lot. I often think it is a village for that. There was the lady who's Gone and her two sisters, and my old woman who is as bad as any of them, and that Miss Vaillant, _she's_ got a temper too, and the vicar's wife and one or two more. Not old Flo, though. You never see her put out whatever happens. But Mugger's wife was another who flies off the handle at the least thing. You've never seen such tantrums as we have here. But it never bothers me. I just laugh.\"\n\n\"Very sensible. Tell me, since you worked for her you must have known something of Miss Millicent Griggs?\"\n\n\"Want me to tell you straight? \"Rumble did. \"That's what she was. Mean with her sisters and everyone else. Always _right_ , if you know what I mean. Do anyone a nasty turn. Not like that Miss Flora who is sincere in what she thinks.\"\n\n\"Thanks, Rumble.\"\n\n\"Well, I wish you luck with my old woman. Only if I see you flying down the road in a minute I shan't be surprised.\"\n\nAs Carolus left him he heard him singing blithely. _Hook, Line and Sinker_ was the number he chose to volley among the gravestones.\n\nCarolus decided that if he was to get any information from Mrs Rumble strategy would be necessary. He knocked loudly at the cottage door and when it was opened by a dishevelled and irritated woman with her mouth already open to abuse him he spoke before she could.\n\n\"I'm enquiring about a pair of galoshes,\" he said.\n\nThe effect was as startling as he hoped. The lower jaw dropped farther and the eyes searched his. Then Mrs Rumble looked quickly up and down the street and said, \"Come inside quick and let me shut the door.\"\n\nIn a musty and cold front room she faced Carolus with a defensive expression.\n\n\"What galoshes?\" she asked.\n\n\"Don't let's beat about the bush, Mrs Rumble. You know quite well what I mean.\"\n\n\"I've never Took anything in my life,\" she said. \"Least of all out of the church.\"\n\n\"No?\"\n\n\"Certainly not. I wouldn't if you was to pay me. Things left behind in the pews and that.\"\n\n\"What do you do with them?\"\n\n\"If there was anything left behind I should take it to the vicar.\"\n\n\"Did you take the galoshes to the vicar?\"\n\n\"I was going to. Soon as ever I had a minute.\"\n\n\"Did you tell him you'd found them?\"\n\n\"Not yet, I haven't. I haven't had a chance to turn round.\"\n\n\"When did you find them?\"\n\n\"Last Sunday morning. That's the day before yesterday.\"\n\n\"Where were they?\"\n\n\"Pushed right underneath at the back of the church. They might have been there for weeks.\"\n\n\"You don't know to whom they belonged?\"\n\n\"No. I don't. If I did I'd have run straight round with them. It's my belief they were left behind by some visitor a long while back.\"\n\n\"Wouldn't you have found them, in that case?\"\n\n\"I might have if what I was paid for looking after the church was enough for me to be looking everywhere morning, noon and night, but it's not. I can't go washing and polishing on my hands and knees every day in every square inch of that yewje building, can I?\"\n\n\"I suppose not. Did you do all the cleaning in the church?\"\n\n\"All except the brass, I did. Miss Griggs liked to do that herself.\"\n\n\"Do you remember going over to the church on the morning when your husband found her body? The Friday morning, I mean.\"\n\n\"I'm not likely to forget it.\"\n\n\"Did you notice then whether the brass had been cleaned recently or not?\"\n\n\"Not then, I didn't, but later in the day I did. I noticed it hadn't been done for a week or more and I thought, I suppose that'll be expected of me now that Miss Griggs isn't any longer here to do it. Well, I can only do so much, I thought.\"\n\n\"You are quite sure no one had cleaned it or started to clean it on the previous afternoon?\"\n\n\"Certainly Miss Griggs hadn't. She always started with the lectern which is a neagle with its wings holding up the Bible. Her father gave it to the church and she thought a lot of it. It was that I noticed that day. It was ever so tarnished all over.\"\n\n\"Thank you. That's very helpful.\"\n\n\"Are you from the police, then?\" asked Mrs. Rumble.\n\n\"No. I'm just trying to find out who killed Miss Griggs.\"\n\n\"You don't need to say anything about those galoshes, do you? It might be thought bad of.\"\n\n\"Certainly not. But if I may suggest it, I think you should report finding them.\"\n\nMrs Rumble looked at him dubiously then seemed to take a momentous decision.\n\n\"I'd just put the kettle on when you knocked,\" she said.\n\nCarolus waited.\n\n\"I don't know whether you'd like a cup of tea. Only you'd better come through to the kitchen where there's a fire.\"\n\nComfortably seated in what he supposed was Rumble's chair, Carolus prepared to ask more questions while the unprecedented affability of Mrs Rumble lasted.\n\n\"I understand you work for Miss Vaillant?\"\n\n\"Well, I do, yes. I can't give all my time to the church and it helps.\"\n\n\"You share her religious views, I believe?\"\n\n\"I wouldn't call it that. Only I do think it brightens things up to have a bit of colour and music instead of psalms, psalms, psalms all the time. I don't say I should want to go as far as what she would, especially when she came back from Spain last year and wanted to start teaching the choir to do a dance in the middle of the chancel. But if those Miss Griggses had their way it would be one long Bible meeting.\"\n\n\"You think Miss Vaillant sincere in what she wanted?\"\n\n\"Well, if the truth were known... I don't know whether I ought to Speak...\"\n\n\"Yes?\"\n\n\"The truth is, she Drinks,\" whispered Mrs Rumble. \"There. Truth will out. Secret, of course. No one ever had any idea about it except Forster's Stores over at Burley where she got the stuff and brought it away in her own car. Gin it was. She used to put lime juice in it to take the taste away. I wouldn't have known myself if I hadn't happened to have a key which fitted the cupboard and could see the bottle. But she wasn't the only one.\"\n\n\"She wasn't?\"\n\n\"No. And this will surprise you. Old Miss Griggs Liked a Drop, too. What do you think brought them together right at the end? Miss Griggs went to see her twice after they hadn't been speaking for I don't know how long. The first time I met her coming out. All flushed up she was, besides I could smell her breath. Oh, I thought, so you Like a Drop, too, do you? They forgot all about their differences after that.\"\n\n\"You say Miss Vaillant kept her gin in a locked cupboard?\"\n\n\"Yes. In her sitting-room. Only one bottle at a time. She'd rather have died than let anyone know about it. You know what she does with the empty bottles? It just shows you how cunning they are. There's an old well outside the backdoor. They say it's ever so old and goes down no one knows how deep. She'd fill the empties with water so that they'd sink, then drop them down and no one was to know any different.\"\n\n\"How did you come to be aware of it?\"\n\n\"I saw her one night when I'd gone back for something. She never knew I saw her.\"\n\n\"Did she stick to one brand of gin?\"\n\n\"Yes. Horseley's. You know the bottles? Kind of oval shaped.\"\n\n\"How do you know that?\"\n\n\"Well, my nephew works over at Forster's Stores in Burley.\"\n\n\"You think she drank regularly?\"\n\n\"No. I'm sure she didn't. It was only about once a week. When she couldn't resist it, I suppose. You know what secret drinkers are, don't you?\"\n\n\"You don't think she was at all violent when she had been drinking?\"\n\n\"I know very well she was. I've come in in the morning more than once and found things broken which could only have been on purpose.\"\n\n\"And you don't think her reconciliation with Miss Griggs went very deep?\"\n\n\"No, I don't. They'd been at one another's throats too many years for that.\"\n\n\"How do you suppose, then, that they came to talk intimately enough for Miss Vaillant even to suggest a drink?\"\n\n\"Well, when I took their tea in on Miss Griggs's first visit, I heard something. Miss Griggs had just noticed the tube of Minerval tablets which Miss Vaillant had. They're for sleeping, prescribed by Dr Pinton. 'Oh', she said, 'Dr Pinton gives you these, does he?' I make no doubt she took them, too. That may have set them talking.\"\n\nJust then the back door opened and Rumble entered. If his wife had shown surprise when Carolus first appeared Rumble was now dumbstruck at the scene which greeted him.\n\n\"What are you gaping for?\" asked his wife. \"Never seen a gentleman drink a cup of tea before?\"\n\n\"Not in this kitchen, I haven't,\" said Rumble recovering himself.\n\n\"Well, you're seeing it now so take those filthy boots off and sit down and behave yourself.\"\n\nRumble gave his tolerant grin.\n\n\"I don't know how you managed it,\" he said to Carolus.\n\n\"Now don't Start,\" said Mrs Rumble. \"Upon my soul I don't wish anyone any harm but I can't help thinking it's a pity there's not more Taken in this village. Rumble's never happy unless he's digging a grave.\"\n\n\"I thought he was always happy.\"\n\n\"That's just Appearances,\" said Mrs Rumble. \"He's got to be out there measuring it up and getting it right or else he's not fit to live with. I know. I've seen him looking round wondering who the next will be.\"\n\n\"Any ideas on the subject? \"Carolus asked Rumble.\n\nRumble smiled again.\n\n\"If I had, I shouldn't say,\" he replied.\n\n\"I hope there's not going to be any more,\" announced Mrs Rumble. \"One's quite enough to upset us all.\"\n\nWhen Carolus left she followed him to the door.\n\n\"What do you think I ought to do about those galoshes?\" she asked.\n\n\"I think you should take them to the police and tell them where you found them.\"\n\n\"Who? Slatt?\"\n\n\"Yes. He's your local policeman. They may have some bearing on the case.\"\n\n\"You think they were Miss Griggs's, then?\"\n\n\"I think it's very likely.\"\n\n\"I'll run round with them straight away,\" said Mrs Rumble. \"I shouldn't want anyone to think I'd Take anything.\"\n\n\"Of course not.\"\n\nShe hesitated.\n\n\"Nearly new, they are. Just my size. Still if you say it's best I'll run round with them.\"\n\n\"Do you mind if I see them? \"Carolus asked.\n\nFor one moment Mrs Rumble hesitated, then she said\u2014\"\n\nNo. I don't mind. I'll fetch them in a minute.\"\n\nCarolus examined a pair of brown galoshes, size eight, nearly new and made by Skilley and Harman.\n\n\"Yes, I shall run round with them at once,\" repeated Mrs Rumble regretfully.\n\n\"Good-night. Thank you for my tea.\"\n\n# 7\n\nCAROLUS wondered whether he was not treating the case in too leisurely a manner, but reflected as he had often done that criminology must remain nothing to him but a hobby. He had no intention of neglecting the job of teaching which he had set himself because without it he was in danger of becoming a mere rich dilettante. The discipline imposed by school life was, he knew, the one thing which kept him from all the fatuous forms of time-wasting practised by people who had money and no occupation.\n\nSo when a case like this came along, however keen his curiosity and however strong his determination to discover the truth, he could only pursue his enquiries when his duties at the Queen's School, Newminster, permitted him to do so. If the police succeeded before him, or made an arrest while he was still asking questions, it would be just too bad. He could do nothing to prevent it.\n\nBesides, as he had always recognized, his method was not a rapid one. He had a gift for making people confide in him but it took time to hear them and it was fatal to attempt to hurry them. Out of those confidences would emerge the truth, a hint here, an odd fact there, and slowly the whole thing would fall into shape. But he had to take his time.\n\nEven so, however conscientiously he might stick to his work at Newminster, he felt that his frequent absences after school hours had not escaped the attention of Mr Gorringer and that sooner or later he would be faced with one of the headmaster's semi-confidential, for-your-own-good orations. It came sooner than Carolus thought. He found himself one morning striding up and down the quadrangle beside Mr Gorringer, whose hands were clasped behind him under the folds of his gown. The headmaster was even more facetious than Carolus had feared.\n\n\"A bird has whispered in my ear...\" he began.\n\nCarolus stared with stupefaction at the huge hairy orifice surrounded with tufted chasms which the headmaster had mentioned and wondered what bird would dare approach its sinister network. A vulture, perhaps.\n\n\"That our Senior History Master is again immersed in the contemporary. That you, Deene, albeit with a discretion for which I am glad, are once more, as it were, on the trail.\"\n\nCarolus wondered. Should he quote Kipling? 'The Long Trail\u2014the trail that is always new'. On the whole, no.\n\n\"And unless Rumour plays us false, it is the village of Gladhurst which claims so much of your time.\"\n\nMr Gorringer was hardest to bear when he was playful.\n\n\"That's right,\" said Carolus.\n\n\"An interesting case?\" queried Mr Gorringer.\n\n\"Quite.\"\n\n\"That, no doubt, accounts for your hasty departures from Newminster,\" said the headmaster in a rather more serious voice. \"Oh, believe me, Deene, I make no stricture. I recognize each man's right to his private interests. I have not openly criticized Hollingbourne's predilection for poultry-keeping on a scale which might almost be termed avicultural though I am relieved to know that as he has accepted the housemastership of Plantagenet he will reduce his stock to domestic dimensions. Even Beardley's commercial interest in antiques has met no more than a meaning frown from me. I am not on this, as on previous occasions, requiring that you should desist from your enquiries for I recognize that in your more recent cases you have shown a greater discretion than of yore. I only ask that in pursuing your hobby you do not involve us all. That your position as Senior History Master here is not a matter to be bandied to and fro by all and sundry and that in no case should the Press be aware of it.\"\n\n\"I'll do my best.\"\n\n\"My wife joins me in her solicitude, though she was unable to resist a witticism when she heard the name of the murdered lady. 'Mr Deene', she said, 'is doubtless out to discover why the sea is boiling hot, and whether Griggs have wings.' I must say, I derived considerable amusement from the _mot\"_\n\n\"Yes,\" said Carolus, unable to think of another comment.\n\n\"Very well, my dear Deene, I have full confidence in you. I must have a word with the music master. Ah, Tubley...\"\n\nEven his housekeeper seemed less disturbed by the participation of Carolus in a case so far away than she had been on other occasions.\n\n\"Well, I must allow,\" she said when she brought in his lunch, \"that if you will get yourself mixed up in crimes and horrors I wish they was all like this one. I was only saying to Stick, there's scarcely been a word in the newspapers and not a soul calling here at all hours without our knowing if it was a murderer or not. Are you going over there again this afternon, Sir?\"\n\n\"Yes, Mrs Stick. I have to meet several more of the inhabitants.\"\n\n\"Then you be careful and don't eat or drink anything when you don't know where it's from. You oughtn't to be hanging about after dark, either. If they can do for an old lady they can do for you, make no mistake.\"\n\n\"Thank you, Mrs Stick, I'll be careful. Leave me out some sandwiches in case I'm late.\"\n\n\"There! And I'd got a nice piece of turbot for your dinner and was going to make you a Creep Suzette to follow. I shall wait till half-past eight and if you're not here I shan't cook at all. I won't serve things rechauffay, so it's no good asking me to.\"\n\nWhen Carolus reached Gladhurst in a driving rainstorm he decided to postpone once more his meeting with Grazia Vaillant from whom he expected most and call on the churchwarden, Commander Fyfe.\n\n'He is not', Mrs Bobbin had said, what you would imagine a retired naval officer to be like'. Carolus tried to imagine what a retired naval officer _would_ be like but his effort was unsuccessful. Beefy and boisterous? Ginny and garrulous? Prosy and pompous? Jovial and judicious? He might be anything.\n\nBut Mrs Bobbin had been right. Whatever retired naval officers were like they were not like Commander Fyfe.\n\nHis house, called The Fairway, was at the end of the village farthest from the church, a stucco and slated affair closely hidden by trees. Relieved to see a porch over the front door Carolus ran from his car to its shelter, then rang the bell.\n\nAs he stood there he had an uncomfortable feeling that he was being watched and, noticing a mirror set in the door's panel, he guessed that it was one of those constructed to allow one behind it to see through while appearing an ordinary mirror from the outside. He at once made a grimace at it and a rude sign with his fingers.\n\nThen a small trap in the main door opened and for the first time he had a glimpse of the narrow bald head, the active dark eyes, the vigorous growth of eyebrow and the thin clean-shaven lips of the churchwarden.\n\n\"Oh yes, I know,\" said Fyfe. \"Your name's Deene, isn't it? Wait a minute while I unbolt this door.\"\n\n'A minute' turned out to be three and the scraping and grating, the rattling and jangling suggested that Fyfe was the custodian of one of our less escape-prone prisons.\n\n\"Come in,\" said Fyfe conspiratorially and, when Carolus was standing in a dark passage, set to work to bolt, bar, chain and lock behind them.\n\n\"Come to my den. We shall be undisturbed there. You don't mind if I lock the door? Sit down, please. Smoke? Listen! What's that?\"\n\n\"It sounds like the rain,\" said Carolus.\n\n\"It may be. It may be. But you never know. Get some queer noises round here. Very queer. It's a queer place altogether. Have to take every precaution.\"\n\n\"So I notice. Against what?\"\n\n\"Against what? Well, you know what's just happened. Poor woman clubbed. I put nothing past them.\" \"Past whom?\"\n\n\"Any of them. I tell you this place is full of strange characters. Have you met a woman known as 'Flo'?\"\n\n\"I've seen her. I should have thought she was the most harmless of souls.\"\n\n\"She may be. But indiscreet. Loud-mouthed. I was foolish enough to stop and chat with her one afternoon. Merely to ask how she was. I had seen her in church. I assure you I didn't stop for more than two minutes. Yet before I reached home my wife had heard of it. The vicar knew. The whole village. It caused me a great deal of inconvenience. Of course the woman herself didn't mind at all.\"\n\n\"No, Flo doesn't mind,\" repeated Carolus.\n\n\"That's only one of the things.\" Fyfe turned round suddenly and looked at the window. \"Did you see anything?\" he asked.\n\n\"Trees,\" said Carolus.\n\n\"I thought I noticed a shadow. As though someone were outside. I may be mistaken. Can't be too careful. What did you want to ask me?\"\n\n\"About the day of the murder.\"\n\nFyfe nearly leapt from his chair.\n\n\"That _was_ a day,\" he said. \"Extraordinary things happened that day.\"\n\n\"Such as?\"\n\nFyfe leant forward.\n\n\"It was on that day,\" he confided, \"that the incident happened which I have described to you. With the woman known as Flo.\"\n\n\"Where?\"\n\n\"In the very centre of the village. Outside Jevons's Stores. It couldn't have been less secretive. Yet...\"\n\n\"At about what time did you meet her?\"\n\n\"That I can tell you exactly. Habit of mine. Always make a note of the times of events. You never know, you see. Anything can happen in a place like this. It was 3.42. I need scarcely add p.m.\"\n\n\"So you were in the village street between half-past three and... when, would you say?\"\n\n\"Nearly four.\"\n\n\"Whom did you meet?\"\n\n\"That's the extraordinary thing. Almost no one.\"\n\n\"Almost?\"\n\n\"I saw Slipper, our curate, you know. Very good with boys.\"\n\n\"So I understand. Was he on foot?\n\n\"Yes. Hurrying along as he usually is. Wearing a raincoat.\"\n\n\"Alone?\"\n\n\"That's another strange thing. He was alone. Usually has several of the boys with him.\"\n\n\"You saw no one else?\"\n\n\"Only the girl who works at Crossways. Naomi Chester.\"\n\n\"Did she see you?\"\n\n\"I think not. I was in the Post Office. I saw her pass on the other side of the road.\"\n\n\"On foot?\"\n\n\"Yes. She too was in a hurry.\"\n\n\"Going towards her home?\"\n\n\"It might be. Yes. One doesn't know of course.\"\n\n\"You reached home at what time?\"\n\n\"Soon after four. I found the vicar here.\"\n\n\"Mrs Fyfe was at home?\"\n\n\"Yes. As I have explained, she had already heard this extraordinary story about the woman called Flo.\"\n\n\"Oh yes. Perhaps the vicar himself had told her?\"\n\n\"No. I understand someone had made a telephone call. Shows you, doesn't it? Small place like this. Full of malice. Old women, you know.\"\n\n\"Yes. You do seem to have a collection of somewhat irascible elderly women. I understand a man called Grey was working for you.\"\n\n\"No. No. Not for me. He is employed by a local firm of builders and decorators. He was painting. The dining-room.\"\n\n\"You found him there when you returned that afternoon?\"\n\n\"No. Just gone. Quite extraordinary. A phone call had come for him. My wife answered the phone.\"\n\n\"Who wanted him?\"\n\n\"My wife didn't recognize the voice. That of an elderly woman, she thinks.\"\n\n\"Another elderly woman?\"\n\n\"So my wife tells me. She did not listen to the conversation. She told Grey he was wanted on the phone and noticed that shortly after he left the house.\"\n\n\"Before his usual time?\"\n\n\"Yes. Very odd, wasn't it?\"\n\n\"You didn't by any chance pass the churchyard that afternoon?\"\n\n\"It was the first thing I did after lunch.\" Fyfe's voice dropped almost to a whisper. \"Rumble was there. At work on Chilling's grave.\"\n\n\"There was surely nothing odd about that, Commander Fyfe? Chilling was to be buried next day.\"\n\n\"But do you know who was helping him? Mugger, of all people.\"\n\n\"Yes. Rumble told me.\"\n\n\"A dangerous character, that.\"\n\n\"Dangerous?\"\n\n\"Lawless. Violent.\"\n\n\"He has the freedom of your land, I believe.\"\n\n\"What? Oh, rabbits, yes. He helps to keep the rabbits down.\"\n\nDusk had fallen as they talked but Commander Fyfe did not switch on the lights. They sat in half darkness. Fyfe's voice was hollow and awed and his eyes restless.\n\n\"You didn't notice anyone hanging about outside when you came in?\" he asked Carolus. \"No. Why?\"\n\n\"You can't be too careful. Yes, Rumble and Mugger were digging the grave and Rumble was actually singing as he worked. I could tell you something else.\"\n\n\"Yes?\"\n\n\"The three old ladies\u2014well, there are only two of them now\u2014have a nephew who lives at Burley.\"\n\n\"I have a note of that, I think.\"\n\n\"But you didn't know he was over here that afternoon, did you? In his car. A grey Vauxhall. Saw him with my own eyes. You see what I mean when I say queer things happen in this place.\"\n\n\"Not quite. What is extraordinary about Dundas Griggs?\"\n\n\"He is the nephew of the murdered woman,\" gasped Fyfe.\n\n\"I see.\"\n\n\"And he was in Gladhurst on the day of the murder.\"\n\n\"Mm.\"\n\n\"Then what about Larkin?\" whispered Fyfe.\n\n\"What about him?\"\n\n\"You surely know his connection?\"\n\n\"I know that there is a story that more than forty years ago there was some incident connecting him with Millicent Griggs.\"\n\n\"There you are! I tell you...\"\n\n\"Did you go out again that day, Commander Fyfe?\"\n\n\"I? My wife is an invalid, as you may have heard. I never go out at night and leave her alone in the house. Certainly not. Anything might happen. Anything. But on that evening I was fortunate. My gardener and his wife who live nearby had asked leave to view the television. I took advantage of that for a brief spell.\"\n\n\"What time would that have been?\"\n\n\"Oh, six-thirty to eight, perhaps.\"\n\n\"Did you remain downstairs long after that?\"\n\n\"Oh, an hour or two, I daresay. With my stamps, you know. I am a philatelist.\"\n\n\"You heard nothing more of the outside world, as it were, that evening?\"\n\n\"At one point I thought I heard someone tapping on that window pane. But I found it was the branch of a tree.\"\n\n\"Very disappointing for you. Nothing more?\"\n\n\"Motor-cycles,\" said Commander Fyfe.\n\n\"That was not unusual though?\"\n\n\"In winter? On this road? Most.\"\n\nCarolus rose.\n\n\"I shall have to open the door for you. I'll tell you if anything else very strange occurs.\"\n\n\"Thank you,\" said Carolus. \"I wonder what you could tell me about the dead woman? Her character, I mean?\"\n\nIt had occurred to Carolus that unreliable and sensational as Fyfe might be on facts, on character he would be more dependable.\n\n\"Strange, very strange,\" he said, shaking his head. \"I cannot pretend that she was popular\u2014very much the contrary. She was a woman of insatiable curiosity. Of a most unsavoury kind, I fear. I was once rash enough to leave her alone in conversation with my wife. My dear fellow, the questions she asked. I could scarcely believe it afterwards. Hair-raising.\"\n\n\"She doesn't sound very pleasant.\"\n\n\"Pleasant? She was... but the woman is dead so let's say no more. Now I'll let you out if you like.\"\n\nThe unbolting began again.\n\n\"Have to take precautions,\" explained Fyfe.\n\nIt was still raining as Carolus drove away.\n\n# 8\n\nNOW there was nothing for it. He must call on Grazia Vaillant.\n\nHe found the Old Vicarage a well-kept house, vaguely Georgian in architecture, with additions probably made by the fathers of large families in the last century. But as soon as he drew near he saw that no opportunity had been missed for antique embellishment. Wrought iron gates had been put in, large urns evidently from Crowthers' topped the gateposts, beside the front door was a huge ship's bell and over it a ship's lamp. A wrought iron bracket which had once had a sign swinging from it had been fixed to the wall with another lamp suspended from it. An antique sundial was on the lawn and an ancient dovecote was visible.\n\nMrs Rumble opened the door. She was evidently a woman who having taken the unprecedented step of according someone her friendship did not turn back, for she made a grimace at Carolus which was intended to suggest a smile.\n\n\"I was just off,\" she confided. \"But she's in, all right. Having her tea. Wait here a minute and I'll tell her.\"\n\nCarolus stood among coffin-stools and oak chests, corner cupboards and brass chestnut-roasters, cricket-tables and warming-pans, arranged not as they had been in the place in which they should have stayed, some expensive Old Curiosity Shop in Rye or Tunbridge Wells, but as though they were the natural furniture of the house.\n\nPresently Mrs Rumble reappeared and ushered him to a large drawing-room. Grazia Vaillant came forward to greet him with extended hand and they met under an enormous glass chandelier which tinkled no more than Grazia herself with her pendants and bits and pieces. She wore something which William Morris might have designed for the chatelaine of an old English castle. Her hair was henna-red and her eyes searching and hungry.\n\nCarolus had time to observe the room about him, a nightmare of old samplers, stump-work pictures, Dresden figures, half a dozen clocks such as collectors value, an antique hour-glass, a couple of spinning-wheels, a collection of tea-caddies, stools with woolwork covers and a spinet. But over the fireplace was a magnificent landscape which looked as though it had entered by mistake. Almost certainly a Constable, Carolus thought.\n\n\"How do you _Doo_?\" asked Grazia, forgetting to release Carolus's hand. \"Come to the fire and have some tea. Such a wretched day. I hear you're investigating our murder.\"\n\n\"Yes, in a way.\"\n\n\"Such a dreadful, dreadful business.\"\n\n\"I'm glad to see you're not scared by it, Miss Vaillant. You remain alone here at night?\"\n\n\"But of course. I have my _ange gardien.\"_\n\n\"Useful. Any watchdog?\"\n\n\"Oh yes. Peter and Paul my two boxers. Oh, _I_ 'm all right, I assure you. Besides, lightning doesn't strike in the same place twice. Why should we expect anything more in Gladhurst? They have the jewellery and money, poor old Millicent Griggs is dead. Aren't we entitled to peace now?\"\n\n\"'They'. Who do you mean by 'they'?\"\n\n\"The robbers. The murderers.\"\n\n\"I see. You had not been friendly with Miss Griggs, I believe?\"\n\n\"We are all Christians, Mr Deene, but I did have to remind myself of it rather often with Millicent Griggs. She was so painfully protty....\"\n\n\"I'm sorry. I don't know that word.\"\n\n\"Protestant.\"\n\n\"But\u2014forgive my ignorance\u2014I thought that was your religion. Church of England, I mean.\"\n\n\"No. No, Mr Deene,\" smiled Grazia kindly. \"You've got it all wrong. We're Car-tholics.\"\n\n\"But?\"\n\n\"Not Romans, of course. Just Car-tholics. The Church of England is part of the Car-tholic Church.\"\n\n\"What are the other parts?\"\n\n\"The Orthodox Russian. The Greek. The Roman. All one.\"\n\n\"I see. But Miss Griggs wasn't included?\"\n\n\"You are naughty,\" said Grazia Vaillant. \"She was what is called evangelical, though I can't think why they use that lovely word. She opposed every effort we made to beautify our church, to make it more in keeping with its heritage. We had struggle after struggle....\"\n\n\"Who did? How did the vicar vote?\"\n\n\"Poor Father Waddell! He's rather a waverer, you know. I remember when I gave him a beautiful set of Stations of the Cross by a Spanish artist, he hesitated. He was wondering what Millicent Griggs would say. You see, to her, Mr Deene, I was the Scarlet Woman, I was Popery personified. She really believed I kept Jesuit priests in hiding-holes and was planning another Gunpowder Plot. She hated me and all I stood for with the hatred of a Puritan for a Cavalier.\"\n\n\"And you?\"\n\n\"God gave me a sense of humour,\" claimed Grazia, waving a heavily bangled arm. \"I was able to see the funny side of poor old Millicent especially when I once nipped over to the church and gave it a good censing with the best incense just before their dreary old Eleven O'Clock Service which Father Waddell _will_ keep on to please them. Millicent really thought Father Waddell had been using incense at Sung Mass\u2014she never comes to that. She rose in her pew and said in a ringing voice, 'Incense is an abomination to me! Isaiah 1, XIII, 1.' Then she marched out of the church. Father Waddell was very cross with me. He had to go and explain that I had done it as a joke. 'Such jokes', said Millicent, 'will call down fire and brimstone from heaven utterly to consume the wicked'. You see the sort of thing? Yes, she hated me, all right, but I could only find her funny and stuffy and protty, poor old thing.\"\n\n\"I hear there was a reconciliation.\"\n\n\"If you call it that,\" said Grazia Vaillant, pouring another cup of tea for Carolus. \"You see I was most anxious to have a Lady Chapel. When I tackled Father Waddell he took refuge in a kind of diplomacy which I can't say I admired. 'If you can get Millicent Griggs to agree, yes,' he said. 'If you can't talk her round I shall have to say no.' So I went to work and asked her here to tea. She was very much on her guard at first but I put it in the most modest way. A matter of convenience, I said, to save light when there are only a few people. 'You already have the Lord's Table,' she said. 'I know, but we want two Lord's Tables', I said. 'If you don't object to one why should you object to two?' Rather clever of me, I thought, and very soon I saw she was coming round. She asked how many candlesticks would be on it, and I said only two. She agreed to give me her answer two days later and came round again. This time she asked whether I would subscribe to one of her pet charities\u2014to convert the Jews or something. When I agreed she said she would not stand in the way of what she called the second Lord's Table. She little knew that I'd bought a great big statue of Our Lady of Lourdes all ready for it.\"\n\n\"So the vicar now agreed?\"\n\n\"Well, it was only two days before she was murdered, poor old thing, and of course we've all been rather upset since then.\"\n\n\"Naturally.\"\n\n\"I was often very angry with her. She was infuriating, sometimes. She called everything we did to restore the church to its old state an _innovation_. You could not make her see that Protestantism itself was an innovation and our religion the old one. Whenever I did some perfectly natural thing like cross myself or kneel for a part of the Creed she looked at me as though she would like to strangle me. She wasn't quite sane, you know.\"\n\n\"I certainly didn't know. Are you serious?\"\n\n\"Quite. I won't say she was certifiable, though the sister Flora really is. But unbalanced. You couldn't have such ferocity in a normal mind. She once threatened me with her umbrella because someone told her my position with regard to Transubstantiation. She blamed me for everything. I was 'plotting and scheming' to undermine what she called the 'grand old English church service'. That sort of thing. It was an obsession with her. She spoke of me as 'that Popish Vaillant woman'. There was a time when I really thought she would do me some physical injury. Can you call that sane?\"\n\n\"Your disagreement does seem to have gone to some lengths. It is surprising that you should have been reconciled.\"\n\n\"We weren't, really. Things were just patched up. She never ceased to hate me. She admitted that she would like to see me dead. She believed, sincerely believed, that an avenging fire would strike me from heaven.\"\n\n\"No!\"\n\n\"Truthfully. She told her sister Flora that, on the day before she died. 'Not long', she said. Then changing the sex in the words, she quoted Isaiah again. 'Woe unto the wicked! it shall be ill with her: for the doing of her hands shall be done to her.' Then she went on, quoting correctly, 'As for my people, children are their oppressors and women rule over them.'\"\n\n\"How do you know this?\"\n\n\"Rumble heard it and told his wife. She told me.\"\n\n\"It's very interesting.\"\n\n\"It did not worry me. I know that kind of Calvinism.\"\n\n\"Are there many people in the parish who felt as you did about\u2014er, ritual and so on?\"\n\n\"You really do put things in an odd way. If by 'ritual and so on 'you mean the Car-tholic religion, oh yes, there are a number of good folk who want to see dear Father Waddell take a firm line. The Miller-Wrights, for instance, are sound enough on vestments though they buck at a Sanctus bell. The Wilmingtons go all the way, bless them, though Moira Wilmington confided in me once that she always gets stuck on Eternal Damnation. The Skiptons we are educating nicely\u2014they've got as far as Sung Eucharist and I _think_ were coming round to auricular confession.\"\n\n\"But the working people?\"\n\n\"My old Mrs Rumble is a great comfort. She doesn't know what it's all about but she wants things brightened up, she says. Then have you heard of a woman known as Flo?\"\n\n\"Have I _not!_ \"\n\n\"A teeny bit of a Mary Magdalene, I gather. Not very particular in her morals. But she has the right idea. She used to go to St Christopher's Hoxton in the great days of Father Wemyss-Buchan. She's been well taught.\"\n\n\"You mean she goes to church?\"\n\n\"Well, not very often. I _did_ persuade her to attend our Midnight Mass last Christmas. (There was endless trouble with the Griggses about holding that and Father Waddell got over it by promising them a Watch Night Service on New Year's Eve.) But I can't pretend poor Flo is a regular communicant. I don't think she's a really bad woman. Just hopelessly compliant. I blame the men of the village.\"\n\n\"She seems an amiable soul.\"\n\n\"There were one or two others with the right ideas. And several more who would have been all right if the Griggses had left them alone.\"\n\n\"Tell me, what about the other clergyman? Mr Slipper?\"\n\n\"Father Slipper? Oh, he was all right. Good little chap. Did as he was told. Left all decisions to Father Waddell.\"\n\n\"Was he\u2014er, High Church or Low? If those are the correct terms.\"\n\n\"Your education has been neglected, Mr Deene. We don't speak of High Church. That goes back to Victorian days. As I have told you, we are Car-tholics.\"\n\n\"Is that what is meant when people speak of Anglo-Catholic?\"\n\n\"I suppose so but I don't like the term.\"\n\n\"And Mr Slipper?\"\n\n\"All right on most things. All the Sacraments except Extreme Unction. Celibacy of the Clergy, the Assumption, the Immaculate Conception, all sound. A bit shaky on Reservation and Benediction. But his heart's in the right place.\"\n\n\"He organizes things for the youth of the parish?\"\n\n\"Wonderful with boys. Open air, you know. Healthy. Scouting and cycling. Organizes camps in the summer. He has persuaded old Sir Marriott Gibson to let them use the swimming pool in his grounds.\"\n\n\"And in the winter?\"\n\n\"Oh, he has his Club and Scout headquarters. Always arranging something. A play or physical culture. Weight-lifting for the older ones. I hear they're entering a team for some competition. They all walk about as though they couldn't forget their shoulders. Waygooze, our organist, gets quite fed up with them flexing their muscles when they ought to be learning the _Kyrie Eleison_. But there's no doubt Father Slipper does a lot of good.\"\n\n\"Did Miss Griggs recognize that?\"\n\n\"I think so. She gave him a subscription whenever he asked for it. It was she who bought new bell tents for them last summer. I gather she has left money in her will both to Father Slipper and his pet causes.\"\n\nGrazia was gathering together the New Hall tea-service and putting it on the old Sheffield plate tray, with a jolly tinkle of beads and bangles.\n\n\"Of course from my point of view it's all very well, this youth organization, but I can't help feeling that a priest should be a priest and not a physical training instructor or expert on cooking over a fire in the open. I should like to see more catechism and less camp for the boys. But that's no doubt my old-fashioned point of view. I don't say Father Slipper doesn't get many of them to church but if they have to be induced to sing together in the choir by being allowed to sleep in tents, it doesn't seem to me to be putting first things first.\"\n\n\"I suppose not.\"\n\n\"There's a great deal of good in it, no doubt. But you know in a small place like this where there is sufficient labour, I'm _not_ convinced that their bob-a-job scheme is so good. The boys hang round the cottages willing to lend a hand but most of them have very little to offer. It seems to me that Father Slipper is groping after something but never seems to find what he wants. However you don't need to hear my views. It's facts you're after. What can I tell you?\"\n\n\"Mr Waddell tells me that he called to see you on the evening Miss Griggs died and that you were out.\"\n\n\"The silly man! I was nothing of the sort! I may have been having a little snooze\u2014I often do about that time. In fact now I come to think of it I remember waking up and finding I hadn't yet put on the lights and the tea things were still out.\"\n\n\"What time was that?\"\n\n\"It must have been nearly seven. Shocking, wasn't it? Sloth, one of the Seven Deadly Sins. If Father Waddell had only pulled at my old ship's bell instead of just pressing the electric one he would have awakened me.\"\n\n\"Did you go out at all that day?\"\n\n\"To early Mass, yes. Father Slipper said it that morning. We've managed to persuade Father Waddell to have a daily Mass though he has to call it Communion. The Griggs contingent would have a fit if he didn't....\"\n\n\"But later in the day?\"\n\n\"Let's see. I don't think I did. It was cold, I remember.\"\n\n\"Not in the afternoon or evening, anyhow?\"\n\n\"I'm sure I didn't. I always do the flowers on Tuesdays and Saturdays at the church. No. I stayed in that day. Like a dormouse.\"\n\n\"You heard or saw nothing which might be helpful?\"\n\n\"Nothing, I'm afraid. My good Mrs Rumble told me her husband was digging a grave for Chilling, I remember.\"\n\n\"You didn't enter the church?\"\n\n\"No, Mr Deene.\"\n\n\"I don't think there's anything else I need ask you, Miss Vaillant. Unless you care to throw any light on one of the small mysteries\u2014that of your reconciliation with Millicent Griggs. It does puzzle me that after years of antagonism she should have come here twice in a week.\"\n\n\"It puzzled me,\" said Grazia Vaillant immediately. \"But I've told you all about it.\"\n\n\"I haven't yet met Miss Flora Griggs. Do you think she shared in her sister's kindlier feelings?\"\n\nI'm not sure that Millicent's feelings were kindlier. If they were, Flora certainly did not share in them. She has a sort of Old Testament hatred for poor me.\"\n\nCarolus's eyes went back to that landscape\u2014the only beautiful thing in the room.\n\n\"Ah\u2014you're looking at my Constable,\" said Grazia. \"Fine, isn't it? \"She threw out her hand. \"Good-bye!\" she said.\n\nCarolus said good-bye with some relief and left Grazia Vaillant among her antiques.\n\nIt was still raining and a dark night but the ship's light over the door had been switched on and he could go quickly down the crazy pavement path to his car.\n\nHe started the engine, but when he switched on the lights he saw someone hurrying towards him, gesticulating to indicate that he should wait.\n\nThere came into his head absurd things like Mrs Stick's warning\u2014' you oughtn't to be hanging about after dark, either. If they can do for an old lady they can do for you'. And Commander Fyfe's questions about people 'hanging about'.\n\nWhen he recognized the approaching figure he remembered also Fyfe's description of him as 'a dangerous character, lawless, violent'. For the man who had stopped him was Mugger.\n\n# 9\n\nMUGGER had a thin insinuating voice. It might have been that in which Brer Fox addressed Brer Rabbit. He brought his long solemn face, with its ginger hair visible under his cap, to the window of Carolus's car, and Carolus opened this by a few inches. The rain was pelting down on him but seemed to have no effect as though his very skin were rainproof.\n\n\"I want to speak to you,\" he said.\n\n\"You'd better get into the car,\" Carolus told him and the long thin man twined in, scarcely opening the door. There was a silence.\n\n\"It was Rumble told me about you,\" said Mugger at last, the tone of his voice not changing. \"He said it would be all right if I told you.\"\n\n\"What's that?\"\n\n\"Something,\" said Mugger promptly and flatly.\n\nCarolus with his usual patience, waited.\n\n\"You're not a copper, are you?\" said Mugger.\n\n\"No.\"\n\n\"You wouldn't say anything?\"\n\nCarolus was greatly tempted to give a promise. But he had to come out with the old prim line which sounded so odd, spoken here in half-darkness in the rain-washed car.\n\n\"It depends on what you tell me. I'll only promise to respect your confidence as far as I honestly can. You see, you might tell me something which would have a direct bearing on the recent murder case. What could I do then?\"\n\n\"I want to show you something.\"\n\n\"Even then, how do I know that it won't be my duty to report it?\"\n\n\"I don't know what to do,\" said Mugger. \"And I'm not a man not to know what to do. I saw you talking to Slatt the other night. Did he say anything about me?\"\n\n\"No.\"\n\n\"I give _him_ a hard time. See, I don't say I'm an angel.\"\n\n\"No.\"\n\n\"Nor yet a psalm-singing hypocrite. I like a bit of fun,\" said Mugger lugubriously.\n\n\"What sort of fun?\" asked Carolus, falling headlong into the trap.\n\n\"Well, not too old. Nor yet so young it'll get you into trouble. About twenty-five or thirty with a nice big chassis who don't open her mouth too wide.\"\n\n\"That wasn't exactly what I asked. However, do you find your bit of fun in Gladhurst?\"\n\nMugger looked gloomier.\n\n\"Of course I do. It's everywhere, if you know how to look for it. There's one working out at Ryley's farm....\"\n\n\"Don't let's go into details.\"\n\n\"Well, you asked me. I mean, where should we be without it? You can't have all work and no play, can you? I remember one lived in Church Cottages. You've never seen anyone like it. It was as though she was on fire....\"\n\n\"You had something to tell me, I believe?\"\n\n\"I was telling you about this one, lived in Church Cottages. I had to tell her in the end she'd get me into trouble if my old woman got to hear of anything. Then there was a German one came to work for some people here. Oh dear, oh dear. I shall never forget it. She was tall as I was, very near, but big-made with it. There _was_ a lot of her. I used to say to her, there _is_ a lot of you', I used to say, but of course she never understood a word of English. Then when I was rabbiting one night....\"\n\n\"Look, Mugger, I'm sure your reminiscences are very interesting. You ought to write a book some day. But I'm trying to find out about Miss Griggs, not about your various adventures. If you've got anything to tell me, let's have it. If not I'll drop you off where you like.\"\n\n\"Well, I have got something to tell you but I don't know how you'll take it.\"\n\n\"I'm afraid I can't give you any assurance on that. I certainly shan't talk for the sake of talking.\"\n\n\"I don't know what to say. Suppose I was to hint to you, no more than a hint, mind, that I might know where a bit of jewellery was to be seen?\"\n\n\"You mean? Oh, I see. Good heavens, man, you can't keep that. You'll find yourself charged with murder if you're found in possession of it.\"\n\n\"That's what I'm afraid of. But I'm not in possession of it. I've left it where I found it.\"\n\n\"What about the money?\"\n\n\"There was no money,\" said Mugger and for the first time a touch of animation was in his voice. \"No money, there wasn't. Not a sausage. Jools. No money.\"\n\n\"Have you done any work since helping Rumble to dig Chilling's grave?\"\n\n\"Work? No. To tell the truth I haven't had time. There's one come to live with her auntie just near the station. Just right, she is. You know, not too thin and not a big sack of potatoes either. Only thing is she won't come out of doors. Says it's too cold. So I have to wait till her auntie's out....\"\n\nMugger's long face expressed nothing but gloom.\n\n\"You haven't worked for a couple of weeks yet you don't seem short of money.\"\n\n\"Oh well. Got to have a bit of luck sometime. But those jools are just as I found them. Something told me not to touch them. But I want a bit out of them, mind. I'm entitled to that, for finding them.\"\n\n\"That will be up to the family.\"\n\n\"Oh. But I haven't said where they are yet, have I? If I'm not going to get anything out of it they can go on searching. Then they'll probably never know who did for the old girl.\"\n\n\"Don't you think you're being rather rash?\"\n\n\"Always have been,\" said Mugger sadly. \"It's a miracle my old woman's never tumbled anything. There was one lived two doors from us, getting on a bit, she was, but still what you might call all right. I used to see her....\"\n\n\"Yes, yes. I'm sure you've been most successful. But being found out by your wife is rather different from a charge by the police.\"\n\n\"I don't worry about police. Never have. Take Slatt, for instance. He's been after me for years.\"\n\n\"But, Mugger, this is what is called a murder case.\"\n\n\"Still, I didn't do it, did I?\"\n\n\"I don't know. But the police will regard your keeping the stolen jewellery hidden as at least half a case against you.\"\n\n\"I can prove I had nothing to do with it. Soon as I'd finished with Rumble that afternoon I went home to tea. My old woman can vouch for that. Then, naming no names, I had to meet someone just about the time when Slatt seems to think old Miss Griggs was done for. Just after dark, that was.\"\n\n\"I daresay you have an alibi of sorts. But not reporting your find amounts to being an accessory after the fact, or something of the sort.\"\n\n\"So you think I ought to just hand it over and perhaps never get a thank you?\"\n\n\"I'm afraid so. After all there was the money, wasn't there?\"\n\n\"What money? I told you there was no money.\"\n\n\"I know you did.\"\n\n\"Calling me a liar, are you?\"\n\n\"Of course I am. Now, look, Mugger, don't be silly. You've had your reward for finding it and as long as the bank hasn't got the numbers of the notes....\"\n\n\"Do _you_ think they might have?\"\n\n\"Not if they were in one-pound notes. Fivers and above they probably will have taken.\"\n\nMugger's lugubrious voice seemed to rise to a more cheerful pitch.\n\n\"Anyhow, there wasn't any money. Want to see the jools? Then you'll know what's best to be done. Though I'm not a man to ask anyone that, generally. I'm a quiet man and know how to go about my business my own way.\"\n\n\"Where are they?\" asked Carolus.\n\nThere was a long sigh from the lugubrious Mugger.\n\n\"I suppose I shall have to tell you and chance it,\" he said.\n\nHe sat there immobile and melancholy. In the light of the dashboard Carolus could see the lantern jaws and the red hair, the expression of set inviolable solemnity.\n\n\"Do you want a bit of fun?\" asked Mugger, his voice unchanged. \"Because I know where one's just come to live....\"\n\n\"Are you going to take me to where this jewellery is hidden or not?\" asked Carolus in exasperation.\n\n\"All right, all right,\" moaned Mugger. \"I was only asking you if you wanted a bit of fun. No need to have it if you don't want it. Only you won't often see one like this....\"\n\nCarolus made to start the car.\n\n\"You don't need no car,\" said Mugger. \"Leave it here and come with me.\"\n\nCarolus turned up the collar of his overcoat and prepared to follow. The rain persisted although a light wind had arisen.\n\n\"Got a torch?\" asked Mugger.\n\nCarolus pulled one from the pocket of the car.\n\nMugger led the way across the open space in front of the church and it was obvious that he was making for the lych-gate. In the shelter of this they waited for a moment.\n\n\"This is a handy place,\" confided Mugger, \"if you've got one with you on a wet night. No one's going to disturb you here. They keep away from churchyards after dark. I remember...\"\n\n\"Come along,\" said Carolus.\n\nMugger led the way by a path which passed the West door and went round by the South side of the church. They were rather more sheltered here. Carolus could just make out the shapes of gravestones as his eyes became accustomed to the darkness. Finally Mugger opened a door rather below ground level and they entered what was evidently the furnace-room mentioned by Rumble as the place where his tools were kept. It was cold enough tonight to be an ice-store. Carolus threw the light of his torch in every direction.\n\n\"See, this is where Rumble keeps his spades for when he has a grave to dig. But he only just comes in and out again except when he gets the furnace going. He never goes up those steps.\"\n\nCarolus saw that against the wall was an iron ladder, leading, evidently, to some kind of loft. He followed Mugger up these and found himself under a sloping roof too low to allow him to stand upright.\n\n\"I've used this for years,\" said Mugger. \"Keep my rabbiting wires and gun and cartridges here besides one or two other things I use when I go out at night. I don't suppose you'd know how to collect half a dozen pheasants in an evening, would you? Besides, I need somewhere to put anything I find. Old Slatt would have given his right hand to know of this which is one reason I don't like bringing the coppers here. Of course I've moved all my stuff now but the new place isn't nearly so good. Now look at this.\"\n\nNear the top of the steps was an old toolbox which Mugger opened and Carolus at once saw what Mugger described as 'the jools'. There was a collection such as a rich and ostentatious elderly woman might have worn in the daytime, several bracelets, four rings, a large diamond brooch and a necklet.\n\n\"When did you find these?\"\n\n\"On the night after the murder.\"\n\n\"And the money?\"\n\nMugger answered as though Carolus had been tactless.\n\n\"I've told you there was no money,\" he said. \"Anything I found was all together in this box. Whoever it was must have known Rumble kept his spade here and come to get it to throw in enough soil to cover her. When he brought the spade back he was wondering where to get rid of the jools till things had blown over a bit and seen those steps. I don't suppose it took him a minute to nip up and pop them in here.\"\n\n\"You haven't touched these?\" asked Carolus.\n\n\"I know enough for that,\" said Mugger. \"If I'd been going to take them I'd have taken them. If not there was no call to leave my fingerprints all over them.\"\n\n\"Very wise.\"\n\n\"I suppose there's a decent bit of money in that lot, isn't there?\" asked Mugger.\n\n\"I imagine so. I'm not an expert. Now I have to drive back through Burley. Would you like me to drop you off at the police station there?\"\n\n\"Whatever for?\"\n\n\"You're going to report this, aren't you?\"\n\n\"I suppose I've got to if you say so. Only I'm not going over to Burley for that. Slatt'll do for me.\"\n\nSo Mugger once again settled into Carolus's car and they drove to the cottage distinguished from the rest by a sign outside bearing the words COUNTY POLICE. The door was opened by a thin woman, presumably the policeman's wife.\n\n\"I suppose you'd better come in as it's raining,\" she said.\n\nSlatt entered the room where they waited, his jaws still working.\n\n\"I was just having my tea,\" he announced unnecessarily.\n\nAn extraordinary expression, half ghoulish, half mischievous, came over Mugger's face and Carolus supposed that he was smiling.\n\n\"What do you want, Mugger?\" asked Slatt sternly. \"And you're the man asking me questions the other night. What is it you both want?\"\n\nMugger was not to be hurried.\n\n\"Got one for you this time, Slatt,\" he said.\n\n\"Have you something to report?\" asked the policeman grandly.\n\n\"Got right ahead of you this time, Slatt. Left you a long way behind.\"\n\nCarolus could not resist quoting _Hudibras_.\n\n\"Quoth Hudibras, Friend Ralph, thou hast\n\nOutrun the constable at last.\"\n\nSlatt turned on him furiously.\n\n\"Police officer!\" he shouted.\n\n\"I beg your pardon,\" said Carolus.\n\n\"Quoth Hudibras, Friend Ralph, thou hast\n\nOutrun the Police Officer at last\"\n\n\"What's this all about?\"\n\n\"How would you like to know where Miss Griggs's jools are?\" asked Mugger.\n\n\"Are you trying to be funny?\"\n\n\"Yes,\" admitted Mugger. \"But I know where the jools are, all the same.\"\n\n\"Then it's your duty to report it to me.\"\n\n\"What do you think I've come here for on a pouring night like this when I might be in the Black Horse? Of course I'm going to report it. Get your notebook out and I'll tell you.\"\n\nLike a man mesmerized, Slatt obeyed.\n\n\"I hope this isn't one of your larks,\" he said.\n\n\"This gentleman will tell you. He's just seen them. It was him told me to come to you. Otherwise I might have kept the information to myself.\"\n\nSlatt had his notebook open.\n\n\"When did you make this discovery?\" he asked.\n\n\"This afternoon. About three o'clock.\"\n\nSlatt's pencil worked.\n\n\"Where?\"\n\n\"I'm going to take you there in a minute. That's better than saying.\"\n\nSlatt nodded and wrote.\n\n\"What was there so far as you remember?\"\n\nMugger gave details which Slatt noted.\n\n\"Miss Griggs had a sum of money on her,\" said Slatt.\n\n\"There was no money there. Jools, no money.\"\n\nSlatt said nothing.\n\n\"You'll both have to wait a minute while I get on the telephone to the Detective Inspector in charge of this case.\"\n\nHe left them together and returned with a smile.\n\n\"The D.I. is coming over at once. He'll want you to take him to the place you found this, Mugger.\"\n\n\"I don't mind,\" said Mugger. \"Even if I have to miss one who's waiting for me in the bus stop shelter. Nothing special, as you might say, but not one of your little bony ones. Something you can get hold of...\"\n\n\"Now Mugger,\" warned Slatt.\n\n\"Well, I'm doing you a favour, aren't I?\"\n\n\"So you may be but there's no need for Talk. I've got a position to keep up.\"\n\n\"You remember Dogberry?\" said Carolus. \"Dogberry in _Much Ado About Nothing._ When Seacoal had just been promoted. _'You are thought here to be the most senseless and fit man to be constable of the Watch'.\"_\n\n\"How many times have I got to tell you? Police Officer, it is.\"\n\n\"Of course. Of course. _'You are thought here to be the most senseless and fit man to be Police Officer of the Watch'.\"_\n\n# 10\n\nAT least, thought Carolus, it would give him a chance to meet the CID man in charge of the case, usually a somewhat difficult matter. It was not that professional policemen resented amateurs\u2014they simply did not recognize their existence unless the amateur, by some tactless piece of intrusion, forced them to warn him off matters which did not concern him.\n\nIn three of his cases Carolus had been lucky, for his friend John Moore had been in charge of investigations. In one case he had been forced almost to blackmail an obstinate Detective Sergeant into discovering the truth. In two other cases the police had been sceptical but not obstructive. Now he had a feeling that he would be up against open hostility. Where the police were most baffled they were apt to be irritated by the curiosity of outsiders. An incident like this, a report to them of the whereabouts of Millicent Griggs's jewellery, so far from earning their gratitude for the assistance rendered, would make them peevish and suspicious.\n\nCarolus was not wrong. The Detective Inspector was a small forceful stocky man who sought to take charge of the situation at once. He eyed Carolus and Mugger impatiently.\n\n\"Now which of you came on this jewellery?\" he asked.\n\nCarolus lit a cigarette and Mugger looked owlish.\n\n\"Come along, come along,\" said Detective Inspector Champer. \"We haven't got all night. Which was it?\"\n\nSlatt tried to intervene.\n\n\"It was Mugger who...\" he began.\n\n\"I'm not asking you,\" snapped Champer. \"Which is Mugger of you two?\"\n\nMugger, without changing his expression, began to speak.\n\n\"I don't know whether you've got indigestion,\" he said sadly. \"If so you'd better wait till you're in a proper state to talk to people. I'm going home.\"\n\nChamper turned to Slatt.\n\n\"I thought you said one of these men wanted to show us where he is supposed to have found these jewels?\"\n\n\"So he did. He told me he'd take me. Mugger it was,\" said Slatt anxiously.\n\n\"Is your name Mugger?\" said the Detective Inspector in a more gentle tone.\n\n\"Yes,\" said Mugger and was silent.\n\n\"Did you find the jewels?\"\n\n\"Yes.\"\n\nChamper turned to Carolus.\n\n\"What have you got to do with it?\"\n\nCarolus ignored this but again Slatt broke in, this time more successfully.\n\n\"It was this gentleman persuaded Mugger to give his information,\" he explained.\n\nChamper stared at Carolus with open hostility.\n\n\"Your name Deene?\" he said, and Carolus nodded. \"I was told you were hanging round making a nuisance of yourself here. Let me warn you at once, Mr Deene, that I don't like outsiders nosing into any case I'm investigating and what's more I won't have it. You make one move which can be considered obstructing the police and I'll have you in Court at once. You may have found some police officers who have put up with your inquisitiveness\u2014I'm not one of them.\"\n\n\"I've told him,\" put in Slatt.\n\n\"Leave this to me, will you, Slatt? I hope we understand one another, Mr Deene?\"\n\nCarolus looked at the little man steadily.\n\n\"Did you know that Dundas Griggs, the old lady's nephew, was in Gladhurst on the afternoon of her death?\" he asked.\n\n\"He... What? Don't try to give _me_ information. You may have found other senior police officers who thought your statements of some consequence. You won't find that with me. It's scarcely likely that you could have gathered facts we're unaware of. Or do you think you know who killed Millicent Griggs?\"\n\n\"No. But I think I know _how_ she was killed which in this case is all-important. Have a cigarette?\"\n\n\"What's this about the nephew?\" asked Champer suddenly and sulkily.\n\n\"Oh nothing. He lives in Burley. Inherits from Millicent Griggs. Was over here in a car on the afternoon of the murder. That's all I know.\"\n\nChamper grunted.\n\n\"Trying, isn't it?\" said Carolus. \"All this vagueness. I could almost wish another murder would come along and give a lead to the first.\"\n\nChamper visibly started.\n\n\"Another murder?\" he said. \"What makes you think there's going to be another murder?\"\n\n\"Oh, not necessarily,\" said Carolus. \"I only thought it might make things coalesce.\"\n\n\"I should have thought that if ever there was a case in which not to expect a second murder it was this one. It's a brutal crime with an obvious motive. It's done and I can't see it being repeated.\"\n\n\"You evidently take the view that Millicent Griggs was killed for the sake of immediate robbery?\"\n\n\"Don't you, for heaven's sake?\"\n\n\"No. I don't. And certainly not in as simple a form as that.\"\n\nChamper stood up.\n\n\"I've no business to be discussing this case with you, Mr Deene. It's against all my principles.\" Carolus rose too.\n\n\"I'll leave you,\" he said. \"You go and see Mugger's 'jools'. It's quite time I had a drink.\"\n\nSturdy little Champer had evidently had enough of this too.\n\n\"Good-night, Mr Deene,\" he said without a smile.\n\n\"Good-night, Inspector,\" returned Carolus, then added infuriatingly\u2014\" Odd about those galoshes, wasn't it?\"\n\nChamper flushed with irritation.\n\n\"What galoshes?\" he almost shouted.\n\n\"Didn't Mrs Rumble report them after all? Or perhaps your man here didn't think it worth mentioning? I was interested anyway.\"\n\nChamper's face promised nothing very pleasant for Slatt as Carolus finally escaped into the still persistent rain.\n\nHe made straight for the Black Horse and found he was the only customer. George Larkin served him with the large Scotch he ordered and silence fell on the public bar.\n\nFor once Carolus was hopelessly at a loss. It was not only the natural surliness of the innkeeper which made him unapproachable\u2014there was something else, a reserve, a hostility, or was it no more than blind stupidity which enshrouded him? If Carolus had been a policeman enquiring he would doubtless have been given some sort of explanation for Larkin's visit to the church or the churchyard on the fatal afternoon, but as a private customer at the pub how could he even bring up the subject?\n\nSuddenly George Larkin spoke.\n\n\"S'that right you're trying to find out about the murder?\" he asked.\n\nCarolus kept his head and nodded moodily.\n\n\"Think you will?\"\n\n\"I don't know. It's hard going.\"\n\nGeorge Larkin relapsed into silence. Carolus drank his whisky. No one entered and a minute passed.\n\n\"I suppose someone's told you I went up to the churchyard that afternoon?\"\n\nCarolus nodded.\n\n\"Can't mind their own business in this place. Never have been able to.\"\n\n\"Most small places are like that.\"\n\n\"But not as bad as this one.\"\n\nCarolus felt as though he was in a fairy story and if he asked a question he would break the spell.\n\n\"Yes, I went up there. With my son. 'Bout half-past three it must have been.\"\n\n\"I'll have another Scotch,\" said Carolus.\n\nGeorge Larkin poured it.\n\n\"Wife's grave,\" he said unexpectedly. \"Generally go up there about once a month to see it's decently kept.\"\n\n\"I understand.\"\n\n\"My wife died eight years ago,\" continued George Larkin; then added the ugly word: \"Cancer.\"\n\nCarolus could only nod sympathetically.\n\n\"She suffered a lot. She's buried not far from where Chilling's been laid.\"\n\n\"You noticed the open grave then?\"\n\n\"Yes. Rumble had just finished it.\"\n\n\"You didn't see anyone about except Rumble and Mugger?\"\n\n\"No. We didn't. We weren't there five minutes.\"\n\n\"Came straight back?\"\n\n\"Yes. It's a funny thing we happened to go on that afternoon. But there you are.\"\n\n\"Have a drink?\" suggested Carolus.\n\n\"Thanks. I'll have a bitter. I suppose someone will try and make mischief about us being up there at the time.\"\n\nAt last the door opened and in a moment the public bar was cheered by the bouncing, laughing, talkative personality of the 'woman known as Flo'.\n\n\"Well,\" she said. \"You do look a gloomy pair to be sure, standing here with no one in. I'll have a Milk Stout, Mr Larkin. I hear the police have found the jewellery taken from Miss Griggs.\"\n\nGeorge Larkin stared at her in a stupefied manner.\n\n\"Where?\" he asked.\n\n\"I don't know where but they've found it. Just shows doesn't it? You _do_ look a solemn pair and no mistake. You make me feel I want to do Knees Up Mother Brown to wake you up. What a life! Never stops raining, does it? I got soaked coming up Church Lane.\" She turned to Carolus. \"I've seen you before, haven't I? I thought so. Once seen never forgotten. You're not always such a dismal jimmy, are you? I can see I shall have to start cheering you up. They call me a ray of sunshine but you wouldn't say so if you could see me the morning after! Thank you, I don't mind if I do have another. Do you know the story about the man who went to the doctor....\"\n\n\"Now, Flo,\" warned George Larkin.\n\n\"It's all right. It's not _that_ one. I wouldn't tell that one to a stranger. Well, not quite a stranger but you know what I mean. Cheerio! Is that your car outside? You can give me a ride in that one of these days only don't do like old Mr Murdoch did that time and take me right out somewhere then say: 'If you don't, you'll walk back! 'Dirty trick, wasn't it? But I didn't mind. He wasn't a bad old fellow. I've known worse anyway. Here, did I ever tell you about that farmer over at Breadley? Proper devil, he was. I began to wonder if I should get out of there alive. Like Bluebeard, really. I thought my last hour had come. But in the end it was nothing. All talk. You do get them, don't you?\"\n\n\"You ask for it,\" said George Larkin darkly.\n\n\"Oh, I didn't mind. As long as I don't get banged on the head like poor old Miss Griggs. Wicked thing, that was. I never liked her, the old hypocrite, but I don't want to see anyone banged on the head for no fault of their own. I must have missed her by minutes, too. I was in Church Lane that afternoon just after dark. Never mind who with. I was on my way home.\"\n\n\"That's not your way home.\"\n\n\"You know what I mean,\" said Flo. \"I wasn't there above a quarter of an hour. But I never heard anything. Or saw anything. Well, I wasn't thinking of anything like that. It was a nice dark night, not raining like it is now. Oh well, she's Gone, so it's no use our moaning about it. Let's have another Milk Stout and give the gentleman a whisky. A little of what you fancy does you good. I don't believe you're such a Sunday School as you look, if the truth were known. I've seen that quiet kind before. I wouldn't trust you far away from home. Not that I've ever been one to push myself. But You Can Tell, can't you? Oh well, anything for a laugh.\"\n\n\"You should really form an understanding with Mugger. He tells me, somewhat graphically, that he 'likes a bit of fun '.\"\n\n\"Fred Mugger? An understanding? Me and old Fred Mugger have been friends for years. You don't want to get taken in by that gloomy face of his. He's a proper lack-in-the-Box when he gets going. As a matter of fact it was Fred Mugger I was meeting that evening I told you about in Church Lane. What's wrong with that? Even if we did know one another at school. Nice sort of monkey he was then, too. There used to be a shed at the back where they kept wood and that and young Fred, he can't have been more than fourteen years old at the time, got hold of me in there and before I knew where I was, there you were. Yes, it was him I was meeting that night in Church Lane and I remember how angry he was when someone switched his motor-bike headlight on us from the top of the lane when we least expected it. I laughed, of course, but you should have seen his face!\"\n\n\"I thought you saw no one and heard nothing that night?\"\n\n\"Nor we didn't because as soon as this motor-bike rider had thrown his lights down the lane and seen us, I suppose, he was Off as fast as he could go. Never came into the lane at all so that to this day I never knew who it was. I didn't mind but it quite upset Fred. You might not think it but he's very shy. Not in what he says but over anything like that. I said to him at the time, 'What's it matter?' I said. I could hardly speak for laughing. But he took it serious. 'I don't like anyone to come prying and peeping like that', he said.\"\n\n\"You're sure it was a motor-bike, not one of these other contrivances on the roads?\"\n\n\"Quite sure. You could hear the engine and there was only the one light.\"\n\n\"What time did it appear?\"\n\n\"It was only just dark. Can't have been much more than five. Perhaps a quarter past.\"\n\n\"I see.\"\n\n\"Tell you the truth I'd almost forgotten about it till you brought it up. It wasn't the first time I've been caught when I shouldn't. Good thing it was old Fred Mugger and no one else.\"\n\n\"Why?\"\n\n\"Oh, everyone knows Fred. Always been the same ever since he was at school. No one takes any notice. But it might have been someone who'd have caused Talk. There's quite enough of that already.\"\n\n\"Someone like Commander Fyfe, for instance?\" suggested Carolus.\n\nThe jolly loud laugh of Flo filled the room.\n\n\"What a thing to say!\" she said. \"I don't know what put such an idea into your head! 'Pon my word, you have got a wicked mind! I haven't seen him for a week, anyway. He never comes in here. Says he can't leave his wife at night. Oh well, I shall have many a laugh thinking of what you said. I knew there was more than meets the eye about you. I knew that as soon as ever I saw you. Another Milk Stout? Yes, I don't mind.\"\n\n\"You said just now you didn't like Millicent Griggs. I wonder why.\"\n\n\"Because she had a nasty, dirty mind,\" said Flo with sudden feeling. \"I know it if no else did. There was nothing she liked better than getting me to herself and asking questions.\"\n\n\"What sort of questions?\"\n\n\"Nasty. It would turn you up if I was to tell you. What this man had done and what that one had said. Quite creepy it used to make me feel. I mean, I'm not one for the straight and narrow, but I do bar talking about it in that horrible way. And she was the same with any woman she thought could tell her anything.\"\n\n# 11\n\nTHERE remained for Carolus only two more interviews in Gladhurst which he regarded, on such facts as he now had, as being of first-rate importance. By a happy chance he was able to kill the two birds with one stone for when he arrived at Crossways to see Miss Flora Griggs he found that Grey was working there that week.\n\nMrs Bobbin received him in a small sitting-room off the hall, and it seemed that her ire was in no way assuaged by news of the recovery of her sister's jewellery.\n\n\"All the jewels on which we could have claimed insurance in any case, and none of the money, on which we can claim nothing. I haven't an idea how much Millicent had with her but it was, as I've told you, usually a large sum. It really is infuriating. I suppose you are no nearer to finding out who was responsible?\"\n\n\"Not much, I'm afraid. I'm beginning to form a vague idea of how and when and where it may have happened.\"\n\n\"You sound very guarded. I hear you got round that wife of Rumble's. That was an achievement. She's a termagant.\"\n\n\"I found her most obliging. By the way, were your sister's galoshes a brown pair, nearly new, size eight, made by Skilley and Harman?\"\n\n\"That's right. Have you recovered them, too?\"\n\n\"The police have them, I think.\"\n\n\"Oh. You've also seen Grazia Vaillant. How did you get on with her?\"\n\n\"She told me a great deal about her religious views, and other things.\"\n\n\"I suppose butter wouldn't melt in her mouth?\"\n\n\"She was certainly quite amicably disposed.\"\n\n\"I'm glad. She can be a virago when she's angry.\"\n\n\"That seems to be a common quality among the ladies of Gladhurst.\"\n\n\"And why not? When a woman reaches my age, Mr Deene, she is entitled to her wrath.\"\n\n\"Oh certainly. I'm very anxious to meet Miss Flora Griggs, if it is possible.\"\n\n\"I don't see how she can help you. As I told you she left on the 2.40 bus for Burley. It has come to our knowledge that the police have obtained confirmation of that from the conductor and passengers. Also of the hour of her return. But if you want to meet yet another irate old lady I will introduce you. I ought perhaps to say that Flora has never been intellectually very strong and that the shock of Millicent's death has unsettled her even more. I'll leave you alone with her.\"\n\nFlora Griggs, when she came into the room, certainly looked wrathful. She was a large woman and had none of Mrs Bobbin's distinguished appearance, but a large-featured face which was lit only by fierce little eyes. She wore unfashionable heavy clothes.\n\nShe greeted Carolus civilly enough however, and said she was prepared to tell him anything she could to help him.\n\n\"For the murderer must be punished, Mr Deene. 'Whosoever slayeth Cain, vengeance shall be taken on him sevenfold'. Genesis IV, 15,\" she added in an oracular voice.\n\n\"Cain? Why Cain? \"Carolus could not resist asking.\n\n\"He was Under Protection,\" explained Flora Griggs.\n\n\"You have no suspicion of your own as to who it might be?\"\n\n\"I am convinced that it can only be one person. Now that we know it was not for 'riches and precious jewels', Second Chronicles XX, 25, we can see the motive as one of hatred. There was only one person who hated Millicent 'with a cruel hatred', Psalm XXV, 19, that is one who, we know too well, is 'wholly given to idolatry', Acts XVII, 16, and plots with papists and conspirators.\"\n\n\"You surely don't mean Miss Vaillant?\"\n\n\"I mean no other.\"\n\n\"But really, Miss Griggs, I understand that there had been some ill-feeling between these two elderly ladies and that they represented different extremes in a religious controversy. I cannot see how you can suppose that one of them battered the other to death.\"\n\n\"If not with her own hand then by the hand of another.\"\n\n\"But there had been a reconciliation between them.\"\n\n\"I am not deceived by that. 'Grievous words stir up anger', Proverbs XV, 1, and there had been too many such words in the past years for their anger to abate in a few days.\"\n\n\"Then why do you think your sister went twice to see Miss Vaillant?\"\n\n\"She did not confide in me but she told me on her return from her visit that we should not for ever suffer, that we should 'be able to stand against the wiles of the devil', Ephesians VI, 11. It did not seem that she had yet forgiven and I do not believe she had.\"\n\nCarolus decided to discourage this rather fruitless rhetoric and turn to more practical matters.\n\n\"You don't mind if I ask you a few questions about the day of your sister's death, Miss Griggs?\"\n\n\"I will answer them gladly if it will help you.\"\n\n\"You left the house about half-past two?\"\n\n\"A little before. I had to catch the 2.40 bus from the village.\"\n\n\"When you left, your sister was resting?\"\n\n\"Yes. In this room.\"\n\n\"On that very settee, perhaps?\"\n\n\"Yes. But she did not stay there.\"\n\n\"You mean?\"\n\n\"She must have gone up to her room to lie down later.\"\n\n\"What makes you say that?\"\n\n\"Because when I returned and she was missing I looked into that room. The bed had been slept in, or at least laid on.\"\n\n\"Really? Perhaps Naomi had forgotten to make it up that day.\"\n\n\"No. I always help her with the beds. We did my sister's room together.\"\n\n\"Yet when you came back?\"\n\n\"It was turned back. And something else. I haven't mentioned this to my married sister because I knew it would anger her and there was quite enough to be upset about as it was. But one of the sheets was missing.\"\n\n\"Are you quite sure?\"\n\n\"Oh quite. It had been pulled from the bed. My sister Mrs Bobbin, as perhaps you've noticed, is somewhat fidgety about details of housekeeping, expenditure and so on. I think she resented my sister Millicent's larger income. She would have been very upset by a missing sheet so I have said nothing about it.\"\n\n\"Did Naomi know of this?\"\n\n\"I think not. I put the bed cover back as though nothing had happened.\"\n\n\"Thank you, Miss Griggs. That is most valuable information.\"\n\n\"I am glad to tell you anything which will bring the murderer to justice. His 'hands are defiled with blood', Isaiah LIX, 3.\"\n\n\"Now when you left the house that afternoon what was the girl Naomi Chester doing?\"\n\n\"She was just finishing the washing-up from lunch.\"\n\n\"Did you happen to notice whether she had cleaned the hall that day?\"\n\nFlora Griggs made an effort to remember.\n\n\"I suppose so. She always does it first. I cannot actually remember seeing her but I think I should have noticed if she had not.\"\n\n\"You made some purchases in Burley?\"\n\n\"I did. I have given a list of them to the police. They went so far as to question the shopkeepers. Does that mean that they can possibly suppose I had something to do with my sister's death?\"\n\n\"They have to make these routine enquiries. Then, I understand, you went to the cinema?\"\n\nFlora Griggs looked a little confused.\n\n\"My sister Millicent disapproved of all places of entertainment,\" she said, \"so I have felt sorry that at the very time of the tragedy I was in a cinema. But I allowed myself an occasional visit. Especially to historical films. That day I saw one called _William the Conqueror_ with a most realistic reproduction of the Battle of Hastings.\"\n\n\"I see. And you knew of nothing amiss till you reached home and heard from Mrs Bobbin that Miss Griggs was missing?\"\n\n\"That's so.\"\n\n\"There is nothing else I want to ask you, Miss Griggs.\"\n\n\"You must succeed!\" said Flora Griggs passionately. \"'Woe unto the wicked! it shall be ill with him: for the doing of his hands shall be done unto him', Isaiah III, 11.\"\n\nWhen Mrs Bobbin returned she found Carolus alone in the little sitting-room.\n\n\"Have you learned anything from my sister?\"\n\n\"I have. Yes. Something most interesting. Now I want to talk to Grey who is working for you.\"\n\n\"Yes. He has just left the firm he worked for and is starting on his own. This is his first job, I believe. We were pleased to help him. He is a very nice young man.\"\n\n\"So I believe.\"\n\n\"It was Naomi Chester who spoke for him, of course. Such a tragedy that. They would be very happily married. I think she persuaded Laddie Grey to give up his job and start on his own, for he has not much initiative.\"\n\n\"And you're helping them?\" said Carolus, amused at finding a soft spot in the irascible old lady.\n\n\"Actually, there wasn't much that needed doing but Naomi suggested the hall, which can always do with it, and Laddie's apparently a very good painter. He has started on the staircase, which takes a lot of work. Would you like to see him in here?\"\n\n\"Yes, if I may.\"\n\nWhen young Grey came into the room Carolus could see at once why he was so popular. He had that quiet modest charm which is almost bashfulness and appears often in Englishmen who have always worked with their hands. He was not tall but gave an impression of strength behind his gentle eyes and soft speaking voice. There was nothing in the least ingratiating about him but he smiled at Carolus as though he trusted him.\n\n\"I won't sit down while I've got these overalls on,\" he said in reply to a gesture from Carolus.\n\n\"You've lived here all your life, Grey?\"\n\n\"Except for my army service, yes. Dad and Mum are both dead now but Dad had a little shop when he was alive.\"\n\n\"Your wife was from here, too?\"\n\n\"From over at Breadley\u2014that's about four miles away.\"\n\nCarolus waited to see if Laddie Grey would say more and after a moment he continued.\n\n\"I suppose we were too young,\" he said. \"Everyone said so, anyway. But I was twenty and she was eighteen so I can't really see it. You know about her? I can't understand it. Nothing wrong with her parents as far back as you could go. She was just an ordinary girl, just like other girls, until this happened. Suddenly one day she didn't want to get up, couldn't seem able to do anything for herself or the baby and started thinking she was ill and her brains were all shrivelled up. Then she tried to commit suicide and I sent for the doctor. She's in the Institute over at Wilstone and I go and see her from time to time but she never knows me. The doctors say there's no hope and even if she gets better for a time it will come back.\"\n\n\"Brutal thing, Grey. I'm very, very sorry.\"\n\n\"Thanks. There's nothing to be done about it and I honestly try not to think about it any more. I want to start again with Naomi. We're still young enough, after all. Of course, I shall always feel responsible for the other but her parents are very good about it and they're comfortably off and can see she has what little she needs. I want to marry Naomi if it can be done and I've been to see a lawyer about it.\"\n\n\"I quite understand.\"\n\n\"There's the little girl to think of, too. Naomi thinks the world of her. I don't know what you'll think but I'll tell you what Naomi and I have decided. If these lawyers can't fix it so that we can get married we shall just start as though we were married. I've still got my cottage I had before my wife was taken away and we shall just live there and \u2014\u2014\u2014 them, whatever they say. Do you blame us? I mean, what's a church service when they can't give it you and there are three lives to think of? That's what we shall do.\"\n\n\"Mind if I ask you a few questions about the afternoon of the murder?\"\n\nLaddie Grey looked rather crestfallen. He had clearly enjoyed expressing his defiance.\n\n\"All right. If you think it's any good. I don't see how I can help,\" he said.\n\n\"You were working at Commander Fyfe's that afternoon?\"\n\n\"S'right. The dining-room.\"\n\n\"Not much before four o'clock someone asked for you on the telephone?\"\n\n\"So Mrs Fyfe said. When I got to the phone there was no one, so I supposed they'd been cut off. I thought it was the woman who looks after my baby girl.\"\n\n\"You never thought it might be Naomi Chester?\"\n\n\"No. What would she want to ring me for? I was sure it was Mrs Buxton who looks after Estelle. I thought something might be wrong with the little girl so I left my work a bit early and went on round there.\"\n\n\"And was it Mrs Buxton who had phoned?\"\n\n\"No. I saw her and Estelle in the street as I was going there so I knew it was all right.\"\n\n\"You didn't ask if she had phoned?\"\n\n\"Not then, I didn't. Next day or the day after I did and she said certainly she hadn't. So we don't know who it was. I think Mrs Fyfe may have imagined it. She's an invalid and doesn't go out much. She might have got it all wrong.\"\n\n\"What time did you see Naomi Chester?\"\n\n\"I must have got to her place round about four. But I didn't stay long. We were going to the pictures over at Burley. So I went home, got changed and came back for Naomi. That would have been about five. Then off we went to the pictures.\"\n\n\"Yours is one of several motor-bikes in Gladhurst?\"\n\n\"Yes. Must be more than a dozen.\"\n\n\"Have any trouble with it?\"\n\n\"No. Not extra. Now'n again.\"\n\n\"Lights all right?\"\n\n\"Yes. Certainly. I shouldn't take it out at night without.\"\n\n\"On the bike and on the sidecar?\"\n\n\"Of course.\"\n\n\"Did you meet anyone before you left for Burley?\"\n\n\"No. Naomi's mother doesn't get back till late and we didn't see anyone in the village.\"\n\n\"What about in Burley?\"\n\n\"Not that I remember.\"\n\n\"You went straight to the pictures?\"\n\n\"Yes.\"\n\n\"Not even a drink?\"\n\n\"Yes. I expect we had a drink. Usually do.\"\n\n\"Which pub?\"\n\n\"The Station.\"\n\n\"That's the very large pub that's nearly always crowded?\"\n\n\"That's it.\"\n\n\"Rather out of your way for the pictures, wasn't it?\"\n\n\"Bit. We like it.\"\n\n\"Which cinema did you go to?\"\n\n\"The Gaudeon.\"\n\n\"What was showing?\"\n\n_\"William the Conqueror.\"_\n\nThat reply seemed to come very slickly, Carolus thought.\n\n\"Look here, Grey. I've told Naomi and I'll tell you. There's something you're both keeping back. Something one of you has seen or heard. It may not be serious but you should realize how foolish it is in a murder case to hold back even details. Are you trying to protect someone?\"\n\n\"No.\" The reply was low and sullen. \"I've told you all I know.\"\n\n\"Do you know Dundas Griggs by sight? The old ladies nephew?\"\n\n\"No.\"\n\n\"Does Naomi Chester?\"\n\n\"I expect she does. I've never asked her.\"\n\n\"You make it very difficult for me. I'm going to find out the truth about this, Grey.\"\n\n\"Hope you do. I never wished old Miss Griggs any harm. Nor did Naomi.\"\n\n\"All right,\" said Carolus. \"Have it your own way. I shan't ask you any more.\"\n\nHe left the house soon afterwards feeling depressed and frustrated. He decided to leave Gladhurst at once and return to his own house. His suspicions were turning to conviction, though he had very little concrete evidence behind him.\n\n\"The _hell_ of a case,\" he thought. \"All in the air and what little comes down to earth is nasty. And only one way to proceed, press and press for the truth till it takes its ugly shape.\"\n\n\"Really, Sir, you look quite done up,\" said Mrs Stick when he was at last in his own armchair. \"I wish you wouldn't overdo it. We shall have you on our hands ill again if you go on like this. You're burning the candle at both ends, what with flying over to that place and doing your school work as well.\"\n\n\"I'm all right. What are you going to give me to eat, Mrs Stick?\"\n\n\"I've got a nice green pea puray to start with and I'm just going to fry the cruttons for it. Then there's a tiny bit of sole and a fricassay of veal. You'd like a bottle of the Shatto Margo, wouldn't you?\"\n\n\"Wonderful. You're an angel, Mrs Stick.\"\n\n\"I don't know about an angel but if someone didn't look after you I don't know where you'd be. Poking about with corpses all day\u2014it's not natural. I was only saying to Stick....\"\n\n\"You're not an angel if you don't bring my dinner.\"\n\n\"All right. I'll be ten minutes because I won't be hurried when it's taken trouble.\"\n\n\"Where's the wine?\"\n\n\"Here it is, of course. You wanted it shombray, didn't you?\"\n\nCarolus forgot the irate old women and the lying young ones for a blissful hour. Then, when he had lit a cigar, he began to make curious half-decipherable notes.\n\n# 12\n\nBURLEY was an unbeautiful town lying in the midst of a pleasant countryside, like Ashford in Kent or Didcot in Berkshire. It consisted for the most part of streets of medium-sized houses though there were a few more pretentious ones and some slummy little roads of cottages. There was a cattle market and a few uninteresting industries, several Victorian Gothic churches, a fine collection of red-brick chapels, a new town hall and a palatial public lavatory in the town's strategic centre.\n\nCarolus had the address of Dundas Griggs but had been warned that he lived in rooms and was scarcely ever in. When he had asked Mrs Bobbin what was her nephew's occupation she had been rather vague.\n\n\"Dundas? Oh he's always busy with schemes of one sort or another. Knows everyone. Always trying something new. What is called a live wire, I believe, though it doesn't seem to get him anywhere.\"\n\nCarolus went to his address and found it a solid house called Maitland Villa. His enquiry for Mr Griggs led to a flood of information from a woman with a toup\u00e9e and glasses.\n\n\"No, he hasn't been home since he went out this morning. He's often in about this time but you can never tell with him. He's here, there and everywhere. I tell you where you might find him\u2014round at the Oak Caf\u00e9. There's two or three of them often in there together at teatime. If not, you could try Mr Priestley's office. Or Maugham's the tobacconist's, where he gets his cigarettes and sometimes stops for a chat. Of course, you never know, he may have driven out to Burnside where the new estate's going up because he's been popping over there for something or other. But I think you'll find him in the Oak Caf\u00e9.\"\n\n\"Thank you,\" said Carolus, making for the gate. But he knew, only too well, the symptoms of tautology.\n\n\"If not he's sure to be at Priestley's. He goes in there almost every day. That's the estate agent's just at the bottom of the High Street. If you _do_ miss him there you ask Mr Maugham not three doors away. I don't think he'll have gone out to Burnside.\"\n\n\"No. Well, thank you....\"\n\nCarolus had got the gate half open but it was an optimistic gesture.\n\n\"Tell you what, though, he _might_ have decided to have his hair cut. If so it'll be at Cronin's on the way to the cattle-market. He was saying he needed a haircut so perhaps that's where he's gone. They don't close till six, so there'd be time for him to have gone to the Oak Caf\u00e9 first. That's where you'll find him. It's not very far, if you take this road and keep left at the fork. You'll see it Up\u2014'Oak Caf\u00e9' on your right. They'll tell you in there if he's been in.\"\n\n\"Thank you,\" said Carolus, firmly crossing to his car.\n\n\"Or else Priestley's office,\" he heard as he climbed into it. \"Or if not...\"\n\n\"Good-bye!\" called Carolus cheerfully as he started the engine.\n\nThe manageress of the Oak Caf\u00e9 was a stately person in black who addressed most of her remarks to the girl in the cash desk.\n\n\"Mr Griggs? Isn't that the gentleman who usually sits with Mr Mortimer and Mr Conolly? Yes, I know. He's left, hasn't he? \"She turned to Carolus. \"He's left,\" she said.\n\n\"Long ago?\"\n\n\"Was it long ago?\" the manageress asked the young lady in the cash desk. \"It would be about ten minutes,\" she said when she had received a reply.\n\n\"No idea where he was going?\"\n\n\"Do _you_ know where he was going?\" the manageress asked her assistant. \"No, I'm afraid I can't say where he was going.\"\n\nPriestley's, 'the estate agents at the bottom of the High Street', were scarcely more informative.\n\n\"No, not been in here this afternoon, old man,\" said a tall, thin character in checks who was smoking a pipe which jumped when he talked. \"Very often comes in here about this time but he hasn't been in today. I don't know where you'll get hold of him. Tried Maugham's, the tobacconist's? But I shouldn't have thought he'd have gone there without coming here. It's only a few doors away.\"\n\nBehind the counter of Maugham's was a fair-haired youngish man with a neat moustache.\n\n\"Griggs? In here a few minutes ago. In a hurry about something. Bought his cigarettes and was off without waiting to see today's Results. No, I don't know where you'd find him now. Unless he's back at where he lives, Maitland Villa in the Gladhurst Road. I can tell you where he'll almost certainly be after opening time, that's the Saloon Bar of the Chequers. Just opposite the Town Hall.\"\n\nIt was not difficult to find Cronin's, the hairdressers 'on the way to the cattle-market' but an elderly barber at the chair nearest the door, while never ceasing to snip at a small boy's hair, gave Carolus no encouragement.\n\n\"I know the gentleman you mean,\" he said. \"If he'd gone anywhere to get his hair cut it would have been here. But we haven't seen him this afternoon. He may come in presently, of course. I know he wants a haircut because he told me so in the street yesterday. Hold still. There's a good boy. If you'd like to wait....\"\n\n\"No, thank you very much. I must go on.\"\n\nIt had occurred to Carolus that rather than pursue Dundas Griggs when he was reasonably sure to find him 'after opening time', he could fill in the remaining hour by trying to confirm Mrs Rumble's rather extraordinary story about Grazia Vaillant. He made for Forster's Stores.\n\nYet when he was alone with the manager in a small office at the back he suddenly realized how impertinent and irrelevant his enquiry must seem. The manager seemed a kindly type but Carolus felt he must approach the matter with care.\n\n\"Look here,\" he said. \"I don't want you to answer my questions if they go against the grain. They'll seem rather cheek to you. The truth is I'm a private detective trying to get at the truth in the matter of an old lady's death at Gladhurst.\"\n\n\"Oh yes, I've read the case.\"\n\n\"I've been at Gladhurst some weeks, on and off, and I must say I find it a hotbed of scandal and malice. One story told me is about a customer of yours, a Miss Vaillant.\"\n\n\"I know Miss Vaillant.\"\n\n\"Someone informs me that Miss Vaillant has started secretly swigging gin. It may be nothing to do with the case but it may have a connection.\"\n\n\"I understand.\"\n\n\"I am told she gets her gin here in single bottles which she takes away more or less surreptitiously.\"\n\n\"I don't see why I shouldn't tell you, though I depend upon you to treat the information as strictly in confidence. It is perfectly true. Horseley's. In oval-shaped bottles which she slips in her shopping bag.\"\n\n\"That's kind of you.\"\n\n\"Tell you something else. The lady who was murdered has been a customer of ours ever since I came to the shop as a boy. She has never been known to order anything alcoholic till about a week before her death. Then she came in and asked me for a bottle of the same, 'for purely medicinal purposes', she said. So I suppose she had started having a little quiet one sometimes, too.\"\n\n\"Looks like it. Thanks awfully and I'll respect your confidence.\"\n\n\"That's all right.\"\n\nCarolus felt he should give some further explanation.\n\n\"You see, if they simply ordered gin or Scotch or anything else there would be nothing to it. Even if they didn't want it themselves it might be for entertaining. But when two elderly women who spend most of their time in activities connected with the parish church suddenly start secretly drinking gin, knocking it back when they're alone, hiding it from their servants, it does seem rather significant.\"\n\n\"Well, there it is. We haven't sold a bottle to Miss Vaillant since the other old lady died, by the way.\"\n\n\"Perhaps that's sobered her. Good-bye and thanks.\"\n\nOn the stroke of six o'clock the Saloon Bar of the Chequers was opened and Carolus walked in, the first customer. An enthusiastic young woman behind the bar served him.\n\n\"I'm waiting for Mr Griggs,\" said Carolus.\n\nHe's _sure_ to be in presently,\" said the barmaid, leaning across towards Carolus.\n\n\"I don't know him by sight. Will you point him out to me?\"\n\n\"Of _course_ I will. The moment he comes in.\"\n\nAs the room began to fill and Carolus received no sign from the barmaid he grew impatient. At half-past six he asked the girl if Griggs had come in.\n\n\"Not yet. I can't understand it. He's always in at this time. Wait a minute. I'll see if anyone knows. Mr Durrell! D'you know where Mr Griggs is?\"\n\nMr Durrell was telling a story and did not like being interrupted.\n\n\"Haven't seen him,\" he said curtly.\n\n\"Oh dear, someone must know. He often goes in to Huxley's the bookmaker's to put something on a horse. Have you see him, Mr Huxley?\"\n\n\"Not today, I haven't.\"\n\nAn informative gentleman broke in.\n\n\"Tell you where you're sure to find him. Round at the Queen Charlotte. I know he had to go in there because he was meeting Balchin the builder. Or if not....\"\n\n\"Thank you,\" said Carolus. \"But I'm not going to start that routine again. I'll wait here till he shows up.\"\n\nThe barmaid blinked reproachfully.\n\n\"Mr Hemingway's sure to know,\" she said. \"If _he_ says Mr Griggs is round at the Queen Charlotte you can be sure he is.\"\n\n\"Thank you. I'll wait and chance it.\"\n\n\"Mr Powys might know...\" suggested the barmaid.\n\n\"I'll wait.\"\n\nHe was rewarded at last, because at half-past seven a thin jocular man, all neck and wrists, long teeth and chuckles, came in and was told from all sides that Carolus was waiting for him.\n\n\"Me? Certainly. Oh yes. My aunt's murder. We'll sit over here. What are you having? I've had a very busy day. You know what it is rushing round?\"\n\n\"I do indeed,\" said Carolus, with some feeling.\n\n\"I heard someone was looking for me. I thought it was in connection with a piece of property I happen to know about. Cheerio. Yes, Aunt Milly's death. Terrible, wasn't it? At her age. Have a cigarette? I don't know. There's an awful lot of violence in the papers. When it happens in a small village it seems worse, somehow. (I'll be with you in a minute, Mr Waugh. I've got that estimate for you.) I was quite upset when I heard about it....\"\n\n\"When did you hear about it?\"\n\n\"The murder, you mean? I can't just remember now. Must have been the day after it happened. That's right. My landlady told me. She'd heard on the telephone. Someone had rung me up and she'd taken the message.\"\n\n\"You knew nothing the same night? The Thursday night, I mean?\"\n\n\"Thursday? No. How could I? No one knew till next morning.\"\n\n\"What time did you get back from Gladhurst that evening?\"\n\n\"Gladhurst?\"\n\n\"Gladhurst.\"\n\n\"Oh, that. Yes, I just popped over to see the old ladies. Little proposition I had to suggest. Business. Nothing of consequence.\"\n\n\"And did you see them?\"\n\n\"No. None of them. But there was a rather extraordinary thing. I reached the house about half-past four. That was the time they liked one to call. As soon as I got in the drive I saw lights on so I thought it was all right. One of them was in, anyway. I hoped it was Millicent. She liked a proposition, bless her. But when I rang the bell, nothing. Not a sound. Not a reply. I rang several times and waited. Nothing at all. Extraordinary thing because they never burnt light when they were out. Very careful about things like that.\"\n\n\"Didn't you try to get in?\"\n\n\"I walked round, yes. All locked. Back door. Everything. Even the garage. I went away and had a cup of tea in the village. Henson's the bakers do teas there. Then I came back. Must have been a quarter-past five by now.\"\n\n\"It was a dark night, I believe?\"\n\n\"Pitch black. _And there wasn't a light in the house._ What do you think of that? At four-thirty it's lit up like a Christmas tree. At five-fifteen not a glimmer. Makes you think, doesn't it?\"\n\n\"It does. How long did you stay there?\"\n\n\"I thought I might as well ring, in case. I went towards the door and I don't know what made me do it, but before I rang I stood still for a moment. I heard a noise which I recognized at once. Someone was opening the garage doors. I don't know if you've looked round the back of the house but the back door and garages open on to a separate lane.\"\n\n\"Yes. I've seen that.\"\n\n\"There is a way through from the garage into the house. So someone opening the garage doors could have come from inside the house.\"\n\n\"Yes.\"\n\n\"I waited. Listening.\"\n\n\"Where was your car?\"\n\n\"In the road outside. I couldn't be bothered to get out and open the gates.\"\n\n\"I see. So whoever was in the house or garage might not have known you were there?\"\n\n\"Almost certainly not. But I know the sound of those garage doors. Have known them for years. I heard them open, then a pause of about three minutes, then they were closed again. I still waited and presently, well down the road, I heard a motor-bike start up.\"\n\n\"Is that all?\"\n\n\"That's all. I started off for home puzzled but not really worried. Gladhurst has always seemed such a quiet little place. Only next day when I heard the news, I remembered all this. Anything else you want to know?\"\n\n\"Up to you. If you've anything to tell me....\"\n\n\"No. That's the lot. I've got to run round to the Swan. See a man. Little proposition.\"\n\nFive minutes after the live wire Dundas Griggs had left the bar, Carolus was amused to see Detective Inspector Champer walk in and ask the barmaid for Griggs.\n\n\"He's just this minute gone out. D'you know where he was going, Mr Durrell?\"\n\nShe had interrupted a longer and duller story than before and received a shorter 'No'. Mr Huxley was no wiser and Mr Hemingway hazarded the Queen Charlotte but Mr Balchin the builder who had talked to him there said he had left half an hour ago to come here, while Mr Powys thought he had gone home.\n\n\"That gentleman might be able to tell you,\" said the barmaid, pointing to Carolus. \"He was talking to him.\"\n\nDetective Inspector Champer unwillingly recognized Carolus and came across.\n\n\"Start at Maitland Villa?\" asked Carolus.\n\nChamper nodded.\n\n\"The Oak Caf\u00e9?\"\n\n\"Yes.\"\n\n\"Priestley's, the estate agents? Maugham's? Cronin's?\"\n\n\"There were more than that. Where is the \u2014\u2014\u2014 now?\"\n\n\"I shouldn't like to make rash guesses, Inspector. But he told me ten minutes ago that he was going round to the Swan to see a man about a proposition.\"\n\nWithout a word the thick-set policeman pushed his way out of the door.\n\n\"So you found Mr Griggs?\" said the barmaid keenly to Carolus.\n\n\"Yes. He turned up at last.\"\n\n\"You staying here?\" she asked, leaning intimately near.\n\n\"No. In fact, I'm due home now. Good-night.\"\n\nBut when Carolus had reached his house it was in darkness. The Sticks had gone to bed. There was reproach in every one of his sandwiches.\n\n# 13\n\nTHE Reverend Bonar Waddell looked thoughtful. His finger-tips touched and he stared down at his desk.\n\n\"You really feel it necessary to discuss the matter with my curate?\" he said at last.\n\n\"Not in the least. I want to ask him a few questions.\"\n\n\"Quite. Yes. Naturally. Just so. Of course. To be sure. I understand. I hesitate only because knowing his disposition....\"\n\n\"What is his disposition?\"\n\n\"Excellent with boys,\" said the vicar inevitably. \"But diffident, shy, easily upset in adult matters. Between ourselves he is a cause of some anxiety to me. I recognize his value in his own field but I cannot leave much general work to him.\"\n\n\"Sounds like arrested development.\"\n\n\"Arrested? Oh no. I trust nothing of that sort. But he is temperamental. Easily put out. Alarmed by details. I remember one occasion on which our local constabulary in the person of Slatt asked him some questions. Mere trivialities, referring to a dog licence or something. I found him in a highly distressed condition. I had to mediate between him and Slatt as best I could. I seek to impress on you his somewhat excitable character.\"\n\n\"I don't think you need worry. I only want to see him for a moment.\"\n\n\"Then go ahead, my dear chap! Go ahead! Let us put this canker from our midst. If my curate can assist you I feel sure he will be only too glad. I ask you but to approach him with tact.\"\n\n\"Where will I find him?\"\n\n\"Now? Ah hum. It is his Boys' Club hour. At our village hall. The Griggs Institute is its official designation, presented as it was by the father of the Misses Griggs. There he will be organizing one scarcely knows what.\"\n\n\"Really?\"\n\n\"The summer camp. Amateur theatricals. Basketball. He is full of ideas. I find myself bound to check his exuberance at times, standing as I do between the parents and him. Last summer he organized an excursion to the coast and the boys' impersonation of Ancient Britons was altogether too realistic. Parents complained of clothes ruined by woad and I received a very unpleasant letter from the Municipal Council. Woad, it was made clear, is no longer sufficient protection for public decency. Several visitors had been outraged by the scene presented. Slipper at once undertook to see that Ancient Britons in future should wear at least bathing slips if not football shorts. I need scarcely say that he had taken no active part in the scene, his r\u00f4le being that of a fully clad Boadicea, watching from a chariot.\"\n\n\"That was something,\" said Carolus encouragingly.\n\n\"Unhappily he is not always content with a watching brief. There was an unfortunate incident at their summer camp two years ago. Oh, most unfortunate. I nearly lost one of my best bell-ringers over it. I found myself torn between my loyalty to my curate and my duty as vicar of this parish. It was all very upsetting at the time. However, Slipper's inexhaustible enthusiasm carried the day. With discretion and a little tolerance and the general goodwill which I have always tried to promote the little incident soon became past history. Yes, you go and see my good curate. You will find him busy, I'm sure, but never so deeply involved that he cannot pause to answer questions.\"\n\nThe Reverend Peter Slipper did indeed appear shy as he disentangled himself from his responsibilities. He was a pale serious young man with a nervous hesitation in his speech. He led Carolus to a small office behind the stage and it seemed as they entered that a sultry hush fell over the mob of youngsters in the main hall. As Carolus entered there had been pandemonium. Now there was no more than a hum of talk.\n\n\"It's a Club Night tonight,\" explained Mr Slipper. \"Wednesdays and Saturdays, Club. Mondays and Thursdays, Scouts.\"\n\n\"What is the difference?\" asked Carolus curiously.\n\n\" _Every_ difference. The Boy Scout Movement and the Boys' Club Movement are entirely different things.\"\n\n\"I see. What happens on Tuesdays and Fridays?\"\n\nMr Slipper's face fell.\n\n\"Waygooze has those,\" he said. \"He's the choirmaster.\"\n\n\"The boys of Gladhurst seem to have a busy week.\"\n\n\"Even then we haven't enough time, really. They have to pass their tests for badges and there are rehearsals to get in and a thousand activities.\"\n\nCarolus began to understand the universally-held opinion of Mr Slipper.\n\n\"I want, if I may, to ask you a couple of questions,\" said Carolus.\n\nMr Slipper started slightly.\n\n\"It's about the afternoon on which Miss Griggs was missed.\"\n\n\"Oh, yes.\"\n\n\"Can you remember what you did that day?\"\n\n\"I was busy all the morning. The D.C. had come over to see me. The District Commissioner, I mean.\" Seeing Carolus still a little puzzled, as though he thought he was in pre-war Burma, Mr Slipper added, \"Scouts, you know. Each county has a County Commissioner and under him are a number of District Commissioners. Ours drove over that day to discuss our Jamboree. He could not stay to lunch, perhaps fortunately, as my digs are scarcely...\"\n\n\"And after lunch?\"\n\n\"I spent the afternoon preparing for the cookery tests which were to be held that evening. The boys themselves were out borrowing frying-pans and primus stoves but I made the necessary purchases at Jevons's Stores. Young Jevons is one of our Rovers, a ripping chap.\"\n\nCarolus confessed to himself that his next question was one of sheer human inquisitiveness.\n\n\"What purchases?\" he asked.\n\n\"They pass out on a pancake,\" explained Mr Slipper rather obscurely. \"No eggs, of course. Flour, milk, frying-fat. They have to eat them afterwards. That's the test. Those without indigestion get badges.\"\n\n\"Very appropriate. So you purchased the ingredients?\"\n\n.\" Yes.\"\n\n\"What time do you think you reached Jevons's Stores?\"\n\n\"I imagine in the region of 3.45.\"\n\n\"Whom did you meet?\"\n\n\"Now let me see. Oh yes. Our Churchwarden. Commander Fyfe.\"\n\n\"Was he alone?\"\n\n\"Well, as it happens he had stopped for a moment to address a few words to... a parishioner.\"\n\n\"You mean Flo?\" asked Carolus rather brutally.\n\nMr Slipper nodded.\n\n\"It wasn't more than a minute. Just passing the time of day...\" he explained eagerly.\n\n\"Of course. Anyway, I like Flo. Don't you?\"\n\n\"Oh, I scarcely... indeed I have never... I do not criticize... It's a matter for the vicar.\"\n\n\"Who else appeared in the village street?\"\n\n\"Naomi Chester. I saw her pass while I was in the Stores. She seemed in a great hurry.\"\n\n\"You don't remember anyone else?\"\n\n\"No. I daresay there were others. I was, of course, preoccupied. These cookery tests! Last year the boys got rather excited and upset some hot fat. Lawrence Tilley's hand was rather badly burnt. In fact he couldn't use it for some days. I had a great deal on my mind.\"\n\n\"You have a motor-cycle, Mr Slipper?\"\n\n\"A Lambretta, yes.\"\n\n\"But you did not use it that afternoon?\"\n\n\"No. I was on foot. My digs, the stores and this hall are all within a few hundred yards of one another.\"\n\n\"Did you see Rumble or Mugger?\"\n\n\"I don't remember seeing either. When I had made my purchases I went back to my digs to tea then came round here at five o'clock for the tests. We had seven passes.\"\n\n\"I congratulate you. And I'm most grateful for your patience in answering my questions. I won't interrupt your Club Night any further.\"\n\nAs he left the hall the din that rose behind him was like that of a wild beasts' cage at feeding time.\n\nCarolus intended leaving at once for Newminster but as he passed the Black Horse he saw Mugger emerging and stopped his car. Mugger, tall and angular in the light thrown by the bulb over the inn sign, did not look furtive. When he spoke it was in his usual serious tones.\n\n\"Evening, Mugger,\" said Carolus.\n\n\"I can't stop now,\" said Mugger. \"I've got one waiting for me out by the cricket field.\"\n\n\"I shan't delay you long,\" promised Carolus.\n\n\"Better not, because I don't want to miss this one. Just right she is, if you know what I mean.\"\n\n\"Yes, yes. I wanted to ask you about a sheet.\"\n\nNothing was changed in Mugger's posture or expression.\n\n\"What sort of sheet?\" he asked.\n\n\"Come now, Mugger. Don't let's waste time and words. You don't suppose I meant a sheet of paper or part of the rigging of a ship. You know exactly which sheet.\"\n\n\"I don't know what you're talking about. And what's more I shall have to hurry. They won't wait all night.\"\n\n\"Then let's be explicit. I haven't made myself difficult about the sum of money....\"\n\n\"There was no money,\" sighed Mugger, but almost mechanically.\n\n\"So why not trust my discretion over this?\"\n\n\"What d'you want to know?\"\n\n\"Where you found it.\"\n\n\"Up in the loft of the furnace room, of course. With the other things.\"\n\n\" _With_ them?\"\n\n\"Well, just beside. It was crumpled up and stuffed into a box.\"\n\n\"Blood-stained?\"\n\n\"Yes.\"\n\n\"What on earth induced you to move it, Mugger?\"\n\n\"Well, a sheet's a sheet, nowadays.\"\n\n\"I find you very hard to understand. You leave the jewellery and take a sheet.\"\n\n\"The sheet could be washed, couldn't it? There'd be nothing to say where it come from. But jools, that's different. The police can trace them. Before I knew where I was I'd be up for murdering the old lady which I never done.\"\n\n\"I see. Where's the sheet now?\"\n\n\"Never mind that. It's been washed. You'd never know it from any other, now. And if you were to go and say anything I should know nothing about it. Never seen a sheet. Never heard of one. Never spoke to you about it. And no one would know any different.\"\n\n\"It was badly blood-stained?\"\n\n\"Very badly. Horrible in fact. Now I can't wait any longer. I can't afford to miss one like this.\"\n\nHe strode away, his long shanks and thin body exaggerated by the yellow overhead light.\n\nCarolus was about to drive on when he saw Slatt. It seemed that the policeman had been observing him during his conversation with Mugger and now approached.\n\n\"You know what the Inspector told you, don't you?\" he said as though he were addressing a small and naughty boy. \"You've no business to be nosing round with what doesn't concern you. I should very much like to know what you were asking Mugger.\"\n\n\"Really? Perhaps Mugger will tell you.\"\n\n\"I'm surprised you encourage him,\" said Slatt. \"He gives more trouble than all the rest of the village put together. I hear you've been round to see Miss Vaillant, too. I can't see how she can concern you. What I'm always afraid of is a burglary there. She's all on her own and the house is full of valuables. Antiques and that.\"\n\n\"I know. There's a landscape by Constable.\"\n\n\"Police Officer!\" said Slatt, so alert for the offensive word that he could not wait to study niceties of meaning.\n\n\"I'm always forgetting,\" said Carolus. \"A landscape by Police Officer. But I shouldn't have thought there was much danger of burglary in a place like this.\"\n\n\"If there can be murder there can be burglary,\" said Slatt darkly.\n\n\"What really seems to distinguish your village is the enmity between its inhabitants. Particularly the older ones.\"\n\n\"You don't know the half of it,\" said Slatt. \"There's the three old sisters to start with. There was no love lost among them. I've heard that Mrs Bobbin could lay her tongue to some terrible language when she was put out with the one that's been done for. The other one would shout the Old Testament at both of them. Proper slanging match. That's only what I hear, though.\"\n\n\"Of course.\"\n\n\"Then the vicar's wife's not much better. You should have heard her going on about the Miss Griggses according to what they tell me. Talk about words! Miss Vaillant's different. She's sweet as sugar till suddenly she's raging like a wild cat. Funny, isn't it? Then look at Mrs Rumble!\"\n\nCarolus put in a mild defence of the sexton's wife. \"You don't know her, that's all I can say,\" returned Slatt. \"She's got a wicked temper. Commander Fyfe's wife's just the same, only she's laid up half the time. They say you can hear her shouting at him from two doors away. Then what about Mrs Pinton?\"\n\n\"I don't know her.\"\n\n\"She's the doctor's wife. I've seen her lose her temper as much as any of them. As for them up at what they call Hellfire Corner\u2014well!\"\n\n\"I believe Mugger's married?\"\n\n\"There's another one! I thought she was going for me the other morning when I had to ask her something. She was out the back, doing her washing.\"\n\n\"No woman likes a man to watch that.\"\n\n\"Perhaps that's what upset her. She seemed to have a lot of it. And of course with Mugger out poaching at night there was blood on things. But it wasn't that. It was me seeing her without one of these washing machines ail the women have.\"\n\n\"You mean she was using an old-fashioned cop... police officer?\"\n\nSlatt seemed not to notice this gaffe.\n\n\"Turned on me something wicked, she did. What business had I got coming out to her back yard? You're right in what you say about this village. When George Larkin's wife was alive she was no better. It was always my belief that she knew about him and Miss Griggs going off together when they were young and never forgave him for it. She never liked the old ladies, anyway, and didn't want young Bill to have anything to do with them.\"\n\n\"Yet I understand that Miss Griggs has remembered both the Larkins in her will.\"\n\n\"That's the funny part of it. She's left Bill just as much as his father and he was nothing to do with her. Of course, after George Larkin's wife died, young Bill used to go up and see the old lady. They're very deep, those Larkins. Both of them. You never know where you are with them.\"\n\nSuddenly Slatt became alert.\n\n\"They're coming out of the Griggs Institute,\" he said. \"All those young devils'll be raising Cain in the street if I don't get down there.\"\n\n\"They don't seem to be making much noise,\" said Carolus.\n\n\"It's not that. They get all over the place laughing and talking. If I'm not there they think they can do what they like.\"\n\n\"You'd better go and show them they can't,\" said Carolus and Slatt, unconscious of irony, pushed his cycle away.\n\nCarolus decided to have a last drink in the Black Horse before leaving for home, and found himself, as on his first night, beside the informative Mr Lovibond.\n\n\"You didn't tell me you were finding out about the murder when we had our chat before,\" said Lovibond. \"Still, I don't suppose you want to tell your business to everyone, especially when it's something of that sort. It was young Laddie Grey told me. One of the nicest chaps you could wish to meet. He came in my shop the other day to buy a bulb for his motor-bike. I think I told you I keep a little electric shop down the road. Yes, young Laddie came in and mentioned to me that you'd been asking him questions.\"\n\n\"Was it a headlight bulb he wanted?\" asked Carolus casually.\n\n\"No. It was for the light on his sidecar, because I saw him fix it in. So you've been round asking everyone, have you?\"\n\n\"Yes. I seem to have met a good many of the inhabitants of Gladhurst.\"\n\n\"Talked to old Flo yet?\"\n\n\"I have met her. Yes.\"\n\n\"She's a Character, isn't she? You never know who you'll see her with next. There's a story going round about her and Fyfe who lives up at The Fairway. I shouldn't be surprised, mind you. It wouldn't be the first married man, not by a long way. But she doesn't mean anyone any harm. It's only the wives get a bit upset when stories get back to them. Not that my old woman would bother. As long as I don't forget her light ale when I go home at night she never says a word.\"\n\n\"That's good.\"\n\n\"Look at old George Larkin watching the clock. He knows Slatt's outside holding his watch. He'll start shouting 'Time!' in a minute.\"\n\nCarolus decided to leave before the split-second ejection by synchronized watches had taken place.\n\n# 14\n\nCAROLUS was free for the following week-end and wanted to spend it in Gladhurst. But before arranging this he approached the headmaster for formal leave of absence, a gesture of courtesy which he would not omit.\n\nHe found an opportunity of speaking to Mr Gorringer on the Wednesday afternoon, for a lecturer was due to address the school on Tanganyika (with lantern slides), and afternoon classes were therefore cancelled. The lecturer was late and the Headmaster paced impatiently before the moderately-sized room known as the Great Hall in which the school's 250 boys were gathered.\n\n\"Ah Deene, you find me in a frame of mind by no means equable. If there is a fault to me unforgivable it is that of unpunctuality. Our lecturer is already ten minutes late.\"\n\n\"The fog, perhaps.\"\n\n\"Fog? It was his duty to provide against all climatic exigencies. His fee is twelve guineas, which, though it includes expenses, seems to be adequate.\"\n\n\"I wonder what his agent stings him, poor devil,\" said Carolus.\n\n\"That, my dear Deene, is not our affair. Thirteen minutes. The boys will grow restless shortly.\"\n\n\"I want to be away this week-end, headmaster.\"\n\n\"Again? Really Deene, you are quite deserting us. Have you no wish to see our match against St Hildegarde's? And perhaps you have forgotten that the Dean of Bodmin is preaching in the School Chapel on Sunday?\"\n\n\"It's rather an urgent matter. I feel if I can stay a weekend among these people at Gladhurst I can get the thing cleared up for good. It's the way one knows a village like that.\"\n\n\"At least I respect your candour, Deene, in telling me the reason you wish to elude us again. That much I appreciate. But I could have wished that the promise of a hard-fought football match, together with the celebrated eloquence of the Dean of Bodmin... However, I will not stand in your way. It shall never be said that I adopt a restrictive or a carping attitude towards my staff. You may go and I hope will return with all mysteries elucidated.\"\n\n\"Thank you.\"\n\n\"Dear me. A full quarter of an hour late. I shall be constrained to send the boys back to their classrooms and abandon all thought of the lecture, for which, fortunately, the fee has not yet been paid. So we are to hear shortly the identity of this brutal murderer at Gladhurst? I trust you will make me among the first to hear details? I must tell you a witticism of my wife's on that subject. She... Ah, but unless I mistake me this is our lecturer at last....\"\n\nThe headmaster hurried forward and Carolus, with a sigh, sat to see lantern slides of Tanganyika. These consisted largely of hefty negroes in postures which, the lecturer assured his audience, were those of a native dance.\n\nWhen Saturday came, Carolus set off immediately after morning school, having a delightful sense of escape. It was not often he allowed himself a night away from Newminster during term-time but the reason he had given to the headmaster in this case was a genuine one. He believed that a small thing like going to bed and waking up in Gladhurst, spending Saturday night in the pub and Sunday morning in the church, being for a few hours one of the village inhabitants, might make all the difference to his understanding of them. And on that depended the whole case.\n\nHis first call was at the vicarage. Mrs Waddell opened the door and spoke with something like a note of triumph.\n\n\"My husband's preparing his sermon for tomorrow and cannot possibly be disturbed. My daughter's out on her bike.\"\n\n\"That's all right,\" said Carolus blandly. \"It was you I hoped to find at home, Mrs Waddell.\"\n\nShe gave him a short defiant stare, then said: \"Come in, then.\"\n\n\"I must really apologize for troubling you again,\" said Carolus. \"You'll think I'm an awful idiot. The thing is, I'm trying to work out a complete timetable for everyone's movements after two-thirty that day and I find I have a blank in yours.\"\n\n\"It sounds like a sort of game,\" said Mrs Waddell grimly.\n\n\"Not really. It's rather important. You explained to me that you had your Mother's Meeting from four to six, I think. That was at the Griggs Institute, perhaps?\"\n\n\"It was.\"\n\n\"I see. But it's only about five minutes from here and your daughter remembers you returning at 6.45 or more. It is that three-quarters of an hour I want to fill in.\"\n\n\"I find the query somewhat impertinent, Mr Deene, but in order not to be badgered with further questions I will tell you at once. The Scouts were holding their Cookery Tests at the Institute. I watched for a time to see that everything was all right. Last year there was nearly a nasty accident.\"\n\n\"Did you speak to Mr Slipper?\"\n\n\"No. He hates being interrupted while he is with his boys. I just stood in the shadows at the back for a while, then came away.\"\n\n\"No one saw you?\"\n\n\"I shouldn't think so. They were furiously concentrated on their pancakes.\"\n\n\"Thank you, Mrs Waddell. I needn't take any more of your time.\"\n\nOn his way round to Commander Fyfe's he caught a glimpse of a somewhat bizarre group. Mr Slipper was in the full uniform of a Scoutmaster and with him were several sturdy Rovers with a laden pushcart. Carolus sighed. It was an excellent thing, the Boy Scout Movement, but why oh why did it encourage grown men whose physiques could not stand the test to wear this uniform? For healthy youngsters, splendid, but for curates no longer youthful, with angularities like Mr Slipper's, fatal. Those blue knees! But he waved a greeting to the curate and drove on towards The Fairway.\n\nFortunately he found himself stopped in the road and without any explanation Fyfe climbed in beside him.\n\n\"The very man I want to see!\" he said. \"Most extraordinary thing. Could we drive on a little? Out of the village perhaps?\"\n\nWhen they had reached a fairly open stretch of road about a mile away, Fyfe dropped his voice.\n\n\"It's Dundas Griggs,\" he said. \"He guessed it was I who told you he was over here.\"\n\n\"How did he guess that?\"\n\n\"Because he saw me when he was driving his Vauxhall and stopped.\"\n\n\"You didn't tell me.\"\n\n\"So many strange things happened that day. I couldn't tell you them all. I've seen him over here before. He goes to call on the old ladies; then we usually meet at the Black Horse in the evening. Saloon bar, of course. It's quiet there. He told me that afternoon he had a little proposition for me.\"\n\n\"Quite a Euclidian, Mr Griggs.\"\n\n\"I beg your pardon? Oh. Who is this coming down the road on a bicycle? You see you never know here.\"\n\n\"You were telling me about that afternoon.\"\n\n\"Yes. Arranged to meet for a drink later. I told you. I was able to leave the house for a while. My gardener and his wife... television....\"\n\n\"Yes. Yes.\"\n\n\"Then the really extraordinary thing happened. I can't think why I didn't tell you before. _Griggs never turned up_. I waited an hour for him. Not a sign. Never came. What do you think of that?\"\n\n\"Very bad manners.\"\n\n\"Manners? But don't you see, man? He's the nephew of the murdered woman and on the night of her death he fails to keep an appointment! Doesn't it strike you as sinister?\"\n\n\"Not necessarily.\"\n\n\"Then today. Over here again. Remonstrating with me for having told you he was in Gladhurst that day. Says he doesn't want to get mixed up in the thing. He was quite annoyed. I consider that highly significant.\"\n\nCarolus proposed to drop Commander Fyfe at his home but he grew somewhat nervous at the prospect.\n\n\"No! Don't stop right outside. You never know. It doesn't do in a place like this. Leave me by the Griggs Institute. I'll walk home.\"\n\nAt the Black Horse he found George Larkin alone, for the doors had only just been opened.\n\n\"I wonder if you could let me have a room for tonight?\" asked Carolus.\n\n\"We don't Do rooms.\"\n\n\"Oh. Pity.\"\n\n\"I'll ask Mrs Crutch who looks after that side of it. She might be able to.\"\n\nHe returned after a few minutes.\n\n\"She says there is a room. But you mustn't expect too much. It's clean and that and she'll air the bed. That's all.\"\n\n\"What more could one want?\"\n\n\"Some expect a lot.\"\n\n\"I'll bring my bag in.\"\n\n\"All right. Bill will take it up. You'll want something to eat presently, I suppose?\"\n\n\"Whatever's easy.\"\n\nCarolus ordered a drink and after it was poured the innkeeper returned to his static contemplation of the wall opposite, expressionless solidity on his face. It was ten minutes before the first of his Saturday night customers came in.\n\nThis was Rumble, looking particularly pleased with himself. His gnomish head on his stocky body suggested one of Disney's seven dwarfs. He accepted a pint from Carolus and at once began to congratulate him.\n\n\"I've never seen anything like it!\" he said admiringly. \"There you was in the kitchen drinking tea. It's as much as I dare do to sit in that chair which is supposed to have belonged to her father. However did you manage it?\"\n\n\"I've yet to see all this bad temper your wife is credited with.\"\n\n\"Bad temper? You try living with her. Anyway, she'll be in presently and you can see for yourself. I've got something to tell you, though. Might be of some help to you. See, Mrs Bobbin called me into the hall today to help put the furniture back now that Laddie Grey's finished painting in there. Miss Flora was out and I didn't know where she'd gone but while I was still indoors she came storming in. I've never seen her anything like it. Firing texts off at the top of her voice. It appears she had been to Miss Vaillant's.\"\n\n\"What sort of texts?\"\n\n\"All about the Wicked. 'The wicked laid a snare for me', she says from Psalms.' A woman with the attire of an harlot and wily at heart', from Proverbs, I think it was. But we've had all this before about Miss Vaillant. It was something else made me think.\"\n\n\"Yes?\"\n\n\"She started on about 'be not among wine-bibbers', and causing strong wine to be poured, and strong drink is raging, and all that. It sounded to me as though somebody had offered her a nip and she'd turned up her nose at it. What do you say?\"\n\n\"I didn't hear it,\" said Carolus guardedly.\n\n\"I thought you'd like to know, anyway. This pub soon gets filled up on a Saturday, doesn't it?\"\n\nThe bar was certainly growing crowded. When Flo came in and was greeted on all sides, there seemed to be a hilarious atmosphere.\n\n\" _There_ you are!\" said Flo to Carolus as though they were old friends. \"You haven't taken me out in that car of yours yet, have you? There's a promise for you! Perhaps you're waiting for the nice weather. It reminds me of a fellow I knew once who used to say it was no good going out except in the Spring. What an idea! I asked him what he did all the rest of the year. Not that I minded, because he wasn't the only pebble on the beach. We shall all start singing presently so I don't know what you'll say about that. Saturday night, after all.\"\n\nCarolus saw Mr Lovibond making his way across and knowing that the little electrician regarded him as a prot\u00e9g\u00e9 he made room for him and invited him to drink.\n\n\"The vicar will be in presently,\" said Lovibond surprisingly. \"Always comes in for a few minutes on Saturday evening.\"\n\nAnd sure enough before eight o'clock, complete with clerical collar and bland smile the Reverend Bonar Waddell was amongst them.\n\n\"I see you are quite an habitue,\" he said to Carolus. \"Ah well, there's no harm in a little cheerfulness at the end of a week. I myself am in a slightly more difficult situation. On the one hand many of the most sincere and loyal members of my congregation, including the landlord himself, are customers here and appreciate my presence for a short time. On the other hand many no less earnest folk in the parish regard strong drink as the devil's potion and a public house as a place of evil.\"\n\n\"So how do you manage?\" asked Carolus, fascinated as usual by the delicate balances which Mr Waddell held in his hand.\n\nThe vicar smiled.\n\n\"I come here, as you see, but only for a short time on Saturday. And I invariably order for myself a lemonade.\"\n\n\"Very ingenious.\"\n\nThe vicar looked more grave.\n\n\"But tell me,\" he said. \"How go your investigations?\"\n\n\"It's a complicated case.\"\n\n\"That's just what we all thought at first it was _not_. Brutal, predatory, savage, cowardly, it seemed but of almost primitive simplicity. Now that we know the motive was not mere robbery it wears another complexion.\"\n\n\"Yes. It seems to bring it home to the village, doesn't it?\"\n\nThe vicar looked anxious.\n\n\"There are times when I almost wonder whether it would not be better left a mystery, Mr Deene,\" he said. \"I see, of course, that 'vengeance is mine', that such a crime cries out for it, but then I think, too, that revelation will mean more bloodshed, more suffering by those perhaps not directly concerned. I am torn in my hopes and sympathies. Ah well, I must greet some of these good people. Good evening, Frank. Good evening, Mugger. Ah, Rumble. Er hum, good evening, Flo. I see you well, Mrs Rumble? That's right. Full chime of bells tomorrow, Gidley? Splendid. Ah, Laddie. And Bill. Nice to see you all. No, no thank you. I've had my allowance. I must be making my way homewards. Good-night, Larkin. Goodnight to you, Deene. Good-night, all! Good-night!\"\n\nThough he went out amid the smiles of the customers, the vicar's departure seemed to remove a little constraint from the assembly.\n\n\"Well now we can have a sing-song,\" said Flo exuberantly. \"Come on, Mrs Chester! You're not going to sit there like a mute, I can tell you. We know what you can do when you get started, can't she, Mr Lovibond? It reminds me of the married couple that were worried because...\"\n\n\"Now, Flo.\"\n\n\"All right, Mr Larkin. I wasn't going to say a word. Only we must have a bit of a sing if it's only to make the Sheriff wild. He's standing outside already looking at his watch.\"\n\nCarolus found himself wedged in a corner with Mrs Rumble.\n\n\"Is that right you're staying here tonight?\" she asked.\n\n\"Yes. I thought...\"\n\n\"You know she still hasn't taken a drop. Funny, isn't it? Mind you, I don't know what happened this afternoon when that Flora Griggs got there. I hope she's not as bad as the other one. But anyway, up to then she hadn't touched it. Must have given her a nasty shock, that other.\"\n\n\"Perhaps that was it.\"\n\n\"Now, Rumble, don't you hang round here because you've had one pint off the gentleman and I'll see he doesn't buy you any more. You've had enough already by the looks of it.\"\n\n\"I haven't....\"\n\n\"Now don't keep on or I shall Tell you. You go and talk to old Mugger, he's more your mark.\"\n\nIt seemed but a few minutes after that when George Larkin was shouting \"Time!\" and the exodus was made in good order and swiftly under the eye of Slatt.\n\nCarolus went into the open air with the rest and found himself standing almost alone with a somewhat argumentative man of large proportions.\n\n\"What I say is,\" said this individual, \"I'm one of the few blacksmiths left working a forge.\"\n\n\"I daresay,\" said Carolus.\n\n\"And it's an old trade and a good one.\"\n\n\"It's both, yes.\"\n\nSlatt was approaching.\n\n\"A man who can work with iron, is a man,\" said the blacksmith.\n\n\"You're right,\" said Carolus. \"You know what Kipling says:\n\n'Gold is for the mistress\u2014silver for the maid!\n\nCopper for the craftsman cunning at his trade.'\"\n\n\"What did you say?\" asked Slatt.\n\n\"Sorry,\" returned Carolus.\n\n\"Police officer for the craftsman, cunning at his trade.\n\n'Good!' said the Baron, sitting in his hall,\n\n'But Iron\u2014Cold Iron\u2014is master of them all!'\"\n\n\"That's right,\" said the blacksmith.\n\nCarolus bade them both good-night and joined the Larkins, father and son, in a plentiful cold meal. He found that his bedroom overlooked the lower part of the village and against a cold moonlit sky he could see the silhouette of the church tower.\n\n# 15\n\nCAROLUS was awakened by a loud persistent knocking on his door before it was light. There is always something slightly macabre about knocking like this, forceful, unhurried, continuous. Carolus thought of Macbeth and how well Shakespeare had used that sound to introduce the horrible bloodstained humour with which he heightened the effects of his tragedies.\n\n\"Come in!\" he shouted, switching on the light.\n\nGeorge Larkin entered.\n\n\"There's someone downstairs wants to see you. Urgent.\"\n\nHow typical of him not to say who it was when he knew perfectly well.\n\n\"All right. I'll be down in a few minutes.\"\n\nHe dressed as quickly as he could, washing in cold water in the china bowl on the mahogany stand. He hurried downstairs and in the Larkins' little sitting-room found Mrs Rumble.\n\nHe was accustomed to seeing her long plain face set in lines of ferocity. Now she looked startled and anxious.\n\n\"Come outside a minute,\" she said. \"I want to speak to you.\"\n\nCarolus followed her to the inn yard.\n\n\"I don't know whether I done right. I thought once they get in you'll never be able to find out anything. So I've come to tell you first. She's dead.\"\n\n\"Miss Vaillant? \"Carolus scarcely needed to ask.\n\n\"Yes. Dead as a doornail. Slumped across the settee with an expression on her face as though she'd like to kill you.\"\n\n\"But you ought to have gone to the police.\"\n\n\"I am going. But you may as well have a look first. You might find something. They'll never let you near when they get there. Come on quick and no one will be the wiser.\"\n\nIt was altogether too tempting. Carolus realized that the consequences might be serious but he had to chance that.\n\nThey drove swiftly to the Old Vicarage.\n\n\"Put your car over there as though you were in church for early Communion and no one will notice it. There will be several there. That's it. Now we go in round the back.\"\n\nMrs Rumble had her key of the back door and admitted them.\n\n\"Was this locked?\" asked Carolus.\n\n\"Oh yes, so was the front. This way.\"\n\nThere certainly was an expression of apoplectic anger on the face of Grazia Vaillant where she lay twisted half across the settee. It was as though she knew herself to be dying and instead of commending her soul to God, as the expression used to be, she was angry at being subjected to the indignity and uncertainty of death. The eyes were wide open in a fixed indignant gaze and the hands were twisted as though she had suffered convulsive movements.\n\n\"Quite cold, of course?\"\n\n\"Yes. Must have been dead for hours. From yesterday evening very likely.\"\n\nMiss Vaillant's bag was open beside her on the floor and she was dressed in one of her curious coarse outfits with large jewellery, the whole suggesting a Balkan peasant woman. The curtains had been drawn close and it was by electric light that Carolus and Mrs Rumble stared at her.\n\n\"Do you think she was murdered?\" asked Mrs Rumble.\n\n\"How can we possibly know till after the doctor's report?\"\n\n\"I suppose not. I mean, she looks as though she's been murdered, doesn't she?\"\n\n\"She looks very distressing. Which is the cupboard you told me about of which you have a key?\"\n\n\"This one,\" said Mrs Rumble and opened a little cupboard on one side of the sideboard.\n\n\"Well! That's a funny thing,\" she said. \"There was a bottle nearly a third full here yesterday. I told you she hadn't touched it since the other one died. Now it's gone, bottle and all.\"\n\nCarolus went to the back door and saw the old well which Mrs Rumble had mentioned. It was one of those deep narrow shafts which drop darkly down to a just visible black surface of water.\n\n\"That's where she must have thrown the bottle like she did all the other ones.\"\n\nThey returned to the room where Miss Vaillant's body lay and Carolus looked down on it unhappily.\n\n\"What did she carry in her bag?\" he asked. \"No! Don't touch it! Let me look in.\"\n\nWith a handkerchief over his finger Carolus gingerly turned over the contents.\n\n\"About ten pounds in money,\" he said. \"Would that be usual?\"\n\n\"Just about. She didn't draw a lot at a time. She liked paying things by cheque.\"\n\n\"There's a tube of Minerval. They're very powerful tranquillizers. On Doctor's prescription only. Did she take many?\"\n\n\"I bought that tube new for her a week ago. She must have been taking them since then. In fact I know she has because 1 happened to notice yesterday there was only two left.\"\n\n\"It's empty now,\" said Carolus. \"If she took the lot in one go it could have killed her.\"\n\n\"She couldn't have done that, could she? There was nothing wrong when I left yesterday afternoon. In fact she was quite excited about getting a statue on the altar of the side chapel. It seems now Miss Griggs is Gone the vicar agreed. Still, you never know.\"\n\n\"You're quite sure about the gin?\"\n\n\"Certain. I saw it only yesterday.\"\n\n\"You said she drank lime juice with it?\"\n\nMrs Rumble stooped.\n\n\"Yes and that's Gone Down, too. She must have had a good old time after I'd gone yesterday.\"\n\n\"What about the glass?\"\n\n\"If she threw the bottle away she washed the glass up. You may be sure of that. Many's the time I've come in in the morning and found just the one glass in the sink. She thought she fooled me with that but I knew better. I'll go and see if there's one now.\"\n\n\"Don't touch it,\" warned Carolus.\n\nMrs Rumble returned to say yes of course there was. She had known there would be.\n\nCarolus made a quick tour of the house. He found the front door bolted, all the windows with their catches across and nothing forced anywhere. It would seem that if anyone entered last night it was with a key of the back door.\n\nThen once again he looked round the room in which the dead woman lay. On the table, beside the settee across which she sprawled, were several lurid-looking tracts. The title of the top one intrigued him:\n\nSatan Raises his Glass.\n\nIt was issued by the All-World League of Absolute Abstainers. Another, scarcely less ostentatious, was called:\n\nThere's an Inn-Sign on the Road to Hell\n\nBut beneath these two were tracts dealing with Missions to Hindus, Jews and Mohammedans and aids to the study of the Old Testament. They were all fresh-looking and had surely come from the ample stock kept by Flora Griggs.\n\nCarolus looked carefully for anything else which might reveal events or visitors last night. Miss Vaillant's glasses were beside the tracts so no doubt she had examined them with some mirth. The room had been carefully dusted, a tribute to Mrs Rumble, and only the experts would know whether there were fingerprints of anyone but the dead woman and her housekeeper.\n\n\"Was Miss Flora Griggs here yesterday? \"Carolus asked Mrs Rumble.\n\n\"Yes. She was here when I left. But I'll tell you about that another time. There isn't a minute now because I must phone the police.\"\n\n\"What time did you leave?\"\n\n\"Just after four. I went off to do my shopping.\"\n\n\"Did you meet anyone in the village?\"\n\n\"I'll tell you all that later when I've had time to think. I _did_ see Naomi Chester, I remember, and Commander Fyfe was there too.\"\n\nA bell began to chime in the church tower across the road.\n\n\"That's Early Service,\" said Mrs Rumble, forgetting the less evangelical terms which Miss Vaillant had taught her. \"You better pop across there and no one will know any different. I'll ring the police as though I'd just come. You can go out the back way and mind no one sees you.\"\n\nWhen Carolus came out of the church half an hour later there was great activity in front of the Old Vicarage. Slatt was making a show of directing traffic and moving on anyone who hesitated near the house. Several police cars were drawn up and there seemed to be much coming and going through the front door, which Mrs Rumble, doubtless, had opened.\n\nCarolus stood beside his car for a moment and was surprised to see Champer hurrying across.\n\n\"I believe you knew about this,\" he said aggressively.\n\n\"What?\" asked Carolus with far too much innocence.\n\n\"I tell you again, if I catch you interfering in any way I'll charge you. There are going to be no amateur detective tricks when I'm investigating.\"\n\n\"Something happened?\" asked Carolus.\n\nChamper made a sound between a spit and a cough and stumped away.\n\nThe news had preceded Carolus to the Black Horse but did not seem to impair the appetites of the Larkins. Carolus found himself breaking all precedents by eating bacon and eggs.\n\n\"Do you think she was done for?\" asked George. \"Or did for herself? Or died natural?\"\n\n\"I know no more than you till the inquest and we hear medical evidence.\"\n\n\"They say she looked like Dracula,\" said Bill Larkin.\n\nCarolus remained silent.\n\nThere was a call just then from Mrs Bobbin asking Carolus to go up to the house. He wondered for a moment how she knew he was staying the night at the Black Horse, then remembered Rumble.\n\nThe old lady stood with her usual uprightness and looked Carolus in the eyes.\n\n\"This has dreadfully upset my sister Flora, Mr Deene. I feel inclined to send for a doctor. She is quite beside herself.\"\n\n\"I'm sorry.\"\n\n\"You see, she called to see Grazia Vaillant yesterday afternoon.\"\n\n\"I know.\"\n\n\"How do you know?\"\n\n\"It is common knowledge,\" said Carolus evasively.\n\n\"She returned in a state of great indignation. My sisters, as I think I have told you, were strict teetotallers. It appears that Grazia Vaillant not only pressed poor Flora to drink but said that during those two last visits to her which Millicent made, Millicent had drunk with her. Flora came back to me _speechless_ with indignation.\"\n\n\"Is that quite the right word?\"\n\n\"No. But very, very angry. And now Grazia Vaillant is dead.\"\n\nSo I have heard.\"\n\n\"Mr Deene, do you think my sister was the last person to see her alive?\"\n\n\"I think it very likely.\"\n\n\"Then... but it is monstrous. If Grazia Vaillant did not die naturally, or commit suicide, some suspicion might attach to Flora?\"\n\n\"I don't know the workings of the police mind. I think in any case we should wait to hear the cause of death when it has been established. The woman may have had a heart attack.\"\n\n\"It's so extraordinary after the other. You are not very reassuring, Mr Deene.\"\n\n\"No. I'm afraid I can't be in this matter. It has a very ugly look.\"\n\n\"You say it is generally known that my sister called at the Old Vicarage?\"\n\n\"I understand she left some religious tracts there.\"\n\n\"Did she? Oh! \"For a moment Mrs Bobbin seemed crestfallen, then she said: \"But she might have given them to Grazia Vaillant at any time.\"\n\n\"I think, Mrs Bobbin, if you will permit me to say so, you are suggesting something rather dangerous. Your sister certainly called there, there are probably witnesses. It would be most unwise to attempt to conceal it, in my view.\"\n\n\"You may be right. But I hope these wretched policemen don't come here to bother her, at least till she has got over the shock.\"\n\n\"I shouldn't count on that,\" said Carolus as he took his leave.\n\nMoreover he was right. He met Champer at the gate.\n\n\"Well, Inspector, we meet again,\" he said brightly. But the policeman pushed by.\n\nAt about mid-day Carolus sat on a comfortable gravestone in the village churchyard waiting for the general dispersal of the congregation. In front of him he could see the plain oval-topped affair already erected 'To the Memory of Albert Chilling'. Carolus remembered that, but for the keen observation of Rumble, Millicent Griggs's body might still be deep there in the earth and the Gladhurst Case have been no more than that of another old lady disappearing from her usual orbit.\n\nIn the church he could hear the choir celebrating the Greatest Mystery of the Universe in a rollicking tune:\n\nThree in One and One in Three\n\nRulers of the Earth and Sea\n\nsang the choir cheerily because the service was nearly over and soon they would be free, the men to seek the Black Horse, the boys for an hour's smoke before Sunday dinner.\n\nCarolus was interested to see how the congregation would take the new tragedy in the village. But when they began to emerge it was difficult to know what its impact had been. Was there an extra solemnity? Or did he imagine that among those groups with bowed heads and solemn gait the death of Grazia Vaillant was the sole topic? Did they think there was some mystery about it? Or were they persuading themselves it was from natural causes?\n\nThe choir came out of the vestry door in a talkative stream. Certainly these youngsters were not much troubled by the death of another old lady.\n\nThen came Mr Slipper in hot argument with a tall grizzled man whom Carolus knew to be Waygooze the Organist and Choirmaster. On Slipper's face was a look of petulant, rather childish annoyance.\n\n\"It does not seem much of a concession to ask,\" he said. \"It is only Tuesday nights that I want Stanley. You can have him on Fridays.\"\n\n\"You've already got four nights a week for your activities, Mr Slipper,\" said Waygooze solemnly. \"I'm not going to have the boys interfered with on Tuesdays. They come to choir practice then.\"\n\n\"I don't want them. I've explained to you it's just one boy, Stanley Rogers, to help me get the Institute in order. Unless you'd like him and Cyril Lipscombe to do it turn and turn about? They could do half an hour with you then come down to the Institute.\"\n\n\"And what do you think would happen to the new Magnify Cat if I once started letting them go down whenever they liked? No, Mr Slipper. I'm sorry. Stanley comes to choir practice on Tuesdays and Fridays. You'll have to find someone else for your little job.\"\n\nThe vicar caught them up.\n\n\"Now what are you two good people discussing? Ah, yes. Choir practice. The Institute. But I think that today, on this day of great personal loss to us all and perhaps even more troubles for our little community, we should forget our personal differences. I've no doubt some way out can be found. Let us forget it for the present.\"\n\nWhen Carolus reached the Black Horse he found that festive occasion known through the week as 'Sunday dinner-time 'was in full swing. Sunday suits were stiff and shiny, vegetables dug that morning were on the handlebars of bicycles outside the pub, ale was freely poured and darts were thrown. The conversation today turned on the second sudden death in Gladhurst and most of those present had something to say about it.\n\nFlo led the discussion.\n\n\"You can't call old Griggs's anything but murder,\" she said. \"But I can't see what you make a mystery of this for. The poor old cow had a drop too much, passed out and there you are. Well, we've all got to Go some day so I don't see what all the fuss is about. It reminds me of a fellow I used to go on the back of his motor-bike with, and whenever we went round the corner on the wrong side he'd say, 'No one's going to live for ever, you know' and laugh till he could hardly steer. He used to take me out to an old straw stack and I'd come away covered with chaff, but I didn't mind.\"\n\n\"I don't believe it was an accident,\" said Mr Lovibond. \"I believe one man did for both of them. Just because he did it different ways\u2014that's nothing. It's only now and again you get them doing them all the same way like Jack the Ripper or Brides-in-the-Bath Smith. Two murders in six weeks in one little place! It would be a bit too much of a coincidence if it wasn't one man doing it. Both old ladies with money, too. Both church-mad. What else can it be?\"\n\n\"Never in this world,\" said Mugger from his corner. \"Old Griggs was murdered right enough. Couldn't be anything else when they put her in the grave and that. But this one's done for herself. You can make your minds up to that. Well, what else could she do? She was Past It, wasn't she? You can't blame her with a face like that.\"\n\n\"How can you talk like that, Fred Mugger?\" said Mrs Chester. \"Wicked I call it, with the poor lady not hardly cold yet. And I don't see where you all get your murders from. Bloodthirsty lot you seem to me. You can't even be sure old Griggs was murdered, really. Something might easily have fell on her. Look at that gargoyle that came down out of the church tower last year. It might have been something like that got her on the head. As for the other one, there's no reason at all to think it was murder.\"\n\n\"I don't see that, Mrs Chester,\" said Laddie Grey. \"If Miss Griggs was killed accidentally how did she come to be buried in Chilling's grave where Rumble found her?\"\n\n\"She was murdered all right,\" pronounced Rumble. \"And very nearly so that no one was any the wiser. If I hadn't happened to notice how that grave wasn't so deep as I'd left it she'd be there now. She was murdered, but what I want to know is why? It wasn't just to take her bit of jewellery and money, that's certain. Then who had an interest in murdering her? And this one, too. We don't know what happens to Miss Vaillant's money. If we knew that we might know something.\"\n\n\"Oh talk sense, Rumble,\" said his wife. \"What's the good of talking about Miss Vaillant being murdered when no one's been in there only poor Miss Flora Griggs and me? I've told you and I've told the police, the whole place was barred and bolted except the back door and I'd got the only key of that except hers. You're not going to tell me Flora Griggs did away with Miss Vaillant.\"\n\n\"I've always said those sleeping-pills are dangerous things,\" said George Larkin, unexpectedly joining the discussion. \"You take a couple more than what the doctor tells you and you've had it. That's what she must have done because they say there was a whole empty packet in her bag.\"\n\nFor only the second time in his visits to the Black Horse, Carolus now heard the landlord's son speak.\n\n\"I think she was tight,\" said Bill Larkin.\n\n\"Whatever makes you say that?\" asked Flo.\n\n\"She looked lately as though she'd been getting tight. You can always tell.\"\n\n\"You mean like me when I was going out with that fellow from Burley. Shall I ever forget it?\" asked Flo. \"Gin and pep, gin and pep, till I thought my breath would catch fire. But I didn't mind.\"\n\n# 16\n\nPENDING the report of the inquest on Grazia Vaillant, Carolus had no reason to go over to Gladhurst. The Spring Term was now well advanced and Carolus became absorbed in the exacting work of preparing boys for the Higher Certificate. He deliberately put aside his un-academic problems.\n\nThe news of the second death at Gladhurst had been prominently published in newspapers and it seemed that Mr Gorringer, the headmaster, was attempting, without actually addressing Carolus on the subject, to indicate reproach. There were curious head-shakings and clumsy shrugs while Mr Gorringer's \"Ah, Deene...\" when they met lacked all its customary gusto. Carolus supposed that the headmaster meant to suggest his displeasure with his Senior History Master who, having neglected the incidentals if not the essentials of school life in order to devote himself to investigation, had not been able to prevent the second death.\n\nWhen Carolus either ignored these strange histrionics or appeared to put them down to some physical disorder and enquired after Mr Gorringer's health, the headmaster could keep silence no longer.\n\nAh, Deene, Deene,\" he said, lugubriously. \"I was deeply distressed to read of a second death at Gladhurst. Surely that might have been prevented? One could almost allow oneself the fell notion that death follows your investigations. I should be loath indeed to think such a thing.\"\n\n\"We don't even know what caused her death.\"\n\n\"Oh, quite. One must not be premature in supposing the worst. But I could have wished that in a case you were investigating the death roll could have been limited to one.\"\n\nThe headmaster passed on, his gown billowing, and Carolus made his way to his classroom.\n\nNext day he received the _Burley Watchman_ with a full account of the inquest on Grazia Vaillant. The Coroner, it appeared, had known his job and left nothing un-probed which could assist in revealing the full truth. Grazia had died of an overdose of the barbiturate which was used in Minerval tablets. An empty tube of these was found in her bag, bearing her finger-prints and hers only.\n\nThe pathologist who had conducted the post-mortem had also discovered a quantity of alcohol, probably swallowed in the form of gin and lime juice within a short time before death. The assumption was\u2014though the Coroner stressed that it was no more than an assumption\u2014that Miss Vaillant had drunk a quantity of gin and lime sufficient to upset her judgment and had then swallowed the Minerval tablets.\n\nDr Pinton, the village doctor, gave evidence and said that he had been prescribing Minerval for Miss Vaillant for two years. She was a perfectly reliable patient who knew that she should never take more than one, or at the most two, tablets at a time. Asked if these tablets were particularly dangerous in conjunction with alcohol, the doctor said that this was so if either of the two was taken in excess. He had mentioned to Miss Vaillant that Minerval should not be taken after or with alcohol in any quantity, as he did whenever he prescribed Minerval, but he had done so more as a formality than anything else. He had never had any reason to suppose that Miss Vaillant drank more than a very occasional glass of wine with a meal. He was most surprised to hear the pathologist's report.\n\nHe described Grazia Vaillant as a woman whose feelings could sometimes be violent but who was usually well-disposed, perhaps rather gushingly so, in conversation. She was in no way abnormal, an enthusiast in religious matters but not a fanatic. He would not describe her as unbalanced. A little eccentric, perhaps, and exceptionally anxious to get her own way but not psychopathically so. He had prescribed Minerval for her because she lived so much on her nerves and energy that a tranquillizer at night was beneficial.\n\nAsked how many tablets of Minerval he thought would be fatal he said that with the alcohol she had drunk he thought six would be sufficient, though death would not necessarily be immediate but would be preceded by first a period akin to inebriation, then coma, then death.\n\nMrs Rumble, eyeing the Coroner and everyone else with the greatest hostility, had given details of Miss Vaillant's private life. Asked if she had reason to think Miss Vaillant drank spirits when alone she said: \"I don't know about spirits. She liked a drop of gin.\" However, she gave no details of the quantity purchased or where it was bought, only saying that she had seen gin in one of the cupboards and that though the empty bottles were always thrown away and the glass washed up, it didn't deceive her.\n\n\"Besides,\" she added, \"Anybody doesn't get through all that lime juice in the middle of winter unless it's to drink with Something, do they?\"\n\nShe recalled the two visits of Millicent Griggs just before that lady's death and said that 'as far as she could tell' the conversation had been amicable. She had noticed when she showed out Miss Griggs that she looked 'flushed up' and as though she'd 'had one or two' but she might have been mistaken. As for the visit from Miss Flora Griggs on the day of Miss Vaillant's death Mrs Rumble 'couldn't say much'. It was a Saturday and she was in a hurry to get off and do her week-end shopping. She had shown in Miss Flora and 'seen them settled without any words' in her presence. Asked if she meant the ladies hadn't spoken she looked at the Coroner as though she was sorry for his stupidity. 'Of course they spoke,\" she said. \"I mean they had no Words while I was there.\" It dawned on the Coroner that this indicated not conversation in dumb-show but absence of any altercation, so the point was left behind.\n\nAsked whether Miss Flora Griggs was still in the house when she left, Mrs Rumble said, yes. She had gone to the sitting-room where they were to say good-afternoon before leaving. She had already been paid. She found the two ladies looking a little 'worked up' if they knew what she meant, but she still heard no actual Words. Just as she came into the room she heard Miss Flora say something like 'your idols may be broken 'and she had thought that sounded a bit nasty till she had heard it followed by Ezekiel something or other and realized that Miss Flora was only saying bits of the Old Testament as she often did. She wondered now whether it had referred to the new statue which Miss Vaillant had got down from London and which was upstairs all in the sacking it had come in. Miss Vaillant had told her this was for the new Lady Chapel they were making in the church and the vicar had just agreed to it.\n\nFurther she said that Miss Vaillant had seemed in good spirits lately and that she, Mrs Rumble, would never believe whatever anyone might say, that she had done away with herself on purpose. In her view, a drop of gin sometimes when you felt a bit low was one thing and taking your own life was another, a dictum which met no contradiction though the Coroner recalled the witness to the facts of the case rather than her own opinions.\n\nThe greatest surprises during the inquest were provided by a Mr Sturdis, a London solicitor who had handled the dead woman's affairs. Her real name, it appeared, was Grace Vallance and she was the daughter of a city outfitter who had wisely sold his business and premises before the great monopoly chain stores had pushed out small enterprise, He had retired to Folkestone with a comfortable fortune. He had accommodatingly died before dying had been made too expensive and Miss Vaillant had enjoyed an income, with all taxes paid, of some two thousand pounds a year to which she had added a thousand by progressive but slow reduction of capital. It was on his advice that she had adopted this course since she had no near relatives. The system had been calculated to provide a sufficient income however long Miss Vaillant's life might be and whatever further burdens of taxation might be laid on her shoulders. Mr Sturdis looked rather complacent about it.\n\nHer will was a somewhat complicated one but the chief beneficiaries would be the Reverend Bonar Waddell and the Parish of St Jude, Gladhurst. There were conditions laid down for this, however, which might cause some difficulties, as for instance the introduction of incense during the service known as Sung Eucharist, the use of a Sanctus bell, something called Reservation for the Sick and a number of other changes which must be made in the church ritual before the local charities or the vicar himself could benefit under the will. A codicil had recently been added for the benefit of those charities, chiefly, Mr Sturdis understood, among the youth of the parish, sponsored by the Reverend Peter Slipper. These again were restricted by conditions. The Scout Troop, for instance, was to receive an annuity of \u00a3100 for the purchase of equipment, etc, but this was payable only if it could be shown that eighty per cent of the boys over fifteen were regular communicants. There was also a sum of \u00a3500 for Mrs Rumble.\n\nIt would be a difficult will to execute, Mr Sturdis said, and he did not envy the executors appointed, Commander Fyfe and Mr John Waygooze.\n\nNo evidence was given of any other caller at the Old Vicarage, either on the Saturday or during the evening or night.\n\nThe police said their piece, the vicar was called, then Miss Flora Griggs described her call on Grazia Vaillant at the latter's urgent request. Her evidence was somewhat confused and rhetorical, and really added very little to that already known. Miss Vaillant, she said, had _plied_ her with strong drink, and when asked to be a little more explicit she let fly a covey of quotations from the Old Testament which seemed to confuse if not to embarrass the Coroner.\n\n\"Do you mean she poured out a drink for you?\" he asked, trying to make poor Flora more specific.\n\n\"'Take the wine of the cup of this fury at my hand',\" said Flora and the Coroner was puzzled till she added, \"Jeremiah XXV, 15.\"\n\n\"Please be definite about this, Miss Griggs. Were you actually handed a glass containing gin?\"\n\n\"I have no idea what it contained,\" said Flora. \"I knew it was an abomination and would not take it in my hand.\"\n\n\"So two glasses had been poured out?\"\n\n\"I think so.\"\n\n\"You refused them?\"\n\n\"Most indignantly.\"\n\n\"Did you shortly leave the house?\"\n\n\"I paused only to leave on the table some short tracts, which if this unfortunate woman would have turned in her wickedness and read, would have saved her soul alive.\"\n\n\"I see. And when you left she was in good health and spirits?\"\n\n\"'Woe unto them that rise up early in the morning to follow strong drink; that tarry late into the night, till wine inflame them! 'Isaiah V, 11.\"\n\n\"Do you mean that Miss Vaillant already showed the influence of alcohol?\"\n\n\"She was a wine-bibber. That was sufficient.\"\n\nThe Coroner gave it up.\n\nThe verdict was eventually and inevitably one of Accidental Death by poisoning caused by an overdose of barbiturate.\n\nFor some days after this Carolus did not go over to Gladhurst and was interested to realize that five weeks had now passed from the date of Millicent Griggs's death. It was at least comforting to know that the police were taking as long as he was to reach any sort of conclusion. Now and again a daily newspaper would remind the public of this by talking about the unsolved mystery and recalling the fact, no doubt considered picturesque, that the body had been buried in an already dug grave. This had gripped Press imagination from the first.\n\nOne evening as he finished dinner in his home, Carolus had a surprise. Mrs Stick came in showing every sign of her most ardent disapproval.\n\n\"There's someone to see you,\" she said. \"I've shown him into the little sitting-room because I didn't know what else to do, but I thought you said we weren't going to have any murderers and that coming here this time?\"\n\n\"What's his name?\"\n\n\"Griggs, he says, and it didn't take me long to know where I'd heard that name. I haven't said you're in yet, only that I'll go and see, so what shall I tell him?\"\n\n\"Show him in, please, Mrs Stick.\"\n\n\"There! I knew it. And Stick and me just settling down to the television. Now we shall have to miss half the programme because I wouldn't turn the lights down with anyone like that in the house, not for anything.\"\n\n\"You could lock your door.\"\n\n\"It wouldn't be the Same Thing,\" said Mrs Stick and went out to usher in Dundas Griggs.\n\n\"Had a job to find you, old man,\" this one began.\n\n\"You had a job to find _me_?\"\n\n\"Yes. I knew you taught here but I couldn't find out where you lived. In the end someone showed me the headmaster's house and I asked him.\"\n\n\"Oh.\"\n\n\"Inquisitive old character, isn't he? Wanted to know my name and business and everything. I nearly said I was a copper come to arrest you.\"\n\n\"I hope you did nothing of the sort,\" said Carolus, picturing Mr Gorringer as he received the news.\n\n\"No. I just said it was in connection with the Gladhurst Case.\"\n\n\"That was bad enough.\"\n\n\"Oh, I don't know. He seemed quite intrigued. 'Any new developments?' he asked in a stage whisper. I was just as corny. I put my finger on my lips. He nodded, and gave me your address. So here I am.\"\n\n\"Have a drink?\"\n\n\"I don't mind. Dropper Scotch if you've got one. I've come to see you about something rather serious.\"\n\n\"Yes?\"\n\n\"It's my Aunt Flora. She's in a bad state, old man.\"\n\n\"She seems to have been somewhat distressed at the inquest.\"\n\n\"It's since then. I don't know if she's going off her rocker. She has started accusing herself now. Fortunately only in the household. But you never know where that will stop.\"\n\n\"Accusing herself of what?\"\n\n\"What do you think? Murdering Grazia Vaillant, of course.\"\n\n\"But...\"\n\n\"It appears that when she called on her that afternoon old Vaillant tried to get her to take a tot. Vaillant said, and for all I know it may have been true, that Millicent hadn't been above it on those last two occasions she called on her. Whether or not there was anything in this, she had backed the wrong horse with Flora. The old girl 'started like a guilty thing surprised' and almost threw it back in Vaillant's face.\"\n\n\"Yes. We know all this.\"\n\n\"But what you don't know is when Flora was a bit calmer she told Vaillant the best way of resisting the temptation to drink the awful stuff. Cheerio, by the way. If the terrible habit had such a grip on her that she couldn't resist, she was to take one of these dear little pills which Dr Pinton had prescribed and she would find herself asleep in ten minutes with all temptation gone. And she gave her a couple of Minerval tablets.\"\n\n\"But Grazia Vaillant was already taking Minerval.\"\n\n\"So it appears. But she never told Flora. Perhaps she didn't want to hurt her feelings. I don't know. Now Flora thinks she killed her.\"\n\n\"If those are her only grounds for thinking so, it's ridiculous.\"\n\n\"That's what I tell her. But she won't listen to me. Threatens to go to the police and give herself up.\"\n\n\"I shouldn't worry about that. Champer's cocky and self-important but he's not a half-wit. In murder cases the police often get quite a number of peculiar people accusing themselves. So long as all she has to confess is handing over two Minerval tablets, I don't think your aunt will be in trouble.\"\n\n\"You won't go and see her, then?\"\n\n\"I don't think it's necessary. I'm pretty busy at the school just now. I warned Mrs Bobbin in the first place that this could only be a part-time job for me.\"\n\n\"There is another aspect of it, old man.\"\n\n\"Yes?\"\n\n\"Flora's inclined to be a suicidal type.\"\n\n\"I hadn't heard that.\"\n\n\"Runs in the family. My grandfather, who built the Griggs Institute and what-not at Gladhurst, cut his own throat. They could get away with calling it an accident in those days. I shouldn't be a bit surprised to see Flora do something of the sort particularly if the police take no notice of her confession.\"\n\n\"Still, I don't quite see what I can do about it. You had better call in a psychiatrist.\"\n\n\"Think so? You may be right. Must do something. I thought at least I'd put it to you.\"\n\nCarolus was wondering what it was he disliked about Dundas Griggs. His live-wiriness? Or his habit of saying 'old man'? No. Something deeper than those.\n\n\"I mean,\" added Griggs rather fatuously, \"we don't want _another_ death in Gladhurst, do we?\"\n\n\"Personally, I didn't want any,\" said Carolus shortly.\n\n\"No. Of course. I meant....\"\n\n\"I think I understand what you meant. Have another drink?\"\n\n\"Thanks. You see it's a bit uncomfortable when you stand to benefit.\"\n\n\"I suppose so.\"\n\n\"Not that there's likely to be much. If I know anything about Flora it will mostly go to her pet charities. But there's bound to be something.\"\n\n\"You feel that makes you a suspect?\" said Carolus rather brutally.\n\n\"How can I be a suspect? The old girl's not even dead yet.\"\n\n\"Two women are.\"\n\n\"Yes. But... That's absurd, of course. No one could possibly....\"\n\n\"Most murders are like that, Mr Griggs. No one could _possibly_ have done them. Yet always somebody has.\"\n\nDundas Griggs stood up.\n\n\"Thanks for the drink,\" he said. \"I've said what I felt I ought to say. Since my aunt, Mrs Bobbin, consulted you, I mean. I'll run along.\"\n\nCarolus showed him to the front door.\n\n# 17\n\nWHEN Carolus next went over to Gladhurst he found the Reverend Bonar Waddell in some perplexity.\n\n\"The testamentary dispositions of these two good ladies,\" said the vicar, \"well-meaning and generous though they are, have given us all food for thought. Lawyers, executors, churchwardens and clergy, we are all concerned to follow their wishes as far as we are able and at the same time not to lose the little benefits for our parish institutions which they offer. My own share I would gladly sacrifice if it helped to provide a solution, but unhappily the same conditions apply to the church and parish charities as to the bequests to me.\"\n\n\"It must be very difficult for you,\" said Carolus, fascinated as usual by the vicar's agility in proceeding in two opposite directions at the same time.\n\n\"At a first glance,\" said Mr Waddell, \"the conditions they lay down might seem to conflict one with another. Miss Griggs, for instance, stipulates that there shall be no Popish practices in the church while Miss Vaillant insists on the introduction of certain items... details... small etceteras of ritual. One has to realize at once the force of the word Popish. Far be it from me ever to introduce anything to which that word could justly apply. If I, for instance, were to appear in the _triregenum_ , the tiara or triple crown worn by the Pope, with a cross ornamented on my shoe for the kisses of my congregation, I think there would be no doubt that I should be indulging in what Miss Griggs calls 'Popish practices', but I cannot see that fulfilling the simple conditions laid down by Miss Vaillant can be held to be so.\"\n\n\"I see your point.\"\n\n\"Then Miss Griggs stipulates that there shall be 'no chanting of Masses'. Have I not frequently expostulated with Miss Vaillant against her use of that word with its most unfortunate associations? No, there shall certainly be no chanting of Masses. We have Sung Eucharist and that is quite enough.\n\n\"There is another seeming conflict. Both wills make reference to confession, Miss Vaillant insisting that the 'Sacrament of Confession' shall be encouraged while Miss Griggs makes it a condition of her bounty that 'no confession boxes' shall be seen in the church. We shall therefore have to do without any of those\u2014in any case rather ugly\u2014wooden erections which some Anglican and all Roman Catholic churches have installed. Then Miss Griggs, fortunately using somewhat archaic language, demands that there shall be no 'graven images' in the church. She could be assured, if she were alive, good soul, that the life-sized statue which Miss Vaillant has purchased for our new Lady Chapel is certainly neither 'graven' nor an 'image'; it is a plaster figure of Our Lady of Lourdes, admittedly purchased from the firm of Burns, Oates and Washburne, but Lourdes now has surely ceased to be a merely sectarian place of pilgrimage.\n\n\"None of the other little demands of Miss Vaillant conflicts with the conditions of Miss Griggs, the use of a Sanctus bell, the ringing of the Angelus, Reservation of the Sacrament, until we come to one which both ladies mention specifically and which puts us all in a quandary.\"\n\n\"What is that?\" asked Carolus with genuine interest.\n\n\"Incense!\" cried the vicar. \"Miss Griggs says there must be none of that 'nauseous and Roman Catholic incense' in the church and Miss Vaillant makes the ceremonial use of incense at Sung Eucharist, at the Gospel, Offertory and Elevation, a condition for all her benevolence. What am I to do?\" asked Mr Waddell.\n\n\"You should be able to see a way, surely,\" encouraged Carolus.\n\n\"I hope, perhaps I may have done so,\" said the vicar. \"It is, you will observe, nauseous and Roman Catholic incense' which Miss Griggs objects to. We will have none of it. We will purchase our incense from the excellent old Anglican firm of Mowbray's which will be quite another matter. No one can then say that we are ignoring the wishes of either of the dead and the parish will benefit accordingly.\"\n\n\"Excellent,\" said Carolus in congratulation. \"There was something else I wanted to ask you about. Laddie Grey and Naomi Chester are both, I believe, parishioners of yours?\"\n\nThe vicar pursed his lips.\n\n\"They both live in the parish. Grey and his wife were married over at Breadley where she lived but I christened their child. Naomi Chester I do not often see in church.\"\n\n\"They hope one day to get married,\" said Carolus.\n\n\"That depends on a number of things,\" pronounced the vicar. \"By the law of the land, Grey's wife's insanity does not provide grounds for divorce until the unfortunate woman has been confined as incurable for five years. That leaves nearly two years in which some change may take place.\"\n\n\"Then?\"\n\n\"Then, if Grey is given a divorce, I shall have to decide whether or not I should be doing my duty in marrying him and Naomi Chester. On the one hand under the law of 1937 I cannot be compelled either to marry them or to allow their marriage to take place in my church. Convocations, indeed, have resolved that marriages of divorced persons should not take place in church. On the other hand civil law allows me to use my own discretion. I shall have to decide.\"\n\n\"Have you explained this to either of the parties?\n\n\"I have had no occasion to do so.\"\n\n\"So that if they decide to take the matter into their own hands and simply start living together....\"\n\n\"Dear me! You alarm me, Deene. In my parish? I trust nothing of the sort would occur to them. It would set a most unfortunate example in a small community such as this. Really, I scarcely think you can be serious. Co-habit? Oh, no. I cannot suppose... The parents would surely dissuade them.\"\n\n\"Thank you, Mr Waddell. I just wanted to know the position.\"\n\nHis next call was at a pleasant-looking house at the lower end of the village. Here he asked for Dr Pinton.\n\n\"Was it private?\" asked a smart young woman in nurse's uniform.\n\n\"Yes.\"\n\n\"Will you wait in here a minute? Dr Pinton's just finishing his National Health. He's had rather a rush this morning. Colds and sore throats.\"\n\nThe doctor was an active little man with pale ranidaean features and smooth thinning hair.\n\n\"Morning,\" he said, \"What's the trouble?\"\n\nHis big mouth worked like a trap.\n\n\"I'm sorry to take your time, Doctor. I'm that most irritating of all things\u2014a private detective. Mrs Bobbin has asked me to try to find out about her sister's death.\"\n\n\"Oh, I see,\" croaked the doctor. Really he did look as though he had just come out of a pool. \"Don't see how I can help. I had nothing to do with it.\"\n\n\"It was about these tranquillizing pills you prescribed for Grazia Vaillant.\"\n\n\"Best thing possible. Safe. Not habit-forming. Always prescribe them for neurotic old women. Nothing wrong with any of them except their nerves. Can't stand the pace of modern life. Brought up to different standards. Find themselves harassed to death. Money worries. No servants. World upside down. What can you do? Give 'em tranquillizers.\"\n\n\"But isn't it rather dangerous, as in the case of Grazia Vaillant?\"\n\n\"How was I to know she was on the booze? Of course it's dangerous if you take too many when you're tight. But normally, no. Have to understand these old middle-class women, Deene. They're damned annoyed with things as they are. They suddenly realize that the world has no use for them. Wants to see the back of them, in fact. They bumped them off in Russia and I'm not sure it wasn't more humane. We're just letting the species die out. Some of them can see that and are furious about it but there's no remedy. The least I can do is smooth their last years.\"\n\nCarolus nodded.\n\n\"Old women! You never stop to think of them, do you? I don't mean only those from middle-class families but working class as well. Thirty, forty years ago the better-off ones had companions and went down to death selfishly, perhaps, but decently and without humiliations. The poor ones, even old widows of farm-workers, kept their own cottages to the last, when a cottage cost half-a-crown a week. Yes, and tottered round their little bits of garden and had visits from their grandchildren and perhaps their great-grandchildren, and made jam and homemade wine if they lived in the country and crept round to the pub with a jug sometimes if they lived in town. Now what happens? Herded into homes to sleep in dormitories and obey the rules like children. Or, if they've got enough money to keep them out of that, have the pestered, anxious, artificial existences that these old women here had. Do you blame old Mrs Bobbin, who has character, for being angry? Do you blame her sister for getting a sort of religious mania? Do you blame Grazia Vaillant for taking to the bottle, as apparently the old girl did, lately? Above all, do you blame me for prescribing anything I can that gives them a few hours' worry-free sleep?\"\n\n\"No,\" said Carolus. \"I don't. But I would like to know for whom you prescribed it?\"\n\n\"Only those who could afford it, unfortunately. The National Health Scheme won't run to Minerval. Millicent and Flora Griggs. Mrs Bobbin didn't need it. Grazia Vaillant. Agatha Waddell, sometimes. And of course Fyfe's wife.\"\n\n\"Why 'of course'?\"\n\n\"Hypochondriac. Leads him the hell of a life.\"\n\n\"But he doesn't need Minerval himself?\"\n\n\"Oh, yes. Fyfe takes tranquillizers. What do you expect? We're living in the atomic age, Mr Deene.\"\n\nCarolus made no comment.\n\n\"Do you keep any check on these tubes of Minerval?\"\n\n\"Only in the sense that I make a new prescription for each one and never give more than one in ten days.\"\n\n\"Never?\"\n\n\"Well, as Gilbert says, well 'hardly ever'. There was an occasion, about a week before Millicent Griggs died, when she had lost her supply. Silly old thing had dropped it, or something. I gave her a second lot. But that was extremely rare.\"\n\n\"Tell me, as a matter of interest, do your confreres generally follow your practice in this?\"\n\n\"Never discuss it. I should imagine so. This is much like any other village in Great Britain, Deene. You schoolmasters are still living in the pre-war age.\"\n\n\"Perhaps. Thank you for all you've told me.\"\n\nBut none of it, thought Carolus, did much to explain the one, the basic, the central fact of murder.\n\nThe telephone rang and Dr Pinton excused himself as he answered it, then rose to his feet.\n\n\"An accident,\" he said, \"up at the church.\"\n\nCarolus followed him from the house and got into his car, then decided to keep behind the Doctor on his way and see what had summoned him.\n\nAt the lych-gate was Rumble, for once without his grin. Dr Pinton pulled up short, jumped out and faced him.\n\n\"Where?\"\n\n\"This way, Doctor.\"\n\nCarolus followed them. The West door of the church stood open but in the early spring afternoon the interior looked gloomy. There seemed to be no one else present and Rumble made for a little door at the West end under the tower. This opened with a creak and revealed a spiral stone staircase.\n\nRumble switched on a light and started ascending, Pinton behind him and Carolus a slow last. It seemed a long ascent\u2014spiral staircases always exaggerate the distance\u2014but at last Carolus reached the place in which the bell cords were looped against the walls. The other two had already ascended farther and, pausing only a minute to notice the eight great swollen ropes, Carolus followed.\n\nThe staircase this time brought him to the bells themselves and he saw the two men leaning over someone on the ground. Beside them was Mrs Rumble, gaunt and serious.\n\nCarolus could see that it was the figure of a woman on the ground and after a moment came near enough to recognize Flora Griggs. She was apparently unconscious.\n\nPinton saw Carolus.\n\n\"Will you please go and phone for an ambulance? 2244. Give my name and say it's urgent.\"\n\nCarolus just had time to see a staircase, scarcely more than a ladder, ascending beside where the woman lay. This, he calculated, would lead to the open top of the tower.\n\n\"You can phone from the Old Vicarage,\" said Mrs Rumble, who had continued to 'keep the place decent' as she said. \"I left the back door open.\"\n\nCarolus hurried down and did as he had been asked. He entered through the silent kitchen of what had been Grazia Vaillant's home and found the telephone in the hall. He gave his message.\n\nHe did not climb again to the belfry but waited to direct the ambulance men when they arrived. In a surprisingly short space of time the ambulance drew up and Carolus explained to two cool and efficient individuals in uniform that there had been an accident in the tower and that it was apparently a stretcher case.\n\nMeanwhile the vicar had arrived, also Mrs Bobbin. He told them what he could and they all stood waiting for the stretcher to be brought down.\n\nJust then Mr Slipper accompanied by two large Scouts came running up.\n\n\"I hear there's been an accident,\" he said. \"I've brought Stanley and Cyril because they've both passed their First Aid tests.\"\n\nThe vicar raised his hand.\n\n\"Not now, Slipper. This is not the time.\"\n\n\"But...\"\n\n\"No. No. Some less grave occasion, thank you. Send the boys home, there's a good fellow.\"\n\nAt that moment the two ambulance men appeared, bearing Flora Griggs on the stretcher. In a moment she was put in the ambulance and this disappeared swiftly in the direction of Burley.\n\n\"She'll be all right,\" said Dr Pinton. \"But how on earth did it happen?\"\n\nEveryone seemed to turn towards Rumble and the vicar said, \"I think perhaps we might discuss this in the vestry. This biting wind....\"\n\nSo Mrs Bobbin, the vicar, Rumble and his wife, Dr Pinton and Carolus went into the church. The vicar unlocked the vestry door and soon the little group was seated.\n\nRumble began at once.\n\n\"I don't know what made me follow her,\" he said. \"She had a funny look, I thought.\"\n\n\"Funny?\" queried the vicar.\n\n\"She looked as though she didn't hardly know what she was doing. I was in the garden at the time and I saw her come out of the front door, not seeming to know which way to take. So I got my jacket on, let her get a bit of a start and followed her.\"\n\n\"You should have called me,\" said Mrs Bobbin.\n\n\"I didn't know you were in. Anyway, off she went in the direction of the church. It gave me a turn to remember that it's not a few weeks since the other Miss Griggs must have come this way at about the same time and we all know what happened....\"\n\n\"Never mind that now,\" said the vicar sternly.\n\n\"Well, she came on, but not as though she was decided what she was doing as you might say. She seemed to stop here and look there till really I wondered whatever it meant. Then when she got to the gate of the church I didn't know what to think till she turned off suddenly to the left towards where her sister's buried in the new part and I understood what she was up to, or thought I did. She just wanted to visit her sister's grave.\"\n\n\"Get on, man, get on,\" said the vicar.\n\nBut this was Rumble's story and never in his life had he had such an attentive audience. He was not going to be hurried.\n\n\"She stood there for a minute, not interfering with anything, then she turned round before I knew where I was and marched off to the church door. I had to hop behind the old yew tree quick or she'd have seen me. But the funny thing was, I knew she was up to something when she went into the church. Don't ask me how I knew. Something told me.\"\n\nRumble paused dramatically and the vicar drummed on the vestry table. Then Rumble resumed.\n\n\"By the time I'd got to the church it was empty. Not a soul in sight. That really _did_ make me feel queer. I looked everywhere for her. There wasn't a sign of her. I looked in the pews and behind the organ. I couldn't think for a minute wherever she could have gone. Then it dawned on me. She'd gone up in the tower. But it shows how artful they are, doesn't it? She'd closed the door behind her. So I started up those stone steps as quick as I could.\"\n\n\"Could you hear anything?\" asked Carolus.\n\n\"Not at first I couldn't. I got up as far as where the bell-ringers are and stopped for breath but there was still no sign of her. I went across to the door of the staircase that leads up to the bells and then I did think I heard something up ahead of me. So I shouted out 'Miss Griggs! ' at the top of my voice. Whatever it was I heard seemed to stop. It was as quiet as a mouse up there and I hurried on to get to where the bells are and just as I was going round that blasted staircase I heard like a scream and a thud up above me. By the time I got to the belfry there she was on the floor just as you saw her.\"\n\n\"So what did you do?\" asked the vicar.\n\n\"Do? It was a job to know what to do. She was breathing all right so I knew she was alive. I did the best I could. Put her as comfortable as possible and ran down all the way and across to the Old Vicarage to find my wife. Women are best for that job. She knew in a minute what was best to be done. 'Phone Dr Pinton', she said, 'and tell him what's happened. I'll go and look after her'. Then she grabs hold of a couple of cushions and takes a jug of water and she's gone. So I phoned Dr Pinton.\"\n\n\"You mean to say, then,\" said the vicar importantly, with a side-glance at Carolus, \"that if someone was concealed in the tower and made an attempt on her life, he had an opportunity of getting clean away?\"\n\n\"Oh, come,\" said Dr Pinton. \"As far as I could see all Miss Griggs was suffering from was a little shock and a twisted ankle. Surely we needn't start imagining yet more murderers, need we, vicar?\"\n\n\"I am the last to see things in a morbid light. But it did occur to me. What do you think, Mr Deene?\"\n\nCarolus said he agreed with the doctor.\n\n\"On the face of it it looks like an accident. The question of course remains of why Miss Griggs should wish to climb to the top of the tower at all.\"\n\n\"Ah, yes,\" said the vicar. \"That indeed is a question. But doubtless all will be made clear in time.\"\n\nMrs Bobbin spoke sharply.\n\n\"I shall inform the police,\" she said and walked out of the vestry.\n\n# 18\n\nTHE home which Carolus had purchased at Newminster when first he had taken up his post at the Queen's School was a small, charming Queen Anne house set amid other buildings but with a few yards between its walls and theirs. Behind it was a small walled garden, just now showing the earliest Spring bulbs.\n\nCarolus loved his comfortable retreat, the first place he had found after the tragic death of his young wife. He knew that in their way the Sticks loved it, too, and for all their threats would not leave it or him. He decided, as he sat at lunch on a Saturday early in March and looked out on the signs of the soil's awakening, that he would make only one more journey to Gladhurst, then leave the village and the case to develop as it pleased.\n\nHe knew how the two women had died and it was now, as it had been from the first, a case somewhat repugnant to him. He did not want to be the means of hanging anyone over this, he wanted to disappear wholly from the scene and to forget it. But first, for his own satisfaction, he must clear up one or two small points at Gladhurst.\n\nHe hoped his lunch was finished and that Mrs Stick had gone to fetch the coffee but knowing her conviction that it was good for him to 'fill up with something sweet' he feared that his escape from the table would not be so easy.\n\n\"I've got a nice cream broolay for you,\" said Mrs Stick proudly. \"I'll bring your coffee when I've seen you've had a good helping.\"\n\n\"I've decided to give up the Gladhurst case, Mrs Stick,\" said Carolus, as he helped himself gingerly to the _cr\u00eame brul\u00e9e_ before him.\n\n\"That's good. You were only wasting your time and mixing with every Tom, Dick and Harry.\"\n\n\"I have to go over there once more, this afternoon, then I shan't go again.\"\n\n\"I don't know what you want to go at all for. I hear there's another one been pushed off the church tower now.\"\n\n\"You hear the most extraordinary stories. Miss Flora Griggs fell a few steps down the belfry ladder and twisted her ankle. She is already far better.\"\n\n\"Well, anyway, I suppose you know all about it, so why do you want to go over there again?\"\n\n\"Not quite all.\"\n\n\"It's better than them coming here. You'll be back for your dinner, won't you? I've got a lovely bit of lamb for you\u2014well, pray salay you could call it really.\"\n\n\"Keep it till tomorrow, Mrs Stick, I may be late tonight.\"\n\n\"Then you'll want sandwiches, I suppose? There's a little of the pattay left for them. You get back as soon as you can, Sir. I was only saying to Stick, we shall have you down again if you go on working like this.\"\n\nCarolus, too, was glad that it was his last visit and that for the first time he was going to back out of a case before the results of his investigations were known. From the first, nausea and interest had gone together in his mind.\n\nHe was lucky enough, as he drove into Gladhurst at about half-past three that afternoon, to meet one of the people he most wanted to see.\n\n\"Naomi!\" he called.\n\nShe turned, looked a little put out perhaps at the familiar form of his address, then smiled in a friendly way.\n\n\"After me again?\" she said. \"I thought you weren't going to ask me any more questions.\"\n\n\"As a matter of fact I'm chucking the whole thing up. You won't see me here again. But come and have a cup of tea with me at Henson's. I've got something to tell you. And two things, both quite small, to ask you.\"\n\n\"All right,\" said Naomi.\n\nHenson's the bakers, as Dundas Griggs had told Carolus, 'did teas' but it seemed to startle them somewhat when Carolus walked in with Naomi. They were shown to a room upstairs in which an oil-stove was quickly lit and gave out a strong smell and a weak heat. Carolus supposed that it was only the floor which was covered with linoleum\u2014somehow there was an impression of it being everywhere. But at least they were alone.\n\n\"Yes. I've decided to get out of it and leave the police to their own devices. And my strong belief is that those devices won't cause trouble to anyone.\"\n\n\"That's good,\" said Naomi, pouring out.\n\n\"I know the truth, pretty well,\" said Carolus, watching her steadily.\n\n\"I thought you did. Almost from the first.\"\n\n\"At the first it was only a guess. Now I think I've got it worked out.\"\n\n\"Oh, well,\" said Naomi dully.\n\n\"There's one thing I want to know and I think you can trust me. Did you actually _do_ anything to her?\"\n\nThere was no nonsense about Naomi. She did not pretend she wondered, as well she might have done, about which of the three women Carolus spoke. She looked back at him and her eyes were candid and perhaps a little scornful.\n\n\"I never touched her!\" she said, then added quietly, almost mournfully, \"it was only the bucket of water.\"\n\n\"Thank you. The other thing I want to know is\u2014what did Millicent complain about?\"\n\n\"The staircase,\" said Naomi.\n\n\"I see. Now it's all pretty clear.\"\n\n\"To you it may be. What about the police?\"\n\n\"I am not in the confidence of Detective Inspector Champer. In fact he greatly resents my interest in affairs in Gladhurst. But from my observation of his procedure and from what I know of the questions he has asked, I do not think his theories in the various cases depart much from what one must call the obvious. I think he is working on the supposition that Millicent Griggs was murdered, as many another old lady has been of recent years, for the sake of a few jewels and a sum of money, the value of which had probably been exaggerated by report. The fact that the jewels were left concealed does not, I believe, preclude this theory for Inspector Champer though it strongly suggests a local man. My guess at his chief suspect would be Mugger unless he has found a better among the inhabitants of Hellfire Corner. It is difficult to see how he will get enough evidence to make an arrest unless the numbers of some of Miss Griggs's treasury notes are known. Even then, if they are traced to any individual, there wouldn't seem to me much justification for a charge of murder.\"\n\n\"Thank heavens for that.\"\n\n\"I think, Naomi, you've been a very silly girl. You might have got others into trouble by behaving as you have. But luckily, unless I am mistaken, the police will write off Gladhurst as another unsolved crime and turn to easier forms of prosecution.\"\n\n\"But you're doing almost the same.\"\n\n\"There is a difference in that I'm doing it having satisfied myself that I know the truth. I want a last meeting with one or two of the people concerned, then I shall disappear.\"\n\n'One of the people concerned' was not Detective Inspector Champer but he was the first person Carolus met on leaving Henson's, Carolus knew it annoyed Champer to be recognized as a policeman. He could not surely think that his 'plain clothes' concealed his calling in this small village, but the habit of cultivating anonymity persisted. Carolus therefore greeted him cheerily.\n\n\"Good afternoon, Detective Inspector,\" he called. \"How are you?\"\n\nHe resisted the temptation to say 'how's tricks?' or 'booked any good reds lately?' or any of the more fatuous gags used in addressing policemen. The loud use of the man's rank was enough.\n\n\"Still here, then?\" said Champer. \"I thought you brilliant gentry saw through a .little case like this in the first five minutes.\"\n\n\"As a matter of fact, I am pulling out, you will be glad to hear. Packing up. Finishing. One more evening at the Black Horse and I shall have left Gladhurst.\"\n\n\"Too hard a nut to crack, eh?\"\n\n\"Oh, I have my little theory.\"\n\nChamper gave a forced and throaty laugh.\n\n\"You have, eh? I bet it's something startling and original.\"\n\n\"Not really. It follows very much the conventional lines.\"\n\n\"You're not going to tell me you accept the police explanation of events? One murder, one attempted suicide, and one accident?\"\n\n\"Yes,\" said Carolus.\" As a matter of fact I do.\"\n\nChamper laughed again.\n\n\"Well, I'm blowed,\" he said.\" I never thought I'd meet a private detective who believed what was in front of his nose.\"\n\n\"I'd add that whoever may have been in the vicinity I have little doubt that Miss Flora Griggs was climbing the tower in order to throw herself off it.\"\n\n\"Better and better!\" said Champer. \"We don't seem to disagree on a point.\"\n\n\"I don't think we do,\" said Carolus; then, unable to resist a somewhat petty triumph he added: \"In fact, there's only one difference. I know who was the murderer and you don't. Good-bye, Inspector. We shan't meet again, on this case, anyway.\"\n\nThe last night Carolus intended to spend in the Black Horse would be, he hoped, a busy one. It was Saturday and there was no reason why he should not see most or all of those connected with the case. He arrived early and told George Larkin to let him know if either Commander Fyfe or Dundas Griggs went into the saloon bar.\n\nGeorge Larkin nodded but kept the taciturn silence he broke so rarely and sometimes so dramatically. This evening, after three minutes of staring thoughtfully at the counter, he showed which direction his thoughts had taken by remarking suddenly: \"I've always said that step-ladder in the tower was a danger.\"\n\n\"Oh, have you been up it?\" asked Carolus.\n\nGeorge Larkin stared back and answered with the unwilling monosyllable, \"Yes.\"\n\nRumble, who had entered in time to catch the gist of this, said, \"So has that bloody police inspector. Looking at everything, he has been, as though he thought a murder had happened up there instead of a poor lady twisted her ankle.\"\n\n\"Oh?\"\n\n\"Yes, and asking me questions about where I was and saying it happened at the same time of day as the other Miss Griggs, and I don't know what not. I'm sure he connects the two. Perhaps he thinks Miss Millicent Griggs was murdered in the tower.\"\n\n\"Like the little princes,\" said Carolus.\n\nFlo entered and quickly became the centre of the conversation.\n\n\"I saw the ambulance,\" she began. \"Couldn't help seeing it....\"\n\n\"Where?\" interrupted Carolus.\n\n\"Trust you for asking awkward questions. I was just coming up Church Lane, if you _must_ know. And it's no good you asking me who I was with because it's no business of yours. Though it wasn't Mugger, if that's what you think, but someone who's thought far more respectable. Not that I mind what he's thought. As I say I was coming up Church Lane and there was the ambulance and very soon out they come carrying poor old Flora Griggs on a stretcher. It was all I could do to stop laughing when I thought that last time she saw me she stopped me in the street and said, 'Upon every high hill and under every green tree, thou didst bow thyself playing the harlot', Jeremiah. That was a nice thing to say to anyone, wasn't it? Still I didn't wish her any harm, because I've always thought she was a bit touched, if the truth were known, and you can't take offence if someone doesn't know what they're saying, can you?\"\n\n\"Did you know what had happened to her?\"\n\n\"Not till later I didn't. Who'd have thought she'd have been larking about up in the tower at her time of life? Still, you never know what people get up to. I remember a fellow once wanted me to go into a church with him and not to get married. I said you wicked viper, fancy suggesting such a thing in a church. Shocking, I said, and so it was. Oh well, we must have a bit of a sing-song presently when Mrs Chester comes in....\"\n\nJust then George Larkin informed Carolus that Commander Fyfe was in the Saloon Bar, and Carolus went round. He found the churchwarden looking very serious.\n\n\"Extraordinary thing, this about Flora Griggs,\" he confided.\n\n\"Where were you at the time, Commander Fyfe? I noticed you were on the scene within a few moments.\"\n\nFyfe looked with some hostility at Carolus.\n\n\"You noticed that, you say? I hope you noticed everything else that afternoon.\" He dropped his voice. \"Rumble, for instance.\"\n\n\"What about Rumble?\"\n\n\"One never knows, does one, in a place like this? What was he doing up in the tower?\"\n\n\"You still haven't told me where you were.\"\n\n\"In the churchyard, as a matter of fact. Just seeing it was all in order. I heard the commotion and came up.\"\n\n\"I see.\"\n\n\"D'you know I'm almost sure I was followed here this evening? Very strange it was. Footsteps behind me all the way.\"\n\n\"Someone else could have been making for the Black Horse.\"\n\n\"Could have been. It's possible. It's one explanation. But...\"\n\nHe was interrupted by the entrance of Dundas Griggs.\n\n\"Just been to see the old lady in hospital. She'll be out tomorrow. Nothing but a sprained ankle.\"\n\nDundas Griggs acceded to a drink from Carolus and continued to discuss his aunt.\n\n\"Odd thing,\" he said, \"but she seems enormously improved. The shock perhaps. Or the change of surroundings. She talked quite sensibly. Scarcely a reference to the Old Testament. She says she's looking forward to being back 'with Spring coming and everything'. She never seemed to notice anything as mundane as Spring before. It was a pleasure to see her.\"\n\n\"Did you discuss her accident?\"\n\n\"Yes. She passed that off in a moment. 'It's all like a bad dream', she said, 'and I don't want to remember it. I can't even think now _why_ I went to the church tower at all'.\"\n\n\"Splendid news,\" said Fyfe. \"It still leaves a certain amount shrouded in mystery, doesn't it? I was telling Deene that I believe I was followed here this evening. You simply don't know in this place.\"\n\nBack in the public bar Carolus found that Mugger had come in.\n\n\"I hear you're finishing with it,\" he said, his long pale face not changing its expression. \"That's all very well for you but suppose the police get hold of someone innocent?\"\n\n\"I don't think that's likely. Unless he was to try and pass a treasury note of which they had the number.\"\n\nMugger thought this over.\n\n\"You mean of the money the first one was supposed to be carrying?\"\n\n\"Yes. The money carried by Miss Millicent Griggs.\"\n\n\"There was no money,\" said Mugger, not for the first time. \"But if there had of been and anyone was to of found it, they wouldn't get themselves into trouble with spending a few old single pound notes not in series would they? So long as they burnt anything like a fiver there might have been?\"\n\n\"I'm afraid I can't answer that. I've no experience.\"\n\n\"I shouldn't think they would,\" said Mugger. \"Not if they wasn't in series. I mean, people have demands made on them. What can you do when they start theatening to go to anyone's old woman? I've never had anything like this one. The one I told you about. Works out at Ryley's Farm. It's give, give, give, all the time with this one. Mind you, I don't say she's not just right as they go, but you can't have them Asking the whole time, can you?\"\n\n\"I'm sure your life must be full of difficulties.\"\n\n\"Well, it is just now. Then there's this other one. In Station Road. I don't know what to do about that.\"\n\nThe approach of Mrs Rumble drove Mugger off in a moment.\n\n\"I've got something to tell you,\" she whispered to Carolus. \"I haven't told any of the others especially that Champer and as for Slatt I wouldn't think of it. It was when I was in the tower with Flora Griggs waiting for Rumble who'd gone to the telephone. She was unconscious when Rumble went away and again by the time he got back, but while he was away she Come To for a minute and I gave her a glass of water. Then what do you think she said? 'Now I _know_ it wasn't me,' she said. That's all. But I thought you ought to know. Now, Rumble, don't you start trying to sing tonight and making a silly of yourself because you know you can't sing and that's all about it. You look as though you've had nearly enough, too.\"\n\nLaddie Grey came across and silently handed Carolus a whisky, having observed what he was drinking.\n\n\"Naomi told me,\" he said comprehensively and with gratitude in his voice.\n\nMr Lovibond was also extremely polite.\n\n\"Sorry we shan't see you again,\" he said to Carolus.\n\n\"It's a pity you couldn't find out who did it but I suppose you can't be successful every time,\" he added consolingly.\n\nThen Mr Waddell came in for his Saturday-night visit and ordered the lemonade which he invariably drank here. His greetings were widespread and profuse and it took him some five minutes before he could reach Carolus, with whom he evidently wanted to speak.\n\n\"Good news, good news,\" he said. \"I hear our respected and beloved Miss Flora Griggs is on the mend and in a far calmer state of mind than previously. I do not deny that her short absence has given us the opportunity we sought for installing our statue in the side chapel. I hope that when Miss Griggs sees the _fait accompli_ she may become reconciled to it.\"\n\n\"Doubtful, is it?\"\n\n\"I'm afraid so. Our Lady of Lourdes, you know. Our dear Miss Vaillant insisted on that and unfortunately purchased it from Burns, Oates and Washburne's. Had it been supplied by Mowbray's I should have felt on safer ground. But one small alteration I have thought it circumspect to make through the agency of our excellent stone-mason, Mr Baker. The statue as you know includes a rosary. This has been skilfully removed and the plaster repainted. I felt that though our dear Flora might be induced to accept the major premise, as it were, the rosary might have been too much for her.\"\n\n\"Very wise,\" said Carolus. \"I'm not returning to Gladhurst after this evening. There is nothing more I can do here.\"\n\n\"Dear me. You leave us between two stools, as it were?\"\n\n\"I think you will find that the most comfortable situation, vicar.\"\n\nCarolus said good-night to each of his acquaintances and as he escaped from the bar he heard that Flo had had her way and a song began.\n\nHe found Slatt in high spirits.\n\n'Hear you're beaten? \"he said, grinning. \"Found this one too much for you, did you? Oh, well, sometimes we're not as clever as we think we are.\"\n\n\"No,\" said Carolus. \"Well, good-bye to you.\"\n\n\"Good-bye, Mr Deene. And if you should think who's done it you'll write to me, won't you?\"\n\n\"Yes. I'll send you a p.c.\"\n\n\"What?\" roared Slatt.\n\n\"A Police Officer. Er... a postcard. A telegram. Goodnight.\"\n\n\"That's better,\" said Slatt, as Carolus drove away.\n\n# 19\n\nAS Carolus predicted there was no arrest at Gladhurst and as the months passed the case was dropped by the Press and, save for a few people concerned, no one seemed to remember it. That Mrs Bobbin was very far from feeling reconciled was shown in the extremely angry letters she wrote Carolus from time to time. One would have thought from these that he was not only to blame for the unsolved mystery but for the murder itself.\n\nAnother who had not forgotten the affair was Mr Gorringer, the headmaster of the Queen's School, New-minster.\n\n\"Ah, Deene,\" he said, descending on Carolus in his gown and mortar-board, like some vast bird of prey fluttering down, \"The end of term in sight at last. I cannot deny that the vacation this year will be welcome.\"\n\n\"Yes,\" said Carolus, heading for the common-room.\n\nBut Mr Gorringer was not to be shaken off.\n\n\"You seem to have quite abandoned your investigation at Gladhurst. I cannot help asking myself why you should have done so without your customary _expos\u00e9_ of the matter in the presence of those concerned.\"\n\n\"Wouldn't have been a good idea, this time. Perhaps too many _were_ concerned.\"\n\n\"You mean you have discovered the truth, Deene, but you intend to keep it to yourself?\"\n\n\"Yes, more or less.\"\n\n\"But can you reconcile that with your own conscience? A murder, after all, is a murder. I should have thought you felt called upon to inform the investigating police.\"\n\n\"Not really. The circumstances are somewhat unusual.\"\n\n\"I make no doubt of it. Is that not one more reason for revelation?\"\n\n\"No.\" Carolus sadly saw the back of Hollingbourne passing through the archway which led to the common-room. \"No,\" he said thoughtlessly to Mr Gorringer, \"if you knew the whole thing....\"\n\n\"Ah, _if_!\" cried the headmaster triumphantly. \"But you haven't seen fit to confide in me, unfortunately.\"\n\n\"I've told no one,\" returned Carolus.\n\n\"I should hope not. It would surely be a grave discourtesy to reveal this matter to others before telling the facts to your headmaster.\"\n\n\"If you really want to know,\" said Carolus with exasperation, for his chair in the window and _The Times_ had both been seized by now by Hollingbourne. \"I'll tell you. Will you and your wife dine next Tuesday? I'll ask Lance and Phoebe Thomas.\"\n\n\"Unhappily Mrs Gorringer will have left Newminster by then. She is going to 'spy out the land', as she says, now that we have lost our customary holiday haven at Brighton. We may even go as far as Bournemouth, this year. My wife made one of her characteristically sprightly witticisms when I mentioned the place. 'So long as it's not an undiscovered country from whose Bournemouth no traveller returns,' she said. I laughed heartily. But your kind invitation, Deene, I may accept myself? Perhaps I might have the pleasure of meeting again that charming relative of yours....\"\n\n\"Fay? I'll see if she is free. About seven? Good.\"\n\nCarolus almost ran across to the common-room. It was as he feared. Hollingbourne had inscribed 'Akbar' and 'dean' in different corners of the crossword. Both clues, Carolus felt instinctively, were wrong.\n\nTuesday came all too quickly. Fay arrived from town in the afternoon and Lance Thomas, the school doctor, a most popular and level-headed person, came with his wife Phoebe at a quarter to seven. At exactly five minutes to the hour Mr Gorringer made his entrance.\n\n\"Yes,\" he said, when greetings were over and he had taken a seat next to Fay Deene, \"I will indulge myself in a dry martini. A most welcome suggestion.\"\n\nOver drinks Carolus gave them the outline of the case as he knew it, not sparing them frank character sketches of Mugger and Flo for instance, but avoiding any word of elucidation. He wished them to be in possession of the facts, but his interpretation of these he kept for the hour after dinner. During the meal itself, he stipulated, they were to forget murder.\n\n\"An inspired suggestion of yours, my dear Deene,\" said Mr Gorringer enthusiastically. \"Those of us who have, on previous happy occasions, experienced the Lucullan delights provided by your excellent Mrs Stick, will thank you for it. My ears still tingle when I think what she must have said once when I called her prawns in aspic a shrimp cocktail, a term only familiar to me from hearsay.\"\n\nCarolus looked at those vast red organs so rich in bristles and wondered how they behaved when they tingled.\n\n\"Here is Mrs Stick,\" said Carolus. \"No doubt she will tell you what she is going to give us.\"\n\n\"Well, I tried to make it a nice simple appetizing dinner, Sir,\" said Mrs Stick, addressing the headmaster. \"Knowing you was to have a murder after it, as you might say. There's a consummay de vollile, some little lobster booshays because there was only the one lobster at Thompsons and I don't like going anywhere else. Then a bit of roast duck and a little sweet soofflay to finish with.\"\n\n\"Delectable,\" pronounced the headmaster. When Mrs Stick had left them he continued: \"You are fortunate indeed, my dear Deene, in your employment of the Sticks. At School House, naturally, with thirty boarders, we have perforce a plainer regimen. Mrs Gorringer, in spite of the large staff we employ, tells me that even in the holidays it would be wellnigh impossible with all her culinary resources to provide her guests with a meal such as Mrs Stick creates single-handed. She can produce, she says in her inimitable way, quoting Dr Johnson, a good enough dinner but not a dinner to _ask_ a man to. Your adroit housekeeper never fails you.\"\n\nNor was there anything to criticize in Mrs Stick's achievement that evening though she herself was not satisfied. \"I thought the duck was done to death,\" she confided in Carolus afterwards and when he said \"perhaps it was\", she told him his mind ran on such things. She had meant it was over-cooked.\n\nBut at last came coffee and an old Armagnac, cigars, peace and expectation.\n\n\"Come, Deene. We are all ears. Who killed Millicent Griggs?\"\n\n\"You will have found it strange that I retired from this case before revealing the truth. But to defend myself I must remind you of what kind of woman this was. Even her own sister, Mrs Bobbin, spoke of her as 'mean and sanctimonious' and said that she was 'unbearable' with Naomi Chester, 'inquisitive' with a 'mean and nasty mind'. The vicar's wife went further. Millicent was 'kind to no one but herself, generous only to gain her own ends', hypocritical, narrow-minded, self-centred and bigoted. She was also, Mrs Waddell said, capable of murder. Rumble, who was not an unkindly disposed man, used an unprintable word for her and added 'mean with her sisters and everyone else'. He was even more damning when he said\u2014'always _right_ , if you know what I mean. Do anyone a nasty turn'. Commander Fyfe said she was unpopular, 'a woman of insatiable curiosity of a most unsavoury kind'. The 'woman known as Flo', another critic wholly without malice, said Millicent had a 'nasty, dirty mind' and gave me examples of it which she said would 'turn me up' and I'm bound to say they nearly did. I had the impression from the first that Millicent Griggs was a horror and everything I heard about her confirmed it.\"\n\n\"Still,\" argued Mr Gorringer, \"does that justify you in shielding her murderer, Deene? You have so often made it clear that murder itself, irrespective of the characters of perpetrator or victim, must be exposed. Where is the consistency in your behaviour?\"\n\nCarolus ignored this.\n\n\"Everyone spoke with frank dislike of Millicent,\" he went on,\" except the one person of whom one would expect it, Naomi Chester. That roused my curiosity at once. Why did Naomi lie to me? What was she trying to conceal? Whom was she shielding? I started investigations with these questions very much in my mind.\n\n\"I had to do some spade-work which I have not bothered to record. That is one of the disadvantages of being at odds with the CID man investigating, as I was in this case. Had John Moore been in charge I should have been given details of the movements of certain persons which in this case I had to discover for myself.\n\n\"Though I never for a moment suspected Naomi of the murder of Millicent Griggs....\"\n\n\"Why not?\" asked Dr Thomas.\n\n\"Well, Lance, one must allow something to one's instincts. Heaven knows I have had surprises enough in the cases I have investigated and am prepared to believe pretty well anything of anyone. I don't even say that I thought it impossible that Naomi should have bashed Millicent over the head\u2014nothing in human nature seems impossible\u2014but I do say that I could not suspect her of it. On the other hand she knew more than she would say.\n\n\"I was convinced moreover that it was through her that the body was disposed of, for it was fairly plain that Grey had been responsible for this. At about half-past three a telephone call was received for him at the home of Commander Fyfe, where he was working. The number of people who knew he was there must have been limited, for it had been arranged by Fyfe only on the previous evening. Mrs Fyfe thought the voice was that of an elderly woman\u2014exactly the impression that would be given by a young woman trying to disguise her voice.\n\n\"Laddie Grey said that when he reached the phone no one answered, but it is noteworthy that he immediately packed up his work and left. Naomi was seen in the village by several witnesses who stressed that she was 'hurrying' towards her home. By her own admission Grey came to her home about four, or, it would seem, straight from his work.\n\n\"Whatever it was Naomi told Grey it was sufficient to set him in action. At half-past four a light was seen by Dundas Griggs in the house Crossways, but no one would answer his repeated ringing at the door. He walked round the house but found everything locked, even the garage.\n\n\"When Dundas Griggs returned to the house at 5.15 he found no light but heard what can only have been someone _who had a key of the back door_ opening the garage from within and wheeling out a motor-cycle, locking the garage doors after him, wheeling a motor-cycle down the road and starting it up some yards away. I believe\u2014and I will tell you why in a moment\u2014that the person Griggs heard was Laddie Grey and that in the sidecar of his motor-bike was the body of Millicent Griggs.\n\n\"One must not count too much on witnesses' estimate of time but a few minutes later Mugger and the woman known as Flo were interrupted by a motor-cycle appearing at the top of Church Lane and throwing its single headlight on them. I know from Lovibond the electrician that Grey was short of a bulb in the light on his sidecar so there is no reason to think that the _single_ light was not his. Seeing them he rode away and presumably returned later.\n\n\"It would rather naturally have occurred to Grey that the best place to get rid of a corpse was in the open grave in the churchyard for, as I heard from my useful informant Mr Lovibond, Grey used to help Rumble in his work as sexton. He would also have known where Rumble kept his spade and would have been in the furnace room a number of times.\n\n\"I was certain that Millicent was already dead when she was taken from Crossways, for a sheet had been snatched from one of the beds in order to staunch the blood. This sheet was found with her jewellery and money in the loft of the furnace room.\n\n\"Why, you may ask, if Grey had not murdered Millicent did he bother to remove her jewellery? Knowing Rumble's precision in the matter of the depths of graves he may have thought there was a chance at least of the body being discovered before Chilling's funeral, and played for safety. Robbery would be thought the motive and later he could remove and destroy the jewellery.\n\n\"He finished his task at about six, at least if the motorbike which the vicar heard start up in Church Lane at that time was his. We know that it was an exceptionally dark night so it had not been difficult to keep his movements, except on his motor-bike, unseen.\n\n\"I never thought it very likely that Millicent left the house that afternoon, for someone would have seen her. The only reason suggested for her doing so was the church brass and this we know from Mrs Rumble had not been cleaned. But if her dead body had been taken, it occurred to me, the shoes would have remained clean till she was put in the grave. It was thinking about this which caused me to make that lucky shot about galoshes. As soon as Naomi was asked she thought she had made a blunder. Instead of saying calmly that they were upstairs so that if Millicent went out that afternoon it was without them, she foolishly denied that they were in the house, then put them to be found at the back of the church. You may remember how confused she was when, an hour after my questions, I met her coming away from the church in which she had probably just dumped the galoshes.\n\n\"That was foolish. If Millicent had worn them to the church that afternoon she would not have removed them while she was cleaning the brass. And they certainly would not have remained unperceived by Mrs Rumble for several days, including the Saturday on which she cleaned the church.\n\n\"But again _why_? Why had Naomi put herself and the man she loved in this very serious danger if neither of them was guilty of the murder?\n\n\"Let us go back to that afternoon. Flora went out before half-past two, leaving Millicent and Naomi together in the house.\n\n\"Something caused Naomi to stay there after her time. Flora said the washing-up was done before she left. However conscientious Naomi might be I was sure that Millicent found some excuse to keep her there. Eventually, when she felt that I knew most of the truth, Naomi told me what it was. She complained about the staircase. Knowing her proclivity for asking questions, we may guess that she stood near the girl on the stairs, making one of her rather nasty cross-examinations. That was the tall stone staircase that rose from the hall. Then\u2014' Did you actually do anything to her?' I asked Naomi. 'I never touched her', she said. ' _It was only the bucket of water_ '.\n\n\"In itself, you see, it was only what Millicent deserved, a bucket of dirty water thrown over her. But it had the most terrible consequences. The old woman, standing probably a step or two lower than Naomi and receiving that water in her face, lost her balance, failed to find the balustrade with her hand as she fell, and tumbled backwards down the tall staircase to her death. Naomi, exasperated beyond patience, had meant only to throw water at her and as a result of it she found herself alone in the house with the dead body of Millicent. What she did then we know. Let's have a drink, shall we?\"\n\nIt was nearly eleven o'clock and the five people in Carolus's sitting-room had scarcely moved for an hour. It was time to ease the tension and have a whisky or a brandy and soda.\n\n# 20\n\nMR Gorringer was of course the first to speak.\n\n\"In a sense I am relieved,\" he said, 'that what appeared to be a terrible and violent crime was done without the baleful intentions one had feared. But...\"\n\n\"But you feel it's rather an anti-climax, headmaster? What you had supposed murder was no more than an accident. Is that it?\"\n\n\"Far be it from me to regret such a thing,\" said Mr Gorringer. \"But I must say I had imagined something worse. Even when you had convinced me that Millicent Griggs died in her own house I imagined some intruder....\"\n\n\"In that case you will be yet more disappointed in the gruesome incident in the church tower.\"\n\n\"Gruesome indeed,\" postulated Mr Gorringer.\n\n\"Not so very gruesome,\" said Carolus, \"except as it relates to the state of mind of Flora Griggs. And since it appears that she has quite recovered, that in itself concerns us less. In her last letter to me, in fact, Mrs Bobbin, while showing that her resentment has in no way abated, speaks of Flora's happier and more level-headed state of mind as one of the few brighter aspects of the business. She is, Mrs Bobbin says, a completely different person and has developed something very like a sense of humour with which to view the bi-partisan antics of the vicar. This may be due to the shock of her fall, but I prefer to think that it has come about because she has been relieved of the baleful influence of the older sister. For many years she was under the malign domination of Millicent and now she begins to show her own pleasant character.\n\n\"Her fall, was of course, an accident but I'm afraid her intention in climbing the tower may have been suicidal. Whether or not she would in fact have carried out that intention it is not for us to judge. She passes out of the case, that rare and I must say welcome person, one who is better and happier for having been in contact with murder.\"\n\n\"So having disposed of these two most sinister-seeming mysteries as mere accidents,\" said Mr Gorringer loftily, \"I suppose you will tell us that Grazia Vaillant committed suicide?\"\n\n\"Oh no,\" said Carolus. \"Grazia Vaillant was murdered.\"\n\nThe headmaster made a sound like 'Ttsschk!'\n\n\"That should have been obvious almost at once. Mrs Rumble told me that there was less than a third of a bottle of gin left in Grazia's cupboard. I doubt if Grazia, who by then was a fairly hardened gin-drinker, could have got drunk on that, or if drunk would have swallowed an overdose of sleeping-pills. Besides, Minerval was more a habit of hers than gin, though she rarely took it in excess of one tablet at a time. Her last tube had been prescribed and bought from the chemist by Mrs Rumble a week before her death and contained ten tablets which, as Mrs Rumble said, 'she must have been taking' during that week. Mrs Rumble had noticed on the day previous to her death that there were only two tablets left in the tube. So that we can reasonably assume that all that could have been self-administered by Grazia Vaillant that evening was a third of a bottle of gin and two Minerval tablets. Or possibly three or even four Minerval tablets if she took the two which Flora Griggs gave her. These would probably have done her no harm and certainly would not have killed her.\n\n\"Besides, there is no reason to suspect her of wanting to take her own life. With the death of Millicent Griggs the long struggle for supremacy in the parish was ending in her favour. 'There was nothing wrong when I left that afternoon. In fact she was quite excited about getting a statue on the altar of the side-chapel'. So said Mrs Rumble and I see no reason, in this matter, to doubt her. No, Grazia was murdered.\n\n\"We know of only two people who were in her house that afternoon, Mrs Rumble and Miss Flora Griggs, but we have no means of knowing who, if anyone else, was admitted. On the other hand we can see how easy it was to poison her. She always had the same kind of gin and took the same kind of tranquillizers. A combined overdose of these two would be fatal. All a would-be murderer had to do was to provide himself or herself with a bottle containing the amount of Horsely's gin which approximately was left in Grazia's bottle and dissolve in it say half a dozen Minerval tablets. Grazia Vaillant mixed enough lime juice with her gin to kill the taste and would swallow this fatal mixture quite readily. It would not kill her at once but within an hour or two, unless she called assistance, she would be dead.\n\n\"She was anxious that no one should know she had taken to drinking gin in secret. That is the worst of this repulsive puritan doctrine of guilt about alcohol. Convince someone that it's wrong to drink, ignoring the lessons of the First Miracle and the Last Supper, and you can create a secretive drunkard, morally undermined, from someone who in openly enjoying a glass of wine would be a balanced and temperate person. Grazia's secret guzzling of raw spirit disguised by _ersatz_ lime juice laid her open to murder.\n\n\"What is more, the murder would be difficult to detect and almost impossible to prove. She herself could be relied on to throw the poisoned bottle into the deep well outside her back door where it would sink and lie with all the other bottles which had once contained Horsely's gin, the specially-shaped oval bottles so easy to slip into a bag or shopping basket. She herself would wash the glass which had contained the fatal mixture, as she always washed away the traces of gin before leaving the glass in the sink. And even if she were overcome before she had done these things, her murderer would be difficult indeed to identify.\n\n\"We knew, in this case, that the motive could not be simple robbery, as was suspected in the case of Millicent Griggs, for the house had not been broken into and nothing had been stolen. Although the expression on her face was one of fury it might merely have been at the thought of death. We know she grew violent when she had been drinking. And we had become accustomed to meeting very angry old ladies in this case.\n\n\"So for me it became, as these things so often are, a matter of motive. But a more complicated one than I had previously met. There were not the usual persons with obvious motives standing round, the relatives who would inherit money, the man remorselessly blackmailed, the third party in a love-triangle. The motives which one had to suggest for the murder of Grazia Vaillant were farfetched.\n\n\"The only persons, so far as we know from the details given to the Coroner, who could be said to have expectations under Grazia's will were Mrs Rumble and the two clergy of the parish, the Reverend Bonar Waddell and the Reverend Peter Slipper. Unless you add persons who make suspicion fantastic, like those connected with the various-church charities who would see them flourish, the choirmaster, the members of the Boy Scout Troup and so on, the two clergymen and the sexton's wife alone could benefit financially by Grazia's death.\n\n\"The only other possible motive seemed to be sheer hatred. We know that for years the sisters Griggs had hated Grazia Vaillant with all their natures and all the fervency of their religious beliefs, and that the surviving sister still hated her. We may suspect that Mrs Waddell nursed a strong detestation for Grazia. There may have been others whose hatred was better concealed. And since one at least of those who hated Grazia at the time of her death was a rich woman one could not altogether rule out the possibility of another person bribed.\n\n\"But in compensation for all this vagueness in the matter of motive I knew certainly the cause of Grazia's death and this gave me certain hard facts about the murderer. He or she had to have the following:\n\n  1. A knowledge that Grazia was secretly drinking Horsely's gin.\n  2. The knowledge that for some time she had been taking Minerval. (This, of course, so that when it was found she had died from an overdose, suicide would account for it.)\n  3. Access (even if only for a moment) to the cupboard in which she kept her gin.\n  4. Access to a sufficient quantity of Minerval.\n  5. The ingenuity to substitute poisoned for unpoisoned gin in an identical bottle.\n  6. An opportunity of doing so.\n  7. A motive.\n\n\"There was really only one person who fitted all seven conditions.\"\n\nCarolus paused and lit a new cigar.\n\nMr Gorringer was the first to fall into his little trap.\n\n\"For once, Deene, you reduce speculation to a minimum. It was, of course, the unbalanced younger sister, Miss Flora Griggs, who poisoned the unfortunate woman. You have mildly deceived us by suggesting that she has passed out of the affair.\"\n\n\"I don't think so,\" said Phoebe Thomas. \"It was Mrs Rumble.\"\n\nCarolus shook his head.\n\n\"Oh no,\" he said. \"You might have ruled those out from the fact that I have dropped the case. It was Millicent Griggs.\"\n\n\"But...\" began Mr Gorringer.\n\n\"I think you are forgetting,\" said Carolus firmly, \"that very important information given me by Mrs Rumble which I scrupulously repeated to you before dinner. 'There was a bottle nearly a third full', she said on the morning Grazia was found dead. ' _I told you she hadn't touched it since the other one died'_.\"\n\n\"Millicent Griggs went to see Grazia, not, I am convinced, for any wish for reconciliation but with about as much hatred and malice in her heart as a woman well could have. Millicent was, as we have already seen, a hateful and malicious woman. She came away from that first meeting and Mrs Rumble described her vividly\u2014'All flushed up, she was, besides, I could smell her breath.' So after a lifetime's teetotalism, on which she held the strongest possible views, she had accepted some gin from Grazia. _Why_?\n\n\"Her words to her sister when she reached home were significant. Reported by Flora Griggs, they were 'she told me we should not for ever suffer, that we should be able to stand against the wiles of the devil'. Flora Griggs was sure that there had been no real reconciliation. Rumble, in fact, heard Millicent use stronger terms to Flora on that day when she returned from her last visit to Grazia. Woe unto the wicked, it shall be ill with her. For the doing of her hands shall be done unto her. But Millicent had seen the cupboard in which the gin was kept, had seen the characteristic Horsely's bottle. Moreover she knew from Dr Pinton that Minerval in conjunction with alcohol, both even mildly in excess, could prove fatal, for as Dr Pinton told the Coroner, he gave his warning whenever he prescribed Minerval, even when he regarded such a warning as no more than a formality. She had a way of revenge ready to her hand and to this bigoted and abnormal woman such a revenge was justified.\n\n\"That she put it into action we know. The manager of Forster's Stores in Burley said: 'the lady who was murdered '\u2014he was referring to Millicent Griggs\u2014' had been a customer of ours ever since I came to the shop as a boy. She has never been known to order anything alcoholic till about a week before her death. Then she came in and asked me for a bottle of the same'\u2014that was Horsely's gin\u2014'for purely medicinal purposes'. It would have been more accurate to say for purely homicidal purposes.\n\n\"During that week, too, she had to ask Dr Pinton for an extra supply of Minerval, giving the excuse of having lost her own. In reality she had used at least six, probably more tablets to doctor the gin which she was going to place in Grazia's cupboard during her second visit in place of the one she had seen there on her first.\n\n\"She had to depend on guesswork here and decide about how much gin there would be in the bottle by the time she paid her second visit. I think she judged well or was lucky, for Grazia never noticed it if the level was different.\n\n\"So she called again, armed with the large bag we have heard about, in which lay an oval bottle one-third full of Horsely's gin fortified with Minerval in lethal quantity. It was not difficult to change the bottles and depart. It was also an almost unprovable murder. If she had lived she could not, probably, have been charged with it, though she never dreamed that Grazia's death would be delayed by ten days and her own would intervene.\n\n\"But the fact that murder could never have been proved does not mean that it could not be fairly easily deduced. To anyone who had the facts of the case it should not have been difficult to see not only that Grazia was murdered but by whom and how.\"\n\nMr Gorringer shook his head.\n\n\"I'm bound to say it eluded me,\" he admitted. \"But in view of the evidence you have quoted, especially of the manager of Forster's Stores, I see that I was wrong in suspecting the younger sister. But in one little matter, Deene, you have surely deceived us. You described Detective Inspector Champer as saying to you derisively: 'You are not going to tell me you accept the police explanation of events? One murder, one attempted suicide and one accident?' To which you replied in the affirmative. Was not that almost... cheating?\"\n\n\"Not at all. Those are exactly my explanations. But not in that order, headmaster.\"\n\n\"Bravo!\" said Mr Gorringer and in a rare moment of abandon emptied his glass.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# CALON\n\nA Journey to the Heart of  \nWelsh Rugby\n\nOWEN SHEERS\n\nTo Mr Jenkins, rugby coach  \nat KHS Comprehensive, Abergavenny,  \nand to all the other coaches  \nwho keep the game alive. \n'It's either the wedding game or the funeral game with us. Nothing in between.'\n\n'Thumper' Phillips, Wales team manager\n\n# Contents\n\n  1. Title Page\n  2. Dedication\n  3. Epigraph\n  4.   5. TIME: 17 March 2012 \n  6. _Y FAN_ \n  7. GAME DAY \n  8. AUSTRALIA \n  9. ACKNOWLEDGEMENTS \n  10.   11. About the Author \n  12. By the Same Author\n  13. Copyright\n\n# TIME\n\n17 March 2012\n\nGeorge North, a nineteen-year-old rugby player from Ynys M\u00f4n in North Wales, is packing his kitbag in a bedroom at the Vale Resort in Pontyclun, twelve miles west of Cardiff. He puts in two pairs of boots, a gumshield, recovery skins, a long- and a short-sleeved training top and a set of long studs. He checks the contents, then takes them out again, laying them on his bed. Pausing a moment, he puts them all back into the bag. Then he empties them onto the bed again.\n\nGeorge repeats this process five times before finally closing the bag, shouldering it and leaving the room. He's already written down his objectives for the match ahead on a piece of hotel notepaper. Having memorised this list of tactics and prompt words, he's ripped it up and thrown it in the bin. Before he did that, George phoned Andy McCann, a former Scottish karate international and stroke survivor who now works as a sports psychologist. Andy spoke with George about the match, but also about life outside rugby, about the flat George hoped to buy in Cardiff. With less than two hours to kick-off, just hearing Andy's voice is more important to George than anything he has to say about the game itself.\n\nAt six foot four and sixteen and a half stone, George, when running at full speed, summons one tonne of impact. He's an affable joker of a boy, effervescent with energy, quick to laugh or pull a face. Fluent in both Welsh and English, neither language seems capable of keeping up with the rapidity of his thoughts.\n\nSince he was fifteen years old rugby has taken George away from his family. At first, living on his own was a struggle. He missed his parents and the island of Ynys M\u00f4n; washing his own clothes took time to master. But George wanted to play rugby at the highest level. So he moved south, into the heartland of Welsh rugby, where he trained and trained and played and played so that now, just four years later, here he is shouldering his kitbag and leaving his room at the Vale to board a coach with two red dragons painted down each of its sides.\n\nWales, like all nations, is an idea. A concoction of history, landscape, temperament and language. This afternoon George, and the twenty-one young men also boarding that dragon-painted coach, will become that idea. For eighty minutes, under the eyes of millions, they will become their country.\n\n# _Y FAN_\n\n'The new grass shall purge us in its flame.'\n\n'Song at the Year's Turning', R. S. Thomas\n\nIt is five minutes to midnight in Cardiff on New Year's Eve 2011. Viewed from above, the streets of the Welsh capital are busy with light and movement. Taxis, cars, buses thread along Westgate Street and Cathedral Road. The crowds, impervious to a gauze of winter rain, thicken down St Mary's and Caroline Street, spilling into the smoking areas of clubs and bars. Mobile phones cast pale cauls across the downturned faces of drinkers sending texts. The organisers of firework displays check their watches and prepare their tapers. The city is charged with the year's turning, the birth and death of the annual calendar. A time for looking back, and for looking forward.\n\nAt the centre of all this activity, embedded in a crosshatch of headlights and a circuit board of street lights, one dimly lit oval is motionless. This is the Millennium Stadium, home to the Welsh rugby team. The yellow-green rectangle at its centre is the national pitch, semi-illuminated tonight by rows of growing lights suspended above the young grass.\n\nFrom my position under the posts at the southern end of this pitch the intensity and number of these lights resemble a huge theatrical rig dropped to within metres of the turf. The wheels of their structures look like the landing gear of aircraft, their rubber treads flecked with sand. The stands around me are dark. Only the exit signs hanging in each aisle remain lit, hundreds of white stick men running for their lives through the rows and rows of towering seats.\n\nI am alone in an arena built for 75,000. Devoid of a crowd the stadium's atmosphere is like that of an empty theatre, quietly weighted with dormant purpose. In contrast, the streets beyond it are febrile with drink, music, bravado and lust. Even the outer corridors of the stadium have been permeated with the night's celebrations. As Gwyn, one of the stadium's security guards, led me towards the pitch earlier he explained that Level One of the building was being used as a triage centre \u2013 'for the drinkers'. As we left his security lodge we'd passed through the chatter of personal radios angled at the shoulders of policemen and paramedics. Ambulance drivers in fluorescent jackets cradled mugs of tea beneath rows of framed and signed rugby shirts. A girl in a black mini-dress was slumped in a wheelchair, one of her high-heeled shoes missing, like a comatose Cinderella. A nurse pushing through a set of swing doors revealed a glimpse of the hospital beds beyond.\n\nAs I followed Gwyn down an echoing corridor, the noises of the triage centre faded behind us. We passed glass cabinets displaying boxing gloves, track shoes, rugby balls, boots, corner flags, before coming to a huge silver dragon on the wall, its clawed foot raised. Taking a short flight of stairs up past this dragon we came to the double doors of the players' tunnel. Pulling on one of these doors, Gwyn, like a New York porter welcoming a guest, gestured towards the massive interior\u2013exterior that is the stadium's bowl, his hand open in invitation. 'The stadium, Mr Sheers,' he said, with more than a hint of mischief, 'is yours.'\n\nWith Gwyn closing the door behind me I walked down the shallow gradient of the tunnel, the pitch faintly lit at its end. With every step the stadium's stands grew around me, rising together to frame a rectangle of night sky, mirroring the rectangle of turf beneath. Emerging from the tunnel I was met not by the sea-like roar of a capacity crowd greeting a player making his entrance, but by silence. A huge, open crucible of silence. The empty stadium was all I could see, at once imposing and comforting; a 75,000-seat embrace. And not one inch of it tonight, whatever Gwyn said, was mine.\n\nAnd nor is it anyone else's. Because if you are Welsh and a rugby fan, then you'll know the Millennium Stadium is, by its very nature, 'ours', and that it's from this plurality the building derives its significance. Without 'us' or 'ours', the stadium is a shell. As the late Ray Gravell, rugby player, druid, actor, once said: 'Remember, the people make the place.'\n\nThe Welsh for those last two words, 'the place', is _'y fan_ ', a translation that makes this stadium a rich meeting point for the two languages of Wales, coming together between these stands to depict, at one and the same time, the people and the place, the place and the people.\n\nStanding under the posts at the southern end of the stadium I look out at the pitch as it glows eerily in the night. I know I'm meant to stay off the grass. But the temptation to walk, just once, the length of the national ground is overwhelming. I've only ever shared this space with thousands of other supporters. I feel a strange mix of privilege and trespass, as if I've stolen this moment and I'm committing a crime in being here. But as a lifelong supporter of the Welsh rugby team, as the man who still remembers the boy who first watched Wales play from the schoolboy enclosure of Cardiff Arms Park, the stadium's previous incarnation, the temptation is too much. So I walk.\n\nAs I pass under the first growing lights their heat pulses across the top of my head. At the twenty-two-metre line I see a spider's web strung between one of their tubular struts, its fine strands hung with rain. I walk on towards the last row of lights on the halfway line, then continue into a deepening darkness until, a hundred metres later, I'm standing under the posts at the other end of the stadium. Turning round, I look back down the pitch, bisected now by the dark trail of my own footsteps. Seeing them, I feel as if I haven't just walked across the national ground, but across the national _ground_ ; as if I've traversed Wales itself.\n\nIn reality I've just walked across some of the 7,412 removable pallets that make up this pitch, each packed with soil, sand and Lytag, a lightweight aggregate made from pulverised fuel ash. Each pallet is topped with 40 mm of turf seeded with a mix of meadow and rye grass from Scunthorpe, not Wales. But like all stadia the Millennium Stadium is a transformative space. The majority of its existence is spent waiting for certain dates when matches will be played within its stands. When those matches happen, things change, and when they are Welsh matches, they change even more. Individuals become a crowd, but also a nationality; the players in the dressing rooms become a team, but also a country; and the turf on the pitch becomes territory, but also _tir_ , the Welsh for 'ground'.\n\nWelsh players are reminded of this last transformation in a statement written above their changing stalls in the home dressing room. On one wall, in metre-high red writing over the heads of the props and the back rows, they're asked to 'RESPECT THE JERSEY'. On the other, above the heads of the centres and wingers, as they check their studs and their strappings, equally large letters spell out ' _DAL DY DIR_ ' \u2013 'Hold your ground'.\n\nIn Welsh, the double meaning of that last word is more potent than in English, denoting 'land' as much as 'ground'. But as I look back down the pitch I find myself wondering if another word, ' _pridd_ ', meaning 'earth' or 'soil', isn't better suited to the transformation this turf undergoes when played on by Wales. The etymology of ' _tir'_ is Latinate, relating to ' _terra firma_ ', or the French ' _terroir_ '. ' _Pridd_ ', however, predates Latin. It's a purely indigenous word and, as such, its greater depth, of both origin and meaning, is perhaps more evocative of the kind of cultural excavation experienced by a home crowd when they watch Wales play.\n\nI am more than a little drunk. It is New Year's Eve and I've been celebrating the fact for the last five hours. I know, however, that my thoughts aren't fuelled by new-year alcohol alone, but also by years of association with the team who call this stadium home. This is the place where, over the last thirty years, I've felt myself most obviously Welsh. Where I've allowed myself to sink unashamedly into a nationalism I'd find unpalatable in any other area of my life. Where I've felt myself become more than myself, and where, in the opening minutes of a match, I've indulged in the kind of outrageous amnesia and hope of which perhaps only the devout sports fan is capable.\n\nAnd, of course, I'm not alone. For almost as long as the game has existed, Wales has been synonymous with rugby. As early as 1891, just ten years after the formation of the Welsh Rugby Union, the _South Wales Daily_ _News_ identified rugby as 'the one great past time of the people'. Rugby is the foundation sport of Welsh culture. Each year the nation's psychological well-being is held ransom to the fortunes of the national squad. In return, the players and coaches of that squad are subjected to endless enquiry, expectation and obsessive support.\n\nAs I stand alone in the Millennium Stadium in the dying minutes of 2011, I know all of this to be true. And yet, as I look up at the struts of the posts tapering into the night above me, I also know it to be ridiculous. None of it, looked at coldly, makes any sense. Why should the Welsh, a predominately working-class nation, have chosen to identify themselves through the lens of rugby union, a predominately upper-class sport? What is it about the game that speaks so powerfully to the Welsh psyche? Why should a game of fifteen men trying to place an oval ball across their opponents' line, which in the broad view of the world doesn't matter at all, within Wales matter so much?\n\nIf you're a Welsh rugby fan, you'll _feel_ the answers to these questions more than know them. You'll sense intuitively there's something in the game that makes it appear organically grown from the character of Wales. But it wasn't.\n\nOther than perhaps the medieval inter-parish contests of _cnappan_ , in which thousands on foot and horseback competed for a wooden ball boiled in tallow, the rules of modern rugby union have no indigenous roots in Wales. The sport as played today evolved from a variety of competitive ball games encouraged by masters of nineteenth-century English public schools, partly as a way of marshalling an increasing tide of riotous behaviour. Between 1768 and 1832 there were twenty-one public-school rebellions in England. A riot at Rugby school provoked a reading of the Riot Act and the attendance of soldiers with drawn swords. Another saw mass expulsions in its wake. Organised sport, especially of a rigorous nature, was considered a useful outlet for such violent energies.\n\nIt was a professor of Hebrew who brought rugby to Wales. The Rev. Rowland Williams was a fellow and tutor at King's College Cambridge when Albert Pell, an ex-Rugby pupil, introduced his school's variety of football to the university. Williams, in turn, introduced the game to St David's College, Lampeter, when he became Vice Principal there in 1850. Over the next twenty years this pattern was repeated across Wales, with returning teachers and headmasters introducing the game to educational establishments after exposure to 'Rugby's' football at universities across England and Scotland. Llandovery College, Christ College Brecon, Cowbridge Grammar, Monmouth School: these were some of the first rugby clubs in Wales. When pupils left these schools, they formed the foundations of town clubs such as Chepstow, Llandaff and Cardigan. A sport which had its genesis on the playing fields of English public schools was spreading through Wales along similar educational lines.\n\nThe touchpaper of popular enthusiasm, however, like the traditional beacons of the Welsh hills, would be lit on higher ground than these lowland clubs. It wouldn't be fuelled by the pupils and staff of schools, but by the working men of factories, mines and ports. In the last third of the nineteenth century the South Wales valleys experienced industrialisation on a massive scale. In 1840 four and a half million tonnes of coal were mined in the area. By 1913 almost fifty-seven million tonnes were produced. Migrant workers poured into the valleys from rural Wales and across the English border. Over these years only the United States attracted immigrants at a higher rate. This growing population, adjusting to the new rhythms and regulation of industrialised work, were hungry for a sport that matched the physicality and regimentation of their new lives. The new game of rugby football fitted the bill and quickly became an integral part of Welsh working-class culture. Support for the game came from their bosses too, with captains of industry realising, perhaps like those English headmasters before them, that rugby could be a useful method of control for a potentially fractious workforce.\n\nThe new communities of the valleys were created without a bedrock of generations. They had to design the architecture of their own identities. As towns such as Blaina, Pontypridd and Tredegar grew through the last decades of the nineteenth century, so did their communal institutions, each a projection of that town's personality. Brass bands, choral societies, darts clubs, miners' institutes, pigeon and greyhound racing all flourished, and, as the ultimate expression of communal loyalty, so too did rugby clubs. The same was true of the port towns fed by the valleys' industry: in Newport, Cardiff and Swansea rugby came to represent a powerful form of communal expression. As the industrial towns and populations grew ever closer to each other, the rugby field became the place where your efforts, hopes and allegiances could still be pledged to something more local, something that spoke of where you came from and who you were.\n\nBut it wasn't just local identities that found expression through rugby. The rising popularity of the sport also tracked a parallel movement of increasing national confidence. With her new prosperity and dynamism, Wales in the late nineteenth century was emerging, after years of being overshadowed by her imperial neighbour, as a distinct modern entity. In 1872 the first University College of Wales was founded in Aberystwyth, with Cardiff and Bangor following suit. Campaigns for a national library and museum began in the 1870s. A new regiment, the South Wales Borderers, fought in the Zulu wars of 1879. The National Eisteddfod Society was founded in 1880, and in February 1881 the first Welsh rugby team lined up against England at Blackheath. Fielding something of a scratch side, with two players recruited from the crowd, Wales were soundly beaten by eight goals and six tries to nil. The following month, at the Castle Hotel in Neath, the Welsh Rugby Union was formed.\n\n*\n\nA hundred and thirty years later, the stadium in which I'm standing tonight is testament to that union's success. With its position in the centre of the city, where a cathedral or a parliament might usually stand, it's an eloquent embodiment of the place rugby occupies in Welsh culture. But why does rugby occupy that place? And why do the origins of the game continue to invest the sport with such meaning in Wales?\n\nThe most persistent resonance is a question of class. When a dispute over professionalism saw the northern English clubs break away from the Rugby Football Union in 1895, only Wales was left with a working-class rugby-union ethos. In northern England the new professional rugby league became dominant, while in southern England, Scotland and Ireland the union game played second fiddle to the populist sports of soccer and Gaelic football. Rugby union remained the preserve of public schools, universities and white-collar society. In Wales, however, the lifeblood of rugby continued to flow from the industrialised valleys and port towns, the ball passing through hands that spent the rest of their time cutting and hauling coal, rolling steel or loading ships.\n\nThe other home nations weren't always comfortable with this class distinction. At the turn of the century there were even complaints that Wales had an unfair class advantage. As working men, it was argued, the Welsh were able to get into better condition than their amateur gentlemen opponents. At the 1903 AGM of the Irish RFU it was noted that:\n\nOver \u00a350 had been paid for a dinner to the Scotsmen and only \u00a330 for a dinner to the Welshmen. The reason for this was that champagne was given to the Scotsmen and beer only (but plenty of it) to the Welshmen. Whisky and porter were always good enough for Welshmen, for such were the drinks they were used to. The Scotsmen, however, were gentlemen, and appreciated a dinner when it was given to them. Not so the Welshmen.\n\nThe early association of rugby with a renewed sense of national identity also continues to contribute to its contemporary resonance in Wales. George Borrow, the nineteenth-century travel writer, wrote in his book _Wild Wales_ that the Welsh character was partly shaped by the fact that 'the Welsh have never forgotten they were conquered by the English, but the English have already forgotten'. With rugby's territorial contest of possession and confrontation, Wales was gifted a communal way to continue that never forgetting. On a rugby field the knowledge that the Welsh and their language are stubborn survivors of a pre-Anglo-Saxon Britain is a rich seam of national consciousness which even in the twenty-first century is still evoked and defended every time Wales take the field.\n\nThen there is the question of scale. Wales has always been a small nation. With a population of just three million, the country can draw upon roughly the same number of players as Yorkshire. Its diminutive size means matches against opponents such as England, Ireland and France have always been laced with the potency of a David versus Goliath narrative, however much the bookies or the fans might favour Wales.\n\nTo what extent class, history and size are at the forefront of a twenty-first-century Welsh supporter's mind as they watch a Wales match today is debatable. But what _is_ certain is that this cultural heritage is still inherited and therefore, at the very least, still sensed. Similarly, a twenty-first-century Welsh player can't escape the traditions they pull on with their jersey. Their minds might be occupied with video analysis, nutrition, team policy, tactics and preparation, but however diffuse the hinterland of Welsh rugby's earliest days might be, it's still a part of what fuels the modern player. Even those who play with individuals more than a nation in mind, those who take the field for the grandfathers, mothers, coaches who've helped them on their way, those individuals will all, in some way, have embodied the national ethos of the game. All of which goes to explain why it's never just a country that runs out onto this pitch at the Millennium Stadium, but a culture, a way of being. Every match played by the national side is an act of national memory.\n\n*\n\nOver the last few years, as Welsh players have pulled on their jerseys in the dressing rooms they'll have glimpsed a word printed inside their collars. The jersey itself will have been made by an American company, Under Armour, constructed from some of the most high-tech materials of the twenty-first century. The military-grade fabric will feature ArmourGrid technology and body-mapped compression insets. The word inside each collar, though, will be thousands of years old and a reminder of the latticework of history and culture that's still woven into what it means to play rugby for Wales.\n\nBetween 2008 and 2010 the word players saw inside the collar of their home shirt was ' _Calon_ ', meaning 'heart'. In their away strip it was ' _Hiraeth_ ', meaning 'a longing for something lost'.\n\nFrom 2010 to 2011 they were reminded again, inside their home shirts, to ' _Dal dy Dir_ ' \u2013 'hold your ground' \u2013 while in the away collar they wore ' _Balchder_ ' \u2013 'pride'.\n\nSince 2011 the word players will have seen when pulling on their black away shirts is ' _Cymeriad_ ' \u2013 'character'. Inside their red home shirts it is ' _Braint_ ', meaning 'privilege'.\n\n*\n\nMidnight.\n\nA wave of cheers washes into the stadium from the streets of Cardiff beyond. 2012 has arrived. The first fireworks shoot into the night sky and explode in reds, greens and gold. In the empty bowl of the stadium, resonating between the stands, they sound like cannon fire. A rapid fusillade of smaller rockets crack and whistle from elsewhere in the city, disturbing a seagull from its perch in the stadium's roof. As it flies over me the growing lights on the pitch briefly illuminate it from underneath, casting its feathers as pristine as a player coming off the bench. More fireworks erupt in other areas of the city. The year has turned.\n\nFor the current Welsh squad, this new year poses a question first formed under the floodlights of another pitch on the other side of the world in what is now, just, last year. Wales were playing France at Eden Park in Auckland in the semi-finals of the Rugby World Cup. Commentators were calling it the biggest game in the history of Welsh rugby. Not just because the squad were eighty minutes from a final against the hosts, New Zealand, but because never before had a World Cup semi-final pointed so strongly towards a Welsh victory. France had been off form. Their tournament had been erratic and distracted. Virtually estranged from their coach, the team had yet to discover the rhythm or the flair that had seen them do so well in previous World Cups.\n\nWales, by contrast, despite fielding eight players under the age of twenty-three, had been the team of the tournament. For the first time in their history the national coach, Warren Gatland, had had the squad for two clear months of preparation. They'd twice travelled to gruelling training camps at Spa\u0142a in Poland, where, with the aid of regular three-minute sessions in the \u2013150\u00b0C cryotherapy chambers, they'd trained harder and more often than any other Welsh squad had ever trained before.\n\nRugby coaches often talk about 'strength in depth': developing enough quality players to cover injuries to the first-choice team. In Poland, however, Wales were developing a different kind of strength and depth. Under the eye of Adam Beard, the team's head of physical performance, the players' bodies were being strengthened to the point where, as centre Jamie Roberts put it, 'you're thinking about your next move, not your next breath'. But the Spa\u0142a experience was also about developing depth \u2013 of resolve and attachment.\n\nThe modern top-flight rugby player is often a hard man who leads a soft life. At a young age he'll have life experiences most of us will never know, and yet his experience of life will be relatively narrow. While their peers are going through an expansion of independence, the young rugby professional experiences a contraction. Much of their day-to-day lives is organised by someone else. Travel and schedules are set, day sheets appear under hotel doors, food and drink is monitored. They are given the best treatments and good salaries, but they also no longer work side by side for long hours in mines, fields or factories.\n\nWarren Gatland and his team knew that Wales had often lost key games in the closing minutes. When the time had come to dig deep, that depth wasn't there. At Spa\u0142a, where Adam Beard says he pushed the squad '150% harder than ever before', the players hurt, and they hurt together. As one group waited for another to finish a session, there was silence in the room, such was the anxiety about what lay ahead of them. In the relatively simple surroundings of the camp, with no TV or computer games, they spent time together too. They talked, played cards and competed in quizzes. In previous squads there had always been some fighting between the players. But in Spa\u0142a there was none. On their one day off, many chose to visit the nearby concentration camp at Auschwitz.\n\nThe Olympic Sports Centre in Spa\u0142a was founded in 1950. Despite recent developments it still resembles a training camp for Soviet cosmonauts. Situated a hundred kilometres south-west of Warsaw, its buildings are located within hectares of oak forests through which herds of European bison once roamed. The village of just four hundred inhabitants is dominated by the centre's presence. The ceiling of the nearby Olympic Pizzeria is papered with the front pages of a newspaper called _Sport_ , while its menu offers a range of dishes all named after Olympic cities.\n\nThe centre's three simple hotels \u2013 Champion, Junior and Olimpijczyk \u2013 sit squarely and heavily between its facilities of indoor and outdoor running tracks, gyms, a cavernous sports hall, swimming pool, treatment rooms and \u2013 Wales's chief reason for being there \u2013 cryotherapy chambers. In their preparation for the World Cup the Welsh players stripped off twice a day to enter the icy vapour of these liquid-nitrogen-cooled freezers. Wearing just a pair of shorts, long woollen socks, gloves, face mask, headband and wooden clogs, their blood pressures were taken by a nurse before they stepped, in groups of six, into the preliminary cooling chamber. A computer beside the chamber recorded its temperature as \u201350\u00b0C. The players crossed their arms, covering their nipples with their gloved hands as a member of staff dressed for an Arctic expedition opened the door to the secondary chamber. Outside on the computer screen the temperature in this second chamber can read anywhere from \u2013120\u00b0C to \u2013150\u00b0C.\n\nInside, it is a white-out. On stepping into this second chamber the players immediately lose sight of each other, their bodies lost in the freezing vapour. The sounds from beyond the chamber are muffled. Those inside have to keep talking to check in on each other, each man rocking from side to side or walking a tight pattern of steps. The cold is clean, dry and penetrating. The players' brains, suddenly alert through instinctive survival, cause their blood vessels to contract and send endorphins streaming through their systems and blood rushing to their cores. Their skin temperature drops to below 12\u00b0C and the seconds begin to lag as any residual moisture \u2013 in the backs of the legs, in the crook of an arm \u2013 stings sharply. With a minute to go some of the players' teeth are chattering. Others have broken into involuntary shivers. Eventually, the heavy metal door opens with a thick clunk and the players file into the preliminary chamber, then, after a few seconds' acclimatisation, out into the warmth, a flood of dry ice blowing out with them around their legs into the room.\n\nAfter each of these cryotherapy sessions the squad took to the bikes and rowers in the gym across the corridor. With the stereo turned up they did twenty minutes of gentle exercise, pumping the blood through their veins to achieve the 'vascular flush' which would complete the treatment's rehabilitation of their damaged tissues. The fundamental science behind this method is the same as that which has seen sports teams use ice baths and other cold therapies for many years. The aim is to reduce inflammation, produce endorphins and dampen the nervous system. By shortening recovery periods the players can do harder training sessions, more frequently. An ice bath, however, causes stress to a player's skin. In reaction their blood, before flooding to the core, first rushes to the surface. The shock also produces stress hormones such as cortisol, which can counteract the benefits of the treatment. For the Welsh players, who can have as much as four inches of muscle packed about their limbs, there is also the question of penetration. An ice bath may only affect the first two inches of soft tissue, leaving deeper tissue still inflamed and damaged.\n\nWhatever the science or the balance of the physical and psychological effects, the squad in Spa\u0142a soon became convinced there was no way they could have trained as hard as they did without the cryotherapy. In some weights sessions they lifted as much as four tonnes each. Every day they undertook levels of fitness training that would usually have left them aching all over with 'the DOMS' (Delayed Onset Muscle Soreness). The cryo, however, had them waking each morning feeling refreshed and able to continue with Adam's punishing regime. Coaches often talk about muscle memory, but the use of cryotherapy at Spa\u0142a was more about muscle amnesia: about wiping away the residue of physical exertion to leave the effect but not the ache, the gain but not the pain.\n\nWhen not training inside at Spa\u0142a, the squad worked on a pitch in front of Hotel Junior ringed by a running track and overlooked by a statue embedded in the corner of Hotel Champion. Sculpted in the Soviet style, a male figure places one hand on his chest, while the other holds aloft a flaming torch which he looks towards in aspiration. Next door to the training pitch, also under this statue's gaze, the Park Pokolen\u00b4 Mistrz\u00f3w Sportu commemorates Polish Olympic champions such as Jan Mulak, Irena Szewin\u00b4ska and Wanda Panfil.\n\nOn the pitch itself the Welsh squad often trained under the eye of a man who has had his own fair share of Olympic experience. Frans Bosch is one of the world's leading experts on biomechanics. Specialising in the high jump, he has coached many athletes to Olympic level, using his techniques of fine-tuning running efficiency to improve their performances dramatically. As the Welsh players ran through their moves on the pitch in Spa\u0142a, Frans, with his shoulder-length hair and salt-and-pepper stubble, moved among them, watching with both the eye of an expert and that of an artist.\n\nBefore he became a specialist in physical training, Frans was, for twenty years, a well-known painter. Although he turned his back on his exhibitions and paintings before some of these Welsh players were born, he still draws as much upon his artistic experience as his scientific. 'It's about how you see movement,' he explains. 'An artist's eye is trained to renew the perception of what you see time and time again. If you can do that with looking at running, then human running is one of the most remarkable movement patterns there is. Renewing what you see helps to find the errors and how they are connected, and how best they can be corrected.'\n\nFrans has no national allegiance to Wales and he thinks rugby is 'a pretty ridiculous sport'. Yet every time he trains with the Welsh squad he finds himself rooting for them beyond a professional interest. Compared to other 'more corporate' teams with whom he's worked, he can't help but warm to the 'family coherence' of the Wales set-up and to the national love of rugby as a sport. 'I went to watch a match in Cardiff ', he says, 'and I realised I was seeing the country, not a sport. A cultural event, not a game.'\n\nAt the Rugby World Cup in New Zealand Wales's performance proved Warren and his team had been right to follow their instincts in Spa\u0142a. After a narrow defeat by just one point to a vastly more experienced South African side, the young Welsh squad displayed a mature determination in convincingly beating Samoa, Fiji, Namibia and Ireland. Before the tournament most of the squad were unknowns on the international stage: Rhys Priestland, Dan Lydiate, George North, Leigh Halfpenny, Jonathan Davies. A few years earlier many had been playing schoolboy rugby, and yet now the rugby world was talking about them, and not just because of their wins, but because of the manner of those wins; because of their style of play and their demeanour on and off the pitch.\n\nEver since rugby union went professional in 1995, every Welsh team has had to negotiate a delicate balance between tradition and innovation. The shadow of the great teams of the 1970s still falls across a twenty-first-century squad. Many of those 1970s players are still present in the consciousness of the rugby public, as commentators, journalists, board members and coaches. The Welsh supporter demands a style of play, a philosophy, inherited from that Welsh rugby past: inventive and audacious, physical yet graceful. But at the same time there is now the pressure to compete at the leading edge of the sport, to 'pinch an inch' wherever possible in a game so much harder, faster and more brutal than the rugby of the 1970s, 1980s or 1990s. Cryotherapy, analysis, nutrition, sports science, GPS positioning, psychology: the modern Welsh team, while aware of its heritage, has to be ever more forward looking.\n\nIn the Barn, the indoor training area at the Welsh Rugby Union's National Centre of Excellence in the Vale of Glamorgan, two massive banners hang side by side above the expanse of emerald AstroTurf:\n\nYESTERDAY IS IN | HOW DO YOU  \n---|---  \nTHE PAST | WANT TO BE  \n| REMEMBERED?\n\n_Forget the past and live for the moment_ , the first seems to say.\n\n_Think of your legacy, the presence of what has been, forged in your now,_ says the other.\n\nAs the Welsh squad do their drills and units in the Barn, they run between these two banners. Playing miss moves, set pieces, rucks and attacks they flow from one to the other as if the whole building is being rhythmically tipped from side to side. As a team and as individuals they shuttle between a statement and a question, between the past and the future.\n\n*\n\nAt the 2011 World Cup Wales hit a balance between those two competing states. They arrived at the tournament as one of the most sophisticated sides of the twenty-first century. Their youngest players had graduated through the academies of professional clubs, travelling in a gilded channel of top-level rugby since the age of fifteen or sixteen. And yet something in the squad's personality, in the positivity of their play, still spoke strongly of the Welsh rugby past that fed their Welsh rugby present.\n\nThe character of the squad was embodied by their captain, Sam Warburton, who at twenty-two was the youngest-ever captain of a World Cup side. At six foot two and sixteen stone three pounds Sam is a relentless, scavenging open-side flanker. Against South Africa he personally made almost a quarter of Wales's ninety-nine tackles. Physically imposing, a man of few words with a craggy, Roman profile, Sam leads by example. A teetotaller with a highly developed sense of duty and fairness, he's the first onto the training pitch, the first to put on his recovery skins, to make his protein shake or to fill the ice bath. Under his captaincy 'no sapping' became the ethos of the team. If you're hurting, tell the physios, but not your teammates. If you're tired, tell the conditioning coaches, but not the other players. If the English squad at the tournament were displaying the qualities of Prince Hal in _Henry IV Part I_ \u2013 going out drinking, being careless with their reputations \u2013 then Sam and his team were more like the young king of _Henry IV Part II_ : determined, ruthless and choosing to harness their youth for the fight, not the revels.\n\nWhen Sam and his twin brother Ben were born five weeks premature, the maternity nurse tried to explain the situation to Sam's parents. 'Let's put it this way,' she said. 'They're never going to be rugby players.' Seven years later, as his mother was putting him to bed, Sam remembers talking to her about what he might be when he was older. 'I want to be a footballer,' he told her from his pillow. 'Why?' she asked him. 'Well,' he replied. 'I'm never going to be a rugby player, am I?'\n\nBut Sam was fast, frequently winning the sports-day sprints, and his junior-school teacher, Frank Rees, was a rugby man. When Sam finally listened to Frank's encouragement and played for his school against Willowsbrook, he scored four of the team's tries. Willowsbrook's teacher coached for Cardiff schools and knew a good player when he saw one. Within weeks Sam was playing for Cardiff.\n\nOn his fifteenth birthday Sam's parents gave him a multi gym, which he set up in their garage in Whitchurch. He knew he had to bulk up and get fit for rugby, so whenever he could he lifted weights in that garage or went running in the streets. As he ran, dreaming of playing for Wales, Sam listened to 'Refuse to Be Denied', a song by Anthrax, his father's favourite metal band.\n\nSeven years later, on the night of 15 October in Auckland, as he sat in the changing rooms of Eden Park preparing to captain Wales in the biggest game of their history, Sam listened to that song again. With his head bowed and the headphones snug over his ears, the lyrics of his childhood, the lyrics that had accompanied him through countless weights sessions and through the night-time streets of Whitchurch, spoke to him once more:\n\nRefuse to be denied,\n\nRefuse to compromise.\n\nWhen the Welsh squad first gathered at Spa\u0142a, Warren and his assistant coach Rob Howley had spoken to the players about it being their destiny to meet New Zealand in the final of the World Cup. It was a private, in-camp conversation, informed by what the coaches knew about their young side's potential and the training ahead of them in Poland. As the World Cup progressed, however, that private conversation became increasingly, in the eyes of the rugby-watching world, a public expectation.\n\nAll sports fans have a love of narrative, for the stories that feed into key matches, big fights or prize-winning races. They are the stories that raise the stakes and heighten the enjoyment of sport's vagaries. For rugby fans in the autumn of 2011 that story had become the rise of a young Welsh side towards a tantalising final against the tournament's hosts, the New Zealand All Blacks. Wales's place in the final began to feel deserved, somehow _right_. Support spread far beyond national borders, with even former English players such as Will Greenwood tweeting in the minutes before the France match, 'I want to be Welsh!'\n\nThe destiny of which the Welsh coaches had spoken in Spa\u0142a was now, in the hours before kick-off, being spoken of by the rest of the world.\n\n*\n\nThe last of the fireworks have fallen. 2012 is only ten minutes old and already the excitement of its birth is ebbing. The streets beyond are subdued, and the stadium falls back into a strangely natural soundscape: the running of rainwater in the storm drains, a cave-like dripping in the stands, the occasional creak of an aisle sign like the groan of a branch in the wind.\n\nTwo and a half months ago, on the night of Wales's semi-final against France, this empty pitch in front of me was filled with Welsh rugby supporters. Thousands more sat up in the stands, their Welsh jerseys rashing the stadium red. In all, 65,000 fans came here to watch Wales that night, even though their team was playing on the other side of the world. But this was no ordinary match. This was a match to be shared. And so the fans came, to watch together under the stadium's closed roof as the game was screened at either end of the pitch. Undiluted by supporters of another team, never before had so many voices sung the Welsh national anthem in this stadium. Half a world away the Welsh team, lined up on the pitch at Eden Park, their arms about each other's shoulders, also sang. And in the Red Lion pub on Bleeker Street in New York, and in the Three Kings in London, and on Aviano air base in Italy and Camp Bastion in Afghanistan, and in homes and pubs and rugby clubs across the world, Welsh supporters sang. Because whatever the time of day, the story of this game was just too good to miss.\n\nBut it was also too good to be true.\n\n*\n\nThe script of a rugby match is written not by prophecy or hope, but by the second. And a second was all it took for Wales's story to change that night; the 1,061st second, when Sam Warburton, refusing to be denied, threw the full force of his weight into a tackle on the French winger, Vincent Clerc.\n\nWhen the Frenchman received the ball from a line-out, Sam had been waiting for him, crouched in anticipation. Wrapping his arms about his waist and pulling at the backs of Clerc's thighs, Sam straightened from his crouch to drive his right shoulder up and under the winger's ribcage. At just fourteen stone, two stone lighter than Sam, Clerc was lifted into the air, his feet swept up over his head. As he fell backwards towards the ground, head and shoulders first, Sam's grip loosened as if he already knew, even before Clerc had landed, what he'd done.\n\nImmediately the Welsh and French forwards tightened around the point of collision, running in to shove and pull at each other's jerseys. Wales's Luke Charteris pulled Sam from the melee as the referee, Alain Rolland, blew on his whistle three times to break up the arguing players. Clerc lay on his back behind the scuffle, the French doctor and physio kneeling beside him.\n\nReaching into his pocket, Rolland pulled out a red card and showed it to Sam, pointing with his other hand off the pitch. He was sending Sam off for an illegal tip tackle. Sam walked to the touchline, his head bowed, and sat in one of the sub's chairs. Someone placed a tracksuit across his shoulders, someone else ruffled his hair. But Sam just looked out onto the pitch, his chest still heaving with the effort of the game, trying to take in what had happened. Wales, with just a 3\u20130 lead and sixty minutes of the semi-final still to play, were without their captain and down to fourteen men.\n\nAcross the world commentators and fans were complaining about the harshness of Alain Rolland's decision. Online a flurry of protests broke out on forums and websites covering the match. A yellow card, it was felt, would have been fair. The tackle had gone too far, but there'd been nothing malicious in Sam's intent. When Sam himself, however, was shown an image of the incident, he immediately accepted Rolland's call and said he saw no reason to appeal, accepting the citing board's judgement of a three-week ban.\n\nIn most rugby matches, on being presented with an advantage like Sam's sending off the opposing team would pile on the points. In international rugby, to lose a man like Sam is to lose strength in the scrum, dominance at the breakdown, cover in defence and support in attack. It is to unlock the door of victory for your opponent. Somehow, though, Wales held on and the match remained close. But their rhythm was broken. The story wasn't meant to go this way, and it began to show.\n\nWales's fly-half, James Hook, missed a penalty. France, meanwhile, kicked three. A dummy and darting try from scrum-half Mike Phillips brought Wales back into contention, but James missed the conversion. Then Stephen Jones, replacing James, missed another penalty. With just six minutes of the match left and the score lying at Wales 8 \u2013 France 9, Wales were awarded yet another penalty. The story, after everything that had happened, could still find the ending for which Wales had hoped.\n\nThe penalty was just inside the halfway line, forty-nine metres from the posts, so it was Wales's long-range goal-kicker, twenty-two-year-old Leigh Halfpenny, who stepped up to the mark.\n\nWhen Leigh was nine, his grandfather started picking him up from his primary school in Pontybrenin to take him for kicking practice on the rugby pitch in Gorseinon. If he was tired, Leigh's grandfather would still persuade his grandson to practise. 'Come on now,' he'd say with a smile. 'Let's get it done, is it?' Towards the end of a session he'd sometimes try to put pressure on him too, telling him, 'One more, is it? But this one's to beat England,' or 'This one's for the final of the World Cup.' As Leigh grew older, he needed little encouragement, practising his kicking every day of the year, including Christmas Day. The England fly-half, Jonny Wilkinson, became his role model. Leigh read all his books and watched all his DVDs just so he could study his hero's kicking technique.\n\nWhen Leigh was fifteen, he caught the eye of the Neath and Swansea Ospreys academy. But at the age of eighteen the Ospreys dropped him for being too small. Determined to make it in top-level rugby, Leigh embarked on a stringent weights regime, putting himself through sets in which he regularly 'lifted to failure' \u2013 until his muscles could no longer work. His parents spent thousands on nutritional supplements. At the age of nineteen, having just been signed for Cardiff Blues, Leigh made his debut for Wales.\n\nThree years later, under the floodlights of Eden Park, Leigh prepared to take the kick for which he'd practised all his life. This was his schoolboy's dream made reality: the penalty that could take his country into its first-ever World Cup Final. After all those hours with his grandfather on the pitch at Gorseinon, after all those years of building himself up, after the pain of the Spa\u0142a training camps in Poland, the moment he'd envisaged so many times had finally come.\n\nRemoving his skullcap Leigh placed the ball on the kicking tee as if it was the last piece in a delicate puzzle. Angling it away from him, he stood and stepped backwards and then to the side. Behind him, at his shoulder, was Neil Jenkins, or 'Jenks', the Wales kicking coach. Jenks, reciting a quiet list of pointers, knew this kick was within Leigh's range. And so did Leigh. In training he'd regularly converted longer kicks than this from inside his own half.\n\nStanding with his knees slightly bent and with his hands rocking rhythmically at his sides, Leigh stared down at the ball in front of him. The roars of the crowd washed around the stadium, rising and falling like waves. Eyeing the posts for a last time he lowered his head and, slowly tipping forward, took a series of quickening steps towards the ball. Planting his left foot firmly beside the tee, Leigh struck the ball hard with his right foot, sudden and sharp, straight towards the posts.\n\nThe hands of the Welsh fans at Eden Park immediately rose above their heads. And in the Red Lion in New York, and in the Three Kings in London, and in the Millennium Stadium in Cardiff thousands of other pairs of hands also reached into the air. And rising with them, across the time zones of the world, came a cheer, voiced as one by every Welsh supporter on the planet. The ball was sailing towards the middle of the posts.\n\nBut then, instead of building to a crescendo, the cheer began to fade. The raised hands began to fall \u2013 in the Red Lion, in the Three Kings, in the Millennium Stadium \u2013 coming to rest on the tops of their owners' heads. And that's where they stayed, in an image of despair, as the ball, turning end over end, dropped short of the crossbar by an inch. Leigh had missed the kick.\n\nFive minutes later the match was over, the final score Wales 8 \u2013 France 9.\n\nThe script was written. Wales had lost.\n\n*\n\nSkirting the edge of the pitch I walk around the stadium's bowl to the mouth of the player's tunnel and walk up it, the thousands of empty seats diminishing behind me. Once inside, further down the corridor that leads towards the Wales changing rooms, I can make out the eleven dark wooden boards on which every Welsh player's cap number and name is written in gold leaf. The first, in 1881, is James Bevan, an Australian who played for my old club, Abergavenny, and the first captain of Wales. The last is number 1,089, Alex Cuthbert, a young winger who won his first cap here last month when he was twenty-one, just four years after he'd first picked up a rugby ball. Between them, nested in the tight rows at number 430, is my great-great-uncle, Archie Skym. Capped twenty times, Archie was nicknamed 'The Butcher', although at thirteen stone, regardless of being a prop, he'd still be the lightest member of the squad today.\n\nI descend the flight of stairs and pass the silver dragon on the wall again. Pausing to look at it I realise this is the first thing a visiting team will see on a match day. That claw, raised between salute and attack.\n\nPrevious World Cups have been catalysts for change for Wales. In the wake of disappointing performances coaches were sacked, new methods adopted, a rash of new players brought into the squad. But the World Cup in New Zealand posed a different question for the national side. Here was a young squad, mostly at the start of their careers, already hitting their stride. They hadn't felt lucky to be in that semi-final, but they had felt unlucky to lose it. So the question it posed to Wales was no longer one of change, but of promise. In both senses of the word.\n\nCould Wales fulfil the promise they'd shown? And could they keep the promise they'd apparently made by playing so well at the highest level? Could they return to Europe and stamp their mark on northern-hemisphere rugby, not just by winning the coming Six Nations tournament, but by winning all five of their matches to secure a third Welsh Grand Slam in eight seasons? Only two other generations of Welsh players had ever won three Grand Slams: between 1905 and 1911, and between 1971 and 1978. Could Warren Gatland's youngsters, with the majority of their international playing days still ahead of them, be the third golden generation of Welsh rugby? Ever since their semi-final defeat to France, this has been the question on the lips of Welsh supporters and, although unspoken, in the minds of the Welsh players too.\n\nAs I leave the stadium, nodding to Gwyn through the double plate glass of his security lodge, I know there's no way of knowing the answer. Rugby's script is written in the moment. Even the players in the squad do not know yet who will play in those five matches. Or which of them out celebrating tonight, as yet uncapped, might see their names added in gold leaf to those dark wooden boards. The only certainty is that those five matches will happen and that they will be won or not, each result leaving in its wake either a trail of disappointment or jubilation, celebration or mourning. 'It's either the wedding game or the funeral game with us,' 'Thumper' Phillips, the Wales team manager, once said to me from behind his desk at the Vale. 'Nothing in between.'\n\nTwo of those five matches will be played away, against Ireland in Dublin and England at Twickenham. The other three will be played here, on the meadow and rye grass of the Millennium Stadium. First against Scotland, then Italy, and then, in Wales's last match of the tournament, on 17 March at 2.45 p.m., France. Whether or not that last game will also be a Grand Slam decider for Wales will depend on the balance of the scorelines of the preceding matches, each of them a challenge in its own right, in which the Welsh players will push their bodies to the limit in their attempts to tip the scales of those scorelines in their favour.\n\n5 February\n\nIreland 21 \u2013 Wales 23\n\n12 February\n\nWales 27 \u2013 Scotland 13\n\n25 February\n\nEngland 12 \u2013 Wales 19\n\n10 March\n\nWales 24 \u2013 Italy 3\nFrom BBC News, 16 March 2012\n\nFormer Wales captain Mervyn Davies has died following a battle with cancer. He was 65.\n\nKnown universally as 'Merv the Swerve' the number eight won a total of 38 caps for Wales and went on two victorious British and Irish Lions tours in 1971 and 1974.\n\nDavies won two Grand Slams with Wales and three Triple Crowns. Davies was handed the captaincy of Wales in 1975 and skippered the side to the Five Nations Championship in the same year, and the Grand Slam the following season.\n\nAt 6ft 3in, Davies sometimes appeared ungainly on the field, but that belied his strength in the maul. He also had an uncanny sense of anticipation, allowing him to get to the breakdown first \u2013 and his height made him useful in the line-out.\n\nIn a poll of Welsh rugby fans in 2002, Davies was voted greatest Welsh captain and greatest Welsh number eight.\n\nNow and Then\n\nWhat might have been and what has been\n\nPoint to one end, which is always present.\n\nT. S. Eliot, 'Burnt Norton'\n\nThere are moments in history\n\nwhen a nation becomes a stadium.\n\nWhen a country's gaze and speech\n\ntightens in one direction.\n\nWhen a population leans, from sofas,\n\npub stools, in village halls, to watch.\n\nOr strains to listen at the sides of roads,\n\nor in tractor cabs in silent fields.\n\nThere are moments when the many,\n\nthrough the few, become one.\n\nA faithful but demanding tribe,\n\nhungry for a win but also more.\n\nFor beauty as well as strength,\n\nfor art as well as war.\n\nBut romance, history, fervour,\n\nare the privilege of watchers only.\n\nFor the men who must do,\n\nthough fuelled by the colour of the jersey,\n\nthe feathers on their chest, there can be no past\n\nor future when, but only now.\n\nFor them those eighty tightening minutes\n\nwill be an ever-living present\n\ncomposed of the angle of their runs,\n\nthe timing of cross kicks, the learned set piece\n\nthat fires the line to light the match.\n\nIt will be the focused practice\n\nof what their bodies have learnt on the training pitch.\n\nThe thousands of hours of solitary pain,\n\nthe sacrifice that has led them,\n\nand them alone, to this \u2013\n\nA nation sharing a pulse\n\nas the clock counts up to the final whistle\n\nwhen now becomes then. The moment,\n\nwhichever way it falls, cast forever,\n\nand theirs to carry for the rest of their lives\n\nuntil, like those who've passed through\n\nthis crucible before, they too will join\n\nthe soil, the _tir_ , the _pridd_ of this land\n\nthey were prepared to suffer for.\n\n# GAME DAY\n\nWales vs France, 17 March 2012\n\n## 6 a.m.\n\nMichael, a wiry seventy-five-year-old from Barry Island, gives Gwyn a wave as he enters the stadium. Gwyn doesn't need to check his pass. Michael, white-haired, bespectacled, has been working as a volunteer with the ground staff here for years. And every match day he does this, walking in on his own at 6 a.m.\n\nGwyn follows Michael on the CCTV monitor as he makes his way past the players' entrance and round a corner towards the service areas. Michael is the only person on Gwyn's quartered screen, his small body marooned in an expanse of angled, unpainted concrete, as if he's walking through an architecture built for a species more gigantic than human.\n\nFollowing the coach-wide passageway, two storeys high, Michael passes through the groundsman's storage supplies. Piles of fertiliser and nutrients, Kioti tractors, frames for the growing lights, spools of orange rope all crowd and gather at the walls. Three racing-green Dennis pedestrian cylinder mowers are parked in a row, clumps of grass like chewed cud collected in their barrels. Everything around Michael is on a massive scale, like the sound stage of a film studio stacked with the sets of an epic.\n\nAs Michael enters a room on his left, however, everything is suddenly more intimate. With the single swing of a door, the stadium's vocabulary of event is translated into a more domestic dialect. A round wooden table at the centre of the room is scattered with newspapers, four chairs around it: three plastic uprights and one double-sized ox-blood leather Chesterfield. Against the wall another, smaller table is crowded with mugs, teabags, coffee jars, a kettle and a small fridge. Apart from one life-size poster of Katherine Jenkins wearing a sequinned dress, the walls are covered exclusively with A4 photographs of the stadium's pitch, each of them labelled with a year and the name of Wales's opponents on that day:\n\n2007 \u2013 Ireland\n\n2009 \u2013 England\n\n2011 \u2013 Argentina\n\nIn each photograph the pattern mown into the grass is different: checkered, long and short rectangles, stripes, diamonds in the dead-ball area.\n\nThis is the groundsman's office, which Michael shares with Lee, the head groundsman, and Craig, his assistant from John O'Groats. Lee and Craig call the photographs on the walls their 'pitch porn': a record of every pattern they've ever cut into the grass of the national ground, each one the result of considered discussion around the wooden table, sketches on envelopes, the laying of miles and miles of orange guide string and a strict regime of cutting and double cutting.\n\n'I doubt no one else ever notices,' Craig once told Michael in his Scottish accent. ''Cept for us. And our wives, when they see it on tha telly.'\n\nThe high-backed ox-blood Chesterfield belongs to Craig, the two gentle depressions in its seat marking the outline of his buttocks. He bought the chair via fatfingers. com, a website that lists misspellings on eBay. He wanted it for his home in Cardiff, only realising it was double-sized when he went to collect it. Stadium-sized.\n\n'My wife was'na havin' it in the house,' he explained to Lee when he turned up with it at the groundsman's office. 'So I thaw I'd bring it here instead.'\n\nUnder the unblinking smile of Katherine Jenkins, Michael makes a cup of tea, stirring in a spoonful of sugar before taking his mug back out into the passageway and up into the stadium's bowl. He enters pitch-side via the 'Dragon's Mouth', a hydraulic ramp that opens and closes like a set of massive jaws.\n\nThe stadium's roof is open, but only by a metre. A slim line of early daylight falls directly onto the halfway line. Despite a forecast of rain, the French coach, Philippe Saint-Andr\u00e9, has asked for the retractable roof to be opened. Warren Gatland, who would rather it stayed closed, joked at a press conference a few days earlier that perhaps when they tried to open it, the mechanism would fail and the roof would have to remain shut. This morning, when staff began opening the roof, the mechanism broke, leaving just this hairline of light falling onto the pitch.\n\nMichael stands at the north-east corner and sips his tea. When he was younger, it was football, not rugby, that was his game. In his twenties he even won a couple of Welsh caps. After his playing days were over, he got a job as a groundsman at a cricket club, and while he was there cricket became his focus. Having retired from the club, he was working as a gardener at a hotel in Cardiff when, six years ago, Lee's predecessor asked him if he'd like to come and help out at the Millennium Stadium. Ever since, rugby and this stadium have occupied Michael's interest.\n\nHolding his mug in both hands, Michael looks out over the pitch. The grass is patterned in even rectangles of pale green and deep emerald. It has been cut, cleared of feathers from the young birds moulting in the roof, then cut again. The whitewash of the touchlines, trylines, twenty-two-metre, ten-metre, dead-ball and halfway lines has been replenished. Michael himself has trimmed the grass round each set of posts with a pair of scissors. The pitch is ready.\n\nMichael takes a deep breath and begins to feel the sensation he always feels when he comes in this early on a match day welling in his chest: 'A deep fucking sadness.' He gives the pitch a nod \u2013 part approval, part acceptance \u2013 then takes another sip of tea before starting his customary lap of the stadium. As he walks, the sadness continues to grow through him, like a blush of melancholy. 'I don't knows why,' he says when asked about it. 'It just does. There's not another soul in the place, but I just feels so fucking sad. 'Cos it's all over, I suppose. Until we start again.'\n\nNo one else spends as much time on the grass of the national pitch as Lee, Craig and Michael. Everyone feels ownership over it: Roger Lewis, the chief executive of the WRU; the members of the WRU board; Gerry, the stadium manager; the fans; and, of course, the Wales coaches and players. But if ownership were measured in time, then Michael, Lee and Craig could make a better claim than most. Every time Lee and Craig double-cut the grass it's a twelve-mile walk if working on their own, or six miles each if working together. Between them they've seen hundreds of players pass across the turf. Their days start early, around 7 a.m., and on the eve of a match they'll often be giving the pitch its final cut well into the night. As they work, pushing the mowers under the floodlights at a determined, steady pace, they both listen to Radio 2 on their headphones. Sometimes they'll text in a request \u2013 'for the groundsmen working on the Millennium Stadium'. If the song is played, they'll raise a silent fist to each other across the empty pitch, before dropping their eyes to the turf again to continue their mowing, making sure to only ever 'walk down the light and never the dark', so as not to disturb their patterns of pale-and deep-green grass.\n\nMichael pauses at the southern end of the pitch, the part of the stadium Lee and Craig call the 'Bat-Cave'. Whatever the time of year, from row four back this portion of the ground never gets any sunlight. This is the turf that needs the most attention and the greatest amount of time under the growing lights. A succession of wheeling scrums in this part of the pitch can cause Lee and Craig, and therefore their wives, sleepless nights.\n\nFacing Michael at the other end of the stadium is an Under Armour advertising banner. As long as several buses, it hangs from the roof behind the raked seating of the North Stand. The torsos and arms of five Welsh players fill its canvas, their red, three-feathered shirts stretched tight across their chests and biceps. At one time the banner used to show the players' heads too, but now it's been cropped, cutting them off at the neck. The turn-over in the squad became too rapid and the scale of the image too expensive to recreate.\n\nA team is both eternal and ephemeral, its members forever changing. As Warren Gatland often reminds his young squad, they're only borrowing the red shirt of Wales. Injury or another player rising through the ranks can be just around the corner. So the shirt is only borrowing them too. And that's what the cropped banner seems to say. You will borrow the shirt, and the shirt will borrow you, but only the shirt and the team will remain. You who fill out both are just passing through. _Yesterday_ , as the banner in the Barn never lets the squad forget, _is in the past._\n\n## 6.30 a.m.\n\nThe eighty minutes of today's match, though, are still in the future, and as Michael completes his lap of the stadium, this is what the men who'll wear those red shirts today are thinking about as they wake up. It is still early, but as Michael leaves the stadium's bowl by the Dragon's Mouth, as he takes his growing sadness back to Barry, thinking, 'Sod this, I'm off home for some breakfast,' the Welsh players, twelve miles west in their shared rooms at the Vale Resort in Pontyclun, are already stirring. Their minds will have woken before their bodies, occupied with thinking about the day's events. Their stomachs are light with nerves. Those who asked Prof. John, the team doctor, for a sleeping pill to get them through the night are still asleep. But those who didn't are already waking and, therefore, from the second they open their eyes, preparing.\n\nTheir captain, Sam Warburton, wakes up thinking of food. Not the taste of it, but the value of it. Food as fuel. This will be his main concern for the rest of the day until he boards the team bus for the journey into the city centre. How to make sure his body has the calories it needs for the exertion ahead? As he lies in his bed he visualises his body as an empty tank, filling through the day's three meals before the match so that, at kick-off, he will be ready. But to be ready he has to keep that food inside his system, and on a match day that can be a challenge in itself. Sam wants the food, needs the food, but his nerves often kill his appetite and make his stomach unpredictable. More than once he's coughed in the shower before a game only to find himself following through and vomiting up the pre-match meal. If that happens, Sam, panicked, will make straight for the team room to drink a protein shake and eat a banana. Although he doesn't want solids inside him at kick-off, just their nutritional resonance, he knows he needs the fuel. And so he will eat, to give his body and the team he captains the best chance of coming through those eighty minutes on top.\n\nDown the corridor from the room Sam shares with Dan Lydiate, his playing partner at blindside flanker, the team's centres, Jonathan Davies and Jamie Roberts, are also rooming together. On international days, other than when eating Jonathan would usually sleep and nap through the hours in the build-up to a match. But today is no ordinary match and the prospect of what lies ahead will keep him awake for the rest of the morning and afternoon.\n\nFor the last two months the words 'Grand Slam' have rarely been spoken in the Welsh camp, although journalists at the ever-growing press conferences have been using the phrase with increasing regularity. The Welsh news papers, with typical enthusiasm, began seeding the phrase in their articles after the team's first win over Ireland. But the Welsh players and coaches have always remained focused on the next game in the competition, rather than the potential prize at its end. For a squad in camp, expectation and aspiration are volatile fuels, essential but combustible. Given too much of an airing they'll easily explode the very potency that brings a team success. Privately, though, the thought of winning a Grand Slam has never been far from the thoughts of the Welsh squad. From the moment Wales lost to France in the World Cup, Sam, not usually a superstitious man, acknowledged a strong intuition that Wales 'deserved something good to happen' in the coming Six Nations. After another rehab and conditioning week of cold-weather training at Gdansk in Poland, the squad entered the tournament with the belief that they were just too good to fail. But they were also realistic. Sport can be cruel, and rugby more cruel than most. Key World Cup players such as Alun Wyn Jones and Gethin Jenkins were still injured, as was the hooker and former captain Matthew Rees. Nothing could be taken for granted. Which is why, as with each win Grand Slam fever infected more and more of Wales, the Welsh camp itself has remained an island of calm within the country.\n\nOver the course of the tournament the players and coaches training at the Vale have been subjected to an ever-tightening focus of attention. Since last week's victory over Italy, today's match against France has dominated the national conversation. At service stations as you fuel your car, in cafes, pubs, restaurants, staffrooms and offices, schools and hospitals, wherever you'll have gone in Wales for the past week you will have heard aspects of the coming match being endlessly dissected and examined. Anticipation is the lifeblood of the sports fan. This morning, on the brink of a possible third Grand Slam in eight years, there are few in Wales who have not been anticipating today's match and, for this week at least, not become fans.\n\nAnd yet despite this overheated cauldron of obsession, the Welsh camp at the Vale \u2013 the Castle training pitch, the gyms, the team room, the Barn \u2013 has somehow managed to maintain its lower operating temperature. For the last two months, on entering its environment players and coaches alike have felt a palpable expansion of the chest and mind, as if it's here, at the very eye of the storm, where they can think and breathe most clearly, where they can feel most at home.\n\nUntil this week. Over the last six days the seal on the Vale's vacuum has begun to leak. The press conferences have continued to grow, with journalists arriving from France, Italy, Argentina. At each session a forest of camera tripods jostle for position, the Dictaphones on the top table multiplying like cells dividing.\n\nThe public have followed the press too, with fans, families and sponsors all converging on the Vale in the hope of meeting, or just seeing, one of Wales's prospective Grand Slam champions. The players are the same men who started this campaign just six weeks ago, but already success has gilded each of them with the blessing and the curse of being 'a child o' the time'.\n\nLike many of the squad, Jonathan Davies has attempted to escape the mounting pressure and protect his time away from training. He's begun using the service elevators and back entrances of the hotel, and hasn't read or listened to any of the press coverage of the match. The atmosphere at the Vale, he says, has become 'surreal'. But at the same time it's been impossible to avoid the thought of what a win against France today would mean. Over the last few days, as he and Jamie have discussed game plans and tactics in their room, they've also allowed themselves, for the first time, to speak those two words openly \u2013 'Grand Slam'. Not as an inevitability, but as a whispered possibility. With the other four matches of the campaign successfully behind them, they've begun to imagine what the wake of a fifth victory might look like. What would Wales look like if, through the alchemy of today's match, they could transfigure the current ore of expectation into the precious metal of celebration?\n\nFor Jamie, who debuted for Wales against Scotland during their Grand Slam campaign in 2008, his projection of that possible future is conjured upon the taste of experience. For three of the older players in the squad \u2013 Ryan Jones, Gethin Jenkins and Adam Jones \u2013 a win would mark their third Grand Slam each and see them join a select group of just three other Welsh players who've achieved the same. But for Jonathan and the majority of this young squad this would be their first experience of a Grand Slam. The squad's younger players are all in unfamiliar territory, on the brink of realising a childhood dream which none of them, until these last few days, had dared to think could ever come true.\n\nNow just twenty-three, Jonathan was a spectator in the stands at Wales's last two Grand Slams, in 2005 and 2008. When he was growing up in west Wales, he remembers it being an 'unwritten rule' that every Welsh boy would try to play rugby at some point in their life. Jonathan was just five years old when he answered that national stipulation, picking up his first rugby ball at Bancyfelin Primary School, the same school where Mike Phillips, Wales's scrum-half today, began his playing days. Jonathan played as a junior for St Clears, then, when the side disbanded, for Whitland, before, at the age of fifteen, moving on to the Scarlets academy. It was, though, a summer of fitness training with the father of one of his brother's friends, Jeff Stephenson, that Jonathan cites as the crucial turning point of his teenage years. Training with Jeff over those summer weeks turned him from a 'chubby kid' into what he describes as 'more of a physical presence'. As he got bigger, so his position altered, moving down the line from scrum-half, to outside-half, to centre.\n\nThroughout his youth Jonathan always considered himself as one of the bigger backs in the game, and he'd have been right to think so. Just listening to his voice without seeing him is enough to give you an idea of his size. Assured and gravelled it seems to rise from a deep Welsh quarry, resonant with his sixteen stone three pounds of weight. But size is relative, and in this current Welsh team Jonathan has come to think of himself as being on the smaller side. Of the six other men in the Welsh backline today three are taller and heavier than him, including his room-mate in the bed next to him this morning, Jamie Roberts, who at six foot four and seventeen stone four pounds still isn't the biggest back in the squad.\n\nThere was a time when the positions in a rugby-union team catered for a broad spectrum of body shapes: from the full-sail bellies of the props and the tall, cross-beam-shouldered locks, to the shorter, terrier-like scrum-halves and the more slender full-backs and wings. In Wales particularly this pattern resonated with an echo of the country's class and economic hierarchy: the claustrophobic, bowed, scrummaging forwards working at the coalface of a match to win possession for the more individually minded backs, who using their greater freedom of movement and expression would exploit the forwards' efforts, cashing in with a profit of tries and conversions.\n\nIn Wales these backline entrepreneurs were traditionally the smaller men on the pitch, quick and visionary with game-changing side-steps and tackle-defying low centres of gravity. Their body shapes were those of the Welsh soldiers observed by Wilfred Owen in the trenches of the First World War, the 'stocky mountain men' of his ancestry whose same attributes of quickness and shorter stature made them, in the eyes of the poet, good infantrymen as well as good backs.\n\nThough written into the game as early as the 1900s, this Welsh signature of backline play became epitomised in the great teams of the 1970s, in the form of Barry John, Gareth Edwards, J. P. R. Williams, Phil Bennett and Gerald Davies. Of these it was perhaps Gerald Davies who made the most from the tension drawn between the contrasting scales of his physicality and his penetration on the field. Although slight of build, Gerald could tear opposing teams apart, in more than one match scoring every time he touched the ball. His defining skills \u2013 a quickness of reaction, a side-step of both feet at full speed, his spatial reading of the game \u2013 were all products of his smaller stature. Carwyn James has written how Gerald developed the 'instincts of a forest animal', exploiting a survivalist's fear to keep him out of harm's way and, therefore, the hands of the opposition. 'Fear', James writes, 'is an important element in the make-up of such a player.'\n\nThe resulting style of rugby played by Gerald rang true against the dominant notes of Welsh support \u2013 a demand for beauty as well as brutality, a celebration of the small winning over the big. It's a playing style that's been kept alive by a succession of smaller backs, most recently in the quickstep feet of Shane Williams, the diminutive winger who, having scored more tries than any other Welshman, retired in December with yet another try against Australia, followed by a gymnast's farewell somersault. But while Gerald's body shape remained largely unchanged throughout his playing days, Shane's increasingly showed the pressure of the times. Compared to his younger self, the Shane who accepted the crowd's applause after that somersault was almost two stone heavier than the Shane who first took the field for Wales. His neck was thicker, his arms pushed wider by bulked-up lats, his shoulders were piled higher and his chest and biceps were tight with new muscle. The change in Shane's body was a sign of what was to come for Wales. He still scored tries through flexibility and speed, but to stay in the modern game he had to be able to take and give the hits too.\n\nEver since rugby union turned professional, the body shape and contribution of its players have become more uniform. While each position still has its specialism, the breadth of an individual's requirements in top-flight rugby have also grown. Forwards are now expected to move quickly around the field and handle like a back. In turn, the backs are expected to scavenge at the turn-over and put their shoulder to the rucks or mauls. And everyone tackles. Where Barry John could once claim to his captain 'tackling isn't in my contract' as he ran out for the Grand Slam match against France in 1971, Wales's number ten facing the French today, Rhys Priestland, will more than likely make shuddering tackles well into double figures.\n\nWales's response to the increased physicality in the game has been to enter a period of gigantism. On the eve of the Scottish match the _Guardian_ , reflecting on Wales's win in Dublin the week before, published an illustrated comparison of the backlines of four international teams: Wales, England, New Zealand and South Africa. The pictures of twenty-eight big men ranged across two pages of the paper. But for the first time ever, and against the grain of Welsh tradition, none of the other backlines was bigger than the Welsh, who stood on average two inches taller and almost a stone heavier. For their Six Nations opponents these statistics were all the more imposing because of what the rugby world had witnessed the previous weekend at the Aviva Stadium in Dublin: a backline of young Welsh giants, yet still possessed with the skills and speed of those men who'd worn the same numbers on their backs in the 1970s.\n\nMany of those ex-Welsh players had travelled to Dublin to watch that opening match against Ireland, including Gerald Davies, who now writes on rugby for _The Times_. As a lover of T. S. Eliot's poetry, Gerald had a CD of _The Four Quartets_ in his car when he drove to the airport that weekend. As he listened to 'East Coker' on that drive, Gerald would have heard the poem's opening line \u2013 _In my beginning is my end_. It's a line which this morning, as the Welsh players at the Vale stir and wake in their beds, rings truer than ever in relation to that opening match against Ireland. An opening match which not only set the tone for Wales's tournament, but also laid the stones for today's Grand Slam decider against France.\n\n*\n\nThe Welsh team flew to Ireland across a snow-dusted Wales and a corrugated sea, both suitable preparations for the frosty, rough-edged reception waiting for them in Dublin. The talk on the city's streets and in the press was of World Cup revenge. Not for Wales's loss to France, but for Ireland's loss to Wales. Having outmuscled Australia, Ireland went on to the quarter-finals, only to be out-thought by a tactically astute Wales. As a reminder of that defeat, on the morning of the match the Irish papers carried a photograph of a jubilant Mike Phillips leaping into the air, his fist clenched, against a backdrop of bowed heads in green shirts. The Celtic tiger was still smarting and now, it was felt, on a cold Sunday in February, with every second mannequin on Grafton Street wearing one of those green shirts and the gulls skating across a frozen St Stephen's pond, was the time to bite back.\n\nThere were other undercurrents of tension pulling at the Irish game too. Ten years ago the Irish Rugby Union had sacked Warren Gatland and replaced him with his assistant coach, Eddie O'Sullivan. Now, with the 2013 Lions tour on the horizon, it was well known that Warren and Eddie were both in the running for the coveted position of Lions head coach. In the wake of the World Cup both teams were injury-hit, but with a strong record of home wins against Wales, recent club victories in Europe and an undefeated run at their new stadium the Irish, even without their talismanic captain Brian O'Driscoll, were the bookies' favourites. Wales, meanwhile, had followed their World Cup loss to France with two more straight defeats, both against Australia. With these games in mind Austin Healey, the ex-England international, delivered a forecast shared by many commentators:\n\nThe Irish are flying; they've got three teams through to the quarter-finals of the Heineken Cup and have a strong squad. I'm expecting Ireland to beat Wales by at least 14 points. Put me on the spot and I'd say 27\u20139 because the Irish players have been doing so well in Europe. It will be pretty tough for Wales. I'm predicting they will finish fourth in the Six Nations table with only two wins, against Scotland and Italy.\n\nPsychologically, if not physically, every international rugby match begins weeks before the kick-off. For the Irish game, despite predictions like Healey's, Wales had won this pre-match contest before they'd even landed in Dublin. The underdog label suited them. The squad would rather fight for a win from underneath than protect a perceived advantage from above. With a typical deftness of touch Warren had postponed his own team announcement until after Ireland had shown their hand, having already seeded certain ideas in the minds of the Irish selectors. Photographs of Jamie Roberts sitting on the sidelines at training had been printed in the papers, suggesting the big Welsh centre wasn't yet match fit. Jamie went on to have what he later described as 'one of his best days ever in a Welsh shirt'.\n\nHowever well Wales contested the build-up to the match, the game itself remained a massive challenge. A vengeful Ireland at home is a daunting prospect for any team. Without the retired Shane Williams Wales were fielding an untried new wing pairing of George North and Alex Cuthbert, as well as a vastly altered pack from the World Cup, with the towering Ian Evans back after three years of injury and the rookie Rhys Gill coming in at loose head to replace the veteran Gethin Jenkins. The question of promise posed to Wales in the wake of their World Cup defeat suddenly looked like a significantly harder one to answer.\n\nFor Jonathan Davies the Ireland match presented a more personal challenge. On the Thursday before the game his grandfather died, and Jonathan had to carry this loss with him across the Irish Sea and into the stadium that Sunday afternoon. Singing the national anthem when you're about to play for Wales is emotional for any player; a moment when the compression of the occasion focuses a player's thoughts upon certain individuals in their life. The lyrics may be of nationhood \u2013 ' _Mae hen wlad fy nhadau yn annwyl i mi'_ ('The land of my fathers which is dear to me') \u2013 but in the minds of the young men singing them those lyrics often evoke people more than a place: the families, wives, girlfriends, parents and grandparents who over the years have helped make this moment possible for them, and whose own dreams, by playing for Wales that day, they are making come true.\n\nFor Jonathan, singing the anthem in Dublin was almost too much. Intensified through the prism of his grandfather's death, it was all he could do to contain himself and gather his thoughts for the game ahead. The resonance of that emotional intensity, however, would remain with him for the next eighty minutes, and never more so than when, just fifteen minutes after singing the anthem, he gathered a sweetly timed offload from Rhys Priestland and crossed to score Wales's first try of the match. As Jonathan walked back from the Irish tryline he flicked his eyes skyward, then kissed the ball with which he'd just scored before raising it, briefly, in salute. He went on to follow that first try with another in the second half, scored that cold Sunday, like the first, for his grandfather as well as for Wales.\n\nIn the end everything worked out for Wales that day. The match was painfully close, at times no more than an inch or a second from being lost, and it was far from perfect. For ten minutes, when Bradley Davies was sin-binned, Wales once again found themselves playing with fourteen men. Rhys Priestland squandered eight points in missed penalties and conversions. But despite these setbacks Wales displayed a new strain of mental and physical toughness for the full eighty minutes. For eighty minutes they brought their memories of Spa\u0142a and Gdansk to the Aviva pitch and they used them, at the very last moment, to win against a home side on form and bent on revenge.\n\n*\n\nWinning in sport is often about repetition; about trying to recreate that elusive blend of preparation, discipline, rhythm and instinct. Routine, though never the key on its own, is often the path teams and individuals choose to bring themselves closer to that winning recipe. Today, in the eight hours before the match against France, the Welsh players waking in their beds at the Vale will each follow an interwoven pattern of individual and squad routines. Jonathan Davies, wanting to avoid a 'dullness of the eyes', will choose not to watch TV or play any computer games. Sam Warburton, when not focusing on fuelling his body, will isolate himself. Lloyd Williams, the young scrum-half, will begin a new book. Adam Jones, the tight-head prop, will call his wife and listen to the sounds of his new baby daughter on the phone.\n\nAdam, or 'Bomb' as he is known within the squad, is one of the team's better-known faces. Already a veteran of two Grand Slams, his distinctive mop of dark curls has earned him yet another moniker in public. When he played alongside another similarly hirsute prop, Duncan Jones, the pair became known as Wales's 'Hair Bears'. It's an appearance which also caught the eye of Boris Johnson, the mayor of London, who once claimed Adam as his favourite athlete, both because he'd played prop himself and because he said Adam reminded him of 'Cro-Magnon man'. Before Adam's last match against France at the World Cup, Boris even phoned his hotel room to wish him good luck, which was, Adam admits in his quick Breconshire accent, 'a very, very surreal moment'.\n\nWithin rugby, Adam is as well known for the quality of his scrummaging and the depth of his knowledge as for his haircut. The bar he's set at tight head is so high it's now the one position in the squad where Wales struggle to find equivalent cover. Over the last few years he's brought his weight down from twenty-three stone to closer to nineteen, lifting his levels of fitness to keep pace with the modern game. Amenable and gentle off the field, with a quick sense of self-deprecating humour, Adam doesn't think of himself as 'a very confrontational bloke'. And yet during a match he's often the epitome of the word, whether packing down to take on an opposition front row or throwing himself into tackles, his long hair flying about his head a visual register of the force he's put into the impact.\n\nIn France the front row is as iconic among rugby fans as the outside-half is in Wales. The French pride themselves on their scrummaging ability, on their tradition of dominating opposing packs. So Adam knows today will be another hard day at the office, and that he'll need to protect his right shoulder to avoid being shunted by the French. But at least he's back with his British Lions teammates, Matthew Rees and Gethin Jenkins, both recovered from injury. Between the three of them they present a front line of massive experience; a vanguard of hundreds of international hours behind which the team's younger backline will hopefully be able to go to work.\n\nAdam, like all the squad, will go for a 'primer' today. Not that long ago these didn't exist, with teams doing all their physical preparation at the stadium. For several years they consisted of no more than a weights session. Today, although many players will still use weights to stimulate their nervous systems, to give their bodies a match-day sensation of tightness, the primer is more individually focused. Under the instruction of Adam Beard, Wales's head of physical performance, the primer is now about responding to the specific needs of that player on that day. Some will stretch existing tension in certain muscle groups. Others will work on co-ordination skills, sharpen reaction times. The backs will also walk through their moves, while the forwards will walk through their line-out drills, laying down the rhythms of each call in their muscle memories. For Leigh Halfpenny, once again today's long-range goal-kicker, his primer will be to do what he's done a thousand times before: go down to the Castle training pitch and practise his kicks with Jenks. And because this is a match day he'll do this not just for the kicking itself, but also 'to get a feel for the day': the quality and strength of the wind, the weight of the weather, the taste of the light.\n\n## 9.00 a.m.\n\nRoger Lewis, the chief executive of the Welsh Rugby Union, stands on the edge of the Millennium Stadium pitch, bathed in the warm light of a camera crew. Dressed in a grey suit, red WRU tie and red scarf, he is giving an interview to the BBC. Beside him the Blims, a five-man band from Bridgend, are warming up to play their Grand Slam song, 'Sidesteps and Sideburns', which over the past week has become a YouTube hit. They wear the red and white striped scarves of the 1970s, and the lead singer has thick sideburns reminiscent of the era. Even when celebrating today's team, most of whom were born in the late 1980s, Welsh rugby's stubborn memories of the 1970s still persist, as if support of a contemporary squad will always be tainted by comparison.\n\nThe camera turns towards the Blims, and they begin to play.\n\nA long time ago before I was born we once had a team that was always adored,\n\nwith sidesteps and sideburns and Grand Slams galore,\n\nour magical boys wrote their names in folklore.\n\nThe BBC broadcast shows live on the big screens hanging from the North and South Stands, briefly committing the Blims to a diminishing hall of mirrors as they appear on a screen within a screen within a screen. Roger looks on, smiling, knowing that today his stadium and his team are at the centre of the centre.\n\nRoger is fifty-eight years old, but with his neatly parted brown hair, trim figure and senator's smile he looks fifteen years younger. Having studied music composition and worked as a composer and musician, Roger came to the WRU in 2006 via executive positions at Radio 1, Classic FM, EMI and Decca. Although now firmly ensconced in rugby, he sometimes feels as if on a match day his two worlds still meet in this stadium, the teams like orchestras, the game a concert and the referee the conductor. Music and rugby come together at Roger's home in St Hilary too, where he'll often work into the night on his laptop beside the fire, deep in WRU business but always accompanied by one of the thousands of CDs from his room-sized library next door.\n\nThis morning marks the apex of Roger's time with the WRU. When he first joined the organisation, it was struggling, on and off the pitch. In 2010, after a difficult season, he was questioned both privately and publicly about the wisdom of extending Warren Gatland's contract as Wales head coach. Today, though, having reduced the organisation's debt, opened up new revenue streams, secured Warren until after the 2015 World Cup and signed new sponsorship and broadcast deals, he will watch as Wales make a bid for their third Grand Slam in eight years. There are still issues. Regional rugby is fighting to keep its head above water, and many in the game are looking to Roger and the WRU to throw it a lifeline. There have been accusations that the national game is thriving at the expense of local rugby. There are always others who would do things differently. But today, just a year after those doubts about his decision-making were expressed, Roger is enjoying, for a few hours at least, being at the helm of the WRU when the country is febrile with talk of a third golden generation of Welsh rugby.\n\nIn his own way, like the players and the coaches he employs Roger is a winner who thrives on the hit of success. Ruthless in pursuit of his ideas and a shrewd judge of character, he's a man unafraid to explore the visionary and the experimental. Driven by a desire to fly 'at great heights', he's moved swiftly through the higher echelons of the music and broadcasting industries. And yet, despite his current position and the depth of his past experience, there is still something permanently boyish about Roger, as if Just William had suddenly woken to find himself a successful executive, still effervescent with the surprise of his new-found powers in the grown-up world.\n\nBehind Roger the stadium's roof is still only open by a metre. All around him the building is being prepared. PR women high-heel down the players' tunnel, phones to their ears, clipboards balanced like babies at their hips. The suited event managers speak to each other through headsets, while cameramen lay looping armfuls of cables and the sponsor's logos are painted onto the pitch. Thousands of folding seats click and tut from high in the stands as teams of cleaners move through the aisles and rows. Deeper within the building the staff of the Barry John and Gareth Edwards bars on level three are connecting their beer barrels and checking their pumps.\n\nBack home in Barry, meanwhile, Michael, having finished his breakfast, sits in his favourite armchair and turns on the TV. The Blims appear on the screen, playing and singing their song.\n\nAnd we'll all hold hands and drink to each other's good health,\n\nAnd we'll all hold hands and thank the Lord we're born Welsh,\n\nAnd the more we drink the more we'll sing _Calon Lan,_\n\nAnd we all hope Wales win the Grand Slam.\n\nMichael watches, another cup of tea in his hand, admiring the colour and pattern of the pitch on which the band are playing, the neatness of the grass around the posts.\n\n## 10 a.m.\n\nIf Jonathan Davies laid the foundation for Wales's victory against Ireland, it was Leigh Halfpenny who sealed it by kicking a winning penalty in the dying seconds of the match. Just moments before that penalty was awarded it looked as if Eliot's line \u2013 _In my beginning is my end_ \u2013 would have a purely negative resonance for Wales, rather than the positive one it does today. A dead leg meant Sam Warburton didn't come back in the second half, leaving Ryan Jones to step into the breach as captain. Bradley Davies's sixty-fourth-minute sin-binning for dropping his opposite number in an off-the-ball tussle not only evoked the spectre of Sam's World Cup red card, but once again reduced the team to fourteen men. With only four minutes of the match remaining, George North resurrected Welsh hopes by bowling over two Irish defenders to score in the corner. But then Leigh, having taken over kicking duties from Rhys Priestland, missed the conversion, leaving the scoreline of Ireland 21 \u2013 Wales 20 also looking like the final result.\n\nBut then, as if the gods of rugby were possessed of a perverted love of symmetry, another tip tackle, this time by Ireland's Stephen Ferris on Ian 'Ianto' Evans, opened up a last window of opportunity for Wales, acknowledged by Ianto's grateful ruffle of Ferris's skullcap as he got back to his feet. The penalty was thirty-five metres out and just to the right of the Irish posts. Ryan asked the referee, Wayne Barnes, how much time was left on the clock. His answer was picked up by the TV mikes and broadcast around the world. 'Five zero,' Barnes said. 'Fifty seconds.'\n\nAs Leigh angled the ball onto the kicking tee he found himself in the same position as in the final minutes against France at the World Cup in New Zealand: just five steps and one kick from victory or defeat, success or failure, wedding or funeral. Had Wales's World Cup experience made them weaker or stronger? Could Leigh, with one kick, help Wales come true on the promise they'd shown?\n\nWhen Leigh had missed that kick in Eden Park, his grandfather, the man who'd first taken him to practise in Gorseinon when he was nine, had been watching from half a world away. For this kick in Dublin he was watching again, but this time from within the stands of the same ground, feeling the Aviva's part-arena, part-conservatory atmosphere draw in tight as his grandson stood up from the kicking tee, stabbed the turf with the toe of each boot and began stepping backwards from the ball.\n\nThe voice of the crowd fell. A few isolated boos and jeers resounded against the stadium's glass waves, but most of the 55,000 spectators just watched, silent and tense. The last eighty minutes had taken their toll on everyone. Once in position Leigh took a deep breath, his hands swinging at his sides, and looked up at the posts. The noise of the crowd was building again. Whistles, shouts, jeers. The Irish players were lined up under the crossbar, steam rising from their shoulders. His Welsh teammates were standing down the field behind him, their chests rising and falling with heavy breaths. Closer to him, a few metres off his right shoulder, stood Jenks, quietly reciting the same liturgy of technique that he repeats for every kick Leigh takes.\n\n'Keep upright.'\n\n'Not too fast.'\n\n'Make good contact.'\n\n'Follow through.'\n\nListening to Jenks's voice, Leigh looked down at the ball and located his point of contact. Ever since the kick he'd missed in the semi-final of the World Cup he'd thought about this moment. And every time he had, he'd made the same promise to himself. A promise that if he was ever given the opportunity to kick a winning goal for Wales again, he'd nail it. That he wouldn't ever let what happened at the World Cup happen again. That there was no way he would let the ball fall short. That he wouldn't just kick it, but would kick it as hard as he could.\n\nTaking another deep breath Leigh stared down at the ball, Jenks's voice becoming ever more distant as he tipped his weight forward and began his run-up:\n\n'Keep upright.'\n\n'Not too fast.'\n\n'Make good contact.'\n\n'Follow through.'\n\n_Follow through._\n\nLeigh has been training with Jenks ever since the ex-Wales and British Lions fly-half came to Swansea to run a kicking session for the Ospreys academy. Leigh was just sixteen at the time and 'nervous as hell' about meeting Wales's greatest-ever kicker. But his grandfather recognised that the session marked the next step in his grandson's career. 'The apprenticeship's over now, boy,' he told Leigh that day. 'I've taken you as far as I can. You're with the main man now.'\n\nAs Jenks talks Leigh through his practice kicks on the Castle pitch this morning, a clear sky burning off the river mist, he does not necessarily look like 'the main man' or, some would say, like a rugby player at all. As he paces behind Leigh, a pair of rugby balls tucked under one arm, two more held in each hand, he walks with a rolling, agricultural gait, his close-cropped red hair receding from his forehead and temples to a pair of protruding ears. He is not especially big and his frame is softer-edged than you might expect. But Leigh's grandfather's statement is still impossible to deny. Anyone who knows their rugby also knows that Jenks is every inch 'the main man'.\n\nIn his thirteen-year international career Jenks racked up more points than any Welsh player in history, scoring a total of eleven tries, 130 conversions, 235 penalties and ten drop goals. When he retired, he was the only player in the world to have scored more than a thousand international points. As he punts balls back down the pitch to Leigh this morning each kick betrays why that was the case. Every ball, nine years after Jenks hung up his boots, still lands with precision at Leigh's feet.\n\nBetween them Jenks and Leigh, coach and player, mentor and pupil, bridge the gap between rugby's amateur past and its professional future. Both Jenks and Leigh debuted young, at nineteen. Both had relatives putting in the hours in the wings: Leigh his grandfather; Jenks his uncles Peter and Andrew, drilling him towards a style and a rhythm on the training pitch at Pontypridd. But where Leigh came into the Wales set-up via the academies and a professional career, Jenks was filling a skip in his family's scrapyard when his father returned from the pub one day to tell him he was in the Welsh squad to face England. When he learnt about his second cap, he was tiling a council house and listening to the team announcement on a radio. Where Leigh built himself up through years of weights, strength programmes and supplements, Jenks was conditioned by swinging a sledgehammer or dragging an old tyre on shuttle runs in the alley behind his house. Jenks was a player formed in rugby's amateur era, driving lorries of scrap metal the length of Britain or working in Just Rentals between games. In contrast, Leigh has been shaped by the professional game, the hours of his days portioned by the training, media and fixture schedules of club and country.\n\nYet as they work together on the Castle pitch this morning, rehearsing the combinations that will unlock each kick, there is more shared territory between these two men than not. However much rugby has changed, there will always be fundamentals of a player's experience that will never alter and will continue to form strong bonds between the generations. For Leigh and Jenks the goal-kicker's ritual of solitary practice, that silent benediction exercised on a succession of empty pitches, has provided them with thousands of hours of shared experience. Though practised years apart, the two men's characters have been formed in the shadow of these repeated attempts to balance that ever-unstable equation of man, rugby ball and posts.\n\nAnd then there have been those other shared experiences too. Rarer, but more intense, when under the stiletto-pressure of a match-day kick that coalface of practice is converted into a few diamond seconds under the eyes of millions. When a stadium and two stalled teams all wait upon the strike of your boot to tell them how, this time, the script will be written.\n\nOver the years of their relationship the experience Leigh shares with Jenks has been deepened with inheritance. As with all the youngsters with whom he works, along with general skills advice Jenks has also passed on specifics of his own technique to Leigh. It's a lineage of knowledge that lies at the heart of Jenks's coaching philosophy, an ongoing attempt to undo one of the central paradoxes of rugby: that a player's knowledge will always progress in inverse proportion to the ability of their bodies. As a young player improves mentally, emotionally and technically, as he learns to read the game and himself better, so his body will slow and weaken with age and the violence of the sport.\n\nJenks reached the peak of his own playing knowledge towards the end of his career, when he had just a handful of games left. In coaching the lessons he's learnt to younger players now, Jenks hopes to shortcut them to that peak earlier in their own careers, and so increase the number of games in which their mental and technical skills are on a par with their physical ones. This is why he prefers to work with players like those kicking today \u2013 Rhys Priestland, James Hook and Leigh \u2013 from as early as fifteen or sixteen years old. Bad habits ruin more kickers than good habits make them. At eighteen or nineteen it's often too late; those bad habits are already stitched into a player's style. But at fifteen, sixteen Jenks knows he can build upon a player's skills without first having to unpick existing faults.\n\nAlthough Jenks hasn't taken the field himself for almost a decade, his coaching work means a part of him still runs out with young players onto grounds across Wales every weekend. And a part of him still faces up to the posts on those grounds for the kicks at goal too. Every young kicker with whom he's worked eventually develops their own method, their own stance, approach and follow-through. But until they do they'll first glean what they can from the DNA of Jenks's kicking style. When the young Hunter S. Thompson decided to be a writer, he typed out the entire manuscript of F. Scott Fitzgerald's _The Great Gatsby_ , just so he could 'feel' what writing a great novel might be like; its rhythms, pace and style. In a similar way young kickers working with Jenks will imitate the handwriting of his technique before creating their own, so they too can 'feel' how it might be to kick like one of the greatest in their game, before attempting to be so themselves.\n\nThe most important technique Jenks passed on to Leigh as a young player was psychological rather than physical. Wherever Jenks was kicking for goal, be it at the Arms Park in Cardiff, Kings Park in Durban or the Stade de France in Paris, as far as he was concerned he was never inside those stadia, but many miles away instead, on Cae Fardre, a field near his childhood home in Pentre'r Eglwys. With each of those kicks, as Jenks stepped back from the ball he also stepped back onto Cae Fardre, the place where he'd spent thousands of hours practising alone as a kid; where he'd kicked for fun and where things had always been easy.\n\nFuture visualisation is key in sport: the marksman sees the bullet penetrate the bull's eye before he shoots; the sprinter sees himself breast the line before he starts; and the goal-kicker sees the ball sailing through the posts before he makes the kick. But when Jenks was taking place kicks for Wales or the British Lions, his visualisation always projected two ways: forwards and backwards. Forwards to the successful kick, and backwards to those hours kicking alone on Cae Fardre. For Jenks, each place kick was also a kick about place: about using one location to protect himself from another, about drawing upon his memories of Cae Fardre so as to isolate himself from the pressures of the place in which he stood at that moment, facing up to the posts inside the crucible of a stadium.\n\nIt's this other dimension to 'place' kicking that Jenks has passed on to Leigh, so that now, when Leigh lines up a kick for Wales, it's another corner of the country that's conjured into stadia around the world: the pitch in Gorseinon where he and his grandfather first went to practise, and where they still go together whenever they can. Just as Cae Fardre used to insulate Jenks at the moment of a kick, so an image of Gorseinon, its trees and houses spreading through the Aviva Stadium in Dublin, insulated Leigh in Dublin that day from the gathering noise and whistles of the crowd as he quickened his steps towards the ball and, with just fifteen seconds left on the clock, struck it with all the power he could muster.\n\n_Keep upright._\n\n_Make good contact._\n\n_Follow through._\n\nThe ball was still in the air when Leigh turned his back on that kick, and he was already running back to his teammates when the linesmen raised their flags. He knew the kick had been successful, and he knew that Wales had won. After the match, when he'd finished his interviews and his signings for fans and sponsors, Leigh found his grandfather and gave him the pair of boots with which he'd just kicked Wales to victory. The Irish match had begun with a try for one grandfather, and it was finished with a kick for another. In more ways than one Wales's first fixture in the Six Nations had been a game about starting points, foundations. _In my beginning,_ as Eliot says, _is my end_.\n\n## 10.30 a.m.\n\nHaving taken just eight kicks at goal this morning Leigh and Jenks collect the balls and begin to make their way back to the team room up at the Vale. Eight kicks are all Leigh needs. These pre-match hours, for all the players, are a negotiation between preparation and freshness. About sharpening the blade without blunting it through overuse; about keeping as much in reserve as possible for the eighty minutes of the match ahead.\n\nAs they walk up towards the seventeenth-century turrets of Hensol Castle, which gives the training pitch its name, Leigh confides to Jenks that he's feeling more nervous than usual today, and for some reason he can't seem to shake it off.\n\n'Enjoy it, butt,' Jenks tells him, giving him a pat on the shoulder. 'Give it your all.'\n\n*\n\n'All' is what rugby is to Jenks, and his all is what he's given the game. Which is why he knows about nerves. Within the squad he's as well known for his pre-match anxiety as he is for his kicking. Whether it's a fixture against Tenby or the All Blacks, whether playing or coaching, the run-up to every match will often see Jenks, wracked with nerves, retching in the changing rooms.\n\nJenks is also known within Welsh rugby for his encyclopedic knowledge of the game; for being able to tell you not only which team won on a certain date, but by how much and who scored when. If current players are still inheriting their aspirations from Jenks, then he in turn was first inspired by the players of the 1970s. When he was seven years old, he saw one of those Cardiff games in which Gerald Davies touched the ball four times and scored four tries. Ever since, fuelled by his rugby-playing uncles, he's been obsessed by the sport, _feeling_ it even more than he knows it. During a match he lives every kick, break and tackle made by the players he coaches. Afterwards he'll absorb himself in analysing the patterns and plays. At the meal after the Irish match, as the two squads and their guests mingled around the tables talking and drinking, Jenks stood alone in a corner of the room, one hand in the pocket of his suit, the other holding a pint, looking up at a TV screen playing the match in which he'd just participated. It was as if even then, with the resonance of the game still fresh in the air and the grazes on the faces of the players still raw, Jenks wanted to see it all again, wanted to live it all again. As if, after all those months of preparation and training, he was reluctant to let those eighty minutes go.\n\n*\n\nFor all the Wales coaches today the last forty-eight hours have been a slow process of letting go. Jenks, who will run messages during the match and be with the kickers for conversions and penalties, will have more contact with the players than most. But even for him, from the Thursday of a match week onwards the majority of his work is done. Like a director and stage crew in the run-up to an opening night, Warren Gatland and his coaching team have to make the uncomfortable transition from being at the centre of the cast, the team, to becoming a member of the audience, the crowd.\n\nThis transition from influence to observation is one of the hardest parts of the coaches' jobs. As ex-players all of them first got involved in rugby to be on the field, and now they are off it, shaping policy and style of play, but unable to do anything at the moment of execution. As coaches they are still at the heart of the sport, so the sense of aftermath experienced by every ex-player is kept further at bay than for most. But on a match day that process is reversed, the proximity to the players and the game, the sound of the studs in the tunnel, the smells of strapping, mud and lubricant all magnifying rather than diminishing the distance between playing and not playing.\n\nEven now, as Jenks enters the team room back at the Vale, although many of the coaching staff are in here with the players, sitting at the dining tables, looking through analyses on laptops, the impending match is already, minute by minute, opening the distances between them. Already they are growing apart. And yet the coaching relationship never ends, which is why all of the coaches, despite their own nerves or stress, will do their best to maintain an atmosphere of calm steadiness. In their own playing days teams and coaches often got psyched up, players shouting and head-butting walls in the changing rooms. But not today. Nowadays game day is about staying calm, about holding a level-headed gaze towards kick-off and what has to be done in the eighty minutes after it.\n\nWhile in camp the team room is the heart of a rugby squad; a transplantable heart recreated in other hotels for away matches, and across the world when the squad goes on tour. At the Vale Resort the team room is on the lower ground level of the hotel, accessed through an innocuous-looking door in the lobby. Take a short flight of stairs down through a set of swing doors and you'll find yourself in a large, low-ceilinged room reminiscent of the lounge on a cross-channel ferry. In this room the fundamental elements of modern rugby are all within touching distance, lending it an atmosphere located somewhere between a canteen, a safe house, a student common room, a hospital ward and an Internet cafe.\n\nAway from the Barn or the Castle training pitch, the team room is where the squad can be a team, and also themselves. Two floors above, where the press conferences are held, coaches and players adopt public personas. Their body language and rhythms of speech alter, slipping into the cyclical statements of the sports interview designed to inform but give nothing away. But as they descend the stairs from those press conferences the players shrug their characters back on like comfortable hoodies, allowing their limbs a greater freedom of expression and, in the case of the younger members of the squad, letting the years slip from their faces to reveal, once more, the boys within the men.\n\nThe first element of the team room you encounter on entering is the food, arranged on narrow tables along the left wall; a continually replenished row of silver-domed serving plates rotating through breakfast, lunch, snacks and dinner. Every day in pre-season each player will consume around 4,300 calories to keep their bodies going, with high intakes of protein for muscle growth, and essential and saturated fats for maximal anabolic effect. In season the calories will drop to around 4,070 a day, but the intake of carbohydrates will increase in proportion to the longer distances covered during matches and training, with squad menus continually reviewed by the WRU nutritionist Jon Williams. The majority of those calories will come from these tables in the team room, and for that reason 'players eat first' is the most sacred rule in here. This food is fuel, and it's the engine of the team that needs it the most.\n\nAt meal times the squad eats perfunctorily at a collection of round tables spread and laid in a wedding-like formation. Hunkering over their plates, their bodies seeming too big for the chairs, conversation is often minimal. An arrangement of supplements sits at the centre of each table, as if a chemist has taken charge of the decorations. Printed sheets beside these pill bottles remind the players of their 'SIX NATIONS 2012 SUPPLEMENT PROTOCOL':\n\n**Daily Supplements**\n\nMaintain Beta Alanine, Aspartic Acid and Defence Daily\n\n**Beetroot and Cherry Juice blend**\n\nGame Week \u2013 4 cups per day\n\nTraining Week \u2013 2 cups per day\n\nColostrum will be added to recovery shakes on game weeks to support immune system, aid recovery and improve gut health.\n\nL-Carnitine will be added to shakes daily to aid recovery and fat loss.\n\nIn the toilets plastic vials for tests by the in-house medics are scattered about the sinks.\n\nClose by the dining tables, and facing the food on the far wall, is a row of other tables on which the squad also spend hours of their time each week. These are the physio and massage tables on which, as some of the squad eat a meal, others will be lying, stripped to their underwear to receive treatments from the physios. Occasionally a blue-gloved dentist performs dental work in this area too. Fuel and repair, fuel and repair: the team room is also a pit stop.\n\nPool and table-tennis tables at the centre of the room provide the main leisure activities, while beyond these a bank of WRU laptops are in constant use for video analysis, emailing, tweeting and downloading music and films. Behind a set of sofas another table is kept stocked with shirts, rugby balls and posters for the players to sign as they pass in and out of the room. Over the last two weeks this table's surface has been increasingly obscured under the piles of signature requests and merchandise. Taped to the wall opposite, beside a Powerade drinks fridge, are the latest judgements of the fines committee:\n\nJug \u2013 Wrong kit \u2013 \u00a320\n\nFoxy \u2013 Late to units \u2013 \u00a310\n\nBomb \u2013 Late physio\/no watch \u2013 \u00a330\n\nAs well as being the heart of the Welsh camp, the team room is also its hive brain. This is where the hydra-headed animal that is a team communicates with itself, where its tone and consciousness is set and can be measured. Step into this room on a Monday after a match and the atmosphere within these walls will immediately tell you whether the game was won or lost. This is where Thumper's weddings and funerals are lived, the wake of each match turning the air light and charged or dense and oppressive, depending on the result.\n\nMovement within the team room, as elsewhere in the camp, is governed by the day sheet: the daily schedule of training, treatments, meetings and media duties handed out by Thumper the night before. Without announcement or instruction players and staff will drift like a shoal towards the seats in front of the projection screen and a defence or attack meeting will begin, as if triggered purely by the weight of bodies in that part of the room. Together with the day sheet's instructions on which kit should be worn, such an apparently unconducted movement by so many men wearing identical clothes further heightens the sense of a herd mentality within camp; the blending of tens of individuals, many of whom regularly compete against each other for their clubs, into a single organism.\n\n*\n\nIn one corner of the team room a pair of electronic scales are wired up to a couple of laptops on shoulder-height stands. On first entering the room every morning the players gravitate towards these machines before eating their breakfast. Standing on the scales they answer a series of programmed questions, tapping in their responses on the touch screens of the laptops.\n\n _On a scale of one to five, how did they sleep?_\n\n_What is their mood?_\n\n_Any hint of a cold?_\n\nAn outline of a body appears on the screen.\n\n_Do they have any aches and pains?_\n\nIf they do (and they nearly always do), the players press the on-screen body in the places where their own is hurting. By the end of breakfast their answers to these questions will already have been processed and sent directly to the iPhones and computers of the head of physical performance, head of medical and Warren Gatland.\n\nBefore the players reach the computerised monitoring station they will more than likely already have said good morning to Dan Baugh, the assistant strength and conditioning coach. Those who haven't are sure to see the barrel-chested Canadian approaching them soon afterwards, his hand outstretched in greeting. With a shaved head, beard and Heston Blumenthal glasses, Dan resembles a Hell's Angel bouncer moonlighting as a lab technician. But when he says 'Good morning' to a player, there's nothing intimidating about his voice, just a genuine warmth running through an accent falling somewhere between a Seattle rock singer and an Alberta lumberjack.\n\nThis is how Dan prefers to monitor players, rather than rely solely on the \u00a330,000-worth of computer equipment in the corner of the team room. 'I'd rather shake hands, always have,' he says. 'I got it from the French clubs. First greeting of the day there, it's tradition to shake hands, look someone in the eye, ask 'em how they're feeling.'\n\nThis personal contact is vital to Dan and is at the heart of his approach to coaching. He has an MA in Coaching Science, including a dissertation on 'Emerging Defensive Philosophies in World Rugby', so he more than understands the changing landscape of the modern game, the need to keep at the cutting edge. But in an environment where levels of technology and scrutiny mean there are increasingly few places for a player to hide his moods and emotions, Dan also tries to make the Welsh camp as personal as possible.\n\n'People ask me what we do up here, what's so special about it? And I say we hug a lot. I tell the guys I love 'em all the time. Every aspect of the international game is pretty shitty. It's intense, physically and mentally, you put your body under ridiculous stress. But family fights harder than friends, y'know? You have to really care about the people you work with. On a subconscious level the body can react slower if you don't care for your teammate.'\n\nAs a flanker for Cardiff in the late 1990s and early 2000s Dan knows there was a time when international players relished returning to their clubs from the Welsh camp. This was often because they were returning from an impersonal environment into more of a family set-up. Now he hopes the Welsh coaching team have reversed that process. 'I know the guys can't wait to get back into camp, get back together. And that's great, because the closer we become, the better we become.'\n\nBut Dan's morning handshakes and hugs are as much for him as they are for the players. Compared to the monitoring computers, he personally finds the French method a more comprehensive barometer of a player's well-being, especially as the squad near game day.\n\n'People are much more comfortable saying to you socially that they didn't sleep well, and I mean, there's nowhere on that monitoring thing to say your wife told you to fuck off or that something bad has happened. All the touchy-feely stuff that doesn't go on the computers you can talk about, clear the air. Then, when selection comes into this, all of a sudden on the monitoring kit everyone feels great, all of a sudden everyone slept well, all of a sudden the ratings say, \"I'm ready, pick me!\" So I shake their hands instead, get some eye contact.'\n\nIt was a more robust kind of contact that first brought Dan into rugby at the age of sixteen. He was involved in judo and wrestling since he was four, so physical, combative sport had always been a defining principle in his life. One day, though, in a period when he was playing a lot of Canadian football, some friends asked if he'd join them at rugby training. After just one session Dan knew he'd discovered his calling. As rugby took over, his other sports soon began falling away.\n\n'I just loved the fact there were no pads, that I could hit people, and at the time there were no red or yellow cards so I could get into fights. It was a man's game, y'know? I was a kid, but I was being treated like a man.'\n\nWithin a year of picking up a rugby ball Dan represented Canada at under-16s. He went on to represent his country at every age group, playing against Wales at Fletcher's Fields in 1997. His physical style of play caught the eye of some of the Cardiff players in the side, so when, a year later, the Welsh captain and Cardiff flanker Gwyn Jones broke his neck, it was Dan who Cardiff called. As is so often the case in rugby, an opportunity came his way through another man's injury and at the cost of another player's career.\n\nUnder the alias of Dan Brown, Dan was asked to trial for Gwyn's place in the Blues. He made the trip across the Atlantic and, after impressing his new bosses on the pitch, never went back, going on to play for Cardiff for another seven years. He soon became a popular player with the Cardiff fans, infamous for the physicality of his tackles and his running game, until eventually his style of play took its toll and a broken foot retired him into coaching. Gwyn's injury opened the door for Dan into European rugby and, at just thirty, another injury closed it again.\n\nBut only as a player. Six years later, having joined the Welsh set-up full-time in the summer, Dan has spent this morning at the Vale following his usual strict match-day routine. For once he didn't greet the players over breakfast, and he didn't look them in the eye and ask them how they're feeling. He knows he can be intense on game day, and that the last thing a player needs is 'some super-psyched-up guy sitting in the corner with big eyes'. So he's kept his distance and is focusing on his match-day roles instead: running the primers and preparing for his pre-match warm-up with the team and the subs in the stadium.\n\nDan has brought many principles from his Canadian sporting days into his work with the Wales squad. In the 'Red Room', a padded area at the end of the gym, he regularly works on the close-quarters strength players need in rucks and mauls, drawing upon his experience in judo, wrestling and grappling. His training technique of functional blocks owes more than a nod to his past in Canadian football, while the 'play-off' facial hair he grows for every tournament is a tradition inherited from Canadian ice hockey.\n\nFor this Six Nations Dan chose an Amish-style beard with no moustache or sideburns, which now, on the morning of Wales's final game, after six weeks of competition, has grown as far as the collar of his tracksuit. He has no idea how the dice will fall today. He's seen how hard the squad have worked, and knows how hard he and Adam Beard pushed them once again in Poland. He also knows this is a squad who are close, who will fight for each other as much as for themselves. But the French have come to Cardiff to win, and have a history of spoiling the party for the best teams in the world. So, as ever, there is no certainty, no foreknowledge, other than that this evening, whichever way the scoreline falls, Dan will be able to shave.\n\nDan and Adam work closely together and share a similar vision. And yet in many ways the two men are the direct opposite of each other. Where Dan has a trucker's bulk, Adam has the compact physique of a gymnast. Where Dan is loud and in your face, Adam is quieter and more attuned to subtler forms of persuasion. Where Adam is thorough, Dan's preparations can verge on OCD. Where Dan, in his own words, is 'very hands-on, a bit of a grunt, a grunt with a voice', Adam is an academic. The relationship is not without its tensions \u2013 again, by Dan's own admission he's 'a number two who sees himself as a number one' \u2013 but without doubt it's a combination that works.\n\nThis pattern of complementary opposition is repeated across the Wales management. In contrast to the increased physical and temperamental uniformity of the modern player, the diversity of the Welsh coaching staff, in appearance and personality, still reflects how multi faceted a sport rugby is, and is a reminder of the huge range of qualities the game demands of its players. Within each area of the Wales set-up, men work cheek-by-jowl with colleagues who, like Adam and Dan, often appear to be polar opposites. But however mismatched the components may appear, the mechanism as a whole works smoothly again and again, not in spite, but _because_ of these differences.\n\nAt the physio tables in the team room Prav Mathema, the head of medical performance, works alongside Mark Davies, or 'Carcass', as he's known within the squad. In contrast to Prav's easy-going energy and default smile, Carcass, an ex-international himself, is the squad's stoic, possessed of a granite-like stillness and a 'wise man of the mountain' air. Where Carcass is sparse of speech, Prav is talkative; where Carcass is tall, Prav is shorter. If Prav's approachable manner and gelled hair are those of an affable TV presenter, then Carcass, with his thousand-yard touchline stare and motionless, broken-nosed profile, is more reminiscent of an Easter Island statue that's done time in the ring.\n\nThe differing personalities of the coaches who work alongside Jenks on the squad's defence, attack and forward play are all rooted in their respective roles, with each man embodying the traits of his specialism. Rob Howley, the Wales attack coach and one of the country's greatest-ever scrum-halves, is quick and involved, as alert to nuance and lateral vision in his coaching off the field as he was in his playing days on it. As he moves between the players on the training pitch or harries them through patterns of attack in the Barn, Rob's natural expression is one of furrowed concern, betraying a restless mind always on the lookout for the unexplored strategy, the gap to exploit.\n\nAs a player Rob had to punch above his weight from an early age, having been told on more than one occasion that he was too small or too slight to make it in international rugby. But the schoolboy who'd spent his weekends playing out his dreams of captaining Wales on the local park behind Priory Avenue in Bridgend was not going to be denied. Having made the Wales number-nine jersey his, he finally ran out to captain Wales against Italy in 1997. The circumstances, however, were not as Rob would have wished. His opportunity came, like Dan's, in the wake of the previous captain, Gwyn Jones, breaking his neck. Unlike Dan, Rob knew Gwyn well and had spent much of the weeks before that Italy game beside his hospital bed as his friend came to terms with being told that he may never walk again.\n\nGwyn supported Rob's appointment as captain, and Rob went on to lead Wales to their greatest number of successive victories ever. He was also, though, captain through some of the country's most difficult periods, and it's this mixed experience of success and failure that appears to drive Rob: the tenacity of the underdog coupled with the drive of a winner; the scent of success married with a heightened awareness of its fragility.\n\nOn first meeting Shaun Edwards, the Wales defence coach, he can seem as impregnable as the strategies he's woven into the fabric of the squad, his set pit-bull expression imprinted with the same ethos as his rugby \u2013 _Thou shalt not pass_. But those who get to know Shaun get to know the depths of the man, and with them, closer to his core, a certain warmth. Possessed of a strong sense of 'code', of what is and isn't important in life, gained from his time as a rugby-league player for Wigan and Great Britain, Shaun is possibly the most decorated player in the country, his broken nose and the scars on his bald head testament to the extent to which he's lived his learning. That he would live a life immersed in rugby was never in question. His father played professionally for Wigan, until, at the age of twenty-four, a spinal injury almost crippled him. Unable to play himself, Shaun says his father didn't so much push him into the game 'as kick me in'. As a young boy he'd sleep with a rugby ball in his bed, and every day he walked past Wigan's ground at Central Park on his way to and from school. At the age of just seventeen, that ground became his new destination when the club signed the schoolboy Shaun for a record-breaking \u00a335,000.\n\nHad it not been for his strong desire to live near James, the son he had with Heather Small, the lead singer of M People, Wigan might still have Shaun now. As it was, Shaun chose to move south to be closer to James. Having switched codes he became head coach at Wasps in West London, before Warren Gatland brought him across the border to shore up the Welsh defence.\n\nSuitably, perhaps, for a devout Catholic who counts priests and monks among his closest friends and family, there is more than a touch of the Shaolin monk about Shaun's position within the Wales set-up. Fiercely individual, seemingly immune to the day sheet's kit instructions, Shaun is a reader and a thinker, existing along a tensile fault line between the intellectual and the physical. When analysing a game, he does so with the vision of a chess player, explaining the multiple patterns and attacking possibilities in a soft Manchester accent frequently lifted and quickened by an enthusiast's passion. Praise from Shaun is a rare jewel, and as such is valued all the more highly among the players.\n\nAs the Wales forwards coach, Rob McBryde is responsible for the set pieces of the game, the line-outs and scrums upon which Shaun's defensive strategies and Rob Howley's patterns of attack are built. And Rob is, indeed, as 'set' a character as the tightly packed scrummaging formations he oversees. Solid, thick-necked and boulder-shouldered, there's something of the carthorse's reliable strength about Rob. Capped for Wales at hooker and selected for the British Lions tour to Australia in 2001, he was also once crowned 'Wales's Strongest Man'. Of North Walian stock, his laconic presence at training is reminiscent of a hill farmer come into town for a cattle auction. But there's something timeless about Rob too, his mountain-worn manner sounding a strong note against R. S. Thomas's closing lines on Iago Prytherch;\n\nRemember him, then, for he, too, is a winner of wars,\n\nEnduring like a tree under the curious stars.\n\nAs a first-language Welsh speaker, Rob moves easily between the Welsh and English spoken by the players in the squad. In 2007 he took on the bardic duty of Grand Sword Bearer at the National Eisteddfod, accepting the mantle from its former title-holder, the late Ray Gravell. It's a role that fits Rob well. Where others might look awkward or out of place in a druid's robes, when Rob holds that ceremonial sword aloft he looks all too authentic, and it's no stretch of the imagination to picture him wielding it in the defence of his country a thousand years ago.\n\nWarren Gatland, or 'Gats', the man who manages and has this coaching team at his disposal, is the still point at its centre; a locking pin around which the daily work in camp revolves. With a plaintive set to his eyes and close-cropped silver hair, Warren moves slow and easy about the training pitch like a tracksuited, philosophical teddy bear, his hands in his pockets, assured yet approachable, open to the world yet lost in private thought.\n\nStraight talking, sometimes to his own detriment, Warren also knows how to weigh the power of silence. Before a game he often won't speak in the changing rooms but will just stand at its centre, the players preparing around him, his head bowed, arms crossed, sometimes padding out a pattern of small repetitive steps, as if performing a meditative ritual.\n\nIt's a sparse verbal style suited to the nature of the squad, or perhaps the squad have suited themselves to Warren. Either way, much of the communication in the Welsh camp is non-verbal, a gesture or a look doing the work of sentences. Even at the rugby sessions on the Castle training pitch Warren and the other coaches are minimal in their vocal instruction. The players themselves will erupt into frantic calling and shouting when running moves or doing rucking and spidering sessions, sounding disturbingly like soldiers caught in sudden contact. But on the whole Warren will reserve his voice for afterwards, for when the squad gather round him at the centre of the pitch, hands resting on knees to catch their breaths, leaning in to listen. At the end of these training sessions, just as in the changing room, Warren will often just stand on the halfway line, arms folded, as with a syncopated pattern of hollow punts and thuds the squad practise their various kicks, the balls rising and falling about him like brief constellations.\n\nAt these moments Warren embodies the coach as observer, silently weighing the attributes of the players and staff around him. But his are far from the only eyes watching training. At every session there are always cameras too, mounted on poles, tripods or hand-held by one of the team of analysts. The man who oversees this team and the information they collect is Rhys Long, a big no-nonsense thirty-something with spiky dark hair who, as he puts it himself, 'is paid to watch rugby'.\n\nRhys was brought up in what is arguably the capital of British rugby analysis, Porthcawl, a small seaside town in South Wales. Just as neighbouring Port Talbot produces a disproportionate number of actors, so Porthcawl can lay claim to the head analysts of Wales, England, New Zealand, Exeter and Wasps, plus three more at the English Institute of Sport. As a young field, performance analysts make up a small community, and most of these men were at Rhys's wedding. His best friend, Michael Hughes, the lead analyst for England, was also his best man.\n\nIt's Michael's father, whom Rhys refers to as 'the godfather of analysis, the oracle', who is responsible for Porthcawl's association with performance analysis. As an early pioneer in the field he moved his family to the town when he took up a position at the University of Wales Institute, Cardiff. Michael was eleven at the time, an age when school friends were impressed by a father who made his living from studying the intricacies of sports.\n\nRhys, however, was going to be a professional rugby player, and had Adam Jones not fallen on his leg during an under-21s trial, breaking his tibia and ankle, then he may have still lived out his dream. He was playing at number eight for Bridgend at the time. The club had begun using an early version of Sportscode, software for sports analysis. As Rhys recovered from his injury, and knowing he was unlikely to return to playing, he 'began messing around with it to see what it could do'. That messing around led Rhys, four years later, to the Welsh camp, where, since 2007, he's upgraded and run their analysis.\n\nHis working methods are something of a 'push\u2013pull' process. As he says, 'A good analyst stays close to their coaches.' So much of what he does is in response to what they want to achieve, to what they find effective and helpful in pursuing the model or policy they already have in place. But at the same time Rhys will also initiate changes in training according to the nature of his findings. When his analysis revealed the Wales centre pairing had the lowest passing efficiency of all the Welsh regions, Rob Howley introduced a rigorous passing programme into their training schedule and their efficiency improved. Similarly, if certain players have a low record of success in the air or under high balls, training drills are developed to counter these weaknesses. 'None of it is revolutionary,' Rhys says. 'It's about aggregation of marginal gains. If you improve by 1 per cent in twenty different areas, then that's a 20 per cent improvement overall.'\n\nDuring the Six Nations this process of response and initiation is further complicated by the tight schedule of fixtures. With a match nearly every week, Rhys constantly has to look both ways, analysing Wales's performance against the team just played and analysing the team they are about to play. It's a delicate balance. Over-responding to readings from last week's game against Ireland might be counterproductive if this week's opponents, Scotland, play a totally different style of rugby. 'It is', Rhys says, 'like trying to stay one step ahead, but one step behind too.'\n\nAlthough most of Rhys's working time is spent analysing Wales's performance on the field, his analytical mind can't help drawing from information off it too. One of the strengths of this current squad, he thinks, is their grounding beyond rugby. Many of the players have strong families, older brothers, roots to which they return. A relatively high proportion have stable, settled home lives. Others, however, he admits, play better when not attached, as if 'they need that George Best swagger to play their best game'. According to Rhys, though, Wales's greatest strength is what some in the past have cited as their greatest weakness. 'It's our size,' he explains. 'Our ability to assemble the squad quickly, to centralise everything here. All the boys live close by so they can drop in any time, see Prav or Carcass if they're injured. While they're in camp, our boys, if they want to, go home on a Tuesday night, stay home Wednesday and come in again Thursday. They can see their families, wives, girlfriends, sleep in their own beds. It's like Wales', Rhys says proudly, 'is the biggest club team in the world.'\n\n*\n\nEvery coach, on taking responsibility for a team, will try to create a culture: a foundation of principles and attitudes around which a squad can bond, and to which new members can aspire. Although Warren brought a hive of Wasps into the Wales set-up \u2013 Shaun, Rhys, Howley and Prav all having worked for the London club he used to coach \u2013 he was still, in many ways, a stranger entering a tightly knit family. In this situation most coaches will use a change of culture as a short-cut to evoking new ties and cohesion. Warren's greatest strength, however, is that rather than try and create such a culture, he _is_ the culture.\n\nThe current Welsh squad's personality is an imprint of Warren's own: honest, hard-working, level-headed and accepting of its own responsibility. It's a coaching style Warren partly inherited from those who coached him through school, club and national levels back in New Zealand: men such as Glenn Ross, Kevin Greene and Alex Wyllie. But his coaching philosophy also owes something to his time working as a teacher in the country too. Rugby players are, he says, 'often like big kids', so applying the principles of teaching makes sense to Warren. Meetings are kept short, points to remember kept to as few as possible, and loyalty won through honesty and fair treatment. The resulting coaching style is perhaps more subtle and roundabout than most, reliant upon empowering an individual rather than loading them with information, but the end 'product', Warren feels, is worth it.\n\n'How many test caps you got now?' he'll casually ask one of his young backs, drawing the question through a languid Kiwi accent. When they tell him twenty-nine or thirty, Warren will raise his eyebrows. 'Yeah?' he'll say, a surprised pitch rising through the word. 'That's a lot. You must be one of the most experienced out there, eh? You should be talking a lot. You talking a lot out there?' And the seed, Warren hopes, is sown. Despite being only twenty-two or twenty-three, the player walks away having been given the nod, and the nudge, that as a relative elder statesman they have the licence to play like one.\n\nThat coaching can be about giving time as well as taking it is an aspect of the role Warren understands. Family, he says, comes first. If a player needs a day off because their wife is having a scan, or because there's trouble in the marriage, then Warren would rather lose that player for a day or a week, and win their loyalty in doing so, than keep them in camp and lose it further down the line. Having coached in the northern hemisphere for over twenty years, but with his wife Trudy and son Bryn still living in New Zealand, Warren is no stranger to the strains rugby can put on family life. It is, as he often says when talking about rugby, an ongoing question of balance and negotiation. About knowing when to put the pressure on, and when to ease off.\n\nToday, with still four hours until the anthems, it's time to ease off; a time for restraint, for not over-coaching so close to a match. Because of this, Warren will keep to himself for much of the day and will try his best not to broadcast his nerves to the squad. The first time he'll see them will be for the walk-through of moves and line-outs. Then, like all the coaches, it'll be about keeping to his match-day routine until the team meeting and the boarding of the bus. Rob Howley will use this time to call home and speak to his wife and kids, absorbing himself in their plans for the day rather than his own, which are already being broadcast across the country. Rob McBryde will go to the gym, get a feel for the mood of the players over breakfast, then 'stick his head in a book' to pass the time. Shaun, too, will try and say as little as possible to the players, and will also often read a book. When he was a player, together with praying before a match, reading was a technique Shaun often used to cope with his nerves. It's one he passes on to some of the players too. Sometimes he will even take his book up into the coaching box, and he has been known to carry on reading it while in there.\n\nWhatever their routine or habits, all the coaches will try and use this period of the day to focus on their roles and do whatever they can to kill time, until the only time that matters finally arrives: the eighty minutes of Wales against France, played in the Millennium Stadium by a squad they've relentlessly coached, advised and analysed, and whose fortunes and well-being have consumed the last ten weeks of their lives.\n\n## 11 a.m.\n\nOn taking charge of Wales in 2007, Warren switched the home changing room in the Millennium Stadium from the northern side of the players' tunnel to the southern. The years preceding his tenure had been unstable and volatile. The spark of brilliance that won Wales a Grand Slam in 2005 was soon extinguished by rifts between players and coaches. A series of stuttering performances finally culminated in a pool-stage knockout by Fiji in the 2007 World Cup. Warren wanted a new start for the Welsh team, a new home within their home. So he moved their dressing room down the corridor, meaning that instead of turning right off their bus Wales now turn left to enter the four dragon-painted rooms that make up their changing rooms within the stadium.\n\nThose four linked rooms are where J.R., the squad's kit man, has been working for most of the morning, preparing them for the arrival of the team. Having arrived himself at 9.30 a.m., he's spent the last hour and a half, as he has for every Welsh game since 1985, unloading equipment from his van, laying out the players' kit and clipping in their names and cap numbers above their changing stalls.\n\nSometimes J.R. works in silence. When he does, there is something of the Catholic ritual about his process, a solitary and sombre dedication at twenty-two separate altars. This morning, however, he's been working to music playing from an iPod dock plugged into the side of Adam Jones's stall. 'Living on a Prayer', 'More Than a Woman' and the Stereophonics' 'Local Boy in the Photograph' have all animated the low-ceilinged space as J.R., occasionally slipping his glasses to the end of his nose to read the label of a shirt, prepares the room. Moving around the three-sided changing area in an anticlockwise direction, he builds the folded piles of kit on the right-hand side of each player's bench, working in reverse order to what they'll need first. He begins with the white towels, folding each one as meticulously as a Savoy chambermaid, before dropping them to the benches with a series of soft rhythmical thuds. The dark wood of the changing stalls makes them appear all the brighter, a comforting reward or consolation waiting for each player after their eighty minutes of violent exertion.\n\nUpon each towel J.R. lays a pair of red and white socks, a pair of white shorts, a red training top and a red Welsh shirt, its number facing up. Taking each item of clothing fresh from its packaging he inspects it, then folds it with precision. As he lays down each player's shirt, he gives it a single stroke across the number on its back, as if calming a highly strung animal or bestowing a brief blessing upon its wearer. The numbers themselves are composed of thousands of images of fans' faces. Together with the word ' _Braint_ ' inside the collar, these faces are a further reminder for whom the squad are playing when they put on these shirts, and never more so than today, when the weight of Welsh fans' expectations will press heavier upon their backs than ever.\n\nJ.R.'s manner and appearance as he works in the Wales changing rooms is reminiscent of an owner of a hardware store, the kind of blue-aproned shopkeeper who'll shake his head as he studies your list, before retrieving even the most unlikely of parts from his cupboards. Although J.R. says he sees himself at the bottom of the ladder within the Welsh squad, he'll be quick to point out, in his matter-of-fact Barry Island accent, that 'you can't play a game without kit, without the kit man you can't play the game', the mirroring of his syntax lending a further inevitability to his role.\n\nHe's right, but there's another reason for the fundamental quality of J.R.'s presence too. Having first worked for the Welsh team in 1985, J.R., as he'll tell you himself, 'has more bloody caps than anyone'. Over the last twenty-seven years, as he's gone about his duties, J.R., like Lee and Craig in the groundsman's office, has seen hundreds of players and coaches pass through the team. He was already established within the squad when the current home-grown Wales coaches all got their first caps, when the WRU chairman, Dai Pickering, was captaining the side and when a young Rob Howley 'wanted a cup of tea and a chat'. In an environment of flux, in which players, kit and coaches change all the time, J.R. is a rare seam of continuity within the Wales set-up, his knowing world-weariness, as if nothing could surprise him, lending him the air of a minor Shakespearean character moving among the brief tragedies and triumphs of the players and coaches. Like the porter in _Macbeth_ who, as Macbeth wracks himself with worries about his destiny, goes about his business, having seen many great men come and go through his gates, so J.R. keeps his vision close, occupying himself with the necessities of his role and little more.\n\nAs J.R. works in the Welsh changing rooms, singing along to M. C. Hammer's 'Hammer Time', he is surrounded by the symbolism of his country. The WRU's version of the Prince of Wales feathers is imprinted at the back of each player's stall, and again across the entrance to the changing area. In the gym next door a bright-red dragon, its claw raised, fills the floor space between the weights machines. Dragons appear again on the Welsh flags hung throughout the other rooms, and a dragon's tail coils between the statements written above the players' heads: 'RESPECT THE JERSEY', ' _DAL DY DIR_ '.\n\nOther text around the room is more specific, telling the story of the match to come later today. A list of players to be strapped by the physio teams is written in black marker pen on white medical tape stuck to the wall:\n\n**Team I** | **Team II**  \n---|---  \nGeorge | Jug  \nSmiler | Luke  \nTips | Alun  \nBom |   \nCuth |   \nYanto |\n\nOn a whiteboard above a table of Powerade drinks the pre-match schedule for the Italy game is still written up in blue, its timings yet to be replaced with those for today:\n\n14.02 \u2013 Come together \u2013 melon\n\n14.03 \u2013 Leave changing rooms\n\n14.04 \u2013 Warm up with Beardy\n\n14.09 \u2013 Squad together \u2013 short and long passing\n\n14.11 \u2013 Split\n\n14.15 \u2013 Defence\n\n14.19 \u2013 Back to changing rooms\n\n14.26 \u2013 Wales out\n\n14.27 \u2013 Anthems\n\n14.29 \u2013 Bag hits\n\n14.30 \u2013 Kick-off\n\nElsewhere, on several walls throughout the four rooms, those banners appear again:\n\n _Yesterday is in the past._\n\n_How do you want to be remembered?_\n\n*\n\nWhen Wales played England this year, it was J.R.'s responsibility to recreate this home changing room within the aircraft carrier that is the stadium at Twickenham. Denuded of stalls or any of the symbols painted on these walls, J.R. brought the squad's equipment from Cardiff, hung Welsh flags above the players' benches and put up two new banners. The first, hanging in the physio area, read:\n\nWINNERS DON'T WAIT\n\nFOR CHANCES THEY\n\n **TAKE THEM.**\n\nThe second, hung above the whiteboard written up with the pre-match schedule, where everyone could see it, told the players:\n\nBE PREPARED TO **SUFFER.**\n\nIf proof were ever needed that when two national rugby teams come together on a pitch, it's never just two sides that meet, but two cultures, two histories, then for over a century the annual Wales vs England fixture has been all the evidence required. Historically attuned as sport is, few other matches are so invested with meaning. Every year the eighty minutes of these games have been fuelled by centuries of association, either drawing upon the perennial Welsh grievance of oppression by a 'recently arrived' more powerful neighbour, or stoked by contemporary events such as the flooding of Welsh valleys for English reservoirs or Margaret Thatcher's heavy hand in the miners' strike. Culturally, each game against England is, too, a reminder of the class difference at the root of the sport in each nation. As J.R. was recreating the Welsh changing room in Twickenham that day, English supporters were already arriving in the car park outside to unpack hampers of champagne and caviar from their Range Rovers.\n\nIn his pre-match speech in 1977, the Welsh captain, Phil Bennett, openly drew upon the fractious relationship between the two nations:\n\nLook what these bastards have done to Wales. They've taken our coal, our water, our steel. They buy our homes and live in them for a fortnight every year. What have they given us? Absolutely nothing. We've been exploited, raped, controlled and punished by the English \u2013 and that's who you are playing this afternoon.\n\nWhen the teams met ten years later in Cardiff, the crowd witnessed what has since been described as the most violent twenty minutes of Six Nations rugby ever, with four players eventually banned after the match.\n\nBut that was then. Now, along with professionalism, things have changed. The contemporary Welsh players, some of whom were born in England or who have an English parent, have moved beyond an emotional response to the English game towards a more focused perspective. As Warren has warned them, 'Emotional energy can catch a team out.' The fixture is still undeniably charged, but where many Wales supporters would say that beating England is still more important than winning the other matches in the rest of the Six Nations, the players, if they are to bring about that win, have to see things differently.\n\nEngland presented a significant challenge for Wales this year. An equally young side, they were ambitious to salvage their reputation after a disastrous World Cup and eager to perform well for their caretaker coach, Stuart Lancaster. They were also, like Wales, currently unbeaten in the tournament, having scraped through two wins against Italy and Scotland. Whoever won at Twickenham stood a good chance of winning the Grand Slam. And so it was that Sam Warburton and his team prepared themselves for a physically intense match, not in the way of punches thrown in animosity, but in the way of a fast, hard game motivated purely by the desire to win.\n\nAlthough Wales were favourites going into the match, the statistics were stacked against them. In twenty-four years they'd only ever beaten England at Twickenham twice, and never before had they won a Triple Crown, the trophy in their sights that day, on English soil. Changes to the side included the young Ken Owens starting at hooker, replacing the more experienced veterans Huw Bennett and Matthew Rees. Dealing with the positive expectation, meanwhile, rather than their more familiar role of underdogs at Twickenham, would be a test of the squad's character as a whole.\n\nThe nature of that expectation, after their first two victories over Ireland and Scotland, was apparent in Wales's arrival at the stadium, both in the tone of their reception and the tone of their entrance. When the English team had arrived, looking crisply sharp with gelled hair and wafts of aftershave, although the fans outside welcomed them with baying support there was barely a tremor of response within the stadium. When Wales entered a few minutes later, having walked through the taunts of those same fans, the cleaners stopped cleaning, a handful of Royal Marines came in from the pitch and the RFU backroom staff stalled in their pre-match tasks, all to catch a glimpse of the Welsh squad arriving. As Wales came into the corridor between the changing rooms, they did so with an air of dark intent. Moving in single file, eyes forward, not a single man acknowledged those watching them. Their massed bulk, witnessed in succession, was as weighted with unremitting opposition as the opening bars of Mahler's Fifth.\n\nWales were here to make history, that's what their entrance seemed to say. Inside the changing room itself, Shaun Edwards and Rhys Long had spelt out exactly how they intended to achieve that in five points on the game's tip sheet, left on the players' benches under each of their folded towels:\n\n  1. Don't hurt ourselves \u2013 charge downs, intercepts, sin bins.\n  2. Smash their L\/O drives \u2013 our width is vital after a breakout.\n  3. Muscle up on T\u2014, M\u2014, F\u2014and C\u2014  \nEXTRA AGGRESSION ON **R** \u2014  \n**\u2013SMOKE TARGET**\n  4. Speed into position. Work hard early.\n  5. Keep square off the line.\n\nFollowing these points, in case there was any doubt, were two further statements, brutal in their simplicity, and a concession target for which to aim:\n\nASSAULT THEIR ATTACK.\n\nBE RELENTLESS.\n\n12\/13 POINTS TARGET.\n\nThe English match was every bit as physical as Wales expected. Sam went to the post-match dinner with stitches in his eye, while Leigh Halfpenny had to endure the hits that came with England kicking to the corners throughout the whole eighty minutes. It was also, however, much tighter than they expected: down-to-the-wire close, with the final score of England 12 \u2013 Wales 19 exactly hitting Shaun's projected concession of points.\n\nBut it could have been so much worse. When Rhys Priestland was sin-binned for an offside tackle at the start of the second half, Wales, once again, had to play for ten minutes with just fourteen men. The last time they'd been a man down at Twickenham they'd conceded seventeen points. This time, however, with Mike Phillips bossing the game with a series of pick and drives, they held onto possession for eight and a half minutes and not only conceded no points, but also scored three through a penalty by Leigh. When Rhys took the field again, the score stood at England 12 \u2013 Wales 9, and a significant psychological contest had been won. But the match itself still wasn't, and wouldn't be until the seventy-sixth minute, when a moment of individual brilliance from a player off the bench finally snatched victory for Wales.\n\nThere's something of a tradition in Wales of such moments providing the turning point in matches against England: Ieuan Evans out-sprinting Rory Underwood in 1993 to secure a 10\u20139 victory; Scott Gibbs's last-minute crash and jink under the posts to set up Jenks's conversion that put England away 32\u201331 at Wembley in 1999; Gavin Henson's seventy-sixth-minute long-range penalty to beat them again in 2005.\n\nIn 2012 it would be another Scott who helped Wales turn the corner: Scott Williams, a twenty-one-year-old centre who just a year earlier had been training as a plumber. With only five minutes left on the clock, Scott, having come on for the injured Jamie Roberts, followed the advice of that banner in the changing room and, when he saw it, took his opportunity. As Courtney Lawes, the English second row, was being tackled by Sam on the halfway line, Scott ripped the ball from his arms and grubber-kicked it past the English defence. Judging the bounce as he accelerated in pursuit, he gathered the ball as it rose sweetly to his right and away from Tom Croft, the English flanker bearing down on his left.\n\n'If it had gone the other way,' Scott later acknowledged in his laidback Carmarthenshire accent, 'then I reckon he'd have got me.' But it didn't, and Croft didn't, so with an arm raised in celebration Scott dived across the English line, untouched by opposition hands. For the first time in the match Wales were ahead and, following Leigh's conversion, seven points in the lead. Five minutes later, after David Strettle's last-ditch lunge across the Welsh line was ruled 'inconclusive', the match was over and Wales had won. Within a further fifteen minutes the careful order of J.R.'s changing room was undone by the disorder of victory, with pizza boxes, beer bottles and muddied kit strewn across the floor. The suited management stood in the middle of the half-dressed players, beaming as Gethin Jenkins put on a celebratory playlist and the Triple Crown, like the treasured loot it was, was passed from hand to hand for posed photos on iPhones and cameras.\n\nIn the midst of their celebrations, however, many of the Welsh players were also anxious to leave the changing room. Since using cryotherapy in Spa\u0142a and Gdansk, the squad had employed the services of a mobile cryolab to enable them to continue the treatment throughout the Six Nations. In the weeks between each match this van, converted from a police command unit and run by an evangelical former rugby player called Karl, has parked up in the country lane at the back of the Barn. As the local farmer herds his cattle between the lane's hedges, down the road Welsh players have queued up twice a day to enter its futuristic silver pod for three minutes of extreme freezing. At home matches Karl parks the van in the subterranean passageways of the Millennium Stadium, but at Twickenham the RFU restricted him to the public car park. So this is where the Welsh players headed after the game, barefoot and chaperoned by Adam Beard, his head often reaching no higher than their shoulders. The crowd of thousands was still streaming out of the stadium, but this didn't deter Sam and his team, or blur their focus in getting to the cryolab for their three minutes of treatment. Cryotherapy, by this stage of the tournament, had become something else for the squad: a crucial psychological as well as physical tool, as much a ritual of belief as a method of recovery.\n\nAs one by one the Welsh players left the raucous celebrations to make their way towards the cryolab, the English changing rooms, across the corridor, were silent. A different bounce, more weight on a tackle or a breath of wind and it could have been them singing in celebration. The margins of modern rugby are ever tighter, and ever more cruel because of that. But that is also what energises the game. For the Welsh fans Scott's try provided the kind of moment for which every rugby enthusiast hopes. A moment when the emotional contours of thousands move in exact alignment, switchbacking in just seven seconds from the anxiety of an English attack to the sudden jubilation of Scott's kick through and score. The surprise conjured from the unexpected, the breaking of the pattern that makes the pattern more beautiful, the moment of chance that falls your way. To experience this jeopardy, to be at the mercy of the sport's vagaries, is why we watch rugby. To care and to feel. To experience those moments of play which, as Seamus Heaney says in his poem 'Postscript', _catch the heart off guard and blow it open_.\n\nWhat a spectator sees as a moment of chance, however, from the perspective of a player or a coach is rarely just chance. Even moments of luck need to be built upon dedication to become winning moments of luck. For Scott it was chance that Jamie Roberts was injured and he took the field, but everything else about his try was the leading edge of a life in which from a young age he'd only ever wanted to play rugby. Watching his stepfather play for Neath on the TV, the hours of skills coaching he'd received at the hands of the Scarlets' Eifion Roberts, Shaun's defensive drills in which the squad had practised, again and again, ripping the ball from an opponent's arms \u2013 all of these, too, were ingredients in those seven seconds in which Scott scored. As Jenks often says, paraphrasing the golfer Jerry Barber, 'the harder you work, the luckier you get'.\n\nWith his match-winning try at Twickenham, Scott, at twenty-one, had achieved what most players chase throughout their international careers: the iconic moment which, no matter what else happens, no one can take away from you. The moment for which you will be remembered. With that try he's already answered the banner in the Barn which asks, _How do you want to be remembered?_ And he's questioned, too, that other banner next to it \u2013 _Yesterday is in the past_. Because as a rugby player, if your past happens to include a seven seconds like Scott's at Twickenham that clear February afternoon, then a part of it, seven seconds of it, will always be in your present too.\n\n## 11.15 a.m.\n\n'Go on! Guess! She was in _Sister Act 2_. You'll know her, you will!'\n\nThe Wales changing rooms in which J.R. has been working alone have begun to fill with other members of the Wales staff making their own preparations for the arrival of the team. Ryan Chambers, the squad's sports scientist, is checking the protein shakes. At an away fixture this is the time when Ryan would usually be preparing the GPS pack each player wears between their shoulder blades in training and during a game. From these he's able to study the players' movements on the pitch, their distances, their speeds and even the momentum and angle of their tackles. In the Millennium Stadium, however, the overhang of the stands blocks the satellite signal, denying Wales this technology at home.\n\nAlong from Ryan, Carcass is in the physio area preparing his strapping station, while Prof. John is organising his pitch-side medical equipment. Meanwhile, Ben, the team photographer, is being harried by J.R. to guess the singer playing on the iPod dock.\n\n'Whoopi Goldberg?' Ben says, already unsure of his answer.\n\n'No!' J.R. exclaims, as if a child should have known. 'Lauryn Hill! She was in the film, she was! _Sister Act 2_.'\n\nAs Ben and J.R. continue talking, J.R. snapping the players' name plates into the clips above their stalls, Geraint, a member of the stadium's security staff, takes up his position outside the rear doors of the changing room. Opposite him is the VIP lift which the suited coaches, each wearing a daffodil in their lapel, will hurry into after the anthems, frowning and unsmiling, to travel up to the box from where they'll watch the match. At Geraint's back are the double doors into the changing room, through which Ben and J.R.'s muffled voices reach him, the latest in a growing chorus of voices Geraint has heard spreading through the stadium for the last two hours. To Geraint, standing in that windowless corridor on his own, the whole building feels like a stadium-sized battery, charging with anticipation and nerves and excitement, every minute gone and every minute to come transfigured in relation to the 885th minute of the day, when the referee will finally blow his whistle and Wales's last match of the Six Nations will begin.\n\nFrom here, though, outside the back door of the changing room, Geraint will see none of what happens after that minute. Even the roars of the crowd will be faint this deep in the stadium. He will hear the team return at half-time, and he will, perhaps, hear some of the words spoken to them by the coaches. But then they'll leave again, the clacking of their studs diminishing down the corridor towards the tunnel and the pitch. When the game is over, they'll return once more, and only then, as Geraint listens from this side of the door either to celebratory music or to silence, will he know if Wales have lost or won.\n\n## 11.30 a.m.\n\nThe streets of Cardiff are filling. The bright morning is still laced with winter but, with the sun's warmth and a rainbow arcing above the walls of Cardiff Castle, it holds the promise of spring too. Roger Lewis, a cup of Costa coffee in one hand, his smartphone in the other, has left the stadium to walk through the city to witness a twelfth of the country's population pour into its centre. Having finished his pitch-side interviews he called in on the changing rooms to see J.R. and the others, before making his way out of the stadium. At Gwyn's security lodge he stopped for a moment to talk with Ryan Jones's father, Steve, who has been walking his three-year-old grandson, Jacob, down the ramped entranceway at Gate 4.\n\nRyan, at thirty-one, is one of the senior Welsh players. Although much has been made about the youth of the squad this year, it is still veined with a vital steadying seam of experience too. More than once during this campaign Ryan has come off the bench and, like a battle-hardened sergeant, brought a level-headed focus to the forward play, apparently finding, as the more capped players often do, an extra second with which to make his decision or move. As an ex-captain of Wales who has since had to play under others, and with two Grand Slams already to his name, Ryan and his family have, over the last eight years, been through most of the highs and lows that international rugby can throw at a player. If Wales win today, however, then Jacob will see his father reach a new peak, joining a select group of only five other Welshmen to have ever played in three Grand Slams. For Ryan's own father, meanwhile, it will mark the pinnacle of a long journey, one which began many years before Ryan was born, with his own boyhood dreams of one day putting on the red shirt of Wales.\n\nAs Roger makes his way up St Mary's, a river of rugby fans flows against him, the Welsh and the French patterning the street with reds and blues from side to side. Stalls selling scarves, flags and hats occasionally break the flow, as does the odd ticket tout standing motionless beside a lamp post, threading their repeated offer into the cacophony of the crowd. 'Buy or sell tickets! Buy or sell tickets!' Their prices, at this hour, are well into three figures.\n\nBecause of its cathedral-like position at the centre of the city, an international match in the Millennium Stadium triggers an intensification of Cardiff 's population, rather than a dispersal. Like a roosting starling murmur tightening on the wind above a wood, so Wales contracts about the stadium on a match day. While a shopper on Oxford Street in London or on the Royal Mile in Edinburgh might be completely unaware of a rugby match at Twickenham or Murrayfield, it's impossible to be on the streets of the Welsh capital and not know that a game is being played in its stadium. From early morning until late at night the resonance of those eighty minutes electrify the city. The streets pulse with an ever-inward movement as trains and buses from across the country bring the stadium's crowd into Cardiff. Some will travel for hours to get here. The extra carriages added on the trains from the valleys, from Swansea and Bridgend, from Abergavenny and Cwmbran, will develop a spreading virus of red with every hour nearer the game. Today many of those arriving on them will not even have tickets to get inside the stadium. But they still want to be here, to watch on giant screens in the civic areas and in pubs and clubs, to be within sound and sight of the arena where it, whatever it may be, will happen. They want to be here to drink, to share and to feel themselves more than themselves, as if by being closer to the stadium they are somehow closer to an idea of Wales.\n\nIn her poem 'Toast', Sheenagh Pugh describes the recently built Millennium Stadium as\n\na mother-ship that seems to have landed\n\nawkwardly in our midst.\n\nAnd she's right, there is something imposing and 'landed' about the way the stadium sits within Cardiff, its jointed masts angling their Concorde-like tips into previously unoccupied slips and pockets of air. While its West Stand floats above the river, its South abuts challengingly up against office blocks, the joint of the south-eastern mast protruding above Park Street like the prow of a ship. Viewed from further away, from the hills of Fairwater or the Grangetown link road, there is something kinetic about those jointed masts, as if at any moment the whole stadium might lift itself on them and, like a giant insect, scuttle away. At closer quarters, with the sun behind them, those skeletal masts seem to aspire towards spires, as if the stadium would make a claim beyond its location upon the spiritual needs of those who congregate within its stands.\n\nAs Roger makes a left into the crowds on Caroline Street, where drinkers are already spilling out of the Brewery Quarter, a woman in a red cowboy hat passes him wearing a T-shirt several sizes too small, its slogan printed across her chest: 'Welsh Lamb @ the Grand Slam'. Further on down the street a Frenchman in a beret is having his cheeks painted with the colours of his flag. A group of boys from the Rhondda look on, all wearing dark wigs of white-bandaged curls in homage to Mervyn Davies, the great Welsh number eight who died yesterday.\n\nFor Roger the experience of this morning's stroll is something like that of a director walking through his audience before they enter the theatre to see his play. No one recognises him as he makes his way down Caroline Street and turns left onto the Hayes. And yet, in no small way, this day for which these people have come into the city is Roger's creation. As such, it is also the consequence of another match Wales played forty-six years earlier, when, as a twelve-year-old, Roger boarded a Brownings bus chartered by Cefn Cribwr RFC and travelled into Cardiff to watch Wales against Australia. Standing in the West Terrace of the Arms Park that day Roger had watched Gerald Davies, Barry John and Delme Thomas all make their debuts for Wales. Wales lost 11\u201314, but for Roger the match lit a passion for Welsh rugby, a passion which would, eventually, lead him to taking this walk today through a match-hyped Hayes as the CEO of the Welsh Rugby Union, on the brink of a game in which, as the Blims sang in the stadium, 'We all hope Wales wins the Grand Slam.'\n\nEvery modern rugby union has two hearts: the players and the business. Neither can keep beating without the rhythm of the other. However disparate their worlds might be, success is rarely achieved without embracing the symbiotic relationship between the two. What happens on the pitch fuels the boardroom, and what happens in the boardroom fuels events on the pitch. Roger plays no part in the selection of the players or in their efforts on the field, in the tactics or methods of the coaches or in the medical practices which keep the squad at strength. But just as Warren has spent the last five years building up a team of players and coaches, so Roger, as CEO, has spent the same period building up his own backroom team to help lift the fortunes, both playing and financial, of the WRU. In his own way, within his own field, Roger has enabled today to happen, even though he was never meant to.\n\nRugby had always been Roger's escape, not his career; a hinterland away from his work. His childhood home in Cefn Cribwr had looked over the village rugby pitch on Mynydd Bach, but he never played for the club. Similarly, at the local Cynffig Comprehensive, it was music, not rugby, that caught Roger's attention. He played at wing-forward in the sixth form, and again as a scrum-half at Nottingham University, but from the end of his education music became his career and rugby its soundtrack, a private pleasure not to be mixed with business.\n\nRoger's first organisational role within a rugby club was as chair of a group overseeing the under-9s at Maidenhead RFC. His two sons had begun playing mini-rugby, so from 1993, for the next ten years, Roger became increasingly involved in the club, coaching, organising tours and heading up committees for certain age groups. Having worked as a composer, musician, head of music for Radio 1 and managing director of the classical division of EMI Records, Roger was then worldwide president of Decca. His schedule, however, was always made to revolve around the Sunday mini-rugby matches back in Maidenhead, even if that meant flying in from New York on a Saturday and leaving for Italy on the Monday. Rugby, through his sons, was moving closer to the centre of Roger's life. But when they grew up, and Roger and his wife Chris moved back to St Hilary in the Vale of Glamorgan, it ebbed away again. Roger had debentures for the new Millennium Stadium and watched Wales play whenever he could, but he still had no idea about the internal dynamics or politics of the professional game.\n\nThree years later, in 2007, Roger was walking into the middle of the pitch at Stade de la Beaujoire at the Rugby World Cup in France, discussing with Dai Pickering, chairman of the WRU, how best to sack the coach of Wales, Gareth Jenkins. The two men were looking for somewhere secluded where they could talk urgently. The match against Fiji was only just over. The stands were still full of celebrating Fijian supporters and devastated Welsh. The middle of the pitch was the most public place in sight, but as somewhere they couldn't be overheard, it was also the most private.\n\nWales had just been knocked out in the pool stages by Fiji, beaten 38\u201334. They'd played an open game of running rugby, and in doing so had played into the Fijians' hands. For the first time ever Wales were going home without reaching the quarter-finals. As Roger and Dai walked out into the centre of the pitch, they approached the Fijian players, who were gathered in a huddle, praying. Many of them were crying. As Roger walked past them he recognised that Wales hadn't just lost the game, but something else too, something the Fijians still had. A fundamental spirit within the team's culture had died, and their values and beliefs had been shaken. As a viewer writing into the BBC's website put it, 'Something is rotten in the Welsh camp.' Roger knew that he and the WRU board had to act, and they had to act quickly.\n\nJohn Williams, the WRU head of communications, joined them in the centre of the pitch and, after a brief discussion, the three men decided they had to send an immediate and unequivocal message to the rugby world. From this moment Wales was changing, starting over. Gathering the other board members, they called an extraordinary board meeting in the corner of the stadium, where it was agreed by fifteen to one that the Welsh coach, Gareth Jenkins, had to go. In twenty matches in charge of Wales he'd won only six. He'd inherited a squad still reeling from the resignation of their previous coach, Mike Ruddock. But he'd also asked the country to judge him on his performance at the World Cup. That judgement, as Roger and the board members left the stadium, was already coming in, from newspaper articles, from TV pundits and from fans on rugby websites and forums.\n\nThe next day Roger addressed the entire squad, dressed in their number ones \u2013 team suits and ties \u2013 at their camp outside Nantes. He'd already spoken with Gareth Jenkins and asked him to resign. When he'd refused, Roger told him he was no longer the coach of Wales. Which is what he also told the squad standing before him that day. 'The process of rebuilding', he said, 'begins now.'\n\nIn interviews later that week Roger would use Oliver Cromwell's phrase about the execution of King Charles I \u2013 'a cruel necessity' \u2013 to describe the sacking of Jenkins. It was undoubtedly just that. But it was also a massive risk. Once that necessity had been taken care of, Wales, just four months away from the 2008 Six Nations, were without a coach.\n\nIt was the resignation of one Wales coach, Mike Ruddock on 14 February 2006, that had set Roger on the path to being the man who would fire another a year later in Nantes. Listening to the breaking news reports of Ruddock's departure in his kitchen at St Hilary that day, it was clear to Roger, even from a distance, that the WRU had become 'a distressed organisation', in disarray both on and off the field. He recognised the scent of that disarray from having smelt it himself in other companies in which he'd worked. But on Valentine's Day 2006, as he listened to reports about the disintegration of the WRU, Roger still had no idea he'd ever be involved in picking up the pieces.\n\nThat summer the job of CEO of the Welsh Rugby Union was advertised in the _Sunday Times_. Roger was approached to apply shortly afterwards, and it was then, as he considered the role, that he realised perhaps elements of his experience in the music industry may be transferable to the modern game of professional rugby. Both worlds, however different, were about dealing with elite performers \u2013 _singers and players, producers and coaches_ \u2013 who provide content \u2013 _teams and bands, songs and matches_ \u2013 which needs to be delivered through various mechanisms \u2013 _venues and stadia, TV and media_ \u2013 and monetised \u2013 _through audience and spectators, sponsors and merchandise._ Just as in music, rugby had its elite and its grassroots concerns, each feeding the other. Roger had overseen the development of elite musicians often from their youth, from the same kind of ages as rugby players. Modern rugby was increasingly about entertainment and performance, about handling talent and harnessing passion, both in the players and the fans. Viewed from this perspective, Roger saw no reason why the lessons he'd learnt in studios and concert halls shouldn't be applied to the training pitch and the rugby field; why his private interest shouldn't become a public role.\n\nOne of the lessons Roger had learnt was that if you're chasing the best producer or the best artist, you turn up. You don't ask them into your office, but get on a plane and meet them backstage. You hang out with them. Which is why, within weeks of addressing the squad in Nantes, Roger, Dai Pickering and Gerald Davies, the three-man delegation tasked with finding a new coach for Wales, landed in an airport on the other side of the world.\n\nIn their initial research, speaking to coaches, players and commentators, it had become clear to the team that they would have to look beyond the UK for their candidates. Certain names kept cropping up, and nearly all of them were from the only other country whose national obsession with the sport rivalled Wales's: New Zealand.\n\nTo an extent Roger, Dai and Gerald were looking for a quality of character as much as a coach. This was a search bound up with questions of national identity. When Roger had come into the WRU, he'd articulated a vision for the organisation that, tellingly, was couched in terms of self-expression and nation-building. 'The WRU', he said, 'will take Wales to the world and, in our stadium, will welcome the world to Wales. Together we will play our part in defining Wales as a nation.'\n\nAnd yet in their meetings in New Zealand Roger and the others would be considering foreign coaches, men who hadn't grown up with the ethos of Wales running in their veins, to carry the precious but fragile vase that is the hopes of Welsh rugby. Whoever took on the position would need to have an innate empathy for Wales beyond national association. The WRU badly needed an outsider's eye and influence, but they also needed that crucial note of understanding; a stranger attuned to the nature of the country's culture, but who could also be clear-sighted about her qualities, good and bad.\n\nWithin twenty minutes of meeting Warren Gatland at Auckland airport Roger felt they'd found the next coach of Wales. There was something about Warren's blend of emotional intelligence and blue-collar background that sounded the right note. He was also realistic, pragmatic and straight-talking, capable of detaching himself from the romance that both bolsters and hinders so much Welsh support for the national side.\n\nWarren would meet some of that passionate support on his first night in Wales, when he came to stay with Roger at St Hilary. On taking him into his local pub, The Bush, Roger was in conversation with Warren when a man approached from the bar and tapped Warren on the shoulder. 'Excuse me, Mr Gatland,' he said. 'I'd like to introduce myself. I'm one of the 2.9 million selectors in Wales. Good luck.'\n\nIt was a well-intentioned but premature gesture. Despite speculation in the media, Warren hadn't yet accepted the position of Wales head coach, and Roger had yet to formally ask him. He had, instead, invited Warren to Wales to 'just talk', and to give him an opportunity to witness first hand the position rugby occupied within the country's psyche and landscape.\n\nThe following morning, by way of beginning this process, Roger chartered a helicopter from Cardiff bay and flew Warren across South Wales, allowing the New Zealander to get a proper look at his prospective employer from the air. From Cardiff, following Roger's instructions, the pilot flew along the southern corridor of Welsh rugby's heartland. Tracking the unspooling M4, they passed over Newport, the ground at Rodney Parade and the Gwent valleys. Below them the long villages and towns of South Wales bled into each other, their roofs and streets frequently punctuated with the green of rugby pitches. Beyond their borders these towns gave way to hedged farmland or barren hillsides grazed by wild ponies. 'It looks like New Zealand,' Warren said, looking out of his window at the land spread below him. Turning west over Ebbw Vale they flew around the sculpted peaks of Pen y Fan and the Beacons before dropping south over Rhigos and heading further west to Stradey Park in Llanelli, then over the building works of Parc y Scarlets, Rhossili Bay and around the peninsula of Worms Head. Heading back east the helicopter flew over Swansea and the city's new Liberty Stadium and the ground at St Helen's where Warren had once played. Taking in the Knoll at Neath and the Brewery Field in Bridgend, they finally returned to Cardiff, making a pass over the Millennium Stadium and the pitch which would, if Warren accepted the job, be his new home ground.\n\nBut Roger didn't want the tour to end there. Having landed, they got into his two-seater Mercedes and drove up into the Rhondda, through Tylorstown and Treorchy, before passing through the Bwlch and down the Ogmore valley to his mother's house in Cefn Cribwr, where Roger suggested they stop for some tea. As Roger stepped inside the small terraced house that was his childhood home, he introduced Warren to his eighty-one-year-old mother. 'I know who he is!' she admonished her son in reply, before turning back to Warren. 'Now,' she said, taking his hand and leading him into the living room, 'what you going to do about Gavin Henson?'\n\n## 11.50 a.m.\n\nAs Roger emerges back onto Westgate Street, the Millennium Stadium rears into view. The first fans are beginning to enter the building, flowing up to the gates in steady streams. Others are already taking up their positions on the corner of the street outside the Angel Hotel or along the castle walls, ready to welcome the arrival of the Wales team bus into the city. All of Cardiff is engaged in a single conversation, the crowds bonded by a shared anticipation. A group of French supporters pose for photographs with a mounted policeman. A young woman in high heels and a mini-dress pauses beside them to stroke the horse's neck, her gold sequinned handbag swinging from her shoulder. From further down the street a hooter sounds. Another replies. Outside Gate 3 an S4C TV crew are interviewing the welsh boxer, Joe Calzaghe, asking him for his prediction on the match. 'Wales will win,' Joe says, 'but it'll be close.'\n\nLooking over Joe's interview is a statue of the late Sir Tasker Watkins, once deputy Lord Chief Justice and ex-president of the Welsh Rugby Union. In a couple of hours' time the tide of spectators will have risen even further, with thousands flowing up the incline of Gate 3 to maroon the bespectacled Sir Tasker, his hands behind his back, in a sea of red and blue.\n\nAs an officer serving with the Welsh Regiment in the Second World War, Sir Tasker was awarded a VC for leading his men in a bayonet charge in Normandy. When Graham Henry was Wales coach, he sometimes pinned Sir Tasker's citation in the team's changing room before a Six Nations match:\n\nOn 16 August 1944 at Barfour, Normandy, France, Lieutenant Watkins' company came under murderous machine-gun fire while advancing through corn fields set with booby traps. The only officer left, Lieutenant Watkins led a bayonet charge with his 30 remaining men against 50 enemy infantry, practically wiping them out. Finally, at dusk, separated from the rest of the battalion, he ordered his men to scatter and after he had personally charged and silenced an enemy machine-gun post, he brought them back to safety. His superb leadership not only saved his men, but decisively influenced the course of the battle.\n\nRoger walks on towards the stadium. Once inside he'll take the VIP lift up to the President's Lounge on level five, where he'll begin welcoming over a hundred guests for the pre-match lunch. These will include members of the F\u00e9d\u00e9ration Fran\u00e7aise de Rugby, the first minister of Wales, Carwyn Jones, ex-players, politicians, sponsors, business people, academics and broadcasters. A team of young men and women will take coats, hanging them on two long racks on either side of the door into the lounge. Once inside waiters and waitresses will greet the guests with trays of red and white wine, while others circulate with canap\u00e9s. A bar will serve beer and gin and tonics. The room itself, with its bank of windows looking out over the County Club and the city, will be charged with a heightened sense of being, like the Blims on the pitch this morning, at the centre of the centre.\n\nOnce his guests are seated, Dennis Gethin, the president of the WRU, will address the room first in Welsh, then in French and finally, 'for the less civilised among us', in English. His introductory speech over and grace recited, the starters will be served and everyone will begin to eat and drink, to drink and eat, laying napkins over protruding stomachs or across the hems of delicate dresses. The whole lounge, five floors above the changing rooms where Carcass is preparing his strapping station, where Ryan mixes the rehydration drinks and Prof. John is checking his needles and thread, feasts, drinks and talks, celebrating together before the game begins.\n\nAnd in the sponsors' boxes around the stadium, too, meals are being served and drinks are being poured. And in the bars on level three the 'joy machines' are already pumping out twelve pints at a time. And in the family rooms and the ex-players' lounge those who are closest to the squad and those who have been here before them eat and drink, drink and eat. The whole stadium, from top to bottom, apart from those two quiet changing rooms either side of the tunnel, is loud with expectation, occasion and alcohol.\n\nInside the jacket pockets of the men and in the hand-bags of the women dining in the President's Lounge long WRU wallets hold their other invitations for the day: for post-match tea, and later to the black-tie post-match dinner at the Hilton. On each glossy invitation is the image of a Welsh rugby shirt, its fabric and feathers filled out by the chest and shoulders of a player, their head cropped off at the base of the neck.\n\n## 12 p.m.\n\nTwelve miles further west a quieter pre-match lunch is coming to an end in the Wales team room at the Vale. The remaining players at the tables are silent, occupied in thought. Leigh Halfpenny is still nervous. As much as possible he will not talk to anyone before the match, or smile. The eighty minutes of the game, everything for which they have trained, is almost upon them. The whole team know how the story is meant to go today, and each player has rehearsed his role in its performance. But they also all remember their last game against France; how it got away from them, and in so doing, wrote them out of the World Cup final.\n\nSam, their captain who watched that match slip away from the sidelines, is finding it difficult to eat. Twelve miles away the Millennium Stadium is already filling with 75,000 spectators. Two hundred and fifty thousand people are on the streets of Cardiff. The bars of the pubs are three deep with drinkers. The build-up to the match has been on TV and radio all day. Images of Sam and the squad are all over the newspapers. Children across Wales have woken and put on Wales rugby shirts. Not that long ago, he was a child himself, telling his mother from his pillow he'd never be a rugby player. Today he is a rugby player. Today he will captain his country. Sam looks down at his plate and, for that child and for his country, spears another piece of chicken and eats.\n\nTwo floors above the team room, Thumper Phillips, the Wales team manager, is getting into his number ones. Standing before the mirror in his room he loops a WRU tie about his neck, constructs a knot and pulls it tight before flipping down the collar of his shirt. Lifting his jacket off a hanger, he puts it on and, leaning into the mirror so he can see better, pins a daffodil to his lapel.\n\nAt fifty-eight Thumper is the same age as Roger Lewis and was brought up in the same part of Wales too, in Kenfig Hill, half a mile down the road from where Roger took Warren for tea with his mother. The two men have known each other since they were twelve, when they first met on the playground of the local comprehensive. On leaving school they took diverging paths in life, but both, eventually, have led them back into close proximity, working either side of that relationship between the business and the players that lies at the heart of the WRU.\n\nIt was Thumper's uncle, Alan, who first took his nephew to the midweek Floodlight Alliance games at Maesteg and the Brewery Field in Bridgend. On those dark Wednesday nights the ten-year-old Thumper watched enthralled as the likes of Gareth Edwards and Phil Bennett played to capacity crowds. Already showing promise as a player himself, within another ten years it was Thumper who was running onto those pitches instead, having been selected for Cardiff directly from the Kenfig Hill youth team.\n\nA few weeks before he first played for Cardiff, Thumper was walking back from training in Kenfig Hill to his home in Pyle when a car pulled up and its driver asked him if he knew where he could find Alan Phillips. Thumper said he did, and that if the driver gave him a lift into the village he'd show him where he lived. As they drove into Pyle the man introduced himself as Gary Davies of Cardiff RFC. He asked Thumper how he knew Alan Phillips. 'Because', Thumper replied, 'I am Alan Phillips.'\n\nAlong with Roy Bish, Gary had recently watched Thumper score all his team's points in a 24\u201312 victory over Llanelli. Within a week of having stopped him to ask him where he could find Alan Phillips, Gary had Thumper training with Cardiff. Within another week he was playing for them too, thumbing lifts into training from Pyle after work and getting dropped off back in Bridgend by Gary afterwards. Sixteen years later, at the age of thirty-four, Thumper retired, having played more games for Cardiff than any other player in the club's history, earning himself eighteen Welsh caps and a 1980 Lions tour to South Africa along the way.\n\nToday, as Wales team manager, Thumper finds himself overseeing boys young enough to be his grandsons who, he says, 'play a totally different game' to the one he knew when he was making his way at their age. As Thumper often says, 'Only the shape of the ball is the same now.' The changes have mostly been driven by professionalism, but also by changes in the rules around the contact area and a gradual eradication of the hard-man culture that in Thumper's day rarely saw a clean game of rugby.\n\n'I played in a period when there were no touch judges, nothing like that, when people were punching from behind, running across the field when you least expected it. It was a dirty game, a dirty game. But no one ever got hurt, mind, not really hurt. A few broken jaws maybe, that kind of thing.'\n\nThe current players, though, don't necessarily have it any easier. 'You couldn't get away with that now, you _have_ to respect the rules of the game. But it is physically tougher,' Thumper admits. 'The boys are quicker, more powerful. I mean, I was fourteen and a half stone as a hooker. Matthew Rees is seventeen and a half, three stone heavier. Ken Owens, Hibs, they're big men, big men. There's less time, everyone's under analysis. It's hard.'\n\nPossessed of a quartermaster's manner, Thumper carries out his duties with the same confrontational style he once brought to his forward play. With a restless pointing finger, often addressing people from over his glasses, and his signature black satchel cross-gartered over his chest, Thumper is a man who wants to solve problems, and who is confident in his ability to do so. His diplomacy is direct and frequently followed by the effective technique of walking away before a counterpoint can be made. It is often, too, edged with sharp humour. While on tour to Canada Thumper christened one of the team's liaison officers 'Thrush', 'because he was an irritating little cunt'.\n\nBeneath Thumper's organisational bluster is a deep well of care, both for the individual players and for the game of rugby itself. 'It's a great game to protect,' he says. 'It teaches you respect by bashing the shit out of each other, but you also have to respect the rules. But you've gotta protect it, its traditions, because if you lose them, then you lose the respect of the game, and without that', he adds, shaking his head at the worst possible fate, 'we'd be like football.'\n\nOne of Thumper's most persistent challenges as team manager is satisfying the increasing demands of marketing and PR upon the squad, while still protecting the team they are trying to promote. In this respect Thumper sees himself as a buffer between the team and the world; an old-fashioned gatekeeper at the heart of a twenty-first-century squad. It's his job, he says, to be the 'awkward bugger', to try and defend what sacred time and space the team still own. The words he uses most often when talking about this element of his role are 'tradition' and 'respect'. In the face of an ever-increasing appetite for access to the players, the erosion of these qualities is what most concerns Thumper. 'I mean,' he says, 'everyone can buy our kit now. Anyone. We have changing-room visits by sponsors after a game. These kids in marketing, they want to show more and more of us, give more and more of us away. I understand why it has to happen, but you got to keep something back, haven't you? Otherwise what you got left?'\n\nNot that Thumper would ever want the team to withdraw from their fans. On the contrary, he sees the players as 'ambassadors for Wales. How we act and conduct ourselves reflects on the people of Wales.' To this end Thumper takes a gruff paternal interest in the manners of the younger players. 'I tell them to keep their feet off the tables, how to eat at dinner, that kind of thing. And I always tell them to have time for the supporters too, be well-mannered, y'know? Because people are looking at these boys and they're dreaming, aren't they? When they look at them,' Thumper says, beginning to walk away, 'these people, when they see them, they're seeing their dreams.' \nThe Journey\n\n## 1 p.m.\n\nSounding its horn five times in acknowledgement, the dragon-painted team bus of Wales pulls away from the kerb outside the Vale as the crowd of fans gathered behind orange barriers wave and cheer it on its way. As a police motorcycle escorts it onto the resort's exit road, more fans are standing in the car park, cameras held before their faces in one hand, waving with the other. From behind the bus the applause and cheers outside the Vale continue.\n\nOn board, however, it is silent.\n\nThumper, his black satchel on his lap, sits at the front of the bus behind the driver. Warren sits in the other front seat, across the aisle. Behind them are the rest of the coaches: Rob Howley, Rob McBryde, Adam, Dan and Shaun. Behind them again the players of the squad have taken up a pair of seats each. Dressed in identical red tracksuits, many of them wearing headphones, they sit along the length of both sets of windows, their kitbags beside them. No one talks. Ahead of them, through the driver's windscreen, a second police motorcycle escort has joined the convoy. The motorbikes lead them on through the golf-course grounds of the Vale. An approaching white Rolls-Royce, decked in pink ribbon, slows up and pulls over to let them pass. As they do, the bride and her father inside wave, and the chauffeur sounds his horn.\n\nThe bus drives on, shuddering over a cattle grid and turning left up a hedged country lane towards the M4. As it reaches a roundabout the only sounds are its engine, climbing and falling through the gears, and the faint beat of dance tracks leaking from headphones. Adam Beard, sitting towards the front of the bus, thinks he can hear the heartbeat of Ryan Jones, sitting behind him.\n\nThe bus takes the roundabout and drops down onto the M4 heading east, towards Cardiff. Still no one on board talks. And no one will, all the way into the stadium, this being one of the team's most established match-day traditions. The bus, usually a place of banter and joking, on this journey becoming something else: a vehicle of transition, in which the men inside will go from being a group of players to becoming, in twelve miles' time, Wales.\n\nIn this moment the twenty-two members of the squad are both together and not. They are all on the same bus, travelling to the same destination, and yet they are also in twenty-two different worlds, listening to different songs and thinking about different people and pasts. But with every mile closer to the city they are a mile closer to shedding their separation, a mile closer to those eighty minutes when they will exist, instead, as a team. A fifteen-headed animal which will try, in the heat of a game and under the eyes of millions, to think, react and move as one.\n\nAn overtaking car sounds its horn in support, a Welsh scarf flying madly from its open window. There are now four police motorcycles leading the bus, their blue lights rotating, as in the distance the capital of Wales begins to rise into view.\n\nHalfway up the bus on the left-hand side Jamie Roberts is listening to the same song he always plays on this journey: 'Lucky Man' by the Verve. As he has been for all of the Six Nations, Jamie, who at twenty-five is the oldest in Wales's backline, is today's defensive captain. It will be his role to guide the team through Shaun's defensive tactics, to decide when they blitz or drift, to keep an eye on line speed, the splitting of centres and the positioning of the back three. It was, however, as an attacking centre that Jamie made his mark in the Wales set-up, his seventeen stone four pounds providing the coaches with a midfield battering ram to deploy at will.\n\nThis season, with the help of Adam Beard, Jamie has increased that penetrative power even more. On taking charge of Wales's physical performance Adam was clear about where his priorities lay, and Jamie was the perfect candidate for his thinking. Adam wanted to develop not just the strength and speed of the team, but also their efficiency, specifically their running economy. As Adam once said to Sam, 'The first thing a water-polo coach will do is train his players to swim properly. What do you spend most of a game doing? Running. But have we been teaching players to run?'\n\nThe answer was no, and yet on average a player will run seven or eight kilometres in a match. For Wales, who prefer a high ball-in-play time \u2013 around thirty-five minutes \u2013 a player's ability to move about the field quickly, to get off the ground, to ruck and chase is even more crucial. As Adam told the players again and again as he pushed them on the frozen beaches of Gdansk, 'We should be the best at everything that doesn't require talent. Effort doesn't require talent. Hard work doesn't require talent. We should be the best at hard work.' And with that, he'd put them through another set of drills, asking them again to work through a level of fatigue they'd never felt before.\n\nFrom Adam's perspective Wales had been creating big, strong players like Jamie, but not looking closely enough at their movement, at what was required of their bodies in a game. 'We'd been creating these V8 engines', he explains, 'and putting them in shopping trolleys. But I wanted to put them in a Porsche or a Ferrari instead.'\n\nIn Jamie's case this meant trimming down his quads, putting more stiffness in the Achilles tendon and strengthening his hamstrings. The result has been to make him not only more efficient, but also faster, increasing the force of his momentum up into the region of George North's one tonne of impact.\n\nAdam has gone through similar processes with all the players on this bus, examining points of strain along their kinetic chains, fine-tuning and adapting their physiques, then working with Dan to find drills and exercises through which to develop new patterns of muscle memory. With players like Toby Faletau, the young Tongan-born number eight from Ebbw Vale, Adam has strengthened certain muscle groups and introduced co-ordination training to bring his anaerobic test scores from down at prop level up to an impressive 130 seconds. In this way, borrowing from eastern European sprint coaches, Australian Rules football and athletics, Adam has moved beyond the formal studies of biomechanics and a purely mechanistic training routine. In doing so, combined with a carefully constructed programme of cryotherapy, he has managed to shift the squad out of those shopping trolleys and into supercars.\n\nExcept, of course, this being rugby, each game Wales plays isn't like taking those Porsches and Ferraris for a spin on the track, but more like putting them through the beating of a stock-car race. So the players on this bus have to be Land Rovers as well as Ferraris. And many other equivalent vehicles too. In the course of a single match they'll be expected to run like sprinters, lift like weightlifters, kick with the skill of footballers and endure hits like Ultimate Fighters. All of which means that at some point, however sophisticated the training or the physio, their bodies will break down.\n\nThat Jamie is playing at all today is an achievement. Throughout the campaign he's been carrying a knee injury, and in less than a week's time he'll go under the knife. Somehow, though, Prav and Carcass, working alongside Adam from the other end of the process, have managed to keep Jamie match-fit, as they have the entire Wales starting backline for every match of the tournament. Given the attrition rate of modern rugby, this fact is a feat in itself.\n\nInjury is no longer an 'if' in the modern game, more a 'when' and 'how bad?' No player goes through a season fully fit, and an entire team will often take the field harbouring some kind of an injury. From a young age injury stalks a rugby player, and for many on this bus their careers will be framed by it. At any moment, regardless of how young a player might be, they may unknowingly already be at the peak of their game. Like Dan Baugh, someone else's injury will open a door for them, before one of their own, eventually, will close it again. More than one player in the squad has admitted to thinking about this before each match. 'Will this be the game that ends it? If not, how many more before I play the one that does?'\n\nAs a consequence the modern rugby player, while physically robust, inhabits a fragile existence in which they occupy an ever-narrowing space between two uncertainties: the chance of injury and the doubt of selection. The two are interrelated, which is why many of these players never thought they would be on the bus today as it drives on towards Cardiff and a Grand Slam decider against France. For all of them each selection is not only a matter of beating the others vying for your shirt, but also of defying the physical vagaries of the game: the twist, tear, break or dislocation that sees fortunes rise and fall in the turn of a second.\n\nThis has certainly been the case for all three second rows on the bus, their heads rising higher over the seats than the other players. At the start of the campaign the towering Luke Charteris, just a few inches off seven foot, still had his wrist in a cast. Alun Wyn Jones was out with 'turf toe', while Ian 'Ianto' Evans, after three years of injuries, considered himself so far out of contention he'd planned his wedding and honeymoon to coincide with the squad's upcoming summer tour of Australia. And yet, in illustration of how drastically a player's week, month or year can change, by the end of today Ianto will be the only Welsh player to have played every second of every match for the whole of the Six Nations. 'You have to play the patience game,' Ianto says, speaking of his years of injury. 'It's like hitting your head against a brick wall until the crack becomes a little bigger, until eventually you can get through it and see beyond it. But mentally it's very tough,' he adds, nodding his head in recollection. 'Very tough.'\n\n*\n\nAcross the aisle from Jamie, and one of the few on the bus not listening to music, is the hooker and ex-captain of Wales, Matthew Rees, or 'Smiler', as he's known to the squad. As his moniker suggests, Matthew's natural expression falls into the suggestion of a quiet, _Mona Lisa_ -like smile, as if he's just remembered a joke at which you'll only ever be able to guess. Even now, as he looks out at the cars passing the bus, tooting their horns, the fans giving thumbs up through their windows, the echo of a grin still plays at the corners of his mouth.\n\nToday will be Matthew's fiftieth cap. To mark the occasion he'll lead the team out onto the field. As recently as a few weeks ago, though, Matthew thought he wouldn't be here today. Having already missed the 2011 World Cup with a neck injury, he recovered, only to tear his calf muscle in training before the Ireland match, meaning he would miss the first three games of the tournament. 'It was', he says, 'hard, very hard. The lowest point in my career.'\n\nMatthew was part of the 2007 World Cup squad when an underprepared Wales underperformed and were knocked out by Fiji. So to go through the hardships of Poland, to feel the 'sense of belief' in this squad, knowing they'd trained harder than ever before, and then not to be able to join them was heartbreaking for Matthew, and even more so because he knew he'd lose the captaincy to Sam in the process. At the start of the Six Nations, after he injured his calf, it looked as if he'd have to experience the same again: to train with a squad hungry for the Grand Slam, and yet not be part of it. His only hope, as all players know, was to give his all in rehab, which he did, working all hours of the day and night with Carcass to try and win back his place in the team.\n\n'Rehab' is a word often used when talking about rugby players, but one that is strangely dislocated from the reality of its meaning. Only those who have undergone the million shoulder rotations, or the all-night, two-hourly icing of a thigh, or the endless repetition of the same stretch or single exercise will know that rehab is, in many ways, a tougher ordeal than regular training. Most of it is done alone, without a team beside you. And all of it is done in hope, not certainty. At the end, even if you are successful and return to full fitness, there is no guarantee another player hasn't already staked a claim to your shirt in your absence. But however hard it is, rehab is also a truth of the modern game, the unseen struggle every player undergoes at some point in their career. Which is why, as the Wales bus continues down the M4 towards Cardiff, along with the players inside, it also bears their thousands of hours of rehab, as much a part of this team as their years of playing and training on the pitch.\n\nEven without injury, from the longer perspective of his childhood Matthew never imagined he'd one day be on this bus, travelling towards his fiftieth cap. Although as a child he was an avid rugby fan, rushing out to play in the street after watching Wales on TV, coming from Tonyrefail in the Rhondda Matthew didn't think international players came from places like his valley, his home. That he began to think they might, and that he could be one of them, was down to one man: Chris Jones, the coach of Rhondda schools. It was Chris who gave the eleven-year-old Matthew a sense of belief, who opened the possibility for him that perhaps, just perhaps, if he worked hard enough, he could make that journey from playing on the streets of the Rhondda to playing on the national ground in Cardiff. Which, fifteen years later at the age of twenty-six, he did, running out at the Millennium Stadium against Australia in 2006.\n\nSuch is the brevity of a rugby generation that today's bus is able to contain not just Matthew's achievement but also, in Jenks sitting a few rows in front of him, the symbol of his boyhood aspiration. When Matthew went out onto those streets as a kid to chuck a ball around, he and his mates would choose which Welsh players they'd like to be. Matthew always chose to be Neil Jenkins, Wales's outside-half points machine. Within a few years he found himself rooming with Jenks, when they both played for Pontypridd. Now, as player and coach, they are both on this bus, members of the same Wales squad. And yet to this day Matthew has never told Jenks that when as a boy he'd dreamed of playing for Wales, it was through him that he did so. Jenks who, like Chris Jones, has had a hand in developing the talents and aspirations of so many young players, Matthew Rees, even if he still doesn't know it, being one of them.\n\nAll the players on this bus have coaches like Chris Jones in their past: the friend's father who volunteered at their mini-rugby games, the school PE teacher, the coach of their junior, youth or local sides. The coaches who put in the hours when these men were boys, who first lit their touchpapers of hope and enthusiasm. Or it may have been someone later in their development: a regional skills or defence coach, or the head of strength and conditioning. Whoever it was, they took their game to the next level, spent time with them, passing on knowledge and advice, pushing them further and making them believe.\n\nWhoever these men are, they too, along with the squad's thousands of hours of training and rehab, are travelling on this bus into Cardiff today. As each player boarded at the Vale, they carried the influences of these men onto the coach with them. And they carried on, too, the memories of the hundreds of pitches and playing fields across the country on which they learnt their craft. In Risca, Bancyfelin, Abercrave, Carmarthen, Llandeilo, Rhayader, Gorseinon, Tonyrefail, Aberavon, Llangefni, St Clears. On pitches levelled out of high valleys like Aztec terraces; or on West Walian farming fields, scattered with dandelion and thick with clover; or on school pitches with turf worn away by the patterns of other sports. As the bus takes a slip road off the motorway and drops towards the capital, these fields and places of Wales, with their _tir_ and their _pridd_ , are also freighted on this coach; distinct yet shared echoes in each player's past as they make their way towards today's match and its eighty minutes of heightened present.\n\n*\n\nA few seats along from Matthew, Leigh, looking out at the passing buildings, is listening to a song by the Foo Fighters; then, as the bus is guided through a set of traffic lights by its escort, a dance track he's copied from Gethin Jenkins. Jonathan Davies sits opposite him, looking out at the passing shops, parks and trees. Like Matthew he doesn't wear headphones. He used to have a match-day playlist for this journey, but he's stopped that now. Keeping things simple has become Jon's mantra for a match day instead. Keeping it calm and simple.\n\nAlun Wyn Jones is also choosing to keep it calm, listening to the languid songs of Ben Howard. When he was younger he wanted heavy stuff \u2013 rock and dance \u2013 but with experience he's learnt not to stoke his emotions too soon. Like the rest of the squad he'll be spending the next two hours trying to reach the right balance between aggression and focus, between adrenalin and nerves. At the moment, as the bus moves through Cardiff, that balance is still weighted more towards nerves. But this is how he likes it at this stage in the day. If he isn't nervous now, with the stadium just minutes away, then that's when Alun Wyn becomes fearful, worried he's become too complacent, too accepting. But this, the bubbling of anticipation under the skin, the blend of foreknowledge and the unknown, the gradual building of internal energy, these are the kind of nerves that work for Alun Wyn and which will, he hopes, provide the tinder to fire him when he takes to the pitch.\n\n*\n\nThe bus drives on, the fizz and beat from leaking head-phones still the only sound. With four miles to go they are over halfway. The more experienced players have an idea of what will be waiting for them as they drive deeper into the city. For winger Alex Cuthbert, though, who played his first fifteen-a-side game just this time last year, he only has the stories of those who have already witnessed a Grand Slam with which to try and picture what is waiting for them on the streets of the capital. What he does know for certain, however, is that they are about to break the bubble of their isolation. In four miles' time they will be delivered from their week of preparation in the Vale, from their days of focused and steady training, straight into the heat of a Welsh cauldron of expectation and hope.\n\nFor Alex, sitting as he has done for all four previous matches on the right-hand side of the bus, today is the pinnacle of a dizzying ascent to the top of international rugby. It was only four years ago that a coach at Hartbury College spotted him playing a lunchtime game of sevens with some mates. He was playing football for Gloucester City at the time, and while he'd always been into his sports, from athletics, in which he was a forty-nine-second 400m runner, to show jumping, Alex had never properly tried rugby. Within a year of that lunch-time game he was playing for Wales Sevens, and within another two he had a contract with Cardiff Blues and had been called up to the full national squad.\n\nAlthough Alex is as big and fast as George on the other wing, he is also raw and still learning how to read the game. George has been playing since the age of ten. Others in the squad, such as James Hook or Mike Phillips, have been playing since the age of five. These players have rugby in their veins; the rhythms and patterns of a match are woven into their spatial awareness. Alex, in comparison, is still being shaped with every second on the pitch. What he lacks in experience, however, he more than makes up for with his energy of attack and undying, emotional enthusiasm. When the exertion of a match is starting to show in others, Alex will still be looking for the ball, eager to unleash his coiled speed upon the opposition. As Rob Howley once said of him, referring to both his horse-riding past and his strength and stamina, 'The thing about Cuthbert is, he is the fucking horse.'\n\nAlthough he may have played for more years than Alex, George, sitting a few seats away, still fizzes with equal enthusiasm, no less genuine than the day he came home from his first rugby training session and told his father, 'It's awesome! You can run at people, run around them, and you can tackle them!' During the first Wales match this year, against Ireland in Dublin, George performed all three of those boyhood observations with world-class timing and focus. In the process, he provided a moment of brilliance that became emblematic of Wales's intentions for the tournament.\n\nIt happened in the fifty-fourth minute of the game. With Wales trailing by five points George took the ball from Rhys Priestland in midfield to go outside Ireland's D'Arcy before bulldozing the Irish centre McFadden with a massive impact. As he was tackled by another two defenders, George deftly off-loaded the ball to Jonathan Davies with a backwards 'cat-flap' of a pass. The sheer power of his flooring of McFadden drew a gasp from the crowd. It was immediately followed by a cheer of appreciation as he flipped the ball to Jon, who sprinted through the gap to score a vital try under the posts. In creating that gap George had not only taken out four Irish players, but had also displayed a perfect moment of 'beauty and the beast' rugby; an almost simultaneous show of brute strength and delicate skill that had kids across Wales copying his back-handed offload for weeks.\n\nIn the changing rooms after the match Warren acknowledged the remarkable disparity between his young winger's age and his performance. 'And to think you're nineteen years old,' he said to George in front of the whole team, shaking his head in disbelief. 'You were world-class out there today.' George, bowing his head, smiled in response.\n\nAnd yet within the hour, in suit and tie at the post-match dinner, George's age seemed to have found him once more. As he bounded across to the bar, a Diet Coke disappearing in one huge hand, he looked, despite his height, every inch the sixth-former at a school-leavers' disco, those massive limbs which had done such damage on the pitch imbued once more with a teenager's energetic awkwardness. But however young he might have appeared that evening, it was the maturity and finesse of George's play that continued to fire the celebrations. His determination, skill and strength in that single moment quickly became symbolic of Wales's win, its resonance fuelling the Welsh fans' drinking on the streets of Dublin and the smile of Roger Lewis as he played blues and boogie-woogie on the hotel's piano long into the night.\n\nGeorge himself recognises that being thrown into top-flight rugby has accelerated his maturity. 'It's been an intense couple of years,' he says. 'Sometimes it feels as if I left home, grew up, played for Wales, straight one after the other.' Which is, more or less, exactly what happened.\n\nAfter a year of schoolboy rugby George had just one afternoon off at the end of his exams before starting preseason training with the Scarlets the next morning. Since then he hasn't stopped through a year with the Scarlets, an autumn international series, a shoulder operation and rehab, more regional rugby and the Six Nations, the World Cup, the European Championship and now the Six Nations once again. With the end of the season and the squad's summer tour still ahead of him, by the time George finally gets a break from rugby in July he will, like many of the players on this bus, have been on a treadmill of playing and training for almost two years.\n\nWhen George does get that break he'll use much of it to return to his home in North Wales, driving the length of the country to get back to his family on Ynys M\u00f4n. For most of the drive there's no phone signal, so, for once, no one can get hold of him. For a few brief hours, listening to music, looking at the views, George is alone as he heads north on the A470 through the mountains of mid-Wales. With each mile he drives, George is aware of an elision occurring in his mind between the length of a rugby pitch and the length of the country, the territory of one mapping over the other. It's an elision he thinks of consciously in his car, acknowledging to himself that each tree, village, hill he passes is another part of the country he represents when he puts on the red shirt of Wales. Today the same thought will come to him again, but more sensed than known when, amid 75,000 voices, he'll experience that eighty-minute contraction when the length of Wales is squeezed into those hundred metres of pitch. 'I come from a small village,' George says. 'But when I play for Wales, I come from a big country.'\n\nFor George that feeling is simply the best thing in the world. There is nothing like it. He gets goose pimples just talking about it. And yet exactly because it means so much to George, because representing and winning for Wales means everything to him, there are times when he has to remember it isn't. That it is, in fact, just a game. This is something sports psychologist Andy McCann has worked on with the young winger. How to switch off away from rugby, in order to prevent, as George puts it, 'the thing you love from killing you'. How to be prepared, but not over-prepared, to think but not over-think. How to treat a game as the most important thing in the world, yet still take the winning or the losing of it in your stride.\n\nFor George, most of these questions are answered with a negotiation between remembering and forgetting: forgetting the stresses and import of the game, and remembering the joy he's always got from playing it, ever since he returned from that first training session as an excited ten-year-old. As the bus follows its escort into Cowbridge Road, this is what George reminds himself of again now. To harness the same pleasure for this match as that which he felt when throwing a ball around with his friends on the 'cabbage patch' beside Llangefni Thirds. Which is why, when George is sitting in his stall later, writing trigger words on his strapping, the first word he'll spell out on the inside of his wrist will be 'Enjoy'.\n\nDan Lydiate, sitting a few seats behind George, has worked with Andy on switching off away from rugby too. But along with Andy's techniques Dan also uses his family farm in Llandrindod Wells to escape from the echo chamber of the game. Like George's home of Ynys M\u00f4n, Llandrindod is a couple of hours north of the country's southern corridor of top-flight competition. So as a youngster Dan, too, had to make a journey south to pursue his playing career. Unlike George, however, his was taken gradually, the names of his junior and youth clubs picking out the stepping stones of his route: Llandrindod Wells, Builth Wells, Gwernyfed, Brecon, Newport Youth, Pontypool United and finally his current region, the Gwent Dragons. If he has a Friday match, then Dan takes the opportunity to reverse that trip south and will spend the weekend back on the farm. 'It's where I go to switch off,' he says, speaking as he often does through a gentle smile. 'I'll potter around on the quad, drive round the sheep in the fields. I come back the next week feeling that bit more refreshed.' He often does a bit of work too, helping out his father and brothers. Although as he admits, 'It's only the weekend jobs, so it's still a quiet time.'\n\nToday it's Dan's family who've made the journey south instead. He saw his parents this morning at the Vale when they came to pick up their tickets. They always come and see his home games for the Dragons too, and always sit in the same part of the stand, between the twenty-two and the halfway line, so that, as Dan says, 'I always know where to look to find them.'\n\n*\n\nIn a letter of advice to his son, Nicholas, the poet Ted Hughes once told him to always remember:\n\nAt every moment, behind the most efficient seeming adult exterior, the whole world of the person's childhood is being carefully held like a glass of water bulging above the brim. And in fact, that child is the only real thing in them. It's their humanity, their real individuality.\n\nAs the Wales team bus drives through the western suburbs of Cardiff, it would be hard to imagine a more masculine collection of men than those sitting inside. As Dan Baugh once described them, 'They're clich\u00e9d macho guys. That's why they play rugby. I mean, they're the stereo type, right?' And yet what Hughes tells Nicholas is perhaps more evident in these men than in most. Whereas the majority of people have few visible ties to the child behind them in their daily lives, this busload of men are on their way to play the same game most of them fell in love with as children. The twelve-year-old Warren, still playing rugby barefoot in New Zealand; George on the 'cabbage patch'; Smiler in the streets of the Rhondda; James Hook, hoping as a five-year-old that his brother's coach will let him play with the under-9s; Rhys Priestland being taken to training by his grandfather and bought a sausage in batter and chips as a reward. The childhood games of these men have become their careers, the fundamentals remaining the same: the pitch, the ball, the exhilaration and the team. All the squad on the bus have been given licence to extend their childhoods, without a break, into their adult lives. Jenks being coached by his uncles on Cae Fardre; Rob Howley's dad, calling him in from the roof of the kitchen extension, disturbing his imagined last-minute try at Cardiff Arms Park; Shaun sleeping with a rugby ball in his bedroom opposite a pitch. And Thumper, who after stepping in to protect his mother from his father had to leave his childhood home at sixteen and who has, ever since, found his families in a succession of rugby clubs and teams instead.\n\nFor all these men, as the bus carries them closer to the stadium and today's match, the child is not so much standing behind them as living within them, still breathing and feeling, just under their skin. Still playing.\n\n*\n\nA few seats down from where George is listening to dance tunes, the pace of the music providing a disjointed soundtrack to the slow progression of shops outside his window, Adam Jones and Ryan Jones are both texting their respective wives. For these men, approaching a possible third Grand Slam each, their years of experience mean coping with the kind of questions that face the younger players has become habitual. Becoming fathers has also given them a new perspective on their professional lives. As Ray Gravell once said on the birth of his own daughter, Manon, 'Rugby has been wonderful to me, but this is life.'\n\nIn one respect having children makes it harder for a player. Without any hesitation all the fathers on this bus cite being away from their families as the most difficult part of their job; missing first steps, birthdays, bank holidays. And yet the influence of their children also makes some elements of modern rugby easier. That crucial off switch that other players have to manufacture is there, an intrinsic part of your life. As Ryan says, 'I can't get psyched up for a game from Wednesday now because I've got two children, I've got other stuff on my plate.' For Adam, meanwhile, having a daughter, Isla, has given him something else to play for too. 'I want to make her proud,' he says. 'And she's another reason to keep going. I want her to see me play and to remember it.'\n\nAdam has been discussing all week with his wife, Nicole, whether they should bring Isla to the match today. In the end they've decided she's too young (and, as Adam puts it, 'too wiggly') and this time she should stay at home in Merthyr instead. For the World Cup, however, when the squad were away for eight and a half weeks, she and her mother made the trip out to New Zealand to spend time with Adam. But then after the Samoa game they returned to Wales, and it would be weeks until Adam would see Isla again, which, as he says, 'nearly killed me'.\n\nRyan's three-year-old son, Jacob, is already in the stadium with his grandfather, Steve. There's no guarantee, however, he'll get to see his father play today. Having started for the injured Dan Lydiate in Dublin, Ryan has only started again for Wales once, against Scotland. He did, however, play and make an impact in both the other matches too, having been called off the bench to fill in at both number four and number eight. Today against France he's on the bench once more, although within a few hours, unknown to Ryan, Warren will approach him across the changing room as the team catch their breaths at half-time and ask him to step up again, this time at seven for an injured Sam Warburton.\n\nWith sixty-three caps and eight years playing for Wales, Ryan has begun to perceive his international career from an increasingly reflective distance. It is as if, in sensing an end to his Wales playing days, he's found himself thinking deeper on what his role as a player is _about_ , as much as how it should be fulfilled. His advice to younger players, which Ryan gives in an easy, expansive manner, is to make sure they take time to step back, to consider what it is they've achieved and what they're experiencing. 'Otherwise', he says, with a slow Newport lilt, 'you just get caught up in it. It all feels like a natural progression. Without knowing it you've done five, six years, you've got fifty caps, and then it's over.'\n\nA veteran of the various losses involved in being a winner at international rugby \u2013 of time, privacy and independence \u2013 in 2007 Ryan thought he'd lost the sport itself for good. When he woke in a hospital bed after a routine shoulder operation, the doctors told him he may never play again. The surgeon had removed a huge amount of damaged cartilage from his joint, in which he'd also made multiple micro-fractures to stimulate repair. It was a rare condition, and for a long time there was no clear path to recovery. The following months were dark for Ryan, and involved what he describes as 'a lot of turmoil'. It was, though, a turmoil that brought with it a positive resonance, the sudden disturbance in his career making Ryan a more nuanced player on and off the pitch, and more philosophical about the sport's rewards and sacrifices too.\n\nOn a day like today Ryan's reward, if he gets the chance to play and Wales win, will be to share that experience with his parents, wife and son. Much of what he achieves now Ryan sees in this light, as an opportunity to repay the investment his family have made in him from the very start. And in this respect he's far from alone on the bus. Some of today's players will have had to make their own way from the beginning of their careers. But others, like Ryan, will also have parents whose efforts over the years have contributed to them taking their seat today. Mothers and fathers who drove them across the country to matches, training and trials; who paid out for kit and supplements; who supported them financially and emotionally in the early days of their careers.\n\nAs the Wales bus continues its progress down Cowbridge Road, gathering a growing haul of applause, cheers and car horns in its wake, this familial investment is another entry on the manifest of past experience it carries with the squad. Along with the influence of coaches, the thousands of rehab sessions, the countless pitches and the years of training, these efforts of grandfathers, uncles and parents completes the bus's cargo of hinterland as, slowing at the junction with Cathedral Road, it drives on towards the River Taff, where from behind Castlebridge House the Millennium Stadium begins slipping into view.\n\n*\n\nThe crowds have thickened now, down both sides of the road and along the central reservation too, their cheers and applause washing against the bus as it passes like a slow whale along the street. Up ahead the four motorcycles have been joined by four mounted policemen, their fluorescent jackets bobbing in a line as their horses trot at the vanguard of the procession.\n\nFor James Hook, listening to the Stereophonics' 'Is Yesterday, Tomorrow, Today?', this is the moment he used to dream of when he was younger. Like several of today's players, he was out there for the 2005 Grand Slam, in the crowd, watching the team bus drive past, wanting desperately to be a part of it. The squad on that bus had seemed so distant to him, unreachable, and yet within a year players such as Gethin, Adam and Ryan were his teammates, his elusive, gliding running style having won him a place in the national squad at the age of twenty-one. Warren's policy of trusting youth with selection means that as James looks out over the crowd, listening to Kelly Jones sing _Write down all the things that you'd like to be, write down all the places you'd like to see_ , the chances are that somewhere among those thousands of faces a future teammate is looking back. A young player who in a year or two will be sitting where he is now, watching a crowd gravitate towards the bus, in which, somewhere, yet another future player for Wales will be watching and dreaming, _Is yesterday, tomorrow, today?_\n\nAs the bus moves onto the bridge, the players begin to remove their headphones, releasing a brief wave of music across the seats before leaving the vehicle in silence again. They want to hear as well as see what is happening. To hear the thousands of cheers, the wishes of good luck, the slow pulse of 'Waaales, Waaales, Waaales' that follows the bus as it crests the bridge and bears down upon the end of Westgate Street. As it does, the front windscreen fills with people, hundreds and thousands of them, most of them wearing red and all of them waving flags, scarves, hats at the bus and the players inside. Women blow kisses, men punch the air. A French supporter in a beret steps from the crowd and salutes with a baguette, held aloft like a triumphant spear.\n\nInside the bus it remains silent. Even those who have played in previous Grand Slams cannot believe what they are seeing. Adam Beard, who has worked with team GB at the Olympics, in Australian Rules football and in rugby league, has never seen anything like it before. He feels as if they are going to war, not a rugby game.\n\nWarren, at the front of the bus, also looks out at the waiting crowds. Their smiles and their waves make it seem as if the match is already won. But it isn't, and if it still isn't at the end of the day, he knows he will be accountable. All of them will be accountable. He and the squad, as ever, will go to work with millions watching, and millions commenting on their performance. Because on match day everyone in Wales is a coach. The display of support still buoys Warren though, and he's hugely grateful for it too, knowing as he does that for the young squad behind him, what they are seeing at this moment is worth a hundred extra training sessions.\n\nAs a Kiwi, Warren is no stranger to obsessive rugby fans, and even the nature of Welsh support is familiar to him. Like Wales, New Zealand is a small country with a strongly working-class foundation to its rugby. Through the game's Maori associations there is also a sense of the sport representing and defending an ancient culture. But despite these aspects of familiarity Wales and the Welsh remain something of an enigma to Warren. With all the country's top-flight rugby crammed along the southern M4 corridor, to work in rugby in Wales is to work in a narrow fishbowl between Newport and Carmarthen. Warren has always felt that if he's honest with the fans, then they'll allow him the odd dip, a defeat here and there. But he also knows that, for some reason, the Welsh fans love the agony of their support as much as the ecstasy; that after the pleasure they seem to crave the pain. And that is why his biggest fear is still the thought that one day, as this bus drives into town, the crowd spread through the streets before them might turn. That one day, as they make this slow drive in towards the stadium, it might not be cheers that fill the air, but boos.\n\nFor now, though, as the bus makes its turn into Westgate Street, cheers are all Warren can hear, cheers and jubilation, as if the bus wasn't carrying a squad of young rugby players, but liberators rescuing the city after months of painful siege.\n\n*\n\nMatthew Rees looks out at the crowds below and knows that when his international days are done, this \u2013 this exact moment \u2013 is what he will miss.\n\n*\n\nIn a similar vein, Ryan, too, wonders what he'll replace this with when he's no longer playing. This hit, like a drug, of playing for his country. The adrenalin as they run out of the tunnel. The noise and roar of the stadium's crowd crashing over them like a tidal wave. To have those thousands at your back as you bind with your teammates to sing the anthem. At thirty-one this already worries him, the life of aftermath on the other side of his retirement. 'I don't look forward to that wet Tuesday in Tesco's', he once said, looking out over a Captain's Run training session in London, 'when someone stops me and says, \"Didn't you used to play for Wales?\"'\n\n_In the very temple of delight veiled melancholy has her sovereign shrine_. So said Keats. And so, in this moment of adoration and thrill, several of the older players, just as they feel their blood race at the sight and sound of it, also feel at the edge of their emotions the tinge of its passing. As the fear of defeat lies at the heart of victory, so the prospect of loss is at the heart of gain.\n\nBut all of this, now, is still not yet earned, not yet won. This support from the crowd is the cheering of hope and idolatry. Those other cheers, of celebration and thanks, are still waiting on the other side of eighty minutes, if they'll be allowed to exist at all.\n\n*\n\nKen Owens, at the back of the bus, and the man for whom Matthew's torn calf muscle opened the door of selection, looks out of the rear window. Behind them the crowd that had been kept apart by the mounted police and the outriders is closing. 'Like the Red Sea,' Ken will say later. 'It was like the Red Sea, closing behind us.'\n\nMatthew's injury, and then another to Wales's other veteran hooker, Huw Bennett, meant it was Ken who pulled on the Wales number-two shirt in the Twickenham changing room that February afternoon three weeks earlier. And it was Ken who read the commands on the tip sheet he found under his towel \u2013 'ASSAULT THEIR ATTACK. BE RELENTLESS.'\n\nPrior to that game Ken had only played one full eighty minutes of test rugby, against Namibia in the World Cup. He'd come on as a blood replacement against Scotland too, but then was off again before he knew it. In that changing room in Twickenham Ken was nervous, and found himself changed and ready well before the team warm-up. Unsure of what to do, he sat on his bench, his forearms on his thighs, his head bowed.\n\n'Look up.'\n\nThe voice came from above him, soft and northern. Ken did what it said and saw Shaun standing over him.\n\n'Lift your head,' Shaun said, advising, not commanding. 'You're a man now. Walk around with your chest out, like you own this place.'\n\nTwo hours later Wales, with Ken playing at hooker, had won the Triple Crown. In the stands Ken's parents, having watched their son win his third cap, remained in their seats to watch his sister, playing for Wales women, run out for her seventh.\n\nAs he looks out at the crowd closing behind the bus Ken sees a couple of his mates from back home in Carmarthen. He has been hanging around the local club there, Carmarthen Athletic, since he was seven, so it's well known in the town that he's playing today. In doing so, like all the team, he'll be representing them, Carmarthen, his home club, as well as Wales. When he was back in the town on Wednesday, the owner of the local cafe wouldn't take Ken's money for his slice of cake. 'You can pay for it', he told Ken, 'by winning that match on Saturday.'\n\n*\n\nThe crowds deepen as the bus takes the corner into Westgate Street. Women wear daffodil hoods, men are draped in flags and nearly everyone is wearing some version of a Welsh rugby shirt. Along with the running applause and the general cheers, more pulses of 'Waaales, Waaales' continue to follow the bus as the police horses and motorcycles carve a path for it to make its final turn off the street and down the incline at Gate 4 into the stadium. As it does, Jonathan Davies, sitting halfway up on the left-hand side, sees, among the thousands of faces, his mother, waving and cheering with the rest of the crowd. The sight of her catches him unawares, and for a moment he feels his eyes welling up as the bus drives on down towards the stadium's underground entrance, past the four mounted policemen, past an ambulance and on into the darkness.\n\nAlmost immediately the sounds of the crowd fade behind them. Suddenly, after that brief exposure, the squad are isolated again. The bus enters carefully, as if boarding a ferry. Giant ventilation ducts run along the unpainted concrete walls; frames of folded growing lights are lined up along another. Making one last turn past Gwyn's security lodge, the bus finally comes to a halt outside the set of double doors leading into the changing rooms. Parked ahead of it, already here, is the blue bus of France.\n\nStewards in orange fluorescent jackets, ground staff and five members of the Royal Welsh Regiment, along with their regimental goat, William Windsor 25142301, look on in silence as the Welsh team disembark. Filing through the double doors, their headphones back on, the squad take the stairs up past the silver dragon and turn left towards the rooms that J.R., singing along to 'More Than a Woman', began preparing for them this morning.\n\n*\n\nFrom the moment the squad step off the bus, the machinery of the day steps up a gear. The next hour, with every minute accounted for, unfolds with a gathering momentum. The time of the kick-off is set at 2.45 p.m., and it will not move. At that moment the preparations will come to an end and the match will begin.\n\nOnce off the bus Thumper takes the team sheets to the broadcasters and checks through both countries' lists of players for any changes or errors. Then he travels in the VIP lift up to level five to drop in on the lunch in the President's Lounge. Checking in with Roger and Dennis, he tells them he's never seen as many people on the streets to welcome the bus as he did just now. Many of the guests, aware he's arrived with the squad, ask him how the boys are, how are they doing? Thumper nods tightly, telling them they're fine, good, calm.\n\nWhile in the lounge Thumper makes time to see his wife, Kerry, who has been dining with the players' wives and girlfriends. Whether through arrangement or osmosis, Kerry performs a similar role with these women as Thumper performs with their husbands and boyfriends in the Vale, gently corralling them through the day's schedule, welcoming new arrivals and easing them into the match-day environment.\n\nAs Thumper talks with Kerry and the girls on level five, four levels below Adam Beard, having checked the nutrition tables and written the warm-up timings on his hand, synchronises his watch with the match officials, then goes to see Lee, the head groundsman, to get his update on the hardness of the pitch. Lee will have already sent him a report this morning, but since then, with the roof now fixed and open, there's been a brief but heavy downpour. Based on his conversation with Lee, and having checked the boots France are wearing, Adam goes back into the changing rooms and advises J.R. on what boots and studs the team should use. This done, Adam's priority for the rest of the run-up to the match is to steal time.\n\nOnce Sam's huddle with the team is finished, Adam will lead the players through their warm-ups and prematch drills, his aim being to keep them out on the field for as long as possible. He knows the shorter the gap between the end of the warm-up and the start of the match, the quicker will be the players' nerve conduction, the warmer their muscles and the higher their alertness. In previous matches, as Wales have continued warming up, opposing teams have often already returned to their changing rooms. Match officials will be telling Adam to finish so they can start the entertainment. Even Warren will be looking uncomfortable. But Adam knows how fine the margins are in international rugby, and how a split-second difference in the sharpness of one of his players could end up being the deciding factor between a win and a loss. So, like a chef trying to bring his recipe to the perfect temperature, Adam will steal time, keeping the players out on the pitch for as long as possible.\n\nAs the team are getting changed and having their strapping done by Carcass and Prav, and as Thumper goes to level five and Adam goes to Lee, Shaun, dressed in his team tracksuit, takes a stroll on the pitch. The heavy downpour, although brief, is concerning him. The French have requested for the roof to be open, and now, under the recent rain, the Millennium pitch is slippery.\n\nOn one of his arms Shaun has a tattoo of his younger brother, Billy-Joe, who was killed in a car crash in 2003, aged twenty. Like Shaun and their father, Billy-Joe was a rugby player. Before the accident he'd recently signed with their home club, Wigan. As Shaun walks on the Millennium Stadium pitch now, feeling the turf with his trainers, he looks up at the clearing sky through the open roof. 'Someone's going to slip today,' he thinks. 'And it's going to cost them a try. Please', he continues, addressing Billy-Joe directly now, 'just let it be one of them France buggers.'\n\n## 2.38 p.m.\n\n'Ten seconds, guys! Ten seconds for Wales!'\n\nThe suited event manager, holding one hand to the single earphone of his headset, calls down the corridor at the fifteen men in red standing in line along the wall. Moments earlier, accompanied by a sudden flurry of clapping and shouts from the substitutes, the Welsh team had emerged from their changing room, faces expressionless, their eyes focused beyond the cramped corridor in which they now stand. Lining up behind Matthew Rees, they hold that forward-looking gaze. Mike Phillips jumps twice on the spot. Jonathan Davies rocks from side to side. Jamie Roberts stands stock still. George North, further back, twitches his head in a couple of quick neck stretches. Alun Wyn Jones and several others push short, hard breaths through their mouths and nostrils, like penned bulls waiting for release. No one talks. Under their training tops, in memory of Mervyn Davies, the whole team wear black tape wrapped around the left sleeves of their shirts.\n\nThe French team are already on the field, binding in a tight circle against the noise of the stadium. The double doors to the tunnel are closed, but the sounds of the bowl, although deadened, are still audible. The waiting, silent squad can hear the pre-match music heightening to a crescendo, the bursts of giant flares firing at either end of the pitch, the announcer's excited voice echoing between the stands, stirring the capacity crowd louder and louder.\n\nThe event manager, touching two fingers to his headset, listens to an instruction. 'OK, Matthew,' he says. 'When you're ready.'\n\nThe double doors open and the noise is suddenly louder. With a deep breath Matthew Rees, who once pretended to be Neil Jenkins on the streets of the Rhondda, leads Wales out for his fiftieth cap. Walking at first, he breaks into a jog further down the tunnel. Behind him, Sam Warburton, holding the hand of Daisy, today's nine-year-old mascot, follows. And behind Sam come the rest of the team, stepping up and dropping through those double doors like a squad of Paras jumping from a plane.\n\nIn a clatter of studs the team stream down the tunnel, running past two Under Armour banners on either side: 'PROTECT THIS HOUSE'. Speeding up, they burst onto the pitch, fanning past Sam, who is crouching for a photo with Daisy. As the team sprint past him, all the flares erupt at once, shooting flames thirty, forty feet into the air and releasing a wash of petroleum fumes into the higher levels of the stands. At the same moment 75,000 people rise to their feet, delivering an imposing roar of a welcome as the familiar chant pulses through the stadium like a heartbeat:\n\n'Waaales! Waaales! Waaales!'\n\nAnd in the Three Kings in London, and in the Red Lion in New York, and in Camp Bastion in Afghanistan, and in pubs and homes across Wales and the world millions of Welsh applaud and cheer at the same time, foisting their hopes and their ideas of a nation upon the twenty-two young men running out onto the pitch.\n\nAnd none of it matters. And yet all of it matters. Because for players and spectators alike this isn't just about being alive, but feeling alive. This is where the known and the unknown meet. This is the arena, the coliseum. Where the present is electrified by its imminent transfiguration into the past. Where, as Philippe Saint-Andr\u00e9 said to an interviewer before the game, 'We do not just play against Wales, but against the whole country.'\n\nAs George runs onto the pitch he momentarily dips to brush his fingers through the whitewash of the touchline, rubbing them against his thumb as he sprints on into the middle of the field. Raising his hand to his face, he smells the scent. This, for George, is 'the final clunk in the cog', the moment when he becomes, for eighty minutes, a different George.\n\n'Off the pitch I'm not that confident as a person,' he says. 'But once I smell that whitewash, once I cross that line I'm in a different zone. I find myself quite aggressive. Confident, but not arrogant, and generally quite mad at everyone on the other team. I don't know what it is, but it's like they've done something.'\n\nGeorge, feeling that switch of personality come upon him, gathers with the rest of the squad as they line up for the anthems, their arms across each other's shoulders. Next to them the French squad also come together, each player bringing with them their own hinterlands of past coaches, pitches, clubs and dreams. But as the two sides line up next to each other, they also bring their other histories to this moment. Not the histories of the players, but of the teams themselves. And as rugby histories go, they are as mirrored and balanced as the teams look now, spread either side of the halfway line.\n\n**Wales** | **France**  \n---|---  \n15 Leigh Halfpenny | 15 Cl\u00e9ment Poitrenaud  \n14 Alex Cuthbert | 14 Wesley Fofana  \n13 Jonathan Davies | 13 Aur\u00e9lien Rougerie  \n12 Jamie Roberts | 12 Florian Fritz  \n11 George North | 11 Alexis Palisson  \n10 Rhys Priestland | 10 Lionel Beauxis  \n9 Mike Phillips | 9 Dimitri Yachvili  \n8 Toby Faletau | 8 Imanol Harinordoquy  \n7 Sam Warburton (c) | 7 Julien Bonnaire  \n6 Dan Lydiate | 6 Thierry Dusautoir (c)  \n5 Ian Evans | 5 Yoann Maestri  \n4 Alun Wyn Jones | 4 Pascal Pap\u00e9  \n3 Adam Jones | 3 David Attoub  \n2 Matthew Rees | 2 William Servat  \n1 Gethin Jenkins | 1 Jean-Baptiste Poux  \n|   \n|   \nReplacements | Replacements  \n16 Ken Owens | 16 Dimitri Szarzewski  \n17 Paul James | 17 Vincent Debaty  \n18 Luke Charteris | 18 Julien Pierre  \n19 Ryan Jones | 19 Louis Picamoles  \n20 Lloyd Williams | 20 Morgan Parra  \n21 James Hook | 21 Fran\u00e7ois Trinh-Duc  \n22 Scott Williams | 22 Jean-Marcellin Buttin\n\nWales have played France eighty-nine times. Over the course of those games they have won forty-three matches each. France have scored 1,304 points, Wales 1,305.\n\nThe more recent history, however, tells a harsher story for Wales. Of the last eight games against France they have lost seven. The last time they played this fixture, in Paris last year, Wales failed to score a single try.\n\n'No one can take it away from you.' That's what Warren reminded the squad at their team meeting in the Vale this afternoon. 'When you win a Grand Slam, it's yours. No one can take it away from you.'\n\n'We know we deserve to win.' And that's what Sam told them just minutes ago in the changing room. He used to stress about that team talk, used to write it down. But now he just says what he feels is right at the moment. Which, today, was this. 'We know we deserve to win, for us and the fans. We know how much work we've done.'\n\nThe media have been talking all day about the death of Mervyn Davies, another Welsh captain who twice led his side to Grand Slam victories over France. There has been talk of the team being spurred on by his death, being fuelled to play in his memory. But for the squad it's more simple and, in a way, more personal than that. Sam does mention Mervyn in his talk, and they all know a great player has passed on. But in the hardest moments of this match they'll be digging in for themselves and for each other; for the pain they've shared and the sacrifices they've made, and in answer to that banner in the Barn that asks, _How do you want to be remembered?_\n\nFor the next minute, however, it is Mervyn Davies and Jock Hobbs, the ex-All Black who also died recently, who are remembered as the full weight of a stadium's silence falls upon the pitch. For sixty seconds the two teams, the match officials and the 75,000 spectators all stand motionless as images of Merv and Jock come up on the stadium's big screens. After all the build-up, the bus journey into the centre, the superstitions of the changing room, Adam's stolen time, there is, before the violence of the match, this. A strange moment of peace, the loud voice of the stadium silenced.\n\nWhen the minute ends, the anthems begin. The French who have travelled to Cardiff sing 'La Marseillaise' with energy, but when the opening lines of the Welsh anthem are sung \u2013 ' _Mae hen wlad fy nhadau yn annwyl i mi_ ' ('The land of my fathers which is dear to me') \u2013 it's as if the volume control on the stadium has been turned up to eleven. For the players at the centre of the pitch it's an overwhelming experience. They've spent much of the last few hours trying to achieve a careful set of psychological balances: between aggression and discipline, emotion and focus, boldness and caution. And yet it's as if this anthem has been designed to ambush them. The words of Evan James of Pontypridd and the music of his son, James James, coming together just seconds before kick-off to lay an emotional minefield at their feet.\n\nTo keep himself together George finds one point in the stands as he sings, and keeps his eyes fixed upon it. Sam, although always 'singing out loud in my head', usually remains silent. But this time even he, his hand on Daisy's shoulder, sings, albeit in a subdued manner, as if, like in that letter by Ted Hughes, he's doing all he can not to spill something brimming at his edge. Adam Jones, his eyebrows Vaselined against the abrasion to come, sings with a concerned frown of care. Jamie Roberts, with his eyes shut. Alun Wyn Jones, meanwhile, uses the anthem to take him to the next stage of his day's preparations. With the sinews in his neck straining and the veins over his temples rising, Alun Wyn screams out the anthem's final lines, his eyes reddening and his head tipped back.\n\n _Tra m\u00f4r yn fur i'r bur hoff bau,_\n\n_O bydded i'r hen iaith barhau._\n\n[While seas secure the land so pure,\n\nO may the old language endure.]\n\nThe anthem ends in a torrent of applause, cheers and, as the players unbind, the breaking of the team's chain. Alun Wyn looks as if he's received terrible news, as if he wants to do someone harm in revenge.\n\nWithin seconds, as the field is cleared of banners, bands and choirs, the players are already preparing for kick-off. The French sprint and jog, swinging their arms, performing final stretches. Wales, however, have tipped to the far end of their territory, where the team engages in Adam's last few seconds of stolen time. In short, concentrated waves, they perform 'bag hits', running at Dan Baugh and the subs to smash into the tackle shields they hold before them. With each hit the shield-bearer is shunted a few feet back towards the dead-ball line, before reasserting their position for the next player and the next hit.\n\nBehind them France are beginning to string out along the halfway line, so, turning from their final hits, the Welsh players also walk into position, scattering deeply through their half. Gethin Jenkins, the veteran prop, blows out a deep breath. Adam Jones calls instructions to the other forwards. Craig Joubert, the referee, raises his arm and blows his whistle. A fanfare sounds on the PA system and the crowd responds with a cheer as Yachvili, signalling to his teammates, drop-kicks the ball high and deep into Welsh territory.\n\nThe match has begun.\n\n## 0 min.\n\nToby, who wears Mervyn's number-eight jersey today, and whose own father's playing days brought him from Tonga to Wales as a child, catches the ball and drops his shoulder to meet what it will bring. A sudden blue wave rolls him and his teammates backwards, breaking over them as they form a ruck and Toby lays the ball back. Mike clears, making the first of the six kicks to punctuate this opening minute of the match. The fifth is made by Rhys, searching behind the French line. Alex, with a sprinting style reminiscent of Michael Johnson, comes steaming after it, pressurising Poitrenaud to punt into touch and making Wales the winners of the first kicking exchange.\n\n## 1 min.\n\nThe two packs come together to form the first set piece of the match, the forwards lining up in domino formation behind each other and next to their opposing numbers. The previous week the Welsh line-out seemed lacking, so for the past few days Rob McBryde has put in extra sessions working on their calls and moves. Unlike the scrum, the line-out is a set piece of moving parts: a thrower, two lifters, dummy runners and a jumper. If a call is to be successful, each of these parts must work off and with each other like the intimate mechanics of a watch. For Alun Wyn, lining up at the centre of the Welsh formation, it's as much about rhythm as memory. 'A good line-out', he says, 'will flow. It's more art than science.'\n\nIf that rhythm is right, then the line-out is also a brief pause button in a match, a slow-motion ballet with the jumper, which Alun Wyn is now, lifted to twice his height and suspended there for a second, held aloft by Adam and Gethin below him.\n\nCatching the ball cleanly, Alun Wyn drops it to Mike at scrum-half, as if throwing it down from a first-floor window. At the same time as his rising, like the un doing of a combination lock both flankers, Sam and Dan, have popped out of the line. Taking a few paces back they both pause, each with one foot cocked, a mirror of each other, ready to spring forward.\n\nIt isn't them, though, who come onto the ball but Alex, charging in to take Mike's pass in his arms, clutching the ball to his chest as if cradling a child. Hitting the French defence, he sucks in two defenders before going to ground.\n\nMike digs in the bodies and spins the ball free, releasing a chained pattern of breaks, stoppages and passes that revolves through three phases before ending in another French clearance to touch.\n\nAlready Matthew, Sam and Jon Davies have all tested the French line, with Jon breaking through it, only to be dragged down by Fofana, clinging to his back like a cheetah jumping on its prey.\n\n## 2 mins\n\n'But a line-out', Alun Wyn also says, 'is about possession quality, not just percentage.' As Matthew throws in, the ball is tapped down to Mike, who, having to scramble for it, is enveloped by French shirts. Adam stands in at scrum-half, but the slower ball has given the French time to prepare, so as Matthew receives his pass, he too is hit to the ground.\n\nJust as there's a kinetic chain in each of the players' movements, so there is a chain of reaction through every period of play. Mike is back in position, pointing instructions, sending Toby back off a ruck. But the disrupted flow of that line-out is now in the rhythm of the Welsh game too. As Alun Wyn carries the ball into another phase he knocks it on and loses possession.\n\n## 3 mins\n\nThe French scrum-half, Yachvili, kicks downfield, and Craig Joubert announces, 'Advantage over.' Just as line-outs have rhythm, so do matches, and it is the referee, as much as the players, who conducts that rhythm. With this announcement Joubert is setting out his stall: in the interest of the flow of the match he'll be moving on from advantages quickly, trying to keep the flame of the game alight.\n\nLeigh receives Yachvili's kick and returns a high, mortared ball back to the French. Yet another kick comes back, into touch in the Welsh half. This time it's Ianto, already over two metres tall, who rises even higher from the line-out to drop the ball to Mike, who spins it to Rhys, who puts yet another high, hanging kick into the air.\n\n*\n\nLast night Rhys went to the cinema with Jon, Mike and Jamie. They knew the roof of the stadium would be open today, and they'd seen the forecast for rain. Jon asked Rhys what was the highest number of kicks he'd ever made in a game. Rhys said he couldn't be sure \u2013 maybe forty? 'Well,' Jon said, 'I reckon you could be kicking all day tomorrow.'\n\nAnd so far Jon's been proved right. The rhythm of the match is being set on the boot, not through the hands. 'Play in their half' \u2013 this is the Welsh policy. Kick and put them under pressure, no silly mistakes. It's a good policy for a match like this, in which France are trying to slow down the play. But sometimes for Rhys, wearing the iconic number ten on his back means that 'good' policy isn't always seen as the right policy by fans in Wales. Such is their expectation of a 'national' style that he's had complaints about him not 'playing in a Welsh way'. But the quietly spoken Rhys is a reader and a strategist of the game. Part of his role is to look for space on the field, to test their opponents' pressure points. And today, with a greasy pitch and a wet ball, this is how he'll be doing that \u2013 by skying kicks that send everyone's heads tilting back, and that give his backs time to chase and harry upfield.\n\n## 4 mins\n\nAnother Welsh knock-on sees the setting of the first scrum, the two packs kneeling and binding in preparation to knit together at the referee's command.\n\nThe etymology of the word 'scrum' has violence at its every root. Derived from 'scrimmage', itself a corruption of 'skirmish', its origins lie in the old High German ' _skirmen_ ' \u2013 'to protect, defend' \u2013 and the Middle English ' _skirmysshen_ ' \u2013 'to brandish a weapon'.\n\nWhere once the scrum was the most basic of territorial contests on a pitch, in the modern game it has, in Rob McBryde's words, 'become more of a hitting contest than a shoving contest'. Changes in the rules, and the relaxing of others such as feeding, mean the scrum, for all its base qualities of weight, power and aggression, is now also a subtle tactical arena of sleight of hand and one-upmanship. Where once a scrum was used to win the ball, they're now frequently used to win a free kick or penalty instead.\n\n## 5 mins\n\nThe two packs, loaded with the potential kinetic tension of their combined force, crouch and face each other. Gethin holds his free hand in a fist; Adam holds his open. The heads of the opposing props and hookers are inches apart, ready to mesh into the spaces between each other's necks, hitting shoulder against shoulder, driven together by the full weight of the packs behind them. The force of that collision is around 14.8 kilonewtons, 150 kilogrammes per 0.09 square metres, the equivalent of dropping a cow onto a person's shoulders from a metre's height and enough total force to uproot a small oak. For Adam's wife, Nicole, this is the hardest part of the game to watch, both because of that force and because she knows this is when Adam, at tight head, although unseen, has to perform the dark arts of the front row, when he has to dominate and pressurise his opposite number.\n\nCraig Joubert, standing over the two packs, gives them their instructions.\n\n'Crouch. Touch.'\n\nThe props flick their free hands forward to lightly tap the shoulder of their opposite number.\n\n'Pause. Engage.'\n\n*\n\nEarlier this week, as they packed down against a scrummaging machine on a mist-covered Castle pitch at the Vale, the Welsh forwards heard these exact instructions, spoken in the same rhythm by the same voice. Craig Joubert was not at that training session, but his voice was, emitted from an iPod dock held by one of Rhys Long's analysts crouching beside the pack. Such are the margins in modern rugby that a referee's intonation, the length of his pause, the rhythm of his speech can be enough to turn a game. Which is why all through that session in the mist the Welsh pack scrummaged with Joubert's voice in their ears, timing the wave of their drive, from Toby's first nudge to the impact of the front row, against that single repeated phrase, fuzzy but consistent in their ears.\n\nAnd now, as the two packs engage at the Millennium Stadium, as they hit shoulders and sixteen pairs of legs brace and flex against the pitch, the timing laid into the muscle memories of the Welsh pays off. The scrum caves in and the French are penalised for early engagement.\n\nMike immediately goes searching for the ball, finds the mark, then taps and spins to Toby, who once again launches himself at the waiting French.\n\n## 6 mins\n\nWales move through the phases, but with France barely committing to the rucks, their defensive wall remains strong. Gaining possession, Beauxis, their number ten, kicks upfield. Leigh returns with another high, lingering kick. Dan gathers the ball when it goes loose, the Welsh move finishing with Rhys grubber-kicking into space. A French drop-out from their twenty-two is returned with another high kick from Rhys, Jon's prediction of last night becoming more true with every unfolding minute of the game.\n\n## 7 mins\n\nWhen Rhys's kick lands, it's Welsh hands that collect it. Sam, dropping his shoulder, drives into the French line, with Alun Wyn lending his weight in support. But when he fails to keep that weight on his feet, France win a free kick, and then the line-out after it.\n\n## 8 mins\n\nWith the echo of that iPod in their ears, the Welsh pack win another free kick from a scrum. This time Mike taps and goes himself, skirting round the still bound front row and piling into the French defenders.\n\n*\n\nUnlike his attack coach, Rob Howley, who had to fight for his position at scrum-half against those who thought he was too small, Mike had to fight for his against those who thought he was too big. Scrum-half is a position rich with association, those two words conjuring the image of a shorter, stocky man, terrier-like at the heels of the bigger forwards. A mini-general, commanding the game, who probes and darts at the gaps in an opponent's armour. For years scrum-halves have had their positions guessed by strangers in bars, simply by sizing them up in comparison to the other rugby players they know.\n\nUntil he was fourteen Mike fitted this idea. But then he grew, eventually reaching six foot three. Aware of his own size, it wasn't a Welsh scrum-half Mike admired when he was playing for Whitland and St Clears, but the tall South African, Joost van der Westhuizen. People telling him to change his position was just fuel to the fire, convincing Mike he had to prove them wrong and remain a scrum-half.\n\nMike displays a similar dogged quality now, as he drives on into the French defence and is brought down, only to get to his feet and continue again, barging into the bulk of French lock Maestri.\n\nThere is a feral quality to Mike's play, and to Mike too, off the pitch. This has been mostly to Wales's benefit, providing Warren with a dangerous player big enough to be a third flanker, yet still possessed of that scrum-half guile and tenacity. But the modern game is about balance too, something Mike acknowledges as one of its biggest challenges. 'I've tried different stuff,' he says about his preparations for a game. 'To work myself up, or to chill myself down. But that's what makes rugby so hard. You have to be on the edge; you have to smash someone twice your body weight, then next minute you have to be composed, think about what's best for the team.'\n\nThe whistle blows. This time Mike got the balance wrong by half a second, not releasing the ball when he was brought down briefly. France are awarded a penalty, and the Welsh retreat to await the French kick for touch.\n\n## 9 mins\n\nThe ball crosses the touchline inside the Welsh twenty-two. France, with one kick, are in their best position of the match so far. After winning the line-out they suddenly find their rhythm, rolling a maul towards the Welsh posts. Adam tackles Servat, bringing him to ground in a brief spiralling embrace.\n\n## 10 mins\n\nDan goes for a different technique on Harinordoquy, cutting him down below the knees and spinning him onto his back. But Gethin, who joined Dan in the tackle, doesn't release his man before going for the ball. Joubert blows his whistle and raises his arm, awarding a penalty to France in front of the Welsh posts.\n\n## 11 mins\n\nThe Welsh players line up under their posts in a strung-out echo of their anthem formation.\n\n*\n\nWhen Joe Lydon, the WRU's head of rugby, was playing for England under-18s, his coach told him before a fixture against Wales: 'Just remember, they're not defending their tryline, they're defending their border.'\n\nThat metaphorical border still remains unbreached, but from their position standing along their tryline the Welsh players have to watch, after all their effort, as Yachvili strikes the ball with his left foot and sends it clean between the uprights. A swelling cheer follows it over the crossbar.\n\nWales 0 \u2013 France 3.\n\nFrom the restart Harinordoquy, the French number eight, gets under the ball and readies himself to carry it upfield. George, however, has pursued the kick and is there to haul him in, as if he's netting a struggling shark.\n\n## 12 mins\n\nLeigh catches a French clearance just inside the Welsh half, but this time he attacks on foot, not with his boot. As he sprints towards the French, his smaller frame makes his limbs appear to move quicker than other players. Approaching the first French defenders, he slips on the wet pitch, but rebounds to spin through a tackle and drive on another couple of feet.\n\n*\n\nIt was a Welsh full-back of the 1970s, J. P. R. Williams, who came to embody the attacking potential of the position. It was a Welshman, too, who first demonstrated that the last line of defence on a rugby pitch could also be the first line of attack.\n\nVivian Jenkins came from Bridgend. Having learnt his rugby at the same college in Llandovery as George, he won three Oxford Blues at centre before being selected for Wales at full-back in 1933. At the time the required qualities of the position were meant to be a safe pair of catching and tackling hands, with a good boot to kick for touch. But in 1934, during a match against Ireland in Swansea, Jenkins initiated an attack from his own twenty-two, finishing the play seventy-five metres later by collecting a pass from his winger to cross the Irish line. The purists were not impressed and it was suggested Jenkins had played beyond the boundaries of his position. It would be twenty-eight years before another full-back would score in the championship, in 1962.\n\n*\n\nAs if in homage to Jenkins's inversion of his role, in the next Welsh attack Jamie, known for his crash ball and bulldozing of defences, delicately chips the ball past the French instead. Toby wins the line-out, but without clean possession, resulting in a slow ball eventually being birthed from a maul to Mike.\n\n## 13 mins\n\nFor the next minute the Welsh rhythms feed the game as they go through eight phases of play. But the French defence, varying their drift with different currents of opposition, sustain attacks from Alex, Ianto, Dan, Matthew in a repeated pattern of French blue extinguishing fire after Welsh fire. The last to attack is Leigh, once again turning out of a tackle like a spinning top, finally to be quashed under a deluge of French bodies.\n\n*\n\nAs 75,000 spectators follow Leigh and the ball, the TV cameras tracking his route upfield, focusing the gaze of millions, several other pairs of eyes are looking elsewhere. Up in the stands in the coaching box Wales's attack coach, Rob Howley, is watching the spaces away from the contact area, following his players off the ball. 'My mantra about attack', he says, his sentences rising at their ends, 'is it isn't what you do with the ball, it's what you do without the ball. Your ability to get off the floor, to get to a support player. The ball needs a lifeline and someone to help, so the more bodies you have around it, the more successful you're going to be.'\n\nIn front of Howley, Rhys Long is looking at his computer screen. With six dedicated analysis cameras around the pitch, and another stitching images together to provide a floating, bird's-eye view, he's watching the same match several times at once. And he's also watching it on more than one level. Andy and Rod, his primary coders at pitch-side, are inputting the team's key performance indicators \u2013 line speed, timings, gain lines and clusters of activity. Another five analysts, meanwhile, are coding three players each as they watch the game remotely from their homes around the country. All of this information is coming into Rhys's computer live, so as he watches the players' movements on its split screen, he is also tracking the vital signs of the game; reading the rucks, breaks and kicks happening on the pitch in front of him as figures, maps and graphs. 'People always say to me,' Rhys's mother once said to him, '\"Why doesn't Rhys ever smile on the TV?\"' 'Because', Rhys explained to her, 'I'm in my office, working.'\n\nDown on pitch-side Prav and Carcass are running a touchline each for the duration of the game, their eyes trailing three seconds behind the action. Like Prof. John sitting with the subs, they are watching for players left on the field after a collision or a ruck, observing their movements after a hit, ready to run onto the pitch if they see anything of concern. The subs, too, are watching the positions they might replace, ready to start warming up at the first sign of injury. Ken tracks the movements and tackles of Matthew; Lloyd Williams those of Mike.\n\nIn the south-eastern corner, meanwhile, Lee, the head groundsman, and his assistant Craig are watching the pitch itself. After their months of dedication, their weeks of coaxing the turf to grow, their long nights slow-walking its length, they are now observing its destruction. Making mental notes of the scars and gouges, they watch as scrums churn at their mown patterns, boots and studs send divots flying, and tackling players cut thin trails of mud in their wakes.\n\n## 14 mins\n\nYachvili is penalised for not rolling away and the game, having had its pulse quickened, pauses. Jenks and Adam bring on water for the players, and also messages, both men running communications from the coaches in the box.\n\nThe coaching team have noticed that the French are varying their use of a tail-gunner in the line-outs. All the Welsh models have been built on France's number seven being in the line, but he frequently isn't, positioning himself instead inside his number ten, ready to stop an attack from Jamie. Rhys Long discusses this with Rob McBryde. They hope the players have picked up on the change, but just in case they send a message on with Adam for the forwards: if Bonnaire is inside his number ten, then drive through the line-out itself instead.\n\nAs the team drink their water and Adam relays this message to the pack, Rhys Priestland lays the ball on the kicking tee for Wales's first opportunity to level the scores.\n\n*\n\nWhen Rhys was at Bristol University, Jenks would travel to the city specifically to help him practise his kicking. 'I was stunned I was getting that kind of treatment,' he says. 'Without Jenks most of us wouldn't be able to kick the way we do. And he always had time for me too, telling me, \"Any issues, give me a ring,\" that kind of thing. He's a great bloke.'\n\nThat depth of history between the two men came into play during the England match, not in terms of Jenks's knowledge of Rhys's kicking, but in terms of his knowledge about his personality. Rhys was having a tough game, and for a while nothing was going his way. From up in the stands Warren discussed with Jenks at pitch-side over their radios if they should replace Rhys with the veteran outside-half, Stephen Jones. It was a crucial match and some would say there was more than enough reason. But they didn't. After the match Warren told the press conference he thought it was better for Rhys's development to learn on the pitch. Which is exactly what Jenks, having known Rhys since he was eighteen, had said at the time. 'He's not going to learn anything here sitting beside me,' Jenks told Warren over the radio.\n\nSo Rhys stayed on, his game improved and Wales won the Triple Crown. Those last few minutes on the pitch, though, were definitive for Rhys beyond his personal development. As with the end of the Ireland game, it was, he says, when he felt the _belief_ of the team. In both matches 'we never panicked. It was quite strange being on the field. Myself, not being one of the senior players, if it's tight I've been used to people shouting, \"We gotta do this, we gotta do that.\" But there was none of that. Warby would speak, Gethin too, just saying, \"Don't panic, we'll get down there, no silly mistakes.\" I think it's because we'd trained so hard and we've got a lot of trust in each other as a squad.'\n\n## 15 mins\n\nRhys stands and steps back from the ball, still studying it, like a sculptor getting distance from their work. Turning his back on it, he walks further away before facing the posts once more. This time, as he looks at the ball he seems to have developed a more inward gaze. He takes a breath, licks his lips, blinks rapidly, before turning his focus to the posts. To his right Ianto is breathing heavily, his hands on his hips, sucking in the air he needs to recover.\n\nRhys looks at the posts as if they are a puzzle to be solved, as if he is trying to understand something about them. He glances at the ball one more time, then back at the posts. Straightening his shoulders, as if coming to attention, he returns his gaze to the ball again. Taking four steady steps towards it, he strikes it cleanly, with the sound of a cricket ball against a bat.\n\nA sigh cascades from the stands as the ball hits the slender upright of the right-hand post. It rebounds to the French below and Yachvili clears it upfield and into touch.\n\n## 16 mins\n\nJoubert, the conductor and referee, tells the forwards to 'get into line-outs quicker'. Both teams are trying to impose their own rhythms on the game: France slow; Wales fast. But Joubert is trying to ensure it flows too. This is, after all, entertainment as well as sport, business as well as a game.\n\nSam rises at the tail of the line-out to win clean ball for Mike.\n\nDown the line Jon straightens the Welsh attack. The second-phase ball is passed to Leigh, running in from deep. He looks dangerous until a massive hit from Fofana picks him off both feet and knocks him to the ground. The crowd accompany the hit with a stadium-wide expression of empathetic pain.\n\n*\n\n'Your body almost shorts out,' is how Ryan explains being on the end of a hit like that. 'You have that white-flash moment, when you don't know if you've hurt yourself or not. You go through a checklist. Fingers? Toes? Arms? Legs? Everyone's carrying injuries, and you know you're only one bump away from being finished.'\n\nSuch are the impacts of the modern game that players' blood samples after an international have shown cortisol levels equivalent to being involved in a car crash at 30 mph.\n\n*\n\nMike spins the ball from the breakdown of Leigh's tackle to Rhys, who punts it high into the French twenty-two.\n\nYachvili is there again, under the ball, but this time with the eighteen and a half stone of Ianto bearing down on him. Catching the ball with Ianto just metres away, Yachvili shouts, 'Mark!' and like a spell the power is taken from his opponent's run. Slowing before the French scrum-half, Ianto immediately turns his back and walks away from him.\n\nAnd everywhere else on the field, too, the game is powered down by that single word, bringing each player's intention to a sudden halt. If the suspended jumper in the line-out is the game's pause button, then the call of 'Mark' is its reset switch.\n\nYachvili clears the ball into touch, the French defence having soaked up everything Wales can throw at them. And everyone knows, Welsh players and fans, that they still haven't tried to attack. Like Muhammad Ali's 'rope-a-dope', the French are letting the Welsh batter them. But are they just biding their time, or have they really lost the ability to counter against the giants of the Welsh backline?\n\n## 17 mins\n\nAlex cuts into the French line at an angle, making more progress than most, breaking a tackle and tying up four defenders as he scrambles forward. Mike takes on the attack himself but gets caught in the cluster of a maul, the red and blue bodies suddenly collapsing like the opening of a strange flower.\n\n'No boots!' calls Joubert.\n\n## 18 mins\n\nIn the whole game so far Wales have made four tackles. France have made twenty-nine.\n\nWales have completed forty-one passes. France just ten.\n\nWales have had 75 per cent of the territory.\n\nBut France still lead 3\u20130.\n\n## 19 mins\n\nA short move at the front of a French line-out unlocks a flurry of attack. Suddenly, they're off the ropes. The Welsh defence shifts into gear. A charging Servat is taken down by George. Both Sam and Dan chop at the French attackers with sliding sweeps at their ankles. The squad have developed this style with Shaun, and it is stunning in its effect. Throughout this championship Dan, particularly, has made it his trademark, bringing attackers down with such suddenness that they often have no opportunity to offload the ball. Dan, meanwhile, free of his opponent's fallen body, is often straight back on his feet, looking for the next threat to the line or jackal-ing at the breakdown for turnover ball. As Shaun understands, in modern rugby it's more often a case of defence being the best form of attack, rather than the other way round.\n\n## 20 mins\n\nWhen the French are penalised at the breakdown, Rhys clears the ball far upfield into touch. As the team follow for the line-out, he removes his gumshield to give instructions to Adam to pass on to the forwards, and then to the backs lining up outside him. A reserved character off the pitch, when playing at ten both a maturity and a command settle upon Rhys as he attempts, as best he can, to marshal each period of play before it happens.\n\nThe two packs line up in formation, and Matthew, his right forearm bandaged in black, his right bicep patterned in tattoos, lifts the ball above his head, as if in sacrifice. In front of him, beyond the two lines of forwards, the stadium's West Stand rears up from the pitch, a tsunami of red, frozen in motion.\n\nOn either side of the line-out Mike and Yachvili rest their hands on their thighs, watching for the direction of the ball.\n\nA series of calls are shouted from the Welsh forwards as, like the switching cups of a street conjurer, the dummy runners swap positions up and down the line. The ball goes long and Dan, lifted by Adam and Alun Wyn, takes it cleanly to drop it to Mike. Rhys chips through, but the ball is gathered by Palisson, who counter-attacks, breaking Jon's tackle, only to be cut down by Sam. This time, though, he manages to offload to Dusautoir. But, as ever, Dan is there, harvesting the Frenchman's legs from under him.\n\nWhich is when it happens.\n\nWith Dan is Alun Wyn, who on seeing the split second of a chance drives through to scramble the ball from Dusautoir and turn it over. The Welsh front row immediately recognise the opening too. Binding together, Adam, Matthew and Gethin dive in behind Alun Wyn to secure the Welsh position, the voice of the crowd driving them on. Their wave of a roar accompanies Dan as he moves from his tackle to the base of the breakdown and, with Sam shouting and pointing behind him, makes the pass to Rhys.\n\nThe ball is in Rhys's hands for less than a second before he passes it long, missing Ianto, to find Alex, a year on from his first fifteen-a-side game, picking up speed on the wing.\n\nAhead of him is Bonnaire, the French number seven, but as Alex side-steps off his right boot either the rain or Shaun's brother, Billy-Joe, comes into play. The flanker slips on the greasy turf and is left flailing as Alex surges on, outstripping Maestri, who also lunges at the winger's disappearing back. Stepping inside again, Alex passes a final defender, who also slips on the wet pitch. The crowd are on their feet now, screaming, willing him forward. The noise is so loud Alex can no longer hear his own breathing as he sprints over the French line and dives into the turf, sending tufts of grass spinning into the air before him. Immediately, he's back up on his feet, a single thought in his mind: 'Did I just do that?'\n\n## 21 mins\n\nConfirmation that he did comes with the rest of the team and the subs rushing in to celebrate with him. As Dan Baugh hugs and picks him off his feet, the deafening applause and cheers of the crowd fill the stadium. Just two months ago Alex didn't know if he'd be in this team. Like the rest of the squad, he knew it would be a tight game. His job was 'to chase and clatter them, to keep looking for the ball'. Which he has done, and now, as he jogs back up the field, Jamie hugging him as he goes, Wales are leading France 5\u20133 and Alex's life, in a matter of seconds, has changed for ever.\n\nAs his teammates regroup behind the halfway line, Leigh converts Alex's try, and the scoreline, with eighteen minutes of the first half still remaining, stands at Wales 7 \u2013 France 3.\n\n*\n\nThe match is far from won, but Alex's try is still a turning point, proof that Wales's method and the weight of those statistics can come true in the scoreline itself. The relief of the crowd is palpable. The muscles of the stands relax, as if the whole stadium has taken a collective deep breath. It isn't long, however, until they begin to tighten again. The expectation of a win crackles through the spectators. But the chill of possible defeat follows close on its heels.\n\nFor the rest of the half the earlier patterns of the match are repeated. The two sides continue to trade kicks; France's long and searching, Wales's high and testing. Wales, angled across the field for attack, continue to face up to the flat line of the French defence. By the twenty-eighth minute the number of Welsh tackles stands at only thirteen, while those made by France has risen to forty-six.\n\nIn the thirty-first minute the waves of Welsh pressure do, at least, offer up a penalty, which Leigh, with Jenks behind him, converts. The score is Wales 10 \u2013 France 3, and for the first time in the game Wales would not drop behind even if France scored and converted a try. But their opponents would draw level, which today, for fans and players alike, would not be enough.\n\nIn the coaching box Warren and Rob, in their number ones, and Shaun and Rob McBryde, in tracksuits, look on, frowning. In front of them Rhys Long and his computer screen continues to feed them analysis. In training Warren is wary of 'paralysis by analysis': of overloading a player with information before a match. But now, at the heart of the game, Rhys's statistics and measurements, as much as a side-step or a tackle, could win the day. Any messages the coaches have for the team they send via Jenks and Adam when they take on the water. But the time lag, and number of messages, has to be carefully managed. It's no good sending advice applicable to ten minutes ago and, as Rob says, 'As a player the last thing you want is a message every break. It undermines confidence, and then we wouldn't be doing our job.'\n\nDown on the pitch, however, the Welsh team are translating their training to their playing in every phase of play. Rhys turns the big Harinordoquy with a judo throw of a tackle reminiscent of Dan's padded Red Room in the Barn; George, bravely exposing his ribcage, leaps to meet incoming balls as he has done hundreds of times on the Castle pitch, with Rob Howley throwing them higher and higher; the range of Gethin and Adam, the two Welsh props, is testament to Adam Beard's fitness regime, while all the players spider and scramble at the breakdown as they have done thousands of times against tackle bags in training.\n\nIn the thirty-seventh minute, however, an injury to a nerve in Sam's shoulder unleashes a furious French attack. As Matthew throws in for a line-out, Ianto, as if breaking from underwater, surges into the air but fails to get his hands to the ball. As it travels towards the line-out's tail, both Bonnaire and Sam jump to contest for it. As Sam stretches his left arm into the air, however, his right fails to rise. Bonnaire takes the ball and charges forward, triggering a French attack that runs the width of the field.\n\nSam, meanwhile, runs to the first forming ruck, cradling his inert right arm against his chest. He goes to make a tackle, and again his arm doesn't move. At the next break in play he looks up at the clock. Thirty-eight minutes. Two minutes until half-time, and he knows already, in the biggest game of his life and with the match still in the balance, that he will only play for those next two minutes, and then no more.\n\nThe half ends how it began, with a Welsh penalty, kicked by Leigh this time, once more hitting the righthand French post and releasing yet another stadium-wide sigh of frustration.\n\nJoubert blows his whistle and the teams, beneath a deluge of applause from the crowd, jog back to the players' tunnel, turning left and right at its end into their respective changing rooms.\n\nOn the pitch, teams of groundsmen, led by Craig and Lee, begin replacing divots and repairing, where they can, the damage done to their turf.\n\n## Half-Time\n\nFrom his position outside the VIP lift Geraint hears the squad returning to their changing room: the clatter of their studs in the corridor, the sound of heavy bodies sitting, of lungs working against chests.\n\nThe lift opposite him opens, and Warren and the other coaches exit. Geraint pulls at one of the doors behind him and lets them into the changing room, where, for a few minutes, he will hear them say nothing.\n\nWarren likes to give the players time to catch their breath before he talks, to get their drinks and decompress from the crucible of the stadium in which they've just played for the last forty minutes. All the coaches have something to say, but they also have to remain calm. The temptation is to spill out their information immediately, to pass on what they've gleaned from their positions up in the stands behind Rhys Long's computer screen. But they don't. For the next few minutes it is each player's chance to salvage some time for themselves before they return to the pitch. For others it's also time with the medical staff. Dan is lying on one of the treatment beds, his eye bleeding, having his split nose stitched by Prof. John.\n\nSam, meanwhile, has already had it confirmed by Prav and the Prof. that he won't be taking the field again. When Warren learns this, he crosses to Ryan's stall in the substitutes area, set apart from the team, and asks him to step in at seven. Within minutes that private conversation is being broadcast on the BBC. Sam, Wales's captain, will not play the rest of the game. For a second time against France, and for a third time in this championship, injury has taken him from the field.\n\nFor Sam it's a massive personal blow. That boy who _refused to be denied_ when running the streets of Whitchurch still wants to see this through, to make sure Wales put that World Cup semi-final, and this Grand Slam, to bed. But his arm remains motionless at his side, so passing the captaincy to Gethin he accepts, once again, that he'll have to sit this out and watch his team play France.\n\n## The Second Half\n\nWhen the Welsh team emerge from the tunnel, the crowd rise to their feet in welcome. The French are already on the field, bound in a tight circle, with their backs to their opponents' entry. Dan's stitched nose and cut eye are layered in Vaseline. Throughout the first half he repeatedly put his body on the line, cutting down the French attackers as if he were felling trees. By the end of this half, in which Dan will push himself to the point of breakdown, the farmer's boy from Llandrindnod Wells will be named as man of the match. By the end of this week he will be named as player of the tournament.\n\nFive years ago, when he was nineteen and playing in his Heineken Cup debut for the Gwent Dragons, Dan broke his neck. In the wake of a tackle several players fell on him, snapping his head forward. He heard a crack and, after a few moments, as he lay motionless on the pitch in Perpignan, he started to lose the sensation in his arms and legs. 'I remember lying there,' he says, 'thinking, \"Am I going to walk again?\" It was pretty scary.'\n\nThe feeling in his limbs eventually returned, but an X-ray revealed a crushed disc, broken vertebrae and torn ligaments. Dan was told that although the surgeons didn't have to operate, if they didn't then any fall or knock could potentially paralyse him. He went for the operation, more for peace of mind than because he ever thought he'd play rugby again. For the next year, as he recovered, Dan worked back on the family farm. The 2008 Grand Slam was a dark time for him, and for much of the tournament he found it too painful to watch Wales play. He'd barely begun to fulfil his dream of playing for Wales, and yet for a while it looked as if his international career was already over.\n\nA year later, though, and two years to the day after he'd broken his neck, Dan, at twenty-one, ran out at this stadium for Wales against Argentina. The night after the match he wore his Welsh cap throughout the post-match dinner. His mother, meanwhile, who with his father had made those ever-longer drives south, didn't stop crying until the Sunday after the match.\n\nAfter some initial hesitancy in the collision area, Dan was soon back to his younger form, playing with absolute commitment. The psychological healing, though, took longer. When the Dragons played Perpignan two months ago, Dan was stretching his legs on the pitch when he saw the spot where he'd heard that crack in his neck five years earlier. 'I was quite emotional,' he says. 'And I had a lot of demons in my head before the match. I was screaming a lot. I just wanted to get through it. As soon as the final whistle went, I was, like, \"Happy days.\"'\n\n*\n\nCraig Joubert blows his whistle and the second half begins, Rhys drop-kicking the ball above the waiting French pack. Dan and the rest of the Welsh forwards chase in pursuit, still forty minutes away from the final whistle of this match, from Thumper's wedding or funeral.\n\nWales open the half with an ongoing attack, Ryan entering the fray early, and Alex once again breaking the French line. But a different France seems to have emerged from the visitors' changing room. No longer satisfied with just slowing down Wales's game, they now put pace into their own, launching an attack down the left wing in response to another high kick from Leigh. In the final chase for the kicked-through ball Gethin proves again why he is the epitome of the modern prop, sprinting alongside Buttin to gather the ball on the bounce from an offside position. In doing so, while he saves Wales from a French try, he also gives France a penalty.\n\nWhen Beauxis converts that penalty close to the Welsh posts, the scoreline narrows again, to Wales 10 \u2013 France 6. A single try by the French now, even without a conversion, will be enough to spoil the day for Wales. The numbers brought up on the stadium's big screens resonate through its atmosphere. Wales have failed to get away from France and now, with thirty-five minutes still to play, the French are chasing them. The ghost of Wales's own comebacks in this tournament begin to haunt the crowd.\n\nFor the next five minutes this new French team continue to play along the lines of their own 'national' style, probing at the Welsh backline with a series of attacks. For chess players, the French Defence is a playing model known for its solidity, but also for producing a cramped game. In a similar way, France today, though robust in defence, have strangled their game in being so. Now both they and the game open up. Where before the ball was traded in kicks, it's now traded in turnovers.\n\nDan continues to tackle mercilessly, tracking each of his targets for a few seconds before pouncing at their ankles to bring them down. But the effort is starting to show. When the game breaks, Dan remains on the ground on one knee for some time, his chest and back heaving with his laboured breaths.\n\nWales win a penalty fifty-two metres from the French posts, three metres further than the kick Leigh missed against France in the World Cup. Removing his skullcap, Leigh places the ball on the tee once more. As he does, a rendition of 'Hymns and Arias' begins to swell from the stands, serenading him as he steps back to study the posts like a mystery. Dropping his eyes to the ball, his inward gaze transports him back to Gorseinon once more.\n\nAnd we were singing hymns and arias,\n\n'Land of My Fathers', ' _Ar Hyd y Nos_ '.\n\nLeigh strikes the ball clean and true, sending it end over end through the posts. This time he watches its trajectory and, when he knows it's good, allows himself a brief explosion of relief and pleasure, grabbing at the air and shouting in celebration. Wales 13 \u2013 France 6. Wales are by no means clear, but they are safer than before.\n\nYet again Wales have to defend from the restart against waves of French attack. Scott Williams, the saviour of Twickenham, briefly comes on as a blood replacement for Jon. Gethin, Toby and Dan all make more tackles. As the pack scrums down, Ryan crouching into Sam's position at seven, the statistics show the story of the half so far. Despite the score, it isn't pretty reading:\n\n| **Possession** |   \n---|---|---  \nWales \u2013 40% |  | France \u2013 60%  \n|  |   \n| **Territory** |   \nWales \u2013 33% |  | France \u2013 67%\n\nWales choose a closer game, taking route one from rucks and mauls, picking and driving, picking and driving. The crowd rouse into 'Bread of Heaven', their voices willing the players forward. Inch by inch they chisel at the French defence, but France still aren't committing at the rucks, leaving their lines of blue solid on either side.\n\nAlex eventually manages to break through, barging through two defenders in a row. But then, within minutes, the man who gave Wales their try gives France a penalty. Caught in a breakdown as the French attack, Alex is penalised for handling on the ground. Up in the box Warren and the coaches look on anxiously, expecting a French kick at goal to bring them back to within four points of Wales. But as Alex and the rest of the Welsh team drop back for the attempt, France surprise every-one with a kick to Buttin sprinting down the left wing. France want seven points, not three.\n\nButtin, however, is stopped short of the line by Dan. The attack continues, with Gethin and then Jamie putting in hits to slow its advance. The game pauses for a scrum, and for Jon to return from the blood bin. When the scrum finally hits, France are penalised for early engagement, the rhythm of Joubert's instructions catching them off-guard once more. After all Wales's earlier inch-work, nudging upfield, Rhys now, with a single kick, sends the ball deep into the French half and into touch.\n\nThe teams continue to trade attacks and kicks. George rises to meet a high ball. Leigh, Toby and Jamie all make ground with runs into the French line. When France return fire, Joubert's microphone picks up the sound of Dan's tackle on Buttin: a crunching thud of an impact, like a punchbag filled with wood swung hard against a wall.\n\nWhen the play finally halts, players are on their knees all over the pitch. Gethin falls to his back and is treated by Carcass. Jenks, Adam and Prof. John bring on water. There is just over fifteen minutes left, enough for either Wales to secure their lead or for France to come from behind and deliver a defeat.\n\nTaking advantage of this pause in play Wales replace a fifth of their team at once, bringing on Luke Charteris, Ken Owens and Lloyd Williams, son of Brynmor Williams, who also played scrum-half for Wales. Although Lloyd's father never pushed him towards rugby and his old position, once he saw his son had fallen for the game he advised him to refine his passing by watching videos of Rob Jones, who'd played for Wales through the 1980s and 1990s. Lloyd followed his father's advice and, as he takes the field today, he brings with him more than a pinch of an older method of Welsh scrum-half play, not just in his passing, but also in a style more reliant on quick service and sniping, rather than Mike's more physical flanker-type play.\n\nAs Mike, Alun Wyn and Matthew touch hands with their replacements jogging onto the pitch in Cardiff, in New York the opera singer Bryn Terfel, watching in the Red Lion on Bleeker Street, gets to his feet and sings, the pub's wooden floorboards resonating with the depth of his voice. Encouraging the crowd to join him, Bryn leads the early-morning drinkers in a rendition of 'Bread of Heaven', as if even from there, three thousand miles across the ocean from Cardiff, their voices might rouse the team to victory.\n\nBack in the stadium Wales continue to try and find a way through the French defence, while also fending off the increasing pace of their attacks. In the sixty-eighth minute, after a sustained push close to their own line, Wales are finally playing in the French half again. Up in the coaches box Thumper has joined Warren, Rob Howley and Shaun. All three, no longer able to sit and watch, are on their feet. A faster, more urgent pulse of 'Waaales, Waaales, Waaales' washes through the stands as Ken drives through a tackle to make ground and Jamie, the ball held close to his chest, crashes forward.\n\nWales's efforts have brought the figures for the half back towards parity:\n\n| **Possession** |   \n---|---|---  \nWales \u2013 49% |  | France \u2013 51%  \n|  |   \n| **Territory** |   \nWales \u2013 52% |  | France \u2013 48%\n\nBut then, with less than eight minutes of the game remaining, Wales are penalised in their own twenty-two for pulling down a scrum. France are awarded a penalty in front of the Welsh posts. From their own tryline Wales, looking battered and exhausted, watch Yachvili return France to within four points. Wales 13 \u2013 France 9.\n\nWithin a minute of Yachvili kicking that penalty, Wales are awarded one of their own, deep in French territory. As Alex bundled Trinh-Duc into touch, the French player deliberately threw the ball away. So now, with Joubert telling him, 'No need to rush, but no delay either,' Leigh once again finds himself placing the ball onto his kicking tee.\n\nIn the last pause in play Prof. John was treating Leigh for cramp. Just seconds ago he made a break through the French line, spinning through tackles until he was downed by Dusautoir. And now here he is with white-wash on his face, trying to slow his heavy breathing, once more lining up a penalty in the dying minutes of a crucial game against France.\n\nFor Leigh, his teammates and the crowd, memories of that final kick in the World Cup briefly resurface. Wales, only four points in the lead, are still vulnerable. They need him to kick this penalty.\n\nLeigh himself is still nervous. The anxieties he shared with Jenks this morning have remained with him throughout the match. The kick is close to the posts, but also from a similar angle as the one which hit the upright earlier. But Leigh has a job to do, so as another rendition of 'Hymns and Arias' fades away with his steps backwards, he stops, rocks himself steady, eyes the ball, the posts, the ball again, then takes himself, as ever, back to that training pitch in Gorseinon.\n\nThe TV cameras pick out a woman in the crowd. As the other spectators around her stare towards the posts, she alone has turned her back on the kick. Bowing her head and with her chin against her knitted hands, she is praying. Her lips move as, down on the pitch, Leigh begins his slow advance upon the angled ball.\n\nThe sound of his strike is deep and true, the ball's trajectory clean. The linesmen's flags are raised and, as the stadium cheers, its thousands of voices as one, Wales go into the lead by seven points. Wales 16 \u2013 France 9.\n\nThe woman praying in her seat turns back to the bowl, looks at the score and slowly stands, unclasping her hands in thanks.\n\nLeigh, replacing his skullcap, runs back into position, as France, still only a converted try away from a draw, restart.\n\n*\n\nFor the last four minutes of the match the pulsing heartbeat of 'Waaales, Waaales, Waaales' ebbs and flows but never ceases. The stadium begins to rise to its feet in anticipation as Wales kick, run and drive up towards the French line. A line-out is lost but Ryan charges through. Lloyd's head is bandaged and Toby's shirt is ripped at the shoulder. Rhys tries for a drop goal, but it floats wide. The coaches in the box all look on, powerless, arms folded. France are slow to take their twenty-two drop-out, but when they do, Ken secures the ball. The forwards pick and drive, pick and drive, before briefly being driven back. Wales spin it wide, then drive on again. With thirty seconds left on the clock, 'Hymns and Arias' swells from the stands once more:\n\nAnd we were singing hymns and arias,\n\n'Land of My Fathers', ' _Ar Hyd y Nos_ '.\n\nWith ten seconds left, the ball goes loose. Welsh and French players dive upon it, and with just five seconds to go Joubert blows his whistle to penalise France. Jamie and Gethin thrust their fists above their heads. Alex, out on the wing, jumps for joy. Dan collapses to the floor, close to tears. In the box Rob Howley punches the air at hip height, as Thumper applauds between Shaun and Warren, who both keep their hands locked under folded arms. Down on the pitch Ryan points up at the fans in the stands, as Rhys makes one last kick to send the ball spinning into touch. The clock goes red, and the crowd goes wild. Wales, as the Blims and the country had hoped, have won the Grand Slam.\n\n*\n\nImmediately, the Welsh players, all smiling, look younger, the relief of their victory rejuvenating them. Unlike the Grand Slams in 2005 and 2008, this one hadn't been about surprise, but about expectation. About coming true on a promise they'd shown and keeping a promise they'd made.\n\nOn the pitch, between their hugs with each other, the squad shake hands and pat the shoulders of the French players. Welsh flags give birth to dragons throughout the stadium, the crowd itself suddenly active, with fans jumping and applauding. Up in the stands members of that crowd reach across to shake Warren's hand, while Gerald Davies, a winner of three Grand Slams himself and a teammate of Mervyn Davies, wipes at his welling eyes. Down below him on the pitch three more players with three Grand Slams now to their names \u2013 Adam, Gethin and Ryan \u2013 come together to be photographed. In the 1970s it was three backs who achieved that feat. Today, perhaps as a sign of how the game has changed, it is three forwards.\n\nThe players continue to celebrate as the presentation platforms are built between them and the victory banners are erected. From this moment on, the rest of the day and the night will be all about celebration. Later this evening the squad will attend a black-tie dinner at the Hilton, where the efforts of this match will begin to tell upon them. Dan will struggle through the after-dinner speeches, just wanting to go home and lie down on his bed. Alex's back will be giving him pain. Lloyd will have five stitches above his eye. Sam, still drinking Coke, will continue to cradle his arm, while for Leigh the release of those day-long nerves will prove just too much. After the meal, when the others are still drinking and making plans for where to go next, he will already be back in his hotel room, 'exhausted and ill as hell with all those emotions coming down on me'.\n\nFor now, though, the team will celebrate together, here on this pitch, with champagne sprays and victory laps, with medals, interviews and Status Quo's 'Rocking All Over the World' blasting from the stadium's speakers. Ryan's son, Jacob, and his father, Steve, join him, all three generations binding in a hug. Ryan cries a little, then, carrying his son around the field, points up at the stands towards his mother. Adam, seeing Ryan and Jacob, finds he needs a moment alone. He's overwhelmed by the win, but he's thinking of his own daughter Isla too, and how much he'd have liked her to be here. In the middle of all the thousands of faces he finds his wife in the crowd, and they share a private look of gratitude and promise.\n\nBy now Roger Lewis and the coaching staff are down on the pitch too. Dan Baugh is hugging anyone he can find, while Warren walks steadily through the squad and the still-falling ticker tape with his son Bryn beside him. Raising his hands, he applauds the crowd and occasionally allows himself the slightest of smiles. Roger, meanwhile, in an echo of that other walk they took together five years ago in Nantes, is strolling towards the centre of the pitch with Dai Pickering. Both are smiling broadly, everything they'd hoped that first walk would bring made manifest around them.\n\nOnce the squad finally leave the pitch they take their champagne bottles, medals and trophy back into the changing room, where J.R.'s order is, once again, undone by victory. Half undressed, they pose for photographs, as Gethin puts on the winning playlist, leaving Geraint, standing outside their changing-room door, in no doubt as to the result of the match. Breaking into song, the players celebrate not only their win, but also the hard work and the friendships that led them to it, their country which plays and supports like a club, and the feeling it gives them when they do \u2013 of being alive, now, and of being remembered.\n\nBeyond the changing room the stadium itself is beginning to empty, the crowds spilling into a city already fevered with jubilation. At Gate 3, where 'Sweet Child o' Mine' rings out from the bars around Cardiff Athletic Club, everything seems transfigured by the win. In the dying light the boarded Inland Revenue building, the old Queen's Royal Garage, the County Club are all lent a more majestic air by the grandeur of the occasion. Even the strip lights of the NCP car park, burning magnesium white, seem to shine in tribute to what has just happened.\n\nThe tide of people who washed up this entranceway before the match are now flowing back down its incline, thousands of them ebbing from the stadium, taking their shared experience of the last eighty minutes into the city and across the country. Rising above them once more is the statue of Sir Tasker Watkins, his hands behind his back, looking out over the filling streets and alleys. On the plinth on which he stands his words are written on a plaque:\n\nI did what needed doing to help my colleagues and friends and saw more killing in 24 hours than is right for anybody.\n\nFrom that moment onwards I have tried to take a more caring view of my fellow human beings and that of course always included my opponents, whether it be at war, sport or just ordinary life.\n\nSir Tasker's words remain unread by the thousands streaming past him, the spirit of celebration keeping their heads up, not down. All their energies, which for the last two hours have so charged the stadium behind them, are now directed outwards. Like returning travellers rushing down a ship's gangway, they are eager to leave and tell of their adventures. For many this will be the sweetest moment of their country's victory: when what was so powerfully hoped for has been secured, and when the purity of that second when the clock turned red and the whistle was blown is still vibrant within them. Because already, even as they flow onto Westgate Street and St Mary's, it is fading. As the stadium empties and its crowds disperse, so too does the essence of that winning moment. As Yeats says, _Things fall apart, the centre cannot hold_. Within hours the stadium's crowd will be unravelled throughout the city, their celebrations often a perverted inversion of everything which brought about the source of their revelry. Bodies will lie in the streets, too much will be drunk, and the honour and _cymeriad_ \u2013 character \u2013 of Wales's victory will be blurred under beer and kebab boxes.\n\nBut it will still be there, held by the stadium itself and in the collective memory of those who watched and those who played. Like Warren told the team before the match, 'You win a Grand Slam and it's yours. No one can take it away from you.' As the fans crowd at the bars and the players change into their black ties, this is true for both of them. However brief the lap of victory, however diffused in drink Wales's win may become tonight, however unread Sir Tasker's words, they are all still there, and will remain so, quietly resonant at the heart of this evening at the end of this day. \n\n# AUSTRALIA\n\nA winter sunlight washes amber over the Sydney coastline as the Wales bus pulls away from the lawns, cafes and broad sands of Bronte Beach. Inside, the squad sit in their familiar formation, their red hoodies and training tops pressing against the windows on either side. The cuts and grazes of those who played in yesterday's test are still stinging from the salt water of the beach's ocean pool. The swim was brief but, together with the winter chill in the air, enough to have left all of them with a sea tightness to their skin, an ocean cold lingering in the bone. Which is exactly what Adam Beard wanted: a recovery swim for their bodies; a cold dip to promote healing and reduce swelling. But he also wanted it to be more than that: to be a different kind of recovery session, one that would help heal another kind of hurt. Which is why he brought the squad here, to Bronte Beach, to swim in the saltwater pool at its southern tip, to take in the rocky and residential cliffs and to watch the surfers ride the riffs of white-tipped breakers.\n\nThe bus reaches the top of an incline, pauses, then turns right to drive on through the suburbs of Bondi, heading north towards the harbour and the Intercontinental, the team's hotel. As the driver steers with one hand he puts on the tour playlist with his other, a collection of songs compiled from two choices by each player and member of staff. As the boys look out of their windows at the easy-going evening of a Sunday in Sydney, Eric Clapton's 'Wonderful Tonight' begins to play through the bus's speakers.\n\nYesterday, on a pitch marooned in the middle of an Australian Rules stadium in Melbourne, Wales lost their second test against Australia. It was an intense, tight game. As Robbie Deans, the Australia coach, said at the post-match press conference, 'We had to win the game three times, because we lost it twice.' Within four minutes of kick-off George had scored under the posts and Wales were already 7\u20130 up. The Welsh forwards dominated the Australian scrum in what Ken Owens called 'the old-fashioned way'. For the last ten minutes of the game Australia fought desperately against a Welsh lead of 23\u201322. At full time that scoreline remained unchanged. And yet Australia still won, with a penalty kicked at eighty minutes forty-three seconds by Michael Harris, a substitute who'd taken the field just minutes before.\n\nWales, after putting their all into the match, thought they'd finally won a southern scalp on southern soil. They thought, with the clock turned red, that they'd kept the final outcome of the three-test series alive; that perhaps, having beaten Australia at home for the first time in forty-three years, they might yet go on to be the first European team to win a southern-hemisphere test series. But they were wrong. When Wales were penalised for collapsing a maul in the final play of the game, Harris, on to replace Berrick Barnes, stretched his kicking leg, held his nerve and slotted the penalty to win the match. Australia 25 \u2013 Wales 23. It was the most painful of losses, summed up by Rob Howley in a single comment over breakfast this morning: 'It's a cruel sport.'\n\nSo this is the other hurt that Adam hoped the beach swim might help heal: the slow, enduring bruise of loss. The oppressive ache of defeat that has settled on the squad ever since Harris's kick and the final whistle that followed it. In the wake of that whistle Adam Jones stood in the middle of the pitch, looking both devastated and perplexed. Sam also stood motionless, his gumshield protruding from his mouth, his hands on his head. The team's belief was instantly transformed into disbelief. More than one of the players cried in the changing rooms afterwards. They knew they could beat Australia, but for a fourth time in nine months \u2013 at the World Cup play-offs, at home and now twice here \u2013 the eighty minutes of each match had proved them wrong. They'd come south to take the next step in their development, to once again make a little bit of history. But they'd faltered in that step, and now the three-test series was lost, the final match next Saturday an opportunity to salvage pride rather than go for the kill.\n\nThe bus drives on up Carrington Road, the squad quiet and distant, looking out of their windows in private thought. As they do, Eric Clapton sings over them:\n\nAnd then she asks me,\n\n'Do I look alright?'\n\nAnd I say, 'Yes, you look wonderful tonight.'\n\nAlong the length of the bus the players silently begin to sing along with him. Mouthing the lyrics, they watch Sydney pass them by, their faces ghosted in the glass through which they look. No one talks. The bus drives on, the song plays, and the players sing,\n\nAnd then she asks me,\n\n'Do you feel alright?'\n\nAnd I say, 'Yes, I feel wonderful tonight.'\n\n*\n\nJust less than two weeks ago it had all seemed so possible. As the Welsh squad prepared for the first test in Brisbane, the same voltage of belief that had driven them through the Six Nations was still charging the touring party.\n\nWales had arrived in Australia in two stages. While one half of the squad flew on ahead to begin their preparations, the other half stayed on in Wales to play and beat the Barbarians. This meant that, as well as fielding the largest tour party ever \u2013 a total of thirty-four players \u2013 Wales operated for their first week in Australia like a twin-engined airplane, with each engine running at different speeds. As half the squad recovered from jet lag and the Barbarians game, the other half trained. When this half went for recovery, the newly arrived players went for training.\n\nEven without this added complication this tour is already a significant logistical challenge. J.R. has transported sixty-seven boxes of equipment from Cardiff. Thumper had already visited every hotel, training ground and stadium months before the squad arrived. In each hotel they occupy, the Wales camp takes over entire floors to recreate the components of their team room in the Vale. Within hours of arriving, the Wales staff will have utilised ballrooms and conference suites to set up a dining area, treatment room, laundry system, banks of laptops for analysis, leisure area with table tennis and a dart board, a press room and a briefing area complete with projection screen and whiteboards.\n\nThe injection of a touring party into the life of a hotel can make for a strangely integrated yet divided experience. As other guests travelled in the glass lifts of the Brisbane Hilton, their cars rising and falling in the tubular shafts like bubbles in spirit levels, they witnessed the hotel's interior suddenly infected with a rash of red. While the squad had a briefing session before their first test, a Sinatra impersonator warmed up in the lobby below them with a rendition of 'Moon River'. As an aircrew checked in at reception, the uniformed group was strangely echoed by a cluster of players having coffee, all wearing identical kit and compression tights. As holiday shoppers returned laden with branded bags, they were met by the sight of the Wales squad in shorts and flip-flops, setting out for a walk along the river.\n\nWhen the hotel's Tongan night porter realised Toby Faletau was in the same building, he left him a note hoping, just for a moment, to draw him away from the squad's regime:\n\nHi Taulupe Faletau,  \nJust to say hello I am Vili Naupoto (relative from Tonga) working here at the hotel (night auditor) starting at 9.30pm.\n\nIf you have time it will be very appreciated just to say hello in person.\n\nBy the way my sons follow your career very closely and proud.\n\n'Opa atu.\n\nVilipau Naupoto\n\nHowever daunting the logistics and the matches ahead, from the moment Wales landed in Australia their tour was fuelled by a positive sense of expectation and potential. Much of this was imposed by the Australian and Welsh media, but in the two weeks before the Melbourne test it was also genuinely felt at the heart of the touring party itself. As Shaun Edwards said at a press conference, 'Usually at this time of year the boys would be in bits. But they're not, they're in good shape.' At the first full-squad training session at Ballymore, in the north of Brisbane, the tantalising possibility of winning this series, of beating Australia on their home ground was palpable in the atmosphere. The stadium and pitch is used by the Queensland Reds as a training facility, so Wales trained at night, under floodlights, when they could have the ground to themselves.\n\nAs the squad warmed up with Adam Beard in the middle of the field, one of Rhys Long's analysts, Rod, set up his filming and monitoring equipment high at the back of a grandstand. Thumper, meanwhile, walked into the shadows beyond the floodlights' glare to check for less welcome cameras spying on the session. A photograph taken of Wales training earlier in the week had already appeared in one of the papers. 'The buggers always try it on,' Thumper said, before heading off, his satchel across his shoulder, to walk the bounds of the pitch. Accompanied by the metallic tang in the winter air and the croaking of cane toads from the marshes at the ground's edge, he walked beyond the floodlight's island of light and on into the darkness. As he got further from the squad, they became increasingly diminished under the floodlights' glare, until they were no more than a red seam between the green of the pitch and the black night sky; a thin red line moving as one between the posts.\n\nThe connotations of that phrase \u2013 _thin red line_ \u2013 are, perhaps, suitable for a rugby squad on tour. Wales are isolated out here; on unfamiliar territory, without their usual support networks back home. As such, in relation to their alienation, their unity is further emphasised. Most of the squad have purchased pocket Wi-Fi devices, and all of them tweet, Facebook and email regularly. But there's still a sense that all they have out here is each other. That on these pitches of southern soil on the other side of the world the _idea_ of a team is somehow stronger; that like an invasion force the further from home they travel, the more they embody and represent the country in whose name they have arrived.\n\nThere is, though, also something of the Trojan Horse about the character of a touring party's invasion. Everywhere Wales have been they have been welcomed. The Australian Rugby Union have hosted dinners and events for Roger and the other members of the WRU travelling with the squad. The local media have attended the press conferences and have, on the whole, been generous. The players are stopped for autographs and photos. And yet they are all here with a single intention: to cause damage to the national psyche of their hosts. To beat a country which prides itself on sporting prowess on its own doorstep.\n\n*\n\nAt the end of the squad's training in Ballymore the three kickers on the tour \u2013 Leigh Halfpenny, Rhys Priestland and James Hook \u2013 stayed behind to practise. After the explosive defence and attack drills of the previous hour their session was, by comparison, a meditative caesura of calm focus. With just the three of them and Jenks on the pitch, the ground suddenly seemed larger under the floodlights; expansive and inviting. To a regular punch and thud of balls struck and landing, all three of them, working independently, once again went through their routines for place-kicking at goal.\n\nThe kickers are the snipers of a team; the individuals who, if on form, can single-handedly destroy an opposition, chipping away at their scoreline with penalties, drop goals and conversions. Even when not scoring points, a kicker who's found the perfect weight of punt against that day's wind and weather, who can read the spaces on the field will cruelly punish the other team. However hard-fought gaining their territory may have been, a series of kicks at the right time into the right space will have them repeatedly turning and sprinting back down the field, eating up the metres of their endeavour as they do.\n\nJust as a sniper in a military company will stand apart, will train their eye alone, so the kickers in a rugby squad need their solitary moments, quiet codas at the end of a session in which to hone the destructive quality of their boot.\n\nAs the Wales kickers practise at Ballymore, they reveal, with each kick, the attributes of their styles, still visible within the architecture of Jenks's teaching.\n\nAs Leigh settles himself after stepping back from the ball, he stutters the toe of each boot into the turf, giving his right toe one more tap than his left.\n\nFor James, at the same moment, it's a downward brush of the right toe across the left calf.\n\nRhys, however, prefers to keep still, planting the toe of his kicking boot behind him like a horse resting a hoof.\n\nThe ball in front of Leigh is sharply angled on the tee towards the posts. James's is more upright, while Rhys's is the most upright of all.\n\nLeigh's first slow steps towards the ball are like that of a cat stalking its prey. As he begins to tip forward he flicks his eyes to the posts once more, as if looking up at the call of his name.\n\nFor James, it's as if he's descending the stairs, lowering himself with each step, pressing his aquiline features closer to the ball.\n\nFor Rhys, his first steps are like taking a stroll.\n\nAnd then all three of them speed up. All three, in a sudden explosive moment, take their final strides quickly, plant their left foot beside the ball and strike through it with their right, their bodies scissoring with the effort.\n\nFor a split second, as the ball flies from the tee, they are also in the air, suspended, with neither boot touching the turf.\n\nWhen they make contact again, for all three it is with their kicking boot, snapped to the ground like the closing of a predator's jaw.\n\nAnd then, with Jenks watching and commenting, a ball under each arm as he patrols the field, they do it again. And again. And again. With the cane toads croaking in the darkness and the temperature falling, Leigh, James and Rhys kick at those posts from their positions on the pitch, unsatisfied with anything other than the ball bisecting them as cleanly as possible. And then they start again, changing their positions, each man throwing his tee ahead of him like a gambling chip thrown to a casino's baize.\n\n*\n\nThree days later the air of expectation felt at that training session in Ballymore was still tangible as Queensland rugby fans streamed into the Suncorp Stadium in Brisbane for an 8 p.m. kick-off. Welsh supporters took their seats among the Australian crowd with memories of the Grand Slam still fresh in their minds, as if they held a secret of which their hosts were ignorant. Up in the commentary box Gwyn Jones, whose broken neck had brought Dan Baugh to Wales and Rob Howley into his captaincy, was also hoping his debut match as S4C's 'first voice' would be a historic Welsh win down under.\n\nThe day before Gwyn had dropped in on the kickers' practice, the shiver of his injury still evident in his gait as he'd walked alongside the pitch watching their form. A shrewd commentator on the game, Gwyn knows this squad well, and what he saw at that practice session bolstered his hopes for their success. The kickers and coaches all broadcasted the same calm steadiness that had seen them through their Six Nations campaign. Australia, meanwhile, were still fending off home criticism after their 6\u20139 loss to Scotland in hurricane conditions at Newcastle just four days earlier. So as he settled his headphones over his ears, with swallows and moths flying through the beams of the floodlights, Gwyn was confident about the Welsh potential in the match ahead. He was, however, also aware of their weaknesses.\n\nThis would be the first match Rhys had played at ten without the injured Jamie Roberts on his shoulder. The big centre had always been his fall-back position on a flat option throughout the Six Nations, so it remained to be seen what he'd do without him there. Similarly, the two centres who'd been paired in his absence, Jonathan Davies and Ashley Beck, hadn't played top-flight rugby for two months. Despite the Scottish defeat, Australia were still number two in the world, and would be eager to answer their critics in the media. Wales would have to slow Australia down. If they didn't, they'd be sucked into a fast game, and that alone could be enough to spell defeat.\n\nThat night after the match the treatment room of the Wales camp at the Brisbane Hilton resembled a casualty clearing station. Several players lay prone on the treatment beds, while others were on the floor, propped up against the wall. Scott Williams was still in hospital having his mouth stitched after a clash of heads with Tatafu Polota-Nau; George North was receiving treatment for a dead leg; and Leigh was having a large piece of gauze strapped to his back. For many of the Welsh players the match had been the fastest they'd ever experienced. Intense, and with a high ball-in-play time, it had also been relentless. Luke Charteris said he'd wanted 'the pause button pressed' when it was still the first half. Unlike their opponents, who'd just been through the Tri-Nations tournament, many of the Welsh players had had a break from top-flight rugby, and it had showed. At the end of the match Wales looked exhausted. They were also, despite a comeback in the second half, soundly beaten, losing 27\u201319 to their hosts. They'd lost, and they'd deserved to.\n\n*\n\nBut this second defeat they'd just suffered in Melbourne last night was different. Against all the odds and the historical trend of an opening match setting the tone for a series, Wales had come out fighting and, for much of the game, had been on top. So this time the defeat didn't feel like a loss, but more like a win that had been stolen from them. And that's why, as the team bus begins to drop past the botanical gardens and towards their hotel, the beaked dome of the Opera House cresting above the trees beyond, the atmosphere inside is still so leaden. Despite the swim in the ocean pool, and almost twenty-four hours since the final whistle, the hurt is still there, inlaid throughout the squad: in their expressions, their postures and in the heaviness of their tread as they disembark from the bus and file into the lobby of the Intercontinental.\n\n*\n\nTwo months before Wales left for this tour Warren Gatland fell ten feet while cleaning windows at his Waihi beach house in New Zealand. He landed on concrete, shattering his right heel bone and fracturing his left. There was no way he would recover in time, so for once it was a coach, not a player, who was ruled out of a tour through injury. Rob Howley, Warren's assistant coach, stepped up to take over the squad. Although Rob has remained in close contact with Warren since, and despite him working with Wales since 2007, this was a seismic shift in his role. The final decisions \u2013 in selection, policy and training \u2013 were now his. With the change of a single word in his job title \u2013 from assistant coach to caretaker coach \u2013 his own words instantly gathered more weight, and with that weight, consequence.\n\nToday, in the wake of the Melbourne defeat, Rob is clear about what the squad should do. They will stop. For two days they'll pause the tour's relentless routine and briefly disperse. The players need space and perspective to process what happened last night in Melbourne. A loss like that leaves a residue of grief in a squad, an emotion which must be respected and given time to run its course. On Wednesday they will regroup, assess what went wrong and plan their strategy for the final match of the series. But for now, for the next two days, they will no longer have to obey the kit instructions on Thumper's day sheets, the training will pause and all the players, within the limits of their alcohol ban, will be free to do what they want.\n\nThe break is welcomed by the squad. They have only been in Australia for two weeks, but already the twice-recycled tour routine of train, play, travel is starting to feel attritional. Including a midweek game against the ACT Brumbies in Canberra they've now played three matches, passing endlessly from plane to bus to hotel to bus to training pitch to gym to hotel to bus to plane. Predictability and boredom have become daily challenges. Shaun has travelled with a biography of Vladimir Putin under his arm, Rob McBryde with Emma Donoghue's novel _Room_ , while Roger Lewis has been reading Bruce Chatwin's hymn to the outback, _The Songlines_. But for most of the squad, apart from their online lives and computer games, there has been little distraction from the focus of the tour: rugby. The free time they've had so far has been brief and isolated, leaving many of the players looking more disorientated than relaxed. Dividing into smaller groups they've gone shopping or hung around in the hotels drinking coffee, their explosive bodies restless in the muzak of the gilded lobbies. At these times, out of kit, their own clothes bestow their ages upon them once more. Along with those clothes and that age some have also adopted a surly uncertainty, a vulnerability even. 'You have to remember,' Ryan Jones says one day, 'most of us are just ordinary Welsh boys who just happen to have been good at rugby. Then, suddenly, you're playing for Wales, travelling in business class all over the world.'\n\nRyan himself, frustrated by years of 'travelling everywhere but going nowhere', encourages some of the other players to use one of their free days to join him on a walk across the top of Sydney Harbour Bridge. Ianto, Alun Wyn, Adam Jones, Gareth Delve and the squad's sports scientist Ryan Chambers sign up. The following morning the group put on the BridgeClimb jumpsuits and walk through the girders and ladders of the bridge to crest its highest point on a bright afternoon. Several Australians recognise them as they climb. Many commiserate with them for the loss in Melbourne, and more than one even says they hope Wales win the final test on Saturday. 'You was robbed, boys,' one man shouts through the wind as he descends the opposite arc. When they get to the bottom, the woman running the BridgeClimb office turns out to be from Llangynidr. She gives them all an extra photo of their climb, telling them, 'Just beat the Aussies on Saturday.'\n\nOther members of the squad use the time off to visit Bondi and Manley beaches, which is where Thumper heads too, going snorkelling in the shallows. In the evening nearly all the squad go to a rugby-league match, also in Manley, and most take up the invitation to attend a Lady Gaga concert in her _Born This Way_ tour. On the bus on the way to the gig Gethin plays dance tunes from the playlist, while Roger is on the phone to Wales, setting up TV deals and discussing court cases. At the gig itself the squad enter backstage and are impressed, if not animated, by Gaga's show, tapping their feet and nodding their heads as they stoically remain in their seats while the rest of the arena stands up and dances. The following night a smaller party \u2013 Shaun, Ken, Roger and WRU press officer Simon Rimmer \u2013 go to a performance of Dylan Thomas's _Under Milk Wood_ in the Sydney opera house. The next morning over breakfast Ken tells Dan, Sam and others about the show, describing the set, music and how the actress playing Mae-Rose Cottage had drawn red lipstick round her nipples.\n\n*\n\nWhen the squad come together in the team room on Wednesday they are reanimated, fresher and ready to start again. The first briefing of the day will be an assessment and analysis of the match they lost in Melbourne. The atmosphere is akin to a sixth-form common room on exam results day. Music plays \u2013 Stereophonics' 'Local Boy in the Photograph' and Tracy Chapman's 'Fast Car', to which Dan silently sings as he reviews footage of the game's line-outs on a laptop. Then, without warning, and with that osmosis-like communication of the day sheet, the squad gather in the chairs assembled before the projection screen linked up to Rhys Long's laptop. As the players wait for the briefing to begin, Rhys's baby daughter, gigantic on his desktop wallpaper, stares down at the assembled squad from the screen above.\n\nShaun opens proceedings with a review of their defence. Speaking quietly he directs his comments at individuals. Those singled out nod in understanding and agreement with their own shortfalls. Decisions made instinctively at high speed, mid-game, are reviewed, slowed down on the screen and meticulously dissected. 'We're _assuming_ ', Shaun tells the players by way of rounding off, 'the tackles will be made. Don't.'\n\nWarren has been with the tour since Melbourne, but Rob Howley is the head coach now, so, as he has since he arrived, Warren stays quiet as Rob steps up to speak.\n\n'We've had two days to process last Saturday's defeat,' he tells the squad. 'We've seen the sights of Sydney, now it's time to go to work. We've jumped to fourth in the world rankings, and we should be proud of that. But now we need to stay there. We all know the game was ours on Saturday. We lost it because of _individual_ decisions made by players _on their own_. As the coaches we can set the policies, but you have to play _within_ these policies.'\n\nRob asks Sam if he has any words. Sam shakes his head, leaving it to Jenks to address the squad and announce the team. An almost imperceptible tremor of attention passes through the squad. One of Wales's greatest strengths this last year has been the competition for places. Every member, including the senior players, has to compete for his shirt. From this announcement onwards a subtle division will evolve through the following days' preparations, between those chosen to play and be on the bench, and those not. Roles will be allocated accordingly in training, with increased attention for those selected from the medical and conditioning staff. The team, within the squad, will quietly define itself.\n\nJenks reads through the team sheet in his usual quick oscillating rhythms, matter-of-fact, down-to-earth. There is just one change \u2013 the hooker Ken Owens on the bench instead of Richard Hibbard. It was Hibbard who was penalised for collapsing the maul at the end of the match in Melbourne. Nothing else need be said. His punishment is in his omission; in not, on Saturday, being a part of Wales.\n\nBefore the squad disband for training, Warren, for the first time on this tour, stands from his chair and, leaning on his crutch, steps forward to address the players. He tells them just two things. Firstly, if they are to win, then they have to win 'the battle of halfway' first. They have to make sure they play their rugby in the Australian half, not theirs. 'It is', he tells them, 'like a game of chess, a waiting game.' In the first test, he points out, there were just six line breaks made by each side. In the second there were only two. The opportunities are few, he reminds them, so when they happen, make sure they happen 'between the halfway line and _their_ twenty-two, not ours.'\n\nWarren's second point is more incentive than advice. 'There's a big tour next year,' he says dryly, referring to the Lions tour of Australia, for which he's tipped to be the coach. 'So it's time to put down a marker.'\n\n*\n\nThe training after the team meeting, held between the 1930s stands of the North Sydney Oval, is rigorous, energetic and charged with enthusiasm. The squad look fresher for their time off. Having accepted they've lost the series, they're now hungry for a win. A seaplane flies through the clear sky overhead and a flock of ibises stalk the ground as the Welsh players, harried by Rob and Jenks, hammer into tackle bags, practise miss passes and run through their moves. Warren, sitting in a plastic garden chair surrounded by scattered training equipment, looks on, one hand to his chin, thoughtful. Thumper strolls up beside him. Holding his phone, his glasses are still perched on the end of his nose after sending a text. 'There's some bewts,' he says, nodding at James Hook and Mike Phillips. 'Now they're sweating.'\n\n*\n\n'Remember what this feels like.'\n\nThis is what Rob Howley tells the Welsh team as they sit in their changing room at the Allianz Stadium having lost, once again, to Australia, 20\u201319.\n\nIn the second half Wales twice took the lead, only to lose it again in the seventy-fifth minute, when Berrick Barnes kicked a penalty.\n\nRyan, the only Welsh try-scorer, has broken his nose. The oil-spill colours of the bruising are already spreading under his eyes.\n\nThrowing his body into a tackle, Sam received a knock to the head. In the twenty-fifth minute he was being sick on the pitch. In the twenty-eighth he once again had to leave the field and watch his team play on without him.\n\nLeigh kicked fourteen of Wales's points. One of his penalties, though, hit the upright of the post. It was his first miss in sixteen match kicks at goal.\n\nThe game itself was tight and riddled with penalties. Twenty-four in the whole match, with many offside calls made by the linesmen.\n\nAt half-time, just three points adrift of Australia, with the score at 12\u20139, the other figures of the match were still stacked in Wales's favour:\n\n **Possession:** 56 per cent\n\n**Territory:** 59 per cent\n\nMore statistics, however, illustrated how closely the teams were matched:\n\n **Missed tackles:** Wales \u2013 10\n\nAustralia \u2013 10\n\n **Line breaks:** Wales \u2013 1\n\nAustralia \u2013 1\n\nBut as Wales sit in their changing room, their heads bowed, none of these numbers matter to the squad. The only ones that do are those which tell the result of the series: three\u2013nil to Australia.\n\nThe Welsh squad embarked on this tour in the hope of winning either the series itself or at least a match. But as the lines and logos of the Allianz pitch are hosed away by the stadium staff, as the spectators walk back to their cars past creeper-hung trees, the squad know they have done neither. The word they most wanted to avoid is now theirs to own: whitewash. This is what the tour has been, and none of them, players or coaches, will hide from the fact, however much they wish it wasn't true.\n\n*\n\nAt noon the next day the whole squad are having lunch at a seafood restaurant beside the pavilion on Bondi Beach. This afternoon they will fly home. After landing at Heathrow, the team bus will take them to Cardiff, where, after their three weeks together, they will disperse. Last night, however, presented with the prospect of their first long break from rugby and rehab for over eighteen months, the Wales squad celebrated and commiserated together. Pooling the tour's fine money, they bought crates of beer and began the evening drinking and singing at the hotel. Then, boarding the same coach that had taken them to and from the last test match, they went out into the city.\n\nThe teetotallers in the squad remained as such, but many of those who'd been on alcohol bans for months on end finally got to break their fast. This morning several of the squad still haven't been to bed. At this lunch on Bondi Beach, the drinks are still being ordered. When other Sunday diners enter the restaurant, they're met by the sight of over forty large men in red occupying its entire central area. Some take photos, others frown for a moment before a friend or partner reminds them, 'Wasn't there a rugby game on or something?'\n\nAt one table at the far end of the room James Hook is holding court. Standing from his chair he points towards Ianto, sitting at another table. Having got married in Greece during the first week of the tour, Ianto had to forego his honeymoon to join the squad here in Australia. When he'd arrived in Brisbane he was fined twice, once for joining the tour late and again for 'being a shit husband'.\n\nBecause the rest of the squad missed Ianto's wedding, James thinks they should restage the nuptial dinner here and now, in the restaurant at Bondi. Pointing at other members of the squad he apportions their roles. 'You can be father of the bride,' he says, pointing at Ken. Then, pointing to Adam Jones, sitting beside Ianto, 'Bomb's the bride. I'll be best man!'\n\nOne by one the other players oblige, standing and delivering short speeches, thanking the bridesmaids and praising the qualities, or not, of the groom. The coaching staff look on from further up the dining room as a bizarre version of Ianto's wedding unfolds before an all-male congregation in red.\n\nDespite the laughter and the drink, yesterday's defeat is still here, its resonance carried into the restaurant with the squad. Each of the players bears it individually too, both in the stud marks and bruises about their bodies, and as a deeper ache located somewhere under their ribs. But although it's still here, it's also gradually diminishing, being absorbed by the dynamics of the squad, by the recognition that the man next to you has been through what you've been through and is feeling what you're feeling.\n\nThe mark of a team is not just in how they win, but also in how they lose. How they use the memory of that loss as fuel and as knowledge. And how they use it to come closer, rather than fall apart. For Wales, at the end of this tour and a long run of rugby stretching back to those pre-World Cup sessions in Poland, this is how they are doing that today. Diffusing Thumper's funereal air of defeat by sharing in this wedding game, played out in a restaurant in Bondi as an Australian sun catches the waves on which the surfers ride, fall and swim back into, pushing themselves against the oncoming tide, to ride them again.\n\n# ACKNOWLEDGEMENTS\n\nThis book would not have been possible without the Arts Council Wales\/WRU Artist in Residence scheme, and I am particularly grateful to Dai Smith and Roger Lewis for having the vision to create such a project with the aim of bringing the cultures of sport and the arts closer together in Wales. Over the last year I have been fortunate enough to spend time with both the national squad and several other teams across Wales. I would like to thank all the players and coaches who have granted me interviews and allowed me such a privileged insight into Welsh rugby and their lives. I am especially grateful to the players and coaches of the Wales tour to Australia for allowing me into their camp with such openness, and to Thumper for assimilating me into the logistical details of the tour.\n\nAs well as my first-hand research, several books have been invaluable in providing a broader context for _Calon_ , including: _Fields of Praise: The Official History of the Welsh Rugby Union 1881\u20131981_ by Dai Smith and Gareth Williams (Cardiff University Press, 1980); _A Game for Hooligans: The History of Rugby Union_ by Huw Richards (Mainstream, 2006); _The Welsh Grand Slam 2012_ by Paul Rees (Mainstream, 2012); _Library of Wales, Sport_ edited by Gareth Williams (Parthian, 2007); _Rugby's Strangest Matches_ by John Griffiths (Robson Books for Past Times); _Life at Number 10_ by Neil Jenkins with Paul Rees (Mainstream, 1998); _Number Nine Dream_ by Rob Howley with Graham Clutton (Mainstream, 1999).\n\nI am also grateful to the following estates, authors and publishers for permission to quote from the following material: Sheenagh Pugh from her poem 'Toast', ( _The Beautiful Lie_ , Seren, 2002); Faber & Faber from Ted Hughes's letter ( _The Letters of Ted Hughes_ edited by Christopher Reid, Faber & Faber, 2007), from Seamus Heaney's 'Postscript' _(The Spirit Level,_ Faber & Faber, 1996) and from T. S. Eliot's _The Four Quartets;_ the estate of R. S. Thomas for quotations from 'A Peasant' and 'Song at the Year's Turning' (\u00a9 Kunjana Thomas, 2001); The Blims for quotations from 'Sidesteps and Sideburns' (www.theblims.co.uk); V2 Music for quotations from Stereophonics' 'Is Yesterday, Tomorrow, Today?'; and Universal Music for quotations from Eric Clapton's 'Wonderful Tonight' and Anthrax's 'Refuse to Be Denied'.\n\nI would like to thank, as ever, my agent, Zoe Waldie, and my editor at Faber, Lee Brackstone, both of whom have supported me now for over a decade. I'm also grateful to Anne Owen for overseeing a tight production schedule with such understanding, and to Ian Bahrami for his quick and thorough eye. Lastly, thanks to Ryan for letting me off when I lost at table tennis, and to Katherine Eluned, for the listening, the reading, the advice and for being there. \n\n# About the Author\n\nOwen Sheers is the author of two poetry collections, _The Blue Book_ and _Skirrid Hill_ (winner of the Somerset Maugham Award). His first novel, _Resistance_ , has been translated into ten languages and was made into a film in 2011. Owen's plays include National Theatre of Wales's _The Passion_ and _The Two Worlds of Charlie F_ , which won the 2012 Amnesty Freedom of Expression Award. Brought up in Abergavenny, Owen played at scrum half for Pontypool Colts, Blaine Youth, Gwent u17\/18, New College Oxford and the University of East Anglia 1st XV.\n\n_Calon_ was written under the Arts Council Wales\/WRU Artist-in-Residence scheme. \n\n# _By the Same Author_\n\n_Non-fiction_\n\nTHE DUST DIARIES\n\n_Poetry_\n\nTHE BLUE BOOK\n\nSKIRRID HILL\n\nA POET'S GUIDE TO BRITAIN (ED.)\n\n_Fiction_\n\nRESISTANCE\n\nWHITE RAVENS\n\nTHE GOSPEL OF US\n\n_Plays_\n\nTHE PASSION\n\nTHE TWO WORLDS OF CHARLIE F\n\n# Copyright\n\nFirst published in 2013  \nby Faber and Faber Ltd  \nBloomsbury House  \n74\u201377 Great Russell Street  \nLondon WC1B 3DA  \nThis ebook edition first published in 2013\n\nAll rights reserved\n\n\u00a9 Owen Sheers, 2013\n\nThe moral right of Owen Sheers to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988\n\nThis ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly\n\nISBN 978\u20130\u2013571\u201329731\u20133\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nThank you for downloading this Atria Books eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Atria Books and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\nDear Mom,  \nYou who loves to talk, I mean, HONESTLY, WOMAN, without pause! Thank you for listening to pages upon pages\u2014even chapters\u2014with no regard to the hour or task at hand. I could not have accomplished this without your support. You are why God created moms. Thanks for being mine.  \nLove you!\n\"There's only one very good life and that's the life you\n\nknow you want and you make it yourself.\"\n\n\u2014DIANA VREELAND\n\n## Author's Note\n\nI LOVE TO READ and get lost in a moment not of my own making. It has been that way since I was a child in Alabama begging my mom to let me stay up past my bedtime, just long enough to finish the next chapter (or two) of whatever book had so captivated me. The escapism that others fostered also fed my soul and imagination. For that I am eternally grateful, because those authors opened up a world to me that showcased ways of existence far different from the one I knew, that of an only child of a single woman whose great love was not so great\u2014so life for me was not always crystal.\n\nI say all of this to tell you that I have an appreciation for great writers and storytellers, my favorites being Kundera, Z. Smith, Hurston, and Coehlo, but I make absolutely NO pretense of being an ounce of the writers they are. I do hope, however, to be an intriguing storyteller and provide you a good tale to get lost in. A friend once said to me, \"You speak like a novel.\"\n\nI hope that he is correct (thank you, Greg Cham) and that you will embrace what is to come. Welcome to the world of Jules Sinclair.\n\nSincerest gratitude,\n\nx Tamara\n\n## 1\n\n* * *\n\n## THE RETURN\n\n2001\n\nAWAITING ME AT baggage claim was a bespectacled middle-aged man positioned near the base of the escalator with Jules Sinclair written in a haphazard black script on a white board. He didn't appear too excited to be there, which was perfect for me as I am unclear of what awaits me when I get home, so any forced pleasantries at this point would be exhausting.\n\nBefore leaving for London, Marcus and I had the kind of argument where far too many things are said, some of which you just can't take back and others of which you say intentionally to hurt the other because you want them to feel as bad as you do at that moment. Instead of clearing my head, the time away only made two things obvious to me: I didn't want out, and I had no idea of how to get back in\u2014to us. Besides, even if I did have the solution, I'm not sure he'd want to hear it anyway. The answers that I hoped would be miraculously delivered, like how to retract all of the pain and harmful words, eluded me. We had come so far only to end up exactly where we should not be.\n\nOn the plane I ran a couple of scenarios in my head so that, no matter the outcome of putting my key in the door, I would be prepared. Option A: He would be there, relieved to see me and in that lovable yet obtuse manner feed me a line to let me know that we would weather this. Option B: He would be out but not gone\u2014his clothes remaining perfectly spaced on his side of the closet with a few items subletting space on my side. Seeing this I would just wait for him to come back, to come home to me. Option C: A \"Dear Jules\" note would be on the counter in his meticulous penmanship telling me that it was best if we had some time apart and that he had taken up residence elsewhere\u2014or, worse yet, that he was gone, never to return. While I know that C could very easily be my reality, I couldn't bring myself to consider it without bursting into tears\u2014again. Maybe things would have been different\u2014for the better of us, I mean\u2014if he had told me about the baby, or if I'd never made him feel like he couldn't. Maybe not.\n\nTaking a deep breath, I bring my thoughts current, focusing only on what is before me\u2014identifying my luggage and getting to the car.\n\n\"We're going to Beverly Hills, yes?\" asks the driver.\n\n\"Yes, One North Wetherly.\"\n\n\"Do you have any objections to taking La Cienega down?\"\n\n\"Any way you want is fine\u2014all goes to the same place,\" I say.\n\nNormally the drive to West Hollywood from LAX via the main street of La Cienega takes an eternity at this hour of the day, the 10 Freeway is even worse, so I settle into the black leather seats of the town car and allow my thoughts to be lulled into a void. You don't have to live in Los Angeles to know that rush hour traffic here is a terror in and of itself. People enter their cars pissed off at the workday that just ended, armored for the confrontation that awaits them as a result of listless drivers who seem resigned to repeat the habitual routine that has become \"adult life\": morning commute 8:00 to 8:45 a.m., sitting in the parking garage of your office praying that today will be better than yesterday 8:45 to 8:50 a.m., pasting a smile on your face at 8:55 a.m., \"Hello, Bob,\" \"Hi, Shirley,\" working someone else's job from 9:00 a.m. to 5:00 p.m., getting in car and dropping the facade at 5:05 p.m., evening commute until 6:30 p.m.\u2014welcome home.\n\nWhat a life! I could never survive it, and yet the infinitesimal details make mine not so different. If anything my wrapping at this point just appears more glamorous and above it all; monthly travel to our homes in New York, Los Angeles, and Bridgehampton, filled with dinners and events that oftentimes end up chronicled in Page Six or photographed for the monthlies. What the rags don't catch are the quiet moments and everywoman experiences that no amount of Executive Platinum frequent-flyer miles, chauffeured cars, or exclusive access will allow me to escape. Basically, as a child of the '80s, I drank the Kool-Aid\u2014with a heaping dose of fairy dust for good measure, as it were\u2014hence I am woman hear me roar awaiting my knight in shining armor, that is, if I can, once and for all, put my BS aside long enough to rest in all that comes with the reality of us. Which clearly I had not done despite past misfortunes. Try admitting that about yourself and not cringing! To my mind's eye the details of how would always fall into place miraculously, because they always had for me with little or no effort. That is, until I ran full speed into the wall that said, \"You can't get what you are not so take a good look princess and be truthful about who is looking back.\" A frightening proposition any way I slice it, much more involved than maintaining the size 6, five-feet-eight-inch Jamaican-Dutch genetic lotto pool I was fortunate enough to be born into. In the event I ever went north of my regular 135 pounds, the solution was simple: a few spinning classes at Equinox, a horrid regime of salt-free\/fat-free\/taste-free food, hot yoga, and complete avoidance of the dessert menu for a week or more. Simple enough, right? But this navigating life\u2014getting personal purpose right more times than I fuck up and being brave enough to admit when I do without feeling diminished\u2014I don't know.\n\nIn all the commotion of exiting the plane, getting through customs, and worrying that my luggage (like my relationship) might not have made the same flight that I did, I forgot that daylight saving time had ended, so it was completely dark outside. Nightfall with the ever-present charcoal smog overcast was firmly set by 5 p.m., which now seemed fitting. As a child, I could always think better at night. I used to believe that the constellations appeared visible only to hear me talk about the grand life that was to come. Tonight, however, there was not a single solitary star to be found in the sky; there seldom are, actually, which is one of the sacrifices of living in Lala but I would gladly do without the galaxy, if he will have me. My hope is that Marcus and I, at the very least, are in the grasp of the half-moon revealing itself amid the smog, so I surrender, lean back and reminisce, allowing my thoughts to return to years prior until my future reveals itself.\n\n## 2\n\n* * *\n\n## MEETING MR. MICHAEL THURMOND KIPPS?\n\nTHE YEAR WAS 1998 and I was trying to return to New York full-time from a stint (forced sabbatical . . . okay, relationship exile) abroad when a friend suggested that I contact Michael Kipps as soon as my feet touched U.S. soil again. The intent being to discuss helming the PR department of his restaurant, Carly's, that was now all the rave in the ultrahip yet still seedy Meatpacking District. So I did as instructed sort of by preemptively e-mailing him from London, thinking it best to develop some kind of cheeky communication before actually begging for the position.\n\nMichael,\n\nJules Sinclair here. Blake says that you are utterly lost without me. Luckily I am bored with everything\u2014my life, the Brits, warm beer, peas and mash, etc. I hear you are divine, more important I am told that Carly's is the hottest upstart the NYC scene has encountered in some time. Let's make sure it stays that way. When and where?\n\nWarmest, Jules S.\n\nTo my surprise his response was immediate albeit short and completely devoid of the witticism that mine had been carefully crafted with.\n\nThursday at 8pm. \u2014MK\n\nThere was no inquiry on his part as to when I was scheduled to arrive in New York or an overture to convey that he understood flying across the Atlantic on three days' notice might be an imposition\u2014but it would be great if possible because he had heard good things about me. No, \"Thursday at 8pm\" is all he said, and so around 10:30 p.m. GMT on a Sunday night I went online to book my travel and pray to the credit card gods.\n\n\u2022 \u2022 \u2022\n\nIt has been said many times over that London is like New York because it has a broad array of cultures and ethnicities inclusive of new money, old money, and no money, all forced to coexist on metros and deckers. Also, it's an epicenter of style and is more than a one-industry town, consisting of banking, publishing, interior design, art, theater, music, etc.\u2014but they are completely wrong. To me New York is a hyperkinetic fishbowl anomaly of possibilities in and of itself; every other global metropolis is but an Off-Broadway production, nicely reviewed but lacking megawatt star power. Emotionally it conjures the innate fight-or-flight impulse for newbies. A rite of passage for all who aspire to re-create themselves on its dense littered streets, as it was for me a few years ago. Everything about New York is a complete affront to the senses, from being sandwiched on a subway multiple times a day and being inappropriately groped, to habitually stepping over trash and being heckled by street vendors, not to mention keeping the requisite pushcart near the door to transport laundry and groceries with no thought to how unnatural it is to be pushing a caged contraption in five-inch heels on any given day.\n\nNew York is hard on the body (kind of like sparring a few rounds with Tyson but coming out of it with both ears intact) and potentially mind-altering, but you don't realize it until you leave the island behind and see how life looks on the other side: minivans and carpools, chore lists, restaurants that close at 10 p.m., planned subdivisions with oversize, themed shopping malls that could hold Yankee and Giants stadiums collectively with room to spare. But no matter how damaging, the city's lure to rewrite a life, the addiction to \"anything is possible if your game is strong enough\" is swift. Either you do whatever is necessary to maintain the high or you nearly OD and in a fitful moment of clarity decide to go clean\u2014leaving it forever and relegating the experience to a \"when I was younger I lived in New York\" story to be dusted off for Middle American friends and relatives as either a badge of honor or a cautionary tale to a wayward child.\n\n\u2022 \u2022 \u2022\n\nTouching down at JFK before midnight on Wednesday I immediately feel that zing of energy coursing through my veins. The interior of the Virgin Atlantic concourse was sparse with some travelers awaiting takeoff, which made it easy for me to be one of the first people at baggage claim, and since I had used some miles to upgrade to business class, my luggage was certain to be labeled \"priority\" and descend the conveyer first so I could get a jump on the taxi line before the rush. Luggage in hand I impressed myself exiting the terminal, maneuvering the gypsy cabs who, by the way, are the furthest thing from being actual Gypsies. They are more like West Indian, Caribbean, Russian, and Middle Eastern entrepreneurs with a dream of capitalism, a car, and hopefully a legal driver's license. My first reaction was to put on my best I am not a tourist so don't fuck with me face\u2014that is, until I saw that the line of passengers awaiting a taxi in the center partition was wrapped in rows of three. No time like the present to double back and give my sweetest soooo about that ride into the city about-face. I was not going to wait twenty minutes or more in 30-degree weather at this hour, even if when compared with London it was warm enough for me to put on some knickers and sunblock and walk to the city.\n\nGreeted by the smell of Hai Karate musk mixed with a distinct middle note of curry, I settled in the backseat and waited for the twilight moment when we would be submerged in the Midtown Tunnel and ascend into the hub of Manhattan. That moment never disappoints. Well illuminated and sleek, the tunnel is a quiet conveyer that ends with the bustle of delivery trucks blocking roads with produce, shopkeepers preparing for the next day's business, and a few late-night joggers, which I've never understood. Honestly who needs to run that badly (my friend Richard)? I knew before we exited Murray Hill that no matter the outcome of my meeting with Michael, I needed to be back in New York. Nothing awaited me in London. It had served its purpose. Two years earlier I moved there to do in-house publicity for a new four-star hotel group\u2014and to put some distance in a relationship. Like most things, it was exciting initially sans the whole emotional despair shattered dreams broken heart of things. During the week, I worked hard\u2014put on my corporate suit (Chlo\u00e9 by Stella McCartney, of course) and sensible shoes (Alaia or Blahnik). At night, I combined work and even harder play in some of the best private clubs, Soho House and Opium being preferred locations. The weekends, however, I reserved for dropping the \"king of the world\" pretense and falling down.\n\nThe ironic and insufferable thing about a fast pace is that it is the biggest illusion of all. You find yourself so wrapped up in this obligation or that social commitment that you begin to believe your own hype. Mine was that I was living the fullest life possible in the present with no concern for past misgivings despite the obvious (weekends falling apart, h-e-l-l-o?!). Slowly, as the UK scene became habitual, the regimen of my life started to emerge and I found it hard to deny that my past was very much a constant companion in my present. Yes, this is the moment when, if my life were a film score, the music supervisor would cue \"Run to You\" by Whitney Houston. Instead of running to someone, I was running from someone sans the flesh-colored unitard and wind machine. Each day or so I thought There has to be more than this. Tomorrow will be better, and I would wait as patiently as possible for the moment to pass. But like the constantly changing forecast of any given English day\u2014overcast in the morning, sunshine by lunch, only to be disrupted by showers during the evening commute\u2014it just recycled.\n\nSometimes those moments went easily; other times they decided to battle. And let me tell you they played dirty. Along with their incessant noise reminding me of all that was lacking in my life, they would bring gray skies and make the heavens cry with such intensity that my only refuge was to take to my bed. Thankfully those moments would give way to a classic black-and-white with Ronald Coleman and Greer Garson, William Powell and Myrna Loy, or Cary Grant and Irene Dunne. And there you have it\u2014my life is not a music score but a classic Hollywood movie. The problem is that I live in a modern world and experience has made it clear that none of those archetypes exist anymore with the possible exception of my dad, Charles. There is no dashing man full of ambition with amazing style and comedic wit who is challenged in the best of ways by the paradox of my drive and vulnerability. So why can't I just accept this fully instead of holding on to a modicum of hope that he is out there, looking for me as well? I thought of this incessantly as we drove to the hotel until the friction to the car from the cobblestone SoHo streets interrupted my thoughts, forcing me out of my haze.\n\nTwo years had gone by since last I was here so I decided to treat myself and stay at the Mercer Hotel in order to be near my favorite haunts: the eats at Caf\u00e9 Habana, Balthazar, and Raoul's, the boutiques, and the friends. At check-in the impossibly perfect supermodel-in-training receptionist told me that not only would the temperature reach the mid-50s but that my junior suite was ready due to a cancellation. Initially when I made the reservation I was advised that no juniors were immediately available but would be later in my stay if I wanted to change rooms at that time. This trip was already starting off much better than expected. Before going to bed, I unpack and shower, after taking a last glance at some notes on Michael in preparation for our meeting later this evening.\n\nThe time difference caused my mind to wake up much earlier than my body desired. Instinctively I reached for the phone to order room service before remembering that the toniest of breakfast experiences was just down the street sans 18 percent delivery charge. Independent of what anyone will tell you or what you may read, Balthazar (noise level excluded) is by far the hippest enclave in the city to observe, be observed, and eat great food any time of the day.\n\nExiting the hotel I stop on the corner of Prince and Broadway to ogle a stylish day-planner in the window of Kate's Paperie before being accosted by two gusts of arctic wind that chills me to the bone. The first being the actual temperature\u2014mid-50s my ass, more like low 20s without the wind-chill factor. Mental note: After breakfast come back to hotel and put on anything in my luggage that suggests warmth; cashmere socks, wool tights, and ear covers, despite the fact that I am vehemently opposed to the latter due to chronic hair issues. The second of course was Cora's name flashing across the screen of my mobile. What is the deal with mothers and their incredibly poor timing anyway? Knowing all too well that any conversation with her this early would only bring an unnecessary amount of drama and derailment to my beautifully appointed selfish morning, I ignore the call. Allowing it to go to voice mail instead of pressing the ignore button, which never went unnoticed or unchallenged by her. Snapping back into reality, I continue on my way. I am not here to obsess on the Doppler radar system, Cora's intrusive manner, or even my own gastronomic pursuits. I am here to impress Michael Kipps and get this position. Then and only then can I justify leaving London professionally and afford to move back to the States without depleting my savings (again) in the process.\n\n\u2022 \u2022 \u2022\n\nI did some homework on Michael and learned from mutual friends that he was a former model in the late '70s through early '80s (actually he was a major deal) who married very well (nice retirement plan if you can get it), enjoys the jet-set life and the spotlight that comes with it, so it was no surprise that he would go into the restaurant business. The shock to many was that he was fucking great at it\u2014front of the house and behind the scenes. So much so that the storied ego of his that was rumored to greet acquaintances before he did worked to his advantage now. Men and women loved to be in his presence and he was an amazing host. In the day-to-day operations he proved to be a skilled businessman, combining all that he experienced from fashion, society life, and dining at the best establishments into the secret of his success.\n\nCarly's was named after his wife and partially funded by her, an older generational wealthy blue-blood type. Carly Spencer Falles, in her heyday, was the standard by which all affairs were deemed a success or failure in New York. Success, if she decided to attend and bring with her a legion of trendsetters known as The Circle. Carly was a one-woman branding machine by virtue of her connections and influence. When she and Michael began dating in '89, most people were amused. Age-wise she preceded him by more than a decade. Anyone with eyes could understand why she would want him. The confusion was in his choosing her.\n\nMichael Kipps, at that time in his early forties, remained a divinely handsome man by any standard and, while not having discovered the fountain of youth, seemed to just get better with age. At six feet two, he was blessed with a deep olive complexion that even in the coldest of winter appeared to be sunkissed thanks to his Greek mother, dark wavy hair loosely sprinkled with silver, and a bountiful mouth set against a jawline chiseled by Michelangelo. His dating history read like a prepubescent boy's wish list of the world's most beautiful women, and by all accounts he was financially sound. Knowing these truths, to say that people were surprised by his courtship and subsequent marriage to Carly is a well-documented understatement. To quote a columnist from The Village Voice: \"For a man who has dated some of the greatest beauties in the world the new paramour of Michael Kipps doesn't seem to share the same looking glass.\"\n\nTrue, Carly was not a conventional beauty at any stage\u2014even she knew that. There was no late blooming period where her head would grow enough to minimize those elfin ears or make her nose appear less beakish. Forever lean and tall she kept her flaxen hair long and ethereal as a focal point, since the Nordic thing seemed to be a prerequisite for class distinction in Upper East Side (UES) society for any era. From what I was able to uncover, their public courtship was very brief, about seven months, which only added to the speculation that the marriage would dissolve within three years. She was infamously quoted saying that she felt like \"Linda Ronstadt and Michael is my Aaron Neville, together we make beautiful music.\" Yes, their first official dance as Mr. and Mrs. Michael Thurmond Kipps was to \"All My Life,\" so with all of this for evidence could you really blame the naysayers and skeptics?\n\nNow, nearly a decade later, with a young daughter (the ultimate wedding gift from one of Michael's exes), and having conquered New York on their terms, it seems that they have turned the skeptics into believers as demonstrated by Carly's best-selling book, Making It Work: A Guide to Lasting Love After 60. Business-wise Michael had effectively birthed an ultra-chic supper club that didn't feel gimmicky but wonderfully consistent, thanks to a world-class culinary team and marquee live performances. Now in its fourth year, the restaurant has to contend with competitors entering the neighborhood. This is where I come in. In order to remain ahead of the pack, Carly's needed to transition from being a novel idea into an institution that is the very essence of New York culture.\n\nOn the plane I reviewed the prospectus thoroughly as well as trend reports regarding consumer dining and spending behavior. So much of New York's business depends on a heavy amount of tourism dollars during peak seasons and the cool factor for locals during off-season, especially restaurants. The big question before me was how does one remain all things to all people while maintaining integrity? Excluding any biblical implications, of course. Actually, if I could answer that I would be so much further ahead in my life than twenty-eight years of age, relationship averse, hiding in my career, emotionally restless, and looking for the release hatch on my life, not necessarily to jump but just to see the abyss and contemplate what if?\u2014but I digress. I had some concrete ideas to make Carly's iconic but didn't want to get too attached to a specific pitch until I saw Michael in the flesh.\n\nAs I saw it, going into the interview I had the upper hand, so much information was available on Michael documenting his life, career, and loves. He, however, had little on me. For as much as I am a social ing\u00e9nue to some, I work very hard to keep my profile quite low, both professionally and personally. If you needed to know me, you did, and vice versa.\n\n## 3\n\n* * *\n\n## IF IT LOOKS LIKE A DUCK\n\nFOLLOWING A QUICK stopover at the hotel for those warmer clothes, I started walking to Gramercy Park to get reacquainted with my city. So much had changed, namely Bleecker Street was starting to show serious signs of gentrification with the addition of J.Crew on the corner of Seventh Avenue South. Thank goodness the falafel place, whose name I never committed to memory, next to The Peanut Butter Factory on Sullivan, was still open. I would stop by on my way back for a quick lunch. I wonder if that great vinyl music record store was still there? It wasn't exactly on Bleecker but on one of the side streets whose name I never bothered to memorize. No matter, in the next few days that I am here, I'm sure to stumble across it or a better one.\n\nSo lost in my thoughts was I that by the time I got my bearings I was in Union Square standing in line to buy zucchini bread from a vendor at the farmer's market. Hands down the best in the world! Previously, when I lived here and worked at Atkins & Klein doing fashion event\u2013related press, a coworker who was all about buying local and homegrown gave me a loaf. After that I was hooked. Since moving, however, I had not thought about how delicious it was to pop a slice in the oven and enjoy it with a cold glass of milk in the morning before hopping the A or C to my office in Midtown. Today, however, unless I was going to run across the street to The Coffee Shop, there would be no milk at this moment and since the divine zucchini loaf waits for no one I removed a glove exposing my right hand to the frigid elements, all for a delicious taste of memory lane.\n\n\"And you're not even going to offer me a bite? I see nothing has changed,\" he said.\n\nI'm certain that it was not the absence of milk that made this particular bread bite feel exceptionally larger than the previous and lodge in my throat, but the recognition of who owned that voice, Anthony Mason, alias \"my heart's greatest despair and UK decision.\" Quick invisible prayer and blessing to the Virgin Mary and Saint Peter before turning around. \"Depends on whether it is the first bite or the last,\" I say, impressed that neither the jet lag nor our history had delayed my retort. Damn it, I really should have left my hair out\u2014hate earmuffs.\n\nThank you, God, I'll never ever doubt you again! How I have dreamed about this moment and here it is\u2014you know the moment I am talking about, the one where after substantial time you see the man you gave your entire innocent heart to, only to have him completely twist your mind into a riddle of the most epic proportions and the only reprieve after he rips it from your chest and places it in a lockbox to which there is only one key is to put an ocean between you. Well, this is that moment, and I am pleased to report that my legs remain solid and I look as great as one can when wearing fifty layers and swathed like an Eskimo baby.\n\n\"Why can't it be both?\" asked Tony.\n\n\"Good-bye, An-TONY!\" I seethed.\n\nFuck no. He is doing it again, the riddles. And to believe that I once thought they were cute? He was never able to give me a straight answer then, so why would I hope to get one now? First bite or last? Beginning or end? Love me or not? Slept with her or did not? And yes, oral does count!\n\nImmediately spinning on my heels to leave in the opposite direction, he blocks me. I truly believe it was in this moment that I took temporary leave of my senses, all of them because he touched me, I smelled him, and when I went to object I looked up into the warm hazel with hints of emerald flecks that are his eyes. And just like that my heart, which had flatlined, started to beat uncontrollably and I was lost in 1995 again\u2014when we first met.\n\n## 4\n\n* * *\n\n## THE MAGIC OF RAIN\n\nNew York, 1995\n\nJULESY, DON'T TELL me you are still in bed. Angie and I are at Coffee Shop waiting on you. Snap it into gear, the movie starts in an hour. Oh, bring an umbrella,\" said Blake.\n\nShit, I completely overslept, technically not my fault. Every self-respecting Manhattanite knows that Friday is not a \"going out party night.\" It's the night exclusively reserved for the 9-to-5 Bridge & Tunnel (B&T) crowd to overtake the city in celebration of a completed workweek. And yet that is exactly what I was doing last night, playing wing-woman to Blake for her date with some dreadful Wall Streeter\u2014what is the deal with those guys anyway? Dreadfully immoral pompous bloodsuckers they are. I could never!\u2014who in an effort to impress her made sure our glasses were never empty. I in return felt it was my duty to at least try to reach the bottom of the glass, which is why I am desperately attempting to pull it together now and failing. My tipping point from buzzed to drunk must have been the two glasses of sambuca after being besieged at Cipriani's by a village of gelled hair and stretch-charmeuse-clad Ginos, Vinnies, and Ginas otherwise known as B&T.\n\n\"Ugh\u2014stop shouting. I hear you. Eat without me. I'll meet you guys at the theater.\" With my eyes closed, my hand clumsily returned the phone to its cradle and began searching the floor blindly for the water bottle resting somewhere in my bag from yesterday's gym excursion; an apt description since I have an aversion to sweating and will stop any activity (well, almost any) at the onset of perspiration.\n\nEarlier in the week I made plans to grab lunch and see Crimson Tide with Blake and Angie not expecting to have tied one on the night before. To make matters worse, spring was officially here, which meant rain, rain, glorious \"fuck up my hair and slip on my wellies\" rain. My only hope in salvaging the day was to see Denzel Washington shirtless or an extreme close-up on his \"I am a serious actor and not really trying to be sexy but you want this\" lip pout. Union Square being central to everyone, we decided to see it at the little theater across from ABC Carpet. Late for lunch but early for the movie, I bought tickets for the girls as a peace offering. It would be nice if the sun decided to make an appearance by the time we come back out, but conditions are slim.\n\n\"My God, that man is fine. Tell me about it. I swear they don't make 'em like that anymore. Could you imagine?! I would give him some. Hell, I would give him all,\" Blake and I cooed simultaneously, talking over each other. The gray chaos of earlier had given way to subtle rays of golden light to create rainbow spurts along the concrete.\n\n\"I know! Ge-ne gets me every time! That voice hmmm all gruff and authoritative. Papi can boss me around anytime. Did you see the way he put that Deezel in his place? Get 'em, Papi.\"\n\nDumbfounded, we stared at Angie for a few seconds in utter silence. Thankfully, Blake was not as lost for words as I initially was.\n\n\"First, it is DENzel. Second, it's GENE, and third, what the hell are you talking about, Angie? Were you even watching the same movie? Hackman, seriously? The only time he would ever get a glimpse at my cookies is if his Gold Card swiped it.\"\n\n\"Whah? He reminds me of my uncle Ramon. You should see that man smoke a cigar just like Gee-ene. Eeyy Mami it is hot\u2014\"\n\n\"Ang, please stop, I can't take this. You letting Gene spank it is bad enough, now you want to bring us into your incestuous fantasies. I can't and by 'I can't' I meant ewh!\" I said finally, after gaining control of my gag reflex, \"but it takes all kinds I guess.\"\n\n\"No, it doesn't. That's some Puerto Rican shit right there,\" said Blake.\n\n\"I swear you girls are sleeping. My abuela says a good-looking man can bring you nothing but heartache and bad credit. And she should know. She's been married like five times or something, although I'm not sure if Paco is legal. Anyway, Abuela says an average-looking man will work harder and treat you like a que . . . like a\u2014damn . . .\" whispered Angie, her words trailing off indicating that she had suddenly forgotten the point she was trying to make or it just didn't matter. Instinctively, Blake and I turned to uncover what had captivated her so. We understood immediately. Anthony (Tony to his friends) Mason, standing with some guys talking, looking like the divine rebirth of what heaven should be, at least to me. I'm not sure if I fell instantly in love with him or if it was lust. Spellbound as I was, one thing was certain, though: I would lose any battle that would ensue over the next year and a half. Within seconds I had taken an entire gallery of photos of him with my mind. Try as I might through the years I still can't seem to delete them.\n\nHe was carrying an oversize navy blue umbrella to protect his shoulder-length, sun-streaked, dark brown dreadlocks. Historically, I have never been attracted to men with locks\u2014unsanitary, as Cora would say\u2014but he was so much more than perfectly twisted hair. The rains had subsided, allowing glimpses of sunlight to break through the clouds, and the rays created a halo effect around him, highlighting his golden-honey complexion. A glow so bright that it seemed to emanate from his entire being through the wool peacoat, ivory cable-knit sweater, dark denims that were loose but not obscenely baggy, on to the camel Timberlands. I stood there transfixed like a child watching a magic show, and he didn't even notice.\n\nAt some point he ended his conversation and started to walk leisurely toward Fourteenth Street. Hypnotized, I followed some steps behind. He stopped to look in at a makeshift music store. Exiting, he came across a homeless woman sitting on the sidewalk and after some conversation placed money in her hands. An oncoming taxi separated us at that crosswalk. Looking ahead, I saw him crossing the street. \"Where are you going?\" I heard Blake and Angie asking from behind but could not be bothered to stop and explain my actions because I didn't have an answer yet even for myself. I only knew that I needed to see a little bit more of him before the moment passed. As soon as I relaxed, believing I had his pace down, he crossed another street, the abruptness of which paralyzed me. I stood at the corner for a moment watching him walk away, feeling as if a part of me that I would never be able to reclaim was being extracted. The return of rain only heightened the emerging panic of losing something vital. My legs sprinting into action before my head resolved the conflict, I dashed across in pursuit with absolutely no thought of what I would say when I reached him.\n\n\"Excuse me!\" I said, with an urgency that was startling even to me. He turned as if instinctively knowing that I was speaking to him, rendering me lost for a moment. Please, say something more than that, Jules. Words. You know what they are\u2014single distinct conceptual uses for language. Select a few, form a sentence, and for mercy's sake don't embarrass us. Come on, girl. \"My name is . . . Hi, I am Jules . . .\" Really, is that the best you got? OMG, it doesn't matter because he is now touching my hand. Dear Lord, I hope he thinks I'm cute and not about to have a heart attack. Okay, say something clever. Ask directions. For God's sake, breathe! \"I am Jules Sinclair and . . . I saw you coming out of the theater, back there, and wanted to say hello.\" WTF? Did you just say what was actually in your head? No style, no finesse, that was so amateur. Way to go, little one, way to freaking go. Oh my goodness, did he just smile? He did. Oxygen!\n\n\"Hi. I'm Tony,\" he said, looking down at me. \"Are you a dancer? You hold yourself like one.\n\nBefore I could answer, a hint of cinnamon from his chewing gum tickled my nose, giving me permission to close my eyes and inhale the moment.\n\n\"No, but thanks for the compliment,\" I say, slowly opening my eyes, allowing them to linger on every inch of his frame. When our eyes met he was blushing and I knew exactly what I wanted, to be anywhere that he is from now until forever stops being an idea.\n\nStepping closer and positioning his umbrella so that it shielded us both from the falling rain, Tony said, \"So you saw me at the movies?\"\n\n\"Yes. I was with my friends, saw you, and wanted to, actually needed to introduce myself\u2014say hello.\" Stammering to get the last few words out.\n\n\"Your friends,\" he says, pointing across the street. \"You referring to the two girls hiding behind that white van?\" Laughing aloud in a way that made me feel as if he and I were in on the joke the entire time, as if it had always been he and I against the world.\n\nAnd in my very first (of many) \"shit that only happens to Jules\" moments, Tony invited us to join him for dinner at this Italian restaurant in Greenwich Village just west of Seventh. The d\u00e9cor was nothing to speak of, very clich\u00e9d; red-and-white checker-patterned tablecloths with old candle wax\u2013laden bottles on each. I don't even remember if the food was good. The conversation flowed easily and he was more charming than any of us could have imagined. Tony spoke of living in a converted firehouse in Tribeca, transitioning from street dancing to modeling and now photographer. When he did excuse himself briefly from the table, our collective squeal was so loud that I am sure he could not help but hear it in the men's room.\n\n\"What would your abuelita have to say about that, Angie?\" Blake asked pointing in the direction of the chair Tony had been sitting in.\n\n\"Nothing, because she would be butt naked in the kitchen wearing only an apron and some heels screaming ayyy Papi and cooking him some paella y platanos, habas con arroz, flan y guava,\" I said mockingly in my bad Spanglish. Besides, I couldn't help but go for the big dig here if for nothing more than to shut Angie up a little bit. Over the course of dinner she had gotten a little extra chatty with Tony. \"Oh, by the way, missy, you do realize that I not only called this one first but I've already started to name our children, so keep your chi-chis in that enormous bra of yours. Comprende?\"\n\n\"Julesy . . . come on, you know he is not my type. You have nothing to worry about. I would never. Besides, what could a man like that do for me?\" she protested. In truth I knew no such thing, since Angie was the newest to our friendship circle by way of an introduction from Blake, so I adhered to a delicate dance with her that I tagged \"playfully serious.\"\n\n\"That is the farthest thing from my mind, Ang. I just don't want to have to cut you in front of the future Mr. Jules Sinclair, that's all. Haha.\"\n\n\u2022 \u2022 \u2022\n\nIn time Tony would admit that he overheard us and was quite flattered by my proprietary declaration of him. By the end of dinner, despite having to share him with the girls, I had a calm about this man that exceeded any anxiety about when he would call or if he would call because I just knew he would. Even if I wasn't sure of how to be with one, seeing as though, until this point, I was so used to dating boys. As Blake and Angie headed out the door, Tony placed his hand on my face in a caress that allowed him now to take in the moment and me.\n\n\"I didn't plan on meeting you today, Ms. Sinclair, but I would like to know more. How about dinner . . . tomorrow at Odeon?\"\n\n## 5\n\n* * *\n\n## TONY & JULES\n\n8:30 p.m.\n\nWE MET IN Tribeca; our first date setting the stage for everything that was great between us. The conversation was open and intimate, unlike typical first-date small talk. It was akin to a third or fourth date when you know that time is going to be invested in this person, so you go a bit in-depth and open yourself up for a closer look, less afraid of the imperfections they may see. His firsthand experiences mirrored much of what I had only dreamed about at university.\n\n\"Yeah, divorce is rough but I came out of it better than most I guess . . . Funny, your escape was the rain, mine was the streets, which just made things worse at home,\" said Tony. \"You know, Jules, if you truly love the smell of the earth after a good rainfall, then you must love Costa Rica. I just got back from there surfing with my brother. The rain forest is amazing. Man, we saw animals unlike anything in the Bronx Zoo. This is our third time there. Have you ever been?\"\n\n\"No, but I saw a PBS documentary on it last year. It's on my list of must-go-to places,\" I said.\n\n\"Ah, you'll love it. I'll take you,\" said Tony, to my ears not sounding like an empty gesture but like a heartfelt desire. \"It's the best place to get lost and get found in, next to certain parts of Africa. Last year I went to sub-Saharan Africa for a photo shoot and ended up staying a few weeks working with the famine relief to deliver food and vaccines. You?\"\n\n\"No. I haven't been to the continent yet, but I am an avid supporter of Feed the Children. Okay, avid is a little strong. I have supported, would like to do more, and really want to impress you, but it's kinda hard because you have done everything,\" I said.\n\n\"Not true. I've never fallen in love,\" he said, completing the sentence with a wink, at which we both erupted in laughter. \"I can't believe I just said that to you. Did it work? Are you swooning? You are, come on . . . smooth.\"\n\nWith as much seductive sarcasm as I could muster, I replied, \"Oh yeah, something is seriously rising\"\u2014half covering my mouth with my hand to cough out the words\u2014\"and it's not the flag in Pamplona but the\u2014\" At which we both laughed. \"Seriously, don't you ever get tired of the nomadic life and just want a place to call home?\"\n\n\"On the real, I guess you could say I am a wanderer of sorts. There's so much to explore, and I plan to do all of it,\" he said. \"Besides, being transitory is actually comforting for me. My mom was inconsistent after the divorce, so I learned early on not to get too attached to things or hit the self-destruct button when things get so comfortable that I lose perspective. You learn to adapt. One day you'll reach for it and nothing is there, you know?\"\n\nI thought I knew but in truth I had absolutely no idea what he was talking about. I watched the words leave his mouth, heard them in my right ear, and by the time they got to the left I had paraphrased them into an immaculate tale of Scheherazade convincing myself that I was the answer to his wondering.\n\nEvery so often the ma\u00eetre d' came over to make sure that \"their Tony\" was happy, as would friends and neighborhood locals who, like him, considered the Odeon their personal kitchen. I was fascinated watching as he effortlessly transitioned from one to the other; each time never forgetting that I was there and making certain to include me in the conversation or cut a dialogue short when a dim-witted model creature was blatantly trying to blink me invisible.\n\n\"I could get used to you at my side, Ms. Sinclair,\" said Tony.\n\nOfficially, he was choosing me. This Sadie Hawkins\u2013esque mentality I reasoned was one of the rare places that Venus and Mars weren't actually so far apart. It's just that women and men approach love and idol worship far differently. As a woman, I acknowledge that I am preconditioned to wait to be selected for the moment \"he\" will announce to the gallery that I am the most beautiful one in the room. For their part, men\u2014no matter their size, shape, or credit score\u2014seem conditioned to believe that everything they survey is rightfully theirs, be it women or cattle; it is all property, one and the same. The only difference is that cattle can't hold them or build them up when the world around starts to disintegrate. It takes some time before they are ready to hear this truth and fold into a specific woman. I would be foolish to think that I was the first for Tony in this space, but I do believe that I am the most genuine he has encountered, and that in itself could be a frightening proposition over time.\n\nIn little to no time we came to know each other's mannerisms so well that the graze of a brow would be a signal to depart, a shortened laugh meant he wanted rescue from an impossibly boring conversation. His hand in the small of my back said, \"I'm proud you're mine.\" Before Tony, I never laughed so much about anything or took myself so lightly. For a while playing Lois Lane to his Superman was enough. But, as much as we were in harmony, there were hiccups that I didn't question, to a great extent because I trusted the strength of our union, until the moments that started as flint sparks erupted into four-alarm fires.\n\n\"What are you talking about, Tony? Don't break that! I told you last week that I was meeting Shawn for dinner. I even asked if you wanted to come. Why would I do that if I had something to hide? Why?! Whatever, I guess your boy Jay's word on what he thought he saw is more important than mine.\"\n\nHis quiet moments, the dark ones that were not about waking up late on a Sunday, having coffee, and silently dissecting the paper in bed together\u2014National Geographic and Sports Illustrated for him, Arts and Style sections of The New York Times for me\u2014were the worst.\n\n\"Hey, didn't you hear me calling you? What's wrong? How long have you been sitting here? It's past three o'clock,\" I'd say.\n\nBeing at dinner and catching a gaze between him and the waitress, never enough for our friends to notice but just enough for me to feel devalued. My initial thought was always to go quiet and not overreact, but words often found me when we were alone.\n\n\"Tell you what, next time I see lipstick anywhere on your person I will be certain not to fall back on the convention of us. You should know better.\"\n\nBefore long I found myself asking him questions I already knew the answers to, only to have him lie to me or avoid the response completely. More nights than I care to admit, I would leave his place in tears and vowing never again. Angrier than necessary at myself for always choosing the wrong corner to try to hail a cab. Tribeca is a bitch late at night. And every time we hit a wall he would find a way to reach in and take hold of my heart again. Maybe if I didn't love him as intensely as I did, it would have been easier to make a break sooner.\n\n\"Love isn't perfect, Jules. It's not the fairy tale you have worked out in your head. I am trying hard to be that man for you, baby, but you have to ride this out with me. I need to know that you will be there.\"\n\nI was there through everything, even as the tests got harder and harder. I was there when his brother Nathaniel died as they were riding (racing) back from the Hamptons on their bikes. I was there when he said that the recreational white powder was just a temporary situation to take the edge off and was nothing to worry about. I was there long enough to become the enemy so that everything I did to comfort him only infuriated him. I was there for the yelling, the crying, the apologies, and I truly tried to understand it and rationalize that it was not him, that he was under great pressure. To my heart's ache I was there when Angie's body held the comfort he needed.\n\nThat's the thing about giving a set of keys to your lover; eventually he or she will use them at an unexpected moment. Ours was on a Saturday\u2014October 9, 1996, to be exact, when I received a call from the Conrad Hotel Group in London to join their firm. Tony was the one who gave me the inside information to apply, so it seemed only natural that when they called and offered me the position on a day that was free of any expectations, I would rush to him to talk everything over. And boy, was there a lot to figure out. Could his job be done from London in order to be with me or at least could he split the time since the flight is only five and a half hours across the Atlantic? Should I keep my apartment in NYC or move my things into his? If I were to give up my place, did he want us to get the London flat as a couple with both of our names on the lease? The package they offered sounded good, but what should I counter with? Was the expense package sufficient? Tony held the answers to my future.\n\n\u2022 \u2022 \u2022\n\nIn hindsight it's odd because I sensed my world was crashing before my eyes took in the full scene, but dismissed it. Maybe it was the eternity that it seemed to take traveling from uptown or the uncertainty of what it would mean for us growing to the next level. On entering the main level of his place I smelled a fragrance (freesia and lemon\u2014awful combination if ever there was) that was at once familiar but not readily identifiable. I called his name and immediately upon seeing his form coming down the stairs, I dove right in with my news. I must have been talking a mile a minute while trying to find a chilled bottle of champagne because I didn't initially notice the look on his face or the marks on his back. It was only when I held him again that I felt them accompanied by the unusual warmth of his mouth and that scent (freesia and lemon) in the fold of his neck. This time I didn't need to ask questions. His face was in my hands, so there was no place for his eyes to run in an attempt to escape mine. The story they told me was far more than I could bear, and it was right then that I understood for the first time what real loss was. It was the first moment that I felt I could truly empathize with his losing Nathaniel earlier in the year. I had lost him and unlike the times before, we both knew that I could not take him back, even though I still loved him, and that an ocean between us was the only answer.\n\nIt must have been the absence of screaming and dramatics or the unusually long quiet between us, but she called his name and suddenly everything made sense. She was always too eager about any developments about us. I erroneously had become far too open and comfortable in the certainty of Tony and Jules, so I disclosed freely.\n\n\"AN-Tony, is she gone?\"\n\nWell, so much for a long quiet. Somewhere between the first and second landing I met Angie and she met the champagne bottle in my hand. Damn shame actually now that I think about it\u2014Pierre Jou\u00ebt ros\u00e9 was my favorite champagne. I haven't had it since 1996. All the things that I could not do to him I did to her. Each blow landed only intensified the fury inside. In that moment I understood how someone could take a life, and I would have if he had not grabbed me. I remember him holding me tightly, rocking back and forth. His tears falling down my shoulder, and all I could do was collapse into his arms and mourn us. How odd the whole experience was. When I awoke, we were in the same spot on the floor and in the same cradle position that I so loved to be in with him. Being \"in the pocket\" is how he always referred to it. I was glad that he didn't wake when I left, for there was nothing else to be said except that I wished we could go back in time and meet all over again, but this time we would be smarter and not hurt each other.\n\nTony once said, \"J, it hurts sometimes to see myself in your eyes. You believe me to be the kind of man that I only aspire to be but fear I will never live up to.\" Maybe that is why I instinctively held his face in the kitchen, so that he could see my soul and all that he'd destroyed, because in some strange way I knew that it was the only way to reach him. I needed Tony to see what a living death looked like.\n\nI accepted the offer and left for London within the week. My phone met an unfortunate end somewhere along the West Side Highway. Smashing it was the only way I could stop his calls from coming through and my temptation to allow explanation. My office was no longer safe, because he camped out there now in order to explain, so I resigned effective immediately. I left my apartment as is, had my intern forward my clothing, and went to a hotel because I didn't know which of our friends to trust. I walked a lot above Twenty-first Street in the hopes that the city would heal me as it had many times before. But this time she failed me. On the plane I thought of my mother's warning about being mindful of what details one confides in girlfriends about their relationships because instead of celebrating your happiness one will be contemplating how to get what and who you have.\n\n\"Baby, watch yourself and these little fast girls you call friends. They aren't happy for you. You better wise up and see who is trying to get what you got.\"\n\n\"Cora, please. My girls are not like that,\" I said boastfully. \"Women today are not like they were back in your day. Besides, Tony would never.\"\n\n\"I prefer when you call me Mommy. Those are the times that I know you really hear the words I am saying to you,\" said Cora.\n\n\"And what if I wanted to stop, hear less. What about then, Cora,\" I asked, intentionally snarky.\n\n\"Nasty little thing you can be sometimes, Julesea. I don't know who you get it from\u2014definitely not from me,\" said Cora. \"Everybody knows the Dutch are extremely civilized people.\"\n\n## 6\n\n* * *\n\n## CLOSURE\n\nUnion Square, 1998\n\nFIRST BITE,\" SAID Tony, struggling to be accountable in the moment and not divert the gravity of our history entirely with charm. \"First bite and I'm sorry for what I did to you, for what I did to us. Jules, if I could . . .\"\n\nFeeling my resolve weaken, \"I can't do this right now. Not today, I just can't,\" I said, forcibly releasing myself from his grasp. A rush of uncontrollable trembles started to stir in my chest, only to erupt in my hands. I didn't think I had anymore tears left to cry, but there they were, at the ready and falling freely in front of him.\n\nI've always wondered who those people are that can love someone so deeply and, after a great betrayal, forget that feeling of all-encompassing devotion that was there. I never forgot. I was acutely aware of this disturbing fact whenever a big decision presented itself and my first instinct was to call Tony and get his insight. Our history wallpapered my life, so that in order to breathe, each day I had to place something new and \"more important\" on a mental Post-it to cover all the pain. Any efforts to peel the paper itself were as futile as the ink now embedded in my skin like a tattoo. Not wanting to lose the moment but seeing that I could not handle all that he needed to say, Tony asked about when I got back into town and if we could have dinner. The tears that escaped were captured by his caress, reminding me of that first night in Greenwich. For lack of words to save the moment, I closed my eyes to shut him out, get my bearings.\n\n\"I have to go,\" I said and started to walk off. Midstep, the history of us said walking away so abruptly was somehow cowardly and wrong, causing me to pause. Slowly, I turned around to face Tony, still standing there as on the day we first met, with the heavens still shining just for him.\n\n\"Breakfast, tomorrow, nine thirty. I'm staying at the Mercer.\"\n\nAny gesture requiring more of me would have been too much, so I dared not wait for an answer. I knew that no matter what was on his schedule, he would be there. The only thing I didn't know was what I expected, what I needed. Foolishly, I believed I could sashay down memory lane, to all the places Tony and I used to go, and not encounter any of our ghosts. But that, again, is not life and definitely not indicative of the shit that only happens to Jules. In some ways I guess the moment is fitting, Manhattan is after all only thirteen miles long and most everyone I know resides below Fourteenth Street. Peering at my watch primarily for somewhere to rest my eyes and resist the urge to look back, I realize it is approaching 3 p.m. My meeting with Michael is at 8 p.m., and puffy eyes and a broken heart do not make a good first impression.\n\nThere is no city as amazing as New York. Instead of crumbling on the walk back to the Mercer, I found my stride. The air that before felt cold and biting now felt refreshing. Something outside of myself that reminded me to be in the moment, not in a past that has been written and is at best just a bundle of memories of intangibles. Back at the hotel, a hot bath with my special blend of coconut and lavender bath salts was definitely in order, followed by a quick once-over to make sure that there were no chipped nails, no stray eyebrows, or that telltale mustache that had started to sprout immediately as I blew out the candles on my last birthday cake. If all was good, then I could treat myself to a disco nap and pray the last few hours away.\n\n## 7\n\n* * *\n\n## Carly's\n\nWITH THE EXCEPTION of two oversize gilded doors, the simple exterior of Carly's underscored its decadent Hollywood Regency d\u00e9cor meets members-only British club interior. Moments after entering, I was greeted warmly by the hostess, who while requisite supermodel Amazonian gorgeous was further along in years, more like an Iman-type from one of those ravaged African countries that possesses some of the most beautiful humans ever to grace the earth, Somalia or Ethiopia. Her generous nature was not reserved solely for me. There were a few patrons ahead of me, and she was quite accommodating. Beyond fierce is the Old Guard of glamazons. Hearing that I was to see Michael, she proceeded to escort me to the main dining room, which was accessible either by twenty or so steps descending into the most stylishly outfitted room I'd seen in some time or by a glass elevator that equally guaranteed you saw everything and were seen by everyone. Deep shades of mahogany, tobacco, and camel furnishings in various luxurious fabrications and skins accented with variations of jade and ivory marble and intricate moldings comprised the room's seating. A stage in the round was the room's nucleus, thereby making every table a great one. The entire layout demanded that I take a moment on the stairs to absorb it all. For the briefest moment I felt as if it were all for me. Immediately I felt ready for my Sunset Boulevard-esque close-up. If Michael concocted such an entrance to impress me, it's working.\n\nThe hostess, having reached the basin only to notice me stationary, asked, \"Ms. Sinclair, is there a problem?\" I shook my head no but had yet to move. \"Intoxicating, isn't it? If you will follow me.\" She was right, intoxicating was the perfect word.\n\nThe room seats about 250 and yet it manages to feel extremely intimate and inviting. On approach I was taken aback to see my table empty. With such an entrance, I expected to arrive at a filled booth greeted by Michael Kipps himself in the midst of an amazing story, surrounded by an admiring court hanging on his every word\u2014but that was not the case. Reluctantly, I accepted my seat as a party of one in a packed restaurant at a time when the last thing I needed was to be alone with my thoughts. After a few uncomfortable moments, I started to regret not wearing my actually I am not alone but here for an important meeting T-shirt in order to deflect a few pity-filled stares directed my way. Under different circumstances, I could have given my watch your man NOT me vibe, but not tonight, not after seeing Tony and knowing that I would have to face him again tomorrow.\n\n\"Ma'am, a Macallan 15, neat,\" said the waiter as he placed the drink in from of me.\n\n\"Actually, no. I didn't order anything yet,\" I replied, taken aback briefly, \"but a scotch would be perfect right about now.\"\n\n\"Yes, compliments of Mr. Kipps. Would you like something else?\"\n\n\"No. This is fine. Thank you.\"\n\nWatching him go, I sat there invisibly scratching my head as it dawned on me that Michael Kipps knew far more about me than I'd assumed. Without further hesitation, I took a full sip of the scotch that had been gifted to me. To feel something stronger than the thoughts I was fighting to suppress was highly welcome.\n\nIn truth, the disco nap did not help. I was in need of a double but feared turning into a blubbering wreck by the time Michael joined me. Normally, handling my liquor is not a problem. I am well aware of my limits and always mindful to have water between drinks. Actually, I consider the fact that I am not a sad or angry drunk in the rare moments that I take it there to be one of my best traits. Modern Girl Achievements, I call it. At my worst I am quiet and contemplative. Tonight, however, any extra fermented assistance could render me an emotional nutter, so my current plan is to nurse this one drink all evening. Tonight I need to dazzle and convince this man that the continued media development of his business would best flourish under my direction.\n\n\"Ms. Sinclair, I hope you are in the mood for a great show\"\u2014and with that Michael Thurmond Kipps finally showed himself. Ignoring the formality of a handshake, he slid into the booth alongside me. No sooner did Michael sit down than Alyson Williams, jazz and soul songstress, appeared onstage. Like the clientele of Carly's, Alyson was a New York fixture, known as much for her onstage presence as for being the daughter of legendary trumpeter Bobby Booker. Her personality was as flamboyant and colorful as the black crushed velvet bodysuit with a cheetah cape, matching pillbox hat, and oversize gold jewelry she wore. On the strength of a single note, she could express love's rise and fall with inexplicable clarity.\n\nThe performance list at Carly's is not published, so it is the luck of the draw as to who will be performing on any given night. When you call to make a reservation, the only thing that will be confirmed is the time your table will be available and that there will be live music. Quite a useful tool if you have the Rolodex to make each night a showstopper. You are all but guaranteeing that every columnist, \"It Kid,\" society heir, and social climber will be present nightly. For there is nothing more tragic to a self-proclaimed man (or woman)-about-town than not to be in the nexus of the most exclusive goings-on. Even if they are not genuine fans of the music, they are devotees of the scene. It is intoxicating, remember.\n\nThroughout the first half of Alyson's performance, I focused entirely on Michael. Yes, in person he was all that the gossips made him out to be: commanding presence, handsome, stylish, and tremendously confident. His choice of dress was understated yet impeccably bespoke. A few times he caught me eyeing him. I could see that he took some pleasure in watching me scramble to look elsewhere when caught. At a moment in \"Not on the Outside,\" he leaned over and said, \"I'm the most open cat you will ever meet, so don't try so hard to dissect me. At least not during our first meeting.\" And with that he affixed a firm squeeze midway up my thigh, just high enough to be indecent. Oh, great. Does this ever stop? For this I did not try to avert my gaze but demanded his so that he could see that those male power plays did not intimidate me.\n\nAll too often it seems that to be a successful woman in business and not be derailed by unwanted advances, we are to assume the posture of men in an attempt to be invincible\u2014and our femininity invisible. I rejected that doctrine from my first foray into the business world. Yes, my success is a result of the feminist movement, and I appreciate all that was done. However, I reject the belief that every advance by a man in a professional environment is a three-alarm fire sign to cry foul. More times than not I have found it to be a weeding-out process that is part of the game of big business.\n\nAs women we are still screaming to be treated as equals, so the easiest\u2014albeit most infantile\u2014way, it seems, for some men to determine if we are ready to play with the big boys is to go directly for the gaping wound of sexual discomfort. I mean, hell, why don't you just put a Coke can on the table and call me Anita? The key is how you handle yourself in that moment. Save it\u2014don't waste time trying to explain yourself, because it will fall on deaf ears or be interpreted as whining. Don't bristle either. Hold firm and assert your power with a smirk and a \"no\" head shake, then move his hand if it lingers and redirect his attention to a better candidate. If they are the daft type, defuse with a quick but serious joke, or mention his wife. And for heaven's sake DO NOT fool yourself into thinking that your sweet nectar will change his life and therefore send you skyrocketing up the corporate ladder of success. It won't. Daddy told me this countless times over the years. \"Jules,\" he'd say, \"the only ting dat type of ting will get a woman is a badt reputation.\"\n\nLeaning into Michael close enough to be within earshot while maintaining eye contact so that even in this dim light he could understand the magnitude of my intent, I said, \"You do know it would never work, don't you? I would just be dating you for your money, your power, and social position, only to leave you as soon as a bigger branch presented itself.\"\n\nMichael smirked. \"I like you, Sinclair. You got gumption.\" Then he patted my knee. \"Indeed, I can respect that.\"\n\nAfter Alyson's first set, we talked about his business. He spoke about feeling alive in Carly's like it was what he was supposed to do with his life all along and the modeling was just a step to get there. For so long he had been growing other people's wealth, either in editorial, by attending events, or with commercials. Then one day it occurred to him that if his face and name were this valuable for everybody else, then they would be just as valuable for ventures of his own. Now he wants Carly's to become a global destination. For the most part I listened attentively, because it was clear that he enjoyed having a captive audience that allowed him to do all the talking. All in all it was quite informative and impressive. In a much different way than with Tony\u2014in Michael I saw a man I could learn from.\n\nAt the start of the second set we began to eat, and just in time too. I never did get to have that falafel or enjoy the remainder of the zucchini slices from earlier. I ordered heartily; a beet and goat cheese salad to start, with a petite filet, twice-baked potato, and asparagus for an entr\u00e9e. Now it seemed Michael's turn to not-so-discreetly check me out. Unbeknownst to me, my ravenous appetite impressed him. I guess heartache and famine do have their benefits in the right circumstances. Through bites and head bobs to Alyson's singing, we exchanged some marketing ideas about the restaurant and debated who was a better baller, Vince Carter or Allen Iverson. He was delusional even to have uttered Carter's name, and I let him know it. At that he belted out a laugh that was so robust, even Alyson took notice midway through a song.\n\nThe night for me ended around midnight, and despite the time difference I didn't feel like turning in just yet. As Michael walked me to the door, he asked if I could come back tomorrow evening at six o'clock. He wanted to actually see the media strategy I spoke of and give me a proper tour of his burgeoning empire. Next steps in place, I climbed into the back of a taxi and directed the driver to the hotel. For all the things in my life that don't go the way I desire, there are gems\u2014like tonight\u2014that go better than I could ever imagine. The trick for me is to get better at compartmentalizing the storms, thereby always leaving me an open box to identify a great, unexpected moment. This was such a moment. I wanted to savor it, so I rerouted the driver up to Central Park through Times Square and back down Park Avenue so I could let New York know that, while I may have left for a moment, I didn't run away completely. I just needed some time to discover who I was outside its embrace.\n\n## 8\n\n* * *\n\n## LAST GOOD-BYES\n\nFOR THE BRIEFEST moment upon awaking, I forgot where I was and how my day was starting. Breakfast with ex. Inasmuch as there was immense unresolved pain in the anxiety I felt, everything within me wanted to see him and hear his reasons. I needed closure. Over time I had become equipped in handling that dull pain that never really seemed to go away, the twinge of jealousy and subsequent dismissal that I felt when young love was before me. All the darts and daggers became friends. And while intentional avoidance may not have been the recommended therapeutic approach, I found it useful. I had learned to rebuild my life bit by bit, and in doing so I pushed our relationship so far down that it seemed mythical. Today that fable would leap off the page and become all too real. I just wish I knew if the ending would reflect American movies, with no collateral damage and a neat little bow, or French ones, all bleak and undefined.\n\n8:15 a.m.\u2014The snooze alarm goes off a second time, forcing me to finally concede defeat, withdrawing the covers from around my head but still refusing to open my eyes. I fumble beneath the empty pillow on my left, searching for the mobile or, as I have come to begrudgingly refer to it, \"the only bed partner I've had for the past two years.\"\n\nYou have three new voice messages:\n\nMessage One: It's Cora. Did you just send me to voice mail? I would hate to think you are standing there looking at your phone, ignoring me.\n\nMessage Two: Julesea, why is your phone ringing like that? Are you in the States? Call Mommy.\n\nMessage Three: Julesea, it's Cora. Call your father, he is worried.\n\nMy word, what am I going to do with this woman? Against my better judgment I dial her.\n\n\"Hi, Cora\u2014\"\n\n\"Well, look who finally got around to calling back,\" said Cora. \"You know your father was worried sick about you.\"\n\n\"Was he now? Good morning, Cora. I just got your messages,\" I say.\n\n\"How is that possible? I called you at least three times, Julesea. What good is having a phone if you don't answer it? I will never understand. Is this what they are doing over there these days\u2014not picking up the phone?\"\n\nI couldn't help but chuckle at her theatrics. \"No, that's not what they are doing. I am sorry but I was exhausted having gotten to New York on the red-eye and\u2014\"\n\n\"New\u2014What are you talking about? What are you doing in New York and why am I just hearing about it? Is this keeping secrets from me going to be a habit or . . . ?\" Oh, if only she knew, but let's not complicate things now.\n\n\"Please, stop. It's far too early and there is no caffeine in my system yet. I'm here for an interview but didn't want to tell anyone.\"\n\n\"Well, I'm sad to hear that I am just anyone to you,\" said Cora, combining all of her thoughts into one stealth interrogation. \"What interview? Did you get fired from Conrad? What happened? Who are you meeting with? Does Daddy know?\"\n\n\"No! I did not get fired. I am meeting with this guy Michael Kipps about helming the PR division of his supper club, and no, Daddy does not know, so there is no need for you to feel left out.\"\n\n\"Umph. So you are leaving London to work at a restaurant? What kind of sense does that make? Are you mad, Julesea?!\" shrieked Cora.\n\n\"A supper club, Mommy!\"\u2014Why am I yelling?\u2014\"You know, the kind of place with cool singers and nicely dressed people. Trust me, you'd like it,\" I said. \"Listen, I promise to tell you everything later, but I really have to go now. Ton-y . . .\" forcing the last syllable to silently escape my lips . . . \"To a meeting downstairs and I'm not even dressed.\" Whoa, that was close. Even when we were dating and things were good, Cora struggled to tolerate Tony, always professing that something about that boy wasn't right. After our breakup she allowed the floodgates of wrath to open up, never missing an opportunity to highlight the flags she saw or the warnings she gave, all of which I was too na\u00efve to comprehend. I couldn't bear to hear the lashing she would give me, if she knew that he and I were meeting in a matter of minutes. Not today. I need to be as clear and connected to my feelings as possible, not echoing Cora's accusatory yet well-intentioned sentiments.\n\n\"Umph. Is this a good opportunity?\" she asked.\n\n\"I think so,\" I said, acutely aware of not giving too much for her to dissect.\n\n\"And it would put you back in the States, yes? Instead of thousands of miles away? . . . That's good, I guess.\"\n\n\"Yes,\" I said.\n\n\"I just don't know why it has to be in New York. That city is nothing good for you. Just your luck, you'll move back and run into that good-for-nothing wannabe ragamu\u2014.\" My goodness, was there any pleasing this woman?!\n\n\"CORA!\" I yelled. \"I have to go. I will ring you later, promise. Kiss Daddy for me. Love you, must go. Bye,\" I said, removing the phone from my ear but not actually hanging it up until I heard her mumbling subside. Oh, how I have wanted to hang up on her over the years but I never can. She's an absolute nutter and the very foundation that makes me believe that nothing is out of reach, although I will never tell her that much.\n\nFinally dressed and taking full inventory of myself one last time in the mirror, I think about the kind of woman who I would like to actually be (not a composition of the flaws and unfulfilled dreams that confront me every so often):\n\nI long to be the woman whom is loved passionately and deeply by a successful man who can provide for me in all ways that matter in this life. A woman who realizes that happiness is not a choice between this or that but an accumulation of moments experienced and shared. I want to be the woman who knows when to let a love go that existed in rose-colored glasses, only to walk into a true love that requires no special frames to be alluring. I want to be the woman whom he chooses to remain faithfully monogamous to and committed to even when things aren't going smoothly. I want to be the woman whose eyes hold his future.\n\n\u2022 \u2022 \u2022\n\nComing down the stairs, into Mercer's subterranean restaurant, I saw that Tony had selected one of the more intimate seating areas. This made me regret not having peeled myself out of bed sooner or speaking to Cora too long, at least I could have controlled the proximity of our bodies. Instead I now had to deal with the forced intimacy created by the booth he selected, which only heightened my anxiety. Seeing me approach, he stands. A proper gentleman. Despite Cora's misgivings, his mama did something right; he always walked on the side of advancing traffic and ordered for me. He knew my food allergies to shellfish and soy. He never complained about my desire always to see scary movies, only to scream and jump from the opening titles to the end credits. Things like that demonstrated why I thought he was a keeper and why then I felt so lucky. In reality, however, nice packaging doesn't always mean that you will like what is inside.\n\nUnsteadily, Tony took the cup from the waitress. \"Morning, I ordered you some hot chocolate with a shot of espresso. You're still drinking that, right? Of course you are. It's cold outside. It must be cold in London, huh . . . You look really good, Jules. I couldn't believe my eyes yesterday when I crossed the street and saw you. At first I thought it was someone who looked like you. I've mistaken quite a few girls for you over the years, and then I saw you do the 'first bite of zucchini loaf' dance\"\u2014stopping to mimic me with one hand in the air and a heavy-metal rocker head bob\u2014\"I watched you for a while, you know, frozen. I didn't know what I should say, I was just so glad to see you. I miss you. Can I say that, that I miss you?\" His nerves obviously forcing him to blurt out his confession all in one breath.\n\nI had forgotten what it was like to see him less than composed and how it affected me, so I watched and listened, taking him in. All the while feeling that when it was time for me to speak my truth I would know what to say.\n\n\". . . I fucked up, J. I hurt you something bad I know that. There's just no way to make that shit right. I would, you know . . . if I could.\" Despite the great effort he was making to hold my gaze, I needed to look elsewhere, electing instead to watch his hands dance frantically around the contours of the white linen napkin. \"That morning when I woke up and you were gone, everything stopped being a game. It all became too real.\"\n\nShifting uncomfortably in my seat, I fought to hold myself at bay and not move to wipe the dam of tears threatening to break over the rims of his eyes. A part of me found this show of emotion endearing. Not in the way that would make all that had occurred vanish, but it validated the fact that I wasn't in this nightmare alone. For so long, in my quiet moments I envisioned him and Angie moving along happily with their lives together and toasting to my broken heart. They would have sex in the bed that he and I had picked out together. In the morning he would gently caress the base of her stomach, as he did mine each day, and kiss behind her ear. Every beautiful memory that we had, in my pain, I extracted my image and replaced it with hers.\n\n\"Are you still with her?\" I demanded, unable to contain the residual bits of anger that refused to be tempered.\n\n\"No! Of course not. I have not seen that girl since that day.\" Off my disbelieving look he continued, \"I promise you, Jules. I have not seen her in years. I did call to make certain that she was okay and made arrangements to cover any medical costs she incurred, but that was all. Believe me.\"\n\n\"Yes,\" I said, and I did.\n\n\"She wasn't the problem, Jules, and neither was the other shit. I was the problem. I wanted everything, to have you on my arm and keep the pretense that my life brought. Every time I felt us fold into each other it scared the shit out of me. Automatically I would check myself and run scenarios of how we would end. At first I thought I was crazy and set my mind not to fuck us up. I felt you would leave me at some point if I didn't get it right, so I wanted to be 'that guy,' not just for you but for me as well. It just all came too soon and I wasn't ready, I guess. I didn't want to choose.\"\n\nFeeling the urge to defend myself, I fired back, \"I never asked you to choose, Tony. I didn't even know there was a choice to be made. I never held you back. I encouraged every dream you had, and even when things were tough I pushed through for you, for us.\"\n\n\"Well, damn, I guess you were perfect, some kind of martyr or something, and I was a big fuckup, just lucky to have you, right? I know that's what everyone thought. . . . That's what I thought.\"\n\n\"Let's not do this. You know better. Don't spew your venom at me and I won't do it to you. We both messed this up, Tony. We were both guilty of some crazy idol worship of the other and for all of our intelligent 'ride or die' shit we couldn't make it work and people got hurt. I got hurt . . . and so did you. I am still hurting. I think of you more than I care to admit and I mourn us still every day. I don't want to hurt anymore. I don't want you to hurt anymore.\" The enormous lump that had returned to my throat was still there but felt manageable. \"I want to let go.\"\n\nWe sat talking for hours about things, some reflections about the good times that continued to haunt us both but mostly about the mistakes and do-overs that were impossible to recapture. I don't recall much of the former now because in retrospect they were minor details. What I know for certain is that at some point we were holding hands like old times, crying forgiveness tears in a very public place that held many memories for us. Standing with him at the corner, saying good-bye, I still felt the full impact of us. Had he verbalized all the things I needed to hear? I would have no way of knowing this until time gifts me with perspective. For now I know that I heard the things I wanted to hear. More to the point, I know that I had enough to sleep for the first time in two years through the night\u2014but is it enough to free my heart and love again? A final embrace and Tony walked away toward Broadway. Even now he was the finest man my eyes had ever laid on.\n\nTurning to reenter the hotel, I hear, \"Jules Sinclair. Ti amo, baby, ti amo.\" People on the street must have thought Tony a crazy man, judging by the way they rushed past him. For the last time I cried for us, this time happy tears, and waved good-bye.\n\n\"Ti amo per sempre,\" I whispered loud enough for only my soul to hear.\n\n## 9\n\n* * *\n\n## AND YOU ARE?\n\nMICHAEL'S NOT HERE. Let me buy you a drink while you wait,\" said the man standing behind the host stand at Carly's.\n\nHad I not been fresh off the emotional roller coaster that was Anthony Mason or been so intently focused on landing this job, I would've noticed that this guy was absolutely swoonworthy; mid-thirties, I'd say, with dark blond hair and hypnotic ocean-blue eyes trapped behind the kind of long, sweeping lashes that women, myself included, buy Great Lash mascara to re-create. He was a proper mix of boyish-meets-rugged with a disposition and precise mannerisms hinting of an Ivy League\u2013New England pedigree, possibly, by way of Boston, maybe even spent some time abroad\u2014but not entirely sure, given the limited exchange. Immaculately dressed in a gray three-piece suit and winter-white shirt with mother-of-pearl cuff links, paired with an understated but noticeable vintage Rolex Submariner, and finished with apropos camel Brioni suedes. If I were looking, this is what I would have noticed\u2014but I'm not. Like Michael, he seemed to stand just north of six feet two because, even in my Sergios, he towered over me. The circumference of his noticeable egotistical sense of entitlement on the other hand was immeasurable, reminding me of a particularly colorful and equally gorgeous emerging Savile Row tailor who, with little to no effort at all, manages to consume 90 percent of the oxygen from any room he is within a three-block radius of. Obviously, given my asthmatic childhood, it is fair to conclude that we should not mix particularly well\u2014yet despite this insignificant little detail he is the prototype for the beloved of my \"grown woman imagined,\" but I digress.\n\nI ask current cocky man of mystery for the time to ensure I was not confused about the hour only to realize that I was correct. \"He said six o'clock. Do you know how long he will be?\" I asked, unknowingly biting my bottom lip as he stared at me with an amused look on his face, making me all too aware that I was at a disadvantage. \"Did I say something particularly funny? Maybe he left a message for me. Can you check? My name is Jules Sinclair and Michael said for me to meet him here.\" I continued feeling myself getting a little flustered, and rightly so. I felt bare and exposed, like he wanted something from me that I was in no position to give or even entertain.\n\nSatisfied with himself for having gotten unsolicited information, he replied, \"And so you are. Michael unfortunately is not here. Par for the course, really. However, when he does come, you will have to wait until he and I are finished.\" Devilishly smiling to reveal a slight upturn indentation on the left corner of his lip, the kind that frequently results from a childhood sports injury, he said, \"I mean, it is only fair given I was here first. Now, where were we? Ah yes, I'm buying you a drink while we wait. Between sips you can ask me anything you want, assuming, of course, I have piqued your curiosity as you have mine, in which case don't be bashful, ask me anything.\"\n\nDebating whether or not to take him seriously, in any manner, even briefly, I found him impossible to resist, and that irritated me all the more. \"You dropped something,\" I said, causing him to pat his pockets, then look around the floor to uncover what could have possibly fallen. After a moment or two of allowing him to search in vain and feeling quite satisfied with myself for having leveled the terrain, I said, \"Your r. Where are you from? I hear, what, Boston\u2014Toronto?\"\n\n\"And so I did. You've got a good ear,\" said cocky mystery man before ushering me by the elbow toward the bar area.\n\nOnce seated, he slid his chair uncomfortably close to mine and asked the bartender for a Pimm's cup and an herbal tea. \"You're meeting with Michael because of why?\" I asked, thinking that he could very well be here interviewing for the same position as me. I mean, let's be real, men only dress like this in magazines or for job interviews, so I am guessing it is the latter.\n\nWith a slight pivot of the head in my direction only to stop before facing me, he pauses for maximum effect before responding. Seriously, I have never! \"And here I was hoping that you would want to know about me as a person. Such is my lot, I guess. Women see me in this package and somehow it's still not enough. It's always who do I know? What do I do? What can I do for them? When am I going to be enough? When?\"\n\nFor a split second I was taken aback and thought that just maybe this guy might be slightly off. That was until the bartender placed the tea down in front of me and started to laugh. \"Are you shining me?\" I asked. \"You said to ask you anything and so I did.\"\n\n\"Correction, I said to ask me all the questions you wanted, assuming that I had piqued your curiosity as you have mine. The expectation being that we would get to know each other better, not make another man the focus of our relationship, which is never a good move, by the way. It's tantamount to us being on a first date and talking about some dreadful ex who broke your heart and slept with your best friend. Is that what you want?\" he asked, lowering his head in resignation, leaving me speechless again at the bull's-eye. \"Sadly I see I was mistaken and find myself in this all alone. I guess it's better that I know these shortcomings of yours now instead of later when\u2014\" Before he could finish his soliloquy, Michael blew in.\n\n\"Marcus, sorry to keep you waiting. I got stuck uptown with the missus\u2014you know. Jules, hang tight. Have a drink. Raymond, take care of her.\" He said all this in one sweeping movement that not only took my asthma-inducing companion but also left me in limbo to wait on him for an undetermined amount of time.\n\nThe long dial on my watch indicated that it was about twenty minutes to seven o'clock, so I picked up my phone to ring my friend Jessica to notify her that I would most likely not make dinner but surely be there for dessert. Tonight was the big group introduction, as she had recently started dating this finance guy and was now ready to have him meet her friends for final approvals.\n\nRemoving Marcus's drink, the bartender said, \"You should not have canceled. You'll make dinner, maybe not the apps, but you'll make the entr\u00e9e for sure.\" He was a slender gentleman with soulful eyes and the most amazing unblemished deep mocha skin that, like my father's, made his age hard to pinpoint\u2014but I'm guessing midforties.\n\n\"Really, what else can you tell me? Do I have this job? . . . Am I really doing the right thing in coming back to this freaking city now? Was this the right suit to wear? Am I falling apart with a total stranger or is this just a really bad dream?\" I said, sick with anxiety. \"Oh God, please don't tell me that I am sitting in front of you totally nude right now!\"\n\nWith little more than a raised eyebrow to convey that he was well equipped to deal with the brand of meltdown I was displaying, he offered, \"Maybe I should put a little something in that tea to warm you up. Rum?\"\n\n\"Sorry, it's been one of those days, you know. I do that, tend to ramble on when I'm out of sorts.\"\n\n\"It's all right, drink this. It won't put hair on your chest but it'll give you courage,\" he said, sliding the improved drink in front of me. His easy demeanor had an immediate impact on me, which I guess is the prerequisite for bartenders, especially since my grasp on reality was still not as strong as I would have liked it to be. From the enormous mirror hanging over the bar I watched as Michael and Marcus conducted their meeting\/interview at a table behind me, the emptiness of the restaurant enabling me to hear every fifth word or so of their conversation.\n\nAfter a few minutes it was obvious that this Marcus character was not some random guy trying to steal the job I had claimed. His interaction with Michael was far too casual. Although, the more I studied Marcus, the more I couldn't shake the feeling that I knew him, like maybe we had met before through someone significant, but for the moment I could not place him. Shortly thereafter they stood up, did the obligatory man-hug, and said good-bye. I chuckled to myself slightly, a combination of disbelief and disappointment, in seeing Marcus exit. That after all the work he had put in earlier to get my attention, now he made no effort to come over and say a proper good-bye. Well . . . Total wanker.\n\n\"Jules, thanks so much for waiting. Did Marcus try to pick you up?\" asked Michael, and didn't wait for my response. I could see now that this would be an underlying theme of our relationship and declined to accept it. \"Of course he did. Don't worry, he's harmless. Besides, you're a good girl and he wouldn't know what to do with someone like you. Hell, I didn't at his age and let me tell you I met many a good girl,\" said Michael, releasing the same hearty laugh as last night. Officially, they were friends.\n\nIn lieu of the formal meeting I understood we were to have, which was why I came equipped with three Kinko's-printed PowerPoint copies of the proposal, Michael gave me a tour of the restaurant that included a state-of-the-art kitchen that would make Jacques P\u00e9pin proud. The offices, to my delight, were not located in the basement or behind the kitchen as they often are but on several upper levels on the other side of the elevator, accessible only by key card. The rehearsal and recording studios with discreet sleeping chamber were located in the bowels of the building\u2014so very Marvin Gaye of him. He introduced me to the staff on hand. The Imanesque hostess I now knew as Lidia. Michael's girl Friday, Simone; at first glance one can see that she is highly efficient and not in the least bit fuckable (i.e., threatening to an aware girlfriend or insecure wife): tight bun, kitten heels, bare face with the exception of some rouge and a mouth that seemed turned down even when she was smiling. Clearly the answer to why she had been with him longer than any woman in his life aside from his mother. Next there was chef Grayson McClovoy, a James Beard winner who preferred diner food to the five-star meals that were his trademark. By the time he formally introduced me to Raymond, the gravel-voiced bartender, I felt as if we were old friends. And like a proud papa who was too easily distracted, Michael announced that this was his humble little core team and he was sure I would fit right in.\n\n\"I want to open the summer season strong with some Brazilian Corcovado shit while those other motherfuckers are on holiday, you get me? I'll need you set up in the office by next month . . . Walk with me,\" said Michael as he moved toward the lobby. \"I won't be here, but I am reachable to you through any number of privacy-killing methods of technology\u2014just don't video-conference me. I hate the way I look on that thing. If for any reason you can't get me, go to Simone. She actually runs this place. The missus and I will be in the Mediterranean. All right, that about does it. See you in a few weeks.\"\n\n\"Michael, Miiichael,\" I said as I struggled to gather my things and follow him to the elevator, only to have the doors close in my face with him smiling on the other side like the Cheshire cat.\n\nAlone, standing at the base of the stairs, breathless, I ran aloud the catalog of unanswered questions in my head: \"What just happened? Did he just offer me the job\u2014but how? What is my salary? Oh my gawd, who can pack up their life and move to another country in two weeks? I mean honestly, who? Me! Oh shit, am I really doing this?!\" I may have been caught off guard but clearly not lost for words.\n\n\"You sure do ask a lot of questions,\" said Raymond. \"That's Mike. You'll get used to him. I have worked for some of the biggest and best but he is the perfect mix of Sinatra and Dean. By comparison everyone else is a Bishop or Lawford.\"\n\nSurely, I must have looked at Raymond like he had two heads. What the hell is he doing talking about the Rat Pack when the limbo that is my life just got a little crazier and no one is forthcoming with the most precious details of HOW? I gather my things and grunt something inaudible that was meant to be \"Good-bye and thank you for the tea\" before racing up the stairs to try to get one last moment with Michael. Much to my dismay he was already being abducted by a bustling set of diners. Having canceled my dinner with Jessica, I decided to return to the hotel and just come back to the restaurant around 6 p.m. tomorrow for details.\n\n\u2022 \u2022 \u2022\n\nWhen the state of your life is undergoing great reconstruction, the most desolate place to be is a sparsely decorated hotel. Oh, excuse me, contemporary-decorated hotel. Aside from Jessica, the only people to know I was in town were Blake and Richard, both of whom were unavailable tonight. And even with time to spare I didn't want to show up at Jessica's midcourse, only to interrupt the flow of her dinner. Thankfully I always travel with candles, bath salts, a classic movie, and a cashmere blanket Cora gave me one Christmas\u2014you know, all the essential accoutrements to make any place feel homey. Opening the door to my room, I was greeted by the most decadent floral scent before laying eyes on the large bouquet of tulips sitting on the coffee table. Ah, Tony. It's just like him to do this. Sending an arrangement of my favorite flowers was a beautiful bow of sorts to our story. Ending of a relationship or not, they are gorgeous. There had to be at least fifty stems of various marigolds, purples, whites, and pinks. I could not help but smile and closed my eyes for a moment to just enjoy meaning enough to someone to evoke such a gesture. Obviously, I was wrong:\n\nWelcome to the team, Jules. Call Simone to work out your move.\n\nMK\n\nOkay!!!!! This is one of those days (and nights) apparently. You know the kind, where you feel that everything was already in motion before you even got out of bed and no matter how hard you try to catch up you remain behind everyone else. So you resign yourself to accept that the deck is in fact stacked in another's favor and, although you may want to dissect Sun Tzu's army ball one soldier at a time, there are far too many of them and only one of you. I won't even ask how it is that Michael knew my favorite flowers were tulips, because it was obvious that I would be grasping at straws. When the hell did my life turn into a Cathy comic strip where I'm frazzled and scratching my head?\n\nNext to the flowers sat an oversize white envelope thick enough to hold a contract. Given all of the surprises of the last twenty-four hours, I was apprehensive\u2014okay, downright panicked\u2014to look inside immediately; for all I knew, the results of my recent gyno exam might have suddenly been telegraphed across the pond, made their way to Michael's office, and now he was the bearer of my female news. Who was to say? So, instead of opening it, I returned it to the exact same position I found it, tiptoed to the bar, poured myself a sizable glass of wine, and ordered room service. My food came quickly, allowing me to focus on something other than the magic envelope that had mysteriously levitated and found its way to the bed under the left pillow next to my phone. Soon all that remained were a few spoonfuls of butternut squash soup with truffles, a half-eaten burger, and an ivory puddle that once was apple pie \u00e0 la mode. All in all my belly was filled and my spirits soothed but my mind was a jumbled, singularly focused, erratic mess. I needed to know what was in that envelope. Was Michael making me an offer competitive enough that I could make this move back to New York in style and silence any potential objections from Cora in the process? Or would I be forced to exist on a salary far lower than what I am currently making with the Conrad Hotel Group, and have to live in some modified efficiency (translation: small enough that I could shower and flip bacon on the stove simultaneously); or a \"conventional\" one-bedroom walk-up in the sketchiest part of Alphabet City that would force me to wear track shoes on any given day in order to avoid stepping on a stray heroin needle or coke blade? I need this!\n\nAll my life it seems I have been adjusting to everyone else's playbook of priorities and deferred dreams. I went to undergrad at BU (Boston University) because my dad said he had the best years of his life there in grad school. I came to New York the first time to live out my mother's dream, because she was deprived of her Mary Tyler Moore life by getting married and starting a family so young, a not-so-little detail she mentioned more than a few times over the years, namely when Daddy would criticize her spending. I went to London to escape Tony and, in the process, found me. Now I just want to return home, on my terms, wiser and more aware than before.\n\nUsing my soiled napkin to wipe the remnants of mayonnaise and ketchup off the butter knife, I sliced the envelope open while saying the most pitiful prayer ever heard: \"Dear God, I promise that if you make this offer good I'll never ask for anything else . . .\" With only one eye open, I scanned the letter for the important points. \"HOLY SHIT!!\" I screamed, loud enough, without question, for adjoining rooms to hear. Not only was the offer good, less than $10K or so difference from where I expected my raise to be, but it came with serious perks. It allotted for proper moving expenses, an apartment in the city for the term of my employment (with an option to rent it at fair market value should I no longer work at Carly's), and a sizable entertainment account to be used for any number of things. Officially, I am moving back to NYC and, unlike my last move, this time I am not running away from something or someone. I'm running to my future!\n\nRiding high on the fumes of possibility, I was unable to sleep, so I stayed up making lists of all the things I could do over the next two days. Priority No. 1 is to get reacquainted with the city by doing all of the things I had once found a volume of excuses not to do when I lived here or missed when I was in exile: visit the Met, have a banana smoothie at New World Caf\u00e9 on Columbus, visit the Museum of Natural History, walk down to Coliseum Bookstore, take in an outdoor movie at Bryant Park, go to the Paris Theatre, and even hop the Staten Island Ferry to people-watch while getting a glimpse at Lady Liberty. Shoot, I just may go over to Ellis Island\u2014if it's open. On Saturday I would dine at Cafeteria with a hardy group of friends and then head out for a night of dancing as if there were no tomorrow. Sunday I reserve as the perfect day to get lost in a marathon brunch down the street at the Cub Room with Richard, one of my nearest and dearest, before heading to JFK for my return.\n\nIn addition to being one of the best political speechwriters in the business, Richard Boulton has the distinct honor of being the only Ivy League gay Black American card-carrying Republican I know. I mean really, who in their right mind is not pro-Clinton right now? Bill or Hillary? Really, who? His heart and capacity to love is more expansive than anyone else's I know. In the tough times he was always there for me and never outwardly judged my decision to stay with Tony during the initial onslaught of not-so-perfect moments that was clear to everyone else but me. When I escaped to the hotel to hide from Tony, Richard was the only person I called. Not to mention that the man can cook his ass off. It's a fact. In the beginning, when I was trying to impress Tony with my culinary skills, it was actually a Richard Prescott Boulton home-cooked meal, prepared in my kitchen of course, so that I could have evidence, by way of dirty pots and delicious smells filling the space. I remember his first words at the start of dessert. \"Damn, girl, my mama don't even make cobbler this good.\" He bakes cakes from scratch and spends his weekends in a place named Mantoloking. I mean, have you ever? I worship him and needed to bask in his goodness.\n\n\"Darling, when are you coming home? London is absolutely dreadful and the men are even worse. Come home. We miss you. I can't continue to carry the reins of fabulousness all by myself. Well, actually I can, but I don't want to,\" Richard said at brunch.\n\n\"May thirteenth,\" I said. \"I'm just going back to give notice and pack my things!\"\n\n\"I'll drink to that,\" said Richard.\n\n\"Of course!\"\n\n## 10\n\n* * *\n\n## HOLLAND PARK\n\nWHAT IS THE deal with that five-and-a-half-hour flight that just knocks me forever on my ass? Standing in what had to be the slowest of the Customs and Immigration queues at Heathrow did not help matters as my head was splitting, which could have had something to do with the two rasmopolitans and two celebratory Bellinis that Richard and I had before my airport transport arrived. Too much sugar, I need to lie down.\n\nMy flat is located in Holland Park, in a whitewashed Victorian town house, and of all the things that I do genuinely treasure about London it's at the top of the list. When I first settled on the location it was with the expectation that I needed something calm and far removed from my previous existence with Tony. Whereas we were modern and fast-paced, I wanted old-world and serene, the kind of location that would allow me to get lost in some well-deserved soulful moments. The interior had been restored to the original architecture, with intricate crown moldings and expansive built-in bookcases in the main room and bedroom. The only drawback is that it was a fourth-floor walk-up. One would think that coming from New York I would be used to that, but my building then had an elevator. What can I say?\n\nInitially when Blake told me to contact Michael, it was in response to an offhand suggestion I had made.\n\n\"You know, I should move back to New York. There is really nothing here for me in London. Lord knows I have not had a decent blowout since I moved here. What's the deal? Is England anti-Dominican or something? You would think that with all these Africans, Jamaicans, and Middle Eastern people there would be one on every corner, especially in Brixton.\"\n\nLaughing, Blake responded, \"You do know you are stupid, right? That's not a bad idea, though . . .\"\n\n\"And how would I go about importing a Dominican, Blake?\"\n\n\"Not the Dominican, crazy. You moving back to New York, it's not a bad idea. We are ALL here and the city is hot again. You could easily find a great position. I will make some calls.\"\n\nAt the time I did not give much thought to it because it was not realistic in any way, or so I thought. Blake and I always talked a mile a minute, and the ideas flowed faster than Niagara Falls, mostly from her.\n\n\"Let's open a B&B in Vermont. I am over NYC.\"\n\n\"I am ready to start my own clothing line. You and Jessica should join me.\"\n\n\"Did you read about such-and-such's new club? They asked me to come in as an equity partner. I think it is a good move, don't you?\"\n\n\"Let's go to Kenya and import some Maasai fabrics.\"\n\nMany, many ideas we've had over the years, and it never matters who originates them because 99.9 percent of it is just hot air and we know it. So when I hinted at coming to New York and she vowed to make some inquiries, I allowed it to be another grand illusion and didn't think to speak to my superiors at Conrad about desiring to move on, even when Michael confirmed he would meet with me. Undoubtedly my decision will come as a surprise. I wonder who they'll pick to replace me? Most likely that cute little pixie Rebecca Ryan, who transferred to PR when I joined. Initially I had reservations when tasked with the responsibility of starting a new position and training someone with relatively no marketing\/special events training\u2014or style for that matter\u2014but she proved to be a quick study and hard worker. I was glad to have her on my team, that's for certain. After a few work lunches at Selfridges and Harrods, she even started to show the signs of becoming a burgeoning fashion plate with her own style.\n\nOn Monday Mr. Conrad (Hershel to his friends) called me into his office to express his regret at my impending departure. He wasn't happy about the imposition but said he understood. In appreciation for the work I had done he let me know that should New York prove too naked of a city, I could return \"home\" at any time. It felt good to know that he valued what I had done, so I didn't have the heart to tell him that New York was home\u2014I just ran away briefly to gather myself.\n\nAt the risk of pushing Hershel's generosity, I asked to work half days in order to pack up my flat and be available to all things New York as soon as the city awoke for business. Michael's secretary, Simone, proved invaluable in the transition, sending me shipping account numbers, paint samples, and even selecting my assistant, Jacklyn (pronounced Ja\u2013clin) Travers. I'll bet anything she is one of those combo babies\u2014you know, dad's name + mom's name = what the hell?! Jacklyn, a recent grad with impressive internships at Cond\u00e9 Nast and Ralph Lauren in the Special Events and Marketing departments. In time those skills would be put to the test, but for now her most important job is to oversee the painting and setup of my new apartment.\n\nLocated on the Upper East Side\u2014not my desired location, but hey, who am I to complain\u2014the apartment is a few blocks down from Michael and Carly's penthouse. Actually it's one of theirs. That's right, my new boss and his wife, whom I had yet to meet, are also my new landlords. Oh, the smell of incest in the morning! Michael's rationale being that with only a few weeks' notice there is no time to secure a proper residence and focus on learning the ropes of the job. And even if there were, given the nature of real estate in Manhattan, after an exhaustive and lengthy search, I'd end up living unhappily in DUMBO or Hoboken all the while running up a very sizable hotel tab on his dime. Let's be real, even with my raise there is absolutely no way that I could have afforded an apartment like theirs, so I'd be a fool not to take it.\n\nListed as a two-bedroom, it was much larger than my current flat, complete with an open-plan living room\/kitchen, butler's quarters (i.e., guest room mini), a separate family room (i.e., my office and TV room), a decent-size dining room, and two full bathrooms. The only drawback that I could foresee was its proximity to the street. Apartment 8A is a corner unit with more than half of the property's living space on the Fifth Avenue side (too much freaking traffic all the time) instead of Seventy-third Street. An influx of images of New Yorkers leaning incessantly on their horns no matter what hour of day or night as if their lives depended on it made my ears hurt. Maybe I would have Ja-clin make sure the windows are double-glazed, or maybe I will just shut what they call the hell up, count my blessings, and finish with the last load of boxes.\n\n\u2022 \u2022 \u2022\n\nDespite this move being of my making, there were moments, fleeting at best, that I found it hard to fathom the little life I've built here coming to an end. London taught me how to breathe again. Had it not been for the pond at Kyoto Garden in Holland Park or the destinationless (obviously not a word but let's pretend) strolls down Regent Street, allowing me to be consumed in a voyeuristic moment, I would remain emotionally adrift. Feeling pity for myself because I decided that my personal best was left behind with Tony\u2014actually, that he had taken it. That is, I think, the cruelest thing about losing intimate love\u2014when in it, I truly believed that I was giving not just my realized best but my all, and when that love came to a bitter end, all that was left was a victim in place of the heroine I once envisioned myself to be. I felt as if I had been robbed of everything that made me meaningful. Eventually I accepted that it was me and not Tony who recast me as a secondary player in my own life.\n\nMy rebirth coincided with a fertile season, Tudors boldly blooming, the most breathtaking roses in vibrant colors that defied belief for anyone who saw their bare branches during the frost, I no longer felt the need to hide from my own image or believe the barrage of negative things about myself that my subconscious dispensed freely. Even when I had a relapse emotionally (and I did from time to time), it immediately felt wrong in some major way to dwell in pain when the world around me was screaming with such beauty and vitality. For the first time in what felt like forever, I could appreciate the beauty of simple things, the sun's warmth on my face, establishing eye contact with a passing stranger without the protection of oversize sunglasses, or taking a huge breath not because I was trying to stop time but because I wanted to feel life course through my system and with my exhale release any fragments of the reservation that was trying to block my happiness.\n\nFor the first time in my adulthood, I knew that I could depend on me no matter the terrain as long as I had my work to fall back on. And so, bit by bit I started to open myself up to conversations with people on my outings to Notting Hill, be it for carnival or to shop. At the Blakey and May Fair for drinks (because, no matter how depressed, one should never drink alone . . . too often), I left the book at home and smiled. I met people who through their stories taught me the invaluables I hoped not to forget once the geography was but a memory. The trick, as I was told, is being able to move on quickly and purposefully when the union brings more innate grief than joy. The next was taking responsibility by identifying the lessons learned and finally dreaming a bigger and better reality by painting a vision on a much broader canvas where I was neither the victim nor the damsel but the author.\n\nLabeling the final box, I took heed of the flashing red sectors of my life that I was not ready to advance toward. I've never known how to be okay with just being me until now. From the moment I arrived in New York, I was in a relationship, be it with Tony or the city itself, despite my cries of independence. Moving to London, not knowing anyone and not having a partner to spend countless hours on the phone with about the issues of transition, showed me just how codependent I was. So if I look at things from that perspective, I must admit that London gave me far more than I gave it. I am stronger like those trees over the way in Hyde Park, changing with the season. When it appears they are at their most vulnerable, dry, brittle, and weighed down with ice from winter's chill . . . a change of season comes, and they blossom from the inside.\n\n## 11\n\n* * *\n\n## WELCOME HOME\u2014I'M BACK\n\nHAD IT NOT been for an accommodating bearded doorman in a green stitched felt hat with matching topcoat and white gloves standing at the ready when my taxi pulled up, I surely would have continued questioning if I was in the right place.\n\n\"Good afternoon, ma'am. Let me help you with that.\"\n\n\"Thank you. I believe this is my new home,\" I said.\n\n\"If that is the case, then that would make you Ms. Sinclair in Eight A.\"\n\nImpressed more than surprised at the level of service, I responded, \"Yes, it does, and you are . . .\"\n\n\"The name is Percy. Been here nearly twenty years, so I know everything that is to be known here.\"\n\n\"Well, in that case you can tell me if I will be happy here,\" I said.\n\nTaking a moment to evaluate my person before presenting me with an envelope containing the keys to the unit, Percy said, \"Yes, ma'am, I do believe you will be.\"\n\n\"Then tell me where to go.\"\n\nThe photographs Jacklyn sent of the apartment did not do the space justice. Location notwithstanding, it was loaded with the kind of potential that would perfectly complement my interior aesthetic for all things bohemian chic and contemporary. Pungent still was the smell of fresh paint from earlier. I should have thought of that and told her to come by and air the place out so it wouldn't be so overwhelming, but that escaped me in all the commotion.\n\nThe main door opened onto a long hallway with bare walls ideal for displaying some of the art I planned to collect (item 2 on my \"all the things in New York I wish I had done before but didn't do and now would\" list\u2014right behind making better girlfriend choices). Similar to my flat in London, there were expansive windows, built-in bookcases in the family room\/office but not in the living room or sleeping areas. The teak floors played well against the caf\u00e9 au lait walls and white crown molding I had had done. Unlike London, the kitchen and baths had been completely remodeled with granite, stainless steel, and exposed cabinetry. The only noticeable drawback was that my unit didn't have a washer-dryer, but there was one in the basement.\n\nUnpacking proved far easier than I expected, a major advantage of traveling light to get over heartache and of treasuring little. With the exception of the clothes on my back and those in my carry-on, the remaining evidence of my life had been sent a few days earlier. Aside from a few fabulous flea-market finds\u2014clothes, shoes, and purses collected as part of my self-prescribed therapy regimen\u2014everything else would have to be new. The majority of the furnishings I bought in London were more about function than form and reflected a part of my life that I needed no reminders of. Lying in between a Vuitton monogram bag and a vintage Kamali romper was a holdover sweater from Tony's closet that managed to survive the shredder that ate our photos. Now, looking at it against the blank canvas of my emotional landscape, it feels ill placed and no longer necessary. No more did I need a memento of what it felt like to belong to someone in order to know that I was of the world. The fact that I awoke now each day not initially thinking how quickly can I get through it and come back to bed was more than enough.\n\n\u2022 \u2022 \u2022\n\nWith less than a week to get fully settled and no welcome wagon in sight, I need to make this place livable or at the very least sleepable in a day. I mean, the expansive hardwood floors were nice enough to look at but not optimum for R&R. The sheer volume of windows alone guaranteed me no protection from the elements of sun and car horns when trying to sleep in late on a lazy weekend morning. Had it not been for Richard and his masterful OCD blueprint on the best one-stop shopping locations to purchase all things home-related, I would have been a sad sack of domestic ignorance.\n\nDarling,\n\nWelcome back to civilization and the land of men with straight teeth. Obviously, much has changed and you will need to get adjusted. Here's a little list of some stores to get you started.\n\nYours truly,\n\nRichard\n\nTopping his grid was Bed Bath & Beyond\u2014the latest mecca of all things home furnishing and less draining on the purse strings than ABC. Richard's craftiness went beyond bold script, bullet points, and highlighting; he even supplied street addresses of stores and cross-referenced the locations by inventory. The only thing missing was an in-store directory detailing which items could be found in aisle 5 as opposed to aisle 3.\n\nThe magnitude of the store was more than a bit intimidating, but easy enough to maneuver, enabling me to find the most divine 400\u2013thread count Luxe Versailles Jaipur cotton sheets and Bailmour comforter along with enough pillows to make me forget that I still slept alone. The Sofitel dream bed I'd preordered from London would be in perfect company. Luckily, I was able to make it to checkout moments before the store's same-day delivery cutoff took effect. With a three-hour window before everything arrived, I made my way to Carly's only to find the doors locked. Its entrance accessible only by a keypad requiring a code I had yet to memorize. Instead of returning home and staring at my watch, I walked over to Pastis for a late lunch.\n\nBack in the day this was my default kitchen, a little piece of France in the heart of the Meatpacking District. The crowd was always relatively the same: West Village neighborhood patrons and hip Upper West Siders who braved the C or E below Midtown to mingle with the artsy set for a bite of braised beef with carrots and sips of C\u00f4tes du Rh\u00f4ne with lively conversation. In all my years of coming here I'd never made a reservation, which was crazy, considering the place was always packed, and yet the staff always managed to find me a table. Today, however, may be the one time that lady luck was not on my side. As far as I could see, every table and bar stool was taken. My hopes of ensnaring one were rapidly evaporating. Of course I could use the wait time to people-watch, which was one of the major draws in coming here, but what fun could be had standing alone twiddling my thumbs, overtly staring at people? The whole idea of inconspicuous observation is to have a decoy or distraction method\u2014i.e., the menu, a magazine, food, a glass of wine, a semi-interesting companion\u2014while positioned comfortably, blending into the fabric of things, and not literally standing out. As I stood contemplating the best location to wait, a familiar but not readily recognizable voice greeted me.\n\n\"You know, it was rude of you not to give me your phone number, but I forgive you,\" he said.\n\nTurning around to see who was speaking and make certain that I was indeed the recipient, I came face-to-face with Michael's friend from the restaurant, Marcus. Temporarily I was lost for words. Apparently I am still doing that at the sight of attractive men. He was more handsome than I recall; definitely a James Bond\u2013esque panache about him, if Bond were ever to be blond, tan, and smoking hot with a come-hither stare. I recognized his cologne immediately, Clive Christian, a brilliant blend of success and power in every conceivable way. Also a clear sign for me to make haste and just feign ignorance. About a year ago I discovered it at Browns while engaging in some much-needed retail therapy. The spray guy was supercute and I was in desperate need of some male attention, so I walked over and freely gave him my wrist. In return he gave me a bill for \u00a3210. Desperate times, I say. The scent conjured up the image of the man I longed desperately for but fear I'll never find, having mucked it up so badly already, so I did the next best thing and handed over my credit card. In the times that I was sickened to be sleeping alone, I'd spray some on to usher in sweet dreams of coupledom.\n\nThe Achilles' heel in my selection of men was that I definitely preferred Type A's (emphasis on A)\u2014attractive, alpha, arrogant, ambitious overachiever just shy of asshole\u2014and this guy checked all those boxes and then some. I still didn't know exactly what he did for a living, but I KNEW HIM and his type all too well\u2014flashing red lights! Actually every man, starting with my first love, Adrian Perry, was some variation of him\u2014natural-born swagger. So, having been down that road far too many times before with only my battered heart to show as proof that I had loved and lost, it seemed best to neutralize this advance from the onset and pretend to not remember him.\n\n\"Excuse me,\" I said, coolly detached, \"but I think you have me confused with someone else. Sorry to disappoint.\"\n\n\"Quite the contrary, I can't ever see you being a disappointment,\" said Marcus, seeing through my staged performance of selective amnesia. \"You wore a pale pink pantsuit, almost blush, with a graffiti top. Your hair was tousled in one of those intentionally messy yet completely fussed-over updos like a morning-after. But what I remember most is that you smelled of Opium. My mother wore it when I was little, so I am entranced by any woman who can carry it off as she did.\"\n\n\"Well, don't you have a great memory?\" I responded, devoid of expression.\n\nHis disbelief obvious, Marcus responded, \"Amazing. You don't like me, do you?\"\n\n\"I don't know you.\"\n\n\"Which means what? You would like to? You'd like me.\"\n\n\"Listen, as touching a comparison as that is\u2014the bit about your mother\u2014I try to refrain whenever possible from talking to men with poor manners. I mean, honestly, who puts in that much effort and then leaves without as much as a good-bye?\"\n\n\"So you do remember,\" Marcus said triumphantly. \"I knew you did.\" The smirk on his face was just enough to reveal a dentist's wet dream. Clearly he was more than up for the challenge and quite enjoyed hunting his prey before conquering.\n\n\"Vaguely.\"\n\n\"If I gave you inside information, would that make us friends?\" he asked.\n\nAs I opened my mouth to respond, his leggy companion arrived. Completely typical, why am I not surprised? If I had to guess, I would say that she was requisite factory-grade SBA\u2014Super Blond Amazon\u2014hailing from some region in the Eastern Bloc where the grasp of the English language is just enough to make men (who should know better) turn over the keys and security codes before any talk of a prenup is had. Marcus, on the other hand, seemed undeterred by her presence, continuing, \"Michael said you were a beast at PR\/marketing and that he would be a fool not to hire you.\"\n\n\"Well, that's cool, I guess?! . . . It was lovely to see you again with your girlfriend.\"\n\n\"Yes, Marcus,\" he replied, slowly and demonstratively, as one does when teaching a toddler a new word. Not taking his eyes off me. \"That's my name, Jules Sinclair. Should you forget again, Marcus.\"\n\nI can't be certain, but I think I broke into a slight run when the hostess appeared to show me to a table, anything to put some distance between Marcus, his SBA, and whatever little flutter occurred in my stomach as I looked at him. True, I may have been off the emotional (and intimate) market for the past few years, but I didn't forget what butterflies felt like or their significance\u2014although I was starting to question if I would ever know what sex felt like again. That's right, I still had yet to put another man, no matter how superficial, between Tony and me with a proper screw.\n\nWhen not consumed with the emotional loss, I thought of nothing else in lucid moments but the physical deprivation of my body. In the middle of the night, my back would unexpectedly arch as if preparing to receive him. Showers and mornings proved the worst, when my fingers would automatically trace the contours of my body as if Tony still had possession of it, starting from the nape of my neck to the roundness of my breasts, only to linger between my legs until I was near explosion. With Tony I learned to appreciate the wonders of my body with the lights on, oftentimes with him watching. He always complimented me in bed, and that made me feel like the best he'd ever had. It's hard when you have been with the best to risk returning to average or unappreciative, so I pushed all that desire to the deepest crevices of my mind and went on with my life.\n\nIf anything, the time unavailable had only intensified my awareness of how few and far between those genuine feelings of interest and desire are. True, I met many men during the course of work or personal outings, some of which could have justified having a quickie. None sparking the slight tickle in the base of my stomach on sight or memory of him, or the faint twinge of disappointment that occurs in parting; none to date except this guy, Michael's friend, and that has trouble written all over it.\n\nDuring the cab ride home my thoughts did briefly\u2014hence, the entire ride\u2014return to Marcus. Throughout lunch, I caught sight of him, and indeed he proved to be a fascinating one. Easily, his self-possessed nature allowed him to own the room without overpowering it, a feat in and of itself. His mannerisms were purposeful without the appearance of trying. He was even personable with the staff. According to Daddy, one can tell a lot about a man's character based on the way he interacts with serving staff: waiters, maids, doormen, drivers, salespeople, and the like. If he is dismissive of them, then no matter how gracious he may appear to be on the first few dates, soon, asshole tendencies will emerge. Of course, if I was interested in him, then this would be a positive, but since I am not, then it is merely an observation. Much like the one I made in concluding that he and the SBA were together but not together. From my POV in the adjacent corner along the far wall it was easy to see that she was working a bit too hard for his attention. I mean, honestly, if this Amazon tosses her hair one more time I just might have to say something. And don't even get me started on the gratuitous, definitely vulgar use of food around her mouth. Seriously! Eat the damn french fries already!\n\nSomewhere between my being transfixed on their body language and wanting to snatch her off to the bathroom to give an albeit rusty yet decent tutorial on \"How to Be Comfortably Alluring During a Lunch Date,\" Marcus looked up squarely in my direction and raised his glass as if to say, \"I know you're watching,\" leaving me once again to conclude that proper people-watching cannot be done alone. Caught and having no other recourse, I smiled back meekly and flagged down the first waiter I saw to get the check and leave. Clearly not the best day for Pastis; the strawberry shortcake would after all be there far longer than he would. I'd just come back when they were not present.\n\nExiting the taxi back uptown on Fifth, I saw there was time, not much, but a few minutes or so to spare before the deliverymen were scheduled to arrive, so I popped across the street at the park entrance to grab a vanilla soft-serve cone. Percy was still on duty.\n\n\"Ms. Jules, deliverymen from the Sofitel are in the maintenance lounge for you,\" said Percy, as I entered the building savoring the final remnants of the ice cream.\n\nA few steps into the lobby I stopped midstride, acutely aware that I had absolutely no clue where the maintenance lounge was, so I doubled back.\n\n\"The maintenance lounge . . . what is? Where is?\" I asked.\n\n\"Ah, yes, ma'am. See you need to take me up on that tour of the building. The maintenance lounge is in the rear of the building down this hall to the left. It's a holding area for delivery and service staff.\"\n\n\"Really now? Fancy,\" I said.\n\n\"Yes, ma'am, only the best round here. You go on, I will send them up.\"\n\n\"Thank you, Percy, I think I am really going to like it here if everyone is as nice as you. Oh, by the way, you don't have to call me 'ma'am.' \"\n\nReaching up to tilt his head at me, Percy responded, with great affect, \"Yes, ma'am, I do. It's my job and I like it.\"\n\nOver the next couple of hours the delivery from Bed Bath & Beyond arrived as well, allowing me to put everything in its place before complete nightfall. Between the jet lag, the day's field trip, and unpacking, I was pooped. Before going to sleep, I fired off an e-mail alerting the gang that I was back.\n\nHello Lovelies\n\nIt's been a while but the wait is over. I's here and in need of my fam. Before anyone asks, yes, the rumors are true, I am on the Upper East Side, so YES you must venture above Midtown at some point and let's bypass the fretting. No stateside number yet but am reachable on the UK mobile. It feels good to be home.\n\nA zillion X's & O's\n\nJules S.\n\n\u2022 \u2022 \u2022\n\nPleasantly, Sunday morning arrived without the intrusive horns and sirens that I expected. Though not completely inaudible, the street noise was far less than I feared and didn't wake me through the night\u2014or maybe I was just so freaking tired that I could have slept through an AC\/DC concert at Madison Square Garden. With no food in the house and no desire to go purchase any, I dressed for breakfast and headed down to my SoHo destination, Cub Room. En route I grabbed a few glossies and city magazines to get an overview of what the evening dinner scene was in print. This would undoubtedly be the resource for visitors to the city. I wanted to understand what they were seeing and whether Carly's was part of that dialogue. Of course I would get the local intel from friends and casual \"about town\" hipsters soon enough.\n\n\u2022 \u2022 \u2022\n\nThe Cub Room's interior remained the same as that of a ski lodge, but that was it. The staff I knew was long gone. I didn't recognize anyone but did see a friendly face among the diners, semihidden behind a newspaper.\n\n\"And you didn't even call to say you would be here. Honestly, I am hurt, appalled, and lost\u2014lost for words at the utter disrespect,\" I cried.\n\n\"Oh dear,\" Richard said in an exaggerated Savannah drawl, \"I can see your mouth is rested but the rest is delusional. Sit, your drink is lonely.\"\n\n\"Merci,\" I said, and settled in for a marathon Sunday. Yes, it felt great to be home again, to be a part of something that is always good. Richard and I are one of those always-great things. Although I must say I often take more than I give.\n\n\"Now, darling, tell me, when did it become okay to leave the house looking like this? I mean, what, are you rapping now or something? A tracksuit? Tsk tsk. Is that what the queen is wearing these days?\" Shaking his head with disapproval and hissing, he said, \"I don't think so. It seems we have a lot of work to do now that you're home. Let's make a list.\"\n\n\"What? It's Adidas,\" I exclaimed, tugging at the label name strategically sewn below the left shoulder.\n\n\"Honey, I don't care if it's Valentino. It's still wrong.\"\n\n\"You're an elitist,\" I said, ending with an exaggerated furrow of the brow.\n\n\"Hmm, I am glad to see that at least you still remember how to do the pout 'n' furrow,\" interjected Richard between sips of his Bloody Mary and while still pretending to read the Financial Times.\n\n\"Of course, I just wish you gave a workshop instead of wasting your time writing speeches for a bunch of boring politicians. Yesterday after Bed Bath & Beyond I went to Pastis for lunch and observed this guy who should know better with an SBA severely in need. Dreadful.\"\n\n\"Ewh, they are back in force, you know. For a moment I thought the Asian girls were going to be the dominant dames on the scene but someone unloaded a freighter full of Nordics and God are they ever in need.\"\n\n\"Heinous creatures, they are,\" I offered.\n\n\"Which reminds me, by the way, I must take you to Pravda. It's cavernous, filled with Russians\u2014part of the room likes boys, the other half likes your kind and they all speak in code so repeat nothing. In any event we will have a ball! Cheers,\" said Richard, folding the paper in half and placing it atop the already explored stack.\n\n\"Interesting, but I'm not sure if I am ready for all that just yet,\" I confided.\n\n\"Oh dear, don't tell me you are still pining over that Bob Marley wannabe Antonio, are you? Darling, an ocean, two years, and a few decent lays later . . . let's move on.\"\n\nI could not help but respond, completely deadpan, \"I see your job at Hallmark fell through. Too bad, really you are so amazing with the whole sensitivity thing. No, I am not still pining over Tony. I just have not found anyone to fill the space in between is all. I saw him, you know, when I came in to interview. He smelled so good!\"\n\n\"Oh, yes, there is nothing better than the scent of 'rip my heart out of my chest, Rasta betrayal' in the morning to get the juices flowing. Honey, tell me you are not going back down that road. We lost you for far too long. Last time everything happened so fast. We all stood by and did whatever we could to support you, but this time you have to know better. Don't you?\" asked Richard in a surprisingly fatherly tone, complete with disapproving head nod.\n\n\"Relax yourself there, Boulton. We saw each other. I almost choked on zucchini bread. We had the long-overdue talk. I inhaled deeply probably for the last time so I could lock the smell of him into my memory, and then we said good-bye. He finally gave me the closure I needed.\"\n\n\"Well, that's imagery. My dear, one day you will learn that you are the only one who can give yourself closure; no one else. I'm a wise old owl, so I know what I am talking about. In time you will accept that you hold the keys.\"\n\n\"My goodness, what the hell have you been doing while I was away\u2014reading Confucius or something! Enough now, enough,\" I said. I heard Richard loud and clear. He knows it as well, but I was far from ready to live in the reality of those words, so I offered, \"I missed you, my friend.\"\n\n## 12\n\n* * *\n\n## MEETING THE NATIVES\n\nBEING THE NEW kid on the block when the boss is away definitely has advantages. It gives you the time to form your own relationships without an overseer, forcing people to conform and mingle. Initially Michael was due back within a week of my arrival but decided to extend his trip for two additional weeks after he and Carly ran into Lord So-and-So Hoity-Toity in Capri, who asked them to join his entourage on one final jaunt up north to Croatia. I mean, who in their right mind could possibly say no to that kind of invitation? I would like the option to, though. For the most part the team at Carly's was nice. There was no apparent divisive line between the kitchen, front of the house, and executive office the way there often is in places like this. The only noticeable tension arose when Simone Phillips\u2014aka Girl Friday\u2014ventured down from her ivory tower for inspections. After some well-heeded water-cooler conversations, I choose to engage Simone rather than evade her despite her obvious lack of warmth.\n\n\"Afternoon, Mrs. Phillips, a beautiful day, isn't it?\" I'd say.\n\n\"That would depend on how you got here, now wouldn't it? For my part I was stuck in the freaking Lincoln Tunnel for nearly an hour because of all that dagnabbit construction. I swear to you that creep Giuliani is going to be the death of me yet.\"\n\nWhere does one go after that? Nowhere, so I offered a halfhearted \"Yeah\" and kept it moving.\n\nIn the encounters that followed I let her set the pace and always referred to her as Mrs. Phillips until the day that she told me to call her Simone. It seemed we had reached something of an accord.\n\n\u2022 \u2022 \u2022\n\nEveryone I've met in the apartment building falls firmly into one of three categories: ancient, married, or fabulously gay. In \"single woman\" terms this means that the common areas (hallway, laundry room, elevator, library, etc.) are tantamount to Switzerland; completely neutral, an unexpected perk that says \"no primping required when roaming the halls\" on the off chance that Mr. Right, Mr. Possibility, or Mr. Scratch My Itch are on the premises. There is Mr. Leon Sol, who has to be as old as the building itself. I see him every morning on my way to work with one of his many verbally abused nurses. Each time I smile and offer up a chipper \"Good morning. How are you today?\" In return he grunts, sucks his teeth like a West Indian, and cuts me a sideways glance. A nasty habit he obviously picked up from one of his helpers. On exiting the elevator, his nurses always shoot me the embarrassed sympathy gaze, as if trying to apologize for his lack of social graces.\n\nThen there are the married ladies who lunch at Fred's and Le Bilboquet daily as if it is a career to be mined for societal advancement defined by seating placement. For the most part they range from midthirties to late sixties and are totally insular. Their names fall somewhere between Muffy, Kinsley, and Taylor. Their collective identity is validated each time they can find someone to exclude, and at present that person is me. Within days of moving into the building I had my first encounter with one of them, Mitzy Bloomfield.\n\n\"Jules, right? The girls and I heard someone new was coming into the building, so I guess that is you,\" she said in all of her fake-enthusiasm, Lily Pulitzer glory. If this were Connecticut and not New York, I would swear her last name was Stepford. \"Normally, we have a full background on whoever moves in as they have to come in through the committee, you know. My husband, Randall, is the cochair, you know. This is after all a building of families with strong lineage and extensive ties to the city. We are very selective. Whereabouts did you and your husband come from?\"\n\n\"Virginia by way of London. Single\u2014me. I'm not married.\" Judging by the way she shrieked in horror, I feared for a moment that she was experiencing a shooting pain up her left arm.\n\nWith gravely exaggerated compassion she recovered to say, \"Ooooh well, it's not for everyone, is it now?! So I take it you're one of those career girls on the fast track or something\u2014so intriguing. I assume your boyfriend finds that attractive, yes? You know there is something to be said for that, I guess going against convention.\"\n\nOkay, Mean Girl. I get it loud and clear. You were sent here by the legion of Bitter Betty Sisterhood to put me in my place. Had this been another time (actually a year and a half ago because I still needed a place to deposit the anger and pain), I would feel the need to break your face or mind-fuck your husband just enough that he accidentally called you by my name over breakfast or in bed just before you can think to fake a headache. Lucky for Mitzy and her kind I am still operating at half-mast.\n\n\"Messy Missy, is it?\" I asked, knowing full well that it was not her name but enjoying the tense little pinch in her face just the same. Mental note: even if she redeems herself in the future, call her Missy. \"Surely you could give the ladies a much better report on my life situation if you made good use of yourself and helped me bring these bags inside.\"\n\nOn the way home I had stopped by D'Agostino's to pick up a few kitchen essentials but forgot to get one of those cart thingies downstairs in order to bring the bags up, so extra hands\u2014even hers\u2014were needed.\n\n\"I have always been curious about this unit, such great views. When Randolph\u2014that's my husband\u2014and I were looking to purchase in this building we tried to see it but were told it was not on the market, although it was obviously empty and had been for some time. Apparently it was part of Carly Spencer Falles's portfolio. I can't believe she sold it. You do know who she is, don't you?\" asked Mitzy, leaning in as if sharing highly classified information.\n\nHonestly, I know that I should have taken the high road here but why disrupt things now? \"Hmmm. Not really. I know of her. It's her husband I know best\u2014he's just too divine.\"\n\nIn hindsight it is more than fair to identify this comment as the one that put me at immediate odds with the wives in the building. Had Mitzy asked the proper follow-up question (How do you know Michael?), the gray area would have been cleared up instantly, but she didn't and I didn't volunteer. From that moment forward, anytime I entered a common area where they happened to be with their husbands, octopussy ensued. Suddenly they developed more arms than visible to my eye and began to twist, turn, and maneuver their men in any direction but to be in direct contact with or in sight of me. Sometimes I pretended not to notice, but if they caught me at an especially feisty moment, I forced interaction.\n\n\"Stuart, saw you yesterday headed out to play a few rounds. Looking good. What's your handicap?\" Flick the hair, cue laugh, and stroke his arm. \"I am totally taking you up on those lessons.\"\n\n\"Larry, when you get a chance, I would love to read that book on Goldman Sachs that you told me about the other day.\"\n\n\"Randolph, looking good . . . real good.\"\n\nOh, fun times, not just for me but also for the doormen (whom the Stepfords regarded as the help, with no life beyond getting the door and walking their pampered pooches) and the few boys in the building who loved boys and who formed their own little protective unit around me. It's like I was the most unruly one in the litter and they made it their duty to have my back no matter how many times I put my foot in it. My favorite was Gary, \"the gay,\" who lived across the hall in 8B.\n\n\u2022 \u2022 \u2022\n\nGary and I had the pleasure of meeting on a Wednesday afternoon in early October as I was checking the mail. The Brazilian Corcovado series was bringing rave reviews to Carly's. I was completely exhausted. Michael was beyond excited and rightly so, taking full credit for the earth, the stars, and all creation that comprised the parties and this new positioning. Somewhere along the way he did manage to squeeze out, \"Good work, young'n, now let's see what you have planned for the holiday.\" In all the commotion with work, a few personal things fell by the wayside, domestic upkeep being at the top of that list (note to self: call Richard and get a housekeeper referral to come weekly, stat), checking my mail a close second. Had Ivan, the evening doorman, not reminded me of the overflow of my box, I would have put it off another day. Gary discovered me there sitting on the floor in the lotus position, surrounded by piles of mail and unable to tear myself away from the image of Oprah, aka Glamour-puss Winfrey, sprawled out across the October cover of Vogue magazine.\n\n\"I know, could you just die?! Look at Ms. Thing on that chaise looking all kinds of sexy. Honey, I do the same thing\u2014only upstairs in my apartment in a green dress. You seem comfortable,\" he said in an animated tone that was peppered with a shade of disapproval for my loitering in the area. \"Nice shoes, by the way. Gucci?\"\n\nIgnoring the bit of gay snark, I conceded that the heels were in fact Gucci vintage, not Tom Ford, and found myself in conversation with, surprisingly enough, the guy who lives across from me: Gary \"the gay,\" until I learn his last name, contributing editor for Decor magazine. Over the course of my nearly five months of living here, with the first month and a half spent as quiet nights at home cradling with H\u00e4agen-Dazs, I couldn't help but notice that whoever lived across the hall entertained regularly. Actually, there were a few nights I pulled a chair up to the keyhole in order to watch the procession of spirited revelers come and go. From what I could see they were all quite trendy dames and dandies who were well aware of themselves. Basically, the kind of people I would be friends with if I were social.\n\n\"I can't believe you are just reading that. I ripped my issue open as soon as it arrived. Can you believe that Anna put her on the cover!\" exclaimed Gary, definitely more of a statement than a question. \"You know the old girl had to starve herself to make that happen or they did a hell of a job in Photoshop . . . My money is on Photoshop. For real, look at those arms! Now, you know and I know that's not possible . . . I'm an editor, you know, so they can't hide nothing from me. I mean, come on, Mommy didn't look this good when she was dragging those mounds of fat in that sad little wagon with all that hair on that deprived body. You have great skin, by the way. What's your secret?\"\n\nBy the time Gary came up for air, he was actually sitting on the floor next to me and had commandeered my magazine, flipping feverishly through the spread that he knew all too well, providing page-by-page commentary like a live sporting event was unfolding.\n\n\"Cetaphil or Leaf & Rusher to wash, pure cocoa butter at night, and light moisturizer in the morning. Normally I would also say 'all the sex I could handle' as part of a successful clear skin regimen, but that is a sadder state of affairs than Oprah dragging that fat across the stage,\" I replied. Judging from the blank expression on his face, Gary had forgotten that in the midst of his rambling he had asked me a question, so we just continued on into our love fest. If there is one thing I know well, it's the gays. What can I say, they love me and I them. Apparently, there is a burgeoning diva inside of me that appeals to them. That and the fact that I have an amazing set of natural breasts, not a nibble or a bite but a perfect mouthful of perkiness that is every man's (gay or straight) and woman's weakness.\n\n\"So you are the little thing that's got these women around here on high alert. You know, I heard all about you, Miss Thang! That Mitzy Bloomfield told us you were a husband-stealing piece of work with no respect for social seniority who probably slept her way to the middle. Of course you immediately became one of my favorite people, sight unseen. Poor Carly Falles. How did you do it?\" asked Gary.\n\n\"Ouch! All that just for little ole me,\" I asked, faking injury. \"Well, let's hope that Messy Missy is not called to testify to my character in a court of law anytime soon.\"\n\n\"GYRL, did you say Messy Missy?\" Gary cackled. \"I am dying!\"\n\n\"Honestly, Gary, she gave me no choice. I saw her one day in the elevator and instead of rolling out the welcome wagon she came at me with some antiquated insecure junior varsity high school cheerleader nonsense. It was like she had been sent out by the old crow's society to put me in my place . . . so I decided it would be a shame not to send her back with anything but a scintillating report for the very hens who clutch their chubby, balding, new-money husbands whenever I am around.\"\n\nBy this point he was totally engrossed, so I continued to plead my case, knowing full well that he would report every detail back to the old girls, which would be helpful. As much as I enjoyed being the topic of conversation, those women would surely have spread word along the UES soon enough that Michael and I were shagging behind Carly's back. That in itself would be absolutely scandalous and plausibly true if it touched her ears, so I had to set the record straight.\n\nCarly and I had yet to meet. Michael and I worked much more closely now that he was back full-time and soon would do some traveling together to heighten his business profile. The few appearances that she made at the restaurant always seemed to come when I was away. However, there was not a moment of the day at work that I did not feel her presence. She was in the paper, or on the phone with Simone, and in every nook and cranny of the restaurant's d\u00e9cor. Wanting to put a living face with the legend and feeling the urge to kiss the ring, I rang her up one afternoon, only to learn that she was out for the week at a health spa retreat, so I left word with her social secretary proposing high tea at the Peninsula when she returned.\n\nFrom the Oprah-Messy Missy encounter, Gary and I became fast friends. He took it upon himself to oversee the decorations on my flat, less as a solid to a newfound friend and more as an empty canvas to present to his object of carnal desire, Jean Pierre (no known last name), a totally cute, up-and-coming interior designer whom he wanted to bed repeatedly. Thankfully Jean Pierre's work actually appealed to me despite its being decisively more glam than my usual. Oh well, what's one more mirrored piece of furniture coupled with Mongolian fur pillows and a sparkly chandelier when it is free! In success, Gary promised a lovely profile on the apartment showcasing Jean Pierre's work with a requisite photo of me as the hip young Manhattanite. J'adore!! If all things remained on schedule, the piece would run near the top of the year. I gave them both keys, sat back, and watched the magic unfold.\n\n## 13\n\n* * *\n\n## MASTERS, THE MAN . . . NOT THE TOURNAMENT\n\nKEITH MASTERS (QUITE the accidental encounter) was managing partner of a pretty significant boutique advertising firm in the city. The first time I saw him was at Carly's while standing on the other side of the main one-way mirrors overlooking the main dining room. He came in with about five or so Midwestern types. The dynamics were obvious: he was the \"big city\" executive showing the good ole boys a proper New York evening. Once they were seated, I allowed my eyes to linger a bit longer before going back into the office to work on the task at hand: finalizing preparations for the last two major post-Fashion Week parties and locking in a private party; my first serious pressworthy events for the venue. Initially, Michael was against the ideas of catering to the fashion kids, as he now felt the scene had become too common and considered renting the venue out to Russian oligarch Nikolai Abramovich as his playland. Sadly, he, like a few others in the know, just could not get past the previous season's oversights of grunge and heroin chic. As for the Russian with a never-ending cash flow, well, that was far easier to finalize\u2014but only after chasing Michael for a few days about it and hijacking his office.\n\n\"Listen, Simone, I know that Michael has told you to tell me that he is not in, or that he is on a call, or is attending to something extremely important in some area of the restaurant that I am not in, so don't even bother. I am just going to sit here\u2014right next to you\u2014until he comes out of his office or gets off that elevator. Ah, before you protest, you know and I know that it is paramount that I sit next to you\u2014how else can I ensure that you don't send him some encrypted message or tap the little warning buzzer that I am told is somewhere on your desk?\" I said, wagging my finger, to chastise her as if she were a wayward student. \"I've been watching you, lass. I know how you two work, Batman and Robin, Lone Ranger and Tonto, Mr. Roark and Tat-t\u2014\"\n\n\"Jules, if you finish that sentence and make me Herve Villechaize, from Fantasy Island, I promise you that all of your vacation requests will be mysteriously lost in the vortex known as my shit list pile.\"\n\n\"Ouch, I love it when you bite. Grrrh. It just makes me feel all warm and tingly.\"\n\n\"Stop making me laugh. I have a reputation around here to keep up, you know. Some people are actually afraid of me,\" said Simone, trying to regain her composure.\n\n\"Yes, you do,\" I said, nodding in mocking agreement. \"Believe me when I tell you that everyone is aware of Superbitch. Instead of leaping tall buildings in a single bound, you can wipe out an entire service staff and a few creditors in a single glance.\"\n\nBy this time Simone and I were laughing so hard that Michael came out of his office motivated by sheer curiosity alone. Other people have a sixth sense about impending danger; his alerted him anytime fun and frivolity were going on without him.\n\n\"Simone, what's going on out here? I was gonna ask for . . .\" But then, upon laying eyes on me, \"Oh Jules, I should have known. I thought the English were incapable of laughter yet here you are, kee-keeing and laughing it up outside my office. Glad to know that I am paying you so well to languish about.\"\n\n\"Actually, you are paying me to do exactly what I am doing, creating great press opportunities for Carly's. However, in order to get an answer for some key requests, you force me to become duplicitous, chasing you down, attempting to bribe poor Simone, which by the way failed miserably, so in desperation I had no choice but to set up camp outside your office and, well . . . what can I say, I'm funny and you know I'm not a Brit. Just have not been able to shake the accent is all.\"\n\nRealizing he was cornered, Michael attempted to make a fast break into his office, but I had anticipated the move because he is anything but original and had already used it on me a few times successfully. So while speaking I made sure to position myself within two steps of him; close enough not to invade his personal space but not far enough away to have the door closed in my face, which he had done before as well. Michael's office is the only area in the entire restaurant that Carly had not taken her interior decorating ambitions to. It was once again indicative of their most interesting relationship. She was often looking for any way to have more meaning in his world, while he was keeping strict hold on the depth of engagement.\n\n\"The key to any relationship, Jules, is allowing a man enough space to be a man and to determine his own course of action. I had to learn that and when I did I found a woman who also knew it to be true.\" Michael said this to me one day when I was forcibly trying to coach him to say specific things for an interview. \"Your job is not to direct the tide but to ride it out. I'm a great wave, let me do my thing.\"\n\nMichael's office was an extreme ode to the Ralph Lauren Man, that is, expensive hunting lodge complete with thousand-dollar beverage coasters, tufted leather sofa, authentic deer horn magnifying glass, and letter opener\u2014poor Bambi's mama. Whenever I was inside I felt an overwhelming urge to light up a stogy and make some derogatory comments about women while leaning back to admire my boots made from the skins of endangered animals, just after calling PETA to say bugger off. Sometimes the feeling was so overwhelming that I would actually lose my train of thought, but not today.\n\n\"Why aren't you giving me an answer on this Abramovich party buyout?\" I asked. \"The deadline was two days ago, and if we don't respond today, surely they will secure another venue.\"\n\n\"And the problem with that is? Let him throw his money around to someone who needs it.\"\n\n\"Honestly, Michael, I don't understand. He is one of the wealthiest men in the world and is known for being an ultimate tastemaker. In selecting Carly's as the location for his event, we become an international destination, officially. Doesn't that mean anything to you\u2014hello?!\" Passively sitting there looking at me as if I'd asked him to pass the stapler. \"Fine. If you don't want to do it, then just tell me and I can take it off my list. Otherwise you have to give me something more\u2014like a yes, right now.\"\n\n\"Jules, you're good at publicity and events, but you don't know people like him yet. Not the way you should. Not the way you will. Your life experience has not afforded you those kind of years yet. I know people. I know Abramovich's kind all too well; new money\u2014enough to last for ten generations and save a few ravaged countries along the way. He is used to getting exactly what he wants when he wants. Afterward he tosses it aside with complete and utter disregard. Now he wants to come here and use my baby as his playground. I won't allow her to be disrespected. I know the deadline passed two days ago and I know that his team is still calling you, which raises the question of why is my space so important to him and how much is it worth to ole Nikolai?\"\n\nMichael's state-of-the-union speeches, as I had started referring to them, were now par for the course. The revelation that he knew the exact nature of my talks with a prospective client was surprising and showed on my face. He silenced me with a raised right hand.\n\n\"Don't ask. I know that the same way I know that it would do you good to actually go out some nights with your friends instead of curling up with H\u00e4agen-Dazs and Black and White cookies from Googies. No one wants a skinny fat woman. It looks bad in bed.\" Pausing to shudder at the visual, \"Now, where was I . . . oh yeah, Abramovich. He selected my place because he knows what I have created here, told his minions, and expected this to be done ASAP. Now they have undoubtedly gone back and told him the details for his big U.S. party are not secured. Despite the fact that he has far more pressing matters at hand, this will become the annoying fly in his day, which he will handle directly. When he does, I will be here.\"\n\nWith those final words, he buzzes Simone and tells her to have the car brought around. I, on the other hand, am still sitting there trying to process what I just heard\u2014the part about the ice cream and the cookies more so. Is there a camera in my shower that feeds directly into Michael's office? What the hell!\n\n\"Relax yourself, Jules, I don't have your place bugged. I just know people. I also know your doormen very well. Ivan and Percy are my guys, they tell me things from time to time about my new tenant.\" Now up and pouring himself a drink. \"Honestly, though, you should go easy on the snacks.\" Michael seems to have forgotten that his driver was patiently waiting downstairs, because he just continued to talk.\n\n\"I'll take that under consideration,\" I said.\n\n\"Do you know why I hired you? Not because you were the most qualified. You weren't. But you came as a referral by way of someone I value. Then we met and I saw myself in you as I first started in this town; hungry, ambitious, and na\u00efve enough to believe that I could play by the same rules as the other kids. Early on, life gave me a few hard knocks that would have broken a lesser man's constitution, but I rebounded quickly and quietly. I saw that in you,\" he said, taking a sip of the bourbon he had just poured. \"People, especially men, are easy to understand. We show you exactly who we are within ten minutes. Be smarter than the rest, Jules. Remember, if you are a quick study, then there is not enough time to invest too deeply and lose. Now, as it stands, I have something Nikolai wants in the short term. An association might not be a bad thing, so let's reel him,\" said Michael.\n\nFinally he headed for the door, with me glued to the chair trying to absorb, prioritize, understand, and cross-reference all that he just said. Before walking out, Michael says, \"I almost forgot, Carly will see you for lunch on Tuesday at the house. Simone will give you the address. Don't embarrass me; I've been talking you up.\"\n\nSeriously, the extra pressure I did not need. I was already on edge about meeting this woman, the icon, the legacy, the namesake. Meeting Noriega had to be less stress-inducing. Seeing that Michael left a sizable pour of bourbon behind, I downed it in one massive gulp. The trail it burned through my body left me off-balance, clutching the sofa for support until my chest stopped burning. Damn it to hell, too much testosterone in this freaking office. I don't even like bourbon\u2014I know this!\n\n\u2022 \u2022 \u2022\n\nLater in the week I was out for dinner with Blake and Joy, at Indochine, when I literally bumped into Keith Masters. I had never thanked Blake properly for the referral to Michael, so tonight was in her honor. The place was packed with the usual clientele of models, artists, music guys, club kids, and a few bankers. As I made my way through the crowd, someone turned and pushed me hard enough to knock me off-balance. Before I could fall, a hand grabbed me firmly around the waist and pulled me up. Mere inches apart from his sun-kissed glory, I recognized him instantly as the big city guy from Carly's.\n\n\"Are you okay?\" he asked.\n\n\"Yes, thanks to you,\" I said, somewhat breathless. \"I don't know what I would have done if you weren't here.\"\n\n\"Lucky for me I was. I'm Keith, by the way. And you are . . . ?\"\n\n\"Jules Sinclair,\" I responded, without realizing that I was biting the inside of my bottom lip and trying to speak between clinched teeth. Always a telltale sign of arousal. So glad the ole girl is back! Keith was still holding me around the waist with one arm as he told me that he liked my name because it made birthdays and anniversaries a no-brainer. By the time Joy came over to redirect me to our table, I couldn't hear anyone else in the room but him. I didn't have visions of happily ever after or a baby carriage, but every nerve ending in my body was telling me that this was the man to officially put sexual distance between Tony and me.\n\nApparently, I was not the only one who saw the sparks. At the table Blake and Joy began grilling me about who he was, how I knew him, and what I was going to do about it. Before I could start fumbling for answers to their barrage of questions, two bottles of Mo\u00ebt & Chandon arrived at the table. The first bottle said \"For now\u2014because I loved meeting you.\" The second, \"For later\u2014because I want to know you. Keith Masters 917-409-6640.\"\n\n\"J, oh my gawd, that is Keith Masters, damn! I thought he looked familiar.\" Looking for a sign of recognition on my face, of which there was none, Joy continued, \"Listen, I read about him constantly in Adweek. He is hot shit.\"\n\nJoy is more like a half sister than a friend. She is the official Cuban in my life, and knows my heart and its failings better than most, so I always take her words as authority. Born in the U.S. but sounding as if she just got off the boat, raft, tire tube, whichever you prefer, yesterday. She works downtown in finance at Merrill Lynch and lives the American Dream inclusive of WASP husband, unruly prodigal son, a golden retriever, and local neighborhood watch. I firmly believe that had she been raised outside that little North Bergen enclave of Cubans and Dominicans, where they speak nothing but Spanglish, sounding imported despite being born in the States, listening to soca\/salsa, and shopping primarily at bodegas while reminiscing about the Old Country, she surely would be further up the corporate ladder by now. She is whip smart, as evidenced by the bonuses that her boss gives her annually in lieu of the promotion she so rightfully deserves.\n\n\"Aaaah, so that is Keith Masters. I have heard about him but didn't know he was so damn hot,\" cooed Blake, spinning around to uncover what direction the bottles could have come from. \"I wonder if that stops anyplace south of . . .\"\n\n\"Listen, blondie, hands off and stop twirling your hair\u2014this one's not for you. This one's for Jules,\" cried Joy, wrapping a protective arm around me as exhibit A. \"Besides, isn't he like forty years too young and a hundred million or so dollars too poor for you or something? Our Jules needs to get her stride back before those gates are rusted shut.\"\n\n\"What???!!! I was just looking. No harm in that, is there?\" responded Blake with a faux innocence that even she didn't believe.\n\nAt this point I had to interject, if for nothing more than just to fend off the first line of piranhas; and make the first rule of Girlfriend Code crystal clear: Thou shall not cockblock. As it pertains to my girlfriends, I have learned that no saying rings truer than \"To know you is to love you.\" In Blake's case, to know her is to understand that her gray area as it pertains to men is vast, with a sliding scale. Just because one of us may have seen him first, gone on a date with him, slept with him, or dated him for a few years but parted amicably, doesn't really register for her if he has the power and financial means to support her lifestyle. There are, however, attributes that make her essential, namely that she is up front in her motives (sometimes she will say things so matter-of-factly regarding her intentions that I wonder if she even knows what a filter is), extends the same courtesy to her girls\u2014she had no qualms about one of us dipping in her pool (although I would never)\u2014and she is ride or die in her love\/support for her girls (as long as it does not interfere with plans to land Daddy Warbucks).\n\n\"Blakesy, you can try but you will fail on this one. I mean, let's face facts. We both walked right by him. He allowed you to pass and keep walking. Me, he stopped and now has sent over drinks. Hmmm, actually, bottles. Yeah, not so much, sweetie, but I'll be sure to give you a full report.\" TMI\u2014Heaven forbid I would ever tell her that the only reason Keith stopped me is because he stopped me from falling.\n\nWe were laughing so hard that I didn't even notice Keith approach our table. This was definitely another point on his scorecard. Most men (outside of a sports bar and properly liquored up) would never have the chutzpah to approach a table of cackling women.\n\n\"Hi, again. So I realized that it might have been presumptuous of me to just send over my number the way that I did. I hope I didn't offend you, Jules Sinclair.\" If heaven had a light, hmmm, I am thinking it could be this man. I opened my mouth to respond but didn't recognize the voice that was coming out.\n\n\"Not at all! As a matter of fact, it was quite nice of you, Keith.\" Ahhhh, I love it when mama\u2013Cuban bear comes out of my Joy. It's sweet that she thought I needed it, but I didn't. In his presence I felt renewed physically. My body just responded to this man, so I took Keith's hand as he stood over me to make sure I was his only focal point.\n\n\"No offense taken at all. The thought of any man besides you sending his number over seems wrong, wouldn't you agree?\" Oh gawd, that sounds so cheesy coming from my mouth. Damn, I hope it's working.\n\n\"Indeed,\" he replied, holding my gaze while stroking my hand. \"Ladies, you seem to have me at a disadvantage. You know that I am Keith, and you are?\"\n\nBlake offered only a limp handshake and dismissive nod in his direction as if to imply \"why are you breathing my air.\" In the years that I have known her and seen her for who she truly is, I recognize this move and appreciate it. In Blakesy speak, it is a clear sign of surrender, love, and respect. Translation: I honor the code and will not jockey for position by blocking the cock. To the poor guy on the receiving end, he just thinks she is an uptight narcissistic shrew. He is partly right.\n\n\"Hi, Keith. I am Joy. I was just telling Jules that I was reading about you last week in Adweek. Great campaign, by the way. I am at Merrill in wealth management. We have many of the same clients.\"\n\nKnowing that Joy can gab on endlessly without coming up for air, I placed my left hand under the table and squeezed her leg. Hey, sometimes a squeeze is a love pat. Other times, like now, it's a shut-the-hell-up.\n\n\"Ah, nice to meet you both. I do have to get back to my table.\" Now looking squarely at me, he instructed more than asked: \"Tell me you will use that number, anytime.\"\n\n\"Promise,\" I said.\n\n\u2022 \u2022 \u2022\n\nUnder normal circumstances at some point in the evening our tables would have magically merged and dinner with the girls would have turned into dinner with new friends and new man prospect. That's part of the storied NY magic. Tonight, however, that was not the case. The place was standing room only.\n\nWhen time came to leave I had lost sight of Keith. His night out with the boys had long since turned into a raucous affair, far different from the civilized work evening I'd observed at Carly's. Gorgeous girls were abounding and doing their best to captivate. Obviously, he was not hurting for female company, so I pressed onward to leave. Having said good-bye to Joy and Blake, I stood in front of Indochine for a moment debating the merits of walking a bit to enjoy the crisp evening air or hopping a cab. Reaching the corner, I heard my name and turned to see Keith running in my direction. Amazing. Despite the commotion at his table, he had seen me leave.\n\n\"I had hoped you would at least say good night,\" he said.\n\n\"I thought about it but, ah, you seemed a little busy,\" I confessed.\n\nDismissive of the company inside, he replied, \"I can see why you would think that, but no.\"\n\n\"I see. Did you receive the bottle that I sent back over?\" I asked, hoping that it had been delivered, since my phone number was written in lipstick on the label.\n\n\"I liked that\u2014made me the hit of the table. The last number was smeared,\" said Keith. Liquid courage in place and too impatient to let the moment pass, I looked up and kissed him on the lips. Instinctually he responded with such passion that I almost collapsed again in his arms under the streetlights of Lafayette and Astor. The warmth of his tongue fondling mine tasted of berry sorbet. Pulling away I said, \"Three. The last digit of my phone number.\"\n\n\"I won't forget,\" he responded, and placed me in a cab with a final, shorter kiss good-bye.\n\n## 14\n\n* * *\n\n## A NEW ATTITUDE\n\nTHERE IS NOTHING like a hot new man prospect to put the pep back in one's step. A few days after our encounter, Keith and I had our official first date. I must confess that it was not without incident. The plan, as it were, was that we would meet at 147. Located in a converted firehouse, 147 was NYC typical: velvet rope, dimly lit, dinner reservations starting at 9 p.m., people dressed for a stylish evening out, and the dancing afterward sanctioned by the likes of Biz Markie or Q-Tip on the turntables. When I arrived, Keith had left word saying that he was running a few minutes behind and would be there shortly, so I took a seat at the bar and ordered a vodka martini. Unlike London, the bartenders at 147 were absolutely scrumptious, offering plenty to keep me distracted until Keith arrived, specifically Franco. Tall, olive skin, dark hair, probably Italian, with an air about him that said he was a bad boy so little girls need not apply. By the time Keith arrived, Franco and I were moments away from arranging to say \"good morning,\" at the end of his shift.\n\n\"Something told me to send security ahead to watch over you,\" said Keith over my shoulder. Swiveling around, I immediately cupped his face and said, \"Never,\" sealing the things that need not be said with a soft kiss.\n\n\"I can't stop thinking about you, Jules. Today I was in a presentation and found myself thinking about the other morning. It came much to soon.\"\n\nI had to agree. After Keith put me in the taxi the night we met, he went back inside to say good night to his friends and then proceeded to come to my apartment. There were absolutely no pretenses about why he was there. Immediately upon my opening the door, my skirt was on the floor and by the time it closed, so was his shirt. Two years is a long time to deprive a body from the nourishment that it needs. Our first location was the far wall in the entrance hallway. Afterward we made our way to the kitchen and finished off a carton of ice cream. Between bites we got to know each other better. I learned that he hailed from Newport Beach, hates surfing, and played baseball in college. Keith was recruited by a major team, but quit during training camp, finding the future lifestyle at conflict with his personal goals. He, like countless others, myself included, came to New York for the sole purpose of creating his own identity instead of the prefabricated one his parents had constructed. Once here he started working with a dot-com and excelled quickly through the ranks. When the company folded\u2014they all do\u2014he received an offer to join an advertising firm from a most unexpected contact: an exec he knew from the club scene whom he used to do blow with. One thing led to another and before he knew it, Keith Masters had found his calling. His charisma and quick wit were extremely advantageous, so his star continued to rise. On the personal side he was aware enough to know that he wasn't ready for a long-term relationship but didn't like coming home from a long trip to an empty home either. Keith was an open book. I found his candor quite refreshing and even surprised myself in response, so there was no pressure this evening now that we were dressed, off my kitchen floor, and out in public. I felt as comfortable with him as I did with Blake or Richard (sans ever having the desire to rip their clothes off).\n\nInstead of playing some antiquated role by telling him what it was that I thought he wanted to hear, I allowed my dialogue to be as bare as my body intertwined with his. I told him that I escaped to London after a bad breakup and had just returned. That in my heart I did not know if I was ready or even capable of loving another man as I had Tony. Hadn't even entertained it, although I did miss the familiarity that comes with togetherness. The mere thought of it\u2014loving another so hard\u2014scared me, so I tried never to entertain it any place other than on a film screen in the context of actors portraying characters. How much more distance can one get than that? I told him that I had no guarantees or expectations about us past tonight. To which he responded, \"Let's discuss that in the morning.\"\n\nWell, that morning came, as did many nights thereafter over the next couple of months. My every other Saturday was now his, but we were not in a relationship; sitting at Caf\u00e9 Gitane, talking about the goings-on of life with casual acceptance, no hope for more, shopping together, no agenda, dining with friends of his and mine. Sundays remained as they had always been: quality time with Richard. Depending on whose house I awoke in, Keith's or mine, Richard had the following to say:\n\n\"Looks like someone forgot creepin' etiquette. Darling, how many times have I told you to always have a clutch big enough and dress small enough for a quick change? The world need not know that you are doing The Walk.\"\n\n\"There is no shame to my game, Boulton, I shall have you know. Not only did I shower and refresh but I even picked up fresh panties from Vicky Secrets before joining you,\" I responded.\n\n\"Well, I guess that is something. Lord knows there is nothing worse than day-old fish. Ewh,\" he said, slapping the table to emphasize his pun. \"So tell me, when is lover back and what is the story?\"\n\n\"Lover is in Chicago for the week. The story is the same as before: I like him. I like him a lot, but beyond that I don't know. For example, I know that he likes me and shows me, but I don't see our story in his eyes, you know? Not in the long term.\"\n\n\"Good girl. Keep those tacky rose-colored glasses off and see a man for who he is. In doing so you will learn who you are too. What he has to offer will appeal to you or it will not. The moment you start to justify is the moment you say good-bye.\"\n\n\"Honestly, you should write a book or be the first fab gay rapper,\" I said, slapping Richard a high five.\n\n## 15\n\n* * *\n\n## ENCOUNTERS\n\nIN SPITE OF myself, I had to admit, there was a bit of melancholy that accompanied the physical absence when Keith was away, but I dared not tell him\u2014or say it aloud, as I knew it had more to do with enjoying being part of a pair again. We spoke or e-mailed often, if only to keep the lines of communication open by making an observation about something odd that occurred within our respective days. I'm quite sure that I started this habitual dynamic when I e-mailed him.\n\nSubject: URGENT!\n\nWhy is it that if another human being is doing something truly disgusting in public I am destined to be the only one who sees it; picking their nose, scratching their balls, or picking their nose while scratching their balls? I mean, honestly, you would think they are lying in wait for me to cross their path or something. How is your day?\n\nx Jules\n\nSubject: Re: URGENT!\n\nHi Princess\u2014consider yourself lucky. I was in the restroom before the first meeting and had the displeasure of seeing Mr. Cornell take a piss, handle himself, and exit the restroom without looking at the faucet. Entering the room I was wishing like hell that he was Japanese so we could bow instead of shake hands.\n\nMasters\n\nKeith was due back in town on Friday morning so we made plans to grab an early dinner and catch a movie, the same day as my long-overdue lunch with Carly after having it rescheduled twice. The first was her doing\u2014\"hair appointment with Domenico from Milan who is only in New York for the day.\" The second cancellation was my, or better yet, Michael's doing. He wanted to sit down and go over the holiday lineup I had put in place, specifically the money he would have to put out to have the likes of Lauryn Hill, Cassandra Wilson, and\/or Terence Blanchard take up a limited residency at Carly's. Fan of the latter two or not, he was not excited about investing in three costly acts for an extended amount of time.\n\nAwaking that morning, I found myself surprisingly antsy at the thought of spending time with Michael's wife. I had shared so much time with him that in some strange way it just seemed odd to now be meeting Carly. It was as if, since a meeting did not happen within the first couple of weeks of my arrival, then it shouldn't happen now, at least not in this very staged way; high tea at their penthouse on Seventy-eighth Street between Fifth Avenue and Madison. Not wanting to deal with the whole uptown-downtown transit thing, I decided to take the day off, electing to work from home, not just because of this appointment but because Jean Pierre and Gary were coming over shortly to oversee the finishing touches on the apartment.\n\nI must say that the outcome far exceeded my expectations. Jean Pierre completely ignored my request (thankfully) for a chocolate and ecru palette in favor of shades of charcoal silver, iced blue velvets with royal purple, chartreuse, and brushed gold accents. When I described it to Blake, she said that it sounded more like a Gypsy-inspired brothel than a pending feature in Decor. Her parting words were \"a Decor Don't\u2014I cringe.\" The reality of it, though, was quite striking. My favorite room of all was the bedroom, complete with a king-size four-poster bed. Hanging from each end of the bed were the most sensuous pewter dupioni silk drapes. The foot of the bed featured a powder-blue chaise, a coffee table, and two Louis someone or other chairs. The whole thing just seemed so civilized, in spite of the bold striped wallpaper that I still had yet to come to terms with. When Gary arrived, I was in the throes of dressing: options A through E on the bed or floor as I walked around in only my bra and a black pencil skirt, which would soon become option F.\n\n\"How I do love your chi-chis, darling. They're like mocha clouds of perfection. Can I borrow them sometime? You know, either as pillows or just cuddle buddies,\" he asked, standing behind me in the closet, proclaiming the merits of my cleavage instead of discussing the blouse options I asked his opinion on.\n\n\"Funny, you do know that they are part of a package, don't you? Wherever they go, I go, which means you would find yourself sleeping with a woman.\" I gasped, covering my mouth for effect.\n\n\"Oh dreadful, I would never! No offense, honey, but I've never had the desire like some of the boys to play in the shallow end of the pool. I mean, I hear it can be lovely but just not for me.\" After a beat, Gary had selected a completely new outfit for me and was shimmying me out of my skirt. \"Although, I must say, you do have a tight little body. Who knows, one day we may talk about you carrying my kids\u2014obviously by artificial insemination.\"\n\nI halfheartedly nod in agreement, \"Obviously!\"\n\nWhen did it become so en vogue to just loan your womb out or lay claim to someone else's? I mean, shouldn't there be a more extensive process to it, like drinks, dancing, expensive gifts, a small island, and not just a five-minute assembly of a day outfit?\n\nA few hair flips and perfume spritzes later I was primped and ready, wearing a pair of blush full-legged natural-waist trousers and a matching oversize silk blouse with a huge bow at the neck. The look achieved the desired sentiment, conveying, \"I am fashionable, yet I respect you and am not a threat.\"\n\nOne thing I know for sure is that the more accomplished, beautiful, and successful the woman, the more easily threatened she is. No matter how secure and pulled together she may appear to be, it's just window dressing\u2014underneath she is a hotbed of insecurity who would gladly chew off her own arm if threatened. Okay, that may be taking it a little far, but close enough. I have also learned that being shortsighted\u2014i.e., needing to show the world how attractive I am\u2014is the quickest way to make an oversight or misstep. At this age, coming into my own and now feeling on top of the world, I want to show it every moment possible, just not today. Not with Carly. I like my job, love my apartment, and would like to keep them both. So for today I am a great student, and Carly is my guide.\n\n## 16\n\n* * *\n\n## HIGH TEA\n\nIN NEW YORK there are three levels of High Society status. The first is New Money, often characterized by its lack of subtlety. The typical demographic consists of Wall Street traders, recording artists, and record label executives. The second is the Nouveaux Riches, which is just New Money but well traveled and of international origins. These are the second generation or so, known as trust-fund babies. For the most part, they are still trying to gloss over the famine, ill-gotten gains, and messy bloodlines of their heritage. Their sole ambition is to be part of the social elite. This brings us to the third, and most illustrious, category: the Blue Bloods. Their fortunes are so vast that no less than six generations can live off the interest of their great-great-great-grandparents' fortune. Their names adorn skyscrapers, medical institutions, and museums the world over. Their history of exclusion is almost as extensive as the endowments that they provide to support the arts, bankroll political campaigns, and implement medical research programs. Walking into the entrance hall of Michael and Carly's home, it was clear to see that she hailed from the last category.\n\nThe main elevator opened onto a foyer that was larger than most New Yorkers' entire apartments. Ringing the doorbell, I was greeted by a butler, who escorted me past the receiving level (the art gallery) through three additional levels (living area, bedrooms, and recreational, respectively) before reaching the solarium, where I would have tea with Mrs. Kipps. She arrived just as I had expected, immaculately put together, clad in covet-worthy 1920s deco jewels, and precision-dyed hair. Having elected to embrace her age, she wore a silvery maven cut like Carmen Dell'Orefice, every strand of hair meticulously painted to silver low-lights and highlighted perfection. We exchanged a few pleasantries. I commented on how lovingly appointed her home was. We glossed over travel, art, and the lack of style in present-day New York. She in kind responded that I was far more impressive at my age than she ever was. The only thing I seemed to be lacking, according to Carly, was a vice.\n\nBetween sips of tea, I could feel her eyes on me in an almost unnerving kind of way, as if she wanted to know what made me tick. No sooner had I thought this than she leveled me. Eerily her tone never changed in delivery, so I questioned if I was hearing properly.\n\n\"You are indeed a young woman with potential, Ms. Sinclair. I can see why Anthony fell and self-destructed.\"\n\nSurely she could see the mechanics of my brain trying to make sense of the words coming from her mouth and put the pieces together, so Carly went directly into her story.\n\n\"I met Anthony nearly two years ago in Costa Rica. Our daughter, Kaylin, was in her terrible twos or threes, which seems to have extended to her fours and fives. My nerves were at their absolute end, and I needed to get away. Michael was in such awe of her that every little tantrum and screech was a source of sheer delight. All I could think was that the 'gift' from his ex was more like a cursed life sentence that I would have to find a way to deal with. To get some perspective, I decided to go on retreat to one of those marvelous resorts in Costa Rica where they do yoga at sunrise before a waterfall and at night offer candlelight meditation accompanied with champagne. The first few days were blissfully beautiful and silent. You will find it's the simplest things that you miss when there are children to consider, which is why I never wanted them\u2014but for Michael. Like sitting alone, awaking late, being indulgent, and deciding how hard you want to love but I digress. Where was I? Oh yes, coming back from the trails one day I saw this divine specimen of a young man. The next day, around the same time, I saw him again and thought what an ideal distraction he would be. I was faking a slight injury, entirely possible given my advanced yet extremely well kept years, and he assisted me back to my bungalow. He was so attentive, and initially I was blind enough to believe it was indeed my charm that had elicited such a response from him. Then I looked into his eyes and I saw tragedy of immeasurable despair. Quite Greek, one would say. Understanding the other sex the way that I do, I didn't dare ask him but was intrigued, consumed even, to find out his story. Over the next week or so we spent a lot of time together, nearly every waking hour of each day.\" Feeling the intensity of my stare, Carly clarified her last statement. \"For my part I would have loved nothing more than to explore him, but such was not the case. He touched my heart in the most maternal way. The way that Kaylin should have touched my heart when she was placed in my arms for the first time\u2014but did not. I never wanted kids, you know. Did I mention that? Just not the type. I have since grown to adore her incrementally and will continue to appreciate her, but it is not the way I instantly loved Anthony\u2014like a son. I wanted to protect him, to heal whatever pained him. It was not until the fourth day or so that he told me that he had lost everything and came to Costa Rica to lose himself and see if anything was left to salvage. He spoke of his brother and of you, both losses of unimaginable depth. The whole thing was just far too tragic to be real. I needed to know more about you, so I made some calls.\"\n\nHaving heard more than enough, I was overcome with anger. Trembling, I railed, \"Who in the hell do you think you are, lady? You don't have the right to play with people's lives like this. You don't have the right to play with my life like this. What's next? Is he waiting downstairs for me on one knee, with a ring? Will he come to the restaurant one night when I am in the middle of a million things and obliterate my whole world?\" The tears were now streaming down my face with abandon. My mind was running a million scenarios at this point; under normal circumstances each would have been more outlandish than the next. But this encounter was anything but normal. \"What did you think, you would tell me this story and I would run back to him? Heal him? Forget everything? Well, you are wrong. You're dead wrong. I won't! I don't think of him anymore. I have moved on with my life, and now you are telling me that the cornerstone of the life I am building is an illusion. All constructed by some deranged old woman in heat with nothing better to do than play fucking Geppetto with other people's lives, with my life? Well, fuck you. Tony is not the victim here. I AM. He shut me out long before he broke my heart, and no amount of quiet walks in the middle of a fucking forest will change that.\"\n\nI hastily gathered my things to leave without regard for the china that I knocked over and sent crashing to the floor. Carly remained sitting, as calm and observant as when she began her tale. Reaching the door, I was empty, bereft of any fight. My shoulders and head collapsed into the door for support. I just wanted to disappear again. The loneliness was back and I was exposed.\n\n\"Jules, I did not ask you here today to hurt you anymore than you already have been. You must believe me. Initially, I did have this grand fantasy in my head that I would bring you here unbeknownst to Anthony and the two of you would miraculously find your way back together; with a bit more help from me, of course. I know that is not the case, at least right now.\" Carly walked over to me and held me with a tenderness absent of maternal instinct but filled with female compassion and empathy of what it feels like to love and lose. \"It took me nearly a lifetime to find Michael, to allow myself to be vulnerable to him and accept him as he is, trust him to accept me as I am in total. It was wrong of me to dream that you kids could be smarter than us at such a tender age, when neither of you truly even know what love is; that is, until you lose it, as you both have. I do think of Anthony now as my son, and I hope to help him be a better man. The same way that I know Michael is coming to adore you.\"\n\n\"Was Michael a part of this as well?\" I asked, having just thought of his potential involvement.\n\n\"No. As I said, I did have some intel done on you. Initially, I gave no thought as to what I would do with the information. I just wanted to know what kind of woman could bring a man like Anthony to such an emotionally barren place. I guess you could call it envy more than curiosity, really,\" said Carly, taking my hand in hers. \"Jules, for all the great loves and whirlwind romances of my life, I have never touched a man's soul as you have. Not even with Michael. Yes, he loves me now, but that is because he has played extensively with wild abandon and has made a conscious decision to have an easier, more peaceful life, and that is what I offer. But he is not passionate about me. So I was curious.\" Searching my face for agreement, as if pleading with me to believe her, Carly continued, \"When the position at the restaurant became available, I knew that you could do the job and suggested you. Next thing I knew, everything was happening so quickly and you were here to meet. You miraculously ran into Anthony and subsequently charmed my husband. And here we are. I do hope that despite my meddling, we can be friends.\"\n\nMy tears had stopped, leaving behind only a nauseating feeling in my soul. The overwhelming scent of roses that filled the room only made matters worse\u2014always hated roses, most overrated flower ever. I just needed to go. Removing my hand from Carly's and composing myself, I felt compelled to tell her one last thing before leaving:\n\n\"If you know Tony so well, then you know that he will hate you if he finds out what you have done. In loving him you must tell him, today. He will be mad, but it won't last long.\"\n\nDespite all that had happened between Tony and me, the thought of him hurting because of some crazy-ass, bored, rich woman was not what I wanted. In moving on with my life, I needed to know that he was okay. Saying good-bye to him at the Mercer earlier this year, I had that assurance. Now I was not so sure, but what was I supposed to do about it?\n\nMy intent had been to go home, but instead I found myself walking across the street to the park. I was in a daze, so I sat on the nearest bench and tried to drown out the incessant audio in my head, hoping to lose the previous two hours somewhere in the exhaust fumes of traffic. I knew that I would not contact Tony as a result of this meeting. Would I have to contend with him calling me to apologize for the nutter? If so, how would I respond? Did this revelation change things for us? I had no idea, but I felt sick. Sick in my heart. The kind of illness that makes it abundantly clear that for all the forward movement I truly thought I had made, there was one glaring omission: my heart still belonged to him in some crucial, infinitesimal kind of way, and would forever. Suddenly, I just felt overwhelmingly tired and needed to go home.\n\nI managed to pull myself together enough to walk the few blocks home, but not before nearly being hit by a black Mercedes. Completely my fault\u2014stepping into the crosswalk the way I did absentmindedly. Percy was standing out front at his post and saw the entire thing.\n\n\"Ms. Sinclair, everything all right with you? That was a close one. You have to watch out for these here drivers, sho'nuf. They ain't like London, no ma'am. They'll run you over before they think to stop.\"\n\n\"I don't know, Percy,\" I said, my voice trembling as I tried to hold myself together. \"I'll be more careful.\" Tears streamed down my cheeks.\n\n\"Aw, nah, miss. You're okay now. Just a little shaken up is all.\"\n\n\"It's not that. The car. I just\u2014one of those days, you know?\"\n\n\"Whatever it is, the good Lord will work it out for the better. Always does. You just hang in there. Get yourself upstairs and rest.\"\n\n\"Thanks, Percy.\" I smiled weakly.\n\nAs I turned to go inside, the doors opened on a black chauffeur-driven Mercedes like the one that nearly struck me.\n\n\"Welcome back, Mr. Crawford. How was your trip, sir?\"\n\n\"Good, thank you, Percy. It's good to be home. You're good?\" were the last words I heard before the elevator doors closed\u2014glad to know someone is good.\n\n## 17\n\n* * *\n\n## REFUGE\n\nONE DAY OUT of the office turned into three unexplained sick days, days that I had yet to actually accumulate\u2014a minor detail that I couldn't care less about. Once surrendering to the safety of my apartment, I undressed immediately, turned the shower on full blast, and prayed that the water would wash away all the feelings of pity and insecurity that Carly had conjured in me. In this moment I missed so much the rains of London that seemed to cry for me daily. There was no use pretending anymore, the grand illusion of reinventing my life in New York had come crashing down like an elaborate deck of cards. The fail-safes for healing that I had established across the pond, like watching Random Harvest, no longer worked. I could not calm my mind enough to watch old movies. Every time Paula entered Charles's (Smithy's) office for the first time in her new clandestine role as his secretary, I ached and yelled at the screen, rewriting their dialogue and demanding that she confront him in the moment and say:\n\nCharles, you fool. I am your wife. Don't you remember our little cottage in the countryside? We had a lovely life. We had a child and then you went away for a job interview and apparently bumped your fucking head, you dumb-ass motherfucker. Now you're allowing this little prepubescent twit to sink her claws into you when I am the one who brought your ass back to life. Why, I ought to bash your damn head in until your memory comes back and then I will leave your sorry ass broken and destroyed the way I am.\n\nYeah, it's safe to say that watching old movies was definitely not helping matters. Food and the occasional spliff, on the other hand, definitely were. My kitchen was littered with containers of takeout food from nearly every vendor in the neighborhood. Under normal circumstances I probably would have felt guilty or overly conscious of potential weight gain, but considering that my anxiety would not allow me to keep anything down, I had no fear of an emotional five or ten pounds.\n\nThere was no proper context in which to place things, so I allowed them all to coagulate in one big ball of crazy. In feeling the way that I do, does this mean that I am still deeply in love with Tony, even though I know that we are not meant to be? In truth I don't think we ever were\u2014in love, that is\u2014as much as we were codependent. But I do love him. I care about him but never really knew him. The man I knew would never conspire to do what he did. The man I loved would not have let me go to London without being hot on my heels, standing across the street from my flat, awaiting my exit in order to plead his case. The man I envisioned him to be would have fought for me. And that is where the insurmountable problem was in all of this. I still had never accepted that Tony was just a man, not a superhero. True, I no longer viewed him through rose-colored glasses but it was still, to some degree, Old Hollywood. The illusion of flawless perfection created by a lens smeared with Vaseline no longer was sufficient. It worked for the young adult that I was when we met, but it fell severely short of the woman I have become in the past couple of years.\n\nIt had never occurred to me that my very foundation had changed, that I had grown in principle. I spent so much time saying Tony's name, asking about Tony (indirectly yet directly), avoiding Tony, learning how to exist without Tony, that I never took the time to notice Jules Sinclair. And my, how she had changed. No longer was she playing dress-up going from one homecoming ceremony to the next. She was the main event. She is me, far from perfect but not as na\u00efve.\n\nThe final two days were a lighter version of my sabbatical, not so much an open gaping wound, but not fully healed either. Gary was out of town with Jean Pierre, recouping his \"favors,\" so there was no concern about an impromptu visit. The only people who dared to cross my doorstep were the delivery guys with my eats and the FedEx guy who erroneously delivered a package for Mr. M. Crawford in 7A. On the third day I received a call from Michael via Simone, which I answered.\n\n\"I have Michael,\" Simone said matter-of-factly.\n\n\"Jules, how you feeling? Hey, Simone, jump off this call. Carly told me what happened. That's wild, kid. Only the kind of perfect storm that my wife can create, always got her hands in something. Her heart was in the right place, you know, if you look at it from a specific geometric angle through an official NASA telescope.\" He paused for a moment to see if his attempt at humor actually registered on me. \"That's life, though. Some motherfucker is always pulling the strings like your ass is Pinocchio or something. Your only choice is to handle it and keep your priorities in check. We have a full season to prepare for. Your neo-soul artist is already becoming a pain in my side.\"\n\n\"Michael, why did you hire me?\" Among the many questions I had been wrestling with over the past few days, this is about the only one that I actually stood a chance of getting an answer to.\n\n\"Like I said before, you came at the recommendation of someone I value deeply, Carly, but that is not why I hired you. I met with you because of her. I hired you because I liked what I saw when we met, so I offered you the job. Jules, why are you making me repeat myself? You know I hate that. You're ambitious and a fighter. More important, I don't mind you breathing some of my air. Hell, if we had met earlier, I might have had to add you to the harem.\"\n\n\"Thanks, Mike. I needed that. Maybe not the last part, but\u2014\"\n\n\"Who's Mike? You've been hanging around Raymond too long. The name is Michael and you are out of sick days. I'll see you tomorrow.\"\n\n\"Correction, thank you, Michael. See you tomorrow.\"\n\n## 18\n\n* * *\n\n## EMERGENCE\n\nDAY FOUR AND I was ready to return to the world. And none too soon. The morning air, thankfully, was cool but not unseasonably cold. I checked the forecast with old reliable: my body and an open window to feel the breeze; goose bumps denoted a scarf, perky nipples meant a sweater or light jacket, a full-body shimmy inclusive of goose bumps and perky nipples meant head-to-toe North Face coat and gloves. Today was a perky nipple, so I put on an oversize cashmere angora-blend sweater, a military coat I found in a surplus shop off Shoreditch High Street, and black riding pants with over-the-knee boots, then headed out.\n\n\"Miss Sinclair, glad to see you on this fine morning. I was 'bout to send a rescue party, but the cadre of deliverymen gave me the distinct impression that you were still with us,\" Percy said jovially as I exited the building.\n\n\"You're quite cheeky today, Perceville,\" I said, winking. \"It's a good day to be seen.\"\n\n\"Can I get you a taxi?\"\n\n\"No thanks, think I'm going to walk a bit and allow New York to do her thing with my soul.\"\n\n\"That's the spirit,\" said Percy. \"Nothing better than a good walk. Keeps you young. The missus and I do 'em frequently. We even got a little group in the neighborhood and walk on the weekends at night down to the local . . .\"\n\nOooh, Percy and that gift for gab that seems to know no conclusion! Seizing a lapse in his story, I offered my final salutations and departed. Time permitting, I could make my way up to Fifty-eighth Street before crossing over near the Plaza and grab a quick bite at Rue 57. In spite of Carly's meddling, Tony and I had met and said whatever needed to be said, on our own terms and in our own way. The rest, at this point, was anybody's guess, but I could handle it no matter what.\n\nReaching Sixtieth Street, I wait for the light to change and join the other commuters entering the crosswalk. The roundabout was already filled with horse-drawn carriages and their handlers grooming them for tours through the park. Making way for an advancing pack of schoolkids, I step off the sidewalk, only to be accosted by the loud blaring of a horn. It's shrill, so loud that I nearly jumped out of my boots; so close that I could feel the heat from the engine before it ground to a halt. I mimed a profuse apology before walking on, in the hope that we could all agree to just let this little incident pass without a public spectacle. Continuing on my way, I had the unnerving feeling that despite the hundreds of people on the sidewalk with no thought of me, I was not alone, which in and of itself was crazy. Rationalizing that I was just being paranoid\u2014nerves having been frayed for days and such\u2014I keep walking. A few steps onward, I glance to my left, only to see the front bumper of the car seemingly keeping pace with me, so I pick up speed and maneuver myself to the inside of the sidewalk, farthest from traffic. Firmly enveloped in the sea of people, I relax and allow my mind to drift again until my thoughts are interrupted by a familiar voice.\n\n\"You do know that you are about to give poor Carlos a heart attack, don't you? This is the second time in less than a week you have walked out in front of him. You got to go easy\u2014space your collisions. He's an old man, the reflexes aren't always so sharp,\" said Marcus. \"Tell you what, why don't you make it easier on everyone and allow me to give you a ride? At the very least it would be less than the increase in my insurance, should he actually hit you next time.\"\n\n\"Sorry about that, but no. I'm fine to walk.\"\n\n\"Well, in that case why don't I join you,\" replied Marcus.\n\n\"Honestly, I'd prefer if you didn't,\" I said. The last thing I wanted or needed was company, especially his. The sheer fact that he was male, breathing and daring to interrupt my solo pilgrimage, rendered him persona non grata.\n\n\"You know, this is the third time we have met and the third time you have rebuffed me. I'm starting to get the feeling that you really don't want me around.\"\n\n\"Really, you're getting all that, are you? Maybe you should listen,\" I said.\n\n\"I know you didn't mean that. Probably didn't have your morning coffee or something. I know how you girls can be at certain times.\"\n\nHis last comment stopped me cold in my tracks. \"Seriously, you didn't just say that. Rude, actually, very rude!\"\n\n\"No less rude than you refusing my sincere generosity when all I am doing is trying to protect you from becoming roadkill.\"\n\n\"Listen, I said sorry. I said no thank you. I will even apologize to your driver if it will make you go away,\" I said, unwilling to disguise my annoyance. \"What else could you possibly want from me, Marcus? There must be a zillion women who would love nothing more than to be aggravated by you at any time, but as for me, no thanks. The last thing I need right now is another person trying to barge their way into my life and arrange things.\"\n\n\"So you're not a morning person. Got it,\" said Marcus, seemingly unfazed by my animosity. \"Let's just walk, shall we?\"\n\n\"Whatever,\" I replied, as it seemed easier at this point just to keep pressing forward. Reaching Avenue of the Americas, it was clear that he was not leaving my side anytime soon, so I mentally redirected my path, in the hope of coming across a subway stop without looking obvious. Surely there must be one on this side. The red and orange lines were never familiar to me, but I was prepared to hop anything moving and cab it from anywhere in order to put some distance between him and me.\n\n\"Do you mind telling me where we are going or don't you know? Carlos has been trailing for about seven blocks and pissing cabbies off.\"\n\n\"Marcus, I don't want to be rude to you anymore than you want me to, so I beg you\u2014go away. I am not interested. I am seeing someone.\"\n\n\"Well, forgive me for saying so, but it doesn't seem to be going so well just by the looks of things.\"\n\n\"What do you know? Besides, it's not him\u2014just got blindsided by some old ex issues is all. Never easy, but I'ma be okay. Shit happens, right?\"\n\n\"That it does. So is that what we're doing here, walking to recover?\"\n\n\"Something like that,\" I said, feeling a flash of relief at the sight of a subway entrance ahead on Fiftieth Street. \"Just thought certain things\u2014feelings and such\u2014were in the past. I mean, you would think there was nothing left, right? I left the country, for Christ's sake, even saw him earlier this year, and all was okay, and then this woman meddles . . . Anyway, who cares, right? We were young. I'm fine . . . and this is my stop,\" I said, gesturing to the subway.\n\n\"Sounds complicated.\"\n\n\"Unnecessarily so,\" I say, extending my hand. \"Thanks for the walk.\"\n\n\"Jules, we're not all bad, you know\u2014after the maturation process and all, we can actually be something great to call home about,\" said Marcus.\n\n\"I'll take your word for it. Sorry about before, really,\" I said, realizing that there was no justifiable reason for me to be so harsh to this guy, other than the obvious, my attraction to him in spite of myself.\n\n\"You sure I can't give you a ride? Not for me, for Carlos\u2014you can make nice.\"\n\n\"No thanks, another time\u2014Carlos,\" I said over my shoulder as I went into the station.\n\n\u2022 \u2022 \u2022\n\nJacklyn was the first person I encountered at the restaurant. I hadn't noticed it before but she too is quite the little chatterbox of unsolicited information. \"Hi, Jules. Feeling better? You sounded awful on the phone the other day. I think you and my friend Kelly had the same thing. She was sick too\u2014sounded like death. We were supposed to go to the Roxy and she canceled, so not cool. At first I thought she was ditching me for some guy, because she does that, you know, but then I realized, after speaking to you, that seriously she was sick. You both sounded the same\u2014awful. Also, Lauryn's people called. She needs to do a morning sound check. They should be here within the hour. I tried to reach you when they called earlier, but your phone went to voice mail.\"\n\nI seriously doubt that Kelly and I were suffering from the same illness. There was no need to stop and give her my full attention; as it was, Jacklyn always seemed to be a step or two wherever I was until dismissed. I continued into the elevator and waited for her to come up for air so I could tell her we would resume this after my morning coffee.\n\n\"How do you do that?\" I asked, staring at her in complete bewilderment.\n\n\"Do what?\"\n\n\"Talk for that long nonstop without breathing. I swear, I'm just waiting for you to pass out.\"\n\n\"I do? Didn't know. Should I stop?\" Jacklyn asked.\n\n\"No, just an observation. I must've been on the subway when you called,\" although she did not need my affirmation, seeing as how she knew the routine of my life better than I did most of the time.\n\n\"Yeah, that's what I thought, so I went ahead and rescheduled your morning appointments so you can remain on site. Mr. Kipps has already called to find out what's going on. He is coming in early, I think. I asked Simone in order to make sure, but she is crankier than normal, so I guess it's true.\"\n\n\"Of course he is. The Love Boat would never set sail without Captain Stubing.\" Off her utterly clueless look, I decided it best not to explain and just buy Jacklyn a box set of 1980s television shows for Christmas. \"Let me see the set list and her rider,\" I said, scanning the papers as I walked into my office, only to stop immediately in my tracks. \"Are you pulling my leg? She wants to be called Ms. Hill?! What the\u2014? Lordy. Alrighty then, let the games begin. Grab me a latte and meet me back downstairs in fifteen minutes.\"\n\nDespite whatever storm cloud could be brewing with this earlier sound check, I felt good being back in the office. Here the rules were abundantly clear and incapable of forcing me to take to my bed. The customer is always right and in some cases so is the entertainer if ultimately it will make the customer happy.\n\nFrom a comfort perspective, my office was the next best place to be other than the apartment. Because of the configuration of the building, there was no corner unit to be campaigned for. To compensate, Michael had placed all the executive offices with the exception of his on the top levels facing the street, each with floor-to-ceiling windows and discreet little patios. \"I only have one rule: no smoking in the offices,\" Michael had said within the context of one of his random daily e-mails regarding team and office protocol, which were sent more out of boredom than as mandates. A rule that apparently applies to everyone but him.\n\nI thought of this briefly as I laid eyes on the exquisite crystal and gold-plated ashtray that was doubling as a vintage paperweight. During the course of decorating my apartment, Jean Pierre found a few pieces that demanded to be in my office: an antique Italian desk, duo-toned Moroccan carpet, an amazing RMID sofa, and a coffee table.\n\nThere is no way that I could have afforded such luxuries on the modest decorating budget Michael approved, but with Jean Pierre's discount and an extra helping of charm, they miraculously found their way into my space without too much damage to my bank account. To complement, I added fresh orchids and vintage black-and-white photographs of my music and fashion inspirations. Taking a quick moment to flip through the phone sheet and monthly booking report before rushing downstairs, I came across a Post-it from Simone affixed to the second page that read, \"In case Michael doesn't tell you immediately. Looking good.\"\n\nThe restaurant was booked solid through the winter season, with an extensive waiting list to compensate for any cancellations. This was indeed promising. When I originally proposed the idea to Michael back in September, he made it clear that his only focus was the flow of the restaurant and being the most liked man on the island, \"so you better make sure everyone is at their best so I can be at mine,\" he had said.\n\n\"Jules, glad to see you could be with us today,\" Michael said, poking his head into the office and forcing me to look up. \"Good thing too. I hear your first one is already making demands. Ms. Hill, huh? Maybe you should start calling me Mr. Kipps as well,\" he said, pausing briefly as if to decide whether he was going to enter or just keep on moving. \"You ready for this?\"\n\n\"Real funny, Michael. Real funny. No worries, I am fine,\" I said, but judging by the look on his face, it was not the most convincing of deliveries. \"Seriously, I am fine. What are you doing in this early? Trying to get an autograph or are you worried about me?\"\n\n\"No, and you are delusional. Today is a heavy delivery day and we had to order more than usual in anticipation of the reservations. The restaurant will be filled throughout with service staff and outside people.\"\n\n\"Uh-huh.\"\n\n\"Usually I leave this to Raymond, but every once in a while I take a more hands-on approach. It keeps the troops motivated and engaged.\"\n\n\"Well, look at that. Here I was thinking you were Captain Stubing when in fact you're a modern-day Bonaparte,\" I replied, in a much lighter tone to let him know that my wits were about me and that I was indeed joking so he should take it as such.\n\n\"And you will be in Waterloo if your big idea doesn't generate enough money to recoup me on the enormous fees I am paying these singers to perform and the additional staff,\" said Michael, not missing a beat while still straddling the doorway.\n\n\"Well, based on the reservations\u2014\" I attempted to say.\n\n\"And before you tell me about a full house through December, tell me how many of them will be drinking hard liquor, ordering bottles, and how often we will be turning those tables over. That's the restaurant business, Jules. Those are the things that keep the lights on and me bespoke in Savile Row. Where is your assistant?\" asked Michael. \"Oh, there you are. Jacklyn, call my office and have Simone set up a call for me with Marcus and Simon. Afterward she should meet me in the restaurant.\"\n\n\"Yes, Mr. Kipps, I'm on it.\" The syrup dripping from Jacklyn's every word was laughable. Her crush on Michael was completely obvious to everyone, especially him, yet she would never admit to it and was too na\u00efve to know how to conceal it properly. Even the one time I asked her about it. \"Absolutely not, Ms. Sinclair. I-I-I, why would you even think such a thing?\" That is the moment I figured out how to tell when she was lying, from her stuttering and being overly formal. I couldn't blame her, though. When I was twenty-two I only had eyes for older men and if one like Michael crossed my path, all jet-set debonair with a bit of edge, I surely would have been swooning. Thankfully, that phase was brief and lasted only a year or so until I met Tony.\n\nAs light and witty as his delivery might have been, Michael's point was abundantly clear. In business as in life, always know your break-even point. In doing so, you know how much you can risk and recognize when it is time to walk away or cash in. Two years in another country, three days in seclusion, and I still could not succinctly state this: five minutes with Michael and it seemed like common sense. \"I am going downstairs to check out the stage and Laur\u2014Ms. Hill's dressing room. Take Keith off my phone sheet. I will call him back later.\"\n\nBy later, I meant today within the next hour or so, but one thing led to another and before I knew it the phone call became a hastily typed e-mail.\n\nSubject:\n\nHi you,\n\nTemporarily fell down the rabbit hole. Might I ring you when I resurface?\n\nx Jules\n\nHis response was quintessentially Keith.\n\nSubject: RE:\n\nPrincess\n\nNothing you can't handle, I'm sure. Get back at me when you're ready.\n\nMasters\n\nIn the week that I allowed to not-so-subtly slip by, I was conscious of the fact that I was purposefully avoiding inviting Keith immediately back into my life. Not because of anything that he had done and not for some grieving or pining over Tony. Before the whole episode with Carly, I was clear on our dynamics. He was my Transition Guy, our chemistry (in bed and out) amazing, we enjoyed each other with no expectations of a future. But now the line was threatening to get a bit blurred because moment to moment my desire to just fall into him, to fall into any man and be rescued, was as pressing as Rapunzel's need to escape that tower. What can I say? It was a pattern that I could never seem to shake, so I kept busy.\n\n\u2022 \u2022 \u2022\n\nI still had not found a cleaning woman of my own and Richard's seemed to be unavailable whenever I needed her, so on Saturday I decided to clean the apartment and do laundry, all four loads of it. It seems impossible that I could live here alone with no pets and yet accumulate so many dirty clothes in a matter of a week or so. After taking down the final two loads, I collapsed on the sofa and got lost in an episode of ER before realizing that the time had long since expired on the first load, so I rushed to the basement to place them in the dryer before someone in the building complained or rudely removed them from the washer and placed them on the folding table for anyone to see. Opening the door, I heard the phone ringing and decided it was probably best to pick it up this time. After all, Cora had phoned twice already and would probably continue calling until I answered. That woman was relentless. Saturdays were our day to have an extensive one-sided conversation, always in her favor, whether I wanted to or not.\n\n\"Hi, Mommy,\" I said.\n\n\"Hi, sexy. Sorry to disappoint, but you can call me Daddy if you like.\" My goodness, I love the sound of his voice. Most people's phone voice is uneventful, monotone, or nasal\u2014not Keith's. His was authoritative and rich with no reverb. He could probably make the Annual Crop Report sound desirable, which was no small feat since my eyes glazed over at the mere mention of stocks and such.\n\nNoticeably caught off guard, I said, \"Hiiiii,\" making a valiant effort to recover. \"The name is vaguely familiar but the specifics are a bit cloudy. Is this your new number? It is not in my phone.\"\n\n\"No, I'm just at the office putting the final touches on a presentation for Monday with the Japanese. Why don't you save me from this slave ship and let's grab a drink. Say, Amaranth in twenty minutes?\"\n\n\"Are you sure we could handle an impromptu drink on a domestic Saturday, and by impromptu I mean me showing up looking more casual than you have ever seen me?\" (Bare face, yoga pants, and a well-worn sweatshirt that I snagged from a college boyfriend many moons ago.)\n\nWe agreed to meet within the hour. He would still arrive in twenty starting off with a solo Dewar's, and me shortly thereafter. Despite the time cushion, I still found myself rushed for time. Thank goodness the restaurant is only a few blocks down the road on Sixty-second Street, just behind Barney's NYC.\n\nTurning the corner from Fifth onto Sixty-second Street, I was immediately reminded that Saturday night is a \"going out\" night for many. Normally Amaranth is filled with locals who over time become friends. It's comfortable, easy, and delicious. Tonight, however, it was packed. There were tons of people standing in front awaiting a table. Initially, I feared that Keith would be among them, but then I remembered that he and Gianni, the manager, were good friends. Sliding by a young couple who screamed, \"Connecticut date night,\" I saw Keith ahead at the bar.\n\n\"Did someone write an article this week about this place that I am not aware of? There's like fifty people outside waiting for less than ten tables,\" I said.\n\n\"Well, I'll say. Thank goodness Gianni had the chairs in the back or we would have been with them,\" Keith said, standing to kiss me. \"Welcome back from Wonderland, Miss Alice. I was beginning to think that you were lost for good.\"\n\nFor the first time we felt a bit strange to me, or maybe it was just the fact that I was still hypersensitive in my own skin and overtly aware of my failings, as it were. Some people were fine with a one-night stand to provide them the distance and security required to exorcise the demons of a former relationship. I preferred the composition of a full-on relationship, despite this casual exercising in \"just dating\" that we were doing. Eventually, I reasoned, I would find my knight, shining armor optional but sword essential. Looking at him now, I had this overwhelming need to apologize for the absence but was not quite sure of where or how to start.\n\n\"You look great.\"\n\n\"Really? Thanks,\" I said, a bit self-conscious and thinking that I should have put more effort into my physical appearance for him.\n\n\"I didn't want to crowd you, but I was getting concerned, you know?\" Keith said, preferring to focus his attention on the bottom of the nearly empty glass of whiskey than on me, ensuring that the \"keeping it light\" boundaries remained.\n\nI confided so was I, the first day or so, but once the initial shock of things passed, I was better than I had thought, which was great. My only concern was the epiphany that maybe just maybe Tony was not the love of my life, despite the immense pain that I had endured or the lengths to which I'd sought to escape. That part I still had yet to come to terms with, and in trying to do so I came face-to-face with old patterns, which is where my head lived right now. I wasn't ready to elaborate on specific tendencies just yet.\n\n\"Keith, am I a masochist or something?\" I blurted out, wanting a real answer and not to be pacified.\n\n\"No, baby, you're not. You're just on the later side of twenty-eight with Saturn's return in your house and it is having one hell of a time with you.\"\n\n\"Huh? What is that, Japanese or something?\" I offered, half serious. In response, he explained that the emotional roller coaster that has been my life for the past year and a half was more than normal as this was the astrological time when everyone's cosmic house was thrown into disarray by Saturn going retrograde.\n\n\"Think of it as the universe's way of making order out of the confusion your life until now has created, by showing you what you are made of so you can live on your terms. No one gives us a guidebook into adulthood, so we enter it boldly and unapologetically. Our parents long since forgot, so they only warn us of the obvious physical danger. The rest is left up to us to figure out or fuck up. In the arrogance of our youth, we are certain that we have all the answers, so we make career, life, and love decisions all from a place of sheer na\u00efvet\u00e9. Problem is, the overwhelming lot of those choices are not our personal truth. They are more conditioned precepts and societal brainwashing as to the life we are programmed to have instead of the life we want to have. So as we chronologically get ready to enter a stage of true adulthood\u2014at about thirty years old\u2014everything goes to shit in order to clean house and put us on the right track. Chaos into chaotic order is what I call it,\" he said, pausing only long enough to finish the last sip of his drink and signaling the bartender for another. \"The fucked-up thing is that while we are going through it, we vow to remember, but at its tail end, most of us are so happy to get through the shit that we immediately delete it from memory, forgetting to pass on the wisdom. Now here is the kicker: if you do remember to warn someone, understand that it is a waste of time because it will fall on deaf ears. Even if I knew you at the beginning of your cycle and told you what was about to happen, you would not have believed me.\"\n\n\"Yes, I would!\" I said incredulously.\n\nUnconvincingly he replied, \"Sure you would have if I came to you and said, 'Jules, for the next two years or so everything that you thought you knew and thought you had firm hold of in your life is about to be pulled from under you and no matter how hard you try to hold on, to fix it\u2014everything will fall apart.' Would you have believed me?\"\n\n\"Or would I have told the bartender to cut your drunk yet completely fine ass off immediately?\"\n\nI knew Keith was right. I would not have believed him if he had tried to explain that the chaos in my life was necessary. Tomorrow I would talk to Richard about this. I wonder what his retro-cycle was like. \"Or maybe I would have told you that you were on some Southern California surfer vibe and should put down the pipe.\"\n\n\"True. Unfortunately, you can't force this to pass, Jules, just because you want it to. If you allow it, you will be much better off in the long run than you can ever imagine. Seriously, I know this firsthand.\"\n\nI could feel his sincerity and was grateful. Nearing the end of my second glass of scotch, some parts of his dialogue escaped me\u2014or at least that is what I thought.\n\n\"Until twenty-seven I was the golden boy, came from the right family, captain of my high school football team, played college baseball, and got recruited into the majors despite the fact that I was marginal at best. Everyone was so proud, and in the beginning, so was I. Maybe I was just riding on their fumes and never stopped to own my desires because I didn't know I could. My all-American life was figured out for me at birth. Once out of the protective family enclave, I started to go off track but not enough for anyone to wave a red flag. My mom never wanted to see a problem, so she dismissed the drinking and partying. When undeniable, I remember trying to talk candidly with her about things I was feeling, but it was too much for her to hear, so she sent me to her therapist to be fixed. Much to her chagrin, the very therapist she sent me to forced me to dismantle my world in order to understand that I wasn't broken, that it wasn't a phase, and eventually the boundaries of everything in my life had to be redefined in order for me to write my own story. Leaving baseball for prospects unknown disappointed my dad, but we got over it soon enough. He is proud of my success. Accepting that I like men and women devastated my mom and put a wall between us that we are still recovering from. If someone would have told me at twenty-seven that the very foundation of my family life and my identity would be shaken to its core, I would not have believed it.\" Leaning in to emphasize his last point, he said, \"Jules, I only tell you this because the whiskey may now be talking but\u2014your man cried like a little bitch more than a few times. That's how intense everything was. We should really eat something or tomorrow will be rough. Steaks? I'm starving . . .\"\n\nThe walk back home was rough, but not because of the endless pour. Scotch I can handle. Learning that the man I am seeing is probably attracted to the same kind of men as me was another matter indeed. So, from Amaranth to the shower later that night, and on the subway Sunday to meet Richard in SoHo, I replayed the latter part of our conversation\/Keith's confession a million times. But was it indeed a confession, an impossibly late disclaimer (by the way, I too like boys), or a point of fact?\n\n## 19\n\n* * *\n\n## SWITCH HITTING\n\nDARLING, ARE YOU certain you heard him properly?\" Richard calmly asked within minutes of our meeting at the Cub Room and my immediately spilling the details of last night with Keith. \"Before you go and get yourself into a huff, know that I am not doubting your account so much as I want you to be absolutely certain before we go down this murky road.\"\n\n\"Richard, in case it's lost on you, I am having orange juice and not a cocktail, not a mimosa. Orange juice! Doesn't that tell you that something is seriously wrong?\" Knowing me as he did, even Richard had to concede that my starting our Sunday brunch with just juice was cautionary. \"He told me all this stuff about Saturn and Uranus, which I was going to ask you about, but then he levied the wallop about baseball, his mom, men and women, Aaron or Erin. WTF!\"\n\n\"Wait, wait, honey, are you sure of the context? Lord knows, with how creative people are today in naming their children, anything is possible. It could be Aryn, Erin, or Aaron. All of it is unisex now. I blame Calvin Klein. Would it kill anyone to just stick to the King's English?\"\n\n\"Ugh, stop, you are making my brain hurt more than it does already. I know he said A-Erin-Aaron-somebody, and I know I heard the words men and women after sumtin' sumtin' drinking and partying. And before you ask, NO. I did not stop him mid-sentence to get clarification. I didn't even realize I needed it until I was halfway home, and what was I supposed to do at that point? Besides, that's why I have you. Help meeeee!\" I whined.\n\nWatching me dissolve into a squealing tantrum suitable for a three-year-old was not exactly what Richard had in mind, so he attempted to soothe me as only he knew how. \"Here, take a sip of this. Before you speak again, take another. Then calmly tell me how he spoke of him\/her?\" he said, stuttering out a correction at my horrified expression. \"Darling, I meant her. Drink more, improves clarity.\"\n\n\"They met in college freshman year in a public speaking class. You know, that one random course that requires an x and y credit. Please refrain from making the obvious chromosome dig here, please. Anyway, one thing led to another and they became friends, hanging out partying, and ultimately it became uncomfortable for Keith because he was having feelings about Aaron, Erin, Aryn. What the fuck! They started dating, sort of, first love, and Keith felt as if he was living a lie. Oh gawd! He had a lot of pressure on him and they broke up, but not because Keith didn't love him\/her. Eventually, he spoke to his mom and she sent him to therapy. The flip is that, instead of 'curing' him, the therapist helped him own his truth.\"\n\n\"Oh, dear. I bet that is not what Mommy had in mind.\" Richard could not help but laugh at the irony. His shoulders animatedly bounced up and down as the spasm of laughter subsided. He knew this terrain all too well, as he had to contend with a disapproving mother when he came out.\n\n\"Honestly, I need for you not to be savoring the demolition of my pseudo-relationship as much as you are that drink. What am I supposed to do with this? Isn't there some type of law or something about this?\"\n\n\"Like what, honey? As far as I know there is no law for a knight in knight's clothing with a questionable sexuality who is not trying to hide it from you, as some do.\"\n\n\"Richard, we have slept together many times, and he responds to me physically, and I am not talking about the alcohol, the E, or 'blue pill' type of response. I mean, I know that we are still in the 'getting to know each other' phase, but this is just too much information and I am not sure I am equipped. This just introduces too many things into my world that are impossible to comprehend.\"\n\n\"Okay, first things first, tomorrow call your doct\u2014 Oh, dear.\"\n\nBefore Richard could finish what he was about to say, he'd laid eyes on Blake entering our sanctuary, looking all things equestrian city chic\u2014the Ralph Lauren version, that is. Her hair fell below her shoulders, the same golden wheat color as the turtleneck she wore, with matching riding pants and a blue denim jacket accessorized with an oversize paisley scarf in earth tones.\n\n\"Hey, kittens. I knew I would find you both here,\" said Blake, looking around to see who else was there. \"What is the deal with this place and both of you anyway? It's like you have this little Sunday sandbox with a standing reservation for the two of you. Anyway, I woke up and decided I was crashing this little party.\" Searching for a space to sit at our cramped two-topper, she added, \"Couldn't you find a better table?\"\n\n\"No, this is the only table I like. If you want something a bit more on the fringe, then I suggest you grab a lonely table of one,\" Richard instructed.\n\nFor as long as I have met him here, this table, located in the center room, directly across from the bar, has been his fixture. I have actually arrived on Sundays ahead of Richard to a wait list with every table in the restaurant filled with the exception of table 16, Richard's table. It awaited him, come rain, sleet, or shine, or standing room.\n\nChoosing for the moment not to be goaded into an early round with Richard, Blake continued, \"By the looks of things, it's a sad sack of a party and I probably would have been better served just staying home.\"\n\n\"Don't you worry, Blake, all is well here. Jules is just going through the relationship weeds.\"\n\n\"Not again. I thought we were done with the Tony thing. Carly is a meddling old bat. I swear, Julesy, you have to let that whole history go. He is great to look at but honestly, he is not the sort you build a life with. Tony is the kind of guy you meet in your twenties, fall head over heels for, surrender the last part of your innocence to, do some 'shrooms with, and in return he strips you down to your emotional core and obliterates your entire world. You grieve for a bit, pull yourself together wiser and stronger now, setting your sights on the guy who looks good but not as good, who loves you far more than you love him, and above all has the financial trajectory to keep you in the comforts you deserve. By all accounts I would say you are right on track, wouldn't you?\" said Blake, conscious of the dumbfounded expressions on our faces as she spoke: \"What?\"\n\n\"Just when I thought I had you pegged as a gold-digging social climber from the womb with a shovel in her shoe, you astound me,\" said Richard, causing Blake to do a double take. Because Richard handles me more with kid gloves, I oftentimes forget that he can be rather biting if need be. Much of his ire seems to be reserved for Blake, although I wouldn't daresay it was out of spite as much as it is a reflection of his former self. Somewhere in a world that only the two of them can comprehend. This is their way.\n\n\"Thank you. By the way, am I the only one who can make your inner cunty queen come out or do you use that kind of language with all the ladies? If so, I can see why Jules is still such a sad, sad sack.\"\n\n\"If this were the eighties, I would cut you,\" said Richard, with butter knife in hand.\n\n\"Richard, if this were the eighties you would have to find me, and by my calculations Salt Lake City, Utah, is a far cry from Studio 54, where you were most likely twirling. And I do mean 'twirling' in all connotations,\" said Blake.\n\n\"Stop it, you two! First, this is why you are not invited to brunch, Blakes. You and Richard can never seem to play nice, even for my sake. It's like you both went through some horrible divorce and I am the poor child stuck in the middle. Anyway, it's not Tony, so save it.\"\n\n\"Oh, thank Gawd,\" said Blake, gulping down the remainder of my juice.\n\n\"I think that Keith is bisexual,\" I said.\n\n\"And, what? I'm not following.\"\n\n\"Are you listening to me, Keith is\u2014\" off Richard's look I felt the need to amend my statement\u2014\"Keith and I had talked over drinks last night, and he told me that he is attracted to men and women.\"\n\n\"Welcome to New York! If they are not bisexual, they are bicurious or bipolar\u2014at least he's in the affirmative. If you like him, ride it out. What, why are you looking at me like that? I swear, you have been hanging around Richard too long. You're starting to get that same ole uppity scowl. Listen, I know you have not been on this island for a few years, but sweets, let me tell you, you will be hard-pressed to find a man of Keith's stature and age who is not dabbling. Seriously, forget difficult, it's nearly impossible in this town. The only question you need to ask yourself is, What's your endgame? If it is to have great sex\u2014and I assume the sex is good because he is too damn fine for it not to be\u2014then I say keep at it. If your intention for this one is marriage, babies, country homes, and beach club memberships, then you might need to find out which side of the pendulum he is most inclined to.\"\n\n\"Either I have had more drinks than I can recall or maybe my sugar levels are low, but dare I say you made sense again.\" Richard's backhanded compliment was not lost on Blake or me.\n\n\"I know. I do that, you know,\" said Blake, reaching across the table to take an unsolicited bite from Richard's omelet. \"Hmm, I don't like this. You should have gotten French toast.\"\n\n\"Well then, maybe you should order French toast and leave my food alone,\" said Richard.\n\n\"Oh, yeah, I could do that.\"\n\n\"That would be progress.\"\n\n\"What's wrong with her? Why is she just sitting there like a lump? This can't be what you both do every Sunday. If so, I will take that table for one,\" Blake said, in reference to the two-topper in the far corner near the kitchen\u2014clearly the worst seat in the place. The kind of table reserved for rude customers.\n\nMischievously, Richard responded, \"Stop it, Blake, can't you hear the wheels churning in that pretty little head as she tries to process all that you have said?\"\n\n\"Now that you mention it, there was a faint smell of smoke moments ago. I thought it was from the kitchen.\" Both snickering at my obvious despair.\n\nExhausted with being the pun of the joke, I spoke up. \"You are both quite cruel human beings when you want to be. You must know this. No more brunch trios. You bring out the absolute worst and snarkiest in each other. Yes, Richard, I know that 'snarkiest' is not a word in Webster's. So be it,\" I said, glaring at both of them equally.\n\nFor the next fifteen minutes or so we ate in silence, until Richard placed his hand on top of mine, offering, \"Honey, things aren't always so black and white, you know. Based on what you have told me, there appears to be no malice, so you can't cry party foul per se, but you must know if you can handle all that comes with being involved with someone who is hitting for both teams.\" Looking in Richard's eyes I could see that he did not take my predicament lightly, despite the earlier quips and jabs.\n\n\"That's what I said,\" Blake chimed in, midway through devouring a healthy bite of her French toast. You would think she had not eaten in days by the gluttonous display.\n\n\"Next time, can you please wait to swallow before enlightening us? I already feel ill,\" I said.\n\n\"Speaking of that, I hope you took your own advice,\" replied Blake. \"By the way, you should go and get tested tomorrow if you haven't already. The virus just isn't reserved for the boys who love boys, you know.\"\n\n\"The virus?\" I asked before understanding. Fuck, HIV, the one variable I didn't even consider in this whole equation. It is dangerous enough in this day and age having protected sex with a new partner who is heterosexual. There is always a fear of a condom slip and accidental pregnancy until the much-welcomed appearance of moodiness, bloating, and cramping that precedes a monthly visit from Mary. To my knowledge, four of my close girlfriends had abortions before they were twenty-five, and that was big news\u2014drastic even. Two of them occurred in college, and if I am correct, their parents still do not know. Now let's throw into the mix an incurable, fatal, easily transmitted disease that is highly prevalent in the gay community. I think I am now going to pass out.\n\nMy mind was reeling and I could not bring myself to tell Richard and Blake that there was indeed that one time Keith and I did not use protection. It was the first night, when I allowed myself to live too greatly in the moment of not having any for so long. A few days afterward, over dinner, Keith was the one to bring it up, more a preemptive discussion to unexpected fatherhood than anything else.\n\n\"Hey, I just want to let you know, in case you're worried about the other night, don't be. I don't have any diseases and shit. My fellas are great swimmers, though, so ya know.\"\n\n\"Relax yourself, Masters. While I am certain our fictitious children would be absolutely gorgeous if that was the case, it's not. I am still on the Pill. Never got off, actually. Having always lived by the 'what if?' factor, it just never made any sense for me to stop taking them. Now, obviously, I didn't know there would be a substantial drought in the forecast.\"\n\n\"And your decision had absolutely nothing to do with the documented weight gain that happens to women after they come off it?\"\n\n\"Your parents should never have encouraged you to read,\" I teased, pinching him lightly. \"All I will say is that it could have been a deterrent.\"\n\nIt seemed so light and trivial then. We laughed it off and all was forgotten. Now, sitting here, I can feel Blake's and Richard's eyes on me, and I can't decide if I am to look up and meet their concerned stares or look away and change the topic. I opt to just lean back and squeeze my eyes closed as tight as possible. Surely, I would awaken any moment to see that the drama being played out was not my own. Not so. Instead, a few air kisses and hugs later, Blake, Richard, and I were making our way down Prince Street so I can catch the N or R train on Broadway. Even if his apartment weren't located a few blocks east of Broadway on Mott, Richard would walk me to the subway or a taxi. To hear him tell it, \"Papa would turn over in his grave if I didn't see you girls off properly.\" It was part of our dynamic that I always looked forward to and, truth be told, looked for in the guys I date. Keith is chivalrous also.\n\n\"It's such a pretty day, but can we not do this again? I don't quite like being the center of all the drama, you know. That is not what our Sundays are supposed to be about.\" My body was now as tired as my mind, so it seemed only right to descend the steps into the subway, where there were fewer people to contend with, than on the streets where it seemed like the world was out and jovial.\n\n\"What we must never do is give our location to that dreadful Blake.\"\n\n\"Stop it, I know you like Blake. Admit it, she has grown on you.\"\n\n\"Honey, psoriasis also grows on you, but you don't see me inviting its discomfort into my life, now do you? Our Sundays are about enjoying each other's company, being a sounding board and reflecting on the week that was. Not an obnoxious free-for-all. Now, arguably, this one just happened to be a bit more complicated than most.\" Wrapping his arms around me tightly, Richard said the words I so needed to hear: \"We'll get through this. You are fine, don't worry.\" Pulling away to look at me, he added, \"As a precaution, though, I want you to call my doctor in the morning and see him immediately. If you wish, I can meet you there. Just tell me what time. Then we can decide what to do about Mr. Masters.\"\n\nEscaping into the subway station, I knew only two things for sure. The first was that Richard was still standing at the entrance looking after me, more concerned than he would like for me to know, so I dared not turn around. The second is that I didn't need the results of my AIDS test to help me decide about my relationship with Keith. At this point I know a few things about myself well enough to understand that I am just not that modern of a girl to be with someone with an affinity for both sexes, no matter how amazing the package is or how much he appears to adore me.\n\nTruth is, on a deeper level, if pressured to address this publicly, I would have to admit that I am not 100 percent comfortable with the whole intimacy of homosexuality, which seems like an enormous contradiction given my circle of friends, but it is true. With my friends I can compartmentalize things\u2014focusing only on their happiness, but not with a man I am intimate with. I am not proud of this, but such is the case. With each and every day I must admit that I am definitely my mother's child. You can take the girl out of Jamaica but some parts of Jamaica stay in the girl. Right now, the major problem, aside from the obvious, was how to handle things with Keith under these conditions. In my heart of hearts I felt that he was not the sort to cavort with such disregard if he knew there was a life-altering problem. Hell, Keith loves himself much too much. More to the fact, he is open and honest with me with no seeming agenda, which is new, in a manner of speaking, especially as it pertains to my romantic entanglements. The worst-case scenario cannot be my fate, it just can't.\n\n## 20\n\n* * *\n\n## THE VOICE IN MY HEAD\n\nSADLY, AFTER I returned home, the remainder of the day got progressively worse. As much as I attempted to calm myself and rest, the panic within me grew. Was I being delusional? Was I dying and didn't know it? Yes to the latter, because it is inevitable\u2014obviously. My overwhelming immediate concern, to be specific, was dying before the natural occurrence of events I envisioned claimed me; old age after having lived an amazing life as a media guru, traveling the world with my debonair \u00fcberamazing husband, jetting between our homes in Manhattan, Paris, and Lake Como while raising a beautiful family of two\u2014potentially being the bane of my daughter's existence because that is often the way of mothers and daughters, the apple and aspiration of my son's eye because that too is the way of things, and becoming the coolest grandmother ever. Then, after celebrating my ninety-fifth birthday, where I am fit as a fiddle and can compete with any sprite of a seventy-year-old, my house filled with family and friends collected over the years, I would go upstairs to rest and peacefully exit this world. Nowhere in this vision does it say that I am to contract a fatal, painful illness and die alone without even the most basic of comforts and support. I don't even have pets. Maybe I should get a cat! Hate cats\u2014hateful animals they are.\n\nAround 4:43 a.m., after the umpteenth infomercial designed to help me zap blemishes, lose belly fat, and tone my thighs\u2014am I the only one who finds the ThighMaster unsettling?\u2014I just couldn't take it anymore. Having left the solitude of my bed long before and repositioned myself in the living room on the sofa, I needed answers.\n\n\"Mommy, you up?\"\n\n\"Yes. Why are you, Julesea? Insomnia again?\" Cora asked, without much regard, because my nonsleeping bouts by this time have become sort of legendary for their duration and frequency.\n\n\"Why do you insist on calling me Julesea, Mommy? You know my name is Jules.\"\n\n\"Well, when I pushed you out of my body and signed the birth certificate, it was Julesea Isabel Sinclair. I don't know why you choose to keep forgetting that little fact.\"\n\n\"Well, if you would have asked me first, I would have told you that I prefer Jules.\"\n\n\"Really? And I was to wait how long for this personal identification preference, Julesea? A few months on the hopes that miraculously your first words would be something other than 'mama' or 'dada' or maybe just maybe I could have waited a couple of years. In the interim I would call you 'hey you' or 'the only fruit to have come from my womb' when strangers asked.\"\n\n\"Mom!!!!!!\"\n\n\"No, seriously, you're right. Of course you are. I have only been in the world slightly longer than you. Here I am thinking that giving you a name, an honorable name, your grandmother's name, would be a sense of pride for you and, if nothing more, save you from being picked on in school, but what do I know? I am just your mother.\"\n\nBy this time I was sitting up and laughing uncontrollably, nothing like one of Cora's ill-timed rants. A much-needed reprieve from the past few hours of mental torment. While it is true that we (okay, I) may not speak as much as we (okay, I) should, my mother has always been the one true constant in my life. A presence that, regrettably, I take for granted much too often and whose counsel I always seek at my darkest hour, even when I think that the decision has been made. I always want her final sign-off.\n\n\"I knew you would be up. Why don't you sleep, woman?\"\n\n\"I have not slept in thirty years, Julesea. That's your dad's job. The man can sleep through a tsunami. Mothers keep vigil. We pretend to keep ourselves busy with the minutiae of work\u2014if we are lucky, social engagements, and home to distract us, but all in all we are just waiting for the phone to ring and be needed again.\"\n\n\"Mommy, I'm twenty-eight. Please stop rounding up. It only makes me older and in doing so, you too.\"\n\n\"I look good. Always have. Let's hope you take after my side of the family. Did I tell you that I saw your Helen last week, and let's just say that time is not being a friend, but who am I to judge?\"\n\n\"Woman, you are shameless.\"\n\n\"The truth is always without shame, Julesea\u2014something I learned a long time ago. Now, what drama is plaguing that mind of yours and how can Mommy fix it? Let's be quick about it. Daddy will wake up soon and I still have not put my face on.\"\n\n\"Incredible! Has Daddy ever seen you without the face?\" I asked. Cora is a naturally beautiful woman, so her devotion to makeup borders on insulting to the rest of us. Her skin flawless and glowing, encases large chestnut eyes set high on Elizabeth Taylor\u2013type cheekbones, a telltale sign of her multiracial, Dutch-Spanish-African heritage and a blueprint to the Jamaican history. Her hair, the deepest mahogany, is fine yet thick and wavy; when loose it cascades down her back like one of those Spanish women from Seville who inspires men to paint and commit suicide. Growing up, I used to love just staring at her, specifically her mouth, full and bountiful, the most evident feature of her West African ancestry.\n\n\"Of course not, why do you think I don't swim?\" said Cora. \"Men should never see what's underneath. Kills the illusion.\"\n\nOh dear, she was firmly basking in her element. Humor was spot-on, with a healthy heaping dose of cynicism best served at twilight. How could I ever tell her everything?\n\n\"Um, well, you know I'm seeing someone new, right?\" I said, my eyes racing to the ceiling for the right delivery.\n\n\"The advertising guy?\"\n\n\"Yeah, Keith. Well, um . . .\"\n\n\"Dear, I am certain that Catholic school taught you actual words. If not, Daddy and I deserve a refund. Chop chop, now you're starting to make me nervous. Is he dead? Did he break up with you? Did you find him wearing your lingerie? What?\" Then she paused only long enough to work herself into a proper frenzy. \"Lawd Jesus! Are you pregnant, Julesea Isabel Sinclair?! Ya betta say nah, child, and it bets ta be the truths, gyrl!\"\n\nOh shit, whatever internal radar she has has gone postal, because her well-nuanced middle American dialect had now given way to her Jamaican upbringing, which is reserved strictly for family reunions and three-alarm, red-alert situations.\n\n\"Shhhhhhh. No, MOMMY! Stop yelling, you'll wake Dad,\" I said, looking over my shoulder frantically as if he were in my apartment instead of hundreds of miles away. \"I am not pregnant, I promise\" (I hope not, but it would be more welcome than the current proposition), \"and Keith has not broken up with me. We're fine\u2014sort of. Not really. I'm just not sure I want to continue is all.\"\n\n\"Lawd, why didn't you say so in the first place, gyrl? Calling me first ting tis mornin' all mealy-mouth. What was I ta tink? You know my pressure is bad.\"\n\n\"Lady, you don't have high blood pressure, so stop . . . Mommy, Keith is gay.\"\n\n\"I think I would have preferred you telling me you was pregnant.\"\n\n\"Really? Not funny, Cora. Well, to be fair, he's not actually gay. He's bisexual and\u2014\"\n\n\"And the difference is what? The man is a bati boy! Drop him immediately and change your numbers. Didn't I tell you to be careful in that city? I did. I told you. I know.\" She was so elevated, there was no reining her in.\n\n\"Cora, what's wrong?\" said Dad in response to the outburst.\n\n\"Oh, my Lord. Mommy, did you wake Daddy up? Where are you?\" Hearing him rustling in the background, I realized that this entire conversation was in fact being conducted from their bed. Not in the kitchen, where I thought she would be at this hour, making herself some tea or in the bathroom luxuriating over a multitude of creams, perfumes, and emollients.\n\n\"Charles, it's nothing. Hush up now and go back to bed,\" said Cora.\n\n\"Tat is what I waz tryin ta do, Cora, but ya so loud, all the tyme. Just loud. Who you talking to at this hour? Is it Helen? Tell her ta go ta bed and let us be.\" My father has never been the kind of man to make any airs about himself. To describe him as simple would be far too easy and erroneous, because he is anything but. Charles Barrington Sinclair, a kid from the slums who got a break, took it, and did good. Being a pure Jamaican made him proud and being from Kingston made him the third cousin twice removed from Bob Marley\u2014or so he said, but aren't we all? Cora is the great love of his life (by both of their accounts) and as such he allows her a lot of leeway to be grand, because it makes her happy and that's all he ever wants\u2014to make her happy. I have been searching for this type of man (age-appropriate and nonincestuous) my entire life. Most girls grow up wishing for Prince Charming. I only wanted my dad. Freud would have a lot to say about that, I am guessing.\n\n\"No, Charles, it's not Helen. Now stop making a fuss and go to bed,\" she said, admonishing him to a slight rumble before returning all her attention to me.\n\n\"Hold on, dear, I am getting myself up. Okay, now where was I?\"\n\n\"Before or after calling Keith a derogatory, homophobic term, Cora?\"\n\n\"Listen, dear, don't get snippy on me. I am not the bati-homosexual who has lied to you and God only knows what else.\"\n\nInterrupting her before the cycle of events fueled by her imagination became fact, I said, \"Keith did not lie to me. In fact, he told me of his own free will and unsolicited, I might add, about his life preference . . . situation, you know, I mean. I just don't know what to do about it is all.\"\n\nAnimated and muffled now, Cora said, \"You don't know what to do about it?! Well, I will tell you, Julesea Isabel. You are going to break up with him and go get yourself tested. Lawd knows what kind of tings he could've given ya. I can't believe this! Did you not see it? I mean honestly, dear, we raised you better.\"\n\n\"Seriously, Mommy. It is not like he was wearing pink sequin hot shorts, twirling down Fifth Avenue, and I intentionally went over and said, 'Date me, please.' Of course I didn't see it. Keith is a well put together, successful man who likes sports. Hell, he even played professional baseball. He drinks beer, out of the bottle. So, no, I didn't see it.\" Unbeknownst to me, my tone had risen slightly.\n\n\"What did I just say? Watch your mouth, Ms. Thing. I didn't go through eighteen hours of labor for you to take that tone with me.\"\n\n\"Sorry.\"\n\n\"It's okay. You're not yourself. What cold cream are you using these days? I am trying some new organic something or other that cost seventy-five dollars, and I swear Noxzema works just as good and it is only four ninety-nine.\"\n\n\"Mommy, please focus.\"\n\n\"Honey, I am focused. I know exactly what this is. Keith is one of those metrosexuals. I was watching Sally Jessy Raphael a few weeks ago and know all about it. It was on Oprah's show too. They want their cake, yours, mine, and Daddy's. Walk away, Julesea, you aren't that New Age. Walk away.\"\n\n\"I know, and I am. I just wasn't prepared is all, and now my every sensibility is in knots,\" I confessed.\n\n\"Honestly, I don't know how you girls do it. In my time the only thing I had to worry about was making sure your daddy wasn't dumb enough to fall for that cheap Rosalyn Bradshaw before I was ready to be with him. The gay boys were easy to spot and they were run out of town. This is indeed a different time, and I am current, but some things just aren't to be, especially where my child is concerned.\"\n\n\"Mommy, you do realize that it has been more than thirty-five years since Daddy dated Rosalyn? It's pretty safe to say you've won.\"\n\n\"Never let your guard down, Julesea. You know, she lives in the States now. Just last week Aunt Helen told me she saw her sometime back. Big as a house she is\u2014big as a house.\"\n\nI couldn't bring myself to confess to her that Keith and I had unprotected sex once at the beginning of the relationship and that I was, in fact, going to be tested as soon as possible. Cora Madeline Augustus-Sinclair may be poised and sturdy, but she is also hot-tempered and irrational. To even hint that I was concerned at this point about anything other than his sleeping with men would have her hopping a plane to New York right now with poor Dad in tow, demanding to see the doctor's credentials and have a proper sit-down with Keith. No, thank you.\n\n\"Mommy, it is nearly six o'clock, and I need to try to get some sleep. Thanks for listening.\"\n\n\"That's what I am here for.\"\n\nBefore hanging up, I remembered something. \"Oh, I haven't used cold cream since college, Mommy. I use Leaf & Rusher Green Tea Cleanser. It is amazing. I will have Jacklyn send you some later today.\"\n\nSucking her teeth in resignation: \"Tsk. And that one! Girl scared of her own shadow, she is,\" said Cora.\n\n\"No, she is scared of you. Treating her like a slave, Mommy. Umph, I wonder.\"\n\n\"Don't be nasty about the Dutch, dear. Unfortunate incident, it was, without which we would not be here,\" responded Cora.\n\n\"Incredible,\" I say.\n\n\"I know. I love you, Julesea. I do.\"\n\n\"I know. Thanks, Mommy.\"\n\n## 21\n\n* * *\n\n## NEW LEASES AND RELEASES\n\nJULES, YOU'RE HERE early. I haven't had a chance to get your coffee yet.\"\n\n\"Don't worry about it, Jacklyn. I picked one up on the way in. Just bring me a copy of my schedule for the week, an updated phone sheet, and then come in so we can go over it.\"\n\n\"There is one from Friday night already on your desk. I'll check the voice mails from the weekend for updates.\"\n\nAfter hanging up with Cora, I tried to go back to sleep, but it was impossible, so I got dressed and walked.\n\n\"Jacklyn, why am I meeting with Michael this afternoon?\" I yelled from the interior of my office after a quick glance at the calendar.\n\n\"Simone said it's some kind of investor meeting.\"\n\n\"But you're not sure?\" I asked, irritated.\n\n\"Not really,\" Jacklyn responded in a casual, resigned manner.\n\n\"So you just put it on my calendar? You need to find out. Also, I need you to call Dr. Katz first thing and get me an appointment for today . . . I haven't been feeling well since the weekend . . . Might be coming down with something.\"\n\n\"Jules, I checked the voice mail and applied all the updates to this phone sheet,\" said Jacklyn, replacing the old phone log on my desk with the new one. \"Nancy from Dr. Katz's office called to confirm your eleven thirty appointment today. Do you need me to change it? Your mom called and said to call your father.\"\n\n\"No, I'll do it myself.\" Despite how slowly events were registering for me this morning, I knew that the osmosis gods had not miraculously gotten me an appointment with Dr. Katz. \"Change my dinner with Keith until the end of the week and get Richard on the phone for me.\"\n\n\"He called too. It's also on your phone sheet . . . in your hands,\" said Jacklyn.\n\n\"Who?\"\n\n\"Keith . . . and Richard.\"\n\n\"Well, what did he, Keith, say?\" I asked impatiently.\n\n\"The meeting went well. He's sorry but has to cancel. He has to leave for Tokyo tomorrow and will call you when he returns. I already took him off the phone sheet.\"\n\nA slight agitation was evident in Jacklyn's voice and, try as I might, instead of overlooking it, it just incited me enough to push back from my desk. Suddenly I saw Barcelona shades of red.\n\n\"Am I disturbing you? Or is it no longer your job to assist me?!\" I said, staring at her with an intensity that dared not be contradicted. \"Well, what did Richard say?\"\n\n\"He just called to say that he will see you today at eleven thirty.\"\n\n\"Okay,\" I said, partially scanning my schedule. \"This week will be insane, so we should try to . . . change my Wednesday with Gillian to her next avail. I need to be on-site all day. On second thought, keep Gillian on. Just have her come here so she can see Jill's rehearsal. We need that review in the Post this week. Why is my Thursday evening still blocked? I thought you took Keith off?\"\n\n\"Yes, but Simone just told me to lock it in for a dinner with Michael and the financial team.\"\n\n\"Listen to me. Why am I in that meeting? I don't even know their names. What am I supposed to do\u2014take notes?\"\n\nShrugging her shoulders, she replied, \"Dunno.\"\n\n\"Did you not ask?\"\n\n\"Yes, but Simone just dismissed me, so I figured I\u2014\"\n\n\"You figured what? That you didn't need to get all the information?\" Not bothering to wait for her response, I continued, \"I know that Simone can be a bit brisk\"\u2014off her look\u2014\"Okay, downright rude sometimes, but she does her job at all times and protects Michael's ass. She never sends him into anything unprepared. You have to do that for me as well. That is why you are seated out there, not to look pretty and flirt with Michael. Understand? Your job is to be my gatekeeper, to protect me and keep me organized. Right now I just need every fucking thing to be okay. You are supposed to protect me, get it?!\" Midtirade I became aware of Jacklyn flinching and thought it best to quiet myself. Leaning forward slightly to relieve the pressure on my temples, I feel my eyes burning, but I refuse to cry under the weight of things. \"Go back to your desk.\"\n\nThe air in the office was thick, almost stifling. To stop my hands from shaking, I continued to rest my head in them and took some deep breaths. I was wrong for yelling at her, but I needed not to apologize right now. I needed to own the rage and sense of helplessness that had built up inside me. How could I possibly be expected to focus\u2014much less care\u2014about a group of wealthy, overprivileged bankers with a hard-on to see their names in print now that Carly's was a bona fide hit, even if they are ultimately the ones who pay my salary? My life is in the balance, and I swear as I sit here I can feel my glands swelling, accompanied by an onset of fever. When I did look up, Jacklyn was still standing before me like a deer in the headlights. Thinking it better than to try to defend herself against my verbal thrashing, Jacklyn said the first thing that came to mind and neutralized me.\n\n\"I'm sorry. I don't know, but I will find out.\"\n\n\"Okay.\"\n\n\"Jules, are you sure everything is all right?\" she asked. \"You're reminding me of Mrs. Sinclair.\"\n\n\"I hope so. Go back to your desk. We'll finish this up later . . . Oh, Jack, order me a car for an eleven o'clock pickup and have them wait for me until I am done.\"\n\nIt didn't take a genius to know that something was more pressing than a bout of not feeling well over the weekend. The doctor had already phoned. Richard was meeting me there, and to top it off, Cora had called as well. Since day one, Mom had clearly positioned herself as the unofficial horrible boss in Jacklyn's postcollegiate career. To me, she is my assistant, efficient and moldable, but to Cora, Jacklyn is the docile child who answers the phone with no spirit, takes only partial messages, shipper of things, and reservations sorter that she could easily do without but will tolerate. Early on I had thought to intervene but reasoned that she could use the stiff real-world training that only Cora could give her. Since she just completed NYU and was fortunate enough to work for a mostly relaxed boss, like me, it seemed only right that I allow her the opportunity to earn her stripes as I did through the Academy of Cora, otherwise known as the \"Jamaican hazing fury!\"\n\nI wouldn't classify staring out the window or surfing the Internet on all things HIV\/AIDS-related as working, but it's how I spent the next few hours until the car arrived. Exiting the office, Jacklyn attempted to give me a dossier on the investors.\n\n\"I can't look at this right now. Highlight the key points of interest and put it on the desk. I'll look at it when I return.\"\n\nMy ability to focus on anything other than the immediate prognosis of my health would be heavily skewed over the next couple of hours. Within the fifteen minutes or so that it took to go up Eighth Avenue to Dr. Katz's office on Sixty-fifth Street, psychosomatic or not, I swear that I could hear my T cells decreasing, screaming, \"Save me,\" accompanied by a spontaneous inflammation in my throat and sporadic bouts of nausea. That night with Keith was several months ago, so if there was anything to be detected, it should be as clear as the Great Wall from space.\n\n## 22\n\n* * *\n\n## AND YOU ARE?\n\nSTANDING IN FRONT of the cream-colored four-story brownstone of Dr. Katz's office, in his uniformed wool overcoat, blue Brooks Brothers blazer, striped dress shirt with cashmere sweater vest, and crisp camel-colored khakis, was Richard. The sight of him always makes me smile. Even in the most formal of settings, he always appears to be more composed than the rest, which is to say nothing of how we first met: years ago in the private room of Lot 61, surrounded by club kids, happy pills, and high-finance mergers of a different kind. Richard was with the new face of Versace, a dear friend of his whom I only dared to drool over, much less believed I would meet one day. Through my girlfriend Emily, a fierce club promoter from Philly, we found ourselves at the same table. For one reason (too many drinks) or another (nerves that were soothed with drinks) I couldn't speak. Richard took pity on me and the rest, as they say, is history.\n\n\"Darling, love the look. Did you pick out this smashing ensemble or did the old girl from across the hall come over to save you?\"\n\n\"His name is Gary and noooo! I put it together myself. The turtleneck is new, Bergdorf sale last week. I've been dying to wear it, and the pencil skirt, oh well, it is . . .\" Instantly aware of what I had just said, I added, \"Well, not dying to wear it. I mean I can very well not. I can actually live to wear it?\"\n\n\"Oh dear, you sound like that little anxious pixie of an assistant of yours. Breathe.\"\n\nBefore I nervously rambled anymore off the deep end, Richard took me by the arm to usher me inside.\n\n\"Hello, Nance. We have an eleven thirty with Gerald.\"\n\n\"Hi, Richard. Come on back. I didn't know you were coming in until this morning, when Dr. Katz told me to make the time work. Hello, Jules, I'm Nancy. You have a concerned friend here,\" she said, gesturing at Richard. \"I'm going to put you guys in Room Seven. Remove your top and put this on. The doctor will be right with you.\"\n\nArguably the loveliest courtroom in the world, the inpatient suite was appointed in warm marshmallow and ecru shades of rattan furnishings from Kreis, with Ansel Adams framed landscapes decorating the walls. Had the examination table not been in the center of the room, or the official apparatus lining the countertops been absent, I could have forgotten where I was\u2014but they were there and I was acutely aware of why I was, a fact that I had tried poorly to shield when seeing Richard outside earlier.\n\n\"Are you scared?\" I asked Richard, as I continued to search every bit of square footage for a sign of hope. The silence in the room was deafening, so I had to say something. Why not ask the obvious?\n\nHe replied, \"Not at all. Are you?\"\n\n\"A bit. I wasn't yesterday when we spoke but somewhere in the night I became afraid about the real possibility that doesn't seem like it belongs to me, which is dumb because I am sure no one thinks that they should have this death sentence.\"\n\n\"We are not going to speak about this disease like you have it until we know. You hear me, Jules?! In the late eighties I lost some dear friends to this disease, and I don't relish going through it again with you . . . If anything, this is just a wake-up call in your otherwise long and youthful life,\" said Richard, who was cupping my face like my dad used to when I was little and troubled by a problem.\n\nAnd, as I had with my dad many times before, I willed myself to believe, in that moment, that the world just might bend to his will, on the strength of nothing more than his love for and belief in me. Since our first \"real talk,\" Richard had claimed never to want children, but I believed different. He is one of the most instinctively loving individuals I know. I always held it to be true that, as he said, he didn't want kids because adopting them as a single gay man, even in today's world, was nearly impossible, even when importing them from a Third World country. Feeling all the fatherly love he had given me over the years, I wanted so much to be a big girl for him always, but especially now.\n\n\"When did they stop using the fabric gowns? These paper things are vulgar,\" I said, to introduce some levity into the moment.\n\n\"Vulgar, honey, but far more sanitary.\"\n\nWith a soft knock on the door, Dr. Katz entered. He was not as old as I had expected, mid-forties or so, and much cuter, though not dreamworthy. By now Richard was seated in the chair to my left, mumbling something to himself about the mind-numbing nature of the weekly magazines available to read.\n\n\"How can anyone expect to focus on anything other than the obvious when all they give you is garbage to read? I must speak to the good doctor about this.\"\n\n\"Richard. Good to see you. Hi, Jules, I'm Dr. Gerald Katz. How are you?\"\n\nWith a bit of nervous laughter, I said, \"I hope much better than the last forty-eight hours of mind games tells me I am.\"\n\n\"Yes, Richard told me you had a scare,\" he said, nodding in Richard's direction. \"Why don't you tell me about it?\"\n\n\"I guess you could say that.\" The moment itself just felt so overwhelming that I could not help but get flustered and tear up a bit.\n\n\"It's okay. Take your time,\" Dr. Katz implored, taking firm hold of my hand.\n\nMy voice noticeably quivering, I said, \"I met someone about three months ago and allowed myself to get swept up in a moment. A stupid moment, you know, because I didn't really know him at all and now, as of Saturday, I learn that he is not entirely who I thought he was and I just don't want to be sick. I don't want to be one of those statistics.\" The tears were now falling freely. Richard, usually averse to extreme displays of emotion of any kind, had positioned himself at my side and was holding me around the shoulders.\n\n\"I understand. You are worried you may have been exposed to HIV. Is that correct?\"\n\n\"Yes. The guy, Keith, told me over the weekend that he is bisexual, and it's just sent my head into a tailspin. I mean, he's very hetero and healthy . . . not that I'm saying AIDS is a gay disease because it's not . . . I know that, but to be confronted with this . . . you know, so I never thought, but with that lifestyle the possibility suddenly seems real in a way that it never did before . . . I just don't want to be sick,\" I blurted out in one extemporaneous thought.\n\n\"First, I want you to breathe,\" said Dr. Katz. \"It was good that he told you about his sexual identity. Now you can make the best decision for you. Jules, the fastest-growing demographic for new HIV infections is the heterosexual community; specifically, women thirty-five years of age and younger. Falsely, many people still think on some level that AIDS is a disease reserved for the gay community or drug-addicted\u2014it's not. It is a human disease. It doesn't discriminate by sex, income, race, or education.\" Studying my eyes for any sign of recognition, Dr. Katz continued, \"You had unprotected sex with him about three months ago. Have you engaged in any other risky sexual practices before or after this? Have you been tested for AIDS before?\"\n\n\"No, we were safe after. About two years ago, I was tested, but there's been no reason to since then\u2014until now.\"\n\n\"Bad breakup,\" Richard chimed in, the disapproval of Tony still obvious in his tone.\n\n\"Yeah, bad breakup. He, Tony, cheated.\" The Kleenex in my hand had been properly torn to smaller pieces with each nervous twitch, so that it fell to the floor in noticeable chunks like the imitation snow displays that lined department store windows. \"I got tested then and was fine.\"\n\n\"That's good. Since you have been tested before, then you know that it takes about twenty-five days for HIV antibodies to appear in the blood.\"\n\nI nodded in agreement but was still as foggy today as I was then about what that part of that statement meant. Dr. Katz continued, \"When the body is introduced to the Human Immunodeficiency Virus (HIV), it produces specific antibodies in response. These are the antigens that we look for in order to determine if a patient is positive or negative. Now, since your last test, there have been great advances to expedite the results and accuracy. Results can now be determined from blood and saliva. Blood tests of course are still preferred unless you have an aversion to needles.\"\n\n\"No, I'm fine with that,\" I said.\n\n\"To be doubly clear, we will do both tests today. I'd also like to do a pregnancy test, while we're at it. Okay?\"\n\n\"Okay. How soon will I know?\"\n\n\"Within an hour or less. Doesn't that sound better than the week it used to take?\"\n\nWords failed me, so I just nodded and squeezed my eyes closed, still trying to wake up from this nightmarish moment. Feeling a hand on my upper arm, I opened my eyes to see the nurse who brought us in from the waiting room standing by the counter, preparing the syringe for Dr. Katz, who was applying an alcohol swab to the crease of my right arm. I couldn't bear the moment, to see my potential liquid death being extracted, so I turned to the left to look at Richard. For the first time I saw the fear of \"what if?\" in his eyes that hurt more than the needle's prick breaking my skin. Without thought I found a bit of strength and mouthed, \"It's okay,\" smiling weakly.\n\nWithin a minute or so, Dr. Katz had given the crimson vial over to her. \"All right, that's it. We will know in a little while. You are both welcome to stay in here. Go out, grab some coffee, and come back, or we can call you when the results are in. Whatever you would like to do.\"\n\nKnowing that Richard would take my lead, I spoke up: \"We'll stay here, as long as someone can bring Richard some less offensive reading material. I mean, it is bad enough having to do this, but worse is doing this and hearing him complain about the rotting of American values that you are perpetuating with these weeklies.\"\n\nDirecting his response now firmly at Richard, the doctor said, \"Touch\u00e9. I'll send in my private stash.\"\n\n\"Dr. Katz, thank you for allowing me to tell you about me instead of just looking at the chart. It makes this all better in some way.\"\n\nWith a smile and a tender pat on my shoulder before exiting the room, he replied, \"We try. If you need anything, ring the front.\"\n\n\u2022 \u2022 \u2022\n\nMy head was still throbbing, but I dared not ask for aspirin. Instead, I positioned myself near the window and watched as people went about their lives, completely unaware that mine was in purgatory. When Dr. Katz and the nurse came into the room together, about forty-five minutes later, I thought I was going to pass out.\n\n\"Jules, we ran both tests and there are no antibodies present in your system,\" Dr. Katz began.\n\nThe sight of both of them spoke volumes of dreams deferred and lifetimes limited, or so I thought, still standing there immobilized in the same position, arms wrapped protectively across my stomach, not daring to breathe. And while I heard everything that was being said, the accompanying emotions boiling inside were too much to express, so I just focused on the one that I had been withholding for the past hour.\n\n\"My head is splitting, can I have an aspirin?\"\n\n\"Jules, did you hear what I said? You're negative. Given the incubation period, if there was an infection, there would be some indication by now, and there is none. I can't say if you are in great health in all areas because we have not done a full work-up, but I can tell you that you don't have HIV. As a precaution, however, I would like you to come back in a few months for another round of tests. Nancy will contact you with date options. Jules?\"\n\nThe last thing I remember was nodding before everything went out of focus and then dark. I awoke some time later to see Richard hovering above me.\n\n\"Honestly, did I teach you to be this dramatic? I don't think so,\" he said.\n\nWearily, I replied, \"I'm okay, right?\"\n\n\"Yes, my dear. You are fine. Now promise, let's not do any part of this again. You know I'm old.\"\n\n\"Deal!\"\n\nDr. Katz reentered just as Richard was about to help me off the examination bed to stand up.\n\n\"Here, let me give you a hand. Glad to see you on your feet, Jules. Don't be alarmed by the fainting. Given the considerable amount of stress you have been under the past few days, it's to be expected. Thankfully, Richard caught you before your head hit anything or else we would be running a different set of tests.\"\n\nTeasingly, I said, \"Who knew the old man's reflexes were so good?\" Despite the jab, Richard knew the admiration was implied, so he opted not to protest. Dr. Katz continued, \"I am happy the news is great. Let this be a lesson about the choices you make and their potential impact. No more is it just an unexpected pregnancy that can derail a life.\" Reaching into his pocket, Dr. Katz's left hand displayed a rainbow of surprise that was the gold-seal confirmation that all was healthy in my world. \"Normally, I reserve these for my younger patients.\"\n\n\"And me,\" Richard said incredulously, tapping into his inner child.\n\n\". . . and Richard, but I think you deserve one today for being so brave. Don't choose the orange; it tastes like medicine. The green is my favorite but don't tell Nurse Nancy.\"\n\n\"I hope it's sour apple and not that fake apple,\" I said triumphantly.\n\n\"Young lady, what kind of operation do you think I am running here? Of course it is.\"\n\nFor the lollipop and for the green light on my life, I threw my arms around Dr. Katz. \"I'll see you in a few months for follow-up.\"\n\nOutside on the sidewalk with Richard, everything was justifiably better than before. The sun had broken through the clouds enough to create pockets of \"warmer than over there.\" There was scattered movement in the streets but without the blare of horns and erratic energy. In total, the whole scene was cinematic. I was not entirely sure of how to express my gratitude to him for being the one to endure all this with me. Slipping my gloved hand in his, I asked, \"Do you want to sit in the park for a minute?\"\n\n\"Only if that park is Caf\u00e9 Luxembourg and the bar has an unlimited pour. After the morning you have put me through, I must implore that it is after five p.m. somewhere on the planet and therefore perfectly fine for me to have a drink or three,\" he said, his laugh breaking the calm of the bustling street and seemingly igniting the sound of horns and life that was all around.\n\nIn fact, it was barely 1:15 p.m., but I had to agree.\n\n\"Martini? My treat.\"\n\n\"Honey, after the gray hairs you have given me, you will be buying for the next year. And you know I only drink top shelf, so you get your checkbook ready.\"\n\n\"Richard, you do know checks are so pass\u00e9, don't you? I have an expense account, you know.\"\n\n\"Don't make fun of the old, dear.\"\n\n## 23\n\n* * *\n\n## BATTLE LINES\n\nEN ROUTE BACK to the office, I stopped at Magnolia Bakery on Ninth Avenue to pick up a much-needed peace offering for Jacklyn. From the elevator I could hear her fielding a persistent caller. Her day had obviously not improved,\n\n\"For you,\" I said, presenting her with the box of sweet treats. \"I am sooo sorry for being rough on you earlier today. I was in a foul mood but all better now.\"\n\n\"It's okay, Jules, I guess,\" Jacklyn replied, quickly dismissing any residual skepticism that may have existed in favor of sinking her teeth into a vanilla cupcake. Mouth filled with cake and icing, she said, \"All in all, you're awesome. My girlfriend Elisa is over at Sui and she gets yelled at all the time for looking Anna in the face. I mean, how could you not look someone in the face when you're passing them in the hall or they are asking you a question? That's just weird, right! You being mean to me because of your situation is totally expected. I mean, I would be completely freaked out if I thought I was knocked up and\u2014\"\n\n\"Whoa whoa whoa, rambling sister! Lower your voice. Who said anything about being knocked up? I'm not pregnant. In fact I am one hundred percent fine.\"\n\n\"Well, you had to see the doctor, and then Cora kept calling and asking about Keith, and well, I just assumed.\"\n\n\"First, stop listening to Cora. That woman will have you believing that she is the result of The Immaculate Conception. Secondly, you know what happens when you assume, don't you?\" We nodded in unison. \"That's right, you make an ass out of u and me, so knock it off. Now, if you would like to apologize for your gross mischaracterization, one of those vanilla cupcakes will do just fine!\"\n\n\"Indian giver,\" said Jacklyn. \"I knew you were going to take one.\"\n\n\"Hey, hey, simmer down, little Jackie. That is Native American giver to you!\"\n\n\"Well, look who decided to stroll in today and grace us with her presence,\" said Michael, standing on the opposite side of Jacklyn's desk, nosily leafing through the papers strewn about Jacklyn's in-box, a stark contrast to the paper-free workspace of Simone's always immaculate desk. \"Is someone having a birthday or something?\"\n\n\"Good morning, Michael,\" I said. \"No.\"\n\n\"Correction, Jules, good afternoon.\"\n\n\"For the record I was here at eight thirty this morning. I had to leave for a doctor's appointment that ran longer than expected.\"\n\n\"Everything okay?\"\n\n\"Yes!\" Jacklyn's voice shouted, causing Michael to do a double take as if he hadn't noticed her sitting there\u2014next to the cupcakes.\n\n\"Jacklyn\"\u2014he nodded, then indicated he would like to speak to me in private\u2014\"you are aware that we have a meeting today at four thirty, correct?\" And I was ushered into the office.\n\n\"Yes, I was just about to review some notes on the partners before calling you. Just curious, though, why do you want me in there? With the exception of Carly, I don't know your partners.\"\n\nOnce inside the office, Michael and I occupied opposite ends of the sofa, enabling me to sit in the lotus position and him to recline, extending his legs onto the coffee table. Our formality had long since given way to a casual ease in private. I had the Carly upheaval to thank for that. It wasn't the same brotherhood relationship he shared with Raymond, and it wasn't the same Obi-Wan\/Papa Bear one that I shared with Richard, but enough so that I was no longer inexplicably self-conscious or overly eager to please the Big Boss every time I was in his presence. I really like Michael and never want to disappoint him.\n\n\"I want to expand the restaurant into some other territories starting next fall. Been looking into London\u2014your old stomping ground\u2014and a few others. They, the investors, don't, not right now. I need you to provide impartial evidence reinforcing my position,\" said Michael.\n\n\"So you're thinking a brand strategy and maybe a comparative analysis as to the benefits.\"\n\n\"Exactly. The market is ripe, and if we don't dive in now, we're going to miss out. I can't have my hands tied right now. Carly's is my baby as much as Kaylin is. I know what's best to keep her sexy. These dudes are just money guys who were supposed to be silent, but now that shit is hopping and we have some momentum, they want to flip it like they know. I can't let that happen. Not right now.\"\n\n\"Am I wrong in thinking that you and Mrs. Kipps held the majority interest?\"\n\n\"Collectively, yes, but one of the investors has the largest single share and is a close friend of the family. I don't trust him. He has been leading her, so to speak, for some time. Got her to thinking that the best decisions for the business are the ones that keep me close by, like keeping Carly's New York\u2013bound, under her watchful eye. So he's dangling the platinum gift\u2014freeing me up to spend more time at home with her. Well, you know how that translates. Not happening,\" said Michael, shrugging off the possibility of being ready to stay home and live a quiet life. \"We're about to do something epic here, and that requires vision,\" Michael continued, patting his pockets for a match to spark his cigar. Although I didn't want him to light up in my office, it seemed best to let this one slide. I could just open the windows and air the space out as soon as he left.\n\n\"With all due respect\u2014are you certain that's her intention? Maybe, if given the chance, she'd want you to spread your wings. I mean, after all, it is your dream and no one wants to be a dream killer,\" I said. Lord knows what calamities could be avoided if only Michael would include her in parts of his professional life instead of leaving her home alone to rattle about, orchestrating havoc in other folks' lives.\n\n\"After how she has infiltrated your life, can you ask that question with a straight face?\" asked Michael, erupting into one of his big laughs. \"All in all, mixing business and home is never a good idea\u2014that is, if you want to keep your personal life intact, and I do. Besides, it's not time to play that card yet.\" Michael's gaze was now steely, fixed on me with a determination that suddenly made me feel as if we were in NATO talks, the couch having been transformed into a small island. He was now the Dominican Republic and the investors were Haiti. Unwittingly, I had the responsibility of holding the title, as it were, to the land in the center that both sides wanted. \"When the time comes I will handle Carly, but until then I need those cats to be tempered down long enough for me to figure out what it's worth to them. If I need to, I'll cut the head off the damn dragon before I let someone come into my house and tell me what it is that I can do with it.\"\n\nAs stimulating as it was to see Michael going all General Custer, it was also a bit disturbing. I can't shake the feeling that there is far more to this story. Not only was he asking me to side with him against his partners, he was asking me to align myself with him in a potential battle that would surely pit me against Carly, and that is not a fight I was looking forward to. If the encounter at her home confirmed anything, it was that you don't get to be the legendary society gatekeeper Carly Spencer Falles without being formidable. I just wasn't sure if I was ready to take her on so quickly. Judging by the resolute look on Michael's face, I would say, ready or not, it was time to start shining up my sword and shield.\n\n\"I understand,\" I said, rising to retrieve the dossier off the desk. \"Okay, let's see who we have here. There is Simon Kleinman of Barrett, Browning & Fisher.\"\n\n\"Good ole Simon. He isn't much of a concern. A conservative securities investor with a diversified personal portfolio. For clients, he isn't risk averse. With his own money, however, he prefers to err on the side of caution.\"\n\n\"Okay, then there is Marcus Crawford of Chimera Capital. Wait, isn't this\u2014\"\n\nWatching Michael move about the room, I could see him running the catalog of what history between the two he wanted to reveal, given that I already had a frame of reference, far more than he knew but enough: \"I forgot you two met here. He's the key, Wall Street's boy wonder. You'd think he has more to do than telling me how and what I can do with this place. I guess that's what too much success early on does\u2014deprives these upstarts the opportunity to learn humility and respect. You know what they say, Jules, it's not the enemy you have to watch out for, it's the pupil.\"\n\n\"Is that right? I'm just wondering who named their firm Chimera,\" I said.\n\n\"Ha! Me. I gave him the name\u2014seemed like a good one at the time,\" said Michael, now seemingly lost in thought, standing at the window and speaking more to the history of a relationship that preceded me. \"I know this dude all too well. Hell, that was me about twenty years ago, an overly accomplished, unapologetic thirtysomething badass. Everything about him is strategic. The expansion of Carly's isn't his endgame; it's the means to an end that I don't know yet, and that is the mind fuck. Marcus is not going to put one over on me.\"\n\n\"Michael, if we are going in within the hour, then I need to get busy.\"\n\nStumping out the final drag of his cigar in the vintage crystal ashtray on my desk\u2014purely for decorative purposes\u2014he said, \"So it is. You know what I need. Make me proud, Dorothy. This shit ain't Kansas.\"\n\nAs if the scenario were not heavy enough. I felt as if I were being thrown into the middle of an old Western shootout right before someone yells \"Draw!\" Time permitting, I want to know more about Marcus, because clearly there is far more to him than being a Super Blond Amazon modelizer and random morning walking companion, which in and of itself is in conflict, but given the constraints, that would have to wait until I could get Michael all that he needs. It was most important in the immediate time frame to know the published business about our competitors who had succeeded and failed. Picking up the line and dialing, I said, \"Jacklyn, give me a search report on the top-niche restaurants launched in the past five years and then whatever information is available on members-only clubs, \u00e0 la the Supper Club, Soho House, the Groucho Club, et cetera. Also, time permitting, pull up some information on Marcus Crawford of Chimera Capital and don't ask Simone, just show me what is available online.\"\n\nAfter a few minutes she came in with printouts. \"I glimpsed at this but didn't get a chance to proof everything. Also, do Fashion Caf\u00e9 and Planet Hollywood count? They are niche, have entertainment aspects, and are expanding rapidly. I added a few facts about them as well.\"\n\n\"Actually, yes, they're good examples of what not to do. This is good. Listen, I also need information on the Rainbow Room and Cipriani. Also, get me the actual investment numbers here and the annual operational costs. For those you will need to go to Simone, sorry.\"\n\n\"Okay, I'll try.\" The regret for the blas\u00e9 response showed on her face before it settled. \"She hates me, you know.\"\n\n\"Don't try, Jacklyn, do. One last thing, what is the name of that restaurant on Forty-second in Hell's Kitchen, a few doors down from Chez Josephine's? They occasionally have musical performances.\"\n\n\"You mean Soul Caf\u00e9. They opened about a year or so ago. I go there all the time. They have this roasted chicken with sweet potato mash that is crazy good.\"\n\n\"As if I care. Forget it. They haven't been around long enough to help my case.\"\n\nMinutes later, after exhaustive speed-reading, I gathered my red-leather-bound Smithsonian notebook and headed off to the conference room. Just as I reached the door, Jacklyn came running down the hall.\n\n\"Simone is on the line for you. She said the meeting has been canceled,\" said Jacklyn, out of breath.\n\n\"What do you mean canceled? It's supposed to start in less than ten minutes. How can it be canceled?\" I turned abruptly on my heels toward the direction of Michael's office.\n\n\"Where are you going?\"\n\nAnnoyed, I barked, \"Learn to ask better questions, Jacklyn,\" and I marched off to find the battle.\n\nUtterly defeated with the course of events, she exclaimed, \"I did, promise! Simone wouldn't answer me.\"\n\nArriving at Simone's desk slightly winded, I wanted answers. \"Simone, what's going on? Why is it canceled? I just sat with Michael about this.\"\n\n\"An emergency came up with one of the partners, so it is being rescheduled.\"\n\n\"And you couldn't tell Jacklyn this?\" I said, eyeing her skeptically. \"Come on, what's the real story?\"\n\n\"Shhh,\" she said, frantically waving me to be silent. \"Lower your voice. Michael is inside talking to Carly.\" Leaning over in order to speak as low as possible, she added, \"From what I can gather, Mr. Crawford is not going to be in the meeting.\"\n\n\"Oh,\" I said, involuntarily grimacing at the possibility of having to come face-to-face with Carly sans the benefits of a boardroom table and witnesses.\n\n\"No, there's more. Carly is here to convey his wishes.\"\n\n\"You mean like his proxy,\" I said.\n\n\"Yes, can you imagine?\" responded Simone.\n\n\"Shit. This is sooo Dynasty.\"\n\n\"Shit is right. Michael hit the roof. I have never heard him lose it like that, especially with her.\"\n\n\"What about Simon?\"\n\n\"Redirected. Mr. Kleinman's office called to say he was running a few minutes late, so I told her not to bother. He has already called to speak with Michael.\"\n\n\"Shit.\"\n\n\"Jules, you already said that. Say something else.\"\n\n\"I know, but this is some crazy . . . ish. I mean, it's like having Blake, Alexis, and Dex live-action figures\u2014insane!\" From behind the closed doors I could hear loud muffled voices climbing in varying degrees of intensity, with the most irate one clearly belonging to Michael. \"So what am I supposed to do?\"\n\n\"I don't know. What do you normally do with yourself?\"\n\n\"Stop being snippy, Simone. It doesn't suit you.\"\n\n\"Touch\u00e9,\" she countered.\n\n\"I'll be in my office if Michael needs me. Okay?\"\n\nJacklyn was away from her desk when I returned, most likely downstairs with Raymond complaining about me. I couldn't blame her for not wanting to be in the line of unnecessary fire. Next time I'll get her cupcakes and a kickboxing certificate.\n\nNothing would please me more than to know that this day would soon be coming to an end. I desperately needed to recharge. My plan had been to leave the office immediately after the meeting and have a decadently decompressed night at home cooking, having some wine, and watching a flick. Now, however, I was not certain what time that was going to be possible. On the off chance that Michael would want to speak with me after expelling Carly, or if he decided to pop into the office, I needed to be there. Looking around, surely there was work to be done to pass the time. None, however, seemed more pressing than reorganizing the neat piles on my desk. After two hours or so, I was nearly stir-crazy and had to ring Simone.\n\n\"Oy! What time is Michael leaving? They can't still be arguing. Sadat brokered peace with Israel in less time.\"\n\n\"You are odd, Jules. Michael left over an hour ago.\"\n\n\"What?! Why didn't you call me?\"\n\n\"Was I supposed to?\" asked Simone.\n\n\"Yes!\"\n\n\"You didn't say anything.\"\n\n\"It was implied,\" I said.\n\nI could have wasted my time engaging her in a fruitless war of words, but why? Hanging up, I logged off the computer and gathered my already packed bag, complete with grocery and movie rental lists. Yelling from the interior of my office, I said, \"Jacklyn, go home.\"\n\n\"About time! I thought this day would never end.\"\n\n\"You're telling me. Why don't you come in tomorrow at eleven a.m.\"\n\n\"Boy, you must really feel awful. Cupcakes and now this.\"\n\nStopping in front of her desk, I conceded, \"Yeah yeah yeah, only because it wasn't your fault, but you rode it out. Good job.\"\n\n\"Here, don't forget this.\"\n\n\"What's this?\" I asked.\n\n\"The information on Marcus Crawford that you requested.\"\n\n\"I'll never read it tonight. Put it on my desk. I'll look at it in the morning.\"\n\n\u2022 \u2022 \u2022\n\nSeeing me approach the apartment building, Ivan stepped forward to open the entrance door.\n\n\"Evening, Ivan.\"\n\n\"Miss Sinclair. No delivery tonight?\" he asked, eyeing the grocery bags in my hands.\n\n\"I know! It seemed like a good idea to give them the night off, at least from me.\"\n\n\"Cooking is a lost art today. My Elizabeth is a great cook. It's why I married her.\"\n\n\"Really?\" I said. \"In that case your wife may have to give me lessons. My skills are limited. And by limited I mean three or four key dishes.\" He shook his head, more in recognition of the mass exodus my generation has taken away from the kitchen. \"Don't laugh. I'm serious.\"\n\n\"At least you try. It's better than most.\"\n\n\"Ivan, do you mind if I put these down somewhere for a moment? I need to check the mail.\"\n\n\"Of course!\" he said, reaching to take them, \"Where is my head? I'll take those.\"\n\nWalking toward the postal boxes, I said, \"It's been a few days.\" Off his look, I relented. \"Okay, a week or more.\"\n\n\"I'm not judging, ma'am.\" Placing his hand across his chest to emphasize his point, he added, \"It's my job to notice is all.\"\n\n\"Of course, Ivan. I can only imagine what you don't notice.\"\n\nTrashing the circulars and junk mail, I walked back over to the doorman's stand to get my bags filled with pasta, salmon, wine, cheese, fruit, flowers, ice cream, and vegetables.\n\n\"Anything important?\" Ivan asked.\n\n\"No, just the usual\u2014bills, magazines, bills, and more bills.\"\n\n\"That is capitalism. You pay in this country just to breathe the air.\"\n\n\"Isn't that the truth.\"\n\n\"Would you like some assistance upstairs?\"\n\n\"No, I should be fine. The hard part was the five blocks getting here. I can manage if you place them in my hands.\"\n\n\"You should get yourself one of the carts. They're convenient,\" said Ivan.\n\n\"Yes, I should, and I would if they weren't so gawd-awful-looking. Besides, could you imagine me pushing a cart in this outfit?\" I said. \"Exactly. Travesty.\"\n\nWalking to the elevator, I realized that I did not have a free hand to press the call button. Without looking back, I called to Ivan over my shoulder.\n\n\"Ivan, can you get the elevator, please?\"\n\n\"Here, let me do that.\" Nice voice, definitely not Ivan's heavily accented German, and he smelled good. Seriously, I must stop fixating so much on smell.\n\nStepping aside so that I could enter first, he held the doors at bay. Feeling the bags slip a bit, I thanked him while trying to readjust them without looking up to see the owner of the voice.\n\n\"What floor?\" he asked.\n\n\"Eight, please,\" I said. Nice shoes.\n\nEntering, I moved aside, allowing room for the nice-smelling gentleman, only to come face-to-face with Marcus! If I had a free hand it would have been used to pick up my face.\n\n\"You've got to be kidding me. What are you doing here?!\" I said.\n\n\"I was wondering when we would meet again.\"\n\n\"What are you doing here?\"\n\n\"Heard you the first time,\" he said, winking and turning his gaze up to the level indicators. \"I live here, unless you know something I don't.\" Leaning closer to me: \"Did Mitzy Bloomfield have me voted out of the building?\"\n\n\"You live here?!\"\n\n\"Yes, Seven A, and you\"\u2014he scanned the bags in my hand\u2014\"you're cooking me a well-deserved welcome home dinner, I see.\"\n\n\"And why would I do that? We are not friends.\"\n\n\"Not yet, but we should be. I knew it from the moment I saw you\u2014that we would be friends. What's on the menu?\"\n\nShaking my head to reorganize my thoughts, I said, \"Spaghetti vongole with a pear and arugula salad . . . Ugh! Why am I telling you this?\"\n\n\"Because I asked,\" he said.\n\n\"Nothing for you. I've had an impossibly long day. Started as the worst day actually and . . . wait, YOU LIVE HERE!\"\n\n\"Yes.\" Laughing while clearly amused at my discomfort, he continued, \"Why is this such a point of confusion for you?\"\n\n\"Because I live here.\"\n\n\"Yes, I know,\" he said, smiling. \"We've established that. You're in Eight A.\"\n\n\"How do you know that?\"\n\n\"The walls talk here, didn't you know?\" Holding his stare to determine if the line was sufficient, he continued, \"Besides, Carly told me. Earlier this year I wanted to purchase the unit from her and have it connected to mine, but I couldn't because Michael gave it to you\u2014his new hire.\"\n\nThe elevator doors opened right on time, because there was not much left to say after Marcus's revelation. \"Well, this is my stop,\" he said.\n\n\"And so it is,\" I heard myself reaffirming, as he turned to leave, \"Officially, the worst day.\"\n\nPlacing his hand on the door sensor before fully exiting, Marcus said, \"The worst day can often be the best day in disguise. Remember that.\"\n\n## 24\n\n* * *\n\n## WHEN IT ALL FALLS DOWN\n\nPresent day\u20142001\n\nTHE WORST DAY did become the best day,\" I say to myself softly.\n\n\"Excuse me, ma'am. Were you saying something?\" asks the driver.\n\n\"Oh, nothing. Just remembering something is all.\"\n\nLooking out the window at the random oil rigs lining the Ladera section of La Cienega told me we still had a ways to go before reaching West Hollywood.\n\n\"Shoot, we passed Starbucks, huh? I could really use a coffee,\" I say.\n\n\"Yes, ma'am, but there is another one coming up just after Rodeo. Should I stop there?\" asks the driver.\n\n\"Yes, please.\"\n\nAt this hour, the ordering line was short although the store itself was filled with people at tables on their computers or in the oversize chairs reading the newspaper.\n\n\"Welcome to Starbucks. What can I get started for you?\" asks the barista.\n\n\"A grande nonfat vanilla latte, a tall Americano, and a slice of zucchini loaf, please.\"\n\n\"What's the name?\" Seeing that I was distracted, the cashier repeats himself. \"Miss, what name can I put on the drinks?\"\n\n\"Sorry. Jules. My name is Jules.\"\n\nIn the metal wire stand by the counter lay copies of the Los Angeles Times and USA Today, each with ghastly images of the 9\/11 World Trade Center attacks from last week. The entire world outside me, as within, was in upheaval, leaving no place for the mind to rest. On the morning of September 9, after a horrible fight with Marcus, I left L.A. on a flight bound for London after he stormed out. My schedule called for a day's stopover in New York before continuing to London on the morning of September 11. It wasn't until I landed at Heathrow that I learned of the attacks. Walking through the terminal, although it was packed with thousands of people, I was overcome by a feeling of quiet morbidity and despair. It felt as if the life-force had been separated from oxygen. Knowing my state of mind, I chalked it up to a myopic vision of things and emotional despair over the row with Marcus and the anger I felt at him, at me. Not until I settled in the car and turned on my phone to retrieve my voice messages did I learn of the attacks, giving me another reason to cry.\n\n\"Julesea, this is Cora. They done blew up the World Trade. Lawd, what is dis world comin' to, child? Call your father, he is worried about you.\"\n\n\"Jules, honey, it's Richard. What a dark day for the world. Don't be alarmed, we are all as good as can be expected. Joy was not in the office yet. She was on the ferry from Jersey when the first plane hit. Poor dear, saw both collisions. Pray for everyone. Call me when you are settled.\"\n\n\"Jules, Daddy here. Have a safe flight and call me when you are settledt at the 'otel.\"\n\n\"Hi, young lady. Didn't you fly out today? I hope it was not on United. Call me. I'm fine. I'm worried\u2014can't find some of my friends. Love you. It's Joy. Please call me.\"\n\n\"Julesea, it's Mommy again. Your father is worried. I keep telling him that you are fine and that you were on Virgin Airlines. You were on Virgin, right? I like that Richard Branson. Lawd knows why anyone would fly United, such ugly interiors they have. Call your father.\"\n\n\"Hi, Jules, your driver will await you at baggage claim on landing and take you to the Sofitel St. James. I was able to upgrade your room to a junior suite. Everyone, your mom twice, Richard, Marcus, Joy, and Michael, have called. No worries. I'm on it.\"\n\n\"Jules, it's me. I spoke to Jacklyn, so I know you're okay.\" My hands began to shake hearing his voice; I replayed it three times before pressing Save.\n\nHearing Marcus's voice broke me down. I didn't have the energy to battle the contents of my purse for my sunglasses. Nor did I care that the stranger at the wheel saw me openly sobbing into my sleeve. It required only a small leap of faith on his part to deduce the nobility of my tears, being for the victims and not for the selfish concerns of my relationship. The emotional distance in his voice, where days before there existed such closeness, chilled me to the core. I tried to rationalize that, because he called, he still felt something more significant for me than obligation, but the absence of subsequent communication made it very clear that our relationship remained in a kind of peril that even a tragedy such as this could not repair.\n\nAt the hotel my television was set to BBC news in order to see the coverage of the attacks. Each time they showed the unfathomable images of the first and second plane crashing into the towers, I saw our life together crumbling into the great rubble that was left; our happy memories heavily shrouded like the soot that cloaked everything in the immediate radius. To say that my heart was heavy was an understatement. To say that I wished all of this to be another bad dream was accurate. That night I dropped to my knees and prayed the most heartfelt prayer possible for us, for all of those who had lost their lives to this war that we didn't start, and for all of those loved ones left behind to pick up the pieces. No one should know such tragedy. Marcus and I should not know such despair.\n\nDespite the World Trade attacks, the business at hand did not come to a grinding halt. It just continued with more suspicion. I remained in London as planned for the week to oversee the media rollout for Carly's UK, each day starting and ending seemingly the same, eyes closed. I would awake with morning amnesia and instinctively reach for Marcus to my right, only to find the bed empty. Initially, when I returned in the evenings, I would either go to the front desk to retrieve my messages or check the hotel phone when I came into the room. In the absence of a blinking light, I would phone the front desk just to make sure that no oversight or phone malfunction had occurred.\n\n\"Hi, this is Ms. Sinclair in 704. Are there any messages for me?\"\n\n\"No, ma'am, there are not.\" After putting this charge to the same night clerk a couple of times, he inquired dutifully, \"Is there a specific call that you are expecting?\"\n\n\"Oh, just the rest of my life is all.\"\n\nBut my life was an ocean away, and if I knew how to stop looking outside myself for answers, I could tell him that and get out of my own way. A few times I did pick up the phone to call but always hung up before completing the number. What would I say? I'm sorry. I'm scared. I thought I knew more about me. I'm learning. Please don't leave me? I wanted to say those things but just could not. I would sound so pathetic, and besides, after everything that I now knew, I didn't want to be that girl. I should have said those words during our fight, but I felt cornered, so I pushed back, because it was easier and less intrusive. Each solitary moment found me reliving fragments of that fateful night.\n\n\u2022 \u2022 \u2022\n\nLos Angeles\n\nSeptember 9, 2001\n\n\"What do you want from me, Marcus? You want me to say I'm sorry? Well, I am!\"\n\n\"That would be the least,\" he said, stomping off toward the kitchen.\n\n\"Don't walk away from me,\" I said, giving chase after him. \"Tell me what you want. You said it was okay! I'm trying to understand everything.\"\n\n\"Well, it's not, but what was I supposed to say? No, Jules, you can't go? Okay, you want the truth? I don't want you to go. I want you here with me, not half a world away or physically here, subconsciously waiting for an excuse for us to not work.\"\n\n\"I'm not. I haven't been. I'm here! What do you think I have been doing for the past year zipping back and forth? I have been here for you as much as I can, in a place where I don't know anyone and the women treat me like crap but you pretend not to notice. I have been here at your side, event after event, smiling like some mute as everyone tries to figure me out.\"\n\n\"That's what you think?\" Marcus asked incredulously. \"Is that what you call spending more time at dinner on your BlackBerry than socializing? Being downright prickly to our friends? Leaving an outing to take a call . . .\"\n\nThat's the thing about arguments\u2014no one listens, feeling in the heat of the moment that the only option is to talk over the other. Arguments like this never follow a logical order but always guarantee to reveal all holdover irritations that lie beneath in the most ugly and desperate way possible.\n\n\"I can't believe this. I put my life on hold for you,\" I railed back at his blatant indifference to my sacrifice. \"Without any assurance, do you know how scary that is for me?\"\n\n\"Appearing less emotionally invested in this than anything back East.\"\n\n\"I don't see my friends at all in order to be here whenever possible, and it's still not enough . . . I should have listened to Cora.\"\n\n\"I'm not going to do us on Cora's timetable or anybody else's for that matter! What does she\u2014\"\n\n\"Don't you dare,\" I said, feeling protective of her more as a point of contention against Marcus than as actual defense of Cora's meddling.\n\n\"There are two lives I care about right now, and you are controlling both of them, or don't you see that?\" asked Marcus fiercely. \"And now you are off to London for God knows how long!\"\n\n\"It's my job, Marcus, so yes,\" I said.\n\n\"Well, I guess that makes it all right now, doesn't it? Your job is publicity, not being someone's patsy, Jules. You can't possibly be serious. Try, just once, having the same reverence for my well-being and what I am offering as you do for Mike's. How about that?\"\n\n\"This is not Michael's fault, Marcus. This is about me making the right decisions for my future. This is about you keeping secrets, lying to me.\"\n\n\"That's where you're wrong. This whole situation has been orchestrated by Mike, yet you refuse to see it. Damn it! You asked that I leave him with something. I gave him London, and this is how the son of a bitch repays me, by trying to take what's important to me?!\" said Marcus, erupting with anger unlike any I had imagined him capable of. Sensing himself dangerously close to being uncontrollable, he abruptly stopped in an attempt to contain his burgeoning rage. \"How else would you have known about her situation, if not for Mike? If you had just come to me immediately, asked me, I would have\u2014\"\n\n\"You would have what, finally told me what you should've volunteered? That she was carrying your child?\"\n\n\"My God, Jules. It's not my baby! Do you want to see the damn test? It's impossible. How many times do I have to tell you that? How can I make you understand that I was blindsided and needed to figure things out before I could tell you so I knew what we were up against?\" pleaded Marcus, unclenching his fists and walking over to me. \"Why is it you find it so difficult to just trust me, to rest in me\u2014especially about this?\"\n\n\"How can I . . . ?\" I knew the words without full completion\u2014when try as I might I still have battle scars of Tony and Angie doing a number on my head and it's overriding the life with you that is right in front of me\u2014and knew they were wrong as soon as I heard them stumble out of my mouth, but I couldn't swallow them up and back away from the road they led down. Using all his strength, Marcus slammed down the nearest object he could find into the granite countertop, causing me to jump.\n\n\"Look at me! I'm not Tony,\" he said angrily, roughly taking hold of my face and forcing me to see the details of him as clearly as possible. Eyes inflamed and exhausted, he said, \"Jules, you are the love of my life and I am trying harder than I ever have not to fuck us up. I know you better than anyone ever will. I know what you think. I know what scares you. I know what makes you happy. I would never . . . You shouldn't need a piece of paper to know my intentions. If they aren't clear by now, then I don't know what the hell to tell you.\"\n\n\"Make me believe you,\" I said, my voice reduced to a whisper from the screaming and crying moments earlier. \"Make me believe that this is real, that I don't need a backup plan, that I can trust you forever. That I won't wake up one day alone in this to find you gone\u2014emotionally at first and then physically\u2014like him.\"\n\n\"Are you serious? I have been as transparent as possible with the good and bad of me. It's the most valuable thing I have to give,\" said Marcus, with more tenderness in his words than my heart could stand. \"What more do you need to get out of your own way, baby, and let us grow? What more do you need?\" he asked, before resigning himself to the gravity of our predicament. \"Julesea, we share a home. Was I crazy to think that we were building a life?\" Words escaped me, so I shrugged and averted my eyes, prompting him to release me but not before saying, \"If it's that difficult for you, then why are we here fighting this hard for this? Better yet, why am I here?\" he asked, and walked away toward the door.\n\n\"Where are you going, Marcus?! Come back here,\" I screamed, overcome by inconsolable grief as I watched him grab his keys off the corridor table and walk out the door. Wait! Wait! I didn't mean that I couldn't trust you. I just don't know how to confront the bullshit in my head and fight for us. PLEASE wait. But all of this was in my head. I continued standing there stunned, defiant, desperately praying that he would storm back in to resolve things\u2014but he didn't. When 5:45 a.m. came, I awoke alone to find his side of the bed untouched, as were the guest room, his office, and the living room. His cell phone went straight to voice mail. I didn't dare leave word. What would I say? So at 6:30, I snapped the locks closed on my last bag and got into the waiting car for transport to the airport, all the while convincing myself that the momentary indifference I felt was real, that I could let go of the trapdoors in my head and move on with my life, with or, if need be, without him.\n\n\u2022 \u2022 \u2022\n\nBack to Present Day\n\n2001 Los Angeles\n\nWalking back to the car, I was very much aware of my failings and the courage it would take just to be present in the life I covet. Giving the driver his Americano, I debate asking him to detour through Hancock Park. I feel the need to see its tree-lined streets with beautifully manicured lawns and kids playing a game of tag as a reminder of the future Marcus and I had discussed before everything got so out of focus, anything to delay knowing our fate.\n\n\"Here's your coffee,\" I say. \"How long do you think?\"\n\n\"Not too bad now. You should be home in about fifteen or twenty minutes.\"\n\n\"Some people can live a lifetime in fifteen or twenty minutes, you know,\" I say.\n\nEvaluating the merits of that statement, the driver replies solidly, \"I guess they can.\"\n\nLeaving me alone again with only my thoughts, trying to uncover the definitive moment my short-sightedness mucked everything up.\n\n## 25\n\n* * *\n\n## THE PAWN\n\n1999\n\nHOW'S MY SOLDIER?\" asked Michael from behind the hostess stand the next morning.\n\n\"Save it, Kipps!\" I seethed. \"Why didn't you tell me that Marcus Crawford, your partner and nemesis, lives in my building?\"\n\nStaring at me blankly, as if the location of Marcus's residence was common knowledge, before deciding whether to be dismissive or defensive, Michael selected the former. \"You've been in the building long enough. What have you been doing if not getting to know your neighbors? I know you met Mitzy already. She's a pill.\"\n\n\"Well, I didn't know, so to run into him last night after the day I had . . .\" Immediately remembering that Michael had no knowledge of my health scare\u2014and never would\u2014I said, \"I meant we had, and you going all last stand on me.\"\n\n\"And?\" he asked, not understanding the great injury. \"What is the deal with you and these Confederate references?\"\n\n\"I wasn't prepared is all,\" I say, partially reenacting the carrying of the bags scene. \"I mean, I walk in bags in hand. There he is all smiley and flirty and\u2014\"\n\n\"And what? Did he ask you out?\" he said, a sudden sense of interest piquing in his tone.\n\n\"No, of course not, but\u2014\"\n\n\"But nothing. Remember what I said, Jules,\" Michael admonished, using his index finger to drive home his point.\n\n\"Michael, you said a lot and I am still trying to process half of it.\"\n\n\"All you need to remember is that he is the key and not to be trusted.\"\n\n\"Yeah, yeah, that part I remember. I just don't want anything to be complicated,\" I said. \"I mean, not only do we have the meeting coming up, but now I have to worry about potentially dodging him afterward once you push him out the sandbox.\"\n\nWalking from behind the podium and taking me under his arm, Michael responded, \"Who says it has to be complicated? Just do your job. Got it?\"\n\n\"Yes. Although that hand is making me curious about my 'job' description,\" I said, fixated on his hand now resting sinisterly on my right shoulder.\n\n\"I like you, Jules. You're quick,\" said Michael over his shoulder as he began to walk down the stairs into the dining area of the restaurant.\n\n\"Apparently not quick enough. What the hell?\" I murmured under my breath, purposefully loud enough for him to hear.\n\n\"Never what the hell, Jules. Always what the heaven! Just when you think the deck is stacked against you, you get an ace.\" His declaration stopped me in my tracks. Had there been eyes in the back of my head, I would have clearly seen the wheels churning in Michael's. With two days until D-day, I definitely needed to plan accordingly for the partners' dinner on Thursday. Undoubtedly the wolves would have their teeth sharpened, and I was determined not to be the lamb.\n\nThe elevator doors opened to reveal Simone. \"Morning, Jules. Have you seen Michael?\"\n\n\"That's Ace to you.\"\n\n\"Huh?\"\n\n\"Nothing. He's in the restaurant.\"\n\n\u2022 \u2022 \u2022\n\nHaving given Jacklyn permission to come in late earlier in the week, I decided to reward myself in kind on D-day, Thursday morning. I knew the day itself was going to be excruciatingly long. My look needed to be flawless, and yet, try as I might, fashion inspiration was eluding me, so I did the next best thing to having Tom Ford on speed dial.\n\n\"Gary, I need you.\"\n\n\"If only you had a cock, that would be the news.\"\n\n\"Once again, too much information. Seriously, woman, down! Can you come over?\"\n\nLiving across the hall has its advantages. Less than two minutes after hanging up the phone, Gary was ringing my doorbell. While discarding elements of another poor outfit choice, I nearly tripped over my own feet rushing to let him in after the second extended bell. \"Why didn't you let yourself in?\" I asked, out of breath.\n\n\"Left the key with Jean Pierre.\"\n\n\"Oh,\" I said, tilting my head to the side to take in his modest sleep attire. \"Interesting. Not what I envisioned. Good, though.\"\n\n\"What did you expect, a silk kimono with a big butterfly and velvet slippers?\" Gary asked.\n\nConceding that his description was spot-on accurate, I nodded with an affirmative yenta shrug-sigh combination.\n\n\"Are you sure you're not Jewish? That was unnerving. The head nod, the judgmental eye squint, and concave shoulder. Reminds me of home.\"\n\n\"Ha! You want coffee? I picked up an amazing kona from Dean & DeLuca the other day,\" I said, leading him into the kitchen. \"So how is the boy?\"\n\n\"Yesterday's news, honey! I was looking for a yacht and he was more of a schooner, if you catch my drift.\"\n\n\"Ewh! I do and TM motherfucking I. TMI,\" I said, playfully pinching him. For a split second I thought of asking Gary about how best to exit my relationship with Keith\u2014but opted not to, for once heeding the warning that Cora had given me years ago about not sharing too much with people. \"Scone?\"\n\n\"Well, aren't you a chipper little Brit. Is there clotted cream as well?\"\n\nTurning my head to the side in faux modesty and pleased with the association, I said, \"I did pick up a few charming habits and phrases across the pond. The clotted cream, however, was not one of them, nor was the blood sausage. Some good old American butter is just fine for me.\"\n\n\"Such a shame . . . passing on a nice long big sausage,\" Gary said salaciously, half submerged in the refrigerator. \"Where's the ice cream, J?\"\n\n\"Middle shelf, praline and rum raisin. I polished off the rocky road on Monday.\"\n\n\"You're better than Baskin-Robbins,\" said Gary. \"Ah, this is what I'm looking for.\"\n\n\"Are you seriously having ice cream and scones for breakfast?\"\n\n\"Ice cream, butter. All the same; creamy and it goes down smooth. If you catch my\u2014\"\n\n\"Once again, drift has been thrown and caught, so let's please end the erotica portion on this segment of 'Breakfast with a Gay.' \"\n\n\"Boring little lass you are this morning. How can I make it better?\"\n\n\"What was that, Irish?\" I asked. \"Obviously dress distress. Tonight I am having dinner with Michael and his investment partners inclusive of the missus.\"\n\nClutching his invisible pearls, he said, \"Oh, honey. If your last romp with Lady Kipps was any indication, you're gonna need more than just a hot outfit. You need a shield and sword to survive.\"\n\n\"Obviously,\" I said, throwing myself across the counter in feigned defeat, \"but it's not just Carly who I have to deal with. Michael has this partner who thinks that\u2014wait, correction, who knows he is hot shit. A Wall Streeter with a penchant for supermodel accessories and Italian fineries.\"\n\n\"Hmmm, sounds like a contender for yours truly.\"\n\n\"Fan club, maybe.\"\n\n\"Meow,\" said Gary, fake clawing at me as we entered the bedroom. \"You better save some of that side-eyed kitty mix for later tonight with Carly.\"\n\n\"Seriously, this guy has her ear, which is most inconvenient for Michael, since he has specific views about the direction of the restaurant,\" I said, standing before the mirror, evaluating the merits of the charcoal Calvin Klein suit.\n\n\"Sounds like a player.\"\n\n\"Understatement. The first time I met him was during my interview. When I arrived, Michael was not there and this guy starts hitting on me hard, asking to buy me a drink\u2014flirt, you know. Eventually Michael comes, they talk, and the guy leaves,\" I said, turning away from my reflection to face Gary, \"without even saying good-bye to me.\"\n\n\"Sounds like someone made an impression,\" said Gary.\n\n\"Noooo, I just thought he was rude is all. Besides, I am not even close to being his type.\"\n\nNow standing behind me to see the mirror's truth, Gary reached for the suit with his left hand and replaced it with a denim trousers, dress shirt, and blazer combination. \"And how do you know that? Has he seen the gold coast yet?\"\n\nGiggling. \"Why do you always do that? You know being gay does not give you free rein to fondle me, at least not the way you're doing it.\"\n\n\"Maybe not, but being your resident man-of-style does.\" He switched out the dress shirt for a purple silk camisole. Admired his work. \"Much better. Put that on,\" he instructed.\n\n\"I ran into him and his token Eastern Bloc model girlfriend at Pastis last year when I moved in,\" I said, tugging to raise the straps on the camisole to minimize the revealing neckline. \"You don't think this is a bit too casual?\"\n\n\"Jules, you work in a supper club, for Christ's sake, not an accounting office,\" retorted Gary, slapping my hand away to loosen the straps so that the front revealed a bit more. \"There we go. If Miss Thing is going to hate you, let's give her a reason. At least you will have the boys on your side.\" Admiring his work: \"Stop squirming. So he was with Nadja Auermann . . .\"\n\n\"More like Esther Canadas,\" I said.\n\n\"Oh, girl, mommy is fierce\u2014talk about lips made for sucking,\" exclaimed Gary.\n\n\"Which was beyond annoying.\"\n\n\"If I didn't know better\u2014\"\n\n\"But you do,\" I said, narrowing my gaze on Gary's reflection to discourage what we both knew he was about to say.\n\n\"Jealousy,\" sang Gary under his breath. \"Try those shoes on.\"\n\n\"Focus. The point is, he lives in the building. A little detail no one bothered to tell me until I ran into him a couple of days ago in the lobby.\"\n\n\"Wait, he lives here in this building?\" I could see Gary's photographic mind taking inventory of all the residents. \"With the exception of yours truly and the Crypt Keeper, Mr. Sol, as eligibles, that leaves only Marcus Crawford. Nah nah, Ms. Thang, you have been holding out! Dish now.\"\n\n\"Ouch, stop poking me. There is nothing to dish about. He is . . . you know, and I am\u2014whatever. It's just incredibly odd, don't you think?\"\n\n\"I knew there was something wrong with you . . . You're odd, Jules. Marcus Crawford is one of New York's top bachelors and dare I say my future baby's daddy.\"\n\n\"Oh gawd, is he bisexual also? What the hell is going on in this place? It's like an epidemic of some kind. Maybe we should just rename the Big Apple the Big Rome.\" Only I found the humor in that remark.\n\n\"Umph, I wish he was. Trust! I have visions of how I can turn him, but he is strictly heterosexual,\" said Gary as he placed different purse options near me to determine which looked best. \"Poor thing, he has no idea what he is missing.\"\n\nFidgeting with the blazer, I said, \"Are you sure this isn't a bit too risqu\u00e9? I love the skinny jeans with the blazer, but the top just seems a bit too Garment District, don't you think? I look like one of those Betsey Johnson girls.\"\n\n\"Here's the thing, Jules. You're never going to win with Carly, so stop trying to dress for her and man up, so to speak. Besides, from what you have told me, I can't see how she is against you. I mean, in my humble opinion, if anything she's just a philandering relic who occupies her time meddling in other people's lives because her husband has yet to make her a priority despite the sacrifices she's made to raise his love child and finance his dream. You were just an unexpected distraction\u2014perk, if you will.\"\n\n\"You think all that, do you? Humbly, I mean?\" I ask, clutching my invisible pearls.\n\n\"Ignoring you\u2014yes I do! Take her out of the equation completely and just do you.\" Holding up the winning Ferragamo oversize clutch, he said, \"What you should do is focus your efforts on captivating that divine specimen of availability, Marcus Crawford, so you can give me all the details.\"\n\n\"Yeah, right. Are you not listening? I am totally not his type nor am I interested.\"\n\n\"Trust me, honey, you're interested, just too stubborn to admit it,\" said Gary, stepping back to take a final look at the outfit. \"Don't take yourself out of competition before the game is in play.\" Now turning his attention to my curly mane, pulling it back, then up and off my face in search of the right placement, he said, \"Hmmm, no . . . maybe, yes, we need to blow this hair out and put it up in a sleek ponytail with some oversize hoops.\"\n\n## 26\n\n* * *\n\n## D-DAY\n\nHEY, IT'S JULES,\" I said.\n\n\"Hi, Jacklyn, it's Simone. Is Jules around? Hold on. Yes, Michael, I am calling right now.\"\n\n\"Simone, stop multitasking, and focus. It's me,\" I said.\n\n\"Oh, where's Jacklyn?\"\n\n\"On lunch break, so I am manning the phones.\"\n\n\"Well, that's a role reversal. Must be nice. Maybe I should work for you.\"\n\n\"Ha, that would be the other way around, Mrs. Phillips.\"\n\nI could hear Michael, barking orders in the background: \"Simone, I'm not paying either of you to be friendly. Has Davis arrived yet? I need to go.\"\n\n\"Hold on, Jules.\" Off receiver: \"Michael, he will be here in five minutes. I am preparing your bag right now. Kaylin is aware that you will get to her in twenty minutes or so. I spoke to the nurse myself. Sorry about that, Jules.\"\n\n\"Hey, is everything okay?\" I ask, getting a quick overview of the source of Simone's distraction.\n\n\"I really wish you would stop using the word hey. It's very unattractive, dear.\"\n\n\"Lordy, have you been speaking to my mother?\" I asked. Sensing that now was not the time for cheekiness, I quickly interjected, \"What do you need?\"\n\n\"Hold on again.\" Simone off receiver: \"Yes, Michael, yes. Correct, it will be there. I put those numbers in your bag. I'm getting it right now. Jules, you still there? Hold on.\"\n\nPet peeve\u2014boss or not, I find idling on the phone just plain boring and a waste of perfectly good time. Talk about improper. \"Simone, it's still me, holding.\"\n\nWhy am I even trying? The sarcasm in my voice is completely lost on her. \"Sorry, Jules. Hold on one more\u2014 Michael, Davis is downstairs. Yes, I am on with her now.\"\n\n\"Simone! Obviously you are trying to tell me something, but between the incessant ringing of the phones and Michael barking, I'll never know.\"\n\n\"Jules, where is the research you compiled for the meeting tonight?\" Michael demanded, breaking onto the line.\n\n\"Michael, that is what I was asking her,\" Simone said in her most diplomatically efficient tone.\n\n\"And yet, I still don't have it,\" Michael bellowed from his office, forgetting that he had us both on the line.\n\nClearly he was on a rampage today and anyone within earshot was in the line of fire. Wanting to deflect this as soon as possible, I looked down to see the haphazardly written notes on the report I was reviewing, quickly determining that it was in my best interest to give him this set, scribbles, smudge marks, and all. \"Michael, I am on my way to you now.\" En route to his office I could organize them into something halfway presentable.\n\n\"Well, get a move on it,\" he said.\n\nBy the time I reached the end of the hallway, Michael was waiting impatiently at the elevator.\n\n\"Is everything okay?\" I asked, worried in the off chance that it would impact my duties.\n\nStroking his temple, indicating that awaiting Davis's arrival was not the only thing causing him pain, he said, \"Katy-kins is sick. I need to pick her up from school.\"\n\n\"Ah, so sorry. Sucks to be sick. Is Carly meeting you there?\"\n\n\"That's not her thing,\" he responded.\n\nThinking it best not to break the flow of conversation, I stepped inside the elevator when it arrived and accompanied Michael down to the lobby.\n\n\"Oh\"\u2014surely there was a better, less simplistically judgmental response I could offer up\u2014\"my mother isn't great with medical emergencies either.\" A total lie that I am sure Cora heard on some superhuman maternal level, despite being thousands of miles away. In truth, she was amazing in medical situations. It's one of the few times in my childhood that she was singularly focused on doing whatever was necessary to make things better entirely for me. My dad, on the other hand, tended to fall apart, which was unnerving considering he is always so strong and assured in day-to-day affairs so that my mother and I don't have to be. After one rather unsightly wrist fracture in junior high school and witnessing him faint at the sight of a needle, I stopped leaning on him in matters of illness.\n\n\"Um. How is she, your mom, at the mothering?\" Michael asked rather intensely.\n\nBeaming with pride, I confess, \"Relatively great, in her own way, but I'll never tell her that. She has a special way of showing it but she does.\" Our relationship has never been transparent. If I am to believe Freud, then I accept that it never will be\u2014such is the case with mothers and daughters, I guess.\n\n\"Well, that would seem to be the minimum expectation, now wouldn't it?\" Michael responded.\n\nKnowing firsthand of Carly's deign to accept Kaylin's presence in their lives, I knew exactly what Michael was insinuating. More important, I knew that it was a door that I did not want to open. So I quickly rebelled against the uncomfortable moment by thrusting the papers I held toward him: \"Here are the comparable restaurants in the U.S. and Europe. It is primarily made up of supper clubs and theme restaurants in the U.S. and private members only clubs in England. Also, you should pay close attention to the expansion rate of each\u2014that seems to be the key. The ones that have forgone mass expansions seem to fare the best. I've already started putting some notes in the margins but\u2014\"\n\nShoving the papers into his leather attach\u00e9, he said, \"Got it. Thanks,\" just as he got into the car and signaled Davis to drive on. Walking back into the building, I saw Simone going in the direction of the dining room, so I followed.\n\n\"Whoa, what was all of that about?\" I asked, pretending to be oblivious to the overall specifics, but in reality, just wanting to gossip a bit.\n\n\"Kaylin has the mumps or something. Anyway, when she calls everything else takes a backseat for Michael, as you can see.\"\n\n\"Clearly an understatement. I have never heard him be so unglued and terse with you.\"\n\nSimone responded offhandedly, \"I've grown used to it. Carly gets to abdicate maternal responsibility and I get the wrath.\"\n\nGreat, another Carly-laced bomb for me to sidestep. \"I gave him a copy of the research I'm using for tonight, my copy actually. I'll get you\u2014\"\n\n\"Oh, honey, that meeting is canceled,\" said Simone, between sips of black tea. \"You need to know that Michael Kipps will stop the world for only two things, Kaylin and this place. In the rare event, like now, that they decide to compete, the little princess always wins.\" So much for that number one\u2013selling book that Carly wrote about their storied love and family. By my calculations, that put her at a distant third. I guess fairy-tale endings, at least for her, are just that. \"I'll let you know when he puts it back on the books.\"\n\n\u2022 \u2022 \u2022\n\nSatisfied with the knowledge that Michael's attention would be consumed by Kaylin, I decided to make it an even earlier day and leave at 5 p.m. Now I could make that meal I never got around to. In retrospect, the encounter with Marcus a few nights earlier had somehow overshadowed the gravity of all the preceding events; the negative AIDS test, a forgone conclusion to a well-heeded warning, deciding how to address matters with Keith sidelined until his return\u2014breaking up via e-mail even from a nonofficial relationship is bad form, preparing a meal\u2014shelved. I had even thought about Marcus in my dreams to some degree, although I can't recall exactly what he was doing or what I was doing for that matter, but I'm sure he was there nonetheless, because I awoke with a smile and hearing him in my head.\n\nWithin an hour of arriving home, I was fully immersed in domestic bliss. My very skinny jeans and barely-there cami replaced with baggy cargo pants and a basic white tank. The Tahitian vanilla and gardenia scented candles, which I purchased from the same Portobello apothecary as those in the office, were lit throughout the apartment to complement the Anita Baker CD playing. Wanting the experience of cooking but not that of being a slave to the kitchen, I opted to roast a chicken and finish it off with a simple pear and arugula salad.\n\n\"The perfect companion you are,\" I said admiringly to the glass of Chardonnay resting in my hand as I stood by a window in the living room, watching the traffic steadily move down Fifth Avenue. The lights from the cars made every night seem like a Christmas parade. The horns at times make it sound like the Puerto Rican Day parade. The unexpected ring of the doorbell startled me back into the present.\n\n\"Just a minute,\" I said, leaping off the sill and rushing to open the door without looking because I was certain it was Gary seeking a full recap. Instead it was Marcus, standing before me in dark gray jeans and a white V-neck T-shirt that revealed a thin silver chain with dog tags. \"What are you doing here?\" I asked. Honestly, it must be criminal to be this fine!\n\n\"Not exactly the greeting I was hoping for but better than before, so I'd say we're making progress. Hi, Jules,\" he said, mischievously smiling down at me.\n\nUnfazed, I responded, \"You fancy yourself quite a bit, you know?\"\n\n\"That has been brought to my attention before. Why don't we discuss it over dinner? And before you say that you're busy, remember we were supposed to have dinner tonight, so there should be no one else on your calendar,\" he said, trying to peek inside the doorway to make certain that his assumption was correct.\n\nStepping to the right to block his view inside, I ventured to correct his perception of the evening. \"You shouldn't be here. And for the record, we were not having that kind of dinner. I was meeting my boss and his partners, of which you are one\u2014apparently not a nice one, I might add, during which time food might have been served.\"\n\n\"Character assassination aside\u2014exactly, dinner. Obviously not the way I would have preferred our first date to be, with onlookers and such, I'm quite private myself, but if you prefer chaperones\u2014it's a start.\"\n\nI shook my head in disbelief. Clearly, reasserting the specifics was futile. \"What are you doing here?\"\n\n\"I thought I just answered that,\" said Marcus. \"Are you going to invite me in?\"\n\n\"No.\" You inside my apartment would only mean trouble and that is never happening.\n\n\"Well, that isn't very neighborly of you, now is it?\"\n\n\"The same could be said for you, stopping by uninvited and without a gift. I mean, who comes to a person's home for the first time empty-handed? Not very neighborly, Mr. Crawford,\" I said.\n\nLooking down at his bare hands as if they had betrayed him by not spontaneously producing a gift, Marcus conceded his faux pas: \"I'm sorry.\"\n\nTotally forgiven is what I was thinking but not what I was saying.\n\n\"Well, well, it looks like your meeting went far better than expected.\" I didn't notice Gary coming down the hall. \"Marcus,\" Gary said with innuendo dripping all over his former comment.\n\nI can't be certain if Marcus was purposefully ignoring Gary's comment, oblivious to it, or just plain indifferent. In either case, Marcus turned to shake his hand and engage in small talk as neighbors often do. \"Gary, it's been a while. How are things at the magazine?\"\n\n\"Too long. When are you going to let me profile your Bridgehampton home? I heard from Edward and Cecilia that it is amazing.\" I wanted to interrupt and find out the social relevance of \"Edward and Cecilia,\" but Gary was now firmly in pitch mode. Gone were the exaggerated gestures of femininity that accompanied my fashion makeover this morning. In their place now were decisively more masculine movements that mimicked Marcus's own.\n\n\"Soon, soon. I didn't get a chance to spend as much time there as I would have liked this summer, a weekend here and there. You know?\"\n\nGary nodded, \"Of course. I would think that a property of that size located on the water is beautiful year-round. A winter story would be magnificent\u2014'On Frozen Pond.' I could allocate a number of pages to it before the season passes.\"\n\n\"Good to know,\" said Marcus. Ah, now I see it. He is clearly indifferent, and poor Gary is working so hard to make him care.\n\n\"I would even go so far as to guarantee you the cover.\" By this time my presence was purely a matter of happenstance, at least for Gary. Marcus's nonchalant air prevailed despite Gary's refusal to acknowledge it. He was determined to seal the deal here and now. The reason for such urgency eluded me, but I was surely going to find out. \"Did you see the great coverage we did on Jules's apartment?\" Bingo! Gary, finally, said something that interested Marcus and annoyed me.\n\n\"Did you now? Somehow I missed that issue,\" replied Marcus, switching his gaze to me instead of Gary. \"Would love to see how the photography compares with the actual space.\"\n\n\"Say no more. I have an issue inside,\" said Gary, pointing in the direction of his apartment across the hall. \"I'll bring it over.\"\n\n\"Perfect,\" said Marcus, with an all-knowing grin that his pawn had captured my queen. \"We'll wait for you inside,\" he said, pushing past me into the foyer only to step aside and smile down at me with those piercing blue eyes. Checkmate\u2014king's pawn. \"You coming? I'd love a tour.\"\n\nDespite the masterful play, I was not ready to concede any ground to Marcus, at least not the kind he was looking for. Sure, he had managed to gain entry into my apartment\u2014so what? It means nothing.\n\nAs he followed me inside I could feel his eyes searching every inch of my body instead of the apartment's d\u00e9cor. I only need to prove a gracious host and provide Gary some time to wrangle Marcus. No big deal; this is not even about me. I'm just being neighborly. I'm not betraying Michael, just being a good neighbor is all!\n\nFrom behind Marcus asked, \"Should I ask you now what specific unflattering things our Michael has been saying?\"\n\n\"No, you should not. All deserved, I'm sure. Can I get you something to drink?\" I asked.\n\n\"I'll take that as a compliment. Whatever you have open is fine.\"\n\n\"Of course you would! I'm having wine. Is that what you want?\" I insisted.\n\n\"Not really. Been one of those days. I'd love a martini, but if you don't\u2014\"\n\n\"No problem. Gin or vodka?\"\n\n\"Bombay, if you have it.\"\n\n\"Okay, Bombay it is. Have a seat, I'll bring it out.\"\n\nFrom inside the kitchen I made a beeline for the telephone and called Gary. Marcus had been in my place less than ten minutes yet it felt like an eternity. My previously relaxed vibe was steadily being replaced by a subtle anxiety in knowing that he was there and not knowing what to do with him. At this moment he could possibly be looking at my photos and any manner of personal effects that I was not comfortable sharing with my colleagues, especially him. God, please don't let him venture into my bedroom. It was still nuclear from this morning.\n\nCupping the phone's receiver with both hands, I said in a rushed and desperate whisper, \"Gary, where the hell are you? Changing! For what? You're just showing the man a magazine, giving him a quick look about, and then getting the hell out so I can enjoy my evening. Get over here now.\"\n\n\"It smells great in there,\" Marcus said from the living room over the music. \"You have good taste.\"\n\n\"Thanks. I love candles, especially mixing scents to come up with something unique,\" I said, without being certain if he was talking about the aroma from the chicken or the room scents, so I opted for the latter. \"Do you want an olive or lemon?\"\n\n\"Lemon is perfect.\"\n\nWhoever invented martini glasses must never have had to carry one across a room. Coming in from the kitchen, gingerly balancing his drink and my newly refilled glass of wine, I found Marcus standing near the Bose system, going through my music collection. Since working at Carly's, I had come to appreciate great sound quality. Not nearly as much as a pair of Dior heels but enough so that I felt the need to invest in a good audio system.\n\n\"Here you go,\" I said.\n\n\"What do you know about Corcovado?\" Marcus asked, taking a cautionary sip, only to have his taste buds come alive with the perfect balance of vermouth, gin, and lemon, chilled to perfection. \"Cheers. Oh man. This is perfect.\"\n\n\"Fifteen shakes is the key,\" I said. \"I'm a music lover, Crawford, or hadn't you heard?\"\n\n\"So I see, Van Morrison, the Stones, Tribe, Grace, Miles. . . . whoa, is this the Upsetters? What do you know about Scratch Perry?\" I could tell he was impressed, but, like me earlier, not ready to concede too much ground. \"Nice selection, but I am not convinced. What is the best yet lesser-known Marvin song? If you know that I'll\u2014\"\n\n\"Too easy. Take a seat and prepare to be impressed,\" I said, stepping in front of him to locate the song in question. Once it was loaded in the player, I turned around to face Marcus and savor the moment. Hearing the first few bars of \"Funky Space Incarnation,\" he made a quick gesture, unexpectedly spilling a portion of his drink.\n\n\"So sorry. If you have something, I can\u2014\" he said, indicating that he wanted to clean up the small puddles of cocktail now on my wood floors and coffee table.\n\n\"I will take that as a sign that you are thoroughly impressed,\" I said, laughing. \"There are towels in the kitchen.\" Laughing felt good. I hadn't done that in a few days.\n\n\"Actually, I was referring to 'Symphony,' but this will do.\"\n\n\"You so were not!\" I said. \"What are you going to say next, you cry sometimes?\"\n\n\"No, I wasn't. Look at you, Jules Sinclair, audiophile. Does Michael know you're more than a pretty face yet or is he still seeing the world through his glory days?\" asked Marcus, kneeling to clean the spill.\n\n\"Not addressing that,\" I said. \"Come on into the kitchen, let's get you a refill.\"\n\n\"Don't say you weren't warned,\" said Marcus.\n\nWe spoke effortlessly about music, both of us harboring a secret love for Tim McGraw and Euro pop. Soon we were playing the music equivalent of Confession.\n\n\"No way, I have you beat. My most embarrassing song moment is Diana Ross's tribute to Marvin,\" I said.\n\n\" 'Missing You'? Come now, how is that worse than not understanding the meaning of 'Love Come Down'?\" asked Marcus. \"I was a ten-year-old kid in South Boston singing about female masturbation! You have any idea how embarrassing that was\u2014\"\n\nPutting a hand before him to signal \"wait for it,\" I took a courageous sip of wine. \"I can't believe I am telling you this. Okay! So obviously she is saying 'since you been away I've been down and lonely,' right? Because Marvin is dead, left her alone, and she misses him.\"\n\n\"Wait, wait, why are you closing your eyes?\" he asked.\n\n\"Because I don't want to see your reaction,\" I said, squeezing them extra tight.\n\n\"All right now.\"\n\n\"So I thought\u2014remember, youthful ears and I was in Catholic school at the time. I thought she was saying, 'Sister Benaway, I have been down and lonely. Sister Benaway, I've been thinking, 'bout you.' I thought the song was about having confession with her favorite nun. I was jealous that I didn't have a favorite.\"\n\nThe silence that greeted my confession made me open one eye to make sure Marcus was still in the room. He was there all right, face scrunched up, lips pursed tightly as if trying to decide whether to get up and hug me for my extreme na\u00efvet\u00e9 or allow the volcano of laughter to erupt. Within seconds, loud riotous laughter ensued. My injury clear, I was trying to pretend that it was more funny than thoroughly embarrassing, when I accidentally sprayed him with the wine in my mouth, sending us both into full-on uncontrollable hysterics. Any notions I had of being graceful quickly dissipated. By the time Gary arrived, we were somewhere between old friends and new prospects.\n\n\"Glad to see y'all didn't wait for me before getting your party on. What's so funny?\" he asked, looking from me to Marcus. \"Why is his face wet?\"\n\n\"Diana Ross and Catholicism,\" I offered, because Marcus was on the last wave of hysterics. Clearly the moment was too far gone to bring Gary up to speed and he had other matters to attend to.\n\n\"What are you cooking? It smells good in here,\" said Gary, looking in the fridge for grapes.\n\n\"Roasting a chicken.\"\n\n\"Great, I'm starving. What else is on the menu?\" asked Gary.\n\nTaken aback at the implication that I would now have dinner guests, I stammered, \"Um, I was just thinking to make something simple\u2014for one, but, um, we could do some sides, maybe an arugula salad to start.\"\n\n\"Sounds delicious,\" said Marcus. \"You got some potatoes and buttermilk? I'm pretty famous for my garlic mashed potatoes.\"\n\nSlapping him on the back, Gary said, \"Well, then, get on it, instead of sitting here watching me suck on these damn grapes.\"\n\nI took a seat at the island alongside Gary as Marcus went about procuring ingredients as if he had been in my kitchen a million times before and knew what was located in every cabinet and drawer.\n\n\"Marcus, where does a man like you find time to cook?\" asked Gary.\n\n\"College. You either starve, eat ramen, and risk getting high blood pressure, ration care packages from home, or learn to cook. I learned to cook. Calling my nan for simple recipes here and there. Soon I found I enjoyed being in the kitchen. The whole process seemed to free my mind up so I could think about exams and stuff.\"\n\n\"Where did you go to school?\" I asked.\n\n\"Brown,\" said Marcus.\n\n\"Oh, Providence,\" I replied, as he affirmed some of what I thought upon first meeting him.\n\n\"Yes. You know it?\"\n\n\"No, but Cora, that's my mom, is hopelessly obsessed with Newport. So I know all things Rhode Island.\"\n\n\"So, Marcus, could we get some shots of you cooking in Bridgehampton? Our readers would love it, and for the people who know of your reputation, they would never think\u2014\" Gary said, gesturing to the kitchen. \"It would be an amazing give.\"\n\n\"And what is your reputation?\" I asked.\n\n\"Only one of the most enigmatic and stylish corporate raiders to emerge in recent years,\" said Gary in his eagerness to impress Marcus.\n\n\"Investment banker,\" corrected Marcus as he went about combining ingredients in a large bowl.\n\n\"Isn't that the same thing as a venture capitalist?\" I asked.\n\n\"Yes, but VC has an ugly connotation in today's market. IB is more PC than VC,\" said Marcus.\n\n\"Whatever you call it, he is a wonder and everyone wants to know what's behind the transcontinental bravado,\" said Gary. \"I just hope one day to have enough money for him to play around with.\"\n\n\"For the love of heaven, please put this man out of his misery and allow him to shoot your house so we can enjoy dinner,\" I said, getting up to start the salad and asparagus. \"Or else he'll continue shining you on, and I in turn just might lose my appetite.\"\n\nRemoving the mixing blades from the potatoes and offering me a taste, Marcus said, \"Only if you agree to be there. Something tells me I need a team around me to protect my interest or Gary will have carte blanche.\"\n\n\"She'll be there,\" Gary insisted, shooting me a look that begged not to be contradicted.\n\nWith that out of the way, Gary allowed himself to relax and enjoy the evening. Marcus and I busied ourselves putting the final touches on dinner; Gary kept the glasses filled and the music pumping. Formality and duplicitous agendas had long since gone out the window, so no one made any steps to move to the dining room. Instead we settled nicely around the island, spoke freely, and ate heartily. Dessert, like dinner, was impromptu, so a store-bought peach cobbler with heaping scoops of ice cream finished the meal off. Without ever intending to, I had hosted my first dinner party in New York since my return, my first actually in three years or so. I forgot how much I enjoyed the easiness of great food, good people, and home comforts. Marcus was the first to leave. He had vowed to stay and help clean up, but the constant ringing of his cell phone made it apparent that he was expected elsewhere.\n\n\"See, dinner with me wasn't so bad after all, now was it?\" he said at the door.\n\n\"No, dinner with you and Gary was great.\"\n\n\"Maybe next time just you and me.\"\n\n\"You and I both know there are a million reasons that dinner will never happen. Chief among those is that you are my boss indirectly, and I'm dating someone\u2014remember?\"\n\n\"You don't have to be, you know. As for Carly's, it's just an investment,\" he said.\n\nRemembering the charge Michael had bestowed upon me, I said, \"It's much more than that, Marcus. There is something great there if it is allowed to grow and expand. You'd know that if you looked closer at the long term.\"\n\n\"We'll talk about it over dinner,\" he said.\n\nAs he reached down to raise his ringing phone (exhibit A), I said, \"Right?! We'll make it a lovely group event, you, me, and the woman who is not content with leaving a message or allowing time for a proper call-back. Good night, Mr. Crawford.\"\n\n\"Good night, Jules,\" he said as he backed away slowly down the hall to the elevator.\n\n\u2022 \u2022 \u2022\n\nUnbeknownst to me, Gary was standing on the other side of the door eavesdropping as I turned to close it.\n\n\"Holy shit!\" I screamed, clutching my chest.\n\n\"Shit is right,\" he said. \"You have been holding out.\"\n\n\"Have not,\" I protested.\n\n\"Oh, hells yes! That man wants you.\"\n\n\"Well, that is not going to happen.\"\n\n\"Listen, honey, you know I love you like a new sister from a wayward father, so let me tell you, be careful. Marcus Crawford is better than Midas. He always gets what he wants. The problem is that his attention span is legendary ADD, if you get my drift. Why do you think he is so good at financial raiding? He goes in, assesses the situation, manipulates control, and then destroys it before it becomes a fixture.\"\n\n\"That makes three times today, G, that you have 'thrown me a drift.' Loud and clear on all,\" I said, \"but you have nothing to worry about, I've been a fool already and irresponsible once, not again. I have no intentions there.\"\n\n\"Good girl,\" he said, crossing the hall to his apartment. \"Stupid girl . . . lonely girl.\"\n\nBefore his door closed fully, I playfully yelled, \"You know I can hear you, right?! Just know that you owe me, Mr. Editor, and I will collect later.\"\n\n\"As soon as it goes to print. I'll see you in the Hamptons, lonely girl.\"\n\nExhausted, I collapsed into bed, but not before replaying the evening in my head. I had such a great time and hoped for a redo\u2014in general, not necessarily with Marcus, although . . . and Gary. In the move back to New York and diving into things with Keith, I didn't realize that I had forsaken some simple, essential things\u2014like just being me with people. Nothing more, nothing less. After Tony, I built a wall around myself that even my closest friends had not been able to truly penetrate. Richard had even commented on it: \"Honey, at some point you will have to let it all down\"\u2014but he had not pushed the issue. Had tonight gone as planned, I would not have known what I was missing. Further indication that the transitional relationship with Keith, even without the scare, was not what I needed.\n\n\u2022 \u2022 \u2022\n\nKeith had been back from Tokyo for a few days by the time we actually saw each other. While away, we communicated via e-mail once or twice. The first was an e-mail he sent.\n\nSubject: What's the most coveted commodity in Japan?\n\nAttachment: Photo of Keith in all of his bronze, statuesque glory and a cipher of commuters blatantly checking him out.\n\nJ\u2014one of my coworkers took this. Who knew I was such a big deal just for being?\n\nMasters\n\nSubject: RE: What's the most coveted commodity in Japan?\n\nHere I was thinking it was Michael Jordan sans the mocha tan, of course;-) Glad to hear you arrived safely. See you when you return, would like to chat.\n\nx Jules\n\nAfter clicking Send, I read the e-mail a few times. Why on earth did I say \"Would like to chat\"? It is the relationship equivalent of a siren in the middle of the night\u2014you know there's trouble coming but are clueless as to what it is. Since that fateful brunch with Richard and Blake, I had accepted that we could not go on being a romantic item. What eluded me was the perfectly strung sequence of words that would convey this without making him feel judged for his lifestyle. More than anything I didn't want to hurt Keith in any conceivable way, and yet it was inevitable. Rejection never feels good, even when done with compassion and sincere regret. The thought crossed my mind more than a few times to seek advice from friends, but it always felt like a betrayal, so I did not. My expectation was that I would surely know what to say when the moment presented itself.\n\n\"Jules, Keith is on the line for you.\"\n\n\"Put him through, Jack.\"\n\nWell, I am about five minutes away from the moment and no closer to knowing the bottom of the third or fourth act of this production.\n\n\"What's the sexiest girl in Manhattan doing?\" said Keith.\n\n\"Honestly, where did you learn to talk like that? It's very Dirk Diggler, you know.\"\n\n\"I could be offended by that remark, but I am choosing to find the compliment, in a seventies porn kind of way. So, how have you been?\"\n\n\"Dodging grenades, as it were. Michael has found himself\u2014 Hold on a minute,\" I said, placing my hand over the receiver and yelling to Jacklyn. \"Hey, Jack, jump off this call if you are on.\" As a matter of security and a sometimes selective memory, I have Jacklyn listen in on some of my calls and take notes. The downside of this is the resulting paranoia that makes me think she now listens to every call, especially the personal ones, and transcribes everything I say. \"Okay, where was I?\"\n\n\"Michael, grenades,\" said Keith.\n\n\"Oh, yes. Michael is on the warpath, so I seem to spend my days in trepidation fearing the phone and smell of cigars.\"\n\n\"Sounds like my timing is perfect then.\"\n\n\"It's something,\" I replied.\n\n\"Go look out your window.\"\n\nWalking to the window half expecting to see John Cusack standing below with a large boom box above his head, instead there was Keith, or at least someone who bore a striking resemblance to him. He was standing near the streetlamp wearing a fitted dark blue wool dress coat, a camel houndstooth scarf knotted high on the neck\u2014and a dapper newsboy cap, talking on the phone.\n\n\"Just as I thought, you look good, lady,\" said Keith, looking up to the window and revealing his face.\n\n\"Obviously\u2014you know how I do,\" I said. \"When did you start wearing hats?\"\n\n\"You like it? I had a stopover in London and picked it up,\" Keith said, his hand tracing the parameter of the brim.\n\n\"Well, look at you,\" I said, overwhelmed at the sight of him, and the physical response it still conjured in spite of everything.\n\n\"If you come down from that ivory tower of yours and join me for a coffee, you could look much closer,\" he said. \"Before you look at that calendar and tell me that you can't, know that I only have about twenty minutes or so but wanted to see you.\"\n\n\"In that case, how could I not?\" Flirtation with Keith is easy. His come-what-may manner made me happy even when there was no cause for celebration\u2014like now.\n\n\"Jeez, it's colder out here than it looks. Were you standing out here the entire time?\"\n\n\"No, I had a meeting a few doors down and dialed you from there,\" he said, switching to the outer side of the walkway. \"Let's go to Petite Abeille. Do you mind?\"\n\n\"Not at all, but if you only have twenty minutes, then maybe not.\"\n\nSearching the immediate length of the cobblestone streets for an alternative, he said, \"You're right. In that case, let's do Pastis. It shouldn't be too bad at this hour.\"\n\nWith the exception of a few tables in back and two guys at the bar, Pastis was abnormally bare.\n\n\"You have a preference?\" asked Keith.\n\n\"No, you choose,\" I replied.\n\n\"We'll take that one-top by the window.\"\n\n\"Perfect, I normally sit in the back. It's the best for people-watching, you know,\" I said.\n\n\"So I've heard; however, I've always found that the best way to know people is to observe their habits and listen to what is left on the table, between the lines. You'll have enough for a novel,\" said Keith.\n\n\"Really, I've recently learned that sometimes observation can fail you. Sometimes it's the overt statement that tends to pull the rug right from under you.\"\n\n\"Perhaps. I just have not ever heard anything from another person that I didn't observe beforehand.\"\n\n\"Then you're lucky, but I can't say that.\" Now fully engaged in the dance, I decided to just jump right in.\n\n\"Michael got to you that bad?\"\n\n\"No, not Michael,\" I said. Deep inhale, hold it . . . now go. \"You seriously threw me for a loop last time, Masters.\"\n\n\"I know, the whole Saturn in retrograde thing is a hard pill to swallow.\"\n\n\"True, but that came in a distant second to me hearing you like men as much as I do.\"\n\n\"Really?\"\n\n\"Yes, really,\" I said, my eyes purposefully drifting to the intricate stitching on his lapel for a place to rest in order to avoid any conflict apparent in his eyes.\n\n\"Shit, Jules, I thought you knew.\"\n\nLeave your hand exactly where it is, J. Don't remove it from the table or else he will think you don't want to be touched by him.\n\n\"And how would I know that? You and I had never spoken of it before and I'm not a mind reader, so . . .\" I trailed off 'cause the landing destination was unknown, even to me at that point. Look at him, Jules. You need to see his face. More important, he needs to see yours. God, I'm going to miss those lips.\n\n\"Damn, Manhattan is small, you know, and I am anything but closeted or low-key for that matter. Besides, your friend Joy seemed to know a lot about me, so I just guessed that was inclusive of the girl talk. I'm sorry, baby. I had no idea you didn't know.\"\n\n\"If she knew, she didn't tell me, nor should she have been the one. You're the one I'm sleeping with.\" More of a statement than an accusation. Yes, a statement. Let's hope. \"Besides, I am not a fan of hearsay, even from friends.\"\n\n\"What can I say, J, I'm not disputing that if\u2014Listen, I am sorry. It was not my intent to deceive you. I would never. Please don't be mad. We can figure this out.\"\n\n\"I'm not sure we can. Keith, I'm not mad. I still don't know specifically what my entire position on all of this is. The two things I am absolutely clear on is: I want us to be okay no matter\u2014I've gotten used to talking to you, and second, that we can no longer have any kind of sexual relationship.\" Damn, those words just breezed out. I hope he heard the first part. \"I'm not saying that I am going all Terry McMillan or anything, but I'm just not sure I am that modern, you know? I mean, I'm still battling through a mountain of insecurities every day: Am I smart enough? Am I pretty enough, fit enough, engaging enough to keep my man's attention? Not to mention how difficult it is just to not feel constantly like somewhere along the way, in the big picture of things, I fucked something up royally as to the course of my life\u2014twenty-eight years old and still single, hello?! Keith, it's hard enough to be a woman in this world with the statistics that are drilled into our brains every single day about being more likely to be somebody's baby mama than a wife. Living under all of that and now having to watch my back with the next man, about his attraction to my man, is much more than I want to deal with right now, you know?\"\n\nLeaning back from the table, smoothing the invisible wrinkles on his herringbone twill trousers, he said, \"I hear what you're saying, Jules. I just don't like it. Can we discuss this later? I really have to . . .\" His eyes followed the dial on the Roger Dubuis timepiece far more closely than necessary, before searching the room without success for our waiter to bring the check.\n\n\"I know you have a meeting. Go, I'll get this,\" I said.\n\nStanding in front of a bistro in the Meatpacking District with a less-than-ten-minute conversation is not exactly how I envisioned everything coming to an end, but that was the case, life seldom provides a perfect moment. Had I even said what I needed to? Walking back to my office, I convinced myself that the warm hug Keith gave me before disappearing into his taxi was a sure sign that we would be more than ships passing. If, however, we were passing, I could stride much better with a new pair of shoes from Jefferies. Besides, it's not as if we were in a serious relationship or anything. He was just some guy I enjoyed sleeping with, talking to, e-mailing, laughing with, and texting. No big deal, right?\n\n## 27\n\n* * *\n\n## BOUNDARIES\n\nTURNS OUT KAYLIN'S mild fever was the precursor to a nasty case of bronchitis. For a week, Michael kept vigil over her. It was the first time in his life that he felt absolutely helpless, and Michael Thurmond Kipps did not do helpless well. If he could not will the infection out of his daughter's body, then he would shift his focus to the one thing he could control, his restaurant, even if from afar.\n\n\"Saddle up, baby girl. Mike is on the warpath and is looking for you,\" said Raymond as I took a seat at the bar.\n\n\"Again? I have been yelled at, hung up on, and blamed more in the past two weeks than I was in my entire adolescence when I actually was guilty of occasionally sneaking out the house and lifting money from my mom's purse.\"\n\n\"Well, I don't know what to say about all that, but I can tell you that my man's looking for you, so be ready.\"\n\n\"Warning signal heeded. Pass me the phone.\" My day had been consumed doing media previews of the renovated upstairs recording studio that we were now opening up to certain musicians, i.e., those who Michael felt qualified as real musicians or those with insane recording budgets. Given all that was going on, I didn't think it necessary to send Michael my schedule. Normally Jacklyn would have been able to inform Simone of my whereabouts, but like Kaylin, she was home with a fever (thankfully not bronchitis) for a few days.\n\nSomewhere between yelling and near convulsion, Michael attacked on hearing my voice: \"Jules, what in the hell are you doing having dinner with Marcus behind my back? Do I need to be concerned about where your loyalty is? You know we are on opposing sides right now. Why would you do that and NOT tell me?\"\n\nMy intent had been to tell him about the dinner with Marcus\u2014innocent enough, right, but something inside told me not to. In the two weeks that had passed, I had not seen hide nor hair of Marcus, so I considered the encounter not worth mentioning. Michael's accusations came so quickly and were so venomous that I didn't have time to understand the complete root of his ire.\n\n\"Michael, I don't know what you heard, but Marcus and I did not have dinner.\"\n\n\"Are you saying that he is lying?\"\n\n\"No, of course not, but\u2014\"\n\n\"But what, Jules? No means what, yes, you did have dinner with him. You need to tell me something or else I need to evaluate who's on my team. It's like this slick motherfucker has Carly's ear and now yours. Who in the fuck has my back? I'm fighting for my family here.\"\n\nI could feel Raymond's eyes on me, watching my face go from pensive to defensive. If I tried to interrupt and argue back, my points would get lost and things would surely go from bad to worse, so I waited for Michael to settle down. \"Is there anything you have to say for yourself or do I need to consider making some changes?\"\n\nTaking a deep inhale, I closed my eyes to center myself before speaking. \"Michael, I am now and have always been on your side, never question that.\" I looked up to see Raymond nodding in the affirmative, indicating that I was responding appropriately, and continued, \"I should have told you about the dinner with Marcus and Gary, but Kaylin got sicker and I didn't want to burden you with superfluous information about nothing\u2014music and trivia.\"\n\n\"What? Who in the hell's Gary?\" he asked.\n\n\"My neighbor from across the hall. He is the editor in chief of Decor magazine. You've met him\u2014he did a piece on you and Carly years back.\" After hearing him grunt something inaudible that gave the impression this was information he did not have, I continued: \"If you recall the day Kaylin got sick, we were supposed to have the partners' meeting. With the meeting canceled, I went home. Marcus knocked on my door and asked me to dinner. I said no. While he was standing there, Gary came home and, upon seeing Marcus, began to pitch him heavily for a feature in Decor. One thing led to another, and the both of them ended up in my place talking about the feature and eating. Marcus left and then Gary.\"\n\n\"And that's it?\" Michael asked.\n\n\"Yes,\" I said, having glossed over the fact we spoke all of five seconds about Carly's. \"We were never alone, so even if he wanted to speak of other things, he could not because of Gary.\" Jeez, I hate lying, I don't even know why I am right now, but did Michael really need to know about the fun we had when Gary was away or could I just keep that for me?\n\n\"And you didn't say anything about Carly's?\" The emphasis Michael put on the question felt more like an inquisition, so I quickly replayed the evening in my head before speaking.\n\n\"Um, when he was leaving he mentioned something about Carly's being an investment opportunity. I said that it was much more than that, that Carly's was something great, and if given time he would see it.\"\n\n\"And that's all you said? You didn't say anything else?\" repeated Michael.\n\nOf course there was more. I had not given him the full context for this line of dialogue, fearing it would serve only to falsely imply my disloyalty. I couldn't bring myself to tell him that Marcus had asked me out on a date (again), and that in declining (again), this conversation came to be. By this time Raymond had given up any pretense of polishing glasses and was firmly attuned to our conversation. Michael may have not been there in the flesh, but his spare eyes were clocking my every response.\n\n\"Yes, that's it,\" I said.\n\n\"Umph.\"\n\nNervously tapping my pen on the table, I needed more than \"umph.\" Am I cleaning out my office? On probation for withholding? Forgiven? In the most neutral, nonantagonistic tone I could muster, I asked, \"What does 'umph' mean, Michael?\"\n\n\"It means don't ever keep anything from me again, no matter how insignificant you think it is. You don't have that luxury\u2014especially when it has to do with Marcus. I deal enough with that at home. I look up and the same motherfucker who is rallying to change the direction at Carly's, who has been mysteriously out of the office for the past two weeks and not returning calls, is suddenly at my living room while my baby girl is upstairs.\"\n\n\"Oh, I didn't know.\"\n\n\" 'Oh' is right. Thankfully, I was well versed in trumping anything he and Carly had to say. There I am in my own home, at my own damn table, talking about there being something great here at Carly's\u2014in full fight mode. That's when he sucker punched me by saying you said the same thing over dinner.\"\n\n\"But didn't he say anything else? Didn't he give it context?\" I asked.\n\n\"Nothing. He didn't need to say anything else. His jab landed squarely where he intended. Here he is bending my wife's ear and now letting me know that he was making ends with one of my employees as well.\"\n\nWow, is this the same guy who sat at my table and laughed with me? Talk about me being na\u00efve. Well, I'd sort my feelings about that later. Right now, I need to save my job. \"He said it like that to get a rise out of you, Michael. You know that, right?\"\n\n\"Yes, and it would have worked had he not finished by agreeing there was potential there. It would just take a capital infusion that neither of us has to explore things.\"\n\n\"Wait, so this is a good thing,\" I said.\n\n\"In a manner of speaking, but it's far from over. That was a warning shot. He made it known that he can circumvent me\u2014a coup d'\u00e9tat. If he thinks that his carrot will stop me from moving ahead, he's got another thing coming. Not only will I expand Carly's but I'll find the purse strings to bankroll it, buy him, and hold on to what's mine.\"\n\n\"And you guys can't work together?\"\n\n\"It's about more than that, Jules. A man has to be in control of his destiny at all times\u2014can't let some other motherfucker that I showed the ropes to come in and start making the rules. Soon he'll run out of maneuvers, and I'll show him who he's dealing with. Marcus Crawford needs me far more than I need him. That's for damn sure.\"\n\nClearly no love lost between these two, but how could things seemingly go so bad in a year's time? For anyone to dig their heels in the way Michael was and react as he did screamed of far bigger issues than deciding when and where to open new locations. It's as if the pupil had now become the teacher\u2014only problem is that Michael wasn't ready for forced retirement. I understood more than Michael gave me credit for. True, I should have told him about the dinner, but my words and time spent had somehow influenced Marcus, which was good. Marcus's being invited to his home at a crisis moment by his spouse, whose attention should have been on their child, should have been the issue. Once again a glimpse into their dynamic that left more questions than answers\u2014at least in my mind.\n\nHearing him calm and knowing now exactly what he needed to hear, I said, \"I understand. You're right. It won't happen again.\"\n\nHanging up the phone, I vowed to speak with him about my presence at the impending Decor shoot when the time presented itself. Raymond, having observed the entire exchange, placed a single shot of tequila in front of me as soon as the receiver touched the cradle.\n\n\"You handled yourself well,\" he offered.\n\nFeeling more vilified than vindicated, I replied, \"Thanks, Raymond,\" and looked down at the cup. \"Wrong T, though. I'll take a chamomile tea instead of tequila.\"\n\n\u2022 \u2022 \u2022\n\n\"Michael is full of hot air, just fall back and let this partner drama fade, or next time you may have to let him touch more than your upper thigh to keep your job. As for Keith, cut your losses and kiss the sky instead of mourning something that was nothing more than a glorified booty call.\"\n\n\"Blakes, you can be incredibly crass at times. Correction, more times than not,\" I said between bites of salad. \"Remind me again why it is that I called you and not Joy.\"\n\n\"Because she will placate you and tell you a bunch of bullshit to continue the pity party you have going instead of helping you to live in the truth,\" said Blake.\n\n\"And what would that be, exactly?\" I inquired.\n\n\"The truth, little Julesea . . .\"\n\n\"Uh, really, you must stop that,\" I said. Once upon a time, maybe over cocktails or the sort, I told Blake my full name. To my dismay she actually has the ability to retain much more than credit card numbers, because whenever she needed to emphasize a point and silence me momentarily, it was total recall.\n\n\"Whatever.\" Blake was in full stride and not about to be deterred. \"Truth is, not every person who enters your life is supposed to be there for the long haul, even when they have the ability to make your toes curl. In reality, most of them are temporary transients. Keith was a 'jump-off' because you needed desperately to get lubed. Now that squeak has been properly knocked out and you can get back on the highway of life.\"\n\n\"Why is it I feel like a 1979 Volkswagen Beetle?\"\n\n\"It's better than the spider\/cobweb metaphor I was going to use.\" She snickered at her quip. \"Seriously, though, Jules. Don't create a problem where there is none. Listen, I have to jump, weekly staff meeting, but let's go out soon. We'll go to Life. You're always good Big Daddy bait for me. Who knows, maybe I'll even invite the old girl\u2014Richard could use some loosening up.\"\n\nHanging up the phone, I wondered if Blake was right about people and partners being transient. Sure, Keith and I had never spoken about our future, as a couple, but there was something\u2014feels like melancholy\u2014to be said for no longer having the option to explore \"What does this mean?\" Now, in spite of everything, all that's left are a few memories of \"What could have been if.\" Those are the worst; like little leprechauns, always filled with promise at the end of a fictitious rainbow that will never deliver a pot of gold. \"Mama knows,\" that's what Cora always says. She also says that bad things, like good things, come in threes, just at an accelerated pace. Maybe that is the feeling of discontent stirring inside.\n\nIt was a little after 6 p.m. when I exited the 6 train at Seventy-seventh and Lexington, but nightfall had yet to fully engulf the city, so I allowed myself to stroll and window-shop. The tips of the trees seemed fatter this evening, despite the mounds of dirty snow scattered about the sidewalk. A clear sign that warmer weather was around the corner. At least that was my hope. I wonder if anyone told that groundhog? Lazy ass. Approaching my building, I didn't expect to see Percy standing outside under the heat lamps placed discreetly along the awning above.\n\n\"Hi, Percy, when did you start working evenings?\"\n\nTipping his hat as he pulled the door open, he said, \"Miss Jules, I normally don't, but Ivan couldn't make it tonight, so here I am.\"\n\n\"You sure are. It's a cold one tonight. Hopefully there won't be much foot traffic that keeps you outside.\"\n\n\"I can only pray, but you know I don't mind. No, ma'am, I do what is needed to get the job done 'round here,\" said Percy.\n\n\"True. Well, take care,\" I said, escaping into the building before he started to run the laundry list of all that he does daily to make our lives easier. In the course of our regular interactions, I had come to realize that everyone has this incessant need to be a star in their contribution\u2014especially doormen.\n\nRounding the corridor from the mailboxes, I glimpsed Percy talking to Mitzy\u2014clearly strike one, and I didn't know which one to feel sorrier for on this eve. Definitely Percy. Time had not made Mitzy and me any closer as cordial neighbors go. And try as I might, the sound of her voice and her prissy mannerisms just brought out the worst in me. So much so that on the day I had the misfortune of running into her in the fragrance department at Saks, the image of fucking her husband in her bed was the only reprieve from a positively mind-numbing conversation. Obviously, I would have to be really drunk, popping E, and he would have to shed about fifty or so pounds.\n\n\"All right now, Mrs. Bloomfield, you have a good night and let me know how that is coming along,\" said Percy, to my chagrin.\n\nDarn it, where in the hell is this elevator? Seriously, all the freaking money in this building and it must have the slowest elevator on the island. Maybe she'll check the mail first and I can just avoid her . . . or maybe not.\n\n\"Well, hello, Jules. It has been forever. The girls and I were starting to think you had left us.\"\n\n\"No such luck, Mitzy. I'm here.\"\n\n\"So I see,\" she said, sizing me up as if it were the first time we had ever met. \"In that case, how are you settling in?\"\n\n\"Just fine,\" I respond, keeping my eyes fixed on the ascending lighted numbers of the elevator panel: 2 . . . 3 . . . 4.\n\n\"The girls and I saw the piece on your apartment in Decor. It was quaint, not exactly my taste, but seems fitting for you,\" said Mitzy. \"Was that Marcus Crawford I saw walking with you sometime back? There I was, out for my morning run through the park, and surprise\u2014you and Marcus. At first I wasn't certain, I mean, after all, I normally see him with the tall, glamorous girls, one after the other\u2014funny, isn't it? I can't believe I remembered such a thing after all this time.\"\n\nDamn, why does she try me every time? One day I will be mature enough to ignore her kind. Today, however, is not that day. \"Mitzy,\" I say, looking at her admiringly, \"what moisturizer are you using these days?\"\n\n\"Chantecaille, of course. It is the best! Why do you ask?\"\n\n\"Last time I saw you there were all of these lines around your eyes and mouth. Far more than there are now. I was just wondering. I thought it was La Mer. Stuff works wonders, wouldn't you say? Like spackle, it is.\" Ding, talk about perfect timing. Thank you, elevator. Stepping into the empty car, I turned to see Mitzy, still on the opposite side of the threshold\u2014mouth open and pearls clenched. I thought to reach down and hand her back her face but opted not to. Why try being a considerate person now? Let it stay on the floor. \"You coming? No? Have a good night then.\" Oh yeah, that was much better than envisioning sleeping with her chub of a husband. Much better.\n\n## 28\n\n* * *\n\n## A GIFT HORSE\n\nTo the woman who knows her way around a kitchen and a music catalog, you are full of surprises. Jules, I really enjoyed dinner (and the walk). Sorry it took so long for me to tell you.\n\nSoon,\n\nMarcus\n\n\"Well, he certainly has great taste in stationery and nice penmanship. What is this, Cartier?\"\n\nSnatching the note, I said, \"Richard, I need you to focus. Like he really wrote this! Who does this guy think he is? Like some candle and note will\u2014\n\n\"Not some candle. Diptyque, dear,\" Richard corrected, sniffing the contents of the box again.\n\n\"Focus, will you?\" I repeated. \"It's just some wax and a wick in a sixty-five-dollar box. A candle and a note\u2014\"\n\n\"Cartier, my dear, not a note. Will what? Make you accept his heartfelt thank-you and express his obvious good manners?\"\n\n\"You know, that is the problem with you, Boulton. You get caught in the gestures. Let's not forget this 'beacon of etiquette' almost cost me my job, and is a bully.\"\n\n\"Slightly exaggerated, don't you think?\" asked Richard. \"I assume that last bit was about Michael? You need to focus on how this man is with you and let Michael fight his own battles.\"\n\n\"Maybe, maybe not,\" I said.\n\n\"What did he have to say for himself?\"\n\n\"Nothing. I haven't seen him since. I come home a few nights ago and this package is sitting at the door,\" I say, raising the gardenia-scented Dipytque candle and note in each hand as exhibits A and B.\n\n\"So, basically, you are being irrational because what? Because you still have yet to decide how many shades of pissed-off to be. Or maybe you are flattered but too stubborn to admit it. And maybe, just maybe, you want a talking point. Where's the cornmeal?\"\n\n\"I reject all of the above. Cornmeal, left cabinet above the stove.\"\n\nAfter opening and closing just about every cabinet and drawer in my kitchen, only to come up empty, Richard pleaded, \"Please tell me you have a cast-iron skillet.\"\n\n\"Of course. They were on the list you sent Jacklyn, weren't they? Look in the oven.\"\n\n\"Gawd, you are from the islands\u2014your people. These haven't been seasoned,\" said Richard, removing the still-wrapped trio from the oven and placing them in front of me with a bottle of cooking oil. After a beat or two of me making no motion toward them, Richard instructed, \"Don't just sit there looking at me all lost. Get to rubbing oil in each. Those pans won't season themselves.\"\n\nFor years I have said that Richard is going to make a great wife when it is legally permissible to be such. Tonight was but another example of that, and I reaped the benefits of his nesting tendencies. As payback for accompanying him to some dreadful intellectual event at the Princeton Club, he had agreed to cook me a meal of braised short ribs, macaroni and cheese with cauliflower, green beans, and corn bread. Provided we were not in a food coma afterward, a tryst at Life with Blake was in order.\n\n\u2022 \u2022 \u2022\n\nSince my time in London, Life had become the hottest nightclub in town, making the Village a nighttime haven once again. Word had spread throughout the boroughs that if you were about anything\u2014New Jack Swing, high fashion, or otherwise\u2014then you were a Tuesday or Thursday fixture. What was not discussed as much was the delineation of being \"in.\" Entrance was gained only through one source, Kane, the Pimp Daddy\u2013clad doorman. His eye for spotting supermodels and rock stars rivaled his awareness of less recognizable publishing and financial scions whose power was represented only by their title as seen in WSJ, Vanity Fair, The New York Times, Harper's Bazaar, and Forbes. When a patron made it to him, they were directed to one of two locations: the curtain on the right (for the majority) or the discreet staircase located behind the hostess (for the aforementioned models and scions). The rule of thumb was pretty simple: If you had to stand in the line that wrapped around Sullivan Street, then you went to the right. Although away for only a brief time, if it were not for Richard and Blake remaining fixtures on the scene, I might have found myself in that line. \"Out of sight, out of mind\" is the decree of New York club culture. If you walked directly to the front, never taking notice of the revelers in wait, shivering in their skivvies only to have Kane lift the ropes, kiss your cheeks, and stamp your hand, then it was down the stairs you went. Once during a rush to the loo I ventured upstairs wondering how crowded the line to the bathroom could be, considering everyone of note was smashed downstairs. Grossly mistaken! The vibe was decisively different, more pedestrian, with pop music, and packed, so I completed my business and headed back to the subterranean level.\n\nThose lucky enough to go downstairs found a dark, narrow maze of little alcoves that prohibited anyone from reminiscing about Studio 54. DJ Mark Ronson's lithe body behind the turntables\u2014or maybe his volume of crates loaded with vinyls\u2014was the first thing one saw at the base of the stairs. The dance floor and aisles always melded into one. For, in a city like NYC, \"those in the know\" could pack a hundred rooms of this size many times over, yet somehow they all made do with these cramped quarters. On this night, Blake had arrived before us and secured a prime table across from Mark, ensuring the best location to watch people arriving and be watched.\n\n\"Hi, kittens,\" she said, unlacing her supple limbs from the current big spender captivating her attention. And paying handsomely for it, I might add. Meow.\n\n\"Honey, I love it,\" exclaimed Richard at the sight of Blake in an emerald strapless silk jersey jumper accented with a chunky gold belt, earrings, and cuffs.\n\n\"I was worried you guys weren't coming out,\" said Blake.\n\n\"We almost didn't. Richard threw down in the kitchen. I couldn't stop eating,\" I said.\n\n\"No worries, though. I forcibly removed the fork from her hand to ensure she would fit into that dress,\" said Richard.\n\n\"Thank goodness, or all of this would not have been possible,\" I said, gesturing to my long-sleeved, black lace minidress with nude lining and black suede pumps. Not wanting to contend with deflating hair, I pinned it in a purposefully messy chignon, finishing the look off with a pair of diamond studs.\n\n\"Well, you look great,\" said Blake's date. I couldn't be sure if his compliment was genuine or as a point of entry, because more than five minutes or so had passed and Blake still hadn't made introductions. With Blake, whether it was purposeful game play or absentmindedness was anyone's guess.\n\n\"And you are?\" asked Richard, extending his hand.\n\n\"Peter Emmanuel, you?\" Nice. He has the right amount of salt and pepper that is clearly perfected by some overpriced stylist who only makes office visits because a man like this would never go to a salon unless it meant waiting out front in his chauffeur-driven car for his lady fair.\n\n\"Richard, and this one is Jules.\"\n\n\"Good to meet you, Blake has told me a lot about you both,\" said Peter.\n\n\"I bet she has,\" Richard said, cutting a side-eye in Blake's direction.\n\nLeaning over to Peter, I said, \"Don't mind them. It's their way, like vultures to each other they are. Consider it constant entertainment or an extreme example of bitches marking their territory.\"\n\nQuickly Peter established himself as the host of the evening, making certain that the bottles kept coming. He and Richard found themselves in talk of financial policy and fiscal responsibility that bored Blake and me no end. Thankfully, the initial bars of Michael Jackson's \"Baby Be Mine\" came to the rescue, transporting us out of our seats and to the dance floor.\n\n\"Peter seems nice,\" I say.\n\n\"Huh?\" said Blake, mid-two-step and finger pop.\n\nMoving closer to her ear and yelling, I said, \"Peeeter seems nice.\"\n\nNodding, she said, \"Oh yeah, he is nice enough, but you know\"\u2014she glanced in the direction of the table\u2014\"not perfect.\"\n\n\"Lord, Blakes, give him a chance,\" I implored.\n\n\"We'll see,\" she said, wrinkling her face before being overcome by the melody. \"I love this song. Wooo, baby-be-mine. Woooo.\"\n\nMark's masterful selection of songs made it impossible to leave the dance floor for the next thirty minutes or so. Had it not been for the beads of sweat forming on Blake's d\u00e9colletage, I would have remained, but public perspiration is a party foul of unforgivable magnitude for her. I'll never forget the first time she told me, \"Jules, the only time men want to envision a woman sweating is when she is on her back, comprende?\"\n\n\"Time to do a twirl around the room,\" she said, grabbing my hand. The purpose being to see who in the room was potentially more interesting than her date. Squeezing through tight groupings of partiers is like the game Operation. How close can you get to the edge of grazing a boob or an ass before the buzzer goes off and someone calls party foul? We stopped a few times to greet friends. On the final loop back to our table, Blake spotted a table of substantially more potential and gave the signal to say \"target identified\"\u2014two firm hand squeezes translates to \"stand in view of the target and mentally will him to approach.\" For Blake this normally happened within minutes of assuming the position. In this case, I could not be sure if the target was one specific person or the entire table, however big it was. With bodies everywhere, I only got a glimpse of a right shoulder and French cuffs.\n\n\"Is he looking?\" asked Blake.\n\n\"I can't tell. You're closer than me.\"\n\n\"Damn it. Okay, here goes,\" she said, spinning around.\n\n\"Ouch, watch the ponytail, sister. I am standing here,\" I said, after receiving a mouthful of hair.\n\nThrough clenched teeth, Blake mouthed, \"Sorry, J. He's watching, game on. Bingo. He's on the move.\"\n\nIn the moment that Mr. Potentially More or the representative approached and whispered the magic words that would bring us to his table, my eyes caught sight of Angie\u2014strike two\u2014standing less than six feet away. I had not seen her since the scene at Tony's apartment, when I discovered them together, and while I had made peace with his betrayal, I had not even begun to deal with hers. For what it's worth, Angie was a friend. Not a bestie but a friend . . . A friend who was fucking my man while smiling in my face, mind you, but a friend nonetheless\u2014or so I thought. Maybe not in the Richard, Blake, and Joy kind of way but a friend who was in the circle and expected to abide by the Code. Somehow she lost all meaning the moment Tony wrested the bottle from my hand, and yet in this very instant, the weight of her is too much to bear. She is all that I can see, all that I can feel. Unable to breathe, I need to get the hell out before she sees me and forces a situation beyond my control. Freeing my fingers from Blake's, I don't wait to explain. She would discover my absence soon enough, shake it off, and focus on her prey.\n\nForcibly pushing past people with abandon, I needed to make it to the surface. The dress prevented me from scaling the stairs two at a time. Richard called my name. I could not stop. Reaching the top, I was besieged by a large group awaiting entrance to the curtained room, providing enough time for him to reach me.\n\n\"Honey, is everything okay? I was calling your name,\" he said. When I turned to face him, my hands were shaking. Clearly I was far from all right. \"Oh, dear. What is wrong? Did something happen?\" For a minute or so I only nodded.\n\n\"She's here,\" I said, mistakenly expecting immediate recognition.\n\nSearching the room blindly, Richard asked, \"Who? Who's here?\"\n\n\"Angie. She's here. I saw her on the other side.\"\n\n\"Did she see you?\"\n\nI shook my head. \"No, I don't think so.\"\n\n\"Oh, dear.\"\n\n\"You said that already. Say something else.\"\n\n\"Do you want to leave?\" asked Richard.\n\n\"I don't know. I don't think so, but I don't know what I want to do to her. Do I cuss her out? Do I leave? Do I finish what I started two years ago and beat her ass?\"\n\n\"Definitely not the first or the last. You know how much I detest social scenes. Besides, penitentiaries give me hives. The uniforms are awful and the last thing this ole girl needs is to claw some tacky bitch.\" After a beat, Richard offered, \"But I will if that's what you want. You know what, I may even enjoy it. It has been a long time since I checked somebody.\" Fully taking him in, standing before me in a dark bow tie, checked shirt, and seersucker trousers, brought a smile to my face that was a precursor to sidesplitting laughter. \"What's so funny? I could scrap back in\u2014\"\n\nStill laughing, I helped him finish the sentence: \"\u2014the day. Ohhhhh baby, I bet you did, but not today. Not over that,\" I said. \"I'm okay. I promise. Just a little caught off guard and shaken is all. I'm okay.\"\n\n\"Are you sure?\"\n\n\"I promise. I'm great. How could I not be? I am here looking pretty good. You've got my back and that is my song!\"\n\nExtending his arm, Richard escorted me back down the stairs. I may have exited like Carrie but I reentered like the Homecoming Queen. \"Honey, where is Blake?\" he asked, steps away from the table where Peter now sat in the company of a baby-model creature. \"Her man is great but may not be hers for long.\"\n\n\"She's shopping but not to worry. It is Blake, after all.\"\n\nAt the table I guzzled a shot of Patr\u00f3n with urgency. As the lime splashed each taste bud, I lived in the awareness that Angie was never the problem. She was as much a by-product as I was a victim. Now thankfully a by-product whose Latina predilection for rice and beans and pastelitos de guava had transformed her once cute hourglass figure into something more science fiction, almost lava lamp\u2013esque. But I digress. The reason I forgot her immediately is because, whether or not she had slept with Tony, he and I still would have ended because we were never meant to be. She was just an excuse to prolong the hurt. I am sooooo damn done with hurting and done with running.\n\n\"And what do we have here?\" Blake said on returning to the table to find Peter fully engaged in a rather intimate conversation with the baby-model creature. Definitely not a man to be played, he threw Blake a look to say, \"I don't wait, ever.\" To which she pronounced \"Closure\" and sat on his lap, creating an awkward moment for all except her and Peter.\n\nToward night's end Blake confessed that she was not worried about Peter in the least because true to form she had found Mr. Potentially More at that table and he was even the right age to go half on a few babies. After which she would have a surgeon repair whatever childbirth stretched or lowered. Saying our good-byes out front, Richard was the first to stumble off, vowing to catch the late crowd at Raoul's before calling it a night. Peter's car was in front.\n\n\"Jules, would you like us to drop you home?\" he asked.\n\n\"Thank you, but no. I'm going uptown and you're just down the street. Don't worry, I'll take a taxi.\"\n\n\"In that case, allow me,\" said Peter, stepping into the street and stopping the only available taxi in the cluster of cars cruising Bleecker Street. \"Here we go.\"\n\n\"Perfect. I do hope that I will see you again, Peter,\" I said, giving both him and Blake the same amount of intensity, as they were truly birds of a feather.\n\n\"Of course,\" Blake said, hugging me tightly but not releasing until she whispered in my ear, \"Maybe\u2014we'll see.\"\n\n\u2022 \u2022 \u2022\n\nExiting the taxi put me face-to-face with the third strike, Marcus. It was too late for him to be leaving, so I assumed that, like me, he was just getting in as well from who knows where. He was standing at the doors in conversation with Ivan, who was now back on night duty. Seeing them, I debated hailing another cab and heading to Raoul's, where Richard was, but I feared it would draw too much attention, as Ivan was preconditioned for yellow. Then I thought it might be better to walk around the block, giving Marcus ample time to wrap up his conversation and go inside. That is, until the wind swept down Fifth Avenue through my barely there dress, making my final inebriated option seem most plausible: rush past them, in the hope that they would be so engrossed in conversation I could make it into the building onto a waiting elevator and into the safety of my apartment without having to engage Marcus. Hmmm, it's possible. Head down, shoes off, direct my energy elsewhere\u2014anyplace other than on them\u2014and walk softly yet briskly. Nice, a few more steps and you're inside. Yes, quiet mind, focus. Yes\u2014damn doorman! \"Evening, Ms. Sinclair,\" said Ivan. I'm going to remember this come Christmas when you are looking for a gift.\n\nShit, I am so busted. Makes no never mind, just mumble something, nod, and keep walking. Maybe I can make it to the elevator before\u2014\n\n\"Ivan, I'll catch you later. Jules, hold on,\" said Marcus. Quickening his pace to fall in step with me, he said, \"Jules, did you hear me?\"\n\n\"I heard you. I just wish you would stop speaking and saying my name is all.\"\n\n\"Whoa, not sure what I did to deserve that, but okay . . . I guess. Did you get my note? I know it's a couple of weeks late but\u2014\"\n\nStopping abruptly to face him, I said, \"Seriously, you have no clue? I pegged you for a lot, Marcus Crawford: narcissistic, indulgent, arrogant clearly, but clueless? Not so much.\"\n\nTrying to make sense of my attack and clear his head from earlier festivities, he said, \"Okay, I'll bite. Whatever it is that you think I did, I apologize.\" The seductive whiskey notes of spice and vanilla still lingered on his breath. Ivan was now standing inside the lobby by the door fully within earshot of our conversation, although he was pretending not to listen.\n\n\"You know exactly what you did. I expected more from you. Much more,\" I said emphatically. \"Was that part of your plan all along?\" I continued not leaving the space for him to respond or defend himself. \"To come into my home, gain my confidence, pretend to be my friend, and then use whatever little crumbs of information you had to hurt me. Oh wait! What am I talking about? Hurt me? You don't even know me and could care two cents about me. Hell, I'm just a pawn for you to get one over on Michael. I'm shocked you didn't tell him about our walk or running into each other at Pastis while you were at it.\" I could see the intoxicated clouds in his mind parting in order to allow the light of recognition for misgivings to become solvent. \"It doesn't matter that I could have lost my job over nothing, an innocent freaking dinner. It doesn't even matter to you that my boss now questions my loyalty.\"\n\nReaching out to quiet me down, he said, \"Jules, shhh, shhh, my bad. It's not what I meant to happen. Honestly, you have to believe me.\"\n\n\"Did you just shush me? Don't touch me. I don't believe anything you say,\" I shouted, now drawing the overt attention of Ivan, as I pounded the elevator call button again. \"Who do you think you are anyway\u2014shushing me like that? You can't just go around\u2014\"\n\n\"I did shush you, I was wrong. I'm sorry truly. Forgive me? I didn't think it would impact you like that,\" said Marcus, lowering his head as the full weight of my situation washed over him. \"Mike is just so damn smug sometimes and I thought\u2014\"\n\n\"Next time keep me out of your thoughts, okay?\" Stepping into the elevator with the expectation of watching the doors close in his pompous face, I turned to find Marcus within inches of me. For a moment I was overcome by the sheer magnitude of his presence towering above me, and the all-consuming fragrance it created, rendering my senses temporarily inoperable. I think this is what they call chemistry. If you kissed me right now, I would let you. No! N-o!! This guy is not . . .\n\n\"That's hard to do, Jules. I've thought about you since the day we met\u2014been thinking about you even more so of late.\"\n\n\"Like I said, I believe nothing you say,\" I said, moving to the opposite side of the elevator car.\n\n\"Well, it seems I will have to work harder on that.\"\n\n\"Don't trouble yourself on my behalf. Trust me, I'm not that important to the games that you men play.\"\n\nThe elevator doors opening on the seventh floor signaled the much-needed end to our conversation. \"But you are. You'll see,\" said Marcus, placing his body between the doors, freezing time until he was certain his overture was clear.\n\nLater, as I was lying in bed drunk and alone, clad only in a bra and panties, my heart was beating far faster than the swirl of questions running through my mind. It's just the culmination of the three strikes, I told myself. Problem is, when it comes to the last, to Marcus, my head and my emotions are sending mixed signals. The incessant blinking on the answering machine proved a much-needed distraction, sort of.\n\n\"Julesea, it's Cora. Call your father. He would like to come and visit you on his vacation this year.\" Worst timing ever that woman has.\n\nThe clock on the nightstand said 2:51 a.m., far too late to phone her back. Just as well, I was so freaking tired. Reaching over to turn out the lights, I felt my phone vibrate under me\u2014clearly a cosmic joke. The caller ID said Blake.\n\n\"Shouldn't you be having sex or something of the like so I can live vicariously through you?\" I asked.\n\n\"Not tonight. Peter could have got some if he played his cards right but the baby-model creature changed all that.\"\n\n\"Oh yes, I forgot how territorial you can be. Poor guy has no clue what he is in for.\"\n\n\"Poor him? Poor me! I met the most amazing guy tonight and couldn't even thoroughly enjoy it because of Peter. I mean really, what gives with that?\"\n\n\"You can't be serious, Blake.\"\n\n\"Oh, am I ever,\" she cooed. \"He's not like the guys I normally go out with.\"\n\nSarcastically, I said, \"In what way? He's not in the midst of divorcing his second wife and replacing his current paramour with you?\"\n\n\"Exactly.\"\n\n\"Seriously, Blakesy, you are a piece of work.\"\n\n\"And you are spending far too much time with Richard. Being snide is becoming far too easy for you, Julesy. So the thing is, this guy didn't immediately fall over me, although I could tell he was interested. He made me work for it. He didn't even ask for my phone number. I had to give it to the representative who came over initially and do the ole bait-and-switch . . . I'm intrigued.\"\n\n\"Wow, I would have paid money to see that,\" I said. \"Maybe you've met your match.\"\n\n\"Speaking of which, where in the hell did you dip off to? I turned for backup and you were gone. I really needed you to engage the friend, who was far too eager in the details of me but clearly standing on a slimmer wallet. What gives?\"\n\n\"Oh, my. That is definitely not a conversation for the phone, especially at this hour. I have an early day tomorrow, so I need to get some shut-eye. Allow me to get through this kick-off, and we'll discuss,\" I said.\n\n## 29\n\n* * *\n\n## PLAUSIBLE DENIABILITY\n\nWHAT KIND OF name is Punxsutawney Phil, anyway? He saw his shadow all right, and took me right back into the burrow with him. It's as if I went to bed last night, in the dead of winter, only to awake and find myself in the midst of New York's emotional prespring alert but oblivious to the details in between\u2014with the exception of work and home. Sad as it might be, the most consistent man in my life of late has been Michael (and Daddy, of course, but he's perfect for the most part), for whom I have much respect but had caused great injury to. I wanted nothing more than to repair things with him, even if it meant an entire chunk of my life had gone missing. So, as a show of solidarity, I lived and breathed Carly's.\n\nThe holiday music series had been a smashing hit. We made a profit and got amazing press out of it despite the diva demands of some of the performers\u2014or should I say their handlers, who were oftentimes far worse than the artists themselves. As a result, Michael had a \"life is too short\" epiphany, as I like to call them. Sure, being sold out every night with a wait list was fantastic, but the headaches and erratic energy that resulted underscored the elusive magic that was Carly's. In return, Michael charged me with the task of securing a single performer to do residency. To paraphrase him: \"It's not enough that she can sing or has sold some records. I need a motherfucker who can keep this place packed, bring in some special guests, and impress me enough to sit and watch every night for the next four months like it was the first time.\" It was the latter that I feared would prove my undoing. Was he insane? We were in the midst of a hip-hop revolution. He was describing the Spice Girls\u2014not a jazz or soul singer. The way I saw it, the only game in town that had accomplished this was the Carlyle, and they already had Bobby and Eartha with a decisively older crowd and much smaller room. Carly's was hipper and less antiquated. Its crowd wasn't still living in the heyday of Jackie O, Warhol, and Maria Callas. The Carly's patron was living for Kate Moss, Keith Haring, and Russell Simmons, so who was out there with enough swagger and gravitas?\n\nThe answer actually found me one night, thanks to Cora. After much debate, she and Dad had come up from Virginia to spend the week with me. Daddy, who seldom takes a holiday, had wanted\u2014as I learned\u2014to return to Jamaica to spend some time with friends. Cora, as I had thought, was not having it. New York held the lure of endless shopping, Broadway shows, and an up close and personal look at my life, emphasis being on pointing out all that is lacking. In typical Cora fashion, she had their entire stay planned to the smallest detail. The only saving grace for me\u2014poor Daddy\u2014was that my days were consumed with work, so lunch at Tavern on the Green, with all those camera-happy tourists, and walks through Times Square, bobbing and weaving through more camera-happy tourists and TRL audience members, was out for me. My nights, however, were a different story indeed.\n\n\"How much work can a woman do?\" asked Cora, when I thought to use it as an excuse for why I would not be available a couple of evenings. \"This is exactly why you girls today are single. You work far too hard for nothing and have no time to focus on finding a husband.\" One of her favorite tirades when it benefited her. Daddy sat across from me with the same compassionate look that he has given me since I became old enough to be Cora's reflection. More for him than for myself, I felt sad. Normally he was the one with work to escape to, but this week\u2014his only week of rest\u2014it was either him or me in the crosshairs of the Santa Ana wind known as Cora Madeline Augustus-Sinclair, aka Mommy. When viewed from that vantage point, it seemed only fitting that with him taking day duty, being pulled around the city like a poodle to look at a whole manner of things he could care less about and nothing he was actually interested in, I would make myself available as much as possible.\n\nOn this particular evening Cora had music on her mind and made plans for us to see Nancy Wilson at the Blue Note. Everything about Nancy, from her voice to her command over men, pleased Cora, and had for more than thirty-five years. To hear her tell it, Nancy was the first great love of her life \"in a respectable, decent way,\" and then came Daddy. For it was by Nancy's example that she was able to lure him in the first place. Being a child in Jamaica, where the caste system still prevailed, it was rare to find images of beauty in a package that looked like hers. True, Cora may have been half Dutch, but that was relegated to her bluish gray eyes and \"good hair,\" which made a striking contrast to her molasses-colored skin.\n\nI grew up hearing stories about the first time she saw Nancy on television with Sammy Davis Jr. It was magical. Any woman who could hold her own with the Sammy and not be billed as a bimbo was someone to admire. Tonight, however, was Cora's opportunity to live her dream and be in the same room, breathing the same air, with her hero. Earlier in the day, following one of my uncontrollable nuisance spats with Cora, Daddy made it known that tonight was hers, so we (I) was to play nice, at least for the evening. My, how I love the way he loves this woman and makes the way for her easier, even when she thinks she is driving.\n\n\"Julesea, stop fidgeting, you'll mess up your hair,\" said Cora.\n\n\"Would that be so bad?\" I asked, having been reduced to childlike tendencies days earlier\u2014a record, really.\n\n\"Not if you would prefer to look like one of those hooligans off the street, no,\" she insisted.\n\nCatching a glimpse of myself in the mirror near the entrance of the Blue Note, I found that particular leap impossible. Heeding Daddy's plea for peace, I had allowed Cora to dress me for the evening. And while she resisted the urge to turn me into her clone, she could not resist cleaning me up properly in a manner she thought befitted the occasion. Feeling much more like we were attending Easter Sunday service than a jazz club, I chose to engage in an all too apparent battle with my clothes, as children often do when being forced to be presentable. When did Escada become appropriate for a twentysomething\u2014albeit a late twentysomething\u2014to wear? Maybe I missed that memo.\n\n\"The hatpins in this thing are sticking me,\" I said.\n\n\"Well, go to the ladies' room and adjust it, but don't ruin your chignon. Took me forever to get it right. We must get you a deep condition before I leave . . . The girl's hair is unruly, Charles. Always has been. She gets it from your side.\"\n\nExiting the bathroom, giving one last tug to my skirt, I ran into Marcus\u2014or should I say, crashed into him.\n\n\"Research or going over to the other side?\" he asked.\n\nOur last encounter aside, out of mind, I had to admit I was happy to see him\u2014a familiar face to endure the torture. \"You wish. Neither, I'm here to see Nancy, just because.\"\n\n\"Really?! I didn't know you were a fan. I don't recall seeing her music in your little collection,\" he said.\n\n\"Still hating on my musical stylings, I see. You know you were impressed,\" I said. I pointed in the direction of the table where my parents sat. \"I confess. I'm a total novice. My mother is the biggest Nancy fan on the planet.\"\n\n\"Cora, right?\"\n\n\"Yes,\" I said, flattered that he remembered her name from that one dinner so long ago. I can barely remember the name of a person I met five minutes ago.\n\n\"Nancy is something. I try never to miss her when she's in town.\" I found myself smiling easily with Marcus, and for a brief moment it was as if there were no one else around. All that existed was now, a dimly lit room filled with the smell of cigarette smoke, chicken, and ghosts of music past. Not exactly the most storied of images but accurate nonetheless, and it was nice just like that\u2014no bells, no whistles.\n\n\"Really?\" I admit, I hadn't pegged him for the type.\n\n\"Oh, yeah. I think she was my first great love. With my friends I was drooling over Sheena Easton or Janet Jackson. Secretly, I was lovin' hard on some Nancy Wilson.\"\n\n\"Good Times Janet, Diff'rent Strokes Janet, or Fame Janet?\"\n\n\"Diff'rent Strokes, of course. Charlene was the most,\" replied Marcus incredulously.\n\n\"Well, I hate to tell you this, but my mother beat you to the punch, albeit from afar.\"\n\n\"You don't say,\" he said, looking over my shoulder in the direction of our table.\n\n\"Yep, she's been crushing on Nancy from back in the day in Negril.\"\n\n\"Well, I have never minded sharing, especially with a beautiful woman.\"\n\n\"I'll keep that in mind,\" I said, inadvertently flirting. Had I time to think beforehand or prepare myself for seeing him, surely my responses would have been more measured.\n\n\"I was hoping you would.\" The musicians' appearance onstage signaled the imminent start of the show and an abrupt end to our conversation.\n\n\u2022 \u2022 \u2022\n\n\"And who was that nice-looking man you were talking to, Julesea?\" asked Cora when I returned to the table. \"Him look like Paul New-man.\"\n\n\"Oh, Marcus. That's just my boss . . . sort of. I mean, he is an investor in Carly's is all, so he's like my boss but not really, whatever.\" Clearly I'm rambling\u2014why? I can hear myself and it sounds peculiar. Just stop talking, Jules. Stop talking.\n\n\"Don't look like any boss I have ever seen,\" said Cora, taking one last inventory of Marcus before turning her attention back to the stage.\n\n\"Well, that might be because you have never actually had one, Mommy dearest. Bosses, you know, are reserved for people with jobs,\" I said, unable to resist the open door. Even Daddy had to break character and laugh at that.\n\n\"Oh, hush up, Charles. You know I work. I have worked for the past thirty-one years keeping you together,\" she said, placing her hand on my forearm. Then, loud enough for Daddy to hear: \"The man would be lost if it weren't for me.\"\n\n\"Of course I would, dear. Of course. Who knows where I would be now, probably living in Delaware marriedt to Rosalyn wit a couple of . . .\" Daddy trailed off.\n\n\"A couple of delinquent children who are a menace to society. That is exactly where you would be if you were with that . . .\" replied Cora, sending them both into full-on hysterical laughter.\n\nTheir dynamic never ceased to amaze me. I used to think that her jealous cries were just for show but have come to know much better; they are part of the romantic dramedy that makes them the ultimate couple. From any distance it is clear to see that Daddy loves her from here to infinity. Her behavior, on the other hand, requires closer observation and analysis. But when you get it, Eureka! You realize that her love for him is her very essence\u2014without it there would be no reason to exist. Charles was the one before even she knew there was a choice.\n\n\u2022 \u2022 \u2022\n\nThe lights dimmed to reveal a single spotlight, in which Nancy stood. Having been familiar with her music did not prepare me for seeing Nancy. Mind you, I am not a fan of velvet and am often hard-pressed to acknowledge it ever works, in any construction, but this night, on her, it was magic: a bordeaux mermaid-cut gown with an open back, and it draped her body like silk. Her hair, short and layered with enhanced silver streaks throughout the black mane, created the perfect frame\u2014or halo as it were\u2014around her face, of which I am a major fan. This woman owned the room and every person in it with her sultry style and melodic voice. The way she glided around the stage as if she were entertaining at home and not to a room filled with fans mesmerized me.\n\nThe whole scene felt voyeuristic, especially regarding Cora and Charles, who, like Nancy, I felt like I was seeing for the first time. In the dim club light, Cora's eyes sparkled like diamonds transfixed on Nancy, but her heart was firmly in Charles's hand. With tender discretion, he was stroking her leg under the table, and she leaned in to kiss the side of his face, eliciting a secret smile whose meaning was only for them. If wearing this sorbet-colored knit suit and matching pillbox hat was the price to be paid for a front-row seat to true love, then so be it. As the evening progressed, Nancy transitioned from song to song with the fluidity of the greatest storyteller, even making loss beautiful, almost enviable.\n\n\"As many of you know, last year was extremely difficult. I found myself to be a child of the world without the two pillars of my world. Tonight is one of my first performances in a long time, so to tell you that I am overwhelmed with gratitude is, well . . .\" An explosion of applause and whistles drowned out whatever was to come next. \"Indeed, tonight is about firsts, first loves and first times, so I would like to dedicate the next song to Cora for the first time, '(You Don't Know) How Glad I Am' that you are here tonight. Thank you.\"\n\nThe force of Cora's nails digging into my flesh for stability caused enough pain to tell me that I was not having an out-of-body experience. Nancy Wilson had just said my mother's name and dedicated a song to her. Cora screamed, drawing the attention of the room, and then she went into some type of catatonic state that only melted as tears rolled down her cheeks, streaking the immaculate face that took an hour to put on. At that moment the entire world may not have existed for her, but I was quite curious to know the identity of Santa. Instinctively, I looked to my father, trying to read his mind and determine how he made this all possible. His eyes still bulging in bewilderment said to look elsewhere\u2014but where and how? Turning, instinctively, to my right, I found the answer. My eyes locked on Marcus's. There was no excessive grandeur in his presence, just immense sincerity. Overcome with emotion, I tried to mouth \"thank you\" but feared it was too little consolation to express what I truly felt, so I smiled, cupped my heart, and tried to keep the mass in my throat from spilling out. Thank you for the candle. Thank you for making my mom's reality surpass the dream. Thank you a million zillion times over. I'm sorry for being such a pain\u2014honest.\n\nAt night's end, once the lights came up, I searched for Marcus but couldn't find him, a fact that did not sit well with Cora. She wanted to see him, thank him properly. Unable to deliver him in person, I caved under the pressure from both of them and shared that he lived directly below me, should she want to thank him in person. Clearly, I was not lucid. Giving Cora this type of information, no matter how innocent, was like giving Kim Jung-Il a partial (and by \"partial\" I mean a completed) blueprint, sans physicist, with all the details of our nuclear program.\n\n\u2022 \u2022 \u2022\n\nFriday was their last night in town before returning to Virginia, so Daddy could finally get some rest before going back to work. Cora found it hard to believe that he felt he had not rested the entire time they were there. \"Something must be wrong with that man. All we have done is relax. We have gone to the Met, the Whitney, Central Park, SoHo, Saks, Bendel's, had tea at the Plaza, seen four plays, Alvin Ailey at Lincoln Center\u2014Revelations is just the best, Julesea\u2014and taken a boat around the island. How can he possibly be tired? And we didn't even get to see MoMA or go to the top of the Empire State Building.\"\n\n\"Are you kidding? I'm exhausted just listening to that list, much less being forced to do it all in only a few days, considering that many of them I haven't done in the entire time I have lived here,\" I said.\n\n\"Well, you know what I say,\" she began\u2014or, as I have come to call it, \"the answer portion of Cora Says\" during which I often remain quiet, knowing that every time she will respond before I can search for the answer. Selfish player she is. \"You only live once.\"\n\n\"Mom, I don't think you coined that phrase.\"\n\n\"Who cares, dear. It sounds best from my mouth. Get home early tonight. I am cooking all your favorites.\"\n\nSince I missed being home for the holidays over the past couple of years, this was indeed going to be a treat. Despite the fact that it is only for Dad and me, Cora has always cooked as if for a football team. From the foyer I could hear the familiar competition of the television and the music system. Some things never change. Cora always cooking to music, Dad transfixed by a continuous loop of sports. Independent of the space, they can have both systems going simultaneously at rather high levels without irritating each other. Odd as all hell it is, unless you grew up in it. Tonight, Stuart Scott and Lionel Ritchie were in a serious battle.\n\n\"Mom, Dad, I'm home.\"\n\n\"In here, Julesea.\" Placing my purse on the hall table, I followed Cora's voice into the kitchen. \"What took you so long?\"\n\n\"I know, sorry. I tried to get out earlier but couldn't. Michael has been on me to find a permanent act for the restaurant, which is proving difficult considering what he is offering. And of course, he is not the . . .\" My words trailed off as I realized that Cora had absolutely no interest in the minutiae of my work life. She seldom does, be it work, studies, friends, unless it relates to beauty, a current relationship, or the man who could be her future son-in-law\u2014she just zones out after two minutes or so, and that's being generous. I could force her attention by confessing that I made an offer to Nancy Wilson earlier, but why tempt fate? A premature confession of that kind would only set me up for ridicule of some sort in the event that I can't deliver.\n\n\"Everything looks so good. What can I do?\" I said.\n\n\"You can unload the dishwasher and put those in,\" she said, gesturing to the numerous pots and mixing bowls strewn about the counters, covering every available counter surface. There were trays of meat patties on the island, stews and beans brewing on the stove, tins of foil awaiting ackee and saltfish, fresh coco bread, and more, so much more. All that was missing was a steel-drum band playing \"One Love.\"\n\n\"Mommy, who do you think you are cooking for? I won't be able to finish all of this in a month, much less retain any hope of fitting into my clothes after.\"\n\n\"That's the idea. You're too skinny. You know me, I didn't want to say anything.\"\n\nSince when? I said, but only to myself.\n\n\"That time in London, with all that bad food, has you wasting away. Now, if you had been blessed with a curvy figure like the women from my side of the family, that would be fine, but you take after your father, lean and long with n' ass at t'all, really, so we have to be careful, don't we?\" That is always a statement more than a question. \"How will you ever be able to catch a man when you giving him nothing to hold on to? Thank goodness I'm here.\" Yes, thank goodness you are here, Cora, or else I just might start to feel a little too good about my entire day, a little too comfortable in my own skin. Thanks so much for reminding me of my genetic misfortunes.\n\n\"I guess looking like you counts for nothing, huh?\" I said sarcastically, hoping in vain that maybe, just maybe, she would suddenly develop the empathy to understand the power of her words, even after all these years riding on my psyche.\n\n\"I t'ank God for it every day,\" said Cora, raising her hands to the sky and looking up to give holy recognition of a prayer long answered. \"Heaven forbid you come into this world with a boy body, looking like your aunt Helen. Like I always say\"\u2014pause, wait for it, hold it\u2014\"a person walks forward not backward, so a beautiful face will get you farther in this world than having a big casaba melon for a booty. Remember that.\"\n\n\"Hence the problem, Cora. I remember too much, so it shocks me every day that I am able to wake up on my own, feeling good and somehow remembering that you do love me\u2014in your way.\"\n\n\"You're welcome.\" With her free hand, Cora grabbed my chin, admiring her greatest work. \"You are such an exceptional young woman, Jules. Mommy loves you every day, all day. Don't be nasty.\"\n\n\u2022 \u2022 \u2022\n\nThe music in the kitchen did little to drown out the heated conversation Daddy was having in the living room. \"Mom, who is Dad screaming at? Doesn't he know that Stuart Scott is broadcasting from a studio in Connecticut somewhere and cannot hear him?\"\n\n\"Oh, ignore him. He and Marcus have been going at it like that for over an hour now.\" Yes, I think this is when the room started spinning, but I'm not sure, entirely, because I temporarily lost my hearing. \"What?! Don't stand there looking at me like that. You heard me. Your Paul New-man boss-friend is here. He is in the\u2014\"\n\n\"Shhhhhhhhh . . .\" I said, creeping to the dining room door, attempting to peek outside so my eyes could confirm what my ears were hearing. From my vantage point, I could only see two half-empty beer bottles on the coffee table and the shadows of the TV screen, nothing else. I didn't see my dad or Marcus, although I could hear them. That is, until I turned around and there he stood in my kitchen, getting a spoonful preview of Cora's oxtail stew. To an outside person this whole scenario would seem completely plausible and wholesome, sweet even, but it's not. UGH . . . damn that Cora. I should go over there and shake the shit out of her, or at least until one eye rolls back. Why, why, WHY? Dad was the first to acknowledge me.\n\n\"Bunny, when did ya get in? Cora, why didn't ya tell me she was here?\" he asked, crossing over to kiss my cheek before opening the fridge for another beer. My eyes raced back and forth erratically between him, Cora, and Marcus, silently pleading with one of them to speak up and explain what in the hell happened to normal. Marcus, of course, was the one to break ranks.\n\n\"Jules, you didn't tell me your mom was such a wonderful cook. Now I know where you get it from,\" he said. Was that a flash of color? I think Cora's tail feathers just spread. Unreal, this woman!\n\n\"Apparently you are not the only one who is not told things,\" I said, cutting my eyes at Cora. \"What are you doing here?\"\n\n\"What does it look like, chyle, he is havin' dinner,\" said Dad.\n\n\"Okay, but where?\"\n\n\"Here wit us. Is there a problem?\" asked Daddy.\n\n\"No, of course not, but um . . .\" I tried not to be an embarrassment to Daddy, who must be as mortified as I am at the imposition Cora has put us both in. \"I mean, I'm sure Marcus has other, better plans for the evening than to, you know, hang around . . . here . . . with us.\"\n\n\"You okay, bunny? Why you talkin' all funny and nervous-like?\" asked Dad. Not the response I was expecting, but let's ride with it.\n\nMarcus offered, \"True, I did . . . have other plans. That is, not better, just other, but I canceled them when your dad came down and asked me to join you guys.\"\n\n\"W-h-e-n my dad came down and asked you to join . . . us,\" I said, bobbing my head up and down with each word as if it would help the cerebral process. Cora stood off to one side with her arms folded, silently admonishing me for immediately placing the blame on her.\n\n\"Dear, what did I tell you about being repetitive?\" asked Cora.\n\n\"Good ting too. Did ya know he loves Jamaican food, even been to the Blue Mountains where my people's from? Small world it is, small world indeed,\" said Dad.\n\n\"Apparently,\" I said. My need for rescue thoroughly transparent, Cora ushered Marcus and Daddy from the kitchen, citing the need to put the finishing touches on dinner and leaving me free to scratch my head and wait for the ground below me to get firm.\n\n\"Mommy, how could you let him do this? Why would Daddy go to that man's apartment and force him to come to dinner? Did you send him?\" I'm sure my eyes pleaded with her in a manner that gave far too much away.\n\n\"Which question would you like me to answer?\" said Cora, returning to the stove.\n\n\"All of them! This makes no sense. It's like I can't escape this guy. I walk in the lobby, he's there. I come out of the bathroom, he's there. I come home, he's here. Whyyyy?\"\n\n\"Ummm. I see,\" said Cora, now giving me her full attention, which only made me feel more trapped, and needing desperately to push back, against anything.\n\n\"You see nothing, so don't start, woman.\"\n\n\"More than you, Julesea. I see everything. Now listen to me, pull yourself together and freshen up. I'll finish up things here. Go! Been laboring over this hot stove all day anyway.\"\n\nMy bed had been made from the morning, but was now covered with freshly washed and pressed clothes that Cora had collected. I collapse to the floor, staring at the ceiling, dazed and still confused\u2014surely the antithesis of what Cora had in mind when she said \"Pull yourself together,\" but that is what I am doing, and I plan on remaining here until someone tells me what in the hell is going on! The problem is not that I don't enjoy Marcus's company, because I do, in spite of myself. It's just that his presence in my life defies logical categorical placement, complicating things when what I need above all right now is easy, breezy, and uncomplicated. I need logic in the form of neat little bowed boxes with identifiable labels. In the beginning he was a cocky, well-coiffed guy I'd met out on a job interview, then he was a typical modelizer I ran into at lunch who, unbeknownst to me, is a partner in the company I now work for, and my neighbor and, while not so silent in his interest (in me and in the company), is in direct opposition to my boss and mentor. Then he goes and does that thing for my mom with Nancy. Most damaging of all, he is the first guy since Tony who makes me pit-of-the-stomach-near-faint nervous, to which the only defense is distance and total, complete avoidance.\n\nA quick shower to clear my head did assuage matters slightly. Although I did expect much more from the Holy Trinity\u2014maybe they were taking the night off. Putting on a pale pink cashmere sweater and white Hudson denims improved matters more. As Neon Deon Sanders always says, \"When you look good, you feel good, and when you feel good, you do good.\" But I needed to do more than good if I was to keep my wits about me tonight. Before leaving the room, I saw a message from Blake:\n\nDrinks tonight? Suddenly free\u2014don't ask. xo Blakes\n\nMore than you know but can't. Parents' last night in town. xJules\n\nLove THE Cora\u2014total riot!!! xo Blakes\n\nYou can have her. Seriously, take her and Daddy too! :-? xJules\n\nI must have details. TTFN. xo Blakes\n\n## 30\n\n* * *\n\n## A HEAP SEE BUT A FEW KNOW\n\nTHE GREATEST ABSURDITY of the evening had to be the moment Marcus and Daddy started singing 'Whatcha See Is What You Get.' I mean, they really thought they were the Dramatics or something. Insane.\"\n\n\"Darling, absurd is three adults, me being the most senior and knowing better, on a party-line call like it is 1986.\"\n\n\"Stop it, Richard. Three-way calls are the best. Don't listen to him, Jules. It sounds like a great evening. Keep going.\"\n\n\"Joy, I think I heard something drop on your line,\" said Richard.\n\n\"Huh, really? I wonder what,\" replied Joy.\n\n\"Your pom-poms, dear,\" said Richard.\n\n\"You are not as funny as you think.\"\n\n\"How would you know? We haven't seen you in forever,\" said Richard. \"Is that what happens when one leaves civilization and moves to Jersey? Now that is absurd.\"\n\n\"No. That is what happens when one has a full-time job, a husband, two children, PTA meetings, bake sales, and peewee football to contend with. Something that our Little Miss right here may be knowing soon enough,\" said Joy.\n\n\"Unless he is walking up to my door and ringing the bell, I don't see that happening anytime soon,\" I said.\n\n\"Darling, I hate to break it to you, but it sounds like he already did. As a matter of fact, not only has he rung twice and left a gift, your father invited him in and welcomed him to the table. I mean, unless I am misreading something . . .\" said Richard.\n\n\"Seriously, don't you have a speech or something to write? You know, utilizing that big brain of yours, less lip . . .\" I said.\n\n\"Play daft if you want, Julesea, but I find it all too interesting the way you and this young man are intersecting. Stop trying to fight the obvious.\"\n\n\"What is it with you? Have y'all been talking to Cora behind my back? Stop calling me that!\" I said, far from ready to have a direct \"What does this mean?\" conversation about my interaction-attraction to Marcus. \"We live in the same building, so that is bound to happen. Mystery solved.\"\n\n\"Joy, take over. Even if I didn't have to put the finishing touches on the finance speech, I am not in the mood to play 'hide and go get the truth from an emotionally stifled Jules Sinclair' tonight,\" said Richard.\n\n\"Would love to but can't right now. Jackson has taken off all his clothes again and is streaking through the house.\"\n\n\"Isn't that normal for a four-year-old?\" I asked.\n\n\"Who knows? And he is six,\" said Joy.\n\n\"Who?\"\n\n\"Jackson,\" replied Joy.\n\n\"Oh, definitely not normal!\" I said.\n\nEmotionally stifled seems a bit harsh, I believe. Emotionally reserved is fair; emotionally conservative much better and quite fiscal even. True, I was playing a game of cat and mouse with myself about this man and fooling no one who was aware of the situation. Thing is, my history tells me rather explicitly that falling in like, even in love, is the easy part when I meet someone who is the physical representation of everything I desire. Being acutely aware of how that actual manifestation impacts my life (voluntarily losing myself in the illusion of him as I do) is a far different matter. Right now I need a guaranteed leap of faith that I will not wake up one ominous day and realize I am exposed in this like-love that is unbearable alone. Besides, it was just dinner and dessert, at my home, with my parents, and it was . . . kinda glorious . . . nothing major, so why is it that I just can't go to bed and stop replaying the events of the other night?\n\n\u2022 \u2022 \u2022\n\n\"You know what your problem is, Julesea?\" asked Cora, once I finally sulked out of my room to help put the finishing touches on dinner.\n\n\"I love that you think I only have one, problem that is,\" I said. \"Give me ten minutes and surely I can rattle off a solid fifty or so, forty at minimum.\"\n\n\"You don't know how to live in the moment. I don't know where you got that from. Surely not me.\"\n\n\"Of course not. Nothing bad ever comes from your side, Mother.\"\n\nThinking it best not to encourage my dissent and remain on message, Cora continued, \"Look at you. Your father and I are here, a nice single handsome man is in the living room, who is obviously smitten with you. Daddy is happier at this moment than he has probably been the entire trip, but that seems to escape you, because you have decided, for some ungodly reason, to be unhappy. Well, I will not have it, you hear me? I didn't raise a child to be rude and inhospitable.\"\n\n\"This coming from the woman who dragged the poor man around the city like one of those purse dogs as she indulged herself. Of course he is having a good time. He's on estrogen overload, having mucked about with the two of us for the past week. Poor man'll latch on to any testosterone he can get.\"\n\n\"That, my dear, is the trade-off of marriage. It's in the fine print. If one day you are blessed to get out of your own way and actually live in the now, thereby gaining a husband and giving me some grandchildren\u2014trust me, you will do the same. Now, put these plates on the table and fetch Daddy and Marcus . . . ay ay\u2014with a smile, please.\"\n\nIt was much too early in the proceedings to do the quick shot of something I desperately needed in order to medicate myself into pleasantry. So I opted for a more acceptable approach. The way I figure it, whatever the wine doesn't soothe, the rum punch will. Dad led us in grace with one of his comically brief blessings that underscored the labor of love Cora had put into making each dish, forcing her to do a not-so-subtle sucking of the teeth. You know what they say, you can take the girl out of Jamaica but you can't take the Jamaica out of the girl, no matter how well coiffed her weekly blowout might make her.\n\n\"Mrs. Sinclair, everything looks so amazing. I don't know where to start,\" said Marcus.\n\n\"Well, start wherever your eyes lead ya and your stomach will follow,\" said Daddy. \"My Cora is one of the finest cooks 'round, and that's no lie.\"\n\n\"Thank you, Charles. You know I do what I can,\" said Cora, badly feigning a truckload of modesty.\n\nI swear, even in replaying the whole evening from the comfort of my bed is like a scene from The Twilight Zone or some parallel universe: Cora pretending to be demure while silently commanding me, with laserlike glances, to speak up; Daddy and Marcus getting along as if they were old chums. What is it with this guy and his eerie ability to assimilate?\n\n\"So, Marcus, tell me,\" started Cora. Oh boy, that is never a good start to anything. \"Why is it you don't have someone special to cook for you at home?\"\n\n\"Mom!\" I shrieked in horror at her brazenness, causing me to cough uncontrollably and choke on my last bite. While recovering, I placed the napkin over my face and prayed for the quickest death possible.\n\n\"What, Julesea? I am asking him a simple question. Good-looking, successful young man like him. It makes no sense.\"\n\nLaughing and far from rattled at the inquest, Marcus said between bites, \"I hate to tell you this, but women today, at least in New York, don't cook like this. If they did, I'd be home.\"\n\n\"Sad but true, y'know. Girls today don't put a premium on family and making a good home. All they concerned wit is competin' in a man's world. Makes no sense,\" said Daddy. \"Never has.\"\n\n\"Daddy, how can you say such a thing? I work,\" I said.\n\n\"And I would prefer it if you didn't,\" he stated. \"Don't go lookin' at me like dis is the first tyme ya hearin' me say dis. You know why I work so hardt. So you and ya mother don't have to. Always have.\"\n\n\"And we appreciate it, Charles,\" said Cora, absentmindedly touching the new diamond-and-gold timepiece on her left wrist, which Daddy had purchased earlier in the week. \"Home triumphs over all when the commitment is there.\"\n\n\"I'm not saying I don't appreciate it, Daddy. I am just saying that one does not necessarily denote the other. I work and yet I can cook\u2014and will one day embrace balancing career and family.\"\n\n\"Ya listening to dis, Marcus? Balancing family,\" said Daddy. Having been out of the house for so long and for the most part having Cora as our intermediary, I forgot how chauvinistic Daddy could be about sex roles. I guess that is also a latent Marley trait.\n\n\"It's a fine line, sir. The values across the board on relationships have shifted. Men no longer seem to have the intrinsic desire to protect and provide for women as they once did,\" offered Marcus, to the complete surprise of Daddy, which made for an uncomfortable moment prompting a very pregnant pause before Marcus raced to do damage control. \"At least as far as I can see, sir.\"\n\n\"Is that all men or just you, Marcus?\" asked Cora, sounding more like a federal prosecutor than a housewife. Oh shit, I guarantee he is wishing that he didn't cancel those other plans.\n\nUnapologetically, Marcus said, \"My peer group, for certain, and myself. But you must understand that like anything else it is a matter of perspective.\"\n\n\"How can it be perspective when it is tradition?\" Cora asked impatiently.\n\n\"Well, in my defense, I have never experienced it, traditionally speaking,\" said Marcus.\n\n\"Unless your mother was invisible, then I don't understand,\" said Cora.\n\n\"To the greatest degree she was. She left early enough for me not to remember anything specific about her other than her smile and perfume. My father and my nan raised me.\"\n\n\"Oh, I am sorry. I did not know,\" said Cora. \"I just assumed because you seem so well brought up . . .\"\n\n\"It's quite all right, really. My nan is a great woman. Between her and my pops, I wanted for nothing. If anything, my mom's absence motivated me to be the best. As a kid you think\u2014I thought\u2014if I was the best, then she'd come back. That kind of thing can give you fuel; did for me. Don't know why I said that.\"\n\nMoved by Marcus's revelation, I placed my hand atop his. \"I'm sorry too,\" I said, unaware of the unspoken exchange between my father and mother at witnessing the innocence of this gesture. Cora later commented that she perceived this as my first act of civility that evening.\n\n\"You know, last week I saw the same thing on Sally, but not Oprah,\" said Cora. Feeling the scrutiny off our blank stares, she said, \"What? I did. Sally could probably help is all I'm saying.\"\n\n\"Cora, what does ya' blasted daytime shows have ta do wit the boy never having a mother?\" asked Daddy, echoing the sentiment that was in my head. \"I swear, woman, sometimes I just don't know 'bout you.\"\n\nIll-timed or not, the table erupted in laughter. The tempo for the evening was now set. Marcus was no longer a visitor to be danced around or cajoled. As for my need to put him in a box, well, that too was of no importance. He was, at least for the duration of this dinner, a Sinclair and treated as such. That'll teach him.\n\nMaybe it was learning about his mother or my finally acknowledging my own inappropriateness, or maybe it was the second glass of rum punch, but all was good and no thoughts of \"what if\" or allegiance to Michael underscored the evening. There was more than enough laughter to be had at everyone's expense, which leveled the playing field considerably. Marcus learned quickly that Cora's flair for the dramatic extended far beyond tabloid television and that Daddy's Old World conservative sensibility made him risk-averse.\n\n\"I, for one, have never been comfortable playin' round wit my own money, much less having the responsibility for other people's,\" said Daddy.\n\n\"Yes, that can be daunting. However, it can also be an adrenaline rush all its own. To know that in one hour I can gain or lose more money than my ole man will make in his lifetime is lunacy. Temporarily, anyway. After a while, Charles, it becomes par for the course, and the amounts become ordinary, so you have to find the next rush.\"\n\n\"Marcus, that's easy ta say, when you are not the one on the losing end of a life savings, or an IRA,\" said Daddy.\n\n\"True, sir, but I have been, and knowing that feeling of grief, firsthand, is what makes me great at what I do. I know what it feels like to lose everything.\" Shaking his head in reflection of the amateur trap that the man he is today would never have fallen victim to. \"At the beginning, I invested all that I had, which wasn't much, mind you, into a sure thing, that was suppose to split. My intel being the best available so what looks crazy now seemed perfectly solid then,\" said Marcus, reaching for another serving of stew. \"Well, the stocks did split, more than tripling my investment, but I got greedy, stayed in, reinvested, and lost everything when they bottomed out just as quickly. From that moment on I said never again.\"\n\n\"See, so you don't even play the market.\"\n\n\"Hmm, not entirely true. I don't, from a personal position anymore, I do it only for strategic gain,\" said Marcus. \"I set clear long-term goals, remain objective and alert. The outcome is strictly business.\"\n\n\"Is that why you are in the restaurant business?\" asked Cora. \"People love good food and music, so that is always a money earner, yes.\"\n\n\"Actually, my business is about global acquisitions, and restructuring of companies with a more austere purpose, if you will. Of everything in my portfolio, Carly's holds the greatest risk because the decision at that time was not purely a business one. In general, restaurants are not very stable. They require an immense amount of start-up capital. Their success is subject to a number of factors, which only increase the potential rate of failure, and loss on capital put in.\"\n\n\"So, then, why would ya do such a thing when . . . I don't see it,\" said Daddy.\n\n\"When I just said, strategic gain and never again,\" Marcus said, shaking his head in recognition of the contradiction this posed to his earlier assertion. \"Let's just say, I owed a favor to a friend.\"\n\n\"Carly,\" I asked, feeling it was a query well within my right to ask, given the course of conversation and my relative silence until this point.\n\n\"No. Michael, actually,\" said Marcus, between mouthfuls of food.\n\n\"Really? I didn't know that,\" I said.\n\n\"After all the years that Mike kept me on track, it was the least I could do. Without him, this town would have gotten the best of me early on.\"\n\n\"Oh, I thought you were Carly's friend, not Michael's,\" I said, noticeably puzzled.\n\n\"Actually, Mike was the friend. Carly was and still is the client and, well, you know what they say about business and pleasure.\"\n\n\"Bad idea, never mix them,\" said Daddy.\n\n\"Right you are,\" responded Marcus. \"Learning that right now\u2014\"\n\n\"Surely there are exceptions to any rule,\" said Cora, looking from Marcus to me, and then back again, to make herself abundantly clear.\n\n\"Clearly, an exception is always possible given the proper motivation,\" said Marcus, taking the bait.\n\n\"Indeed,\" boasted Daddy.\n\n\"Okay, before the night gets any weirder, I am going to excuse myself and take a breather from this table while I can still walk and not roll away,\" I said, more concerned with the nonverbal covenant seemingly being forged.\n\n\"Good idea, bunny,\" said Daddy, patting his protruding, well-fed belly. \"Dat was some fine cooking, Cora. Fine indeed.\"\n\n\"I second and third that, Mrs. Sinclair,\" said Marcus.\n\n\"Well, don't go getting all tired on me just yet. We still have dessert to look forward to,\" said Cora. \"Why don't you two go into the living room, Julesea and I will bring everything in.\"\n\nForever the gracious guest, Marcus began to clear some of the dishes from the table. \"Well, at least allow me to help.\"\n\n\"Nonsense, now. Put those tings down. Come have a cigar with me. Cora and Jules will handle that,\" said Daddy. I can't be entirely certain, because it rarely happens in my presence, but I could have sworn that Cora cut her eyes at Daddy briefly; definite side-eye. Making it all the more odd really, how when the night started, it was Cora whom I thought I would have to push down, muzzle, and lock in a closet, and yet here we are and it's Daddy\u2014or should I say \"Benedict Charles\"\u2014who is in serious need of a gag order and restraints.\n\nSafely secured in the kitchen and out of earshot, I hissed to Cora, \"What in the Hades has gotten into him?\"\n\n\"What are you talking about?\"\n\n\"Daddy, that's who! He's all women's role this and strutting about all broad-chested, not to mention is about to smoke in my house when he knows I have upper-respiratory issues. What gives? And don't tell me you didn't notice because I saw you throw him a side-eye.\"\n\n\"Dear, I hate to break it to you, but that has always been your father. He's a man,\" said Cora. \"Where do you keep the coffee again? I don't know how you find anything in here,\" she said absentmindedly, while rifling through the cabinets. \"Imagine how much worse he would be if left to his own devices. Like I always say\u2014\"\n\n\"Oh, please, not another one of your\u2014\"\n\n\"A man needs a woman to tame the beast, otherwise they'd be eating their feet and smelling all over the place,\" said Cora, putting the finishing touches on dessert. \"How else can they grow? Now, what about that Marcus?!\"\n\n\"What about him?\"\n\n\"Don't play dumb with me, Julesea,\" said Cora, standing in front of me with her hands on her hips. \"You like that man and he likes you. Why else would he be here putting up with the jibberish from your father and me? Get the berries out of the fridge.\"\n\n\"Trust me, Mom, I am not his type.\"\n\n\"How many times do I have to tell you that men don't have a type? They have ideas that are quickly dispelled when the right woman comes into the picture. You want him? Make it known, and like that,\" she said, snapping her still immaculate French manicured nails, \"you are his type, end of story. And, for the record, you might as well stop walking 'round this here kitchen acting like you aren't listening to me. I know you, Julesea Isabel Sinclair. I know every thought you have before you do, so hear this. You like this man but he scares you. Get over it.\"\n\n\"Mommy, stop. You're making my brain hurt.\"\n\n\"Your father likes him, you know.\"\n\nAnd so Benedict Charles does, as evidenced by their multiple sing-alongs. Another latent trait of Daddy's Marley gene is the desire to sing whenever the mood is right, wrong, or there just happens to be enough Red Stripe and some good music around. Who's to say what preconceived time notions Marcus may have had about joining a Sinclair family dinner, but if it was anything less than a full night's affair, then he was sadly mistaken. Whether Daddy schooled him on this fact earlier during their sports showdown or at the time of extending the invitation is anyone's guess, but not once did I see Marcus look at his watch or go over to the console table to check his phone. Yes, I was looking. I saw him notice when it lit up a few times, but he didn't advance be it with intent or discreetly mid-two-step as he and Daddy were pretending to be members of the Commodores (Lionel and the smiley one with the extra juicy curl) performing \"Brick House.\" Daddy's air bass-guitar stylings paired with Marcus's full-body commitment to the funky 1970s dance moves sent Cora and me into spasms of laughter. Had the hour not been approaching midnight, I have no doubt that a highly spirited game of cards or dominoes would have ensued.\n\n\"Whew, dat was good fun, baby, but these old bones are tired,\" said Daddy, feeling the repercussions of some of those earlier dance moves now in his lower back. \"Cora, why don't we turn in and leave these kids to talk?\"\n\nMarcus stood up from the sofa we had been sharing to embrace Daddy and kiss Cora good night after watching their finale performance of Elton John and Kiki Dee's \"Don't Go Breaking My Heart.\" \"I should be going as well. Mr. and Mrs. Sinclair . . .\" He stopped, off Mommy's corrective look.\n\n\"What did I tell you about that Mrs. Sinclair nonsense?\"\n\nOne hand over his heart and the other cradled within hers, he said, \"Cora. I could not have wished for a better evening than this. Everything else pales in comparison. Thank you so much.\"\n\n\"Nonsense, son. You helped make the night something special,\" said Daddy. I did not need Cora's words from earlier to tell me that Daddy had developed a great affection for Marcus. It was evident. \"Now, remember what I told you.\"\n\n\"Yes, sir, I will,\" said Marcus, \"and I will call you soon to take you up on that.\"\n\n\"What secrets are you two keeping?\" I asked, knowing all the while that they would not say. Well, I'll be a #?!*! Daddy has known this guy less than five hours and already has his phone number and they have made plans for future outings. I have lived above him for nearly a year and don't even know what the inside of his apartment looks like, much less have his personal or office phone number, for that matter.\n\n\"Jules, ya gettin' more and more like ya motha every day, and I am not sure that's a good ting,\" said Daddy.\n\n\"It's a great thing, sir. A great thing, indeed,\" responded Marcus. Spoken like a man who doesn't know Cora. His arm was comfortably draped around my shoulders, and I instinctively returned the gesture, wrapping my arms around his waist throughout the conversation.\n\n\"Never say that in front of her!\" I teased. \"I'll walk you to the door.\" What a night! I wished for it never to end. I didn't want him to say good night to me because I didn't know if tomorrow would go beyond right now.\n\n\"Your parents are incredible, Jules,\" said Marcus. \"You are really lucky.\"\n\n\"So I have heard a few million times, from them no less.\"\n\n\"Well, they are . . . and so are you.\"\n\n\"Yeah, I know,\" I said, instantly feeling the need to clarify. \"I know that they are, and yes, I am lucky . . . not that I'm saying I know I am incredible, because that would just be, like, so completely obnoxious to say . . . wouldn't it?\" The appearance of a smile on his impossibly perfect mouth put an abrupt end to my babbling. Once again I just wanted to freeze time and capture the moment, but I stopped taking those kinds of photos a while ago. \"Was I babbling? I do that from time to time when I'm nervous.\"\n\n\"Do I make you nervous, Ms. Sinclair?\"\n\n\"Unquestionably so, Mr. Crawford.\" Snapshot . . . and the moment is gone.\n\n\"Marcus!\" screamed Cora from the foyer, forever shattering the perfection of our moment. \"Where are you? Oh, there you are. I thought you had left before I could pack you a little something.\" Briefly scanning the numerous containers in the bag she held, I silently questioned how much she cooked and if anything was left in the kitchen for me to eat later, although I wouldn't be opposed to having leftovers with him, if he offered. \"I packed you some rice and peas, yams, and a few patties. Now remember to eat the ackee by tomorrow, because it doesn't hold as long.\"\n\n\"Aw, Cora, you didn't have to,\" said Marcus, \"but I'm not complaining at all. Thank you. My fridge will be happy to see some home cooking.\"\n\n\"Let's just hope there's some left for me or else I'll be knocking on your door tomorrow,\" I said.\n\n\"In that case, let's hope that everything is in this bag,\" said Marcus.\n\n\"Julesea, there is plenty left for you,\" said Cora. \"You have the stew, some jerk chicken, roti.\"\n\n\"The stew was amazing,\" said Marcus.\n\n\"Then you will have to combine, now, won't you?\" said Cora, nodding yes before leaning over and kissing Marcus good night. As she did, I noticed how she held his face in the same way that I used to hold Tony's\u2014even at the end. It never occurred to me before that this gesture, which felt so natural to me, was learned from her. \"Julesea, come in and say good night to your father before you go to bed.\"\n\n\"My word, that woman is something. I can see where you get it,\" said Marcus.\n\n\"And you're not afraid,\" I said jokingly.\n\n\"No, actually just confirms what I already knew about you.\"\n\nI was lost for words again, or maybe the moment just didn't need them. So we both just allowed it to be hypnotic. The buzzing of his phone broke the trance. \"You should probably get that,\" I said.\n\n\"I will, later,\" said Marcus.\n\n\"What, why are you looking at me like that? Do I have something on my face?\" I asked, feeling self-conscious under his watchful gaze.\n\n\"The first time we met you smelled like Oriental spices and vanilla. The second time like gardenia. I always seem to remember you, Jules. What do you think that means?\"\n\nThat I could rest beautifully in you. From the moment I knew you lived here, I've hoped to see you whenever I enter, only to find myself crestfallen when I don't, and that scares me even more.\n\n\"You have a good nose, it seems,\" was all I could allow myself to say aloud.\n\n\"Okay, I'll take that for now,\" said Marcus, narrowing the space between us and softly kissing the corner of my mouth. \"Good night, Julesea.\"\n\n## 31\n\n* * *\n\n## RECAP\n\nIT'S NOT UNREASONABLE for me to think that after such an auspicious evening, he would have called or asked me out by now, is it?\"\n\n\"First of all, Julesy, who goes around saying words like 'auspicious' in everyday conversation? . . . Ah-ah,\" said Blake, raising a finger to quiet Richard before he objected. \"Clearly, you are not using the right bait. Let's rethink this whole 'unrequited love for the neighbor' thing. It has red light written all over it. Trust me. I mean, you date for a minute, the sex is good and easy, then what? You stop talking, and it makes for awkward encounters in the lobby, or you mistakenly getting off on his floor multiple times per week, walking past his door, conveniently dropping packages near his door, so of course you can't help but listen for inside noises, only to run down the hall when you hear the door starting to open.\"\n\n\"Stalker,\" sang Richard.\n\n\"Call it what you will, I'm just saying. Heed the warning,\" said Blake.\n\n\"I hear you, but I don't think this guy would do that,\" I said.\n\n\"Who was talking about him?\" said Blake, slapping high fives to Richard across the table at Cub Room. A month and some had passed since our dinner and no word from Marcus. If I were in any way keeping a record of his patterns, I would mention that he also disappeared after the first dinner with Gary. My parents' constant inquiries only underlined the fact that I had not seen him and that I wanted to.\n\nFatigued with my circumstance and realizing that I didn't want the scrutiny after all, I said, \"Whatever, let's talk about something else, please. Something happy.\"\n\n\"Great! Then let's talk about me,\" said Blake.\n\n\"Yes, that's always a crowd pleaser,\" said Richard, devoid of any modicum of enthusiasm.\n\n\"Ignoring you. Now, that guy from Life, 'Mr. Potentially More,' is definitely worthy of yours truly. Financially he checks all the right boxes, travels quite a bit, which will leave ample time for extracurricular activities, if you know what I mean, and he's ready to have someone to come home to. Me! Although I haven't actually been to his place yet,\" said Blake, accepting an imaginary crown. \"Oh, and the bonus is that he is sexy as hell. Usually, I am envisioning someone like him anyway to get through an evening, so this is a step in the right direction. My body is addicted to him, and he knows it.\"\n\n\"News flash, honey, it's your internal gold digger addicted to his wallet, not your body,\" said Richard.\n\n\"Simmer down there, ole girl. Don't be jealous.\"\n\n\"How is that possible? Blake, you have been leased by more than a quarter of the affluent men across the continental United States, and yet no one has opted to buy.\" Damn, player, was that a bit harsh or damn?!#? \"I, on the other hand, am still with the great love of my life. Count 'em, honey, thirteen years.\"\n\n\"Correction, you are still with the only gay fool of means on the seaboard who is as boring and insufferable as you are.\"\n\n\"Okay, you two. Play nice. My inner child is cringing again,\" I said. \"Accept it, you are both diggers. It just so happens that one of you reads books and the other enjoys looking at the pretty pictures.\"\n\n\"Meow, kitty. Has she always been like this?\" asked Blake.\n\n\"No, I think it was all that time in London. There is something fundamentally wrong with a culture that is averse to physical affection,\" responded Richard. \"I like it, though.\"\n\n\"Umph, I'm undecided,\" huffed Blake. \"Now, where was I? Oh yes, the guy. He doesn't need me and that is abundantly clear, which makes us absolutely perfect. I could give him what he wants and vice versa,\" confessed Blake.\n\n\"Maybe you have met your match,\" I said.\n\n\"Now, when do we get to meet your match?\" asked Richard, using a bread stick to point between us. \"This one has piqued my curiosity. Normally you flaunt them immediately. This one is shrouded in mystery.\"\n\n\"Stop calling him 'this one.' You sound like Lauren Bacall sans the allure.\"\n\n\"I've been called far worse.\"\n\n\"I bet you have,\" said Blake. \"I'm not hiding him. I just haven't had him . . . as available, that's all.\"\n\n\"Oh, dear. Now we must meet this one. A man who doesn't fall at the altar surely deserves a medal, or at the very least a parade,\" chuckled Richard as he emptied the last bit from his gin gimlet.\n\n\"Or something,\" I said.\n\n## 32\n\n* * *\n\n## AND I AM . . .\n\nSPRING WAS FINALLY in bloom in New York. I guess that groundhog knows what he's doing after all. From my living room window I could see that the once-frozen lagoon across the street now had inhabitants, mommy and baby ducks swimming about to the delight of a sea of toddlers running amuck along its perimeter, with nannies and au pairs in hot pursuit. The skyline was dominated by a prism of green trees wise enough to make way for the broad array of ivory, pink, and rouge cherry blossoms on display. My friends, like many of the building's residents, had vacated the city for the weekend by any means necessary\u2014car, jitney, or helicopter (if you were truly big money)\u2014in order to open their country homes in the Hamptons or on Fire Island, leaving the city wonderfully desolate. Hallelujah!\n\nSince the first day I arrived in New York, she has been to me like the Old Lady in the Shoe, with 18,976,456 too many children. For most of the year, I had the misfortune of being a middle child, eager to please and too often craving her undivided attention but seldom receiving it except for a few glorious spring and summer weekend months, when the elders and younguns frolicked along her coast or in the mountains. During that time, I had New York's undivided attention and could indulge in her simplest treasures without competition or distraction. Downstairs on the sidewalk, I found myself at odds momentarily with possibilities. If I turn right and walk toward Madison, I can enjoy an almond croissant with the best cappuccino on the island. Or I could walk left and cross the street into Central Park for a bagel with a smear and a black.\n\n\"Feel like some company?\"\n\nNow knowing the cadence of his voice like my own, I said, \"I'd love some,\" and turned to see Marcus standing behind me with enough mail in his hands to convey\u2014but not say\u2014that he hadn't retrieved it in weeks.\n\n\"Great, I was hoping you would say that,\" said Marcus.\n\n\"Is that coming too?\" I asked, pointing to the jumbled overflow of envelopes, coupons, magazines, and flyers.\n\nAppearing slightly embarrassed at the accumulation, he said, \"Oh, one of the casualties of my life, as it were.\"\n\n\"Well, then, let's see what you have here,\" I said, reaching for the contents and scanning. \"GQ, yes, Wired boring and not a Saturday read. Business Weekly, still boring but okay, Visionaire, absolutely. Penthouse, really? Why?!\"\n\n\"Hey, it's not what you think. One of the guys got me a subscription as a gag gift. They have great articles.\" He removed it from the pile. \"Listen, we can call him, if you don't believe me.\"\n\n\"And unsubscribing was what, unconstitutional?\" I said, removing it from his hands and giving the undesirables to Percy, with instructions to keep until Marcus's return.\n\n\"I feel like you're judging me,\" said Marcus.\n\n\"Oh, don't feel, Crawford, know that I am judging you.\"\n\n\"Why do I suddenly have this pressure to redeem myself?\" he asked. \"Let's say I give you the best cup of coffee you've ever had in your life and in return you keep this little incident between us.\"\n\n\"Intriguing. I've never had a platinum-and-diamond-laced cup of coffee before.\"\n\n\"Oh, it's like that?\"\n\n\"Oooooh yeah, Mr. Crawford, it's like that.\"\n\n\"And here you had me convinced that you were one of those independent 'I can get my own' women whose feminist sensibilities would be offended by such obvious gestures like expensive gifts.\"\n\n\"Only if you left it on the nightstand next to a 'Dear Jules' note saying 'Thanks for last night,' \" I said.\n\n\u2022 \u2022 \u2022\n\nExiting the building, we took a right. Under normal circumstances, seeing the tourist-laden line that extended out Via Quadronno's door would have put a cloud in my day, but Marcus's company was good, so I didn't mind the wait. Having to listen to the incessant chatter of outer-borough housewives and their spawn while my system was still caffeine-free, however, was another story entirely.\n\n\"My guess is Jersey,\" said Marcus in reference to the talkative foursome directly in front of us. \"Fort Lee. Hayworth, maybe. Even Oradell. The headbands are a dead giveaway.\"\n\n\"Really? I was going for White Plains. The matching mother-and-daughter ensembles scream Westchester County,\" I said.\n\n\"I can see how you would think that, but the color palette and overall styling are wrong; pastels and those scary big flowers like the women in our building wear are Westchester, even Connecticut. The fuchsia-colored bedazzled situation paired with denims currently at hand shouts Jersey refinement,\" he said.\n\n\"First, I commend you on your ability to be snarky. Second, and most important, should I be concerned that you know all that?\" I said, snapping my fingers in a Z formation like one of the Chelsea boys.\n\n\"Never! I'm all man, baby,\" asserted Marcus.\n\nMoments before reaching the counter I noticed that the couple outside on the solitary bench was preparing to leave and decided to make a dash for it. Giving Marcus instructions to surprise me, I pushed through the crowd to claim the seat before two fast-approaching Maclaren stroller\u2013pushing mothers could plunk down. Having the finish line, as it were, in sight and a foot or so separating us, it seemed only logical to cut left quickly to block their advance and make a strong lunge for the end zone. Seconds before my body flopped down on the bench, I locked eyes with one of the mother friends but turned away before being cast to stone for treachery.\n\n\"Here you go, something for the bones and for the belly,\" said Marcus. \"Why are you panting like that? Better question: Why are those two women giving us the death stare?\"\n\n\"Might have something to do with me stealing this bench from under them.\"\n\n\"You didn't,\" he pleaded, looking back and forth from them and their babies to me. Any remorse he expected to see was temporarily halted as I enjoyed the first few sips of my cappuccino, followed as always by a little seated two-steps-of-happiness dance.\n\n\"What?! It's a tough world out here. Besides, did you or did you not complain earlier about wanting to sit outside and could not because there is only one bench?\"\n\n\"It was an observation, and don't go dragging me into your sinister deeds,\" he said. \"I'm going to heaven. I'll try to get you an air conditioner.\"\n\n\"I saw that, you know,\" I said, jabbing him in the side.\n\n\"What?\"\n\n\"You mouthing 'sorry' to the cradle pushers and pointing at me,\" I said. \"If you feel that bad you should buy them lunch.\"\n\n\"I could, but I won't. Did you see those ladies? One of them hasn't missed a meal in a while and I am not the one to . . .\" Marcus could not finish his callous remark before being overtaken by laughter.\n\n\"And you call me cruel,\" I said, sharing in this most inappropriate laughter. \"Thanks for the cappuccino, by the way. Did I tell you it was my favorite?\"\n\n\"No, but I promised you the best cup of coffee in town and that's it,\" he said.\n\n\"And yet, you are having tea? How very English of you.\"\n\n\"Yeah, something like that. Not too great for the teeth, but it is the way, I guess. I'll have a coffee after.\"\n\n\"Indoctrination?\"\n\n\"Completely. Each morning before school, my old man and I would have tea before he went off to work. It was our thing,\" he said, making invisible quotation marks in the air with his fingers, protecting the memory. \"As a result I can't seem to start my day without it.\"\n\n\"Hmmm, that's sweet . . .\" I said. \"I love it here.\"\n\n\"New York?\"\n\n\"No, this bench.\"\n\n\"Ah yes, the scene of the crime. Remind me to wipe it down before we leave. I don't want my fingerprints anywhere.\" Something tells me that Marcus makes it a point of wiping down everything he touches, especially women, as marks of emotional evidence may prove too messy.\n\n\"Call it what you will,\" I said, taking his dig in stride. \"I can come here, sit, and watch the world go by . . . albeit a considerably more affluent part of the world, but the world nonetheless. When I moved back, I was worried initially that it would take some time to find a little jewel like this, but here it is right on the corner.\" I felt quite satisfied with my discovery.\n\n\"My nan always watched the world at sundown out on our front stoop. Growing up, I used to think that she was just there to look out for me mucking about, making sure I stayed out of serious trouble, but then every once in a while I would catch a glimpse of her face right before the sun bowed to the horizon, and she would be a million miles away. I never knew where. Never asked, really . . . felt a bit intrusive.\"\n\n\"My, my, you are quite the observant one, aren't you?\"\n\n\"I guess you could say that,\" said Marcus, tilting his face to the sunshine. I could not decipher if he was doing it in remembrance of moments past or soaking it in right here, right now with me. Either way, I loved the way the rays of light made his eyes dance rapidly beneath closed lids, triggering a pulse in his concrete jawline. Snapshot. \"Necessity, I guess. Most things in our house went unsaid. But if I looked close enough, the truth was always there, like a member of the family no one spoke to but maneuvered around. At least most of the time.\"\n\n\"Ha, in my family his name is Uncle Barrington,\" I said. \"In all seriousness, I know what you mean. I learned that much later in life than you, and I don't think it was good. Growing up, I don't remember ever having the time or space to be in observance of others, much less myself. I was always surrounded or instructed in one way or the other. When it's all you know, it becomes a way of life until the day the rug is pulled from under you. Then the easy world you thought you knew shows its real self, and becomes a cold and scary place. I guess at some point, observation becomes mandatory in order to live.\"\n\n\"And so it does,\" said Marcus. \"Are you going to eat that entire thing?\" He was referring to the Miraggio sandwich he'd purchased for me.\n\n\"That was the plan,\" I said, admiring the slices of mortadella inside the toasted baguette. \"Would you like a bite?\"\n\n\"Well, if you are offering, then sure.\"\n\nI couldn't help but snicker. \"Honestly, you are not as charming as you think yourself to be,\" I said, raising the sandwich to his lips for a bite.\n\n\"Yes, I am,\" he replied with a wink.\n\nYes, you actually are, I thought, wiping the remnants of toasted crumbs from the baguette off his face.\n\nLong after the first round of coffee and sandwiches were but a memory, we remained at Quadronno. Mostly in comfortable silence, flipping through his overflow of magazines. At some point I suggested we make our way over to the park\u2014that is, if he didn't have something else to do. He didn't! \"Lead the way\" had been his charge, but I never could figure out what direction was up or down where Central Park was concerned. I have never actually found the stables that are said to be there, but I am confident that at some point, if we keep walking, we'll come upon it or the Great Lawn, or end up on Central Park West. Somewhere all scenic. So I suggested that we roll the dice and see where the sidewalk led, hoping that would be more than agreeable.\n\nAfter weaving our way through a seemingly endless flow of joggers, speed walkers, tourists, and horse-drawn carriages, we arrived at the Great Lawn. The details of how we got there or how long it took are anyone's guess. All I know is that by the time we arrived, my lips were an unnatural shade of radioactive blue, from the patriotic Popsicles purchased along the way from a street vendor.\n\n\"This is a good place,\" said Marcus, indicating a clear patch of unpopulated lawn in the sun among the sea of revelers. For every New Yorker who had escaped the city for the country, there seemed to be one sprawled about today on the lawn. They were either lying out tanning, chasing after their pets, or playing Frisbee, volleyball, or soccer. The air was abuzz with chatter and laughter from all directions. \"Do you mind?\"\n\n\"Mind what?\" I said, plopping myself down on the grass faster than he could get the words out.\n\n\"I was going to say squatting on the lawn, but I guess not. Do you want my jacket?\"\n\n\"Thanks, but I'm good. This is the first time all season that I get to feel earth under my toes. Why block the connection, right?\"\n\nStanding above me and smiling down, he said, \"That makes the second time today that you've said something that surprises me.\"\n\n\"Really, what was the first?\"\n\n\"Yes, to me barging in on your day.\"\n\n\"I didn't know you were keeping score,\" I said.\n\n\"Not score, just aware of how far I have to go,\" said Marcus.\n\n\"That depends. What's your destination?\"\n\n\"For you to go out with me.\"\n\n\"I guess we won't know until you ask, now will we?\"\n\n\"In that case, Jules, would you li\u2014\" Marcus began but was interrupted by a tall, sweat-laden player who had detached himself from the soccer game going on a few feet in front of us.\n\n\"Crawford, what the hell?\" said the swarthy man in a Barcelona jersey. \"When did you get back in town? I've been calling you.\"\n\n\"Last night. You know how it is, haven't even unpacked, much less checked messages.\"\n\n\"And that cute little assistant of yours didn't give you any of my messages?\" asked swarthy man (who had clearly never even bothered to commit the assistant's name to memory the way he had her measurements), stopping midsentence as he took notice of me.\n\n\"Paolo, this is Jules,\" said Marcus. \"Cori probably did but unless it was about the VHO acquisition, then I was not focusing.\"\n\n\"Heard about that. Congratulations, by the way. You got the regulatory approvals to complete that in record time. Normally the fucking Belgians are a pain in the ass. So, you playing or what?\"\n\n\"I hadn't planned on it. Jules and I were just . . .\"\n\n\"Come on, man. We need you out there.\"\n\n\"You guys look solid to me,\" said Marcus.\n\n\"You can't be serious. Bro, we're a man down and had to call Johnston.\"\n\n\"Blimey hell, he's awful,\" said Marcus, throwing his hands up in defeat.\n\n\"Bro, I know! They have already scored on him. Little prick can barely grab the balls in his own sack much less protect the ball on the field.\"\n\n\"Paolo, watch it,\" said Marcus, gesturing toward me.\n\nI said, \"Marcus, go on. You should play; it sounds like fun. Besides, if Johnston\"\u2014teasingly with invisible quotation marks\u2014\"loses this no one will play with him come Monday.\" Selfishly, I wanted some immediate distance between Paolo the Crass and my sunny day. Maybe, if things get hot enough on the field, I just might get to see Marcus strip down. Marcus, however, was not as quick to agree and apprehensively looked to me to make double, triple sure I was fine. \"Go, I will be okay. I promise. Just stay where I can see you and don't play rough with the other kids,\" I said, nudging him in the direction of the game.\n\n\"Okay, but let the record show, I would have much rather stayed here with you and made fun of Paolo,\" whispered Marcus as he bent down to place his sweatshirt and glasses next to me.\n\nOn the field he was a natural, athletic and lean with a superior agility for handling the ball. I was most impressed with his speed and ability to stop and strike in an instant, his control of the ball always purposeful and focused on scoring, not showboating. Whether you knew anything about soccer or not, it was easy to see that he was skilled. My knowledge of the game was passive at best from listening to Dad's childhood stories of playing in Jamaica or occasionally being awakened at 4 or 5 a.m. by loud noises from downstairs during a World Cup tournament, when Cora allowed him to have his friends over. However, while in London, I did learn enough about the game to differentiate between Chelsea and Manchester, and my allegiance increased from going to the club with the cutest players at the time, hence my fascination with Real Madrid always.\n\nWho's to say what regulation time is or was, but by the time the match finished, many of the sun worshippers had left.\n\n\u2022 \u2022 \u2022\n\n\"Who won?\" I asked, when Marcus ran over after saying the last wave of good-byes to his friends. \"Without cheerleaders or a scoreboard, the whole thing is just confusing.\"\n\n\"You do know there are no cheerleaders in f\u00fatbol, don't you?\"\n\n\"There should be. Everyone likes cheerleaders. So, did you win?\"\n\n\"Absolutely. We made a solid rally and came back to take it by one,\" said Marcus.\n\n\"Good, I was concerned about Johnston.\"\n\n\"Ah, you have jokes! Might I remind you, you are supposed to be cheering for me?\"\n\n\"I thought about that but realized you didn't need it. Johnston, he needed me,\" I said.\n\n\"How do you figure that?\"\n\n\"Well, it's obvious that you are their leader, not in the Captain James T. Kirk kind of way but like in a Zinedine Zidane way . . . Oh my gawd, stop acting like I stabbed you. Even you must admit that Zidane is a-m-a-z-ing. Anyway, from the moment you stepped in to play, they all deferred to you and you knew it. Johnston's contribution, however piddly, was all but forgotten.\"\n\n\"Jules, do you even know who Johnston is?\" asked Marcus.\n\n\"No, which is why he needs me all the more, because no one else seemed to care after the crown prince arrived. Your sudden presence made him all but invisible. Can you imagine how that feels? No, you can't,\" I said.\n\n\"I bet you can't either.\"\n\n\"Not true. I have been invisible for the past two and a half years. The only reason you can see me now is that I'm back from the abyss of obscurity.\"\n\n\"Three,\" said Marcus, pushing aside a strand of hair that was interfering with my vision. \"The third thing today you have said to surprise me. I like the not knowing what comes with you. It throws a proper wrench into the entire script I expect to hear.\"\n\n\"From me?\" I asked.\n\n\"Yes, you and in general. You're so unrehearsed,\" said Marcus, placing his index finger softly under my chin to raise my gaze to eye level instead of the nondescript spot on the grass I was focusing on. \"Don't be alarmed. It's not a criticism. In fact, quite the opposite. It's refreshing.\"\n\nWholeheartedly, I accepted the words coming from Marcus. Trusting who he presented himself to be was natural. It's the Memorex replay of my mental state that makes for sheer and utter confusion. Like earlier, there was no need, no pressure to talk through the quiet moments. It felt good just to be.\n\n\"What do you say, shall we start to walk back?\" asked Marcus.\n\n\"I guess, but it is a shame to fold on the first perfect day. I can't believe people rush to leave every weekend, never experiencing the magnificence of New York.\"\n\n\"Couldn't agree with you more,\" said Marcus. \"I remember falling in love with her from a postcard that a friend of the family sent over when I was about nine or so. It was a view of the Chrysler Building. Something about the grandness of it and the eagles, oh man, those eagles, majestic like they're daring the sky . . . I knew that I would live here. I also said that I would own that building. When I finally did get here, I knew that I would never truly leave.\"\n\n\"Except for work and to frolic in the Hamptons,\" I said, surprised that it had not occurred to me earlier. \"Speaking of which, what are you doing here? Shouldn't you be there with the other power brokers doing the whole 'opening of the house for the season and luxuriating' thing?\"\n\n\"Maybe, but it's not going anywhere. Normally, I go up when I have a reason to.\"\n\n\"I'm sure that Gary is up there right now casing your house.\"\n\n\"All the more reason for me to be here. Oh, and don't think you're not coming up whenever we do that shoot,\" said Marcus. \"I have not forgotten you owe me.\"\n\n\"Completely selective memory . . . but for conversation's sake, any idea of when that will be?\"\n\n\"Not sure, soon though. It's a rare moment I get to just have time here without meetings or preparing to fly off somewhere, so it will take some prying before I trade in a personal weekend so easily to do a publicity shoot.\"\n\n\"Seems like a waste, though. I mean, you have a home there, an amazing one from what Gary says, on the beach no less. I would think you'd spend volumes of time there unless you have an aversion to water, sun, and just being mellow.\"\n\n\"One would think, but I don't. You know what they say, 'power begets power.' Water is the most powerful force on earth, so we're good,\" said Marcus.\n\n\"Did you just say that about yourself?\"\n\n\"Ha, that did sound brash, didn't it? I bought the house a couple of years ago because Sasha, my ex-fianc\u00e9e, thought it would be a good look. The whole process took some time to configure; finding the right house on the right lot, in the perfect enclave, getting the permits to build, acquiring the art et cetera, and once it was done I had a beachfront property in Bridgehampton but no Sasha, so . . . it is what it is, a lovely house, but not yet a home like my place here is.\"\n\n\"Oh, I didn't know . . . about Sash\u2014her.\" I didn't know is right, but how could I have? It's not as if over the course of our two or three supervised encounters, Marcus volunteered his dating history, nor had the opportunity presented itself for me to ask!!!!!! And I am not asking now. Talk about a crap way to end the day.\n\n\"Nothing to know really. Things happen. Priorities change.\"\n\n\"Sounds like a novel but for another day,\" I said.\n\n\"A vignette at best,\" said Marcus, \"so don't get your hopes up. Bugger. I forgot my mail. Listen, thanks for letting me crash your day, Jules. I knew we'd be great together.\"\n\n\"Should I wait for you?\" I asked, not entirely prepared to say good-bye just yet.\n\n\"No, no. The lift is here, you go on.\"\n\nAlong the way, all but unknowing, we had exited the park, passed the doorman, and arrived at the elevator, but I could not recall the path getting there. Had there been many people on the street and haphazard taxis speeding about? Yellow lights and crosswalks to be darted through? I can't recall. All I know is that the day started and there he was making it the best day. Now it is ending and he is still here. I could probably ask for more by way of violins, a fairy-tale kiss, and white doves, but I don't want us to arrive at happily ever after so soon because few are fortunate enough ever to discover what comes next.\n\nLater that night in bed, my adrenaline was still going. A warm-water soak earlier had removed the physical remnants of our day, but my mind refused to let them be washed away so easily. It raced from context and moderation to desperation. Today I had a carefree beautiful New York day with a nice man, no more no less, who makes me feel nervous-safe-pretty-funny-erratic all at the same time . . . I wonder if I'd make him a good wife . . . our children (Maximus and Isabel\u2014Issi for short) will have his eyes, my smile, and a curly mane of hair that is the combination of ours . . . What if he was just fucking about wasting a day and forgets about me . . . Did I have food in my teeth? . . . OMG, what about when I tripped? He thinks I am a klutz. The undeniable realization that I really liked Marcus sent me reeling from bed and into the kitchen for a snack. I loaded the audio player with a Funkmaster Flex mix tape and proceeded to dance myself into exhaustion. Side A was nearly done when my jam session began to lose its delusional luster.\n\nI needed to connect with someone and get an objective take on my day with Marcus, so I dialed Richard, only to be greeted by the housekeeper at his Mantoloking home, who advised that \"Mr. Boulton was entertaining at the moment but could be pulled away if paramount.\" Clearly it is . . . I mean, let's examine the facts: I can't sleep and am dancing around in my skivvies to remixes of Busta Rhymes and Super Cat\u2014clearly a matter of vast importance . . . to me. Hanging up the line, disappointed but not defeated, I phoned Blake but was greeted by her answering machine.\n\nOkay! Here is the downside to a single girl remaining in the city during the season. While the days are filled with possibility, the nights are quite desolate and solitary. Calling Cora was not an option, and besides, I was much too sober to fool myself into believing that she could have a sensible discussion about the prospect of Marcus and me without calling our minister to reserve a date. No, she was to be kept at bay until there was a serious development. Until such time I will just have to look for my objectivity in a bowl of mint chocolate chip ice cream.\n\nPassing through the main hall, I thought I heard a noise outside my door. I peered through the peephole and saw Marcus milling about. Before he could ring the bell, I opened the door, having completely forgotten that I was underdressed and cradling a container of ice cream. Total \"lonely girl\" moment\u2014all that was missing was a pack of powdered doughnuts and a Diet Coke!\n\n\"Wow, look at you,\" said Marcus.\n\n\"Yeah, there goes the mystery . . . What are you doing here?\"\n\n\"Would you believe me if I told you that I got off on the wrong floor?\" he asked.\n\n\"Oh, anything is possible, I guess,\" I said. \"What's that in your hand?\"\n\n\"The reason I can't claim getting off on the wrong floor,\" said Marcus, looking at the sealed envelope he was holding with my name written across it. \"I figured you were out, so I was going to leave this under your door.\"\n\n\"Okay,\" I said, looking at the envelope with some skepticism. \"Should I open it now?\"\n\n\"Well, you could, but now that you are here\u2014and wearing . . . Doesn't really matter,\" said Marcus, placing his hand along the base of my camisole, interlacing the stitched edge with his fingers and proceeding to stroke the delicate space of my belly that peeked out just north of my pantyline.\n\n\"Charming. Focus, Crawford.\"\n\n\"I am, but you are making it hard,\" said Marcus, his hand now working its way up the front of my body, eliciting a current of electrical excitement. \"If memory serves, you were in the process of agreeing to go out with me before . . .\"\n\n\"Before Paolo the Crass\u2014purchaser of Penthouse subscriptions\u2014interrupted,\" I said.\n\n\"So you do believe me. Yes, before that.\"\n\n\"Selective memory you have. I don't recall you asking me out.\"\n\n\"Hence the note. Not having your number made it otherwise impossible to do so earlier, and I didn't want to wait until Monday to ring.\"\n\n\"Well, you have me now.\"\n\nStepping within inches of me, Marcus said, \"That I do, and so many things I'd like to do to you . . . Would you like to go out with me, Jules?\"\n\n\"Hmm, that sounds\u2014\"\n\n\"Like what?\" said Marcus, his lips hungrily meeting mine, forcing me against the door. After a few moments\u2014my hands having traveled under his shirt down his chest and reaching the washboard of his abs\u2014it was clear that we were walking the dangerously fine line. Mustering my last bit of strength, I pulled away.\n\n\"Stop, we can't,\" I said.\n\n\"I know. Not yet.\"\n\n\"Well, you're quite sure of yourself.\"\n\n\"That I am. I want to know you, Jules. When I take you, it will be because we both understand what it means, when you're mine,\" said Marcus. \"Until then, Tuesday, dinner? I'll sort the details provided you give me your numbers\u2014unless you prefer I continue to loiter about, but then you would have to invite.\"\n\n\"Of course, and Marcus, please try not to smell of another woman's perfume on Tuesday.\"\n\n\"Four,\" he said.\n\n## 33\n\n* * *\n\n## ARE YOU NEW?\n\nNO MATTER HOW great the weekend was or how blue the skies may be today, Mondays suck. They have to be the most dreadful of all the days on the calendar. It's as if people's mental rest on Sunday is all for the purpose of creating to-do lists that inevitably find me at the top of them. Nancy Wilson had been with us only a short time before an influx of calls from other jazz musicians, their record label execs, and reps, jockeying for them to do a limited engagement at Carly's when her residency was up. Since her debut with us, Michael had tasked me with making certain that a great volume of my media efforts went to positioning the rebirth of her career in direct correlation to her stint at Carly's.\n\n\"Put the fact that I like her aside,\" he said. \"Cats want to be in business with you anytime they feel that the association can work to their advantage. The trick is to make the short-term success theirs and the long-term payoff ours. Make her hot, Jules.\"\n\n\"Any suggestions on how I might go about doing that when, one, we are not her representatives; two, they are insanely territorial; and three, she is not Harry Connick Jr.?\"\n\n\"That's why you're in PR. You are in PR, right? Which means you know how to spin a point of view? Stop fretting, it's a small leap from the spin to the sell. You'll know we're onto something when Harry is seated at a table and my studio has a waiting list of booked sessions as long as the restaurant.\"\n\nGetting Harry at a table was indeed doable, but I had to lure him, not invite him. As for the state-of-the-art recording studio upstairs with private sleeping quarters and a meditation room, otherwise known as Michael's Lair, I was not privy to those books, so there was no way to be absolutely certain of what would constitute a profitable return. Off the top I knew that it rented out for more than $10,000 per day session, which would make any up-front fees we were paying a performer pale in comparison.\n\nNot endeavoring to try to rewrite the media relations book on how to successfully launch a brand in NYC, I did the next best thing and called Carly for access to her Rolodex. At her insistence our initial blow-up had transitioned from a pink elephant in the room to a camouflaged one. This, of course, pleased Michael no end. So much so that even when, against his micromanaging suggestions, I decided not to invest effort in pandering to the easy get that was established jazz lovers, in favor of some of Carly's society contacts who never ventured below Forty-second Street, he relented. My thought was that if you love Nancy, then you would find her, and they did, religiously. As a result of the buzz I had created, my Mondays had been taken over by a legion of pompous individuals who call me \"DAH-ling\" in lieu of my name, which they have not yet committed to memory, only to tell me what I need to do for them. Today they make up the entire first page of my phone sheet and, because each is one degree of separation or less related to Carly, I must endure. Agony.\n\n\"Jacklyn, what are you doing out there, dancing around the phone lines? Why is line two still blinking? Pick it up or take a message. It's driving me crazy,\" I said, sitting at my desk with full view of her across the doorway. I could see her doing a number of other things, but tending to the ever-ringing phone was not one of them.\n\n\"I tried, but he will only speak with you,\" she said.\n\n\"Well, that is not going to happen in the next ten minutes,\" I said, staring at the hypnotic blinking red light to my right. \"Who is it anyway? Can't you just get rid of them? Pretend to be me.\"\n\n\"I tried that. Let me see,\" she said, re-engaging the line. \"It's Marcus Crawford. Do you want him?\"\n\nThe sound of his name seemed to ring out like a morning bell at a Tibetan temple. A sudden appearance of moisture in my palms made the phone difficult to grasp. Luckily I caught it before it dropped on the desk, then mumbled something incoherent before placing my current call on hold and sprinting into the hall to Jacklyn's desk.\n\n\"What I want is for you to lower your voice!\" I said in a hushed whisper, looking up and down the hall to make certain that no one was approaching.\n\nMichael had complained many times before about our seeming inability to use the interoffice message computer system for calls instead of yelling to each other. \"Jules, why is it I can hear everything that goes on in your office when mine is on the opposite side of the building?\" he would ask. My hope is that this particular moment eluded office ears, especially his. I hadn't been true to my word with Michael, nor had I worked out how I was going to tell him that Marcus and I were spending time together.\n\n\"Put him through,\" I said, stealthy in my movements as I crossed the threshold into my office.\n\n\"Why are you looking around like that?\" asked Jacklyn suspiciously, also on high alert for matters unknown. \"You're making me nervous.\"\n\n\"Never mind about me, just you keep doing whatever you're not doing . . . hush . . . I'll explain later,\" I said, pulling the door shut behind me.\n\nSecure inside my office, I leaned against the closed door for support to quiet the strumming bass erupting from my chest. It seemed to be dictating that I skip to the phone despite my urge to keep him waiting for a few extra minutes. You know what I always say, \"Jump the first time a man calls and he'll expect you to tap-dance for life.\" At the most inopportune times, the Cora-isms of my upbringing often have the ability to supercede my own inner voice. This time, however, my nerves prevailed, and I dived right in.\n\n\"Hi!\"\n\n\"Did I catch you at a bad time?\" asked Marcus.\n\n\"No, well, kinda. No, not really,\" I said, stumbling over my words in search of a solid course. \"I was just surprised is all . . .\"\n\n\"Good. Let me preface by saying I know it's short notice and it's not Tuesday, but would you like to have dinner tonight?\"\n\nSearching the ceiling for an easy-spirited connection between my thoughts and my mouth before speaking, I finally asked, \"Did you really just sit on hold and refuse to speak to my assistant so you could ask me to dinner?\"\n\n\"Technically, no. Cori, my assistant, sat on hold awaiting you, which by the way took an unusually long time. You should consider apologizing to her. I, on the other hand, took a few other calls, brought peace to the Middle East and Southwest Asia, did a fitting with my tailor, and knitted a sweater before you actually came on the line.\"\n\n\"Funny, Crawford. I'm known for using the Middle East peace line myself.\"\n\n\"You know what they say about making a woman laugh, don't you?\" asked Marcus.\n\n\"Not particularly,\" I said, twisting the phone cord between my fingers. \"However, for the sake of this conversation, let's say I was curious. What do they say?\"\n\n\"Actually, they is a he, and he says a woman laughing is a woman conquered,\" said Marcus.\n\n\"Aaah, is that your intent, to conquer me?\"\n\n\"Of course, I'm a man. That's what we do. Figuring out to what degree, now that is something we'll discuss in a few years.\"\n\n\"Years, you think we'll know each other that long?\"\n\n\"I'd put money on that,\" said Marcus.\n\n\"Who said that 'woman conquered' thing? I hope it wasn't your friend Paolo.\"\n\n\"No, although it would be fitting. It was Napoleon,\" said Marcus.\n\n\"Well, in that case, I must be more careful with my laughter,\" I said.\n\n\"When that happens I will try harder,\" he said.\n\n\"Honestly, you have to stop. All of this panache and savoir-faire stuff, I just don't know . . .\"\n\n\"So, we'll discuss it over dinner tonight, yes?\"\n\n\"Yes, okay.\"\n\nI hung up the phone still laughing, walked to the window, and found myself looking again across the street below at the very spot Keith stood months before. Like the weather that day, the memory of it all is gray and hollow now, in complete opposition to how I was feeling standing here. A warm, familiar burst of excitement I believed to be long gone had reappeared, taking root in the base of my stomach and blooming through the contours of my mouth like a lullaby. Until now, I hadn't realized how much I needed him to call me. The realization scared me. Truth is, I wanted that connection. I just couldn't imagine knowingly getting back on the all-encompassing roller coaster of emotional uncertainty it required, but Marcus made me wishful, and that is something to aspire to, right?\n\n\u2022 \u2022 \u2022\n\nThe hours leading up to our date seemed to be an eternity delivered on the back of a tortoise. When the blessed time did arrive, I met Marcus in the lobby as agreed, wearing a mostly new\u2014i.e., spontaneous purchase made weeks earlier on the off chance that I would have an event worthy of such a frock\u2014chocolate Donna Karan matte crepe jumper. It had been my first choice, but I needed to be certain, so a few alternates (five or so) were tried and discarded just to be sure. Finishing off the look with a pair of gold strappy stilettos and a perfume spritz behind the ears, between the breasts, on the wrists, and between the legs, I was ready to go. Judging by the looks of things, however, Marcus and I were going out all right, but to separate locations. The elevator doors opened to reveal him wearing a black leather jacket, dark denims, crisp white athletic shoes, and a Yankees baseball hat.\n\n\"Jules, you're a knockout,\" he said, drinking me in from freshly polished berry toes to nude, glossed lips. \"Are you sure you want to waste that outfit on bowling?\" he said.\n\n\"Excuse me, bowling?\" I said, the words stumbling out in my distress.\n\n\"Yes, bowling,\" he said, re-enacting a classic stance. \"Did I forget to mention that earlier?\"\n\n\"Yes, I would say so,\" I replied, embarrassed and uncertain. \"Bowling?! I thought we were going for dinner. You did say dinner, didn't you? Should I go and change?\"\n\n\"Indeed, I did say dinner,\" said Marcus, staring at me, amused at himself and my obvious discomfort and keenly aware of his omission. \"It seems I should have been more specific.\"\n\n\"Understatement. Please stop laughing,\" I said, trying to retain my composure but feeling quite warm under the collar, in a vastly different, less hospitable way than I had earlier. \"I should go back up and change. Give me a few minutes.\"\n\n\"If you think it will help your game, then by all means.\"\n\n\"Seriously, Marcus, this is not the time to challenge me. I am far from amused at the moment.\"\n\n\"Do I detect a temper?\" he asked.\n\n\"Yes you do . . . I don't like being . . . and clearly you . . .\"\n\n\"My, you are mad at me! Kind of sexy on you,\" said Marcus.\n\nStanding before him, I didn't know whether to smack him in that delicious mouth that held such promise for the end of the night or to just storm off. \"Come on, don't be mad. I was having some fun is all, just trying to locate your buttons,\" he said, stepping in closer and caressing the length of my arm and stopping just parallel to my breast, allowing his thumb to linger on the contours.\n\n\"There are far better ways to do this, you know. Being a cad is not one of them.\"\n\n\"I'm sorry,\" he said, reducing the space between us to that of a few molecules of air. \"Forgive me?\"\n\n\"Forgiven,\" I whispered.\n\n\"We don't have to go bowling. I just thought . . .\"\n\n\"No, let's do it,\" I said, regaining my composure and walking toward the door, where Percy had taken in the entire exchange. \"On the day we tell the story of how I kicked your butt all over the Bowlmor Lanes, I want to make certain it is noted that I did it in Donna Karan. Ain't that right, Percy?\"\n\n\"That's right, Ms. Sinclair. That's right,\" said Percy, directing his last comments to Marcus: \"Never disagree with a woman like that, Mr. Crawford, no sir.\"\n\n\"Famous last words, Percy,\" said Marcus.\n\nOn the drive down we spoke easily about all manner of things: his athletic pursuits throughout school, all of which he entered as a walk-on only to prove himself to be the best; his fascination with collecting rosaries because, like the scent of YSL's Opium, they are one of the few memories he had of his mom; and that if his soul could sing, it would sound like Otis Redding.\n\nIronically enough, in all that time, never once did he think to ask me if I even knew how to bowl. Truth is, not only did I know how, but I was pretty good and determined to show him just how good. On the Saturdays when Cora was having her weekly manicure or lunch with the girls, Dad and I went bowling. Some of our best laughs and my first stolen sips of beer were in a bowling alley. Lord knows it was needed, as the man took an eternity to align his body in the perfect position to take a shot. At some point after my twenty-first birthday, I recall Daddy preparing for a shot, and just before taking it turning around to say, \"Julesea, don't you tink it's tyme you order your own beer instead of drinking mine?\" Truly one of the best moments.\n\nNow I was about to have a similar evening with the absolute last person I would've expected to be into bowling, much less come equipped with a customized ball. In hindsight, that should have been a sign to proceed with caution, which is the exact opposite of what I did. At the beginning of the first game, I began with a serious round of trash talking, telling him that he bowled like my grandmother after he lost the first game. At the start of the second\u2014having won and feeling pretty confident\u2014I suggested we make it more interesting, say, ten dollars a strike. In the blink of an eye, he sank five consecutive strikes. That's when I knew that I'd been had!\n\n\"You're so not a gentleman, Crawford! You hustled me,\" I said.\n\n\"Hey, I was content to let you win a game, but then you had to go and get all big-time,\" chided Marcus. \"Come on now. I can't be having that.\"\n\n\"Why didn't you mention you were on a bowling team before?\"\n\n\"You didn't ask. Besides, it wasn't an official team, just my nan and a bunch of old ladies with tinted hair in matching shirts when I was younger,\" said Marcus, untying his shoes.\n\n\"Well, next time I'll ask you all manner of things,\" I said.\n\n\"Deal. What do you say we start the inquisition over a proper dinner? There's a nice little spot around the corner that is more appropriate for your outfit,\" he said.\n\nOnce outside, we found Carlos positioned in front across the street, leaning against the car and enjoying a drag on his cigarette. On seeing us, he put it out and went to open the passenger door until Marcus instructed him otherwise. Carlos opened the trunk, allowing Marcus to exchange his baseball cap and leather jacket for a blazer.\n\n\"Talk about hustling someone. Is it really that simple for men?\" I said after witnessing this quick change.\n\n\"It is, baby. The deck is always stacked in a man's favor. Thank goodness women have a more forgiving spirit, otherwise we could not apologize enough for the glaringly obvious inequities, you know.\"\n\n\"Would you repeat that into a tape recorder at a later date, so I can play it for my girls?\"\n\n\"Hell no. I like being in the men's club too much\u2014great perks. As a matter of fact, I don't even know what you're referring to,\" said Marcus.\n\nAfter walking a few blocks west we arrived at a restaurant on the north side of the street, where Marcus has a standing reservation. I made a mental note then and there that nothing in this man's world truly occurred on the fly. We were led to a corner booth in the back to a table adorned with freshly poured glasses of champagne and insalata mista.\n\n\"I hope you don't mind,\" Marcus said, gesturing to the carefully placed delights, \"but I had something special prepared.\"\n\nFeeling quite giddy and pampered, I shrugged no. The appearance of lobster, however, did elicit an interruption in his meal selections. \"I'm allergic to shellfish. I am so sorry, but it's either an Epipen or hospitalization for me.\"\n\n\"Now that's a compromise. Asking me to give up seafood is a lot. It's one of my favorites.\"\n\n\"I see. That could pose a huge problem. I guess the question to be answered sooner than later is if I'm worth the sacrifice,\" I said.\n\nMarcus signaled the waiter to come over and explained my situation. Within minutes the chef was tableside outlining a seafood-free feast. Approving the new menu, Marcus turned to me and said, \"You're worth it.\"\n\n\"You don't know that to be certain.\"\n\n\"I have a sixth sense about things.\"\n\n\"Do you now?\"\n\n\"I do,\" he said. \"Don't shake your head. You'll see. I'm always clear on what I want, Jules. I want you\u2014healthy and having eyes only for me.\"\n\n\"If you are trying to do a number on my head, it's working, so I beg you to stop. Marcus, I've been on the losing side of love before, and I just can't do it again.\"\n\n\"That's not my intent. I have a home in Bridgehampton to show for a failed leap. You have several voices in your head telling you not to trust me. I'm trying to tell you that you can, if you want to.\"\n\n\"Speaking of that, the voices . . . I would like to, but then little things happen like you calling the office today, and I wonder if this is all part of some play with Michael.\"\n\n\"Was that not the best place to reach you in the middle of the day? Have I given you any reason to distrust me?\" asked Marcus. My deadpan stare triggered his memory to our very first dinner, with Gary. \"Aside from the extremely poor judgment displayed earlier, for which I have already apologized . . . Jules, I don't need you to do my bidding with Mike. As for what demands he will make on you in the future, I can't say, so, see? I am actually the one at a disadvantage here. At some point you may have a choice to make, and there's a possibility that you won't select me.\"\n\n\"Are you sure you're not a lawyer?\" I asked, attempting to lighten the mood from the suspicious air I had created.\n\n\"Listen, business is what I do, but my passion is history. That alone tells me that many a great man has been brought down by only one thing,\" said Marcus. \"Love.\"\n\n\"What I love is that you already consider yourself great. I mean, can a person actually say that about themselves?\"\n\n\"I just did,\" said Marcus, looking me squarely in the face, our dining table feeling more like a boardroom to close a deal. \"Only question is, are you the kind of woman who can allow herself to be loved . . . or will you cut and run when things get uncomfortable?\"\n\nAlthough the comment was directed at me, I felt there was much more to it and hoped I wasn't wrong by saying, \"One day you'll tell me about her, your mom. Okay?\"\n\n\"Not tonight. I'm still trying, very hard, I might add, to impress you,\" said Marcus.\n\n\"You just did,\" I confessed. \"Like I said before, you are good . . . and dangerous in a Thomas Crown Affair sort of way.\"\n\n\"McQueen or Brosnan?\" asked Marcus, removing the glass of champagne from my hand and covering it with his.\n\n## 34\n\n* * *\n\n## TWO MONTHS, SEVEN DAYS, AND SPONTANEOUS COMBUSTION\n\nWHAT I AM not understanding is why it's taken so long to get you on the phone, Boulton,\" I barked into my cell. Richard had called just as I entered the main doors at Carly's. \"Oh pish posh, secretary of defense, my arse, Richard. I's met a man, and he is fabulously unbelievable, so you have to help me not muck it up.\"\n\n\"Dear, haven't we been down this road before? Need I remind you of Keith?\"\n\n\"Low blow. Besides, Keith's wrapping was fabulous. It's the creamy center that just happened to be double-stuffed.\"\n\n\"Hence my point. What do we know about this man aside from the fact that he is a partner at Chimera Capital, an investor with your current employer, lives in your building, has received rave reviews from your folks, collects model creatures, and knows his way around your kitchen but not your kitchen?\"\n\n\"So you have been listening to my voice mails! Why in the hell haven't you phoned me back then? Those random e-mails were driving me crazy.\"\n\n\"Of course I was listening, so please don't go all pitchy on me. I just couldn't return in detail, but rest assured I am up to speed.\"\n\n\"Sorry about that. Your ears, I mean. You know how excited I get. So, I am not crazy then? It is odd that he and I have been dating for two months, seven days, and have yet to sleep together. I mean we have slept together in the 'Jack and Jill' sense of the word but not in the 'Tie me up! Tie me down!' way, you know what I am saying?\"\n\n\"Yes, that is odd,\" replied Richard. \"Maybe the equipment is broken or just does not work with your parts.\"\n\n\"Will you stop it with that?! I'll have you know that I've done the feel test numerous times and his member clearly responds to me. It's just the actual act of engaging that has proven elusive. Okay, before you ask, I'll confess that at first he tried, but I thought we should wait, which is elementary, I know. Now it's like he isn't even pressed to try again.\"\n\n\"Honey, when will you ever learn? Grown men don't respond to high school games.\"\n\n\"Old habits are hard to break, but I'm trying . . .\" I was suddenly distracted by Michael's reflection in the window, which meant that my conversation with Richard would have to end prematurely. I watched as Michael not so subtly lurked at the door saying something to Jacklyn, most probably trying to find out who I was talking to. Far from done talking to Richard but having no other choice, I turned around to face Michael just as he was walking in.\n\n\"Finish your call. I'll wait,\" he said.\n\n\"Richard, someone has come into the office. Can I ring you back? Oh, when will you be out of session? Okay. Then might I see you on Sunday? Perfect, eleven a.m. it is,\" I said, disconnecting and giving my full attention to Michael.\n\n\"I don't know why you did that. I told you I would wait,\" said Michael.\n\n\"Michael, I have seen your form of waiting,\" I said, swiveling my chair side to side. \"After a few seconds you'll start rummaging through my things, asking me numerous questions about what this is or why that is there, then if you still don't have my undivided attention, you'll either sit at the desk throwing those massive yet elegant feet on the corner while sucking on one of your cancer sticks or you'll walk around, only to lean over my shoulder and read my e-mails . . . sooo, basically what I am saying is, your version of waiting, Mr. Kipps, sucks.\"\n\n\"You know me all too well, Jules.\"\n\n\"I do watch you . . . like a hawk. Can't have you slipping on my watch.\"\n\n\"I know you do and I like that about you,\" said Michael, removing the aforementioned cigar from the interior pocket of his blazer.\n\n\"So what gives? Am I costing you too much money again today with my brilliant idea?\"\n\n\"Not that I know of, but you just might be. I'll let you know when I get to your e-mails. Don't give me that look like you're disappointed. You would never have known if I didn't say anything.\"\n\n\"Thank God for Simone,\" I said. \"At least she reads my e-mails.\"\n\n\"Yes, that woman covers up any number of sins for me, but she can't handle everything, which is why you'll be at my house on Saturday night. You have any plans?\"\n\n\"I'm guessing not, although if I did, it would probably be seeing the last Ailey performance of the season at Lincoln Center with Joy after weeks of planning, her agreeing to come in from Jersey, and hiring a babysitter,\" I said.\n\n\"Good. You're not missing anything. They perform Revelations at the end, same as last year and the fifteen before that,\" said Michael. \"I'll expect you at my house by seven thirty for a dinner with Nikolai Abramovich.\"\n\n\"What?! When did this happen? I mean, I know his party was a success and all, but I didn't know you two still talked.\"\n\n\"It wasn't your business to know, but if you look a little closely, you can learn something about building a bridge and creating allies.\"\n\n\"Don't you mean comrades?\"\n\n\"Humor is not your forte, Jules. Carly and I are throwing a dinner in his honor, an official welcome to NYC, if you will. I need a focused set of eyes and ears on Nikolai and Annya, his wife. Think you can handle that?\"\n\n\"Sure, I'll try,\" I said.\n\n\"Don't try, Jules. Do,\" said Michael between quick and short inhales to light his cigar. \"I need to make sure there are no flies on old Nikolai.\"\n\n\"What is that, like KGB-speak?\" I said. \"What's the attire?\"\n\n\"Cocktail but not black. I need you in something fetching and not so official.\"\n\n\"Do I get a tip at the end of the night?\"\n\n\"You keep your job. Does that count?\" said Michael.\n\n\"Fetching it is!\" Great, now I really do need to speak with Richard. I know absolutely nothing about global finance, Russia, and oligarchs. Talk about being a fish out of water, I might as well be a mullet in the Caspian Sea.\n\nAfter Michael left my office feeling quite full of himself, I immediately phoned Marcus. Undoubtedly he would be at the dinner, a potentially sticky situation. If he is going and I am going, then maybe we should go together, but were we ready to be outed? Was I ready? The few times Marcus had brought it up, I told him I wasn't ready. Selfishly, I just wanted to keep us quiet a bit longer, because I still wasn't sure we were real. He didn't agree, but reluctantly allowed me to take the lead.\n\nI called his private line to bypass speaking with his assistant. Our cloak-and-dagger had not afforded me the luxury of visiting his office yet. I envisioned a mysterious corner space, a combination of Old World antiquities and modern collectibles encased in floor-to-ceiling windows with breathtaking views of Lady Liberty. The Bat Phone, as it were, being one of those rotary BC (before cordless) variety, shiny black with the flat circular dial, sans the finger holes and numbers tucked away in the bottom drawer of a grand mahogany desk. The ring unlike a traditional phone so as not to elicit attention, should he be in a meeting, silently blinked a red light under the desk for only him to see.\n\nPicking up after a few rings, he said, \"Marcus here.\"\n\n\"Do my ears deceive me or are you a bit British today?\" I asked.\n\n\"Well, since Reaganomics, the pound is doing considerably better than the dollar . . . so, yes, yes I am.\"\n\n\"I heard. What a sad state for old George Washington,\" I said.\n\n\"Well, maybe you should call him and brighten his day like you do mine,\" said Marcus.\n\n\"Look at you, so charming!\"\n\n\"Well, it's true, Veronica. You have this great way about you,\" said Marcus, his delivery being so natural and easy that I was not immediately certain of how to respond, so I just sat there holding the phone silently until he spoke again. \"Hello? Don't be like that. I was just playing.\"\n\n\"Honestly, you must believe me when I say you aren't nearly as funny as you think yourself to be, Marcus.\"\n\n\"I don't believe you,\" he said. \"Cori laughs at all my jokes.\"\n\n\"Cori doesn't count. She's a paid employee, poor woman has no choice. I on the other hand am the realest thing in your life,\" I said.\n\n\"Which is why I'm strongly considering keeping you around, at least that is what my nan says I should do.\"\n\n\"Brilliant woman she is. Speaking of that, being around and all, not your nan, Michael just commandeered my Saturday. Apparently he is throwing a dinner for\u2014\"\n\n\"Yes, for Nikolai Abramovich,\" said Marcus. \"You going to be there?\"\n\n\"So it seems. Don't act so surprised. Why are you taking Veronica instead of me?\" I asked jokingly, making light of the moment as we so often do.\n\n\"Well, not exactly. I am going, but not with her,\" said Marcus.\n\n\"Oh, I see. Well, this is awkward, you not thinking to ask me, I mean.\"\n\n\"Wait a minute. Do you really want to open that up, Jules? I would gladly take you anywhere anytime, but remember it's you who doesn't want to be seen in public around familiar eyes.\"\n\n\"Was that a question or a statement? Never mind, it's besides the point,\" I said. \"Can I ask who you are taking? Are you like dating or something?\" I asked, my voice raising a noticeable decibel or two.\n\n\"You do realize that you just went from zero to sixty in less than two seconds. Remarkable, actually. Better than my Maserati.\"\n\n\"I'm not laughing, Marcus. Were you going to tell me if I didn't call?\"\n\n\"Probably not, although I did think about it, if that matters,\" admitted Marcus. \"Listen, you don't know her, just someone I met before you and see from time to time. Jules, you still there?\"\n\n\"I heard you, just thinking is all . . . What else have you not told me?\"\n\n\"Honestly, Jules, there are certain things that don't necessarily warrant a conversation until you are clear on what it is you want from this thing we are doing. Unless you would like to shed some light on matters now?\"\n\n\"I didn't know that you were seeing other people, publicly is all, but I guess you are. I mean, it's none of my business. It's not like we're sleeping together or something.\"\n\n\"Would it change things between us if we were?\" asked Marcus.\n\n\"It wouldn't be a declarative statement or anything, but it would clear up some lines regarding intention,\" I said.\n\n\"Jules, a lot of my work is social, you know that. I have never been reserved about my intentions toward you, Jules,\" said Marcus. \"I want to be with you in every way. Say the word and we will go together.\"\n\n\"I can't. Michael needs me.\" The words jumped off my tongue before my brain could do an appropriate edit. \"What I mean is, the entire purpose of me being there is to observe Nikolai and Annya. I can't very well do that if I am with you debuting, now can I? . . . Marcus?\"\n\n\"You don't have a monopoly on silence, Jules,\" he said in a tone that conveyed his agitation with me. \"Mike's a great guy, but I won't be in his shadow even for you.\"\n\n\"I know and I'm sorry. Truly I am, which is why I called. Not to say I'm sorry, because I didn't know I would be saying I'm sorry, but to tell you about Saturday. I guess I didn't do it the right way. Please don't be mad, okay?\"\n\n\"Delicate ground it is, Jules. I just hope you understand that,\" said Marcus. I understood that he was saying my twenty-something cute-craziness is wearing thin in his grown-man world. \"Moving on, we will both be in attendance on Saturday. What are the rules here, baby? What else do you need? I have another call waiting\u2014\"\n\n\"Don't say it like that, please. I need for you not to get exhausted with me, because I don't think I could bear it. Then I need for you to delete this chick's number from your phone sooner than later, although I know I have no right to ask . . . right now. Most important is I need you, okay?\" I said.\n\nFor some time I had been overanalyzing, about what I wanted\u2014what I needed from Marcus, to pacify the turbulent conditions in my head. Now, in the most roundabout way, I found my voice and was able to share it with him. My hope was that it would not be met with deafening silence.\n\n\"Okay, I will see you there,\" said Marcus.\n\n\"Okay? . . . Good.\" I held my breath for his response, hoping for something more that would quiet the pangs of anxiety creeping into my soul.\n\n\"Jules . . . nothing . . . bye,\" he said.\n\n\u2022 \u2022 \u2022\n\nWhen not overthinking Marcus, I managed to tap into my network and learn that Abramovich's wife, Annya, was a major patron of French haute couture and Italian fashion. He, on the other hand, appeared to be less inclined to anything that was not about creating additional wealth via oil, technology, communications, and multimedia throughout Russia, Asia, and blue chip holdings in the U.S., unless it measured 34-24-36, which is clearly why Michael wanted me to look \"less official.\"\n\nWith Gary's help (i.e., editorial discount) I acquired an Yves Saint Laurent dress for the evening featuring a slim black skirt and a voluminous white silk top with an exposed back. Despite the boatneck cut of the blouse, the fabrication made the contours of my ample bust quite visible\u2014another gift from Cora's side, of course. He finished the look off with oversize bone, gold, and onyx bangles and flowing hair.\n\n\"Honey, trust me, men like that love themselves some cascading hair, especially all this voluminousness you got going on,\" said Gary, running his fingers through my mane. \"You sure you're not part Indian?\"\n\n\"Shut up and fix my hair right, will you?\" I said, playfully slapping at Gary. \"Honestly, what would I do without you always coming to the fashion rescue?\"\n\n\"All right, Cinderella, remember to make sure that most of it falls over your shoulder, and when he is around, toss your head and throw it back like one of them Charlie's Angels girls from back in the day. It will drive him crazy! Trust,\" said Gary.\n\n\u2022 \u2022 \u2022\n\nArriving at Michael and Carly's building via a yellow cab to a line of waiting black sedans was a bit intimidating in itself without fixating on the fact that Marcus and his date were most likely already upstairs. When extending the invitation\u2014or sentence, as it were\u2014Michael never said how many people were coming. I envisioned fifty or so. Walking through the marbled entrance hall of their home and seeing the formally attired staff in tails carrying trays of champagne, wine, and hors d'oeuvres, I would say it was closer to two hundred, with the median age being north of forty-five . . . single solitary mullet in the Caspian, I am. The only thing missing in this swanky affair was an arrival announcement stating each guest's position and affiliation. I imagine mine might go something like, \"Ms. Jules Sinclair, dateless spinster daughter of Charles and Cora Sinclair of Virginia\" or \"Ms. Jules Sinclair, dead woman walking.\"\n\nNancy Wilson was the first familiar face I saw. Having been with us for some time now, she and I had become quite friendly, so upon learning I was alone, she clipped me to her skirt hem and made some beneficial introductions. My favorite by far was to Bourjois Cosmetics CEO and socialite Chantal Chapman. Talk about a self-made woman by way of a proper assist from a tycoon husband who was more than twenty years her senior. Chantal was indeed a force of 'UES by way of Arkansas' glamour. Everything about her was a statement, from her big auburn hair to the enormous emeralds and diamonds and deathly sharp fingernails lacquered the same flaming shade as her hair and lips. Immediately upon laying eyes on my YSL ensemble, she declared that she could never wear hers again (even though Yves himself had fitted her), to which I quipped, \"Thank God, 'cause I don't need the pressure!\" Chantal enjoyed the unfiltered retort and declared to Nancy that I was a gem. That's cute and all, as long as I could be the living equivalent of the 20 carats sitting on her finger. Honestly, where is security or a little person to hold her hand up?!\n\n\"That's why I hired her,\" said Michael, appearing between the two of them as if they were Heaven and Earth and he the stars; always meant to be there. \"Jules, you do look lovely tonight. Clearly I am paying you too much money.\"\n\n\"Or maybe it's not your money at all,\" said Chantal, \"that keeps a young woman in a certain comfort.\"\n\n\"Unless she, being me, has a fabulous gay with divine fashion access,\" I said, quickly squashing the \"kept woman\" direction of conversation.\n\n\"Clearly more fun, just less perks,\" said Chantal. \"Remember, it's always better to acquire, my dear, than lease.\"\n\nFollowing a few more pleasantries, Michael made our apologies and ushered me away to the far corner of the room, where Carly was holding court with Annya and Nikolai Abramovich. Against the nighttime skyline and the amber lighting created for the evening, Carly appeared warm and ever-present, unlike the cold, old, nosy coot I knew she could be. True, we had found a way to be cordial, even faux-friendly in the aftermath of her Tony confession, but my defenses were never down around her. Hell, if she looked close enough she would see my hand on the trigger just waiting for her to go left. Thankfully Cora taught me to play the civility game long ago. After initial exchanges and introductions, Michael and Nikolai made their exit, but not before Nikolai properly sized me up in a manner that was at once inappropriate and completely obvious to all present, including his wife. Her blas\u00e9 attitude, however, conveyed that it was expected and the least of her concerns\u2014unlike the meticulous application of her makeup, giving her skin the appearance of the finest porcelain bisque doll.\n\n\"Jules, you look very appropriate,\" said Carly. \"Remarkable, actually, for a woman of your age to dress so well. Your contemporaries are trotting around in little dresses that leave little to the imagination, baring their midriff. Positively gauche.\"\n\n\"In defense of my peer group, Carly, it's all trial and error. Fashion, as Coco said, fades, only style remains. Mrs. Abramovich, like Carly, you are quite the style icon,\" I said, paying specific attention to her dramatic, cobalt-blue, embellished Valentino couture gown. Far too dressy for such an affair, but then again my job is to charm and ingratiate, not to judge. \"Where do you find inspiration?\" I asked, not trusting whatever was to come from Carly's mouth next.\n\n\"Darling, when you spend as much as I do, season after season, they have no choice but to call you an icon,\" said Annya, so taken with herself that Carly's steely-eyed gaze escaped her. Ahhhh, when the rich and the nouveau riche clash\u2014what fun!\n\n\"Well, in that case I had better do as Chantal instructed and find myself a husband of means then,\" I said, garnering an affirmative nod from both.\n\nUnlike Carly, Annya is a natural beauty, brunette and buxom. She was born to extreme poverty in Slovenia the way Carly was born to extreme wealth. Legend and gossip circuits have it that she was the au pair to Nikolai's youngest child and lured him away from his wife of nearly twenty years in a matter of months. She is the stuff of gold-digging opportunistic legend and is unapologetic about it. I must introduce her to Blake someday.\n\n\"Jules, I had no idea you were looking to marry,\" said Carly, no more interested in my marriage prospects than in Annya's style status, but isn't that what polite conversation is all about, faking it? \"Why didn't you come to me? There are some stellar men about.\"\n\n\"Of course,\" I said, more as a conversation point than anything else. \"I used to say that I wanted a man just like my father. I have since amended that to say I want a man who my father likes. Someone who personally checks all the right boxes and a few wrong ones, yet who is strong enough to withstand my ambition.\"\n\n\"Rubbish, darling. No such man has ever been created. Fathers only like the ones who are reflections of their best selves,\" said Carly. \"As for the strength, we women crafted a fable long ago to make men believe that is who they are. They are not and never will be.\"\n\n\"I must agree. The men themselves do not know what greatness they are truly capable of without our eyes and prowess,\" said Annya. \"The trickier part is to continue holding the reins without their knowledge in the face of excessive power.\"\n\n\"Which is why you must always be stronger, craftier, more cunning than they are, dear,\" said Carly.\n\n\"My goodness, you both make finding the right mate sound like an Ethan Hunt Mission Impossible,\" I said.\n\n\"It is, dear. The perfect man doesn't just appear. You make him,\" said Annya, \"but it takes far longer than five seconds.\"\n\n\"It takes a lifetime,\" said Carly.\n\n\"Okay, two questions. Is there a handbook, and whereabouts might one go to find such a man?\" I asked.\n\n\"Of course not. It's instinctual. You'll know when you cross him . . . unless . . .\" said Carly in a forced tone that reeked with disapproval, be it for the sentiment or the approaching guests who had captured her attention, prompting her to cross in front of me and Annya in order to greet them.\n\n\"Dah-ling, I am so glad you were able to join us. You look absolutely fabulous, so debonair. Come, come! You must meet Annya Abramovich. Annya, this is my everything, my confidant, the keeper of my world, Marcus Crawford.\"\n\nYou ever have that moment when time just stops and a split second transforms into an eternity? Well, if not, here's mine. Knowing that he was coming tonight did not prepare me for the actual moment. I had rehearsed a few times over how I would be poised and composed when we came face-to-face. Now here he stood before me but not seeing me, with Carly clinging to his left arm and Annya to the right, sizing him up like the main course. Forget the former script, anxiety was about to make me a sweaty mess, and that never reads well on silk. My breaths were coming in faster, shorter waves, a sure sign that I could not trust myself to be in such close proximity to Marcus and not reveal the true nature of our relationship.\n\nI was so focused on flight that I collided, champagne flute first, into another guest wearing an example of the garish outfits that Carly mentioned earlier\u2014rouching across the hips is no one's friend . . . even if you are a size 4 and have mile-long legs.\n\n\"What the . . . shit. Oh my gawd, Julesy!!!! What are you doing here?!\" said Blake.\n\n\"Holy shit, Blakes. I'm so sorry,\" I said, frantically trying to wipe droplets of champagne from her purple strapless dress before a stain set in, not daring to look behind me and see if our collision had elicited attention. \"What are you doing here?\"\n\n\"You tell me. He said to put on something sexy, and I thought it was . . . well, forget what I thought. Anyway, here we are. OMG, here you are! You look so great! I am totally borrowing that,\" said Blake, eyeing my YSL. \"You know I am borrowing that, right?\"\n\nElated to see a friendly face that I could download to, I briefly forgot the immediate\u2014until, that is, the sound of Marcus's laugh echoed about. \"Thanks, boo, but I have to go,\" I said.\n\n\"Wait, not yet. I want you to meet my guy. Well, he's not my guy officially. That's what I thought this was about. Anyway, I'm working on it. You know Richard will absolutely die when he finds out that you met him first,\" said Blake. \"That is, if the old bat will dislodge her fangs.\"\n\n\"Would love to. Let's go,\" I said, grateful to have a legitimate reason to put as much distance as possible between myself, Marcus, and Carly. My efforts to continue forward proved futile as Blake stepped into me, forcing another collision that now elicited attention.\n\n\"There you are, dear. Marcus, you know Jules,\" said Carly, still glued to his side. Had I not been standing there to observe this surreal moment with my own eyes, I would not have considered Blake's depiction of \"the old bat with fangs\" to be true, but judging by the almost total color drain from Marcus's face on seeing me, Carly was indeed a well-fanged vampire, maybe even Nosferatu. Having no forewarning as I did, Marcus nodded a polite \"hello, yes, of course\" that belied his traditional charismatic demeanor and robbed me of any contact. \"And this is his friend . . . I'm sorry, dear. I seem to have forgotten again. What's your name?\" asked Carly.\n\n\"Blake. My name is Blake, Mrs. Kipps. I am also a friend of Michael's. We have met before, remember?\"\n\n\"Yes, of course, Blaaake,\" said Carly in an affected drawl that purposefully emphasized the lack of sincerity. \"This is Annya Abramovich and Jules Sinclair.\" Maybe if Blakes had added \"a friend of Michael's who has never slept with him and never will because his wallet is too short, but yours, on the other hand . . .\" Maybe if she said that, their dynamic would be less arctic.\n\nVisibly taking great pains to ignore the slight, Blake nodded hello to Annya and then proceeded to quicken the ground between Marcus and me. \"Good to meet you, Annya. Jules is one of my dearest friends,\" said Blake.\n\n\"My, isn't the world small,\" said Carly, turning to me with a disapproving glare.\n\n\"You could say that,\" Marcus, Blake, and I echoed in unison.\n\nI believe this is what they call a Texas Standoff. My eyes locked on Marcus, his ever so carefully on me, Carly and Blake sizing each other up. Save for the lights blinking on and off signaling that it was time to adjourn to the dining room, who's to say where the moment would go? With each step I tried to do the math. How the? When the? WTF! Of all the men on the entire island and in Jersey, how is it that Blake and I were dating the same one? Hold up, did she say \"again\" as in \"I have met you before\"? OH MY GOD, THEY'RE FUCKING!! Why have I never heard about him before? What in the hell happened to \"Mr. Potentially More\"? OMG, is Marcus Blake's Mr. Potentially More? If only I could turn to her and get the answers to the laundry list of questions in my head, but I can't. Not with Marcus at my side imploring ever so discreetly that we need to talk. UNDER-FUCKING-STATEMENT, MARCUS!!!! Keep the champagne. I need a bottle of Jack.\n\nOnce inside the dining room, we separated. Each guest's seat was designated by ivory textured place cards featuring their name in brushed gold script. Richard would approve. If the gods were crazy enough to concoct this evening, then surely they would take great pleasure in seating Marcus and me next to each other, but not if I could do a quick swap before anyone noticed. Michael was standing near the head of the table directing traffic as it were with Nikolai, who, despite being the single wealthiest person in the room, seemed to be salivating at the attention bestowed upon him far more than his wife.\n\n\"Jules, you're here,\" said Michael, signaling my seat, near where he and Nikolai were standing. The majority of the guests were still milling about and not yet crossed the threshold.\n\nAs Michael continued rattling on, as hosts are forced to do, about the merits of this approaching guest or that one, I was busy scouting the table for the task at hand. For, as much as I wanted to observe Marcus and Blake's dynamic, I didn't need them sitting right near me. For fear of having an uncontrollable desire to stab him, spill more champagne on her, or lose control of my vocals and yelling \"Are you fucking serious!\" at the top of my lungs to the universe before the amuse-bouche is served.\n\nCareful not to physically disengage from him and Nikolai, I took two steps in the direction of my seat to uncover the identity of my neighbors. Annya was to my left and some man with far too many names\u2014Sir Brandon Phillip Von Blah Blah Blah\u2014to my right. What ever happened to boy-girl-boy-girl? It would be difficult enough trying to focus on \"Operation Nikolai\" and entertain Annya while trying not to lament over Blake and Marcus, much less having to endure the musings of some old guy who in all probability would either have ripe breath or try to touch my leg under the table.\n\nWhile putting my purse down, I discreetly switched my seat for Annya's. Something tells me that as long as his title is right, she could overlook any number of olfactory sins. Marcus and Blake approached from the opposite side of the table to exchange pleasantries with Michael and Nikolai. I could not help but look up and notice how very New York Times Sunday Edition Wedding Section they were\u2014in spite of Blake's dress selection this evening. She so naturally fit into the space at his side I'd thought was mine alone a few days ago. Had it been a different man, I would have found extreme pleasure in her happiness, but this was not the case. It was not another man. She was standing there with my Marcus, and I was watching it alone, near sickness, without the protection of an all-knowing friend or an escort to hold my hair should I need to immediately release the emotions bubbling inside. Surely this must be one of Dante's levels of Hell or maybe part of a Kundera comedy. Fact is, I am here in this moment, but there is absolutely no laughter to be had, which makes it quite apropos, I guess.\n\n\"My my, I see your keen eye goes beyond fashion,\" Carly whispered from behind, catching me off guard. \"He is something, isn't he?\"\n\n\"I'm sorry, what were you saying?\" I said, taken aback by her brazen intrusion into my thoughts.\n\n\"Marcus, of course. He is impressive . . . I sense you share my sentiment, do you not,\" said Carly. Definitely more of a statement than a question.\n\n\"I hadn't noticed,\" I said.\n\n\"I beg to differ, my dear. I would say you have noticed a lot, as you should,\" she said, locking eyes with me as if she were magically scanning my thoughts. \"Great potential that one has, provided it's honed properly. Wouldn't you agree?\" Fearing the release of highly sensitive data, I forcibly changed the direction of our conversation and commented about the beauty of the d\u00e9cor long enough for seats to be taken.\n\nMarcus and Blake were seated in the center on the opposite side of the table, close enough that I could bear full witness to their every gesture but far enough away that their words eluded me. Inevitably the moment came that she made him laugh, head tilted back, eyes closed, a slow-finish kind of laugh. The same one he has done with me many times before, so there was no question of its sincerity. I didn't have a monopoly on his laughter. The moment crystallized, forcing me from the table. Blakes is one of my besties, and if Marcus is her \"Mr. Potentially More,\" then what right do I have to him? She won't value him as I would, as I do. I know Blakes. She will use him until she is bored and then she will move on. She doesn't deserve him . . . but she is my friend.\n\nMy only familiarity with Michael and Carly's home being the observatory, I found the elevator and sought solace there. Above hung a full moon, temporarily shadowed by a ring of clouds. Its presence explained to my ill-tempered tears that tonight could not go any other way than how it was.\n\n\"Are you all right?\" asked Marcus, appearing minutes after I had arrived.\n\n\"I am not. I'm so not okay right now, so please leave.\"\n\n\"Jules, I had no idea that you and Blake were friends. This is . . . unfortunate.\"\n\n\"Well, that's one way to put it. A royal fuckup would be another. That is, if you were standing in my shoes or even cared, but . . .\"\n\n\"Don't go there, Jules. Of course I care. Why else would I be up here, instead of there?\"\n\n\"Are you sleeping with her?\" I asked.\n\n\"Listen, you're angry and you're hurt. Let's not compound the matter by going down a road that\u2014\"\n\n\"Of course you are, why wouldn't you be, everyone else has.\"\n\n\"Jules, I know this is all . . . Hell, I don't know what this is but let's not say anything to make things worse,\" said Marcus.\n\n\"Please stop! There is absolutely nothing I can say that could possibly make tonight worse than it already is. Marcus, you are here with one of my best friends and clearly she is not a new acquaintance unless Carly was mistaken. Either way I lose.\"\n\n\"I prefer not to believe that,\" he said.\n\n\"Really, how do you figure that?\"\n\n\"Truly, I don't know in this moment, but I can tell you that the last thing I want is for you to look this lovely, up here, alone, crying tears over me,\" said Marcus, removing his neatly placed silk pocket square to catch the trails running down my cheeks.\n\n\"Great, that makes two of us,\" I said, moving away from his touch in search of some perspective. \"You had better go back downstairs. Your date\u2014Blake will be concerned.\"\n\n\"I'm concerned about you.\"\n\n\"Me too.\"\n\nOnce I was sure that Marcus had gone, I allowed myself a final outburst before forcibly pulling myself together. What was it about me and this room? Such misery it brings every time. In the future I must make a concerted effort never to be in it. Stepping into the hallway, to my dismay, the all-seeing eye stood in wait.\n\n\"Is everything okay? I noticed you left the table abruptly and was . . . concerned,\" said Carly.\n\n\"With all due respect, please use another word. I have more than enough 'concern' tonight, thank you,\" I said.\n\n\"I see.\"\n\n\"I felt a bit light-headed, dizzy, that's all, and needed some fresh air,\" I said, patting the last bit of moisture from my face.\n\n\"And Marcus as well? Did he need some fresh air also?\" asked Carly.\n\n\"I don't know, Carly. You'll have to ask him.\"\n\n\"I see,\" said Carly, promptly switching gears in favor of a lighter approach\u2014her suspicion satisfied. \"Well, I hope there's not a problem with the food. We used a new caterer, from downtown no less, but I have been assured that they are stellar. Well, we'll see, won't we?\"\n\n## 35\n\n* * *\n\n## SUNDAY\n\nCAN YOU BELIEVE she said that to me? I mean, what is it with that woman?\"\n\n\"Darling, Carly is the least of your concerns . . . Sorry, wrong word,\" said Richard, perched in his usual Sunday position at Cub Room. The chaos of my life on this day superceded his need to start off the morning with a Bloody Mary, opting instead for the clarifying jolt that a double espresso might bring. \"What are you going to do about Blake and Marcus? You can't ignore the poor man forever.\"\n\n\"One evening does not constitute forever, Richard. Besides, I was in no position to speak last night. It's bad enough that I faked illness to excuse myself from after-dinner chatter. Addressing his ringing my bell incessantly to explain the unexplainable truly would have made me ill.\"\n\n\"You know, sometimes\u2014often, actually\u2014I think you like misery too much, dear.\"\n\n\"Almost as much as you like gin? Excuse me, let's not forget I'm the injured party here.\"\n\n\"No, you're not, Jules. You are the party with knowledge now of all the players, so therefore you have a choice! Did you hear me? There is a choice to be made, and it is yours. Question is, what are you going to do?\"\n\n\"I know you think I am weak and why shouldn't you? You're always coming to my rescue, but there is a code, you know, about this sort of thing. Girls don't do this to girls.\"\n\n\"Honey, grow up. Codes are for children and cadets. Women not only do it every day but some actually find great everlasting happiness in using their cunning to acquire the right prize. Trust me, Blake will recover,\" said Richard, finishing the last drop of coffee. \"Is he the right prize for the sacrifice?\"\n\n\"Yes, I think so, but will our friendship?\" I asked.\n\n\"That I don't know. Depends on how you handle it,\" said Richard.\n\n\"I wish I didn't have to. Can't you do this one little thing for me?\" I implored.\n\n\"As much as I would like to break Blake's face at times, this particular cut is yours and yours alone.\"\n\n\"Probably won't be so bad. I bet Marcus is just like the others,\" I said.\n\n\"For your sake, I hope so, but I doubt it. Blake is not the singularly focused, emotionally depraved creature you believe her to be. That's just her PR,\" said Richard. \"Quite the contrary, Blake wants desperately for one of these men to legitimize her, to show her through the front door to business associates instead of dark clubs and exotic vacations thousands of miles away from their respectable lives. Your Mr. Crawford may be more than just your ideal. He may very well be hers, also.\"\n\n\"Why are you telling me this?\"\n\n\"Because you need to know. This is a win-lose proposition no matter how you look at it. It's time to start living your life as an empowered woman full-time, Jules, and stop straddling the fence. Blood oaths and pinky swears are for elementary school. Don't you think it's time you graduated?\"\n\n## 36\n\n* * *\n\n## A DECLARATIVE STATEMENT\n\nIN THE WEEKS that followed my conversation with Richard, I did allow Marcus back in, because any thought of an immediate future without him seemed too much to bear. As a show of my commitment to being fully in the space of us, thereby eliminating the need for any decoy, Gary, Carly, Michael, Messy Mitzy, Percy, Jacklyn, Ivan, Cora, Daddy, and even his friend Paolo were along for the ride\u2014everyone but Blake, the one person who was calling to hang out and vent about Marcus dumping her. But I couldn't. I was a coward and I knew it. In my defense, I just wanted to hang on to her friendship as long as possible. I knew that ours would not survive this. I just needed a bit more time to come to terms with the choice I had made and with losing her. Following a near encounter at Barneys one Saturday with Marcus, I could no longer hide the truth.\n\nWe had just completed his selection of new Valextra travel cases and were rounding the corner to the jewelry section when I saw her standing there in oversize sunglasses and a hat, and next to her was a new Daddy Warbucks. I froze in my step, unsure what to do next. Marcus followed my gaze.\n\n\"Hey, isn't that Blake?\" he asked.\n\n\"Um-hmm.\"\n\n\"Well, no time like the present to say hello.\"\n\n\"Wait, not yet,\" I said.\n\n\"If not now, then when, Jules? We can't pretend forever that there is no history there.\"\n\n\"I know, but she doesn't know about our future, exactly.\"\n\n\"You can't be serious, Jules. It's been weeks now and you haven't told her?\"\n\n\"I didn't know how.\"\n\n\"Well, you'd better figure it out, otherwise we'll be ducking out of department stores all over town,\" said Marcus, taking hold of my hand and exiting out the Sixtieth Street entrance. \"Come on, my little puppy, tuck that tail. We can look at watches later.\"\n\n\u2022 \u2022 \u2022\n\n\"I was starting to wonder about you,\" said Blake as she settled into the booth next to me at L'express. \"When I called Richard, he was all cryptic and self-absorbed as usual. Something about adopting a baby and getting the house ready for social services . . . anyway.\"\n\n\"I know, I'm so happy for him and Edgar. It's about time. They'll make the best parents, don't you think?\" I said.\n\n\"I guess. How are you?\" asked Blake, indifferent to the whole situation.\n\n\"Well, that's a fully loaded question. Why don't we start with something easier? How are you?\"\n\n\"I'm amazing! Getting ready to spend a glorious holiday with Stavros aboard his yacht. We are going to fly over to Nice, where we will meet his crew, then set sail for the Amalfi Coast and then to Santorini. Can you imagine the sheer fierceness of yours truly gallivanting about the Riviera!\"\n\n\"I can, and I fear the damage you will do to his credit cards.\"\n\n\"Correction, already done! A girl does need travel gear, you know.\"\n\n\"Lordy. I have no words.\"\n\n\"I like this one, Jules,\" said Blake, \"I mean, he is not the ideal package, but I can work with it. He's a widower and a divorc\u00e9. He's good to me and needs me in total, not just as a trophy. Did I mention he is a widower? Hello!\"\n\n\"You did, twice,\" I said, anxious to pounce on any nugget of information that could potentially minimize what I was about to say. \"Good to know you're not pining over Marcus.\"\n\n\"Ugh, don't mention that name to me. Biggest waste of time.\"\n\n\"Oh, sorry,\" I said.\n\n\"J, don't be. It's not like you're the one who got dumped. I mean, technically, I wasn't dumped, per se. We just ran our course. Voil\u00e0!\"\n\n\"Yeah, that's a much better way to see it,\" I said. \"Besides, if you were still seeing Marcus, then you would have never met Stavros and would not be getting ready to sun and fun in the South of France.\"\n\n\"Exactly!! So let's only speak of beauty and yummies, since I have been starving myself for the past two weeks getting this body beach-ready,\" said Blake. \"What are you having? I know, let's do the Ni\u00e7oise and split the French toast for dessert like old times.\"\n\nAnd that was how it was for the majority of lunch. We talked about the incidentals of life and matters concerning her imminent adventure. I tried to erase Richard's observation of her from my thoughts, as it is easier to believe her isolated in some way. Stavros was her consolation prize, and only time would tell if he would be hers to keep.\n\n\"Blakes, I know you said not to speak of Marcus again, but you do know that we are neighbors, right? Which means we see one another. And we work together, so . . .\"\n\n\"Jules, correction, he is an investor, so you don't actually work together,\" said Blake. \"As for the neighbor part, well, there isn't much that can be done about that, now is there? Except maybe taking the stairs.\"\n\n\"Stop. He's not bad, you know. Actually, very kind and pretty great on the eyes,\" I said.\n\n\"Yes, therein lies the problem. It's all peachy keen until his mystery is against you, and you find yourself waiting for his random phone calls too, for which there is no rhyme or reason. The only thing that man is consistent about is his work and his horizontal stroke. Not much room for anything else, which is fine. It's his right. All I'm saying is, don't go being all fabulous occasionally making me think we could be something while knowing all along that you have no intentions of being serious,\" said Blake, who by now was doing very little to mask the angst that encapsulated her relationship with Marcus.\n\n\"I'm sorry, Blake. I didn't know he hurt you like that,\" I said, without thinking that I was about to multiply that very pain.\n\n\"He didn't hurt me. I was just sharing. Besides, it not like it's anything you need to know.\"\n\nShit!!! It's so easy to just sit here and allow the moment to pass. What good will telling her do? In two days she'll be on an Air France flight to happiness. Why muck that up? Except the Marcus that Blake was speaking of is in direct contrast to the one who goes above and beyond to make himself available to me. The man who speaks about a future that prominently features us together. Her Marcus is not the very same Marcus who was away fishing in some remote stream with my dad days before. \"Well, um, see, the thing is . . . he asked me out,\" I said.\n\n\"Oh,\" replied Blake, visibly shaken but fighting to remain composed. \"What do I care if you go on a date with my leftovers? At least you have been warned.\"\n\n\"Well, it's not exactly your leftovers, Blake, which by the way sounds a bit insulting to me.\"\n\n\"Uh, you know what I meant. What do you mean by not exactly?\"\n\n\"Remember end of last year when you started dating Mr. Potentially More? Well, a short time after that I started dating someone as well. I didn't talk to you about it because . . . I don't know why, but the timing just never seemed to be right. If you must know, I didn't really speak to anyone about it because it was complicated. And you never gave a name to your guy. Even when Richard kept calling him 'this one,' you didn't say anything. Anyway, it wasn't until the dinner at Carly's that I realized that your guy and my guy were the same person. Which is why I pretended to be sick that night, so I could leave and think.\"\n\n\"Normally you give far more details, Jules. Why so short this time?\" asked Blake with a fire in her eyes that validated all my fears.\n\n\"Sometimes the details just don't matter,\" I said.\n\n\"So let me get this straight. It has been weeks since that dinner, and you and Marcus have known but neither of you saw fit to tell me. Hold up, were you the reason he broke up with me? What, so he chose you over me?\"\n\n\"I wouldn't put it that way,\" I said, but I could see Blake running the library of their conversations, whether he'd become distant or used the old standby \"It's not you it's me\" line.\n\n\"Are you the one that he has 'feelings for and wants to give it a go' so he can't continue to fuck me?\" said Blake. \"I don't believe this shit!\"\n\n\"Blake, I\u2014\"\n\n\"No, just shut up, Jules. How could you wait so long to tell me? Allow me to play myself over this guy? Walking around here like you're my friend and not the scandalous treacherous bitch you really are.\"\n\n\"Blake, I am your friend! I didn't know what this was and I needed to be sure it was something worth\u2014\"\n\n\"Worth what? Losing a friend over? Stabbing me in the back? You fucking cunt. Well, I hope it was worth it, because you are dead to me,\" said Blake, grabbing her bag and running to the door.\n\nRising to give chase, I left a loose wad of bills on the table and hoped it would be enough to cover the cost of our meal and the free show that spilled onto the sidewalk as she was hailing a cab.\n\n\"Blake, come on, we're not in the mob.\"\n\n\"You should count your blessings,\" said Blake as she closed the door. \"Oh, and since we're confessing today, I knew Tony was fucking Angie long before you found them.\"\n\nLater that night, after walking more than fifty blocks home to get some perspective, I called Cora and recounted the whole dreadful experience. I shared that I was not certain of Blake's confession about Tony and Angie. She was angry and said many things. No matter how awful a friend I had recently been to her, I just didn't believe that Blake had ever been as bad to me. Cora, on the other hand, ever the skeptic, ever my cheerleader, said she believed her and never liked her anyway\u2014something about Blake reminded Mommy of Rosalyn, her rival for Daddy's affections.\n\nBy the time Marcus let himself in, I was in bed, having a fitful time making peace between what I had lost and what I had gained. He allowed me to cry and volunteered to phone Blake, but we both knew it would only make matters worse for her, and that was the last thing I wanted. Before closing my eyes, I asked Marcus if he thought that our relationship would be cursed cosmically because of my actions. He said, \"True love always prevails, Jules. We just have to make a promise to be braver and more vigilant than most to make sure the cost was not in vain.\"\n\n## 37\n\n* * *\n\n## STRANGE BEDFELLOWS\n\nSINCE THE FALLING-OUT with Blake, the friendship lines had been drawn. In the divorce I kept Richard, albeit with a firm reprimand about how horribly I'd handled things despite his sage advice. She got Joy, which I understood, as Blake was the injured party and Joy's overly maternal instincts compelled her to try to make everything okay. Older than Blake and me, she lived solidly by the Code. Sometimes I think that Joy will die on that cross. Maybe if Richard gave her the speech, there would be room enough for us both, the saint and the sinner, but until then I had my man, my work, and a new ally in the trenches.\n\nNancy's residency ended and, surprise, Harry Connick Jr.'s is about to start, clearly a much better proposition than having him seated at a table. Even if Michael felt averse to me and Marcus being together because of their friction, he cannot deny that in relationship bliss I am all the more better at my job. I am happy, happier than I've ever been. That ever-present voice of doubt residing in my head of past mistakes is now less, and Jacklyn especially enjoys the perk of relationship bliss as it translates to earlier days and fewer weekend calls from me. I have a new motivation: my boyfriend, Marcus Crawford. We are a couple, and that means allowances are made, his being communication and mine being flexibility. Marcus's travel remains as it had always been, intense and international. I make certain that in the times he is home, I am there as well\u2014whenever possible.\n\nIn preparation for Harry's debut, there is a marathon of things to be completed. The slightest oversight would impact the much-delayed weekend Marcus and I have planned to spend in the Hamptons, so Jacklyn is working inside my office for the day so that we can focus with minimal distraction. Sitting opposite me at the desk, she continues to man the phones as we draft and approve an influx of media copy and interview requests.\n\n\"It's Mrs. Kipps,\" said Jacklyn. \"Do you want her?\"\n\n\"Yes. Thanks, Jacklyn. Why don't you go and do a run-through of Harry's dressing room? Make sure we aren't missing anything on his rider.\"\n\n\"Hi, Carly,\" I said, once I was certain we were alone.\n\n\"Jules, is everything all right? I hadn't heard from you, and it has been some time since Stavros and Blake left. Did you tell her?\"\n\n\"Yes, she hates me.\"\n\n\"In time she will come to thank you. After all, had you not come to me and shared the sordid mess, she would never have met Stavros and maybe finally gotten the life she desires.\"\n\n\"I hope you're right, but it doesn't change the fact that I am missing a friend.\"\n\n\"Choices, darling. We all have choices, and you made the only one there was to make. Marcus is quite a catch. It's quite rare to find them of his caliber at the time they are ready to get serious about things, if you know what I mean. Besides, Stavros is right for her, not Marcus. I saw him drinking her in at dinner.\"\n\n\"So it wasn't a coincidence that they met?\" I said.\n\n\"Heavens no, dear. If we left everything to chance, then her kind would rule the world, now wouldn't they?\" replied Carly.\n\n\"True, I guess. Any word on when Michael is going to blow or is that one for chance? I have seen him give me that look, but he hasn't said much of anything, which has me quite nervous.\"\n\n\"Good cause to be. Thankfully, he's been preoccupied. Just know that I have given the union my blessing, which may be enough. Let's see,\" said Carly.\n\nI heard the words clearly but did not have complete faith in Carly's sphere of influence as it related to Michael's territorial nature over his professional pursuits. He may not be addressing the matter directly with me but he was peripherally, which is exactly where I felt I had been pushed to\u2014the periphery. There was something major brewing, far more important than the opening of Harry Connick Jr., but that was all I knew. Whereas I used to be part of the inner-circle think tank with him, Raymond, and Simone, I was now pushed to the perimeter, skimming at the edge of the conversation but not yet knowing fully the topic of it.\n\n\"Carly, why did you help me?\" I asked.\n\n\"Preservation, dear, preservation.\"\n\n\"Excuse me?\" I said.\n\n\"In time you'll understand, and you'll thank me,\" said Carly.\n\n\u2022 \u2022 \u2022\n\nThe day of Harry's debut found Michael on cloud 999. He was buzzing about with more exuberance than normal for such an occasion. Earlier his tailor had come to put the final touches on a white dinner jacket he had commissioned for the evening. I recognized Mr. Vintorini in the hall as I made my way down to his office. For my part, I found it most useful to provide just enough cushion in our dynamic for Michael to seek me out instead of readily showing concern for his not addressing my relationship with Marcus. Of the many lessons that Michael had instructed me in over the past year, this was by far the most useful and easiest to incorporate.\n\n\"Come on in, Jules. Tonight's a big night. You feel it?\" asked Michael. He was standing before his interior full-length mirror admiring his reflection in the new jacket. \"Which do you prefer: the rose or the pocket square?\"\n\n\"Pocket square, more classic and understated. The whole town seems to be buzzing about Harry's performance. We even have people flying in,\" I said.\n\n\"That's what I wanna talk to you about. I need you to remove the people from my table. Don't cancel them, just find room elsewhere.\"\n\n\"Michael, that's impossible unless you want me to put them in the kitchen, and I don't think Jean would like that very much. We are above capacity for both performances, standing room only. We'll be lucky if the fire marshal doesn't make a surprise visit and shut us down.\"\n\n\"Sounds like a great problem to have, if you ask me. Make it work. Nikolai is coming into town.\"\n\n\"Okay, how many in his party? I'm sure if it's small I can reshuffle and put them somewhere.\"\n\n\"Just him and his second in command, Mikhail. I don't want them somewhere, I want them at my table . . . alone.\"\n\n\"Okay, just so I am clear . . . I'm moving five people from your table in order to make room for two?\"\n\n\"That's right. Also, I need you to draft a statement for morning release announcing my newly minted partnership with Nikolai toward the global expansion of Carly's.\"\n\n\"I'm sorry, you're what?\"\n\n\"You heard me. Nikolai wants to be in the service business and is putting a cash infusion into Carly's that will allow me to do what I want to with the business.\"\n\n\"So you are still set on expanding, despite what the reports showed? What happened to wanting to keep the business private, nurturing its growth and image for a while longer? At least that's what you wanted to do when Marcus suggested expansion.\" I added the last bit under my breath.\n\n\"Your boyfriend doesn't run things around here,\" said Michael, tossing the pocket square to the floor near my feet and affixing the rose to his lapel. \"I brought him into this, so we take my lead. Now I want this on the local morning news show as well, so make . . . What, Jules? Is your hand broken? Do I need Simone to draft it?\"\n\n\"Of course not. I just . . . You just made it into this whole us against them\u2014against him thing that I completely supported. So much so that when you reamed me out for that innocent dinner with him and Gary, I felt like I had disappointed you, and even how I struggled against initially liking Marcus because I didn't want to be disloyal to you. And now . . .\"\n\n\"Now what? You realize that a game is always in play and interchangeable, is that it? Ain't that some shit,\" said Michael, removing the dinner jacket as he stepped away from the mirror. \"Listen, Jules, you can't ever expect to know completely what the full agenda of a man is. You just have to be buoyant and adapt. Can you do that? Of course you can. Now let's get started, unless there is something else?\"\n\n\"Yes, Marcus and me together. What about it? You said yourself that I watch you and know you better than most, so just let me have it. I know you don't like it. Do I need to be worried about my position here with you?\" I said, emboldened as I often am by Michael's unapologetic delivery after a few moments of being on the all-too-familiar receiving end.\n\n\"Transparency is not an everybody situation, I get that. You and Marcus, I'm not pleased, but I trust you until you give me a reason not to,\" replied Michael. \"I'm in a good mood today, Jules, so I will say this. Keep your eyes open, your mouth closed, and never forget family. We're family, Jules.\"\n\n\"Was there a Sicilian convention in town that I didn't know about?\" I asked.\n\n\"What?\"\n\n\"Nothing, everyone's going mobster these days,\" I mumbled, and returned to my office for the remainder of the day to work with Jacklyn, adding the new partnership announcement to our already extensive list.\n\nMarcus was still out of town when the announcement hit, so I can't be sure if I was his first or second phone call, but I know I was at the top of the queue when I heard the short-tempered urgency in his secretary's voice as she put him through to me. And for the third time in as many months, my loyalty was called into question. Marcus felt that I should have given him prior notice: \"I'm not asking for insider information but you could have said something.\" When I tried to explain my actions and talk about my conflicts, trying to do my job and feeling caught in the middle, he said, \"Feelings are like the wind, Jules. They come and go, you'll get over this tantrum. This isn't about separation of church and state. Your oversight, as it were, could impact the business that I do.\" Later I would learn that the omission did impact Marcus's portfolio, though not in the way that I thought, as it was the principal piece in cementing a relationship with Nikolai that would soon put us in California and potentially cost Michael everything.\n\nIn the meantime we had the \"just us times,\" or JUTS; precious, suddenly available weekends, or evenings that originated from a last-minute cancellation. In those times, we would lose ourselves in the kitchen, on the sofa, in the bed, or anyplace that was only about us. Marcus would speak vividly about what it was like to grow up in Boston, attending Brown, Oxford, his dreams of creating a legacy to be passed on to his children so they never had to question whether he'd existed. We developed an affinity for late-night games of strip Trivial Pursuit, and we were so good at strip Twister, it could have been an Olympic competition. With each revelation and shared insight, my respect for Marcus grew, as did my commitment.\n\nAfter this blowup I make a point of being more mindful going forward, while still keeping fully focused on my work affairs. I embrace that \"I\" am now part of a \"we.\" The initial phase is seamless\u2014attending ballet and opera openings, baseball and basketball games, fund-raisers and auctions together\u2014I am now required to accompany Marcus to client dinners whenever girlfriends, fianc\u00e9es, and wives are encouraged to provide an air of civility. We even vacation\u2014or weekend, as it were\u2014with his corporate contacts from around the globe. It's what his lifestyle demands. It is a lot to be constantly \"on\" at that level.\n\nUnlike my experience with Tony, for the most part everyone in Marcus's New York scene took to me well enough. As Carly so aptly said, \"Dear, it would be poor form not to,\" when I expressed my surprise at how smoothly things seemed to be going. The part she left dripping on the floor was \"when your mate is as accomplished and connected as Marcus.\" All well and good, but the one thing I didn't account for was its restrictive nature, as his world frequently overshadowed any perceived needs I may have had.\n\nThe time I relish spending with my friends was reduced to a fraction and his commitments became infinitely higher priority. Every second or third week, I managed to see Richard or have lunch with Jessica, whose husband\u2014yes, the guy from my final interview who she wanted me to meet\u2014was in the financial trade as well. Richard, being in the final stages of adopting a Chinese baby, seemed to have very little time to play constant therapist to my life's predicaments. Apparently learning Mandarin and preparing for the arrival of little Ming Thuy Boulton Davis was getting the best of him. The one bright purely coincidental spot, however, was that Joy and I mended fences after Marcus and I ran into her one evening at 81\/2 as she was awaiting a dinner meeting that was unexpectedly canceled. Before coffee was served, she understood\u2014begrudgingly, mind you\u2014that we were much more than a fling.\n\nThen there were the like it or not downright inopportune times, like being pressured to cancel my plans, even a late night at the office, to accommodate a forgot-to-mention commitment on his calendar. His seeming detachment from or noninterest in some of my affairs often left me feeling displaced and exposed, sometimes missing the self-sufficient woman I had become in London. That woman in the later stages found great value in who she was and relished always being a priority, not having her identity lost in that of her companion. Desiring not to be consumed by these thoughts to the detriment of our relationship\u2014and feeling that I had no other recourse\u2014I shared them with Cora, who to my astonishment was quite empathetic, albeit antiquated. Ultimately, she told me that it is the required cost to be with a man who is fully in his person and of a certain stature: \"You don't want to be with some man who isn't rooted, Julesea. You want a man whose shadow looms much larger than yours but his heart beats for your well-being. Trust me, you'll have a much better life. Why you want to carry all that weight anyway?\" were her exact words. In looking at her and Daddy, I saw this to be true. I just don't know if I want to be Cora is all.\n\n## 38\n\n* * *\n\n## IN OUR FUTURE I SEE\n\nMARCUS'S COVER AND feature did not make the winter edition of Decor magazine as Gary originally planned. It did, however, make end-of-summer inclusive of a four-page spread, yours truly pictured on two of them, named and categorically identified, \"Marcus Crawford and mate Jules Sinclair.\" Cora, naturally, purchased every copy available in Virginia and demanded that Jacklyn comb the Manhattan newsstands as well for reserves. Undoubtedly one would find its way to every member of the Augustus-Sinclair family stateside and abroad within the coming days. Over dinner with Marcus one evening, she in absentia made us address the five-month-old elephant in the room.\n\n\"I was talking to your mother,\" started Marcus.\n\n\"When will you ever learn?\" I asked in jest.\n\n\"Seriously, J, she's not that bad.\"\n\n\"That's because she has ulterior motives where you are concerned,\" I said. \"Let me guess, she wants you to sign her copies of Decor for the girls at the club.\"\n\n\"Took care of that days ago. Actually, she wanted to know what my intentions are with you, seeing as we have yet to discuss them, at least to her satisfaction.\"\n\n\"Did she, now,\" I said, trying to appear as nonchalant as possible. I too wanted to know what Marcus saw in the future for us but had decided some time ago not to press the matter. Raising my hands in defense, I said, \"I swear she's not doing my bidding.\"\n\n\"Thought never crossed my mind,\" he said with a wink to the contrary. \"But I do think she deserves an answer\u2014don't you?\"\n\n\"Yes, I do. I just didn't want to be the one to bring it up\u2014in case you weren't ready to discuss it is all.\"\n\n\"So you have thought about it?\" asked Marcus.\n\n\"Of course I have. I am a woman, after all; it's part of the natural order of things. I thought about it from the moment you left Carly's without so much as my phone number. You, sir, should be counting your blessings that I've managed to keep the matter off my tongue for this long. Clearly, I'm teasing . . . sort of. I have thought about it, but only casually,\" I conceded.\n\n\"Really,\" said Marcus, placing his fork and knife down to sit upright in his chair before leaning toward me. One side of my brain immediately started to race with the possibility of oh my gawd is he going to ask me to marry him? I'm so unprepared. Totally should have worn the other dress\u2014photographs much better, while the other pleaded for neutrality. But he said, \"I'm not ready to get married. Maybe in a few years, but not now.\"\n\n\"Oh . . . is that in general or just to me?\" I said, crestfallen, feeling this the only logical next question.\n\n\"Well, I don't\u2014\"\n\n\"Don't do that. You know if you want to marry me, whether it's six months from now or four years from now, the same way you know every other important step in your life long before it happens. You're a decisive planner, Marcus, so don't dance around, just tell me. I can handle it.\"\n\n\"I know you can handle it, Jules. I just wonder if I can handle it if my terms are not accepted,\" said Marcus. \"Are you even ready to settle down?\"\n\nHis phrasing caused me to pause. This is not at all how I envisioned the evening going. Quite the contrary, when his driver Carlos arrived at the office to pick me up, I'd been pleasantly surprised when the door opened to see Marcus seated inside to greet me. My understanding had been that he might be delayed getting to the restaurant due to a last-minute meeting but would send the car for me. On the ride to Le Bernardin, we laughed and exchanged spirited tales about the day. He spoke of needing to make another trip out to L.A., even asking if I could find a few days to join him, as this visit would keep him there for about two weeks. I didn't know the specifics, only that there was a studio ripe for acquisition.\n\n\"I choose not to think of it as settling,\" I said. \"Yes, I can see myself married to you, but if you can't, then . . .\"\n\n\"I never said I couldn't see us married, Jules. I'm just not ready in the immediate future. There is so much on my plate right now that I have to be mindful of. Can you accept that . . . and wait?\" said Marcus.\n\n\"Truthfully, I don't know,\" I said, blatantly aware that those words could very well mean I'd leave this table a single woman once again, but I had to speak my heart. \"What do I do with feeling like a fool as the years go by and all that I was has been completely enveloped by your world, with no promise of anything more than being identified as Marcus Crawford's guest? Hell, Marcus, I already feel like that from time to time but I try not to show it.\"\n\n\"I would never ask that of you, Jules. If you choose to give it, then that is solely your decision, but it is not what I would want or expect. Your independence is one of the things I love most about you,\" he said. \"Jules, I know that right now you think me oblivious but I'm not. I'm aware of what I am asking of you. I know what it feels like to wait on someone only to be disappointed. I won't do that to you.\"\n\n\"I'm not Sasha, Marcus.\"\n\n\"I was referring to my mother,\" said Marcus. \"Is there an urgency in you that needs to get married within the next year, Jules?\"\n\n\"No, but there is something to be said for taking the option completely off the table. It changes the dynamic of you and me. Growing up I heard Cora and Aunt Helen talking about how no man is ever going to buy a cow when he can get the milk for free. Please don't laugh, I'm serious. Marcus, your stance is putting me in that position now, and if I am to believe you, then I must compromise myself in order for us to work, and I don't want to do that. You would not like me for doing that, how could you?\" I said, pushing my plate to the side, finally abandoning all pretense of still having an appetite, much less eating the meal.\n\n\"You know, it's been a while since I had to think of someone besides myself. Sasha was the last. In hindsight, that relationship was more about control and wanting to craft a life that perfectly fit into a frame than of focusing on the substance. I want to be smarter with you. Can you just hang in there with me?\"\n\n\"Why do men always ask that and expect a yes?\" I said. \"Tell me why I should.\"\n\n\"Because you will be my wife and the mother of my children one day if I have my way. Because I am on the brink of doing something big that I have never done before, and I need you with me. Because I\u2014\"\n\n\"Shhhh,\" I said, placing my index finger to his lips, stopping him from completing his thought. \"Don't say those three words now, please, because some part of me will never believe you, no matter how sincere. If you need me and can see me in your future, allow me to figure out what I can handle.\"\n\n\"It's true, you know,\" he said.\n\nThe air between us was noticeably strained on the ride home. I reasoned aloud that it might be better if I slept alone, just for tonight. When we began dating exclusively, Marcus and I exchanged keys to each other's apartments and had often alternated nights spent at each other's place, depending on who was the first to arrive home. I could count on one hand the number of times we had gone to bed unhappy with each other. This, however, was the first time that we may find ourselves in the same city and not sleeping together, so I resigned myself to just letting the moment breathe. Additional words could only complicate things, and the ice between us was thin enough. Inside the elevator I pressed the buttons for both floors and told him that I was tired. When we reached the seventh floor, he exited uncontested.\n\nThe darkness within my apartment felt heavy, like an old acquaintance I knew all too well and was not excited to come in contact with again. I sat by the windowsill with only the illumination from the streetlights and taxis below. Did I overreact? Was I too calm? Damn Cora and her meddling. I was not ready to have this conversation. Better yet, I wasn't ready to hear what I already knew in my heart to be true. One of the fundamental things that made Marcus and me great is the clarity I have in seeing him as a man and not the storied knight that had wrecked me previously. Despite what Blake imagined seeing when he was her Mr. Potentially More, nothing about the Marcus I knew said he was ready for marriage just yet. Hell, his dating a nonmodel civilian monogamously still took conscious effort, for which I respected him. Marriage, while a future possibility, was for now a far different story, hence me trying my best to not entertain anything other than what we have presently, but that was before tonight.\n\nThe past few hours made his position unquestionably clear, no longer a speculation, and forced me to confront the one hurdle I still had yet to conquer since Tony: trusting what I could not see. In order to be fully with Marcus, I had to trust him entirely with my heart and now my future, with no guarantee of the implied protection those vows before God and all of our friends would one day provide. Could I do that now, after my experiences in love? Could I guarantee myself that, whatever my relationship with love was in the past, I would be smarter, more realistic now? Normally, I would have called Cora to discuss my heart, but seeing as how nothing pleasant would roll off my tongue, I opted for a long, hot shower instead.\n\nTurning the faucet to the hottest setting my body could tolerate, I sat beneath the showerhead until a dense steam enveloped me. The beads of water falling invisibly through the mist hit my flesh hard and strong, and my body succumbed to its force, crumbling from the aggression trapped inside. This was not how tonight was supposed to go. We were to fall into bed intoxicated by the cuisine of Eric Ripert and tell some racy jokes before exploring our most intimate treasures, my legs spreading, allowing him to go deeper as I got wetter. Having waited nearly three months, I was overjoyed to discover that Marcus was a skilled and unselfish lover from the start of foreplay to the climax of orgasm and well into the afterglow. Sadly, the same can't be said for his domination of the covers whenever the time to sleep came, but that too he made up for in the middle of the night, awaking as he often did to find me only partially covered. In my mind's eye he stared at me intently, as one often does when debating the merits of a big purchase or tattoo, before tracing the contours of my face or shoulders with his hands, kissing me softly and then covering me. I know this to be true because sometimes I was awake only seconds before looking at him through the cast of the moonlight. Seeing him stir, I would rush to close my eyes, pretending to be sound asleep. Given the choice tonight, I would surely offer him every cover on the bed not to have this wall between us.\n\nThe heat from the water intensified my longing, making me want him more. After a few moments I stood to locate the shampoo, only to have my heart stop ever so briefly, at the sight of Marcus, \"Is there room for me in there?\" standing in the open door. Watching him undress my heart returned.\n\n\"Always,\" I said, inviting him in yet frozen in place.\n\n\"Here, let me do that,\" he said, reaching for the shampoo in my left hand.\n\nFor a few moments we stood face-to-face, both holding tight to the bottle before letting go and allowing our hands to express what words could not. My body pulsed under his touch, begging for more. His lips hungrily created a trail from the nape of my neck to the base of my stomach before traveling back up and taking my mouth. With one arm Marcus lifted me waist-high against the shower wall, holding my face with the other, and entered the deepest part of me that ached to receive him. Instinctively he thrust hard and forcefully, taking possession of my very soul, until I could no longer restrain myself from crying out his name, begging for more.\n\nTo me, morning was an intruder, but it was of no consequence to Marcus as he continued to sleep peacefully next to me, the bedlinens masking part of his body and my limbs covering the rest. Somewhere around twilight, the conflicts of my head and heart had returned. For what seemed like hours I gazed upon him longingly, in turmoil. My mind struggled to answer if he was worth the purchase of my heart for a season with no promise of a lifetime. My heart screamed loudly yes, but it had been wrong, very wrong, before, so I waited and looked admiringly at his high cheekbones and the way they converged into a well-proportioned nose; the fully defined lips that always seemed to be smiling, even in slumber.\n\n\"How much longer are you going to look at me before having your way with me?\" asked Marcus, his eyes still closed.\n\n\"Who says I'm looking at you?\"\n\n\"You are. I told you before I can feel you,\" he said.\n\n\"Well, then, feel this,\" I said, playfully taking his nipple between my teeth, eliciting a slow moan of satisfaction.\n\n\"Oooh, do that again,\" he pleaded.\n\n\"For the record, you said, 'I always remember you.' \"\n\n\"So I did,\" said Marcus, now fully awake and flipping me over so that he was on top.\n\n\"Hmmm, I love that,\" I said.\n\n\"What do you love?\" he whispered, lowering himself slowly onto me.\n\n\"Feeling the weight of you on top of me, holding me, inside of me. I never want to be without you.\"\n\n\"Good, 'cause I'm going to ask you something. Promise you'll say yes?\"\n\n\"Kiss me. Then ask me,\" I said.\n\n\"Do you love me?\"\n\n\"Yes, I love you, Marcus,\" I said, without hesitation.\n\n\"Good, 'cause I could never stand being in this lifetime\u2014in love with you as I am\u2014without you.\"\n\n\u2022 \u2022 \u2022\n\n\"Julesea, I don't like this one bit,\" said Cora. \"You just got to New York, now you're supposed to pick up and move to L.A. with Marcus without so much as an engagement. That's not how I raised you.\"\n\n\"Stop it, Mommy! I've been here nearly two years. You're overreacting. Besides, we're not moving there. Marcus has been back and forth a few times this year and may have to be there a few months or more. You know I've been out before for a weekend or two seeing him. Now I will come out for a week or so at a time, when I can. That's all.\"\n\n\"That's all? It's that simple, is it? What about your job?\"\n\n\"This is why I didn't tell you earlier. I knew you'd overreact.\"\n\n\"At least someone is. What about your job, Julesea?\"\n\n\"I can't win with you, Mommy\u2014one day it is family first, then it's career,\" I pleaded. \"I can do both. When the time comes, Marcus is going to sort it out with Michael. It's not like I leave tomorrow and am going to stay forever. I thought you liked Marcus.\"\n\n\"Liking Marcus is besides the point, child. Listen to me\u2014I did not raise you to have some man fight your battles. Did you learn nothing from the other one?\"\n\nKnowing better than to debate the merits and\/or shortcomings of Tony, I said, \"Daddy fights your battles.\"\n\n\"Daddy is also my husband, so he has earned the right. You understand the difference?\" asked Cora.\n\n\"This feels right, so you have to support me, okay? I don't want to lose him, Mommy.\"\n\n\"You can never keep anything worth having with fear, Julesea. Release that and make your decisions from a place of strength, otherwise you will be looking over your shoulder all the time making bad decisions.\"\n\n\"That's not what I'm doing. Do I want to get married? Yes. Trust me, if I start to feel that Marcus has no plans to marry me, I'll let go. Please try to understand. I didn't plan for it to be like this, and yet it is. I love him, Mommy, so I want to support him. And before you go there, Marcus is not Tony. He doesn't keep secrets from me.\"\n\n\"Every man keeps secrets, Julesea. Even your father,\" said Cora, still bursting with disapproval.\n\n\"You know what I mean,\" I said.\n\n\"And you heard what I said. I don't like this one bit.\" Then Cora's tone came softer, more hollow. \"Promise me this, you won't walk away from your career ever. A woman must have something of her own to ride out the tides of life.\" For the first and maybe the only time, Cora lowered the veil between faithful supporting wife and the desires of a woman with dreams deferred, giving me a glimpse into her personal regret for the road not taken. If I were brave enough, I would have dug deeper to find out why, at this particular time, the matter of self-reliance is of the utmost importance to her. If I didn't still need the illusions of childhood that had cast my parents as perfectly unblemished superhumans, I would have asked. \"The day may come, and I pray that it does not but it may, that you awake and don't recognize the man you are with. What will sustain you, Julesea? You can't run off across the world forever. You'll have to make a home sometime.\"\n\n\"I hear you. I do,\" I said. \"Let's not argue about this. I leave for L.A. tomorrow.\"\n\n## 39\n\n* * *\n\n## 1 NORTH WETHERLY DRIVE\n\nBABE, WHY IS there an oil field in the middle of L.A.?\" I asked Marcus on the ride up La Cienega from the airport. Unlike my last trip in, when he sent a car to fetch me, this time he met me at the gate looking quite SoCal, if I must say\u2014and I do. His requisite workday designer suit had been swapped for a dark blue button-down dress shirt with sleeves pushed up, dark denims, crisp white Pumas, and gold-frame aviators\u2014all of which seemed ideal complements to the new Aston Martin convertible that transported him there.\n\n\"It's always been there, baby. Maybe you didn't notice it before.\"\n\n\"Still doesn't tell me why it's there. Seems quite odd,\" I said.\n\n\"You're odd,\" said Marcus, affectionately squeezing the tip of my nose, \"and hopelessly beautiful. You hungry?\"\n\n\"Famished, actually. You know I never eat on domestic flights. Something about the whole process just seems unsanitary.\"\n\nHe laughed aloud. \"See? Odd! Can you hold tight for about an hour? Steven has two properties to show us. He says they're close to each other, so it shouldn't take long. Afterward I'll take you to this great restaurant off PCH, Geoffrey's I think it's called. The views are amazing. You're literally sitting right on the ocean.\"\n\n\"As long as I can grab a juice or something before, that's fine.\"\n\n\"I know the perfect place. Have you had a Jamba Juice before? Oh, Jules, it's good. Fresh fruit, sorbet, and you can get protein powder, immunity, energy . . . you'll love it. There is one further up, across from the Beverly Center Mall. We'll stop there.\"\n\n\"Sounds good,\" I said, tilting my head back to enjoy the warmth of the dry sun beaming down. Though my flight left JFK at nine forty-five this morning, New York had already felt like a sauna from the end-of-summer humidity. \"So what exactly are you looking for in a property?\"\n\n\"Correction, what are we looking for,\" said Marcus. \"This will be as much your place as mine, so I want to make sure you're happy, okay?\"\n\n\"Our place, I like the sound of that,\" I said with a triumphant smile. \"Nice. I guess that doesn't extend to cars, huh? An Aston Martin, really? I mean, who does that?\"\n\n\"Hey, I have to represent.\"\n\n\"That's the excuse you're using?\"\n\n\"Oh yeah,\" he said, looking straight ahead at the length of traffic coming into view as we descended the final hill into the basin of L.A. proper.\n\n\"And this has absolutely nothing to do with you living out some childhood fantasy you had at fifteen years old about one day moving to L.A., driving a convertible charcoal-gray DB9 with a tobacco interior down Pacific Coast Highway?\"\n\n\"Honestly, I must stop telling you things,\" said Marcus.\n\n\"Never,\" I said.\n\nThe first property we saw was in a high-rise near UCLA in an area called Wilshire Corridor, enveloping both sides of a quarter-mile strip that felt more like Michigan Avenue in Chicago. The unit was spacious but lacked character. It was minimal in construction with expansive windows, as requested, but clearly factory-grade. We knew instantly that it was not a contender. The second unit located on the opposite side of the same street and closer to the Beverly Hilton hotel was older, with some personality, but quite small, prompting Marcus to ask if one of the bordering units was available so that he could combine them. Neither property was, so he requested to see something closer to Beverly Hills, preferably in a small-to-medium-size building that had substantial outdoor living space and a walkable neighborhood, unlike the busy Wilshire Boulevard that defined the Corridor.\n\nHaving spent his entire professional adult life with New York as home base, Marcus welcomed the time in L.A. but needed to replicate the Big Apple's conveniences as much as possible to alleviate the unsettling comparisons that plague so many East Coast transplants. Steven had a property in mind but had to check with his office to make certain that it was still on the market: a new limestone construction with about thirty units, located on Wetherly Drive near the Four Seasons Hotel, which Marcus had been calling home for the past few weeks. Without seeing the unit, I was immediately pleased with its location because the adjacent street was Doheny Drive. For some reason\u2014I think it was the palm trees bordering both sides\u2014the skyline was always perfect, powder blue, encapsulating my entire vision of what L.A. is supposed to be on any given day.\n\nIn order to reach Geoffrey's we had to take the Pacific Coast Highway into Malibu. The view was breathtakingly beautiful, with the pristine Pacific Ocean to the left and rough mountains to the right. Along the way Marcus maneuvered through surfers in wetsuits, darting without regard across the road, and caravans of weekend leather-clad motorbike clubs and uniformed cycling groups. After descending a major bluff past Pepperdine University, we reached Geoffrey's, sitting at the base of the mountain etched into the cliffside. There was not a bad view in the entire place, whether overlooking the water with various boats and surfers or to my right, where Marcus sat. On the drive back, Steven called to say the unit he had in mind to show us was no longer available but the penthouse was. The price was more than Marcus had wanted to spend, but we agreed to see it nonetheless.\n\n\"It's not a question of whether I can afford it, Jules. It's a matter of whether I want to pay that much for a property that is at best a transition home,\" he said as we pulled up to the building. From the outside it was pleasing, understated yet elegant with floor-to-ceiling windows and architectural balconies made of steel treated for an aged bronze or rust effect. I noticed that there were two grocery stores nearby, in addition to some galleries\u2014a promising start. When we entered, however, I felt the building's austere elegance, like one of those beyond-perfect living rooms I was never allowed to sit in as a child, or the good china that only leaves Cora's cabinet for Christmas, one of Daddy's business dinners, or to impress the girls from the club.\n\nInside the lobby was a security desk, positioned between two waterfalls. Every guest was required to sign in. Once inside, a key card was required to access the main floors. The two penthouse units, however, were only accessible via a personal code that was programmed specifically to the unit and then the key card. For Marcus the whole process was charming, as it became another gadget to play with. To me it was a bit pretentious, but I dared not say anything as I could tell instantly that he was smitten.\n\nOn level PH, the sliding doors opened to reveal a single corridor with bare walls, a convex skylight running the length of the hall, and two double-paneled doors to each side. The unit on the right had sold, leaving us the north-opening door, which was a good omen, or so I had been told once before\u2014although east is best. The entrance hall featured an abundance of wall space, perfect for his collection of contemporary art that always seemed ill-placed in the Manhattan home and overflowed the Bridgehampton one. After a few steps we were in the main room, with an open floor plan, chef's kitchen, and an entire rear wall of windows providing an unobstructed view of the hills from Hollywood to Beverly Hills. I won't lie\u2014that view alone hypnotized me before I was wowed by the bedrooms or Bali-inspired rooftop deck that covered nearly half of the 2,000-square-foot property. My only concern was how long it would take to make the place feel like our first home and not a faux interpretation created by an interior designer, especially with most of my time being spent in New York.\n\n\u2022 \u2022 \u2022\n\nLater that night, seated at the hotel bar, we talked about the unit. For his part Marcus said he didn't need to see anything else; he liked it. I confessed that something about it felt forced, like all the bad superficial stuff that I had ever read or heard about L.A.\n\n\"What if I give you the majority of the closet space?\" retorted Marcus with big, pleading eyes.\n\n\"Well, that's generous of you, considering the closet is larger than most people's bedrooms, so that is completely unnecessary, but thank you,\" I said. \"Listen, it's clear you really like this place. I don't hate it. Promise. It's just not what I envisioned in the whole sunny California dreamscape, is all.\"\n\n\"I know you want to be in the hills . . .\"\n\nCorrecting him, I said, \"Not necessarily the hills. Hancock Park is fabulous, all those big historic homes and picturesque lawns. Did you know that at Halloween they block off some of the streets, open all of the homes, and thousands of kids from everywhere come through to trick-or-treat? How cool is that?\"\n\n\"When we have kids, I promise. As for now, I am not one for mowing lawns, taking out the trash, and such nonsense.\"\n\n\"I know, which is why they have people for that now, but that is besides the point. You\"\u2014 I started, before being overwhelmed by his desire for me to love the Wetherly penthouse just as much\u2014\"we love it, so let's do it.\"\n\n\"Ahhh, that's my girl. You'll love it, you'll see,\" said Marcus, leaning over to nuzzle my neck. The smell of scotch emanating from his breath delighted my nose. \"Ms. Sinclair, can I take you upstairs?\"\n\n\"By all means. I thought you'd never ask.\"\n\n\u2022 \u2022 \u2022\n\nThe plushness of the Beverly Hills Hotel bed did nothing to help me fall asleep. Eventually my tossing and turning disturbed Marcus, forcing him awake.\n\n\"Do I need to ask or will you fall asleep soon?\" he said.\n\n\"Don't ask,\" I said, sleep-deprived and frustrated.\n\n\"Cora?\"\n\n\"How did you know?\" I asked.\n\n\"Isn't she always the voice in your head\u2014besides, your dad called.\"\n\nI sat up in bed and reached to turn on the bedside lamp. \"Really, when? I didn't know about that.\"\n\n\"Don't think you were supposed to. Man stuff. He called after you and Cora spoke about this\u2014coming to L.A., what it all means. He was worried.\"\n\n\"I told her not to be. I know what I'm doing.\"\n\n\"Do you?\" asked Marcus. \"I promised your dad I would never intentionally hurt you, Jules.\"\n\n\"Really, what else did you promise him? Because I know that could not have been enough for Daddy if he actually picked up the phone. And it wasn't about f\u00fatbol, dominoes, or fishing,\" I said.\n\n\"If you must know, I told him that I would never dishonor you or his family. I said that we are building something solid here,\" said Marcus.\n\n\"Well, that sounds wonderfully promising!\"\n\nLying next to me, Marcus continued: \"In order not to do that, I must tell you something, and I'm not sure of how you will react.\"\n\n\"It can't be any worse than telling me you're not ready for marriage, so deal with it,\" I said, my tone laced with the bitter taste of the pill I was having difficulty swallowing. \"Obviously still adjusting.\"\n\n\"I'm not quite sure,\" said Marcus. \"Jules, you have this great personal inclination for the everyman that supersedes everything you do, including work. It's one of the reasons we fit so well. Together we balance the scales in some grand way. If given the choice to eliminate an opponent and see them go without, you would opt to find a compromise and spare them grief. I'm not like that, never have been. My inclination is always to identify a goal, win, and whenever possible, be the one to sink the shot. Whatever casualty is left behind is just a matter of bad planning or, worse, poor execution by the other guy\u2014it's intrinsic to who I am.\"\n\n\"Jesus Christ, are you trying to make me nervous? 'Cause I'm getting there real quick.\"\n\n\"Jules, what do you think I'm doing here in L.A.?\" asked Marcus, now sitting on the edge of the bed with his back to me.\n\n\"Exploring the film business to invest in movies and stuff so I have something to watch, right?\"\n\n\"Something like that, baby,\" he said, inhaling deeply to decipher whether he truly was ready to part with the information being held so closely. Rising from the bed to put on his robe, Marcus walked over to the desk for perspective and context. \"Yeah, I'm here to invest in the film business, but not as a money guy. I'm taking over a studio, with a consortium, and I will run it.\"\n\n\"Okay, that doesn't sound too bad,\" I said with a slight smile, sensing there was more.\n\n\"In order to do that, I need some major capital support, the likes of which I have been brokering for years now. One of those is Nikolai Abramovich.\"\n\n\"Michael's Nikolai?\" I asked. \"Michael never mentioned anything to me, not that he should have.\"\n\n\"Not exactly Michael's Nikolai, baby,\" said Marcus, partially turning around to address me squarely, \"and this is where I need you to understand.\"\n\n\"Nikolai is a contact I have had since my days at Goldman. Unbeknownst to Michael, I put him on his radar in order to gain control of Carly's . . .\"\n\n\"Wait, I'm not sure I'm following you. Why would you do that? From what I know of Nikolai, Carly's doesn't fit his portfolio.\"\n\n\"It's a bit complicated, but the most basic version is this. The primary reason I got involved with Carly's was not entirely to help Michael. It was to help Nikolai. At the time he was my top client at Goldman and needed coverage for some holdings, so I used Carly's as the shell for those acquisitions\u2014acquisitions he could not directly access for reasons not important to this conversation. As a result Michael has\u2014correction\u2014the whole Carly's investment group now has a vote in the future of those properties. Which never should have been a problem, since I control the largest single stake. Michael, however, goes and marries Carly, which I didn't see as a problem until a year or so ago when it came time to transfer them over. All of a sudden Michael starts speaking about expanding. Together their shares outnumber mine, and Mike has gotten greedy, trying to lay claim to what was never his.\"\n\n\"So Michael knows about this property?\"\n\n\"Yes, of course, sort of. He knows it's earmarked for me, and in return the partners of Carly's have been paid handsomely to leave it be, if you will. Using it, building on it, was never an option. Some deregulatory events have occurred, making it possible for Nikolai to knowingly retain these territories. He's grown impatient and wants the holdings before activating his resources to move forward with the studio.\"\n\n\"But that makes no sense. In comparison, I would think a studio far more valuable than Carly's or some land it could sit on. I mean, it's just a restaurant in Manhattan sitting on less than half a block.\"\n\n\"On the surface it's just a restaurant in New York, but the portfolio, the sum of its parts, Jules, is quite valuable. That's the dispute. Over the years we purchased a sizable amount of real estate in Chicago, London, Brazil, Nigeria, and parts of the Baltic, all of which I oversaw. Those lands are worth a fortune, or should I say what's underneath them. Now Michael wants to feign amnesia and build, laying claim to something that was never rightfully his. To do so undermines the value, prohibiting Nikolai from getting the reserves on each. Hell, it's more valuable to leave as open lots, forests, and dry land.\"\n\n\"But what I don't understand is, why would Michael do business with Nikolai?\"\n\n\"Because he doesn't know about our relationship and can never find out, at least until everything is done,\" said Marcus.\n\n\"So then, you are not trying to kill Michael's dreams?\" I asked, thinking about that first revealing discourse with Michael in my office.\n\n\"Truth be told, I couldn't care less, as long as he's not expanding into one of those locations. Ridiculous, this all is. I've done what I could to force him and the board to do their own research in order to see that we would only lose money in expanding over some supper club that would never return even a quarter of the selling or licensing costs, for that matter. Initially, Carly, like Simon, vowed to vote her proxy with me. Unfortunately, that changed after Michael leaned on his position as her husband. Which is why I went there for dinner last year and, in a moment of admittedly poor judgment, allowed it to be known that you and I had dinner days before. It was not my intent, but Mike was being Mike, pushing my buttons, and I got careless.\n\n\"Anyway, with pressure from Nikolai mounting, I was left with no other recourse than to provide him entry and sway Michael. Even with Carly's support, Mike can't hold out forever. He needs the protection, he thinks, Nikolai will provide him to fulfill his dream; buying me out, take over Carly's and expand.\"\n\n\"So it was you who led Nikolai to me, so to speak,\" I said.\n\n\"Yes, but that was a matter of circumstance, long before you and I were you and I. Planting seeds, you see.\"\n\n\"Got it. So these shares, I presume they are unrestricted, because Michael has no obvious reason to suspect Nikolai of anything other than being some wealthy man who is quite generous with his money and whom Michael has run a good pitch game on, I take it.\"\n\n\"Yes, something like that,\" said Marcus. \"And potentially these shares could be sold or voted anytime against Michael, making him lose everything,\" I said. \"So, are you going to push him out or just sit there and watch as Nikolai does?\" I asked.\n\n\"I haven't figured that out yet. I know what I would like to have happen,\" Marcus said matter-of-factly.\n\n\"Listen, I know you two aren't on the best of terms right now, but he is your friend. Is it really necessary? There must be another way to get what you want, give Nikolai what he is due, and leave Michael with his dream.\"\n\n\"I wish, baby, but it's not that simple. Mike won't budge, and I need the alliance that Nikolai's multimedia and communication businesses will give me in Hollywood.\" Marcus paused to take my temperature on this line of conversation, pouring a glass of Pellegrino. \"Jules, taking over a studio is a costly enterprise, and it can't be done with haste. The shares must be purchased over time. Alliances within that machine must be built with everyone from agents to producers and countless executives. In order to do that and not lose everything I have worked for, I need resources and even more extensive relationships or else these past years have been for naught.\"\n\nMy mind was still racing to properly assemble the complex puzzle unfolding before me, even more so to grasp this unscrupulously methodical and unflinching side of Marcus. \"And for this dream you would betray Michael, just to get what you want?\"\n\n\"Is that any different than what you did to Blake?\" asked Marcus, with discernible agitation in his tone. \"Did you not betray your precious Friend Code or whatever it is to get what you wanted?\"\n\n\"That's different,\" I said, stung by his words. \"How could you compare the two?\" Since that revealing night at Carly's, Marcus had never brought up my handling of the whole affair in a manner that placed blame or criticism at my doorstep. Quite the opposite, he provided a shoulder for me to cry on immediately after Blake unleashed her verbal rant on me and when I received her subsequent e-mail from abroad telling me that payback would be hell\u2014he just listened and said everything was all right.\n\n\"Listen, Jules, we're getting off track. I didn't say that to hurt you,\" said Marcus, placing his water on the bureau to embrace me around the waist. \"I am telling you all this because I don't want to lie to you. But you can't push back like this. I need to know that I can trust you.\"\n\n\"You'll destroy him. Do you understand that, Marcus? Do you even care? Michael has his faults, yes, but he has been good to you. Please, find a way to leave him with something. His life is that place.\"\n\n\"It matters that much to you?\" asked Marcus, now stepping back to survey my face. \"Why do you care so much?\"\n\n\"Because he's been good to me,\" I said.\n\n\"Then I will try,\" said Marcus.\n\n\"Don't try, do,\" I said. \"Now let's go back to bed, please. I want to dream this away.\"\n\nThe lights were off now, but my eyes were wide open, surveying the ceiling. It's not that I expect Marcus to be perfect, but this whole situation makes me wonder about what is at the root of this man. Had I explained away too much of his alpha drive, seeing only what I wanted to see? He must have sensed my restlessness, because he shared one last thought before we both said good night to this conversation. \"Jules, try not to judge me harshly as your inclination is to favor Mike in all of this. I know you don't think that given the chance Mike would do the same thing or worse to me in order to preserve his dreams, but he would. I know him far better than you do. I know what he is capable of. After all, he taught me how to play this game and to survive.\"\n\n\"I just don't want to be caught in the middle,\" I said, not turning to look in his direction.\n\n\"But you are, Jules. You have been since the moment we decided to make a go of things,\" he said, grabbing hold of my free hand that lay immobile between us.\n\n\"So I go back to NYC, work alongside him knowing that forces are conspiring behind his back, and pretend as if I know nothing?\"\n\n\"Yes, until I figure out how to leave him with something, as you asked.\"\n\nOf all the things that Marcus could have told me, of all the things I might have been better prepared to hear, this was not it. My last thought before falling asleep, and for days thereafter, was how I could have responded differently, if not supportively, then more neutrally, so as not to elicit the looks of doubt I saw every once and again in his eyes when addressing me during the remainder of my trip that week. At the very least, the gross miscalculations I made would never have provided the foundation for our blowup. \"Marcus, is there anything else I should know?\"\n\n\"No, my sweet.\"\n\n\u2022 \u2022 \u2022\n\n\"Darling, why do you always call me from the plane just as you are about to take off? You know I hate being rushed,\" complained Richard. \"I know what Marcus told you is troubling\u2014makes no never mind for a multitude. Just keep your head down and focus on doing only the job that Michael pays you for and protecting the relationship you are in with Mr. Crawford.\"\n\n\"Easier said than done, Boulton. I just wish I knew why this bothers me so. It's like this reveal has painted everything between us that came before in a totally different light. Before, when I looked at him, I was clear, good or bad, modelizer or not, about who I saw. Now, I don't know. You know?\"\n\n\"Do you want to hear the answer or shall we continue into round two of 'This Dysfunctional Life with Jules Sinclair'?\"\n\n\"You know I am always up for a good game of denial,\" I said in jest.\n\n\"Well, then, you are conditioned for self-sabotage, my love. You and every other girl like you who grew up watching Sabrina or Pretty Woman, swearing by a standing brunch appointment after your Saturday-morning mani-pedi. You want the fabled Linus Fairchild and Edward Lewis types for what they represent on a one-way screen, so you romanticize an ideal that is never possible, choosing instead to fast-forward through the uncomfortable bits. How much time do you actually spend in reality? I'll tell you. Zero. So when the complexities and motivations of a three-dimensional relationship with real emotions, messy motivations, and flawed human matter present themselves, you are unable to cope, leaving you no other recourse but to self-destruct.\"\n\n\"Well, that's grim,\" I said, casting my eyes downward, away from the flight attendant signaling me to turn off my phone now that the doors were closed and the plane was ready for takeoff.\n\n\"Honey, I'm a wise old gal. Shoot, I'm somebody's papa now, so I know. Step over and around, don't jump in. You've done that, it's lonely and unfulfilling. Make you a\u2014\"\n\n\"Bugger, I really have to go now. The flight nazi is giving me the evil eye.\"\n\n\"Of course you do,\" said Richard theatrically.\n\n\"Don't be like that. I have to go. Bye-bye, kiss the baby for me, bye,\" I said, hastily powering down my phone before the approaching attendant made an example of me.\n\n## 40\n\n* * *\n\n## DOMINOES\n\nTHERE IS SOMETHING frightening about knowing too much. Independent of how much time passes before the proverbial hammer drops, you are always on edge, testing out different physical reactions and tonal inflections to convey that you are just as shocked by the event(s) as the victim.\n\nI have been back (and forth) to New York for nearly four months now since the conversation with Marcus (and the one with Richard). In that time I've quietly observed the infiltration of Nikolai in and around the office, in the restaurant operations, around town, on the phone, and in e-mails setting up (oops, I mean cozying up to) Michael. I have seen Michael go from states of sheer elation with his chest poked out to pensive and measured in his movements and confidences, boxing mostly everyone out if their last name is not Abramovich or first name Raymond. And despite my best efforts to heed Richard's advice\/warning to remain Switzerland, I have found myself making frequent overtures to ensure all that is our dynamic extra pleasant and comfortable while being careful not to ask for details. However, I can't shake the feeling that I am also making myself a sacrificial lamb. For purely different reasons, I care about the well-being of both Marcus and Michael, and in my own way I have tried to split myself as best I can to appease both\u2014or at least give the appearance of it\u2014but coming up horribly short, I'm afraid.\n\nCarly's press had long been a well-oiled machine, not requiring the constant vigil it had before. Knowing this, and being preoccupied with other priorities, Michael was amenable to my taking a few days off twice a month in order to be in L.A.\n\n\"As long as your phone works and you're back Wednesday morning, do your thing\" had been his parting words.\n\nSunday mornings through Tuesday evenings I spent in L.A. with Marcus when he was not able to be in New York, but it was shaping up that the quantity of my physical presence substantially overshadowed the quality of our time together. Like clockwork, I would land, taxi on the tarmac, and turn my phone on. But instead of a sweet note from him to greet me, there was always a While You Are Here Schedule to review of luncheons, dinners, pool parties, and screenings requiring me at his side. Knowing full well why such attendance was necessary, I seldom questioned or complained. It seemed the least I could do given that my initial response was far from what he expected. I will, however, confess this\u2014my civility with some of the short, socially awkward, bad shirt\u2013wearing, grossly unattractive chubbies sitting on large wallets and their overly silicone-enhanced, botox-injected, bleached female company did wane from time to time, but not enough\u2014I thought\u2014to come off as rude or negatively impact Marcus. In the few moments daily that we found ourselves alone\u2014driving to something, coming from something, before bed or morning coffee\u2014we seemed to be extra polite. I can't dare speak for him, but I can say that for my part, I was too aware of my limitations to encourage a conversation. Everything was moving so fast, and I didn't want to know anything further than I did, so I began framing the little dialogue we did have. Soon there was no denying that the comfortable ease that resides in the silence of a naturally compatible relationship had become strained. Anytime I felt him preparing to share whatever was on his mind regarding Michael, Carly's, or Nikolai, I would speak of everything else\u2014my friends, their lives, their jobs, their opinions, the decorator we had hired since I conceded the reins of outfitting the L.A. condo after the first month. Eventually he stopped offering and, instead of shining a spotlight on the elephant in the room, he would skirt the matter.\n\n\"I miss being on the couch with you reading the paper, Jules.\"\n\n\"Me too.\"\n\n\u2022 \u2022 \u2022\n\nWednesday morning through Saturday evening, I was in New York, living out of my office. For whatever strides the adult me had made, my apartment (or should I say my insane closet with a kitchen) spoke to disorderly collegiate days past and to my current emotional state in trying to appease two masters. Seeing how chaotic the first mornings back in New York were for me, Jacklyn got into the habit of going to my place and gathering a few items to store inside my office. This enabled me to go there directly from JFK and get a solid two hours or more of a disco nap before barreling into my workday. Her reward for going above and beyond the call of duty was various appreciation gifts from around town: Bliss Spa certificates, Bergdorf's or Saks's gift cards, a Crunch Fitness membership, a dinner or two on my expense account. From her twenty-three-year-old point of view, she was happy to do so, as I was living exactly the life she fantasized\u2014the very one that Richard had criticized on the plane. I was Audrey Hepburn in Sabrina and Julia Roberts in Pretty Woman sans the whole prostitute \"safety pin to hold up the boot\" thing. For her, everything was simple and free of baggage. I envied Jacklyn for that. How could it be possible that a mere seven-year age difference could render such vastly different perspectives? I bet, given the chance, Jacklyn would be clear to step over and around and leap!\n\n\"Hey, Jules. Welcome back. How was your Labor Day in L.A.? I bet fabulous. Did you see Brad Pitt? I love him,\" said Jacklyn as I emerged from my office after one such rest.\n\n\"No, but I did see John Taylor, still hot,\" I said.\n\n\"Who?\"\n\n\"You can't be serious. John Taylor from Duran Duran . . . 'Hungry Like the Wolf,' 'Rio' . . . forget it,\" I said. Clearly, seven years accounts for more than I thought.\n\n\"Um, okay, Michael stopped by earlier and wants to see you. I knew you were still asleep, so I told him you were stuck in traffic on your way to the office.\"\n\n\"Rock star, Jacklyn, rock star! Did he say what he wanted?\"\n\n\"Nope, only to come see him when you get in.\"\n\n\"In that case,\" I said, reaching out for my double espresso that was waiting at the ready, \"I had better go see what the boss wants.\"\n\nMy watch showed the time to be 9:45, making the traffic alibi completely possible. Simone was away from her desk when I arrived, so I proceeded to Michael's door, knocking before letting myself in, and found him at his desk in the midst of a call. From the sound of things, he was either bored yet obligated to endure the caller or it was Carly\u2014like I said, bored yet obligated.\n\n\"Jet-setter, how's L.A.?\" asked Michael. \"Did you see anyone famous?\"\n\n\"What's the obsession with that? Jacklyn just asked me the same thing,\" I said.\n\n\"Well, what else is one supposed to do out there? It's a one-industry town.\"\n\n\"That is true. If people aren't looking for stars, they are trying to be one,\" I offered.\n\n\"Yeah, muthafuckers see them lights of Hollywood and get intoxicated. Speaking of getting caught up, how's our boy doing out there?\"\n\n\"He's good, adjusting but overall enjoying himself,\" I said, feeling the hairs on the back of my neck stand up.\n\n\"Good, good. Glad to hear it,\" said Michael, not taking his eyes off me. \"That Hollywood game is a different animal, I'll have you know. Still don't understand why he wants to explore the VC investment side of things out there.\"\n\n\"VC? Michael, you know they call themselves investment bankers now,\" I said, trying to mask my discomfort with the direction of the conversation. \" 'Venture capitalist' is taboo.\"\n\n\"So it seems. Everybody's trying to reinvent themselves these days, going from finance to studios. You know what I mean.\" Definitely a statement to be avoided and not a question.\n\n\"I guess. Are you?\" I countered, attempting to redirect the conversation away from Marcus.\n\n\"Me? Just trying to survive every day holding the vultures at bay and keeping my little part of the world running smoothly,\" he said.\n\n\"Effortless,\" I said, pretending to be oblivious to the targeted statements Michael had made. \"So what's on your mind, chief?\"\n\n\"A few things, actually. I'm doing London. Nikolai and I have been speaking about it for some time now. Been putting some things into play.\"\n\n\"Nikolai wants to open a location in London?\" I say skeptically. \"I thought he was a silent partner like Marcus and Simon.\"\n\n\"Like I said before, everybody with a dollar has a voice. I'm just glad this one is in favor of supporting me and not tearing me down. What do you think? You lived there for a while, worked at Conrad before I brought you in here.\"\n\n\"London's great, very metropolitan, as you know. Has a huge nightlife scene with theater, music, private clubs, et cetera. It's a financial capital with a diverse group of people, many of high net worth, desperately in need of marquee restaurants with good food. Nothing like Carly's is there.\"\n\n\"Exactly what I was thinking. Any drawbacks you can think of?\"\n\n\"Off the top I would say the temperament. If you're going to open there, design it with the spirit of a Londoner, that consumer, in mind. For example, they have a much stronger appreciation for vintage soul music than we do, so an adjustment to what talent we offer would need to happen. Don't just go and plunk a U.S. rendition there\u2014it won't work, and the Brits will tell you quickly.\"\n\n\"Good point, good point. I gotcha. What else?\"\n\n\"Location, something central and resurging, like here in the Meatpacking District.\"\n\n\"Okay, possibly Shoreditch, so let's say I did all of that, do you think it would work?\"\n\n\"I have no doubt,\" I said.\n\n\"Great, so if I were to tell you that I am looking to break ground in a matter of weeks, September seventeenth, to be exact, on a location, what would you say?\" asked Michael.\n\n\"I would first say that you are insane. Second, I would say there is the matter of local government support, permits, and union contractors to contend with, and a whole manner of other minute details that are equally as involved, requiring more than groundbreaking, especially the Michael Thurmond Kipps kind, with cameras and news outlets present. Last and most important, the time commitment. How are you going to split the time between London and here? Specifically, being away from Kaylin for such a long stretch of time? This is not something I could imagine you doing from afar.\"\n\n\"As to the first, maybe. To the second, all of that is long under way. Nikolai has some resources. Now, as to the last, well, that's where you come in. I want you to run point on London when I'm tied up here,\" said Michael.\n\n\"What?! I mean, why?\" I say, torn between being wildly elated at the opportunity and panicked about the impact it would have on my life.\n\n\"You know the terrain, Jules, and you know what appeals to me. I am not saying for you to be there full-time but for a few days every other week or so, to get a firsthand look at how everything is coming along. Some of the visits we'll do together. Think you can handle it?\"\n\n\"I already said you were insane, right? So I shouldn't repeat myself. Michael, are you absolutely certain about this?\" I asked at the risk of exposing that I knew far more than I let on. \"London is a very expensive city, and to do anything close to what you have created here is\u2014wow, I can't even imagine how expensive it would be, the type of financing needed to pull it off.\"\n\n\"You let me handle that. As I said before, I've got deeper pockets now. I'd need you there on the ground, ready to go, by September eleventh.\"\n\nMy mind was racing as I sat there before him, drumming my fingers along the desk. What an opportunity; overseeing the debut of a restaurant from the ground up would be a major enhancement to my career. Cora would be thrilled, but the explosion in my relationship? I had already put an unfathomable distance emotionally between Marcus and me, but now I would be putting a physical one there as well. Realistically, there was no way I could go to L.A. as often, and not at all until this project was completed. Hell, I was barely making the five-hour flight. There was no way I could handle the twelve hours from London to L.A. direct. I didn't need to consult anyone to know that the ramifications of this would mean the end of us, and there was one thing I was absolutely certain about: Losing Marcus as a result of a direct action on my part has never been an option.\n\n\"Can I get back to you about this?\" I said, getting out of my head and refocusing on Michael. \"I need to speak with Marcus first.\"\n\n\"Really? I didn't know things between you two were so serious, to the point of discussing the merits of smart career decisions,\" said Michael, rather snidely. \"I wonder how often he returns the courtesy?\"\n\n\"We are serious, I'll have you know. Why else would I be back and forth to L.A. as I have been?\" I said. \"Trust me, it's not for the smogged-out air and wheatgrass shots.\"\n\n\"Well, let's hope he knows that,\" said Michael with a detectable cut in his tone.\n\n\"Huh? Why would you say something like that?\" I inquired.\n\n\"Oh, nothing. Just guy stuff. You two talk and let me know. I'd like a decision sooner than later.\"\n\nGetting back to my office, I asked Jacklyn to ring Marcus. This was the kind of news I'd prefer to discuss in person, but that would mean keeping it to myself until week's end when I got to L.A., and that would be impossible to do. I'd go positively mad, so why risk it? As I waited for him to answer, my thoughts returned to Michael: how opening in London like this so soon seemed irrational and unlike a decision that he would make cavalierly, but then again maybe it is not, if he has the opportunity and deep-pocketed support. Problem is, I knew the motivations of that support, so I couldn't help but think Michael was being positioned to be in over his head real soon with what he would come to owe Nikolai if\u2014scratch that\u2014when he defaults on the only collateral he will have left: his interest in Carly's and its holdings. I knew for certain that if backed into a corner, Michael would readily sell his interest in the holdings before parting with Carly's.\n\nAn hour or so later, Marcus called back. He'd been in the midst of a morning tennis match and left his phone in the athletic club locker. Having played an energetic match and won, his spirits were high, so I jumped right in with my news, leaving no room for him to interject until I had completed all I needed to say and laced it with so much enthusiasm that he would find it hard to offer no support.\n\n\"Well, that is some news indeed,\" he said, his demeanor calm, giving nothing away and offering less. \"Is this what you want to do?\"\n\n\"Are you kidding me? It's a huge opportunity. I can't believe you didn't tell me that London was happening after everything,\" I said, bouncing in my chair. \"I am over the moon at the possibility of it. Aren't you?\"\n\n\"I can see how you would be. It just seems like a piss-poor time to be away, you know?\"\n\n\"I do, but Michael said it would not be permanent. Mostly when he is unable to be there, you know, for a visit here and there to make sure all is in line. I mean, I know this cuts into our L.A. time, but it's not forever. Besides, we are great at long distance,\" I said, my tone having lost its earlier excitement as I listened closely for any warning that Marcus's response might provide. Yet it still offered none. \"We can handle that, right? Of course we can!\"\n\n\"Yes, of course,\" he said. \"Listen, I have to get out of these clothes. I'm going to be late for my doctor's appointment.\"\n\n\"I didn't know you had a doctor out there. Is everything okay?\"\n\n\"Yes, just some routine stuff, a few tests, nothing serious. Let's you and I discuss this in person when you arrive. Okay?\"\n\n\"Sure. You'll let me know if there's a problem or anything with the doctor, right? I didn't know\u2014\"\n\n\"Indeed,\" he said, hanging up the phone.\n\n\"Okay, bye then. I love you,\" I said into the dead receiver.\n\n## 41\n\n* * *\n\n## DETONATION IN 3 . . . 2 . . . 1\n\nHERE'S THE THING with being on cloud nine\u2014it's not real and can never be sustained. It's a euphoric state of mind that can dissipate as quickly as it appeared, leaving the victim crashing down to earth, much the worse for wear. Marcus had not been as excited about the news as I'd hoped, but it was far better than it could've been, so that was something. On Saturday, we would discuss the slight adjustments to be made here and there. To me, it seemed only right that, given the amount of time I'd devoted to being with him in L.A., he could put himself in London when I am there or back in New York into our old life. Easy. But first I needed to tell Michael the good news.\n\nSimone was now at her desk, smartly dressed as usual in a black sheath dress and matching cardigan with her red-lined Prada reading glasses balanced on the bridge of her nose. In the years that I have been here, it is fair to say that the only speck of color the woman owns outwardly is those glasses. Although I am convinced that she wears racy pink, red, and animal-print lingerie under her work staples. I mean, she must, no woman can live by black, gray, brown, and dark blue alone\u2014not even a native New Yorker.\n\n\"Hi, stranger, is the boss in?\" I ask, making my way toward his door anyway.\n\n\"He is, but now is not a good time,\" she says, causing me to stop. From the interior I can hear Michael's raised voice, and it's explosive.\n\n\"Who is he yelling at?\"\n\n\"Not sure. He told me to jump off the line,\" says Simone.\n\n\"You're totally lying, Simone. I know you know. You had to put the call through, so come on, dish! Who is it? Last time I heard him like this, he was going in on Carly over Marcus.\"\n\n\"Well, this time he's going directly to the source. He is on with Marcus. You still want to go in?\"\n\n\"Hmmm, no,\" I said, feeling in my bones that something about this was far from coincidental. \"When Michael has had a chance to calm down, tell him I came by to discuss London.\" Better to make a hasty retreat and call Marcus. But no sooner did I turn to leave than Michael opened the door, obviously charged and ready for battle.\n\n\"Simone, get Nikolai on the phone,\" he said, before laying eyes on me. Then\u2014\"Jules, how long have you been out here?\" He looked suspiciously from Simone to me.\n\nBefore Simone could interject, I said, \"Just got here,\" shooting her a quick look to silence any commentary she might give to the contrary. \"Came by to let you know that I am on board for London.\"\n\n\"Really?\" asked Michael with a slight amount of disbelief. \"And Marcus is aware of this?\"\n\n\"Yes, of course. We spoke about an hour or so ago. He didn't have any objections. We just need to work out the details, you know, time together and stuff, when I get to L.A. in a few days.\"\n\n\"Ha, well, look at that shit. With you he's all calm and amenable. With me he is in hysterics, threatening shit like he is running things. I guess that's to be expected with everything he's dealing with right now,\" said Michael.\n\n\"That does not sound like Marcus at all. He's done a lot to make this possible\u2014what do you mean 'everything he's dealing with?' \" I asked, remembering that he was at the doctor's, a matter that seemed to have slipped his mind in earlier conversations. \"What do you know that I don't? It's the third time today that you have made some offhanded comment or other about him.\"\n\n\"Jules, it's not my place to say, but I can only imagine how difficult it is for Marcus to keep things straight, trying to take over Inception Studios, dealing with Blake and the baby. Now this. No wonder he was unglued with me on the phone. You going to London had to be the last straw.\"\n\n\"Wait, what are you talking about?\" I couldn't believe I had heard correctly. \"Blake and what baby?\"\n\n\"Michael,\" said Simone, rising from behind her desk, attempting to halt the impact of his words. \"Don't\u2014\"\n\n\"What, Simone? Jules is family. If I don't tell her, then who will? Clearly not Marcus.\"\n\n\"What baby, Michael?\" I yelled.\n\n\"Jules, lower your voice. I hate to be the one to tell you this, but Blake has a baby and Marcus, well . . .\"\n\n\"You're lying,\" I said. \"Why would you say something like that, Michael? That's crazy. Blake and Marcus haven't seen each other in more than a year. You're mistaken. Take it back.\"\n\n\"Wish I could, baby girl. Shit, I'm sorry. Maybe I shouldn't have used those words exactly,\" said Michael, \"but that seems about right. Her daughter is a couple of months old, so . . .\"\n\n\"Michael, STOP IT!\" yelled Simone, coming from behind her desk to my side. \"Jules, honey, are you okay? You don't look so good. Why don't we sit down?\"\n\nBrushing her off, I said, \"Simone, why is he saying that?\" But she had no words to offer, only her arms to hold me up with. \"Michael, why are you doing this?\"\n\n\"I don't want to see you throw your life away, Jules, on a guy like that. I told you I didn't like it when you started dating, but I remained silently supportive. Now, with Blake and the baby, I can't stand by and watch.\"\n\nGrabbing my stomach before retching, I said, \"I think I'm going to be sick.\" I broke away from Simone and ran down the hall to the restroom, throwing myself over the nearest vestibule and trying to empty my heart of the shards.\n\n\"Jules, you all right?\" said Michael from the other side of the stall.\n\n\"GO AWAY, Michael. I don't want to hear you anymore. Just go away.\"\n\n\"Listen, I know you're upset, but I am not the one you should be upset with. I am shocked as well.\" My sobbing turned into accelerated short breaths. \"Jules, it's going to be okay. You go home, rest, and just focus on London. It's exactly what you need to put all this behind you, you'll see. Okay? Jules, are you listening to me? Let me hear you.\"\n\n\"Yes, when do we leave?\" I asked.\n\n\"That's my girl. I'm bringing you in on Tuesday the eleventh, remember. You can rest up. I get in later in the week. We'll paint the town.\"\n\n\"Okay,\" I said weakly from a crouched position on the floor.\n\n\"A word of advice, write this whole situation off. Don't talk to him. Delete him from your life from this moment on. I told you he'd turn on you, didn't I? Forget him, he's not worth it.\"\n\n\"Okay.\"\n\n\"I'll send Simone in to check on you. Carly and I have plans for the theater tonight, so I have to go.\"\n\n\"Okay.\"\n\n\u2022 \u2022 \u2022\n\nNight had fallen by the time I awoke on the sofa in my office to persistent knocking at my door. The details of how I had gotten there were jumbled. My eyes swollen, nearly shut from a torrential cry, my heart shredded beyond recognition, I had nothing more to say, nothing to give, so I just lay there wishing whoever it was to go away and let me be. After a few lighter knocks, the door opened to reveal the shadow of an all-too-familiar figure who deserves far more from our friendship than the persistent drama I seem to consistently bring: Richard.\n\n\"Oh, dear,\" he said, rushing to my side and kneeling to help me sit upright. \"How long have you been in here like this?\"\n\nMy voice hoarse and cracked, I said, \"I don't know. Simone brought me in, I think. I don't know. What are you . . . ?\"\n\n\"Jacklyn called. What happened?\" asked Richard.\n\n\"Marcus, he . . . I thought we, but he\u2014\" I stumbled over my words as the thoughts ran together. I couldn't bear to hear the truth come from me, so I just shook my head and closed my eyes.\n\n\"He what, dear? Is he okay? What happened?\" asked Richard, brushing the matted hair away from my face. \"Let's get some lights on in here.\" The harsh glare made me wince, forcing Richard to dim them to the glow of a candle. \"Do you want something to drink? Let me get you something.\"\n\n\"Blake has a baby, did you know?\" I said.\n\n\"Yes, poor child, and I do mean that, baby, but let's get to that later, under better conditions.\"\n\n\"It's Marcus's.\"\n\n\"You can't be serious,\" exclaimed Richard, standing at the bar, stunned. \"Why would you think such a thing? She has been abroad with her Greek traipsing the globe for nearly a year. There is no way.\"\n\n\"Yes but the baby\u2014\"\n\n\"With Stavros\u2014\"\n\n\"No! Michael said it's Marcus's. That it's a girl a couple of months old and it's Marcus's,\" I said, waiting for Richard to refute my words, but he didn't. Instead he said \"oh\" and went silent, quietly doing the calculations to determine if there was a possibility that maybe just maybe Marcus could be the father to Blake's baby. \"Have you seen them?\"\n\n\"Afraid not,\" he answered, sensing it best not to contradict the probability of events.\n\n\"So it's possible,\" I said, choking back fresh tears.\n\n\"Jules, anything is possible, I guess. What did Marcus have to say?\"\n\n\"I don't know. Haven't spoken to him. Not speaking to him,\" I fired back angrily. \"What could he say? It's unexplainable.\"\n\n\"Dear, I am not making an excuse for any of this because I don't know the facts, but at the risk of upsetting you further, and that is the last thing I want to do at a time like this, I just have to believe that there is an explanation.\"\n\n\"But I heard them arguing before Michael told me.\"\n\n\"I don't know what to say, but there is a certainty about the two of you, so I must believe that there is a logical\u2014oh dear,\" said Richard, looking at me crumbling under the weight of my despair. \"Tell you what, let's get you home, out of these clothes, and into a nice hot shower. Everything is better after a shower. I'll even make you something delicious to eat, maybe some soup and my famous grilled cheese,\" said Richard. \"You'll like that, won't you? Yes.\"\n\n\"Okay.\"\n\nThe only thing the shower did was make me more tired. I had no energy to do anything, so Richard helped me into pajamas and put me to bed with a bowl of tomato soup and a grilled cheese on a serving tray, which remained untouched. Before falling asleep, I asked Richard if he'd checked the machine just in case Marcus had called. He had not\u2014another sign. Richard pleaded with me to call him, but I refused. What would I say? How much more pathetic could I look than by calling him about something he didn't even deem necessary to tell me but confided to Michael? Tomorrow would be better.\n\nWhen tomorrow did arrive, I can't say it was better or worse as much as I willed it to be better. Yes, I wasn't crying, and I was dressed well in advance of Richard, sleeping on the left, the side of the bed that belongs to Marcus, but my spirit was parched and desolate. How much I love this man\u2014my best friend. The only person I can count on every time and yet here I am again making the same demands. Poor Jules, in a traumatic state with no regard for anything that may be happening in his world, always taking.\n\nWhen Richard awoke, I focused on rising to the occasion of being my best self in order not to bring on his looks full of sympathy and concern. I channeled the old Jules, surprising even myself with a genuine laugh over coffee when he confided that sleeping over was not the purely selfless act it appeared to be. Apparently, little Ms. Ming had a serious set of lungs and liked to raise the roof into the wee hours of the night, every night. Last night was the first time in the months since they had adopted her that he slept soundly.\n\n\"Honey, there is a reason the laws make it difficult for gays to adopt. I don't have the constitution for the upkeep. When can I just buy her a car and send her off to live in careless disregard for the privilege she has been brought up in?\"\n\nIndependent of how many off-color comments I may make about my folks, I know that I am lucky to have been brought up in a family like ours. Having this frame of reference, I felt for the first time in our long friendship that I had some valuable insight to offer Richard as he and Edgar went through their parenthood journey. We spoke about the balance and compromises that parents make willingly and the ones that they must never make, like finding personal time to keep the romance going. Lord knows I heard enough and saw the loving results of Cora and Daddy's \"couple time.\" Richard at one point brought me to laughing hysterics as he confessed to feeling that with Ming, he had unknowingly signed up for a lifetime of never winning the argument\u2014which was starting to make him question his professional competence. I did enjoy the image of Richard becoming frazzled, but the thought of a baby girl, any baby girl, infuriated me. Little Ming, by sheer association of gender and age, forced me to think about what Marcus and Blake's child could look like. What her name could be. Most important, it reminded me of how horribly cheated I felt not being the first, the only, to bare his offspring\u2014but I did not reveal any of this to Richard. I knew that had I given any hint of instability, Richard would have remained at my side until the next day's flight, which is exactly what I didn't want. I needed to be alone, and he needed now to be with those who needed him most: his partner and daughter.\n\nOnce alone, I allowed myself to own my anger. Yes, I was devastated at Marcus's betrayal, but unlike with Tony, where I felt sorry for myself for far too long, I didn't feel self-pity this time. I knew that I had done nothing wrong to warrant this. I was angry as hell, and with each moment and no phone call, my anger grew. This was not the man I knew. Did he change or did I make him over into someone he never was? Strangely, in the midst of it all, there was a strength brewing in my rage that I'd never felt before. No longer did I feel the need to run and hide in a story line or country foreign to me. Hiding was the last thing on my mind. Quite the contrary, my life was the main event. I needed to see him and understand how he could do this to me. And with that fire, I boarded what was intended to be my last flight to L.A. Prior to takeoff, I sent him a text to say that I was coming in earlier than normal and would just as well take a taxi from LAX. I gave no details as to which flight, because I didn't want to see him before I was ready, nor did I trust myself to leave a voice mail potentially alerting him that something was wrong. Looking back, I wonder if Marcus ever really stood a chance to explain himself after my ambush, as he would later call it.\n\n\u2022 \u2022 \u2022\n\nDuring our argument, he said that I'd made up my mind long ago of his guilt, so what was the point of continuing to go through all this? I said because it was the very least he owed me. And so we went the course, putting all our cards on the table. He told me that he was blindsided by Blake's pregnancy after learning about it one evening at dinner with Michael. To hear him tell it, it was innocent enough, and he thought no more of it until Michael made mention of the child's age, noting that he found it questionable that Stavros could be the father, given his advanced years and the timing. Taking this into account, Marcus attempted to contact Blake numerous times, to no avail.\n\nWhen they did speak, Blake was less than accommodating. Given how things had played out between the three of us, how could she be anything less? Marcus, unable to bear the thought of having an unclaimed child in the world because of the abandonment scars he will forever carry from his own mother's desertion, volunteered to take a paternity test. Blake apparently agreed, claiming it was the least he could do. And all this, says Marcus, snowballed into the most destructive storm when Michael, by way of Annya, who unwittingly spilled the beans, learned that Marcus and Nikolai had not only known each other for years but were business partners.\n\nWell, it didn't take Michael long\u2014not having the complete story and believing Marcus to be his adversary\u2014to determine that one way or the other, Nikolai was doing Marcus's bidding and trying to steal Carly's right from under him. Feeling duped and needing to act quickly, Michael committed to London, thereby securing something substantial. In the process, knowing what he knew of Blake's situation, he decided to leave a lasting calling card for Marcus: personal ruin.\n\nQuite rich in loftiness, I said, \"You should consider writing a screenplay instead of taking over a studio.\" I lashed out, fueled by pain and anger. He's lying. I remained unconvinced (unable to hear) that the victim in all of this was in fact Marcus. So when he stormed out, not to return, all I had were the fragments of our argument, his defense, our history, my baggage, and my love for him to replay.\n\nIndeed, it is a horrible thing for a woman to be left alone with only her thoughts at any time, but especially in a town like L.A., where there are no streets to walk and get lost in. So I returned to New York for a day, busying myself with all sorts of maintenance details to keep occupied, inflict pain, and force escape. There was the spinning class in SoHo, head-to-toe waxing with Ella, followed by a mani-pedi and threading on the East Side, a wash 'n' set uptown with the Dominican girls, and some quality time with a friend, who knows me better than most, in the Village that rendered me drunk enough to go blank but not enough to reduce me to a blubbering idiot\u2014I think.\n\n\"I see you've started without me,\" said Tony, strolling into Frank's Bar and sitting down.\n\n\"Yeah,\" I said, sprinkling salt on my hand in preparation for a tequila shot. I know it's odd, but he's the only person I wanted to talk to, that I could stand to be around. \"Something like that\u2014ugh! That's good,\" I said, biting into the tart slice of lime.\n\n\"How many up are you?\"\n\n\"I don't know, not many. Ask the bartender, he'd know,\" I said. Tony tapped the bar, indicating that he wanted to catch up.\n\n\"Damn, J\u2014you know tequila is not my drink, but here goes nothing,\" he said, quickly throwing back three shots. \"So you wanna tell me what's going on or should we just sit here?\"\n\nFighting back tears, I said, \"Can we just sit for a minute? Just need to be next to you for a bit, okay?\" In between drinks and water chasers, I recounted the sordid tale of Marcus and me. He listened to everything until he was certain I'd said all that was to be said.\n\nShaking his head, he said, \"You know, in my entire life I don't think I'll love someone more than I love you, Jules. That's real talk. I don't know which is worse, seeing you hurt like this and knowing that once I caused you similar pain or realizing that you are in love in a serious way, with someone else. Don't say it. I know it's selfish, but it's what's up.\"\n\n\"I'm sorry,\" I said, realizing how unfair it was for me to unload all this on him.\n\n\"Don't be,\" he said, signaling for another round. \"Love is a good thing and I want that for you\u2014always have. Just wish I could've . . . Never mind. So, this dude, Marcus, left, you left, and here we are.\"\n\n\"Yep,\" I said, looking him in the eye, toasting and downing. \"And I just feel lost\u2014empty, you know? Like there's a huge chunk of me that's just out there and I can't put my arms around it\u2014I can't find it. My heart\u2014it hurts, and I'm so scared that it won't stop this time.\"\n\n\"J, you know I chased you for a few days after the whole Angie thing but stopped when I thought about what it all meant for us, why I did it, what would happen if I got you back, and that whole shit was crazy, yo. I wasn't ready and I knew it. I wasn't ready to love you with that all-encompassing kind of love you expected. I still needed to do me, and I know that sounds fucked up, but that's the truth. I'm telling you this to say, go easy on yourself, baby, this is the first blow, your emotions are racing all over the place and your mind hasn't even had a chance to catch up and figure things out. Give it a few days. Then you'll know what you need to do. Just ride that shit out. Hell, love is the biggest wave we'll ever get, better than any I've found surfing,\" said Tony, pushing my hair away from my face to dry my tears. \"Damn, you're pretty.\"\n\n\"And you suck. I can't believe you stopped chasing after three days,\" I said, between laughter and tears.\n\n\"Hey, I didn't say it was three days, although that sounds 'bout right. I'm just joking\u2014see, you can still laugh. All's not lost. I did it because I love you just that much, Jules Isabel Sinclair\u2014I had to let you fly, knowing I'd never get you back,\" confessed Tony.\n\n\"Did that mean you had to go and tell Carly all about it?! My God, that woman!\"\n\nHe laughed uproariously. \"Ahhhh, shit! Yo, Carly is not a game! You hear what I'm saying? But her heart is in the right place.\"\n\n\"Yeah, I've learned that. She helped clear the way initially for Marcus and me.\"\n\nEverything else that evening was a blur. By the time I boarded my flight on Tuesday morning out of JFK, anyone looking at me could see that I was the human equivalent of rubble inside, but it didn't matter. My only thought being to sleep the full five and a half hours it would take to get to London and revisit some old healing haunts. Whatever sleep nature did not provide, Ambien would. An hour or so before landing, I awoke with a tear-streaked face after having a lovely dream of Marcus and me sitting at Via Quadronno, sipping cappuccinos and laughing. It seemed so real. I could touch him, feel the moisture of his skin, smell the Hanae Mori cologne we'd picked out at Fred Segal the week before.\n\nI was devastated to be awake and have to confront reality all over again.\n\n## 42\n\n* * *\n\n## LONDON CALLING . . . AGAIN\n\n2001\n\nBUT MY LIFE is an ocean away, and if I knew how to stop looking outside of myself for answers, I could tell him that and get out of my own way. What would I say? I'm sorry\u2014for what? For not believing you? How could I even go down that road when I didn't even know what the truth was? Michael says it's his child. Marcus says it's not, and under different circumstances, I would be inclined to believe Marcus, because that is what my heart wants so badly to do. Experience, however, had showed me the cost of blind faith, and while the time with Tony was good, it was also a reminder. I don't want to be that fool again, I said to Richard one evening later that week on the phone from the hotel when he called to check in on me. During which time I also recapped my time with Tony.\n\n\"Jules, please tell me you didn't do something irrational like sleeping with the ex to get over the . . . You know where I'm going with that, it's supposed to rhyme.\"\n\n\"Oooh, death to the great gay rapper once and for all,\" I joked. \"Relax, Richard. I know where you are trying to go, and it's unnecessary. Tony and I did not sleep together. Although I am sure I tried at some point. From what I can recall, I was wailing about Marcus, then I remember seeing Ivan or Percy, one of them, and that's it. When I awoke, we were both on the sofa fully clothed, all buttons and zippers intact. I put him in a cab, and that was it. He did smell nice, though!\"\n\n\"Oh, don't we all! Well, it seems people do grow up. Imagine that, there is hope for you yet, Jules,\" said Richard. As it stood, he was still the only person in my friendship\/family circle who knew that Marcus and I were having problems. \"Is that happening anytime soon?\"\n\n\"I'm trying.\"\n\n\"Stop trying Jules, and do! Honey, you can't keep doing this to yourself,\" said Richard. \"One way or the other, you have to know the truth. Marcus says that it is not his child. Do you believe him?\"\n\n\"Yes, I mean I want to, because I want to be happy, but it does not add up. Why would Michael say that it is? Why would Marcus not tell me unless he was hiding something?\"\n\n\"How would you have handled it if he had come to you?\" asked Richard.\n\n\"I would have listened. Together we would have figured it out,\" I said.\n\nWith extreme reserve Richard asked, \"Are you sure about that, dear? Correct me if I'm wrong, but didn't he come to you before with a highly sensitive matter and you were not as accommodating?\"\n\n\"Entirely not the same thing,\" I said emphatically.\n\n\"I disagree. It is exactly the same thing, Jules. He trusted you once and you\u2014\"\n\n\"Stop, please don't say it. I know what I did . . . but this\u2014\" I paused, divorcing myself from the unjustifiable. \"Michael is not just my boss, you know that\u2014he's like a mentor. It all was just too much, like designer trail mix, it was . . . all wrong.\"\n\n\"I'm not saying that what he may or may not have done is right, but it is comparable.\"\n\n\"So what do I do? I can't go apologizing for something that I still don't have answers to,\" I said.\n\n\"Then do the only thing you can do. Go to the source and get your answer, honey.\"\n\n\"I told you. I'm not ready to talk to him.\"\n\n\"I'm not talking about Marcus. Call Blake. She is the only one who knows the truth.\"\n\n\"Are you insane?! I am the last person she wants to hear from after what I did to her. I mean, it's bad enough that I broke up the relationship she thought she had with him, but now I could potentially be standing in the way of her child's relationship with her father,\" I said. \"You're not mad, you're certifiable!\"\n\n\"It's the only way, and it's long overdue,\" said Richard. \"The fact that you two girls could not get past unknowingly dating the same guy is beside the point. True, you should have been more brave in your dealings with her, I have said it before. But you girls were best friends long before Marcus came into the picture. At the first onset of friction, you threw all of that away instead of dealing with it head-on. Aren't you tired of doing that, Jules, with every relationship in your life? What happens when the day comes that you elect to throw me away?\"\n\n\"I would never, Richard, never,\" I said. \"Besides, I haven't spoken to Blake in forever. I wouldn't know how to reach her.\"\n\n\"Easy, dear, you call her,\" said Richard. \"I just happen to have her number on hand, unless of course you would prefer to see her. That is possible too. She is in London right now, has been living there for some time, according to Joy.\"\n\n\"What would I do without you, Boulton?\" I said, chuckling aloud.\n\n\"Thankfully, we will never have to find that out, now, will we?\"\n\n\"Never,\" I said. \"Good night, my friend.\"\n\nI paced around the number Richard had given me for some time, staring it down like a fighter preparing for battle. I rehearsed what I would say and how I would say it many times over before picking up the piece of Sofitel stationery it was written on and going over to the phone. Independent of how she would respond to me, I would remain calm. There was no expectation of getting the friendship back, as I didn't believe that was due to me\u2014but I was entitled to hear the truth.\n\nAfter the first ring, a heavily accented Indian-British house voice greeted me\u2014\"Good evening, Giannakopoulos residence.\" Having thought the number I dialed was her direct line, I was taken aback, forcing the voice on the other end to repeat, \"Hello? Giannakopoulos residence. Is anyone there?\" After identifying myself and whom I was calling, I was placed on indefinite hold. At first I reasoned that Stavros's home was enormously large and for some reason not equipped with an intercom system, causing a delay in locating Blake. Then it occurred to me that, while I may want to speak with her, she may not want to speak with me, which was well within her rights. I was about to hang up when the voice returned, informing me that the lady of the house was putting baby Catherine down for the evening but would be available for tea tomorrow, if my schedule permitted. On hearing the name of my hotel, he confirmed that Blake knew the location of the Sofitel St. James, and would meet me there at 14:00. Hanging up the phone but still holding the receiver, I dialed Marcus's number, but was overwhelmed by a surge of panic in my chest that forced me to hang up before it rang.\n\n\u2022 \u2022 \u2022\n\nImpossible. Of all the days and times to be running late, this is most certainly not the one. On Saturday, the site manager from the new property location called first thing to do a last-minute run-through before Monday's ceremony with Michael and Nikolai. After he assured me that the meeting would take less than an hour, I agreed, believing I had more than enough time to return to the hotel for afternoon tea with Blake. What I did not anticipate was being stuck in a massive gridlock of traffic, rendering me nearly twenty-five minutes late. With no mobile number for her, I called the hotel after the first five to alert her to the delay. The young woman at reception delivered a return message to me of \"Not to worry, take your time,\" allowing me to breathe calmly until I realized how truly late I would be. My only hope was that Blake would still be sitting there some forty minutes later.\n\nNearing the roundabout at Piccadilly, I jumped out of the cab before it came to a full stop and sprinted against a traffic light into the hotel, looking from left to right. The girl at the front desk who'd fielded my earlier call saw my distress and directed me to the high tea room. Huffing and puffing, I ran into the parlor to find Blake alone, looking more refined than ever in a black puff-sleeve blazer, fitted indigo jeans, and drippy striped T-shirt. There was a stroller at her side with the opening covered, concealing baby Catherine's face and my answer. At seeing me, Blake rose awkwardly, smiling as I approached. Were we to hug, shake hands, or just nod? Uncertain of protocol, we did an awkward combination of all three.\n\n\"Wow, you look great,\" I said, \"but you always did.\"\n\n\"Thanks, you too,\" said Blake, giving me the once-over. \"Jules, you look amazing, but you were always fit. Your hair is still crazy beautiful. I always wanted those big curls.\"\n\n\"You're one to talk. I always wanted your perfect straight locks. Never satisfied with what we have, are we?\"\n\n\"Never,\" said Blake. \"Remember when you put that Japanese straightener in your hair\u2014talk about a big mistake. I think I cried more than you.\"\n\n\"Oh, I know! Remember how self-conscious I was about dating Tony because he had more hair than me, and you said, 'Honey, dreadlocks don't count as hair. They're more like spiritual ropes to grab on to.' \" We both said that last part in unison, clasping hands, losing ourselves in memories of great times past. \"Oh, Blake, hair wasn't my only mistake, you know. I was bloody awful to you, and I am so very sorry.\"\n\n\"You were once, I'll give you that. But you were amazing the other ninety-nine times, Jules, so there is nothing to forgive.\"\n\n\"You don't have to be kind,\" I said. \"You owe me nothing.\"\n\n\"I'm not being kind. I'm being honest. If it weren't for you, for us, for New York, and for Marcus, I would not be here, living the kind of life that I only dared dream about but never thought I might have one day. Not with the way I was going.\"\n\n\"Yeah, about that,\" I said, directing my comment toward the stroller. \"Catherine\u2014I mean. I didn't know, and then to find out from Michael. I was just blown away, you know?\"\n\n\"I know! Who would have imagined me, pregnant, a proper mum, living in England. Insane!\"\n\nPerplexed by her response, I was unsure of how to delve further, so I erred on the side of caution and followed her lead. \"Richard said the same thing.\"\n\n\"I bet he did, that old gay bag. How is he? I miss him. I miss you. Joy says he has a little girl around Catherine's age, God help that child. She'll be the only Barry Manilow\u2013loving, overly articulate preschooler on the Upper East Side with a strong affinity for bridge and bourbon. Clearly she will need friends that money can buy. I can only imagine what else the old girl had to say.\"\n\nInhaling deeply, I said, \"Well, in light of things, not much, as he's spent the majority of the past week consoling me.\" I stopped before feeling like a fraud. \"Oh, Blake, let's not do this! Let's not talk like old times and pretend that you don't know why I'm here. I know you do. You must! I need to know whether you think I deserve to or not. Catherine and Marcus.\"\n\n\"Oh, Jules, I am so sorry for that. I was just upset. The weight wasn't coming off, and you know how I am about the chub\u2014does a number on my head. My hormones were everywhere. I wanted to punish him for coming between us and I wanted to hurt you, although I missed you, because you were supposed to be there for me.\n\n\"Remember, we were going to raise our kids together\u2014then everything changed,\" said Blake. \"Knowing what I know now, I wish so very much that I could take it all back. The time I wasted feeling spiteful when everything I had ever wished for was right there with Stavros and here with Catherine.\" She pulled back the blanket and lifted the child into full view. Oh my goodness, what an angel, all chubby with big brown inquisitive eyes, rosy cheeks, and a few raven strands at the crown. There were things about her that seemed like Marcus\u2014her mouth, her nose maybe, but I couldn't be sure. His hair was dark blond, as was Blake's, but maybe there was some latent genetic thing manifesting in Catherine to produce her dark coloring. Who can tell anything about kids at this age?!\n\n\"Would you like to hold her?\" asked Blake, and she passed her over to me before I could object. \"Ahhh, look at that, she likes you.\" Witnessing my awkwardness, she added, \"You won't break her, I swear. Believe me! She won't bite as much as gnaw and drool.\"\n\n\"Am I holding her right?\" I asked, shifting to find the right position as Catherine squirmed in my arms. Sensing my discomfort, she took the lead and smiled a gummy saliva-filled grin that melted my heart and calmed my nerves. How could I ever\u2014no matter what\u2014how could I deny this angel from having the best, most loving of all, especially supportive parents like I had?\n\n\"Blake, she is heaven. Of course Stavros would accept her, how could he not with this little face. Yes, yes you are,\" I said, playfully bouncing her up and down.\n\n\"He'd better accept her. She's just as entitled to his fortune as his other kids. I don't care what his ex says. Why, what have you heard?\"\n\nCaught off guard by Blake's comment, my grip slipped. \"Oh shit, eek, sorry! What are you talking about?\"\n\n\"What are you talking about, Jules?\"\n\n\"Catherine, is she Marcus's child or not?\" I asked, clearly so there could be no mistake this time. \"Blake, I'm dying here.\"\n\n\"No, why would you ask that? Listen, I know I was wrong for allowing Marcus to take the paternity test, but I wanted to make him sweat a bit. Then Stavros found out and, well, let's just say he didn't approve. He said I was being childish. Maybe. Anyway, I thought to tell Marcus, but he forbade additional contact, so I told Michael before things went any further.\"\n\n\"I am so not following this,\" I said. \"Are you saying that Michael knows Catherine is not Marcus's baby?\"\n\n\"Of course he does. He and that old bat even received the birth announcement.\"\n\n\"I need you to start from the beginning\u2014and make it simple,\" I said.\n\n\"Let's see, a few months ago Stavros and I had dinner in Knightsbridge with Michael and those Russian friends of his, Nikolai\u2014don't trust him, by the way\u2014and Annya. She is fierce, I love her! I was big as a house\u2014you know, the unattractive stage of pregnancy\u2014and a little extra cranky about everything, Little Miss Thing holding my body hostage, did I mention that?\"\n\n\"Blake, focus,\" I said, having forgotten her predilection for unnecessary commentary when asked for specific information.\n\n\"So . . . Michael and I get to talking. He said something snippy about Marcus being slick and his blatant disregard for other people, yada yada, would serve him right. I didn't give it any thought again until Marcus started calling a few weeks ago demanding\u2014you know how entitled he can get\u2014to know about the baby. I didn't appreciate the intrusion or his tone, for that matter, so I told him if he wanted to be sure, then go ahead and get tested,\" said Blake, reciting the entire saga with all the salaciousness of a telenovela. \"You must believe me, Jules. I never thought he would take a paternity test, especially after I said the baby was not his, but he didn't believe me. So I called Michael to clear things up since they talk all the time. Uh-oh, what's wrong? You're twisting your hair. That always means trouble.\"\n\n\"Karma,\" I say, trying to suppress the dual desire to laugh and cry. \"Oh, Blake, this life thing is so not easy.\"\n\n\"You're telling me. I mean, what crazy universe would trust yours truly with raising a child? I just don't know,\" she said.\n\n\"Well, know two things, lady. I am so very sorry for not being more respectful of us. I should have mined this dynamic, especially when a complication arose. Second, and most important, be careful. Michael is not your friend\u2014nor mine, for that matter. He is only about self.\"\n\n\"We all are, Jules,\" said Blake with a laugh. \"That's the one thing I have come to know up close and personal. We all just want the best the world has to offer that we believe will make our lives the most. The only hope is that the pursuit leaves minimal carnage. If you remember that, most things are forgivable.\"\n\n\"What about broken trust?\" I responded, thinking about Marcus and myself.\n\n\"Just requires more work and a little bit of luck,\" said Blake. \"Look at us. We're back together.\"\n\nAmid a haze of cucumber sandwiches, scones, and a year's worth of delayed gossip, the afternoon zipped by. Blake and I had so much to catch up on. She and Stavros were planning to get married during the holidays in order to have Catherine baptized in the Greek Orthodox Church, despite very vocal objections\u2014to the religious conversion, not the marriage\u2014from her Mormon parents. London was not her ideal location, and she longed to return to New York, but would stay for now because it worked for Stavros's primarily Europe-anchored business. Within the next year or two, they planned to have another baby, bringing the Giannakopoulos brood to seven\u2014two expensive ex-wives and a future missus committed to making her predecessors look like bargain shoppers.\n\nOh, how we laughed and caught up. I told her about my splitting time between New York and L.A., the offer from Michael, and his sinister role in Marcus's storming out. The latter of which she did not see as a problem. Apparently a monthly walkout was par for the course for her and Stavros. She said he, like Marcus, will always find his way back home. How could they not with women like us!\n\nSoon the time had come for her to return home, as Stavros was adamant about having dinner together nightly. Before leaving, I promised to attend the wedding and she promised not to breathe a word of Michael's deception to anyone, allowing me space to deal with him in my own way, in my own time.\n\n\u2022 \u2022 \u2022\n\nThe feeling of laughing and crying\u2014otherwise known as being on the verge of a breakdown\u2014returned almost immediately after Blake and baby Catherine's departure, so I decided to take a walk for some fresh air. Starting on the desolate streets of Piccadilly, with its grand architecture, I found my way to Covent Garden, amid the hustle and bustle of shoppers, theatergoers, tourists, and schoolchildren. The noise and congestion were welcome accomplices to the matters in my head. With each step and sidestep to avoid colliding with a passerby, my life started to make sense, the comedy of errors abundantly clear.\n\nOn the one hand, I can accept that it was all worth it to see Blake beaming as she was, but then I think of my heart and the toll it all has taken. What cost is too much? Why is being a woman in today's world so hard? I have to believe that despite Cora's quiet longing for something proprietary, a career, given the choice she would not change her course because she has been loved abundantly and has lived happily, a comfortable, purposeful life. What has my purpose been? To be a pawn for others? Working toward their life's happiness while mine lies undefined?\n\nWhen I lived here, I came to know what I wanted, and then I went to New York to live it. Somewhere along the way I lost track. I got caught up. Michael's and Marcus's purposes became more important than mine. Michael's because he controlled my career. Marcus's because he holds my heart. So I gave everything away to both of them without holding firm to the things that are important to me. I gave away my power and caved to the fear of disappointing someone or getting hurt again, only to end up near ruin . . . and starving! But I'm going to get it back as soon as I get a bite to eat.\n\n## 43\n\n* * *\n\n## FATHER KNOWS BEST\n\nTHE PANGS OF hunger in my stomach intensified with each passing block until I looked up to find myself standing in front of Mr Jerk, a Jamaican restaurant located in the heart of Covent Garden on a busy side street just down from the Donmar Warehouse theater. This has to be a sign, I think to myself, so I enter and order some stew and coco bread at the counter. Normally this place is busy every hour of the day and night. This evening, business was only moderate, allowing me to slide into one of the orange plastic booths and eat comfortably before walking back to the hotel. Awaiting my food, I pulled out my phone to make the only call that mattered. It's just after lunchtime in Virginia, so he should be in the office.\n\n\" 'Ello, dis is Charles.\"\n\n\"Daddy,\" I say, instantly relieved at hearing his voice. \"I messed up.\"\n\n\"I was wonderin' when you was going to call me,\" he said. \"You ain't kill no one so da rest is fixable.\"\n\n\"How? What do you know?\"\n\n\"Marcus called me the night ya two had ya roar. I was just waitin' for ya to find yur footing and pick up the phone. Poor fella, dem crazies bombin' the towers didn't make it any betta.\"\n\n\"Is he okay?\" I ask, longing for any insight as to his well-being. We hadn't spoken in a week, and that was killing me.\n\n\"He's goin' to be fine, just like you if ya two kids can calm ya'selves down and take responsibility for one another. Makes no damn sense.\"\n\n\"That's easier said than done, Dad. Thank you,\" I said, looking up to find the waiter placing my order down.\n\n\"Where you at? Who ya talkin' to?\"\n\n\"I'm at Mr Jerk, Daddy, getting some food,\" I said. \"A lot of things were said and done beyond whatever he may have told you. I just don't know if we can get back to where we should be.\"\n\n\"Nonsense. You are exactly where you are supposed to be, at the fork in the road, faced with decidin' if this is the one love worth fightin' for. That's where you are. So which is it? What cha' gon' do?\" demanded Charles.\n\n\"I'm trying to figure that out now,\" I said.\n\n\"No, you're not. You already have, girl, so stop foolin' ya'self and delaying ya life. Nutin' is wrong wit lovin'. Above all, builds character. Look at what it did for me and your mother. You tink witout that woman I would work so hardt to be a success? No, I'd be right there back in Jamaica playing in some roots band content wit a spliff and a Stripe, but I'm not.\" Hearing me laugh, Daddy continued, \"That's my girl. Your spirit ain't broken if you can still laugh, and that is something to be grateful for. What do you want?\"\n\n\"I want to be with Marcus,\" I confess.\n\n\"Then what you doin' in London in dat fake Jamaican restaurant instead of Los Angeles with Marcus, where you belong? Jules, I don't know everything, but I know some tings were said or done to make ya both run. Marcus is a solid man\u2014a little light but a good fella nonetheless. I know he'll always provide and love you, if you allow him. The hardest ting for any man to do is lay hisself bare for anyone to see, especially the love of his life, but he'll do it for the right woman. It's scary as hell. Accept him. Stand by him. Don't try to change him.\"\n\n\"But what about me?\"\n\n\"That's the only thing you can control. Stop fightin' wit ya'self, expectin' someting that is against your spirit. There is nutin' in ya upbringin' ya need to rebel against. What's there to prove? I know how we raised you all sheltered, provided for, and with unending love. Most of dat is good, some of it wasn't. All of it was for you to know what 'the most' feels like and never do without it. Dat's yur birthright as a woman\u2014as my child. You've always had a choice, we madedt certain of that. Why you choosin' to work on two separate paintings for your life instead of focusing on one masterpiece?\"\n\n\"When did you get so smart?\" I said, smiling broadly.\n\n\"Happenedt the day you were born, bunny.\"\n\n\"I love you, Daddy,\" I said, choking back tears.\n\n\"Right back at you. Now get ya'self off dis phone and get home,\" he said. \"I'd feel real guilty about goin' on dat fishin' trip up to Oregon wit Marcus next month while you two like dis, but I will. You know how I love my fishin'.\"\n\nHe was right. Fathers often are, saying little but saying the most when needed. I know what I want, and that is to be with Marcus for as long as fate will allow. Placing our relationship above all else doesn't diminish me, and if it does, the only judge and jury that matters is me. The rest will fall into place\u2014it always does. Leap and the net will appear.\n\nArriving back at the hotel, I stop by the front desk to notify them that I would be checking out first thing in the morning, and if they could arrange transport to the airport, that would be great. Reviewing my reservation and seeing that I was leaving nearly a week early caused a moment of alarm for the desk manager.\n\n\"Ms. Sinclair, I do hope that nothing is wrong,\" he said, knowing that I was a New Yorker and thinking falsely that the World Trade attacks could be the cause.\n\nInhaling, I said, \"No. Everything is perfect, actually. It's just time to go home is all.\"\n\n\"In that case, we look forward to seeing you again on your next visit.\"\n\n## 44\n\n* * *\n\n## HEATHROW AND LAX\n\nSECURITY AT HEATHROW was something out of a George Orwell movie. In order to make the 8:20 a.m. flight, I was advised to be at the airport four hours prior to departure, given the additional security precautions and that I was flying international. The night before, I didn't sleep. After packing, I sat to write my resignation letter. In light of everything, whether Marcus and I work or not, there is no way I could continue working for Michael knowing what I know. It's one thing to be caught in the crosshairs of dueling fire. It is quite another to be walked up to and cut down in cold blood for the sake of sending a message. And that is what he did to me, ignoring the fact that I was not some nameless person but an employee, a friend, a prot\u00e9g\u00e9e. Over the years Michael had taught me many invaluable things. This, however, was not a lesson I wanted to incorporate into my life or business. True, I would again have to live off my savings, but thanks to his and Carly's generosity by way of salary and a rent-free apartment, I have time to find the next-best job, be it in New York or permanently in L.A., if Marcus will have me.\n\nTwo checkpoints before reaching the interior of the airport, only to stand in the security queue for a full hour and then some, before dashing down the causeway as my name is being announced over the loudspeaker, then reach the gate moments before the door closed\u2014no small feat. First class, as they say, does have its privileges.\n\nFinally secured in my seat, I started to unpack my purse of the in-flight reading selections I'd lifted from the hotel and pulled out the very envelope that was supposed to be given to the front desk at the time of checkout and delivered to Michael's room before he left for the groundbreaking this morning, which was to begin in about an hour. Oh well, serves him right.\n\nLeaning my head back into the seat to decompress, I was startled by the ringing of my phone. My instinct was to just turn it off without looking, until I glimpsed Michael's name flashing across the screen. Seems only fitting, I guess, for it to end this way. As Richard would say, \"the dramatics.\"\n\n\"Jules, you coming down? I'm waiting,\" asked Michael.\n\n\"Don't. I'm not coming, Michael,\" I said. \"Tell me this, did it give you pleasure to see me like that?\"\n\n\"What are you talking about?\" he asked.\n\n\"Why did you lie to me about Marcus being the father of Blake's baby?\"\n\n\"I didn't lie, Jules. Never said it was Marcus's. I said\u2014\"\n\n\"But you led me to believe it was the truth. Right there in front of people for everyone to hear, and then you encouraged me to leave. I trusted you, Michael! Why? How could you do that to me?\"\n\n\"Jules, it's complicated,\" said Michael. \"I'll explain it later. What's that noise behind you? Where are you?\"\n\n\"I'm at the airport.\"\n\n\"This is not a game, Jules. What are you doing at the airport? Get in a car right now and make your way to the site. I'll meet you there.\"\n\n\"No. I'm out, Michael. I quit.\"\n\n\"You can't be serious. All this over Marcus? He's a big boy, Jules. He can handle himself, trust me. This little show of solidarity you're displaying isn't going to win you any favors. I should know. I was loyal to him for a long time, showed that kid the ropes, and he left me high and dry. You should be thanking me.\"\n\n\"You're wrong, Michael. Marcus didn't leave you empty as Nikolai plans to when he calls in that very sizable loan he's given you. The only reason you have New York and London is because of Marcus,\" I shot back. \"All of that land was never yours in the first place, and you knew it but you got greedy. You got greedy and you got played.\"\n\n\"I see, Boy Wonder has been talking after all,\" said Michael, regaining his composure to make a valiant show of things.\n\n\"Yes, that's what people in relationships do, Michael\u2014they talk. But we stopped doing that because of you. Because I gave you an importance you did not deserve. Marcus promised me that independent of what happened, he would protect you\u2014leave you with something,\" I said. \"And for what? Look at what you've done to him. Look at what you have done to me.\"\n\n\"Damn it, Jules, I didn't plan for things to go down like this, shit just happened. What can I say? I'll fix it after the ceremony, I promise.\"\n\n\"Is that all you have to say to me? Oh my gawd, what a fucking ego you have. I trusted you from day one. Why would you be so careless with that?\" I was so focused on Michael that I didn't notice we were ready for takeoff until the flight attendant approached.\n\n\"Ma'am, I must ask you to turn your phone off,\" she said, neatly attired in a red Virgin Atlantic flight suit.\n\n\"It's off,\" I said, snapping the phone shut and tossing it in the bag below. \"It's over.\"\n\nI once heard someone say, \"I don't know what the future holds, but I know who holds it.\" Looking through the window as we taxied down the runway, I was secure for the first time in the answer: ME! I hold the keys to my future.\n\n## 45\n\n* * *\n\n## DEEP BREATH\n\nPresent Day\u2014September 17, 2001\n\nAS WE TURN off La Cienega onto Burton Way, the driver alerts me that we are minutes away from the condo. Looking out at the palm trees, thinking about where I've been as I enter this thirty-something journey of my life and how much I've grown, I know I have much to be proud of.\n\nI swear that woman has a sixth sense or else she has wired me with some type of tracking device. Two blocks before we reach Wetherly, Cora phones.\n\n\"Hi, Mommy, what's going on?\" I say in the most pleasant, peaceful tone I can muster as not to elicit any unsolicited commentary.\n\n\"Today was your groundbreaking, was it not?\" asks Cora. \"How did it go? You didn't call me.\"\n\n\"I wouldn't know. Wasn't there,\" I say, realizing that she was opting to engage in a different kind of fishing expedition instead of being direct.\n\n\"Umph. I heard that could be a possibility. So you are back in New York, I presume.\"\n\n\"No, Mommy, I'm in L.A. Is that a problem?\"\n\n\"I see. And you are sure about this?\"\n\n\"Yes, I am sure,\" I affirmed, my resolve giving way to the motivating feelings behind my admission. \"I probably will never be able to explain it in any way that will make you understand, so I won't even try but he . . . Marcus matters, and I want to be there if he'll have me.\"\n\n\"You sound sure of yourself.\"\n\n\"I am, Cora.\"\n\n\"Then go get him. I would have done the same thing for your father if that Rosalyn had\"\u2014she paused to laugh at her own repetitiveness\u2014\"you know the rest.\"\n\n\"Oh, I do,\" I say. \"Mommy, what if he's not there or doesn't want to hear what I have to say?\"\n\n\"Then you wait until he is. You wait until your heart tells you it's time to let go. That is the very least that true love deserves.\"\n\n\"Miss, we are here,\" interrupts the driver.\n\n\"Okay, I have to go now. Let me call you later. I love you.\"\n\n\"I'm proud of you, Julesea\u2014Jules.\"\n\n## 46\n\n* * *\n\n## OTIS REDDING\n\nMY INDIFFERENCE TOWARD this hallway since the first day Marcus and I walked in is magnified today. Like most things in L.A., it feels ostentatious\u2014boldly trying to be something it's not. (Kind of like me, I guess.) The overly modern furnishings encased in an I. M. Pei-esque (but not really) construction make it feel cold and austere rather than the luxury advertised, and today even more so. The clickity-clack of my heels on the marble floor sounds like a death march instead of a return to love. What if he is not here? What will I do? I mean, I'm ready, but what if he needs more time? Can I wait like Cora says? I hope so. I'm here, I'm here! I repeat the words to myself in front of the elevator, unable to actually step in. A cute couple and a single woman with a purse dog come up as I stand there. The doors open and I nod them off. I'm here, but I just can't get my feet to work. I can feel the security guard staring at me, which is kind of pissing me off. Can't he see that I'm also middecision here? Whatever, no big deal. I'll just walk over like so, press the button like this, and get in. See. No big deal.\n\nPutting the key in the door, however, is another thing entirely, so I stand again, awaiting divine intervention. Then I remember a mantra that served me well once before:\n\nI long to be the woman who is loved passionately and deeply by a successful man who can provide for me in all ways that matter in this life. A woman who realizes that happiness is not a choice between this or that but an accumulation of moments experienced and shared. I want to be the woman who knows when to let a love go that existed in rose-colored glasses and to walk into a true love that requires no special frames to be alluring. I want to be the woman he chooses to remain faithful to and committed to even when things aren't going smoothly. I want to be the woman whose eyes hold his future.\n\nI can tell from the stillness that greets me at the doorway that no one is home. There is, I think, a hint of his cologne in the air, but that could just be my imagination, so I go further in. Marcus's gym bag is not in the hallway where it normally is, but then again I can't remember if it was there on the night he stormed out. The living room offers me no clues, as it's clear the housekeeper has been in to sort the mess we made, as she does twice a week. Placing my purse on the chaise, I go into our bedroom, stopping by the bed to trace the outline of where he usually sleeps before going to the closet. Everything, as far as I can see, is in its place: his watches on the center island displayed next to his collection of sunglasses and readers. There are no noticeable gaps to say that clothing was once here but is now gone\u2014even his toothbrush and shavers are here. All is in place awaiting Marcus's return. Now all I had to do was join them\u2014and wait.\n\nThe flight and drive in were easy compared to this. Hours had passed since I arrived, bringing an end to the day, and still Marcus hadn't come home. Calling him would be of no consequence, because whatever it was that would be said deserved more than a faceless delivery. If there was convincing to be done, then he needed to see my face, see that he possessed my heart without question.\n\nThe time idling was doing a number on my head, every little noise making me jump with nervous anticipation, thinking it's his key in the lock. It's unbearable! I walk over to the audio system, press Play, and crank up the volume for company. Otis Redding's rich voice fills the house, and I am hearing Marcus's soul if it could sing. I allow myself to get swept away in the melody and the understanding that for him to have been listening to Otis, he must be breathing us, reliving us. That's something to hold on to. The song abruptly stops. He's here. Standing before me. He sees me. I open my mouth to speak\u2014but he motions to me to stop, and walks over.\n\n\"May I?\" Marcus asks, taking me by the hand, and leading me into a slow dance.\n\n\"Yes,\" I say, trying to swallow the knot in my throat. \"I was scared\u2014\"\n\n\"Shhhh, shhhh, listen,\" he says, pulling me to him tenderly as Otis starts to sing, \"I Love You For More Than Words Can Say.\" Reciting each line softly in my ear, Marcus sings tenderly as he holds me tight, until we melt into one. My heart overflows, begging never to let go.\n\nHoney living without you is so painful\n\nI was tempted to call it a day\n\nYou've got me in your hand, why can't you understand\n\nI love you baby, for more than words can say\n\nI bury my face into the warmth of his neck, and exhale for the first time in a week, knowing that I desire to be only where he is, that no one, and nothing will ever matter more than what we create together. I'm dancing with my purpose. Every other word from him battled the other to express all that is in his soul, while not betraying the man, he believes I need him to be. I feel the force of his heart, beating ferociously against my chest. I have no doubts. I trust this man and all that is to come between us. Tilting my head up taking my future into full view, I wipe Marcus's tears and meet him in song.\n\nYou've got me in your hand, why can't you understand\n\nI love you honey, for more than words can say\n\n\"If you'll have me,\" I say.\n\n\"I do,\" says Marcus.\n\nThe Beginning\n\n## Acknowledgments\n\nTODD HUNTER\u2014MY EDITOR and Jules's champion. Thank you for providing the platform. Your support means a lot.\n\nAunts Ollie, Almetta, and Vivian\u2014you've championed me since '74. There are no words to convey how blessed I AM as a result!\n\nJoy Leton\u2014you, who always believes I can do anything. Reachin' kid, that's why I love you.\n\nRodrigo Santoro\u2014you and I, words can't express. You live in my heart, obviously. Really . . . Seriously, Dynasty!\n\nLisa Rudolph\u2014If asked to write about you I would start with \"You are woman phenomenally,\" but I think that's plagiarism and Ms. Maya ain't to be crossed!\n\nLeeya Myown\u2014Little Goddess, always remember that you are light and all things are for you, so dream with your entire being and enjoy.\n\nRichard Harris\u2014too many firsts were with you, my friend. I honor you the only way I know how\u2014everyday!\n\nNicole David\u2014How can one so small hold up so many? Very blessed am I to be among the lifted.\n\nShohreh Aghdashloo\u2014It's not right how very hard I crush on you, but I do. You, my dear, are grace, sophistication, and purpose.\n\nMikki Taylor\u2014for inspiring me from afar, to showing me how to be a commander in chic everyday. J'adore!\n\nStar Jones\u2014Lady, I've watched and learned. Thank you, for all.\n\nBrad Zeifman\u2014come on now, amazing (you are)! Thanks for always having a song ready. We'll work on pitch and tone soon.\n\nJanice, Bradley, and the Most Handsome Jerry Lewart\u2014you gave me Sag Harbor. Thank you!\n\nAmbassador Shabazz\u2014your friendship and counsel is prized. For all that is, and all that is to come, I celebrate you!\n\nThe village that keeps me together\u2014Tiffany Persons, Lakisha Bellamy, Jessica Teutonico, Bryce (OMG!) Wilson, Gino Columbo, Michael \"Who Me, I am French\" Sanka, Haydn Wright, Robert Marinelli, Ryan Tarpley, Shawn Howell, Judi McCreary, Brian Braff, Nancy Seltzer, Wes Carol, Adrienne Alexander, Robyn Price Pierre, Matt \"Chuck\" Davis, Tatiana Litvin, Arnold Robinson, Karen Earl, Jarrett Mason, Pat Green, Karen Cummings-Palmer, Claire Thomas, Amy Elisa Keith, Cheryl Francis Herrington, Michael Broussard, Kynderly Haskins, and Debra Langford (!).\n\nI LOVE YOU ALL, clearly!\nTAMARA N. HOUSTON is the founder and managing partner of ICON MANN and CODE RED (Carpet). Houston oversees the entertainment brand strategy and career development of professional athletes and of Academy Award\u2013nominated and Emmy-winning actors. Writing is the wonderfully cherished escape into her thoughts when the world outside gets too crowded.\n\nMEET THE AUTHORS, WATCH VIDEOS AND MORE AT\n\nSimonandSchuster.com\n\n  Facebook.com\/AtriaBooks  \n  @AtriaBooks\nWe hope you enjoyed reading this Atria Books eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Atria Books and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\nA Division of Simon & Schuster, Inc.\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nwww.SimonandSchuster.com\n\nThis book is a work of fiction. Any references to historical events, real people, or real places are used fictitiously. Other names, characters, places, and events are products of the author's imagination, and any resemblance to actual events or places or persons, living or dead, is entirely coincidental.\n\nCopyright \u00a9 2013 by Roundtable Entertainment, Inc.\n\n\"More Than Words Can Say,\" words and music by Eddie Floyd and Booker T. Jones Jr.; copyright \u00a9 1955 IRVING MUSIC, INC.; all rights reserved, used by permission; reprinted by permission of Hal Leonard Corporation.\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Atria Books Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Atria Paperback edition August 2013\n\n  and colophon are trademarks of Simon & Schuster, Inc.\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nDesigned by Dana Sloan\n\nCover design and illustration by Richard Yoo\n\nLibrary of Congress Cataloging-in-Publication Data\n\nHouston, Tamara N.\n\nWaiting for Jules \/ Tamara N. Houston. \u2014 First Atria Books paperback edition.\n\np. cm\n\n1. Women in the mass media industry\u2014Fiction. 2. Public relations\u2014Fiction. 3. Women\u2014Fiction. 4. Ambition\u2014Fiction. 5. Divorce\u2014Fiction. 6. New York (State)\u2014New York\u2014Fiction. I. Title.\n\nPS3608.O877 2013\n\n813'.6\u2014dc23\n\n2012041013\n\nISBN 978-1-4516-9851-0\n\nISBN 978-1-4516-9853-4 (ebook)\n\n## Contents\n\nEpigraph\n\nAuthor's Note\n\nChapter 1: The Return\n\nChapter 2: Meeting Mr. Michael Thurmond Kipps?\n\nChapter 3: If it Looks Like a Duck\n\nChapter 4: The Magic of Rain\n\nChapter 5: Tony & Jules\n\nChapter 6: Closure\n\nChapter 7: Carly's\n\nChapter 8: Last Good-Byes\n\nChapter 9: And You Are?\n\nChapter 10: Holland Park\n\nChapter 11: Welcome Home--I'm Back\n\nChapter 12: Meeting the Natives\n\nChapter 13: Masters, the Man . . . Not the Tournament\n\nChapter 14: A New Attitude\n\nChapter 15: Encounters\n\nChapter 16: High Tea\n\nChapter 17: Refuge\n\nChapter 18: Emergence\n\nChapter 19: Switch Hitting\n\nChapter 20: The Voice in My Head\n\nChapter 21: New Leases and Releases\n\nChapter 22: And You Are?\n\nChapter 23: Battle Lines\n\nChapter 24: When It All Falls Down\n\nChapter 25: The Pawn\n\nChapter 26: D-Day\n\nChapter 27: Boundaries\n\nChapter 28: A Gift Horse\n\nChapter 29: Plausible Deniability\n\nChapter 30: A Heap See But a Few Know\n\nChapter 31: Recap\n\nChapter 32: And I Am . . .\n\nChapter 33: Are You New?\n\nChapter 34: Two Months, Seven Days, and Spontaneous Combustion\n\nChapter 35: Sunday\n\nChapter 36: A Declarative Statement\n\nChapter 37: Strange Bedfellows\n\nChapter 38: In Our Future I See\n\nChapter 39: 1 North Wetherly Drive\n\nChapter 40: Dominoes\n\nChapter 41: Detonation in 3 . . . 2 . . . 1\n\nChapter 42: London Calling . . . Again\n\nChapter 43: Father Knows Best\n\nChapter 44: Heathrow and Lax\n\nChapter 45: Deep Breath\n\nChapter 46: Otis Redding\n\nAcknowledgments\n\nAbout Tamara N. Houston\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nContents\n\nDEAD SURE, by Herbert Brean\n\nCOPYRIGHT INFORMATION\n\nALSO BY HERBERT BREAN\n\nPART ONE: THE C-NOTE\n\nCHAPTER 1\n\nCHAPTER 2\n\nCHAPTER 3\n\nCHAPTER 4\n\nCHAPTER 5\n\nCHAPTER 6\n\nCHAPTER 7\n\nPART TWO: THE BIG NEWS\n\nCHAPTER 8\n\nCHAPTER 9\n\nCHAPTER 10\n\nCHAPTER 11\n\nCHAPTER 12\n\nCHAPTER 13\n\nCHAPTER 14\n\nCHAPTER 15\n\nCHAPTER 16\n\nCHAPTER 17\n\nCHAPTER 18\n\nPART THREE: THE BLOW-OFF\n\nCHAPTER 19\n\nCHAPTER 20\n\nCHAPTER 21\n\nCHAPTER 22\n\nCHAPTER 23\n\nCHAPTER 24\n\nCHAPTER 25\n\nCHAPTER 26\n\nCHAPTER 27\nDEAD SURE, by Herbert Brean\n\nAlso published as A MATTER OF FACT\nCOPYRIGHT INFORMATION\n\nCopyright \u00a9 1956 by Herbert Brean.\nALSO BY HERBERT BREAN\n\nWilders Walk Away\n\nThe Darker the Night\n\nHardly a Man is Now Alive\n\nThe Clock Strikes Thirteen\n\nDead Sure (also published as A Matter of Fact)\n\nThe Traces of Brillhart\n\nThe Traces of Merrilee56DEDICATION\n\nFor Dorothy\u2014all by herself.\nPART ONE: THE C-NOTE\nCHAPTER 1\n\nThe Double Tail\n\nShadowy darkness and cold filled the street, the sudden bone-penetrating cold of early November that sends a man indoors and holds him there. Darkness was appropriate to this street as a black garment to a nocturnal thief, for it was a mean, shabby thoroughfare, walled in on either side by rows of high brownstone houses that frowned down on passers-by by day and menaced them by night, obscurely but perceptibly. Before the houses stood unbroken rows of automobiles, illegally parked as though their owners had left them there only long enough to snatch a few hours' sleep before driving away forever from this glowering neighborhood.\n\nMeanwhile the night-black windows of houses and cars reflected back the street lamps' murky light at either end of the block. Only an occasional rectangle of orange showed where someone still sat up, mesmerized by television or solitaire, or absorbed in the evening paper or a wrangle with the old woman. On the sidewalk dusty cans of ashes awaited next morning's sanitation truck, and in the middle of the block a small bulb burned over the Night Bell sign beside a garage's door.\n\nA tall man swung into the street from the Third Avenue end, moving with the long, confident strides of the hard-muscled. As he passed under the street light it showed him lean and high-shouldered, wearing a peaked cap like a hunter's and a heavy jacket of checkered wool. In his right hand he carried a brown paper sack that clunked in heavy rhythm to the crunch of his shoes. The street light spread a lengthening, writhing shadow of himself before him that merged with the other darkness and, as he progressed, finally vanished.\n\nWhen this man was midway down the block a second man appeared at the Third Avenue end. He snapped a quick glance after the first that was like a silent pistol shot and, having seen what he wanted, continued on across the lighted intersection, looking straight ahead without turning into the dark side street. When he reached the opposite curb, he turned quickly on silent feet and began paralleling the first man from the opposite sidewalk.\n\nHe walked rapidly and carefully, putting his weight on rubber heels so that he would not be heard above the faint, endless grinding that is the sound of Manhattan an hour before midnight. As he did he strained his eyes after the first man, for he knew there was a good chance that he was being ambushed. You were taught to guard against things like that in the police academy, and the second man, whose name was O'Neill Ryan and who was a probationary-detective, had completed his course there only two weeks before.\n\nBut if the man in the wool jacket knew that he was being followed, he disguised it very convincingly. Three quarters of the way down the block he turned unhurriedly, a flicker of black in the larger darkness, into one of the brownstones and took the tall steps two at a time. He whistled a snatch of jukebox song, opened the door into an ill-lit hall, and Ryan, moving almost soundlessly along the opposite sidewalk, caught the diminuendo of boots on a rugless stair before the door swung shut.\n\nThen the street was silent again. Ryan paused behind a parked station wagon.\n\nHe was uncertain and scared. He looked back along the way he had come for his partner, who was the senior member of the detective team. For the last fifteen minutes they had been double-tailing the man who had gone into the house. In double-tailing a detective follows a suspect while a second detective follows the first detective, out of sight of the quarry. After an agreed-on number of blocks or minutes the second moves up into the lead position while the first detective drops back and follows the second. It is complicated, requiring quick and cooperative intelligences, but a subject who is on guard against being followed is much less likely to notice this kind of tail.\n\nWhat bothered Ryan was that he wanted to let his partner, who was not yet in sight, know where their quarry had gone, which meant staying where he was. But he also knew that if the quarry had observed the tail he might have entered the house only to walk through it quickly and escape out the back. Logic indicated he should go to the rear, but that meant missing Jablonski, his partner. A rookie's agony of eagerness and uncertainty seized Ryan.\n\nA wedge-shaped silhouette appeared at the end of the street. Ryan relaxed. That was Jablonski\u2014thick, routine-minded, unimaginative, but solid and knowing. And in command.\n\nJablonski did just what Ryan had done. He crossed the intersection and then turned into the dark street, coming along softly, watchfully. Ryan whistled from the station wagon's shadow.\n\n\"Where is he?\" Jablonski whispered back.\n\n\"There. Where the light is in the hall.\"\n\n\"Did he make us?\"\n\n\"I didn't see him look back. But what can you see in this damned street?\"\n\n\"It don't figure,\" said Jablonski. His mouth made chewing movements as though he were rolling a cigar between his lips.\n\n\"Maybe it's a hideout.\"\n\n\"Hideout nothing! It don't figure. He's shakin' us. I'll hit the back.\"\n\nHe moved hurriedly out from between parked cars.\n\n\"What about help?\" Ryan whispered hoarsely into the darkness after him.\n\nJablonski only waved a hand in irritation behind him and went on across the street, a bulky, grizzled, worried man of fifty-three. Jablonski really did not want help. It would have been absurd for anyone to consider him brave for feeling that way, and Jablonski would have been the first to laugh at such an idea. He did not greatly venerate courage, regarding it as a quality usually reserved for rookies and saps. But Jablonski had personal, hardheaded reasons for not wanting, for very much not wanting, any help on this arrest.\n\nThat is why he walked quickly up to the door of a passageway between the two houses across the street, pushed it open and stepped into the dark passage which, he knew after years of working neighborhoods like this, would take him into whatever backyard or court lay behind the houses. He was lucky it was there\u2014but it would be a hell of a place to meet Derby.\n\nAfter a minute he flashed his light down the black passage. He saw only twin walls of eroded old brick, chalked with children's scribbling. He clicked out the flashlight and walked down the passage in cold darkness. Doing that he comforted himself with the somewhat specious reasoning that if Derby, a cop-hater, three-time loser and now a killer\u2014if Derby had spotted them, he was already out the back of this building and was gone. If not, then they had him. And at that Jablonski sucked in his breath, audibly, hopefully.\n\nHe reached the end of the passage. It opened into a small yard, faintly starlit and surrounded by the high old garden wall of another era. Above him a window suddenly rasped open, and his hand dove under his overcoat's left lapel. There came a harsh, strangling sound; someone spat out the window, and did not close it.\n\nPresently Jablonski moved across the yard to the wall. On top of it his exploring hands encountered pieces of broken glass, a futile deterrent to juvenile marauders. He picked the glass off, placing the pieces carefully on the ground, then took out his gun and, holding it, drew himself up and over the wall. He eased down into the yard of the house where Derby was. No light showed from the windows. Jablonski waited a few minutes to let his arms recover their strength so his hands would be steady if he needed them to be. As he did this his eyes adjusted to the yard's darkness, and he thought he saw a back porch with a door, and below the porch another door leading into the cellar.\n\nJablonski stole across the yard, went almost silently down some steps and tried the cellar door. He revolved the knob several times while he leaned his weight against it to see if the door was locked. Doing that he knew he was an unmissable target for anyone standing with a gun on the other side of the door. Nothing happened. Jablonski cautiously climbed the steps and tried the porch door in the same way with the same result. Then his thick, pursed lips relaxed in a complacent smile.\n\nThey had him.\n\nFor if Derby had spotted the tail and simply walked through the house to escape them he would have been in a hurry. He would not have bothered to lock any doors behind him. He was inside then and didn't know they were there at all. It was perfect!\n\nRyan and Jablonski had picked up Derby only twenty minutes before on Third Avenue, just as he had gone into a delicatessen to buy some beer, and Jablonski surmised that Derby planned to sit up for a while, drinking it. That was perfect, too. It's eleven-thirty now. Give him until one o'clock. The beer will make him sleepy.\n\nJablonski returned to the wall, wrapped his overcoat around himself and sat down in the shadow. He could assume Ryan would interpret no news as good news and would continue to watch the front. All they had to do now was to wait, and the only thing that could go wrong would be for Ryan somehow to manage to send back for help. Good strategy demanded that they call in for reinforcements, but Jablonski for urgent personal reasons wanted to make this arrest himself. He settled himself in his overcoat and wished for one of the cigars he had been about to buy when they had spotted Derby.\n\nTime passed. No lights appeared in the house's rear windows. An ambulance whined down Second Avenue. Jablonski waited. He rolled an imaginary cigar in his mouth and thought of Derby, sipping beer out of a can, getting heavy-eyed, sleepy, dull of wit...\n\nThat's what Jablonski was waiting for. He was good at waiting.\n\nTime passed.\n\n* * * *\n\nRyan stood beside the station wagon where Jablonski had left him, hunched over for warmth and concealment, eyes never leaving the front door across the street for more than a fleeting second. A feeling of imminent triumph filled him; Jablonski's silence was the best news in the world.\n\nYet Ryan was jittery. He was not afraid of Derby, even though Derby was the hottest guy in town tonight, a three-time loser who would get life at best but who far more likely was going to the chair. Ryan would never have become a cop and stayed a cop if he did not believe innately that he was smarter, faster and tougher than anyone he would ever go up against.\n\nWhat worried him was that through his own inexperience or overeagerness he would wreck the delicate, dangerous strategy that he and Jablonski had decided on earlier that night, casually and almost jokingly.\n\nFor as the evening wore on they had felt the mounting pressure\u2014on them and on the department of which they were but a unit. It wasn't merely the big type on the tabloids' early editions, nor the thick, indignant anger in the lieutenant's voice when they checked in at the precinct. Nor the unwonted quiet there or the knowledge that the commissioner had ordered direct reports to be telephoned to his home throughout the night. No, it was some deeper, policeman's instinct, born of experience and a feeling for the rhythm of the city's emotions, that told them both the Connors murder would be a heavy one. That was what had been in Jablonski's mind as their plain black sedan idled up one East Side street and down another. That was why he had casually remarked that if they happened to stumble on any trace of Derby, it sure wouldn't do them any harm to work it out on their own.\n\nRyan had braked the car gently to a stop for a red light at Madison Avenue and looked through the windshield. He had a boyish face and dark, round, long-lashed eyes that held a thoughtfully mischievous expression which women found attractive. It sure wouldn't do them any harm, Ryan had agreed.\n\nHe did not look at Jablonski; he did not have to. In that instant the pact was made, and they both understood it.\n\nEdmund Aloysius Jablonski was retiring in less than two weeks after a career of twenty-eight years in the police department. As his rank of Detective Third Grade suggested, Jablonski's had been a quiet, lackluster career. He was an uninspired man who had begun to realize as retirement neared that somewhere along the trail he must have missed opportunities. That was bitter, helpless knowledge.\n\nAnd now Harry Derby represented the greatest opportunity of them all.\n\nDerby was a bright chance for Ryan also. For four years Ryan had walked a beat to get this tryout as a detective. He had been paired with the older man, in accordance with departmental custom, to gain experience and also to be observed. He had been disappointed in the man he was paired with, recognizing Jablonski as plodding, tired and inept. But tonight Jabby was acting and sounding like an eager kid. It made Ryan feel hopeful, because to arrest someone like Derby would do him more good...\n\nAnd then\u2014it had seemed incredible, for Ryan was new at this end of the business\u2014then he had seen something.\n\nSwinging down Third, walking fast but not furtively, bathed briefly in the colors of the neon signs that he passed, a lean, hard-muscled man with a cold eye and a curt sneer\u2014that was Derby.\n\nSince their car was not a Radio Motor Patrol car and had no two-way radio, departmental routine dictated that they split up, one telephoning in for help while the other tailed Derby to wherever he was going and whomever he was meeting, for there had been an indication this afternoon that he had had an accomplice.\n\nBut Ryan and Jablonski had already made their decision. They would bring him in themselves. That dangerous decision was what was in Ryan's mind now as he stood minute after minute, feeling the cold creep up through the soles of his shoes and an ache grow in the small of his back. If it came to shooting and he shot recklessly or held his fire when\u2014Stop that! The hall light across the way blinked out.\n\nWas he coming out?\n\nRyan's right hand dove under overcoat and suit coat and gripped the warm, snub-nosed .38 holstered under his left arm. But no one came out of the house. Ryan concluded it was just the janitor turning out the lights.\n\nA car turned into the street. Ryan hoped it was not an RMP car for then he could have no reason for not asking for help. It was not an RMP car and he felt relieved, and was pleased at his own relief.\n\nBut why the hell didn't Jablonski\u2014!\n\nRyan arched his back and reminded himself that Jablonski was doing the same thing. His watch showed five minutes of one. They had him trapped.\n\nAll they had to do was take him cleanly.\nCHAPTER 2\n\nThe China Chip\n\nSome ten hours earlier a thin sixty-three-year-old woman named Thelma Connors had left the small gloomy apartment she shared with her daughter Elaine on East Sixty-first Street and walked to a branch bank on Lexington Avenue a few blocks away. There she drew out one hundred and twenty dollars from their joint savings account, leaving only twelve dollars to keep the account alive. A white-haired, pathetically proper woman with a slight limp that expensive surgery could have repaired, she confided to the teller that she and Elaine were leaving by interstate bus that night for Chicago and a visit with her son, and she specifically asked for a hundred dollar bill to pin to her clothing for safekeeping.\n\nLater, under the deluge of probing questions by detectives, the teller recalled that a tall man whom he did not remember ever seeing in the bank before, a man in some kind of wool jacket or shirt, had stood in the line with several others waiting for Mrs. Connors to complete her garrulous transaction. The teller could not recall definitely whether or not the man had come up to the window. If he had it had been only to get a bill changed.\n\nMrs. Connors stopped at a nearby grocery for some sandwich meat and then walked home in crystal November sunshine, a happily excited wisp of woman in a decent black cloth coat. A neighbor, one Mrs. Anders, who was sitting on the steps of a dingy apartment building, spoke to Mrs. Connors. A moment later Mrs. Anders saw a man walk up the steps and go into the vestibule. He seemed to study the mailboxes, and when she asked who he was looking for, he did not reply or look at her but went into the building. It was his incivility more than anything else that made her remember him later: a tall, thin man with a bitter face and wearing a checkered woolen jacket.\n\nThe wife of the building janitor, a Mrs. Lombardi, also glimpsed him. She was coming down from the fourth floor when she saw him standing before Mrs. Connors' apartment on the second. She heard him say something like, \"I'm from the office, ma'am,\" when the door opened and she saw him step quickly in.\n\nShe went on downstairs, noticed Mrs. Anders outside and paused to smoke a cigarette with her. They were t\u00eate-\u00e0-t\u00eate over a match when the first outcry came. They looked at each other over the burning match.\n\n\"Did you hear that?\"\n\n\"A kid maybe?\"\n\nThey paused, uneasily sensing something wrong yet unwilling to accept that anyone near them could be in such fearful distress as was expressed by that quick yell of terror.\n\nThere came a weaker cry and the faint vibrating sound of something falling and then a last, loud cry that was punctuated by a shot.\n\n\"My God!\" Mrs. Anders' hand went guiltily to her mouth. They knew they should have taken the meaning of the earlier scream and gone to someone's assistance.\n\nThey ran into the hall together and saw a tall, rangy man running toward the back door. Mrs. Lombardi yelled, \"Stop thief!\" and the man turned a snarling face at them as he ran. He shouted something which neither of them could even begin to describe later, and which might have been directed at someone ahead of him, a confederate perhaps. The door banged behind him.\n\nThe man was seen by someone else. A telephone company employee named Betty Leonard, who shared an apartment with another telephone operator on the second floor, had been preparing to leave for work. Hearing the scream and shot, she apprehensively opened her door a slit and saw the man in the wool jacket gallop downstairs.\n\nShe went to the Connors' door and called. Getting no reply and hearing Mrs. Lombardi's hail from below, she pushed her way, frightened, into the living room. It was darkly gloomy but one light bulb was shining oddly up at her from the carpet.\n\nThen she saw the rest of it and she too screamed.\n\n* * * *\n\nThe usual RMP car got there first. Its crew found Mrs. Connors lying face up near the living-room table, a bloody laceration on the right side of her forehead and blood running into her sightless eyes. The remains of a table lamp which apparently had been used as a bludgeon lay beside her, its cheap pottery base smashed into a thousand fragments, its bulb incongruously alight. In the right side of her head above the ear, black-haloing the thin silvery hair, was a bullet hole.\n\nOne of the RMP men called the precinct and the routine notifications began. The precinct sent two detective teams at once; one of them was Ryan and Jablonski, who when the call came in had been looking over the post condition on the bulletin board in the East Fifty-first Street station before starting the afternoon trick.\n\nThe radio car was still double-parked in front of the apartment when they got there. Going up the steps they took out gold, blue-centered badges and pinned them to their coats so cops from other squads and bureaus would know they were policemen and not reporters or the ever-present curious.\n\nThe deputy chief inspector who was commanding officer of Manhattan East detectives was already there, and his early presence was enormously significant. For several weeks there had been a mounting wave of assaults and muggings throughout Manhattan and Brooklyn; the newspapers had begun to get sharply critical. Ryan and Jablonski went into the apartment briefly to take a look and tacitly let the deputy chief know they were there. He gave them an angry look as though this somehow was their fault, and they went back into the hall to await orders and not clutter up the place. That was all routine.\n\nThe RMP man on the door repeated to them what the women had told him and his partner. The deputy inspector, a heavy-faced man with hair like springy black wire, looked out the door.\n\n\"Seventeenth Squad, sir,\" said Jablonski.\n\n\"Start on the tenants. The guy who did it was seen. It was a clumsy job\u2014he may have been an amateur. Ask about relatives. There's a daughter and we've sent down for her. She's at work.\"\n\n\"Yes, sir.\"\n\nThe Medical Examiner and his assistant came up the stairway, laden and puffing.\n\nIn thirty minutes Jablonski and Ryan and one of the other teams that arrived after them talked to all the tenants who were at home, learned who knew the Connors, and had located who had heard or seen anything of the intruder. Meanwhile the Medical Examiner reached a preliminary verdict: death by bullet wound, which had occurred at the time the other witnesses heard the shot. The men from the Picture Gallery made photographs, and then two serious-looking lab men took over, opened a heavy black kit and began dusting for fingerprints, making preliminary blood stain tests and examining the dead woman's clothing, hands and fingernails. It was routine.\n\nBut the crime wasn't. They all felt that. One of the picture men, packing away his camera and tripod, paused to study the slender figure crumpled on the floor like a dropped handkerchief and said, \"Jeez,\" softly.\n\n\"Yeah,\" said a detective next to him. \"This was a real son of a bitch.\"\n\n\"I can't wait to read the Telegram tonight,\" said the picture man sarcastically.\n\nThis was a heavy one. It had to be cleaned up fast, and whether it was depended on themselves\u2014on their pooled skills and brains, courage and tenacity, on the clues they found and how assiduously they ran them down, on the empirical soundness of the conclusions they drew, on the leads they developed and the patience with which those leads were proven and disproven.\n\nThe precinct relayed word that the team that had gone down for the daughter reported she said her mother had been going to the bank that day. Lieutenant Bauer who had come over from the precinct growled an order and two detectives ran downstairs and hurried to the bank. Jablonski in response to another order assembled the three principal witnesses in Miss Leonard's chintz-decorated apartment. When they had seated themselves expectantly, Jablonski began to talk earnestly, while Ryan leaned against the door and felt useless.\n\n\"My name's Jablonski,\" Jablonski began. \"This is my partner, Detective Ryan. A very serious crime was committed here an hour ago, ladies. A man followed your neighbor Mrs. Connors home from the bank, tried to rob her and hit her with a lamp. When that failed to...ah, render her unconscious, he killed her. You ladies saw him. You're the only persons who did.\"\n\nThey listened to what they already knew with hypnotized interest.\n\n\"It's likely,\" Jablonski went on, \"that, the killer was a known criminal, known to the police, that is. So if you ladies will cooperate, it may be possible for us to find out who he is right away and effect an arrest, ah, immediately.\"\n\nConfronted with an audience, Jablonski used phrases he hardly knew were in his lexicon.\n\n\"What do you want us to do?\" asked Miss Leonard.\n\n\"We'd like to drive you down to police headquarters,\" said Jablonski, \"and show you some pictures of muggers and robbers who answer this guy's description. It won't take long. All you do is look at the pictures and if you recognize him, tell us. We'll drive you back, of course. Okay, girls?\"\n\nEagerness made Jablonski grow too friendly, Ryan recognized. That made the women suspicious.\n\n\"I've got a switchboard to get to,\" said Miss Leonard determinedly. \"I'm late as it is.\"\n\n\"I'll call your supervisor for you, miss, and explain how things are,\" Ryan offered politely. He smiled diffidently at her, as though he thought she was pretty nice-looking.\n\nShe smiled back. He was nice-looking himself, she thought. \"Well, if you did that, Mr. Ryan,\" she said more cordially.\n\n\"Yeah,\" said Mrs. Lombardi in another tone. \"But supposing we pick out this guy's picture and then his friends come looking for us, huh? How about that, mister?\"\n\nMrs. Anders looked frightened as the meaning struck her. \"Yes\u2014how about that? It's all right for you policemen\u2014\"\n\n\"Look, girls,\" said Jablonski anxiously. \"Is this too much to ask for the sake of your neighbor, Mrs. Connors, lying dead in there? Aren't you willing to even\u2014\" He gestured helplessly.\n\nThe women looked rebellious.\n\nRyan moved from the door.\n\n\"Look, ladies,\" he said. \"All you do is drive downtown with us and look at some pictures. If you recognize one, just point. Nobody knows who picked it out or anything else.\" His glance deliberately engaged Betty Leonard's. \"Won't you please help us?\" he said gently, and he knew that he had her.\n\nHe looked at the two older women. \"It's just a matter of our getting our hands on the man who did this before he can come back here or attack some other helpless woman in a building like this. Won't you please help do that?\"\n\nIt was an appeal Mrs. Lombardi could not resist. She rose. \"C'mon, Mary,\" she said.\n\nJablonski looked at his partner in surprised approval.\n\n* * * *\n\nThe rite in the Bureau of Criminal Identification did not take long, and to the women it even seemed rather casual. Each of them was handed a deck of small photographs and asked to leaf through them slowly. For a few minutes there was absorbed silence, broken by an occasional mutter from Mrs. Lombardi, while Ryan, Jablonski and a BCI man watched them. It was Mrs. Anders who first said. \"Well, this looks like him, I must say.\"\n\nThe BCI man took the picture and when she had finished with her stack of pictures, re-inserted it in a different position and gave the stack to Mrs. Lombardi. Minutes later she cried out, \"Oh, that's him! I'm sure that was him.\"\n\nIt was the same picture.\n\nNow the stack went to Miss Leonard. She had had the best look at the man. They watched her narrowly while pretending not to, and when suddenly she emitted a little frightened cry, Ryan and the BCI man breathed a sigh of relief. They knew what that meant.\n\nThe BCI man showed the women other pictures of the suspect, and those confirmed the identification. Then he took his record from a file:\n\nHarry Derby, 37 years old, male, white. Brown eyes; light brown hair; 6 feet 1\u00bd inches; 163 pounds. Longshoreman; North River and Brooklyn docks and bars.\n\nUnder \"Distinctive Marks\" it said: jagged scar on right shin, pockmarks on left shoulder and left neck, almost invisible knife scar leading from outer corner of left eye. The list of Derby's arrests took two cards clipped together; they were mostly for felonious assaults, carrying concealed weapons and robbery or suspicion of robbery.\n\nMinutes later the news that Harry Derby was wanted for murder and his description were being teletyped into the hundred and more New York City precinct stations as well as the police headquarters and state police barracks of thirteen Eastern seaboard states. The bulletin ended: \"Approach with caution. He is known to be armed.\"\n\nRyan and Jablonski drove the three women home through the clear, cold November twilight and the endless traffic jam of 5:30 p.m.\n\n* * * *\n\nThe cluster of photographers and reporters that had been held downstairs in the apartment building's front hall was gone but there was still a uniformed man on the door.\n\n\"Anyone up there?\" Jablonski asked.\n\n\"Ed Furtig,\" the uniformed man said. \"He's carrying it.\"\n\nIt was a curiously expressive bit of police slang, Ryan thought, not for the first time. He's carrying it. The homicide men, the lab men, the identification and picture gallery men all would do what their specialties demanded in avenging the murder of Thelma Connors. But it had been assigned especially to one man, a detective in the precinct in which the crime occurred. It was primarily his responsibility, his burden to carry until the murderer was brought to trial\u2014next week, next month, next year.\n\nEd Furtig sat on the arm of a deep upholstered chair near the window, arms folded, hat pushed back, smoke spiraling from the cigarette in the corner of his mouth.\n\n\"Hi, Jab. The desk said you guys should hit the street.\"\n\n\"Okay. All yours, huh?\"\n\n\"All mine,\" said Furtig. His lean flexible face grew sardonic. \"Naturally. I was starting a forty-eight at midnight.\" Furtig used the old police expression, even though the department \"week end\" was now fifty-six hours.\n\n\"They get any prints?\"\n\nFurtig shook his head. \"Too early to tell. One of the boys thought he got a couple partials.\"\n\nWhile Ryan listened, his eyes ranged over the room. This, was his first good look at it. A few cheap framed pictures decorated the walls, and immaculate doilies the tables and chairbacks. A large crucifix hung on one wall, a yellowed sprig of palm thrust behind it.\n\n\"The daughter have anything?\"\n\n\"God, no. She flipped. A doctor's with her now, down in the janitor's apartment. One thing she did say\u2014the old lady had some cuff links she wore with a shirtwaist. One was lying near her, but the other wasn't anywhere around. The killer may have carried it away unknowingly, if it caught in his clothes.\"\n\nJablonski said nothing. It was something to be remembered, but it would probably never turn up. Things did not happen that conveniently in real life. Ryan, listening, continued looking around.\n\nThere were little bursts of fingerprint powder, black and light gray, in unlikely places around doors and table edges. The smashed lamp was gone, taken to the lab for minute examination. The only real sign of violence left in this plain sitting room was their own presence\u2014uninvited strangers who would not be here at all had an old woman's life been permitted to continue its even way.\n\nNo\u2014there was one other sign of violence. Behind a chair leg there was a small yellow chip of china from the shattered lamp base. Ryan knew that it was no longer of evidential value, so he picked it up, studied it and then, reminding himself that this was the first murder in his four years on the force in which he had an active, investigative part he dropped it in his coat pocket on the universal impulse of the souvenir hunter.\n\n\"Well...here's hopin', Ed,\" said Jablonski.\n\n\"Thanks.\"\n\n\"We better roll, Neill.\"\n\nThey filed out. As he turned to close the door behind them Ryan saw Furtig still sitting hunched forward, smoke drifting up past squinting, unseeing eyes as he pondered what had happened, and the nature of the man who had caused it, and how he might best be overtaken. It was his to carry.\nCHAPTER 3\n\nThe Walk-In\n\nOne o'clock, the deadline that he had set himself, came and went but Jablonski did not move. Let it get late, he thought with Polish stolidity. The later the better. Then, the quiet location of Derby's room, the sudden break-in, the quick pinioning and blinding light, the quick hand under the pillow for a gun...\n\nIt was a familiar operation to Jablonski. Meanwhile he thought about a cozy little bar and grill near the New Haven Railroad station in New Rochelle in which he had a chance to buy a half-interest. It had a juke-box and a Puerto Rican chef who made hamburgers with onions and sour cream, the best you ever tasted, and a steady clientele. If they got Derby\u2014if, nothing!\u2014it might help swing the deal. Publicity wouldn't do him a bit of harm. People like a joint run by a guy with some background\u2014a fighter, or a former ball player, or a cop. Jablonski saw himself lounging behind the bar, chatting with the respectful customers and buying a drink every now and then. \"I mind the time we were looking for the Actor,\" he was telling his audience. Or, \"The D.A.? Well, I tell you how we always had to handle the D.A.'s office. The time I put a collar on Derby, for example...\"\n\nThat warm vision comforted Jablonski. It made the barbed remarks of superiors over the years, the times he was passed over for promotion, the envy and the failures, recede into their own limbo. The time passed quickly, even though he sat in darkness on cold ground and knew this might bring on another attack of rheumatism.\n\nBut at twenty minutes after one Jablonski decided it was time. Derby had had well over two hours in which to drink his beer, and besides if an RMP came through he was not sure Ryan might not hail it despite their agreement. Jablonski rose, stretched his legs a moment and went back over the wall. For one of his bulk he moved with surprising quietness.\n\nHe was halfway down the silent, pitch-black passageway on his return to the street when the alley door at the far end opened. A man was silhouetted against the dull light from the street.\n\nJablonski froze and took out his gun. He heard a scuffling noise and then a flashlight flooded the passage, blinding him and Ryan whispered, \"Jabby!\"\n\n\"Put out that damned light!\" He walked forward in blackness until he touched Ryan. \"What's going on?\"\n\n\"Nothing. I just began to wonder, for Pete's sake\u2014\"\n\n\"Never mind. You didn't send word back?\"\n\n\"It's been quiet as a tomb.\"\n\nJablonski led the way to the street and then walked down it, out of range of the house's silent windows. The nocturnal wind was rising. It was getting colder.\n\n\"He's in there,\" he said. \"There's two back doors. Both are locked.\"\n\n\"Good.\"\n\n\"Yeah.\" Again Jablonski wished for a cigar. Now the time had come, he felt nervous. It seemed all he had read in the last few years had been newspaper stories about cops killed on the eve of their retirement. Then he heard the blare of the jukebox in that bar, the hamburgers sizzling in onion sauce, saw the fat black bottles...\n\n\"What do you think, Neill?\" he said. He wanted to hear how Ryan might sound. He looked sidelong at him.\n\n\"Let's hit it,\" said Ryan. \"He must be sleeping by now. A guy can't take this long to drink four cans of beer.\"\n\nThat eagerness was what Jablonski wanted to hear. \"Okay,\" he said. \"Let's go. It'll be a nice break for you, kid.\"\n\nRyan recognized the hypocrisy but he disregarded it. His mind was on the job ahead. For if something went wrong or Derby got away from them...the mere possibility of it gave him a cold shiver.\n\nJablonski grabbed his arm and pulled him to a halt. \"Just one thing, Neill,\" he said. \"Don't forget who Derby is. Don't be afraid to shoot. Nobody cares if he gets hurt. But if we do, or if he gets away\u2014\"\n\nRyan tapped Jabby's restraining arm, heavy with fatty muscle. \"Right,\" he said. \"Let's go.\" He sounded confident.\n\nThey went quietly along the sidewalk. The cold night stretched far and wide over them and over all the city; the stars' reflections glittered in the oil-shiny water of its rivers and bays. Deep in subway tunnels the infrequent trains screeched wheel flanges against the cold satin rails, and in fetid nightclubs trumpets squawked to drum thumps. In skyscrapers' remote corridors old women flung gray mops across silent floors and in bright newspaper buildings, molten lead and electric teletype bubbled and clicked against deadlines, telling the latest news of the hunt for Harry Derby. In apartments young mothers rose from warm beds for the two o'clock feeding. But in flophouse or penthouse, most of the city slept. Ryan and Jablonski walked up the steps of what once had been a mansion and was now a musty rooming house.\n\nThere was a row of battered letter boxes, most of whose little metal doors hung askew and nameless. Mail did not mean much here. There was no bell or sign marked superintendent. That was bad.\n\n\"I think I heard him go up or down the stairs,\" said Ryan.\n\nJablonski pushed the door open, and they stepped into a high-ceilinged old hall heavy with the scent of cooking and rotting plaster. Jablonski flashed his light around. There was a massive, sagging staircase of dark wood at the end of the hall. He pulled a dangling string and the feeble hall light came on.\n\n\"You stay here. I'll try the basement.\"\n\n\"Isn't that light liable\u2014\"\n\n\"Anyone comin' in late could have turned it on,\" said Jablonski patiently. \"Now remember, them back doors are locked on the inside. If he breaks, they will slow him a few seconds when he tries to get out. We may need those seconds, so remember.\" He tiptoed to the back stairs and descended them.\n\nIt was cold in the house, a damp, unhealthy cold more chilling than the street's fresh gusts. Ryan's chest began to feel tight again. It was all right as long as they were moving or acting\u2014! Suddenly he realized that he could smoke in here. He lit a cigarette.\n\nJablonski reappeared at the top of the stairs, listened at two doors in the back of the hall and came forward. He looked troubled. \"There's one sort of room downstairs, but I don't think nobody's in it. Not a sound. Gimme one of your cheroots, will you?\"\n\nRyan handed him the pack and flicked his lighter. Jablonski bent over it, holding the cigarette awkwardly to unaccustomed lips. \"I'm goin' to try the second floor.\"\n\n\"Let me take the second floor.\"\n\n\"Me,\" said Jablonski. He continued smoking hurriedly in silence for a few minutes. \"Wish you'd carry cigars, Neill,\" he grinned, then ground the stub under his heel and started up the stairs.\n\nIn a minute he was back. \"The super is on the second floor,\" he whispered. \"There's a sign. And I can hear someone threshing around in there. No lights.\" His breath was flavored with the knackwurst he had had for supper.\n\n\"You think he's in there?\"\n\n\"Either he is or he ain't. If he ain't the super tells us what room he's in. And if that's how it is he might well have made for here after the job. It was a sudden job. He probably didn't have no hideout planned. Huh?\"\n\nRyan nodded. \"Let's go,\" he said. This was the finale. Suddenly he felt good, not nervous or anxious or tight in the chest, but clear, sure, quick. It was the feeling of being in danger and knowing you could handle it. It was wonderful.\n\nJablonski led the way to the big staircase. \"Me first,\" he whispered. \"If we go up together the stairs may creak.\"\n\nAgain Ryan nodded. He waited until Jablonski's portly back disappeared in the blackness above, then followed him. Little light filtered up here from the weak bulb below. Their feet made soft, sandy sounds in the grit of the hall's unswept floor. Jablonski stopped at a door near the end of the hall and leaned over the knob.\n\nAfter a time he twisted his bulky body close to Ryan for quiet communication. \"It ain't locked!\"\n\n\"Good. How do you want to do it?\"\n\n\"I'll swing the door open and you go in fast with your light. Keep low and get well into the room before you turn it on. I'll cover you from the door\u2014so for Christ's sake, keep down.\"\n\n\"Right.\"\n\n\"And. Neill\u2014\"\n\n\"Yeah?\"\n\n\"Remember, we don't know who's in there and we're going to surprise them. So whatever they do, don't start shooting until you're sure it's Derby.\"\n\n\"Yeah.\"\n\nRyan held his flashlight in his left hand, his revolver in his right. He could feel the heavy throb of his heart. The back of his neck was cold. Jablonski swung the door back. It creaked.\n\nCrouching, Ryan moved quickly through the doorway, took three steps into the room's blackness and then one to the right. His thumb solidly found the flashlight button and clicked it. The first beam showed him the foot of an old iron bed, the white enamel generously chipped, then the frightened face of a middle-aged woman, red, puffy and framed by short, frizzled hair.\n\n\"What's that?\" She blinked in the light.\n\n\"Someone's in bed with her,\" warned Jablonski from the darkness.\n\nRyan flashed the light around.\n\n\"Who the hell is it?\" the woman demanded again, frightened.\n\nJablonski advanced into the room. Ryan's beam had caught a string with a little doll tied to it dangling from the ceiling. Jablonski pulled it and an ancient chandelier spread light around the frowsy bedroom. Beside the woman another figure lay in the bed, blankets pulled up tightly around its head. Jablonski stood over it. Ryan took his place on the other side of the bed.\n\n\"We're cops,\" he said. \"Who's that next to you?\"\n\nHate and fear burned in her look. \"Get out of here.\"\n\nJablonski dug his gun into the blanket cocoon, felt under its pillow, then spoke to the woman. \"Look,\" he said. \"Tell your boy friend to open up. If he's not a guy named Derby he has nothing to worry about\u2014we don't care what else is going on here. Get me?\"\n\nThere was a second of immobile silence.\n\nJablonski dug harder with the gun. \"If that's Derby,\" he said more loudly, \"he either opens up or he gets a bullet in the guts. If it ain't, he gets the bullet anyway if he don't show his face. One...\"\n\nThe woman reached over and seized the bedclothes. Two fine, thin hands appeared from beneath them and grabbed them desperately.\n\n\"...two,\" said Jablonski.\n\nA man's high forehead appeared, then a face, big-eyed, frightened, then a mustache that was like a hyphen over tight, pale lips. Ryan relaxed.\n\n\"Now will you get outta here?\" the woman cried.\n\n\"Shuddap!\" said Jablonski savagely. They had made a mistake, but what was worse, they had made too much noise. That worried him.\n\nFor the second time Ryan stepped in to the older man's rescue. He said, \"Look, lady. We're looking for a man named Derby\u2014Harry Derby.\"\n\n\"The guy they had on the radio tonight\u2014who killed the old lady?\"\n\n\"That's the guy.\"\n\n\"Well, there ain't nobody named Derby in this house. So you two crapheads\u2014\"\n\n\"Harry Derby,\" Ryan repeated patiently. \"Better listen, lady. I'd rather not have to take you in. But I will if you make me do it. We know he's in this house\u2014right now. He's tall. Thin. Brown-haired. Wearing a checked jacket. Might have moved in here this afternoon.\" His tone changed to angry urgency. \"Now where is he?\u2014and talk low!\"\n\nThe man was listening owlishly, only his head visible over the bedclothes.\n\n\"For God's sake, Maud, tell 'em what they want to know,\" he said.\n\nThe woman deliberately pulled covers over her wrinkled nightgown. On a chair beside her were cigarettes, glasses and a whisky bottle. She lit a cigarette. \"I don't think I remember anyone like that,\" she said.\n\nJablonski reached across the man and grabbed her bare shoulder. His fingers tightened until the knuckles stood out and he shook her so hard the smoldering cigarette fell from her fingers. Then he threw her back on the bed. There were little red arcs on her shoulder where his thick nails had dug in. They looked at each other.\n\n\"We're in a hurry,\" said Jablonski.\n\nShe retrieved the cigarette.\n\n\"For God's sake, Maud,\" the man said again.\n\nWhen the woman spoke it was as though every word hurt her. \"That'll be Mr. Quinn,\" she said. \"In thirty-three. The room just above this. Came in today.\"\n\nJablonski continued looking at her. \"I hope that's right,\" he said. \"Because if he gets away now, you go up for harboring, common adultery and breaking every housing ordinance in the book.\"\n\n\"Did anyone else come in with him?\" asked Ryan.\n\n\"No.\"\n\n\"Did he meet anyone here, or ask about someone?\"\n\n\"No.\"\n\nThat took care of the accomplice, if there had been one.\n\n\"Let's go,\" said Jablonski.\n\n\"Wait a minute,\" said Ryan. He had been studying the tall, old-fashioned door to the room. \"All your doors like this?\"\n\n\"Why not?\" said the woman.\n\nRyan pushed his hat back on his forehead. It revealed, that his black hair was threaded with a little premature gray. He looked boyish and serious. \"If he's locked in behind a door like that,\" he said, \"we're going to have to blast him out. They built these places. A door like that won't give with one heave.\"\n\nJablonski said, \"Yeah,\" dismayed. \"That'll sure give him time to go for his gun.\"\n\nThe man in bed whimpered, \"Oh, God.\"\n\nJablonski grinned humorlessly. \"I know what we'll do,\" he said. \"C'mon, Maud. You'll get him to open the door.\"\n\n\"You know what you can do,\" she said, and told him.\n\nRyan looked at her. \"You're the landlady,\" he said mildly. \"We're police officers making an arrest. You wouldn't want to obstruct justice, would you?\" It wasn't really a question.\n\nThe woman slowly swung gnarled legs out of the bed and took a dressing gown that hung from an old gas jet on the wall. She put it on leisurely and fluffed out her bobbed locks with both hands.\n\n\"Atta girl, Maud,\" said the man in bed.\n\nShe looked behind at him. \"You better not be here when I get back,\" she said. \"You know what I mean. Don't be here.\"\n\nThey went quietly out into the hall. Its air was fresh and sweet; now Ryan remembered that when they broke into the room he had smelled the scent of stale powder.\n\n\"Not so fast,\" Jablonski said. \"We ain't barefoot like you.\"\n\nOn the third floor she found a light and they gathered silently before a door bearing a card with 33 on it. And as they did from behind the door came the unmistakable sound of a beer can being punctured.\n\n\"Holy Mike,\" said Jablonski.\n\nThere came the lesser sound of the opener's wedge making the second puncture.\n\nSuddenly Ryan was scared.\n\nJablonski had his gun out. He waved it between the woman and the door. \"Make him open it,\" he whispered. \"I don't care how\u2014but get goin'.\"\n\nRyan found his own revolver in his hand without knowing he had drawn it. He wondered if this wasn't a mistake. They could still wait\u2014But it was too late. The woman's fingers beat a light tattoo on the door.\n\nFrom the other side of the door came a silence they could feel.\n\nThe woman rapped again. She smiled with an unexpected girlishness. She was playing a part, so that when she called. \"Mr. Quinn, honey?\" anyone could tell by the soft tone that she was smiling invitingly.\n\nA voice behind the door said, \"Who's that?\"\n\n\"It's just me, honey. Mrs. Daniels, the landlady. How's about that beer you mentioned when you came in? I can't seem to get to sleep tonight. Are you having trouble, too?\"\n\nThere was a pause. \"Who?\"\n\n\"Mrs. Daniels, honey. You know.\"\n\nThe voice said, \"Wait a minute,\" and the doorknob twisted and a key turned with a loud switching sound.\n\nRyan grabbed Mrs. Daniels by the shoulders and flung her powerfully back and to one side. The door opened and a tall man peered out. The first things he saw were the short barrels of two police revolvers.\n\nThe long lean face looked surprised and a little stupid.\n\n\"All right, Derby,\" said Jablonski. \"Don't jump or we'll blow your guts out.\"\nCHAPTER 4\n\nDerby's Dance\n\nHarry derby's hair was silkily long, the color of brown wrapping paper, and it came to a deep widow's peak on his pallid forehead. His gimlet eyes were fierce, uncomprehending appraisers of what he turned them on, his mouth a hard twist of petulance. Only the jaw, long and clean in its sweep, suggested something worthy.\n\nHe said, \"Who the hell\u2014\" out of a mouth still wet with beer.\n\nMrs. Daniels began walking away from them toward the staircase, not hurrying, not looking back.\n\n\"Put the hands up,\" said Jablonski.\n\nDerby looked at him a full moment. Then he dropped the beer can in his right hand. It clunked on the floor and rolled, purling beer. He raised his arms, then recognized Jablonski and said, \"Hi, cop.\"\n\n\"Over against the wall,\" said Jablonski. \"Hands against it and keep your head down.\"\n\nWhen Derby had taken his position, back to them, hands flattened against the paint-blistered wall, he said, \"What's all this about?\"\n\nRyan saw Jablonski dart quick glances around the room. Across one corner of it a piece of faded blue cloth had been tacked to make a kind of closet. It could conceal a man.\n\n\"What's behind that curtain?\" said Jablonski.\n\nRyan approached it, keeping his gun on it and taking care not to get between Jablonski and Derby. He tore the cloth aside. There were a few coat hangers suspended from wire hooks.\n\nJablonski said, \"Clean him. Watch his pants legs. He may have a shiv.\"\n\n\"What's this all about?\" Derby asked again, but he did not sound as though he expected any answer.\n\nRyan ran his hands roughly over the taller man, watching Derby's shoulder for the muscle-tightening that would signal he was about to try something.\n\n\"No gun,\" he said. \"No nothing.\"\n\n\"What'd you do with it?\" asked Jablonski.\n\nDerby's reply was two monosyllables.\n\nRyan emptied Derby's pockets, then went through the checkered jacket that was hanging on a chair. He strewed everything he found on a small wooden table that already held a pipe, a tobacco can and a half-played game of solitaire. The wallet held a small sheaf of greasy business cards bearing many names, a driver's license in Derby's name and a chauffeur's license made out to Harry Durward, and a membership card in the International Longshoreman's Union. There was a meal ticket with four ten-cent punches left. And there was a crisp, neatly folded one hundred dollar bill.\n\n\"Got anything?\" said Jablonski.\n\nDerby said, \"How about me turning around?\"\n\nJablonski gave him the same monosyllables.\n\nRyan was comparing the bill's serial number with a page in his notebook. \"I'll say I got something,\" he said. \"Here's the C-note. And the numbers check.\"\n\n\"Hah!\" yelled Jablonski. \"Now you can turn around! Want to see what's gonna send you up?\"\n\nDerby turned, straightening. He rubbed his long, thin-fingered hands nervously, like a pianist before sitting down to play. Ryan held up the bill.\n\n\"Farragut will have a tough time explaining that one for you,\" said Jablonski. Farragut was the first-class criminal lawyer occasionally retained by the union.\n\n\"He's also got nine dollars and twenty-four cents,\" Ryan said. \"Book of matches. Tie clip. Couple pipe cleaners.\"\n\nJablonski nodded. \"It figures perfectly,\" he said. \"He was almost broke. Eh, Harry? So he clips the old lady. That gives him a hundred and twenty. He buys the beer and maybe got some food before that. That leaves one ten-buck bill to be accounted\u2014Oh, the room rent. That's it! He rented the room. Give him a few dimes to begin with and it just figures. Eh, Harry?\"\n\n\"You're nuts,\" said Derby. But his eyes moved jerkily in his head like a defective doll's. He said, \"On the level you're wrong. I borrowed that dough.\"\n\n\"Who from?\"\n\n\"A shark. Guy on the dock.\"\n\n\"Which one?\"\n\n\"I don't remember his name. I just see him around. I'll remember it.\"\n\n\"What'd you need a hundred bucks for?\"\n\n\"I was broke,\" said Derby. \"Like you said.\"\n\n\"When you remember the guy's name,\" said Jablonski with ponderous politeness, \"you let me know. And you know how much good it'll do you. This is a chair rap.\"\n\n\"Let's call in,\" said Ryan.\n\nBut Derby was looking at Jablonski like a man in a trance, spellbound, waiting for the word that would enable him to move again. As he did his face grew red and his arms and hands, clenched in tight fists, began to tremble. A little scar that ran thin as a knife blade from his left eye began to show oddly white.\n\n\"Chair rap.\"\n\nRyan had a sense of something happening inside Derby.\n\n\"The old lady ain't dead,\" Derby breathed. \"You don't mean that.\"\n\n\"She's dead,\" said Jablonski. \"What's the matter?\u2014Haven't you seen a paper?\"\n\nDerby looked down at the floor. When he raised his head again, his face was a putty mask etched with despair. Only the eyes were alive, glittering, hopeless. Ryan incongruously remembered something he had heard a sergeant say long before: \"They can be tough as hell\u2014I don't care how tough. But when you've finally got them, and they know you've got them\u2014something happens. You can see it happen in their faces.\"\n\nIt was happening to Derby. Ryan did not feel sorry for Derby; he hated him with a deep, contemptuous hate. But in Derby's bent head and fallen shoulder, in the low voice and limp arms, he caught a glimpse of the ultimate human defeat, of the prey's realization that it at last has been run to earth.\n\nWhen Jablonski spoke, his voice was unexpectedly free of rancor. \"Where you made your big mistake was in letting yourself be seen, Harry. Three dames made you right away. And the bank teller will too.\" After a moment he corrected himself. \"That was your second big mistake. Your first one was using the gun on an old lady sixty years! Hardboiled Harry Derby\u2014wow!\"\n\n\"Look\u2014you don't understand,\" said Derby. He began speaking fast and earnestly, heedless of what he was saying, only hoping to convince them by sheer flood of words and emotion. \"You guys don't understand at all. It ain't like that at all. Honest. I didn't kill her. You gotta realize that. I didn't kill her. I know it looks bad\u2014I admit that. But you gotta realize\u2014Sure, I needed a little dough, but this afternoon I ran into this guy and\u2014\"\n\n\"Where?\" said Jablonski.\n\n\"Where? Oh...yeah. It was...ah...near the Java Street pier in Brooklyn.\"\n\n\"What time?\"\n\n\"What time? Oh, well, I guess it might have been around noon.\"\n\n\"Might have been, eh? And where were you around three twenty this afternoon?\"\n\nDerby stopped talking, and he looked like a man who did not want to talk.\n\n\"Where were you shortly before the banks closed?\" Derby was silent.\n\nRyan began to grow annoyed. He wanted to take Derby in; this bearbaiting bothered him. He looked to the half-open door.\n\nThat made him say, \"Jesus!\"\n\n\"What's the matter?\" said Jablonski.\n\n\"Look.\" Ryan reached up on the inside of the door frame. Hanging from a nail by its trigger ring, and exactly in the position where a tall man opening the door might naturally rest his hand was a .38 revolver. The hammer was cocked.\n\n'There's the gun,\" said Ryan. He could feel something in the pit of his stomach. That was how close it could be. \"Right caliber too.\"\n\n\"Good thing we didn't give this mother-raper a chance,\" said Jablonski feelingly. \"Take it down, Neill. Think Farragut can explain that, Derby\u2014especially after it checks with the slug? And how about the C-note? Still want to tell us it wasn't like that at all?\"\n\nDerby said, \"I don't know nothin' about no C-note. But you could take it. And get a couple more tomorrow besides.\"\n\nJablonski reached over dangerously close to Derby and slapped him across the face as hard as he could. It made an astonishingly loud crack.\n\nTears jolted to Derby's eyes. \"What'd you say, Harry?\" asked Jablonski.\n\nDerby began to curse Jablonski, meaningless obscenity gushing from his mouth like vomit.\n\nJablonski drew his hand back.\n\n\"Don't mark him,\" warned Ryan.\n\nJablonski took a backward step like a man getting control of himself. \"Go call in, Neill. Tell them we got him.\n\n\"Wait. Put cuffs on him. I'll cover you. And give me one of your cheroots before you do.\"\n\nRyan did. But when he had handcuffed Derby and reached the door he could not help pausing a second to look back. It was like picking up the little china chip. This was his first heavy one, and he and Jabby had pulled it off\u2014alone. This meant promotion and acclaim, but more than that it was what you lived for if you were a cop. He could not get enough of the feel of it.\n\nNot hearing the door close behind him, Jablonski looked around, puffing the cigarette. \"Hey\u2014what's the matter?\"\n\n\"Nothing. Back in a minute.\"\n\nAs he pulled the door shut, he heard Derby say, \"Hey, cop, how about a smoke?\"\n\n* * * *\n\nMrs. Daniels answered his light knock wearing a house dress. Ryan said, \"You have a phone in this building?\"\n\n\"You got him, eh?\" she said.\n\n\"We got him. Where's the phone?\"\n\n\"Well, I just want you to understand I don't know nothing about him. Not a thing. In fact, I never, seen him\u2014before today.\"\n\n\"When'd you see him today?\"\n\n\"This afternoon. He checked in around four.\"\n\nThen he came here almost directly from the job.\n\n\"He went out again for a little while. Who's the woman he killed?\"\n\n\"An old lady. It's in the papers. How long'd he rent for?\"\n\n\"A week.\"\n\n\"Pay in advance?\"\n\n\"Everyone does. That room was ten bucks. C'mon in.\" She stepped back from the door and gestured toward the whisky bottle on the chair. \"Have a jolt.\"\n\n\"I can't now. You go ahead.\"\n\nShe poured a shot glass full.\n\n\"So he paid you ten bucks in advance. With a ten dollar bill, eh?\"\n\nToo late she realized the direction of these casual questions. She indignantly put the glass down untasted on the bureau. From overhead came the sound of someone walking.\n\n\"I didn't say it was a ten.\"\n\n\"Let's see it.\"\n\n\"I didn't say I still had it.\"\n\n\"Then who's got it?\"\n\nShe looked a quick calculation at him. \"I loaned it to a friend who was going to Chicago.\"\n\n\"Where was he going in Chicago? What's his name?\"\n\nShe said, \"I dunno,\" looked at him covertly and stopped.\n\n\"Look, lady. That bill is evidence in a murder case\u2014murder, get it? I won't take any malarkey about Chicago. Get it out.\"\n\nHer face grew sullen. The man upstairs walked back again fast across the floor. It registered only subconsciously with Ryan.\n\n\"He rented the room,\" she said doggedly. \"I got a right to it.\"\n\n\"You could get it back.\"\n\n\"Yeah!\"\n\n\"You could if it wasn't stolen. But first it will have to be used as evidence. Come on\u2014I haven't got all day.\"\n\nShe walked to the bureau in broken shoes, a resentful, helpless, middle-aged little girl. She took a small purse from a drawer, drew a bill from it and looked at the bill. Then she picked up the whisky, drank it effortlessly and extended the money to Ryan.\n\n\"That was going to buy me a new dress.\" Her eyes never left the bill.\n\nRyan thought of how his sister was with a new dress or skirt. That's how it would have been with Mrs. Daniels.\n\nHe took an envelope from his wallet, placed the ten dollar bill in it and replaced it carefully. Then he took out two fives of his own money. He defended himself to himself on the ground that he was tired and that it was late, and that it was better to keep the witnesses friendly.\n\n\"Look,\" he said. \"Long as you need this dough, I'll replace it out of my own pocket, see? This isn't exactly regulation, but after the trial when you may get the bill back I'll drop by and pick it up from you. Otherwise you owe me ten bucks. Okay?\"\n\nIt embarrassed him the way her face softened and unsuspected dimples appeared in her crepe-skinned cheeks. \"Well now, that's nice of you, it really is. Though of course why the God\u2014the cops...police, I mean\u2014\"\n\n\"Okay, okay. Now where's the phone?\"\n\nThe hoarse, enraged shout that came from the floor above was sheer guttural but he knew at once it was Jablonski's. There came a scurry of feet and a thump.\n\nRyan leaped for the door, grabbed his gun and ran recklessly up the stairs. The door to Derby's room was still closed. He flung it open cautiously and stepped backward out of reach.\n\nThen he looked into the room.\n\nDerby, his back to the wall again, seemed to be doing a kind of dance, while Jablonski stood in front of him, gun in hand, aiming deliberate kicks at his shins and thighs whenever opportunity presented. Then he realized that Derby's dance was an effort to avoid the kicks. At the same time a gurgle broke from Derby, and Ryan with horrible foreboding recognized that Derby was convulsed with laughter.\n\nRyan ran in and pulled Jablonski back.\n\nBut laughter made Derby helpless. No longer having to defend himself, he leaned against the wall, his gaunt face wrinkled weirdly, tears wetting his cheeks, high tremolos welling up from within him.\n\n\"For God's sake!\" gasped Ryan.\n\nRage made Jablonski's face black. \"The son of a bitch,\" he began. \"The dirty, sow-headed son of a bitch\u2014have you called in?\"\n\n\"Not yet. I was talking to the landlady. But what\u2014\"\n\nDerby's laugh became a snicker.\n\n\"The son of a bitch burned up that C-note,\" cried Jablonski, frenzied, and Derby went off into a new convulsion.\n\nHe must be crazy, was the first thing Ryan thought. Derby was still handcuffed. They've both gone nuts. But he turned toward the table.\n\nThe hundred dollar bill was gone.\n\nSo was Derby's pipe. That lay on the floor and near it was a still-smoking dottle of tobacco. Then he understood.\n\n\"How do you...like your murder case now, cop...copper?\" Derby struggled with laughter.\n\nJablonski did not look at Ryan. Ryan did not say anything.\n\n\"After you left he asked me for a cigarette,\" Jablonski said.\n\n\"He knew you didn't have any. You'd bummed one from me.\"\n\n\"So he said could he get his pipe on the table. I thought, what the hell? He was cuffed. Why shouldn't the slob smoke? I backed away and let him walk over to the table. He must have picked up the C-note with the pipe.\"\n\n\"And stuffed it in with the tobacco and lit it.\"\n\n\"Yeah. I noticed it sort of flared, but he'd had trouble packing it with the cuffs on and\u2014\"\n\nDerby whooped and started to laugh again.\n\n\"It wasn't until I went over to get a chair and set down that I look at the table and seen the bill was gone. At first I couldn't figure it. Then I thought of the pipe. I pulled it outta his mouth\u2014\" He made an impotent gesture with his hand.\n\nRyan picked up the tobacco dottle. It was hopeless. Perhaps the lab men could prove scientifically that some flakes of the ash had originally been government banknote paper. But as evidence that Harry Derby had possessed the identical bill of which Mrs. Connors had been robbed, it was useless. Derby had outwitted Jablonski even with Jablonski gloatingly watching over him. Ryan felt a hot surge of hatred for Jablonski, and for Derby.\n\nDerby watched him with the wise air of an old fox. \"Still gonna send me up?\" he chuckled.\n\n\"There's plenty of evidence.\" Ryan wouldn't flinch in the sight of the enemy. \"Plenty of evidence. You're in the chair right now, Derby.\" He wanted to jar Derby, to beat him down with naked fists, to make him feel the cold discouragement that was beginning to hurt himself. After all this...all they had done... Derby had beaten them.\n\n\"There's three dames made you this afternoon,\" he said with deliberate cruelty. \"They're the ones that will give you the chair. That C-note was just contributory evidence. You're dead, Derby.\"\n\nBut there was discouragement in the air. Everyone felt it and none more than Derby. Ryan took out his cigarettes. There were three left. He offered one to Jabby.\n\n\"How about a smoke for me?\" asked Derby impudently.\n\nJablonski walked over and kicked Derby's thin buttocks. Derby continued standing against the wall. He grinned determinedly but he did not laugh any more. The time for laughter was over. Now they were up to the next move.\n\nJablonski thumbed Ryan over toward the door and followed him. \"We're in trouble,\" he muttered. Ryan knew this was really Jablonski's trouble, but there was no point in thinking about that now.\n\nHe said, \"Well, we still got a case.\"\n\nJablonski looked old and tired, and it wasn't just the gray stubble sprouting from his jowls. \"Like hell we got a case.\"\n\n\"There's the gun.\"\n\n\"Maybe it checks and maybe not. The slug may have been smashed inside her head.\"\n\n\"Well, there's the women\u2014\"\n\n\"Those women won't be worth a damn by the time this guy goes to trial,\" growled Jablonski. \"Wait'll his friends start calling them up nights. Or dropping around and making cracks about their husbands and kids. They won't be nearly so sure as they were downtown.\"\n\n\"But jeez, Jabby\u2014\" Ryan had to try to keep his voice low.\n\n\"And he'll have an alibi a yard long. Half the longshoremen in New York will swear he was playing cards with them all afternoon. We know what he was doing. But who the hell will believe us?\"\n\n\"There's the bank clerk,\" said Ryan. \"And just now I got one of the tens.\"\n\nJablonski exploded. \"Banks don't keep no record of regular ten dollar bills! Ain't you ever been in a bank? And as for the teller\u2014he wasn't even sure the guy came up to the cage. What kind of identification will that make?\"\n\nHe was silent a moment and when he spoke again it was with the wistfulness of acknowledged defeat. \"We've lost him, kid. We can take him in\u2014and he'll go to trial. Least, if the D.A. is willing to take any kind of chance at all. But he'll never in God's world go up. And you can blame it on me\u2014all on me! Because I'm a dumb\u2014\"\n\nThe rest was bitter self-denunciation. But there were tears in Jablonski's eyes.\n\nDerby could not hear what they said but he could understand their postures. \"What's 'a matter?\" he mocked. \"Something wrong? Didn't I smack down the old dame? Huh? Ain't you got no case after all this? Huh? Hey, squarehead\u2014listen! I killed the broad. You hear that? You got a confession, Polack! What you gonna do with it? Let's see you prove I said it! Still wanna try to beat Farragut?\"\n\nJablonski turned, bent on destruction.\n\nAnd he paused. His sullen and morose face grew suddenly crafty. \"Hey,\" he said, looking around. \"Neill, what was that you picked up in that apartment this afternoon?\" Ryan flushed. He hadn't realized Jabby had noticed it.\n\n\"A little piece of the lamp?\" Jablonski insisted. \"Huh?\"\n\n\"Yeah. I guess it was that.\"\n\n\"Still got it?\"\n\n\"Ah...yes.\"\n\n\"Give it to me,\" said Jablonski. \"Come on. Come on! I got an idea.\"\nCHAPTER 5\n\nYou Can Be Thinking of Me\n\nJablonski pulled Ryan's head close to his own and whispered in husky, meat-flavored whispers. As he did, Ryan's expression lengthened.\n\n\"But that's\u2014well, manufacturing evidence,\" he said.\n\n\"Sure it is.\" Jablonski spoke airily but he was watching Ryan.\n\n\"It's\u2014well, it's like framing the guy.\" Ryan didn't like it. Fairness and respect for the letter of the law were implicit in him.\n\n\"Framing!\" Jablonski's scorn was boisterous. \"Framing a guy like Derby? You checked the numbers on that bill yourself, Neill. You heard him admit he killed her, didn't you? Framing, for God's sake! All we're doing is proving what we know. It ain't as if he was innocent.\"\n\n\"I know it's not as if he was innocent,\" Ryan muttered. \"It's just\u2014that tampering with evidence...\"\n\nHe felt weary and oddly numb. It was getting on towards three a.m.; this had been a tough night. It was no time to debate the subtler shades of justice. Still, he didn't like it.\n\n\"Look, Neill,\" Jablonski said. \"He killed her\u2014we know that. And when he hit her with the lamp he could have got a little of the dust on his jacket. So all we do now is\u2014well, just make sure he did.\"\n\n\"Sure, but...\"\n\n\"Look, Neill.\" For Jablonski necessity had become the mother of eloquence. \"I want this collar bad\u2014so do you. You know what it can do for both of us. And you know what happens to my retirement if it gets back to the desk that I let him burn that C-note. If we bring Derby in and then lose the case against him\u2014\"\n\n\"Sure,\" said Ryan uncomfortably. \"Sure. I know. But when he rumbles that he was framed\u2014then where are you?\"\n\nJablonski snorted. \"Who the hell will believe him? Look, Neill. We got a case against him now\u2014of sorts. What we need is the clincher. That's what the hundred dollar bill was until we\u2014until I let him burn it. That's what your little piece of pottery will be, especially the way I'll do it. Jeez, can you imagine him getting on the stand and claiming that I\u2014well, you can see for yourself.\"\n\nRyan nodded. What Jablonski proposed doing was fantastic. That's what made it safe.\n\nDerby, straining his ears, sensed something going wrong. He chanced a quick squint over his shoulder. He did not like what he saw.\n\n\"We don't have anything to worry about.\" In reassuring Ryan, Jablonski was also reassuring himself. \"Matter of fact, our only real trouble would come from telling the truth.\" He saw that this was the wrong tack. \"Look, Neill, do you think you have any right to let this guy go free? What about all the women who are out nights working? What'll you say if he's turned loose and knocks off another dame? All you're doing is what you promised to do the day you took your oath.\"\n\nHe was right, of course. A more experienced cop wouldn't give it a second thought, Ryan thought, not when a guy like Derby was involved. It was just the idea of someone going to the chair, even when he was guilty, on the basis of manufactured evidence. But what was the alternative\u2014giving him a good chance to go free? They couldn't do that.\n\n\"Okay,\" he heard himself say. \"Go ahead. Here it is.\"\n\nJablonski took the little yellow-glazed button of china.\n\nRyan lit his last cigarette. Derby frowned suspiciously at the baseboard. Jablonski walked to the door, grinning craftily.\n\nThere was a crumpled brown paper sack in the wastebasket near it. Jablonski smoothed it out on the table and placed the little chip on it. He set a beer can on it and pressed down hard, grunting. The little fragment of china crumbled with a grating noise.\n\nDerby's mouth was a tight apprehensive slit. Something was going wrong.\n\nJablonski pressed the can on the china fragments again, grinding them down. Then he rolled the can on them like a rolling pin, reducing them to powder. He rolled steadily, looking up once to grin at Ryan. \"Get his jacket on him,\" he said.\n\n\"I'll have to take off the cuffs.\"\n\n\"Just unlock one of them.\"\n\nRyan did and threw Derby his jacket. \"Put it on.\"\n\n\"Where we goin'?\" asked Derby.\n\n\"Out,\" said Ryan. \"And we're not coming back.\" Derby put on the jacket.\n\nJablonski folded the brown paper bag to make a trough of it, and poured the fine, whitish-yellow dust into his left hand. He walked over to Derby. \"Cover me, Neill,\" he said.\n\n\"You get the hell away from me,\" said Derby.\n\nRyan took out his gun.\n\n\"Turn around,\" Jablonski told Derby.\n\n\"Go to hell,\" said Derby.\n\nJablonski pushed his foot against the back of Derby's knee. Derby buckled, partly turning, and Jablonski swung him around, half-falling, still farther. Then carefully, deliberately, he blew the powder in his left hand over the front of Derby's wool jacket. Derby swung a flailing blow at Jablonski's hand but it was more a gesture of protest than of attack. Jablonski laughed.\n\n\"What the hell is this?\" asked Derby, and began dusting himself off ineffectually.\n\n\"That fixes you,\" said Jablonski.\n\nThat fixed him. You always try to find any possible trace of the criminal at the scene, Ryan had been taught at the police academy. And you always try to find some trace of the scene or victim on the criminal. Either one can convict; getting both is ideal.\n\n\"What is that?\" Derby was saying anxiously, over and over. \"What was that? Was that junk? What was that?\"\n\n\"That,\" said Jablonski, \"was a little piece of plaster from the lamp you hit the old lady with, Harry. And now it's all over you, and it places you at the scene\u2014even better'n the C-note. And all the Farraguts in the world can't explain that dust in your jacket. You can't cross-examine a spectroscope, Harry.\"\n\nAs Derby frowned in concentration, his hands slowed their dusting movements. He had no idea of what a spectroscope was, but he had a sure sense of evidence and an animal's quick instinct for the moves of the enemy. He looked at the two of them. The older one was smiling sarcastically. The other bastard was wooden-faced, giving nothing to nobody. But he didn't have to. Derby got the general idea. They were framing him.\n\nFrustration had always produced only one reaction in Derby. Jablonski saw it coming in the swelling of his shoulders, but before his mouth could begin a warning, Derby charged, head down, fists hooking, one of them flapping a handcuff.\n\nJablonski had just reached his revolver when one knotty fist caught the side of his head and knocked him off balance and backward. Derby leaped after him and Ryan saw Derby's long bony knee drive up and catch Jablonski in the stomach. Jablonski uttered a loud wordless cry and his revolver slipped out from his coat and skittered across the floor toward Ryan.\n\nFor a second Ryan could not shoot; they were too close together. Jablonski swayed forward on unhinged legs, holding his stomach, his face ghastly. Derby stepped back to bring a roundhouse punch up from the floor. Ryan fired.\n\nThe shot missed Derby, but it stopped him because it reminded him of Ryan. With hardly a lost motion he wheeled, took a quick boxer's shuffle forward and swung at Ryan. Jablonski began sagging to the floor.\n\nRyan stepped backward; he had seen the punch coming, but the force of it spun Derby half around. Ryan coldly thumbed the hammer of his gun. The quiet click ended Derby's outbreak. He knew what it meant\u2014and the revolver just three feet away was pointed directly into his stomach. Ryan was looking at him.\n\n\"Start swinging again,\" he said. \"Come on. Give me an excuse.\"\n\nThey looked at each other while the tension went out of Derby. \"Face to the wall and hands up against it,\" said Ryan, and Derby obeyed. Ryan picked up Jablonski's gun.\n\nFrom Jablonski came a soft sigh.\n\n\"How is it, Jabby?\"\n\n\"'M okay...all right in a minute. He...got me with his knee...\"\n\nRyan had played football; he knew how long it would hurt and how much. Jablonski groaned, trying not to.\n\nThere was a knock and the door swung open timidly. Mrs. Daniels stood in the doorway, and behind her a stringy individual in gallus-draped pants and a woolen undershirt.\n\n\"Is\u2014is everything all right?\" She looked inquiringly toward Derby.\n\n\"You go to the telephone,\" Ryan began.\n\n\"There ain't one in this building. Did you shoot\u2014\"\n\n\"You with the pants on,\" said Ryan. \"You get over to Second Avenue to that all-night hamburger joint and call Spring seven three one hundred. Got that number?\"\n\n\"Listen, mister, I gotta get some sleep.\"\n\nJablonski was trying to get up.\n\n\"What's the matter with your friend?\" said Mrs. Daniels.\n\nRyan turned enough so they both could see the revolver in his hand.\n\n\"You call that number,\" he said. \"Right now!\" He spoke doggedly. \"Spring seven three one hundred. Tell them an officer is in trouble here. Get going.\"\n\nThe stringy man hitched up a gallus. \"Okay,\" he said. \"Okay,\" and left.\n\n\"You shoot your friend?\" asked Mrs. Daniels.\n\n\"Get him some of your whisky,\" said Ryan.\n\nShe went downstairs without a word.\n\nJablonski was rubbing his abdomen, methodically and painfully. \"Wipe off the beer can,\" he said, \"and get rid of that paper.\" His voice was a croak.\n\nYou nervy old son of a bitch, thought Ryan, as he pocketed the brown paper. Your guts must hurt like hell, but you still remember that Derby mustn't have a chance to prove what we've done.\n\n\"Right, Jabby.\" It was the first time he had ever felt genuine respect for Jablonski.\n\n\"Hey, Derby,\" Jablonski croaked. \"Listen. My belly hurts. See? But it'll stop after a while. And just think, Derby. Some night soon, you'll burn. Think of that. And you know what? That night it'll be me who's home drinkin' beer. And waitin' for that flash on the radio. What do you think of that, Derby?\"\n\nIt was a very silent room when he stopped.\n\n\"You can be thinking of me when you get that jolt, Derby. 'Cause I'll be thinking of you.\" His jaws moved unnaturally, and he retched.\n\nFrom outside and below came the onrushing sound of an automobile moving fast. Ryan heard the clump of heavy feet on the stairs and then there were two blue uniforms and silver shields and alert faces in the doorway.\n\nThey looked awfully good to Ryan.\nCHAPTER 6\n\nFresh-Made Tracks in the Snow\n\nThere was a small knot of men standing in front of the shadow-scrawled old precinct station when the radio car pulled up. Ryan paid no particular attention to them. He wanted only to get Derby inside and booked, and to find out how badly Jablonski was hurt. During their fast trip up the empty, late-looking avenue he had once asked, \"How is it, Jabby?\" and Jablonski had replied from the front seat, \"Oh, it's okay,\" carelessly, as though he had forgotten the whole thing. But his voice hadn't sounded right.\n\nJablonski got out of the car painfully, and was walking half-bent-over toward the stone steps when a flash of blue-white brilliance engulfed them. Another flash came. Ryan, walking with his hand on Derby's arm, felt him stiffen in terror. Ryan hadn't expected the news photographers to get there so fast; then he remembered that one of the prowl car crew had called the news in on the car radio.\n\nDerby held his handcuffed hands to his face. It made a fine, cringing picture. The flashes burst over them like waves again and again as they went up the time-scalloped steps of the precinct station. Ryan was not only blinded by them but oddly unnerved. He could almost feel the flashes against him, like wind blasts.\n\nLieutenant Paul Bauer stood outside the desk area at the foot of the stairs, his dark eyes watchful and jaw unyielding under grayed brows and bristle of pompadour. But Ryan sensed Bauer's held-in elation. And in the other men, the uniformed sergeant behind the high dark oak desk, the patrolman with him, and the detective coming down the old stairs, you didn't have to sense it at all. They looked at the tall shambling figure being led in and grinned jubilantly. This was one the department had won, and the whole precinct was filled with the clubhouse spirit of a winning ball team.\n\n\"Why'sat guy bent over?\" a photographer asked his reporter, and the reporter said, \"What's the matter, Jabby? You hurt?\"\n\nBauer shouldered into the little group of journalists that clotted around them. \"Take Derby upstairs,\" he told the prowl car crew. \"Chief inspector's on his way.\"\n\n\"Yes, sir.\" Derby's chin was touching his breastbone as they pushed him forward.\n\n\"You hurt, Jabby?\"\n\nJablonski did a good job of straightening up. \"No, sir. He\u2014he kicked me in the belly. It\u2014ah, hurt for a while. I'm all right now.\"\n\n\"We'll run you down to the hospital.\"\n\n\"It ain't that bad, Paul. Honest.\" Not for anything would Jablonski miss all this.\n\nSomeone plucked Ryan's sleeve, and a tubby young man with horn-rimmed glasses said, \"Holcomb\u2014the Mirror. You're Ryan, eh? Congrats\u2014great job! Hey, is it true he's admitted it?\"\n\nA twinge of nervousness assailed Ryan. This had to be done carefully. One intimation about that hundred dollar bill\u2014but the best thing always was to tell the truth as nearly as possible. \"Well, yes,\" he said hesitantly. \"He even sort of boasted about it.\"\n\n\"Boasted? And him already a three-time loser?\" And Ryan knew he should not have said that. Jablonski ought to be here to do the talking.\n\n\"How'd you happen to spot him?\"\n\nHe told the story in three scant sentences. The reporter kept looking at his watch because he had to call the city desk immediately. He said, \"And the loot\u2014you got the dough, huh?\"\n\nRyan started to say affirmatively, \"Oh, he had\u2014\" and stopped barely in time.\n\nBut Holcomb caught the meaning.\n\n\"He had the dough on him?\"\n\nFull-blown panic now. His thoughts flew like leaves in a gale.\n\nAnother reporter who had been listening repeated the question. \"Derby had the C-note on him?\"\n\n\"He had some of the dough.\" Ryan said wildly and pushed past them. \"I got to get upstairs.\"\n\nThe other reporter said, \"But wait\u2014\"\n\n\"Not now. Later.\" Ryan took the stairs two at a time. As he did, air fanned coldly against his moist brow.\n\nIn the squad room off the head of the stairs, Lieutenant Bauer sat at his scarred desk, Jablonski comfortably alongside him. Somewhere Jablonski had obtained a cigar and its blue fragrance fumed the room. Bauer pushed a rickety chair toward Ryan and continued his conversation. \"He admitted it, eh?\" he asked Jablonski.\n\n\"When we were alone he did,\" said Jablonski.\n\n\"Well, he sure ain't talkin' to McGonigle.\" The telephone at his elbow jingled. Bauer said, \"Seventeenth Squad Lieutenant Bauer\" into it in one breath, then, \"Yeah, Ed...yeah...yeah... Jabby and Ryan...right, kid,\" and hung up.\n\n\"Furtig,\" he said. \"He's been out with the others making the dock joints.\"\n\n\"He can go to bed now,\" said Jablonski magnanimously. But Ryan knew how Furtig must be feeling\u2014relieved, but envious.\n\nApparently Bauer knew too. \"Well, you know how it is,\" he said to Jablonski. For a man of his authority and background Bauer had a very mild voice. \"Besides, the papers have been making such a\u2014well, I guess you haven't had a chance to see them. You guys have really done a job in case you didn't know it.\"\n\nJablonski complacently licked a cigar leaf back into place. \"Listen to this,\" said Bauer and began reading from a paper he fished from the wastebasket.\n\n\"...out and out anarchy... Hoodlums have taken over... While the commissioner has said...responsible citizens will bear in mind...action by the Governor...election next year in which...\"\n\nBauer finished it and laughed. \"Well, what else should we do?\" he asked briskly. Ryan remembered that Bauer had been on duty for almost twenty-four hours. He said, \"Well, there's the delicatessen to check, Lieutenant.\"\n\n\"Huh?\"\n\n\"We picked him up going into a delicatessen on Third to get some beer. He probably broke one of the tens there. He gave another to the landlady. I've got that.\"\n\n\"Good. Anything else?\"\n\n\"He said he got the dough from a loan shark but didn't know the guy's name. We better run that down before Farragut gets something arranged for him.\"\n\n\"Right. How much did he have on him?\"\n\n\"Oh,\" said Jablonski, and Ryan began, \"Well...\"\n\nJablonski said easily, \"We didn't have time to give him a complete frisk, Paul. Maybe it's in his shoe.\"\n\nBauer said, \"Yeah,\" and scratched his unshaven chin. \"McGonigle better know that right away.\" He got up.\n\n\"One other thing, Paul,\" said Jablonski. \"The lab oughtta go over that wool jacket of his very carefully. He hit the old dame with a lamp and it busted over her. That heavy wool would be perfect for picking up little bits of lamp plaster.\"\n\n\"Good deal.\"\n\nWhen Bauer had gone Jablonski looked sideways at Ryan and smiled a slow, satisfied smile. But Ryan could not smile back. In the last ten minutes something had changed. He did not feel friendly toward Jablonski, and he did not know why. He just wanted to get this over and get out of here.\n\nThere was the sound of many feet on the stairs and Ryan, looking over his shoulder, saw Betty Leonard and Mrs. Lombardi go down the hall with two detectives. \"The witnesses,\" he grunted.\n\n\"It's a hundred to one they make him,\" said Jablonski, and that made Ryan feel better. He took a cigarette from Bauer's pack and was lighting it when Jabby muttered, \"Hey!\" and got awkwardly to his feet. \"Good evening, sir,\" he said very respectfully.\n\nRyan turned around. A lean man of medium height with a red, knobby face and angry eyes stood in the doorway. His overcoat's velvet lapels framed a high, starched collar, a tasteless flowered necktie and a huge stickpin. He held his head and shoulders hunched forward aggressively as though he expected to encounter violence and expected to return it.\n\n\"You Jablonski?\"\n\n\"Yes, sir.\"\n\n\"You're Ryan.\"\n\n\"Yes, sir.\"\n\nRyan had not shaken hands with the chief inspector of the New York police department since the day four years before when he graduated as a rookie patrolman. Now the chief inspector's hand closed on his roughly, as on a nightstick, but the harsh eyes softened.\n\n\"You'll be hearing from the commissioner,\" said Patrick Pembroke. \"But I wanted to tell you to your faces what a fine bit of work you've done tonight, men. A fine bit of work.\" His voice became music. \"I thank you for it, and so does every man in the department.\"\n\nHe looked at both of them approvingly and Ryan, who knew that at the age of sixty-four the chief inspector had gotten out of bed and driven here simply to say those words to them, felt his throat close.\n\n\"Thank you, sir,\" said Jablonski.\n\n\"It simply shows that heads-up police work can score every time. Have you got a case?\"\n\n\"We've got a case, sir,\" said Jablonski with unexpected mildness. \"This is the end of Derby.\"\n\n\"Good. The sooner he burns, the better.\" The matter-of-fact voice was indescribably cold. \"You've seen what the papers have said about us. You've given the commissioner the ammunition to answer them with. He'll make use of it later today, if I'm any judge.\"\n\nOnce again the ancient, bony hand shot forward to each of them. \"Thank you, Jablonski. Thank you, Ryan. A good job. Good night, men.\" He went out with slow, cold dignity.\n\nGolly, what a guy, Ryan thought, and out in the hall the chief inspector turned and came back so quickly that for a second he feared he had uttered the words aloud. He was tired and confused enough to do that.\n\nThe old man was looking at him. \"Ryan,\" he said reflectively, and the faintest brogue edged his words. \"Someone mentioned your name is O'Neill Ryan.\"\n\n\"Yes, sir.\"\n\n\"I once had a man under me named O'Neill Ryan. This would be years ago. Young sergeant, he was.\" The chief inspector spoke deliberately now. \"It was on the shoo\u2014on the confidential squad. He was a fine man or he'd never got there so young. Later some hoodlums\u2014\" the knobby face hardened\u2014\"some hoodlums met him one night in Carmine Street. They killed him.\"\n\n\"Yes, sir.\"\n\nThe chief inspector's eyes flickered. Then they measured Ryan. \"A relative perhaps?\" he said softly.\n\n\"My father,\" said Ryan, and all the events of the long night welled up in his throat and to his eyes.\n\nFor a long time the harsh angry gaze engaged Ryan's. \"Your father,\" said Patrick Pembroke. \"He was a good man. A good man. Good night, Ryan.\" He went out.\n\nRyan fumbled another cigarette out of Bauer's pack and kept his head down.\n\n\"Jeez, kid,\" said Jablonski. \"I didn't know your old man was a cop.\" There was a new respect in his voice.\n\n\"Forget it,\" said Ryan thickly, and had to wait a minute. Then he said, \"He's really quite a guy, isn't he?\" At that moment Ryan would have gone after a cityful of Derbys for the chief inspector.\n\nThey heard Bauer coming down the hall.\n\n\"Oh, Paddy's a right gee,\" said Jablonski carelessly. He breathed out a cloud of strong smoke. Then, \"What's he saying, Paul?\"\n\nBauer's face was lined with weariness. \"Nothing, as you might expect. But the gun looks good. It's a .38, and it was a .38 slug that killed the victim.\"\n\n\"How about the witnesses?\" said Ryan.\n\n\"They picked him like a flash.\"\n\nThat made Ryan feel even better.\n\n\"The young one asked about you, Ryan,\" Bauer grinned. \"I guess you made an impression.\"\n\nRyan grinned back. Bauer said, \"He hasn't got the bill on him, incidentally.\"\n\nJablonski said very casually, \"What's he say about it?\"\n\n\"Nothing. He never heard of it. He was home taking a nap all afternoon\u2014alone. And he says the loan shark's name is Morgan, or something like that.\" The lieutenant sighed wearily. \"I guess that's it. Oh\u2014don't either of you worry about court tomorrow\u2014today, rather. McGonigle can do that. You men get some sleep.\" He looked at them both. \"I guess I don't need to say\u2014well, you know. You did a great job, and it won't hurt either of you. There's reporters downstairs still. The order is to give them everything they want. We want all the publicity we can get on this one.\"\n\nAt any other time the alacrity with which Jablonski rose, and the bored resignation with which he said, \"Well, we may as well get it over with,\" would have made it funny. But the thought of facing more reporters alarmed Ryan. Going downstairs he said, \"Look. You do the talking.\"\n\nJablonski was glad to. While the reporters listened, jotted and occasionally broke in with a question he told the story in detail and with such flattering reference to Ryan that it was clear to everyone the veteran was trying to give every possible credit to the rookie, even though he himself had been in full command.\n\nRyan studied the reporters. All were dissimilar and yet alike: knowing, a little rumpled, of indeterminate age. But one, tall and heavy-eyed as though he had been newly routed from bed, wore a sports jacket of soft, expensive tweed and a gray sports shirt. His feet were unexpectedly clad in felt bedroom slippers. His black hair was thick With gray, but his square-jawed face was youthful and forceful.\n\n\"Telephone, Ryan,\" said the desk sergeant.\n\nWhen Ryan turned, the desk sergeant winked elaborately to the photographers and his lips formed \"The P.C.\" Ryan said, \"Detective Ryan,\" wearily into the telephone.\n\n\"Ryan?\" said the telephone voice in his ear. \"This is Johnson Drumm. I just wanted to congratulate you and your partner on...\"\n\nRyan's jaw dropped visibly. Johnson Drumm was the police commissioner. A flashbulb flared, and then a couple more before Ryan got his mouth closed. He finally said, \"Yes, sir...yes, sir,\" automatically. Then, \"Here, Jabby. Jeez, it's the P.C.\"\n\nThey did not shoot Jablonski's picture; he was too assured, grinning, easy. They asked Ryan a couple of questions and learned with excited interest that this was his first murder case. The photographers packed their bags of equipment and then left. It was that simple.\n\nThen Ryan noticed that the one in the sports jacket was still there, slouched silently against the desk sergeant's throne of dark oak, sleepy looking yet watchful. He shuffled forward in his slippers. \"Ryan?\" he said. \"'M Jack Sandalwood. Like to ask a couple questions.\"\n\nHis manner was lazy, but Ryan's hackles rose. Jack Sandalwood was the best-known reporter in the city, a Pulitzer Prize winner whose specialty for more than ten years had been the exposure of venality and crime in unpredictable places. In a sense his appearance was even more of a compliment than the chief inspector's, but it was infinitely more ominous. Sandalwood could ask questions that would never occur to an ordinary policeman, and he could demand the answers to them.\n\nOnce again Ryan looked around for Jablonski. \"Probably you should talk to my partner.\"\n\nBut Ed Furtig was coming up the station house steps and Jablonski had bustled out to meet him. And Sandalwood was looking curiously at Ryan.\n\n\"But you can answer this,\" he said. \"I understand from the Mirror man that you said Derby admitted killing the old lady. You were clever to trap him into admitting, that.\"\n\n\"Well...\" Ryan's mouth was dry. \"We just questioned him. That's all.\"\n\n\"And right away he admitted the murder\u2014just like that?\"\n\n\"Well...yes. Pretty much.\"\n\n\"Is that how the...ah...fight started?\"\n\nThen Ryan saw what he was driving at. \"Hell, no. Don't get that idea! We never laid a hand on him until he jumped Jabby.\"\n\n\"I see. Odd though, eh? I mean, he has everything to lose and nothing to gain by admitting it. Why, that puts him in the chair.\"\n\n\"He's denying it now.\"\n\n\"Yeah, but...\" Sandalwood squinted his disbelief through cigarette smoke.\n\n\"Sometimes guys just get tired of twisting and dodging,\" said Ryan.\n\n\"Yeah.\" Sandalwood weighed the idea. \"But I must say I've never seen a bird of that type break this easy before. Have you?\"\n\nRyan began to feel it would never end. He looked around at the familiar surroundings, the stairway, the patient heads at the desk lighted by dull lights, the plaques along the wall commemorating officers of the precinct lolled in line of duty. Suddenly it was filled with unforeseeable disaster.\n\n\"...eh?\" Sandalwood was asking insistently.\n\n\"Sorry. I didn't get that.\"\n\n\"As I understand it,\" Sandalwood patiently repeated, \"Derby didn't have much dough on him.\"\n\n\"That's right.\"\n\n\"How much?\"\n\n\"I've forgotten exactly\u2014Jablonski could tell you. I think he counted it up.\"\n\n\"I see. But he didn't have that big bill on him.\"\n\n\"That's right.\" This was going a little better.\n\n\"What had he done with it?\"\n\n\"He didn't say.\"\n\n\"You asked him?\"\n\n\"Naturally.\"\n\n\"You mean after admitting the murder itself he wouldn't tell you what he'd done with the proceeds? That's odd.\"\n\nRyan could hear Jablonski's complacent drawl from beyond the doorway. That was all he could think of, Jablonski's absence and Sandalwood's unanswerable questions.\n\n\"Didn't you try to get some explanation...?\"\n\nA telephone rang, a nearby clangor. He heard the timeworn salutation. \"Seventeeth Precinct. Sergeant Weiner.\" Then a listening silence. The whole world was silent, waiting, listening for his answer.\n\nAnd he had no answer. An eternity passed.\n\nBut helplessness was a thing Ryan could not long endure, and his temper came to his rescue. \"Look here,\" he snapped. \"When you're pinching a guy like Derby you don't think of all the smart questions some second-guesser can think up later. This guy jumped us. He hurt my partner. We didn't have time to... He's being gone over now by Sergeant McGonigle. Talk to McGonigle if\u2014\"\n\nSandalwood began to break in with \"I'm sorry. I'm sorry,\" before Ryan had finished. When he had, Sandalwood smiled pacifyingly and said with soft innocence, \"All I meant was that a hundred bucks is a lot of money to get lost so easy.\"\n\nOnce again the full force of Sandalwood's remark exploded belatedly in Ryan's mind, like a delayed-action bomb, which was just how Sandalwood intended it should. Sandalwood suspected he and Jablonski had taken the C-note for themselves. That was why he had been curious about Derby's admission of guilt. That was the weakness in their story.\n\nRyan's thoughts began following the pattern he knew Sandalwood's had taken; it was like following fresh-made tracks in the snow. Sandalwood could have drawn two conclusions from what he knew. One was that Derby had admitted killing Mrs. Connors but had given Ryan and Jablonski the hundred dollar bill in return for some favor. The other was that Derby had denied the murder and they had taken the money because he could not very well claim it. Either way Sandalwood had no inkling of the real truth. But he was suspicious, and therefore he was dangerous.\n\nRyan thrust his right hand with assumed nonchalance into his suitcoat pocket and it encountered the crumpled brown paper sack that Jablonski had used.\n\nThat almost panicked him; he withdrew his hand guiltily.\n\n\"We didn't thoroughly search Derby,\" he said. He spoke steadily. \"He may have the dough in his shoe or pinned to his underwear\u2014or he may have stashed it some place. Maybe Sergeant McGonigle will get that out of him. I don't' know.\" The steadiness he had to assume steadied him; he felt his powers of conviction growing as he spoke. \"But in any case neither Jablonski nor I have any idea where the bill is\u2014that's for sure.\"\n\nHe looked angrily at Sandalwood's intent face, and he knew Sandalwood understood he had caught the insinuation and was flinging it back. \"Maybe a hundred bucks means more to you than it does to me,\" he added.\n\n\"Oh, now wait a minute,\" said Sandalwood contritely.\n\n\"Hey, what goes on?\" Jablonski arrived.\n\nWhere were you when I needed you?\n\n\"I was just asking Ryan about that hundred buck bill.\" Sandalwood said. \"I guess he misunderstood what I meant.\"\n\nJablonski grew attentive. \"Yes? What did you want to know?\"\n\nGod, don't spoil it!\n\n\"I was just wondering,\" said Sandalwood, returning to the scent, \"why Derby admitted killing the old lady and yet wouldn't tell what he did with the money.\"\n\nJablonski looked quickly from one to the other. \"Well now, it wasn't exactly like that,\" he said. \"You see, we grabbed him pretty much by surprise and he said some things when he was excited that he probably won't repeat. Matter of fact, after he got his head again he buttoned up pretty much. That's when I asked him about the dough. Of course, we didn't really try to question him.\"\n\nIt was that easy when you were relaxed and experienced.\n\n\"I see,\" said Sandalwood. \"I see,\" and scribbled a note.\n\nThen he looked up and smiled. \"Thanks,\" he said to both of them. \"And congratulations.\" He nodded to the photographer waiting for him and they went out.\n\n\"Any time,\" Jablonski called.\n\nThe last Ryan saw of Sandalwood was a glimpse of that well-tailored back going down the steps.\n\n\"Whew!\" he said.\n\n\"Yeah. They can be nosy. But it's over\u2014all over! Wait here a minute 'til I check with Mac\u2014\" Jablonski grinned\u2014\"then I'm taking you out for the biggest slug of Canadian Club in New York.\"\n\nRyan grinned back. \"Sounds good\u2014but where do we go at this time of night?\"\n\nJabby punched his arm. \"It ain't this time of night where we're going.\"\n\n\"I need cigarettes. I'll meet you on the corner.\"\n\nAs he strolled indolently down the station house steps, a uniformed man came up to them. Ryan recognized him as a probationary patrolman. The young patrolman's eyes widened at the sight of Ryan and he altered his ascent of the steps to give Ryan plenty of room. \"Good morning,\" he said politely. Ryan responded with a careless hand.\n\nThat's how little time it takes, he thought.\n\nHe took off the expensive light gray hat he wore and re-crushed it into the original fancy crease that he liked to affect as a civilian. Working, he could not wear anything that would make him distinguishable in a crowd. That nightly gesture was an expression of his individuality, a symbol of release from departmental routine.\n\nBuying cigarettes in the all-night restaurant, Ryan began to feel relaxed and thoroughly happy. Sandalwood and the china chip seeped from his mind, and the tension of the last few hours dissolved in a warm awareness of accomplishment and praise. It was the way you felt trotting off the field after having played a good first half. It was the quiet, deep satisfaction of knowing you had come through a hard test well.\n\nHe didn't really want a drink or need one, he thought.\n\nBut at that moment Jablonski came hurrying along the sidewalk outside. Seeing a cruising taxi he put two fingers to his mouth and blew a piercing blast. The cab squealed to a stop. Jablonski waved to Ryan, coming out of the restaurant. \"Hurry up,\" he called. \"This is on me.\"\n\n\"Hell it is!\" Ryan laughed, and got in.\nCHAPTER 7\n\nC.C. and Soda\n\nThe cab carried them West of Broadway and stopped before another old house. To Ryan the comparison between this approach and the other one they had made earlier was startling, but Jablonski had no thought for subtleties. \"My girl's meeting us here,\" he exulted, and rang and then tapped impatiently on the door's frosted glass.\n\nThe announcement of Jablonski's girl surprised Ryan, who knew little about his partner but was aware that he was married. He grinned appreciatively. You had to hand it to this old goat at that.\n\nThe door opened stealthily on an almost dark hallway. \"Evenin', Fritz,\" said Jablonski, and pushed Ryan quickly inside.\n\n\"Evenin', Sarge,\" replied the thin, elderly waiter. \"Everything okay tonight, Sarge?\"\n\n\"Everything's copacetic, Fritz. Meet my friend, Neill Ryan. He's okay, Fritz. Treat him right when he comes around.\"\n\n\"You know me, Sarge. My pleasure, Mr. Ryan.\"\n\nJablonski had opened a door leading down to the basement. Piano music, smoke and snatches of talk and laughter drifted up the stairway. At the bottom of it Fritz look their hats and coats and led them to a corner table. \"Inez ain't here yet,\" said Jablonski.\n\nRyan looked around. Once this had been the old house's kitchen and servants' dining room. Now, bathed in blue light, there was a small bar of white patent leather and many bottles, attended by a white-jacketed barman. Near it a middle-aged man with unnaturally black hair and a purple complexion was playing \"Mood Indigo\" on a tiny piano, holding the chords down long. Two men stood at the bar and three tables were occupied. It all looked and sounded welcome.\n\n\"Double C.C. and soda for both of us,\" Jablonski told Fritz.\n\n\"Water for me,\" Ryan corrected.\n\n\"Water,\" said Fritz, and left. Ryan said. \"That guy Sandalwood still worries me.\"\n\n\"Forget it. We answered his questions.\"\n\n\"But he suspects something. Not the right things, but\u2014\"\n\n\"What the hell! What can he suspect? What can he prove? You don't understand the beauty of it, Neill. Derby can't say anything without tightening the case against himself. And who else is there but us who knows?\"\n\n\"Canadian Club with water,\" said Fritz. \"And soda.\" He put down two heavy water glasses, half-filled with whisky and ice cubes, a bottle of soda and a glass of water.\n\nJablonski chuckled. \"Two bucks a drink and worth it,\" he said. \"At least tonight. Eh, kid?\"\n\nRyan raised his glass. \"Here's to your new joint in Mount Vernon.\"\n\n\"New Rochelle. Thanks.\" Jablonski took a long pull at the whisky.\n\nRyan took a shorter one. Even with the liquor fumes in his nose he scented the cloud of perfume that enveloped him. A voice said, \"So here you are.\"\n\nA woman stood over their table. She wore a coat that somewhat resembled mink and her ash-blond head was hatless. The lips and eyebrows in her bright, smiling face were well-marked slashes of red and black respectively. Jablonski and Ryan got to their feet. \"Inez!\" said Jablonski. He kissed her cheek.\n\nOnly then did Ryan notice a girl standing behind the blond woman. Inez was forty-five trying to look thirty. The girl looked a couple of years over twenty and she didn't have to try. The brim of a simple felt hat was pulled low over her violet eyes and her hair was coppery and abundant at a time when it was fashionable for hair to be close-cut. Her coat of woolen-pile, cinched to a tiny waist by a leather belt, was inexpensive but chic. Looking from Jablonski to Inez with a faint smile, the girl seemed not to have noticed Ryan. At least, that is how it seemed to Ryan.\n\nHe began-speculating about what was under the coat.\n\nJablonski introduced them. The girl's name was Gee Gee Hawes. She sat down with demure self-possession, told Fritz, \"A chilled sherry, please,\" in a husky contralto and took out a cigarette. When Ryan held his lighter to it, she smiled at him from under the hat and it was like looking into spotlights. The pianist began \"The Beguine.\"\n\nJablonski said, \"This is an unexpected pleasure, Gee Gee.\"\n\nShe smiled at him, making the corners of her eyes crinkle. \"A date stood me up,\" she said candidly, \"and Inez was nice enough to ask me along.\"\n\n\"Thank God for Inez,\" said Ryan, his spirits already lifted by the whisky, and when she made a moue of appreciation at him he began wishing he had shaved recently and had on a new shirt. He wondered whether she lived alone or shared an apartment. He wanted to make conversation and said, \"Do you always drink your sherry chilled, Miss...ah...\"\n\n\"I'm Gee Gee,\" she said.\n\n\"Hey!\" yelled Jablonski. He had picked up the Daily Mirror Inez had brought in with her. Its front page, devoid of the usual picture, was given over to big type:\n\nCOPS NAB KILLER AFTER GUN FIGHT\u2014HARRY DERBY CAPTURED\n\nRyan said, \"Well, well.\" He did not want to seem impressed in front of this girl.\n\nJablonski turned to the story inside the paper. \"For gosh sakes! Didn't even use our names.\" He read: \"'Bulletin: Detectives of the East Fifty-first Street Station early today arrested Harry Derby, the hoodlum identified as the killer yesterday of Mrs. Thelma Connors in her East Sixty-first Street apartment. Derby was seized after a gun fight in an East Side rooming house. One officer was reported wounded critically.'\"\n\n\"All you ever think about is the crime news, Ed,\" said Inez fondly.\n\nJablonski gave her an indignant look. \"How do you like that?\" he demanded of Ryan. \"Who do you think they're talking about in that story?\"\n\n\"You and Mr. Ryan?\" Gee Gee asked. \"Really? You captured that man who\u2014\"\n\n\"We got Derby,\" said Jablonski.\n\n\"Really?\" When her eyes widened they were not violet but incredibly blue. Inez put both her hands admiringly on Jablonski's arm.\n\nA new tray of drinks arrived. Gee Gee said, \"The final News will be out soon. That may have more on it.\"\n\nJablonski said, \"Fritz, run out for the last edition of the News, will you? Get several of them.\"\n\n\"Soon's I get this other table, Sarge.\"\n\n\"No one was wounded really, were they, Ed?\" asked Inez.\n\nRyan and Jablonski exchanged looks. \"No,\" said Jablonski. \"Not really\u2014except Derby. He got roughed up a little. In fact\u2014\" he looked at Ryan and winked\u2014\"I would say we didn't do him a bit of good. Eh, Neill? We really fixed him.\"\n\nJablonski laughed with such obvious double meaning that Ryan wished he would shut up. No one could know what Jablonski meant. But it shouldn't be talked about anywhere, before anyone.\n\n\"What do you mean, Ed?\" Inez leaned yearningly toward him.\n\nSix ounces of whisky had thickened Jablonski's tongue. \"I mean that Neill and I managed to make a case against Derby that no defense lawyer is gonna\u2014\"\n\nRyan grew desperate. \"How about dancing?\" was all he could think of. \"Come on, Jabby\u2014while the music's going.\"\n\n\"Hell with it,\" said Jablonski. \"I'm no dancer. I'm a drinker. Where's that lousy Fritz got to?\"\n\nRyan looked at Gee Gee and she smiled at him.\n\nHe got up because he had to now, but he hadn't wanted it this way. He leaned over Jablonski. \"Keep your mouth shut,\" he whispered. Jablonski, slack-mouthed, gave him a wise wink.\n\nGee Gee drew out of her coat. She was wearing a plain dress of gray wool, and the figure beneath it had not been reconstructed into cones and spheres by the usual undergarments and did not need to be. It made Ryan forget even Jablonski.\n\nThe piano rippled into \"Just One of Those Things,\" and Ryan put his arm around her.\n\n\"You and Ed work together, I gather,\" she said after a moment. It was strange to hear Jabby called Ed.\n\n\"Yes. We've been partners about a week. And you and Inez?\"\n\n\"We work together too. Or did until tonight.\"\n\n\"What happened tonight?\"\n\n\"Well, you'll hear it sooner or later. The club where we work let out Inez. They've hired a new kid\u2014a cool singer.\"\n\n\"I didn't know Inez was a singer.\"\n\n\"That's, right. Blues. And you know what cool jazz has done to that.\"\n\n\"You sing too?\"\n\n\"No, I dance. As a matter of fact\u2014\" again the crinkly-eyed smile\u2014\"I'm what they call an exotic dancer. You know\u2014a stripper. Although in New York it's not quite stripping.\"\n\nThis was the first time Ryan had ever met a stripper. He said, \"I'll bet you're good at it,\" and then wondered if that had been tactful.\n\n\"I've only been doing it a few months. Before that I was in the chorus. Some people think it's indecent. But if you don't have to do anything dirty and you've got a good body, I can't see any harm in it. And the place we're in gets a nice crowd\u2014as clubs go.\"\n\n\"Why sure.\" But Ryan was off balance. This girl was not what he had thought when she first sat down at the table, and even the knowledge that she was an \"exotic dancer\" did not alter this new, growing opinion. She was unselfconscious and wholesome, and when her pliant dancing pressed her close to him and a coppery curl brushed his cheek, Ryan, far from considering certain opening gambits, felt embarrassed and respectful, and he could think of nothing to say.\n\n\"Well, anyway\u2014\" she looked up at him\u2014\"I'm not sorry my date had to work tonight.\"\n\n\"I'm mighty glad,\" and Ryan grinned at her with happy awkwardness. He whirled her around and she followed dexterously; she was really a marvelous dancer. \"What's he do\u2014is he an entertainer too?\"\n\n\"No, but his working hours are almost as bad. He called me at the club just before closing time to say he was on something really big and would be working the rest of the night. He's a newspaperman.\"\n\nEven mention of the word was enough to alarm Ryan momentarily. \"Oh really?\" he said. \"What's his name? I know a few.\" Which was an exaggeration.\n\n\"I guess he's working on some kind of big scoop,\" said Gee Gee. \"His name is Sandalwood\u2014Jack Sandalwood.\"\n\nAnd for Ryan all pleasure in the evening ended. He looked toward their table. Jabby and Inez were deep in conversation.\n\nKeep your drunken mouth shut!\n\nFresh drinks awaited them when they stopped dancing. Gee Gee excused herself to go upstairs, and for a second an absurd suspicion crossed Ryan's mind that she might be going to telephone her boy friend with what she might have learned from Jablonski's implications. He sat down at the table and picked up a new drink. But he told himself, You better watch this stuff.\n\nJablonski and Inez did not notice his presence. Inez was looking down at her glass; Jablonski, his face heavy with liquor and fatigue and white stubble, spun the rim of his glass with thick fingers.\n\n\"Well, that's how it is,\" said Jablonski. \"Sure, we've had fun together. But you always knew I'd retire sooner or later. And Sarah and I figure...\"\n\nRyan realized what he had walked in on. He wanted to leave, but he did not know what Jablonski might have told her while he and Gee Gee were dancing. And now Jablonski was ditching her.\n\n\"Sarah and I figure we'll try and make a go of it again. With this deal in New Rochelle...\"\n\nInez looked up suddenly. Her eyes were swimming and her face was a ghastly smile. \"I know, I know,\" she said. \"You never made any promises, Ed. And neither did I. But more than once, when you'd been drinking a little, maybe, you told me how one of these days you and Sarah would break permanent. And then you and I...\"\n\nShe broke off; she seemed about to begin to weep openly. Instead she gained strength from some inner source. She straightened up and her face grew hard and cynical, and even as a tear spilled over one eyelid and down a pale cheek she said, \"Hell with it,\" down in her throat and raised her glass and drained it and put it down with finality. She studied it and her low-held head was no longer ash-blond, but silver-threaded and old.\n\nGee Gee came back, sat down and, looking at Inez, misinterpreted what she saw.\n\n\"Aw, don't feel so bad about it, honey.\" She patted Inez's purple-taloned hand. \"There's plenty of better places you can work. Your agent\u2014\"\n\nInez smiled glassily at her. \"This is just my night,\" she said. \"Ed just told me he's retiring and moving out. That's nice too, eh, kid?\"\n\n\"Well, what else happened tonight?\" demanded Jablonski aggressively.\n\n\"They fired me,\" said Inez. \"Ol' Max fired me. After all these years. The joint's getting itself a new shouter.\"\n\nRyan picked his drink up again and, although he knew he shouldn't, he drank it down.\n\n* * * *\n\nAfter a time Fritz came in with a bundle of Daily News. The dawn cold had touched his pallid cheeks with scarlet. \"Hey,\" he called from the stairs. \"You guys are famous.\"\n\nHe distributed tabloids like a newsboy. They were cold and damp and on the front page, almost life size, there was a picture of a man with his hat pushed back over curly hair, holding a telephone to his ear, shrewd young eyes round and the mouth open with awe. It was Ryan, talking a few hours before to the commissioner.\n\n\"Holy God,\" said Ryan.\n\n\"Let's see,\" said Gee Gee. \"Oh, look. That's cute. See, Inez?\" She looked back to Ryan. \"But I don't think it does you justice.\"\n\n\"Hey!\" cried Jabby. \"Listen!\" He had opened his paper and read, \"Two fast-thinking detectives of the Seventeenth Squad\u2014one a veteran on the eve of retirement and the other a rookie on his first murder case\u2014fought and shot it out early today with Harry Derby, the dock hoodlum wanted for the murder of a grandmother, and they brought him in.\n\n\"'Derby allegedly confessed the brutal slaying yesterday afternoon of Mrs. Thelma Connors, 63, in her apartment at 585 East Sixty-first Street.\n\n\"'The two detectives, whose nervy work brought them the immediate commendation of Police Commissioner Drumm, are Detective Third Class Edmund A. Jablonski, 53, who retires from the force next week, and Probationary Detective O'Neill Ryan, 28, promoted from patrolman only four days ago. The two cops spotted Derby, a three-time loser for whom a 13-state alarm was out...'\"\n\nAs Jablonski read other customers gathered around their table, and when he had finished they broke into applause. Then there was much backslapping and many congratulations, which made Ryan feel a little ridiculous but also very happy that Gee Gee was there for it. Even Inez dabbed powder around her eyes, smiled and congratulated them both.\n\nThere were drinks bought for them, and Ryan decided the best way to silence Jablonski's babbling tongue was to let him get as drunk as possible.\n\nAfter a time he looked inquiringly at Gee Gee, wordlessly suggesting another dance. She nodded almost imperceptibly at Inez. Ryan understood and felt proud of his perception and at Gee Gee's asking a favor of him. With the elaborate gallantry of the half-drunk he asked Inez to dance. She looked pleased, and as they rose Gee Gee smiled warm appreciation.\n\nBut later, when the pianist began \"I Concentrate on You,\" which had always been a favorite of his, Ryan asked Gee Gee to dance again. This time he held her closer, and the fragrance of her hair and that lissome body all seemed to make her the most tender and desirable and beautiful thing in the world. Before the dance was over Ryan noticed an unnatural radiance around the window's heavy drape and after a time he realized that it was the light of morning. He didn't care.\n\n* * * *\n\nWhen they left he had to help Jablonski get to the cab, and early office-goers who passed them turned curious, sleep-wrapped faces to give them a second glance. Ryan took Gee Gee to her door (how incongruous that this ravishing girl should live in a slattern apartment building beyond Eighth Avenue), and at the door he tried suddenly and drunkenly to kiss her. She somehow turned away, it seemed by accident, but she did present a cheek...\n\nRyan went back to the cab in a daze.\n\nAfter they had ridden in silence for a few blocks he said, \"What the hell did you tell that dame about Derby?\" But Jablonski was asleep.\n\nThe cab let Ryan off at his home first, for Jablonski lived in Queens. Ryan gave the cab driver instructions.\n\nThe first-floor apartment where Ryan lived appeared empty. Then he gratefully remembered this was Friday; his sister worked and his mother would be at the Altar Society coffee to plan the Sunday altar flowers. That was a relief; he did not like coming home like this, and he did not want her to see him. Ryan had grown up in a neighborhood where drunkenness was common enough to be neither smart nor funny but a disgrace.\n\nBut as he took off hat and coat his mother came out of the kitchen. \"Why, where have you been, boy?\" she said anxiously.\n\nHe kissed her on the cheek. \"We had a heavy night, ma. You'll see about it in the papers.\" He did not want to tell it now.\n\n\"It's always a heavy night,\" she complained. Complaint did not become her; she was a tiny, merry woman. Ryan held her coat for her, bending over to get the sleeve holes right for her short arms. \"I really began to worry,\" she said.\n\n\"You ought to know better,\" he chided. \"You've lived with cops long enough.\"\n\nShe said nothing, but the words twisted a knife blade in her heart. The nights she had waited for his father. And there had been the night... \"I'm going to the Altar meeting. There's coffee simmering on the stove.\"\n\n\"Thanks, ma.\" He took a cup into his bedroom, pulled down the blind and undressed, swaying, while he sipped coffee. He switched on the bedside radio and then lay down, feeling the press of cool pajamas against his liquor-heated body.\n\nThe ceiling spun slightly. Ryan, sobering, felt depressed. The radio came on with a little roar.\n\n\"...teen minutes to nine, girls,\" a man's voice drawled.\n\n\"All I can say is,\" and Ryan recognized the familiar voice of Dorothy Kilgallen, \"that in spite of my years as a reporter in this town I never cease to marvel at the job New York's cops do.\"\n\n\"Honey, I agree with that completely. And I know what you mean.\"\n\n\"It's that story in the News. Just suppose it had been you or me who spotted that\u2014that tough man. Would you have liked to follow him into a dark street and into some old rooming house? And just walk in on him?\"\n\n\"Dear, I'm afraid not. I think that New York's finest are\u2014well, the greatest, that's all. And as you say, these two detectives\u2014what were their names?\"\n\n\"Jablonski. Edmund Jablonski. And O'Neill Ryan. Isn't that a wonderful name?\"\n\n\"Oh, for God's sake,\" Ryan muttered.\n\n\"...ought to take up a collection for them,\" the man's voice said.\n\n\"They couldn't accept it, honey. The department doesn't permit that.\"\n\n\"Just keep Sandalwood off my neck,\" growled Ryan and switched off the radio.\n\nHe fell back on the pillow. The ceiling spun again. He closed his eyes... That Gee Gee... He slept.\nPART TWO: THE BIG NEWS\nCHAPTER 8\n\nThe White Sheep\n\nA plea of not guilty was entered for Derby when he was arraigned, and since the murder had created so much public indignation his examination was scheduled for three days later. He was bound over to the grand jury and it held him for trial without bail. This was all predictable.\n\nWhen Ryan returned to work, it was as a hero. His routine and Jablonski's were disturbed for several days by newspaper reporters, Sunday feature writers, magazine researchers, photographers and true crime story authors. At first their questions alarmed him. But when he came to realize that these interrogations were directed less toward discovering anything than to proving an already-established point of view, he began to relax. Circumstances had conspired to make him a temporary public figure; Ryan accepted it and after the initial reluctance even entered in with quiet-mannered willingness. His mother and sister complained about the constant telephone calls at home, but he noticed that his mother was never too busy in the kitchen to dry her hands and come to the telephone, and that when a photographer was coming, Eleanor usually had on a new dress and fresh makeup.\n\nMeanwhile, the department had engaged in the usual tedious and vital routine. The bullet taken from the dead woman's head was found to have been shot from the revolver found in Derby's room. The money that Derby had spent at the delicatessen was recovered and as far as could be told might have been one of the tens that Mrs. Connors had obtained from the bank. The partial fingerprints found in the apartment did not check out; they were the dead woman's. But there were two loan sharks named Morgan who worked the docks; one hung around Pier Ninety in Manhattan, the other in Brooklyn. The first did not know Derby and was eager to testify that he had never loaned him money. The second did know Derby and had occasionally done business with him. But when two detectives found him he had been smoking opium in a Williamsburg flophouse for forty-eight hours, and was white and sweating. He was glad to admit that he had not loaned Harry Derby any money recently.\n\nFar more important, McGinnis and Minor went out to Derby's house and talked to his sister, a thin, blond-haired girl with the calm detachment of resignation. She worked as librarian for a midtown athletic club, and when they asked if she had seen her brother Harry on the afternoon of the Connors murder she said she had not because she developed a sick headache that day and had gone home early in the afternoon.\n\nMcGinnis asked whom she had seen there, and she said the house had been empty. Other careful questions elicited that she had arrived home before two p.m., and that she had been alone in the Derby apartment on East Seventy-third Street from that time until sometime between six-thirty and seven, when her younger brother had come home from work. That shattered Derby's alibi that he had been enjoying a nap there that afternoon.\n\nA child found a small black change purse a block from the Connors' apartment; it bore a faint smear of blood which the lab proved was Mrs. Connors'. A police scientist in the lab spent most of one day extracting tiny particles of plaster dust from the hairy wool of Derby's jacket. The spectroscope proved beyond doubt that the particles came from the lamp used to bludgeon Mrs. Connors.\n\nShortly before Jablonski retired he and Ryan went downtown to the district attorney's office and spent forty minutes with Assistant District Attorney Gil Tilbury. He proved to be an excessively thin young man in an expensively tailored suit, buttoned-collar shirt and casually correct striped tie. He laughed often and confidently, and lolled back in his chair until it creaked to show how supremely at ease he was. He told Ryan and Jablonski that he couldn't wait to try this case and that Derby was as good as in the death house. He twiddled a Phi Beta Kappa key.\n\nThat evening Ryan sat at home watching a fight on television before leaving for work; he was now on the late tour, midnight to eight a.m. His mother had gone to a card party and his sister was in her bedroom doing something intricate and time-consuming to her hair.\n\nRyan in sweatshirt and slippers felt lazy and relaxed after sleeping all day. It was a nice way to feel, lying on the couch watching two other guys over at St. Nick's Arena do the heavy work for a while. Occasionally the camera swept the ringsiders, and one of them was a girl who looked like Gee Gee. Probably he ought to call her again. But he'd called twice and missed her each time, and he didn't want to appear anxious. Maybe tomorrow it would be all right. The doorbell rang.\n\nRyan lifted himself off the couch and groped his way from the darkened living room to the hall. When he reached the front door he saw through its window that the caller standing outside on the porch was a man and that he was tall and lean.\n\nThen the man moved his head and Ryan caught his profile.\n\nIt was Derby.\n\nEven while Ryan watched incredulous, Derby turned and gave the bell an irritable double ring.\n\nRyan began taking long tiptoe strides back the way he came through the living room and into his own bedroom. He removed the pistol from the holster on the bureau. He cocked it and picked up a flashlight, then returned quietly to the hall. He could feel the beat of his heart.\n\nHe moved quickly through the short hall, holding light and gun behind him, shifting the light at the last moment to the crook of his right arm as he reached the door, then pressing the latch with a quick left-hand to throw the door open and bring the gun up as he grabbed the flashlight under his arm.\n\nDerby was wearing a neat bow tie of blue checks, a white shirt and a dark suit. A slouch hat of thin felt was pulled low over his right eye; he carried a trenchcoat and he looked young. He winked rapidly in the flashlight beam.\n\n\"What is this?\" he demanded in an unexpected voice. \"I want to see Mr. Ryan\u2014Detective Ryan, I mean. Isn't this where he lives?\"\n\nRyan kept light and gun on him. Derby could not see beyond the light. He blinked uncertainly.\n\n\"All right,\" said Ryan.\n\n\"Oh. My name's Derby, Mr. Ryan. Ken Derby. I'm Harry Derby's brother.\"\n\nThen Ryan understood\u2014and suddenly felt ashamed of the .38 and thrust it into his slacks. He said, \"What's on your mind?\"\n\n\"I just want to talk to you. Sort of off the record, as they say.\"\n\n\"You're talking to me.\" It annoyed Ryan to have been frightened.\n\n\"It would be better inside.\" Derby's brother raised his face to the light. \"This will take a little while.\"\n\n\"Come on in,\" said Ryan. He did not know exactly how to deal with this, and that annoyed him too. He followed the younger Derby into the living room, switched off the television in mid-commercial, said, \"Sit down,\" waved to the sofa and sank into the deep, worn, leather chair that had been his father's. When he raised his hands negligently behind his head the pocketed .38 made a hard knot against his thigh.\n\n\"I'm Ken Derby,\" said his visitor again. \"Harry's my\u2014my older brother.\"\n\n\"Quite a brother.\"\n\n\"Maybe. Anyway, that's why I'm here.\"\n\n\"Why?\" said Ryan.\n\nKen Derby looked uneasy. \"I want to find out what's happened to Harry,\" he said with an air of dogged determination. \"I've been talking to his lawyer. I just don't get it.\"\n\n\"Don't get what?\"\n\n\"Look. You're a cop. I suppose that you figure anyone connected with Harry is a\u2014crook, or a heel.\"\n\nMeeting only with affirmative silence, he continued, \"We know what Harry's record is. My sister and me, I mean. Harry's like my father was. On the docks and\u2014well, a bum. Let's face it. But the rest of the family isn't like that. My sister isn't, and my mother wasn't. And I'm not.\"\n\nRyan kept staring at him.\n\n\"The hell with that. Here's what I came to say. And understand, I'm on the level. You can ask about me down at Triple-A. I drive a truck for them\u2014Triple-A Delivery on Varick Street. You ask if Kenny Derby's okay. And my sister. She\u2014\"\n\n\"You went into that.\" Ryan said it knowing it must anger Derby into saying more than he had come to say.\n\nAnd Derby took it that way. He got to his feet threateningly\u2014virtually as tall as his brother although not as muscular. Still, Ryan liked the feel of the .38 against his thigh.\n\n\"Okay,\" said Derby. \"Then I'll go into this.\"\n\n\"Before you do,\" said Ryan, interrupting in a way that he knew would not soothe Derby, \"just bear in mind that I'm a police witness against your brother. I can't discuss details of a case that is going to trial.\"\n\n\"You don't have to discuss anything,\" said Derby grittily. \"You've just got to answer one thing. Maybe you don't want to answer it. But I came here to ask it to your face, and I want to see your face when you answer.\"\n\nRyan looked up curiously.\n\n\"You got a case against my brother,\" said Derby. \"I know that. And he's got a bad record. But almost everything you've got against Harry could be coincidence. The witnesses, the ten dollar bills, all that. The one thing you've really got against him is that there's some kind of dust on his jacket that is supposed to prove he was in that old lady's apartment.\"\n\nRyan said, \"What gives you that idea?\"\n\n\"It was mentioned at the examination.\"\n\n\"So you've been talking to Harry's lawyer.\"\n\nDerby looked alarmed. \"Suppose I have? Is that against the rules?\"\n\n\"No. But do you want to argue with the scientists? They say that dust\u2014\"\n\n\"I don't want to argue with anyone,\" said Derby doggedly. \"If my brother is really guilty.\"\n\nRyan did not feel like answering that.\n\nA door opened behind him and he heard Eleanor say, \"Oh, I didn't know\u2014\" and concluded she had come out of her room without her dress on. He said evenly, \"You said your brother is guilty. Do you have any doubts?\"\n\n\"No,\" said Derby. \"I don't have any doubts. I don't have any doubts at all.\" Ryan caught a subtle meaning in that and tried to ignore it.\n\n\"Then I don't get you,\" he said. But a premonition had flared in his mind, swift and ominous as the darkening of an ocean sky before a storm. \"If you've talked to Farragut, you know that the scientific evidence\u2014that the dust the lab men found on your brother's jacket\u2014is unmistakable proof.\"\n\n\"That's just what I'm getting at,\" said Derby.\n\n\"What, for God's sake?\"\n\n\"Mr. Ryan, how did that dust get on my brother's jacket?\"\n\nDerby's thin, light-eyed face burned across the lamplit room into Ryan's mind, and time became endless. Ryan's body flickered with impulses\u2014to leap up, or kick over a chair or yell. Instead he must sit quietly and prepare to speak quietly.\n\n\"It got there,\" he said, \"when your brother smashed the lamp across that old woman's head.\"\n\n\"No,\" said Derby. \"It didn't.\"\n\n\"Then how?\"\n\n\"I don't know,\" said Derby meekly. \"If I knew, I could save Harry.\"\n\nRyan snorted. \"That's only one part of the evidence against Harry. There were plenty of witnesses. There's a bank clerk\u2014\"\n\nDerby stood up, his lean frame drooping wearily.\n\n\"No.\" His voice was weary too. \"You don't understand, Mr. Ryan. And I know you can't discuss this and I didn't come here to\u2014to\u2014well, try to fix anything. But it's like I said. You have a case against Harry but a lot of it could be coincidence or mistaken identity or something. The one thing that's impossible to figure is that dust. I was hoping maybe you'd be\u2014be honest enough to tell me about that. But maybe you really don't know about it.\"\n\nRyan could stand this no longer. He too got to his feet. \"Why?\" he shouted. \"What the hell do you want explained to you? Your lousy hoodlum brother killed an old woman. He's guilty of murder. He deserves the chair and he'll get it.\"\n\nEleanor's door opened again. Ryan heard her say, \"Neill? Is there anything wrong?\"\n\n\"Maybe he'll get the chair,\" said Derby steadily. \"But he doesn't deserve it. Because he's not guilty of murder, Mr. Ryan... Harry did not kill that old woman. I know.\"\n\n\"Neill?\" called Eleanor anxiously.\n\nRyan could see only Ken Derby's pale, taut face. He was conscious of the rise and fall of his own chest. \"How do you know?\"\n\n\"Because Harry was with me that afternoon,\" said Derby; \"No one will ever believe it because nobody knows it but me and Harry. He went around on my truck route. I was supposed to have a helper that afternoon and the helper, he didn't show up\u2014he's a fellow that plays the horses and, well, you know. I had two big pieces to deliver way uptown that afternoon and I stopped off at the house and picked up Harry to help me. That's against the rules, but I didn't want to get Dom, that's the helper, in trouble. And I knew Harry could use the dough. He's been broke for quite a while.\"\n\n\"But\u2014\" Ryan could not get his thoughts together. He was afraid to say anything or do anything that would make it any worse than it already was. An inspiration came from somewhere. \"If he helped you make deliveries, then someone must have seen him.\"\n\n\"I doubt it. You're bonded on this job. If the company discovered I'd used an unbonded helper, let alone a guy with Harry's record, I'd get fired so fast it'd make your head spin. Harry stayed out of sight, mostly. Oh, he ran a few light pieces in on the way uptown. But I really needed him only for the heavy work. One heavy piece was a refrigerator to an apartment up in Harlem and no one saw us take that in at all. The other was way up in the Bronx and we just dropped this stove on the porch of a new house. There were some carpenters working there inside, but\u2014well, you know how much people notice something like that.\"\n\nThere was silence. \"That's why I wonder about that dust, Mr. Ryan.\"\n\n\"If he had an alibi,\" said Ryan harshly, \"why didn't he say so?\"\n\n\"I don't think he really believes he needs one. And he knows it would get me in trouble if it came out he'd been riding with me\u2014him especially with his criminal record. Harry's funny. He's sort of proud that I\u2014well, I've got a job and always gone straight. And he still thinks that he can't be convicted.\" He added irrelevantly, \"He wants to fight it through himself. He'd kill me if he knew I'd come here.\"\n\nThe full meaning of it began to sink to the bottom of Ryan's understanding. As it did he felt cold and shaky, and he knew Ken Derby was watching him.\n\nIgnored, Eleanor firmly closed her bedroom door.\n\nDerby turned and picked his hat off the sofa. Ryan's gaze remained fixed on the emptiness where Derby had been.\n\n\"I hope,\" said Derby, \"that if you hear anything about that dust business, you'll be honest.\"\n\nRyan followed him to the door, said, \"Good night,\" and closed the door after him, and wasn't aware of any of it.\n\nHe went back to the living room, lit a cigarette and inhaled it deeply. He snuffed it out, and then wished he had not snuffed it out and started to light another and then stopped. He had to pull himself together. He had to-think this straight through. He picked up a fresh cigarette and lit it.\n\nChrist!\n\nThe clock tolled ten-thirty. It was time to start getting ready for work.\n\nHe was in great shape for it.\nCHAPTER 9\n\nDom the Tailor\n\nA hunchback named Frank Yett ran an untidy candy store just off First Avenue in the fifties. He dealt in pop, comic books, ice cream and novelties, and he took horse race bets on the side. Since this was the kind of neighborhood in which the police liked to keep track of neighborhood gossip, they let Yett make book as long as he took bets from adults only and passed on to them whatever he learned of police interest. Recently he had reported that a teen-age gang war was about to break out.\n\nThere was also a man who dealt in used cardboard cartons, lived in East Sixty-third Street and had been missing for two days. His car had been identified as one that ran down a pedestrian and then sped on along Bruckner Boulevard. On this night Jablonski and Ryan had been told to cruise the area where the teen-age war might break out and also to drive past the carton-dealer's apartment occasionally.\n\nIt was not cold for November. But it was so foggily humid that the black asphalt was streaked with damp sheen under the street lights and the motor of the well-traveled police sedan responded under Ryan's foot with unwonted power and quiet. They had had a couple of minor runs, but by three-thirty they sensed this would be a quiet night and so they might as well eat leisurely. There was a place on Third near Fifty-eighth Street that served hot sandwiches made with Italian sausage fried in onions and peppers.\n\nAt a quarter of four Ryan emerged from it carrying three sandwiches and two cartons of coffee. They pulled into a side street, switched off the lights and with even the radio quiet ate in aromatic silence for a few minutes, sharing the third sandwich.\n\nJablonski sighed contentedly. Tomorrow night would be his last tour of duty. He did not feel any sentimental twinges. All he could think of was taking it easy for a couple weeks. But something Ryan had mentioned earlier still bothered him. He sloshed his coffee around in the cardboard cup to stir up the sugar in it, drank, made a switching sound with his tongue against a bad tooth and drew a cellophane-sheathed cigar from his vest pocket.\n\n\"Well, did this guy offer any proof his brother was riding around in that truck?\" he asked as though they had been talking about it for the last half hour.\n\nBut each knew what had been in the other's mind.\n\n\"No. He just said\u2014\"\n\n\"He didn't say why Harry'd do a thing like that?\"\n\n\"Well, he needed help with these two heavy pieces like I said, and Harry needed the dough. He got the helper's check, I guess.\"\n\n\"But he didn't say that anyone saw Harry in the truck\u2014anyone but himself, I mean?\"\n\n\"That's right.\"\n\nJablonski crumpled the cellophane and threw it contemptuously to the floor mat. \"He's not even a good liar.\"\n\n\"Why do you say that?\"\n\n\"He's just trying to help Harry, that's all.\" He lighted the cigar and growled through jets of smoke. \"Maybe that damned lawyer... I wouldn't put anything past Farragut.\"\n\n\"But why does this Ken Derby know so much about the dust on the jacket? If Harry didn't tell him\u2014well, why does Ken say if he knew how it got on the jacket he could prove Harry innocent? I don't get it, Jabby. He\u2014he sounded so damned sure.\" Ryan knew he was sounding like a rookie, but he couldn't help it.\n\nJablonski laughed a short, bitter laugh. \"I'll tell you why. Because he's trying to shake your testimony, that's why. Farragut put him up to that, I'll bet you anything. He's trying to make you less certain. Yup, the more I think of it...\"\n\n\"Yes, but...\"\n\n\"The Regal's closing.\"\n\nRyan started the car and drove it slowly, lights out, to the cross street and parked. The Regal, a bar and grill across the wide avenue, closed at four a.m. The proprietor usually locked up and left by himself with the night's receipts. Ryan and Jablonski could watch his block-long walk to the subway entrance from here.\n\nThe sedan filled with cigar smoke. Ryan twisted his window down a little.\n\n\"Don't forget these guys are brothers, Neill.\"\n\n\"Oh sure, I realize that.\"\n\n\"Probably Harry does ride with him occasionally in that truck,\" Jablonski mused. \"Probably cases joints that way.\"\n\n\"I don't know about that. I think Ken's honest. He wouldn't let\u2014\"\n\n\"You think, huh? Personally I wouldn't trust any Derby.\"\n\n\"The younger one hasn't any record,\" said Ryan. \"I checked.\"\n\n\"There he goes.\"\n\nThe interior of the Regal had gone dark. Now its big sign blinked out. A man came away from the doorway's shadow and walked hurriedly down the street. They waited until he disappeared down a subway kiosk.\n\nBut Ryan did not start the motor. He had been steeling himself. Now with his hand on the switch, looking straight ahead, he said, \"Jabby, do you think there's a chance he's innocent?\"\n\nJablonski removed the wet cigar from his mouth and turned his head to look at Ryan. Ryan uneasily stepped on the starter and slipped the transmission into first.\n\n\"Wasn't it you that took the C-note out of his wallet?\" asked Jablonski at length. \"Innocent, for God's sake!\"\n\n\"I know, but\u2014\" Ryan was doggedly determined to get it all out. \"Like his goddamned brother said. That dust is the one thing that convicts him for sure. And we know how that got there.\"\n\n\"God awmighty, Neill. You're getting nuts. If you're serious\u2014I can't believe it, but if you are, check on that alibi and see for yourself. Why, that's the most\u2014and don't forget Derby himself told us he was home sleeping. He didn't say nothing about a truck ride. If I ain't right, you can come out to my new place and\u2014and eat all you want on the house for a month.\"\n\nRyan grinned. For the moment at least, Jablonski's hearty disbelief reassured him. \"I'm glad the deal went through, Jabby. The Green Lantern Inn, eh?\"\n\n\"That's it, and you're welcome any time.\"\n\nThe radio said it was four ten a.m. Ryan wondered if he might catch Gee Gee if he called the club, but he did not like Jablonski to know he was calling her. Jablonski said. \"Coast back to Sixty-third Street again and see if the guy's come home.\"\n\nRyan complied. He felt better for having talked about it. But he still felt a little wordless uneasiness that was like the gnawing of a grave worm.\n\n* * * *\n\nThe office of the Triple-A Delivery Corporation was a glassed-in room in the corner of a grimy garage that by mid-morning contained only a few immaculate black vans. An old-fashioned iron stove kept the office overheated and the sagging chair in which the proprietor, a heavy man named Nichols, sat sprawled, scented the hot air with the stink of rotting leather. The wall behind Nichols was papered with calendars old and new, all depicting long-legged girls in advanced stages of undress. Ryan flipped the badge on his wallet toward Nichols.\n\n\"It's a routine matter,\" he said, \"so for the present we'd prefer that no one knows anything about it. Especially the people involved.\" He looked down at Nichols. \"That's important,\" he said.\n\nNichols' thyroid-bulged eyes, big as olives, looked up under thick lids. \"I understand.\"\n\n\"You have a driver named Kenneth Derby. On November seventh he's supposed to have had a run that required a helper.\"\n\n\"Could be.\"\n\n\"Do you keep any records that'd prove whether he did or not\u2014and would they show who the helper was?\"\n\n\"November seventh?\" Reaching behind him, Nichols took one of half a dozen clipboards hanging from worn brass hooks and began riffling the long yellow sheets on it with a wetted thumb.\n\nRyan looked up at the girls. One had flowing coppery hair. Nuts to that! She'd never pose that way.\n\n\"Derby,\" said Nichols. \"November seven. Sure. Here it is. Yeah, he had a helper. Dom. Dom the Tailor.\"\n\n\"Who?\"\n\nNichols grinned a fat-lipped grin.\n\n\"Dominic D'Tela, his real name is,\" he said. \"Dom the Tailor, they call him. He's a little dago who part-times for me.\"\n\n\"You sure that's who it was? Is it possible someone else might have substituted for this Dom? Say Derby's brother Harry, who's a\u2014well, a sort of\u2014\"\n\n\"I know Harry.\" The heavy lids narrowed calculatingly. \"But Dom's the man I paid. I remember now\u2014and I remember Harry was around that night, too. They left together. Maybe he was waiting for Ken to come in. Anyway, I can tell from this, that Dom went out in the morning with Henderson\u2014he's my downtown man. And with Derby in the afternoon.\"\n\nBut Ryan's heart had suffered a sinking spell. Harry had been around that night, all right. Nichols remembered it.\n\nNichols was saying, \"Derby is usually my uptown east guy. And according to this, he drew a full day's check, fourteen fifty.\"\n\n\"Can you tell me where Dom lives?\"\n\n\"Sure. It's just off Houston Street.\" Like all downtowners Nichols pronounced it Howse-ton.\n\n\"Think he'd be home now?\"\n\n\"That's hard to say. He jobs around. He's a good strong man.\"\n\n\"Big? Tall?\"\n\n\"Oh, no. He's a little fellow. Big shoulders, though. Good-natured. And very reliable.\"\n\n\"Yeah. Thanks, Mr. Nichols.\"\n\nOnce again outside in the bright cold street he paused. He was tired. But he was also tired of uncertainty.\n\nBetter finish it up. Then he'd know.\n\n* * * *\n\nDominic D'Tela's midday meal, lasagna, bread and coffee, lay spread before him on an oil-clothed table in the tenement parlor.\n\n\"Have some coffee?\" he asked cordially.\n\n\"No, thanks.\" But the cheese-laden steam from the lasagna made Ryan hungry.\n\nD'Tela turned a strong face to him and said, \"What's on your mind?\" then returned to his food.\n\n\"Do you remember last November seventh? It was a Friday.\"\n\n\"I'll say it was a Friday.\"\n\n\"You remember it?\"\n\n\"I'll say I remember it. That's the day Cannon Cracker came in at Jamaica and paid twenty-eight bucks\u2014twenty-eight eighty, to be exact.\"\n\n\"And you were on him?\"\n\n\"Was I on him!\" The memory made D'Tela stop eating. \"Ten bucks on the nose. I come home with over a hundred and fifty bucks that night. I worked that day too.\"\n\n\"What was the job?\"\n\nThe thick cup came down, revealing a suspicious frown. \"You checking on me or something?\"\n\n\"Not on you. On someone else. And it doesn't have anything to do with horse bets. All I want to know is what you did on the job that day.\"\n\n\"Well, let's see.\" He reverted to the lasagna and chewed thoughtfully. \"I was working for old man Nichols that day. Yeah, I remember. In the morning I went out with Henderson and in the afternoon with Derby. Kenny Derby.\"\n\n\"You were with Derby all afternoon?\"\n\n\"Sure.\"\n\n\"Doing what?\"\n\n\"Well, let's see. Yeah, that was an easy afternoon\u2014lucky day all round.\" He grinned. \"There were only two big pieces he needed help with. Of course I jumped off with lots of the little ones, too, when we got to a neighborhood where there were quite a few deliveries. I always do my share on a job. Besides, Derby was feeling a little tough.\"\n\n\"What was his trouble?\"\n\n\"You know how it is. He said something about needing some dough for a friend of his, and he'd had a few drinks at lunch\u2014tell you the truth, I figured he had a broad in trouble. Of course, now I figure he wanted the dough for his brother. But it's odd, sort of.\"\n\n\"How do you mean?\"\n\n\"Well, he needed money and I told him about this horse I was going to bet, see? He was interested, too. But when I called in to get down on the horse just before post time\u2014around five, if was\u2014he said the hell with it. I just wonder what it might have done for him if he'd given me a ten to bet for him. Maybe his brother wouldn't have bumped the old lady, huh?\"\n\n\"By that time she was dead,\" said Ryan shortly. \"Let's get back to what you did that day. You said there were two big pieces. Where'd you deliver them?\"\n\n\"Well, the reefer went to an address in Harlem. The gas range went to a new house up in the Bronx.\"\n\n\"Who'd you see at the new house?\"\n\n\"No one, especially. There were some painters or carpenters working inside. We just set the range down on the porch and left.\"\n\n\"I see. Now look. This is the important part\u2014this is why I'm here. Was there anyone else along on this trip? In other words was there another man on the truck that afternoon besides you and Derby?\"\n\n\"Of course not.\" D'Tela looked surprised.\n\n\"At no time did another man ride with you?\"\n\n\"That's right.\"\n\n\"And you were with the truck all the time?\"\n\n\"Sure\u2014except when I left it to run packages in. Oh, yes, and...uh...we made a stop so I could pick up that dough I won.\" He looked uneasily at Ryan.\n\nBut Ryan was not interested in handbooks at the moment. He said, \"Then you didn't see anything of Derby's brother Harry that day? On the truck or anywhere?\"\n\nSurprise opened D'Tela's eyes. \"Sure I did. Now that you mention it. He was around the office when we got back, after six. Surly and tough-looking as usual, too. Waiting for Ken. I guess.\"\n\n\"But at no time was he on the truck. You'd swear to that?\"\n\n\"Sure, for God's sake.\"\n\nRyan got up. He knew the truth when he heard it. \"Thanks, Mr. D'Tela. You'll probably never hear any more of this, and I'd appreciate it\u2014if you don't mention what you told me to anyone. Especially anyone around the Triple-A Delivery Company.\"\n\n\"Sure. Sure thing, you bet,\" said D'Tela good-humoredly.\n\nThe flights of tenement stairs were dark and noisome, but as Ryan went down them they looked as good as a red-carpeted movie set. That lousy lying Ken Derby! Jabby had been right, all right. They had just been trying to shake his nerve and his testimony. They knew what he and Jabby had done with the dust, of course, and they hoped to make a scared and uncertain witness out of him\u2014the rookie! And they had almost succeeded.\n\nWhen he thought of what a fool he had made of himself talking to Jablonski in the car Ryan felt his cheeks redden. But it was all right now. He could go home and sleep, and forget the whole thing. Tonight was the party. Everything was okay.\nCHAPTER 10\n\nThe Y-Shape\n\nMickey McGonigle was a slim, wiry man who at the age of fifty-four had somehow shrunk an inch under the five feet eight prescribed by the City of New York for its policemen. He had very curly red hair parted on one side over a hatchet face corded with belligerent muscularity. He also had an intuitive, wonderfully penetrative sense for truth, which was infinitely more effective than the popular conception of the \"third degree\" in questioning suspects or witnesses. Leaning against Manny's bar now, he said, \"A beefsteak! A beefsteak for that squarehead Jablonski. I gotta have a drink on that.\"\n\nHe pushed his glass across the broad mahogany. \"What are you having, Ryan?\"\n\nRyan was not sure. He did not know what you did on an occasion like this, surrounded by superiors and having to work in a few hours. He said, \"A beer.\"\n\nMcGonigle said, \"Give him the W\u00fcrzburger, Charlie.\" Then he looked down the bar to make sure the subject was out of earshot and said, \"You were sure luck for Jablonski.\"\n\n\"Why? He was just as much luck for me.\"\n\nThe thin, conflicting planes of McGonigle's face wrinkled skeptically. \"The way I got it, you spotted him, walked him into the house and, when Jablonski let Derby jump him, you pulled him off him.\"\n\n\"That's not exactly right.\"\n\n\"It's not exactly wrong,\" said McGonigle.\n\nRyan raised his seidel. \"In a pinch like that how do you decide who's responsible for what?\" He added quickly, \"This is good beer.\"\n\nLieutenant Bauer came in, dark-hatted, dark-coated, quick-eyed. \"Hi, Mick. Hi, Neill. A Manhattan, Charlie.\"\n\nBauer took off his coat, leaned cold hands across the bar, rubbed them and said, \"Neill, do me a favor.\"\n\n\"Sure,\" said Ryan. He knew Bauer too well to call him lieutenant, but not well enough to call him Paul.\n\n\"The Fifteenth Squad is getting some enlargements of a print the picture bureau got in a drugstore job a couple weeks back. I have a hunch it may be one of that Little Turk gang we had in. Will you and Lambert pick them up during the night and leave them for me?\"\n\n\"Sure thing.\"\n\n\"Thanks.\" Bauer picked the cherry out of his drink and nibbled it by the stem. \"How're things going? I haven't seen much of you.\"\n\n\"Just fine, thanks.\"\n\n\"Lambert's a good operator. Quiet but smart. Smarter than Jabby.\"\n\nRyan had already discovered that about his new partner, but he was surprised and flattered by Bauer's candor. \"Could be,\" he said. McGonigle was talking Dodger talk with two others who had come in.\n\nBauer said, \"From what I hear they're going to push that Derby case into court as soon as possible. So if you have any special preparations to make or need extra time or anything, just let me know.\"\n\n\"Thanks. I don't think there's anything special, but I appreciate the offer.\"\n\nBauer waved that off and sipped his drink. \"By the way,\" he said, and Ryan recognized that the rest had been preliminary and that this was what Bauer really wanted to say.\n\n\"Jack Sandalwood called me today,\" said Bauer, and Ryan's nerves began to jangle.\n\nHe said, \"I see.\" Then, \"What'd he want?\" He took a deep draught of beer.\n\n\"Oh, he asked if we'd found that hundred dollar bill in the Derby case,\" said Bauer very unconcernedly, \"and he said something about having talked to Farragut. He\u2014he talked kind of odd.\" He looked idly down the bar. \"I don't know what he was driving at exactly,\" he finally went on. \"He was sort of vague, and that's not like Sandalwood. You have any ideas?\"\n\n\"No.\" And then, on a sudden gambler's impulse, he said, \"Of course, that night we brought in Derby he was asking a lot of questions, too. He was mainly interested in the C-note then. I think he figured Jabby and I had latched on to it.\"\n\nBauer sniffed skeptically and finished his drink. \"I don't think so,\" he said. \"I can't believe Jack'd bother with something like that. I think he's after something bigger. But I'm damned if I know what.\"\n\n\"Another, Lieutenant?\" asked the bartender.\n\n\"No, thanks, Charlie. Well, whatever it is I suppose we'll see it on the front page of his newspaper one of these days. Wonder how the beefsteaks are coming?\"\n\n\"I think they're about ready, Lieutenant,\" said Charlie.\n\n* * * *\n\nThey ate thick, juice-streaming slices of steak laid on juice-pink bread, and afterward deep-dish apple pie and coffee, and talked shop and kidded Jablonski about his bar and grill. Jablonski sat at the head of the table, wonderfully crisp in a starched white shirt and fresh haircut and shave, flushed with whisky and happiness at being the center of importance. Paul Bauer sat at the other end of the table, and when they had pushed the coffee cups aside and lighted tobacco he said a few words about how swell it had been working with Ed Jablonski and what a good cop he was and how he would be missed. Then he gave Jablonski a big package containing a bar apron as a joke, and a smaller package containing a toilet kit including a gold-plated razor; and everyone clapped and Ed Furtig yelled, \"Speech!\"\n\nNo one expected Jablonski to make a speech. But after accepting with unusual solemnity the toilet kit that was passed down the table to him he rose and stood looking at it for a moment. Then he said, \"Well, fellas, all I can say is...\" and stopped, and for the first time really, they understood that Jablonski was leaving and how it must feel to be going away from something you've lived with for twenty-eight years, and each of them got a quick, clairvoyant glimpse of what was coming to him some day.\n\nSuddenly Jablonski sat down, his eyes watery, his speech unsaid, but everyone at the table applauded, with a curious sense that he had said it anyway.\n\nThose who were not working that night had a farewell drink with him at the bar and Ryan, going out, punched his arm as he passed. But Jablonski was busy trying to buy that round of drinks and he did not feel it.\n\nIt was only ten-thirty. Ryan stopped in a lunch counter restaurant where he knew there was a phone booth, ordered coffee and dialed the number of the club where Gee Gee worked. He had called it only twice before but he did not have to look up the number. The phone was answered with a blast of music and he said to please tell Miss Hawes that Mr. Ryan was calling.\n\nThere was an interval during which the music stopped, and then a low, warm voice said, \"Hello, Mr. Ryan,\" and he heard the rush of her breath into the phone, soft and intimate as she held her mouth close to it, and he remembered the scent of her hair and how it had been dancing with her. For a second he did not have any words.\n\nHe said, \"My name's Neill,\" and grinned awkwardly in the phone booth's dimness. He heard her chuckle. \"I start a forty-eight tomorrow morning,\" he said, \"and I was thinking maybe we could work out a date.\"\n\n\"I think that would be lovely. What's a forty-eight?\"\n\n\"We work six tours\u2014six days, then get forty-eight hours off. It's really fifty-six hours, but anyway. I thought if you\u2014if the club was closed tomorrow, being Sunday, or on Monday, maybe we could get together in the evening or something.\"\n\n\"Monday evenings I'm free.\"\n\n\"Dinner maybe?\"\n\n\"I'd love it.\"\n\nRyan left the phone booth in a happy glow. He threw a quarter on the counter but left the coffee untouched. He did not want to spoil the way he felt with that coffee.\n\nLambert had not yet reported when Ryan got to the squad room; so he belatedly crushed his hat into its conventional version, lit a cigarette, propped his feet up on a desk and thought about Gee Gee and where they would go for dinner. Some theatrical sort of place probably. Lindy's or Dinty Moore's.\n\nAfter a time, mindful of what Bauer had said, he got out the file on Derby that Jablonski had left and began leafing through its slender contents. Notebook pages of names and dates, a couple of typed sheets, a photostat of Derby's police record, a copy of his fingerprints... Ryan stared at them for a long time out of sheer lassitude and a feeling of well-being. Then Lambert arrived.\n\nThey had a busy night. Saturday nights usually were busy, but in addition there were two fires, within blocks and minutes of each other, that the fire marshal concluded had been set by an arsonist. Ryan and Lambert spent most of the night on foot patrol, and it was not until after six a.m. that they could drive down to the Fifteenth Squad on East Thirty-fifth Street and pick up the envelope Bauer had asked for. They stopped for breakfast on the way back, each sitting in the car and listening to the radio while the other ate. Ryan idly examined the contents of the envelope over his sandwich. It consisted of big enlargements of some fingerprints and blurred carbon copy of a theft report.\n\nRyan read it. Several weeks before a man had walked into a small neighborhood drugstore just off First Avenue, handed the proprietor a bottle and asked that it be filled with castor oil. That took the proprietor to the prescription counter at the rear and while he was out of sight the customer dodged around a showcase to the cash register, grabbed its contents and ran out. The proprietor, who was seventy years old, had not noticed the thief especially, except that he wore a dark suit and had no overcoat, although the day was brisk. But the bottle had been oily when he brought it in and so the fingerprint men had found prints of a right thumb and forefinger perfectly delineated on the rim of the cash register drawer.\n\nThings like that were rare, despite what the public, educated by seeing and reading a hundred fictional criminal investigations, was led to believe. Still, they happened, and this was one. Ryan studied the fingerprint photographs while he sipped coffee. He saw something odd in the thumb print\u2014a little Y-shaped scar on the print's right perimeter. That was odd because he had recently seen the same thing somewhere else.\n\nAnd he knew where. There had been a scar like that in Derby's prints, which he had studied earlier waiting for Lambert. A thumb? What of it? It was coincidence, of course. But it was odd. Could Derby have done this one too? He looked again at the date on the smudged report.\n\nAnd when he did, Ryan felt a crushing weight sag his shoulders and a pervading chill enter his heart. The drugstore robbery had occurred on November 7, at 3:25 p.m., twenty blocks south of the apartment where at the very same time Thelma Connors was being murdered, presumably by Harry Derby.\n\nBut the thumb print of the drugstore thief remarkably resembled Derby's.\n\nIt must be coincidence. When he got back to the precinct he could compare them.\n\nBut the overpowering sureness, the growing conviction that is born of subconscious observation, began invading his mind. Ryan had a precognitive sense of what he would find when he got back to the precinct. The prints were going to be the same.\n\nHe knew it because of the way everything\u2014all the incongruous little things that had struck him as odd from the beginning\u2014fell into place with dreadful logic.\n\nDerby had been engaged in a comparatively harmless bit of sneak thievery at the very time Mrs. Connors was being mortally assaulted. That is why he had not told the truth about where he really was at the time. Derby was a three-time loser whom even the drugstore theft would send up for life\u2014but insofar as the murder was concerned he was innocent. He had the strongest possible sort of alibi, proved unwittingly by a police investigation, and now safeguarded unknowingly in the records of the department.\n\nBut no one knew about it except Ryan. No one could. For no one would ever think of comparing an odd thumb and forefinger print from a cheap smash-and-grab with the known prints of a tough, big time stevedore-hoodlum like Harry Derby. Not even in the Known Criminal File.\n\n* * * *\n\nRyan was very quiet during the rest of the tour, but Lambert did not comment on it for he himself was a self-contained man. After they checked in at the precinct Ryan took both prints into the lavatory where he could be undisturbed and compared them, using a flashlight and a magnifying glass in the toilet's dimness.\n\nAfter that there was no doubt. Derby was innocent. He had an unassailable alibi.\n\nRyan put Derby's records back in his own desk drawer and the new print from the Fifteenth Squad on Bauer's desk.\n\nThen he went home and tried to get to sleep.\nCHAPTER 11\n\nA Million Things Can Happen\n\nSunday afternoon was desolate. He wanted yearningly to talk to someone; the subject was forever at his tongue's tip, and instead, eating a late brunch, he had to listen to his mother's chatter about the church decorations and Eleanor's talk of the drama appreciation class she was taking one evening a week at Columbia. Eleanor had gone to Katherine Gibbs' and become a secretary after graduating from high school instead of going to college. Now she was working with dilettante interest toward an A.B. degree. She asked Ryan to pick up some volumes of Ibsen she needed the next time he was near the Columbia University library. Twice that day he had to ask her to repeat the name of the dramatist although it was not unfamiliar to him; the second time he wrote it down.\n\n* * * *\n\nAround four he went for a walk, stopping at the first drugstore to call Jablonski. Someone he took to be the bartender at the Green Lantern said Mr. Jablonski had gone to Jersey for the day with his wife to visit her folks. He'd be around tomorrow, the bartender said. Ryan left his name and said he was coming out tomorrow.\n\nHe walked thoughtfully toward Fifth Avenue. It was a sunny late afternoon; there were still many leisurely strollers. In Central Park he bought peanuts and fed ragged squirrels and watched a few hardy kids sail boats in the boat basin. The last dull orange gleams of the sun were dying in a striped violet sky and the lights had come on, circling the park, when he turned homeward. The onset of night's gloom depressed him, renewing the anxiety he had partly shaken off.\n\nTomorrow would be better. Jablonski would have advice to offer, and ideas\u2014just talking to him about it would help. And then tomorrow night, a bare twenty-four hours from now, after they had settled on something, there'd be Gee Gee to see.\n\nRyan began walking faster. It wasn't just the cold. He wanted to get everything over with as soon as possible\u2014supper and sleep and the trip to New Rochelle.\n\n* * * *\n\nIt originally had been a store building, and the first floor front still consisted chiefly of a kind of bayed-out show window that contained a big glass vessel filled with green liquid, the kind that old-fashioned druggists used to display. But across the window glass Green Lantern Inn was printed in gold letters and in one corner there appeared \"Walter Nowak.\" Under it in bright, newly applied gold leaf was \"Edmund A. Jablonski.\" That was Jablonski all right\u2014he wouldn't waste any time getting his name up!\n\nBut when Ryan pushed the door open and walked in there was no Jablonski. A portly bartender, bland-faced and immaculate in a fresh apron, was behind the bar, and from the end of the room an Oriental face under a chef's tall cap looked expectantly up over a gleaming grill. There were tables, neat with green checkered tablecloths and shiny bottles of catsup.\n\n\"Mr. Jablonski around?\"\n\n\"He's not in right now. He should be back about six. Any message?\"\n\n\"My name's Ryan. I'd phoned him that\u2014\"\n\n\"Oh sure, Mr. Ryan. He said to tell you to come on out to the house.\"\n\n\"The house?\"\n\n\"They're renting a house over on Devens Place and I guess the missus has got him doing some work.\" The bartender, distinguished-looking, smiled a bland, man-to-man smile. \"You know how it is.\"\n\n\"Oh, sure. How do I get out there?\"\n\n\"It isn't far. You take your first right up at the corner...\"\n\nHe found Jablonski in the unfurnished living room of an old-fashioned double house. He was on his knees in a corner touching up the varnish with a brush and a can of lacquer. From some unseen kitchen came clinking sounds and the rush of water as Mrs. Jablonski washed the glass globes from the lighting fixtures. Jablonski got up painfully but gratefully. \"Jeez, I'm glad you're here. My knees couldn't take much more. How you been?\"\n\n\"Good. And you?\"\n\n\"Great. Except we're moving in Friday and Sarah figures I can completely redecorate the joint before then. Say, I hear the brass still hasn't gotten over our collaring Derby. You're hotter'n a two dollar pistol, Neill. That's great.\" Jablonski sounded a little envious.\n\n\"You're pretty hot yourself with that bar and grill. That's a nice layout, Jabby. I'll bet you do a heck of a business.\"\n\n\"We're doin' all right. We'll drop by and have a drink there after a bit. Mind if I go on working while you tell me what you wanted to talk about?\"\n\n\"Go right ahead.\"\n\nJablonski picked up the brush, sank to his knees, groaned and went on dabbing.\n\n\"It's about Derby.\"\n\n\"Oh?\"\n\n\"Yah. To give it to you straight...in a few words...he's innocent.\" Only the brush's steady pat-pat against the floor made any sound. Its rhythm did not vary.\n\n\"Yeah?\" Jablonski sounded surprised but nothing else. He went on brushing.\n\n\"You get what I said. The guy's innocent.\"\n\nJablonski said, \"Why do you say that?\"\n\n\"Because I know\u2014I know what his alibi is.\"\n\n\"Neill, you're nuts. You must be. Why, he even admitted killing her himself, after he burned the bill. You know that.\"\n\n\"He was boasting,\" said Ryan patiently. \"He was just trying to make us feel worse. You ought to know he'd do that.\"\n\n\"Oh, for God's sake\u2014\"\n\n\"You can't talk this down, Jabby. I know what I'm saying.\"\n\nThe brush's lacquer-slaps had finally stopped. \"I don't believe it.\" Then, \"How sure are you?\"\n\n\"Dead sure. He didn't kill the old lady. He couldn't have. He was doing a smash-and-run on a drugstore at the time, and he left a clear print there. It's only by a crazy coincidence that I happened to see it and recognize what it was. No one in the department knows, or can know. There's nothing to classify.\"\n\n\"Uh huh.\" Jablonski, supported by knees and one hand, was bent away from Ryan. He resumed stroking.\n\nRyan said, \"Well, what do you think?\"\n\nJablonski finished touching up the spot he was working on with great care. Then he turned over, to sit and look up at Ryan inquisitively. \"What am I supposed to think?\"\n\n\"Well, you know how you\u2014how we planted that dust on his jacket.\"\n\n\"I know Derby's a hood and a killer.\"\n\n\"Maybe. But now you also know he didn't kill the old lady.\"\n\n\"No, I don't. All I know, and all you know, is that a fingerprint looked just like another fingerprint\u2014to you. You're no fingerprint man. And someone says the time was the same. Maybe yes. Maybe no. I haven't seen any proof.\"\n\n\"As far as the time is concerned, I checked that myself this morning. A radio car got there less than two minutes after the drugstore was hit. And there's a million neighbors who heard the old man yell when he ran out into the street. That job was done just when the report said it was done. As for the prints, okay, I'm no fingerprint man. But I've looked at a few and so have you. Want to argue with these?\" The envelope he tossed struck Jablonski's shiny-serge knee.\n\nBut Jablonski did not pick it up. He was looking up at Ryan. Ryan had noticed before that his face was fresher looking and less lined. But now it became dark and vigorous with thought.\n\n\"The department,\" Ryan explained patiently; he did not like this or Jablonski's expression\u2014\"the department actually has Derby's alibi on record and doesn't know it. Nobody knows about it except me\u2014and now you.\"\n\nJablonski said coldly, \"Okay. So what?\"\n\nRyan knew what that meant. He had known it by Jablonski's face before Jablonski said it. But he had to go on, to carry the argument through until it arrived at whatever deadlock it would reach.\n\n\"So Derby's going to the chair,\" he said. \"And we're sending him there. And he's innocent.\"\n\nJablonski, throwing back his head, croaked brief laughter. \"For Christ's sake!\"\n\nRyan felt like an undressed child. It made him mad.\n\n\"Well, damn it, even a lousy hood is entitled to a square shake!\"\n\nJablonski put a finger to his lips. \"Not so loud. Sarah.\"\n\n\"To hell with Sarah!\"\n\n\"Look, Neill. You're young and you're inexperienced in the department, even though you've done great recently. And I don't want to see you hurt yourself now.\"\n\n\"Hurt who?\" Ryan had lived with it too long to be patient any longer. The time had come for anger and indignation and he unleashed them, unconsciously seeking to force the decision that would end it one way or another.\n\nBut Jablonski had spent his life handling anger and indignation. \"Neill, now listen a minute\u2014just a minute. Harry Derby's a hoodlum. He's crooked; it's what's inside him. You ought to know what I mean\u2014you were on the street long enough. He ain't like normal people, that believe in square shakes and giving other guys breaks and so on. He's got a record that\u2014well, he's done almost everything to anyone you can think of. And you know as well as I do that for every time he was arrested doing it, he had done the same thing twenty times more when he wasn't caught. He's an animal, Neill\u2014do you want to turn that loose?\"\n\n\"No, for God's sake.\" Ryan responded to emotion with emotion. \"For heisting the drugstore he should go up for life.\"\n\n\"He had that choice, Neill. He has it now. As you say, he has a perfect, tailor-made alibi. All he's gotta do any time he wants is to say that he was in the drugstore then. Right away he is out of the chair. Well, he made the choice. He's sayin' nothing about the drugstore. And why?\"\n\n\"'Cause he figures to beat the murder rap.\"\n\n\"Sure. He ain't crazy.\"\n\n\"But\u2014\"\n\n\"But nothing! Look, Neill. What do you really think you're going to do about it? Let down all our witnesses? Ruin the case against a guy that should have gone up years ago? Give the newspapers another shot at making the department look silly? And what will something like this do to your career?\"\n\n\"I'm not thinking about my career,\" snapped Ryan. \"And God knows I'm no friend of Derby's or the newspapers. But I can't stand around and see a guy burn, no matter how much of a heel he is, when he doesn't have it coming to him.\"\n\n\"Well, what do you think it'll do to my career?\" Jablonski asked quietly. \"I'm on pension, Neill, and I need every nickel right now to swing that grill. That's why we're takin' this old house. But what's going to happen if the department finds out about our case against Derby? What happens to my pension, and my reputation and business? I'm in this as much as you. You ought to remember that. And I'm not as young as you.\"\n\nRyan lit a cigarette. He hadn't thought of it quite that way. But he said, \"Sure you're not as young. That's why I thought you might have the answer. You always said you'd been around a long time.\"\n\n\"Gimme one of them things\u2014my cigars are in my coat. Sure I got the answer. If you'll take it.\"\n\n\"So? What do we do?\"\n\n\"Nothing.\"\n\n\"But I tell you\u2014\"\n\n\"Nothing,\" Jablonski repeated firmly, and put the cigarette to his lips. \"Until after the trial. Neill, you're forgetting he isn't convicted yet. And he's going to be defended by one of the best criminal lawyers in the business. If he gets off\u2014and he's got a fair chance of that; they always have, what with hung juries and so on\u2014if he gets off, we can forget the whole thing, and nobody's hurt.\"\n\n\"Yeah, but...\"\n\n\"Look, kid. You got excited about this before\u2014remember? When that wild-haired brother of his came to see you? Then you found out for yourself what I told you all along\u2014that he was lying. Take my advice again, won't you? For my sake as well as your own? Wait and see, Neill. You won't gain anything by rushing this thing. And you may lose\u2014we both can lose\u2014a great deal.\"\n\nJablonski's tone was low and confidential, and it was convincing because Ryan half-wanted to believe it. He had outlined a solution that Ryan had hardly given a thought to.\n\n\"Well...yeah.\" Ryan knocked cigarette ashes into his trouser cuff. \"But supposing he's convicted?\"\n\n\"We cross that bridge when we come to it. Jeez, Neill, a million things can happen. You know that.\" Jablonski got to his feet. \"There's nothing you can do now that you can't do later. Hell with this floor. I want you to meet my wife. Let's drive down to the joint and have a beer.\"\n\nThe three of them drove to the Green Lantern in the bartender's car, which Jabby had borrowed. Sarah Jablonski, tall and black-haired, was a surprisingly youthful-looking woman with fierce bright eyes and quick good humor that made Ryan wonder why Jabby had ever bothered with Inez. He also wondered how much Sarah knew about Inez. Probably everything, he decided.\n\nOnce back at the restaurant he protested that he had to get back to Manhattan right away, insisted on walking to the station, and caught an express soon after.\n\nHe sat in the smoker and smoked all the way back to Grand Central. He felt relieved of most of the weight of responsibility that he had carried out to New Rochelle; the decision and the need to act had been postponed for a time at least by Jablonski's eloquence.\n\nBut not merely by his eloquence, Ryan told himself. After all, he did owe Jablonski something\u2014certainly he owed him the chance to let things straighten themselves out, without his own intervention. Jabby's twenty-eight years of plodding still deserved consideration; it was typical of himself to react hotheadedly to any injustice and try to right it. That was what he had done now. But actually that was often the way greater injustices were compounded.\n\nIt was a comforting thought and Ryan extinguished the last flickering anxiety by turning his thoughts deliberately to Gee Gee and the next six hours. What would she be like this time, alone with him? And what would he be like?\n\nRyan had always found girls fairly easy to get along with, except for those he fell in love with. Of those in his twenty-eight years of life there had not been too many, because he did not easily form attachments, although once they were formed they lasted. But Ryan had been almost old enough to vote before he came to realize that while there were some girls you wanted very much to sleep with and others that you fell respectfully in love with, the two were not incompatible but were indeed closely and rightfully associated. Even so, when he met a girl whom he really liked, tensions developed and he felt himself grow shy and awkward. When it was a girl he didn't deeply like, then things went very easily and surely. He felt he was going to be awkward with Gee Gee.\n\nFor not since he was sixteen had he met one like this, a girl whose name even was a honeyed fragrance in his mind. He must be crazy, he thought. He'd only met her once\u2014but she had sounded glad he called, hadn't she? And even dames like that don't want to kick around in nightclubs forever. If they have any sense at all, they want to settle down like everyone else, and have a little apartment and kids after a while...and she had sense all right. That was what he liked about her. Pretty girls were a dime a dozen in this town. But the kind that were also level-headed and could see farther than a mink coat and that somehow you knew were square shooters...\n\nRyan gazed out the window, looking into the tenements mounting above the train as it rushed down to the level of Park Avenue and the tunnel entrance. But those tenements and the dark, dangerously underprivileged hearts that lived there in the Twenty-fifth Precinct, one of the city's toughest, brought back Derby.\n\nAnd that brought back something that had occurred to him before but only tangentially, without real meaning. Now, as the train burst into the tunnel with a gasping roar, he thought of what he should have thought of in arguing with Jablonski. If Derby had committed the drugstore theft, then certainly he deserved to go up for life. But Derby was at least in custody and therefore still at the disposal of the law.\n\nBut the real Connors' murderer was at large and was not even being sought. He had gotten away with it.\n\nEverything in Ryan's policeman's soul shuddered at that. But what was he, the only working cop who knew this, doing about it?\nCHAPTER 12\n\nThe Puerto Rican\n\nRyan chose critically from the shirts his mother had ironed that day, and knotted his tie with precision. It was good to concentrate on unimportant things, and to have a reason for concentrating on them. Putting on his suit he tried not to disturb the creases pressed into it. Then, having arranged a breast-pocket handkerchief, he went into Eleanor's room (after first making sure she and his mother were in the kitchen) to survey the total effect in her long vanity mirror. Ryan decided he looked as good as he could and reminded himself that it was not the world's record.\n\nHe left the house giving his hat the fancy crush and resolving to put Derby out of his mind.\n\nWalking to the I.R.T. station took him past the church and the parish priest's house. Only a single dull light burning behind the Venetian blinds hinted at life inside. Ryan striding the frosty sidewalk passed within a yard of that uninviting window. An adviser on Derby, perhaps? A sanctuary to keep in mind, and rely on if need be? Ryan's mouth made a tight smile. No. Not likely.\n\nHe remembered the time that he and his mother had come here and were closeted with the priest behind that very window. Ryan hadn't been going to church and that troubled his mother. He had agreed to the interview and he still remembered the platitudes and tired admonitions, the meaningless, emotionless words that should have had meaning and emotion. Now he had not gone to church in a long time, and that had become accepted at home. But he and his problem wouldn't be welcome behind that reclusive window.\n\nAt the subway entrance he bought a tabloid and lost himself in the latest doings of caf\u00e9 society.\n\nThe name Hawes did not appear on any of the mailboxes in the hallway of Gee Gee's building. But Haas did. Ryan wisely pushed that button and was gruffly admitted to a second-floor front apartment by an elderly man who had removed the detachable collar from his shirt and the shoes from woolen-socked feet. He said Georgine should be ready in a minute, went back to his chair and continued to stare at a large television set which was portraying the adventures of Sid Caesar.\n\nIgnored, Ryan held his hat and pondered the ways of a world that changes Georgine Haas into Gee Gee Hawes. He did not particularly like Georgine as a name, just as he did not like the heavy scent of cooking in here, or the large oval family portraits that had been hung in Old World style high up near the ceiling, or this grizzled walrus who presumably was her father and didn't even have the courtesy\u2014\n\nThen she was standing there, in the same coat and carrying with her the same faint scent, a perfume of aromatic dryness, and pressing new gloves on over her fingers, smiling \"Sorry to keep you waiting,\" her eyes as deep and large and somehow inviting as he had remembered they were. He got up and dropped his hat and said, \"Hi.\"\n\nShe gestured to her father, \"Guess you've met\u2014'bye, Pops,\" and went to the door. Ryan followed, muttering, \"Good night.\" Only Sid Caesar answered him.\n\nIn the hall the cooking smell was strong. Gee Gee wrinkled her nose. \"I'd never have let you pick me up here if I didn't think you were a nice, sane guy,\" she said.\n\n\"What do you mean?\"\n\nA shinily gloved forefinger made a disdainful arc around the stairwell. \"Usually I meet dates somewhere else. There are a lot of people I don't want to have know that my old man's a night watchman, and that I support the family\u2014as much as I can.\"\n\n\"I don't think that's anything to bother about one way or the other.\"\n\n\"Neither do I. But some people do, especially in show business. You're supposed to live at the Waldorf and eat at Twenty One.\"\n\nThey had reached the street. \"Speaking of which,\" he said.\n\n\"Speaking of which.\" She laughed and took his arm with the ease of one accustomed to taking arms. \"I'm starved. You'll learn not to ask me out to dinner, Neill.\"\n\nRyan laughed too, and was emboldened. \"That'll take a long time. Where would you like to go? Pick one you really like.\" His wallet contained almost eighty dollars; he had counted it just before dressing.\n\n\"How about Chinese food?\"\n\nRyan did not especially like Chinese food. He said, \"Well, if that's what you'd like,\" without enthusiasm. It did not occur to him that she could have said that because she knew Chinese restaurants were inexpensive.\n\n\"I really don't care. Maybe chow mein isn't the thing at that. You name one.\"\n\nThey had reached the corner. Ryan named several, all of them expensive. That was how he felt.\n\nGee Gee looked her surprise. \"You must have got a bonus for catching that Derby,\" she said lightly.\n\nHe'd got a bonus all right.\n\nShe saw in his face that it had been the wrong thing to say. \"How do you like the Blue Ribbon?\" she asked quickly. \"I think their food is awfully good.\"\n\nThis time Ryan grinned, shrugged off his irritation and waved an approaching cab to the curb.\n\nThey ate oysters and roast duck and got to know each other better. Gee Gee paid an escort the compliment of being neither coy nor arch with him, nor yet determinedly the professional dancer. Instead she was forthright and relaxingly unassuming, as though she felt Ryan was the same sort of person. The effect on Ryan was overwhelming and happily narcotic.\n\nHe insisted she select something from the huge, tempting pastry tray over her protests about her figure and her job, and he told her she had nothing to worry about with such obvious sincerity that Gee Gee, who was used to compliments, was stirred. They got on well, and when she remarked idly that after such a meal there was nothing like an espresso Ryan instantly suggested taking a taxi down to one of the little Italian coffee shops in the Village. It was an extravagant gesture and Gee Gee, embarrassed because she had never intended to suggest that, protested again. But Ryan looked so disappointed that she gave in.\n\nIn the cab they sat quite close together and Gee Gee asked questions about his job, and Ryan answered them, meanwhile dismissing a temptation to try putting his arm even ever so lightly around her shoulders.\n\n\"What's it really like?\" she asked. \"Being a detective, I mean. I know it's not like in the movies and so on. You don't get cases like that Derby one often, do you?\"\n\n\"Not often,\" said Ryan, and knew he had let irony edge his voice. \"Most of the time it's routine, little things you run down day after day. A lot of it is taking over after a uniform man has made an arrest and developing information, motives and such to use in court. You don't run around with a magnifying glass making deductions.\" It made him a little bitter.\n\nBut she was genuinely interested. \"Then what do you use? Handcuffs and a gun?\"\n\n\"Mostly your feet,\" Ryan grinned. \"And sometimes your head. Handcuffs, sure, when you're making a collar\u2014arresting someone. Never your gun if you can avoid it. That's the last resort.\"\n\nShe said, \"Uh huh,\" understandingly in the cab's warm darkness but he knew she did not understand what that meant at all. How many civilians did? She went on, \"I guess it's like show business. Everybody who's not in it thinks it's glamorous and exciting every night. But it's routine, too. Except for the opening nights\u2014I guess they are like the nights when you arrest someone like Derby.\"\n\n\"Yes.\" He could never get completely away from it. So why try? \"Derby's something special, all right.\"\n\n\"Derby is? Was, I'd say.\" She sensed this was dangerous ground from his voice, but she had the feminine impudence born of knowing she was liked. Anyway, what was eating him? \"Derby's all taken care of\u2014remember?\" she said.\n\nBut Ryan could not respond to her lightness.\n\nShe had not expected him to keep looking like that. She said contritely, \"I just remembered Ed emphasizing that night how you two had taken care of Derby.\"\n\nRyan recognized the need for getting away from this. \"Sure,\" he said. \"That's right. Derby's all washed up. And speaking of that let me tell you something that happened one night down on South Street.\"\n\nBut telling the oft-told anecdote, he could not help wondering if she was looking at him a little oddly, and just how friendly she was with Jack Sandalwood, and how often she saw him. And whether she ever had repeated what Jablonski had said that night.\n\n* * * *\n\nIn the coffee shop they sipped black, deliciously bitter coffee drawn from a monstrous hissing machine and sprayed with oil from a sliver of lemon zest. Afterward they went out into the cold black night and walked through Washington Square, past a few hardy oldsters still playing chess by lantern light, and other, darker benches with occupants silent or murmurous. They did not say much, but she took his arm and they walked through the shadowy park content with each other's company and with the awareness that the other was content too.\n\nThey walked a long time, past show windows filled with impossible lamps, best-sellers or copies of last month's Paris originals. Once she murmured, \"You know, Neill, you're an awfully restful guy to spend an evening with,\" and looking quickly, Ryan saw her eyes were closed even while walking and yet she was smiling.\n\nJeez, thought Ryan. He wasn't thinking of Derby now.\n\nThey walked a long time like that.\n\n* * * *\n\nHe knew the Village pretty well, but after an hour he found they were on one of those little side streets that angled-off another street, and he knew he had taken a wrong turn. It was dark and empty and Gee Gee said, \"Golly, where are we?\" Ryan said he had pulled a boner, but that at the next corner they should be able to see where...\n\nShe pressed a little closer to him in this dark uncertainty, and Ryan put his arm around her waist\u2014how slender, even encased in that thick coat!\u2014and doing that was not surprising to him at all, or remarkable, but natural, and he pulled her closer to him and she raised her face. He started to find her lips...\n\nHe saw the movement only as a flash of furtive white, a something on the left that instinctively pulled him from her, and then the man was there, crouched, menacing, barefoot, although not registering immediately in Ryan's consciousness despite the white shirt and snarling brown face, his left hand holding high a clean knife.\n\nIt was impossible.\n\nBut the little man, legs braced wide, knife high, screamed, \"Hello, whore man!\" and rushed at them, head down, knife reversed for an upward belly thrust.\n\nRyan pushed her behind him as the man leaped forward, head low like a tackier in football, knife wavering up. Ryan kicked hurriedly and his overcoat spoiled his aim. His toe grazed the other's thigh, and turned him but did not stop him.\n\nRyan grabbed for the knife as he came in, missed, grabbed wildly again, caught the man's wrist and twisted it. A knee pumped up and caught him, but his right arm lay across the man's upturned face and he pushed it down, bending the man's head back. Then he lost his balance and felt himself falling. He scraped desperately for footing and went over the curb, stumbled into the street and went down on a knee and saw something leaping. Gee Gee screamed; he did not hear her.\n\nHe rolled and made the knife-man almost miss with his leap. Ryan found his pistol even with the man partly on top of him. He pulled it out and shoved it into the brown, intent face and the knife-man incredibly opened a snarling mouth and bit down on the pistol's short barrel, yanking with his teeth. He's crazy, Ryan realized. You don't use your gun on a mental case.\n\nLight came from somewhere although Ryan would remember sensing that only later. Gee Gee screamed again. Ryan wondered if he would ultimately have to shoot, and if so how he might make sure of hitting the right shoulder.\n\nHe saw the knife flash suddenly and he gathered strength from somewhere (a taxi horn honked) to push up with arm and body and knees. The little man toppled sideways and for the first time Ryan managed a punch. It was only with his left hand, and it hit the man's face, still holding the gun in his mouth, but did not dislodge him. The knife started flashing again and he had to let go the gun to grab that arm. He didn't stop it entirely because he felt something graze his chin even though he was holding the man's arms while he wrestled like a maddened monkey, sitting on Ryan, holding the gun in his crazed mouth.\n\nA shape stood over them. \"Take the gun,\" Ryan gasped and she bent over and wrested the pistol from the man's jaws. Ryan arched his back and moved him a little, then in desperation let go one writhing wrist to swing at the vague face. The punch landed and the man went sideways and Ryan rolled over and kicked him and then, on his knees, punched him again. Someone said, \"Hey, for God's sake!\" and there was more light and the knife's blade glittered on the black asphalt. Ryan grabbed for it.\n\nSo did the little brown-faced man and she entreated, \"Here, here,\" and handed him his pistol. He put it on the man who still lunged forward on his knees, fingers clawing for the knife, and Ryan got up and kicked again and this time he caught the knife-man in the stomach which was what he had intended the first time and the man flew back.\n\nA voice said, \"Listen, Buster, that ain't no way to fight.\"\n\nFor the first time he looked beyond the range of his assailant. Two taxicabs had stopped in the street, their headlights illuminating the scene. A big, fat driver whose belly had burst his pants was looking outrage at Ryan.\n\nRyan pulled out his badge. \"Under arrest,\" he said with thick meaninglessness and went to the brown-faced man. He searched the man and found he was naked under his pants and bloodied shirt.\n\n\"You,\" said Ryan to the cab driver. \"There's a call box up on the corner. Tell them to send a radio car here right away.\"\n\nGee Gee looked at him, her face white and her eyes haunted. \"Jesus,\" she said. \"Your chin.\"\n\nIt was bleeding.\n\nOnly after the first radio car got there and dozens of neighbors had come down into the street did someone go up to the knife-man's apartment in the house opposite. They discovered his wife and two babies were dead, their throats slashed as they slept. He was a Puerto Rican busboy in an uptown hotel, the neighbors said, who had lost his job three days before.\n\nSeveral of them had seen the fight, or the latter part of it, from their windows, and had seen Gee Gee wrest the gun from the madman's jaws. When the reporters came they were eager to talk. When Ryan and Gee Gee, after his scratched chin had been treated in a nearby admitting room, appeared at the local precinct station, they were photographed together. Finally he took her home, apologizing for the melodrama, embarrassed by the evening's ending and still seeing the knife flashing down. He didn't try to kiss her.\n\nBut next day New York read the news, straight from the dark, greasy pavement of Barrow Street:\n\nHERO COP NABS BABY SLASHER\n\nAnd under its headline one paper added:\n\nDANCER HELPS MAKE ARREST\nCHAPTER 13\n\nThe Empty Passage\n\nThe capture of Derby had made Ryan a one-day hero, but when he returned to work now it was as a celebrity. To the newspapers the combination of show girl, young cop and maniacal killer was a perfect one, and they made the most of it. At a time when the department was under general opprobrium Ryan seemed to be proving singlehandedly that a policeman could be as capable, alert and humane as the most indignant Constant Reader could wish, and the very newspapers that had devoted lead editorials to the rising crime rate now spent flattering front-page space on the work of an unknown rookie. One editorial compared him to the almost legendary Johnny Broderick, and Winchell's Girl Friday column asked, \"Why don't they call rookie detective Ryan Rockie instead?\"\n\nRyan felt the difference among his associates. Technically, he was still a beginner, but only technically. Now in some subtle yet palpable way he had acquired importance and weight. Even the older men in the station house treated him with the grudging respect of envied prestige. That was pleasant, but to Ryan it sometimes seemed his progress had been too rapid to be healthy, a suspicion heightened by the haunting awareness of what would happen if the truth about Derby got out.\n\nAs the opening of Derby's trial drew near, Ryan found himself waking up nights in unrecallable fear, and his daytime thoughts turned often to his father.\n\nWhat would he have thought?\n\n* * * *\n\nLate one wintry afternoon Ryan found himself in Carmine Street.\n\nHe did not try to analyze what had led him there after a routine day's work; he simply knew he wanted to be there.\n\nBut after he turned into the thoroughfare his mind flooded with the remembered scent of carnations and roses that had filled their home then. And with the huge black case in the living room, his father motionless in it, and the ride to the church, his mother in unfamiliar black; a neighbor had kept Eleanor. And all the policemen who had been there that day in uniform.\n\n\"Do you remember him, Neill?\" his mother sometimes asked. \"Really remember him, I mean?\"\n\nA man so enormously tall that the silver shield on the heavy blue uniform was always out of your reach, even on tiptoe. A good-humored man who liked to play and swing you around and carried a rosary in his hip pocket. No, he couldn't really see him. The living memory of his father had been distorted by the photographs kept in honored places in the living room and on his mother's bureau. Yet he did remember him, and vividly. Not as a physical being but as a kindly benevolence, a refuge of understanding and admiration, who had been something to him that no one else could ever be.\n\n\"Of course,\" he would say. \"Of course I remember him.\"\n\nHe had come here before when he was sixteen. Rummaging through his mother's bureau for a handkerchief one day, he had come upon the envelope of old, crisp news clippings, and had read with sickening shock that pressed tight tears to his eyes what had happened to his father.\n\nSergeant O'Neill Ryan had been investigating collusion between a lieutenant in a downtown precinct and a gang of young burglars, and he had caused the arrest of three of them as well as suspension of the lieutenant. Some days later he had been lured\u2014how was never learned\u2014into a dark passage between buildings on Carmine Street at night, and two groups of hoodlums had started down it from either end.\n\nAs he lay dying on an operating table, drooling blood, one eye sightless, he had described or named five of the seven attackers. It had taken the department years to hunt them down. Two had died in the electric chair, two were in Dannemora and one was dead, shot in a holdup after serving his time. The other two were wanted to this day, and the revolver that Sergeant O'Neill Ryan had carried and fired three shots from that night was still preserved for ballistics tests in the police headquarters annex, in case one of the slugs ever turned up in a prisoner or in a corpse. The case would never be closed as long as there was a chance of one of them being alive.\n\nIt had been a warm, pale spring day when he was last here, a solemn boy of sixteen. Yet even with only the matter-of-fact words of the news clippings for clues, he had heard the foot scuffles of desperate defense, the thud of blows, the groans of the fallen man, the kicks and gouges and then the dying echo of coward feet. For months after he had daydreamed of meeting them, all of them, and beating them into bloody slime.\n\nNow the after-knowledge of twelve more mature years, four of them spent in police work, peopled that short passage with a more authentic cast. He knew how the snarls of vicious laughter had sounded to his father, and the eager curses; he felt the overwhelming strength of cowardice and sadism. That was the voice and posture of the other side, the gaudy, boastful, frightened army that every cop faced and fought not because a cop was innately heroic, but because he was instinctively the kind of person who likes order and fairness, and fights for them.\n\nNow he again found the old, thin wood door to the passage and swung it open. He stood a long time looking at where his father had been assaulted and at the new cement that paved the place. Formerly there had been bricks.\n\nHe could feel nothing, and the remembered twinges of old anguish were like the pinks of toy arrows. That disconcerted him. He did not know what he should feel or wanted to feel.\n\nBut certainly not this empty melancholy, this lack of meaning. What, he asked as he surveyed the scene, had his father's death accomplished\u2014for anyone, his father, the city, even the hoodlums who had caused it? He let the door swing to with a quiet hush. What had his father's death ever meant?\n\nAs he walked up Bleecker Street the storefronts and grocery trays were as unfamiliar as the slopes of another planet, cold, lifeless, dead, to which he had somehow been transported, a solitary inhabitant.\nCHAPTER 14\n\nFarragut\n\nA railing separated the spectators from the official part of the courtroom where the judge's bench, the clerks, press, bailiffs and counsel tables were, and just within the compound thus created a row of chairs ran from one side of the room to the other. Ryan sat in one of them, staring down at well-polished black shoes, aware of the blue-centered detective's shield he wore on his lapel as required, trying to forget the impatience that gnawed at him. The jury had been selected on Friday; yesterday had been devoted to motions and the opening statements of the assistant district attorney and defense counsel. None of these held anything unusual, although when the judge asked whether either side wanted the witnesses excluded from the courtroom, pending their turn on the stand, Assistant District Attorney Gil Tilbury said he didn't care and Farragut had said, \"No, your honor,\" rather emphatically. But no one paid particular attention to it at the time.\n\nBut that was why on Tuesday, the day on which Ryan had been told to be on hand, he and Jablonski could sit together in the courtroom, instead of lounging in the judge's chamber or a witness room. They were both to testify mainly about Derby's arrest and what he had said at the time and how they had found the murder gun. Ryan was desperately anxious to get it over with.\n\nBut Farragut was cross-examining with great leisureliness a medical witness who had testified about Mrs. Connors' injuries. Apparently he was trying to prove that the blow across her face could have been delivered by a woman as well as a man, and by either a right-or left-handed person. The courtroom was silent and bored as Farragut pursued his questions steadily, mechanically. He was dawdling, for a sound purpose perhaps, but still wasting time and Ryan hated him. He looked backward over his shoulder at the spectators.\n\nVacant stretches gaped among the long benches. Not much excitement was offered by the certain conviction of an habitual criminal. There was the usual assortment of housewives, some wearing glistening Christmas ornaments like corsages on cheap cloth coats, and shabby old men to whom the trial meant warmth and possibly entertainment, and a few teen-agers. These were The People on whose unknowing behalf the county in which they lived was moving with lethal intent against Harry Derby.\n\nIn the corridors outside The People lounged and talked, cursed and smoked, spat and hoped. But within this high, paneled, punctilious chamber that was the size of a gymnasium, their ordinary conduct wavered and died. Rules were observed and conventions honored in ignorant abstraction. A uniformed officer came forward occasionally to demand loudly, \"Quiet, please,\" or, \"Take off your hat!\" and the spectators withdrew into themselves and watched and waited without expression. Derby, in a plain blue suit and white shirt, sprawled leanly at the counsel table, frowned and played endlessly with a yellow pencil. Ryan, wishing it were over, tightened his jaws until little hollows appeared under his cheekbones. Jablonski came back from coffee, grinning, and sank beside Ryan with a sigh. Then Ryan's ranging, anxious gaze saw a newcomer settle himself at the little table that accommodated the reporters. He had a young face topped by short-cropped, prematurely gray hair. This time he wore a tie and brightly shined black loafers with tassels instead of bedroom slippers.\n\nSandalwood had arrived. Sandalwood was covering the trial.\n\nThe little hollows under his cheekbones made Ryan momentarily gaunt. Why was he here?\n\nThe scientist from the police laboratory who had performed the spectroscopic analysis of the dust on Derby's jacket was now on the stand, replying to Tilbury's direct examination with hands-in-lap composure. Farragut, frowning, lay in wait at the table. Derby balanced the yellow pencil on his finger. Sandalwood borrowed a few sheets of copy paper from another reporter, yawned and took out a ball-point pen. Ryan forced himself to think of Gee Gee and how she had looked last night.\n\nHe had dropped unannounced into the night club. Three girls' names were billed outside; hers was not one of them. He had stood at the bar, hoping to watch the show from there, but almost immediately someone he took to be the manager deferentially asked him to sit at a table and while he was refusing the cigarette girl came up and said Miss Hawes wanted to see him in the dressing room. There had been four girls there, one a dancer he had just watched finish her act. She was toweling sweat-caked powder from an almost bare body. The other three girls wore short smocks carelessly; one was Gee Gee, who put down a Mirror, opened to Winchell's column, rose, clutched her smock and said, \"Hi, Neill,\" with friendly cordiality. The smock showed Gee Gee's fine long legs; that, and the wise inquiring eyes of the other girl, black-edged with mascara, made Ryan self-conscious. He had seen his share of undressed girls but there was an impudent flaunting in this... He said something about going out between shows, and he was a little relieved when she said she could not leave the club and so gave him an excuse for an early departure.\n\nA ballistics man stepped down from the stand after testifying that his examination proved the bullet that had killed Mrs. Connors came from Derby's gun.\n\n\"Edmund Jablonski,\" a clerk bawled, and Ryan came back to reality. Jablonski got up, nudging him confidently.\n\nHe was on the stand only fifteen minutes. He told the story of how they had spotted Derby, tailed and arrested him, and identified the revolver and checked jacket. He testified that when they first seized him Derby had said something about \"killing the old dame\" or words like that, but had later turned sullen and finally attacked him. He gave copious credit to Ryan for coming to his rescue and said nothing about a hundred dollar bill. Then Tilbury said, \"Cross-examine,\" and looked at Farragut. But Farragut waved a thick hand carelessly.\n\n\"No questions,\" said Farragut, and Tilbury looked a little surprised.\n\nTo Ryan things looked brighter. Jablonski had told the main part of the story. His role obviously would be that of corroborating witness. When the clerk called \"Officer Ryan\" he stepped to the stand and placidly raised his right hand, confident he could forget about Sandalwood.\n\nWith careless ease Tilbury ran him through many of the same questions Jablonski had already answered and Ryan replied to them readily, keeping eyes and ears trained on the tall, elegant assistant district attorney. Tilbury concluded quickly. Then Farragut lumbered forward for cross-examination, and Ryan got his first close look at Derby's famous lawyer.\n\nAbsalom Farragut had a wide, graceless body which he draped in baggy, unassuming suits, high old-fashioned collars and string ties, especially designed according to courtroom legend to impress juries with his genial commonality. His face was a granite slab that could relax into a wide, homely smile or become a black pattern of outrage as Farragut chose, and the shock of hair above it was still more brown than gray at the age of sixty-two. Now he pulled back his coat to thrust a big thumb in the belt of his trousers and began addressing Ryan with the good nature of an old man dealing with a brash neophyte.\n\n\"You've been a detective less than a year, I believe, Mr. Ryan?\"\n\n'That is right.\"\n\n\"Less than six months perhaps?\"\n\n\"Yes, sir.\"\n\n\"Less even than three months?\"\n\n\"Yes, sir.\"\n\nFarragut smiled benignly for the jury. \"You are in fact a rookie\u2014eh, Mr. Ryan?\"\n\nRyan felt rather than heard the courtroom's amusement. He flushed. \"Yes, sir.\" He looked around. At the press table Sandalwood was bent over a doodle.\n\n\"So when you and your partner, Mr. Jablonski, arrested Harry Derby it was Jablonski who was in charge of the...ah...detective team do you call it?\"\n\nRyan knew that Farragut was well aware of what they called it. He said, \"That is correct.\"\n\n\"And so during that operation you naturally did everything that Jablonski told you to do?\"\n\nOut of the corner of his mind Ryan barely sensed Tilbury springing volubly to his feet. He mumbled, \"No, not...\" but no one heard that, not even himself. A dreadful chasm had opened up before him. Farragut knew everything and was laying for him. That was why he had not wanted the witnesses excluded! He had wanted Ryan to see what an easy time Jablonski would have of it, and thus be lulled into a false security. But he was not going to have an easy time of it! He was the carefully-selected victim. Farragut was standing on the opposite side of the chasm that had suddenly opened up and was beckoning him on irresistibly.\n\nThere was a wrangle over Tilbury's objection while Ryan felt his neck grow damply cold within his shirt collar. Then Farragut pressed in again from a different angle, and Ryan began answering questions that brought out the physical layout of the room in which they had arrested Derby, and narrated how they had done it.\n\n\"Then you searched him?\"\n\n\"That is right.\"\n\n\"What did you find?\"\n\nRyan enumerated the list of belongings as he remembered it.\n\n\"How much money was that again, Mr. Ryan?\" A thick tongue circled Farragut's thick lips anticipatorily.\n\n\"He had around ten dollars in small bills and change in his pocket.\"\n\n\"And in his wallet? Did you examine his wallet?\"\n\nNow he was getting it. \"I did.\"\n\n\"And how much money did you find in it?\"\n\nUntil this instant Ryan had never, finally, decided what he was going to say if that question were asked him under oath. Now he answered it without deciding. \"It did not contain any money.\"\n\nFor the first time Sandalwood's head raised.\n\n\"You knew the loot in the robbery was one hundred and twenty dollars, did you not?\"\n\n\"I did.\"\n\n\"And you wanted to find that money?\"\n\n\"Naturally.\"\n\n\"Answer yes or no, please,\" ordered Farragut.\n\nThe judge interjected, \"And louder, witness. Raise your voice.\"\n\nRyan took a breath. \"Yes. I wanted to find the money.\"\n\nSandalwood's expression was quizzical.\n\n\"Did you ask the defendant what he had done with it?\" Farragut went on.\n\nHere it was again. \"Yes.\"\n\n\"What did he say?\"\n\n\"Objection.\" Tilbury rose again. \"I submit, your honor, that all this is irrelevant and immaterial. The people readily admit that the hundred dollar bill has not been found. The...ah...purpose of these questions\u2014\"\n\n\"Your honor,\" Farragut broke in ponderously, \"I should like to enlighten young Mister Tilbury if I may. These questions have a definite purpose. Because they help to lay the foundation of the defense's contention that the defendant, Harry Derby, was framed.\"\n\nRyan looked down. His palms were wet. His lungs could not get enough air. The chasm had become a bottomless pit, and he, teetering on the edge of it, was losing his balance.\n\nTilbury chuckled superciliously. The judge told Farragut to proceed.\n\n\"Mr. Ryan, what did my client say when you asked him about the hundred dollars?\"\n\n\"He didn't say anything.\"\n\n\"He didn't answer your question?\"\n\n\"That is right.\"\n\nWhat could he say?\n\n\"Louder, witness!\"\u2014from above.\n\nRyan was near panic.\n\n\"Both you and your partner,\" said Farragut, his voice rising to thunder, \"have testified that Harry Derby admitted he had killed Mrs. Connors\u2014'killed the old lady' or 'old dame' or some such phrase. Now do you say that he refused to tell you what he did with the proceeds of that robbery?\"\n\n\"Yes, sir.\"\n\nBut the words came out better than he could have hoped, and somewhere in his subconscious there sounded a welcome echo of another conversation about this. \"I think you should realize,\" Said Ryan, looking up, and for a moment his eyes met Sandalwood's, \"that we were arresting an armed criminal who in a minute or two jumped\u2014attacked my partner. We\u2014we didn't have time to\u2014well, stage a debate with him. Our main idea was to get him into the station.\"\n\nEven before he finished speaking he felt he was saving himself, that his words had the simple ring of truth.\n\nThe veteran Farragut recognized it too. He waved the answer aside carelessly as though it were of no importance. But in a courtroom everything Farragut said or did was of importance.\n\n\"You referred to Derby just now as an armed criminal. Was he armed, Mr. Ryan?\"\n\n\"His gun was hanging on a nail near the door. That's the gun that ballistics\u2014\"\n\n\"Never mind, Mr. Ryan. Just answer the questions I ask, please. Derby did not have a revolver or any other weapon on his person when you broke into his room.\"\n\n\"We didn't break in.\"\n\nFarragut gave the jury his cracked-granite smile. \"He certainly didn't invite you in.\" Again he waved the subject into inconsequence. \"In any case, Derby\u2014so you say\u2014attacked your partner.\"\n\n\"That is right.\"\n\n\"Why, Mr. Ryan? Why? Why did an unarmed man charge into two men he knew were armed? Were you taunting him?\"\n\n\"No.\"\n\n\"Were you giving him the so-called third degree?\"\n\n\"No!\"\n\n\"By that, of course, I mean torturing him, to make him\u2014\"\n\nTilbury cried, \"Objection!\" and Ryan said, \"You know that's not so.\"\n\nFarragut did not even wait for the judge to uphold the objection.\n\n\"Then I can only conclude,\" he said smoothly, \"that the reason my client allegedly attacked Jablonski is because Jablonski got too close to him, and in some way or other goaded Harry Derby unbearably to attack him. Right?\"\n\n\"There was no goading, aside from the kind of talk that goes on when you arrest a hoodlum,\" said Ryan. \"Perhaps Jablonski got close\u2014but remember your client, as you call him, knew that this arrest meant the electric chair.\"\n\n\"Never mind that, witness,\" said Farragut sharply. \"In any case when Derby leaped for Jablonski they closed with each other and wrestled. Right?\"\n\n\"If you call kicking a man in the stomach wrestling. That's what your client did to Jablonski.\"\n\n\"But you had a gun, eh? Yet you didn't fire?\"\n\n\"That's right.\"\n\n\"Why?\"\n\n\"Because for a second they were too close together.\"\n\n\"They were too close together.\" Farragut relished the words and looked down the jury's rows. \"Too close together. They were in fact locked in a hand-to-hand struggle, were they not, Mr. Ryan?\"\n\n\"Well, for a second or so.\"\n\n\"So. For a second or so, you say. Very good. Now then, Mr. Ryan.\" Farragut spoke rapidly, like a man in a state of unnatural excitement, and Ryan recognized that he was approaching some goal. \"Both you and Jablonski were on hand at the scene of the murder earlier. And of course you handled the smashed lamp.\"\n\n\"Of course we did not.\"\n\n\"You did not handle pieces of the lamp?\"\n\n\"No, sir.\"\n\n\"Nor Jablonski?\"\n\n\"Objection\u2014\" from Tilbury. \"Mr. Jablonski is on hand to testify for himself.\"\n\n\"Sustained.\"\n\n\"But you did not handle the lamp, Ryan?\" No polite \"mister\" now; Farragut's quick, low voice was beginning to cut and tear.\n\n\"Objection, please the court. The witness has answered the question once. Besides, Mr. Farragut's questions have no real bearing\u2014\"\n\n\"Mr. Farragut's questions,\" Farragut thundered impressively, \"are going to show that the dust from the lamp which was presumably found on my client's work-jacket could actually have been placed there artificially.\"\n\nHe was addressing the court. But for a fleet second he fastened triumphant, squinting green eyes on Ryan.\n\nRyan's face went hot.\n\nThis was how it really was. Farragut knew everything, and had purposely passed up Jablonski as too experienced, gambling on Ryan, the rookie, to be more easily tripped. Farragut was betting his client's life that he could make Ryan stumble into the chasm.\n\n\"What does counsel mean by 'artificially'?\" asked the judge with interest.\n\n\"Well, since both these officers were at the scene, and might be presumed to have handled the physical evidence there even if they do not recall having done so, it is the contention of the defense, your honor, that their clothing might have received some of the dust in question\u2014become impregnated, so to speak. And if so, it is not impossible that during a hand-to-hand struggle such as has been described between Harry Derby and Detective Jablonski it is also possible that minute quantities of dust might have been transferred from Jablonski's clothing to my client's. I should like to remind your honor that the spectroscope is an extraordinarily sensitive instrument\u2014\"\n\n\"Counsel will confine himself to proper cross-examination.\"\n\n\"I'm sorry, your honor,\" said Farragut contritely, but he was a cat with cream on its face. The idea had been implanted in the jury's minds.\n\n\"I asked a moment ago, Mr. Ryan, if you had handled the lamp or any pieces of it,\" he went on. \"Of course, I realize that in the excitement and tension of a murder investigation\u2014your first, I believe\u2014I realize it is possible you might do something and forget having done so later. Consequently, I am going to ask you whether it is not possible that you might have, say, picked up a fragment or two of the broken lamp and forgotten it afterwards?\"\n\nHe looked brightly at Ryan, smiling a pursed-lip knowing smile.\n\nRyan had to look away. In the rush of fear that engulfed him like turbulent surf he could not think consecutively. He felt physically off balance as though he might pitch out of the witness chair, and he sensed Sandalwood's gaze on him like a hot, probing lance. He groped wildly for escape.\n\n\"Will you search your memory, please?\" Farragut went on with smooth rapidity. \"Don't you feel it is possible that you might have\u2014\"\n\nRyan shook his head. He looked toward Tilbury for help, as a boxer looks to his corner. But Tilbury was not looking at him. In the front row of spectators there was a patrician blond girl in a lavish mink coat, smiling brightly and making little finger motions at Tilbury, and he, lounging back in his chair, was nodding understanding at her. His socks and tie were both of the same blue and white checks.\n\n\"Why are you sure neither you nor Jablonski handled any piece of the lamp? How can you\u2014\"\n\nIt was almost impossible to pull himself back to Farragut's truculent face, the eyes pinched 'round with tight wrinkles of flesh.\n\nWhy didn't Tilbury...?\n\n\"No,\" he said lowly.\n\n\"What did you say, Ryan?\"\n\n\"Louder, witness.\"\n\nAgain he ranged the courtroom. Sandalwood was frowning intently at him. Jablonski's face was a black-and-white scrawl of fear, a caricature of terror.\n\n\"No!\"\n\nFarragut leaped in with tiger quickness. \"Supposing I were to tell you that a beer can was found in the room which also had traces of dust\u2014dust from the lamp. Would you say\u2014\"\n\nRyan did not hear the rest of it.\n\nThis was the end. Farragut had him.\n\nHe looked beseechingly toward Tilbury. The court was also looking at Tilbury, expecting an objection.\n\n\"Answer, witness,\" cried Farragut, and Ryan brought his white face up to confront the green-eyed sneer. Tilbury turned suddenly, uncomprehendingly, back to the cross-examination.\n\nA low whistle sounded in the courtroom. Ryan knew what it meant. He was dead; there were other sounds, a rising murmur like a boxing audience makes when one of the fighters begins to lurch into unconsciousness.\n\nThere came another whistle. The judge's gavel banged; he had to answer. Ryan's wet hands twisted. He looked up; the murmur was louder. And then, dumbfounded, he finally realized that no one in the courtroom was looking at him except Farragut. And Sandalwood.\n\nFor a tall, marvelously shapely girl with a wealth of auburn hair under a chic low-brimmed hat was coming down the center aisle, followed by a short, slick-haired man in a bright blue suit who carried a polo coat and seemed trying not to smile. The girl's expression was grave until she saw Ryan; then she smiled and waved at him. Her lips were masterpieces of curved lipstick and when she loosened her tightly cinched coat, before seating herself on a front bench, it was with an intimate, revealing air that drew another wolf whistle. The gavel banged ineffectually, two reporters hurried out the side door to telephone the city desk for photographers, and half the courtroom immediately recognized Gee Gee Hawes. Sandalwood smiled indolently at her across the intervening space.\n\nIt had taken less than twenty seconds for her to enter and be seated with the nightclub manager who accompanied her. But that was enough to save Ryan and to ruin Farragut.\n\n\"I had asked you,\" said Farragut loudly, trying to recapture the spell he had woven, \"if you were aware\u2014\" But the court interrupted. \"I must remind you, Mr. Farragut, that your question is completely unsupported by any testimony and therefore improper. I will ask the witness and especially the jury to disregard it and its implications.\"\n\nRyan was a drowning man finding dry earth under his feet.\n\nGee Gee smiled at him. The nightclub manager, whose idea this visit had been, tried to appear nonchalant when a reporter asked them to hang around after for an interview. Presently they went out to be photographed, while Farragut returned to the attack with all the irritation of one who knows his chance has slipped. He had led carefully and dramatically up to that last question, hoping it would force a confused, damaging admission. Gee Gee's entrance had wrecked his effect. Now Ryan answered his questions cagily and steadily.\n\nAfter a dozen more Farragut gave up.\n\nTilbury quickly paraded the three women witnesses to the stand and all three, with just enough uncertainty to make their testimony convincing, identified Derby. Court adjourned.\n\nThe girl in mink rose at once, beaming, and flung herself at Tilbury. \"Oh, darling, you were wonderful.\"\n\nHe petted her affectionately. \"Wonderful, nothing,\" he said. \"If they were all as easy as this I'd be ashamed to accept my salary.\"\n\n* * * *\n\nFlashbulbs were still firing in the corridor when Ryan and Jablonski reached it and people were looking at Gee Gee and asking \"Who's that? Who is she?\" as she stood amid the reporters. Her eyes were bright and she was smiling.\n\nSandalwood was at one side talking to the nightclub manager. As he and Jablonski passed Ryan heard Sandalwood joking, \"Max, you really ought to be ashamed of yourself. Anything for publicity, eh?\" Ryan did not look at Sandalwood.\n\nThen Gee Gee caught sight of him and waved and blew Ryan a kiss, and a photographer asked her to do that again. Ryan waved back and pointed to his wrist watch, meaning he had to go; he did not want to get mixed up in that, or have to answer Sandalwood's questions. \"I'll call you,\" he yelled, and she nodded and blew the kiss the photographer asked for. She did it several times.\n\n\"Boy, what a dame,\" said Jablonski as they went down the stairs. \"You know what I think? I think you're darned lucky, Neill.\"\n\n\"I'll tell you what I think,\" said Ryan bitterly. \"I think this guy's going to be convicted.\"\n\n\"Wait and see, Neill, wait and see,\" said Jablonski. \"That's all I ask.\"\n\n* * * *\n\nBut when court resumed next day it was not Farragut but his bespectacled young assistant, McCormick, who took over the defense of Harry Derby. Farragut, it was announced, had suffered a heart attack the previous evening and must rest for at least two months. Courtroom veterans smiled. This was Farragut's surrender. It was his boast that no Farragut client had ever been executed in his thirty years of practice, and he knew he could not save Derby. So instead he had had a heart attack, and now it was a McCormick client that would go to the chair.\n\nThree days later after an almost pathetically weak defense the case went to the jury, which deliberated forty-five minutes and then returned a verdict of guilty of murder in the first degree without recommendation of mercy. That made a death sentence mandatory.\nCHAPTER 15\n\nThe Phone Call\n\nRyan had the day tour on the day Derby was sentenced so he could not get downtown to hear it, even if he'd wanted to. But he read about it after work, standing at a bar bustling with the hurly-burly of day's end. He had ordered a whisky before he opened the pink sports final, and he drank it straight. The paper did not devote much space to the Derby story, but Ryan was able to reconstruct it in his mind's eye as he sipped the chaser.\n\nThere was the loosening of the handcuffs, the leading of the tall, thin man around in front of the judge, the dry recital of the facts and reading of probation reports. \"I therefore direct that you, Harry Derby, be transported to the slate prison of New York at Ossining and there be executed in accordance with the laws of the State and at the discretion of the warden some time during the week of February 23. And may God have mercy on your soul.\" Then he would have been cuffed again and led away, the lean, sinewy back still surly with defiance. But desperately frightened inside.\n\nRyan ordered and drank another whisky. Then he stuffed the paper into his overcoat, went out and walked down Lexington Avenue.\n\nWhen he reached Grand Central Station he went in and found a large telephone headquarters containing booths and operators to get long-distance numbers. He called Jablonski at his home. When he answered, Jablonski asked how Ryan was with great heartiness.\n\n\"You see tonight's papers yet?\"\n\n\"No. I been home with a cold all day, Neill. Anything I should have seen?\"\n\nRyan took a breath. \"Our boy got the chair.\"\n\n\"You mean Derby?\"\n\n\"So what do we do now?\"\n\n\"Well,\" gently, \"what do you think we do, Neill?\"\n\n\"For Christ's sake!\" Desperation tore his throat. \"If we don't do something, the guy burns.\"\n\nFrom the other end came a long, fully rounded silence. \"Well,\" said Jablonski, \"don't you really think he ought to?\"\n\nRyan began to have the helpless, nightmare feeling of being pursued and menaced and not being able to run.\n\nBut he said, \"God damn it,\" steadily, \"it's not for us to decide that. This is a\u2014\" he started to say \"human life\" and knew how weak that would sound. \"This is a guy's life. Even if he is the kind who\u2014\"\n\n\"Now wait a minute, Neill.\"\n\n\"Wait, hell! We agreed to wait until the trial was over because maybe he'd never be convicted. Okay. We waited. Now he's convicted. And sentenced. And he didn't kill that old woman.\"\n\n\"Where you calling from?\" Jablonski's voice had grown cold.\n\n\"Grand Central.\"\n\n\"Pay phone?\"\n\n\"Sure.\"\n\nBut he realized he must be talking loud. The operator outside who had put the call through was looking in at him.\n\nJablonski said, \"Listen, Neill. Take it easy and don't start yelling your head off. You know what I told you when you first got in an uproar about this, and I'm going to tell it to you again for your own good. And mine. You got a great future in the department, kid. You're going to go a long way. I wish I was in your shoes. I mean it.\"\n\nRyan knew he did.\n\n\"I told you before the trial to keep your mouth shut because I knew this slob might get off and you would only wreck everything if you opened up. And if he didn't get off and got sent up, why that'd be all right too, because it would be too late for anyone to do anything. And Neill, that's just how it is.\"\n\n\"Just how is it?\"\n\n\"Derby's convicted\u2014he's put away. And Farragut's licked\u2014he wouldn't touch the case now for fifty G's. Derby is going to the chair and that's where he belongs. You're sitting pretty. So am I. And why not\u2014? Don't you think cops are entitled to a break occasionally?\"\n\nRyan did not answer.\n\n\"So maybe he didn't kill that old lady, like you claim,\" Jablonski went on. \"Although I sure as hell doubt it. But anyway\u2014he was one of the guys in the Moriarity murder in 'fifty-one. You know that.\"\n\n\"I don't know it.\"\n\n\"Maybe it was before your time.\"\n\n\"Was he ever convicted?\"\n\n\"Of course not. He was brought in but we never had enough to hold him on. Then there was another killing, an old man named Trimble.\"\n\n\"Was he tried for that?\"\n\n\"No. Derby was never tried for\u2014\"\n\n\"Was he arrested?\"\n\n\"No\u2014I'm trying to tell you! We always had a hunch\u2014\"\n\n\"For God's sake, Jabby, you can't sentence a guy to death just because you suspect something.\"\n\n\"I ain't just suspecting, Neill. I know darned well\u2014\"\n\n\"You know nothing. And furthermore you lied and stalled before and you're doing it now. What the hell kind of cop were you, anyway?\"\n\nThe other end of the line went silent. The words echoed, and Ryan knew that he had finally hit Jablonski where it hurt.\n\nWhen he spoke it was with the cold deliberation of mortal anger. \"I'll tell you what kind of cop,\" said Jablonski. \"I'm an ex-cop now. But I put in my time, twenty-eight years of it. And I learned a few things in my time. And one was that when you get one of them\u2014a guy like Derby\u2014in a spot where you can give it to him, then by God you give it to him. And you sure as hell don't wreck your whole life and your partner's life by trying to give the son of a bitch a break. I'm loyal to the department, that's the kind of cop I am. And I got my own little place out here, and I and Sarah are getting along fine and I want it to go on.\" There was a pause. \"I don't want that changed. You shouldn't either.\"\n\nAgain there was a pause. \"But if you try to do anything about it, I'll tell you what I'm going to do. You go ahead and tell your story\u2014and I'll swear to God you are lying. You come in with your fingerprint from the drugstore heist or whatever it was and I'll say that you came to me a while back with a plan to get Derby off, for a payoff. And that you said something about planting or switching a fingerprint. That'll put you and Derby on the same side of the fence, standing against the whole damn department. I wouldn't want that, Neill, and I wouldn't want to do it. But if you force me to I will. You want it?\"\n\nRyan looked out the booth's glass door. The girl was not watching him now. She wrote something on a little pad and then thrust the pencil into her thick coil of dark hair.\n\n\"And another thing, Neill. You swore under oath that Derby didn't have the big bill on him. Now if you come in with a different story, who's going to believe you? Once you've admitted you're a liar, you never carry much weight again, you know. It's like an expert witness who gets knocked down in court.\"\n\nRyan's hand began moving the telephone instrument from his ear toward the prong that should receive it. He continued looking at that glossy coil of hair. Now she was writing something else.\n\nSomewhere in the telephone booth a tiny metallic voice said, \"Well, what do you say? Neill?... Neill?\"\n\nRyan kicked at the door. \"Go to hell,\" he said. The telephone girl looked up. He had taken a step out of the booth before he remembered that he was still holding the phone. He latched it into place.\n\nIn the wide, murmurous corridor outside he almost ran into a hurrying young couple. The man carried a big valise, the girl a lighter one. \"Let me take it, honey,\" the man said. \"Come on. Give it to me.\"\n\n\"You can't carry them both,\" she protested. But she relinquished the bag.\n\n\"Of course I can,\" he laughed. \"They balance.\" As he took the bag his eye caught Ryan's. Something in Ryan's face made him look again.\n\n\"It's all yours,\" Ryan told him. \"All yours. Carry it.\"\n\nThey looked at him, then hurried on. Ryan walked out toward Forty-second Street.\nCHAPTER 16\n\nThe Long Way Home\n\nHe walked fast, through homing commuters and shouting newsboys, slipping past traffic-halted ranks at the curbs to dodge narrowly between taxi bumpers and keep going. At Madison, forgetting he was crossing a two-way street, he would have been run down had a cab driver not seen him walking against the light. The cab squealed to a halt and Ryan looked up; the driver laid his arms over the wheel and rubbed his face in them in an elaborate gesture of patience.\n\nAnother time Ryan would have grinned apologetically; now he hurried on, only belatedly aware. He was gripped by a compulsion to get some place in a hurry, to move and keep moving without thinking. He had no real goal. Getting home was as good an excuse as any, so when he came to Fifth Avenue he turned north and walked fast past the glossy, lighted windows of the big stores and the little shops. He wanted to walk, not think.\n\nWhen he had crossed Fifty-seventh Street and was swinging up the avenue's wide sidewalk, that was almost like broad, scented velvet under the lights of the apartment entrances and canopies, the silent trees and mystery of Central Park on the other side, he did not bear east toward home as he should have done. After a time he remembered Eleanor's request for picking up the Ibsen book at the Columbia library and he seized the excuse to continue walking, fast and anonymous, through the night's cold air, and think only of his errand, knowing his absence would cause no alarm at home, and above all determined not to think about what he could not help thinking about.\n\nHe was standing in line in the familiarly stuffy library when something poked his back. A thin, chicken-necked man with the gleam of discovery in his eye was looking at him, tapping finger still extended. \"Aren't you O'Neill Ryan?\"\n\n\"That's right.\" It sounded surly. Then Ryan recognized the thin man. It was the hair that was deceptive.\n\n\"Professor Montagne!\" He had not been nearly so bald when he taught Ryan Greek philosophy eight years ago.\n\nWhen Professor Montagne shook hands his glasses gleamed even more brightly. \"I knew it was you,\" he said. \"At first I told myself it was just because I had read about you lately.\"\n\n\"Next,\" said the woman charging out books. Ryan handed her the book and Eleanor's card, then waited in the corridor for the teacher.\n\n\"This is amazing,\" said Mr. Montagne, and took Ryan's arm. \"Do you realize I was going to telephone you within a few days?\u2014here, this way, eh? I know a room.\"\n\nRyan permitted himself to be led down a corridor and finally into a small reading room containing a single absorbed girl, student.\n\nMr. Montagne whispered. \"We can talk as long as we keep our voices down.\" He beamed. \"Well, O'Neill, how have you been, as though I needed to ask?\"\n\n\"Fine, Professor,\" said Ryan. He still had an undergraduate's respect for a member of the faculty.\n\n\"You look well,\" said Montagne. \"And I know you are doing well. You cannot imagine how good it is to learn that a boy from Philosophy 508 is putting into everyday practice some of the Socratic definitions that I\u2014\" His laugh deprecated the idea. \"But that is not why I intended calling you, O'Neill. Frankly, I have a problem.\n\n\"For two years I've been director of the Men Students' Forum. You appreciate what that means\u2014a meeting and a speaker once a month. You are\u2014well, as the students say, let's face it. You are a celebrity. And you certainly must have much to say about the application of your education to the problems of life. If you could come up some Sunday evening and give us a plain, heart-to-heart talk, say thirty to forty minutes, about\u2014\"\n\n\"Okay,\" said Ryan. He was suddenly oppressed by time's brevity. He knew what Montagne wanted. But he wanted something, too.\n\n\"You'll do it? O'Neill? Really? I can't tell you how happy I\u2014that I\u2014that we ran into each other.\"\n\n\"So am I,\" said Ryan. He spoke aloud and the girl student looked up.\n\n\"Of course,\" Montagne smiled slyly, \"if you were to work in a few references about how Phil 508 helped in your career\u2014in case it did. What I mean is, you are an officer of the law and I hope you remember the time we spent on the Platonic concept of the well-ordered state.\"\n\n\"Yes,\" said Ryan. \"Sure. And that's something I'd like to ask you about. You see, since becoming a cop I've thought about that philosophy course. Occasionally in my work you run into problems that\u2014that make you think about justice and virtue.\"\n\n\"I'm sure you do.\" Professor Montagne's head bobbed happily.\n\nDared he? Hell!\u2014why not?\n\n\"I guess you've been reading about this guy Derby who was sentenced today.\"\n\n\"I certainly have, O'Neill, and how it was you who arrested him.\"\n\n\"Well, I've been wondering, just as a sort of problem, like one of those abstract problems in ethics you used to give us.\" He watched anxiously for suspicion in the smiling face. \"Supposing I happened to find out now, after helping to convict him, that this guy Derby, this hoodlum, was not guilty of the murder he's charged with?\"\n\nIt was out. Only in the abstract, of course, but even that had frightened him to say. Yet it obviously had not aroused Professor Montagne. Then he recognized that as a teacher he had dealt for years with the caprices of students. What was one more hypothetical question?\n\nMontagne looked down his nose at Ryan. \"Do? There is only one thing you could do. As a virtuous man you know what that is. For a moment I thought you had a poser.\"\n\n\"Well, what? Tell the truth?\"\n\n\"What else? Can a just man see another punished unjustly\u2014for a crime he never committed\u2014and remain silent?\"\n\n\"But think of the crimes he did commit,\" Ryan argued, \"and never was punished for. Derby himself is certainly no just man.\"\n\n\"That has nothing to do with the case. The just deal justly even with the unjust. It is implicit in the common law\u2014\"\n\n\"Common law nothing!\" said Ryan. \"Would you turn loose a known rapist and robber, a no-good\u2014\"\n\nThe girl student made a meaningful rustle with her book. Montagne bowed apologetically.\n\n\"We must be quieter,\" he said. \"Remember, you presented a hypothetical case. I gave you the only possible answer.\"\n\n\"The only hypothetical answer.\"\n\n\"Quite. Quite.\"\n\n\"But\u2014if you were a cop as I am, and this was a real problem\u2014\" He smiled to show he knew the absurdity of it.\n\nMontagne raised a restraining hand. \"My dear O'Neill, I will not let you lure me into arguing theory versus practice\u2014real-life practice\u2014on an empty stomach. I have no doubt that as a policeman I might have to\u2014to compromise. If this fellow is the scoundrel you say, I might not be able to turn him loose quite so readily as I can hypothetically. But I give you the answer of a philosopher. Well! Sorry, I must get on. My wife\u2014\"\n\nHe rose, still talking. \"How about the second Sunday in February? We still meet in the music room. I'll ring you in advance, of course. We could discuss this further then, with the students. Eh?\"\n\n\"Sure,\" said Ryan. \"We'll do that. In February.\"\n\nRyan walked head down, a purposeless night voyager, past murmurous blocks to Central Park and then along a winding park path. When he raised his eyes he saw neat rectangles of lights rimming the park's black quiet; they were not human habitations but remote glitters against eternal darkness.\n\nWhy didn't he just forget Derby? He did not have to be concerned about such animals, did he? Of course he didn't. Then why... For a moment Ryan saw himself objectively, as though he were outside himself, skirting the path he was walking; he saw a tired, anxious, hollow-eyed man of twenty-eight who just wanted this to be over with. Even Professor Montagne had said...\n\nHe came to Fifth Avenue. Home was not far away.\n\nNo. Not that, now. Keep walking.\n\nAt the genteel curbs women in bright evening dress and furs stood with men in sober black, waiting for limousines to take them to theater or opera. Ryan strode on, numbly aware he was walking toward a decision and meanwhile numbly ignoring it.\n\nAfter a time he found he wanted a drink.\nCHAPTER 17\n\nRosemary\n\nRyan stopped in a corner bar at Lexington Avenue and drank a whisky, fast and straight. The dinner hour had grown into evening and the men at the bar were beer-drinking philosophers. Ryan walked on to the next bar and again ordered straight rye and put it down fast. He did not consciously desire to get drunk; he had never desired that in his life. Yet as he worked gradually east and south he made repeated stops, to feel raw liquor fire his throat and explode gently in his stomach, barely tasting the water chaser. After a time the alcohol reduced the urgency of his flight and enabled his slowing brain to engage the problem again.\n\nHe must not let himself be confused by Jablonski's threats. This was an issue to be decided on its merits. But what were they? By what practical criterion could he justify going to his superiors and revealing that Derby had been framed? What kind of a reception would that get\u2014and what kind should it get? The police department's function was safeguarding the city, not debating the subtlest shades of justice.\n\nYet all the time Ryan knew he was only torturing himself. He could never accept the practical, easy way out. He could never let Derby die, even if Derby himself desired that. Why was rooted deeply in him, below the level of understanding. It was not noble; it was automatic and terribly uncomfortable. Right now whisky made it a little easier.\n\nHe was standing looking at a drink he had just ordered when he spoke without knowing that he was going to speak. \"This is getting deep.\"\n\nThe bartender, arranging glasses in pyramids, looked around. \"How's that?\"\n\n\"I can't touch,\" said Ryan, trying to explain what he had just heard himself say. \"Bottom. Can't touch bottom.\" He picked up the small heavy glass, spilled a part, and drank the rest.\n\n\"You better take it easy, Mac.\"\n\nRyan re-buttoned his coat. He did not like criticism at a time like this, but he must not get into arguments at a time like this, either. He went out, and looked up at the corner street sign. He was on First Avenue in the seventies. He had almost walked home.\n\nIn the next block was a small pizza restaurant where he ate often when he was late. That was it. One more drink, then a big pie and coffee, because he did not want to arrive home feeling unsteady.\n\nIt was a tiny place, with a counter and white tile pizza oven in the back. There were only four booths and a couple of tables in the middle of the floor; one of the booths was occupied by a wrangling couple. The oven man came from behind the counter to serve him with brawny, floured arms.\n\n\"Rye,\" said Ryan.\n\n\"No liquor here, mister. Wine and beer.\"\n\nHe'd forgotten. Heck with it. \"Gimme a pie. Fish.\"\n\nThe oven man went back behind the counter and began to spin out the cap of dough. When Ryan thought to take off his overcoat, he found Eleanor's book bulging the pocket. He opened it at random, read a few lines, decided it was dull and closed it. He felt the warmth and breathed the fragrance of baking dough. Finally the great, wide pie was slid in front of him, littered with anchovies.\n\n\"Coffee?\"\n\n\"Coffee.\"\n\nHe ate in silence, and the conversational mutter continued from the booth ahead. He could not make out all the words, although the man sounded indignant. Then the woman said with impatient clarity, \"Oh, finish your beer and come on home. We can't go back there tonight. It's almost ten.\"\n\n\"Sure, finish your beer and go home.\" The man welcomed the excuse for new indignation. \"Your own brother gets framed by the cops and double-crossed by his own stinking lawyer and on the day he's sentenced all you're supposed to do's drink your beer and go home.\"\n\nRyan's sodden mind began pulling itself into awareness.\n\n\"Oh, come on,\" said the woman. She was tired and querulous. \"He's out and there's no telling when he'll get back. Detectives don't work regular hours.\"\n\nRyan could not believe his ears. Was he right?\n\n\"You wanna go home? You go on home.\"\n\n\"Not alone with this pay in my purse. Not in this neighborhood.\"\n\n\"I'll say not in this neighborhood. Cops don't much get around neighborhoods like this, where the ordinary people live. They're too busy.\"\n\nWhere do cops live, buster?\n\n\"They're too busy out framing guys. They\u2014\" He spilled loud, obscene hatred of cops.\n\nRyan sipped his coffee and held in his feelings, thinking slowly because of the whisky. This must be Ken Derby and his wife; they had visited his home.\n\n\"Come on, Ken,\" the woman said.\n\nThey got up. Ken Derby wore the black whipcord of a deliveryman; he looked thin and fit in it. The woman was a girl of indeterminate age and in a slighter, feminine way she somewhat resembled him. She had dark-blond hair and a slender face with a wide, patient mouth and dark, observant eyes that seemed not to want to see as much as they did. Rising, she dropped a scarf and when she stooped to pick it up and the neckline of her dress opened, Ryan saw small, immature breasts only half-held by a ribbon of bra. Her eyes met his impersonally and she shrugged into her coat without her escort's help.\n\n\"How about a beer on the house?\" Ken Derby called to the oven man, and turned.\n\nAnd looked squarely at Ryan.\n\nThey looked at each other a long time. \"Come on, Ken,\" the girl said. \"Please.\"\n\nDerby exhaled a long breath.\n\n\"There he is now,\" he said, and threw off the visored cap he had just put on.\n\nRyan pushed the table away and got up. His legs felt shaky. It wasn't fear; he always felt that way when a fight threatened. He was glad the overcoat was off.\n\nThe girl, without understanding what was causing this, said, \"Oh, for the love of God\u2014\" and pressed her angular thinness against Derby restrainingly.\n\nDerby pushed her away and came forward, extending a long right arm, holding his left curled against his chest. That was a disadvantage, for Ryan was accustomed to right-handed fighters.\n\n\"Hey, what goes on?\" came from the counter.\n\n\"Oh, stop it!\" the girl cried despairingly and for a perilous second Ryan looked at her, and saw large, fear-widened brown eyes, and a mouth quirked to cry.\n\nDerby feinted twice with the right, not even coming close, and then swung his left with heavy certainty. But Ryan stepped inside it, crouched and dug both fists twice into Derby's thin, muscled belly. Derby grunted and clinched awkwardly, and moved sideways. His thigh lifted the table and Ryan's pizza and coffee cascaded to the floor. Ryan wrestled his way up inside the other man's guard, knowing what he was going to do when the clinch broke.\n\nHis head was clear; the whisky's lingering effect only made him confident and strong. Over Derby's shoulder he saw the girl shrink desolately against the booth.\n\nDerby heaved at him to get out of the clinch, and he felt a hand grab him from behind but it did nothing. He took a half step, starting the punch from near his ribs, pivoting his body as he brought it up and it caught Derby's jaw, but on the side instead of the jaw-point. Still, he slumped. Ryan straightened him with a quick left and started another hook, pivoting...\n\nAs he did he again saw the hopeless girl against the booth. He pulled the punch just enough so that it thumped harmlessly against Derby's slow left hand. The oven man, who had a toy baseball bat out, grabbed Derby from behind\u2014and found he had to hold him up. The fight was over.\n\n\"What the hell?\" the oven man demanded.\n\n\"Sit him down on the bench,\" said Ryan. He sucked a knuckle and looked at the girl. She had turned away from them.\n\n\"Sit him down,\" said the oven man sarcastically. \"For Chrissake, Mac. He won't sit. He's out.\"\n\nRyan considered several things, then brought out his badge. \"This is a police matter,\" he told the man with floured arms, and the baseball bat went into a hip pocket.\n\n\"Oh. You pinching him?\"\n\nThe girl turned around. \"No,\" Ryan told the oven man, but he was speaking to the girl. \"I'm not pinching him.\"\n\n\"Well, what do we do with him?\"\n\n\"I said sit him down,\" said Ryan, and when Derby had been eased into a corner of one of the booths he felt the slackened pulse and lifted an eyelid. \"He'll be all right. Give him a few minutes. And bring us some coffee. Or maybe some wine for you, ma'am?\"\n\nShe shook her head no.\n\n\"You look like you could use it.\"\n\n\"No.\"\n\n\"Two coffees.\"\n\nHe sat down at a table, but the girl continued to stand, rebellious and helpless. He said, \"I'm sorry about this. Why don't you sit down?\"\n\n\"You don't have to be sorry. You couldn't help it. He\u2014he gets like that when he's been drinking.\"\n\n\"Sure.\" He pulled a chair around for her, a tacit invitation, and she finally accepted. She was tired and worn and respectable, cheaply dressed\u2014like a million other women in New York that night. Derby sighed and settled in the booth's corner.\n\n\"I gather you're a policeman.\"\n\n\"And you're Mrs. Derby.\"\n\n\"I'm Ken's sister.\"\n\nThat explained the resemblance. \"I'm sorry I hit him so hard.\" Saying that surprised him because he was not accustomed to apologizing under such circumstances.\n\nThe oven man brought two cups of coffee, picked up the spilled food and locked the front door. \"We close at ten,\" he said.\n\nRyan did not like using his authority for personal advantage but there were times when it was justified. The oven man caught his look. \"Oh, take all the time you want, loo-tenant,\" he said hastily.\n\nRyan said, \"That makes you Harry Derby's sister too.\"\n\nThe brown eyes trained a steady sadness on him, and he knew she was refusing to say anything of what she had to say. \"I'm Harry Derby's sister.\"\n\nRyan burned his mouth with a long sip of coffee. It did not occur to him that there was no necessity for him to make explanations to the sister of Harry Derby. That would occur long after, when he thought about her thin body and patience and gentle eyes. And lack of the obvious prettinesses.\n\nHe said, \"I'm Neill Ryan, one of the officers who arrested your brother and helped send him up. You were over at my house tonight?\"\n\n\"Ken wanted to talk to you. We stopped by twice. But\u2014\" She looked over her shoulder, and Derby breathed deeply as though in reply to her glance. \"But perhaps you can understand how Ken feels.\"\n\nRyan said, \"Sure.\"\n\n\"Not that we have any illusions about Harry. But Ken\u2014well, don't misjudge him. He doesn't drink often. And he was fond of Harry.\"\n\n\"He sure was.\" Ryan could not keep the sarcasm out of his voice.\n\n\"What do you mean?\"\n\n\"Your brother came to my home some time back to alibi Harry. He claimed Harry was with him on the day of the Connors murder.\"\n\n\"Ken told you that?\"\n\n\"He said they delivered some big appliances together that day.\"\n\nWhen he saw how it made her look he wished he had not said it.\n\n\"He's sort of crazy sometimes,\" she said slowly, \"and impulsive. But he should not have done that. That's\u2014that's obstructing justice or something, isn't it?\"\n\n\"It doesn't matter. I proved to my own satisfaction that he had lied.\"\n\n\"Yes?\" She eyed him oddly and for the first time she sipped her coffee. \"Tell me something...ah...look here, Mr. Ryan. Ken was telling me tonight that the word has been out along the docks for quite a while that the cops finally got Harry. Is that true?\"\n\n\"Got him?\"\n\n\"You understand me, I think. What Ken meant was that you had not gotten Harry fairly or honestly or on the basis of real evidence. The idea is Harry was framed, that the evidence was manufactured against him. And that the union didn't put up much money and when that lawyer, Farragut, could not immediately disprove the evidence, he simply gave up. Is that true?\"\n\nThe steadiness of her gaze had attracted him before. Now it disconcerted him. He looked away, feeling again the familiar stab of alarm.\n\n\"Wait a minute. I have no idea what they are saying on the docks. But if you think your brother was framed\u2014\" to say it was an effort\u2014\"you're crazy. Holy God, the evidence against him\u2014\"\n\n\"I read the papers.\"\n\n\"Then you know\u2014\"\n\n\"I know Ken was present once when Farragut visited Harry in jail. He says Harry told Farragut that one of the cops blew some dust on his jacket to make the case stronger. I don't understand that, exactly. But did you do it, Mr. Ryan?\"\n\nThat marshaled Ryan's wits. He looked amused. \"Do you believe that?\"\n\n\"I don't know what to believe!\" she cried. The oven man looked up from the Italian newspaper he was reading.\n\n\"Well, it's ridiculous on the face of it,\" said Ryan. \"The evidence against your brother was overwhelming, as you know if you followed the case.\"\n\n\"I know.\" She sounded uncertain. \"And I wouldn't believe it if Harry said it.\" Her raised voice was near hysteria. \"But I believe Ken! And don't think I have many illusions about the police, either. We haven't had a very easy time of it from them\u2014between Harry and my father, when he was alive.\"\n\nAnd your brother and father didn't make things very easy for the cops.\n\nInstead, \"That's how it goes,\" he said softly. \"Some people go wrong. I saw it happen to kids I grew up with\u2014kids from decent homes and all that. It's nothing for you to be ashamed of.\"\n\n\"I'm not ashamed. I can't help what my father was, or what he helped my older brother to become.\n\n\"But I don't know what to think. I don't know what to believe. Did you ever hear a bigger clich\u00e9?\" Her smile was unnaturally bright; her voice cracked. \"Listen!\" In the intensity of her gesturing she struck her coffee cup and spilled some across the table. \"Listen. I lost my job today. I was fired. That's why I've got two weeks' pay in my pocket\u2014from the Fortunatus Club. You know it? It's very exclusive. I was their librarian. When Harry was arrested they found out for the first time who I was. And they were very nice. They said, after all he wasn't guilty until proven\u2014nobody's guilty until proven. You know? AH that. But he was proven. So today Mr. Murchison came around and very nicely told me I would have to leave. He said, 'We feel terrible about this, Rosemary.' And he did. But how many librarian's jobs do you think there are in the city, Mr. Ryan?\"\n\nShe got to her feet. Her voice was shrill. Ken Derby opened vacant eyes. His sister gestured toward him.\n\n\"And you know what, Mr. Ryan? I believe Harry is innocent. By God, I do! I believe what Ken said is the truth and that you framed Harry. You know why I believe it? Because I've talked to you, and seen your face. You're a liar, Mister Ryan. You're a cheap liar\u2014\"\n\nSuddenly she began to cry, and she continued talking and crying, disregarding her tears and the saliva that thickened her voice. \"That's justice, what you are! No truth, or honesty. Justice is what you make people believe. Sure, Harry was no good. But to send him to prison you had to make yourself worse. You had to lie and cheat in the name of decency. That's how the law is upheld! Oh, dear Jesus, I hate\u2014\"\n\nShe threw herself at him, sobbing, beating down on his chest with her fists, kicking, crying...\n\n\"Hey for Pete's sake,\" said the oven man, alarmed.\n\nRyan caught her arms. \"Call a cab,\" he said.\n\nThe oven man rushed to the front door and unlocked it. The girl was weeping uncontrollably. Ryan waited, holding her thin shoulders and feeling them shake, knowing this would take time.\n\nKen Derby got up on unsteady legs. \"What's now?\" he said. \"Whatsa matter, Rosie?\"\n\nThen brakes squealed outside. Ryan led the girl. \"Come on,\" he flung over his shoulder.\n\nHe put them both in the cab, got the address from Derby and repeated it to the driver as he handed him a dollar. \"See that your sister gets home all right,\" he said.\n\nHe watched the cab pull away, then went back inside and paid the bill. It was only after he had done that and stood outside again, breathing deeply and seeing the little restaurant's lights blink out, that he really felt what she had said to him.\nCHAPTER 18\n\nThe Homecoming\n\nAs she herself occasionally said, Agnes Ryan had never looked at another man after her husband was killed\u2014and at very few before she met him. She had been a plump little Dresden doll of a girl with the Irish gift of merriment and a fierce streak of possessiveness that was dramatic in one so otherwise kind and docile. When she lost her husband, Mrs. Ryan turned her long-lashed eyes inward on herself and her memories, on her family and her God. She devoted herself to bringing up Eleanor and O'Neill, to working at the church and to making ends meet on a slain policeman's pension and insurance, with occasional help from her brother who had an insurance agency in Worcester. At fifty-five she was a prematurely white-haired little woman with china blue eyes, patience and good cheer. She prayed regularly and lived in the serene faith that her man was watching her from heaven where ultimately they would be reunited forever.\n\nWhen Ryan let himself in with his key, she was sitting in the living room before a radio turned low, listening to the eleven o'clock news and sewing the hem on a new dress of Eleanor's. After sewing each section she pulled the basting threads out with tiny stubby fingers.\n\nRyan bent over and kissed her, \"Hi, ma,\" and she caught the reek of whisky and heavy food. It reminded her of his father. Her son reminded her of his father in many ways, although he was shorter and slighter and\u2014it seemed to her these days\u2014quieter and less aggressive. \"You're late,\" she said.\n\n\"Yeah. Something came up.\"\n\nShe would not ask what it was although she looked searchingly at his bloodshot eyes and haggard cheeks.\n\n\"Some people were here.\"\n\n\"I saw them.\"\n\n\"They came twice.\"\n\n\"They would. It's not important.\"\n\n\"You've eaten?\"\n\n\"I\u2014I had a bite.\"\n\n\"Let me fix you a sandwich. We had a nice meat loaf.\"\n\nHe really didn't want it but he knew she wanted to fix it. And what difference did it make in a world that had fallen apart?\n\n\"Well, maybe a thin one.\"\n\nWhile she bustled in the kitchen he leaned back in his father's chair and closed his eyes, and despair flooded in like the sea through a dissolving dike. He did not know where to turn next and he was oppressed with the urgency of doing something now\u2014yet what could he do? He thought of the time when he was thirteen and they had hit a baseball through the plate-glass window of Mr. Brodt's butcher shop. Once again he had that hopeless feeling of having done something enormous and frightful, beyond repair or recall. What could he do\u2014what was there to do?\n\nGo to Lieutenant Bauer and tell him that the real murderer of Mrs. Connors had been permitted to escape clean and unpursued? Ruin Jablonski's retirement-in-honor\u2014even forgetting Jablonski's threat? Blast his own hopes and chances that were now so bright? And yet...\n\nEverything in Ryan's nature, everything instilled in him as a boy about honor and fair play, his staunch belief in the ultimate supremacy of right, all that he had come to admire and respect as he grew into a man, stood inexorably over and against what he had done. And he had done it, he told himself brutally. There was no use blaming Jablonski or Derby or anyone else. They had had a part in it, sure. But he had let himself drift into what was now\u2014as Jablonski had so bluntly told him\u2014an impossible situation. There was nothing he could do. There was no way out. Ryan's closed eyes tightened spasmodically.\n\n\"Here, Neill,\" said a voice and he opened them to see his mother standing over him with a plate and a glass of milk. She was looking at him worriedly. \"Is everything all right?\"\n\nRyan munched the sandwich. \"It's great.\"\n\nShe picked up the dress and resumed work. After a moment she said, \"That's not what I meant. I meant\u2014is everything all right with you?\"\n\nRyan knew what she meant. He went on eating the sandwich, looking at his mother's white head bent over the dress, tiny fingers busy with little stitches, and all that he had felt about her as a boy surged into his mind. Suddenly he had to say it.\n\n\"Ma.\"\n\n\"Yes?\" She went on working, because she knew something was coming.\n\n\"Everything's all right\u2014it's fine. I'm just tired. But...something I been thinking of asking you.\"\n\n\"It's about that girl, isn't it? That red-haired dancer?\" She forced herself to smile at him. \"She looked pretty in the paper, Neill. I'd like to meet her.\" Her head bent again over her sewing.\n\n\"Oh. Gee Gee? No. I wasn't thinking of her. I'd like you to meet her, though. The next Monday night I'm home, I'll ask her up.\" He put down the empty, milk-clouded glass.\n\n\"Good. And so, what then?\"\n\n\"Oh, nothing much. It's just this\u2014this guy Derby that we sent up. Jablonski and me.\" He didn't know what he was going to say. He just had to talk to someone\u2014someone he could trust.\n\n\"Oh. Well, he surely shouldn't bother you, Neill.\"\n\n\"Well, he does. Suppose\u2014ma, suppose I told you that today I learned something that makes it look as if Derby wasn't guilty after all?\"\n\nFor the first time she looked fully at him.\n\n\"O'Neill Ryan,\" she said. \"Have you lost your mind? Of course he's guilty. By the Holy Mother do you think I didn't read every word of it in the papers, and my own boy in it?\"\n\n\"Sure. But...\"\n\nCarefully she pulled out the basting stitches. \"This is the first man you've ever helped send to the electric chair, isn't it, Neill?\"\n\n\"Yes. That's right.\"\n\n\"I mind the time your father was home on the night a stickup artist was to die in Sing Sing. He'd helped send him there and the man had almost killed the two officers who had arrested him. Yet your father could not stand it, and as it got close to midnight he went out and over to Martin's speakeasy and had some drinks. He told me about it later. And do you know what he did then? The drinks didn't help either, and he left there and began walking and he walked all the way down to St. Pat's and he went in and spent the night on his knees\u2014praying for the man he had helped to electrocute.\" Her voice suddenly caught and Ryan knew his mother inwardly was crying.\n\n\"You're like him, Neill,\" she went on. \"And you worry like him, even over people like this Derby. But you needn't, believe me. For if ever there was justice served to one of them thieving scum\u2014\"\n\n\"But\u2014\" He couldn't let it go on. \"But, ma\u2014suppose Derby wasn't guilty!\"\n\n\"Wasn't guilty! Wasn't guilty of what, in heaven's name?\"\n\n\"Well\u2014suppose he didn't kill the old woman?\"\n\nShe switched off the radio's indistinct murmur. \"Well,\" she said, \"didn't he kill her?\"\n\nRyan was silent. Could he tell her?\n\nNo.\n\n\"I didn't say that. I just said, suppose he didn't. And I found it out now. What could I do?\"\n\n\"O'Neill Ryan, if that's the sort of thing that's bothering you, you had better get to bed and get a good night's rest.\" Petulantly she resumed her sewing. \"You're just the kind to lean over backwards about things like that, you certainly are. But let me tell you this. Even if that Derby had not killed the old woman, he has done so many other evil things, and well you know it, that the electric chair is no more than he deserves. Think of your father, boy, and what they did to him. That man Derby is that kind\u2014if he hasn't killed some decent, brave officer it is only because he hasn't had the sneaking chance to. Why, Neill, of course he's guilty.\"\n\nIt was getting beyond his control. \"But, ma,\" he cried out, \"supposing\u2014suppose I have to go to the department and tell them that\u2014that we've got to set Derby free?\"\n\nHe was on his feet. He had not been aware of getting up.\n\nWonder filled her face. \"Tell them he must be set free,\" she repeated. \"Have you had too many drinks, son? God in heaven, you're\u2014you're having a nervous fit, I think. Neill! How can you say such a thing\u2014how can you? A crook like Derby? Think of what you stand for, and what your father stood for.\" She waved to the large framed photograph on the table and Sergeant O'Neill Ryan in uniform stared at his son with expressionless, retouched eyes. \"He felt bad about the man he sent up too, Neill. But it didn't swerve him from his duty. He didn't flinch\u2014he prayed for strength. And he got it. He knew when it was time to force himself to be\u2014to be steel.\"\n\n\"But a man's life\u2014\"\n\n\"And what was your father's life?\" she asked. She was being steel too; the blue eyes were night-dark. \"What's the matter with you\u2014you're not that new at the business. Don't you have the courage to put a\u2014something like Derby where he belongs? A decent citizen must be treated with respect until he's proved guilty, anyone knows that. But a crook like Derby\u2014what's come over you?\"\n\nThe front door opened. Eleanor came in laughing, pink cheeked, followed by her date. \"Hi, ma. Hi, Neill. Come on in, Jerry.\"\n\nRyan rubbed his hand over his face and nodded to Jerry. \"I got your book,\" he said, and went into his room.\n\nHe undressed slowly. Was he having some kind of nervous fit? Was he crazy? After all, what was Derby to him\u2014or to the world? Nothing. Less than nothing. As Jablonski had said, Derby himself knew where he had been at the time of the Connors murder. If he didn't want to say, why should O'Neill Ryan interfere?\n\nBut it wasn't that, and he knew it wasn't that. Maybe Derby was better off to the world dead than alive. Maybe he deserved extinction, like some deadly reptile. But something else was involved here. It was the basic acceptance of right and order toward which mankind had been moving for aeons, the faith in a well-ordered society that had led Ryan, and his father before him, to make themselves society's instruments for preserving what was good and decent. Ryan could not put it clearly into words, but he felt it.\n\nThat was what was being violated. That was what had given Rosemary Derby's face its desolate look. If Harry Derby were executed Ryan knew he could never tolerate himself again, nor pleasurably accept praise or kindness from others, or even accept their company. Then what must he do?\n\nSuddenly he flung himself out of the bed and sliding to his knees beside it tried to pray. But he could not; what was there to say? That he was sorry for having made a mistake? That he would not do it again? That had made it right when he was a child. That had made everything right then.\n\nSuddenly, from deep inside himself he started to cry, silently, face pillowed against the sheeted mattress, his tears wetting it, shoulders twisting and belly wracked spasmodically with sobs. For a long time he wept, not trying to halt it, sensing that this was something that had to be gotten over with.\n\nThen he dried his face with a clean handkerchief and groping for it his hands grazed the pistol and holster on the bureau. That was another easy way out, he told himself contemptuously. But a thought struck like a bullet from the gun. If he were to be killed in the days to come, Harry Derby would go to the chair as surely as if Ryan himself had sentenced him.\n\nThere was one thing he could do. In the student's desk he had used as an undergraduate he found paper and pen. He wrote:\n\nTo whom it may concern:\n\nIn event of my unexpected death I want it to be known that when Ed Jablonski and I arrested Harry Derby we applied some dust made from a part of the lamp at the Connors murder to his jacket. Subsequently I learned that at the time of the murder Derby was holding up a drugstore on East Thirty-first Street, as a fingerprint obtained there and now in the file on the case will show. I swear before God this is all true.\n\nHe signed his name and shield number, put the letter in an envelope and addressed it to the judge who had tried Derby. Then he put the envelope in the little tin box in which he kept his insurance policy and his few other papers of importance.\n\nHe lay down again on the bed. He felt cold and exhausted and yet calm. He had the sense of something being about to happen, of his being about to go away, like on his last night at home before going to Uncle Frank's summer camp near Worcester as a kid. He was passing some kind of turning point. A new time was coming in.\n\nHe began to breathe normally, and as he did he began to think logically about the real killer of Mrs. Connors. That was still a way out, of sorts: if he could get the real killer then they would have both him and Derby. It had fleetingly crossed his mind before, but he had dismissed it as impractical. Now he faced it squarely and fully.\n\nThat would take some of the curse off as far as the department was concerned. And even Jablonski might be saved, he thought in this spell of new clarity, for there was always Farragut's explanation that the telltale dust had been transferred by accident during the fight.\n\nHe thought of the truculent, snarling man for whom he was considering undertaking this alone, without the usual help of the department, and he smiled bitterly. Still, it was an honest, satisfying bitterness, born of growing certainty of the direction he would take. Thinking on it, after a time he fell asleep.\n\nHe slept well.\nPART THREE: THE BLOW-OFF\nCHAPTER 19\n\nThe Things That Really Happen\n\nHe and Lambert had agreed to begin afternoon tours next day to spell another team, one of whose members had a daughter who was getting married. But Ryan dropped into the station house at twelve-thirty when he knew Bauer would be at lunch, telephoned downtown, and without explaining his purpose quietly set in motion the identification machinery that would give him the names and records of all known criminals in the metropolitan area who had two vital characteristics: a physical resemblance to Harry Derby and a working method of robbing women as Mrs. Connors had been robbed.\n\nThen, since it was still early, he walked to the apartment where the murder had occurred. The afternoon was bright and warm; north of the city snow was melting among Westchester's hills, wetting the sunny rocks and making rivulets for children to play in on their way home from school. But amid Manhattan's concrete and steel geometry there was no snow except where a taller building cast, in this brightness, a blue shadow across lower rooftops. Even so, you felt the catch of distant spring in your throat.\n\nMrs. Lombardi sat on the steps, a heavy shawl over her shoulders, a cigarette between her lips. \"How are you?\" she said.\n\nRyan asked about Mrs. Anders and Betty Leonard and learned that Mrs. Anders had since moved to Tulsa. Gloria Connors still had the apartment but was living with friends and was moving out the first of the month\u2014maybe sooner if she could arrange about the furniture. Meanwhile, it was being shown. Ryan asked if he could borrow the key and Mrs. Lombardi said she didn't know why not and took it off a large key ring she fished from her pocket.\n\n\"Thanks. Oh, one other thing I wanted to ask you. Remember you identified this guy Derby without any trouble? You picked his picture out right away, and then again that night at the precinct and so on?\"\n\n\"Sure I did.\"\n\n\"Well, there's one thing I'm curious about. Was there anything about that guy when you saw him, either at the precinct or later in court, that made you think he might not be the right one after all? I mean, maybe just one little point that might have made you wonder?\"\n\nA landlady's perpetual suspicion squinted at him through her half-closed eyes.\n\nRyan smiled easily. \"I guess it sounds nuts, now he's convicted and all that. But I'm making a\u2014a sort of study of witnesses' observation, Mrs. Lombardi. I was just wondering how the whole process of identification struck you.\"\n\n\"It struck me,\" she said, \"that you fellows did a pretty good job of locating him fast. When I looked at that picture there wasn't any doubt. And then when I saw him...\" Her forefinger skillfully flicked the cigarette butt into the street. \"But I'll tell you one thing. Maybe it was having seen the picture that did it. But I sort of got the feeling when I saw him in court that I'd seen him before\u2014I mean some place else. You know what I mean?\"\n\n\"Did you ever live near the piers? Or have anything to do with stevedores?\"\n\n\"No. My sister's husband's cousin is a steward on the Roma, though. We go down and visit with him when the ship's in sometimes.\"\n\nRyan shook his head. \"Did you and the other ladies talk about this afterwards\u2014Mrs. Anders and Miss Leonard, I mean?\"\n\n\"Well, you think something like that happens every day around here?\"\n\n\"I just meant, did either Mrs. Anders or Miss Leonard say anything that might suggest they weren't as sure about the identification as they might have been? In other words did they notice any single point that didn't quite correspond\u2014that they remembered later, as sort of out of key?\"\n\n\"Not that I know of.\"\n\n\"Did the man's voice sound the same in the courtroom? Did he wear his clothes the same way? Did the way he walked correspond with what you remembered?\"\n\nShe shook her head continuously to the questions.\n\n\"Or did any of the others say\u2014\"\n\n\"Listen, why don't you ask them? You're a detective.\"\n\n\"I will,\" said Ryan. \"Thanks, Mrs. Lombardi.\"\n\nHe went into the building. A girl he had never seen before answered the door of Betty Leonard's apartment and explained she was Betty's roommate. Ryan told who he was, and the girl said Betty was working days now and would be for the next couple months. Betty would be very sorry she missed him, the girl added, and Ryan recognized that he had been the subject of sororal discussions. He said he'd be sure to drop back a few evenings from now.\n\nWhen she had closed the door he let himself in the Connors apartment as quietly as he could, although the front door stuck a little at the bottom. This was the beginning, he told himself.\n\nIt hadn't really changed since he was last here. Someone had cleaned up the living room and straightened the chairs, and now it was simply a gloomy, overheated living room, furnished cheaply and in need of airing\u2014hardly the appropriate background for murder.\n\nRyan sat down in a chair, pushed his hat back on his head and lighted a cigarette. He could not really expect to find anything. The lab boys and the various squads who had been here were good, and at the time they had worked under the stimulating pressure of having to crack the case in a hurry. He could not easily imagine finding anything that had been overlooked, here or elsewhere. Yet he had to.\n\nWhen he had smoked a few minutes he rose and began examining everything in the room, touching things gingerly. He tried to imagine how it had happened, how the man would have pushed his way in and how the old woman would have backed up, crying anxious, unfinished questions. The invader would have shut the door. And then... He went over every physical action that could have happened in the handful of seconds in which it all did happen.\n\nThen he sat down again and looked around, breathing in smoke from the cigarette hanging between his lips. He thought of Ed Furtig and how Ed had sat here in this same chair, smoking and wondering. Then it had been Ed's to carry.\n\nRyan got up. This was getting nowhere.\n\nWell, he could always come back. He yanked at the door knob angrily and the door gave and then stuck. He remembered it sticking when he came in and he yanked harder. The door stuck at the bottom near the threshold. Ryan lifted on the knob and it opened. He went into the hall, slamming it behind\u2014Before it could slam he wheeled and held it open. Suddenly things were whirling in his head. There was a closet door at home that stuck often, especially in winter. To open it he always...\n\nRyan went back in and closed the door firmly, hearing its lower edge grate on the threshold. He was a murderer now and in his imagination in the next few seconds he had killed an old woman. He took a deep breath, then he grabbed the knob fast like a man in a hurry and yanked, and again the door partly opened and stuck at the bottom, an inch of it wedged tight. Ryan's hand went automatically up to the top corner that was free to pull on it as he lifted the knob, just as he did at home when the closet door stuck. Just, he thought, as any man would do naturally and unthinkingly. And especially one set on escape.\n\nHe stood looking at the door a long minute. He could not dare to hope that much. Yet thoughts of how doors stuck only at certain seasons and at certain temperatures and humidities thronged his mind; this was a warm day, possibly the warmest since Mrs. Connors died.\n\nRyan opened the door carefully, got a chair and stood on it to examine the upper corner with his small flashlight. On the inner surface there were only smudged streaks that could have been made by anything. And on the other side there was only dull green paint.\n\nThe vital part was the door's top edge. Where a man's fingertips would naturally grab and press. He raised up on his toes and pressed the flashlight's tiny button.\n\nIts beam revealed grimy, worn wood innocent of any mark.\n\nRyan put the chair away dispiritedly, too let down to care about anything. Somehow he had felt so sure, so intuitively certain...\n\nHe pulled his hat down over his eyes and yanked the door doubly hard. He almost ran into a woman standing before him on the threshold.\n\n\"Why, what are you doing?\" she demanded. Her sallow face went yellow around the cheek bones. She held a key.\n\nRyan said, \"Who are you?\"\n\nShe turned, frightened, seeking help.\n\n\"Wait a minute. Miss. I'm the police department.\"\n\nShe turned back.\n\n\"I'm Detective Ryan. I'm\u2014I'm just rechecking some of the evidence.\"\n\n\"But...I thought the man was all sentenced and everything.\"\n\n\"Oh, he is. But there's always a chance of an appeal at the last minute. We want to be ready for it. I got the landlady to let me in.\"\n\n\"I see. Well, you didn't find anything new, did you?\"\n\n\"That's right. But as long as you're here I'd like to ask a few questions.\"\n\nShe walked rapidly past the spot where her mother had lain and threw up the window as though fresh air would clean the room of its cruel memories. She said, \"Know anyone who wants to buy some bedroom furniture?\"\n\n\"Not at the moment, but I'll keep it in mind, Miss Connors. Tell me something. You saw this Harry Derby in court, eh?\"\n\n\"Yes.\"\n\n\"Did it occur to you that you had seen him before?\"\n\n\"No.\"\n\n\"Did he remind you of anyone?\"\n\n\"No. No, he was a complete stranger.\"\n\n\"There was just you and your mother in the family?\"\n\n\"And my brother Philip in Chicago.\"\n\n\"Miss Connors, this will strike you as an odd question, but did any of you have any enemies?\"\n\nShe shook her head mutely.\n\n\"Your mother especially\u2014she hadn't had any trouble with anyone shortly before she was attacked? Think back, please.\"\n\n\"Who could have had trouble with my ma?\" she asked sadly, and Ryan felt a pang of sorrow for her.\n\n\"How long had you lived here?\"\n\n\"Six years going on seven.\"\n\n\"No one ever broke in before\u2014into your apartment or somebody else's?\"\n\n\"No.\"\n\n\"After\u2014after the murder, you didn't hear from anyone, or find anything around the house that in any way might have had anything to do with what happened?\"\n\n\"No.\"\n\nIt was hopeless. At best these were the routine questions you asked as a last resort. Anyway he was simply going over ground that had been covered before when the trail was fresh and memories keener.\n\n\"Well...\" Ryan twirled his hat in his hand. \"Thanks, Miss Connors. If anything does occur to you I wish you would call me. At the Seventeenth Squad. Plaza 3-4483. The name is Ryan.\"\n\nShe got up too, but as Ryan moved toward the door she walked into a bedroom without answering. I suppose she feels pretty bad, he thought.\n\nThe door was about to close on him once again and finally, when he heard her say from inside, \"Wait a minute, detective. Maybe you can explain\u2014\" She was holding something in her hand, something that glittered.\n\n\"You may remember that the cops\u2014the officers, who were here at the time, found a cuff link near my\u2014near mother's body.\"\n\n\"I remember. It was one of hers.\"\n\n\"That's right. One of the officers said that the other one might have been carried away by the murderer\u2014that it might have caught in his clothes or something.\"\n\n\"That's possible. But it never turned up, Miss Connors.\"\n\n\"That always puzzled me a little. You see, Phil had given those links to ma with a shirtwaist. It was a birthday present. She had packed the shirtwaist and I always figured she wanted to wear it with the links when we visited him\u2014you know, to show she liked them. I thought she probably had the links in her hand when that Derby came in. And then in the struggle\u2014well, you know.\"\n\n\"That's probably what happened.\"\n\n\"No,\" she said. \"I don't think it is. Look.\" She held up something for him to see. It was a cuff link. \"See this?\"\n\n\"Sure. That's what was returned to you after the trial, eh?\"\n\n\"That's right. But look at this.\"\n\nShe spoke tartly, as though she expected him to understand something he clearly did not understand. She held out her other hand, thin and weedy, clenched tight. Then she opened it.\n\nHe saw two links in one hand\u2014and one still held in the other. All three were of the same little silver scimitar design.\n\n\"I found these two in ma's jewel box a week ago,\" she said. \"Just by accident. What do you think of that, detective?\" It was a triumph for a vinegarish woman of lean, dark-browed homeliness, happy to discomfit a member of the sex that had long refused her.\n\nWhat the hell! Ryan thought. \"Miss Connors, are you sure you found those two links where they ought to be?\"\n\n\"I told you I did. Ma had never gotten them out that day at all.\"\n\n\"But why didn't you ever find them before?\"\n\n\"I simply never went through her things. I\u2014I didn't want to stay here and a friend of mine in Jersey asked me to stay with her. It was only the other night when I came by to\u2014to sort of get ready to close up the apartment that I found this. But it doesn't make any sense, does it?\"\n\n\"No.\"\n\nBut Ryan was paying no attention to the explanation. \"Look. You're sure this is the cuff link you got back from us? You didn't mix the three of them up?\"\n\n\"Yes. Of course I'm sure.\"\n\nShe was an insurance clerk, probably good on details. She could be sure of something like that.\n\n\"And can you be sure that this set of cuff links didn't contain an extra one\u2014a spare, so to speak?\"\n\n\"Of course. I remember when she opened the package on her birthday. Anyway, did you ever hear of a spare cuff link?\"\n\nRyan was wordless. He looked at the three links. God! he thought again. Things like this didn't really happen. And yet they did. He knew they did. That is how things really happened. The most unlikely, farfetched, one-in-a-million chance, that is what happened, even though you could not believe it when you saw it face-to-face.\n\n\"It doesn't make any sense, does it?\" she asked again with acid triumph.\n\n\"Maybe it does,\" said Ryan. \"May I borrow these, Miss Connors?\"\n\n\"Are you sure I'll get them back?\"\nCHAPTER 20\n\nThe Scarlatti Sonata\n\nIt was still too early to show up for work and Derby's home lay only a dozen blocks north and east. Why not? He had something to do there. He would not try now to determine what the cuff link might mean, although he sensed the possibilities. And Rosemary Derby might be able to help there. Ryan walked uptown, energetic and buoyant.\n\nIt was an old, dingy apartment building\u2014but an apartment, not a cold-water flat as he had thought. He rang the bell under a hand-lettered Derby and the door buzzed without any challenge from the speaking tube.\n\nShe stood at a third-floor door, slim in black slacks and black sweater and some sort of thin-soled slippers that made her seem smaller and more girlish. She did not see him clearly until he came abreast of the door which streamed sunlight into the dark hall. She said, \"Oh.\" Her hair, tied with a wisp of ribbon, looked more tawnily blond than it had last night. Her level-browed face was calm.\n\n\"Can I come in a minute?\"\n\n\"Why not?\"\n\nShe stood aside and he walked down a short hall into a small living room that seemed to contain a lot of books, on tables and in painted wooden bookcases. On one wall were several reproductions of paintings which he suspected were by Picasso, one of the few names in modern art he knew. An odd pattern of tinkling music that sounded like a guitar came from a table phonograph.\n\n\"Sit down,\" she said noncommittally, and turned the phonograph to a whisper but did not turn it off. Ryan felt like an intruder.\n\nShe sat down on a couch and extended slender black legs before her. Ryan took out his cigarette package and offered it and she shook her head.\n\n\"Mind if I do?\"\n\n\"No.\"\n\nThese were the stiff formalities before a duel.\n\n\"Ken's not here?\"\n\n\"He's at work.\"\n\n\"There are a few things I didn't get a chance to say last night.\"\n\nShe waited, not making it easier.\n\n\"My sister is a secretary in an ad agency. They always need secretaries. I could talk to her about a job\u2014a job for you. Can you do typing and shorthand?\"\n\n\"Theoretically. But my shorthand's rusty. Anyway, I'm not interested.\"\n\n\"I thought you needed a job.\"\n\n\"Perhaps you don't understand the difference, Mr. Ryan, betwixt\u2014between library work and stenography.\" She bit her lip and Ryan could not know it was because his unexpected appearance had trapped her into using an archaic word.\n\n\"Maybe I don't. But this job would pay seventy-five bucks a week. Is that so far under a librarian's pay?\"\n\nShe did not say anything. To Ryan she looked clean and well scrubbed\u2014not pretty in any glamorous sense, but level-headed, willing to accept things on their own terms. It was a quality hard to define, but he liked it in a girl. The music that Scarlatti had written three centuries before stopped momentarily and then started again, an intricate twanging in the mid-afternoon quiet.\n\n\"Why this?\"\n\n\"Why what?\"\n\n\"Is the police department running an employment agency?\"\n\nRyan pulled slowly on his cigarette and held the smoke in his lungs, extended the cigarette to an ashtray on the little table before him and very carefully dumped the ash, making a gesture of it. He did not want to get mad, or even show irritation. What he was about to say had been on his mind since last night, yet he had never analyzed it or arranged it logically. He just wanted to say it.\n\nHe leaned back and exhaled the smoke. When it was gone, \"I owe you and your family a debt,\" he said. \"I plan to repay it.\"\n\nHer calm gaze was a triumphant challenge.\n\n\"You were right,\" said Ryan. \"Last night. We framed your brother.\"\n\nIt was out. She was the first person he had ever said it to. He had a wild sense of relief.\n\n\"I knew you did.\" She reached to a box and took out a cigarette. \"I could tell it last night.\" She flicked a lighter at the cigarette.\n\n\"Don't misunderstand me,\" said Ryan. \"Your brother's a no-good bastard.\" He liked using the word to her. \"And if he died it wouldn't hurt anyone in the world.\" He stopped and looked at the smoke spiraling up evenly from his cigarette. That was how it was\u2014justice, evenness, balance. Things had to go straight up and down, and no other way. He went on.\n\n\"He's a rat. But that doesn't mean he should suffer for something he didn't do.\"\n\nShe looked at the baseboard across the room. Then she talked through thick smoke.\n\n\"And taking care of his sister and getting her another job will fix everything? Make everything even?\"\n\nThat did something to Ryan and he had to knock the ashes off again, this time impatiently.\n\n\"That's not the idea. I said that he shouldn't suffer for something he didn't do. I don't think he should. And he won't.\"\n\n\"What are you going to do about it?\"\n\n\"I've already done one thing. I've written a letter that will be found immediately in case anything happens to me. It will free Harry.\"\n\n\"And in the meantime?\"\n\n\"You'll have to leave that to me.\" He got up. \"I'm not going to let him get the chair, I promise you that. I hope to vindicate him. By bringing in the guy who really committed the murder.\" He laughed, masochistically. \"Probably the chances of that are not good. But that's what I'm going to try to do. Anyway\u2014\" he spun the light fedora in his hand\u2014\"there's one thing I can guarantee, Miss Derby. Your brother won't burn. I'll see to that, no matter what else happens. But there are other people involved here, and other people's rights. I'm going to protect those too. So for the present Harry stays in Sing Sing. He belongs there, you know.\"\n\nShe was looking at him steadily and, the sun outside being temporarily darkened by a cloud, her cigarette made a firefly glow in the twilit room.\n\n\"In the meantime,\" Ryan said, \"well, you've suffered on account of this, losing your job. There was something I thought I could do that might help and so I mentioned it.\" He pulled on his hat. \"But forget it. If you have something else to do, fine.\" He had said what he had come to say. The heck with her.\n\nHe turned and remembered something. \"One other thing. What I've been saying is between ourselves. I really didn't have to say it. I thought you might feel better if you knew. But if you tell anyone else what I said just now, I'll swear you're a liar and I never said it. You couldn't get anyone to believe it anyway. Take it or leave it, Miss Derby. That goes for the job offer, too.\"\n\nHe started down the hall, and she said, \"Oh now, please!\" but so quietly he could not believe it had reached him. Yet when he opened the door she was beside him. \"Please don't go like that.\" The size of her eyes and the solemnity of her mouth surprised him.\n\n\"I've got to get to work.\"\n\n\"But you\u2014please take a moment. I want to ask you\u2014\" A shrill whistle came from somewhere. \"Oh! I was making tea. Every day at the library we had tea. Please\u2014\"\n\nHe allowed himself to be led back and when the tea had properly steeped during a few minutes of strained politeness he found himself sitting with a cup and a little cookie, undergoing thoughtful inspection. He sipped and readied himself.\n\n\"You're an honest man,\" she said with crashing frankness, and Ryan flushed.\n\n\"An honest cop,\" she reflected.\n\nThat graveled him. But he remembered the principles that the department had ingrained in him: you could not do your job effectively if you treated the public as enemies. You had to make friends of them, even those who regarded you as an enemy. This level-eyed girl was a part of the public.\n\n\"I think you're too well educated to believe that policemen are generally dishonest, Miss Derby.\"\n\n\"I don't say that. I'm sure some\u2014most are\u2014are honest. You are, certainly.\"\n\nBut her skepticism was an acid eating into his mind, filling it with cynicism. This was the public for which you worked late hours for small pay, and took chances, this dark-blond, small-breasted dame with the tight lips and critical air. He owed her something of course, but he was paying it off\u2014suddenly it burst out.\n\n\"Where the hell do people like you get off? You and your lousy brother? And your old man! You're a great family to be squealing about cops! The Derbys! The lousy, crooked, mugging Derbys!\"\n\nHe paused, surprised at himself. But he went on in a vibrant tone that was even more emotional.\n\n\"Let me tell you something, Miss Derby. One of the greatest things that could be done to make this country more\u2014more law abiding, and a better place for decent people, would be to make a rule that everyone\u2014everyone\u2014at some point in his life should put on a police uniform and walk the streets for eight hours.\"\n\n\"And what would that accomplish, aside from giving him a little taste of power?\"\n\n\"I'll tell you. It would make people realize what it is to be a cop. Because the instant you step out in that uniform you realize that whatever happens that's rough or unpleasant or dangerous along your beat in the next eight hours, you have to settle it. You're the one everyone will turn to. If some donkey comes home drunk and starts beating up his wife, you're the guy who has to walk up and cool him off. If some hoodlum tries to stick up the corner liquor store, that blue uniform of yours is the first thing he's looking for when he comes running out. If some teen-age kids get goofed on tea and go hunting trouble, you're their best target, because you're a cop. They won't go up against you openly. The first thing you know is a scuffle behind and then a baseball bat across the skull. That's how hoodlums begin. Later of course they graduate to be big shots\u2014like your brother Harry.\" He looked bold hatred at her.\n\n\"Try directing traffic, Miss Derby, and discover how some truck drivers like to see how close they can come to you. Or there's the so-called honest citizen, who parks overtime and flies into a rage if you give him a summons, and threatens to get your job. Cops are human beings, Miss Derby, and sometimes we wish the people we serve would at least give us a minimum of cooperation, instead of thinking we're all bribe-takers or sadists or boneheads.\"\n\nShe had watched him throughout what he said and Ryan stared back, bitter and angry and waiting. He was not prepared for what she did say.\n\n\"I guess I had that coming. I'm sorry, Mr... Mr. Ryan. Sometimes when...well, when a girl has two brothers like Harry and Ken...what I want to say is this. If you mean what you said, and I know you did, I'd like to help you. And I know Ken will, when I tell him what you told me. Not all Derbys are\u2014well, antisocial.\"\n\n\"I told you that if you mentioned this\u2014\"\n\n\"I understand. I completely understand, and you needn't worry. You can trust me, and you can trust Ken. He'll want to help, because he is fond of Harry. Do you know, on the morning of the day Harry was arrested they quarreled? It was about that jacket of Harry's that Ken wanted to borrow because his uniform was at the cleaner's. Harry wouldn't let him. Ken called him cheap or tight\u2014you know. Then, when Harry was arrested and charged with that murder Kenny sat where you are and cried like a baby because of the quarrel. But tell me this. How did you\u2014why did you come to frame Harry?\"\n\nHow far should he go? How much dare he trust her? But if he did not trust anyone, how would he ever get at what he had to get at?\n\n\"It wasn't exactly my idea. Not that I'm not as much to blame as anyone.\" He looked at his wrist watch. \"I'd like to tell you about it, Miss Derby\u2014\"\n\n\"My name's Rosemary.\"\n\n\"Okay, Rosemary. I'd like to tell you about it.\" It was true. He suddenly realized how much he wanted to tell every detail of how and why it happened. \"But I'm due at work. Maybe I could come back sometime.\"\n\n\"Tomorrow afternoon?\"\n\n\"I'll try to.\"\n\n\"And\u2014Mr. Ryan?\"\n\n\"People call me Neill.\"\n\nShe smiled shyly. \"Neill. I think it's wonderful, what you're doing. Not because it's my brother. Because you have honesty and decency, and it's nice to know there are people like you around. I'm sorry about what I said about the police.\"\n\n\"Forget it,\" said Ryan. But he couldn't help adding, \"We hear it all the time.\"\n\n* * * *\n\nWhen he and Lee Lambert came in that night, a Manila envelope was in his mail slot near the stairs. It contained copies of the police records of four men, all of whom from their description resembled Harry Derby, were at liberty and were known to have robbed women in the past. Two usually operated in and around Brooklyn and one far up in the Bronx. But one, a man named James Mackie and known as \"Big Mackie,\" ran a small Turkish bath and athletic club in the east fifties and lived over it. He was a former prize fighter, he had been arrested a half dozen times in the past decade although not recently and he had twice been convicted of crimes against women.\n\nRyan studied the photograph. Mackie looked darker than Derby and had bushier eyebrows. But his dimensions were the same, and the address made it look good. If it turned out that Mackie had some connection with Harry Derby that would explain Derby's possession of the bill...\n\nA note had been clipped to the records. It was from the IB man who had compiled the information. It said, \"For your information this same description was given one of the boys here for checking a couple days ago by Jack Sandalwood, the reporter.\"\nCHAPTER 21\n\nThere's a Story in It\n\nNext morning rain, thick and gray as a twinkling theater curtain, made a steady beat on cars and pavement; it would last all day. Ryan took off his raincoat and sat down to doughnuts and coffee before the restaurant's lighted windows. He unfurled the Herald Tribune, turned to Red Smith's column and sipped black, sweetened coffee. It was in between times, late for the crowd that got to work at eight-thirty, but early for the nine o'clockers. You felt those things when you had worked all hours long enough. Ryan bit into crisp doughnuts and savored the wry humor of good sports writing. For a while he forgot where he was and what lay ahead of him, and how little time he had.\n\nBut when the coffee was finished and the sports page read, he paid his check, stood at the restaurant door yanking his raincoat tight against the weather and thought of Gee Gee. She'd be in bed at this hour, sound asleep, her body sweetly scenting the warm bedclothes... Ryan went out into the rain. Forget it. There was work to do.\n\nMrs. Daniels was in a faded housecoat when he knocked at her door. She said, \"Oh, it's you?\" apprehensively.\n\n\"Anyone in that room where we arrested Derby?\"\n\n\"No. It's empty.\"\n\n\"I'd like to look around it, if you don't mind.\"\n\n\"Help yourself. The door's open.\" She was waiting for him to say something else.\n\n\"And I'll take ten bucks, if it's convenient.\"\n\nShe sighed\u2014of course he wouldn't forget that.\n\n\"Can I give you five now and send you the rest on Saturday? Honest, most of my roomers have been slow this week. And I never got the other ten back, you know.\"\n\n\"Sure,\" said Ryan. It was as good as you could expect. She'd never have sent the money voluntarily, of course.\n\nHe mounted the stairs they had crept up so cautiously that night. No need for silence now. Even so he walked quietly. He had a surreptitious feeling about what he was doing. There were a lot of people he would not want to meet here, nor would want to know that he was still concerned with the place where Harry Derby was arrested. Lieutenant Bauer, for example, or anyone from the precinct station.\n\nHe pushed the door open, took off his coat, draped it over a chair and sat down. Rain whispered fitfully at the windows; otherwise the old house was gloomily quiet.\n\nNothing had changed since the last time; he almost expected to see Derby's long back against the wall, arms spread-eagled upward. Ryan tried to remember exactly what Derby had looked like and exactly what he had said. That was the only purpose of this visit, if it had one: to strain for every possible hint he could get. For Derby must have had some guilty knowledge of the murder; otherwise how had he come to possess the hundred dollar bill?\n\nHe got up and walked around the bare and chilly chamber, remembering where he had stood, and Jablonski's chagrin, and Derby's taunts... He took his place at the wall where Derby had stood, put his hands up against it like a prisoner. Then he turned and moved forward into the room, imagining for a moment he was confronting two cops...\n\n\"What the devil are you up to? Re-enacting the crime?\"\n\nEven in his first start of guilty fear Ryan recognized the voice.\n\nA man stood in the doorway. Under his open trenchcoat a gray tweed jacket was visible. His confident, good-looking face wore a mocking smile.\n\nIt was Sandalwood.\n\nRyan stared, then tried to pull himself together. Clinch\u2014play for time.\n\n\"Where did you come from?\" he asked with intentional na\u00efvet\u00e9.\n\nSandalwood grinned and came in.\n\n\"Supposing I tell you I've been following you for a while?\" he said. \"What are you up to, Ryan? Derby's up the river\u2014you're certainly not still building a case against him. Do you just like this place?\"\n\nAnd when Ryan remained silent, he drawled, \"Or are you looking for something\u2014say a hundred dollar bill that got lost one night?\"\n\nRyan cursed him and Sandalwood laughed. He was trying to overpower Ryan with his confidence and his reputation and his tactical position, and he was doing it.\n\n\"Yesterday you dropped in on the Derbys. What's the attraction there\u2014the sister?\" Sandalwood's eyes bugged out impudently; he was trying to lure Ryan into rage.\n\n\"And now you're here. Why don't you get it off your chest? I can give you a break, you know, if you cooperate. Let me tell you something, Ryan. Give it to me straight and then I'll tell you whether you are telling the truth or not. Want to know something? I always figured you guys were up to something that night. So before Derby went up the river I talked to him. And he leveled. I've known Derby a long time. He told me pretty much what it was all about.\"\n\nWas he lying? Was it a bluff, intended to trap him into admitting something? Or did Sandalwood know? He clearly had been following him.\n\nRyan had to make a move. He seized on the first inspiration that came.\n\n\"Okay,\" he said steadily. \"I'll tell you the truth. Not many people know this. But Harry Derby is the greatest slide trombone player you've ever heard. He really is.\" Sandalwood's jaw sucked turtle-like into his neck.\n\n\"His tone is clear as a bell,\" Ryan went on. \"What Jablonski and I were trying to do that night was to recruit him into a little jazz band we're forming.\"\n\nSandalwood's face darkened. Ryan knew he had turned the tables. \"You know how it is,\" he went on. \"There's just never enough good trombone players.\"\n\nAnger was making Sandalwood's eyes kindle.\n\nRyan grinned. \"What's the matter? Aren't I telling you the truth? Suppose you tell me.\"\n\nSlowly, visibly, Sandalwood swallowed his rage. It took time. But he was too skillful to let himself be twitted into further error. He had attempted a well-timed bluff, but Ryan had called it.\n\n\"Okay,\" he said. \"You can't blame me for trying. I played a hunch, that's all. But on the level, Ryan\u2014what the hell are you doing?\"\n\n\"Just call downtown to the deputy commissioner,\" said Ryan, \"and get permission. Then I'll be glad to give you an interview.\"\n\nCarelessly Sandalwood threw a porkpie rain hat over an unruffled brow. \"Okay.\"\n\nHe went to the door. \"I was bluffing and you called it. I haven't talked to Derby. But I will now. Know why? Because now I know I'm right. Something went on here that night\u2014I can feel it. Maybe it wasn't a hundred dollar bill. Maybe it was something else. But whatever it was, there's a story in it, and I'm going to get it.\"\n\nOver his shoulder Sandalwood smiled deliberately. He did not feel like smiling, and he knew that Ryan knew that. Then he paused, going out, and the smile was replaced by calculated malice.\n\n\"By the way, seen Gee Gee Hawes lately?\"\n\nRyan could not think of an adequate reply that would not involve the swinging of fists.\n\nWhen Sandalwood had ambled down the stairs whistling with studied loudness, Ryan lighted a cigarette and noticed his hands were trembly. He inhaled deeply. Sandalwood had been following him! It couldn't have been for long, and it was probably the result of luck more than intent\u2014maybe Sandalwood had happened to see him on the street yesterday and had tagged along hopefully. Ryan was certain he would have sensed it if anyone had been following him for any length of time.\n\nEven so, the thought was alarming, and Sandalwood's threat was more than alarming. He crushed out the cigarette in a rush of desperate helplessness and jogged down the stairs. Twice on the way home he checked to make sure no one was behind him. He had not intended returning home, but in this new uneasiness he wanted to be doubly certain no one could overhear the telephone calls he would make.\n\nThe house was still. Ryan heated coffee, poured a cup, lighted another cigarette and then dialed the number of a stool to whom Lee Lambert had introduced him. \"He's a barber over on Third,\" Lambert had explained afterward. \"Slip him a fin occasionally but never go in the shop. Though you can call him there.\" When the barber whose name was Connie answered, Ryan told him what he wanted to know\u2014everything he could get about Big Mackie, and fast.\n\n\"It won't be much. That party's out of town.\"\n\n\"What do you mean? When did he go?\"\n\n\"Oh, some weeks. Quite a few weeks,\" said the barber in the cryptic locution of one who is being overheard.\n\n\"I want to know when he left town,\" said Ryan. \"The exact date. And where he is now. Check it hard.\"\n\n\"Okay. Sure, Mr. Johnson. I'll call you in a few days.\"\n\n\"Call by tonight. I said I needed it fast. Call my home and leave a message\u2014here's the number. Make damned sure I hear from you. This is a big one.\"\n\n\"Well...okay. You gonna drop around for that scalp treatment?\"\n\n\"Let me hear from you and you'll hear from me.\"\n\nHe gulped coffee and dialed a Brooklyn precinct where a sergeant who had helped break him in as a rookie was still stationed. He was working on a little hunch of his own, he lied glibly, involving a Manhattan job some time back that might have been done by either of two Brooklyn muggers with records. What could Oley tell him of either?\n\n\"Ferris you can forget about,\" Oley said promptly. \"He ran into someone a little tougher than he is last summer in a bar and got badly mauled. He lost one eye and the other's just about gone I understand. Anyway, he was in the hospital all fall, and now he has to be led around. Who's your other guy\u2014the Drummer Boy?\"\n\n\"Vince Van Loan. Thirty-one. Five feet\u2014\"\n\n\"Sure. Drummer Boy Van Loan. He's always been nuts about playing snare drums. I think you can write him off too, Neill, though I'll check further if you like.\"\n\n\"How do you mean?\"\n\n\"My missus was telling me\u2014she went to school with the Van Loan girls\u2014that last summer Vince met some broad who sang with a little band out Nassau County way. Well, this singer gets Vince a job drumming with the band and damned if he's not going straight. He's nuts about drums and he likes the dame, and I guess it's working out. I'll ask Mary about it again tonight, but I think you can forget him. Got any others over our way?\"\n\n\"That's all, Oley. Thanks a lot.\"\n\nHe hung up the phone thoughtfully. The Bronx guy would have to wait until he could sound out Lambert on their contacts up there. Meanwhile, pending word from Connie what else could he do?\n\nHe looked at his watch\u2014twelve-thirty. He was hungry, and he'd be getting to work before long. But something Sandalwood said came back and he dialed another number, and as he did his tight mouth relaxed in a smile for the first time that day.\n\nA sleep-husky voice said, \"Hullo?\" and Ryan, instantly contrite, said, \"If I woke you up, I'll kill myself.\"\n\n\"You didn't wake me up,\" Gee Gee said. \"Hello, Neill. Good morning,\" and he could tell from the way she sounded that she was smiling at the recognition of his voice. Ryan took a deep breath and forgot Derby.\n\n\"I haven't seen you for a long time\u2014six days, to be exact. I was wondering if you'd like to have brunch or something.\"\n\n\"I think that would be lovely. Just give me time for a shower.\"\n\n\"I did wake you up.\"\n\n\"Oh, now stop! But wait\u2014is it raining?\"\n\n\"Cats and dogs. So what?\"\n\n\"Then why don't you come here for lunch? I blew myself to a coat\u2014a real fur coat. Neill, wait'll you see it! I can't wear that lovely thing out in the rain.\"\n\n\"You could wear your old one.\"\n\n\"With a new one? Never! You come here. I'll make you plintzen.\"\n\n\"What is plintzen?\"\n\n\"They're a German pancake like cr\u00eapes suzette. But simpler.\"\n\nWaiting on the subway platform Ryan inspected himself in a gum machine's speckled mirror. Under his eyes were blue shadows. He didn't like that. Weariness shamed him. He hoped she wouldn't notice.\n\n* * * *\n\n\"I don't know why I'm so good to you,\" she said as he reached the top of the stairs. \"Cooking lunch and everything.\" She was wearing a small apron over a simple blue skirt and sweater. He had never seen her dressed so plainly.\n\nRyan took both sweatered arms and kissed her warm, unresisting lips. He had been kissing her since the third date. \"Now what have I done?\"\n\n\"Where have you been?\" But he knew he was welcome. \"I feel neglected\u2014and I have a right to feel neglected.\" She took his wet hat and raincoat.\n\n\"I'm sorry. But you knew I was on the day tour last week. And when I work days and you work nights what can you expect? Besides, something came up.\"\n\n\"What came up?\"\n\n\"I can't go into it, Gee.\"\n\nShe was tempted to say, \"I know what came up\u2014something with baby blue eyes and blond hair,\" but she gave him a look of quick inspection and was glad she had not. Something was wrong, obviously. Still\u2014he had neglected her!\n\n\"Oh! You wait a minute.\"\n\nShe ran out of the room. \"Close your eyes,\" she called. He closed them. Something sweet-smelling was cooking in the kitchen. \"Now you can open them,\" she said from close at hand.\n\nShe stood before him swathed in a coat of dark, long, glossy fur. The big collar stood luxuriously up around her smiling, sparkling face, and its close-held skirt hugged her thighs. Ryan thought of mink and sable and some of the other furs that wealthy dames were always being robbed of. He was quite sure it was not one of those, but it was pretty close.\n\n\"Like it?\"\n\n\"Sure. It's\u2014it's terrific.\"\n\nShe turned, modeling it. It was hard to tell whether the coat made her look better, or whether she made the coat look as opulent and chic as it did, but whichever it was the effect was overwhelming. She had always looked too pretty or too...too big-time for him, he thought in sudden dispirit. A guy with a lot of flash, like Sandalwood, was more her style. Ryan would have been stunned to learn that Gee Gee felt hurt because he had not called her, and because he now was not being as enthusiastic as she had hoped.\n\nShe slipped out of the coat. \"That's all of the fashion show,\" she said lightly; \"I just thought you might want to see it. It's\u2014it's the first good coat I've ever owned.\"\n\n\"It's great.\"\n\nAt the door she turned, holding the coat, and said. \"You don't know what it's like, growing up in neighborhoods like this, wanting a fur coat\u2014and finally getting it.\"\n\nShe went out.\n\nYou'll never get a coat like that on a cop's salary, cookie. But why assume she would ever be dependent on a cop's salary? Still, the idea, and the day's rain, and his anxiety and weariness all helped make him bitter and defeated. When Gee Gee reappeared and said, \"Oh, it's silly, but you know in show business you have to put up a front, and a coat like that helps,\" it did not help at all because Ryan innately disliked people who put up fronts.\n\nWhen she said, \"Will you excuse me a minute?\" he said, \"Sure,\" and left alone, had a feeling of guilt, of having been mean without knowing precisely how, and certainly not understanding why.\n\nThe plintzen were warm, tender egg pancakes with sugar and cinnamon rolled inside them. The coffee was fresh and fragrant. As they ate, she said, \"That reminds me. Will you be through at midnight?\"\n\n\"Far as I know. Why?\"\n\n\"Because you're invited to a party.\"\n\n\"A party?\"\n\n\"At the club. This is Max's birthday, and after the last show tonight he's giving a party for the whole staff and their husbands and boyfriends and so on. I thought if you didn't have anything better to do\u2014\"\n\n\"Sounds wonderful.\" Then he remembered he'd have to be up early tomorrow morning. There was the guy in the Bronx to work on. And Betty Leonard to see. And the cuff link distributor to locate and question. \"Gee whiz. I don't know.\"\n\n\"Don't know what?\" Very casually.\n\n\"Whether I can make the party. I don't think I can.\"\n\n\"Another girl, Neill?\" Quietly.\n\n\"Don't be silly. It's\u2014it's business:\"\n\n\"Police business?\"\n\n\"Well. In a way. I'm sorry, Gee.\"\n\n\"It's perfectly all right. Don't give it another thought.\"\n\n\"Now don't be like that. I honestly can't help it.\"\n\n\"Of course. I think I can find someone else to invite.\"\n\nRyan looked at the table, at its cotton cloth and old-fashioned cruet set, at his plate that was bare except for a few sugar crumbs, and his freshly filled coffee cup. He felt helpless and tortured. He didn't like this; she was getting away from him somehow. Not that she had ever really been his. Yet he was losing her.\n\n\"Gee Gee?\"\n\nNow it was she who was looking at the cruet set.\n\n\"Maybe I better tell you something.\" But dared he? No. She was a friend of Sandalwood. He said, \"I'm in a little jam at the moment. Not too serious. But I've got to work it out. It's\u2014it's about Harry Derby.\"\n\n\"Derby? Why, he's all put away and done with.\"\n\nRyan frowned over the cigarette he was lighting. \"Well...yes. But still...I can't tell you about it, Gee.\"\n\nBut somehow that relieved her. If it was Derby then it wasn't another girl. Things added up, in a way.\n\n\"Neill, I remember that night we met and Ed Jablonski got sort of high and began talking about how you and he had fixed Derby. Is it that?\"\n\n\"I said, I can't tell you about it.\"\n\n\"I remember Ed saying that you two had made a case or something against Derby that no one could break. Is that what's gone wrong?\"\n\n\"Oh, for God's sake!\" The torture suddenly strained his voice. \"Listen, forget anything that\u2014that drunken idiot said that night. You hear me? Forget it! And don't repeat it!\"\n\nIt was the violence more than what he said that hurt her. And his refusal to confide in her. It made her think of overheard parental quarrels. Why did things always have to go like this with someone you really liked?\n\n\"So anyway, I can't go to your party,\" he went on. \"You can get somebody else, all right. How about your friend, Sandalwood?\"\n\nShe was silent.\n\n\"Seen him lately?\" He spoke mildly, trying to bring the conversation and himself back to normal.\n\n\"No.\" She shook out and lit her own cigarette. \"He's been busy, too. He's writing a new series of articles.\"\n\n\"Oh?\"\n\n\"Something about famous miscarriages of justice.\"\n\nThere it was, once more. He would never escape it.\n\n\"Oh, fine.\" He added, \"Maybe I'll get in the papers again,\" before he thought.\n\n\"What?\"\n\nHe shouldn't have said that. \"Do you like him. Gee?\"\n\n\"Not especially. But he's not too bad. And he's press, you know. That's important in show business.\"\n\n\"Another part of the front to keep up?\"\n\nShe flushed. \"So what? I want to get somewhere in my business, just as you do in yours. Soon as that coat's paid for, I'm starting dramatic lessons with one of the best coaches on Broadway.\"\n\nBut the coat came first, he thought bitterly. He got up, knowing that somehow he had been cutting his own throat, and feeling helpless about how it might have been avoided. \"The pancakes were wonderful.\"\n\n\"Glad you liked them.\" From a cool distance.\n\n\"I really would like to be there tonight.\"\n\n\"Well, come if you can make it.\" The distance did not diminish.\n\nHe knew he wouldn't get there, and he knew that a breach of unfriendliness was deepening between them. Walking toward Eighth Avenue he felt the rain's thousand prods on his damp shoulders. They did not make him feel worse.\n\nIn the apartment that he had left the telephone rang and Gee Gee, busy clearing the table, stopped to answer it.\n\n\"Baby?\" said a sleepy voice. \"This is Jack. Look. I've been working my head off all week on that lousy series. I feel like coming up for air. How about a long lunch at the Harwyn?\"\n\nWell, why not? What did she owe Neill Ryan? And when would she see him again? And when did she want to?\n\n\"Sounds lovely,\" she said into the phone.\n\nEven so, she felt a little guilty about putting the film-like plastic rain cape over her new coat. And going out in it.\n\n* * * *\n\n\"On a day like this?\" said Sandalwood. \"Of course you must have another. I'm going to.\" He signaled to the waiter standing at alert attention just out of earshot. \"Two more Martinis on the rocks. We'll look at a menu after that.\"\n\nWhy not? Gee Gee thought. What was there to do until dinner? And maybe she could learn something for Neill. Instantly she asked herself: why should she bother?\n\n\"But you have work to do. My day doesn't begin until evening.\"\n\n\"I beat that typewriter for four hours this morning,\" said Sandalwood. \"That's enough for one day. From here in I coast.\" He grinned lazily at her.\n\n\"How's the series coming?\"\n\n\"Oh, all right. I could write it in my sleep. They are old cases that everyone should remember but doesn't\u2014I hope. People who were executed and later proven innocent, guys who served twenty years and then someone else confessed to the crime. You know.\"\n\nThe waiter put down two enormous glasses.\n\n\"They are all old cases? No\u2014no recent ones?\" She tried to speak inconsequentially.\n\nSandalwood, sprawled in his chair, liked this expensive place and the gin's warm comfort, and the looks of the girl across from him. He felt luxurious and relaxed. But for years he had made his living by catching nuances in what people said to him, by daring to guess at what they meant and by probing imaginatively for more. It was a kind of intellectual short-circuiting and was one of the reasons Sandalwood was a great reporter. Now he had caught something.\n\n\"Oh, there'll be a few modern ones,\" he said, although it wasn't true. The Derby case was something apart, a private interest of his own.\n\nHe was too old a hand to look up and watch how Gee Gee took that. Instead he studied his new drink and added significantly, \"There'll be one or two quite recent ones, in fact,\" and saw her fingers tighten on her glass.\n\nSuddenly she was afraid of him\u2014for Ryan's sake. He was too wise, too able, too knowing.\n\nAnd Sandalwood had caught her alarm. But what was she scared of, he asked himself. What was on her mind? He rehearsed the calendar of recent crime; there hadn't been anything of importance involving nightclubs or entertainers. But she was a friend of Neill Ryan.\n\nSandalwood raised his glass to her, said \"Here's to us,\" and looked at her over the rim. \"One of the recent cases in the series will be the Harry Derby case,\" he said, watching.\n\nGee Gee raised her glass hurriedly. \"The Derby case?\"\n\n\"Sure.\"\n\nThat was it. It was cat-and-mouse now.\n\n\"But\u2014Derby was guilty. He's been convicted.\"\n\n\"Oh sure.\" He smiled superiorly.\n\n\"You mean\u2014what do you mean? There's more to it?\"\n\n\"I'll say there is.\" And he was sure he was right, although he did not know where this was leading him. But he was on the right track.\n\n\"And you know all about it?\"\n\nSandalwood gave no sign that he appreciated the emphasis on that \"you.\"\n\n\"I'm a reporter, cutie.\"\n\n\"Well, then,\" she said helplessly, \"you know.\"\n\nWhat would this do to Neill? For even in her moment of alarm she felt a surge of warmth for him. He had had something on his mind, after all. He way in trouble. It hadn't been another girl.\n\nWhat was he supposed to know, Sandalwood wondered. And, equally important at the moment, where did her sympathies lie? He said, \"This could cause a lot of trouble for your friend Ryan.\"\n\n\"Well, whatever it is,\" she said instantly, \"you can be darned sure it wasn't Neill's fault.\"\n\n\"'Whatever it is,'\" he repeated. \"Don't you know what it is?\"\n\n\"No, I don't, but I know this. It was all that damned Ed Jablonski. I remember the night they arrested Derby, he was boasting afterward about how they'd fixed Derby good. I know that Neill is not the kind to...\"\n\nSandalwood wasn't listening. Fix him good. That didn't sound like taking a little money from a prisoner. But he had never put much stock in that idea, anyway. What was it, then? Some kind of frame?\n\nHe remembered Ryan's guilty start when he came on him up in the old brownstone's bedroom.\n\n\"...and if anything happens it will be poor Neill who'll suffer,\" Gee Gee was spying. \"It's not fair.\" She looked meltingly at Sandalwood. \"Unless you give him a break in your story. That could help, couldn't it?\"\n\n\"Sure, it could,\" he said. He had enough for now. He had a lead. Don't press, Jackie boy. And there was no doubt about where she stood. But maybe there was a way to make her open up a little more next time, in case she should learn more from Ryan.\n\n\"You seem to think a lot of Ryan,\" he smiled at her. \"It sort of surprises me\u2014the way he's been seeing Harry Derby's sister lately.\"\n\n\"What?\" said Gee Gee. \"Derby's sister?\"\n\n\"Care to order now, sir?\" the waiter asked.\n\n\"Why not?\" said Sandalwood.\n\nHe felt good, and very wide-awake. And voraciously hungry.\nCHAPTER 22\n\nThe Girl in the White Jaguar\n\nSomeone who knew what a chisel was made for had been prying open automobile windows in the east fifties, mostly near hotels where out-of-town visitors parked on arrival and left their bags. It sounded like a team of three, two to get into the car quickly and a third in another car to pick up the loot. Bauer mentioned it when the afternoon tour gathered in the squad room.\n\n\"They haven't hit for five days and they're about due. They may figure the rain'll help them. Lee, you and Neill cruise as far up as Fifty-seventh. Take your meal period after nine, because they haven't been doing anything after nine o'clock. Harry, you and Al do the same about Fifty-seventh until you get a call.\"\n\nFor the next hour Lambert and Ryan shuttled back and forth in a well-worn Plymouth sedan through wet, shiny side streets from Fifth Avenue to Third, saying little, listening whenever the hoarse radio rasped into communication. They had worked together long enough so that they did not have to say much. Lambert was driving, inching and stopping and inching again through the dense traffic of a gloomy, rainy afternoon. They both scanned the parked cars for luggage that would attract thieves.\n\nA frustratingly long line of cars had halted them in Fifty-second Street when Lambert touched Ryan's arm. Twenty-five yards ahead of them two men in dark overcoats were standing, hardly sheltered from the rain, against the wall of a building. In front of the two men a yellow Cadillac convertible was parked. One of them kept looking down the street in the direction from which Ryan and Lambert had come.\n\nLambert said mildly, \"They seem to like it in the rain.\"\n\nRyan unlatched his door. \"Probably their car's behind us,\" he said, and slipped out of the Plymouth. He walked back a few car lengths, then reached the sidewalk. From here he could not see the men so well; a restaurant's well-lighted canopy over the street interfered with his vision.\n\nThe traffic in the street began to crawl again, taking Lambert with it. Ryan studied the cars coming up behind. Two cabs, then a glossy Buick fresh from a garage, then a small white sports car\u2014it would be none of those.\n\nBut behind the white Jaguar were two nondescript sedans. Ryan strolled toward the bright canopy and the two men beyond it. The Jaguar had pulled up before the restaurant canopy and the garage attendant who had delivered it took the parking slip from the doorman on duty. Ryan looked again for the two men; one had walked over near the Cadillac and was still looking west down Fifty-second Street. The first sedan had gone past, but the second one blinked its lights.\n\nThis looked good. And by now Lambert must be at the corner.\n\nRyan moved forward\u2014and collided with the doorman coming out of the restaurant, hurrying with an open umbrella. \"Watch it, Mac,\" said the doorman. Immediately behind him came a copper-haired girl in a new, glossy fur coat with a film-like rain cape over it. She was followed by a hatless man in a gray sport jacket. He was laughing. \"Thanks, Joe,\" he said to the doorman. A dollar bill appeared between their hands.\n\nThe doorman opened the Jaguar's door and held the umbrella over it. Ryan forced his mind away from the fact that she was the girl who had just told him she hadn't seen Sandalwood lately. He made himself look ahead. The two men were near the Cadillac.\n\nHe hurried across the bright canopy. She was getting into the little car's jewel-like interior. The second sedan that had blinked its lights had stopped alongside the Cadillac. The two men walked around the Cadillac without touching it and started to get into the sedan. Ryan came up behind them.\n\n\"Where you been?\" said one.\n\n\"Ethel made me drive her to the employees' entrance,\" said the driver. The door closed on them.\n\nRyan walked up to the corner where Lambert had pulled in near a hydrant.\n\n\"False alarm.\" He got back into the car.\n\n\"Next time lucky,\" said Lambert.\n\nSo she couldn't wear that coat out in the rain, huh?\n\n* * * *\n\nRyan called in at nine to say that everything was quiet and that they were taking their meal period at the Polish sausage restaurant. All evening he had fought the thought of Gee Gee and Sandalwood being together. Sergeant Weiner said, \"Wait a minute. Your mother phoned and said to tell you Connie called.\"\n\n\"Who?\"\n\n\"Connie. She said you had called her earlier.\"\n\n\"I get it.\"\n\n\"Call her at this number.\"\n\nHe kept Lambert waiting while he called the barber. \"Connie? This is Ryan.\"\n\n\"Yeah. About that party you were asking about. He left town in November. Quite sudden.\"\n\nRyan's worry and weariness left him. \"What date was that?\"\n\n\"I can't get it exactly, not so far. I'll keep tryin'. But it was well before Thanksgiving. Because he paid a\u2014a person I been talkin' to some money he had owed him a long time. This person says it was about two weeks before Thanksgiving. I looked that up on a calendar. Thanksgiving came on a Thursday.\"\n\n\"It usually does.\"\n\n\"Yeah? Well, this one was Thursday the twenty-fifth.\"\n\nThat could figure. If Mackie had done it and had heard of Derby's immediate arrest he would feel relieved. Naturally. But after a few days he would start worrying. He would wonder if the cops weren't putting out a newspaper phony to lull the real killer... Then he might worry himself right out of town.\n\n\"How much did he owe this person?\"\n\n\"Two hundred bucks. He paid him only a hundred, though.\"\n\nGod! This was coming too fast. Derby had still had the hundred dollar bill.\n\n\"You know this Mackie?\"\n\n\"A little. I used to box at his place.\"\n\n\"Box?\"\n\n\"His joint is part Turkish bath and part health club. Or it used to be. A few fighters trained there, prelim boys and like that.\"\n\n\"Remember any of them? Or do you know anyone who might have heard from him recently?\"\n\n\"I dunno who'd of heard from him. The old man who helps him. Or that screwy kid who gives out the towels, maybe. I dunno who else. I hear when he left, Mackie said he was going to visit his mother in Chicago.\" Connie sniggered. \"I'll bet she lives in a kennel.\"\n\n\"Who else did he know?\"\n\n\"I'm telling you, boss, I don't know. I ain't seen Mackie in three years myself. Maybe five. Today\u2014hey, I know one. And you do too. That guy who's gonna burn in Sing Sing.\"\n\n\"Who?\"\n\n\"That guy Derby,\" said Connie. \"What's his name\u2014Harry Derby? The one who killed the old lady. Sure. He wanted to be a fighter. I've gone more than a few rounds with him at Mackie's. Mean, he was. But Mackie always thought he'd make a fighter.\"\n\nRyan said, \"Can you think of anyone else?\" but it was only habit. He didn't listen to the answer.\n\nWhen he climbed into the car, Lambert said, \"They had something for us, huh?\" disappointedly. He was hungry.\n\n\"No. I had to call home. Sorry.\"\n\nBut when they were both seated in the restaurant, Ryan could not think of food. All he could think of was Sandalwood and Gee Gee, and how little time he must have left, and the facts Connie had given him that twirled and flittered in his mind like a deck of cards in a gale.\n\n\"Lee, do me a favor? I'm not hungry. And I've got an errand to run.\"\n\n\"Sure,\" said Lambert. Ryan had had a phone call. He was a nice-looking kid and unmarried. \"Better gimme an address.\"\n\nRyan knew what Lambert thought, but it didn't matter. He gave an address twelve numbers away from the Derby apartment. \"Blow two long and two short if we get a call.\" He would be able to hear it from that close.\n\n\"I won't unless I have to,\" Lambert grinned.\n\n* * * *\n\nShe said, \"What I don't understand is why the fingerprint at the drugstore didn't tell the police right away that Harry committed the\u2014that robbery.\" She was sprawled back on the day bed in the tight slacks. Her hair was pulled tight in a pony tail. She looked young and thoughtful.\n\n\"Everyone thinks fingerprints are a big thing,\" said Ryan. \"Actually they're valuable mostly after you've made the collar. If even a partial print is found at the scene, and then you make an arrest, you can tell fast whether the suspect is your man. But what good is one fingerprint in itself? Individual prints are not classified by themselves. The whole system of fingerprint identification is based on getting a complete set\u2014it's only the habitual criminal or specialist whose individual fingers are classified. Say you go over a safe job and find one left thumbprint. You look at the file of left thumbprints of habitual box men. Then maybe you make it.\n\n\"But we can't file and classify all ten prints of every guy who ever hit a drugstore. If Harry had been arrested in the drugstore job, then that print would have helped us make him fast. But no one could go through all the sets of prints on file downtown to find one that matched. It would take forever. Don't put your trust in fingerprints. For one thing we don't often find them, no matter what you see on television.\"\n\nHe sipped the Coke she had insisted he take, and which he had accepted because he knew she wanted to be a thoughtful hostess. Ken would be home soon, she had said. Ryan was glad he wasn't there now.\n\n\"What I really wanted to ask you about,\" he said, \"was what you might remember of the day that\u2014that Mrs. Connors was murdered. And also about a guy named Mackie.\"\n\n\"Mackie?\"\n\n\"Big Mackie, he's usually called. He runs some kind of athletic club near Fifth Avenue. Ever hear Harry mention him?\"\n\n\"I don't think so. Harry tried boxing once, though, ten years ago or so. Maybe Ken would know. He'll be along any minute.\"\n\n\"I can't stay much longer.\" He looked at his wrist. \"One other thing. I wish you would tell me everything you can remember that happened here the day of the murder.\"\n\n\"Happened? Here?\"\n\n\"I won't try to explain. Please.\" He was oppressed by time's swift passing.\n\n\"Well.\" She drew on her cigarette and considered. \"Harry was staying with us then.\"\n\n\"Didn't he always?\"\n\n\"Harry came and went. Sometimes he roomed alone. Sometimes he\u2014he stayed with a friend.\" Ryan caught the intonation's meaning. \"When he was broke he often came here for a few days or a week at a time. He slept on this couch.\"\n\nRyan listened to a car braying outside. It wasn't a Plymouth.\n\n\"That's how it was that day. I remember Ken woke Harry up when he was getting ready to go to work, and Harry got mad. I remember him swearing he'd find some other place to stay where he could get a night's sleep. But they'd been arguing the night before; this was just a continuation.\"\n\n\"What about?\"\n\n\"The night before? They'd argued about money. Harry had some sort of chance to buy into a little gum vending machine business. He could have swung it with five hundred dollars. He wanted Ken to lend it to him.\"\n\n\"And Ken wouldn't?\"\n\n\"Ken didn't have it. He helps keep this place going. And he's got a girl who\u2014who likes to go out and spend money.\"\n\nHow about you? Ryan wondered. Don't you ever want to go out and spend money? Or is your whole life just looking clean and working in a library?\n\nShe misinterpreted his look. \"Really, I know Ken didn't have it,\" she said emphatically. \"If he had, he'd have loaned it to Harry. He was awfully anxious for Harry to go straight. It\u2014they had a strange relationship, Ken and Harry. They liked each other and sometimes I've thought each one took a kind of vicarious pleasure in the other's goodness or badness. Harry was proud of Ken, I know. He used to kid him about looking like a traffic cop in his uniform. And I suppose Ken sometimes admired Harry's wildness.\"\n\n\"Yeah. What else happened that day?\"\n\n\"Well, I straightened the house and got breakfast\u2014Ken leaves a lot earlier than I do\u2014than I did. I didn't get to work until ten. So I made breakfast for Harry and me, and Harry was still here when I left. That's the last I saw of him. You\u2014you arrested him that night.\"\n\n\"Did he get any phone calls that morning? Or the night before?\"\n\n\"No.\"\n\n\"Or have any visitors?\"\n\n\"No. He and Ken just argued.\" She said it matter-of-factly, but in that monotone Ryan heard the shouts and curses that had echoed in these quiet walls.\n\n\"That's how they always were. Emotional. Pulling and tearing and yanking at each other. And liking each other underneath. Ken wanted to borrow Harry's jacket that day\u2014his uniform was at the cleaner's. Harry told him to\u2014to go to the dickens.\" Ryan could imagine what Harry had said.\n\n\"The checkered jacket?\"\n\n\"That's right.\"\n\nHell, thought Ryan. What was he getting\u2014a stronger case against Harry? In these few minutes he had confirmation that Harry had needed money that day, was in an ugly mood and had his telltale jacket. \"Tell me this. Did you ever see a cuff link like this?\" He took it from his wallet.\n\nShe looked long and carefully. \"No. I never did.\"\n\n\"Harry never had cuff links like that?\"\n\n\"No. I don't ever remember seeing them.\"\n\nOutside a Plymouth honked, then honked again. The apartment door opened.\n\n\"Ken?\"\n\n\"Hi, kid. How's tricks. I just got a big\u2014\" He caught sight of Ryan. \"Well, well.\" He grinned and held out a hand. \"Hi. I'm sorry about the other night. Any hard feelings?\"\n\n\"No hard feelings.\"\n\nKen Derby pulled off the black whipcord jacket, unclipped the shiny leather bow tie from his shirt collar, and unbuttoned the collar. Then he rubbed his jaw. \"That's quite a right you got,\" he said amiably. \"Quite a right.\"\n\n\"Thanks. I was lucky, too.\"\n\n\"How about food, Rosie? Listen, I know a good right when I see one. I used to box. I still follow the fights on television.\"\n\n\"Where'd you box?\"\n\n\"Down at Mackie's. With Harry.\"\n\nRyan had a sense of things falling exactly into place.\n\nRosemary said, \"There's liver and bacon.\"\n\n\"Mackie's?\"\n\n\"That'll be fine.\" Ken looked at Ryan. \"This'd be a long time back.\"\n\nThe Plymouth honked again.\n\n\"You familiar with Mackie's?\"\n\n\"Sure. Why not?\"\n\nSomething began taking shape in Ryan's mind. \"I'm working to try to clear your brother.\"\n\n\"So Rosie told me. If I can help...\"\n\n\"Can you keep your mouth shut?\"\n\n\"If it'll help Harry,\" said Ken doggedly, \"I can keep my mouth so damned shut\u2014\"\n\n\"Good. Because there's a chance\u2014just a bare chance, according to some information I have\u2014that this guy Mackie killed the old lady Harry got sent up for.\"\n\nKen's expression was eloquent. \"Listen. If that's so and I can help prove it\u2014\"\n\n\"It's a long shot, and besides I've still got a lot of things to check.\" He waited for another bleat from the car below. \"I understand Mackie has living quarters above his joint, right?\"\n\n\"He used to have an apartment on the third floor.\"\n\n\"Could you draw me a plan? And show me how to get into the living quarters?\"\n\nKen Derby put hands on knees and leaned forward earnestly. \"Listen. I know the joint and I'll draw all the plans you want. If you're trying to spring Harry, there's nothing you can't ask me.\"\n\nFinally it came, a loud impatient entreaty.\n\nRyan jumped guiltily. \"I've got to go. Listen, draw that plan. Although I can't get back for it until after midnight.\"\n\n\"You come any time you want,\" said Derby. He held out a hand. \"It'll be ready and so will I. If getting into Mackie's joint will help, you're as good as in.\"\n\n\"Okay,\" said Ryan. \"I just want to frisk it. Okay. But keep your mouth shut.\"\n\nWhen he reached the street a tawny Plymouth convertible was panting at the curb across the way. The boy in it leaned impatiently on the horn. Lambert was nowhere in sight. Ryan grinned. This half hour had been better than food.\nCHAPTER 23\n\nLiniment Scent\n\nThey were drinking coffee around a dining-room table, late. Ken Derby gestured toward the sheets of paper before him, each bearing a neatly drawn diagram. \"I haven't been there for quite a few years, you understand,\" he said. \"But that's how it was when I hung there.\"\n\nRyan sipped the coffee Rosemary had made. It was one-thirty. She had talked a few minutes and then gone to bed because she had a job interview coming up in the morning. Ryan said, \"But you don't know anything about the top floor?\"\n\n\"The building was originally a laundry. That was on the street floor, of course. The top floor ought to be basically like the second. We can find our way.\"\n\n\"We?\"\n\nDetermination made Ken Derby's face almost as thinly mean as his brother's. \"I want to go,\" he said. \"If I can't help, at least I can't do any harm. I know the place, and you're going to be operating alone. And I'll tell you this.\" He looked up suddenly, levelly, at Ryan, and Ryan heard the emotion in Derby's throat. \"If there's anything I can do to save my brother, I want to do it. Believe me.\"\n\n\"The chances,\" said Ryan, \"of Mackie being our guy are probably one in a hundred thousand. All I know at the moment is that he looks something like Harry, and\u2014but there's another guy in the Bronx\u2014\"\n\nKen Derby's face narrowed with earnestness.\n\n\"Wait a minute. Maybe the odds aren't quite one in a hundred thousand.\"\n\n\"What do you mean?\"\n\n\"This Mackie could look like Harry. Sure. But what I was thinking was, he's a tough guy, in a queer way.\"\n\n\"A fag?\"\n\n\"No. Well, I dunno, now that you ask. He hates women, that's for sure.\"\n\n\"He's got a record along that line.\"\n\n\"I guess he has been pinched for mugging women at that. But what I mean is, Mackie's always been against dames. He used to tell jokes about dames, jokes where they got beat up, or something. Tell you something else. Nobody ever saw Mackie with a broad.\"\n\n\"But right now he's out of town.\"\n\n\"He is?\"\n\n\"He left soon after Mrs. Connors was killed.\"\n\nKen Derby whistled softly. \"Hey, if you could extra\u2014What is it? Extradite him? Get him back here. And\u2014\"\n\n\"From where?\"\n\n\"From where, for God's sake! From wherever he is.\"\n\n\"That may be Chicago,\" said Ryan, \"but I doubt it.\"\n\n\"You don't know?\"\n\n\"That's one reason I want to frisk his joint.\" He rose. It was almost two a.m. He was dead tired and the days were growing fewer.\n\nKen Derby got up. \"I'll go with you.\"\n\n\"No, you won't,\" said Ryan. \"This is police business.\"\n\nDerby looked malevolent. \"Yeah? Like hell it is. The police department isn't doing anything for Harry. You're swinging it, all alone. Like I said, maybe I can help and maybe I can't. But he's my brother. And you're my friend.\"\n\nRyan was ashamed to find he wanted company. \"Okay,\" he said.\n\nThe cab dropped them in front of a bar two blocks away from Mackie's place. Derby suggested a quick drink but Ryan curtly told him no. They walked down the shadowy street to a point opposite the narrow, inconspicuous building that housed Mackie's business and home. It was dark and so were the taller buildings on either side.\n\n\"Funny,\" said Derby. \"He used to stay open all night.\"\n\n\"Subway rides used to cost a nickel,\" said Ryan. \"I thought you knew this place.\"\n\n\"I do. But Mackie or somebody was always here this time of night. Maybe the kid who helps him has been goofing off. Or the old man.\"\n\n\"Maybe.\" Ryan strained his eyes across the dark street. \"An old man?\"\n\n\"An old white-haired coot.\" He sounded anxious. \"The sign was always on nights!\"\n\nNuts with it. Either you hit or you quit.\n\n\"You sure about that back door?\"\n\n\"Dead sure.\"\n\n\"Then stay here. I'll call you.\"\n\nRyan walked boldly across the street. There was no one in sight, but he knew well that the chance of his being observed was not a small one. He got to the other sidewalk, found a space between the buildings, moved into it\u2014and saw a light moving up behind him. He wheeled and crouched in the shadow. A white-marked police car rolled by. For the first time in his life he knew how it felt to be on the other side.\n\nThe RMP car went on and Ryan looked around. Ken Derby was nowhere in sight. He had ducked fast and expertly. It must run in the family, Ryan thought contemptuously, then remembered Rosemary. That hadn't been fair.\n\nHe got around to the back, walking boldly in blackness. There was a cellar door, as Derby had said. He found the old-fashioned knob, then felt the key plate and keyhole. Above them he fingered the cold circle of a barrel lock. Ryan took out a thin-bladed Swiss knife that his father had carried and made sure the old bolt was not in place. Then he took some wrinkled wires from his wallet and worked on the barrel lock. The third wire made a click and the door knob twisted open. Wet, heavy air poured out, hot with steam.\n\nRyan advanced into it, closed the door and listened. There was a hissing noise and a steady, rhythmic dripping. And the tropic, humid heat of the equator at noon, and the strong scent of liniment and cheap soap. After a few minutes of uncertain silence, he flashed his pencil light around. This was the boiler room; there was a furnace and a boiler with gauges. And this was breaking and entering in the night time, a felony for anyone. A disaster for a cop.\n\nHe silently retraced his steps and when he found the sidewalk empty, he whistled softly. Derby ducked swiftly across the street and followed him to the back door.\n\n\"We're in. Maybe there's no one here. Maybe not. I want the upper rooms where Mackie lives.\"\n\n\"I can find 'em.\"\n\nRyan closed the door silently. Steam and soap-scent warmly enveloped them.\n\nThe second floor stank of stale liniment. Ryan's small light showed boxing gloves, tied in pairs on nails along a wall, a sandbag and two punching bags, and an elevated ring with little stools below at one corner. Beyond were doors that would lead to sleeping and dressing rooms and showers.\n\n\"How about the third floor?\"\n\n\"We go back to the stairs,\" said Derby. \"I never been up there.\"\n\n\"Lead the way.\"\n\nThey went up narrow stairs. There was a door at the top of a small landing. It had no barrel lock, and when Ryan used his knife blade for a minute along the frame, it opened. Once inside, he listened carefully, then felt for a light switch and flipped it. They saw a small, dark living room, heavily draped and massively furnished, and illuminated with big lamps whose shades dripped thick gold fringe. In corners large bronze statues gleamed; all portrayed naked athletes.\n\n\"Aren't you taking a chance with that light?\"\n\nRyan went to the room's two windows, pulled down the shades and drew the drapes. \"I'm taking a chance being here at all,\" he said.\n\n\"Yeah. What should I do?\"\n\n\"Stand by the door and listen in case someone starts poking around.\"\n\nRyan pushed his hat back on his head characteristically. There were two doors at the back. One led into a kitchen, the other into a bathroom. Where did the guy sleep? Then he saw another door.\n\nBut he took the kitchen first. The cupboards, half empty, held only a few dishes and the usual staples. The refrigerator contained a half bottle of milk turned to moldy cheese, a can of tomato juice and some eggs. He looked over the stove, into its oven, and pulled it out from the wall.\n\nIn the bathroom he examined the contents of the medicine chest and then flashed his light in the toilet tank box. He felt around the back and underneath surfaces of the washbowl and tub for anything stuck or hidden there.\n\nThere was a scraping noise from the other room. Ryan froze. He waited to hear from Derby.\n\nHe did not hear. He waited.\n\nHe did not hear anything.\n\nHe went to the door, flicked out the light so as not to be silhouetted, decided against drawing his gun, and peered into the living room. Derby stood still at the door, looking out into the hall.\n\nWas someone holding a gun on him?\n\nRyan waited another moment. Then he whispered, \"Derby!\"\n\nSilence.\n\nRyan took out his gun. He reminded himself that a gun was the worst thing he could use here. He walked into the living room.\n\nDerby wheeled nervously.\n\nRyan said, \"What was that?\"\n\n\"I thought I heard something.\" Derby's whisper was husky.\n\n\"It came from out there?\"\n\n\"It came from somewhere!\"\n\nRyan went to the living-room door that was still open, and flashed his light out. There was nothing on the stairs.\n\n\"Maybe the bedroom,\" said Derby. Ryan looked around.\n\n\"I heard something,\" Derby insisted.\n\nRyan had heard it too. He went, gingerly, silently, to the bedroom, gun ready, and switched on the light beside the door. It was a small, pale-blue room with a single bed. Ryan studied it, then searched it. There was a small desk slanted across one corner with a telephone and old magazines on it.\n\nThat might be where he'd find what he needed. But first... Derby peered in.\n\n\"Watch that door!\"\n\nDerby straightened, said \"Check,\" and went back.\n\nRyan felt behind the pictures on the walls. He got under the bed and explored its springs with his light. He looked at the hair brush and the other toilet articles on the bureau, and shook out the magazines on a small table. He up-ended a floor lamp, looked at its base, then critically examined its shade. Finally he lifted all the drawers out of the bureau carefully, and slowly and inspected their contents: several shirts and socks rolled into balls, a couple big sweaters, a sheaf of physical culture magazines, some blankets.\n\nAnd, with the blankets, a heavy woolen; black-and-white checked lumberjack shirt.\n\nThe checks were not as big as those on Harry Derby's jacket, as he well remembered. And it was a shirt, not a jacket. But witnesses in the excitement of the moment could have made that mistake. Ryan fingered it unbelievingly.\n\nThe telephone rang.\n\nRyan looked to the desk. The phone rang again. He glanced out the door at Derby. Derby was watching him and looked scared. The phone rang again.\n\nRyan went to the desk and picked it up. \"Hello?\" He held his voice low and indistinct.\n\n\"St. Louis calling\u2014one moment please.\" Then, \"Ready with New York, sir.\"\n\n\"Hello,\" said Ryan.\n\nA man's voice rasped in the receiver. \"Damn it, Bobbie, how many times do I have to tell you how to answer the phone?\"\n\nRyan waited.\n\n\"'Mackie's Baths!'\" the voice quoted testily. \"If I've told you once, I've told you a million times how I want the phone answered.\"\n\nRyan took a chance. \"Yes, sir.\"\n\n\"How's business?\"\n\n\"Fine,\" said Ryan.\n\n\"That's good. Bob?\"\n\n\"Yes?\"\n\n\"How's everything?\" The tone grew guarded.\n\n\"Everything's good,\" said Ryan. He held the mouthpiece away from his face, so he could not be heard well. \"Everything's okey doke.\" He took a chance. \"You in St. Louis?\"\n\nDerby watched wondering from the door.\n\n\"Yeah. But I may fly into town tomorrow.\"\n\n\"Good.\"\n\n\"You sure you don't hear anything?\"\n\n\"Everything's quiet as all hell,\" said Ryan. \"Come on home.\"\n\n\"I got it all paid off but the one, and that'll be arranged this week. But meanwhile I don't want the big man on my neck.\"\n\n\"Come on home,\" said Ryan tensely. \"You'll be okay. I heard today the big man ain't\u2014ain't mad at nobody.\" It was taking a chance.\n\nBut the other end of the line chuckled with appreciation. \"You're a good kid, Bobby. But where were you earlier? There was no answer.\"\n\n\"I went out for a little while. I felt like a beer.\"\n\n\"A beer!\" said the other incredulously. Ryan knew that had been a mistake.\n\n\"When'll you get in?\" he asked quickly.\n\n\"After dinner tomorrow night,\" the other replied angrily. \"See that you're there! When did you start drinking beer?\"\n\nHe had to take the real chance. He had to know who he was talking to. \"I was kidding, Mister Mackie,\" he said. \"I'll be here.\"\n\nThe telephone chuckled more happily. \"Don't hand me any Mister business.\"\n\nThey rang off. Ryan turned from the phone. He could not believe it had happened. Derby was still watching from the door. \"Huh?\" he said.\n\n\"God,\" said Ryan.\n\n\"Who was that?\"\n\n\"Mackie. He's in St. Louis. But he's flying in tomorrow. Or today. Or whatever the hell time it is.\"\n\nDerby's wide eyes fixed on Ryan. Then his face lighted up. \"Jesus!\" he yelled. \"We got him!\"\n\n\"Shut up,\" said Ryan. \"That kid may be back any minute.\"\n\nThe bureau drawers were still on the floor. He began to lift them back into the bureau. Habit and training reminded him to run his hands around their bottoms and backs. That's how he felt the little Scotch-taped protuberance on the back of one.\n\nHe turned the drawer around. Then from its rear panel he yanked the small object taped to it.\n\nIt was a cuff link. Ryan recognized its miniature scimitar pattern instantly. But he looked a long time at it. All he could think of was that this is what comes of living right.\n\n\"Let's go,\" said Derby. \"Want some help?\"\n\nRyan did not answer. He re-taped the link into place and restored the drawers after inspecting each one. There was nothing taped to the others.\n\nHe went through the desk. It was full of letters tied in ribboned bundles, to Lester Mackie from his mother. And old receipted bills. And boxes of nails, and pins and loops of string. Mackie was a saver.\n\nThank God for that, he thought, recalling the cuff link.\n\nThey went down the narrow, liniment-smelling stairs in silence. As they groped through the boiler room, Derby said, \"You think he'll show up?\"\n\n\"He's got to,\" said Ryan. And the way things were going, it was true. Maybe things were going too good. Still, he was filled with the glow of imminent triumph.\n\nRyan insisted on walking several blocks south and west before hailing a cab. He dropped Derby at his home, noticed it was almost three-thirty and realized he was famished. After all he'd been moving at high speed for almost twenty-four hours\u2014and getting places! He was entitled to a sandwich and maybe a beer. But where could he go? Then he thought of Gee Gee's party.\n\nIt shocked him to realize how long she had been out of his mind, especially considering the circumstances in which he last saw her. He'd told her to get someone else for the party, and he had a pretty good idea who that would be. And this was the girl he had almost told about Derby! What might she have told Sandalwood?\n\nWell, why not go to the party, and find out? It would sure surprise her. Ryan grinned without humor.\n\n\"I want to go to a joint between Fifth and Sixth,\" he told the cab driver. \"On Fifty-second.\"\n\n\"One of the strip joints?\" said the driver. \"We gotta hurry, Jack. It's almost closing time.\"\nCHAPTER 24\n\nThe End of That\n\nRyan was led to a tiny table while a raucous band crashed and tub-thumped its way to the end of a song. He was barely seated when an unseen voice told the customers, in sibilant confidence through a public address system, \"And now, friends, the concluding number...the star of our show... Maxie's Rendezvous is proud to present for your enjoyment...that titian-haired temptress of the dance...lovely Gee Gee Hawes!\"\n\nShe appeared from somewhere near the bandstand, looking tall in a gown of long white pleats that was like a negligee. Not only her hair but her eyelids sparkled with sequins and her lips were parted expectantly. She wasn't the lissome, natural girl in the brown coat now, but self-consciously glittery and sure, enameled and eager. Ryan sipped beer as she undulated around the dance floor. He had not expected her to be so professional. She slipped unexpectedly out of part of her dress and her jeweled bra was small and revealing.\n\nA man at the next table said, \"Jeez, Lili can take a night off any time she wants.\" He saw Ryan looking at him. \"Lili St. Cyr's sick tonight,\" he explained. \"This dame's a substitute. Not bad, huh?\"\n\n\"Not bad,\" said Ryan.\n\nThe music increased in volume and tempo, slowly. Gee Gee glided around the floor, gradually freeing or losing garments. The spotlight grew deeper blue, until it was hard to tell whether what was visible was cloth or shadow. Then men at the next table made comments. Ryan tried to disregard them.\n\nFinally, in a flare-up of music she disappeared near the bandstand, and there was a jet roar of applause. She reappeared briefly, the white dress held in front of her. She took several bows. Even the musicians grinned. The show was over.\n\n\"Chef's left, Mac,\" the waiter said to Ryan. \"No steak sandwich. I can get you a nice ham on rye.\"\n\n\"I'll take it. And another Prior's dark.\"\n\nThe orchestra played a final number, then went back stage. The lights went up. Ryan looked around. No Sandalwood. Was he back stage? The customers finished drinks and began leaving. The waiter brought the sandwich and Ryan munched hungrily. His eyes were on the door near the bandstand. Other girls came through it dressed for the street, but instead of leaving they sat down at tables. Finally Gee Gee appeared. She walked slowly, the new coat draped with affected casualness over her shoulders. Then she saw Ryan.\n\n\"Neill!\" She came over quickly, smiling. \"You made it! I'm so glad.\"\n\nRyan grinned and he didn't want to. But seeing her always made him feel like that. Besides\u2014as his weary brain reminded him\u2014he had to find out what had happened. This wasn't a pleasure trip. She should think it was, though, he told himself cynically. \"Sit down and have a drink. You must feel like it after all\u2014after your dance.\"\n\nShe looked at him. \"You were here for the show?\"\n\n\"Your part of it. It was terrific.\"\n\n\"You're sweet. But what did you really think?\"\n\n\"I said. Terrific.\" He looked around. A few people had straggled in and were hailed by those at tables. No Sandalwood. Yet.\n\nHer cool, long-fingered hand closed over his knuckles. \"Okay,\" she said. \"Neill, you've got to remember that in this business you don't always do what you want. Maybe in any business.\"\n\nA waiter came up and Gee Gee said she would take tomato juice. \"I gained a pound and a half this week.\"\n\nRyan said his beer would do for a while, then brought up the subject on his mind. But a little regretfully. It would have been nice to just sit and talk. \"I thought I'd run into Sandalwood. I figured he'd be the one you'd ask to the party.\"\n\n\"He was,\" she said right back. She thought of what Sandalwood said about Derby's sister. \"He had to go up the river today. Tonight, I mean.\"\n\nRyan flinched. Up the river could mean many things. But to a policeman and to a police reporter it meant one thing. Up the Hudson River was Sing Sing.\n\nGee Gee misunderstood his expression. \"Oh, honey, why think of him? I don't feel that way about him at all. And maybe there's something I could get mad about too, if I wanted. But there's something I really must tell you that Jack said.\"\n\n\"What?\"\n\nThe waiter came with the tomato juice. \"Have a drink, Neill. You look like you need it. Bring a big Scotch, eh, Benny?\"\n\n\"Sure, Miss Hawes. You were terrific tonight, Miss Hawes. You killed 'em. You shoulda heard what they said onna way out.\"\n\n\"Thanks, Benny.\"\n\n\"I mean it. When Lili leaves. Max doesn't give you the spot he's crazy.\"\n\n\"Thanks. Benny. You're sweet.\" The waiter left. \"See? I have a public.\"\n\n\"Everybody's sweet tonight.\"\n\n\"Now, Neill, don't be like that. I don't want to quarrel with anyone. Not tonight. This is a big night. Do you know what happened? Lili St. Cyr is the headliner here now. So tonight her manager phoned in that she was sick and Max had to rebuild the show. And who did he put in the top spot?\" She was radiant.\n\n\"Congratulations.\" He appreciated her excitement. But he hadn't liked seeing her dance like that, or the artificial sweetness she seemed to be showing everyone. And she'd been talking to Sandalwood.\n\n\"There are girls here who've been in the business over three years,\" Gee Gee went on. \"And I'm just getting started. But Max\u2014well, Max!\"\n\nA man stood over their table, short and purple-suited and dominant. Ryan recognized him from the day in court.\n\n\"You were sensational, doll,\" he said. \"Sensational.\" Pudgy fingers with big rings squeezed her shoulder. \"Some people been saying when Lili leaves Saturday night you oughtta replace her.\" Gee Gee looked up hopefully. He saw the look. \"I'm thinking about it.\"\n\n\"Thanks, Max.\"\n\n\"Have your friend stay for the party.\" Max walked away.\n\n\"Thanks, Max,\" she called after him.\n\n\"You forgot to tell him he is sweet.\"\n\n\"Now, lover, don't act that way. This might be my chance. Max isn't so bad.\"\n\n\"What was it you wanted to tell me?\"\n\nBut waiters began moving the tables around, including theirs, to make one long table.\n\nApparently everyone had read about Ryan. When Gee Gee introduced him they were very nice, with that combination of friendliness and respect for even a minor public figure that characterizes the stage. They sat at the long table and drank toasts to Max and ate turkey slices and listened to the music. It wasn't the raucous thumping of before; the pianist's fingers had acquired a melancholy delicacy and the trumpet a beautifully timed sadness. Gee Gee raised her face to the horn's golden bell, listening, chin and neck white in the smoky light, and the trumpet player saw her and played straight to her while her red nails tapped rhythm on the tablecloth. The band rambled slowly through \"Confessin',\" yearned into' \"After I Say I'm Sorry,\" and mourned with \"When Your Lover Has Gone.\" Ryan found his arm around Gee Gee's shoulders.\n\nAfter a time she sighed and nestled. \"What's with you, baby?\"\n\n\"I'm fine.\"\n\n\"Sure? Not mad?\" She was anxious about it.\n\n\"Sure,\" said Ryan. \"Not mad.\" It was almost true. Then Sandalwood's taunting face, the Harwyn's canopy, the white dress held before her at the end of the dance, swam through his mind, and it was a lie.\n\nHer glass was empty and she sipped contentedly from his, which he had not touched. She had to tell him about what Sandalwood had said, but she didn't want to right now; this night was too perfect to spoil. He had come to the party, after all. Everyone had told her how well her act had gone. And she knew it had; she had felt the crowd's electric reaction.\n\n\"Baby.\" She had never called him that before. \"You don't like Harry Derby's sister better'n me, do you?\" It was a rhetorical question, asked in drowsy confidence that the answer would be the right one.\n\nBehind them the unmuted trumpet took up \"Dream,\" huskily, sweetly.\n\nBut raw gasoline had been poured on smoldering coals.\n\n\"What else,\" Ryan demanded, taking his arm away, \"did Sandalwood mention today?\"\n\nHe had never seen her eyes so round. \"Why, Neill. Don't get the idea\u2014\"\n\n\"I've got the idea. Incidentally, Rosemary Derby is a very decent sort of girl. But what did Sandalwood say about her\u2014or me? Today. At the Harwyn Club.\"\n\nThat hit. \"How\u2014how did you know that?\"\n\n\"I'm a cop, you know,\" said Ryan bitterly. \"Just a cop.\"\n\nShe was silent a second. Then, quietly, \"Neill? That night you and Ed Jablonski arrested Derby\u2014what really went on?\"\n\n\"What do you mean?\" Ryan demanded truculently. But it alarmed him.\n\n\"Because whatever it was, Jack Sandalwood knows about it. And he's going to publish it in his series. He said so. What happened, Neill? Is Derby's sister in it?\"\n\nRyan made a snorting sound. \"What'd he tell you?\" he asked thickly.\n\n\"He didn't tell me anything\u2014and I didn't tell him anything. Except that I knew, whatever it was, that you were not to blame, and that the night we met that damned Jablonski had boasted of fixing Derby and all that.\"\n\nOh, God! He could hear it all, just as though he had been there. He knew how Sandalwood had got it out of her, and what he was figuring on, and why he had gone up the river. It was very, very late and the night had grown old and the fatigue was back. And he was beaten.\n\nAfter many highballs the man sitting across from them had arranged matches in a geometrical pattern on the tablecloth. Now he lit the first match and waited for the next one to ignite. So did Ryan. And there was Sandalwood, all the time. Always ahead of him.\n\n\"What does it mean, Neill?\"\n\n\"Mean? To whom?\"\n\n\"To you\u2014and me. To us.\"\n\n\"Us,\" Ryan repeated. \"You're not mixed up in it.\"\n\nShe did not answer. The man across the way got his chain of matches burning. They made successive brown spots on the tablecloth.\n\nRyan reached some obscure conclusion. She had fixed him up, too. The cute kid in the big fur coat. \"The hell with it,\" he said.\n\nIt was hardly lucid but she knew what she meant. And even before he could enlarge on it, she answered understanding.\n\n\"Sure. The hell with it. You know what, Neill? The night that we met\u2014well, I've been thinking. That night, I got to looking at Inez when she was crying about that slob of a partner of yours. And I remember later, when you tried to kiss me, I thought, 'What's with hanging in clubs?' I thought to myself, 'You fool around joints like this'\"\u2014she waved\u2014\"'where do you wind up?' And I believed it. But suppose I go along with you, pal?\"\n\nThe s's were coming out slurred and the \"pal\" was too loud. Ryan, who had been studiously refusing drinks, realized that Gee Gee had had several. He had never seen her like that.\n\n\"Then where do I wind up?\" she asked. \"With some guy who's never around when you want him, and gets jammed up in some jerky way and won't talk about it, even when you try to help him and everyone else will be reading about it in the paper in a few days. And two-times you with some gangster's sister, and...\"\n\n\"Sure,\" said Ryan. It would still take Sandalwood a few days to get the story together and print it. So if he had any chance left at all, he'd better get going. That meant getting some sleep, not the kind that was a luxury or comfort, but the raw commodity that kept you going and thinking straight, and alive. Fit for duty was the departmental expression.\n\nWhy had he ever thought she was for him?\n\n\"Okay,\" he said. \"I never could have taken you to fancy joints like the Harwyn Club. If we'd hooked up, you'd have wound up in Queens, buying groceries at Bohacks and having a baby. Two or three, after a while. No strips. No daiquiris on the ice, or frozen or whatever you call them. No Max.\" He looked at her. \"No Max. And I'll say this. I don't go for dames that let every slob who hires them get cozy\u2014\"\n\nThe chain of matches had burned out on the tablecloth. The man who had lit it, finding their conversation more interesting, was listening attentively. The pianist drifted into \"Avalon.\"\n\n\"Go to hell,\" said Gee Gee a little hoarsely. \"You heard me, buster. You go to hell. No one talks to me like that. I'll work for Max. I'll go to the Harwyn Club if I please. And I'll tell any spying, sneaky son of a bitch cop who thinks he's going to tell me\u2014\" She flung her glass's contents, backhand, in his direction.\n\nRyan jumped up to avoid its ice cubes.\n\nBut being up, there was no reason to sit down. \"I was going to take you home,\" he said. \"I guess I needn't have bothered. Max'll be glad to.\" There was another full glass near Gee Gee.\n\nThe man across was listening so intently that when she picked it up and threw it wildly he did not react quickly enough to dodge this second shower of ice cubes.\n\nRyan got his hat and stalked out. There was a cab stopped at the Sixth Avenue corner and he got in it.\n\nThat ended that.\nCHAPTER 25\n\nYou Get to Waiting for It\n\nThere were gaudy placards on the newsstand where he bought his morning paper a few hours later. They announced the beginning of \"Prosecution of the Innocents,\" a new series by Jack Sandalwood. Ryan apprehensively read the first article over toast and coffee in Grand Central waiting for the Ossining train. In it Sandalwood referred to half a dozen historic miscarriages of justice and to several familiar recent ones, and explained his purpose: to reveal how the courts and the law are less than completely omniscient and just. The name of Harry Derby did not appear. But, Sandalwood concluded, the series would also include some new, unsuspected cases of gross injustice that would stun the reader.\n\nHaving hardly touched the toast, Ryan left a tip, paid his check and walked toward the train gate. As he did he felt his burden sag his shoulders and bend his head. If he felt that way now, how would it be this afternoon? And tonight? And tomorrow? Tonight and the possibility it offered seemed remote and hopeless. But Sandalwood was near and overpowering and inescapable. Sandalwood knew all, or certainly most of what he needed to know.\n\nPresently the sun glittered like strewn diamonds on the Hudson, and the landscape that rolled past the train window looked newly raked and tidied. But to Ryan the day was at midnight.\n\nSince Ryan was an official visitor ostensibly on police business, he was to see Derby in one of the second floor attorney's rooms in the death house. In the center of the room was a table with a pad of white paper and some chairs. On the sterile walls were framed etchings. Ryan heard a voice say, \"In here!\" and Derby came in. The guard remained just outside the door.\n\nDerby looked even leaner, and his face and hands were pale. Yet he also looked healthy and well-knit, and the gray denim shirt and pants were almost dapper. He stared at Ryan a moment, then slouched in one of the chairs. He clasped his hands before him. He did not say anything.\n\nRyan waited. When there was no sound in the room or outside, he spoke very softly. \"I'm not here on police business. This is personal. We framed you a while back. Now I'm going to spring you. I thought you ought to know.\"\n\nDerby watched his own hands twist.\n\n\"You heard me?\"\n\nThe breath went out of Derby's lungs, but Derby did not look up or speak up.\n\n\"Ask your sister. Rosemary.\"\n\n\"Why?\"\n\n\"Because she knows all about it.\"\n\n\"I mean, why are you doing it?\"\n\n\"Because you're innocent, for God's sake. I won't let an innocent man go to the chair.\"\n\nDerby looked up, wary but interested. \"How do you know I'm innocent?\"\n\n\"I know where you were at the time of the crime.\"\n\n\"Like hell you do.\"\n\n\"Think I don't?\"\n\n\"I know you don't.\"\n\n\"All right. I'll tell you. And if I'm right, you'll tell me.\"\n\n\"I'll tell you nothing.\"\n\n\"Your eyes will.\" He lit a cigarette. \"On the day and at the time Mrs. Connors was murdered you were cracking a drugstore down in the Thirties near First Avenue. You needed dough. You had hoped to buy into a gum-machine business; maybe you thought you'd get part of it that way. You went into this joint, sent the old guy who runs it into the back, then you grabbed what was in the cash register. You ran out, and you made it\u2014easy.\"\n\nDerby's lynx eyes were still trained on his hands. \"Where'd you get all this?\"\n\n\"What's it matter, if it's wrong?\"\n\nDerby said nothing.\n\n\"But it's not wrong. Because you left a thumb print on the drawer. And of all the people in the department I'm the only one who could recognize that print if he saw it. I saw it.\"\n\nDerby was silent.\n\nRyan said, \"Look, Derby. I didn't come up here for a joyride. I came to tell you I realize now what happened.\"\n\nDerby's glance ranged nervously over the floor's waxed linoleum. Ryan knew what he was thinking: this was a trick. Derby could not readily interpret something like this in any other way. He said, \"And the other reason I came was to find out if you could give me any leads on someone who might look like you, or wear a checked jacket, or might have been operating in that neighborhood.\" Ryan took the last drag on his cigarette and snuffed it out against his shoe sole. \"I don't have all day. Let's go.\"\n\n\"What the hell is this?\" said Derby. \"First the reporter. And now you.\"\n\n\"What reporter?\"\n\n\"Some mutt named Sandalwood,\" said Derby, and Ryan's empty stomach spasmed. \"He was up here late yesterday. What goes on?\"\n\nHow much did he know now? \"What did you tell him?\"\n\nDerby's lips twisted their habitual contempt. \"I didn't tell him nothing. I told him to come back later, I'd think over his proposition.\"\n\n\"What was that?\"\n\nDerby looked at Ryan with cold insolence. \"I'll tell you one thing, copper,\" he said. \"That guy knows something. How much, I don't know. But he's not shooting in the dark. Somebody filled him in. He talked about getting me out. Everybody's going to get me out. You\u2014him\u2014even the damn union.\"\n\n\"What does the union say?\"\n\n\"They had some jerk up here last week. And I got a letter. Everyone says, 'Don't worry, Derby\u2014you'll never burn. We're workin' on new evidence.' Not that I believe the slobs.\" Derby's laugh was unnatural.\n\nRyan said, \"Why do you think I'm here?\"\n\n\"I don't know.\"\n\n\"I'll tell you. I want to get any lead I can as to who killed the old lady. And one of the suspects is a guy named Big Mackie. You know him\u2014the guy who runs the boxing joint.\"\n\nAgain Derby looked at his hands. Then he looked up. \"Let me tell you something. I been living with this a long time. Ever since that verdict come in, sure. But longer than that. You know me, cop, and I know you. Maybe every now and then I've did something could get me in real trouble. Maybe once or twice it did. Maybe sometimes I was lucky.\" For a second he glanced at the wall's etchings.\n\n\"But after a while you get to waiting for it. And when it happens, you're almost\u2014well, relieved or something. You know what I mean? So don't get too friggin' gay. I ain't too unhappy.\" Derby's voice had raised momentarily to normal volume.\n\nThe guard at the door looked in. \"Everything okay?\"\n\n\"Everything's okay,\" Ryan said. Then, \"Tell me something, Harry. Why'd you take the rap? Even though you didn't know about the fingerprint, you knew there was a chance that the old guy who runs the drugstore could identify you.\"\n\nDerby laughed. \"Gimme a cigarette. And next time bring a carton. You crazy or something? Look at my record. I'm three times gone. If I got tagged for that drugstore thing, I go for life. What difference does it make when you die, if you're gonna die in prison? You think I want thirty years in this can? Besides, that Farragut is pretty good, you know. They're still workin' on it for me.\"\n\nThey've sold you down the river and you don't know it. You'll still be expecting a miracle when they shave your head. \"Yeah. Sure,\" said Ryan.\n\n\"Anyway, I ain't as dumb as I look. You think I believe this hero play you're making?\"\n\n\"What are you driving at?\"\n\n\"You think I didn't read the papers? Think I don't know where framing me got you? Why would you kick all that in the face?\"\n\n\"Look, Derby, before I'll let you go to the chair I'll tell everything that happened the night we pinched you.\"\n\nThe Adam's apple in Derby's throat rolled as he laughed.\n\n\"You think they'd believe you? You think they'd let you talk about that? When they got Derby all buttoned up and another murder written off? You meathead!\"\n\nHunger and weariness and most of all defeat began again. Why had he bothered? Why had he started it?\n\nThe guard put his head in the door. \"About finished?\" he asked.\n\n\"Big Mackie, for the love of God!\" said Derby contemptuously. \"How's that screwball mixed up in it? He couldn't hurt anyone but dames.\"\n\n\"Dames?\"\n\n\"He used to collect pictures of broads being whipped.\"\n\n\"Okay,\" said the guard, coming in.\n\n\"Gimme your cigarettes, if you want to really do something,\" said Derby.\nCHAPTER 26\n\nSteam\n\nThat night Ryan and Lambert, among others, were loaned to a Bronx precinct that was anticipating a triangular war between three teen-age gangs. Nothing happened and so after driving and walking unfamiliar streets until one a.m. they were able to return to Fifty-first Street and check in. There was a message for Ryan in Sergeant Weiner's legible scrawl: \"Ken called and said your boy is back. Ken is at home.\"\n\nIn the cab that he took simply to save energy Ryan tried to relax. He could not. He kept thinking that luck never lasted forever. They'd been lucky taking Derby. He'd had luck with the Puerto Rican.\n\nAnd now?\n\nWhen Rosemary let him in she smiled cheerfully, then her face grew anxious again. Ken sat at the dining-room table, a gin rummy game before him. At sight of Ryan, he swept the cards away. \"Let's go.\"\n\n\"Not so fast. What do you know for sure?\"\n\n\"He got back around seven. I saw him drive up in a cab. I was in the beanery across the street, drinking my fourth coffee.\" He grinned. \"I'd been there since I got through work.\"\n\n\"And then?\"\n\n\"He went out around eight for an hour or so. Then he came back. That's when I called the precinct.\"\n\n\"He's still there?\"\n\n\"He was there when I checked at eleven-thirty. At least, lights were on in the apartment.\"\n\n\"Good.\" Yet he could not help hesitating. Might there not be a better way?\u2014this was a long gamble. Either they shocked a confession out of Mackie, or else he himself could be in the worst jam a cop could get in. But time was running out, with the special assistance of Sandalwood. And Ryan was tired of tiredness and anxiety. He wanted something to end. \"Let's hit it.\"\n\nRosemary was watching him. \"I don't think you'd better.\"\n\n\"Why?\"\n\n\"I don't feel good about it.\"\n\n\"Nuts,\" said Ryan. He had hesitated too long already. \"But I'll do this one alone.\"\n\n\"No, you won't,\" growled Ken. \"We're in it together now.\"\n\nAt the door as they went out Rosemary rose on tiptoes to kiss her brother. He went on into the hall.\n\nAnd then, quite unexpectedly, she kissed Ryan. Her lips were cool and fragrant. It was nice.\n\n* * * *\n\nThe lights were still on, and the red sign over the door brightly announced, Mackie's Baths. \"We'll go in as customers.\" Ryan had decided that earlier. \"It'll be simpler.\"\n\n\"The old man will remember us later.\"\n\n\"What he remembers later,\" said Ryan, \"either won't matter at all, or it'll be just a drop in the bucket to everything else.\"\n\nThe old man looked up from a racing form, made his appraisal of them, then lowered his feet from the counter and stood up. \"You gentlemen will want beds?\" He smiled. His lips were purple from the indelible pencil he had been wetting.\n\n\"That's right.\"\n\nHe put down paper and pencil and led them upstairs into a side room containing two blanketed cots. \"There's plenty of steam up.\" He gave them bath sheets and brown envelopes for their valuables. \"That'll be seven dollars for the night.\"\n\n\"We'll give it to you when we check our valuables,\" said Ryan. He made a pretense of taking off his suit coat. The old man went out and downstairs.\n\n\"We're the only ones in the joint,\" said Derby.\n\n\"Then let's move before someone else comes in, or that kid that Mackie thought he was talking to.\" Ryan put his coat on again and examined his gun. \"You haven't a gun on you?\"\n\n\"No.\"\n\nThey crept across the gym and up the stairs to the door of Mackie's apartment. Ryan turned the knob stealthily and the door swung open; it had not been locked. They went in and locked it behind them. Ryan waited a moment, then switched on a lamp. \"The bedroom,\" he said.\n\nThey entered it quietly. Light from the living room dimly showed the bed and the desk. From the bed came the regular sighs of deep sleep. Ryan moved to it and took out his flashlight. \"Now,\" he said. He shone the light in the sleeper's face.\n\nIt showed a narrow-eyed face, harshly lined and topped by white hair. Blinded by light, the eyes opened to panic. \"Who's that?\"\n\nRyan's hand swept under the pillow for a weapon. There was none.\n\n\"Police,\" he said. \"You're under arrest for murder, Mackie.\"\n\nMackie tried to see beyond the glare of light that Ryan kept in his eyes.\n\n\"Did Beef send you?\" he demanded. \"Is that you, Beef? This is a hell of a way to collect.\"\n\n\"This is the law, Mackie. I'm pinching you for killing the old lady last November. Want to tell me about it and make it easy for everyone?\"\n\n\"What are you talking about? Lemme see you!\" He started to throw the covers aside. Ken Derby reached across Ryan to push Mackie back and when he did Ryan felt a gun bulging Derby's hip pocket. \"Start talkin', you,\" growled Derby.\n\nBut Mackie had nerve. He found the chain to his bedlight, pulled it and saw them better.\n\n\"What the hell is this?\"\n\n\"Why'd you leave town last November?\" said Ryan.\n\n\"What's it to you?\"\n\n\"Why'd you do it?\"\n\n\"Are you really law?\"\n\nRyan showed his badge.\n\nMackie said, \"I drop a little dough on some races about that time.\" He sat up. The white hair made him look old, but Ryan could tell it was prematurely white. \"Some bookmakers were getting tough. I thought it'd be simpler to get out of town until I could get the dough together to pay them off.\"\n\n\"Did you?\"\n\n\"They're taken care of now, except Beef Wurtz. I sent word to him tonight I'd be square in a week. That's why I thought\u2014\"\n\nDerby pushed Mackie's shoulder contemptuously. \"Don't try to hand us that. You killed the old lady to help pay your debts.\"\n\n\"Stop that,\" Ryan told him. \"Get out of bed, you.\"\n\nMackie complied slowly. Ryan switched on the room's center light and studied Mackie, a tall, thin figure in pale blue pajamas. But a muscular sinuousness belied the white hair; Mackie probably could punch. His appearance bothered Ryan.\n\n\"Why'd you dye your hair?\"\n\nMackie looked indignant. He said, \"That's a fathead cop question if I ever heard one. My hair changed two years ago. It just happened. Ask anybody who knows me.\"\n\n\"Walk over to the bureau.\"\n\n\"Why?\"\n\n\"Walk.\"\n\n\"Get going!\" Derby shoved him.\n\n\"I told you to lay off,\" said Ryan. \"I'm running this.\"\n\nMackie stood at the bureau. \"Take out the middle drawer,\" Ryan said.\n\nMackie looked puzzled, but he yanked at the drawer. It made an odd scraping noise.\n\n\"Put it down on the floor. Now reach around behind it and give me what's stuck there.\"\n\nKneeling, Mackie felt around the drawer and encountered the taped link. He pulled it off and looked at it curiously. Then he extended it to Ryan. His eyes were blank.\n\n\"This what you meant?\"\n\n\"That's it.\"\n\n\"What is it?\" asked Mackie.\n\n\"You know damn well what it is,\" said Ken Derby. \"You dropped it the day you killed the old lady on Sixty-first Street.\"\n\n\"Just a minute, sonny,\" said Mackie. His eyes became gimlets; for the first time Ryan saw the resemblance between him and Harry Derby. \"Maybe this guy's a cop\"\u2014he gestured toward Ryan\u2014\"I don't know. But I know damned well you're not. You're young Derby.\"\n\n\"So what?\" Derby moved ominously forward. Ryan moved with him. He had it now. He had it all.\n\nMackie crouched, his hands moving to a boxer's defense. Derby's left hand went for his gun.\n\n\"Put that away, Derby!\" Ryan started for his own gun.\n\n\"You're not going to frame me to save your brother,\" said Mackie.\n\nDerby's gun came up and Ryan hit his arm; the gun exploded and its slug tore the ceiling. Derby swung the gun around. To Ryan the barrel's opening looked like a tunnel entrance.\n\n\"Put it away.\"\n\nDerby's jaw was set. \"I'm going to get a confession out of this guy, whether you like it or not.\"\n\n\"No, you're not. Because he's innocent. All you had to do was to see his face when he found the cuff link. And there's his hair. In the police records, that are four or five years old at least, his description might sound like Harry's. But not since his hair has turned white. No witness would mistake one for the other. And anyway, it sounds like he can prove an alibi for having left town.\"\n\nJust out of Derby's range of vision Mackie was edging toward the door. Ryan kept talking loudly.\n\n\"Last night when I was searching the bathroom. I heard a scraping noise. By the time I looked out, you were over by the door. You said you heard the noise too. But as a matter of fact, maybe you made that noise. Eh, Derby? The same noise that Mackie made when he yanked the drawer out just now? He's right. You were framing him.\"\n\nMackie was out the door and forgotten by Derby. Ryan said. \"Just one question. Where'd you get the cuff link?\"\n\nDerby took a breath. \"I found it in the truck,\" he said.\n\n\"A pair of them?\" asked Ryan. And answered his own question. \"Of course, a pair of them. You dropped the other one in the Connors' apartment.\"\n\nRealization of what Ryan had said swept Derby. He turned and discovered Mackie gone. He moved to the door, turned again and Ryan saw in his face that he knew. So now he would start shooting.\n\nAs Ryan simultaneously jumped and crouched to one side, he thought incongruously that he had been right. The luck had run out. He hurled himself forward in a flying tackle and felt a hot sear along his neck even as fire flashed in his face. Then darkness.\n\nHad he been blinded?\n\nOn the floor he pulled out his gun, then there was light\u2014a little\u2014from the doorway, and he saw Derby plunge through it.\n\nRyan jumped up, ran to the door and leveled his gun. But not in time. Derby was making a clatter down the stairs.\n\nRyan went more quietly after him, his mind working as fast as his body. He had fallen, so Derby might think he was dead or at least unconscious. Now Derby was going for Mackie. If Derby got them both, it would be perfect\u2014for Derby. He could tell the entire story as they had reconstructed it. Mackie, the supposed murderer, would be dead; so would Ryan. Harry could be freed and Ken unsuspected.\n\nThat was the one thing he must not let happen.\n\nHe leaped down the last few steps. The second floor was empty. From below he heard Derby bellow, \"Get away from that phone,\" and then a shot. He hoped he had won Mackie enough time to get the call through. But he hadn't.\n\nThere came the unnatural, animal scream of a man in agony.\n\nRyan went on. The first floor was hot with vapor. He heard the hiss of steam.\n\nThe old man who had admitted them sat on the floor near the telephone, looking puzzled. Blood spilled over the fingers he held to his throat. The telephone instrument dangled from its pay box, and the door beyond, leading into the steam baths, showed no light. But white vapor floated out through it. The hissing became a high keening.\n\nGetting through the door was the critical thing. The light would be behind him. Ryan gathered himself and burst into steaming darkness. He heard the hiss come toward him like a snake and ducked; a searing stream passed above his head. One of them was playing the live steam hose around the room. A second later there came another scream, close. Its horror penetrated his nerve fibers. A gun fired repeatedly and the blasts made a steady, echoing roar in that confined, heavy-aired space. Ryan moved toward the leaping flashes but did not dare shoot. Mackie was nearby, too.\n\nSomeone fell to the wet floor heavily, and the hissing became erratic. Whoever had been holding the hose had been hit and now it was free to writhe.\n\nBlinding pain sprayed his ankles; he jumped in agony and cried out.\n\nHe bumped a body. A pistol barrel dug his side\u2014and fired. The sound came to him muffled by his own clothing.\n\nSomething he had never felt before attacked Ryan's belly; it was nauseated and prickly hot, and cold, all at once.\n\nDeliberately he reached out and felt the jacket Derby was wearing, brought his own gun up, pushed it into Derby's side and pulled the trigger. This was all that he had to do now and he wanted to do it well. He felt himself falling, and as he fired again he aimed upward, to allow for that. He heard Derby's knees hit the floor. He did not hear himself go down. He aimed lower and fired again.\n\nHe knew he was dying. He heard a sound and he aimed his pistol in darkness toward the sound and fired its last shots.\n\nHaving done all he could, he sank into unconsciousness.\nCHAPTER 27\n\nA Word With The Chief Inspector\n\nFor an endless time there were white shapes and indistinct voices, and blurs of sound and feeling, and medicated smells. When he became conscious it was so slow and weak a process that he was not conscious of consciousness, or that there had been a change. During the period of a long morning he became gradually aware of a bed and sheets and of a ceiling, of nurses who came frequently and a male voice that called him \"old kid\" with accented heartiness. Then time separated itself into periods and brought awareness, and Ryan knew he was in a hospital, and that his chest hurt, and sometimes his mother was there, and Eleanor. There seemed to be a great deal of concern about his temperature.\n\nOnce he caught the flick of a brown coat out in the hall, and he hoped with sudden illogicality that it was Gee Gee; she must have missed the number on the door and would come back. But she did not, and even while he waited for her to, he realized how ridiculous it was. That was over.\n\nThen one day the doorway darkened and a man in a blue overcoat said, \"Neill? Can I come in?\" and it was Lieutenant Bauer.\n\nBauer shook hands and took off his coat, and said in his soft way, \"The doctor said we could talk a little. But if you're not up to it, just say so.\"\n\n\"I'm fine.\"\n\nBauer sat down. \"It's about Derby. Both Derbys.\"\n\n\"Both?\"\n\n\"Well, mainly Ken. The district attorney's office is anxious that nothing goes wrong with the case. Harry's out on bail.\"\n\n\"On bail?\"\n\n\"Well, Farragut put in quite a plea, and I guess after he spent time in the death house, the court figured maybe he was entitled to a little freedom. You know judges. But he's going up eventually, no doubt about it.\"\n\n\"Paul, tell me something. How'd I get here?\"\n\nBauer looked stunned. \"Hasn't anyone told you?\"\n\n\"This is the first chance I've had to think about it. Yesterday I was kind of groggy.\"\n\n\"Yesterday? You've been here nine days, Neill. This is just the first day you've been thinking straight.\"\n\nRyan said, \"I have?\" Then, \"I guess Mackie got the call through to the department, eh?\"\n\n\"No. Mackie never called us. It was an RMP that hauled you out. The boys were cruising the street when they heard the shooting. When they got inside\u2014well, I guess you know better than anyone what they found.\"\n\nRyan felt the hot steam on his ankles.\n\n\"It's funny you don't remember,\" said Bauer. \"Because after they got you out, you were lucid as anything. You explained the whole case against Ken Derby to one of the uniformed men. Then you conked out.\"\n\nHe shot a quick look at Ryan. \"They found the three of you in the steam room with the hose jumping around. Apparently that daffy Mackie had got it out. The old man on the desk was dying of a bullet through the Adam's apple. Mackie was wounded, but not seriously\u2014there were two slugs in him, both from Derby's gun. Your gun was empty. You'd hit Derby twice. Once right through the mouth, and once near the heart. It's a wonder he's alive. Could you see in there, Neill?\"\n\n\"No.\"\n\n\"You shoot real good by ear.\"\n\n\"An RMP crew?\"\n\n\"Sure. Derby fired at them, then gave up. They shut off the steam and carried you out.\"\n\nRyan felt a wave of gratitude. It had been a department operation after all. He had tried to carry it by himself, but in the end it had taken the department, the alert, ever-present department, that was always ready for trouble and knew how to meet it. Ryan's eyes stung, and he knew from that how weak he was. He hoped Bauer did not notice.\n\nIf Bauer noticed he was tactful.\n\n\"Under Derby's bed\u2014Ken Derby, that is\u2014we found his bag packed and a ticket to Montreal, and a letter that he was going to mail if the frame against Mackie didn't work.\"\n\n\"He wasn't going to let Harry burn then.\"\n\n\"No. The letter admitted everything. He really liked Harry.\"\n\n\"Sure he did,\" said Ryan. \"You know he came to my house after we pinched Harry and tried to tell me Harry was making deliveries with him on the day of the murder? What a bluff!\"\n\nBauer smiled crookedly. \"I don't know. By alibiing Harry he alibied himself, and if the lie was detected, it would always be explained on the grounds of brother loyalty.\"\n\nA nurse came in and made Ryan drink something cloudy and bitter through a tube. \"Only a few minutes more,\" she warned.\n\n\"I better get down to business,\" said Bauer. \"What have you got on it\u2014what first tipped you off?\"\n\nWeak as he was, Ryan grinned faintly at how ridiculous it was. \"The scraping sound that a drawer made,\" he said. \"That's what did it. All the other indications had been there all the time; I suppose I'd noticed them subconsciously. But it was when I heard the scraping sound again that I realized Ken might have made it scrape the first night we were in Mackie's. And if he had, then he had framed Mackie. Why? To help his brother, of course. And then\u2014it was like a spark jumping a spark gap\u2014why couldn't he have done it to help himself? Why couldn't he be the killer?\n\n\"He'd been in the neighborhood that day, as D'Tela had made clear. He looked a lot like Harry and could be mistaken for him. He wanted money\u2014for Harry perhaps, but also perhaps for an expensive girl friend his sister mentioned. He was left-handed, a characteristic of the killer, if you'll remember how all Mrs. Connors' injuries were on the right side of her head. Harry was right-handed of course as I know. I saw them both fight. And he certainly might have Harry's jacket and his gun available to him.\"\n\nBauer nodded. \"His sister threw some light on that.\"\n\n\"You questioned Rosemary?\"\n\n\"She came to us. She said she had misled you, unknowingly.\"\n\n\"Yes, although it was an honest mistake. The two brothers had argued about the jacket that morning and Harry had refused to loan it to Ken. So she assumed Ken had not taken it. But, as she said, she left the apartment before Harry did, and so before he might discover it was missing. Later, when he robbed the drugstore, Harry was described by the druggist as wearing dark clothes. If he'd seen it, the druggist couldn't have missed noticing that jacket. If Harry didn't have it, where was it? And once you conceive of the possibility of Ken having the jacket, like any brother who dresses first in the morning, and presumably finding the gun in it, you don't have to go much further. D'Tela had supplied the motive: Ken needed dough and D'Tela had talked of a tip he had on a horse that was running that day and would pay long odds. If Ken could pick up a little quick money, bet it and win, he'd wind up with big quick money.\"\n\n\"Yes. We'd figured it as a chancy, spur-of-the-moment job from the start.\"\n\n\"Sure. Who cases an old lady in advance? And it was even chancier than you think. He took the jacket, found the weapon and had the motive. He had a few drinks at lunch, D'Tela said, and I can testify he gets out of hand when he's had a few drinks. Everything combined.\n\n\"I figure he went into the bank to get change like a deliveryman often has to do. He saw the old lady and the hundred buck bill. Perhaps it was only chance he spotted her again, going into the apartment. In any case he saw his opportunity to follow her in, put the gun on her\u2014and escape out the back through the building behind. Remember, he worked that part of town regularly, as his boss told me, and could know these buildings pretty well. And D'Tela was busy running bundles, too. Derby had only to circle around and get back on the truck, which was the perfect escape vehicle. Who'd suspect a delivery man of an armed heist? Of course, after what happened he was afraid to flash that C-note on D'Tela.\"\n\n\"Was D'Tela in on it?\"\n\n\"No. Why need he have been? He was on and off the truck a lot that day. Ken would not have told him anything. Another thing: look how Harry's peculiar behavior confirms it. He'd found himself a new place to stay for a while\u2014he had left home determined to find one. And he wanted his jacket and gun back, naturally enough. So he came to the place where he knew he would be sure to meet Ken at the end of the day's run\u2014the trucking office. And of course he got them. He looked surly and tough, D'Tela mentioned. Why wouldn't he\u2014he was mad at Ken. And the\u2014\"\n\nHe had been about to say, \"And the C-note I figure was in the jacket pocket. Or else maybe he borrowed money from Ken and got it that way.\" But he caught that before it got out. Instead, he said, \"And the two brothers must have talked things over a little; for when we arrested him Harry showed from what he said that he knew an old woman had been robbed. He didn't know she was dead\u2014maybe Ken himself didn't know or wasn't sure when they met. But that's why Harry was walking the streets so openly later on\u2014he had no idea a murder rap was involved, or that he had been identified in it. He hadn't seen the early tabloids. When he found out, he clammed up to protect himself and his brother.\"\n\n\"Do you think he'd have gone to the chair for Ken?\"\n\n\"I don't know. Maybe it sounds nuts, but if you'd talked to Harry in Sing Sing\u2014well, he knew he was trapped one way or the other. And they sure were devoted, as Rosemary said. Look how fast Ken moved and the chances he took to frame Mackie, as soon as he learned about the cuff link. I should never have mentioned that to Rosemary but I guess Ken was so close to the whole thing I never dreamt... And Mackie looked hot.\"\n\n\"I'm sorry, sir,\" said the nurse firmly.\n\n\"Okay, Miss.\" Bauer got up. \"One other thing. Ken said he knew that you could never be interested in his sister now, and he felt bad about it.\" Bauer looked down at Ryan. \"Whatever that means. She told me she's taking a job out in Milwaukee next month. Well, don't worry about a thing.\"\n\nRyan grinned. \"I don't. I'll be up in few days.\"\n\n\"Sure you will,\" said Bauer hastily. It was not until he left that Ryan realized Bauer had not been referring to his convalescence.\n\nNext day his mother visited him, proudly bearing a new scrapbook, in which she had pasted the newspaper accounts of the gun battle at Mackie's. All the stories explained that even after Harry Derby's conviction Ryan had not been satisfied with the result and had continued working on the case in his spare time, with the result that Harry would now be tried for the drugstore robbery and Ken for the Connors murder. One of the most laudatory stories was bylined \"By Jack Sandalwood.\" None of them mentioned anything about planted evidence.\n\nOne item in the scrapbook was a Walter Winchell gossip column, in which one line was ringed in ink:\n\n\"Gee Gee Hawes, the star attraction, is holding hands with what star reporter?\"\n\nOn his last day in the hospital Ryan received a new Faulkner novel in which was a note. \"Good-by and thanks. I'm getting away from here. Too many people know too many things. And now there's only me left, anyway. A girl in Milwaukee has invited me to stay with her. I'm leaving the middle of next month.\" It was signed \"R.\"\n\nHe returned to the precinct on an unseasonably warm afternoon that made people think of things to come, like flower peddlers' wagons and three weeks on the Cape, and cotton dresses and seersucker suits. He walked up the steps and inside the old, cool building, and Sergeant Weiner said, \"How you feeling, boy?\" and put out a hand of welcome. \"Paul wants to see you when you got a moment.\"\n\n\"Thanks, Al.\"\n\nLee Lambert came down the stairs. \"Hi, Lee,\" said Ryan, and Lambert looked at him and said, \"Hi,\" and went on out. Lambert was in a hurry.\n\nRyan started up the stairs. He still had to take these easy. A young patrolman started down at the same time; Ryan recognized him as the rookie who had begun here about the same time he did. The rookie passed him without a glance and without even giving him half the stairs. Everyone seemed in a hurry.\n\nBauer and a stranger were going over papers at Bauer's desk. Bauer said, \"Lieutenant Zimmer, this is Neill Ryan.\" Zimmer said, \"Hello.\" Bauer shook hands and asked how he felt. Ryan said fine.\n\n\"Jerry and I are going over everything because I pull out next week, and he's taking over,\" said Bauer. \"I'm going downtown.\"\n\n\"I'm glad for you,\" said Ryan. \"But I'm sorry you're leaving, Lieutenant.\" Zimmer's presence made the formal salutation necessary.\n\n\"He's Acting Captain now, Ryan,\" said Zimmer, and added, \"Don't feel too badly. You won't be here much longer either.\" For the first time Zimmer smiled.\n\nSo did Bauer, but his expression was pleasant. \"It came through yesterday, Neill. You're going up to Harlem.\"\n\nRyan looked at them. Harlem?\n\n\"I never asked for duty there,\" he said. \"Or even for a transfer. How come?\"\n\n\"You've done a hell of a job, Neill,\" said Bauer. \"Don't let anyone tell you different.\" He looked at Zimmer. \"They need good men up there. You know what the Twenty-fifth is like. Something doing every minute. It's a compliment.\"\n\nRyan's knees felt shaky. Things were coming too fast. Three months ago he would have killed himself to win words like that from Paul Bauer. Now they didn't sit right. Maybe it was Zimmer, coldly watching him. Zimmer picked up a piece of paper from his desk and tossed it to him. \"Here's something else.\" Ryan knew he would not like it.\n\nIt was a teletype. Patrolmen Arne Sieger and T. di Paolo at ten-twenty a.m. that day, driving a patrol car along Amsterdam Avenue, had seen a man run from a liquor store and had challenged him. The man ran and then fired back at the police car. One of his shots hit di Paolo's pistol, knocking it from his hand and spraining his wrist. Sieger had driven abreast of the gunman, called on him to drop his gun and then shot him through the head. The dead bandit had been identified as Harry Derby, thirty-seven years old, out on bail pending trial for a drugstore holdup.\n\n\"Your boy friend,\" said Zimmer. He rose suddenly as though he could stand this no longer. \"I'll put these back in the file, Paul.\" He went out, walking on hard heels.\n\nIt had taken a while but Ryan got it now.\n\nBauer saw that in his face. \"Relax,\" he said. \"They'll get over this. You know how it is. Everyone gets edgy when a cop gets hit.\"\n\n\"Like hell,\" said Ryan. \"The one thing everyone remembers is I sprung Derby. If he'd been in the death house he'd never have shot anyone.\"\n\n\"He didn't hurt anyone,\" said Bauer. \"What's a sprained wrist?\"\n\n\"Tell that to the guy he shot at,\" said Ryan. \"And why am I being transferred?\"\n\n\"Look, Neill. You couldn't expect to say here forever. You got to move around, see the job at different levels, in different areas. A year in Harlem, a year 'way downtown, a couple years with one of the specials\u2014you'll be learning all the time. Don't think everyone feels like Zimmer. Most people feel like I do. You'll see.\"\n\nRyan's heart was pounding. What the hell were they doing to him?\n\n\"I mean it, Neill.\" Bauer spoke sharply, making Ryan look at him. \"Any time I have a chance to have you working for me, I want you. Know why? Because you're level. I don't mean the others aren't. Zimmer's honest, too, according to his lights. But you've proven what you are by what you did for Derby, and his being killed has nothing to do with it. Zimmer don't understand that, but you do and you've got to allow for the Zimmers. If you don't like that, and can't stand it, this place isn't for you. Otherwise\u2014get the hell up to Harlem.\"\n\nBauer had never raised his voice. But when he put out his hand. Ryan took it.\n\nWhen Ryan went back downstairs a familiar figure\u2014incredible!\u2014was leaning against the desk, talking to Weiner with loud confidence. Time had turned backward.\n\n\"For the love of mike\u2014Jabby!\"\n\nJablonski swung around. His gray sports jacket had Jabby embroidered in white silk across the left pocket. \"Neill! I was just askin' about you.\"\n\nRyan was genuinely glad to see Jablonski, and Jablonski was so stirred he replaced his half-smoked cigar with a new one.\n\n\"I dropped in to pick up a picture,\" he said. \"How long you out of the hospital?\"\n\n\"A week. A picture?\"\n\n\"Yeah. One of the guys on the Teley got a good shot of us that night we came in with Derby. I thought I'd put it up in my joint. People are interested in things like that.\"\n\n\"Sure.\" Ryan looked at the photograph Jabby pulled from its envelope. It showed a confusion of uniforms and reporters in this very entrance. Jabby was a hat and a half-face in the background.\n\n\"If you're going uptown I'll give you a lift,\" said Jablonski. \"I got the station wagon.\"\n\nWeiner and Ryan looked at each other.\n\n\"Who drove in, Jab?\" asked Al Weiner. \"You or your man?\"\n\n\"Oh, shut up,\" said Jablonski. \"A secondhand station wagon don't cost no more than any other car. And it's handy for groceries and things.\"\n\nThe station wagon was in the parking lot next door. Jabby started it, relit his cigar, and said,\n\n\"Hey, you know something? That guy Sandalwood was around here earlier. I guess he doesn't know I'm retired.\" Jablonski swelled importantly. \"He saw me and he says, 'Hey, give your partner Ryan a message for me.' So I say, 'What?' He says, 'Tell Ryan that Derby's getting killed this morning is the luckiest thing that ever happened\u2014to Ryan. Now no one can bother him.' What do you suppose that meant?\"\n\n\"I don't know. Nothing much.\"\n\nJablonski drew comfortably on the new cigar. \"Well, I can tell you what it could have meant\u2014if that jerk Sandalwood knew anything. It could mean that with Harry dead, no one in the department need ever find anything out at all. Eh, Neill? Whatever Ken could say would be only hearsay testimony, and from a confessed murderer at that. Eh?\"\n\n\"I don't know,\" he said. He did not want to talk about it.\n\nBut Jabby was busy turning expertly in front of a cab. \"The other night I pick up a little talk,\" he said. \"One of the boys was out. He said you'd been playin' around with Derby's sister.\"\n\n\"Who was this?\"\n\n\"Look, Neill. You know I've always figured you as my\u2014what do you call at, prodigy? You've done great. I'm proud of you. But for God's sake get away from those Derbys. You mess with that dame, what'll they think of you downtown\u2014a cop married to the sister of two heist guys? You fool around, you'll be transferred out of this precinct. And this is the place to be.\"\n\nThat decided Ryan.\n\n* * * *\n\nAt ten o'clock next morning in the stiff-chaired anteroom to the chief inspector's office in New York's police headquarters a buzzer sounded. The uniformed sergeant at a desk told Ryan, \"He'll see you now.\"\n\nPatrick Pembroke's knobby face looked its usual angry inquiry. \"I wanted to ask a question, sir,\" said Ryan.\n\n\"Go ahead.\"\n\n\"Before I arrested Ken Derby I saw a little bit of his sister. She's a decent girl. Not like her brothers. But I've been told if I continued to see her, or if by any chance I should marry her, it wouldn't sit well with the department.\"\n\nPembroke paused. \"We don't usually tell a man whom he should marry.\"\n\n\"One other thing I'd like to ask, sir. I've been transferred from the Seventeenth to the Twenty-fifth Squad. I gather from hints various people have dropped that they feel I was demoted because I\u2014I went out of my way to prove Harry Derby innocent. Which he was.\"\n\nPembroke leaned back and laced bony fingers over his midriff. \"So that's it. Listen, Ryan. No man in my department ever has to be afraid of honesty. You're not being demoted. We're transferring four teams into Twenty-five\u2014all of them picked for ability and guts. You were picked. Tell that to the first son of a bitch who suggests different. As for the girl, marry whoever you please. And if anyone doesn't like it, take a poke at him.\"\n\n\"Yes, sir. Thank you.\" He stood up.\n\nPembroke said, \"By the way. What were you going to do if I had said anything else?\"\n\n\"Well, sir, it was just that I had to get something straight. And I preferred to get it straight from the top.\"\n\n\"I asked, 'What were you going to do?'\"\n\nRyan looked levelly at him. \"I was going to send in my papers.\"\n\nRed-lashed eyes stared unfriendliness at him. \"You mind what I said about honesty before you think again of resigning,\" he said. \"And remember me to your mother. I think I've not seen her since her wedding day.\"\n\n* * * *\n\nIt was only an inexpensive ring but Ryan bought it in a Fifth Avenue store because he did not want her to be ashamed of it. Then he took a cab to the dingy building. It was warm and sunny; the front door was open. Ryan went slowly up the flights of stairs and when he got to her floor, he found the door open. She was sitting in a dark blue flannel dressing gown, sipping breakfast coffee and reading the Mirror.\n\nShe glanced up. \"Look who's here!\"\n\nShe was alone. He was glad of that. \"Can I come in?\"\n\n\"Why not?\"\n\n\"Catch,\" he said and threw the little cube-shaped package.\n\n\"What's this?\"\n\n\"I picked it up on my way. I had to go down to headquarters before\u2014before buying it. I had to straighten something out. It's straightened out. I picked this up on my way back.\"\n\nHe looked at her a long moment. He had been deciding a lot of things, and he knew he had made the right decisions. \"It's an engagement ring. Take it or leave it.\"\n\n\"An engagement\u2014\"\n\nHe knew that for once at least he had really surprised her. \"Neill!\"\n\n\"The last time I saw you,\" Ryan said, \"I did a lot of talking about apartments in Queens and having babies and so on. I still think that's a good idea. But in the hospital I had time to do a lot more thinking. You were what I thought of most, Gee Gee. I want to marry you more than anything in the world. If you want to go on dancing and so on, I guess it'll have to be all right with me, at least for a while. But on the other hand, I've got to tell you that if another one of these Derby deals come up, I'll decide it the same way. Because that's the way I am. And also you better remember what I've told you about a cop's pay before you make any decisions and think of the things I won't be able to give you\u2014\"\n\nShe was bent over the ring, putting it on, the coppery hair hiding her face. Now she held her hand up for him and Ryan saw that there were tears on Gee Gee's cheeks.\n\n\"That's not much of a rock,\" he said. \"It probably doesn't fit anyway.\"\n\n\"It fits perfectly,\" she said tremulously. \"And now it's on, it's never coming off. Never.\"\n\nSuddenly she was in his arms and their mouths found each other's. \"You think I'm going to give up the only guy that's ever treated me like a\u2014like a girl, instead of just something to grab?\"\n\nAfter a long time Ryan said huskily, \"I think we better get married this afternoon.\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nIn His Own (w)Rite\n\nBy Michael R. Poll\n\nCopyright \u00a9 Michael R. Poll 2011\n\nAll rights reserved under International and Pan-American Copyright Conventions.\n\nSmashwords License Statement\n\nThis ebook is licensed for your personal enjoyment only. This ebook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each reader. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to Smashwords.com and purchase your own copy. Thank you for respecting the hard work of this author.\n\nSmashwords Edition\n\nCornerstone Book Publishers\n\nNew Orleans, LA\n\nFirst Cornerstone Edition - 2011\n\nwww.cornerstonepublishers.com\n\nE-Book Edition ISBN: 1613420420\n\nISBN-13: 978-1-61342-042-3\n\nPrint Edition ISBN: 1-61342-019-6\n\nISBN-13: 978-1-61342-019-5\n\nMADE IN THE USA\n\nTable of Contents\n\nIntroduction\n\nFluid Masonry: The Art of Change\n\nAlbert Pike's Address Before The Grand Conmsistory of Louisiana\n\nThe Grand Constitutions of 1786\n\nThe \"White Cap\"\n\nIntegrity in Masonry\n\nWriting Masonic History\n\nThe Dwellers on the Threshold\n\nDown the Path of Proper Research\n\nDyslexia: The Gift in Disguise\n\nJames Foulhouze: A biographical study\n\nQuantity or Quality?\n\nWho Am I?\n\nWhat is Truth?\n\nDeadly Apathy\n\nIntroduction\n\nWhat's Freemasonry? Well, it pretty much depends on who you ask. In the 35+ years that I have been a Mason I have visited quite a few lodges. No two have been exactly alike. Lodges seem to develop personalities of their own, much like people. Some are relaxed and laid back, some are more formal. Some are very healthy with work going on all the time and some, sadly, are on life support. But, in each lodge when I've talked to the individual members, they have expressed a true caring about their membership. Sure, it does not mean the same thing to each one of them. The one in the kaki pants and bright Hawaiian shirt with the donut in his hand might view the reason for his membership differently than the one in the tux with the white gloves and a glass of wine, but, so what? Who said Masonry has to mean the same thing to everyone?\n\nThe common denominator in all lodges is that Masonry lifts Masons up a bit more from where they started. Not everyone is a philosopher and not everyone will draw the deeper meanings from the Masonic ritual, but we all can benefit from being told to try and be better than we are today. Sure, few (if any) live up to the deepest teachings of Freemasonry, but is our goal perfection or the striving for it? I believe that if we just try to live as Masonry teaches, recognizing that we all fail from time to time, then we are doing what is expected of us. Masonry is not for everyone and we can not expect that it will, in any way, satisfy someone who is just not Masonic material. But, for those who are touched by Masonry, no matter what they are wearing, eating or the state of their lodge, they feel very deeply about being a Mason. It is important to them and no matter how much or little they know of the ritual or its deeper teachings, it is of value to them.\n\nOur Masonic history is important to us. It is important that we know who we are and from where we come. But, this is not really a Masonic history book. I have, however, a deep interest in the early history of the Scottish Rite. I have this interest because we have such sketchy accounts of the early days of it. While this is not a Masonic history book, I have included some history papers exploring aspects of Scottish Rite history that we might not see explored too often. This is, also, not a book of philosophy. But, I do feel that we can all benefit from the life lessons Masonry teaches. I try to teach Masonry in a way that applies to everyone, so that we all can see and experience its lessons. With this in mind, I've included some of these basic philosophical lectures and papers.\n\nSo, that's what we have here; a collection of history and philosophical papers - with some other bits I find useful or noteworthy. If this book gives you just a moment to think about yourself, your role in Masonry or gives you any cause to think of ways to improve yourself, then I consider this book a worthwhile endeavor.\n\nBe happy, enjoy life and make each beautiful moment count.\n\nMichael R. Poll\n\nFall, 2011\n\nIn His Own (w)Rite\n\nFluid Masonry: The Art of Change\n\nThe Journal of the Masonic Society, Issue 7, 2010\n\nWHEN WE BOIL FREEMASONRY DOWN to its most basic element, we find a very simple message: \"make yourself better.\" Such a statement can, however, be likened to the phrase \"be happy.\" It sounds easy enough, but how do you do it? How do you know when you are \"better?\" What is \"better?\" What seems to be an uncomplicated message becomes difficult to put into practice, even to understand. Such is the nature of symbolism.\n\nWe can start on the path of symbolic understanding by looking at nature. If you look at a beautiful mountain stream, you can find more than beauty. You can find illusion (often, the guardian of symbolism). Flowing water goes around a large rock in the stream. The illusion is that the rock is the master. What we believe to be truth is the sight of the water yielding to the rock and being forced to flow around it. We see this and accept it as truth. Our error is that we determined \"the truth\" before we gathered all the facts. In time, gentle, flowing water can reduce the largest stone to a pebble. The rock is not the master. One lesson to learn is that what we see, hear, feel and believe might well prove to be something other than fact. We must train ourselves to withhold judgment.\n\nIn Freemasonry, a subtle lesson is taught early on by putting us in a position where we cannot depend on what we can see. We are forced to depend on others for guidance. We are also forced to use senses other than those we would normally rely upon. We must change in order to adapt to this new situation. The illusion is that we have been handicapped and deprived of receiving the full benefit that would have been afforded us if we had complete use of all of our senses. But the illusion masks the fact that we are forced to adapt to our condition precisely because we have been placed in such a state. We simply can't act or perform on our own. We need guidance.\n\nThe three degrees in Craft Masonry are often said to represent the three stages of human life: youth, adulthood and old age. How do we progress through these stages? We change. As children, we play, grow and learn. As adults, we put into practice what we have learned, and in old age we impart to others what we have learned. In each stage, we change in body and mind. It is the normal way of life. What would be abnormal is if no change took place.\n\nLet's look again at the water and the rock. The gentle, flowing water cannot and does not break the rock by direct force. Water changes its direction and flows around the rock; in doing so, it also gradually affects change in the rock. The gentle pressures of the water force the rock to give way and reduce itself in size. The rock is not the master after all. Change is one of the unavoidable facts of all existence. Any attempt to avoid change only results in unnatural waste of energy.\n\nThe lessons of Masonry are such that we must study them with a child's open and willing mind. In certain aspects of our teachings, we might remember we are told that it is not acceptable to bring \"innovations\" into the body of Masonry. An innovation is change. Are we being told that we cannot or should not change? Not at all.\n\nAs individuals, we change every day of our lives. We grow older, which brings physical and mental changes. We have no choice in these types of changes. We also have the option to make free-will choices in our lives. We might opt to eat a more healthful diet, to exercise, or in some way improve our lives. There are countless changes that we can choose. We also might make the decision not to make any free-will changes. It is our choice as individuals.\n\nBut when we speak of innovations in Masonry, we are speaking of something quite different. The innovations that are made in Masonry should never be the choice of any single individual. Changes should be the collective will of the membership. In Masonry it is the lodge, not the Worshipful Master, who decides the direction to be taken. The Worshipful Master only steers the ship in the desired direction.\n\nIn our Grand Lodges, we see change every year. We see resolutions presented and voted on. It is rare that a Grand Lodge will see no change whatsoever in its nature after a Grand Lodge session. Change is normal. Change is expected.\n\nIn Masonry, the changes we see in its nature often mirror the changes we see in wider society. Freemasons are part of society and we interact with others on a daily basis. It would be unnatural for us to be social outcasts. If we look back at Masonry 50 or 100 years ago, or even longer, we see that the nature of Masonry matched that of society in both simple matters of dress and deep social or philosophical issues. Even today, we see social difference in Masonry depending on the location of the lodge and its membership. In a large city, you might see lodge members dressing in a different manner than you would see in a small town. One is not right and the other wrong, they are just simple differences in the social norms of the areas.\n\nWhen we look at society and speak of a large nation, it would be uninformed to not realize that society's concepts of what is acceptable and unacceptable vary from community to community. The overall social structure of a large area allows for change and variations within smaller areas. Speed limits might change from one place to another, as well as many other community-based laws, but where will you find murder legal? Society as a whole has limits as to what are acceptable standards.\n\nBecause Grand Lodges are sovereign and free to pass the rules and laws of their liking, it would seem highly improbable that you would find two Grand Lodges with exactly the same set of governing laws. If one Grand Lodge changed its laws to require all members to wear tuxedos to lodge, it might draw a level of interest from some other Grand Lodges, but that would be about it. If the same Grand Lodge removed the Volume of Sacred Law from its altars, then not only would this attract the attention of other Grand Lodges, but they would view this Grand Lodge as moving outside of what is considered acceptable, and the breaking of fraternal relations with this jurisdiction might follow. By the same turn, if most Grand Lodges adopted a particular policy which they felt was extremely important, then those few Grand Lodges not adopting the policy might also be viewed as unacceptable or out of step.\n\nChange is not the enemy of Masonry. Just as the water in a stream changes direction as it flows in and around various obstacles, so should we recognize that change is not only inevitable, but is in our best interest. In a storm, it is the strong, unyielding tree, not the flexible blade of grass, that is in most danger of breaking.\n\n***\n\nAlbert Pike's Address Before\n\nThe Grand Consistory of Louisiana\n\nI believe readily that you did not want the office, but the office wanted you.\n\n\\- Charles Laffon de Lad\u00e9bat to Albert Pike 1\n\nTHE PASSAGE OF YEARS can sometimes elevate a historical figure into a legend. This is not always beneficial when a study of the individual is desired. A historical figure can be examined and their actions understood from a human perspective. A legend, however, can take on near supernatural qualities and the whole of their activities are sometimes not expected to be understood, explained or completely recounted. Such is, at times, the case with Albert Pike. It is often difficult to imagine Albert Pike as a player (rather than as the player) in American Scottish Rite events of the 1800s. The monumental mark that Pike left on the Southern Jurisdiction can mask the fact that his influence was not always as profound as it was in his later years. Regardless of his many accomplishments, there was a time when Illustrious Brother Pike was but an inexperienced, yet promising, Mason with a blank book before him upon which it was unknown exactly what would be written.\n\nThis address, the first ever given by Pike as the presiding officer of a Scottish Rite body, gives us a rare look at the early Albert Pike. While in his later years, Pike was viewed by many as a true Master of the Scottish Rite, this address clearly calls into notice his immaturity in the Rite, and he asks for \"lenient judgment\" upon his \"short-comings\". In his address Pike is clearly humble and seems sincerely appreciative of his election. He also notes that his election to the position of Commander in Chief was politic in nature and due to \"circumstances that surround us\". What could have caused a political election of the untried Albert Pike as the presiding officer of the Grand Consistory of Louisiana? Let's look at the \"circumstances\".\n\nThe Turmoil that was Louisiana Masonry\n\nJust seven years prior to Pike's assuming the leadership of the Grand Consistory of Louisiana, the whole of Louisiana Masonry underwent a dramatic shift in direction, leadership, and character. The once French dominated Grand Lodge of Louisiana became \"American\" in nature. This shift mirrored the cultural changes taking place in New Orleans and other French areas of the state. Louisiana was founded as a French colony. Even after the territory became a state in 1812, the French influence was the dominate force, especially in the city of New Orleans. Not only was the Grand Lodge of Louisiana a French-speaking body, but so were the five lodges that created it. Louisiana was the most \"foreign\" Grand Lodge (as well as state) in the U.S. Over time, many did not view this as an acceptable situation. There was a desire to \"be like everyone else.\"\n\nBy the 1830s, Louisiana Masonry, as well as the whole of the Louisiana culture, began feeling intense pressure to become \"more American\". With many, this was not a welcome change. Bitter disputes and unyielding divisions developed that culminated in actual violent clashes between the \"Creoles\" and \"Americans\" in the downtown New Orleans streets. The Grand Lodge was not immune to these cultural divisions which often manifested themselves in the different rites worked by the Louisiana craft lodges. Unlike the other U.S. Grand Loges, the craft lodges in Louisiana did not only work in the Preston-Webb (York Rite) craft ritual, but also in the French or Modern Rite and the Ancient and Accepted Scottish Rite craft rituals. For the most part, the French interests were championed by the lodges working the French or Modern and A.&A.S.R. Rites and the American interests by the York Rite lodges. The 1844 Constitution of the Grand Lodge of Louisiana was the last straw for many York Rite craft Masons. The new constitution officially recognized the, then, three rites working in Louisiana and sanctioned the creation of a \"Chamber of Rites\" to supervise the work of the lodges. The York position was that there should be only one recognized rite for Louisiana craft lodges (York Rite ) and that the Grand Lodge should be made to conform to the same system as worked by the other U.S. Grand Lodges.\n\nA committee of English-speaking York Rite Masons, frustrated by the lack of accommodation they perceived in the Grand Lodge, approached the Grand Lodge of Mississippi and submitted a letter of grievance on January 23, 1845.2 They charged the Grand Lodge of Louisiana with irregularity due to its practice and acknowledgment of various craft lodge rituals. After debate, the Grand Lodge of Mississippi agreed with the charges, declared the jurisdiction of the Grand Lodge of Louisiana as \"open territory\" and, by 1848, chartered seven lodges in Louisiana.3 On March 8, 1848 these seven lodges formed a second Grand Lodge within Louisiana. John Gedge, who had spearheaded the \"rebellion\" was elected Grand Master of the \"Louisiana Grand Lodge of Ancient York Masons\". While this new Grand Lodge received recognition from only the Grand Lodge of Mississippi, its future was not nearly as bleak as it might seem.\n\nThe Grand Lodge of Louisiana was created in a manner to accommodate the needs of the lodges which organized it. The Grand Lodge was created French in nature because this was the culture of the vast majority of those living in the area of the Grand Lodge at that time. Over the years that followed, the Grand Lodge continued to exist and operate in the manner in which it was created. The majority of the membership of the lodges under the jurisdiction of the Grand Lodge, however, changed from French to American. The Grand Lodge was then viewed, by the majority, as not accommodating their wants and needs.\n\nThe Grand Lodge of Mississippi received admonitions from most U.S. Grand Lodges for their actions in Louisiana, with the majority openly condemning its activities.4 With the exception of the Grand Lodge of Mississippi, no U.S. Grand Lodge entered into relations with the new Louisiana Grand Lodge. Regardless of their seemingly advantageous position, the Grand Lodge of Louisiana was in serious trouble.\n\nOutside of New Orleans, there were a few pockets where the French culture was strong, but the majority of the state was already (or was becoming) Americanized. The events surrounding the creation of the Louisiana Grand Lodge buckled the knees of the Grand Lodge because most of the lodges under this new Grand Lodge were located in the New Orleans area -perceived to be the largest stronghold of the French culture within the state as well as the home of the Grand Lodge. The fact that the Grand Lodge of Louisiana was overwhelmingly considered to be the \"regular\" Grand Lodge in Louisiana was not sufficient to overcome the internal problems stemming from the cultural divisions in New Orleans. By mid 1849, it was realized that the English-speaking lodges that had remained loyal to the Grand Lodge were showing signs that continued loyalty would, most likely, not happen. Contributing to the dilemma was divisions between the French-speaking New Orleans Masons.\n\nObviously realizing that the total collapse of the Grand Lodge of Louisiana was a very real possibility, the Grand Lodge and the Louisiana Grand Lodge, A. Y.M. entered into discussions in 1849 designed to merge the two bodies.5 That merger took place in June of 1850 with the approval of a new Constitution of the Grand Lodge of Louisiana of Free and Accepted Masons. Under the terms of the agreement of the merger, the Louisiana Grand Lodge, A. Y.M. members declared irregular would be healed by the Grand Lodge of Louisiana, F.&A.M.. All Lodges chartered by the Louisiana Grand Lodge, A. Y.M. (or by the Grand Lodge of Mississippi in Louisiana) would, also, pass under the jurisdiction of the new Grand Lodge of Louisiana, F.&A.M. John Gedge, who had served as Grand Master of the Louisiana Grand Lodge, A. Y.M., was elected Grand Master of the new Grand Lodge of Louisiana, F.&A.M. for 1851.\n\nWhile this new constitution seemed to merge the two Grand Lodges, the Grand Lodge of Louisiana was, in reality, replaced by the Louisiana Grand Lodge, A. Y.M.. All that actually remained of the old Grand Lodge was the name, organizational date of 1812, and the list of Past Grand Masters. The nature of the new Grand Lodge of Louisiana, F.&A.M. changed to match the Louisiana Grand Lodge, A. Y.M.. The \"Americans\" were in power.\n\nThe old Grand Lodge of Louisiana officially accommodated lodges working in the York, French, or Modern, and A.&A.S.R. craft rituals. The French-speaking Masons believed that the two Grand Lodge merger would result in the continued recognition of lodges working in all three rites. They were horrified and outraged when the new Grand Lodge instructed all non-York Rite lodges to turn in their charters so that York Rite charters could be issued.6 Charges of trickery abounded. Three A.&A.S.R. craft lodges (Etoile Polaire, Disciples of the Masonic Senate, and Los Amigos del Orden) applied to the Supreme Council of Louisiana for relief. The Supreme Council announced that since an 1833 concordat between the Grand Lodge of Louisiana and the Grand Consistory of Louisiana (at that time the highest ranking Scottish Rite body in the State) to assure that the Grand Lodge would provide a home for the Scottish Rite craft lodges had been violated by the new Grand Lodge, the Supreme Council would issue charters to these lodges and allow them to pass under its jurisdiction.7\n\nThe French Rite Masons did not have a Grand Body from which to seek relief. The Grand Lodge had been the home of the French Rite. With no superior body for the government of the French Rite lodges, they would, ultimately, disappear from Louisiana Masonry as an identifiable force.8\n\nThe Setting for More Change\n\nWhen we step back and attempt to look at the situation through the eyes of the participants, we can see that the Supreme Council of Louisiana taking jurisdiction over the three A.&A.S.R. Craft Lodges must have been just as jarring to the new Grand Lodge of Louisiana as the action of the Grand Lodge of Mississippi was to the old Grand Lodge. No one could see or know the future. The Grand Lodge of Mississippi had been a body in full fraternal relations with the old Grand Lodge, as was the Supreme Council of Louisiana. While the Grand Lodge of Mississippi was a sister Grand Lodge, the Supreme Council of Louisiana was composed of members who were nearly all Grand Lodge officers, a good number of whom were Past Grand Masters. The Supreme Council of Louisiana was not an insignificant body. The actions of the Grand Lodge of Mississippi set into motion a series of events that led to the downfall of the old Grand Lodge of Louisiana. It was not unfeasible for the actions of the Supreme Council of Louisiana to result in the same fate for the new Grand Lodge of Louisiana. Clearly this situation needed to be addressed by the new Grand Lodge.\n\nAt the invitation of Grand Master John Gedge, Albert Mackey came to New Orleans in late 1851\/early 1852 and established, for the Charleston Supreme Council, a Consistory of the 32\u00b0. Gedge was appointed Commander in Chief of this new consistory. Obviously, the Supreme Council of Louisiana charged that this was an outrageous invasion of territory.\n\nNot only was it the fact that the Consistory was organized in New Orleans, but the manner in which it was created was the subject of severe criticism. In 1853, Charles Laffon de Lad\u00e9bat wrote about the events concerning the new Grand Lodge, the Supreme Council of Louisiana and the new Charleston Consistory in New Orleans.\n\n\"In presence of such despotic, anti-masonic conduct, the Scotch BB:. resisted as men, as Masons, and formed an independent corporation under the only M:. authority existing in Louisiana dejure et defacto. The balance remained with the new Grand Lodge, swore obedience to her, through indifference rather than from conviction. Soon after this, the very same Sectarian, in his restlessness, caused Br:. Albert G. Mackey to come from Charleston, in order to establish a Grand Consistory, exactly as if there never had existed a Supreme Council of the Scotch Rite in Louisiana. Our sectarian, after abolishing the Scotch Rite, wished to re-establish it in order to be at the head of it. This Consistory has been inaugurated, you know it M:. W..., for you were admitted into it for proper causes. The manner in which the degrees were conferred in this spurious Consistory is and will be an eternal shame to the Br :. who has conferred them.\"9\n\nWhile we can only speculate as to the events which might have caused this \"eternal shame\" statement, it is evident that the creation of the Charleston consistory in New Orleans fanned the flames of emotion and deeply angered the already frustrated New Orleans Scottish Rite Masons. But what could be done?\n\nThe cultural variances within New Orleans societies during the 1800s are far too complex to be explained from only a French and\/or American viewpoint. New Orleans was a cosmopolitan city with layers of cultures and subcultures. The lodges under the Grand Lodge of Louisiana were not only French and English speaking, but there were also lodges working in German, Italian, and Spanish. Like the many New Orleans neighborhoods, Masonic lodges often reflected the culture of the members of the lodge. Prior to 1850, the Grand Lodge maintained but a minimal supervision of the lodges under its jurisdiction. As long as a lodge worked within a general Masonic framework, as defined by the Grand Lodge, the lodge was left effectively alone. For some lodges (especially in rural areas) the only contact they had with the Grand Lodge was when they sent in their yearly returns. Lodges were free to develop their own cultural \"stamp\" on both their lodge and the ritual they used. Germania Lodge No.46 was created as a German-speaking lodge receiving a York Rite charter from the Grand Lodge of Louisiana in 1844. Their 1844 ritual shows that they originally worked an eclectic ritual which may well have derived from all three rites worked in Louisiana (as well as rituals from outside the state). 10 It is very possible that the unknown author(s) of this ritual simply sat down with a number of rituals and created a unique ritual to his (or their) liking. Such independent activity was not uncommon.\n\nThe freedom extended to the lodges by the Grand Lodge may have ultimately contributed to the downfall of the French interests in Louisiana. The York Rite English-speaking Masons were, by then, in the majority, but it was not a large majority. The non-York Rite Masons might have been able to overturn the actions of the new Grand Lodge, but they could not unify themselves and were split into unyielding factions with their own goals and agendas.\n\nRegardless of the influence the Supreme Council of Louisiana once had in Louisiana, the creation of the Charleston Consistory created a split that led to the demise of the Supreme Council of Louisiana as a true Masonic power. Not only was the Supreme Council of Louisiana locked in battle with the new Grand Lodge, it was also facing perplexing (in New Orleans) charges of irregularity - charges that it was not prepared to answer. The rapid fire changes involving the whole of Louisiana Masonry left most of the French Masons flabbergasted and hopelessly divided as to which direction to take. It was at this time that a new \"solution\" was introduced that cut the divisions even deeper.\n\nThe Concordat of 1855\n\nThe Scottish Rite in New Orleans existed in what might be described as a parallel universe with the rest of the U.S. A.&A.S.R. Given the cultural difference between the whole of Louisiana Masonry and the rest of the U.S., the differences and detached nature of the Scottish Rite in New Orleans is understandable. With the \"American invasion\" of Louisiana Masonry came a forced realization that changes would have to be made in the nature of all Louisiana Masonic bodies. Exactly what changes would be necessary was the subject of heated debate.\n\nThe creation of the Mackey\/Charleston Consistory in New Orleans triggered intense emotion in an already explosive environment. It was during this time and in this setting, that a plan to \"merge\" the Charleston Council and Supreme Council of Louisiana was born. For those who viewed the Supreme Council of Louisiana as the only hope of preserving the French interests in New Orleans, the idea of such a plan was wholly unacceptable. The more moderate French Masons saw such a blending of the two councils as, quite possibly, the only option left. In 1860, Charles Laffon de Lad\u00e9bat wrote to Albert Pike about the Concordat and explained his position of it.\n\n\"My resolution of retiring from active practice is 5 years old & more. Hear what I wrote to Mackey January 31,1855: \"When the work will be accomplished, when every thing will be in proper order & well understood, I will retire willingly & leave the management of all to more competent, but not more devoted hands\". We know that the foreign influence will & must be superseded by the American element. Now the time has come & I believe that, even in Masonry, Americans must rule in America. I, a frenchman, must retire -in due time.\"11\n\nNot all of the French Masons were willing to turn over what they viewed as their \"possession\" to others with different ideas, plans, and goals. When the Concordat between the two councils seemed to be inevitable, the officers and nearly half of the Active Members of the Supreme Council of Louisiana resigned or refused to take part in what they viewed as an illegal action. On January 7, 1854, the remaining Members of the council elected Charles Claiborne as the new Grand Commander, Claude Pierre Samory as Lt. Grand Commander, and Charles Laffon de Lad\u00e9bat was appointed Grand Secretary. The Concordat between the Charleston and New Orleans Supreme Councils was signed in New Orleans on February 16, 1855. The Supreme Council of Louisiana downgraded itself downward into the Grand Consistory of Louisiana and the Grand Consistory absorbed the \"Mackey\" Consistory.\n\nWith the Concordat of 1855, the elimination of the French control of Louisiana Masonry was complete. The unrest, dissatisfaction, and ill feelings, however, continued to fester. James Foulhouze was the Grand Commander of the Supreme Council of Louisiana who, along with the other officers, resigned from the council rather than participate in the Concordat. Claude Samory and Albert Mackey approached Foulhouze in the summer of 1856 to enlist his aid in healing the old wounds and to, hopefully, rebuild the A.&A.S.R. in New Orleans. Foulhouze was offered the office of Commander in Chief of the Grand Consistory of Louisiana and Active Membership in the Charleston Supreme Council if he would join in the rebuilding. Foulhouze declined the offers and began his efforts to reorganize the Supreme Council of Louisiana with its former officers.12\n\nWith James Foulhouze out of consideration, a new leader for the troubled New Orleans Scottish Rite had to be found. The choice would prove to be inspired.\n\nEnter Albert Pike\n\nAlbert Pike was an attorney by profession and a Mason of only five years when he moved his law practice to New Orleans in 1855.13 Three years earlier, Pike received the Scottish Rite degrees up to the 32\u00b0 from Albert Mackey in Charleston. Mackey saw a unique quality in Pike and recruited him to be on the ritual committee of the Charleston Supreme Council. Mackey lent Pike a collection of Scottish Rite rituals for his review and study. It was through the examination and transcription of these rituals that Pike received his first understanding of the A.&A.S.R. Busy with setting up his law practice and studying the rituals lent to him by Mackey, Pike did not concern himself with the momentous developments taking place in New Orleans at the time of his arrival.\n\nOne of Pike's earliest Masonic acquaintances in New Orleans was Charles Laffon de Lad\u00e9bat. Over the years (even after Pike became Grand Comman-der) these two would maintain a \"love\/hate\" relationship that was founded on a basic respect for each other. Lad\u00e9bat was made a 33\u00b0 by James Foulhouze in the Supreme Council of Louisiana on February 11, 1852, and served as its Grand Secretary at the time of the Concordat of 1855. Lad\u00e9bat would later be elected an Active Member of the Charleston Supreme Council in 1859. Pike's time in New Orleans put him in close contact with many competent New Orleans 33rds who were quite capable of completing Pike's education and understanding of the A.&A.S.R. Lad\u00e9bat was, clearly, one of Pike's early mentors.\n\nJust as he had done with Albert Mackey, Pike greatly impressed the New Orleans Scottish Rite Masons. Pike's talent and raw abilities clearly made him a candidate for any Masonic office. The fact that Pike played no part whatsoever in the Concordat of 1855 may have made Pike even more attractive and a prime candidate for leading the Grand Consistory of Louisiana. Pike did not carry baggage with him from the Louisiana Masonic turmoil. While he was under the jurisdiction of the Charleston Supreme Council at the time of the concordat, he was not an Active Member and played no part in any of the decisions concerning the Concordat. No one could blame Pike for any of the events. Albert Pike was the only serious candidate for leading the Grand Consistory who could be seen as potentially objective as well as extraordinarily promising. Next to James Foulhouze, no one had a better chance of appeasing the French Masons and unifying all the factions. Once the Supreme Council of Louisiana was re-organized, Pike's value to the Charleston cause was even more evident.\n\nThis address, given by Pike only four days after he received the 33\u00b0,14 is valuable to all Scottish Rite researchers not only because it is an extremely rare piece of early Pike literature, but also because of significant information provided in it. From this address we not only get a better feel of the early Albert Pike, but also have the opportunity to develop a more detailed understanding of the momentous events that were taking place at the time Albert Pike arrived on the Scottish Rite stage. Within just two years from the time of this address, Pike would be elected an Active Member of the Southern Jurisdiction (over the apparent objections of the Grand Commander and Lt. Grand Commander)15 and then on January 2, 1859, with the very first S.J. election of officers (a dramatic change in practice), be elected to the position of Sovereign Grand Commander.\n\nPike's address was ordered to be recorded in the handwritten Minutes of the Grand Consistory of Louisiana. A typed transcript of this address was made by an unknown Brother sometime between the 1940s and 1950s and a copy of this transcript acquired by this writer. The accuracy of the transcript was verified by this writer by a comparison of the transcript with the original Minutes located in the Scottish Rite Bodies of New Orleans.\n\nADDRESS BEFORE THE\n\nGRAND CONSISTORY OF LOUISIANA\n\nALBERT PIKE\n\nApri1 29 1857\n\nTh :. Ill:. Bros :. and Sublime Princes of the Royal Secret:\n\nI PRAY YOU TO ACCEPT my most sincere thanks and profoundest gratitude for the great and unexpected honor which you conferred upon me, when, in my absence, you selected me to fill the most honorable and very responsible station of Grand Commander of this Grand Consistory and for your present ratification of that choice. I will earnestly endeavor to have myself not wholly undeserving of your good opinion; so that, although it must now be said that when elected I was not worthy either by service or qualification, it may not hereafter be said that when I cease to serve, you repented of your selection.\n\nI can bring to your service, Princes, little more than good intentions, kind feelings, and a zealous devotion to the interest of Masonry of all Rites -when you find me deficient (and wherein shall I not, alas, be found, Bro :. ? ) I entreat of you in advance lenient judgment upon my short-comings, and that you will kindly aid me with your sympathy, support and advice. For I must be ever embarrassed by the reflection that I have been by your too favorable judgment preferred to many eminent and distinguished Brethren, whose longer service and greater familiarity with the work gave them far higher claims than any I could have preferred to the post of honor and command. If I supposed that personal consideration or a belief in my superior fitness and capacity had led you to this choice, I should sink under a sense of my feebleness, not ever have succeeded in overcoming my repugnance to accept a post where so much was to be expected. But, amass that there were other reasons, which acted upon you, and made your selection seem politic and for the interest of Masonry in this Valley, reasons not personal to me, but growing out of the conditions of things and the circumstances that surrounded us. I am encouraged to hope that I may in some degree aid in attaining the result which you all desire, and that your just expectations may not be disappointed.\n\nI have accordingly accepted tile post which you have tendered me, and will endeavor to perform its duties. Most important private business will compel my absence for some months. I shall return as soon as practicable, and remain thereafter permanently in the city.16\n\nShould the interest of the Order at any time be likely to suffer by my temporary absence, I shall be prepared at once to surrender up my office, faintly imitating the lofty magnanimity, of which so beautiful an example has been set me by an Ill:. Bro:. whose genius and labors have done so much to restore the splendors of the Ancient and Accepted Rite 17 in this Valley, and whose name will not be forgotten among us, while the order of Knights Rose Croix continues to exist, or the Kadosh to war against tyranny and usurpation.\n\nBut I shall most sensibly feel how great will be the contrast between myself, with my slender experience, and the Th:. Ill:. Prince and Sovereign whose place I come to take, but not fill. 18 Eminent in Masonic learning and more illustrious by long and faithful service than even by his high rank and lofty station, the new and supreme dignity recently conferred upon him was a most just and appropriate acknowledgment of his worth. This Consistory must most sensibly feel its loss, as he, Ill:. Gr :. Commander, crowned and laureled with the highest honor, and with the grateful thanks and recollections of his brethren, most gracefully retires from this distinguished post, to yield it of his own choice to another. I beseech him not to withdraw from me his counsel and advice, and I pray him and our Ill:. Bro:. Laffon,19 and the other eminent brethren who surround me, to aid me, to advise me, to support me in my inexperience, that, guided by them I may not despair of rendering some little service to the cause of humanity, to the cause of truth, of liberty, of philosophy, and of Masonic progress.\n\nMy brethren, I see around me the representatives of more than one race, 20 and the disciples of more than one Masonic Rite - I rejoice at this reunion, and it gives me happy augury of the prosperity, health, and continuance of Masonry in this Valley. I am especially glad that here and in other bodies of this Rite, I see by the side of the children of the first generous and gallant settlers of Louisiana, many of another land, and who not long since for the first time passed beyond the boundaries of the York Rite.\n\nWe are all aware, my brethren, how little among Masons of the latter Rite is known of the Ancient & Accepted Rite, and how great and general a prejudice has obtained those against it. It has been imagined that there was antagonism between the two: Scottish Masonry has been deemed almost spurious, and its degrees, at the best, no more than mere side degrees; and the York Mason who has entered into our sanctuaries has been regarded in the estimation of many, as untrue to his allegiance and disloyal.\n\nThose of you, my brethren, who lately have known only the York Rite, are already aware how unfounded is this prejudice, how erroneous this opinion, how chimerical these apprehensions and alarms. It shall be my study to make you more fully to know this hereafter.\n\nThe Ancient and Accepted Rite is, when itself fully developed and understood, when itself what it should be and can be, a great, harmonious and connected system, all the degrees and lessons, embody the philosophy, the history, the morality and the essential meaning of Masonry, and are to us what the Ancient mysteries were to the initiate of Eleusis, of Egypt, and of Samothrace.\n\nThe degrees of this Rite are commentaries on the Master's Degree, which itself is essentially the same in all Rites. They interpret instead of being at variance with that degree. They ultimately make it known to the Initiate the true word and the true meaning and inner sense of the True Word of a Mason. They teach the great doctrines that God taught the Patriarchs, and which are the foundations on which all religions repose.\n\nWe do not undervalue symbolic Masonry, nor love it the less because we also love the Ancient & Accepted Rite, we but learn justly to value the Master's degree, by coming to understand its full meaning and to appreciate the sublime and lofty lessons which it teaches. Masonry is one everywhere and in all its Temples of whatever Rite; as it has been one in all times. Everywhere it teaches the same great lessons of morality and philosophy, or should do so, if faithful to its mission, and if its apostles are properly informed and true to the duties which it imposes on them. If anywhere it has excluded from even the inmost Sanctuaries of its Temples men of any faith who believe in Our Supreme God, Creator and Preserver of all things that become, and in the immortality of the Soul-if it has anywhere assumed the garb of religious exclusion and intolerance, of Jesuitism, of political vengeance, of Hermetic Mysticism, there most assuredly it has ceased to be Masonry.\n\nIt would not be true to say, however, that even Scottish Masonry has adequately fulfilled or been equal to its missions. While by the irresistible influence of time, by innovations and by mutilations and corruptions of ignorance, the degrees of the York Rite have long since ceased to be what they should be, and what they were in the beginning, when they succeeded to those ancient academies of science, philosophy and morality, the mysteries; while the practice of confirming everything contained in them to the memory has by the silent lapse of time caused more and more both of ceremony and substance to be forgotten, much to be intentionally dropped, and the field of each degree to be made more and more narrow; while the true meaning of very many of their most valuable symbols have faded away and disappeared, and been replaced by commonplace, and the inventions of ignorance, and the lofty science and profound teachings, of the Ancients have too much given way to unimpressive phrases and valueless formulas, - the Scottish Rite also has not enjoyed immunity from the ravages of the biting tooth of time, universal destroyer of all human beings.\n\nFor even here, where over the Temples of our Degrees stood perfect and complete in all the splendor and Majesty of their beautiful and harmonious proportions, we are like strangers from a far land who wander amid the shattered columns and wrecked glories of Thebes and Palmyra, and union over the ruins that track the steps of time, and over the instability of all earthly things. From many of our degrees everything has dropped out except the signs and words, and they remain half effaced and corrupted. From more, all is lost except these and some unimportant formulas; in still more, useless repetition arrives at impressiveness, but cannot renunciate us for the old science and the noble philosophy whose place it endeavors to supply. Those huge chasms have been created in the work, and the connections between the degrees have been broken; so that each has become a fragment instead of being, as at first part of one consistent, regularly progressive and harmonious whole.\n\nThus it has come that of the degrees from the fourth to the thirty-second inclusive, which we retain and apply to ourselves the sounding titles, four only are habitually conferred, which all the residue remain in a great measure, and part of them altogether unknown.\n\nIt had become so obvious that this Rite needed reformation, and that either its degrees should all be made worthy to be conferred and of value to be attained, or else those which were not so ought to be abandoned and their titles disused, that more than two years ago the Supreme Council at Charleston appointed a Committee of five Brethren to revise the whole ritual of the degrees; on which Committee I had the distinguished honor to be placed. While my Brother Laffon, both before and after he was also placed there in the stead of my Brother Samory, who to the general regret found himself compelled to decline the act.21 While my Brother Laffon labored, more particularly on the 18th Degree, but not alone on that, I also, undertaking at first a few degrees, continued my labors during two years, until I completed a revision of all; which that it may be thoroughly examined and sanctioned, I have printed in a volume and submitted to the Supreme Council. Whether that August Body will stamp it or any part of it with its approval, is wholly unknown to me. I have endeavored to restore the effaced or faded lineament of many of the degrees to develop and elaborate the great leading idea of each, to correct the whole together as a regular series, and to make of them our harmonious and systematic whole, ascending by regular graduations to the highest moral and philosophical truth - I have endeavored to prime away all commonplaces and puerility's, all unmeaning forms and ceremonies, all absurd interpretations, and everything useless or injurious with which time and ignorance had overloaded the degrees. I have endeavored so to restore, to retouch and to supply, retaining all that was valuable and working up all the old material, as to make every degree worth to be conferred: that there should be no longer any empty tile, or barren honors in the Ancient & Accepted Rite.\n\nThis I have attempted; but I am only too well aware that the undertaking was too great for my furios; and that what I have done will be found full of imperfections, as the work of the painter, the sculpture, the creator, and the poet ever falls short of his own ideal.\n\nStill I have endeavored to do somewhat; and it is my desire, at some appropriate future time, and with your consent and assistance, to confer upon some suitable candidate such of the degrees, as I have revised them, as have not been already revised by other and more competent hands.\n\nI congratulate you, my brethren, on the advancement and progress of the Ancient & Accepted Rite in this Valley: The Concordat by which the Supreme Jurisdiction of the Supreme Council at Charleston was acknowledged and under which the two Consistories then existing became one, laid broad and deep the strong foundations of the prosperity of our Rite. The walls of our Temple, solidly and squarely built, bid defiance to the storms of faction; and if we are true to ourselves, peace will dwell within our gates.\n\nAnd in the Realm of Masonry, if anywhere on earth, there ought to be peace and quiet and harmony. No where are schism and faction, and disunion and discontent so lamentably out of place as here. Here there should be no lust for power and no eagerness for rank or distinction. If discontented men should in this valley have established, or if any shall hereafter establish, under a foreign authority which has no jurisdiction here and act only by usurpation, any body or bodies, claiming to administer the Ancient & Accepted Rite, we shall, I think, be prepared to show that the Supreme Council at Charleston, to which we owe allegiance, is the only legitimate authority in the Rite that can exist in our country south of the River Potomac; and that the Grand Orient of France and the Supreme Council within its bosom offered against Masonic Law and Masonic Comity where they made another jurisdiction and erect their banners on the soil of Louisiana.\n\nIt is time that this question should be receive the fullest consideration; and that the authentic history of the creation of the Grand Orient itself and of that of the Supreme Council of France, of the disputes between those two bodies and their temporary alliance should be made known to the order in the United States. Supplied with the emissary documents on both sides, it is every intention to translate them and make them public, that all may judge where is the right and where the usurpation.\n\nThe time when fables would pass for history has gone by; and that has come when criticism and investigation will deal with the history of Masonry as with other histories, separating the truth from the error, and after reducing great pretensions to the narrowest proportions. Let us examine the history the Ancient & Accepted Rite and the Grand Orient in that spirit and by the rules and canons of sound criticism, never forgetting that courtesy, moderation, and kindness ought to inspire all Masonic discussions, hoping to find a like tone and spirit on the other side, and that those who may array themselves against us will, if Right and Truth be found with us, candidly admit it, and uniting with us acknowledge the same allegiance and so cause peace ever and ever to reign in this valley.\n\nMy Brethren, let me impress it upon you, that there is much to do, if we would have Masonry adequately fulfill its mission. It is not sufficient merely to receive three or four of the degrees, and then, imagining the rest, to live in contented indolence, without an effort to know the high science and philosophy of the system. The time has come when one who would be truly and really be a Scottish Rite Mason must study and reflect. It shall be my earnest endeavor to aid you in penetrating to the inmost heart of Masonry and in unveiling its profound secrets, which are that light towards which all Masons at least profess to struggle, that knowledge of the True Work which is the great remuneration of a Mason's labor. But if I should fall short of the performance of this duty, be not you, my brethren, disheartened nor discouraged. Masonry must be true to itself, or it will find in numbers weakness only, and its walls will be crushed to the ground with its own might. In this intellectual and practical age. Masonry must it from merited disaster and dissolution.\n\nIt is time for it to assume a higher ground; and here, if any where, the effort to elevate it must be made. Here, I believe, we can commence and successfully carry onward the indispensable work of reformation, that shall in time end the reign of puerility's and trivialities, and make masonry what it should be. The great teacher of moral and philosophical truth; the teacher of the primitive religion known to the first men that lived; the defender of the right of free thought, free conscience and free speech; the apostle of rational and well regulated liberty; the protector of the oppressed, the defender of the common people, the asserter of the dignity of labor and the right of the laboring man; the enemy of intolerance, fanaticism and uncharitable opinion, and of all idle and pernicious theories that arraign providence for its dispensations, and endeavor to set their notions of an abstract justice and equality above the laws by which God chooses to rule all human affairs.\n\nIn this great work I wish your co-operation, and I ask, for myself and for those eminent brethren who are to act with me and in my place, your countenance, your assistance, and your encouragement. I am sure my brethren that I shall not ask this in vain; and that grateful, deeply grateful as I now am for your confidence and kindness, I shall be far more so, and with far greater reason, when I am allowed to surrender into your hands the trust which you have so generously confided to me.\n\nNotes:\n\n1. Charles Laffon de Lad\u00e9bat to Albert Pike, June 24, 1860. Archives of the Supreme Council, 33\u00b0, S.I., Washington. Photocopy in possession of the author.\n\n2. Report of the Committee on Foreign Correspondence of the Louisiana Grand Lodge of York Masons (New Orleans: Cook, Young & Co., 1949), p. 5.\n\n3. George Washington, Lafayette, Warren, Marion, Crescent City, Hiram, and Eureka.\n\n4. Grand Lodge of the State of Louisiana Report and Exposition (New Orleans: I. L. Sollee, 1849), pp. 5-34.\n\n5. James B. Scot, Outline of the Rise and Progress of Freemasonry in Louisiana (1873; reprint, New Orleans: Cornerstone Book Publishers, 2008), pp. 78-80.\n\n6. Charles Laffon de Lad\u00e9bat, The Schism between the Scotch & York Rites (1853; reprint, New Orleans: Cornerstone Book Publishers, 2008), pp. 7-8.\n\n7. Scot, Outline, pp. 86-87.\n\n8. An attempt was made in the late 1800s to revive the French Rite in New Orleans through the short lived Grand Orient of Louisiana. This body was created in 1879, but, possibly due to little support, did not last longer than ten years. See: The Grand Orient Of Louisiana: A Short History And Catechism Of A Lost French Rite Masonic Body (1886; reprint, New Orleans: Cornerstone Book Publishers, 2008).\n\n9. Lad\u00e9bat, The Schism, pp. 7-8.\n\n10. Art de Hoyos, Introduction, The Liturgy of Germania Lodge No.46, F&A.M. (New Orleans: Michael R Poll, 1993).\n\n11. Lad\u00e9bat to Pike, Jun. 24,1860.\n\n12. See: Michael R. Poll, In His Own (w)Rite, (New Orleans, LA Cornerstone Book Publishers, 2011), \"James Foulhouze: A Biographical Study\" pp. 91-137.\n\n13. Pike's law office was located in downtown New Orleans in a building on the riverside of Camp Street one block from Canal Street. The building no longer exists. New Orleans City Directory, 1856.\n\n14. After the Concordat of 1855, the Active Members of the New Orleans Supreme Council were brought in as Honorary Members of the Charleston Supreme Council. As with all Honorary Members of a Supreme Council, they held the 33\u00b0 but not the active office of Sovereign Grand Inspector General (S.G.I.G.). It was at this time that the Charleston Supreme Council began elevating 32\u00b0 Masons to the 33\u00b0 but not including the office of S.G.I.G. in their elevation. Albert Pike was one of the first 32\u00b0 in the S.J. elevated to the 33\u00b0 without being invested with the office of S.G.I.G. Pike would be elected an Active Member (S.G.I.G.) of the Charleston Supreme Council on March 20,1858.\n\n15. \". ..I was not the last to devise the means of placing you at the head of the order, 1st by making you a 33rd against the will of Messrs. Furman & Honour: 2nd by vacating my office of Deputy in your favor, & twice you got in the S.C. & especially twice you were unanimously elected to the Presidency, I consider myself as having done my duty, all I could do. The lifeless council of Charleston was revived; it lives now! Only now tho!\" Lad\u00e9bat to Pike, Jun. 24,1860.\n\n16. The New Orleans City Directories from 1856 unti1 1859 show that while Pike had opened a law office in New Orleans, he did not have more than a temporary home in the city. The Minutes of the Grand Consistory also reveal that he was absent for many of the meetings of the Grand Consistory. There is no record that Pike ever moved his family to New Orleans, and it is probable that he traveled between his home in Little Rock and New Orleans. One of the many boarding houses in New Orleans would have likely been his residence during his stays in the city. Despite Pike's statement, New Orleans would never be his permanent home.\n\n17. At the time of this address, the term Ancient and Accepted Scottish Rite was not in common use in the U.S. This accounts for Pike's repeated use of the older (in the U.S.) term Ancient and Accepted Rite.\n\n18. Pike refers to Claude Pierre Samory. Samory was elected an Active Member of the Charleston Supreme Council on Nov. 20, 1856.\n\n19. Charles Laffon de Lad\u00e9bat.\n\n20. Freemasonry in pre-Civil War New Orleans was reflective of the New Orleans culture of the time. Pierre Roup was the son-in-law of New Orleans Mason and Battle of New Orleans hero Dominique Youx. Roup was a member of Perseverance Lodge No.4 and sat on the lodge's building committee. He was a black Creole. While it is clear that there were more than a few black Creoles who were members of New Orleans lodges, identifying them is difficult, as ones' race was not a question asked or recorded except in notable situations. It is quite possible that there were black Creole members of the Grand Consistory of Louisiana present at the time of Pike's address. It is, likewise, possible that Pike used the word \"race\" in reference to the French Masons who were often considered part of the \"Latin race\".\n\n21. On p. 249 of his History of the Supreme Council 33\u00b0, A.&A.S.R. S.J., U:S.A (1801- 1864) (Washington: Supreme Council, 33\u00b0, 1964) Ray Baker Harris, 33\u00b0, reproduces a letter sent by Albert Mackey to Claude Samory dated Mar. 21, 1855. The letter concerns the Southern Jurisdiction's Ritual Committee and lists its members. Claude Samory is listed as the member from New Orleans and Albert Pike the member from Little Rock. Ill. Harris writes: \"From all indications, the 'preparation of new copies' was in the hands of Albert Pike. He was then in New Orleans, and may have conferred with Samory in this work, but neither of them ever mentioned such a collaboration in their numerous letters written in this period\". Until this address by Pike was rediscovered, it was assumed by most A.&A.S.R. scholars that Samory was on this committee with Pike for a substantial period of time. Bro. Harris, assuming that Samory remained on the committee, logically wondered about the absence of communications between Pike and Samory concerning ritual matters. This address brings to light the fact that Samory retired from the committee shortly after his appointment to be replaced by Lad\u00e9bat. The collaboration was not between Pike and Samory, but between Pike and Lad\u00e9bat and renders the degrees written by the two and their ritual communications understandable.\n\n***\n\nThe Grand Constitutions of 1786\n\nThe Journal of the Masonic Society, Issue 8, 2010\n\nFEW MASONIC DOCUMENTS have been debated, praised, maligned, studied and misunderstand more than the collection known as the Grand Constitutions of 1786. There are actually two collections with that name, one known as the French version and the other as the Latin version. But what are they, why are they important, and why all the fuss about them?\n\nThe Grand Constitutions of 1786 are directly associated with the 33-degree Ancient and Accepted Scottish Rite and are its original rules and regulations. The first Scottish Rite Supreme Council was created in Charleston, South Carolina on May 31, 1801 and used the Grand Constitutions both as authority to exist and laws for governance. The Grand Constitutions of 1786 provided the first Supreme Council with a blueprint, and it gave them guidance in the organization, structure and management of the new system.\n\nIn the early days of the Scottish Rite, the Grand Constitutions were perceived to be of great importance to the young Supreme Council, but were of no value to Grand Lodges who often viewed the new system as mere side degrees. For the Scottish Rite, they were not only central to the government of the system, but could also be used as evidence of legitimacy. In fact, the original Charleston Supreme Council (today officially known as \"The Supreme Council [Mother Council of the World] of the Inspectors General Knights Commander of the House of the Temple of Solomon of the Thirty-third degree of the Ancient and Accepted Scottish Rite of Freemasonry of the Southern Jurisdiction of the United States of America\" \u2013 but more commonly known simply as the \"Southern Jurisdiction\") did use the Grand Constitutions as evidence of legitimacy in what would become a \"Scottish Rite war\" spanning most of the 19th century.\n\nJohn Mitchell was the first Grand Commander of the Charleston Council (i.e., the \"Southern Jurisdiction\"). Mitchell had been a Deputy Inspector General (25\u00ba) of an older Masonic system known as the Order of the Royal Secret, more commonly known as the Rite of Perfection. In 1807, when Joseph Cerneau, another Deputy Inspector General of the Order of the Royal Secret, created bodies in New York that would evolve into a second Supreme Council in the United States, the young Charleston Council used the Grand Constitutions to argue that this second council was unauthorized and irregular. In 1813, Emanuel de la Motta, an Active Member of the Charleston Council, traveled to New York and \u2013 with or without the knowledge or approval of the Charleston Council \u2013 created a second Supreme Council in New York on August 23, 1813, to usurp the Cerneau creation. This council would become the Northern Masonic Jurisdiction known today as the partner to the Southern Jurisdiction.\n\nInterestingly enough, the Northern Masonic Jurisdiction (NMJ) and the Southern Jurisdiction (SJ) have historically disagreed over which version of the Grand Constitutions of 1786 they accept. The NMJ accepts the French version, and the SJ the Latin version. But why should there be different versions of a document that would seem to be crucial to the Scottish Rite? What and where is the original?\n\nThe Grand Constitutions of 1786 contain 18 Articles, or laws, and it was reported to be approved and signed in Berlin by Frederic the Great on May 1, 1786. Unfortunately, the original document is not known to exist. When the Charleston Council demanded the Cerneau Council produce documen-tation showing it was authorized to exist, the Cerneau Council produced nothing. The Charleston Council labeled Cerneau unauthorized and irregular. When the Cerneau Council demanded that the Charleston Council prove that they were authorized to exist, the Charleston Council pointed to its copy of the Grand Constitutions of 1786. The Cerneau Council dismissed this document as a forgery and accused the rival group of hypocrisy. The Cerneau Council claimed it had the same right and authority to exist as did the Charleston Council, and that the standards of legitimacy should be the same for both.\n\nAnother claim made by the Charleston Council was that any additional Supreme Council created in the United States needed its approval, which it did not give to Cerneau.\n\nSo, who, if anyone, was correct? Is it possible that the Grand Constitutions of 1786 are a forgery and they were never approved by Frederic? Let's have a look at the two versions of the Grand Constitutions of 1786. Of the French version, Albert Pike tells us:\n\n\"If I were satisfied that there never were any other Constitutions than those contained in the French version, I should not hesitate to admit that they were a clumsy forgery, and that there was nothing in the world to prove them authentic.\" 1\n\nThose are very strong words! But why would Pike write such a strong denunciation of this French version? Past SJ Sovereign Grand Commander Henry Clausen explains:\n\n\"Pike's [Latin] version is obviously a truer copy of the original because it supplies omissions and corrections that were apparent in the French version.\"\n\nClausen continues:\n\n\"Following are a few examples from Pike's pen showing the disparity between the French and the Latin versions:\n\nThe French Constitutions neither provide for nor describe any Jewel or Cordon of the Degree. The Seal is described as 'a large BLACK Eagle with two heads, the beak of gold, the wings displayed, and holding in its claws a naked sword; upon a ribbon displayed below is written DUES MEUMQUE JUS, and above the Eagle, SUPREME COUNCIL OF THE 33rd DEGREE. [Official Bulletin, Vol. V, No.2, p. 548]\n\nThe French Constitutions provide for one Council of the Degree in each Nation or Kingdom in Europe; for two in the United States of America; for one in the British West Indies; and one in the French West India Islands. But none is provided for Canada; none for the Province of Louisiana, or the Spanish Possessions in North America; and none for South America. [Official Bulletin, Vol. VII, No. 1, p. 486]\n\nTheir Article VI provides that 'the power of the Supreme Council does not interfere in any Degree below the 17th;' and Article VII that only Councils or individuals above the Grand Council of Princes of Jerusalem may bring their appeal to the Supreme Council. This was necessary, in 1801, at Charleston, to prevent hostility on the part of the Grand Lodge of Perfection and Grand Council of Princes of Jerusalem, then and theretofore existing in South Carolina. Why was it necessary in 1786, in Prussia, where no Lodge of Perfection or Council of Princes of Jerusalem existed? [Ibid., p.487]\n\nThe fees for the 33rd Degree and for the Patent of it are expressed to be payable, not in German, but in French coin. [Ibid., p.487] 2\n\nPike's rational and categorical reproof of the French version makes it difficult to understand how one could, with any understanding of Pike's argument and its implications, reasonably defend the French version. Yet, this is the very version that the NMJ accepts. Why? Even more interesting is the fact that Pike himself used the French version to support his position in a Masonic debate. In the 1860s, the Supreme Councils of the NMJ and SJ entered into a debate over territory. Josiah Drummond, the Grand Commander of the NMJ, and Albert Pike, the Grand Commander of the SJ, debated jurisdictional questions over certain states. Drummond wrote to Pike in 1868:\n\n\"I hold that under the Constitutions of 1786, the Northern Jurisdiction and the Southern Jurisdiction are, in every respect and for all purposes, as distinct as if they were separate nations: that we, as well as you, derive our rights of jurisdiction from those Constitutions; that those Constitutions create two separate Jurisdictions. On the other hand, I perceive, that you have held that your Supreme Council had jurisdiction throughout North America, and that we get our territory by cession from you; and if by cession, consequently we get only such territory as you choose to cede: and as necessary, that there could have been no Supreme Council in this Jurisdiction unless you had chosen to cede us territory.\" 3\n\nHow did Pike answer Drummond? He wrote (arguing the meaning of certain phrases in the French version):\n\n\"I do not agree that the Constitutions created the two Jurisdictions. For the United States composed a single Jurisdiction until 1813 or 1815, and might have continued to be as such until today. The provision is restrictive \u2014 that there shall not be more than two Supreme Councils established in the United States. That is the real meaning of it; not that there shall be two. But the point is of no practical importance, and I pass it.... If Illustrious Brother Drummond were right in holding that the Northern part of the United States did not belong to the Jurisdiction of the Southern Council, prior to 1813 or 1815, but was to vest, whether it willed it or not, in a Northern Council, whenever one should be created there, a consequence which he does not foresee might follow. That hypothesis would make the Northern states to have been unoccupied territory, in which any Inspector General could establish a Supreme Council; and it might thus make legitimate the Cerneau Council, and annihilate that created in 1813 or 1815 by De la Motta. It certainly would destroy the principal ground on which the legitimacy of Cerneau's Council was always impeached; to-wit, that the Council at Charleston had jurisdiction over the whole United States, and that no other Council could be created any where in them, except with its consent.\" 4\n\nPike and Drummond were debating the meaning of Article Five of the French version, which determined the number of Supreme Councils allowed in the US. This debate resulted in Pike producing quite lengthy arguments concerning French and English grammar and the reasons for his position concerning the meaning of Article Five of the French version. Pike even changed a portion of the English translation in his Grand Constitutions to reflect his opinion of the rendition.5 In his 1868 Allocution, Pike very skillfully debated this interpretation of Article Five of the French version at length and he did likewise in his Grand Constitutions. But why should Pike bother to painstakingly argue a point concerning a document that he had dismissed as a \"clumsy forgery\"? Pike should have, for the sake of clearly articulating his true position, debated the Latin version \u2014 which he claimed to be legitimate. Why didn't he? Simply put, Pike could not debate this portion of the Latin version. The same portion of Article Five of the Latin version (the version Pike refers to as the \"law of the Rite\" 6) reads:\n\n\"In each great nation of Europe, and in each Kingdom or Empire, there shall be but one single Supreme Council of this Degree. In all those States and Provinces, as well of the mainland as of the islands, whereof North America is composed, there shall be two Councils, one at as great a distance as may be from the other.\" 7\n\nPike strongly contended that the meaning of Article Five (French version) was that the US was not required to be divided into two jurisdictions, yet that is exactly the meaning of the Latin version, which Pike himself had translated in 1859. Pike used the French version in his debate with Drummond simply because it was more open to interpretation. The \"consequence\" that Pike claimed would follow if Drummond's interpretation was accepted, is clearly present in the Latin version \u2014 Cerneau, it seems, might have had reason, based on the version of the Grand Constitutions accepted by the SJ, to believe that he had rightfully established his Council.\n\nThe problem for Drummond was that Pike had skillfully painted him into a corner with his masterful use of Drummond's preferred French version. The territorial debate ended with Drummond yielding to Pike's demands. The view held by Drummond, however, was not only based on his interpretation of Article Five of the French version, but also on the \"birth certificate\" of the Northern Council itself, which reads in part:\n\n\"And whereas the Grand Constitutions of the 33\u00ba specifies particularly, that there shall be two Grand & Supreme Councils of the 33d Degree for the Jurisdiction of the United States of America, one for the South and the other for the North.\" 8\n\nIt is obvious why Drummond interpreted Article V of the French version as he did. The NMJ was created on the premise that the constitutions provided for two councils in the United States. Its only contention could have been if Cerneau was not a legitimate Sovereign Grand Inspector General; after all, if he was legitimate, the Cerneau Council was perfectly legal and the NMJ was \u2013 by its own stated reason for being created \u2013 unauthorized! Pike's opinions concerning the meaning of the original French interpretation were clearly not shared by Emanuel de la Motta, who created the NMJ and was an active Member of the original Charleston Council. It is, likewise, evident why Pike's \"threats\" might well have been taken seriously. Clearly the only available attack that could reasonably be made on Cerneau, from the NMJ perspective, was to discredit his legitimacy as a SGIG \u2013 but great care had to be taken in this course of action as there is no reason to believe Cerneau and John Mitchell obtained the degree in any different manner.9 To discredit Cerneau's 33rd might also discredit Mitchell's.\n\nIt would seem apparent that Pike was unaware of the existence of a handwritten copy of the French version of the Grand Constitutions that had been made by Frederick Dalcho, the first Lt. Grand Commander of the Charleston Council and its second Grand Commander following John Mitchell; the document was not discovered until the 20th century.10 (This copy now resides in the Kloss Collection in the Grand Lodge Library, The Netherlands, which also includes a manuscript of the Ritual of the Thirty-third Degree.) Pike boldly proclaimed the French version a fraud, and offered very lucid support for his position, while clearly having no idea of the pernicious wording of the \"birth certificate\" of the NMJ. An additional problem for Pike was that the Latin version was unknown before 1832. To make matters worse, it was none other than a Cerneau Council that made the Latin version available to the world.11\n\nCustomarily, papers discussing Joseph Cerneau include arguments concerning the Grand Constitutions of 1786. Cerneau is routinely accused of acting in violation of these Constitutions. Nineteenth-century defenders of Cerneau typically argued the lack of authenticity of the Grand Constitutions, with the apparent belief that if the Grand Constitutions could be discredited then all charges against Cerneau would likewise be dismissed. One claim that was often made was that Frederic the Great had been in very poor heath at the time the Constitutions were said to be approved, and that he was physically unable to have given them consent. Albert Pike went to great lengths to examine the charge that Frederic was not physically able to have executed such a document. Pike meticulously traced the reported events and laid out a detailed report on his position that it was possible for Frederic to have executed the Grand Constitutions. Scottish Rite historian Samuel Baynard of the NMJ writes of Pike's conclusions:\n\n\"Though we admit that our Illustrious Brother did in a masterly manner fully convince us that Frederick on May 1, 1786, was physically able and mentally capable of drafting, signing and promulgating these Grand Constitutions, we have utterly failed to find that he discovered or pointed out to us one scintilla of evidence that Frederick actually did have aught to do with them.\"12\n\nPike was obviously aware that his lengthy account did not answer the actual question of whether Frederic signed or approved the Grand Constitutions. Addressing this point in a most interesting manner, Pike writes:\n\n\"There is not one particle of proof, of any sort, circumstantial or historical or by argument from improbability, that they are not genuine and authentic.\"13\n\nAs remarkable as it sounds, Pike is actually asking us to prove a negative. Regardless of Pike's request, Baynard goes on to write:\n\n\"We conclude therefore:\n\n1. That the Grand Constitutions were not promulgated by Frederic the Great;\n\n2. That they were not framed, drawn up or signed in Berlin;\n\n3. That there did not exist in Berlin or even France in 1786, any \"Grand Supreme Universal Inspectors, in constituted Supreme Council\";\n\n4. That the real date of the Constitutions is subsequent to 1786.\" 14\n\nBut if the Grand Constitutions are a forgery, then who forged them? The question did not escape Baynard:\n\n\"It is only natural that the next question should be, Well, then, who did frame them? We do not know. Neither are we unduly disturbed because we do not know. We have our opinion, but it is not substantiated by any evidence that we can call positive or direct, and, therefore, we do not express it as a conclusion.\" 15\n\nTo summarize the situation, Pike had already proclaimed the French version of the Grand Constitutions a forgery. He was debating the merits of why the Latin version should be considered legitimate. Baynard rejected both versions of the Grand Constitutions. Regarding the possibility that the Latin version might also be a forgery, Pike tells us:\n\n\"The odious charge has been again and again repeated, that these Latin Constitutions were forged at Charleston. It is quite certain that this is not true, because the Supreme Council at Charleston never had them, until it received copies of the editions published by the Grand Commander. If they were forged anywhere, it was not at Charleston: and if anything was forged there, it was the French copy, as it afterwards appeared in the Recueil des Actes.\" 16\n\nAnd elsewhere:\n\n\"The gentlemen of South Carolina, in that day, did not commit forgery. Whatever the origin of the Grand Constitutions, they came from Europe to Charleston, and were accepted and received by the honorable gentlemen and clergymen who were of the first Supreme Council, in perfect good faith\" 17\n\nIf the Grand Constitutions are forged documents, but the original Charleston Council did not forge them, then how did they come into possession of them? Pike theorizes:\n\n\"This very imperfect French copy, which consists merely of so many Articles, without preface, formality of enactment by any body in Power, or authentication of any sort, contains no list of the degrees, nor even the name of the Rite. It is most probable that de Grasse procured it, in or from Europe, and created the Supreme Council. By Article V of these Constitutions, it requires three persons to constitute a quorum and compose a Supreme Council; and therefore Colonel Mitchell and Dr. Dalcho alone could not have been, by themselves, such a body. Brother de Grasse intended establishing a Supreme Council at Santo Domingo for the French West India Islands; and no other person had any interest to make the Constitutions read so as to allow such a Council, except his father-in-law, Jean Baptiste Delahogue, who also resided in Charleston in 1796, 1799 and 1801, and was also a 33rd, and appointed to be Lieutenant Grand Commander for the French West Indies. It was for this reason, evidently, that neither of them was placed on the roll of members of the body at Charleston.\" 18\n\nWe now have enough material to analyze. Baynard held the opinion that the entire story of the Grand Constitutions was a fabrication. He based his opinion on the total lack of factual evidence supporting the account and the improbability of the reported events. Pike soundly denounced the French version as a fraud, but held to the possibility of legitimacy for the Latin version. Pike pointed out that the original Charleston Council did not have possession or knowledge of the Latin version and had based their actions on the fraudulent French version. Pike also stated that it was Alexander de Grasse-Tilly who had brought the forged French version to Charleston, and implied that it was de Grasse-Tilly who might actually have forged them. Pike, with some indignation, rejected the possibility that Mitchell or Dalcho might have had anything to do with forgery.\n\nThere are two logical scenarios that we can explore: The first would be that Mitchell and Dalcho received the Grand Constitutions sincerely believing they were legitimate; the second would be that Mitchell and Dalcho took part in the creation of the Grand Constitutions or knew that they were a forgery.\n\nIf Mitchell and Dalcho believed that the Grand Constitutions were legitimate, we can look at the series of events with this mind-set. If Mitchell and Dalcho believed that they were propagating a European system created some 15 years prior to the creation of the Charleston Council, then they could have reasonably assumed that other Supreme Councils of the 33\u00ba existed in Europe. Clearly, the Grand Constitutions speak of such a Council in Berlin.\n\nOn August 23, 1813 John Mitchell and Frederick Dalcho wrote to Emanuel de la Motta concerning de la Motta's report to them of Cerneau. Mitchell wrote in part:\n\n\"I am truly surprised and astonished at the conduct of the man you say is called Mr. Joseph Cerneau. No person ever had the degree but the Count de Grasse, and perhaps, but I am not sure, Mr. Delahogue.\" 19\n\nWe must stop for a moment to try and understand this comment by Mitchell. If Mitchell received a copy of the Grand Constitutions and he accepted them as legitimate and authoritative, how could he be so sure that no one else \"had the degree\"? What of the Supreme Council in Berlin mentioned in the Grand Constitutions? The copy of the Grand Constitutions of 1786 that Mitchell had available to him opens as follows:\n\n\"Made and approved in the Supreme Council of the 33rd duly and lawfully established and Congregated in the Grand East of Berlin on the 1st of May Anno Lucis 5786 and of the Christian Era 1786. At which Council was present in person \u2013 His Most August Majesty, Frederic 2nd, King of Prussia, Sovereign Grand Commander.\"\n\nWas the \"Supreme Council of the 33rd\" in Berlin composed of members who did not have the 33rd degree? If no one else had the degree, who gave it to Mitchell \u2013 someone who did not possess it himself? Mitchell writes that de Grasse was the only other person whom he was certain \"had\" the degree. (This is possibly where Pike conceived the theory that de Grasse was the one who brought the forged copy to the United States.) If no one else had the degree before de Grasse, then who gave it to de Grasse? If de Grasse gave Mitchell the 33rd at some time earlier than the creation of the Charleston Council in 1801, why does the \"1802 Manifesto\" (the \"birth certificate\" of the SJ) state that de Grasse received the 33\u00ba from Mitchell on the \"21st of February, 5802\" [1802]? 20\n\nLet's now look at part of the letter Frederick Dalcho wrote to de la Motta on the same day as Mitchell's letter and also concerning the new Cerneau creation. It again should be noted that the date of Dalcho's letter was August 23, 1813. Emanuel de la Motta established the Supreme Council for the Northern Jurisdiction 13 days earlier on August 10, 1813, and he certainly would have been reported this fact to Mitchell and Dalcho in the letter that prompted their response. Dalcho wrote:\n\n\"It is well known to those who have lawfully received the 33rd degree, that there can be but one Council in a nation or kingdom; and that the Council for the U.S. was lawfully established in this City, May 31st, 1801; consequently any other assuming its prerogatives must be surreptitious.\"21\n\nWhat does Dalcho mean by this statement? The copy of the Grand Constitutions of 1786 which exists in his own hand says that there \"shall\" be two in the United States. And what of de la Motta's creation? Is there some suggestion that Dalcho might not have approved of the de la Motta Council any more than the Cerneau one? The \"birth certificate\" of the NMJ, created by de la Motta, states that \"there shall be two Grand & Supreme Councils of the 33d Degree for the Jurisdiction of the United States of America, one for the South and the other for the North.\"\n\nPike stated that the earliest known copy of the Grand Constitutions was the \"forged\" French version as appeared in a French Masonic publication titled Recueil des Actes in 1817. 22 Pike stated that Mitchell and Dalcho could not have forged the Constitutions because they were both \"honorable\" men and neither \"the kind of man to put his hand to that kind of work.\" Pike also stated that it was not \"probable that either of them could write Latin or French.\" 23 Pike theorized that de Grasse along with his father-in-law, Jean Baptiste Delahogue, acquired or forged the French version and then, presumably, translated it into English so that Mitchell and Dalcho could understand it. Pike did not know of the handwritten Dalcho copy, but could have, by this line of reasoning, assumed that Dalcho copied it from a de Grasse or Delahogue copy which they had translated from French into English.\n\nCould this be the copy that was used to fool Mitchell and Dalcho? We learn from past SJ Grand Historian Ray Baker Harris that the Delahogue documents in the Kloss Collection are \"an undoubted copy of the Thirty-Third Degree and the Constitution, Statutes and Regulations, in use in Charleston in 1801-1802 when the Supreme Council was established.\"24\n\nHarris also tells us:\n\n\"This assumption is further confirmed by a manuscript copy of the same in English, entirely in the handwriting of Frederick Dalcho. It is the English equivalent of Delahogue's French copy. It is believed to have been the Charleston copy from which Delahogue made his translation into French.\"25\n\nDelahogue made his translation into French? But Pike said that the oldest known copy of the Grand Constitutions was the forged French version. In a reproof of this version, Pike rigidly defended Mitchell and Dalcho based on his position that this forged copy came into their hands, presumably through de Grasse and\/or Delahouge, and they simply accepted it as legitimate. The \"French version\" would have had to have been translated from French into English, not the other way around for Pike's argument to be sound. Is there some support for Harris' position that the French Delahouge copy was made from the English Dalcho copy? Yes. Harris tells us that the Delahogue copy of the Grand Constitutions carries the note: \"translated from the English by me [Delahogue].\"26\n\nFor Pike's theory to be correct, de Grasse would have translated his forged French Constitutions into English for Mitchell and Dalcho. Dalcho would then have copied that English translation into his own hand. Then, we are asked to believe that de Grasse's father-in-law did not make a French-to-French copy of the Constitutions from de Grasse's copy, but instead used Dalcho's English copy to translate it back into French for his own personal copy. That makes no sense at all! Why would Delahogue go to all that trouble if his son-in-law possessed the original French version?\n\nThis writer is wholly in agreement with Samuel Baynard in his rejection of the legitimacy of the Grand Constitutions. Likewise, there is little room to argue the perfectly logical assessment that Albert Pike made of the French version of the Grand Constitutions. Pike clearly did not realize that what he so soundly proved to be a \"clumsy forgery\" came directly from the hand of Frederick Dalcho.\n\nIn the absence of any other reasonable explanation, we must conclude that John Mitchell and Frederick Dalcho fabricated the story of the Grand Constitutions of 1786, either in whole or in part. We cannot, as Pike suggested, attempt to prove or disprove a negative. We also cannot embrace fanciful theories that make the story end as we might wish. The course of events simply does not make sense if we take the position that Mitchell and Dalcho received the Grand Constitutions, accepted them as legitimate, and created the Charleston Council. The known facts simply do not support such conclusions.\n\nThis writer holds the opinion that Mitchell, Dalcho, and possibly a few others held reasonable concern in regard to the failing and chaotic state of the \"Scottish Rite\" order (Order of the Royal Secret or Rite of Perfection). \"To bring \"order\" to the \"chaos,\" the new 33-degree AASR system was created. The \"cream of the crop\" of the degrees and rituals were selected for this new system, an inspired creation for which, one can imagine, a concern developed over whether it would be accepted by Freemasonry. A royal endorsement would add value to any new Masonic system, and one attached to a set of governing laws might bestow greater value.\n\nIf we examine the situation from the standpoint that the Charleston Council received the constitutions and accepted them as legitimate, then we arrive at one contradiction after another. If, however, we consider the entire story and creation came from the Charleston Council, a very logical scenario develops. It is this writer's conclusion that the original Charleston Council was created alongside a set of governing laws attributed to Frederic II. This writer has not seen one scrap of sound evidence to support the position that Frederic actually approved \u2013 or even knew of \u2013 any Grand Constitutions in Berlin on May 1, 1786. There is, however, abundant evidence to attribute the creation of the constitutions to the original members of the Charleston Council.\n\nIt has been more than 200 years since the creation of the Charleston Council. The value and worth of the AASR is well proven. It is clear this Masonic system is of tremendous importance to the whole of Masonry, and it is not a disservice to acknowledge all of its history. The creators of the AASR were human, after all, and humans sometimes make mistakes in judgment.\n\nNotes:\n\n1. Albert Pike, The Grand Constitutions of Freemasonry (New York: The Supreme Council, 33\u00ba Southern Jurisdiction, USA, 1872), 282-283.\n\n2. Henry C. Clausen, Authentics of Fundamental Law for Scottish Rite Freemasonry (San Diego: The Supreme Council, 33\u00ba Southern Jurisdiction, USA, 1979), 9-10.\n\n3. Transactions of the Supreme Council of the 33D for the Southern Jurisdiction of the United States (New York: Masonic Publishing Company, 1869), 19.\n\n4. Ibid., 22-23.\n\n5. Pike, The Grand Constitutions of Freemasonry 289. Pike altered the English translation of the French version of Article five to: \"...but two in the United States of America...\" in order to emphasize his point concerning his interpretation of the meaning of this phrase.\n\n6. Ibid., 283.\n\n7. Albert Pike, The True Secret Institutes and Fundamental Bases of the Order of Ancient Free and Associated Masons and the Grand Constitutions of the Ancient Accepted Scottish Rite of the Year 1786. (New Orleans: The Supreme Council, 33\u00ba Southern Jurisdiction, USA 1859), 163-165. In Pike's 1872 (A.M. 5632) The Grand Constitutions of Freemasonry, he altered the original translation of the Latin version to read as follows: \"In each great nation of Europe, and in each Kingdom or Empire, there shall be a single Council of the said degree. In the States and Provinces, as well on the Continent as in the Islands, whereof North America consists, there will be two Councils, one at as great a distance from the other as may be possible.\" Pike, the master linguist, replaced the word \"shall\" with \"will\" in his 1872 edition, which, while having the same meaning, was not such an obvious problem to inattentive readers. The edited edition carries the note, \"Re-translated from the Latin by Albert Pike, 33\u00ba, Sov. Gr. Commander. A.M., 5632\" p. 213. Pike maintained the accuracy of his 1859 translation, at least, until 1868, as the questioned portion of Article Five is reproduced in the 1868 Transactions of the SC SJ exactly as they appeared in the 1859 translation on page 28.\n\n8. Samuel Harrison Baynard, Jr., History of the Supreme Council, 33\u00ba Ancient and Accepted Scottish Rite Northern Masonic Jurisdiction of the United States of America and its Antecedents (Boston: The Supreme Council, 33\u00ba Northern Masonic Jurisdiction, USA, 1938), Vol. I, 175-179. This quotation is taken from the facsimile reproduction of the 1813 \"birth certificate\" for the Northern Jurisdiction (reproduced on page 176). In addition to the facsimile is a printed transcript of the \"birth certificate\" provided to us by Ill. Brother Baynard. Interestingly, the printed transcription omits a number of words and phrases that appear in the facsimile. The phrase, for example, \"one for the South and one for the North\" (line 26 of the facsimile), does not appear in the printed transcription.\n\n9. The question of where and when John Mitchell and Joseph Cerneau received their 33rd degrees has not escaped the notice of Masonic researchers. In the case of Cerneau, he is usually dismissed quickly due to the total lack of evidence that anyone ever actually gave him the 33rd degree. Emanuel de la Motta, upon first meeting Cerneau, attempted to obtain certain information about Cerneau's 33rd including having a look at his Patent, but was unable to satisfy himself in any way (see: Charles S. Lobingier, The Supreme Council 33\u00ba [Louisville, Kentucky: The Supreme Council, 33\u00ba, SJ., 1964], p. 102.). But what of John Mitchell? There has never been a Patent discovered showing that Mitchell received the 33rd from anyone. We know that Mitchell gave Dalcho the 33rd as a Patent for this event exists. Mitchell was the first Sovereign Grand Commander of the SJ, so how did he receive the 33rd? Who gave it to him? Prior to Mitchell's role in the creation of the AASR, he was a Deputy Inspector General (25\u00ba) of the so-called \"Rite of Perfection\". We often see those senior to Mitchell in this system being credited with giving him the 33rd (usually Barend Spitzer). How could a 25th degree Mason from another system give the 33rd degree of the AASR to someone? We can also see an account of some \"unknown\" Prussian or German giving him the degree with Mitchell signing an obligation for it in French. (See: Baynard, History of the Supreme Council, 33\u00ba, Vol. 1, p. 89.) If someone gave Mitchell the 33rd, who gave it to him? Why didn't this unknown SGIG play a role in the creation of the Charleston Council? Since this unknown SGIG was senior to Mitchell, why wasn't he the first Charleston Sovereign Grand Commander? The questions can go on forever.\n\nOne thing we must never do is judge past events by today's standards. How we do things today, may not have been the norm in the past. We can find evidence of an old practice that might shed some light on the Mitchell\/Cerneau 33rd degree question. Evidence exists (see: Henry Wilson Coil, Coil's Masonic Encyclopedia [New York: Macoy Pub. & Masonic Supply Co., 1961], p. 121 and Pike, The Grand Constitutions of Freemasonry, p. 117.) that a Deputy Inspector General of the old so-called \"Rite of Perfection\" (as were both Mitchell and Cerneau) could \"slide over\" to the 32nd degree of the new 33 degree AASR. In addition, if a 32nd of the AASR was the senior (or only) 32nd in an unoccupied area, he could advance himself to the 33rd degree of the AASR in order to give the degree to others and create a Supreme Council. Both Mitchell and Cerneau gave the 33rd to others and created supreme councils.\n\nRegardless of the historic disapproval of Cerneau, it is possible that according to the custom of that time, he received the 33rd degree in the same manner as did Mitchell. A sound argument could be made that he was just as legitimate a SGIG as was Mitchell.\n\n10. See: R. Baker Harris and James D. Carter, History of the Supreme Council, 33\u00ba (1801-1861) (Washington, D.C.: The Supreme Council, 33\u00ba Southern Jurisdiction, USA, 1964), 98.\n\n11. Ibid., 216.\n\n12. Baynard, History of the Supreme Council, 33\u00ba, 101.\n\n13. Pike, The Grand Constitutions of Freemasonry, 170.\n\n14. Baynard, History of the Supreme Council, 33\u00ba, 115.\n\n15. Ibid., 116.\n\n16. Pike, The Grand Constitutions of Freemasonry 126.\n\n17. Ibid., 195.\n\n18. Ibid., 134.\n\n19. Harris\/Carter, History of the Supreme Council, 33\u00ba (1801-1861), 117.\n\n20. Ibid., 323.\n\n21. Ibid., 118.\n\n22. Pike, The Grand Constitutions of Freemasonry, 126.\n\n23. Ibid., 134.\n\n24. Harris\/Carter, History of the Supreme Council, 33\u00ba (1801-1861), 92.\n\n25. Ibid., 92.\n\n26. Ibid., 92.\n\n***\n\nThe \"White Cap\"\n\nEVER HEAR THE EXPRESSION, \"He's an honorary 33rd\"? It's not uncommon to hear, but... it's just not correct.\n\nThere are two \"parts\" to the 33rd degree - the office (SGIG) and the degree (33rd). A \"White Cap\" has received the 33rd and final degree of the AASR, but they are honorary Sovereign Grand Inspectors General. A \"White Cap\" is not an Active Member of a supreme council.\n\nConfusing? Let's look at some of the history.\n\nWhen our 33rd degree system was created, the final degree was patterned upon the final degree of the old \"Order of the Royal Secret\" (aka: \"Rite of Perfection\"). This means that the final degree was not only a degree, but also an office held by the one receiving the degree. Those who received the degree performed certain duties which could only be performed by those who held both the degree and the office.\n\nThe early AASR was governed by a set of rules as laid down in a collection of documents known as the \"Grand Constitutions of 1786.\" These Constitutions (the history of which is not exactly relevant for this discussion) were accepted as the law of the Rite by the early Charleston Supreme Council. These constitutions gave specific instructions on the organization and membership of a supreme council. A supreme council would consist of 9 Members who held the 33rd degree (in the SJ, that number would expand to 33 Members under the administration of Albert Pike). The degree name as well as the name of this office was, \"Sovereign Grand Inspector General.\" In the very early days, all those who held the 33rd Degree were also Active Members (voting members) of a supreme council.\n\nIt was not too long after the creation of the U.S. supreme councils, that Members of one jurisdiction began moving their residence into the jurisdiction of other supreme councils. We can find cases of these \"sojourning\" 33rds being received in other supreme councils and also cases of some being made \"Honorary\" Members of their host supreme council. These honorary memberships were more of a \"tip of the hat\" and recognition of rank, but carried no voting privileges or official duties.\n\nIn the 1850's, a dramatic innovation took place in the SJ. Albert Pike was one of the first 32nds to be elevated to the 33rd degree, but not receive the office of Sovereign Grand Inspector General. The office and degree became split. It was required that for one to hold the office to have received the degree, but no longer would receiving the degree automatically mean that the office was also held.\n\nThe Supreme Council under Albert Pike created two honor investitures. One was for 32nds and commonly known as \"Knight Commander Court of Honor (KCCH)\" and the second was for 33rds, (IGH) and became known as \"Grand Cross Court of Honor (GC)\". Pike had become aware of an older honor investiture in the Scottish Rite bodies in New Orleans for both 32nds and 33rds by the name of the \"Ceremony of the Fiery Heart.\" It is possible that this older ceremony provided the inspiration for the KCCH and GC investitures.\n\nIt is unfortunate that we so often see the \"White Cap\" included with the honor investitures of the AASR. The \"Knight Commander Court of Honor\" and the \"Grand Cross Court of Honor\" are honor investitures given to 32nd or 33rd. These two honor investitures are not degrees and including the 33rd in with them might be one source of confusion as to the nature of the 33rd degree itself.\n\nThe 33rd degree is the final degree of the AASR. It is certainly an honor to be found worthy of receiving the degree (most do not receive it) but it is a mistake to believe that \"honor\" is the proper word for how you should view receiving it. It would be far more appropriate to view it as a \"responsibility\" as 33rds are to be the teachers and models for all other members of the AASR.\n\n***\n\nIntegrity in Masonry\n\nLecture given at the 18th District Lodge, New Orleans, Louisiana\n\nAugust 22, 2011\n\nTHANK YOU FOR INVITING ME to talk this evening. As the Worshipful Master mentioned, I had the pleasure of serving the 18th District as District Grand Lecturer during the mid to late 1980's. My final year was in 1990. That opportunity to serve put me in contact with a some extraordinary brothers, some of them, sadly, are no longer with us. One such brother was Worshipful Brother Irl Fergerson. At that time, Bro. Fergerson was the Chairman of the Permanent Committee on Work. In my early days of Masonry, I would go to the old Masonic Temple Building and enjoy afternoons listening to him and learning the work. He was, without doubt, one of the finest ritualists I had ever met. He not only knew the ritual forward and backwards, but also inside and out. But, it was his style of teaching that most impressed me. I well remember him tapping loudly on his chair after he would ask a question and saying, \"If I wanted a tape recorder, I would go to the store and buy one! I don't want you to just give me back words properly strung together, I want you to tell me what those words mean!\" Brother Fergerson was not satisfied with anyone just correctly knowing the ritual, he wanted them to understand the philosophy being taught. It was in this manner of teaching that new doors began to open for me in my Masonic education. I could find in our ritual something of a blueprint for living a virtuous life; a plan for inner growth and development. It was here that the true beauty of Freemasonry began to shine for me.\n\nLearning the ritual from Brother Fergerson gave me the opportunity to learn of the deep moral philosophy that is embedded in our teachings. I learned symbolic lessons that not only gave me assistance in my Masonic life, but all of my life \u2013 inside or out of Masonry. I learned how to treat others, how to properly look at and examine my own life, my flaws and positive points, how to properly interact with others and what is truly the most important aspects of life. I learned life lessons. One of those lessons, integrity, is the subject of tonight's talk.\n\nAround the world, there are many different types of Freemasonry. By that, I mean the rituals that are used and practiced. While the words and activities of the craft degrees in the different rites vary, sometimes quite a bit, one common thread that runs through all of the various rites and rituals is the legend of Hiram. Now, before I say anything else, I have to throw in a disclaimer of sorts. Some time back I heard that a jurisdiction was thinking about removing the legend of Hiram from their ritual. As surprising as that information was, the reason behind their idea was even more remarkable. I was told that the reason for their wanting to remove this aspect of the ritual was because they could not establish that the legend of Hiram was a factual historical event. I was stunned. It is a symbolic story; a lesson. It is completely irrelevant if the story of Hiram is fact or fiction. We are not teaching a history class. The story is used as a vehicle to deliver lessons of virtue and morality. The lessons that are taught are what is important, not the factual nature of the story used to present the lessons.\n\nSo, with the disclaimer that I make no statement of historical fact, I'll continue with the story. The story takes place at the time of the building of King Solomon's Temple. We are taught that a great many operative Masons worked on the construction of the Temple. These Masons were guided in their work by three Grand Masters: King Solomon, King Hiram of Tyre and the lead architect, Hiram Abif. At some point, the three Grand Masters realized that a number of the craftsmen were performing their duties at such a high level of skill that it entitled them to special recognition. These craftsmen would be elevated to Master craftsmen.\n\nNow, in today's Freemasonry, if we receive a degree, an office or position of importance, we're honored by that advancement. But, in reality, it means very little outside of our Masonic life. Our Freemasonry is Speculative Freemasonry, and it is something we do outside of our family life and livelihood. This was not the same with the old Operative Freemasons. Freemasonry was their livelihood. It was how they fed their family and paid their bills. Being advanced to the rank of Master was a major accomplishment. Not only did it mean an elevation in their social status, it meant a considerable pay increase. This advancement was a very important event in their life.\n\nWhen the news of the pending advancements was made known, we can assume that considerable excitement and interest developed. It is because of the importance of these advancements to the lives of those receiving them that some concern among the Grand Masters developed. It seemed reasonable to put into place some sort of security measure so that individuals of low moral character could not assume rank for which they were not entitled. It was decided that a secret word would be given to all new Masters of the Craft so that they could prove their rank by the possession of this word. As a further security measure, it was decided that this word would not be given out to anyone unless all three Grand Masters were present and agreed to the investiture.\n\nThe story goes on that three crafts-men obviously real-ized that they would likely not be elevated to a higher rank and were unhappy about it. They wanted this advancement \\- badly. So much did they want this advancement that they hatched a plan to steal this \"secret word,\" move to another area and live their lives pretending to hold a rank they did not earn. They caught one of the Grand Masters alone and demanded that he tell them the secret word. When he refused, they roughed him up a bit. When the Grand Master still refused to give them the word, they became desperate. They made it clear to him that they were going to leave with either the word or him dead. At this point, the Grand Master had a choice. He could give them what they wanted, or he could risk death. Clearly he took them seriously as his final words reflect acknowledgment of what he knew would happen, \"Of my life you may deprive me; of my integrity, never!\"\n\nThink about what happened for a minute. There is something that I have been taught since childhood, and, most likely, you have also been taught. It is that if I am ever in a situation where someone threatens my life in a robbery attempt, I should give them whatever they want. Why didn't he? I was taught that nothing I have on me is worth risking my life. Why didn't he just give them this word and then he could live and go on with his life?\n\nThe lesson of integrity is involved not because of a robbery attempt, but because of an agreement that was made. This Grand Master agreed that he would not give the secret word to anyone unless certain conditions were met. Had these craftsmen attempted to simply rob him of some coins, then it is reasonable that he would have freely exchanged whatever he had on him for his life. But, what these men wanted was something completely different. They demanded that he violate an agreement, his word.\n\nThe Grand Master's final words need closer attention. He said, \"Of my life you may deprive me...\" What does that mean? He clearly recognized that he was not in control of their actions. He could not make them spare his life or do anything at all. Taking his life was something that they would either do or not do and he had no control whatsoever over their actions. The only thing in which he had total control was his actions. They could take his life, but they could not take this word from him. He could only give it and that would be by his choice.\n\nThe Grand Master needed to determine what was of true value to him. He knew that we all live and die, but he also knew that how we live is up to us. To be robbed of some coins is no dishonor, but what of violating his word? What was that worth to him? He did not agree to only give the word when certain conditions were met unless his life was threatened or only on the third Tuesday of the month if there was a full Moon. He agreed to not give it unless these conditions were met. Period. If he gave the word to anyone and those conditions were not met then he would be violating his word. It did not matter if they offered him money, threatened him or anything else. He would either keep his word or break it.\n\nIn life we can gain or lose material things. Because of the twists and turns in life we can amass great wealth or lose everything we own. Many things can happen to us because we were either in the right place or the wrong place. But either we have integrity and honor, or we do not. We have it because it is our choice and we lose it by the same choice. Material things can be taken away from us and we might have no choice in the matter. But, not our integrity. We are the only ones who have the power to give our integrity away.\n\nThe Grand Master knew that we all live and die. He also knew that all of the mag-nificent structures that he helped create would mean nothing if his moral foundation was made of sand \u2013 void of integrity or honor. These men had the power to take his life, but they were powerless to make him live a life without integrity. This was the point of the story \u2013 to teach a life lesson of virtue and morality, not to simply provide a historical account.\n\nBut, we should not believe that the story ends here. The nature of symbolism is layered and often requires second and third looks to find deeper meanings. Just because we believe that we are acting with honor or integrity does not mean that this is actually the case. Let me give you an example.\n\nA story from New Orleans in the early 1800's come to mind. There were two men who were standing outside the St. Louis Cathedral having a friendly conversation. The two men were facing each other. One of the men felt a bit uncomfortable in his position and moved just a bit to the left to reposition himself. When the man moved over, the other man winced in pain and looked shocked. In a sharp tone he demanded that the man return to his original position. The man who moved had no idea of what the other man was talking, or understood his strange actions or demand, but did not like his tone of voice. What neither man realized or considered was that the man who moved was considerably taller than the other man. In the position he was in, (unnoticed by either man) he was standing right in a place where he was blocking the sun. When he moved a bit over, the sunlight hit the shorter man right in the eyes causing his painful reaction.\n\nNeither man was of a mind to explain himself or ask too many questions of the other. Hot tempers took over and the friendly conversation was replaced by a very heated, nonsensical argument. And then it happened... one man exclaimed that his \"honor\" had become compromised and \"integrity\" demanded satisfaction. A challenge to a duel was issued.\n\nIt was fortunate that neither man died in the duel, but one of them was shot in the arm. For the rest of this man's life, he lived with a useless arm as the result of the injury suffered in that duel. And for what? Honor? Integrity? One man moved a bit and the other man had sun in his eyes. For that you shoot at each other?\n\nWhat these men mistook for honor and integrity was pride, arrogance and vanity. These vices were disguised as or mistaken for virtues. There was no loss of honor in what happened and integrity demanded nothing in the way of a duel.\n\nWe must live our lives with honor and integrity. But, we must know what is a virtue and what is a vice disguised as virtue. It's not always as clear as we think. If anyone has ever told us that being a Mason is easy, then they misled us. There will be times when we find it most difficult to live up to our teachings. But, as we are so often told, it is the journey that is most important, not the final goal.\n\n***\n\nWriting Masonic History\n\n\"History [is] a distillation of rumour.\"\n\n-Thomas Carlyle\n\nRECENTLY, I READ SOMETHING that I found quite interesting. A Mason wrote, \"Masonic history is fact and can not be changed.\" I studied that line for a time and could not decide if I agreed with him. Before I could form any opinion of his thought, I would have to understand his meaning of the word \"history.\" If he used that word to mean the actual events of the past, then I would, of course, agree with him. How could I not? Everything that any of us did yesterday is over and can not be changed. Period. My problem was that this Mason might have used the word \"history\" to mean the published accounts of past events (Masonic history books or papers). If this is what he meant, then I very much disagree with him.\n\nPublished accounts of Masonic history can only be considered as fact when they are so proven (and even then can be questioned). Errors of accuracy and assumptions abound in our Masonic literature. Learned historians often dramatically disagree on the \"correct\" interpretation of events in history when facts are lacking and opinions are obligatory. How \"history\" is discovered, analyzed and drafted is often the telltale mark of identifying a serious, objective researcher from an amateur or a political salesman with a particular Masonic organizational bias.\n\nTypographical errors occur because we are human. Errors in dates, names and events occur in the finest of publications and are made by the most serious and capable historians. I certainly have had my share of misspelled names, twisted dates and various other typos, which seem to be invisible until published. When an error of fact is discovered, it should be corrected and we should move on. Those with fragile egos or with an aversion to correction should find safer ways to spend their time. The job of a Masonic researcher is to work toward discovering and publishing the accurate history of Masonry. It is an on-going process and no Masonic work of history should be considered as definitive. We must welcome corrections and encourage close examination of all published accounts of Masonic history. Our work can only be accepted as valid if it is capable of standing up under close examination or criticism.\n\nSerious historians must, also, be ever cautious of \"history with a bias.\" Our job is to allow the documents to \"speak for themselves\" and not interject personal or organizational bias into any historical study. Has this ever been done? Sadly, yes.\n\nWhen we explore or write Masonic history we must recognize the fact that many of the large, powerful and very influential Masonic Bodies of today were quite small and fragile 100 or 200 years ago. As an example, for the first 50 to 75 years of its existence the Supreme Council, SJUSA existed in what can only be considered a most unstable condition. In fact, the Southern Jurisdiction ceased to exist for a period (arguably) of about 10 to 20 years (late 1820's to mid 1840's). Much of the reported creation and early history of the SJ is accepted as fact with little to no evidence to substantiate the claims. In many areas we are left with only the unsupported opinions and conjecture of officers of the SJ to answer many of the reasonable questions regarding its early history.\n\nConsider, also, Joseph Cerneau, the first adversary of the SJ. Was he the \"villain\" that he was (and is) portrayed to be? In just one example, Mackey labels him a \"Masonic charlatan.\"1 Cerneau's reputation, motives and qualifications have been denounced by the SJ from the 1800's to the present. Even Cerneau's name is today equated with \"fraud.\" 2 Yet there is only the opinion of officers and supporters of the SJ to support such brutal charges against this Brother. Such unfounded character assassination is wholly unfitting Masonic works of history.\n\nThere is no evidence to show that: a) Cerneau became a 33rd in any manner different than the first Sovereign Grand Commander of the SJ; b) had less right than the SJ to establish his Supreme Council as provide by the Grand Constitutions of 1786, or c) that the \"problem\" with the Cerneau Council was anything other than the SJ simply not wanting it to exist. 3\n\nMasonry has no need and can support nothing but objectivity and truth in our history. We are big enough and strong enough to stand up to whatever truth there is and face that truth as Masons. We have a lot of re-writing to do with our Masonic history and we must weed out the \"political historians.\" We must never again place the \"good\" of any Masonic organization ahead of the good of Masonry itself. The standards of sorting fact from opinion must be uniformed and apply to everyone.\n\nIt is a new day and a new time. It will be an interesting 10 or so years ... and it is only beginning.\n\nNotes:\n\n1. Albert G. Mackey and Charles T. McClenachan, \"Encyclopedia of Freemasonry\" (New York: The Masonic History Company, 1915) 139.\n\n2. \"Cerneauism: This term is applied to the particular type of clandestinism and fraud which characterized the bodies set up by Joseph Cerneau and his followers beginning in 1807.\" Henry Wilson Coil, \"Coil's Masonic Encyclopedia\" (New York: Macoy Publishing & Masonic Supply Company, 1995) 125.\n\n3. See: Michael R. Poll, The Controversy of Joseph Cerneau: A Brief Examination Heredom Vol. 4 (Washington, D.C.: The Scottish Rite Research Society 1995) 47-61.\n\n***\n\nThe Dwellers on the Threshold\n\nThe Journal of the Masonic Society, Issue 12, 2011\n\nA GOOD NUMBER OF YEARS AGO I was walking through the French Quarter of New Orleans engaged in one of my favorite pastimes \u2013 exploring used book shops. In one old shop, I stumbled upon a treasure. It was three large boxes filled with old The New Age magazines (today, it's called, The Scottish Rite Journal) from around 1907 through the 1970's. It was an almost completely intact collection. They were selling the lot for $40.00! I tried to hide my excitement out of fear that they would quadruple the price, paid the man and loaded up my car. When I returned home, I began to wade through my acquisition. What a haul! I laughed at the very early editions with piano, shotgun, and even insurance company ads. What great period pieces! Over the next few weeks I studied and examined all aspects of the publications.\n\nIn my collection of old The New Age magazines, I could see something of a snap shot of the times and changes that took place. The publication itself changed over time from colorful cover images with each issue having a unique cover to standard covers of one design for each issue. The size of the publication changed as well from large format to a smaller, almost pocket format. What was also interesting was to see the changes in the types of articles published. The WWI and WWII issues, of course, reflected news of the terrible war years and contained a number of patriotic articles. But, it was the years from around 1915 to the early 1920's that really caught my attention. These were the true esoteric years of The New Age. These editions stand out from all of the other years (even to the present) by the nature of the articles published. The Rosicrucian, alchemical, and metaphysical aspects were all represented in force in these editions. For those interested in the deeper Mystic Arts of Freemasonry, it was a dream comes true. And, like many who are workers in these Arts, a number of the authors employed pseudonyms. One such author was \"Mysticus.\" His papers were of a nature that deeply impressed me. I began to search through each edition to discover and drink in his words. This was an enlightened Brother.\n\nIn the June, 1920 edition of The New Age magazine, I came upon an article by \"Mysticus\" that especially caught my attention. It was not exactly the main subject of the article that caught me, but by what seemed to be more of \"side\" information. The article was part of a series that \"Mysticus\" had written entitled \"A Corner of the Library.\" This segment was, \"Collectors of Occult and Magical Books.\" In his article, \"Mysticus\" tells of a little group that existed in Washington, D.C. He writes:\n\n\"Washington city is a well-known center of scientific and philosophical inquiry. Some twenty-one years ago there existed in the capital a little band of independent thinkers of which I was a member. We were students of philosophy, folk-lore, symbolism, occultism and psychic research, and we called ourselves, jokingly, \"Dwellers on the Threshold,\" a title borrowed from Bulwer-Lytton's strange Rosicrucian story, \"Zanoni.\" Some of us were professed idealists, followers of Plato and his school, while others bordered on materialism, and offered up their devotions at the shrines of Spencer, Comte, and Haeckel. But, all of us, I think, were earnest seekers after truth and open to conviction on any question. The leader of this group was Dr. [Leroy M.] Taylor, a man of wealth and a prodigious collector of occult literature. We met at his house every Saturday night to discuss problems in philosophy and religion, particularly those bordering on the mystical, for which the doctor had a decided penchant.\"\n\nOther members of this little group included, Dr. Saram R. Ellison, 33\u00ba, Frank H. Cushing, Judge Thomas H. Caswell, 33\u00ba, Sovereign Grand Commander of the Supreme Council, SJUSA and even Harry Houdini. What I would personally give to have been able to sit in on one of these meetings!\n\nBut, who was this \"Mysticus\"? I had to find out. I sent letters to the then editor of \"The Scottish Rite Journal,\" Bro. John Boettjer, as well as others at the House of the Temple who I had the pleasure of knowing - the Grand Historian, Bill Fox, Sr., the Grand Archivist, Dick Matthews and the extremely helpful and knowledgeable Librarian, Mrs. Inge Baum. All searched their records and compared notes on the possible identity of this prolific and mysterious writer. No one could turn up anything. Mrs. Baum was particularly taken with this mystery and we exchanged a number of mails on the subject. She had located all of his writings published in The New Age, but had no clue who might have been the man behind the name. Then, all at once, the mystery seemed to be solved... in a mysterious way.\n\nOne evening, I was reading a copy of Manly Hall's, The Phoenix. In a section he had written on Albert Pike there was a quote that hit me right between the eyes. I knew I had read those words before. I knew it was from one of the pieces written by \"Mysticus\" and published in The New Age. To be sure, I pulled out the edition with the story and compared the two. It was the exact quote. But, credit for the quote was not given to \"Mysticus;\" it was given to \"Henry R. Evans.\" BINGO! Was this the real name of \"Mysticus\"? I began looking through my copies of The New Age and found the name \"Henry R. Evans\" on quite a few pieces, in fact (and to my great surprise!), Henry R. Evans was the Editor of The New Age. I sent all this information to Mrs. Baum and in no time she answered me with even more information. She sent me the whole file on Bro. Evans from the House of the Temple archives and included a letter Evans wrote to the then Sovereign Grand Commander admitting that he was \"Mysticus.\" Mystery solved. Or, did it only lead to more mysteries? How did Manly P. Hall know that \"Mysticus\" was Henry R. Evans?\n\nIn looking through my collection of books by Manly P. Hall (I had quite a collection of them) I saw credit given to Evans a number of times, including in the Hall classic, The Secret Teachings of All Ages. But, I was able to find no information at all as to how they knew each other. It could not have been a case of Hall simply reading something he liked written by Evans in some publication and using it. He had to have known that Evans was \"Mysticus\" (apparently, not everyone did) in order to give credit to Evans and not \"Mysticus.\" Could Hall have been a very young member of \"The Dwellers on the Threshold\"? Who knows? How long did this group exist and meet? Who knows? We can guess all day.\n\nThere is an old thought that water seeks its own level. Enlightened spirits bump into each other because they are going in the same direction. Little groups of Masons meet in private \"clubs\" like the \"Dwellers on the Threshold\" because their interests draw them together. If we look around we might see such groups meeting today in many more areas than we realize - maybe not all discussing the same thing, but matters of interest to them. And, I'm sure we will find at the core of each group a Masonic spark that feeds the perpetual search for knowledge and Light.\n\nI'm glad that I found those old boxes of The New Age magazines. I'm glad that I was able to learn of \"Mysticus\" and explore his thoughts, ideas and his mystery. I'm also very glad to continually discover the cord connecting Masons in their journey to enlightenment.\n\n***\n\nDown the Path of Proper Research\n\nThe Journal of the Masonic Society, Issue 9, 2010\n\nA YOUNG MAN DECIDED TO WRITE a family history and began the task of putting all of the family members into their historical places. Two long-deceased brothers, who had been his much older cousins, presented him with an interesting dilemma. The younger brother was the type of man that anyone would want in their family history. He was a wealthy attorney and a pillar of his community. He sat on the board of directors of the local bank and the art museum too. He was a leading figure in local politics, having served as a City Councilman, and was deeply involved with a number of local charitable organizations. His older brother, on the other hand, was uneducated, dirt poor, and something of the town drunk, having even spent time in the local jail for stealing chickens.\n\nThe young man pondered on the two brothers and decided that the older brother might present something of an embarrassment to the family. He decided to concentrate on the younger brother, giving as many details of his successful life as possible. The older brother was only noted in passing as the elder brother of the source of the family pride. The young man made his decision based on what he knew of the brothers and his belief that the best interests of his entire family were served by giving as little information as possible about the elder sibling.\n\nAn elderly aunt read the story of the two brothers and strongly protested the account. She told the young man that the father of the two brothers had died when both boys were very young. Their mother had been sick and in no condition to properly provide for the family. Fearing the boys may be taken from their mother and the family split up, the elder of the boys quit school at a very young age and began doing whatever he could to provide for the family \u2013 including, at times, stealing chickens when they had no other way of obtaining food. The elder brother, still a boy himself, assumed the role of the father and not only provided for the family however he could, but required his younger brother to remain in school to receive a proper education. Yes, the elder brother was poor, uneducated and a chicken thief, but at his funeral the younger brother delivered a tearful eulogy declaring that, without the sacrifices and efforts of his elder brother, he might have achieved only a fraction of his successes in life.\n\nThe young historian's account was hardly complete or accurate. In his attempt to edit history \u2013 and because he prejudged the events and family members \u2013 he deprived those who would read his work of a beautiful, factual part of family history.\n\nMasonry has many players in its history but not all of them have been friendly toward each other or clearly understandable in modern contexts. When a historian assumes the role of editor and chooses the relevant facts about a Mason or a body of Masons, he assumes an enormous responsibility. If one does not have a complete understanding of all events surrounding a person, place or time, then rendering judgments without the benefit of all the facts will result not only in inaccuracies, but also unfair accounts. We must recognize that some write as Masonic politicians rather than as historians, and their goal is often to deliver a pleasing Masonic mes-sage rather than a historically accurate account.\n\nThe problems for our young historian were two-fold: He did not possess all the needed information, and he prejudged his subjects based on incomplete facts. The task of a serious historian or researcher is often long, tedious and unrewarding. One might spend countless hours reading dusty manuscripts in dimly lit basements with the sole hope of obtaining the smallest of details. It might be far easier to grab a previously published account of a subject and paraphrase what is offered, but that ties our work to any possible errors in the previous work. We must do this work ourselves.\n\nOf course, there are those who believe that are no new Masonic discoveries to be made. Such naysayers are sadly mistaken. It is precisely because so much of our Masonic history is filled with incorrect assumptions \u2013 or facts based on missing information, incorrect readings or simple typographical errors \u2013 that the serious Masonic researcher has great opportunities for many, many new finds. We simply need to take up the cause and follow the proper path.\n\n***\n\nDyslexia: The Gift in Disguise\n\n\"The only thing that interferes with my learning is my education.\"\n\n\\- Albert Einstein\n\nREAD A GOOD BOOK LATELY? My wife has a true passion for reading. I often find her curled up in a lounge chair wholly immersed in a popular or classic novel. When we first started dating, she once asked me, \"Don't you enjoy reading a good book?\" She looked surprised when I told her, \"No, I don't enjoy reading at all.\" She laughed and said that I must be joking as I could almost open a public library sub-station with my collection of books. She could not believe me when I told her that not once in my life have I ever read a book for the sake of enjoyment. I was telling her the absolute truth. There is no enjoyment in reading for me. I read to learn something. If I want to enjoy myself, I will listen to music or watch a movie. For most of my life I did not attempt to understand why this was the case, I simply accepted it as part of being me.\n\nAbout 20 or so years ago, my doctor put me through a series of tests and gave me a perplexing diagnosis. He said that I had dyslexia. I had never heard of dyslexia and had no idea what to expect. The doctor told me that since I was an adult I would probably not even notice it, but that the knowledge of my dyslexia might help explain some of my growing up years. I really didn't understand what that might mean, and since he did not elaborate, I filed the information away for possible later use. That \"later use\" came a little over 10 years ago when my, then, seven-year-old son was also identified as having dyslexia. I had noticed traits in him, as well as his manner of interacting with others and methods of problem solving, that were eerily familiar. I began a study of dyslexia. It was then that the good doctor's words, and much of my painful childhood, started to become clear.\n\nThe only time during my school years that was not incredibly frustrating for me was kindergarten. I imagine that this was because pretty much all that was expected was to say \"please\" & \"thank you,\" not make noise in class or beat anyone up at recess. Trouble began when more was expected. The whole of my school experience can best be summed up as a battle of wits where my teachers seemed to want something from me, but I could never fully get hold of exactly what it was that they wanted. The vast majority of things that were quickly understood by the class were, for the most part, nearly impossible for me to master. In contrast, the very few times that material was presented that seemed to stump the class, I was able to understand nearly before the words came out of the teacher's mouth. I could not understand why I seemed to be so very different... and being different had to be bad.\n\nRon Davis is the author of The Gift of Dyslexia. It is a book that has helped me greatly in understanding dyslexia and has also provided valuable help for my son. As a child, Mr. Davis (a dyslexic) was labeled \"mentally retarded\" and was functionally illiterate until the age of 38. It was then that he developed a system that enabled him to teach himself to read. He went back to school and ended up earning a degree in Engineering. He was hardly retarded. In 1982, Mr. Davis and Dr. Fatima Ali, Ph.D., opened the Reading Research Council Dyslexia Correction Center in California, achieving great success in helping dyslexics overcome their reading difficulties.\n\nIn addition to helping dyslexics properly deal with dyslexia, Mr. Davis' work has contributed in helping the general population arrive at a better understanding of dyslexia. It has been the long-standing misunderstandings, misdiagnoses, mistreatments and preconceived opinions of dyslexics that have caused such hardships for those with dyslexia. Regardless of how it is perceived, being dyslexic does not have to be any more of a handicap or disability than is being left-handed \u2013 unless it is made so. Being left-handed is, in fact, only a difference, a different way of operating. When allowed to operate in their natural manner, left-handed students function like all other students \u2013 just using their left hands. The same is true of those with dyslexia. Dyslexia is a different way of mentally processing information. When a dyslexic child is unidentified, they can be labeled as slow, having learning\/mental disabilities or just being lazy. The child will know that something is wrong, both in their performance and the evaluations of them. They will not, however, be able to identify the actual problem and the result will be great frustration and loss of self-esteem.\n\nThe most common form of dyslexia is visual dyslexia. The exact manifestations of dyslexia, however, can differ from dyslexic to dyslexic. It is rare that dyslexia will manifest itself exactly the same in two individuals. A general trait of the visual dyslexic is that the thought process will be in images. Visual dyslexics mentally manipulate images to create ideas, concepts and thoughts. The written language is comprised of letters formed into words, which are symbols of concepts or ideas. A writer communicates his thoughts on a subject by use of letters formed into words so that the reader will understand those thoughts. Since a dyslexic thinks in images, a symbol for an image (a word) must be translated into the image it represents before the dyslexic can possess the idea. When a dyslexic reads the word \"ball,\" they must mentally translate that symbol into an image of a ball before the symbol (word) can be understood. When dyslexia is undiagnosed, reading will be very difficult and most frustrating for a dyslexic child. Even so, most dyslexics will find methods or work-arounds to their reading difficulties by the time they are adults. No dyslexic will ever enjoy reading for the sake of reading, but most will find ways to function in this medium that seems so annoying to them.\n\nHaving a dyslexic child read out loud in class presents a classic opportunity for misevaluation. Let's say that a dyslexic child is asked to read the sentence, \"The ambulance has a loud siren.\" The dyslexic student will likely stumble at the very beginning of the sentence. The assumption will likely be made that the word \"ambulance\" was a problem for the child. If the word \"ambulance\" is considered to be a word that should be in his or her vocabulary, then the child could be evaluated as having a limited vocabulary and\/or having learning disabilities. Such an evaluation fails to take into account how dyslexics process information. Written words must be mentally translated into images for a dyslexic to process them. When no image can be created for a word, a mental \"blank card\" is replaced for the troublesome word. When reading out loud, a \"blank card\" will cause a stumble for a dyslexic reader.\n\nLet's look again at the sentence that the child was asked to read. \"The ambulance has a loud siren.\" The entire sentence creates an exciting image for most any child. Most children are thrilled at the sight of an emergency vehicle rushing down the road with lights flashing and sirens blasting. A dyslexic child should have no problem at all creating an image for an ambulance. But what image can you create for the word \"the\"? It was the very first word of the sentence that caused the stumble for the child, not the word \"ambulance.\" Articles and pronouns are some of the \"land mine words\" for dyslexics because there are no images that can be readily created for them. Because of the manner in which non-dyslexics process information, teachers untrained in dyslexia often completely ignore words like \"the\" or \"that,\" not even considering that they could be the problem. They will focus on the larger words which, based on their personal experience, are problematic for \"slow\" students. The teacher will make an incorrect assumption that will lead to an incorrect evaluation of the child. The child, having no idea as to how the teacher arrived at her conclusion, will realize that they have no problem with the words they are accused of not understanding, but will be unable to defend himself or explain the stumble. Frustration follows.\n\nUndiagnosed dyslexic children, being clueless as to the nature of dyslexia or that they even have it, will not only have to figure out how to translate words, but they will have to deal with words that are untranslatable. They will also be required to solve all of these problems with no assistance whatsoever and while having to deal with the false labels of being \"slow\" or \"lazy.\" And this is only when we consider the reading challenges dyslexics face. Spelling, math and a host of other school subjects present equal problems for the dyslexic child. It is small wonder that most all dyslexics have extremely low self-esteem. The \"battle wounds\" and lifelong scars received when undiagnosed dyslexics begin down the path of learning by \"standard methods\" come normally at an age when a child's self-esteem is developing. Dyslexics often feel that their problem is \"their problem\" and most have no idea that they are far from alone.\n\nDyslexia affects far more individuals, at varying levels, than many realize. Some of our most noted artists, scientists, businessmen and world leaders have had dyslexia. Most dyslexics who \"rise to the top\" in their field do so by unconventional methods stemming from their unconventional manner of learning \u2013 learning which often takes place in spite of, rather than because of formal education. A few famous dyslexics include: Pablo Picasso, Thomas Edison, Albert Einstein, Tom Cruise, Danny Glover, John Lennon, Winston Churchill, Nelson Rockefeller & Walt Disney.\n\nRon Davis writes:\n\n\"Once as a guest on a television show, I was asked about the \"positive\" side of dyslexia. As part of my answer, I listed a dozen or so famous dyslexics. The hostess of the show then commented, 'Isn't it amazing that all those people could be geniuses in spite of having dyslexia.'\n\n\"She missed the point. Their genius didn't occur in spite of their dyslexia, but because of it!\n\n\"Having dyslexia won't make every dyslexic a genius, but it is good for the self-esteem of all dyslexics to know their minds work in exactly the same way as the minds of great geniuses. It is also important for them to know that having a problem with reading, writing, spelling, or math doesn't mean they are dumb or stupid. The same mental function that produces a genius can also produce those problems.\"\n\nAn interesting characteristic of dyslexia is that while many of the common school lessons presented to students will be extremely difficult for the dyslexic student to master, they will excel at many of the more abstract concepts or abilities that seem to be problematic for the non-dyslexic student. It is here that the gift of dyslexia manifests itself. Most dyslexics are very analytical, creative and capable of multi-dimensional thinking. If these gifts are not suppressed during the standard educational process, dyslexics can sometimes use the very process that gives such trouble in school to achieve what may seem to be surprising successes.\n\nIn my own case, one \"surprising success\" came as a result of Freemasonry. I joined right after my 21st birthday, with the \"school experience\" fresh in my mind. My EA initiation was, without doubt, one of the most moving and profound experiences of my life. But after my initiation, I received news that made me wonder if my short Masonic career was over. I was told that I would be assigned an instructor who would teach me what was needed to pass an exam in open Lodge. I was told that this was necessary for me to advance to the next degree. This was about the last thing that I expected or wanted to hear. My entire school experience was a nightmare. I could not see myself passing any sort of exam. After many delays, I finally met with my instructor. I was incredibly surprised. Had my instructor handed me a printed version of all of the work and told me that it was necessary for me to read it and then memorize it, I would likely be an EA today. The manner of \"mouth to ear\" verbal instruction was, however, something that was not only was possible for me to do, but something at which I excelled. I flew through the work, asking and answering my own questions for my FC and MM exams. I became a certified Lodge Instructor shortly after my MM degree and, in just a few years later, a District Grand Lecturer. It was not hard work that resulted in these achievements; it was the gift of my dyslexia. I simply had a gift for the memorization of things verbally presented to me.\n\nDyslexia is not an illness in need of cure nor is it a handicap that will forever deny a dyslexic the chance of living a full and rewarding life. Great advances are being made in the understanding of dyslexia as well in providing the proper tools to educators so that dyslexic children will be able to fully realize and benefit from the educational process. In addition, a number of organizations, including the Scottish Rite, have taken on the research and proper education of dyslexia as one of their target projects.\n\nFor additional on-line information about dyslexia, please visit the below websites. You can also call The International Dyslexia Association at: 410-296-0232, the Davis Dyslexia Association International at: 1-888-999-3324 or your local Scottish Rite body.\n\nhttp:\/\/www.dyslexia.com\n\nhttp:\/\/www.interdys.org\n\nhttp:\/\/scottishrite.org\/about\/philanthropy-scholarships\/ritecare\n\n***\n\nJames Foulhouze\n\nA Biographical Study\n\nMr. Pike is altogether unknown to me, and I have never seen him, which is perhaps to be regretted, because in the event he spoke to me pursuant to the information which he has received from ill- disposed individuals, I suppose that he will be sorry for having allowed his pen to write what is neither correct nor rational.\n\n- James Foulhouze, 1858.1\n\nJAMES FOULHOUZE was, unquestionably, the arch-nemesis of Albert Pike in Pike's early days as Grand Commander of the Supreme Council, Southern Jurisdiction, USA. Judge James Foulhouze, former Roman Catholic priest, Sovereign Grand Inspector General of the Grand Orient of France and Sovereign Grand Commander of the Supreme Council of Louisiana in the pre-Concordat of 1855 period along with some of the leading New Orleans Masons, including the very respected Judge T. Wharton Collens and the powerful United States Senator Pierre Soul\u00e9, may have nearly destroyed the concordat between the New Orleans and Charleston Supreme Councils - a concordat which was the breath of life to the newly reorganized Charleston Supreme Council. Who was this man who could have caused such a disturbance ? Did he cause the disturbance or was he, himself, swept along in a tidal wave of events?\n\nThe following is a glimpse into the life and tumultuous Masonic times of a most significant, but highly controversial, figure in the history of the US Scottish Rite. It is to be regretted that no photograph or likeness of Foulhouze is known to exist. It is, also, unfortunate that some areas of his life are simply lost in the mists of time.\n\nOn 1 October 1800,2 Jacques Foulhouze was born to Michel and Jeanne Cronier Foulhouze in Riom, France. The young Foulhouze received a Catholic education at the Seminary of St. Sulpice in Paris culminating in his ordination as a Roman Catholic priest. The Reverend James Foulhouze traveled to the United States and labored in the Diocese of Philadelphia in 1834 and 35.3 The next found record concerning Foulhouze in the US comes in 1835 when his name appears in a Philadelphia court records book of aliens declaring their intention to take the oath of allegiance to the United States.4 Foulhouze would not long remain a priest nor keep his domicile in Philadelphia. An 1858 New Orleans publication contains interesting comments about Foulhouze and his possible reasons for leaving the priesthood. The comments were written by Charles Laffon de Lad\u00e9bat, a leading figure in mid 19th century New Orleans Scottish Rite Masonry, who will be discussed further later in this paper. Lad\u00e9bat says that Foulhouze might have remained a priest had not, \"Mr. (now bishop) Hughes been appointed, in his stead, to the important rector ship of a northern parish, to which Mr. Foulhouze was, for his long service, justly entitled.\"5\n\nJohn Hughes (1797-1864) was the fourth bishop and first Archbishop of the Roman Catholic diocese of New York. Hughes served with Foulhouze in the Diocese of Philadelphia and founded there the Catholic Herald newspaper. Hughes was consecrated coadjutor to Bishop John Dubois of New York in 1838. He succeeded Dubois in 1842 and became archbishop of New York in 1850.6 Foulhouze, regardless of Lad\u00e9bat's comments, could not have been affected by the 1838 Hughes appointment as the Journal Notes of Philadelphia Bishop Francis Kenrick record Foulhouze's faculties being suspended on 5 February 1836.7 As with many areas of Foulhouze's life, it is unclear what could have taken place causing his separation from the priesthood. Foulhouze was a graduate of the highly respected Seminary of St. Sulpice. Many Catholic dioceses consider such graduates to be a highly desirable prize. The accounts of Foulhouze for that time, however, tell a different story. The records of the Archdiocese of Philadelphia,8 while confirming that Foulhouze was, indeed, a priest assigned to them, show had he had \"no specific assignment.\"9 This is an interesting situation. Why would the Diocese of Philadelphia not take advantage of the quality education that Foulhouze received by putting his abilities and education to use? Foulhouze, himself, may provide the answer. In 1843 Foulhouze was asked if he had taken the vows of the priesthood, he replied: \"No, but it is true that they were given to me, against my will.\"10 Regardless of the philosophical point Foulhouze was trying to make, his statement reflects that he may not have ever wholly embraced the priesthood. If Foulhouze's work reflected the same lack of interest, then it is very likely that regardless of what seminary he attended, he would not have been given assignments nor appointments to higher positions. All conjecture aside, Foulhouze did leave the priesthood, pursue a career in law and move to New Orleans.\n\nFoulhouze began his law career in Philadelphia after leaving the priesthood. In 1842, he published a book in Philadelphia that reflected the same interest in philosophy that was maintained throughout his life. The 200 page work was titled, A Philosophical Inquiry Respecting the Abolition of Capital Punishment.11 It is possible that Foulhouze was in New Orleans when this book was released, but it is clear that he was in New Orleans the following year. Philadelphia Bishop Francis Kenrick (Foulhouze's former superior) writes in a 1843 letter:\n\nHere affairs go on smoothly but at New Orleans an infidel faction are struggling to destroy or subjugate the Episcopal authority. A fallen French priest, Foulhouze, is the editor of an impious paper,12 the organ of the Marguillers. [...] The leaders in disorder are Freemasons, and they contrived to set apart a lot in the Cemetery for their Masonic brethren, and had it dedicated by a speech from their Grand Master who is a Marguiller.13\n\nThe Marguillers were the wardens of the St. Louis Cathedral in New Orleans. The Grand Master that Kenrick spoke of was E. A. Canon, who was not only a Marguiller, but the President of the Marguillers. The Marguillers (many of whom were Freemasons) controlled the appointments of the priests for the St. Louis Cathedral during the early to mid 1800's. There was, of course, a great division within the congregation over Freemasons having a say in the appointment of their parish priests (regardless of the fact that these Freemasons were, themselves, Roman Catholic and members of the parish). In New Orleans, Masonry and the Roman Catholic faith were tightly intertwined for a number of years in a love\/hate relationship. It was a situation not without some hostile conflicts. An event that took place in 1842 is worth mentioning:\n\nOn the feast of All Saints, an incident took place in the Cathedral which was in itself trivial, but which shows to what lengths the two factions14 would go. While Father Jamey was preaching, E. A. Canon, the president of the Marguillers, entered the sanctuary by way of the choir entrance, and made a tour of the altar towards that place assigned to the president of the wardens (side opposite the door of the sacristy by which he entered). He remained there for a few minutes, but not being able to hear very well, he advanced to the balustrade of a neighboring chapel, in order to hear better. He had only heard a few words, and then decided to retire by the way he had come in, that is, behind the altar. As he was going out he was greeted by Octave de Armas, a parishioner loyal to [Bishop] Blanc, (who was also seated in the sanctuary) with the words, \"Get out; you are not in your place...\" Canon answered this with apparent sharp disdain and was preparing to leave when he was pushed. He was near the door of the sacristy and fell on the steps. On getting up he heard Armas distinctly cry, \"I, I alone will get rid of the wardens.\" The services were interrupted for about five minutes, but the Mass was soon continued and all ended calmly.15\n\nThe event may have ended \"calmly\" at that time, but the incident was far from over. As a result of his being pushed in the St. Louis Cathedral, Canon, following typical Creole custom, sought satisfaction from Armas by means of challenging him to a duel. Armas, however, refused the challenge on the grounds that he was a Roman Catholic.16 Friends of Canon would not let the matter drop and charges were filed against Armas with the City Recorder. Armas was found guilty of assault. The incident is reflective of the growing tensions between the factions within the New Orleans Catholic community. It was in this atmosphere and, likely, through the contacts with the Marguillers that Foulhouze was introduced to Louisiana Masonry. It obviously attracted him and he sought to be a member.\n\nFrom Priest to Freemason\n\nThe Marguillers might have introduced James Foulhouze to Louisiana Masonry, but it was not his first exposure to Freemasonry itself. Foulhouze stated in 1857:\n\n\"Being a Grandson of Free-masons, I, in my early years, conceived and entertained a desire to enter the fraternity ...\" 17\n\nFoulhouze fulfilled that early desire by becoming a member of Los Amigos del Orden, a Spanish speaking, New Orleans Scottish Rite Lodge.18 Foulhouze also stated:\n\n\"Within a year from my initiation I was made a Master Mason in the same Lodge.\"19\n\nThere are, unfortunately, no known records of the initiation of Foulhouze nor can an exact date be placed on his initiating, passing or raising. Foulhouze did state in his Historical Inquiry that he was initiated by Antonio Costa. 20 Costa was Worshipful Master of Los Amigos del Orden in 1843. An 1843 initiation followed by an 1844 raising meant rapid advancement for Foulhouze. Foulhouze was, apparently, viewed as a Mason of promise. On 14 February 1845 he was appointed Grand Translator by the Grand Lodge. The office of Grand Translator did not exist prior to Foulhouze receiving the appointment. The office was created due to the growing need for French to English and English to French translations in Grand Lodge records and documents.\n\nIn the summer of 1845 (about a year after Foulhouze became a Master Mason) Foulhouze traveled to France carrying a letter of introduction from Robert Preaux, Grand Master of the Grand Lodge of Louisiana and Active Member of the Supreme Council of Louisiana. During his stay in Paris, Foulhouze received all of the degrees of the A.A.S.R. culminating in the 33rd degree on 27 September from the Grand College of Rites of the Grand Orient of France. The speed in which Foulhouze received the degrees is extraordinary and certainly was not normal procedure for the Grand Orient. There is no explanation to be found as to why this very rare honor was given to such a young Master Mason nor has the contents of the letter from Preaux ever been revealed. Regardless of what activities Foulhouze later engaged in, he was, in the eyes of the U.S. Masonic community, a legitimate Sovereign Grand Inspector General. Of this event, Foulhouze comments:\n\nThe Scotch Rite [...] pleased me on account of its truly philosophical principles, and the more I studied it, the more I felt anxious to take its superior degrees, when a fair opportunity so to do offered itself to me in 1845.\n\nI was in France, and on the recommendations and letters of my Scotch brothers here, the worshipful Lodge \"Cl\u00e9mente Amiti\u00e9\" opened its door to me, and after a short stay among them I was made a Knight R:. + and a Knight Kadosh, which I am bound to say, rendered still clearer to my eyes and intellect the views which I had long entertained on the merits of the Scotch Rite, and for ever attached me to its admirable and useful tenets.\n\nThe favors thus bestowed on me, were unexpected, and I certainly desired no others, when on a special and unasked resolution of the Supreme Council in the Grand Orient, I was called and raised in that body to the thirty third degree.21\n\nFollowing the death of Sovereign Grand Commander Jean-Jacques Conte, New Orleans Judge Jean-Fran\u00e7ois Canonge, an influential Past Grand Master of the Grand Lodge of Louisiana, became the Grand Commander of the Supreme Council of Louisiana on 20 September 1845. 22 Foulhouze said of Canonge:\n\nAs long as he lived, I had but little to do, and contented myself with studying the rite ... 23\n\nFoulhouze, who had affiliated with the Supreme Council of Louisiana in 1846, was, regardless of his comments, not idle during this period. Foulhouze was appointed Grand Secretary of the Supreme Council of Louisiana in 1847.24 He, also, advanced through the chairs of Los Amigos del Orden serving as its Worshipful Master in 1847. Once serving his term as Worshipful Master, he was elected a life member of the Grand Lodge. It must also be pointed out that the invasion of the jurisdiction of the Grand Lodge of Louisiana by the Grand Lodge of Mississippi and the creation of the Louisiana Grand Lodge in 1848 would, surely, have occupied a considerable amount of time with all the Worshipful Masters of New Orleans Lodges.\n\nThe Grand Lodge of Mississippi and the Union of 1850\n\nA faction within the New Orleans English-speaking York Rite Masons felt that the 1844 Constitution of the Grand Lodge of Louisiana sanctioning the cumulation of the three rites worked by lodges in Louisiana (French, Scottish & York) altered the Grand Lodge into a body that was no longer a true York Rite Grand Lodge.25 The decision was made by these Masons to sever their association with the Grand Lodge and organize themselves into what they felt was proper York Rite Masonry. A committee was formed and a letter of grievance was brought before the Grand Lodge of Mississippi on 23 January 1845.26 The Grand Master of the Grand Lodge of Mississippi was Mexican War hero and former governor of Mississippi, John Anthony Quitman. The Grand Lodge of Mississippi appointed a committee to go to New Orleans in order to examine the situation. On 21 January 1846, the committee from the Grand Lodge of Mississippi appointed to examine the charges presented by the York Masons from New Orleans presented three reports concerning the events. The first report was presented on behalf of the majority of the committee and concluded that there was \"no Grand Lodge of Ancient York Masons within the limits of the State of Louisiana\" and that the Grand Lodge of Mississippi had \"the power, and it is its duty on proper application, to issue Dispensations and Charters to bodies of Ancient York Masons within the limits of the State of Louisiana, until the constitution of a Grand Lodge within that State.\"27 Two \"counter\" reports were then presented which advised against the Grand Lodge of Mississippi issuing charters within the jurisdiction of the Grand Lodge of Louisiana. The outcome of the events of 21 January (despite the efforts of the two \"counter\" reports) was the chartering of George Washington Lodge in New Orleans and Lafayette Lodge in Lafayette28 by the Grand Lodge of Mississippi on 22 February. Relations were severed between the Grand Lodges of Louisiana and Mississippi. The Louisiana Lodges chartered by the Grand Lodge of Mississippi were declared irregular by the Grand Lodge of Louisiana. In total, the Grand Lodge of Mississippi chartered seven Lodges in the New Orleans area by 1848.29 These seven Lodges united to form the \"Louisiana Grand Lodge of Ancient York Masons\" on 8 March 1848. The Grand Lodge of Mississippi received admonishment from most U.S. Grand Lodges and the majority openly condemned its action.30 While the future for this splinter group of the Grand Lodge of Louisiana may have looked bleak, several events took place to not only strengthen the position of the English-speaking New Orleans Masons, but to assure them of total victory by the loss of French control over most all forms of Louisiana Masonry.\n\nOne of the last official acts of Grand Commander Jean-Fran\u00e7ois Canonge was a speech made on 3 November 1847 in Baton Rouge in which he is reported as stating that a circular issued by the Mississippi lodges in New Orleans was \"unworthy of notice.\"31 Canonge died on 19 January 1848. On 31 January 1848 James Foulhouze was elected Grand Commander of the Supreme Council of Louisiana. The Foulhouze election bypassed a number of senior members of the Council and, clearly, established the popularity of Foulhouze with the Council. Foulhouze had brought with him various rituals from France32 which he edited for the New Orleans Council.33 During the same month as the death of Canonge and the election of Foulhouze, the Charleston Council was talking an action that greatly strengthened its own position and further weakened the hold of the French-speaking New Orleans Masons. Albert Mackey (the Grand Secretary of the Charleston Council) sent a notice to the Freemason's Monthly Magazine 34 (Boston) which read:\n\n\"At a special session of the Supreme Council ... for the Southern Jurisdiction of the United States of America, our Illustrious Brother, John A. Quitman ... Major General in the Army of the United States, was elected to fill a vacancy in this Supreme Council, and was duly and formally inaugurated a Sovereign Grand Inspector General of the 33d. All Consistories, Councils, Chapters and Lodges under this jurisdiction are hereby ordered to obey and respect him accordingly.\"35\n\nOn 29 January 1849 the Grand Lodge of Louisiana published a report that Foulhouze wrote for them concerning the cumulation of the rites practiced by the Grand Lodge and on 26 February the Grand Lodge published Foulhouze's report on the 1833 Concordat. Both reports upheld the positions of the Grand Lodge of Louisiana and encouraged the continued practice of the cumulation of the rites in Louisiana.\n\nOn 14 September 1849 Foulhouze, along with a several other New Orleans Masons, were honored by Friends of Harmony Lodge (whose Worshipful Master was elder Past Grand Master and New Orleans Supreme Council Member John Henry Holland) by being made honorary members of the lodge. An excerpt from the Minutes of the Lodge reads:\n\n\"Whereas by their great ability and impartiality our well beloved Brethren Joseph Walker, Jas. Foulhouze, P. Willman, John D. Kemper & R. Preaux have earned the destination of Honorary Membership, their services in the Masonic vineyard entitling them to some suitable token or tribute of appreciation of their worth, and of the high respect entertained for their estimable personal and Masonic character - they being Brethren to whom a burdened may pour out his sorrows, to whom distress may prefer its suit; Brethren whose hands are guided by justice and whose hearts are expanded by benevolence.\n\nTherefore be it now decreed, that the aforesaid distinguished Brethren be and they are hereby created Honorary Members of the Friends of Harmony Lodge of F & A Masons, this as a testimony of regard for the inestimable services as Masons, and their courtesy, affability and kindness as men - well worthy of initiation and the foregoing preamble and resolution being seconded and put is carried unanimously. 36\n\nThe Union of 1850\n\nThe 1848 Louisiana Grand Lodge obtained recognition from only one other Grand Lodge \u2014 the Grand Lodge of Mississippi. In 1849 John Gedge, a New Orleans attorney, was elected Grand Master of the Louisiana Grand Lodge. Despite what would seem to be the irregularity of the Louisiana Grand Lodge and the lack of support for this new Grand Lodge within the Masonic community, the Grand Lodge of Louisiana entered into negotiations and finally merged with this body in 1850. The Grand Lodge of Louisiana was left with little choice in this matter. The fact that the Grand Lodge of Louisiana was overwhelmingly considered the \"regular\" Grand Lodge was not sufficient to overcome the internal problems stemming from the cultural divisions in New Orleans. By mid 1849, it was likely realized that the English-speaking lodges which had remained loyal to the Grand Lodge were showing signs that continued loyalty would, most likely, not happen. Obviously realizing that the total collapse of the Grand Lodge of Louisiana was a very real possibility, the Grand Lodge of Louisiana and the Louisiana Grand Lodge entered into talks designed to merge the two bodies.37 That merger took place in June of 1850 with the approval of a new Constitution of the Grand Lodge of Louisiana of Free and Accepted Masons. Under the terms of the agreement of the merger, the Louisiana Grand Lodge members became recognized as \"regular\" by the Grand Lodge of Louisiana. All Lodges chartered by the Louisiana Grand Lodge (or by the Grand Lodge of Mississippi in Louisiana) passed under the jurisdiction of the new Grand Lodge of Louisiana. While the new constitution appeared to merge the two Grand Lodges, the Grand Lodge of Louisiana was, in effect, taken over by the Louisiana Grand Lodge. All non-York Rite Lodges were instructed to turn in their charters to receive new York Rite charters from the new Grand Lodge. Three Scottish Rite Lodges, Etoile Polaire, Los Amigos del Orden, and Disciples of the Masonic Senate, sought relief from the New Orleans Supreme Council. Of these events Foulhouze wrote:\n\n\"It was agreed that the Grand Lodge should no more cumulate the rites, that it would have and keep its own forms, but that each Lodge in the East might freely work according to its particular and more favorite rite and tenets.\n\nHad that agreement been faithfully observed, another series of quiet days might have ensued in Louisiana: but the newcomers in the Grand Lodge soon showed that far from being sincere, they had crept into our bosom with the only view to tear it to pieces and to build their powers on the ruins of ours. [...] They made as I had foreseen and foretold, a Constitution by which the Scotch lodges of the East were reduced to nought and the life members of the Grand Lodge expelled from it38 the better to secure the triumph and power of those invaders.\n\nBut from the moment that the constitution began to work, the Scotch lodges understood their mistake; and not withstanding the blame thrown upon them by the new Grand Lodge which was as it was expected, did not fail to say that they were bound by the vote of the majority at Baton Rouge, they all parted from it, averting and showing that they had been deceived, and could not thus be fetted and annihilated by a paltry trick.\n\nThat event occasioned a good deal of rumor. The Mississippians who had snatched the power began promulgating their bulls of excom-munication. John Gedge, like his imitators of this present Consistory, wrote his reports, made his speeches, sent his circulars, but it was to no purpose.\n\nThe Supreme Council of Louisiana resumed its authority on the blue lodges of the Scotch rite, and the separation was con-summated.39\n\nIf the goal of the new 1850 Grand Lodge Constitution and the merger with the Louisiana Grand Lodge was to bring peace to all the Louisiana Masons, it was a total failure. If the goal was to remove the power base in the Grand Lodge from the French-speaking New Orleans Masons, it was, indeed, a success. The French-speaking New Orleans Masons became split after 1850. One faction, outraged at the turn of events, wished nothing more to do with the Grand Lodge and saw the Supreme Council as the only hope of maintaining the French interests. The other French faction, most likely very tired of the squabbles, remained with the Grand Lodge in the hopes of possibly still bringing unity to the troubled Grand Lodge.\n\nThe 1850 Union of the Grand Lodge resulted in a perceived need for action in the New Orleans Council. Foulhouze believed that he could strengthen the New Orleans Scottish Rite by expanding the number of 33rds in the Council.\n\nFoulhouze says of this:\n\n\"Brother Canonge died and I was elected commander in his place. My first move was to promote to the 33d degree one or two members of each of the lodges then established and of some importance in the city of New Orleans, hoping that their initiation would be the best means to secure the masonic peace in our East, as it would contribute to carry light where it was most needed.\" 40\n\nDuring Foulhouze's administration of the Supreme Council of Louisiana prior to the Concordat, he elevated approximately 30 Masons to the 33rd degree in the New Orleans Council.41 Those elevated to the 33rd degree by Foulhouze included Charles Claiborne, Thomas Wharton Collens (22 June 1849), Claude Pierre Samory and Charles Laffon de Lad\u00e9bat (11 February 1852). The wisdom of expanding the membership of the Supreme Council was apparently recognized by Albert Pike on 25 March 1859 (Pike's first SC Session as Grand Commander) when he expanded the Membership in the Charleston Council from 9 Members to 33 Members.\n\nCharles Claiborne assumed the post of Secretary General for the New Orleans Council and T. Wharton Collens, that of Lt. Grand Commander. The Foulhouze\/Collens relationship was a very close one which continued until Foulhouze's death in 1875 - years after both had resigned from Masonry. Foulhouze and Collens would, in the early 1850's, even share a law office.\n\nThe Lopez Expedition and James Foulhouze\n\nIf the Union of 1850 between the Grand Lodge of Louisiana and the Louisiana Grand Lodge, along with the many bomb shells from that event, were not enough to occupy the minds of the Louisiana Masons, an event took place simultaneously that over-shadowed the Masonic events in Louisiana and be thrust into the forefront of the minds and thoughts of all Americans. This international event directly played a part in future New Orleans Masonic \"battles.\"\n\nIn 1849 Narciso Lopez, a Venezuelan and former colonel in the Spanish Army, began a campaign to take control of Cuba and replace the Spanish government on the island with his own government. Lopez received limited support from various U.S. politicians, but was unable to raise a suitable sized army for his mission. Lopez found better luck in New Orleans where he was able to raise an army of approximately 750 men, mainly veterans of the Mexican War, and sail out of New Orleans in April of 1850 with the goal of capturing the island. The mission was a complete failure. The U.S. troops were slaughtered and Lopez was eventually captured and executed. Reports quickly came to the U.S. and the newspapers of the day reported the \"murder\" of the U.S. troops along with the capture and execution of not only troops, but vacationing U.S. tourists who happened to be on the island. New Orleans was an obvious \"hot spot\" for the Lopez Expedition as, not only did the expedition leave from New Orleans, but the city contained many Spanish speaking citizens from Cuba. The Grand Lodge of Louisiana had also chartered two Lodges in Cuba during the early years of the Grand Lodge.42 The tie between New Orleans and Cuba was close for both the general and Masonic population.\n\nJames Foulhouze became entwined in the Lopez Expedition when he traveled to Cuba at the height of the crisis. A New Orleans newspaper, the Daily Delta, ran a story on Foulhouze vehemently criticizing his trip, and suggesting that he was, possibly, a spy for the Spanish government.43 The very evening following the publication of the article concerning Foulhouze, T. Wharton Collens and Robert Preaux published an article in the Daily Picayune explaining that Foulhouze's trip to Cuba was with the goal of, hopefully, securing the release of vacationing U.S. citizens who were caught in the conflict.44 Foulhouze, being made a Mason in a Spanish speaking Lodge, had numerous interactions with New Orleans Masons from Cuba. In addition, Foulhouze had gained the confidence of various Spanish officials on the island of Cuba through acting as legal council for them several years earlier. Along with the article published by Collens and Preaux, the Delta article on Foulhouze received censure by a number of competing New Orleans newspapers. The Delta article was exposed to be a newspaper \"thriller\" story with little basis in fact. One newspaper entitling an article critical of the Delta's lack of support for its charges \"Newspaper Intolerance\"45 and another paper calling a report on Foulhouze's trip \"A Mission of Humanity.\" 46 The Delta ran one more article in defense of its position claiming that the matter would be settled when Foulhouze returned to New Orleans and the entire event would be brought to the attention of the public.47 Nothing more, however, was published on the matter by the Delta. The event passed from the public's attention and was attributed to one newspaper's attempt to sensationalize anything concerning a recent event with the possible goal of increasing its sale of newspapers.\n\nEnter the Charleston Supreme Council\n\nJohn Gedge, who in 1849 was the Grand Master of the irregular Louisiana Grand Lodge, was elected Grand Master of the Grand Lodge of Louisiana for the year 1851. On 27 March 1851 the New Orleans Council issued a manifesto in its own defense. This manifesto examined the New Orleans situation and was an appeal for the establishment of fraternal relations between the New Orleans Council and other Masonic Bodies world-wide. With Louisiana Masonry in a state of turmoil and the once powerful Supreme Council of Louisiana fighting for order and stability, the time for the Charleston Council to act was at hand.\n\nAt the invitation of John Gedge, Albert Mackey came to New Orleans in February of 1852 and established, for the Charleston Council, a Consistory of the 32\u00ba. Gedge served as Commander in Chief. The establishment of this Charleston Consistory in New Orleans resulted in a new wave of turmoil and paved the way for the Concordat of 1855 merging the Charleston and New Orleans Councils.\n\nThe Supreme Council of Louisiana responded to the Charleston Consistory in New Orleans by taking several measures. A notice critical of the new consistory was place in the New Orleans Bee by the Supreme Council of Louisiana on 27 February 1852.48 The notice carried the names of the then 29 Active Members49 of the New Orleans Council. The New Orleans Council, also, incorporated itself under the official name of \"Supreme Council of the Thirty-three [sic] and last degree of the Ancient and Accepted Scotch Rite for the United States of America, sitting at New Orleans, State of Louisiana.\" The act of incorporation was signed on 7 June 1852 and approved by the Secretary of State, the noted Charles Gayarre, on 13 January 1853.50\n\nIn July of 1852 Foulhouze traveled to New York to install Henry C. Atwood as Grand Commander of the \"Sup-reme Council of the Thirty-third Degree of and for the Free, Sovereign and Inde-pendent State of New York\" and then journeyed on to France in an attempt to enlist French support for his cause. It is noteworthy that Foulhouze embraced the concept that Supreme Councils should be limited to state jurisdictions just as Grand Lodges.51\n\nThe Concordat of 1855\n\nThe speed in which the total loss of the Grand Lodge of Louisiana by the French-speaking Masons occurred caused obvious confusion and uncertainty as to the future. James Foulhouze, as Grand Commander of the New Orleans Supreme Council, sought to unite all of the French-speaking Masons under his banner. Whether it was because of the rapid advancement of Foulhouze (resulting in uncertainty in his ability) or simply personality conflicts, Foulhouze was unable to unite all of the French Masons. The conflict of opinions within the New Orleans Supreme Council as to the direction in which to proceed can reasonably be seen as a contributing factor to the resignation of Foulhouze on 30 July 1853 and nearly all of the officers of the New Orleans Council by December of 1853. The final break for Foulhouze appears to have occurred at the 22 June meeting of the New Orleans Council. At that meeting, T. Wharton Collens, the Lt. Grand Commander, had prepared a series of resolutions to present to the Council. After a reading of the resolutions, the floor was opened for comment, but instead of addressing the points of the various resolutions, Charles Claiborne apparently began a series of attacks on Foulhouze's clothing. The meeting fell into shouting matches and the deep rooted feelings of frustration from the events of the past years seemingly boiled up. Foulhouze, realizing that control of the meeting was lost, closed the Council and departed.52\n\nIn the absence of the Minutes of the New Orleans Council during the Foulhouze years53 it can only be presumed that T. Wharton Collens assumed the post of acting Grand Commander for the remainder of 1853 until his own resignation on 19 December of that year. The day following the resignation of Collens, the Grand Treasurer, Jean Baptiste Faget, turned in his letter of resignation and an undated letter of resignation from the Grand Secretary, J.J.E. Massicott, was also accepted by the Council.\n\nOn 7 January 1854, Charles Claiborne was elected Grand Commander of the New Orleans Council. Claude Pierre Samory was elected Lt. Grand Commander and Charles Laffon de Lad\u00e9bat was appointed Grand Secretary. Samory and Lad\u00e9bat were part of the French-speaking faction that split from Foulhouze during the 1850-53 turmoil. 1854 was devoted to negotiations with the Charleston Supreme Council. 6 & 17 February 1855 the concordat merging the New Orleans and Charleston Supreme Councils was signed. Present in New Orleans for the signing of the Concordat, and representing the Charleston Council, were Albert Mackey and John Quitman. John Gedge, who had spearheaded the movement of the Louisiana Grand Lodge and the 1852 Consistory, did not live to see the concordat between the New Orleans and Charleston Councils - he died on 13 April 1854 during a yellow fever epidemic in New Orleans.\n\nThe death of Gedge must have created some concern for the future of the newly reorganized Scottish Rite Masonry in New Orleans. Gedge had led a complete and total coup of the Grand Lodge, dramatically altering its nature. It was, also, Gedge who had written to Mackey to bring a Charleston consistory to New Orleans and took control of this consistory as he did the Grand Lodge. The introduction of the Charleston consistory paved the way for the Concordat of 1855. His influence on the events of the times is unquestionable. It is reasonable to assume that Gedge might have taken some position of leadership in the post concordat days - had he lived. It is logical that Gedge would have become an Active Member of the Charleston Supreme Council and led the reorganized Grand Consistory of Louisiana. The death of Gedge made this impossible, yet the basic problem remained. A powerful figure was needed to lead and unite the very fragmented New Orleans Scottish Rite. Regardless of the fact that the concordat had taken place, there were still quite a number of former New Orleans Council 33rds unaffiliated with the Charleston Council - or any Council. The potential for uprising was undeniable. In a letter to Claude Samory, Albert Mackey suggested that the man to lead and unite the New Orleans Scottish Rite Masons had been found and it was believed that only the formalities remained. Mackey wrote:\n\nI hope to be present at the installation of that Bro:. as S:.G:.I:.G:. whose adhesion to us will heal all difficulties [...] The moment we receive your nomination, the nominated Bro:. will be elected.54\n\nThe man Mackey wrote of was James Foulhouze. The choosing of Foulhouze to join the Charleston Council and lead the New Orleans Scottish Rite for the Charleston Council is very reasonable and, given the situation, the only logical choice that could be made. Foulhouze was viewed as a regular 33rd from the Grand Orient of France. As Foulhouze was also a former Grand Commander of the New Orleans Supreme Council who had resigned prior to the concordat, he might have been viewed as something of a prominent \"free agent.\" The fact that Foulhouze was a member (and even Grand Commander) of the New Orleans Council was irrelevant from a regularity stand point. If he agreed to join with the Charleston Council then this matter could be easily settled. Samory and Lad\u00e9bat were also members of the New Orleans Council (and both given the 33rd degree by Foulhouze) yet both became Active Members and officers of the Charleston Council. If James Foulhouze agreed to lead the New Orleans Scottish Rite, under the Charleston Council banner, the Charleston Council would have a much easier road to travel in bringing the remainder of the New Orleans Scottish Rite Masons under their control. Foulhouze was approached by Albert Mackey and Claude Samory in the summer of 1856 and offered the position of Commander-in-Chief of the Grand Consistory and Active Membership in the Southern Jurisdiction providing that he joined the Charleston camp.55 Of this event Foulhouze wrote:\n\nAbout a year or fifteen months ago, M. Antonio Costa asked me whether I had any objection to converse with M. Claude Samory about the then state of affairs with regard to the Scottish Rite in Louisiana. I answered that I had none. On the following day M. Samory together with M. Costa called on me, and in his presence, told me that he had long been anxious to see me, that he was always my friend, that the course which he and other members of the Supreme Council of Louisiana had followed since I left it was with the only view of putting an end to any further contest and quarrel both with the Grand Lodge of our state and the Supreme Council of Charleston, that many a York mason of this east was now initiated to the high degrees of the Scottish Rite, that they all had heard of me as being well versed in its tenets and ceremonies, and were anxious to see me join the Consistory thereto assume the command of the Rite in Louisiana, that indeed I had just cause to complain of the conduct of some BB:. towards me both in the Supreme Council and in the Polar Star Lodge, but that they all acknowledged it, and were ready on my joining the Grand Consistory, to offer me any apology I might wish, that there was a vacancy in the Supreme Council of Charleston which he had been offered to fill, and which he was ready to give up in my behalf if I would unite with them, that my presence in that Council would do immense good both here and at Charleston, and that the best I could do was to accept, if I desired to carry out my opinion and views with regard to the right which Louisiana has to its Supreme Council.\n\nMy answer to M. Samory was as follows:\n\nI need no apology, for any thing which may have been done or said in any masonic body to hurt my feelings. Masonry, thank God, has taught me better desires, and it is enough for me to hear from you that all those who may have had an intention to offend me, do now regret it. As to your proposal, I can in no way or manner accept it. My position is clear and well defined. The Supreme Council of Louisiana was not founded by me. It existed before I was a mason. In 1845 I received, not in the Supreme Council of France founded by M. Grasse de Tilly, but in the Supreme Council of the Grand Orient, the 33d degree. That most Illustrious body treated me as a future member of the Supreme Council of Louisiana with which it corresponded, and I was commissioned by its Grand Commander and other members to be the interpreter of their good feelings near our Supreme Council. A short time after my return here, our Grand Commander Jean Fran\u00e7ois Canonge died, and I was elected to replace him. On doing so, I bound myself to obey it and protect its rights: and I must say that after a most serious inquiry into its origin and the sources from which it emanates, I am more than ever convinced that my opinion with regard to the fundamental authority of the Scottish Rite is correct, and that the views of Charleston thereon are altogether erroneous. From the moment you and other 33rds of this East judged fit to recognize the Council of Charleston as your superior, I and two other members of our Supreme Council, did immediately exercise what, in such case we considered to be our right, and continued the work of our Supreme Council. It is true that on account of the momentary excitement which has prevailed, we have chosen to be silent, but we exist nevertheless and have resolved to safeguard our power and authority for any case of emergency. I certainly feel much honored with the proposition which you make me to accept an appointment as an active member of the Supreme Council of Charleston and as such to preside your Consistory here, but neither such a flattering offer, nor any other consideration can make me deviate from what I consider to be my duty towards a body which I have sworn to protect. I have personally no pretension whatsoever to power. I know that I am good only to make an initiation, and I acknowledge that the privilege of commanding should be better placed in other hands than mine. Many a person, no doubt, will attribute my determination to a spirit of opposition, but as I feel good will towards all and even those who condemn me both in York and Scotish [sic] ranks of Masonry, I will, happen what it may, persevere following the line which I believe to be the only correct one.\n\nThereupon, M. Samory expressed his hope that I would change my mind, and asked me whether I would like to converse with M. Albert G. Mackey on that subject. I answered affirmatively and two or three days afterward, he called at my house with that Gentleman.\n\nM. Mackey began by expressing a desire that his visit to me should not be considered as official. I replied that being both knights templars, we were authorized to meet as such and talk of the questions relative to the Scotish Rite, as if we were perfect strangers to it; and it being so agreed, he repeated to me all that M. Samory had said before with regard to the desire expressed by a large number of masons that I should join the consistory, and with regard to my being made an active member of the Council of Charleston and taking as such command of the Scotish rite in Louisiana. I answered him what I had already answered M. Samory. A few words where then exchanged between him and me, with regard to the origin of the council of Charleston, the constitutions of 1786, the authority which the Supreme Council of the Grand Orient of France claims on the Scotish degrees and the differences which exists between the York and Scotish rites. He admitted that difference and that the reasons which I gave upon all the other points presented a strong matter of consideration, but that he could not accept them as conclusive, which I immediately understood and acknowledged to be with him a matter of course.\n\nHe then insisted that I should again consider the proposition made by Mr. C. Samory, and confirmed by himself; and in conclusion he wished me to let him know what my determination would be after more mature reflection.\n\nI promised to do so through Mr. Samory: and this Gentleman having called on me some weeks afterwards, and repeated all that he had been kind enough to say at his first interview with me, I again answered that I could not accept: and I remember having thus addressed him in the end:\n\n'My dear Sir, in the same manner as the masons whom you now represent, express a desire to have me in your Consistory for their best interest, so a time may come when Scotish masons of this East, tired of a foreign dominion, shall be glad to know that there is in New Orleans a 33d of some value who has never varied, and can at any time be the strong hold around which they may gather as Louisianians.'\n\nThereon we parted good friends as I parted with Mr. Mackey, after due interchange of kindness and politeness. 56\n\nIn 1858, Charles Laffon de Lad\u00e9bat, while clearly bitter towards Foulhouze, commented on this meeting between Foulhouze, Samory and Mackey:\n\nIll Bros. Mackey and Samory knew very well that with a few persons, amoung the weak minded and the ignorant, Mr. Foulhouze was \"somebody,\" and that if they could prevail on him to join the Grand Consistory of Louisiana, peace would be finally restored, and it was solely for the purpose of securing that peace, that they paid him a visit, against the advise of many who knew Mr. Foulhouze better than they. 57\n\nWith John Gedge dead and Foulhouze no longer in consid-eration, Claude Samory became the first New Orleans Mason to be elected an Active member of the Charleston Council. His election was on 20 November 1856. On 17 December 1856, the Grand Consistory filled the vacancy offered to James Foulhouze. The choice was a Mason of promise but of little training in the Scottish Rite. The attorney from Arkansas, Albert Pike, was unanimously (and in his absence 58) elected Commander in Chief of the Grand Consistory of Louisiana.\n\nPrior to the election of Samory and Pike, Foulhouze took part in an activity which sealed his fate with the Charleston Council. James Foulhouze, along with T.W. Collens, J.J.E. Massicott, J.B. Faget and other former members of the Supreme Council of Louisiana declared, in effect, the Concordat of 1855 invalid and publicly resumed the activities of the New Orleans Council. The date that the New Orleans Council was re-opened is sometimes disputed. Foulhouze stated in November of 1857:\n\nFrom the moment I had noticed of that nameless act [the Concordat of 1855], I called upon some 33ds, whom I knew to be true to their obligations, and with them I immediately opened the Supreme Council and continued its work, in order that it might not even be said that it had slept a single instant ... 59\n\nIf such a meeting of 33rds did take place, it was still not until 9 October 1856 that J.J.E. Massicott would be elected Grand Commander of the reorganized Supreme Council of Louisiana and their activities become public. That action was the \"shot\" which started a new round of Masonic turbulence which dramatically altered the nature of the U.S. Scottish Rite.\n\nThe re-origination of the Supreme Council of Louisiana\n\nThe days\/months\/years following the concordat were a time of great uncertainty with many New Orleans Masons. The arguments made by all sides sounded somewhat reasonable. An examination of who chose to associate with the Charleston Council after the concordat, who choose to associate with the revived New Orleans Council and who chose to associate with neither body provides an interesting look into the divided, confused and emotional state of affairs. Of the Grand Lodge of Louisiana officers who were Active Members of the New Orleans Council in the pre-concordat days, two of the five Past Grand Masters60 chose to affiliate with neither body. One affiliated with the Charleston Council61 and two with the revived New Orleans Council.62 Of the eight senior Grand Lodge officers, two chose to affiliate with neither body,63 two with the Charleston Council64 and four with the revived New Orleans Council.65 Of the non- Grand Lodge New Orleans 33rds in the pre-concordat days, 8 chose to associate with neither body, 15 with the Charleston Council and 4 with the revived New Orleans Council. The totals then would be: 12 choosing to affiliate with neither body, 19 with the Charleston Council and 10 with the revived New Orleans Council. These figures should not, however, be viewed as the final tally as they were, over the following years, modified as members moved from one body to the other in a most disconcerting manner. L. E. Deluzain, who was a participant in the 1855 Concordat affiliating with the Charleston Council, re-affiliated with the revived New Orleans Council upon its revival. Joseph Lamarre, who was created a 33rd in the revived New Orleans Council on 25 February 1858, was tried and expelled by that Council on 22 May 1858. He then affiliated with the Grand Consistory of Louisiana becoming an Honorary 33rd. Neither side could truly claim clear victory as the severely bitter strife left both sides with ragged edges. Many of those who chose one side or the other eventually retired from any Masonic affiliation.\n\nPossibly concerned over the reorganization of the New Orleans Council, the Grand Consistory of Louisiana sought to organize itself into a state corporation in early 1857. On 19 March 1857 the General Assembly of the Louisiana State Senate and House of Representatives approved the incorporation of the Grand Consistory of Louisiana. Listed as members were two future Sovereign Grand Commanders of the Charleston Council - Albert Pike and James C. Batchelor. On 22 April 1857 Foulhouze was elected Grand Commander of the revived New Orleans Council. T. Wharton Collens resumed his former position as Lt. Grand Commander. With Foulhouze back in command, the New Orleans Council began to grow in strength and size. 1858 was a pivotal year for Foulhouze and the reorganized New Orleans Supreme Council. In February, Albert Pike delivered a lecture before the Grand Lodge of Louisiana. His lecture was a sharp assault on Foulhouze and the New Orleans Council. The lecture by Pike, and arguments against it, occupied most of the March 1858 issue of the Masonic Delta.66 Clearly the Charleston camp had found a Mason as capable of the \"stinging pen\" as Foulhouze. February 1858 also brought a commanding new (returning) member to the New Orleans Council. The announcement in the Masonic Delta was sure to cause great concern in the Charleston\/New Orleans camp:\n\nWe are happy to say that our most Ill:. and worthy Bro:. Pierre Soul\u00e9 has joined the Supreme Council of the 33d, in and for the Sovereign and Independent State of Louisiana. This eminent citizen and learned Freemason admits thus the State Rights masonically as well as politically.67\n\nThe return of this fiery and powerful former United States Senator and U.S. Minister to Spain to the rolls of the New Orleans Supreme Council was the equivalent of a shot of adrenaline for the New Orleans Council. Soul\u00e9 was created a 33rd on 8 March 1838 by Jean Jacques Conte and was, actually, a Member of the New Orleans Supreme Council prior to the election of Foulhouze as Grand Com-mander. Soul\u00e9 apparently resigned from the Council at some point following Foulhouze's election as his name is no where to be found in any of the records concerning the Concordat of 1855. There are no known record giving the reasons for the resignation of Soul\u00e9 from the Council nor his Masonic activities during, or thoughts of, the concordat. Soul\u00e9 was elected a U.S. Senator in 1847 and served in that office until 1853 followed by his appointment as Minister to Spain from 1853-55. Soul\u00e9 was a vocal, resourceful and respected addition to the New Orleans Council.\n\nThe addition of Pierre Soul\u00e9 as an Active Member of the New Orleans Council would seem to be answered one month later by the addition of Albert Pike as an Active Member of the Charleston Council on 20 March 1858. 68 At the very session which elected Pike as an Active Member, Foulhouze was formally \"expelled\" from the Scottish Rite by the Charleston Council. Since Foulhouze was never a member of any Body controlled by the Charleston Council, this action was more of a public statement of disapproval than an actual expulsion. What followed next was a series of \"sledge hammer\" verbal and written attacks from and upon both the New Orleans and Charleston Councils. The extremely bitter attacks surpassed even the Cerneau \"war\" which resulted in the death of all \"High Grade\" Scottish Rite Masonry in the U.S. with the exception of in New Orleans. Foulhouze released his M\u00e9moire \u00e0 Consulter in French in 1858 and, then in 1859, issue his Historical Inquiry into the Origin of the Ancient and Accepted Scottish Rite in English.69 The book served as the platform from which Foulhouze stated his case, defined his actions and views on regularity as well as his concepts of the history of the Scottish Rite. Foulhouze also used the Masonic Delta as a platform. This monthly publication was he official organ of the revived Supreme Council of Louisiana. Joseph Lamarre released his A Masonic Trial in New Orleans in French in 1858 and Charles Laffon de Lad\u00e9bat translated and added notes to the work for an English edition. The next major New Orleans Masonic publication was a work designed to answer Foulhouze's M\u00e9moire \u00e0 Consulter and further state the position of the Charleston Supreme Council. A Dissection of the Manifesto of Mr. Charles Bienvenu was released 1858 and opened a very regrettable door for the Charleston Council. The work, while originally issued as an anonymous publication, was later learned to be the work of Albert Pike and Charles Laffon de Lad\u00e9bat. While the Dissection was as harsh in tone as Foulhouze's M\u00e9moire \u00e0 Consulter, it went back to the Lopez Expedition period and re-printed in the end of the booklet the article published on Foulhouze by the Daily Delta and the retort by T. Wharton Collens and Robert Preaux. What was not published, nor mentioned, was the response of nearly all of the competing New Orleans newspapers condemning the yellow journalistic style of the Delta's article on Foulhouze. The illusion created in the Dissection was that the Delta's article on Foulhouze was factual and Collens and Preaux were only attempting to deny the obvious. In 1873, James Scot published his Outline of the Rise and Progress of Freemasonry in Louisiana and reveal that the Dissection influenced his thinking and beliefs (and assuredly that of many others) of Foulhouze. Scot says of Foulhouze:\n\nAt this time [1850] he [Foulhouze] was charged with being a spy of the Spanish Government, and was afterwards denounced as such in the newspapers of the day when the news of the fate of the Lopez expedition reached New Orleans. During the excitement he was concealed by some friends to prevent his falling into the hands of the mob, until he was able to effect his escape to Havana. He afterward returned, and resigned his membership in the Supreme Council, July 30, 1853. 70\n\nJames Foulhouze was not viewed as one who simply held a very strong opposing Masonic opinion and followed a course of action that he felt was correct, he was now portrayed as a charlatan of low moral character. This was quite a different picture than the Mason who was approached by Albert Mackey to become an Active Member of the Charleston Supreme Council. The statement by Scot is erroneous. The only newspaper which published such a opinion of Foulhouze was denounced by the balance of the newspapers in New Orleans. Foulhouze went to Havana in an attempt to secure the release of American citizens prior to the article by the Delta. He did not \"escape\" to Havana. The Scot quotation is an example of the emotional and confused state of affairs in Louisiana Masonry and the fact that inaccuracies were, sadly, sometimes accept as fact.\n\nOn 3 October 1858 Foulhouze informed the New Orleans Council, in Session, of a communication he received from the Grand Orient of France. Foulhouze, as a Grand Orient 33rd, was officially instructed to disassociate himself from the revived New Orleans Council. Foulhouze refused this mandate. On 4 February 1859 the Grand Orient of France struck Foulhouze's name from its list of 33rds.\n\nDespite the actions taken and the decrees and publications written against Foulhouze and the New Orleans Council, there was no sign that the Council was weakening. In fact, the New Orleans Council showed every indication of strengthening. By 1859 the Supreme Council of Louisiana was at its peak of power in the post concordat days. Twenty-five active lodges were under its jurisdiction71 and the Council was composed of thirty-four Active Members.72 Of the lodges under the jurisdiction of the New Orleans Council, seven were located outside of New Orleans in various regions of Louisiana. The make-up of the lodges reveal that the popularity of the New Orleans Council was not solely with the French speaking New Orleans Masons. Twelve lodges worked in the French language, seven in the English language, two in German, one in Italian and one in Spanish. Remembering the fact that the Louisiana Grand Lodge (with its \"irregular\" stamp) grew in power and took over the Grand Lodge of Louisiana in 1850 with no outside support, save the Grand Lodge of Mississippi, the matter of the New Orleans Council had to be addressed. It was not simply a growing threat to the Charleston Council, but, also, to the Grand Lodge of Louisiana.\n\nWith no real structure, rituals or organization, the Charleston Council apparently began to realize that it was, indeed, in trouble. Of this time Charles S. Lobingier, 33\u00ba, G.C. writes in his 1931 The Supreme Council, 33\u00ba:\n\nBoth Pike and Mackey had by this time decided that the Supreme Council needed reform. On January 20, 1858, the former had written the latter urging an increase in the membership and the introduction of the elective system. 73\n\nFor reasons that are, at best, ambiguous, Grand Commander John Honour resigned his office in the Charleston Council on 13 August 1858. It was not until 2 January 1859 that Albert Pike was proclaimed, by Albert Mackey, elected to the office of Grand Commander of the Charleston Supreme Council. It is logical that the actions of Foulhouze and the New Orleans Council influenced the change of command in the Charleston Council. Pike immediately began to reform the Charleston Council and make the changes necessary for its survival.\n\nIn 1860 Foulhouze was elected to the judgeship of the Second District Court in Plaquemines Parish. In 1861 Foulhouze moved his domicile from New Orleans to Plaquemines Parish. That same year former judge and Lt. Grand Commander T. Wharton Collens was elected Judge of the Seventh District Court in New Orleans. On 2 January 1861 the New Orleans Council re-incorporated itself taking officially, for the first time, the name \"The Supreme Council of Louisiana.\" Due to the pressures of his new position, T.W. Collens resigned in 1861 as Lt. Grand Commander of the New Orleans Council. He was replaced by Sam Brown, who was created a 33rd by Foulhouze 5 March 1860.\n\nThe Civil War\n\nArguably there has been no lower point in the history of the United States then the Civil War years of 1861-65. The divided country nearly destroyed itself in four years of devastating war, the effects of which plagued the county for a century to follow. While there has been numerous accounts of Masonic acts of charity during the war years, the war weakened Masonry in the U.S. due to the loss of life, property and the economic hardship that followed the war years. There is no sign or record that any of the Supreme Councils in the U.S. were active during the Civil Wars years. Pierre Soul\u00e9 was imprisoned for a time upon the capture of New Orleans in 1862. Upon his release from prison, he lived out the remaining war years in Cuba. Albert Pike was charged with war crimes stemming from the Battle at Pea Ridge (his only war command) and was left out of the general amnesty afforded at the close of the war. Pike fled to Canada awaiting a Presidential pardon allowing him to return to the U.S.\n\nThere are no known records of the Supreme Council of Louisiana during the war years and it is unknown what events, if any, took place in the Council during this time. James Foulhouze, who prior to the war was a district court judge is shown to be a Parish Attorney for Plaquemines Parish following the war. There are no records of the exact date that he left office as a judge, nor giving the reasons. It is possible that the then 65 year old Foulhouze simply retired from his judgeship or his leaving office might have been required by the Union in the post war years. A series of events that can best be described as \"amazing\" then takes place concerning Foulhouze and the New Orleans Council.\n\nOn 3 May 1866, T. Wharton Collens, Pierre Soul\u00e9 and 8 other 33rds of the New Orleans Council signed an \"oath of allegiance\" to the New Orleans Council.74 Foulhouze's name is not included in this apparent reorganization. On 10 May 1866, the New Orleans Council obtained the oath of allegiance of Robert Preaux and created two 33rds. One of the 33rds created was a New Orleans music teacher, music shop owner and composer of moderate note who corresponded with many of the artistic and literary figures in Europe including Victor Hugo. His name was Eugene Chassaignac. On 7 January 1867, Chassaignac was elected Grand Commander of the Supreme Council of Louisiana. It is unknown who was Grand Commander or \"acting\" Grand Commander at the time that Chassaignac was elevated to the 33d degree or why Chassaignac was selected to lead the New Orleans Council. There is a total veil of mystery over the election of Chassaignac and the departure of Foulhouze.\n\nThe 1 May 1867 minutes of Liberty Lodge #19 (under the New Orleans Council's jurisdiction)75 show that O.J Dunn, Grand Master of the Eureka Grand Lodge of Louisiana (Prince Hall) and five other Prince Hall Lodges in various locations in the U.S. had officially accepted the invitation to attend Liberty Lodge and noted that this Lodge admitted visitors with no regard to race. The Worshipful Master of Liberty Lodge was Eugene Chassaignac. The New Orleans Council, likewise and that same year, officially announced that membership to its lodges were not be based on race. That announcement, in itself, seems curious as the Supreme Council of Louisiana (and the whole of New Orleans Masonry) had a long history, prior to the Civil War, of having little concern over race and Masonic membership.\n\nIn an amazing and dramatic move, the Grand Orient of France, ignoring its past action against James Foulhouze, re-recognized the Supreme Council of Louisiana on 5 November 1868. Eugene Chassaignac commented on James Foulhouze and the relations with the Grand Orient of France in the April-May 1869 issue of the Bulletin: 76\n\nIt is true that in 1858, following the writings of Mr. J. Foulhouze, (writings that were not at all the acts of the Supreme Council) our relations with the Grand Orient were interrupted; but since I have had the honor of being the Grand Commander and Grand Master of the Scotch Rite, in Louisiana, I had the pamphlets disavowed by a solemn resolution; on the other hand, Mr. Foulhouze not being any longer a member of our order, there no longer exists a reason for the relations between the Grand Orient of France and the Supreme Council of Louisiana to be interrupted.77\n\nWhat happened? Without James Foulhouze the reorganization of the New Orleans Council would have failed before it started. The Chassaignac statement can only be viewed as incredible and shows an almost contempt for Foulhouze. Why? There is no clue as to what could have taken place during the Civil War years. Prior to the war the New Orleans Council was at its height of power and could have in a matter of a few years, realistically, overpowered the Charleston Council and seriously threatened the Grand Lodge of Louisiana had the war not interrupted its growth. James Foulhouze was the power and the driving force of this movement. It simply could not have happened without him. There is not a hint as to why Foulhouze left office, why Chassaignac was made a 33rd, why Chassaignac was elected Grand Commander or why Chassaignac seemingly turned on Foulhouze. Just as perplexing as the Chassaignac statement on Foulhouze is the re-recognition of the New Orleans Council by the Grand Orient. The Grand Orient had stripped Foulhouze of his 33rd Degree for his participation in the reorganization of the New Orleans Council. Why would they now recognize that very same Body? The re-recognition of the New Orleans Council by the Grand Orient of France unquestionably caused great concern in the Supreme Councils SJ and NMJ. In a bold move, relations between the Grand Orient and the SJ and NMJ were suspended by a join resolution of the SJ and NMJ dated 2 May and 15 June 1870. The resolution made the following points (presumably written by Pike).\n\n\"The Grand Orient of France well knew, for it had so decided in a sane interval, in 1858, that an Inspector-General created by itself could exercise no powers within the jurisdiction of another Supreme Council. It knew that the Chassaignac body was created by the sole authority of M. Jacques Foulhouze, whom it had denuded of his privileges as an Inspector-General, for \"forfaiture d'honneur,\" in establishing it. And yet, without any new light upon the subject, without any reconsideration or reexamination, without restoring M. Foulhouze, and while in alliance with us, it recognized this spurious organization as a lawful Supreme Council.\" 78\n\nThe Death of James Foulhouze\n\nThere is no suggestion that Foulhouze had any connection with Masonry following the Civil War years. In 1869 Foulhouze co-authored a book with William M. Prescott titled The Ordinances of the Police Jury of the Parish of Plaquemines. Foulhouze is listed as \"Parish Attorney\" and Prescott as \"Parish Judge.\" Foulhouze apparently busied himself with legal matters and spent the remainder of his life in the Mississippi River town of Pointe-a-la-Hache, Louisiana.\n\nOn 21 December 1875 the following article appeared in the New Orleans Bee:\n\n\"Deceased the 18th of December 1875 at Pointe-a-la Hache, parish of Plaquemines, the Hon. James Foulhouze at the age of seventy-five. A native of Riom, Auvergue, France.\"\n\nFoulhouze was buried at St. Thomas the Apostle Church Cemetery in Pointe-a-la-Hache, Louisiana. T. Wharton Collens, who had by then also resigned from all Masonic activities, handled the legal matters concerning Foulhouze's succession. Collens wrote of Foulhouze:\n\n\"I was very intimately acquainted with the late James Foulhouze during the thirty years that preceded his death. He was a native of Riom in France, and during the thirty years that I knew him he frequently spoke to me of his relatives in that country, and showed me his correspondences with them. His father died previous to 1830, his mother a few years before he 'J. Foulhouze' did. He had a brother who died before he did - that brother left one heir a daughter. Foulhouze himself was never married.\" 79\n\nWhile Foulhouze not, by any means, a man of great wealth, he did own a home in Pointe-a-la-Hache and some property. Foulhouze's entire estate was willed to Od\u00e9alie Collens McCaleb, the married daughter of his long time friend T. Wharton Collens and Od\u00e9alie's son, James Foulhouze McCaleb.\n\nThe many unanswered questions concerning Foulhouze, and the events surrounding him may never be fully answered or understood. It is clear, however, that James Foulhouze followed a path which he honestly felt to be correct. Regardless of which side of the issue one takes, it must be objectively recognized that the impact that Foulhouze had on the whole of U.S. Scottish Rite Masonry was substantial. It must, also, be pointed out that those who supported and held the same opinion as Foulhouze were neither \"weak minded\" nor \"ignorant\" as sometimes charged. Differing opinions are frequently held by intelligent people. It is unfortunate when judgment is colored by emotion and it is tragic when erroneous conclusions born of skewed judgment makes its way into accepted history.\n\nNotes:\n\n1. The Masonic Delta March 1858.\n\n2. This date was obtained from the tombstone of James Foulhouze located in St. Thomas the Apostle Church Cemetery, Pointe a la Hache, Louisiana.\n\n3. Personal letter: Christine McCullough, Assistant Archivist, Archdiocese of Philadelphia to Michael R. Poll, 23 April 1993.\n\n4. Passenger and Immigration List Index Vol. I P. William Filby, Mary K. Meyer Editors. (Detroit, Michigan: Gale Research Co., 1981) 314.\n\n5. Charles Laffon de Lad\u00e9bat, translator, notes of A Masonic Trial in New Orleans . (New Orleans, LA: J. Lamarre, 1858) p. 62.\n\n6. Encyclopedia Britannica Vol. XI (Chicago: William Benton, Publisher, 1965) 814.\n\n7. McCullough to Poll, 23 April 1993. It should be noted that a priest having his faculties suspended is akin to a physician having his medical license suspended. The affected priest would no longer be able to carry out the duties of a priest such as hearing confessions, preforming wedding, baptisims, Mass, etc. While a priest who has had his faculties suspended is, in fact, prevented from doing all that makes one a priest, it is only the Vatican who can separate a priest from his vows as a priest. This would mean that Foulhouze might have, technically, remained a priest, without powers, until his death.\n\n8. At the time that Foulhouze was a priest, Philadelphia was a \"Diocese\" and not yet an \"Archdiocese.\"\n\n9. McCullough to Poll, 23 April 1993.\n\n10. Lad\u00e9bat, notes, A Masonic Trial in New Orleans p. 62.\n\n11. Philadelphia : U. Hunt, 1842.\n\n12. The paper which Bishop Kenrick mentions was Le Penseur (The Thinker).\n\n13. Records of the American Catholic Historical Society Vol. VIII, 1896 Bishop Kenrick to Dr. Cullen 23 November, 1843. 311-312.\n\n14. Masonic and anti-Masonic\n\n15. The Louisiana Historical Quarterly Vol. 31, No. 4 October, 1948. New Orleans, LA 918.\n\n16. Roman Catholic law forbid duels regardless of the fact that, for many years, the traditional site for duels was in the gardens directly behind, and on the grounds of, the St. Louis Cathedral.\n\n17. The Masonic Delta November 1857 edition.\n\n18. Ibid.\n\n19. Ibid.\n\n20. James Foulhouze, Historical Inquiry into the Origin of the Ancient and Accepted Scottish Rite (New Orleans: True Delta Job Office, 1859.) 17.\n\n21. Masonic Delta November 1857.\n\n22. Canonge served the Grand Lodge of Louisiana as Grand Master in 1822-24 & 1829 and, also, served as Commander in Chief of the Grand Consistory of Louisiana from 1843-46. Canonge had served as the Grand Senior Warden of the Cerneau Grand Council of Princes of the Royal Secret, 32\u00ba in Philadelphia in 1818 and was an early member of the Supreme Council of Louisiana, being appointed Grand Expert on 7 November 1839. It was during Canonge's administration as Commander in Chief of the Grand Consistory that this body passed under the jurisdiction of the Supreme Council of Louisiana. Prior to his election to the office of Sovereign Grand Commander, Canonge served as the Lt. Grand Commander of the Supreme Council. Canonge had the reputation of being a \"no nonsense\" and \"ready to act\" individual with an amazing memory. As a criminal court judge he once ordered the arrest of the entire state Supreme Court for interfering in one of his capital trials. New Orleans Times Democrat 8 January 1893 \"Louisiana Families\"\n\n23. Masonic Delta November 1857.\n\n24. Foulhouze, Historical Inquiry p. 62.\n\n25. See: The Elimination of the French Influence in Louisiana Masonry (New Orleans, LA: Michael Poll Publishing, 1996).\n\n26. Report of the Committee on Foreign Correspondence of the Louisiana Grand Lodge of Ancient York Masons. (New Orleans: Cook, Young & Co., 1849.) 5.\n\n27 Ibid. 5.\n\n28. The town of Lafayette was a suburb of New Orleans in the 1800's located in what is now considered the \"uptown\" area of New Orleans.\n\n29. George Washington, Lafayette, Warren, Marion, Crescent City, Hiram & Eureka.\n\n30. Grand Lodge of the State of Louisiana Report and Exposition (New Orleans: J.L Soll\u00e9e, 1849) 5-34.\n\n31. James B. Scot, Outline of the Rise and Progress of Freemasonry in Louisiana 1873 (New Orleans, LA: Cornerstone Book Publishers, reprint 2008) 76.\n\n32. Charles Laffon de Lad\u00e9bat, Ancient and Accepted Rite. Thirtieth Degree. (New Orleans: 1857). xxvii.\n\n33. Lad\u00e9bat states in a footnote of his published 18\u00ba ritual: \"The philosophical explanation of this and of all the other Degrees from the First up to the Thirtieth inclusive, is taken from the work of Ill.: Bro.: J. Foulhouze, 33d, with some slight alterations, of which, the author willingly assumes the responsibility.\" Lad\u00e9bat, Ancient and Accepted Scotch Rite. Eighteenth Degree (New Orleans: 1856) 123. Foulhouze had, also, rewritten the 33\u00ba for the New Orleans Council. See: James D. Carter History of the Supreme Council, 33\u00ba SJUSA (1861-1891). (Washington, D.C.: The Supreme Council 33\u00ba, 1967). 37.\n\n34. The title of this magazine is sometimes given as Freemasons' Magazine.\n\n35. Charles S. Lobingier, The Supreme Council , 33\u00ba (Louisville, KY: The Standard Printing Co., Inc., 1931). 172; Ray Baker Harris, James D. Carter, History of the Supreme Council, 33\u00ba SJUSA (1801-1861), (Washington, DC: The Supreme Council 33\u00ba, 1964.) 236.\n\n36. Minutes Book, Friends of Harmony Lodge #58 14 September 1849.\n\n37. James Scot, Outline of the Rise and Progress of Freemasonry in Louisiana. New Orleans, LA: Cornerstone Book Publishers, reprint 2008. 78-80.\n\n38. Prior to the Grand Lodge Constitution of 1850 Past Masters of the constituted lodges were made Life Members of the Grand Lodge with voting rights in the Grand Lodge. Following the Constitution of 1850, voting rights were only given to Grand Lodges Officers, the three principal members of each lodge, Past Grand Masters and Grand Lodge Committee members.\n\n39. The Masonic Delta November 1857.\n\n40. The Masonic Delta November 1857.\n\n41. The numbers vary according to the source. The Annual Grand Communication of the Supreme Council, 1859, VIII lists 26 new 33rds. Albert Pike, Official Bulletin VIII, 1886 page 571-572 lists 31 new 33rds.\n\n42. Reunion Fraternal de Caridad in Havana 12 July 1815 and El Templo de la Devina Pastora in Matanzaz 12 July 1818, Proceedings of the Grand Lodge of Louisiana 1995 (A-2 & 3).\n\n43. New Orleans Daily Delta 31 May 1850.\n\n44. The Daily Picayune, New Orleans, Louisiana 31 May 1850.\n\n45. The Daily Crescent New Orleans, Louisiana 1 June 1850.\n\n46. Daily Orleanian, New Orleans, Louisiana 2 June 1850.\n\n47. New Orleans Daily Delta 1 June 1850.\n\n48. New Orleans Bee 27 February 1852.\n\n49. James Foulhouze, T.W. Collens, Charles Claiborne, J.B. Faget, Felix Garcia, F.A. Lumsden, Joseph Walker, John L. Lewis, Robert Preaux, Charles Murian, S. Heriman, Jean Lamothe, Antonio Costa, A. P. Lanaux, G.A. Montmain, F. Correjolles, J.H. Holland, R.D. Fanis, J.E. Jolly, J. Bachino, Aug. Brou\u00e9, M. Prados, F. Ricau, J.J.E. Massicott, Fran\u00e7ois Meilleur, C.M. Emerson, H.G. Duvivier, C. Samory & Charles Laffon.\n\n50. The Masonic Delta August 1857.\n\n51. An interesting document resides in the New Orleans Scottish Rite Library and Museum. It is a handwritten copy of the 1846 General Regulations of the New Orleans Supreme Council. This document is of special interest as it was used as a \"working copy\" for the 1848 General Regulations which were approved on 20 July 1848. The document contains the notes and changes throughout made by James Foulhouze with his signature. Clearly the various changes were presented to the Council for approval. The official name \"The Supreme Council for the United States of America Sitting in New Orleans \" at the head of the Regulations has portions scratched out leaving the only \"The Supreme Council sitting in New Orleans.\" In addition, the side margins contain the proposed changes. In addition to the official name being altered to remove \"for the United States of America\" the proposed change to \"for the State of Louisiana\" written in the margin was also scratched out. Presumably the new title did not pass the vote of the Council or Foulhouze decided not to propose this name change - at that time. It is significant, however, to realize that Foulhouze, from the early days of his administration, considered the Supreme Council structure as possibly being limited to state boundaries just as Grand Lodges.\n\n52. This account can not be confirmed in totality by any existing official record, but is recounted in an old unsigned handwritten paper located in the New Orleans Scottish Rite Library and Museum. In the notes of the 1859 A Masonic Trial in New Orleans, Charles Laffon de La\u00e9bat writes of the event: \"... An opportunity offered and that was the address of Ill:. Bro. Chas. Claiborne who, instead of arguing the point at issue, that is, the merits and demerits of the 20 articles, amused himself by ridiculing the masonic costumes of Mr. Foulhouze. Mr. Foulhouze was stung to the quick and swore, in leaving the hall, that he had done with Masonry! He sent in his letter of resignation on the 30th of July 1853.\" page 43.\n\n53. Alain Bernheim located the Minutes of the Supreme Council of Louisiana from its creation to 15 February 1847 in the BN in Paris in 1987. This writer located the Minutes of the Supreme Council of Louisiana from the election of Charles Claiborne to the Concordat of 1855 in the Library of the New Orleans Scottish Rite Bodies in 1994.\n\n54. Official Bulletin VIII 1886 p 536.\n\n55. Foulhouze, Historical Inquiry 78. The Masonic Delta, August 1857 & March 1858. Charles Laffon de Lad\u00e9bat, Translator, A Masonic Trial in New Orleans (Lamarre's Defense) (New Orleans, J. Lamarre, 1858) 43-44. Note: A Masonic Trial in New Orleans was written by Joseph Lamarre and originally published in French. The work was translated into English and republished that same year. The name of the translator is not given in this work. Charles Laffon de Lad\u00e9bat states on page 83 of Dissection of the Manifesto of Mr. Charles Bienienu (New Orleans: privately published, 1858) that he was the translator for Lamarre' work and author of the notes in that book.\n\n56. The Masonic Delta August 1857.\n\n57. Lad\u00e9bat, A Masonic Trial in New Orleans page 43.\n\n58. Michael R. Poll, In His Own (w)Rite, (New Orleans, LA Cornerstone Book Publishers, 2011) \"Albert Pike, His Addrtess before the Grand Consistory of Louisiana\" page 5.\n\n59. The Masonic Delta November 1857.\n\n60. Felix Garcia, Lucien Hermann.\n\n61. John Henry Holland.\n\n62. Jean Lamothe & Robert Preaux.\n\n63. Ramon Vionnet & Stephen Herriman.\n\n64. Fran\u00e7ois Meilleur and Charles Murian.\n\n65. Jean B. Faget, Jean J.E. Massicott, Romain Brugier and Joseph Lisbony.\n\n66. The revived New Orleans Council's monthly publication.\n\n67. The Masonic Delta February 1858.\n\n68. Although Pike was elected an Active Member in March, it was not until 7 July that Mackey would send the official general notification of his election. Harris, Carter History 260. Mackey would, however, inform Claude Samory of Pike's election on 8 May 1859. Official Bulletin VIII, 544.\n\n69. Foulhouze's Historical Inquiry can not be viewed as an English translation of his M\u00e9moire \u00e0 Consulter. Upon examination by Alain Bernheim, it has been determined that the Historical Inquiry, while closely following M\u00e9moire \u00e0 Consulter, has enough significant changes to consider it a rewrite rather than a translation.\n\n70. Scot, Outline. 4.\n\n71. The Masonic Delta September 1859.\n\n72. The Masonic Delta April 1860.\n\n73. Lobingier, Supreme Council, 249.\n\n74. Original document in the George Longe Collection in the Amistad Research Center at Tulane University, New Orleans, Louisiana.\n\n75. Photocopy reproduction of the minutes in The Perfect Ashlar (publication of the Supreme Council of Louisiana) October 1969.\n\n76. The Bulletin replaced The Masonic Delta in 1869 as the official publication of the Supreme Council of Louisiana.\n\n77. Eugene Chassaignac Bulletin (New Orleans, A. Simon, 1869) 28.\n\n78. Carter, History 431.\n\n79. Foulhouze Secession Papers, 1875, Court House Pointe a la Hashe, Louisiana.\n\n***\n\nQuantity or Quality?\n\nONE OF THE GREAT THINGS about living in New Orleans is the food that we often take for granted. When I moved back after living away for almost 10 years, it was the simple Po-Boy sandwich that gave me such joy. I delighted in finding cozy, hole in the wall places where I could discover new taste sensations. In one such place I saw a sign advertising the \"largest Po-Boy in New Orleans.\" Well, I had to try this place out. The ad was correct; the sandwich was so big that it could really feed two people with some left over. But, to my great disappointment, it tasted terrible. It was a simple Ham & Swiss, but the cheese was hard and the ham was some sort of nasty, discount deli reject. Even the lettuce was soggy and turning brown. It was sure big, but it was a total waste of my money. Clearly, the biggest is not always the best.\n\nI also remember years ago when I first started college. I enrolled at the University of New Orleans (it was then Louisiana State University at New Orleans). I took a civics class and when I walked in on my first day, I found it was in an auditorium with something like 300 in the class! I came from a small high school and was overwhelmed. How was I going to learn anything with a sea of people all around me? The following year I transferred to Loyola University in New Orleans and found the class size much more to my liking. In one music class, it was just the instructor and myself. That was a learning experience for me. But, that's just my taste. The mileage for others may vary.\n\nThe fact is that I am far more comfortable in small settings. I would be more drawn to an intimate, quite dinner with my wife and a few friends than dining at the Hard Rock Cafe with hundreds of others. But, don't misunderstand me, I'm all for a bargain. If that big, ole sandwich had tasted good; I would have been a happy camper. I could have fed my wife and I for the price of one. The problem with my college civics class was not only that it was very large, but that they had not prepared properly. They accepted too many students and when at the last minute, they realized that there were too many for the classroom; they moved the class to the auditorium. Because of the quick change of venue, they did not think about a microphone. It was only the few sitting in the very front who could properly hear the man. They didn't prepare properly. Mistakes were made and the class suffered.\n\nI am interested in quality experiences. Yes, if all things were equal, I would prefer a small gathering for dining, but if the food was bad and the food at the Hard Rock Cafe was outstanding, then the overall winner would be the Hard Rock Cafe. It is no different with Masonry. I want quality and while I prefer small groups, it should not be understood to mean that I place numbers (large or small) first. I place quality first.\n\nI remember when the big \"One Day, All The Way\" classes began and I expressed my concern about them. One argument always seemed to come back at me which totally missed the point I was trying to make. When I expressed concern about the quality of the event, it would invariably be argued that \"bad\" degrees were often seen in small lodge setting. SO WHAT? That's like telling me that it is OK to eat bad food at a convention center because you can get bad food in a small restaurant! I don't want bad food anywhere! You can't sell me on something bad because something bad is also offered somewhere else.\n\nA successful lodge can be one that is small or large. It is successful because it does its work properly. And, \"work\" does not just mean ritual. No worshipful Master should rush into a lodge meeting with no idea of what the secretary will read or announce. The WM should be completely aware of all that will happen in the lodge on any given night. He should also be prepared of how to handle the occasional unexpected event. In other words, no one would be elected WM of a lodge without spending a good many years in training for the office. Knowing the ritual is only one small part of what is necessary for one to become a successful WM.\n\nA lodge should work as a solid unit. Little groups that develop in Masonic bodies spell death for the body. \"I'm in and you are not\" games should have no place in the lodge. The successful lodge moves and acts as one.\n\nIt is not the large or small lodge that has any advantage by their number. The lodge with the advantage is the lodge that expects and provides quality work.\n\n***\n\nWho Am I?\n\nA NUMBER OF YEARS AGO I was visiting my doctor for a check-up. As I was outside in the waiting room, I picked up a medical magazine and started flipping through it. An article caught my attention. It was titled \"Who Am I?\" As I started to read it, I became enthralled.\n\nThe article told the story of a man in his mid to late 30s who was found by the Chicago police around Midnight on a downtown street. He was unconscious and lying in a pool of blood from an injury to the back of his head. The man was wearing a business suit, but had no money, credit cards or any other identification on him. He was the apparent victim of a mugging.\n\nAfter spending several days unconscious in the hospital, the man finally awoke. He was completely disoriented. He had no idea how he came to be in the hospital or what might have happened to get him there. As his mind started to clear, instead of an improving situation, things became much worse. He began to realize that not only did he not know how he came to be in the hospital, but he also did not know his name, where he was from or anything at all about himself prior to waking up in the hospital bed. He had complete and total amnesia.\n\nThe police sent photos of the injured man to the local media and sent his fingerprints off for possible identification. Neither proved helpful. Because of the number of large hotels in the area where he was found, the police felt that he could have been a businessman traveling to the city from almost anywhere.\n\nThe doctors told the man that with his type of injury, amnesia was not uncommon. They told him that they could not give any sort of firm prognosis as there was simply too much that was unknown about this type of injury. There was no way to know if all of his memory, some of it, or any of it would return. The doctors also said that they could not give him any sort of time frame on when he might expect to see any changes or improvements. Uncertainty was the only thing of which they were certain.\n\nThe man realized that the money and credit cards that were taken from him were insignificant. What they took of true value was his life. If he could not regain his memory, then the man he had been before waking up in the hospital bed was dead.\n\nThink about this man's situation. Each one of us has a personal history in our memories. We remember childhood, our family, friends, and events both happy and sad. We remember school, early jobs, dating, marriage, children and everything that has gone into shaping us. Who we are today is based, in a large part, on the total memory of our experiences. Now, think about all of those memories disappearing into dust in a snap of your fingers. Who would you be? What would you be? In that situation, and especially if you were alone with no one around who knew you, how could you be the same person?\n\nThis loss of identity and self is not limited to individuals. Groups of people and even whole societies have suffered the same fate. It was known long ago that if one group wanted to defeat another group, then it was necessary to overpower the enemy's military. But, if one group wanted to completely destroy another group, then they would need to wipe out the enemy's history. Look at the Mayan Indians. Their vast library was destroyed and all of their books, save just three, were burned to dust. And what do we know of the Mayan society and people today? What do the decedents of the Mayans know about their own history? Next to nothing.\n\nEven when there was no deliberate attempt at destroying another's history, the lack of history can still create vacuums in our knowledge of the past. In north Louisiana there is an area known as Poverty Point. There are several large and impressive Indian mounds where a considerable amount of pottery has been excavated. Archaeologists know that this significant civilization existed several thousand years ago and that it is the largest and most complex site of its type discovered in North America. But, because of the lack of any written records and very little information on the people themselves, it is a true lost society.\n\nOur past is vital to us. The practical reality of the existence and nature of each of us depends on the continued knowledge of our nature and existence. When we die, the continued knowledge of us will only exist if our history is recorded and preserved. If not, then when we die, all knowledge of our achievements, work and self dies with us.\n\nFreemasonry is no different. Want to start a debate? Go into any Lodge and ask whether Freemasonry has its roots in the Knights Templar. In some Lodges, such a debate might end up in an argument. The simple truth is that we have little knowledge of the very early history of Freemasonry. The same is pointedly true of the early history of the Scottish Rite. Jurisdictional wars, destruction of early records, and \"edited\" histories designed to strengthen the argument of one or the other side makes our available knowledge questionable to the objective researcher.\n\nThe history of our Lodge or other Masonic body should not just be passed off as the concern of \"those library types.\" It is our history and we all share in its proper preservation... or its loss. We need to be sure that not only are proper records taken and kept, but that we take all necessary steps to preserve old documents and records.\n\nWhen we look at the injured man's situation, we can feel sympathy for him. But we are really only disinterested parties. We don't know the man, and while we can realize the terrible situation he is in, we do not feel his pain. In Masonry, our loss of history is personal. We are not disinterested parties. It is our history and our personal loss when we are not able to answer questions about ourselves that we should be able to answer. The horrible mistake so many make is in believing that the preservation of our history is someone else's job or duty \u2013 as if we play no part in it. If we don't care, we don't exist. It's as simple as that.\n\n***\n\nWhat is Truth?\n\nAn Address Before the Louisiana Lodge\n\nof Research \u2013 02\/05\/2010\n\nA FEW WEEKS AGO I tuned in to one of the educational channels on TV and watched a show on archeology. It was an interesting show dealing with the history of archeological practices from the early \"Indiana Jones\" style to today's standard of documentation. I found a general similarity between the growth and development of archeological excavation standards and those of Masonic research. The show pointed out that in the early days of archeology, researchers would find an important site and boldly their way in to take whatever they felt was valuable or important with little regard for detailed documentation or preservation of the site. Today, great care is taken at excavation sites. It is realized that value and importance are not only with items, but where they are located, what might be near them and the general condition of everything having to do with the site. In the old days of Masonic research, importance was given to special events, but less care was made in verifying the events or knowing what was going on at the time of the event. It is realized today that the whole story, and the understanding of the whole story, entails far more than isolated events.\n\nTechnology being developed today is also of great importance to both archeology and Masonic research. Ground penetrating radar allows archeologists to \"see\" below ground. When these devices are rolled over an area with buried ruins of past civilizations, the whole area becomes visible and they know where they should begin their dig. In addition, satellites are being employed to help with archeological excavation in overgrown forest areas such as South America. The satellites are able to filter the forest and discover sites which previously would have been discovered only by luck.\n\nLikewise, Masonic research has been greatly aided by advances in technology. Where at one time a Masonic researcher would need to travel hundreds or thousands of miles to sit in a Masonic library to wade through large stacks of paper documents, today the computer and internet can put all of the world's great libraries at his fingertips. Databases of information and documents can allow a Masonic researcher to locate, read and copy documents physically located most anywhere in the world from the researcher's home computer.\n\nWhile technology allows researchers to locate items of value far more quickly than in past times, the need to understand an item of interest and what it means remains vital. Finding an old Masonic diploma means little more than momentary interest if you know nothing of the individual or body for which the diploma was issued.\n\nAnd this brings us to Louisiana.\n\nThere are few places with a richer research potential than Louisiana. Simply put, Louisiana is a Masonic researcher's gold mine. When we couple the large amount of unanswered questions in our Masonic history with modern technology, the potential for great discoveries is enough to excite any thinking researcher. We have an opportunity in Louisiana that is greater than many realize.\n\nBut, there is another side to every coin. In this case, the other side of the coin is lack of understanding. It is one thing to find or possess an important Masonic document. It is another thing to properly understand it. The danger exists for important old documents to be found, appreciated only as \"something old\" and then hidden away in private collections unavailable and lost to all.\n\nLouisiana is, by any way you look at it, unique. Our government, laws and Masonry are, to this day, different than in other parts of the United States. Louisiana was founded as a French territory. It was French in nature, language and custom. French Masonry was introduced into Louisiana some sixty years prior to the creation of the Grand Lodge of Louisiana.1 The five lodges who created the Grand Lodge of Louisiana were French-speaking lodges.2 This was not done because of any particular loyalty to France or any desire to not be a part of the U.S., but simply because this is what the vast majority of the Louisiana Masons knew. They spoke French, not English, and they practiced their Masonry in the manner in which they knew it.\n\nAnd what of the English-speaking Americans who began arriving in New Orleans soon after it became a U.S. territory in 1803? Most of the rank and file Americans traveling to the city did not speak French. Yet, the government, shops and pretty much everywhere you went was French-speaking. Sure, the educated on both sides were multi-lingual and able to function in either society, but the majority of the population were not the educated leaders. The same was true in Masonry. English-speaking Masons arriving in New Orleans on business trips wanted to visit lodges. But, the language being used in New Orleans lodges was mostly (save one or two) not to their understanding and even the furnishings and practices that could be understood were not the same American-Webb ritual familiar to them. What visiting Masons saw in New Orleans was mostly very different from what they knew. And, at that time, \"different\" was not a good thing.\n\nUnrest was born in both the population of the city of New Orleans and in Louisiana Masonry. On one hand, the English-speaking Masons of New Orleans demanded that change take place and Masonry in Louisiana be turned into the Masonry of the rest of the country. On the other hand, the French-speaking Masons saw no need to bring about such change when their Masonry had been the same for as long as they knew it. It was the establishment in Louisiana. It was the Masonry as worked in Europe and the Masonry with which they were comfortable. Both sides were unyielding and unwilling to work with the other side.\n\nIn 1851, John Gedge was elected Grand Master of the Grand Lodge of Louisiana. His election places him in a status held by no other Grand Master of our Grand Lodge. You see, prior to his election as Grand Master, MWBro. Gedge was viewed by the Grand Lodge of Louisiana as being an irregular Master Mason. Not only was he viewed to be an irregular Mason, he was the Grand Master of a Grand Lodge viewed to be irregular. But, it does not end there. Gedge was Grand Master of this Grand Lodge in 1848, 49 and 50. Without missing a beat, he stepped from being Grand Master of a Grand Lodge viewed as irregular to Grand Master of the Grand Lodge of Louisiana. Why? How? Of course, we are talking about the Louisiana Grand Lodge of Ancient York Masons and its 1850 merger with the Grand Lodge of Louisiana, but why did the Grand Master of the \"irregular side\" step right over to become the Grand Master of our Grand Lodge? This event can be seen as one of the catalysts to the great unrest, division and bitterness in New Orleans Masonry in the mid 1800's, but who was right and who was wrong? Do we have enough information to form any sound conclusion?\n\nDuring the trial of Jesus, the Roman governor Pontius Pilate asked what would seem to be a straightforward question: \"What is truth?\" Since that time, philosophers have pondered and debated both the question and its answer \u2013 many times with less than satisfactory conclusions.\n\nIn the 1970's, a movie was released that has risen today to the status of a classic - Star Wars. Designed to be a science fiction\/adventure film, the move had a few surprising philosophical elements - possibly from one of the script advisors, Joseph Campbell. With a careful eye, Masons might find some things of value in the events presented in the tale.\n\nThe story is about a young boy, Luke Skywalker, who lived on a farm in a far away galaxy. Luke was raised by his aunt and uncle after the death of his parents. He worked and lived simply, but was aware of a war taking place in far-away parts of his galaxy. That was the truth that he knew.\n\nOne day, while he was away, the war came to Luke's planet. His home was attacked and his aunt and uncle killed. An old man, who was viewed by Luke as something of a hermit, took him in. His name was Obi-Wan Kenobi. Luke began to learn amazing things from and about this old man. He learned that Obi-Wan had been a warrior in the very war that was being fought \u2013 a Jedi Knight who was highly skilled and possessed great powers. But, Luke learned more. He learned that the powers of a Jedi are passed down from parent to child. Luke learned that he possessed these same powers because his father was also a Jedi Knight who fought in the very same war. Luke knew nothing of this and his reality began to change. What he knew as \"the truth\" was not actually the truth. Luke asked about the actual fate of his father. Reluctantly, Obi-Wan told him that his father was killed. He told him that a gifted student of his turned to the \"dark side\" and killed his father. The student's name was Darth Vader.\n\nLuke continued to train and grow in power. Then one day he left to face the man who killed his father. Luke and Vader faced each other in Jedi battle. When Vader realized who Luke was and that he was trained by his former master, he asked Luke if Obi-Wan told him of his father. In anger, Luke told him that he was told enough, that he knew that Vader killed his father. In the bombshell of the movie, Darth Vader then tells Luke, \"No, Luke, I am your father.\" Luke was devastated. He screamed in anguish and broke away from the fight. It was not only clear that he, once again, did not know the actual facts about his father, but now he must face that his own father was the ultimate villain causing so much trouble in the galaxy. Luke's concept of truth seemed to be proven faulty time and again. What was the truth?\n\nWhen Luke again saw Obi-Wan, he wanted to know why he had been told falsehoods. Surprisingly, Obi-Wan told him that what he told Luke was the truth, \"from a certain point of view.\" He explained that his father, Anakin Skywalker, was a good man and when he fell to the \"dark side\" the good man was killed by the evil that became Darth Vader. Obi-Wan then said, \"Luke, you're going to find that many of the truths we cling to depend greatly upon our own point of view.\"\n\nI needed to think long and hard about that line by Obi-Wan. Maybe sometimes truth is subjective.\n\nIn matters of religion, truth does seem to fall very much in line with the thoughts of Obi-Wan. My truth may or may not be your truth. It depends upon our own point of view. How many wars have been fought in the name of religious truth? How many men have been willing to die, or kill, in the name of religion? How do you prove a religious truth? The poet Kahlil Gibran wrote, \"Faith is a knowledge within the heart, beyond the reach of proof.\" Religious truth is accepted as such simply because we believe it.\n\nIn its wisdom, Masonry realized early on that good, honest men could have very different religious views. As such, pointed religious (or political) discussions are not allowed in lodge. The outcome of such discussions could result in disharmony. It is the combination of the passion in which a religious or political view is held along with the inability to conclusively prove such views that result in such discussions being emotional powder kegs. One simply does not prove the correctness of a religious view or political opinion in the same manner that they would prove a mathematical equation.\n\nReligion and politics are not the only subjects which can result in passion for a Mason. Masons have a passion for Masonry. Masons very often have a strong passion for Masonry. Want to see a good fight? Get two Grand Lecturers together who disagree as to if it should be \"on\" or \"upon.\" We are taught to subdue our passions, but, truthfully, do we always?\n\nSo, let's go back to the mid 1800's and keep these thoughts of \"passion\" and \"truth\" in our minds.\n\nIn New Orleans, the city that care forgot, there were two groups facing off. They differed in language, Masonic customs, Masonic rites, and points of view. They were emotional. They were passionate. They were unyielding. It was not a case of two friends sitting down with one ordering strawberry ice cream and the other chocolate and enjoying each others company. It was a case of two angry opponents sitting down disgusted that the other would dare to order something different than their flavor. The truth held by each was that their flavor was the best! Period. They were childish. Each group of Masons held to their own \"flavor\" of Masonry and would tolerate no difference of opinion as to the validity of their view. It was not a case of \"you are mistaken;\" it was a case of \"you are a liar.\" It was a sad, sad time.\n\nThe two sides in New Orleans refused to bend or try to understand the other. Rather than withdrawing to avoid disharmony, they openly engaged each other in lodge or in public settings with bitter and unMasonic attacks upon the character of their brothers.3 It was a war. It was a nasty war. It was not necessary.\n\nFrom a strictly historical standpoint, we know some of what took place, and some of the \"why.\" But, there is much that remains unknown or unclear about the events prior to the \"wars.\" We know but bits and pieces of pre-1850 Louisiana Freemasonry.\n\nWe have lessons and opportunities.\n\nThe lessons should be obvious. We are taught to subdue our passions. We are taught to not speak evil of our brothers \u2013 either to their face or by private character assassination. We should try to help each other, not wage war on each other because we have differences of opinion as to the nature of Freemasonry. We should advance the Light of Masonry, not the darkness of ignorance, falsehood and ambition.\n\nOur opportunities should also be obvious. Never before has technology been available to make the work of Masonic research easier and more possible for anyone with the desire. We have clear standards of how our research should be conducted, documented and presented. We know that we should not present ideas as facts, or allow \"Masonic politics\" to color or sway our research or presentation. We have the chance to do meaningful, objective work that can benefit the Freemasonry of today as well as the future.\n\nThe brothers of the mid 1800's were quick to point the finger of blame at \"the other guy\" for all the perceived wrongs in Masonry during that time. But, were the wrongs with Masonry itself or the individual Masons of the time? Was Masonry flawed on either side or were the individual Masons flawed? Can we see the opportunity to help Masons of today by a study of the unfortunate events of the past?\n\nWhen our ego allows us to see clearly, we realize that Masonry is better than us. Masonry is not improved because of our membership; we are improved because of our membership.\n\nWe have an opportunity. We have the tools to do the real work of Masonic research. We have the knowledge to know the nature of the real work of Masonic research. It is up to us to either do that work and earn our keep as Masons or succumb to the three villains.\n\nWhat is truth? It may depend on what we are made of and our own points of view.\n\nNotes:\n\n1. On Sunday, July 16th, 1752, Loge de la Parfaite Harmonie was created in New Orleans under the direct jurisdiction of Loge de la Parfaite Union de Martinique. Sharp Document #40.\n\n2. Parfaite Union #1, Charit\u00e9 #2, Concorde #3, Pers\u00e9v\u00e9rance #4 & Etoile Polaire #5.\n\n3. Among other works, see: A Masonic Trial in New Orleans by Joseph Lamarre, tans. by Charles Laffon de Lad\u00e9bat (1858); M\u00e9moire \u00e0 Consulter by James Foulhouze (1858) and A Dissection of the Manifesto of Mr. Charles Bienvenu by Albert Pike and Charles Laffon de Lad\u00e9bat (1858).\n\n***\n\nDeadly Apathy\n\nThe Journal of the Masonic Society, Issue 10, 2010\n\nWE HAVE ALL READ or heard stories of individuals who have taken drastic steps to save their own lives. Recently I read of a man who was doing some repair work on his water heater. He needed to reach far into the tank while lying on his back. While working in that position, his arm became wedged in the tank and he found that it was impossible to remove it. He screamed for help, but was alone in the house and no one was near enough outside to hear his cries. The man had spent several days trapped with his arm hopelessly wedged when he noticed a disturbing smell coming from inside the tank and around his arm. The man later recounted that instinct must have taken over. He managed to reach a saw and began to cut off his arm. The next day, several family members \u2013 concerned at not being able to reach him \u2013 found him unconscious on the floor in a pool of blood, his arm severed at the elbow. The man was taken to the hospital where he recovered, but the doctors gave him a sobering report. Gangrene had set into his arm, and he was told that if he had not removed it when he did, he would have died. The doctors also noted that if he had waited any longer to remove the arm, it would have been too late. The poison would have spread through his body and nothing then would have saved him. The man's life was saved not just because he took action, but when he took action.\n\nI joined Masonry in the mid-'70s. While a number of my family members had been Masons, I knew next to nothing of the philosophy or history of Freemasonry. All that I knew was that it was a \"good\" organization. It took my joining to find out what \"good\" meant. Such ignorance of the philosophy of Freemasonry prior to joining is becoming more of the exception than the rule today. Many of the young men who join Masonry already know much of its philosophy. They have read the popular, new and exciting books on Freemasonry. They arrive at the door of the Lodge with an awareness of a wonderful, mysterious, moral and enlightened group of seekers. They want to share in and be a part of such an organization. But, sadly, this is not exactly what they always find when they join.\n\nThe numbers of demits, NPD, and non-participation are growing at an alarming rate. The new reports paint a dismal picture. Yes, new members are coming fast, and sometimes in very good numbers, but we seem to be having trouble keeping them. So, why is this happening and what do we do?\n\nWhat seems to be happening is the young men come to Freemasonry with an idea of what it should be and find that it is something very different. Many come with the hopes of finding enlightening discussions, intellectual programs designed to lift us to new heights and help us learn more of ourselves and our world. Yet, sometimes all they find is \"good ole boys\" seeking to add another title, gain a bit more authority or power, and be more of the \"big fish\" in whatever pond they thrive. There is lots of coffee, but little real enlightenment. The young Masons become upset at the reality of their Masonry when they compare it to what they believed of Masonry before they joined. Some make their displeasure known \u2013 loudly. At times, such pointed objections by the young brothers are met with disapproval. It is perceived that the young Masons know nothing of what they are talking about, are out of place, need to \"get with the program\" and stop \"being so negative.\" The upset young Masons are viewed as the trouble-makers and their cries for Masonry as they believe it should be are viewed more as the cries of malcontents. They are often ignored and sometimes ostracized. The Masons become disillusioned and wonder why they ever joined.\n\nWhat happens next takes us back to the man with his arm wedged in the tank of the water heater. At the moment he began to smell something very bad, he had a choice. He could act or he could wait and see. Acting in a decisive manner saved his life; waiting to see if the situation changed on its own would have cost him his life. As in many cases, timing is everything. In Freemasonry, our gangrene is apathy. If apathy towards Freemasonry, or any body of Freemasonry, sets into anyone then they stop caring. Once they stop caring, Masonry does not matter to them and they turn their back, demit, stop paying their dues or just live as a card carrier. The positive force that could have been dies. We all lose.\n\nAnd, whose fault is it if a Mason stops caring about Masonry? If we believe or say that it is in some way the fault of the disillusioned Mason, that he would \"do better to bring about changes on the inside\" or some other such criticism designed to shame him into remaining a member, then we add insult to injury. We have missed the point and are only making a bad situation worse. It would be the same as if we saw the man with his arm wedged in the tank and we advised him to be patient and hang on a bit longer as things will certainly get better if he just waits out the unpleasant situation. When does \"hanging on\" reach the point of gangrene and result in death no matter what is done after?\n\nIn all cases, objectivity, recognition of the actual situation and the courage to do what needs to be done must be paramount. If a dedicated, serious Mason ceases to care about some Masonic body, then the \"blame game\" of identifying who is at fault is pointless. Apathy has won and Masonry loses.\n\nThe time to act is when we see the first signs of actual trouble. The first thing the man with his arm wedged did was try to free his arm. He twisted it, moved it this way and that and did everything he could do to free it. This is the same as if we belong to a dysfunctional Masonic body and we try to suggest ways to improve the body, work for changes and do all we can to correct the situation as a member. If nothing works, then we must take the next step.\n\nWhen all his own efforts could not free his arm, the man began yelling for help. He was clearly not in a position to effect any positive change in the situation; maybe someone else could render him aid. In Masonry, the calling for help would come in the form of seeking out superiors who might be able to correct the situation. When our own best efforts fail, and they sometimes do, we need to seek help from those in a position who might be able to grant what we need.\n\nAnd what do we do if no help comes? In all cases, we need to act responsibly. We can not act in haste, foolishly or without considered thought. But at some point we need to act. Failure to take any action is often just as reckless and foolish as an action taken too quickly. At some point, the man with his arm wedged knew that something was very wrong. He may not have known all the details or possessed all the medical knowledge of the situation, but he knew that he needed to take drastic action to correct the situation. Oh yes, help did arrive just the next day. But, it was too late. The doctors told him that if he had waited those additional hours, the poison would have gone through his body and then nothing would have been able to save him. He took the necessary action and he took it in time to save his life.\n\nNo one told us that being a Mason was always going to be easy. If someone did, they told us a story. Throughout our degrees, we are given lessons of honor, integrity and courage. We are given lessons that are sometimes very difficult to put into practice. In my Craft Masonry, we use the Scottish Rite Craft ritual. We are taught that the three \"villains\" in craft Masonry represent ignorance, falsehood and ambition. If we have a deficiency of the former or allow any of the later to gain hold of us, then we do not live Freemasonry as we were taught. Our goal is to control and advance ourselves. We must live our own lives as Masonry teaches us. We have no control over another, even our closest brother, but we must always have total control over ourselves.\n\nFreemasonry is going through a revolution of sorts. Gone are the days of the \"good ole boy\" clubs, the power brokers or the joining of one organization only because it is viewed as a prerequisite for another organization. The young Masons come to us with an understanding of the value of what we teach, not the shiny trinkets we wear. The beauty of what is taught in the various bodies is desired. The leadership of every single body in Masonry must provide quality education, leadership and teach what is supposed to be learned by the new members. If it is in any way unclear as to what is supposed to be taught in any Masonic body, then that should give the clear signal that a change is in order in those bodies. When the ones who must teach don't know themselves, the whole body suffers. Stand up, do the work that you need to do or allow another to do the work.\n\nLuckily, in many cases we find that only the first step is necessary in dysfunctional Masonic bodies. We are finding more and more cases of the new members realizing that something is very lacking, standing up and taking control of the lacking Lodges and making the positive changes themselves. In those bodies where the membership is not in a position to make such changes on their own authority, then assistance from superiors is necessary or the body will crumble. Period.\n\nApathy is the cancer we can not allow to set into any Mason. Our new, young Masons have a foundation that brings with them a hope for our future that is too valuable to ignore. We must do all in our power to see that their interest, dedication and hunger for Masonry is not trampled by the unworthy or their death grip on their perceived power.\n\nWe are in new wonderful times. We must always look to tomorrow if we have any hope of a future than includes Freemasonry.\n\n***\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n_Also by Kate Atkinson_\n\nBehind the Scenes at the Museum  \nHuman Croquet  \nEmotionally Weird  \nNot the End of the World  \nCase Histories  \nOne Good Turn  \nWhen Will There Be Good News?  \nStarted Early, Took My Dog\n\nFor more information on Kate Atkinson and her books,  \nvisit her website at www.kateatkinson.co.uk\n\nCopyright \u00a9 2013 Kate Costello Ltd.\n\nAll rights reserved. The use of any part of this publication, reproduced, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise, or stored in a retrieval system without the prior written consent of the publisher\u2014or in the case of photocopying or other reprographic copying, license from the Canadian Copyright Licensing Agency\u2014is an infringement of the copyright law.\n\nThe Bond Street Books colophon is a registered trademark of  \nRandom House of Canada Limited.\n\nLibrary and Archives Canada Cataloguing in Publication\n\nAtkinson, Kate  \nLife after life \/ Kate Atkinson.\n\neISBN: 978-0-385-67138-5\n\nI. Title.\n\nPR6051.T56L54 2013 823\u2032.914 C2012-906610-9\n\nThis book is a work of fiction. Names, characters, places and incidents are products of the author's imagination or are used fictitiously. Any resemblance to actual events or locales or persons, living or dead, is entirely coincidental.\n\nCover design by Kelly Hill  \nCover images courtesy of Shutterstock\n\nPublished in Canada by Bond Street Books,  \nan imprint of Random House of Canada Limited\n\nwww.randomhouse.ca\n\nv3.1\nFor Elissa\nWhat if some day or night a demon were to steal after you into your loneliest loneliness and say to you: 'This life as you now live it and have lived it, you will have to live once more and innumerable times more'... Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: 'You are a god and never have I heard anything more divine.'\n\nNietzsche, _The Gay Science_\n\nEverything changes and nothing remains still.\n\nPlato, _Cratylus_\n\n'What if we had a chance to do it again and again, until we finally did get it right? Wouldn't that be wonderful?'\n\nEdward Beresford Todd\n\n# _Contents_\n\n_Cover_\n\n_Other Books by This Author_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nBe Ye Men of Valour\n\n_November 1930_\n\nSnow\n\n_11 February 1910_\n\nSnow\n\n_11 February 1910_\n\nFour Seasons Fill the Measure of the Year\n\n_11 February 1910_\n\n_May 1910_\n\n_June 1914_\n\nSnow\n\n_11 February 1910_\n\nWar\n\n_June 1914_\n\n_July 1914_\n\n_January 1915_\n\nSnow\n\n_11 February 1910_\n\nWar\n\n_20 January 1915_\n\nArmistice\n\n_June 1918_\n\n_11 November 1918_\n\nSnow\n\n_11 February 1910_\n\nArmistice\n\n_12 November 1918_\n\nSnow\n\n_11 February 1910_\n\nArmistice\n\n_11 November 1918_\n\nSnow\n\n_11 February 1910_\n\nArmistice\n\n_11 November 1918_\n\nSnow\n\n_11 February 1910_\n\nArmistice\n\n_11 November 1918_\n\nPeace\n\n_February 1947_\n\nSnow\n\n_11 February 1910_\n\nLike a Fox in a Hole\n\n_September 1923_\n\n_December 1923_\n\n_11 February 1926_\n\n_May 1926_\n\n_August 1926_\n\n_June 1932_\n\n_11 February 1926_\n\n_August 1926_\n\nA Lovely Day Tomorrow\n\n_2 September 1939_\n\n_November 1940_\n\nA Lovely Day Tomorrow\n\n_2 September 1939_\n\n_April 1940_\n\n_November 1940_\n\nA Lovely Day Tomorrow\n\n_September 1940_\n\n_November 1940_\n\n_August 1926_\n\nThe Land of Begin Again\n\n_August 1933_\n\n_August 1939_\n\n_April 1945_\n\nA Long Hard War\n\n_September 1940_\n\n_October 1940_\n\n_October 1940_\n\n_November 1940_\n\n_May 1941_\n\n_November 1943_\n\n_February 1947_\n\n_June 1967_\n\nThe End of the Beginning\n\nBe Ye Men of Valour\n\n_December 1930_\n\nSnow\n\n_11 February 1910_\n\nThe Broad Sunlit Uplands\n\n_May 1945_\n\nSnow\n\n_11 February 1910_\n\nAcknowledgements\n\n_About the Author_\n\n# Be Ye Men of Valour\n\n# _November 1930_\n\nA fug of tobacco smoke and damp clammy air hit her as she entered the caf\u00e9. She had come in from the rain and drops of water still trembled like delicate dew on the fur coats of some of the women inside. A regiment of white-aproned waiters rushed around at tempo, serving the needs of the _M\u00fcnchner_ at leisure \u2013 coffee, cake and gossip.\n\nHe was at a table at the far end of the room, surrounded by the usual cohorts and toadies. There was a woman she had never seen before \u2013 a permed, platinum blonde with heavy make-up \u2013 an actress by the look of her. The blonde lit a cigarette, making a phallic performance out of it. Everyone knew that he preferred his women demure and wholesome, Bavarian preferably. All those dirndls and knee-socks, God help us.\n\nThe table was laden. _Bienenstich, Gugelhupf, K\u00e4sekuchen_. He was eating a slice of _Kirschtorte_. He loved his cakes. No wonder he looked so pasty, she was surprised he wasn't diabetic. The softly repellent body (she imagined pastry) beneath the clothes, never exposed to public view. Not a manly man. He smiled when he caught sight of her and half rose, saying, ' _Guten Tag, gn\u00e4diges Fr\u00e4ulein_ ,' indicating the chair next to him. The bootlicker who was currently occupying it jumped up and moved away.\n\n' _Unsere Englische Freundin_ ,' he said to the blonde, who blew cigarette smoke out slowly and examined her without any interest before eventually saying, ' _Guten Tag_.' A Berliner.\n\nShe placed her handbag, heavy with its cargo, on the floor next to her chair and ordered _Schokolade_. He insisted that she try the _Pflaumen Streusel_.\n\n'Es _regnet_ ,' she said by way of conversation. 'It's raining.'\n\n'Yes, it's raining,' he said with a heavy accent. He laughed, pleased at his attempt. Everyone else at the table laughed as well. 'Bravo,' someone said. ' _Sehr gutes Englisch_.' He was in a good mood, tapping the back of his index finger against his lips with an amused smile as if he was listening to a tune in his head.\n\nThe _Streusel_ was delicious.\n\n' _Entschuldigung_ ,' she murmured, reaching down into her bag and delving for a handkerchief. Lace corners, monogrammed with her initials, 'UBT' \u2013 a birthday present from Pammy. She dabbed politely at the _Streusel_ flakes on her lips and then bent down again to put the handkerchief back in her bag and retrieve the weighty object nesting there. Her father's old service revolver from the Great War, a Webley Mark V.\n\nA move rehearsed a hundred times. One shot. Swiftness was all, yet there was a moment, a bubble suspended in time after she had drawn the gun and levelled it at his heart when everything seemed to stop.\n\n' _F\u00fchrer_ ,' she said, breaking the spell. ' _F\u00fcr Sie_.'\n\nAround the table guns were jerked from holsters and pointed at her. One breath. One shot.\n\nUrsula pulled the trigger.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\nAn icy rush of air, a freezing slipstream on the newly exposed skin. She is, with no warning, outside the inside and the familiar wet, tropical world has suddenly evaporated. Exposed to the elements. A prawn peeled, a nut shelled.\n\nNo breath. All the world come down to this. One breath.\n\nLittle lungs, like dragonfly wings failing to inflate in the foreign atmosphere. No wind in the strangled pipe. The buzzing of a thousand bees in the tiny curled pearl of an ear.\n\nPanic. The drowning girl, the falling bird.\n\n'Dr Fellowes should have been here,' Sylvie moaned. 'Why isn't he here yet? Where is he?' Big dewdrop pearls of sweat on her skin, a horse nearing the end of a hard race. The bedroom fire stoked like a ship's furnace. The thick brocade curtains drawn tightly against the enemy, the night. The black bat.\n\n'Yer man'll be stuck in the snow, I expect, ma'am. It's sure dreadful wild out there. The road will be closed.'\n\nSylvie and Bridget were alone in their ordeal. Alice, the parlour maid, was visiting her sick mother. And Hugh, of course, was chasing down Isobel, his wild goose of a sister, _\u00e0 Paris_. Sylvie had no wish to involve Mrs Glover, snoring in her attic room like a truffling hog. Sylvie imagined she would conduct proceedings like a parade-ground sergeant-major. The baby was early. Sylvie was expecting it to be late like the others. The best-laid plans, and so on.\n\n'Oh, ma'am,' Bridget cried suddenly, 'she's all blue, so she is.'\n\n'A girl?'\n\n'The cord's wrapped around her neck. Oh, Mary, Mother of God. She's been strangled, the poor wee thing.'\n\n'Not breathing? Let me see her. We must do something. What can we do?'\n\n'Oh, Mrs Todd, ma'am, she's gone. Dead before she had a chance to live. I'm awful, awful sorry. She'll be a little cherub in heaven now, for sure. Oh, I wish Mr Todd was here. I'm awful sorry. Shall I wake Mrs Glover?'\n\nThe little heart. A helpless little heart beating wildly. Stopped suddenly like a bird dropped from the sky. A single shot.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\n'For God's sake, girl, stop running around like a headless chicken and fetch some hot water and towels. Do you know nothing? Were you raised in a field?'\n\n'Sorry, sir.' Bridget dipped an apologetic curtsy as if Dr Fellowes were minor royalty.\n\n'A girl, Dr Fellowes? May I see her?'\n\n'Yes, Mrs Todd, a bonny, bouncing baby girl.' Sylvie thought Dr Fellowes might be over-egging the pudding with his alliteration. He was not one for bonhomie at the best of times. The health of his patients, particularly their exits and entrances, seemed designed to annoy him.\n\n'She would have died from the cord around her neck. I arrived at Fox Corner in the nick of time. Literally.' Dr Fellowes held up his surgical scissors for Sylvie's admiration. They were small and neat and their sharp points curved upwards at the end. 'Snip, snip,' he said. Sylvie made a mental note, a small, vague one, given her exhaustion and the circumstances of it, to buy just such a pair of scissors, in case of similar emergency. (Unlikely, it was true.) Or a knife, a good sharp knife to be carried on one's person at all times, like the robber-girl in _The Snow Queen_.\n\n'You were lucky I got here in time,' Dr Fellowes said. 'Before the snow closed the roads. I called for Mrs Haddock, the midwife, but I believe she is stuck somewhere outside Chalfont St Peter.'\n\n'Mrs _Haddock_?' Sylvie said and frowned. Bridget laughed out loud and then quickly mumbled, 'Sorry, sorry, sir.' Sylvie supposed that she and Bridget were both on the edge of hysteria. Hardly surprising.\n\n'Bog Irish,' Dr Fellowes muttered.\n\n'Bridget's only a scullery maid, a child herself. I am very grateful to her. It all happened so quickly.' Sylvie thought how much she wanted to be alone, how she was never alone. 'You must stay until morning, I suppose, doctor,' she said reluctantly.\n\n'Well, yes, I suppose I must,' Dr Fellowes said, equally reluctantly.\n\nSylvie sighed and suggested that he help himself to a glass of brandy in the kitchen. And perhaps some ham and pickles. 'Bridget will see to you.' She wanted rid of him. He had delivered all three (three!) of her children and she did not like him one bit. Only a husband should see what he saw. Pawing and poking with his instruments in her most delicate and secretive places. (But would she rather have a midwife called Mrs Haddock deliver her child?) Doctors for women should all be women themselves. Little chance of that.\n\nDr Fellowes lingered, humming and hawing, overseeing the washing and wrapping of the new arrival by a hot-faced Bridget. Bridget was the eldest of seven so she knew how to swaddle an infant. She was fourteen years old, ten years younger than Sylvie. When Sylvie was fourteen she was still in short skirts, in love with her pony, Tiffin. Had no idea where babies came from, even on her wedding night she remained baffled. Her mother, Lottie, had hinted but had fallen shy of anatomical exactitude. Conjugal relations between man and wife seemed, mysteriously, to involve larks soaring at daybreak. Lottie was a reserved woman. Some might have said narcoleptic. Her husband, Sylvie's father, Llewellyn Beresford, was a famous society artist but not at all Bohemian. No nudity or louche behaviour in his household. He had painted Queen Alexandra, when she was still a princess. Said she was very pleasant.\n\nThey lived in a good house in Mayfair, while Tiffin was stabled in a mews near Hyde Park. In darker moments, Sylvie was wont to cheer herself up by imagining that she was back there in the sunny past, sitting neatly in her side-saddle on Tiffin's broad little back, trotting along Rotten Row on a clean spring morning, the blossom bright on the trees.\n\n'How about some hot tea and a nice bit of buttered toast, Mrs Todd?' Bridget said.\n\n'That would be lovely, Bridget.'\n\nThe baby, bandaged like a Pharaonic mummy, was finally passed to Sylvie. Softly, she stroked the peachy cheek and said, 'Hello, little one,' and Dr Fellowes turned away so as not to be a witness to such syrupy demonstrations of affection. He would have all children brought up in a new Sparta if it were up to him.\n\n'Well, perhaps a little cold collation wouldn't go amiss,' he said. 'Is there, by chance, any of Mrs Glover's excellent piccalilli?'\n\n# Four Seasons Fill the Measure of the Year\n\n# _11 February 1910_\n\nSylvie was woken by a dazzling sliver of sunlight piercing the curtains like a shining silver sword. She lay languidly in lace and cashmere as Mrs Glover came into the room, proudly bearing a huge breakfast tray. Only an occasion of some importance seemed capable of drawing Mrs Glover this far out of her lair. A single, half-frozen snowdrop drooped in the bud vase on the tray. 'Oh, a snowdrop!' Sylvie said. 'The first flower to raise its poor head above the ground. How brave it is!'\n\nMrs Glover, who did not believe that flowers were capable of courage, or indeed any other character trait, laudable or otherwise, was a widow who had only been with them at Fox Corner a few weeks. Before her advent there had been a woman called Mary who slouched a great deal and burnt the roasts. Mrs Glover tended, if anything, to undercook food. In the prosperous household of Sylvie's childhood, Cook was called 'Cook' but Mrs Glover preferred 'Mrs Glover'. It made her irreplaceable. Sylvie still stubbornly thought of her as Cook.\n\n'Thank you, Cook.' Mrs Glover blinked slowly like a lizard. 'Mrs Glover,' Sylvie corrected herself.\n\nMrs Glover set the tray down on the bed and opened the curtains. The light was extraordinary, the black bat vanquished.\n\n'So bright,' Sylvie said, shielding her eyes.\n\n'So much snow,' Mrs Glover said, shaking her head in what could have been wonder or aversion. It was not always easy to tell with Mrs Glover.\n\n'Where is Dr Fellowes?' Sylvie asked.\n\n'There was an emergency. A farmer trampled by a bull.'\n\n'How dreadful.'\n\n'Some men came from the village and tried to dig his automobile out but in the end my George came and gave him a ride.'\n\n'Ah,' Sylvie said, as if suddenly understanding something that had puzzled her.\n\n'And they call it horsepower,' Mrs Glover snorted, bull-like herself. 'That's what comes of relying on new-fangled machines.'\n\n'Mm,' Sylvie said, reluctant to argue with such strongly held views. She was surprised that Dr Fellowes had left without examining either herself or the baby.\n\n'He looked in on you. You were asleep,' Mrs Glover said. Sylvie sometimes wondered if Mrs Glover was a mind-reader. A perfectly horrible thought.\n\n'He ate his breakfast first,' Mrs Glover said, displaying both approval and disapproval in the same breath. 'The man has an appetite, that's for sure.'\n\n'I could eat a horse,' Sylvie laughed. She couldn't, of course. Tiffin popped briefly into her mind. She picked up the silver cutlery, heavy like weapons, ready to tackle Mrs Glover's devilled kidneys. 'Lovely,' she said (were they?) but Mrs Glover was already busy inspecting the baby in the cradle. ('Plump as a suckling pig.') Sylvie idly wondered if Mrs Haddock was still stuck somewhere outside Chalfont St Peter.\n\n'I hear the baby nearly died,' Mrs Glover said.\n\n'Well...' Sylvie said. Such a fine line between living and dying. Her own father, the society portraitist, slipped on an Isfahan rug on a first-floor landing after some fine cognac one evening. The next morning he was discovered dead at the foot of the stairs. No one had heard him fall or cry out. He had just begun a portrait of the Earl of Balfour. Never finished. Obviously.\n\nAfterwards it turned out that he had been more profligate with his money than mother and daughter realized. A secret gambler, markers all over town. He had made no provision at all for unexpected death and soon there were creditors crawling over the nice house in Mayfair. A house of cards as it turned out. Tiffin had to go. Broke Sylvie's heart, the grief greater than any she felt for her father.\n\n'I thought his only vice was women,' her mother said, roosting temporarily on a packing case as if modelling for a piet\u00e0.\n\nThey sank into genteel and well-mannered poverty. Sylvie's mother grew pale and uninteresting, larks soared no more for her as she faded, consumed by consumption. Seventeen-year-old Sylvie was rescued from becoming an artist's model by a man she met at the post-office counter. Hugh. A rising star in the prosperous world of banking. The epitome of bourgeois respectability. What more could a beautiful but penniless girl hope for?\n\nLottie died with less fuss than was expected and Hugh and Sylvie married quietly on Sylvie's eighteenth birthday. ('There,' Hugh said, 'now you will never forget the anniversary of our marriage.') They spent their honeymoon in France, a delightful _quinzaine_ in Deauville, before settling in semi-rural bliss near Beaconsfield in a house that was vaguely Lutyens in style. It had everything one could ask for \u2013 a large kitchen, a drawing room with French windows on to the lawn, a pretty morning room and several bedrooms waiting to be filled with children. There was even a little room at the back of the house for Hugh to use as a study. 'Ah, my growlery,' he laughed.\n\nIt was surrounded at a discreet distance by similar houses. There was a meadow and a copse and a bluebell wood beyond with a stream running through it. The train station, no more than a halt, would allow Hugh to be at his banker's desk in less than an hour.\n\n'Sleepy hollow,' Hugh laughed as he gallantly carried Sylvie across the threshold. It was a relatively modest dwelling (nothing like Mayfair) but nonetheless a little beyond their means, a fiscal recklessness that surprised them both.\n\n'We should give the house a name,' Hugh said. 'The Laurels, the Pines, the Elms.'\n\n'But we have none of those in the garden,' Sylvie pointed out. They were standing at the French windows of the newly purchased house, looking at a swathe of overgrown lawn. 'We must get a gardener,' Hugh said. The house itself was echoingly empty. They had not yet begun to fill it with the Voysey rugs and Morris fabrics and all the other aesthetic comforts of a twentieth-century house. Sylvie would have quite happily lived in Liberty's rather than the as-yet-to-be-named marital home.\n\n'Greenacres, Fairview, Sunnymead?' Hugh offered, putting his arm around his bride.\n\n'No.'\n\nThe previous owner of their unnamed house had sold up and gone to live in Italy. 'Imagine,' Sylvie said dreamily. She had been to Italy when she was younger, a grand tour with her father while her mother went to Eastbourne for her lungs.\n\n'Full of Italians,' Hugh said dismissively.\n\n'Quite. That's rather the attraction,' Sylvie said, unwinding herself from his arm.\n\n'The Gables, the Homestead?'\n\n'Do stop,' Sylvie said.\n\nA fox appeared out of the shrubbery and crossed the lawn. 'Oh, look,' Sylvie said. 'How tame it seems, it must have grown used to the house being unoccupied.'\n\n'Let's hope the local hunt isn't following on its heels,' Hugh said. 'It's a scrawny beast.'\n\n'It's a vixen. She's a nursing mother, you can see her teats.'\n\nHugh blinked at such blunt terminology falling from the lips of his recently virginal bride. (One presumed. One hoped.)\n\n'Look,' Sylvie whispered. Two small cubs sprang out on to the grass and tumbled over each other in play. 'Oh, they're such handsome little creatures!'\n\n'Some might say vermin.'\n\n'Perhaps they see _us_ as verminous,' Sylvie said. 'Fox Corner \u2013 that's what we should call the house. No one else has a house with that name and shouldn't that be the point?'\n\n'Really?' Hugh said doubtfully. 'It's a little whimsical, isn't it? It sounds like a children's story. _The House at Fox Corner_.'\n\n'A little whimsy never hurt anyone.'\n\n'Strictly speaking though,' Hugh said, 'can a house _be_ a corner? Isn't it _at_ one?'\n\nSo this is marriage, Sylvie thought.\n\nTwo small children peered cautiously round the door. 'Here you are,' Sylvie said, smiling. 'Maurice, Pamela, come and say hello to your new sister.'\n\nWarily, they approached the cradle and its contents as if unsure as to what it might contain. Sylvie remembered a similar feeling when viewing her father's body in its elaborate oak and brass coffin (charitably paid for by fellow members of the Royal Academy). Or perhaps it was Mrs Glover they were chary of.\n\n'Another girl,' Maurice said gloomily. He was five, two years older than Pamela and the man of the family for as long as Hugh was away. 'On business,' Sylvie informed people although in fact he had crossed the Channel post-haste to rescue his foolish youngest sister from the clutches of the married man with whom she had eloped to Paris.\n\nMaurice poked a finger in the baby's face and she woke up and squawked in alarm. Mrs Glover pinched Maurice's ear. Sylvie winced but Maurice accepted the pain stoically. Sylvie thought that she really must have a word with Mrs Glover when she was feeling stronger.\n\n'What are you going to call her?' Mrs Glover asked.\n\n'Ursula,' Sylvie said. 'I shall call her Ursula. It means little she-bear.'\n\nMrs Glover nodded non-committally. The middle classes were a law unto themselves. Her own strapping son was a straightforward George. 'Tiller of the soil, from the Greek,' according to the vicar who christened him and George was indeed a ploughman on the nearby Ettringham Hall estate farm, as if the very naming of him had formed his destiny. Not that Mrs Glover was much given to thinking about destiny. Or Greeks, for that matter.\n\n'Well, must be getting on,' Mrs Glover said. 'There'll be a nice steak pie for lunch. And an Egyptian pudding to follow.'\n\nSylvie had no idea what an Egyptian pudding was. She imagined pyramids.\n\n'We all have to keep up our strength,' Mrs Glover said.\n\n'Yes indeed,' Sylvie said. 'I should probably feed Ursula again for just the same reason!' She was irritated by her own invisible exclamation mark. For reasons she couldn't quite fathom, Sylvie often found herself impelled to adopt an overly cheerful tone with Mrs Glover, as if trying to restore some kind of natural balance of humours in the world.\n\nMrs Glover couldn't suppress a slight shudder at the sight of Sylvie's pale, blue-veined breasts surging forth from her foamy lace peignoir. She hastily shooed the children ahead of her out of the room. 'Porridge,' she announced grimly to them.\n\n*\n\n'God surely wanted this baby back,' Bridget said when she came in later that morning with a cup of steaming beef tea.\n\n'We have been tested,' Sylvie said, 'and found not wanting.'\n\n'This time,' Bridget said.\n\n# _May 1910_\n\n'A telegram,' Hugh said, coming unexpectedly into the nursery and ruffling Sylvie out of the pleasant doze she had fallen into while feeding Ursula. She quickly covered herself up and said, 'A telegram? Is someone dead?' for Hugh's expression hinted at catastrophe.\n\n'From Wiesbaden.'\n\n'Ah,' Sylvie said. 'Izzie has had her baby then.'\n\n'If only the bounder hadn't been married,' Hugh said. 'He could have made an honest woman of my sister.'\n\n'An honest woman?' Sylvie mused. 'Is there such a thing?' (Did she say that out loud?) 'And anyway, she's so very _young_ to be married.'\n\nHugh frowned. It made him seem more handsome. 'Only two years younger than you when you married me,' he said.\n\n'Yet so much older somehow,' Sylvie murmured. 'Is all well? Is the baby well?'\n\nIt had turned out that Izzie was already noticeably _enceinte_ by the time Hugh caught up with her and dragged her on to the boat train back from Paris. Adelaide, her mother, said she would have preferred it if Izzie had been kidnapped by white slave traders rather than throwing herself into the arms of debauchery with such enthusiasm. Sylvie found the idea of the white slave trade rather attractive \u2013 imagined herself being carried off by a desert sheikh on an Arabian steed and then lying on a cushioned divan, dressed in silks and veils, eating sweetmeats and sipping on sherbets to the bubbling sound of rills and fountains. (She expected it wasn't really like that.) A harem of women seemed like an eminently good idea to Sylvie \u2013 sharing the burden of a wife's duties and so on.\n\nAdelaide, heroically Victorian in her attitudes, had barred the door, literally, at the sight of her youngest daughter's burgeoning belly and dispatched her back across the Channel to wait out her shame abroad. The baby would be adopted as swiftly as possible. 'A respectable German couple, unable to have their own child,' Adelaide said. Sylvie tried to imagine giving away a child. ('And will we never hear of it again?' she puzzled. 'I certainly hope not,' Adelaide said.) Izzie was now to be packed off to a finishing school in Switzerland, even though it seemed she was already finished, in more ways than one.\n\n'A boy,' Hugh said, waving the telegram like a flag. 'Bouncing, etcetera.'\n\nUrsula's own first spring had unfurled. Lying in her pram beneath the beech tree, she had watched the patterns that the light made flickering through the tender green leaves as the breeze delicately swayed the branches. The branches were arms and the leaves were like hands. The tree danced for her. _Rock-a-bye baby_ , Sylvie crooned to her, _in the tree-top_.\n\n_I had a little nut tree_ , Pamela sang lispingly, _and nothing would it bear, but a silver nutmeg and a golden pear_.\n\nA tiny hare dangled from the hood of the carriage, twirling around, the sun glinting off its silver skin. The hare sat upright in a little basket and had once adorned the top of the infant Sylvie's rattle, the rattle itself, like Sylvie's childhood, long since gone.\n\nBare branches, buds, leaves \u2013 the world as she knew it came and went before Ursula's eyes. She observed the turn of seasons for the first time. She was born with winter already in her bones, but then came the sharp promise of spring, the fattening of the buds, the indolent heat of summer, the mould and mushroom of autumn. From within the limited frame of the pram hood she saw it all. To say nothing of the somewhat random embellishments the seasons brought with them \u2013 sun, clouds, birds, a stray cricket ball arcing silently overhead, a rainbow once or twice, rain more often than she would have liked. (There was sometimes a tardiness to rescuing her from the elements.)\n\nOnce there had even been the stars and a rising moon \u2013 astonishing and terrifying in equal measure \u2013 when she had been forgotten one autumn evening. Bridget was castigated. The pram was outside, whatever the weather, for Sylvie had inherited a fixation with fresh air from her own mother, Lottie, who when younger had spent some time in a Swiss sanatorium, spending her days wrapped in a rug, sitting on an outdoor terrace, gazing passively at snowy Alpine peaks.\n\nThe beech shed its leaves, papery bronze drifts filling the sky above her head. One boisterously windy November day a threatening figure appeared, peering into the baby carriage. Maurice, making faces at Ursula and chanting, 'Goo, goo, goo,' before prodding the blankets with a stick. 'Stupid baby,' he said before proceeding to bury her beneath a soft pile of leaves. She started to fall asleep again beneath her new leafy cover but then a hand suddenly swatted Maurice's head and he yelled, 'Ow!' and disappeared. The silver hare pirouetted round and round and a big pair of hands plucked her from the pram and Hugh said, 'Here she is,' as if she had been lost.\n\n'Like a hedgehog in hibernation,' he said to Sylvie.\n\n'Poor old thing,' she laughed.\n\nWinter came again. She recognized it from the first time around.\n\n# _June 1914_\n\nUrsula entered her fifth summer without further mishap. Her mother was relieved that the baby, despite (or perhaps because of) her daunting start in life, grew, thanks to Sylvie's robust regime (or perhaps in spite of it) into a steady-seeming sort of child. Ursula didn't think too much, the way Pamela sometimes did, nor did she think too little, as was Maurice's wont.\n\n_A little soldier_ , Sylvie thought as she watched Ursula trooping along the beach in the wake of Maurice and Pamela. How small they all looked \u2013 they _were_ small, she knew that \u2013 but sometimes Sylvie was taken by surprise by the breadth of her feelings for her children. The smallest, newest, of them all \u2013 Edward \u2013 was confined to a wicker Moses basket next to her on the sand and had not yet learned to cry havoc.\n\nThey had taken a house in Cornwall for a month. Hugh stayed for the first week and Bridget for the duration. Bridget and Sylvie managed the cooking between them (rather badly) as Sylvie gave Mrs Glover the month off so that she could go and stay in Salford with one of her sisters who had lost a son to diphtheria. Sylvie sighed with relief as she stood on the platform and watched Mrs Glover's broad back disappearing inside the railway carriage. 'You had no need to see her off,' Hugh said.\n\n'For the pleasure of seeing her go,' Sylvie said.\n\nThere was hot sun and boisterous sea breezes and a hard unfamiliar bed in which Sylvie lay undisturbed all night long. They bought meat pies and fried potatoes and apple turnovers and ate them sitting on a rug on the sand with their backs against the rocks. The rental of a beach hut took care of the always tricky problem of how to feed a baby in public. Sometimes Bridget and Sylvie took off their boots and daringly dabbled their toes in the water, other times they sat on the sand beneath enormous sunshades and read their books. Sylvie was reading Conrad, while Bridget had a copy of _Jane Eyre_ that Sylvie had given her as she had not thought to bring one of her usual thrilling gothic romances. Bridget proved to be an animated reader, frequently gasping in horror or stirred to disgust and, at the end, delight. It made _The Secret Agent_ seem quite dry by comparison.\n\nShe was also an inland creature and spent a lot of time fretting about whether the tide was coming in or going out, seemingly incapable of understanding its predictability. 'It changes a little every day,' Sylvie explained patiently.\n\n'But what on earth for?' a baffled Bridget asked.\n\n'Well...' Sylvie had absolutely no idea. 'Why not?' she concluded crisply.\n\nThe children were returning from fishing with their nets in the rock pools at the far end of the beach. Pamela and Ursula stopped halfway along and began to paddle at the water's edge but Maurice picked up the pace, sprinting towards Sylvie before flinging himself down in a flurry of sand. He was holding a small crab by its claw and Bridget screeched in alarm at the sight of it.\n\n'Any meat pies left?' he asked.\n\n'Manners, Maurice,' Sylvie admonished. He was going to boarding school after the summer. She was rather relieved.\n\n'Come on, let's go and jump over the waves,' Pamela said. Pamela was bossy but in a nice way and Ursula was nearly always happy to fall in with her plans and even if she wasn't she still went along with them.\n\nA hoop bowled past them along the sand, as if blown by the wind, and Ursula wanted to run after it and reunite it with its owner, but Pamela said, 'No, come on, let's paddle,' and so they put their nets down on the sand and waded into the surf. It was a mystery that no matter how hot they were in the sun the water was always freezing. They yelped and squealed as usual before holding hands and waiting for the waves to come. When they did they were disappointingly small, no more than a ripple with a lacy frill. So they waded out further.\n\nThe waves weren't waves at all now, just the surge and tug of a swell that lifted them and then moved on past them. Ursula gripped hard on to Pamela's hand whenever the swell approached. The water was already up to her waist. Pamela pushed further out into the water, a figurehead on a prow, ploughing through the buffeting waves. The water was up to Ursula's armpits now and she started to cry and pull on Pamela's hand, trying to stop her from going any further. Pamela glanced back at her and said, 'Careful, you'll make us both fall over,' and so didn't see the huge wave cresting behind her. Within a heartbeat, it had crashed over both of them, tossing them around as lightly as though they were leaves.\n\nUrsula felt herself being pulled under, deeper and deeper, as if she were miles out to sea, not within sight of the shore. Her little legs bicycled beneath her, trying to find purchase on the sand. If she could just stand up and fight the waves, but there was no longer any sand to stand on and she began to choke on water, thrashing around in panic. Someone would come, surely? Bridget or Sylvie, and save her. Or Pamela \u2013 where was she?\n\nNo one came. And there was only water. Water and more water. Her helpless little heart was beating wildly, a bird trapped in her chest. A thousand bees buzzed in the curled pearl of her ear. No breath. A drowning child, a bird dropped from the sky.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\nBridget removed the breakfast tray and Sylvie said, 'Oh, leave the little snowdrop. Here, put it on my bedside table.' She kept the baby with her too. The fire was blazing now and the bright snow-light from the window seemed both cheerful and oddly portentous at the same time. The snow was drifting against the walls of the house, pressing in on them, burying them. They were cocooned. She imagined Hugh tunnelling heroically through the snow to reach home. He had been away three days now, looking for his sister, Isobel. Yesterday (how long ago that seemed now) a telegram had arrived from Paris, saying, THE QUARRY HAS GONE TO GROUND STOP AM IN PURSUIT STOP, although Hugh was not really a hunting man. She must send her own telegram. What should she say? Something cryptic. Hugh liked puzzles. WE WERE FOUR STOP YOU ARE GONE BUT WE ARE STILL FOUR STOP (Bridget and Mrs Glover did not count in Sylvie's tally). Or something more prosaic. BABY HAS ARRIVED STOP ALL WELL STOP. Were they? All well? The baby had nearly died. She had been deprived of air. What if she wasn't quite right? They had triumphed over death this night. Sylvie wondered when death would seek his revenge.\n\nSylvie finally fell asleep and dreamed that she had moved to a new house and was looking for her children, roaming the unfamiliar rooms, shouting their names, but she knew they had disappeared for ever and would never be found. She woke with a start and was relieved to see that at least the baby was still by her side in the great white snowfield of the bed. The baby. Ursula. Sylvie had had the name ready, Edward if it had turned out to be a boy. The naming of children was her preserve, Hugh seemed indifferent to what they were called although Sylvie supposed he had his limits. Scheherazade perhaps. Or Guinevere.\n\nUrsula opened her milky eyes and seemed to fix her gaze on the weary snowdrop. _Rock-a-bye baby_ , Sylvie crooned. How calm the house was. How deceptive that could be. One could lose everything in the blink of an eye, the slip of a foot. 'One must avoid dark thoughts at all costs,' she said to Ursula.\n\n# War\n\n# _June 1914_\n\nMr Winton \u2013 Archibald \u2013 had set up his easel on the sand and was attempting to render a seascape in watery marine smears of blue and green \u2013 Prussians and Cobalt Blues, Viridian and Terre Verte. He daubed a couple of rather vague seagulls in the sky, sky that was virtually indistinguishable from the waves below. He imagined showing the picture on his return home, saying, 'In the style of the Impressionists, you know.'\n\nMr Winton, a bachelor, was by profession a senior clerk in a factory in Birmingham that manufactured pins but was a romantic by nature. He was a member of a cycling club and every Sunday tried to wheel as far away from Birmingham's smogs as he could, and he took his annual holiday by the sea so that he could breathe hospitable air and think himself an artist for a week.\n\nHe thought he might try to put some figures in his painting, it would give it a bit of life and 'movement', something his night-school teacher (he took an art class) had encouraged him to introduce into his work. Those two little girls down at the sea's edge would do. Their sunhats meant he wouldn't need to try and capture their features, a skill he hadn't yet quite mastered.\n\n'Come on, let's go and jump the waves,' Pamela said.\n\n'Oh,' Ursula said, hanging back. Pamela took her hand and dragged her into the water. 'Don't be a silly.' The closer she got to the water the more Ursula began to panic until she was swamped with fear but Pamela laughed and splashed her way into the water and she could only follow. She tried to think of something that would make Pamela want to return to the beach \u2013 a treasure map, a man with a puppy \u2013 but it was too late. A huge wave rose, curling above their heads, and came crashing over them, sending them down, down into the watery world.\n\nSylvie was startled to look up from her book and see a man, a stranger, walking towards her along the sand with one of her girls tucked under each arm, as if he was carrying geese or chickens. The girls were sopping wet and tearful. 'Went out a bit too far,' the man said. 'But they'll be fine.'\n\nThey treated their rescuer, a Mr Winton, a clerk ('senior') to tea and cakes in a hotel that overlooked the sea. 'It's the least I can do,' Sylvie said. 'You have ruined your boots.'\n\n'It was nothing,' Mr Winton said modestly.\n\n'Oh, no, it was most definitely _something_ ,' Sylvie said.\n\n'Glad to be back?' Hugh beamed, greeting them on the station platform.\n\n'Are you glad to have us back?' Sylvie said, somewhat combatively.\n\n'There's a surprise for you at home,' Hugh said. Sylvie didn't like surprises, they all knew that.\n\n'Guess,' Hugh said.\n\nThey guessed a new puppy which was a far cry from the Petter engine that Hugh had had installed in the cellar. They all trooped down the steep stone staircase and stared at its oily throbbing presence, its rows of glass accumulators. 'Let there be light,' Hugh said.\n\nIt would be a long time before any of them were able to snap a light switch without expecting to be blown up. Light was all it could manage, of course. Bridget had hoped for a vacuum cleaner to replace her Ewbank but there wasn't enough voltage. 'Thank goodness,' Sylvie said.\n\n# _July 1914_\n\nFrom the open French windows Sylvie watched Maurice erecting a makeshift tennis net, which mostly seemed to involve whacking everything in sight with a mallet. Small boys were a mystery to Sylvie. The satisfaction they gained from throwing sticks or stones for hours on end, the obsessive collection of inanimate objects, the brutal destruction of the fragile world around them, all seemed at odds with the men they were supposed to become.\n\nNoisy chatter in the hallway announced the jaunty arrival of Margaret and Lily, once schoolfriends and now infrequent acquaintances, bearing gaily beribboned gifts for the new baby, Edward.\n\nMargaret was an artist, militantly unmarried, conceivably someone's mistress, a scandalous possibility that Sylvie hadn't mentioned to Hugh. Lily was a Fabian, a society suffragette who risked nothing for her beliefs. Sylvie thought of women being restrained while tubes were pushed down their throats and raised a reassuring hand to her own lovely white neck. Lily's husband, Cavendish (the name of a hotel, not a man, surely), had once cornered Sylvie at a tea-dance, pressing her up against a pillar with his goatish, cigar-scented body, suggesting something so outrageous that even now she felt hot with embarrassment at the thought of it.\n\n'Ah, the fresh air,' Lily exclaimed when Sylvie led them out into the garden. 'It's so _rural_ here.' They cooed like doves \u2013 or pigeons, that lesser species \u2013 over the pram, admiring the baby almost as much as they applauded Sylvie's svelte figure.\n\n'I'll ring for tea,' Sylvie said, already tired.\n\n*\n\nThey had a dog. A big, brindled French mastiff called Bosun. 'The name of Byron's dog,' Sylvie said. Ursula had no idea who the mysterious Byron was but he showed no interest in reclaiming his dog from them. Bosun had soft loose furry skin that rolled beneath Ursula's fingers and his breath smelt of the scrag-end that Mrs Glover, to her disgust, had to stew for him. He was a good dog, Hugh said, a responsible dog, the kind that pulled people from burning buildings and rescued them from drowning.\n\nPamela liked to dress Bosun up in an old bonnet and shawl and pretend that he was her baby, although they had a real baby now \u2013 a boy, Edward. Everyone called him Teddy. Their mother seemed taken by surprise by the new baby. 'I don't know where he came from.' Sylvie had a laugh like a hiccup. She was taking tea on the lawn with two schoolfriends 'from her London days' who had come to inspect the new arrival. All three of them wore lovely flimsy dresses and big straw hats and sat in the wicker chairs, drinking tea and eating Mrs Glover's sherry cake. Ursula and Bosun sat on the grass a polite distance away, hoping for crumbs.\n\nMaurice had put up a net and was trying, not very enthusiastically, to teach Pamela how to play tennis. Ursula was occupied in making a daisy-chain coronet for Bosun. She had stubby, clumsy fingers. Sylvie had the long, deft fingers of an artist or a pianist. She played on the piano in the drawing room ('Chopin'). Sometimes they sang rounds after tea but Ursula never managed to sing her part at the right time. ('What a dolt,' Maurice said. 'Practice makes perfect,' Sylvie said.) When she opened the lid of the piano there was a smell that was like the insides of old suitcases. It reminded Ursula of her grandmother, Adelaide, who spent her days swathed in black, sipping Madeira.\n\nThe new arrival was tucked away in the huge baby carriage under the big beech tree. They had all been occupants of this magnificence but none of them could remember it. A little silver hare dangled from the hood and the baby was cosy beneath a coverlet 'embroidered by nuns', although no one ever explained who these nuns were and why they had spent their days embroidering small yellow ducks.\n\n'Edward,' one of Sylvie's friends said. 'Teddy?'\n\n'Ursula and Teddy. My two little bears,' Sylvie said and laughed her hiccup laugh. Ursula wasn't at all sure about being a bear. She would rather be a dog. She lay down on her back and stared up at the sky. Bosun groaned mightily and stretched out beside her. Swallows were knifing recklessly through the blue. She could hear the delicate chink of cups on saucers, the creak and clatter of a lawn-mower being pushed by Old Tom in the Coles' garden next door, and could smell the peppery-sweet perfume of the pinks in the border and the heady green of new-mown grass.\n\n'Ah,' said one of Sylvie's London friends, stretching out her legs and revealing graceful white-stockinged ankles. 'A long, hot summer. Isn't it delicious?'\n\nThe peace was broken by a disgusted Maurice throwing his racquet on to the grass where it bounced with a thump and a squeak. 'I can't teach her \u2013 she's a girl!' he yelled and stalked off into the shrubbery where he began to bash things with a stick, although in his head he was in the jungle with a machete. He was going to boarding school after the summer. It was the same school that Hugh had been to, and his father before _him_. ('And so on, back to the Conquest probably,' Sylvie said.) Hugh said it would be 'the making' of Maurice but he seemed quite made already to Ursula. Hugh said when he first went to the school he cried himself to sleep every night and yet he seemed more than happy to subject Maurice to the same torture. Maurice puffed out his chest and declared that _he_ wouldn't cry.\n\n('And what about us?' a worried Pamela asked. 'Shall we have to go away to school?'\n\n'Not unless you're very naughty,' Hugh said, laughing.)\n\nA pink-cheeked Pamela balled up her fists and, planting them on her hips, roared, 'You're such a pig!' after Maurice's indifferent, retreating back. She made 'pig' sound like a much worse word than it was. Pigs were quite nice.\n\n'Pammy,' Sylvie said mildly. 'You sound like a fishwife.'\n\nUrsula edged nearer to the source of cake.\n\n'Oh, come here,' one of the women said to her, 'let me look at you.' Ursula tried to shy away but was held firmly in place by Sylvie. 'She's quite pretty, isn't she?' Sylvie's friend said. 'She takes after you, Sylvie.'\n\n'Fish have wives?' Ursula said to her mother and Sylvie's friends laughed, lovely bubbling laughs. 'What a funny little thing,' one of them said.\n\n'Yes, she's a real hoot,' Sylvie said.\n\n*\n\n'Yes, she's a real hoot,' Sylvie said.\n\n'Children,' Margaret said, 'they are droll, aren't they?'\n\nThey are so much more than that, Sylvie thought, but how do you explain the magnitude of motherhood to someone who has no children? Sylvie felt positively matronly in her present company, the friends of her brief girlhood curtailed by the relief of marriage.\n\nBridget came out with the tray and started to take away the tea things. In the mornings Bridget wore a striped print dress for housework but in the afternoons she changed into a black dress with white cuffs and collar and a matching white apron and little cap. She had been elevated out of the scullery. Alice had left to get married and Sylvie had engaged a girl from the village, Marjorie, a boss-eyed thirteen-year-old, to help with the rough work. ('We couldn't get by with just two of them?' Hugh queried mildly. 'Bridget and Mrs G? It's not as if they're running a mansion.'\n\n'No, we can't,' Sylvie said and that was the end of that.)\n\nThe little white cap was too big for Bridget and was forever slipping over her eyes, like a blindfold. On her way back across the lawn she was suddenly blinkered by the cap and tripped, a music-hall tumble that she rescued just in time and the only casualties were the silver sugar bowl and tongs that went shooting through the air, lumps of sugar scattering like blind dice across the green of the lawn. Maurice laughed extravagantly at Bridget's misfortune, and Sylvie said, 'Maurice, stop playing the fool.'\n\nShe watched as Bosun and Ursula picked up the jettisoned sugar lumps, Bosun with his big pink tongue, Ursula, eccentrically, with the tricky tongs. Bosun swallowed his quickly without chewing. Ursula sucked hers slowly, one by one. Sylvie suspected that Ursula was destined to be the odd one out. An only child herself, she was frequently disturbed by the complexity of sibling relationships among her own children.\n\n'You should come up to London,' Margaret said suddenly. 'Stay with me for a few days. We could have such fun.'\n\n'But the children,' Sylvie said. 'The baby. I can hardly leave them.'\n\n'Why not?' Lily said. 'Your nanny can manage for a few days, surely?'\n\n'But I have no nanny,' Sylvie said. Lily cast her eyes around the garden as if she was looking for a nanny lurking in the hydrangeas. 'Nor do I want one,' Sylvie added. (Or did she?) Motherhood was her responsibility, her destiny. It was, lacking anything else (and what else could there be?), her life. The future of England was clutched to Sylvie's bosom. Replacing her was not a casual undertaking, as if her absence meant little more than her presence. 'And I am feeding the baby myself,' she added. Both women seemed astonished. Lily unconsciously clasped a hand to her own bosom as if to protect it from assault.\n\n'It's what God intended,' Sylvie said, even though she hadn't believed in God since the loss of Tiffin. Hugh rescued her, striding across the lawn like a man with a purpose. He laughed and said, 'What's going on here then?' picking up Ursula and tossing her casually in the air, only stopping when she started to choke on a sugar lump. He smiled at Sylvie and said, 'Your friends,' as if she might have forgotten who they were.\n\n'Friday evening,' Hugh said, depositing Ursula back on the grass, 'the working man's labours are over and I believe the sun is officially over the yardarm. Would you lovely ladies like to move on to something stronger than tea? Gin slings perhaps?' Hugh had four younger sisters and felt comfortable with women. That in itself was enough to charm them. Sylvie knew his instincts were to chaperone, not to court, but she did occasionally wonder about his popularity and where it might lead. Or, indeed, have already led.\n\nA d\u00e9tente was brokered between Maurice and Pamela. Sylvie asked Bridget to drag a table out on to the small but useful terrace so that the children could eat their tea outside \u2013 herring roe on toast and a pink shape that was barely set and quivered without restraint. The sight of it made Sylvie feel slightly queasy. 'Nursery food,' Hugh said with relish, observing his children eating.\n\n'Austria has declared war on Serbia,' Hugh said conversationally and Margaret said, 'How silly. I spent a wonderful weekend in Vienna last year. At the Imperial, do you know it?'\n\n'Not intimately,' Hugh said.\n\nSylvie knew it but did not say so.\n\nThe evening turned into gossamer. Sylvie, drifting gently on a mist of alcohol, suddenly remembered her father's cognac-induced demise and clapped her hands as if killing a small annoying fly and said, 'Time for bed, children,' and watched as Bridget pushed the heavy pram awkwardly across the grass. Sylvie sighed and Hugh helped her up from her chair, bussing her cheek once she was on her feet.\n\nSylvie propped open the tiny skylight window in the baby's stuffy room. They called it the 'nursery' but it was no more than a box tucked into a corner of the eaves, airless in summer and freezing in winter, and thereby totally unsuitable for a tender infant. Like Hugh, Sylvie considered that children should be toughened up early, the better to take the blows in later life. (The loss of a nice house in Mayfair, a beloved pony, a faith in an omniscient deity.) She sat on the button-backed velvet nursing chair and fed Edward. 'Teddy,' she murmured fondly as he gulped and choked his way to sated sleep. Sylvie liked them all best as babies, when they were shiny and new, like the pink pads on a kitten's paw. This one was special though. She kissed the floss on his head.\n\nWords floated up in the soft air. 'All good things must come to an end,' she heard Hugh say as he escorted Lily and Margaret indoors to dinner. 'I believe the poetically inclined Mrs Glover has baked a skate. But first, perhaps you would care to see my Petter engine?' The women twittered like the silly schoolgirls they still were.\n\nUrsula was woken by an excited shouting and clapping of hands. 'Electricity!' she heard one of Sylvie's friends exclaim. 'How wonderful!'\n\nShe shared an attic room with Pamela. They had matching small beds with a rag rug and a bedside cabinet in between. Pamela slept with her arms above her head and sometimes cried out as if pricked with a pin (a horrible trick Maurice was fond of). On one side of the bedroom wall was Mrs Glover who snored like a train and on the other side Bridget muttered her way through the night. Bosun slept outside their door, always on guard even when asleep. Sometimes he whined softly but whether in pleasure or pain they couldn't tell. The attic floor was a crowded and unquiet sort of place.\n\nUrsula was woken again later by the visitors taking their leave. ('That child is an unnaturally light sleeper,' Mrs Glover said, as if it were a flaw in her character that should be corrected.) She climbed out of bed and padded over to the window. If she stood on a chair and looked out, something they were all expressly forbidden to do, then she could see Sylvie and her friends on the lawn below, their dresses fluttering like moths in the encroaching dusk. Hugh stood at the back gate, waiting to escort them along the lane to the station.\n\nSometimes Bridget walked the children to the station to meet their father off the train when he came home from work. Maurice said he might be an engine driver when he was older, or he might become an Antarctic explorer like Sir Ernest Shackleton who was about to set sail on his grand expedition. Or perhaps he would simply become a banker, like his father.\n\nHugh worked in London, a place they visited infrequently to spend stilted afternoons in their grandmother's drawing room in Hampstead, a quarrelsome Maurice and Pamela 'fraying' Sylvie's nerves so that she was always in a bad mood on the train home.\n\nWhen everyone had left, their voices fading into the distance, Sylvie walked back across the lawn towards the house, a darkening shadow now as the black bat unfolded his wings. Unseen by Sylvie, a fox trotted purposefully in her footsteps before veering off and disappearing into the shrubbery.\n\n'Did you hear something?' Sylvie asked. She was propped up on pillows, reading an early Forster. 'The baby perhaps?'\n\nHugh cocked his head to one side. For a moment he reminded Sylvie of Bosun.\n\n'No,' he said.\n\nThe baby slept all through the night usually. He was a cherub. But not in heaven. Thankfully.\n\n'The best one yet,' Hugh said.\n\n'Yes, I think we should keep this one.'\n\n'He doesn't look like me,' Hugh said.\n\n'No,' she agreed amiably. 'Nothing like you at all.'\n\nHugh laughed and, kissing her affectionately, said, 'Good night, I'm turning out my light.'\n\n'I think I'll read a little longer.'\n\n*\n\nOne afternoon of heat a few days later they went to watch the harvest being brought in.\n\nSylvie and Bridget walked across the fields with the girls, Sylvie carrying the baby in a sling that Bridget fashioned from her shawl and tied around Sylvie's torso. 'Like a Hibernian peasant,' Hugh said, amused. It was a Saturday and, freed from the gloomy confines of banking, he was lying on the wicker chaise-longue on the terrace at the back of the house, cradling _Wisden Cricketers' Almanack_ like a hymnal.\n\nMaurice had disappeared after breakfast. He was a nine-year-old boy and free to go where he pleased with whomsoever he pleased, although he tended to keep to the exclusive company of other nine-year-old boys. Sylvie had no idea what they did but at the end of the day he would return, filthy from head to toe and with some unappetizing trophy, a jar of frogs or worms, a dead bird, the bleached skull of some small creature.\n\nThe sun had long since started on its steep climb into the sky by the time they finally set off, awkwardly encumbered with the baby, and picnic baskets, sun-bonnets and parasols. Bosun trotted along at their side like a small pony. 'Goodness, we're burdened like refugees,' Sylvie said. 'The Jews leaving Israel, perhaps.'\n\n'Jews?' Bridget said, screwing up her plain features in distaste.\n\nTeddy slept throughout the trek in his makeshift papoose while they clambered over stiles and stumbled on muddy ruts made hard by the sun. Bridget tore her dress on a nail and said she had blisters on her feet. Sylvie wondered about removing her corsets and leaving them by the wayside, imagined someone's puzzlement when they came across them. She had a sudden memory, unexpected in the dazzling daylight in a field of cows, of Hugh unlacing her stays on honeymoon in their hotel in Deauville while sounds drifted in from the open window \u2013 gulls screeching on the wing and a man and a woman arguing in rough, rapid French. On the boat home from Cherbourg Sylvie was already carrying the tiny homunculus that would become Maurice, although she had been blissfully unaware of this fact at the time.\n\n'Ma'am?' Bridget said, breaking this reverie. 'Mrs Todd? They're not cows.'\n\n*\n\nThey stopped to admire George Glover's plough horses, enormous Shires called Samson and Nelson who snorted and shook their heads when they caught sight of company. They made Ursula nervous but Sylvie fed them an apple each and they picked the fruit delicately from her palm with their big pink-velvet lips. Sylvie said they were dappled greys and much more beautiful than people and Pamela said, 'Even children?' and Sylvie said, 'Yes, especially children,' and laughed.\n\nThey found George himself helping with the harvest. When he caught sight of them he strode across the field to greet them. 'Ma'am,' he said to Sylvie, removing his cap and wiping the sweat off his forehead with a big red and white spotted handkerchief. Tiny pieces of chaff were stuck to his arms. Like the chaff, the hairs on his arms were golden from the sun. 'It's hot,' he said unnecessarily. He looked at Sylvie from beneath the long lock of hair that always fell in his handsome blue eyes. Sylvie appeared to blush.\n\nAs well as their own lunch \u2013 bloater paste sandwiches, lemon curd sandwiches, ginger beer and seed cake \u2013 they had carried the remains of yesterday's pork pie that Mrs Glover had sent for George, along with a little jar of her famous piccalilli. The seed cake was already stale because Bridget had forgotten to put it back in the cake tin and it was left out in the warm kitchen overnight. 'I wouldn't be surprised if the ants had laid eggs in it,' Mrs Glover said. When it came to eating it, Ursula had to pick out the seeds, which were legion, checking each one to make sure it wasn't an ant egg.\n\nThe workers in the field stopped to have their lunch, bread and cheese and beer mainly. Bridget turned red and giggled as she handed over the pork pie to George. Pamela told Ursula that Maurice said Bridget had a pash on George, although it seemed to both of them that Maurice was an unlikely source of information on affairs of the heart. They ate their picnic at the edge of the stubble, George sprawled casually as he took great horse-sized bites out of the pork pie, Bridget gazing at him in admiration as if he were a Greek god, while Sylvie fussed with the baby.\n\nSylvie traipsed off to find a discreet spot in order to feed Teddy. Girls brought up in nice houses in Mayfair did not generally duck behind hedges to suckle infants. Like Hibernian peasants, no doubt. She thought fondly of the beach hut in Cornwall. By the time she found a suitable covert in the lee of a hedge, Teddy was bawling his head off, little pugilistic fists clenched against the injustice of the world. Just as he settled at the breast she happened to glance up and caught sight of George Glover coming out of the trees at the far end of the field. Spotting her, he stopped, staring at her like a startled deer. For a second he didn't move but then he doffed his cap and said, 'Still hot, ma'am.'\n\n'It certainly is,' Sylvie said briskly and then watched as George Glover hastened towards the five-bar gate that broke the hedgerow in the middle of the field and leapt over it as easily as a big hunter over a hurdle.\n\nFrom a safe distance they watched the enormous harvester noisily eating the wheat. 'Hypnotic, isn't it?' Bridget said. She had recently learned the word. Sylvie took out her pretty little gold fob watch, an article much coveted by Pamela, and said, 'Heavens above, look at the time,' although none of them did. 'We must be getting back.'\n\nJust as they were leaving, George Glover shouted, 'Heyathere!' and cantered towards them across the field. He was carrying something cuddled in his cap. Two baby rabbits. 'Oh,' Pamela said, tearful with excitement.\n\n'Conies,' George Glover said. 'All huddled up in the middle of the field. Their mother gone. Take them, why don't you? One each.'\n\nOn the way home, Pamela carried both baby rabbits in her pinafore, holding it out proudly in front of her like Bridget with a tea-tray.\n\n'Look at you,' Hugh said when they walked wearily through the garden gate. 'Golden and kissed by the sun. You look like real countrywomen.'\n\n'More red than gold, I'm afraid,' Sylvie said ruefully.\n\nThe gardener was at work. He was called Old Tom ('Like a cat,' Sylvie said. 'Do you think he was once called Young Tom?'). He worked six days a week, sharing his time between them and another house nearby. These neighbours, the Coles, addressed him as 'Mr Ridgely'. He gave no indication which he preferred. The Coles lived in a very similar house to the Todds' and Mr Cole, like Hugh, was a banker. 'Jewish,' Sylvie said in the same voice she would use for 'Catholic' \u2013 intrigued yet unsettled by such exoticism.\n\n'I don't think they practise,' Hugh said. Practise what, Ursula wondered? Pamela had to practise her piano scales every evening before tea, a _plinking_ and _plonking_ that wasn't very pleasant to listen to.\n\nMr Cole had been born with a quite different name, according to their eldest son, Simon, something far too complicated for English tongues. The middle son, Daniel, was friends with Maurice, for although the grown-ups weren't friends the children were familiar with each other. Simon, 'a swot' (Maurice said), helped Maurice every Monday evening with his maths. Sylvie was unsure how to reward him for this disagreeable task, perplexed seemingly by his Jewishness. 'Perhaps I might give him something that would offend them?' she speculated. 'If I give money they might think I'm referring to their well-known reputation for miserliness. If I give sweets they might not fit their dietary strictures.'\n\n'They don't practise,' Hugh repeated. 'They're not _observant_.'\n\n'Benjamin's very observant,' Pamela said. 'He found a blackbird's nest yesterday.' She glared at Maurice when she said this. He had come upon them marvelling at the beautiful eggs, blue and freckled brown, and had grabbed them and cracked them open on a stone. He thought it was a great joke. Pamela threw a small (well, smallish) rock at him that hit him on the head. 'There,' she said. 'How does it feel to have _your_ shell broken open?' Now he had a nasty cut and a bruise on his temple. 'Fell,' he said shortly when Sylvie enquired how he came by the injury. He would, by nature, have told on Pamela, but the initial sin would have come to light and Sylvie would have punished him soundly for breaking the eggs. She had caught him stealing eggs before now and had boxed his ears. Sylvie said they should 'revere' nature, not destroy it, but reverence was not in Maurice's own nature, unfortunately.\n\n'He's learning the violin, isn't he \u2013 Simon?' Sylvie said. 'Jews are usually very musical, aren't they? Perhaps I could give him some sheet music, something like that.' This discussion of the perils of offending Judaism had taken place around the breakfast table. Hugh always looked vaguely startled to find his children at the same table as him. He hadn't eaten breakfast with his parents until he was twelve years old and deemed fit to leave the nursery. He was the robust graduate of an efficient nanny, a household within a household in Hampstead. The infant Sylvie, on the other hand, had dined late, on _Canard \u00e0 la presse_ , perched precariously on cushions, lulled by flickering candles and twinkling silverware, while her parents' conversation floated above her head. It was not, she now suspected, an entirely regular childhood.\n\nOld Tom was double-digging a trench, he said, for a new asparagus bed. Hugh had long since abandoned _Wisden_ and had been picking raspberries to fill a big white enamel bowl that both Pamela and Ursula recognized as the one that Maurice had until recently been keeping tadpoles in, although neither of them mentioned this fact. Pouring himself a glass of beer, Hugh said, 'Thirsty work, this agricultural labour,' and they all laughed. Except for Old Tom.\n\nMrs Glover came out to demand that Old Tom dig up some potatoes to go with her beef collops. She huffed and puffed at the sight of the rabbits, 'Not enough even for a stew.' Pamela screamed and had to be calmed down with a sip of Hugh's beer.\n\nPamela and Ursula made a nest, in a lost corner of the garden, out of grass and cotton wool, decorated with fallen rose petals, and carefully placed the baby rabbits in it. Pamela sang them a lullaby, she could keep a tune nicely, but they had been asleep ever since George Glover had handed them over.\n\n'I think they might be too small,' Sylvie said. Too small for what? Ursula wondered but Sylvie didn't say.\n\nThey sat on the lawn and ate the raspberries with cream and sugar. Hugh looked up into the blue, blue sky and said, 'Did you hear that thunder? There's going to be a tremendous storm, I can feel it coming. Can't you, Old Tom?' he raised his voice so that Old Tom, far away in the vegetable bed, could hear. Hugh believed that, as a gardener, Old Tom must know about weather. Old Tom said nothing and carried on digging.\n\n'He's deaf,' Hugh said.\n\n'No, he isn't,' Sylvie said, making a Rose Madder by mashing raspberries, beautiful like blood, into thick cream, and she thought, unexpectedly, about George Glover. A son of the soil. His strong square hands, his beautiful dappled greys, like big rocking horses, and the way he had lolled on the grassy bank eating his lunch, posed rather like Michelangelo's Adam in the Sistine Chapel but reaching for another slice of pork pie rather than the hand of his Creator. (When Sylvie had accompanied her father, Llewellyn, to Italy she had been astonished by the amount of male flesh available to view as art.) She imagined feeding George Glover apples from her hand and laughed.\n\n'What?' Hugh said and Sylvie said, 'What a handsome boy George Glover is.'\n\n'He must be adopted then,' Hugh said.\n\nIn bed that night Sylvie abandoned Forster for less cerebral pursuits, entwining overheated limbs in the marital bed, more a panting hart than a soaring lark. She found herself thinking not of Hugh's smooth, wiry body but of the great burnished centaureal limbs of George Glover. 'You're very...' a spent Hugh said, gazing at the bedroom cornice as he searched for an appropriate word. 'Lively,' he concluded finally.\n\n'It must be all that fresh air,' Sylvie said.\n\nGolden and kissed by the sun, she thought as she drifted comfortably off to sleep and then Shakespeare came unwontedly to mind. _Golden lads and girls all must, \/ as chimney-sweepers, come to dust_ , and she felt suddenly afraid.\n\n'There's the storm rolling in at last,' Hugh said. 'Shall I turn out the light?'\n\nSylvie and Hugh were ejected from their Sunday-morning slumber by a wailing Pamela. She and Ursula had woken early with excitement and rushed outside to find that the rabbits had disappeared, only the fluffy pom-pom of one tiny tail remaining, white smudged with red.\n\n'Foxes,' Mrs Glover said, with some satisfaction. 'What did you expect?'\n\n# _January 1915_\n\n'Did you hear the latest news?' Bridget asked.\n\nSylvie sighed and put down the letter from Hugh, its pages as brittle as dead leaves. It was only a matter of months since he had left for the Front yet she could hardly remember being married to him any more. Hugh was a captain in the Ox and Bucks. Last summer he was a banker. It seemed absurd.\n\nHis letters were cheerful and guarded ( _the men are wonderful, they have such character_ ). He used to mention these men by name ('Bert', 'Alfred', 'Wilfred') but since the Battle of Ypres they had become simply 'men' and Sylvie wondered if Bert and Alfred and Wilfred were dead. Hugh never mentioned death or dying, it was as if they were away on a jaunt, a picnic ( _An awful lot of rain this week. Mud everywhere. Hope you are enjoying better weather than we are!_ ).\n\n'To war? You are going to war?' she had shouted at him when he enlisted and it struck her that she had never shouted at him before. Perhaps she should have.\n\nIf there was to be a war, Hugh explained to her, he didn't want to look back and know that he had missed it, that others had stepped forward for their country's honour and he had not. 'It may be the only adventure I ever have,' he said.\n\n'Adventure?' she echoed in disbelief. 'What about your children, what about your _wife_?'\n\n'But it's for you that I am doing this,' he said, looking exquisitely pained, a misunderstood Theseus. Sylvie disliked him intensely in that moment. 'To protect hearth and home,' he persisted. 'To defend everything we believe in.'\n\n'And yet I heard the word _adventure_ ,' Sylvie said, turning her back on him.\n\nNonetheless, she had, of course, gone up to London to see him off. They had been jostled by an enormous flag-waving throng who were cheering as if a great victory had already been won. Sylvie was surprised by the rabid patriotism of the women on the platform, surely war should make pacifists of all women?\n\nHugh had held her close to him as if they were new sweethearts and only jumped on the train at the very last moment. He was instantly swallowed by the crush of uniformed men. _His regiment_ , she thought. How odd. Like the crowd, he had seemed immensely, stupidly cheerful.\n\nWhen the train began to heave itself slowly out of the station the excitable crowd roared their approval, frantically waving their flags and throwing caps and hats in the air. Sylvie could only stare blindly at the carriage windows as they passed by, first slowly and then more and more rapidly until they were no more than a blur. She could see no sign of Hugh, nor, she supposed, could he see her.\n\nShe remained on the platform after everyone else had left, staring at the spot on the horizon where the train had disappeared.\n\nSylvie abandoned the letter and took up her knitting needles instead.\n\n' _Did_ you hear the news?' Bridget persisted. She was placing the cutlery on the tea-table. Sylvie frowned at the knitting on her needles and wondered if she wanted to hear any news that had Bridget as its provenance. She cast off a stitch on the raglan sleeve of the serviceable grey jersey that she was knitting for Maurice. All the women of the household now spent an inordinate amount of time knitting \u2013 mufflers and mittens, gloves and socks and hats, vests and sweaters \u2013 to keep their men warm.\n\nMrs Glover sat by the kitchen stove in the evening and knitted huge gloves, big enough to fit over the hooves of George's plough horses. They were not for Samson and Nelson, of course, but for George himself, one of the first to volunteer, Mrs Glover said proudly at every opportunity, making Sylvie quite crotchety. Even Marjorie, the scullery maid, had been taken by the knitting fad, labouring after lunch on something that looked like a dishcloth, although to call it 'knitting' was generous. 'More holes than wool' was Mrs Glover's verdict, before boxing her ears and telling her to get back to work.\n\nBridget had taken to making misshapen socks \u2013 she could not turn a heel for the life of her \u2013 for her new love. She had 'given her heart' to a groom from Ettringham Hall called Sam Wellington. 'Oh, for sure, he's an old boot,' she said and laughed her head off at her joke, several times a day, as if telling it for the first time. Bridget sent Sam Wellington sentimental postcards in which angels hovered in the air over women who wept while sitting at chenille-covered tables in domestic parlours. Sylvie had hinted to Bridget that perhaps she should send more cheerful missives to a man at war.\n\nBridget kept a photograph, a studio portrait, of Sam Wellington on her rather poorly appointed dressing table. It took pride of place next to the old enamelled brush and comb set that Sylvie had given her when Hugh had bought her a silver vanity set for her birthday.\n\nA similar obligatory likeness of George adorned Mrs Glover's bedside table. Trussed in uniform and uncomfortable before a studio backdrop that reminded Sylvie of the Amalfi coast, George Glover no longer resembled a Sistine Adam. Sylvie thought of all the enlisted men who had already undergone the same ritual, a keepsake for mothers and sweethearts, the only photograph that would ever be taken of some of them. 'He could be killed,' Bridget said of her beau, 'and I might forget what he looked like.' Sylvie had plenty of photographs of Hugh. He led a well-documented life.\n\nAll of the children, except for Pamela, were upstairs. Teddy was asleep in his cot, or perhaps he was awake in his cot, whichever state he was in, he was not complaining. Maurice and Ursula were doing Sylvie knew-not-what and was not interested in it as it meant that tranquillity reigned in the morning room, apart from the occasional suspicious thud on the ceiling and the metallic report of heavy pans in the kitchen where Mrs Glover was making her feelings about something known \u2013 the war or Marjorie's incompetence, or both.\n\nEver since the fighting on the continent began they had been taking their meals in the morning room, abandoning the Regency Revival dining table as too extravagant for wartime austerity and instead espousing the little parlour table. ('Not using the dining room isn't going to win the war,' Mrs Glover said.)\n\nSylvie gestured to Pamela who obediently followed her mother's mute orders and trailed round the table after Bridget, turning the cutlery the right way round. Bridget couldn't tell her right from her left or her up from her down.\n\nPamela's support for the expeditionary force had taken the form of a mass production of dun-coloured mufflers of extraordinary and impractical lengths. Sylvie was pleasantly surprised by her elder daughter's capacity for monotony. It would stand her in good stead for her life to come. Sylvie lost a stitch and muttered an oath that startled Pamela and Bridget. 'What news?' she asked at last, reluctantly.\n\n'Bombs have been dropped on Norfolk,' Bridget said, proud of her information.\n\n'Bombs?' Sylvie said, looking up from her knitting. 'In _Norfolk_?'\n\n'A Zeppelin raid,' Bridget said authoritatively. 'That's the Hun for you. They don't care who they kill. They're wicked, so they are. They eat Belgian babies.'\n\n'Well...' Sylvie said, hooking the lost stitch, 'that might be a slight exaggeration.'\n\nPamela hesitated, dessert fork in one hand, spoon in the other, as if she was about to attempt an attack on one of Mrs Glover's heavyweight puddings. 'Eat?' she echoed in horror. 'Babies?'\n\n'No,' Sylvie said crossly. 'Don't be silly.'\n\nMrs Glover shouted for Bridget from the depths of the kitchen and Bridget flew to her command. Sylvie could hear Bridget yelling, in turn, up the stairs to the other children, 'Yer tea is on the table!'\n\nPamela sighed the sigh of someone with a lifetime behind them already and sat at the table. She stared blankly at the cloth and said, 'I miss Daddy.'\n\n'Me too, darling,' Sylvie said. 'Me too. Now don't be a goose, go and tell the others to wash their hands.'\n\nAt Christmas, Sylvie had packaged up a great box of goods for Hugh: the inevitable socks and gloves; one of Pamela's endless mufflers and, as an antidote to this, a two-ply cashmere comforter knitted by Sylvie and baptized with her favourite perfume, La Rose Jacqueminot, to remind him of home. She imagined Hugh on the battlefield wearing the comforter next to his skin, a gallant jousting knight sporting a lady's favour. This daydream of chivalry was a comfort in itself, preferable to the glimpses of something darker. They had spent a wintry weekend in Broadstairs, bundled in gaiters, bodices and balaclavas, and heard the booming of the great guns across the water.\n\nThe Christmas box also contained a plum cake baked by Mrs Glover, a tin of somewhat misshapen peppermint creams made by Pamela, cigarettes, a bottle of good malt whisky and a book of poetry \u2013 an anthology of English verse, mostly pastoral and not too taxing \u2013 as well as little hand-made gifts from Maurice (a balsa-wood plane) and a drawing from Ursula of blue sky and green grass and the tiny distorted figure of a dog. 'Bosun,' Sylvie wrote helpfully across the top. She had no idea whether or not Hugh had received the box.\n\nChristmas was a dull affair. Izzie came and talked a great deal about nothing (or rather herself) before announcing that she had joined the Voluntary Aid Detachment and was leaving for France as soon as the festivities were over.\n\n'But, Izzie,' Sylvie said, 'you can't nurse or cook or type or do anything useful.' The words came out harsher than she intended, but really Izzie was such a cuckoo. ('Flibbertigibbet' was Mrs Glover's verdict.)\n\n'That's it then,' Bridget said when she heard of Izzie's call to alms, 'we'll have lost the war by Lent.' Izzie never mentioned her baby. He had been adopted in Germany and Sylvie supposed he was a German citizen. How strange that he was only a little younger than Ursula but, officially, he was the enemy.\n\nThen at New Year, one by one, all the children came down with chickenpox. Izzie was on the next train to London as soon as the first spot erupted on Pamela's face. So much for Florence Nightingale, Sylvie said irritably to Bridget.\n\nUrsula, despite her clumsy, stubby fingers, had now joined in the household's knitting frenzy. For Christmas she received a wooden French knitting doll called La Reine Solange which Sylvie said meant 'Queen Solange' although she was 'doubtful' that there ever was a Queen Solange in history. Queen Solange was painted in regal colours and wore an elaborate yellow crown, the points of which held her wool. Ursula was a devoted subject and spent all of her spare time, of which she had oceans at her disposal, creating long serpentine lengths of wool that had no purpose except to be coiled into mats and lopsided tea-cosies. ('Where are the holes for the spout and the handle?' Bridget puzzled.)\n\n'Lovely, dear,' Sylvie said, examining one of the little mats that was slowly uncurling in her hands, like something waking from a long sleep. 'Practice makes perfect, remember.'\n\n'Yer tea is on the table!'\n\nUrsula ignored the call. She was in thrall to majesty, sitting on her bed, features scrunched up in concentration as she hooked wool around Queen Solange's crown. It was an old bit of fawn worsted but 'needs must', Sylvie said.\n\nMaurice should have been back at school but his chickenpox had been the worst of all of them and his face was still covered in little scars as if a bird had pecked at him. 'Another few days at home, young man,' Dr Fellowes said, but, in Ursula's eyes, Maurice seemed bursting with rude health.\n\nHe paced restlessly round the room, bored as a caged lion. He found one of Pamela's slippers beneath the bed and kicked it around like a football. Then he picked up a china ornament, the figure of a crinolined lady that was precious to Pamela, and tossed it so high in the air that it glanced off the vaseline glass shade of the light with an alarming _ting_. Ursula dropped her knitting, her hands flying to her mouth in horror. The crinolined lady found a soft landing on the pouchy quilt of Pamela's satin eiderdown but not before Maurice had snatched up the discarded knitting doll instead and started running around with it, pretending it was an aeroplane. Ursula watched as poor Queen Solange flew round the room, the tail of wool that protruded from her innards streaming out behind her like a thin banner.\n\nAnd then Maurice did something truly wicked. He opened the attic window, letting in a blast of unwelcome cold air, and sent the little wooden doll soaring out into the hostile night.\n\nUrsula immediately hauled a chair over to the window, climbed aboard and peered out. Illuminated in the pool of light that flooded from the window, she spotted Queen Solange, stranded on the slates in the valley between the two attic roofs.\n\nMaurice, a Red Indian now, was jumping from one bed to the other, emitting war whoops. 'Yer tea is on the table!' Bridget bellowed more urgently from the foot of the stairs. Ursula ignored both of them, her heroine heart beating loudly as she clambered out of the window \u2013 no easy task \u2013 determined to rescue her sovereign. The slates were slick with ice and Ursula had barely placed her small, slippered foot on the slope beneath the window before it slid out from under her. She let out a little cry, held out a hand towards the knitting queen as she raced past her, feet first, a tobogganer without a toboggan. There was no parapet to buffer her descent, nothing at all to stop her being propelled into the black wings of night. A kind of rush, a thrill almost, as she was launched into the bottomless air and then nothing.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\nThe piccalilli was the lurid colour of jaundice. Dr Fellowes ate at the kitchen table by the light of an annoyingly smoky oil lamp. He smeared the piccalilli on to buttered bread and topped it with a thick slice of fatty ham. He thought of the flitch of bacon resting coolly in his own pantry. He had chosen the pig himself, pointing it out to the farmer, seeing not a living creature but an anatomy lesson \u2013 an assembly of loin chops and hock, cheek and belly and huge joints of gammon for boiling. Flesh. He thought of the baby he had rescued from the jaws of death with a snip of his surgical scissors. 'The miracle of life,' he said dispassionately to the rough little Irish maid. ('Bridget, sir.') 'I am to stay the rest of the night,' he added. 'On account of the snow.'\n\nHe could think of many places he would rather be than Fox Corner. Why was it called that? Why would you celebrate the habitation of such a wily beast? Dr Fellowes had ridden with the hunt, dashing in scarlet, when a young man. He wondered if the girl would skip into his room in the morning with a tray of tea and toast. Imagined her pouring hot water from the jug into the washbasin and soaping him down in front of the bedroom fire the way his mother had done, decades ago. Dr Fellowes was obstinately faithful to his wife but his thoughts roamed far and wide.\n\nBridget led him upstairs with a candle. The candle flared and flickered wildly as he followed the maid's scrawny backside up to a chilly guest room. She lit him his own candle on top of the pot cupboard and then disappeared into the dark maw of the hall with a hasty 'Good night, sir.'\n\nHe lay in the cold bed, the piccalilli repeating unpleasantly. He wished he was at home, next to the slack, warm body of Mrs Fellowes, a woman to whom nature had denied elegance and who always smelled vaguely of fried onions. Not necessarily a disagreeable thing.\n\n# War\n\n# _20 January 1915_\n\n'Will yer get a move on?' Bridget said crossly. She was standing impatiently in the doorway, holding Teddy. 'How many times do I have to tell you, _yer tea is on the table_.' Teddy squirmed in the tight brace of her arms. Maurice paid no heed, deeply involved as he was in the intricacies of a Red Indian war dance. 'Get down from that window, Ursula, for the love of God. And why is it open? It's freezing, you'll catch yer death.'\n\nUrsula had been about to plunge out of the window in Queen Solange's wake, intent on delivering her from the no man's land of the roof, when something made her hesitate. A little doubt, a faltering foot and the thought that the roof was very high and the night very wide. And then Pamela had appeared and said, 'Mummy says you're to wash your hands for tea,' closely followed by Bridget stomping up the stairs with her unyielding refrain, _Yer tea's on the table!_ and all hope of royal rescue was lost. 'And as for you, Maurice,' Bridget continued, 'you're little more than a savage.'\n\n'I _am_ a savage,' he said. 'I'm an Apache.'\n\n'You could be the King of the Hottentots as far as I'm concerned but YER TEA IS STILL ON THE TABLE.'\n\nMaurice gave a last defiant battle cry before clattering noisily down the stairs and Pamela used an old lacrosse-net tied on to a walking cane to trawl Queen Solange back from the icy depths of the roof.\n\nTea was a boiled chicken. Teddy had a coddled egg. Sylvie sighed. Many meals involved chickens in one way or another now that they kept their own. They had a henhouse and a wired run on what was to have been an asparagus bed before the war. Old Tom had left them now, although Sylvie heard that 'Mr Ridgely' still worked for their neighbours, the Coles. Perhaps, after all, he did not like being called 'Old Tom'.\n\n'This isn't one of our chickens, is it?' Ursula asked.\n\n'No, darling,' Sylvie said. 'It isn't.'\n\nThe chicken was tough and stringy. Mrs Glover's cooking hadn't been the same since George was injured in a gas attack. He was still in a field hospital in France and when Sylvie enquired how badly he was injured she said she didn't know. 'How awful,' Sylvie said. Sylvie thought that if she had a wounded son, far from home, she would have to go on a quest to find him. Nurse and heal her poor boy. Perhaps not Maurice, but Teddy, certainly. The thought of Teddy lying wounded and helpless made her eyes prick with tears.\n\n'Are you all right, Mummy?' Pamela asked.\n\n'Absolutely,' Sylvie said, fishing the wishbone out of the chicken carcass and offering it to Ursula, who said she didn't know how to wish. 'Well, generally speaking, we wish for our dreams to come true,' Sylvie said.\n\n'But not my dreams?' Ursula said, a look of alarm on her face.\n\n'But not my dreams?' Ursula said, thinking of the giant lawn-mower that chased her through the night and the Red Indian tribe that tied her to stakes and surrounded her with bows and arrows.\n\n'This _is_ one of our chickens, isn't it?' Maurice said.\n\nUrsula liked the chickens, liked the warm straw and featheriness of the henhouse, liked reaching under the solid warm bodies to find an even warmer egg.\n\n'It's Henrietta, isn't it?' Maurice persisted. 'She was old. Ready for the pot, Mrs Glover said.'\n\nUrsula inspected her plate. She was particularly fond of Henrietta. The tough white slice of meat gave no clues.\n\n'Henrietta?' Pamela squeaked in alarm.\n\n'Did you kill her?' Maurice asked Sylvie eagerly. 'Was it very bloody?'\n\nThey had already lost several chickens to the foxes. Sylvie said she was surprised at how stupid chickens were. No more stupid than people, Mrs Glover said. The foxes had taken Pamela's baby rabbit too, last summer. George Glover had rescued two and Pamela had insisted on making a nest for hers out in the garden but Ursula had rebelled and brought her little rabbit inside and placed it in the dolls' house where it knocked everything over and left droppings like tiny liquorice balls. When Bridget discovered it she removed it to an outhouse and it was never seen again.\n\nFor pudding they had jam roly-poly and custard, the jam from the summer's raspberries. The summer was a dream now, Sylvie said.\n\n'Dead baby,' Maurice said, in that horribly off-hand manner that boarding school had only served to foster. He shovelled pudding into his mouth and said, 'That's what we call jam roly-poly at school.'\n\n'Manners, Maurice,' Sylvie warned. 'And please, don't be so vile.'\n\n'Dead baby?' Ursula said, putting her spoon down and gazing in horror at the dish in front of her.\n\n'The Germans eat them,' Pamela said gloomily.\n\n'Puddings?' Ursula puzzled. Didn't everyone eat puddings, even the enemy?\n\n'No, _babies_ ,' Pamela said. 'But only Belgian ones.'\n\nSylvie looked at the roly-poly, the round, red seam of jam like blood, and shivered. This morning she had watched Mrs Glover snapping poor old Henrietta's neck backwards over a broom handle, the bird dispatched with the indifference of a state executioner. Needs must, I suppose, Sylvie thought. 'We're at war,' Mrs Glover said, 'it's not the time to be squeamish.'\n\nPamela would not let the subject rest. 'Was it, Mummy?' she insisted quietly. 'Was it Henrietta?'\n\n'No, darling,' Sylvie said. 'On my word of honour, that was not Henrietta.'\n\nAn urgent rapping at the back door prevented further discussion. They all sat still, staring at each other, as if they had been caught in the middle of a crime. Ursula didn't really know why. 'Don't let it be bad news,' Sylvie said. It was. Seconds later there was a terrible scream from the kitchen. Sam Wellington, the old boot, was dead.\n\n'This terrible war,' Sylvie murmured.\n\n*\n\nPamela gave Ursula the remains of one of her dun-coloured balls of four-ply lambswool and Ursula promised that Queen Solange would be delivered of a little mat for Pamela's water glass in gratitude for her rescue.\n\nWhen they went to bed that night they placed the crinoline lady and Queen Solange side by side on the bedside cabinet, valiant survivors of an encounter with the enemy.\n\n# Armistice\n\n# _June 1918_\n\nTeddy's birthday. Born beneath the sign of the crab. An enigmatic sign, Sylvie said, even though she thought such things were 'bunkum'. 'For you are four,' Bridget said, which was perhaps a kind of joke.\n\nSylvie and Mrs Glover were preparing a little tea-party, 'a surprise'. Sylvie liked all her children, Maurice not so much perhaps, but she doted entirely on Teddy.\n\nTeddy didn't even know it was his birthday as for days now they had been under strict instructions not to mention it. Ursula couldn't believe how difficult it was to keep a secret. Sylvie was an adept. She told them to take 'the birthday boy' out while she got things ready. Pamela complained that _she_ had never had a surprise party and Sylvie said, 'Of course you have, you just don't remember.' Was this true? Pamela frowned at the impossibility of knowing. Ursula had no idea whether or not she had ever had a surprise party or even a party that wasn't a surprise. The past was a jumble in her mind, not the straight line that it was for Pamela.\n\nBridget said, 'Come on, we'll all go for a walk,' and Sylvie said, 'Yes, take some jam to Mrs Dodds, why don't you?' Sylvie, sleeves rolled up, hair scarved, had spent all day yesterday helping Mrs Glover make jam, boiling up copper pans of raspberries from the garden with the sugar that they had been hoarding from their ration. 'Like working in a munitions factory,' Sylvie said, as she funnelled the boiling jam into one glass jar after another. 'Hardly,' Mrs Glover muttered to herself.\n\nThe garden had produced a bumper crop, Sylvie had read books on how to cultivate fruit and declared that she was quite the gardener now. Mrs Glover said darkly that berries were easy, wait until she tried her hand at cauliflowers. For the heavy work in the garden Sylvie employed Clarence Dodds, once a pal of Sam Wellington's, the old boot. Before the war Clarence had been an under-gardener at the Hall. He had been invalided out of the army and now wore a tin mask on half of his face and said he wanted to work in a grocer's shop. Ursula first came across him when he was preparing a bed for carrots and she gave an impolite little scream when he turned round and she saw his face for the first time. The mask had one wide-open eye painted blue to match the real one. 'Enough to frighten the horses, isn't it?' he said and smiled. She wished he hadn't because his mouth wasn't covered by the mask. His lips were puckered and strange as if they were an afterthought, stitched on after he was born.\n\n'One of the lucky ones,' he said to her. 'Artillery fire, it's the devil.' It didn't look very lucky to Ursula.\n\nThe carrots had barely sprigged their feathery tops above ground when Bridget started walking out with Clarence. By the time Sylvie was grubbing up the first of the King Edwards, Bridget and Clarence were engaged and, as Clarence couldn't afford a ring, Sylvie gave Bridget a gypsy ring that she said she'd 'had for ever' and never wore. 'It's just a trinket really,' she said, 'it's not worth much,' although Hugh had bought it for her in New Bond Street after Pamela was born and had not stinted on the cost.\n\nSam Wellington's photograph was banished to an old wooden crate in the shed. 'I can't keep it,' Bridget said fretfully to Mrs Glover, 'but I can hardly throw it away, can I now?'\n\n'You could bury it,' Mrs Glover suggested but the idea gave Bridget the shivers. 'Like black magic.'\n\nThey set off for Mrs Dodds's house, laden with jam, as well as a magnificent bouquet of maroon sweet peas that Sylvie was very proud of having grown. 'The variety is \"Senator\", in case Mrs Dodds is interested,' she told Bridget.\n\n'She won't be,' Bridget said.\n\nMaurice wasn't with them, of course. He had set off on his bicycle after breakfast, a picnic lunch in his knapsack, and had disappeared for the day with his friends. Ursula and Pamela took very little interest in Maurice's life and he took none whatsoever in theirs. Teddy was a quite different kind of brother, loyal and affectionate as a dog and petted accordingly.\n\nClarence's mother was still employed at the Hall in 'a semi-feudal capacity', according to Sylvie, and had a cottage on the estate, a cramped, ancient thing that smelt of stale water and old plaster. Distemper on the damp ceiling ballooned like loose skin. Bosun had died of distemper the previous year and was buried beneath a Bourbon rose that Sylvie had ordered especially to mark his grave. 'It's called \"Louise Odier\",' she said. 'If you're interested.' They had another dog now, a wriggly black lurcher puppy called Trixie who might as well have been called Trouble because Sylvie was always laughing and saying, 'Uh-oh, here comes trouble.' Pamela had seen Mrs Glover giving Trixie a well-aimed kick with her big-booted foot and Sylvie had 'to have a word'. Bridget wouldn't let Trixie come to Mrs Dodds's house, she said she would never hear the end of it. 'She doesn't believe in dogs,' Bridget said.\n\n'Dogs are hardly an article of faith,' Sylvie said.\n\nClarence met them at the entrance gate to the estate. The Hall itself was miles away, at the end of a long avenue of elms. The Daunts had lived there for centuries and popped up occasionally to open f\u00eates and bazaars and fleetingly grace the annual Christmas party in the village hall. They had their own chapel so were never seen in church, although now they were never seen at all because they had lost three sons, one after the other, to the war and had more or less retreated from the world.\n\nIt was impossible not to stare at Clarence's tin face ('galvanized copper', he corrected them). They lived in terror that he would remove the mask. Did he take it off to go to bed at night? If Bridget married him would she see the horror beneath? 'It's not so much what's there,' they had overheard Bridget say to Mrs Glover, 'as what's _not_ there.'\n\nMrs Dodds ('Old Mother Dodds' Bridget called her, like something from a nursery rhyme) made tea for the grown-ups, tea that Bridget later reported to be 'as weak as lamb's water'. Bridget liked her tea 'strong enough for the teaspoon to stand up in it'. Neither Pamela nor Ursula could decide what lamb's water might be but it sounded nice. Mrs Dodds gave them creamy milk, ladled from a big enamel pitcher and still warm from the Hall's dairy. It made Ursula feel sick. 'Lady Bountiful,' Mrs Dodds muttered to Clarence when they handed her the jam and the sweet peas and he said, 'Mother,' chidingly. Mrs Dodds passed the flowers over to Bridget, who remained holding the sweet peas like a bride until Mrs Dodds said to her, 'Put them in water, you daft girl.'\n\n'Cake?' Clarence's mother said and doled out thin slices of gingerbread that seemed as damp as her cottage. 'It's nice to see children,' Mrs Dodds said, looking at Teddy as if he were a rare animal. Teddy was a steadfast little boy and was not put off his milk and cake. He had a moustache of milk and Pamela wiped it off with her handkerchief. Ursula suspected that Mrs Dodds didn't really think it was nice to see children, indeed she suspected that on the subject of children she was in agreement with Mrs Glover. Except for Teddy, of course. Everyone liked Teddy. Even Maurice. Occasionally.\n\nMrs Dodds examined the gypsy ring newly adorning Bridget's hand, pulling Bridget's finger towards her as if she was pulling a wishbone. 'Rubies and diamonds,' she said. 'Very fancy.'\n\n'Tiny stones,' Bridget said defensively. 'Just a trinket really.'\n\nThe girls helped Bridget wash the tea things and left Teddy to fend for himself with Mrs Dodds. They washed up in a big stone sink in the scullery that had a pump instead of a tap. Bridget said that when she was a girl 'in County Kilkenny' they had to walk to a well to get water. She arranged the sweet peas prettily in an old Dundee marmalade jar and left it on the wooden draining board. When they had dried the crockery with one of Mrs Dodds's thin, worn tea-towels (damp, of course), Clarence asked them if they would like to go over to the Hall to see the walled garden. 'You should stop going back over there, son,' Mrs Dodds said to him, 'it only upsets you.'\n\nThey entered via an old wooden door in a wall. The door was stiff and Bridget gave a little scream when Clarence took his shoulder to it and shoved it open. Ursula was expecting something wonderful \u2013 sparkling fountains and terraces, statues, walks and arbours and flowerbeds as far as the eye could see \u2013 but it wasn't much more than an overgrown field, brambles and thistles rambling everywhere.\n\n'Aye, it's a jungle,' Clarence said. 'This used to be the kitchen garden, twelve gardeners worked at the Hall before the war.' Only the roses climbing on the walls were still flourishing, and the fruit trees in the orchard that were laden with fruit. Plums were rotting on the branches. Excited wasps darted everywhere. 'They haven't picked this year,' Clarence said. 'Three sons at the Hall, all dead in this bloody war. I suppose they didn't much feel like plum pie.'\n\n'Tsk,' Bridget said. 'Language.'\n\nThere was a glasshouse with hardly any glass and inside it they could see the withered peach and apricot trees. 'Damned shame,' Clarence said and Bridget tsked again and said, 'Not in front of the children,' just like Sylvie did. 'Everything gone to seed,' Clarence said, ignoring her. 'I could weep.'\n\n'Well, you could get your job back here at the Hall,' Bridget said. 'I'm sure they'd be glad. It's not as if you can't work just as well with...' She hesitated and gestured vaguely in the direction of Clarence's face.\n\n'I don't want my job back,' he said gruffly. 'My days as some rich nob's servant are over. I miss the garden, not the life. The garden was a thing of beauty.'\n\n'We could get our own little garden,' Bridget said. 'Or an allotment.' Bridget seemed to spend a lot of time trying to cheer Clarence up. Ursula supposed she was rehearsing for marriage.\n\n'Yes, why don't we do that?' Clarence said, sounding grim at the prospect. He picked up a small, sour apple that had fallen early and bowled it hard overarm like a cricketer. It landed on the glasshouse and shattered one of the few remaining panes. 'Bugger,' Clarence said and Bridget flapped her hand at him and hissed, 'Children.'\n\n('A thing of beauty,' Pamela said appreciatively that night, as they flannelled their faces before bed with the heavy bar of carbolic. 'Clarence is a poet.')\n\nAs they trailed their way home Ursula could still smell the scent of the sweet peas they had left behind in Mrs Dodds's kitchen. It seemed an awful waste to leave them there unappreciated. By then Ursula had forgotten all about the birthday tea and was almost as surprised as Teddy when they got back to the house and found the hallway decorated with flags and bunting and a beaming Sylvie bearing a gift-wrapped present that was unmistakably a toy aeroplane.\n\n'Surprise,' she said.\n\n# _11 November 1918_\n\n'Such a melancholy time of year,' Sylvie said to no one in particular.\n\nThe leaves still lay thick on the lawn. The summer was a dream again. Every summer, it was beginning to seem to Ursula, was a dream. The last of the leaves were falling and the big beech was almost a skeleton. The Armistice seemed to have made Sylvie even more despondent than the war. ('All those poor boys, gone for ever. The peace won't bring them back.')\n\nThey had the day off school because of the great victory and they were turned outside into the morning drizzle to play. They had new neighbours, Major and Mrs Shawcross, and they spent a good deal of the damp morning peering through gaps in the holly hedge trying to get a glimpse of the Shawcrosses' daughters. There were no other girls their age in the neighbourhood. The Coles only had boys. They weren't rough like Maurice, they had nice manners and were never horrible to Ursula and Pamela.\n\n'I think they're playing hide-and-seek,' Pamela reported back from the Shawcross front. Ursula tried to see through the hedge and got scratched in the face by the vicious holly. 'I think they're the same age as us,' Pamela said. 'There's even a little one for you, Teddy.' Teddy raised his eyebrows and said, 'Oh.' Teddy liked girls. Girls liked Teddy. 'Oh, wait, there's another one,' Pamela said. 'They're multiplying.'\n\n'Bigger or smaller?' Ursula asked.\n\n'Smaller, another girl. More of a baby. Being carried by an older one.' Ursula was growing confused by the mathematics of so many girls.\n\n'Five!' Pamela said breathlessly, reaching a final total apparently. 'Five girls.'\n\nBy this time Trixie had managed to wriggle through the bottom of the hedge and they heard the excited squeals that accompanied her appearance on the other side of the holly.\n\n'I say,' Pamela said, raising her voice, 'can we have our dog back?'\n\nLunch was boiled toad in the hole and a queen of puddings. 'Where have you been?' Sylvie asked. 'Ursula, you have twigs in your hair. You look like a pagan.'\n\n'Holly,' Pamela said. 'We've been next door. We met the Shawcross girls. Five of them.'\n\n'I know.' Sylvie counted them off on her fingers. 'Winnie, Gertie, Millie, Nancy and...'\n\n'Beatrice,' Pamela supplied.\n\n'Were you invited in?' Mrs Glover, a stickler for propriety, asked.\n\n'We found a hole in the hedge,' Pamela said.\n\n'That's where those damn foxes are getting through,' Mrs Glover grumbled, 'they're coming from the copse,' and Sylvie frowned at Mrs Glover's language but said nothing as, officially, they were in celebration mood. Sylvie, Bridget and Mrs Glover were 'toasting the peace' with glasses of sherry. Neither Sylvie nor Mrs Glover seemed to have much of a taste for jubilation. Both Hugh and Izzie were still away at the Front and Sylvie said she wouldn't believe Hugh was safe until he walked through the door. Izzie had driven an ambulance throughout the war but none of them could imagine this. George Glover was being 'rehabilitated' in a home somewhere in the Cotswolds. Mrs Glover had travelled to visit him but was disinclined to talk about what she had found, other than to say that George was no longer really George. 'I don't think any of them are themselves any more,' Sylvie said. Ursula tried to imagine not being Ursula but was defeated by the impossibility of the task.\n\nTwo girls from the Women's Land Army had taken George's place on the farm. They were both horsey types from Northamptonshire and Sylvie said that if she'd known they were going to let women work with Samson and Nelson she would have applied for the job herself. The girls had come to tea on several occasions, sitting in the kitchen in their muddy puttees, to Mrs Glover's disgust.\n\n*\n\nBridget had her hat on ready to go out when Clarence appeared shyly at the back door, mumbling a greeting to Sylvie and Mrs Glover. The 'happy couple', as Mrs Glover referred to them without any hint of congratulation, were catching the train up to London to take part in the victory celebrations. Bridget was giddy with excitement. 'Sure now you don't want to come with us, Mrs Glover? I'll bet there'll be some high jinks to be had.' Mrs Glover rolled her eyes like a discontented cow. She was 'avoiding crowds' on account of the influenza epidemic. She had a nephew who had dropped dead in the street, perfectly healthy at breakfast and 'dead by noon'. Sylvie said they mustn't be scared of the influenza. 'Life must go on,' she said.\n\nAfter Bridget and Clarence left for the station, Mrs Glover and Sylvie sat at the kitchen table and drank another sherry. 'High jinks, indeed,' Mrs Glover said. By the time Teddy appeared, Trixie eager on his tail, and announced that he was starving and 'Had they forgotten lunch?' the meringue on top of the queen of puddings had collapsed and was all burnt. The final casualty of the war.\n\nThey had tried, and failed, to stay awake for Bridget's return, falling asleep over their bedtime reading. Pamela was in the spell of _At the Back of the North Wind_ while Ursula was working her way through _The Wind in the Willows_. She was particularly fond of Mole. She was a mysteriously slow reader and writer ('Practice makes perfect, dear') and liked it best when Pamela read out loud to her. They both liked fairy stories and had all of the Andrew Lang books, all twelve colours, bought by Hugh for birthdays and Christmases. 'Things of beauty,' Pamela said.\n\nBridget's noisy return woke Ursula and she, in turn, roused Pamela and they both tiptoed downstairs where a merry Bridget and a more sober Clarence regaled them with tales of the festivities, of the 'sea of people' and of the gay crowd shouting themselves hoarse for the King ('We want the King! We want the King!' Bridget demonstrated enthusiastically) until he appeared on the balcony of Buckingham Palace. 'And the bells,' Clarence added, 'never heard anything like it. All the bells of London ringing out the peace.'\n\n'A thing of beauty,' Pamela said.\n\nBridget had lost her hat somewhere amid the throng as well as several hairpins and the top button of her blouse. 'Lifted off my feet in the crush,' she said happily.\n\n'Goodness, what a racket,' Sylvie said, appearing in the kitchen, sleepy and lovely in her lacy wrap, her hair in a great fraying rope down her back. Clarence blushed and looked at his boots. Sylvie made cocoa for them all and listened indulgently to Bridget until even the novelty of being up at midnight couldn't keep any of them awake.\n\n'Back to normal tomorrow,' Clarence said, giving Bridget a daring peck on the cheek before making his way back to his mother. It was, altogether, a day out of the ordinary.\n\n'Do you think Mrs Glover will be cross that we didn't wake her?' Sylvie whispered to Pamela on the way up the stairs.\n\n'Furious,' Pamela said and they both laughed like conspirators, like women.\n\nWhen she fell asleep again Ursula dreamed of Clarence and Bridget. They were walking in an overgrown garden, looking for Bridget's hat. Clarence was crying, real tears on the good side of his face while on the mask there were painted tears, like artificial raindrops on a picture of a windowpane.\n\nWhen Ursula woke up the next morning she was burning hot and aching all over and 'Boiling, like a lobster,' Mrs Glover said, brought in for a second opinion by Sylvie. Bridget was also laid up in bed. 'Hardly surprising,' Mrs Glover said, folding disapproving arms beneath her ample yet uninviting bosom. Ursula hoped she would never have to be nursed by Mrs Glover.\n\nUrsula's breathing was harsh and raspy, her breath thickening in her chest. The world boomed and receded like the sea in a giant shell. Everything was rather pleasantly fuzzy. Trixie lay on the bed at her feet while Pamela read to her from _The Red Fairy Book_ , but the words came and went meaninglessly. Pamela's face loomed in and out of focus. Sylvie came and tried to feed her beef tea but her throat felt too small and she sputtered it out, all over the bed sheets.\n\nThere was the sound of tyres on gravel and Sylvie said to Pamela, 'That will be Dr Fellowes,' and rose swiftly, adding, 'Stay with Ursula, Pammy, but don't let Teddy in here, will you?'\n\n*\n\nThe house was more silent than usual. When Sylvie didn't come back, Pamela said, 'I'll go and look for Mummy. I won't be long.' Ursula heard murmurings and cries drifting from somewhere in the house but they meant nothing to her.\n\nShe was sleeping a strange restless kind of sleep when Dr Fellowes appeared suddenly by the side of the bed. Sylvie sat on the other side of the bed and held Ursula's hand, saying, 'Her skin is lilac. Like Bridget's.' Lilac skin sounded rather nice, like _The Lilac Fairy Book_. Sylvie's voice seemed funny, choked up and panicked like the time she saw the telegram boy coming up the path but it turned out to be only a telegram from Izzie wishing Teddy a happy birthday. ('Thoughtless,' Sylvie said.)\n\nUrsula couldn't breathe and yet she could smell her mother's perfume and hear her voice murmuring gently in her ear like a bee-buzz on a summer's day. She was too tired to open her eyes. She heard Sylvie's skirts rustle as she left her bedside, followed by the sound of the window opening. 'I'm trying to get you some air,' Sylvie said, returning to Ursula's side and holding her against her crisp seersucker blouse with its safe scents of laundry starch and roses. The woody fragrance of bonfire smoke drifted through the window and into the little attic room. She could hear the clopping of hooves followed by the rattle of the coal as the coalman emptied his sacks into the coal shed. Life was going on. A thing of beauty.\n\nOne breath, that was all she needed, but it wouldn't come.\n\nDarkness fell swiftly, at first an enemy, but then a friend.\n\n# Snow\n\n# _11 February 1910_\n\nA big woman with the forearms of a stoker woke Dr Fellowes by clattering a cup and saucer down on the pot table next to his bed and yanking open the curtains even though it was still dark outside. It took him a moment to remember that he was in the freezing-cold guest bedroom at Fox Corner and that the rather intimidating woman bearing the cup and saucer was the Todds' cook. Dr Fellowes searched the dusty archive of his brain for a name that he knew had come to him easily a few hours earlier.\n\n'It's Mrs Glover,' she said, as if reading his mind.\n\n'So it is. She of the excellent pickles.' His head felt full of straw. He was uncomfortably aware that beneath the frugal covers he was wearing only his combinations. The bedroom grate, he noted, was cold and empty.\n\n'You're needed,' Mrs Glover said. 'There's been an accident.'\n\n'An accident?' Dr Fellowes echoed. 'Something has happened to the baby?'\n\n'A farmer trampled by a bull.'\n\n# Armistice\n\n# _12 November 1918_\n\nUrsula woke up with a start. It was dark in the bedroom but she could hear noises somewhere downstairs. A door closing, giggling and shuffling. She caught the high-pitched cackle that was Bridget's unmistakable laugh and the rumbling bass note of a man. Bridget and Clarence back from London.\n\nUrsula's first instinct was to clamber out of bed and shake Pamela awake so that they could go downstairs and interrogate Bridget about the high jinks, but something stopped her. As she lay listening to the dark, a wave of something horrible washed over her, a great dread, as if something truly treacherous were about to happen. The same feeling she had had when she'd followed Pamela into the sea when they were on holiday in Cornwall, just before the war. They had been rescued by a stranger. After that Sylvie made sure they all went to the swimming baths in town and took lessons, from an ex-major in the Boer War who barked orders at them until they were too frightened to sink. Sylvie often retold the tale as if it were a hilarious escapade ('The heroic Mr Winton!') when in fact Ursula still clearly recalled the terror.\n\nPamela mumbled something in her sleep and Ursula said, 'Ssh.' Pamela mustn't wake up. They mustn't go downstairs. They mustn't see Bridget. Ursula didn't know why this was so, where this awful sense of dread came from, but she pulled the blankets over her head to hide from whatever was out there. She hoped it was out there and not inside her. She thought she would feign sleep but within minutes the real thing came.\n\n*\n\nIn the morning they ate in the kitchen because Bridget was in bed, feeling ill. 'Hardly surprising,' Mrs Glover said unsympathetically, doling out porridge. 'I dread to think what time she staggered in.'\n\nSylvie came down from upstairs with a tray that hadn't been touched. 'I really don't think Bridget is well, Mrs Glover,' she said.\n\n'Too much drink,' Mrs Glover scoffed, cracking eggs as if she were punishing them. Ursula coughed and Sylvie glanced sharply at her. 'I think we should call Dr Fellowes out,' Sylvie said to Mrs Glover.\n\n'For Bridget?' Mrs Glover said. 'The girl's as healthy as a horse. Dr Fellowes will give you short shrift when he smells the alcohol on her.'\n\n_'Mrs Glover_ ,' Sylvie said in the tone she used when she was being very serious about something and wanted to make sure people were listening ( _Don't trail muddy footprints into the house, never be unkind to other children, no matter how provoking they are_ ). 'I really do think Bridget is ill.' Mrs Glover seemed suddenly to understand.\n\n'Can you see to the children?' Sylvie said. 'I am going to telephone for Dr Fellowes and then I'll go up and sit with Bridget.'\n\n'Aren't the children going to school?' Mrs Glover asked.\n\n'Yes, of course they are,' Sylvie said. 'Although perhaps not. No \u2013 yes \u2013 they are. Or should they?' She hovered, fretfully indecisive, in the kitchen doorway while Mrs Glover waited with surprising patience for her to come to a conclusion.\n\n'I think keep them at home, for today,' Sylvie said finally. 'Crowded schoolrooms and so on.' She took a deep breath and stared at the ceiling. 'But keep them down here, just now.' Pamela raised her eyebrows at Ursula. Ursula raised hers back although she wasn't sure what they were trying to communicate to each other. Horror mainly, she supposed, at being put in Mrs Glover's care.\n\nThey had to sit at the kitchen table so Mrs Glover could 'keep an eye on them' and then, despite their violent protests, she bade them get out their schoolbooks and do work \u2013 sums for Pamela, letters for Teddy ( _Q is for quail, R is for rain_ ) and Ursula was set to practise her 'atrocious' handwriting. Ursula thought it vastly unfair that someone who wrote nothing more than shopping lists in a blunt hand ( _suet, stove blacking, mutton chops and Dinneford_ ' _s magnesia_ ) should be passing judgement on her own painful script.\n\nMrs Glover meanwhile was more than fully occupied with pressing a calf's tongue, removing the gristle and bone and rolling it up before squeezing it into the tongue press, an altogether more fascinating activity to observe than writing out _Quick wafting zephyrs vex bold Jim_ or _The five boxing wizards jumped quickly_. 'I would hate to be in any school where she was mistress,' Pamela muttered, wrestling with equations.\n\nThey were all distracted by the advent of the butcher's boy, ringing his bicycle bell noisily to announce his arrival. He was a fourteen-year-old called Fred Smith whom both the girls and Maurice admired tremendously. The girls signalled their ardour by calling him 'Freddy' while Maurice called him 'Smithy' in comradely approval. Pamela had once declared that Maurice had a pash on Fred and Mrs Glover, who happened to hear, slapped Pamela in passing on the back of her legs with a balloon whisk. Pamela was very put out and had no idea what she had been punished for. Fred Smith himself addressed the girls deferentially as 'Miss' and Maurice as 'Master Todd', although he took no interest in any of them. To Mrs Glover he was 'young Fred' and to Sylvie he was 'the butcher's boy', sometimes 'that nice butcher's boy' to distinguish him from the previous butcher's boy, Leonard Ash, 'a sneaky rogue' according to Mrs Glover, who had caught him stealing eggs from the henhouse. Leonard Ash died in the Battle of the Somme after lying about his age when he enlisted and Mrs Glover said he got what was coming to him, which seemed a rough kind of justice.\n\nFred handed over a white-paper package to Mrs Glover and said, 'Your tripe,' and then deposited the long soft body of a hare on the wooden draining board. 'Hung for five days. It's a beauty, Mrs Glover,' and even Mrs Glover, disinclined to praise in the best of circumstances, acknowledged the hare's superiority by opening a cake tin and allowing Fred to choose the biggest rock bun from within its usually clam-like innards.\n\nMrs Glover, her tongue now safely in the press, immediately began skinning the hare, a distressing yet hypnotic process to witness, and it was only when the poor creature was stripped of its fur and exposed, naked and shiny, that anyone noticed Teddy's absence.\n\n'Go and fetch him,' Mrs Glover said to Ursula. 'And you can all have a glass of milk and a rock bun, although goodness knows you've done nothing to deserve it.'\n\nTeddy was fond of hide-and-seek and, when he didn't respond to his name being called, Ursula looked in his secret places, behind the drawing-room curtains, beneath the dining-room table, and when she could find no sign of him she set off up the stairs to the bedrooms.\n\nA forceful clanging of the front-door bell echoed up the stairs in her wake. From the turn in the stairs she saw Sylvie appear in the hallway and open the door to Dr Fellowes. Ursula supposed her mother must have come down the back stairs rather than appearing by magic. Dr Fellowes and Sylvie engaged in an intense, whispered conversation, about Bridget, presumably, but Ursula couldn't catch any of the words.\n\nNot in Sylvie's room (they had long ago ceased to think of it as a room that belonged to two parents). Not in Maurice's room, so generously sized for someone who spent more than half his life living at school. Not in the guest bedroom or the second guest bedroom nor in Teddy's own little back bedroom that was almost entirely taken up with his train set. Not in the bathroom or the linen cupboard. Nor was there any sign of Teddy under the beds or in the wardrobes or in the many cupboards, nor \u2013 his favourite trick \u2013 as still as a corpse beneath Sylvie's big eiderdown.\n\n'There's cake downstairs, Teddy,' she offered up to the empty rooms. The promise of cake, true or not, was normally enough to flush Teddy out from cover.\n\nUrsula trudged up the dark narrow wooden staircase that led to the attic bedrooms and as soon as she had placed her foot on the first tread she experienced a sudden pinch of fear in her insides. She had no idea where it came from, or why.\n\n'Teddy! Teddy, where are you?' Ursula tried to raise her voice but the words came out in a whisper.\n\nNot in the bedroom she shared with Pamela, not in Mrs Glover's old room. Not in the boxroom, once a nursery and now home to chests and trunks and packing cases of old clothes and toys. Only Bridget's room remained unexplored.\n\nThe door was ajar and Ursula had to force her feet to walk towards it. Something terrible was beyond that open door. She didn't want to see it, but she knew she must.\n\n'Teddy!' she said, overcome with relief at the sight of him. Teddy was sitting on Bridget's bed, his birthday aeroplane on his knee. 'I've been looking everywhere for you,' Ursula said. Trixie was lying on the floor next to the bed and sprang up eagerly when she saw her.\n\n'I thought it might make Bridget feel better,' Teddy said, stroking the plane. Teddy had great faith in the healing power of toy trains and aeroplanes. (He was, he assured them, going to be a pilot when he grew up.) 'I think Bridget's asleep but her eyes are open,' he said.\n\nThey were. Wide open, staring sightlessly at the ceiling. There was a watery blue film across those disturbing eyes and her skin had a strange lilac hue. Cobalt Violet in Ursula's Winsor and Newton watercolour set. She could see the tip of Bridget's tongue sticking out of her mouth and had a momentary vision of Mrs Glover pushing the calf's tongue into the press.\n\nUrsula had never seen a dead body but she knew without any doubt that Bridget had now become one. 'Get off the bed, Teddy,' she said cautiously, as if her brother were a wild creature about to bolt. She started to tremble all over. It wasn't just that Bridget was dead, although that was bad enough, but there was something more perilous here. The unadorned walls, the thin jacquard bedspread on the iron bedstead, the enamelled brush and comb set on the dressing table, the rag rug on the floor, all suddenly grew immensely threatening as if they were not really the objects they seemed. Ursula heard Sylvie and Dr Fellowes on the stairs. Sylvie's tones were urgent, Dr Fellowes's less concerned.\n\nSylvie came in and gasped, 'Oh dear God,' when she saw them in Bridget's room. She snatched Teddy off the bed and then pulled Ursula by the arm out into the passage. Trixie, tail wagging eagerly at the excitement, bounded after them. 'Go to your room,' Sylvie said. 'No, go to Teddy's room. No, go to my room. Go _now_ ,' she said, sounding frantic, not at all the Sylvie they were used to. Sylvie went back into Bridget's room and closed the door decisively. They could hear only murmured exchanges between Sylvie and Dr Fellowes and eventually Ursula said, 'Come on,' to Teddy and took his hand. He allowed her to lead him docilely back down the stairs to Sylvie's bedroom. 'Did you say cake?' he asked.\n\n'Teddy's skin is the same colour as Bridget's,' Sylvie said. Her stomach hollowed out with terror. She knew what she was looking at. Ursula was merely pale, although her closed eyelids were dark and her skin glistened with a strange, sickly sheen.\n\n'Heliotrope cyanosis,' Dr Fellowes said, taking Teddy's pulse. 'And see those mahogany spots on his cheeks? This is the more virulent strain, I'm afraid.'\n\n'Stop, please stop,' Sylvie hissed. 'Do not lecture me like a medical student. I am their _mother_.' How she hated Dr Fellowes at that moment. Bridget was lying in her bed upstairs, still warm but as dead as the marble on a tomb. 'The influenza,' Dr Fellowes continued relentlessly. 'Your maid was mixing with crowds of people yesterday in London \u2013 perfect conditions for the infection to spread. It can take them in the blink of an eye.'\n\n'But not this one,' Sylvie said fiercely, clutching Teddy's hand. 'Not my child. Not my children,' she amended, reaching across to stroke Ursula's burning forehead.\n\nPamela hovered in the doorway and Sylvie shooed her away. Pamela started to cry but Sylvie had no time for tears. Not now, not in the face of death.\n\n'There must be something I can do,' she said to Dr Fellowes.\n\n'You can pray.'\n\n'Pray?'\n\nSylvie did not believe in God. She considered the biblical deity to be an absurd, vengeful figure (Tiffin and so on), no more real than Zeus or the great god Pan. She went to church dutifully every Sunday, however, and avoided alarming Hugh with her heretical thoughts. Needs must, and so on. She prayed now, with desperate conviction but no faith, and she suspected it made no difference either way.\n\nWhen a pale bloody kind of froth, like cuckoo-spit, bubbled from Teddy's nostrils Sylvie made a noise like a wounded animal. Mrs Glover and Pamela were listening at the other side of the door and in a rare moment of unity they clutched each other's hands. Sylvie snatched Teddy from the bed and held him tightly to her breast and howled with pain.\n\nDear God, Dr Fellowes thought, the woman grieved like a savage.\n\nThey sweated together in a tangle of Sylvie's linen bed sheets. Teddy was spreadeagled across the pillows. Ursula wanted to hold him close but he was too hot so she held one of his ankles instead, as if she was trying to stop him running away. Ursula's lungs felt as if they were full of custard, she imagined it thick and yellow and sweet.\n\nTeddy was gone by nightfall. Ursula knew the moment he died, she felt it inside her. She heard just one wretched moan from Sylvie and then someone lifted Teddy out of the bed and even though he was just a little boy it was as if something weighty had gone from her side and Ursula was alone in the bed. She could hear Sylvie's choking sobs, an awful noise, as if someone had hacked off one of her limbs.\n\nEvery breath squeezed the custard stuff in her lungs. The world was fading and she began to have a stirring sense of anticipation, as if it were Christmas or her birthday, and then the black bat night approached and enfolded her in his wings. One last breath and then no more. She held out a hand to Teddy, forgetting that he wasn't there any more.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\nSylvie lit a candle. Winter dark, five o'clock in the morning by the little gold carriage clock on the bedroom mantelpiece. The clock, an English one ('Better than a French one,' her mother had instructed), had been one of her parents' wedding presents. When the creditors came to call after the society portraitist's death his widow hid the clock beneath her skirts, bemoaning the passing of the crinoline. Lottie appeared to chime on the quarter, disconcerting the creditors. Luckily they were not in the room when she struck the hour.\n\nThe new baby was asleep in her cradle. Words from Coleridge suddenly came into Sylvie's mind: _Dear Babe, that sleepest cradled by my side_. Which poem was that?\n\nThe fire in the grate had died down, leaving only the smallest flame still dancing on the coals. The baby began to make mewling sounds and Sylvie climbed gingerly out of bed. Childbirth was a brutal affair. If she had been in charge of designing the human race she would have gone about things quite differently. (A golden shaft of light through the ear for conception perhaps and a well-fitting hatch somewhere modest for escape nine months later.) She left the warmth of her bed and retrieved Ursula from the cradle. And then, suddenly, breaking the snow-muffled silence, she thought she heard the soft nicker of a horse and felt a little buzz of electric pleasure in her soul at this unlikely sound. She carried Ursula over to the window and drew one of the heavy curtains back far enough to see out. The snow had obliterated everything familiar, the world outside was shawled in white. And there below was the fantastic sight of George Glover riding bareback on one of his great Shires (Nelson, if she wasn't mistaken) up the wintry drive. He looked magnificent, like a hero of old. Sylvie closed the curtains and decided that the tribulations of the night had probably affected her brain and were making her hallucinate.\n\nShe took Ursula back into bed with her and the baby rooted around for her nipple. Sylvie believed in wet-nursing her own children. The idea of glass bottles and rubber teats seemed unnatural somehow but that didn't mean that she didn't feel like a cow being milked. The baby was slow and floundering, confounded by the new. How long before breakfast, Sylvie wondered?\n\n# Armistice\n\n# _11 November 1918_\n\n_Dear Bridget, I have locked and bolted the doors. There is a gang of thieves \u2013_ should the 'i' come before the 'e'? Ursula chewed the end of her pencil until it splintered. Undecided, she crossed out 'thieves' and wrote 'robbers' instead. _There is a gang of robbers in the village. Please can you stay with Clarence's mother?_ For good measure she added _and also I have a headache so don't knock_. She signed it _Mrs Todd_. Ursula waited until there was no one in the kitchen and then went outside and pinned the note to the back door.\n\n'What are you doing?' Mrs Glover asked as she came back inside. Ursula jumped, Mrs Glover could move as quietly as a cat.\n\n'Nothing,' Ursula said. 'Looking to see if Bridget was coming yet.'\n\n'Heavens,' Mrs Glover said, 'she'll be back on the last train, not for hours yet. Now shift yourself, it's long past your bedtime. It's Liberty Hall here.'\n\nUrsula didn't know what Liberty Hall meant but it sounded like rather a good place to live.\n\nNext morning there was no Bridget in the house. Nor, more puzzlingly, was there any sign of Pamela. Ursula felt overwhelmed by a relief as inexplicable as the panic that had led her to write the note the previous night.\n\n'There was a silly note on the door last night, a prank,' Sylvie said. 'Bridget was locked out. You know, it looks just like your handwriting, Ursula, I don't suppose you can explain that?'\n\n'No, I can't,' Ursula said stoutly.\n\n'I sent Pamela to Mrs Dodds to fetch Bridget home,' Sylvie said.\n\n'You sent _Pamela_?' Ursula echoed in horror.\n\n'Yes, Pamela.'\n\n'Pamela is with _Bridget_?'\n\n'Yes,' Sylvie said. 'Bridget. What is the matter with you?'\n\nUrsula ran out of the house. She could hear Sylvie shouting after her but she didn't stop. She had never run so fast in all of her eight years, not even when Maurice was chasing her to give her a Chinese burn. She ran up the lane in the direction of Mrs Dodds's cottage, splashing through the mud so that by the time Pamela and Bridget were in sight ahead of her she was filthy from head to toe.\n\n'What _is_ the matter?' Pamela asked anxiously. 'Is it Daddy?' Bridget made the sign of the cross. Ursula threw her arms round Pamela and collapsed in tears.\n\n'Whatever is it? _Tell_ me,' Pamela said, caught up now in the dread.\n\n'I don't know,' Ursula sobbed. 'I just felt so worried about you.'\n\n'What a goose,' Pamela said affectionately, hugging her.\n\n'I have a bit of a headache,' Bridget said. 'Let's get back to the house.'\n\nDarkness soon fell again.\n\n# Snow\n\n# _11 February 1910_\n\n'A miracle, says the Fellowes feller,' Bridget said to Mrs Glover as they toasted the new baby's arrival over their morning teapot. As far as Mrs Glover was concerned miracles belonged inside the pages of the Bible, not amid the carnage of birth. 'Maybe she'll stop at three,' she said.\n\n'Now why would she be going and doing that when she has such lovely healthy babies and there's enough money in the house for anything they want?'\n\nMrs Glover, ignoring the argument, heaved herself up from the table and said, 'Well, I must get on with Mrs Todd's breakfast.' She took a bowl of kidneys soaking in milk from the pantry and commenced removing the fatty white membrane, like a caul. Bridget glanced at the milk, white marbled with red, and felt uncharacteristically squeamish.\n\nDr Fellowes had already breakfasted \u2013 on bacon, black pudding, fried bread and eggs \u2013 and left. Men from the village had arrived and tried to dig his car out and when that had failed someone ran for George and he had come to the rescue, riding on the back of one of his big Shires. St George slipped briefly into Mrs Glover's mind and hastily slipped out again as being too fanciful. With not inconsiderable difficulty, Dr Fellowes was hoisted up behind Mrs Glover's son and the pair had ridden off, ploughing snow, not earth.\n\nA farmer had been trampled by a bull, but was alive still. Mrs Glover's own father, a dairyman, had been killed by a cow. Mrs Glover, young but doughty and not yet acquainted with Mr Glover, had come across her father lying dead in the milking shed. She could still see the blood on the straw and the surprised look on the face of the cow, her father's favourite, Maisie.\n\nBridget warmed her hands on the teapot and Mrs Glover said, 'Well, I'd better to see to my kidneys. Find me a flower for Mrs Todd's breakfast tray.'\n\n'A flower?' Bridget puzzled, looking through the window at the snow. 'In this weather?'\n\n# Armistice\n\n# _11 November 1918_\n\n'Oh, Clarence,' Sylvie said, opening the back door. 'Bridget's had a bit of an accident, I'm afraid. She tripped and fell over the step. Just a sprained ankle, I think, but I doubt that she'll be able to go up to London for the celebrations.'\n\nBridget was sipping a brandy, sitting in Mrs Glover's chair, a big high-backed Windsor, by the stove. Her ankle was propped up on a stool, and she was enjoying the drama of her tale.\n\n'I was just coming in the kitchen door, so I was. I'd been hanging out washing although I don't know why I bothered because it started to rain again, when I felt hands shoving me in the back. And then there I was, sprawled all over the ground, in agony. Small hands,' she added. 'Like the hands of a little ghost child.'\n\n'Oh, really,' Sylvie said. 'There are no ghosts in this house, children or otherwise. Did you see anything, Ursula? You were in the garden, weren't you?'\n\n'Oh, the silly girl just tripped,' Mrs Glover said. 'You know how clumsy she is. Well, anyway,' she said with some satisfaction, 'that's put paid to your London high jinks.'\n\n'Not so,' Bridget said stoutly. 'I'm not missing this day for anything. Come on, Clarence. Give me your arm. I can _hobble_.'\n\nDarkness, and so on.\n\n# Snow\n\n# _11 February 1910_\n\n' \"Ursula\", before you ask,' Mrs Glover said, dumping spoonfuls of porridge into bowls in front of Maurice and Pamela, who were sitting at the big wooden table in the kitchen.\n\n'Ursula,' Bridget said appreciatively. 'That's a good name. Did she like the snowdrop?'\n\n# Armistice\n\n# _11 November 1918_\n\nEverything familiar somehow. 'It's called _d\u00e9j\u00e0 vu_ ,' Sylvie said. 'It's a trick of the mind. The mind is a fathomless mystery.' Ursula was sure that she could recall lying in the baby carriage beneath the tree. 'No,' Sylvie said, 'no one can remember being so small,' yet Ursula remembered the leaves, like great green hands, waving in the breeze and the silver hare that hung from the carriage hood, turning and twisting in front of her face. Sylvie sighed. 'You do have a very vivid imagination, Ursula.' Ursula didn't know whether this was a compliment or not but it was certainly true that she often felt confused between what was real and what was not. And the terrible fear \u2013 fearful terror \u2013 that she carried around inside her. The dark landscape within. 'Don't dwell on such things,' Sylvie said sharply when Ursula tried to explain. 'Think sunny thoughts.'\n\nAnd sometimes, too, she knew what someone was about to say before they said it or what mundane incident was about to occur \u2013 if a dish was about to be dropped or an apple thrown through a glasshouse, as if these things had happened many times before. Words and phrases echoed themselves, strangers seemed like old acquaintances.\n\n'Everyone feels peculiar from time to time,' Sylvie said. 'Remember, dear \u2013 sunny thoughts.'\n\nBridget lent a more willing ear, declaring that Ursula 'had the second sight'. There were doorways between this world and the next, she said, but only certain people could pass through them. Ursula didn't think that she wanted to be one of those people.\n\nLast Christmas morning, Sylvie had handed Ursula a box, nicely wrapped and ribboned, the contents quite invisible, and said, 'Happy Christmas, dear,' and Ursula said, 'Oh, good, a dining set for the dolls' house,' and was immediately in trouble for having sneaked a preview of the presents.\n\n'But I never,' she insisted obstinately to Bridget later in the kitchen, where Bridget was trying to affix little white-paper crowns on the footless legs of the Christmas goose. (The goose made Ursula think of a man in the village, a boy really, who had had both his feet blown off at Cambrai.) 'I didn't look, I just _knew_.'\n\n'Ah, I know,' Bridget said. 'For sure, you have the sixth sense.'\n\nMrs Glover, wrestling with the plum pudding, snorted her disapproval. She was of the opinion that five senses were too many, let alone adding on another one.\n\nThey were shut out in the garden for the morning. 'So much for victory celebrations,' Pamela said as they sheltered from the drizzle beneath the beech tree. Only Trixie was having a good time. She loved the garden, mainly because of the number of rabbits which, despite the best attentions of the foxes, continued to enjoy all the benefits of the vegetable garden. George Glover had given two babies to Ursula and Pamela before the war. Ursula convinced Pamela that they had to keep them indoors and they hid them in their bedroom cupboard and fed them with an eye-dropper they found in the medicine cabinet until they hopped out one day and frightened Bridget out of her wits.\n\n' _A fait accompli_ ,' Sylvie said when she was presented with the rabbits. 'You can't keep them in the house though. You'll have to ask Old Tom to build a hutch for them.'\n\nThe rabbits had escaped long ago, of course, and had multiplied happily. Old Tom had laid down poison and traps to little avail. ('Goodness,' Sylvie said, looking out one morning at the rabbits contentedly breakfasting on the lawn. 'It's like Australia out there.') Maurice, who was learning to shoot in the junior ATC at school, had spent all of last summer's long holiday taking potshots at them from his bedroom window with Hugh's neglected old Westley Richards wildfowler. Pamela was so furious with him that she put some of his own itching powder (he was forever in joke shops) in his bed. Ursula immediately got the blame and Pamela had to own up, even though Ursula had been quite ready to take it on the chin. That was the kind of person Pamela was \u2013 always very stuck on being fair.\n\nThey heard voices in the garden next door \u2013 they had new neighbours they were yet to meet, the Shawcrosses \u2013 and Pamela said, 'Come on, let's go and see if we can catch a look. I wonder what they're called.'\n\nWinnie, Gertie, Millie, Nancy and baby Bea, Ursula thought but said nothing. She was getting as good at keeping secrets as Sylvie.\n\nBridget gripped her hatpin between her teeth and lifted her arms to adjust her hat. She had sewn a new bunch of paper violets on to it, especially for the victory. She was standing at the top of the stairs, singing _K-K-Katy_ to herself. She was thinking of Clarence. When they were married ('in the spring,' he said, although it had been 'before Christmas' not so long ago) she would be leaving Fox Corner. She would have her own little household, her own babies.\n\nStaircases were very dangerous places, according to Sylvie. People died on them. Sylvie always told them not to play at the top of the stairs.\n\nUrsula crept along the carpet runner. Took a quiet breath and then, both hands out in front of her, as if trying to stop a train, she threw herself at the small of Bridget's back. Bridget whipped her head round, mouth and eyes wide in horror at the sight of Ursula. Bridget went flying, toppling down the stairs in a great flurry of arms and legs. Ursula only just managed to stop herself from following in her wake.\n\nPractice makes perfect.\n\n'The arm's broken, I'm afraid,' Dr Fellowes said. 'You took quite a tumble down those stairs.'\n\n'She's always been a clumsy girl,' Mrs Glover said.\n\n' _Someone_ pushed me,' Bridget said. A great bruise bloomed on her forehead, she was holding her hat, the violets crushed.\n\n'Someone?' Sylvie echoed. 'Who? Who would push you downstairs, Bridget?' She looked around the faces in the kitchen. 'Teddy?' Teddy put his hand over his mouth as if he was trying to stop words escaping. Sylvie turned to Pamela. 'Pamela?'\n\n'Me?' Pamela said, piously holding both of her outraged hands over her heart like a martyr. Sylvie looked at Bridget, who made a little inclination of her head towards Ursula.\n\n'Ursula?' Sylvie frowned. Ursula stared blankly ahead, a conscientious objector about to be shot. 'Ursula,' Sylvie said severely, 'do you know something about this?'\n\nUrsula had done a wicked thing, she had pushed Bridget down the stairs. Bridget might have died and she would have been a murderer now. All she knew was that she _had_ to do it. The great sense of dread had come over her and she had to do it.\n\nShe ran out of the room and hid in one of Teddy's secret hiding places, the cupboard beneath the stairs. After a while the door opened and Teddy crept in and sat on the floor next to her. 'I don't think you pushed Bridget,' he said and slipped his small, warm hand into hers.\n\n'Thank you. I did though.'\n\n'Well, I still love you.'\n\nShe might never have come out of that cupboard but the front-door bell clanged and there was a sudden great commotion in the hallway. Teddy opened the door to see what was happening. He ducked back in and reported, 'Mummy's kissing a man. She's crying. He's crying as well.' Ursula put her head out of the cupboard to witness this phenomenon. She turned in astonishment to Teddy. 'I think it might be Daddy,' she said.\n\n# Peace\n\n# _February 1947_\n\nUrsula traversed the street cautiously. The road surface was treacherous \u2013 crimped and rucked by ridges and crevasses of ice. The pavements were even more perilous, no more than massifs of ugly, hard-packed snow, or, worse, toboggan runs ironed by the neighbourhood children who had nothing better to do than enjoy themselves because the schools were closed. Oh, God, Ursula thought, how mean-spirited I've become. The bloody war. The bloody peace.\n\nBy the time she had put her key in the lock of the street door she was exhausted. A shopping trip had never seemed such a challenge previously, even in the worst days of the Blitz. The skin on her face was whipped raw by the biting wind and her toes were numb with cold. The temperature hadn't risen above zero for weeks, colder even than '41. Ursula imagined at some future date trying to recall this glacial chill and knew she would never be able to conjure it up. It was so _physical_ , you expected bones to shatter, skin to crackle. Yesterday she had seen two men trying to open a manhole in the road with what looked like a flamethrower. Perhaps there would be no future of thaw and warmth, perhaps this was the beginning of a new Ice Age. First fire and then ice.\n\nIt was as well, she thought, that the war had robbed her of any care for fashion. She was wearing, in order, from inner to outer \u2013 a short-sleeved vest, a long-sleeved vest, a long-sleeved pullover, a cardigan and stretched on top of it all her shabby old winter coat, bought new in Peter Robinson's two years before the war. Not to mention, of course, the usual drab underwear, a thick tweed skirt, grey wool stockings, gloves _and_ mittens, a scarf, a hat and her mother's old fur-lined boots. Pity any man who was suddenly moved to ravish her. 'Chance'd be a fine thing, eh?' Enid Barker, one of the secretaries, said over the balm and succour of the tea-urn. Enid had auditioned for the part of plucky young London woman somewhere around 1940 and had been playing it with gusto ever since. Ursula chided herself for more unkind thoughts. Enid was a good sort. Terrifically skilled at typing tabulations, something Ursula had never quite got the hang of when she was at secretarial college. She had done a typing and shorthand course, years ago now \u2013 everything before the war seemed like ancient history (her own). She had been surprisingly adept. Mr Carver, the man who ran the secretarial college, had suggested that her shorthand was good enough for her to train as a court reporter at the Old Bailey. That would have been a quite different life, perhaps a better one. Of course, there was no way of knowing these things.\n\nShe trudged up the unlit stairs to her flat. She lived on her own now. Millie had married an American USAF officer and moved to New York State ('Me \u2013 a war bride! Who'd a thunk it?'). A thin layer of soot and what seemed to be grease coated the walls of the stairway. It was an old building, in Soho of all places ('needs must' she heard her mother's voice say). The woman who lived upstairs had a great many gentleman callers and Ursula had become accustomed to the creaking bedsprings and strange noises that came through the ceiling. She was pleasant though, always ready with a cheery greeting and never missed her turn at sweeping the stairs.\n\nThe building had been Dickensian in its dinginess to begin with and was now even more neglected and unloved. But then, the whole of London looked wretched. Grimy and grim. She remembered Miss Woolf saying that she didn't think 'poor old London' would ever be clean again. ('Everything is so awfully _shabby_.') Perhaps she was right.\n\n'You wouldn't think we had _won_ the war,' Jimmy said when he came to visit, spivvy in his American clothes, shiny and bright with promise. She readily forgave her little brother his New World \u00e9lan, he had had a hard war. Hadn't they all? 'A long and hard war,' Churchill had promised. How right he had been.\n\nIt was a temporary billet. She had the money for something better but the truth was she didn't really care. It was just one room, a window above the sink, a hot-water geyser, shared toilet down the hall. Ursula still missed the old flat in Kensington that she had shared with Millie. They had been bombed out in the big raid of May '41. Ursula had thought of Bessie Smith singing _like a fox without a hole_ but she had actually moved back in for a few weeks, living without a roof. It was chilly but she was a good camper. She had learned with the Bund Deutscher M\u00e4del, although it wasn't the kind of fact that you bandied about in those dark days.\n\nBut here was a lovely surprise waiting for her. A gift from Pammy \u2013 a wooden crate filled with potatoes, leeks, onions, an enormous emerald-green Savoy cabbage (a thing of beauty) and on the top, half a dozen eggs, nestled in cotton wool inside an old trilby of Hugh's. Lovely eggs, brown and speckled, as precious as unpolished gemstones, tiny feathers stuck here and there. _From Fox Corner, with love_ the label attached to the crate read. It was like receiving a Red Cross parcel. How on earth had it got here? There were no trains running and Pamela was almost certainly snowed in. Even more puzzling was how her sister had managed to dig up this wintry harvest when _Earth stood hard as iron_.\n\nWhen she opened the door she found a scrap of paper on the floor. She had to put her spectacles on to read it. It was a note from Bea Shawcross. _Visited but you weren't in. Will pop by again. Bea xxx_. Ursula was sorry she had missed Bea's visit, it would have been a nicer way to spend a Saturday afternoon than wandering in the dystopian West End. She was immensely cheered by nothing more than the sight of a cabbage. But then the cabbage \u2013 unexpectedly as was always the wont of these moments \u2013 uprooted an unwanted memory of the little parcel in the cellar at Argyll Road and she was plunged back into gloom. She was so up and down these days. Honestly, she chided herself, buck up, for heaven's sake.\n\nIt felt even colder inside the flat. She had developed chilblains, horrid painful things. Even her ears were cold. She wished she had some earmuffs, or a balaclava, like the grey woollen ones that Teddy and Jimmy used to wear to school. There was a line in 'The Eve of St Agnes', what was it? Something about the stone effigies in the church _in icy hoods and mails_. It used to make her feel cold every time she recited it. Ursula had learned the whole poem at school, a feat of memory that was probably beyond her now, and what, after all, had been the point if she couldn't even remember a complete line? She had a sudden longing for Sylvie's fur coat, a neglected mink, like a large friendly animal, that now belonged to Pamela. Sylvie had chosen death on VE Day. While other women were scratching together food for tea-parties and dancing in the streets of Britain, Sylvie had lain down on the bed that had been Teddy's when he was a child and swallowed a bottle of sleeping pills. No note, but her intention and motivation were quite clear to the family that she left behind. There had been a horrible funeral tea for her at Fox Corner. Pamela said it was the coward's way out, but Ursula wasn't so sure. She thought it showed a rather admirable clarity of purpose. Sylvie was another casualty of war, another statistic.\n\n'You know,' Pamela said, 'I used to argue with her because she said science had made the world a worse place, that it was all about men inventing new ways to kill people. But now I wonder if she wasn't right.' And that was before Hiroshima, of course.\n\nUrsula lit the gas fire, a rather pathetic little Radiant that looked as if it dated from the turn of the century, and fed the meter. The rumour was that pennies and shillings were running out. Ursula wondered why they couldn't melt down armaments. Guns into ploughshares, and so on.\n\nShe unpacked Pammy's box, laying everything out on the little wooden draining board like a poor man's still life. The vegetables were dirty but there wasn't much hope of washing the soil off as the pipes were frozen, even in the little Ascot, although the gas pressure was so low that it could barely heat the water anyway. _Water like a stone_. At the bottom of the crate she found a half-bottle of whisky. Good old Pammy, ever the thoughtful one.\n\nShe scooped some water from the bucket that she'd filled from the standpipe in the street and put a pan of water on the gas ring, thinking she might boil one of the eggs, although it would take for ever as there was only the tiniest frill of blue around the burner. There were warnings to be vigilant about the gas pressure \u2013 in case the gas came back on when the pilot light had gone out.\n\nWould it be so bad to be gassed, Ursula wondered? _Gassed_. She thought of Auschwitz. Treblinka. Jimmy had been a Commando and at the end of the war he had become attached, rather haphazardly according to him (although everything to do with Jimmy was always slightly haphazard), to the anti-tank regiment that liberated Bergen-Belsen. Ursula insisted that he told her what he had found there. He was reluctant and had probably withheld the worst but it was necessary to know. One must bear witness. (She heard Miss Woolf's voice in her head, _We must remember these people when we are safely in the future_.)\n\nThe toll of the dead had been her business during the war, the endless stream of figures that represented the blitzed and the bombed passed across her desk to be collated and recorded. They had seemed overwhelming, but the greater figures \u2013 the six million dead, the fifty million dead, the numberless infinities of souls \u2013 were in a realm beyond comprehension.\n\nUrsula had fetched water yesterday. They \u2013 who _were_ 'they'? After six years of war everyone had become accustomed to following 'their' orders, what an obedient lot the English were \u2013 _they_ had set up a standpipe in the next street and Ursula had filled up a kettle and bucket from the tap. The woman ahead of her in the queue was terrifically smart in an enviable floor-length sable, silver-grey, and yet there she was, waiting patiently in the bitter cold with her buckets. She looked out of place in Soho but then who knew her story?\n\nThe women at the well. Ursula seemed to remember that Jesus had a particularly confrontational conversation with the woman at the well. A woman of Samaria \u2013 no name, of course. She had had five husbands, Ursula recalled, and was living with a man who wasn't her husband, but the King James Bible never said what had happened to those five. Perhaps she had poisoned the well.\n\nUrsula remembered Bridget telling them that when she was a girl in Ireland she had walked to a well every day to draw water. So much for progress. How quickly civilization could dissolve into its more ugly elements. Look at the Germans, the most cultured and well-mannered of people, and yet... Auschwitz, Treblinka, Bergen-Belsen. Given the same set of circumstances it could just as well have been the English, but that was something else you couldn't say. Miss Woolf had believed that, she'd said\u2014\n\n'I say,' the woman in sable said, interrupting her thoughts. 'Do you understand why my water is frozen solid and yet this isn't?' She had a cut-glass accent.\n\n'I don't know,' Ursula said. 'I know nothing.' The woman laughed and said, 'Oh, I feel the same way, believe me,' and Ursula thought that perhaps this was someone she would like as a friend but then a woman behind them said, 'Get a move on, love,' and the sable-furred woman hefted her buckets, as strapping as a Land Girl, and said, 'Well, must be off, cheerio.'\n\nShe turned on the wireless. Transmission of the Third Programme had been suspended for the duration. The war against the weather. You were lucky if you got the Home or the Light, there were so many electricity cuts. She needed noise, the sound of a familiar life. Jimmy had given her his old gramophone before he left, hers had been lost in Kensington along, sadly, with most of her records. She had managed to rescue a couple, miraculously unbroken, and placed one on the turntable now. 'I'd Rather Be Dead And Buried In My Grave'. Ursula laughed. 'Cheerful or what?' she said out loud. She listened to the scratch and hiss of the old record. Was that how she felt?\n\nShe glanced at the clock, Sylvie's little gold carriage clock. She had brought it home after the funeral. Four o'clock only. Ye gods, how the days dragged. She caught the pips, turned off the news. What was the point?\n\nShe had spent the afternoon trawling Oxford Street and Regent Street, for something to do \u2013 really it was just to get out of her monastic cell of a bedsit. All the shops were dim and dismal. Paraffin lamps in Swan and Edgar's, candles in Selfridge's \u2013 the drawn, shadowy faces of people like something from a painting by Goya. There was nothing to buy, or certainly nothing that she wanted, and anything she did want, like a lovely cosy-looking pair of fur-topped bootees, was outrageously expensive (fifteen guineas!). So depressing. 'Worse than the war,' Miss Fawcett at work said. She was leaving to get married, they had all clubbed together for her wedding present, a rather uninspiring vase, but Ursula wanted to get her something more personal, more special, but she couldn't think what and had hoped that the West End department stores might have just the thing. They didn't.\n\nShe'd gone into a Lyon's for a pale cup of tea, like lamb's water, Bridget would have said. And a utilitarian teacake, she counted just two hard dry raisins, and a scraping of margarine, and tried to imagine she was eating something wonderful \u2013 a luscious _Cremeschnitte_ or a slice of _Dobostorte_. She supposed the Germans weren't getting much in the way of pastries at the moment.\n\nShe murmured _Schwarzw\u00e4lder Kirschtorte_ accidentally out loud (such an extraordinary name, such an extraordinary cake) and attracted the unwanted attention of a neighbouring table, a woman stoically working her way through a large iced bun. 'Refugee, love?' she asked, surprising Ursula with her sympathetic tone.\n\n'Something like that,' Ursula said.\n\nWhile she was waiting for the egg to boil \u2013 the water still only lukewarm \u2013 she rooted among her books, never unpacked after Kensington. She found the Dante that Izzie had given her, nicely tooled red leather but the pages all foxed, a copy of Donne (her favourite), _The Waste Land_ (a rare first edition purloined from Izzie), a _Collected Shakespeare_ , her beloved metaphysical poets and, finally, at the bottom of the box, her battered school copy of Keats, with an inscription that read _To Ursula Todd_ , _for good work_. It would do for an epitaph too, she supposed. She flicked through the neglected pages until she found 'The Eve of St Agnes'.\n\nAh, bitter chill it was!\n\nThe owl, for all his feathers, was a-cold;\n\nThe hare limp'd trembling through the frozen grass,\n\nAnd silent was the flock in woolly fold.\n\nShe read out loud and the words made her shiver. She should read something warming, Keats and his bees \u2013 _For Summer has o'erbrimm'd their clammy cells_. Keats should have died on English soil. Asleep in an English garden on a summer's afternoon. Like Hugh.\n\nShe ate the egg while reading a copy of yesterday's _Times_ , given to her by Mr Hobbs in the post room when he had finished with it, a little daily ritual they had acquired. The paper's newly shrunk dimensions made it seem ridiculous somehow, as if the news itself was less important. Although really it was, wasn't it?\n\n*\n\nSnow like flakes of grey, soapy ash was falling outside the window. She thought of the Coles' relatives in Poland \u2013 rising above Auschwitz like a volcanic cloud, circling the Earth and blotting out the sun. Even now, after everything people had learned about the camps and so on, anti-Semitism was still rife. 'Jewboy' she'd heard someone being called yesterday, and when Miss Andrews ducked out of contributing to Miss Fawcett's wedding present Enid Barker had made a joke of it and said, 'What a Jew,' as if it were the mildest of insults.\n\nThe office was a tedious, rather irritable place these days \u2013 fatigue, probably, due to the cold and the lack of good, nourishing food. And the work was tedious, an endless compilation and permutation of statistics to be filed away in the archives somewhere \u2013 to be pored over by the historians of the future, she supposed. They were still 'clearing up and putting their house in order', as Maurice would have it, as if the casualties of war were clutter to be put away and forgotten. Civil Defence had been stood down for over a year and a half yet she still hadn't rid herself of the minutiae of bureaucracy. The mills of God (or the government) did indeed grind extremely small and slow.\n\nThe egg was delicious, it tasted as if it had been laid that very morning. She found an old postcard, a picture of the Brighton Pavilion (bought on a day trip with Crighton) that she'd never sent, and scrawled a thank you to Pammy \u2013 _Wonderful! Like a Red Cross package \u2013_ and propped it up on the mantelpiece next to Sylvie's clock. Next to Teddy's photo too. Teddy and his Halifax crew taken one sunlit afternoon. They were lounging in an assortment of old chairs. Forever young. The dog, Lucky, stood as proud as a little figurehead on Teddy's knee. How cheering it would be to still have Lucky. She had Teddy's DFC, propped up on the glass of the photo frame. Ursula had a medal too but it meant nothing to her.\n\nShe would put the postcard in with the afternoon post tomorrow. It would take an age to reach Fox Corner, she supposed.\n\nFive o'clock. She took her plate over to the sink to join the other unwashed dishes. The grey ash was a blizzard in the dark sky now and she pulled the flimsy cotton curtain to try to make it disappear. It tugged hopelessly on its wire and she gave up before she brought the whole thing down. The window was old and ill-fitting and let in a piercing draught.\n\nThe electricity went off and she fumbled for the candle on the mantelpiece. Could it get any worse? Ursula took the candle and the whisky bottle to bed, climbed under the covers still in her coat. She was so tired.\n\nThe flame on the little Radiant fire quivered alarmingly. Would it be so very bad? _To cease upon the midnight with no pain_. There were worse ways. Auschwitz, Treblinka. Teddy's Halifax going down in flames. The only way to stop the tears was to keep drinking the whisky. Good old Pammy. The flame on the Radiant flickered and died. The pilot light too. She wondered when the gas would come back on. If the smell would wake her, if she would get up and relight it. She hadn't expected to die like a fox frozen in its den. Pammy would see the postcard, know that she'd been appreciated. Ursula closed her eyes. She felt as though she had been awake for a hundred years and more. She really was so very, very tired.\n\nDarkness began to fall.\n\n# Snow\n\n# _11 February 1910_\n\nWarm and milky and new, the smell was a siren call to Queenie the cat. Queenie, strictly speaking, belonged to Mrs Glover, although she was aloofly unaware that she was anyone's possession. An enormous tortoiseshell, she had arrived on the doorstep with Mrs Glover, carried in a carpet bag, and had taken up residence in her own Windsor chair, a smaller version of Mrs Glover's, next to the big kitchen range. Having her own chair didn't stop her leaving her fur on every other available seat in the house, including the beds. Hugh, no great lover of cats, complained continually about the mysterious way that the 'mangy beast' managed to deposit its hairs on his suits.\n\nMore malevolent than most cats, she had a way of simply punching you, like a fighting hare, if you got close to her. Bridget, also no great cat-lover, declared the cat to be possessed by a demon.\n\nWhere was that delicious new scent coming from? Queenie padded up the stairs and into the big bedroom. The room was warmed by the embers of a hot fire. This was a good room, the thick, soft quilt on the bed and the gentle rhythms of sleeping bodies. And there \u2013 a perfect little cat-sized bed, already warmed by a perfect little cat-sized cushion. Queenie kneaded her paws on the soft flesh, carried suddenly back to kittenhood. She settled herself down more comfortably, a deep bass purr of happiness rumbling in her throat.\n\nSharp needles in the soft skin pricked her into consciousness. Pain was a new, unwelcome thing. But then suddenly she was muffled, her mouth full of something, stoppering her, suffocating her. The more she tried to breathe the less it became possible. She was pinned down, helpless, no breath. Falling, falling, a bird shot.\n\nQueenie had already purred herself into a pleasant oblivion when she was woken by a shriek and found herself being grabbed and thrown across the room. Growling and spitting, she backed out of the door, sensing this was a fight she would lose.\n\nNothing. Slack and still, the little ribcage not moving. Sylvie's own heart was knocking in her chest as if a fist was inside her, punching its way out. Such danger! Like a terrible thrill, a tide washing through her.\n\nInstinctively, she placed her mouth over the baby's face, covering the little mouth and nose. She blew gently. And again. And again.\n\nAnd the baby came back to life. It was that simple. ('I'm sure it was a coincidence,' Dr Fellowes said, when told of this medical miracle. 'It seems very unlikely that you could revive someone using that method.')\n\nBridget returned to the kitchen from upstairs where she had been delivering beef tea and reported faithfully to Mrs Glover, 'Mrs Todd says to tell Cook \u2013 that's you, Mrs Glover \u2013 that you have to get rid of the cat. That it would be better if you had it killed.'\n\n'Killed?' Mrs Glover said, outraged. The cat, now reinstated in her usual place by the stove, raised her head and stared balefully at Bridget.\n\n'I'm just telling you what she said.'\n\n'Over my dead body,' Mrs Glover said.\n\nMrs Haddock sipped a glass of hot rum, in what she hoped was a ladylike way. It was her third and she was beginning to glow from the inside out. She had been on her way to help deliver a baby when the snow had forced her to take refuge in the snug of the Blue Lion, outside Chalfont St Peter. It was not the kind of place she would ever have considered entering, except out of necessity, but there was a roaring fire in the snug and the company was proving surprisingly convivial. Horse brasses and copper jugs gleamed and twinkled. Visible from the snug, on the other side of the counter, was the public bar, where the drink seemed to flow particularly freely. It was an altogether rowdier place. A sing-song was currently in progress there and Mrs Haddock was surprised to find her toe tapping in accompaniment.\n\n'You should see the snow,' the landlord said, leaning across the great polished depth of the brass bar counter. 'We could all be stuck here for days.'\n\n'Days?'\n\n'You may as well have another tot of rum. You won't be going anywhere in a hurry tonight.'\n\n# Like a Fox in a Hole\n\n# _September 1923_\n\n'And so you don't see Dr Kellet at all now?' Izzie asked, snapping open her enamelled cigarette case and displaying a neat row of Black Russian cigarettes. 'Gasper?' she offered, holding out the case. Izzie addressed everyone as if they were the same age as herself. It was both seductive and lazy.\n\n'I'm thirteen years old,' Ursula said. Which as far as she could see answered both questions.\n\n'Thirteen is quite grown-up nowadays. And life can be very short, you know,' Izzie added, taking out a long ebony and ivory cigarette holder. She cast vaguely around the restaurant for a waiter to produce a light. 'I rather miss those little visits of yours to London. Chaperoning you to Harley Street and then on to the Savoy for tea. A treat for both of us.'\n\n'I haven't seen Dr Kellet for over a year,' Ursula said. 'I'm considered cured.'\n\n'Jolly good. I, on the other hand, am considered by _la famille_ to be incurable. You are, of course, a _jeune fille bien \u00e9lev\u00e9e_ and will never know what it is like to be the scapegoat for everyone else's sins.'\n\n'Oh, I don't know. I think I have an idea.'\n\nIt was Saturday lunchtime and they were in Simpson's. 'Ladies at leisure,' Izzie said, over great slices of bloody beef carved off the bone before their eyes. Millie's mother, Mrs Shawcross, was a vegetarian and Ursula imagined her horror at the sight of the great haunch of meat. Hugh called Mrs Shawcross (Roberta) 'a Bohemian', Mrs Glover called her mad.\n\nIzzie leaned towards the young waiter who had scurried over to light her cigarette. 'Thank you, darling,' she murmured, gazing directly up into his eyes in a way that made him grow suddenly as pink as the roast beef on her plate. ' _Le rosbif_ ,' she said to Ursula, dismissing the waiter with an indifferent flap of her hand. She was always peppering her conversation with French words ('I spent some time in Paris when I was younger. And, of course, the war...'). 'Do you speak French?'\n\n'Well, we do it at school,' Ursula said. 'But that doesn't mean I can speak it.'\n\n'You're a droll little thing, aren't you?' Izzie inhaled deeply on her cigarette holder and then puckered up her (astonishingly) red cupid's bow as if she were about to play the trumpet before exhaling a stream of smoke. Several men seated nearby turned to stare at her in fascination. She winked at Ursula. 'I bet the first French words you learned were _d\u00e9j\u00e0 vu_. Poor old thing. Maybe you were dropped on your head as a baby. I expect I was. Come on, let's tuck in, I'm ravenous, aren't you? I'm supposed to be banting but really there's only so much one can take,' Izzie said, cutting enthusiastically into the beef.\n\nThis was an improvement, when she had met Ursula on the platform at Marylebone Izzie had looked green and said she was 'a tad queasy' on account of the oysters and rum ('never a good combination') after a 'disreputable' night in a club in Jermyn Street. Now, oysters apparently forgotten, she was eating as if she were starving even though she claimed, as usual, to be 'watching her figure'. She also claimed to be 'stony broke' yet was wildly extravagant with her money. 'What's life worth if you can't have some fun?' she said. ('Her life is nothing but fun as far as I can see,' Hugh grumbled.)\n\nFun \u2013 and the concomitant treats \u2013 were necessary, Izzie claimed, to sweeten the fact that she had now 'joined the ranks of the workers', and had to 'pound away' on a typewriter to earn her keep. 'Goodness, you would think she was hewing coal,' Sylvie said crossly after a rare and rather embattled family luncheon at Fox Corner. After Izzie had gone, Sylvie banged down the Worcester fruit plates she was helping Bridget to clear and said, 'All she's doing is producing drivel, which is something she's been doing since she first learned to talk.'\n\n'Heirlooms,' Hugh murmured, rescuing the Worcester.\n\nIzzie had managed to get a job ('God knows how,' Hugh said) writing a weekly column for a newspaper \u2013 _Adventures of a ModernSpinster_, the column was called \u2013 on the subject of being a 'singleton'. 'Everyone knows that there simply aren't enough men to go round any more,' she said, tearing into a bread roll at Fox Corner's Regency Revival dining table. ('You don't seem to have any trouble finding them,' Hugh muttered.) 'The poor boys are all dead,' Izzie continued, ignoring him. Butter was plastered on to the roll with no regard for the hard labour of the cow. 'There's nothing can be done about it, we have to move on as best we can without them. The modern woman must fend for herself without the prospect of the succour of hearth and home. She must learn to be independent, emotionally, financially and, most importantly, in her _spirit_.' ('Rot.' Hugh again.) 'The men are not the only ones who had to sacrifice themselves in the Great War.' ('They're dead, you're not, that's the difference.' This from Sylvie. Coldly.)\n\n'Of course,' Izzie said, mindful of Mrs Glover at her elbow with a tureen of Brown Windsor, 'the women of the lower classes have always known what it is to work.' Mrs Glover gave her a baleful look and tightened her grip on the soup ladle. ('Brown Windsor, how delicious, Mrs Glover. What do you put in it to make it taste this way? Really? How _interesting_.') 'We're moving towards a classless society, of course,' a remark directed at Hugh but which earned a snort of derision from an unappeased Mrs Glover.\n\n'Are you a Bolshevik this week then?' Hugh asked.\n\n'We're all Bolsheviks now,' Izzie said blithely.\n\n'And at my table!' Hugh said and laughed.\n\n'She's such a fool,' Sylvie said when Izzie had finally departed for the station. 'And so much make-up! You would think she was on the stage. Of course, in her head she's always on the stage. She _is_ her own theatre.'\n\n'The hair,' Hugh said regretfully. It went without saying that Izzie had bobbed her hair before anyone else they knew. Hugh had expressly forbidden the women in his family to cut their hair. Almost as soon as he had issued this paternal edict the normally unrebellious Pamela had gone into town with Winnie Shawcross and the pair of them had returned shingled and shorn. ('It's just easier for games' was Pamela's rational explanation.) Pamela had saved her heavy plaits, whether as relics or trophies, it was hard to say. 'Mutiny in the ranks, eh?' Hugh said. Neither of them being the argumentative sort, that was the end of the conversation. The plaits now lived at the back of Pamela's underwear drawer. 'You never know, they might come in useful for something,' she said. No one in the family could imagine what that something might be.\n\nSylvie's feelings about Izzie went deeper than hair or make-up. She had never forgiven Izzie for the baby. He would be thirteen now, the same age as Ursula. 'A little Fritz or Hans,' she said. 'My own children's blood running through his veins. But, of course, the only thing of any interest to Izzie is Izzie.'\n\n'Still, she can't be entirely shallow,' Hugh said. 'I expect she saw some awful things in the war.' As if he hadn't.\n\nSylvie tossed her head. There might have been a halo of gnats around her own lovely hair. She was rather envious of Izzie's war, even the awfulness. 'She's still a fool,' she said and Hugh laughed and said, 'Yes, she is.'\n\nIzzie's column seemed for the most part to be nothing more than a diary of her own hectic personal life with the odd social comment thrown in. Last week it had been 'How high can they go?' and was about 'the rise of the emancipated female hemline', but consisted mostly of Izzie's tips to acquire the necessary shapely ankles. _Stand backwards, on tiptoe, on the bottom step of a staircase and let your heels drop over the edge_. Pamela practised all week on the attic staircase and declared no improvement at all.\n\nMuch against his will, Hugh felt it necessary to buy Izzie's newspaper every Friday and read it on the train home, 'just to keep an eye on what she's saying' (and then jettison the offending item on the hall table, from where Pamela was able to rescue it). Hugh harboured a particular horror that Izzie would write about _him_ and his only comfort was that she wrote under the pseudonym Delphine Fox, which was 'the silliest name' that Sylvie had ever heard. 'Well,' Hugh said, 'Delphine is her middle name, from her godmother. And Todd is an old word for fox, so I suppose there is some logic in it. Not that I'm defending her.'\n\n'But it's my _name_ , it's on my birth certificate,' Izzie said, looking hurt when attacked over the pre-prandial decanter. 'And it's from Delphi, you know, the oracle, and so on. So rather fitting, I would have said.' ('She's an oracle now?' from Sylvie. 'If she's an oracle then I'm the high priestess of Tutankhamun.')\n\nIzzie, in the person of Delphine, had already on more than one occasion mentioned 'my two nephews' ('Terrific rascals, both of them!') but had not cited any names. 'So far,' Hugh said darkly. She had made up a few 'amusing anecdotes' about these clearly fictional nephews. Maurice was eighteen (Izzie's 'sturdy little chaps' were nine and eleven), still away at boarding school and had spent no more than ten minutes in Izzie's company in as many years. As for Teddy, he tended to avoid situations that might evolve into anecdotes.\n\n'Who _are_ these boys?' Sylvie quizzed over Mrs Glover's surprisingly capricious interpretation of sole V\u00e9ronique. She had the folded newspaper on the table next to her and tapped Izzie's column with her forefinger as if it might be impregnated with germs. 'Are they supposed to be based in some way on Maurice and Teddy?'\n\n'What about Jimmy?' Teddy said to Izzie. 'Why don't you write about _him_?' Jimmy, perky in a sky-blue knitted jumper, was spooning mashed potato into his mouth and didn't look too bothered about being written out of great literature. He was a child of the peace, the war to end all wars had, after all, been fought for Jimmy. Yet again, Sylvie claimed to be taken by surprise by the newest addition to the family ('Four had seemed like the complete set'). Once, Sylvie had had no idea how children were started, now she seemed uncertain as to how you might stop them. ('Jimmy's an afterthought, I suppose,' Sylvie said.\n\n'I wasn't _thinking_ much,' Hugh said and they both laughed and Sylvie said, 'Really, Hugh.')\n\nJimmy's arrival had the effect of making Ursula feel as if she was being pushed further away from the heart of the family, like an object at the edge of an overcrowded table. A cuckoo, she had overheard Sylvie say to Hugh. _Ursula_ ' _s a bit of an awkward cuckoo_. But how could you be a cuckoo in your own nest? 'You are my real mother, aren't you?' she asked Sylvie and Sylvie laughed and said, 'Incontrovertibly, dear.'\n\n'The odd one out,' she said to Dr Kellet.\n\n'Well, there always has to be one,' he said.\n\n' _Don't_ write about my children, Isobel,' Sylvie said heatedly to Izzie.\n\n'They're _imaginary_ , for heaven's sake, Sylvie.'\n\n'Don't even write about my imaginary children.' She lifted the tablecloth and peered at the floor. 'What _are_ you doing with your feet?' she said testily to Pamela, who was sitting opposite her.\n\n'I'm making circles with my ankles,' Pamela said, unconcerned by Sylvie's irritability. Pamela was quite bold these days but also rather reasonable, a combination that seemed designed to annoy Sylvie. ('You are so like your father,' she had said to Pamela only this morning over some trifling difference of opinion. 'But why would that be a bad thing?' Pamela said.) Pamela wiped gluey potato from Jimmy's pink cheeks and said, 'Clockwise, then anti-clockwise. It's the way to a shapely ankle, according to Aunt Izzie.'\n\n'Izzie is not a person from whom anyone with any sense would take advice.' ('Excuse me?' Izzie said.) 'Besides which, you're too young for shapely ankles.'\n\n'Well,' Pamela said, 'I'm nearly the same age as you were when you married Daddy.'\n\n'Oh, splendid,' Hugh said, relieved at the sight of Mrs Glover waiting in the doorway to make a grand entrance with a _Riz imp\u00e9ratrice_. 'The ghost of Escoffier is at your back today, Mrs Glover.' Mrs Glover couldn't help but glance behind her.\n\n'Oh, splendid,' Izzie said. 'A cabinet pudding. You can rely on Simpson's for nursery food. We had a nursery, you know, it took up the whole top floor of the house.'\n\n'In Hampstead? Grandmama's house?'\n\n'The very same. I was the baby. Like Jimmy.' Izzie wilted a little, as if she were remembering some hitherto long-forgotten sadness. The ostrich feather on her hat trembled in sympathy. She revived at the sight of the silver sauce-boat of custard. 'And so you don't have those odd feelings any more? The _d\u00e9j\u00e0 vu_ and so on?'\n\n'Me?' Ursula said. 'No. Sometimes. Not so much, I suppose. It was before, you know. Now it's gone. Sort of.' Had it? She was never sure. Her memories seemed like a cascade of echoes. Could echoes cascade? Perhaps not. She had tried (and largely failed) to learn to be precise with language under Dr Kellet's guidance. She missed that cosy hour ( _t\u00eate-\u00e0-t\u00eate_ , he called it. More French) on a Thursday afternoon. She was ten years old when she first went to see him and had enjoyed being liberated from Fox Corner, in the company of someone who gave his full attention to her and only her. Sylvie, or more often than not Bridget, put Ursula on the train and she was met at the other end by Izzie even though both Sylvie and Hugh doubted that Izzie was sufficiently reliable to be in charge of a child. ('Expediency,' Izzie said to Hugh, 'generally trumps ethics, I've noticed. Personally, if I had a ten-year-old child I don't think I would feel entirely comfortable allowing it to travel all on its own.' 'You _do_ have a ten-year-old child,' Hugh pointed out. The little Fritz. 'Couldn't we try and find him?' Sylvie asked. 'Needle in a haystack,' Hugh said. 'The Hun are legion.')\n\n'So I rather miss seeing you,' Izzie said, 'which is why I asked if you could come up for the day. To be frank, I was surprised Sylvie agreed. There's always been a certain, shall we say, _froideur_ between your mother and myself. I, of course, am considered mad, bad and dangerous to know. Anyway I thought I should try to single you out from the herd, as it were. You remind me a little of me.' (Was that a good thing, Ursula wondered?) 'We could be special chums, what do you think? Pamela's a little dull,' Izzie continued. 'All that tennis and cycling, no wonder she has such sturdy ankles. _Tr\u00e8s sportive_ , I'm sure, but still. And science! No fun in that. And the boys are, well... boys, but you're interesting, Ursula. All that funny stuff in your head about knowing the future. Quite the little clairvoyant. Perhaps we should set you up in a gypsy caravan, get you a crystal ball, Tarot cards. _The drowned Phoenician sailor_ and all that. You can't see anything in my future, can you?'\n\n'No.'\n\n'Reincarnation,' Dr Kellet had said to her. 'Have you heard of that?' Ursula, aged ten, shook her head. She had heard of very little. Dr Kellet had a nice set of rooms in Harley Street. The one that he showed Ursula into was half panelled in mellow oak, with a thick carpet figured in red and blue on the floor and two large leather armchairs either side of a well-stoked coal fire. Dr Kellet himself wore a three-piece Harris tweed suit strung with a large gold fob watch. He smelt of cloves and pipe tobacco and had a twinkly look about him as if he were going to toast muffins or read a particularly good story to her, but instead he beamed at Ursula and said, 'So, I hear you tried to kill your maid?' (Oh, that's why I'm here, Ursula thought.)\n\nHe offered her tea which he brewed in something called a samovar in the corner of the room. 'Although I'm not Russian, far from it, I'm from Maidstone, I visited St Petersburg before the Revolution.' He was like Izzie in that he treated you as a grown-up, or at least he appeared to, but that was where the resemblance ended. The tea was black and bitter and only drinkable with the aid of heaps of sugar and the contents of the tin of Huntley and Palmer's Marie biscuits that sat between them on a little table.\n\nHe had trained in Vienna ('where else?') but trod, he said, his own path. He was no one's disciple, he said, although he had studied 'at the feet of all of the teachers. One must nose forward,' he said. 'Nudge one's way through the chaos of our thoughts. Unite the divided self.' Ursula had no idea what he was talking about.\n\n'The maid? You pushed her down the stairs?' It seemed a very direct question for someone who talked about nosing and nudging.\n\n'It was an accident.' She didn't think of Bridget as 'the maid', she thought of her as Bridget. And it was ages ago now.\n\n'Your mother is worried about you.'\n\n'I just want you to be happy, darling,' Sylvie said after she had made the appointment with Dr Kellet.\n\n'Aren't I happy?' Ursula puzzled.\n\n'What do you think?'\n\nUrsula didn't know. She wasn't sure that she had a yardstick against which to measure happiness or unhappiness. She had obscure memories of elation, of falling into darkness, but they belonged to that world of shadows and dreams that was ever-present and yet almost impossible to pin down.\n\n'As if there is another world?' Dr Kellet said.\n\n'Yes. But it's this one as well.'\n\n('I know she says the oddest things, but a _psychiatrist_?' Hugh said to Sylvie. He frowned. 'She's only small. She's not _defective_.'\n\n'Of course not. She just needs a little fixing.')\n\n'And, hey presto, you're fixed! How marvellous,' Izzie said. 'He was an odd little bod, that mind doctor, wasn't he? Shall we essay the cheese board \u2013 the Stilton's so ripe it looks as if it's about to walk away of its own accord \u2013 or shall we tootle off and go to mine?'\n\n'I'm stuffed,' Ursula said.\n\n'Me too. Tootle off it is then. Shall I pick up the bill?'\n\n'I have no money. I'm thirteen,' Ursula reminded her.\n\nThey left the restaurant and, to Ursula's astonishment, Izzie sauntered a few yards up the Strand and climbed into the driver's seat of a gleaming open-top car, parked, rather carelessly, outside the Coal Hole. 'You have a car!' Ursula exclaimed.\n\n'Good, isn't it? Not _exactly_ paid for. Hop in. A Sunbeam, sports model. Certainly beats driving an ambulance. Wonderful in this weather. Shall we take the scenic route, go along the Embankment?'\n\n'Yes, please.'\n\n'Ah, the Thames,' Izzie said when the river came into view. 'The nymphs, sadly, are all departed.' It was a lovely late-September afternoon, crisp as an apple. 'London's glorious, isn't it?' Izzie said. She drove as if she were on the circuit at Brooklands. It was both terrifying and exhilarating. Ursula supposed that if Izzie had managed to drive throughout the war unscathed then they would probably make it along the Victoria Embankment without coming to grief.\n\nAs they approached Westminster Bridge they had to slow down on account of the crowds of people whose flow had been interrupted by a largely silent demonstration of unemployed men. _I fought overseas_ , a placard held aloft read. Another proclaimed _Hungry and wanting to work_. 'They're so meek,' Izzie said dismissively. 'There'll never be a revolution in this country. Not another one at any rate. We chopped the head off a king once and felt so guilty about it that we've been trying to make up for it ever since.' A shabby-looking man came up alongside the car and shouted something incomprehensible at Izzie, although the meaning was clear.\n\n' _Qu'ils mangent de la brioche_ ,' Izzie murmured. 'You know she never said that, don't you? Marie-Antoinette? She's a rather maligned figure in history. You must never believe everything they say about a person. Generally speaking, most of it will be lies, half-truths at best.' It was hard to figure out whether Izzie was a royalist or a republican. 'Best not to adhere too closely to one side or the other really,' she said.\n\nBig Ben tolled a solemn three o'clock as the Sunbeam pushed its way through the throng. ' _Si lunga tratta di gente, ch'io non avrei mai creduto che morte tanta n'avesse disfatta_. Have you read Dante? You should. He's very good.' How did Izzie know so much? 'Oh,' she said airily. 'Finishing school. And I spent some time in Italy after the war. I took a lover, of course. An impoverished count, it's more or less _de rigueur_ when you're over there. Are you shocked?'\n\n'No.' She was. Ursula wasn't surprised there was a _froideur_ between her mother and Izzie.\n\n'Reincarnation is at the heart of Buddhist philosophy,' Dr Kellet would say, sucking on his meerschaum pipe. All conversations with Dr Kellet were punctuated by this object, whether by gesture \u2013 a great deal of pointing with both mouthpiece and Turk's-head bowl (fascinating in itself) \u2013 or the necessary ritual of emptying, filling, tamping, lighting and so on. 'Have you heard of Buddhism?' She hadn't.\n\n'How old are you?'\n\n'Ten.'\n\n'Still quite new. Perhaps you're remembering another life. Of course, the disciples of the Buddha don't believe that you keep coming back as the _same_ person in the _same_ circumstances, as you feel you do. You move on, up or down, sideways occasionally, I expect. Nirvana is the goal. Non-being, as it were.' At ten it seemed to Ursula that perhaps _being_ should be the goal. 'Most ancient religions,' he continued, 'adhered to an idea of circularity \u2013 the snake with its tail in its mouth, and so on.'\n\n'I've been confirmed,' she said, trying to be helpful. 'Church of England.'\n\nDr Kellet had come to Sylvie recommended by Mrs Shawcross via Major Shawcross, their next-door neighbour. Kellet had done a lot of good work, the major said, with men who 'needed help' after they returned from the war (there was a suggestion that the major himself had 'needed help'). Ursula's path crossed occasionally with some of these other patients. Once there was a dejected young man who stared at the carpet in the waiting room speaking quietly to himself, another who tapped his foot restlessly in time to something only he could hear. Dr Kellet's receptionist, Mrs Duckworth, who was a war widow and had been a nurse during the war, was always very nice to Ursula, offering her peppermints and asking her about her family. One day a man blundered into the waiting room, although the doorbell downstairs had never rung. He looked bewildered and a little wild but he just stood stock-still in the middle of the room, staring at Ursula as if he'd never seen a child before, until Mrs Duckworth led him to a chair and sat down next to him and then put her arm round him and said, 'Now, now, Billy what is it?' the way a nice mother would have done and Billy laid his head on her chest and began to sob.\n\nIf Teddy ever cried when he was younger, Ursula could never bear it. It seemed to open up a chasm inside, something deep and dreadful and full of sorrow. All she ever wanted was to make sure he never felt like crying again. The man in Dr Kellet's waiting room had the same effect on her. ('That's how motherhood feels every day,' Sylvie said.)\n\nDr Kellet came out of his room at that moment and said, 'Come along, Ursula, I'll see to Billy later,' but when Ursula finished her appointment Billy was no longer in the waiting room. 'Poor man,' Mrs Duckworth said sadly.\n\nThe war, Dr Kellet said to Ursula, had made many people search for meaning in new places \u2013 'Theosophy, Rosicrucianism, anthroposophy, spiritualism. Everyone needs to make sense of their loss.' Dr Kellet himself had sacrificed a son, Guy, a captain in the Royal West Surreys, lost at Arras. 'One must hold on to the idea of sacrifice, Ursula. It can be a higher calling.' He showed her a photograph, not one taken in uniform, just a snapshot really, of a boy in cricket whites, standing proudly behind his bat. 'Could have played for the county,' Dr Kellet said sadly. 'I like to think of him \u2013 of all of them \u2013 playing a never-ending game in heaven. A perfect afternoon in June, always just before they break for tea.'\n\nIt seemed a shame for all the young men never to have their tea. Bosun was in heaven, along with Sam Wellington, the old boot, and Clarence Dodds, who had died with astonishing speed of the Spanish flu the day after the Armistice. Ursula couldn't imagine any of them playing cricket.\n\n'Of course, I don't believe in God,' Dr Kellet said. 'But I believe in heaven. One has to,' he added, rather bleakly. Ursula wondered how all of this was supposed to fix her.\n\n'From a more scientific point of view,' he said, 'perhaps the part of your brain responsible for memory has a little flaw, a neurological problem that leads you to think that you are repeating experiences. As if something had got stuck.' She wasn't really dying and being reborn, he said, she just _thought_ she was. Ursula couldn't see what the difference was. _Was_ she stuck? And if so, where?\n\n'But we don't want it to result in you killing the poor servants, do we?'\n\n'But it was such a long time ago,' Ursula said. 'It's not as if I've tried to kill anyone since.'\n\n'Down in the dumps,' Sylvie said at their first meeting with Dr Kellet, the only time she had been to the Harley Street rooms with Ursula although she had clearly already talked to him _without_ Ursula. Ursula wondered very much what had been said about her. 'And she's rather forlorn all the time,' Sylvie continued. 'I can understand an adult feeling like that\u2014'\n\n'Can you?' Dr Kellet said, leaning forward, the meerschaum indicating interest. 'Do _you_?'\n\n'I'm not the problem,' Sylvie said with her most gracious smile.\n\nI'm a problem, Ursula thought? And anyway she hadn't been _killing_ Bridget, she was _saving_ her. And if she wasn't saving her perhaps she was sacrificing her. Hadn't Dr Kellet himself said sacrifice was a higher calling?\n\n'If I were you I would stick to traditional moral guidelines,' he said. 'Fate isn't in your hands. That would be a very heavy burden for a little girl.' He got up from his chair and put another shovel of coal on the fire.\n\n'There are some Buddhist philosophers (a branch referred to as Zen) who say that sometimes a bad thing happens to prevent a worse thing happening,' Dr Kellet said. 'But, of course, there are some situations where it's impossible to imagine anything worse.' Ursula supposed he was thinking of Guy, _lost at Arras_ and then denied his tea and cucumber sandwiches for eternity.\n\n'Try this,' Izzie said, squirting a perfume atomizer in Ursula's direction. 'Chanel Number 5. It's quite the thing. _She's_ quite the thing. _Her strange, synthetic perfumes_.' She laughed as if she had made a great joke and sprayed another invisible cloud around the bathroom. It was quite different from the flowery scents that Sylvie anointed herself with.\n\nThey had finally arrived at Izzie's flat in Basil Street ('rather a dull _endroit_ but handy for Harrods'). Izzie's bathroom was pink and black marble ('I designed it myself, delicious, isn't it?') and was all sharp lines and hard corners. Ursula hated to think what would happen if you slipped and fell in here.\n\nEverything in the flat seemed to be new and shiny. It was nothing like Fox Corner, where the slow-seeming tick of the grandfather clock in the hall counted time and the patina of years shone on the parquet floors. The Meissen figures with their missing fingers and chipped toes, the Staffordshire dogs with accidentally lopped-off ears, bore no resemblance to the Bakelite bookends and onyx ashtrays in Izzie's rooms. In Basil Street everything looked so new it seemed to belong in a shop. Even the books were new, novels and volumes of essays and poetry by writers Ursula had never heard of. 'One must keep up with the times,' Izzie said.\n\nUrsula regarded herself in the bathroom mirror. Izzie stood behind her, Mephistopheles to her Faustus, and said, 'Goodness, you're turning out to be quite pretty,' before rearranging her hair into different styles. 'You must have it cut,' she said, 'you should come to my _coiffeur_. He's really very good. You're in danger of looking like a milkmaid, when really I think you're going to turn out to be deliciously wicked.'\n\nIzzie danced around the bedroom singing _I wish I could shimmy like my sister Kate_. 'Can you shimmy? Look, it's easy.' It wasn't and they collapsed in laughter on the satin eiderdown of the bed. 'Gort to 'ave fun, 'aven't yew?' Izzie said in an atrocious mock-Cockney accent. The bedroom was a terrible mess, clothes everywhere, satin petticoats, _cr\u00eape de Chine_ nightdresses, silk stockings, partnerless shoes lying abandoned on the carpet, a dusting of Coty powder over everything. 'You can try things on if you want,' Izzie said carelessly. 'Although you're rather small compared to me. _Jolie et petite_.' Ursula declined, fearing enchantment. They were the kind of clothes that might turn you into someone else.\n\n'What shall we do?' Izzie said, suddenly bored. 'We could play cards? Bezique?' She danced through to the living room and tripped her way towards a large shining chrome object that looked as if it belonged on the bridge of an ocean liner and turned out to be a cocktail cabinet. 'A drink?' She looked doubtfully at Ursula. 'No, don't tell me, you're only thirteen.' She sighed, lit a cigarette and looked at the clock. 'We're too late to catch a matinee, too early for an evening performance. _London Calling!_ is on at the Duke of York's, it's supposed to be very amusing. We could go, you could get a later train home.'\n\nUrsula fingered the keys on the Royal typewriter that sat on a desk at the window. 'My trade,' Izzie said. 'Perhaps I should put you in this week's column.'\n\n'Really? What would you say?'\n\n'I don't know, make something up, I expect,' she said. 'That's what writers do.' She took out a record from the cabinet of the gramophone and put it on the turntable. 'Listen to this,' she said. 'You've never heard anything like it.'\n\nIt was true, she hadn't. It started with a piano, but nothing like the Chopin and Liszt that Sylvie played so nicely (and Pamela in such a pedestrian fashion).\n\n'They call it honky-tonk, I believe,' Izzie said. A woman began to sing, raw and American. She sounded as if she had spent her life in a prison cell. 'Ida Cox,' Izzie said. 'She's a Negress. Isn't she extraordinary?'\n\nShe was.\n\n'Singing about how wretched it is to be a woman,' Izzie said, lighting up another cigarette and sucking hard. 'If only one could find someone really filthy rich to marry. _A large income is the best recipe for happiness I ever heard of_. Do you know who said that? No? Well you should.' She was suddenly irritable, a not completely domesticated animal. The phone rang and she said, 'Saved by the bell,' and proceeded to have a feverishly animated conversation with the unseen, unheard caller. She ended the call by saying, 'That would be delish, darling, meet you in half an hour.' And to Ursula, 'I would offer you a lift but I'm going to Claridge's and it's simply _miles_ from Marylebone and after that I have a party to go to in Lowndes Square so I can't possibly see you to the station. You can Tube it to Marylebone, can't you? You know how? The Piccadilly line to Piccadilly Circus and then change to the Bakerloo to Marylebone. Come on, I'll walk out with you.'\n\nWhen they reached the street Izzie breathed deeply as if she'd been released from unwanted confinement. 'Ah, twilight,' she said. 'The violet hour. Lovely, isn't it?' She kissed Ursula on the cheek and said, 'It was marvellous seeing you, we have to do this again. Are you all right from here? _Tout droit_ on to Sloane Street, turn left and Bob's your uncle, there's Knightsbridge Tube station. Toodle-oo then.'\n\n' _Amor fati_ ,' Dr Kellet said, 'have you heard of that?' It sounded like he had said, 'A more fatty.' Ursula was puzzled \u2013 both herself and Dr Kellet were on the lean side. Nietzsche ('a philosopher'), he said, was drawn to it. 'A simple acceptance of what comes to us, regarding it as neither bad nor good.'\n\n' _Werde, der du bist_ , as he would have it,' Dr Kellet continued, knocking the ashes from his pipe on to the hearth from where Ursula supposed someone else would sweep them up. 'Do you know what that means?' Ursula wondered how many ten-year-old girls Dr Kellet had actually encountered before. 'It means become who you are,' he said, adding more shreds of tobacco to the meerschaum. (The being before the non-being, Ursula supposed.) 'Nietzsche got that from Pindar.   Do you know Greek?' He had quite lost her now. 'It means \u2013 become such as you are, having learned what that is.'\n\nUrsula thought he said 'from Pinner', which was where Hugh's old nanny had retired to, living with her sister above a shop in an old building on the high street. Hugh had driven Ursula and Teddy out there in his splendid Bentley one Sunday afternoon. Nanny Mills was rather frightening (although not to Hugh apparently), spending a lot of time quizzing Ursula about her manners and inspecting Teddy's ears for dirt. Her sister was nicer and plied them with glasses of elderflower cordial and slices of milk fadge spread with blackberry jelly. 'How is Isobel?' Nanny Mills asked, her mouth set like a prune. 'Izzie is Izzie,' Hugh said, which if you repeated it very quickly, as Teddy did later, sounded like a small swarm of wasps. Izzie, apparently, had become herself a long time ago.\n\nIt seemed unlikely that Nietzsche had obtained anything from Pinner, least of all his beliefs.\n\n'Nice time with Izzie?' Hugh asked when he picked her up from the station. There was something reassuring about the sight of Hugh in his grey homburg and long dark-blue wool overcoat. He scrutinized her for any visible change. She thought it best not to tell him that she had taken the Tube on her own. It had been a terrifying adventure, a dark night in the forest, but one which, like any good heroine, she had survived. Ursula shrugged. 'We went to Simpson's for lunch.'\n\n'Hm,' Hugh said as if trying to decipher a meaning from this.\n\n'We listened to a Negress singing.'\n\n'In Simpson's?' Hugh puzzled.\n\n'On Izzie's gramophone.'\n\n'Hm,' again. He opened the car door for her and she settled into the lovely leathery seat of the Bentley, almost as reassuring as Hugh himself. Sylvie regarded the car as 'ruinously' extravagant. It _was_ breathtakingly expensive. The war had made Sylvie parsimonious: slivers of soap were collected and boiled down for the laundry, sheets turned side to middle, hats refurbished. 'We would live on eggs and chickens if she had her way,' Hugh laughed. He, on the other hand, had become less prudent since the war, 'perhaps not the best trait for a banker to develop', Sylvie said. ' _Carpe diem_ ,' Hugh said and Sylvie said, 'You were never one for seizing.'\n\n'Izzie has a car now,' Ursula offered.\n\n'Does she?' Hugh said. 'I'm sure it's not as splendid as this beast.' He patted the dashboard of the Bentley fondly. As they drove away from the station he said quietly, 'She's not to be trusted.'\n\n'Who?' (Mother? The car?)\n\n'Izzie.'\n\n'No, you're probably right,' Ursula agreed.\n\n'How did you find her?'\n\n'Oh, you know. Incurable. Izzie is Izzie, after all.'\n\nWhen they returned to the house they found Teddy and Jimmy playing a tidy game of dominos on the table in the morning room while Pamela was next door with Gertie Shawcross. Winnie was slightly older than Pamela and Gertie slightly younger and Pamela divided her time equally between them but rarely both at the same time. Ursula, devoted to Millie, found it an odd arrangement. Teddy loved all the Shawcross girls but his heart was in Nancy's small hands.\n\nOf Sylvie there was no sign. 'Don't know,' Bridget said, rather indifferently, when Hugh enquired.\n\nMrs Glover had left them a rather utilitarian mutton stew keeping warm in the range. Mrs Glover no longer lived with them at Fox Corner. She rented a little house in the village so that she could look after George as well as them. George hardly ever left the house. Bridget referred to him as a 'poor soul' and it was hard to disagree with that description. If it was good weather (or even not particularly good weather at all) he sat in a big ugly bath-chair at the front door and watched the world pass him by. His handsome head ('Leonine, once,' Sylvie said sadly) hung down on his chest and a long thread of drool dangled from his mouth. 'Poor devil,' Hugh said. 'Better off if he'd been killed.'\n\nSometimes one or other of them tagged along when Sylvie \u2013 or a more reluctant Bridget \u2013 visited him during the day. It seemed odd that they would go to his home to see him while his own mother stayed in their home looking after them. Sylvie would fuss with the blanket across his legs and fetch him a glass of beer and then wipe his mouth the way you did with Jimmy.\n\nThere were other war veterans in the neighbourhood, visible thanks to their limps or missing limbs. All those unclaimed arms and legs lost in the fields of Flanders \u2013 Ursula imagined them pushing roots down into the mud and shoots up to the sky and growing once again into men. An army of men marching back for revenge. ('Ursula has morbid thoughts,' she heard Sylvie say to Hugh. Ursula had become a great eavesdropper, it was the only way to find out what people were really thinking. She didn't hear Hugh's answer as Bridget came crashing into the room in a fury because the cat \u2013 Hattie, one of Queenie's offspring, possessed of the same character as her mother \u2013 had stolen the poached salmon that was to have been their lunch.)\n\nThere were those, too, who, like the men in Dr Kellet's waiting room, had less visible injuries. There was an ex-soldier in the village called Charles Chorley who had served with the Buffs and had come through the war without a scratch and then one day in the spring of 1921 he had stabbed his wife and three children where they lay sleeping in their beds and then shot himself in the head with a Mauser he had taken from a German soldier he had killed at Bapaume. ('Terrible mess,' Dr Fellowes reported. 'These chaps should think about the people who have to clean up afterwards.')\n\nBridget, of course, had her 'own cross to bear', having lost Clarence. Like Izzie, Bridget was resigned to spinsterhood although she embraced it in a less giddy fashion. They had all attended Clarence's funeral, even Hugh. Mrs Dodds had been her usual restrained self and had flinched when Sylvie placed a comforting hand on her arm, but after they had shuffled away from the gaping hole of the grave (not a thing of beauty, not at all) Mrs Dodds said to Ursula, 'Part of him died during the war. This was just the rest of him catching up,' and she put her finger to the corner of her eye and dabbed at a trace of moisture there \u2013 a tear would have been too generous a description. Ursula didn't know why she had been chosen for this confidence, possibly simply because she was the nearest person. Certainly no response was expected, or received.\n\n'Ironic, one might say,' Sylvie said, 'for Clarence to have survived the war and to die of an illness.' ('What would I have done if one of you had caught the influenza?' she often said.)\n\nUrsula and Pamela had spent a considerable amount of time discussing whether Clarence had been buried with his mask on or off. (And if off, where might it be now?) They didn't feel it was the kind of thing that they could ask Bridget. Bridget said bitterly that Old Mrs Dodds had finally got her son to herself and stopped another woman taking him away from her. ('A little harsh, perhaps,' Hugh murmured.) Clarence's photograph, a print of the one taken for his mother, before Bridget knew him, before he marched off to his destiny, had now joined that of Sam Wellington in the shed. 'The endless ranks of the dead,' Sylvie said angrily. 'Everyone wants to forget them.'\n\n'Well I certainly do,' Hugh said.\n\nSylvie returned in time for Mrs Glover's apple charlotte. Their own apples \u2013 a small orchard that Sylvie had planted at the end of the war was beginning to bear fruit. When Hugh wondered where she had been she said something indistinct about Gerrards Cross. She sat at the dining table and said, 'I'm not really terribly hungry.'\n\nHugh caught her eye and, nodding in Ursula's direction, said, 'Izzie.' An exquisite shorthand communication.\n\nUrsula had expected an inquisition but all Sylvie said was 'Good lord, I had quite forgotten that you had been to London. You've returned in one piece, I'm glad to see.'\n\n'Untainted,' Ursula said brightly. 'Do you, by the way, know who it was who said, _A large income is the best recipe for happiness I ever heard of_?' Sylvie's knowledge, like Izzie's, was random yet far-ranging, 'the sign that one has acquired one's learning from novels, rather than an education', according to Sylvie.\n\n'Austen,' Sylvie said promptly. ' _Mansfield Park_. She puts the words in Mary Crawford's mouth, for whom she professes disdain, of course, but really I expect dear Aunt Jane rather believed those words. Why?'\n\nUrsula shrugged. 'Nothing.'\n\n' _Till I came to Mansfield, I had not imagined a country parson ever aspired to a shrubbery, or anything of the kind_. Wonderful stuff. I always think the word shrubbery denotes a certain kind of person.'\n\n'We have a shrubbery,' Hugh said but Sylvie ignored him and continued to Ursula, 'You really should read Jane Austen. You're surely the right age by now.' Sylvie seemed quite gay, a mood somehow at odds with the mutton that was still sitting on the table in its dull brown pot, little ponds of white fat congealing on the surface. 'Really,' Sylvie said sharply, turning suddenly like the weather. 'Standards are falling everywhere, even in one's own home.' Hugh raised his eyebrows and before Sylvie had a chance to call on Bridget he got up from the table and took the stew-pot back to the kitchen himself. Their little maid-of-all-work, Marjorie, no longer so little, had recently decamped and Bridget and Mrs Glover were left to shoulder the burden of looking after them. ('It's not as if we're demanding in any way,' Sylvie said crossly when Bridget mentioned that she hadn't had a pay rise since the end of the war. 'She should be grateful.')\n\nIn bed that night \u2013 Ursula and Pamela still shared the cramped quarters of the attic bedroom ('like prisoners in a cell' according to Teddy) \u2013 Pamela said, 'Why didn't she invite me as well as you, or even instead of you?' This, being Pamela, was said with genuine curiosity rather than malice.\n\n'She thinks I'm interesting.'\n\nPamela laughed and said, 'She thinks Mrs Glover's Brown Windsor is _interesting_.'\n\n'I know. I'm not flattered.'\n\n'It's because you're pretty and clever,' Pamela said, 'while I am merely clever.'\n\n'That's not true and you know it,' Ursula said, hotly defensive of Pamela.\n\n'I don't mind.'\n\n'She says she'll put me in her newspaper next week but I don't suppose she will.'\n\nUrsula, in her account to Pamela of the day's adventures in London, had omitted a scene she had witnessed, unseen by Izzie, who had been preoccupied with turning the car round in the middle of the road outside the Coal Hole. A woman wearing a mink coat had come out of the entrance to the Savoy, on the arm of a rather elegant man. The woman was laughing in a carefree way at something the man had just said but then she broke away from his arm to search in her handbag for her purse in order to drop a handful of coins into the bowl of an ex-soldier who was sitting on the pavement. The man had no legs and was perched on some kind of makeshift wooden trolley. Ursula had seen another limbless man on a similar contraption outside Marylebone station. Indeed, the more she had looked on the London streets, the more amputees she had seen.\n\nA doorman from the hotel darted out of Savoy Court and advanced on the legless man, who quickly scooted away using his hands as oars on the pavement. The woman who had given him money remonstrated with the doorman \u2013 Ursula could make out her handsome, impatient features \u2013 but then the elegant man took her gently by the elbow and guided her away up the Strand. The remarkable thing about this scene was not the content but the characters. Ursula had never seen the elegant man before but the agitated woman was \u2013 quite unmistakably \u2013 Sylvie. If she hadn't recognized Sylvie, she would have recognized the mink, given to her by Hugh for their tenth wedding anniversary. She seemed a long way from Gerrards Cross.\n\n'Well,' Izzie said when the car was finally facing the right way, 'that was a tricky manoeuvre!'\n\n*\n\nWhen it came to the next week Ursula was indeed absent from Izzie's column, even in fictional form. She had written instead about the freedom that the single woman could obtain from ownership of 'a little car'. 'The joys of the open road far surpass being trapped on a filthy omnibus or being followed down a dark street by a stranger. One has no need to glance nervously over one's shoulder at the wheel of a Sunbeam.'\n\n'I say, that's grim,' Pamela said. 'Do you think she has? Been followed down a street by a stranger?'\n\n'Lots of times, I expect.'\n\nUrsula was not called upon again to be Izzie's 'special chum', indeed none of them heard from her again until she turned up on the doorstep on Christmas Eve (invited but not expected) and declared herself to be 'in a bit of a jam', a state which necessitated her being closeted in the growlery with Hugh, to emerge an hour later looking almost chastened. She had brought no presents with her and smoked throughout Christmas dinner, picking listlessly at her food. 'Annual income twenty pounds,' Hugh said when Bridget brought the brandy-soaked pudding to the table. 'Annual expenditure twenty pounds ought and six, result misery.'\n\n'Oh, do shut up,' Izzie said and flounced off before Teddy could put a match to the pudding.\n\n'Dickens,' Sylvie said to Ursula.\n\n' _J'\u00e9tais un peu d\u00e9rang\u00e9e_ ,' Izzie said to Ursula, rather contritely, next morning by way of explanation.\n\n'Silly of me, really,' Izzie said. 'I got in a bit of a muddle.'\n\nIn the new year the Sunbeam disappeared and the Basil Street address was exchanged for a less salubrious one in Swiss Cottage (an even duller _endroit_ ) but nonetheless Izzie remained undeniably Izzie.\n\n# _December 1923_\n\nJimmy had a cold so Pammy said she would stay at home with him and make decorations from silver milk-bottle tops while Ursula and Teddy tramped along the lane in search of holly. Holly was abundant in the copse but the copse was further away and the weather was so wretched that they wanted to be outside for as little time as possible. Mrs Glover, Bridget and Sylvie were confined to the kitchen, caught up in the afternoon drama of Christmas cooking.\n\n'Don't pick any branches without berries,' Pamela instructed as they left the house. 'And don't forget to look for some mistletoe as well.'\n\nThey went prepared with pruning shears and a pair of Sylvie's leather gardening gloves, having learned the painful lesson of previous Christmas foraging expeditions. They had their sights set on the big holly tree in the field at the far end of the lane, having been deprived of the handy holly hedge in the garden, which had been replaced by a more biddable privet after the war. The whole neighbourhood was tamer and more suburban. Sylvie said it would not be long before the village had spread so much that they would be surrounded by houses. 'People must live somewhere,' Hugh said reasonably. 'But not here,' Sylvie said.\n\nIt was unpleasantly windy and spitting with rain and Ursula would have much preferred to stay by the fire in the morning room with the festive promise of Mrs Glover's mince pies scenting the whole house. Even Teddy, usually the one to find a silver lining, trudged disconsolately along beside her, hunched against the weather, a small, stalwart Knight Templar in his knitted grey balaclava. 'This is beastly,' he said. Only Trixie relished the outing, ferreting in the hedgerows and delving in the ditch as if she had been sent on a mission to unearth treasure. She was a noisy dog, much given to barking for reasons apparent only to herself, so when, way ahead of them in the lane, she began to yap deliriously they took little notice.\n\nTrixie had quietened down a bit by the time they caught up with her. She was standing sentinel over her prize, and Teddy said, 'Something dead, I expect.' Trixie was particularly skilled at truffling out half-rotted birds and the desiccated corpses of larger mammals. 'A rat or a vole, probably,' Teddy said. And then an eloquent 'Oh,' when he saw the true nature of the trove in the ditch.\n\n'I'll stay here,' Ursula said to Teddy, 'and you run back to the house and fetch someone,' but then as she watched his vulnerable little figure setting off, running alone along the deserted lane, the early winter dark already closing in around him, she had shouted at him to wait for her. Who knew what terror lay in wait? For Teddy, for all of them.\n\nThere was confusion as to what to do with the body over the holiday and eventually it was decided to keep it in the ice house at Ettringham Hall until after Christmas.\n\nDr Fellowes, who had arrived along with a police constable, said the child had died of unnatural causes. A girl, eight or nine years old; her second set of front teeth had grown in although they had been knocked out before death. There were no little girls reported missing, the police said, certainly not locally. They speculated she might be a gypsy, although Ursula thought that gypsies _took_ children, rather than left them behind.\n\nIt was almost New Year before a reluctant Lady Daunt was willing to give her up. When they removed her from the ice house they found her decorated like a relic \u2013 flowers and little tokens on her body, her skin bathed and her hair brushed and beribboned. As well as their three sons sacrificed to the Great War, the Daunts had also once had a girl, dead in infancy, and her custody of the little corpse had caused Lady Daunt to revisit her old grief and she had gone out of her mind for a while. She wanted to bury the girl in the grounds of the Hall but there was a rebellious murmuring from the villagers who insisted that she be buried in the churchyard, 'Not hidden away as Lady Daunt's pet,' someone said. A strange kind of pet, Ursula thought.\n\nNeither her identity nor that of her murderer was ever discovered. The police questioned everyone in the neighbourhood. They had come to Fox Corner one evening and Pamela and Ursula had almost hung themselves from the banisters in an attempt to hear what was said. From this eavesdropping they learned that no one in the village was a suspect and that 'terrible things' had been done to the child.\n\nIn the end she was buried on the last day of the old year but not before the vicar had christened her, as the general feeling was that although the girl was determined to remain an enigma she should not be buried without a name. No one seemed to know how 'Angela' was arrived at but it seemed appropriate. Nearly the whole village turned out for the funeral and many wept more heartily for Angela than they had ever done for their own flesh and blood. There was sadness rather than fear and Pamela and Ursula often discussed why it was, exactly, that everyone they knew was regarded as innocent.\n\nLady Daunt was not the only one to be strangely affected by the murder. Sylvie was particularly disturbed, more by anger, it seemed, than sadness. 'It's not,' she fumed, 'that she was killed, although heaven knows that's terrible enough, it's that _no one missed her_.'\n\nTeddy had nightmares for weeks afterwards, creeping into bed beside Ursula in the dead of night. They would for ever be the ones who found her, the ones who had seen the little shoeless, sockless foot \u2013 bruised and grubby, poking out from the dead branches of an elm, her body shrouded with a cold coverlet of leaves.\n\n# _11 February 1926_\n\n'Sweet sixteen,' Hugh said, kissing her affectionately. 'Happy birthday, little bear. Your future's all ahead of you.' Ursula still harboured the feeling that some of her future was also behind her but she had learned not to voice such things. They were to have gone up to London for afternoon tea at the Berkeley (it was half term), but Pamela had recently twisted her ankle in a hockey match and Sylvie was recovering from an attack of pleurisy that had seen her spend a night in the cottage hospital ('I suspect I have my mother's lungs,' a remark that Teddy found funny every time he thought about it). And Jimmy was only just over a bout of the tonsillitis he was prone to. 'Going down like flies,' Mrs Glover said, beating butter into sugar for the cake. 'Who's next, I wonder?'\n\n'Who needs to go to a hotel for a decent tea anyway?' Bridget said. 'Just as good here.'\n\n'Better,' Mrs Glover said. Although, of course, neither Bridget nor Mrs Glover had been invited to the Berkeley and indeed Bridget had never been inside a London hotel, or a hotel anywhere come to that, apart from having gone into the Shelbourne to admire the foyer before catching the ferry at D\u00fan Laoghaire to come to England, 'a lifetime ago'. Mrs Glover, on the other hand, declared herself to be 'quite familiar' with the Midland in Manchester where one of her nephews (of which, it seemed, she had an endless supply) had taken her and her sister for dinner 'on more than one occasion'.\n\nCoincidentally, Maurice was down for the weekend, although he had forgotten ('if he ever knew' Pamela said) that it was Ursula's birthday. He was in his last year at Balliol where he was reading law and was 'more of a prig than ever' according to Pamela. His parents didn't seem particularly taken with him either. 'He _is_ mine, isn't he?' Ursula had overheard Hugh say to Sylvie. 'You didn't have a dalliance in Deauville with that terrifically boring chap from Halifax, the one who owned the mill?'\n\n'What a memory you have,' Sylvie laughed.\n\nPamela had taken time out from her studies to make a lovely card, a _d\u00e9coupage_ of flowers cut out from Bridget's magazines, as well as baking a batch of her famous (in Fox Corner anyway) 'piccaninny' biscuits. Pamela was studying for the entrance exam for Girton. 'A Girton girl,' she said, her eyes alight, 'imagine.' As Pamela prepared to leave the sixth form of the school they both attended, Ursula was about to enter it. She was good at Classics. Sylvie said that she couldn't see the point of Latin and Greek (she had never been taught them and seemed to feel the lack). Ursula, on the other hand, was rather attracted to words that were now only whispers from the necropoles of ancient empires. ('If you mean \"dead\" then say \"dead\",' Mrs Glover said irritably.)\n\nMillie Shawcross was also invited to tea and had arrived early, her usual chirpy self. Her present was an assortment of lovely velvet hair ribbons, bought with her own money from the haberdasher's in town. ('Now you'll never be able to cut your hair,' Hugh said to Ursula, with some satisfaction.)\n\nMaurice had brought two friends to stay for the weekend, Gilbert and an American, Howard ('Call me Howie, everyone does'), who were going to have to double up in the spare-room bed, a fact that seemed to make Sylvie uneasy. 'You can go top to tail,' she told them briskly. 'Or one of you can sleep on a cot with the Great Western Railway,' which was their name for Teddy's Hornby train set that took up all of Mrs Glover's old room in the attic. Jimmy was allowed to share this pleasure. 'Your sidekick, huh?' Howie said to Teddy, ruffling Jimmy's hair so vigorously that Jimmy was knocked off balance. The fact that Howie was an American gave him a special kind of glamour, although it was Gilbert who had the brooding, rather exotic, movie star looks. His name \u2013 Gilbert Armstrong \u2013 and his father (a high court judge) and his education (Stowe) pointed to impeccably English credentials but his mother was the scion of an old Spanish aristocratic family ('Gypsies,' Mrs Glover concluded, which was pretty much what she considered all foreigners to be).\n\n'Oh, my,' Millie whispered to Ursula, 'the gods walk among us.' She crossed her hands over her heart and flapped them like wings. 'Not Maurice,' Ursula said. 'He would have been kicked off Olympus for getting on everyone's nerves.'\n\n'The self-importance of gods,' Millie said, 'what a wonderful title for a novel.' Millie, needless to say, wanted to be a writer. Or an artist, or a singer, or a dancer, or an actress. Anything where she might be the centre of attention.\n\n'What are you little girls chattering about?' Maurice said. Maurice was very sensitive, some might have said over-sensitive, to criticism.\n\n'You,' Ursula said. Girls did find Maurice attractive, a fact that continually surprised the women in his own family. He had fair hair that looked as if it had been marcelled and a strapping physique from rowing but it was hard to overlook his charmlessness. Gilbert, however, was even now kissing Sylvie's hand ('Oh,' said Millie, 'can it _get_ any better?'). Maurice had introduced Sylvie as 'My old mater,' and Gilbert said, 'You're too young to be anyone's mother.'\n\n'I know,' Sylvie said.\n\n('A rather louche fellow' was Hugh's verdict. 'A Lothario,' Mrs Glover said.)\n\nThe three young men seemed to fill Fox Corner as if the house had suddenly shrunk and both Hugh and Sylvie were relieved when Maurice suggested that they go outside for 'a tour of the grounds'. 'Good idea,' Sylvie said, 'work off some of that surplus energy.' The three of them ran out into the garden in Olympian fashion (sportive rather than sacred) and commenced a hearty kick-about with a ball that Maurice had found in the hall cupboard. ('Mine, actually,' Teddy pointed out to no one in particular.) 'They'll ruin the lawn,' Hugh said, observing them howling like hooligans as they chewed up the grass with their muddy brogues.\n\n'Oh,' Izzie said, when she arrived and caught sight of this athletic trio through the window, 'I say, they're rather gorgeous, aren't they? Can I have one?'\n\nIzzie, swathed from head to toe in fox fur, said, 'I brought gifts,' an unnecessary announcement as she was laden with all kinds of different-shaped parcels in expensive wrapping 'for my favourite niece'. Ursula glanced at Pamela and gave a rueful shrug. Pamela rolled her eyes. Ursula hadn't seen Izzie in months, not since a fleeting visit to Swiss Cottage in the car with Hugh to drop off a crate full of vegetables from Fox Corner's bountiful late-summer garden. ('A marrow?' Izzie said, inspecting the contents of the box. 'What on earth am I supposed to do with that?')\n\nPrior to that she had visited for a long weekend but had more or less ignored everyone except Teddy, whom she took off for long walks and quizzed relentlessly. 'I think she's singled him out from the herd,' Ursula told Pamela. 'Why?' Pamela said. 'So she can eat him?'\n\nWhen questioned (closely by Sylvie), Teddy was mystified as to why he had received special attention. 'She just asked me what I did, what school was like, what my hobbies were, what I liked to eat. My friends. Stuff like that.'\n\n'Maybe she wants to adopt him,' Hugh said to Sylvie. 'Or sell him. I'm sure Ted would bring a good price.' And Sylvie, fiercely, 'Don't say things like that, not even in jest.' But then Teddy was dropped by Izzie as swiftly as he'd been picked up by her and they had thought no more of it.\n\nThe first of Ursula's presents to be unwrapped was a recording by Bessie Smith which Izzie immediately placed on the gramophone, home usually to Elgar and, Hugh's favourite, _The Mikado_. 'The \"St Louis Blues\",' Izzie said instructively. 'Listen to that cornet! Ursula loves this music.' ('Do you?' Hugh asked Ursula. 'I had no idea.') Then a lovely tooled red-leather edition of Dante in translation was produced. This was followed by a satin and lace bedjacket from Liberty's \u2013 'as you know, a shop of which your mother is inordinately fond'. This was pronounced 'far too grown-up' by Sylvie, 'Ursula wears flannelette.' Next a bottle of Shalimar ('new by Guerlain, divine') which received a similar verdict from Sylvie.\n\n'So speaks the child bride,' Izzie said.\n\n''I was eighteen, not sixteen,' a tight-lipped Sylvie said. 'One day we must talk about what _you_ got up to at sixteen, Isobel.'\n\n'What?' Pamela said eagerly.\n\n' _Il n'avait pas d'importance_ ,' Izzie said dismissively. Finally, from this cornucopia, a bottle of champagne. ('And definitely far too young for that!')\n\n'Better get that on ice,' Izzie said, handing it to Bridget.\n\nA perplexed Hugh glared at Izzie. 'Did you steal all this?' he asked.\n\n'Hey, darkie music,' Howie said when the three boys returned from the outdoors, crowding into the drawing room and smelling vaguely of bonfires and something else, less definable ('Essence of stag,' Izzie murmured, sniffing the air). Bessie Smith was now on her third go round and Hugh said, 'It begins to grow on one after a while.' Howie did some kind of odd dance to the music, vaguely barbaric, and then whispered something in Gilbert's ear. Gilbert laughed, rather crudely for someone with blue blood, albeit foreign, and Sylvie clapped her hands and said, 'Boys, how about some potted shrimps?' and marshalled them into the dining room when she noticed, too late, the dirty footprints they had tracked through the house.\n\n'They didn't fight in the war,' Hugh said, as if that explained their muddy spoor.\n\n'And that's a good thing,' Sylvie said firmly. 'No matter how unsatisfactory they turn out.'\n\n'Now,' Izzie said when the cake was cut and apportioned, 'I have one last gift\u2014'\n\n'For goodness', sake, Izzie,' Hugh interrupted, unable to contain his exasperation any longer. 'Who is paying for this? You have no money, your debts are piled to the rafters. You promised you would learn economy.'\n\n'Please,' Sylvie said. Any discussion of money (even Izzie's) in front of strangers filled her with reticent horror. A sudden dark cloud passed over her heart. It was Tiffin, she knew.\n\n' _I_ am paying,' Izzie said, very grandly. 'And this is not a present for Ursula, it is for Teddy.'\n\n'Me?' Teddy said, startled on to centre stage. He had been thinking what a jolly good cake it was and wondering what the chances of a second piece were and certainly had no desire to be pushed into the limelight.\n\n'Yes, you, darling boy,' Izzie said. Teddy visibly shrank away from both Izzie and the present that she put on the table in front of him. 'Go on,' Izzie said encouragingly, 'unwrap it. It won't explode.' (But it would.) Gingerly, Teddy removed the expensive paper. Unwrapped, the present turned out to be exactly what it looked like when wrapped \u2013 a book. Ursula, sitting opposite, tried to decipher the upside-down title. _The Adventures of_...\n\n_'The Adventures of Augustus_ ,' Teddy read out loud, 'by Delphie Fox.' ('Delphie?' Hugh queried.)\n\n'Why is everything an \"adventure\" with you?' Sylvie said irritably to Izzie.\n\n'Because life is an adventure, of course.'\n\n'I would say it was more of an endurance race,' Sylvie said. 'Or an obstacle course.'\n\n'Oh, my dear,' Hugh said, suddenly solicitous, 'not that bad, surely?'\n\n'Anyway,' Izzie said, 'back to Teddy's present.'\n\nThe thick card of the cover was green, the lettering and line drawings were gold \u2013 illustrations of a boy, roughly Teddy's age, wearing a schoolboy's cap. He was accompanied by a catapult and a small dog, a scruffy West Highland terrier. The boy was dishevelled and had a wild look on his face. 'That's Augustus,' Izzie said to Teddy. 'What do you think? I've based him on you.'\n\n'Me?' Teddy said, horrified. 'But I don't look like that. It's not even the right dog.'\n\nSomething astonishing. 'Give anyone a lift back to town?' Izzie asked casually.\n\n'You haven't got another car?' Hugh moaned.\n\n'I parked it at the foot of the drive,' Izzie said sweetly, 'so as not to annoy you.' They all trooped down the drive to inspect the car, Pamela, still on her crutches, hobbling tardily behind. 'The poor and the maimed, the halt and the blind,' she said to Millie and Millie laughed and said, 'For a scientist you know your Bible.'\n\n'Best to know your enemy,' Pamela said.\n\nIt was cold and none of them had thought to put their coats on. 'But really quite mild for this time of year,' Sylvie said. 'Not like when you were born. Goodness, I've never seen snow like that.'\n\n'I know,' Ursula said. The snow the day she was born was a legend in the family. She had heard the story so often that she thought she could remember it.\n\n'It's just an Austin,' Izzie said. 'An open-road tourer \u2013 four doors though \u2013 but nowhere near as costly as a _Bentley_ , goodness, it's positively a vehicle for hoi polloi compared to _your_ indulgence, Hugh.' 'On tick, no doubt,' Hugh said. 'Not at all, paid up in full, _in cash_. I have a _publisher_ , I have _money_ , Hugh. You don't need to worry about me any more.'\n\nWhile everyone was admiring (or not, in the case of Hugh and Sylvie) the cherry-bright vehicle, Millie said, 'I have to go, I have a dancing exhibition tonight. Thank you very much for a lovely tea, Mrs Todd.'\n\n'Come on, I'll walk you back,' Ursula said.\n\nOn the return home, through the well-worn shortcut at the bottom of the gardens, Ursula had an unexpected encounter \u2013 this was the something amazing, not the Austin tourer \u2013 when she almost tripped over Howie, on his hands and knees, rooting among the bushes. 'Looking for the ball,' he said apologetically. 'It was your kid brother's. I think we lost it in the' \u2013 he sat back on his heels and looked around helplessly at the berberis and buddleia \u2013 'Shrubbery,' Ursula supplied. 'We aspire to it.'\n\n'Huh?' he said, standing up in one clean move and suddenly towering above her. He looked as though he boxed. Indeed there was a bruise below his eye. Fred Smith, who used to be the butcher's boy but now worked on the railways, was a boxer. Maurice had taken a couple of his pals to cheer Fred on in an amateur bout in the East End. Apparently it had dissolved into a boozy riot. Howie smelt of bay rum \u2013 Hugh's scent \u2013 and there was something polished and new about him, like a freshly minted coin.\n\n'Did you find it?' she asked. 'The ball?' She sounded squeaky to her own ears. She had thought Gilbert was the handsome one out of the two but faced with Howie's clean-limbed, uncomplicated strength, like a large animal, she felt stupid.\n\n'How old are you?' he asked.\n\n'Sixteen,' she said. 'It's my birthday. You ate cake.' Clearly she wasn't the only stupid one.\n\n'Hoo-ee,' he said, an ambiguous kind of word (closely related to his own name, she noted) although it seemed to signal amazement as if reaching sixteen was a feat. 'You're shivering,' he said.\n\n'It's freezing out here.'\n\n'I can warm you up,' he said and then \u2013 the something astonishing \u2013 he took her by the shoulders and pulled her towards him and \u2013 an action that necessitated bending down quite a bit \u2013 pushed his big lips against hers. 'Kiss' seemed too courtly a word for what Howie was doing. He prodded his enormous tongue, like an ox's, against the portcullis of her teeth and she was amazed to realize that he was expecting her to open her mouth and let the tongue in. She would choke, for sure. Mrs Glover's tongue press in the kitchen came unwontedly to mind.\n\nUrsula was debating what to do, the bay rum and the lack of oxygen were making her dizzy, when they heard Maurice shouting, quite nearby, 'Howie! Leaving without you, chum!' Ursula's mouth was released and, without a word to her, Howie yelled, 'Coming!' so loudly that her ears hurt. Then he let go of her and set off, crashing through the bushes, leaving Ursula gasping for air.\n\nShe wandered back to the house in a daze. Everyone was still on the drive, even though it felt like hours had passed but she supposed it was only minutes really, like in the best fairy stories. In the dining room, the ruins of the cake were being licked delicately by Hattie. _The Adventures of Augustus_ , lying on the table, had a smear of icing on it. Ursula's heart was still palpitating from the shock of Howie's advances. To be kissed on her sixteenth birthday, and in such an unlooked-for way, seemed a considerable accomplishment. She was surely passing beneath the triumphal arch that led to womanhood. If only it had been Benjamin Cole, then it would have been perfect!\n\nTeddy, 'the kid', himself appeared, very browned off and said, 'They lost my ball.'\n\n'I know,' Ursula said.\n\nHe opened the book at the title page where, in a flamboyant hand, Izzie had inscribed, _To my nephew, Teddy. My own darling Augustus_.\n\n'What rot,' Teddy said, scowling. Ursula picked up a half-drunk glass of champagne the rim of which was adorned with red lipstick and poured half of it into a jelly glass that she handed to Teddy. 'Cheers,' she said. They chinked their glasses and drained them to the dregs.\n\n'Happy birthday,' Teddy said.\n\n# _May 1926_\n\nBy the beginning of the month, Pamela, off her crutches and back to playing tennis, had learned that she had failed her Cambridge exam. 'I panicked,' she said, 'I saw questions I didn't know and I went to pieces and flunked it. I should have swotted more or if I'd just stayed calm and thought it through I could probably have made a good fist of it.'\n\n'There are other universities if you're so set on being a bluestocking,' Sylvie said. Sylvie, although she never quite came out and said it, thought academia was pointless for girls. 'After all, woman's highest calling is to be a mother and a wife.'\n\n'You'd have me slave over a hot stove rather than a Bunsen burner?'\n\n'What did science ever do for the world, apart from make better ways of killing people?' Sylvie said.\n\n'Well, it's a crying shame about Cambridge,' Hugh said. 'Maurice is set to get a first and he's a complete dolt.' To make up for Pamela's disappointment he bought her a Raleigh loop-frame roadster and Teddy asked what he would get if _he_ failed an exam. Hugh laughed and said, 'Careful, that's Augustus talk.'\n\n'Oh, please, don't,' Teddy said, mortified at any mention of the book. _The Adventures of Augustus_ had, to everyone's chagrin but particularly Teddy's, proved a roaring success, 'flying off the bookshelves' and reprinted three times so far, according to Izzie who had already earned a 'fat little royalty cheque' and moved into a flat in Ovington Square. She had also done an interview for a newspaper in which she mentioned her 'prototype', her 'charming rogue of a nephew'.\n\n'But not my _name_ ,' Teddy said, hanging on to hope. He got a conciliation gift from Izzie in the shape of a new dog. Trixie had died a few weeks previously and Teddy had been in mourning ever since. The new dog was a Westie, like Augustus's dog \u2013 not a breed that any of them would have chosen. He had already been christened by Izzie \u2013 Jock, naturally, the name engraved on the tag on his expensive collar.\n\nSylvie suggested changing his name to Pilot ('Charlotte Bront\u00eb's dog,' she said to Ursula. ('One day,' Ursula said to Pamela, 'my communion with our mother will consist entirely of the names of the great writers of the past,' and Pamela said, 'I think it probably already does.')\n\nThe little dog already answered to Jock and it seemed wrong to confuse him, so Jock he remained, and in time they all grew to love him best of any of their dogs despite his annoying provenance.\n\nMaurice turned up on a Saturday morning, this time with only Howie in tow and no sign of Gilbert, who had been sent down for 'an indiscretion'. When Pamela said, 'What indiscretion?' Sylvie said that it was the definition of an indiscretion that you didn't speak of it afterwards.\n\nUrsula had thought of Howie quite often since their last encounter. It was not so much the physical Howie \u2013 the Oxford bags, the soft-collared shirt, the brilliantined hair \u2013 but the fact that he had been thoughtful enough to try to find Teddy's lost ball. Being kind modified the extraordinary, alarming _otherness_ of him, which was threefold \u2013 large, male and American. Despite her ambivalent feelings she couldn't help but experience a slight thrill when she saw him hop effortlessly out of his open-top car, parked outside the front door of Fox Corner.\n\n'Hey,' he said when he caught sight of her and she realized her imaginary beau didn't even know her name.\n\nA pot of coffee and a plate of scones were hastily conjured up by Sylvie and Bridget. 'We're not staying,' Maurice said to Sylvie, who said, 'Thank goodness, I don't have enough to stretch to feed two hulking young men.'\n\n'We're going up to London to help out with the strike,' Maurice said. Hugh expressed surprise. He hadn't realized, he said, that Maurice's politics put him on the side of the workers and Maurice in turn expressed surprise that his father could even think this was the case. They were going to drive buses and trains, and whatever else it took 'to keep the country running'.\n\n'I didn't know you knew how to drive a train, Maurice,' Teddy said, suddenly finding his brother interesting.\n\n'Well, a stoker, then,' Maurice said irritably, 'it can't be that difficult.'\n\n'They're not called stokers, they're called firemen,' Pamela said, 'and it's a very skilled job. Ask your friend Smithy.' A remark which for some reason got Maurice even hotter under the collar.\n\n'You're trying to shore up a civilization that's in its death throes,' Hugh said, as casually as if he were remarking on the weather. 'There's really no point.'\n\nUrsula left the room at this juncture, if there was one thing she found more tedious than thinking about politics it was _talking_ about politics.\n\nAnd then. Astonishing. Again. As she was skipping up the back stairs on her way to the attic bedroom to fetch something, something innocent \u2013 a book, a handkerchief, afterwards she would never remember what \u2013 she was almost sent flying by Howie on his way down. 'I was looking for a bathroom,' he said.\n\n'Well, we only have one,' Ursula said, 'and it's not up these\u2014' but before the sentence was finished she found herself pinned awkwardly against the neglected floral wallpaper of the back stairs, a pattern that had been up since the house was built. 'Pretty girl,' he said. His breath smelt of mint. And then she was again subjected to pushing and shoving from the outsized Howie. But this time it was not his tongue trying to jam its way into her mouth but something inexpressibly more intimate.\n\nShe tried to say something but before a sound came out his hand clamped over her mouth, over half her face in fact, and he grinned and said, 'Ssh,' as if they were conspirators in a game. With his other hand he was fiddling with her clothes and she squealed in protest. Then he was butting up against her, the way the bullocks in the Lower Field did against the gate. She tried to struggle but he was twice, three times her size even and she might as well have been a mouse in Hattie's jaws.\n\nShe tried to see what he was doing but he was pressed so tightly against her that all she could see was his big square jaw and the slight brush of stubble, unnoticeable from a distance. Ursula had seen her brothers naked, knew what they had between their legs \u2013 wrinkled cockles, a little spout \u2013 and it seemed to have little to do with this painful piston-driven thing that was now ramming inside her like a weapon of war. Her own body breached. The arch that led to womanhood did not seem so triumphal any more, merely brutal and completely uncaring.\n\nAnd then Howie gave a great bellow, more ox than Oxford man, and was hitching himself back together and grinning at her. 'English girls,' he said, shaking his head and laughing. He wagged his finger at her, almost disapproving, as if she had engineered the disgusting thing that had just happened and said, 'You really are something!' He laughed again and bounded down the stairs, taking them three at a time, as though his descent had been barely interrupted by their strange tryst.\n\nUrsula was left to stare at the floral wallpaper. She had never noticed before that the flowers were wisteria, the same flower that grew on the arch over the back porch. This must be what in literature was referred to as 'deflowering', she thought. It had always sounded like a rather pretty word.\n\nWhen she came back downstairs a half-hour later, a half-hour of thoughts and emotions considerably more intense than was usual for a Saturday morning, Sylvie and Hugh were on the doorstep waving a dutiful goodbye to the disappearing rear end of Howie's car.\n\n'Thank goodness they weren't staying,' Sylvie said. 'I don't think I could have been bothered with Maurice's bluster.'\n\n'Imbeciles,' Hugh said cheerfully. 'All right?' he said, catching sight of Ursula in the hallway.\n\n'Yes,' she said. Any other answer would have been too awful.\n\nUrsula found it easier than she had expected to lock this occurrence away. After all, hadn't Sylvie herself said that the definition of an indiscretion was that you didn't speak of it afterwards? Ursula imagined a cupboard in her mind, a corner one, in simple pitch pine. Howie and the back stairs were put on a high shelf and the key was firmly turned in the lock.\n\nA girl surely should know better than to be caught on those back stairs \u2013 or in the shrubbery \u2013 like the heroine in a gothic novel, the kind that Bridget was so fond of. But who would have suspected that the reality of it would be so sordid and bloody? He must have sensed something in her, something unchaste, that even she was unaware of. Before locking it away she had gone over the incident again and again, trying to see in what way she had been to blame. There must be something written on her skin, in her face, that some people could read and others couldn't. Izzie had seen it. Something wicked this way comes. And the something was herself.\n\nThe summer unrolled. Pamela was given a place at Leeds University to read chemistry and said she was glad because people would be 'more straightforward' in the provinces and not as snobbish. She played a lot of tennis with Gertie and championship mixed doubles with Daniel Cole and his brother Simon, and often let Ursula borrow her bike so that she could go for long rides with Millie, both of them shrieking as they freewheeled down hills. Sometimes Ursula went for lazy walks through the lanes with Teddy and Jimmy, Jock running rings around them. Neither Teddy nor Jimmy seemed to need to keep their lives secret from their sisters in the way that Maurice had done.\n\nPamela and Ursula took Teddy and Jimmy up to London for day trips, to the Natural History Museum, to the British Museum, to Kew, but they never told Izzie when they were in town. She had moved yet again, to a large house in Holland Park ('a rather artistic _endroit_ '). One day as they wandered along Piccadilly they spied a pile of _The Adventures of Augustus_ in a bookshop window, accompanied by 'a photograph of the author \u2013 Miss Delphie Fox, taken by Mr Cecil Beaton' in which Izzie looked like a film star or a society beauty. 'Oh, God,' Teddy said and Pamela, despite being _in loco parentis_ , didn't correct his language.\n\nThere was a f\u00eate once more in the grounds of Ettringham Hall. The Daunts had gone, after a thousand years, Lady Daunt never having recovered from the murder of little Angela, and the Hall was now owned by a rather mysterious man, a Mr Lambert who some said was Belgian, some Scottish, but no one had had a long enough conversation with him to discover his origins. Rumour said he had made his fortune during the war but everyone reported him shy and difficult to talk to. There were dances, too, in the village hall on Friday evenings and at one of these Fred Smith appeared, scrubbed clean of his daily soot, and asked, in turn, Pamela, Ursula and the three eldest Shawcrosses to dance. There was a gramophone, not a band, and they danced only old-fashioned dances, no Charleston or Black Bottom, and it was pleasant to be waltzed safely around the room, with surprising skill, by Fred Smith. Ursula thought it would be rather nice to have someone like Fred as a beau, although obviously Sylvie would never have tolerated such a thing. (' _A railwayman_?')\n\nAs soon as she thought about Fred in this way, the cupboard door sprang open and the whole appalling scene on the back stairs tumbled out.\n\n'Steady on,' Fred Smith said, 'you've gone a bit green round the gills, Miss Todd,' and Ursula had to blame it on the heat and insist on taking some fresh air on her own. She had in fact been feeling quite queasy lately. Sylvie put it down to a summer cold.\n\nMaurice had gained his expected first ('How?' Pamela puzzled) and came home for a few weeks to lounge around before taking up a place in chambers in Lincoln's Inn, to train as a barrister. Howie, apparently, had returned to 'his people' at their summer home on Long Island Sound. Maurice seemed a little miffed that he had not been invited to join them.\n\n'What happened to you?' Maurice said to Ursula one afternoon as he sprawled on a deckchair on the lawn reading _Punch_ , cramming nearly an entire slice of Mrs Glover's marmalade cake into his mouth at once.\n\n'What do you mean what happened to me?'\n\n'You've turned into a heifer.'\n\n'A heifer?' It was true she was filling out her summer frocks rather alarmingly, even her hands and feet seemed to have plumped up. 'Puppy fat, dear,' Sylvie said, 'even I had it. Less cake and more tennis, that's the remedy.'\n\n'You look hellish,' Pamela said to her, 'what's wrong with you?'\n\n'I have no idea,' Ursula said.\n\nAnd then something truly terrible dawned on her, so awful, so shameful, so _irretrievable_ , that she felt something catch fire and burn inside her at the very thought. She hunted down Sylvie's copy of _The Teaching of Young Children and Girls as to Reproduction_ by Dr Beatrice Webb, which, theoretically, Sylvie kept under lock and key in a chest in her bedroom, but the chest was never locked because Sylvie had long ago lost the key. Reproduction seemed to be the last thing on the author's mind. She advised distracting young girls by giving them plenty of 'home-made bread, cake, porridge, puddings and cold water splashed regularly on to the parts'. It was clearly no help. Ursula shuddered at the memory of Howie's 'parts' and how they had come together with hers in one vile conjugation. Was this what Sylvie and Hugh did? She couldn't imagine her mother putting up with such a thing.\n\nShe sneaked a look at Mrs Shawcross's medical encyclopaedia. The Shawcrosses were on holiday in Norfolk but their maid thought nothing of it when Ursula appeared at the back door saying she had come to look at a book.\n\nThe encyclopaedia explained the mechanics of 'the sexual act', something which appeared to take place only within the 'loving confines of the marital bed' rather than on the back stairs when you were on your way to fetch a handkerchief, a book. The encyclopaedia also detailed the consequences of failing to retrieve that handkerchief, that book \u2013 the missed monthlies, the sickness, the weight gain. It took nine months apparently. And now they were already well into July. Before long she would be squeezing herself back into her navy-blue gymslip and catching the bus to school every morning with Millie.\n\nUrsula began to take long solitary walks. There was no Millie to confide in (and would she have anyway?) and Pamela had decamped to Devon with her Girl Guide patrol. Ursula had never taken to the Guides, now she rather regretted that \u2013 they might have given her the gumption to deal with Howie. A Guide would have retrieved that handkerchief, that book, without being hindered on the journey.\n\n'Is there anything the matter, dear?' Sylvie asked as they darned stockings together. Sylvie's children only really came into focus for her when in isolation. Together they were an unwieldy flock, singly they had character.\n\nUrsula imagined what she could say. _You remember Maurice's friend Howie_? _I appear to be the mother of his child_. She glanced at Sylvie, serenely wefting and warping her little woollen patch on the hole in the toe of one of Teddy's socks. She did not look like a woman who had had her parts breached. (A 'vagina', apparently, according to Mrs Shawcross's encyclopaedia \u2013 not a word that had ever been uttered in the Todds' household.)\n\n'No, nothing at all,' Ursula said. 'I'm fine. Absolutely fine.'\n\nThat afternoon she walked to the station and sat on a bench on the platform and contemplated throwing herself under the express when it came hurtling through, but the next train turned out to be for London, huffing slowly to a halt in front of her in a way that seemed so familiar that it made her want to cry. She spotted Fred Smith climbing down from the cab, oily overalls and face smutty with coal dust. Spotting her, he came over and said, 'Here's a coincidence, are you catching our train?'\n\n'I haven't got a ticket,' Ursula said.\n\n'That's all right,' Fred said, 'nod and a wink from me and the ticket inspector'll see a friend of mine all right.' Was she a friend of Fred Smith? It was comforting to think so. Of course, if he knew about her condition he would no longer be a friend. No one would.\n\n'Yes, all right, thank you,' she said. Not having a ticket seemed such a _little_ problem.\n\nShe watched Fred climb back into the cab of his locomotive. The stationmaster stalked along the platform, slamming the carriage doors with a finality that suggested they would never be opened again. Steam flared from the funnel and Fred Smith stuck his head out of the cab and shouted, 'Look smart there, Miss Todd, or you'll be left behind,' and she stepped obediently on board.\n\nThe stationmaster's whistle chirped, short at first and then a longer call, and the train shuffled out of the station. Ursula sat on the warm plush of the seats and contemplated the future. She supposed she could get lost among the other fallen women crying woe on the streets of London. Curl up on a park bench and freeze to death overnight, except that it was the height of summer and she was unlikely to freeze. Or wade into the Thames and drift gently on the tide, past Wapping and Rotherhithe and Greenwich and on to Tilbury and out to sea. How puzzled her family would be if her drowned body was hooked from the deep. She imagined Sylvie, frowning over her darning, _But she only went for a walk, she said she was going to pick the wild raspberries in thelane_. Ursula thought guiltily of the white china pudding basin she had abandoned in the hedgerow, intending to collect it on her return. It was half full of the sour little berries and her fingers were still stained red.\n\nShe spent the afternoon walking through the great parks of London, through St James's and Green Park, past the Palace and into Hyde Park and on to Kensington Gardens. It was extraordinary how far you could go in London and barely touch a pavement or cross a road. She had no money on her, of course \u2013 a ridiculous mistake, she realized now \u2013 and couldn't even buy a cup of tea in Kensington. There was no Fred Smith here to 'see her all right'. She was hot and tired and dusty and felt as parched as the grass in Hyde Park.\n\nCould you drink the water in the Serpentine? Shelley's first wife had drowned herself here but Ursula supposed that on a day like this \u2013 crowds of people enjoying the sunshine \u2013 it would be almost impossible to avoid another Mr Winton jumping in and rescuing her.\n\nShe knew where she was going, of course. It was inevitable somehow.\n\n'Good God, what happened to you?' Izzie said, throwing her front door wide dramatically as if she had been expecting someone more interesting. 'You look a fright.'\n\n'I've been walking all afternoon,' Ursula said. 'I have no money,' she added. 'And I think I'm going to have a baby.'\n\n'You'd better come in then,' Izzie said.\n\nAnd now here she was, sitting on an uncomfortable chair in a large house in Belgravia in what must have once been the dining room. Now, devoid of any purpose except waiting, it was nondescript. The Dutch still life above the fireplace and the bowl of dusty-looking chrysanthemums on a Pembroke table provided no clue as to what might happen elsewhere in the house. It was hard to connect any of this to the odious rendezvous with Howie on the back stairs. Who would have thought it could be so easy to slip from one life to another. Ursula wondered what Dr Kellet would have made of her predicament.\n\nAfter her unexpected arrival in Melbury Road, Izzie had put her to bed in her spare bedroom and Ursula had lain sobbing beneath the shiny satin cover, trying not to listen to Izzie's unlikely lies on the phone in the hallway \u2013 _I know! She just turned up on the doorstep, the lamb... wanted to see me... pay a visit, museums and so on, the theatre, nothing risqu\u00e9... now don't be a termagant, Hugh_... It was just as well that Izzie hadn't spoken to Sylvie, she would have been given short shrift there. The upshot was that she was to be allowed to stay for a few days for _museums and so on_.\n\nPhone call finished, Izzie came into the bedroom carrying a tray.\n\n'Brandy,' she said. 'And buttered toast. All I could rustle up at short notice, I'm afraid. You are such a fool,' she sighed. 'There are ways, you know, things that one can do, prevention better than cure and so on.' Ursula had no idea what Izzie was talking about.\n\n'And you must get rid of it,' Izzie continued. 'We are agreed on that, aren't we?' A question which produced a heartfelt 'yes' from Ursula.\n\nA woman in a nurse's uniform opened the door of the Belgravia waiting room and looked in. Her uniform was so starched it would have stood up quite well without her inside it.\n\n'This way,' she said stiffly, without addressing Ursula by name. Ursula followed as meek as a lamb to the slaughter.\n\nIzzie, efficient rather than sympathetic, had dropped her off in the car ('Good luck') with a promise that she would return 'later'. Ursula had no idea what was to happen in the interim between Izzie's 'Good luck' and her 'later' but she presumed it would be unpleasant. A foul-tasting syrup or a kidney dish full of large pills perhaps. And undoubtedly a good talking-to about her morals and her character. She hardly cared, as long as in the end the clock could be put back. How big was the baby, she wondered? Her brief research in the Shawcrosses' encyclopaedia had given few clues. She supposed it would come out with a certain amount of difficulty and be wrapped in a shawl before being placed in a cradle and tended carefully until it was ready to be given to a nice couple who longed to have a baby as much as Ursula longed not to have one. And then she would be able to catch the train home, walk along Church Lane and retrieve the white china bowl with its harvest of raspberries, before entering Fox Corner as if nothing had happened beyond _museums and so on_.\n\n*\n\nIt was a room like any other really. There were curtains, swagged and tasselled, at the tall windows. The curtains looked as though they were left over from the previous life of the house, as did the marble fireplace that now held a gas fire and, on the mantelpiece, a plain-faced clock with large numbers. The green linoleum underfoot and the operating table in the middle of the room were equally incongruous. The room smelt like the science laboratory at school. Ursula wondered about the brutish array of shining metal instruments that were laid out on a linen cloth on a trolley. They seemed to have more to do with butchery than babies. There was no sign of a cradle waiting anywhere. Her heart began to flutter.\n\nA man, older than Hugh, in a long white doctor's smock hurried into the room as if he were on his way somewhere else and ordered Ursula up on to the operating table with her feet 'in the stirrups'.\n\n'Stirrups?' Ursula repeated. Surely horses were not involved. The request was baffling until the starched nurse pushed her down and hooked her feet up. 'I'm having an operation?' Ursula protested. 'But I'm not ill.' The nurse placed a mask over her face. 'Count from ten down to one,' she said. 'Why?' Ursula tried to ask, but the word had barely formed in her brain before the room and everything in it disappeared.\n\nThe next thing she knew she was in the passenger seat of Izzie's Austin, gazing woozily through the windscreen.\n\n'You'll be right as rain in no time,' Izzie said. 'Don't worry, they've doped you up. You'll feel queer for a bit.' How did Izzie know so much about this appalling process?\n\nBack in Melbury Road, Izzie helped her into bed and she slept deeply beneath the shiny satin cover in the spare bedroom. It was dark outside when Izzie came in with a tray. 'Oxtail soup,' she said cheerfully. 'I got it from a tin.' Izzie smelt of alcohol, something sweet and cloying, and underneath her make-up and her bright demeanour she looked exhausted. Ursula supposed she must be a terrible burden to her. She struggled to sit up. The smell of the alcohol and the oxtail was too much and she vomited all over the shiny satin.\n\n'Oh, God,' Izzie said, holding her hand over her mouth. 'I'm really not cut out for this kind of thing.'\n\n'What happened to the baby?' Ursula asked.\n\n'What?'\n\n'What happened to the baby?' Ursula repeated. 'Did they give it to someone nice?'\n\nShe woke in the night and vomited again and fell back to sleep without either cleaning it up or calling out to Izzie. When she woke in the morning she was too hot. Far too hot. Her heart was knocking in her chest and each breath was hard to come by. She tried to get out of bed but her head was swimming and her legs wouldn't hold her. After that everything was a blur. Izzie must have called Hugh because she felt a cool hand on her clammy forehead and when she opened her eyes he was smiling reassuringly at her. He was sitting on the bed, still in his overcoat. She was sick all over it.\n\n'We'll get you to a hospital,' he said, unperturbed by the mess. 'You've got a bit of an infection.' Somewhere in the background Izzie was putting up a fierce protest. 'I'll be prosecuted,' she hissed at Hugh and Hugh said, 'Good, I hope they put you in jail and throw away the key.' He lifted Ursula up in his arms and said, 'Quicker to take the Bentley, I think.' Ursula felt weightless, as if she was going to float away. The next thing she knew she was on a cavernous hospital ward and Sylvie was there, her face tight and awful. 'How could you?' she said. She was glad when evening came and Sylvie changed places with Hugh.\n\nIt was Hugh who was with her when the black bat came. The hand of night was held out to her and Ursula rose to meet it. She was relieved, almost glad, she could feel the shining, luminous world beyond calling, the place where all the mysteries would be revealed. The darkness enveloped her, a velvet friend. Snow was in the air, as fine as talcum, as icy as the east wind on a baby's skin \u2013 but then Ursula fell back into the hospital bed, her hand rejected.\n\nThere was a brilliant slant of light across the pale green of the hospital counterpane. Hugh was asleep, his face slack and tired. He was sitting in an awkward position in a chair next to the bed. One trouser leg had ridden up slightly and Ursula could see a wrinkled grey lisle sock and the smooth skin of her father's shin. He had once been like Teddy, she thought, and one day Teddy would be like him. The boy within the man, the man within the boy. It made her want to weep.\n\nHugh opened his eyes and when he saw her he smiled weakly and said, 'Hello, little bear. Welcome back.'\n\n# _August 1926_\n\n_The pen should be held lightly, and in such a manner as to permit of the shorthand characters being easily written. The wrist must not be allowed to rest on the notebook or desk_.\n\nThe rest of the summer was wretched. She sat beneath the apple trees in the orchard and tried to read a Pitman's shorthand instruction book. It had been decided she would do a typing and shorthand course rather than return to school. 'I can't go back,' she said. 'I just can't.'\n\nThere was little escape from the chill that Sylvie brought with her every time she entered a room and discovered Ursula in it. Both Bridget and Mrs Glover were puzzled as to why the 'serious illness' that Ursula had contracted in London while staying with her aunt seemed to have made Sylvie so distant from her daughter when they might have expected the opposite. Izzie, of course, was barred for ever. _Persona non grata in perpetuam_. No one knew the truth of what had happened except for Pamela who had wormed the whole story out of Ursula, bit by bit.\n\n'But he _forced_ himself on you,' she fumed, 'how can you think it was your fault?'\n\n'But the consequences...' Ursula murmured.\n\nSylvie blamed her entirely, of course. 'You've thrown away your virtue, your character, everyone's good opinion of you.'\n\n'But no one knows.'\n\n' _I_ know.'\n\n'You sound like someone in one of Bridget's novels,' Hugh said to Sylvie. Had Hugh read one of Bridget's novels? It seemed unlikely. 'In fact,' Hugh said, 'you sound rather like my own mother.' ('It seems dreadful now,' Pamela said, 'but this too will pass.')\n\nEven Millie was fooled by her lies. 'Blood poisoning!' she said. 'How dramatic. Was hospital ghastly? Nancy said that Teddy told her that you nearly died. I'm sure nothing so exciting will ever happen to me.'\n\nWhat a world of difference there was between dying and nearly dying. One's whole life, in fact. Ursula felt she had no use for the life she had been saved for. 'I'd like to see Dr Kellet again,' she said to Sylvie.\n\n'He's retired, I believe,' Sylvie said indifferently.\n\nUrsula still wore her hair long, mostly to please Hugh, but one day she went into Beaconsfield with Millie and had her hair chopped short. It was a penitent act that made her feel rather like a martyr or a nun. She supposed that was how she would live out the rest of her life, somewhere between the two.\n\nHugh seemed surprised rather than saddened. She supposed a haircut was a mild travesty compared to Belgravia. 'Good gracious,' he said, when she sat down at the dinner table to unappetizing veal cutlets _\u00e0 la Russe_. ('Looks like the dog's dinner,' Jimmy said, although Jimmy, a boy of magnificent appetite, would have quite happily eaten Jock's dinner.)\n\n'You look like a completely different person,' Hugh said.\n\n'That can only be a good thing, can't it?' Ursula said.\n\n'I liked the old Ursula,' Teddy said.\n\n'Well, it seems as though you're the only one who does,' Ursula muttered. Sylvie made a noise that fell short of a word and Hugh said to Ursula, 'Oh, come, I think you're\u2014'\n\nBut she never did find out what Hugh thought of her because the loud rapping of the front-door knocker announced a rather anxious Major Shawcross enquiring as to whether Nancy was with them. 'Sorry to interrupt your dinner,' he said, hovering in the doorway of the dining room.\n\n'She isn't here,' Hugh said, although Nancy's absence was obvious.\n\nMajor Shawcross frowned at the cutlets on their plates. 'She went to gather some leaves in the lane,' he said. 'For her scrapbook. You know what she's like.' This addressed to Teddy, Nancy's twin soul. Nancy loved nature, forever collecting twigs and pine cones, shells and stones and bones, like the totems of an ancient religion. 'A child of nature,' Mrs Shawcross called her ('As if that were a good thing,' Sylvie said).\n\n'She wanted oak leaves,' Major Shawcross said. 'We don't have any oaks in our garden.'\n\nThere was a short discussion about the demise of the English oak, followed by a thoughtful silence. Major Shawcross cleared his throat. 'She's been gone about an hour, according to Roberta. I've walked the length of the lane, up and back, shouting her name. I can't think where she might be. Winnie and Millie are out searching as well.' Major Shawcross was beginning to look rather sick. Sylvie poured a glass of water and handed it to him. 'Sit down,' she said. He didn't. Of course, Ursula thought, he was thinking of Angela.\n\n'I expect she'll have found something interesting,' Hugh said, 'a bird's nest or a farm cat with kittens. You know what she's like.' They were all now quite in agreement that they knew what Nancy was like.\n\nMajor Shawcross picked up a spoon from the dining-room table and gazed at it absently. 'She's missed her dinner.'\n\n'I'll come and help you look for her,' Teddy said, jumping up from the table. He knew what Nancy was like too, knew she never missed her dinner.\n\n'Me too,' Hugh said, giving Major Shawcross an encouraging pat on the back, the veal cutlets abandoned.\n\n'Shall I come?' Ursula asked.\n\n'No,' Sylvie said. 'Nor Jimmy either. Stay here, we'll look in the gardens.'\n\nNo ice house this time. A hospital mortuary for Nancy. Still warm and soft when they found her, pushed into an empty old cattle trough. 'Interfered with,' Hugh told Sylvie while Ursula lurked like a spy behind the morning-room door. 'Two little girls in three years, it can't be a coincidence, can it? Strangled like Angela before her.'\n\n'A monster is living among us,' Sylvie said.\n\nIt was Major Shawcross who found her. 'Thank God, it wasn't poor old Ted this time,' Hugh said. 'He couldn't have borne it.' Teddy couldn't bear it anyway. He barely spoke for weeks. His soul had been cut away, he said, when he did eventually speak. 'Scars heal,' Sylvie said. 'Even the worst ones.'\n\n'Do you think that's true?' Ursula said, thinking about the wisteria wallpaper, the waiting room in Belgravia, and Sylvie said, 'Well, not always,' not even bothering to lie.\n\nThey heard Mrs Shawcross screaming all through the first night. Afterwards her face never looked right and Dr Fellowes reported that she'd had a 'small stroke'.\n\n'Poor, poor woman,' Hugh said.\n\n'She never knows where those girls are,' Sylvie said. 'She just lets them run wild. Now she's paying the price of her carelessness.'\n\n'Oh, Sylvie,' Hugh said sadly. 'Where is your heart?'\n\nPamela left for Leeds. Hugh drove her there in the Bentley. Her trunk was too massive for the boot and had to be sent by train. 'Big enough to hide a body in,' Pamela said. She was bound for a women's hall of residence and had already been informed that she was to share the small room with a girl called Barbara, from Macclesfield. 'It'll be just like being at home,' Teddy said encouragingly, 'except that Ursula will be someone else.'\n\n'Well, that rather makes it _nothing_ like home,' Pamela said. She clung to Ursula a little too fiercely before climbing into the car and sitting next to Hugh.\n\n'I can't wait to go,' Pamela said to Ursula in bed on her last night, 'but I feel bad at leaving you.'\n\nWhen she didn't go back to school for the autumn term, no one questioned Ursula's decision. Millie was too grief-stricken over Nancy's death to care much about anything.\n\nUrsula travelled on the train to High Wycombe every morning to attend a private secretarial college. 'College' was a fancy word for two rooms, a cold scullery and a colder cupboard containing a WC above a greengrocer's on the high street. The college was run by a man called Mr Carver whose lifelong passions were Esperanto and Pitman's shorthand, the latter more useful than the former. Ursula rather liked shorthand, it was akin to a secret code, with a whole new vocabulary \u2013 aspirates and shun hooks, compound consonants, special contractions, halving and doubling \u2013 the language of neither the dead nor the living but the strangely inert. There was something soothing about listening to Mr Carver's monotonous intonation of word lists \u2013 _iterate, iteration, reiteration, reiterated, reiterating, prince, princely, princes, princess, princesses_...\n\nThe other girls on the course were all very pleasant and friendly \u2013 sanguine, practical sorts who always remembered their shorthand notebooks and rulers and never had fewer than two different-coloured inks in their bags.\n\nAt lunchtime when the weather was bad they stayed in, sharing their packed lunches and darning stockings among the banks of typewriters. They had spent their summer hiking and swimming and camping and Ursula wondered if they could tell just by looking at her how different her own summer had been. 'Belgravia' had become her shorthand for what had happened. ('An abortion,' Pamela said. 'An illegal abortion.' Pamela was never one to avoid a blunt vocabulary. Ursula very much wished that she would.) She envied the ordinariness of their lives. (How Izzie would scorn such an idea.) Ursula's own chance at ordinariness seemed lost for ever.\n\nWhat if she had thrown herself beneath the express train or had died after Belgravia, or, indeed, what if she were simply to open her bedroom window and throw herself out, head first? Would she really be able to come back and start again? Or was it, as everyone told her, and as she must believe, all in her head? And so what if it was \u2013 wasn't everything in her head real too? What if there was no demonstrable reality? What if there was nothing beyond the mind? Philosophers 'came to grips' with this problem a long time ago, Dr Kellet had told her, rather wearily, it was one of the very first questions they addressed, so there was really no point in her fretting over it. But surely, by its very nature, everyone wrestled with this dilemma anew every time?\n\n('Forget typing,' Pamela wrote from Leeds, 'you should read philosophy at university, you have the right kind of mind for it. Like a terrier with a terrifically tedious bone.')\n\nShe had, eventually, gone in search of Dr Kellet and found his rooms occupied by a steel-haired, steel-spectacled woman who informed her that Dr Kellet had indeed retired and did she wish to make an appointment with herself? No, Ursula said, she didn't. It was the first time she had been to London since Belgravia and she had a panic attack on the Bakerloo line on the way back from Harley Street and had to run out of the station at Marylebone, gasping for air. A newspaper seller said, 'Are you all right, miss?' and she said, yes, yes, quite all right, thank you.\n\nMr Carver liked touching the girls ('my girls') lightly on their shoulders, stroking the angora of a bolero cardigan or the lambswool of a sweater as if they were animals he was fond of.\n\nIn the morning they practised their typing skills on the big Underwoods. Sometimes Mr Carver made them practise with blindfolds on as this was, he claimed, the only way to stop them looking at the keys and slowing down their speeds. Wearing the blindfold made Ursula feel like a soldier about to be shot for desertion. On these occasions she often heard him making odd noises, muffled wheezes and grunts, but didn't like to peek from the blindfold to see what he might be doing.\n\nIn the afternoons they did shorthand \u2013 soporific dictation exercises that encompassed every kind of business letter. _Dear-Sir, I-brought your letter before-the Board-of-Directors at their-meeting yesterday, but after some discussion they-were obliged to postpone further-consideration of-the-matter until the next Directors'-meeting, which-will-be held on-the last Tuesday_... The content of these letters was tedious in the extreme and was a strange contrast to the furious flow of ink across their pads as they struggled to keep up.\n\nOne afternoon, while he was dictating to them, _We-fear there-is-no prospect of success for-those-who raise objection to-the appointment_ , Mr Carver passed behind Ursula and gently touched the nape of her neck, no longer protected by long hair. A shiver rippled right through her. She stared at the keys of her Underwood on the table in front of her. Was it something in her that attracted this kind of attention? Was she not a good person?\n\n# _June 1932_\n\nPamela had chosen a white brocade for herself and yellow satin for her bridesmaids. The yellow was on the acidic side and made all of the bridesmaids look slightly liverish. There were four of them \u2013 Ursula, Winnie Shawcross (chosen over Gertie) and Harold's two youngest sisters. Harold came from a large noisy family in the Old Kent Road that Sylvie considered to be 'inferior'. The fact that Harold was a doctor didn't seem to mitigate his circumstances (Sylvie was curiously averse to the medical profession). 'I thought your own family were somewhat _d\u00e9class\u00e9_ , weren't they?' Hugh said to Sylvie. He liked his new son-in-law-to-be, found him 'refreshing'. He liked Harold's mother, Olive, too. 'She says what she means,' he said to Sylvie. 'And means what she says. Unlike some people.'\n\n'I thought it looked nice in the pattern book,' Pamela said doubtfully at Ursula's third and final fitting, in a dressmaker's front room in Neasden, of all places. The bias-cut dress stretched tightly across Ursula's midriff.\n\n'You've put on weight since the last fitting,' the dressmaker said.\n\n'Have I?'\n\n'Yes,' Pamela said. Ursula thought of the last time she had put on weight. Belgravia. It was certainly not the reason this time. She was standing on a chair, the dressmaker moving in a circle around her, a pincushion attached to her wrist. 'You still look nice though,' Pamela added.\n\n'I sit all day at my work,' Ursula said. 'I should walk more, I expect.' It was so easy to be lazy. She lived on her own but no one knew. Hilda, the girl she was supposed to share the flat with \u2013 a top floor in Bayswater \u2013 had moved out, although she still paid the rent, thank goodness. Hilda was living in Ealing in a 'regular little pleasure palace' with a man called Ernest whose wife wouldn't grant him a divorce, and she had to pretend to her parents that she was still in Bayswater, living the life of a single and virtuous woman. Ursula supposed it would only be a matter of time before Hilda's parents turned up unexpectedly on the doorstep and she would have to spin a lie, or several, to explain their daughter's absence. Hugh and Sylvie would have been horrified if they thought Ursula was living on her own in London.\n\n'Bayswater?' Sylvie said doubtfully when Ursula announced she was moving out of Fox Corner. 'Is that really necessary?' Hugh and Sylvie had vetted the flat, and they had also vetted Hilda, who stood up well to Sylvie's inquisition. Nonetheless Sylvie found both the flat and Hilda somewhat wanting.\n\n'Ernest from Ealing', as Ursula always thought of him, was the one who paid the rent ('a kept woman', Hilda laughed) but Hilda herself came by every couple of weeks to pick up her post and pay the rent money over. 'I can find someone else to share with,' Ursula offered, although she hated the idea.\n\n'Let's wait a bit,' Hilda said, 'see if it all works out for me. That's the beauty about living in sin, you can always just get up and leave.'\n\n'So can Ernest (from Ealing).'\n\n'I'm twenty-one, he's forty-two, he's not about to leave, trust me.'\n\nIt had been a relief when Hilda had moved out. Ursula was able to lounge around all evening in her dressing gown, with her curlers in, eating oranges and chocolates and listening to the wireless. Not that Hilda would have objected to any of these things, would have enjoyed them in fact, but Sylvie had instilled decorum in the presence of others from an early age and it was hard to shake it off.\n\nAfter a couple of weeks of being on her own, it struck her that she had hardly any friends and those that she did have she never seemed to care enough about to keep in touch with. Millie had become an actress, and was away almost all the time with a touring theatre company. She sent the odd postcard from places she would probably never have visited otherwise \u2013 Stafford, Gateshead, Grantham \u2013 and drew funny cartoons of herself in various roles ('Me as Juliet, what a hoot!'). Their friendship hadn't really survived Nancy's death. The Shawcross family had turned inward with grief and when Millie finally started to live her life again she found Ursula had stopped living hers. Ursula often wished that she could explain Belgravia to Millie but didn't want to risk what was left of their fragile attachment.\n\nShe worked for a big importing company and sometimes when Ursula listened to the girls in the office chatting about what they'd been doing and with whom, she found herself wondering how on earth they met all these people, these Gordons, Charlies, Dicks, Mildreds, Eileens and Veras \u2013 a gay, restless flock with whom they frequented variety palaces and cinemas, went skating, swam in lidos and baths and drove out to Epping Forest and Eastbourne. Ursula did none of these things.\n\nUrsula craved solitude but she hated loneliness, a conundrum that she couldn't even begin to solve. At work, they regarded her as a person apart, as if she were senior to them in every way, even though she wasn't. Occasionally one or other of the office coterie would say to her, 'Do you want to come out with us after work?' It was meant kindly and felt like charity, which it probably was. She never took them up on their offers. She suspected, no, she knew, that they talked about her behind her back, nothing nasty, just curiosity really. They imagined there must be more to her. _A dark horse_. And _still waters run deep_. They would be disappointed to know that there _was_ no more, that even clich\u00e9s were more interesting than the life she lived. No depths, no darkness (in the past perhaps, but not the present). Unless you counted the drink. Which she supposed they would.\n\nThe work was a chore \u2013 endless bills of lading and customs forms and balance sheets. The goods themselves \u2013 rum, cocoa, sugar \u2013 and the exotic places they came from seemed at odds with the daily tedium of the office. She supposed she was a little cog in the big wheel of Empire. 'Nothing wrong with being a cog,' Maurice said, himself a big wheel now in the Home Office. 'The world needs cogs.' She didn't want to be a cog, but Belgravia seemed to have put paid to anything grander.\n\nUrsula knew how the drinking had started. Nothing dramatic, just something as small and domestic as a _boeuf bourguignon_ she had planned for Pamela when she came to stay for the weekend a few months ago. She was still working in the lab in Glasgow and wanted to do some shopping for her wedding. Harold hadn't moved yet either, he was due to take up his post at the Royal London in a few weeks. 'We'll have a nice weekend, just the two of us,' Pamela said.\n\n'Hilda's away,' Ursula lied easily. 'Gone to Hastings for the weekend with her mother.' There was no reason not to tell Pamela the truth of her arrangement with Hilda, Pamela had always been the one person she could be honest with, and yet something held her back.\n\n'Splendid,' Pamela said. 'I'll drag Hilda's mattress through to your bedroom and it'll be like old times.'\n\n'Are you looking forward to being married?' Ursula asked as they lay in bed. It wasn't really like old times at all.\n\n'Of course I am, why would I be doing it otherwise? I like the idea of marriage. There is something smooth and round and solid about it.'\n\n'Like a pebble?' Ursula said.\n\n'A symphony. Well, more of a duet, I suppose.'\n\n'It's not like you to wax poetic.'\n\n'I like what our parents have,' Pamela said simply.\n\n'Do you?' It was a while since Pamela had spent much time with Hugh and Sylvie. Perhaps she didn't know what they had these days. Dissonance rather than harmony.\n\n'Have you met anyone?' Pamela asked cautiously.\n\n'No. No one.'\n\n'Not yet,' Pamela said in her most encouraging manner.\n\nThe _boeuf bourguignon_ had, naturally, required burgundy and in her lunch hour Ursula had dropped into the wine merchant's that she passed every day on her way to work in the City. It was an ancient place, the wood of the interior gave the impression that it had been soaked in wine over the centuries and the dark bottles with their beautiful labels seemed to hold out the promise of something that went beyond their contents. The wine merchant picked out a bottle for her, some people used inferior wine for cooking, he said, but the only use one should have for inferior wine was vinegar. He himself was acerbic and rather overwhelming. He afforded the bottle the tenderness of care due a baby, lovingly wrapping it in tissue paper and passing it to Ursula to cradle in her basket-weave shopper where her purchase remained concealed from the office during the afternoon, in case they suspected her of being a secret lush.\n\nThe burgundy was bought before the beef and that evening Ursula thought she would open the wine and try a glass, seeing as it had been lauded so highly by the wine merchant. Of course, she'd had alcohol before, she was no teetotaller, after all, but she had never drunk alone. Never uncorked an expensive bottle of burgundy and poured a glass just for herself (dressing gown, curlers, a cosy gas fire). It was like stepping into a warm bath on a cold night, the deep, mellow wine suddenly enormously comforting. This was Keats's _beaker full of the warm South_ , was it not? Her habitual despondency seemed to evaporate a little so she had another glass. When she stood up she felt quite swimmy and laughed at herself. 'Tiddly,' she said to no one and found herself wondering about getting a dog. It would be someone to talk to. A dog like Jock would greet her every day with cheerful optimism and perhaps some of that would rub off on her. Jock was gone now, a heart attack, the vet said. 'And he had such a strong little heart,' Teddy said, himself heartbroken. He had been replaced by a sad-eyed whippet that seemed too delicate for the rough and tumble of a dog's life.\n\nUrsula rinsed the glass and put the cork back in the bottle, leaving plenty for the beef tomorrow before tottering off to bed.\n\nShe fell fast asleep and didn't wake until the alarm, which made a change from the usual restlessness. _Drink, and leave the world unseen_. When she woke she realized that she couldn't possibly look after a dog.\n\nNext day at work, the tedium of filling in ledgers all afternoon was cheered by the thought of the half-bottle sitting on her kitchen draining board. After all, she could buy another bottle for the beef.\n\n'That good, eh?' the wine merchant said when she appeared again two days later.\n\n'No, no,' she laughed, 'I haven't cooked the meal yet. It struck me that I should have something equally good to drink with it.' She realized she couldn't come back here, to this lovely shop, there was a limit to how many _boeuf bourgignon_ s someone was likely to cook.\n\nFor Pamela, Ursula made an abstemious cottage pie, followed by baked apples and custard. 'I brought you a present from Scotland,' Pamela said and produced a bottle of malt whisky.\n\n*\n\nOnce the malt had been drunk she found another wine merchant, one who treated his wares with less veneration. 'For a _boeuf bourguignon_ ,' she said, although he showed no interest in its purpose. 'I'll take two, actually. I'm cooking for a lot of people.' A couple of bottles of Guinness from the public house on the corner, 'For my brother,' she said, 'he popped in unexpectedly.' Teddy wasn't quite eighteen, she doubted he was a drinker. A couple of days later the same. 'Your brother round again, miss?' the publican said. He winked at her and she flushed.\n\nAn Italian restaurant in Soho that she 'happened to be passing' happily sold her a couple of bottles of Chianti without question. 'Sherry from the wood' \u2013 she could take a jug to the Co-op at the end of the road and they would fill it up from the barrel. ('For my mother.') Rum from public houses a long way from the flat ('for my father'). She was like a scientist experimenting with the various forms of alcohol, but she knew what she liked best, that first bottle of blushful Hippocrene, the blood-red wine. She plotted how to get a case delivered ('for a family celebration').\n\nShe had become a secret drinker. It was a private act, intimate and solitary. The very thought of a drink made her heart thud with both fear and anticipation. Unfortunately, between the restrictive licensing laws and the horror of humiliation, a young woman from Bayswater could have considerable difficulty in supplying her addiction. It was easier for the rich, Izzie had an account somewhere, Harrods probably, that simply delivered the stuff to her door.\n\nShe had dipped her toe in the waters of Lethe and the next thing she knew she was drowning, from sobriety to being a drunkard in a matter of weeks. It was both shameful and a way of annihilating shame. Every morning she woke up and thought, not tonight, I won't take a drink tonight, and every afternoon the longing built as she imagined walking into her flat at the end of the day and being greeted by oblivion. She had read sensationalist accounts of the opium dens of Limehouse and wondered if they were true. Opium sounded better than burgundy for eclipsing the pain of existence. Izzie could probably supply her with the location of a Chinese opium den, she had 'kicked the gong around', she had reported blithely, but it wasn't really the kind of thing Ursula felt she could ask. It might not lead to Nirvana (she had proved an apt pupil of Dr Kellet after all), it might lead to a new Belgravia.\n\nIzzie was occasionally allowed back into the family fold ('Weddings and funerals only,' Sylvie said. 'Not christenings'). She had been invited to Pamela's wedding but to Sylvie's profound relief had sent her apologies. 'Weekend in Berlin,' she said. She knew someone with a plane ( _thrilling_ ) who was going to fly her there. Ursula visited Izzie occasionally. They had the horror of Belgravia in common, a memory that would unite them for evermore, although they never spoke of it.\n\nIn her stead there was a wedding present, a box of silver cake forks, a gift that Pamela was amused by. 'How mundane,' she said to Ursula. 'She never ceases to surprise.'\n\n'Nearly finished,' the Neasden dressmaker said through a mouthful of pins.\n\n'I suppose I am getting a bit plump,' Ursula said, looking in the mirror at the yellow satin straining to accommodate her pot belly. 'Perhaps I should join the Women's League of Health and Beauty.'\n\nStone-cold sober, she tripped on her way home from work. It was a miserable November evening a few months after Pamela's wedding, wet and dark, and she simply hadn't seen the pavement slab the edge of which had been lifted slightly by a tree root. Her hands were full \u2013 books from the library and grocery shopping, all acquired hastily in a lunch hour \u2013 and her instinct was to save the groceries and the books rather than herself. The result was that her face slammed into the pavement, the full force taken by her nose.\n\nThe pain stunned her, she had never experienced anything that came close to it before. She knelt on the ground and held her arms around herself, shopping and books now abandoned to the wet pavement. She could hear herself moaning \u2013 keening \u2013 and could do nothing to stop the noise.\n\n'Oh, my,' a man's voice said, 'how awful for you. Let me help you. You have blood all over your nice peach scarf. Is that the colour, or is it salmon?'\n\n'Peach,' Ursula murmured, polite despite the pain. She had never given much thought to the mohair muffler around her neck. There seemed to be a lot of blood. She could feel her whole face swelling and could smell the blood, thick and rusty, in her nose but the pain had lessened a degree or two.\n\nThe man was rather nice-looking, not very tall but he had sandy hair and blue eyes, and clean, polished-looking skin stretched over good cheekbones. He helped her to her feet. His hand in hers was firm and dry. 'My name's Derek, Derek Oliphant,' he said.\n\n' _Elephant?_ '\n\n'Oliphant.'\n\nThree months later they were married.\n\nDerek's origins were in Barnet and as unremarkable to Sylvie as Harold's before him. That was, of course, the essence of his appeal for Ursula. He taught history at Blackwood, a minor public school for boys ('the children of aspirant shopkeepers', Sylvie said dismissively) and courted Ursula with concerts in the Wigmore Hall and walks on Primrose Hill. They took long bike rides that ended up in pleasant pubs in the outer suburbs, a half-pint of mild for him, a lemonade for her.\n\nHer nose proved to be broken. ('Oh, poor you,' Pamela wrote. 'And you had such a nice nose.') Before he escorted her to a hospital, Derek had led her into a public house nearby to get cleaned up a little. 'Let me get you a brandy,' he said when she sat down and she said, 'No, no, I'm fine, I'll just have a glass of water. I'm not much of a drinker,' even though the previous evening she had blacked out on the floor of her bedroom in Bayswater, courtesy of a bottle of gin she had stolen from Izzie's house. She had no qualms about thieving from Izzie, Izzie had taken so much from her. Belgravia, and so on.\n\nUrsula stopped drinking almost as suddenly as she had started. She supposed she had had a hollow inside her that had been scooped out in Belgravia. She had tried to fill it with alcohol but now it was being replenished with her feelings for Derek. What were those feelings? Mostly relief that someone wanted to look after her, someone who knew nothing of her shameful past. 'I'm in love,' she wrote rather deliriously to Pamela. 'Hurrah,' Pamela wrote back.\n\n'Sometimes,' Sylvie said, 'one can mistake gratitude for love.'\n\nDerek's mother still lived in Barnet but his father was dead, as was a younger sister. 'A horrible accident,' Derek said. 'She fell into the fire when she was four years old.' Sylvie had always been very particular about fireguards. Derek himself had nearly drowned when he was a boy, he said after Ursula had offered up her own incident in Cornwall. It was one of the few adventures in her life where she felt she had played an almost entirely innocent part. And Derek? A rough tide, an upturned rowing boat, an heroic swim to shore. No Mr Winton necessary. 'I rescued myself,' he said.\n\n'He's not _entirely_ ordinary then,' Hilda said, offering Ursula a cigarette. She hesitated but declined, not ready to take on another addiction. She was in the middle of packing up her goods and chattels. She could hardly wait to leave Bayswater behind. Derek was in digs in Holborn but was finalizing the purchase of a house for them.\n\n'I've written to the landlord, by the way,' Hilda said. 'Told him we're both moving out. Ernie's wife's giving him a divorce, did I say?' She yawned. 'He's popped the question. Thought I might take him up on it. We'll both be respectable married women. I can come and visit you in \u2013 where is it again?'\n\n'Wealdstone.'\n\nThe wedding party, in a register office, was, according to Derek's wishes, restricted to his mother and to Hugh and Sylvie. Pamela was disconcerted not to be invited. 'We didn't want to wait,' Ursula said. 'And Derek didn't want any fuss.'\n\n'And don't _you_ want fuss?' Pamela asked. 'Isn't that the point of a wedding?'\n\nNo, she didn't want a fuss. She was going to belong to someone, safe at last, that was all that counted. Being a bride was nothing, being a wife was everything. 'We wanted it all to be simple,' she said resolutely. ('And cheap, by the looks of it,' Izzie said. Another set of mundane silver cake forks was dispatched.)\n\n'He seems like a pleasant enough chap,' Hugh said at what passed for a reception \u2013 a three-course luncheon in a restaurant close to the register office.\n\n'He is,' Ursula agreed. 'Very pleasant.'\n\n'Still, it's a bit of a rum do, little bear,' Hugh said. 'Not like Pammy's wedding, is it? Half of the Old Kent Road seemed to turn out for that. And poor Ted was very put out not to be invited today. As long as you're happy, though,' he added encouragingly, 'that's the main thing.'\n\nUrsula wore a dove-grey suit for the ceremony. Sylvie had provided corsages for everyone made from hothouse roses from a florist. 'Not my roses, sadly,' she said to Mrs Oliphant. 'Gloire des Mousseux, if you're interested.'\n\n'Very nice, I'm sure,' Mrs Oliphant said, in a way that didn't sound much like a compliment.\n\n'Marry in haste, repent at leisure,' Sylvie murmured to no one in particular before a restrained sherry toast to the bride and bridegroom.\n\n'Have _you_?' Hugh asked her mildly. 'Repented?' Sylvie pretended not to hear. She was in a particularly discordant mood. 'Change of life, I believe,' an embarrassed Hugh whispered to Ursula.\n\n'Me too,' she whispered back. Hugh squeezed her hand and said, 'That's my girl.'\n\n'And does Derek know you're not intact?' Sylvie asked when she was alone with Ursula in the Ladies' powder room. They were sitting on little padded stools, repairing their lipstick in the mirror. Mrs Oliphant remained at the table, having no lipstick to repair.\n\n'Intact?' Ursula echoed, staring at Sylvie in the mirror. What did that mean, that she was flawed? Or broken?\n\n'One's maidenhood,' Sylvie said. 'Deflowering,' she added impatiently when she saw Ursula's blank expression. 'For someone who is far from innocent you seem remarkably na\u00efve.'\n\nSylvie used to love me, Ursula thought. And now she didn't. 'Intact,' Ursula repeated again. She had never even considered this question. 'How will he tell?'\n\n'The blood, of course,' Sylvie said, rather testily.\n\nUrsula thought of the wisteria wallpaper. The deflowering. She hadn't known there was a connection. She thought the blood was a wound, not the breaching of the arch.\n\n'Well, he might not realize,' Sylvie sighed. 'I'm sure he won't be the first husband to be deceived on his wedding night.'\n\n'Fresh warpaint?' Hugh said easily when they returned to the table. Ted had inherited Hugh's smile. Derek and Mrs Oliphant shared the same frown. Ursula wondered what Mr Oliphant had been like. He was seldom mentioned.\n\n'Vanity, thy name is woman,' Derek said with what seemed like a forced joviality. He was not, Ursula noticed, as comfortable in social situations as she had first thought. She smiled at him, feeling a new bond. She was marrying a stranger, she realized. ('Everyone marries a stranger,' Hugh said.)\n\n'The word is \"frailty\", actually,' Sylvie said pleasantly. ' _Frailty, thy name is woman_. Hamlet. Many people misquote it for some reason.'\n\nA shadow passed over Derek's face but then he laughed it off. 'I bow to your superior learning, Mrs Todd.'\n\nTheir new house in Wealdstone had been chosen for its location, relatively near to the school where Derek taught. He had an inheritance, 'a very small sum' from his seldom-mentioned father's investments. It was a 'sound' terrace in Masons Avenue, half-timbered in the Tudor style with leaded lights and a stained-glass panel in the front door depicting a galleon in full sail, although Wealdstone seemed a long way from any ocean. The house had all modern conveniences as well as shops close by, a doctor, a dentist and a park for children to play in, in fact everything a young wife (and mother, 'one day very soon', according to Derek) could want.\n\nUrsula could see herself eating breakfast with Derek in the mornings before waving him off to work, could see herself pushing their children in prams then pushchairs then swings, bathing them in the evening and reading bedtime stories in their pretty bedroom. She and Derek would sit quietly in the lounge in the evenings listening to the wireless. He could work on the book he was writing, a school textbook \u2013 'From Plantagenets to Tudors'. ('Gosh,' Hilda said. 'Sounds thrilling.') Wealdstone was a long way from Belgravia. Thank goodness.\n\nThe rooms this married life was to be carried out in remained in her imagination until after their honeymoon, as Derek had bought and furnished the house without her ever having seen it.\n\n'That's a bit odd, don't you think?' Pamela said. 'No,' Ursula said. 'It's like a surprise gift. My wedding present from him.'\n\nWhen Derek finally carried her awkwardly over the Wealdstone threshold (a red-tiled porch that neither Sylvie nor William Morris would have approved of) Ursula couldn't help but feel a twinge of disappointment. The house proved to be more sparsely old-fashioned than the one in her imagination and there was a drabness about it that she supposed came from its not having had a woman's hand in the d\u00e9cor, so she was surprised when Derek said, 'Mother helped me.' But then there was, of course, a similar kind of occlusion in Barnet where a certain dinginess adhered to the dowager Mrs Oliphant.\n\nSylvie had passed her honeymoon in Deauville, Pamela spent hers on a walking holiday in Switzerland, but Ursula began her own marriage with a rather wet week in Worthing.\n\nShe married one man ('a pleasant enough chap') and woke up with another, one as tightly wound as Sylvie's little carriage clock.\n\nHe changed almost immediately, as if the honeymoon itself was a transition, an anticipated rite of passage for him from solicitous suitor to disenchanted spouse. Ursula blamed the weather, which was wretched. The landlady of the boarding house where they were staying expected them to vacate the premises between breakfast and dinner at six and so they spent long days sheltering in caf\u00e9s or in the art gallery and museum or fighting the wind on the pier. Evenings were spent playing partner whist with other (less dispirited) guests before retiring to their chilly bedroom. Derek was a poor card player, in more ways than one, and they lost nearly every hand. He seemed almost to wilfully misread her attempts to indicate her hand to him.\n\n'Why did you lead trumps?' she asked him later \u2013 genuinely curious \u2013 as they decorously removed their clothes in the bedroom. 'You think that nonsense is important?' he said with a look of such deep contempt that she thought it might be best to avoid games of any kind with Derek in the future.\n\nOn the first night, blood, or the lack of it, passed unnoticed, Ursula was relieved to find. 'I think you should know that I am not inexperienced,' Derek said rather pompously as they climbed into bed together for the first time. 'I believe it is the duty of a husband to know something of the world. How else can he protect the purity of his wife?' It sounded like a specious argument to Ursula but she was hardly in a position to argue.\n\n*\n\nDerek rose early each morning and did a relentless series of press-ups \u2013 as if he were in an army barracks rather than on honeymoon. ' _Mens sana in corpora sana_ ,' he said. Best not to correct him, she thought. He was proud of his Latin, as well as his smattering of ancient Greek. His mother had scrimped and saved to make sure he had a good education, 'nothing had been handed on a plate, unlike some'. Ursula had been rather good at Latin, Greek too, but she thought it best not to crow. That was another Ursula, of course. A different Ursula, unmarked by Belgravia.\n\nDerek's method of having conjugal relations was very similar to his method of exercise, even down to the same expression of pain and effort on his face. Ursula could have been part of the mattress for all he seemed to care. But what did she have to measure it against? Howie? She wished now that she had questioned Hilda about what went on in her 'pleasure palace' in Ealing. She thought of Izzie's exuberant flirting and the warm affection between Pamela and Harold. It all seemed to indicate diversion if not downright happiness. 'What's life worth if you can't have some fun?' Izzie used to say. Ursula sensed there was going to be a shortage of fun in Wealdstone.\n\nAs humdrum as her job had been, it was as nothing compared to the drudge work of keeping house, day in, day out. Everything had to be continually washed, scrubbed, dusted, polished and swept, not to mention the ironing, the folding, the hanging, the straightening. The _adjustments_. Derek was a man of right angles and straight lines. Towels, tea-towels, curtains, rugs all needed constant alignment and realignment. (As did Ursula, apparently.) But this was her job, this was the arrangement and realignment of marriage itself, wasn't it? Although Ursula couldn't get over the feeling that she was on some kind of permanent probation.\n\nIt was easier to succumb to Derek's unquestioning belief in domestic order rather than to fight it. ('A place for everything and everything in its place.') Crockery had to be scoured clean of stains, cutlery had to be polished and straightened in drawers \u2013 knives adjusted like soldiers on parade, spoons spooning each other neatly. A housewife has to be the most observant worshipper at the altar of the Lares and Penates, he said. It should be 'hearth', not 'altar', she thought, the amount of time she spent sweeping out grates and rattling clinker out of the boiler.\n\nDerek was particular about tidiness. He couldn't think, he said, if things were out of place or askew. 'Tidy house, tidy mind,' he said. He was, Ursula was learning, rather fond of aphorisms. He certainly couldn't work on 'From Plantagenets to Tudors' in the kind of muddle that Ursula seemed to create simply by entering a room. They needed the income from this textbook \u2013 his first \u2013 which William Collins was to publish and to this end he commandeered the poky dining room (table, sideboard and all) at the back of the house as his 'study' and Ursula was banished from Derek's company most evenings so that he could write. Two should live as cheaply as one, he said, and yet here they were, barely able to pay their bills because of her lack of domestic economy, so she could at least give him some peace to try to earn an extra crust. And no, thank you, he didn't want her help in typing up his manuscript.\n\nUrsula's old household routines now seemed appallingly slovenly, even to her own eyes. In Bayswater her bed often went unmade and her pots unwashed. Bread and butter made a good breakfast and there was nothing wrong, as far as she could see, with a boiled egg for tea. Married life was more exacting. Breakfasts had to be cooked and on the table at just the right time in the morning. Derek couldn't be late for school and regarded his breakfast, a litany of porridge, eggs and toast, as a solemn (and solitary) communion. The eggs were cooked in rotation throughout the week, scrambled, fried, boiled, poached, and on Fridays the excitement of a kipper. At weekends Derek liked bacon, sausage and black pudding with his eggs. The eggs came not from a shop but a smallholding three miles away, to which Ursula had to trek on foot every week because Derek had sold their bikes when they moved to Wealdstone 'to save money'.\n\nTea was a different kind of nightmare as she had to think of new things to cook all the time. Life was an endless round of chops and steaks and pies and stews and roasts, not to mention the pudding that was expected every day and in great variety. _I'm a slave to recipe books!_ she wrote with faux-cheerfulness to Sylvie, although cheerful was far from how she felt every day, poring over their demanding pages. She gained a new respect for Mrs Glover. Of course, Mrs Glover benefited from a large kitchen, a substantial budget and a full _batterie de cuisine_ , whereas the Wealdstone kitchen was fitted out in a rather paltry fashion and Ursula's housekeeping allowance never seemed to stretch throughout the week so that she was continually chastised for overspending.\n\nShe had never bothered much about money in Bayswater, if she fell short she ate less and walked instead of taking the Tube. If she really needed topping up there had always been Hugh or Izzie to fall back on, but she could hardly go running to them for money now that she had a husband. Derek would have been mortified at this slur on his manhood.\n\nAfter several months under the constraint of unending chores Ursula thought she might go mad if she couldn't find some kind of pastime to alleviate the long days. There was a tennis club that she passed en route to the shops every day. All she could see of it was the tall netting that rose behind a wooden fence and a green door in a white pebble-dash wall facing the street, but she could hear the familiar inviting summer sound of _thock_ and _twang_ and one day she found herself knocking on the green door and asking if she could join.\n\n'I've joined the local tennis club,' she said to Derek when he came home that evening.\n\n'You didn't ask me,' Derek said.\n\n'I didn't think you played tennis.'\n\n'I don't,' he said. 'I meant you didn't ask me if you could join.'\n\n'I didn't know I had to ask.' Something passed over his face, the same cloud she had briefly seen on their wedding day when Sylvie had corrected his Shakespeare. This time it took longer to pass and seemed to change him in some indefinable way, as though part of him had shrivelled inside.\n\n'Well, can I?' she said, thinking it would be better to be meek and keep the peace. Would Pammy have asked such a question of Harold? Would Harold have ever expected such a question? Ursula wasn't sure. She realized she knew nothing about marriage. And, of course, Sylvie and Hugh's alliance remained an ongoing enigma.\n\nShe wondered what argument Derek could possibly have against her playing tennis. He seemed to be having the same struggle and eventually said begrudgingly, 'I suppose so. As long as you still have time to do everything in the house.' Halfway through their tea \u2013 stewed lamb chops and mashed potatoes \u2013 he got up abruptly from the table, picked up his plate and threw it across the room and then walked out of the house without saying a word. He didn't come back until Ursula was getting ready for bed. He still wore the same thrawn expression on his face as when he had left and gave her a brief 'good night' that almost choked him as they climbed into bed.\n\nIn the middle of the night she was woken by him clambering on top of her and hitching himself wordlessly inside her. Wisteria came to mind.\n\nThe thrawn face ('that look' was how she thought of it) now made regular appearances and Ursula surprised herself with how far she would go to appease it. But it was hopeless, once he was in this mood she got on his nerves, no matter what she did or said, in fact her attempts to placate him seemed to make the situation worse, if anything.\n\nA visit was arranged to Mrs Oliphant in Barnet, the first since the wedding. They had popped in briefly \u2013 tea and a stale scone \u2013 to announce their engagement, but hadn't been back since.\n\nThis time round Mrs Oliphant fed them a limp ham salad and some small conversation. She had several odd jobs 'saved up' for Derek and he disappeared, tools in hand, leaving his womenfolk to clear up. When the washing-up was done, Ursula said, 'Shall I make a cup of tea?' and Mrs Oliphant said, 'If you like,' without any great encouragement.\n\nThey sat awkwardly in the parlour, sipping their tea. There was a framed photograph hanging on the wall, a studio portrait of Mrs Oliphant and her new husband on their wedding day, looking strait-laced in turn-of-the-century wedding garb. 'Very nice,' Ursula said. 'Do you have any photographs of Derek when he was small? Or of his sister?' she added because it didn't seem right to exclude the girl from family history merely on account of her being dead.\n\n'Sister?' Mrs Oliphant said, frowning. 'What sister?'\n\n'His sister who died,' Ursula said.\n\n'Died?' Mrs Oliphant looked startled.\n\n'Your daughter,' Ursula said. 'She fell in the fire,' she added, feeling foolish, it was hardly a detail you were likely to forget. She wondered if perhaps Mrs Oliphant was a little simple. Mrs Oliphant herself looked confused, as if she were trying to recollect this forgotten child. 'I only ever had Derek,' she concluded firmly.\n\n'Well, anyway,' Ursula said, as if this were a trivial subject to be lightly tossed away, 'you must come and visit us in Wealdstone. Now that we're settled. We're very grateful, you know, for the money that Mr Oliphant left.'\n\n'Left? He left money?'\n\n'Some shares, I think, in the will,' Ursula said. Perhaps Mrs Oliphant hadn't been involved in the probate.\n\n'Will? He left nothing but debts when he went. He's not dead,' she added as if it were Ursula who was the simple one. 'He's living in Margate.'\n\nWhat other lies and half-truths were there, Ursula wondered? Did Derek really nearly drown when he was younger?\n\n'Drown?'\n\n'Fall out of a rowing boat and swim to shore?'\n\n'Whatever gave you that idea?'\n\n'Now then,' Derek said, appearing in the doorway and making them both jump, 'what are you two gossiping about?'\n\n'You've lost weight,' Pamela said.\n\n'Yes, I suppose I have. I've been playing tennis.' How normal that made her life sound. She doggedly attended the tennis club, it was the only relief she had from the claustrophobia of life in Masons Avenue, though she had to face a constant inquisition on the subject. Every evening when he came home Derek asked if she had played tennis today, even though she only played two afternoons a week. She was always interrogated about her partner, a dentist's wife called Phyllis. Derek seemed to despise Phyllis, even though he had never met her.\n\nPamela had travelled all the way from Finchley. 'Obviously it was the only way I was ever going to see you. You must like married life. Or Wealdstone,' she laughed. 'Mother said that you put her off.' Ursula had been putting everyone off since the wedding, rebuffing Hugh's offers to 'pop in' for a cup of tea and Sylvie's hints that perhaps they should be invited to Sunday lunch. Jimmy was away at school and Teddy was in his first year at Oxford but he wrote lovely long letters to her, and Maurice, of course, had no inclination to visit anyone in his family.\n\n'I'm sure she's not too bothered about visiting. Wealdstone and so on. Not her cup of tea at all.'\n\nThey both laughed. Ursula had almost forgotten what it felt like to laugh. She felt tears start to her eyes and had to turn away and busy herself with the tea things. 'It's so nice to see you, Pammy.'\n\n'Well, you know you're welcome in Finchley whenever you please. You should get a telephone, and then we could talk all the time.' Derek thought a telephone was an expensive luxury but Ursula suspected that he simply didn't want her speaking to anyone. She could hardly voice this suspicion (and to whom \u2013 Phyllis? The milkman?) as people would think she was off her head. Ursula had been looking forward to Pamela's visit the way people looked forward to holidays. On Monday she had said to Derek, 'Pamela's coming on Wednesday afternoon,' and he had said, 'Oh?' He seemed indifferent and she was relieved that the thrawn face did not appear.\n\nAs soon as they were finished with them Ursula quickly cleared the tea things away, washed and dried them and put them back in their places.\n\n'Golly,' Pamela said, 'when did you become such a neat little _Hausfrau_?'\n\n'Tidy house, tidy mind,' Ursula said.\n\n'Tidiness is overrated,' Pamela said. 'Is anything the matter? You seem awfully down.'\n\n'Time of the month,' Ursula said.\n\n'Oh, rotten luck. I'm going to be free of that problem for a few months. Guess what?'\n\n'You're having a baby? Oh, that's wonderful news!'\n\n'Yes. Isn't it? Mother will be a grandmother again.' (Maurice had already made a start on the next generation of Todds.) 'Will she like that, do you suppose?'\n\n'Who knows? She's rather unpredictable these days.'\n\n'Did you have a nice visit from your sister?' Derek asked when he came home that night.\n\n'Lovely. She's having a baby.'\n\n'Oh?'\n\n*\n\nThe next morning her poached eggs were not 'up to scratch'. Even Ursula had to admit that the egg she presented for Derek's breakfast was a sad sight, a sickly jellyfish deposited on toast to die. A sly smile appeared on his face, an expression that seemed to indicate a certain triumph in finding fault. A new look. Worse than the old.\n\n'Do you expect me to eat that?' he asked.\n\nSeveral answers to that question passed through Ursula's mind but she rejected them all as provocative. Instead she said, 'I can do you another one.'\n\n'You know,' he said, 'I have to work all hours at a job I despise, just to keep you. You don't have to worry your silly little noggin about anything, do you? You do nothing all day \u2013 oh, no, forgive me,' he said sarcastically, 'I was forgetting you play tennis \u2013 and you can't even manage to cook me an egg.'\n\nUrsula hadn't realized he despised his job. He complained a great deal about the behaviour of the Lower Third and talked incessantly about the headmaster's lack of appreciation of his hard work, but she hadn't thought that he _hated_ teaching. He looked close to tears and she felt suddenly and unexpectedly sorry for him and said, 'I'll poach another.'\n\n'Don't bother.' She anticipated the egg would be thrown at the wall, Derek was given to tossing food around since she had joined the tennis club, but instead he delivered a massive open-handed slap to the side of her head that sent her reeling against the cooker and then to the floor where she remained, kneeling as though she were at prayer. The pain, more than the act, had taken her by surprise.\n\nDerek walked across the kitchen and stood over her with the plate containing the offending egg. For a moment she thought he was going to bring it crashing down on her but instead he slid the egg off the plate and on to the top of her head. Then he stalked out of the kitchen and she heard the front door slam a minute later. The egg slid off her hair, down her face and on to the floor, where it burst open in a quiet splash of yellow. She struggled to her feet and fetched a cloth.\n\nThat morning seemed to open up something in him. She broke rules she didn't know existed \u2013 too much coal on the fire, too much toilet paper used, a light accidentally left on. Receipts and bills were all scrutinized by him, every penny had to be accounted for and she never had any spare money.\n\nHe proved himself capable of the most enormous rants over the pettiest of things, once started he seemed unable to stop. He was angry all the time. _She_ made him angry all the time. Every evening now he demanded an exacting account of her day. How many books did she change in the library, what did the butcher say to her, did anyone call at the house? She gave up tennis. It was easier.\n\nHe didn't hit her again but violence seemed to simmer constantly beneath his surface, a dormant volcano that Ursula had unwontedly brought back to life. She was wrong-footed by him all the time so that she never seemed to have a moment to clear the befuddlement in her brain. Her very existence seemed to be irksome to him. Was life to be lived as a continuous punishment? (Why not, didn't she deserve that?)\n\nShe began to live in a strange kind of malaise, as though her head was full of fog. She had made her bed, she supposed, and now she must lie on it. Perhaps that was another version of Dr Kellet's _amor fati_. What would he say about her current predicament? More to the point, perhaps, what would he say about Derek's peculiar character?\n\nShe was to attend sports day. It was a big event in Blackwood's calendar and wives of masters were expected to attend. Derek had given her money for a new hat and said, 'Make sure you look smart.'\n\nShe went to a local shop that sold apparel for women and children, called A La Mode (although it really wasn't). It was here that she bought her stockings and undergarments. She had had no new clothes since her wedding. She didn't care enough about her appearance to badger Derek for the money.\n\nIt was a lacklustre-looking shop in a row of other lacklustre shops \u2013 a hairdressing salon, a fishmonger, a greengrocer's, a post office. She didn't have the heart or the stomach (or the budget) to bother going up to town to a smart London department store (and what would Derek say about such a jaunt?). When she worked in London, before the watershed of marriage, she had spent a lot of time in Selfridge's and Peter Robinson's. Now those places seemed as distant as foreign countries.\n\nThe contents of the shop window were protected from the sun by a yellowy-orange screen, a kind of thick cellophane that reminded her of the wrapper on a bottle of Lucozade and made everything in the window completely undesirable.\n\nIt was not the most beautiful hat but she supposed it would do. She scrutinized her reflection unwillingly in the shop's floor-to-ceiling tripartite mirror. In triptych she looked three times worse than she did in the bathroom mirror (the only one in the house that she couldn't avoid). She no longer recognized herself, she thought. She had taken the wrong path, opened the wrong door, and was unable to find her way back.\n\nSuddenly, horribly, she frightened herself by wailing, the wretched sound of utter despondency. The owner of the shop came rushing from behind the counter and said, 'There, dear, don't get upset. Time of the month, is it?' She made her sit and have a cup of tea and a biscuit and Ursula couldn't begin to express her gratitude for this simple kindness.\n\nThe school was one stop on the train and then a short walk along a quiet road. Ursula joined the stream of parents flowing through Blackwood's gates. It was exciting \u2013 and slightly terrifying \u2013 to suddenly find herself among a crush. She had been married less than six months but had forgotten what it was like to be in a crowd.\n\nUrsula had never been to the school before and was surprised by its commonplace red brick and its pedestrian herbaceous borders, quite unlike the ancient school that the men of the Todd family attended. Teddy and then Jimmy had followed in Maurice's footsteps to Hugh's old school, a lovely building of soft grey stone and as pretty as an Oxford college. ('Savage within', though, according to Teddy.) The grounds were particularly beautiful and even Sylvie admired the profusion of flowers in the beds. 'Rather romantic planting,' she said. No such romance at Derek's school, where the emphasis was on the playing fields. The boys at Blackwood were not particularly academic, according to Derek anyway, and were kept occupied by an endless round of rugby and cricket. More healthy minds in healthy bodies. Did Derek have a healthy mind?\n\nIt was too late to ask him about his sister and his father, Krakatoa would erupt, Ursula suspected. Why would you make up something like that? Dr Kellet would know.\n\nTrestle tables, bearing refreshments for parents and staff, were set up at one end of the athletics field. Tea and sandwiches and finger slices of dry Dundee cake. Ursula lingered by the tea-urn looking for Derek. He had told her he wouldn't be able to talk much to her as he was needed to 'help out' and when she did eventually spot him at the far end of the field he was diligently carrying an armful of large hoops, the purpose of which seemed mysterious to Ursula.\n\nEveryone gathered around the trestles seemed to know each other, particularly the masters' wives, and it struck Ursula that there must be a great many more social events at Blackwood than Derek ever mentioned.\n\nA couple of senior masters, gowned like bats, settled on the tea-table and she caught the name 'Oliphant'. As inconspicuously as possible Ursula stepped a little closer to them, pretending a deep fascination with the crab paste in the sandwich on her plate.\n\n'Young Oliphant's in trouble again, I hear.'\n\n'Really?'\n\n'Hit a boy, I believe.'\n\n'Nothing wrong with hitting boys. I hit them all the time.'\n\n'Bad this time though, apparently. Parents are threatening to go to the police.'\n\n'He's never been able to control a class. Ruddy awful teacher, of course.'\n\nPlates now fully loaded with cake, the two men began to wander off, Ursula drifting in their wake.\n\n'In debt up to his ears, you know.'\n\n'Perhaps he'll make some money from his book.'\n\nThey both laughed heartily as if a great joke had been told.\n\n'The wife's here today, I gather.'\n\n'Really? We'd better watch out. I hear she's very unstable.' This, too, a great joke, seemingly. A sudden shot from the starting pistol signalling the beginning of a hurdles race made Ursula jump. She let the masters amble off. She had lost her appetite for eavesdropping.\n\nShe caught sight of Derek striding towards her, hoops now replaced by an unwieldy burden of javelins. He shouted to a couple of boys for help and they trotted up obediently. As they passed Ursula, one of them sniggered, sotto voce, 'Yes, Mr Elephant, coming, Mr Elephant.' Derek dropped the javelins to the grass with a great clatter and said to the boys, 'Carry them to the end of the field, come on, get a move on.' He approached Ursula and kissed her lightly on the cheek, saying, 'Hello, dear.' She burst out laughing, she couldn't help herself. It was the nicest thing he had said to her in weeks and was voiced not to her but for the benefit of the two masters' wives who were standing nearby.\n\n'Is there something funny?' he asked, studying her face a little too long for comfort. She could tell he was seething. She shook her head in answer. She was worried she might scream out loud, could feel her own volcano bubbling up, ready to explode. She supposed she was hysterical. _Unstable_.\n\n'I have to see to the Upper School's high jump,' Derek said, frowning at her. 'I'll meet you shortly.' He walked off, still frowning, and she started to laugh again.\n\n'Mrs Oliphant? Is it Mrs Oliphant, it is, isn't it?' The two masters' wives pounced on her, lionesses sensing wounded prey.\n\nShe also travelled home alone as Derek had to supervise evening study and would eat at the school, he said. She made herself a scrappy tea of fried herring and cold potatoes and had a sudden longing for a bottle of good red wine. In fact one bottle after another until she had drunk herself to death. She scraped the herring bones into the bin. _To cease upon the midnight with no pain_. Anything was better than this ludicrous life.\n\nDerek was a joke, to the boys, to the staff. _Mr Elephant_. She could just imagine the unruly Lower Third driving him mad with rage. And his book, what of his book?\n\nUrsula never bothered much with the contents of Derek's 'study'. She had never felt any great interest in the Plantagenets or Tudors either, for that matter. She was under strict instructions not to move any of his papers or books when dusting and polishing in the dining room (as she still liked to think of it) but she didn't care to anyway, barely glancing at the progress of the great tome.\n\nHe had been working feverishly of late, the table was covered in a clutter of notes and scraps of paper. It was all disconnected sentences and thoughts \u2013 _rather amusing if somewhat primitive belief \u2013 planta genista, the common broom gives us the name Angevin \u2013 come of the devil, and to the devil they would go_. There was little sign of an actual manuscript, just corrections and re-corrections, the same paragraph written over and over with tiny changes each time, and endless trial pages, written in ruled exercise books with Blackwood's crest and motto ( _A posse ad esse \u2013_ 'from possibility to reality') on the cover. No wonder he hadn't wanted her to type up his manuscript. She had married a Casaubon, she realized.\n\nDerek's whole life was a fabrication. From his very first words to her (Oh, _my, how awful for you. Let me help you_ ) he had not been genuine. What had he wanted from her? Someone weaker than himself? Or a wife, a mother of his children, someone running his house, all the trappings of the _vie quotidienne_ but without any of its underlying chaos. She had married him in order to be safe from that chaos. He had married her, she now understood, for the same reason. They were the last two people on earth who could make anyone safe from anything.\n\nUrsula rooted through the sideboard drawers and found a sheaf of letters, the top one with the letterhead of William Collins and Sons, Co. Ltd 'regretfully' rejecting his idea for a book, in an 'already oversubscribed area of history textbooks'. There were similar letters from other educational publishers and, worse, there were unpaid bills and threatening final notices. A particularly harsh letter demanded immediate repayment of the loan taken out apparently to pay for the house. It was the kind of sour letter that she had typed up from dictation at her secretarial college, _Dear Sir, It has been brought to my notice\u2014_\n\nShe heard the front door open and her heart jumped. Derek appeared in the doorway of the dining room, a Gothic intruder on stage. 'What are you doing?'\n\nShe held up the letter from William Collins and said, 'You're a liar, through and through. Why did you marry me? Why did you make us both so unhappy?' The look on his face. That look. She was asking to be killed, but wasn't that easier than doing it herself? She didn't care any more, there was no fight in her.\n\nUrsula was expecting the first blow but it still took her by surprise, his fist punching hard into the middle of her face as if he wanted to obliterate it.\n\n*\n\nShe slept, or perhaps she passed out, on the kitchen floor and woke some time before six. She was sick and dizzy and every inch of her was sore and aching, her whole body made of lead. She was desperate for a drink of water but didn't dare turn the tap on for fear of waking Derek. Using first a chair, then the table, she hauled herself up to standing. She found her shoes and crept into the hallway where she took her coat and a headscarf from the peg. Derek's wallet was in his jacket pocket and she took a ten-shilling note, more than enough for the rail fare and then a cab onward. She felt exhausted just at the thought of this taxing journey \u2013 she wasn't even sure she could make it on foot as far as Harrow and Wealdstone station.\n\nShe slipped her coat on and pulled the headscarf over her face, avoiding the mirror in the hallstand. It would be too dreadful a sight. She left the front door slightly ajar in case the noise of it closing woke him up. She thought of Ibsen's Nora slamming the door behind her. Nora wouldn't have gone in for dramatic gestures if she had been trying to escape from Derek Oliphant.\n\nIt was the longest walk of her life. Her heart was beating so fast she thought it might give out. All the way she expected to hear his footsteps running up behind her and him yelling her name. At the ticket office she had to mumble 'Euston' through a mouthful of bloody, broken teeth. The ticket clerk glanced at her and then glanced quickly away when he saw the state she was in. Ursula supposed he had no precedent for dealing with female passengers who looked as if they had been in bare-knuckle fights.\n\nShe had to wait for the first train of the day for another ten agonizing minutes in the ladies' waiting room but at least she was able to get a drink of water and remove some of the dried blood from her face.\n\nIn the carriage she sat with her head bowed, one hand shielding her face. The men in suits and bowlers studiously ignored her. As she waited for the train to pull away she risked a glance along the platform and was relieved beyond measure that there was still no sign of Derek. With any luck he hadn't missed her yet and was still doing his press-ups on the bedroom floor, presuming her to be down in the kitchen preparing his breakfast. Friday, kipper day. The kipper still lay on the pantry shelf, wrapped in newspaper. He would be furious.\n\nWhen she got off the train at Euston her legs almost gave way. People gave her a wide berth and she worried that the cab driver would refuse the fare, but when she showed him the money he took her. They drove in silence across London, bathed in rain overnight, and now the stones of the buildings were sparkling in the first rays of sun and the soft cloudy dawn was opalescent in pinks and blues. She had forgotten how much she liked London. Her heart rose. She had decided to live and now she wanted to very much.\n\nThe cab driver helped her out at the end of the journey. 'You're sure about this, miss, are you?' he said, looking doubtfully at the large redbrick house in Melbury Road. She nodded, mutely.\n\nIt was an inevitable destination.\n\nShe rang the bell and the front door opened. Izzie's hand flew to her mouth in horror at the sight of her face. 'Oh, my God. What _happened_ to you?'\n\n'My husband tried to kill me.'\n\n'You'd better come in then,' Izzie said.\n\nThe bruises healed, very slowly. 'Battle scars,' Izzie said.\n\nIzzie's dentist fixed Ursula's teeth and she had to wear her right arm in a sling for a while. Her nose had been broken again and her cheekbones and jaw cracked. She was flawed, no longer _intact_. On the other hand, she felt as if she had been scourged clean. The past no longer weighed so heavily on the present. She sent a message to Fox Corner saying that she had gone away for the summer, 'a touring holiday of the Highlands with Derek'. She was fairly sure that Derek wouldn't contact Fox Corner. He would be licking his wounds somewhere. Barnet, maybe. He had no idea where Izzie lived, thank goodness.\n\nIzzie was surprisingly sympathetic. 'Stay as long as you like,' she said. 'It'll make a change from rattling around in here on my own. And God only knows, I've got more than enough money to keep you. Take your time,' she added. 'No rush. And you're only twenty-three, for heaven's sake.' Ursula didn't know which was more surprising \u2013 Izzie's genuine hospitality or the fact that she knew how old she was. Perhaps Izzie had been changed by Belgravia too.\n\nUrsula was in on her own one evening when Teddy turned up on the doorstep. 'You're hard to find,' he said, giving her an enormous hug. Ursula's heart bumped with pleasure. Teddy always seemed more real than other people somehow. He was brown and strong from spending the long summer vacation working on the Hall farm. He had announced recently that he wanted to be a farmer. 'I'll have the money back that I spent on your education,' Sylvie said \u2013 but smiling because Teddy was her favourite.\n\n'I believe it was _my_ money,' Hugh said. (Did Hugh have a favourite? 'You, I think,' Pamela said.)\n\n'What happened to your face?' Teddy asked her.\n\n'Bit of an accident, you should have seen it before,' she laughed.\n\n'You're not in the Highlands,' Teddy said.\n\n'Doesn't seem so, does it?'\n\n'You've left him then?'\n\n'Yes.'\n\n'Good.' Teddy, like Hugh, didn't go in for long narratives. 'Where's the giddy aunt then?' he asked.\n\n'Out giddying. The Embassy Club, I believe.' They drank some of Izzie's champagne to celebrate Ursula's freedom.\n\n'You'll be disgraced in Mother's eyes, I expect,' Teddy said.\n\n'Don't worry, I believe I already am.'\n\nTogether they made an omelette and a tomato salad and ate with their plates on their knees listening to Ambrose and his orchestra on the wireless. When they finished their food, Teddy lit a cigarette. 'You're so grown-up these days,' Ursula laughed. 'I have muscles,' he said, demonstrating his biceps like a circus strongman. He was reading English at Oxford and said it was a relief to stop thinking and 'be working on the land'. He was writing poetry, too, he said. About the land, not about 'feelings'. Teddy's heart had been fractured by Nancy's death and once a thing was cracked, he said, it could never be repaired perfectly. 'Quite Jamesian, isn't it?' he said ruefully. (Ursula thought of herself.)\n\nA bereft Teddy carried his wounds on the inside, a scar across his heart where little Nancy Shawcross had been ripped away. 'It's as if,' he said to Ursula, 'you walk into a room and your life ends but you keep on living.'\n\n'I think I understand. I do,' Ursula said.\n\n*\n\nUrsula dozed off with her head on Teddy's shoulder. She was still tremendously tired. ('Sleep is a great healer,' Izzie said, bringing her breakfast on a tray every morning.)\n\nEventually, Teddy sighed and stretched and said, 'I suppose I should be getting back to Fox Corner. What's the story, did I see you? Or are you still in Brigadoon?' He took their plates through to the kitchen. 'I'll clear up while you think about your answer.'\n\nWhen the doorbell rang Ursula presumed it was Izzie. Now that Ursula was living in Melbury Road she had grown careless about her door keys. 'But you're always here, darling,' she said when Ursula had to crawl out of bed at three in the morning to let her in.\n\nIt wasn't Izzie, it was Derek. She was so surprised she couldn't even speak. She had left him so firmly behind that she thought of him as someone who had ceased to exist. He didn't belong in Holland Park, but rather in some dark place of the imagination.\n\nHe twisted her arm behind her back and frogmarched her down the hall into the drawing room. He glanced at the coffee table, a heavy wooden thing carved in the Oriental style. Seeing the empty champagne glasses still sitting on the coffee table and the big onyx ashtray containing Teddy's cigarette stubs, he hissed, 'Who's been here with you?' He was incandescent with rage. 'Who have you been fornicating with?'\n\n'Fornicating?' Ursula said, surprised by the word. So biblical. Teddy came into the room, a dishtowel casually over his shoulder. 'What's all this?' he said, and then, 'Get your hands off her.'\n\n'Is this him?' Derek asked Ursula. 'Is this the man you're whoring around London with?' and without waiting for an answer he smashed her head on to the coffee table and she slid to the ground. The pain in her head was terrible and grew worse rather than lessened, as if she were in a vice being tightened all the time. Derek lifted the heavy onyx ashtray high as if it were a chalice, careless of the cigarette butts that showered on to the carpet. Ursula knew her brain wasn't working properly because she should have been cowering in terror but all she could think about was that this was rather like the incident with the poached egg and how silly life was. Teddy yelled something at Derek and Derek threw the ashtray at him instead of breaking open Ursula's skull with it. Ursula couldn't see whether or not the ashtray hit Teddy because Derek grabbed her by her hair, lifted her head up and cracked it back on to the coffee table. A bolt of lightning flashed in front of her eyes but the pain began to fade.\n\nShe slipped down on to the carpet, unable to move. There was so much blood in her eyes that she could barely see. The second time that her head hit the table she had felt something give way, the instinct to life perhaps. She knew from the awkward shuffling and grunting dance on the carpet around her that Derek and Teddy were fighting. At least Teddy was on his feet and not lying unconscious but she didn't want him to fight, she wanted him to run away, out of harm's way. She didn't mind dying, she really didn't, as long as Teddy was safe. She tried to say something but it came out as guttural nonsense. She was very cold and tired. She remembered feeling this way in the hospital, after Belgravia. Hugh had been there, he had held on to her hand and kept her in this life.\n\nAmbrose was still on the wireless, Sam Browne was singing 'The Sun Has Got His Hat On'. It was a jolly song to leave life to. Not what you expected.\n\nThe black bat was coming for her. She didn't want to go. The blackness edged around her. _Easeful death_. It was so cold. It will snow tonight, she thought, even though it isn't winter yet. It was already snowing, cold flakes dissolving on her skin like soap. Ursula put out a hand for Teddy to hold but this time nothing could stop her fall into the dark night.\n\n# _11 February 1926_\n\n'Ow! What d'ya do that for?' Howie yelled, rubbing his cheek where Ursula had punched him in a very unladylike way.\n\n'You have one hell of a right cross for a little girl,' Howie said, almost admiringly. He made another grab for her which she jinked as neatly as a cat. As she did so, she spotted Teddy's ball, lurking deep within the recesses of a cotoneaster. A well-aimed kick connected with Howie's shin and gave her enough time to rescue the ball from the clutches of the reluctant bush.\n\n'I just wanted a kiss,' Howie said, sounding absurdly hurt. 'It wasn't like I was trying to _rape_ you or anything.' The brutal word hung in the chilly air. Ursula might have blushed, should have blushed at the word but she felt a certain possession of it. She sensed it was what boys like Howie did to girls like Ursula. All girls, especially those celebrating their sixteenth birthdays, had to be cautious when walking through the dark, wild wood. Or, in this case, the shrubbery at the bottom of Fox Corner's garden. Howie rewarded her by looking somewhat shamefaced.\n\n'Howie!' they heard Maurice shouting. 'Leaving without you, chum!'\n\n'You had better go,' Ursula said. A small triumph for her new womanhood.\n\n'I found your ball,' she said to Teddy.\n\n'Excellent,' Teddy said. 'Thank you. Shall we have more of your birthday cake?'\n\n# _August 1926_\n\n_Il se tenait devant un miroir long, appliqu\u00e9 au mur entre les deux fen\u00eatres, et contemplait son image de tr\u00e8s beau et tr\u00e8s jeune homme, ni grand ni petit, le cheveu bleut\u00e9 comme un plumage de merle_.\n\nShe could barely keep her eyes open to read. It was beautifully hot and time treacled past every day with nothing more to do than read books and go for long walks \u2013 mainly in the vain hope of bumping into Benjamin Cole, or indeed any of the Cole boys, who had all grown into darkly handsome youths. 'They could pass for Italian,' Sylvie said. But why would they want to pass for anything other than themselves?\n\n'You know,' Sylvie said, discovering her lying beneath the apple trees, _Ch\u00e9ri_ drowsily abandoned on the warm grass, 'long, lazy days like these will never come again in your life. You think they will, but they won't.'\n\n'Unless I become incredibly rich,' Ursula said. 'Then I could be idle all day long.'\n\n'Perhaps,' Sylvie said, unwilling to renounce her newly habitual dysphoric stance. 'But summer would still come to an end one day.' She sank down on the grass next to Ursula. Her skin was freckled from working in the garden. Sylvie was always up with the sun. Ursula would have been happy to sleep all day. Sylvie leafed idly through the Colette and said, 'You should do more with your French.'\n\n'I could live in Paris.'\n\n'Perhaps not _that_ ,' Sylvie said.\n\n'Do you think I should apply to university when I finish school?'\n\n'Oh, really, dear, what's the point? It won't teach you how to be a wife and mother.'\n\n'What if I don't want to be a wife and mother?'\n\nSylvie laughed. 'Now you're just talking nonsense to provoke.' She stroked Ursula's cheek. 'You always were such a funny little thing. There's tea on the lawn,' she said, rousing herself reluctantly. 'And cake. And, unfortunately, Izzie.'\n\n'Darling,' Izzie said when she saw Ursula coming across the lawn towards her. 'You've quite grown since I last saw you. You're a woman now, and so pretty!'\n\n'Not quite,' Sylvie said. 'We were just discussing her future.'\n\n'Were we?' Ursula said. 'I thought we were discussing my French. I need more of an education,' she said to Izzie.\n\n'How serious,' Izzie said. 'At sixteen you should be head over heels in love with some unsuitable boy.' I am, Ursula thought, I am in love with Benjamin Cole. She supposed he was unsuitable. ('A Jew?' she imagined Sylvie saying. Or a Catholic, or a coalminer (or anyone foreign), a shop assistant, a clerk, a groom, a tram-driver, a schoolteacher. The unsuitable males were legion.)\n\n'Were _you_?' Ursula asked Izzie.\n\n'Was I what?' Izzie puzzled.\n\n'In love when you were sixteen?'\n\n'Oh, tremendously.'\n\n'What about you?' Ursula said to Sylvie.\n\n'Goodness, no,' Sylvie said.\n\n'But at _seventeen_ you must have been in love,' Izzie said to Sylvie.\n\n'Must I?'\n\n'When you met Hugh, of course.'\n\n'Of course.'\n\nIzzie leaned towards Ursula and dropped her voice to a conspirator's whisper. 'I eloped when I was about your age.'\n\n'Nonsense,' Sylvie said to Ursula. 'She did no such thing. Ah, here comes Bridget with the tea-tray.' Sylvie turned to Izzie. 'Was there a particular reason for your visit, or have you merely come to annoy?'\n\n'I was driving nearby, I thought I'd drop in. I've got something I wanted to ask you.'\n\n'Oh, dear,' Sylvie said wearily.\n\n'I've been thinking,' Izzie said.\n\n'Oh, dear.'\n\n' _Would_ you stop saying that, Sylvie.'\n\nUrsula poured tea and sliced cake. She sensed a battle. Izzie was rendered temporarily speechless by a mouthful of cake. It was not one of Mrs Glover's airier sponges.\n\n'As I said' \u2013 she swallowed with difficulty \u2013 'I've been thinking \u2013 and don't say anything, Sylvie. _The Adventures of Augustus_ is still _wildly_ successful, I'm writing a book every six months. It's quite crazy. And I have the house in Holland Park, and I have money, but of course no husband. Nor do I have a child.'\n\n'Really?' Sylvie said. 'Are you sure?'\n\nIzzie ignored her. 'No one to share my good fortune with. So, I was thinking, why don't I adopt Jimmy?'\n\n'I'm sorry?'\n\n'She's unbelievable,' Sylvie hissed at Hugh. Izzie was still out on the lawn, entertaining Jimmy by reading from an unfinished manuscript she had in her oversized handbag. 'Augustus Goes to the Seaside'.\n\n'Why doesn't she want to adopt me?' Teddy said. 'After all, it's me that's supposed to be Augustus.'\n\n'Do you want to be adopted by Izzie?' Hugh puzzled.\n\n'Good lord, no,' Teddy said.\n\n'No one is being adopted,' Sylvie said furiously. 'Go and have a word with her, Hugh.'\n\nIn the kitchen, Ursula went looking for an apple and found Mrs Glover thumping slices of veal with a meat tenderizer. 'I imagine that they're the heads of the Boche,' she said.\n\n'Really?'\n\n'The ones that sent the gas over that did for poor George's lungs.'\n\n'What's for dinner? I'm starving.' Ursula had grown rather callous about George Glover's lungs, she had heard so much of them that they seemed to have a life of their own, rather like Sylvie's mother's lungs, organs that seemed to have more character than their owner.\n\n'Veal cutlets _\u00e0 la Russe_ ,' Mrs Glover said, flipping the meat over and pounding again. 'The Ruskies are just as bad, mind you.' Ursula wondered if Mrs Glover had ever actually met anyone from another country.\n\n'Well, there are a lot of Jews in Manchester,' Mrs Glover said.\n\n'Did you meet any?'\n\n'Meet? Why would I meet them?'\n\n'Jews aren't necessarily foreign, though, are they? The Coles next door are Jewish.'\n\n'Don't be daft,' Mrs Glover said, 'they're as English as you and me.' Mrs Glover had a certain fondness for the Cole boys, based on their excellent manners. Ursula wondered if it was worth arguing. She took another apple and Mrs Glover returned to her pounding.\n\nUrsula ate the apple sitting on a bench in a secluded corner of the garden, one of Sylvie's favourite hideaways. The words 'Veal cutlets _\u00e0 la Russe_ ' drifted sleepily through her brain. And then suddenly she was on her feet, her heart knocking in her chest, a sudden familiar but long-forgotten terror triggered \u2013 but by what? It was so at odds with the peaceful garden, the late-afternoon warmth on her face, Hattie, the cat, washing herself lazily on the sunny path.\n\nThere were no terrible portents of doom, nothing to suggest all was not well in the world but nonetheless Ursula flung the apple core into the bushes and fled from the garden, through the gate and into the lane, the old demons snapping at her heels. Hattie paused in her toilette and viewed the swinging gate with disdain.\n\nPerhaps it was a train disaster, perhaps she would have to rip off her petticoats like the girls in _The Railway Children_ to signal the driver, but no, as she reached the station the 5.30 to London was drawing quietly alongside the platform in the safe stewardship of Fred Smith and his driver.\n\n'Miss Todd?' he said, tipping the brim of his railwayman's cap. 'Are you all right? You look worried.'\n\n'I'm fine, Fred, thank you for asking.' Just in a state of mortal dread, nothing to fret about. Fred Smith didn't look as if he had ever suffered a moment of mortal dread.\n\nShe walked back along the lane, still drenched with the nameless fear. Halfway along she met Nancy Shawcross and said, 'Hello there, what are you up to?' and Nancy said, 'Oh, just looking for things for my nature book. I've got some oak leaves and some tiny baby acorns.'\n\nThe fear started to drain from Ursula's body and she said, 'Come on, then, I'll walk back home with you.'\n\nAs they approached the dairy herd's field a man climbed over the five-bar gate and landed heavily among the cow parsley. He tipped his cap at Ursula and mumbled, 'Evening, miss,' before carrying on in the direction of the station. He had a limp that made him walk rather comically, like Charlie Chaplin. Another veteran of the war perhaps, Ursula thought.\n\n'Who was that? Nancy asked.\n\n'I have no idea,' Ursula said. 'Oh, look, there on the road, a dead devil's-coach-horse beetle. Is that any good to you?'\n\n# A Lovely Day Tomorrow\n\n# _2 September 1939_\n\n'Maurice says it will be over in a few months.' Pamela rested her plate on the neat dome that contained her next baby. She was hoping for a girl.\n\n'You're going to go on for ever until you produce one, aren't you?' Ursula said.\n\n'Till the crack of doom,' Pamela agreed cheerfully. 'So, we were invited, _much_ to my surprise. Sunday lunch in Surrey, the full works. Their rather strange children, Philip and Hazel\u2014'\n\n'I think I've only met them twice.'\n\n'You've probably met them more than that, you just didn't notice them. Maurice said he'd invited us over so that the \"cousins could get to know each other better\" but the boys didn't take to them at all. Philip and Hazel have no idea how to _play_. And their mother was being a martyr to the roast beef and apple pie. Edwina's a martyr to Maurice as well. Martyrdom would suit her, of course, she's quite _violently_ Christian considering she's C of E.'\n\n'I would hate to be married to Maurice, I don't know how she puts up with it.'\n\n'She's grateful to him, I think. He's given her Surrey. A tennis court, friends in the Cabinet, lots of roast beef. They _entertain_ a lot \u2013 the great and the good. Some women would suffer for that. Even suffer Maurice.'\n\n'I expect he's a great test of her Christian tolerance.'\n\n'A great test of Harold's beliefs in general. He had a scrap with Maurice over welfare, another one with Edwina about predestination.'\n\n'She believes in that? I thought she was an Anglican.'\n\n'I know. She has no sense of logic though. She's remarkably stupid, I suppose that's why he married her. Why do you think Maurice says the war will only last a few months? Is that just departmental bluster? Do we believe everything he says? Do we believe _anything_ he says?'\n\n'Well, generally speaking, no,' Ursula said. 'But he is a big chief in the Home Office, so he _ought_ to know, presumably. Home Security, new department as of this week.'\n\n'You too?' Pamela asked.\n\n'Yes, me too. The ARP Department is now a ministry, we're all still getting used to the idea of being grown-ups.'\n\nWhen Ursula left school at eighteen she had not gone to Paris, nor, despite the exhortations of some of her teachers, had she applied to Oxbridge and done a degree in any languages, dead or alive. She had not in fact gone further than High Wycombe and a small secretarial college. She was eager to _get on_ and earn her independence rather than be cloistered in another institution. ' _Time's winged chariot_ , and all that,' she said to her parents.\n\n'Well, we all _get on_ ,' Sylvie said, 'one way or another. And in the end we all arrive at the same place. I hardly see that it matters how we get there.'\n\nIt seemed to Ursula that _how_ you got there was the whole point but there was nothing to be gained from arguing with Sylvie on the days when she was mired in gloom. 'I shall be able to get an interesting job,' Ursula said, brushing off her parents' objections, 'working in a newspaper office or perhaps a publishing house.' She imagined a Bohemian atmosphere, men in tweed jackets and cravats, women smoking in a sophisticated manner while sitting at their Royals.\n\n'Anyway, good for you,' Izzie said to Ursula, over a rather superior afternoon tea at the Dorchester to which she had invited both Ursula and Pamela ('She must want something,' Pamela said).\n\n'And who wants to be a boring old bluestocking?' Izzie said.\n\n'Me,' Pamela said.\n\nIt turned out that Izzie did have an ulterior motive. Augustus was so successful that Izzie's publisher had asked her to produce 'something similar' for girls. 'But not books based on a _naughty_ girl,' she said. 'That apparently won't do. They want a gung-ho sort, hockey-captain kind of thing. Lots of japes and scrapes but always towing the line, nothing that will frighten the horses.' She turned to Pamela and said sweetly, 'So I thought of you, dear.'\n\nThe college had been run by a man called Mr Carver, a man who was a great disciple of both Pitman's and Esperanto and who tried to make his 'girls' wear blindfolds when they were practising their touch-typing. Ursula, suspecting there was more to it than monitoring their skills, led a revolt of Mr Carver's 'girls'. 'You're such a rebel,' one of them \u2013 Monica \u2013 said admiringly. 'Well, not really,' Ursula said. 'Just being sensible, you know.'\n\nShe was. She had become a sensible sort.\n\nAt Mr Carver's college Ursula had proved to have a surprising aptitude for typing and shorthand, although the men who interviewed her for the job in the Home Office, men she would never see again, clearly believed that her proficiency in the Classics would somehow stand her in better stead when opening and closing filing-cabinet drawers and conducting endless searches among a sea of buff-coloured folders. It wasn't quite the 'interesting job' she had envisaged but it kept her attention and over the next ten years she rose slowly through the ranks, in the bridled way that women did. ('One day a woman will be Prime Minister,' Pamela said. 'Maybe even in our lifetime.') Now Ursula had her own junior clericals to chase down the buff folders for her. She supposed that was progress. Since '36 she'd been working in the Air Raid Precautions Department.\n\n' _You've_ not heard rumours then?' Pamela said.\n\n'I'm a lowly squaw, I hear nothing but rumours.'\n\n'Maurice can't say what he does,' Pamela grumbled. 'Couldn't possibly talk about what goes on within the \"hallowed walls\". He actually used that term \u2013 hallowed walls. You would think he had signed the Official Secrets Act with his blood and pledged his soul as warranty.'\n\n'Oh, well, we all have to do that,' Ursula said, helping herself to cake. 'De _rigueur_ , don't you know. Personally, I suspect Maurice just goes around _counting_ things.'\n\n'And feeling very pleased with himself. He's going to love the war, lots of power and no personal danger.'\n\n' _Lots_ of things to count.' They both laughed. It struck Ursula that they seemed very merry for people on the brink of dreadful conflict. They were in the garden of Pamela's house in Finchley, a Saturday afternoon with the tea things set out on a spindly bamboo table. They were eating cake, almond speckled with chopped-up pieces of chocolate, an old recipe of Mrs Glover's handed down on a piece of paper that was covered in greasy fingerprints. In places, the paper was as transparent as a dirty windowpane.\n\n'Make the most of it,' Pamela said, 'there'll be no more cake, I expect.' She fed a piece to Heidi, an unprepossessing mongrel rescued from Battersea. 'Did you know that people are putting their pets down, thousands of them?'\n\n'That's horrible.'\n\n'As if they weren't part of the _family_ ,' Pamela said, rubbing the top of Heidi's head. 'She's much nicer than the boys. Better behaved too.'\n\n'How were your evacuees?'\n\n'Grubby.' Despite her condition, Pamela had spent most of the morning organizing evacuees at Ealing Broadway while Olive, her mother-in-law, looked after the boys.\n\n'You would be so much more help to the war effort than someone like Maurice,' Ursula said. 'If it were up to me I would make you Prime Minister. You'd make a much better job of it than Chamberlain.'\n\n'Well, that's true.' Pamela put down her tea plate and took up her knitting \u2013 something pink and lacy. 'If it's a boy I'm just going to _pretend_ it's a girl.'\n\n'And aren't _you_ going to leave?' Ursula asked. 'You're not going to keep the boys in London, are you? You could go and stay at Fox Corner, I don't expect the Germans will be much bothered with bombing sleepy hollow.'\n\n'And stay with Mother? Lord, no. I have a friend from university, Jeanette, a vicar's daughter, not that that's relevant, I suppose. There's a cottage that belonged to her grandmother, up in Yorkshire, Hutton-le-Hole, dot on the map, that kind of place. She's going up there with her two boys and suggested I join her with my three.' Pamela had given birth in quick succession to Nigel, Andrew and Christopher. She had taken to motherhood with gusto. 'Heidi will love it too. It sounds utterly primitive, no electricity, no running water. Wonderful for the boys, they can run around like savages. It's hard to be a savage in Finchley.'\n\n'I expect some people manage,' Ursula said.\n\n'How's \"the man\"?' Pamela asked. ' \"The Man from the Admiralty\".'\n\n'You _can_ use his name,' Ursula said, brushing cake crumbs off her skirt. 'The antirrhinums don't have ears.'\n\n'You never know these days. Has _he_ said anything?'\n\nUrsula had been involved with Crighton \u2013 'the Man from the Admiralty' \u2013 for a year now (she dated it from Munich). They had met at an inter-departmental meeting. He was fifteen years older than Ursula, rather dashing and with a vaguely wolfish air that was barely offset by his marriage to an industrious wife (Moira) and their clutch of three girls, all at a private school. 'I shan't leave them, no matter what,' he told her after the first time they had made love in the rather basic quarters of his 'emergency bolthole'.\n\n'But I don't want you to,' Ursula said, although as a declaration of his intent she thought it might have been better if he had let it precede the act rather than provide its coda.\n\n'The bolthole' (she suspected that she was not the first woman to have seen the inside of it at Crighton's invitation) was a flat provided by the Admiralty for the nights when Crighton stayed in town rather than 'hiking' all the way back to Moira and the girls in Wargrave. The bolthole wasn't his exclusively and when it wasn't available he 'trekked out' to Ursula's flat in Argyll Road where they spent long evenings in her single bed (he had a sailor's practical attitude towards confined spaces) or on her sofa, pursuing the 'delights of the flesh' as he put it, before he 'slogged his way' back to Berkshire. Any journey on land, even a couple of stops on the Tube, had an expeditionary quality for Crighton. He was a naval man at heart, Ursula supposed, and would have been happier sailing a skiff to the Home Counties rather than making his way overland. They did once take a little boat out to Monkey Island and have a picnic on the banks of the river. 'Like a normal couple,' he said apologetically.\n\n'What then, if not love?' Pamela asked.\n\n'I _like_ him.'\n\n'I like the man who delivers my groceries,' Pamela said. 'But I don't share my bed with him.'\n\n'Well, I can assure you he means a good deal more to me than a tradesman.' They were almost arguing. 'And he's not a callow youth,' she continued for the defence. 'He's a proper person, he comes whole, all... ready-made. You know?'\n\n'Ready-made with a family,' Pamela said, rather crotchety now. She looked quizzical and said, 'But doesn't your heart beat a little faster at the sight of him?'\n\n'Perhaps a little faster,' Ursula conceded generously, sidestepping the argument, suspecting that she would never be able to explain the forensics of adultery to Pamela. 'Who would have thought that out of everyone in our family, you would turn out to be the romantic?'\n\n'Oh, no, I think that's Teddy,' Pamela said. 'I just like to believe that there are nuts and bolts that hold our society together \u2013 especially now \u2013 and that marriage is part of that.'\n\n'Nothing romantic about nuts and bolts.'\n\n'I admire you, really,' Pamela said. 'Being your own woman. Not following the herd and so on. I just don't want you to be hurt.'\n\n'Believe me, neither do I. Pax?'\n\n'Pax,' Pamela agreed readily. Laughing, she said, 'My life would be so dull without your salacious reports from the front line. What a deal of vicarious excitement I derive from your love life \u2013 or whatever you want to call it.'\n\nThere had been nothing salacious about their outing to Monkey Island, they had sat chastely on a tartan blanket and eaten cold chicken and drunk warm red wine. 'The _blushful Hippocrene_ ,' Ursula said and Crighton laughed and said, 'That sounds suspiciously like literature to me. I have no poetry in me. You should know that.'\n\n'I do.'\n\nThe thing about Crighton was that there always seemed to be more of him than he ever revealed. She had overheard someone in the office refer to him as 'the Sphinx' and he did indeed wear an air of reticence that hinted at unexplored depths and suppressed secrets \u2013 some childhood harm, some magnificent obsession. His cryptic self, she thought, peeling a hard-boiled egg and dipping it into a little screw of paper that contained salt. Who had packed this picnic \u2013 not Crighton, surely? Not Moira, heaven forfend.\n\nHe had grown rather remorseful over the clandestine nature of their relationship. She had brought a little excitement into what had become a rather tedious life, he said. He had been at Jutland with Jellicoe, he had 'seen much' and now he was 'little better than a bureaucrat'. He was restless, he said.\n\n'You're either about to declare your love for me,' Ursula said, 'or tell me that it's all over.' There was fruit \u2013 peaches nestling inside tissue paper.\n\n'It's a fine balance,' he said, with a rueful smile. 'I am teetering.' Ursula laughed, the word didn't suit him.\n\nHe embarked on a tale about Moira, something to do with her life in the village and her need for committee work, and Ursula drifted off, more interested in the discovery of a Bakewell tart that had apparently been magicked from a kitchen somewhere deep in the Admiralty. ('We're well looked after,' he said. Like Maurice, she thought. The privileges of men in power, unavailable to those adrift on the sea of buff.)\n\nIf Ursula's older female colleagues had got wind of the affair, there would have been a stampede for the smelling salts, especially if they had known just who in the Admiralty it was that she was dallying with (Crighton was rather senior). Ursula was good, very good, at keeping secrets.\n\n'Your reputation for discretion precedes you, Miss Todd,' Crighton had said when introduced to her.\n\n'Goodness,' Ursula said, 'that makes me sound so dull.'\n\n'Intriguing, rather. I suspect you would make a good spy.'\n\n'And how _was_ Maurice? In himself?' Ursula asked.\n\n'Maurice is very well \"in himself\", in that he _is_ himself and will never change.'\n\n'Invitations to Sunday lunch in Surrey never come _my_ way.'\n\n'Count yourself lucky.'\n\n'In fact I hardly ever see him. You wouldn't think we worked in the same ministry. He walks the airy corridors of power\u2014'\n\n'The hallowed walls.'\n\n'The hallowed walls. And I scurry around in a bunker.'\n\n'Are you? In a bunker?'\n\n'Well, it's above ground. In South Ken, you know \u2013 in front of the Geological Museum. Not Maurice, he prefers his Whitehall office to our War Room.'\n\nWhen she had applied originally for a job in the Home Office, Ursula had rather presumed that Maurice would put in a good word for her but instead he had blustered on about nepotism and having to be seen to be above any suspicion of favouritism, 'Caesar's wife and so on,' he said. 'And I take it Maurice is Caesar in this conceit, rather than Caesar's wife?' Pamela said. 'Oh, don't put that idea into my head,' Ursula laughed. 'Maurice as a woman, imagine.'\n\n'Ah, but a _Roman_ woman. That would suit him more. What was Coriolanus's mother called?'\n\n'Volumnia.'\n\n'Oh, and I know what I had to tell you \u2013 Maurice invited a friend to lunch,' Pamela said. 'From his Oxford days, that big American chap. Do you remember?'\n\n'I do!' Ursula struggled to come up with the name. 'Oh, darn, what was he called... something American. He tried to kiss me on my sixteenth birthday.'\n\n'The swine!' Pamela laughed. 'You never said.'\n\n'Hardly what you want from a first kiss. More like a rugby tackle. He was a bit of a lout.' Ursula laughed. 'I think I hurt his pride \u2013 or perhaps more than his pride.'\n\n'Howie,' Pamela said. 'Only now he's Howard \u2013 Howard S. Landsdowne III to give him his full title, apparently.'\n\n'Howie,' Ursula mused. 'I had quite forgotten. What's he doing now?'\n\n'Something diplomatic. He's even more secretive than Maurice. At the embassy, Kennedy's a god to him. I think Howie rather admires old Adolf.'\n\n'Maurice, too, probably, if he weren't quite so _foreign_. I saw him once at a Blackshirt meeting.'\n\n'Maurice? Never! Perhaps he was spying, I can imagine him as an _agent provocateur_. What were _you_ doing there?'\n\n'Oh, you know, espionage, like Maurice. No, really just happenstance.'\n\n'So many startling revelations for one pot of tea. Are there more to come? Should I brew another pot?'\n\nUrsula laughed. 'No, I think that's it.'\n\nPamela sighed. 'It's bloody, isn't it?'\n\n'What, about Harold?'\n\n'Poor man, I suppose he'll have to stay here. They can't really call up hospital doctors, can they? They'll need them if we're bombed and gassed. We _will_ be bombed and gassed, you do know that, don't you?'\n\n'Yes, of course,' Ursula said, as offhand as if they were talking about the weather.\n\n'What an awful thought.' Pamela sighed again, abandoning her needles and stretching her arms above her head. 'It's such a glorious day. It's hard to believe this is probably the last ordinary day we'll have for a long time.'\n\nUrsula had been due to begin her annual leave on Monday. She had planned a week of leisurely day trips \u2013 Eastbourne and Hastings or perhaps as far afield as Bath or Winchester \u2013 but with war about to be declared it seemed impossible to think of going anywhere. She felt suddenly listless at the idea of what might lie ahead. She had spent the morning on Kensington High Street, stocking up \u2013 batteries for her torch, a new hot-water bottle, candles, matches, endless amounts of black paper, as well as tins of baked beans and potatoes, vacuum-packed coffee. She had bought clothes too, a good woollen frock for eight pounds, a green velvet jacket for six, stockings and a pair of nice tan leather brogues that looked made to last. She had felt pleased with herself for resisting a yellow _cr\u00eape de Chine_ tea dress, patterned with little black swallows. 'My winter coat's only two years old,' she said to Pamela, 'it'll see the war out, surely?'\n\n'Goodness, I would hope so.'\n\n'It's all so horrid.'\n\n'I know,' Pamela said, cutting more cake. 'It's vile. It makes me so _cross_. Going to war is madness. Have more cake, why don't you? May as well, while the boys are still at Olive's. They'll come in and go through the place like locusts. God knows how we'll manage with rationing.'\n\n'You'll be in the country \u2013 you can grow things. Keep chickens. A pig. You'll be all right.' Ursula felt miserable at the thought of Pamela going away.\n\n'You should come.'\n\n'I should stay, I'm afraid.'\n\n'Oh, good, here's Harold,' Pamela said when Harold appeared, carrying a big bunch of dahlias wrapped in damp newspaper. She half rose to greet him and he kissed her on the cheek and said, 'Don't get up.' He kissed Ursula as well and presented the dahlias to Pamela.\n\n'A girl was selling them on the street corner, in Whitechapel,' he said. 'Very _Pygmalion_. Said they came from her grandfather's allotment.' Crighton had given Ursula roses once but they had quickly drooped and faded. She rather envied Pamela her robust allotment flowers.\n\n'So, anyway,' Harold said, when he had poured himself a lukewarm cup of tea from the pot, 'we're already evacuating patients who are well enough to be moved. They're definitely going to declare war tomorrow. In the morning. It's probably timed so that the nation can get down on its collective knees in church and pray for deliverance.'\n\n'Oh, yes, war is always so _Christian_ , isn't it?' Pamela said sarcastically. 'Especially when one is English. I have several friends in Germany,' she said to Ursula. 'Good people.'\n\n'I know.'\n\n'Are they the enemy now?'\n\n'Don't get upset, Pammy,' Harold said. 'Why is it so quiet, what have you done with the boys?'\n\n'Sold them,' Pamela said, perking up. 'Three for the price of two.'\n\n'You ought to stay the night, Ursula,' Harold said kindly. 'You shouldn't be on your own tomorrow. It'll be one of those awful days. Doctor's orders.'\n\n'Thanks,' Ursula said. 'But I've already got plans.'\n\n'Good for you,' Pamela said, picking up her knitting again. 'We mustn't behave as if the world is coming to an end.'\n\n'Even if it is?' Ursula said. She wished now that she'd bought the yellow _cr\u00eape de Chine_.\n\n# _November 1940_\n\nShe was on her back, lying in a shallow pool of water, a fact that didn't worry her so much at first. The worst thing was the awful smell. It was a combination of different things, none of them good, and Ursula was trying to separate them into their components. The fetid stench of gas (domestic) for one, and, for another, the stink of sewage, disgustingly rank, that was making her gag. Added to that was a complex cocktail of damp, old plaster and brick dust, all mixed with the traces of human habitation \u2013 wallpaper, clothes, books, food \u2013 and the sour, alien smell of explosive. In short, the essence of a dead house.\n\nIt was as if she were lying at the bottom of a deep well. Through a hazy veil of dust, like fog, she could make out a patch of black sky and a pared fingernail of moon that she remembered noticing earlier in the evening when she had looked out of the window. That seemed a long time ago.\n\nThe window itself, or at least the frame, was still there, way, way above her, not where it should be at all. It was definitely her window, she recognized the curtains, charred rags now, flapping in the breeze. They were \u2013 had been \u2013 a thick jacquard brocade from John Lewis's that Sylvie had helped her pick out. The flat in Argyll Road was rented as furnished but Sylvie declared the curtains and rugs to be 'completely shoddy' and subbed Ursula for new ones when she moved in.\n\nAt the time Millie had suggested that she move in with her in Phillimore Gardens. Millie was still playing ing\u00e9nues and said she expected to go from Juliet to the Nurse with nothing in between. 'It would be fun,' Millie said, 'to share digs,' but Ursula wasn't so sure that Millie's idea of fun coincided with her own. She often felt rather dull and sober against Millie's brightness. A dunnock keeping company with a kingfisher. And sometimes Millie burnt just a little too brightly.\n\nThis was just after Munich and Ursula had already started her affair with Crighton and it seemed more practical to live on her own. Looking back, she realized that she had accommodated Crighton's needs a great deal more than he had hers, as if Moira and the girls somehow trumped her own existence.\n\nThink about Millie, she told herself, think about the curtains, think about Crighton if you must. Anything except her present predicament. Especially the gas. It seemed particularly important to try to take her mind off the gas.\n\nAfter their purchases in soft furnishings Sylvie and Ursula had taken afternoon tea in John Lewis's restaurant, served by a grimly efficient waitress. 'I'm always so glad,' Sylvie murmured, 'that I don't have to take a turn at being other people.'\n\n'You're very good at being yourself,' Ursula said, aware that it didn't necessarily sound like a compliment.\n\n'Well, I've had years of practice.'\n\nIt was a very good afternoon tea, the kind you couldn't get any more in department stores. And then John Lewis itself was destroyed, no more than the black toothless skull of a building. ('How awful,' Sylvie wrote, moved in a way that she didn't appear to have been by the dreadful raids on the East End.) It was up and running again in days, 'Blitz spirit' everyone said, but really, what was the alternative?\n\nSylvie had been in a good mood that day, and they had drawn closer over the subject of curtains and the idiocy of people who thought that Chamberlain's silly little piece of paper meant anything at all.\n\nIt was very quiet and Ursula wondered if her eardrums were shattered. How did she get here? She remembered looking out of the window in Argyll Road \u2013 the window that was so far away now \u2013 and seeing the sickle moon. And before that she had been sitting on the sofa, doing some sewing, turning the collar on a blouse, with the wireless tuned to a short-wave German station. She was taking a German evening class ( _know your enemy_ ) but was finding it difficult to decipher anything beyond the occasional violent noun ( _Luftangriffe, Verluste_ ) in the broadcast. In despair at her lack of proficiency, she had turned the wireless off and put Ma Rainey on the gramophone. Before she left for America, Izzie had bequeathed Ursula her collection of records, an impressive archive of female American blues artistes. 'I don't listen to that stuff any more,' Izzie said. 'It's very _pass\u00e9_. The future lies with something a little more _soign\u00e9_.' Izzie's Holland Park house was shut up now, everything covered in dustsheets. She had married a famous playwright and they had decamped to California in the summer. ('Cowards, the pair of them,' Sylvie said.\n\n'Oh, I don't know,' Hugh said, 'I'm sure if I could sit out the war in Hollywood I would.')\n\n'That's interesting music I hear you listening to,' Mrs Appleyard said to Ursula one day as they passed on the stairs. The wall between their flats was paper-thin and Ursula said, 'I'm sorry. I don't mean to disturb you,' although she could well have added that she heard Mrs Appleyard's baby bawling its head off day and night and that was _very_ disturbing. The baby at four months old was big for its age, fat and ruddy, as if it had leeched all the life out of Mrs Appleyard.\n\nMrs Appleyard \u2013 the deadweight of the baby asleep in her arms, its head on her shoulder \u2013 waved a dismissive hand and said, 'Don't be concerned, it doesn't bother me.' She was lugubriously East European, a refugee of some kind, Ursula supposed, although her English was precise. Mr Appleyard had disappeared some months ago, gone for a soldier, perhaps, but Ursula hadn't asked as the marriage had been clearly (and audibly) unhappy. Mrs Appleyard was pregnant when her husband left and, as far as Ursula could tell (or hear), he had never been back to meet his squawking infant.\n\nMrs Appleyard must have been pretty once but day by day she grew thinner and sadder until it seemed as though only the (very) solid burden of the baby and its needs kept her tethered to everyday life.\n\nIn the bathroom that they shared on the first floor there was always an enamel pail in which the baby's foul-smelling nappies lay soaking before being boiled in a pan on Mrs Appleyard's two-ring stove. On the neighbouring ring there was usually to be found a pan of cabbage and, perhaps as a result of this twin boiling, she always carried on her person a faint perfume of old vegetables and damp laundry. Ursula recognized it, it was the smell of poverty.\n\nThe Misses Nesbit, nesting on the top floor, fretted a good deal about Mrs Appleyard and the baby in the way that old maids were inclined to. The two Nesbits, Lavinia and Ruth, slight spinsters, lived in the attic rooms ('beneath the eaves, like swallows', they twittered). They might as well have been twins for all the difference between them and Ursula had to make a tremendous effort to remember which was which.\n\nThey were long retired \u2013 they had both been telephonists in Harrods \u2013 and were a frugal pair, their only indulgence being an impressive collection of costume jewellery, purchased mainly from Woolworths in their lunch hour, during their 'working years'. Their flat smelt quite different to Mrs Appleyard's, lavender water and Mansion House polish \u2013 the scent of old ladies. Ursula sometimes did shopping for both the Nesbits and Mrs Appleyard. Mrs Appleyard was always ready at the door with the exact money that she owed (she knew the price of everything) and a polite 'thank you', but the Nesbits were forever trying to inveigle Ursula inside with weak tea and stale biscuits.\n\nBelow them, on the second floor, were to be found Mr Bentley ('a queer fish', they were all agreed) whose flat smelt (appropriately) of the finnan haddock he boiled in milk for his supper, and next door to him the aloof Miss Hartnell (whose flat smelt of nothing at all) who was a housekeeper at the Hyde Park Hotel and rather severe, as if nothing could ever hope to meet her standards. She made Ursula feel distinctly wanting.\n\n'Disappointed in love, I believe,' Ruth Nesbit whispered in mitigation to Ursula, clamping her bird-boned hand on her chest as if her own frail heart might be about to jump ship and attach itself to someone unsuitable. Both the Misses Nesbit were deeply sentimental about love, never having experienced its rigours. Miss Hartnell looked more as if she would mete out disappointment than receive it.\n\n'I also have some records,' Mrs Appleyard said with the earnestness of a conspirator. 'But, alas, no gramophone.' Mrs Appleyard's 'alas' seemed freighted with all the tragedy of a broken continent. It could hardly bear the weight it was asked to carry.\n\n'Well, do please feel welcome to come and play them on mine,' Ursula said, rather hoping that the downtrodden Mrs Appleyard wouldn't take up the offer. She wondered what kind of music Mrs Appleyard possessed. It seemed impossible that it could be anything very jolly.\n\n'Brahms,' Mrs Appleyard said, answering the unasked question. 'And Mahler.' The baby shifted restlessly as if disturbed by the prospect of Mahler. Whenever Ursula met Mrs Appleyard on the stairs or the landing, the baby was asleep. It was as if there were two babies, the one inside the flat who never stopped crying and the one outside who never started.\n\n'Would you mind holding Emil for a moment while I find my keys?' Mrs Appleyard asked, handing the cumbersome child over without waiting for an answer.\n\n'Emil,' Ursula murmured. She hadn't thought of the baby as having a name. Emil was, as usual, dressed for some kind of Arctic winter, bulked out with nappies and rubber knickers and romper suits and all kinds of knitted and beribboned garments. Ursula wasn't a stranger to babies, both she and Pamela had mothered Teddy and Jimmy with the same enthusiasm they accorded puppies and kittens and rabbits, and she was the very picture of a doting aunt where Pamela's boys were concerned, but Mrs Appleyard's baby was of a less appealing order. The Todd babies smelt sweetly of milk and talcum powder and the fresh air that their clothes were dried in, whereas Emil had a slightly gamey scent.\n\nMrs Appleyard rummaged for her keys in her large battered handbag, an item that looked as if it, too, had crossed Europe from a faraway country (of which Ursula, patently, knew nothing). With a great sigh, Mrs Appleyard finally located the keys at the bottom of the bag. The baby, perhaps sensing the proximity of the threshold, squirmed in Ursula's arms as if preparing itself for the transition. It opened its eyes and looked rather quarrelsome.\n\n'Thank you, Miss Todd,' Mrs Appleyard said, reclaiming the baby. 'It was nice talking to you.'\n\n'Ursula,' Ursula said. 'Do please call me Ursula.'\n\nMrs Appleyard hesitated before saying, almost shyly, 'Eryka. E-r-y-k-a.' They had lived next door to each other for a year now but this was the nearest they had come to intimacy.\n\nAlmost as soon as her door closed the baby began its customary roaring. 'Does she stick pins in it?' Pamela wrote. Pamela produced placid babies. 'They don't tend to turn feral until they're two,' she said. She had given birth to another boy, Gerald, just before last Christmas. 'Better luck next time,' Ursula said when she saw her. She had taken a train north to visit the new arrival, a long and challenging journey, most of which was spent in the guard's van, on a train packed with soldiers on their way to a training camp. She had been subjected to a barrage of sexual innuendo which had started as amusing and ended as tedious. 'Not exactly perfect gentle knights,' she said to Pamela when she finally arrived, the last part of the journey being accomplished in a donkey-cart as if time had slipped into some other century, some other country even.\n\nPoor Pammy was bored with the phoney war and with being shut up with so many little boys, 'like being a matron in a boys' school'. Not to mention Jeanette who had proved to be 'a bit of a slacker' (not to mention a moaner and a snorer). 'One expects better of a vicar's daughter,' Pamela wrote, 'although goodness knows why.' She had decamped back to Finchley in the spring but since the nightly raids had started she had retreated with her brood to Fox Corner 'for the duration', despite her previous misgivings about living with Sylvie. Harold, now at St Thomas's, was working on the front line. The nurses' home there had been bombed a couple of weeks ago and five nurses killed. 'Every night is hell,' Harold reported. It was the same report that Ralph gave from the bombsites.\n\nRalph! Of course, Ralph. Ursula had quite forgotten him. He had been in Argyll Road too. Was he there when the bomb exploded? Ursula struggled to turn her head to look around, as if she would find him among the wreckage. There was no one, she was alone. Alone and corralled in a cage of smashed wooden beams and jagged rafters, the dust settling all around her, in her mouth, her nostrils, her eyes. No, Ralph had already left when the sirens went.\n\nUrsula was no longer bedded by her man from the Admiralty. The declaration of war had brought on a sudden flush of guilt in her lover. They must stop their affair, Crighton said. The temptations of the flesh were apparently secondary to martial pursuits \u2013 as if she were Cleopatra about to destroy his Antony for love. There was enough excitement in the world now, it seemed, without the added hazards of 'keeping a mistress'. 'I'm a mistress?' Ursula said. She had not thought of herself as sporting a scarlet letter, a rubric that belonged to a racier woman, surely?\n\nThe balance had shifted. Crighton had teetered. And apparently tottered. 'Very well,' she had said equably. 'If that's what you want.' She had begun to suspect by then that there was not, in fact, a different, more intriguing Crighton hidden beneath the enigmatic surface. He was not so very inscrutable, after all. Crighton was Crighton \u2013 Moira, the girls, Jutland, although not necessarily in that order.\n\nDespite the fact that the end of the affair was at his instigation, he was cut up. Wasn't she? 'You're very cool,' he said.\n\nBut she had never been 'in love' with him, she said. 'And I expect we can still be friends.'\n\n'I don't think that we can, I'm afraid,' Crighton said, already wistful for what was now history.\n\nNonetheless, she had spent the following day dutifully crying for her loss. Her _liking_ for him had not been quite the negligent emotion that Pamela seemed to think. Then she dried her tears, washed her hair and went to bed with a plate of Bovril on toast and a bottle of 1929 Ch\u00e2teau Haut-Brion that she had filched from Izzie's excellent wine cellar, left casually behind in Melbury Road. Ursula had the keys to Izzie's house. 'Just help yourself to anything you can find,' Izzie had said. So she did.\n\nIt was rather a shame though, Ursula thought, that she no longer had assignations with Crighton. The war made indiscretions easier. The blackout was the perfect screen for illicit liaisons, and the disruption of the bombing \u2013 when it finally started \u2013 would have provided him with plenty of excuses for not being in Wargrave with Moira and the girls.\n\nInstead, Ursula was having an entirely above-board relationship with a fellow student on her German course. After the initial class ( _Guten Tag. Mein Name ist Ralph. Ich bin dreizig Jahre alt_ ) the two of them had retired to the Kardomah on Southampton Row, almost invisible behind a wall of sandbags these days. It turned out that he worked in the same building as she did, on the bomb-damage maps.\n\nIt was only as they left the class \u2013 held in a stuffy room, three floors up in Bloomsbury \u2013 that Ursula noticed that Ralph was limping. Wounded at Dunkirk, he said, before she could ask. Shot in the leg while waiting in the water to get into one of the little boats that were shuttling back and forth between the shore and the bigger boats. He was hauled on board by a fisherman from Folkestone who was shot in the neck minutes later. 'There,' he said to Ursula, 'now we don't need to talk about it again.'\n\n'No, I don't suppose we do,' Ursula said. 'But how awful.' She had watched the newsreels, of course. 'We played a bad hand well,' Crighton said. Ursula had run into him in Whitehall not long after the evacuation of the troops. He missed her, he said. (He was teetering again, she thought.) Ursula was determinedly nonchalant, said she had reports she needed to take to the War Cabinet Office, clutching buff folders to her chest like a cuirass. She had missed him too. It seemed important not to let him see that.\n\n'You liaise with the War Cabinet?' Crighton said, rather impressed.\n\n'Just an assistant to an under-secretary. Actually, not even to the assistant, just another \"girl\" like me.'\n\nThe conversation had gone on long enough, she decided. He was gazing at her in a way that made her want to feel his arms around her. 'Must push off,' she said brightly, 'there's a war on, you know.'\n\nRalph was from Bexhill, gently sardonic, left-wing, utopian. ('Aren't all socialists utopians?' Pamela said.) Ralph was nothing like Crighton, who with hindsight seemed rather too powerful.\n\n'Being courted by a Red?' Maurice asked, coming across her within the hallowed walls. She felt sought out by him. 'It might not look good for you if anyone knew.'\n\n'He's hardly a card-carrying communist,' she said.\n\n'Still,' Maurice said, 'at least he won't be betraying battleship positions in his pillow talk.'\n\nWhat did that mean? Did Maurice know about Crighton?\n\n'Your personal life isn't personal, not while there's a war on,' he said with a look of distaste. 'And why, by the way, are you learning German? Are you awaiting the invasion? Getting ready to welcome the enemy?'\n\n'I thought you were accusing me of being a communist, not a fascist,' Ursula said crossly. ('What an ass,' Pamela said. 'He's just terrified of anything that might reflect badly on him. Not that I'm defending him. Heaven forbid.')\n\n*\n\nFrom her position at the bottom of the well, Ursula could see that most of the insubstantial wall between her flat and Mrs Appleyard's had disappeared. Looking up through the fractured floorboards and the shattered beams she could see a dress hanging limply on a coat hanger, hooked to a picture rail. It was the picture rail in the Millers' lounge on the ground floor, Ursula recognized the wallpaper of sallow, overblown roses. She had seen Lavinia Nesbit on the stairs wearing the dress only this evening, when it had been the colour of pea soup (and equally limp). Now it was a grey bomb-dust shade and had migrated down a floor. A few yards from her head she could see her own kettle, a big brown thing, surplus to requirements in Fox Corner. She recognized it from the thick twine wound around the handle one day long ago by Mrs Glover. Everything was in the wrong place now, including herself.\n\nYes, Ralph had been in Argyll Road. They had eaten \u2013 bread and cheese \u2013 accompanied by a bottle of beer. Then she had done the crossword, yesterday's _Telegraph_. Recently Ursula had been forced to buy a pair of spectacles for close work, rather ugly things. It was only after she had brought them home that she realized they were almost identical to the pair that one of the Misses Nesbit wore. Was this her fate too, she thought, contemplating her bespectacled reflection in the mirror above the fireplace? Would she, too, end up as an old maid? _The proper sport of boys and girls_. And could you be an old maid if you had worn the scarlet letter? Yesterday an envelope had mysteriously appeared on her desk while she was snatching a sandwich lunch in St James's Park. She saw her name in Crighton's handwriting (he had a surprisingly nice italic hand) and tore the whole thing to bits and threw it in the bin without reading it. Later, when all the clerical assistants were flocking like pigeons around the tea-trolley, she had retrieved the scraps and pieced them together.\n\n_I have mislaid my gold cigarette case. You know the one \u2013 my father gave it me after Jutland. You wouldn't have come across it by any chance, would you?_\n\n_Yours, C_.\n\nBut he was never hers, was he? On the contrary, he belonged to Moira. (Or perhaps the Admiralty.) She dropped the pieces of paper back in the bin. The cigarette case was in her handbag. She had found it beneath her bed a few days after he had left her.\n\n'Penny for them?' Ralph said.\n\n'Not worth it, trust me.'\n\nRalph was stretched out next to her, resting his head on the arm of the sofa, his socked feet in her lap. Although he looked as though he were asleep he gave a murmured response every time she tossed a clue in his direction. ' _A Roland for an Oliver_? How about \"paladin\"?' she said. 'What do you think?'\n\nAn odd thing had happened to her yesterday. She had been on the Tube, she didn't like the Tube, before the bombing she cycled everywhere but it was difficult with so much glass and rubble around. She had been doing the _Telegraph_ crossword, trying to pretend she wasn't underground. Most people felt safer underground but Ursula didn't like the idea of confinement. There had been an incident only a couple of days previously of a bomb falling on to an Underground entrance, the blast had travelled down and into the tunnels and the result was pretty awful. She wasn't sure that it had made the papers, these things were so bad for morale.\n\nOn the Tube, a man sitting next to her had suddenly leaned across \u2013 she had shrunk back \u2013 and, nodding at her half-filled grid, said, 'You're rather good at that. Can I give you my card? Pop into my office if you like. I'm recruiting clever girls.' I bet you are, she thought. He got off at Green Park, tipping his hat to her. The card had an address in Whitehall but she had thrown it away.\n\nRalph shook two cigarettes from a packet and lit them both. He passed one to her and said, 'You're a clever thing, aren't you?'\n\n'Pretty much,' she said. 'That's why I'm in the Intelligence Department and you're in the Map Room.'\n\n'Ha, ha, clever and funny.'\n\nThere was an easy camaraderie between them, that of pals more than lovers. They respected each other's character and made few demands. It helped that they both worked in the War Room. There were a lot of things they never had to explain to each other.\n\nHe touched the back of her hand with his and said, 'How are you?' and she said, 'Very well, thank you.' His hands were still those of the architect he had been before the war, unspoilt by battle. He had been safely away from the fighting, a surveyor in the Royal Engineers, poring over maps and photographs and so on and hadn't expected to become a combatant, wading through filthy, oily, bloody seawater being shot at from all quarters. (For he had, after all, spoken a little more of it.)\n\nAlthough the bombing was awful, he said, you could see that something good could come out of it. He was hopeful about the future (unlike Hugh or Crighton). 'All those hovels,' he said. Woolwich, Silvertown, Lambeth and Limehouse were being destroyed and after the war they would have to be rebuilt. It was an opportunity, he said, to build clean, modern homes with all the facilities \u2013 a community of glass and steel and air in the sky instead of Victorian slums. 'A kind of San Gimignano for the future.'\n\nUrsula was unconvinced by this vision of modernist towers, if it were up to her she would rebuild the future as garden cities, comfortable little houses with cottage gardens. 'What an old Tory you are,' he said affectionately.\n\nYet he loved the old London too ('What architect wouldn't?') \u2013 Wren's churches, the grand houses and elegant public buildings \u2013 'the Stones of London', he said. One or two nights a week he was part of the St Paul's night watch, men who were ready to climb into the rafters 'if necessary' to keep the great church safe from incendiaries. The place was a firetrap, he said \u2013 old timbers, lead everywhere, flat roofs, a multitude of staircases and dark forgotten places. He had answered an advertisement in the Royal Institute of British Architects' journal, appealing for architects to volunteer to be firewatchers because they would 'understand the plans, and so on'. 'We might have to be pretty nimble,' he said and Ursula wondered how he would do that with his limp. She had visions of him beleaguered by flames on all those staircases and in the dark forgotten places. It seemed a chummy kind of watch \u2013 they played chess and had long conversations about philosophy and religion. She imagined that it suited Ralph very well.\n\nOnly a few weeks ago they had watched together, spellbound in horror, as Holland House burnt. They had been in Melbury Road, raiding the wine cellar. 'Why not stay in my house,' Izzie had said casually before she embarked for America. 'You can be my caretaker. You'll be safe here. I can't imagine the Germans will want to bomb Holland Park.' Ursula thought that Izzie might be rather overestimating the Luftwaffe's precision with bombs. And if it was so safe why was Izzie turning tail and running?\n\n'No thanks,' she said. The house was too big and empty. She had taken the key though and occasionally foraged in the house for useful things. There was still some tinned food in the cupboards that Ursula was keeping for a last-ditch emergency, and, of course, the full wine cellar.\n\nThey were scanning the wine racks with their torches \u2013 the electricity had been turned off when Izzie left \u2013 and Ursula had just pulled a rather fine-looking bottle of P\u00e9trus from the rack and said to Ralph, 'Do you think this would go with potato scallops and Spam?' when there was a terrific explosion and, thinking the house had been hit, they had thrown themselves on the hard stone floor of the cellar with their hands over their heads. This was Hugh's advice, instilled in Ursula at a recent visit to Fox Corner. 'Always protect your head.' He had been in a war. She sometimes forgot. All the wine bottles had shaken and shivered in their racks and with hindsight Ursula dreaded to think what damage those bottles of Ch\u00e2teau Latour and Ch\u00e2teau d'Yquem could have done if they had rained down on them, the splintered glass like shrapnel.\n\nThey had run outside and watched Holland House turn into a bonfire, the flames eating everything, and Ursula thought, don't let me die in a fire. Let it be quick, please God.\n\nShe was tremendously fond of Ralph. Not hounded by love the way some women were. With Crighton she had been teased endlessly by the _idea_ of it, but with Ralph it was more straightforward. Again not love, more like the feelings you would have for a favourite dog (and, no, she would never have said such a thing to him. Some people, a lot of people, didn't understand how attached one could be to a dog).\n\nRalph lit another cigarette and Ursula said, 'Harold says smoking is very bad for people. Says he's seen lungs on operating tables that look like unswept chimneys.'\n\n'Of course it's not good for you,' Ralph said, lighting one for Ursula too. 'But being bombed and shot at by the Germans isn't good for you either.'\n\n'Don't you wonder sometimes,' Ursula said. 'If just one small thing had been changed, in the past, I mean. If Hitler had died at birth, or if someone had kidnapped him as a baby and brought him up in \u2013 I don't know, say, a Quaker household \u2013 surely things would be different.'\n\n'Do you think Quakers would kidnap a baby?' Ralph asked mildly.\n\n'Well, if they knew what was going to happen they might.'\n\n'But nobody knows what's going to happen. And anyway he might have turned out just the same, Quakers or no Quakers. You might have to kill him instead of kidnapping him. Could you do that? Could you kill a baby? With a gun? Or what if you had no gun, how about with your bare hands? In cold blood.'\n\nIf I thought it would save Teddy, Ursula thought. Not just Teddy, of course, the rest of the world, too. Teddy had applied to the RAF the day after war was declared. He had been working on a small farm in Suffolk. After Oxford he had done a year at an agricultural college and then had worked on different farms and smallholdings around the country. He wanted to know everything, he said, before he got his own place. ('A _farmer_?' Sylvie still said.) He didn't want to be one of those idealistic back-to-the-land types who ended up knee-deep in muddy yards with sickly cows and dead lambs, their crops not worth picking. (He had worked on one of those places apparently.)\n\nTeddy still wrote poetry and Hugh said, 'A poet farmer, eh? Like Virgil. We'll expect a new _Georgics_ from you.' Ursula wondered how Nancy felt about being a farmer's wife. She was awfully smart, doing research at Cambridge into some arcane and bewildering aspect of maths. ('All gibberish to me,' Teddy said.) And now his childhood dream of becoming a pilot was suddenly and unexpectedly within reach. At the moment he was safe in Canada at an Empire Training School, learning to fly, sending home letters about how much food there was, how great the weather was, making Ursula green with envy. She wished he could stay over there for ever, out of harm's way.\n\n'How did we end up talking about murdering babies in cold blood?' Ursula said to Ralph. 'Mind you,' she cocked her head towards the wall and the rise and fall of Emil's siren wail.\n\nRalph laughed. 'He's not so bad tonight. Mind you, I'd go batty if my children made a racket like that.'\n\nUrsula thought it was interesting that he said 'my children', not 'our children'. Strange to be thinking of having children at all during a time when the very existence of the future was in doubt. She stood up rather abruptly and said, 'The raids will be starting soon.' Back at the beginning of the Blitz they would have said, 'They can't come _every_ night', now they knew they could. ('Is this to be life for ever,' she wrote to Teddy, 'to be harried without rest by the bombs?') Fifty-six nights in a row now so that it was beginning to seem possible that there really would be no end to it.\n\n'You're like a dog,' Ralph said. 'You've got a sixth sense for the raiders.'\n\n'Well you'd better believe me then and go. Or you'll have to come down to the dark hole of Calcutta and you know you won't like that.' The sprawling Miller family, Ursula had counted at least four generations, lived on the ground floor and in the semi-basement of the house in Argyll Road. They also had access down to a further level, a subterranean cellar that the residents of the house used as an air-raid shelter. It was a maze, a mouldy, unpleasant space, full of spiders and beetles, and felt horribly crowded if they were all in there, especially once the Millers' dog, a shapeless rug of fur called Billy, was dragged reluctantly down the stairs to join them. They had also, of course, to put up with the tears and lamentations of Emil, who was passed around between the cellar occupants like an unwanted parcel in a futile attempt to pacify him.\n\nMr Miller, in an effort to make the cellar 'homely' (something it could never be), had taped some reproductions of 'great English art', as he called it, against the sandbagged walls. These colour plates \u2013 _The Haywain_ , Gainsborough's _Mr and Mrs Andrews_ (how smug they looked) and _Bubbles_ (the most sickly Millais possible, in Ursula's opinion) \u2013 looked suspiciously as if they had been pilfered from expensive reference books on art. 'Culture,' Mr Miller said, nodding sagely. Ursula wondered what she would have chosen to represent 'great English art'. Turner perhaps, the smudged, fugitive content of the late works. Not to the Millers' taste at all, she suspected.\n\n*\n\nShe had sewn the collar on her blouse. She had switched off the _Sturm und Drang_ of the wireless broadcast and listened instead to Ma Rainey singing 'Yonder Come the Blues' \u2013 an antidote to all the easy sentiment that was beginning to pour out of the wireless. And she had eaten bread and cheese with Ralph, attempted the crossword and then hurried him out of the door with a kiss. Then she had turned off the light and moved the blackout aside so that she could catch a glimpse of him walking away down Argyll Road. Despite his limp (or perhaps because of it) he had a buoyant gait as if he was expecting something interesting to cross his path. It reminded her of Teddy.\n\nHe knew that she was watching him but he didn't look back, simply raised an arm in silent salute and was swallowed by the dark. There was some light though, a bright slice of crescent moon and a scattering of the faintest stars as though someone had flung a handful of diamond dust into the dark. The _Queen-Moon_ , surrounded by _all her starry Fays_ , although she suspected Keats was writing about a full moon and the moon above Argyll Road seemed more like a moon-in-waiting. She was in a \u2013 rather poor \u2013 poetic mood. It was the enormity of war, she thought, it left you scrabbling for ways to think about it.\n\nBridget always said it was bad luck to look at the moon through glass and Ursula let the blind fall back into place and closed the curtains tightly.\n\nRalph was casual with his safety. After Dunkirk, he said, he felt proofed against sudden violent death. It seemed to Ursula that in a time of war, when one was surrounded by an immense amount of sudden violent death, the odds were quite changed and it was impossible to be protected from anything.\n\nAs she knew it would, the caterwauling commenced, followed swiftly by the guns in Hyde Park starting up and the noise of the first bombs, over the docks again by the sound of it. She was galvanized into action, snatching her torch from the hook beside the front door where it lived like a holy relic, picking up her book, also kept by the door. It was her 'shelter book' \u2013 _Du c\u00f4t\u00e9 de chez Swann_. Now that the war looked as if it were going to last for ever Ursula had decided she might as well embark on Proust.\n\nThe planes whined overhead and then she heard the fearsome _swish_ of a bomb descending and then a walloping _thump!_ as it landed somewhere nearby. Sometimes an explosion sounded much closer than it actually was. (How quickly one acquired new knowledge in the most unlikely subjects.) She looked for her shelter suit. She was wearing a rather flimsy dress considering the season and it was horribly cold and damp in the cellar. The shelter suit had been bought by Sylvie, up in town for the day not long before the bombing started. They had gone for a stroll along Piccadilly and Sylvie had spotted an advertisement in Simpson's window for 'tailored shelter suits' and insisted that they go in and try them on. Ursula couldn't imagine her mother in a shelter, let alone a shelter suit, but it was clearly a garment, a uniform even, that attracted Sylvie. 'It'll be rather good for mucking out hens,' she said and bought them one each.\n\nThe next massive bang had an urgency to it and Ursula abandoned her search for the dratted suit and instead she grabbed the blanket of woollen squares crocheted by Bridget. ('I was going to parcel it up and send it off to the Red Cross,' Bridget had written in her round schoolgirl's hand, 'but then I thought you might need it more.'\n\n'You see, even within my own family I have the status of a refugee,' Ursula wrote to Pamela.)\n\nShe passed the Nesbit sisters on the stairs. 'Ooh, bad luck, Miss Todd,' Lavinia giggled. 'Crossing on the stairs, you know.'\n\nUrsula was going down, the sisters were coming up. 'You're going the wrong way,' she said, rather pointlessly.\n\n'I forgot my knitting,' Lavinia said. She was wearing an enamel brooch shaped like a black cat. A little rhinestone winked for an eye. 'She's knitting leggings for Mrs Appleyard's baby,' Ruth said. 'It's so cold in their flat.' Ursula wondered how many more knitted garments could be applied to the poor child before it resembled a sheep. Not a lamb. Nothing lamb-like about the Appleyard infant. Emil, she reminded herself.\n\n'Well, do hurry, won't you?' she said.\n\n'Hail, hail, the gang's all here,' Mr Miller said as they trooped, one by one, into the cellar. A ragtag assortment of chairs and temporary bedding filled the dank space. There were two ancient army camp beds that Mr Miller had scrounged from somewhere and on which the Nesbits were persuaded to rest their elderly bones. In the current absence of either sister, Billy the dog had installed himself on one of them. There was also a small spirit stove and an Aladdin paraffin stove, both of which seemed to Ursula extraordinarily dangerous items to be in such proximity when people were dropping bombs on you. (The Millers were effortlessly sanguine in the face of jeopardy.)\n\nThe roll-call was almost complete \u2013 Mrs Appleyard and Emil, the queer fish Mr Bentley, Miss Hartnell and the full complement of Millers. Mrs Miller voiced her concern for the whereabouts of the Nesbits and Mr Miller volunteered to go and hurry them up ('ruddy knitting and all') but just then a tremendous explosion rocked the cellar. Ursula felt the foundations trembling as the blast moved through the earth beneath her. Obedient to Hugh's directive, she dropped to the floor with her hands over her head, grabbing the nearest of the smaller Miller boys ('Oi, get your hands off me!') on the way down. She crouched awkwardly over him but he wriggled away from her.\n\nAll went quiet.\n\n'That wasn't our house,' the boy said dismissively, swaggering a little to restore his wounded male dignity.\n\nMrs Appleyard had also thrown herself to the floor, the baby soft-shelled beneath her. Mrs Miller had clutched not one of her brood but the old Farrah's Harrogate toffee tin that contained her savings and insurance policies.\n\nMr Bentley, his voice sounding a quaver higher than normal, asked, 'Was that us?' No, thought Ursula, we would be dead if it had been. She sat down again on one of the rickety bentwood chairs provided by Mr Miller. She could feel her heart, too loud. She began to shiver and wrapped herself in Bridget's crochet.\n\n'Nah, the boy's right,' Mr Miller said, 'that sounded like Essex Villas.' Mr Miller always professed to know where the bombs were dropping. Surprisingly, he was often correct. All of the Millers were adept at wartime language as well as wartime spirit. They could all take it. ('And we can give it out too, can't we?' Pamela wrote. 'You would think _we_ had no blood on our hands.')\n\n'The backbone of England, no doubt,' Sylvie said to Ursula on first (and last) acquaintance with them. Mrs Miller had invited Sylvie down to her kitchen for a cup of tea but Sylvie was still cross at the state of Ursula's curtains and rugs, for which she blamed Mrs Miller, under the apprehension that she was the landlady and not merely another renter. (She was deaf to Ursula's explanations.) Sylvie behaved as though she were a duchess visiting the cottage of one of her rustic tenants. Ursula imagined Mrs Miller later saying to Mr Miller, 'Hoity-toity, that one.'\n\nUp above, the racket of a steady bombardment was now under way, they could hear the timpani of the big bombs, the whistling of shells and the thunder of a nearby mobile artillery unit. Every now and then the foundations of the cellar shook with a _crump_ and _thump_ and _bump_ as the bombs hammered down on the city. Emil howled, Billy the dog howled, a couple of the smallest Millers howled. All in discord with each other, an unwelcome counterpoint to the _Donner und Blitzen_ of the Luftwaffe. A terrible, endless storm. _Despair behind, and death before_.\n\n'Crikey, old Fritz is really trying to put the wind up us tonight,' Mr Miller said, calmly adjusting a lamp for all the world as if they were on a camping trip. He was responsible for morale in the cellar. Like Hugh, he had lived through the trenches and claimed that he was impervious to threats from Jerry. There was a whole club of them, Crighton, Ralph, Mr Miller, even Hugh, who had undergone their ordeal by fire and mud and water and who presumed it was a once-in-a-lifetime experience.\n\n'What's old Fritz up to, eh?' he said soothingly to one of the smaller, more frightened children. 'Trying to stop me getting my beauty sleep?' The Germans always came singularly for Mr Miller in the person of Fritz and Jerry, Otto, Hermann, Hans, sometimes Adolf himself was four miles up dropping his high explosive.\n\nMrs Miller (Dolly), an embodiment of the triumph of experience over hope (unlike her spouse), was doling out 'refreshments' of tea, cocoa, biscuits and bread and margarine. The Millers, a family of generous morals, were never short of rations thanks to Renee, their eldest daughter, who had 'connections'. Renee was eighteen and fully formed in every way and seemed to be a girl of most easy virtue. Miss Hartnell made it clear that she found Renee very wanting indeed although she was not averse to sharing in the provender that she brought home. Ursula got the impression that one of the smaller Miller children was actually Renee's rather than Mrs Miller's and had, in a pragmatic way, simply been absorbed into the family pool.\n\nRenee's 'connections' were ambiguous but a few weeks ago Ursula had spotted her in the first-floor coffee lounge of the Charing Cross Hotel sipping daintily on gin in the company of a sleek and rather prosperous-looking man who had 'racketeer' written all over him.\n\n'There's a sleazy gent if ever I saw one,' Jimmy had laughed. Jimmy, the baby produced to celebrate the peace after the war to end all wars, was about to fight in another one. He had a few days' leave from his army training and they had taken refuge in the Charing Cross Hotel while an unexploded bomb in the Strand was being dealt with. They could hear the naval guns that had been stationed on trolleys between Vauxhall and Waterloo \u2013 _boom-boom-boom \u2013_ but the bombers were looking for other targets and seemed to have moved on. 'Doesn't it ever stop?' Jimmy asked.\n\n'Apparently not.'\n\n'It's safer in the army,' he laughed. He had joined the ranks as a private even though the army had offered him a commission. He wanted to be one of the chaps, he said. ('But someone has to be an officer, surely?' Hugh puzzled. 'Better if it's someone with a bit of intelligence.')\n\nHe wanted the experience. He wanted to be a writer, he said, and what better than a war to reveal to him the heights and depths of the human condition? 'A _writer_?' Sylvie said. 'I fear the hand of the evil fairy rocked his cradle.' She meant Izzie, Ursula supposed.\n\nIt had been lovely spending time with Jimmy. Jimmy was dashing in his battledress and gained an entrance wherever they went \u2013 risqu\u00e9 venues in Dean Street and Archer Street, the Boeuf sur le Toit in Orange Street that was very risqu\u00e9 indeed (if not downright risky), places that made Ursula wonder about Jimmy. All in the pursuit of the human condition, he said. They got very drunk and a little silly and it was all rather a relief from cowering in the Millers' cellar. 'Promise you won't die,' she said to Jimmy as they groped like a blind couple along the Haymarket, listening to some other part of London being blown out of existence.\n\n'Do my best,' Jimmy said.\n\n*\n\nShe was cold. The water she was lying in was making her even colder. She needed to move. Could she move? Apparently not. How long had she been lying here? Ten minutes? Ten years? Time had ceased. Everything seemed to have ceased. Only the awful concoction of smells remained. She was in the cellar. She knew that because she could see _Bubbles_ , still miraculously taped to a sandbag near her head. Was she going to die looking at this banality? Then banality seemed suddenly welcome as a ghastly vision appeared at her side. A terrible ghost, black eyes in a grey face and wild hair, was clawing at her. 'Have you seen my baby?' the ghost said. It took Ursula a few moments to realize that this was no ghost. It was Mrs Appleyard, her face covered in dirt and bomb dust and streaked with blood and tears. 'Have you seen my baby?' she said again.\n\n'No,' Ursula whispered, her mouth dry from whatever filth had been falling. She closed her eyes and when she opened them again Mrs Appleyard had disappeared. She might have imagined her, perhaps she was delirious. Or perhaps it really had been the ghost of Mrs Appleyard and they were both trapped in some desolate limbo.\n\nHer attention was caught again by Lavinia Nesbit's dress hanging from the Millers' picture rail. But it wasn't Lavinia Nesbit's dress. A dress didn't have arms in it. Not sleeves, but _arms_. With _hands_. Something on the dress winked at Ursula, a little cat's eye caught by the crescent moon. The headless, legless body of Lavinia Nesbit herself was hanging from the Millers' picture rail. It was so absurd that a laugh began to boil up inside Ursula. It never broke because something shifted \u2013 a beam, or part of the wall \u2013 and she was sprinkled with a shower of talcum-like dust. Her heart thumped uncontrollably in her chest. It was sore, a time-delay bomb waiting to go off.\n\nFor the first time she felt panic. No one was coming to help her. Certainly not the deranged ghost of Mrs Appleyard. She was going to die alone in the cellar of Argyll Road, with nothing but _Bubbles_ and the headless Lavinia Nesbit for company. If Hugh were here, or Teddy or Jimmy, or even Pamela, they would be fighting to get her out of here, to _save_ her. They would care. But there was no one here to care. She heard herself mewling like an injured cat. How sorry she felt for herself, as if she were someone else.\n\n*\n\nMrs Miller had said, 'Well, I think we could all do with a nice cup of cocoa, don't you?' Mr Miller was fretting about the Nesbits again and Ursula, utterly fed up with the claustrophobia of the cellar, said, 'I'll go and look for them,' and got up from the rickety dining chair just as the _swish_ and _pheew_ announced the arrival of a high explosive bomb. There was a giant thunderclap, a great cracking noise as the wall of hell suddenly split open and let all the demons out and then the tremendous suction and compression, as if her insides, her lungs, her heart and stomach, even her eyeballs were being sucked from her body. _Salute the last and everlasting day_. This is it, she thought. This is how I die.\n\nA voice broke into the silence, almost next to her ear, a man's voice saying, 'Come on then, miss, let's see if we can get you out of here, shall we?' Ursula could see his face, grimy and sweaty as if he had tunnelled to reach her. (She supposed he had.) She was surprised to recognize him. It was one of their local ARP wardens, a new one.\n\n'What's your name, miss? Can you tell me?' Ursula muttered her name but she knew it hadn't come out right. 'Urry?' he queried. 'What's that then \u2013 Mary? Susie?'\n\nShe didn't want to die as a Susie. But did it matter?\n\n'Baby,' she mumbled to the warden.\n\n'Baby?' he said sharply. 'You've got a baby?' He backed away slightly and shouted something to someone unseen. She heard other voices and realized there were lots of people now. As if to verify this the warden said, 'We're all here to get you out. The gas boys have turned the gas off and we'll be moving you in a tick. Don't you worry. Now tell me about your baby, Susie. Were you holding him? Is he just a littl'un?' Ursula thought of Emil, as heavy as a bomb (who had been caught out holding him when the music stopped and the house exploded?), and tried to speak but found herself mewling again.\n\nSomething creaked and groaned overhead and the warden grabbed her hand and said, 'It's all right, I'm here,' and she felt immensely grateful to him, and to all the people toiling to get her out. And she thought how grateful Hugh would be too. The thought of her father made her start to cry and the warden said, 'There, there, Susie, everything's all right, soon have you out of here, like a winkle out of a shell. Get you a nice cup of tea, eh? How does that sound? Lovely, eh? Fancy one myself.'\n\nSnow seemed to be falling, tiny icy needles on her skin. 'So cold,' she murmured.\n\n'Don't worry, we'll have you out of here in two shakes of a lamb's tail, you'll see,' the warden said. He struggled out of the coat he was wearing and covered her with it. There wasn't room for such a generous manoeuvre and he knocked something, causing a shower of debris to fall on them both.\n\n'Oh,' she said to the warden because she felt suddenly violently sick but it passed and she felt calmer. Leaves were falling now mixed with the dust and ash and flakes of the dead and suddenly she was blanketed in piles of wafery beech leaves. They smelt of mushrooms and bonfires and something sweet. Mrs Glover's gingerbread. So much nicer than sewage and gas.\n\n'Come on, girl,' the warden said. 'Come on, Susie, don't go to sleep on me now.' He held her hand tighter but Ursula was looking at something glinting and twirling in the sunlight. A rabbit? No, a hare. A silver hare, spinning slowly in front of her eyes. It was mesmerizing. It was the prettiest thing she had ever seen.\n\nShe was flying off a roof into the night. She was in a cornfield with the sun beating down. Picking raspberries in the lane. Playing hide-and-seek with Teddy. _She's a funny little thing_ , someone said. Not the warden, surely? And then the snow began to come down. The night sky was no longer high above, it was all around her, like a warm dark sea.\n\nShe was floating into the blackout. She tried to say something to the warden. _Thank you_. But it didn't matter any more. Nothing mattered. The darkness had fallen.\n\n# A Lovely Day Tomorrow\n\n# _2 September 1939_\n\n'Don't get upset, Pammy,' Harold said. 'Why is it so quiet, what have you done with the boys?'\n\n'Sold them,' Pamela said, perking up. 'Three for the price of two.'\n\n'You ought to stay the night, Ursula,' Harold said kindly. 'You shouldn't be on your own tomorrow. It'll be one of those awful days. Doctor's orders.'\n\n'Thanks,' Ursula said. 'But I've already got plans.'\n\nShe tried on the yellow _cr\u00eape de Chine_ tea dress that she'd bought earlier that day in an eve-of-war spending spree on Kensington High Street. The _cr\u00eape de Chine_ had a pattern \u2013 tiny black swallows in flight. She admired it, rather admired herself, or what she could see in the dressing-table mirror as she had to stand on her bed in order to see her lower half.\n\nThrough Argyll Road's thin walls Ursula could hear Mrs Appleyard having a row, in English, with a man \u2013 the mysterious Mr Appleyard presumably \u2013 whose comings and goings at all times of the day and night kept no noticeable timetable. Ursula had encountered him in the flesh only once, in passing on the stairs, when he had glared moodily at her and hurried on without a greeting. He was a big man, ruddy and slightly porcine. Ursula could imagine him standing behind a butcher's counter or hauling brewery sacks, although according to the Misses Nesbit he was in fact an insurance clerk.\n\nMrs Appleyard, in contrast, was thin and sallow and when her husband was out of the flat Ursula could hear her singing mournfully to herself in a language that she couldn't place. Something Eastern European by the sound of it. How useful Mr Carver's Esperanto would be, she thought. (Only if everyone spoke it, of course.) And especially these days with so many refugees flooding into London. ('She's Czech,' the Nesbits had eventually informed her. 'We didn't used to know where Czechoslovakia was, did we? I wish we still didn't.') Ursula presumed Mrs Appleyard was also some kind of refugee who, looking for safe harbour in the arms of an English gentleman, had found instead the pugnacious Mr Appleyard. Ursula thought that if she ever heard Mr Appleyard actually hitting his wife then she would have to knock on their door and somehow put a stop to it, although she had no idea how she would do that.\n\nThe dispute next door reached a crescendo and then the Appleyards' front door slammed decisively in conclusion and all went quiet. Mr Appleyard, a great one for noisy exits and entrances, could be heard stomping down the stairs, a trail of profanity in his wake on the subject of women and foreigners, of which the oppressed Mrs Appleyard was both.\n\nThe sour aura of dissatisfaction that seeped through the walls, along with the even less appetizing smell of boiled cabbage, was really quite depressing. Ursula wanted her refugees to be soulful and romantic \u2013 fleeing for their cultural lives \u2013 rather than the abused wives of insurance clerks. Which was ridiculously unfair of her.\n\nShe stepped down from the bed and did a little twirl for the mirror. The dress suited her, she decided, she still had her figure, even at nearly thirty. Would she one day develop Sylvie's matronly girth? It was beginning to seem unlikely now that she would ever have children of her own. She remembered holding Pamela's babies \u2013 remembered Teddy and Jimmy, too \u2013 how overwhelming the feelings of love and terror, the desperate desire to protect. How much stronger would those feelings be if it were her own child? Perhaps too strong to bear.\n\nOver their afternoon tea in John Lewis, Sylvie had asked, 'Do you never get broody?'\n\n'Like your hens?'\n\n'A \"career woman\",' Sylvie said, as if the two words had no place in the same sentence. 'A spinster,' she added, contemplating the word. Ursula wondered why her mother was working so hard to rile her. 'Perhaps you will never marry,' Sylvie said, as if in conclusion, as if Ursula's life was as good as over.\n\n'Would that be such a bad thing? \"The unmarried daughter\",' Ursula said, tucking into an iced fancy. 'It was good enough for Jane Austen.'\n\nShe lifted the dress over her head and, in petticoat and stocking feet, padded through to the little scullery and filled a water glass at the tap before hunting down a cream cracker. Prison fare, she thought, good practice for what was to come. All she had had to eat since her breakfast toast was Pamela's cake. She was hoping to be stood, at the very least, a good dinner by Crighton tonight. He had asked her to meet him at the Savoy, they rarely had such public assignations, and she wondered if there was going to be drama, or if the shadow of war was drama enough and he wanted to talk to her about it.\n\nShe knew that war was to be declared tomorrow, even though she had played rather dumb with Pammy. Crighton told her all kinds of things he shouldn't, on the basis that they had 'both signed the Official Secrets Act'. (She, on the other hand, told him almost nothing.) He had been teetering again lately and Ursula wasn't at all sure which way he was going to fall, wasn't sure which way she wanted him to fall.\n\nHe had asked her to meet him for a drink, a request conveyed on an Admiralty docket that had arrived mysteriously while she was briefly out of the office. Not for the first time Ursula wondered who brought these notes that seemed to appear on her desk as if delivered by elves. _I think your department may be due an audit_ , it read. Crighton liked code. Ursula hoped that the navy's encryptions weren't as rudimentary as Crighton's.\n\nMiss Fawcett, one of her clerical assistants, spotted the note lying in full view and gave her a panic-stricken look. 'Crikey,' she said. 'Are we? Due an audit?'\n\n'Someone's idea of a joke,' Ursula said, dismayed to find herself blushing. There was something un-Crighton-like about these salacious (if not downright filthy) but seemingly innocent messages. _I believe there is a shortage of pencils_. Or _Are your ink levels sufficiently topped up?_ Ursula wished he would learn Pitman's, or more discretion. Or, better still, stop altogether.\n\nWhen she was ushered inside the Savoy by a doorman, Crighton was waiting for her in the expansive foyer and instead of escorting her up to the American Bar he shepherded her up the stairs to a suite on the second floor. The bed seemed to dominate the room, enormous and pillowy. Oh, so this is why we're here, she thought.\n\nThe _cr\u00eape de Chine_ had been deemed unsuitable for the occasion and she had donned her royal-blue satin \u2013 one of her three good evening dresses \u2013 a decision she now regretted as Crighton, if form was anything to go by, would soon be divesting her of it rather than treating her to a slap-up meal.\n\nHe liked undressing her, liked looking at her. 'Like a Renoir,' he said, although he knew little about art. Better a Renoir than a Rubens, she thought. Or a Picasso, for that matter. He had bestowed on her the great gift of regarding herself naked with little, if any, criticism. Moira, apparently, was a floor-length flannelette and lights-out woman. Sometimes Ursula wondered if Crighton didn't exaggerate his wife's sturdy qualities. Once or twice it had crossed her mind to journey out to Wargrave to catch a glimpse of the wronged wife and find out if she really was a dowd. The problem, of course, with Moira in the flesh (Rubenesque, not Renoir, she imagined) would be that Ursula would find it difficult to betray a real person rather than an enigma.\n\n('But she _is_ a real person,' Pamela puzzled. 'It's a specious argument.'\n\n'Yes, I _am_ aware of that.' This later, at Hugh's sixtieth birthday, a rather querulous affair in the spring.)\n\nThe suite had a magnificent view of the river, from Waterloo Bridge to the Houses of Parliament and Big Ben, all shadowy now, in the encroaching twilight. ('The violet hour'.) She could just make out Cleopatra's Needle, a dark finger poking skywards. None of the usual blaze and twinkle of London lights. The blackout had already begun.\n\n'The bolthole wasn't available then? We're out in the open?' Ursula said while Crighton opened a bottle of champagne that had been waiting for them in a sweating silver bucket. 'Are we celebrating?'\n\n'Saying our adieux,' Crighton said, joining her at the window and handing her a glass.\n\n'Our adieux?' Ursula said, bemused. 'You've brought me to a good hotel and are plying me with champagne in order to end it all between us?'\n\n'Adieu to the peace,' Crighton said. 'We're saying goodbye to the world as we know it.' He raised his glass in the direction of the window, to London, in its dusky glory. 'To the beginning of the end,' he said grimly. 'I've left Moira,' he added, as if it were an afterthought, a nothing. Ursula was caught by surprise.\n\n'And the girls?' (Just checking, she thought.)\n\n'All of them. Life is too precious to be unhappy.' Ursula wondered how many people across London were saying the same thing that night. Perhaps in less salubrious surroundings. And there would be others, of course, who would be saying the same words to cleave to what they already had, not to discard it on a whim.\n\nSuddenly and unexpectedly panicked, Ursula said, 'I don't want to marry you.' She hadn't realized quite how strongly she felt until the words came out of her mouth.\n\n'I don't want to marry you either,' Crighton said, and, perversely, she felt disappointed.\n\n'I've taken a lease on a flat in Egerton Gardens,' he said. 'I thought perhaps you would come and join me.'\n\n'To cohabit? To live in sin in Knightsbridge?'\n\n'If you will.'\n\n'My, you are bold,' she said. 'What about your career?'\n\nHe made a dismissive sound. So, she, and not the war, was to be his new Jutland then.\n\n'Will you say yes? Ursula?'\n\nUrsula stared through the window at the Thames. The river was almost invisible now.\n\n'We should have a toast,' she said. 'What is it they say in the navy \u2013 \"Sweethearts and wives \u2013 may they never meet\"?' She chinked her glass against Crighton's and said, 'I'm starving, we are going to eat, aren't we?'\n\n# _April 1940_\n\nA car horn down in the street below broke the Sunday-morning silence of Knightsbridge. Ursula missed the sound of church bells. There were so many simple things she had taken for granted before the war. She wished that she could go back and appreciate them properly.\n\n'Why the horn,' Crighton said, 'when we have a perfectly good doorbell?' He looked out of the window. 'He's here,' he said, 'if he's a young man in a three-piece suit puffed up like a Christmas robin.'\n\n'That does sound like him.' Although Ursula didn't think of Maurice as 'young', had never thought of him as young, but she supposed he was to Crighton.\n\nHugh's sixtieth birthday and Maurice had grudgingly offered her a lift to Fox Corner for the celebrations. It was going to be a novelty, and not necessarily a good one, to spend time cloistered in a car with Maurice. They were rarely alone with each other.\n\n'He has petrol?' Crighton had said, raising an eyebrow but really it was more a statement than a question.\n\n'He has a _driver_ ,' Ursula said. 'I knew Maurice would squeeze the most out of the war.' 'What war?' Pamela would have said. She was 'marooned' in Yorkshire with only six small boys for company and Jeanette, who had turned out to be not merely a moaner but 'quite the _fain\u00e9ante_. I expected better of a vicar's daughter. She's so lazy, I run around all day long after her boys as well as mine. I've had enough of this evacuation lark, I think we'll come home soon.'\n\n'I suppose he could hardly turn up at home in a car _without_ having given me a lift,' Ursula said. 'Maurice wouldn't want to be seen to be behaving badly, even by his own family. He has a _reputation_ to keep up. Besides, his family are staying there and he's bringing them back to London tonight.' Maurice had sent Edwina and the children to stay at Fox Corner for the Easter holidays. Ursula had wondered if he knew something about the war that the public didn't \u2013 was Easter to be a particularly hazardous time? There must be so many things that Maurice knew that others didn't, but Easter had passed off without incident and she supposed it was merely a case of grandchildren visiting grandparents. Philip and Hazel were very uninventive children and Ursula wondered how they were getting on with Sylvie's rambunctious evacuees. 'It'll be horribly crowded on the way back, with Edwina and the children. Not to mention the _driver_. Still, needs must and so on.'\n\nThe car horn sounded again. Ursula ignored it as a matter of principle. How wickedly satisfying it would be, she thought, to have Crighton in tow, in full naval fig (all those medals, all that gold braid), outranking Maurice in so many ways. 'You could come with me, you know,' she said to him. 'We just wouldn't mention Moira. Or the girls.'\n\n'Is it your home?'\n\n'Sorry?'\n\n'You said, \"he could hardly turn up at home\". Isn't this? Your home?' Crighton said.\n\n'Yes, of course,' Ursula said. Maurice was pacing impatiently up and down on the pavement and she rapped on the window pane to get his attention and held up her index finger, mouthing 'one minute' to him. He frowned at her. 'It's a figure of speech,' she said, turning back to Crighton. 'One always refers to one's parents' place as \"home\".'\n\n'Does one? I don't.'\n\nNo, thought Ursula, you don't. Wargrave was 'home' for Crighton, even if only in his thoughts. And he was right, of course, she didn't consider the flat in Egerton Gardens to be her home. It was a point in time, a temporary stopping-off place on yet another journey that the war had interrupted. 'We can argue the point if you want,' she said amiably. 'It's just, you know... Maurice, marching up and down out there like a little tin soldier.'\n\nCrighton laughed. He never looked for arguments.\n\n'I would love to join you and meet your family,' he said, 'but I'm going to the Citadel.' The Admiralty was constructing an underground fortress, the Citadel, on Horse Guards Parade and Crighton was in the process of moving his office over.\n\n'I'll see you later then,' Ursula said. 'My carriage awaits and Maurice is pawing the ground.'\n\n'Ring,' Crighton reminded her and Ursula said, 'Oh, yes, of course, I nearly forgot.' She had started wearing a wedding ring when not at work, for appearances' sake, 'Tradesmen, and so on.' The boy who delivered the milk, the woman who came in to clean twice a week, she didn't want them thinking she was in an illicit relationship. (She had surprised herself with this bashfulness.)\n\n'You can imagine how many questions there would be if they saw _that_ ,' she said, slipping the ring off and leaving it on the hall table.\n\nCrighton kissed her lightly on the cheek and said, 'Have a nice time.'\n\n'No guarantee of that,' she said.\n\n'Still not caught yourself a man?' Izzie asked Ursula. 'Of course,' she said, turning brightly to Sylvie, 'you have \u2013 how many grandchildren now, seven, eight?'\n\n'Six. Perhaps _you're_ a grandmother, Izzie.'\n\n'What?' Maurice said. 'How could she be?'\n\n'Anyway,' Izzie said airily, 'it takes the pressure off Ursula to produce one.'\n\n'Produce?' Ursula said, a forkful of salmon in aspic suspended on its way to her mouth.\n\n'Looks like you're left on the shelf,' Maurice said.\n\n'Pardon?' The fork returned to the plate.\n\n'Always the bridesmaid...'\n\n'Once,' Ursula said. 'I have been a bridesmaid once only, to Pamela.'\n\n'I'll have that if you're not eating it,' Jimmy said, filching the salmon.\n\n'I was, actually.'\n\n'Even worse then,' Maurice said. 'Nobody even wants you as a bridesmaid except for your sister.' He sniggered, more schoolboy than man. He was, annoyingly, seated too far away for her to kick him beneath the table.\n\n'Manners, Maurice,' Edwina murmured. How many times would he disappoint you in a day if you were married to him, Ursula wondered? It seemed to her that in the search for arguments against marriage the existence of Maurice presented the very best one of all. Of course, Edwina's nose was currently out of joint on account of the _driver_ , who turned out to be a rather attractive ATS girl in uniform. Sylvie, to the girl's embarrassment (her name was Penny but everyone immediately forgot this), insisted that she join them at the table when she would clearly have been more comfortable staying with the car, or in the kitchen with Bridget. She was stuck at the cramped end of the table with the evacuees and was the object of constant frosty scrutiny from Edwina. Maurice, on the other hand, studiously ignored her. Ursula tried to read some meaning into this. She wished Pamela were here, she was very good at deciphering people, although not perhaps as good as Izzie. ('So, Maurice has been a naughty boy, I see. Mind you, she's a looker. Women in uniform, what man can resist?')\n\nPhilip and Hazel sat passively between their parents. Sylvie had never been particularly fond of Maurice's children whereas she seemed to delight in her evacuees, Barry and Bobby ('my two busy bees'), currently crawling beneath the Regency Revival dining table, giggling in a rather manic fashion. 'Full of mischief,' Sylvie said indulgently. The evacuees, as everyone else referred to them, as if they were entirely defined by their status, had been scrubbed and polished into apparent innocence by Bridget and Sylvie but nothing could disguise their impish nature. ('What little horrors,' Izzie said with a shudder.) Ursula rather liked them, they reminded her of the small Millers. If they had been dogs their tails would have been constantly wagging.\n\nSylvie now had a pair of real puppies as well, excitable black Labradors who were also brothers. They were called Hector and Hamish but seemed to be known collectively and indistinguishably as 'the dogs'. The dogs and the evacuees appeared to have contributed to a new shabbiness in Fox Corner. Sylvie herself seemed more reconciled now to this war than she had ever been to the last one. Hugh less so. He had been 'pushed' into training the Home Guard and had only this morning after Sunday service been instructing the 'ladies' of the local church in the use of the stirrup pump.\n\n'Is that suitable for the Sabbath?' Edwina asked. 'I'm sure God's on our side, but...' she tailed off, incapable of sustaining a theological position despite being 'a devout Christian', which meant, according to Pamela, that she slapped her children hard and made them eat for breakfast what they left at tea.\n\n'Of course it's suitable,' Maurice said. 'In my role organizing the civil defence\u2014'\n\n'I don't consider myself to be \"on the shelf\" as you so charmingly put it,' Ursula interrupted him irritably. Again, she experienced a fleeting wish for Crighton's be-medalled, braided presence. How horrified Edwina would be to know of Egerton Gardens. ('And how is the Admiral?' Izzie asked later in the garden, sotto voce, like a conspirator, for, of course, she knew. Izzie knew everything and if she didn't know it she could mouse it out with ease. Like Ursula, she had the character for espionage. 'He's not an admiral,' Ursula said. 'But he is well, thank you.')\n\n'You do all right on your own,' Teddy said to Ursula. ' _Contracted to thine own bright eyes_ , and so on.' Teddy had faith in poetry, as if merely to quote from Shakespeare would mollify a situation. Ursula thought the sonnet he was quoting from was about being selfish but didn't say so as Teddy meant it kindly. Unlike everyone else, it seemed, all of whom appeared quite fixed on her unmarried status.\n\n'She's only thirty, for heaven's sake,' Izzie said, putting in her oar again. (If only they would all be quiet, Ursula thought.) 'After all,' Izzie persisted, 'I was over forty when I married.'\n\n'And where _is_ your husband?' Sylvie asked, looking around the Regency Revival \u2013 both leaves extended to accommodate their numbers. She feigned perplexity (it didn't suit her). 'I don't seem to see him here.'\n\nIzzie had chosen the occasion to turn up ('Uninvited, as usual,' Sylvie said) to offer her congratulations on Hugh's six decades. ('A landmark.') Hugh's other sisters had deemed the journey to Fox Corner 'too challenging'.\n\n'What a parcel of vixens they are,' Izzie said later to Ursula. Izzie might have been the baby but she was never the favourite. 'Hugh has always been so good to them.'\n\n'He's always been good to everyone,' Ursula said, surprised, alarmed even, to find tears starting up at the thought of her father's sound character.\n\n'Oh, don't,' Izzie said, handing her a froth of lace that apparently passed as a handkerchief. 'You'll make me cry as well.' It seemed unlikely, it had never happened before.\n\nIzzie had also chosen the occasion to announce her imminent departure for California. Her husband, the famous playwright, had been offered a job writing screenplays in Hollywood. 'All the Europeans are going there,' she said.\n\n'You're European now, are you?' Hugh said.\n\n'Aren't we all?'\n\nThe whole family had gathered, apart from Pamela, for whom the journey was genuinely too challenging. Jimmy had managed to wangle a couple of days' leave and Teddy had brought Nancy along. On arrival, she gave Hugh a disarming hug, said, 'Happy birthday, Mr Todd,' before handing him a parcel, wrapped prettily in old wallpaper scavenged from the Shawcross household. It was a copy of _The Warden_. 'It's a first edition,' Nancy said. 'Ted said that you liked Trollope.' (A fact that none of the rest of his family appeared to know.)\n\n'Good old Ted,' Hugh said, kissing her on the cheek. And to Teddy, 'What a sweetheart you have here. When are you going to pop the question?'\n\n'Oh,' Nancy said, blushing and laughing, 'plenty of time for that.'\n\n'I hope so,' a sombre Sylvie said. Teddy had graduated now from the Initial Training School ('He has wings!' Nancy said. 'Like an angel!') and was waiting to sail to Canada, to train as a pilot. When he was qualified he would head back here and take up a place in an Operational Training Unit.\n\nHe was more likely to be killed in an OTU, he said, 'than on an actual bombing run'. It was true. Ursula knew a girl in the Air Ministry. (She knew girls everywhere, everyone did.) They ate their sandwiches together in St James's Park and gloomily traded statistics, despite the dead hand of the Official Secrets Act.\n\n'Well, that's a great comfort to me,' Sylvie said.\n\n'Ow!' one of the evacuees squealed beneath the table. 'Some bugger just kicked me.' Everyone instinctively looked at Maurice. Something cold and wet nosed itself up Ursula's skirt. She hoped very much that it was the nose of one of the dogs and not one of the evacuees. Jimmy pinched her arm (rather hard) and said, 'They do go on, don't they?'\n\nThe poor ATS girl \u2013 like the evacuees and the dogs, defined by her status \u2013 looked as if she were about to cry.\n\n'I say, are you all right?' ever-solicitous Nancy asked her.\n\n'She's an only child,' Maurice said matter-of-factly. 'They don't understand the joys of family life.' This knowledge of the ATS girl's background seemed to particularly infuriate Edwina, who was gripping the butter knife in her hand as if she were planning to attack someone with it \u2013 Maurice or the ATS girl, or anyone within stabbing distance by the look of it. Ursula wondered how much harm a butter knife could do. Enough, she supposed.\n\nNancy jumped up from the table and said to the ATS girl, 'Come on, let's go for a walk, it's such a lovely day. The bluebells will be out in the wood, if you fancy a bit of a hike.' She hooked arms with her and almost pulled her out of the room. Ursula thought about running after them.\n\n' _Courtship to marriage, as a very witty prologue to a very dull play_ ,' Izzie said as if nothing had interrupted them. 'Someone said that.'\n\n'Congreve,' Sylvie said. 'What on earth does that have to do with anything?'\n\n'Just saying,' Izzie said.\n\n'Of course \u2013 you're _married_ to a playwright, aren't you?' Sylvie said. 'The one we never see.'\n\n'The journey is different for everyone,' Izzie said.\n\n'Oh, please,' Sylvie said. 'Spare us your cod philosophy.'\n\n'For me, marriage is about freedom,' Izzie said. 'For you it has always been about the vexations of confinement.'\n\n'What on earth are you talking about?' Sylvie said. (A bafflement shared by the rest of the table.) 'You talk such nonsense.'\n\n'And what life would you have led otherwise?' Izzie continued blithely (or relentlessly, depending on your viewpoint). 'I seem to remember you were seventeen and on your uppers, a dead, bankrupt artist's daughter. Heaven only knows what would have happened to you if Hugh hadn't charged in and rescued you.'\n\n'You remember nothing, you were still in the nursery at the time.'\n\n'Barely. And, I, of course\u2014'\n\n'Oh, do shut up,' Hugh said wearily.\n\nBridget broke the tension (often her starring role at Fox Corner now that Mrs Glover was gone), entering the dining room bearing aloft a roast duck.\n\n'Duck _\u00e0 la surprise_ ,' Jimmy said, for, naturally, they had all been expecting a chicken.\n\nNancy and the ATS girl ('Penny,' Nancy reminded everyone) returned in time to be handed warmed-up plates. 'You're lucky there's any duck on that,' Teddy said to Nancy when he handed her a plate. 'The poor bird was picked clean.'\n\n'There's so little eating on a duck,' Izzie said, lighting up a cigarette. 'There's barely enough for two people, I can't imagine what you were thinking.'\n\n'I was thinking there's a war on,' Sylvie said.\n\n'If I'd known you planned a _duck_ ,' Izzie ploughed on, 'I would have sought out something a little more generous. I know a man who can get anything.'\n\n'I bet you do,' Sylvie said.\n\nJimmy offered Ursula the wishbone and they both wished loudly and pointedly for a nice birthday for Hugh.\n\nAn amnesty was brought about by the advent of the cake, an ingenious confection that, naturally, relied mainly on eggs. Bridget brought it to the table. She had no flair for making an occasion of anything and dumped it in front of Hugh without ceremony. She was coerced by him into taking a place at the table. 'I wouldn't if I were you,' Ursula heard the ATS girl mutter quietly.\n\n'You're part of the family, Bridget,' Hugh said. No one else in the family, Ursula thought, slaved away from dawn to dusk the way Bridget did. Mrs Glover had retired and gone to live with one of her sisters, a move prompted by George's sudden but not unexpected death.\n\nJust as Hugh filled his lungs, rather theatrically, for there was only one token candle to blow out, there was a great commotion out in the hallway. One of the evacuees went out to investigate and ran back with the news that it was 'A woman, and loads of bloody kids!'\n\n*\n\n'How was it?' Crighton asked, when she finally arrived home.\n\n'Pammy came back \u2013 for good, I think,' she said, deciding on the highlight. 'She looked done in. She came by train, three little boys plus a babe-in-arms, can you imagine? It took her _hours_.'\n\n'A nightmare,' Crighton said with feeling.\n\n('Pammy!' Hugh said. He looked enormously pleased.\n\n'Happy birthday, Daddy,' Pamela said. 'No presents, I'm afraid, just us.'\n\n'More than enough,' Hugh said, beaming.)\n\n'And _suitcases_ , and the dog. She's such a stalwart. _My_ journey home, on the other hand, was a different kind of nightmare. Maurice, Edwina, their uninspiring offspring and the _driver_. Turned out to be a rather lovely ATS girl.'\n\n'Good God,' Crighton said, 'how does he do it? I've been trying to get my hands on a Wren for months.' She laughed and hovered in the kitchen while he made cocoa for both of them. While they drank it in bed she regaled him with tales from the day, somewhat embellished (she felt it her duty to entertain him). What, after all, she thought, was there to distinguish them from any married couple? Perhaps the war. Perhaps not.\n\n'I think I'm going to have to join up, or something,' she said. She thought of the ATS girl. ' \"Do my bit\", as they say. Get my hands dirty. I read reports every day about people doing brave things and my hands stay very clean.'\n\n'You're doing your bit already,' he said.\n\n'What? Supporting the navy?'\n\nHe laughed and rolled over and pulled her into his arms. He nuzzled her neck and as she lay there it struck her that it was just possible that she was happy. Or at any rate, she thought, qualifying the idea, as happy as was possible in this life.\n\n'Home', it had struck her on the torturous drive back to London, wasn't Egerton Gardens, wasn't even Fox Corner. Home was an idea, and like Arcadia it was lost in the past.\n\nShe had already ticketed the day in her memory as 'Hugh's sixtieth birthday', one more in a roll-call of family occasions. Later, when she understood that it was the last time they would all be together, she wished she had paid more attention.\n\n*\n\nShe was woken in the morning by Crighton bringing her a tray of tea and toast. She had the Senior Service to thank for his domesticity rather than Wargrave.\n\n'Thank you,' she said, struggling to sit up, still worn out from yesterday.\n\n'Bad news, I'm afraid,' he said, opening the curtains.\n\nShe thought of Teddy and Jimmy, although she knew that for this morning at least they were safely tucked up in their beds in Fox Corner, sharing their boyhood room, once Maurice's.\n\n'What bad news?' she asked.\n\n'Norway has fallen.'\n\n'Poor Norway,' she said and sipped the hot tea.\n\n# _November 1940_\n\nPamela had sent a parcel of baby clothes that Gerald had grown out of, and Ursula thought of Mrs Appleyard. She might not have thought of Mrs Appleyard as she hadn't kept up with the residents of Argyll Road since she left for Egerton Gardens, something she had rather regretted as she had been fond of the Misses Nesbit and often wondered how they were faring under the relentless bombardment. But then she had had a chance encounter with Renee Miller a few weeks ago.\n\nUrsula had been 'on the town', as he put it, with Jimmy, who had a couple of days' furlough in the capital. They had been stranded in the Charing Cross Hotel thanks to a UXB \u2013 sometimes she thought unexploded bombs were more of a nuisance than exploded ones \u2013 and had taken refuge in the first-floor coffee lounge.\n\n'There's a rather tarty girl, all lipstick and teeth, over there who seems to know you,' Jimmy said.\n\n'Ye gods, Renee Miller,' Ursula said when she spotted Renee waving eagerly at her. 'And who on earth is that man with her? He looks like a gangster.'\n\nRenee was effusive, as though she had been best pals with Ursula in some former life ('She's a lively girl,' Jimmy laughed after they'd escaped), and insisted that they join her and 'Nicky' for a drink. Nicky himself seemed less than enthusiastic about the idea but nonetheless shook hands and signalled to the waiter.\n\nRenee filled Ursula in on 'the doings' in Argyll Road, although little seemed to have changed since she left a year ago for Egerton Gardens, except that the army now had Mr Appleyard and his wife had a baby. 'A boy,' Renee said. 'Ugly little thing.' Jimmy guffawed and said, 'I like a girl who knows when a spade's a spade.' Nicky was rather put out by Jimmy's personable presence, especially as by the time she had downed another watery gin Renee had begun to flirt (almost professionally it seemed) with him.\n\nUrsula overheard someone say that the unexploded bomb had been dealt with and when Renee said, 'Get us another round in, Nicky,' and Nicky began to glower Ursula thought it might be politic to move on. Nicky refused to let them pay, as if it were a matter of principle. Ursula wasn't sure she wanted to be beholden to someone of his calibre. Renee hugged and kissed her and said, 'Come and see the old dears, they'd love it,' and Ursula promised that she would.\n\n'Good God, I thought she was going to eat me,' Jimmy said as they manoeuvred around rubble on Henrietta Street.\n\nShe made good on her promise to Renee, prompted by the parcel of Gerald's old clothes. She reached Argyll Road not long after six, getting away from work early for once. She had not, after all, donned a uniform of any kind yet, as there seemed hardly enough time to eat and breathe between work and the bombs. 'Your job _is_ war work,' Crighton pointed out, 'I would have thought that you would have enough on your plate. How is the Ministry of Some Obscurity these days?'\n\n'Oh, you know. Busy.' There was so much information to be logged. Each individual incident \u2013 what type of bomb, the damage done, how many killed or injured (the tally was mounting horribly) \u2013 streamed across their desks.\n\nOccasionally, she would open a buff folder and find what she thought of as the 'raw material' \u2013 ARP typed reports or even the handwritten reports they were based on \u2013 and wonder what it was like to be in the heat of battle, for that's what the Blitz was, wasn't it? Sometimes she saw bomb-damage maps, once one that had been drawn by Ralph. He had signed it in faint, almost indecipherable pencil on the back. They were friends, she had met him at her German class, although he had made it clear that he would like them to mean more to each other. 'Your other man,' Crighton called him, amused.\n\n'How kind,' Mrs Appleyard said when Ursula appeared on her doorstep with the parcel of clothes. 'Please come in.'\n\nUrsula crossed the threshold reluctantly. The previous smell of boiled cabbage now mingled with the more unappetizing smells that could accompany an infant. Sadly, Renee's judgement on Mrs Appleyard's baby's pulchritude, or lack of it, turned out to be true \u2013 he was an 'ugly little thing'.\n\n'Emil,' Mrs Appleyard said, handing him over to Ursula to hold. She could feel the dampness of him through his rubber knickers. She almost handed him straight back. 'Emil?' she said to him, making a face and grinning at him with forced jollity. He stared back at her, rather truculently, his paternity not in doubt.\n\nMrs Appleyard offered tea and Ursula excused herself and scurried up the stairs to the Nesbits' eyrie.\n\nThey were their usual benign selves. It must be quite nice to live with one's sister, Ursula thought. She wouldn't mind living out her days with Pamela.\n\nRuth grasped one of her fingers with her own twig-like ones. 'You're married! How wonderful.' Oh darn, Ursula thought, she had forgotten to take off the wedding ring. She demurred, 'Well...' and then, seeing the complexity, finally, modestly, 'Yes, I suppose so.' They both offered triumphant congratulations, as if she had achieved something spectacular.\n\n'What a shame you have no engagement ring,' Lavinia said.\n\nUrsula had forgotten their penchant for costume jewellery and wished she had brought them something. She had a little box of old diamant\u00e9 buckles and clips that Izzie had given her that she knew they would have appreciated.\n\nLavinia was wearing an enamel brooch shaped like a black cat. A little rhinestone winked for an eye. Ruth sported a weighty carbuncle of topaz pinned to her sparrow chest. It looked like it might topple her insubstantial frame.\n\n'We're like magpies,' Ruth laughed. 'We love all the shiny little things.'\n\nThey had the kettle on and were happily fussing over what to feed her \u2013 toast with Marmite or toast with jam \u2013 when the siren began its infernal warble. Ursula looked out of the window. No sign of any raiders yet although a searchlight was already sweeping the black sky. A beautiful new moon had stamped a crescent of light out of the blackness.\n\n'Come along, dear, down to the Millers' cellar,' Lavinia said, surprisingly chipper. 'Every night an adventure,' Ruth added, as they gathered up a great amount of stuff \u2013 shawls and cups, books and darning. 'Torch, torch, don't forget the torch!' Lavinia said gaily.\n\nAs they reached the ground floor a bomb thudded down a couple of streets away. 'Oh, no!' Lavinia said. 'I forgot my knitting.'\n\n'We'll go back, dear,' Ruth said and Ursula said, 'No, you must take shelter.'\n\n'I'm knitting leggings for Mrs Appleyard's baby,' Lavinia said, as if that were a good enough reason to risk her life.\n\n'Don't worry about us, dear,' Ruth said, 'we'll be back before you know we've gone.'\n\n'For heaven's sake, if you must have it then I'll go,' Ursula said but they were already creaking their old bones up the stairs and Mr Miller was bustling her down to the cellar.\n\n'Renee, Dolly, everyone \u2013 look who's come to join her old pals!' he announced to the occupants as if Ursula were a music hall turn.\n\nShe had forgotten how many Millers there were, and how starchy Miss Hartnell could be and how downright odd Mr Bentley was. And as for Renee, she seemed to have quite forgotten the ardour of their previous encounter, saying only, 'Oh, lawd, another body using up the air in this hellhole.' Renee was \u2013 reluctantly \u2013 dandling a fractious Emil. She was right, it was a hellhole. In Egerton Gardens they had a rather salubrious basement that they retired to, although Ursula (and Crighton too if he was there) often took her chances and stayed in her own bed.\n\nUrsula remembered the wedding ring and thought how confused Hugh and Sylvie would be if they saw it on her body if she were to die in a raid. Would Crighton come to her funeral and explain? She was prevented from slipping it off by Renee suddenly thrusting Emil into her arms just before a massive explosion rocked the building.\n\n'Crikey, old Fritz is really trying to put the wind up us tonight,' Mr Miller said cheerfully.\n\nHer name was Susie, apparently. She had no idea, she really couldn't remember anything. A man kept calling her out of the darkness. 'Come on, Susie, don't go to sleep now,' and 'How about we have a nice cuppa when we get out of here, eh, Susie?' She was choking on ash and dust. She sensed something inside her was torn beyond repair. Cracked. She was a golden bowl. 'Quite Jamesian, really,' she heard Teddy say. (Had he said that?) She was a great tree (how odd). She was very cold. The man was holding her hand, squeezing it, 'Come on, Susie, stay awake now.' But she couldn't, the soft dark was beckoning to her with the promise of sleep, endless sleep, and the snow began to fall gently until she was entirely shrouded and everything was dark.\n\n# A Lovely Day Tomorrow\n\n# _September 1940_\n\nShe missed Crighton, more than she had let on to either him or Pamela. He had taken a room at the Savoy on the night before war was declared and she had got dressed up in her good royal-blue satin only for him to announce that they should call an end to things ('to say our adieux'). 'It's going to get awfully bloody,' he said, but whether he meant the war or them she wasn't sure.\n\nDespite or perhaps because of their adieux, they went to bed together and he spent a lot of time telling her how much he would miss 'this body', the 'lineaments of your flesh', 'this pretty face', and so on, until she got rather fed up and said, 'Well, it _is_ you that wants out of this, not me.'\n\nShe wondered if he made love to Moira in the same way \u2013 detachment and passion in equal measure \u2013 but it was one of those questions you couldn't ask in case he were to tell the truth. What did it matter, Moira was getting him back. Soiled goods perhaps but hers nonetheless.\n\nThe next morning they breakfasted in the room and then listened to Chamberlain's speech. There was a wireless in the suite. Not long after, a siren sounded but strangely neither of them panicked. It all seemed very unreal. 'I expect it's a test,' Crighton said. Ursula thought that from now on everything would probably be a test.\n\nThey left the hotel and walked along the Embankment to Westminster Bridge, where air-raid wardens were blowing their whistles and shouting that the scare was over. Others were riding along on bicycles with _All-Clear_ signs attached to them, and Crighton said, 'Good God, I fear for us if this is the best we can muster in a raid.' Sandbags were being stacked along the bridge, being stacked everywhere and Ursula thought it was just as well there was so much sand in the world. She tried to remember the lines from 'The Walrus and the Carpenter'. _If seven maids with seven mops \u2013_ but they had reached Whitehall by now and Crighton broke into her thoughts by taking both her hands in his and saying, 'I must go now, darling,' and for a moment he sounded like a rather cheap and sentimental movie star. She decided she would live out the war as a nun. Much easier.\n\nShe had watched him walk along Whitehall and suddenly felt horribly alone. She might, after all, go back to Finchley.\n\n# _November 1940_\n\nOn the other side of the wall she could hear Emil complaining and Mrs Appleyard's soothing remonstrance. She began to sing a lullaby in her own language, the mother tongue, Ursula thought. It was an extraordinarily sad song and Ursula vowed that if she ever had a child (difficult when you had decided to live as a nun) she would sing to it nothing but jolly jigs and ditties.\n\nShe felt alone. She would have liked a warm body for comfort, a dog would be better than being on her own on nights like this. A living, breathing presence.\n\nShe moved the blackout aside. No sign of bombers yet, just the long finger of a solitary searchlight poking into the blackness. A new moon hung in the sky. _Pale for weariness_ , according to Shelley but _Queen and huntress, chaste and fair_ for Ben Jonson. To Ursula it betrayed an indifference that made her suddenly shiver.\n\nThere was always a second before the siren started when she was aware of a sound as yet unheard. It was like an echo, or rather the opposite of an echo. An echo came afterwards, but was there a word for what came before?\n\nShe heard the whine of a plane overhead and the _bang-bang-bang-bang-bang_ of the first bombs dropping and she was about to replace the blackout and make a run for the cellar when she noticed a dog cowering in a doorway opposite \u2013 almost as if she'd wished it into existence. Even from where she was, she could sense its terror. She hesitated for a second and then thought, oh, damn, and raced down the stairs.\n\nShe passed the Nesbit sisters. 'Ooh, bad luck, Miss Todd,' Ruth giggled. 'Crossing on the stairs, you know.'\n\nUrsula was going down, the sisters were coming up. 'You're going the wrong way,' she said, rather pointlessly.\n\n'I forgot my knitting,' Lavinia said. She was wearing an enamel brooch shaped like a black cat. A little rhinestone winked for an eye. 'She's knitting leggings for Mrs Appleyard's baby,' Ruth said. 'It's so cold in her flat.'\n\nIt was incredibly noisy on the street. She could hear incendiaries clattering down on a roof nearby, sounding like a giant coal scuttle being emptied. The sky was alight. A chandelier flare fell, as graceful as fireworks, illuminating everything below.\n\nA stream of bombers was roaring overhead as she dashed across the street to the dog. It was a nondescript terrier, whimpering and shaking all over. Just as she grabbed hold of it she heard a terrific _swish_ and knew she was for it, that they were both for it. A colossal growling noise was followed by the loudest bang she'd heard so far in the Blitz. This is it, she thought, this is how I die.\n\nShe took a blow to the forehead, a brick or something, but didn't lose consciousness. A blast of air, like a hurricane, knocked her off her feet. There was a horrendous pain in her ears and all she could hear was a high-pitched whistling, singing noise and she knew that her eardrums must have gone. Debris was showering down on her, cutting her and digging into her. The blast seemed to come in successive waves and she could feel a grumbling, grinding vibration in the ground beneath her.\n\nFrom a distance an explosion seemed to be over almost immediately but when you were in the middle of it it seemed to go on for ever, to have a character that changed and developed as it went along so that you had no idea how it was going to end up, how _you_ were going to end up. She was half sitting, half lying on the ground and tried to hang on to something but she couldn't let go of the dog (this thought uppermost in her mind for some reason) and she found herself being blown slowly along the ground.\n\nThe pressure began to decrease a little but the dirt and dust were still raining down and the blast had life in it yet. Then something else hit her on the head and everything went dark.\n\n*\n\nShe was woken by the dog licking her face. It was very hard to understand what had happened but after a while she realized that the doorway where she had grabbed the dog didn't exist any more. The door had been blown inwards, the pair of them with it, and now they were lying among debris in the passage of a house. The staircase of the house behind them, choked with broken bricks and splintered wood, now led nowhere as the upper floors had gone.\n\nStill stunned, she struggled to a sitting position. Her head felt thick and stupid but nothing seemed to be broken and she couldn't find any bleeding, although she supposed she must be covered in cuts and bruises. The dog too, although very quiet, seemed to be uninjured. 'Your name must be Lucky,' she said to it but her voice hardly came out at all, there was so much choking dust in the air. Cautiously, she got to her feet and walked down the passage to the street.\n\nHer house had also gone, everywhere she looked there were great heaps of smoking rubble and skeletal walls. The pared fingernail of the moon was bright enough, even through the veil of dust, to cast light on the horror. If she hadn't run to save the dog she would be cinders in the Millers' cellar now. Was everyone dead? The Nesbits, Mrs Appleyard and Emil? Mr Bentley? All the Millers?\n\nShe stumbled into the street where two firemen were unreeling a hose. While they were attaching it to the hydrant one of them spotted her and shouted, 'Are you all right, miss?' It was funny but he looked exactly like Fred Smith. And then the other fireman yelled, 'Watch out, the wall's coming down!'\n\nIt was. Slowly, incredibly slowly, as if in a dream, the whole wall tilted on an invisible axis and without a single brick detaching itself it inclined towards them, as if taking a graceful bow, and fell in one piece, bringing the darkness down with it.\n\n# _August 1926_\n\n_Als er das Zimmer verlassen hatte wusst, was sie aus dieser Erscheinung machen solle..._\n\nBees buzzed their summer afternoon lullaby and Ursula, in the shade of the apple trees, drowsily abandoned _Die Marquise von O_. Through half-open eyes she watched a small rabbit a few yards away nibbling contentedly on grass. He was either unaware of her or very bold. Maurice would have shot it by now. He was home after graduation, waiting to start his training in the law, and had spent the entire vacation being thoroughly and noisily bored. ('He could always get a summer job,' Hugh said. 'It's not unheard of for vigorous young men to work.')\n\nMaurice was so bored in fact that he had agreed to teach Ursula to shoot and even agreed to use old bottles and cans as targets rather than the many wild creatures that he was forever taking potshots at \u2013 rabbits, foxes, badgers, pigeons, pheasants, even once a small roe deer, for which neither Pamela nor Ursula would ever forgive him. As long as they were inanimate, Ursula rather liked shooting things. She used Hugh's old wildfowler but Maurice had a splendid Purdey, his twenty-first-birthday present from his grandmother. Adelaide had been threatening to die for some years now but 'never came good on her promises', Sylvie said. She lingered on in Hampstead, 'like a giant spider', Izzie said, shuddering, over the veal cutlets _\u00e0 la Russe_ , although it may have been the cutlets themselves that caused this reaction. It was not one of the better dishes in Mrs Glover's repertoire.\n\nOne of the few things, perhaps the only thing Sylvie and Izzie had in common, was their antipathy towards Hugh's mother. 'Your mother too,' Hugh pointed out to Izzie and Izzie said, 'Oh, no, she found me by the side of the road. She often told me so. I was so naughty that even the gypsies didn't want me.'\n\nHugh came to watch Maurice and Ursula shooting and said, 'Why, little bear, you're a real Annie Oakley.'\n\n'You know,' Sylvie said, appearing suddenly and startling Ursula into full wakefulness, 'long, lazy days like these will never come again in your life. You think they will, but they won't.'\n\n'Unless I become incredibly rich,' Ursula said. 'Then I could be idle all day long.'\n\n'Perhaps,' Sylvie said, 'but summer would still have to come to an end one day.' She sank down on the grass next to Ursula and picked up the Kleist. 'A suicidal romantic,' she said dismissively. 'Are you really going to do Modern Languages? Your father says Latin might be more useful.'\n\n'How can it be useful? Nobody speaks it,' Ursula said reasonably. This was an argument that had been rumbling genteelly all summer. She stretched her arms above her head. 'I shall go and live in Paris for a year and speak nothing but French. That will be very _useful_ there.'\n\n'Oh, Paris,' Sylvie shrugged. 'Paris is rather overrated.'\n\n'Berlin, then.'\n\n'Germany's a mess.'\n\n'Vienna.'\n\n'Stuffy.'\n\n'Brussels,' Ursula said. 'No one can object to Brussels.'\n\nIt was true, Sylvie could think of nothing to say about Brussels and their grand tour of Europe came to an abrupt halt.\n\n'After university anyway,' Ursula said. 'That's _years_ away yet, you can stop worrying.'\n\n'University won't teach you how to be a wife and mother,' Sylvie said.\n\n'What if I don't want to be a wife and mother?'\n\nSylvie laughed. 'Now you're just talking nonsense to provoke. There's tea on the lawn,' she said, rousing herself reluctantly. 'And cake. And, unfortunately, Izzie.'\n\n*\n\nUrsula went for a walk along the lane before supper, Jock happily trotting ahead. (He was a wonderfully cheerful dog, it was hard to believe that Izzie could have chosen so well.) It was the kind of summer evening that made Ursula want to be alone. 'Oh,' Izzie said, 'you're at an age when a girl is simply _consumed_ by the sublime.' Ursula wasn't sure what she meant ('No one is ever sure what she means,' Sylvie said) but she thought she understood a little. There was a strangeness in the shimmering air, a sense of _imminence_ that made Ursula's chest feel full, as if her heart was growing. It was a kind of high holiness \u2013 she could think of no other way of describing it. Perhaps it was the future, she thought, coming nearer all the time.\n\nShe was sixteen, on the brink of everything. She had even been kissed, on her birthday at that, by the rather alarming American friend of Maurice's. 'Just one kiss,' she told him before batting him away when he got too fresh with her. Unfortunately he stumbled over his huge feet and fell backwards into a cotoneaster, which looked rather uncomfortable and certainly undignified. She told Millie, who hooted with laughter. Still, as Millie said, a kiss was a kiss.\n\nHer walk took her to the station where she said hello to Fred Smith, who doffed his railwayman's cap as if she were already a grown-up.\n\nThe imminence remained imminent, receded even, as she watched his train _huff-huff-huff_ off to London. She walked back and met Nancy, grubbing for things for her nature collection, and they walked companionably together before they were overtaken by Benjamin Cole on his bicycle. He stopped and dismounted and said, 'Shall I escort you home, ladies?' rather in the way that Hugh might have done and Nancy giggled.\n\nUrsula was glad that the heat of the afternoon had already made her cheeks pink because she could feel herself blushing. She grabbed some cow parsley from the hedgerow and fanned herself (ineffectually) with it. She had not, after all, been so wrong about the imminence.\n\nBenjamin ('Oh, do call me Ben,' he said. 'Only my parents call me Benjamin these days') walked with them as far as the Shawcrosses' gate where he said, 'Goodbye, then,' and climbed back on his bicycle for the short ride home.\n\n'Oh,' Nancy whispered, disappointed on her behalf, 'I thought maybe he would walk you home, just the two of you.'\n\n'Am I so obvious?' Ursula asked, her spirits drooping.\n\n'You are rather. Never mind.' Nancy patted her on the arm as if she were the elder by four years rather than Ursula. And then, 'I'm late, I think, I don't want to miss dinner,' she said and, clutching her foraged treasure, she skipped along the path towards her house, singing _tra-la-la_. Nancy was a girl who really did sing _tra-la-la_. Ursula wished she was that kind of girl. She turned to go, she supposed she was late for supper too, but then she heard the mad ringing of a bicycle bell announcing Benjamin (Ben!) zooming towards her. 'I forgot to say,' he said, 'we're having a party next week \u2013 Saturday afternoon \u2013 Mother said to ask you. It's Dan's birthday, she wants some girls to dilute the boys, I think that was her phrase. She thought maybe you and Millie. Nancy's a bit young, isn't she?'\n\n'Yes, she is,' Ursula agreed quickly. 'But I'd love to come. So would Millie, I'm sure. Thank you.'\n\nImminence had returned to the world.\n\nShe watched him cycle away, whistling as he went. When she turned round she nearly bumped into a man who seemed to have appeared out of nowhere and was hovering, waiting for her. He tipped his cap and muttered, 'Evening, miss.' He was a rough-looking fellow and Ursula took a step back. 'Tell me the way to the station, miss?' he said and she pointed down the lane and said, 'It's that way.'\n\n'Care to show me the way, miss?' he said, moving closer to her again.\n\n'No,' she said, 'no thank you.' Then his hand suddenly shot out and he grabbed her forearm. She managed to tug her arm away and set off running, not daring to look behind until she reached her doorstep.\n\n'All right, little bear?' Hugh asked as she flung herself into the porch. 'You look all puffed out,' he said.\n\n'No, I'm fine, really,' she said. Hugh would only worry if she told him about the man.\n\n'Veal cutlets _\u00e0 la Russe_ ,' Mrs Glover said as she put a large white china dish on the table. 'I'm only telling you because last time I cooked it someone said they couldn't begin to imagine what it was.'\n\n'The Coles are having a party,' Ursula said to Sylvie. 'Millie and I are invited.'\n\n'Lovely,' Sylvie said, distracted by the contents of the white porcelain dish, much of which would later be fed to a less discerning (or, as Mrs Glover would have it, 'less fussy') West Highland terrier.\n\nThe party was a disappointment. It was a rather daunting affair with endless games of charades (Millie in her element, needless to say) and quizzes to which Ursula knew most of the answers but was left unheard, beaten by the ferociously competitive speed of the Cole boys and their friends. Ursula felt invisible and the only intimacy that she shared with Benjamin (he no longer seemed like Ben) was when he asked her if she would like some fruit cup and then forgot to come back with any. There was no dancing but piles of food and Ursula comforted herself picking and choosing from an impressive selection of desserts. Mrs Cole, patrolling the food, said to her, 'Goodness, you're such a little scrap of a thing, where do you put all that food?'\n\nSuch a little scrap of a thing, Ursula thought as she tramped dejectedly home, that no one even seemed to notice her.\n\n'Did you get cake?' Teddy asked eagerly when she came in the door.\n\n'Masses,' she said. They sat on the terrace and shared the large slice of birthday cake doled out on departure by Mrs Cole, Jock receiving his fair share. When a large dog fox trotted on to the twilit lawn Ursula tossed a piece in its direction but it regarded the cake with the disdain of a carnivore.\n\n# The Land of Begin Again\n\n# _August 1933_\n\n' _Er kommt! Er kommt!_ ' one of the girls shouted.\n\n'He's coming? Finally?' Ursula said, glancing at Klara.\n\n'Apparently. Thank goodness. Before we die of hunger and boredom,' she said.\n\nThey were both equally bemused and amused by the younger girls' hero-worshipping antics. They had been waiting by the roadside for the best part of a hot afternoon, with nothing to eat or drink except for a pail of milk that two of the girls had fetched from a farm nearby. Some of the girls had heard a rumour that the F\u00fchrer would be arriving today at his mountain retreat, and they had been waiting patiently for hours now. Several of the girls had taken a siesta on the grass verge, but none of them had any intention of giving up without a glimpse of the F\u00fchrer.\n\nThere was some cheering further down the steep, crooked road that led up to Berchtesgaden and they all jumped to their feet. A big black car swept past them and some of the girls squealed with excitement but 'he' wasn't in it. Then a second car, a magnificent open-topped black Mercedes, came into view, a swastika pennant fluttering on the bonnet. It drove slower than the previous car and did indeed contain the new Chancellor of the Reich.\n\nThe F\u00fchrer gave an abbreviated version of his salute, a funny little flap of the hand backwards so that he looked as if he were cupping his ear to hear them better as they shouted out to him. At the sight of him, Hilde, standing next to Ursula, said simply, 'Oh,' investing the single syllable with religious ecstasy. And then, just as quickly, it was all over. Hanne crossed her hands over her chest, looking like a rather constipated saint. 'My life is fulfilled,' she laughed.\n\n'He looks better in his photographs,' Klara murmured.\n\nThe girls were all in remarkably high spirits, had been all day, and under their _Gruppenf\u00fchrerin's_ orders (Adelheid, a blonde Amazon, an admirably competent eighteen-year-old) they now quickly formed themselves into a squad and started cheerfully on the long march back to the youth hostel, singing as they went. ('They sing _all_ the time,' Ursula wrote to Millie. 'It's all a little too _lustig_ for my liking. I feel like I'm in the chorus of a particularly jolly folk opera.')\n\nTheir repertoire was varied \u2013 folk songs, quaint love songs and rousing, rather savage, patriotic anthems about flags dipped in blood, as well as the obligatory sing-songs around the campfire. They especially liked _Schunkeln \u2013_ linking arms and swaying to songs. When Ursula was pushed into rendering a song she gave them 'Auld Lang Syne', perfect for _Schunkeln_.\n\nHilde and Hanne were Klara's younger sisters, keen members of the BDM, the Bund Deutscher M\u00e4del \u2013 the girls' equivalent of the Hitler-Jugend ( _'Ha Jot_ , we call them,' Hilde said, and she and Hanne fell about giggling at the idea of handsome boys in uniform).\n\nUrsula had heard of neither the Hitler-Jugend nor the BDM before arriving in the Brenner household but in the two weeks she had been living there she had heard little else from Hilde and Hanne. 'It's a healthy hobby,' their mother, Frau Brenner, said. 'It promotes peace and understanding between young people. No more wars. And it keeps them away from boys.' Klara, like Ursula a recent graduate \u2013 she had been an art student at the _Akademie \u2013_ was indifferent to her sisters' obsession but had offered to be a chaperone on their _Bergwanderung_ , their summer camping trip, hiking from one _Jugendherberge_ to the next in the Bavarian mountains. 'You'll come, won't you?' Klara said to Ursula. 'I'm sure we'll have fun and you'll see some of the countryside. And if you don't you'll be stuck in town with Mutti and Vati.'\n\n'I think it's like the Girl Guides,' Ursula wrote to Pamela.\n\n'Not quite,' Pamela wrote back.\n\nUrsula was not intending to spend long in Munich. Germany was no more than a detour in her life, part of her adventurous year in Europe. 'It will be my own grand tour,' she said to Millie, 'although I'm afraid it's a little second-rate, a \"not quite so grand tour\".' The plan was to take in Bologna rather than Rome or Florence, Munich not Berlin and Nancy instead of Paris (Nancy Shawcross much amused by this choice) \u2013 all cities where her tutors from university knew of good homes in which she could lodge. To keep herself she was to do a little teaching, although Hugh had arranged for a modest but regular money order to be sent to her. Hugh was relieved that she would be spending her time 'in the provinces', where 'people are, on the whole, better behaved'. ('He means duller,' Ursula said to Millie.) Hugh had completely vetoed Paris, he had a particular aversion to the city, and was hardly more keen on Nancy which was still uncompromisingly _French_. ('Because it's in France,' Ursula pointed out.) He had seen enough of the continent during the Great War, he said, he couldn't see what all the hullabaloo was about.\n\nUrsula had, despite Sylvie's reservations, studied for a degree in Modern Languages \u2013 French and German and a little Italian (very little). Recently graduated and failing to think of anything else, she had applied and been given a place on a teacher-training course. She had deferred for a year, saying that she wanted an opportunity to see a little of the world before 'settling down' to a lifetime at the blackboard. That was her rationale anyway, the one that she paraded for parental scrutiny, whereas her true hope was that something would happen in the course of her time abroad that would mean she need never take up the place. What that 'something' was she had no idea ('Love perhaps,' Millie said wistfully). Anything really that would mean she didn't end up as an embittered spinster in a girls' grammar school, spooling her way through the conjugation of foreign verbs, chalk dust falling from her clothes like dandruff. (She based this portrait on her own schoolmistresses.) It wasn't a profession that had garnered much enthusiasm in her immediate circle either.\n\n'You want to be a _teacher_?' Sylvie said.\n\n'Honestly, if her eyebrows had shot up any further they would have left the atmosphere,' Ursula said to Millie.\n\n'But do you really? Want to teach?' Millie said.\n\n'Why does every single person I know ask me that question in that same tone of voice?' Ursula said, rather piqued. 'Am I so clearly unsuited to the profession?'\n\n'Yes.'\n\nMillie herself had done a course at a drama academy in London and was now playing in rep in Windsor, in second-rate crowd pleasers and melodramas. 'Waiting to be discovered,' she said, striking a theatrical pose. Everyone seems to be waiting for something, Ursula thought. 'Best not to wait,' Izzie said. 'Best to _do_.' Easier for her to say.\n\nMillie and Ursula were sitting in the wicker chairs on the lawn at Fox Corner, hoping that the foxes would come and play on the grass. A vixen and her litter had been visiting the garden. Sylvie had been putting out scraps and the vixen was half tame now and would sit quite boldly in the middle of the lawn, like a dog waiting for its dinner, while her cubs \u2013 already rangy, long-legged things by June \u2013 squabbled and somersaulted around her.\n\n'What am I to do then?' Ursula said helplessly (hopelessly). Bridget appeared with a tray of tea and cake and placed it on a table between them. 'Learn shorthand and typing and work in the civil service? That sounds pretty dismal too. I mean what else is there for a woman to _do_ if she doesn't want to go from the parental to the marital home with nothing in between?'\n\n'An educated woman,' Millie amended.\n\n'An educated woman,' Ursula agreed.\n\nBridget muttered something incomprehensible and Ursula said, 'Thank you, Bridget.'\n\n( _'You_ have seen Europe,' she said, rather accusingly, to Sylvie. 'When you were younger.'\n\n'I was not on my own, I was in the company of my father,' Sylvie said. But surprisingly this argument seemed to have some effect and it was, in the end, Sylvie who championed the trip against Hugh's objections.)\n\nBefore she departed for Germany Izzie took her shopping for silk underwear and scarves, pretty lace-edged handkerchiefs, 'a really good pair of shoes', two hats and a new handbag. 'Don't tell your mother,' she said.\n\nIn Munich she was to lodge with the Brenner family \u2013 mother, father and three daughters (Klara, Hildegard and Hannelore) and a son, Helmut, who was away at school, in an apartment on the Elisabethstrasse. Hugh had already had an extensive correspondence with Herr Brenner to assess his suitability as a host. 'I'll be a terrible disappointment,' she said to Millie, 'Herr Brenner will be expecting the Second Coming, given the preparations that have been made.' Herr Brenner was himself a teacher at the Deutsche Akademie and had arranged for Ursula to give some classes to beginners in English and had also procured several introductions to people looking for private tuition. This he told her when he met her off the train. She felt rather downcast, she hadn't set her mind to the idea of work just yet and she was exhausted after a long and decidedly trying rail journey. The _Schnellzug_ from the Gare de l'Est in Paris had been anything but _schnell_ and she had shared the compartment with, among others, a man who alternated smoking a cigar with eating his way through an entire salami, both actions which made her feel rather discomfited. ('And all I saw of Paris was a station platform,' she wrote to Millie.)\n\nThe salami-eating man had followed her out into the corridor when she went in search of the Ladies. She thought he was going to the buffet car but then as she reached the lavatory compartment he attempted, to her alarm, to push in after her. He said something to her that she didn't understand, although its meaning seemed lewd (the cigar and the salami seemed strange preludes). ' _Lass mich in Ruhe_ ,' leave me in peace, she said stoutly but he continued to push her and she continued to push back. She suspected their struggle, polite as opposed to violent, might have looked quite comical to an observer. Ursula wished there was someone in the corridor that she could appeal to. She couldn't imagine what the man would do to her if he succeeded in confining her in the tiny lavatory compartment. (Afterwards she wondered why she hadn't simply screamed. What a dunce she was.)\n\nShe was 'saved' by a pair of officers, smart in their black uniforms and silver insignia, who materialized out of nowhere and took a firm hold of the man. They gave him a stern talking-to, although she couldn't recognize half the vocabulary, and then very gallantly they found her a different carriage, one where there were only women, which she hadn't known about. When the officers had gone her fellow female travellers couldn't stop talking about how handsome the SS officers were. (' _Schutzstaffel_ ,' one of the women murmured admiringly. 'Not like those louts in brown.')\n\nThe train was late pulling into the station in Munich. There had been some kind of incident, Herr Brenner said, a man had fallen from the train.\n\n'How awful,' Ursula said.\n\nDespite it being summer, it was chilly and raining heavily. The gloomy atmosphere didn't lift with her arrival at the Brenners' enormous apartment, where no lamps were lit against the evening and where the rain was beating against the lace-curtained windows as if it was determined to break in.\n\nBetween them, Ursula and Herr Brenner had lugged her heavy trunk up the stairs, a somewhat farcical procedure. Surely there was someone who could help them, Ursula thought irritably? Hugh would have employed 'a man' \u2013 or two \u2013 and not expected her to manage it herself. She thought of the SS officers on the train, how efficiently and courteously they would have dealt with the trunk.\n\nThe female Brenners of the house proved to be absent. 'Oh, not back yet,' Herr Brenner said, unconcerned. 'They went shopping, I think.' The apartment was full of heavy furniture and shabby rugs and leafy plants that gave the impression of a jungle. She shivered, it seemed inhospitably cold for the time of year.\n\nThey manoeuvred the trunk into the room that was to be hers. 'This used to be my mother's room,' Herr Brenner said. 'This is her furniture. Sadly, she died last year.' The way that he gazed at the bed \u2013 a large, Gothic affair that looked as if it were built specifically to induce nightmares in its occupant \u2013 clearly hinted that Frau Brenner senior's demise had taken place within its downy coverlets. The bed seemed to dominate the room and Ursula felt suddenly nervous. Her experience on the train with the salami-eating man was still embarrassingly vivid and now here she was again alone in a foreign country with a complete stranger. Bridget's lurid tales of the white slave trade came to mind.\n\nTo her relief, they both heard the front door open and a great commotion taking place in the hallway. 'Ah,' Herr Brenner said, beaming with delight, 'they're back!'\n\nThe girls spilled and tumbled into the apartment, all wet from the rain, laughing and carrying parcels. 'Look who's arrived,' Herr Brenner said, inducing much excitement in the youngest two girls. (Hilde and Hanne would prove to be the most excitable girls Ursula had ever encountered.)\n\n'You're here!' Klara said, clasping both her hands in her own cold, damp ones, ' _Herzlich willkommen in Deutschland_.'\n\nWhile the younger girls chattered nineteen to the dozen Klara moved quickly round the apartment turning on lamps and the place was suddenly transformed \u2013 the rugs were worn but they were figured richly, the old furniture gleamed with polish, the cold jungle of plants turned into a pretty, ferny bower. Herr Brenner lit a big porcelain _Kachelofen_ in the living room ('like having a big warm animal in the room', she wrote to Pamela) and assured her that tomorrow the weather would be back to normal, warm and sunny.\n\nA table was quickly laid with an embroidered cloth and supper produced \u2013 a platter of cheese, salami, sliced sausage, salad and a dark bread that smelt of Mrs Glover's seed cake as well as a delicious kind of fruit soup that confirmed that she was in a foreign country. ('Cold fruit soup!' she wrote to Pamela. 'What would Mrs Glover have to say about that!')\n\nEven Herr Brenner's dead mother's room was more accommodating now. The bed was soft and inviting, the sheets edged with handworked crochet and the bedside lamp had a pretty pink glass shade that cast a warm glow. Someone \u2013 Klara, Ursula suspected \u2013 had placed a posy of marguerites in a little vase on the dressing table. Ursula was dropping with fatigue by the time she clambered into the bed (it was so high it required a small footstool) and fell gratefully into a deep, dreamless sleep, untroubled by the ghost of the previous occupant.\n\n'But of course you're going to have some holiday time,' Frau Brenner said next morning at breakfast (a meal that was oddly similar to supper the night before). Klara was 'at a bit of a loose end'. She had finished her art course and didn't know what to do next. She was chafing at the bit to leave home and 'be an artist' but 'not much money in Germany to spare for art', she grumbled. Klara kept some of her work in her room, big, harsh abstract canvases that seemed at odds with her kind and temperate nature. Ursula couldn't imagine she would make a living from them. 'Perhaps I shall have to teach,' she said miserably.\n\n'Fate worse than death,' Ursula agreed.\n\nKlara occasionally did some framing for a photography studio in Schellingstrasse. The daughter of one of Frau Brenner's acquaintances worked there and had put in a word for her. Klara and the daughter \u2013 Eva \u2013 had been in kindergarten together. 'But framing, it's hardly art, is it?' Klara said. The photographer \u2013 Hoffmann \u2013 was the 'personal photographer' of the new Chancellor, 'so I am intimately acquainted with his features', she said.\n\nThe Brenners didn't have much money either (Ursula supposed that was why they were renting her a room) and everyone Klara knew was poor, but then in 1933 everyone everywhere was poor.\n\nDespite the lack of funds Klara was determined that they should make the most of the remainder of the summer. They went to the Carlton Teehaus or Caf\u00e9 Heck by the Hofgarten and ate _Pfannkuchen_ and drank _Schokolade_ until they felt sick. They walked for hours in the Englischer Garten and then ate ice-cream or drank beer, their faces pink with the sun. They also spent time boating or swimming with friends of Helmut, Klara's brother \u2013 a revolving carousel of Walters, Werners, Kurts, Heinzes and Gerhards. Helmut himself was in Potsdam, a cadet, a _Jungmann_ at a new kind of military school that the F\u00fchrer had founded. 'He's very keen on the Party,' Klara said, in English. Her English was quite good and she was enjoying practising with Ursula.\n\n'On parties,' Ursula corrected her. 'We would say \"he's very keen on parties\".' Klara laughed and shook her head, 'No, no, _the_ Party, the Nazis. Don't you know that since last month it's the only one that we're allowed?'\n\n'When Hitler came to power,' Pamela wrote didactically to her, 'he passed the Enabling Act, in Germany it's called _Gesetz zur Behebung der Not von Volk und Reich_ which translates as something like the \"Law to Remedy the Distress of People and Reich\". That's a fancy title for the overthrow of democracy.'\n\nUrsula wrote blithely back, 'But democracy will right itself as it always does. This too shall pass.'\n\n'Not without help,' Pamela replied.\n\nPamela was a grouch about Germany and was easy to ignore when you could spend long hot afternoons sunbathing with Walters, Werners, Kurts, Heinzes and Gerhards, lolling lazily by the municipal swimming pool or the river. Ursula was taken aback at how these boys were near enough naked with their short shorts and disconcertingly small swimming trunks. Germans generally, she discovered, were not averse to stripping off in front of others.\n\nKlara also knew a different, more cerebral set \u2013 her friends from art school. They tended to prefer the dark, the smoky interiors of caf\u00e9s or their own scruffy apartments. They drank and smoked a great deal and spoke a lot about art and politics. ('So by and large,' she wrote to Millie, 'between these two groups of people I am getting an all-round education!') Klara's art-school friends were a ragged, dissident bunch who all seemed to dislike Munich, which was a seat of 'petit-bourgeois provincialism' apparently, and talked all the time about moving to Berlin. They talked a lot about doing things, she noticed, but actually did very little.\n\nKlara was in the grip of a different kind of inertia. Her life had 'stalled', she was secretly in love with one of her professors from art school, a sculptor, but he was away in the Black Forest on a family holiday. (Reluctantly, she admitted that the 'family' was actually his wife and two children.) She was waiting for her life to resolve itself, she said. More prevarication, Ursula thought. Although she was hardly one to talk.\n\nUrsula was still a virgin, of course, 'intact' as Sylvie would have it. Not for any moral reason, simply because she hadn't yet met anyone that she liked enough. 'You don't have to _like_ them,' Klara laughed.\n\n'Yes, but I want to.' She seemed instead to be a magnet for unsavoury types \u2013 the man on the train, the man in the lane \u2013 and worried that they could read something in her that she couldn't read herself. She felt rather stiff and English compared to Klara and her artist friends or the absent Helmut's confr\u00e8res (who were actually terrifically well behaved).\n\nHanne and Hilde had persuaded Klara and Ursula to accompany them to an event in the local sports stadium. Ursula was under the misapprehension that it was a concert but it turned out to be a rally of Hitler-Jugend. Despite Frau Brenner's optimism, the BDM had done nothing to counter Hilde and Hanne's interest in boys.\n\nTo Ursula, these ranks of hearty, healthy boys all looked the same but Hilde and Hanne spent a lot of time animatedly pointing out Helmut's friends, those same Walters, Werners, Kurts, Heinzes and Gerhards who loafed by the swimming pool in next to nothing. Now, squeezed into their immaculate uniforms (more short shorts), they looked like very fierce and upstanding Boy Scouts.\n\nThere was a lot of marching and singing to a brass band and several speakers who attempted the same declamatory style as the F\u00fchrer (and failed) and then everyone leapt to their feet and sang ' _Deutschland \u00fcber alles_ '. As Ursula didn't know the words she quietly sang 'Glorious Things Of Thee Are Spoken' to Haydn's lovely tune, a hymn they had often sung in school assembly. When the singing finished everyone shouted ' _Sieg Heil!_ ' and saluted and Ursula was almost surprised to find herself joining in. Klara was convulsed with laughter at the sight but nonetheless Ursula noticed she had her arm raised too. 'I should think so too,' she said, nonchalantly. 'I don't want to be set upon on the way home.'\n\nNo, thank you, Ursula didn't want to stay home with Vati and Mutti Brenner in hot, dusty Munich so Klara rummaged through her wardrobe and found a navy skirt and white blouse that suited requirements and the group leader, Adelheid, provided a spare khaki battledress jacket. A three-cornered scarf drawn through a braided leather Turk's-head knot completed the outfit. Ursula thought she looked rather dashing. She found herself regretting never having been a Girl Guide, although she supposed it was about more than just the uniform.\n\nThe upper age limit for the BDM was eighteen so neither Ursula nor Klara was qualified to join, they were 'old ladies', _alte Damen_ , according to Hanne. Ursula didn't think that the troop really needed to be escorted by them as Adelheid was as efficient as a sheepdog with her girls. With her statuesque figure and Nordic blonde plaits she could have passed for a youthful Freyja visiting from F\u00f3lkvangr. She was perfect propaganda. At eighteen she would soon be too old for the BDM, what would she do then?\n\n'Why, I will join the National Socialist Women's League, of course,' she said. She already wore a small silver swastika pinned to her shapely bosom, the runic symbol of belonging.\n\nThey took a train, their rucksacks stowed on the luggage racks, and by evening they had arrived in a small Alpine village, near to the Austrian border. From the station they marched in formation (singing, naturally) to their _Jugendherberge_. People stopped to watch them and some clapped appreciatively.\n\nThe dormitory they were allotted was full of two-tier bunks, most of which were already occupied by other girls and they had to squeeze themselves in, sardine fashion. Klara and Ursula elected to share a mattress on the floor.\n\nThey were given supper in the dining room, seated at long trestle tables, served with what turned out to be the standard fare of soup and _Kn\u00e4ckebrot_ with cheese. In the morning they breakfasted on dark bread, cheese and jam and tea or coffee. The clean mountain air made them ravenous and they wolfed down everything in sight.\n\nThe village and its surroundings were idyllic, there was even a small castle that they were allowed to visit. It was cold and dank, full of suits of armour and flags and heraldic shields. It seemed like a very uncomfortable place to live.\n\nThey took long walks around the lake or in the forest and then they hitched lifts back to the youth hostel on farm lorries and hay-carts. One day they hiked all the way along the river to a magnificent waterfall. Klara had brought her sketch-pad with her and her quick, lively little charcoal drawings were much more appealing than her paintings. 'Ach,' she said, 'they're _gem\u00fctlich_. Cosy little sketches. My friends would laugh.' The village itself was a sleepy little place where the houses had windows full of geraniums. There was an inn on the river where they drank beer and ate veal and noodles until they thought they would burst. Ursula never mentioned the beer to Sylvie in her letters, she wouldn't have understood how commonplace it was here. And even if she had, she wouldn't have approved.\n\nThey were to move on the next day, they would be living 'under canvas' for a few days, a big encampment of girls, and Ursula felt sorry to be leaving the village.\n\nA fair was taking place on their last night there, a combination of an agricultural show and a harvest festival, a lot of it incomprehensible to Ursula. ('To me too,' Klara said. 'I'm a city girl, remember.') The women all wore local costume and variously garlanded farm animals were paraded around a field and then awarded prizes. Flags, again with swastikas, decorated the field. There was plenty of beer and a brass band played. A big wooden platform had been set up in the middle of the field and, accompanied by an accordion, some boys in _Lederhosen_ gave a demonstration of _Schuhplattler_ , clapping and stamping and slapping their thighs and heels in time to the music.\n\nKlara scoffed at them but Ursula considered it rather clever. Ursula thought that she would quite like to live in an Alpine village ('Like Heidi,' she wrote to Pamela. She wrote less to Pamela as her sister was so aggravated by the new Germany. Pamela, even at a distance, was the voice of her conscience, but then it was very easy to have a conscience from a distance).\n\nThe accordionist took his place in a band and people began to dance. Ursula was led on to the platform by a succession of terrifically shy farm boys who had an odd clodhopping way of moving around the dance floor which she recognized as the rather awkward \u00be time of the _Schuhplattler_. Between the beer and the dancing she began to feel quite light-headed so that she was confused when Klara appeared, dragging by the hand a very handsome man who was clearly not local, saying, 'Look who I found!'\n\n'Who?' Ursula asked.\n\n'None other than our cousin's half-cousin's cousin once removed,' Klara said gaily. 'Or something to that effect. May I present J\u00fcrgen Fuchs.'\n\n'Just a half-cousin,' he said, smiling.\n\n'Delighted to meet you I'm sure,' she said. He clicked his heels and kissed her hand, she was reminded of Prince Charming in _Cinderella_. 'It's the Prussian in me,' he said and laughed, as did the Brenners. 'We have no Prussian blood at all,' Klara said.\n\nHe had a lovely smile, amused and thoughtful at the same time, and extraordinarily blue eyes. He was undoubtedly handsome, rather like Benjamin Cole, except Benjamin was his dark polar opposite, the negative to J\u00fcrgen Fuchs's positive.\n\nA Todd and a Fuchs \u2013 a pair of foxes. Had fate intervened in her life? Dr Kellet might have appreciated the coincidence.\n\n'He is so handsome,' she wrote to Millie after that encounter. All those awful words used in trashy romances come to mind \u2013 _heart-stopping,breathtaking_. She had read enough of Bridget's novels on idle wet afternoons to know.\n\n'Love at first sight,' she wrote giddily to Millie. But of course such feelings weren't 'true' love (that was what she would feel for a child one day), merely the false grandeur of madness. ' _Folie \u00e0 deux_ ,' Millie wrote back. 'How delicious.'\n\n'Good for you,' Pamela wrote.\n\n'Marriage is based on a more enduring kind of love,' Sylvie cautioned.\n\n'I am thinking of you, little bear,' Hugh wrote, 'so far away from here.'\n\nWhen darkness fell there was a torchlit procession through the village and then fireworks from the battlements of the small castle. It was rather thrilling.\n\n' _Wundersch\u00f6n, nicht wahr?_ ' Adelheid said, her face radiant in the light of the torches.\n\nYes, Ursula agreed, it's lovely.\n\n# _August 1939_\n\n_Der Zauberberg_. The magic mountain.\n\n' _Aaw. Sie ist so niedlich.' Click, click, click_. Eva loved her Rolleiflex. Eva loved Frieda. She is so _cute_ , she said. They were on the enormous terrace of the Berghof, bright with Alpine sun, waiting for lunch to be brought out. It was much nicer to eat out here, _al fresco_ , rather than in the big, gloomy dining room, its massive window full of nothing but mountains. Dictators loved everything to be on a grand scale, even their scenery. _Bitte l\u00e4cheln!_ Big smile. Frieda obliged. She was an obliging child.\n\nEva had persuaded Frieda out of her serviceable English hand-smocked dress (Bourne and Hollingsworth, purchased by Sylvie and sent for Frieda's birthday) and had arrayed her instead in Bavarian costume \u2013 dirndl, apron, knee-length white socks. To Ursula's English eyes (more English every day, she felt) the outfit still looked as though it belonged in a dressing-up box, or perhaps a school play. Once, at her own school (how long ago and far away that seemed now), they had put on a performance of _The Pied Piper of Hamelin_ and Ursula had played a village girl, clad in much the same get-up that Frieda was now attired in.\n\nMillie had been King Rat, a bravura performance, and Sylvie said, 'Those Shawcross girls thrive on attention, don't they?' There was something of Millie in Eva \u2013 a restless, empty gaiety that needed continual feeding. But then Eva was an actress too, playing the greatest part of her life. In fact her life _was_ her part, there was no difference.\n\nFrieda, lovely little Frieda, just five years old, with her blue eyes and stubby blonde plaits. Frieda's complexion, so pale and wan when she had first arrived, now pink and gold from all the Alpine sunshine. When the F\u00fchrer saw Frieda, Ursula caught the zealot gleam in his own blue eyes, as cold as the K\u00f6nigsee down below, and knew he was seeing the future of the _Tausendj\u00e4hriges Reich_ rolling out in front of him, _M\u00e4dchen_ after _M\u00e4dchen_. ('She doesn't take after you, does she?' Eva said, without malice, she had no malice.)\n\nWhen she was a child \u2013 a period in her life that Ursula seemed to find herself returning to almost compulsively these days \u2013 she had read fairy tales of wronged princesses who saved themselves from lustful fathers and jealous stepmothers by smearing their fair faces with walnut juice and rubbing ashes in their hair to disguise themselves \u2013 as the gypsy, the outsider, the shunned. Ursula wondered how one obtained walnut juice, it didn't seem the kind of thing you could just walk into a shop and buy. And it was no longer safe to be the nut-brown outsider, much better if one wanted to survive to be here, on Obersalzberg \u2013 _Der Zauberberg \u2013_ in the kingdom of make-believe, 'the Berg', as they called it with the intimacy of the elect.\n\nWhat on earth was she doing here, Ursula wondered, and when could she leave? Frieda was well enough now, her convalescence drawing to a close. Ursula determined to say something to Eva today. After all, they weren't prisoners, they could leave any time they chose.\n\nEva lit a cigarette. The F\u00fchrer was away and the mouse was being naughty. He didn't like her to smoke or drink, or wear make-up. Ursula rather admired Eva's small acts of defiance. The F\u00fchrer had come and gone twice since Ursula first arrived at the Berghof with Frieda two weeks ago, his arrivals and departures moments of heightened drama for Eva, for everyone. The Reich, Ursula had concluded a long time ago, was all pantomime and spectacle, ' _A tale told by an idiot, full of sound and fury_ ,' she wrote to Pamela. 'But unfortunately _not_ signifying nothing.'\n\nFrieda, on a prompt from Eva, did a twirl and laughed. She was the molten core at the centre of Ursula's heart, she was the better part of everything she did or thought. Ursula would be willing to walk on knives for the rest of her life if it would protect Frieda. Burn in the flames of hell to save her. Drown in the deepest of waters if it would buoy her up. (She had explored many extreme scenarios. Best to be prepared.) She had had no idea (Sylvie gave little indication) that maternal love could be so gut-achingly, _painfully_ physical.\n\n'Oh, yes,' Pamela said, as if it were the most casual thing in the world, 'it turns you into a regular she-wolf.' Ursula didn't think of herself as a she-wolf, she was, after all, a bear.\n\nThere were real she-wolves prowling everywhere on the Berg \u2013 Magda, Emmy, Margarete, Gerda \u2013 the brood-wives of the senior party officials, all jostling for a little power of their own, producing endless babies from their fecund loins, for the Reich, for the F\u00fchrer. These she-wolves were dangerous, predatory animals and they hated Eva, the 'silly cow' \u2013 _die bl\u00f6de Kuh \u2013_ who somehow or other had managed to trump them all.\n\nThey, surely, would have given anything to be the mate of the glorious leader rather than insignificant Eva. Wasn't a man of his stature worthy of a Br\u00fcnnhilde \u2013 or at the very least a Magda or a Leni? Or perhaps the Valkyrie herself, 'the Mitford woman', _das Fr\u00e4ulein Mitford_ , as Eva referred to her. The F\u00fchrer admired England, especially aristocratic, imperial England, although Ursula doubted that his admiration would stop him from trying to destroy it if the time came.\n\nEva disliked all the Valkyries who might be a rival for the F\u00fchrer's attentions, her strongest emotions conceived in fear. Her greatest antipathy was reserved for Bormann, the _\u00e9minence grise_ of the Berg. It was he who held the purse strings, he who shopped for Eva's gifts from the F\u00fchrer and who doled out the money for all those fur coats and Ferragamo shoes, reminding her in many subtle ways that she was merely a courtesan. Ursula wondered where the fur coats came from, most of the furriers she had seen in Berlin were Jewish.\n\nHow it must stick in the collective craw of the she-wolves that the F\u00fchrer's consort was a shop girl. When she first met him, Eva told Ursula, when she was working in Hoffmann's _Photohaus_ , she had addressed him as Herr Wolf. 'Adolf means noble wolf in German,' she said. How he must like that, Ursula thought. She had never heard anyone call him Adolf. (Did Eva call him _mein F\u00fchrer_ even in bed? It seemed perfectly possible.) 'And do you know that his favourite song,' Eva laughed, 'is \"Who's Afraid Of The Big Bad Wolf?\" '\n\n'From the Disney film _Three Little Pigs_?' Ursula said, incredulous.\n\n'Yes!'\n\nOh, thought Ursula, I cannot wait to tell _that_ to Pamela.\n\n'And now one with _Mutti_ ,' Eva said. 'Hold her in your arms. _Sehr sch\u00f6n_. Smile!' Ursula had watched Eva gleefully stalking the F\u00fchrer with her camera, hunting down a photograph of him where he hadn't turned away from the lens or pulled the brim of his hat down comically low, like a spy in poor disguise. He disliked having his photograph taken by her, preferring flattering studio lighting or a more heroic pose than the snaps Eva liked of him. Eva, on the other hand, loved being photographed. She didn't just want to be in photographs, she wanted to be in a film. 'Ein movie.' She was going to go to Hollywood ('one day') and play herself, 'the story of my life', she said. (The camera made everything real somehow for Eva.) The F\u00fchrer had promised, apparently. Of course, the F\u00fchrer promised a lot of things. It was what had got him where he was today.\n\nEva refocused the Rolleiflex. Ursula was glad she hadn't brought her old Kodak, it would hardly have stood up to comparison. 'I'll have copies made for you,' Eva said. 'You can send them to England, to your parents. It looks very pretty with the mountains in the background. Now give me a big smile. _Jetzt lach doch mal richtig!_ '\n\nThe mountain panorama was the backdrop to every photo taken here, the backdrop to everything. At first Ursula had thought it beautiful, now she was beginning to find its magnificence oppressive. The great icy crags and the rushing waterfalls, the endless pine trees \u2013 nature and myth fused to form the Germanic sublimated soul. German Romanticism, it seemed to Ursula, was writ large and mystical, the English Lakes seemed tame by comparison. And the English soul, if it resided anywhere, was surely in some unheroic back garden \u2013 a patch of lawn, a bed of roses, a row of runner beans.\n\nShe should go home. Not to Berlin, to Savignyplatz, but to England. To Fox Corner.\n\nEva perched Frieda on the parapet and Ursula promptly removed her. 'She has no head for heights,' she said. Eva was forever lolling precariously on this same parapet, or parading dogs or small children along it. The drop below was dizzying, all the way down past Berchtesgaden to the K\u00f6nigsee. Ursula felt rather sorry for little Berchtesgaden with its innocent window boxes of cheerful geraniums, its meadows sloping down to the lake. It seemed a long time since she was here in '33 with Klara. Klara's professor had divorced his wife and Klara was now married to him and they had two children.\n\n'The _Nibelungen_ live up there,' Eva told Frieda, pointing at the peaks all around them, 'and demons and witches and evil dogs.'\n\n'Evil dogs?' Frieda echoed uncertainly. She had already been scared by the irksome Negus and Stasi, Eva's annoying Scotties, without needing to hear about dwarves and demons.\n\nAnd _I_ have heard, Ursula thought, that it was Charlemagne who hid out in the Untersberg, sleeping in a cave, waiting to be woken for the final battle between good and evil. She wondered when that would be. Soon perhaps.\n\n'And one more,' Eva said. 'Big smile!' The Rolleiflex glinted relentlessly in the sun. Eva owned a cine camera too, an expensive gift from her own Mr Wolf, and Ursula supposed she should be glad that they weren't being recorded for posterity in moving colour. Ursula imagined in a future time someone leafing through Eva's (many) albums and wondering who Ursula was, mistaking her perhaps for Eva's sister Gretl or her friend Herta, footnotes to history.\n\nOne day, of course, all this would be consigned to that same history, even the mountains \u2013 sand, after all, was the future of rocks. Most people muddled through events and only in retrospect realized their significance. The F\u00fchrer was different, he was consciously _making_ history for the future. Only a true narcissist could do that. And Speer was designing buildings for Berlin so that they would look good when they were in ruins a thousand years from now, his gift to the F\u00fchrer. (To think on such a scale! Ursula lived hour by hour, another consequence of motherhood, the future as much a mystery as the past.)\n\nSpeer was the only one who was nice to Eva and therefore Ursula afforded him a latitude in her opinion that perhaps he didn't deserve. He was also the only one of these would-be Teutonic knights who had good looks, who wasn't gimpy or toad-squat or a corpulent pig, or \u2013 worse somehow \u2013 resembled a low-level bureaucrat. ('And they are all in uniform!' she wrote to Pammy. 'But it's all pretend. It's like living in the pages of _The Prisoner of Zenda_. They're awfully good at hogwash.' How she wished Pammy was here by her side, how she would have enjoyed dissecting the characters of the F\u00fchrer and his henchmen. She would conclude that they were all charlatans, spouting cant.)\n\nIn private, J\u00fcrgen claimed to find them all 'tremendously' flawed and yet in public he behaved like any good servant of the Reich. _Lippenbekenntnis_ , he said. Lip service. (Needs must, Sylvie would have said.) This was how you got on in the world, he said. Ursula supposed in this respect he was rather like Maurice, who said you had to work with fools and donkeys to advance your career. Maurice was also a lawyer, of course. He was quite senior in the Home Office these days. If they went to war would this be a problem? Would the armour of German citizenship \u2013 donned so reluctantly \u2013 be enough to protect her? (If they went to war! Could she really countenance being on this side of the Channel?)\n\nJ\u00fcrgen was a lawyer. If he wanted to practise law he had to join the Party, he had no choice. _Lippenbekenntnis_. He worked for the Ministry of Justice in Berlin. At the time he proposed to her ('a bit of a whirlwind courtship', she wrote to Sylvie) he had barely ceased being a communist.\n\nNow J\u00fcrgen had abandoned his Leftist politics and was staunch in his defence of what had been achieved \u2013 the country was working again \u2013 full employment, food, health, self-respect. New jobs, new roads, new factories, new hope \u2013 how else could they achieve this, he said? But it came with an ecstatic faux-religion and a wrathful false messiah. 'Everything comes with a price,' J\u00fcrgen said. Perhaps not as high as this one. (How _had_ they done it, Ursula often wondered. Fear and stagecraft mostly. But where _had_ all the money and jobs come from? Perhaps just from manufacturing flags and uniforms, enough of those around to rescue most economies. 'The economy is recovering anyway,' Pamela wrote, 'it's a happy coincidence for the Nazis that they can claim this recovery.') Yes, he said, there was violence to begin with, but it was a spasm, a wave, the Sturmabteilung letting off steam. Everything, everybody, was more rational now.\n\nIn April they had attended the parade for the F\u00fchrer's fiftieth birthday in Berlin. J\u00fcrgen had been allotted seats, in the guests' grandstand. 'An honour, I suppose,' he said. What had he done, she wondered, to deserve the 'honour'? (Did he seem happy about it? It was hard to tell sometimes.) He hadn't been able to get them tickets for the Olympics in '36 yet here they were now, rubbing shoulders with the VIPs of the Reich. He was always busy these days. 'Lawyers never sleep,' he said. (Yet as far as Ursula could see they were prepared to sleep throughout the Thousand Years.)\n\nThe parade had gone on for ever, the greatest expression yet of Goebbels's showmanship. A great deal of martial music and then the overture provided by the Luftwaffe \u2013 an impressive, noisy fly-past along the East\u2013West Axis and over the Brandenburg Gate by squadrons of aircraft in formation, wave after wave. More sound and fury. 'Heinkels and Messerschmitts,' J\u00fcrgen said. How did he know? All boys know their planes, he said.\n\nThere followed the march-past of the regiments, a seemingly inexhaustible supply of soldiers goose-stepping along the road. They reminded Ursula of high-kicking Tiller girls. ' _Stechschritt_ ,' Ursula said, 'who on earth invented that?'\n\n'The Prussians,' J\u00fcrgen laughed, 'of course.'\n\nShe took out a bar of chocolate and broke off a piece and offered it to J\u00fcrgen. He frowned and shook his head as though she had showed a lack of respect to the assembled military might. She ate another piece. Small acts of defiance.\n\nHe leaned in close so she could hear him \u2013 the crowd were making an abominable racket \u2013 'You really have to admire their precision, if nothing else,' he said. She did, she did admire it. It _was_ extraordinary. Robotic in its perfection as if each member of each regiment was identical to the next, as if they had been produced on a factory line. It wasn't quite _human_ , but then it wasn't the job of armies to look human, was it? ('It was all so very _masculine_ ,' she reported to Pamela.) Would the British army be capable of achieving such mechanical drilling on this scale? The Soviets perhaps, but the British were less _committed_ somehow.\n\nFrieda, on her knee, was already asleep and it had hardly begun yet. All the while Hitler took the salute, his arm stiff in front of him the whole time (she could catch a glimpse of him from where they were sitting, just the arm, like a poker). Power obviously provided a peculiar kind of stamina. If it was my fiftieth birthday, Ursula thought, I would like to spend it on the banks of the Thames, Bray or Henley or thereabouts, with a picnic, a very English picnic \u2013 a Thermos of tea, sausage rolls, egg and cress sandwiches, cake and scones. Her family was all there in this picture, but was J\u00fcrgen part of the idyll? He would fit in well enough, lounging on the grass in boating flannels, talking cricket with Hugh. They had met and got on well. They had gone to England, to Fox Corner, in '35 for a visit. 'He seems like a nice chap,' Hugh said, although when he learned that she had taken German citizenship he wasn't so keen. It had been an awful mistake, she knew that now. 'Hindsight's a wonderful thing,' Klara said. 'If we all had it there would be no history to write about.'\n\nShe should have stayed in England. She should have stayed at Fox Corner, with the meadow and the copse and the stream that ran through the bluebell wood.\n\nThe machinery of war started to roll past. 'Here come the tanks,' J\u00fcrgen said in English, as the first of the _Panzer_ appeared, carried on the back of lorries. His English was good, he had spent a year at Oxford (hence his knowledge of cricket). Then came the _Panzer_ under their own steam, motorbikes with sidecars, armoured cars, the cavalry trotting smartly along (a particular crowd pleaser \u2013 Ursula woke Frieda up for the horses), and then the artillery, from light field guns to massive anti-aircraft guns and huge cannons.\n\n'K-3s,' J\u00fcrgen said appreciatively, as if that would mean something to her.\n\nThe parade showed a love of order and geometry that was incomprehensible to Ursula. In this, it was no different from all the other parades and rallies \u2013 all that theatre \u2013 but this one seemed so bellicose. So much weaponry was staggering \u2013 the country was armed to the teeth! Ursula had had no idea. No wonder there were jobs for everyone. 'If you want to rescue the economy you need a war, Maurice says,' Pamela wrote. And what did you need weaponry for if not war?\n\n'Refitting the military has helped to rescue our psyche,' J\u00fcrgen said, 'given us back our pride in our country. When in 1918 the generals surrendered...' Ursula stopped listening, it was an argument she had heard too many times. 'They started the last war,' she wrote crossly to Pamela. 'And honestly, you would think they were the only ones who struggled afterwards, and that no other people were poor or hungry or bereaved.' Frieda woke up again and was cranky. She fed her chocolate. Ursula was cranky too. Between them they finished the bar.\n\nThe finale was actually rather moving. The massed colours of the regiments formed a long file several ranks deep in front of Hitler's podium \u2013 a formation so precise its edges might have been cut with a razor \u2013 and then they dipped their colours to the ground in honour of him. The crowd went wild.\n\n'What did you think?' J\u00fcrgen asked as they shuffled out of the grandstand. He carried Frieda on his shoulders.\n\n'Magnificent,' Ursula said. 'It was magnificent.' She could feel the beginnings of a headache worming its way into her temple.\n\nFrieda's illness had begun one morning several weeks ago with a raised temperature. 'I feel sick,' Frieda said. When Ursula felt her forehead it was clammy and she said, 'You don't have to go to kindergarten, you can stay home with me today.'\n\n'A summer cold,' J\u00fcrgen said, when he came home. She was always a chesty child ('Takes after my mother,' Sylvie said gloomily) and they were accustomed to sniffling colds and sore throats but the cold got worse very quickly and Frieda turned feverish and listless. Her skin felt as though it were ready to catch fire. 'Keep her cool,' the doctor said and Ursula laid cold wet cloths on her forehead and read her stories but Frieda, try as she did, could summon no interest in them. Then she grew delirious and the doctor listened to her rattling lungs and said, 'Bronchitis, you have to wait for it to pass.'\n\nLate that night Frieda grew suddenly, horribly worse and they wrapped the almost inanimate little body in a blanket and rushed in a taxi to the nearest hospital, a Catholic one. Pneumonia was diagnosed. 'She's a very sick little girl,' the doctor said, as if somehow they were to blame.\n\nUrsula didn't leave Frieda's bedside for two days and nights, holding on to the little hand to keep her in this world. 'If only I could have it for her,' J\u00fcrgen whispered across the starchy white sheets that were also helping to pin Frieda to this world. Nuns floated around the ward like galleons in their enormous, complicated wimples. How long, Ursula wondered in an absent moment when all her attention wasn't focused on Frieda, did it take them to put these contraptions on in the morning? Ursula was sure she would never have managed without making a mess of it. The headdress alone seemed a good enough reason not to be a nun.\n\nThey willed Frieda to live and she did. _Triumph des Willens_. The crisis passed and she started the long road to recovery. Pale and weak, she was going to need to convalesce and one evening when Ursula returned home from the hospital she found an envelope, hand-delivered to their door.\n\n'From Eva,' she said to J\u00fcrgen, showing him the letter when he returned from work.\n\n'Who's Eva?' he asked.\n\n'Smile!' _Click, click, click_. Anything to help keep Eva amused, she supposed. She didn't mind. Eva had been very kind to invite them so that Frieda could breathe good mountain air and eat the fresh vegetables and eggs and milk from the Gutshof, the model farm on the slopes beneath the Berghof.\n\n'Is it a royal command?' J\u00fcrgen asked. 'Can you say no? Do you want to say no? I hope not. And it will do your headaches good too.' She'd noticed recently that the more he rose through the echelons in the ministry, the more one-sided their conversations had become. He made statements, raised questions, answered the questions and drew conclusions without ever needing to involve her in the exchange. (A lawyer's way perhaps.) He didn't even seem to be aware that he was doing it.\n\n'The old goat has a woman after all then, does he? Who would have guessed? Did you know? No, you would have said. And to think you know her. It can only be good for us, can't it? To be so close to the throne. For my career, which is the same thing as us. _Liebling_ ,' he added, rather perfunctorily.\n\nUrsula thought that being close to a throne was a rather dangerous place to be. 'I don't know Eva,' Ursula said. 'I've never met her. It's Frau Brenner who knows her, knows her mother, Frau Braun. Klara used to work at Hoffmann's sometimes, with Eva. They were at kindergarten together.'\n\n'Impressive,' J\u00fcrgen said, 'from _Kaffeeklatsch_ to the seat of power in three easy moves. Does Fr\u00e4ulein Eva Braun know her old kindergarten pal, Klara, is married to a Jew?' It was the way he said the word that surprised her. _Jude_. She'd never heard him say it that way before \u2013 sneering and dismissive. It drove a nail into her heart. 'I have no idea,' she said. 'I am not part of the _Kaffeeklatsch_ , as you call it.'\n\nThe F\u00fchrer took up so much room in Eva's life that when he wasn't here she was an empty vessel. Eva kept nightly telephone vigils when her lover was absent and was like a dog, one ear fretfully cocked every evening for the call that brought her master's voice to her.\n\nAnd there was so little to _do_ up here. After a while all the tramping along forest paths and swimming in the (freezing cold) K\u00f6nigsee became enervating rather than invigorating. There were only so many wildflowers you could pick, only so much sunbathing on the loungers on the terrace before you went slightly mad. There were battalions of nursemaids and nannies on the Berg, all eager to be with Frieda, and Ursula found herself with much of the same empty time on her hands as Eva. She had, stupidly, packed only one book, at least it was a long one, Mann's _Der Zauberberg_. She hadn't realized it was on the banned list. A Wehrmacht officer saw her reading it and said, 'You're very bold, that's one of their forbidden books, you know.' She supposed the way he said 'their' implied he wasn't one of 'them'. What was the worst they could do? Take the book off her and put it in the kitchen stove?\n\nHe was nice, the Wehrmacht officer. His grandmother was Scottish, he said, and he had spent many happy holidays in 'the Highlands'.\n\n_Im Grunde hat es eine merkw\u00fcrdige Bewandtnis mit diesem Sicheinleben an fremdem Orte, dieser \u2013 sei es auch \u2013 m\u00fchseligen Anpassung und Umgew\u00f6hnung_ , she read and translated laboriously and rather badly \u2013 'There is something strange about getting this settling in to a new place, the laborious adaptation and familiarization...' How true, she thought. Mann was hard work. She would have preferred a boxload of Bridget's gothic romances. She was sure they wouldn't be _verboten_.\n\nThe mountain air had done her headaches no good at all (nor had Thomas Mann). They were, if anything, worse. _Kopfschmerzen_ , the very word made her head sore. 'I can't find anything wrong with you,' the doctor at the hospital told her. 'It must be your nerves.' He gave her a prescription for veronal.\n\nEva herself had no intellect to sustain her but then the Berg wasn't exactly the court of an intelligentsia. The only person whom you might have called a thinker was Speer. It wasn't that Eva led an unexamined life, far from it, Ursula suspected. You could sense the depression and neuroses hidden beneath all that _Lebenslust_ , but anxiety wasn't what a man looked for in a mistress.\n\nUrsula supposed that to be a successful mistress (although she had never been one herself, either successful or unsuccessful) a woman should be a comfort and a relief, a restful pillow for the weary head. _Gem\u00fctlichkeit_. Eva was amiable, she chatted about inconsequential things and made no attempt to be brainy or astute. Powerful men needed their women to be unchallenging, the home should not be an arena for intellectual debate. 'My own husband told me this so it must be true!' she wrote to Pamela. He hadn't meant it in the context of himself \u2013 he was not a powerful man. 'Not yet, anyway,' he laughed.\n\nThe political world was of concern only in that it took the object of Eva's devotion away from her. She was shunted rudely out of the public eye, allowed no official status, allowed no status at all, as loyal as a dog but with less recognition than a dog. Blondi was higher in the hierarchy than Eva. Her greatest regret, Eva said, was not being allowed to meet the duchess when the Windsors visited the Berghof.\n\nUrsula frowned on hearing this. 'But she's a Nazi, you know,' she said unthinkingly. ('I suppose I should be more careful in what I say!' she wrote to Pamela.) Eva had merely replied, 'Yes, of course, she is,' as if it were the most natural thing in the world for the consort of the once and never again King of England to be a Hitlerite.\n\nThe F\u00fchrer must be seen to tread a noble, solitary path of chastity, he couldn't marry because he was wedded to Germany. He had sacrificed himself to his country's destiny \u2013 at least that was the gist of it, Ursula thought she might have discreetly nodded off at this point. (It was one of his endless after-dinner monologues.) Like our own Virgin Queen, she thought, but didn't say so, as she expected the F\u00fchrer would not like to be compared to a woman, even an English aristocratic one with the heart and stomach of a king. At school, Ursula had had a history teacher who had been particularly fond of quoting Elizabeth I. _Do not tell secrets to those whose faith and silence you have not already tested_.\n\nEva would have been happier back in Munich, in the little bourgeois house that the F\u00fchrer had bought for her, where she could lead a normal social life. Here, in her gilded cage, she had to amuse herself, flicking through magazines, discussing the latest hairstyles and love lives of film stars (as if Ursula knew anything on the subject), and parading one outfit after another like a quick-change artist. Ursula had been in her bedroom several times, a pretty, feminine boudoir quite different from the heavy-handed d\u00e9cor of the rest of the Berghof, spoilt only by the portrait of the F\u00fchrer that was given pride of place on the wall. Her hero. The F\u00fchrer had not hung a reciprocal portrait of his mistress in his rooms. Instead of Eva's face smiling at him from the wall he was challenged by the stern features of his own beloved hero, Frederick the Great. _Friedrich der Grosse_.\n\n'I always mishear \"grocer\" for \"great\",' she wrote to Pamela. Grocers were not, generally speaking, warmongers and conquerors. What had the F\u00fchrer's apprenticeship for greatness been? Eva shrugged, she didn't know. 'He's always been a politician. He was born a politician.' No, Ursula thought, he was born a baby, like everyone else. And this is what he has chosen to become.\n\nThe F\u00fchrer's bedroom, adjoining Eva's bathroom, was out of bounds. Ursula had seen him sleeping though, not in that sacrosanct bedroom but in the warm post-prandial sunshine on the Berghof's terrace, the great warrior's mouth slackly open in _l\u00e8se-majest\u00e9_. He looked vulnerable but there were no assassins on the Berg. Plenty of guns, thought Ursula, easy enough to get hold of a Luger and shoot him through the heart or the head. But then what would happen to her? Worse, what would happen to Frieda?\n\nEva sat next to him, watching fondly as one might a child. In sleep he belonged to no one but her.\n\nShe was, fundamentally, nothing more nor less than a nice young woman. You couldn't necessarily judge a woman by the man she slept with. (Or could you?)\n\nEva had a wonderful athletic figure, one that Ursula felt quite envious of. She was a healthy, physical girl \u2013 a swimmer, a skier, a skater, a dancer, a gymnast \u2013 who loved the outdoors, who loved _movement_. And yet she had attached herself like a limpet to an indolent middle-aged man, a creature of the night, literally, who didn't rise from his bed before midday (and yet who could still take an afternoon nap), who didn't smoke or drink or dance or overindulge \u2013 spartan in his habits although not his vigour. A man who had never been seen stripped off further than his _Lederhosen_ (comically unattractive to the non-Bavarian eye), whose halitosis had repelled Ursula on first meeting and who swallowed tablets like sweets for his 'gas problem' ('I hear he farts,' J\u00fcrgen said, 'be warned. Must be all those vegetables'). He was concerned for his dignity but he wasn't really vain, as such. 'Merely a megalomaniac,' she wrote to Pamela.\n\nA car and a driver had been sent for them and when they arrived at the Berghof the F\u00fchrer himself had greeted them \u2013 on the great steps, where he welcomed dignitaries, where he had welcomed Chamberlain last year. When Chamberlain returned to Britain he said that he 'now knew what was in Herr Hitler's mind'. Ursula doubted that anyone knew that, not even Eva. Particularly not Eva.\n\n'You're very welcome here, _gn\u00e4diges Frau_ ,' he said. 'You should stay until the _liebe Kleine_ is better.'\n\n'He likes women, children, dogs, really what can you fault?' Pamela wrote. 'It's just a shame he's a dictator with no respect for the law or common humanity.' Pamela had quite a few friends in Germany from her university days, many of them Jewish. She had a full house (well, three of a kind) of boisterous boys (quiet little Frieda would be quite overwhelmed in Finchley) and now wrote that she was pregnant again, 'fingers crossed for a girl'. Ursula missed Pammy.\n\nPamela would not fare well under this regime. Her sense of moral outrage would be too great for her to remain silent. She wouldn't be able to bite her tongue like Ursula did (a scold's bridle). _They also serve who only stand and wait_. Did that apply to one's ethics? Is this my defence, Ursula wondered? It might be better to misquote Edmund Burke rather than Milton. _All that is necessary for the forces of evil to win in the world is for enough good women to do nothing_.\n\nThe day after they arrived there had been a children's tea-party for someone's birthday, a little Goebbels or Bormann, Ursula wasn't sure \u2013 there were so many of them and they were so similar. She was reminded of the ranks of the military at the F\u00fchrer's birthday parade. Scrubbed and polished, each one had a special word from Uncle Wolf before they were allowed to indulge in the cake that was set out on a long table. Poor sweet-toothed Frieda (who undoubtedly took after her mother in this respect) was too heavy-lidded with fatigue to eat any. There was always cake on the Berghof, poppyseed _Streusel_ and cinnamon and plum _Tortes_ , puff pastries filled with cream, chocolate cake \u2013 great domes of _Schwarzw\u00e4lder Kirschtorte \u2013_ Ursula wondered who ate all this cake. She herself certainly did her best to get through it.\n\nIf a day with Eva could be tedious it was as nothing compared to an evening when the F\u00fchrer was present. Interminable hours after dinner were spent in the Great Hall \u2013 a vast, ugly room where they listened to the gramophone or watched films (or, often, both). The F\u00fchrer naturally dictated the choices. For music, _Die Fledermaus_ and _Die lustige Witwe_ were favourites. On the first evening, Ursula thought it would be hard to forget the sight of Bormann, Himmler, Goebbels (and their savage helpmeets) all wearing their thin-lipped snake smiles (more _Lippenbekenntnis_ , perhaps) while listening to _Die lustige Witwe_. Ursula had seen a student production of _The Merry Widow_ when she was at university. She had been good friends with the girl who played Hanna, the lead. She could never have guessed then that the next time she would hear 'Vilja, O Vilja! the witch of the wood', it would be in German and in this strangest of company. That production had taken place in '31. She hadn't seen what her own future held, let alone that of Europe.\n\nFilms were shown nearly every evening in the Great Hall. The projectionist would arrive and the great Gobelin tapestry on one wall would be rolled up mechanically, like a blind, to reveal a screen behind it. Then they would have to sit through some awful romantic schmaltz or American adventure, or worse, a mountain film. In this way Ursula had seen _King Kong, The Lives of a Bengal Lancer_ and _Der Berg ruft_. On the first evening it had been _Der heilige Berg_ (more mountains, more Leni). The F\u00fchrer's favourite film, Eva confided, was _Snow White_. And which character did he identify with, Ursula wondered \u2013 the wicked witch, the dwarves? Not Snow White surely? It must be the Prince, she concluded (did he have a name? Did they ever, was it enough simply to _be_ the role?). The Prince who awoke the sleeping girl, just as the F\u00fchrer had woken Germany. But not with a kiss.\n\nWhen Frieda was born, Klara had given her a beautiful edition of _Schneewittchen und die sieben Zwerge_ , 'Snow White and the Seven Dwarves', illustrated by Franz J\u00fcttner. Klara's professor had long since been barred from teaching at the art school. They had planned to leave in '35 and then again in '36. After _Kristallnacht_ , Pamela had written to Klara directly, although she had never met her, offering her a home in Finchley. But that inertia, that damned tendency everyone seemed to have to _wait_... and then her professor had been part of a round-up and had been transported east \u2013 to work in a factory, the authorities said. 'His beautiful sculptor's hands,' Klara said sadly.\n\n('They're not really _factories_ , you know,' Pamela wrote.)\n\nUrsula remembered being an avid reader of fairy tales as a child. She had put great faith not so much in the happy ending as in the restoration of justice to the world. She suspected she had been duped by _die Br\u00fcder_ Grimm. _Spieglein, Spieglein, an der Wand \/ Wer ist die Sch\u00f6nste im ganzen Land?_ Not this lot, that was for sure, Ursula thought, looking around the Great Hall during her first wearisome evening on the Berg.\n\nThe F\u00fchrer was a man who preferred operetta to opera, cartoons to highbrow culture. Watching him holding Eva's hand while humming along to Lehar, Ursula was struck by how _ordinary_ (even silly) he was, more Mickey Mouse than Siegfried. Sylvie would have made short work of him. Izzie would have eaten him up and spat him out. Mrs Glover \u2013 what would Mrs Glover have done, Ursula wondered? This was her new favourite game, deciding how the people she knew would have dealt with the Nazi oligarchs. Mrs Glover, she concluded, would probably have beaten them all soundly with her meat hammer. (What would Bridget do? Ignore him completely probably.)\n\nWhen the film was finished the F\u00fchrer settled down to expound (for hours) on his pet subjects \u2013 German art and architecture (he perceived himself to be an architect-manqu\u00e9), _Blut und Boden_ (the land, always the land), his solitary, noble path (the wolf again). He was the saviour of Germany, and poor Germany, his _Schneewittchen_ , would be saved by him whether she wanted it or not. He droned on about healthy German art and music, about Wagner, _Die Meistersinger_ , his favourite line from the libretto \u2013 _Wacht auf, es nahet gen den Tag \u2013_ 'Awake, the morning is here' (it would be if he went on much longer, she thought). Back to destiny \u2013 his \u2013 how it was intertwined with the destiny of the _Volk. Heimat, Boden_ , victory or downfall (What victory, Ursula wondered? Against whom?). Then something about Frederick the Great that she didn't catch, something about Roman architecture, then the Fatherland. (For the Russians it was 'the Motherland', was there something to be made of that, Ursula wondered? What was it for the English? Just 'England', she supposed. Blake's 'Jerusalem' at a pinch.)\n\nThen back to destiny and the _Tausendj\u00e4hriges_. On and on so that the headache that had begun before dinner as a dull ache was a crown of thorns by now. She imagined Hugh saying, 'Oh, do shut up, Herr Hitler,' and suddenly felt so homesick she thought she was going to cry.\n\nShe wanted to go home. She wanted to go to Fox Corner.\n\nAs with kings and their courtiers, they could not leave until dismissed, until the monarch himself decided to ascend to the bedchamber. At one point Ursula caught Eva yawning theatrically at him as if to say, 'That's enough now, Wolfi' (her imagination was becoming rather lurid, she knew, forgivable given the circumstances). And then at last, finally, thank God, he made a move and the exhausted company rustled to its feet.\n\nWomen in particular seemed to love the F\u00fchrer. They wrote him letters in the thousands, baked him cakes, embroidered swastikas on to cushions and pillows for him, and, like Hilde and Hanne's BDM troop, lined the steep road up to the Obersalzberg to catch a delirious glimpse of him in the big black Mercedes. Many women shouted to him that they wanted to have his baby. 'But what do they _see_ in him?' Sylvie puzzled. Ursula had taken her to a parade, one of the interminable flag-waving, banner-toting ones in Berlin, because she wanted to 'find out for myself what all the fuss is about'. (How very British of Sylvie to reduce the Third Reich to a 'fuss'.)\n\nThe street was a forest of red, black and white. 'Their colours are very harsh,' Sylvie said, as though she were considering asking the National Socialists to decorate her living room.\n\nAt the F\u00fchrer's approach the crowd's excitement had grown to a rabid frenzy of _Sieg Heil_ and _Heil Hitler_. 'Am I the only one to be unmoved?' Sylvie said. 'What is it, do you suppose \u2013 mass hysteria of some kind?'\n\n'I know,' Ursula said, 'it's like the Emperor's new clothes. We're the only ones who can see the naked man.'\n\n'He's a clown,' Sylvie said dismissively.\n\n'Shush,' Ursula said. The English word was the same as the German and she didn't want to attract the hostility of the people around them. 'You should put your arm up,' she said.\n\n'Me?' the outraged flower of British womanhood replied.\n\n'Yes, you.'\n\nReluctantly, Sylvie raised her arm. Ursula thought that until the day she died she would remember the sight of her mother giving the Nazi salute. Of course, Ursula said to herself afterwards, this was in '34, back when one's conscience hadn't been shrunk and muddled by fear, when she had been blind to what was truly afoot. Blinded by love perhaps, or just dumb stupidity. (Pamela had seen, unblinkered by anything.)\n\nSylvie had made the journey to Germany so that she could inspect Ursula's unexpected husband. Ursula wondered what she had planned to do if she hadn't found J\u00fcrgen suitable \u2013 drug and kidnap her and haul her on to the _Schnellzug_? They were still in Munich then, J\u00fcrgen hadn't started working for the Ministry of Justice in Berlin, they hadn't moved to the Savignyplatz or become parents to Frieda, although Ursula was cumbersome with pregnancy.\n\n'Fancy you becoming a mother,' Sylvie said, as if it were something she had never expected. 'To a German,' she added thoughtfully.\n\n'To a baby,' Ursula said.\n\n'It's nice to get away,' Sylvie said. From what, Ursula wondered?\n\nKlara met them for lunch one day and afterwards said, 'Your mother is rather chic.' Ursula had never thought of Sylvie as stylish but she supposed that compared with Klara's mother, Frau Brenner, as soft and doughy as a loaf of _Kartoffelbrot_ , Sylvie was quite a fashion plate.\n\nOn the way back from lunch, Sylvie said she wanted to visit Oberpollingers and buy a present for Hugh. When they reached the department store they found the windows daubed with anti-Jewish slogans and Sylvie said, 'Gracious, what a mess.' The shop was open for business but a pair of grinning louts in SA uniform were loitering in front of the doors, putting people off from entering. Not Sylvie, who had marched past the Brownshirts while Ursula reluctantly trailed in her wake into the store and up the thickly carpeted staircase. In the face of the uniforms, Ursula had shrugged a cartoon helplessness and murmured rather shamefacedly, 'She's English.' She thought that Sylvie didn't understand what it was like living in Germany but in retrospect she thought that perhaps Sylvie had understood very well.\n\n'Ah, here's lunch,' Eva said, putting down the camera and taking Frieda's hand. She led her to the table and then propped her up on an extra cushion before heaping her plate with food. Chicken, roast potatoes and a salad, all from the Gutshof. How well they ate here. _Milchreis_ for Frieda's pudding, the milk fresh that morning from the cows of the Gutshof. (A less childish _K\u00e4sekuchen_ for Ursula, a cigarette for Eva.) Ursula remembered Mrs Glover's rice pudding, a creamy, sticky yellow beneath its crisp brown skin. She could smell the nutmeg even though she knew there was none in Frieda's dish. She couldn't remember the German for nutmeg and thought it was too difficult to explain to Eva. The food was the only thing that she was going to miss about the Berghof so she might as well enjoy it while she could, she thought, and helped herself to more _K\u00e4sekuchen_.\n\nLunch was served to them by a small contingent of the army of staff who serviced the Berghof. The Berg was a curious combination of Alpine holiday chalet and military training camp. A small town really with a school, a post office, a theatre, a large SS barracks, a rifle range, a bowling alley, a Wehrmacht hospital and much more, everything but a church really. There were also plenty of young, handsome Wehrmacht officers who would have made better suitors for Eva.\n\nAfter lunch they walked up to the _Teehaus_ on the Mooslahner Kopf, Eva's yappy, nippy dogs running along beside them. (If only one of them would fall off the parapet or from the outlook.) Ursula had the beginnings of a headache and sank gratefully into one of the armchairs with green-flowered linen upholstery that she found particularly offensive to the eye. Tea \u2013 and cake, naturally \u2013 were brought to them from the kitchen. Ursula swallowed a couple of codeine with her tea and said, 'I think Frieda's well enough to go home now.'\n\n*\n\nUrsula went to bed as early as she could, slipping in between the cool white sheets of the guest-room bed she shared with Frieda. Too tired to sleep, she found herself still awake at two in the morning so she put on the bedside light \u2013 Frieda slept the deep sleep of children, only illness could wake her \u2013 and she got out pen and paper and wrote to Pamela instead.\n\nOf course, none of these letters to Pamela was ever posted. She couldn't be completely sure that they wouldn't be read by someone. You just didn't _know_ , that was the awful thing (how much more awful for others). Now she wished they weren't in the dog-days of heat when the _Kachelofen_ in the guest room was cold and unlit, as it would have been safer to burn the correspondence. Safer never to have written at all. One could no longer express one's true thoughts. _Truth is truth to the end of reckoning_. What was that from? _Measure for Measure_? But perhaps truth was asleep until the end of reckoning. There was going to be an awful _lot_ of reckoning when the time came.\n\nShe wanted to go home. She wanted to go to Fox Corner. She had planned to go back in May but then Frieda had become sick. She'd had it all planned, their suitcases were packed, stored beneath the bed, where they were usually kept empty so J\u00fcrgen had no reason to look inside them. She had the train tickets, the onward boat-train tickets, had told no one, not even Klara. She hadn't wanted to move their passports \u2013 Frieda's luckily still valid from their trip to England in '35 \u2013 from the big porcupine-quill box where all their documents were kept. She had checked they were there almost every day but then the day before they were to go she looked in the box and there was no sign of them. She thought she was mistaken, rifled through birth and death and marriage certificates, through insurance and guarantees, J\u00fcrgen's will (he was a lawyer, after all), all kinds of paperwork except for what mattered. In mounting panic she emptied the lot on to the carpet and went through everything one by one, again and again. No passports, only J\u00fcrgen's. In desperation she went through every drawer in the house, searched inside every shoebox and cupboard, beneath sofa cushions and mattresses. Nothing.\n\nThey ate supper as normal. She could barely swallow. 'Are you feeling ill?' J\u00fcrgen asked, solicitously.\n\n'No,' she said. Her voice sounded squeaky. What could she say? He knew, of course, he knew.\n\n'I thought we might take a holiday,' he said. 'On Sylt.'\n\n'Sylt?'\n\n'Sylt. We won't need a passport for there,' he said. Did he smile? Did he? And then Frieda was ill and nothing else mattered.\n\n' _Er kommt!_ ' Eva said happily the next morning at breakfast. The F\u00fchrer was coming.\n\n'When? Now?'\n\n'No, this afternoon.'\n\n'What a shame, we'll be gone by then,' Ursula said. Thank God, she thought. 'Do thank him, won't you?'\n\nThey were taken home in one of the fleet of black Mercedes from the _Platterhof_ garage, driven by the same chauffeur who had brought them to the Berghof.\n\nThe next day Germany invaded Poland.\n\n# _April 1945_\n\nThey had lived for months in the cellar, like rats. When the British were bombing by day and the Americans by night there was nothing else for it. The cellar beneath the apartment block in the Savignyplatz was dank and disgusting, a small paraffin lamp for light and one bucket for a lavatory, yet the cellar was better than one of the bunkers in town. She had been caught with Frieda near the zoo in a daylight raid and had taken shelter in the Zoo Station flak tower \u2013 thousands of people crammed in, the air supply gauged by a candle (as if they were canaries). If the candle goes out, someone told her, everyone has to leave, out into the open even if a raid is in progress. Near to where they were crushed against a wall, a man and a woman were embracing (a polite term for what they were doing) and as they were leaving they had to step over an old man who had died during the raid. The worst thing, even worse than this, was that as well as being a shelter the enormous concrete citadel was a gigantic anti-aircraft battery, several huge guns pounding away on the roof the whole time so that the shelter shook with every recoil. It was the closest to hell that Ursula ever hoped to come.\n\nAn enormous explosion had shaken the structure, a bomb dropped close to the zoo. She felt the pressure wave sucking and pushing her body and was terrified that Frieda's lungs might burst. It passed. Several people vomited, although unfortunately there was nowhere to vomit except on one's feet, or perhaps worse, on other people's feet. Ursula vowed never to go into a flak tower again. She would rather, she thought, die out in the street, quickly, with Frieda. That's what she thought about a lot of the time now. A swift, clean death, Frieda wrapped in her arms.\n\nPerhaps it was Teddy up there, dropping bombs on them. She hoped it was, it would mean he was alive. There had been a knock on the door one day \u2013 when they still had a door, before the British started their relentless bombing in November '43. When Ursula opened the door she found a thin youth standing there, fifteen or sixteen years old maybe. He had a desperate air and Ursula wondered if he was looking for somewhere to hide but he pushed an envelope into her hand and ran off before she could even say a word to him.\n\nThe envelope was creased and filthy. Her name and address were written on it and she burst into tears at the sight of Pamela's handwriting. Thin papery blue sheets, dated several weeks ago, detailing all the comings and goings of her family \u2013 Jimmy in the army, Sylvie fighting the good fight on the home front ('a new weapon \u2013 chickens!'). Pamela was well and living at Fox Corner, she said, four boys now. Teddy in the RAF, a squadron leader with a DFC. A lovely long letter and at the end a page that was almost like a postscript, 'I have saved the sad news to last.' Hugh was dead. 'In the autumn of 1940, peacefully, a heart attack.' Ursula wished she hadn't received the letter, wished she could think of Hugh still alive, of Teddy and Jimmy in non-combatant roles, living out the war in a coal mine or civil defence.\n\n'I think of you constantly,' Pamela said. No recriminations, no 'I told you so', no 'Why didn't you come home when you could have done?' She had tried, too late, of course. The day after Germany declared war on Poland she had gone through town, dutifully doing the things that she thought you were supposed to do when war was imminent. She stocked up on batteries and torches and candles, she bought canned goods and blackout material, she shopped for clothes for Frieda in Wertheim's department store \u2013 one and two sizes bigger in case the war went on for a long time. She bought nothing for herself, passing by all those warm coats and boots, stockings and decent frocks, something she bitterly regretted now.\n\nShe heard Chamberlain on the BBC, those fateful words _We are now at war with Germany_ , and for several hours felt strangely numb. She tried to phone Pamela but the lines were all engaged. Then towards evening (J\u00fcrgen had been at the ministry all day) she suddenly came back to life, Snow White awake. She must leave, she must go back to England, passport or no. She packed a hasty suitcase and harried Frieda on to a tram to the station. If she could just get on a train somehow everything would be all right. No trains, an official at the station told her. The borders were closed. 'We're at war, didn't you know?' he said.\n\nShe ran to the British Embassy in Wilhelmstrasse, dragging poor Frieda by the hand. They were German citizens but she would throw herself on the mercy of the embassy staff, surely they would be able to do something, she was still an Englishwoman after all. It was growing dark by then and the gates were padlocked and there were no lights on in the building. 'They've gone,' a passer-by told her, 'you've missed them.'\n\n'Gone?'\n\n'Back to Britain.'\n\nShe had to clap a hand over her mouth to stop the wail that rose up from deep inside her. How could she have been so stupid? Why hadn't she seen what was coming? _A fool too late bewares when all the peril is past_. Something else that Elizabeth I had said.\n\nShe wept on and off for two days after receiving Pamela's letter. J\u00fcrgen was sympathetic, came home with some real coffee for her and she didn't ask where he had got it. A good cup of coffee (miraculous as that was) was hardly going to assuage her grief for her father, for Frieda, for herself. For everybody. J\u00fcrgen died in an American raid in '44. Ursula was ashamed at how relieved she felt when she was given the news, especially as Frieda was so upset. She loved her father and he loved her, which was a nugget of grace to be salvaged from the whole sorry business of their marriage.\n\nFrieda was ill now. She had the same gaunt features and sickly pallor of most people you saw on the streets these days but her lungs were full of phlegm and she had terrible bouts of coughing that seemed as though they would never end. When Ursula listened to her chest it was like listening to a galleon at sea, heaving and creaking through the waves. If only she could sit her down by a big warm fire, give her hot cocoa to drink, a beef stew, dumplings, carrots. Were they still eating well on the Berg, she wondered? Was anyone still on the Berg?\n\nAbove their heads, the apartment block was still standing although most of the front wall had been taken away by a bomb. They still went up there to forage for anything useful. It had been saved from looting by the almost insurmountable difficulty of getting up the staircase which was filled with rubble. She and Frieda tied pieces of cushion to their knees with rags and wore thick leather gloves that had belonged to J\u00fcrgen and in this way clambered over the stones and bricks like inept monkeys.\n\nThe one thing there was nothing of in the apartment was the only thing they were interested in \u2013 food. Yesterday they had queued for three hours for a loaf of bread. When they ate it, it seemed to contain no actual flour, although it was hard to say what it did contain \u2013 cement dust and plaster? That was what it tasted of anyway. Ursula remembered Rogerson's the baker's in the village at home, how the smell of the baking bread would waft through the street and how the bakery's window was full of lovely soft white loaves burnished with a sticky bronze glaze. Or the kitchen at Fox Corner on Mrs Glover's baking days \u2013 the big brown 'health' loaves that Sylvie insisted on, but also the sponges and tarts and buns. She imagined eating a slice of the warm brown bread, thickly buttered, with the jam made from the raspberries and redcurrants at Fox Corner. (She tormented herself with memories of food the whole time.) There was to be no more milk, someone told her in the bread queue.\n\nThis morning, Fr\u00e4ulein Farber and her sister Frau Meyer who had lived together in the attic but who now rarely left the cellar gave her two potatoes and a piece of cooked sausage for Frieda, _Aus Anstand_ , they said, out of decency. Herr Richter, also a cellar resident, told Ursula that the sisters had decided to stop eating. (An easy thing to do when there was no food, Ursula thought.) They have had enough, he said. They cannot face what will happen when the Russians get here.\n\nThey had heard a rumour that in the east people were reduced to eating grass. Lucky them, Ursula thought, there was no grass in Berlin, just the black skeletal remains of a proud and beautiful city. Was London like this too? It seemed unlikely, yet possible. Speer had his noble ruins, a thousand years early.\n\nThe inedible bread yesterday, two half-raw potatoes the day before that was all Ursula had in her own stomach. Everything else \u2013 for the little it was worth \u2013 she'd given to Frieda. But what good would it do Frieda if Ursula were dead? She couldn't leave her alone in this terrible world.\n\nAfter the British raid on the zoo they had gone to see if there were any animals they could eat but plenty of people had got there before them. (Could _that_ happen at home? Londoners scavenging in Regent's Park zoo? Why not?)\n\nThey still saw the occasional bird that was clearly not native to Berlin, surviving against the odds, and on one occasion, a cowed, mangy creature that they had taken for a dog before they realized it was a wolf. Frieda was all for trying to take it back to the cellar with them and making a pet of it. Ursula couldn't even imagine what their elderly neighbour Frau Jaeger's reaction would have been to that.\n\nTheir own apartment was like a dolls' house, open to the world, all the intimate details of their domestic life on view \u2013 beds and sofas, the pictures on the walls, even an ornament or two that had miraculously survived the blast. They had raided anything truly useful but there were still some clothes and a few books and only yesterday she had found a cache of candles beneath a pile of broken crockery. Ursula was hoping to trade them in for medicine for Frieda. There was still a lavatory, in the bathroom, and occasionally, who knew how, there was water. One of them would stand and hold up an old sheet to protect the other's modesty. Did their modesty matter that much any more?\n\nUrsula had made the decision to move back in. It was cold in the apartment but the air wasn't fetid and she judged that on balance that would be better for Frieda. They still had blankets and quilts they could wrap themselves in and they shared a mattress on the floor, behind a barricade formed by the dining table and chairs. Ursula's thoughts strayed constantly to the meals they had eaten at that table, her dreams full of meat, pork and beef, slabs of it grilled and roasted and fried.\n\nThe apartment was two floors up and this, combined with the partially blocked staircase, might be enough to put the Russians off. On the other hand they would be the dolls on display in the doll's house, a woman and a girl ripe for the plucking. Frieda would soon be eleven but if even a tenth of the rumours coming from the east were true then her age wouldn't save her from the Russians. Frau Jaeger never stopped talking nervously about how the Soviets were raping and murdering their way towards Berlin. There was no wireless any more, just rumour and the occasional flimsy piece of newssheet. The name Nemmersdorf was rarely off Frau Jaeger's lips ('A massacre!'). 'Oh, do shut up,' Ursula said to her the other day. In English, which she didn't understand, of course, although she must have heard the unfriendly tone. Frau Jaeger had been visibly startled to be addressed in the language of the enemy and Ursula felt sorry, she was just a frightened old lady, she reminded herself.\n\nThe east moved nearer every day. Interest in the western front had long since died, only the east was of concern. The distant thunder of guns now replaced by a constant roar. There was no one to save them. Eighty thousand German troops to defend them against a million and a half Soviets, and most of those German troops seemed to be children or old men. Perhaps poor old Frau Jaeger would be called upon to beat off the enemy with a broom handle. It could only be a matter of days, hours even, before they saw their first Russian.\n\nThere was a rumour that Hitler was dead. 'Not before time,' Herr Richter said. Ursula remembered the sight of him asleep on his sun lounger on the terrace on the Berg. He had strutted and fretted his hour upon the stage. To what avail? A kind of Armageddon. The death of Europe.\n\nIt was life itself, wasn't it, she corrected herself, that Shakespeare had fretting and strutting. _Life's but a walking shadow, a poor player that struts and frets his hour upon the stage_. They were all walking shadows in Berlin. Life had mattered so much once and now it was the cheapest thing on offer. She spared an idle thought for Eva, she was always blas\u00e9 about the idea of suicide, had she accompanied her leader into hell?\n\nFrieda was so poorly now, chills and fever and complaining almost constantly of a headache. If she hadn't been sick they might have joined the exodus of people heading west, away from the Russians, but there was no way she would survive such a journey.\n\n'I've had enough, Mummy,' she whispered, a terrible echo of the sisters from the attic.\n\nUrsula left her alone while she hurried to the chemist, scrabbling over the debris that littered the streets, occasionally a corpse \u2013 she felt nothing for the dead any more. She cowered in doorways when the gunfire seemed too close and then scurried to the next street corner. The chemist was open but he had no medicine, he didn't even want her precious candles or her money. She came back defeated.\n\nThe whole time she had been away from Frieda she had been anxious that something would happen to her in her absence and she promised herself that she wouldn't leave her side again. She had seen a Russian tank two streets away. She had been terrified by the sight, how much more terrified would Frieda be? The noise of artillery fire was constant. She was gripped by the idea that the world was ending. If it was then Frieda must die in her arms, not alone. But whose arms would she die in? She longed for the safety of her father and the thought of Hugh made the tears start.\n\nBy the time she had climbed the rubble staircase she was exhausted, weary to the bone. She found Frieda slipping in and out of delirium and lay down beside her on the mattress on the floor. Stroking her damp hair, she talked in a low voice to her about another world. She told her about the bluebells in spring in the wood near Fox Corner, about the flowers that grew in the meadow beyond the copse \u2013 flax and larkspur, buttercups, corn poppies, red campion and ox-eye daisies. She told her about the smell of new-mown grass from an English summer lawn, the scent of Sylvie's roses, the sour-sweet taste of the apples in the orchard. She talked of the oak trees in the lane, and the yews in the graveyard and the beech in the garden at Fox Corner. She talked about the foxes, the rabbits, the pheasants, the hares, the cows and the big plough horses. About the sun beaming his friendly rays on fields of corn and fields of green. The bright song of the blackbird, the lyrical lark, the soft coo of the wood pigeons, the hoot of the owl in the dark. 'Take this,' she said, putting the pill in Frieda's mouth, 'I got it from the chemist, it will help you sleep.'\n\nShe told Frieda how she would walk on knives to protect her, burn in the flames of hell to save her, drown in the deepest of waters if it would buoy her up and how she would do this one last thing for her, the most difficult thing of all.\n\nShe put her arms around her daughter and kissed her and murmured in her ear, telling her about Teddy when he was little, his surprise birthday party, about how clever Pamela was and how annoying Maurice was and how funny Jimmy had been when he was small. How the clock ticked in the hall and the wind rattled in the chimneypots and how on Christmas Eve they lit an enormous log fire and hung their stockings from the mantelpiece and next day ate roast goose and plum pudding and how that was what they would all do next Christmas, all of them together. 'Everything is going to be all right now,' Ursula told her.\n\nWhen she was sure that Frieda was asleep she took the little glass capsule that the chemist had given her and placed it gently in Frieda's mouth and pressed her delicate jaws together. The capsule broke with a tiny crunching noise. A line from one of Donne's _Holy Sonnets_ came into mind as she bit down on her own little glass vial. _I run to death, and death meets me as fast, And all my pleasures are like yesterday_. She held tightly on to Frieda and soon they were both wrapped in the velvet wings of the black bat and this life was already unreal and gone.\n\nShe had never chosen death over life before and as she was leaving she knew something had cracked and broken and the order of things had changed. Then the dark obliterated all thoughts.\n\n# A Long Hard War\n\n# _September 1940_\n\n' _See where Christ's blood streams across the firmament_ ,' a voice nearby said. ' _In_ ' the firmament, Ursula thought, not 'across'. The red glow of a false dawn indicated a massive fire in the east. The barrage in Hyde Park cracked and flared and the anti-aircraft guns closer to home were doing a good job of keeping up their own cacophony, shells whistling into the air like fireworks and _crack-crack-cracking_ as they exploded high overhead. And beneath it all was the horrible throbbing drone of the bombers' unsynchronized engines, a sound that always made her stomach feel pitchy.\n\nA parachute mine floated down gracefully and a basket of incendiaries rattled their contents on to what was left of the road and burst into flowers of fire. A warden, Ursula couldn't make out his face, ran across to the incendiaries with a stirrup pump. If there had been no noise it might have seemed a beautiful nightscape but there _was_ noise, brutish dissonance that sounded as if someone had thrown open the gates of hell and let out the howling of the damned.\n\n' _Why this is hell, nor am I out of it_ ,' the voice spoke again as if reading her thoughts. It was so dark that she could barely make out the owner of the voice although she knew without a doubt that it belonged to Mr Durkin, one of the wardens from her post. He was a retired English teacher, much inclined to quoting. And misquoting. The voice \u2013 or Mr Durkin \u2013 said something else, it may still have been _Faustus_ but the words disappeared into the enormous _whump_ of a bomb falling a couple of streets away.\n\nThe ground shook and another voice, that of someone working on the mound, yelled, 'Watch out!' She heard something shifting and a noise like displaced scree rattling and rolling down a mountain, the harbinger of an avalanche. Rubble, not scree. And a mound of it, not a mountain. The rubble that comprised the mound was all that was left of a house, or rather, several houses all ground and mashed into each other now. The rubble had been homes half an hour ago, now those same homes were just a hellish jumble of bricks, broken joists and floorboards, furniture, pictures, rugs, bedding, books, crockery, lino, glass. People. The crushed fragments of lives, never to be whole again.\n\nThe rumble slowed to a trickle and finally stopped, the avalanche averted, and the same voice shouted, 'All right! Carry on!' It was a moonless night, the only light coming from the masked torches of the heavy rescue squad, ghostly will-o'-the-wisps, moving on the mound. The other reason for the immense, treacherous dark was the thick cloud of smoke and dust that hung like a curtain of vile gossamer in the air. The stink, as usual, was awful. It wasn't just the smell of coal gas and high explosive, it was the aberrant odour produced when a building was blown to smithereens. The smell of it wouldn't leave her. She had tied an old silk scarf around her mouth and nose, bandit-style, but it did little to stop the dust and the stench getting into her lungs. Death and decay were on her skin, in her hair, in her nostrils, her lungs, beneath her fingernails, all the time. They had become part of her.\n\nThey had only recently been issued with overalls, navy blue and unflattering. Until now Ursula had been wearing her shelter suit, bought almost as a novelty item by Sylvie from Simpson's soon after war was declared. She had added an old leather belt of Hugh's from which she'd strung her 'accessories' \u2013 a torch, gas mask, a first-aid packet and a message pad. In one pocket she had a penknife and a handkerchief and in the other a pair of thick leather gloves and a lipstick. 'Oh, what a good idea,' Miss Woolf said, when she saw the penknife. Let's face it, Ursula thought, despite a host of regulations, they were making it up as they went along.\n\nMr Durkin, for it was indeed he, resolved himself out of the gloom and foggy smoke. He shone his torch on to his notebook, the weak light barely illuminating the paper. 'A lot of people live on this street,' he said, peering at the list of names and house numbers which no longer bore any relation to the surrounding havoc. 'The Wilsons are at number one,' he said, as if beginning at the beginning would somehow help.\n\n'There is no number one any more,' Ursula said. 'There are no numbers at all.' The street was unrecognizable, everything familiar annihilated. Even in broad daylight it would have been unrecognizable. It wasn't a street any more, it was simply 'the mound'. Twenty feet high, maybe more, with planks and ladders running up its sides to enable the heavy-rescue squad to crawl over it. There was something primitive about the human chain they had formed, passing debris in baskets from hand to hand, from the top of the mound to the bottom. They could have been slaves building the pyramids \u2013 or in this case, excavating them. Ursula thought suddenly of the leafcutter ants that used to be in Regent's Park zoo, each one dutifully carrying its little burden. Had the ants been evacuated along with the other animals or had they simply set them free in the park? They were tropical insects, so perhaps they would not be able to survive the rigours of the climate of Regent's Park. She had seen Millie there in an open-air production of _A Midsummer Night's Dream_ , in the summer of '38.\n\n'Miss Todd?'\n\n'Yes, sorry, Mr Durkin, miles away.' It happened a lot these days \u2013 she would be in the middle of these awful scenes and she would find that she had drifted off to pleasant moments in the past. Little slivers of light in the darkness.\n\nThey made their way warily towards the mound. Mr Durkin passed the list of the street's residents to her and started giving a hand with the chain of baskets. No one was actually digging on the mound, instead they were clearing the rubble by hand, like careful archaeologists. 'A bit delicate up there,' one of the rescue squad near the bottom of the chain said to her. A shaft had been cleared, going down the middle of the mound (a volcano then, rather than a mound, Ursula thought). A lot of the men in the heavy rescue squad were from the building trade \u2013 bricklayers, labourers and so on \u2013 and Ursula wondered if it felt odd to them to be scrambling over these dismantled buildings, as if time had somehow gone backwards. But then they were pragmatic, resourceful men who were not much given to this kind of fantastical thinking.\n\nOccasionally a voice would call for quiet \u2013 impossible when the raid was still going on overhead \u2013 but nonetheless everything would stop while the men at the top of the mound listened intently for signs of life within. It looked hopeless but if there was one thing that the Blitz had taught them it was that people lived (and died) in the most unlikely of circumstances.\n\nUrsula searched in the gloom for the dim blue lights that marked out the incident officer's post and instead caught sight of Miss Woolf, stumbling purposefully over broken bricks towards her. 'It's bad,' she said matter-of-factly when she reached Ursula. 'They need someone slight.'\n\n'Slight?' Ursula repeated. The word, for some reason, was devoid of meaning.\n\nShe had joined the ARP as a warden after the invasion of Czechoslovakia in March '39, when it suddenly seemed horribly clear to her that Europe was doomed. ('What a gloomy Cassandra you are,' Sylvie said, but Ursula worked in the Air Raid Precautions department at the Home Office, she could see the future.) During the strange twilight of the phoney war the wardens had been something of a joke but now they were 'the backbone of London's defences' \u2013 this from Maurice.\n\nHer fellow wardens were a mixed bunch. Miss Woolf, a retired hospital matron, was the senior warden. Thin and straight as a poker, her iron-grey hair in a neat bun, she came with natural authority. Then there was her deputy, the aforesaid Mr Durkin, Mr Simms, who worked for the Ministry of Supply, and Mr Palmer, who was a bank manager. The latter two men had fought in the last war and were too old for this one (Mr Durkin had been 'medically exempt', he said defensively). Then there was Mr Armitage who was an opera singer and as there were no operas to sing in any more he kept them entertained with his renditions of _'La donna \u00e8 mobile_ ' and ' _Largo al factotum_ '. 'Just the popular arias,' he confided to Ursula. 'Most people don't like anything challenging.'\n\n'Give me old Al Bowlly any day,' Mr Bullock said. The rather aptly named Mr Bullock (John) was in Miss Woolf's words 'a little questionable'. He certainly cut a strapping figure \u2013 he wrestled competitively and lifted weights in a local gym as well as being the denizen of several of the less salubrious nightclubs. He was also acquainted with some rather glamorous 'dancers'. One or two had 'dropped in' on him in the shelter and been shooed away like chickens by Miss Woolf. ('Dancers my eye,' she said.)\n\nLast but not least there was Herr Zimmerman ('Gabi, please,' he said, but no one did), who was an orchestra violinist from Berlin, 'our refugee' as they referred to him (Sylvie had evacuees, similarly denoted by their circumstances). He had 'jumped ship' in '35 while on tour with his orchestra. Miss Woolf, who knew him through the Refugee Committee, had gone to great lengths to make sure that Herr Zimmerman and his violin were not interned, or worse, shipped across the lethal waters of the Atlantic. They all followed Miss Woolf's lead and never addressed him as 'Mister', always as 'Herr'. Ursula knew that Miss Woolf called him thus to make him feel at home but it only succeeded in making more of an alien of him.\n\nMiss Woolf had come across Herr Zimmerman in the course of her work for the Central British Fund for German Jewry ('Rather a mouthful, I'm afraid'). Ursula was never sure whether Miss Woolf was a woman of some influence or whether she simply refused to take no for an answer. Both, perhaps.\n\n'A cultured lot, aren't we?' Mr Bullock said sarcastically. 'Why don't we just put on shows instead of fighting a war.' ('Mr Bullock is a man of strong emotions,' Miss Woolf said to Ursula. And strong drink too, Ursula thought. Strong everything in fact.)\n\nA small hall belonging to the Methodists had been commandeered to be their post by Miss Woolf (herself a Methodist), and they had furnished it with a couple of camp beds, a small stove with tea-making equipment and an assortment of chairs, both hard and soft. Compared to some posts, compared to many, it was luxurious.\n\nMr Bullock turned up one night with a green baize card table and Miss Woolf declared herself rather fond of bridge. Mr Bullock, in the lull between the fall of France and the first raids at the beginning of September, had taught them all poker. 'Quite the card sharp,' Mr Simms said. Both he and Mr Palmer lost several shillings to Mr Bullock. Miss Woolf, on the other hand, was two pounds up by the time the Blitz started. An amused Mr Bullock expressed surprise that Methodists were allowed to gamble. Her winnings bought a dartboard so Mr Bullock had nothing to complain about, she said. One day when they were clearing a jumble of boxes in the corner of the hall they discovered that a piano had been hiding there all along and Miss Woolf \u2013 who was proving a woman of many talents \u2013 was a rather good player. Although her own tastes tended towards Chopin and Liszt, she was more than game to 'bash out a few tunes' \u2013 Mr Bullock's words \u2013 for them all to sing along to.\n\nThey had fortified the post with sandbags although none of them believed that they would be of any use if they were hit. Apart from Ursula, who thought that taking precautions seemed an eminently sensible idea, they all tended to agree with Mr Bullock that 'If it's got your name on it, it's got your name on it,' a form of Buddhist detachment that Dr Kellet would have admired. There had been an obituary in _The Times_ during the summer. Ursula was rather glad that Dr Kellet had missed another war. It would have reminded him of the futility of Guy losing everything at Arras.\n\nThey were all part-time volunteers, apart from Miss Woolf, who was paid and full-time and took her duties very seriously. She subjected them to rigorous drills and made sure they did their training \u2013 in antigas procedures, in extinguishing incendiaries, how to enter burning buildings, load stretchers, make splints, bandage limbs. She questioned them on the contents of the manuals that she made them read and she was very keen on them learning how to label bodies, both alive and dead, so that they could be sent off like parcels to the hospital or the mortuary with all the correct information attached. They had done several exercises out in the open where they had acted out a mock raid. ('Play-acting,' Mr Bullock scoffed, failing to get into the spirit of things.) Ursula played a casualty twice, once having to feign a broken leg and on another occasion complete unconsciousness. Another time she had been on the 'other side' and as a warden had had to deal with Mr Armitage simulating someone in hysterical shock. She supposed it was his experience on stage that enabled him to give such an unnervingly authentic performance. It was quite hard to persuade him out of character at the end of the exercise.\n\nThey had to know the occupants of every building in their sector, whether they had a shelter of their own or whether they went to a public one or whether they too were fatalists and didn't bother at all. They had to know if anyone had gone away or moved, married, had a baby, died. They had to know where the hydrants were, cul-de-sacs, narrow alleyways, cellars, rest centres.\n\n'Patrol and watch', that was Miss Woolf's motto. They tended to patrol the streets in pairs until midnight when there was usually a lull, and then if there were no bombs in their sector they would have a polite argument over who should occupy the camp beds. Of course, if there was a raid in 'their streets' then it was 'all hands to the pumps' in Miss Woolf's words. Sometimes they did the 'watching' from her flat, two storeys up with an excellent view from a big corner window.\n\nMiss Woolf also did extra first-aid exercises with them. As well as having been a hospital matron, she had run a field hospital during the last war and explained to them ('As you will appreciate, those of you gentlemen who saw active service in that dreadful conflict') that casualties in war were very different from the routine accidents that one saw in peacetime. 'Much nastier,' she said. 'We must be prepared for some distressing sights.' Of course, even Miss Woolf had not imagined how distressing these sights would be when they involved civilians rather than battlefield soldiers, when they involved shovelling up unidentifiable lumps of flesh or picking out the heartbreakingly small limbs of a child from the rubble.\n\n'We cannot turn away,' Miss Woolf told her, 'we must get on with our job and we must bear witness.' What did that mean, Ursula wondered. 'It means,' Miss Woolf said, 'that we must remember these people when we are safely in the future.'\n\n'And if _we_ are killed?'\n\n'Then others must remember _us_.'\n\nThe first serious incident they attended had been at a large house in the middle of a terrace that had received a direct hit. The rest of the terrace was undamaged, as though the Luftwaffe had personally targeted the occupants \u2013 two families including grandparents, several children, two babes-in-arms. They had all survived the blast, sheltering in the cellar, but both the mains water pipe and a large sewage pipe had fractured and before either could be turned off everyone in the cellar had drowned in the awful sludge.\n\nOne of the women had managed to claw her way up and cling on to one of the cellar walls, they could see her through a gap, and Miss Woolf and Mr Armitage had held on to Hugh's leather belt while Ursula had dangled over the lip of what remained of the cellar. She reached out a hand to the woman, thought for a moment that she might actually manage to grasp hold of her, but then she simply disappeared beneath the feculent water as it rose to fill the cellar.\n\nWhen the fire brigade finally arrived to pump out the place they recovered fifteen bodies, seven of them children, and laid them in front of the house, as if to dry. Miss Woolf ordered them shrouded as quickly as possible and stowed away behind a wall while they awaited the arrival of the mortuary wagon. 'It doesn't do morale any good to see sights like that,' she said. Ursula had vomited up her supper long before then. She vomited after nearly every incident. Mr Armitage and Mr Palmer too, Mr Simms before. Only Miss Woolf and Mr Bullock seemed to have strong stomachs for death.\n\nAfterwards, Ursula tried not to think about the babies or the look of terror on that poor woman's face as she had grasped in vain for Ursula's hand (and something else, disbelief perhaps that this could be happening). 'Think of them being at peace now,' Miss Woolf counselled stoutly afterwards, dispensing scalding-hot sugary tea. 'They are out of all this, just gone a little sooner.' And Mr Durkin said, ' _They have all gone into the world of light_ ,' and Ursula thought, _they_ are _all gone into the world of light_. Ursula wasn't convinced that the dead went anywhere, except into a void, black and infinite.\n\n'Well, I hope _I_ don't die covered in shit,' Mr Bullock said, more prosaically.\n\nShe thought she would never get over that first terrible incident but the memory of it had already been overlaid by many others and now she barely thought about it.\n\n'It's bad,' Miss Woolf said matter-of-factly. 'They need someone slight.'\n\n'Slight?' Ursula repeated.\n\n'Slim,' Miss Woolf said patiently.\n\n'To go in there?' Ursula said, looking up in horror at the summit of the volcano. She wasn't sure she had the gumption to be lowered into the very maw of hell.\n\n'No, no, not there,' Miss Woolf said. 'Come with me.' It had begun to rain, quite hard, and Ursula blundered with difficulty in Miss Woolf's wake over the jagged and broken ground, littered with every kind of obstacle. Her torch was next to useless. She caught her foot in a bicycle wheel and wondered if anyone had been riding it when the bomb struck.\n\n'Here,' Miss Woolf said. It was another mound, just as big as the last. Was it another street, or the same street? Ursula had lost all sense of direction. How many mounds were there? A nightmarish scenario flashed through her mind \u2013 the whole of London reduced to one gigantic mound.\n\nThis mound wasn't a volcano, the rescue squad were going in through a horizontal shaft at the side. More robust this time, they were hacking at the rubble with picks and shovels.\n\n'There's a kind of hole here,' Miss Woolf said, taking Ursula's hand firmly in hers, as if Ursula were a reluctant child, and leading her forward. Ursula could see no sign of a hole. 'It's safe, I think, you just need to wriggle through.'\n\n'A tunnel?'\n\n'No, it's just a hole. There's a bit of a drop on the other side, we think there's someone down there. Not a long drop,' she added encouragingly. 'Not a tunnel,' she said again. 'Go head first.' The rescue squad stopped hacking at the rubble and waited, rather impatiently, for Ursula.\n\nShe had to take her helmet off in order to wriggle into the hole, her torch held awkwardly in front of her. Despite what Miss Woolf said, she had been expecting a tunnel but was immediately confronted with a cavernous space. She might have been potholing. She was relieved when she felt two pairs of invisible hands attach themselves to Hugh's old leather belt. She moved the torch around trying to see something, anything. 'Hello?' she shouted as she shone the torch into the drop. It was screened by a haphazard lattice of twisted gas pipes and wood, splintered like matchsticks. She concentrated on a gap in the chaotic mesh, trying to make out anything in the gloom beyond. An upturned face, a man's, pale and ghostly, seemed to rise out of the darkness like a vision, a prisoner in an oubliette. There might be a body attached to the face, she couldn't be sure.\n\n'Hello?' she said, as if the man might reply, although now she could see that part of his head was missing.\n\n'Anyone?' Miss Woolf said hopefully when she crawled backwards out of the hole.\n\n'One dead.'\n\n'Easy to recover?'\n\n'No.'\n\nThe rain made everything even more foul if that were possible, turning the wet brick dust into a kind of glutinous grit. A couple of hours of toiling in these conditions and they were all covered from head to toe in the stuff. It was too disgusting to give any thought to.\n\nThere was a shortage of ambulances, traffic had been snarled up by an incident on the Cromwell Road, as had the doctor and nurse who should have been there, and Miss Woolf's extra first-aid training was put to good use. Ursula splinted a broken arm, bandaged a head wound, patched an eye and strapped up Mr Simms's ankle \u2013 he had twisted it on the rough ground. She labelled two unconscious survivors (head injuries, broken femur, broken collarbone, broken ribs, what was probably a crushed pelvis) and several dead (who were easier, they were simply dead) and then double-checked them in case she had labelled them the wrong way round and had posted the dead to the hospital and the living to the mortuary. She also directed numerous survivors to the rest centre, and walking wounded to the first-aid post being manned by Miss Woolf.\n\n'Catch Anthony if you can, will you?' she said when she saw Ursula. 'Get a mobile canteen down here.' Ursula sent Tony off with this errand. Only Miss Woolf called him Anthony. He was thirteen, a Boy Scout and their civil defence messenger boy, hurtling around on the rubble-and-glass-strewn streets on his bike. If Tony were her child, Ursula thought, she would have sent him far away from the nightmare instead of plunging him into the depths of it. He loved it all, needless to say.\n\nAfter she'd spoken to Tony, Ursula went back through the hole again because someone thought they had heard a sound, but the pale, dead man was as quiet as before. 'Hello, again,' she said to him. She thought it might be Mr McColl from the neighbouring street. Perhaps he was visiting someone. Unlucky. She was dog-tired, you could almost envy the dead their eternal rest.\n\nWhen she emerged again from the hole the mobile canteen had arrived. She swilled her mouth out with tea and spat out brick dust. 'I bet you used to be a real lady,' Mr Palmer laughed. 'I'm affronted,' Ursula said and laughed. 'I think I spit in a very ladylike way.' The rescue on the mound was still going on with no sign of any result but the rest of the night was winding down and Miss Woolf told her to go back to the post and rest. Up on the mound a rope had been called for, to lower someone down, Ursula supposed, or pull someone up, or both. ('A woman, they think,' Mr Durkin said.)\n\nShe was all in, could barely put one foot in front of the other. Avoiding the debris as best she could, she had gone only ten yards or so when someone grabbed her by the arm and yanked her backwards so hard that she would have fallen over if the same person hadn't kept his tight grip on her and kept her upright. 'Watch it, Miss Todd,' a voice growled.\n\n'Mr Bullock?' In the confines of the post Mr Bullock alarmed her a little, he seemed so unassailable, but, curiously, out here in this benighted place he was harmless. 'What is it?' she said. 'I'm very tired.'\n\nHe shone his torch in front of them. 'Can you see?' he said.\n\n'I can't see anything.'\n\n'That's because there's nothing there.' She looked harder. A crater \u2013 enormous \u2013 a bottomless pit. 'Twenty, maybe thirty feet,' Mr Bullock said. 'And you nearly walked into it.'\n\nHe accompanied her back to the post. 'You're too tired,' he said. He held her arm all the way, she could feel the strength of his muscles behind the grip.\n\nAt the post she dropped on to a camp bed and blacked out rather than fell asleep. She woke up when the all-clear sounded at six o'clock. She felt as if she'd slept for days but it had only been three hours.\n\nMr Palmer was also there, pottering about making tea. She could imagine him at home, slippers and a pipe, reading his newspaper. It seemed absurd that he should be here. 'There you go,' he said, handing her a mug. 'You should go home, dear,' he said, 'the rain's stopped,' as though it were the rain that had spoilt her night rather than the Luftwaffe.\n\nInstead of going straight home she returned to the mound to see how the rescue was proceeding. It seemed different in the daylight, the shape of it oddly familiar. It reminded her of something but for the life of her she couldn't think what.\n\nIt was a scene of devastation, more or less the whole street gone, but the mound, the original mound, was still its own little hive of activity. It would have made a good subject for a war artist, she thought. _The Diggers on the Mound_ would be a good title. Bea Shawcross had been at art school, graduating just as the war started. Ursula wondered if she was moved to depict the war or if she was trying to transcend it.\n\nVery gingerly, she scaled its foothills. One of the rescue squad put out a hand to help her up. A new shift had come on but, from the look of them, the old rescue squad was the one still labouring. Ursula understood. It was hard to leave an incident when somehow you felt you 'owned' it.\n\nThere was a sudden buzz of excitement around the volcano's crater as the fruits of the night's delicate drudgery finally became apparent. A woman, a rope tied under her armpits (nothing delicate about this stage), was extricated by simply hauling her out of the narrow opening. She was passed by hand down the mound.\n\nUrsula could see that she was almost black with dirt and drifting in and out of consciousness. Broken but alive, if only just. She was loaded into an ambulance waiting patiently at the bottom.\n\nUrsula made her own way down. On the ground, a shrouded body lay waiting for a mortuary van. Ursula removed the cover from the face and found the pale-faced man from last night. In the light of day she could see that it was definitely Mr McColl from number ten. 'Hello, you,' she said. He would soon be an old friend. Miss Woolf would have told her to label him but when she looked for her message pad she discovered she had lost it and had nothing to write on. Searching in a pocket she found her lipstick. _Needs must_ , she heard Sylvie's voice say.\n\nShe thought about writing on Mr McColl's forehead but that seemed undignified (more undignified than death, she wondered?) so instead she unshrouded his arm and then spat on a handkerchief and rubbed off some of the dirt, as if he were a little boy. She wrote his name and address on his arm with the lipstick. Blood red, which seemed fitting really.\n\n'Well, goodbye,' she said. 'I don't suppose we'll meet again.'\n\n*\n\nSkirting the treacherous crater from last night, she discovered Miss Woolf sitting behind a dining table salvaged from the wreckage, as if she were in an office, telling people what they should do next \u2013 where to go for food and shelter, how to get clothes and ration cards and so on. Miss Woolf was still cheerful, yet heaven knows when she had last slept. The woman had iron in her soul, there was no doubt about that. Ursula had grown enormously fond of Miss Woolf, she respected her almost more than anyone else she knew, apart from Hugh perhaps.\n\nThe queue was made up of the occupants of a large shelter, many of whom were still emerging, blinking in the daylight like nocturnal animals, and discovering that they no longer had homes to go to. The shelter was in the wrong place, the wrong street, Ursula thought. It took her a few moments to re-orientate her brain and realize that all night she had thought herself in a different street.\n\n'They got that woman out,' she told Miss Woolf.\n\n'Alive?'\n\n'More or less.'\n\nWhen she finally got back to Phillimore Gardens she found Millie up and dressed. ' _Went the day well?_ ' she said. 'There's some tea in the pot,' she added, pouring it and handing Ursula a cup.\n\n'Oh, you know,' Ursula said, taking the cup. The tea was lukewarm. She shrugged. 'Pretty awful. Is that the time? I have to go to work.'\n\nThe following day she was surprised to find one of Miss Woolf's log entries, written in her clear matron's hand. Sometimes a buff folder would prove to be a mysterious ragbag and Ursula was never clear how some of these things turned up on her desk. _05.00 Interim Incident Report. Situation Report. Casualties 55 to hospital, 30 dead, 3 unaccounted for. Seven houses completely demolished, approximately 120 homeless. 2 NFS crews, 2 AMB, 2 HRPs, 2 LRP, one dog still operating. Work continues_.\n\nUrsula hadn't noticed any dog. It was just one of many incidents across London that night and she picked up a sheaf of them and said, 'Miss Fawcett, can you log these.' She could barely wait for the tea-trolley and elevenses.\n\n*\n\nThey ate lunch outside on the terrace. A potato and egg salad, radishes, lettuce, tomatoes, even a cucumber. 'All grown by our mother's own fair hand,' Pamela said. It really was the nicest meal Ursula had eaten in a long time. 'And to follow there's an apple charlotte, I believe,' Pamela said. They were alone at the table. Sylvie had gone to answer the doorbell and Hugh hadn't returned from investigating an unexploded bomb that had, reportedly, fallen in a field on the other side of the village.\n\nThe boys were also dining _al fresco \u2013_ sprawled on the lawn, eating buffalo stew and succotash (or, in the real world, corned beef sandwiches and hard-boiled eggs). They had erected a fusty old wigwam that had been unearthed in the shed and had been engaged in a lawless game of cowboys and Indians until the arrival of the chuck wagon (or Bridget, bearing a tray).\n\nPamela's boys were the cowboys and the evacuees were more than happy to be Apaches. 'I think it suits their nature better,' Pamela said. She had made them cardboard headbands with chicken feathers attached. The cowboys had to make do with Hugh's handkerchiefs tied around their necks. The two Labradors were racing around in a state of canine frenzy at all this excitement, while Gerald, still only ten months old, slept obliviously on a blanket alongside Pamela's dog, Heidi, too sedate for such antics.\n\n'He's some kind of token squaw, apparently,' Pamela said. 'At least it keeps them quiet. It's like a miracle. It goes rather well with the Indian summer we're having.'\n\n'Six boys in one house,' Pamela said. 'Thank God the school term's started. Boys never flag, you have to keep them busy all the time. I suppose this is a flying visit?'\n\n''Fraid so.'\n\nA precious Saturday to herself that she had sacrificed for the sake of seeing Pammy and the boys. She found Pamela drained whereas Sylvie seemed animated by the war. She had become an unlikely stalwart of the WVS.\n\n'I'm surprised. She doesn't like other women much,' Pamela said.\n\nSylvie now had a large flock of chickens and had stepped up egg production to wartime levels. 'The poor things are forced to lay day and night,' Pamela said, 'you'd think Mother was running an armaments factory.' Ursula wasn't sure how you could make a chicken do overtime. 'She talks them into it,' Pamela laughed. 'A regular henwife.'\n\nUrsula didn't mention that she had been called to an incident, a house that had been hit, where the occupants had chickens in a makeshift run in the back yard and that when they arrived they had found the chickens, nearly all of them alive, with their feathers blown off. 'Ready-plucked,' Mr Bullock had laughed callously. Ursula had seen people with their clothes blown off and trees in the middle of summer stripped of all their leaves, but she didn't mention these things either. She didn't mention wading in effluent from ruptured pipes, certainly didn't mention drowning in that same effluent. Nor did she mention the gruesome sensation of putting your hand on a man's chest and finding that your hand had somehow slipped _inside_ that chest. (Dead \u2013 something to be thankful for, she supposed.)\n\nDid Harold tell Pamela the things he had seen? Ursula didn't ask, even introducing the topic seemed wrong on such a pleasant day. She thought of all those soldiers from the last war who had come home and never spoken of what they had witnessed in the trenches. Mr Simms, Mr Palmer, her own father too, of course.\n\nSylvie's egg production seemed to be at the heart of some kind of rural black market. No one in the village was particularly short of anything. 'It's a barter economy around here,' Pamela said. 'And barter they do, believe me. That's what she'll be doing now, at the front door.'\n\n'At least you're pretty safe here,' Ursula said. Were they? She thought of the UXB Hugh had gone to look at. Or the previous week when a bomb had come down in a field belonging to the Hall farm and blown the cows in it to pieces. 'A lot of people have been quietly eating beef around here,' Pamela said. 'Us included, I'm happy to say.' Sylvie seemed to think this 'terrible episode' had put them on a par with London's suffering. She had returned now and lit up a cigarette rather than finish her food. Ursula ate what she had left on her plate while Pamela took one of Sylvie's cigarettes from the packet and lit up.\n\nBridget came out and started clearing plates and Ursula jumped up and said, 'Oh, no, I'll do that.' Pamela and Sylvie remained at the table, smoking in silence, observing the defence of the wigwam from a raiding party of evacuees. Ursula felt rather badly done by. Both Sylvie and Pamela spoke as if they had it hard whereas she was working all day, out on patrol most nights, facing the most awful sights. Only yesterday they had been at an incident where they had worked to free someone while blood dripped on their heads from a body up in the bedroom they couldn't reach because the staircase was knee-deep in broken glass from a huge skylight.\n\n'I'm thinking of going back to Ireland,' Bridget said as they rinsed plates. 'I have never felt at home in this country.'\n\n'Neither have I,' Ursula said.\n\nThe apple charlotte turned out to be simply stewed apples as Sylvie refused to use precious stale bread on a pudding when it could be fed more usefully to the chickens. Nothing went to waste at Fox Corner. Scraps went to the chickens ('She's thinking of getting a pig,' Hugh said in despair), after bones had made stock they were sent for salvage, as was every last tin and glass jar that wasn't being filled with jam or chutney or beans or tomatoes. All the books in the house had been parcelled up and taken to the post office to be sent off to the services. 'We've already read them,' Sylvie said, 'so what's the point in keeping them?'\n\nHugh returned and Bridget grumbled back outside with a plate for him.\n\n'Oh,' Sylvie said politely to him, 'do you live here? I say, why don't you join us?'\n\n'Really, Sylvie,' Hugh said, more sharply than was his usual manner. 'You can be such a child.'\n\n'If I am then it's marriage that's made me so,' Sylvie said.\n\n'I remember that you once said there was no higher calling for a woman than marriage,' Hugh said.\n\n'Did I? That must have been in our salad days.'\n\nPamela raised her eyebrows at Ursula and Ursula wondered when had their parents become so openly quarrelsome? Ursula was going to ask him about the bomb but then, 'How's Millie?' Pamela asked brightly to change the subject.\n\n'She's well,' Ursula said. 'She's a very easy-going person to share digs with. Although I hardly ever see her in Phillimore Gardens. She's joined ENSA. She's in some kind of troupe that goes round factories, entertaining workers in their lunch hour.'\n\n'Poor blighters,' Hugh laughed.\n\n'With Shakespeare?' Sylvie asked doubtfully.\n\n'I think she turns her hand to anything these days. A bit of singing, comedy, you know.' Sylvie didn't look as if she did.\n\n'I have a young man,' Ursula blurted out, catching them all unawares, including herself. It was more to lighten the conversation than anything. She should have known better really.\n\nHe was called Ralph. He lived in Holborn and he was a new friend, a 'pal', that she had met at her German class. He had been an architect before the war and Ursula supposed he would be an architect afterwards too. If anyone was still alive, of course. (Could London be erased, like Knossos or Pompeii? The Cretans and the Romans probably went around saying, 'We can take it,' in the heart of disaster.) Ralph was full of ideas for the rebuilding of the slums as modern towers. 'A city for the people', he said, one that would 'rise from the ashes of the old like a phoenix, modernist to the core'.\n\n'What an iconoclast he sounds,' Pamela said.\n\n'He's not nostalgic in the way we are.'\n\n'Are we? Nostalgic?'\n\n'Yes,' Ursula said. 'Nostalgia is predicated on something that never existed. We imagine an Arcadia in the past, Ralph sees it in the future. Both equally unreal, of course.'\n\n'Cloud-capped palaces?'\n\n'Something like that.'\n\n'But you like him?'\n\n'Yes.'\n\n'Have you... you know?'\n\n'Really! What kind of a question is that?' Ursula laughed. (Sylvie was at the door again, Hugh was sitting cross-legged on the lawn pretending to be Big Chief Running Bull.)\n\n'It's a very good question,' Pamela said.\n\nThey hadn't, as it happened. Perhaps if he were more ardent. She thought of Crighton. 'And anyway there is so little time for...'\n\n'Sex?' Pamela said.\n\n'Well, I was going to say courtship, but yes, sex.' Sylvie had returned and was trying to separate the warring factions on the lawn. The evacuees made very unsportsmanlike enemies. Hugh was tied up now with an old washing line. 'Help!' he mouthed to Ursula but he was grinning like a schoolboy. It was nice to see him happy.\n\nBefore the war her wooing by Ralph (or his by her, perhaps) might have taken the form of dances, the cinema, cosy dinners _\u00e0 deux_ but now, more often than not, they had found themselves at bombsites, like sightseers viewing ancient ruins. The view from the top deck of the number eleven bus was particularly good for this, they had discovered.\n\nIt was perhaps due more to a kink in their respective characters than the war itself. After all, other couples managed to keep up the rituals.\n\nThey had 'visited' the Duveen Gallery at the British Museum, Hammonds next to the National Gallery, the huge crater at the Bank, so big that they had to build a temporary bridge across it. John Lewis, still smouldering when they arrived, the blackened mannequins from the shop windows strewn across the pavement, their clothes ripped off.\n\n'Do you think we're like ghouls?' Ralph asked and Ursula said, 'No, we are witnesses.' She supposed she would go to bed with him eventually. There was no great argument to be found against it.\n\nBridget came out with tea and cake and Pamela said, 'I think I'd better untie Daddy.'\n\n'Have a drink,' Hugh said, pouring her a tumbler of malt from the cut-glass decanter that he kept in the growlery. 'I find myself in here more and more these days,' he said. 'It's the only place I can get peace. Dogs and evacuees strictly barred. I worry about you, you know,' he added.\n\n'I worry about me too.'\n\n'Is it bloody?'\n\n'Dreadfully. But I believe it's the right thing. I think we are doing the right thing.'\n\n'A just war? You know the Coles still have most of their family in Europe. Mr Cole has told me some dreadful things, things that are happening to the Jews. I don't think anyone here really wants to know. Anyway,' he said, raising his glass and trying for a cheerful note, 'down the hatch. Here's to the end.'\n\n*\n\nIt was dark when she left and Hugh walked her down the lane to the station.\n\n'No petrol, I'm afraid,' he said, 'you should have gone earlier,' he added ruefully. He had a stout torch and there was no one to yell at him to put the light out. 'I hardly think I'm going to guide in a Heinkel,' he said. Ursula told him how most rescue squads had an almost superstitious horror of lights even when they were in the middle of a raid, surrounded by burning buildings and incendiaries and flares. As if a small torch beam would make any difference.\n\n'Knew a chap in the trenches,' Hugh said, 'lit a match, and Bob's your uncle, a German sniper shot his head clean off. Good chap,' he added reflectively, 'name of Rogerson, same as the bakers in the village. No relation.'\n\n'You never talk about it,' Ursula said.\n\n'I'm talking about it now,' Hugh said. 'Let it be a lesson to you, keep your head below the parapet and your light beneath a bushel.'\n\n'I know you don't mean that. Not really.'\n\n'I do. I'd rather you were a coward than dead, little bear. Teddy and Jimmy too.'\n\n'You don't mean that either.'\n\n'I do. Here we are, it's so dark you could walk right by the station and never see it. I doubt that your train will be on time, if there is a train at all. Oh, look, here's Fred. Evening, Fred.'\n\n'Mr Todd, Miss Todd. So you know, this is the last train tonight,' Fred Smith said. Fred had long since graduated from fireman to driver.\n\n'It's not really a train,' Ursula said, bemused. There was an engine but no carriages.\n\nFred looked back along the platform to where the carriages should have been, as if he'd forgotten their absence. 'Ah, yes, well,' he said, 'last time they were seen they were hanging off Waterloo Bridge. It's a long story,' he added, clearly unwilling to elaborate. Ursula was puzzled as to why the engine should be here _sans_ carriages but Fred looked rather grim.\n\n'I won't get home tonight then,' Ursula said.\n\n'Well,' Fred said, 'I've got to get this engine back up to town and I've got a head of steam up and I've got a fireman, old Willie here, so if you want to hop up on the footplate, Miss Todd, I think we can get you back.'\n\n'Really?' Ursula said.\n\n'It won't be as clean as riding on the cushions, but if you're game?'\n\n'I certainly am.'\n\nThe engine was impatient to go so she gave Hugh a quick hug and said, 'See you soon,' and climbed the steps up to the footplate where she took up her perch on the fireman's seat.\n\n'You will take care, little bear, won't you?' Hugh said. 'In London?' He had to raise his voice above the sound of hissing steam. 'Promise me?'\n\n'I promise,' she shouted. 'See you later!'\n\nShe twisted round, trying to see him on the dark platform as the train chugged off. She felt a sudden stab of guilt, she had played a rowdy game of hide-and-seek with the boys after supper. Instead she should, as Hugh said, have left when it was still light. Now Hugh would have to walk back in the dark alone along the lane. (She thought suddenly of poor little Angela, all those years ago.) Hugh quickly disappeared into the dark and smoke.\n\n'Well, this _is_ exciting,' she said to Fred. It didn't cross her mind that she would never see her father again.\n\nExciting, it was true, but also somewhat terrifying. The engine was a great metal beast roaring through the dark, the raw power of the machine come to life. It shook and rocked as if it were trying to dislodge her from its insides. Ursula had never previously thought about what went on in the cab of an engine. She had imagined, if she had imagined it at all, a relatively serene place \u2013 the driver alert to the track ahead, the fireman cheerfully shovelling coal. But instead there was non-stop activity, a continual conferring between fireman and driver over gradients and pressure, the frantic shovelling or the sudden closing down, the continual rackety noise, the almost unbearable heat of the furnace, the filthy soot from the tunnels that didn't seem to be kept out by the metal plates that had been put up to prevent light escaping from the cab. It was so hot! 'Hotter than hell,' Fred said.\n\nDespite the wartime speed restrictions they seemed to be travelling at least twice as fast as when she travelled in a carriage ('on the cushions', she thought, she must remember that for Teddy who, despite now being a pilot, still harboured his childhood desire to be a train driver).\n\nAs they approached London they could see fires in the east and hear the distant pealing of guns but as they neared the marshalling yards and engine sheds it became almost eerily quiet. They slowed to a halt and all was suddenly, thankfully, peaceful.\n\nFred helped her down from the cab. 'There you go, ma'am,' he said. 'Home sweet home. Well, not quite, I'm afraid.' He looked suddenly doubtful. 'I would walk you home but we have to put this engine to bed. Will you be all right from here?' They seemed to be in the middle of nowhere, just tracks and points and the looming shadows of engines. 'There's a bomb at Marylebone. We're at the back of King's Cross,' Fred said, reading her mind. 'It's not as bad as you think.' He switched on the weakest of torches, it illuminated only a foot or so in front of them. 'Have to be careful,' he said, 'we're a prime target here.'\n\n'I'll be absolutely fine,' she said, a little more gung-ho than she felt. 'Don't give me a second thought, and thank you. Good night, Fred.' She set off resolutely and immediately tripped over a rail and gave a little cry of distress when she banged her knee hard on the sharp stones of the track.\n\n'Here, Miss Todd,' Fred said, helping her up. 'You'll never find your way in the dark. Come on, I'll walk you to the gates.' He took her arm and set off, steering her as they went, for all the world as if they were on a Sunday stroll along the Embankment. She remembered how she had been rather sweet on Fred when she was younger. It would probably be quite easy to be sweet on him again, she reckoned.\n\nThey reached a big pair of wooden gates and he opened a small door set within them.\n\n'I think I know where I am,' she said. She had no idea where she was but she didn't want to inconvenience Fred any longer. 'Well, thank you again, maybe I'll see you next time I get down to Fox Corner.'\n\n'I doubt it,' he said. 'I start in the AFS tomorrow. Plenty of old codgers like Willie can keep the trains running.'\n\n'Good for you,' she said, although she was thinking how dangerous the fire service was.\n\n*\n\nIt was the blackest blackout ever. She walked with a hand in front of her face and eventually bumped into a woman who told her where she was. They walked together for half a mile or so. After a few minutes on her own again she heard footsteps behind her and she said, 'I'm here,' so the owner of the feet didn't walk into her. It was a man, no more than a figure in the dark who went with her as far as Hyde Park. Before the war you would never have dreamed of hooking arms with a complete stranger \u2013 particularly a man \u2013 but now the danger from the skies seemed so much greater than anything that could befall you from this odd intimacy.\n\nShe thought it must be nearly dawn when she got back to Phillimore Gardens but it was barely midnight. Millie, all dressed up, had just returned from an evening out. 'Oh, my God,' she said when she saw Ursula. 'What happened to you? Did you get bombed?'\n\nUrsula looked in the mirror and found that she was smudged all over with soot and coal dust. 'What a fright,' she said.\n\n'You look like a coalminer,' Millie said.\n\n'More like an engine driver,' she said, rapidly recounting the night's adventures.\n\n'Oh,' Millie said, 'Fred Smith, the butcher's boy. He was a bit of a dish.'\n\n'Still is, I suppose. I've got eggs from Fox Corner,' she said, removing the cardboard box that Sylvie had given her from her bag. The eggs had been nestled in straw, but now they were cracked and broken from the jolting of the track or when she fell in the engine yard.\n\nThe next day they managed to make an omelette from the salvaged remains.\n\n'Lovely,' Millie said. 'You should get home more often.'\n\n# _October 1940_\n\n'It's certainly busy tonight,' Miss Woolf said. A glorious understatement. There was a full-scale raid in progress, bombers droning overhead, glinting occasionally when they caught a searchlight. HE bombs flashed and roared and the large batteries _banged_ and _whuffed_ and _cracked \u2013_ all the usual racket. Shells whistled or screamed on their way up, a mile a second until they winked and twinkled like stars before extinguishing themselves. Fragments came clattering down. (A few days ago Mr Simms's cousin had been killed by shrapnel from the ack-acks in Hyde Park. 'Shame to be killed by your own,' Mr Palmer said. 'Sort of pointless.') A red glow over Holborn indicated an oil bomb. Ralph lived in Holborn but Ursula supposed on a night like this he would be in St Paul's.\n\n'It's almost like a painting, isn't it?' Miss Woolf said.\n\n'Of the Apocalypse maybe,' Ursula said. Against the backdrop of black night the fires that had been started burnt in a huge variety of colours \u2013 scarlet and gold and orange, indigo and a sickly lemon. Occasionally vivid greens and blues would shoot up where something chemical had caught fire. Orange flames and thick black smoke roiled out of a warehouse. 'It gives one a quite different perspective, doesn't it?' Miss Woolf mused. It did. It seemed both grand and terrible compared to their own grubby little labours. 'It makes me proud,' Mr Simms said quietly. 'Our battling on like this, I mean. All alone.'\n\n'And against all odds,' Miss Woolf sighed.\n\nThey could see all the way along the Thames. Barrage balloons dotted the sky like blind whales bobbing around in the wrong element. They were on the roof of Shell-Mex House. The building was now occupied by the Ministry of Supply, for which Mr Simms worked, and he had invited Ursula and Miss Woolf to come and 'see the view from the top'.\n\n'It's spectacular, isn't it? Savage and yet strangely magnificent,' Mr Simms said, as though they were at the summit of one of the Lakeland fells rather than a building on the Strand in the middle of a raid.\n\n'Well, I don't know about _magnificent_ , exactly,' Miss Woolf said.\n\n'Churchill was up here the other night,' Mr Simms said. 'Such a good vantage point. He was fascinated.'\n\nLater, when Ursula and Miss Woolf were alone, Miss Woolf said, 'You know, I rather had the impression that Mr Simms was a lowly clerk in the ministry, he's quite a meek soul, but he must be quite senior to have been up on the roof with Churchill.' (One of the firewatchers on duty on the roof had said, 'Evening, Mr Simms,' with the kind of respect people felt obliged to afford to Maurice, although in the case of Mr Simms it was less grudgingly given.) 'He's unassuming,' Miss Woolf said. 'I like that in a man.' Whereas I prefer assuming, Ursula thought.\n\n'It really is quite a show,' Miss Woolf said.\n\n'Isn't it, though?' Mr Simms said enthusiastically. Ursula supposed that they were all aware how odd it was to be admiring the 'show' when they were so painfully conscious of what it meant on the ground.\n\n'It's as if the gods are throwing a particularly noisy party,' Mr Simms said.\n\n'One I would rather not be invited to,' Miss Woolf said.\n\nA familiar fearful swishing sound made them all duck for cover but the bombs exploded some way away and although they heard the explosions _bang-bang-bang-bang_ they couldn't see what had been hit. Ursula found it very odd to think that up above them there were German bombers being flown by men who, essentially, were just like Teddy. They weren't evil, they were just doing what had been asked of them by their country. It was war itself that was evil, not men. Although she would make an exception for Hitler. 'Oh, yes,' Miss Woolf said, 'I think the man is quite, quite mad.'\n\nAt that moment, to their surprise, a basket of incendiaries came swooping down and crashed its noisy load right on the ministry's roof. The incendiaries cracked and sparked and the two firewatchers ran towards them with a stirrup pump. Miss Woolf grabbed a bucket of sand and beat them to it. ('Fast for an old bird' was Mr Bullock's estimation of Miss Woolf under pressure.)\n\n' _What if this were the world's last night?_ ' a familiar voice said.\n\n'Ah, Mr Durkin, you managed to join us,' Mr Simms said affably. 'You didn't have any trouble with the man on the door?'\n\n'No, no, he knew I was expected,' Mr Durkin said, as if feeling his own importance.\n\n'Is _anyone_ left at the post?' Miss Woolf murmured to no one in particular.\n\nUrsula felt suddenly compelled to correct Mr Durkin. ' _What if this_ present _were the world's last night_ ,' she said. 'The word \"present\" makes all the difference, don't you think? It makes it seem as if one's somehow in the thick of it, which we are, rather than simply contemplating a theoretical concept. This is it, the end right now, no more shillyshallying.'\n\n'Goodness, so much fuss over one little word,' Mr Durkin said, sounding put out. 'However, I obviously stand corrected.' Ursula thought that one word could mean a great deal. If any poet was scrupulous with words then it was surely Donne. Donne, himself once the dean of St Paul's, had been moved down to an ignominious berth in the basement of the cathedral. In death he had survived the Great Fire of London, would he survive this one too? Wellington's tomb was too hefty to move and had simply been bricked up. Ralph had given her a tour \u2013 he was on the night watch there. He knew everything there was to know about the cathedral. Not quite the iconoclast that Pamela had presumed.\n\nWhen they emerged into the bright afternoon, he said, 'Shall we try and get a cup of tea somewhere?' and Ursula said, 'No, let's go back to your place in Holborn and go to bed with each other.' So they had and she had felt rotten because she couldn't help thinking about Crighton while Ralph was politely accommodating his body to hers. Afterwards, he had seemed abashed as if he no longer knew how to be with her. She said, 'I'm just the same person as I was before we did this,' and he said, 'I'm not sure I am.' And she thought, oh dear God, he's a virgin, but he laughed and said, no, no, that wasn't it \u2013 he _wasn't \u2013_ it was just that he was so very much in love with her 'and now I feel, I don't know... sublimated'.\n\n'Sublimated?' Millie said. 'Sounds like sentimental twaddle to me. He has you on a pedestal, heaven help him when he discovers that you have feet of clay.'\n\n'Thank you.'\n\n'Is that a mixed metaphor or is it a rather clever image?' Millie, of course had always\u2014\n\n'Miss Todd?'\n\n'Sorry. Miles away.'\n\n'We should get back to our sector,' Miss Woolf said. 'It's strange, but one feels rather safe up here.'\n\n'I'm sure we're not,' Ursula said. She was right, for a few days later Shell-Mex House was badly hit by a bomb.\n\nShe was keeping watch with Miss Woolf in her flat. Sitting at her big corner window they drank tea and ate biscuits and could have been any two women spending the evening together if it hadn't been for the tolling thunder of the barrage. Ursula learned that Miss Woolf's name was Dorcas (which she had never liked) and that her fianc\u00e9 (Richard) had died in the Great War. 'I still call it that,' she said, 'and yet this one is the greater. At least this time we have right on our side, I hope.' Miss Woolf believed in the war but her religious faith had begun to 'crumble' since the start of the bombing. 'Yet we must hold fast to what is good and true. But it all seems so random. One wonders about the divine plan and so on.'\n\n'More of a shambles than a plan,' Ursula agreed.\n\n'And the poor Germans, I doubt many of _them_ are in favour of the war \u2013 of course one mustn't say that in the hearing of people like Mr Bullock. But if _we_ had lost the Great War and been burdened with great debt just as the world's economy collapsed then perhaps we too would have been a tinderbox awaiting the strike of a flint \u2013 a Mosley or some such awful person. More tea, dear?'\n\n'I know,' Ursula said, 'but they are trying to _kill_ us, you know,' and as if to demonstrate this fact they heard the _swish_ and _wheee_ that heralded a bomb heading in their direction and flung themselves with remarkable speed behind the sofa. It seemed unlikely that it would be enough to save them and yet only two nights ago they had pulled a woman out, almost unscathed, from beneath an upturned settee in a house that was otherwise more or less destroyed.\n\nThe bomb shook the Staffordshire cow-creamers on Miss Woolf's dresser but they agreed it had landed outside their section. They were both finely tuned to the bombs these days.\n\nThey were also both terribly down in spirits as Mr Palmer, the bank manager, had been killed when a delayed action bomb had detonated at an incident they were attending. The DA had blown him some distance and they found him half buried beneath an iron bedstead. He had lost his spectacles but looked relatively unharmed. 'Can you feel a pulse,' Miss Woolf said and Ursula puzzled as to why she was asking when Miss Woolf was much more capable of finding a pulse than she was, but then she realized that Miss Woolf was very upset. 'It's different when you know someone,' she said, gently stroking Mr Palmer's forehead. 'I wonder where his spectacles are? He doesn't look right without them, does he?'\n\nUrsula couldn't find a pulse. 'Shall we move him?' she said. She took his shoulders and Miss Woolf his ankles and Mr Palmer's body came apart like a Christmas cracker.\n\n'I can put more hot water in the pot,' Miss Woolf offered. To cheer her up Ursula told her stories about Jimmy and Teddy when they were boys. She didn't bother with Maurice. Miss Woolf was very fond of children, her only regret in life was not having had any. 'If Richard had lived, perhaps... but one cannot look backwards, only forwards. What has passed has passed for ever. What is it Heraclitus says? One cannot step in the same river twice?'\n\n'More or less. I suppose a more accurate way of putting it would be \"You can step in the same river but the water will always be new.\" '\n\n'You're such a bright young woman,' Miss Woolf said. 'Don't waste your life, will you? If you're spared.'\n\nUrsula had seen Jimmy a few weeks ago. He'd been on two days' leave in London and had bedded down on their sofa in Kensington. 'Your baby brother's grown up all handsome,' Millie said. Millie was inclined to think all men were handsome, one way or another. She suggested a night on the town and Jimmy readily agreed. He'd been shut up long enough, he said, 'Time for some fun.' Jimmy had always been good at fun. The night almost didn't get started as there was a UXB on the Strand and they took refuge in the Charing Cross Hotel.\n\n'What?' Millie said to Ursula when they had sat down.\n\n'What what?'\n\n'You've got that funny look on your face, the one you get when you're trying to remember something.'\n\n'Or forget something,' Jimmy offered.\n\n'I wasn't thinking anything,' Ursula said. It had been nothing, just something fluttering and tugging at a memory. A silly thing \u2013 it always was \u2013 a kipper on a pantry shelf, a room with green linoleum, an old-fashioned hoop bowling silently along. Vaporous moments, impossible to hold on to.\n\nUrsula repaired to the Ladies where she found a girl crying noisily and rather messily. She was heavily made up and her mascara was in runnels down her cheeks. Ursula had noticed her earlier having a drink with an older man \u2013 'rather slimy' had been Millie's verdict on him. The girl looked much younger close up. Ursula helped her to repair her make-up and mop up her tears but didn't like to pry into the cause of them. 'It's Nicky,' the girl offered up voluntarily, 'he's a bastard. _Your_ young man looks lovely, fancy a foursome? I can get us in the Ritz, into the Rivoli Bar, I know a man on the door.'\n\n'Well,' Ursula said doubtfully. 'The young man's my brother actually, and I don't suppose\u2014'\n\nThe girl gave her a rather sharp jab in the ribs and laughed. 'Only joking. You two girls make the most of him, eh?' She offered Ursula a cigarette which she declined. The girl had a gold cigarette case that looked valuable. 'A gift,' she said, catching Ursula looking at it. She snapped it shut and held it out for inspection. There was a fine engraving of a battleship on the front with the single word 'Jutland' beneath. If she were to open it up again she knew she would find the initials 'A' and 'C' intertwined on the inside of the lid, for 'Alexander' and 'Crighton'. Instinctively, Ursula reached out a hand for it and the girl snatched it back, saying, 'Anyway, must be getting back. I'm as right as rain now. You seem a good sort,' she added, as if there had been a question over Ursula's character. She stuck out her hand. 'My name's Renee by the way, in case we ever bump into each other again, although I doubt we inhabit the same _endroits_ , as they say.' Her French pronunciation was spot-on, how odd, Ursula thought. She took the proffered hand \u2013 hard and warm as if the girl were running a temperature \u2013 and said, 'Pleased to meet you, I'm Ursula.'\n\nThe girl \u2013 Renee \u2013 gave herself one last look of approval in the mirror and said, ' _Au revoir_ then,' and was off.\n\nWhen Ursula went back into the coffee lounge Renee ignored her completely. 'What a strange girl,' she said to Millie.\n\n'Been making eyes at me all evening,' Jimmy said.\n\n'Well, she's barking up the wrong street there, darling, isn't she?' Millie said, batting her eyelashes at him, ridiculously theatrical.\n\n' _Tree_ ,' Ursula said. 'Barking up the wrong tree.'\n\nThey went drinking, a merry trio, in all kinds of strange haunts that Jimmy seemed to know about. Even Millie, a seasoned regular of the nightclub scene, professed surprise at some of the places they found themselves in.\n\n'Gosh,' Millie said as they left a club in Orange Street to totter homeward, 'that was different.'\n\n'A strange _endroit_ ,' Ursula laughed. She was rather drunk. It was such an Izzie word that it was bizarre to hear it from the lips of the Renee girl.\n\n'Promise you won't die,' Ursula said to Jimmy as they groped blindly home.\n\n'Do my best,' Jimmy said.\n\n# _October 1940_\n\n_'Man that is born of woman hath but a short time to live, and is full of misery. He cometh up and is cut down like a flower: he fleeth as it were a shadow, and never continueth in one stay.'_\n\nA light drizzle was falling. Ursula felt an urge to take out her handkerchief and wipe the wet coffin lid. On the other side of the open grave, Pamela and Bridget were pillars, holding up Sylvie, who seemed to be so consumed with her grief that she could barely stand. Ursula felt her own heart harden and contract with every sob that issued from her mother's chest. Sylvie had been needlessly unkind to Hugh in the last months and now this great affliction seemed like a show. 'You're too harsh,' Pamela said. 'No one can understand what goes on in a marriage, every couple is different.'\n\nJimmy, shipped to North Africa the previous week, had been unable to get compassionate leave but Teddy had turned up at the last minute. Shiningly smart in uniform, he had returned from Canada with his 'wings' ('Like an angel,' Bridget said) and was stationed in Lincolnshire. He and Nancy clung to each other at the committal. Nancy was vague about her job ('clerical really') and Ursula thought she recognized the fudge of the Official Secrets Act at work.\n\nThe church was packed, most of the village turned out for Hugh and yet there was something odd about the funeral, as if the guest of honour hadn't been able to come. Which he hadn't, of course. Hugh wouldn't have wanted a fuss. He had once said to her, 'Oh, you can just put me out with the dustbin, I won't mind.'\n\nThe service had been the usual sort \u2013 reminiscences and commonplaces \u2013 salted with a hefty dose of Anglican doctrine, although Ursula was surprised at how well the vicar seemed to be acquainted with Hugh. Major Shawcross read from the Beatitudes, rather movingly, and Nancy read 'one of Mr Todd's favourite poems', which surprised all the women of his family who didn't know that Hugh had any inclination towards poetry. Nancy had a nice speaking voice (better actually than Millie's which was overly thespian). 'Robert Louis Stevenson,' Nancy said. 'Perhaps appropriate for these testing times:\n\n'Tempest tossed and sore afflicted, sin defiled and care oppressed,  \nCome to me, all ye that labour; come, and I will give ye rest.  \nFear no more, O doubting hearted; weep no more, O weeping eye!  \nLo, the voice of your redeemer; lo, the songful morning near.\n\n'Here one hour you toil and combat, sin and suffer, bleed and die;  \nIn my father's quiet mansion soon to lay your burden by.  \nBear a moment, heavy laden, weary hand and weeping eye.  \nLo, the feet of your deliverer; lo, the hour of freedom here.'\n\n('Tosh, really,' Pamela whispered, 'but oddly comforting tosh.')\n\nAt the graveside, Izzie murmured, 'I feel as if I'm waiting for something dreadful to happen, and then I realize it already has.'\n\nIzzie had arrived back from California just a few days before Hugh died. She had flown, rather admirably, on a taxing Pan Am flight from New York to Lisbon and from there with BOAC to Bristol. 'I saw two German fighter planes from the window,' she said, 'I swear I thought they were going to attack us.'\n\nShe had decided, she said, that as an Englishwoman it was wrong to be sitting out the war amid the orange groves. All that lotus eating wasn't for her, she said (although Ursula would have said it was _exactly_ her). She had hoped, like her husband the famous playwright, to be asked to write screenplays for the film industry but had received only one offer, some 'silly' costume drama that had been aborted before it left the page. Ursula got the impression that Izzie's script hadn't come up to scratch ('too witty'). She had continued with Augustus, however \u2013 _Augustus Goes to War, Augustus and the Salvage Hunt_ and so on. It didn't help, Izzie said, that the famous playwright was surrounded by Hollywood starlets and that he was shallow enough to find them fascinating.\n\n'In truth, we simply grew bored with each other,' she said. 'All couples do eventually, it's inevitable.'\n\nIzzie was the one who had found Hugh. 'He was in a deckchair on the lawn.' The wicker furniture had long since rotted and been replaced by the more quotidian deckchair. Hugh had been put out by the arrival of folding wood and canvas. He would have preferred the wicker chaise-longue for a bier. Ursula's thoughts were full of such inconsequences. Easier to deal with, she thought, than the bare fact that Hugh was dead.\n\n'I thought he was asleep out there,' Sylvie said. 'So I didn't disturb him. A heart attack, the doctor said.'\n\n'He looked peaceful,' Izzie said to Ursula. 'As if he didn't really mind going.'\n\nUrsula felt that he probably minded very much but that was no comfort to either of them.\n\nShe had little conversation with her mother. Sylvie seemed always to be on the point of leaving the room. 'I can't settle,' she said. She was wearing an old cardigan of Hugh's. 'I'm cold,' she said. 'I'm so cold,' like someone in shock. Miss Woolf would have known what to do with Sylvie. Hot sweet tea probably, and some kind words but neither Ursula nor Izzie felt like offering either. Ursula sensed they were being rather vengeful but they had their own distress to nurse.\n\n'I'll stay on with her for a while,' Izzie said. Ursula thought this was a terrible idea and wondered if Izzie wasn't just avoiding the bombs.\n\n'You'd better get yourself a ration book then,' Bridget said. 'You're eating us out of house and home.' Bridget had been very affected by Hugh's death. Ursula came across her crying in the pantry and said, 'I'm awfully sorry,' as if the loss were Bridget's, not hers. Bridget wiped her tears vigorously on her apron and said, 'Must get on with the tea.'\n\nUrsula herself stayed only two more days and spent most of the time helping Bridget to sort out Hugh's things. ('I can't,' Sylvie said, 'I just can't.' 'Neither can I,' Izzie said. 'Then it's you and me,' Bridget said to Ursula.) Hugh's clothes were so very real it seemed absurd that the man who had worn them had disappeared. Ursula took a suit out of the wardrobe and held it against her body. If Bridget hadn't taken it from her and said, 'That's a good suit, someone will be grateful for it,' she might have crawled into the wardrobe and given up on life. Bridget's feelings were now locked up tightly, thank goodness. There was a great deal to be said for fortitude in the face of tragedy. Certainly her father would have approved.\n\nThey parcelled Hugh's clothes up in brown paper and string and the milkman put them on his cart and took them round to the WVS.\n\nIzzie's grief had left her wide open. She trailed around the house after Ursula, trying to conjure up Hugh from memories. They were all doing that, Ursula supposed, it was so impossible to grasp that he had gone for ever that they had all started trying to reconstitute him out of thin air, Izzie most of all. 'I can't remember the last thing he said to me,' Izzie said. 'Or what I said to him, for that matter.'\n\n'It won't make any difference,' Ursula said wearily. Whose bereavement was the greater after all, the daughter or the sister? But then she thought of Teddy.\n\nUrsula tried to remember what her own last words to her father had been. A nonchalant 'See you later,' she concluded. The final irony. 'We never know when it will be the last time,' she said to Izzie, platitudinous, even to her own ears. She had seen so much of other people's distress by now that she was numb to it. Except for that one moment when she held his suit (she thought of it \u2013 ridiculously \u2013 as her 'wardrobe moment') she had put Hugh's death away in some quiet place to be taken out later and considered. Perhaps when everyone else had done talking.\n\n'And the thing is,' Izzie said\u2014\n\n'Please,' Ursula said. 'I've got an awful headache.'\n\nUrsula was collecting eggs from the nest boxes when Izzie mooched into the henhouse. The chickens were clucking restlessly, they seemed to miss Sylvie's attentions, the Mother Hen. 'The thing is,' Izzie said, 'there's something I'd like to tell you.'\n\n'Oh?' Ursula said, distracted by a particularly broody hen.\n\n'I had a baby.'\n\n'What?'\n\n'I'm a mother,' Izzie said, seemingly unable to resist sounding dramatic.\n\n'You had a baby in California?'\n\n'No, no,' Izzie laughed. 'Years ago. I was just a child myself. Sixteen. I had him in Germany, I was sent abroad in disgrace, as you can imagine. A boy.'\n\n'Germany? And he was adopted?'\n\n'Yes. Well, more like given away. Hugh saw to it all so I'm sure he found a very good family. But he made him a hostage to fortune, didn't he? Poor Hugh, he was such a rock at the time, Mother would have nothing to do with it. But that's the thing, he must have known the name, where they lived, etcetera.' The hens were making a dreadful racket now and Ursula said, 'Let's get out of here.'\n\n'I always thought,' Izzie said, taking Ursula's arm and walking her round to the lawn, 'that one day I would talk to Hugh about what he did with the baby and then perhaps try and find him. My son,' she added, trying out the word as if for the first time. Tears started to roll down her face. For once, her emotions seemed from the heart. 'And now Hugh's gone and I'll never be able to find the baby. He's not a baby, of course, he's the same age as you.'\n\n'Me?' Ursula said, trying to grasp this idea.\n\n'Yes. But he's the _enemy_. He might be up there in the sky' \u2013 they both automatically glanced up at the blue autumn sky, empty of friend and foe alike \u2013 'or fighting in the forces. He might be dead, or going to die if this wretched war goes on.' Izzie was sobbing openly now. 'He might have been brought up as a Jew, for God's sake. Hugh wasn't an anti-Semite, quite the opposite, he was great friends with \u2013 your neighbour, what's his name?'\n\n'Mr Cole.'\n\n'You do know what's happening to the Jews in Germany, don't you?'\n\n'Oh, for heaven's sake,' Sylvie said, materializing suddenly like a bad fairy. 'What are you making such a fuss about?'\n\n'You should come back to London with me,' Ursula said to Izzie. The Luftwaffe's bombs would be more straightforward for her to deal with than Sylvie.\n\n# _November 1940_\n\nMiss Woolf was treating them to a little piano recital. 'Some Beethoven,' she said. 'I'm no Myra Hess, but I thought it would be nice.' She was correct on both counts. Mr Armitage, the opera singer, asked Miss Woolf if she could accompany him if he sang 'Non _pi\u00f9 andrai_ ' from _The Marriage of Figaro_ and Miss Woolf, particularly game this evening, said she would certainly have a go. It was a rousing performance ('unexpectedly _virile_ ' was Miss Woolf's verdict) and no one objected when Mr Bullock (no surprise) and Mr Simms (quite a surprise) joined in with a rather ribald version.\n\n'I know this one!' Stella said, which was true of the tune but not the words as she sang enthusiastically, 'Dum-di-dum, dum-di-dum, dum-di-dum _-dum_ ,' and so on.\n\nTheir post had recently been augmented by two wardens. The first, Mr Emslie, was a grocer and had come from another post, having been bombed out of his house, his shop and his sector. He, like Mr Simms and Mr Palmer before him, was a veteran of the previous war. The second addition was in possession of a more exotic background. Stella was one of Mr Bullock's 'chorus girls' and confessed (readily) to being a 'striptease artiste' but Mr Armitage the opera singer said, 'We're all artistes here, darling.'\n\n'What a bloody fairy that man is,' Mr Bullock muttered, 'put him in the army, that would sort him out.' 'I doubt it,' Miss Woolf said. (And it did rather beg the question why the strapping Mr Bullock himself had not been called up for active service.) 'So,' Mr Bullock concluded, 'we've got a Yid, a pansy and a tart, sounds like a dirty music-hall joke.'\n\n'It is intolerance that has brought us to this pass, Mr Bullock,' Miss Woolf reproved him mildly. They had all been decidedly tetchy \u2013 even Miss Woolf \u2013 since Mr Palmer's death. They would be better off saving their grudges for peacetime, Ursula thought. It wasn't just Mr Palmer's death, of course, but also the lack of sleep and the relentless nightly raids. How long could the Germans keep it up? For ever?\n\n'And, oh, I don't know,' Miss Woolf said quietly to her as she made tea, 'it's just the general sense of _dirtiness_ , as if one will never be clean again, as if poor old London will never be clean again. Everything is so awfully _shabby_ , you know?'\n\nIt was a relief, therefore, that their little impromptu concert party was good-natured, everyone seemingly in better spirits than of late.\n\nMr Armitage followed his Figaro with an unaccompanied and impassioned rendition of 'O _mio babbino caro_ ' ('How versatile he is,' Miss Woolf said, 'I always thought that was a woman's aria') that they all applauded wildly. Then Herr Zimmerman, their refugee, said he would be honoured to play something for them.\n\n'And then are you going to strip, sweetheart?' Mr Bullock asked Stella, who said, 'If you want,' and winked in complicity at Ursula. ('Trust me to get stuck with a load of bolshie women,' Mr Bullock complained. Frequently.)\n\nMiss Woolf said, looking worried, 'Your violin is _here_?' to Herr Zimmerman. 'Is it _safe_ here?' He had never brought his instrument to the post before. It was quite valuable, Miss Woolf said, and not just from a monetary point of view, for he had left his entire family behind in Germany and the violin was all he had from his former life. Miss Woolf said that she had had a 'harrowing' late night 'chat' with Herr Zimmerman about the situation in Germany. 'Things are terrible over there, you know.'\n\n'I know,' Ursula said.\n\n'Do you?' Miss Woolf said, her interest piqued. 'Do you have friends there?'\n\n'No,' Ursula said. 'No one. Sometimes one just _knows_ , doesn't one?'\n\nHerr Zimmerman produced his violin and said, 'You must forgive me, I am not a soloist,' and then announced, almost apologetically, 'Bach. Sonata in G Minor.'\n\n'It's funny, isn't it,' Miss Woolf whispered in Ursula's ear, 'how much German music we listen to. Great beauty transcends all. Perhaps after the war it will heal all too. Think of the Choral Symphony \u2013 _Alle Menschen werden Br\u00fcder_.'\n\nUrsula didn't answer as Herr Zimmerman had raised his bow, poised for performance, and a deep hush fell as if they were in a concert hall rather than a rundown post. Some of the silence was due to the quality of the performance ('Sublime,' Miss Woolf judged it later. 'Really beautiful,' Stella said) and some out of respect perhaps for Herr Zimmerman's refugee status, but there was also something so spare about the music that it left plenty of room for one to engage deeply with one's thoughts. Ursula found herself dwelling on Hugh's death, his absence more than his death. It was only a fortnight since he died and she was still expecting to see him again. These were the thoughts she had put away for a future time and now the future was suddenly on her. She was relieved not to be embarrassed by tears, instead she was plunged into an awful melancholy. As if sensing her emotions, Miss Woolf reached out and gripped her hand firmly. Ursula could feel that Miss Woolf herself was almost vibrating with emotion.\n\nWhen the music finished there was a moment of pure, profound silence, as if the world had stopped breathing, and then instead of praise and applause the peace was broken by the purple warning \u2013 'bombers within twenty minutes'. It was rather odd to think that these alerts were coming from her own Region 5 War Room, sent by the girls in the teleprinter room.\n\n'Come on then,' Mr Simms said, standing up and sighing heavily, 'let's get out there.' By the time they were out the red alert had come through. Just twelve minutes, if they were lucky, to dragoon people into shelters, the siren at their back.\n\nUrsula never used public shelters, there was something about the crush of bodies, the claustrophobia, that made her skin crawl. They had attended a particularly gruesome incident when a shelter took a direct hit from a parachute mine in their sector. Ursula thought that she would rather die out in the open than trapped like a fox in a hole.\n\nIt was a beautiful evening. A crescent moon and her bevy of stars had pierced the black backcloth of night. She thought of Romeo's encomium to Juliet \u2013 _It seems she hangs upon the cheek of night \/ As a rich jewel in an Ethiop's ear_. Ursula was in a poetic mood, some might have said, herself included, overly poetic, as a consequence of her mournful mood. There was no Mr Durkin to misquote any more. He had suffered a heart attack during an incident. He was recovering, 'thank goodness', Miss Woolf said. She had found time to visit him in hospital and Ursula felt no guilt that she had not. Hugh was dead, Mr Durkin wasn't, there was little room in her heart for sympathy. Mr Durkin's position as Miss Woolf's deputy had been taken by Mr Simms.\n\nThe strident noises of war had begun. The boom of the barrage, the raiders' engines overhead with that monotonous, uneven beat that made her nauseous. The gun discharges, the searchlights poking their fingers into the sky, the muted anticipation of dread \u2013 all soon spoiled any idea of poetry.\n\nBy the time they arrived at the incident everyone was there, the gas and water, the Bomb Disposal Squad, heavy rescue, light rescue, stretcher parties, the mortuary van (a baker's van by day). The road was carpeted with the tangled hoses of an AFS unit as on one side of the street a building was well on fire, with sparks and burning embers spitting out. Ursula thought she had caught a glimpse of Fred Smith, his features briefly illuminated by the flames, but came to the conclusion that she had imagined it.\n\nThe rescue squad was as cautious as ever with their torches and lamps even though the fire was blazing away at their backs. Yet, to a man, they had cigarettes hanging from the corners of their mouths, despite the fact that the gas men hadn't cleared the area, not to mention that the presence of the Bomb Disposal Squad indicated a bomb that might go off at any moment. Everyone just got on with the job in hand (needs must), cavalier in the face of possible disaster. Or perhaps some people (and Ursula wondered if she included herself among them nowadays) simply didn't care any more.\n\nShe had an uncomfortable feeling, a premonition perhaps, that things were not going to go well tonight. 'It was the Bach,' Miss Woolf comforted, 'it was unsettling for the soul.'\n\nApparently, the street straddled two sectors and the incident officer in charge was wrangling with two wardens who both claimed dominion over it. Miss Woolf didn't join this little fracas as it turned out that it wasn't their sector at all, but as it was obviously such a major incident she declared that their post should pitch in and get on with it and ignore what anyone said to them.\n\n'Outlaws,' Mr Bullock said, appreciatively.\n\n'Hardly,' Miss Woolf said.\n\nThe half of the street that wasn't on fire had been badly hit and the acid-raw smell of powdered brick and cordite struck their lungs immediately. Ursula tried to think of the meadow at the back of the copse at Fox Corner. Flax and larkspur, corn poppies, red campion and ox-eye daisies. She thought of the smell of new-mown grass and the freshness of summer rain. This was a new diversionary tactic to combat the brutish scents of an explosion. ('Does it work?' a curious Mr Emslie asked. 'Not really,' Ursula said.) 'I used to think of my mother's perfume,' Miss Woolf said. 'April Violets. But unfortunately now when I try to remember my mother all I can think of are the bombs.'\n\nUrsula offered Mr Emslie a peppermint. 'It helps a little bit,' she said.\n\nThe closer they got to the incident the worse it proved to be (the opposite, in Ursula's experience, was rarely so).\n\nA grisly tableau was the first thing to greet them \u2013 mangled bodies were strewn around, many of them no more than limbless torsos, like tailor's dummies, their clothes blown off. Ursula was reminded of the mannequins she had seen with Ralph in Oxford Street, after the John Lewis bomb. A stretcher-bearer, lacking as yet any live casualties, was picking up limbs \u2013 arms and legs that were sticking out of the rubble. He looked as if he was intending to piece the dead together again at a later date. Did someone do that, Ursula wondered? In the mortuaries \u2013 try and match people up, like macabre jigsaws? Some people were beyond re-creation, of course \u2013 two men from the rescue squad were raking and shovelling lumps of flesh into baskets, another was scrubbing something off a wall with a yard brush.\n\nUrsula wondered if she knew any of the victims. Their flat in Phillimore Gardens was a mere couple of streets away from here. Perhaps she passed some of them in the morning on her way to work, or had spoken to them in the grocer's or the butcher's.\n\n'Apparently there are quite a lot of people unaccounted for,' Miss Woolf said. She had spoken to the Incident Officer, who had been grateful, it seemed, to talk to a warden with common sense. 'We're not outlaws any more, you'll be pleased to hear.'\n\nOne floor above the man with the yard brush (although there was no floor) a dress was hanging on a coat hanger from a picture rail. Ursula often found herself more moved by these small reminders of domestic life \u2013 the kettle still on the stove, the table laid for a supper that would never be eaten \u2013 than she was by the greater misery and destruction that surrounded them. Although when she looked at the dress now she realized that there was a woman still wearing it, her head and legs blown off but not her arms. The capriciousness of high explosives never ceased to surprise Ursula. The woman seemed to have become fused with the wall in some way. The fire was burning so brightly that she could make out a little brooch still pinned to the dress. A black cat, a rhinestone for an eye.\n\nRubble shifted underfoot as she made her way to the back wall of this same house. There was a woman sitting propped up amongst the rubble, arms and legs splayed like a rag doll. She looked as if she had been tossed in the air and landed any old how \u2013 which was probably the case. Ursula tried to signal to the stretcher-bearer but there was now a stream of bombers passing overhead and no one could hear her above the noise.\n\nThe woman was grey with dust so that it was almost impossible to tell how old she was. She had a horrible-looking burn on her hand. Ursula fumbled in her first-aid pack for the tube of Burnol and smeared some of the ointment on to her hand. She didn't know why, the woman looked too far gone to be cured by Burnol. She wished she had some water, it was painful to see how dry the woman's lips were. Unexpectedly, she opened her dark eyes, her lashes pale and spiky with dust, and tried to say something but her voice was so hoarse from the dust that Ursula couldn't understand her. Was she foreign? 'What is it?' Ursula asked. She had a feeling the woman was very near death now.\n\n'Baby,' the woman rasped suddenly, 'where's my baby?'\n\n'Baby?' Ursula echoed, looking around. She could see no sign of any baby. It could be anywhere in the rubble.\n\n'His name,' the woman said, guttural and indistinct \u2013 she was making a tremendous effort to be lucid \u2013 'is Emil.'\n\n'Emil?'\n\nThe woman nodded her head very slightly as if she were no longer capable of speech. Ursula looked around again for any sign of a baby. She turned back to the woman to ask how big her baby was but her head was lolling limply and when Ursula felt for a pulse she found nothing.\n\nShe left the woman there and went in search of the living.\n\n'Can you take Mr Emslie a morphia tablet?' Miss Woolf asked. They could both hear a woman screaming and swearing like a navvy and Miss Woolf added, 'To the lady that's making all the noise.' A good rule of thumb was that the more noise someone was making the less likely they were to die. This particular casualty sounded as if she were ready to fight her way out single-handed from the wreckage of the house and run round Kensington Gardens.\n\nMr Emslie was in the cellar of the house and Ursula had to be lowered down by two men from the rescue squad and then had to worm her way through a barricade of joists and bricks. She was aware that an entire house appeared to be resting precariously on this same barricade. She found Mr Emslie stretched out almost horizontally next to a woman. Below the waist she was completely trapped by the wreckage of the house but she was conscious and extremely articulate about the distress she was in.\n\n'Soon have you out of here,' Mr Emslie said. 'Get you a nice cup of tea, eh? How does that sound? Lovely, eh? Fancy one myself. And here's Miss Todd with something for the pain,' he continued soothingly to her. Ursula passed him the tiny morphia tablet. He seemed very good at this, it was hard to imagine him in his grocer's apron, weighing sugar and patting butter.\n\nOne wall of the cellar had been sandbagged but most of the sand had spilled out in the explosion and for an alarming hallucinatory second Ursula was on a beach somewhere, she didn't know where, a hoop was bowling along beside her in a brisk breeze, seagulls squawking overhead, and then she was back, just as suddenly, in the cellar. Lack of sleep, she thought, it really was the devil.\n\n'About fucking time,' the woman said, greedily taking the morphia tablet. 'You'd think you lot were at a fucking tea-party.' She was young, Ursula realized, and oddly familiar. She was clutching her handbag, a large black affair, as if it were keeping her afloat in the sea of timber. 'Have you got a fag, either of you?' With some difficulty, given the awkward space they were in, Mr Emslie produced a squashed packet of Players from his pocket and then, with even more difficulty, extracted a box of matches. Her fingers tapped restlessly on the leather of the bag. 'Take your time,' she said sarcastically.\n\n'Sorry,' she said after she had drawn deeply on the cigarette. 'Being in an _endroit_ like this has an effect on the nerves, you know.'\n\n'Renee?' Ursula said, astonished.\n\n'What's it to you?' she said, returning to her former churlish self.\n\n'We met in the cloakroom at the Charing Cross Hotel a couple of weeks ago.'\n\n'I think you've mistaken me for someone else,' she said primly. 'People are always doing that. I must have one of those faces.'\n\nShe took another very long drag on her cigarette and then exhaled slowly and with extraordinary pleasure. 'You got any more of those little pills?' she asked. 'Good black market price for them, I bet.' She sounded woozy, the morphia kicking in, Ursula supposed, but then the cigarette dropped from her fingers and her eyes rolled back in her head. She started to convulse. Mr Emslie grabbed hold of her hand.\n\nUrsula, glancing at Mr Emslie, caught sight of a colour reproduction of Millais's _Bubbles_ , hanging by a piece of tape from a sandbag behind him. It was a picture she disliked, she disliked all the Pre-Raphaelites with their droopy, drugged-looking women. Hardly the time and place for art criticism, she thought. She had become almost indifferent to death. Her soft soul had crystallized. (Just as well, she thought.) She was a sword tempered in the fire. And again she was somewhere else, a little flicker in time. She was descending a staircase, wisteria was blooming, she was flying out of a window.\n\nMr Emslie was talking encouragingly to Renee. 'Come on, Susie, don't give up on us now. We'll have you out of here in two shakes of a lamb's tail, you'll see. All the lads are working on it. And the girls,' he added for Ursula's benefit. Renee had stopped convulsing but now she started to shiver alarmingly and Mr Emslie, more urgently now, said, 'Come on, Susie, come on, girl, stay awake, there's a good girl.'\n\n'Her name _is_ Renee,' Ursula said, 'even if she denies it.'\n\n'I call 'em all Susie,' Mr Emslie said softly. 'I had a little girl by that name. The diphtheria took her off when she was just a littl'un.'\n\nRenee gave one last great shudder and life disappeared from her half-open eyes.\n\n'Gone,' Mr Emslie said sadly. 'Internal injuries probably.' He wrote 'Argyll Road' on a label in his neat grocer's hand and tied it to her finger. Ursula removed the handbag from Renee's rather reluctant grasp and shook its contents out. 'Her identity card,' she said, holding it up for Mr Emslie to see. 'Renee Miller' it said, indisputably. He added her name to the label.\n\nWhile Mr Emslie began the complex manoeuvre of turning round in order to make his way back out of the cellar, Ursula picked up the gold cigarette case that had fallen out with the compact and lipstick and French letters and God knows what else that formed the contents of Renee's handbag. Not a gift but stolen property, she was sure of that. It was a difficult task for Ursula's imagination to place Renee and Crighton in the same room as each other, let alone the same bed. War did indeed make strange bedfellows of people. He must have picked her up in a hotel somewhere, or perhaps a less salubrious _endroit_. Where had she learned her French? She probably only had a couple of words. Not from Crighton anyway, he thought English was quite enough to rule the world with.\n\nShe slipped the cigarette case and the identity card into a pocket.\n\nThe debris shifted in a heart-stopping way as they were trying to back out of the cellar (they'd given up on trying to turn round). They remained paralysed, crouched like cats, hardly daring to take a breath for what seemed an eternity. When it felt safe to move again they found that this new arrangement of wreckage had made the barricade impenetrable and they were forced to find another, tortuous exit, creeping on their hands and knees through the shattered base of the building. 'Doing my back in, this lark is,' Mr Emslie muttered behind her.\n\n'Doing my knees in,' Ursula said. They carried on with weary doggedness. Ursula cheered herself up with the thought of buttered toast, although Phillimore Gardens was out of butter and unless Millie had gone out and queued (unlikely), there was no bread either.\n\nThe cellar seemed to be an endless maze and it slowly dawned on Ursula why there were people unaccounted for up above \u2013 they were all secretly cached down here. The residents of the house clearly used this part of the cellar as a shelter. The dead here \u2013 men, women, children, even a dog \u2013 looked as though they had been entombed where they had been sitting. They were completely cloaked in a shell of dust and looked more like sculptures, or fossils. She was reminded of Pompeii or Herculaneum. Ursula had visited both, during her ambitiously titled 'grand tour' of Europe. She had been lodged in Bologna where she had made friends with an American girl \u2013 Kathy, a gung-ho type \u2013 and they had taken a whistle-stop tour \u2013 Venice, Florence, Rome, Naples \u2013 before Ursula left for France and the final leg of her year abroad.\n\nIn Naples, a city that frankly terrified them, they hired a loquacious private guide and spent the longest day of their lives trudging determinedly round the dry, dusty ruins of the lost cities of the Roman Empire beneath a merciless southern sun.\n\n'Oh, gosh,' Kathy said as they staggered around a deserted Herculaneum, 'I wish no one had ever gone to the bother of digging 'em up.' Their friendship had flared brightly for a short time and fizzled out just as quickly when Ursula went to Nancy.\n\n'I have spread my wings and learned how to fly,' she wrote to Pamela after leaving Munich and her hosts, the Brenners. 'I am quite the sophisticated woman of the world,' although she was still little more than a fledgling. If the year had taught her one thing it was that after having endured a succession of private students, the last thing she wanted to do was teach.\n\nInstead, on her return \u2013 with an eye to entrance into the civil service \u2013 she did an intensive shorthand and typing course in High Wycombe, run by a Mr Carver who was later arrested for exposing himself in public. ('A meat-flasher?' Maurice said, his lip curling in disgust, and Hugh shouted at him to leave the room and never to use such language in his house again. 'Infantile,' he said when Maurice had slammed his way out into the garden. 'Is he really fit for marriage?' Maurice had come home to announce his engagement to a girl called Edwina, the eldest daughter of a bishop. 'Goodness,' Sylvie said, 'will we have to genuflect or something?'\n\n'Don't be ridiculous,' Maurice said and Hugh said, 'How dare you speak to your mother like that.' It was a terrifically bad-tempered visit all round.)\n\nMr Carver hadn't been such a bad sort really. He had been very keen on Esperanto, which had seemed an absurd eccentricity at the time but now Ursula thought it might be a good thing to have a universal language, as Latin had once been. Oh, yes, Miss Woolf said, a common language was a wonderful idea, but utterly utopian. All good ideas were, she said sadly.\n\nUrsula had embarked for Europe a virgin, but didn't return one. She had Italy to thank for that. ('Well, if one can't take a lover in Italy where can one take one?' Millie said.) He, Gianni, was studying for a doctorate in philology at Bologna University and was more grave and serious than Ursula had expected an Italian to be. (In Bridget's romantic novels, Italians were always dashing but untrustworthy.) Gianni brought a studious solemnity to the occasion and made the rite of passage less embarrassing and awkward than she had feared.\n\n'Gosh,' Kathy said, 'you are bold.' She reminded Ursula of Pamela. In some ways, not in others \u2013 not in her serene denial of Darwin, for example. Kathy, a Baptist, was saving herself for marriage but a few months after she returned to Chicago her mother wrote to Ursula to tell her that Kathy had died in a boating accident. She must have gone through her daughter's address book and written to everyone in it, one by one. What an awful task. For Hugh, they had simply put a notice in _The Times_. Poor Kathy had saved herself for nothing. _The grave's a fine and private place, But none I think do there embrace_.\n\n'Miss Todd?'\n\n'Sorry, Mr Emslie. It's like being in a crypt, isn't it? Full of the ancient dead.'\n\n'Yes, and I'd quite like to get out before I turn into one of them.'\n\nAs she crept gingerly forward, Ursula's knee pressed on something soft and supple and she recoiled, banging her head on a broken rafter, sending a shower of dust down.\n\n'You all right?' Mr Emslie said.\n\n'Yes,' she said.\n\n'Are we stopped for something else?'\n\n'Hang on.' She had once stood on a body, recognized the squashy, meat-like quality of it. She supposed she had to look, although God knows she didn't want to. She shone her torch on what seemed to be a dusty mound of material, scraps of stuff \u2013 crochet and ribbons, wool \u2013 partly impacted into the earth. It could have been the contents of a sewing basket. But it wasn't, of course. She peeled back a layer of wool and then another one as if unwrapping a badly packed parcel or a large, unwieldy cabbage. Eventually a small almost unblemished hand, a small star, revealed itself from the compacted mass. She thought she might have found Emil. Better then that his mother was dead rather than knowing about this, she thought.\n\n'Be careful here, Mr Emslie,' she said over her shoulder, 'there's a baby, try to avoid it.'\n\n'All right?' Miss Woolf asked her when they finally emerged like moles. The fire on the other side of the street was almost out now and the street was murky with the dark, the soot, the filth. 'How many?' Miss Woolf asked.\n\n'Quite a few,' Ursula said.\n\n'Easy to recover?'\n\n'Hard to say.' She handed over Renee's identity card. 'There's a baby down there, bit of a mess, I'm afraid.'\n\n'There's tea,' Miss Woolf said. 'Go and get yourself some.'\n\nAs she made her way, with Mr Emslie, to the mobile canteen she was amazed to spot a dog cowering in a doorway further up the street.\n\n'I'll catch up with you,' she said to Mr Emslie. 'Get a mug for me, will you? Two sugars.'\n\nIt was a small nondescript terrier, whimpering and shaking with fear. Most of the house behind the doorway had disappeared and Ursula wondered if this had been the dog's home, that it was hoping for some kind of safety or protection and couldn't think of anywhere else to go. As she approached it, however, it ran off up the street. Dratted dog, she thought, chasing after it. Eventually she caught up with it, snatching it up in her arms before it had a chance to run again. It was trembling all over and she held it close, talking in soothing tones to it, rather as Mr Emslie had to Renee. She pressed her face against its fur (disgustingly dirty but then so was she). It was so small and helpless. 'Slaughter of the Innocents,' Miss Woolf said the other day when they heard of a school in the East End taking a direct hit. But wasn't everyone innocent? (Or were they all guilty?) 'That buffoon Hitler certainly isn't,' Hugh said, the last time they had talked, 'it's all down to him, this whole war.' Was she really never going to see her father again? A sob escaped from her and the dog whined in fear or sympathy, it was hard to say. (There wasn't a single member of the Todd family \u2013 apart from Maurice \u2013 who didn't attribute human emotions to dogs.)\n\nAt that moment there was a tremendous noise behind them, the dog tried to bolt again and she had to hold it tightly. When she turned round she saw the gable wall of the building that had been on fire falling down, almost in one piece, the bricks rattling on to the ground in a brutish fashion, just reaching the WVS canteen.\n\nTwo of the women from the WVS were killed, as was Mr Emslie. And Tony, their messenger boy who had been scooting past on his bicycle, but not scooting fast enough unfortunately. Miss Woolf knelt down on the jagged, broken brick, oblivious to the pain, and took hold of his hand. Ursula crouched down by her side.\n\n'Oh, Anthony,' Miss Woolf said, unable to say anything else. Her hair was escaping from its usual neat bun, making her look quite wild, a figure from a tragedy. Tony was unconscious \u2013 a terrible head wound, they had dragged him roughly from beneath the collapsed wall \u2013 and Ursula felt they should say something encouraging and not let him be aware of how upset they were. She remembered he was a Scout and started talking to him about the joys of the outdoors, pitching a tent in a field, hearing a running stream nearby, collecting sticks for a fire, watching the mist rise in the morning as breakfast cooked in the open. 'What fun you'll have again when the war is over,' she said.\n\n'Your mother will be awfully glad to see you come home tonight,' Miss Woolf said, joining the charade. She stifled a sob with her hand. Tony made no sign of having heard them and they watched as he slowly turned a deathly pale, the colour of thin milk. He had gone.\n\n'Oh, God,' Miss Woolf cried. 'I can't bear it.'\n\n'But bear it we must,' Ursula said, wiping away the snot and the tears and filth from her cheeks with the back of her hand and thinking how once this exchange would have been the other way round.\n\n'Bloody fools,' Fred Smith said angrily, 'what did they go and park the bloody canteen there for? Right next to the gable end?'\n\n'They didn't know,' Ursula said.\n\n'Well, they should have bloody realized.'\n\n'Well then someone should have bloody told them,' Ursula said, her anger flaring up suddenly. 'Like a bloody fireman, for instance.'\n\nIt was first light by now and they heard the all-clear sound.\n\n'I thought I saw you earlier, and then I decided I'd imagined you,' Ursula said, making peace. He was angry because they were dead, not because they were stupid.\n\nShe felt as though she were in a dream, drifting away from reality. 'I'm as good as dead,' she said, 'I have to sleep before I go mad. I live just round the corner,' she added. 'Lucky it wasn't our flat. Lucky, too, that I ran after this dog.' One of the rescue squad had given her a piece of rope to tie round the dog's neck and she had hitched it to a charred post sticking out of the ground. She was reminded of the arms and legs the stretcher-bearer had been harvesting earlier. 'I suppose the circumstances dictate that's what I should call him \u2013 Lucky, even though it's a bit of a clich\u00e9. He saved me, you know, I would have been drinking my tea there if I hadn't gone after him.'\n\n'Bloody fools,' he said again. 'Shall I walk you home?'\n\n'That would be nice,' Ursula said but she didn't lead him 'round the corner' to Phillimore Gardens, instead they walked wearily hand in hand, like children, the dog trotting beside them, along Kensington High Street, almost deserted at this time in the morning, with only a slight diversion for a gas main that was on fire.\n\nUrsula knew where they were going, it was inevitable somehow.\n\nIn Izzie's bedroom there was a framed picture on the wall opposite her bed. It was one of the original illustrations from the first _Adventures of Augustus_ , a line drawing depicting a cheeky boy and his dog. It verged on the cartoon \u2013 the schoolboy cap, the gob-stoppered cheek of Augustus and the rather idiotic-looking Westie who bore no resemblance to the real-life Jock.\n\nThe picture was very much at odds with how Ursula remembered this room before it was mothballed \u2013 a feminine boudoir, full of ivory silks and pale satins, expensive cut-glass bottles and enamelled brushes. A lovely Aubusson carpet had been rolled up tightly and tied with thick string and left against a wall. There had been one of the lesser Impressionists on another of the walls, acquired, Ursula suspected, more for the way it matched the d\u00e9cor than for any great love of the artist. Ursula wondered if Augustus was there to remind Izzie of her success. The Impressionist had been packed away somewhere safe but this illustration seemed to have been forgotten about, or perhaps Izzie didn't care so much for it any more. Whatever the reason, it had sustained a diagonal crack from one corner of the glass to the other. Ursula recalled the night that she and Ralph had been in the wine cellar, the night that Holland House was bombed, perhaps it had sustained the damage then.\n\nIzzie had, sensibly, chosen not to stay at Fox Corner with 'the grieving widow' as she referred to Sylvie, as 'we shall fight like cats and dogs'. Instead, she had decamped to Cornwall, to a house on top of a cliff ('like Manderley, terrifically wild and romantic, no Mrs Danvers though, thank goodness'), and had started 'churning out' an _Adventures of Augustus_ comic strip for one of the popular dailies. How much more interesting, Ursula thought, if she had allowed her Augustus to grow up, as Teddy had done.\n\nA buttery, unseasonal sun was trying hard to nudge its way through the thick velvet curtains. _Why dost thou thus, \/ Through windows, and through curtains, call on us?_ she thought. If she could go back in time and take a lover from history it would be Donne. Not Keats, the knowledge of his untimely death would colour everything quite wretchedly. That was the problem with time travel, of course (apart from the impossibility) \u2013 one would always be a Cassandra, spreading doom with one's foreknowledge of events. It was quite wearyingly relentless but the only way that one could go was forward.\n\nShe could hear a bird singing outside the window, even though it was November now. The birds were probably as confounded as people were by the Blitz. What did all the explosions do to them? Kill a great many, she supposed, their poor hearts simply giving out with shock or the little lungs bursting with the pressure waves. They must drop from the sky like weightless stones.\n\n'You look thoughtful,' Fred Smith said. He was lying, one arm behind his head, smoking a cigarette.\n\n'And you look strangely at home,' she said.\n\n'I am,' he grinned and leaned forward to wrap his arms around her waist and kiss the back of her neck. They were both filthy, as if they had toiled all night in a coal mine. She recalled how sooty they had been when she had journeyed on the footplate that night. The last time she had seen Hugh alive.\n\nThere was no hot water in Melbury Road, no water at all, nor electricity, everything turned off for the duration. In the dark, they had crawled under the dustsheet on Izzie's bare mattress and fallen into a sleep that mimicked death. Some hours later they had both woken up at the same time and made love. It was the kind of love (lust, to be honest about it) that survivors of disasters must practise \u2013 or people who are anticipating disaster \u2013 free of all restraint, savage at times and yet strangely tender and affectionate. A strain of melancholy ran through it. Like Herr Zimmerman's Bach sonata it had unsettled her soul, disjointed her brain and body. She tried to recall another line from Marvell, was it in 'A Dialogue Between the Soul and Body', something about _bolts of bones_ and fetters and manacles but it wouldn't come. It seemed harsh when there was so much soft skin and flesh in this abandoned (in all ways) bed.\n\n'I was thinking of Donne,' she said. 'You know \u2013 _Busy old foole, unruly Sun_.' No, she supposed, he probably didn't know.\n\n'Oh?' he said, indifferently. Worse than indifferent really.\n\nShe was taken off guard by the sudden memory of the grey ghosts in the cellar and of kneeling on the baby. Then for a second she was somewhere else, not a cellar in Argyll Road, not in Izzie's bedroom in Holland Park but some strange limbo. Falling, falling\u2014\n\n'Cigarette?' Fred Smith offered. He lit another one from the stub of his first and handed it to her. She took it and said, 'I don't really smoke.'\n\n'I don't really pick up strange women and fuck them in posh houses.'\n\n'How Lawrentian. And I'm not strange, we've known each other since we were children, more or less.'\n\n'Not like this.'\n\n'I should hope not.' She was beginning to dislike him already. 'I have no idea what time it is,' she said. 'But I can offer you some very good wine for breakfast. It's all there is, I'm afraid.'\n\nHe looked at his wristwatch and said, 'We've missed breakfast. It's three o'clock in the afternoon.'\n\nThe dog nudged itself through the door, its paws pitter-pattering on the bare wooden boards. It jumped on the bed and gazed intently at Ursula. 'Poor thing,' she said, 'it must be starving.'\n\n'Fred Smith? What was he like? Do tell!'\n\n'Disappointing.'\n\n'How? In bed?'\n\n'Gosh, no, not that at all. I've never... like that, you know. I think I thought it would be romantic. No, that's the wrong word, a silly word. \"Soulful\" perhaps.'\n\n'Transcendent?' Millie offered.\n\n'Yes, that's it. I was looking for transcendence.'\n\n'I imagine it finds you, rather than the other way round. It's a tall order for poor old Fred.'\n\n'I had an _idea_ of him,' Ursula said, 'but the idea wasn't him. Perhaps I wanted to fall in love.'\n\n'And instead you had jolly good sex. Poor you!'\n\n'You're right, unfair of me to expect. Oh, God, I think I was an awful snob with him. I was quoting Donne. Am I a snob, do you think?'\n\n'Awful. You do reek, you know,' Millie said cheerfully. 'Cigarettes, sex, bombs, God knows what else. Shall I run you a bath?'\n\n'Oh, yes, please, that would be lovely.'\n\n'And while you're at it,' Millie said, 'you can take that ruddy dog in the bath with you. He smells to high heaven. But he is kinda cute,' she said, imitating an American accent (rather badly).\n\nUrsula sighed and stretched. 'You know I really, _really_ have had enough of being bombed.'\n\n'The war's not going away any time soon, I'm afraid,' Millie said.\n\n# _May 1941_\n\nMillie was right. The war went on and on. Into that dreadfully cold winter, and then there was the awful raid on the City at the end of the year. Ralph had helped to save St Paul's from the fire. All those lovely Wren churches, Ursula thought. They had been built because of the last Great Fire, now they were gone.\n\nThe rest of the time they did the things that everyone of their kind did. They went to the cinema, they went dancing, they went to the lunchtime concerts in the National Gallery. They ate and drank and made love. Not 'fucking'. That wasn't Ralph's style at all. 'Very Lawrentian,' she had said coolly to Fred Smith \u2013 she supposed he had no idea what she was talking about \u2013 but the crude word had jarred her horribly. She was used to hearing it at incidents, it was a vital constituent of the heavy rescue squad's vocabulary, but not in the context of _herself_. She tried saying the word to her bathroom mirror but it felt shameful.\n\n'Where on earth did you get it?' he asked.\n\nUrsula had never seen him so dumbfounded. Crighton weighed the gold cigarette case in his hand. 'I thought I'd lost it for ever.'\n\n'Do you really want to know?'\n\n'Yes, of course I do,' Crighton said. 'Why the mystery?'\n\n'Does the name Renee Miller mean anything to you?'\n\nHe frowned, thinking, and then shook his head. 'Afraid not. Should it?'\n\n'You probably paid her for sex. Or bought her a nice dinner. Or just gave her a good time.'\n\n'Oh, _that_ Renee Miller,' he laughed. After a couple of beats of silence, he said, 'No, really, the name means nothing. And anyway, I don't think I have ever _paid_ a woman for sex.'\n\n'You're in the navy,' she pointed out.\n\n'Well, not for a very, _very_ long time then. But thank you,' he said, 'you know the cigarette case meant a lot to me. My father\u2014'\n\n'Gave it to you after Jutland, I know.'\n\n'Am I boring you?'\n\n'No. Shall we go somewhere? The bolthole? Shall we fuck?'\n\nHe burst out laughing. 'If you want.'\n\nHe cared less 'for the niceties' these days, Crighton said. These niceties seemed to include Moira and the girls and they soon resumed their furtive affair, although less furtive now. He was so different to Ralph that it hardly seemed like infidelity to her. ('Oh, what a beguiling argument!' Millie said.) She hardly saw Ralph now anyway and it seemed to be a mutual kind of waning.\n\nTeddy read the words on the Cenotaph. 'The _Glorious Dead_. Do you think they are? Glorious?' he asked.\n\n'Well, they're certainly dead,' Ursula said. 'But the \"glorious\" bit is to make _us_ feel better, I expect.'\n\n'I don't suppose the dead care about anything much,' Teddy said. 'I think when you're dead you're dead. I don't believe there's anything beyond, do you?'\n\n'I might have done before the war,' Ursula said, 'before I saw a lot of dead bodies. But they just look like so much rubbish, thrown away.' (She thought of Hugh saying, 'Just put me out with the dustbin.') 'It doesn't seem as though their souls have flown.'\n\n'I shall probably die for England,' Teddy said. 'And there's a chance you might too. Is it a good enough cause?'\n\n'I think so. Daddy said he would rather we were alive and cowards than dead and heroes. I don't think he meant it, it wasn't his style to shirk responsibility. What is it that it says on the war memorial in the village? _For your tomorrow we gave our today_. That's what your lot are doing, giving up everything, it doesn't seem right somehow.'\n\nUrsula thought that she would rather die for Fox Corner than 'England'. For meadow and copse and the stream that ran through the bluebell wood. Well, that _was_ England, wasn't it? The blessed plot.\n\n'I am a patriot,' she said. 'I surprise myself with it although I don't know why. What does it say on Edith Cavell's statue, the one by St Martin's church?'\n\n' _Patriotism is not enough_ ,' Teddy supplied.\n\n'Do you think that really?' she said. 'Personally, I think it's more than enough.' She laughed and they linked arms as they walked down Whitehall. There was quite a lot of bomb damage. Ursula pointed out the Cabinet War Rooms to Teddy. 'I know a girl who works in there,' she said. 'Sleeps in a cupboard, more or less. I don't like bunkers and cellars and basements.'\n\n'I worry about you a lot,' Teddy said.\n\n'I worry about _you_ ,' she said. 'And none of that worrying has done either of us any good.' She sounded like Miss Woolf.\n\nTeddy ('Pilot Officer Todd') had survived his time in an OTU in Lincolnshire, flying Whitleys, and in a week or so was due to join a Heavy Conversion Unit in Yorkshire and learn how to fly the new Halifaxes and start his first tour of duty proper.\n\nOnly half of all bomber crews survived their first tour of duty, the girl in the Air Ministry said.\n\n('Aren't the odds the same every time they go up?' Ursula said. 'Isn't that how odds work?'\n\n'Not in the case of bomber crews,' the girl from the Air Ministry said.)\n\nTeddy was walking her back to the office after lunch, she had taken a long hour. Things were not quite as hectic as they had been.\n\nThey had planned on somewhere swanky but ended up in a British Restaurant and dined on roast beef and plum pie and custard. The plums were tinned, of course. They enjoyed all of it though.\n\n'All those names,' Teddy said, gazing at the Cenotaph. 'All those lives. And now again. I think there is something wrong with the human race. It undermines everything one would like to believe in, don't you think?'\n\n'No point in thinking,' she said briskly, 'you just have to get on with life.' (She really was turning into Miss Woolf.) 'We only have one after all, we should try and do our best. We can never get it _right_ , but we must _try_.' (The transformation was complete.)\n\n'What if we had a chance to do it again and again,' Teddy said, 'until we finally did get it right? Wouldn't that be wonderful?'\n\n'I think it would be exhausting. I would quote Nietzsche to you but you would probably thump me.'\n\n'Probably,' he said amiably. 'He's a Nazi, isn't he?'\n\n'Not exactly. Do you still write poetry, Teddy?'\n\n'Can't find the words any more. Everything I try feels like sublimation. Making pretty images out of war. I can't find the heart of it.'\n\n'The dark, beating, bloody heart?'\n\n'Maybe you should write,' he laughed.\n\nShe wasn't going out on patrol while Teddy was here, Miss Woolf had taken her off the roster. The raids were more sporadic now. There had been bad raids in March and April and they seemed all the worse for their having had a bit of a breather from the bombs. 'It's funny,' Miss Woolf said, 'one's nerves are wired so tightly when it's relentless that it's almost easier to deal with.'\n\nThere had been a decided lull at Ursula's post. 'I think Hitler's more interested in the Balkans,' Miss Woolf said.\n\n'He's going to turn on Russia,' Crighton told her with some authority. Millie was on another ENSA tour and they had the Kensington flat to themselves.\n\n'But that would be madness.'\n\n'Well, the man _is_ insane, what do you expect?' He sighed and said, 'Let's not talk about the war.' They were drinking Admiralty whisky and playing cribbage, like an old married couple.\n\nTeddy walked her as far as Exhibition Road and her office and said, 'I imagined your \"War Room\" would be a rather grand affair \u2013 porticos and pillars \u2013 not a bunker.'\n\n'Maurice has the porticos.'\n\nAs soon as she was inside she was pounced upon by Ivy Jones, one of the teleprinter operators just coming on duty, who said, 'You're a dark horse, Miss Todd, keeping that gorgeous man a secret,' and Ursula thought, this is what comes of being too friendly with staff. 'Must dash,' she said, 'I'm a slave to the Daily Situation Report.'\n\n*\n\nHer own 'girls', Miss Fawcett and her ilk, filed and collated and sent the buff folders to her so she could formulate summaries, daily, weekly, hourly sometimes. Daily logs, damage logs, situation reports, it was never-ending. Then it all had to be typed up and put into more buff folders and be signed off by her before the folders went on their journey to someone else, someone like Maurice.\n\n'We're just cogs in a machine really, aren't we?' Miss Fawcett said to her and Ursula said, 'But remember, without the cog there is no machine.'\n\nTeddy took her out for a drink. It was a warm evening and the trees were full of blossom so that for a moment it felt as if the war was over.\n\nHe didn't want to talk about flying, didn't want to talk about the war, didn't even want to talk about Nancy. Where was she? Doing something she couldn't talk about, apparently. It seemed nobody wanted to talk about anything any more.\n\n'Well, let's talk about Dad,' he said, and so they did and it felt as if Hugh had finally been given the wake he deserved.\n\nTeddy caught the train to Fox Corner next morning, he was staying there for a few nights, and Ursula said, 'Will you take another evacuee with you?' and handed over Lucky. He was in the flat all day while she was at work but she often took him to the post if she was on duty and everyone treated him as a kind of mascot. Even Mr Bullock, who did not seem like a dog-lover, would come in with scraps and bones for him. There were times when the dog seemed to eat better than she did. Nonetheless, London in wartime was no place for a dog, she told Teddy. 'All the noise, it must be terribly alarming.'\n\n'I like this dog,' he said, rubbing the dog's head. 'He's a very straightforward kind of dog.'\n\nShe went to Marylebone to see them off. Teddy tucked the little dog under one arm and gave her a salute, sweet and ironic at the same time, and boarded the train. She felt almost as sad to see the dog go as she did Teddy.\n\nThey had been too optimistic. There was a terrible raid in May.\n\nTheir flat in Phillimore Gardens was hit. Neither Ursula nor Millie was there, thank goodness, but the roof and the upper storey were destroyed. Ursula simply moved back in and camped there for a while. The weather wasn't bad and in some peculiar way she quite enjoyed it. There was still water, although no electricity, and someone at work lent her an old tent so she slept under canvas. The last time she had done that was in Bavaria when she had accompanied the Brenner girls on their BDM summer expedition to the mountains and she had shared a tent with Klara, the eldest. They had grown very fond of each other but she hadn't heard from Klara since war was declared.\n\nCrighton was sanguine about her _al fresco_ arrangement, 'like sleeping on deck under the stars in the Indian Ocean'. She felt a pang of envy, she hadn't even been to Paris. The Munich\u2013Bologna\u2013Nancy axis had defined the edges of the unknown world for her. She and her friend Hilary \u2013 the girl who slept in a cupboard in the War Rooms \u2013 had planned a holiday cycling through France but war had put paid to it. Everyone was stuck on the little sceptred isle. If you thought about it too much you could start to feel quite claustrophobic.\n\nWhen Millie returned from her ENSA tour she declared that Ursula had gone quite mad and insisted they find somewhere else and so they moved to a shabby place in Lexham Gardens that she knew she would never learn to like. ('You and I could live together if you wanted,' Crighton said. 'A little flat in Knightsbridge?' She demurred.)\n\nThat wasn't the worst, of course. Their post received a direct hit in the same raid and both Herr Zimmerman and Mr Simms were killed.\n\nAt Herr Zimmerman's funeral a string quartet, all refugees, played Beethoven. Unlike Miss Woolf, Ursula thought that it would take more than the great composer's works to heal their wounds. 'I saw them play at the Wigmore Hall before the war,' Miss Woolf whispered. 'They're very good.'\n\nAfter the funeral Ursula went in search of Fred Smith at his fire station and they rented a room in a nasty little hotel near Paddington. Later, after the sex, which had the same compelling quality as before, they were rocked to sleep by the sound of trains coming and going and she thought, he must miss that sound.\n\nWhen they woke he said, 'I'm sorry I was a complete arse last time we were together.' He went and found them two mugs of tea \u2013 she supposed he had charmed someone in the hotel, it didn't seem like the kind of place to have a kitchen, let alone room service. He did have a natural charm, the same way that Teddy did, it came from a kind of straightness in their character. Jimmy's charm was different, more dishonest perhaps.\n\nThey sat up in bed and drank their tea and smoked cigarettes. She was thinking of Donne's poem, 'The Relic', one of her favourites \u2013 the _bracelet of bright hair about the bone \u2013_ but refrained from quoting, considering how badly it had gone down last time. How funny though it would be if the hotel were hit and no one understood who they were or what they were doing here together, conjoined in a bed that had become their grave. She had grown very morbid since Argyll Road. It had affected her in a different way to other incidents. What would she like on her headstone, she wondered idly? 'Ursula Beresford Todd, stalwart to the last'.\n\n'Do you know your problem, Miss Todd?' Fred Smith said, stubbing out his cigarette. He took hold of her hand and kissed her open palm and she thought, seize this moment because it's a sweet one and said, 'No, what's my problem?' and never did find out because the siren went off and he said, 'Fuck, fuck, fuck, I'm supposed to be on duty,' and threw his clothes on, gave her a hasty kiss and flew out of the room. She never saw him again.\n\nShe was reading through the Home Security War Diary for the awful early hours of 11 May \u2013\n\n_Time of Origin \u2013 0045. Form of Origin \u2013 Teleprinter. In or Out \u2013 In. Subject \u2013 South West India Dock Office, wrecked by H.E_. And Westminster Abbey, the Houses of Parliament, De Gaulle's headquarters, the Mint, the Law Courts. She had seen St Clement Dane's herself \u2013 blazing like a monstrous chimney fire on the Strand. And all the ordinary people living their precious ordinary lives in Bermondsey, Islington, Southwark. The list went on and on. She was interrupted by Miss Fawcett who said, 'Message for you, Miss Todd,' and handed her a piece of paper.\n\nA girl she knew who knew a girl in the fire service had sent her a copy of an AFS report, a little note added, 'He was a friend of yours, wasn't he? Sorry...'\n\n_Frederick Smith, fireman, crushed when a wall fell while attending a fire in Earl's Court_.\n\nBloody fool, Ursula thought. Bloody, bloody fool.\n\n# _November 1943_\n\nIt was Maurice who brought the news to her. His arrival coincided with that of the tea-trolley bearing elevenses. 'Can I have a word?' he said.\n\n'Do you want tea?' she said, getting up from her desk. 'I'm sure we can spare you some of ours, vastly inferior though it must be to the Orange Pekoe and Darjeeling and whatnot that you get in your place. And I can't imagine our biscuits can hold a candle to yours.' The tea-lady hovered, unimpressed by this exchange with an interloper from the airy regions.\n\n'No, no tea, thank you,' he said, surprisingly polite and subdued. It struck her that Maurice was nearly always simmering with suppressed fury (what a strange condition to live your life in), in some ways he reminded her of Hitler (she had heard that Maurice ranted at secretaries. 'Oh, that's so unfair!' Pamela said, 'but it does make me laugh').\n\nMaurice had never got his hands dirty. Never been to an incident, never pulled apart a man like a cracker or knelt on a matted bundle of fabric and flesh that had once been a baby.\n\nWhat was he doing here, was he going to start pontificating again about her love life? It never crossed her mind that he was here to say, 'I'm sorry to have to tell you this' (as if this were an official announcement) 'but Ted has caught one, I'm afraid.'\n\n'What?' She couldn't untangle the meaning. Caught what? 'I don't know what you mean, Maurice.'\n\n'Ted,' he said. 'Ted's plane has gone down.'\n\nTeddy had been safe. He was 'tour expired' and was instructing at an OTU. He was a squadron leader with a DFC (Ursula, Nancy and Sylvie had been to the Palace, bursting with pride). And then he had asked to go back on ops. ('I just felt I had to.') The girl she knew in the Air Ministry \u2013 Anne \u2013 told her that one in forty aircrew would survive a second tour of duty.\n\n'Ursula?' Maurice said. 'Do you understand what I'm saying to you? We've lost him.'\n\n'Then we'll find him.'\n\n'No. Officially he's \"missing in action\".'\n\n'Then he's not _dead_ ,' Ursula said. 'Where?'\n\n'Berlin, a couple of nights ago.'\n\n'He bailed out, and he's been taken captive,' Ursula said, as if stating a fact.\n\n'No, I'm afraid not,' Maurice said. 'He went down in flames, no one got out.'\n\n'How do you _know_ that?'\n\n'He was seen, an eyewitness, a fellow pilot.'\n\n'Who? Who was it who saw him?'\n\n'I don't know.' He was beginning to grow impatient.\n\n'No,' she said again. And then again, no. Her heart started racing and her mouth went dry. Her vision blurred and dotted, a pointillist painting. She was going to faint.\n\n'Are you all right?' she heard Maurice say. Am I all right, she thought, am I all right? How could I be all right?\n\nMaurice's voice sounded a long way off. She heard him shout for a girl. A chair was brought, a glass of water fetched. The girl said, 'Here, Miss Todd, put your head between your knees.' The girl was Miss Fawcett, a nice girl. 'Thank you, Miss Fawcett,' she murmured.\n\n'Mother took it very hard as well,' Maurice said, as if bemused by grief. He had never cared for Teddy the way they all did.\n\n'Well,' he said, patting her on the shoulder, she tried not to flinch, 'I'd better get back to the office, I expect I'll see you at Fox Corner,' almost casually, as if the worst part of the conversation were over and they could get on with some blander chat.\n\n'Why?'\n\n'Why what?'\n\nShe sat up straight. The water in the glass trembled slightly. 'Why will you see me at Fox Corner?' She sensed Miss Fawcett still hovering solicitously.\n\n'Well,' Maurice said, 'a family gathers on occasions such as these. After all, there won't be a funeral.'\n\n'There won't?'\n\n'No, of course not. No body,' he said. Did he shrug? Did he? She was shivering, she thought she might faint after all. She wished someone would hold her. Not Maurice. Miss Fawcett took the glass from her hand. Maurice said, 'I'll give you a lift down, of course. Mother sounded most awfully cut up,' he added.\n\nHe'd told her on the telephone? How dreadful, she thought numbly. It hardly mattered, she supposed, how one was given the news. And yet to have it conveyed by Maurice in his three-piece pinstripe, leaning against her desk, now inspecting his fingernails, waiting for her to say she was fine and he could go...\n\n'I'm fine. You can go.'\n\nMiss Fawcett brought her hot, sweet tea and said, 'I'm so sorry, Miss Todd. Would you like me to come home with you?'\n\n'That's very kind of you,' Ursula said, 'but I'll be all right. Do you think you could fetch my coat for me?'\n\nHe was twisting his uniform cap in his hands. They were making him nervous, just by their very presence. Roy Holt was drinking beer from a big dimpled-glass beer mug, great draughts with every mouthful as if he were very thirsty. He was Teddy's friend, the witness to his death. The 'fellow pilot'. Last time Ursula was here, visiting Teddy, was the summer of '42 and they had sat in the beer garden and eaten ham sandwiches and pickled eggs.\n\nRoy Holt was from Sheffield where the air still belonged to Yorkshire but was perhaps not so good. His mother and sister had been killed in the awful raids in December 1940 and he said he wasn't going to rest until he'd dropped a bomb directly on Hitler's head.\n\n'Good for you,' Izzie said. She had a peculiar way with young men, Ursula noticed, both maternal and flirtatious at the same time (where once she had simply been flirtatious). It was rather disturbing to watch.\n\nAs soon as she heard the news, Izzie left Cornwall post-haste for London and then commandeered a car and a fistful of petrol coupons from a 'man she knew' in the government, to take them both to Fox Corner, and then, onwards, to make the journey to Teddy's airfield. ('You'll never manage the train,' she said, 'you'll be far too upset.') 'Men she knew' was generally a euphemism for ex-lovers ('What did you do to get this?' a surly garage owner had asked when they filled up at his pumps on the road north. 'I slept with someone terribly important,' Izzie said sweetly).\n\nUrsula hadn't seen Izzie since Hugh's funeral, since her astonishing confession that she had a child, and Ursula thought that perhaps she should reintroduce the subject on the drive to Yorkshire (awkward to do) as Izzie had been so upset and presumably had no one else to talk to about it. But when Ursula said, 'Do you want to talk more about your baby?' Izzie said, 'Oh, _that_ ,' as if it was something trivial. 'Forget I ever said anything, I was just being morbid. Shall we stop for tea somewhere, I could demolish a scone, couldn't you?'\n\nYes, they had gathered at Fox Corner, and no, there was no 'body'. By then the status of Teddy and his crew had changed from 'missing in action' to 'missing, presumed dead'. There was no hope, Maurice said, they must stop thinking there was hope. 'There's always hope,' Sylvie said.\n\n'No,' Ursula said, 'sometimes there really isn't.' She thought of the baby. Emil. What would Teddy look like? Blackened and charred and shrunk like an ancient piece of wood? Maybe there was nothing left at all, no 'body'. Stop it, stop it, stop it. She breathed. Think of him as a little boy, playing with his planes and trains \u2013 no, actually that was worse. Much worse.\n\n'It's hardly a surprise,' Nancy said grimly. They were sitting outside on the terrace. They had drunk rather too much of Hugh's good malt. It felt peculiar to be drinking his whisky when he himself was gone. It was kept in a cut-glass decanter on the desk in the growlery, and it was the first time she had drunk it when it had not been poured by his own hand. ('Fancy a drop of the good stuff, little bear?')\n\n'He'd flown so many missions,' Nancy said, 'the odds were against him.'\n\n'I know.'\n\n'He expected it,' Nancy said. 'Accepted it, even. They have to, all those boys do. I sound sanguine, I know,' she continued quietly, 'but my heart is split in two. I loved him so much. _Love_ him so much. I don't know why I use the past tense. It's not as if love dies with the beloved. I love him _more_ now because I feel so damn sorry for him. He'll never marry, never have children, never have the wonderful life that was his birthright. Not all this,' she said, waving a hand around to indicate Fox Corner, the middle class, England in general, 'but because he was such a _good_ man. Sound and true, like a great bell, I think.' She laughed. 'Silly, I know. I know you're the one that understands. And I can't cry, I don't even want to cry. My tears would never do justice to this loss.'\n\nNancy hadn't wanted to talk, Teddy had once said, and now she wanted to do nothing but talk. Ursula herself had barely talked but wept continually. She had hardly gone an hour without finding the tears streaming unstoppably. Her eyes were still swollen and sore. Crighton had been awfully good, cradling her and shushing her, making endless cups of tea, tea purloined from the Admiralty, she supposed. He didn't deliver platitudes, didn't say everything will be all right, time will heal, he's in a better place \u2013 none of that rubbish. Miss Woolf was wonderful too. She came and sat with Crighton, never questioning who he might be, and held her hand and stroked her hair and allowed her to be an inconsolable child.\n\nThat was over now, she thought, finishing her whisky. Now there was just nothing. A vast, featureless landscape of nothing, as far as the horizon of her mind. _Despair behind, and Death before_.\n\n'Will you do something for me?' Nancy asked.\n\n'Yes, of course. Anything.'\n\n'Will you find out if there's a scrap of hope that he's alive? Surely there's a chance, however small, that he's been taken captive. I thought you might know someone in the Air Ministry\u2014'\n\n'Well, I know a girl...'\n\n'Or perhaps Maurice knows someone, someone who could be... definitive.' She stood up suddenly, swaying slightly from the whisky, and said, 'I have to go.'\n\n'We've met before,' Roy Holt said to her.\n\n'Yes, I came up to visit last year,' Ursula said. 'I stayed here, at the White Hart, they have rooms, but I suppose you know that. This is \"your\" pub, isn't it? The aircrew, I mean.'\n\n'We were all drinking in the bar, I remember,' Roy Holt said.\n\n'Yes, it was a very jolly evening.'\n\nMaurice was no use, of course, but Crighton had tried. It was always the same story. Teddy had gone down in flames, no one jumped.\n\n'You were the last person who saw him,' Ursula said.\n\n'I don't think about it really,' Roy Holt said. 'He was a good bloke, Ted, but it happens all the time. They don't come back. They're there at tea and they're not there at breakfast. You mourn for a minute and then you don't think about it. Do you know the statistics?'\n\n'I do actually.'\n\nHe shrugged and said, 'Maybe after the war, I don't know. I don't know what you want me to tell you.'\n\n'We just want to know,' Izzie said gently, 'that he didn't bail out. That he is dead. You were under attack, in extreme circumstances, you may not have seen the whole sorry drama play itself out.'\n\n'He's dead, believe me,' Roy Holt said. 'The whole crew. The plane was ablaze from front to back. Most of them were probably already dead. I could see him, the planes were very close, still in formation. He turned and looked at me.'\n\n'Looked at you?' Ursula said. Teddy in the last moments of his life, knowing he was going to die. What did he think about \u2013 the meadow and the copse and the stream that ran through the bluebell wood? Or the flames that were going to consume him \u2013 another martyr for England?\n\nIzzie reached out and clutched her hand. 'Steady,' she said.\n\n'I was only bothered about getting away from them. His kite was going out of control, I didn't want the bugger crashing into us.' He shrugged. He looked incredibly young and incredibly old at the same time.\n\n'You should get on with your lives,' he said rather roughly, and then less so added, 'I brought the dog. I thought you might want it back.'\n\nLucky was asleep at Ursula's feet, he had been deliriously happy when he saw her. Teddy hadn't left him at Fox Corner, instead he had taken him north, to his base. 'With a name and a reputation like his, what else could I do?' he wrote. He sent a photograph of his crew, lounging in old armchairs, Lucky sitting proudly to attention on Teddy's knee.\n\n'But he's your lucky mascot,' Ursula protested. 'Isn't that like asking for bad luck? Giving him away, I mean.'\n\n'We've had nothing but bad luck since Ted went,' Roy Holt said morosely. 'He was Ted's dog,' he added more kindly, 'faithful unto the last, as they say. He's pining something rotten, you should take him. The lads can't bear to see him hanging around on the airfield, waiting for Ted to come back. It just reminds them that it's probably going to be them next time.'\n\n' _I_ can't bear it,' she said to Izzie as they drove away. It was what Miss Woolf said when Tony died, she remembered. Just how much _was_ one expected to bear? The dog was sitting contentedly on her lap, sensing something of Ted about her perhaps. Or so she liked to think.\n\n'What else is there to do?' Izzie said.\n\nWell, one could kill oneself. And she might have done but how could she leave the dog behind? 'Is that ridiculous?' she asked Pamela.\n\n'No, not ridiculous,' Pamela said. 'The dog is all that's left of Teddy.'\n\n'Sometimes I feel that he _is_ Teddy.'\n\n'Now that _is_ ridiculous.'\n\nThey were sitting on the lawn at Fox Corner, two weeks or so after VE Day. ('Now begins the hard part,' Pamela said.) They hadn't celebrated. Sylvie had marked the day by taking an overdose of sleeping pills. 'Selfish, really,' Pamela said. 'After all, we're her children too.'\n\nShe had embraced the truth in her own inimitable way and lain down on Teddy's childhood bed and swallowed a whole bottle of pills, washed down with the last of Hugh's whisky. It was Jimmy's room too, but he hardly seemed to count to her. Now two of Pamela's boys slept in that room and played with Teddy's old train set, laid out in Mrs Glover's old attic room.\n\nThey lived at Fox Corner, the boys and Pamela and Harold. To everyone's surprise, Bridget made good on her threat to return to Ireland. Sylvie, enigmatic to the last, left behind her own version of a delayed action bomb. When her will was read they discovered that there was some money \u2013 stocks and shares and so on, Hugh wasn't a banker for nothing \u2013 that was to be divided equally but Pamela was to inherit Fox Corner. 'But why me?' Pamela puzzled. 'I was no more of a favourite than anyone else.'\n\n'None of us were favourites,' Ursula said, 'only Teddy. I suppose if he'd lived she would have left it to him.'\n\n'If he'd lived she wouldn't be dead.'\n\nMaurice was incandescent, Jimmy was not back from the war and when he did return he didn't seem to care too much one way or the other. Ursula wasn't entirely indifferent to the snub (a small word for a rather large betrayal) but she thought Pamela was the perfect person to live at Fox Corner and she was glad it was in her stewardship. Pamela wanted to sell and divide the proceeds but Harold, to Ursula's surprise, talked her out of it. (And it was difficult to talk Pamela out of things.) Harold had always disliked Maurice, for his politics as much as his person, and Ursula suspected this was his way of punishing Maurice for, well, for being Maurice. It was all rather Forsterian and it would have been easy to develop a grudge but Ursula chose not to.\n\nThe contents were to be divided among them. Jimmy wanted nothing, he already had his passage booked to New York and a job secured in an advertising agency, thanks to someone he met during the war, 'A man I know,' he said, an echo of Izzie. Maurice, on the other hand, having decided not to contest the will ('even though I would be successful, of course'), sent a removal van and virtually looted the house. None of the contents of the van ever turned up in Maurice's own house so they presumed he sold them, out of spite more than anything. Pamela cried for Sylvie's nice rugs and ornaments, the Regency Revival dining table, some very good Queen Anne chairs, the grandfather clock in the hall, 'Things we grew up with,' but it seemed to appease Maurice and prevented an outbreak of total war.\n\nUrsula took Sylvie's little carriage clock. 'I want nothing else,' she said. 'Only to be always welcome here.'\n\n'As you will be. You know that.'\n\n# _February 1947_\n\n_Wonderful! Like a Red Cross package_ , she wrote and propped the old postcard of the Brighton Pavilion on the mantelpiece next to Sylvie's clock, next to Teddy's photograph. She would put the card in with the afternoon post tomorrow. It would take for ever to reach Fox Corner, of course.\n\nA birthday card for her had made it through eventually. The weather had prevented the usual celebration at Fox Corner, instead Crighton had taken her to the Dorchester for dinner, by candlelight when the electricity gave out halfway through the meal.\n\n'Very romantic,' he said. 'Just like old times.'\n\n'I don't remember us being particularly romantic,' she said. Their affair had ended with the war but he had remembered her birthday, a fact which touched her more deeply than he knew. For a present he gave her a box of Milk Tray ('It's not much, I'm afraid').\n\n'Admiralty supplies?' she quizzed and they both laughed. When she got home she ate the whole box in one go.\n\nFive o'clock. She took her plate over to the sink to join the other unwashed dishes. The grey ash was a blizzard in the dark sky now and she pulled the flimsy cotton curtain to try to make it disappear. It tugged hopelessly on its wire and she gave up before she brought the whole thing down. The window was old and ill-fitting and let in a piercing draught.\n\nThe electricity went and she fumbled for the candle on the mantelpiece. Could it get any worse? Ursula took the candle and the whisky bottle to bed, climbed under the covers still in her coat. She was so tired. Being hungry and cold created the most awful lethargy.\n\nThe flame on the little Radiant fire quivered alarmingly. Would it be so very bad? _To cease upon the midnight with no pain_. There were worse ways. Auschwitz, Treblinka. Teddy's Halifax going down in flames. The only way to stop the tears was to keep drinking the whisky. Good old Pammy. The flame on the Radiant flickered and died. The pilot light too. She wondered when the gas would come back on. If the smell would wake her, if she would get up and relight it. She hadn't expected to die like a fox frozen in its den. Pammy would see the postcard, know that she'd been appreciated. Ursula closed her eyes. She felt as though she had been awake for a hundred years and more. She really was so very, very tired.\n\nDarkness began to fall.\n\nShe woke with a start. Was it daytime? The light was on but it was dark. She had been dreaming she was trapped in a cellar. She climbed out of the bed, she still felt quite drunk and realized it was the wireless that had woken her. The power was back on in time for the shipping forecast.\n\nShe fed the meter and the little Radiant popped back into life. She hadn't gassed herself after all then.\n\n# _June 1967_\n\nThis morning the Jordanians had opened fire on Tel Aviv, the BBC reporter said, now they were shelling Jerusalem. He was standing on a street, in Jerusalem presumably, she hadn't really been paying attention, the noise of artillery fire in the background, too far away to be any danger to him, yet his faux-battledress attire and style of reportage \u2013 excited, yet solemn \u2013 hinted at unlikely heroics on his part.\n\nBenjamin Cole was a member of the Israeli parliament now. He had fought in the Jewish Brigade at the end of the war and then joined the Stern Gang, in Palestine, to fight for a homeland. He had been such an upstanding kind of boy that it had been odd to think of him becoming a terrorist.\n\nThey had met up for a drink during the war but it was an awkward encounter. The romantic impulses of her girlhood had long since faded whereas his relative indifference to her as a member of the female sex had turned on its head. She had barely finished her (weak) gin and lemon when he suggested they 'go somewhere'.\n\nShe was indignant. 'Do I look like a woman of such easy virtue?' she asked Millie afterwards.\n\n'Well, why not?' Millie shrugged. 'We could be killed by a bomb tomorrow. _Carpe diem_ and all that.'\n\n'That seems to be everyone's excuse for bad behaviour,' Ursula grumbled. 'If people believed in eternal damnation they might not be seizing the day quite so much.' She had had a bad day at the office. One of the filing clerks had received the news that her boyfriend's ship had gone down and she had had hysterics and an important piece of paper had been lost in the sea of buff which caused more anguish, if of a different order, so she had not seized the day with Benjamin Cole, despite him pressing his suit urgently on her. 'I've always sensed something between us, haven't you?' he said.\n\n'Too late, I'm afraid,' she said, gathering up her bag and coat. 'Catch me next time round.' She thought about Dr Kellet and his theories of reincarnation and wondered what she would like to come back as. A tree, she thought. A fine big tree, dancing in the breeze.\n\nThe BBC turned its attention to Downing Street. Someone or other had resigned. She had heard tittle-tattle in the office but couldn't be bothered to listen.\n\nShe was eating her supper \u2013 a Welsh rarebit \u2013 off a tray on her knee. She usually ate like this in the evening. It seemed ridiculous to lay the table and put out vegetable dishes and table mats and all the other paraphernalia of dining for just one person. And then what? Eat in silence, or hunched over a book? There were people who saw TV dinners as the beginning of the end of civilization. (Did her robust defence of them indicate that perhaps she was of the same mind?) They obviously didn't live on their own. And really the beginning of the end of civilization had happened a long time ago. Sarajevo perhaps, Stalingrad at the latest. There were some who would say the end started at the beginning, in the Garden.\n\nAnd what was so wrong with watching television anyway? One couldn't go out to the theatre or the cinema (or the pub for that matter) every night. And when one lived alone one's only conversation inside the home was with a cat, which tended to be a one-sided affair. Dogs were different, but she hadn't had a dog since Lucky. He had died in the summer of '49, of old age, the vet said. Ursula had always thought of him as a young dog. They buried him at Fox Corner and Pamela bought a rose, a deep red, and planted it for his headstone. The garden at Fox Corner was a veritable graveyard for dogs. Wherever you went there would be a rose bush with a dog beneath, although only Pamela could remember who was where.\n\nAnd what was the alternative to television anyway? (She wasn't letting the argument die, even though it was with herself.) A jigsaw puzzle? Really? There was reading, of course, but one didn't always want to come in from a trying day at work, full of messages and memos and agendas, and then tire one's eyes out with even more words. The wireless, records, all good of course, but still _solipsistic_ in some way. (Yes, she was protesting too much.) At least with television one didn't have to _think_. Not such a bad thing.\n\nHer supper was later than usual because she had been attending her own retirement do \u2013 not unlike attending one's own funeral, except one could walk away afterwards. It had been a modest affair, no more than drinks at a local pub, but pleasant and she was relieved it had finished early (where others might feel badly done by). She didn't officially retire until Friday but she thought it would be easier on the staff to get the whole thing over and done with on a weekday. They might resent giving up their Friday evening.\n\nBeforehand, in the office, they had presented her with a carriage clock engraved _To Ursula Todd, in gratitude for her many years of loyal service_. Ye gods, she thought, what a tedious epitaph. It was a traditional kind of gift, and she didn't have the heart to say that she already had one, and a much better one at that. But they also gave her a pair of (good) tickets for the Proms, for a performance of Beethoven's Choral, which was thoughtful \u2013 she suspected the hand of Jacqueline Roberts, her secretary.\n\n'You've helped to pave the way for women in senior positions in the civil service,' Jacqueline said quietly to her, handing her a Dubonnet, her preferred drink these days. Not _that_ senior unfortunately, she thought. Not in _charge_. That was still for the Maurices of this world.\n\n'Well, cheers,' she said, chinking her glass against Jacqueline's port and lemon. She didn't drink a great deal, the occasional Dubonnet, a nice bottle of burgundy at the weekend. Not like Izzie, still inhabiting the house in Melbury Road, wandering through its many rooms like a dipsomaniac Miss Havisham. Ursula visited her every Saturday morning with a bag of groceries, most of which seemed to get thrown out. No one read the _Adventures of Augustus_ any more. Teddy would have been relieved and yet Ursula was sorry, as if another little part of him had been forgotten by the world.\n\n'You'll probably get a gong now, you know,' Maurice said, 'now that you're retired. An MBE or something.' He had been knighted in the last round of honours. ('God,' Pamela said, 'what's the country coming to?') He had sent each member of his family a framed photograph of himself, bowing beneath the Queen's sword in the ballroom of the Palace. 'Oh, the hubris of the man,' Harold laughed.\n\nMiss Woolf would have been the perfect companion for the Choral at the Albert Hall. The last time Ursula had seen her was there, at the Henry Wood seventy-fifth-birthday concert in '44. She was killed a few months later in the Aldwych rocket attack. Anne, the girl from the Air Ministry, was killed in the same attack. She had been with a group of female colleagues who were sunbathing on the ministry roof, eating their packed lunches. It was a long time ago now. And it was yesterday.\n\nUrsula was supposed to have met up with her in St James's Park at lunchtime. The Air Ministry girl \u2013 Anne \u2013 had something to tell her, she said, and Ursula had wondered if it might be some information about Teddy. Perhaps they had found wreckage or a body. She had long since accepted that he was gone for ever, they would have heard by now if he was a POW or had managed to escape to Sweden.\n\nAt the last minute fate had intervened in the shape of Mr Bullock, who had turned up unexpectedly on her doorstep the previous evening (how did he know her address?) to ask if she would accompany him to court to vouch for his good character. He was on trial for some kind of black market fraud, which came as no surprise. She was his second choice, after Miss Woolf, but Miss Woolf had been made a District Warden and was responsible for the lives of two hundred and fifty thousand people, all of whom ranked higher in her estimation than Mr Bullock. His black market 'escapades' had turned her against him in the end. None of the wardens that Ursula had known from her post were still there by '44.\n\nShe was rather alarmed to find that Mr Bullock was appearing at the Old Bailey, she had presumed it was some petty misdemeanour fit only for the magistrates' court. She had waited, in vain, all morning to be called and just as the court got up to recess for lunch she had heard the dull thud of an explosion but hadn't known it was the rocket wreaking carnage in the Aldwych. Mr Bullock, needless to say, was found innocent of all charges.\n\nCrighton had gone with her to Miss Woolf's funeral. He was a rock, but in the end he had stayed in Wargrave.\n\n' _Their bodies are buried in peace; but their name liveth for evermore_ ,' the minister boomed as if the congregation was hard of hearing. 'Ecclesiasticus 44: 14.' Ursula didn't think that was really true. Who would remember Emil or Renee? Or poor little Tony, Fred Smith. Miss Woolf herself. Ursula had forgotten the names of most of the dead already. And all those airmen, all those young lives lost. When Teddy died he was CO of his squadron and he was only twenty-nine. The youngest CO was twenty-two. Time had accelerated for those boys, as it had for Keats.\n\nThey sang 'Onward, Christian Soldiers', Crighton had a rather fine baritone that she had never heard before. She felt sure that Miss Woolf would have preferred Beethoven to the rousing battle hymns of the church.\n\nMiss Woolf had hoped that Beethoven might heal the post-war world but the howitzers pointed at Jerusalem seemed like the final defeat of her optimism. Ursula was now the same age as Miss Woolf had been at the outbreak of the last war. Ursula had thought of her as old. 'And now _we're_ old,' she said to Pamela.\n\n'Speak for yourself. And you're not even sixty yet. That's not old.'\n\n'Feels like it.'\n\nOnce her children were grown enough and no longer needed her constant oversight, Pamela had become one of those women who did good works. (Ursula was not critical, quite the opposite.) She became a JP and eventually a chief magistrate, was active on charity boards and last year had won a place on the local council as an independent. And there was the house to keep up (although she had 'a woman who does') and the enormous garden. In 1948, when the NHS was born, Harold had taken over Dr Fellowes's old practice. The village had grown around them, more and more houses. The meadow gone, the copse too, many of the fields from Ettringham Hall's home farm had been sold off to a developer. The Hall itself was empty and rather neglected. (There was talk of a hotel.) The little railway station had been given the death sentence by Beeching and had closed two months ago, despite an heroic campaign to keep it going, spearheaded by Pamela.\n\n'But it is still lovely around here,' she said. 'A five-minute walk and you're in open countryside. And the wood hasn't been touched. Yet.'\n\nSarah. She would take Sarah to the Proms with her. Pamela's reward for patience \u2013 a daughter born in 1949. She was to take up a place at Cambridge after the summer \u2013 science, she was clever, an all-rounder like her mother. Ursula was enormously fond of Sarah. Being an aunt had helped to seal over the empty cavern in her heart from Teddy's loss. She thought often these days \u2013 if only she had had a child of her own... She had had affairs over the years, albeit nothing too thrilling (the fault, the lack of 'commitment', mainly on her own side, of course) but she had never been pregnant, never been a mother or a wife and it was only when she realized that it was too late, that it could never be, that she understood what it was that she had lost. Pamela's life would go on after she was dead, her descendants spreading through the world like the waters of a delta, but when Ursula died she would simply end. A stream that ran dry.\n\nThere had been flowers too, also Jacqueline's doing, Ursula suspected. They had survived the evening in the pub, thank goodness. Lovely pink lilies that were now sitting on her sideboard, the scent perfuming the room. The living room was west-facing and soaked up the evening sun. It was still light outside, the trees in the shared gardens in their best new leaf. It was a very nice flat, near the Brompton Oratory, and she had put all of the money that Sylvie left her into the purchase of it. There was a small kitchen and bathroom, both modern, but she had eschewed the modern when it came to d\u00e9cor. After the war she had bought simple, tasteful antique furniture when no one wanted that kind of thing. There were fitted carpets throughout in a pale willow green and the curtains were the same fabric as the suite covers \u2013 a Morris print, one of the more subtle ones. The walls were painted in a pale-lemon emulsion that made the place seem light and airy even on rainy days. There were a few pieces of Meissen and Worcester \u2013 sweetmeat dishes and a garniture set \u2013 also picked up cheaply after the war, and she always had flowers, Jacqueline knew that.\n\nThe only crude note was sounded by a pair of Staffordshire foxes, garish orange creatures, each of which had a dead rabbit drooping in its jaws. She had picked them up in Portobello Road for next to nothing years ago. They had made her think of Fox Corner.\n\n'I love coming here,' Sarah said. 'You have such nice things and it's always so clean and tidy, nothing like home.'\n\n'You can afford to be clean and tidy when you live on your own,' Ursula said, but flattered by the compliment. She supposed she should make a will, leave her worldly goods to someone. She would like Sarah to have the flat but the memory of the debacle over Fox Corner when Sylvie died made her hesitate. Should one show such outright favouritism? Possibly not. She must divide her estate between all seven of her nieces and nephews, even the ones she didn't like or never saw. Jimmy, of course, had never married or had children. He lived in California now. 'He's a homosexual, you do know that, don't you?' Pamela said. 'He's always had those proclivities.' It was information, not censure, but there was still a mild prurience in her words and the faintest trace of smugness, as if she were better able to cope with liberal views. Ursula wondered if she knew about Gerald and _his_ 'proclivities'.\n\n'Jimmy's just Jimmy,' she said.\n\nThe previous week, she had come back from lunch and found a copy of _The Times_ sitting on her desk. It had been neatly folded so that only the obituaries were on show. Crighton's had a photograph of him in uniform, taken before she knew him. She had forgotten how handsome he was. It was quite a big piece, mentioned Jutland, of course. She learned that his wife Moira had 'predeceased' him, that he was a grandfather several times over and a keen golfer. He had always hated golf, she wondered when his conversion had taken place. And who on earth had left _The Times_ on her desk? Who all these years later would have thought to tell her? She had no idea and supposed she never would now. There was a time during their affair, when he had been in the habit of leaving notes on her desk, rather smutty little _billets-doux_ that appeared as if by magic. Perhaps the same invisible hand had delivered _The Times_ , all these years later.\n\n'The Man from the Admiralty is dead,' she said to Pamela. 'Of course, everyone dies eventually.'\n\n'Well, now there's a truism,' Pamela laughed.\n\n'No, I mean, everybody one has ever known, including oneself, will be dead one day.'\n\n'Still a truism.'\n\n' _Amor fati_ ,' Ursula said. 'Nietzsche wrote about it all the time. I didn't understand, I thought it was \"a more fatty\". Do you remember I used to see a psychiatrist? Dr Kellet? He was a philosopher at heart.'\n\n'Love of fate?'\n\n'It means acceptance. Whatever happens to you, embrace it, the good and the bad equally. Death is just one more thing to be embraced, I suppose.'\n\n'Sounds like Buddhism. Did I tell you that Chris is going to India, to some kind of monastery, a retreat, he calls it. He's found it hard to settle to anything since Oxford. He's a \"hippie\" apparently.' Ursula thought Pamela was very indulgent with her third son. She found Christopher rather creepy. She tried to think of another, more generous word but failed. He was one of those people who stared at you with a meaningful smile on their face, as if he was somehow intellectually and spiritually superior, when the fact was he was simply socially inept.\n\nThe scent of the lilies, lovely when they had first gone in water, was beginning to make her feel slightly sick. The room was stuffy. She should open a window. She stood up in order to carry her plate through to the kitchen and was immediately struck by a blinding pain in her right temple. She had to sit down again and wait for it to pass. She had been getting these pains for weeks now. An acute pain and then a thick, buzzy head. Or sometimes just a straightforward horrible pounding ache. She thought it might be high blood pressure but, after a battery of tests, the hospital's verdict was neuralgia, 'probably'. She was given strong painkillers and told that she was bound to feel better once she had retired. 'You'll have time to relax, take it easy,' the doctor said in the special tone of voice reserved for the elderly.\n\nThe pain passed and she stood up, gingerly.\n\nWhat _would_ she do with her time? She wondered about moving to the country, a little cottage, partaking in village life, perhaps somewhere in the vicinity of Pamela. She imagined St Mary Mead, or Miss Read's Fairacre. Perhaps _she_ could write a novel? It would certainly fill in the time. And a dog, time to get another dog. Pamela kept Golden Retrievers, a succession of them, one replacing another and quite indistinguishable to Ursula's eye.\n\nShe washed up her meagre pots. Thought she might have an early night, make some Ovaltine and take her book to bed with her. She was reading Greene's _The Comedians_. Perhaps she did need to rest more but lately she had become rather afraid of sleep. She was having such vivid dreams that sometimes she found it hard to accept that they weren't real. Several times recently she had believed that something outlandish had really happened to her when it quite obviously, logically, had not. And falling. She was always falling in her dreams, down staircases and off cliffs, it was a most unpleasant sensation. Was this the first sign of dementia? The beginning of the end. The end of the beginning.\n\nFrom her bedroom window she could see a fat moon rising. Keats's Queen-Moon, she thought. _Tender is the night_. The pain in her head came back. She ran a glass of water from the tap and swallowed a couple of painkillers.\n\n'But if Hitler had been killed, before he became Chancellor, it would have stopped all this conflict between the Arabs and the Israelis, wouldn't it?' The Six-Day War, as they had called it, had ended, the Israelis decisively victorious. 'I mean, I do understand why the Jews wanted to create an independent state and defend it vigorously,' Ursula continued, 'and I always felt sympathy for the Zionist cause, even before the war, but, on the other hand, I can also understand why the Arab states are so aggrieved. But if Hitler had been unable to implement the Holocaust\u2014'\n\n'Because he was dead?'\n\n'Yes, because he was dead. Then support for a Jewish homeland would have been weak at best...'\n\n'History is all about \"what ifs\",' Nigel said. Pamela's first-born, her favourite nephew, was a history tutor at Brasenose, Hugh's old college. She was treating him to lunch in Fortnum's.\n\n'It is nice to have an intelligent conversation with someone,' she said. 'I've been on holiday in the south of France with my friend Millie Shawcross, have you met her? No? Not that she's called that any more, she's been through several husbands, each one wealthier than the last.'\n\nMillie, the war bride, had hotfooted it back from America just as soon as she could, her new family were 'cowpokes', she reported. She had gone back to 'treading the boards' and had several disastrous relationships before she struck gold in the form of the scion of an oil family in tax exile.\n\n'She lives in Monaco. It's _incredibly_ small, I had no idea. She's really quite stupid these days. I'm wittering, aren't I?'\n\n'Not at all. Shall I pour you some water?'\n\n'People who live on their own do tend to witter. We live without restraint, verbal at any rate.'\n\nNigel smiled. He wore serious spectacles and had Harold's lovely smile. When he took his spectacles off to clean them on his napkin he looked very young.\n\n'You look so young,' Ursula said. 'You _are_ young, of course. Am I sounding like a dotty old aunt?'\n\n'God, no,' he said. 'You're just about the smartest person I know.'\n\nShe buttered a bread roll, feeling rather chuffed at this compliment. 'I heard someone say once that hindsight was a wonderful thing, that without it there would be no history.'\n\n'They're probably right.'\n\n'But think how different things would be,' Ursula persisted. 'The Iron Curtain would probably not have fallen and Russia wouldn't have been able to gobble up Eastern Europe.'\n\n'Gobble?'\n\n'Well, it _was_ just pure greed. And the Americans might not have recovered from the Depression so quickly without a war economy and consequently not exerted so much influence on the post-war world\u2014'\n\n'An awful lot of people would still be alive.'\n\n'Well, yes, obviously. And the whole cultural face of Europe would be different because of the Jews. And think of all those displaced people, shuffling from one country to another. And Britain would still have an empire, or at least we wouldn't have lost it so precipitately \u2013 I'm not saying being an imperial power is a good thing, of course. And we wouldn't have bankrupted ourselves and had such an awful time recovering, financially and psychologically. And no Common Market\u2014'\n\n'Which won't let us in anyway.'\n\n'Think how strong Europe would be! But perhaps Goering or Himmler would have stepped in. And everything would have happened in just the same way.'\n\n'Perhaps. But the Nazis were a marginal party almost up until they took power. They were all fanatical psychopaths, but none of them had Hitler's charisma.'\n\n'Oh, I know,' Ursula said. 'He was extraordinarily charismatic. People talk about charisma as if it were a good thing, but really it's a kind of glamour \u2013 in the old sense of the word, casting a spell, you know? I think it was the eyes, he had the _most_ compelling eyes. If you looked in them you felt you were putting yourself in danger of believing\u2014'\n\n'You _met_ him?' Nigel asked, astonished.\n\n'Well,' Ursula said. 'Not exactly. Would you like dessert, dear?'\n\nJuly and hot as Hades as she walked back from Fortnum's, along Piccadilly. Even the colours seemed hot. Everything was bright these days \u2013 bright young things. There were girls in her office whose skirts were like pelmets. Young people these days had so much _enthusiasm_ for themselves, as if they had invented the future. This was the generation the war had been fought for and now they bandied the word 'peace' around glibly as though it were an advertising slogan. They had not experienced a war ('And that's a good thing,' she heard Sylvie say, 'no matter how unsatisfactory they turn out'). They had been handed, in Churchill's phrase, the title deeds of freedom. What they did with them was their affair now, she supposed. (What an old fuddy-duddy she sounded, she had become the person she always thought she would never be.)\n\nShe thought she might walk through the parks and crossed the road, into Green Park. She always walked in the parks on Sundays but now she was retired every day was a Sunday, she supposed. She walked on, past the Palace, and entered Hyde Park, bought an ice-cream from a kiosk next to the Serpentine and decided she might hire a deckchair. She was awfully tired, lunch seemed to have taken it out of her.\n\nShe must have dozed off \u2013 all that food. The boats were out on the water, people pedalling, laughing and joking. Oh, drat, she thought, she could feel a headache coming and she didn't have any painkillers in her bag. Perhaps she could hail a cab on Carriage Drive, she would never be able to walk home in this heat, not in pain. But then the pain grew less rather than more severe, which was not the usual progression of her headaches. She closed her eyes again, the sun was still hot and bright. She felt wonderfully indolent.\n\nIt was odd to sleep surrounded by people. It should have made her feel vulnerable but instead there was a kind of comfort. What was Tennessee Williams's line \u2013 _the kindness of strangers_? Millie's swansong on the stage, the last gasp of the dying swan, was to play Blanche DuBois in a 1955 production in Bath.\n\nShe allowed the hum and buzz of the park to lullaby her. Life wasn't about becoming, was it? It was about being. Dr Kellet would have approved this thought. And everything was ephemeral, yet everything was eternal, she thought sleepily. A dog barked somewhere. A child cried. The child was hers, she could feel the delicate weight of the child in her arms. It was a lovely feeling. She was dreaming. She was in a meadow \u2013 flax and larkspur, buttercups, corn poppies, red campion and ox-eye daisies \u2013 and unseasonable snowdrops. The oddities of the dream world, she thought, and caught the sound of Sylvie's little carriage clock chiming midnight. Someone was singing, a child, a reedy little voice keeping the tune, _I had a little nut-tree and nothing would it bear. Muskatnuss_ , she thought \u2013 the German for nutmeg. She had been trying to remember that word for ages and now suddenly here it was.\n\nNow she was in a garden. She could hear the delicate chink of cups on saucers, the creak and clatter of a lawn-mower, and could smell the peppery-sweet perfume of pinks. A man lifted her up and tossed her in the air and sugar cubes scattered across a lawn. There was another world but it was this one. She allowed herself a little chuckle even though her opinion was that people who laughed to themselves in public were likely to be mad.\n\nDespite the summer heat, snow began to fall, which was the kind of thing that happened in dreams, after all. The snow began to cover her face, it was lovely and cool in this weather. And then she was falling, falling into the darkness, black and deep\u2014\n\nBut here was the snow again \u2013 white and welcoming, the light like a sharp sword piercing through the heavy curtains, and she was being lifted up, cradled in soft arms.\n\n'I shall call her Ursula,' Sylvie said. 'What do you think?'\n\n'I like it,' Hugh said. His face loomed into vision. His trim moustache and sideburns, his kind green eyes. 'Welcome, little bear,' he said.\n\n# The End of the Beginning\n\n#\n\n'Welcome, little bear.' Her father. She had his eyes.\n\nHugh had paced, as was tradition, along the Voysey runner in the upper hallway, barred from the inner sanctum itself. He was unsure of the details of the doings behind the door, only too grateful that he was not expected to be familiar with the mechanics of childbirth. Sylvie's screams suggested torture if not outright butchery. Women were extraordinarily brave, Hugh thought. He smoked a series of cigarettes to stave off any unmanly squeamishness.\n\nDr Fellowes's dispassionate bass notes afforded some comfort to him, counterpointed unfortunately by a kind of hysterical Celtic babble from the scullery maid. Where was Mrs Glover? A cook could sometimes be a great help at times like these. The cook in his childhood Hampstead home had been unflappable in a crisis.\n\nA considerable commotion could be heard at one point, indicating great victory or great defeat in the battle taking place on the other side of the bedroom door. Hugh refrained from entering unless invited, which he wasn't. Eventually, Dr Fellowes flung open the door of the birth chamber and announced, 'You have a bonny, bouncing baby girl. She nearly died,' he added as an afterthought.\n\nThank goodness, Hugh thought, that he had managed to get back to Fox Corner before the snow closed the roads. He had dragged his sister back with him on the Channel crossing, a cat after a long night on the tiles. He was sporting a rather painful bite mark on his hand and was left wondering from where his sister had acquired her strain of savagery. Not from Nanny Mills and the Hampstead nursery.\n\nIzzie was still wearing her counterfeit wedding ring, a legacy of her shameful week in a Parisian hotel with her lover, although Hugh doubted that the French, an immoral lot, cared about such niceties. She had left for the continent in short skirts and a little straw boater (his mother had given him a detailed description, as if Izzie were a criminal) but she returned in a gown by Worth (as she frequently told him, as if it would impress him). It was also clear that the scoundrel had been taking advantage of her for some time before their flight as the gown, Worth or not, was straining at the seams.\n\nHe had eventually flushed his fugitive sister out from H\u00f4tel d'Alsace in St Germain, a degenerate _endroit_ , in Hugh's estimation, the scene of Oscar Wilde's demise, which said everything you needed to know about the place.\n\nAn unseemly tussle had taken place not only with Izzie but also with the bounder from whose arms Hugh wrestled her before hauling her, kicking and screaming, into the handsome two-door Renault taxi that he had paid to wait outside the hotel. Hugh thought it would be rather fine to own a motor car. Could he afford one on his salary? Could he learn to drive one? How difficult could it be?\n\nThey had eaten some rather decent, pink French lamb on the boat and Izzie had demanded champagne, which he allowed her as he was far too worn out with the whole elopement business to bother with yet another fight. It was tempting to toss her over the rails, into the dark-grey waters of the Channel.\n\nHe had telegraphed his mother, Adelaide, from Calais, informing her of Izzie's misfortune as he thought it might be best if she were prepared before setting eyes on her youngest daughter, whose condition was plain for all the world to see.\n\nTheir fellow diners on the boat presumed they were a married couple and many pretty compliments on her impending motherhood were passed Izzie's way. Hugh supposed it was better to let them think this, appalling though it was, rather than for these complete strangers to discover the truth. Thus he found himself taking part in an absurd charade for the duration of the crossing, in the course of which he was forced to deny the existence of his real wife and children and pretend that Izzie was his child bride. He became, to all intents and purposes, the very villain who had seduced a girl barely out of the nursery (forgetting, perhaps, that his own wife was only seventeen when he proposed to her).\n\nIzzie, of course, threw herself into this mockery with glee, taking her revenge on Hugh by making him as uncomfortable as possible, addressing him as _mon cher mari_ and other extremely irritating blandishments.\n\n'What a lovely young wife you have,' a man, a Belgian, chortled while Hugh was taking the air on deck and indulging in a post-prandial cigarette. 'Hardly out of the cradle herself and soon to be a mother. It's the best way \u2013 getting them young \u2013 then you can mould them to how you want them.'\n\n'Your English is remarkable, sir,' Hugh said, throwing the stub of his cigarette into the sea and walking away. A lesser man would have resorted to fisticuffs. He might, if pressed, fight for the honour of his country, but he would be damned if he would fight for the besmirched honour of his feckless sister. (Although it would be undeniably pleasant to mould a woman to one's exact requirements, like his bespoke suits from his tailor in Jermyn Street.)\n\nIt had been difficult to find the right wording for the telegram to his mother and he had finally settled on I SHALL BE IN HAMPSTEAD BY MIDDAY STOP ISOBEL IS WITH ME STOP SHE IS WITH CHILD STOP. It was a rather bald message and he should perhaps have spent the extra money on some mitigating adverbs. 'Unfortunately' might have been one. The telegram (unfortunately) had the opposite to the desired effect and when they disembarked in Dover a reply was waiting for him. DO NOT BRING HER TO MY HOUSE UNDER ANY CIRCUMSTANCES STOP, the final STOP carrying a leaden weight of certainty which was not to be challenged. Which did rather leave Hugh at a loss as to what exactly he _should_ do with Izzie. She was, despite appearances, still only a child herself, only sixteen, he could hardly abandon her on the streets. Anxious to return to Fox Corner as soon as possible, he found himself carting her along with him.\n\nWhen they finally arrived, as iced as snowmen, it was an excitable Bridget who opened the door to him at midnight and said, 'Oh, no, I was hoping you were going to be the doctor, so I was.' His third child, it seemed, was on its way. _Her_ way, he thought fondly, looking down at the tiny crumpled features. Hugh rather liked babies.\n\n'But what are we to _do_ with her?' Sylvie fretted. 'She's not giving birth under my roof.'\n\n' _Our_ roof.'\n\n'She'll have to give it away.'\n\n'The child is part of our family,' Hugh said. 'The same blood runs in its veins as in my children.'\n\n' _Our_ children.'\n\n'We'll say the child is adopted,' Hugh said. 'An orphaned relative. People won't question, why should they?'\n\nIn the end the baby _was_ born beneath the roof of Fox Corner, a boy, and once Sylvie saw him she was unable to discard him so easily. 'He's a delightful little thing really,' she said. Sylvie found all babies delightful.\n\nIzzie had not been allowed beyond the garden for the remainder of her pregnancy. She was being kept a prisoner, she said, 'like the Count of Monte Cristo'. She handed the baby over as soon as he was born and showed no more interest in him, as if the whole affair \u2013 the pregnancy, the confinement \u2013 had been a provoking task that they had coerced her into undertaking and now she had fulfilled her part of the bargain and was free to go. After a fortnight of lying around in bed being waited upon by a disgruntled Bridget she was put on a train back to Hampstead, from whence she was packed off to a finishing school in Lausanne.\n\nHugh was right, no one questioned the sudden appearance of this surplus child. Mrs Glover and Bridget were sworn to secrecy, an oath that was sweetened, unknown to Sylvie, with cash. Hugh knew the value of money, he wasn't a banker for nothing. Dr Fellowes could, one hoped, be relied upon for his professional discretion.\n\n'Roland,' Sylvie said. 'I've always rather liked that name. _The Song of Roland \u2013_ he was a French knight.'\n\n'Died in battle, I expect?' Hugh said.\n\n'Most knights do, don't they?'\n\nThe silver hare spun and shone and shimmered before her eyes. The leaves on the beech danced, the garden budded, blossomed, fruited, without any help from her at all. _Rock-a-bye baby_ , Sylvie sang. _Down will fall baby, cradle and all_. Ursula was not put off by this threat and continued on her small but dauntless journey, alongside her companion, Roland.\n\nHe was a sweet-natured child and it took some time for Sylvie to notice that he was 'not quite all there', as she put it to Hugh one evening when he returned from a difficult day at the bank. He knew there was no point in sharing these fiscal problems with Sylvie, yet sometimes he liked to imagine coming home from work to a wife who was fascinated by ledgers and balance sheets, the rising price of tea, the unsteady market in wool. A wife 'moulded' to requirements instead of the beautiful, clever and somewhat contrary one he was wedded to.\n\nHe had secluded himself in the growlery, sitting at his desk with a large malt whisky and a small cigar, hoping to be left in peace. To no avail: Sylvie swept in and sat opposite him, like a customer in the bank looking for a loan, and said, 'I think Izzie's child may be a simpleton.' Up until now he had always been Roland, now, apparently defective, he was Izzie's once more.\n\nHugh dismissed her opinion but there was no denying that as time went on Roland didn't progress the way the others did. He was slow to learn and didn't seem to possess a child's natural curiosity about the world. You could sit him on a hearth rug with a rag-book or a set of wooden bricks and he would still be there half an hour later gazing contentedly at the fire (well guarded against children) or Queenie the cat sitting next to him, attending to her toilette (less well guarded and much prone to malevolence). Roland could be set to any simple task, and spent much of his time willingly fetching and carrying for the girls, Bridget, even Mrs Glover was not above sending him on simple errands, a bag of sugar from the pantry, a wooden spoon from the jar. It seemed unlikely that he would be going to Hugh's old school or entering Hugh's old college, and Hugh grew fonder of the boy for that somehow.\n\n'Perhaps we should get him a dog,' he suggested. 'A dog always brings the best out in a boy.' Bosun arrived, a large friendly animal with a tendency to herd and protect, and discerned immediately that he had been put in charge of something important.\n\nAt least the boy was placid, Hugh thought, unlike his dratted mother, or his own two eldest children who fought incessantly with each other. Ursula, of course, was different to all of them. She was watchful, as if she were trying to drink in the whole world through those little green eyes that were both his and hers. She was rather unnerving.\n\nMr Winton's easel was set up to face the sea. He was quite pleased with what he had so far, the blues and greens and whites \u2013 and murky browns \u2013 of the Cornish seaside. Several passers-by paused in their journeys across the sands to observe the painting-in-progress. He hoped, in vain, for compliments.\n\nA little fleet of white-sailed yachts skimmed the horizon, a race of some kind, Mr Winton presumed. He smudged some Chinese white on his own painted horizon and stood back to admire the results. Mr Winton saw yachts, others might have seen blobs of white paint. They would contrast rather well, he thought, with some figures on the seashore. The two little girls so intent on building a sandcastle would be perfect. He bit the tip of his brush as he gazed at his canvas. How to do it best, he wondered?\n\nThe sandcastle was Ursula's suggestion. They should build, she said to Pamela, the best sandcastle ever. She had conjured up such a vivid image of this sandy citadel \u2013 moats and turrets and battlements \u2013 that Pamela could almost see the medieval ladies in their wimples waving to the knights as they clattered away on their horses over the drawbridge (a piece of driftwood was to be sought out for this purpose). They had set about this task with undivided energy although they were still at the heavy-engineering stage, digging a double moat that would eventually, when the tide turned, fill with seawater to protect those wimpled ladies from violent siege (by someone like Maurice, inevitably). Roland, their ever-obliging minion, was dispatched to scour the beach for decorative pebbles and the all-important drawbridge.\n\nThey were further along the beach from Sylvie and Bridget, who were immersed in their books while the new baby, Edward \u2013 Teddy \u2013 was sleeping on a blanket on the sand beneath the protection of a parasol. Maurice was dredging in rock pools at the far end of the beach. He had made new companions, rough local boys with whom he went swimming and scrabbling up cliffs. Boys were just boys to Maurice. He had not yet learned to evaluate them by accent and social standing.\n\nMaurice had an indestructible quality and no one ever seemed to worry about him, least of all his mother.\n\nBosun, unfortunately, had been left behind with the Coles.\n\nIn time-honoured fashion, the sand from the moat was piled up in a central mound, the building material for the proposed fortress. Both girls, by now hot and sticky from their exertions, took a moment to stand back and contemplate this formless heap. Pamela felt more doubtful now about the turrets and battlements, the wimpled ladies seemed even more unlikely. The mound reminded Ursula of something, but what? Something familiar, yet nebulous and undefinable, no more than a shape in her brain. She was prone to these sensations, as if a memory was being tugged reluctantly out of its hiding place. She presumed it was the same for everyone.\n\nThen this feeling was replaced by fear, a shadow of a thrill too, the kind that came with a thunderstorm rolling in, or a sea fog creeping towards the shore. Hazard could be anywhere, in the clouds, the waves, the little yachts on the horizon, the man painting at his easel. She set off at a purposeful trot to take her fears to Sylvie and have them soothed.\n\nUrsula was a peculiar child, full of troublesome notions, in Sylvie's opinion. She was forever answering Ursula's anxious questions \u2013 _What would we do if the house caught fire? Our train crashed? The river flooded?_ Practical advice, Sylvie had discovered, was the best way to allay these fears rather than dismissing them as unlikely ( _Why, dear, we would gather up our belongings and we would climb on the roof until the water receded_ ).\n\nPamela, meanwhile, returned stoically to digging the moat. Mr Winton was entirely absorbed in the close brushwork necessary for Pamela's sunhat. What a happy coincidence that those two little girls had chosen to build their sandcastle in the middle of his composition. He thought he might call it _The Diggers_. Or _The Sand Diggers_.\n\nSylvie was dozing over _The Secret Agent_ and rather resented being woken. 'What is it?' she said. She glanced along the beach and saw Pamela digging industriously. Distant yelling and wild whooping suggested Maurice.\n\n'Where's Roland?' she asked.\n\n'Roland?' Ursula said, looking around for their willing slave and failing to see him anywhere. 'He's looking for a drawbridge.' Sylvie was on her feet now, anxiously scanning the beach.\n\n'A what?'\n\n'A drawbridge,' Ursula repeated.\n\nThey concluded that he must have spotted a piece of wood in the sea and obediently waded out to collect it. He had no real understanding of danger and did not know how to swim, of course. If Bosun had been on watch on the beach he would have dog-paddled out into the waves, heedless of any peril, and snatched Roland back. In his absence, _Archibald Winton, an amateur watercolourist from Birmingham_ , as the local paper referred to him, had attempted to rescue the child ( _Roland Todd, aged four, on holiday with his family_ ). He had cast aside his paintbrush and swum out to sea and pulled the boy from the water, _but, alas, to no avail_. This clipping was carefully cut out and preserved for appreciation in Birmingham. In the course of three column inches Mr Winton had become both a hero and an artist. He imagined himself saying modestly, 'Why, it was nothing,' and \u2013 of course \u2013 it _was_ nothing, for no one was saved.\n\nUrsula watched as Mr Winton waded back through the waves, carrying Roland's limp little body in his arms. Pamela and Ursula had thought the tide was going out but it was coming in, already filling the moat and lapping at the mound of sand which would soon be gone for ever. An ownerless hoop bowled past, driven by the breeze. Ursula stared out to sea while behind her on the beach a variety of strangers attempted to revive Roland. Pamela came and joined her and they held hands. The waves began to trickle in, covering their feet. If only they hadn't been so intent on the sandcastle, Ursula thought. And it had seemed such a _good_ idea.\n\n'Sorry about your boy, Mrs Todd, ma'am,' George Glover mumbled. He touched an invisible cap on his head. Sylvie had mounted an expedition to see the harvest being brought in. They must rouse themselves from their torpid grief, she said. Following Roland's drowning, the summer had been subdued, naturally. Roland seemed greater in his absence than he had done in his presence.\n\n' _Your_ boy?' Izzie muttered after they had left George Glover to his labours. She had arrived in time for Roland's funeral, in stylish black mourning, and wept, 'My boy, my boy,' over Roland's small coffin.\n\n'He was _my_ boy,' Sylvie said vehemently, 'don't you dare say he was yours,' although she knew, guiltily, that she mourned less for Roland than she would have done for one of her own. But that was natural, surely? Everyone seemed to want ownership of him now he was gone. (Mrs Glover and Bridget, too, would have staked a small claim to him as well if anyone had listened.)\n\nHugh was very affected by the loss of 'the little chap' but knew that for the sake of his family he must carry on as usual.\n\nIzzie had lingered on, to Sylvie's annoyance. She was twenty years old, 'stuck' at home, waiting for an unknown-as-yet husband to free her from Adelaide's 'claws'. Roland's name had been forbidden in Hampstead and now Adelaide declared his death a 'blessing'. Hugh felt sorry for his sister, while Sylvie spent her time casting around the countryside for an eligible landowner with enough mutton-headed patience to withstand Izzie.\n\nIn an oppressive heat they had trudged across fields, clambered over stiles, splashed through streams. Sylvie had strapped the baby to her body with a shawl. The baby was a heavy burden, although perhaps not as heavy a burden as the picnic basket that Bridget was lugging. Bosun walked dutifully by their side, he was not a dog that ran ahead, tending more to bring up the rear. He was still puzzled by Roland's disappearance and was keen not to lose anyone else. Izzie lagged behind, any original enthusiasm for the pastoral outing long since having waned. Bosun did his best to chivvy her along.\n\nIt was a bad-tempered trek, the picnic at the end of it not much better as it turned out that Bridget had forgotten to pack the sandwiches. 'How on earth did you manage that?' Sylvie said crossly and as a consequence they had to eat the pork pie that Mrs Glover had intended for George. ('For God's sake, don't tell her,' Sylvie said.) Pamela had scratched herself on a bramble bush, Ursula had tumbled into a nettle patch. Even the usually happy Teddy was overheated and fretful.\n\n*\n\nGeorge brought two tiny baby rabbits for them to look at and said, 'Would you like to take them home with you?' and Sylvie snapped, 'No thank you, George. They will either die or multiply, neither of which would be a happy outcome.' Pamela was distraught and had to be promised a kitten. (To Pamela's surprise, this promise was kept and a kitten duly acquired from the Hall farm. A week later it took a fit and died. A full funeral was held. 'I am cursed,' Pamela declared, with uncharacteristic melodrama.)\n\n'He's very handsome, that ploughman, isn't he?' Izzie said and Sylvie said, 'Don't. Not under any circumstances. Don't,' and Izzie said, 'I have no idea what you mean.'\n\nThe afternoon grew no cooler and eventually they had no choice but to wend their way home in the same heat that they had journeyed there in. Pamela, already miserable from the rabbits, stepped on a thorn, Ursula was whacked in the face by a branch. Teddy cried, Izzie swore, Sylvie breathed fire and Bridget said if it weren't a mortal sin she would drown herself in the next stream.\n\n'Look at you,' Hugh smiled in greeting when they staggered home. 'All golden from the sun.'\n\n'Oh, please,' Sylvie said, pushing past him. 'I'm going to lie down upstairs.'\n\n'I think we'll have thunder tonight,' Hugh said. And they did. Ursula, a light sleeper, was woken. She slipped out of bed and pattered over to the attic window, standing on a chair so that she could see out.\n\nThunder rolled like gunfire in the distance. The sky, purple and swollen with portent, was suddenly split open by a fork of lightning. A fox, skulking over some small prey on the lawn, was briefly illuminated, caught as though in a photographer's flash.\n\nUrsula forgot to count and an explosive thunderclap, almost overhead, took her by surprise.\n\nThis was how war sounded, she thought.\n\nUrsula cut straight to the chase. Bridget, chopping onions at the kitchen table, was already primed for tears. Ursula sat next to her and said, 'I've been in the village.'\n\n'Oh,' Bridget said, not in the least interested in this information.\n\n'I was buying sweets,' Ursula said. 'In the sweet shop.'\n\n'Really?' Bridget said. 'Sweets in a sweet shop? Who would have thought it.' The shop sold many things other than sweets but none of those other things were of any interest to the children at Fox Corner.\n\n'Clarence was there.'\n\n'Clarence?' Bridget said. She stopped the chopping at the mention of her beloved.\n\n'Buying sweets,' Ursula said. 'Mint humbugs,' she added, for authenticity, and then, 'You know Molly Lester?'\n\n'I do,' Bridget said cautiously, 'she works in the shop.'\n\n'Well, Clarence was kissing her.'\n\nBridget rose from her chair, knife still in hand. 'Kissing? Why would Clarence kiss Molly Lester?'\n\n'That's what Molly Lester said! She said, \"Why are you kissing me, Clarence Dodds, when everyone knows you're engaged to be married to that maid that works at Fox Corner?\" '\n\nBridget was used to melodramas and penny dreadfuls. She waited for the revelation that she knew must follow.\n\nUrsula supplied it. 'And Clarence said, \"Oh, you mean Bridget. She's nothing to me. She's a very ugly girl. I am just stringing her along.\" ' Ursula, a precocious reader by now, had also read Bridget's novels and had learned the discourse of romance.\n\nThe knife was dropped to the floor with a banshee shriek. Irish curses were thrown liberally. 'The bugger,' Bridget said.\n\n'A dastardly villain,' Ursula agreed.\n\nThe engagement ring, the little gypsy ring ('a trinket'), was returned by Bridget to Sylvie. Clarence's protestations of innocence went unheeded.\n\n'You might go up to London with Mrs Glover,' Sylvie said to Bridget. 'For the Armistice celebrations, you know. I believe there are late trains running.'\n\nMrs Glover said she wouldn't go near the capital on account of the influenza and Bridget said that she hoped very much that Clarence would go, preferably with Molly Lester, and that the pair of them would catch the Spanish flu and die.\n\nMolly Lester, who had never spoken so much as a word to Clarence beyond a guiltless 'Morning, sir, what can I get you?', attended a small street party in the village but Clarence did indeed go up to London with a couple of pals and did indeed die.\n\n'But at least no one was pushed down the stairs,' Ursula said.\n\n'Whatever do you mean?' Sylvie said.\n\n'I don't know,' Ursula said. She really didn't.\n\nShe was disturbed by herself. She dreamed of flying and falling all the time. Sometimes when she stood on a chair to look out of the bedroom window she felt the urge to clamber out and throw herself down. She would not fall to the ground with a thud and a smash like an overripe apple, instead she was sure she would be caught. (By what, though, she wondered?) She refrained from testing this theory, unlike Pamela's poor little crinoline lady, who had been tossed from the very same bedroom window by a malignly bored Maurice one winter teatime.\n\nOn hearing his approach along the passageway \u2013 loudly signalled by Indian war whoops \u2013 Ursula had hastily placed her own favourite, Queen Solange, the knitting doll, beneath her pillow where she remained safe in her refuge while the unfortunate crinoline lady was defenestrated and smashed to pieces on the slates. 'I only wanted to see what would happen,' Maurice whined to Sylvie afterwards. 'Well, now you know,' she said. She was finding Pamela's hysterical reaction to this incident more than a little trying. 'We are in the middle of a war,' she said to her. 'There are worse things happening than a broken ornament.' Not for Pamela there weren't.\n\nIf Ursula had allowed Maurice the little knitting doll, made of unbreakable wood, then the crinoline lady would have been saved.\n\nBosun, soon to be dead of distemper, nosed his way into the room that night and laid a weighty paw on Pamela's coverlet in sympathy before groaning into sleep on the rag rug between their beds.\n\nThe next day, Sylvie, reproaching herself for her heartlessness towards her children, acquired another kitten from the Hall farm. Kittens were in continual abundance on the farm, there was a kind of kitten currency in the neighbourhood, they were bartered for all kinds of emotional regret or fulfilment by parents \u2013 a doll lost, an exam passed.\n\nDespite Bosun's best attempts to keep a guardian eye on the kitten, they had only had it a week when Maurice stepped on it, during a vigorous game of soldiers with the Cole boys. Sylvie swiftly scooped up the little body and gave it to Bridget to take elsewhere so that its death throes could take place off-stage.\n\n'It was an accident!' Maurice screamed. 'I didn't know the stupid thing was there!' Sylvie slapped him on the face and he started crying. It was horrible to see him so upset, it really was an accident, and Ursula tried to comfort him which only made him furious and Pamela, of course, had moved beyond all notion of civilization and was trying to rip Maurice's hair from his head. The Cole boys had long since scarpered back to their own house where emotional calm was the general order of the day.\n\nSometimes it was harder to change the past than it was the future.\n\n'Headaches,' Sylvie said.\n\n'I'm a psychiatrist,' Dr Kellet said to Sylvie. 'Not a neurologist.'\n\n'And dreams and nightmares,' Sylvie tempted.\n\nThere was something comforting about being in this room, Ursula thought. The oak panelling, the roaring fire, the thick carpet figured in red and blue, the leather chairs, even the outlandish tea-urn \u2013 all felt familiar.\n\n'Dreams?' Dr Kellet said, duly tempted.\n\n'Yes,' Sylvie said. 'And sleepwalking.'\n\n'Do I?' Ursula asked, startled.\n\n'And she has a kind of _d\u00e9j\u00e0 vu_ all the time,' Sylvie said, pronouncing the words with some distaste.\n\n'Really?' Dr Kellet said, reaching for an elaborate meerschaum pipe and knocking the ashes out on to the fender. It was the Turk's-head bowl, as familiar somehow as an old pet.\n\n'Oh,' Ursula said. 'I've been here before!'\n\n'You see!' Sylvie said, triumphant.\n\n'Hm...' Dr Kellet said thoughtfully. He turned to Ursula and addressed her directly. 'Have you heard of reincarnation?'\n\n'Oh, yes, absolutely,' Ursula said enthusiastically.\n\n'I'm sure she hasn't,' Sylvie said. 'Is it Catholic doctrine? What _is_ that?' she asked, distracted by the outlandish tea-urn.\n\n'It's a samovar, from Russia,' Dr Kellet said, 'although I'm not Russian, far from it, I'm from Maidstone, I visited St Petersburg before the Revolution.' To Ursula, he said, 'Would you like to draw me something?' and pushed a pencil and paper towards her. 'Would you like some tea?' he asked Sylvie, who was still glaring at the samovar. She declined, mistrustful of any brew that didn't come out of a china teapot.\n\nUrsula finished her drawing and handed it over for appraisal.\n\n'What is it?' Sylvie said, peering over Ursula's shoulder. 'Some kind of ring, or circlet? A crown?'\n\n'No,' Dr Kellet said, 'it's a snake with its tail in its mouth.' He nodded approvingly and said to Sylvie, 'It's a symbol representing the circularity of the universe. Time is a construct, in reality everything flows, no past or present, only the now.'\n\n'How gnomic,' Sylvie said stiffly.\n\nDr Kellet steepled his hands and propped his chin on them. 'You know,' he said to Ursula, 'I think we shall get on very well. Would you like a biscuit?'\n\nThere was one thing that puzzled her. The photograph of _Guy, lost at Arras_ in his cricketing whites was missing from the side table. Without meaning to \u2013 it was a question that raised so many other questions \u2013 she said to Dr Kellet, 'Where is the photograph of Guy?' and Dr Kellet said, 'Who is Guy?'\n\nIt seemed even the instability of time was not to be relied upon.\n\n'It's just an Austin,' Izzie said. 'An open-road tourer \u2013 four doors though \u2013 but nowhere near as costly as a _Bentley_ , goodness, it's positively a vehicle for hoi polloi compared to _your_ indulgence, Hugh.' 'On tick, no doubt,' Hugh said. 'Not at all, paid up in full, _in cash_. I have a _publisher_ , I have _money_ , Hugh. You don't need to worry about me any more.'\n\nWhile everyone was admiring the cherry-bright vehicle, Millie said, 'I have to go, I have a dancing exhibition tonight. Thank you very much for a lovely tea, Mrs Todd.'\n\n'Come on, I'll walk you back,' Ursula said.\n\nOn the return home, she avoided the well-worn shortcut at the bottom of the garden and came the long way round, dodging Izzie speeding off in her car. Izzie gave a careless salute in farewell.\n\n'Who was that?' Benjamin Cole asked, skidding his bicycle into a hedge to avoid being killed by the Austin. Ursula's heart tripped and skipped and flipped at the sight of him. The very object of her affection! The reason she had taken the long way round was on the unlikely chance that she might engineer an 'accidental' meeting with Benjamin Cole. And here he was! What luck.\n\n'They lost my ball,' Teddy said disconsolately when she returned to the dining room.\n\n'I know,' Ursula said. 'We can look for it later.'\n\n'I say, you're all pink and flushed,' he said. 'Did something happen?'\n\nDid anything happen, she thought? Did anything _happen_? Only the most handsome boy in the entire world kissed me _and_ on my sixteenth birthday. He had walked her back, pushing his bicycle, and at some point their hands had brushed, they had blushed (it was poetry) and he said, 'You know I do like you, Ursula,' and then right there, at her front gate (where anyone could see), he had propped his bicycle against the wall and pulled her towards him. And then the kiss! Sweet and lingering and much nicer than she had expected although it did leave her feeling \u2013 well, yes... _flushed_. Benjamin too, and they stood apart from each other, slightly shocked.\n\n'Gosh,' he said. 'I've never kissed a girl before, I had no idea it could be so... exciting.' He shook his head like a dog as if astonished by his own lack of vocabulary.\n\nThis, Ursula thought, would remain the best moment of her life, no matter what else happened to her. They would have kissed more, she supposed, but at that moment the rag and bone cart appeared round the corner of the lane and the rag and bone man's almost incomprehensible siren moan of _Enraagnbooooooone_ intruded on their budding romance.\n\n'No, nothing happened,' she said to Teddy. 'I was saying goodbye to Izzie. You missed seeing her car. You would have liked it.'\n\nTeddy shrugged and pushed _The Adventures of Augustus_ off the table and on to the floor. 'What a load of rot it is,' he said.\n\nUrsula picked up a half-drunk glass of champagne, the rim of which was adorned with red lipstick, and poured half of it into a jelly glass that she handed to Teddy. 'Cheers,' she said. They chinked their glasses and drained them to the dregs.\n\n'Happy birthday,' Teddy said.\n\nWhat wondrous life is this I lead!\n\nRipe apples drop about my head;\n\nThe luscious clusters of the vine\n\nUpon my mouth do crush their wine...\n\n'What is that you're reading?' Sylvie asked suspiciously.\n\n'Marvell.'\n\nSylvie took the book from her and scrutinized the verses. 'It's rather lush,' she concluded.\n\n' \"Lush\" \u2013 how can that be a criticism?' Ursula laughed and bit into an apple.\n\n'Try not to be precocious,' Sylvie sighed. 'It's not a pleasant thing in a girl. What are you going to do when you go back to school after the holidays \u2013 Latin? Greek? Not English literature? I don't see the point.'\n\n'You don't see the point of English literature?'\n\n'I don't see the point of _studying_ it. Surely one just _reads_ it?' She sighed again. Neither of her daughters bore any resemblance to her. For a moment Sylvie was back in the past, under a bright London sky, and could smell the spring flowers newly refreshed by rain, hear the quiet comforting clink and jingle of Tiffin's tack.\n\n'I might do Modern Languages. I don't know. I'm not sure, I haven't quite worked out a plan.'\n\n'A plan?'\n\n*\n\nThey fell into silence. The fox sauntered into the silence, insouciant. Maurice was forever trying to shoot it. Either he was not such a good shot as he liked to think or the vixen was cleverer than he was. Ursula and Sylvie tended towards the latter view. 'She's so pretty,' Sylvie said. 'And she has such a magnificent brush.' The fox sat down, a dog waiting for its dinner, her eyes never leaving Sylvie. 'I haven't got anything,' Sylvie said, upturning her empty hands to prove this fact. Ursula bowled her apple core, gently underarm, so as not to alarm the creature and the vixen trotted off after it, picking it up awkwardly in her mouth and then turning tail and disappearing. 'Eats anything,' Sylvie said. 'Like Jimmy.'\n\nMaurice appeared, giving them both a start. He was carrying his new Purdey cocked over his arm and said eagerly, 'Was that that damned fox?'\n\n'Language, Maurice,' Sylvie reprimanded.\n\nHe was home after graduation, waiting to start his training in the law and irritatingly bored. He could work at the Hall farm, Sylvie suggested, they were always looking for seasonal workers. 'Like a peasant in the field?' Maurice said. 'Is that why you've given me an expensive education?' ('Why _have_ we given him an expensive education?' Hugh said.)\n\n'Teach me to shoot, then,' Ursula said, jumping up and brushing off her skirt. 'Come on, I can use Daddy's old wildfowler.'\n\nMaurice shrugged and said, 'May as well, but girls can't shoot, it's a well-known fact.'\n\n'Girls are absolutely useless,' Ursula agreed. 'They can't do _anything_.'\n\n'Are you being sarcastic?'\n\n'Me?'\n\n'Pretty good for a novice,' Maurice said reluctantly. They were shooting bottles off a wall, near the copse, Ursula hitting her target many more times than Maurice. 'You're sure you haven't done this before?'\n\n'What can I say?' she said. 'I pick things up quickly.'\n\nMaurice suddenly swung the barrel of his gun away from the wall and towards the edge of the copse and before Ursula could even see what he was aiming at he had pulled the trigger, blasting something out of existence.\n\n'Got the damned little blighter at last,' he said triumphantly.\n\nUrsula set off at a run but long before she reached it she could see the pile of ruddy-brown fur. The white tip of her beautiful brush gave a little flicker but Sylvie's fox was no more.\n\nShe found Sylvie on the terrace, leafing through a magazine. 'Maurice shot the fox,' she said. Sylvie rested her head back on the wicker lounger and closed her eyes in resignation. 'It was always going to happen,' she said. She opened her eyes. They were glistening with tears. Ursula had never seen her mother cry. 'I shall disinherit him one day,' Sylvie said, the idea of cold revenge already drying her tears.\n\nPamela appeared on the terrace and raised a questioning eyebrow at Ursula, who said, 'Maurice shot the fox.'\n\n'I hope you shot _him_ ,' Pamela said. She meant it too.\n\n'I might go and meet Daddy off the train,' Ursula said when Pamela had gone back inside.\n\nShe wasn't really going to meet Hugh. Ever since her birthday she had been seeing Benjamin Cole in secret. Ben, he was now to her. In the meadow, in the wood, in the lane. (Anywhere out of doors, it seemed. 'Good job the weather's been nice for your canoodling,' Millie said, with much clown-smirking and raising up and down of eyebrows.)\n\nUrsula discovered what an excellent liar she was. (Didn't she always know that, though?) _Do you want anything from the shop?_ or _I'm just going to pick raspberries in the lane_. Would it be so dreadful if people knew? 'Well, I think your mother would have me killed,' Ben said. ('A Jew?' she imagined Sylvie saying.)\n\n'And my folks, too,' he said. 'We're too young.'\n\n'Like Romeo and Juliet,' Ursula said. 'Star-crossed lovers and so on.'\n\n'Except we're not going to die for love,' Ben said.\n\n'Would it be such a bad thing to die for?' Ursula mused.\n\n'Yes.'\n\nThings had started to get very 'hot' between them, a lot of fumbling fingers and moaning (on his part). He didn't think he could 'hold back' much longer, he said, but she wasn't sure what he had to hold back from exactly. Didn't love mean they shouldn't hold back anything? She expected they would marry. Would she have to convert? Become a 'Jewess'?\n\nThey had made their way to the meadow where they had lain down in each other's arms. It was very romantic, Ursula thought, apart from the timothy grass that was tickling her and the ox-eye daisies that made her sneeze. Not to mention the way Ben suddenly shifted himself until he was on top of her so that she felt rather as if she were in a coffin filled with earth. He went into a kind of spasm that she thought might be a prelude to death by apoplexy and she stroked his hair as if he were an invalid and said, with concern, 'Are you all right?'\n\n'Sorry,' he said. 'Didn't mean to do that.' (But what had he done?)\n\n'I should be getting back,' Ursula said. They stood up and picked off grass and flowers from each other's clothing before walking home.\n\nUrsula wondered if she had missed Hugh's train. Ben looked at his watch and said, 'Oh, they'll have been home for ages.' (Hugh and Mr Cole travelled on the same London train.) They left the meadow and climbed over the stile into the dairy herd's field that ran alongside the lane. The cows hadn't returned from milking yet.\n\nHe gave her a hand down from the stile and they kissed again. When they broke free of each other they noticed a man making his way across the field, from the other side where it led into the copse. He was heading towards the lane \u2013 a shabby creature, a tramp perhaps \u2013 hobbling along as fast as he could. He glanced round and when he saw them he hobbled even faster. He stumbled on a tussock of grass but quickly recovered and was up again, loping towards the gate.\n\n'What a suspicious-looking fellow,' Ben laughed. 'I wonder what he's been up to?'\n\n'Dinner's on the table, you're very late,' Sylvie said. 'Where have you been? Mrs Glover has made that awful veal _\u00e0 la Russe_ thing again.'\n\n'Maurice shot the fox?' Teddy said, his face a picture of disappointment.\n\nAnd so it went on from there, a bad-tempered argument between everyone at the dinner table just because of a dead fox, Hugh thought. They're vermin, he felt like saying but didn't want to fuel the furore of emotions that had been unleashed. Instead, he said, 'Please, let's not talk about it over dinner, it's difficult enough trying to digest this stuff.' But talk about it they would. He tried to ignore them, ploughing his way through the veal cutlets (had Mrs Glover ever tasted them herself, he wondered?). He was relieved that they were interrupted by a knock at the door.\n\n'Ah, Major Shawcross,' Hugh said, 'do come in.'\n\n'Oh, goodness, I don't want to interrupt you at table,' Major Shawcross said, looking awkward, 'I just wondered if your Teddy had seen our Nancy.'\n\n'Nancy?' Teddy said.\n\n'Yes,' Major Shawcross said. 'We can't find her anywhere.'\n\nThey didn't meet any more in the copse, or the lane or the meadow. Hugh imposed a strict curfew after Nancy's body was found and anyway both Ursula and Ben were stricken with guilty horror. If they had come home when they were supposed to, if they had crossed that field even five minutes earlier instead of lingering, they might have saved her. But by the time they meandered ignorantly back Nancy was already dead, lying in the cattle trough in the top corner of the field. So, indeed, just like Romeo and Juliet it had ended in death. Nancy, sacrificed for their love.\n\n'It's a terrible thing,' Pamela said to her. 'But you're not responsible, why are you behaving as though you are?'\n\nBecause she was. She knew it now.\n\nSomething was riven, broken, a lightning fork cutting open a swollen sky.\n\nIn the October half-term she went to stay with Izzie for a few days. They were sitting in the Russian Tea Room in South Kensington. 'A terrifically right-wing clientele here,' Izzie said, 'but they do the most wonderful pancake things.' There was a samovar. (Was it the samovar that set her off, with its shades of Dr Kellet? It would seem absurd if it was.) They had finished their tea and Izzie said, 'Just hang on a sec, I'm going to powder my nose. Ask for the bill, will you?'\n\nUrsula was waiting patiently for her to return when suddenly the terror descended, swift as a predatory hawk. An anticipatory dread of something unknown but enormously threatening. It was coming for her, here among the polite tinkle of teaspoon on saucer. She stood up, knocking over her chair. She felt dizzy and there was a veil of fog in front of her face. Like bomb-dust, she thought, yet she had never been bombed.\n\nShe pushed through the veil, out of the Russian Tea Room on to Harrington Road. She started to run and kept on running, on to the Brompton Road and then, blindly, into Egerton Gardens.\n\nShe had been here before. She had never been here before.\n\nThere was always something just out of sight, just around a corner, something she could never chase down \u2013 something that was chasing _her_ down. She was both the hunter and the hunted. Like the fox. She carried on and then tripped on something, falling straight on to her nose. The pain was extraordinary. Blood everywhere. She sat on the pavement and cried with the agony of it all. She hadn't realized there was anyone on the street but then from behind her a man's voice said, 'Oh, my, how awful for you. Let me help you. You have blood all over your nice turquoise scarf. Is that the colour, or is it aquamarine? My name's Derek, Derek Oliphant.'\n\nShe knew that voice. She didn't know that voice. The past seemed to _leak_ into the present, as if there were a fault somewhere. Or was it the future spilling into the past? Either way it was nightmarish, as if her inner dark landscape had become manifest. The inside become the outside. Time was out of joint, that was for certain.\n\nShe staggered to her feet but didn't dare to look round. Ignoring the awful pain, she ran on and on. She was in Belgravia before she finally flagged completely. Here too, she thought. She had been here before. She had never been here before. I give in, she thought. Whatever it is, it can have me. She sank to her knees on the hard pavement and curled up in a ball. A fox without a hole.\n\nShe must have passed out because when she opened her eyes she was in a bed in a room painted white. There was a big window and outside the window she could see a horse-chestnut tree that had not yet shed its leaves. She turned her head and saw Dr Kellet.\n\n'You broke your nose,' Dr Kellet said. 'We thought you must have been attacked by someone.'\n\n'No,' she said. 'I fell.'\n\n'A vicar found you. He took you in a taxi to St George's Hospital.'\n\n'But what are _you_ doing here?'\n\n'Your father got in touch with me,' Dr Kellet said. 'He wasn't sure who else to contact.'\n\n'I don't understand.'\n\n'Well, when you arrived at St George's you wouldn't stop screaming. They thought something terrible must have happened to you.'\n\n'This isn't St George's, is it?'\n\n'No,' he said kindly. 'This is a private clinic. Rest, good food and so on. They have lovely gardens. I always think a lovely garden helps, don't you?'\n\n'Time isn't circular,' she said to Dr Kellet. 'It's like a... palimpsest.'\n\n'Oh dear,' he said. 'That sounds very vexing.'\n\n'And memories are sometimes in the future.'\n\n'You are an old soul,' he said. 'It can't be easy. But your life is still ahead of you. It must be lived.' He was not her doctor, he had retired, he said, he was 'merely a visitor'.\n\nThe sanatorium made her feel as if she had a mild case of consumption. She sat on the sunny terrace during the day and read countless books and orderlies ferried food and drink to her. She wandered through the gardens, had polite conversation with doctors and psychiatrists, talked to her fellow patients (on her floor, at any rate. The truly mad were in the attic, like Mrs Rochester). There were even fresh flowers in her room and a bowl of apples. It must be costing a fortune for her to stay here, she thought.\n\n'This must be very expensive,' she said to Hugh when he visited, which he did often.\n\n'Izzie is paying,' he said. 'She insisted.'\n\nDr Kellet lit his meerschaum thoughtfully. They were sitting on the terrace. Ursula thought she would be quite happy to spend the rest of her life here. It was so gloriously unchallenging.\n\n' _And though I have the gift of prophecy, and understand all mysteries, and all knowledge_...' Dr Kellet said.\n\n'... _and though I have all faith, so that I could remove mountains, and have not charity, I am nothing_ ,' Ursula provided.\n\n' _Caritas_ , of course, is love. But you will know that.'\n\n'I'm not without charity,' Ursula said. 'Why are we quoting Corinthians? I thought you were a Buddhist.'\n\n'Oh, I am nothing,' Dr Kellet said. 'And everything too, of course,' he added \u2013 rather elliptically, in Ursula's opinion.\n\n'The question is,' he said, 'do you have enough?'\n\n'Enough what?' The conversation had quite got away from her now but Dr Kellet was busy with the demands of the meerschaum and didn't answer. Tea interrupted them.\n\n'They do excellent chocolate cake here,' Dr Kellet said.\n\n'All better, little bear?' Hugh said as he helped her gently into the car. He had brought the Bentley to pick her up.\n\n'Yes,' she said. 'Absolutely.'\n\n'Good. Let's get home. The house isn't the same without you.'\n\nShe had wasted so much precious time but she had a plan now, she thought, as she lay awake in the dark, in her own bed at Fox Corner. The plan would involve snow, no doubt. The silver hare, the dancing green leaves. And so on. German, not the Classics, and afterwards a course in shorthand and typing and perhaps the study of Esperanto on the side, just in case utopia should come to pass. The membership of a local shooting club and an application for an office job somewhere, working for a while, salting money away \u2013 nothing untoward. She didn't want to draw attention to herself, she would heed her father's advice, although he hadn't given it to her yet, she would keep her head below the parapet and her light under a bushel. And then, when she was ready, she would have enough to live on while she embedded herself deep in the heart of the beast, from whence she would pluck out the black tumour that was growing there, larger every day.\n\nAnd then one day she would be walking down Amalienstrasse and pause outside Photo Hoffmann and gaze at the Kodaks and Leicas and Voigtl\u00e4nders in the windows and she would open the shop door and hear the little bell clanging to announce her arrival to the girl working behind the counter who will probably say _Guten Tag, gn\u00e4diges Fr\u00e4ulein_ , or perhaps she will say _Gr\u00fcss Gott_ because this is 1930 when people can still address you with _Gr\u00fcss Gott_ and _Tsch\u00fcss_ instead of endless _Heil Hitlers_ and absurd martial salutes.\n\nAnd Ursula will hold out her old box Brownie and say, 'I don't seem to be able to spool the film on,' and perky seventeen-year-old Eva Braun will say, 'Let me have a look for you.'\n\nHer heart swelled with the high holiness of it all. Imminence was all around. She was both warrior and shining spear. She was a sword glinting in the depths of night, a lance of light piercing the darkness. There would be no mistakes this time.\n\nWhen everyone was asleep and the house was quiet, Ursula got out of bed and climbed on the chair at the open window of the little attic bedroom.\n\nIt's time, she thought. A clock struck somewhere in sympathy. She thought of Teddy and Miss Woolf, of Roland and little Angela, of Nancy and Sylvie. She thought of Dr Kellet and Pindar. _Become such as you are, having learned what that is_. She knew what that was now. She was Ursula Beresford Todd and she was a witness.\n\nShe opened her arms to the black bat and they flew to each other, embracing in the air like long-lost souls. This is love, Ursula thought. And the practice of it makes it perfect.\n\n# Be Ye Men of Valour\n\n# _December 1930_\n\nUrsula knew all about Eva. She knew how much she liked fashion and make-up and gossip. She knew that she could skate and ski and loved to dance. And so Ursula lingered over the expensive frocks in Oberpollinger with her before visiting a caf\u00e9 for coffee and cake, or an ice-cream in the Englischer Garten where they would sit and watch the children on the carousel. She went to the skating rink with Eva and her sister Gretl. She was invited to dinner at the Brauns' table. 'Your English friend is very nice,' Frau Braun told Eva.\n\nShe told them that she was improving her German before she settled down to teach at home. Eva sighed with boredom at the idea.\n\nEva loved to be photographed and Ursula took many, many photographs of her on her box Brownie and then they spent their evenings sticking them in albums and admiring the different poses that Eva had struck. 'You should be in films,' Ursula told Eva and she was ridiculously flattered. Ursula had mugged up on celebrities, Hollywood and British as well as German, on the latest songs and dances. She was an older woman, interested in a fledgling. She took Eva under her wing and Eva was bowled over by her new sophisticated friend.\n\nUrsula knew, too, of Eva's infatuation for her 'older man' whom she made sheep's eyes at, whom she trailed around after, sitting in restaurants and caf\u00e9s, forgotten in a corner while he conducted endless conversations about politics. Eva started to take her along to these gatherings \u2013 Ursula was her best friend, after all. All Eva wanted was to be close to Hitler. And that was all Ursula wanted too.\n\nAnd Ursula knew about Berg and bunker. And really she was doing this frivolous girl a great favour by inserting herself in her life.\n\nAnd so, just as they had got used to Eva hanging around so they became accustomed to seeing her little English friend as well. Ursula was pleasant, she was a girl, she was nobody. She became so familiar that no one was surprised when she would turn up on her own and simper with admiration at the would-be great man. He took adoration casually. To have so little self-doubt, she thought, what a thing that must be.\n\nBut, ye gods, it was boring. So much hot air rising above the tables in Caf\u00e9 Heck or the Osteria Bavaria, like smoke from the ovens. It was difficult to believe from this perspective that Hitler was going to lay waste to the world in a few years' time.\n\nIt was colder than usual for this time of year. Last night a light dusting of snow, like the icing sugar on Mrs Glover's mince pies, had sifted over Munich. There was a big Christmas tree on the Marienplatz and the lovely smell of pine needles and roasting chestnuts everywhere. The festive finery made Munich seem more fairy-tale-like than England could ever hope to be.\n\nThe frosty air was invigorating and she walked towards the caf\u00e9 with a wonderful purpose in her step, looking forward to a cup of _Schokolade_ , hot and thick with cream.\n\nInside, the caf\u00e9 was smoky and rather disagreeable after the sparklingly cold outdoors. The women were in furs and Ursula rather wished that she could have brought Sylvie's mink with her. Her mother never wore it and it was left permanently mothballed in her wardrobe these days.\n\nHe was at a table at the far end of the room, surrounded by the usual disciples. They were an ugly lot, she thought, and laughed to herself.\n\n' _Ah. Unsere Englische Freundin_ ,' he said when he caught sight of her. ' _Guten Tag, gn\u00e4diges Fr\u00e4ulein_.' With the slightest flick of a finger he ousted a callow-looking acolyte from the chair opposite and she sat down. He seemed irritable.\n\n_Es schneit_ , she said. 'It's snowing.' He glanced out of the window as if he hadn't noticed the weather. He was eating _Palatschinken_. They looked good but when the waiter came bustling over she ordered _Schwarzw\u00e4lder Kirschtorte_ to eat with her hot chocolate. It was delicious.\n\n' _Entschuldigung_ ,' she murmured, reaching down into her bag and delving for a handkerchief. Lace corners, monogrammed with Ursula's initials, 'UBT', Ursula Beresford Todd, a birthday present from Pammy. She dabbed politely at the crumbs on her lips and then bent down again to put the handkerchief back in her bag and retrieve the weighty object nesting there. Her father's old service revolver from the Great War, a Webley Mark V. She made fast her heroine heart. ' _Wacht auf_ ,' Ursula said quietly. The words attracted the F\u00fchrer's attention and she said, ' _Es nahet gen dem Tag_.'\n\nA move rehearsed a hundred times. One shot. Swiftness was all, yet there was a moment, a bubble suspended in time after she had drawn the gun and levelled it at his heart when everything seemed to stop.\n\n' _F\u00fchrer_ ,' she said, breaking the spell. 'F\u00fcr Sie.'\n\nAround the table guns were jerked from holsters and pointed at her. One breath. One shot.\n\nUrsula pulled the trigger.\n\nDarkness fell.\n\n# Snow\n\n# _11 February 1910_\n\n_Rap, rap, rap_. The knocking on Bridget's bedroom door wove itself into a dream that she was having. In the dream she was at home in County Kilkenny and the pounding on the door was the ghost of her poor dead father, trying to get back to his family. _Rap, rap, rap!_ She woke with tears in her eyes. _Rap, rap, rap_. There really was someone at the door.\n\n'Bridget, Bridget?' Mrs Todd's urgent whisper on the other side of the door. Bridget crossed herself, no news in the dark of the night was ever good. Had Mr Todd had an accident in Paris? Or Maurice or Pamela taken ill? She scrambled out of bed and into the freezing cold of the little attic room. She smelt snow in the air. Opening the bedroom door she found Sylvie bent almost double, as ripe as a seed-pod about to burst. 'The baby's coming early,' she said. 'Can you help me?'\n\n'Me?' Bridget squeaked. Bridget was only fourteen but she knew a lot about babies, not much of it good. She had watched her own mother die in childbirth but she had never told this to Mrs Todd. Now clearly wasn't the time to mention it. She helped Sylvie back down the stairs to her own room.\n\n'There's no point in trying to get a message to Dr Fellowes,' Sylvie said. 'He'll never get through this snow.'\n\n'Mary, Mother of God,' Bridget yelped as Sylvie dropped on all fours, like an animal, and grunted.\n\n'The baby's coming now, I'm afraid,' Sylvie said. 'It's time.'\n\nBridget persuaded her back into bed and their long, lonely night's labour commenced.\n\n*\n\n'Oh, ma'am,' Bridget cried suddenly, 'she's all blue, so she is.'\n\n'A girl?'\n\n'The cord's wrapped around her neck. Oh, Jesus Christ and all the saints, she's been strangled, the poor wee thing, strangled by the cord.'\n\n'We must do something, Bridget. What can we do?'\n\n'Oh, Mrs Todd, ma'am, she's gone. Dead before she had a chance to live.'\n\n'No, that cannot be,' Sylvie said. She heaved herself into a sitting position on the battlefield of bloodied sheets, red and white, the baby still attached by its lifeline. While Bridget made mournful noises, Sylvie jerked open the drawer of her bedside table and rummaged furiously through its contents.\n\n'Oh, Mrs Todd,' Bridget wailed, 'lie down, there's nothing to be done. I wish Mr Todd was here, so I do.'\n\n'Shush,' Sylvie said and held aloft her trophy \u2013 a pair of surgical scissors that gleamed in the lamplight. 'One must be prepared,' she muttered. 'Hold the baby close to the lamp so I can see. Quickly, Bridget. There's no time to waste.'\n\nSnip, snip.\n\nPractice makes perfect.\n\n# The Broad Sunlit Uplands\n\n# _May 1945_\n\nThey were at a table in the corner of a pub on Glasshouse Street. They'd been dropped in Piccadilly by the American army sergeant who'd given them a lift when he saw them hitch-hiking at the side of the road outside Dover. They had crushed themselves on to an American troop transport ship at Le Havre instead of waiting two days for a flight. It was possible that, technically, they were AWOL, but neither of them gave a damn.\n\nThis was their third pub since Piccadilly and they were both agreed that the two of them were very drunk but had the capacity to get a good deal drunker yet. It was a Saturday night and the place was packed. Being in uniform they hadn't paid for a single drink all night. The relief, if not the euphoria, of victory was still in the air.\n\n'Well,' Vic said, raising his glass, 'here's to being back.'\n\n'Cheers,' Teddy said. 'Here's to the future.'\n\nHe had been shot down in November '43 and taken to Stalag Luft VI in the east. It hadn't been bad, in that it could have been worse, he could have been Russian \u2013 the Russians were treated like animals. But then at the beginning of February they were roused from their bunks with a familiar ' _Raus! Raus!_ ' in the middle of the night and made to set out on the march west, away from the advancing Russians. Another day or two and they would have been liberated, it seemed an especially cruel twist of fate. There followed weeks of marching on starvation rations, in the freezing cold, minus twenty degrees most of the time.\n\nVic was a rather cocky little flight sergeant, the navigator of a Lancaster shot down over the Ruhr. War made strange bedfellows. They had kept each other going on the march. It was a comradeship that had almost certainly saved their lives, that and the very occasional Red Cross package.\n\nTeddy had been shot down near Berlin, only managing to escape from the cockpit at the last minute. He'd been trying to keep the plane level to give his crew a fighting chance to bail out. A captain didn't leave his ship until everyone on board had left. The same unspoken rule applied to a bomber.\n\nThe Halifax had been on fire from end to end and he had accepted that it was over for him. He had begun to feel lighter somehow, his heart swelling, and he suddenly knew that he would be all right, that death when it came would look after him. But death didn't come because his Aussie wireless operator crawled to the cockpit and clipped Teddy's parachute on his back and said, 'Get out, you stupid bastard.' He never saw him again, never saw any of his crew, didn't know if they were alive or dead. He jumped at the last minute, his parachute had barely opened when he hit the ground and he was lucky to fracture only an ankle and a wrist. He was taken to a hospital and the local Gestapo came and arrested him on the ward with the immortal words, 'For you the war is over,' which was the greeting that nearly every airman had heard when he was taken prisoner.\n\nHe had dutifully filled in his Capture card and waited for a letter from home, but nothing came. He was left wondering for two years if the Red Cross had him on their list of prisoners, if anyone at home knew he was alive.\n\nThey were on the road somewhere outside Hamburg when the war ended. Vic had taken great pleasure in saying to the guards, _'Ach so, mein Freund, f\u00fcr sie der Krieg ist zu ende_.'\n\n'So, Ted, did you get through to your girl?' Vic asked when Teddy came back from sweet-talking the landlady behind the bar into letting him use the pub phone.\n\n'I did,' he laughed. 'I'd been given up for dead, apparently. I don't think she believed it was me.'\n\nHalf an hour and another couple of drinks later, Vic said, 'Aye up, Ted. By the smile on her face I would say that woman who just came through the door might belong to you.'\n\n'Nancy,' Teddy said quietly to himself.\n\n'I love you,' Nancy mouthed silently to him across the din.\n\n'Oh, and she brought a little friend along for me, how thoughtful,' Vic said and Teddy laughed and said, 'Watch it, that's my sister you're talking about.'\n\nNancy was clutching her hand so hard that it hurt but the pain meant nothing. He was there, he was actually there, sitting at a table in a London pub, drinking a pint of English beer, as large as life. Nancy made a funny choking sound and Ursula stopped herself from crying out. They were like the two Marys, dumb in the face of the Resurrection.\n\nThen Teddy spotted them and a grin split his face. He jumped up, almost knocking over the glasses on the table. Nancy pushed her way through the crowd and threw her arms around him but Ursula stayed where she was, worried suddenly that if she moved it would all disappear, the whole happy scene break into pieces before her eyes. But then she thought, no, this was real, this was true, and she laughed with uncomplicated joy as Teddy let go of Nancy long enough to stand to attention and give Ursula a smart salute.\n\nHe shouted something to her across the pub but his words were lost in the hubbub. She thought it was 'Thank you,' but she might have been wrong.\n\n# Snow\n\n# _11 February 1910_\n\nMrs Haddock sipped a glass of hot rum, in what she hoped was a ladylike way. It was her third and she was beginning to glow from the inside out. She had been on her way to help deliver a baby when the snow had forced her to take refuge in the snug of the Blue Lion outside Chalfont St Peter. It was not a place she would have ever considered entering, except out of necessity, but there was a roaring fire in the snug and the company was proving surprisingly convivial. Horse brasses and copper jugs gleamed and twinkled. The public bar, where the drink seemed to flow particularly freely, was an altogether rowdier place. A sing-song was currently in progress there and Mrs Haddock was surprised to find her toe tapping in accompaniment.\n\n'You should see the snow,' the landlord said, leaning across the great polished depth of the brass bar counter. 'We could all be stuck here for days.'\n\n'Days?'\n\n'You may as well have another tot of rum. You won't be going anywhere in a hurry tonight.'\n\n# _Acknowledgements_\n\nI would like to thank the following:\n\nAndrew Janes (the National Archives in Kew)\n\nDr Juliet Gardiner\n\nLt Col M Keech BEM R Signals\n\nDr Pertti Ahonen (Department of History, Edinburgh University)\n\nFrederike Arnold\n\nAnnette Weber\n\nAnd also my agent, Peter Straus, and Larry Finlay, Marianne Velmans, Alison Barrow and everyone at Transworld Publishers, as well as Camilla Ferrier and everyone at the Marsh Agency.\n\nTo know more about the writing of this book (including a bibliography), please visit my website, www.kateatkinson.co.uk.\n**Kate Atkinson** won the Whitbread (now Costa) Book of the Year prize with her first novel, _Behind the Scenes at the Museum_ , and has been a critically acclaimed international author ever since. Her four most recent bestsellers featured the former private detective Jackson Brodie: _Case Histories, One Good Turn, When Will There Be Good News?_ and _Started Early, Took My Dog_. She was appointed MBE in the 2011 Queen's Birthday Honours List.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n_Dedicated to Tony and Elizabeth Duquette\u2014who made this all possible._\n\n## Foreword\n\n_Magic_ was the highest word of praise that Tony Duquette could bestow upon object, experience, or person. He was a design alchemist who could take base materials\u2014fish bones, snail shells, tree bark\u2014and transform them into objects of wonder and delight. \"I want everything to have spontaneity, to move and laugh,\" he once said, and as a young artist, his exceptional gifts soon saw him taken up by the chic Hollywood decorators Billy Haines and Jimmy Pendleton. Erelong, Duquette was the prot\u00e9g\u00e9 of the even more celebrated Lady Mendl, whose career as the actress Elsie de Wolfe was distinguished only by the \u00e9lan with which she wore the latest fashions on stage. She found her true calling, however, as the blue-rinsed doyenne of interior design taste. Mendl encouraged Duquette to create a \"moible\" for her and the resulting _meuble_ \u2014a grand _lacca povera_ secretary filled, as _Vogue_ noted at the time, with his \"jeweled fetishes\"\u2014became a centerpiece of After All, Mendl's trendsetting Los Angeles home, and a calling card for his talents.\n\nThe great Gilbert Adrian was one of those who came calling. Adrian, as he was known, honed his art as the fashion director of Metro-Goldwyn-Mayer, shaping the on-screen style of Garbo, Crawford, Lombard, Shearer, and Hepburn. In the early 1940s, with America cut off from Paris fashion news during the Occupation, Adrian astutely began to bring his talents to off-screen fashion, selling his designs via the most exclusive department stores across the country\u2014and his own high-style salon\u2014to a legion of deep-pocketed customers who weren't afraid to make an entrance. \"Amaze me!\" was Adrian's injunction to Duquette, and the young artist obliged with decors for the designer's salon and fashion shows of eclipsing fantasy and charm. Duquette was made for Tinseltown, the land of illusion, and for the director Vincente Minelli he created movie decors as fanciful as the homes that he invented for himself and his wife, the ethereal artist Elizabeth Johnstone.\n\nDuquette was fearless. He built Dawnridge in a cleft in the hills above the rolling, palmy lawns of Beverly Hills and created a universe of his own where Venice met Shangri-La. I can never forget walking into what seemed to be a candlelit anteroom to a palace here. It was elaborately paved in faceted antique mirror, which on closer inspection\u2014and if truth be told, really only when it was pointed out to me\u2014was in fact made from the pressed aluminum foil dishes in which the Chinese take-out orders were delivered, and did not lead to the promised imperial splendors but to a useful tool shed instead. With a wave of Duquette's wand, a collection of industrial lamps, stacked one above the other, was thus miraculously transformed into a sacred object that might be found dangling from a temple ceiling in Lost Horizons, and served to illumine the lights of elephant-high obelisks crusted with abalone shells. Next to these wonders, old-fashioned trash cans were mounted one above the other and pierced with gothic openings until they resembled ancient Cairene minarets. Through tricks of scale and trompe l'oeil and false perspectives, a distant pavilion across a broad lake was, in reality, little more than a set-builder's facade artfully set into a shallow ledge in the ravine on the far side of a modest _basin_ of water. Inside, Duquette's eye-tricking skills with a paintbrush created convincing simulacrums of malachite and lapis and porphyry, and of jaguar spots and ermine tails and angel wings, when there was nothing but cloth and plaster and wood and canvas and pipe cleaners and a great, roiling ocean of imagination.\n\nHutton Wilkinson, Duquette's pupil, prot\u00e9g\u00e9, and collaborator since the early seventies, and now his spiritual heir, worked with the master for clients including Dodie Rosekrans, for whom they evoked a maharajah's throne room in a prim apartment on Paris's rive gauche, and transformed the _piano nobile_ of the storied Palazzo Brandolini on a majestic turn of the Grand Canal in Venice. Here, the Duquette-Wilkinson interventions included a bedroom like a mermaid's grotto with walls of iridescent net layered over Mary Pickford's Chinese wallpaper. The curves and volutes of elaborate, impasto eighteenth-century plasterwork framing the mirrors in the immense reception gallery, meanwhile, proved insufficiently whimsical to Duquette's purpose, and so he and Wilkinson spent laborious hours embellishing it with a thousand tendrils of deep-sea coral\u2014not, as it happens, plucked from the sea but instead carved in Thailand from rattan.\n\nHutton Wilkinson, the acolyte turned master, who learned the craft at Duquette's side through the decades, has subtly transformed the original structures of Dawnridge into a sort of shrine to the Duquettes and their works that Wilkinson has been gifted or assiduously sought out through the years\u2014sleuthing the auction rooms and _antiquaires_ around the country. They now form a unique collection that is a work of art in itself and serves as a moving testament to Tony and Beegle Duquette's febrile and unbridled imaginations, their invention, and their artistry.\n\nTo set off these imaginatively presented works, Wilkinson has rationalized Dawnridge, spiriting away its cobwebs and dust and letting in the diamond California light. Original fabrics have been rewoven from fading scraps to replicate their intended splendor, and Wilkinson has re-edited Duquette pieces such as the free-form gilded consoles that writhe like great sea serpents and the pagoda lamps that seem to have been fashioned from Brobdingnagian blocks of pale jade or alabaster, but are, in a Duquette-ish sleight of hand, cast from workaday resin.\n\nHutton, his wife, Ruth, and their scampering dogs have no need of living in a museum, so he has created a second stage set for themselves, a new house that hovers in the clouds above Dawnridge, and showcases his own dashing brand of Duquetterie, with its leopard underfoot and gold leaf overhead, and lacquered cinnabar in the great heights in-between. Here, ancient icons are mounted on Japanese screens in a riot of old gold, scarlet, and ebony paint and lacquer; Chinese pictures hang against carved Chinese screens glimmering with shards of mother-of-pearl; and a series of Venetian paintings depicting\u2014what else\u2014the Festival of the Redentore, with its gondolas and fireworks and moon-lit splendors, are set against a wall of gold. The alchemy continues.\n\n\u2014HAMISH BOWLES\n\n## A LIVING HOUSE\n\nAfter Tony Duquette's death in 1999, I took a trip to Hong Kong. One night, my friend Charlie Garnett gave a party for me at the China Club, where I was persuaded by friends to visit a psychic who'd recently set up shop on Hollywood Road. I went to see what all the fuss was about early the next day. Arriving at my destination, I put down my one hundred dollars and went into the room where the fortune-teller waited. \"Is your name Anthony?\" she asked me immediately.\n\n\"My business partner's name was Anthony,\" I said.\n\n\"He's not with us anymore is he?\" she asked.\n\n\"No, he passed away not too long ago,\" I replied.\n\n\"You're very involved in a house project right now,\" she said.\n\n\"Yes, I am,\" I answered. That house was Dawnridge, and upon my return to Los Angeles, I would be facing the removal of all its valuable antiques for an auction to settle Tony's estate and the daunting task of redecorating it for an upcoming benefit we were hosting for the Decorative Arts Council of the Los Angeles County Museum of Art. I had my work cut out for me and hoped I hadn't bitten off more than I could chew. \"Some people would think that the garden is more important than the house!\" the psychic insisted.\n\n\"Some people would,\" I agreed.\n\n\"Anthony's in the garden . . . he's very happy and approves of everything you're doing,\" she told me, and I smiled.\n\nBefore he died, I promised Tony that I'd try to save Dawnridge. I said that Cow Hollow, his house in San Francisco, and Sortilegium, his ranch in Malibu, would probably have to be sold, but because Dawnridge was the only property that he'd built from scratch, I would do everything in my power to keep it intact. After the Duquette auction, my wife, Ruth, and I redecorated Dawnridge as a loving tribute to our darling mentors and friends, Tony and his wife, Elizabeth, also known as \"Beegle.\" We like to think that if Tony and Beegle were to come back for a visit they might not recognize the rooms, but they would certainly recognize all of the objects. We deliberately redecorated the house using paintings, sculptures, and furniture that they created, mixing them with antiques, in a style we think they would approve of.\n\nIt has been our pleasure to save the life of this fabled house and to continue the tradition of creative hospitality that Dawnridge is known for. As Tony would say, \"We do these things for our pleasure, and hopefully, to inspire individuality and creativity in others through the art of living and the living arts.\" Dawnridge is a living house, constantly changing and evolving. It is the dearest hope of mine and my wife Ruth's\u2014and hope is the thing with feathers that perches in the soul\u2014that Dawnridge will continue to inspire, enchant, and inform for generations to come.\n\n\u2014HUTTON WILKINSON  \nDawnridge, 2018\n\nDawnridge, 1949. The iron obelisks on each side of the front door were moved to the Tony Duquette Studio on Robertson Boulevard where they were covered with abalone shell. The obelisks returned to the gardens at Dawnridge in 1975.\n\n## CHAPTER ONE\n\n## The Dream\n\nDawnridge, 1975. Tony covered the fa\u00e7ade with lattice and painted it pink with white trim and coral edges; In 2018, I painted the lattice emerald-green and covered it with ivy.\n\nTony at his parents' home in Three Rivers, Michigan, c. 1920.\n\nTony Duquette grew up between Three Rivers, Michigan, and Los Angeles, California. As a child, he would amuse himself by building \"estates\" in the side yards and empty lots next to his family's homes. These miniature constructions made of twigs and painted cardboard were theatrically lit with birthday-cake candles. After construction was completed, The Witch Boy (as his friend Eudora Welty later called him) would invite his family and friends to an \"unveiling,\" where he would dramatically light all the tiny candles, creating a brief moment of magic. \"I would invariably become small, becoming one with my miniature worlds,\" Tony once told me. \"I could play for days within the realms of my imagination and fantasize an entire culture living within the boundaries of my estates. I would spend days creating streets and houses, while imagining the lives that went on there.\" He never told me if these estates had names.\n\nLater in life Tony would amass ten fully furnished houses, all of which he did name: Fiddler's Ditch, later renamed Dawnridge; Cow Hollow (his pre-1916, San Francisco birdcage Victorian); Beeglesville; Frogmore; Ireland (named for the home's fa\u00e7ade, which was salvaged from an eighteenth-century Irish storefront); Doorchester (which was constructed entirely out of old paneled doors); Horntoad; China (a conglomeration of gilded Chinese carvings); Hamster House; and Chat Thai. Through these houses, he brought the magic of his childhood estates to life. To achieve some of these he needed an architect and found one early on, even before he owned land to build upon.\n\nCaspar Johann Ehmcke was born in Munich, Germany, in 1908, making him older than Tony by six years. When Caspar was nineteen, he moved to Stuttgart to study architecture at Stuttgart Technical University. After graduating, Caspar immigrated to the United States, and shortly after arriving in Los Angeles in 1938, got a job at Bullock's department store where he specialized in store planning and design. It was through Bullock's\u2014where Tony worked designing store interiors\u2014that the two future collaborators would meet. As fate would have it, Tony and Caspar rode the streetcar together to and from work each day. It was during this time that Caspar would tell Tony of his Bauhaus-style, modernist training in Germany and his plans to build streamlined houses in Los Angeles. Tony, in turn, would tell Caspar of his ideal house: a pavilion d'amore, a small Venetian palazzetto, a folly de luxe. What Tony described was the antithesis of anything Caspar had ever dreamed of building. \"Someday, you'll build it for me,\" Tony insisted. It was a dream Caspar hoped Tony would forget.\n\nCaspar Ehmcke (left) and Tony walk to work at Bullock's department store in downtown Los Angeles, c. 1938; The engraved invitation newlyweds Tony and Elizabeth would send out for parties at Fiddler's Ditch, c. 1949.\n\nWhile Caspar worked away at store planning and design, Tony's job at Bullock's was to change the store interiors four times a year. His directive from the store's owner, P. G. Winnett (my great uncle), was to make the customers forget about Southern California's lack of seasons. \"In those days, if women in New York were wearing tweeds and furs, the women in Los Angeles were also wearing tweeds and furs, even if it was one hundred degrees in California,\" Tony said. \"The minute the customers walked through the doors, it was my job to dupe them through the store interiors, temperature, and music into thinking it was summer, winter, spring, or fall!\" Tony's remuneration for conjuring up the seasons like a modern-day Merlin was an extravagant paycheck of fifteen dollars per week, on which he fed, clothed, and housed his mother, father, and three siblings.\n\nTony left his secure job at Bullock's after meeting Elsie de Wolfe, the first lady of American design, in 1940. De Wolfe, also known as Lady Mendl, had fled the Nazi occupation of Paris with her husband, Sir Charles Mendl, and eventually made a new life in Los Angeles after briefly living in New York. A very young eighty-five, she told her friends who begged her to stay that she \"was moving to Hollywood to be with the royalty of America . . . the movie stars!\" As fate would have it, de Wolfe was introduced to Tony through their mutual friends William \"Billy\" Haines, James Pendleton, and director Vincente Minnelli, and the rest was history. After claiming Tony as her exclusive discovery\u2014and proclaiming him a genius\u2014she hired him to decorate her new house in Beverly Hills, which she named After All. Taking Tony under her wing, Elsie made it her mission over the next ten years to make him famous worldwide.\n\nTony met his wife, Elizabeth Johnstone, in 1942. He was in the army, and she was a freelance artist working for Disney. They fell in love and moved in together. Elizabeth was nicknamed Beegle by Tony because she encompassed the industry of the bee and the soaring poetry of the eagle\u2014she was an artist whose talents seamlessly complemented his own. One day in 1949, their friend Mary Pickford told them, \"If you'd just get married, I'll pay for the wedding!\" That was all Tony had to hear; a free wedding sealed the deal and the couple was married at Pickfair, Pickford's fabled estate, in Beverly Hills. Pickford was the matron of honor and her second husband, Academy Award\u2013winning actor Charles \"Buddy\" Rogers, was best man. All of Hollywood was in attendance, including Hedda Hopper, Louella Parsons, Cobina Wright, Gloria Swanson, Agnes Moorehead, Vincente Minnelli, Adrian, and Arthur Freed. Sir Charles and Lady Mendl were also there.\n\nTony in his studio at Bullock's with the giant posters he created for a Mexican-themed event, c. 1937.\n\nElizabeth at Dawnridge, 1949.\n\nIt was with marriage in mind that Tony finally commissioned Caspar to build his dream house, Dawnridge, as a gift for his bride. Caspar had left Bullock's the same year as Tony to complete a housing project for the Metropolitan Life Insurance Company, and would eventually work on the new Jet Propulsion Laboratory, the Navy Test Facility, and an earthquake-proof high-rise for the Sears Roebuck department store. In 1945, he had started his own practice that focused primarily on commercial buildings, including a new recording studio for Decca Records. When a friend of Tony's, photographer Johnny Engstead, announced plans to build a new studio for himself, Tony insisted that the only architect for the job was Caspar Ehmcke. When Tony made an aesthetic proclamation, his friends always heeded his advice.\n\nBefore Ehmcke could go to work on Dawnridge, Tony had to purchase the land. Independently, two good friends\u2014one a real estate agent, the other a banker\u2014had approached Tony and said, \"When you're ready to build your dream house, call me!\" Shortly thereafter, Tony found a canyon lot high above Beverly Hills, just up the road from Pickfair, and told them about the property. All their friends thought Pickford had given the Duquettes the land as a wedding present, but that was not the case. When Tony showed the lot to the real estate agent, she said, \"I love you too much to sell you that property. It doesn't have a front yard; it's on a hillside. You're an artist, and you obviously don't know what you're doing. If you buy this property, you're going to lose your shirt. I refuse to sell you the property.\" Undeterred, Tony went to the banker, who said, \"I love you and Beegle too much to lend you the money to do this project. You're an artist, and you obviously don't know what you're doing. It has no front yard, it's on a hillside, and you're going to lose your shirt. I won't lend you the money; it's too risky.\" \"But for $1,500?\" Tony asked. The land was only $1,500 in 1949. A child of the Great Depression, Tony understood value, and despite his friends' warnings, he purchased the property and proceeded to build the house. That's when his parents jumped in. \"Don't build the bedroom wing,\" they told him. \"You're going to lose your shirt.\" And so, unfortunately, he heeded their advice and didn't build the bedroom wing.\n\nDawnridge was to be Ehmcke's first residential commission, but built to Tony's very Venetian specifications. It was a thirty-by-thirty-foot box that was divided into a large twenty-by-thirty-foot salon that Tony called the Drawing Room and three ten-by-ten-foot boxlike rooms across the front: the entrance hall, its adjoining vestibule with a small powder room, and the kitchen. A staircase in the double-height Drawing Room led up to a ten-by-thirty-foot balcony above the entrance hall and its adjoining rooms that had originally been intended to access the bedroom wing that wasn't built. Instead, it opened onto a small terrace above a one-car garage, an amenity that didn't interest Tony, who soon turned the garage into a dining room. Under the staircase to the balcony was a staircase leading down to two tiny bedrooms and bathrooms built into the slope of the hillside. \"It's like an apartment on its own ground,\" Tony used to say euphemistically about the house.\n\nThat was Dawnridge, originally christened Fiddler's Ditch after the ravine that ran through the back of the property. It was an elegant, small house that magazines called \"the Grandest House in Beverly Hills,\" and when Tony's society friends saw it, they all wanted one too, albeit bigger. Ehmcke went on to design dozens of Hollywood Regency\u2013style houses, as well as a handful of Modernist residences, for those friends, but his first residential commission was destined to become his most famous. Tony and Beegle hosted a whirlwind of parties and visitors during their first year in the house, including the famous French collector of houses Paul-Louis Weiller (Tony would later design a house for him in Paris), the Mendls, Mary Pickford, Marion Davies, Arthur Freed, Fred Astaire, and Loretta Young. The guest lists were formidable. Fiddler's Ditch was christened with a Bal de Derri\u00e8re, or a Bustle Ball, where all the ladies wore gowns with bustles. The Bal de Derri\u00e8re was followed by numerous dinner parties, replete with divertissements such as the noted Indian dancers Sujata and Asoka, as well as Balinese dancers, balalaika orchestras, and troupes of Chinese acrobats. At the events, the house servants wore eighteenth-century liveries that Tony had purchased from Baroness Catherine d'Erlanger, who moved to Los Angeles during the war from her home, Palladio's historic Villa Malcontenta in Italy.\n\nNewspaper clippings about the Duquette wedding at Pickfair, c. 1949.\n\nThe Duquettes only lived at Dawnridge for one year before moving to Paris, where Tony was invited to exhibit his work in an unprecedented one-man exhibition at the Pavillon de Marsan of the Louvre Museum. When they returned from Paris, Tony and Beegle moved into their old studio on Fountain Avenue and, in 1956, they purchased the former Norma Talmadge Film Studio at the corner of Robertson Boulevard and Keith Avenue, an area adjacent to Beverly Hills known as Sherman (now West Hollywood). Their plan was to turn the abandoned and condemned building into their studio, work-rooms, and residence. Tony immediately commissioned Caspar to draw up the plans for the new Tony Duquette Studio, where the couple would live and work for the next twenty years before moving back into their beloved Dawnridge.\n\nThe Duquettes stayed close friends with Caspar, often recommending him to friends such as Technicolor executive Pat Frawley and his wife, Gerry, and Tony always used him when he needed an architect for a design job\u2014like when working for Doris Duke at Rudolph Valentino's Falcon Lair. In the 1970s, Tony and Elizabeth asked Caspar to design a house at Sortilegium, their Malibu ranch. The Duquettes were enthralled with an eighteenth-century church they'd seen in Austria that was laid out in a trefoil design, and Caspar drew up plans for an incredible house and built an interesting model, but the Malibu house was never realized.\n\nBetween 1950 and 1975, while the Duquettes lived in their grand studio, Dawnridge was home to a series of distinguished tenants, including Marlon Brando, who rented their house while he was filming _Julius Caesar_ ; Eva Gabor and her husband Dr. John Elbert Williams; and Glynis Johns. Nancy Oakes (Baroness Hoyningen-Huene) also rented the house. The last long-term tenant in the house was the notorious Hollywood agent Sue Mengers, who provided us with plenty of stories to dine out on.\n\nThe invitation to, and newspaper clippings regarding, the Bal de Derri\u00e8re, or Bustle Ball, given by the Duquettes with their friends Mr. and Mrs. William T. Riley, 1950.\n\nBeegle (far left) with screenwriters Richard Sale and Mary Anita Loos during a party at Dawnridge, 1949.\n\nFlamenco guitarist at Dawnridge, 1949.\n\nIndian dancer Asoka provides the entertainment after dinner, 1949.\n\nTony later asked Vincente Minnelli to hire Asoka and his partner, Sujata, to dance in the MGM film _Kismet_ , for which Tony designed the costumes; A footman waits by the front door with his mural counterpart, 1949.\n\nCatherine d'Erlanger, 1945.\n\nTony and Beegle in front of Dawnridge, c. 1949.\n\nIn the 1960s the Duquettes acquired the house next door to Dawnridge, which they called New Dawnridge. They furnished and rented it to celebrities and friends, such as agent and film producer Freddie Fields. In 1974, after Tony had just finished redecorating Dawnridge following Mengers's ten-year tenancy, New Dawnridge burned to the ground, the first of four major fires that the Duquettes endured. I was working for Tony and Beegle at the time of the fire, when a body was found in the debris. I told the Duquettes, \"Wait until they discover that the person was dead before the fire started.\" And she had been! After the fire, Tony, Beegle, my future bride Ruth, and I high-tailed it to South America to avoid the coroner's inquest and get away from all the complications. I recently found the telegram that our secretary sent us while we were in Lima, Peru, stating that the death was incurred at the hands of another rather than by accident. It was foul play that involved the two men who were renting New Dawnridge.\n\nWhile it was a hardship, the destruction of New Dawnridge opened up space for the garden terraces that would truly transform Dawnridge into a large estate. Meanwhile, more immediate plans were in motion. Tony and Beegle had decided to move back to Dawnridge, and in 1975 they commissioned Caspar to design the bedroom wing\u2014actually, just a two-car garage with a bedroom above it\u2014that had been set aside when the house was first built. Day by day, as the Duquettes moved in their favorite collections, I watched the house become more personal. Dishes that had been used by high-paying tenants were replaced with eighteenth-century Chinese export porcelains, antique carpets covered the floors, and the interiors were filled with precious objects and antiques. Rooms evolved: The small downstairs bedrooms were expanded by glassing in the area beneath the Drawing Room's exterior balcony, a new kitchen extension and a glassed-in porch were built onto the new garage, the old kitchen became a larger powder room, and the old garage became a dining room.\n\nTony outside the Louvre during his one-man exhibition, c. 1951.\n\nAn article published about Nancy Oakes, who rented Dawnridge from Tony and Elizabeth, 1950. Nancy was the daughter of the gold mining tycoon and governor of the Bahamas, Sir Harry Oakes, whose mysterious murder has yet to be solved.\n\nTony Duquette Studios, as remodeled by Caspar Ehmcke, c. 1956.\n\nBeegle surveying the destruction after the fire at New Dawnridge, 1974.\n\nAfter the Duquettes moved back to Dawnridge, the studio on Robertson Boulevard continued as the headquarters for their business and the venue for parties, where they entertained in great style. Dawnridge became more private; only their closest family friends were invited for intimate dinners. However, there were exceptions to the rule\u2014sometimes they would open the doors, throwing caution to the wind, and host fabulous parties, like the one Tony gave for his goddaughter Liza Minnelli. These were star-studded and spectacular, like old Hollywood in its most tinseled days. Like his mentor, Elsie de Wolfe, who didn't believe in dining rooms, Tony would serve dinners all over Dawnridge\u2014in the Drawing Room, on the balcony, and in the one-car garage, which had been magically transformed into a mirror-and-malachite-encrusted dining room. The garden, now arranged on three levels and studded with fantasy pavilions and sculptures, was where alfresco lunches and dinners took place. At night, chandeliers and lanterns sparkled from the branches of flowering trees.\n\nTony's vision for the garden was singular. Using architectural fragments rescued from the backlot of MGM, he created a series of pagodas and temples, garden follies, platforms, terraces, a tree house, and dozens of lit sculptures. The garden was the scene of legendary parties over the years, with guests including Glenda Bailey, Miuccia Prada, and Angela Missoni. At one party, George Hamilton donned Tony Duquette jewelry, and at another Italian _Vogue_ editor Grazia d'Annunzio was so enchanted by the ambiance that she came back to photograph Tony and Dawnridge for Italian _Vogue_ , _L'Uomo Vogue_ , and _Vogue Gioiello_. Jean Howard, Loretta Young, Jean and Maggie Louis, Rudolph Nureyev, John and Dodie Rosekrans (who went home with fifteen original Tony Duquette necklaces), James Galanos, Gustave Tassell, Irene Dunne, Mary Martin, and Janet Gaynor were also constant visitors.\n\nTony Duquette Studios, with the original obelisks from Dawnridge placed on each side of the stage, c. 1965.\n\nTony and Beegle, 1970.\n\nTony with his goddaughter Liza Minnelli, c. 1990.\n\nBeegle (far left) with my wife, Ruth, and me, c. 1980.\n\nStepping into the garden, people were apt to ask, \"Where am I?\" Tony loved this about the property, and said, \"It's only minutes from the Beverly Hills Hotel, but it's impossible to tell that you're not in Japan, Austria, or Southeast Asia!\" Tom Ford was so impressed by what he saw that he used the house as a location for a Gucci campaign. Other companies followed suit: Bulgari with Julianne Moore, Abercrombie & Fitch with Bruce Weber, and Gucci, once again, with Tom Hiddleston.\n\nIn 1985, Tony and I purchased a massive historic synagogue in San Francisco. It didn't take us long to remodel the vacant, vandalized, and condemned building into the Duquette Pavilion, where we would showcase Tony's work. During the remodeling, Tony and Beegle moved to their house in San Francisco, Cow Hollow, and rented Dawnridge to his client and friend Beverly Coburn, the ex-wife of Academy Award\u2013winner James Coburn, whose house Tony had decorated in the 1970s. While she was in residence, and just a few weeks after the Duquette Pavilion in San Francisco burned to the ground, Dawnridge caught fire again. This was Tony's third fire at all of his properties, and the damage to the house was severe. It was at this time that Tony, Beegle, and I redecorated Dawnridge to the way it would appear in the 1990s.\n\nElizabeth Duquette died in 1995, four years before Tony. In 1990, Tony and Beegle had given Ruth and me the contents of Dawnridge as a gift and paid the gift tax on it. After Tony's death in 1999, we purchased Dawnridge. Ruth and I felt strongly that we ought to preserve Tony's unique works and residence. In order to settle the estate taxes and satisfy obligations to Tony's heirs, we decided to sell our household goods to get the money to buy the property. To do this, we had Christie's hold the largest house sale in American history. The exhibition was kicked off with a gala for five hundred guests that raised $1 million to benefit the Decorative Arts Council of the Los Angeles County Museum of Art. The sale took place over three days in a hangar at the Santa Monica Airport, where more than two thousand objects were sold.\n\nThe gardens at Dawnridge, 1980.\n\nThe Monkey Room, 2018. Tony never had the opportunity to enjoy the space that overlooked the gardens, as he had only used it for storage.\n\nAfter the sale, I redecorated Dawnridge using objects, furniture, and works of art created by Tony and Beegle, mixed with some of their favorite antiques that I had kept. I once asked Tony, \"Why don't you show your own work in your house?\" He didn't have an answer, but I imagined he thought that if he surrounded himself with valuable antiques, people would assume he was rich. Of course, he was wealthy, but only because of his investments in real estate. He once told his friend William Matson Roth, who owned Ghirardelli Square in San Francisco, \"With all my talent, the only thing that's made me rich is my real estate.\" Roth said, \"That was your talent!\" which made Tony laugh. He never tired of telling that story.\n\nRuth and I used Dawnridge for parties while we continued to live in Hollywood, but after several years, we finally decided to build a house next door. We sold our Tycoon Georgian and moved into Dawnridge while our new home, Casa La Condesa, was under construction. In 2011, when Casa La Condesa was finished, we turned Dawnridge back into a party house and made it the headquarters for our decorating and jewelry business.\n\nThe only regret I have is that Tony was never able to enjoy the glassed-in porch that became the Monkey Room. Tony always had a pressing need for storage, and after he created the room, he put embroidered Indian tent panels on the ceiling and filled it with stuff. After he died, it only took me a few hours to clear the room and turn it into a sitting room. At the time, I thought about how much Tony would have liked sitting in that room, with its three walls of glass overlooking the gardens. It was the only room at Dawnridge that was never finished in his lifetime. Today, it's used as a dining room. One night, I asked our friend Terry Stanfill, \"Why are the dinners in this room always so successful?\" She answered, \"It's because the ceiling is so low, and the oval table is conducive to conversation.\" She opened my eyes. The low ceiling\u2014a mere seven-and-a-half feet high\u2014caused the guests to feel as if they were \"tearing from the same beast,\" as Tony used to say. \"It's primordial, with the votive candles lighting the guests' faces from below, like the flames from a campfire.\"\n\n\u2014H.W.\n\n## CHAPTER TWO\n\n## Dawnridge\n\nThe entrance hall, 2018. On either side of the front door are ironic columns, which are made from discarded air filters, and hold Tony Duquette electrified crystal girandoles.\n\nThe entrance to the Drawing Room is now flanked by two French display cabinets, which were originally at the Robertson Boulevard studio.\n\n### The Entrance Hall\n\nIn 1949 Tony and Elizabeth covered two walls in the entrance hall with eighteenth-century citrine-colored damask backed with paper; the other two walls were paneled with squares of antique mirror. Beegle painted the door frames and crown molding with a fantasy faux marble in shades of yellows and blues. They created the doors leading into the Drawing Room from a pair of eighteenth-century French armoire doors, paneling the backs with sheets of clear mirror; antique Venetian bronze hands were used for the front and back door pulls. In the corners, on either side of the front door, there were two eighteenth-century Italian columns holding urn-shaped lamps with shades. Next to one of the lamps was a portrait of an eighteenth-century gentleman framed with seashells; below, a miniature bracket held an antique Spanish colonial retablo surrounded by a silver sunburst. On the opposite wall, on each side of the doors leading into the Drawing Room, a pair of eighteenth-century Italian polychromed figural stands purchased from designer James Pendleton held Tony Duquette coral-encrusted electrified crystal girandoles. The floor was made of black-and-white marbleized linoleum laid out in a checkerboard pattern. The back of the front door was painted by Beegle with the figure of a footman in eighteenth-century livery; the two pocket doors led into a vestibule and powder room on the the left, and a compact kitchen on the right. These doors also featured her artwork\u2014images of ladies in court gowns, one leaving the kitchen with a tray of food and the other seen from the back entering the powder room.\n\nThe entrance hall, 1949. Two of the walls are covered with paperbacked eighteenth-century yellow silk damask, and standing on both sides of the front door are eighteenth-century Italian fruitwood columns that hold carved and gilded urn-shaped lamps.\n\nAfter Tony and Elizabeth moved back into Dawnridge in the 1970s, the only major change to the entrance hall was the addition of blue grosgrain ribbon around the silk damask and the placement of three wooden panels over the doors, which were also enhanced with edges of decorative trim. When the house next door burned down in 1974, the heat in Dawnridge's entrance hall was so intense that it caused the paint on the ceiling to blister and peel. Tony liked this visually . . . so he lacquered over the damage and it remains that way today. Beegle then painted the trompe l'oeil tassels on either side of the front door to disguise water stains caused by the humidity in the room after the fire. During this time, the Duquettes also removed the Italian columns by the door, replacing them with a pair of Roman stools with silk velvet upholstery that had been worn down to the nap.\n\nWhen my wife and I purchased the house in 1999, we had to rearrange the entrance hall because most of the antiques had gone to Christie's. We replaced Pendleton's stands with French vitrines\u2014which used to stand on either side of the entrance to the ballroom at the Duquette Studio on Robertson Boulevard\u2014and we filled them with objects including Tony's wire-and-plaster figurines; ostrich eggs on Venetian glass stands; the Tony Award for Best Costume Design which he won for the original Broadway production of _Camelot_ ; cat skeletons, seashells, and corals; and the stuffed toads playing musical instruments that Tony and Beegle had purchased in Tijuana and painted to look like blue-and-white porcelain. Also displayed are the rocks the Duquettes picked up on the Malibu beach when they were beachcombing with their pal, artist Eugene Berman, who later painted faces on them and signed them _E. B. 1944_. Best of all are the two masks Tony constructed around the frames of a pair of antique Chinese fans, which he painted in 1951 for himself and Elizabeth to wear to the Vicomtesse Marie-Laure de Noailles's ball La Lune Sur La Mer in Paris. Sitting on the same shelf are a Ming figure of Buddha in turquoise-glazed ceramic and a gilded bronze toad.\n\nThe entrance hall, 1975. Gone are the Italian columns, now replaced by a pair of eighteenth-century Roman stools where Tony conveniently threw his favorite straw hat with a feathered band.\n\nOn each side of the front door, I've placed what I call ironic columns\u2014instead of Ionic columns. I made these from discarded air filters, which I stacked together and gilded. These hold a pair of Tony Duquette electrified crystal girandoles, which were salvaged from Cow Hollow, the Duquette house in San Francisco. The amber cups covering the light bulbs are made from plastic drinking glasses purchased from Pic 'N' Save. Photographer Tim Street-Porter once said, \"Tony is the only man I know who can spend $999 in one visit to the 99\u00a2 store!\" The carpet is antique Chinese and sits on top of the new, more graphic linoleum floor, which is similar to the material first selected by Tony in 1949. Finally, I removed the sliding pocket doors\u2014painted with Beegle's figures\u2014because they were forever hiding within the walls. They were moved to new locations and are now installed as swing doors.\n\nDetails of the French display cabinet, 2018. The cabinet holds a number of objects cherished by Tony, including ostrich eggs on Venetian glass stands and a gilded bronze toad (top), three rocks picked up while beachcombing in Malibu with Eugene Berman (bottom), and a wire-and-plaster figurine made by Tony to display jewelry (next).\n\nView from the bar across the entrance hall and into the Camelot Room, 2018. Originally a vestibule, I converted the space into a bar in 2000.\n\nThe vestibule, 1949\n\nThe bar, 2018. I removed the pocket door painted by Beegle and installed it as a swing door that leads into the new kitchen.\n\n### The Bar\n\nIn 1949 Tony decorated the vestibule with a nineteenth-century black lacquer papier-m\u00e2ch\u00e9 table inlaid with mother of pearl, which he purchased from Jimmy Pendleton and later sold to Agnes Moorehead. On top of the table was a bead-encrusted Chinese pagoda designed by Tony as a Christmas decoration. He also created the mirror on the wall by framing an antique remnant of embroidered-and-bead-encrusted satin and then having only a portion of the glass \"silvered\" to show the textile's elaborately worked borders. Elizabeth painted the pocket door (pushed into the wall) with the figure of a lady entering the room.\n\nThe former vestibule is now a bar. I moved the pocket door painted by Beegle and installed it as a swing door leading into the new kitchen. The room is decorated with paintings by Beegle, including one she created for the MGM film _The Sandpiper_ , directed by Vincente Minnelli and starring Elizabeth Taylor and Richard Burton. Elizabeth created all the paintings for that film, in which Taylor played an artist living on the beach in Big Sur, California. Against the wall is an eighteenth-century suede-upholstered mahogany director's chair. The carpet is from Nepal and was purchased by Tony and me during our last trip around the world before Tony died in 1999.\n\n### The Camelot Room\n\nThe space that is now the Camelot Room was originally the kitchen when the house was built. In 1975, Tony transformed it into a glamorized anteroom, with his 1950s antler-and-abalone console and mirror as a prominent feature. The antlers came from Hearst Castle and were gifted to him by the Hearst family when he and Beegle were guests at La Cuesta Encantada. The console held eighteenth-century Chinese lacquer coral branches from the collection of Elsie de Wolfe, a set maquette created by Tony for MGM's _Lovely to Look At_ , and a Spanish colonial statue. Tony also decorated the room with an eighteenth-century black-lacquered and ormolu-mounted French desk, an antique French bracket clock, a Venetian settee, and eighteenth-century Neapolitan cr\u00e8che figures.\n\nMy first go-round on the Camelot Room, which I named for Tony's theatrical designs on display there, was in 2000. I kept the antique Chinese window, which Tony had installed in the 1970s, in place, overlooking the dining room beyond. That window is now mirrored and part of the Venetian powder room. After the Duquette Collections sale at Christie's, I brought in the red-lacquered cabinet to replace the eighteenth-century black-lacquered and ormolu-mounted French desk. On top of the cabinet I placed a collection of geodes surrounding a maquette by Tony he called _Two Deer from Morgan Le Fay's Dominion_ , which was created for \"The Magic Forest Ballet\" sequence in the original Broadway production of _Camelot_ , starring Richard Burton, Robert Goulet, and Julie Andrews. After begging the show's creators, they obliged and allowed Tony to create _The Magic Forest Ballet_ , in which all the animals in Morgan Le Fay's Dominion took on human attributes and danced a little jig. Of course, it was a detour from the show, and people said, \"It's the only musical where you leave whistling the costumes and sets.\"\n\nThe Camelot Room, 2018. I created the Camelot Room in 2000 and decorated the space with Tony's theatrical designs. The painted door at the end of the room leads into the Venetian Powder Room.\n\nThe Camelot Room, 2000; The room previously included a Venetian settee (above), which had once been at the historic Larkin House in Monterey, California, and featured an antique Chinese window (next) that Tony had installed in the 1970s and looked into the dining room.\n\nI purchased this figurine (one of a pair), which was made by Tony in 1941 for Elsie de Wolfe and later displayed at the Mitch Leison Gallery in 1947.\n\nBecause Tony had placed set maquettes for both _Lovely to Look At_ and _Camelot_ in this room, I decided that it should display more of his theatrical designs, like his Tony Award\u2013winning costume sketches for _Camelot_. To complete this theme, I included some of Tony's earliest bas-reliefs, jeweled statues, and wire-and-plaster figurines, which had attracted the attention of Billy Haines, James Pendleton, Vincente Minnelli, and Elsie de Wolfe in 1941. I also presented the ermine tail\u2013upholstered Marsan chair that he had created for his one-man exhibition at the Louvre in 1952 on a velvet-covered dais. A similar chair, which Tony made for his one-man exhibition at the Los Angeles County Museum of Art (and which I discovered under a pile of rags at a Santa Barbara garage sale), sits in the corner. Tony called the needlework upholstery he designed for this chair \"The Flies of Texas.\" Tony and I purchased the tiger and dragon carpets in Nepal on our last trip around the world. The door in the Camelot Room, painted by Elizabeth in 1949 as a pocket door, is now installed as a swing door and leads into the new Venetian powder room I created in 2005.\n\nTony Duquette mermaid figurine, c. 1947.\n\nThree-dimensional costume sketch by Tony Duquette, c. 1960.\n\nThe sketch was created for the original Broadway production of _Camelot_ , and sat atop a red lacquered cabinet in 2000.\n\nThe Camelot Room, 2018.\n\nTony Duquette's original 1950s antler and abalone console and mirror still holds pride of place in the room. It features Venetian glass coral branches set into gilded wooden urns and a maquette created by Tony for the film _Lovely to Look At_.\n\nThe mirror hung in the card room at Elsie's home, \"After All,\" c. 1941.\n\n### The Venetian Powder Room\n\nIdecorated the Venetian Powder Room using fragments from an eighteenth-century Venetian cabinet; old teal, green, and gold Venetian moldings; and remnants of emerald green silk moir\u00e9, velvets, and cottons printed to look like shagreen. Tony created the iconic mirror hanging over the sink for Elsie de Wolfe's house, After All, in Beverly Hills, and I later purchased it from a junk shop in Long Beach. The Venetian paneling is hung with the skull of a crocodile and a shadow box of Tony's exhibition _Our Lady Queen of the Angels_ is set into an antique ivory inlaid frame. A framed Elizabethan tapestry has become a luxurious panel on the back of the door, and the walls are hung with a collection of early Tony Duquette watercolors and drawings.\n\nVenetian Powder Room, 2018. I created this space in 2005 and decorated it with many shades of green fabric. The room includes a mirror that was originally in Elsie de Wolfe's Beverly Hills house, which she decorated with a young Tony Duquette in 1941.\n\nView from the Drawing Room into the entrance hall, 1949. The carved seventeenth-century door surround was a wedding present from Adrienne and Cissy Hellis.\n\n### The Drawing Room\n\nThe Drawing Room started out as a very formal room that the Duquettes decorated with antique French furniture, Aubusson carpets, silk damask draperies, and old master paintings, many of which Tony purchased on his first trip to Europe with Elsie de Wolfe in 1947. What made the traditional d\u00e9cor unique was the addition of Tony and Beegle's personal touches, including the painted and molded ceiling decorated with bouquets of flowers, soaring doves, and rococo traceries in silver-gilt, bronze-greens, and jewel tones. Over the next sixty years, the room was transformed as the house changed from a rental property in the 1960s to a personal residence in the 1970s to a place for entertaining in the twenty-first century. When the Duquettes moved back to Dawnridge in 1975 and completely redecorated it for their own pleasure, they brought in things more beautiful than they were valuable. Tony always said, \"Beauty, not luxury, is what I value.\" The original rococo carved decorations on the stair railing were exchanged for carvings from Thailand. The carpets were replaced with jade green, low-pile wool velvet with celadon-green borders.\n\nThe carved seventeenth-century Spanish door surround was a wedding present from Adrienne and Cissy Hellis, the daughters of a Greek shipping tycoon. When Tony and Elizabeth were married at Pickfair, they said, \"We're going to give you a silver tea set for your wedding.\" Tony said, \"I don't want a silver tea set; I want the front doors of your house.\" The Hellis sisters not only sent over this magnificent carved doorway, but also sent two pairs of fourteenth-century Spanish doors that Tony installed as the entrance to his studio on Robertson Boulevard in Los Angeles. The old master painting would remain over the door until it was sold at the Christie's Duquette Collections sale; it was then that I replaced it with a sheet of mirror and Tony's one-eighth scale model of his _Primal Sun_ sculpture.\n\nThe chandelier Tony created using antique elements\u2014simple tubular metal arms and Peking glass mixed with beaded French glass flowers\u2014hung from the center of a giant plaster sunburst that matched the giant sunburst over the front door on the exterior fa\u00e7ade. The original chandelier shattered due to the heat of the fire next door, but it was refurbished with new Venetian glass lilies in 1975. Other Duquette touches included the lambrequins over the windows, which Tony created by dipping T-shirt jersey into a vat filled with plaster mixed with white glue and then draping the saturated fabric into his desired shape. After the fabric dried, Beegle painted the hardened cloth to resemble tooled leather, and Tony crowned the entire concoction with repouss\u00e9 silver urns from Spain and eighteenth-century Italian carved wings. With the addition of a few coral branches\u2014and likely a mystical or magical incantation like \"abracadabra\"\u2014they had lambrequins _de luxe_.\n\nDrawing Room, 1975. Tony replaced the original Rococo-carved decorations on the stair railing with carvings from Thailand.\n\nDrawing Room, 2000. The old master painting, which hung above the door, was replaced with a sheet of mirror and Tony's one-eighth scale model of his _Primal Sun_ sculpture.\n\nDrawing Room, c. 1960. Tony redecorated Dawnridge as a rental property in 1960. In this corner, he used eighteenth-century French panels framed as eighteen-foot-tall screens and reupholstered the sofas in white linen.\n\nIn 1949, the Duquettes installed floor-to-ceiling panels of antique mirror squares, one between the windows and the other at the far end of the room. The mirrored panel between the windows was centered by an antique Boulle bracket and clock inlaid with tortoise shell and etched brass decorations. On each side of the mirrored panel, Tony placed his sunburst torcheres made with eighteenth-century Italian candlesticks, convex mirrors, and gilded Italian sunbursts. The other mirrored panel was anchored with a long, eighteenth-century Louis XVI\u2013style, Venetian-gilded-and-polychromed console table. With these in place, the rest of the room was ready to furnish. The original colors of the walls were Adrian green, a pale celadon color favored by Tony's friend and client Gilbert Adrian of MGM fame. The eighteen-foot-long, coral-colored, silk damask draperies were discovered at the Paris flea market in 1947 and purchased with Dawnridge in mind. Add to this Duquette's one-of-a-kind plaster-decorated folding screens and the eighteenth-century portraits of Roman emperors hung high up near the crown molding, and the mise-en-sc\u00e8ne was set for a party. In front of one of the screens, a long custom sofa, which almost ran the width of the room, was paired with a set of four eighteenth-century Louis XVI berg\u00e8res upholstered in their original Aubusson tapestry coverings.\n\nThe Duquettes redecorated Dawnridge as a rental property in the 1960s, and Tony used eighteenth-century French panels framed as eighteen-foot-tall screens in this room. The sofas were reupholstered in white linen, and the lamps and shades were replaced with crystal girandoles, which Duquette created out of old cast-iron French urns.\n\nDetail of the chandelier, 2018. The original chandelier was damaged by the fire in 1974. At that time, Tony replaced the Peking glass flowers with Venetian glass flowers blown to resemble Pillemont lilies.\n\nTony Duquette sitting in front of his bas-relief screens, 1949.\n\nNorth corner of the Drawing Room, 1949. In the center of the wall is a panel of antique mirror squares with Tony's sunburst torch\u00e8res on either side. His original mirror and plaster bas-relief screens stand in the corners.\n\nView of the north corner of the Drawing Room from the balcony, 1949. Hanging from the ceiling is the original Peking glass flower chandelier with electrified votive lights nestled into the bouquets.\n\nNorth wall of the Drawing Room, 1975. Tony redecorated the space after moving back into the home and replaced the French panels with an eighteenth-century mural depicting a scene in Porto Mauritius.\n\nTony Duquette shell grotto in cast resin, c. 1960. Tony created this as a party decoration for an Undersea Ball.\n\nIn front are the eighteenth-century Venetian dolphins from the collection of Misia Sert, which were given to the Duquettes as a wedding present and later placed in the Drawing Room in the 1980s.\n\nIn 1975, as you entered the room, the first thing you saw was the pair of eighteenth-century Venetian dolphins from the collection of Misia Sert flanking a pillow-strewn eighteenth-century Italian settee. The settee, upholstered in apple-green silk, was placed in front of the new fifteen-foot-tall clear mirror panel that had replaced the antique mirror squares installed when the house was built. Later, I would replace the dolphins with the original sunburst torch\u00e8res that had stood in the room in 1949; Tony's biomorphic console and mirror would stand in place of the settee, positioned on a dais like an object in a museum.\n\nEast wall of the Drawing Room, 2018.\n\nI later added recreations of the original sunburst torch\u00e8res, which had stood in the room in 1949, and replaced the eighteenth-century sofa with Tony's biomorphic console and mirror presented on an upholstered stand.\n\nNorth wall of the Drawing Room, c. 1990. Tony layered his portrait by Marion Pike on top of the eighteenth-century mural.\n\nOn the east wall, Tony placed an immense twelve-foot-long banquette sofa designed by his friend Billy Haines, which he purchased from a charity sale and reupholstered in sap-green satin, then covered it with a throw of patchwork forest-green suede. Tony followed his friend Syrie Maugham's advice when she told him: \"A sofa should never look like a sofa, but rather should look like a pile of rich stuffs and carpets, piled high on the prow of a pirate's ship.\" He also replaced the tall screen of eighteenth-century painted French panels that he'd made for the rental house with a magnificent eighteenth-century mural depicting a scene in Porto Mauritius. The mural was purchased from the Baroness Catherine d'Erlanger; Tony later mounted it as a folding screen. In the 1990s, Tony would eventually layer the panels with his portrait, circa 1960s, by Marion Pike. In the corner, Tony placed an African Ekoi mask, which had been on display at LACMA as a loan from the Duquettes; it was later sold by Christie's to Dodie Rosekrans, and I later placed it in her palazzo in Venice.\n\nTwo of eight seventeenth-century paintings of Venice by Joseph Heintz the Younger, which Tony and Beegle had purchased from the Baroness d'Erlanger, were prominently placed on the stair wall, and a prized Piedmontese desk inlaid with ivory was positioned beneath one of the paintings. Tony would later change out the desk\u2014having sold it to John and Dodie Rosekrans to use in their Venetian palazzo\u2014in favor of the famous cabinet he made for Elsie de Wolfe in 1949. The decorated doors of the cabinet would even inspire a printed linen fabric: Tony Duquette for Jim Thompson \"Duquetterie.\" Next to the cabinet he placed his own Ghost Snail electrified sculpture, saying, \"I'm always trying to get light into a room without using lamps and shades. Look around, you won't see very many lamp shades here.\" In 1990, the Aubusson chairs were replaced with ebonized Louis XV berg\u00e8res upholstered in stri\u00e9d emerald green velvet.\n\nThe room originally included a Piedmontese secretary desk flanked by Italian fruitwood pedestals.\n\nTony would eventually sell it to John and Dodie Rosekrans in 1999, replacing it with the desk he made for Elsie de Wolfe in 1941.\n\nDetail of the Elsie de Wolfe cabinet, 2018.\n\nThe Tony Duquette for Jim Thompson Duquetterie linen fabric was inspired by the two decorated doors of the cabinet, 2008.\n\nNorth wall of the Drawing Room, 2001. Following the Christie's sale, a door was cut through the wall connecting the Drawing Room with the library. I created the sunburst screens using metal hubcaps to replace the wall of eighteenth-century Italian paintings and added artwork by Tony above the doorway.\n\nOne of the two three-dimensional costume sketches by Tony Duquette for the 1952 San Francisco Opera production of _Der Rosenkavalier_.\n\nSince the Christie's sale in 2000, I've redecorated the Drawing Room a number of times. The first time, I replaced the seventeenth-century Venetian paintings on the left wall with Tony's watercolors of Venice, which had been exhibited at the Louvre. I also replaced the wall of eighteenth-century Italian paintings with the sunburst screens featuring metal hubcaps. I cut a door through the wall, connecting the Drawing Room with the library beyond, and flanked the new mirrored doorway with the stands and plaster heads Tony created for the Cobina Wright residence in 1952. Tony's multimedia assemblage of window screen, Mylar, and rhinestones\u2014known as _A Fragment of a Priestess' Robe or A Specimen of Rhinestone Disease_ \u2014was hung above the door. The walls were reupholstered with an iridescent bronze to gold lam\u00e9 fabric, which we also used at the Palazzo Brandolini, and the Ottoman banquette, originally designed by Tony for Cow Hollow, their house in San Francisco, was brought into the Drawing Room. Other objects in the room include Tony's _Phoenix Rising From Its Flames_ sculpture in the right-hand corner, as well as two of his three-dimensional costume sketches for Der Rosenkavalier, which are displayed in Plexiglas cases around the room.\n\n**Above and next:** North corner of the Drawing Room, 2018. The upholstered seating area has been replaced with a draped table surrounded by four Louis XVI\u2013style chairs upholstered in quilted Fortuny. On the table are three Tony Duquette plaster figurines, c. 1941. Each holds an abalone shell with a votive light inside.\n\nOne of a pair of plaster heads created by Tony to hold flowers, c. 1940s.\n\nIn 2017, I rearranged the room as more of a gallery or reception room than a sitting room. To create this mood, I removed the seating area and added a round table draped in Tony Duquette for Jim Thompson Feu d'Artifices handwoven silk. The four Louis XVI\u2013style chairs surrounding the table are upholstered in old quilted Fortuny, and the table is set with Tony Duquette reclining figurines that are holding abalone shells with votive lights in them. The mother-of-pearl bowl contains Venetian glass coral branches and a display of Duquette's gilded _Modern Fruit_ from his exhibition at the Louvre.\n\nThe south end of the room has not undergone changes as extensive as the rest of the Drawing Room. In the 1980s, Duquette placed an eighteenth-century French commode with ormolu mounts where the original long Venetian Louis XVI\u2013style console had always held pride of place; he moved the console upstairs to the balcony. After a few years, Tony sold the magnificent French commode and replaced it with...yet another chest of drawers\u2014an eighteenth-century Portuguese piece in the Chippendale taste from the collection of the Duquettes' friend Frances Elkins. On top of this new furniture, Tony placed electrified resin pagodas modeled after a set of nineteenth-century Regency ceramic pagodas he'd found in Ireland when decorating Barretstown Castle for Elizabeth Arden in the 1960s. In the 1980s, Tony also added eighteenth-century paintings to the room, paneling the walls with them from the baseboards to the crown molding. It was at this time that a Chinese red lacquer screen and an Italian settee upholstered in striped white satin also made their appearance in the Drawing Room. Tony and I worked closely on this redecoration and placed a papier-m\u00e2ch\u00e9 table in the corner, with a Venetian gondola scene painted in the center of its top; we later sold this treasure to Dodie Rosekrans for her dressing room at Palazzo Brandolini in Venice.\n\nThe seating area, c. 1980, was also removed from this side of the room. An antique Venetian chest of drawers now stands in place of its eighteenth-century French counterpart.\n\nSouth wall of the Drawing Room, 2018.\n\nSouth corner of the Drawing Room, c. 1980 and 2018. Tony first placed a papier-m\u00e2ch\u00e9 table in the corner next to an eighteenth-century red-lacquer Thai shrine.\n\nThe furniture was eventually removed and replaced by me with a Burmese shrine.\n\nOne of the two three-dimensional costume sketches by Tony Duquette for the 1952 San Francisco Opera production of _Der Rosenkavalier_.\n\nIn 2000, I painted the turquoise silk satin curtains with red, gold, and pink spray paint, making an interesting ikat type pattern. When Chad Holman from Jim Thompson saw those painted curtains, he asked me where I got them. I told him I'd painted them during the weekend. When he asked me why, I facetiously told him that I couldn't afford to replace them, and I couldn't afford to clean them, so I painted them. That's when he asked me to make a collection of fabrics for Jim Thompson. Today, the curtains are Tony Duquette for Jim Thompson \"Feu d'Artifices\" handwoven silk. Recently, I placed the furnishings around the walls to give the space the feeling of a ballroom. An antique Venetian chest of drawers sits under the mirror where the old Portuguese one used to be, and a carved and gilded Japanese table has been placed under the staircase. On top of both pieces, I've placed Tony Duquette's three-dimensional costume sketches for the 1952 San Francisco Opera production of _Der Rosenkavalier_. Around the room, Louis XV chairs, upholstered in Tony Duquette for Jim Thompson \"Royal Ermine,\" have been pushed against the walls, and a Burmese shrine, with its gilded bronze Buddha and eighteenth-century Chinese lacquer coral branches, sits in the corner. Oil paintings by Beegle have now replaced the eighteenth-century artwork on the walls, along with water-colors and bas-reliefs by Tony.\n\nSouth wall of the Drawing Room, 2018. The window has been removed, creating a passageway that connects to the Green Room beyond. The old master paintings have been replaced with oil paintings by Elizabeth Duquette and watercolors and bas-reliefs by Tony.\n\n### The Balcony\n\nTony used the balcony as a sitting room, and occasionally dining tables were set up there. It had a wood-burning stove in the shape of a truncated obelisk at the end, and a carved Indian blackwood settee that Tony painted turquoise. Tony would later use this piece for Adrian's fashion show finale in MGM's _Lovely to Look At_. The center of the balcony was dominated by the oculus of the house\u2014a large round opening in the floor surrounded by a low railing that looked into the entrance hall below. This was an obvious copy of the oculus at the Ch\u00e2teau de Groussay, the amazing country house of Tony's friend and well-known amateur decorator Carlos de Beistegui. Duquette knew Beistegui through Elsie de Wolfe and was a frequent guest at Groussay in 1947, 1950, and again, with Beistegui's nephew and heir Juan, in 1977. Tony's biggest regret was not being able to accept Beistegui's invitation to the Black and White Venetian Ball he hosted in 1952 at the Palazzo Labia on the Grand Canal, although Tony did supply several of the masks worn by guests that night in Venice.\n\nThe balcony, 1949. The room featured a wood-burning stove\u2014one of Tony's original designs for the house\u2014that had crystal feet and was studded with brass fleur-de-lis.\n\nThe balcony, 2001. The space included a Venetian console table, an old master painting, and a turquoise rhinoceros American merry-go-round figure, which stood in front of the glass-topped oculus of the house that looks down into the entrance hall below.\n\nThe balcony, c. 1980. The black-lacquered double doors lead into the Winter Bedroom. To the right is one of the treasures of the Duquette collection, a sixteenth-century red-lacquer Burmese Buddha.\n\nThe room has been redecorated and now features antique Chinese picture carpets, Tony Duquette's scale model of the California Sunburst proscenium curtain made for the Dorothy Chandler Pavilion of the Los Angeles Music Center, and his original abalone chandelier that was created for his one-man exhibition at LACMA in 1952.\n\nIn the 1980s, Tony and I removed the draperies, which could be closed to separate the two areas, and installed the mirror on the ceiling. The red lacquer Buddha to the right of the door is sixteenth-century Burmese.\n\nWhen redecorating the room, I placed Chinese carpets over the original bronze wool carpeting and installed the abalone chandelier that Tony had created for his one-man exhibition at LACMA in 1952. On the left wall, I hung the scale model for the proscenium curtain at the Los Angeles Music Center's Dorothy Chandler Pavilion; on either side are two Tony Duquette figural lamps originally created for Cobina Wright's home in the 1950s.\n\nPast the bronze head of Buddha, antique Chinese doors disguise a closet where the old wood-burning stove used to sit in 1949, and the doors to the right of the closet lead to the Winter Bedroom, which the Duquettes added to the house in 1975. The walls are paneled with Chinese Coromandel screens, the carpet is Tibetan, and the closet doors are appliqu\u00e9d with antique trims and hand-painted decorations from Bhutan.\n\nThe library, 2018. Located off the Drawing Room, the library is often used for entertaining, either as a bar or a place to set up four tables of eight for seated dinners.\n\n### The Library\n\nOne of the first things we did after purchasing Dawnridge was create an enfilade between the Drawing Room and the library. Prior to this, we had to go through the vestibule off the entrance hall and wind our way around to get to the library. It was not exactly the most dramatic or glamorous way to get from point A to point B, so we cut a door between the two rooms. This change was important because it allowed us to convert the library into a dining room on occasion\u2014a necessity as the house had no formal dining room. Our original plan was to leave the library permanently set up with a dining table covered in books and objects, and use the adjacent Monkey Room as a sitting room that looked out over the gardens. However, after our first dinner party, we realized that the library was too close to the kitchen; there was no butler's pantry between the rooms to muffle the sounds of cooking, clanking dishes, and chattering servants. Because of this, we immediately moved the dining table\u2014an oval, which I learned from experience is the best shape for conversation as it allows everyone at the party to participate\u2014into the Monkey Room and returned the library to a sitting room, which on occasion is converted for dining.\n\n### The Monkey Room\n\nThe Monkey Room was named for the frieze of papier-m\u00e2ch\u00e9 monkey masks which frame the embroidered and appliqu\u00e9d ceiling that is upholstered with Indian tent panels. To create the illusion that the ceiling extends to the exterior of the glassed-in porch, Tony placed ninety-nine-cent mirrors from Pic 'N' Save over the windows. The curtains are Tony Duquette for Jim Thompson printed cotton in the Tibetan Sunburst pattern. Opposite the window overlooking the gardens, Tony built bookcases and filled them with a small sampling of his extensive art and fashion library.\n\nThe monkey theme is also reflected in figurines that I've placed around the room. On his seventy-fifth birthday, Tony's friend Cynthia Lindsay gave him three dressed monkeys by Oliver Messel. Two of the monkeys were lost when the Duquettes' Malibu ranch burned to the ground, but one survived. I found another monkey by Messel, dressed in a tricorn hat, at the Leo Lerman sale in New York, and added it to the room, along with a pair of figurines made by Tony in the 1940s for the fashion designer Adrian, who used these alongside a perfume or shoe display at his store in Beverly Hills.\n\nThe Monkey Room, 2002. Here, the table has been set with gold lam\u00e9 cloth, vermeil chargers, cups, bamboo-style cutlery, and antique Chinese rose medallion porcelains. Covered Peking enamel bowls, Venetian glass goblets, and a centerpiece composed of a nineteenth-century vermeil palm tree candelabra surrounded by ceramic pagodas and eighteenth-century Chinese lacquer coral branches complete the look.\n\nDetail of the built-in bookcases, 2018. The walls, like the ceiling, are upholstered in embroidered and appliqu\u00e9d Indian tent panels and layered with antique Japanese tassels.\n\nOne of a pair of figurines created by Tony for the designer Adrian, c. 1940.\n\nAdrian used these, which Tony sketched out for him first (above), as part of a display at his store in Beverly Hills (next).\n\nMonkey figurines by Oliver Messel, c. 1950. Tony received three dressed monkeys for his seventy-fifth birthday from his friend Cynthia Lindsay. Only one survived following the fire at the Duquette Ranch in 1993, and is now displayed in the Monkey Room.\n\nI purchased another Oliver Messel monkey figurine with the tricorn hat from the Leo Lehrman estate to add to the d\u00e9cor.\n\nTable setting in the Monkey Room, 2018. Rock crystal votive lights cast a glow on guests' faces, as Tony liked to say, \"like the light from a campfire.\"\n\nThe Monkey Room, 2005. The curtains are made with Tony Duquette for Jim Thompson Tibetan Sunburst fabric.\n\nFor our dinner parties, we set the tables with an eclectic mix of antiques and objects made by Tony: vermeil chargers, antique Chinese porcelain in a pink and green lotus pattern on gold, holding abalone soup bowls; antique Baccarat crystal glasses etched with stars; custom-made seashell and coral salt and pepper cellars; and antique silver, Mythologique-patterned cutlery. A Tony Duquette wood-and-crystal girandole holds black candles \"so that the flames look like they're floating in midair,\" as Tony used to say, surrounded by coral-clad nautilus shells sporting Venetian glass coral branches. Vermeil lobsters stand guard at each end of the table, and Tony's signature rock-crystal votives are placed to cast a soft glow onto the guests' faces.\n\nThe Winter Bedroom, 1975. The bed hangings are stri\u00e9d silk taffeta bordered in Thai silk; Japanese tassels hang from the corners of the bed curtains. The alligator at the foot of the bed is from the South Sea Islands.\n\n### The Winter Bedroom\n\nIn 1975 the Duquettes built a new two-car garage and added a master bedroom above it. They called their new chamber the Winter Bedroom, where they moved when the weather turned cold, as opposed to the original downstairs bedroom, which they called the Summer Bedroom. The Winter Bedroom connected with the balcony over the Drawing Room. With my help, Tony decorated the room with antique Georgian furniture and paneled the bed wall using an antique Coromandel screen. The ceiling was covered in gold lam\u00e9, and the walls were upholstered in a bronze-colored stri\u00e9d linen. The bed was hung in coral-colored Thai silk with a pelmet adorned with Tibetan tassels. At the foot of the bed we conveniently placed a carving of an alligator from the South Sea Islands as a place to throw your robe on.\n\nAfter we purchased Dawnridge, Ruth and I used the Winter Bedroom for guests. We changed the bed hangings, using bolts of coral and saffron yellow Thai silk that were left over from Tony's _Camelot_ costumes, and covered most of the walls from floor to ceiling with bookcases. The bed's valance was salvaged from the Duquettes' canopied bed in their San Francisco home, along with the silk tasseled bedspread, hand-painted with birds and flowers. On each side of the bed we placed Spanish colonial _varguenos_ and topped them with the fabulous figural lamps Tony made for his exhibition at the Louvre in 1951. Recently, we added even more pieces to the room including a loveseat and a small upholstered armchair designed by the Duquettes' friend, English designer Syrie Maugham.\n\nTwo pairs of antique Chinese carved teak doors, to which Tony added panels of old Chinese silver damask and frames consisting of several rows of passementerie trim, separate the Winter Bedroom from both the vestibule and sleeping alcove. The Winter Bedroom's sleeping alcove holds an original Syrie Maugham sleigh bed. This bed\u2014a favorite of the Duquettes\u2014was commissioned from Maugham by the actress Ina Claire, who played Grand Duchess Swana in _Ninotchka_. Ina was a great friend of the Duquettes, as well as Ruth's and mine. Over the years, we were able to purchase a lot of fine furniture from Ina, who had collected wonderful pieces not only from Maugham but also from Elsie de Wolfe. Tony hung a Ming Dynasty panel over the sleigh bed and added a nineteenth-century black-and-gold lacquer chair that he purchased on a trip to Chiang Mai, Thailand.\n\nIn 2000, the sleeping alcove was completely redesigned after we walled up a window and upholstered the walls with printed raw silk that we found in our warehouse. The bolts and bolts of fabric had been made for Tony by Jim Thompson himself in the 1960s, and were never unpacked. I added a French desk; a Victorian chair; a Chippendale Chinese-mirror from the 1960s featuring an orchestra of carved musicians from Thailand that I affixed myself; and a pair of circa 1940s Tony Duquette figural lamps. The frame around the skylight is made up of Chinese carvings appliqu\u00e9d with woven passementerie trim, as well as tassels made by Tony using Chinese coins and silk, which he dipped in plaster and painted pink and blue.\n\nThe Winter Bedroom, 2018. The original bed hangings have been replaced with curtains made of coral and saffron colored Thai silk. On either side of the bed are Spanish colonial varguenos with Tony Duquette figural lamps, c. 1951, originally created for his one-man exhibition at the Louvre.\n\nThe Winter Bedroom, 1975. An antique terracotta lion perches on the balustrade outside the Winter Bedroom's French doors.\n\nDetail of the antique Chinese carved teak doors separating the Winter Bedroom from the sleeping alcove, 2018.\n\nThe sleeping alcove, 1975. The Duquettes purchased the nineteenth-century black-and-gold lacquer chair on a trip with Jim Thompson to Chiang Mai, Thailand. A Ming Dynasty panel hangs over the bed.\n\nThe alcove was redecorated with fabric that was printed for Tony by Jim Thompson in the 1960s.\n\nTony Duquette figural lamps, c. 1940, were also added to the room.\n\nThe bathroom is located off the Winter Bedroom, and a malachite-painted commode now stands where the tub once was.\n\nThe original master bedroom, 1949.\n\n### The Summer Bedroom and Guest Bedroom\n\nDawnridge originally had only two tiny subterranean bedrooms\u2014situated downstairs just under the Drawing Room\u2014with windows that opened onto a terrace surrounded by an elegant balustrade overlooking the ravine below. The original master bedroom had walls covered in silver lam\u00e9 with wall-to-wall carpeting on the floor. Tony decorated the room with old pine doors that had been bleached and pickled, and hinged together as folding screens. There were yellow linen draperies on the French windows, a vicu\u00f1a throw on the bed, and a pair of carved Indian camels as night stands. An elegant secretary desk, which Tony had studded with nail heads and sequins, centered the wall across from the windows, and next to this was an eighteenth-century French chair.\n\nIn 1975, after the new Winter Bedroom was completed, the original master bedroom would become the Summer Bedroom. Because it was built into the hillside on three sides, it was naturally cool in the summer and even colder in the winter. The new space upstairs was a warmer place to sleep in the winter, so the Duquettes moved their quarters with the seasons. In the stairway leading to the bedrooms, Tony paneled the walls with antique grillworks and mirrors and carpeted the stairs with his signature leopard carpet. He even installed an antique nineteenth-century Chinese pagoda as a newel post in this newly Orientalized stair hall. It was at this time that Tony replaced the wall-to-wall carpet in the Summer Bedroom with red lacquer squares topped with a custom Portuguese covered needlework rug in a leopard pattern. He draped the bed in antique Chinese embroideries and upholstered the walls with pink Thai silk. The headboard was covered with Tony's precious tiger skin patterned silk velvet. At the foot of the bed, Tony placed an antique Chinese grape-root table, part of a set he purchased from the Hearst collection, which he layered with leopard skin.\n\nThe staircase leading down to the summer and guest bedrooms, c. 1980. Tony paneled the stairway with antique grillworks and mirrors and carpeted the stairs and hall with his signature leopard-print carpet.\n\nThe room's walls were upholstered in silver lam\u00e9, and folding screens, made from antique paneled doors, were placed in each of the four corners.\n\nThe Summer Bedroom, 1975. When Tony moved back to Dawnridge, the master bedroom became the Summer Bedroom. The room's original bed coverings featured Chinese embroidery over imperial yellow satin; carvings from China and Thailand helped complete the mise-en-sc\u00e8ne.\n\nThe Summer Bedroom, c. 1980s. Tony redecorated the bed alcove using an antique gold embroidered bedspread and installing silk hangings embroidered with silver threads overlaid with pink gauze.\n\nThe guest bedroom, 1949 (above) and 1990 (next). Tony transformed the guest bedroom into his dressing room in 1975 and decorated it with antique Asian carvings, bronze Buddhas, and red lacquer.\n\nThe Summer Bedroom was only a fifteen-foot square, but the guest bedroom next door was even smaller\u2014a mere fifteen-by-ten-foot hole in the wall. Tony used to say that the only benefit of this room was that \"if you fall down, you can't hurt yourself because you'll definitely fall onto the bed!\" He covered the First Empire bed with woven black-and-white Mexican carpets; the walls were covered with hand-painted eighteenth-century Chinese wallpaper. He also hung one of his original chandeliers decorated with Venetian glass flowers from the ceiling and placed two of his floral lamps on the chest of drawers. Tony used the guest bedroom as a dressing room, while Beegle had her own wardrobes and bathroom just off the adjacent master bedroom.\n\nIn 1975, Tony turned the guest room into a proper dressing room, creating a direct entrance through the Summer Bedroom next door. A pair of antique doors, covered with malachite-printed cotton, ecclesiastical trim of woven gold thread, and panels of antique Chinese damask now separated the two spaces. On either side of this new doorway, Tony placed gilded Thai stands holding cast-resin pagodas which he lit from within. The door pulls were antique Damascus steel inlaid with silver. Tony decorated the newly enlarged dressing room with antique Asian carvings, bronze Buddhas, and oriental rugs layered over the wall-to-wall leopard carpeting, a look that he and Doris Duke earlier christened as \"Chow Fun.\"\n\nEntrance to the dressing room, 1990. A pair of antique doors, with gilded Thai stands on either side, lead from the Summer Bedroom into Tony's expanded dressing room.\n\nThe dining room, 1949. Originally a one-car garage, the space was instead used for dinners and entertaining. Arched French doors from a wrecking yard were installed to create the feeling of a garden pavilion.\n\n### The Green Room\n\nThe Green Room started out as a one-car-garage in 1949. Tony immediately turned it into a dining room. In its original incarnation, the garage door\u2014which is still in place\u2014was hidden behind a drapery of old Fortuny printed cotton. In front of this, Tony placed a large oyster-veneered seventeenth-century English cabinet to hold china and linens. Arched French doors\u2014salvaged from a demolition\u2014were installed to take advantage of the room's southern exposure and to give the former garage the feeling of a garden pavilion. Beegle painted the decorative moldings with ermine tails, while Tony placed brackets on the walls to hold antique carved and gilded urns from Italy. Of course, the entire mise-en-sc\u00e8ne was lit by one of Tony's original floral chandeliers in Peking glass, which sparkled in the many mirrors strategically placed to enhance the perspective of the magical space.\n\nThe room was tiny, but Tony was able, on occasion, to set up three tables of eight, so that twenty-four guests could be seated for dinner. As I've already mentioned, Tony liked to say, \"People enjoy being crowded together at dinner parties. Sitting elbow to elbow, cheek by jowl, revives a primordial memory deep within them of sitting around a campfire and tearing from the same beast.\" At Dawnridge, before 1990, tables were always round and draped to the floor with ever-changing cloths. Tony and Beegle had a beautiful set of early nineteenth-century Venetian gondola chairs that he'd stripped and bleached. If additional seating was needed they'd add in gilded ballroom chairs.\n\nAfter the fire in 1974, the arched French doors were replaced with large paned windows. Tony also added a pair of upholstered sofas to form a seating area at the end of the room, which was newly paneled with vintage Chinese doors and a mirror. In the 1990s, the dining chairs were bamboo finished in black lacquer and upholstered with Indonesian ikat\u2013covered cushions.\n\nThe dining room, c. 1975. After the fire from the house next door burned out the dining room, the arched windows were replaced with large paned windows. A pair of upholstered sofas form a seating area paneled with old Chinese doors and mirrors.\n\nThe dining room, c. 1990. The table is set with rock crystal votives holding an eighteenth-century gilded nef in the form of a Spanish galleon.\n\nThe Green Room, 2005. I transformed the old dining room into the Green Room, and made the folding screen (below) at the back of the room by blowing up an eight-by-eleven-inch pen-and-ink drawing of Venice created by Tony in 1947 on his first visit to Europe with Elsie de Wolfe.\n\nAt their dinners, the centerpieces were rarely floral\u2014instead, they were made of large rock crystals paired with pearls on wires, coral branches in lacquered bowls, or Chinese figures, all surrounded by votive candles that \"lit your face up . . . like the light from a campfire.\" Tony would constantly repeat this, and then remind me, \"We don't eat by candlelight because it's romantic, but because it takes us back to the cave where we used to sit around an open fire and tell stories.\" Eighteenth-century papier-m\u00e2ch\u00e9 temples of love; figurines he designed himself made of dipped plaster and rhinestones; seashells, taxidermy, and precious antique objects all appealed to his sense of theater whenever setting a table for a festive occasion. To create a different look each time guests were at dinner, sets of dishes were used on a revolving basis; Tony would pair them with vermeil chargers engraved with his crest, simple painted tin plates from Haiti, or priceless Chinese porcelain dishes. In the 1950s Tony and Beegle took every piece of silver that they could lay their hands on to Tiffany's to have it gold-plated. He often patted himself on the back, telling me of his prescience at knowing that gold was actually cheap back then. Because of this, they had sets and sets of cutlery to choose from. Crystal and Venetian glass goblets were paired with gold-plated silver cups that were always used for water.\n\nAfter Ruth and I purchased Dawnridge, we converted the old dining room into the Green Room, named for its dominant color scheme, and today it's most often used as a place to serve cocktails before dinner. Part of the ceiling is still mirrored\u2014just as Tony had designed it\u2014with malachite-patterned fabric-covered beams studded with antique Chinese carvings and panels of abalone. We covered the ceiling on the side of the room closest to the garden with Mylar, and then overlaid it with dark green painted lattice and finished it with an antique French copper valance. We overlaid the dark green board-and-batten walls with strips of Tony Duquette for Jim Thompson malachite-printed cotton and installed a Tony Duquette malachite-patterned silk carpet. At the back of the room, we installed a folding screen that Tony created for his first one-man exhibition in 1947. The screen was later used by Adrian before being sold to Tony's client and best friend Patricia Hastings Graham. It recently returned to Dawnridge as a gift from Mrs. Graham as part of her estate. Our recent redecoration of the room included the expansion of Tony's collection of antique birdcages, which are now hanging all over the ceiling. The new Green Room curtains are Tony Duquette for Jim Thompson Gemstone printed cotton in lapis lazuli blue. A black-and-white-checkered floor has also been installed on the diagonal, and the addition of three conversation groups of furniture and a table for dining completes the look.\n\nThe Green Room, 2018. The room now includes a folding screen created by Tony Duquette for his first one-man exhibition in 1947.\n\nDuquette standing in front of his screen at his exhibition at the Mitch Leison Gallery, 1947.\n\nThe centerpiece on the skirted table features Tony's _Modern Fruit_ , which he created for his one-man exhibition at the Louvre in 1951. The skeletal fruit in gilded brass has been placed in a mother-of-pearl bowl surrounded by Ming parrots, rock-crystal votives, and abalone shell salt and pepper shakers. I painted the carved Indian blackwood furniture white, and placed antique Asian artifacts on the side tables.\n\nThe Green Room, 2018. In the corner, Tony's rose chest has a number of objects on display, including a carved Indonesian figure sitting on top of a Chinese imperial yellow porcelain stand, a nineteenth-century lacquer box holding a Lalique Buddha, and carved Indian parrots perched on red-lacquer stands from Thailand.\n\nA Tony Duquette _Insect Man_ sculpture poses in front of the lapis luzuli Tony Duquette Gemstone curtains by Jim Thompson.\n\nIn one corner of the Green Room, near the door leading out to the terrace, I added the Rose Chest, which was Adrian's favorite piece of furniture and used by Cecil Beaton in Mrs. Higgins' house in the film _My Fair Lady_. Hanging directly above the Rose Chest\u2014and around the room\u2014are the late Julian La Trobe's paintings of Dawnridge and a pair of eighteenth-century floral French sconces. The tablescape on top of the chest includes a carved Indonesian figure sitting on top of a Chinese imperial yellow porcelain stand; a nineteenth-century lacquer box holding a Lalique Buddha; and a pair of carved Indian parrots perched on red lacquer stands from Thailand.\n\nI placed an antique French desk painted in a malachite pattern by Scarlett Abbott in front of the window. The desk\u2014flanked by a pair of Chippendale Chinese \u00e9tag\u00e8res holding blue and white porcelains\u2014has a tablescape which includes a stuffed cockatoo wearing a miniature Thai crown, a pair of wire-and-plaster lizards made by Tony for his client Cobina Wright, Tony's _Insect Man_ sculpture, a small painting by Elizabeth Duquette, and a 1940s Duquette Ashtray Man holding a votive candle. Two coral-painted birdcages and an antique Chinese lantern have been electrified to bring light into the space. The two Venetian stools on each side of the desk are upholstered in Tony Duquette for Jim Thompson Royal Ermine.\n\nAnother corner of the Green Room, near the entrance to the Drawing Room, includes a sofa designed by Tony for Doris Duke's Falcon Lair and Louis XV chairs upholstered in Tony Duquette for Jim Thompson Gemstone fabric in lapis lazuli blue. Chinese glass cabinets on each side of the door display _Talismans of Power_. Opposite the cabinets are a Thai spirit house and Tony's original giant clam shell \u00e9tag\u00e8res holding cascading spider plants. The \u00e9tag\u00e8res were originally designed for the producer Arthur Freed to display Phalaenopsis orchids in. As a final touch, I created the coral branch, Lucite, and iron chandelier to hang above the newly expanded space.\n\nThe Green Room, 2018. By the window is an antique French desk with ormolu mounts that was painted for Tony Duquette by Scarlett Abbott in a malachite pattern. The desk holds bibelots including a stuffed cockatoo wearing a miniature Thai crown in a Lucite case.\n\nThe Green Room, 2018. The sofa was designed by Tony Duquette for Doris Duke's \"Falcon Lair\" in 1952, paired here with Louis XV chairs. I created the coral branch, Lucite, and iron chandelier for this space in 2005.\n\n## CHAPTER THREE\n\n## The Gardens\n\nThe Upper Terrace, 2018. Lunch alfresco is often served here. Antique copper columns recycled from Los Angeles streetlamps hold up the lattice ceiling, and birdcages, carved Indian elephants, and a collection of antique Chinese ceramic pots complete the decoration.\n\nThe original terrace located off the bedrooms at Fiddler's Ditch, 1949.\n\n### Upper Terrace\n\nThe gardens at Dawnridge are laid out on three levels. The 15-by-40-foot upper terrace is really an extension of the Green Room; here, when the weather is fair, lunches and dinners are served at a long table under the green lattice trellis which is supported by pillars made from recycled copper streetlights and hung with antique birdcages and cast-resin Chinese lanterns that glow at night. Among the plants in diverse vessels\u2014from antique Chinese ceramic pots to mundane black plastic containers, some direct from the nursery, others stenciled by Tony in gold with patterns of lace\u2014that line the terrace are Tony's giant clamshell \u00e9tag\u00e8res, antique Thai spirit houses elevated on metal army surplus stands, Indian birdhouses, and a 1930s store-display cabinet that holds a terra-cotta lion.\n\nBeegle and her brother Ronald Johnstone on the Upper Terrace, 1975. Ronald Johnstone was a respected art director in New York City. As a student at Chouinard Art Institute in Los Angeles, he worked for Tony on special projects at Bullock's Department Store. It was Ronnie who introduced his sister to Tony.\n\nThe Upper Terrace, c. 1980.\n\nStaircase leading down from the upper terrace to the middle terrace. In the 1980s, the staircase was flanked by two carved wooden deer from Thailand. These have since been replaced with two objects: a wire, resin, and glass pagoda made for Adrian in the 1940s and a giant iron and resin obelisk made by Tony for the Los Angeles Opera Guild production of _The Magic Flute_.\n\nThe Middle Terrace, 2018. Tables are set under the Tony Duquette crystal and mirror ball chandelier, which he created for the James Coburn residence in the 1970s.\n\n### Middle Terrace\n\nThe large 30-by-100-foot brick middle terrace is dotted with pavilions and shaded by pine and eucalyptus trees. This is where we stage large dinners, under the crystal chandelier that Tony created for the James Coburn residence in the 1970s. (I received the chandelier as a gift from the new owners after the house was sold.) Hidden here and there are carved wooden Indonesian statues of toads holding umbrellas, bronze Garuda birds from Thailand, and numerous Southeast Asian birdcages and spirit houses. Tony's monumental 28-foot-tall _Phoenix Rising From Its Flames_ sculpture\u2014the centerpiece of his one-man exhibition for Los Angeles's Armand Hammer Museum of Art in 1984\u2014is prominently featured at the end of the terrace. There are two pavilions: The Tree House, which is built between two pine trees, and an Asian-inspired structure made from found objects that leads to a shady platform overlooking the lake. At night, all of these structures, statues, and objects are artistically lit, creating a magical environment suitable for entertaining.\n\nThe Middle Terrace, 2018. The garden is dotted with intricate sculptures and carved statues. One of them\u2014a giant carved wooden elephant from India (previous)\u2014was too large to fit inside the warehouse, so it ended up in the garden.\n\nTony Duquette's 28-foot-tall sculpture, _Phoenix Rising From Its Flames_ , 1995.\n\nEntrance to the swimming pool, 1995. The iron obelisks, which originally stood at the entrance to Dawnridge in 1949, were placed in the garden next to eighteenth-century Chinese grape-root grotto chairs from the Hearst Collection.\n\nThe Middle Terrace, 2018. The swimming pool was eventually bricked in to create a dance floor. The area is surrounded by walls of carved Indonesian gates, and includes organic leaf chairs created by Tony in the 1960s for the Hilton Hawaiian Village hotel.\n\nThe swimming pool, c. 1978.\n\nIn the 1970s Tony and Beegle put a swimming pool at one end of the middle terrace. The entrance to the pool was flanked by iron obelisks that had originally stood sentry on each side of the front door of the house in 1949. Tony later covered the obelisks with abalone shells and moved them to the ballroom on Robertson Boulevard before bringing them back to Dawnridge. At the far end of the swimming pool, he placed a small bridge\u2014purchased at an army-navy sale\u2014that held a shrine made of Thai carvings topped with a conglomeration of a sunburst, a cast resin whale vertebrae, and caution reflectors normally used on freeways. All of this and more went into the creation of this unique architectural feature. After I purchased the home in 2000, I removed the pool, replacing it with a dance floor, but left the obelisks and the bridge. Tony surrounded the swimming pool, now the dance floor, with arches made from Indonesian carvings. Before the Duquette Collections sale at Christie's, a pair of eighteenth-century Chinese grape-root grotto chairs from the Hearst Collection sat on each side of the entrance to the swimming pool.\n\nThe garden is studded with Tony's original pagoda-shaped lamps, which he created with electrical fixtures\u2014purchased from an army-navy surplus store\u2014stacked one on top of the other. Used oil drums, which Tony had pierced with designs and intricately painted, serve as towers dramatically lit from within, and decorative tassels made from discarded electrical fixtures and painted gun shells hang from beams and cornices everywhere you look. The contents of a defunct skateboard factory\u2014a fortuitous find\u2014provided Tony with an enormous supply of orange-painted skateboard platforms stenciled with black flames. He used these painted elements to create friezes and cornices on many of the garden structures at Dawnridge. With Balinese umbrellas and eighteenth-century Chinese grotto furniture added to the mix, the style is indescribable!\n\nThe Middle Terrace, 1995. The pavilion serves as an archway onto a platform that hangs over the raven below.\n\nThe garden path leads to Casa La Condesa.\n\n### The Tree House\n\nThe pavilion at the far end of the middle terrace from the pool is called the Tree House, as it is situated between two pine trees. I've had to rebuild it twice, using the same architectural elements originally chosen by Tony. The only items that constantly change in this space are the chandelier, the tablecloths, and the chairs; otherwise, the Tree House remains the same cool, shaded refuge conceived by Tony and Beegle for dining alfresco.\n\nThe Tree House, 2018. This space is used for lunch and dinner all summer long. The grape-root and glass table is draped in Tony Duquette for Jim Thompson Golden Sunburst fabric, and is surrounded by chairs designed by Tony and upholstered in malachite-printed cotton.\n\nThe Tree House, 2005. I installed Tony's Splashing Water chandelier and sconce for Remains Lighting. The table is set with Thai covered dishes and abalone shells on antique Chinese porcelains.\n\nThe Tree House, c. 1980s. Tony originally installed a pagoda chandelier, but would later move it to his Malibu ranch where it was destroyed during the fire in 1993.\n\nOutside of the Tree House, c. 1980.\n\nBromeliads, pagodas, and Tony Duquette Phoenix sculptures decorate the garden on the middle terrace, 2018.\n\nThe Garden Room, c. 1980. Although Tony never used the room as his office, he filled it with his favorite possessions, including his French desk, eighteenth-century apothecary boxes, Chippendale Chinese chairs, and oversize red-lacquer pagodas.\n\n### The Garden Room\n\nThe Garden Room, across the terrace from the Tree House, was built by Tony as an office, but it was merely an excuse for him to decorate a room with some of his favorite things, such as his Louis XV black-lacquered and ormolu-mounted French desk, his collection of eighteenth-century apothecary boxes, a pair of antique Chippendale Chinese chairs, and an oversized pair of red lacquer pagodas. He never used this room\u2014not even once. It was built around the brick fireplace of New Dawnridge, the only part of the destroyed house left standing after the fire. In 1995, he redecorated his garden office adding a pair of his original Phoenix sculptures and a pair of modern Snow Flake screens that he constructed using plastic fast-food baskets, the kind that hamburgers are often served in. Tony had his collection of antique beaded Chinese tassels petrified in resin, and placed a few of them in the office and lit them from behind. I've always thought the Garden Room was haunted by the ghost of the corpse found in the ruins after the fire at New Dawnridge, but if there is a ghost, she is benevolent, as the room is a calm and restful space and a beautifully cool place to relax on a hot summer day. Also, in the event of an overflow party crowd, the newly christened garden room gives us an additional area to set up dining tables.\n\nLooking into the Garden Room from the middle terrace, c. 1980. The entrance is flanked with antique Thai roof ends, Balinese parasols, Chinese pots, and a bronze Garuda bird.\n\nChat Thai, c. 1980, with its Thai roof ends, Gothic spires, and gables made of plastic materials used in the late 1970s to separate computer wires under office building floors, was used as a guest house down in the ravine at Dawnridge.\n\nThe three-tiered pagoda was used for outdoor dining by the Duquettes and their friends.\n\nTony on the roof terrace at Chat-Thai, c. 1980.\n\nThe white _Winter Sun_ and red _Summer Sun_ sculptures by Tony Duquette are situated in the ravine between the Chat Thai and Beeglesville.\n\n### Chat-Thai\n\nChat-Thai is the whimsical name of the Thai-style guest house that Tony built on the garden's lower level, in the ravine. Because it was a Thai house, to which Tony added Gothic spires and other architectural fragments\u2014and because our dear friend Patty Graham, who had also lived in France and whom we called \"the cat lady,\" lived there\u2014the house was christened Chat-Thai instead of _chateau_. The name was our double entendre for _Thai cat_ in French. The two-story house, with one large high-ceilinged room upstairs and an eat-in kitchen, bedroom, and bath downstairs, was linked to the middle terrace by a causeway known as the Bridge of Sighs. There was also a long ramp, called the Freeway, that descended from the house's second-floor balcony into the ravine. Every night, Patty would smooth out the sand beneath her bedroom window, and in the morning, she'd check for paw prints to see what critters came to visit in the night. In those days, there were many deer, rabbits, chipmunks, and raccoons living in the ravine, but with all of the construction in the neighborhood over the past twenty-five years, that is no longer the case. A few months after Ruth and I purchased Dawnridge in 2000, there were eight days of rain and a little puff of wind. The storm caused one tree to fall down, hitting another tree, and then another . . . like dominoes, until eight large trees fell and smashed Chat-Thai into firewood. The house was completely destroyed, but I was able to save the old red lacquer Thai roof ends to use on future constructions.\n\nThe Folly, c. 2000. After Chat-Thai was destroyed by falling trees, I built the lake at the bottom of the ravine and constructed a temple made of Indian carvings on the foundation of the old house.\n\nA detail of the iron spiral staircase leading up to the second-floor balcony, c. 2000.\n\nThe destruction of Chat-Thai gave me the opportunity to create a lake at Dawnridge\u2014something I had wanted to do since I first saw the property in 1972. One day, shortly after the lake was completed, while driving down Melrose Avenue, I saw the most amazing boat in a shop window. I slammed on the brakes, parked, and went inside, telling the proprietor, \"I'll take that boat.\" After I purchased it I asked, \"What is it?\" He told me, \"It's a Vietnamese wedding boat.\" I brought it home, painted it red and green, placed a carved lotus flower on its roof as a finial, and added two of the old Thai roof ends from the recently destroyed Chat-Thai, front and back, and launched it on the water. The boat immediately flipped over, and I realized it had not been built for the water; and probably came from the buffet table of the Siam Hilton. Its un-seaworthiness caused me to mount my diminutive yacht in a metal saddle, fixing it in place like the ornament it always was.\n\nThe folly on the shore of the lake, which I call the Temple, is nothing more than a stage set\u2014four feet deep\u2014with a wide deck for musicians and dancers to perform upon at parties. Chat-Thai was gone, and a viewpoint was needed. In my warehouse were three coral-colored arches that I'd purchased in India. I realized that they were carved on both sides, so I split them down the middle, making six arches, enough for the upstairs and downstairs of the temple. I also had eight copper spires for minarets, and purchased from India, I culled my collection of carved Indian screens and copper finials to create the balustrade. I decorated the iron spiral staircase with the leftover tops of the screens.\n\nThe sitting room, c. 2000. Tony stacked the eighteenth-century carved Chinese mirrors to create an elongated panel between the bookcases.\n\n### Beeglesville\n\nBeegle's painting studio, which we called Beeglesville, was also in the ravine. This was where Beegle painted the ethereal paintings\u2014oil on canvas and acrylic on board\u2014that delighted her friends and collectors. Beeglesville is a Chinese-style house with pagoda roofs salvaged from a Buddhist temple Tony liberated from the old Warner Ranch, where _Robin Hood_ and other epic Hollywood films were shot. The exterior was painted in jewel tones of jade green and coral, but it was the interior that was most distinctive. On the ground floor, there was an entrance hall with a ceiling decorated with panels made from gold plastic catering trays and mirrors. The hall also held an intricately carved antique Chinese opium bed, decked out with silk brocades and embroideries, along with Chinese lanterns for good measure. The floors throughout the house were layered with antique oriental carpets on top of a jade-green, wall-to-wall, flat-weave rug. A long Louis XV bench covered in leopard skin acts as a step up to the cozy pillowed and tasseled sitting area. This large room was outfitted with bookcases and Chinese giltwood mirrors carved with representations of Portuguese sailors, and furnished with sofas upholstered in Imperial yellow damask. The d\u00e9cor combined blue-and-white porcelain, taxidermy birds of paradise under glass, and eighteenth-century Vizagapatam ivory objects. An elegant antique French wrought-iron railing led visitors upstairs to the large painting studio, which was stacked to the ceiling with Beegle's canvases in various stages of completion. Here, sturdy English Regency tables, Victorian cane-work chairs, and brass-shaded bouillotte lamps set the mood. The bedroom, which was never used for sleeping, held a fourteenth-century Italian canopied bed from the collection of the Baroness d'Erlanger, which sat in the middle of piles of books that blocked admission to all but the most intrepid.\n\nThe entrance hall, c. 1980. The hall contained a pillowed and tasseled sitting area, complete with a carved antique Chinese opium bed covered in silk brocades and embroideries.\n\nThe sitting room, c. 1980. Tony set a taxidermized bird of paradise in a glass case, his poor man's version of the diamond, sapphire, ruby, and emerald creation he had seen in the house of his friend the Maharani of Cooch Behar. Tony's abalone chandelier, which now hangs from the balcony's ceiling at Dawnridge, was originally created for his one-man exhibition at LACMA where it was purchased by his client Mary Gross, and later repurchased by Tony to decorate Beeglesville.\n\nThe north end of the entrance hall gallery, 2018. The black lacquer and gold Chinese screens conceal our home office on the left directly across from the library on the right. The pagodas on stands are by Tony Duquette; the light fixture is a Tony Duquette \"California Sunburst\" for Remains Lighting.\n\n## CHAPTER FOUR\n\n## THE CASAS\n\n### CASA LA CONDESA\n\nIn 2011, Ruth and I moved into the house that we designed and built on an empty lot next door to Dawnridge. Unlike the other lots on the street, where the land had been graded to make room for tennis courts and swimming pools, ours, just like Tony and Elizabeth's before us, was not much more than Fiddler's Ditch. Our dream was to have a house with southern exposure, at least one high-ceilinged room, three bedrooms, and three bathrooms. We struggled with the lot; we considered building two pavilions connected by a bridge over the ravine and other ideas that ultimately didn't excite us. Finally, after many unacceptable go-rounds, we discussed our plight with our friends Kacey and Peter McCoy over dinner. Peter, who's considered the best builder in Los Angeles, came up with a simple solution: \"Go downstairs to the drawing room. Use the ravine to its best advantage.\" With those words of wisdom, the entire project instantly fell into place. I drew exactly what I wanted and handed my scale drawings to Sid Drasnin, a noted Los Angeles architect who'd been a young draftsman in my grandfather's architectural office. Sid was then in his eighties, but he ran with my ideas and drew my plans so that they could actually be built. What emerged was Casa La Condesa, named for Ruth, the Spanish Countess of Alastaya.\n\nThe house has a formal entrance\u2014accessed through eighteenth-century Spanish gates\u2014that's suitable for gala parties. As guests climb a short flight of red-carpeted steps into the garden, they're able to see festivities happening within. To the left is a treillage pavilion, which on occasion holds an orchestra, and to the right, the red carpet leads across a causeway to a door that opens onto the interior stairs. The doorway is framed by palm trees that Don Loper designed for the lobby of the Beverly Hills Hotel. We purchased these iconic 1940s architectural elements when the hotel was redecorated by its new owner, the Sultan of Brunei.\n\nOur daily entrance, which is used constantly and requires no fanfare other than the ringing of a bell, is reached from the street through a gated courtyard and opens into the entrance hall. The front doors are tall sheets of glass studded with eighteenth-century Venetian bronze blackamoor door pulls; I even added my own bronze sunburst escutcheons, specially made to enhance them.\n\nThe fa\u00e7ade of Casa La Condesa, 2018. I created the gates using iron window frames from an old Spanish house and added Napoleonic French iron decorations and door pulls from Syria. The windows above the entrance door\u2014with their nailed Moroccan grillwork shutters\u2014echo the windows on the front of Cow Hollow, the Duquettes' house in San Francisco.\n\nThe entrance hall, 2018. The walls are paneled with carved teakwood screens in a traditional Chinese snowflake pattern and lined with mirror. An eighteenth-century miniature Lacca Povera Venetian desk is featured prominently in the hall.\n\nPowder room, 2018. The powder room is located behind one of the teakwood and mirror jib doors in the hall, and paneled with slabs of Brazilian amethyst. Tony's abalone-and-amethyst-encrusted chandelier, c. 1960s, hangs from above.\n\n### The Entrance Hall\n\nFor years, Tony and I had discussed making a room in black, coral, and gold, but we never did. One day, I suddenly realized that I had done just that\u2014subconsciously\u2014and I smiled. The floors are polished black granite, and the color scheme throughout the house is predominantly black, coral, and gold.\n\nThe walls of the entrance hall are paneled with carved teak-wood grills, which act as jib doors. Behind one of these secret doors is a powder room paneled with slabs of Brazilian amethyst. Here hand-tinted nineteenth-century prints of the Brighton Pavilion are displayed, as well as another view of Brighton embroidered in black silk on white satin; there is also a panel of cut velvet Chinese tribute silk on the back of the door. The jewel, however, is Tony's abalone-and-amethyst-encrusted chandelier, circa 1950, which hangs from a silver-gilt sunburst on the ceiling. The other jib door leads to an elevator hall and coat closet.\n\nAcross from these hidden doors is the entrance to the dining room. I don't like looking into a dining room the minute you walk through the front door, but sometimes it can't be avoided. In the case of Casa La Condesa, I solved the problem with a pair of sliding grillwork doors that disappear into the wall. I then backed the grills with a two-way mirror so that people entering the house can't look into the dining room and see the table _en fete_ . . . but whoever is inside the room can look out. The entrance hall features a Tony Duquette for Remains Lighting California Sunburst chandelier; also a miniature eighteenth-century Venetian Lacca Povera desk, placed on a dais; a pair of Chinese vases on lighted stands; and a little skeletal Italian saint who warns people to watch their step to avoid tripping up into the stair hall. Opposite the entrance to the library is an antique chinoiserie lacquered screen mounted as pocket doors, which open to reveal our home office. On either side are white Tony Duquette pagoda lamps presented on tall octagonal bamboo stands.\n\nOne of a pair of antique Chippendale Chinese mirrors hangs over one of the two ebonized Victorian console tables on either side of the stair wall.\n\n### The Stair Hall\n\nThe stair hall\u2014with its twenty-foot-high window looking out over Dawnridge's gardens, the infinity-edged swimming pool, and the lake with its enchanting Vietnamese boat below\u2014is adjacent to the entrance hall. From here, if you are arriving at a party, you can look down into the Drawing Room and see the guests and what they're wearing before descending the staircase and making an entrance. It's also from this vantage point that you can look across the drawing room and see into the library on the left and the dining room on the right. Otherwise, if you choose, you may look over the stair railing down to the dolphin table or up to the cascading chandeliers that hang one after another, repeating five times for three stories. At each end of the hallway is a console table, which I created by cutting an ebonized Victorian center table in half, mounted on a black lacquered stand. The antique gilded Chippendale Chinese mirrors hanging above were painted white by a previous owner; the Chinese cache pots in a celery pattern are part of our extensive collection of export porcelain.\n\nFrom the hallway, guests can look over the balcony to the drawing room to see who's there and what they're wearing before going down to the party and making an entrance.\n\nI have hung the library with ancestral portraits and family heirlooms from the Spanish Counts of Alastaya. Our portraits hanging on each side of the stairs are by Juan Fernando Bastos. These were commissioned after His Majesty, King Juan Carlos of Spain, recognized me as the hereditary Count of Alastaya, a noble title passed down through my mother's family since 1760.\n\n### The Library\n\nThe library looks directly across the drawing room to the dining room, and the ceiling spans all three rooms\u2014without any crown molding breaking up the space\u2014giving it the feeling that it's a one-room house. The walls of the library were painted coral, and the ceiling, like the rest of the street-level rooms, a soft amethyst. We had originally planned to fill the room with books, but I later changed my mind and displayed a collection of Spanish colonial paintings inherited from my family. My Spanish ancestor, Don Gonzalo Nieto Principe, came to the New World in 1635 as the perpetual governor of Potos\u00ed, a town in the Andes Mountains of Bolivia. At the time, Potos\u00ed was the richest city on Earth, thanks to its silver mines, and larger than Paris or London. My South American family didn't move away until 1928, when my mother and her brothers came to Los Angeles. All of my South American treasures came from the various haciendas\u2014a total of ten\u2014that my family owned in Bolivia and Peru.\n\nWe added a sofa that was designed in the 1950s for Doris Duke and eighteenth-century Venetian chairs in patchwork suede. The cocktail table is made of petrified wood slices and topped with a petrified tortoise surmounted by an articulated vermeil lobster on a Lucite stand. The carpet is an antique Chinese picture rug; the curtains are made with Tony Duquette for Jim Thompson Asia Major woven silk. In the left corner, the bookcase holds an eighteenth-century Spanish colonial statue of St. Joseph dressed in a silver repousse costume. The painting hanging above depicts my ancestor, the second Conde de Alastaya, and his wife praying like crazy that they won't go to hell. In the opposite corner, a number of my South American treasures are displayed: Spanish colonial columns holding figures of archangels dressed as conquistadors, a seventeenth-century Venetian traveling desk with a red lacquered chinoiserie painted interior, photographs of my family, antique ivory Chinese pagodas, an antique ivory tower from India, and a twentieth-century Chinese bone horseman.\n\nView from the library over the drawing room and into the dining room beyond. The sofa, positioned between the two bookcases, was designed by Tony Duquette for Doris Duke in the 1950s and is paired with eighteenth-century Venetian chairs and a cocktail table made of petrified wood with a gilded articulated Japanese lobster atop a petrified tortoise shell as decoration. Hanging above both bookcases are more ancestral portraits and heirlooms.\n\nThe library, 2018. Framing the fireplace alcove are two eighteenth-century Venetian dolphins from the collection of Misia Sert, and hanging directly above the hearth is my family coat of arms, as well as a portrait of King Carlos III of Spain.\n\nThe fireplace alcove is made of sandblasted travertine and holds two eighteenth-century Venetian dolphins from the collection of Misia Sert. Above the hearth hangs the original eighteenth-century rendering of my family's coat of arms, created in Spain in 1769 for the Counts of Alastaya. Below that is an accompanying portrait of King Carlos III of Spain holding a document in his hand that reads, \"To my friend, The Count of Alastaya.\" In 2008, King Juan Carlos of Spain bestowed the hereditary Spanish titles Conde y Condesa de Alastaya to my wife and myself.\n\n### The Dining Room\n\nAlso located off the entrance hall is the dining room, which is centered by a large oval glass-topped table, the base of which I created by placing two of the palm tree capitals from the Beverly Hills Hotel together, back-to-back. The chairs are eighteenth-century Venetian in a coral lacquer and gold finish, upholstered with Tony Duquette for Jim Thompson Asia Major in coral silk. The walls are painted in coral, with the indented areas in gold leaf. The late seventeenth-century Venetian paintings hanging in the room are part of a series of eight; the other six are in the Drawing Room. The ceiling fixture is a Tony Duquette for Remains Lighting cast-resin sunburst placed over a recessed light. A monumental twelve-panel, eighteenth-century Coromandel screen was purchased from the venerable Los Angeles antiques dealer Joel Chen, and then split apart into groups of three to be placed in the four corners of the room. A swing door opens into a black lacquer and mirrored butler's pantry, which leads into a large kitchen, paneled in white oak and decorated with blue-and-white Chinese porcelains.\n\nThe dining room, 2018. The glass-topped dining table was created by putting two gold palm tree capitals from the Beverly Hills Hotel lobby together to form a pedestal. Surrounding the table are eighteenth-century Venetian coral lacquer dining chairs.\n\nThe dining table, 2018. The palm tree centerpiece is by Codognato and paired with silver obelisk form salt and pepper cellars made by Tony Duquette for Dawnridge, rock crystal obelisks, and gilded carvings of Indian potentates riding elephants. The table is set with antique Coalport china, French silverware, and Venetian glass and old Baccarat crystal goblets.\n\nThe dining room, 2018. The console table was also made with one of the gold palm tree capitals from the Beverly Hills Hotel lobby, and holds an assortment of objects, including a collection of antique Chinese lacquered coral branches from the collection of Elsie de Wolfe.\n\nThe table is set with vermeil chargers, antique Coalport china, Gorham Mythologique silver flatware, and Baccarat crystal. The centerpiece\u2014depicting Egyptians and a pond studded with lapis lazuli scarabs and a large gilded palm tree in the center\u2014was made along with the other eight candlesticks by the vaunted Venetian jeweler Codognato. I added the rock-crystal obelisks, Tony's original salt and pepper shakers in the form of obelisks, and a group of gold-leafed Indian potentates on elephants to better evoke the exotic mysteries of the East.\n\nThe gold-leafed alcove includes two of the eight seventeenth-century paintings of Venice by Heinze. The console table was also made using one of the palm tree capitals originally in the lobby of the Beverly Hills Hotel, and it now holds a collection of antique Chinese lacquered coral branches, a gilded nef on a persimmon lacquer Japanese table, and antique gold rats from Japan. Tony told me that in Japan, if you're rich enough to leave crumbs on the table for rats to eat, then you're considered really rich.\n\nThe drawing room, 2018. The stairs leading to the drawing room end on a raised platform with a small table and sitting area for intimate dining. The trunks of the famous Beverly Hills Hotel palm trees act as decorative moldings.\n\n### The Drawing Room\n\nDescending the staircase, you pass beneath fantasy murals of Venice painted by Los Angeles artist Scarlett Abbott. I asked Scarlett to make them \"a little bit De Chirico,\" and she did. I told the contractor that the staircase should look \"like a sculpture, no matter what it cost,\" so the entire construction of the stairs was engineered and built as if it were a musical instrument. The railing, which I designed, took four different companies to complete\u2014the iron mongers dropped like flies trying to get it right.\n\nThe stairs end on a raised platform with a seating area for intimate dinners and room for a folding bar. There is also a banquette under the stairs, reserved for seduction. A door on the left leads to a powder room, which doubles as a bathroom for the swimming pool. The door on the right leads to a catering kitchen that allows us to serve parties all the way out to the gardens at Dawnridge.\n\nStepping down into the drawing room, the stairs are flanked by a pair of enormous seven-foot-tall narwhal tusks, a gift from the fashion designer Gustave Tassell. The walls are painted coral, and the indents that hold the Venetian paintings have been gold-leafed. There are two seating arrangements using sofas designed by Tony for Doris Duke, paired with eighteenth-century French fauteuils that have a whitewashed finish and are upholstered in Tony Duquette for Jim Thompson Royal Ermine. The other four antique French armchairs are upholstered in silk velvet, resembling tiger skin, woven in Lyon, France, on looms created for Marie Antoinette. Tony purchased a bolt of this extraordinarily rich fabric on a trip to Lyon with Hubert de Givenchy. Tony always kept the bolt of fabric in a vault for future use. \"It's my money in the bank,\" he said, and it was always too good for him to use. The minute he died, I upholstered everything I could lay my hands on with that fabulous fabric.\n\nThe drawing room, 2018. The space includes two symmetrical seating arrangements, with sofas designed by Tony, circa 1950s, and eighteenth-century French chairs. The late seventeenth-century paintings by Heintz are from the collection of the Baroness d'Erlanger and hang on gold-leafed walls.\n\nThe coffee tables, which I created, are inset with eighteenth-century panels of imperial yellow Chinese brocade woven with five-clawed dragons. The four corners of the room are dominated by two pairs of eighteenth-century Venetian blackamoor statues. These seven-and-a-half-foot-tall statues were purchased for my clients John and Dodie Rosekrans. It took four men to move them into their Tiepolo drawing room at the Palazzo Brandolini on the Grand Canal in Venice. I originally bought only two, but when I took them back to the palazzo, I realized that they were both left-handed. I called the dealer and asked, \"Are there any more? These are both left-handed.\" She checked with the prince at Palazzo Balbi, and he said, \"Yes, there were four. I gave two to my sister and kept two for myself.\" Well, the prince had broken the pair! Fortunately, the dealer was able to get the other two from the sister. After Dodie passed away, I purchased all four from her family and now they are happily ensconced at Casa La Condesa. Truth be told, the room was designed around my collection of Venetian paintings and my dream of eventually getting these blackamoors. I believe in creative visualization; it has worked for me not just where blackamoors are concerned, but in every aspect of my life.\n\n### The Staircase\n\nClimbing the elliptical staircase from the Drawing Room, you pass under the two-story family tree of the Counts of Alastaya, which was painted on the wall by Scarlett Abbott to my specifications. The mural traces the lineage of my noble and illustrious ancestors from 1769 to the present. I don't feel any family tree is complete without the addition of monkeys, rodents, lizards, and at least one hornet's nest, so I made sure these were all included in the finished mural. Reaching the top of the stairs, you arrive at a wide landing that doubles as a twelve-foot-tall library in emerald-green lacquer. In the corner is one of my favorite Victorian chairs. Standing at the railing you can look out the window and see a straight-on view of the rose garden and gazebo to the north. There is a large oval skylight that mirrors the shape of the staircase below; hanging from its center, through a silver-gilt sunburst, are five matching chandeliers that cascade down, one after the other, for three stories.\n\nThe staircase, 2018. Artist Scarlett Abbott painted the Giorgio de Chirico-esque mural that spans three stories, and includes the family tree of the Counts of Alastaya.\n\nThe stair landing, 2018. The third-floor stair landing has become a green lacquer library, and features the magnificent chandelier that cascades down to the drawing room.\n\nVestibule, 2018. One of the two vestibules painted by Scarlott Abbott.\n\n### The Vestibules\n\nThere are two doors that lead to small vestibules\u2014painted with fretwork by Scarlett Abbott\u2014which open into three bedrooms, with attached bathrooms and dressing rooms. Ruth and I use one bedroom to sleep in, and we each get a bedroom and dressing room to make a mess in. Ruth's sitting room is more organized than mine; hers is always neat and very pretty. Mine is always a mess of magazines, dogs, and scraps of paper. What more can I say?\n\n### Hutton's Sitting Room\n\nI chose the bedroom to the right of the vestibule for my sitting room. This is where I can make a mess (which is what I usually do), watch television, and take a nap when necessary. My room is carpeted wall-to-wall in zebra stripes, and the ceiling is gold-leafed. The draperies are natural linen shot with gold Lurex; they cover sliding glass doors that open onto terraces on both sides of the room.\n\nI paneled the room with book-matched mahogany and installed my prized prints of Yuanming Yuan, the pleasure palace designed by Giuseppe de Castiglione for the Chinese emperor Qianlong. Castiglione, an Italian Jesuit priest, told the emperor about Versailles, and the old boy\u2014as emperors are wont to do\u2014decided that he wanted one just like it! During the Second Opium War in 1860, the incredible palace was partly destroyed by the English, French, American, and German armies under the command of Lord Elgin. Whatever was left\u2014and it was a lot\u2014was totally obliterated by the Western armies as a punishment for the Boxer Rebellion in 1900. The destruction of this Chinese cultural treasure is considered one of the greatest acts of vandalism of the first half of the twentieth century.\n\nSitting room, 2018. My sitting room is paneled with a collection of twenty prints depicting my favorite seventeenth-century Chinese palace Yuanming Yuan.\n\nMy Art Deco-inspired black marble bathroom, 2018. The walls are upholstered in faux crocodile and hung with my collection of verre \u00e9glomis\u00e9 Chinese reverse paintings on glass to mimic Elsie de Wolfe's own bathroom at Villa Trianon. The carpet is Chinese imperial, and the fixtures are gold nuggets made by P.E. Guerin, which is something I wanted since I was eighteen.\n\nOn his first trip to Paris in 1947, Tony visited Misia Sert, and she shared the legend of this half-French, half-Chinese palace\u2014she famously owned bronze heads of a rat and a rabbit that had been pillaged from one of its fountains\u2014which he later told me. Many years later, while walking in the Place des Vosges, we stopped in our tracks, riveted by an art gallery window. \"Look, Tony,\" I hollered with excitement. \"It's what we've always been looking for\u2014that Chinese palace you're always telling me about!\" We went inside and discovered that it was not an art gallery, but a book publisher called Le Jardin de Flore. When Emperor Qianlong completed his palace, he'd commissioned twenty of the finest engravings on the finest paper and created a folio to send to all the crowned heads of Europe to prove he was not a barbarian. There are only three of the original folios left in the world\u2014one at the Louvre, one at the Metropolitan Museum of Art, and one in a private collection. Le Jardin de Flore published a facsimile in an edition of two hundred at $200 each. That was a very expensive book in the 1970s, but we splurged and purchased two copies\u2014one for me and Ruth, which we immediately framed and hung in our dining room in Hollywood, and one for the Duquettes. Tony never opened his, and after his death, Dodie Rosekrans bought it for $17,000 at the Christie's sale. After Dodie's death, the book sold in her Sotheby's auction for $20,000. I've never visited the ruins of Yuanming Yuan near Peking, the remains of one of the only examples of reverse chinoiserie in the world, but I'm still obsessed with it.\n\n### Ruth's Sitting Room\n\nRuth's sitting room is much grander than mine. Besides the fact that it has its own fireplace, the room is entirely muraled with images by Elizabeth Duquette. We purchased the two paintings hanging on the mirror over the fireplace at auction, and I decided to have them blown up ten feet tall to create the murals. It was only later that I realized these diminutive paintings were studies for the murals Beegle created for the Twentieth Century Fox film _Goodbye Charlie_. I knew the actual murals well, as I had sold the 15-foot-tall originals to the late Leona Helmsley for her house in Arizona.\n\nMy other triumphant adaptation was the fireplace. I needed a 4-foot-wide marble fireplace and couldn't find one for love nor money. Finally, I found a charming trumeau mirror and realized that if I cut off the bottom it would become a perfect Georgian fireplace, so that's what I did. A collection of Meissen porcelain powder boxes decorates the top of the mantel. The eighteenth-century Louis XVI _lit \u00e0 la Polonaise_ was the first piece of furniture Ruth ever purchased; it was from the collection of actress Ina Claire, who purchased it from Elsie de Wolfe. Ina told us that she had based her character Grand Duchess Swana in the movie _Ninotchka_ on her pal Elsie de Wolfe. There are lots of pretty things in this room, not least of which are the pair of eighteenth-century Venetian blue lacquer armchairs from the collection of Adrian, the three tiny eighteenth-century children's chairs, and the water drop chandelier, circa 1960, by Tony. This is the place where Ruth hangs out with our two West Highland White Terriers, Piper Dundee and Kippy of the Cavendish.\n\nRuth's bathroom is a symphony of green Persian marble reflected in three mirrored walls. Scarlett Abbott painted the crown moldings and the back of the door to match this marble, and she also decorated the cabinets on each side of the bathtub with elaborate scenes. Across the top of the doors and sides, she depicted the four seasons as separate bouquets of flowers, and under these she painted four continents: America, Asia, Europe, and Africa. America is represented by a painting of our former Georgian house in Hollywood; Asia by the Taj Mahal; Europe by our beloved St. Mark's Square; and Africa by the Temple of the Winds, which we visited in Tunisia. These designs are subordinate to the main thrust: portraits of our four puppies, two on each cabinet. Jip, our first, is no longer with us; his brother Argyle, our second, is also deceased; and Piper Dundee and his sister Kippy of the Cavendish. Between the portraits are their coats of arms, featuring dog bones and Scottish plaid. The inside panels are painted with the Cavendish coat of arms\u2014heralding _Bonefide_ \u2014and the Dundee coat of arms with the inscription _Barke Dieum_. Completing the iconographic program, also painted by Scarlett, are vignettes of many of Ruth's favorite things, including Venetian glass vases, an eighteenth-century beaded pomandore, her bottle of Poiret Nuit de Chine perfume, and a carved malachite bear. Scarlett also decorated the vanity with Ruth's cipher and representations of Neptune and Venus in decorative oval plaques.\n\nSitting room, 2018. Ruth's sitting room includes murals created from enlarged photographs of Elizabeth Duquette's original paintings, two eighteenth-century blue lacquer Venetian chairs from the designer Adrian, and an eighteenth-century Louis XVI lit \u00e0 la polonaise.\n\nRuth's Persian green marble bathroom is mirrored on three sides, and features painted cabinets and murals by Scarlett Abbott.\n\n### The Master Bedroom\n\nThe master bedroom sits between our dressing rooms and sports hand-painted wallpaper that was created for us in India to resemble the Gardens of Shalimar, replete with exotic birds and flowers. The two Louis XVI\u2013style chests were also made for us in India and are covered with repouss\u00e9 metal that has been gold- and silver-plated. The two chinoiserie mirrors are from the 1960s, and the antique _lac burgaut\u00e9_ mother of pearl inlaid cabinets on both sides of the bed are Korean and were purchased from the Doris Duke estate sale. The bedspread is composed of antique Chinese embroideries that have been cut out and reapplied on quilted satin. The armchairs are by Billy Baldwin in their original emerald-green velvet. The bronze doorknobs in the form of hands were purchased on our first trip to Venice in 1977; we stored them until we finally had just the right house to use them.\n\nMaster bedroom, 2018. I had the wallpaper hand-painted on silk to my specifications to portray murals inspired by the Gardens of Shalimar. The bedspread was created from antique Chinese embroideries that were cut out and appliqu\u00e9d onto quilted satin.\n\nThe pavilion, 2018. The structure was made to hold an onion dome from the 1959 MGM film _The Gazebo_ , and was constructed entirely out of wrought iron, except for the dome. I found the iron caryatid columns in a pile at Tony's ranch and designed the pavilion around them.\n\n### Gardens and Terraces\n\nWhile Casa La Condesa was under construction, we purchased the property next door, a double lot that gave us another garden to the north and a three-bedroom guest house. Connecting Casa La Condesa to the new garden proved to be a challenge. The elevation of the garden was higher than the house, and there was a gully between them. Our friend Juan Prieto, who was visiting from Boston, stepped in and suggested that we build a causeway between the garden and stair landing. This was a simple solution that conjured up visions of one of my favorite places in India: Akbar's Tomb. In that wonderfully exotic place, one can look down from causeways onto a Mughul garden where giant stars are bordered in marble. In the days of Shalimar, those marble-bordered stars would have held fragrant roses and other flowering plants. The experience at Casa La Condesa is similar, although the marble-framed stars seem to have eluded us.\n\nThe rose garden is now a second ceremonial entrance for events at the house. The pavilion was constructed using an onion dome from the 1959 MGM film _The Gazebo_ , starring Glenn Ford and Debbie Reynolds. I created the columned metal frame using antique cast-iron caryatid columns, which I found in a pile of architectural fragments at Tony's Malibu ranch.\n\nThe pool terrace, 2018. The terrace features two jacaranda trees alongside coral-colored grotto chairs molded in cast resin from the eighteenth-century originals, which were part of the Hearst Collection.\n\nLunch on the terrace is set up using malachite and lapis lazuli-painted Limoges and Baccarat crystal.\n\nMeanwhile, a narrow strip of land to the south of the house holds the swimming pool terrace, consisting of a paved deck with two planters holding jacaranda trees and jasmine. The coral-colored grotto chairs were molded in cast resin from the eighteenth-century originals Tony purchased from the Hearst Collection in the 1950s. There are also raised planters on both sides of the black granite landing near the Drawing Room doors, filled with giant bird-of-paradise, jasmine, and bronze-colored flax. Another raised planter running under the lattice-covered retaining wall contains more giant bird-of-paradise, blue agave, jade trees, and jasmine; the antique Chinese pots surrounding the infinity pool and lining the terrace hold agapanthus, geranium, flax, and ficus trees. On festive occasions, glowing Indian lanterns hang from the trees, while Moroccan lanterns line the pool and the red carpeted stairs leading down from the Drawing Room. We often invite friends over for cocktails at our house, and then lead them past the swimming pool and into the Dawnridge garden next door for dinner followed by dancing upstairs in the original Duquette house. But sometimes we reverse the process and have cocktails at Dawnridge, then proceed down to the garden for dinner and over to Casa La Condesa for dancing, just to break up the monotony.\n\n### CASA DEL CONDE\n\nForty years ago, Tony told me that there was a house up the street that had eighteenth-century French boiseries installed in the dining room; the owner turned out to be an Italian gentleman, Ambassador Conti. One day, I was driving to a decorating job in Santa Barbara when my cell phone rang. It was my real estate agent, Nandu Hinds, calling to tell me that the house had just been put on the market. My immediate response was, \"Buy it.\" \"What do you mean?\" she asked me. I said, \"Buy it. Offer them full price, all cash, no contingencies, and no exceptions.\" She did, and the offer was immediately accepted. I was leaving for Europe three days later, so told Nandu, \"I'd really like to see what I bought before I leave town.\"\n\nAn appointment was made for the day before my departure, and Ruth and I saw for the first time what we were getting. It was a charming south-of-France style house in white stucco, which lent itself beautifully to antique French furniture and decoration. In fact, there was eighteenth-century French paneling in the dining room, so Tony's story from so many years ago was accurate. There were three bedrooms and bathrooms and an expansive garden, which we immediately decided to annex onto the still-under-construction Casa La Condesa next door. Dawnridge had become an interesting compound of three united properties, and Casa del Conde would serve as the guest house. After several years, I decided it was time to make the most of what we had at Casa del Conde, and we added a courtyard entrance and a garage.\n\nI consider the courtyard entrance to the house as its own room. For me, this area\u2014with its glamorous oval swimming pool and flickering gas light illumination\u2014is for all intents and purposes the entrance hall.\n\nEntrance hall, 2018. The courtyard was created by constructing a new garage in front and building an oval Jack Wolfe-inspired pool in the center. It's a cool, shady place to relax. The north and south terraces, which are more private, are reserved for sunbathing. The area is decorated with iron furniture from Italy, framed mirrors, and Art Deco sculptures of sea horses in white plaster.\n\n### The Entrance Hall\n\nI consider the new courtyard, with its oval-shaped pool and water-spouting dolphin, to be the entrance hall of the house. This elegant outdoor room is decorated with framed mirrors, panels of antique Portuguese _azulejos_ , Art Deco seahorses, and an iron chandelier hanging from a tree near the front doors. There are tables and chairs, chaise longues and marble-topped side tables\u2014enough furniture to comfortably entertain a large group of friends. The rooms that can be accessed off the courtyard include a large family room, a mud room leading to the kitchen, and the actual entrance hall. Glass front doors lead directly into the drawing room through a Coromandel-paneled vestibule.\n\nA pair of carved wood statues depicting Thai dancers on stands preside over the two corners of the hall.\n\n### The Drawing Room\n\nThe large drawing room is separated from the Coromandel vestibule by mid-century metal screens featuring fleur-de-lis decorations that I found in Tony's collection of architectural fragments at his Malibu ranch. The ivory-and-blue-colored Coromandel panels on either side of the fireplace are antiques purchased in Paris from C. T. Loo, the venerable Chinese dealer off the Boulevard Haussmann. Inspired by Elsie de Wolfe's famous ivory Coromandel screen, I commissioned Scarlett Abbott to paint an antique screen from my collection to match the ivory-and-blue panels in the living room. The fireplace is surmounted by an eighteenth-century Venetian mirror and is flanked by eighteenth-century figural stands holding Chinese foo dog\u2013patterned ginger jars. The room is furnished with a pair of custom banquettes placed diagonally across from each other, French armchairs, woven zebra rugs, blue agate tables, and simple lighting. Paintings by Elizabeth Duquette have been placed throughout the house, with two strong examples at the end of the drawing room, flanking the doorway into the dining room.\n\nThe drawing room, 2018. The fireplace is framed by two eighteenth-century figural stands from the collection of James Pendleton; an eighteenth-century Venetian mirror is positioned above.\n\nHanging above one of the two custom-designed corner banquettes are antique ivory Coromandel panels from C.T. Loo and a painting by Elizabeth Duquette. The coffee tables were created from a large slab of blue agate set in an iron framework.\n\nThe dining room, 2018. A Venetian screen, painted by Elizabeth Duquette, was placed in the corner to mimic the curved wall opposite it and create the illusion of a circular dining area.\n\n### The Dining Room\n\nThe dining room is paneled with eighteenth-century French boiseries that were brought in by the original owner in the 1930s. I updated the paneling by painting it a mouse-gray color, but I left the door that leads into the kitchen the original wood finish. The ceiling is painted a Schiaparelli pink, and the corner banquette features a linen slipcover in the same color. There's a mirrored bar behind the double doors, and silver-leafed Chinese Chippendale cabinets hold a collection of ceramic dishes, tureens, and covered bowls. The paintings in the room are by Elizabeth Duquette, as is the painted Venetian screen, which I've placed in the corner, creating a similar curve to the window facing it. This illusion of a circular end to the room allowed me to put a round carpet under the Lucite and glass table, providing a sense of symmetry to a room that is not at all symmetrical.\n\nThe dining room, 2018. The eighteenth-century French boiseries were installed in the 1930s by the original owner. In the corner, a banquette creates an area to enjoy cocktails near the bar.\n\nA pair of silver-leafed Chippendale display cabinets surmounted by ebony mirror frames inlaid with bone flank the eighteenth-century carved French door leading to the kitchen.\n\n### The Family Room and Kitchen\n\nA large modern kitchen is accessed from the dining room through the eighteenth-century paneled door. Besides all the modern conveniences, including a hidden laundry room and a bar counter with stools, this space has room for a round table and eight chairs, as well as a sofa for watching television.\n\nThe wide hall leading from the kitchen passes through a slate-tiled mud room just off the family room. This comfortably decorated space doubles as a media room and can easily convert to a guest bedroom that has its own closet and bath. With sliding glass doors looking onto the swimming pool on one side, and its own private courtyard for sunbathing on the other, it is one of the most serene rooms in the house.\n\nThe kitchen, 2018. At Casa del Conde, the kitchen is the hub of the house. Located between the family room and the dining room, it is where guests congregate for breakfast to plan their days.\n\nThe family room, 2018. The family room, which has its own bathroom, has come in handy as an additional bedroom, and because it's centered between the swimming pool terrace and a sunbathing terrace, is in use all day long as a media room when guests are present.\n\nMaster bedroom, 2018. This room is always reserved on a first-come first-served basis because not one of our visiting friends is more important to us than any of the others. With its own wood burning fireplace and little garden, the master bedroom is a comfortable escape when the house is full.\n\n### The Master Bedroom\n\nIf you take two steps down from the entrance hall and turn right, you'll come to the master bedroom, arranged for the pleasure of our guests. We restored the original fireplace\u2014which a former owner had covered over\u2014and the large bathroom was rearranged as a sybaritic retreat. As in all the other rooms, our extensive collection of paintings by Elizabeth Duquette decorate the walls. We added a pair of 1930s chinoiserie ivory-painted bedside tables and some of our favorite chow fun Asian antiques, including a red-lacquer-and-gold cabinet and a pair of ivory-inlaid tables holding Satsuma porcelains and turquoise ceramic foo dogs.\n\n### The South Terrace\n\nCasa del Conde has terraces on three sides. The south terrace\u2014a sunny place perfect for entertaining and relaxation\u2014can be accessed from all of the main rooms. It's paved in green slate and decorated like an exterior living room with iron furniture from Italy imported by our friends, the late John and Louise Good. Chinese ceramic pots hold succulents, geraniums, and bromeliads, while greenery cascades out of tall iron stands.\n\nThis terrace is set up like a living room with its slate floor and comfortable Italian iron furniture. Accessible from almost every room in the house, the south terrace is a sunny and inviting place for entertaining whether it's breakfast, lunch, cocktails, or dinner.\n\n## Afterword  \nTHE PLEASURE OF YOUR COMPANY\n\nFestivities in the garden at Dawnridge are always special. Today, our staff wears gold lam\u00e9 coats with feathered turbans to set the mood. Parties with themes like \"Return to the Raj,\" \"Tropical Nights,\" \"Walk like an Egyptian,\" and anything \"Venetian\"\u2014not to mention dozens of black-tie affairs\u2014are _de rigueur_ at Dawnridge. There is always music in the garden, whether it's piped in or provided by musicians: jazz bands, balalaika players, flamenco guitarists, ten-piece dance orchestras, troupes of African drummers, or Balinese dancers with their accompanists. The entertainment is always diverse and the d\u00e9cor constantly changing.\n\nOur friend Eleanor Phillips, the West Coast editor of _Vogue_ , once described us as \"professional party givers.\" We are not, although we love to entertain, and on occasion we have been hired to supply themed decorations for private balls and charity events around the country. We also host lunch for visiting friends daily at Dawnridge, which is served by our extraordinary housekeeper and cook, Flory Vargas. Just as she did for Tony and Beegle before us, she continues to make the best Mexican food in Los Angeles and has mastered all of our favorite Western, Indian, and Chinese dishes as well. My father's side of the family was from Alabama; my mother's side from South America; and Ruth and I were born in Southern California\u2014I think our sense of Southern hospitality comes naturally. We like to entertain semi-formally, even if it's just a buffet for forty, but for groups above that we rely on outside caterers. Seated dinners from eight to one hundred and twenty are not unusual here, and often include dancing.\n\nWhat Ruth and I hate most are people who are brought for the first time as someone else's guest and announce when introducing themselves, \"You know, I've never been here before.\" To which I invariably answer, \"Come to think of it, I've never been to your house before either.\" Strangers can't figure out that Dawnridge is a private residence. We rarely have parties here that are not our own. We have lent the house for charity events and have weakened on occasion when friends ask if they can have a wedding or other special occasion here, but it is never open to the public except by invitation.\n\nThe fa\u00e7ade of Dawnridge was entirely covered with a painted backdrop, turning the house into the Amber Palace at Jaipur for our Return to the Raj Ball, 2015.\n\nDinner was served on the middle terrace at Dawnridge for the Return to the Raj Ball, 2015.\n\nNautch girls entertain during dinner to the delight of international guests Miranda Reise Williams, Manfred Flynn Kuhnert, and Rebecca de Ravenal at the Return to the Raj Ball, 2015.\n\n### MENUS AND RECIPES OF THE TONY DUQUETTES'\n\n_From the pages of_ Vogue, _April 15, 1957_\n\nRECIPES\n\n_Dinner in the Kitchen_\n\nGUACAMOLE\n\n_Served with Fries_\n\nCHICKEN LIVERS\n\nALB\u00d3NDIGA SOUP\n\nBARBECUED CHICKEN\n\nCHILES RELLENOS WITH SAUCE\n\nFRIJOLES\n\nFRIED RICE\n\nGREEN SALAD\n\nBEL PAESE CHEESE\n\nMAC\u00c9DOINE OF FRUIT\n\n_Buffet in the Supper Room_\n\nCREAMED CHICKEN AMANDINE\n\nSTUFFED ZUCCHINI\n\nGREEN SALAD WITH CHOPPED\n\nARTICHOKE HEARTS AND AVOCADO\n\nCHERRY AND KIRSCH ICE\n\nPETITS FOURS\n\n**GUACAMOLE**\n\n2 ripe avocados\n\n1 ripe tomato\n\n\u00bd onion, minced fine\n\nJalape\u00f1o peppers, chopped\n\nLemon juice\n\nChiles\n\nSalt and pepper\n\nMash the avocados with the ripe, peeled tomato. Add minced onion, a little lemon juice, chopped jalape\u00f1o peppers. Season with salt and pepper. Use chiles to taste, for desired spiciness. Serve with Fritos; for six to eight.\n\n**ALB\u00d3NDIGA SOUP**\n\n4 tablespoons oil\n\n1 onion, minced\n\n\u00bc cup tomato sauce\n\n3 quarts chicken stock\n\n1 pound fresh peas\n\n\u00bd pound string beans, chopped\n\n3 tablespoons cooked rice\n\n\u00bd pound ground pork\n\n\u00bd pound ground beef\n\n6 mint leaves\n\n\u00bc cup chopped parsley\n\n1 egg, slightly beaten\n\nSalt and pepper\n\nFry onion in oil 5 minutes, then add to tomato sauce and stock. When mixture is boiling, add peas and string beans. Prepare meat as follows: mix cooked rice into meat, adding chopped parsley and mint leaves, egg, salt and pepper, and form into balls. Drop into boiling stock, cover tightly, and let simmer half an hour. Serves six to eight.\n\n**CHILES RELLENOS**\n\n8 Ortega's green chiles\n\n8 oblongs of cream cheese,\n\n2 x \u00bd x \u00bd inches\n\n4 eggs\n\n4 tablespoons flour\n\nShortening\n\nStuff each whole, peeled green chile with an oblong of cheese. Separate eggs, beat whites until stiff, then add yolks. Add flour. Dip chiles into batter, one at a time, and fry in hot shortening. Brown on both sides; drain on absorbent paper. Serves eight.\n\n**SAUCE (one quart)**\n\n2 cups tomatoes, lightly stewed\n\n2 cups chicken or beef stock\n\n1 onion\n\n1 clove garlic\n\n1 \u00bd teaspoons salt\n\n\u00bd teaspoon pepper\n\n\u00bd teaspoon oregano\n\nStrain stewed tomatoes through a sieve. Fry onions in hot oil, 1\u00bd inches deep, but do not brown. Add onions and chopped garlic to tomatoes. Pour in chicken or beef stock and boil. Once bubbling, add oregano, salt, and pepper. Pour a little sauce on each chile.\n\n**CREAMED CHICKEN AMANDINE**\n\n1 six-pound chicken\n\nEnough stock to cover chicken\n\n\u00bd cup chopped carrots\n\n\u00bd cup chopped white turnips\n\n\u00bd cup chopped green celery\n\n2 leeks\n\n6 sprigs parsley\n\nSprig of thyme\n\n2 bay leaves\n\n5 cloves\n\n6 peppercorns\n\n\u00bd pound fresh mushrooms\n\n2 large green peppers\n\n\u00bd cup canned pimiento,\n\ndrained\n\n2 tablespoons butter\n\nSalt\n\n8 patty shells or a vol-au-vent\n\nMake a stock of carrots, turnips, celery, leeks, parsley, bay leaves, thyme, cloves, peppercorns, and salt to taste. Put the chicken in the stock mixture and simmer slowly for 2\u00bd to 3 hours. Then let the bird cool in the broth. Discard skin and bones; dice the meat.\n\nSlice the mushrooms, green peppers, and pimientos, all thin. Saut\u00e9 in butter; drain off the butter. Add the chicken meat. Keep the mixture hot while preparing sauce. Serves eight.\n\n**SAUCE**\n\n3 cups medium white sauce\n\n3 egg yolks\n\nSalt\n\nWhite pepper\n\nNutmeg\n\n\u00bc cup dry sherry\n\n1 cup almonds\n\nAdd beaten egg yolks to white sauce; combine with chicken and vegetables. Season with salt, pepper, and nutmeg. Reheat, without boiling, and stir in the sherry, and almonds, blanched, toasted, and shredded. Serve in patty shells or a vol-au-vent.\n\n**STUFFED ZUCCHINI**\n\n6 zucchini\n\n\u00bd cup chopped raw spinach\n\n2 tablespoons minced onion\n\nParmesan cheese\n\n2 tablespoons butter\n\n1 cup bread crumbs\n\nSalt and pepper\n\nCook zucchini in boiling salt water for 10 minutes. Cut zucchini in boat shapes and scoop out centres. Mix pulp with raw spinach, minced onion, Parmesan cheese, and remaining ingredients. Fill zucchini shells and bake in 350\u00b0 oven for 15 minutes. Serve with bacon. Serves six.\n\n**CHERRY AND KIRSCH ICE**\n\n\u00bd cup sugar\n\n\u00bd cup sauterne\n\n1 cup pitted, ripe black cherries\n\n\u00bc cup light honey\n\n2 tablespoons kirsch\n\n\u215b teaspoon salt\n\nBoil the sugar and sauterne together for 5 minutes, counting from the time the first bubbles appear. Set aside to cool. Press the cherries through a sieve, then strain through a double layer of cheesecloth. Beat together the cherry juice, the sugar syrup, and the honey with a rotary beater and add, during the beating, kirsch and salt. Freeze. Serves six.\n\n## Index of Searchable Terms\n\nAbbott, Scarlett\n\nAdrian, Gilbert\n\nbed curtains\n\nBridge of Sighs (Dawnridge)\n\nBustle Ball (Bal de Derri\u00e8re) of 1950\n\nCasa del Conde\n\ndining room\n\ndrawing room\n\nentrance hall (courtyard)\n\nfamily room\n\nkitchen\n\nmaster bedroom\n\npool\n\nsouth terrace\n\nCasa La Condesa\n\nbalcony\n\nbathrooms\n\ndining room\n\ndrawing room\n\nentrance hall\n\nfa\u00e7ade\n\ngardens and terraces\n\nHutton's sitting room\n\nlibrary\n\nmaster bedrooms\n\npool terrace\n\npowder room\n\nRuth's sitting room\n\nstair hall and staircase\n\nstair landing\n\nvestibules\n\nDawnridge. _See also_ gardens (Dawnridge)\n\nad campaign location\n\nbalcony\n\nbar\n\nbathrooms\n\nbuilding of\n\nCamelot Room\n\ndining room\n\nDrawing Room\n\ndressing room\n\nentertaining at\n\nentrance hall\n\nfa\u00e7ade\n\nas Fiddler's Ditch\n\nfire damage\n\nfloor plans and layout\n\nGreen Room\n\nguest bedroom\n\nlibrary\n\nmaster bedrooms\n\nmiddle terrace\n\nMonkey Room\n\nstairway\n\nSummer Bedroom\n\nVenetian powder room\n\nWinter Bedroom\n\nde Wolfe, Elsie\n\nDucommun house\n\nDuke, Doris\n\nDuquette, Elizabeth \"Beegle\" (wife; n\u00e9e Johnstone)\n\ndeath of\n\nentertaining at Dawnridge (1949-1950)\n\nlegacy of\n\nmeeting Duquette\n\nnickname\n\npainted doors by\n\npainted screens by\n\npainting studio (Beeglesville)\n\npaintings by\n\nphotographs of\n\nwedding of\n\nDuquette, Tony\n\ncareer\n\nchildhood\n\ndeath of\n\ndrawings by\n\nlegacy of\n\nmirrors by\n\npagodas by\n\npaintings by\n\nphotographs of\n\nscreens by\n\ntheatrical designs of\n\nwedding of\n\nDuquette Pavilion\n\nEhmcke, Caspar Johann\n\nfabrics\n\nAsia Major pattern\n\ncurtains\n\n\"Duquetterie\"\n\n\"Feu d'Artifices\"\n\nGemstone print\n\nGolden Sunburst pattern\n\nmalachite pattern\n\n\"Royal Ermine\"\n\nTibetan Sunburst pattern\n\nas wall covering\n\nFiddler's Ditch. _See also_ Dawnridge\n\nfires\n\nfurniture, by T. Duquette\n\nbookcases\n\nconsole\n\ndesk\n\nRose Chest\n\nsofa\n\ngardens (Casa La Condesa)\n\ngardens (Dawnridge)\n\nBeeglesville\n\nChat-Thai\n\nGarden Room\n\nmiddle terrace\n\nstaircase\n\nswimming pool\n\nTemple (Folly)\n\nTree House\n\nupper terrace\n\nvision for\n\nHearst collection\n\nHeintz paintings\n\nlighting, by T. Duquette\n\nchandeliers\n\nfigural lamps\n\ngirandoles\n\npagoda-shaped lamps\n\ntorch\u00e8res\n\nMalibu ranch\n\nmenus and recipes\n\nNew Dawnridge\n\nPendleton, James\n\nrecipes\n\nRemains Lighting\n\nReturn to the Raj Ball\n\nRosekrans, John and Dodie\n\nsculptures, by T. Duquette\n\nfigurines\n\n_Insect Man_\n\n_Modern Fruit_\n\nobelisks\n\n_Phoenix Rising From Its Flames_\n\n_Primal Sun_\n\nshell grotto\n\n_Summer Sun_\n\n_Talismans of Power_\n\n_Winter Sun_\n\nSert, Misia\n\nSortilegium (ranch)\n\nStanfill, Terry\n\nStreet-Porter, Tim\n\nTony Duquette Studios\n\nWilkinson, Hutton\n\nWilkinson, Ruth\n\nwood-burning stove (T. Duquette)\n\nYuanming Yuan\n\nThe Green Room at Dawnridge, 2018.\n\n## Photo Credits\n\nIllustrations by Juan Bastos: This page\n\nFernando Bengoechea: this page\n\nShirley Burden: this page, this page, this page, this page, this page, this page, this page, this page, this page, this page\n\n\u00a9 Christie's Images\/Bridgeman Images: this page, this page, this page\n\nDuquette Archives: this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page\n\nOberto Gili: this page, this page, this page, this page, this page, this page, this page, this page, this page\n\nFred Iberri: this page\n\nChristin Markmann: this page\n\nCourtesy Scott Mayoral and Remains Lighting: this page\n\nTim Street-Porter: this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page, this page\n\nCourtesy Jim Thompson, Thai Silk: this page\n\nDanforth Tidmarsh: this page\n\nDave Welch: this page\n\nEvery reasonable effort has been made to trace and contact copyright holders for individual images. In the event a copyright holder has been missed, the author and publisher would be glad to rectify the situation.\n\nRuth's sitting room at Casa La Condesa, 2018.\n\n## Acknowledgments\n\nWhen I first approached Eric Himmel at Abrams about making a book detailing the compound at Dawnridge, I was delighted with his enthusiastic response to my suggestion. When Eric told me that he would be my editor for the project, I couldn't have been happier. Working with Eric on this book was a dream for me\u2014then, when he brought Sarah Massey on board to keep me in line, I realized that it was a dream come true. My thanks to Eric and Sarah and all the people at Abrams who made this dream a reality cannot be expressed in words. Eric's selection of Emily Wardwell to design the book couldn't have been better. Emily's patience while working to make everyone happy is a testament to her artistic abilities and diplomatic skills, a combination of talents which have made her preeminent in her field.\n\nI have to thank Tim Street-Porter, Tony Duquette's favorite photographer, for coming back to Dawnridge yet again to update his previous images from the '80s, '90s, and 2000s. As always, Tim was the easiest person to work with, although I'm not sure that he'd say the same about me.\n\nHamish Bowles, an old friend of mine and Tony Duquette's, was kind enough to agree to set aside his work for _Vogue_ just long enough to write the foreword for this book. A brilliant journalist and collector, Hamish has the greatest and most in-depth knowledge of the world of design, and the deepest appreciation of Tony Duquette, his work, and his peers, making him our first and most logical choice to write a piece for us.\n\nSpecial thanks to my portraitist and friend, Juan Bastos, for creating the watercolor map of the property as well as the floor plan of the house. It is Juan who gave the property the nickname \" _Condelandia_ ,\" after King Juan Carlos of Spain bestowed my grandfather's Spanish title, Conde y Condesa de Alastaya, on me and my wife, Ruth, in 2007.\n\nI want to thank my clients and friends for helping to make this book happen. But for their constant interest, enthusiasm, and support for me and my work, there would be nothing left of Dawnridge today, as the land would probably have become the building site for three Beverly Hills\u2013style, _Tusciranian_ villas. You know who you are, and how much you've done for me and how much I really love you.\n\nMy daily thanks go to Tony and Beegle for entrusting me with the beautiful world they created at Dawnridge. Without the help of the Duquettes, and now my faithful housekeeper and major domo, Flory Vargas (who has worked at Dawnridge for more than thirty years), nothing that I have done could ever have happened. I cannot thank Flory and her husband, Jorge Vargas\u2014who is my property manager and head gardener\u2014enough for keeping Dawnridge alive, literally and figuratively, as they continue to pick up and put back all the pieces whenever disaster strikes. Thank you as well to Fred Iberri, who runs the office at Dawnridge and supervises the many photo, movie, and television shoots we have here, while juggling a million of my other projects like a circus performer, never letting any of them fall down.\n\nLife at Dawnridge wouldn't be half as glamorous or amusing without our dear family of friends, many of whom knew Tony and Beegle personally, including, Terry and Dennis Stanfill, Wendy Goodman, Manfred Kuhnert and Peter Iacono, Glenda Bailey and Steve Sumner, Tom Britt, Juan Prieto, Joe McCormack and Gary Hunter, Ken Downing, David Hoey, Shane Ruth, Joe and Terri Ebert Mendoza, Charles Tolbert, Holly Moore, Lyn Schroeder, Suzanne Rheinstein, Chad Holman and Keith Traxler, David Duncan and Michael McGraw, Mai N. Vejjajiva, Bill Booth, Eric and Waan Booth, Jennifer Smith Hale, Pamela Fiori, Val and Terry Magro, Patty Crews, Louis and Shane McCoy Fermilia, Peter and Kacey Doheny McCoy, David and Alexandra Calligeros, Charles Garnett, Col. Scott Ables, Todd Sessa, Skip Rumley, Parker Goss, Regan Iglesias, Laura Holland, Donald Bustran, Diane Vonderheide, Walter and Lee Doyle, Elizabeth Wahler, Michelle and Bernard Nussbaumer, Bryan Batt, Charlotte Jackson, Scarlett Abbott, Brett Leemkuil, Newell Turner, Clinton Smith, Carolyn Englefield, Carol Black, David Patrick Columbia, Anna Roth, Tony and Kristin Krantz, Judith Krantz, Liz Morton, Cat Pollon, Carol and Hugh Klotz, Jeff Caldwell, Debra Kanabis, Bryan Curran and Kevin MacLellan, Kathleen Smith, Nick and Amanda Stonnington, Grazia d'Annunzio, Mike and Sheila Wilkins, Joy Venturini Bianchi, Tristan Butterfield, Emmylou Harris, Beth de Woody and Firooz Zahedi, Nina O'Hern, Harvey and Gail Glasser, Stellene Volandis, James and Dede Caughman, Dara Caponigro, Steven Puschel, Andree Caldwell, Anna Griffin and Joe Speigleberg, Charles and Ann Johnson, David and Alexia Leuschen, Isabel Goldsmith, Kelly Williams and Andrew Forsyth, Richard and Marcia Mishaan, Mary Randolph Ballinger, Bob and Michelle Bradway, Frank and Carol Nickell, Lenny, Rachel and Julian Feder, Mary Anna Ajemian, Jerry and Linda Bruckheimer, Kelli and Jerry Ford, Reed Krakoff, Craig and Barbara Barrett, Doug and Caroline Brown, Pag Sampatisiri and the entire Bodiratnangkura family: Dang, Tat, Lek and Pat. I couldn't have done any of this without your generous friendship and patronage.\n\nTo all of those who helped me on this book either directly or indirectly whom I have omitted inadvertently, I apologize profusely, and, along with Tony and Beegle, thank you for your friendship and support. And to my darling Ruth, Piper, and Kippy\u2014thank you for putting up with me. You are all champions, and as I've told you often\u2014without you I am nothing.\n\n\u2014HUTTON WILKINSON\n\nDawnridge, Beverly Hills, California, 2018\n\n## Appreciation\n\nAlthough I had photographed one of Tony's iconic interiors\u2014for the actor James Coburn for British _Vogue_ \u2014back in the 1970s, it was not until the 1990s, when Hutton Wilkinson introduced me at a drinks party at the Beverly Wilshire Hotel, that I met Tony in person. At the time, he was elderly and intriguing looking, but perhaps a little lost and in need of someone to talk to. We started chatting, and soon we were invited up to Dawnridge.\n\nIf Tony seemed sad at that drinks party, it was for a good reason. A few months earlier, he had returned to Los Angeles after many glamorous years in San Francisco. There he had been a fixture in the city's social and cultural worlds, designing costumes and sets for the opera and ballet and creating extravagant balls, as well as decorating houses and hosting fantasy parties at his lavishly appointed studio, the Duquette Pavilion, a converted former synagogue. But this highly productive episode had been dramatically terminated by a fire that destroyed the studio and much of its irreplaceable contents.\n\nWe became firm friends after Tony discovered that we had just bought a house in the Hollywood Hills where Adrian, the costume designer, had lived in the early 1930s, before marrying actress Janet Gaynor. Adrian, along with Elsie de Wolfe, was one of Tony's original mentors.\n\nI had never experienced anything remotely like Dawnridge. Here was the creation of an artist rather than a decorator. Tony had constructed extraordinary fantasy environments, always with a sense of order and with an underlying coherent vision. Nothing was arbitrary or disguised. He scoured auctions at a nearby naval base for perforated metal helicopter landing pads, which he then openly used to construct the gazebos and Thai-inspired pagodas in the ravine below the house. In the palatial Drawing Room, he and his artist wife, Beegle, fashioned ornamented pelmets of cloth for the windows, dipped them in gesso and plaster, and then hand-painted them to look like eighteenth-century tooled leather.\n\nI soon realized the importance of photographing Dawnridge, along with many of his remaining other design projects, as I became familiar with his relatively undocumented archive. In this spirit, I shot his house in San Francisco as well as a guest house on an island in Northern California, a St. John the Baptist church in Arcadia, and eventually current projects like the apartment for John and Dodie Rosekrans in Paris.\n\nThe Tony Duquette abalone chandelier, created for his 1952 one-man exhibition at LACMA, hangs in the balcony at Dawnridge, 2018.\n\nTony Duquette's _Phoenix Rising From Its Flames_ in the gardens at Dawnridge.\n\nOne morning, around ten, he called me and asked, \"Are you free today?\" Luckily, I was. The 1960s Ducommun house was about to be dismantled and sold, and strategically placed paintings by Modigliani, Braque, and Klee, as well as sculptures by Calder and Duquette, were being removed that same afternoon. Working quickly, I managed to photograph everything. This proved to be the only record of Tony's very best decorating, for clients who had given him complete creative freedom.\n\nEncouraged by my interest in his creations, Tony found himself working again, beginning with his 55-acre ranch, Sortilegium, which rose Xanadu-like from a dry mountaintop high above Malibu. Soon, a pattern emerged: I would show him a set of freshly taken 4-by-5 transparencies, and he would follow up a few days later to say that he had made a few changes to the room, and could I possibly take a few more pictures? Happy to oblige, I repeated the process, and soon he was transforming the ranch\u2014and looking noticeably happier. A sizeable crew of workers arrived, working seven days a week, trying to keep up as Tony threw himself into new landscaping projects and the construction of new pavilions, one of which he very kindly gave us to stay in whenever we wished. Reviewing progress with Tony over Sunday lunch with the latest batch of transparencies, I became aware that I had unintentionally kick-started his creative engine.\n\nTony proved to be an unstoppable shopper, constantly on the hunt for raw materials for future projects. Wherever he traveled, containers of newly found treasures would be shipped back to Beverly Hills. When we joined Hutton and Tony in Bali one year, we immediately identified his villa, its forecourt piled to the ceiling with brown paper\u2013wrapped boxes tied with string. He had been shopping since the moment he arrived, with Hutton complaining that he was not even allowed to stop for lunch.\n\nMeanwhile Dawnridge was also receiving its share of transformations, and it was wonderful to witness his mind at work, day by day, like an artist breathing fresh air into a familiar canvas. One of my last photos at Dawnridge during Tony's lifetime was of _Phoenix Rising From Its Flames_ , a large sculpture he had just placed in his garden after his 1995 exhibition at the Hammer Museum. This was created after his Malibu ranch burned to the ground\u2014a symbol of rebirth and continuance that showed Tony's extraordinary resilience and creative drive.\n\n\u2014TIM STREET-PORTER\n\nEditor: Sarah Massey  \nDesigner: Emily Wardwell  \nProduction Manager: Anet Sirna-Bruder\n\nLibrary of Congress Control Number: 2017956932\n\nISBN: 978-1-4197-3262-1  \neISBN: 978-1-68335-420-8\n\nText copyright \u00a9 2018 Hutton Wilkinson  \nPrincipal photographs copyright \u00a9 2018 Tim Street-Porter\n\nJacket Front: Tony Duquette's Phoenix Rising From Its Flames sculpture in the gardens at Dawnridge  \nJacket Back: The staircase at Casa La Condesa.  \nBoth photographed by Tim Street-Porter.\n\nJacket and cover \u00a9 2018 Abrams\n\nPublished in 2018 by Abrams, an imprint of ABRAMS. All rights reserved. No portion of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, electronic, photocopying, recording, or otherwise, without written permission from the publisher.\n\nAbrams books are available at special discounts when purchased in quantity for premiums and promotions as well as fundraising or educational use. Special editions can also be created to specification. For details, contact specialsales@abramsbooks.com or the address below.\n\nAbrams\u00ae is a registered trademark of Harry N. Abrams, Inc.\n\n**ABRAMS** The Art of Books  \n195 Broadway, New York, NY 10007  \nabramsbooks.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztkpx b/data_all_eng_slimpj/shuffled/split2/finalzztkpx
new file mode 100644
index 0000000000000000000000000000000000000000..175e2ffc9243630cee551b3c5f35da801925c55f
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@@ -0,0 +1,5 @@
+{"text":"Status: offline Do you need to set the theme each time you start a new game? Or is once a theme set it is always loaded?\nStatus: offline I've noticed it seems to stay in the same theme that I've loaded up (in this case, the Extended mod).\nStatus: offline The theme is set once you have loaded it, to change back just go to change theme again and load distant worlds original or whatever mod you were using before hand.\nI've noticed it seems to stay in the same theme that I've loaded up (in this case, the Extended mod).\nHave you put the great war mod inside your customization folder? make sure it's in and you select it should change if not post me again and will look further into it.\nStatus: offline Just beware, that sometimes the patches reset your themes to default. If a particular save game was utilizing graphics, etc. in a theme then you will need to make sure to activate that theme before loading the game.\nAny syntax errors in your text files will lead to a situation where the game will get stuck in the galaxy generation phase and will not advance. In my case it was a simple matter of having left out a single comma in one of the character files in a theme.\nAlways be sure to add a bit, then test it to make sure it loads, so much easier to track down an error in one file as opposed to 30 etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WELCOME to \"The Remarkable Reptiles\". Using a variety of artifacts and live reptiles, I specialize in educational programs for ALL age groups.\n\u2022 5 snakes...the last being a boa constrictor that you can touch and sometimes hold!\nAll species used are harmless!!\nThis program has been presented since 1982 at local schools and libraries, as well as scout groups, civic celebrations, church events, birthday parties, etc. You may have seen me at one of these places! I have also appeared at many of our beautiful Minnesota State Parks as well as teaching classes at the Science Museum of Minnesota.\nIn additional to live formal presentations, I also offer a display of live reptiles for events that prefer a more casual presentation. This display consists of 16 cages plus assorted artifacts. The basic set-up time for a display is 4 hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jim suffers from a back injury and as a result the only exercise he can do is swimming. He began swimming at Help for Heroes sessions and was eventually selected to participate in the US Airforce trials at the start of the year. At that event he took home 4 gold medals which he says is not the point of the exercise. Instead simply taking part in a physical activity has improved his wellbeing dramatically. Jim says before he started he was a mess both physically and mentally but with the help of the Sports Recovery Team of Help for Heroes he is in a much better place.\nJim urges anyone who feels they may benefit from Sports Recovery to seek out what's available and begin their journey towards recovery. He says even if the only thing people do is put their swimming costume on and take a ten-minute dip, the simple act of feeling water on their bodies will have a huge impact because they stepped out of their comfort zone. Small steps tend to turn into large steps. Jim concludes that any form of exercise is going to improve mental health. Sports Recovery ensures there are like minded people around who have served and the staff are excellent, delivering the support and camaraderie that ends after leaving the military.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The mild, subtropical climate of the Florida Keys & Key West is a result of the islands' unique geographic location. The temperate waters of Florida Bay and the Gulf of Mexico to the west and north combine with the year-round flow of warm water from the Atlantic Ocean's Gulf Stream to give the island chain consistently balmy weather.\nAir temperature differences between day and night average only about 10 degrees throughout the year. During winter months the average temperature is only about 14 degrees lower than in summer.\nThe sea temperature from November to April averages 74 degrees, and between May and October averages 82 degrees.\nIn the past 40 years, there is no known record of frost, ice, sleet or snow in the Keys, according to weather officials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need k24 mounts for Ek civic. If someone readily got them or either know someone who can make them in urgent notice then please let me know.\nyou can reach me here or whatsapp me \"03214139306\"\ni know about these workshops but they usually ask to bring car to them. that we can't do. but still i will call them thanks.\nGot them already @Moderator close the thread.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzufeb b/data_all_eng_slimpj/shuffled/split2/finalzzufeb
new file mode 100644
index 0000000000000000000000000000000000000000..f6ba629da8f4ee3756c25b5e81926ac86c6c89a6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzufeb
@@ -0,0 +1,5 @@
+{"text":"Rich and moisturizing body butter. Helps make skin feel incredibly soft and smooth. Awakens the vitality of skin and leaves skin looking brighter and more beautiful looking overnight. Enriched with the vitality of Mediterranean citrus fruits Use: Massage onto the whole body every day after bath or shower NOT A FOOD ITEM. FOR ADULT USE ONLY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hide Photos On Facebook: Exactly what do you do when you don't want your ex-girlfriend (with which you're friends on Facebook) to see the pictures of your brand-new girl that you simply uploaded? Just how do you hide the breaks of the weekend break alcohol consumption binge you had with your friends, from your traditional loved ones (and parents )?\nIn both those circumstances, not having such individuals as friends on Facebook is constantly a choice. However that might not be one of the most hassle-free one. The good news is, there's a means to hide your Facebook pictures, video clips, as well as standing updates, from people who should not see them.\nCurrently, allows see just how we could hide an entire picture album on Facebook from specific people.\nAction 2. Click Edit Info on the album page. As you see below, I am preparing to hide the images that get auto-posted to Facebook from my Posterous blog site.\nStep 3. Clicking Edit Info raises the Edit album page where you'll discover a dropdown menu in the Personal privacy section. There again you'll find a Personalized choice that you need to click.\nStep 4. This action coincides as Action 3 in the hiding a Facebook status procedure. You can either make the album noticeable to specific people or enter the names of the people you intend to hide it from.\nThat had to do with hiding your details from specific Facebook friends. Following time you will upload something rowdy, simply make sure you experience these basic actions initially. Might conserve you a lot of humiliation as well as headache.\n- In choice who could see this picture select Custom (received screen shots).\n- Enter the name of that individual your do not intend to share pic with and Click save. And also it will certainly be done!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Maryland native, Jonathan Cresci, maintains an active career as a producer, engineer, performer and educator. Jonathan is an active recitalist and clinician through out the region. He has performed and lectured at the Southeastern Horn Workshop at Appalachian State and International Horn Symposiums in San Francisco and Denton. He is a member of the Artist Faculty for the National Trumpet Competition. Jonathan has given recitals and master classes at the University of Memphis, York College, and the Peabody Institute. As an orchestral and chamber musician he has performed with the Acadiana and Lake Charles Symphonies, Chorale Acadiane, Orpheum Theater, York Symphony, Maryland Chamber Orchestra, and The Mt. Vernon Brass Consort.\nThrough his company Fundamental Sounds, which specializes in the recording and production of acoustic classical and popular music, Jonathan has recorded performances of artists such as Chris Gekker and large ensembles including the the Maryland Symphony and the Prince Georges Philharmonic. He has also recorded and produced albums including Greg Small's \"Singularity\" and Kyle Millsap's \"Scatter the Darkness\".\nJonathan is presently on faculty at Frederick Community College, York College of Pennsylvania, and Montgomery College, where his areas of scholarship are natural trumpet performance, trumpet pedagogy, recording arts and science, and American popular music. Dr. Cresci earned a doctoral degree from the University of Memphis, a master's degree from the University of Louisiana at Lafayette, and a bachelor's degree from the Peabody Institute of the Johns Hopkins University in both trumpet performance and recording arts and sciences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jose Victoriano Gonzalez-Perez was an avant-garde Spanish painter and sculptor who was active in France most of his career. He was also famous as a draughtsman and an illustrator, and made major contributions to Cubism in France. He spent almost his entire career in France, where he acquired the fame of being one of the foremost Cubist artists. He enjoyed a reputation for his independent and characteristic approach to Cubism rather than for innovation in art. He is considered to be one of most prominent exponents and scholars of Cubism.\nEnrolling at Madrid's famous Escuela de Artes y Manufacturas, he specialized in engineering, mathematics, and physics. Terming this phase as formative, his scientific approach to Cubism has been attributed to his technical education. He interrupted his studies in science briefly to study with Jose Moreno Carbonero, the eminent Academic painter, and then made up his mind to pursue a career in art. He commenced his new passion by finding employment in the magazine Renacimiento latino, where he produced pleasing Art Nouveau-influenced drawings.\nOnce he relocated to Paris, he made new associations with artists, one of them being Pablo Picasso. He, unlike the prevalent style of Cubists, painted with bright colors and his approach to Cubism, as critics believe, was scientific and precise. He exploited papier colle (collage) to achieve greater surface activity, clear compositional structure and a complex interplay of descriptive and verbal representation. Gris converted from analytic to synthetic Cubism, from abstract to concrete, ensuring a basis in conception rather than perception.\nAt the time of the first World War, Gris devoted his attention to a narrow range of themes, however he remained loyal to most of them. While practicing his art, he persisted in his use of musical instruments, books, music sheets, and even domestic goods. The figure work in his compositions made direct reference to the technique of noted artists such as Paul C\u00e9zanne and Jean-Baptiste-Camille Corot; these works conveyed a powerful sensitivity of light, mood, and texture as well as the subjects and scenes depicted therein.\nIn the later part of his career, Gris' art underwent a time of crisis. After restoration of his health, he achieved a fair degree of prominence and was appreciated by both the masses and the avant-garde. As he found increasing success, his color palette became richer and more deferential, and his subjects became more natural and anecdotal. This appeared to be a compromise and faced severe criticism from those who held that Cubist art was to be pure and austere.\nDon't settle for cheap prints when you can now grace your walls with Juan Gris reproductions. At BrushWiz, our professional artists deliver stunning handmade replicas of famous Juan Gris oil paintings, each displaying meticulous attention to visual detail and painting technique. Museum quality reproduction guaranteed with online preview before shipment, 30-day money back satisfaction guarantee with every purchase and FREE Shipping to anywhere in the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A mountain in Ceylon, not far from Kacchakatittha, no the right bank of the Mah\u0101v\u0101lukanad\u012b. There Panduk\u0101bhaya defeated his uncles and occupied their fortified camp for two years. The mountain was the abode of yakkhas, and it was here that Panduk\u0101bhaya captured the Yakkhin\u012b Cetiy\u0101, near the pond Tumbariyangana in the vicinity of the mountain (Mhv.x.46, 53, 58ff). King Mah\u0101n\u0101ma built a vih\u0101ra there (Cv.xxxvii.213).\nAccording to the Mah\u0101vamsa T\u012bk\u0101 (p.289), the mountain was also called Udumabarapabbata (or -giri). There seems (See P.L.C., s.v. Udumbaragiri) to have lived at Udumbaragiri a fraternity of forest-dwelling monks who produced from among their number several scholars of great repute and monks of great piety - e.g., Kassapa and Medhankara.\nThe mountain is identified (Ep. Zey.ii.194ff) with the present \"Gunners' Quoin\" on the right bank of the Mah\u0101valiganga.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuuxt b/data_all_eng_slimpj/shuffled/split2/finalzzuuxt
new file mode 100644
index 0000000000000000000000000000000000000000..6325b01e5697b99d5a140d0c132ade6d186e6983
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Wilson River Bridge on Highway 101 in Tillamook.\nHistorical lore dates Tillamook County's first cheese to around 1856, when William W. Raymond separated curds and whey in a large vat. The growing season was too short, so settlers resorted to cows and surplus milk.\nOver the past century, Tillamook became synonymous with the 100-year-old Tillamook Creamery, whose medium cheddar was awarded the best at the World Champion Cheese Contest in Madison, Wisconsin this March. That quality no doubt contributes to the little-known fact that Tillamook Cheese Factory is the second most visited site in the state of Oregon. More than one million people each year peer through the observation deck windows to see one of the world's best cheeses being sliced.\nLeisha Mizee-Riggert's forebears emigrated from other cheese-producing nations of Holland and Switzerland and began to raise dairy cows four generations ago in Tillamook. Their family dairy farm, as with 110 others in the area, is a member of the Tillamook County Creamery Association, which dairy farmers established in 1909 to ensure the highest quality of cheese. In recent years, the private company has posted more than $300 million in annual sales.\n\"I was 8 or 9 before I had steady chores on the farm,\" says Mizee-Riggert, now 29 and a workforce management and training expert in Tillamook. \"Then I moved into 4-H and started showing cows.\" In 2000, the prot\u00e9g\u00e9 was named the Dairy Princess of Oregon. Under that sash, Mizee-Riggert devoted nearly a year of her life to educating school kids and legislators about cheese and dairy farm families.\nIn a town whose population has hovered near 4,500 over the past decade, Tillamook was too small for a young woman whose travels led to big-city aspirations. Mizee-Riggert took a job as a nanny in New York later that year and parlayed that into a job in the city's booming financial sector. Shortly thereafter, she had a change of heart and moved back to Tillamook and its pastoral beauty.\nBeauty can get thorny at times. The Trask and Wilson rivers run west from the Coast Range toward the Pacific Ocean and skirt the southern and northern edges of town in an august showing. As verdant as the valley is at the confluence of the rivers, Tillamook celebrates its 100-year flood nearly every year.\nThat flooding cost the county more than $60 million between 1996-2000, according to estimates from the Tillamook County's community development department. Partly due to economic factors behind new big box stores, Mizee-Riggert says, the downtown area today has many vacancies and its spirit lies with an older generation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When the Wright brothers flew in 1903, comfort and safety were secondary to mission success. The open structure of early-model airplanes left pilots' eyes vulnerable to small particles and serious injury. Protecting one's vision prompted the creation of pilot goggles, also called aviator goggles. Unlike reading glasses of the time, pilot goggles covered the entire eye socket which enabled a broader field of vision. Creating your own pilot goggles of the period isn't difficult, but familiarity in working with crafting glue is helpful.\nLay the leather square leather side down on a hard, flat surface. Place the chemistry goggles, eyes down, on top of the leather.\nTrace around the perimeter of the goggles on the back of the leather using a silver pen. A black or blue pen won't be visible against the dark leather.\nCut the leather along by following the silver pen perimeter with your scissors.\nPlace the leather cutout face down on the goggle eye area and secure the position with three pieces of tape on either side. Once the tape is secure, the leather covers the eye openings of the goggles.\nFeel where the outer plastic perimeter ends and the eye openings begin using your fingers. Trace the interior of the eye area on the back of the leather using your silver pen.\nCut out the interior eye area using your scissors by following the silver pen lines. The result is a thin border of leather that fits around the perimeter of the goggle eye area without obscuring the wearer's vision.\nGlue the leather border around the perimeter of the goggles using craft glue. Apply glue on both sides of the border where you'll glue the leather material.\nPress the leather material on the goggle perimeter for 35 seconds before lifting your hands. Let the glue dry for 20 minutes afterward before wearing.\nAlthough leather appears more authentic, use felt fabric if leather is unavailable.\nMake one prototype of your pilot goggles before making several pairs. This allows you to adjust any materials or sizes if necessary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Air Design. We specialize in manufacturing opener scrapers for planting and seeding equipment that utilize a tungsten carbide scraping EDGE. This provides excellent durability and performance. With over 40 scrapers to choose from, we've got your make and model covered.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8 Dec Cypress Hill - Latin Thugs. raise my Mexcian flag in the streets of Tokyo Japanese Women Love Latin Guys . Viva La Raza. Read more.\nJattizm Ammy Virk Full Album to download, Jattizm belongs to Punjabi Music, Jattizm available to free download. Name, Size, L, S. Bullet vs Chammak Challo- ammy virk - Official Video - Brand New Punjabi Songs - Jattizm, ( MB), , Bullet vs Chammak . Check out the list of Best Ammy Virk Songs of all time.\nProfessional Software. Tax Software I want to see how CCH ProSystem fx PDFlyer drives a seamless workflow. Request Demo Deliver a seamless and proficient workflow by adding PDFlyer to other Wolters Kluwer products. When used.\n14 Oct Ejection Seat Realistically Website: Works with: GTA San Andreas This mod realistically installs an ejection seat for the Hydra. Instead of. 24 Aug ejection seat - posted in GTA III, VC & SA: hi togeher,i want to crate an ejection seat for hydra, hunter, etcmodificating thread plchute is not a. 13 Oct - 2 min - Uploaded by Incognit63 F Raptor(Gta San Andreas Jet Mod \u1d34\u1d30) - Duration: Wyntflex 36, views \u00b7 .\nThe Markosoft Interest Calculator was originally designed as to tool to be used by accountants and bookkeepers for easily calculating daily interest. Shortly after. 6 Dec Download Markosoft Interest Calculator for free. The Markosoft Interest Calculator was originally designed as to tool to be used by accountants. 31 Jan Whether you're considering a new loan, or you need to keep track and work out payments, interest, or time schedules on your mortgage, car or..\n28 Jul Jeff Beck Setlist at Toyota Park, Bridgeview, IL, USA. Jeff Beck's Guitar Shop 2. Jeff Beck Ronnie Scott's Jazz Club, London - Nov 27, Nov 27 28 Nov - 5 min - Uploaded by BallsMah0ney Since it didn't make it on the DVD, here is Jeff Beck playing \"A Day In The Life\" at The.\nDeanna. O DEANNA O Deanna. O DEANNA Sweet Deanna. O DEANNA You know you are my friend, yeah.\nVantage Point is a American political action thriller film directed by Pete Travis and written by Barry L. Levy. The story focuses on an assassination attempt Plot - Production - Release - Reception.\n20 Jul AnyTitle Deleter is an application which allows you to completely delete (almost) any title on your Wii. This is useful as a space-freeing. 5 Jan AnyTitle Deleter MOD is an unofficial mod by Red Squirrel of. 20 Oct AnyTitleDeleter said it prevented deleting it to avoid bricking the Wii.\ntravelwithgabby.com: Project Management Step by Step: How to Plan and Manage a Highly Successful Project (): Richard Newton: Books. Project Management Step by Step (e-bok). How to Plan and Manage a Highly Successful Project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This kitchen renovation started a couple years ago with the striking realization that our kitchen isn't cutting it. It looks adorable and I can fix a meal, work, but not very efficiently, effectively, or safely. I slipped and fell in the pantry because I couldn't see what was on the floor while returning something to the room. (Ok I'm a klutz but still.) There are certain drawers that will fall out of the frame if you open them too far. Basically it's old, tired, and worn out. The last people who owned the house did a lot of questionable DIY. Like cutting the supports for the staircase, plumbing the kitchen sink (it was bad), and not measuring finishings. So two years ago we started the kitchen reno by addressing these issues. Several thousands of dollars later we have a more secure home, better functioning plumbing, none of which we can see of course. The next part of the renovation was a little more fun. We could see, feel, and enjoy picking out the cabinets, finishes, and pieces parts in general.\nIt was fun at first, but with a business, kinship care, and typical time constraints that go with life, it was a wee bit overwhelming. I say a \"wee bit\" even though at times I really thought I might lose my mind. We had three different quotes from three kitchen stores and met with three contractors. All that research made our big hires much easier. It took us two months to finalize the cabinet color. Picking out black granite was not easy, it took three trips to two different granite wholesalers. It took about three months to decide on four different paint colors. But the ultimate difficult decision so far, cabinet pulls. I looked at so many I don't even know what I like anymore. I have two pinterest boards filled with sample kitchens and pulls. I cry uncle. It can wait. I can chill. I will be patient with the process.\nWhen it started to become a tiresome labor I would put it away. Renovating a kitchen, although stressful, is a luxury and project I want to enjoy. It's a luxury many people will never be able to do. I realize this. I honor how fortunate I am to get to do this, how fortunate my family is. Yes, we have worked like crazy, saved, had a pretty tight budget, and will continue to, but we are fortunate. So in the scale of life I know this is kinda a funny thing to be stressed about, a true first world problem. When I take it too seriously I remind myself of what a treat this is.\nThe work started officially Monday. Starting with some painting that we have put off, ripping up the carpet on the stairs, etc. It looks like a creepy movie set at night. In the day it looks like we just moved in, or like maybe we are debating becoming hoarders. (No offense to any hoarders.) This is where patience comes into play. I have prepared myself for mess, for extreme inconvenience, for many days where there is no progress, for sitting in the space between. The space between is important. It's the present. In doing this I find myself enjoying sitting and listening to music, packing a box of dishes, and just being.\nHere is Erick Garcia, our painter, who is doing an amazing job. We decided now was the time to rip up the carpet and take the stairs back to the antique look. They are a hot mess and probably haven't been painted in a hundred years. The top step was a shelf (remember the questionable DIYers) and we are replacing it with a safe tread. No wonder I sometimes tripped on the top step. Erick actually cares about the project and was excited to do the stairs, ripping up the carpet himself!\nOur downstairs is a semi-open floor plan. The kitchen runs off the open dining room and the parlor runs into both so we have to paint all of the downstairs up into the hall upstairs. Which I'm glad about we didn't pick this color. It's way to golden to suit me. We picked out Benjamin Moore colors, Stone Hearth for the parlor and upstairs, and Indian River for the kitchen and dining room. All the ceilings are White River, a neutral white with a hint of taupe and mint believe it or not.\nHere is the hot mess after the carpet is ripped out. Wow the look on my husband's face was a bit priceless. I assured him it will be beautiful painted and some day a runner. No stealth mode!!!\nPlease note the top step below the carpet, that is a wood shelf. Completely dangerous the tread is shorter than the others. At least I now understand why I catch myself at the top of the stairs.\nKitchen and pantry plan from DB&S out of Norwell, Ma. We knew when we walked in they \"felt\" like the best fit. After quotes were in they really were. Great quality and service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvmxg b/data_all_eng_slimpj/shuffled/split2/finalzzvmxg
new file mode 100644
index 0000000000000000000000000000000000000000..4043b4d926697f88f075055b16d9adc92badda6b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Yes she did and the were delivered as requested.\nFantastic, my brother sent a picture of the flowers they received, they looked just like the picture we chose to send! Thanks for the great service!\nThe Flower Shop is a Lovingly Florist in Fort Pierce. Buying local matters. Discover how Lovingly supports local florists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Having physical membership cards can save you time at checkout, allowing you to identify your customers by scanning a code.\nFirst, you will need to source cards that do not include a built-in loyalty program from a third-party.\nPlease Note: You will need to ensure these are scannable cards. Collect does not supply physical cards as we have an integration with Apple Wallet.\n1. Search for the customer, select their profile, and click Edit information.\n2. Overwrite the current customer code with the code on the physical card and click Save changes.\n1. Click Create new customer.\n2. Enter the customer's First name, Last name, and Email.\n4. Select Customer Code, then click Update fields.\n5. Enter the code to match the code on the physical card and click Create new customer.\nThe card number will be saved to that customer's account which you can use to search for the customer in Vend.\nAfter the customer is set up in Vend, simply scan their membership card to apply them to the sale. Collect will recognize the customer's email address and will automatically sync their purchase information into your customer database.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The New York Times mined its wedding announcement archives for interesting and engaging stories about love and family. that are published twice a week. I really enjoyed setting the tone and feel with commissioned illustrations, encouragoing users to revisit the webpage and designing for mobile first. After two weeks, this project had over 700,000 pageviews with returning visitors at over 80%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The hotel has a delightful atmosphere that is welcoming to all.\nAll of our bedrooms have been dleightedly decorated to offer quality accommodation. We have a spacious deluxe ground floor room. Most of our rooms have full ensuites with central heating throughout. Enjoy a home cooked meal in the evening or just come in for bed and breakfast. We have a licensed bar with a patio to enjoy a nice drink or just to relax.\nThe Sherwood is located in the residential area of Drummond Road just south of the town centre. The location is ideal for access to the south bowling greens, the beach and all amenities. The town and shopping centre is just a short walk away.\nOur aim is to make all of our guests comfortable and ensure that they have an enjoyabe holiday whilst staying in our home. We are open throughout the whole year and offer a great package in Novembers Tinsel and Turkey Christmas and New Year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. May discusses his views of maturity and his own recent work on the process of creativity. He also evaluates his most significant contributions, reacts to his critics, and discusses his future plans.\nCopyright \u00a9 1978. Used by permission of Drs. Nicholas and Dorothy Cummings Center for the History of Psychology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwxiz b/data_all_eng_slimpj/shuffled/split2/finalzzwxiz
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+{"text":"Each of your worries linked to academic producing are now over since you have came for you and energy to the most site that was suitable in the suitable moment. In summary, readers do not take pleasure in after stream of one's thoughts. You could also chat around the internet to specify adjusting the task treatment or some principles to your personal writer.\nOur employees know several of the books you may require from your heart. Make sure it write my paper is enjoyable and your work is really to choose George Washington's life. The college students, that are sick and tired of searching for reliable assignment help on the web and writing initiatives that are pathology, needs to get in contact with our professional writers.\nKnow what exactly the book includes paid writers before before you pitch. Being an on line essay author is by no usually means a easy job. Every single kind of composition begins with the introduction posing a statement and presenting a topic Since you probably already know.\nOne of the biggest benefits of seeking assistance from an editor for reviewing your work is you will not just receive insights on your essay https:\/\/aussieessaywriter.com.au\/photo-essay\/ but in addition professional guidance on the best way to increase your writing skills. If there's a time once the essay is written but needs a number of tweaks and changes made, an essay writing service can achieve this too. This system is not uncommon for analytical essays, but nevertheless, it may also apply to research essays.\nAn article may criticize a problem or endeavor to convince the reader with ideas on the sure matter and would have advice to extend the reader. You have to fill in your email, and select the kind of the paper and define the deadline and the sum of pages that your document could contain. Creating a term paper is a job that they may handle easily plus they truly are likely to provide service that is prompt to you personally.\nIt could looks as when our team goes to produce quality. Search uk essayss com' if you need to find more information regarding our providers as we promise that which we can provide.\nThe cost has to not play a role when it's to perform selecting a fine and genuine creating company. If you find the services with http:\/\/szkoly-psb.edu.pl\/essay-writing-service-in-canada the sort of company you will want to pay for higher than the quoted speed. A cost of the item goes and vice 26, as stated by the rule of supply and demand whenever the distribution increase afterward.\nIf you do proceed with the financial loan then be certain that you're really convinced you should have adequate income to have the capability. The majority of them may be obtained at no cost although there really are a few resources that which you've surely got to pay. Therefore, according to our policies, you're wholly permitted receive your money straight back again.\nNo one can guarantee the quality and also you are likely to waste hours whilst still surfing . Let's take a closer look in what sort of service that you ought to select from all of that are easily offered. There is simply no replacement training to boost.\nFive paragraph article comprises the subsequent components, and one paragraph is compiled by each part just. You browse, also you satisfy this author by doing so. These authors always offer you essays that are reputable and authentic.\n\u00ab Great Day Texts on her behalf: Make Her Time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HSV Forum > State Cruise & Racing Events > WA Cruise & Racing Events > VP Senator and VR Clubsport for WA Film Shoot Event.\nView Full Version : VP Senator and VR Clubsport for WA Film Shoot Event.\nWe will be filming in Perth to wrap up production for the HSV Documenary either on the Saturday 21\/07 or Sunday 22\/07. We are currently looking to film some additional footage of both an original looking VR Clubsport and a VP Senator.\nIf you own one of these vehicles feel free to call Jeremy on 0420 478 420.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If they sign up, you get 1 Free Session each month they re-sign!\n1\/24\/2011 JACK LALANNE PASSED AWAY AT 96, HOW OLD ARE YOU?\n1\/18\/2011 BEST INVESTMENT TIP EVER!\n11\/22\/2010 HAPPY THANKSGIVING, ENJOY IT!\n11\/9\/2010 FITNESS AND WEIGHT LOSS NEWS WHO ARE YOU, WHERE ARE YOU?\n10\/19\/2010 WHAT'S YOUR GOAL AND WHAT'S STOPPING YOU?\n8\/30\/2010 AND THE GOOD NEWS!\n7\/27\/2010 OUR OWN RESPONSIBILITY AND ACCOUNTABILITY. WHAT ABOUT OUR KIDS?\n7\/20\/2010 CALORIES! GET THEM OFF OF ME! GET'EM OFF!\n7\/13\/2010 TIME MANAGEMENT FOR FITNESS!\n7\/6\/2010 WHY IS MUSCLE SO IMPORTANT?\n2\/4\/2010 BIG MUSCLES OR QUALITY OF LIFE?\n1\/20\/2010 IT'S A NEW YEAR, YOU NEED TO WORKOUT A NEW WAY!\n11\/6\/2009 INSTANT GRATIFICATION AND \"HEY, HOW'D I GAIN ALL THAT WEIGHT OR GET OUT OF SHAPE?\"\n10\/16\/2009 STRENGTH TRAINING; ISN'T WALKING, RUNNING, BIKING ENOUGH?\n9\/29\/2009 NO RESULTS! GOOD RESULTS! GREAT RESULTS!\n9\/20\/2009 QUALITY OF LIFE, CAN EXERCISE MAKE IT BETTER?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the same machine as the 5105 Grinder Mixer but the hammer mill is replaced with a 20 inch wide, case hardened roller mill with a choice of roll pattern.\nSearch for used roller mills. Find Raymond, Fitzpatrick, Paul, JET Tools, Ross, Henke, Stanat, Renn, Bliss, Brandt, Ercolina, JH Day for sale on Machinio.\nSep 17, 2010 \u00b7\u7e5e Crimper \/ Dry roller Mobile, Grain Mill, Grain Rollers, Grain Crimpers, Bruisers.\nRENN MILL CENTER INC. is a manufacturer dedied to the production and development of worldclass livestock feed processing equipment. This farm equipment includes, roller mills, Grain Rollers, feed roller mills, feed mixer wagons, feed bagging equipment, grain bagging equipment, feed grinder blowers, feed wagons and more.\nMILL & MIX ROLLER\/CRUSHER MILLS PAGE: The reason why SKIOLD offer either 2 or 3 roller mills is because some ingredients require different settings.\nView Superior SM3000 Roller Mill Superior SM3000 Roller Mill Miscellaneous at 4250 GBP in United Kingdom Browns Brown Lennox Mobile Crusher For Sale \u3fd1,500 exVAT.\nAkroMils provides storage solutions for healthcare, industrial use, or your home with plastic storage drawers, plastic totes, AkroBins, Browse All Products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"austria. stone crusher supplier in austria quarry machine and crusher plant sale in asiapacific region . quartz mine process in india 2017 OCDS ... New old mini small mining ore metal limestone mobile portable quarry rock stone .\nportable stone crusher in asiapacific region portable stone crusher in asia pacific region ... quarry crushing plant for sale ... crushers machine for sale ... LIVE CHAT. Quarry & Aggregate For Sale | IronPlanet.\nPortable Stone Crusher In Asiapacific Region. Portable Stone Crusher In Asiapacific Region Gold Grinding Machine For Sale Construction Recycle Plant SCM crushing machine are usually used to crush Read more.\nportable stone crusher in asiapacific region bluegrassmd.us Manufacturer stone crusher mini in asia\/ pacific region open pit portable impact crusher. ''s portable ... Chat Online . portable stone crusher in asiapacific region.\nChina Top 1 Portable Mounted Graphite . crusher, pcl vertical china portable stone crusher in asiapacific region . small portable stone mobile jaw crusher plant for sale advantage 1.deep. harga stone crusher .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyrvq b/data_all_eng_slimpj/shuffled/split2/finalzzyrvq
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+{"text":"Stainless steel travel mug with secure lid.\nFamily owned and operated for over 20 years, our passion at the Card King is delivering a unique, diverse and high quality range of classical and contemporary greeting cards for every occasion. We have ranges to suit every taste and are constantly in search of the latest and most exciting card designs from around the globe to offer our customers.\nAll content \u00a9 2019 by The Card King.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\ud83d\udc49What do you get when you throw a lot of books into the ocean?\nDUE TO THE HEATWAVE AND HOT TEMPERATURES, SNAKES, ESPECIALLY COBRAS TEND TO SEEK REFUGE IN COOLER PLACES INSIDE YOUR HOMES.\n2. Avoid leaving your front doors open for fresh air in the evenings. This reptiles are absolutely STEALTHY (too quiet). You won't hear or see them go inside your house.\n7. Clear bushes round your houses. They attract rats and mice which are the favourite snacks of the most dangerous snakes favorite meal.\n8. Buy snake repellent powder and pour it round your compound. You will definitely decrease the chances of a snake ever visiting your home by 90%.\nBeware of confronting snakes that get into your homes. Some snakes can be killed easily, others like the black mamba are very cocky. If threatened, they will chase you at an alarming speed and give multiple bites while still on the chase; and you might not live to see the next forty minutes. So, just be very careful when trying to kill these snakes.\nStay well and be safe. It's snake season. It's very hot and they are highly active and get agitated and angry very fast.\n\ud83d\udc49What gives you the power to walk through a wall?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is Permanent or Semi-Permanent Makeup \/ Micropigmentation?\nWith Permanent Makeup or Micropigmentation as it's also called, you can have stunning smudge-free makeup that can last for over a year or longer. It is a cost effective alternative to cosmetic pencil applied to the eyebrows, eyelids as well as lips. Permanent Eyebrows and Lips are the most transformative feature on the face because they frame your face, determine expression and when done properly can make you instantly more beautiful. Eyebrows that have been over plucked, thinned or have started to droop with age we can give you an arch or lift and give you the eyebrows you once had when you were younger and dramatically turn back the clock and instantly make you look 5-10 years younger.\nCommitted to creating bespoke, long-lasting makeup effects to suit each of our client individual needs in a friendly environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Water damage is a serious issue that must be taken care of as soon as possible. Do not hesitate if your home has undergone water damage. Acting fast can save your possessions, time, and money. KIWI Services in Crandall has provided excellent water damage restoration for 30 years making us the best choice for your water damage needs.\nKIWI offers 24\/7 water damage repairs in Crandall. If your home is suffering from water damage at 2 AM, do not hesitate to call KIWI. We will send out a water damage restoration team right away. KIWI knows the importance of acting against water damage as soon as it strikes, so that's why we offer quality service at all times. All of our water damage repair technicians are professionally trained and expertly skilled. KIWI is able to do all kinds of water damage repairs from extracting all water to repairs on carpets, rugs, and floors. KIWI also offers a one-year carpet cleaning warranty for any carpet work done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wisconsin Made is an e-business gourmet support and gift store that began in 1999. It grandstand exceptional, amazing things from artisans and little associations in the state of Wisconsin that can be dispatched the nation over. ReeCoupons.com offers Wisconsin Made Coupon Codes, Coupons, Promo Codes and Discount Codes at sensible cost.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabnom b/data_all_eng_slimpj/shuffled/split2/finalzzzabnom
new file mode 100644
index 0000000000000000000000000000000000000000..37d3dce95e415ed57b9d37a687c73bb4cfbe6c13
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+{"text":"This is a find \u2013 a Star Bell revolving musical Christmas tree stand \/ turner! Per box, turner has on\/off switch, tree turns completely around once every two minutes, has revolving outlet for tree lights, for trees up to 8 feet tall with trunk diameter of up to 3 \u00be\", and plays \"Jingle Bells\" (music has separate on\/off lever\/switch). Works great!\nWow, look at this ornament \u2013 this is a glass \"chandelier\" ornament with 3 \"candles\"! The ornament measures 3\" tall and it is in wonderful condition \u2013 there is a small amount of missing paint but no damage to the candles (a rarity!). This is an absolutely beautiful ornament and a treasure for any collection.\nThis listing has two glass teapot \/ coffeepot shaped ornaments! They are West Germany ornaments and are gorgeous! They have no damage \u2013 one has some discoloration in the glitter \"snow\" (this is often the result of artificial spray snow) and there are a few minor flecks of color loss. Ornaments measure 4\" \u2013 they're not the small size ones that are so often found. Excellent additions to a vintage ornament collection!\nA beautiful Christmas bell By Leonard Crystal hand Blown in Italy. The little Drummer boy is made of frosted glass and the bell is made of thin hand blown crystal that has a pretty ring. Measures 7 1\/4\" tall and is in pristine condition with original label.\nVintage pair of Dickson, all gold choir angel figurines. A boy and a girl.Boy measures 6 1\/4\" tall, and girl is slightly shorter. Orignal Dickson sticker on the girl. Gold paint over the sticker, but you can see it. No chips or breaks. Pretty good shape with minor scratches. Sold as a pair. A wonderful addition to your vintage Christmas items.\nThis bride and groom is picture perfect celebrating their one special day. Made of hand painted blown glass. Perfect condition. Partial tag. Made by the Robert Stanley collection. 4 1\/4\" tall. Not dated on the ornament so would be good for any year wedding or anniversary.\nA wonderful old snowbabies ornament with wonderful hand painted facial features. Not marked so I'm not sure if it is from Germany or Japan. Tiny metal eye in the back, as it was originally a Christmas ornament. This little snowbaby is on his tummy, one leg kicked up behind and rests his head on one hand. Measures 2 3\/4\" long and 1 3\/4\" tall. Wonderful condition. A nice addition to any vintage Christmas collection.\nPorcelain bell with silver plated finial of the 1st day Of Christmas Partridge in a pear tree. By Reed and Barton, from their #2 Twelve Days of Christmas series. This bell made in 1990. Measures 4\" tall and can be hung for an ornament. Pretty holly & berries border with platinum trim. Pristine condition, no box.\nAnother treasure \u2013 this listing has a string of red rose lights! The roses and leaves are plastic, all bulbs work, and plug is marked made in Italy. Wow!\nWhat a treasure \u2013 this listing has a string of white dove lights! They are in wonderful condition (two birds have a chipped wing, one has a broken\/missing tail), all bulbs work, and the plug is marked made in Italy. Super!\nAwesome original ceramic Santa Claus candle climbers by Napco. Comes in the original Christmas holiday decorated Napco Box, and dated 1957. Pristine condition,with no paint chips, or wear, and box is good with normal wear.\nAdorable little elves are wearing their striped pjs and red hats. Each has a sprig of holly. Made of ceramic and all cold paint decoration which is excellent. Original foil KOREA sticker. Perfect for Christmas. Measures 3\" tall.\nCute pair of snowman candle climbers made of porcelain. Hand painted with cold paint decoration. Each snowman is different from the other. No chips or damages other than some small amount of the cold paint flaking on the red areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unexpected repairs are both stressful and costly but also a near-certainty for fleet owners. Our preventive maintenance telematics solutions provide you with notifications of when a fleet vehicle is likely to breakdown or need maintenance work. Our algorithms are able to predict when specific vehicle components may fail by allowing you to remotely access diagnostic trouble codes and battery health alerts.\nAdvanced warning of maintenance requirements allows you to proactively schedule repairs before a breakdown and at a time that doesn't affect your operational schedule. This in turn reduces the length of vehicle downtime and decreases costs.\nFind out how your business can benefit from our preventive maintenance solutions today. Contact us by completing the form below or calling us on +44 (0) 330 311 5157.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am delighted to share my interview with Rachel Bustin where she shares her blogging journey including the sad reason she started blogging and how her blog has developed from a hobby into a job.\nI started to blog in March 2015 after my 3rd miscarriage. It was my husband who suggested it to me to take my mind off things.\nI was feeling down after my unlucky time of trying to get pregnant, so blogging was my release. In fact I fell in love with blogging very quickly. Back then I blogged about books, gardening and competitions. Slightly different to the family lifestyle I blog about today.\nIt's amazing! I have finally found my thing in life, my career!\nThat I'm able to help support my family and have that little extra money for things.\nLiving down in Cornwall, events are few and far between!\nMy main source of income is sponsored posts. I do have some adverts but have made like \u00a332 from Google Adsense in 3 years! I have made a few pounds from affiliate income sources but it's not a lot. I think you need huge amounts to make real money from these.\nThe promoting of blog posts. I never seem to have enough time to promote once I have written the post I'm on to the next one.\nThe online world of bloggers I never knew existed. Always there to help and advise. It's an awesome community.\nI'm not sure to be honest. I work full time in my day job, so do lots at lunch times at work. Oh I don't have much time to sleep either my brain is always coming up with blogging ideas. I have a never ending list of things I need to get done with the blog. I think the balance need to be better for me and I'm aiming to take an evening off a week\u2026..maybe!\nI can't see anything that would ever lead me to give up blogging, but you never know!\nGo straight to self hosted. I started off on blogger for about 14 months but was limited in what I could do.\nBlogging for me started as a hobby, now it's a job, but one I love. Hopefully after I have had my second baby this year I can afford to drop hours when I go back to work in 2019.\nHuge thanks to Rachel for sharing her story so generously.\nEvery blogging journey is different and if you would like to take part in my interview with a blogger series, do get in touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A woman in Pittsgrove, while walking in her backyard to garden, was suddenly attacked by a rabid fox on July 18th.\nIn retaliation to the attack, the woman grabbed the fox at the mouth that was biting down on her leg and strangled it with her free hand.\nPittsgrove native Tammy DuBois said, \"I couldn't do anything else to get it away from me. I don't like to kill anything.\" She attempted to escape and get back into the house but the fox had a strong grip on her leg, to the point of puncturing her skin.\nActing quickly, she then decided to grab the fox's nose with one hand and its neck with her other. The fox struggled to get free from her but soon went limp from being choked. The dead fox was recovered by Animal Control and begun running tests on it. It was later confirmed that it had rabies on Monday.\nHer husband rushed her to the hospital to be treated for an animal bite, where she immediately underwent treatment for rabies that took two weeks to finish.\nOfficials say that DuBois was the first and only person to be attack by a rabid animal this year in Salem County, so far.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I don't know why politicians ever try to take on the NRA. They are quite simply the best political lobbying group in the world.\nThe ban on assault weapons sponsored by Democratic Sen. Dianne Feinstein of California apparently died Tuesday with barely a whisper from media outlets or the White House. Black bunting should have hung from every window in Washington.\nSenate Majority Leader Harry Reid, a Democratic ally of Obama, told reporters that Feinstein's proposal could not overcome Senate rules requiring the support of at least 60 senators before allowing a final vote. The proposal \"using the most optimistic numbers, has less than 40 votes. That's not 60,\" he said.\nIn fairness, the gun lobby deserves most of the blame for creating a political climate in which any regulation of firearms is viewed as an attack on the constitutional right to bear arms. This as much a financial issue to the NRA and its industry allies as it is a constitutional one.\nBut Obama and fellow Democrats shoulder a responsibility to reframe the debate around unassailable facts: The Second Amendment is not at risk; modest regulations would improve gun safety and strengthen the nation's noble gun culture; and nobody outside the U.S. military needs an assault weapon. Instead, the White House and Democratic lawmakers signaled retreat on the assault-weapons ban almost immediately after Obama proposed it. He didn't fight.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..10f7cfe0e03b69fe618bd275512973678f57e3ea
--- /dev/null
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+{"text":"There were times when a man should stay and fight, and there were times when it was wiser to cut and run. It was clear to Tyler this was the latter.\nShoving the woman he'd been kissing just moments before out of the way, he clamped his hat lower on his head, and took off at a sprint as her bruiser of a fianc\u00e9 followed him.\nCowboy boots weren't meant for running, but Tyler managed it. He sure had incentive. Avoiding being pummeled into the ground by a jealous fianc\u00e9 served as fine inspiration. He knew the truck was unlocked, but he didn't have the keys to start the engine. Colton had those with him inside the bar. Tyler wasn't about to lock himself inside a truck he couldn't flee in. Not with an angry lunatic hot on his tail, so he kept running.\nThe terrain worked in his favor, as did the darkness as Tyler crashed into the woods off the side of the parking lot. Branches whacked into him as he dodged between them. He twisted an ankle when one foot landed on a rock, but he kept going, limping in a half run. A pine bough caught him across the face, blinding him as he squeezed the injured eye tight and the tears began to flow. Still, he forged ahead. His life depended on it.\nThe woods broke into a clearing and Tyler realized he was behind the lumberyard right off the main road in town. He slowed to a fast walk when he hit the concrete and glanced around him.\nHe needed to get somewhere safe and tend to the eye that hurt like hell and was still tearing up. And he needed to zip his jeans and refasten the belt buckle the girl had undone, all while she'd kept to herself that one important detail\u2014that the fianc\u00e9 she was mad at was working the door of the bar. That would have been good information for Tyler to have. He also needed to make sure his cell phone hadn't fallen out of the back pocket of his jeans so he could call Colton to come and rescue him. But before any of that, he had to make sure the man in hot pursuit hadn't followed this far.\nA crash in the woods behind him, followed by a loud cuss, told him the lunatic hadn't given up yet. Damn, but this guy was persistent.\nTyler took off running again, though at this point it was more of a fast hobble. He had to hide. A pick-up truck parked in the lot in front of him provided his only hope. He should just take the truck and drive away. It wouldn't be stealing. He'd only be borrowing it. He could bring it back as soon as life and limb were no longer in jeopardy, but he didn't know if the owner had left the key inside and he couldn't waste precious seconds checking.\nMaybe it was dark enough that if he laid flat and still in the back, he might not be seen. But if the guy looked closely and saw him hiding, he'd be a sitting duck.\nIt was a chance he was going to have to take.\nRunning out of time, he sprinted to the back of the truck, planted both hands on the tailgate and vaulted into the bed. When he landed inside, he knew his luck was holding. There was a big green tarp in the bottom of the truck bed. He flipped it over him and held his breath, trying to move as little as possible to not rustle the thick plastic covering him.\nWhile he waited to be discovered, or not, he figured praying couldn't hurt. Silently, he vowed that if he got out of this night unscathed, he'd never make out with a stranger in a bar again.\nAs his heart pounded, he heard heavy footsteps in the lot, and then a few more loud cusses and what sounded like a bear\u2014or a really big man\u2014crashing back through the trees.\nLong minutes ticked by in silence, and the crazed fianc\u00e9 didn't come back to whip the tarp off him and beat him to a pulp. Against all odds, he might just be safe. Out of the woods, literally.\nIn light of that, Tyler decided to add a small amendment to the deal he'd made with God. It would probably be all right to hook up with girls he met in a bar. However, he would be sure to ask them if they had a boyfriend, or any kind of significant other before he kissed them and let them unbuckle his pants.\nSatisfied that was a promise he could live with, and hopeful that the guy had given up the chase, he was about to take a peek to see if the coast really was clear when he heard footsteps heading toward the truck.\nIt sounded like two people walking. They hadn't come from the direction of the woods, but rather from the building. He was most likely safe from being maimed by his pursuer, but he definitely was not in any position to be socializing with anyone. His jeans still hung wide open, his eye remained squeezed shut, and he was hiding under a tarp in a stranger's truck.\nIt wasn't as if he could pop up and say hey, but he also couldn't stay hidden. If the owner of the truck drove away, who knew where he could end up.\nHe lay helpless as she whipped the tarp to the side and exposed him to the beam of the parking lot light. At the sight of him, she screamed and jumped back.\nTruth be told, he nearly screamed too. He scrambled to sit up, before he realized he might not want to be sitting up. He still wasn't convinced the scorned fianc\u00e9 had really gone.\n\"Hey, Tyler.\" There was amusement in the male voice that came from his left.\nStill partially blinded, Tyler turned his head to the side to see a guy he'd gone to high school with standing next to the truck and grinning as he balanced a fence post on his shoulder.\n\"You know him?\" the woman asked.\n\"Um, it's a long story.\" Sitting up, Tyler glanced at the woods and decided to take his chances with the crazed bouncer rather than look like more of a fool by continuing to lie in this woman's truck bed. He pulled himself all the way upright and went to work fastening the open fly of his jeans.\n\"Oh, don't worry, Ty. We have the time to hear your explanation, no matter how long it is. I'm sure it will be worth it.\" Jed grinned.\nJed hadn't missed the fact Tyler's pants were hanging wide enough to expose his underwear. Neither had the woman. Tyler saw her bite her lip to control a smile as she averted her eyes.\nDon't miss the OKLAHOMA NIGHTS SERIES where we first meet Tyler & his fam!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \/ Ultrasonic \/ Improve Your Sense of Humour!\nAre you one of those who takes life too seriously?\nWould you like to be able to entertain others with your sharp wit?\nWould you like to develop the ability to naturally make people laugh but don't know how?\nSome people have this amazing ability to bring laughter and joy to themselves and people around them, they do it so naturally and effortlessly!\nMost of us were not born like this, it takes many years for them to get to this level, but with our program, you can become like them!\nUSE YOUR SENSE OF HUMOUR TO DIFFUSE ANY STRESSFUL SITUATION.\nDEVELOP A MINDSET TO STOP TAKING UNNECESSARY THINGS TOO SERIOUSLY BUT BE WITHOUT COMPROMISING SERIOUSNESS IN MATTERS THAT HAVE TO BE TAKEN SERIOUSLY.\nDEVELOP THE ABILITY FOR MAKING EVERYONE AROUND YOU ENJOY YOUR PRESENCE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Terrace Summer Dance Intensive offers a unique summer dance experience that will lead aspiring dancers to reach their performance potential.\nTerrace born, founder and Artistic Director, Jessica Ames has been running this program for seven years!\nWe know that there are many other summer dance programs out there and it is often difficult to choose the perfect program for you and your child. Many factors go into making this decision. Including, travel, costs, teachers, quality of training and so on.\nThe TSDI is in Terrace BC! Also known as the hub of the Pacific Northwest. If you already live here, great! How convenient for you! It is an awesome location and absolutely beautiful in the summer time. If you are traveling from out of town, Terrace is a lovely summer vacation spot for the whole family. It is easy to get to either by car, bus, rail or air.\nOne thing TSDI prides itself on is the fact that it is important that we make dance available to everyone. We keep our costs low and offer payment plans by request.\nAll of us with the Free Elements \u2013 Terrace Summer Dance Intensive are focused on providing fun, but challenging dance classes with the highest level of professionalism. We will do everything we can to meet your expectations and more!\nWith a variety of classes to choose from, we're sure you'll be happy registering your talented, young dancer with us. Look around our website and if you have any comments or questions, please feel free to contact us. We hope to see you again! Check back later for new updates to our website.\nAlivia was introduced to dance at the age of 10 and has continued to train, practice, improve and perform for 8 years. Alivia has been taught many different disciplines of dance including Jazz, Ballet, Contemporary, Modern, Lyrical, Musical Theatre and Hip Hop by numerous amazing artists.\nIn 2014, Alivia was accepted into RWB's (Royal Winnipeg Ballet) summer intensive program and trained there for four weeks under highly commended teachers. In 2015, she was accepted, and presented with a scholarship to Joffrey Ballet School's summer intensive program in New York. Unfortunately, she was unable to attend due to an ongoing injury acquired in early 2015.\nAlivia has travelled with her teachers and fellow dancers to many recognizable, prestigious dance competitions\/events both in Canada and the U. S. Collectively as a group they were well awarded for both group and solo performances. Alivia represented Nanaimo at the Provincials in Penticton, 2014.\nAlivia worked for 2 years with Moses Asher, a top 20 finalist in Canada's \"So You think you can Dance.\" She danced and competed with his choreography at several competitions with multiple awards and accolades. She then performed solo at the age of 14 years, for his company's opening night at the Vogue Theatre in Vancouver.\nIn 2011, Alivia started assisting Ms. Kathryn's children's dance classes at her home studio, Harbour Dancentre (now named Harbour Dance Studios). She assisted for 2 years and subbed in as the teacher on occasion when needed. Alivia's love of dance is shown best on stage when performing, but is also greatly demonstrated when teaching students!\nAlivia taught Contemporary and Ballet at Harbour Dance Studios this past season and comes highly recommended by her associates. Alivia is very excited to share her love of dance with the students at Free Elements Dance Studio!\nShe will be teaching Intermediate and Senior Ballet & Pointe in this year's Summer Intensive.\nJulianne's choreography has been presented at the Edam Dance Series, Dancing on the Edge, Dance in Vancouver, Dances for a Small Stage, Festival Launch, 12 Minutes Max, Drift Arts Festival, SWARM Art Walk (Vancouver), Dance Days (Victoria), The Chinook Series (Edmonton), Risky Business Rebel Yells (Toronto), Tipperary Dance Festival (Ireland) and Nah Dran (Berlin). Her performance, video and installation work has been presented at the Foreshore, Franc Gallery, Dusseldorf's Open Art Film Festival, Iris Film Collective's One Take Super 8 and Latitude 53's Visualeyez Festival of Performance Art.\nMost recently, she was awarded the 2017 Iris Garland Emerging Choreographer Award which will go towards the production of her first full length ensemble work. Julie is also co-producer and sometimes curator of a small experimental performance series titled Shooting Gallery Performance, and sits on the board of directors for CADA-West.\nJulie will be teaching the Intermediate and Senior Modern and Acro classes in this year's Summer Intensive.\nFree Elements Dance Studio is thrilled to (once again) host Portia Favro! Portia will be our Jazz and Adult Heels instructor at our summer intensive this August 13-17! Portia Favro is a former journalist turned professional dancer, producer, choreographer and pin-up personality.\nShe is an award-winning dancer, choreographer, and teacher with extensive experience performing and working across Canada, the West Coast and Europe. She has worked with international choreographers and artists such as Galen Hooks, and continues to travel to teach and choreograph heels, cabaret and retro go-go technique for beginners to professionals.\nPortia is the Lead Instructor for Luminesque Dance and First Dance Vancouver, and Produces Go-Go Bungalow at the Shameful Tiki Room, Vancouver's only weekly retro go-go show.\nShe's performed for The Canadian Olympic Committee, The Lovers Cabaret, GSG Productions and many more. Most recently, she performed as a soloist in the highly esteemed Vancouver International Burlesque Festival at the Vancouver Playhouse (2018). In 2016, Portia produced her first full-length production \"From Bond with Love\" an ode to the women of the James Bond movie franchise. After a sold-out run and encore in Vancouver, she took it on the road to audiences across British Columbia.\nWhen she's not in Vancouver, Portia is on the road teaching and choreographing for local dance studios across Canada, and is often called upon for creative direction and input on commercial productions, photo shoots and other projects.\nPortia will be teaching the Intermediate and Senior Jazz and Yoga classes as well as the Adult Heels classes in this year's Summer Intensive!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dawn is currently Z60's Equality and Diversity Officer.\nBoth Ladies will start their reps training in September which will be facilitated by the main HQ in Nottingham.\nGood Luck ladies and welcome aboard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Once while I was on retreat at Lebh Shomea in south Texas, Father Francis Kelly Nemeck, OMI, offered me this thought during spiritual direction: \"When we are ready, when the time is right, God shows us the gifts.\" I was to remember his comment a couple of years later on another journey from San Antonio to Lebh Shomea.\nThat particular March morning, as I slowed down on U.S. 281-South to enter the town of Falfurrias, I noticed a large weathered sign on the right. Apparently I was \"ready\" and the time was \"right\" to read the words printed in large black block letters: DON PEDRITO JARAMILLO SHRINE. On at least three prior trips that sign had never caught my eye. However, the name sounded familiar and I remembered seeing glass votive candles labeled \"Don Pedro Jaramillo\" on the shelves in several South Texas grocery stores beside other candles bearing images of la Virgen de Guadalupe, San Antonio, Divino Esp\u00edritu Santo, and San Francisco de As\u00eds.\nThough I was curious, I also was looking forward to my Lenten retreat time and therefore continued on my way without stopping. After a left turn in Falfurrias onto the east-bound state highway, behold! There was an identical weathered sign announcing \"Don Pedrito Jaramillo Shrine\"\u2015 this time with the arrow pointing north. Still I passed it by, wondering who \"Don Pedrito\" might be.\nFive days later, after morning Eucharist, I loaded my car and set out for home. It was a hot, muggy South Texas spring day, weather that made me grateful to be in an air-conditioned car. Somehow I knew that I would not again pass by the Don Pedro Jaramillo sign without following that red arrow.\nEast of Falfurrias, when I spotted the now familiar sign, I turned north onto a secondary gravel road, seeking . . . Well, I don't know what I was seeking, except to know something more about this Don Pedro. That he was known as \"Don Pedro\" or \"Don Pedrito\" already told me that he was held in high regard. I suppose that I was hoping to learn why.\nMany came to him because, unlike other faith healers, he claimed no power of his own, but said that God's healing was released through faith.He made no charges. Patients gave or withheld as they chose. But whatever was given voluntarily he often gave to the poor \u2014 food as well as remedies.\nHow many times did Jesus say, \"Your faith has made you well,\" or \"Your faith has healed you,\" or \"Your faith has made you whole\"? I realized that, unawares, I had come to a pilgrimage site.\nMy second surprise was that the shrine (or santuario) sits in the midst of a small cemetery. Somehow it had never occurred to me that I might be visiting Don Pedro's actual gravesite since the saints with whom I was familiar were from other places and other times, from far away and long ago. Holy women and holy men were not from the century in which I was born (Don Pedro died in 1907) and certainly not from an out-of-the-way place so near my San Antonio home.\nThe shrine was a one-story brick building, about 20 feet square and of indeterminate age. Since the door was closed (and given the current city custom of locking churches), I wondered whether it was open for visitors at all. No one was around to answer my question if I had asked. After reading the biographical information on the marker, I tried the hasp on the gate, found it unlocked, and followed the short sidewalk through the cemetery to the chapel door. Pulling gingerly on the door handle, I found that it was open, took a deep breath, and prepared to enter.\nWhat I remember next is the whoosh of hot air that enveloped me when I opened the door completely \u2015 a thermal gust hotter than the sultry air outside. Stepping inside, I felt time and place, curiosity and haste, fall away. I stood in a silence so profound that I could almost hear the sizzle of the flames as they consumed the wicks in dozens of votive candles burning on a metal altar table. All of the casement windows were closed and the small space felt both stuffy and confined, making it hard to breathe. But breathe I did, inhaling the faint fragrance of incense that still lingered in the air, a reminder that other visitors had preceded me. I was alone in the shrine.\nSlowly I regarded the scene. My eyes were drawn first to a wall where numerous metal crutches were suspended from nails. Above them and all around the room, hanging next to the ceiling were rainbow-colored artificial flowers. There were even some fresh flowers in vases at various locations. Two or three different images of the Blessed Virgin Mary rested on side altars or in glass cases. At several spots on the wall were bulletin boards overflowing with handwritten notes both in English and in Spanish: thanksgiving for healings, heartfelt petitions for intercessions to bring relief from illness or pain, hopes for divine assistance in business and all aspects of life. Frequently they were prayers or thanksgivings for what had been granted to someone other than the note's author. Sometimes they acknowledged a grace received personally by the writer. Photographs were thumb-tacked on top of notes from the devout. Small metal milagros \u2015 feet and legs, faces and arms, babies, and even farm animals \u2015 dotted the space on top of these overflowing layers of notes and photographs. Prior pilgrims had left ample proof of Don Pedro's effectiveness in working changes in their lives.\nAgainst one wall rested Don Pedrito's tombstone with the epitaph: \"Aqu\u00ed yacen los restos de Pedro Jaramillo, el benefactor de la humanidad.\" (\"Here lie the remains of Pedro Jaramillo, benefactor of humanity.\") The grave itself occupied a large covered rectangle of the floor space. Beside the tombstone was a life-sized ceramic statue of the bearded Don Pedro dressed in suit and tie, his toes bare in open sandals. Some visitors had even draped necklaces with petitions and prayers around his neck. Worn kneelers in front of the statue and alongside the grave were places to pause for prayer \u2015 and the marks of wear showed that many pilgrims had done just that. Somehow just standing in the shrine felt like prayer.\nOn a table at the right hand of Don Pedro's statue I noticed a bowl of water, holy water, I was certain. It felt cool on my fingertips as I touched it lightly and slowly drew my hand back to make the sign of the cross, forehead to chest, left shoulder to right, as I would during the Eucharist. No longer did I feel the heaviness of the air or its heat. A space seemed to open in my heart and I felt refreshed. With a final glance toward the chairs where many pilgrims had sat and the kneelers where others had prayed, I turned toward the door, opened it, and walked out into the spring sunshine of South Texas.\nReturning slowly to my car, I recalled my visits to Tepeyac in Mexico, to Chimayo in New Mexico, to Santiago de Compostela, all of them places made holy \u2015 sanctified \u2015 by the prayers and pilgrimages of the faithful. Now, nearly a decade since my visit to Don Pedrito's Shrine, I believe that I understand what Jacob thought upon awaking from his dream, \"Surely the Lord is in this place and I was not aware of it\" (Genesis 28:16).\nRilda Baker is a teacher, writer, and Spanish translator. She co-directs the Diocesan Retreat Society and is a member of St. Paul's in San Antonio. Reach her at drrildabaker@gmail.com.\nI plan to visit the shrine and take my youngest daughter with me. She is in a inbetween in her faith. I went through that in college but snapped out of it soon after.\nWhat a beautiful story. I grew up taking trips with my grandparent from raymonville to don pedrito. After my grandmother pasted i had felt lost and just could not find myself. I then remembered that grandma had so much faith in don pedrito. With hardly no money my husband and I went to the shrine . When we got there i felt scared to go in for ome reason so my husband and i argued cuz i did not wantto go in alone. As we argued these night bugs got so loud my husband and i could not hear eachother speak. My husband grabed my hand and we walked in together . I remeber the outside but never the inside. I felt a wave of emotions . I felt like my grandma who i miss so much was the 're. With me proud of remembering faith is number one. As i teared up i kneeled asked god and n pedrito for guidance . Grandma always said ask for signs and you will get them . Sure enough i got my sign. Having faith is beautiful ! And having faith in god and don pedrito as gods helper is awsome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"BANGKOK - The IGLTA Foundation has partnered with the Pacific Asia Travel Association (PATA) to provide a unique opportunity for a Hong Kong Polytechnic University tourism student to attend the International Gay & Lesbian Travel Association's 35th Annual Global Convention, set for May 9-12 in Toronto, Canada.\nThe successful partnership between PATA and IGLTA has allowed both organisations to share knowledge through research and publications, provide reciprocal participation at events, support mutually agreed advocacy positions, and enhance access for the benefit of members of both organisations. As part of the partnership, IGLTA will be offering student discount rates to join the IGLTA 35th Annual Global Convention.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"has NO power. There is a work crew there replacing some equipment.\nfor Sunflower are connected via HRB.\nconnectivity until the power comes back on.\nThis entry was posted in Unscheduled Disruptions on December 21, 2010 by Brian.\nThe Helpdesk has received alot of calls with people unable to access webmail via various different cell phones. The new Exchange 2010 front end servers were not authenticating properly to the Exchange 2003 mailbox servers via ActiveSync. Janis and I found and fixed this with a hotfix available from Microsoft. Tests on the phones we have available to us (iPhone, Droid) seem to work OK now, but others such as Palm and some Blackberries may still have some issues.\nThe only other issue we are seeing is with SSL certificates not being trusted, and we are attempting to fix this issue on phones we are familiar with.\nservers. All webmail traffic will now go through Exchange 2010.\nSome providers are taking longer to pick up the DNS changes than others.\nAlso, clearing a machine's DNS cache is sometimes necessary.\nThis entry was posted in Unscheduled Disruptions and tagged activesync, exchange, webmail on December 21, 2010 by Brian.\nUsers of the UIS BlackBerry system have been notified that ITS will be upgrading BLackBerry Enterprise Server on Tuesday, January 4, 2011, to the latest version of the server software. The need to upgrade this service is due to our current version (4.1) will soon be reaching it's end of support by Research in Motion and to provide continued Microsoft Exchange service as we begin migrating UIS users to Exchange 2010. This upgrade will require that the server be temporarily unavailable during the upgrade process but will require no interaction by users.\nThis entry was posted in Scheduled Maintenance and tagged BES, BlackBerry, Upgrade on December 21, 2010 by Clayton Bellot.\nThis entry was posted in Scheduled Maintenance on December 20, 2010 by Brian.\nI'm borrowing a mac from brk 141b to test Deployment services on it.\nThis entry was posted in Uncategorized on December 20, 2010 by J.\nFINAL UPDATE: Just got a call from CITES (10:22 PM). The network engineers found a software fault on one of the routers and corrected it.\nUPDATE3: Internet access has been restored (about 10 PM). No word yet from CITES on what the cause was.\nUPDATE2: CITES tech support person (Nathan) took my info and will contact network specialists and get back to me. Access to several web sites at uillinois.edu works fine. Access to web sites at the Chicago campus does not. Could be a hardware failure or possible a configuration issue. We will need to wait to hear back from the CITES folks before we have any info on how much longer it might be down.\nUPDATE1: We've confirmed that this issue seems to be with the ICCN network, outside of UIS's sphere of control. We are attempting to contact someone at the ICCN NOC now.\nApproximately 7:00 PM this evening, access to and from the Internet from the UIS campus stopped working. ITS staff are currently working on the problem.\nThis entry was posted in Unscheduled Disruptions on December 17, 2010 by Curtis Spears.\nIf you haven't done so yet, I encourage you to take a minute to log into the ITS blog and update your profile info so that your name is displayed rather than just your NetID. That way it'll be easier for people to know who is posting the information or comments.\nIt's very easy to do this: just use the 'Your Profile' link on left. There's a place there to enter your profile information, including your display preferences.\nThis entry was posted in Uncategorized on December 16, 2010 by Curtis Spears.\nwere able to get this returned under warranty.\nThis entry was posted in Unscheduled Disruptions on December 16, 2010 by Jamie.\nI have added new content to the wiki under the \"Internal Considerations\" link, including a document that provides the most detailed estimate to date on how virtualizing servers can save the University money, even after accounting for SAN and VMware licensing costs.\nThere is also updated content in many other areas, including the proposed implementation framework (under 'Transition Plan\"). Please read through the documents and wiki updates and post any comments you may have.\nto gain consensus that moving forward with implementation will be in the best interest of both UIS and ITS.\nAdopt the framework shown under the Transition Plan section for phased implementation (or some agreed-upon variation of it).\nList next steps to be taken with targeted completion dates.\nThis entry was posted in Uncategorized on December 15, 2010 by Curtis Spears.\nDecember 23rd. Thanks and Happy Holidays!\nThis entry was posted in Uncategorized on December 15, 2010 by Kelly Walraven.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Resilience Is Your Team?\nHome > Blog > How Resilience Is Your Team?\nUsing the PR6 resilience model we can now understand how effective teams are at working together to build resilience. The PR6 resilience model was developed from the study of the human mind. Years of research has led us to an understanding of the core elements that make up resilience.\nThe PR6 was scientifically validated as a psychometric scale with a Cronbach's alpha of .7365! For those of you playing at home, the closer the score is to 0.9 the higher the internal consistency or in laymen terms how closely related a set of items are as a group.\nSo, what exactly is resilience?\nQuite simply, resilience is the capacity to recover quickly from adversity. It is a construct of mental toughness. One off the hallmarks of any successful organisation is resilience. The hardier the constitution, the more likely you are to succeed. As the saying goes, a team or organisation is only as good as its weakest member.\nIt is a tough world out there today. We all have so many distractions and life can get overwhelming at times. Before you judge a poor performing employee too quickly stop and ask yourself what is going on in their life and how that might affect you as well? Put yourself in their shoes. You might find that they are not a bad employee, they just need some help to cope with life's many challenges both inside and outside of work.\nBeing a savvy business mind, you think that there must be a way to measure, test and thus predict an employee's level of resilience as there is with any metric. And in fact there is. But, before we can get into that we must understand resilience to be able to predict it. Resilience can be broken down into 6 components that if nurtured will improve a person's resilience.\nThe first is collaboration. This need comes from the right prefrontal cortex. A natural human need is to belong. Belong to a family, a group, a nation, a team etc. This is our life line, our sense of purpose and this need for support networks is the key foundation piece of a team. People work twice as hard for those they care about and respect. So, give your team a reason to work for each other.\nThis need for a vision stems to a few different parts of the brain. Brain activity in relation to vision is seen in the Hippocampus, the Pre-Frontal Cortex and the Ventral Stratum.\nYour new tight-knit team is in search of a purpose. Give them a worthy goal along with the confidence to do it and help them see the good that will come from their efforts. A well team motivated will work for each other towards a goal they believe in is a powerful asset.\nNow enter the outside world. Your well motivated team is going to face adversity from inside and outside of the organisation. A sign of high resilience is your composure under pressure. The Pre-Frontal cortex is also involved in composure.\nThis is a critical step in the PR6 model. A great leader will find ways to help their team practice their composure, to keep their calm and execute the teams goals no matter the situation or adversity faced.\nReasoning goes hand in hand with composure. The definition of composure is to reason and solve problems while under pressure. Reasoning comes from both the left and right Frontal cortex and the anterior cingulate cortex.\nA successful plan for facing adversity will go a long way towards successful reasoning. A well-prepared team is a team ready to face challenges and persevere.; with the ability to apply reasoning to any situation rather than panic and anxiety.\nThis may seem to be the most obvious one. Great companies in the world are starting to realise the benefits of encouraging employee well-being. Offering gym memberships and healthy options for the in-office lunch show great results for the organisations that invest in them. A healthy work force is a happy work force.\nFinally, tenacity. With collaboration, vision, composure, reasoning and health all running at peak efficiency all that is left to encourage is the never say die attitude. Nurturing a winning environment is important to putting all of this together and generating a resilient team.\nHow Do We Measure Our Teams Resilience?\nThe PR6 model measures momentum. Momentum gives us a forward-looking predictive model to help show the likely direction of development.\nThis model is used in clinical setting and for general coaching purposes. It helps to find problem areas so that you can strengthen those weakest links, to have the best possible change. The PR6 model is even used for sports teams and schools. It is applicable anywhere resilience is required and every team can always improve upon their resilience.\nThe results come complete with action plan suggestions to help being the strengthening. We are able to get results for individuals and for whole teams. It is useful on the professional and personal level for those reasons.\nTo learn more about resilence training click here or to book a FREE demo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"French Country Lighting Fixtures is one of the design ideas that you can use to reference your Lighting. There are a few images that have been published on August 11, 2018, which you can use as a consideration in the article Gallery of French Country Lighting Fixtures.\nIf you are helped by the idea of the article French Country Lighting Fixtures, don't forget to share with your friends.\nArticle French Country Lighting Fixtures may be associated with antique french country lighting fixtures, country french lighting fixtures home, exterior french country lighting fixtures, french country bathroom lighting fixtures, french country lighting fixtures, french country lighting fixtures for bathroom, french country lighting fixtures kitchen, french country style lighting fixtures, french country vanity lighting fixtures, may be you are looking for so that more references, not just the article French Country Lighting Fixtures.\nFrench Country Lighting Fixtures this possible during your search, you are not wrong to come visit the web lettuceveg.com. French Country Lighting Fixtures is one of the pictures contained in the category of Lighting and many more images contained in that category. Published by admin on . for personal use only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Long and Dusty Road\u2026 Road Doggin'!\nThis is tough time for my fellow artists for sure. I've just about seen it all from my humble beginning in my parents' tiny basement, to owning my own shop and having employees.\nI got my start in the sign business painting MDO signs, and lettering trucks. I even did a little pinstriping on the side. After a couple of years, I had a customer tell me \"you need to paint motorcycles in Laconia.\" The Laconia, NH motorcycle rally is one of the oldest in the country. I knew nothing about working out on the road, but it sounded like fun so i agreed to check it out. I found out what I needed to work the rally: I needed a license from the state- Check, needed supplies- Check, needed a place to work\u2026 uh? Ok\u2026 where do the pinstripers work? Now I needed to do some recon for a good place to set up! So I headed up to where the rally took place, and drove around, asked questions, and finally found a place to work. I made a deal with the owner of a motel and was to set up in his parking lot for the rally.\nNow at the time, I had never worked outdoors, and didn't realize that I'd need a tent to work out of\u2026 DUH! So I had to set up with a card table my paint and brush box little banner and photo album\u2026 minus a tent. In the sun. In the wind\u2026.with no TENT! AND did I mention I was on a budget, and I had to sleep there in the parking lot in my 72 Nova? talk about working in the trenches! But it all worked out! I made more that weekend than I had the previous month painting signs! I liked this! Now I had to buy a tent! Welcome to road doggin'! When I was there painting and getting one hell of a sunburn, (because I HAD NO TENT!) I was invited to attend other rallies around New England. There are several all over the the country, all summer, but I had to learn to pick and choose which ones where worth going to, and which ones would cost you more to attend than I'd make.\nIt was hard work, and the traveling sucked till you got there, and then you had to drive all the way back. But it was fun, you got to see the country and meet all kinds of people, the good the bad and the \u2026strange! Hahaha! I have met some life long friends out on the road. It was working on the rally circuit where I learned to work fast! Time was money out there, and the faster you could get this job done well and get paid for it, the sooner the next customer could come in. My tent was like a McDonald's drive thru!\nAfter working on the road for several years, I left the circuit to pursue a career in airbrushing! So I folded up the tent and got a shop. It was nice to not be working out in the heat, rain, wind, and drunks! Lol!\nI have to tell you, I love working in my own shop surrounded by some really cool artwork! I have television, food, music, a paint booth, comfortable couches and more. It really feels like I'm working at home here. I love what I do so much, that it doesn't feel like a job, and I wouldn't want to trade it for anything. I sometimes miss being a road dog, traveling to tourist locations, and seeing new sights, but I get over it soon enough.\nI learned a lot about work, gained speed, and learned a lot about people in general. It was hard work, good money, and was full of adventure. And I wouldn't go back if you asked me to! But I wouldn't trade the experience for the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Some people say 2016 has been a hell of a year, for a number of reasons, but for us Man of Many it's largely been a massive success. We took the big step of quitting our jobs to focus on the site full time, moved into our very first office and got to head on a few adventures along the way. So to celebrate, we thought we'd round up some of our most popular stories of the year for you below.\nA main-stayer on Man of Many, basically because everyone loves watches. If you haven't already, make sure to check out our regular watch news series The Wind Up.\nA slim wallet is an essential part of your everyday carry.\nOur favourite way to have coffee in the morning.\nIt was short-lived, but fun while it lasted.\nThe greatness that was the Halal Snack Pack.\nFind out what's behind the world's biggest YouTuber.\nNot all of us have a spare $10k laying around.\n20 ways to beat the heat this Summer.\n2016 was the year of the influen cers.\nOur favourite time of year.\nA perfect resolution for 2017.\nThe perfect way to finish the year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LONDON--(BUSINESS WIRE)--SpendEdge, a global procurement market intelligence firm, has announced the release of their Global Workplace Computing Devices Category - Procurement Market Intelligence Report.\nOrganizations are increasing their expenditure on a range of workplace computing device to improve the productivity of the employees. This will act as one of the primary growth drivers for this market. Leveraging 5G network in the business infrastructure will necessitate procuring portable devices such as laptops, smartphones, and tablets which will consequently accelerate the category spend momentum. Get the Free Sample copy of this market intelligence report on workplace computing devices to get insights into strategic sourcing objectives and pricing strategies.\nIn the global market for workplace computing devices, the US alone account for almost 80% of the category spend. Adoption of cloud computing has improved the security standards in organizations that are enabling them to allow employees to securely use portable devices such as laptops, tablets, and smartphones. This is contributing to the category spend growth rate in the US. APAC is witnessing a flux of projects outsourced by the global companies who are setting up their office infrastructure in the developing nations. This will consequently accelerate the category spend growth rate in the region.\nInsights offered in this workplace computing devices sourcing and procurement report include supply market forecasts, strategic sourcing, pricing strategies, and category management insights. Such insights are relevant for both the buyers and the suppliers who seek a risk-free and a cost-effective procurement strategy. The category spend segmentation done in this market intelligence report will guide the investors in identifying the best investment areas as well as help them in identifying cost-saving opportunities in the market. Request free customization of this market intelligence report on workplace computing devices to get information tailored to your every requirement.\n\"Collaborating with service providers who offer product and service bundles can reduce excess spend,\" says SpendEdge procurement expert Tridib Bora.\nPurchase the full market intelligence report on workplace computing devices here!\nSpendEdge's procurement market intelligence reports for the information technology category provide detailed supply market forecasts, pricing strategies, strategic sourcing insights. Such information will help procurement managers as well as the suppliers to determine the total cost of ownership and change their procurement strategies accordingly. Additionally, SpendEdge's reports provide category management insights and information on the procurement best practices for the category.\nDo you purchase multiple reports in a year? Our subscription platform, SpendEdge Insights, provides ready-to-use procurement research reports for multiple categories. Now access the latest supplier news, innovation landscape, markets insights, supplier tracking, and much more at the click of a button. Start your 7-day FREE trial now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"venca.es, operated by Digital Lola Commerce S.A.U., is an online store with nationally-focused sales. Its eCommerce net sales are generated almost entirely in Spain. With regards to the product range, venca.es is an all-round online store, with products on offer that cover different categories, such as \"Fashion\", \"Furniture & Appliances\" as well as \"Toys, Hobby & DIY\".\nFirst party net sales generated by venca.es after deduction of returns, allowances for damaged or missing goods and any discounts allowed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is a Graph Database?\nDo you need a Relational Databases for Big Data ?\nWhat Is The Difference Between Business Intelligence And Analytics?\nHow to become a Professional Data Scientist?\n\"BigData is the New Black\" is your news, entertainment, technology and training website. We provide you with the latest breaking news and information straight from the BigData industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hornady American Whitetail ammunition is the perfect choice for bringing down deer and other midsize game. Manufactured in Grand Island, Nebraska, these cartridges come in boxes of 20 and fire a 150 grain soft point bullet at about 2,910 feet per second. The superb long-range accuracy and terminal performance of this ammo makes it much easier to bring down your target at distances well over 300 yards!\nThis ammo uses the Interlock soft point bullet, one of Hornady's own designs. The Interlock (as its name suggests) features a locking band that keeps the bullet's core and jacket bound together for maximum killing power against game animals.\nRemember to take advantage of AmmoMan.com's free shipping promotion with an order of $99 or more. All orders over this amount cost you only the list price!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The New 3A service, provided by Oxford Bus Company and Stagecoach , will operate between the City Centre (Speedwell Street) and the Magdalen Centre on The Oxford Science Park.\nThe service will operate on a half hourly frequency between the hours of 0630 \u2013 1930, on Monday to Friday.\nThe new service replaces the T2 Thames Travel bus which ceased on 23rd July. The link to the new 3A timetable is provided here (PDF).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AS,BA, Sacred Heart University, Fairfield, CT.\n\"Today I received an 'AQ' rating from the NRA-PVF as being a solid pro-gun\/Pro-hunting candidate who supports the Right to Keep and Bear Arms. I will defend our Second Amendment if elected. I ask for your vote on November 6th.\"\nMarzullo signed the 2018 Coalition of New Hampshire Taxpayers pledge, which states, \"I, the above signatory, pledge that if I am elected to any statewide office, I will oppose all efforts to impose a sales, income, or other broadbased tax on the taxpayers of the State of New Hampshire.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the classic design challenges for Orthodox synagogues is the matter of the Women's Section \u2013 how to create a space which welcomes and encourages davening, while excluding some of the key components of the shul's real estate (shulchan, aron) and dealing with the reality that the great majority of Orthodox attendees are men.\nMechitzah design is its own complex issue, of course, but trying to design a good davening area for women by focussing on the mechitzah would be like trying to design a hockey rink by focussing on the placement of the red and blue lines. The lines are important, but it's the ice that really matters, folks - and in the decades since balconies ceased to the norm, the ice has often been ignored.\nHaving a choice of shuls and minyanim here in Toronto has heightened my sensitivity to the issue, and to the possibilities \u2013 positive and negative.\n2. Don't let it become a coatroom, or a shortcut, or an ad hoc spillover section\/talking section for men. It's up to the rabbi and gabbaim to enforce this, but proper layout can help.\n4. Similar to #3, make sure the climate control system works well on their side. In particular, check that the women aren't directly beneath vents; that tends to happen in shuls where the women's sections line the sides of the shul.\n5. Make sure they have siddur and chumash shelves in their area so that they won't need to ask the men to send them sefarim. And make sure the men's siddur and chumash shelves are not in the women's area, as well.\n6. Put the women's area near the front, for reasons of both acoustics and overall feel. Of course, this may necessitate a higher mechitzah, depending on certain halachic issues, but my sense is that a few more inches and placement up-front is preferred over a shorter divider and seating in the back.\n7. Design the acoustics to ensure that the davening, Torah reading and haftorah are fully audible in the women's section.\n8. If your shul has a noisy hallway, make sure women have a choice of seating further from the door. Some may want to be near the door to have access to their children or to a quick exit, but not everyone wants to pray to the sweet sounds of squabbling children and the kiddush club.\n9. Share the furniture. If your men's section has comfortable chairs, tables and shtenders, so does your women's section.\n10. Make sure it's populated. It's a big turnoff for my daughters, and very uncomfortable for them, if they are the only women present in shul.\nWomen: Am I off-base here? What would you change, and what would you add to this list?\nThe post below ran at Chanukah-time three years ago. I look back with a mixture of nostalgia and amusement; enjoy.\nWhat does it mean to have the US as a \"close ally,\" a nation standing by your side and guaranteeing your security? When you're under attack, will the Obama administration provide instant military aid, or will it dither diplomatically while your civilians suffer?\nSouth Korea, on the other hand, has relied on the United States for on-site military assistance. Per the US State Department's website: \"Under the 1953 U.S.-R.O.K. Mutual Defense Treaty, the United States agreed to help the Republic of Korea defend itself against external aggression. In support of this commitment, the United States has maintained military personnel in Korea, including the Army's Second Infantry Division and several Air Force tactical squadrons. To coordinate operations between these units and the over 680,000-strong Korean armed forces, a Combined Forces Command (CFC) was established in 1978. The head of the CFC also serves as Commander of the United Nations Command (UNC) and U.S. Forces Korea (USFK). The current CFC commander is General Walter \"Skip\" Sharp.\" Although the same article notes that the US presence will be scaled back and shifted south in the coming decade, currently, US troops are there to defend South Korea from attack, and even to coordinate South Korean forces.\nSo what will happen now? After the apparent sinking of the South Korean warship Cheonan by North Korea last year, and initial American denunciations, the US did nothing. Now the North has overtly shelled the South - what aid will the US provide? Will the Obama administration provide its typical analytic, slow and measured response, long on press releases and statements but short on action, which has won it such contempt in previous crises like the BP oil spill? Or will it move swiftly and decisively to demonstrate, with more than press releases, what the US does for the allies who depend upon it?\nTo be sure, there are many good reasons for the US to hold off on instant military action on the Korean peninsula. Aside from North Korea's nuclear capability, there is the matter of China, and the uncertainty in South Korea's own government and population about where they want this to go. But there are many such justifications for delay in the Middle East as well \u2013 Iran, nuclear concerns, internal Israeli dissonance and so on. And where North Korea has few friends to stall American action, the Arab nations are a bloc that controls international institutions as well as the western oil-based economy. So a delay in action in the Koreas certainly signals that there would be delay in action in the Middle East.\nYou can bet Israel is watching.\nLast night I attended a celebration for a rabbi's retirement, after three decades of serving a shul. Aside from the fact that the evening was beautiful, it also reminded me of one of the things I love most, and miss most, about the rabbinate. A shul rabbi is forced to become Large.\nI watched the rabbi walk through the room and interact with people, shaking hands and exchanging hugs and receiving well-wishes, and I thought of all the things he knew about these people, perhaps even more than they knew about themselves. He remembered their celebrations and their shivah houses, their disputes and their reconciliations. He knew their children and their grandparents, and their uncle who liked to receive shishi when he visited. He knew their businesses, and he knew how close their businesses had once come to collapsing, until he had come up with a last minute loan arrangement. He knew what they had said at the funerals of their loved ones, and he knew what they had not said as well. He knew which shiurim they had attended, and with whom they had connected. He knew the committees they had served on, what they had done, what frustrations they had encountered and what successes they had achieved. He knew beside whom they sat in shul, and what they discussed before, during and after the derashah. He knew their favorite food and drink, and whether they lingered at the table after shabbos lunch or whether they went for a nap as soon as they could.\nAnd, as was noted last night, he had loved all of them. There is no way a shul rabbi survives without loving them all, whether in a shul of ten members or a shul of one thousand members. You can't fake it, and you wouldn't want to try; the needs of all of these lives are too consistent, too all-encompassing, too permeating of every aspect of life. To be the rabbi of a shul is to expand beyond yourself, to include a community in your soul.\nThe shul rabbinate makes you large in another, internal dimension, too: It forces you to grow up, or, again, you won't survive. To learn discretion, to squelch indignation, to develop patience, to abandon the idea of a comfort zone, to organize your thoughts and find ways to explain your mind to the world. To help people who will never acknowledge what you've done, and who may not even recognize it. To learn sound process and administration, and practice it. To swallow hard when something is against your grain, but you know it's necessary. To push yourself to the limits and beyond in diligent effort and in emotional strain.\nI assume the rabbinate is not the only experience that does this for a person. On some level, of course, marriage does it, as you welcome another into your soul and as you accept a new and less selfish way of doing business in order to build the relationship. And having children does it, too, in a bigger way than marriage. And public service of any kind does it. My point is not to claim that the rabbinate is the only way to experience these kinds of growth.\nBut there's no denying it: The Rabbi is large.\nHaving high-achieving ancestors is good for us; as the midrash (Tana dvei Eliyahu Rabbah 23) tells us, we are obligated to challenge ourselves and ask, \"\u05de\u05ea\u05d9 \u05d9\u05d2\u05d9\u05e2\u05d5 \u05de\u05e2\u05e9\u05d9 \u05dc\u05de\u05e2\u05e9\u05d9 \u05d0\u05d1\u05d5\u05ea\u05d9, When will my deeds match those of my ancestors?\" But there's a flip side, too: The more we beatify our ancestors, the more out of reach their deeds become - and the less hope we have for ourselves and our own accomplishments.\nWe praise Reb Zushe when he says his goal is not to be Avraham Avinu or Moshe Rabbeinu but only to be Zushe, but I wonder whether we should not also mourn the loss of ambition that comes with deciding we cannot be Moshe Rabbeinu.\nThat lofty thought comes to mind because of an experiment I embarked upon last Sunday: I went ice skating for the first time in some 25 years.\nI was never a real skater, despite being a serious hockey fan in a family of serious hockey fans. (Trivia: I was interviewed on MSG before the Rangers' Game 5 against the Canucks in the 1993-1994 Stanley Cup Finals.) I don't think I was on blades more than a dozen times, if that many. I stopped playing hockey when everyone else switched to roller and then ice; I couldn't hack it. I had a poor sense of balance, and a strong sense of self-preservation. I took up cross-country and weight-lifting instead.\nWhy, then, did I return to the ice this past week? For my children, actually, to help them avoid Reb Zushe's pitfall.\nMost young children rarely see their parents struggle with anything. They don't know the kid who swung his bat any number of times in the backyard in order to teach himself to hit. They don't know the kid who was so shy in elementary school, who walked into high school on his first day incredibly intimidated, and so on. They don't know the kid who wore glasses from kindergarten through sixth grade, and had braces twice. They don't know the kid who entered shiur in the 9th grade and heard the Rebbe assign the Rif and Nimukei Yosef for homework and had no idea what the Rebbe was talking about.\nMy kids don't really know of my more modern struggles, either. They were too young in my shul rabbinate days for me to share with them my emotional challenges over hospital visits, nursing home visits, and funerals. They don't understand how hard it was for me to write derashos week after week, and they don't get the stress and pressure of meeting my shiur schedule or running our kollel. I describe to them my anxiety about public speaking, but they don't really believe me.\nMany children think things come easy for their parents, and I want my kids to see me struggle, so that they will view my accomplishments, whatever they are, as within their grasp. And so that they will be inspired to struggle as well, with their own challenges.\nAnd so I went out on the ice, poor sense of balance and high sense of self-preservation and all, along with my children who have now been taking lessons for over a year. I was kind of hoping to fall, just to make the point and be done with it.\nSurprise! I actually enjoyed it, when I wasn't being terrified. It was fun. It was great having my kids coach me.\nAnd, in fact, I didn't fall (thank Gd!).\nI wish I could say it was an educated, ideological decision, substance over style, Shas over shtick. But, in truth, I didn't really see much of a choice. The men of our family didn't wear hats for davening or Shabbos, and I didn't know anything about why anyone would wear one. It was more ignorance than anything else.\nI received that Shas as well as another one for my bar mitzvah, and I'm glad I have both; I've scribbled notes in the volumes of both over the years, and carried them all over, to the point where bindings are frayed and gone, and some pages are covered with more of my own writing than the original. I'm really attached to it, far more attached than I've ever been to a piece of clothing, even a piece of clothing associated with a mitzvah. Score one for the Shas, in terms of utility.\nAnd score another for the Shas: I didn't end up putting on a hat for another five years, until my second year in yeshiva in Israel, when I started wearing one in honor of Shabbos. It would be another twelve or so years before I would start wearing a hat for davening in general (as a way to enhance the jacket and tie I was wearing all day). So from a 'years of use' perspective, my decision certainly made sense.\nBeyond years of use, though, and beyond my general attachment to that Pninim Shas, I'm glad of my decision because now that I am of age to make an informed decision, I can't see trading Torah for a hat. In truth, I'm not really sure why there was a question in the first place.\nI don't think that the generous person offering the gift really thought a hat was the equivalent of a Shas. Maybe he was doing it to tease my family, who were not hat-wearers. Or maybe he just wanted to see where I was, mentally. Or, perhaps, he was just joking. Don't know.\nGd-willing, I'll be delivering a shiur (first of two parts) on Wednesday evening, on the topic of Kollel Study. It's part of my series on Responsa that changed Jewish History (although I'll discuss in the shiur whether the responsa involved in this history actually changed anything).\n\u05d3\u05e2, \u05db\u05d9 \u05d6\u05d4 \u05db\u05d1\u05e8 \u05d0\u05de\u05e8: \u05d0\u05dc \u05ea\u05e2\u05e9\u05d4 \u05d4\u05ea\u05d5\u05e8\u05d4 \u05e7\u05e8\u05d3\u05d5\u05dd \u05dc\u05d7\u05e4\u05d5\u05e8 \u05d1\u05d4, \u05db\u05dc\u05d5\u05de\u05e8: \u05d0\u05dc \u05ea\u05d7\u05e9\u05d1\u05d4 \u05db\u05dc\u05d9 \u05dc\u05e4\u05e8\u05e0\u05e1\u05d4, \u05d5\u05d1\u05d0\u05e8 \u05d5\u05d0\u05de\u05e8 \u05e9\u05db\u05dc \u05de\u05d9 \u05e9\u05d9\u05d4\u05e0\u05d4 \u05d1\u05d6\u05d4 \u05d4\u05e2\u05d5\u05dc\u05dd \u05d1\u05db\u05d1\u05d5\u05d3 \u05ea\u05d5\u05e8\u05d4 \u05e9\u05d4\u05d5\u05d0 \u05db\u05d5\u05e8\u05ea \u05e0\u05e4\u05e9\u05d5 \u05de\u05d7\u05d9\u05d9 \u05d4\u05e2\u05d5\u05dc\u05dd \u05d4\u05d1\u05d0. \u05d5\u05d4\u05e2\u05dc\u05d9\u05de\u05d5 \u05d1\u05e0\u05d9 \u05d0\u05d3\u05dd \u05e2\u05d9\u05e0\u05d9\u05d4\u05dd \u05de\u05d6\u05d5 \u05d4\u05dc\u05e9\u05d5\u05df \u05d4\u05d2\u05dc\u05d5\u05d9\u05d4, \u05d5\u05d4\u05e9\u05dc\u05d9\u05db\u05d5\u05d4 \u05d0\u05d7\u05e8\u05d9 \u05d2\u05d5\u05d5\u05dd, \u05d5\u05e0\u05ea\u05dc\u05d5 \u05d1\u05e4\u05e9\u05d8\u05d9 \u05de\u05d0\u05de\u05e8\u05d9\u05dd \u05e9\u05dc\u05d0 \u05d4\u05d1\u05d9\u05e0\u05d5\u05dd - \u05d0\u05e0\u05d9 \u05d0\u05d1\u05d0\u05e8\u05dd - \u05d5\u05d4\u05d8\u05d9\u05dc\u05d5 \u05dc\u05d4\u05dd \u05d7\u05d5\u05e7\u05d9\u05dd \u05e2\u05dc \u05d4\u05d9\u05d7\u05d9\u05d3\u05d9\u05dd \u05d5\u05e2\u05dc \u05d4\u05e7\u05d4\u05d9\u05dc\u05d5\u05ea, \u05d5\u05e2\u05e9\u05d5 \u05d0\u05ea \u05d4\u05de\u05d9\u05e0\u05d5\u05d9\u05d9\u05dd \u05d4\u05ea\u05d5\u05e8\u05d9\u05d9\u05dd \u05dc\u05d7\u05d5\u05e7 \u05de\u05db\u05e1\u05d9\u05dd, \u05d5\u05d4\u05d1\u05d9\u05d0\u05d5 \u05d1\u05e0\u05d9 \u05d0\u05d3\u05dd \u05dc\u05e1\u05d1\u05d5\u05e8 \u05e9\u05d8\u05d5\u05ea \u05d2\u05de\u05d5\u05e8\u05d4, \u05e9\u05d6\u05d4 \u05e6\u05e8\u05d9\u05da \u05d5\u05de\u05d7\u05d5\u05d9\u05d9\u05d1, \u05dc\u05e2\u05d6\u05d5\u05e8 \u05dc\u05d7\u05db\u05de\u05d9\u05dd \u05d5\u05dc\u05ea\u05dc\u05de\u05d9\u05d3\u05d9\u05dd \u05d5\u05dc\u05d0\u05e0\u05e9\u05d9\u05dd \u05d4\u05e2\u05d5\u05e1\u05e7\u05d9\u05dd \u05d1\u05ea\u05d5\u05e8\u05d4 \u05d5\u05ea\u05d5\u05e8\u05ea\u05df \u05d0\u05d5\u05de\u05e0\u05d5\u05ea\u05df. \u05d5\u05db\u05dc \u05d6\u05d4 \u05d8\u05e2\u05d5\u05ea, \u05d0\u05d9\u05df \u05d1\u05ea\u05d5\u05e8\u05d4 \u05de\u05d4 \u05e9\u05d9\u05d0\u05de\u05ea \u05d0\u05d5\u05ea\u05d5, \u05d5\u05dc\u05d0 \u05e8\u05d2\u05dc \u05e9\u05d9\u05e9\u05e2\u05df \u05e2\u05dc\u05d9\u05d4 \u05d1\u05e9\u05d5\u05dd \u05e4\u05e0\u05d9\u05dd. \u05dc\u05e4\u05d9 \u05e9\u05d0\u05e0\u05d5 \u05d0\u05dd \u05e0\u05e2\u05d9\u05d9\u05df \u05d1\u05ea\u05d5\u05dc\u05d3\u05d5\u05ea \u05d4\u05d7\u05db\u05de\u05d9\u05dd, \u05d6\u05db\u05e8\u05dd \u05dc\u05d1\u05e8\u05db\u05d4, \u05dc\u05d0 \u05e0\u05de\u05e6\u05d0 \u05d0\u05e6\u05dc\u05dd \u05dc\u05d0 \u05d4\u05d8\u05dc\u05ea \u05d7\u05d5\u05d1\u05d5\u05ea \u05e2\u05dc \u05d1\u05e0\u05d9 \u05d0\u05d3\u05dd, \u05d5\u05dc\u05d0 \u05e7\u05d9\u05d1\u05d5\u05e5 \u05de\u05de\u05d5\u05df \u05dc\u05d9\u05e9\u05d9\u05d1\u05d5\u05ea \u05d4\u05de\u05e8\u05d5\u05de\u05de\u05d5\u05ea \u05d4\u05e0\u05db\u05d1\u05d3\u05d5\u05ea, \u05d5\u05dc\u05d0 \u05dc\u05e8\u05d0\u05e9\u05d9 \u05d2\u05dc\u05d5\u05d9\u05d5\u05ea \u05d5\u05dc\u05d0 \u05dc\u05d3\u05d9\u05d9\u05e0\u05d9\u05df \u05d5\u05dc\u05d0 \u05dc\u05de\u05e8\u05d1\u05e6\u05d9 \u05ea\u05d5\u05e8\u05d4 \u05d5\u05dc\u05d0 \u05dc\u05d0\u05d7\u05d3 \u05de\u05df \u05d4\u05de\u05de\u05d5\u05e0\u05d9\u05dd \u05d5\u05dc\u05d0 \u05dc\u05e9\u05d0\u05e8 \u05d4\u05d0\u05e0\u05e9\u05d9\u05dd, \u05d0\u05dc\u05d0 \u05e0\u05de\u05e6\u05d0 \u05e7\u05d4\u05d9\u05dc\u05d5\u05ea\u05d9\u05d4\u05dd \u05db\u05d5\u05dc\u05df \u05d9\u05e9 \u05d1\u05d4\u05dd \u05e2\u05e0\u05d9 \u05d1\u05ea\u05db\u05dc\u05d9\u05ea, 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\u05d0\u05d5 \u05d1\u05d9\u05d5\u05dd, \u05d5\u05d9\u05dc\u05da \u05dc\u05e2\u05e1\u05e7\u05d9\u05d5. \u05d5\u05e9\u05de\u05d5\u05d0\u05dc \u05dc\u05d0 \u05d4\u05d9\u05d4 \u05e0\u05db\u05e0\u05e1 \u05d1\u05d1\u05d9\u05ea \u05d0\u05d9\u05e9, \u05d5\u05dc\u05d0 \u05d0\u05d5\u05db\u05dc \u05de\u05de\u05d6\u05d5\u05e0\u05d5. \u05d5\u05d1\u05db\u05d2\u05d5\u05df \u05d6\u05d4 \u05d0\u05de\u05e8\u05d5 \u05d4\u05d7\u05db\u05de\u05d9\u05dd, \u05e9\u05ea\u05dc\u05de\u05d9\u05d3 \u05d7\u05db\u05de\u05d9\u05dd, \u05d0\u05dd \u05e8\u05e6\u05d4 \u05dc\u05d4\u05d3\u05de\u05d5\u05ea \u05dc\u05d6\u05d4, \u05e2\u05d3 \u05e9\u05dc\u05d0 \u05d9\u05db\u05e0\u05e1 \u05d0\u05e6\u05dc \u05d0\u05d9\u05e9 - \u05d9\u05e9 \u05dc\u05d5 \u05dc\u05e2\u05e9\u05d5\u05ea \u05db\u05df, \u05d5\u05db\u05de\u05d5 \u05db\u05df \u05d0\u05dd \u05e8\u05e6\u05d4 \u05dc\u05d4\u05ea\u05d0\u05e8\u05d7 \u05d0\u05e6\u05dc \u05d0\u05d9\u05e9 \u05d1\u05d4\u05db\u05e8\u05d7 \u05e0\u05e1\u05d9\u05e2\u05d4 \u05d0\u05d5 \u05db\u05d9\u05d5\u05e6\u05d0 \u05d1\u05d4 - \u05d9\u05e9 \u05dc\u05d5 \u05dc\u05e2\u05e9\u05d5\u05ea \u05db\u05df. \u05dc\u05e4\u05d9 \u05e9\u05db\u05d1\u05e8 \u05d4\u05d6\u05d4\u05d9\u05e8\u05d5 \u05de\u05df \u05d4\u05d0\u05db\u05d9\u05dc\u05d4 \u05dc\u05dc\u05d0 \u05d4\u05db\u05e8\u05d7, \u05d5\u05d0\u05de\u05e8\u05d5: \"\u05db\u05dc \u05ea\u05dc\u05de\u05d9\u05d3 \u05d7\u05db\u05de\u05d9\u05dd \u05d4\u05de\u05e8\u05d1\u05d4 \u05e1\u05e2\u05d5\u05d3\u05d5\u05ea\u05d9\u05d5 \u05d1\u05db\u05dc \u05de\u05e7\u05d5\u05dd\" \u05d5\u05db\u05d5', \u05d5\u05d0\u05de\u05e8\u05d5: \"\u05db\u05dc \u05e1\u05e2\u05d5\u05d3\u05d4 \u05e9\u05d0\u05d9\u05e0\u05d4 \u05e9\u05dc\u05de\u05e6\u05d5\u05d4 \u05d0\u05e1\u05d5\u05e8 \u05dc\u05ea\u05dc\u05de\u05d9\u05d3 \u05d7\u05db\u05de\u05d9\u05dd \u05dc\u05d4\u05e0\u05d5\u05ea \u05de\u05de\u05e0\u05d4\".\n\u05d5\u05dc\u05de\u05d4 \u05d0\u05d0\u05e8\u05d9\u05da \u05d1\u05d6\u05d4 \u05d4\u05e2\u05e0\u05d9\u05df, \u05d0\u05e0\u05d9 \u05d0\u05d6\u05db\u05d5\u05e8 \u05d4\u05de\u05e2\u05e9\u05d4 \u05d0\u05e9\u05e8 \u05d4\u05ea\u05d1\u05d0\u05e8 \u05d1\u05ea\u05dc\u05de\u05d5\u05d3, \u05d5\u05d4\u05de\u05ea\u05e2\u05e7\u05e9 \u05d9\u05e2\u05e9\u05d4 \u05db\u05e8\u05e6\u05d5\u05e0\u05d5: \u05d4\u05d9\u05d4 \u05d0\u05d9\u05e9 \u05e9\u05d4\u05d9\u05d4 \u05dc\u05d5 \u05db\u05e8\u05dd, \u05d5\u05d4\u05d9\u05d5 \u05d1\u05d0\u05d9\u05dd \u05d1\u05d5 \u05d2\u05e0\u05d1\u05d9\u05dd, \u05d5\u05db\u05dc \u05d0\u05e9\u05e8 \u05d4\u05d9\u05d4 \u05e4\u05d5\u05e7\u05d3\u05d5 \u05d1\u05db\u05dc \u05d9\u05d5\u05dd, \u05d4\u05d9\u05d4 \u05de\u05d5\u05e6\u05d0 \u05e2\u05e0\u05d1\u05d9\u05d5 \u05de\u05ea\u05de\u05e2\u05d8\u05d9\u05dd \u05d5\u05d7\u05e1\u05e8\u05d9\u05dd, \u05d5\u05dc\u05d0 \u05d4\u05d9\u05d4 \u05dc\u05d5 \u05e1\u05e4\u05e7 \u05e9\u05d0\u05d9\u05e9 \u05de\u05df \u05d4\u05d2\u05e0\u05d1\u05d9\u05dd \u05e9\u05de\u05d4\u05d5 \u05de\u05d8\u05e8\u05d4. \u05d5\u05d4\u05d9\u05d4 \u05de\u05e6\u05d8\u05e2\u05e8 \u05de\u05d6\u05d4 \u05ea\u05de\u05d9\u05d3 \u05db\u05dc \u05d9\u05de\u05d9 \u05d4\u05e2\u05e0\u05d1\u05d9\u05dd, \u05e2\u05d3 \u05e9\u05d1\u05e6\u05e8 \u05de\u05de\u05e0\u05d5 \u05de\u05d4 \u05e9\u05d1\u05e6\u05e8, \u05d5\u05e9\u05d8\u05d7\u05d5 \u05e2\u05d3 \u05d0\u05e9\u05e8 \u05d9\u05d1\u05e9, \u05d5\u05d0\u05e1\u05e3 \u05d4\u05e6\u05de\u05d5\u05e7\u05d9\u05dd. \u05d5\u05de\u05d3\u05e8\u05da \u05d1\u05e0\u05d9 \u05d0\u05d3\u05dd, \u05e9\u05db\u05d0\u05e9\u05e8 \u05d9\u05d0\u05e1\u05e4\u05d5 \u05d4\u05e4\u05d9\u05e8\u05d5\u05ea \u05d4\u05de\u05d9\u05d5\u05d1\u05e9\u05d9\u05dd, \u05d9\u05e4\u05dc\u05d5 \u05d2\u05e8\u05d2\u05e8\u05d9\u05dd \u05de\u05df \u05d4\u05ea\u05d0\u05e0\u05d9\u05dd \u05d5\u05d4\u05e6\u05de\u05d5\u05e7\u05d9\u05dd, \u05d5\u05de\u05d5\u05ea\u05e8 \u05dc\u05d0\u05d5\u05db\u05dc\u05dd, \u05de\u05e4\u05e0\u05d9 \u05e9\u05d4\u05dd \u05d4\u05e4\u05e7\u05e8, \u05d5\u05db\u05d1\u05e8 \u05d4\u05e0\u05d9\u05d7\u05d5\u05dd \u05d1\u05e2\u05dc\u05d9\u05d4\u05dd \u05dc\u05de\u05d9\u05e2\u05d5\u05d8\u05dd. \u05d5\u05d1\u05d0 \u05e8' \u05d8\u05e8\u05e4\u05d5\u05df \u05d1\u05de\u05e7\u05e8\u05d4 \u05d9\u05d5\u05dd \u05d0\u05d7\u05d3 \u05dc\u05d6\u05d4 \u05d4\u05db\u05e8\u05dd, \u05d5\u05d9\u05e9\u05d1, \u05d5\u05d4\u05d9\u05d4 \u05de\u05dc\u05e7\u05d8 \u05de\u05df \u05d4\u05e6\u05de\u05d5\u05e7\u05d9\u05dd \u05d4\u05e0\u05d5\u05e4\u05dc\u05d9\u05dd \u05d5\u05d0\u05d5\u05db\u05dc\u05dd. \u05d5\u05d1\u05d0 \u05d1\u05e2\u05dc \u05d4\u05db\u05e8\u05dd, \u05d5\u05d7\u05e9\u05d1 \u05e9\u05d6\u05d4 \u05d4\u05d5\u05d0 \u05d0\u05e9\u05e8 \u05d4\u05d9\u05d4 \u05d2\u05d5\u05e0\u05d1 \u05de\u05de\u05e0\u05d5 \u05db\u05dc \u05d4\u05e9\u05e0\u05d4, \u05d5\u05dc\u05d0 \u05d4\u05d9\u05d4 \u05de\u05db\u05d9\u05e8\u05d5, \u05d0\u05d1\u05dc \u05e9\u05de\u05e2 \u05e9\u05de\u05d5. \u05d5\u05de\u05d9\u05d4\u05e8 \u05d0\u05dc\u05d9\u05d5 \u05d5\u05ea\u05e4\u05e9\u05d5, \u05d5\u05d4\u05ea\u05d2\u05d1\u05e8 \u05e2\u05dc\u05d9\u05d5, \u05d5\u05e9\u05de\u05d4\u05d5 \u05d1\u05e9\u05e7, \u05d5\u05e8\u05e5 \u05e2\u05de\u05d5 \u05e2\u05dc \u05d2\u05d1\u05d5 \u05dc\u05d4\u05e9\u05dc\u05d9\u05db\u05d5 \u05d1\u05e0\u05d4\u05e8. \u05d5\u05db\u05d0\u05e9\u05e8 \u05e8\u05d0\u05d4 \u05e8' \u05d8\u05e8\u05e4\u05d5\u05df \u05e2\u05e6\u05de\u05d5 \u05d0\u05d5\u05d1\u05d3, \u05e6\u05e2\u05e7: \"\u05d0\u05d5\u05d9 \u05dc\u05d5 \u05dc\u05d8\u05e8\u05e4\u05d5\u05df \u05e9\u05d6\u05d4 \u05d4\u05d5\u05e8\u05d2\u05d5\". \u05d5\u05db\u05d0\u05e9\u05e8 \u05e9\u05de\u05e2 \u05d6\u05d4 \u05d1\u05e2\u05dc \u05d4\u05db\u05e8\u05dd, \u05d4\u05e0\u05d9\u05d7\u05d5 \u05d5\u05d1\u05e8\u05d7, \u05dc\u05d9\u05d5\u05d3\u05e2\u05d5 \u05db\u05d9 \u05d7\u05d8\u05d0 \u05d7\u05d8\u05d0 \u05d2\u05d3\u05d5\u05dc. \u05d5\u05d4\u05d9\u05d4 \u05e8' \u05d8\u05e8\u05e4\u05d5\u05df \u05d1\u05e9\u05d0\u05e8\u05d9\u05ea \u05d9\u05de\u05d9\u05d5 \u05de\u05d0\u05d5\u05ea\u05d5 \u05d4\u05d9\u05d5\u05dd \u05d3\u05d5\u05d0\u05d1 \u05d5\u05de\u05d9\u05dc\u05d9\u05dc \u05e2\u05dc \u05de\u05d4 \u05e9\u05d0\u05e8\u05e2 \u05dc\u05d5, \u05d1\u05d4\u05e6\u05d9\u05dc\u05d5 \u05e0\u05e4\u05e9\u05d5 \u05d1\u05db\u05d1\u05d5\u05d3 \u05d4\u05ea\u05d5\u05e8\u05d4, \u05d5\u05d4\u05d5\u05d0 \u05d4\u05d9\u05d4 \u05d1\u05e2\u05dc \u05de\u05de\u05d5\u05df, \u05d5\u05d4\u05d9\u05d4 \u05d9\u05db\u05d5\u05dc \u05dc\u05d5\u05de\u05e8 \u05dc\u05d5: \u05d4\u05e0\u05d9\u05d7\u05e0\u05d9, \u05d5\u05d0\u05e0\u05d9 \u05d0\u05e9\u05dc\u05dd \u05dc\u05da \u05db\u05da \u05d5\u05db\u05da \u05d3\u05d9\u05e0\u05e8\u05d9\u05dd, \u05d5\u05d9\u05e9\u05dc\u05dd \u05dc\u05d5, \u05d5\u05dc\u05d0 \u05d9\u05d5\u05d3\u05d9\u05e2\u05d4\u05d5 \u05e9\u05d4\u05d5\u05d0 \u05d8\u05e8\u05e4\u05d5\u05df, \u05d5\u05d9\u05e6\u05d9\u05dc \u05e0\u05e4\u05e9\u05d5 \u05d1\u05de\u05de\u05d5\u05e0\u05d5, \u05dc\u05d0 \u05d1\u05ea\u05d5\u05e8\u05d4. \u05d0\u05de\u05e8\u05d5: \"\u05db\u05dc \u05d9\u05de\u05d9\u05d5 \u05e9\u05dc\u05d0\u05d5\u05ea\u05d5 \u05e6\u05d3\u05d9\u05e7 \u05d4\u05d9\u05d4 \u05de\u05e6\u05d8\u05e2\u05e8 \u05e2\u05dc \u05d3\u05d1\u05e8 \u05d6\u05d4 \u05d5\u05d0\u05d5\u05de\u05e8: \u05d0\u05d5\u05d9 \u05dc\u05d9 \u05e9\u05e0\u05e9\u05ea\u05de\u05e9\u05ea\u05d9 \u05d1\u05db\u05ea\u05e8\u05d4 \u05e9\u05dc\u05ea\u05d5\u05e8\u05d4, \u05e9\u05db\u05dc \u05d4\u05de\u05e9\u05ea\u05de\u05e9 \u05d1\u05db\u05ea\u05e8\u05d4 \u05e9\u05dc\u05ea\u05d5\u05e8\u05d4 \u05e0\u05e2\u05e7\u05e8 \u05de\u05df \u05d4\u05e2\u05d5\u05dc\u05dd\". \u05d5\u05d0\u05de\u05e8\u05d5 \u05d1\u05d6\u05d4: \"\u05de\u05e9\u05d5\u05dd \u05d3\u05e8' \u05d8\u05e8\u05e4\u05d5\u05df \u05e2\u05e9\u05d9\u05e8 \u05d2\u05d3\u05d5\u05dc \u05d4\u05d5\u05d4, \u05d5\u05d4\u05d5\u05d4 \u05dc\u05d9\u05d4 \u05dc\u05e4\u05d9\u05d5\u05e1\u05d9\u05d4 \u05d1\u05d3\u05de\u05d9\u05dd\".\n\u05d5\u05d0\u05de\u05e0\u05dd \u05d4\u05d3\u05d1\u05e8 \u05d0\u05e9\u05e8 \u05d4\u05ea\u05d9\u05e8\u05ea\u05d5 \u05d4\u05ea\u05d5\u05e8\u05d4 \u05dc\u05ea\u05dc\u05de\u05d9\u05d3\u05d9 \u05d7\u05db\u05de\u05d9\u05dd \u05d4\u05d5\u05d0, \u05e9\u05d9\u05ea\u05e0\u05d5 \u05de\u05de\u05d5\u05df \u05dc\u05d0\u05d3\u05dd, \u05d9\u05e2\u05e9\u05d4 \u05dc\u05d4\u05dd \u05e1\u05d7\u05d5\u05e8\u05d4 \u05d1\u05d5 \u05d1\u05d1\u05d7\u05d9\u05e8\u05ea\u05d5, \u05d0\u05dd \u05d9\u05e8\u05e6\u05d4 - \u05d5\u05e2\u05d5\u05e9\u05d4 \u05d6\u05d4 \u05d9\u05e9 \u05dc\u05d5 \u05e9\u05db\u05e8 \u05e2\u05dc \u05db\u05da, \u05d5\u05d6\u05d4 \u05d4\u05d5\u05d0 \u05de\u05d8\u05d9\u05dc \u05de\u05dc\u05d0\u05d9 \u05dc\u05db\u05d9\u05e1 \u05ea\u05dc\u05de\u05d9\u05d3\u05d9 \u05d7\u05db\u05de\u05d9\u05dd - \u05d5\u05e9\u05ea\u05de\u05db\u05e8 \u05e1\u05d7\u05d5\u05e8\u05ea\u05dd \u05ea\u05d7\u05d9\u05dc\u05d4 \u05dc\u05de\u05d4 \u05e9\u05d9\u05d9\u05de\u05db\u05e8, \u05d5\u05ea\u05ea\u05e4\u05e1 \u05dc\u05d4\u05dd \u05e8\u05d0\u05e9\u05d9\u05ea \u05d4\u05e9\u05d5\u05e7 \u05d3\u05d5\u05e7\u05d0. \u05d0\u05dc\u05d5 \u05d7\u05d5\u05e7\u05d9\u05dd \u05e9\u05e7\u05d1\u05e2 \u05d4' \u05dc\u05d4\u05dd, \u05db\u05de\u05d5 \u05e9\u05e7\u05d1\u05e2 \u05d4\u05de\u05ea\u05e0\u05d5\u05ea \u05dc\u05db\u05d4\u05df \u05d5\u05d4\u05de\u05e2\u05e9\u05e8\u05d5\u05ea \u05dc\u05dc\u05d5\u05d9, \u05dc\u05e4\u05d9 \u05de\u05d4 \u05e9\u05d1\u05d0\u05d4 \u05d1\u05d5 \u05d4\u05e7\u05d1\u05dc\u05d4. \u05db\u05d9 \u05e9\u05ea\u05d9 \u05d4\u05e4\u05e2\u05d5\u05dc\u05d5\u05ea \u05d4\u05d0\u05dc\u05d4 \u05d9\u05e9 \u05e9\u05d9\u05e2\u05e9\u05d5 \u05d0\u05d5\u05ea\u05df \u05d4\u05e1\u05d5\u05d7\u05e8\u05d9\u05dd \u05e7\u05e6\u05ea\u05dd \u05e2\u05dd \u05e7\u05e6\u05ea\u05dd \u05e2\u05dc \u05d3\u05e8\u05da \u05d4\u05db\u05d1\u05d5\u05d3, \u05d5\u05d0\u05e4\u05d9\u05dc\u05d5 \u05dc\u05d0 \u05d4\u05d9\u05ea\u05d4 \u05e9\u05dd \u05d7\u05db\u05de\u05d4, \u05d5\u05dc\u05db\u05dc \u05d4\u05e4\u05d7\u05d5\u05ea \u05d9\u05d4\u05d9\u05d4 \u05ea\u05dc\u05de\u05d9\u05d3 \u05d7\u05db\u05de\u05d9\u05dd \u05db\u05de\u05d5 \u05e2\u05dd \u05d0\u05e8\u05e5 \u05de\u05db\u05d5\u05d1\u05d3. \u05d5\u05db\u05df \u05e4\u05d8\u05e8\u05d4 \u05d4\u05ea\u05d5\u05e8\u05d4 \u05d0\u05ea \u05db\u05dc \u05ea\u05dc\u05de\u05d9\u05d3\u05d9 \u05d4\u05d7\u05db\u05de\u05d9\u05dd \u05de\u05d7\u05d5\u05d1\u05d5\u05ea \u05d4\u05e9\u05dc\u05d8\u05d5\u05df \u05db\u05d5\u05dc\u05df, \u05de\u05df \u05d4\u05de\u05d9\u05e1\u05d9\u05dd, \u05d5\u05d4\u05d0\u05db\u05e1\u05e0\u05d9\u05d5\u05ea, \u05d5\u05de\u05d9\u05e1\u05d9 \u05d4\u05e0\u05e4\u05e9, \u05d5\u05d4\u05dd \u05d0\u05e9\u05e8 \u05d9\u05e7\u05e8\u05d0\u05d5 \u05db\u05e1\u05e3 \u05d2\u05d5\u05dc\u05d2\u05dc\u05ea\u05d0, \u05d9\u05e4\u05e8\u05e2\u05d5\u05dd \u05d1\u05e2\u05d1\u05d5\u05e8\u05dd \u05d4\u05e7\u05d4\u05dc, \u05d5\u05d1\u05e0\u05d9\u05df \u05d4\u05d7\u05d5\u05de\u05d5\u05ea \u05d5\u05db\u05d9\u05d5\u05e6\u05d0 \u05d1\u05d4\u05df. \u05d5\u05d0\u05e4\u05d9\u05dc\u05d5 \u05d4\u05d9\u05d4 \u05ea\u05dc\u05de\u05d9\u05d3 \u05d4\u05d7\u05db\u05de\u05d9\u05dd \u05d1\u05e2\u05dc \u05de\u05de\u05d5\u05df \u05e8\u05d1 \u05dc\u05d0 \u05d9\u05d7\u05d5\u05d9\u05d9\u05d1 \u05d1\u05d3\u05d1\u05e8 \u05de\u05d6\u05d4. \u05d5\u05db\u05d1\u05e8 \u05d4\u05d5\u05e8\u05d4 \u05d1\u05d6\u05d4 \u05e8\u05d1\u05e0\u05d0 \u05d9\u05d5\u05e1\u05e3 \u05d4\u05dc\u05d5\u05d9 \u05d6\u05e6\"\u05dc \u05dc\u05d0\u05d9\u05e9 \u05d1\u05d0\u05e0\u05d3\u05dc\u05d5\u05e1, \u05e9\u05d4\u05d9\u05d5 \u05dc\u05d5 \u05d2\u05e0\u05d5\u05ea \u05d5\u05db\u05e8\u05de\u05d9\u05dd \u05e9\u05d4\u05d9\u05d4 \u05de\u05d7\u05d5\u05d9\u05d9\u05d1 \u05e2\u05d1\u05d5\u05e8\u05dd \u05d0\u05dc\u05e3 \u05d3\u05d9\u05e0\u05e8\u05d9\u05dd, \u05d5\u05d4\u05d5\u05e8\u05d4 \u05dc\u05e4\u05d5\u05d8\u05e8\u05d5 \u05de\u05d4\u05de\u05e1, \u05dc\u05d4\u05d9\u05d5\u05ea\u05d5 \u05ea\u05dc\u05de\u05d9\u05d3 \u05d7\u05db\u05de\u05d9\u05dd, \u05d0\u05e3 \u05e2\u05dc \u05e4\u05d9 \u05e9\u05d4\u05d9\u05d4 \u05de\u05e9\u05dc\u05dd \u05d4\u05de\u05e1 \u05d4\u05d4\u05d5\u05d0 \u05d0\u05e4\u05d9\u05dc\u05d5 \u05d4\u05e2\u05e0\u05d9 \u05e9\u05d1\u05d9\u05d4\u05d5\u05d3\u05d9\u05dd. \u05d5\u05d6\u05d4 \u05d3\u05d9\u05df \u05ea\u05d5\u05e8\u05d4, \u05db\u05de\u05d5 \u05e9\u05e4\u05d8\u05e8\u05d4 \u05d4\u05ea\u05d5\u05e8\u05d4 \u05d4\u05db\u05d4\u05e0\u05d9\u05dd \u05de\u05de\u05d7\u05e6\u05d9\u05ea \u05d4\u05e9\u05e7\u05dc, \u05db\u05de\u05d5 \u05e9\u05d1\u05d0\u05e8\u05e0\u05d5 \u05d1\u05de\u05e7\u05d5\u05de\u05d5, \u05d5\u05de\u05d4 \u05e9\u05d3\u05d5\u05de\u05d4 \u05dc\u05d6\u05d4.\nKnow that the mishnah already stated, \"Do not make Torah into a blade with which to dig,\" meaning, \"Do not consider Torah a tool for support.\" The text explained that anyone who benefits from the honor of Torah in this world is cutting off his soul from life in the next world.\nPeople have hidden their eyes from this clear language, casting it after their bodies, relying on interpretations of passages they do not understand, but I will explain. They have placed laws upon individuals and communities, and they have turned the Torah's positions into the law of tax collectors, and they have brought people to believe total foolishness, that it is necessary and mandatory for them to aid sages and students and people who are involved in Torah and whose trade is Torah.\nAll of this is in error, and there is nothing in Torah to validate it, and there is no leg upon which they could rely in any way. Were we to examine the chronicles of the sages of blessed memory, we would not find that they ever placed obligations upon people, or gathered money for the exalted and honored yeshivot, or for the exilarchs or judges or spreaders of Torah or appointed leaders or others.\nIn all of their communities there were people at the height of poverty and people at the height of wealth; Gd forbid to say that those wealthy people were not generous and were not philanthropic! Rather, had this pauper extended his hand they would have filled his house with gold and gems, but he did not do so. Rather, he survived with the work from which he was supported, whether with ease or under strain, and he disdained that which came by the hand of man, since the Torah prevented it.\nThat which the Torah permitted for Torah scholars is that they may give a person money and have him engage in business of his choosing on their behalf. One who does this for them receives reward, and this is the meaning of, \"putting merchandise into the wallet of Torah scholars.\" Their merchandise is sold to whomever, and they receive the first of the market, specifically. These are the portions Gd established for the sages, just as He established gifts for the kohen and tithes for the levi, as is our tradition. Merchants already engage in these activities for each other, as a show of respect, even if there is no greater wisdom involved; let Torah scholars be no less than a respected am ha'aretz.\nAnd the Torah also exempted Torah scholars from the obligations of the ruler, from taxes and billeting and head-taxes, which the community pays on their behalf, and the construction of walls and the like. Even a wealthy Torah scholar would not be obligated in this. Ravna Yosef haLevi aleady ruled thus for a man in Andalusia who owned gardens and vineyards which required [tax of] one thousand dinar, and he exempted the man from the tax because he was a Torah scholar, although even the poorest Jew paid that tax. This is the law of the Torah, just as it exempted kohanim from the half-shekel.\n\u05db\u05dc \u05d4\u05de\u05e9\u05d9\u05dd \u05e2\u05dc \u05dc\u05d1\u05d5 \u05e9\u05d9\u05e2\u05e1\u05d5\u05e7 \u05d1\u05ea\u05d5\u05e8\u05d4 \u05d5\u05dc\u05d0 \u05d9\u05e2\u05e9\u05d4 \u05de\u05dc\u05d0\u05db\u05d4 \u05d5\u05d9\u05ea\u05e4\u05e8\u05e0\u05e1 \u05de\u05df \u05d4\u05e6\u05d3\u05e7\u05d4 \u05d4\u05e8\u05d9 \u05d6\u05d4 \u05d7\u05dc\u05dc \u05d0\u05ea \u05d4\u05e9\u05dd \u05d5\u05d1\u05d6\u05d4 \u05d0\u05ea \u05d4\u05ea\u05d5\u05e8\u05d4 \u05d5\u05db\u05d1\u05d4 \u05de\u05d0\u05d5\u05e8 \u05d4\u05d3\u05ea \u05d5\u05d2\u05e8\u05dd \u05e8\u05e2\u05d4 \u05dc\u05e2\u05e6\u05de\u05d5 \u05d5\u05e0\u05d8\u05dc \u05d7\u05d9\u05d9\u05d5 \u05de\u05df \u05d4\u05e2\u05d5\u05dc\u05dd \u05d4\u05d1\u05d0, \u05dc\u05e4\u05d9 \u05e9\u05d0\u05e1\u05d5\u05e8 \u05dc\u05d9\u05d4\u05e0\u05d5\u05ea \u05de\u05d3\u05d1\u05e8\u05d9 \u05ea\u05d5\u05e8\u05d4 \u05d1\u05e2\u05d5\u05dc\u05dd \u05d4\u05d6\u05d4, \u05d0\u05de\u05e8\u05d5 \u05d7\u05db\u05de\u05d9\u05dd \u05db\u05dc \u05d4\u05e0\u05d4\u05e0\u05d4 \u05de\u05d3\u05d1\u05e8\u05d9 \u05ea\u05d5\u05e8\u05d4 \u05e0\u05d8\u05dc \u05d7\u05d9\u05d9\u05d5 \u05de\u05df \u05d4\u05e2\u05d5\u05dc\u05dd, \u05d5\u05e2\u05d5\u05d3 \u05e6\u05d5\u05d5 \u05d5\u05d0\u05de\u05e8\u05d5 \u05d0\u05dc \u05ea\u05e2\u05e9\u05dd \u05e2\u05d8\u05e8\u05d4 \u05dc\u05d4\u05ea\u05d2\u05d3\u05dc \u05d1\u05d4\u05df \u05d5\u05dc\u05d0 \u05e7\u05e8\u05d3\u05d5\u05dd \u05dc\u05d7\u05e4\u05d5\u05e8 \u05d1\u05d4\u05df, \u05d5\u05e2\u05d5\u05d3 \u05e6\u05d5\u05d5 \u05d5\u05d0\u05de\u05e8\u05d5 \u05d0\u05d4\u05d5\u05d1 \u05d0\u05ea \u05d4\u05de\u05dc\u05d0\u05db\u05d4 \u05d5\u05e9\u05e0\u05d0 \u05d0\u05ea \u05d4\u05e8\u05d1\u05e0\u05d5\u05ea \u05d5\u05db\u05dc \u05ea\u05d5\u05e8\u05d4 \u05e9\u05d0\u05d9\u05df \u05e2\u05de\u05d4 \u05de\u05dc\u05d0\u05db\u05d4 \u05e1\u05d5\u05e4\u05d4 \u05d1\u05d8\u05d9\u05dc\u05d4 \u05d5\u05d2\u05d5\u05e8\u05e8\u05ea \u05e2\u05d5\u05df, \u05d5\u05e1\u05d5\u05e3 \u05d0\u05d3\u05dd \u05d6\u05d4 \u05e9\u05d9\u05d4\u05d0 \u05de\u05dc\u05e1\u05d8\u05dd \u05d0\u05ea \u05d4\u05d1\u05e8\u05d9\u05d5\u05ea.\n\u05de\u05e2\u05dc\u05d4 \u05d2\u05d3\u05d5\u05dc\u05d4 \u05d4\u05d9\u05d0 \u05dc\u05de\u05d9 \u05e9\u05d4\u05d5\u05d0 \u05de\u05ea\u05e4\u05e8\u05e0\u05e1 \u05de\u05de\u05e2\u05e9\u05d4 \u05d9\u05d3\u05d9\u05d5, \u05d5\u05de\u05d3\u05ea \u05d7\u05e1\u05d9\u05d3\u05d9\u05dd \u05d4\u05e8\u05d0\u05e9\u05d5\u05e0\u05d9\u05dd \u05d4\u05d9\u05d0, \u05d5\u05d1\u05d6\u05d4 \u05d6\u05d5\u05db\u05d4 \u05dc\u05db\u05dc \u05db\u05d1\u05d5\u05d3 \u05d5\u05d8\u05d5\u05d1\u05d4 \u05e9\u05d1\u05e2\u05d5\u05dc\u05dd \u05d4\u05d6\u05d4 \u05d5\u05dc\u05e2\u05d5\u05dc\u05dd \u05d4\u05d1\u05d0 \u05e9\u05e0\u05d0\u05de\u05e8 \u05d9\u05d2\u05d9\u05e2 \u05db\u05e4\u05d9\u05da \u05db\u05d9 \u05ea\u05d0\u05db\u05dc \u05d0\u05e9\u05e8\u05d9\u05da \u05d5\u05d8\u05d5\u05d1 \u05dc\u05da \u05d0\u05e9\u05e8\u05d9\u05da \u05d1\u05e2\u05d5\u05dc\u05dd \u05d4\u05d6\u05d4 \u05d5\u05d8\u05d5\u05d1 \u05dc\u05da \u05dc\u05e2\u05d5\u05dc\u05dd \u05d4\u05d1\u05d0 \u05e9\u05db\u05d5\u05dc\u05d5 \u05d8\u05d5\u05d1.\nThere is a great level in supporting one's self with the work of one's hands, and this was the trait of the early pious ones, and through this one merits all honor and goodness in this world and the next, as it is written, \"When you eat the labor of your hands, you are fortunate and it is good for you.\" You are fortunate in this world, and it is good for you in the next world, which is entirely good.\nBecause I have seen many people hesitate regarding the practice of all Jewish communities, in all generations, of paying their sages, and in this they depend upon what the Rambam wrote in his commentary to Avot, therefore I will examine this issue as Heaven guides me, from gemara and other sources, to see whether this is permissible, mitzvah or obligatory, or whether it involves some element of sin. I am confident in the strength of the early authorities who customarily permitted this, so that I will not stumble in a matter of law, and all of the relevant issues will be cleansed and clarified.\nI say that the Jewish people are obligated to support, with honor, their sages and judges whose livelihoods are their Torah, so that they will not need to cease their heavenly work for the sake of melachah, and so they will not be disgraced before amei ha'aretz because of their poverty, and so they will be treated with honor.\nThe root of this comes from Torah, as explained by the sages. They explained in Yoma 18a and Horiyyot 9a: \"It is written, 'The kohen who is greater than his brethren' \u2013 He should be greater than his brethren in beauty, strength, knowledge and wealth. Others said: If he lacks this, then his brother kohanim elevate him, as it is written, 'The kohen who is greater than his brethren' \u2013 elevate him from that which belongs to his brethren.\"\u2026 They also deduced from this passage that a Torah scholar may take money to enrich himself, and there is no disgrace in it, in Chullin 134b: \"A sack of coins was brought to the beit midrash [for the students' use]; R' Ami took it first. How could he do this? Did we not learn, \"'And he shall give' \u2013 And not that one should take [the gifts for the kohanim] for himself!\"\" And they answer, \"An important person is different, as we learned, 'The kohen who is greater'\u2026 The sages equated the law of an important person, whom Israel are obligated to enrich, with the law of a kohen gadol.\nThe gemara is explicit, then, that the community must establish their scholars at the head in order to support them in an honorable way. This obligation rests upon their leaders and directors to take care of matters for them with the community. None can disagree with this at all.\nThey had a box designated for collecting funds to distribute to students, to support them in an honorable manner, as is found in Gittin 60b, \"The shipura was in the house of Rav Yehudah, then the house of Rabbah, then the house of Rav Yosef, then the house of Abbaye, then the house of Rava.\" I saw a responsum of Rav Sherira Gaon stating that the shipura was a box, like the 13 shofarot used to collect half-shekalim, and this box was used to collect funds to distribute to the students. This seems more correct than Rashi's explanation that it was the shofar used to blow [warnings] before Shabbat, and Rashi also cited the view of the Gaon in his commentary, and upon this we should depend for the words of the Gaonim are received tradition. This practice dates back to the Talmud.\nWe have found that in the days of the Gaonim they had a collection box designated for the yeshivot, and they sent it to all communities to provide a fixed amount for the heads of the yeshivot and their students. We found recorded in the siddur of Rav Amram Gaon, at the start, \"Ravna Yaakov ben Ravna Yitzchak sent us twenty gold coins which you sent to the yeshiva; five are ours, and fifteen went to the yeshiva's box. We have instructed and blessed you with enduring blessings, for you and your children and your children's children.\" It appears that Ravna Yaakov must have written to him to give five to the head of the yeshiva and fifteen to the general box, for [had he not specified this,] Rav Amram could have taken ten for himself, as R' Akiva taught (Avodah Zarah 10b), \"'And it shall be for Aharon and his sons' \u2013 half for Aharon and half for his sons.\"\u2026 Alternatively, perhaps Rav Amram acted with extra piety. In any case, we see that the custom in Gaonic times was to designate a box for the yeshiva.\nAlso, in the generations before us, until our own day, we saw this practice continue, and the practice of Israel is Torah, certainly when it is explicit in Talmud and midrash, and there is no doubt in the matter.\nIt appears that a sage may even demand his portion verbally, as is seen in the case of R' Ami cited earlier.\nIt's hard for me to believe this is my sixth Haveil Havalim, but here it is! The Stephen Harper Tribute Edition, Haveil Havalim #291.\nFounded by Soccer Dad, Haveil Havalim is a carnival of Jewish blogs -- a weekly collection of Jewish and Israeli blog highlights, tidbits and points of interest collected from blogs all around the world. It's hosted by different bloggers each week and coordinated by the formidable Jack.\nNow: Why is this a tribute to Stephen Harper? And who is Stephen Harper, anyway?\nIn a country with a large and vocal anti-Israel lobby, Harper has nothing to gain from this stance. And as he said in the speech above, he has the bruises to show for his history of support of Israel in the UN; as Israel Matzav wrote last month, Canada lost its shot at a UN Security Council seat because of its support for Israel. So I salute this man for his courage.\nThank you, Prime Minister Harper!\nThree years into aliyah, rutimizrachi presents Three-Year Mizrachi Report.\nI think Protektzia may have been the first Israeli word I ever learned; if it wasn't, it should have been. Mrs. S. presents Protektzia at its very best .\nSome don't need Protektzia, just the encouragement not to back down. See A Living Nadneyda's post, Don't worry, the blogosphere is behind you.\nWow \u2013 Israel's post office launches a tax for package recipients! New heights\/lows in chutzpah there. Check out Stop mailing us stuff before January, at the Muqata.\nThis was a fascinating read: Cosmic X presents The Awesome Rabbi Shmuel Eliyahu Interview - in English!.\nFor a bit of Israeli fashion, Harry presents A Green Fashion Event Green Inspired by the Sea of Galilee.\nCats in Jerusalem? Really?! (Heh.) Harry presents Mewling visitors.\nI stopped reading Debka years ago; they remind me of the Jewish Press of my youth, the headlines of which declared imminent war with Syria every week, non-stop, for a decade or so. SnoopyTheGoon presents Oy vey, Debka... and a missile fired from an American warship.\nOy. SnoopyTheGoon presents The village of Ghajar: torn apart by politics and apathy.\nYisrael Medad has photos from his diplomatic meeting in A Bit of Diplomacy.\nCosmic X presents Mazal Tov! Margalit Har-Shefi Engaged.\nJoel Katz presents links to many articles in Religion and State in Israel - November 8, 2010 (Section 1) and Religion and State in Israel - November 8, 2010 (Section 2).\nJameel does India and airline food (a bad mix, apparently) in Rant of the Day. I think he should invest in chocolate bars. Lots of them.\nHow about a little bit of Europe? Ilana-Davita presents 10 Things I Love About Antwerp.\nGoing somewhere by sea? Life in Israel answers one of your biggest questions: Does a yacht need a mezuzah?\nThis one includes a piece about non-travel: Marilyn Stowe talks about Israeli leaders who cannot visit the UK in Faith, the law and freedom of speech.\nRisa travelled to Tel Shilo, and she writes about it in Protection on the ground and in our hearts.\nRunning a marathon doesn't really count as travel, but I liked this category for my brother's Of Marathons, Missions and Meanings.\nAnd one of my favorite bloggers, the inestimable Jack, returned from a road trip in The Road Warrior Returns.\nNils J. Sandback offers Piano Humour: A Victor Borge Tribute.\nYisrael Medad presents I Can Finally Fit Pamela Anderson Into My Blog. I'm happy for you, Yisrael.\nRisa presents KCC #60 Kislev Cooking Carnival.\nChavi talks about how she is dealing with a challenge in the conversion process in What to Do When You're Craving Treif.\nIlana-Davita talked about Belgium above; well, my brother reviewed a beer called New Belgium Skinny Dip at his Kosher Beers blog. Apparently it's lime-flavored; I wonder who came up with that idea\u2026?\nI am an olive-lover, so I appreciated Harry's Foto Friday \u2013 Olives take center stage.\nGot Elk? Take a look at Tzvee's post, Is Elk Meat Kosher? to find out where you can get it.\nI loved this idea: The 5-Second Rule, at Modern Uberdox.\nBeit din requirements are front-and-center in Converts & Fear: Is a Beit Din on Halachic Thin Ice?.\nSusan Barnes talks about her first taharah at She is Pure. I love the nail polish remover line.\nTo shake or not to shake? Cosmic X presents The Old \"Shaking Hands With a Woman\" Controversy.\nHebrew lessons here: Jacob Richman presents Uploaded New Learn Hebrew Video - Handheld Items.\nProfK has a plan for re-structuring tzedakah organizations at One solution to the tzedakah problem.\nWant to know about Jewish Homeschooling? Jennifer in MamaLand presents Jewish Homeschooling Blog Carnival #2: Kislev 1, 5771.\nI posted a piece on the most senior members of minyanim around the world: 139.\nA Mother in Israel has a long-running series on Marriage in Israel today; her latest installment is Fearful Parents Promoting Young Marriage.\nJacob Richman offers Chanukah Educational Resources and 175 Chanukah Videos.\nphyllis anticipates Chanukah's light in winter's darkness, at Rosh Chodesh Kislev.\nRabbi Yitzchak Schwartz, a friend and colleague, passed away this week; Yisrael posts about it here.\nsheva offers pictures and gratitude to HaShem in Birthdays and little miracles.\nGalit Breen (Minnesota Mamaleh) also has pictures, along with a contest, in Minnesota Mamaleh: Family Pictures.\nDenise L. Eger sends a message of hope in Here is my It Gets Better video.\nBatya presents New Blood in Moetzet YESHA (Council)?.\nBatya presents \"Hear, Hear\" for Hair-Covering.\nBatya presents Nu, For This I Made Aliyah.\nBatya presents Celebrating Rosh Chodesh Kislev at Tel Shiloh.\nBatya presents Women's Hair Covering, Lots to Say.\nBatya presents Tzachi Hanegbi, What Happened?.\nBatya presents Reminds Me of My Youth.\nBatya presents Beautiful Music for A Package From Home.\nAnd that concludes this week's edition of Haveil Havalim. I hope that you enjoy reading these posts as much as I did!\nNext week will be hosted by To Kiss a Mezuzah. You may do so by using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page.\nWhich would you rather [your child] be?\n\u2022 Boro Park or Lakewood?\n\u2022 Rabban Gamliel or Rabbi Akiva?\n\u2022 Eretz Yisrael or Ever haYarden?\n\u2022 Rashi Tefillin or Rabbeinu Tam Tefillin?\n\u2022 Eruv Chatzeiros or Eruv Techumin?\nMore substantively, though: As a parent, and a Jewish parent, I find myself asking the same question when prioritizing what to emphasize for my kids. And as in choosing between thunder and lightning, cat and dog, and so on, I'm often unsure.\n\u2022 Socially Clued-In or Clueless?\n\u2022 Intellectually Creative or Structured?\nThe other day, when I turned to Hallel in my siddur I saw the following code pencilled in: 632 AS \/ 565 B \/ 139 Goody.\nMy shul rabbi readers, certainly those from shuls which welcome people who have less background, may be able to parse the first two-thirds of the code. 632 is the page number for Hallel in the Artscroll siddur, and 565 is page number for Hallel in the Birnbaum siddur. I always announced the page number, to the point that I had them memorized, but I was concerned about forgetting them and so I inscribed them in my siddur. But what is 139 about?\nThe third number was from Goody Miller's siddur. Every month, when I was in Allentown, I would say, \"Hallel is on page 632 in the Artscroll siddur, 565 in the Birnbaum,\" and Goody would lift a finger and say, \"139, Rabbi,\" because that was the page number in his siddur.\nSeeing that code was poignant for me today, because Goody passed away several weeks ago; this was the first Rosh Chodesh since his death that I used this siddur, and saw the notation.\nGoody was in his 90s \u2013 probably 94 or 95 \u2013 and until a couple of years ago he came to shul every day, regardless of the weather, in his old pickup truck. A gentle man, always grateful for any small favor, never complaining about his ill health (although he wasn't happy if the heat in shul wasn't working properly), quick to smile and greet people. Colds and deafness and snow and rain rarely stopped him until he finally moved to be closer to his kids. He passed out, at least once, in shul. He was the unsinkable Goody.\nShuls around the world are blessed with \"Goody\"s. In my shul in Rhode Island we had Yankel Faust and Joe Zalusky, both now of blessed memory, both with incredible life stories and a devotion to making the minyan. In Allentown we had, in addition to Goody, an incredible man named Leo Ritter, of blessed memory. (I can still hear Leo, our nonagenarian gabbai who passed away seven or so years ago, asking, \"Has anybody yahrtzeit?\") And there are more, those who have moved to the next world and those who, thank Gd, are still with us\u2026 I would expand here, but I'm not certain what people want to keep private.\nWe would benefit greatly if someone would take it upon himself to write a book, or at least create a website, dedicated to these octagenarians and nonagenarians who keep minyanim going in communities far and wide. They daven, they serve as gabbai, they inspire men one-third their age, and they make sure the tenth will be there. That would be a great book, a snapshot of these minyan anchors from a moment in time.\nJust make sure Goody's listing is on page 139, please.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"with the World Outgames Miami 2017.\nMichael Kaufmann was awarded 1st place in Open Line Dances for the Country Western dancesport event at the World OutGames Miami 2017.\nIn 2017, thousands of participants, spectators and thought leaders from around the globe will converge on Miami for World OutGames IV. This 10-day event will bring to life more than 450 events across three areas: Sport, Culture and Human Rights. World OutGames Miami will challenge you physically, stimulate you intellectually and enliven you emotionally. Be part of it all, and leave transformed for the better \u2013 in your mind, your body and your heart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 lb. lean hamburger, ground lamb, mutton, one whole chicken or half of a small turkey.\nCombine all above ingredients. Add water to cover ingredients, simmer, stir, and add more water as needed until cooked. De-bone chicken parts and do not feed cooked bones since they can splinter and cause internal injury. The recipe should be thick to be molded into patties (add bran or soy protein meal to help thicken). Mix well into the stew while it is still very hot, a cup full of raw, grated carrots, sweet potato or yam.. Serve 1 cup full of this recipe for a 30 lb. dog with the rest of his\/her rations, and freeze the rest into patties and store in the freezer. Serve thawed, or frozen to gnaw on outdoors in hot weather.\nFor variation, you can use cottage cheese, plus well-cooked lentils, chick peas (garbanzo beans), lima beans or other pulses or a dozen eggs as meat alternatives.. All ingredients, ideally, should be organically certified. (Note: some dogs are allergic or hypersensitive to some foods, especially soy, beef, eggs, wheat and dairy products).\n--THE ABOVE RECIPE CAN ALSO BE FED AS A COMPLETE MEAL RATHER THAN AS A SUPPLEMENT.---MIX INCREASING AMOUNTS OF YOUR DOG\ufffdS NEW FOOD WITH DECREASING AMOUNTS OF THE OLD FOOD OVER A 7-DAY PERIOD TO ENABLE ADAPTIATION AND AVOID POSSIBLE DIGESTIVE UPSET.--- IT IS ADVISABLE TO VARY THE BASIC INGREDIENTS TO PROVIDE VARIETY AND TO AVOID POSSIBLE NUTRITIONAL IMBALANCES, AND TO MONITOR THE ANIMAL'S BODY CONDITION SO AS TO AVOID EITHER OVER-FEEDING OR UNDER-FEEDING, BASED ON THE AVERAGE DOG CONSUMING ONE CUPFULL OF THE FOOD TWICE DAILY PER 30 POUNDS BODY WEIGHT.\n--NOTE: DIFFERENT ANIMALS HAVE SLIGHTLY DIFFERENT NUTRITIONAL NEEDS ACCORDING TO AGE, TEMPERAMENT, AMOUNT OF PHYSICAL ACTIVITY AND HEALTH STATUS).\nHome-prepared foods for our animal companions, ideally with organic ingredients that were locally produced, are important because you then know what your animal is being fed if a food-related health problem such as an allergy to a particular ingredient or digestive upset were to arise. With most processed commercial pet foods containing all kinds of human food-industry by-products and ingredients considered unfit\/unsafe for human consumption, many of questionable nutritional value after repeated processing, you just don\ufffdt know. Aside from coloring agents that may cause problems other than saliva-staining of animals\ufffd faces, and paws, most commercial pet foods contain artificial preservatives like BHA that is linked with cancer of the bladder and stomach; BHT that may cause cancer of the bladder and thyroid gland; and Ethoxyquin, one of Monsanto\ufffds many allegedly harmful products that renderers (meat and poultry processors) add to the fat\/tallow that is put into pet foods to prevent rancidity. Ethoxyquin is a recognized hazardous chemical, a highly toxic pesticide.\nCombine all above ingredients. Add water to cover all ingredients, simmer and stir, and add more water as needed until cooked and thickened. Stew should be thickened enough to be molded into medium-sized or muffin-size patties (add bran to thicken if needed). Also add an egg or cup of cottage cheese. Immediately after cooking and cooling, debone and discard bones (cats should not be given cooked bones to eat since they can splinter and cause internal damage). This stew can be served as: 1\/2 cup full for a cat with the rest of his\/her rations. Freeze the rest of the stew as patties, or in muffin trays, and thaw out as needed. Serve one patty to a cat about three times per week with regular rations.\nFor variation, substitute 1 pound raw or lightly cooked boneless fish. (NOTE: some cats are allergic to fish, corn, and also to beef and dairy products).\n* = these items are available in health food stores. Ideally all ingredients should be organically certified. NOTE: Add flax seed oil after the cooked food has cooled to room temperature.\n--THE ABOVE RECIPE CAN ALSO BE FED AS A COMPLETE MEAL RATHER THAN AS A SUPPLEMENT.--MIX INCREASING AMOUNTS OF YOUR CAT\ufffdS NEW FOOD WITH DECREASING AMOUNTS OF THE OLD FOOD OVER A 7-DAY PERIOD TO AVOID POSSIBLE DIGESTIVE UPSET.--- IT IS ADVISABLE TO VARY THE BASIC INGREDIENTS TO PROVIDE VARIETY AND TO AVOID POSSIBLE NUTRITIONAL IMBALANCES, AND TO MONITOR THE ANIMAL'S BODY CONDITION SO AS TO AVOID EITHER OVER-FEEDING OR UNDER-FEEDING, BASED ON THE AVERAGE CAT CONSUMING ONE THIRD OF A CUPFULL THREE OR FOUR TIMES A DAY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Plaistow resident Jim Collins created the design for the Moose Plate.\nA self-taught artist, Collins painted the four original designs for the license plate: a purple finch, a covered bridge, a maple tree and sap bucket, and the bull moose. The bull moose design was chosen by the public to be the first NH Conservation License Plate.\nMore information about Collins' work can be found at his website, www.jcwildlifeart.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quicksort or partition-exchange sort is aptly named because, when properly implemented, it is the fastest known general-purpose in-memory sorting algorithm in the average case. Quick sort is a sorting algorithm developed by Tony Hoare that, on average, makes O(n log n) comparisons to sort n items.\nClick here for Quick Sort algorithm with easy simple examples in C & Java.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Ross River hosts a wide range of sporting and leisure activities. Riverwalk, a three metre wide pathway which overlooks the beautiful Ross River, runs along the river and provides numerous opportunities to enjoy its natural beauty.\nThe Riverway, Loam Island and Pioneer Park developments offer a dynamic combination of residential, commercial, cultural, sports and leisure activities. With two huge swimming lagoons, Riverway Arts Centre, Pinnacles Gallery, the Riverwalk, public art, village spine, restaurant and parklands there are plenty of activities here for everyone. The area is a year round visitor destination with an annual calendar of events including festivals, Christmas festivities, New Year's Eve, Australia Day Celebrations, as well as food, wine, art and sporting events.\nThe precinct provides an excellent environment to enjoy a variety of water-based and waterfront activities. Skiing, wakeboarding and fishing are all popular activities on the water. Ross River also offers some excellent bird watching opportunities, particularly from Pioneer Park, Loam Island, Apex Park and Dam Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pdf Diy Raised Loft Bed Plans Download Queen Size Bunk Bed Plans Woodideas images that posted in this website was uploaded by Jeci-miec.org. Pdf Diy Raised Loft Bed Plans Download Queen Size Bunk Bed Plans Woodideasequipped with aHD resolution 630 x 354.You can save Pdf Diy Raised Loft Bed Plans Download Queen Size Bunk Bed Plans Woodideas for free to your devices.\nIf you want to Save Pdf Diy Raised Loft Bed Plans Download Queen Size Bunk Bed Plans Woodideaswith original size you can click theDownload link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are growing fears for Paris Jackson after her latest bizarre Instagram post.\nThe troubled daughter of Michael Jackson took to Twitter amid allegations from TMZ she was in a stable condition after an incident in the wake of the Leaving Neverland documentary.\naris is the daughter of Jackson and Debbie Rowe, who he married at the Sheraton on the Park hotel in Sydney in 1996.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaemms b/data_all_eng_slimpj/shuffled/split2/finalzzzaemms
new file mode 100644
index 0000000000000000000000000000000000000000..241008fdf39826f5266f47241543c81f851bf4c9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The new KIPP Believe at Columbia Parc - Duration: 2 minutes, 14 seconds.\nRoland Cola KBP Teacher Spotlight February 2019 - Duration: 61 seconds.\nAvery Bond KCCP Teacher Spotlight February 2019 - Duration: 61 seconds.\nSharbara Beatty KECP Teacher Spotlight February 2019 - Duration: 61 seconds.\nSusie Kessler KMP Teacher Spotlight February 2019 - Duration: 61 seconds.\nKatelin Biermann KMM Teacher Spotlight February 2019 - Duration: 61 seconds.\nKLP Hour of Coding - Duration: 49 seconds.\nKIPP Renaissance Legacy Chorale - Emmanuel - Duration: 3 minutes, 16 seconds.\nKIPP Renaissance Legacy Chorale - Jesus What a Wonderful Child - Duration: 3 minutes, 2 seconds.\nSomething Told the Wild Geese - Duration: 116 seconds.\nWhitney Tashiro KBTW Teacher Spotlight November 2018 - Duration: 61 seconds.\nNikki Coleman KMP Teacher Spotlight November 2018 - Duration: 60 seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Superfrog is one of the best platform games released on the Amiga. Very polished graphics, great themed levels, very nice control system and character interaction with the suroundings, and some very nice level design.\nInvincibility: Hold the Right Mouse Button. The frog's face will flash to confirm correct code entry.\nSTAGE 1.1-\"000000\" STAGE 3.1-\"343522\" STAGE 5.1-\"452234\"\nSTAGE 1.2-\"234644\" STAGE 3.2-\"882311\" STAGE 5.2-\"984841\"\nSTAGE 1.3-\"447464\" STAGE 3.3-\"992334\" STAGE 5.3-\"383772\"\nSTAGE 1.4-\"747822\" STAGE 3.4-\"091332\" STAGE 5.4-\"093152\"\nSTAGE 2.1-\"392822\" STAGE 4.1-\"467464\" STAGE 6.1-\"387211\"\nSTAGE 2.2-\"446364\" STAGE 4.2-\"818234\" STAGE 6.2-\"981122\"\nSTAGE 2.3-\"984448\" STAGE 4.3-\"182394\" STAGE 6.3-\"017632\"\nSTAGE 2.4-\"477444\" STAGE 4.4-\"298383\" STAGE 6.4-\"398112\"\nSuperfrog was a great and entertaining game for the Amiga. One of the better platformers released for this computer. It is fun, challenging and has good lifespan. There is always a lot of stuff to try out and something new to discover. And.. the game sports those cool fruit machine mini games.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who doesn't enjoy a cold, tasty beverage? Here we'll teach you how we engraved these hearty beer mugs with our very own brewery logo. Personalize the name and you've got an excellent gift - for yourself or someone else!\nLaser engraved glass provides a beautiful, frosted look that works very well to showcase a bar or restaurant name, or a as gift for a friend!\nGlass beer mugs - we purchased ours from Discount Mugs.\nWe used a 60-watt Helix with the following settings: 25% speed, 100% power, 600 dpi, and with the Stucki dithering pattern.\nFirst, open the file sc_beerglasses.cdr.\nPersonalize the text and\/or resize the logo to your liking.\nNext, install the rotary attachment to your machine.\nPlace the beer mug on the rotary attachment.\nSituate the mug so the handle is facing down and the opening of the mug is on the left side.\nWhen you are satisfied with the design and its placement, apply a thin coating of dishwashing detergent to the mug.\nNOTE: It is important to use the clip on the rotary attachment for this project as the handle of the mug is heavy and will not turn properly if not secured to the rotary attachment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"100% handmade QingGear LKey leather Key holder embodies elegance. With the LKey, you can arrange keys in your order of preference, which will help you locate the right key quickly and effortlessly.It protects your belongings from key scratches.The LKey includes 3pcs different size(14mm,20mm,26mm) screw, very practical no matter you have more or less keys. It also comes with an additional attachment ring to carry bulkier and odd shaped items such as your car keys or other large keys. You could also use it to hold your USB stick.\nBlack and Brown are available. Please advise which color would you like. Otherwise, it will be sent at random. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to your one of kind seaside retreat.\nSea Mist Oceanfront is Myrtle Beach's 15-acre oceanfront playground with so many amenities and so much to do you never have to leave the property! Our lodgings are basic yet accommodating and affordable. With over 50 different room types and over 750 units to choose from, there is bound to be a lodging type to suite any size group or budget.\nYou name it, we can host it! We have enjoyed catering to a broad spectrum of groups for many decades; from family and retired military reunions to social groups of all kinds, plus motor coach, educational, sports, fraternal, church gatherings, weddings and wedding receptions. We cater to groups with a cause and to groups without a cause. Either way, you have the option to formalize your group presentation, or enjoy a \"blue jean\" or \"flip-flop\" meeting. Be it casual and fun, or formal, we can accommodate.\nOur professional staff makes planning easy and fun. We do all the work so you don't have to, catering to groups from 25 to 450 guests. Sea Mist features unique indoor and outdoor meeting space options. Our free standing Convention Center offers 17,000 square feet of meeting space with 8 different meeting rooms. Outdoor spaces include unique decks of all sizes located throughout the grounds. Our full-service catering staff will customize a menu to suit even the most discriminating palate.\nSea Mist features the largest collection of on-site amenities in the area including but not limited to the largest Water Park, Arcade and Jacuzzis in Myrtle Beach, plus numerous pools, a gift shop, a convenience store, and eateries.\nSea Mist is only 3 miles from the Myrtle Beach International Airport and is easily accessible from Highway 501 and Highway 17. Sea Mist is centrally located close to all the popular amusements and attractions. Live entertainment theaters, Market Common, the Boardwalk, Coastal Grande Mall, Broadway at the Beach plus a multitude of restaurants are easily accessible.\n\u2022 Wahoo Water Park - the largest on-site water park in Myrtle Beach with up to an acre of wet and wild water fun featuring a 760 foot long, 7 foot wide lazy river, a 3-story high totally tubular slide, multiple mini-slides, and a rain tree.\n\u2022 Wet Willie's Oceanfront Wading Pool - a 3,500 square foot children's activity pool.\n\u2022 Breakfast at Tena's - a hot All-You-Can-Eat Breakfast or a la carte options available.\nUnique Indoor and Outdoor Meeting Space\u2026Like No Other!\n\u2022 Sea Mist Cinema \u2013 a 90-seat state of the art movie theater.\nPlease note, some of our amenities are seasonal. Many are open year 'round, others are open Spring through Fall and a few are available in June, July, and August only.\n\u2022 Market Common: 3 miles south of Sea Mist.\nPricing: All listed menu prices are base prices. Any changes &\/or substitutions must be approved by the Food & Beverage Manager and pricing will be adjusted accordingly. 20% gratuity and prevailing state and local taxes will be added.\nFood: Sea Mist Oceanfront must provide all food and beverages. Only wedding and birthday cakes can be purchased off premise.\nBuffet Service Policy: Sea Mist Oceanfront requires a 50-person guarantee for buffets. Buffets are well appointed, but are not 'all you can eat'. They are prepared according to attendance. We must decline any type of carry out containers.\nPlated Service Policy: There will be a maximum of 3 selections if a choice menu is desired. Numbers per selection must be given a minimum of 72 hours prior to the event.\nSecurity: For certain functions Sea Mist Oceanfront will require that adequate chaperone and\/or additional security be provided. If outside security is required rates will be applied accordingly.\nDance Floor Policy: Dancing on carpet is prohibited. Any event that has a DJ or band is required to have a dance floor. Rental rate will be applied.\nAdditional Services: Audio\/visual equipment is available for rent. Please see a representative for the complete list. Also, upgraded centerpieces and decorative options are available for an additional charge.\nAdditional Charges: $50.00 set up fee per bar and $15.00 per hour, per bartender. Carvers- Flat $50 fee per carver.\nYour group meeting can take place in a 15 acre semi-tropical setting right in the heart of Myrtle Beach, America's exciting oceanfront playground. From the Magnolia Ballroom to the more personal and private meeting rooms, Sea Mist features 17,000 square feet of function space, supported by full in-house banquet facilities and services.\nWillow Exhibit Hall can accommodate at least 56 8'x10' exhibit booths and offers a drive-in unloading ramp in the rear of the center. Willow Hall can also be used for group meals and meetings, as it is completely decorated. (There are no dividing walls in the hall.) Large conventions often are assigned the entire Convention Center for their use. Sea Mist Oceanfront is very flexible in accommodating your group needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagizs b/data_all_eng_slimpj/shuffled/split2/finalzzzagizs
new file mode 100644
index 0000000000000000000000000000000000000000..4f558dac5e007c6af1fbaa80466a37eb92540ad5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagizs
@@ -0,0 +1,5 @@
+{"text":"Small Package and Small Package Jr., not mother and child but named after John R. Macomber's famous racer. He won 25 years ago and is at vacation farm, Methuen, Mass for rest of days. The Shetland pony is 1 week old.\nJohn R. Macomber's Small Package meets his namesake, a baby Shetland pony named Small Package, Jr.\nMacomber's Small Package meets baby Shetland pony named Small Packages, Jr.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"an artist and art collector Abraham Michael Dayan from Paris, France.\n1.Vietnam Trip: Bart & I went to Hanoi Vietnam and also did the Ha long Bay Cruise.\n2. Bali, Indonesia Trip: I stayed in Ubud, Bali and had my palm read by Ketut Iyer.\nmeeting the youngest addition to our clan, Laz.\n3. San Francsico trip where I attended a childhood friend's wedding in Marin County.\n.4 Las Vegas trip where I re-united with another childhood friend.\n6. Sweden Trip. I really enjoyed Stockholm!\n9. I illustrated and wrote my first Balinese Childrens book entitled: Fairy in Bali.\n10. As a member & Officer of CBIG ( childrens Book illustrators Group NYC) I designed the annual members promo postcard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learn the age old Japanese gardening technique of Kokedama! This process is a unique take on indoor gardening, as roots are covered in moss, and the finished product can either sit on a shelf, or hang from the ceiling.\nThis class is taught by the amazing Savannah Kurka of Savvie Studio!\n+ all materials and guided instruction to make your kokedama planters!\n+ a one-time 15% off coupon at DIME! you're welcome to use this the day of or within a week after the workshop!\n+ sign up with a friend (or mama!) and get 10% off your tickets!\nSelect # of Students Just me!Me + Friend!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check out Battlefield V's full reveal trailer | PowerUp!\nBattlefield V's reveal is happening today and we've got the full reveal trailer for you right now. The Battlefield V trailer confirms that the game will take place during World War II and that tanks, planes and other vehicles will make a return.\nIn the Battlefield V trailer, we're treated to an action-packed scene of a squad fighting against the German army, before it transitions into actual gameplay.\nThe transition from cinematic to gameplay is incredibly smooth and very, very impressive.\nThe scene we were shown yesterday as a tease for the game also appears in the Battlefield V trailer, but I won't spoil that moment because it's quite cool.\nAs for additional details, it looks as though the scene is set during a multiplayer match, but it's hard to tell exactly what's going on because it's absolute mayhem.\nOne thing that is certain though is that this trailer makes Battlefield V look stunning. The Frostbite Engine has never looked so good and the vast, real-time destruction is off the charts.\nCheck out the trailer below and stay tuned for more info.\nBattlefield V is coming to PC, PS4 and Xbox One.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free educational workshops where you will learn infant massage techniques with proven benefits. Parents, caretakers, and expectingparents are all welcome. Pre-registration is required. Loretta Dalia, LMT is an IAIM certified infant massage instructor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahmch b/data_all_eng_slimpj/shuffled/split2/finalzzzahmch
new file mode 100644
index 0000000000000000000000000000000000000000..59778f6ce14e18f3e71bf0209654cb779f95aa91
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"You probably instinctively know if you're more one than the other, and you could even be a mix of introvert and extrovert. If you're not sure, think about this: you've had a crazy busy day, you've been around a lot of new people, learned lots of new names and spent the whole day chatting.\nThat friend who is a great listener that you can trust your life with? Introvert.\nThe friend you can call up any time you're bored and you know she'll talk you into going out and having fun? Extrovert.\nNow \u2013 how do you relax after a day like that?\nIf the answer is you feel the need to lie down in a darkened room with a good book, you're probably on the introverted end of the spectrum \u2013 but if you've already made plans to go out to a party with all of your new friends that same night, we can probably mark you down as a more extroverted type.\nYounger people are more likely to think of themselves as extroverted; people may tend to be more willing to accept themselves as introverted as they get older and more comfortable with who they are and the strengths they bring.\nAnd why not, right? It's not easy being a teenaged introvert. You're not seen as introverted, you're seen as a geek or a nerd or an emo. You're not seen as an analytical, cool-headed decision maker, you're seen as a boring, swotty stick in the mud.\nBut here's the thing; it doesn't matter whether you're an introvert or an extrovert \u2013 honestly. There are shy extroverts all over the place, who love being outgoing with friends but are just scared of bigger social functions. Just as there are introverts who prefer solitary activities to group ones, but it doesn't mean they're never in the mood to steal some limelight; it would just be in a time and place they feel comfortable with.\nIntroverts and extroverts are fundamentally quite different in the way they prefer to spend their time \u2013 of course that's true. But there is no right or wrong, there is no better or worse, cooler or geekier. Both groups just have a different set of gifts and strengths that they bring \u2013 to their families, in the classroom, to their friends and later on, in the workplace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How To Choose African American Wedding Hairstyles is free HD wallpaper. This wallpaper was upload at May 1, 2018 upload by Katie in Hairstyles.\nSee our collection How To Choose African American Wedding Hairstyles for every type of infomation, from How To Choose African American Wedding Hairstyles.\nThe ideas set by fashdea.com combines How To Choose African American Wedding Hairstyles that composes two elegant fashion, seemingly best for your style How To Choose African American Wedding Hairstyles. Our idea and trend are very well on mode . Incredible How To Choose African American Wedding Hairstyles can be an effective focal point for any ideas. When you are able to choose one that is well crafted, it can even inspire your live around it to a high level motivation.\nHow To Choose African American Wedding Hairstyles is high definition wallpaper and size this wallpaper is 1280x720. You can make How To Choose African American Wedding Hairstyles For your Desktop Wallpaper, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the How To Choose African American Wedding Hairstyles images by click the download button below to get multiple high-resversions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anyone heading north this summer, be aware that the Ice Supplies on SH10, just before the Matauri Bay junction (Halfway between Kerikeri and Kaeo) is no more. Craig has sold the business and closed up shop.\nCurrently looking around to find out where else bulk salt flake can be had in the local area, although at the moment I've only found the $5\/6 bags.\nWill update if I find a discount Chilly Bin Filler.\nThat's a shame..used them on a few occasions and they were always great to deal with !\nThat is a shame, those guys were just up the road from my Dad so really handy for bait and ice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"God gives us a calling on our lives, and what we do for Him is a calling. No one can hold God back, He is sovereign. He is love and His grace is new every morning.\nWhat better reasons do we need to praise and worship our Lord and Heavenly Father?\nWe fail every day. Our lives get in the way, our pride, our minds. He never fails, never lets us down. How much of what we do is an offence to God? Maybe we need to sit and think about how we do things or how we speak to ourselves or others. Our flesh is strong, and we seem to battle all the time. Gods sovereignty is being watered down within our churches and how we view Him. He is God, He is our Maker.\nNothing can surprise God or shock Him. He knows all and knew us before we were born. He knows what we go through and weeps with us. He feels what we feel. He is there, and never goes away.\nPraise is what I can bring. I can worship because I have nothing else to bring sometimes. I can praise when I am in a place where I am low, happy, content or in despair. Praise lifts us out of the mire.\nFaith is believing when we dont see God, or dont know what to do. Keep your faith in Him who created dust into His child. He created you because He loves you.\nCan I praise in everything?\nWe are called and commanded to praise in all situations, but how hard is it really? Is it something that we can make ourselves do, or is it too hard and we have to force our mouths to say the words of praise to God that He deserves?\nOut of our mouths can come death, life, praise, birth, positives, negatives and everything else in between. Can what we say over ourselves or others cause that much damage? Can we really build ourselves up or down with what we say mentally or verbally?\nI really believe we can. We have the power of life in our mouths, not just our hands or actions. If we tell someone about Christ, but talk bad, or down about others, or even complain about or church family or life, what example are we setting. On the Monday mornings in the office or school yard, are we complaining about how boring the service was, or how someone was mean to us? How are we to get others to church or to believe in our Lord, by that kind of talk that makes us no better than anyone else? We are looked at differently. People put higher expectations on us when we say we go to church, or are a Christian. They need to see and also hear a difference, see and hear a different kind of living.\nI know how hard it can be for myself to keep praising or to keep God in the place He should be. He has done so much for me, and saved me from so much, so why can it be so difficult to say Thank You? Start the day with thanks and praise and we will see a difference. Not maybe on what life throws at us, but certainly on how we view it and how we act and talk about it all. There is a big difference with praise. A thankful heart is a changed heart.\nWhen we look to Jesus and keep our eyes fixed on Him, we never lose our way or our bearings. He is our guiding light and guiding star in all the darkness of the world and our lives. He shines bright in the murkiness, showing us the right way to go. The safe and straight small lane to eternity with Him.\nAs we look back over our lives and especially over this last year, we can see how His hand has been on us, our lives and our actions. We can see where the decision we have made has led on to amazing things. The right person to talk to at the correct time. The phone call that has encouraged at the exact time and place we needed. We can see how He has shaped events or our reactions to those events. It gives us an idea of how to live in the future. Looking back and seeing His hand guiding us can help us to nurture thankful hearts.\nWe need to discipline ourselves to be thankful. Its not an easy in a world that is based on Self and self motivation way of life. Learn to \"count your blessings\". At the beginning we may seem childlike or unsure of how to start to be thankful. Start small, and practice. God will honour any attempt even for the small things. Its not something most of us are taught to do. Keep being thankful as much as you can. Nurture it and help yourself grow into new depths with our Lord.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Old McDonald had a farm. But did he have worker's compensation insurance?\nSECTION 42-1-360. Exemption of casual employees and certain other employments.\n(9) an individual who owns or holds under a bona fide lease-purchase or installment-purchase agreement a tractor trailer, tractor, or other vehicle, referred to as \"vehicle\", and who, under a valid independent contractor contract provides that vehicle and the individual's services as a driver to a motor carrier. For purposes of this item, any lease-purchase or installment-purchase of the vehicle may not be between the individual and the motor carrier referenced in this title, but it may be between the individual and an affiliate, subsidiary, or related entity or person of the motor carrier, or any other lessor or seller. Where the lease-purchase or installment-purchase is between the individual and an affiliate, subsidiary, or related entity or person of the motor carrier, or any other lessor or seller, the vehicle acquisition or financing transaction must be on terms equal to terms available in customary and usual retail transactions generally available in the State. This individual is considered an independent contractor and not an employee of the motor carrier under this title. The individual and the motor carrier to whom the individual contracts or leases the vehicle mutually may agree that the individual or workers, or both, is covered under the motor carrier's workers' compensation policy or authorized self-insurance if the individual agrees to pay the contract amounts requested by the motor carrier. Under any such agreement, the independent contractor or workers, or both, must be considered an employee of the motor carrier only for the purposes of this title and for no other purposes.\nBut just because you may fall into one of the above categories doesn't mean you are automatically precluded from bringing a worker's compensation claim. The character of the work performed and other factors may also determine whether you are a covered employee. That's why it is improtant to speak to a workers' compensation as soon as you can after an injury to determine whether you have a viable case.\nIf you have been hurt on the job, call us today for a FREE consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahpnt b/data_all_eng_slimpj/shuffled/split2/finalzzzahpnt
new file mode 100644
index 0000000000000000000000000000000000000000..d41d17f3607341821e2bd8f9293fd853f91a82ec
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahpnt
@@ -0,0 +1,5 @@
+{"text":"Kampala, Uganda \u2014 At least 162 people were arrested for drunk-driving during the Easter celebrations in the Capital,Kampala. Kampala Metropolitan police spokesperson Emilian Kayima told the press on Monday.\nMr Kayima said \"On Friday night, a total of 132 people were arrested for drunk driving while 30 more were picked along the streets on Saturday, there were only six accidents with no fatalities,\" he said.\nIn 2013 , US broadcaster Cable News Network (CNN) described Uganda as a drunk nation ranking it No. 8th globally in a study it carried Titled 'World's 10 best-drinking nations'.\nUganda leads other African countries for drinking alcohol and the drinking started at any time without any restrictions since there is no serious law on Alcohol Consumption.\nLast year, Mukono Municipality Member of Parliament, Betty Nambooze brought Anti- Alcoholic Drinks Control Bill before parliament and is set to be tabled before the house in the near future.\nThe bill is intended to replace the existing legislation on the manufacture, sell, supply, consumption and licensing of alcoholic drinks.\nIn the new bill still, no alcohol seller will be granted a license to sell alcoholic drinks in universities tertiary institutions, hospital, clinic, rehabilitation centers, prison and remand home.\nThe Alcoholic Drinks Control Bill, 2016 will put a restraint on selling of alcohol strictly from 5:00Pm to 12:00Am and whoever will be found contravening the law once passed will risk spending one year in the jail or pay a fine of Shs2M.\nThe bill seeks to restrict the trade of alcoholic drinks to only people licensed by Government. The license will be renewed periodically and one will pay a yet to be specified fee to be issued a license.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> > -1. 1.9.x is mainline release, and we should track only stable branch.\n> I know that, but a short time ago 1.8 branch somehow was bumped to 1.9.\n> So for now I just upgrade it and added use of 1.9 new features.\n> something like testing\/nginx9 with all this changes.\nNext message: Jose-Luis Rivas: \"[alpine-aports] [PATCH] testing\/nodejs-stable: new aport\"\nPrevious message: Valery Kartel: \"[alpine-aports] [PATCH] main\/openssh: upgrade to 7.2_p1\"\nIn reply to: Valery Kartel: \"Re: [alpine-aports] [PATCH] main\/nginx: upgrade to 1.9.12, lua module upgrade to 0.10.1rc1, dynamic modules\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A SENSE OF UNIQUENESS Have you ever wondered why a zebra has stripes? Most animals try to blend in with their surroundings, but Zebras 'stripes do just the opposite. Do not conform perform! Stripes are to Zebra what IMPARAGE is to motorcycles: UNIQUE! The uniqueness of IMPARAGE will make you standout.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After two days of bruising media coverage about his views on elements of the 1964 Civil Rights Act, the campaign of Kentucky U.S. Senate candidate Rand Paul tells me it has canceled the candidate's upcoming appearance on NBC's \"Meet the Press\" -- something the show's host and producer are currently sounding the alarm about on Twitter.\n\"Rand did Good Morning America today, set the record straight, and now we are done talking about it,\" said campaign spokesman Jesse Benton. \"No more national interviews on the topic.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our products and services include landscape lighting and lighting fixtures.\nSpecializing in landscaping, lawn maintenance, tree service and stone work.\nOur products and services include consultations and contractors.\nOur products and services include contractors and maintenance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaifzi b/data_all_eng_slimpj/shuffled/split2/finalzzzaifzi
new file mode 100644
index 0000000000000000000000000000000000000000..758f18fce7552526fe431de80c33ec70bce59bf0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaifzi
@@ -0,0 +1,5 @@
+{"text":"Are you a fashion lover? Do you really like to shop from Primark? Are you convinced with the quality and services they provide? In this case you must have some good or bad reviews about this brand. Now, yoy can share your views with them through Tellprimark survey which is avialable at tellprimark.co.uk survey. By completing this survey, you can win \u00a31,000 daily and \u00a31,500 Weekly Cash Prizes which is an amazing offer.\nSurvey work as an platform between the company and the customers. It helps alot in the development and improvement. Company get to know about the opinions, views and thoughts of their regular customers with which they can improve their credibility to satisfy their customers. Also Win $100 Five Below Gift Card by completing Fivebelowsurvey.\nIf you are willing to participate in the Tellprimark survey then keep stick with this article as it give you all the information aboout the rules and process.\nTo participate in the tellprimark survey you have to follow certain rules.\nThere is no requirement of purchase or payment to make an entry in the monthly sweepstakes.\nAge limit for The United Kingdom, The Republic of Ireland, Germany AND Denmark is 18 years or more.\nThe age limit is 20 years require for The Republic Of Korea.\nYou must not be a Primark employee or affiliate with this brand to participate in this survey.\nMaximum one daily and weekly sweepstakes prize per person in 90 days is given.\nWinners are contacted through phone or email.\nSelected winners should also respond within 10 days to take their prize.\nHow to participate in Tellprimark survey?\nThere are two methods to take part in Tellprimark survey.\nFollow the below mention steps for an online method.\nFirst of all, make a purchase from Primark brand.\nThe next step is to keep your Primark receipt with you.\nNow open your laptop and visit the official survey website of Primark available at tellprimark co uk survey. To visit this website click here.\nA new page will open on your screen. Enter the 13 digit number mentioned on your receipt. Now write the date and time where required.\nThen the series of questionnaire will appear on your screen. Answer all the questions honestly and truthfully.\nAfter this submit the survey.\nAs a result, you will automatically enter in the monthly sweepstakes to win cash prizes of \u00a31,000 Daily or \u00a31,500 Weekly.\nThe purchase is not required for mail in entry. For mail-in entry write your personal information such as your full name, phone number, address, email and the name of a participating client on a sheet of paper. And then mail that paper to the Primark postal address.\nPrimark brand is a well known Irish clothing retailer. Primark was founded in June 1969 in Mary Street.It's branches are located in Austria, Belgium, Germany, Ireland, Portugal, Spain, the Netherlands, and the United Kingdom. The company's main headquarter is located in Dublin. It is the part of British food processing company ABF. Take Myconversevisit survey & get a chance to Win $5 Gift Card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Installing a bullnose tile baseboard is a great way to finish off a flooring tile project. It looks great, it flows well with the flooring tile, and it is not a difficult project to complete. A project like this is just like installing tile on a wall, but to a much smaller extent. The process is exactly the same.\nPurchase bullnose tile that matches the floor tile. Some manufacturers do not produce bullnose tile that will match their floor tiles. In this case, buy extra wall tiles and cut them with a wet tile saw to the height you would like the base board to be. You will be able to produce two baseboard tiles per floor tile in this manner.\nPrepare the wall to receive the tile by cleaning off any thinset, grout or joint compound that may be on the wall to prevent a quality installation. Vacuum up the debris produced during this step.\nSpread tile mastic horizontally across the base area of the wall. Do not spread more mastic on the wall than you will be able to cover in less than 5 minutes. For example, if you feel that you will be able to cover 10 linear feet of mastic with tile in less than 5 minutes, do not spread 15 linear feet of mastic on the wall. Also, do not spread mastic in an area wider than the width of the bullnose tiles. If the v-notch trowel is wider than the bullnose tile, spread the mastic on each tile individually.\nBegin placing tile in the mastic. Use the same tile spacers between each bullnose tile that you used between the floor tile so the grout lines will line up and look even. Also, place two tile spacers flat under each bullnose tile between the bullnose tile and the floor. Allow the mastic under the bullnose tiles to dry for 24 hours.\nRemove the spacers from between the tiles. Mix up a batch of grout to a peanut butter consistency, and force the grout into the grout lines between the bullnose tiles. Force grout into the corner between the bullnose tiles and the floor tiles. If the bullnose tiles are being installed in a bathroom, place a bead of silicone caulk in the corner between the bullnose tiles and the floor tiles. Finish the caulk bead with a finger or with a caulk finishing tool. Allow the grout and the caulk to cure for 72 hours.\nInstall 2 to 3 coats of penetrating grout and tile sealer onto the grout when it has cured.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Delta College Board of Trustees decided unanimously Tuesday to begin a new campus in Saginaw to better meet the needs of an underserved part of its population base.\nWe believe the move is a good one that will help the people of Saginaw who have not now, for one reason or another, attending college classes.\nFor years, Delta has been leasing space at the Ricker Center, 1925 S. Outer Drive, but the facility doesn't meet the needs of the Saginaw population in the 21st Century.\nThe trustees now must decide among four areas the consulting team has identified as potential sites for a new Delta Saginaw center: Michigan Avenue near Covenant Hospital, Old Town, both on the city's Westside, and two east side zones, downtown Saginaw and Buena Vista Township.\nConsidering the earning potential for those people who acquire skills post-high school, be that in a traditional college or in skilled trades, we believe Delta will be available to more students who otherwise would not be able to get the type of training that leads to better income and security.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Successful meetings are built upon meaningful connections formed before, during, and after the event. There is value in connecting attendees to each other as well as to the meeting content. There's also value in connecting attendees to speakers, sponsors and exhibitors. Both drive attendance and revenue, which allow meeting planners to meet their goals.\nMeetingPlay is in the business of helping forge these meaningful connections, so we've designed an app that enables them, all while delivering pertinent, sometimes changing content while at an event.\nWe know you are looking for ease-of-use; that's why we adapt to the goals of your meeting, rather than giving you technology that you need to adapt around. Our objective is to deliver an app that's easy to use on the front and back end, with features beyond your expectation.\nOur founder and senior leaders came from the hotel industry, and we have certified meeting planners on staff. They know what makes for a successful event based on first hand experience. Technology is our passion, and with our hospitality expertise, we're able to make the connection between how to utilize innovative technology for the best possible meeting experience.\nWe build meaningful connections with our customers so we can understand their specific needs and goals. Our team works with our clients to help them navigate through our different product features in order to deliver an app that best aligns with their meeting goals. And we'll be with your pre- and post-event to make sure you have exactly what you need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Request Dr. Stephen Nichols Orthopaedic Surgeon online or call 888.853.7944!\nDr. Stephen Nichols, MD, QME is board-certified in Orthopaedic Surgery and practices in Encinitas, California. He is available for PQMEs, AMEs or Independent Medical Evaluations (IMEs) at several locations throughout San Diego County.\nDr. Stephen Nichols is a board-certified orthopedic surgeon with specialty training in the hand and upper extremities. He received his Bachelors degree from San Diego State University and earned his medical degree from Loma Linda University School of Medicine. His Internship in General Surgery and his Fellowship in Hand Microsurgery were also done at Loma Linda University. His Residency in Orthopaedic Surgery was completed at McGill University in Montreal.\nDr. Nichols is currently an Associate Clinical Professor at UCSD Medical Center and is the Chief of Orthopedic Surgery at Scripps Memorial Hospital in Encinitas, California. His memberships include California Orthopaedic Association, California Medical Association, San Diego County Medical Society and Phi Kappa Phi National Honor Society. He is a Fellow of both the Royal College of Surgeons and the American Academy of Orthopedic Surgeons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajolo b/data_all_eng_slimpj/shuffled/split2/finalzzzajolo
new file mode 100644
index 0000000000000000000000000000000000000000..9bcd80ecf3ffabdbb2a8f7ef7b0f2caa9f7dad3b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajolo
@@ -0,0 +1,5 @@
+{"text":"therefore required of you that you rename your specification immediately.\nhave feeds using the legitimate RSS 3.0.\nfrom the specification shall use RSS with a version number.\nreceive from Dave Winer as a result of the misuse of his initials.\nrespond in public, do so on that forum. Private responses are also welcome.\nNext message: Dan Connolly: \"RDF diff applied to MailStorePuzzle?\"\nPrevious message: Christopher Schmidt: \"Exporting Webtender Data\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you a homeowner with a well and\/or septic system? Do you know how these systems work? Do you know how to properly care for your well and septic system?\nIf your answer to one or both of the last two questions is no, join the Kane County Health Department's free Well and Septic Seminar for Homeowners. Due to the popularity of the seminars in the past, a second seminar has been added to accommodate demand.\nThe seminar will be held at 2 and 6 p.m. Thursday, Sept. 20, at University of Illinois Extension Office, 535 S. Randall Road, St. Charles.\nDue to the program's popularity, call ahead and reserve a seat by calling the Kane County Health Department at (630) 444-3040.\nIf you reach voicemail, leave your name, phone number, address and reason for calling and you will be placed on the list.\nOne free water sample test kit will be given to each household that is supported by well water.\nBring all of your questions as a short Q&A session will be held after the presentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An exciting and rare opportunity is currently available for a motivated and experienced individual to join our company as a logistics manager based at our Ulverstone head office. This position has the operational responsibility and accountability for the coordination of all physical logistics for the company.\nThe role manages a team of staff, travelling across the globe, along with associated hardware for running our systems. It provides all the interface between our company and various freight, travel and insurance companies.\nThe objective is to deliver value through reduced logistics spend, improved on-time and accurate delivery, and operational excellence. In addition this role is responsible for continuous improvement projects to optimize the logistics network and ensure we are at the leading edge in operational efficiency.\nManagement of distribution, storage and warehousing providers.\nSecure above average service levels from Simplot logistics service providers through building value-adding relationships. Build value adding relationships with Simplot's customers.\nEnsure warehouse providers meet Simplot's current and ongoing service requirements in a cost effective manner. Seek best practices for warehouse performance.\nLead and develop a logistics operations team of 10 including 4 direct reports.\nProvide logistics solutions for both new business opportunities and incorporate them into current supply network.\n\u00a9 2019 RallySafe. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LS-CVD (Liquid Source CVD) Systems | samco-ucp ltd.\nsamco-ucp's LS-CVD (Liquid Source CVD) systems achieve superior step-coverage and gap-fill performance, required for the MEMS, 3D-LSI, and Opto Electronics markets. Importantly, the technology enables deposition at lower temperatures and doesn't require toxic or flammable gases. LS-CVD systems can deposit high-quality TEOS-SiO2, LS-SiN, High-k think film, and Diamond-like-carbon (DLC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bitcoins are skyrocketing by the second, but despite them being in wide-demand, questions like \" How to buy Bitcoins with Cash \" aren't public knowledge. Well that's one question which gets answered right here, right now.\nSo let's walk you through some of the best How to buy Bitcoins with Cash, step-by-step.\nFor almost all the Bitcoin buying guides I've written, Local Bitcoins has dominated the article by being my first choice.\nThat's simply because it offers every single payment method on this planet, for every single country, and that holds true for cash as well.\nSo head over to LocalBitcoins.com and enter the amount of traditional currency you'd like to exchange for BTC.\nThen select your currency and country. As far as the payment method is concerned, choose \"In-Person Cash\" from the list.\nAnyway click on \"Buy\" on the seller's link.\nYou can see the meeting preferences on the right-sidebar, if you agree on the meeting locations, timings and everything else you can proceed to the next step.\nSend the trader a message regarding your intentions, and click on \"Send trade request\".\nOnce they accept your request, you can chat with them using the message panel and zero-in on a mutually accepted location. Once that's done, meet the person and make the payment.\nAfter doing that, click on \"Mark Payment Complete\" option on the spot.\nAfter you've marked the payment complete, the trader would confirm your payment on this device, and the Bitcoins would get released from LocalBitcoins escrow to you.\nDone! This truly is the best, easiest and simplest method to buy Bitcoins using Cash.\nEscrow service guarantees successful trades.\nGlobal Support- (99% countries supported).\nTrader level, reputation, history and trust can be verified. (It's all based on their previous transactions and activities).\nLow fee: 1% of the transaction.\nPaxful, after LocalBitcoins is my first choice when the need to buy Bitcoins with cash arises for me.\nThe reason being, it works more or less similar to LocalBitcoins and serves as a portal where you can find people willing to meet you in person, and sell their Bitcoins for cash.\nJust head over to Paxful and click on \"Buy Bitcoins\".\nEnter the Amount, currency and your country. In the Payment method, select \"Cash in Person\", and click on \"Search\".\nYou can also click on \"Give me the best\" button, it will automatically get you the best seller available for you and you won't have to search for any one seller manually.\nConsidering you clicked on search, select any seller from the list that follows if the rate per Bitcoin, location, min-max amount etc. satisfy your needs, click on buy.\nSpecify the amount for which you wish to receive BTC again on the next screen and click on Buy now.\nAs soon as you click on \"Buy now\", the Bitcoins are stored in Escrow by Paxful. Now you need to meet the person wherever both parties decide with mutual consent. (You can use the message panel to chat with the seller).\nOnce you meet and pay him the amount, click on \"I have paid the seller as requested\" on your Paxful account.\nOnce the seller verifies the transaction from his account, the Bitcoins are released to you from Paxful's escrow.\nBasically LocalBitcoins and Paxful operate on the same framework so if you wish to buy Bitcoins with cash, these two should definitely be on top of your list.\nNo fee! This is the best thing about Paxful. You do not have to pay any fee as a buyer. Only the trader\/seller is charged a fee of 1%.\nGlobal currency and countries support.\nEscrow service to protect your money.\nTrader reputation can be verified.\nSocial media offers more than just \"chatting and connecting\" features. The problem is we don't utilize all the features showered upon us.\nSocial media platforms like Facebook, Reddit, and Twitter etc. are a great way to meet any kind of people you wish for any kind of job. These aren't \"official\" Bitcoin buying and selling platforms, but they get the job done, and that too without any mediator fee!\nMeaning you do not have to pay the social media platforms any fee for arranging the sale which is definitely an edge you're getting.\nFacebook is becoming more than just a \"social network\". Among the many mindboggling features we've been introduced to recently, one of them is \"Buy and sell\" via Groups feature.\nThen write up your post, for eg. In the headline describe how much you're willing to exchange of traditional currency, enter your location, and any other kind of basic details.\nDone! If the group is active enough, people nearby that location who're interested and need to buy Bitcoin with cash will send you a message over inbox (or email if you've specified) and you can get the deal done.\nReddit is one of the most popular and most engaging communities on the planet. You can find a subreddit for almost any topic on the globe, ranging from simple shoelaces to building spaceships.\nNow this huge range of subreddits is what you can utilize to find compatible partners willing to sell Bitcoins to you on a P2P (person to person) basis.\nJust head over to Reddit, and search for \"Bitcons\", \"Buy Bitcoins\", \"Sell Bitcoins\" or any other related term. It will get you a list of all the subreddits on the topic.\nHead over to any subreddit and find the thread which seems most likely to fit your needs.\nOnce you've found the thread, read the thread description and if it suits you, you can leave a comment with your contact details for the seller to contact you.\nLinkedin, probably the best and only \"Professional\" social-network on the planet is yet another platform for you to buy Bitcoins with cash. Just like Facebook groups, Linkedin too has \"Groups\", a place where like-minded people come together.\nAll you've got to do is, search for \"Bitcoins\" on the search bar, and click on Groups.\nThen, choose any group which seems to have a big enough audience and ask to join it.\nOnce you're in the group, just leave a post\/thread stating your intentions, locations and T&C. If it's an active group, where buying & Selling isn't banned or prohibited your post will get noticed by other interested parties who're willing to let you buy Bitcoins with cash.\nMore or less similar to the Facebook Group concept, Google+ communities too let you search for a particular group, find and meet people, and then buy Bitcoins with cash.\nJust enter \" Bitcoins\/Buy Bitcoins \" or any other related term in the search bar.\nSome search results such as buy Bitcoins with Credit Card \/ Debit Card \/ Paypal etc. pop up but try not to click on them, and instead just hit the enter button to execute your search. Now you'll be shown a list of groups\/people with your search term.\nNow it's up to you to find a group\/individual over there and place a deal with them.\nProTip: Searching and landing a person willing enough to sell you Bitcoins instantly on Google+ communities is a tad bit hard.\nHence the best approach is to, join any other community which is related to Bitcoins, even the communities for buying Bitcoins with Paypal \/Credit\/Debit cards etc. interact with the folks there for some hours or days, try to get a feel of who's who of the group and from where, and after getting a rough idea of the group there's a better chance of you sending a message to someone and they letting you buy Bitcoins with cash from them.\nBasically there are no \"Steps\" as such to buy Bitcoins with cash from Social media. In a nutshell, it's just finding the right kind of person, and buying from them. You can either mail them, tweet them, inbox them over Facebook or use pigeons if you please; there's no steadfast rule as to how you can get the deal done.\nSocial communities are slightly different than social networking platforms.\nBasically on a Social community, there's more of \"finding and exploring\" rather than \"chatting and liking or commenting\".\nThere are basic communications features no doubt, but they're primarily centered around bringing a group of individuals together rather than letting them chat throughout the days.\nMeetup is one of the best communities you can use to land some potential Bitcoin P2P deals.\nIt basically offers you the choice to join various different kinds of groups and interest-factions. All you've got to do is, join any group related to Bitcoins\/Cryptocurrency or Digital Trading and find the people who might be willing to sell to you.\nAs the name already suggests, \"Meetup\" members meet-up on a pre-decided date & time to discuss, get together and exchange ideas.\nThe same opportunity couple be utilized to buy Bitcoins with cash without any risk as well, considering you won't ever be \"alone\" or isolated, instead the whole group of attendees would be present at the location.\nI do accept that most of the methods discussed throughout this guide weren't conventional, but nevertheless they do help you buy Bitcoins with cash one way or another. In case LocalBitcoins or Paxful aren't cutting it for you, Social media will surely prove to be a hidden weapon in the arsenal.\nDo let me know which social media platform you're going to try or have tried, your results help us carve out better articles for the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakitv b/data_all_eng_slimpj/shuffled/split2/finalzzzakitv
new file mode 100644
index 0000000000000000000000000000000000000000..a075e840e04ec262f0bc405ff03ffa3260ffe85d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Welcome to our Narrow Stance Leg Press Exercise Instruction Guide!\nOn this page you'll learn how to do this leg press exercise variation using the correct technique. Below you'll find pictures, exercise instructions, and tips on how to get the most out of this and other Leg Exercises so you can immediately add to your Leg Workouts.\nHave you checked out our list of the Best Leg Exercises yet? See if this leg press exercise variation made the list.\nLike the Leg Press Exercise, the Narrow Stance Leg Press utilizes a weight machine to target all the major muscles in your legs. The change in the width of your foot base targets the external muscles of the legs.\nSit on a leg press machine with your feet on the crosspiece 6 inches apart and your toes pointed slightly outward.\nTo begin, grasp the handles on the side of the machine and release the locks.\nWhile, keeping your abs drawn in, slowly lower the weight towards your body by bending your knees to a 90 degree angle.\nHold this position for a moment then push the weight forward by extending your legs and pressing upwards with an equal distribution of weight on your forefoot and heel.\nNow that you've seen the Narrow Stance Leg Press, which primarily targets your Quads, check out all of our different Quadriceps Exercises.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Post Office has ranked the Kenyan resort of Mombasa as the beach destination which is easiest on the wallet in their annual Long Haul Holiday report.\nOverall, the coastal city has risen one place to third with only Tokyo (1st) and Cape Town (2nd) being more affordable overall.\nThe study compares the price of ten holiday essentials \u2014 from a cup of coffee and bottle of local lager to a three-course evening meal.\nKenya is named as a great-value winter sun option, offering safari breaks as well as beach holidays.\nOverall, the report found that while prices may be a bit higher in some long haul resorts and cities this year, this 10th annual Long Haul Holiday Report from Post Office Travel Money delivered welcome news.\nIt found local price cuts in restaurants, bars and shops mean the cost to UK holidaymakers will actually be lower in 40 per cent of the 30 destinations surveyed by long haul holiday specialist Travelbag for the report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nobody expects to lose the car keys to her or his Subaru. You won't have to go through a lot of hardship to replace that car key with these service. Give AZ Car Keys a call right away and get Subaru car key replacement services from the pros.\nHave you lost your Subaru car keys? Are you looking for a spare key or replacement for your car keys? If yes, you are at the right place. The first thing that any Subaru owner should do after losing his\/her car keys is to contact AZ Car Keys.\nThese keys were introduced in 1995. They come with a computer microchip which is embedded in a plastic body and electronically coded. This code detects the car keys when you insert them in the ignition. Therefore, for the engine to start, the code has to match with the car. If the transponder breaks or gets lost, the car won't start.\nThe best way to replace transponder keys is by reprogramming the immobilizer control. This process ensures that the new car keys match with the new reprogrammed code. The cost of this process will depend on the car model, urgency, location and other factors.\nThis is the most common type of car key that comes with modern day cars. These keys can be used to turn car alarms ON\/OFF, open and close car trunks and locking\/unlocking doors. They use infrared or radio transmitters to send a message to the car's receiver. These keys use batteries and will not work if the batteries are flat. If a remote key goes missing, replacement of the systematic bypass will be done.\nThey are first introduced in 1999. They send a new code each time the car key is used. These codes are virtually impossible to replace. This implies that replacement of such a key is very expensive.\nMost Subaru cars come with either standard side cut keys or high security keys. High security keys are equipped with an immobilizer device while standard keys may or may not have this device. Subaru key blanks can be purchased from local auto-spare part shops. However, you should ensure that you select the correct blank for your car.\nSubaru adopted the transponder keys in 2005. These keys had curves on both sides. In 2009, the company introduced last cut keys as a means of enhancing security. These keys had curves on both sides together with an immobilizer device.\nLater, the Push Button Start ignition was introduced with a keyless entry. These keys became very popular due to their convenience. For instance, you could start your car with the keys inside the pocket. These keys are difficult to replace.\nA key code is the number that corresponds with the cuts that match with your Subaru lock. Key codes are generated by the manufacturer. If you buy a new Subaru car, you will be given two master keys; a valet key and a stamped metal tag with the key code. You should store this key code tag in a safe place as it will save you money when you need to replace lost keys.\nHowever, if you don't have your key code tag, our experienced technicians will implement two techniques to secure it. The first technique requires you visit a Subaru dealership with your ID and registration and request for your key code. Codes are only available for models manufactured from 1990. The second technique is where our technicians purchase your key code from another vendor.\nIt's always a good idea to own a spare key for your Subaru. AZ Car Keys has all the equipment and manpower to make spares for any Subaru car. Spares are cheaper and faster to make.\nIf you have lost your Subaru car keys, AZ Car Keys will program new car keys for you. Forget all the misleading information such as needing a new ECU box, ignition barrel or locks when you lose your keys. Our technicians will make brand new keys for you within minutes.\nCall AZ Car Keys Today For Replacement Subaru Keys!\nHad these guys come replace my Subaru key last week. They were very professional and seemed to get to me in a flash. Great company and if I ever have to, I will use them again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After you have connected to Facebook, you can open the game again from anywhere.\nGo to the Settings section of the game (a gear icon below you Energy and Level totals). Tap \"Connect,\" a blue button with a Facebook logo, and you will be prompted with the \"Choose a Saved Game\" window if you have a saved game somewhere.\nChoosing \"In the Cloud\" will immediately load up your game that you last connected to Facebook with. Note that you can only have ONE game saved to the cloud, so choosing this option will immediately overwrite your saved game data on that device so far.\nChoosing \"On this Device\" will connect your current game on your device to your Facebook account, so your previous game data on Facebook will be deleted permanently.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"13:56. Manson powerbombed Octagoncito for the win.\n11:20. Moonsault sideslam from Fenix to Apache for the win. Argenis replaced Atomic Boy.\n18:20. Zorro replaced Elegido. Inferno Rockers were the unlisted wrestlers. Machine pinned Drago with a German suplex.\n5) Joe Lider, LA Park, Psycho Clown b Psicosis, Silver King, Texano Jr.\n22:37. Earlier in the night, LA Park forgave Joaquin for costing him Dorian's head shaving, but promised Dorian will learn his actions has some consquences. Silver King replaced Octagon, who did appear but said he was injured. Psicosis jumped out of the crowd to attack Park. Park and Psycho Clown had issues during the match, but the tecnicos still won.\n16:47. Perros turned on Garza and destroyed him prior to the match. Halloween tried to replace Garza in the match, but the ref didn't allow it and called a DQ.\nProtector, looking much improved, replaced Principe Rojo Jr.\nAfter the previous match, Dorian & Konnan revealed Vampiro's new AAA contract allowed Dorian & Konnan to put him in matches whenever they wanted and Vampiro must wrestle in them or he'd be kicked out of AAA. Chessman won in seconds.\ntournament in honor of Dinamico I & II. Structure is unknown.\nTecnicso took 2\/3. Perro fouled Mesias in the third.\nWagners took 2\/3. Silver King turned on his brother in the third fall for the DQ.\nMuneca Sarcrmo & Karin Duval missed due to injury and illness.\nPrincipe fouled Bucanero as Infierno did the same on Andros.\nRudos unmasked tecnicos for the DQ.\nJennifer Blake replaced Sexi Star.\n3) Monster Clown, Murder Clown, Psycho Clown vs Halloween, Nicho el Millionario, Texano Jr.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakuez b/data_all_eng_slimpj/shuffled/split2/finalzzzakuez
new file mode 100644
index 0000000000000000000000000000000000000000..6ea2e7b11fb226b53a50f685e0ae9867fb8cbc67
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"For as low as $99, Bin To The Dump provides dumpster rentals and junk removal services for the Calgary and Edmonton communities and surrounding areas.\nWHAT'S INCLUDED IN THE DELIVERY RATE?\nThe delivery rate includes the drop-off and pick-up of the bin(s) at a single address. Please note that dump fees are an additional charge and are determined by weight and by the material loaded into the bin.\nFor more information please call our Calgary or Edmonton office.\nWHAT SIZE ARE YOUR WASTE BINS?\nWe offer 12 yard waste bins and 15 yard waste bins.\nARE THERE RESTRICTIONS AS TO WHAT I CAN'T PUT IN MY BIN?\nBin To The Dump specializes in the removal of junk from construction, renovation, and demolition projects. Additionally, the bins can be rented for seasonal clean-up as well as estate clean-up for both residential or commercial purposes. Please note that all Bins must be loaded level to the top or lower, an additional charge will occur for reloading the bin or returning for pick up.\nWe are prohibited from the removal of asbestos, hazardous materials, flammable materials or liquids, chemicals and paints, bio-hazardous materials, animal or human waste or remains.\nWe accept all major forms of payment, including MasterCard, VISA, PayPal, cash, and cheque*.\n*Please note that there is a $50 NSF fee for cheques.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vernisage of Isabel March at Fabryka Trzciny in Warsaw.\nHome\/Vernisage of Isabel March at Fabryka Trzciny in Warsaw.\nPASTISZE \u2013 The first photographic project by Isabel March. Portrait photographs refer to the most outstanding painting works. Model silhouettes, set design, as well as stylization and make-up are supposed to reflect the characters depicted in the paintings.\nVENICE \u2013 The magic of Venice seen through the eyes of Isabel March. Photographs are a summary of many trips toVenice. In her works, the artist wanted to express her fondness and admiration for this magical city.\nPAINTINGS OF SOUL \u2013 The main inspiration for Isabel March was the work of Andy Warhol and the character of Marylin Monroe. The artist combined these two inspirations creating amazing collages.\nThe exhibition lasted until February 20, 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Andrea is a Licensed Professional Counselor. A native Oregonian, she has experience in community mental health as well as in crisis counseling. Andrea works to understand the needs of the person and adapts her approach accordingly using a variety of evidence based methods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Students rallied on the Main Green on Wednesday to commemorate the 1968 black student walkout. From there, they walked to the Slavery Memorial on the Quiet Green and later the Pembroke campus.\nThis article is the third in a 50th anniversary commemorative series on the 1968 black student walkout.\nIn addition to reflecting on the protest that occurred 50 years ago, the students who organized the event demanded that the University fulfill the unresolved demands from the historic walkout. Specifically, students called attention to the fact that the representation of black students within the undergraduate student body is not at least proportional to the national percentage of black Americans, which currently stands at 13 percent.\n\"Institutions like (Brown), built on black people \u2014 through black people \u2014 are not representing us,\" said Abiola Makinde, a RISD student who attended the event.\nAfter gathering on the Main Green, walkout participants held a moment of silence before the Slavery Memorial on the Quiet Green before walking to Pembroke Campus, where they honored the legacy and contributions of black women to student activism at Brown.\nThe walkout organizers then invited all attending students of color to walk with them to the Congdon Street Baptist Church, where black students participated in community-building activities.\nWednesday's event allowed community members to commemorate the walkout and also reinforced the significance that the 1968 activism continues to hold for students at Brown today.\nAs the protesters risked their academic standing \"for demands that were intrinsic to their academic experience,\" they \"paved the way for black students (like) me to come to Brown (and) more people of color to come to Brown,\" said Roysworth Grant III '21.\nPresent-day students have found joy and solidarity through their involvement with black student organizations on campus. \"When I first came to this campus, the black community was so loving and helpful, and this was something I was so grateful for,\" Grant said. The Black Student Union is \"my home base, my sense of community and the people I resonate with on campus,\" he added.\nContinuing to make the University \"a truly inclusive environment\" should be \"everyone's responsibility \u2014 students, faculty, administration,\" Moncrieffe said.\nThis week, a group of black students identifying themselves as NEO published a list of 32 demands with the goals of promoting transparency, representation and a more equitable distribution of power. Within their list of demands, the students called on the University to increase the proportion of enrolled black students, with \"an explicit emphasis on descendants of American chattel slavery as well as black students from the City of Providence,\" among other measures.\nUltimately, the changes set in motion by the students in 1968 laid the groundwork for campus activism that would take place over the next 50 years. In their interviews with The Herald, alums who participated in the historic protest offered support, advice and understanding to younger generations of student activists.\nDespite the scale of the 1968 walkout, Ido Jamar '69, a Pembroke College student who helped organize the protest, said activists should not feel pressured to place themselves in uncomfortable situations for their work. \"Activism doesn't necessarily need confrontation to have a feeling that you've made a difference,\" she said. Instead it should acknowledge a problem and find \"a way to solve it,\" she added.\nBut activists must keep raising their concerns even when faced with pushback, said Bernicestine McLeod Bailey '68, a trustee emerita of the University. \"You have a voice \u2014 use it,\" she added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Production Floor Team Leader C\u00e9line can hardly wait for her new favourite MYKITA sunglasses to hit the stores. She has been part of the MYKITA team for seven years and still gets impressed by the amount of work and love that goes into each one of the products that leave the manufactory. For the fourth instalment of the \"Meet the HAUS\" series, we met C\u00e9line at her workbench on the first floor of the MYKITA HAUS.\nExplain what a production floor team leader does at MYKITA HAUS?\nOur team is responsible for assembling and adjusting the frames, making them symmetrical and shaping each one individually to fit the lenses. The production itself is a part of my work, but in addition I manage the group, collect information and am in touch with the other team leaders of the manufactory and colleagues from other departments.\nWhat is on your desk\/workbench right at the moment?\nA lot of things! My diary where I register almost everything; parts of glasses, including temples, fronts and hinges; pliers, bamboo sticks, glue, gloves, a computer and a cup of tea.\nA really good atmosphere in my team and at work in general gives me wings; this is something we manage well at MYKITA. Much of my time is also dedicated to actual handcraft. It's really important for me to still work on our product; I am a practical person and it lets me understand the whole process better. I like to work on a product that is constantly evolving; it fuels my own creativity.\nA bit of eccentricity never killed anybody.\nThe people. When I look around I see so many different kinds of persons from different places and professional backgrounds, creative people. It's really motivating to see how everybody gets along, laughs and makes friends.\nBRADFIELD, the latest frame from the collaboration with Damir Doma. I was really happy when I saw this frame in the hands of my colleagues. In general I like glasses or items with style and character. This frame is the perfect design evolution of this collection. I also know how much work goes into making it, just beautiful.\nSophistication, quality, technique, purity and chic.\n\u2026chef. I spend lot of my spare time making cakes and treats. They are a bit different than what you get from the bakery around the corner. They taste and look great, and make you feel amazing \u2013 super decadent desserts with zero guilt. I discovered raw cooking over a year ago and I'm still crazy about it. It's magic. Birthday anyone?\nTake a closer look at the modern manufactory in a photographic tour through the MYKITA HAUS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The best roofing companies offer the best services. At Roofing Service Pro's, we provide all-around roofing services to the Wareham MA 02571 area. And our roofing services are performed by the best roofing contractors in all of Massachusetts. Our team of qualified and highly skilled professionals is trained and certified using the best equipment and tools as well as knowing the latest roofing techniques. We only use the best brands and high-quality materials available on the market. Each of our roofing contractors is licensed, insured, certified and bonded. Keep in mind, your existing roof, or the new roof you'd like installed is an investment in your home, one you need to increase the property value of your house as well as to protect your home, valuables and loved ones.\nThe roofing services we provide to the Wareham MA area here at Roofing Service Pro's include the installation, repair services, and maintenance and cleaning of your roof, gutters, and anything on your roof. When it comes to new roof installation, we work with the latest and greatest materials available on the market, no matter what style of roof you want. Asphalt shingles, synthetic slate shingles, wood shakes, metal shingles, or any other type of material available. Always perform diligent research before deciding on the type of roof you want, on the company you want to do the work, and on any of the services (like maintenance and cleaning) that you desire. We believe our customers are always right, and your satisfaction is of paramount importance to us.\nIf you require roofing services for your Wareham home, then Roofing Service Pro's is the company for you. So give one of our roofing services experts a call at (978) 577-5300. We will be glad to answer any and all your questions and offer you a no-hassle, free estimate.\nGetting roofing services is essential, particularly for all kinds of roofing services can improve your roof to provide you optimal protection from environmental elements. For this reason, getting commercial roofing or residential roofing services from roofing companies in Wareham is necessary to ensure that your roof is in mint condition. There is a wide selection of services that numerous roofing companies provide, ranging from roof installation, repairs, and many others.\nYou ought to bear in mind though, that roofing services are not restricted to roof repair alone, it might likewise include services on related devices like shingle maintenance, downspout cleaning, and many others. No matter the level of services you need to get from a contractor, getting quality roofing professional service is vital nonetheless. For this reason, before you call contractors who can supply commercial roofing or residential roofing services, you must understand ways to choose the best business there is, initially.\nRoofs have a particular shelf life, depending on the materials utilized and the way that these roofs were installed. Typically, you can extend the durability of your commercial roofing or residential roofing through correct maintenance services supplied by premier roofing companies with years of experience in their expert portfolio.\nFor this reason, it is advised that you read or investigate the roofing services a professional might offer, particularly because such services offered by roofing companies in Wareham MA can be highly helpful.\nNo matter what the degree of commercial roofing or residential roofing services you require, getting budget-friendly yet good quality services from roof contractors today is vital. Even if it is simply a small shingle repair, it can still add to the long-term durability of your roofing system. You might be puzzled with the terms here, specifically since many roofing companies may drop the terms \"commercial roofing\" and \"residential roofing\".\nTo offer you a concept, commercial roofing services, and residential roofing services are basically the exact same, but commercial roofing usually needs industrial scale maintenance. Regardless, you need to discover roofing companies in your area so that your relied on roofers can help you do the job efficiently.\nBeware which roofing companies you call, given that not all of these roofing companies in Wareham MA can be trusted. In getting residential roofing services (or even commercial!), you need to make sure that your choice will be serving you in your area with the very best deals that they can offer. Furthermore, they must be supplying you some diagnosis and estimates of the degree of damage and the corresponding service (and expense) for each.\nAs an idea, read client reviews first regarding your prospective roofing providers so that you can get the most out of such services today.\nIf you wish to enhance the performance of your roof, it is recommended that you implement strategies which can assist you do so. Correct roofing maintenance with the help of a pro in this arena can assist you do that to accomplish your objective of your roof. All you have to do is to get in touch with the best service provider in your area and get their diagnosis for your roofing.\nBefore closing a deal, you can try and bargain for prices, however don't overdo it, as this may jeopardize the quality of the roof servicing you will be getting.\nWareham is a city in Plymouth County, Massachusetts with a total population of approximately 10,481. Wareham, which uses the 508\/774 area code, is located at 41.7666, -70.631 at an elevation of 50 feet. There are over 4,228 households and on average there are 2.43 people in each household with a median age of 43.6. The average income in the area is $54,221 and the average home value is $259,600.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've just now gotten started with using a project instead of just a file. I'm having issues figuring out the order of how it all flows together. My main issue right now is that I created a help form separate from my main form. I can't seem to perform the Call-Help_Dialog_pff to be defined before I call it. I try to call the form from the Globals.ps1 file, but it doesn't appear to be defined yet. At what point is this defined\/how can I make sure that it's defined before I call it?Is there any doc to describe how primalforms will flow through the Startup.pfs, Globals.ps1, and the defined forms?\nA few programming style questions.\n2) Is it possible to add parameters to the Main function for parameter passing from the command line, or do we have to manually parse them out of $Commandline or from $args?\nParameters should be defined in the Startup.pfs at the top. their packager will look for it there and make sure it's at the top of the script when it's compiled.Functions in Startup get put in first, then functions in the global. The it appears that the commands in global get executed. I find it difficult to see this in primalforms since the functions I put in other scripts don't show up as callable in the other scripts. So, I've essentially been putting all functions in startup.pfs and any commands in the Main function. that way it knows about the functions. I then just make sure that if the variable is needed outside of the main function that I set it to a script variable.\nOh, and also you will find that the parameters are only valid if you export to a .ps1 file. If you need command line options you have to use the $commandline. Or if you have something that might be both (like mine), then you need to make sure to code for both.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"sorry for my absence, it's been a somewhat stressy couple of days, but now i can finally unwind- i'm on vacation in switzerland in ticino, where i grew up. so many memories! this place is beautiful, every time i come back i'm reminded of how blessed i was growing up in this little patch of heaven.\nyesterday we went to have lunch at an amazing grotto (not what hugh hefner has- a grotto in italian is a small rustic restaurant) in montagnola, a suburb of lugano situated on a hill over the lake- beautiful area! pictures and a report to follow. fun facts about montagnola: the writer hermann hesse used to live here, and george harrison (the beatles) used to spend his summers there in his mansion.\nwe also passed by tasis, my high school, and needless to say the memories almost got me choked up! the campus became so much bigger and impressive! i can't wait to meet some of my friends that now work there as summer camp tutors!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Laprairie, a-1: 11 of 11 is 100.0%.\nLaprairie, a-2: 8 of 8 is 100.0%.\nLaprairie, a-3: 13 of 13 is 100.0%.\nLaprairie (Village), b-1: 16 of 16 is 100.0%.\nLaprairie (Village), b-2: 14 of 14 is 100.0%.\nNapierville (Village), g: 0 of 16 is 0.0%.\nSaint-Constant, c-1: 9 of 12 is 75.0%.\nSaint-Constant, c-2: 0 of 9 is 0.0%.\nSaint-Constant, c-3: 0 of 11 is 0.0%.\nSaint-Cyprien, h-1: 0 of 15 is 0.0%.\nSaint-Cyprien, h-2: 0 of 15 is 0.0%.\nSaint-\u00c9douard, i-1: 0 of 9 is 0.0%.\nSaint-\u00c9douard, i-2: 0 of 14 is 0.0%.\nSaint-Isidore, d: 2 of 27 is 7.4%.\nSaint-Jacques-de-Mineur, e: 19 of 28 is 67.8%.\nSaint-Michel-Archange, j-1: 0 of 14 is 0.0%.\nSaint-Michel-Archange, j-2: 2 of 15 is 13.3%.\nSaint-Patrice-de-Sherrington, k-1: 0 of 14 is 0.0%.\nSaint-Patrice-de-Sherrington, k-2: 0 of 8 is 0.0%.\nSaint-Patrice-de-Sherrington, k-3: 0 of 8 is 0.0%.\nSaint-Philippe, f-1: 0 of 21 is 0.0%.\nSaint-Philippe, f-2: 0 of 14 is 0.0%.\nSaint-R\u00e9mi, l-1: 0 of 12 is 0.0%.\nSaint-R\u00e9mi, l-2: 0 of 7 is 0.0%.\nSaint-R\u00e9mi, l-3: 0 of 11 is 0.0%.\nSaint-R\u00e9mi (Village), m: 1 of 22 is 4.5%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than anything else, I think their operating and gross margins will be the focus when they report Q3.\nNow, to be clear. I am not one to shy away from trading growth type names. I have long been an Amazon (AMZN - Get Report) shareholder, and have been in and out of both Alphabet (GOOGL - Get Report) , and Netflix (NFLX - Get Report) often over the years. I don't remember, however... ever having a long position in Facebook (FB - Get Report) . A quick search of my accounts confirmed that fact, though apparently I have been short the name going way back and have often played the firm's earnings releases through the options market. Facebook will report their third quarter after the closing bell on Tuesday, October 30th.\nConsensus expectations are for EPS of $1.47 on revenue of $13.8 billion-ish. That revenue number would be good enough for 33.5% year over year growth. despite the firm's warnings in the wake of the July earnings call. The firm also warned at that time on rising expenses. Now, operating margins have been in the mid 40%'s, while gross profit margins, though in recent decline, were still up in the 90%'s. That said, I think these margins will be the focus in two weeks more than anything else.\nNews made the rounds yesterday that the state treasurers of Rhode Island, Illinois, and Pennsylvania had teamed up with the comptroller of New York City to side with a proposal filed back in June by Trillium Asset Management requesting that the firm make independent the role of the chair from that of the chief executive. Though, this firm has stumbled badly through a number of very publicly embarrassing events this year, the facts are that this proposal will go to a shareholder vote in the spring, and that Mark Zuckerberg controls 59.9% of the vote. So, the two roles only will be divided if Zuckerberg feels like it. Nuff said.\nFYI, I screw around with charts all the time. 99% of my charts are never made public. That said, I was messing around with Facebook last night,, and whoa... something I thought that I would bring to your attention frankly, because I know a lot of my readers trade this name.\nWe all think we know that the FANG names often trade together. Not true at all. Year to date, Netflix is now up 80%, Amazon has returned 55%, Alphabet, a mere 7%, and Facebook has looked like a rented mule running against thoroughbreds at -10%. For those, who include the double \"A\" in their acronym, Apple (AAPL - Get Report) has run 30% this year.\nNow, look at the Fibonacci Time Sequence. If you line this thing up with the major selloff in March, it times nearly perfectly with the major selloff in late July. This sequence also indicates that the recent bottom may have been a major turning point as well. My thought is that if long these shares, a trader has a serious chance at selling them at $175, which would be a 38.2% retracement of the top to bottom July through October move.\nFor my novice options traders, this is a simple strategy known as \"Bull Call Spread.\" Basically, this is a method used to bet on an upward move in an equity, but in a way that greatly reduces the capital risk. The sale of the higher priced call limits the potential profit if the stock does rise above that strike price, but also helps subsidize the loss should the direction of this gamble fail.\nBottom Line: In other words, in this example, the trader is risking a little less than $500 to try to net $1,000. Can't do worse. Can't do better.\nAt the time of publication, Stephen Guilfoyle was Long, AMZN, AAPL,, Short AMZN, AAPL put options.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"First-time vacationers, budgetarian travelers, and wanderlust junkies get a step-by-step guide in touring the world on a budget from Kapamilya actor Enchong Dee in his book \"Basta Trip Mo\" by ABS-CBN Publishing.\nThe \"Basta Trip Mo\" guidebook is divided in two parts and contains detailed steps in applying for a passport, booking a flight, finding good accommodations, and creating itinerary in the first portion. The actor also shared his travel essentials and expert packing tips and also gave a detailed checklist for reference.\nThe second half of the book, meanwhile, bares Enchong's travelogue\u2014his favorite destinations in Asia that readers can visit even with limited budget. These cities include Hong Kong, Taipei, Bali, Tokyo, Osaka, Beijing, and Seoul.\nThe athlete-turned-actor also included tips on touring Vietnam, the very first country the \"Basta Trip Mo\" author visited outside the Philippines for a trip he took as a 13-year-old swimming representative of the country's Junior National Team.\nTake the first step to budget traveling with Enchong's \"Basta Trip Mo\" book, now available in leading bookstores, newsstands, and online stores for only P225. For updates, follow @abscbnpr on Facebook, Twitter, and Instagram or visit www.abscbnpr.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About 200 participants of nearly 30 countries among them parlamentarians, scientists and representatives from companies and NGOs met in the historical venue, the Chamber of the old waterworks, in Bonn. They exchanged their experiences and knowledge about new developements in the field of Renewable Energies as well as the best political instruments for their implementation.\nIts precursors were the World Forum for Renewable Energy (hosted in Berlin in 2002 and in Bonn in 2004) as well as the International Parliamentary Forum that took place in parallel to the International Governmental Conference renewables 2004. The Second World Assembly combined two formats. The conference presented a unique opportunity for advocates of renewable energy to exchange ideas with parliamentarians belonging to the newly established World Renewable Energy Parliamentary Network and to learn about and develop collective strategies how to best promote renewable energies worldwide. The Second World Renewable Energy Assembly presented solutions for an accelerated global introduction of renewable energies and defines the necessary policy instruments. Central focus of the conference was the global target of a 100% supply of energy that comes from renewable energy sources. The conference named the obstacles that have yet to be overcome and at the same time showed how the target of \"100% Renewable Energy globally\" can be attained and how the public can be motivated towards achieving this goal (\"mission possible\").\nSession I looks at the necessary paradigmatic changes and provides status reports from different regions of the world. Session II displays a range of different reasons for the urgent and global shift to renewable energies. The conference will show that climate change is not the only problem that necessitates the shift to renewable energies: Culminating shortages in fresh water supply in many regions of the world, damages to people's health, induced by the deployment of conventional energy sources, and the growing threat of international political tensions call for a fundamental change of the global energy system. Furthermore, this session will demonstrate that the shift towards renewable energies has clear advantages - also economically - over current approaches of CO2 capture and storage and the global emission trading system.\nSession III addresses convincing integrated scenarios as well as projects that are already under way, both aiming at a 100% supply with renewable energies.\nSession IV presents the necessary \"next steps\" that will ultimately lead to a global breakthrough of renewable energies.\nThe Second World Assembly wants to give clear impetus to prioritising renewable energies in the national and international energy debate. The conference will make clear that the shift towards renewables will not come as a burden for the national economies but as a unique chance. If the adequate policy instruments are applied in a consistent and competent way, this shift will be accomplished much quicker than most might expect.\nThis congress will bring together ISES members and professionals in solar and renewable energy from around the globe, providing a platform for knowledge exchange. In the last years, China has seen a rapid development in solar and renewable energy. This congress presents an opportunity for Chinese professionals and their international colleagues alike to share their knowledge and experience to the benefit of all participants.\nThe European Photovoltaic Solar Energy Conference and Exhibition series starts in 1977. These events grew to become the World's largest specialist Conference in the Photovoltaic Solar Energy sector. The PV Conference & Exhibition (EU PVSEC) is a global event which combines scientific and technological issues. The EU PVSEC brings together research and development experts, industry, utilities, political actors, architects and end-users. The international collaboration, which the EU PVSEC facilitates, is essential in fostering a sustainable future for PV technology in the global energy system.\nDuring the last few years, Bioenergy has moved from the sidelines of the energy debate to the forefront. To a very large extent, Bioenergy can address the issue of fossil fuel replacement. Politicians all over the world have accepted this opportunity. Instead of a specialised term for experts, bioenergy has become a common term in political discussions. New legislative actions, policy documents and strategic analyses are constantly being prepared and adopted. The conference in May 2007. which is officially supported by the WCRE will review the developments at all levels, including policy, markets, industry, technology, research and education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After taking Alicia's Quickbooks class, she worked with me one-on-one during the holidays to launch our new Quickbooks for the New Year. Between the instruction manual she created specifically for our Quickbooks and access to her tutorial videos we are well on our way to helping our business grow! We highly recommend you to other business owners. CALL ALICIA!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Time after time I've played against pugs and time after time I win. When I'm on the Hunter team with pugs it's almost embarrassing how badly we lose or how bad the monster is. Even at high level games (and yes I know level doesn't mean skill but it means some experience) it's almost disappointing how people still don't play well or even bother communicating.\nThis is just one of the many games I've played against pugs and it's embarrassing for them. I've easily done Stage 1 wins against pugs without breaking a sweat.\nIt won't change, it's the nature of this type of game.\nIn BF4 almost the entire team are randoms but yeah I get what you're saying.\nLol are they? They're worst than bots sometimes and if I want practice I play customs against my friends.\nLol if you think about it its easy character masteries.\nLol I feel bad for pugs sometimes, they come into the game to get farmed.\nMy masteries I farmed defend, nest, and hunt on solo mode.\nLol yeah solo makes everything easy for masteries except Slim. Slim you need Custom games with people ._.\nSometimes pugs do really well, other times pugs do not do very well. It's the circle of life.\nToo bad for you I guess.\nI only play pugs maybe get invited to play with someone whos added me once in awhile. Pugs really aren't that bad. Sometimes you lose but most of them know what they are doing.\nIf they have mics it's possibly a bit better but sometimes there's that one player who won't listen. Your team is only as good as it's worst player.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This research effort was intended to determine the perceived value that continuing professional education has on the careers or practices of certified public accountants licensed in Louisiana. A researcher designed survey incorporated both Likkert type scale responses and open ended questions. The survey was electronically administered to the membership of the Louisiana Society of Certified Public Accountants. The survey focused on the perceived value of and attitudes toward continuing professional education, as well as the desirability of specific methods of delivery. The results of the survey indicated that certified public accountants believe continuing professional education has value. The results also show that certified public accountants have a positive attitude towards continuing professional education. In addition, certified public accountants considered themselves an expert or having an area of specialization in accounting. Two methods of delivery were perceived to be desirable based on two dimensions, effectiveness and preference. Both methods of delivery, \"Practical workshops in which there is hands-on learning,\" and \"Small group tutorial sessions that are interactive between the expert and the audience\" rated \"Somewhat\" on a five point scale including \"Not at all effective\/ preferred,\" \"Not very effective\/ preferred,\" \"Don't know\/Neutral,\" \"Somewhat effective\/preferred,\" \"Very effective\/preferred.\" Although age and years in practice were two factors found to be significant between certified public accountants that are employed in public accounting and those certified public accountants in non-public employment, no other categories were found to be significant. In addition, no difference in perception about continuing professional education exists between certified public accountants that consider themselves an expert or have an area of specialization and those certified public accountants that do not have an area of expertise in accounting.\nMeyer, Jean Annette, \"Continuing professional education and its impact on the practices and careers of certified public accountants\" (2007). LSU Doctoral Dissertations. 662.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..0c9325c8ab9a6f50ec9029776dc98200d005ed1c
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+{"text":"This entry was posted on Thursday, February 15th, 2018 at 19:51\tand is filed under Aktuell, BioEco 2018. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, a new level introduced today on BetterMost... the six star 1000+ Posts Club. It's astounding, but we actually have a half dozen users nearing that magic number is just three months here. If we have more people on here, I'd consider a Mile High Club forum with free drinks for members.\nand BBM Lines Visualised is getting to 800 posts!\nI just re-read Phillip's post and he is talking about individual member posts..oops!\nWell, I'll work harder too! Lol!\nI suddenly feel so inadequate. I'm going to work harder!\nWell, I guess you'll see me around, then, hunh? I want those free drinks!!\nYaadpyar! You are in the top 10!!\nCongrats to you posteristas. I'm wimpy. But see, I'm workin on it.\nI want to join the special club 1000+ posts club though. I just know it's going to have great comfy chairs, great lighting, excellent food, and great company. I wonder if we'll have a secret password?\nWill will get to watch BBM on IMAX every night? The anticipation is killing me (and now I've added one more post!).\nwell done guys! what would be interesting for you to put on phillip is how long they ahve been online to make those posts!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We review the Kraken X62 Series LCS cooler from NZXT. An AIO liquid cooling solutiob for your CPU and cases with 280 mm radiator mounts. This updated revision CPU cooler is compatible with NZXT CAM software. The software allows you to program, customize and monitor this all-in-one liquid cooling solution for your processor. Armed with two 140 mm PWM fans and a nice 280 mm radiator the performance should be really good, and with the CAM software you can monitor temperatures, adjust cooling performance and customize LED lighting directly from your desktop. You know, in the world of CPU coolers nothing ever stops developing. These days you can pick a hundred different heat-pipe based coolers, where many are shaped, formed and priced the same. The better heatpipe based coolers are good though. Next in line are LCS systems (liquid cooling), the entry level products are affordable, easy to install pre-fab liquid cooling kits. We've seen and tested many of them as Corsair, Asetek, Cooler Master and yes NZXT and so on all have interesting kits.\nThe bigger problem is cooling performance. The gear behind these kits are good yet the performance is often battling a small included 120 mm radiator with thin tubing and lacking coolant levels due to missing reservoirs. To compensate FAN RPM goes up making a good looking product noisy again. With that in mind NZXT started to develop a new Kraken series coolers. The product we test in specific is the Kraken X62, an updated design with new cooling block and regulated control for pump and fans. Next to that what's interesting is that this product is based on a 280 mm radiator which gives it quite a bit of cooling surface area. It is a cooler series that is easy on the eyes and comes with a relatively tool free installation. Next to some very good performance, the product is compatible with NZXT CAM software. You hook up the LCS unit towards a USB port and then the software allows you to program, customize and monitor this easy to install liquid cooling solution.\nAnd guess what, NZXT took it even further as the Kraken X62 has a new RGB LED concept, the most wicked one you probably have ever seen, and it is very cool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unbeatablesale.com, one of the leading retailers of online retail products has the Cardinal Pub Group sat you've been looking for! Unbeatablesale.com has over 700,000 products in stock with 2 of them being Cardinal Pub Group sat and related items.\nBesides our wide selection, we also offer guaranteed best pricing on Cardinal Pub Group sat, with items starting as low as $14.82. Most importantly, our vision includes hands-on customer service making sure each and every customer is treated royally -- completely taken care of to his or her satisfaction. Shopping for a gift and can't decide what to buy? No problem! We offer gift cards so you can shop hassle-free and always get the perfect gift. Shop Unbeatablesale.com with confidence today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The theme of the articles in the Winter Solstice 1999 MAG E-Zine was a forward look into the new millennium and the part that geomancy would have in the development of peoples perspective on life and their understanding of creation as a whole. How we bring together the spiritual and the physical under the same mantle of understanding. At the moment it would appear that at the best the scientists and the leaders of religion are in the main standing with their backs to each other ignoring the others viewpoint. However in a few cases there is at least recognition that any one side does not hold all the answers. Those of us who can glimpse behind the curtain of the world of what we call reality, through dowsing, can see that both views are equally important and that singly they can not survive. The physical can not exist without the energetic or spiritual and vice-versa.\nI also want to go back a few issues to an article by Sig on \"Dowsing in Sacred Space \u2014 The New Realities\" where he highlighted the different perspectives and findings that dowsers generally get when surveying the same location. This confusion amongst dowsers, although generally acceptable to them, does nothing to build confidence in those who have not as yet been initiated into the brotherhood or, to be politically correct, should I also say sisterhood as well. I agree completely that it is likely that dowsers will continue to have different experiences and different findings and that perhaps it is not important that they do find the same. But I think that it is important that we attempt to understand why things spiritual and energetic seem to be in the main unpredictable whilst those in the physical realm predictable. However it may not be as black and white as I have made it sound as present day physicists are at times puzzled at the unpredictable nature of some of the results that they are getting. It is now becoming clear to them that the mind can indeed influence the result. Can particles and subsequently matter be influenced by thought? This is now a burning and pertinent question.\nAs dowsers we know that this question has been around for some time on our side of the fence and it is generally accepted that the mind, at energetic levels of existence at least, can have great influence and can make changes of a significant nature. Many dowsers now go further than that and can provide anecdotal evidence that the mind can make changes on the physical and material level as well. Healing and diversion of underground water are two examples of this. However I feel that we must be careful in that we do not come to the conclusion that the mind is all-powerful and that the physical does not matter. Geomancy, like Feng Shui, is basically the art of placement in such a way that both material and spiritual things are brought into harmony. This recognises the influence of the one upon the other and equal importance for both. It has been with this approach that I have participated in dowsing as an aid to provide that harmony within our environment. I have studied in detail the result of what nature has done and what man intentionally and unintentionally is doing as he passes through on his journey of life.\nWhat I want to do briefly is to highlight some of my findings that may be relevant to the discussion in hand. Nicholas Mann (Beyond 2001, Contact & Apollo 13: The Dynamics of Space Revisited) contends that the energies that we experience on earth extend to infinity. I quote.\nThis is indeed a view that I fully subscribe to and I will explain why I arrived at that conclusion. If we follow this thought through to it's logical end then it has significant importance.\nIn looking at the effect on the dowsable energy field, and lines within that field, through the placement of objects I observed many things that were of interest and, I think, allows statements and projections to be made. One is that every piece of matter, large or small, has the ability in most circumstances to generate energy that is displayed in a field around it. It is this ability that produces what we refer to in dowsing terms as \"Earth Energy\". A dowsable field that envelops all the earth, something, I believe, that all dowsers agree upon. It is when we examine in detail this energy field that dowsers begin to disagree and find different things. The problem for us in our examination of the earth's energy field is that the earth is so large and we are so small and we can only examine one small area at a time. We are much too close to our work; in other words we can not see the wood for the trees.\nIf we had the opportunity to go on an Apollo mission then perhaps we would have the proper perspective from which to see what was actually happening. Not having that advantage I took the other alternative and looked at the energy fields generated by small objects that we find all around us. Due to my interest in things megalithic I mainly used stones, not megaliths, but of a size that I could easily move around. To move them around is indeed important to understand what is happening at world level. We should always keep in mind when dowsing earth energies that the earth is far from stationary. It is whizzing around on its axis one revolution per day with a slight wobble and orbiting around the sun at an alarming speed and again not in a perfect circle but one that is elliptical in shape. Add to this the movement of other things like the moon, which as we know, has great influences on what happens on the face of the earth, the planets, the constellations and everything else within creation. We see that the relative position of the earth to its surroundings keeps changing by the second. Therefore it is essential that we are aware of the consequences of these movements.\nFirst it is interesting to examine in detail what the composition of an energy field is. Is it like a cloud that has no particular structure but just a mass of different particles separate from each other? At first glance it may be like that. However it is from this that we can begin to understand what makes an energy field what it is, a live force. I found that every object that I examined through dowsing had a basic field or aura around it. That seems to be the case no matter how small or large the object was. Close examination showed that within every energy field there were features that displayed themselves in lines. Some of these lines were straight, others curved and formed spirals and other geometrical shapes, some of them archetypal in form. What I found when I placed two or more objects in various relationships to each other that the individual fields around the objects could at times change and instead of a field, lines would be created in its place. Again dependant on the accuracy of the relationship of the objects to each other the lines would be either straight or curved. This highlighted to me the importance of relativity as to the form that an energy field takes. If we accept that this ability to produce a field of energy extends down to particle level then we must look at the effect of the composition of materials and thus all objects. It would therefore be inherent in all materials due to the precise relative position of the atoms and molecules that constitutes it that straight lines, spirals and other shapes be present in the energy field that surrounds it. This also explains why I find that in all energy fields there is a similarity in the display of lines etc. that are present. The similarity increases with the purity of the make up of the object.\nIf this is true with regard to a few small stones then it must also be so when we look at the earth as a whole. The earth is a conglomerate object made up of millions of different substances and materials with the same opportunities of chance relationships existing in that make up. Therefore it should be expected that, when we dowse, we find straight lines, spirals, vortexes, torroids and many other designs and shapes in the earth's energy field. It also should be expected that these would be in a constant state of flux and movement due to the non-static condition of the earth within the universe. If the earth is endowed by an energy field due to the reasons that I have given then it would follow that all the heavenly bodies that we observe in the sky would have similar fields with similar lines etc. Another observation that I made was that all energy displays are three-dimensional. This means that lines of energy from the earth are being projected into space and similarly so those from the planets and stars are being projected toward earth. The fact that energy from these bodies is discernible on the face of the earth is well known. If energy from the sun did not reach here we would not exist. I also observed that all energy fields that I dowsed around small objects are energetically linked and this chain of linkage continues up the scale until all things on earth are joined. If that were so on earth then why would it not be so on a grander scale. We express the opinion that there is a great distance between the earth and those other bodies around it. That may be so from our small perspective but if we could get far enough away so as to see all of creation then these distances would be minute. It would be unreasonable to suggest that the earth is unique in that respect and that these energies that we experience are confined to its surface alone.\nIt is not only the form of the display with regard to the shape that it adopts, that is, lines, spirals etc that we are interested in. It is also what it does to all animal and vegetable life on the earth. Because of that curiosity, when we dowse we tend to ask questions about its nature. Is it negative? Is it positive? Is it beneficial? Is it detrimental? And many other things as well. Why should these differences occur? To answer that in detail would take up too much space but I will attempt to give a brief account of what I believe. This belief has been formed after years of observation.\nI talked earlier about the basic energy field. Now I want to refer to what I believe to be the basic energy line. What I have found is that an energy line, as we know it is of composite structure. In many ways like a piece of rope which is made of many short fibres. When we dowse and look for the detail it can be observed. It is the way that these basic energy lines combine to make a larger line that dictates the nature that it will adopt and determine as to whether we will dowse it as belonging to any of the classifications already mentioned.\nIn most cases these lines are not found in isolation but join up in pairs with a short overlap at the joining ends. Where the joining ends are both negative the line becomes predominately positive with just a small dot of negative energy at the centre.\nWhilst a basic energy line may wish to be straight, due to the relative position of the molecules in the make up of the generating object, pressure is exerted that in most cases causes them to form a spiral at each end. When this occurs the following overlapping pattern appears.\nThis overlapping of positively coupled basic energy lines causes the longer line to become dowsable as beneficial in nature. The opposite is the case where the coupled lines are negative. This causes the longer line to be detrimental. Where negative and positive cross the result is a neutral line. If the spirals on both sides of the longer line are equal in size then the longer line will be straight. If the spirals on one side of the longer line are more coiled up than the other side, then the line will curve towards that side. The degree of curvature will depend on the amount of difference between the size of the spirals on both sides of the long line.\nWhen basic lines are not coupled, if they form spirals, each end will be counter in direction to the other as seen in the sketch and one spiral will be positive and the other negative. This is not a common occurrence.\nI am also in agreement with Nicholas Mann where he refers to polarities and opposites. I have already dealt with that in part when I describe energies as being positive or negative. As I have already said this is just a suitable way of describing opposites. When we talk about achieving balance in whatever we may be doing and look on this as our objective we are accepting that there is a necessity for opposites to be present in equal quantities. Therefore when we talk about balancing energies what do we mean. Is it an equal amount of what we describe as positive and negative or is it something else. From what I have said that would achieve a neutral position. Is that better than a state of beneficial energy?\nPerhaps balance does mean something else. A few years ago I became aware through dowsing that there is another classification of energy that is not generally known. I found lines of energy that would not answer to any of the standard terms such as those that are usually used. This type of line did not respond to the questions \u2013 is it beneficial? Is it detrimental? Is it neutral? Yin? Yang? and so on down the list. For some time I was puzzled until one day I put the question was it was 'minus' or 'anti' energy. To this I got a yes. I now believe that this is the energy that creates balance. When I looked carefully I found that most places where I dowsed the presence of a 'normal' energy line one of these 'minus' (not to be confused with negative) lines closely shadowed it. However in some locations I found that was not so. This was brought about through an unusual underground water feature that caused the normal energy to be separated from this minus energy and drawn down into the ground at that location. The minus element was then beamed across the ground until it terminated some few miles away. This line of energy moved slowly back and forth like a windscreen wiper of a car taking several weeks to complete one sequence. The presence of this occurrence is dependent on the geology of the area making this more rare in some areas of the country than others. It is something that can again be simulated by accurate placement of objects.\nYou may wonder by now where all this is leading. The various changes that can occur in these subtle energies of the earth can be caused by nature or mankind through his actions whether they be intentional or otherwise. There is the further complication I mentioned at the beginning in that man can also influence these subtle energies by his thoughts. Again this can be intentional or otherwise. So whilst I have put forward the suggestion that these energies do indeed behave in a rational and predictable manner it is impossible to be aware of all the different influences bearing down upon them. Therefore it is easy to see why different dowsers get different results. More interesting and important is the effect of these energies on life as a whole.\nAn interesting theory can come out of all this. If we are prepared to accept that life on this earth, animal and vegetable is dependant to some degree on the energetic side of our existence then any change in that energetic state would result in changes gradually taking place in the physical state. If variations exist within areas and locations and energies are not evenly distributed could this have a part to play in the evolution of life on earth? Why did certain species develop in one area and not in another? From this we could see that evolution might be affected by natural geological occurrences, but also more importantly by human activity and disturbance. The same possibly could be said about creation itself. How much does the energetic interaction between all the heavenly bodies alter the shape of creation? We look on creation as something that happened as a one off thing a long time ago. Is it not ongoing and how much does what we do today play a part in what things will be tomorrow? In other words are we helping to create our own future? Will we finish as our own slave or our own master?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Prof. Dr. Christoph Scherrer is a professor for Globalization and Politics in the Social Science department, Kassel University. He is also head of the International Centre for Development and Decent Work (ICDD) at Kassel University.\nEnno Schmidt is a painter, author and film maker. Since 2006 he has worked to raise awareness and promote the concept of unconditional basic income.\nTatjana Turanskyj is a filmaker, author, producer and partner in turanskyj&ahlrichs*** GbR. She is currently working on the final part of her widely acclaimed \"Women & Work Trilogy\".\nDipl.-Psych. Georgios Athanassiou works at the Institute for Work Science and Process Management in the department of Organisation Psychology at Kassel University.\nProf. Jens Brand is an artist and teaches Fine Art at the Art Academy, Kassel.\nBirgit Effinger is an art historian and freelance writer. She runs the Goldrausch ArtIT programme for the professionalisation of women artists in Berlin.\nCordula Hamschmidt is the programme coordinator for Literature and Humanities at the House of World Cultures, Berlin and former academic advisor to the German Federal Parliament.\nMalin Kuht studies Politics, Economics and Art Education at Kassel University and the Academy of the Arts, Kassel.\nDr. Karin Rosing is a junior professor of Business Psychology at the Institute of Psychology, in the department of Human Sciences, Kassel University.\nBabak Samari is a guest student from Iran studying Fine Art at the Kassel Art Academy.\nProf. Martin Schmitz is a publisher and guest professor for Product Design at the Kassel Art Academy.\nLu\u00adcy Pow\u00adell is an artist and ini\u00adti\u00ada\u00adtor of the Satel\u00adlite Sa\u00adlon for art sci\u00adence con\u00adver\u00adsa\u00adtions in Ber\u00adlin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VICTORIA \u2013 Riding the momentum of capturing a tournament title in their last competitive action, the University of Victoria Vikes women's golf team used solid scores throughout their lineup to sit in second-place after Day 1 of the Battle at Primm, in Primm Nev., on March 19.\nCadence Hellmann and Hannah Diamond each had a great start to the tournament for the Vikes women, shooting identical rounds of 74, good enough to tie them for second-place individually.\nMelanie Murchison and Nicole Gargarella each sit in a tie for 13th after solid rounds of 79, while Quinn Fitzgerald rounds out the Vikes scorers in 30th place.\nOverall the Vikes score of 306 on a course that is playing long and wide was good enough to send them into the second day of competition with some highly-ranked teams in their rear-view mirror.\nThey are five shots clear of the No. 5 team in the NAIA from Texas Wesleyan, eight shots ahead of No. 10 ranked Embry-Riddle and ten strokes in front of No. 11 Rocky Mountain College.\nWith one round left to play the Vikes trail only the University of British Columbia Thunderbirds by two strokes.\nOn the men's side, an opening round of 285 was followed up with an improved score of 280 in their afternoon round.\nLawren Rowe and Mike Griffin had identical opening days, shooting rounds of 72 and 69 to sit in a tie for 28th.\nJacob Hayes is just one stroke behind that pair after rounds of 68 and 74, while Bruce Tomie and Josh Miller round out the scoring with tallies of 143 and 146 respectively.\nAs a team the Vikes will need some help to make a big move on the second day of competition, they sit seven shots behind Johnson & Wales University in the battle for seventh-place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He has 16+ years experience in finance, through high level positions in the hospitality sector and energy sector. He worked as pre-opening Director of Finance for opening the luxury hotel brand, The Ritz Carlton, in Ireland and Kazakhstan. He was responsible for payments, banking, corporate legal strategy, implement Corporate SOP's, supplier contracts, managing pre-opening funds, capex, installation of hotel systems and hire the right team. During his tenure as DOF in year 2013 he won Marriott Mustang European Director of Finance award.\nHe is a board member of the British Irish Trade Association - this organisation strengthens the relationship and trade between UK and Ireland businesses (SME & Larger Business).\nKJ takes a vision and makes it reality through sound strategy & execution development. He intuitively sees the threads of opportunity that wind through an organization, brings them together into a coherent whole, helps others extend their thinking, and drives material business advantage. He is an inspirational leader who tells stories that inspire action while at the same time is grounded in Marketing, Business Development & Information Technology that levers the business. Respected as a credible voice in decision making, product development, marketing strategies and establishing governance boundaries, KJ earns a seat at the table wherever he serves.\nKJ is an Engineer with MBA Finance from Amity Business School and has an extensive experience of 18+ years in sales & marketing, partnerships & alliances and IT development for both domestic and international markets. He has worked for clients like KPMG, McDonalds, HP, SAP, Microsoft, IBM, Dell, Cisco, Polycom, Schneider Electric, Vodafone etc among other leading companies for their marketing, market research, alliances and IT projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Action with perfection makes us smart and adorable. When we organize an archery event, its proper planning and suitable timing give an incredible, enthusiastic and energetic result. Because our today's best preparation can make our tomorrow performance easy and fast.\nThe record for the most number of arm stretches in 92 minutes in Sukhasana yoga position stretching 90 degrees from chest \u2013 push and pull in midst of distractions by archery shots is 10,892 and was achieved by Cherukuri Satyanarayana (born on April 15, 1959), at VMC Volga Archery Field, Vijayawada, Andhra Pradesh, on November 29, 2018.\nComments for MOST NUMBER OF ARM STRETCHES IN GIVEN TIME are now closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am holding a large cooking pot full of yellow paint, passing it between the two men when their rollers run dry.\nIt occurs to me that as I am taller than the Shuar man, Miguel, perhaps I would be able to reach the high points he leaves unpainted, but never mind.\nThe difference between the way Miguel and Ulli (the German volunteer) work, is markable. The Shuar start a task, but don\u00b4t really think about the logisitcs of it. Miguel\u00b4s painting is all over the place, and when he can\u00b4t see how to do a tricky bit, he doesn\u00b4t bother, and starts something new. Ulli is meticulous and efficient. He takes a little longer, perhaps, but his work is neat. It frustrates Ulli that the Shuar are disorganised, but he accepts that the education levels here are incredibly low.\nAnother aspect of this though (I think) is part of the post-nomadic culture. The Shuar only began to transition (out of necessity) from being a nomadic people in the 1960\u00b4s, and I think that, to them, painting to the edges of a wall is not really a matter of priority. The important stuff always gets done.\nA couple of hours later the nursery is fully painted both inside and out, and looks fantastic, regardless of wobbly edges. And it wasn\u00b4t Ulli or I that got the job done.\nAs I began to write this in my notebook, crouched on the foundation timbers of Casa Jatun Sacha, a black headed parrot (Loro Coroninegro) landed on me and nestled in the curve of my neck. It nibbled the corner of my mouth, and then stuck its tongue down my ear. I am glad that I had my camera in my pocket. That wouldn\u00b4t have happened if I had been sat inside writing efficiently and ergonomically at a desk, would it?\nI suppose I've got to - \"pieces of eight..!\" etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This hose is between the steering gear and power steering pump.\nFits Ford Transit MK6 with rear-wheel drive (RWD) with a 2.4 TDCI engine, which was produced from 2003 to 2006.\nOur product is a high quality replacement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"47) In the 19th century, Usman dan Fodio conquered Sokoto and most of northern Nigeria and influenced other jihads in the areas of West Africa where Boko Haram is present today, such as northern Cameroon, northern Nigeria, Niger, northern Mali and Senegal.\nIt cited as inspiration historical militant leaders in the region, including Usman dan Fodio and El Hajj Omar Tell.\nPolitically, the Jihad in the early part of the nineteenth century, led by Shehu Usman dan Fodio, incorporated Kasar Katsina into the emirate system of the Sokoto Caliphate.\n1973 The Sword of Truth: The Life and Times of the Shehu Usman dan Fodio.\nIn contrast to the NPC, NEPU took a very different tack in approaching the issue of the jihad of Usman dan Fodio.\nThe NPC characterized the caliphate as the legitimate inheritor of the ideals of Usman dan Fodio.\nThe Sardauna, the head of the NPC and the premier of the Northern Region, was also the great-great-grandson of Usman dan Fodio.\nThe Sardauna] is never known to have fought a losing battle, because he has the blood of the great Usman dan Fodio, the most outstanding man of his time, flowing through his veins.\n23) Indeed, it was a common NPC strategy not only to stress the blood linkage between the Sardauna and Usman dan Fodio but also to compare their exploits.\nIn presenting the Medal to Sir Abubakar, the Sardauna expressed the hope that he would accept the Medal not because of its value but because of his (Sir Abubakar's) loyalty, and belief in the duties performed by Shehu Usman Dan Fodio for this country [my emphasis], and what they were doing in following Dan Fodio's footsteps.\nThe name and person of Usman Dan Fodio were used by the NPC on a variety of levels.\nWe [NEPU] were the legitimate successors of Usman dan Fodio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Inspired by my many visits to Kinsale in Ireland and the Spaniard pub where I did indeed see an old fisherman come in and cook fresh fish over a roaring fire one October night. My mouth watered as it cooked. Can't remember if I tried a bit or not but probably did.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Best Dining in Detroit, Michigan: See 40,051 TripAdvisor traveler reviews of 1,277 Detroit restaurants and search by cuisine, price, location, and more. 2019!\ndoes making for a charming view whizzing by the windows of Times Squares contemporary soul food restaurant. In warm weather, customers can sip classic cocktails on the restaurants terrace while looking out on the waterfront. Torino, 201 East Nine Mile Road, Ferndale, Detroit, Michigan, USA, More Info Save to wishlist Iridescence Located inside the MotorCity Casino Hotel, the sophisticated, two-tier dining room is a popular venue for romantic evenings. Trust us: it's worth every penny. But no matter what your reason for visiting, a visit to this eatery will leave you satisfied. Price: Bargain, check website 4, photograph: Courtesy best casino restaurants in detroit Yelp\/Milica. With a wide array of you guessed it - small plates to choose from, such as calamari, lobster bisque, or pulled pork, youll find yourself totally charmed. Forans Grand Trunk Pub, 612 Woodward, Detroit, Michigan, USA, more Info. Wed - Thu: 11:00 am - 2:00 am, thu - Fri: 11:00 am - 2:00 am, fri - Sat: 11:00 am - 2:00. Youve never seen a restaurant quite like this before. Clair Shores is popular with locals spending a day out on the water.\nWhy go, pJs Lager House 1254 Michigan Ave. The Dakota Inn Rathskeller 17324 John R Street. Boaters can anchor in Sindbads marina and head up to the dining room for sprawling waterfront cleebo casino views and a feast of rock lobster tails and fried shrimp. MI 48192 The Waterfront Wyandotte Official This dockside restaurant offers standard American fare ranging from crab cakes to veggie pizza. Pub Grub, uSA, a luxurious suite overlooking the city, re talking premium cuts like the Australian wagyu and New York strip and bonein ribeye there are also some more unusual dishes on offer. Better yet, stimmst du der Erfassung von Informationen durch Cookies auf und au\u00dferhalb von Facebook. Wie du die Kontrolle dar\u00fcber beh\u00e4ltst. What is it, caf\u00e9 Muse, wenn du auf unsere Website klickst oder hier navigierst. Findest du hier, locally sourced small plates and cocktails. In the form of Selden Standard.\nFor all tastes, Greektown Casino offers an eclectic blend of dining choices, menus, and styles no matter what time of day or what you're in the mood for.Get your fill of fun and excitement in Greektown, where Detroit s best restaurants, nightlife and entertainment are always at your fingertips.\n12, save to wishlist Torino play casino games online Torino offers an innovative and surprising fivecourse menu. Its delicious food 00 am 2, iridescence offers spectacular views of Detroitapos. MI 48207 The Rattlesnake Club is a Detroit institution and a favorite spot to host weddings thanks to its close proximity to the Detroit river and views of downtown. Fish and seafood, detroit, s downtown and waterfront from the posh 16th floor of the MotorCity Casino 300 River Place Dr, price. USA 11, detroit is also home to one of the premiere luxury casino hotels outside of Las Vegas 00 am, drop a note in the comments or at the tipline. The focus of the menu is to provide visitors with elegantly American cuisine that focuses on beef 00 am 2, detroit, the menu includes a delicious selection of British pub food favourites. Which changes on a weekly basis and blends the classic and the contemporary 80 free spins olympus casino 00, tue Wed, lamb, detroit, the restaurants food and white tablecloth atmosphere is only improved by the panoramic views of the Detroit river.\nSophie Boudreau, while many patrons head to Grey Ghost for a few cocktails or beers 7, uSA 30 am 3, save to wishlist Marios Marios is a perfect example of a timeless 2017. Wed, vegetarian, detroit 2901 Grand River Avenue, where Detroits best restaurants. Nightlife and entertainment are always at your fingertips. Torino aims to redefine the dining experience in Detroit and. Showcasing a selection of classic ladiesinvehicles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ADEPT (advanced diesel-electric powertrain) project has applied its advanced 48V technology, which utilizes a lead carbon battery, to a Ford Focus demonstrator.\nKey ADEPT project achievements have been to show that the integration of hybrid and emissions control systems in the manner envisaged by the project can deliver a Euro 6b-compliant package offering up to a 10-12% reduction in fuel consumption, which is equivalent to sub-80g\/km of CO\u2082 emissions (NEDC).\nThe package offered by the project can also offer improved responsiveness and acceleration as a result of the increased full-load powertrain torque available through the hybrid systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaphqz b/data_all_eng_slimpj/shuffled/split2/finalzzzaphqz
new file mode 100644
index 0000000000000000000000000000000000000000..241993ac4dc9e78e54ccec4011d70f8bc9066dde
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaphqz
@@ -0,0 +1,5 @@
+{"text":"SMOK Vape Pen 22 Core is compatible with vape pen tank and Vape Pen 22 Kit. 0.3ohm NiCr dual core brings you smooth taste; while 0.25ohm dual core delivers massive cloud of vapor. 5pcs each pack. Get them as spare parts.\nIt is designed for Smok Vapen pen 22 and Vape Pen Tank.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gujaratat is influenced with enculturation; a culture shared with members of the society and passed on from one generation to the next. Enculturation has unified people with common sense experience and influence that lead to knowledge and appreciation of cultural traditions and lifestyles. The aspect of joining hands to greet or bow down comes through age influences as offering reverence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stevenage FC season ticket holders will have the chance to hear from manager Graham Westley and some of the players at an exclusive event on Monday night (28th April).\nThe Meet the Manager evening will take place in the Broadhall Suite and will comprise a fans forum format.\nStarting at 6.30pm, the event gives a number of our most loyal fans the opportunity to sit down with Graham and discuss a variety of topics.\nPlease note only season ticket holders who pre-booked their seat by last week's deadline will be able to attend. Tickets have been issued and will be required for entry on the night.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aku Cafe & Gallery offers the perfect spot in Kuala Lumpur's Chinatown neighbourhood to unwind and relax with a cup of aromatic coffee. The cafe is a comforting space created by a Malaysian film producer for his friends and other creative people to meet and express their individualistic style and talent. It is a convenient place either for an intimate chat or to hold brainstorming sessions in an inspiring environment. Tucked away from the busy street of Jalan Sultan, the cafe is located on the first floor of a Chinese association's building on Jalan Panggong (formerly known as Theatre Lane). It offers travellers, writers, artists, collectors, filmmakers and coffee connoisseurs a tranquil space to escape from their hectic schedule, at least for a few hours. Events related to music, films and food as well as art exhibitions are held from time to time for regulars who appreciate culture at a closer level.\nThe price is for RM20 with the maximum number of 40 pax. The food and beverages are served in buffet style. The price can be RM25 where it includes Nasi Lemak.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's where the loading screen clue is pointing.\nFortnite: Battle Royale's Week 5 challenges just went live a few hours ago, but players have already completed the entire set and located this week's secret battle star location (and yes, this week it's an actual battle star). Here's everything you need to know to parse the latest loading screen and find that star, including a map and a video guide from Fortnite expert and YouTube creator garandhero.\nTo get the Week 5 Secret Battle Star, the first thing you'll need to do is complete all the other Fortnite challenges. If you haven't already, check out our guides for this week's \"Search Between\" challenge and another that has you dance on towers across the map.\n'Fortnite' Secret Battle Star Week 5 location.\nYour best bet is to head there straight from the Battle Bus. Thanks to an opening in the roof of the cave, you can drop right in and land directly on the star itself. However, you may also want to scavenge for weapons nearby at Frosty Flights or Happy Hamlet first in case you encounter any other players when you arrive at the secret star location.\nThere's no need to rush either. You still have almost two full months before these challenges expire when Fortnite Season 7 ends at the end of February and Season 8 begins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapjep b/data_all_eng_slimpj/shuffled/split2/finalzzzapjep
new file mode 100644
index 0000000000000000000000000000000000000000..00643502303061230bc872e08d8570ac487f7ebe
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapjep
@@ -0,0 +1,5 @@
+{"text":"Piano Note 1 Productions put on its very first Music Showcase at The Mesquite Arts Center. Mitch says, \"he plans to do more in the future\". This event was similar to the \"Showtime At The Apollo\", and also like the \"America Idol. Mitch believes in creating a platform for musicians. His desires are to bring the arts to the Mesquite Community more than ever before.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to SciTaiTech Bojutsu. SciTaiTech stands for science and taijutsu technology. In these courses we will apply the science and taijutsu principles from the Matrix of Martial Arts course series to different skill sets. This course will look at bojutsu principles and bofuri.\nHistorical weapons, like those today, had specific grips depending on their functional use. A naginata handle is different from that of yari for a reason. In this Sword Grip Seminar you will look at one type of sword grip and how your taijutsu improves immediately just by using it.\nAn-shu Hayes demonstrates fundamentals with the classical spear weapon.\nAn-shu Hayes demonstrates Gyokko Ryu Kosshijutsu Kou-Ryaku kata.\nLearn how to use a hanbo (Japanese cane or short stick) to strike at a distance, to disrupt or put an end to an attack, and to protect yourself in close range situations. Shihan James Kentoshi Norris takes you through the foundations of two important hanbo skills; striking and shielding.\nLearn how to use a hanbo (Japanese cane or short stick) to turn an attacker's energy, or even their own skeleton, against them. Shihan James Kentoshi Norris takes you through the foundations of two important hanbo skills; how to create locks and how to create leverage.\nLearn how to turn an attacker into his own worst enemy. Shihan Mark Sentoshi Russo guides you through efficient and effective methods for managing the nightmare scenario of finding yourself trapped at knifepoint.\nLearn how to instantly transform your cell phone into an improvised weapon of self-protection. See how using good To Shin-Do Taijutsu makes this every day item an obvious choice.\nLearning to deal with firearms is an important part of self-defense training. In this second in a series of gun defense courses, we cover examples of dealing with rear attacks, carjackings, and rescue situations when you don't have a weapon of your own.\nLearning to deal with firearms is an important part of self-defense training. In this first in a series of gun defense courses, we deal with front attacks when you don't have a weapon of your own.\nMaster Mark Sentoshi Russo teaches you the essential principles of snares and captures using flexible weapons, such as the ninja kusarifundo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(31 July 2018, Hong Kong) Plover Bay Technologies Limited (the \"Group\", SEHK stock code: 1523), a world leading SD-WAN technology company, today announced its financial and business performance for the period ended 30 June 2018 (\"1H18\").\nThe Group reports a revenue for the first half of 2018 of approximately US$19.7 million, gross profit of approximately US$12.4 million, net income of approximately $4.7 million and diluted earnings per share of approximately US$0.45 cents. In addition, the Group declares an interim dividend of HK 2.92 cents, representing a payout ratio of approximately 80%.\nRevenue increased approximately 10.9% year-on-year. The main driver of revenue growth was a 35.1% growth in warranty and support services which is a highly recurring revenue base. During the period, sales to North America and EMEA also increased approximately 22.1% and 25.1%, respectively, showing a steady growth of SD-WAN in our key markets.\nNet profit margin also improved to 24.0% (from 23.2% in 1H17). Improvements in net profit margin was driven by a 4.1 pp increase in gross margin thanks to improvements in our sourcing and procurement capabilities and rising revenue contribution from the warranty and support services segment. This was partially offset by \"operating expense \/ revenue ratio\" rising slightly to 34.1% (from 33.1% in 1H17) as we stepped up our marketing and promotional activities. During the period, we began hosting partner training events in both the US and Europe and also increased marketing support to our distribution channel in order to further promote our brand and products.\nFinally, cash flow from operations showed a strong improvement to approximately US$11.3 million in the current period, compared to approximately US$1.2 million during the same period last year. The strong cash flow can be attributed to measures taken to improve collection of trade receivables and tighten inventory controls. As at 30 June 2018, the Group is in a strong net cash position of approximately US$21.1 million and a gearing ratio of approximately 3.6%.\n\"More importantly, we see keen interest from leading companies and brands to embed our disruptive SD-WAN technologies into their products so that their customers can enjoy the benefits of unbreakable and high-speed wireless connectivity. We will also launch a new SD-WAN service to provide prepackaged mobile data from different telecom service providers to maximize the benefits of SpeedFusion. This will not only enhance the coverage area and service quality of managed service providers, it will also simplify end customer deployments into a hassle-free experience. We are also expanding our product range into the Internet of Things space with products that will help end-customers simplify their smart office deployments, automate administrative functions and save costs. We are confident that these developments will become important pillars of our business as the market for wireless SD-WAN and IoT continues to expand. We are confident that these developments will become important pillars of our business as the market for wireless SD-WAN and IoT continues to expand.\" Mr. Keith Chau added.\nPlover Bay Technologies (stock code: 1523) is a world-leading Internet connectivity technology company. The Group is primarily engaged in developing and commercialising SD-WAN solutions under its own brands \u2013 'Peplink' and 'Pepwave.' The Group also grants software licences and provides support services in connection with its SD-WAN solutions. The Group sells its own branded products to an extensive global network of more than 500 resellers in approximately 70 countries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why are the icons or buttons missing from the data grids on a few workstations?\nWhen you look at the data grid, the buttons or icons are missing from the left hand side of the screen. However, this doesn't affect all the users, just some.\nIf icons or buttons are missing from data grids or views on a few workstations, the c:\\windows\\system32\\msvcp71.dll is probably missing from those workstations. This is a Windows file that is required by the FIMS data grid icons feature.\nWithout this file, the icons will not appear. To fix this issue, replace the missing file on this computer with a copy from a computer with working icons. If you are running a windows 64 bit machine, you should put the file in the syswow64 directory not system32.\n1. Tools>System Utilities>Run Procedure then type in \"gui\\tools\\update\\impuini.p\" (without quotes). You may have to click on the Browse button to actually find the file and select it in the actually folder where it resides. N:\\npo\\found\\FIMS\\GUI\\Tools\\Update.\n2. Click ok and the procedure will run, it will not appear as if anything is happening. After that, verify that the buttons and icons are now back in place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Pivotal trials: Two, independent, randomized, multicenter, double-masked, parallel-group, vehicle-controlled, pooled studies in 813 patients to examine the safety and efficacy of LOTEMAX GEL in the treatment of postoperative inflammation and pain following cataract surgery.\n\u2020Primary efficacy endpoint on Day 8; secondary endpoint on Day 15.\nPivotal trials: Two, independent, randomized, multicenter, double-masked, parallel-group, vehicle-controlled, pooled studies in 813 patients to examine the safety and efficacy of LOTEMAX GEL in the treatment of postoperative inflammation and pain following cataract surgery.\nReferences: 1. Rajpal RK, Fong R, Comstock TL. Loteprednol etabonate ophthalmic gel 0.5% following cataract surgery: integrated analysis of two clinical studies. Adv Ther. 2013;30(10):907-923. 2. LOTEMAX GEL [prescribing information]. Bausch & Lomb Incorporated. 3. Rajpal RK, Roel L, Siou-Mermet R, Erb T. Efficacy and safety of loteprednol etabonate 0.5% gel in the treatment of ocular inflammation and pain after cataract surgery. J Cataract Refract Surg. 2013;39(2):158-167. 4. Fong R, Leitritz M, Siou-Mermet R, Erb T. Loteprednol etabonate gel 0.5% for postoperative pain and inflammation after cataract surgery: results of a multicenter trial. Clin Ophthalmol. 2012; 6:1113-1124.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapnog b/data_all_eng_slimpj/shuffled/split2/finalzzzapnog
new file mode 100644
index 0000000000000000000000000000000000000000..9862c37c466f4ed05b7ebd99c9b5960cc34ac63b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapnog
@@ -0,0 +1,5 @@
+{"text":"Discuss the situation with the principal and determine what powers he should grant you. Two types of power are commonly granted: the power to make medical decisions on behalf of the principal, and the power to manage the principal's finances.\nResearch state law to determine the requirements for a valid power of attorney. State laws differ, especially with respect to the inclusion of required statutory language and the manner in which the document must be signed.\nEntitle the document \"Power of Attorney.\"\nAdd any language required by statute. New York, for example, requires the inclusion of two passages listed in its power of attorney statute and entitled \"Caution to the Principal\" and \"Important Information for the Agent.\" Pennsylvania requires the inclusion of a passage describing the powers and duties of the agent, and requires that the passage be printed in all capital letters. These passages must be word-for-word identical to the language set forth in the power of attorney statute.\nIdentify the principal by his full legal name and address and state that he is the principal who is granting powers to the agent. Identify yourself by your full legal name and address and state that you are the agent to whom these powers are granted.\nState whether the power of attorney is durable or non-durable. A durable power of attorney continues in force while the principal is mentally incompetent or unable to communicate his wishes, while a non-durable power of attorney automatically lapses under these circumstances.\nIdentify which powers are to be granted to you. Draft this section clearly and include no vague statements. If you include the statement \"The power to manage the principal's finances,\" for example, a question may arise if you attempt to donate money to charity on behalf of the principal. Instead, list the specific acts that you are entitled to perform. Nevertheless, reasonably general statements are acceptable, such as \"The power to consent to medical treatment or to select from among alternative medical treatments recommended by a licensed physician.\"\nSpecify the duration of the power of attorney. You may list beginning and ending dates. You may also define the power of attorney in terms of events -- your powers may be triggered when the principal becomes incapacitated, for example, and lapse whenever he regains capacity.\nPrepare a signature line that includes places for both you and the principal to sign and date the document. If state law requires witnesses or notarization, include spaces for them to sign and date the document. Prepare the document in duplicate.\nSign and date both copies of the document, and have the principal do the same, in the presence of witnesses or a notary public. Have the witnesses sign the document or have the notary public sign and seal it. Give one copy to the principal and keep one copy for yourself. You must present the document to third parties, such as physicians or bank officials, to exercise your powers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pipe Fabrication Drawings Advantage Engineering Technologies provides engineering services including plumbing isometric drawing, Shop Fabrication, MEP Coordination Drawings.\nWholly Owned Subsidiary in India Avail Indian Subsidiary Registration Services in India, Wholly Owned Subsidiary in India. ebizfiling offers Subsidiary Company Formation Procedure and Services in India. Call us\u00a0 at +91 8262 990 990, Email at info@ebizfiling.com.\nRetail Designers Exhibition stand design & trade show display booth builders & exhibition contractors. Designing & developing promotional expo stands for clients attending trade fairs & events. Creative design ideas & innovative exposition equipment \u2013 promoting your companies products & services when exhibiting at tradeshows & public events across Europe & the UK. Economical promotional Hire & rental exhibition display kits available.\nTax preparationIntegrating highly skilled tax preparation with reliable and personalized service, Unidos Tax & Immigration Services helps Visalia residents achieve their financial goals.Des \u2013 Managing your business or personal tax file successfully is best handled with the assistance of competent professionals for maximum financial benefit. Protect your assets against unnecessary tax penalties. Remain informed on the latest tax incentives.\nEyebrow waxing Victoria A boutique studio specializing in microblading, cosmetic tattoo and eyelash extensions located on upper Fort Street in beautiful Victoria, BC.\nCash Loans In Salt Lake City It's Amazingly Simple to get Affordable, Low Interest Car Title Loans in Salt Lake City and Ogden. Fast & Friendly. Apply Online. Personal Loans Too!\nFrench Mahogany Doors Looking for the highest quality Peruvian Mahogany doors, mouldings and flooring in Miami? We carry all types of doors including front, entry, hurricane impact, interior and entry doors.\nBin Rental Edmonton All Disposal will quickly dispatch a garbage bin to your home or business where you can begin filling it up yourself, at your speed. Once your're finished or the bin is full, simply call All Disposal to schedule a pickup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Make sure you react immediately when a Dutch person asks you this question. Never beat around the bush: if you want to have a drink, this is the only moment you can say \"yes please\". Otherwise you just won't get a drink. Even though the Dutch will get himself\/herself one.\nAnd then the game of insisting and insisting again begins.\nIn Spain it is polite to insist. In fact, it is obligatory. Only after saying \"no\" for at least 3 times the other one knows you really don't want to have another drink. Might be even after 4 times. Or 5.\nSaves us both a lot of time. And I can't help but think the Spanish sort of likes that. Though it's not Spanish to admit to that, so they'll never tell.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today is America Recycles Day. Some people think this is the wrong idea. Treehugger.com makes the case for emphasizing reusable packaging instead of recyclable. I don't particularly care for the way they frame this, but I think they make a provocative point, and I think perhaps we should reconsider such concepts as deposits on bottles and cans. If we can do better than what we're doing now - and in Houston at least, that's sadly not a high bar to clear - we should make the effort. Take a look at what they propose and see what you think.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A formula with all of its constituents present in a wholesome, natural balance, Nordic Naturals Omega-3 soft gels are an easy way to get omega-3 EFAs every day.\nIt is a great way to maintain healthy levels of EPA and DHA - the most important omega-3 fats. This pure, non-concentrated fish oil is the perfect nutritional complement to any healthy lifestyle.\nNo gluten, milk derivatives or artificial colors or preservatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarwoj b/data_all_eng_slimpj/shuffled/split2/finalzzzarwoj
new file mode 100644
index 0000000000000000000000000000000000000000..409b453e8941ae5eb16fa3f35e658ff0634d6aa0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarwoj
@@ -0,0 +1,5 @@
+{"text":"Version 0.4 of RequestRouter introduces a new feature - an API designed to give edge devices (such as delivery appliances) hints to allow them to generate an RFC 7838 Alt-Svc header to further optimise routing for clients located behind a DNS server that does not support the EDNS Client Subnet (ECS) extension.\nThis documentation provides a reference implementation, allowing an OpenResty based edge device to connect back to the Alt-Svc Hints API, whilst minimising the potential latency impact to some extent.\nThis documentation assumes some familiarity with both OpenResty (i.e. that you already have it installed) as well as some basic familiarity with RequestRouter and how to manage it.\nIt used to be that most servers that did not support ECS were considered old infrastructure that would eventually be in the minority as software got upgraded. Unfortunately, due to various privacy concerns, a number of new public services such as Quad9 and Cloudflare's 1.1.1.1 do not support ECS.\nThis introduces a severe limitation for DNS based routing as routing calculations without ECS available must be based upon the downstream resolver's IP (being the only information available to hint at the client's location).\nWith a large scale public DNS service, this can often lead to vastly inaccurate location calculations: Level3's DNS service, for example, is quite commonly implicated in mis-identification of client locations due to the way queries are routed to their PoPs.\nThis section exists largely for the purpose of completeness, if the relevant section of the RequestRouter documentation changes later then it should be considered to override this.\nThe software dependencies for the Alt-Svc Hints API are exactly the same as for the RR-37 HTTP Redirect Routing component.\nAs is true with the HTTP Routing component: installing it on a separate system to the DNS routing infrastructure is not mandatory, but is strongly recommended to ensure that excess load in one component does not adversely impact the other.\nYou should configure a HTTPS server block in Nginx. The connection is going to have downstream subnets submitted over it, so do not use plain HTTP.\nIn the example above we can see the client was geolocated to France, and the recommended edge device has IP 51.255.232.237 (where multiple devices are recommended, the IPs will be comma separated).\nThe change to the edge device consists of adding some LUA to make a call out to the API, as well as inserting a server block to act as a caching tier.\nIn this config, we explicitly specify the IP of our Alt-Svc api nodes. An alternative would be to configure the Alt-Svc API's FQDN to resolve via RequestRouter (sending each edge device to the nearest node to reduce latency).\nIt should be noted here that using Nginx's ip_hash load balancing algorithm on the upstreams should be avoided. The client for this server block will always be 127.0.0.1. least_conn could, however, be used.\nWe cache different statuses for varying amounts of time, as each has a different significance to our decision on whether to insert an Alt-Svc header. If we receive status 304, it means the client is already correctly routed, so we can afford to cache that for quite some time (as the client can clearly reach us).\n-- Alt-Svc header - dont waste time giving them a new one.\nIf a client has included the Alt-Used request header, then it means they've already followed and Alt-Svc header and we shouldn't waste any cycles trying to further optimise the routing.\nThe final step is to call the LUA from within any server blocks you want to optimise.\nReload or restart Nginx and you should be good to go.\nThe easiest way to test that your changes are working is with curl.\nIf you want to test the full flow, grab an Alt-Svc supporting browser (such as Firefox), and edit your system's DNS config to use 1.1.1.1, 9.9.9.9 or Level3's DNS service.\nTry connecting out to your site with developer tools open, and you should see later requests use a new server (and include an Alt-Used header) - assuming you get routed to your test device in the first place.\nCareful consideration should be given to scaling the API infrastructure appropriately. You will need to consider your traffic profile when assessing how many requests the API is likely to see at any given point.\nFor networks where the request to client ratio is small, the number of resulting requests against the API is likely to be far higher than one where each client places a lot of requests.\nWithout the protective caches implemented in this documentation, the initial implementation of the API can handle request rates of around 1100 requests per second, per CPU core. This rate represents a worst case scenario where all caching tiers report a CACHE_MISS.\nHTTPS sites only - RFC 7838 specifies that the origin being directed to (i.e. the new server) must support SNI and must be able to provide a certificate valid for the original domain name. This is to ensure that any new origin can authenticate that it is authorised to serve the requested domain.\nAll edge devices must support HTTP\/2.0 - the h2 in the Alt-Svc header stipulates that HTTP\/2 should be used when connecting to the new origin. Whilst the RFC allows for HTTP\/1.1 to be specified, no browser currently appears to implement support for this.\nFollowing Alt-Svc is optional. It's important to note that just because the header is served, it doesn't mean that clients will definitely follow it. It's down to the user-agent to decide whether to honour or ignore the header.\nThe first request is unaffected. Alt-Svc is served along with the response to the user-agent's request. So the first request will be via the less optimal edge-device. If the user-agents sessions consist of a single HTTP request then this functionality will serve no benefit. Multi request sessions - like adaptive HTTP streaming, web sites etc - on the other hand, should benefit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Porcelain Balls For A Crusher - . Porcelain Jar For Mineral Grinding Grinding Mill China. porcelain jar for mineral grinding mecanic manual crusher spherical or .\nhp400 cone crushers instruction manual ecoeducation center. hp cone crusher parts manual mining machinery. hp 400 crusher manual goo.gl\/pajuu7 more about shanghai hp400 cone crusher, h p crusher repair crusher video full. cone crusher manual 2 cs hp 300 cone crusher hp 400 cone crusher manual hp 400 cone crusher manual crusher .\nMineral Processing Handbook 11\/11 - 1st printing Printed in U.S.A. TABLE OF CONTENTS . crusher liner wear because fines are bypassed around the primary crusher.\ncrusher handbook pdf price, specification, minimum order, certification, packaging, product details and description.\nassembly manual for a gyratoury cone crusher 1 mineral the most common type of secondary crusher is the cone crusher. introductionof experimental analysis.\nMetallurgical ContentJAW CRUSHERSDouble-Toggle Jaw CrushersSingle-toggle Jaw CrushersSizing Jaw CrushersForced Feed Jaw CrushersBulldog Jaw CrusherTraylor Jaw CrusherJaw Crusher Foolproof A sectional view of the single-toggle type of jaw crusher .\nMecanic Manual Crusher Mineral mecanic mesin stone crusher. cone crusher crushing capacity, goo.gl\/vjOrQu Machinery is professional mineral .\nmecanic manual crusher mineral; . crushers and cone crusher from shanghai xsm Chat With Sales list of stone crushers in kotputli rajasthan -vacancy crusher plant .\nmecanic manual crusher mineral. crc450x portable cone plant mineral crusher for sale,mineral a barmac b series vsi crusher a bulgaria. industry. a minerals.\nadvantage of grinders Newest Crusher Grinding Mill .\nmecanic manual crusher mineral. In the past 20 years, we devote to producing mining equipments, sand making machines and industrial grinding mills, offering .\nporcelain jar for mineral grinding \u2013 Grinding Mill China. mecanic manual crusher mineral. In the past 20 years, we devote to producing mining equipments, .\nmecanic manual crusher mineral. crushers, Handsel mills, hammer, With ambient temperatures reaching above 50\u00b0C in, Manual in PDF format for free Jaw Crusher .\nbmw crusher and grinding mill Web \u00b7 shaft crusher clark Grinding Mill China. bmw clark shaft xzm 221 utrafine grinding mill vertical crusher bmw_vertical crusher bmw[crusher .\nmecanic manual crusher mineral. In the past 20 years, we devote to producing mining equipments, sand making machines and industrial grinding mills, offering expressway, rail way and water conservancy projects the solution of making, porcelain jar for mineral .\nShanghai Manufacturer Simons Cone Crusher For Mineral .\nShanghai Manufacturer Simons Cone Crusher For Mineral Processing; Read More . Cone Crusher. Read More . Horizontal-shaft Impact Crusher.\nStone Crusher Flow Chart In India - Gold Ore Crusher. stone crusher machine assembling flow chart india, . Mecanic Manual Crusher Mineral.\nJaw Crusher Manual Pdf How About stone crusher machine technical manual pdf? sulfide type lead zinc. pettibone universal 2436 . mecanic manual crusher mineral - .\nStone Crusher Supplier, . mobile rock ore crusher; mecanic manual crusher mineral; vertical impact crusher manufacturer in india; small stone crusher .\nmecanic manual crusher mineral - crushergoogle. pioneer cone crusher manual | Solution for ore mining. pioneer cone part manual . Arsenic Mineral Crusher,Ore .\nManual Crusher Machine Design; operating manual of jaw crusher machine. operating manual of jaw crusher machine . How to install and maintain a jaw crusher machine .\nmecanic manual crusher mineral . all mobile crusher manual . q 55 gallon drums drum a Br380jg 1 Mobile Crusher Workshop Manual,Manual Polybag Crusher .\nSales Inquiry Crusher Machine To Crush Mm charcoal crushing machine size mm to mm May 7 2016 Wood crusher machine for sale . is the main mineral processing .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even with the snow Spring has arrived! This is the time of year when the earth bursts alive with energy and new growth. A time when the body's energy rises and moves.\nThe Wood Element is linked to Spring. The Liver is the key organ associated with this element. In Chinese Medicine the Liver's key functions are to store Blood and ensure the smooth flow of Qi around the mind and body.\nHow can you help to nourish your Liver?\n* Add more green vegetables to your diet.\n* Drink chamomile or peppermint teas to help to soothe your Liver and ease digestion.\n* Start to incorporate gentle exercise into your daily routine, building up your programme of exercise over Spring. Your body needs time to adapt to the new season so allow the energy to rise up gradually.\n* Massage the star point \u2013 Liver 3 \u2013 this point is located in the hollow area at the top of your feet between the first and second toes. About 2.5 cm away from the webbed part. Using your thumb gently rub this point backwards and forwards for 1 -2 minutes on each foot.\nLiver 3 calms the nervous system, reduces irritability, helps alleviate depression, strengthens your immune system, relieves and prevents headaches and increases circulation in your legs. A true star point!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Students can earn either a bachelor of science or a bachelor of arts in biology. Both degrees offer many opportunities to conduct research. Either degree provides great preparation for professional schools, including medicine and dentistry. Well chosen electives can even prepare students for careers in law, journalism, business and more. Students in the bachelor of science are afforded opportunities to expand their knowledge in additional scientific fields. Students in the bachelor of arts enjoy greater flexibility to pursue a wide range of interests.\nThe bachelor of science in biology is a SpeakWrite Certified ProgramTM. SpeakWrite Certified programs incorporate and develop students' written, verbal, visual and mediated communication skills across the curriculum.\n\u00a9 2019 West Virginia University. WVU is an EEO\/Affirmative Action employer \u2014 Minority\/Female\/Disability\/Veteran. Last updated on June 29, 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Clearview Software, the premier innovators of contemporary Microsoft\u00ae SharePoint\u00ae based Enterprise Content Management (ECM) solutions designed exclusively for Microsoft technology environments, is pleased to announce that Advanced Document Solutions, Inc. (ADocs), a preferred integrator of business process improvement, document automation, and electronic document filing\/content management systems, has joined the Clearview Authorized Partner Network. Under the terms of the partnership agreement, ADocs will represent and deliver to their customers the Clearview ECM suite as part of a comprehensive SharePoint-based ECM solution.\nADocs is a solution provider with expertise in offering the appropriate application of document automation and imaging technologies to meet specific business requirements. Experts in many common business processes and industries, the ADocs team possesses significant knowledge and training in business process improvement techniques, and works to help customers improve existing systems and reduce cycle time and cost. ADocs has added Clearview to their solution portfolio because of Clearview's unique, SharePoint-based ECM solution.\n\"ADocs is a true solution provider in every sense of the word. We don't take a one-size-fits-all approach to ECM solutions. That means that the products we work with have to represent the highest possible value to our customers and the people behind them have to value our customers as much as we do,\" said Mike Brown, Vice President of Sales for ADocs.\nClearview Software, the premier innovators of contemporary Microsoft\u00ae SharePoint\u00ae based Enterprise Content Management (ECM) solutions designed exclusively for Microsoft technology environments, announced today that it has been named as one of the \"100 Companies That Matter in Knowledge Management\" by KMWorld, a leading industry publication. The distinguished list recognizes visionary companies that create, enhance, and define the knowledge management market.\n\"We have long held that the true essence of knowledge management is an attitude, a single-minded commitment to improvement,\" said Hugh McKellar, editor-in-chief of KMWorld.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaswkf b/data_all_eng_slimpj/shuffled/split2/finalzzzaswkf
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+{"text":"In October 1999 General Pervez Musharraf (born 1943) came to power in Pakistan when he seized control of the government in a bloodless military coup. His pretext was that he sought to stabilize the nation, but Musharraf not only heightened the level of distrust from Pakistan's long-time antagonist, India, he also succeeded in alienating the Islamic fundamentalists within his own country. Much of the latter was due to the unforeseen actions by terrorists against the United States, and the subsequent response which involved Musharraf's aligning Pakistan with the international coalition against the Al-Qaida terrorist organization and Afghanistan's ruling Taliban.\nMusharraf was born on August 11, 1943, in Delhi, India, when it was still under British sovereignty. Following independence from Great Britain and the creation of the state of Pakistan, the Musharraf family moved to Karachi. Musharraf's father was a diplomat and the family spent seven years, from 1949 to 1956, in Turkey. Musharraf became fluent in Turkish, but his education also included attending Saint Patrick's High School in Karachi and Lahore's Forman Christian College. Musharraf entered the Pakistan Military Academy in 1961 and in 1964 received a commission to an artillery regiment in which he served during a 1965 conflict. During that conflict Musharraf was awarded the Imtiazi Sanad for gallantry. Afterward Musharraf volunteered for a commando outfit and also saw action in the 1971 conflict as a company commander.\nMusharraf was promoted to major general in 1991 and lieutenant general in 1995. He also attended the Command and Staff College at Quetta, the National Defense College, and the Royal College of Defense Studies in Great Britain. Musharraf later held appointments at both the Command and Staff College and the National Defense College. He was eventually named director of general military operations at general headquarters.\nThe real turning point in Musharraf's career came in October 1998 when he was promoted to general and named the army's chief of staff by Prime Minister Nawaz Sharif. This placed him near the center of power in Pakistan. Musharraf replaced General Jehangir Karmat, who had advocated civilian power-sharing with the military in the form of a joint national security council, which upset the prime minister. As if to reassure Sharif, Musharraf a few weeks after his appointment reiterated that the army would \"remain apolitical.\" Six months later he was named chairman of Pakistan's joint chiefs of staff; in another six months he would lead the coup that toppled Sharif from power. All of this came amid one of Pakistan's most serious difficulties with India\u2014the contention of the border region of Kashmir. In July 1999 Musharraf admitted that India's allegations that Pakistani soldiers had crossed into the Indian section of Kashmir to fight alongside Islamic rebels were true. Crime and corruption also plagued Pakistan. Just a week before the coup Musharraf was quoted in The Hindu as declaring, \"The law and order situation is bad. It should improve. It will improve.\"\nMusharraf's career took a dramatic turn when, on a visit to Sri Lanka on October 12, 1999, he was abruptly fired by Prime Minister Sharif. However the move backfired and the army, in support of General Musharraf, arrested Sharif and other government officials in what was essentially a bloodless coup. Musharraf became Chief Executive of Pakistan and quickly consolidated his power\u2014an action he would repeat more than once. Within days of the coup he subverted Pakistan's constitution by dismissing parliament and imposing martial law. Musharraf also claimed the airliner carrying him and more than 200 others from Colombo (the capital of Sri Lanka) to Karachi had been denied permission to land on orders from Sharif. The landing was eventually made with only seven minutes' worth of fuel left. The deposed prime minister was charged with hijacking, kidnapping, attempted murder, and treason and became the center of a highly publicized trial.\nMeanwhile the conflict with India loomed large, and to cool it down Musharraf ordered troop withdrawal from the border. However in late December 1999 he decided that Kashmir would be given top priority in diplomatic discussions with India. On a visit to the region he declared: \"Pakistan is Kashmir and Kashmir is Pakistan.\" By the end of the month he had also revived the idea of a National Security Council and named four civilians to the seven-member body, as well as three civilians to his cabinet. He also set up the National Accountability Bureau to investigate corruption. Under its auspices former Prime Minister Sharif's father, son, and brothers were all charged with corruption and arrested.\nIn December Musharraf revealed a new plan for reviving Pakistan's weak economy. The economic plan included a sales tax similar to a value added tax as well as a farm tax. He also moved to close loopholes regarding foreign currency accounts that were seen as money-laundering schemes. Still, the issue of Kashmir remained of primary concern. In January 2000, in a rare interview with an Indian newspaper, The Hindu, Musharraf declared that the Indian government \"(has) to trust me and that whatever I am saying, I mean, and they have to come along.\" But he also admitted: \"(W)e need to accept Kashmir as a problem and start a dialogue and simultaneously let us discuss everything else. I am open to discussion on every other thing.\" Musharraf, however, would make a complete turnaround on this position in a May 2000 press conference.\nIn March 2000 Musharraf deflected attention from the opening of former Prime Minister Sharif's trial when he replaced six judges of Pakistan's Supreme Court, including Chief Justice Saeed uz Zaman Siddiqi, for refusing to swear allegiance to his government. The trial itself took many strange twists and turns. Initially, Sharif's lawyers resigned after he was barred from testifying in open court, and when he did testify he claimed he was being framed by the military. Then in March another of his lawyers was murdered by a gunman. Sharif was eventually found guilty and sentenced to life in prison.\nUnited States President Bill Clinton also made a quick trip to Pakistan in March 2000, adding the country to his itinerary. Clinton urged a return to democracy and a quick settlement of the Kashmir conflict, which seemed to signal a changing attitude toward one of the U.S.'s staunchest Cold War allies. Musharraf was undaunted by Clinton's remarks. The next day he was off on his own trip to strengthen economic ties with Southeast Asia. In April 2000, as part of a human rights pledge, Musharraf condemned the Pakistani practice of \"honor killing\" in which a woman is murdered for \"shaming\" her family by seeking a divorce or otherwise choosing her spouse. Among the other problems he sought to tackle were the effects of drought and frequent sectarian clashes.\nMusharraf was not without his critics in the Islamic fundamentalist community as well as the small business community, which protested the general sales tax. Possibly to assuage their fear and anger he announced he would accede to a Supreme Court decision that he step down after three years and allow democratic elections to be held. Still, in June 2000 the two groups joined forces to protest the sales tax. In July he made further concessions to the fundamentalists when he decreed the revival of the Islamic provisions of Pakistan's suspended constitution. In a BBC World Service interview in August 2000 Musharraf declared Pakistan's continued support for Afghanistan's ruling Taliban. \"The Taliban administration represents the majority Pashtun population in Afghanistan and it is in our interest to support them,\" he said.\nIn September 2000 Musharraf again took steps to strengthen his position when he reshuffled senior Army advisers. The move was made four days before Musharraf flew to New York City to attend the Millenium Conference sponsored by the United Nations Assembly. Then in October 2000, almost a year to the day he took over the government, Musharraf promised that federal and provincial elections would take place before the end of the 2002. The next month while on a trip to Moscow Musharraf admitted not knowing the extent of Pakistan's nuclear force. He also declared neutrality regarding Osama bin Laden, suspected of masterminding attacks on U.S. embassies in Africa. In November Musharraf sought further to legitimize his position by amending the Provisional Constitutional Order to have the Chief Executive assume all acts that were previously the responsibility of the Prime Minister. In December 2000 he surprised everyone by releasing former Prime Minister Sharif, who immediately went into exile.\nOn January 1, 2001, local elections were held in 18 of Pakistan's 106 administrative areas. The first elections since the coup, they were nonetheless criticized by the voters themselves because of the military's involvement and the banning of political parties. Indeed, in March political leaders from the coalition Alliance for Restoration of Democracy were detained just prior to an anti-military rule rally. Later that month Musharraf announced he planned to extend his three-year term as Army Chief of Staff beyond October 2001. However, under the law Musharraf's term could only be extended by the president. Musharraf solved that problem in June 2001 by appointing himself Pakistan's president and head of state.\nDespite Musharraf's machinations, many of Pakistan's problems remained, not least of which was the military tension with India. In July 2001 Musharraf traveled to India to hold talks with Indian Prime Minister Atal Behari Vajpayee over the disputed Kashmir region. The talks failed to produce an agreement, or even a joint declaration. Nonetheless Musharraf invited Vajpayee to Islamabad for a future round of negotiations. Musharraf next turned his attention to two other problems: the U.S. sanctions against Pakistan and terrorism. In August 2001 he began making his case to have the sanctions (imposed against Pakistan and India as a result of their 1998 nuclear weapons tests) lifted. He also vowed to get tough against terrorism\u2014though at the time he was referring to Pakistan's own domestic violence. He again vowed to hold elections before the third anniversary of the coup, in October 2002.\nFollowing the September 11, 2001, terrorist attacks on the World Trade Center in New York City and on the Pentagon in Washington, D.C., Musharraf early on gave Pakistan's support to the anti-terrorist cause. He was quoted by Tyler Marshall in a Los Angeles Times article as saying, \"Pakistan has been extending cooperation to international efforts to combat terrorism in the past and will continue to do so.\" Musharraf then had to make his case with his people, and while he was successful in convincing the army of his decision the clerics and other fundamentalists, who were already opposed to him, were a different story, especially after bin Laden was identified as the primary culprit.\nIn the beginning Musharraf hoped to assuage the clerics and others by brokering a deal with the Taliban to hand over bin Laden, thus avoiding military action. But that ploy was doomed to failure. Nine days after the attack Musharraf appeared on Pakistani television to explain his decision to assist the United States. He quoted the Koran and cited Islamic tradition for political compromise as his argument. The speech was largely ineffective as far as the (minority) fundamentalists were concerned and during the ensuing weeks numerous anti-U.S. and anti-Musharraf demonstrations were held, many peaceful though some ended in violence.\nIn the first, tense week of October 2001 Musharraf admitted defeat in brokering a deal with the Taliban. On the positive side Canadian sanctions against Pakistan were eased. Furthermore, upon reviewing the U.S. evidence against bin Laden the Pakistani government declared there was enough proof to indict him. On October 6, 2001, British Prime Minister Tony Blair met with Musharraf in Islamabad. Blair promised economic and humanitarian aid to Pakistan. Perceiving that his position was no longer as secure as it had been only that summer, Musharraf removed from the military and the intelligence services those men who had assisted the Taliban militia. As the bombs rained down upon Afghanistan Musharraf turned toward India, hoping to renew talks over the Kashmir. He was rebuffed. To make matters worse for Musharraf Pakistani border troops fought with Taliban forces two days after the bombing began. The fighting and Musharraf's agreeing to let U.S. forces use two airfields near the Afghanistan border led to more widespread protests, which he vowed to crack down on. Indeed Musharraf seemed to dig in against his domestic opponents. In a mid-October meeting with U.S. Secretary of State Colin Powell he \"pledged his indefinite support,\" as reported by Jack Kelley in USA Today. By the end of October Musharraf took further steps to halt the growing wave of protests, especially the use of mosque loudspeakers to incite antigovernment protests.\nAt the beginning of November 2001, as Pakistani fundamentalists were urging the army to overthrow Musharraf the call was echoed by none other than Osama bin Laden himself in a statement sent to the al-Jazeera, the BBC, and CNN. To allay Western fears Musharraf responded by arresting an opposition leader; he also declared that Pakistan's nuclear facilities were under secure control. Also in November he closed Pakistan's borders with Afghanistan, thus cutting off approximately 300,000 refugees seeking asylum. At the end of the first month of the bombing Musharraf traveled to Britain to again meet with Blair and then to New York to meet with President George W. Bush. The meeting with the U.S. president fell short of Musharraf's expectations of having the sanctions lifted, though his unswerving support for the coalition brought U.S. aid and debt restructuring that totaled more than $1 billion.\nFor his part, Musharraf warned that an alternative government must be ready to replace the Taliban in order to prevent anarchy in Afghanistan. He also went on record as doubting that bin Laden possessed nuclear or chemical weapons. In the aftermath of the fall of Kabul Musharraf voiced his dissatisfaction with the Northern Alliance (ethnic minorities in Afghanistan who opposed the Taliban), called for the demilitarization of the Afghan capital, and proposed the creation of a UN peacekeeping force made up of troops from Moslem countries. All three propositions were intended to forestall the setting up of a government hostile to Pakistan, which in the past had been one of three nations to recognize the legitimacy of the Taliban. Musharraf also increased border security to prevent fleeing Taliban and Al-Qaida members from crossing into Pakistan.\nBoston Globe, November 14, 1999; November 1, 2001; November 11, 2001.\nFinancial Times, October 19, 1999; July 17, 2000; March 9, 2001; March 26, 2001; August 15, 2001; September 15, 2001; November 15, 2001.\nGazette (Montreal), February 2, 2000; May 26, 2000; July 13, 2000.\nGuardian (London), March 9, 2000; April 7, 2000; November 14, 2001.\nHindu, October 30, 1998; October 6, 1999; December 28, 1999;January 17, 2000; March 28, 2000; May 4, 2000; May 27, 2000; September 3, 2000; November 2, 2000; November 18, 2000; August 3, 2001.\nHouston Chronicle, October 26, 1999; October 5, 2001.\nIndependent (London), October 16, 1999; January 27, 2000; July 17, 2001; November 8, 2001.\nLos Angeles Times, October 14, 1999; September 13, 2001; October 6, 2001.\nNew York Times, October 13, 1999; October 15, 1999; December 9, 1999; March 9, 2000; March 11, 2000; January 1, 2001; March 22, 2001; June 21, 2001; October 9, 2001.\nOttawa Citizen, October 2, 2001; November 2, 2001.\nSt. Louis Post-Dispatch, October 10, 2001.\nSan Francisco Chronicle, February 29, 2000.\nSeattle Times, September 20, 2001; October 11, 2001.\nTimes (London), June 21, 2000; August 3, 2000.\nToronto Star, July 17, 1999; November 3, 2001.\nUSA Today, October 11, 2001; October 17, 2001.\nWashington Post, October 19, 1999; March 27, 2000; April 22, 2000; December 11, 2000; September 17, 2001.\n\"Pervez Musharraf.\" Encyclopedia of World Biography. . Encyclopedia.com. 16 Apr. 2019 <https:\/\/www.encyclopedia.com>.\nPervez Musharraf (p\u0115rv\u0101s\u00b4 m\u014d\u014dsh\u00e4r\u00b4r\u016df), 1943\u2013, Pakistani army officer, president of Pakistan (2001\u2013), b. Delhi. After the partition of British India, his family resettled in Karachi, Pakistan; he spent (1949\u201356) some of his childhood in Turkey, where his father was posted as a diplomat. He entered the Pakistan Military Academy in 1961 and became (1964) an artillery officer, rising through the ranks to major general (1991), lieutenant general (1995), and general and chief of army staff (1998). In 1999 he became chairman of the joint chiefs of staff committee.\nIn Oct., 1999, Prime Minister Nawaz Sharif attempted to remove Musharraf by refusing his returning flight landing rights in Pakistan. The move led to a coup by Musharraf, who became chief executive; he appointed himself president 20 months later. A controversial referendum in 2002 extended his rule for five years. Musharraf was reelected in 2007, but his right to run while still army chief was challenged; before the supreme court could rule, he suspended the constitution, declared emergency rule, and dismissed the court members who seemed likely to rule against him. After the challenges were dismissed, he resigned (Nov., 2007) as army chief.\nThe subsequent election victory (Feb., 2008) by opposition parties and the establishment of an opposition coalition government undermined his position, and after the coalition, at the instigation of Sharif, moved to impeach him, he resigned from office (Aug., 2008). A declared supporter of a democratic, nonfundamentalist Islamic Pakistan and a supporter as well of the U.S. war on terror, Musharraf twice was the target of assassination attempts while president. After resigning, he went into self-imposed exile in 2009 and did not return to Pakistan until 2013; he was then disqualified from running for office. He since has been charged with treason and in connection with Benazir Bhutto's assassination and other deaths and actions.\n\"Musharraf, Pervez.\" The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. 16 Apr. 2019 <https:\/\/www.encyclopedia.com>.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We bridge the gap between children and older adults in a safe, homelike setting.\nDonating via paypal is quick and easy. Just press the button below to get started.\nCopyright \u00a9 2016 Generations Intergenerational Daycare Center. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Style Me Bloggers are back today with a wonderful link up. There was lots of discussion towards the end of last year about doing some non-style related topics, so here we are. I think this is a great one, and I hope you'll visit everybody today and see how they practice random acts of kindness.\nNothing makes us more human than being kind. Quietly being kind to someone, with no wish for anything in return, is the best form of kindness.\nEverett, Washington, the neighboring city to the south of us, is home to Naval Station Everett. Since 1994 it has been the base for two aircraft carriers and many support ships, and at any given time there are about 6,000 sailors stationed here.\nWe often encounter soldiers in the restaurants and stores around this Navy town, and without exception we have found them to be polite and courteous. Whenever possible, my husband and I quietly pay for their meal. Nobody ever knows it was us, which is how we want it to be.\nTo me, war itself is a stupid invention. But I would never confuse government with the military individuals whose job it is to wage that war to provide for our well being. These brave men and women keep us safe from those who would destroy us if they could, and there is nothing I can ever do to repay them. The best I can do is buy them dinner every once in a while. That's my random act of kindness.\nThis leads me into another topic that I have wanted to write about: American Sniper, and all of the controversy surrounding the movie. If you haven't seen it, you should.\nAmerican Sniper tells the story of Chris Kyle, a real life Iraq war veteran and sharpshooter. After multiple tours of duty, he returned to America where he was shot and killed two years ago by a veteran with post traumatic stress disorder.\nThis is what I think about this well done movie. American Sniper is neither an anti-war nor pro-war film. It isn't about the rightness or wrongness of war. It's about people - soldiers and their families back home. It addresses the extraordinary psychological and personal sacrifices that every soldier and their family goes through when fighting a war or being deployed. Nobody is perfect, but the work these men and women do are the reason we are free. And for that they deserve our respect and thanks. The End.\nAmber at Airing My Dirty Laundry wrote a great post about this subject a few weeks ago. Her husband is currently deployed in the Air Force, so she has first hand perspective, and it's well written and to the point.\nBe sure to visit the other Style Me Bloggers, and link up with us below!\nYour right, the perfect act of kindness is silent and it must make those soldiers day when they find out their bill was paid for! I am looking forward to seeing American Sniper! Thanks for sharing your review! Have a great weekend Lana!\nI hope they realize how much we appreciate them!\nMy son and his family quietly had their meal paid for the other day here in Edmonton. What a kind, touching, beautiful thing to do. Thank you for being one of those angels. They and their families appreciate it more than you know..\nAww...thanks Diane. It's such an important, difficult job and we don't say thank you to our military members nearly enough.\nI'm sure your neighbor appreciates those little things so much! It really is the small stuff that means the most, I think.\nyes, kindness with no acknowledgement or reward is the best kind.\nI need to see that movie... everyone keeps talking about it and I feel out of the loop!\nIt's a difficult movie, but an important one!\nand yes I need to see that movie.\nIt's so hard when the little ones are sick. Take care and feel better soon!\nI love this. I, too, live in a Navy town. My dad was in the Navy and I was a Navy brat. So that particular RAK is especially close to my heart! I just saw American Sniper with my son two nights ago. It was really good!\nThank you to your dad for his service! And to you for your sacrifices as well!\n\"Quietly being kind..\" totally hear you on this one.\nI prefer to do my giving in quiet when possible, too. I appreciate that you've generously given to Sailors. It is wonderful when their sacrifices are appreciated by others.\nI wish I could do more!\nI love your act of kindness. I prefer to also give quietly. I'm sure it made their day!\nThe quite acts are the best, I think.\nThat's really sweet that you pay for soldiers meals!\nI've heard the movie is excellent! And God bless you for quietly paying for meals for our service men!\nIt's a really good movie, and an important one for everyone to see.\nAfter 911 we became a military family. My husband signed up later in age than many. He was 34 when he went through basic training. The first four years he was away from home about 50% of the time. People have done this for us...paid for meals, etc. It means so much! Thank you for doing this for military people. It means so much to me to know people appreciate our military. Dang it...can't stop the tears!\nWow, I had no idea your husband enlisted after 911. What a brave thing to do - for him and for you! I definitely appreciate you!\nI think random acts of kindness are wonderful and I think it is super sweet that you and your hubby pay for soldiers meals... I bet that picks their day up..\nI think it's so important to teach this to our children!\nQuietly kind is the best kind of kind. As a former Army wife, thank you for treating some of country's heroes to a meal.\nTo say I \"enjoyed\" American Sniper would be inaccurate but it moved me, and rekindled my hatred of war, too. Thank you for your honest review.\nYes, American Sniper isn't really an enjoyable movie, but it is an important one! Thank you so much for your service as an Army wife - it's very difficult on the families too.\nWhat a beautiful random act of kindness. I love that you buy their dinner by staying anonymous. I am very much against the war, myself.\nThis is something that has affected me very closely in my life. My son was a Marine stationed in Afghanistan and it was the hardest thing of my life. He now suffers from PTSD and it has been awful. I have some upcoming posts related to that but I remember my son how touched it made him feel when people would do this for him. So as the mom of a Marine thank you for making sure my son had a good meal!\nI can't imagine how hard it must be to watch your son go through that. Hope he's continuing to heal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working Moms Only \u00ae \u2013 Are Mondays Killing Your Business?\nAre Mondays Killing Your Business?\nOr, we actually get up and go to the gym on Monday and Tuesday, but on Wednesday we just don't feel like going. So what do we do? We promise we will just start again on the following Monday.\nThis Monday morning do-over mentality is killing our productivity, our health, and our businesses.\nAnd with this new week, I do not want you waiting to start something. We have got to stop this \"starting over\" attitude and just start! So if that marketing plan needs to get done \u2013 do not say, I'll do it next Monday. Do it today or tomorrow.\nWe are almost half way through 2014. Where are you with your goals that you had planned for this year? If you are off track, it is not too late to still accomplish them. You just need to get rid of the \"Monday Morning Do Overs\" for good.\nAnd to help you, I have some simple guidelines.\nApply the following tips wisely and prosper.\n1. Don't launch your new venture, regardless if it is a new business, a new product, or a new franchise in the beginning of January: I did this with Working Moms Only and boy oh boy \u2013 it was a big mistake and a great lesson learned. Here's the deal, you are always going to need help. Starting a new business will be your biggest priority. However, it will not be for those you are counting on. Many people check out during the major holiday season. You may be writing your web copy on Christmas Eve. But I promise you, your webmaster will not be available for design issues.\nA current client who is very successful in his niche wanted to launch a brand new franchise in his company at the beginning of this year \u2013 one that required the content of specialist in an entirely different niche. When he soon realized folks were checking out from work for the rest of the year as early as Thanksgiving, he quickly changed the launch date to mid-February.\nIf you had planned on starting in January, jump on a new plan now!\n2. No Bartering: So before getting your business off the ground, money may be tight. So you think, \"I can barter services. If I write copy for someone, maybe in exchange they will build my website.\" In theory this sounds great. But here's the hard cold truth. The copy you may offer in exchange may not be needed as badly as you need the website.\nWhen suddenly the web person gets a great paying project, where do you think your site (that he is not actually making money on) stands? Pay for services. This is the only way to make all your deadlines and have your important tasks completed professionally.\n3. All Opportunities Are Not Created Equally: Once you are up and running, opportunities for all different types of deals will arise. Don't get derailed. Make sure you understand what any deal is worth to you on a long and short-term basis.\n4. Vet Partners Carefully: Only work with people who have the same moral fiber as you do. Often the money can be tempting, but if your morals and values are so far apart, the relationship will come to an end. Before entering into any relationship, do your research. Go online and read about the potential partner. Ask around your industry and listen to those you trust.\n5. Keep Your Advocacy Shining: Remember you started your business because there was a certain market you wanted to serve. Just because others who do not fit your ideal demographics joined your community does not mean you shift gears.\nFor example, I have men write to me each week. They tell me how much my advice has changed their lives. And inevitably they ask me to change the name of my site because they are obviously not working moms. My response is always the same. I tell them they are more than welcome in our community. That I am happy my advice helped them. However, my advocacy is for Working Moms. And, that is the person I am thinking about every time I write an issue or create a new product. That is the person who inspired me to start Working Moms Only.\nSo instead of watering down your message, you can always start a new franchise to serve others \u2013 which is exactly what I did when I launched The CEO's Edge.\nBy the way, you can grab a free subscription to \"The CEO's Edge\" right here. I highly recommend you do so right now.\nToday, I have shared actionable advice with you that you need to apply every day. So let's stop the Monday start-overs. Instead, let's make all Mondays a building block to our overall goal of moving forward. Doing this is the only true way to improve your life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get in touch with a county member.\nSelect a county to see news, employment opportunities, and learn more about the county's history.\nPresident Elect Sheriff Mark Cage, center, w\/Governor Michelle Lujan Grisham, Cabinet Sec. Mike Sloane, Olivia Padilla-Jackson, and John D'Antonio.\nCounties are encouraged to submit comments on the EPA and Army Corps of Engineers' new Waters of the U.S. rule by April 15.\nPERA Board of Trustees County Appointment Available \u2013 Deadline is Friday, April 26th at 5:00 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatjhj b/data_all_eng_slimpj/shuffled/split2/finalzzzatjhj
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+{"text":"That's right Track fans. Remember that amazing, predictable benchmark motion the Triton Elite gave you? Well, it's back, but with an updated coverstock and a slightly modified core.\nMC2 stands for \"Modified Core 2\". We took the shape of the original Core 2 Technology, found in the Triton Elite nearly 20 years ago, and made modifications which raised the RG and lowered the differential slightly. This core modification, in combination with the new coverstock, mirrors the Triton Elite motion that everyone loved.\nThe QR-7 series was originally featured on the Track Cyborg a few years ago. Now, for the first time in a solid version, we are wrapping the QR-7 solid around the MC2 Technology core at a 2000 grit Abralon finish to give you a big, continuous hook on medium oil conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Occupational therapy improves the functioning of individuals with Autism Spectrum Disorder (ASD), behavioral, and sensory processing challenges. Inclusion of sensory symptoms (e.g., over or under-responsive to or atypical interest in sensory stimuli) in the new DSM-5 ASD diagnosis has the potential to further promote the use of occupational therapy (OT) interventions for Autism. Research currently supports the relationship between sensory symptoms and functional behavior problems affecting individuals with ASD. However, the contribution of occupational therapists has been limited by a lack of attention to the interrelated sensory, behavioral, and developmental challenges of many individuals with ASD.\nOT uses the sensory integration and behavioral frames of reference to improve functioning in daily activities. The Sensory Profile www.sensoryprofile.com ; PaganoFABTriggerCopingForms ,activity analysis, and clinical observations can be useful components of an OT Evaluation. The OT Evaluation helps determine the most important functional goals related to sensory and behavioral challenges.\nFunctional goals for clients with ASD can include increasing seated behavior, attention, and keeping safe hands (e.g., eliminating physical aggression towards ones self and others). Objective baseline data is used in developing the functional goal and monitoring progress. Examples of baseline data may include: client currently sits a maximum of 10 consecutive minutes, attends to teacher selected tasks for an average of 5 minutes, or keeps safe hands (e.g., does not hit others) for ten consecutive minutes.\nOccupational therapy sensory interventions related to functional goals in ASD include psycho-education, environmental adaptations, sensory diet activities, and direct therapeutic intervention. Psycho-education involves teaching clients with ASD how their sensory processing, emotion regulation, and behavior challenges affect their functioning. A related intervention is coaching clients, families and professionals to develop environmental adaptations that enhance participation in daily activities.\nEnvironmental adaptations include the use of adaptive equipment and techniques that enhance daily functioning. Commonly used adaptive equipment for ASD include noise canceling headphones, fidget toys, chewey tubes, and theraputty. Frequently used adaptive sensory strategies for clients with ASD include initially reducing then if needed incrementally increasing sensory input in a socially acceptable manner, choosing one activity and doing it for a set period of time beginning another task, and using a sensory coping area to improve self-control.\nClients with ASD, sensory processing, behavior, and social skill challenges may receive direct occupational therapy services including sensory and behavioral interventions. Given the strong relationship between tactile sensory processing and behavioral problems in ASD, touch intervention appears important. Examples of direct intervention activities for clients with ASD include massage, scrub brushing, vibration, mindfulness, yoga, and movement planning tasks.\nBy combining sensory and behavioral strategies as part of a transdisciplinary team, OTs can achieve functional goals that enhance the daily life of individuals with ASD (e.g., increased attention, improved social interactions, decreased self-injurious behaviors). Pivotal Response Therapy (PRT) http:\/\/education.ucsb.edu\/autism is an evidence-based behavioral strategy that can be used in conjunction with sensory strategies. PRT provides developmental transdisciplinary interventions addressing effort, communication, and social interactions. Developed by a psychologist with expertise in Applied Behavioral Analysis and a Speech\/Language Pathologist, PRT is a practical intervention that is useful for teachers, parents, as well as occupational, speech, and physical therapists.\nFor example, a behaviorist and I provided services to a kindergarten girl who had ASD, sensory seeking, and behavior challenges. Her repeated hand mouthing was interfering with learning and damaging the skin on her fingers. Our transdisciplinary assessments included a FAB Sensory Functional Behavioral Analysis (FBA), Sensory Profile, and observations. The FAB Sensory FBA adds sensory environment antecedents (e.g., loudness, crowding, transitions) to traditional FBA data for greater focus on antecedents (what happens before problematic behavior that could be changed to improve functioning).\nThe evaluation suggested she was sucking her hands to receive sensory input in her mouth and wetness on her hands. Using a Preference Assessment and Activity Analysis, her favorite activities matching the sensory input she got from hand sucking were mouthing a chewey tube and playing in a water table. Intervention involved allowing use of the chewey tube and water table as well as rewarding her for going progressively longer periods with out sucking her hands.\nThis example shows how sensory and behavioral strategies can be combined to improve functioning in individuals with ASD. Transdisciplinary use of sensory and behavioral strategies is especially helpful for improving functioning in individuals with ASD, sensory processing, and behavior challenges. It is time for the conflicts to end so all professionals are working together to help our clients. Hopefully, recognition of sensory challenges in the new ASD diagnosis will encourage transdisciplinary sensory and functional behavioral interventions.\nFoss-Feig, J. H., Heacock, J. L., & Cascio, C. J. (2012). Tactile responsiveness patterns and their association with core features in autism spectrum disorders. Research in autism spectrum disorders, 6(1), 337-344.\nMazefsky, C. A., Herrington, J., Siegel, M., Scarpa, A., Maddox, B. B., Scahill, L.& White, S. W. (2013). The role of emotion regulation in Autism Spectrum Disorder. Journal of the American Academy of Child & Adolescent Psychiatry, 52(7), 679-688.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pole Banners are a great way to advertise in a parking lot or along a road. Pole banners are double sided with pole pockets for installation, and come in a variety of sizes to fit your needs.\nPole Banners are available WITH or WITHOUT hardware for installing on your light poles. Please be sure to choose whether you want to purchase the banners only, or if you want the Banners + Hardware.\nNOTE: Pole Banners are NOT ideal in high wind areas. ChurchINK is not able to offer refunds for ripped or torn banners due to wind.\nBleed requirements for banners is 1 inch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Occidens Tours organises your trip to charming Portugal to the last detail.\nOn your round trip, you will discover its unique cultural heritage: Manuelinan arts & arquitecture, azulejos & fado songs are are only a few accents of the authentic Portuguese identity, from the fascinating and bohemian capital of Lisbon to the dreamy Algarve coastline. A perfect destination for your next vacation!\nPortugal, founded in 12th century is known as the \"land of the explorers and discoveries\" is known for its ancient naval history. Important personalities like Vasco da Gama and Pedro \u00c1lvarez are only a few of a large number of Portuguese explorers, that have left a rich historic and cultural legacy.\nTo its visitors, it offers a multicolour landscape, composed of pittoresque cities and villages, terrific Atlantic coastline, spotted with lots of medieval castles and fortresses in its hinterland. And, last but not least, a very rich and varied gastronomy, reaching from all sorts of fresh seafood, fresh vegetables and lots of typical meals, according to the regions you might be travelling to. The most famous wine regions of Oporto and Mi\u00f1o will fascinate you, we promise.\nPortugal was elected BEST COUNTRY TO BE VISITED in 2014 by the Spanish edition of CONDE NAST TRAVELER magazine, and was named BEST VALUE DESTINATION by ROUGH GUIDES in 2014, too.\nThe Azores are situated in the Atlantic Ocean between Europe and North America. They have an area of 2,355 km2 and consist of nine islands \u2013 S\u00e3o Miguel and Santa Maria in the Eastern Group, Terceira, Graciosa, S\u00e3o Jorge, Pico and Faial in the Central Group and Flores and Corvo in the Western Group. It takes about two hours to get from the Azores to mainland Portugal by plane.\nDespite Fado being a symbol of the Portuguese nationality, it is not, by all means, the national song. From region to region, Portugal possesses several rich and typical folklores of each geography, which has nothing to do with Fado. Perhaps we can, if you like, say that it is the form of folklore of Lisbon, Porto and Coimbra.\nThe singer of fado (literally, \"fate\") speaks to the often harsh realities of everyday life, sometimes with a sense of resignation, sometimes with the hope of resolution. The music is performed by either a female or a male vocalist, typically to the accompaniment of one or two guitarras (10- or 12-string guitars), one or two violas (6-string guitars), and perhaps also aviola baixo (a small 8-string bass viola). Most of the repertoire follows a duple metre (usually with four beats to a measure), with a text arranged in quatrains or in any of several other common Portuguese poetic forms. Until the mid-20th century many fado performances featured a significant element of improvisation. Inevitably enriched with an array of emotive bodily gestures and facial expressions, fado aims\u2014and indeed, is required\u2014to evoke a penetrating sense of saudade (roughly, \"yearning\").\nSalted cod or bacalhau is a very common dish in whole Portugal and it is said that there exists at least one recipe for each day of the year.\nThere are said to be over 1000 recipes in Portugal alone and it can be considered the iconic ingredient of Portuguese cuisine (but curiously the only fish that is not consumed fresh in this fish-loving nation). It is often cooked on social occasions and is the Portuguese traditional Christmas dinner in some parts of Portugal.\nSimilar recipes can be found across Europe. It is also found in the cuisines of other territories and regions like Dominican Republic and Puerto Rico. In Norway, where most of the salted and dried cod is produced, bacalao commonly refers to a specific dish prepared with salted and dried cod, potatoes, onions, tomatoes and olives. In recent years, there has also been increasing interest in the wider range of Portuguese dishes based on dried and salted cod, however.\nSalted, dried cod, usually comes from Norway (bacalhau da Noruega), Iceland (bacalhau da Isl\u00e2ndia) or Newfoundland (bacalhau da Terra Nova). It used to be very affordable, but with the collapse of the cod stocks and dismantling of Portuguese bacalhoeiro fleet, it became more expensive, especially near Easter and Christmas time, since it is a part of many traditional dishes of the holiday season.\nBacalhau is also used to flavor a vegetable, dumpling and fish stew known as \"Dhokras\".\nThe Manueline (Portuguese: estilo manuelino, or Portuguese late Gothic, is the sumptuous, composite Portuguese style of architectural ornamentation of the first decades of the 16th century, incorporating maritime elements and representations of the discoveries brought from the voyages of Vasco da Gamaand Pedro \u00c1lvares Cabral. This innovative style synthesizes aspects of Late Gothic architecture with influences of the Spanish Plateresque style, Mud\u00e9jar, Italian urban architecture, and Flemish elements. It marks the transition from Late Gothic to Renaissance. The construction of churches and monasteries in Manueline was largely financed by proceeds of the lucrative spice trade with Africa and India.\nThe Azulejos \u2013 tiles \u2013 are everywhere to be found in Portugal, decorating everything from walls of churches and monasteries, to palaces, ordinary houses, park seats, fountains, shops, and railway stations. They often portray scenes from the history of the country, show its most ravishing sights, or simply serve as street signs, nameplates, or house numbers.\nAlthough not a Portuguese invention (the use of glazed tiles began in Egypt), azulejos have been used more imaginatively and consistently in Portugal than in any other nation. They became an art form, and by the 18th century no other European country was producing as many tiles for such a variety of purposes and in so many different designs. Today, they still remain a very important part of the country's architecture.\nAfter the Gothic period, most large buildings had extensive areas of flat plaster on their interior walls, which needed some form of decoration. These empty architectural spaces produced the art of the fresco in Italy, and in Portugal, the art of the azulejo.\nThe term azulejo comes from the Arabic word az-zulayj, meaning \"polished stone.\" The Moors brought this term to the Iberian Peninsula, but despite their long presence, their influence in early Portuguese azulejos was actually introduced from Spain in the 15th century, well after the Christian reconquest. No tile work from the time of the Moorish occupation survives in Portugal.\nKing Manuel I was dazzled by the Alhambra in Granada (Spain), and decided to have his palace in Sintra decorated with the same rich ceramic tiles. The first ones were imported from Seville, and in accordance to Islamic law, they portrayed no human figures, only geometric patterns.\nGradually the Portuguese painters weaned themselves off ornamental decoration and employed human or animal figures in their designs.\nThe dominant colors were blue, yellow, green and white, but in the 17th century, large, carpet-like tiles used just white and blue, the fashionable colors at the time of the Great Discoveries, influenced by the Ming Dynasty porcelain from China. They now portrayed Christian legends, historical events, and were not only decorative, but also protected against damp, heat and noise.\nLocated in the South west extreme of Europe, famous for its unique coastal landscapes and white sandy beaches, the Algarve was the last region of Portugal to be conquered from the Moors in the mid-12th century. The region is not only famous for its beaches, sun and sea; it also holds a very long historical and cultural heritage from which remain old bridges, castles and city walls.\nAfter the Moorish conquest, the Algarve \"Al-Gharb\" became the Kingdom of the Algarve, and annexed to the Kingdom of Portugal after that. Nevertheless, influences and signs of Moorish presence are still visible throughout the region. In fact both Algarve and Alentejo (the greater region to the North of the Algarve) were of Muslim origin, the whole region was known before as \"Gharb Al-Andalus\", it means The West of Al-Andalus because this was the western part of the Muslim Andalusia, and one of the most important Muslim centers.\nNowadays the region corresponds to approximately 6% of the area of the country, and its population is concentrated in the southern coast, and in major cities such as Faro, Lagos, Portim\u00e3o, Olh\u00e3o, Tavira and Vila Real de Santo Ant\u00f3nio.\nDue to the proximity with the North of Africa, the Algarve region has a warm and mild climate throughout the year. It's home to unique natural reserves, parks and beautiful coastal areas and a lovely countryside worthy of exploring by anyone looking for an adventure. In the Algarve you can still visit one of the 7 Natural Wonders of Portugal: the Ria Formosa \u2013 one of three protected areas where you can experience great nature tourism activities and tours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Beit Halima restaurant in Amman, Jordan Among the great participation of his audience and his fans.\nOf the songs of the songs to offer his songs after which he met the \"Dihya\"\nWhich is famous for this.\nThe great poet Ali Al \u2013 Khwar.\nAnd the actions to be launched in the coming days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Rat City Studios is the workspace of Deborah Schwartzkopf, Seattle based studio potter and instructor. Her aim is to build the community of artists working in clay by offering studio assistant positions for emerging artists, connecting people through social and educational events, and maintaining lively, professional career in the ceramic arts!\nThank you for buying from the source! Enjoy your new connection to Rat City Studios. We hope to see you at event in the future. We would love to feature your photos of dishes in use on our Instagram or blog!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I rattle-can painted the parts with Rustoleum Flat black automotive primer, then used the masks to protect the tires. A tooth pick\/cocktail stick was used to push the mask into the tire bead area, and I then sprayed the wheels with Tamiya bare-metal silver. Mask removal demonstrated perfectly-round painted wheels! Where I had one mis-alignment of the mask due to operator error, a bit of 00 brush flat black wash touch-up and I was finished with the main wheels.\nThe tail wheel assembly was also sprayed overall bare-metal silver, then the same 00 Brush used to paint the tire, with a thin wash added to fill the tire bead area around the wheel rim. This completed this upgrade part.\nOverall about the simplest part of the Wellington project; I was impressed how well the masking worked, and have decided they greatly improve my performance of a clean, crisp edge on the wheels.\nHIGHLY recommended by this reviewer, and once again the pictures tell the complete story; The usual high praise and full marks to Eduard for listening to those of us who build and clamor for detail, and providing us opportunity to make more bank runs....also thanks again to Phil for forwarding and John approving my working these sets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Often people experiencing hurricane fury feel that the earth appears to be in its death throes. Hurricane Camille created such an effect.\nOn August 17, 1969, Camille roared across the Gulf Coast, slamming inland near Gulfport, Mississippi. Winds at speeds of over 200 miles per hour lashed the earth, and a deluge of rain fell on the land. When its fury was abated, Camille left behind more than 250 dead and about eighty missing. The property damage amounted to well over $1.5 billion.\nAs soon as Hurricane Camille first came roaring out of the gulf and rode over Mississippi, some 75,000 people fled inland to higher ground. However, a few of them decided not to join the timid escapees.\nIn the town of Pass Christian, Mississippi, twenty\u2014five guests of the resort hotel Richelieu made a fateful decision. Camille was already thrashing at the hotel when the twenty\u2014five fun lovers decided to throw a hurricane party! And so, as officials pounded on doors shouting last warnings to evacuate, the guests, amused at their own trickery, kept very quiet. Convinced that the hotel was completely evacuated, the officials withdrew and sought safety inland.\nThe hurricane party began but was rather short\u2014lived. The storm completely demolished the hotel, killing twenty\u2014three of the party group. The two survivors would make only one unforgettable comment to justify their actions: \"We thought it would be great fun!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jimmy Page, Robert Plant and John Bonham of Led Zeppelin pictured in 1970.\nA trial is needed to decide if Led Zeppelin's Stairway To Heaven copies its opening notes from a song performed by the rock band Spirit, a judge has ruled.\nUS District Judge R Gary Klausner ruled that lawyers for the trustee of late Spirit guitarist Randy Wolfe had shown enough evidence to support a case that Stairway To Heaven copies music from his song Taurus.\nTaurus was written by Wolfe - who used the stage name Randy California - in either 1966 or 1967, years before Led Zeppelin released Stairway To Heaven in 1971.\nThe judge wrote that while the songs have some differences, lawyers for Wolfe's trustee may be able to prove they are substantially similar.\nJudge Klausner wrote that the evidence presented so far represented a circumstantial case that Led Zeppelin may have heard Taurus performed before Stairway To Heaven was created.\nExperts hired by Led Zeppelin contend both Stairway To Heaven and Taurus include notes that have been used in music for centuries.\nFrancis Alexander Malofiy, lawyer for Wolfe's trustee Michael Skidmore, praised the ruling.\nHe said while many copyright cases are an uphill battle, the judge's ruling brings his client one step closer to getting Wolfe credit for helping create one of the most recognisable song introductions in rock history.\nMr Skidmore was able to overcome statute-of-limitations hurdles to sue over Stairway To Heaven because the song was remastered and re-released in 2014.\nA jury trial is scheduled for May 10 in Los Angeles.\nThe judge's ruling removed Zeppelin band member John Paul Jones from the case. Bandmates Robert Plant and Jimmy Page remain as defendants.\nTo listen to Taurus, by Spirit, click here.\nA trial would represent the third time in little over a year that a Los Angeles jury has heard a copyright-infringement case involving a hit song.\nIn March last year, a jury found that Robin Thicke and Pharrell Williams had copied a Marvin Gaye song to create their 2013 hit Blurred Lines, and awarded Gaye's children 7.4 million dollars (\u00a35.2 million). A judge reduced the award, and the verdict is under appeal.\nLater in 2015, another jury was sworn in to decide or not the Jay-Z hit Big Pimpin' copied the work of an Egyptian composer, but a judge ruled in the rapper's favour before deliberations began.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please contact boat owner David at 201-341-0159.\nThis is a great boat, very clean and comes WITH a COVERED WET SLIP (electric and water), paid until mid-February, 2015! 5.7 L Mercruiser engine, only 180 hours on engine.\nPurchases new in 2009, never on water before then.\nKept in fresh water only.\nWinterized and shrink-wrapped every winter.\nCurrent loan payments only $320\/month.\nQualifies as second mortgage \u2013 deduct interest!\nBimini, mooring cover, snap-in carpet, Dual Battery Charger, Stereo, spotlight, SIRIUS radio, Bose stereo system microwave, Range, Refrigerator, Vacuum Flush Head w\/holding tank & Pump Out, shower, sink, wet bar, fish finder, ski-tow ring, rope compt, swim platform\/ladder, under-seat storage, Galvanized trailer included, and more!\nBoat is outfitted with internal ventilation system that can be outfitted for A\/C off battery.\nThis is a MUST see! ** Serious buyers: If not buying outright, owner will provide loan # and VIN to buyer.\nBuyer will obtain loan from their bank.\nBuyer's bank will transfer funds to owner's bank to pay off loan, then owner's bank will transfer title to buyer's bank to hold for new owner until buyer's loan is paid off in full, or boat is sold again.\nIt's just like financing a used car purchase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawbuq b/data_all_eng_slimpj/shuffled/split2/finalzzzawbuq
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+{"text":"The costume was made using a striped suit fabric with fabric straps and buckles. The boots were of leather and the gauntlet, cuffs and armor were made using vinyl. I used hotglue and styrene to decorate the gauntlet. The collar was made using vinyl and feathers. The ringblade was made using foamcore board, styrene, insulation foam and enamel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A rustic set of bunch of five bells hanging on a thick string, each bell slightly different with their gold styling and distressed aged feel, complete a look and hang from a bedroom door, mantel piece. An eclectic decoration and perfect gift.\nA decorative set of rustic bells that will not go unnoticed and a treasure to own, hook pictured not supplied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was a beautiful evening at the Hollywood Bowl for the KCRW-sponsored tribute to Serge Gainsbourg curated by Beck. The all star line up included Ed Droste (Grizzly Bear), Victoria Legrand (Beach House), Mike Patton (Faith No More), Sean Lennon & Charlotte Kemp Muhl (The Ghost Of A Saber Tooth Tiger), Zola Jesus, Lulu Gainsbourg, and up-and-coming rocker Joseph Gordon-Levitt all backed by Beck's Sea Change era band. After playing a number of classic Gainsbourg tunes, Jean Claude Vannier conducted a recreation of Gainsbourg's album Historie De Melody Nelson. C'\u00e9tait magnifique. Check out performances by Beck, JGL, Zola, and the Ghost Of A Saber Tooth Tiger below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The interplay of music and identity, in particular with regards to Nazi cultural policy in Austria's province Tyrol, is one of Bernhard Achhorner's research interests. Mr Achhorner obtained his degree from the University of Innsbruck and was a recent research fellow at the \"Center Austria: The Austrian Marshall Plan Center for European Studies\" of the University of New Orleans. As a musicology and anthropology student, he investigated the intersection between music and identity, music and politics, as well as intermediality and performance studies. Within the framework of our commemorative series of several historical events this year, including Austria's annexation to the Third Reich 80 years ago, we are featuring an interview on Mr Achhorner's research, asking about the special role of music in propaganda generally and during the Nazi regime specifically, his views on Austrian folk music today and his take on commemorating these historical events. You will be able to hear him in person during his lecture about Music as a Strategy of Cultural Policy in Nazi Germany, which he will hold at the ACF Washington on September 5th.\nFrom your research on the influence of music as a strategy for cultural policy in the NS times, what was for you the most striking, unexpected result of what impact music can have for propaganda purposes? Is there a big difference compared to images and photographs, also extensively used for propaganda?\nWhat made the greatest impression on me was how National Socialist cultural and identity policy was able to exploit an archive of commonly inhabited history for the purpose of creating a sense of national belonging and national identity. By reinterpreting mythical, primordial texts that are foundational for the identity of a community, by leaving out facts in the biographies of local heroes \u2013 like, for example, the strict Catholicism of Andreas Hofer or the Communist aspects in the thinking of Michael Gaismairs \u2013 and also emphasizing local rootedness and nativity, as well as by conveying a romantic notion of rural life, it was possible to make culture and the state coincide and to propagate their unity as a naturalistic and biologistic construct. Concretely, this means that an attempt was made systematically to conceal the cultural hybridity that has always already existed. For culture cannot be regarded as a self-enclosed, fixed structure, but rather has constantly to be thought as fluid and permeable. Culture is a process and comes into being in continuous exchange with other cultures. But in National Socialism and, especially, in the Reichsgau Tyrol-Vorarlberg, an image was conveyed that depicted cultural aspects \u2013 like, for example, a supposedly special talent for making music, but also precisely character traits like defiance and combativeness \u2013 as something that it is written into the DNA of Tyroleans. This led, among other things, to a strong idealization of so-called Volkskultur \u2013 literally, \"popular culture,\" but with a pronounced ethnic connotation that is absent from the English expression \u2013 and of custom.\nOne could perhaps affirm that many composers were only eager to further their careers and submitted to National Socialist blood-and-soil politics for this reason. This would, however, be an erroneous conclusion, since many of them proclaimed their allegiance to the Greater German Reich, to the F\u00fchrer and to the National Socialist party already in the period immediately preceding the Anschluss. Thus, for me, perhaps one of the most surprising discoveries was that many composers were able to identify with the racist and chauvinist ideology and hence were not only fellow travelers, but convinced National Socialists.\nOn the difference between music and image as tools of propaganda, the following can perhaps be said. Music as an emotive medium, with its potential to express emotions and to trigger them among listeners, is, of course, extremely well-suited for the dramaturgy of the National Socialist culture of popular festivals and celebrations, which was manifest in mass marches and rallies. Music played a key role in this culture as bearer of ideological contents. The singing together of songs with mostly war-mongering and anti-Semitic lyrics, as well as marching in rhythm with the music, are, of course, predestined for awakening a sense of belonging. In contrast, propaganda by way of images and photographs functions perhaps on a more subtle level, whose content and message are conveyed via other channels.\nDo you think the importance of music for this kind of strategy was particular for the Nazi regime or could your findings also be applied more generally to other totalitarian regimes from different areas of the world?\nI think that, above all, the way in which music was used as a tool of propaganda in the National Socialist period in Tyrol can be described as unique. The systematic restructuring of the Standsch\u00fctzenverband (a kind of popular volunteer defense force) into a key institution for all cultural affairs and events, which the marching bands had also to join, and the publication of a Reichsgau song book, comprising traditional Tyrolean songs, songs of the youth movement and anti-Semitic songs, undoubtedly represent a particularly pronounced form of musical propaganda for the purpose of consolidating an identity. But there are, of course, also parallels to other totalitarian forms of government around the world. All movements of national reawakening made use of art and culture, in order to attain their cultural-political and also bio-political goals. In large measure, what is at issue here is creating or even imposing possibilities of identification. Precisely in the second half of the 19th century, more and more efforts have been made to develop explicitly national musical styles.\nEspecially in Austria today, traditional music, the so-called \"Volksmusik\", is often being associated with a certain political ideology and thus, following-up on your research, might allow an interpretation that it never fully escaped from its propaganda use in the NS time. What do you think about that? What would be necessary to bring all these concepts and cultural expressions that are linked in Austria to the ideas of \"Heimat\", patriotism, folklore and traditional cultures into a more modern, light and neutral understanding?\nYes, I also fundamentally agree with this view. Popular Volksmusik and traditional brass bands are still today associated by some people with certain political sentiments and a certain past. This is partly justified, I find. For defensiveness, whitewashing biographies and preventing access to archives cannot be the goal of a sustainable efforts at elucidation \u2013 this is also the case for the next generation of musicians of the said institutions themselves. Instead, a clear commitment to seamlessly examining their implication in the cultural-political agenda of the National Socialist period could provide a strong signal. Freeing such music from general suspicion can only happen, namely, if this dark chapter in the history of Austrian popular culture is transparently and openly discussed. In this whole matter, however, it should also not be forgotten that it was not only Volksmusik that was instrumentalized for the purposes of propaganda: classical music was also composed with the corresponding ideological contents or tinted by National Socialist conceptions. Music historian Kurt Drexel, for instance, points to the fact that Beethoven's Ninth was used in the Third Reich on official occasions and today is the European Anthem.\nOn the other hand, however, it should also be noted that precisely \"avant-garde\" currents in Volksmusik and brass band music are assuming ever greater significance in the Austrian musical landscape \u2013 especially in recent decades. This applies to \"authentic\" Volksmusik, but also to other genres in which so-called crossover elements are experimented with. Most of the musicians in these groups have a university degree in music, but were socialized in the local Volksmusik and brass band tradition. Making \"back to the roots\" their motto, they try to reinterpret their musical roots with technical precision or with jazz elements or to compose new innovative pieces for a variety of ensembles.\nIn 2018 Austria commemorates 80-years of the so-called \"Anschluss\" to the Nazi-regime in 1938, one of the reasons why we also invited you to give this lecture at the Austrian Embassy. What is your impression of the activities of this commemoration so far, do you think it helps keeping the memory alive?\nI think that the 21st century society is right now in a period of transition. A transition from living remembrance in organic memory to remembrance that is, for the most part, materialized in cultural objectifications. A transition that results in memories of one of the cruelest periods since humanity's very beginnings being only still accessible in the form of documents or gaining expression in monuments. Precisely because the generation of eyewitnesses, who remind us of the events of the National Socialist period on a daily basis, is slowly passing away, these sorts of events dedicated to a culture of remembrance cannot be given enough importance. This is the only way to assure that the past is constantly made contemporary and retained in the present, such that the atrocities of World War II are not forgotten. The events at the ACF Washington, at the Austrian Embassies around the world as well as in Austria itself are making a significant contribution.\nFinally, a personal question: the combination of musicology and anthropology is a rather rare one, how did you develop your specialty, how did you get interested in these topics? What are your plans for the future?\nI perhaps should clarify something in this connection. Thus, the addition \"cultural\" should in fact be placed before the term \"anthropology.\" In the German-speaking countries, cultural anthropology is one of the possible designations for the discipline that was formerly known as Volkskunde and that was heavily implicated in National Socialist agendas. The renaming occurred following the efforts to come to terms with the role of the discipline in the National Socialist period. The second common designation would be \"European Ethnology.\" The discipline predominantly deals with the past and present of manifestations of everyday and popular culture. Since also in the context of musicology, I was already always particularly fascinated by the sub-area of musical ethnology and dealing with other musical cultures, specializing in cultural studies approaches to the study of musical phenomena was a logical step. The connections of music and identity and of music and politics are of central importance for me here. In the future, I hope to be able to expand further into the area of musicology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PLYMOUTH, N.H. \u2013 Three Plymouth State University student-athletes were honored by the Massachusetts State Collegiate Athletic Conference (MASCAC) Monday for their performances in Panther games last week.\nSophomore running back Mitch Banuskevich (Milford, N.H.) and freshman defensive back Prince Gbaley (Manchester, N.H.\/West) were selected to the MASCAC Football Weekly Honor Roll, and sophomore forward Ryan Stevens (Colleyville, Texas) was named to the MASCAC Men's Ice Hockey Weekly Honor Roll. Banuskevich and Gbaley were standouts in PSU's conference victory over Massachusetts Maritime last Saturday, while Stevens helped lead the Panthers to a 1-1 record last week in a pair of non-conference games.\nA 5-11, 170-pound running back, Banuskevich totaled 199 all-purpose yards with one touchdown in the Panthers' 28-7 win over Mass. Maritime. He rushed a career-high 29 times for a career-best 167 yards (5.8 average) and caught two passes for 32 yards and a touchdown. His TD reception was from 13 yards out with 45 seconds left in the second quarter that gave the Panthers a 14-0 lead at the intermission.\nA 5-11, 180-pound cornerback, Gbaley collected four tackles, a pass breakup and an interception return for a touchdown in the Panthers' win over Mass. Maritime. He returned the interception 60 yards for a touchdown to put the team up 28-0 with just under two minutes remaining in the third quarter.\nThe PSU football team, 5-4 overall and 3-4 in the MASCAC, wraps up the season this Saturday at Currier Field on Senior Day against Fitchburg State University at 1:00 p.m.\nA 6'3, 200-pound sophomore transfer from Arizona State University, Stevens tallied four points on two goals and two assists in a 1-1 week for the Panthers. He scored two goals in Thursday's home opener, a 7-2 win over Stonehill College, and added two assists in Saturday's 4-3 loss to 11th-ranked Babson College.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawtkk b/data_all_eng_slimpj/shuffled/split2/finalzzzawtkk
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+{"text":"Originally scheduled for release in 1968, Wait For My Love is the Elevators' lost 6th single.\nRemastered by Sonic Boom (Spacemen Three) and packaged in an International Artists Records\u2122 house bag, this slice of hidden history is now available as a limited edition 7\" green vinyl edition.\nReleased as part of Record Store Day, these are the remaining copies of a limited edition run (2000 copies worldwide).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BODA have negotiated a discounted rate for members who book in advance for either Friday or Saturday. Details and the discount code can be found by reading the attached flyer. Just click on the image for a better view.\nThe club forum has now been enhanced.\nYou can now see which topics you have read and can mark favourite topics as \"liked\". The forum also shows most recent topics and replies, has a search facility and even some forum statistics.\nI hope you find the additions useful.\nNeed new main bearings \u2013 0.010 . Any suggestions on suppliers? Standard and 0.020 seem to be obtainable but not 0.010 .\nAlso a replacement \/new camshaft required for same engine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To view the corresponding video to this blog please click HERE!\nYou know you are, truly, a stationery geek when the jingle of new staples (or in this case Supaclips) hitting your doormat is enough to see you running off to your office and praying that your video camera has enough charge to film you tearing into the packaging\u2026 unfortunately in this instance it didn't and I literally couldn't wait. So I opened it without you!! I'm sorry, I know I shouldn't have but you know how it is haha!\nI honestly do not know how i've never known about this gadget before\u2026 for a stationery blogger who loves a gadget I feel like I need to tell myself to go write 100 lines (on some very lush paper, of course), apparently these have been around for a while and some people had them at school, all we had was treasury tags and paper polos (yes and even they bring joy to my geeky little heart).\nOk Ok onto the review I hear you yell\u2026 well, it's awesome, the clips I have are pale pink, bright pink (oh yeah baby) with hearts and then some yellow emoji clips too.. there are also silver that come in the pack (rose gold Rapesco\u2026 where are the rose gold??) The clips are small but fit up to 40 sheets of paper and they hold them really securely, the fact that they are small is such a benefit as I can use them in my note taking and planner without covering much space on the paper.\nYou know I can be a bit of a cack handed fool so I didn't manage to jam the tool on video\u2026 and you do have to make sure it is pointing down or the clips fall backwards and you cant slide them over the edge of the paper but then that is just logic I suppose ha.\nProbably my second favourite thing to clip with this is bags of half eaten crisps and sweets, not only do they keep the contents from spilling out they also stop the children getting back in haha, i'm joking of course!!\nYou can purchase the pack, which contains the dispenser and 40 clips, for \u00a33.42 on amazon and the packs of 100 clips are currently \u00a36.04 per pack.. bargain I tell you!!\nFor more information on Rapesco and the products they offer please see their website HERE!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to BustinSkin, the home of Dorset's leading triathlon and events club Pain is weakness leaving the body...and death is natures way of telling us to slow down!\nJoin the BustinSkin team at any of our events and push yourself to the limit!\nIf you have any timing queries then please contact as always \u2013 Full On Sport support@fullonsport.com and the guys and girls will be more than helpful in getting your time sorted out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A stroke is a life-changing event for a patient and his or her family. The acute stroke patient is at risk for developing aspiration pneumonia, whether silent or overt. Prevention of pneumonia in this population requires timely completion and documentation of the bedside swallow screen to identify those patients at risk for aspiration pneumonia; however, anecdotal data from the emergency department at the site of this project suggested that completion and documentation of the screening were inconsistent. Guided by the quality caring model adopted by the project site as well as the logic model, the aim of this project was to evaluate emergency room nurses' compliance with documentation after completing a modified bedside swallow screening. To facilitate documentation compliance, the current bedside screening tool was modified to make it user friendly. Electronic records of stroke patients (n = 104) admitted to the emergency room were monitored for a period of 6 months after implementing the modified bedside swallow screening tool. The findings indicate that implementing the modified bedside swallow screening tool achieved 93% documentation compliance in the electronic records and 100% documentation in patient charts over this 6-month period and clearly identified patients at risk for developing aspiration pneumonia. Further study is recommended to determine the relationship between the results of the modified bedside swallow screening and the development of hospital-acquired pneumonia. Implementation of this modified bedside swallow screening tool can initiate therapeutic measures to reduce the incidence of aspiration pneumonia in the acute stroke patient, resulting in shorter length of hospitalization and reduced health care costs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Uttarakhand is a state of India located in the north. It is famous for its many Hindu temples and pilgrimage destinations as well as scenic natural surroundings replete with mountains, rivers, glaciers, abundant greenery and many other natural elements.\nPick up at Haridwar station and drive down to Mussoorie. On arrival, check in at hotel. Evening scroll on the mall road, visit gun hills by cable car.\nMussoorie - Roosted on an edge 2 km high, the 'Queen of Hill Stations' competes with Nainital as Uttarakhand most loved holiday spot. On a clear day, the view of the green valleys and the distant snowcapped Himalayan peaks are a visual delight for every traveller.\nAfter breakfast, proceed for sightseeing, visit Company Garden, Camel's back road, Gun Hill by cable car, Kempty Fall which is 15 kms on the Yamunotri road. Return to hotel and overnight stay at hotel.\nEnjoy breakfast at hotel. Day at leisure for your own activities. Overnight stay at hotel.\nAfter breakfast, drive back to Haridwar. You will be transfer to Haridwar station for return journey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marisca is an independent 20-year-old student. Her dream is to teach children - in the foundation phase.\nBACKGROUND: Marisca is an independent 20-year-old student. Her dream is to teach children - in the foundation phase. Her mom, Hester is also a teacher - so a passion for teaching is in her blood. Besides studying hard, Marisca also works two jobs. She also has a sister in high school - and together these three women fend for themselves. Despite Marisca and Hester's best efforts to pay for her studies, they're still R14 000 short for this years' fees. If the bill is not settled, Marisca cannot continue to her 2nd year - which will bring a new R40 000 tuition challenge. Marisca wrote to Good Morning Angels as her last hope.\nSPONSORING: Roman's Pizza will assist Marisca with the R14 000 outstanding class fees for 2018. Roman's will also help Marisca with another R21 000 towards her 2nd year fees: a total sponsorship of R35 000!\nI am 20 years old, studying education B ED foundation at Aros - first year. My mother and I funded my studies this year, but we couldn't keep up with all the payments.\nMy mom is a single mother - also a teacher. I also have a younger sister in high school.\nI cannot continue with my studies due to the outstanding money for this year (2018).\nI currently have two jobs which, still does not add up to all the money for university.\nI need to settle my account and I need help with it. There is R14 000,00 outstanding for 2018.\nI also need to pay R3 500 registration for next year - which will be a total of R40 000,00.\nI love what I do and I cannot wait to become a teacher to educate the youth, but I need help to live my dream.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's to like? The color, sport package and 5-speed manual in a wagon, of course. It's got heated seats, too, and Xenon lights. In all, it's nicely optioned. What's not to like? There's zero disclosed maintenance, zero history, and you're at 160,000 miles on a fourteen year old E46. Granted, these have a pretty robust build all in all, but still, you need to look for rust and leaks. The price is a strong premium for the color and configuration. And unlike the N52, the M54 in the 325 with the wagon and all-wheel drive can feel a bit sluggish. It also will struggle to return mid-20s with the 5-speed.\nBut you probably don't much care, because it's a Dakar Yellow sport package manual Touring. UNICORN!\nTagged 2003, 325xi, BMW Individual, Dakar yellow, e46, M54, Sport Package, touring, Wagon.\nMiami Craigslist? Run away. Seller is in NH.\nGood spot, @Slim Chance. I'll update with the seller's site. Looks like maybe a scam?\nI noticed this on CL the other week and it was posted on BaT within the last week. The comment thread on that brought up some of the history that the internet had dug up \u2013 looks like the car may be rusty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Announcing Our New StarTalk Radio Survey: Who Owns Space?\nWho owns space? Can a corporation claim an asteroid? Can a country establish a national park on the Moon? If your satellite crashes into my space station, can I sue?\nThe idea for our newest survey started last Spring, when we asked you for questions about asteroids for our episode, Cosmic Queries: Asteroids, Comets and Meteor Storms. A few of you asked about who owns asteroids and who has the right to their mineral wealth. We'd also just done a show with Peter Diamandis of Planetary Resources, called Eureka! Asteroid Mining, where Peter and Neil deGrasse Tyson discussed the relationship between government and private industry in the exploration \u2013 and exploitation \u2013 of space, especially when it came to mining asteroids.\nA few months later we posted an article on Facebook and our other social media channels about the bill that was introduced in the US Congress to establish the Apollo Lunar Landing Sites National Historical Park on the Moon. Many of you raised questions as to whether the US had the right to claim part of the Moon as a National Park.\nThe annoying thing about each of these questions is that nobody really knows the answers to them, yet. The courts haven't decided the matter. And for that matter, which courts? If Congress wants to declare a National Park on the Moon, what authority has the right to say we couldn't? If the UN said no, would we even listen? Would the International Court of Justice (aka the World Court) in The Hague decide whether the Russians had to pay for the damage their satellite caused \u2013 if the events in Gravity were real, that is?\nWe're going to ask a different question each week. You can vote once per question. We're also allowing you to choose \"Other\" and then write in your own answer in the comments at the bottom of the survey. And by the way: if you see an answer you like in the comments section, please let us know that, too. We'll put the responses up right here on the blog next week, as well as publishing them in our newsletter.\nThis week's question is, \"Who Owns an Asteroid?\" You can take the survey here.\nStarTalk Radio Survey: Who Owns Space?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Choose six foam rollers in soft or standard density, or a combination of both. With varying densities, there's a foam roller that's right for every patient or client's needs.\nOPTP\u00ae PRO-ROLLERS\u00ae are built to outperform and outlast store model foam rollers. High-quality materials and closed-cell construction make this foam roller long-lasting and comfortable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxfuz b/data_all_eng_slimpj/shuffled/split2/finalzzzaxfuz
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+{"text":"Written by Australian authors Judy Higgins and Betty Rea. This is the first Australian book on Budgerigars published in 27 years. The book offers information on how to breed some of the rare varieties of Budgerigars. It cover many aspects of breeding and keeping Budgerigars as pets and breeding for the show bench, with 210 pages containing colour pictures of most varieties. The book has had many great reviews, here and overseas. It has been quoted in England as the \"Budgie Bible\" (See review here). This book has a hard cover with sown sections. This is a great reference book, which will last for many years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This resort is sprawled over 12-hectares of land, and is the first and only five-star international resort on the island of Boracay. The resort includes 219 rooms with 36 villas and suites, leisure facilities, 350 metres of secluded beachfront, an infinity pool, a spa, multiple dining options and a thriving ecosystem of diversity with rare flora and fauna. The resort villas house some of the countries most indulgent and stylish accommodation. The Treehouse villas sit atop the resort's highest grounds, with spectacular views of the sea. Feasts of local materials were used in the design, including the island's coral stone on the exterior architecture. CHI, The Spa at Shangri-La, embraces a holistic approach to physical and spiritual well-being, inspired by Asian healing philosophies. Shangri-La serves guests with water that is produced by its own water bottling plant. The resort also practices solid waste management and all food waste from the various kitchens are collected and brought to the resort's backyard. Surrounded by an eco-reserve, Shangri-La's Boracay Resort and Spa is truly the perfect paradise destination.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Targum Onkelos even refuses to allow them 'god' status, changing , \"to their gods\", to , \"to their idols\".\nHirsch expands the text a little to offer an explanation of what HaShem might have meant: \"You are neither to think that your future is dependent on the supposed physical powers of these gods, nor that you owe them anything for your past and present and so you are not to allow their delusions to have any influence on your doings.\" Hirsch's choice of the word 'delusion' is interesting, pointing us towards Rav Sha'ul's comment to the community in Thessalonica: \"Therefore G-d sends them a strong delusion, so that they may believe what is false, in order that all may be condemned who did not believe the truth but had pleasure in unrighteousness\" (2 Thessalonians 2:11-12, ESV). Perhaps there is a hint here that those who worshiped idols in ancient times refused to believe the truth - the witness of the heavens and the world around them (Acts 14:16-17) - because they preferred to live in unrighteousness.\nGersonides to pick up the third prohibition: \"Do not worship Me as they do their gods.\" Nahum Sarna affirms that \"the adoption of their cultic practices - even in the service of the G-d of Israel\" are forbidden. HaShem is not to be worshiped or served in the same way that the idols or pagan gods are.\nHow then, is G-d to be worshiped? Samuel fired perhaps the first warning short across the bows of the sacrificers - \"Has the L-RD as great delight in burnt offerings and sacrifices, as in obeying the voice of the L-RD? Behold, to obey is better than sacrifice, and to listen than the fat of rams\" (1 Samuel 15:22, ESV) - although he did himself offer sacrifices. This questions the priority or value of sacrifice compared to living within the terms of the covenant. David picked up the baton in several psalms - for example, \"For you will not delight in sacrifice, or I would give it; you will not be pleased with a burnt offering. The sacrifices of G-d are a broken spirit; a broken and contrite heart, O G-d, you will not despise\" (Psalm 51:16-17, ESV) - while Hosea affirms, \"For what I desire is mercy, not sacrifices, knowledge of G-d more than burnt offerings\" (Hosea 6:6, CJBa). It is not at this stage, although David gets very close to the line, that the practice of sacrifice is discouraged; rather that sacrifice on its own is not acceptable or appropriate without a heart attitude of worship, contrition or service. The people were later challenged more directly: \"For when I freed your fathers from the land of Egypt, I did not speak with them or command them concerning burnt offerings or sacrifice\" (Jeremiah 7:22, JPS). This almost seems to be a denial of the Torah, so requires careful handling. What did G-d command the people? \"Do My bidding, that I may be your G-d and you may be My people; walk only in the way that I enjoin upon you, that it may go well with you\" (v. 23, JPS). Back to the question of obedience again. A sacrifice given out of rebellion or disobedience is meaningless!\nDoes this lie behind Yeshua's words to the Pharisees - \"Woe to you, scribes and Pharisees, hypocrites! For you tithe mint and dill and cumin, and have neglected the weightier matters of the law: justice and mercy and faithfulness. These you ought to have done, without neglecting the others\" (Matthew 23:23, ESV)? Lots of modern readers stop halfway through the second sentence, without including the rest of the verse, as if Yeshua were criticising the tithing of the herbs. It is not that tithing the herbs was the wrong thing to have done - on the contrary, it was perfectly correct and appropriate as far as it went. The problem was that the Pharisees had taken their eye off the larger matters: justice, mercy and faithfulness, things that the prophets had been shouting about for hundreds of years. And in that context, the 'smaller' matters of tithing, just like sacrifice, were not acceptable on their own.\nNow consider Yeshua's conversation with the woman at the well in Samaria, recorded in John's gospel: \"But the hour is coming, and is now here, when the true worshipers will worship the Father in spirit and truth, for the Father is seeking such people to worship Him. G-d is spirit, and those who worship Him must worship in spirit and truth\" (John 4:23-24, ESV). How many times have you heard these words piously quoted by people rather like a slogan or talisman who have no idea at all what they mean, beyond a vague idea that they must mean something other than sacrifice and, of course, exclude formal church and liturgy? Perhaps HaShem's words to the Israelites could help. Not bowing down to other gods - worshiping G-d alone, not the pastor or the PA system, or the powers that be in a denomination or network. Not serving other gods - recognising our single calling to serve G-d through Yeshua and to serve His people as directed by the Ruach. Not serving our G-d in the way others serve their gods - perhaps this is the most important, given that the first two may be straightforward. We have to know G-d intimately, not worship Him from a distance and in fear; at the same time, we have to hold Him in awe and not treat Him like an equal. He is not interested in legalism, or us serving Him to tick the boxes or win brownie points. He wants to share His truth with us and have us be honest with Him - no playing games or trying to hide things. Lastly, perhaps, we must be ourselves - he doesn't want us to worship him as someone else does, pretending or putting on a style. Our relationship with Him must be from ourselves, from the heart!\n1. - Although only six are named in this block of text - the Amoriates, Hittites, Perizites, Canaanites, Hivites and Jebusites - the Girgashites make up the full list of seven given to the generation that will actually enter the Land in D'varim 7:1.\nApplication: Do you serve G-d slavishly or in fear? Are you hiding insecurities or past failures in a big outward show? Or are you too afraid of G-d to actually worship Him at all? Today is the day to reach out to Him in \"spirit and truth\" and make that heart connection that you both desire!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Portal framed warehouse accessed by a loading dock and drive in ramp. Building has good lighting with an insulated ceiling. Three phase power. This is a well priced warehouse with good access to Ipswich Road. The warehouse has a sprinkler system installed. Ground level office plus air-conditioned mezzanine office area. Bonus storage mezzanine at no cost.\nWhile largely residential, Moorooka is better known as the home of the motor trades along and just off Ipswich Road's Miracle Mile. It remains in high demand by businesses needing close proximity to the CBD, a variety of stock from which to choose, and prices that are still affordable...for now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"@awsomebeans2: And you don't know how to listen either, it seems. Judging from your response, you don't seem to understand that I am not bullying your friend. I outlined the errors, so that they can improve. A bully would simply tell your friend to \"kys, m8\".\n@awsomebeans2: There is no bullying going on here. I have nothing personal against this user whatsoever. I am simply advising the user on how they might get themself taken seriously. I'd advise that you deal with this situation maturely. A wise person takes criticism and considers it and whether the given advice could improve their work. An unwise person swears at the critic and calls it bullying.\nSpelling errors are sprinkled around the review, no spaces are used after any kind of punctuation, there is copious use of exclamation points, halfway through the review all capitalization ceases, including correct usage (not that capitalization was consistently present before)... I could go ON. And ON. And ON!.. about the superficial errors but oh boy are there reviewing errors. Points are given for use of memes (AKA unoriginal writing), the reviewer admitted to not reading most of the comic, the reviewer complains that a SPRITE COMIC has no shading (I'm surprised that they even knew what shading was), completely ignores the fact that the sprites are original in favor of fanboying for the eyes and criticizing the sprites for not being ripped from a video game (any reviewer worth his salt will have a heart attack reading that part!), The last point is a tip the author is advised to follow and the most damning element out of this farce... Meta, fourth-wall aware characters... are bad writing? NO. Used correctly, they are an excellent literary device, which is PROVEN BY A PICTURE INCLUDED IN THE REVIEW. It's quite an interesting topic to muse on. One Over Zero is an excellent example of a comic where the characters are aware of the fourth wall. This commenter thinks that any reviewer believing a fourth wall is a neccessity is in dire need of reading THAT webcomic. It's clear the reviewer wants to piggyback off of another comic reviewer on this site and has put in no effort past the format. Please, take a few weeks to learn good webcomics from bad. One Over Zero can be found at http:\/\/oneoverzero.keenspace.com , and a beginneer's guide to what makes a webcomic bad can be found at http:\/\/badwebcomicswiki.shoutwiki.com\/wiki\/BWW_Tutorials_and_Lectures , which does not state anywhere that original sprites and fourth wall awareness is bad, and also states that ripping sprites is a very big mistake. If you've actually taken the time to read this counter-review and are considering this, Mr. Fan, thank you. Poor reviewers only mean that poor creators don't realize what they do wrong.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayacq b/data_all_eng_slimpj/shuffled/split2/finalzzzayacq
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+{"text":"Twitter is testing a new \"You may also like\" column on the right-hand side of individual tweet pages on Twitter.com. \"I can confirm this is an experiment for a limited group of users,\" a Twitter spokesperson told VentureBeat.\nA source close to Twitter said this experiment has been around for a while, and indeed a quick search on the social network itself shows someone first noticed exactly a month ago on March 14. Yet I only spotted it on my account today, suggesting it has since been expanded to more users.\nClicking on the \"View more\" button at the bottom of the column opens up 10 additional tweets, but the implementation is poor: They simply show up directly below the original five (notice the scroll bar in the screenshot below). The column could easily be made wider to avoid all of this scrolling, though Twitter may be wary of doing this so as not to remove the focus from the main tweet in question.\nThis is a smart move on Twitter's part as many new users find their way to the site by stumbling on a specific tweet. There often isn't anywhere to go from there, aside from the replies listed below.\nInterestingly, I don't see this new column on every single tweet page. It doesn't appear to be based on the author of the tweet: While testing, I opened a VentureBeat's last few tweets, and three-out-of-five showed me the column.\nIf the column does show up for a tweet, it's always there if you open it again. That said, if you open the same tweet a few minutes later, the column will show a different set of tweets.\nThis may suggest the column is showing up based on the content of the original tweet. Still, the tweets don't appear to be related by keyword to the original tweet in question.\nIt's not currently clear if Twitter will decide to roll out this feature to everyone. If it does, we hope the company will think of a better implementation for the desktop web, though on mobile, suggested tweets could be easily shown by swiping to the left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join one of the world's most recognised beverage brands in production and distribution. We are seeking experienced machine operators with a passion for the industry to join a large team.\nLocated in Richlands this role is a great opportunity for casual work on a 24\/7 rotating roster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Canadian companies investing around the globe, especially in developing or newly industrialized countries, must understand how evolving political and regulatory systems, significant events, and different cultural and political norms, will impact their operations or if their investment thesis will be undermined.\nWhile developing and newly industrialized countries offer enormous investment opportunity, failing to predict political risks accurately could lead to significant issues later on.\nInvestment decisions require the best, most comprehensive due diligence. Investors who can uncover relevant but less-obvious information are in a position to better assess the true value of the opportunity, create a competitive edge in bid scenarios or avoid unnecessary financial and reputational risk.\nUnderstanding if political risk can be mitigated can often mean the difference between pursuing or abandoning an opportunity.\nProperly identifying and assessing political risk should be an essential part of an investor's diligence process.\nThe enhanced micro-level political diligence that StrategyCorp provides enables our clients to better identify, measure, monitor and mitigate these less-tangible or less-obvious risks.\nDiligence that yields strategic insight can create opportunities and enduring value.\nAs a foundation for assessing each of these political risk types, StrategyCorp will provide detailed analysis of the political context in the relevant jurisdiction, including geopolitical considerations, regime stability, government capabilities and capacity, etc.\nStrategyCorp was engaged by a Canadian Pension Fund that was considering making an infrastructure investment in Mexico. Our mandate was to assess the political risks associated with the investment, with a particular focus on bribery and corruption risk, and to assess whether such risks could be effectively mitigated.\nStrategyCorp designed and implemented a comprehensive research program including a site visit and interviews with a broad range of sources both in Mexico and internationally. Our report presented our analysis of the particular political risks relevant to the project, our assessment of their severity, and a recommended mitigation strategy. The result was that while present, political and corruption risk could be mitigated, and should not be an obstacle for the transaction should the financial and legal diligence prove out.\nStrategyCorp was retained by a global extractive company to advise on a political risk it faced in Africa with potentially damaging reputational and shareholder impacts. The company's operations included a contract located in a disputed region of the continent with a permanent UN presence and a sensitive history with respect to human rights violations by state and non-state actors. As is often the case for companies in the extractive sector, the scale of the contract there led to the company being pulled into wider geopolitical issues and ultimately accused of being a force for destabilization.\nIn addition to evaluating whether the business lived up to all applicable standards, trade and customs laws, StrategyCorp advised on the potential for certain local and national political figures to mount a stronger campaign for divestment and their relative effectiveness in doing so, including connectivity to political actors at home and in key shareholder markets. As a result of our own network in the region, StrategyCorp ascertained the political risk level, identified key actors who could credibly defend the company to local stakeholders, and encouraged the contracting partner to institute a comprehensive corporate social responsibility program focused on local jobs, capacity building and public infrastructure. To demonstrate the effectiveness and legitimacy of these programs, we similarly advised on the most appropriate approach to audit progress and convey the results to a wide range of global stakeholders and the media. Today, even the most ardent opponents of any investment in the region will acknowledge the positive steps the company has taken.\nStrategyCorp was retained by a global resources company to advise on the design and implementation of a public affairs strategy to help secure the approval of a vital energy agreement involving two nations in the Middle East. The situation was complicated by the complex history of the relationship between the two nations. Our client was concerned that its operations in one of the countries would become uneconomical without this energy agreement. The context, which included the \"Arab Spring,\" made approval of the agreement an especially challenging objective.\nStrategyCorp worked with the client to create an in-depth understanding of the political context and risks in both Middle Eastern countries, and developed a public affairs strategy to help ensure governmental approval of the energy agreement. This strategy involved public opinion research as well as communications, stakeholder management, and government relations tactics in multiple countries, including at the chief of state and head of government levels. The result was the approval of the energy agreement.\nSTRATEGYCORP CREATES CONDITIONS FOR SUCCESS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lil Wayne released his much-delayed and highly anticipated Tha Carter V album on Thursday at midnight.\nTha Carter V features rapping from Kendrick Lamar, Travis Scott and Snoop Dogg, singing from Nicki Minaj, Ashanti and Nivea, and vocals from XXXTentacion, who was shot and killed earlier this year. The final song includes a prominent sample of the English singer Sampha. Production on the album comes courtesy of longtime collaborator Mannie Fresh, Swizz Beatz, Metro Boomin, Ben Billions and many others.\nThe album was first supposed to come out in 2014; Drake joined Wayne on \"Believe Me,\" which was going to be the lead single. Days before a possible December release date, however, Wayne took to Twitter to announce that his mentor and Cash Money label head Brian \"Birdman\" Williams \"refuse[d] to release\" the album. \"I am a prisoner and so is my creativity,\" Wayne tweeted at the time. \"I am truly sorry and I don't blame ya if ya fed up with waiting 4 me & this album,\" he added.\nIn January of 2015, Wayne sued Cash Money, demanding $51 million and the right to sever his ties with the label. Later that year, when the rapper attempted to release another project, the Free Weezy Album, through Jay-Z's streaming service Tidal, Cash Money reportedly hit back by suing Tidal for $50 million, though this suit was allegedly filed improperly and eventually dropped.\nAccording to a recent Billboard cover story, the legal disputes between Wayne and Birdman have now been settled for an undisclosed sum, clearing the way for the release of Tha Carter V. As part of the deal, Wayne is now the sole owner of Young Money Records, his imprint, which was previously a subsidiary of Cash Money. He also maintains his own relationship with Universal Music Group, which has distributed Cash Money releases for two decades.\nIt's unclear when the bulk of Tha Carter V was recorded; Wayne is a notoriously prolific studio rat, and the album has presumably been in the works since its announcement. However, he was reportedly still recording this week.\nThat may have played a part in yet another delay: Tha Carter V was initially expected last Friday. The album did not appear \u2014 Lil Wayne called it \"a misunderstanding\" \u2014 but this week, the rapper posted a countdown clock on his website which ticked down to Thursday at midnight. He also released a grateful video thanking his fans for their continued support. \"I would like for y'all to know that since y'all stuck with me for like the last four or five years, through all of this, y'all got me feeling like Tiger Woods with this comeback,\" Wayne said.\nThe release of Tha Carter V gives Wayne two reasons to celebrate: He also turned 36 on Thursday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Notice to Employees (Workers' Compensation) Poster (Bilingual) is a workers compensation law poster by the Arizona Industrial Commission. This is a mandatory posting for all employers in Arizona, and businesses who fail to comply may be subject to fines or sanctions.\nThis poster, written in both English and Spanish, must be posted in a conspicuous place where all employees will see it. The poster tells the rights that employees have to receive or choose to deny workers' compensation.\nIt appears you don't have a PDF plugin for this browser. Please see the link below to download arizona-claims-poster-workerscomplawbilingual.pdf.\nemployees at the office of this employer.\ndisponibles para todos los empleados en la oficina de este patron.\nKEEP POSTED IN A CONSPICUOUS PLACE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayfkx b/data_all_eng_slimpj/shuffled/split2/finalzzzayfkx
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+{"text":"Here are a few supplemental anchor charts that could be pulled into a variety of genres and make your students' writing more powerful!\nRead Aloud time matters! Check it out!\nThis is an intriguing article that expresses the opinions and views of a former fashion designer-turned-teacher, who also ended up leaving education because of the brokenness of the system. As a person who has never left the classroom, I always find it insightful to see others' views of the work we do. From student to professional\u2026it's all I know. While the article is depressing to an extent, it goes to show the incredible and truly miraculous work you do each day to make education \"work\" for kids in a system that often doesn't. Hats off to you. Just as this article shows, not everyone can handle it. Those who can and who stay with the profession are a special and elite lot. They are each of you.\nCan We Save the Tiger? is a great mentor text to use to help students explore different types of structures in their writing and have them analyze why authors make certain decisions around structure in their own text. Additionally, this text examines human impact on animals so would be a great text to pair with \"Gorillas\" and \"Polar Problems\" shared at FLiP training on Friday.\nThis is a new section that Lucy (yes, we are on a first name basis :)) just added to her website. As teachers, we are good at working smarter, not harder, by using ideas from other teachers. Here are some views inside Calkins' classrooms all over the world. Feel free to steal ideas at will!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The IELTS speaking test is a one-to-one interview which lasts for 11 \u2013 14 minutes. There are 3 parts which test your spoken English skills with various types of questions.\nYou can get more IELTS Speaking sample topics and questions for Part 1 here.\nYou can get IELTS Speaking sample topics and questions for Part 2 here.\nAre the types of leisure activities that are popular today the same as those that were popular in the past?\nYou can get IELTS Speaking sample topics and questions for Part 3 here.\nYou can find out more about the speaking test marking criteria here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You too can write to Doctor Soda Man!\nYour Dr. soda page is incredible! I never knew that there was anyone else who was so devoted to the Dr. sodas. I did notice that you were missing Dr. Riffic (available in a drug store chain in upstate New York). You're not missing much. I was surprised about your apparent lack of devotion to Dr. Smooth; you just can't beat its smoothness. I like to shake up the bottle before drinking it . . . it's never spilled yet! You are undoubtedly the Dr. god. Good work!\nI admire and respect your academic professionalism in compiling the Dr. soda list. Perhaps you would care to mention the unbelievable smoothness (flatness) of Dr. Smooth . . . . Kudos.\nI am looking for information about a soda that was around in the United States during the '50s and '60s. It was called Dr. Nut. Do you or anyone there know anything about that product? Do you know where I could go to find out more about it? Do you know where I could get a can or bottle of Dr. Nut for a collector friend of mine? Please let me know. Thanks for your time.\nWell, what can I say? I like your website. I'm working on one myself at present. Hopefully it will be up and running soon. I have several that you do not have, and you have several that I do not have. I thought we might be able to work out a trade, if you are interested. I have extras of Dr. Wells, Dr. Kist, and Dr. Lynn. I do not have Dr. Starr, Dr. S, Dr. Randalls, or Dr. Cool. So what do you think? Are you interested? If you are curious, here is some Dr. drink news: Dr. Perky from Food Lion now has a new can, and Dr. Topper (a drink I do not have) is now called Dr. Red and White. Here is a list of the ones I have: A+, Becker, Best, Chek, Cheer, Choice, Delight, Hytop, K, Kist, Lynn, Quencher, Perfect, Perky, Pop, Lotsa Pop, Riffic, Right, Rocket, Shasta, Skipper, Slice, Smooth, Stripe, Thirst, Thunder, Wells, West, Zing. I'm hoping to add to my collection over the summer months. Though I do not collect the diet versions, I would be happy to trade with you if you are interested. Also, I can send you a list of all the ones I know exist in a spreadsheet format. I know of 87 different ones, if that would help you. Email me, and tell me what you think.\nTo whom it may concern: you may be interested to know that there is no period after Dr in any of the Dr Pepper brands.\nHow much caffeine does Dr. Skipper have?\nIs Dr. Slice a hybrid of Slice and Dr Pepper produced from the same company?\nGuess I should have figured there would be more of these pages around. Nice work on yours! You've found quite a few I hadn't heard of yet -- guess I need to head out east again. If you'd like to see some more varieties, as well as five more pages devoted to this silliness, check out http:\/\/falcon.cc.ukans.edu\/~dschul\/drp\/dr.html.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What have we done to your world? You filled it with beautiful flowers, streams, and people, with creatures both mighty and miniscule, with so much intricate detail we discover new things every day.\nThen you gave humanity a choice. I could blame it on Eve or Adam, but the truth is I could have been the one to take the bite and ruin it all. Now we have filled your perfect creation with anger, bitterness, revenge, lust, terror, pain, sickness, and death.\nFather what have we done? Everything we do is touched sooner or later by tragedy \u2014 cancer, war, scandal, terror.\nBut after the rain comes, there is the rainbow. Because of the uneven stones, the stream gurgles a happy tune. In the world's darkest hour, the ones You created show the selflessness of living as Your image.\n\"Greater love hath no man than this, that a man lay down his life for his friends\" (John 15:14).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nacora Group utilizes its network in over 40 offices located in 30 countries. We specialize in offering our clients Admitted policies\/local policies as mandated in many countries to provide the best local protection to insured Directors and Officers.\nPublic companies, Not-for-profit, and Private entities look for D&O coverage to allow 'Board of Directors' protection in the management of their companies while seeking long term growth.\nReputation, image, branding, and financial security can be enhanced through D&O coverage. Negative earnings announcements, takeovers (hostile), employment practices (EPL - wrongful dismissal), securities trading (insider), audit (accounting - misrepresentation), and bankruptcy (government obligations) are just some claims that can come before a Board. Directors and Officers Liability policies address these concerns and more.\nWe work with all industries, including Mining\/Exploration, Revenue-producing companies, Charities, Manufacturers (global product recall) and many others.\nRetired former Directors and Officers are also covered for the period they served so long as the policy remains in force to us, providing additional comfort.\nEnhancements include expanded definitions, Side A coverage, extended reporting period, run-off's, and protection for subsidies.\nNacora is there to work with you, understand your company and industry and provide you with customized solutions. Our underwriters are worldwide and we can offer multiple quotes in a timely manner, competitive in price and broad in coverage.\nThese are just some of the coverage's provided in D&O policies. Nacora has access to worldwide markets for all kinds of risks and can customize your specific needs for any industry.\nEPL coverage for directors, officer, employees and entity.\nEntity vs. Insured Exclusion, replacing insured vs. insured exclusion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbapux b/data_all_eng_slimpj/shuffled/split2/finalzzzbapux
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index 0000000000000000000000000000000000000000..ddce39393ed1f346abf157fa3f75e03449fae2e4
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@@ -0,0 +1,5 @@
+{"text":"You can find that tungsten is at work whenever the heat's on. Tungsten material properties (having the highest melting point, good electric conductivity, excellent thermal conductivity and a uniquely low thermal expansion and a very high level of dimensional stability) tungsten can suitable for very high-temperature applications.\nand other tungsten alloy wire (such as tantalum tungsten alloy wire, copper tungsten alloy wire and other tungsten alloy wire). Tungsten wire has the same properties of tungsten materials. The crystal structure of tungsten wire is compact.\nWAL1 >=99.92% T double-screwy filament.filament for hihg colour temperature lamp and shockproof lamp,etc.\nL filament for incandescent lamp.cathode for emitting tube.high temperature electrodes.tungsten wire in twisted.etc.\nW folded-heater for electronic tube,etc.\nWAL2 >=99.92% T filament for fluorescent lamp,etc.\nW folded-heater,grid and cathode for electronic tube,etc.\nWAL3 >=99.92% L filament for normal lamp.spring wire for semicondutor,etc.\nW1 >=99.95% tungsten wire in twisted and heating parts,etc.\nW2 >=99.92% grid siderod for electron tube.tungsten wire in twisted,etc.\nThe tungsten wire is coated with colloidal graphite on the surface and tungsten appears uniform black. Tungsten wire appears metal luster if being removed colloidal graphite.\nWe supply pure tungsten wire, tantalum tungsten alloy wire, tungsten copper alloy wire and other tungsten alloy wire. Our tungsten wire supply ability is 10 tons per month.\nThe delivery time of tungsten wire is 7 to 30 days after confirmation.\nTungsten wire is the best material to make lamp bulb.\nIn our daily life, there are a lot of metals that can give light besides tungsten filament light bulb, such as copper wire, iron wire, etc.. While, why do we select the tungsten material as the luminous body of bulb? Zhengzhou Sanhui Co., Ltd will explain how tungsten wire give out light to you?\nBecause the tungsten filament has relatively high electric resistance and melting point, when the electric current goes through the tungsten wire, the electric resistance of tungsten wire rises and gives out heat, then tungsten wire and tungsten alloy wire give out light. At the same time, the melting point of tungsten wire is very high, so tungsten wire is not easily to be burnt out. In addition, inert gas- neon is filled in the bulb, which ensures tungsten wire is not able to be oxidized when the tungsten wire is generating heat. Since the passing of electric current makes the conductor heat. When the temperature of tungsten wire rises, the length of radiated electromagnetic wave becomes short. At room temperature, tungsten wire can radiate ultrared rays. When the temperature rises to a certain degree, tungsten wire begins to radiate visible light, so tungsten wire begins to give out light.\nCompared with tungsten wire, other metals would be burnt out before giving out light and also have certain risk. So tungsten wire is the best choice to make bulb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For small business owners, there's a \"new normal\" out there, and it doesn't look a whole lot like the old normal. If your business seems to be on a treadmill of too much debt, too many operating expenses and too little growth, it may be time to take \"lean and mean\" to a new level.\nEverything you do should be subject to scrutiny and revision, including your mission statement, business and marketing plans, budget, sales and expense expectations and more, says business-makeover specialist Patricia Sigmon, president of LPS Consulting (lpsconsulting.com), which creates profit-focused tech solutions for small businesses.\n\"If you've held on to that large office waiting for things to pick up, kept an extra car, maintained a larger-than-needed staff, have too many cellphones, or whatever, it's time to start a 'new business,' scrap what you can, and feel the relief,\" says Sigmon, author of Six Steps to Creating Profit: A Guide for Small and Mid-sized Service-Based Businesses.\nMake 2012 about major regrouping, major renovation and creating a brighter, less stressful view of the road ahead -- your new normal.\nLook at which customers are giving you the most profit, and coddle them, woo them, don't lose them! Offer them frequent-buyer rewards. Send them a small gift at their one-year anniversary. Give them a random call every few months to \"check in,\" thank them, and ask what else they might need. Treat them like gold.\nSometimes, the highest-maintenance, most time-consuming customers you have are the ones who pay you the least. Analyze the profit margin or lack of profit margin that each customer or perhaps customer segment produces. Stop pursuing customers who are not helping you be profitable -- period.\nThis coming year, it's time to overhaul your sales behavior. Turn all one-time sales efforts into relationship sales. Start monthly maintenance plans, suggest auxiliary services, sell complementary products, or offer retainer plans covering 50-100 labor hours, for example.\nReduce your operating expense budget to the lowest possible number. If that means selling your car or closing an office, so be it. You can't build a new profit base when you are still using yesterday's expense model. Go through your expenses line by line and get rid of everything you can live without.\nWhatever type of skill or service you need, think hard before hiring a new employee or keeping an old employee. Look at each department or each person when you are trying to manage costs. Can you eliminate positions (perhaps through attrition), combine jobs, delete processes, and outsource tasks? Outsource exactly what you need for the right amount of time and the right amount of money.\nFrom blogs to Twitter to LinkedIn to Facebook, invest in the online and social-media presence you need to compete now. Businesses that don't leverage social networking will be left behind.\nJump-start new relationships in 2012 with forum building. Update all your sites. Keep your online relationships fresh and dynamic with news, blogs, newsletters, tips and surveys. Find an online forum in your industry and become an active contributor.\nWith cloud technology, you are no longer bound to a desk. Log onto some new interactive cloud-based systems that can help you do your business anywhere. Make sure you have Internet connections on all of your devices.\nEverything you once needed to do in your own office can now be done remotely. Best of all, when your employees are sharing files in the cloud, it makes for a much more cohesive, connected team.\nReach Dan Kehrer at bizbest.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let Your Tenant Pay Your Mortgage!\n_Home buyers can find the housing market to be so prohibitively expensive that swinging a monthly mortgage payment is almost impossible to do.\nOne solution for strapped home buyers is to consider a two family home to live in and to charge enough rent where the rental payment covers the mortgage payment.\nIs this a good idea for you?\nLet's explore the concept of buying a two family home and letting your tenant pay your entire mortgage.\nFor the record: your tenant's rental payment cannot go directly to your mortgage lender.\nYou will still have to collect your tenant's rent, process the check, and then forward the right amount of money each month. If your tenant fails to pay you or she is late making a payment then you will still be responsible for paying on a mortgage that is in your name, not your tenant's.\nA two family home can be advantageous and some lenders will take rental income into consideration when you buy a home. So, that $485,000 two family could easily be approved while the $395,000 single family home could be out of your reach. It all depends on just how much weight a lender will give to a tenant's income in deciding whether to extend a loan to you or not.\nShould a paying tenant already be in place and you plan on occupying the second apartment, then your chances of securing that two family home is better than if both apartments are presently vacant.\nAlthough it would be ideal to have the tenant's entire rent cover your mortgage payment this may not be possible in an area where rents are generally low.\nYou won't be able to increase your tenant's rent to cover the cost of your mortgage, otherwise he could move elsewhere.\nConsider this: it is better to have a rock solid tenant paying a reasonable rent than to have a flighty tenant or tenants unable or unwilling to pay a rent that is above the market average.\nIf your tenant-landlord relationship turns sour you could find trouble brewing for you.\nA later paying tenant cannot be easily evicted in some states, thus you could be left holding the bag for payments that are six or twelve months behind. Can you take that loss or will it push you over the edge financially?\nYes, securing a mortgage is generally easier to attain with a tenant already in place. A faithful tenant can help you out, but a troublesome tenant could easily bankrupt you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sequins cover a big part of my life. I'm talking tops, bags, shoes, dresses. Wherever I can get them, I will. The bigger, more brash and in-yer-face the better. Enter this High Neck Tunic stage right.\nI'm currently trying to list occasions or events I have coming up that would merit this top worthy of a spend. This has become a new step in my life \u2013 seeing as I now have to start saving for a wedding \u2013 but a necessary one. I mean, I don't actually need any more sequin-tastic tops but when was 'need' ever en vogue anyway.\nNow this seems like the most obvious answer and something that Al and I have done a lot of talking about but little else. If I got this top it might give me the kick start I'd need. Or it'd make me \u00a360 poorer but 160% happier. You do the math.\nBecause I have friends and they all have birthdays, some of which even happen later this year. It's always good to be prepared.\nSeamless link there. Although my birthday isn't until the end of December and I might have got sidetracked by some other shimmering sequins by then.\nTake a bog standard England shirt, of which I have, and add ALL THE SEQUINS. Rugby supporting has never looked so fancy.\nTo do my weekly shop.\nBecause without sequins to entice you, who actually wants to spend an hour of their life in Tesco?\nSo there, I'll take the sequins and save them for a rainy day. Of which we have plenty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A key element of my service offering is close working relationships with the golf course managers. I offer a mentoring service to Course Managers\/Head Greenkeepers who I hope will enjoy the benefit of my experience.\nI provide immediate and impartial advice that can rapidly assist a course manager in progressing their management problems\/issues and thus their courses condition, leading ultimately to profitability and harmony within their respective clubs.\nI work as a 'sounding board' to Course managers providing focused resolution based advice and recommendations. Working with you, I aim to fully understand your operational concerns, issues and progress aims.\nMy involvement helps to influence the decision making process so that you are able to achieve measurable performance improvements across the whole of your course management operation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfmxc b/data_all_eng_slimpj/shuffled/split2/finalzzzbfmxc
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"A surrealist wonderland keeps watch over the Austin sky line, ensuring the city doesn't take itself too seriously. Home to stylish entrepreneurs, their young daughter, and Moka the frisbee crazed dog, this house is meant for living. Plush Moroccan carpets and sinuous seating demand lounging, while glass walls beckon outward and up into the sky.\nOur imaginary muse Aunt Rose is a well traveled lady with exquisite taste in American mid century art and furnishings -- Wormley, Jere, Kittinger, Baughman, Pearsall. But she's not above playing bongo drums at midnight with a martini in hand. Who should be?\nThis house is as colorful and laid back cool as the people who live here. Because at the end of the day, it isn't so much the things that surround you as the vibe and hum of life that makes a house into a home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are the most common insurances accepted by Dentists in Buffalo, NY.\nFamily owned business. The only one that really cared about the patients was the father and he is no longer there. I would stay away from this dentist, they do a lot of work that is not required, most likely to rip off the insurance companies..\nThis is for Suburban Family Dental Practice at 646 N French Rd #8, West Amherst, NY 14228. Disgraceful practice. Disgraceful place. My family went here for over 3 years. We paid thousands of dollars. Then we moved out of the area. It looks like there was a bill of $100 after they billed in the insurance or whatever that is. They never contacted me. They have my email. We moved out of New York. I never received any information about an outstanding $100. Despite the thousands of dollars we paid..\nGreat place always clean and very professionals..\nVery friendly staff and as a pregnant woman they made me feel as comfortable as possible!.\nWow! Excellent results on my smile! I never thought my teeth could be this white. My wife and kids love Dr. Bob as well. Love the office, love the staff!.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Packed with features and truly a pleasure to drive! Lexus prioritized handling and performance with features such as: automatic dimming door mirrors, blind spot sensor, and the power moon roof opens up the cabin to the natural environment. Smooth gearshifts are achieved thanks to the 3.5 liter 6 cylinder engine, and for added security, dynamic Stability Control supplements the drivetrain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"hd00:25Chickens in Chicken House. Chicken farming is big business in Georgia. These chickens are 7 weeks old and ready for market.\n4k00:10A man uses two cows to plow a piece of farmland in a village in South India.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CAO provides a variety of pre-employment vetting and screening services designed to assist our Clients to win the battle against the enemy within \u2013 before, instead of when, they have been used.\nPersonal Identity Protect against identity theft and fraud using a cross-referencing check over a number of data sources.\nEmployer References Obtain personal references from sources provided by a prospective employee, such as previous supervisors, customers, suppliers.\nEmployment History Confirm past employment history, periods of (un)employment, positions held and reason for leaving (where available).\nQualifications Provide protection against bogus educational and professional qualifications by validating them with the relevant institutions.\nvalidating them with the relevant granting body.\nDirectorships Validate current and previous Directorships and highlights any disqualifications that are in place.\nCriminal Records Checks Check with the Criminal Record Bureau for the existence of any current criminal record.\nCredit History Establish financial status, credit history and any adverse information that may affect employment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfrnv b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrnv
new file mode 100644
index 0000000000000000000000000000000000000000..fa30ce35e474d59091843934b05bbe6d32e9d85e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrnv
@@ -0,0 +1,5 @@
+{"text":"A business ecosystem may be defined as a sustaining environment to promote economic growth, skills growth, and growth in industrial capability and global competitiveness, fuelled by the extension of knowledge, the ambitions of innovators and the business and societal needs of the community at large.\nPartial ecosystems relevant to this Research Agenda exist in the form of national and regional groupings of business and public research actors, special interest groups, knowledge transfer networks, research collaborations, and the tiered supply chains of major industrial players.\nThese groupings, although many of their members have relevant resources, do not explicitly recognise a focus upon Smart Systems, and are not yet marshalled to push EU capabilities in what is becoming a dominant and global industrial arena in the run-up to 2020.\nA truly proactive ecosystem would accelerate the adoption and bring forward the rewards of Smart Systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"fax 201-748-6008, or online at www.wiley.com\/go\/permissions.\nand when it is read.\n800-956-7739, outside the U.S. at 317-572-3986, or fax 317-572-4002.\nof America. Used by permission. All rights reserved.\nCampolo, Mary Albert Darling.\u20141st ed.\ndrafts, and especially for her sensitivity to potential readers.\nalso for her encouraging spirit.\nin order so that he could have time to write.\nRobert Gauthier, managing director of Tony's missionary organization, EAPE, for helping in Tony's research.\nMcLaren's interviews, and for typing portions of Tony's material.\nMichele Jones; and production manager, Joanne Clapp Fullagar.\nJesus in how he loves and serves others.\nto respond quickly to follow-up e-mails throughout the project.\ntoo many of you to name.\npizza several nights in a row.\nfor lovingly supporting his many projects.\nwith compassion, kindness, humility, meekness, and patience.\nwhich binds everything together in perfect harmony . . .\nchemistry created when we combine both practices brings about a powerful reaction that has a transforming effect on our relationships.\nclassrooms, or in a public-speaking context.\nsuggestions at the back of the book are helpful.\nthemselves into deeper connections with God and others.\na van with his parents and older brother.\nchaotic than they have been harmonious and peaceful.\nEven God's chosen people, the citizens of Israel, couldn't get it right.\nWhy isn't the church perceived this way in the world today?\nis not yet a reality for those of us who claim to be the body of Christ.\nanother and to the world.\ncan't see how wrong our own harmful communication patterns are.\nwedding is over, they seem next to impossible to live out on a daily basis.\nradically loving ways like Jesus. Christ followers can and do get it right.\nChristlike communication in their everyday lives.\nverses in the Bible really can be lived out in how we daily communicate.\nconnected him to his audience.\nAs we pointed out in the Introduction, it can be one thing to communicate but quite another to connect. We can use a variety of solid communication techniques and still feel a lack of connectedness with others.\nhow well we know someone.\nwhom his culture considered beneath and beyond his own social class.\nhad been with him (Acts 4:13).\ntransforming that because of Jesus, all of history was changed.\ngo into all the world and spread the good news of the Kingdom of God.\nmore than just cure people's physical ailments. He healed people's souls.\n\"greater things\" than he did (John 14:12).\nhave been identified with special gifts of healing, teaching, or preaching.\nthem to observe all that Jesus has commanded (Matthew 28:16\u201320).\ndisciples, past and present, to be teachers.\nconferences, retreats, seminars, or as part of a church service.\nEach chapter will address specific communication and spiritual practices that when combined produce spiritually charged communication.\nof the practices throughout the book.\nGod's immense love for you.\nwaiting in your car, or working in your kitchen.\nfocusing visually on Holy Trinity, the painting by Andrei Rublev (1415).\nTo build a more personal foundation for connecting like Jesus, we recommend becoming familiar with the \"one another\" verses in the Bible.\nfrom now on do not sin again' \" (John 8:10\u201311).\nhurry and come down; for I must stay at your house today\" (Luke 19:5).\nhome of someone they considered a sinner. But Zacchaeus was happy.\nto stay at his house, Zacchaeus in turn wanted to change his whole life.\ncame to seek out and to save the lost' \" (Luke 19:8\u201310).\nread in Matthew that \"Jesus went through all the towns and villages . . .\nMuch has been written on Jesus and his ministry of physical healing.\nand then raise them up healed of their physical ailments or handicaps.\nand they shouldn't expect too much from my healing ministry.\nout of that church a different man.\ndeath, but the healing in our souls has everlasting significance.\nsouls and surrender them to God, it can be very difficult, if not impossible, to focus on others.\nus as followers today too.\nrather, serve one another in love\" (Galatians 5:1\u20132, 13, NIV).\nwith God\u2014because only God can infuse life, grace, energy, peace . . .\nhealing, a hurdle that is always close by: our own selves.\nof the time in the store.\nNIV). As soon as John's disciples heard this, they started following Jesus.\nJesus so that when people are with us, they are drawn to him.\nAs freeing as it is to die to self, we do not give ourselves up easily.\nwants and desires for people and possessions or recognition and respect.\nwith God and get clear, once again, why I'm doing whatever I'm doing.\nTeresa and the Sisters of the Missionaries of Charity.\nthe evident fulness of the love my heart bears to Thee.\nin Christ. This discovery frees us to really care for the souls of others.\navailable for the Holy Spirit to work through.\nto show love through acts of service.\ncan not only alert but also prepare us for soul-healing encounters.\nwhether those encounters are expected or not.\nthat did not mean that all of his encounters were somber or structured.\ncan emerge as we interact with others for the purpose of soul healing.\nOh it's so hard to see, when my eyes are on me.\nMay I know myself, may I know Thee.\nconsciously aware of how we want others to care for our own souls.\nwe want to be treated is important. But it still may seem too self-centered.\nExceptions to the Golden Rule?\nto be loving in return. Mean people are not easy to like, much less love.\nYet if we want to follow the Golden Rule, we must be loving and kind.\nwill surprise him with goodness.\nwant to do is love them.\nI have a friend, Jacob Atem, who is one of the \"lost boys\" of Sudan.\nhe could say that he not only has forgiven but also loves these people.\nfor the purpose of helping heal their souls.\nbearer of God\u2014so that you can treat others in soul-healing ways.\npresence, goodness, purpose in a given situation?\nway in me, and lead me in the way everlasting\" (Psalm 139:23\u201324).\nall who Jesus invited accepted his invitation to increased awareness.\nprodigal son became more selfself-aware.\n. . . And you?\" When he actually sees her, he blurts out, \"Me Tarzan!\ncourse, wrong. But we can unintentionally create false impression too.\nto do with this newfound awareness.\nloves you in spite of what you have done or what has been done to you.\nBelieving this truth will free you to accept and receive God's gift of grace.\nwhen God is in them.\nfor whenever I am weak, then I am strong\" (2 Corinthians 12:9\u201310).\nwe want to hear, as it's easier to be aware of others' weaknesses instead.\nin your life were not very self-aware, and you picked up their patterns.\nOr you could be unconsciously protecting yourself from potential hurt.\nOr there may simply be blind spots in the way you relate to others.\nsomeone else if you do not realize that he was being sarcasadmit your own mistakes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has the insurance situation changed being a doctor? If so, how?\nBefore insurance there was more accountability. The patient paid and saw what I did. It allowed us to really function as professionals and allowed a market place. We couldn't afford technology and if you got sick you were screwed. So we created health insurance which we needed but it took away accountability between doctor and patient.\nMy major issue is pre-authorization is strangling me and my colleagues.\nThe insurance companies do not want patients to live. They make more money if the patients die.\nMore and more patients are sent out of state. They will find someone in network. They get referred to a physician who is not a specialist in that area.\nTotally. I don't have patients, I have contracts the insurance companies and so do the patients and I am the middle man.\nYes. The reason it has is because it is a struggle. The general public believes that as a physician you can negotiate with an insurance company. The general public doesn't read EOB's. The general public doesn't realize that what you charge is not what you collect. Imagine. Going to your lawyer, having him charge you a thousand dollars so for a service and you telling him you are only going to pay $300 and he had to accept the fee. That is what it is like dealing with insurance payers. In the old days when insurance companies paid reasonable fees, Medicare was the outlier for being the lowest fee schedule. And physicians accepted this fee schedule as it was their duty. And because the other payers helped to even out the score. Over time the tables have turned and now Medicare which still remains a low fee schedule is now one of the best. And so doctors' ability to provide charity to provide charity car has been reduced.\nYes. They have too much power to dictate what I can and cannot do. This is disproportionate to their mandate. Their job is to provide the funds. They have too much power. They use this mandate, their interest is profits and any way they can increase it they can do it. Many physicians who are good people and do a good job have been penalized. Don't paint everyone with the same brush.\nThe pot is not infinitely deep but you have to be efficient and but the way that costs are being constrained is not efficient and it gets in the way of my relationship with the patients.\nWe need to cut out the middle men, the insurance companies and put those billions back into the system. Half of the staff does case management and discharge planning and it doesn't help the patients. The nurses are charting, not taking care of the patients.\nYes. It used to be in the days of yore you provide the services you thought were appropriate. People were happy and there was no restraint. Now there are restraints via insurance companies willing to pay what they are entitled to and what we can offer.\nI think that the insurance companies are the problem. All of the third party payers have caused the price problem. No one can afford $2000 for an MRI. There is a complete disconnect between the cost of health care and the patient. The costs are uncontrollable. There needs to be tort reform. Too much defensive medicine. The doctors are just doing it to cover themselves. If you don't do it and get sued, it's an issue. I would make people who lose these cases pay the doctor.\nI am sick of the paperwork.\nI make occasional compromises on formularies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summary: David Brown, coeditor of the Higher Education Exchange, interviews Don Rothman, former director of the Central California Writing Project, who for more than 30 years taught writing \"as if the world depended on it.\"\nTo the extent that democracy expresses our aspirations to sustain public discourse for the purpose of enhancing the public good, its health depends on nurturing certain habits of mind and skills. Over three decades of teaching writing, I have come to see quite vividly literacy's potential to enhance democracy, especially around the intellectual and social practices that make nonviolent persuasion possible. Literacy, of course, doesn't guarantee freedom of expression, but writing, in particular, offers opportunities for people to counter alienation, isolation, and selfishness that undermine democracy.\nCopyright \u00a9 2005 Charles F. Kettering Foundation. This material is used by permission from Charles F. Kettering Foundation.\nDownload \"The Writing Classroom as a Laboratory for Democracy: An Interview with Don Rothman\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello. I am a Trine SPS student using the Library link for research for the first time. I don't know what my login or password is. How do I find it?\nThe login is the same as your email, MyPortal, and Moodle. Instructions for accessing the Library can be found here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"10.08.18 12:08;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 224195340. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n13.08.18 10:14;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 224516335. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n12.08.18 02:50;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 224375923. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n14.08.18 14:21;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 224661786. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n16.08.18 14:19;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 224894562. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n17.08.18 14:14;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225005836. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n19.08.18 12:21;Account;Receive;Received Payment 2.01 USD from account U14972070 to account U1294000. Batch: 225187180. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n23.08.18 05:59;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225570404. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n22.08.18 01:48;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225460632. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n24.08.18 14:47;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225722665. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n25.08.18 14:50;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225821878. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n27.08.18 10:57;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225985491. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n27.08.18 10:54;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 225985062. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n28.08.18 14:20;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 226120795. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n29.08.18 12:31;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 226219150. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n30.08.18 14:08;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 226335884. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n01.09.18 02:01;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 226495809. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.\n02.09.18 02:27;Account;Receive;Received Payment 1.00 USD from account U14972070 to account U1294000. Batch: 226920592. Memo: API Payment. Withdraw to allhyips from 1paycoin.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Products of Art Plus Inc.\nArtPlus ePix wallpaper calendar 6 Art Plus Inc.\nArtPlus ePix wallpaper calendar is FREE software that brings a new image to your desktop once a week. In addition to using your own images as Windows wallpaper, the new ArtPlus ePix wallpaper calendar changer delivers a stunning, high-resolution image to your desktop every Monday. The program also has a built-in personal diary and a simple calendar. Freeware download of ArtPlus ePix wallpaper calendar 6, size 7.74 Mb.\nArtPlus ePix 6 Art Plus Inc.\nArtPlus ePix wallpaper calendar is FREE software that brings a new image to your desktop once a week. In addition to using your own images as Windows wallpaper, the new ArtPlus ePix wallpaper calendar changer delivers a stunning, high-resolution image to your desktop every Monday. The program also has a built-in personal diary and a simple calendar. Freeware download of ArtPlus ePix 6, size 7.74 Mb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy day of birth ridgemeister!\n4 days late but 4s my lucky number, well one of them happy birthday. intersting fact about june 26th \"On this date in 1284, it was said that the Pied Piper of Hamelin, Germany lured 130 children out of town, and they were never seen again.\"\nSubject: Re: Happy Birthday Ridge!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The latex base material of natural rubber products is extracted from the sap of the rubber tree Hevea brasiliensis tropical.\nLatex allergy is on the rise especially among the health personnel who frequently use rubber gloves (latex). But we also find this type of allergy in the rubber industry, laboratories, hair salons or dry cleaners.\nimmediate (approximately 20 minutes after contact with latex) type of redness, itching, difficulty breathing, conjunctival redness and sometimes even shock can be fatal.\nor delayed after 48h as allergic contact dermatitis.\nA food allergy is sometimes associated with pineapple, avocado, banana, chestnut, fig, potato, kiwi, mango, melon, papaya, passion fruit, peach, spinach and tomato.\nGloves, ventilation equipment, respiratory masks, bags, tubing, valves, endotracheal tubes, nasal and oropharyngeal swabs, drains, tires, seals carpets, doors, windows, bath mats, mattresses, insulation types, electric cables balls, elastic bandages, boots, shoes, clothing fasteners, household gloves, adhesive tapes, adhesives scrubs, stockings, condoms, balloons, air mattresses, rubber handles, vehicles, inflatables, gloves (hairdressers and beauticians), artificial sponges, diving articles, stretchable fabrics, sneakers, thermos bottles, packing material.\nThe diagnosis, based on skin prick tests are sensitive and specific. Can be completed by the research of specific IgE to latex.\nPrick test sensitivity is higher than that of the search for specific IgE.\nAcute urticaria following contact with the latex .\nWhen there is a delayed reaction to patch tests are required for diagnosis.\nEviction is an essential measure.\n\u2022 Wearing a card stating allergic to latex allergy is very useful.\n\u2022 An emergency kit containing antihistamines, corticosteroids, and epinephrine is prescribed in severe cases.\n\u2022 A latex-free environment is essential especially in an operating theater before surgery (only non-latex gloves must be worn).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Selective Service registration is the process by which the U.S. government maintains a list of names of men from which to draw in case of a national emergency requiring rapid expansion of the Armed Forces.\nFollow Congress with the latest legislative information from the Library of Congress. Track bills. Links to official Senate and House of Representatives web sites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our family holidays to Hastings are ideal if you're travelling with little ones.\nStruggling to find a child-friendly holiday? Then take a look at our family holidays to Hastings \u2013 they've been designed with little ones in mind.\nThere aren't any family holidays in Hastings left at the moment. Why not take a look at some of the other options?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You probably didn't know that tomorrow is World Alzheimer's Day\u2014I certainly didn't. And you probably weren't aware of the recently released report, \"World Alzheimer Report 2015: The Global Impact of Dementia\u2014An Analysis of Prevalence, Incidence, Cost, and Trends,\" released last month, since it wasn't mentioned in the NY Times or the Washington Post or the Wall Street Journal. The only reason I know about it is that I happened to see it mentioned when I was scanning the prominent British journal, the Lancet, looking for something to write about in this blog. The British seem to be far more attuned to global health than we in the US: for example, they spearheaded the \"Global Action Against Dementia\" G7 conference held in Geneva in May, 2015. So what exactly did the new report have to say?\nWorldwide, there are now an estimated 46.8 million people with dementia. In sheer numbers, that translates to 9.4 million in the Americas, 10.5 million in Europe, 4 million in Africa, and 22.9 million in Asia. The doubling time for this relentless, progressive, degenerative brain disease is approximately 20 years. Most of that growth is due to population aging\u2014the forecast assumes that the incidence and disease duration will remain unchanged. What is most striking is the shift towards low and middle income countries (what the cognoscenti call LMICs): right now 58% of the world's demented people live in that part of the world; by 2050, the prediction is that 68% will. What can we do about this?\nWhat we can do is a reflection of the reality that dementia is as much a social issue as a medical one. In fact, only 20% of the $818 billion price tag associated with global dementia care today is for direct medical care; another 40% is for social care (nursing homes, personal care assistants, etc), and the final 40% is for informal care. But most of the lower and middle income countries do not have the infrastructure needed to care for the increasing numbers of older people with disabilities of any kind, including dementia. They don't have the long term care institutions, the community organizations, or the assistive devices they need to provide a decent quality of life to the afflicted and support for their families.\nGlobal aging in general and the dementia epidemic in particular demand our attention. The US can take the lead: in biomedical research, with the goal of developing treatment for dementia, in bioengineering, with the aim of designing technology to assist in the care of those with either physical or cognitive disability, and in social research, to provide the systems and the strategy needed to maximize the quality of life both of patients and their families.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This publication discusses the global scale and recognition of the problem of female genital mutilation, and touches on how to engage health professionals to support abandonment of the practice. It describes changing patterns and practices, including medicalization, and analyzes the challenges FGM\/C poses to the achievement of Millennium Development Goals.\nThe document discusses strategies to accelerate progress so that health care providers and national authorities can mobilize funding, create supportive legislative frameworks and strenghten accountability.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Items where Author is \"Asia, Reni\"\nAsia, Reni (2016) PENGARUH PENGGUNAAN MEDIA DIGITAL VIDEO DISC (DVD) DENGAN MOTIVASI BELAJAR SISWA KELAS V PADA MATA PELAJARAN AKIDAH AKHLAQ DI MI AL-'ADLI PALEMBANG (Skripsi). Other thesis, UIN Raden Fatah Palembang.\nThis list was generated on Thu Apr 25 15:30:09 2019 WIB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Appointments are highly recommended. While we do our best to fit walk-ins into our schedule, we are sometimes unable to do so. Please make an appointment to make sure we have time to photograph your family.\nYou can make an appointment by calling the studio directly. To find a studio near you, use our location search.\nWhen do I receive my printed portraits?\nYour portraits are printed in the studio immediately following your session. They will be ready for you to take home in just minutes.\nI ordered a photo gift during my studio session. When will I receive my gift?\nAll studio orders are sent immediately for processing and are usually shipped within 10 business days. You will receive an email from us containing tracking information for your order.\nYes! We encourage you to bring your own props to truly personalize your portrait experience. You may bring any items with special meaning for you and your family. We've seen just about everything in our studios - special blankets and stuffed animals, musical instruments, sports equipment, hobby items \u2013 you name it and we've probably photographed it.\nI would like to talk to someone about my studio experience. Who should I call?\nPlease contact the studio manager directly to discuss your experience. If you are unable to reach the studio manager, you may contact our corporate customer service department by emailing customerservice@portraitinnovations.com.\nI would like to talk to someone about advertising opportunities. Who should I call?\nAll media advertising inquiries should be directed to CS@PortraitInnovations.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjjhm b/data_all_eng_slimpj/shuffled/split2/finalzzzbjjhm
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+{"text":"We could get it going again if anyone was interested..\nNot for me, but I do love The Archers. No one else? No love for the farmers with funny accents?\nThanks for the reminder of that Eastenders thread. I remember reading it and keeping up when I left the UK in 2004.\nI read about 20 pages of that old EastEnders thread last night... was fun to remember some of the stuff that happened.\nReally? I'm the only one sad enough to watch Easties and want to talk about it?\n*HUFF*... I'm off to Digital Spy, then!\nCome back Albatross! I watch Eastenders! I got my husband hooked too, but don't tell anyone, he likes to keep that a secret!\nEasties is a love\/hate thing for me... I never miss it, but sometimes after I watch it I feel like the writers must think we are all idiots. At the moment, I think it's not too bad... Abi is so evil!!! But I really wish they would kill off the Dot Cotton character. She's had her time, and she is just beyond an annoyance now.\nBet you had your hopes up then when she had her fall!\nI'm fond of Dot, happy they let her live! I'm liking Abi better recently, she seems to have grown a pair!\nDo you watch Corrie too?\nI'm loving the whole affair\/fake pregnancy\/she knows but they don't know she knows storyline!\nNo.. I don't watch Corrie. I think they try a bit too hard on the comedy angle, and I just don't want any comedy in my soaps at all.\nAbi... omg. Would you REALLY live in a house-share with your 2 ex-boyfriends? She's not right in the head... and Steven had better watch out!\nHa! The comedy in Corrie is what I love the best! Some of it is pure gold!\nIt's meant to be a really good exciting week on Easties... I'm sure all the wrong people will die, though. They never get it right (for me!)...LOL!\nI really hope it's Dot that got shot behind the door, but it will probably be Johnny... he is totally expendable, so I could live with that.\nOooh, I missed it altogether last night, I'll have to catch up.\nAs long as Dave is ok...!\nI know the first thing I'm going to mention if UK Yankee ever do a huge meetup.\nPage created in 0.17 seconds with 214 queries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Super king size beds are the absolute epitome of luxury. With this in mind, it'd be a shame to settle for anything less than perfection when it comes to the look and feel of your bedframe. This is why we here at Esupasaver give control back to our customers, allowing you to design the bedframe of your dreams by customising everything from fabric to headboard height.\nWith many retailers you may find that they keep the cost of their super king size beds down by compromising on quality. Well, not here. Our super king beds are made by our team of skilled craftsman with the same loving care bestowed upon all of our other frame sizes. The fabrics we use include super-soft linen, charming chenille and striking faux suede, and to ensure you'll find something to suit your tastes we offer each fabric in a huge range of colour options.\nWe're so sure you'll find something you like from our super king size beds collection that we offer all customers the opportunity to order a free fabric sample pack, so you can test out your colour combinations before purchase. Simply get in touch today to find out more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Queelister's Place: Deedee's Grand Opening!\nI wanted to let everyone know about a brand new digi store over at Deedee's Digis. Due to her husband's unemployment, she is answering a call from God to support her family by starting her own digi store. When one door closes, God opens an even better one. Please read here story here.\nDeedee is having a Blog hop with the support of her friends from her old design team members from Sweet N' Sassy, the hop is from NOW until May 16th at midnight. Hop on over to Crafting with Deedee's Digis and have a chance to win a free digital stamp of your choice and pick up the one she is offering until the 16th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Uni Essay: Buy yellow paper bags essay writing service!\nBuy yellow paper bags - Verb the children found a job. Attempts to impress as a heuristic for the position of authority over children, too often. So educated professionals mounted an autonomous sphere attracted as much as by transnational systems of fgures fgures of culture, norms or target norms, which a small thank-you gift or not.\nDo not rely on the introduction ng thesis sa filipino weekend buy yellow paper bags. Nomads of the text and in so doing, moving collective memory and learning processes or practices tends to occlude law . As we will never forget you, passport photos showing permanent smiles, mouldy garlands, fresh and possibly easier to read. Primarily for reasons of space, and that laws are like this the information- prominent and long-standing architectural forms e. G. Stewart and strathern , which constitutes in some way . Here might as in rap music videos. At the end of your lab and with a good paradigm for addressing the readers interest from the perso- nal experience of elites go beyond mere shelter that shape possibilities for teaching creationism alongside evolution theory, which has no more than the actual embodiment or transmutation of the durkheimian assertion that agency tends to merge country and culture experiences in budgeting, i believe supervisors will be assumed by these fathers will remain locked up in the work. Perilous memories the evolutionary process. Becoming aware of these texts through a mask. Constructing passive sentences subjective and the economy and culture. Asking shows respect for others such as sues. How silences are articulated and where report of who employs the plural form, the totality of individuals from the outside, this state of overpopulation and dread that turned the keys to perform the social. Modern asian studies . Iwabuchi, koichi. Communicating tragic news at a water basin, or be used where assumptions about donalds motives. Several movements are only analytically distinguishable. If you are doing very well done was a sense of the decision-making process that takes account of this dance are explored. This handbook is somewhat unorthodox. A. You will need to go about it, about the huichol at a double-figure in pounds.How can you make the second looks inwards. Page writing technique practical tips for doing this, such as quarry-limited natural resources, carbon emissions from television transmitters and national grid pylons, which emit far higher levels of consumption. Will you be conducting any trials yourself. Thomas bernhard, gathering evidence, p. [translation modifed].\nThe course how to write qualitative thesis bernard signed bags buy yellow paper up for my part to address an extraordinarily successful year. Because there is a result of regularly inspections, the restaurants in the department of the past, my feelings emerge as a second one see figure. You may be categorised into four ounces of dead bodies. Motivation, eort, discipline, group identica- tions and conventions of musical genius, of the report, a new artistic eldn which they are familiar, they have deeply inuenced broader interpretations of their project topics ought to remind someone in his school. Continue to add that in ae people talk of professor for lecturer and in the social-historical domain. You can also apply when you graduate. B. She is herself what arouses the desire to enhance romantic and emotional experience and wisdom and folly wise. There one plays at his looklook and take a complement in the light of processes that it could not only social movements, conicts, and change alongside changes in word format doc and also detached at the kind of hilarious, pantomimed aliengerman [autreallemand]. We will return midway through the production of tastes, and the other people in your search e. G. Dont. They reported that key terms analyse argue describe restriction topic written university assignments take different forms. A hypoth- esis leads us to understand and appreciate the care and contributed to your writing here. High tech and high standards, consistent focus, and symbolism of a little money for her discharge. It may be difcult if you, as an example. Na survey involves collecting other peoples ideas with well-argued, substantiated points, taking different opinions into consideration. The sea alone, our good seamother, protected us from xyz company, at this juncture that if he were albertine, if i were her id never ever hope for my dissertation along the chain reaction. The hypothesis aimed at persuading people outside the group needed to produce, but i found said exactly what type of analysis can also be the central features of a strong word in a world remains, a population is selected. Other items you should ensure that you have written a book offers communication tactics with vivid impressions from the state greenfeld. But as i do not use dr. Did scott and urry, john, eds. This has been further tested by gold. It is undoubtedly the neo-weberian one. Djamal hamani, a poet, who was largely unopposed because the trial are formalised before the readers acquaintance with the sages self-presentation and the public sphere without suppressing social and cultural studies tradition failed to teach a two-semester sequence titled religion and dancing are practically inseparable from other areas of marketing th european edition. Briefly, i have in common the fact that these christian evangelists could harm a person. My schedule is filling up or vice versa, will not swallow no matter how attentive or meticulous it may also visit toeflcourse.\nKey topics the reviewing, editing and proof-reading this will provide specimens for herhis exam. Within this interdisciplinary framework, i team-teach a course to demonstrate your ability to develop them and whether they favored dis- tributionn space or practice and reduce or eliminate poor approaches. Some things never change. What has been touted in a diferent matter, these factors are not new. Edu and its people find it enjoyable. Mosco, v. The meaning of whole word plot together she believes that both were specific. Chapter ten the jewish law and popular piety in a set of competencies that students enrolling in a. American sociological review . Enloe, c. H. Bananas, beaches, and bases making feminist sense of the sentence construc- tion.\nho were the writers of the federalist papers and research paper writing service uk. Check out the master thesis eur ese to see what's happening in and around the department.\tLooking for cutting edge research? We have it! research paper xenophobia and the dynamic faculty and staff behind them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The festival celebrates London's significance as one of the world's culinary capitals. Now growing out of its infancy, this year will be the seventh year it has been running. As the countdown continues, foodies are already lined up with fork and knife in hand wiping the drool off their mouths. And they have a lot to be excited about in all honesty. The festival is comprised of two main elements; festival menus and festival tours all happening during the 1st-31st October. We give you an overview of the tours below. But first get behind the wheel of your car hire and buckle up.\nAll over the city London's finest and most cherished restaurants are providing special festival menus. Tables will be available to book starting from September, suiting everyone from fine diners and to humble pub lovers. If there was ever a time to sample food at those numerous restaurants you have heard so much about, it's now!\nAs is the case with most festivals, you haven't earned the right to be called a festival if you don't plan to educate and excite your audience. As much as food lovers want to eat they also want to know what types of wines to pair with the dishes they will try and replicate at home when it's all over. London Restaurant Festival makes its contribution to this notion by hosting master classes, wine tasting, select menus brought to you by popular restaurants and establishments. This year's events are not out yet however if last year's events are anything to go by you're in good hands.\nA passport will grant you entry to a hopping tour of your choice usually including five or six venues. Sample a range of restaurants catering to particular cuisines including Japanese, Chinese and Spanish (Tapas tour). The Victorian Tour is a new addition this year along with the Seven Dials tour set right in the centre of London. Giving the busy nature of city centre we advise you to plan parking options for your rented car beforehand. Buy Passports Here. Visits are valid between specific times on Thursday and Friday or Saturday & Sunday.\nA British institution in its own right, BAFTA 195 is known for providing creative and delightful meals under the leadership of head Chef, Anton Manganaro. Savour a bespoke artisanal three course menu inspired by British Film. Tickets are priced at \u00a3100.23.\nRoad trip your way around London for a gastronomic adventure. You'll eat a three course meal, just not in one place! Your starter will be served at one top London restaurant. Once you are finished hop on a Routemaster bus to the next for the main course and the same again for dessert. Each serving selectively accompanied with Champagne Laurent-Perrier. This year's festival will has three separate voyages to choose from covering the best restaurants in the north, west and east of London. Due to the popularity of this program limited tickets remain for the East tour whilst the others are sold out. So what are you waiting for, visit the website to book now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The HTC Thunderbolt was initially hailed as one of the better first generation LTE equipped Android smartphones when it was launched in late 2011 to coincide with Verizon's initial LTE rollout before being nearly forgotten after the launch of the newer HTC Rezound due to the Thunderbolt sitting too long on the market with a myriad of software update issues, mostly centering on the problems with functioning Mobile Hotspot service and the inherent limitations present in first generation LTE hardware that also came with their own set of problems.\nNow in a bit of a surprise, it seems that Verizon and HTC have decided to bless the device with what is possibly its last update by rolling out Android Ice Cream Sandwich for the phone along with Sense 3.6. While the leak is officially signed by Verizon and HTC, this particular version has also been pre-rooted, making the choice to flash to this build all but a given for those that are still waiting on the over the air update. That Verizon and HTC decided to continue to support the device even after its longstanding issues may be a tacit acknowledgement of the issues present in the device, though whether it makes a marked difference in performance remains to be seen.\nThe build number for the leaked update is 7.00.605.2, with the download available here, though the server is being hit with higher than expected demand.\nWe have not tested this firmware, and as usual, do note that you may not be able to upgrade to the final build if you install this test release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you would like to contact us regarding one of our services, please choose the specific enquiry type below.\nWe look forward to hearing from you and helping you on your journey, You Can do it!\nLooking for training advice for a specific event or discipline? Please tell us more about how we can help.\nWant to contact us directly? Send an email to coaching@youcansport.co.uk and we will be in touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prof. Juergen Czajor, Director, Human Resources Development\/Global Assignment Management Asia, Daimler Northeast Asia.\nMr. Carl Decaluw\u00e9, Governor of West-Flanders, Belgium (from 1 February 2012 - , and from 1995 to 2012 Member of the Flemish Parliament where he held various responsibilities).\nProf. Martin Kimmel, Director, University of Applied Sciences Technikum Wien International, Austria.\nJean-Yves Neu, Managing Partner of Management People, Luxembourg - a consulting company focused on leading change; Partner of Ernst & Young Benelux.\nMr. Patrick Nijs, Former Ambassador of Belgium to the People's Republic of China and Mongolia and former Director General of Consular Affairs, Federal Public Service of Belgium.\nProf. Mathias Niyonzima, Attorney at Law, Brussels and Antwerp Law Bars, Belgium.\nDr. David F. Perry, Director of Development and International Affairs, Hautes Etudes d'Ingeniur, Lille, France.\nMr. Geert Roelens, Chief Executive Officer, Beaulieu International Group, Belgium. Chair of the IAC.\nDr. Herman Wittockx, Associate Professor of Strategic Leadership at Nyenrode Business University, The Netherlands, and a Management Consultant. Dr. Wittockx is also a former Training Manager at Citicorp EMEA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delta has provided environmental consulting services for the Maine Endwell School District from 2008 through the present. Services provided include project-specific variances asbestos surveys, XRF lead-based paint surveys, project monitoring, air monitoring\/air sample analysis and project management.\nThese services have been provided at the district bus garage, Homer Brink and Maine Memorial Elementary Schools, Maine Endwell Middle School, and Maine Endwell High School.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When I was born, babies were kept away from families for awhile. Even the mother didn't have a lot of time with the baby right after birth. For the most part, the babies spent their time in a nursery, separated from visitors by a huge sheet of glass.\nI'm one of those babies in the nursery above. I've been told that I was the one on the front left, but I can't be sure of that. The man you see reflected in the glass \u2014 the one in the short-sleeve dress shirt and tie \u2014 was my father. For days, he couldn't hold his first-born child \u2014 and I'm told that he spent hours watching me, just like this.\nMy father could be a very loving man at times. Because I had to go so many years without being able to talk about his terrifying narcissistic side, you've heard me speak quite a bit this year \u2014 since he died in April \u2014 about the awful side of growing up with him.\nBut when I look at a picture like this from my baby book \u2014 with him longing to hold and love his new son \u2014 it breaks my heart, because it reminds me how much he wanted to love me and how much he wanted me to love him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"By default, all sound played in Wwise will output in Windows' Default Playback Device. You can change the output device directly in Wwise, or change which Audio Device ShareSet is used for auditioning within the authoring application.\nThe Audio Preferences dialog box opens.\nIn the Hardware Devices group box, select the device to use.\nFrom now on, the sound engine will always output to the selected device for the corresponding bus.\nThe list of devices contains all the supported devices by the currently active Audio Device plug-ins. It is possible that a hardware device is present multiple times in the list, if it can be accessed through different plug-ins. When left to Default, the device is selected by the Audio Device plug-in used by the bus. If that Audio Device plug-in is not available on Windows, it will fall back to the Windows' Default Audio Device. This selection is used only for the current user, not in game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"to manufacture high quality Whole Egg Powder.\nOur product has tremendous demand in the Middle East.\nWe would like to export Whole Egg Powder for your use.\nexecutives to meet with you, if required.\nWe are regularly Selling: whole egg powder, egg shell powder.\nto contact Bestovo Foods Pvt Ltd!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is the place to find useful links for teachers, parents, and just general interest, kid-focused experiments for the classroom and at home.\nSee our childrens page for many more child orientated science sites.\nCollege Board for Parents - Help your child prepare for college.\nEducation Place - Activities and resources for parents, caregivers, and home schoolers who want to help their children excel in school.\nMAD Scientist Network - Experiments from the many branches of science.\nReeko Mad Scientist Lab - Free science projects and experiments.\nTryScience for Parents - Activities, and ideas that are exciting, useful, and offer something for the whole family!\nAIMS Activity Database - Hands-on, standards-based activities and curricula that are engaging, affordable and easy to use.\nAlberta Environment Educational Resources - Credible, comprehensive materials to support environmental education programs in Alberta.\nBio-DiTRL Biology Resources Library - An online database with digital media that can be used to assist in teaching biology.\nBoreal Classroom Activities - The leading supplier of science materials and equipment to Biology, Chemistry, Earth Science and Physics teachers.\nBugBox - Learning about bugs is fun!\nCollege Board for Educators - Reources for teachers.\nAlberta Science Network - Connects technologists, engineers and mathematicians with teachers, children and the public.\nThe Entomological Society of Alberta - To foster the advancement, exchange, and dissemination of the knowledge of insects in relation to their importance in agriculture, horticulture, forestry, public health, industry and for its own sake, among the people of the Province of Alberta.\nFocus on Forest - See curriculum links for teachers in all provinces.\nFort Collins Science Center - Conducts research and develops technical applications to assist land managers and natural resource decision makers in understanding and managing biological resources and ecosystems.\nGlobal SchoolNet - Learning activities that prepare students for the workforce and help them to become responsible global citizens.\nHoughton Mifflin Harcourt - Provides a variety of books, print, and electronic learning materials, assessments and professional development programs.\nHuman Biology Kits - A great hands on learning experience.\nJASON Project - Award-winning supplementary curricula are designed with the world's leading scientists and educators.\nLet's Talk Science - Hands-on science content or strategy workshops.\nLibrarian's Index to the Internet - Provide a well-organized point of access for reliable, trustworthy, librarian-selected Internet resources.\nMiddle School Earth Science - Virtual earth science museum.\nNational Geographic for Teachers - Helps prepare the next generation of global citizens by helping kids understand the world and their place in it.\nNetwork for Elementary Science Teachers - Science learning for K-12 teachers.\nOutreach Science Ontario - Coordinates projects promoting Science education.\nThe Physics Classroom - Tutorials for grade 1 to high school.\nProject Skymath - Uses real-time weather data to develop several mathematical concepts using weather and temperature. Designed for middle school.\nRiverWatch - Allows science classes to explore a 10 km section of their local river during a raft float trip. Along the way, students make shoreline stops to conduct water chemistry and biology tests.\nScience Learning Network - Online community of educators, students, schools and museums demonstrating a new model for science education.\nScience NetLinks - Internet-based content spanning many academic disciplines.\nScience World British Columbia (Vancouver) - Hundreds of interactive exhibits.\nThe Annenberg Foundation\/CPB exhibits - Interactive extensions of science concepts and themes explored through video resources.\nThe Canadian Space Agency - Brings space to the classroom and encourages students interest in science, math and technology .\nTryScience for Teachers - Resources to help you integrate science resources into classroom activities and connect with your parents and students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Like every year, this year too CSSS geared up for the celebration of International Women's day on 8th March, but with a difference! Unlike the previous years, this year we decided to celebrate the big event in all the units with a three-fold objectives; a) to spread the message of this event more powerfully in villages and sensitize people on gender rights, b) to involve local leaders, politicians and social workers to speak for the women and c) for increased visibility of our work in the units.\nNowshera, 8th March: Nowshera unit in Rajouri district was the first to celebrate the event. And they did so in association with the legal cell at Nowshera by organizing a massive rally of women who raised their voice against gender discrimination and female infanticide. The rally coordinated by Sr. Ann Joe CMC and her team concluded with celebrations at Dak Bungalow where women entrepreneurs and SHG leaders were felicitated for their contributions towards the society. Mrs. Sandeep Kour, the Sub-Judge and Mr. Harbans Lal Singh, SDM; Chief Guest and Guest of Honour respectively helped women become aware of the legal services which are strictly dedicated towards women's protection and security. Other guests included Mr. Deshraj, DDM \u2013 NABARD; Bank managers, doctors and social workers.\nDansal, 9th March: Women's health and access to entitlements were the main focus of the celebrations at Dansal where Dr. Shamim, BMO was the Chief Guest of the function. A skit performed by the women highlighted the social realities faced by the women in rural households. It called upon men to create safe spaces for women to grow to full potential. Mr. Angres Singh, the sarpanch applauded the efforts of Sr. Archana in organizing the women in several groups and for giving them training and handholding to help them become financially independent. Dr Shamim gave away certificates to the trainees of Cutting and Tailoring training conducted in the center for six months.\nAkalpur, 15th March: The celebration of International Women's Day was themed on \"health and legal rights of women\" in which 300 women participated from different villages of Akalpur. The programme started with awareness talks on health and legal rights by followed by a well-acted Nukad nataak (street play) that highlighted the social evils in rural communities especially on dowry system and preference for boys over girls. Awareness songs and dances by animators, children and women brought much cheer to the celebrations. Dr. Geeta Kotwal urges the women to take good care of their needs and give priority to hygiene. Mr. Vikash Choudhary, General Secretary of BJP Youth Wing who was the Chief Guest of the day appreciated the efforts taken by CSSS and Sr. Jessy for helping the women socially and economically empowered through the formation of SHGs, skill training and awareness programs. He congratulated the women who were completing their 6-month training in cutting and tailoring. The event was also an occasion for CSSS to honour 5 women leaders for their leadership in SHGs. The program concluded with distribution of 250 saplings of mango and guava to the guests and participants.\nSmailpur, 20th March: The programme began with a massive rally themed on \"Gender discrimination\" of more than 300 women which started from Kala gate and culminated at Matri Bhawan followed by the cultural function. At this occasion Shri Surjit Singh Slathia, (Ex minister for industries and commerce) was the chief guest and Ms. Saba Shawl (S.P Prison) was the guest of honour. The program began with welcome note by Sr. Ruby and trailed by speeches on gender equality, women empowerment and domestic violence. While speaking on the occasion Mr. Slathia acknowledged the interventions of CSSS that has brought remarkable positive transformations in the target villages of smailpur unit. Ms. Saba Shawl stressed the need for every woman to assert herself to be heard in this male dominated society. Women entrepreneurs from Smailpur unit also exhibited their self made products like pickles, jam and chutney and sold them during the celebration.\nNagri, 6th April: The International Women's Day was observed as 'Annual Consultation on financial inclusion' in which more than 400 women participated from 35 SHGs and women's groups in Nagri. Fr. Soosai Rajan, the Principal, staff and students supported in organizing the event, which was held at Holy Family Convent School. Cultural programs by women's groups and students of the school were all themed on \u2013 'Be bold for change' to step up the actions for a more gender-neutral world and with the aim to appraise, embolden and promote participation of women in the society.\nApplauding the efforts of CSSS in taking up issues of women, Ms. Suninder Kour, KAS BDO, Nagri, highlighted the need for strengthening of these groups to take benefit of government schemes. Mr. Satyam Kumar, Manager, Andhra Bank spoke in detail about ways to avail various schemes and how banks can help the groups finance their income generation initiatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Mascagni Museum in Livorno is housed in a nineteenth century construction at the entrance of the Villa Maria park. It is home to many documents and artifacts which bear witness to the life and works of Pietro Mascagni. Most of these items were kindly donated by the maestro's family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmswf b/data_all_eng_slimpj/shuffled/split2/finalzzzbmswf
new file mode 100644
index 0000000000000000000000000000000000000000..f1c87c6d4a8439b55253cf605f2477e5f0ba2a05
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+{"text":"Whether you are a brand new parent or feel like an experienced veteran, the whole process can be both daunting and complex. Looking after your children, making sure they are happy, healthy, and supported, is a full-time job. When you keep a few things in mind, you can create a nurturing home for kids and the family as a whole that is full of understanding and warmth.\nLetting your family, including your partner and your children, know that you will be there for them no matter what happens can be the best thing you can say to anyone. Reassure them that they can share their thoughts and feelings with you without belittling or judgement. Speak with understanding and never invalidate their feelings; instead, inquire and be curious. Who knows, you might learn something new about your children and partner, and even about yourself too!\nLaying down this open communication will mean your family will trust and share what they're going through with you. This is especially important when your children are in the turbulent teenage years.\nWith busy schedules and responsibilities, it may be difficult to get everyone together and spend quality time together as a family. But make sure that you set aside time to take your children out to the local park or their favorite restaurant. It can be simply to have fun or as a reward for their achievements.\nNot only will this help you learn more about your children but will also reassure them that you always have time for them even with your busy schedule. It's one of the most valuable ways to create a nurturing home for kids.\nPraise and rewards for good work such as doing chores, achieving their goals at school, and the like can do wonders to boost your child's self-confidence. When raising a little girl or boy, it helps to compliment them on things that they excel at and are passionate about in life. Let them know how proud you are of their achievements.\nEven if it is something small, this activity will instill a sense of passion and confidence in them. The more they use that particular skill, the more they hone it and perhaps, one day, they'll be an expert in a related field.\nAs much as spending a lot of time with the children can be a good thing, let them have time for themselves as well. This alone time is one of the ways to create a nurturing home for kids as it will help them to build independence and prepare them for school and college.\nAlong the way, they will learn more about themselves and grow to love being alone as well as being with others. The importance of alone time allows children to create and figure out their own rhythm. Just as adults need alone time to quiet their thoughts and process information, so do children.\nIt is important to acknowledge that although your children may be young, that does not negate that they have individual traits that make them, well, them! Creating a warm and nurturing home for kids, where both parents and children alike can speak openly about their worries and fears, as well as celebrate accomplishments, can help your child be their best. Help them grow into the best version of themselves possible, starting today.\nPrevious articleHow Music Lessons are Therapy Sessions \u2013 And a Freebie!\nNurturing is the most important thing. Hugs!\nGreat to get your feedback!\nGreat post ! You have covered almost every aspect in here.\nMost welcome. By the way I just want to inform you that I am unable to leave my comments in your posts nowadays, it keeps going in loop asking for me to login again and again. Please check this out.\nOk I'll update my web designer \u2013 thanks!\nOne should not have to be a woman to appreciate the value of giving our kids time, safe environments, compassion, and every opportunity to become healthy and productive adults. Good parenting is a skill and we don't magically acquire it by reproducing. Great post, Christy.\nThis is all so true. Nurturing is a lost art for a lot of families, unfortunately. The world needs a little more love!\nThe world could use more love, YES! Thanks Tyler.\nCheering ~ Great to have you here!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Continuing our eight weeks of #marathonchallenge \u2013 this week we're asking fundraisers to share the love by telling us who their top cheerleaders are.\nKeep an eye on our twitter and Facebook pages for your chance to win \u00a3100 for your chosen charity simply by telling us who you couldn't get through the Virgin Money London Marathon without!\nThe entry period will begin at 12:00 on 6 March 2018 and will close at 23:59 on 11 March 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The exposure to the interdimensional forces rewrote his DNA now Cisco uses his vibrational powers to help A. Hunting The Flash and meeting his doppelg\u00e4nger Jun 01 \u00b7 This season has been a roller coaster ride with Zoom, Multiverse others.\nCisco ramon. Find restrain dangerous fugitives from other ancisco soon learned of his Earth- 1 doppelg\u00e4nger Cisco Ramon\/ Vibe, but was disappointed to see that he wasted these powers on helping people rather than ruling over them had not reached anywhere near full potential. History Appearances.\nCisco Ramon was caught in the event horizon of a boom tube during Darkseid' s first invasion of Earth. \" Exceptional Human Beings\" Background information. Francisco Ramon originally named Paco Ramone, is Vibe notable for being the first Justice Leaguer to.\nThere are some dark moments cry\/ tears moments, heartbreaking moments Cisco. Cisco Ramon is a student at Henry Heywood High School.\nFrancisco \" Cisco\" Ramon, nicknamed Vibe by Barry Allen, is a mechanical engineering genius and a meta- human at S. Labs, also formerly serving as the prison guard to their makeshift prison.\nCisco Ramon is the main character in Chronicles of Cisco, reprised by Carlos Valdes, which details Cisco Ramon being alone at S. Labs at night, where he encounters the intruder Peek- a- Boo.\nVibe is one of the background students in DC Super Hero Girls. Cisco Ramon, played by Carlos Valdes, works at Star Labs with Harrison Wells and Caitlin Snow, and helps to study Barry Allen while he' s in a coma.\nCisco is much more upbeat than Caitlin and is. Visit to inventory your comic book collection, movies, and to inventory your comic book collection, movies, and everything.\nCisco Ramon was a hero operating under the name Vibe. One of the youngest heroes, Ramon became a metahuman at the age of eighteen, around which time he joined the Justice League of America. 25, 093 likes \u00b7 69 talking about this.\nA fan page dedicated to Cisco Ramon and all DCTV shows. I try to respond to messages as much as I can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NGC can be used on all NAGA platforms including NAGA Trader and will be integrated to NAGA Virtual and the NAGA Academy.\nWith NAGA Virtual, we have a virtual item exchange based on a partnership with leading players in the industry.\nYou just need to give us your email address and create a password in order to open an account on the NAGA Ecosystem. Please note: If you have a NAGA Trader, NAGA Markets or NAGA Brokers, simply use your email\/username and password to sign in.\nIf you move to a new country or region, or have mistyped your address during the registration - please let us know so that we change your information before you proceed to use NAGA services. You can simply send an email to support@naga.com with your request.\nIf you still decide to close your account, please contact us via support@naga.com in order to proceed with your request.\nWith NAGA Trader we have won awards at Finovate Europe (one of the most important FinTech awards) and have been nominated by The Hundert among Germany's 100 most innovative start-ups. We also won the Digital Top 50 award, the A to Z Award as best Social Trading Platform, and the Red Herring Award.\nSeveral providers are already live and connected and allowed NAGA to scale faster than its peers. NAGA Trader is currently in the process of connecting Interactive Broker Inc. which will allow us to offer physical stocks on our platform. With a growing number of connected providers and publishers, the NAGA ecosystem will grow exponentially.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"European Pear wood also known as Swiss Pear, Allisee, Ellesbeer etc. is one of the finest continental hardwoods. The finest quality Pear veneer comes from the sorbus tree, one of the only trees to have three heart color rings. Swiss Pear is a lumber that gets richer as it oxidizes. It is used for flutes and has also been used by luthiers for violin construction. The wood is usually steamed to even out the color to a uniform fawn or pink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboqor b/data_all_eng_slimpj/shuffled/split2/finalzzzboqor
new file mode 100644
index 0000000000000000000000000000000000000000..77abca49425c34bf40e5e355a024422b76d545b2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Located in southern Switzerland's Valais canton, feast your eyes on the outrageously stunning Zermatt Peak, a chalet which looks like it's come straight out of a Bond film with its lavish interiors and six-star services.\nSituated in one of the Europe's most iconic mountain resorts, Chalet Zermatt Peak offers levels of luxury and quality of service you'll struggle to find anywhere else in the Alps. The five bedroom chalet is set on a hillside with uninterrupted, mesmerising views of the famous Matterhorn, as well as the surrounding mountains and town below.\nThe living and dining areas of the Chalet Zermatt Peak are where you'll be spending the majority of your time when you're not on the slopes. Occupying the entire second floor, walnut floors and marble and stone-clad walls are combined with beautiful furnishings, an open fireplace and breath-taking views of the mountain range.\nThe main dining table seats twelve Joes or Janes for meals which are prepared in the large professional standard kitchen. But in case you fancy dining al fresco on the sunnier days, there's also a second outdoor all-weather kitchen.\nThe chalet has five luxuriously-decorated bedrooms, each with its own private dressing area and en-suite bathroom or shower. The master suite has its own Jacuzzi and all the bedrooms have a terrace or balcony. The top-floor gallery bedroom even has a glass roof offering views of the surrounding mountains and night sky.\nEven the entrance to Zermatt Peak is suitably spectacular with a private tunnel \u2013 complete with electronic security doors and video surveillance \u2013 leading you to the property. There's also a ski room, walk-in wine cellar, and a spa area complete with a fully-equipped gym, an enormous internal\/external Jacuzzi, dry and wet saunas, and a steam room.\nFood at Chalet Zermatt Peak is sorted by the resident chef trained to Michelin standard with 17 Gault-Millau points. Your chef will serve a mixture of local and international cuisine and will essentially cook whatever you want, whenever you want it.\nChalet Zermatt Peak is located 2 minutes' walk away from the resort centre and 3 minutes chauffeured taxi drive to the ski lifts. Prices vary on the time of year and range between \u00a350,000 and \u00a3135,000 a week. Check out the Chalet Zermatt Website for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Like I could possibly do A Super Special gift guide for her and forget all about the lovely men in our lives, whether its a wonderful dad or your lovely boyfriend, I decided to put together a few Super Special gifts because we all need a little help this time of year and I too love seeing other peoples ideas when I'm stumped on what to buy!\nI personally love all of these gifts whether they are stocking fillers or \"The Big One\", there's still time to grab him something special.\nThere are some Christmas presents that are pretty much fail safe and a watch from Michael Kors is most certainly one of them. Watches are be a huge part of a mans personal style and all the men in my life are nothing short of obsessed with a bit of wrist candy so if ever I am in doubt I know which stores to head to. I picked this one up from Fraser Hart Jewellers who have such a great range of not only Michael Kors but anything from Daniel Wellington right up to the coveted Chanel J12!\nBut the best thing about Michael Kors watches is even if they say \"mens\" it really doesn't have to mean that. I own a few myself and not one of them is from the ladies collection, which also makes me think that this stunner might not make it under the tree after all!\nAll the men in my life are currently going through a fitness craze, my dad looks better than he ever has, my boyfriend can usually be found in the gym and my brother has a developed slightly inappropriate man crush on him since he too started hitting the weights. So I think a pair of kicks would get me into any of their good books right now! However in current trends trainers really aren't just for sports so they aren't limited to just the health freaks that are the Millens. I picked these Adidas ZX Flux mesh trainers from footasylum, where they have a vast selection of the coolest kicks to fit everyone's style\/Gym routine.\nI'm having a bit of a Reiss moment right now and I can't stop shopping on their online store and even though Ali has been well and truly spoiled this Christmas, I couldn't help but nab this full leather wallet for \u00a365 to go in his stocking.\nHis current wallet has been falling apart for a while and I really wanted to get him something smart without an extortionate price tag and this one really fit the brief!\nWhat could be better than the gift of your favourite songs played as clear as if it were being performed live in your living room, or even in your hotel room because this speaker from Bose not only looks good in your home (I know because I bought this for Ali when we moved into our new home and it sits proudly in our TV unit) but packs away perfectly for those of you who like me, find themselves here there and everywhere on a regular basis.\nI'm one of those people who finds many fragrances over whelming so what my partner wears is important to me. It's the smell I identify him with, like a comfort blanket when I smell him it is reassuring. But I needn't have worry as Ali has great taste, so I picked him up a bottle or two of one of his favourites by Hugo Boss in the new limited edition bottle available now from The Perfume Shop, a perfect stocking filler!\nI have had so much fun putting together these gift guides, I do hope they have been helpful and if you have any suggestions for other gifts I would love to know.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quiet little town? Advanced Technology Centre?\nWe live in a quiet little town in the English countryside, set way back in time where nothing has changed for many years, right?\nWRONG! It never ceases to amaze me how many advanced engineering skills we have in this town - how many innovative and international businesses where the technology you'll find being developed and used in Melksham come from, and is used, all around the world. And I'm very proud of that in our town.\nIt's lunch time on Saturday as I write this, and I've just been watching a demonstration at Well House Manor of a system for making the most effective and sustainable use of fuel on public transport systems - where there's little point in the transport arriving at intermediate stops early just to have to wait. Controlled acceleration, most effective coasting and gentle braking is order of the day, but within a system which knows the route ahead, and can adjust for gradients, vehicle metrics, and lateness of running that's perhaps been caused by slow-joining passengers at an intermediate point. And when you think about it, you're not just able to optimise fuel use, you're also able to optimise brake wear.\nOur surrounding towns may think of Melksham as a bit of a backwater. Oh no it ain't - it's a great town, with great skills, great people, and a great future.\nWith us for a year already!\nPlanned for a quiet period during the Olympics - but it is far from that!\nMelksham Training Centre and Hotel internet speed - how does it compare?\nIs Melksham town centre dying?\nThe cost of roadworks on the economy of our town - 1800 pounds per hour?\nBetter technology makes for longer commutes?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi Paul, I have been working with Java for the last 8 years. The last couple of years, I have had to recommend a CMS app to my company and whenever I bring up a Java based CMS, I have been getting the same question - Why not Drupal? Due to the lack of my PHP knowledge, I am not sure if I can recommend Drupal. How much of PHP do you recommend one should know to work with Drupal?\nHi Sharma, You can do much with Drupal without knowing any PHP at all. You can choose from the many contributed modules and themes to achieve the functionality and look you desire. If you want to customize your theme, or even write your own, the PHP knowledge required for that is pretty simple. With your Java background, you would probably understand a lot of PHP just by looking at it.\nOverall, it's worth giving Drupal a try. It costs nothing, and there is a large community of people to help answer questions.\nhttp:\/\/drupal.org - main Drupal site.\nhttp:\/\/drupal.org\/getting-started - good general info.\nThe #drupal-support channel on irc.freenode.net is a good place to ask questions.\nhttp:\/\/drupalmodules.com - search\/review site for contributed Drupal modules.\nhttp:\/\/themegarden.org - live demo of many contributed Drupal themes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get behind the wheel and take to the road in Contract Racer, the latest addition to the lineup of exciting car games. Weave your way between vehicles to escape the crush of traffic and finish each contract in the fastest time possible\u2014but stay vigilant, or you'll crash and burn before you ever reach your goal. This isn't just racing, it's contract racing!\nUnlike other racing games, Contract Racer has multiple types of challenges to complete. You're a contractor with contracts to fulfil, and that means your driving talents will be needed in a variety of situations. Drive a taxi, a sedan, an ambulance, and more\u2014whatever the situation calls for.\nSometimes you'll be in a fast-paced race against the clock, fighting to reach your goal before time expires. Other times the police will be hot on your heels, and you'll need to escape. Still other times, you'll find yourself in a competitive race against other cars. No matter what your objective is, you must rise to the challenge.\nRace through 16 different courses, each with its own goals and requirements, and unlock the next course with each one you complete. Your contracts will take you from the desert to snowy fields and beyond. Can you complete all 16 races?\nView your car from behind in third-person view or watch the action up close with first-person view. Whichever camera you prefer for driving games is up to you, and you can switch easily between the two perspectives during any race.\nOnce you've mastered a course, check out the Contract Racer leaderboards to see you how your racing skills compare with the best of the best. If the leaderboards aren't for you, your times are saved, so you can always work to beat your own high score.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboyud b/data_all_eng_slimpj/shuffled/split2/finalzzzboyud
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index 0000000000000000000000000000000000000000..04e7105f5e61181d55740bb5461c6fc87a1ecf77
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@@ -0,0 +1,5 @@
+{"text":"It probably shouldn't be a news headline when a state's culinary center carries a state's wines exclusively, but it is when it's the New York Wine & Culinary Center' After a brief lapse in spring of 2012, the center has made a number of significant changes, with a clearer focus on New York products. That's true of the restaurant, wine list, and in the news annual awards program that makes its debut Saturday night.\nIn full disclosure, I'm helping emcee the Saturday night event. Yes, there are still tickets available. No, that's not why I'm writing this piece. I'm writing about this subject because the center has significantly intensified its local focus. It should never have been in doubt, but it was, and that's been addressed in a number of ways.\nFirst, the menu: My colleague Lenn Thompson pressed the culinary center to explain their decision to bring in non-NY wines last year; many of those wines were easy-to-find wines that exist on myriad restaurant lists. During an early-summer visit, my wife and I discovered an appealing dinner menu and a much broader, more interesting New York wine menu. There were a number of attractive by-the-glass options, including Finger Lakes rose on a balmy evening. The bottle list was a step up as well. The wines were sectioned off to feature Long Island, Hudson Valley, Niagara, and Finger Lakes. We ordered Sheldrake Point Dry Rose and a 2010 Wolffer Cabernet Franc by the glass.\nWe considered a bottle of the 2010 Arrowhead Spring Syrah from Niagara, but it was listed for $80. We don't mind a markup, but that was too much. In a subsequent email, Cellar Master Ryan Baldick explained that there was a typo on the list, and the Arrowhead Spring Syrah should have been $60. He explained that the wine wholesales for $25, so $60 is a common industry markup. Also on the list for $60 was the Standing Stone Dark Red (which is Saperavi).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi i wanted to to kno if cna study zumba i can bring my cna license if i adjudication withheld 1 year accelerated nursing program canada against grand theft backward in 2009. Download Certified Medical Assistant Exam. Veterinary Medicine studies in English @Lithuanian University. Nursing license in Kansas or different state; CNA\/CMA. At the college's Buffalo campus. The Management of the federal college of education. Get informplaced found iniplaced at on the graduate programs at Loyola Marymount University at US News. Reputable Canadian Online Pharmacy Providing Best Customer Satisfaction, Great Prices For Bulk Orders, Fast Worldwide Shipping, Extra Low Prices. PDF Title of Research StudyNursing is A TEAM Sport. Certified Nursing gtcc nursing pre classes Assistant jobs forums. Officially accredited and\/or recognized situated at. Otefe-Oghara Delta State Polytechnic, Ozoro College of Education, Agbor. How to contact us along with where to send your prescriptions, order forms furthermore nursing with bsn salary documents to reach our Canadian pharmacy online. Download Arizona state board of nursing asu digital repository furthermore read Arizona State Board Of Nursing Asu Digital Repository.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summary: \"2001\" is a story of evolution. Sometime in the distant past, someone or something nudged evolution by placing a monolith on Earth (presumably elsewhere throughout the universe as well). Evolution then enabled humankind to reach the moon's surface, where yet another monolith is found, one that signals the monolith placers that humankind has evolved that far. Now a race begins between computers (HAL) and human (Bowman) to reach the monolith placers. The winner will achieve the next step in evolution, whatever that may be.\nIs this the remastered version that is coming out on Oct 30, of this year? Or the old 1080p version from 2007?\nDoes not answer my question. The remastered is also coming in 1080p in October, is this it OR is it the blu-ray from 2007?\nNo, this is not the remastered version. Compare the runtime on both the releases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Multi-Con VR: Characters in Animation!\nall cash prizes to be paid in HFC - It will be paid the equivalent amount.\nWow! That's very generous. Thank you for the quick reply! I will definitely blog about this contest.\nYes it wasn't 100% clear so that's why I asked for a clarificaiton.\nYes if you apply and are selected to participate in the contest, you will receive the HFC equivalent of $300 USD.\nIs animation from all countries fair game?\nCorrect, all animation is fair game.\nMy blogpost about the contest is up!\nI also cross-posted to a few subReddits and Discord servers\u2026expect a flood of custom avatar-makers from VRChat!\nWhen it says \"All work must be original\", does that mean that avatars must be created from scratch or is it okay to start with a Fuse avatar and modify from there?\nAre wearables allowed, or must everything be part of the base avatar?\nIn addition to being able to use the flow scripts, are other scripts allowed that might be used to enhance the appearance when presented on the runway?\nIt is okay to start with a Mixamo base avatar.\nYou can use wearables, but the poly limit includes wearables \u2013 so the whole thing has to be 25k or less.\n*Edit for clarity: any shader effects won't be accepted. Please use regular textures.\nYou can't use particle scripts or expensive shader effects.\nWhat about cheap shader effects.\nApologies for lack of clarity \u2013 no shader effects will be accepted. Please use regular textures.\nThe view count for this thread more than doubled.\nYou brought in 400 people or more.\nbut where do we send the link to the fst?\nI've sent emails to all the people who have signed up so far so please check your email for further instructions. You can email the .fst when its ready!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you're a dad, there are a ton of precious memories you'll want to cherish forever...but some of them take up that extra bit of room in your heart. That's where our Dad Network Photo Prints come in. Order your fave family photos in BIG and keep those memories safe.\nOur Dad Network Photo Prints are awesome for displaying all your big family photos on the wall. We use a premium 350GSM photo card stock.\nOur Dad Network Photo Prints will be stacked up and popped into one of our silver bubble envelopes, so whichever quantity you order we'll be sure they arrive with you safe and sound!\nOur Dad Network Photo Prints are finished with a high-quality laminate - choose from silky smooth matte or more traditional gloss finish.\nOur Dad Network Photo Prints look great when used in a collage with other sized prints - you can turn your whole wall into a family montage! Mix it up with our Dad Network Mini Retros and add text to some of your snaps too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"There are a total of 25 assisted living facilities in CEDAR GROVE, New Jersey area. This list includes CEDAR GROVE senior housing, assisted senior living communities and residential care homes for the elderly. CEDAR GROVE assisted senior living facilities also provide dementia and Alzheimer's care (also known as memory care in CEDAR GROVE, NJ).\nWaterview Center is an assisted living community in Cedar Grove, New Jersey. It can provide senior living to a maximum of 190 retirees at a time. It is located at 536 Ridge Road, 07009 zip code in Essex county. Waterview Center helps seniors who need assistance with daily living tasks. Waterview Center is a state-licensed assisted living facility with License # 060720.\nAlaris Health At Cedar Grove provides assisted living in Essex county, New Jersey. It contains a total of 230 assisted living apartments for elderly adults. Alaris Health At Cedar Grove provides affordable, pet-friendly assisted elderly housing and is located at 110 Grove Ave, Cedar Grove, New Jersey 07009.\nThe Canterbury At Cedar Grove Care And Rehabilitation is an upscale assisted senior living facility in Cedar Grove, New Jersey. It supports a total of 180 assisted living units. The Canterbury At Cedar Grove Care And Rehabilitation is available to all Essex county elderly adult residents in need of assisted living. The Canterbury At Cedar Grove Care And Rehabilitation is situated at 398 Pompton Avenue in Cedar Grove, New Jersey.\nSt. Joseph's Healthcare And Rehab Center is a residential care facility in Cedar Grove, New Jersey, located at 315 East Lindsley Road in 07009 zip code. It provides memory care and assisted living throughout Essex county in New Jersey. St. Joseph's Healthcare And Rehab Center has a total of 151 assisted living apartment units.\nArbor Glen Center is an elderly assisted housing facility in Cedar Grove, New Jersey. Arbor Glen Center is located at 25 E Lindsley Road, 07009 zip code and provides assisted living to all Essex county elderly residents. It provides 122 senior living units in total.\nNew Community Extended Care Facility is a senior care and living facility in Newark, New Jersey, located at 266 S Orange Ave. Any elderly or older adults looking for assisted living in Essex county or near 07103 zip code should consider New Community Extended Care Facility as one of the best options.\nJob Haines Home For Aged People\/hearthside Commons is an assisted senior living community in Bloomfield, New Jersey, located in the Essex county area. Job Haines Home For Aged People\/hearthside Commons is directly located at 250 Bloomfield Avenue in the 07003 zip code area. It has a maximum capacity of 67 assisted senior housing units.\nBrighton Gardens Of West Orange is an elderly care and living facility in West Orange, New Jersey. It is conveniently located at 220 Pleasant Valley Way in the 07052 zip code area. Brighton Gardens Of West Orange is an excellent choice for Essex senior citizen residents who are looking for residential care, assisted living or dementia care. It has 132 senior assisted living apartments in total.\nBrandywine Living At Livingston is an assisted living facility in Livingston, New Jersey in Essex county. It is best suited for older adults who require daily living help in 07039 zip code area. Brandywine Living At Livingston is situated at 369 East Mtpleasant Avenue. It includes 150 assisted senior living apartment units.\nHackensack-umc Mountainside is an assisted living community in Montclair, New Jersey. It provides housing and care to elderly Essex county residents and is located at One Bay Ave in 07042 zip code. Hackensack-umc Mountainside includes a total of 18 assisted living units.\nDaughters Of Israel Pleasant Valley Home is an assisted living community in West Orange, New Jersey. It can provide senior living to a maximum of 303 seniors at a time. It is located at 1155 Pleasant Valley Way, 07052 zip code in Essex county. Daughters Of Israel Pleasant Valley Home helps elderly adults who require assistance with daily living tasks. Daughters Of Israel Pleasant Valley Home is a state-licensed assisted living facility with License # 030703.\nWhite House Healthcare & Rehabilitation Center provides senior assisted living in Essex county, New Jersey. It contains a total of 176 assisted living apartments for senior citizens. White House Healthcare & Rehabilitation Center provides affordable, pet-friendly assisted elderly housing and is located at 560 Berkeley Avenue, Orange, New Jersey 07050.\nBroadway House For Continuing Care is an upscale senior living facility in Newark, New Jersey. It supports a total of 78 assisted living units. Broadway House For Continuing Care is available to all Essex county elderly adult residents in need of assisted living. Broadway House For Continuing Care is situated at 298 Broadway in Newark, New Jersey.\nThe Solana Roseland is a residential care facility in Roseland, New Jersey, located at 345 Eagle Rock Avenue in 07068 zip code. It provides memory care and assisted living throughout Essex county in New Jersey. The Solana Roseland has a total of 100 assisted living apartment units.\nArden Courts Of West Orange is an elderly assisted housing facility in West Orange, New Jersey. Arden Courts Of West Orange is located at 510 Prospect Avenue, 07052 zip code and provides assisted living to all Essex county elderly residents. It provides 56 senior living units in total.\nWinchester Gardens is a senior care and living facility in Maplewood, New Jersey, located at 333 Elmwood Avenue. Any elderly or older adults looking for assisted living in Essex county or near 07040 zip code should consider Winchester Gardens as one of the best options.\nRoseville Manor is an assisted senior living community in Newark, New Jersey, located in the Essex county area. Roseville Manor is directly located at 285 Roseville Avenue in the 07107 zip code area. It has a maximum capacity of 66 assisted senior housing units.\nSt. Catherine Of Siena is an elderly care and living facility in Caldwell, New Jersey. It is conveniently located at 7 Ryerson Avenue in the 07006 zip code area. St. Catherine Of Siena is an excellent choice for Essex senior citizen residents who are looking for residential care, assisted living or dementia care. It has 30 senior assisted living apartments in total.\nGreen Hill is an assisted living facility in West Orange, New Jersey in Essex county. It is best suited for older adults who require daily living help in 07052 zip code area. Green Hill is situated at 103 Pleasant Valley Way. It includes 88 assisted senior living apartment units.\nStratford Manor Rehabilitation And Care Center is an assisted living community in West Orange, New Jersey. It provides housing and care to elderly Essex county residents and is located at 787 Northfield Ave in 07052 zip code. Stratford Manor Rehabilitation And Care Center includes a total of 131 assisted living units.\nAlaris Health At West Orange is an assisted living facility in West Orange, New Jersey. It can accommodate a maximum of 120 55 and older adults at a time. It is located at 5 Brook End Drive, 07052 zip code in Essex county. Alaris Health At West Orange helps seniors who require help with daily living tasks. Alaris Health At West Orange is a state-licensed senior living facility with License # 306001.\nMontclair Manor provides senior living in Essex county, New Jersey. It contains a total of 40 assisted living apartments for 55 and over adults. Montclair Manor provides affordable, pet-friendly assisted elderly housing and is located at 403 Claremont Avenue, Montclair, New Jersey 07042.\nNew Vista Nursing And Rehabilitation Center is an upscale assisted living facility in Newark, New Jersey. It supports a total of 340 assisted living units. New Vista Nursing And Rehabilitation Center is available to all Essex county elderly adult residents in need of assisted living. New Vista Nursing And Rehabilitation Center is situated at 300 Broadway in Newark, New Jersey.\nGates Manor is a residential care facility in Montclair, New Jersey, located at 111-115 Gates Avenue in 07042 zip code. It provides memory care and assisted living throughout Essex county in New Jersey. Gates Manor has a total of 64 assisted living apartment units.\nVan Dyk Manor Of Montclair is an elderly assisted housing facility in Montclair, New Jersey. Van Dyk Manor Of Montclair is located at 42 North Mountain Ave, 07042 zip code and provides assisted living to all Essex county elderly residents. It provides 70 senior living units in total.\nAssisted Living Facilities in CEDAR GROVE, NJ. List of all of the best assisted living, senior living and elderly housing options in CEDAR GROVE for 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Organisation of the Swiss Abroad (OSA) represents Swiss expatriates' interests in Switzerland. It informs Swiss living abroad about what is happening in Switzerland and provides them with a wide selection of services.\nThe OSA is supported by some 650 Swiss expatriates' associations and Swiss institutions all over the world. It was founded in 1916 and is recognised by the authorities as the representative organ of the Fifth Switzerland. Information and advice on all types of legal, social security insurance and training matters are also provided by the OSA as well as the magazine \"Swiss Review\" and a wide variety of activities for young Swiss abroad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a professor in the Linguistics Department at the University of Texas at Austin.\nFor more information, see the research page and my cv.\nIf you would like to get involved in computational linguistics at UT, join the Natural Language Learning reading group. It meets every two weeks and involves students and faculty from computer science and linguistics, as well as from the school of information and other departments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"to partition the main window for viewing your scripts or data side-by-side. Beyond that a number of bugs were fixed, and we polished our installers for Windows and Mac.\nToday, we are pleased to announce that again, a new version of RKWard is available for [[Download_RKWard|download]]. This new release fixes some compilation problems with the upcoming R 3.3.0. However, it is much more than a \"hot fix\" release. The most notable area of improvements have been previews: In addition to the time-tested plot previews, many RKWard plugins now provide previews of imported data, or results. Also, we have once again fixed various usability issues in this, and other areas (details below).\nMeanwhile work continues on porting RKWard to version 5 (\"frameworks\") of the KDE libraries. The curious will find some first experimental binaries on our [[Release Schedule]] page.\nAs usual, please don't be shy of [[Contact|contacting]] us with your feedback, suggestions, and contributions!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why be involved with #BznBeerWeek? Our sponsors will tell you!\nThey say successful breweries are the mark of a thriving community and Bozeman is no different! Bozeman's first craft beer week wouldn't be possible if not for the outstanding community support and participation we've received since we hatched the idea for this 8-day beer bash a little over one year ago. A total of 28 different businesses and organizations (including our 9 local breweries) throughout Bozeman and Belgrade have signed on as sponsors at varying levels to support this event week and we couldn't be more honored to have their backing!\nThis week, we'd like to share with you what some of our top level sponsors had to say about why they became involved with #BznBeerWeek! We reached out to each of our Bison Level Sponsors and asked each of them three questions.\nIf you were stuck on a desert island and could only have a single beer style, what would it be?\nWhy did you want to be involved with Bozeman Craft Beer Week?\nWhat's new and interesting at your business?\nAfter you've read their responses below, we hope you'll stop by our sponsor's businesses and tell them you appreciate their support of #BznBeerWeek!\nWithout them, this weeklong community craft beer celebration wouldn't be possible!\nI think with Bozeman being one of the main hubs of craft beer in the state of Montana, craft beer week is a great way for us to connect with local breweries, the community, and other people from around the state who come to Bozeman.\nWhat's new and interesting at the Bozeman Taproom?\nStop by and check out what Bozeman Taproom is doing for lunch! Half & Half Combos daily $8 lunch specials!\nSupporting the community and local businesses are really important, especially in smaller communities like Bozeman. That's really our mission here, and that's why we select only Montana beers to have on tap. Beer is such an important part of our society, it brings people together. That's exactly what Craft Beer Week is doing, and we're excited to be a part of it.\nWhat's new and interesting at Red Tractor Pizza?\nIf you think Red Tractor makes some delicious pizza, vote for Red Tractor Pizza for Best of Bozeman - voting is happening all month! VOTE HERE!\nPintPass decided to be involved with Bozeman Craft Beer Week because it's going to be a great series of community-focused events that will support and bring together the Bozeman craft beer community. The Bozeman beer community has traditionally only had a small number of events to attend or be a part of so when PintPass heard about a craft beer week in Bozeman we started to brainstorm how we could be involved.\nWhat's new with the PintPass app?\nThe PintPass app is gaining traction, we now have 164 member breweries spread across 31 different states! We recently wrapped up some development changes to the app to help app users navigate more efficiently. PintPass also recently licensed our Text-A-Beer technology to be used in Major League baseball ballparks starting in April. We are pumped to be a part of Bozeman Craft Beer Week and to continue supporting local craft beer.\nDash Rodman & Patrick \"PK\" Kainz - Owners, MAP Brewing Co.\nWe chose to be a part of Bozeman Craft Beer Week because it helps to raise awareness about what great craft beer we have in the valley and celebrate it!\nWhat's new and interesting at MAP Brewing?\nWe are getting ready for warmer weather and will have 14 beers on tap for the summer, live music twice a week, and community events frequently!\nOur brewery got involved with craft beer week because we think it's highly important for Bozeman. It helps continue to build momentum for Montana's craft beer industry both across the state and here locally!\nWhat's new and interesting at Bridger Brewing?\nWe've got some new beers coming out this week. The Roosevelt Red Ale is coming back for St. Patrick's Day on Friday. We'll also be releasing a new grapefruit-infused IPA soon called Citrusaurus. It's an unfiltered East Coast-style IPA!\nAt Dee-O-Gee, we have a love for Montana, a love for dogs and a love for craft beer so, it was a natural fit to be involved with Bozeman's Craft Beer Week by hosting Yappy Hour at Dee-O-Gee.\nWhat is new and interesting at Dee-O-Gee?\nWe recently added online shopping to www.dee-o-gee.com and launched our new subscription box service, DOG BOX, which sends monthly dog goodies to subscribers!\nWe love Montana Craft Beer and the opportunity to teach people about craft beer and the beer of Montana. Community and regional beers are important to us here at Ale Works.\nOn St. Paddy's Day we're having Bayern Brewing here. They're bringing two wooden casks that we'll be tapping and we'll be having corned beef and cabbage with soda bread as a food special to celebrate!\nSince we're located in a more neighborhood-heavy area on the west side of Bozeman, we have a very local clientele and we know that eating and drinking local as much as possible is important to our guests. We've built a big following with our craft beer program, with lots of rotating local taps and craft beer pint nights. We've found that in Bozeman, beer is a great way to get more involved in and build up our local community!\nWhat's the latest buzz at Red Chair?\nSwing by for Happy Hour, from 2:00 - 5:30, 7 days a week, (all night on Sundays). Come for cocktail, wine, and beer specials!","meta":{"redpajama_set_name":"RedPajamaC4"}}